diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/LICENSE.md b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/LICENSE.md new file mode 100644 index 0000000000000000000000000000000000000000..1b5163d8435976c24988afbd39ded304947178cb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/LICENSE.md @@ -0,0 +1,25 @@ +Copyright (C) 2010-2019 Max-Planck-Society +All rights reserved. + +Redistribution and use in source and binary forms, with or without modification, +are permitted provided that the following conditions are met: + +* Redistributions of source code must retain the above copyright notice, this + list of conditions and the following disclaimer. +* Redistributions in binary form must reproduce the above copyright notice, this + list of conditions and the following disclaimer in the documentation and/or + other materials provided with the distribution. +* Neither the name of the copyright holder nor the names of its contributors may + be used to endorse or promote products derived from this software without + specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND +ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE +DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR +ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES +(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; +LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON +ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT +(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..0671484c9a0780df353b9b783813b6fa7492d38d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/__init__.py @@ -0,0 +1,9 @@ +""" FFT backend using pypocketfft """ + +from .basic import * +from .realtransforms import * +from .helper import * + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/basic.py new file mode 100644 index 0000000000000000000000000000000000000000..bd2d0d33958021c431171b72f72c37363ac98e03 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/basic.py @@ -0,0 +1,251 @@ +""" +Discrete Fourier Transforms - basic.py +""" +import numpy as np +import functools +from . import pypocketfft as pfft +from .helper import (_asfarray, _init_nd_shape_and_axes, _datacopied, + _fix_shape, _fix_shape_1d, _normalization, + _workers) + +def c2c(forward, x, n=None, axis=-1, norm=None, overwrite_x=False, + workers=None, *, plan=None): + """ Return discrete Fourier transform of real or complex sequence. """ + if plan is not None: + raise NotImplementedError('Passing a precomputed plan is not yet ' + 'supported by scipy.fft functions') + tmp = _asfarray(x) + overwrite_x = overwrite_x or _datacopied(tmp, x) + norm = _normalization(norm, forward) + workers = _workers(workers) + + if n is not None: + tmp, copied = _fix_shape_1d(tmp, n, axis) + overwrite_x = overwrite_x or copied + elif tmp.shape[axis] < 1: + message = f"invalid number of data points ({tmp.shape[axis]}) specified" + raise ValueError(message) + + out = (tmp if overwrite_x and tmp.dtype.kind == 'c' else None) + + return pfft.c2c(tmp, (axis,), forward, norm, out, workers) + + +fft = functools.partial(c2c, True) +fft.__name__ = 'fft' +ifft = functools.partial(c2c, False) +ifft.__name__ = 'ifft' + + +def r2c(forward, x, n=None, axis=-1, norm=None, overwrite_x=False, + workers=None, *, plan=None): + """ + Discrete Fourier transform of a real sequence. + """ + if plan is not None: + raise NotImplementedError('Passing a precomputed plan is not yet ' + 'supported by scipy.fft functions') + tmp = _asfarray(x) + norm = _normalization(norm, forward) + workers = _workers(workers) + + if not np.isrealobj(tmp): + raise TypeError("x must be a real sequence") + + if n is not None: + tmp, _ = _fix_shape_1d(tmp, n, axis) + elif tmp.shape[axis] < 1: + raise ValueError(f"invalid number of data points ({tmp.shape[axis]}) specified") + + # Note: overwrite_x is not utilised + return pfft.r2c(tmp, (axis,), forward, norm, None, workers) + + +rfft = functools.partial(r2c, True) +rfft.__name__ = 'rfft' +ihfft = functools.partial(r2c, False) +ihfft.__name__ = 'ihfft' + + +def c2r(forward, x, n=None, axis=-1, norm=None, overwrite_x=False, + workers=None, *, plan=None): + """ + Return inverse discrete Fourier transform of real sequence x. + """ + if plan is not None: + raise NotImplementedError('Passing a precomputed plan is not yet ' + 'supported by scipy.fft functions') + tmp = _asfarray(x) + norm = _normalization(norm, forward) + workers = _workers(workers) + + # TODO: Optimize for hermitian and real? + if np.isrealobj(tmp): + tmp = tmp + 0.j + + # Last axis utilizes hermitian symmetry + if n is None: + n = (tmp.shape[axis] - 1) * 2 + if n < 1: + raise ValueError(f"Invalid number of data points ({n}) specified") + else: + tmp, _ = _fix_shape_1d(tmp, (n//2) + 1, axis) + + # Note: overwrite_x is not utilized + return pfft.c2r(tmp, (axis,), n, forward, norm, None, workers) + + +hfft = functools.partial(c2r, True) +hfft.__name__ = 'hfft' +irfft = functools.partial(c2r, False) +irfft.__name__ = 'irfft' + + +def hfft2(x, s=None, axes=(-2,-1), norm=None, overwrite_x=False, workers=None, + *, plan=None): + """ + 2-D discrete Fourier transform of a Hermitian sequence + """ + if plan is not None: + raise NotImplementedError('Passing a precomputed plan is not yet ' + 'supported by scipy.fft functions') + return hfftn(x, s, axes, norm, overwrite_x, workers) + + +def ihfft2(x, s=None, axes=(-2,-1), norm=None, overwrite_x=False, workers=None, + *, plan=None): + """ + 2-D discrete inverse Fourier transform of a Hermitian sequence + """ + if plan is not None: + raise NotImplementedError('Passing a precomputed plan is not yet ' + 'supported by scipy.fft functions') + return ihfftn(x, s, axes, norm, overwrite_x, workers) + + +def c2cn(forward, x, s=None, axes=None, norm=None, overwrite_x=False, + workers=None, *, plan=None): + """ + Return multidimensional discrete Fourier transform. + """ + if plan is not None: + raise NotImplementedError('Passing a precomputed plan is not yet ' + 'supported by scipy.fft functions') + tmp = _asfarray(x) + + shape, axes = _init_nd_shape_and_axes(tmp, s, axes) + overwrite_x = overwrite_x or _datacopied(tmp, x) + workers = _workers(workers) + + if len(axes) == 0: + return x + + tmp, copied = _fix_shape(tmp, shape, axes) + overwrite_x = overwrite_x or copied + + norm = _normalization(norm, forward) + out = (tmp if overwrite_x and tmp.dtype.kind == 'c' else None) + + return pfft.c2c(tmp, axes, forward, norm, out, workers) + + +fftn = functools.partial(c2cn, True) +fftn.__name__ = 'fftn' +ifftn = functools.partial(c2cn, False) +ifftn.__name__ = 'ifftn' + +def r2cn(forward, x, s=None, axes=None, norm=None, overwrite_x=False, + workers=None, *, plan=None): + """Return multidimensional discrete Fourier transform of real input""" + if plan is not None: + raise NotImplementedError('Passing a precomputed plan is not yet ' + 'supported by scipy.fft functions') + tmp = _asfarray(x) + + if not np.isrealobj(tmp): + raise TypeError("x must be a real sequence") + + shape, axes = _init_nd_shape_and_axes(tmp, s, axes) + tmp, _ = _fix_shape(tmp, shape, axes) + norm = _normalization(norm, forward) + workers = _workers(workers) + + if len(axes) == 0: + raise ValueError("at least 1 axis must be transformed") + + # Note: overwrite_x is not utilized + return pfft.r2c(tmp, axes, forward, norm, None, workers) + + +rfftn = functools.partial(r2cn, True) +rfftn.__name__ = 'rfftn' +ihfftn = functools.partial(r2cn, False) +ihfftn.__name__ = 'ihfftn' + + +def c2rn(forward, x, s=None, axes=None, norm=None, overwrite_x=False, + workers=None, *, plan=None): + """Multidimensional inverse discrete fourier transform with real output""" + if plan is not None: + raise NotImplementedError('Passing a precomputed plan is not yet ' + 'supported by scipy.fft functions') + tmp = _asfarray(x) + + # TODO: Optimize for hermitian and real? + if np.isrealobj(tmp): + tmp = tmp + 0.j + + noshape = s is None + shape, axes = _init_nd_shape_and_axes(tmp, s, axes) + + if len(axes) == 0: + raise ValueError("at least 1 axis must be transformed") + + shape = list(shape) + if noshape: + shape[-1] = (x.shape[axes[-1]] - 1) * 2 + + norm = _normalization(norm, forward) + workers = _workers(workers) + + # Last axis utilizes hermitian symmetry + lastsize = shape[-1] + shape[-1] = (shape[-1] // 2) + 1 + + tmp, _ = tuple(_fix_shape(tmp, shape, axes)) + + # Note: overwrite_x is not utilized + return pfft.c2r(tmp, axes, lastsize, forward, norm, None, workers) + + +hfftn = functools.partial(c2rn, True) +hfftn.__name__ = 'hfftn' +irfftn = functools.partial(c2rn, False) +irfftn.__name__ = 'irfftn' + + +def r2r_fftpack(forward, x, n=None, axis=-1, norm=None, overwrite_x=False): + """FFT of a real sequence, returning fftpack half complex format""" + tmp = _asfarray(x) + overwrite_x = overwrite_x or _datacopied(tmp, x) + norm = _normalization(norm, forward) + workers = _workers(None) + + if tmp.dtype.kind == 'c': + raise TypeError('x must be a real sequence') + + if n is not None: + tmp, copied = _fix_shape_1d(tmp, n, axis) + overwrite_x = overwrite_x or copied + elif tmp.shape[axis] < 1: + raise ValueError(f"invalid number of data points ({tmp.shape[axis]}) specified") + + out = (tmp if overwrite_x else None) + + return pfft.r2r_fftpack(tmp, (axis,), forward, forward, norm, out, workers) + + +rfft_fftpack = functools.partial(r2r_fftpack, True) +rfft_fftpack.__name__ = 'rfft_fftpack' +irfft_fftpack = functools.partial(r2r_fftpack, False) +irfft_fftpack.__name__ = 'irfft_fftpack' diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/helper.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/helper.py new file mode 100644 index 0000000000000000000000000000000000000000..ab2fbc553ccc46a4b337060a62702ec28cb8b254 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/helper.py @@ -0,0 +1,221 @@ +from numbers import Number +import operator +import os +import threading +import contextlib + +import numpy as np + +from scipy._lib._util import copy_if_needed + +# good_size is exposed (and used) from this import +from .pypocketfft import good_size, prev_good_size + + +__all__ = ['good_size', 'prev_good_size', 'set_workers', 'get_workers'] + +_config = threading.local() +_cpu_count = os.cpu_count() + + +def _iterable_of_int(x, name=None): + """Convert ``x`` to an iterable sequence of int + + Parameters + ---------- + x : value, or sequence of values, convertible to int + name : str, optional + Name of the argument being converted, only used in the error message + + Returns + ------- + y : ``List[int]`` + """ + if isinstance(x, Number): + x = (x,) + + try: + x = [operator.index(a) for a in x] + except TypeError as e: + name = name or "value" + raise ValueError(f"{name} must be a scalar or iterable of integers") from e + + return x + + +def _init_nd_shape_and_axes(x, shape, axes): + """Handles shape and axes arguments for nd transforms""" + noshape = shape is None + noaxes = axes is None + + if not noaxes: + axes = _iterable_of_int(axes, 'axes') + axes = [a + x.ndim if a < 0 else a for a in axes] + + if any(a >= x.ndim or a < 0 for a in axes): + raise ValueError("axes exceeds dimensionality of input") + if len(set(axes)) != len(axes): + raise ValueError("all axes must be unique") + + if not noshape: + shape = _iterable_of_int(shape, 'shape') + + if axes and len(axes) != len(shape): + raise ValueError("when given, axes and shape arguments" + " have to be of the same length") + if noaxes: + if len(shape) > x.ndim: + raise ValueError("shape requires more axes than are present") + axes = range(x.ndim - len(shape), x.ndim) + + shape = [x.shape[a] if s == -1 else s for s, a in zip(shape, axes)] + elif noaxes: + shape = list(x.shape) + axes = range(x.ndim) + else: + shape = [x.shape[a] for a in axes] + + if any(s < 1 for s in shape): + raise ValueError( + f"invalid number of data points ({shape}) specified") + + return tuple(shape), list(axes) + + +def _asfarray(x): + """ + Convert to array with floating or complex dtype. + + float16 values are also promoted to float32. + """ + if not hasattr(x, "dtype"): + x = np.asarray(x) + + if x.dtype == np.float16: + return np.asarray(x, np.float32) + elif x.dtype.kind not in 'fc': + return np.asarray(x, np.float64) + + # Require native byte order + dtype = x.dtype.newbyteorder('=') + # Always align input + copy = True if not x.flags['ALIGNED'] else copy_if_needed + return np.array(x, dtype=dtype, copy=copy) + +def _datacopied(arr, original): + """ + Strict check for `arr` not sharing any data with `original`, + under the assumption that arr = asarray(original) + """ + if arr is original: + return False + if not isinstance(original, np.ndarray) and hasattr(original, '__array__'): + return False + return arr.base is None + + +def _fix_shape(x, shape, axes): + """Internal auxiliary function for _raw_fft, _raw_fftnd.""" + must_copy = False + + # Build an nd slice with the dimensions to be read from x + index = [slice(None)]*x.ndim + for n, ax in zip(shape, axes): + if x.shape[ax] >= n: + index[ax] = slice(0, n) + else: + index[ax] = slice(0, x.shape[ax]) + must_copy = True + + index = tuple(index) + + if not must_copy: + return x[index], False + + s = list(x.shape) + for n, axis in zip(shape, axes): + s[axis] = n + + z = np.zeros(s, x.dtype) + z[index] = x[index] + return z, True + + +def _fix_shape_1d(x, n, axis): + if n < 1: + raise ValueError( + f"invalid number of data points ({n}) specified") + + return _fix_shape(x, (n,), (axis,)) + + +_NORM_MAP = {None: 0, 'backward': 0, 'ortho': 1, 'forward': 2} + + +def _normalization(norm, forward): + """Returns the pypocketfft normalization mode from the norm argument""" + try: + inorm = _NORM_MAP[norm] + return inorm if forward else (2 - inorm) + except KeyError: + raise ValueError( + f'Invalid norm value {norm!r}, should ' + 'be "backward", "ortho" or "forward"') from None + + +def _workers(workers): + if workers is None: + return getattr(_config, 'default_workers', 1) + + if workers < 0: + if workers >= -_cpu_count: + workers += 1 + _cpu_count + else: + raise ValueError(f"workers value out of range; got {workers}, must not be" + f" less than {-_cpu_count}") + elif workers == 0: + raise ValueError("workers must not be zero") + + return workers + + +@contextlib.contextmanager +def set_workers(workers): + """Context manager for the default number of workers used in `scipy.fft` + + Parameters + ---------- + workers : int + The default number of workers to use + + Examples + -------- + >>> import numpy as np + >>> from scipy import fft, signal + >>> rng = np.random.default_rng() + >>> x = rng.standard_normal((128, 64)) + >>> with fft.set_workers(4): + ... y = signal.fftconvolve(x, x) + + """ + old_workers = get_workers() + _config.default_workers = _workers(operator.index(workers)) + try: + yield + finally: + _config.default_workers = old_workers + + +def get_workers(): + """Returns the default number of workers within the current context + + Examples + -------- + >>> from scipy import fft + >>> fft.get_workers() + 1 + >>> with fft.set_workers(4): + ... fft.get_workers() + 4 + """ + return getattr(_config, 'default_workers', 1) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/realtransforms.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/realtransforms.py new file mode 100644 index 0000000000000000000000000000000000000000..5a0c616742305444d51258e650344c060129dfab --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/realtransforms.py @@ -0,0 +1,109 @@ +import numpy as np +from . import pypocketfft as pfft +from .helper import (_asfarray, _init_nd_shape_and_axes, _datacopied, + _fix_shape, _fix_shape_1d, _normalization, _workers) +import functools + + +def _r2r(forward, transform, x, type=2, n=None, axis=-1, norm=None, + overwrite_x=False, workers=None, orthogonalize=None): + """Forward or backward 1-D DCT/DST + + Parameters + ---------- + forward : bool + Transform direction (determines type and normalisation) + transform : {pypocketfft.dct, pypocketfft.dst} + The transform to perform + """ + tmp = _asfarray(x) + overwrite_x = overwrite_x or _datacopied(tmp, x) + norm = _normalization(norm, forward) + workers = _workers(workers) + + if not forward: + if type == 2: + type = 3 + elif type == 3: + type = 2 + + if n is not None: + tmp, copied = _fix_shape_1d(tmp, n, axis) + overwrite_x = overwrite_x or copied + elif tmp.shape[axis] < 1: + raise ValueError(f"invalid number of data points ({tmp.shape[axis]}) specified") + + out = (tmp if overwrite_x else None) + + # For complex input, transform real and imaginary components separably + if np.iscomplexobj(x): + out = np.empty_like(tmp) if out is None else out + transform(tmp.real, type, (axis,), norm, out.real, workers) + transform(tmp.imag, type, (axis,), norm, out.imag, workers) + return out + + return transform(tmp, type, (axis,), norm, out, workers, orthogonalize) + + +dct = functools.partial(_r2r, True, pfft.dct) +dct.__name__ = 'dct' +idct = functools.partial(_r2r, False, pfft.dct) +idct.__name__ = 'idct' + +dst = functools.partial(_r2r, True, pfft.dst) +dst.__name__ = 'dst' +idst = functools.partial(_r2r, False, pfft.dst) +idst.__name__ = 'idst' + + +def _r2rn(forward, transform, x, type=2, s=None, axes=None, norm=None, + overwrite_x=False, workers=None, orthogonalize=None): + """Forward or backward nd DCT/DST + + Parameters + ---------- + forward : bool + Transform direction (determines type and normalisation) + transform : {pypocketfft.dct, pypocketfft.dst} + The transform to perform + """ + tmp = _asfarray(x) + + shape, axes = _init_nd_shape_and_axes(tmp, s, axes) + overwrite_x = overwrite_x or _datacopied(tmp, x) + + if len(axes) == 0: + return x + + tmp, copied = _fix_shape(tmp, shape, axes) + overwrite_x = overwrite_x or copied + + if not forward: + if type == 2: + type = 3 + elif type == 3: + type = 2 + + norm = _normalization(norm, forward) + workers = _workers(workers) + out = (tmp if overwrite_x else None) + + # For complex input, transform real and imaginary components separably + if np.iscomplexobj(x): + out = np.empty_like(tmp) if out is None else out + transform(tmp.real, type, axes, norm, out.real, workers) + transform(tmp.imag, type, axes, norm, out.imag, workers) + return out + + return transform(tmp, type, axes, norm, out, workers, orthogonalize) + + +dctn = functools.partial(_r2rn, True, pfft.dct) +dctn.__name__ = 'dctn' +idctn = functools.partial(_r2rn, False, pfft.dct) +idctn.__name__ = 'idctn' + +dstn = functools.partial(_r2rn, True, pfft.dst) +dstn.__name__ = 'dstn' +idstn = functools.partial(_r2rn, False, pfft.dst) +idstn.__name__ = 'idstn' diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/tests/test_basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/tests/test_basic.py new file mode 100644 index 0000000000000000000000000000000000000000..feffc37944c24f81bba3352329cbf75743e4e280 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/tests/test_basic.py @@ -0,0 +1,1013 @@ +# Created by Pearu Peterson, September 2002 + +from numpy.testing import (assert_, assert_equal, assert_array_almost_equal, + assert_array_almost_equal_nulp, assert_array_less, + assert_allclose) +import pytest +from pytest import raises as assert_raises +from scipy.fft._pocketfft import (ifft, fft, fftn, ifftn, + rfft, irfft, rfftn, irfftn, + hfft, ihfft, hfftn, ihfftn) + +from numpy import (arange, array, asarray, zeros, dot, exp, pi, + swapaxes, cdouble) +import numpy as np +import numpy.fft +from numpy.random import rand + +# "large" composite numbers supported by FFT._PYPOCKETFFT +LARGE_COMPOSITE_SIZES = [ + 2**13, + 2**5 * 3**5, + 2**3 * 3**3 * 5**2, +] +SMALL_COMPOSITE_SIZES = [ + 2, + 2*3*5, + 2*2*3*3, +] +# prime +LARGE_PRIME_SIZES = [ + 2011 +] +SMALL_PRIME_SIZES = [ + 29 +] + + +def _assert_close_in_norm(x, y, rtol, size, rdt): + # helper function for testing + err_msg = f"size: {size} rdt: {rdt}" + assert_array_less(np.linalg.norm(x - y), rtol*np.linalg.norm(x), err_msg) + + +def random(size): + return rand(*size) + +def swap_byteorder(arr): + """Returns the same array with swapped byteorder""" + dtype = arr.dtype.newbyteorder('S') + return arr.astype(dtype) + +def direct_dft(x): + x = asarray(x) + n = len(x) + y = zeros(n, dtype=cdouble) + w = -arange(n)*(2j*pi/n) + for i in range(n): + y[i] = dot(exp(i*w), x) + return y + + +def direct_idft(x): + x = asarray(x) + n = len(x) + y = zeros(n, dtype=cdouble) + w = arange(n)*(2j*pi/n) + for i in range(n): + y[i] = dot(exp(i*w), x)/n + return y + + +def direct_dftn(x): + x = asarray(x) + for axis in range(x.ndim): + x = fft(x, axis=axis) + return x + + +def direct_idftn(x): + x = asarray(x) + for axis in range(x.ndim): + x = ifft(x, axis=axis) + return x + + +def direct_rdft(x): + x = asarray(x) + n = len(x) + w = -arange(n)*(2j*pi/n) + y = zeros(n//2+1, dtype=cdouble) + for i in range(n//2+1): + y[i] = dot(exp(i*w), x) + return y + + +def direct_irdft(x, n): + x = asarray(x) + x1 = zeros(n, dtype=cdouble) + for i in range(n//2+1): + x1[i] = x[i] + if i > 0 and 2*i < n: + x1[n-i] = np.conj(x[i]) + return direct_idft(x1).real + + +def direct_rdftn(x): + return fftn(rfft(x), axes=range(x.ndim - 1)) + + +class _TestFFTBase: + def setup_method(self): + self.cdt = None + self.rdt = None + np.random.seed(1234) + + def test_definition(self): + x = np.array([1,2,3,4+1j,1,2,3,4+2j], dtype=self.cdt) + y = fft(x) + assert_equal(y.dtype, self.cdt) + y1 = direct_dft(x) + assert_array_almost_equal(y,y1) + x = np.array([1,2,3,4+0j,5], dtype=self.cdt) + assert_array_almost_equal(fft(x),direct_dft(x)) + + def test_n_argument_real(self): + x1 = np.array([1,2,3,4], dtype=self.rdt) + x2 = np.array([1,2,3,4], dtype=self.rdt) + y = fft([x1,x2],n=4) + assert_equal(y.dtype, self.cdt) + assert_equal(y.shape,(2,4)) + assert_array_almost_equal(y[0],direct_dft(x1)) + assert_array_almost_equal(y[1],direct_dft(x2)) + + def _test_n_argument_complex(self): + x1 = np.array([1,2,3,4+1j], dtype=self.cdt) + x2 = np.array([1,2,3,4+1j], dtype=self.cdt) + y = fft([x1,x2],n=4) + assert_equal(y.dtype, self.cdt) + assert_equal(y.shape,(2,4)) + assert_array_almost_equal(y[0],direct_dft(x1)) + assert_array_almost_equal(y[1],direct_dft(x2)) + + def test_djbfft(self): + for i in range(2,14): + n = 2**i + x = np.arange(n) + y = fft(x.astype(complex)) + y2 = numpy.fft.fft(x) + assert_array_almost_equal(y,y2) + y = fft(x) + assert_array_almost_equal(y,y2) + + def test_invalid_sizes(self): + assert_raises(ValueError, fft, []) + assert_raises(ValueError, fft, [[1,1],[2,2]], -5) + + +class TestLongDoubleFFT(_TestFFTBase): + def setup_method(self): + self.cdt = np.clongdouble + self.rdt = np.longdouble + + +class TestDoubleFFT(_TestFFTBase): + def setup_method(self): + self.cdt = np.cdouble + self.rdt = np.float64 + + +class TestSingleFFT(_TestFFTBase): + def setup_method(self): + self.cdt = np.complex64 + self.rdt = np.float32 + + +class TestFloat16FFT: + + def test_1_argument_real(self): + x1 = np.array([1, 2, 3, 4], dtype=np.float16) + y = fft(x1, n=4) + assert_equal(y.dtype, np.complex64) + assert_equal(y.shape, (4, )) + assert_array_almost_equal(y, direct_dft(x1.astype(np.float32))) + + def test_n_argument_real(self): + x1 = np.array([1, 2, 3, 4], dtype=np.float16) + x2 = np.array([1, 2, 3, 4], dtype=np.float16) + y = fft([x1, x2], n=4) + assert_equal(y.dtype, np.complex64) + assert_equal(y.shape, (2, 4)) + assert_array_almost_equal(y[0], direct_dft(x1.astype(np.float32))) + assert_array_almost_equal(y[1], direct_dft(x2.astype(np.float32))) + + +class _TestIFFTBase: + def setup_method(self): + np.random.seed(1234) + + def test_definition(self): + x = np.array([1,2,3,4+1j,1,2,3,4+2j], self.cdt) + y = ifft(x) + y1 = direct_idft(x) + assert_equal(y.dtype, self.cdt) + assert_array_almost_equal(y,y1) + + x = np.array([1,2,3,4+0j,5], self.cdt) + assert_array_almost_equal(ifft(x),direct_idft(x)) + + def test_definition_real(self): + x = np.array([1,2,3,4,1,2,3,4], self.rdt) + y = ifft(x) + assert_equal(y.dtype, self.cdt) + y1 = direct_idft(x) + assert_array_almost_equal(y,y1) + + x = np.array([1,2,3,4,5], dtype=self.rdt) + assert_equal(y.dtype, self.cdt) + assert_array_almost_equal(ifft(x),direct_idft(x)) + + def test_djbfft(self): + for i in range(2,14): + n = 2**i + x = np.arange(n) + y = ifft(x.astype(self.cdt)) + y2 = numpy.fft.ifft(x.astype(self.cdt)) + assert_allclose(y,y2, rtol=self.rtol, atol=self.atol) + y = ifft(x) + assert_allclose(y,y2, rtol=self.rtol, atol=self.atol) + + def test_random_complex(self): + for size in [1,51,111,100,200,64,128,256,1024]: + x = random([size]).astype(self.cdt) + x = random([size]).astype(self.cdt) + 1j*x + y1 = ifft(fft(x)) + y2 = fft(ifft(x)) + assert_equal(y1.dtype, self.cdt) + assert_equal(y2.dtype, self.cdt) + assert_array_almost_equal(y1, x) + assert_array_almost_equal(y2, x) + + def test_random_real(self): + for size in [1,51,111,100,200,64,128,256,1024]: + x = random([size]).astype(self.rdt) + y1 = ifft(fft(x)) + y2 = fft(ifft(x)) + assert_equal(y1.dtype, self.cdt) + assert_equal(y2.dtype, self.cdt) + assert_array_almost_equal(y1, x) + assert_array_almost_equal(y2, x) + + def test_size_accuracy(self): + # Sanity check for the accuracy for prime and non-prime sized inputs + for size in LARGE_COMPOSITE_SIZES + LARGE_PRIME_SIZES: + np.random.seed(1234) + x = np.random.rand(size).astype(self.rdt) + y = ifft(fft(x)) + _assert_close_in_norm(x, y, self.rtol, size, self.rdt) + y = fft(ifft(x)) + _assert_close_in_norm(x, y, self.rtol, size, self.rdt) + + x = (x + 1j*np.random.rand(size)).astype(self.cdt) + y = ifft(fft(x)) + _assert_close_in_norm(x, y, self.rtol, size, self.rdt) + y = fft(ifft(x)) + _assert_close_in_norm(x, y, self.rtol, size, self.rdt) + + def test_invalid_sizes(self): + assert_raises(ValueError, ifft, []) + assert_raises(ValueError, ifft, [[1,1],[2,2]], -5) + + +@pytest.mark.skipif(np.longdouble is np.float64, + reason="Long double is aliased to double") +class TestLongDoubleIFFT(_TestIFFTBase): + def setup_method(self): + self.cdt = np.clongdouble + self.rdt = np.longdouble + self.rtol = 1e-10 + self.atol = 1e-10 + + +class TestDoubleIFFT(_TestIFFTBase): + def setup_method(self): + self.cdt = np.complex128 + self.rdt = np.float64 + self.rtol = 1e-10 + self.atol = 1e-10 + + +class TestSingleIFFT(_TestIFFTBase): + def setup_method(self): + self.cdt = np.complex64 + self.rdt = np.float32 + self.rtol = 1e-5 + self.atol = 1e-4 + + +class _TestRFFTBase: + def setup_method(self): + np.random.seed(1234) + + def test_definition(self): + for t in [[1, 2, 3, 4, 1, 2, 3, 4], [1, 2, 3, 4, 1, 2, 3, 4, 5]]: + x = np.array(t, dtype=self.rdt) + y = rfft(x) + y1 = direct_rdft(x) + assert_array_almost_equal(y,y1) + assert_equal(y.dtype, self.cdt) + + def test_djbfft(self): + for i in range(2,14): + n = 2**i + x = np.arange(n) + y1 = np.fft.rfft(x) + y = rfft(x) + assert_array_almost_equal(y,y1) + + def test_invalid_sizes(self): + assert_raises(ValueError, rfft, []) + assert_raises(ValueError, rfft, [[1,1],[2,2]], -5) + + def test_complex_input(self): + x = np.zeros(10, dtype=self.cdt) + with assert_raises(TypeError, match="x must be a real sequence"): + rfft(x) + + # See gh-5790 + class MockSeries: + def __init__(self, data): + self.data = np.asarray(data) + + def __getattr__(self, item): + try: + return getattr(self.data, item) + except AttributeError as e: + raise AttributeError("'MockSeries' object " + f"has no attribute '{item}'") from e + + def test_non_ndarray_with_dtype(self): + x = np.array([1., 2., 3., 4., 5.]) + xs = _TestRFFTBase.MockSeries(x) + + expected = [1, 2, 3, 4, 5] + rfft(xs) + + # Data should not have been overwritten + assert_equal(x, expected) + assert_equal(xs.data, expected) + +@pytest.mark.skipif(np.longdouble is np.float64, + reason="Long double is aliased to double") +class TestRFFTLongDouble(_TestRFFTBase): + def setup_method(self): + self.cdt = np.clongdouble + self.rdt = np.longdouble + + +class TestRFFTDouble(_TestRFFTBase): + def setup_method(self): + self.cdt = np.complex128 + self.rdt = np.float64 + + +class TestRFFTSingle(_TestRFFTBase): + def setup_method(self): + self.cdt = np.complex64 + self.rdt = np.float32 + + +class _TestIRFFTBase: + def setup_method(self): + np.random.seed(1234) + + def test_definition(self): + x1 = [1,2+3j,4+1j,1+2j,3+4j] + x1_1 = [1,2+3j,4+1j,2+3j,4,2-3j,4-1j,2-3j] + x1 = x1_1[:5] + x2_1 = [1,2+3j,4+1j,2+3j,4+5j,4-5j,2-3j,4-1j,2-3j] + x2 = x2_1[:5] + + def _test(x, xr): + y = irfft(np.array(x, dtype=self.cdt), n=len(xr)) + y1 = direct_irdft(x, len(xr)) + assert_equal(y.dtype, self.rdt) + assert_array_almost_equal(y,y1, decimal=self.ndec) + assert_array_almost_equal(y,ifft(xr), decimal=self.ndec) + + _test(x1, x1_1) + _test(x2, x2_1) + + def test_djbfft(self): + for i in range(2,14): + n = 2**i + x = np.arange(-1, n, 2) + 1j * np.arange(0, n+1, 2) + x[0] = 0 + if n % 2 == 0: + x[-1] = np.real(x[-1]) + y1 = np.fft.irfft(x) + y = irfft(x) + assert_array_almost_equal(y,y1) + + def test_random_real(self): + for size in [1,51,111,100,200,64,128,256,1024]: + x = random([size]).astype(self.rdt) + y1 = irfft(rfft(x), n=size) + y2 = rfft(irfft(x, n=(size*2-1))) + assert_equal(y1.dtype, self.rdt) + assert_equal(y2.dtype, self.cdt) + assert_array_almost_equal(y1, x, decimal=self.ndec, + err_msg="size=%d" % size) + assert_array_almost_equal(y2, x, decimal=self.ndec, + err_msg="size=%d" % size) + + def test_size_accuracy(self): + # Sanity check for the accuracy for prime and non-prime sized inputs + if self.rdt == np.float32: + rtol = 1e-5 + elif self.rdt == np.float64: + rtol = 1e-10 + + for size in LARGE_COMPOSITE_SIZES + LARGE_PRIME_SIZES: + np.random.seed(1234) + x = np.random.rand(size).astype(self.rdt) + y = irfft(rfft(x), len(x)) + _assert_close_in_norm(x, y, rtol, size, self.rdt) + y = rfft(irfft(x, 2 * len(x) - 1)) + _assert_close_in_norm(x, y, rtol, size, self.rdt) + + def test_invalid_sizes(self): + assert_raises(ValueError, irfft, []) + assert_raises(ValueError, irfft, [[1,1],[2,2]], -5) + + +# self.ndec is bogus; we should have a assert_array_approx_equal for number of +# significant digits + +@pytest.mark.skipif(np.longdouble is np.float64, + reason="Long double is aliased to double") +class TestIRFFTLongDouble(_TestIRFFTBase): + def setup_method(self): + self.cdt = np.complex128 + self.rdt = np.float64 + self.ndec = 14 + + +class TestIRFFTDouble(_TestIRFFTBase): + def setup_method(self): + self.cdt = np.complex128 + self.rdt = np.float64 + self.ndec = 14 + + +class TestIRFFTSingle(_TestIRFFTBase): + def setup_method(self): + self.cdt = np.complex64 + self.rdt = np.float32 + self.ndec = 5 + + +class TestFftnSingle: + def setup_method(self): + np.random.seed(1234) + + def test_definition(self): + x = [[1, 2, 3], + [4, 5, 6], + [7, 8, 9]] + y = fftn(np.array(x, np.float32)) + assert_(y.dtype == np.complex64, + msg="double precision output with single precision") + + y_r = np.array(fftn(x), np.complex64) + assert_array_almost_equal_nulp(y, y_r) + + @pytest.mark.parametrize('size', SMALL_COMPOSITE_SIZES + SMALL_PRIME_SIZES) + def test_size_accuracy_small(self, size): + rng = np.random.default_rng(1234) + x = rng.random((size, size)) + 1j * rng.random((size, size)) + y1 = fftn(x.real.astype(np.float32)) + y2 = fftn(x.real.astype(np.float64)).astype(np.complex64) + + assert_equal(y1.dtype, np.complex64) + assert_array_almost_equal_nulp(y1, y2, 2000) + + @pytest.mark.parametrize('size', LARGE_COMPOSITE_SIZES + LARGE_PRIME_SIZES) + def test_size_accuracy_large(self, size): + rng = np.random.default_rng(1234) + x = rng.random((size, 3)) + 1j * rng.random((size, 3)) + y1 = fftn(x.real.astype(np.float32)) + y2 = fftn(x.real.astype(np.float64)).astype(np.complex64) + + assert_equal(y1.dtype, np.complex64) + assert_array_almost_equal_nulp(y1, y2, 2000) + + def test_definition_float16(self): + x = [[1, 2, 3], + [4, 5, 6], + [7, 8, 9]] + y = fftn(np.array(x, np.float16)) + assert_equal(y.dtype, np.complex64) + y_r = np.array(fftn(x), np.complex64) + assert_array_almost_equal_nulp(y, y_r) + + @pytest.mark.parametrize('size', SMALL_COMPOSITE_SIZES + SMALL_PRIME_SIZES) + def test_float16_input_small(self, size): + rng = np.random.default_rng(1234) + x = rng.random((size, size)) + 1j*rng.random((size, size)) + y1 = fftn(x.real.astype(np.float16)) + y2 = fftn(x.real.astype(np.float64)).astype(np.complex64) + + assert_equal(y1.dtype, np.complex64) + assert_array_almost_equal_nulp(y1, y2, 5e5) + + @pytest.mark.parametrize('size', LARGE_COMPOSITE_SIZES + LARGE_PRIME_SIZES) + def test_float16_input_large(self, size): + rng = np.random.default_rng(1234) + x = rng.random((size, 3)) + 1j*rng.random((size, 3)) + y1 = fftn(x.real.astype(np.float16)) + y2 = fftn(x.real.astype(np.float64)).astype(np.complex64) + + assert_equal(y1.dtype, np.complex64) + assert_array_almost_equal_nulp(y1, y2, 2e6) + + +class TestFftn: + def setup_method(self): + np.random.seed(1234) + + def test_definition(self): + x = [[1, 2, 3], + [4, 5, 6], + [7, 8, 9]] + y = fftn(x) + assert_array_almost_equal(y, direct_dftn(x)) + + x = random((20, 26)) + assert_array_almost_equal(fftn(x), direct_dftn(x)) + + x = random((5, 4, 3, 20)) + assert_array_almost_equal(fftn(x), direct_dftn(x)) + + def test_axes_argument(self): + # plane == ji_plane, x== kji_space + plane1 = [[1, 2, 3], + [4, 5, 6], + [7, 8, 9]] + plane2 = [[10, 11, 12], + [13, 14, 15], + [16, 17, 18]] + plane3 = [[19, 20, 21], + [22, 23, 24], + [25, 26, 27]] + ki_plane1 = [[1, 2, 3], + [10, 11, 12], + [19, 20, 21]] + ki_plane2 = [[4, 5, 6], + [13, 14, 15], + [22, 23, 24]] + ki_plane3 = [[7, 8, 9], + [16, 17, 18], + [25, 26, 27]] + jk_plane1 = [[1, 10, 19], + [4, 13, 22], + [7, 16, 25]] + jk_plane2 = [[2, 11, 20], + [5, 14, 23], + [8, 17, 26]] + jk_plane3 = [[3, 12, 21], + [6, 15, 24], + [9, 18, 27]] + kj_plane1 = [[1, 4, 7], + [10, 13, 16], [19, 22, 25]] + kj_plane2 = [[2, 5, 8], + [11, 14, 17], [20, 23, 26]] + kj_plane3 = [[3, 6, 9], + [12, 15, 18], [21, 24, 27]] + ij_plane1 = [[1, 4, 7], + [2, 5, 8], + [3, 6, 9]] + ij_plane2 = [[10, 13, 16], + [11, 14, 17], + [12, 15, 18]] + ij_plane3 = [[19, 22, 25], + [20, 23, 26], + [21, 24, 27]] + ik_plane1 = [[1, 10, 19], + [2, 11, 20], + [3, 12, 21]] + ik_plane2 = [[4, 13, 22], + [5, 14, 23], + [6, 15, 24]] + ik_plane3 = [[7, 16, 25], + [8, 17, 26], + [9, 18, 27]] + ijk_space = [jk_plane1, jk_plane2, jk_plane3] + ikj_space = [kj_plane1, kj_plane2, kj_plane3] + jik_space = [ik_plane1, ik_plane2, ik_plane3] + jki_space = [ki_plane1, ki_plane2, ki_plane3] + kij_space = [ij_plane1, ij_plane2, ij_plane3] + x = array([plane1, plane2, plane3]) + + assert_array_almost_equal(fftn(x), + fftn(x, axes=(-3, -2, -1))) # kji_space + assert_array_almost_equal(fftn(x), fftn(x, axes=(0, 1, 2))) + assert_array_almost_equal(fftn(x, axes=(0, 2)), fftn(x, axes=(0, -1))) + y = fftn(x, axes=(2, 1, 0)) # ijk_space + assert_array_almost_equal(swapaxes(y, -1, -3), fftn(ijk_space)) + y = fftn(x, axes=(2, 0, 1)) # ikj_space + assert_array_almost_equal(swapaxes(swapaxes(y, -1, -3), -1, -2), + fftn(ikj_space)) + y = fftn(x, axes=(1, 2, 0)) # jik_space + assert_array_almost_equal(swapaxes(swapaxes(y, -1, -3), -3, -2), + fftn(jik_space)) + y = fftn(x, axes=(1, 0, 2)) # jki_space + assert_array_almost_equal(swapaxes(y, -2, -3), fftn(jki_space)) + y = fftn(x, axes=(0, 2, 1)) # kij_space + assert_array_almost_equal(swapaxes(y, -2, -1), fftn(kij_space)) + + y = fftn(x, axes=(-2, -1)) # ji_plane + assert_array_almost_equal(fftn(plane1), y[0]) + assert_array_almost_equal(fftn(plane2), y[1]) + assert_array_almost_equal(fftn(plane3), y[2]) + + y = fftn(x, axes=(1, 2)) # ji_plane + assert_array_almost_equal(fftn(plane1), y[0]) + assert_array_almost_equal(fftn(plane2), y[1]) + assert_array_almost_equal(fftn(plane3), y[2]) + + y = fftn(x, axes=(-3, -2)) # kj_plane + assert_array_almost_equal(fftn(x[:, :, 0]), y[:, :, 0]) + assert_array_almost_equal(fftn(x[:, :, 1]), y[:, :, 1]) + assert_array_almost_equal(fftn(x[:, :, 2]), y[:, :, 2]) + + y = fftn(x, axes=(-3, -1)) # ki_plane + assert_array_almost_equal(fftn(x[:, 0, :]), y[:, 0, :]) + assert_array_almost_equal(fftn(x[:, 1, :]), y[:, 1, :]) + assert_array_almost_equal(fftn(x[:, 2, :]), y[:, 2, :]) + + y = fftn(x, axes=(-1, -2)) # ij_plane + assert_array_almost_equal(fftn(ij_plane1), swapaxes(y[0], -2, -1)) + assert_array_almost_equal(fftn(ij_plane2), swapaxes(y[1], -2, -1)) + assert_array_almost_equal(fftn(ij_plane3), swapaxes(y[2], -2, -1)) + + y = fftn(x, axes=(-1, -3)) # ik_plane + assert_array_almost_equal(fftn(ik_plane1), + swapaxes(y[:, 0, :], -1, -2)) + assert_array_almost_equal(fftn(ik_plane2), + swapaxes(y[:, 1, :], -1, -2)) + assert_array_almost_equal(fftn(ik_plane3), + swapaxes(y[:, 2, :], -1, -2)) + + y = fftn(x, axes=(-2, -3)) # jk_plane + assert_array_almost_equal(fftn(jk_plane1), + swapaxes(y[:, :, 0], -1, -2)) + assert_array_almost_equal(fftn(jk_plane2), + swapaxes(y[:, :, 1], -1, -2)) + assert_array_almost_equal(fftn(jk_plane3), + swapaxes(y[:, :, 2], -1, -2)) + + y = fftn(x, axes=(-1,)) # i_line + for i in range(3): + for j in range(3): + assert_array_almost_equal(fft(x[i, j, :]), y[i, j, :]) + y = fftn(x, axes=(-2,)) # j_line + for i in range(3): + for j in range(3): + assert_array_almost_equal(fft(x[i, :, j]), y[i, :, j]) + y = fftn(x, axes=(0,)) # k_line + for i in range(3): + for j in range(3): + assert_array_almost_equal(fft(x[:, i, j]), y[:, i, j]) + + y = fftn(x, axes=()) # point + assert_array_almost_equal(y, x) + + def test_shape_argument(self): + small_x = [[1, 2, 3], + [4, 5, 6]] + large_x1 = [[1, 2, 3, 0], + [4, 5, 6, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]] + + y = fftn(small_x, s=(4, 4)) + assert_array_almost_equal(y, fftn(large_x1)) + + y = fftn(small_x, s=(3, 4)) + assert_array_almost_equal(y, fftn(large_x1[:-1])) + + def test_shape_axes_argument(self): + small_x = [[1, 2, 3], + [4, 5, 6], + [7, 8, 9]] + large_x1 = array([[1, 2, 3, 0], + [4, 5, 6, 0], + [7, 8, 9, 0], + [0, 0, 0, 0]]) + y = fftn(small_x, s=(4, 4), axes=(-2, -1)) + assert_array_almost_equal(y, fftn(large_x1)) + y = fftn(small_x, s=(4, 4), axes=(-1, -2)) + + assert_array_almost_equal(y, swapaxes( + fftn(swapaxes(large_x1, -1, -2)), -1, -2)) + + def test_shape_axes_argument2(self): + # Change shape of the last axis + x = numpy.random.random((10, 5, 3, 7)) + y = fftn(x, axes=(-1,), s=(8,)) + assert_array_almost_equal(y, fft(x, axis=-1, n=8)) + + # Change shape of an arbitrary axis which is not the last one + x = numpy.random.random((10, 5, 3, 7)) + y = fftn(x, axes=(-2,), s=(8,)) + assert_array_almost_equal(y, fft(x, axis=-2, n=8)) + + # Change shape of axes: cf #244, where shape and axes were mixed up + x = numpy.random.random((4, 4, 2)) + y = fftn(x, axes=(-3, -2), s=(8, 8)) + assert_array_almost_equal(y, + numpy.fft.fftn(x, axes=(-3, -2), s=(8, 8))) + + def test_shape_argument_more(self): + x = zeros((4, 4, 2)) + with assert_raises(ValueError, + match="shape requires more axes than are present"): + fftn(x, s=(8, 8, 2, 1)) + + def test_invalid_sizes(self): + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[1, 0\]\) specified"): + fftn([[]]) + + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[4, -3\]\) specified"): + fftn([[1, 1], [2, 2]], (4, -3)) + + def test_no_axes(self): + x = numpy.random.random((2,2,2)) + assert_allclose(fftn(x, axes=[]), x, atol=1e-7) + + def test_regression_244(self): + """FFT returns wrong result with axes parameter.""" + # fftn (and hence fft2) used to break when both axes and shape were used + x = numpy.ones((4, 4, 2)) + y = fftn(x, s=(8, 8), axes=(-3, -2)) + y_r = numpy.fft.fftn(x, s=(8, 8), axes=(-3, -2)) + assert_allclose(y, y_r) + + +class TestIfftn: + dtype = None + cdtype = None + + def setup_method(self): + np.random.seed(1234) + + @pytest.mark.parametrize('dtype,cdtype,maxnlp', + [(np.float64, np.complex128, 2000), + (np.float32, np.complex64, 3500)]) + def test_definition(self, dtype, cdtype, maxnlp): + rng = np.random.default_rng(1234) + x = np.array([[1, 2, 3], + [4, 5, 6], + [7, 8, 9]], dtype=dtype) + y = ifftn(x) + assert_equal(y.dtype, cdtype) + assert_array_almost_equal_nulp(y, direct_idftn(x), maxnlp) + + x = rng.random((20, 26)) + assert_array_almost_equal_nulp(ifftn(x), direct_idftn(x), maxnlp) + + x = rng.random((5, 4, 3, 20)) + assert_array_almost_equal_nulp(ifftn(x), direct_idftn(x), maxnlp) + + @pytest.mark.parametrize('maxnlp', [2000, 3500]) + @pytest.mark.parametrize('size', [1, 2, 51, 32, 64, 92]) + def test_random_complex(self, maxnlp, size): + rng = np.random.default_rng(1234) + x = rng.random([size, size]) + 1j * rng.random([size, size]) + assert_array_almost_equal_nulp(ifftn(fftn(x)), x, maxnlp) + assert_array_almost_equal_nulp(fftn(ifftn(x)), x, maxnlp) + + def test_invalid_sizes(self): + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[1, 0\]\) specified"): + ifftn([[]]) + + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[4, -3\]\) specified"): + ifftn([[1, 1], [2, 2]], (4, -3)) + + def test_no_axes(self): + x = numpy.random.random((2,2,2)) + assert_allclose(ifftn(x, axes=[]), x, atol=1e-7) + +class TestRfftn: + dtype = None + cdtype = None + + def setup_method(self): + np.random.seed(1234) + + @pytest.mark.parametrize('dtype,cdtype,maxnlp', + [(np.float64, np.complex128, 2000), + (np.float32, np.complex64, 3500)]) + def test_definition(self, dtype, cdtype, maxnlp): + rng = np.random.default_rng(1234) + x = np.array([[1, 2, 3], + [4, 5, 6], + [7, 8, 9]], dtype=dtype) + y = rfftn(x) + assert_equal(y.dtype, cdtype) + assert_array_almost_equal_nulp(y, direct_rdftn(x), maxnlp) + + x = rng.random((20, 26)) + assert_array_almost_equal_nulp(rfftn(x), direct_rdftn(x), maxnlp) + + x = rng.random((5, 4, 3, 20)) + assert_array_almost_equal_nulp(rfftn(x), direct_rdftn(x), maxnlp) + + @pytest.mark.parametrize('size', [1, 2, 51, 32, 64, 92]) + def test_random(self, size): + rng = np.random.default_rng(1234) + x = rng.random([size, size]) + assert_allclose(irfftn(rfftn(x), x.shape), x, atol=1e-10) + + @pytest.mark.parametrize('func', [rfftn, irfftn]) + def test_invalid_sizes(self, func): + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[1, 0\]\) specified"): + func([[]]) + + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[4, -3\]\) specified"): + func([[1, 1], [2, 2]], (4, -3)) + + @pytest.mark.parametrize('func', [rfftn, irfftn]) + def test_no_axes(self, func): + with assert_raises(ValueError, + match="at least 1 axis must be transformed"): + func([], axes=[]) + + def test_complex_input(self): + with assert_raises(TypeError, match="x must be a real sequence"): + rfftn(np.zeros(10, dtype=np.complex64)) + + +class FakeArray: + def __init__(self, data): + self._data = data + self.__array_interface__ = data.__array_interface__ + + +class FakeArray2: + def __init__(self, data): + self._data = data + + def __array__(self, dtype=None, copy=None): + return self._data + +# TODO: Is this test actually valuable? The behavior it's testing shouldn't be +# relied upon by users except for overwrite_x = False +class TestOverwrite: + """Check input overwrite behavior of the FFT functions.""" + + real_dtypes = [np.float32, np.float64, np.longdouble] + dtypes = real_dtypes + [np.complex64, np.complex128, np.clongdouble] + fftsizes = [8, 16, 32] + + def _check(self, x, routine, fftsize, axis, overwrite_x, should_overwrite): + x2 = x.copy() + for fake in [lambda x: x, FakeArray, FakeArray2]: + routine(fake(x2), fftsize, axis, overwrite_x=overwrite_x) + + sig = (f"{routine.__name__}({x.dtype}{x.shape!r}, {fftsize!r}, " + f"axis={axis!r}, overwrite_x={overwrite_x!r})") + if not should_overwrite: + assert_equal(x2, x, err_msg=f"spurious overwrite in {sig}") + + def _check_1d(self, routine, dtype, shape, axis, overwritable_dtypes, + fftsize, overwrite_x): + np.random.seed(1234) + if np.issubdtype(dtype, np.complexfloating): + data = np.random.randn(*shape) + 1j*np.random.randn(*shape) + else: + data = np.random.randn(*shape) + data = data.astype(dtype) + + should_overwrite = (overwrite_x + and dtype in overwritable_dtypes + and fftsize <= shape[axis]) + self._check(data, routine, fftsize, axis, + overwrite_x=overwrite_x, + should_overwrite=should_overwrite) + + @pytest.mark.parametrize('dtype', dtypes) + @pytest.mark.parametrize('fftsize', fftsizes) + @pytest.mark.parametrize('overwrite_x', [True, False]) + @pytest.mark.parametrize('shape,axes', [((16,), -1), + ((16, 2), 0), + ((2, 16), 1)]) + def test_fft_ifft(self, dtype, fftsize, overwrite_x, shape, axes): + overwritable = (np.clongdouble, np.complex128, np.complex64) + self._check_1d(fft, dtype, shape, axes, overwritable, + fftsize, overwrite_x) + self._check_1d(ifft, dtype, shape, axes, overwritable, + fftsize, overwrite_x) + + @pytest.mark.parametrize('dtype', real_dtypes) + @pytest.mark.parametrize('fftsize', fftsizes) + @pytest.mark.parametrize('overwrite_x', [True, False]) + @pytest.mark.parametrize('shape,axes', [((16,), -1), + ((16, 2), 0), + ((2, 16), 1)]) + def test_rfft_irfft(self, dtype, fftsize, overwrite_x, shape, axes): + overwritable = self.real_dtypes + self._check_1d(irfft, dtype, shape, axes, overwritable, + fftsize, overwrite_x) + self._check_1d(rfft, dtype, shape, axes, overwritable, + fftsize, overwrite_x) + + def _check_nd_one(self, routine, dtype, shape, axes, overwritable_dtypes, + overwrite_x): + np.random.seed(1234) + if np.issubdtype(dtype, np.complexfloating): + data = np.random.randn(*shape) + 1j*np.random.randn(*shape) + else: + data = np.random.randn(*shape) + data = data.astype(dtype) + + def fftshape_iter(shp): + if len(shp) <= 0: + yield () + else: + for j in (shp[0]//2, shp[0], shp[0]*2): + for rest in fftshape_iter(shp[1:]): + yield (j,) + rest + + def part_shape(shape, axes): + if axes is None: + return shape + else: + return tuple(np.take(shape, axes)) + + def should_overwrite(data, shape, axes): + s = part_shape(data.shape, axes) + return (overwrite_x and + np.prod(shape) <= np.prod(s) + and dtype in overwritable_dtypes) + + for fftshape in fftshape_iter(part_shape(shape, axes)): + self._check(data, routine, fftshape, axes, + overwrite_x=overwrite_x, + should_overwrite=should_overwrite(data, fftshape, axes)) + if data.ndim > 1: + # check fortran order + self._check(data.T, routine, fftshape, axes, + overwrite_x=overwrite_x, + should_overwrite=should_overwrite( + data.T, fftshape, axes)) + + @pytest.mark.parametrize('dtype', dtypes) + @pytest.mark.parametrize('overwrite_x', [True, False]) + @pytest.mark.parametrize('shape,axes', [((16,), None), + ((16,), (0,)), + ((16, 2), (0,)), + ((2, 16), (1,)), + ((8, 16), None), + ((8, 16), (0, 1)), + ((8, 16, 2), (0, 1)), + ((8, 16, 2), (1, 2)), + ((8, 16, 2), (0,)), + ((8, 16, 2), (1,)), + ((8, 16, 2), (2,)), + ((8, 16, 2), None), + ((8, 16, 2), (0, 1, 2))]) + def test_fftn_ifftn(self, dtype, overwrite_x, shape, axes): + overwritable = (np.clongdouble, np.complex128, np.complex64) + self._check_nd_one(fftn, dtype, shape, axes, overwritable, + overwrite_x) + self._check_nd_one(ifftn, dtype, shape, axes, overwritable, + overwrite_x) + + +@pytest.mark.parametrize('func', [fft, ifft, fftn, ifftn, + rfft, irfft, rfftn, irfftn]) +def test_invalid_norm(func): + x = np.arange(10, dtype=float) + with assert_raises(ValueError, + match='Invalid norm value \'o\', should be' + ' "backward", "ortho" or "forward"'): + func(x, norm='o') + + +@pytest.mark.parametrize('func', [fft, ifft, fftn, ifftn, + irfft, irfftn, hfft, hfftn]) +def test_swapped_byte_order_complex(func): + rng = np.random.RandomState(1234) + x = rng.rand(10) + 1j * rng.rand(10) + assert_allclose(func(swap_byteorder(x)), func(x)) + + +@pytest.mark.parametrize('func', [ihfft, ihfftn, rfft, rfftn]) +def test_swapped_byte_order_real(func): + rng = np.random.RandomState(1234) + x = rng.rand(10) + assert_allclose(func(swap_byteorder(x)), func(x)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/tests/test_real_transforms.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/tests/test_real_transforms.py new file mode 100644 index 0000000000000000000000000000000000000000..38b3f0a7367a9e97a97133f62ddb5dfb223581ed --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_pocketfft/tests/test_real_transforms.py @@ -0,0 +1,505 @@ +from os.path import join, dirname +from collections.abc import Callable +from threading import Lock + +import numpy as np +from numpy.testing import ( + assert_array_almost_equal, assert_equal, assert_allclose) +import pytest +from pytest import raises as assert_raises + +from scipy.fft._pocketfft.realtransforms import ( + dct, idct, dst, idst, dctn, idctn, dstn, idstn) + +fftpack_test_dir = join(dirname(__file__), '..', '..', '..', 'fftpack', 'tests') + +MDATA_COUNT = 8 +FFTWDATA_COUNT = 14 + +def is_longdouble_binary_compatible(): + try: + one = np.frombuffer( + b'\x00\x00\x00\x00\x00\x00\x00\x80\xff\x3f\x00\x00\x00\x00\x00\x00', + dtype=' decimal +dec_map: DecMapType = { + # DCT + (dct, np.float64, 1): 13, + (dct, np.float32, 1): 6, + + (dct, np.float64, 2): 14, + (dct, np.float32, 2): 5, + + (dct, np.float64, 3): 14, + (dct, np.float32, 3): 5, + + (dct, np.float64, 4): 13, + (dct, np.float32, 4): 6, + + # IDCT + (idct, np.float64, 1): 14, + (idct, np.float32, 1): 6, + + (idct, np.float64, 2): 14, + (idct, np.float32, 2): 5, + + (idct, np.float64, 3): 14, + (idct, np.float32, 3): 5, + + (idct, np.float64, 4): 14, + (idct, np.float32, 4): 6, + + # DST + (dst, np.float64, 1): 13, + (dst, np.float32, 1): 6, + + (dst, np.float64, 2): 14, + (dst, np.float32, 2): 6, + + (dst, np.float64, 3): 14, + (dst, np.float32, 3): 7, + + (dst, np.float64, 4): 13, + (dst, np.float32, 4): 5, + + # IDST + (idst, np.float64, 1): 14, + (idst, np.float32, 1): 6, + + (idst, np.float64, 2): 14, + (idst, np.float32, 2): 6, + + (idst, np.float64, 3): 14, + (idst, np.float32, 3): 6, + + (idst, np.float64, 4): 14, + (idst, np.float32, 4): 6, +} + +for k,v in dec_map.copy().items(): + if k[1] == np.float64: + dec_map[(k[0], np.longdouble, k[2])] = v + elif k[1] == np.float32: + dec_map[(k[0], int, k[2])] = v + + +@pytest.mark.parametrize('rdt', [np.longdouble, np.float64, np.float32, int]) +@pytest.mark.parametrize('type', [1, 2, 3, 4]) +class TestDCT: + def test_definition(self, rdt, type, fftwdata_size, + reference_data, ref_lock): + with ref_lock: + x, yr, dt = fftw_dct_ref(type, fftwdata_size, rdt, reference_data) + y = dct(x, type=type) + assert_equal(y.dtype, dt) + dec = dec_map[(dct, rdt, type)] + assert_allclose(y, yr, rtol=0., atol=np.max(yr)*10**(-dec)) + + @pytest.mark.parametrize('size', [7, 8, 9, 16, 32, 64]) + def test_axis(self, rdt, type, size): + nt = 2 + dec = dec_map[(dct, rdt, type)] + x = np.random.randn(nt, size) + y = dct(x, type=type) + for j in range(nt): + assert_array_almost_equal(y[j], dct(x[j], type=type), + decimal=dec) + + x = x.T + y = dct(x, axis=0, type=type) + for j in range(nt): + assert_array_almost_equal(y[:,j], dct(x[:,j], type=type), + decimal=dec) + + +@pytest.mark.parametrize('rdt', [np.longdouble, np.float64, np.float32, int]) +def test_dct1_definition_ortho(rdt, mdata_x): + # Test orthornomal mode. + dec = dec_map[(dct, rdt, 1)] + x = np.array(mdata_x, dtype=rdt) + dt = np.result_type(np.float32, rdt) + y = dct(x, norm='ortho', type=1) + y2 = naive_dct1(x, norm='ortho') + assert_equal(y.dtype, dt) + assert_allclose(y, y2, rtol=0., atol=np.max(y2)*10**(-dec)) + + +@pytest.mark.parametrize('rdt', [np.longdouble, np.float64, np.float32, int]) +def test_dct2_definition_matlab(mdata_xy, rdt): + # Test correspondence with matlab (orthornomal mode). + dt = np.result_type(np.float32, rdt) + x = np.array(mdata_xy[0], dtype=dt) + + yr = mdata_xy[1] + y = dct(x, norm="ortho", type=2) + dec = dec_map[(dct, rdt, 2)] + assert_equal(y.dtype, dt) + assert_array_almost_equal(y, yr, decimal=dec) + + +@pytest.mark.parametrize('rdt', [np.longdouble, np.float64, np.float32, int]) +def test_dct3_definition_ortho(mdata_x, rdt): + # Test orthornomal mode. + x = np.array(mdata_x, dtype=rdt) + dt = np.result_type(np.float32, rdt) + y = dct(x, norm='ortho', type=2) + xi = dct(y, norm="ortho", type=3) + dec = dec_map[(dct, rdt, 3)] + assert_equal(xi.dtype, dt) + assert_array_almost_equal(xi, x, decimal=dec) + + +@pytest.mark.parametrize('rdt', [np.longdouble, np.float64, np.float32, int]) +def test_dct4_definition_ortho(mdata_x, rdt): + # Test orthornomal mode. + x = np.array(mdata_x, dtype=rdt) + dt = np.result_type(np.float32, rdt) + y = dct(x, norm='ortho', type=4) + y2 = naive_dct4(x, norm='ortho') + dec = dec_map[(dct, rdt, 4)] + assert_equal(y.dtype, dt) + assert_allclose(y, y2, rtol=0., atol=np.max(y2)*10**(-dec)) + + +@pytest.mark.parametrize('rdt', [np.longdouble, np.float64, np.float32, int]) +@pytest.mark.parametrize('type', [1, 2, 3, 4]) +def test_idct_definition(fftwdata_size, rdt, type, reference_data, ref_lock): + with ref_lock: + xr, yr, dt = fftw_dct_ref(type, fftwdata_size, rdt, reference_data) + x = idct(yr, type=type) + dec = dec_map[(idct, rdt, type)] + assert_equal(x.dtype, dt) + assert_allclose(x, xr, rtol=0., atol=np.max(xr)*10**(-dec)) + + +@pytest.mark.parametrize('rdt', [np.longdouble, np.float64, np.float32, int]) +@pytest.mark.parametrize('type', [1, 2, 3, 4]) +def test_definition(fftwdata_size, rdt, type, reference_data, ref_lock): + with ref_lock: + xr, yr, dt = fftw_dst_ref(type, fftwdata_size, rdt, reference_data) + y = dst(xr, type=type) + dec = dec_map[(dst, rdt, type)] + assert_equal(y.dtype, dt) + assert_allclose(y, yr, rtol=0., atol=np.max(yr)*10**(-dec)) + + +@pytest.mark.parametrize('rdt', [np.longdouble, np.float64, np.float32, int]) +def test_dst1_definition_ortho(rdt, mdata_x): + # Test orthornomal mode. + dec = dec_map[(dst, rdt, 1)] + x = np.array(mdata_x, dtype=rdt) + dt = np.result_type(np.float32, rdt) + y = dst(x, norm='ortho', type=1) + y2 = naive_dst1(x, norm='ortho') + assert_equal(y.dtype, dt) + assert_allclose(y, y2, rtol=0., atol=np.max(y2)*10**(-dec)) + + +@pytest.mark.parametrize('rdt', [np.longdouble, np.float64, np.float32, int]) +def test_dst4_definition_ortho(rdt, mdata_x): + # Test orthornomal mode. + dec = dec_map[(dst, rdt, 4)] + x = np.array(mdata_x, dtype=rdt) + dt = np.result_type(np.float32, rdt) + y = dst(x, norm='ortho', type=4) + y2 = naive_dst4(x, norm='ortho') + assert_equal(y.dtype, dt) + assert_array_almost_equal(y, y2, decimal=dec) + + +@pytest.mark.parametrize('rdt', [np.longdouble, np.float64, np.float32, int]) +@pytest.mark.parametrize('type', [1, 2, 3, 4]) +def test_idst_definition(fftwdata_size, rdt, type, reference_data, ref_lock): + with ref_lock: + xr, yr, dt = fftw_dst_ref(type, fftwdata_size, rdt, reference_data) + x = idst(yr, type=type) + dec = dec_map[(idst, rdt, type)] + assert_equal(x.dtype, dt) + assert_allclose(x, xr, rtol=0., atol=np.max(xr)*10**(-dec)) + + +@pytest.mark.parametrize('routine', [dct, dst, idct, idst]) +@pytest.mark.parametrize('dtype', [np.float32, np.float64, np.longdouble]) +@pytest.mark.parametrize('shape, axis', [ + ((16,), -1), ((16, 2), 0), ((2, 16), 1) +]) +@pytest.mark.parametrize('type', [1, 2, 3, 4]) +@pytest.mark.parametrize('overwrite_x', [True, False]) +@pytest.mark.parametrize('norm', [None, 'ortho']) +def test_overwrite(routine, dtype, shape, axis, type, norm, overwrite_x): + # Check input overwrite behavior + np.random.seed(1234) + if np.issubdtype(dtype, np.complexfloating): + x = np.random.randn(*shape) + 1j*np.random.randn(*shape) + else: + x = np.random.randn(*shape) + x = x.astype(dtype) + x2 = x.copy() + routine(x2, type, None, axis, norm, overwrite_x=overwrite_x) + + sig = (f"{routine.__name__}({x.dtype}{x.shape!r}, {None!r}, axis={axis!r}, " + f"overwrite_x={overwrite_x!r})") + if not overwrite_x: + assert_equal(x2, x, err_msg=f"spurious overwrite in {sig}") + + +class Test_DCTN_IDCTN: + dec = 14 + dct_type = [1, 2, 3, 4] + norms = [None, 'backward', 'ortho', 'forward'] + rstate = np.random.RandomState(1234) + shape = (32, 16) + data = rstate.randn(*shape) + + @pytest.mark.parametrize('fforward,finverse', [(dctn, idctn), + (dstn, idstn)]) + @pytest.mark.parametrize('axes', [None, + 1, (1,), [1], + 0, (0,), [0], + (0, 1), [0, 1], + (-2, -1), [-2, -1]]) + @pytest.mark.parametrize('dct_type', dct_type) + @pytest.mark.parametrize('norm', ['ortho']) + def test_axes_round_trip(self, fforward, finverse, axes, dct_type, norm): + tmp = fforward(self.data, type=dct_type, axes=axes, norm=norm) + tmp = finverse(tmp, type=dct_type, axes=axes, norm=norm) + assert_array_almost_equal(self.data, tmp, decimal=12) + + @pytest.mark.parametrize('funcn,func', [(dctn, dct), (dstn, dst)]) + @pytest.mark.parametrize('dct_type', dct_type) + @pytest.mark.parametrize('norm', norms) + def test_dctn_vs_2d_reference(self, funcn, func, dct_type, norm): + y1 = funcn(self.data, type=dct_type, axes=None, norm=norm) + y2 = ref_2d(func, self.data, type=dct_type, norm=norm) + assert_array_almost_equal(y1, y2, decimal=11) + + @pytest.mark.parametrize('funcn,func', [(idctn, idct), (idstn, idst)]) + @pytest.mark.parametrize('dct_type', dct_type) + @pytest.mark.parametrize('norm', norms) + def test_idctn_vs_2d_reference(self, funcn, func, dct_type, norm): + fdata = dctn(self.data, type=dct_type, norm=norm) + y1 = funcn(fdata, type=dct_type, norm=norm) + y2 = ref_2d(func, fdata, type=dct_type, norm=norm) + assert_array_almost_equal(y1, y2, decimal=11) + + @pytest.mark.parametrize('fforward,finverse', [(dctn, idctn), + (dstn, idstn)]) + def test_axes_and_shape(self, fforward, finverse): + with assert_raises(ValueError, + match="when given, axes and shape arguments" + " have to be of the same length"): + fforward(self.data, s=self.data.shape[0], axes=(0, 1)) + + with assert_raises(ValueError, + match="when given, axes and shape arguments" + " have to be of the same length"): + fforward(self.data, s=self.data.shape, axes=0) + + @pytest.mark.parametrize('fforward', [dctn, dstn]) + def test_shape(self, fforward): + tmp = fforward(self.data, s=(128, 128), axes=None) + assert_equal(tmp.shape, (128, 128)) + + @pytest.mark.parametrize('fforward,finverse', [(dctn, idctn), + (dstn, idstn)]) + @pytest.mark.parametrize('axes', [1, (1,), [1], + 0, (0,), [0]]) + def test_shape_is_none_with_axes(self, fforward, finverse, axes): + tmp = fforward(self.data, s=None, axes=axes, norm='ortho') + tmp = finverse(tmp, s=None, axes=axes, norm='ortho') + assert_array_almost_equal(self.data, tmp, decimal=self.dec) + + +@pytest.mark.parametrize('func', [dct, dctn, idct, idctn, + dst, dstn, idst, idstn]) +def test_swapped_byte_order(func): + rng = np.random.RandomState(1234) + x = rng.rand(10) + swapped_dt = x.dtype.newbyteorder('S') + assert_allclose(func(x.astype(swapped_dt)), func(x)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_realtransforms.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_realtransforms.py new file mode 100644 index 0000000000000000000000000000000000000000..1c7a3d683dd78d3227a7de88f5c47569d2f4e17f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_realtransforms.py @@ -0,0 +1,693 @@ +from ._basic import _dispatch +from scipy._lib.uarray import Dispatchable +import numpy as np + +__all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn'] + + +@_dispatch +def dctn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, + workers=None, *, orthogonalize=None): + """ + Return multidimensional Discrete Cosine Transform along the specified axes. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DCT (see Notes). Default type is 2. + s : int or array_like of ints or None, optional + The shape of the result. If both `s` and `axes` (see below) are None, + `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is + ``numpy.take(x.shape, axes, axis=0)``. + If ``s[i] > x.shape[i]``, the ith dimension of the input is padded with zeros. + If ``s[i] < x.shape[i]``, the ith dimension of the input is truncated to length + ``s[i]``. + If any element of `s` is -1, the size of the corresponding dimension of + `x` is used. + axes : int or array_like of ints or None, optional + Axes over which the DCT is computed. If not given, the last ``len(s)`` + axes are used, or all axes if `s` is also not specified. + norm : {"backward", "ortho", "forward"}, optional + Normalization mode (see Notes). Default is "backward". + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + workers : int, optional + Maximum number of workers to use for parallel computation. If negative, + the value wraps around from ``os.cpu_count()``. + See :func:`~scipy.fft.fft` for more details. + orthogonalize : bool, optional + Whether to use the orthogonalized DCT variant (see Notes). + Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. + + .. versionadded:: 1.8.0 + + Returns + ------- + y : ndarray of real + The transformed input array. + + See Also + -------- + idctn : Inverse multidimensional DCT + + Notes + ----- + For full details of the DCT types and normalization modes, as well as + references, see `dct`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fft import dctn, idctn + >>> rng = np.random.default_rng() + >>> y = rng.standard_normal((16, 16)) + >>> np.allclose(y, idctn(dctn(y))) + True + + """ + return (Dispatchable(x, np.ndarray),) + + +@_dispatch +def idctn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, + workers=None, orthogonalize=None): + """ + Return multidimensional Inverse Discrete Cosine Transform along the specified axes. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DCT (see Notes). Default type is 2. + s : int or array_like of ints or None, optional + The shape of the result. If both `s` and `axes` (see below) are + None, `s` is ``x.shape``; if `s` is None but `axes` is + not None, then `s` is ``numpy.take(x.shape, axes, axis=0)``. + If ``s[i] > x.shape[i]``, the ith dimension of the input is padded with zeros. + If ``s[i] < x.shape[i]``, the ith dimension of the input is truncated to length + ``s[i]``. + If any element of `s` is -1, the size of the corresponding dimension of + `x` is used. + axes : int or array_like of ints or None, optional + Axes over which the IDCT is computed. If not given, the last ``len(s)`` + axes are used, or all axes if `s` is also not specified. + norm : {"backward", "ortho", "forward"}, optional + Normalization mode (see Notes). Default is "backward". + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + workers : int, optional + Maximum number of workers to use for parallel computation. If negative, + the value wraps around from ``os.cpu_count()``. + See :func:`~scipy.fft.fft` for more details. + orthogonalize : bool, optional + Whether to use the orthogonalized IDCT variant (see Notes). + Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. + + .. versionadded:: 1.8.0 + + Returns + ------- + y : ndarray of real + The transformed input array. + + See Also + -------- + dctn : multidimensional DCT + + Notes + ----- + For full details of the IDCT types and normalization modes, as well as + references, see `idct`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fft import dctn, idctn + >>> rng = np.random.default_rng() + >>> y = rng.standard_normal((16, 16)) + >>> np.allclose(y, idctn(dctn(y))) + True + + """ + return (Dispatchable(x, np.ndarray),) + + +@_dispatch +def dstn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, + workers=None, orthogonalize=None): + """ + Return multidimensional Discrete Sine Transform along the specified axes. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DST (see Notes). Default type is 2. + s : int or array_like of ints or None, optional + The shape of the result. If both `s` and `axes` (see below) are None, + `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is + ``numpy.take(x.shape, axes, axis=0)``. + If ``s[i] > x.shape[i]``, the ith dimension of the input is padded with zeros. + If ``s[i] < x.shape[i]``, the ith dimension of the input is truncated to length + ``s[i]``. + If any element of `shape` is -1, the size of the corresponding dimension + of `x` is used. + axes : int or array_like of ints or None, optional + Axes over which the DST is computed. If not given, the last ``len(s)`` + axes are used, or all axes if `s` is also not specified. + norm : {"backward", "ortho", "forward"}, optional + Normalization mode (see Notes). Default is "backward". + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + workers : int, optional + Maximum number of workers to use for parallel computation. If negative, + the value wraps around from ``os.cpu_count()``. + See :func:`~scipy.fft.fft` for more details. + orthogonalize : bool, optional + Whether to use the orthogonalized DST variant (see Notes). + Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. + + .. versionadded:: 1.8.0 + + Returns + ------- + y : ndarray of real + The transformed input array. + + See Also + -------- + idstn : Inverse multidimensional DST + + Notes + ----- + For full details of the DST types and normalization modes, as well as + references, see `dst`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fft import dstn, idstn + >>> rng = np.random.default_rng() + >>> y = rng.standard_normal((16, 16)) + >>> np.allclose(y, idstn(dstn(y))) + True + + """ + return (Dispatchable(x, np.ndarray),) + + +@_dispatch +def idstn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, + workers=None, orthogonalize=None): + """ + Return multidimensional Inverse Discrete Sine Transform along the specified axes. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DST (see Notes). Default type is 2. + s : int or array_like of ints or None, optional + The shape of the result. If both `s` and `axes` (see below) are None, + `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is + ``numpy.take(x.shape, axes, axis=0)``. + If ``s[i] > x.shape[i]``, the ith dimension of the input is padded with zeros. + If ``s[i] < x.shape[i]``, the ith dimension of the input is truncated to length + ``s[i]``. + If any element of `s` is -1, the size of the corresponding dimension of + `x` is used. + axes : int or array_like of ints or None, optional + Axes over which the IDST is computed. If not given, the last ``len(s)`` + axes are used, or all axes if `s` is also not specified. + norm : {"backward", "ortho", "forward"}, optional + Normalization mode (see Notes). Default is "backward". + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + workers : int, optional + Maximum number of workers to use for parallel computation. If negative, + the value wraps around from ``os.cpu_count()``. + See :func:`~scipy.fft.fft` for more details. + orthogonalize : bool, optional + Whether to use the orthogonalized IDST variant (see Notes). + Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. + + .. versionadded:: 1.8.0 + + Returns + ------- + y : ndarray of real + The transformed input array. + + See Also + -------- + dstn : multidimensional DST + + Notes + ----- + For full details of the IDST types and normalization modes, as well as + references, see `idst`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fft import dstn, idstn + >>> rng = np.random.default_rng() + >>> y = rng.standard_normal((16, 16)) + >>> np.allclose(y, idstn(dstn(y))) + True + + """ + return (Dispatchable(x, np.ndarray),) + + +@_dispatch +def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, + orthogonalize=None): + r"""Return the Discrete Cosine Transform of arbitrary type sequence x. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DCT (see Notes). Default type is 2. + n : int, optional + Length of the transform. If ``n < x.shape[axis]``, `x` is + truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The + default results in ``n = x.shape[axis]``. + axis : int, optional + Axis along which the dct is computed; the default is over the + last axis (i.e., ``axis=-1``). + norm : {"backward", "ortho", "forward"}, optional + Normalization mode (see Notes). Default is "backward". + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + workers : int, optional + Maximum number of workers to use for parallel computation. If negative, + the value wraps around from ``os.cpu_count()``. + See :func:`~scipy.fft.fft` for more details. + orthogonalize : bool, optional + Whether to use the orthogonalized DCT variant (see Notes). + Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. + + .. versionadded:: 1.8.0 + + Returns + ------- + y : ndarray of real + The transformed input array. + + See Also + -------- + idct : Inverse DCT + + Notes + ----- + For a single dimension array ``x``, ``dct(x, norm='ortho')`` is equal to + MATLAB ``dct(x)``. + + .. warning:: For ``type in {1, 2, 3}``, ``norm="ortho"`` breaks the direct + correspondence with the direct Fourier transform. To recover + it you must specify ``orthogonalize=False``. + + For ``norm="ortho"`` both the `dct` and `idct` are scaled by the same + overall factor in both directions. By default, the transform is also + orthogonalized which for types 1, 2 and 3 means the transform definition is + modified to give orthogonality of the DCT matrix (see below). + + For ``norm="backward"``, there is no scaling on `dct` and the `idct` is + scaled by ``1/N`` where ``N`` is the "logical" size of the DCT. For + ``norm="forward"`` the ``1/N`` normalization is applied to the forward + `dct` instead and the `idct` is unnormalized. + + There are, theoretically, 8 types of the DCT, only the first 4 types are + implemented in SciPy.'The' DCT generally refers to DCT type 2, and 'the' + Inverse DCT generally refers to DCT type 3. + + **Type I** + + There are several definitions of the DCT-I; we use the following + (for ``norm="backward"``) + + .. math:: + + y_k = x_0 + (-1)^k x_{N-1} + 2 \sum_{n=1}^{N-2} x_n \cos\left( + \frac{\pi k n}{N-1} \right) + + If ``orthogonalize=True``, ``x[0]`` and ``x[N-1]`` are multiplied by a + scaling factor of :math:`\sqrt{2}`, and ``y[0]`` and ``y[N-1]`` are divided + by :math:`\sqrt{2}`. When combined with ``norm="ortho"``, this makes the + corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(N)``). + + .. note:: + The DCT-I is only supported for input size > 1. + + **Type II** + + There are several definitions of the DCT-II; we use the following + (for ``norm="backward"``) + + .. math:: + + y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right) + + If ``orthogonalize=True``, ``y[0]`` is divided by :math:`\sqrt{2}` which, + when combined with ``norm="ortho"``, makes the corresponding matrix of + coefficients orthonormal (``O @ O.T = np.eye(N)``). + + **Type III** + + There are several definitions, we use the following (for + ``norm="backward"``) + + .. math:: + + y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right) + + If ``orthogonalize=True``, ``x[0]`` terms are multiplied by + :math:`\sqrt{2}` which, when combined with ``norm="ortho"``, makes the + corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(N)``). + + The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up + to a factor `2N`. The orthonormalized DCT-III is exactly the inverse of + the orthonormalized DCT-II. + + **Type IV** + + There are several definitions of the DCT-IV; we use the following + (for ``norm="backward"``) + + .. math:: + + y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right) + + ``orthogonalize`` has no effect here, as the DCT-IV matrix is already + orthogonal up to a scale factor of ``2N``. + + References + ---------- + .. [1] 'A Fast Cosine Transform in One and Two Dimensions', by J. + Makhoul, `IEEE Transactions on acoustics, speech and signal + processing` vol. 28(1), pp. 27-34, + :doi:`10.1109/TASSP.1980.1163351` (1980). + .. [2] Wikipedia, "Discrete cosine transform", + https://en.wikipedia.org/wiki/Discrete_cosine_transform + + Examples + -------- + The Type 1 DCT is equivalent to the FFT (though faster) for real, + even-symmetrical inputs. The output is also real and even-symmetrical. + Half of the FFT input is used to generate half of the FFT output: + + >>> from scipy.fft import fft, dct + >>> import numpy as np + >>> fft(np.array([4., 3., 5., 10., 5., 3.])).real + array([ 30., -8., 6., -2., 6., -8.]) + >>> dct(np.array([4., 3., 5., 10.]), 1) + array([ 30., -8., 6., -2.]) + + """ + return (Dispatchable(x, np.ndarray),) + + +@_dispatch +def idct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, + workers=None, orthogonalize=None): + """ + Return the Inverse Discrete Cosine Transform of an arbitrary type sequence. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DCT (see Notes). Default type is 2. + n : int, optional + Length of the transform. If ``n < x.shape[axis]``, `x` is + truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The + default results in ``n = x.shape[axis]``. + axis : int, optional + Axis along which the idct is computed; the default is over the + last axis (i.e., ``axis=-1``). + norm : {"backward", "ortho", "forward"}, optional + Normalization mode (see Notes). Default is "backward". + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + workers : int, optional + Maximum number of workers to use for parallel computation. If negative, + the value wraps around from ``os.cpu_count()``. + See :func:`~scipy.fft.fft` for more details. + orthogonalize : bool, optional + Whether to use the orthogonalized IDCT variant (see Notes). + Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. + + .. versionadded:: 1.8.0 + + Returns + ------- + idct : ndarray of real + The transformed input array. + + See Also + -------- + dct : Forward DCT + + Notes + ----- + For a single dimension array `x`, ``idct(x, norm='ortho')`` is equal to + MATLAB ``idct(x)``. + + .. warning:: For ``type in {1, 2, 3}``, ``norm="ortho"`` breaks the direct + correspondence with the inverse direct Fourier transform. To + recover it you must specify ``orthogonalize=False``. + + For ``norm="ortho"`` both the `dct` and `idct` are scaled by the same + overall factor in both directions. By default, the transform is also + orthogonalized which for types 1, 2 and 3 means the transform definition is + modified to give orthogonality of the IDCT matrix (see `dct` for the full + definitions). + + 'The' IDCT is the IDCT-II, which is the same as the normalized DCT-III. + + The IDCT is equivalent to a normal DCT except for the normalization and + type. DCT type 1 and 4 are their own inverse and DCTs 2 and 3 are each + other's inverses. + + Examples + -------- + The Type 1 DCT is equivalent to the DFT for real, even-symmetrical + inputs. The output is also real and even-symmetrical. Half of the IFFT + input is used to generate half of the IFFT output: + + >>> from scipy.fft import ifft, idct + >>> import numpy as np + >>> ifft(np.array([ 30., -8., 6., -2., 6., -8.])).real + array([ 4., 3., 5., 10., 5., 3.]) + >>> idct(np.array([ 30., -8., 6., -2.]), 1) + array([ 4., 3., 5., 10.]) + + """ + return (Dispatchable(x, np.ndarray),) + + +@_dispatch +def dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None, + orthogonalize=None): + r""" + Return the Discrete Sine Transform of arbitrary type sequence x. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DST (see Notes). Default type is 2. + n : int, optional + Length of the transform. If ``n < x.shape[axis]``, `x` is + truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The + default results in ``n = x.shape[axis]``. + axis : int, optional + Axis along which the dst is computed; the default is over the + last axis (i.e., ``axis=-1``). + norm : {"backward", "ortho", "forward"}, optional + Normalization mode (see Notes). Default is "backward". + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + workers : int, optional + Maximum number of workers to use for parallel computation. If negative, + the value wraps around from ``os.cpu_count()``. + See :func:`~scipy.fft.fft` for more details. + orthogonalize : bool, optional + Whether to use the orthogonalized DST variant (see Notes). + Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. + + .. versionadded:: 1.8.0 + + Returns + ------- + dst : ndarray of reals + The transformed input array. + + See Also + -------- + idst : Inverse DST + + Notes + ----- + .. warning:: For ``type in {2, 3}``, ``norm="ortho"`` breaks the direct + correspondence with the direct Fourier transform. To recover + it you must specify ``orthogonalize=False``. + + For ``norm="ortho"`` both the `dst` and `idst` are scaled by the same + overall factor in both directions. By default, the transform is also + orthogonalized which for types 2 and 3 means the transform definition is + modified to give orthogonality of the DST matrix (see below). + + For ``norm="backward"``, there is no scaling on the `dst` and the `idst` is + scaled by ``1/N`` where ``N`` is the "logical" size of the DST. + + There are, theoretically, 8 types of the DST for different combinations of + even/odd boundary conditions and boundary off sets [1]_, only the first + 4 types are implemented in SciPy. + + **Type I** + + There are several definitions of the DST-I; we use the following for + ``norm="backward"``. DST-I assumes the input is odd around :math:`n=-1` and + :math:`n=N`. + + .. math:: + + y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(n+1)}{N+1}\right) + + Note that the DST-I is only supported for input size > 1. + The (unnormalized) DST-I is its own inverse, up to a factor :math:`2(N+1)`. + The orthonormalized DST-I is exactly its own inverse. + + ``orthogonalize`` has no effect here, as the DST-I matrix is already + orthogonal up to a scale factor of ``2N``. + + **Type II** + + There are several definitions of the DST-II; we use the following for + ``norm="backward"``. DST-II assumes the input is odd around :math:`n=-1/2` and + :math:`n=N-1/2`; the output is odd around :math:`k=-1` and even around :math:`k=N-1` + + .. math:: + + y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(2n+1)}{2N}\right) + + If ``orthogonalize=True``, ``y[-1]`` is divided :math:`\sqrt{2}` which, when + combined with ``norm="ortho"``, makes the corresponding matrix of + coefficients orthonormal (``O @ O.T = np.eye(N)``). + + **Type III** + + There are several definitions of the DST-III, we use the following (for + ``norm="backward"``). DST-III assumes the input is odd around :math:`n=-1` and + even around :math:`n=N-1` + + .. math:: + + y_k = (-1)^k x_{N-1} + 2 \sum_{n=0}^{N-2} x_n \sin\left( + \frac{\pi(2k+1)(n+1)}{2N}\right) + + If ``orthogonalize=True``, ``x[-1]`` is multiplied by :math:`\sqrt{2}` + which, when combined with ``norm="ortho"``, makes the corresponding matrix + of coefficients orthonormal (``O @ O.T = np.eye(N)``). + + The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up + to a factor :math:`2N`. The orthonormalized DST-III is exactly the inverse of the + orthonormalized DST-II. + + **Type IV** + + There are several definitions of the DST-IV, we use the following (for + ``norm="backward"``). DST-IV assumes the input is odd around :math:`n=-0.5` and + even around :math:`n=N-0.5` + + .. math:: + + y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right) + + ``orthogonalize`` has no effect here, as the DST-IV matrix is already + orthogonal up to a scale factor of ``2N``. + + The (unnormalized) DST-IV is its own inverse, up to a factor :math:`2N`. The + orthonormalized DST-IV is exactly its own inverse. + + References + ---------- + .. [1] Wikipedia, "Discrete sine transform", + https://en.wikipedia.org/wiki/Discrete_sine_transform + + """ + return (Dispatchable(x, np.ndarray),) + + +@_dispatch +def idst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, + workers=None, orthogonalize=None): + """ + Return the Inverse Discrete Sine Transform of an arbitrary type sequence. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DST (see Notes). Default type is 2. + n : int, optional + Length of the transform. If ``n < x.shape[axis]``, `x` is + truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The + default results in ``n = x.shape[axis]``. + axis : int, optional + Axis along which the idst is computed; the default is over the + last axis (i.e., ``axis=-1``). + norm : {"backward", "ortho", "forward"}, optional + Normalization mode (see Notes). Default is "backward". + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + workers : int, optional + Maximum number of workers to use for parallel computation. If negative, + the value wraps around from ``os.cpu_count()``. + See :func:`~scipy.fft.fft` for more details. + orthogonalize : bool, optional + Whether to use the orthogonalized IDST variant (see Notes). + Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. + + .. versionadded:: 1.8.0 + + Returns + ------- + idst : ndarray of real + The transformed input array. + + See Also + -------- + dst : Forward DST + + Notes + ----- + .. warning:: For ``type in {2, 3}``, ``norm="ortho"`` breaks the direct + correspondence with the inverse direct Fourier transform. + + For ``norm="ortho"`` both the `dst` and `idst` are scaled by the same + overall factor in both directions. By default, the transform is also + orthogonalized which for types 2 and 3 means the transform definition is + modified to give orthogonality of the DST matrix (see `dst` for the full + definitions). + + 'The' IDST is the IDST-II, which is the same as the normalized DST-III. + + The IDST is equivalent to a normal DST except for the normalization and + type. DST type 1 and 4 are their own inverse and DSTs 2 and 3 are each + other's inverses. + + """ + return (Dispatchable(x, np.ndarray),) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_realtransforms_backend.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_realtransforms_backend.py new file mode 100644 index 0000000000000000000000000000000000000000..2042453733bec54860974cc1e20ba908e8c9b94d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/_realtransforms_backend.py @@ -0,0 +1,63 @@ +from scipy._lib._array_api import array_namespace +import numpy as np +from . import _pocketfft + +__all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn'] + + +def _execute(pocketfft_func, x, type, s, axes, norm, + overwrite_x, workers, orthogonalize): + xp = array_namespace(x) + x = np.asarray(x) + y = pocketfft_func(x, type, s, axes, norm, + overwrite_x=overwrite_x, workers=workers, + orthogonalize=orthogonalize) + return xp.asarray(y) + + +def dctn(x, type=2, s=None, axes=None, norm=None, + overwrite_x=False, workers=None, *, orthogonalize=None): + return _execute(_pocketfft.dctn, x, type, s, axes, norm, + overwrite_x, workers, orthogonalize) + + +def idctn(x, type=2, s=None, axes=None, norm=None, + overwrite_x=False, workers=None, *, orthogonalize=None): + return _execute(_pocketfft.idctn, x, type, s, axes, norm, + overwrite_x, workers, orthogonalize) + + +def dstn(x, type=2, s=None, axes=None, norm=None, + overwrite_x=False, workers=None, orthogonalize=None): + return _execute(_pocketfft.dstn, x, type, s, axes, norm, + overwrite_x, workers, orthogonalize) + + +def idstn(x, type=2, s=None, axes=None, norm=None, + overwrite_x=False, workers=None, *, orthogonalize=None): + return _execute(_pocketfft.idstn, x, type, s, axes, norm, + overwrite_x, workers, orthogonalize) + + +def dct(x, type=2, n=None, axis=-1, norm=None, + overwrite_x=False, workers=None, orthogonalize=None): + return _execute(_pocketfft.dct, x, type, n, axis, norm, + overwrite_x, workers, orthogonalize) + + +def idct(x, type=2, n=None, axis=-1, norm=None, + overwrite_x=False, workers=None, orthogonalize=None): + return _execute(_pocketfft.idct, x, type, n, axis, norm, + overwrite_x, workers, orthogonalize) + + +def dst(x, type=2, n=None, axis=-1, norm=None, + overwrite_x=False, workers=None, orthogonalize=None): + return _execute(_pocketfft.dst, x, type, n, axis, norm, + overwrite_x, workers, orthogonalize) + + +def idst(x, type=2, n=None, axis=-1, norm=None, + overwrite_x=False, workers=None, orthogonalize=None): + return _execute(_pocketfft.idst, x, type, n, axis, norm, + overwrite_x, workers, orthogonalize) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/mock_backend.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/mock_backend.py new file mode 100644 index 0000000000000000000000000000000000000000..48a7d2b3b50501b84f7c18e366ad2e66782b4ab6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/mock_backend.py @@ -0,0 +1,96 @@ +import numpy as np +import scipy.fft +import threading + +class _MockFunction: + def __init__(self, return_value = None): + self.number_calls = threading.local() + self.return_value = return_value + self.last_args = threading.local() + + def __call__(self, *args, **kwargs): + if not hasattr(self.number_calls, 'c'): + self.number_calls.c = 0 + + self.number_calls.c += 1 + self.last_args.l = (args, kwargs) + return self.return_value + + +fft = _MockFunction(np.random.random(10)) +fft2 = _MockFunction(np.random.random(10)) +fftn = _MockFunction(np.random.random(10)) + +ifft = _MockFunction(np.random.random(10)) +ifft2 = _MockFunction(np.random.random(10)) +ifftn = _MockFunction(np.random.random(10)) + +rfft = _MockFunction(np.random.random(10)) +rfft2 = _MockFunction(np.random.random(10)) +rfftn = _MockFunction(np.random.random(10)) + +irfft = _MockFunction(np.random.random(10)) +irfft2 = _MockFunction(np.random.random(10)) +irfftn = _MockFunction(np.random.random(10)) + +hfft = _MockFunction(np.random.random(10)) +hfft2 = _MockFunction(np.random.random(10)) +hfftn = _MockFunction(np.random.random(10)) + +ihfft = _MockFunction(np.random.random(10)) +ihfft2 = _MockFunction(np.random.random(10)) +ihfftn = _MockFunction(np.random.random(10)) + +dct = _MockFunction(np.random.random(10)) +idct = _MockFunction(np.random.random(10)) +dctn = _MockFunction(np.random.random(10)) +idctn = _MockFunction(np.random.random(10)) + +dst = _MockFunction(np.random.random(10)) +idst = _MockFunction(np.random.random(10)) +dstn = _MockFunction(np.random.random(10)) +idstn = _MockFunction(np.random.random(10)) + +fht = _MockFunction(np.random.random(10)) +ifht = _MockFunction(np.random.random(10)) + + +__ua_domain__ = "numpy.scipy.fft" + + +_implements = { + scipy.fft.fft: fft, + scipy.fft.fft2: fft2, + scipy.fft.fftn: fftn, + scipy.fft.ifft: ifft, + scipy.fft.ifft2: ifft2, + scipy.fft.ifftn: ifftn, + scipy.fft.rfft: rfft, + scipy.fft.rfft2: rfft2, + scipy.fft.rfftn: rfftn, + scipy.fft.irfft: irfft, + scipy.fft.irfft2: irfft2, + scipy.fft.irfftn: irfftn, + scipy.fft.hfft: hfft, + scipy.fft.hfft2: hfft2, + scipy.fft.hfftn: hfftn, + scipy.fft.ihfft: ihfft, + scipy.fft.ihfft2: ihfft2, + scipy.fft.ihfftn: ihfftn, + scipy.fft.dct: dct, + scipy.fft.idct: idct, + scipy.fft.dctn: dctn, + scipy.fft.idctn: idctn, + scipy.fft.dst: dst, + scipy.fft.idst: idst, + scipy.fft.dstn: dstn, + scipy.fft.idstn: idstn, + scipy.fft.fht: fht, + scipy.fft.ifht: ifht +} + + +def __ua_function__(method, args, kwargs): + fn = _implements.get(method) + return (fn(*args, **kwargs) if fn is not None + else NotImplemented) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_backend.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_backend.py new file mode 100644 index 0000000000000000000000000000000000000000..933e9c0302d46faf33b3dc015d58996e3e46c058 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_backend.py @@ -0,0 +1,98 @@ +from functools import partial + +import numpy as np +import scipy.fft +from scipy.fft import _fftlog, _pocketfft, set_backend +from scipy.fft.tests import mock_backend + +from numpy.testing import assert_allclose, assert_equal +import pytest + +fnames = ('fft', 'fft2', 'fftn', + 'ifft', 'ifft2', 'ifftn', + 'rfft', 'rfft2', 'rfftn', + 'irfft', 'irfft2', 'irfftn', + 'dct', 'idct', 'dctn', 'idctn', + 'dst', 'idst', 'dstn', 'idstn', + 'fht', 'ifht') + +np_funcs = (np.fft.fft, np.fft.fft2, np.fft.fftn, + np.fft.ifft, np.fft.ifft2, np.fft.ifftn, + np.fft.rfft, np.fft.rfft2, np.fft.rfftn, + np.fft.irfft, np.fft.irfft2, np.fft.irfftn, + np.fft.hfft, _pocketfft.hfft2, _pocketfft.hfftn, # np has no hfftn + np.fft.ihfft, _pocketfft.ihfft2, _pocketfft.ihfftn, + _pocketfft.dct, _pocketfft.idct, _pocketfft.dctn, _pocketfft.idctn, + _pocketfft.dst, _pocketfft.idst, _pocketfft.dstn, _pocketfft.idstn, + # must provide required kwargs for fht, ifht + partial(_fftlog.fht, dln=2, mu=0.5), + partial(_fftlog.ifht, dln=2, mu=0.5)) + +funcs = (scipy.fft.fft, scipy.fft.fft2, scipy.fft.fftn, + scipy.fft.ifft, scipy.fft.ifft2, scipy.fft.ifftn, + scipy.fft.rfft, scipy.fft.rfft2, scipy.fft.rfftn, + scipy.fft.irfft, scipy.fft.irfft2, scipy.fft.irfftn, + scipy.fft.hfft, scipy.fft.hfft2, scipy.fft.hfftn, + scipy.fft.ihfft, scipy.fft.ihfft2, scipy.fft.ihfftn, + scipy.fft.dct, scipy.fft.idct, scipy.fft.dctn, scipy.fft.idctn, + scipy.fft.dst, scipy.fft.idst, scipy.fft.dstn, scipy.fft.idstn, + # must provide required kwargs for fht, ifht + partial(scipy.fft.fht, dln=2, mu=0.5), + partial(scipy.fft.ifht, dln=2, mu=0.5)) + +mocks = (mock_backend.fft, mock_backend.fft2, mock_backend.fftn, + mock_backend.ifft, mock_backend.ifft2, mock_backend.ifftn, + mock_backend.rfft, mock_backend.rfft2, mock_backend.rfftn, + mock_backend.irfft, mock_backend.irfft2, mock_backend.irfftn, + mock_backend.hfft, mock_backend.hfft2, mock_backend.hfftn, + mock_backend.ihfft, mock_backend.ihfft2, mock_backend.ihfftn, + mock_backend.dct, mock_backend.idct, + mock_backend.dctn, mock_backend.idctn, + mock_backend.dst, mock_backend.idst, + mock_backend.dstn, mock_backend.idstn, + mock_backend.fht, mock_backend.ifht) + + +@pytest.mark.parametrize("func, np_func, mock", zip(funcs, np_funcs, mocks)) +def test_backend_call(func, np_func, mock): + x = np.arange(20).reshape((10,2)) + answer = np_func(x.astype(np.float64)) + assert_allclose(func(x), answer, atol=1e-10) + + with set_backend(mock_backend, only=True): + mock.number_calls.c = 0 + y = func(x) + assert_equal(y, mock.return_value) + assert_equal(mock.number_calls.c, 1) + + assert_allclose(func(x), answer, atol=1e-10) + + +plan_funcs = (scipy.fft.fft, scipy.fft.fft2, scipy.fft.fftn, + scipy.fft.ifft, scipy.fft.ifft2, scipy.fft.ifftn, + scipy.fft.rfft, scipy.fft.rfft2, scipy.fft.rfftn, + scipy.fft.irfft, scipy.fft.irfft2, scipy.fft.irfftn, + scipy.fft.hfft, scipy.fft.hfft2, scipy.fft.hfftn, + scipy.fft.ihfft, scipy.fft.ihfft2, scipy.fft.ihfftn) + +plan_mocks = (mock_backend.fft, mock_backend.fft2, mock_backend.fftn, + mock_backend.ifft, mock_backend.ifft2, mock_backend.ifftn, + mock_backend.rfft, mock_backend.rfft2, mock_backend.rfftn, + mock_backend.irfft, mock_backend.irfft2, mock_backend.irfftn, + mock_backend.hfft, mock_backend.hfft2, mock_backend.hfftn, + mock_backend.ihfft, mock_backend.ihfft2, mock_backend.ihfftn) + + +@pytest.mark.parametrize("func, mock", zip(plan_funcs, plan_mocks)) +def test_backend_plan(func, mock): + x = np.arange(20).reshape((10, 2)) + + with pytest.raises(NotImplementedError, match='precomputed plan'): + func(x, plan='foo') + + with set_backend(mock_backend, only=True): + mock.number_calls.c = 0 + y = func(x, plan='foo') + assert_equal(y, mock.return_value) + assert_equal(mock.number_calls.c, 1) + assert_equal(mock.last_args.l[1]['plan'], 'foo') diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_basic.py new file mode 100644 index 0000000000000000000000000000000000000000..4ed32d54c8936b8b36ff52ef7b639d05338a783c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_basic.py @@ -0,0 +1,502 @@ +import queue +import threading +import multiprocessing +import numpy as np +import pytest +from numpy.random import random +from numpy.testing import assert_array_almost_equal, assert_allclose +from pytest import raises as assert_raises +import scipy.fft as fft +from scipy.conftest import array_api_compatible +from scipy._lib._array_api import ( + array_namespace, xp_size, xp_assert_close, xp_assert_equal +) + +pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends")] +skip_xp_backends = pytest.mark.skip_xp_backends + + +# Expected input dtypes. Note that `scipy.fft` is more flexible for numpy, +# but for C2C transforms like `fft.fft`, the array API standard only mandates +# that complex dtypes should work, float32/float64 aren't guaranteed to. +def get_expected_input_dtype(func, xp): + if func in [fft.fft, fft.fftn, fft.fft2, + fft.ifft, fft.ifftn, fft.ifft2, + fft.hfft, fft.hfftn, fft.hfft2, + fft.irfft, fft.irfftn, fft.irfft2]: + dtype = xp.complex128 + elif func in [fft.rfft, fft.rfftn, fft.rfft2, + fft.ihfft, fft.ihfftn, fft.ihfft2]: + dtype = xp.float64 + else: + raise ValueError(f'Unknown FFT function: {func}') + + return dtype + + +def fft1(x): + L = len(x) + phase = -2j*np.pi*(np.arange(L)/float(L)) + phase = np.arange(L).reshape(-1, 1) * phase + return np.sum(x*np.exp(phase), axis=1) + +class TestFFT: + + def test_identity(self, xp): + maxlen = 512 + x = xp.asarray(random(maxlen) + 1j*random(maxlen)) + xr = xp.asarray(random(maxlen)) + # Check some powers of 2 and some primes + for i in [1, 2, 16, 128, 512, 53, 149, 281, 397]: + xp_assert_close(fft.ifft(fft.fft(x[0:i])), x[0:i]) + xp_assert_close(fft.irfft(fft.rfft(xr[0:i]), i), xr[0:i]) + + @skip_xp_backends(np_only=True, reason='significant overhead for some backends') + def test_identity_extensive(self, xp): + maxlen = 512 + x = xp.asarray(random(maxlen) + 1j*random(maxlen)) + xr = xp.asarray(random(maxlen)) + for i in range(1, maxlen): + xp_assert_close(fft.ifft(fft.fft(x[0:i])), x[0:i]) + xp_assert_close(fft.irfft(fft.rfft(xr[0:i]), i), xr[0:i]) + + def test_fft(self, xp): + x = random(30) + 1j*random(30) + expect = xp.asarray(fft1(x)) + x = xp.asarray(x) + xp_assert_close(fft.fft(x), expect) + xp_assert_close(fft.fft(x, norm="backward"), expect) + xp_assert_close(fft.fft(x, norm="ortho"), + expect / xp.sqrt(xp.asarray(30, dtype=xp.float64)),) + xp_assert_close(fft.fft(x, norm="forward"), expect / 30) + + @skip_xp_backends(np_only=True, reason='some backends allow `n=0`') + def test_fft_n(self, xp): + x = xp.asarray([1, 2, 3], dtype=xp.complex128) + assert_raises(ValueError, fft.fft, x, 0) + + def test_ifft(self, xp): + x = xp.asarray(random(30) + 1j*random(30)) + xp_assert_close(fft.ifft(fft.fft(x)), x) + for norm in ["backward", "ortho", "forward"]: + xp_assert_close(fft.ifft(fft.fft(x, norm=norm), norm=norm), x) + + def test_fft2(self, xp): + x = xp.asarray(random((30, 20)) + 1j*random((30, 20))) + expect = fft.fft(fft.fft(x, axis=1), axis=0) + xp_assert_close(fft.fft2(x), expect) + xp_assert_close(fft.fft2(x, norm="backward"), expect) + xp_assert_close(fft.fft2(x, norm="ortho"), + expect / xp.sqrt(xp.asarray(30 * 20, dtype=xp.float64))) + xp_assert_close(fft.fft2(x, norm="forward"), expect / (30 * 20)) + + def test_ifft2(self, xp): + x = xp.asarray(random((30, 20)) + 1j*random((30, 20))) + expect = fft.ifft(fft.ifft(x, axis=1), axis=0) + xp_assert_close(fft.ifft2(x), expect) + xp_assert_close(fft.ifft2(x, norm="backward"), expect) + xp_assert_close(fft.ifft2(x, norm="ortho"), + expect * xp.sqrt(xp.asarray(30 * 20, dtype=xp.float64))) + xp_assert_close(fft.ifft2(x, norm="forward"), expect * (30 * 20)) + + def test_fftn(self, xp): + x = xp.asarray(random((30, 20, 10)) + 1j*random((30, 20, 10))) + expect = fft.fft(fft.fft(fft.fft(x, axis=2), axis=1), axis=0) + xp_assert_close(fft.fftn(x), expect) + xp_assert_close(fft.fftn(x, norm="backward"), expect) + xp_assert_close(fft.fftn(x, norm="ortho"), + expect / xp.sqrt(xp.asarray(30 * 20 * 10, dtype=xp.float64))) + xp_assert_close(fft.fftn(x, norm="forward"), expect / (30 * 20 * 10)) + + def test_ifftn(self, xp): + x = xp.asarray(random((30, 20, 10)) + 1j*random((30, 20, 10))) + expect = fft.ifft(fft.ifft(fft.ifft(x, axis=2), axis=1), axis=0) + xp_assert_close(fft.ifftn(x), expect, rtol=1e-7) + xp_assert_close(fft.ifftn(x, norm="backward"), expect, rtol=1e-7) + xp_assert_close( + fft.ifftn(x, norm="ortho"), + fft.ifftn(x) * xp.sqrt(xp.asarray(30 * 20 * 10, dtype=xp.float64)) + ) + xp_assert_close(fft.ifftn(x, norm="forward"), + expect * (30 * 20 * 10), + rtol=1e-7) + + def test_rfft(self, xp): + x = xp.asarray(random(29), dtype=xp.float64) + for n in [xp_size(x), 2*xp_size(x)]: + for norm in [None, "backward", "ortho", "forward"]: + xp_assert_close(fft.rfft(x, n=n, norm=norm), + fft.fft(xp.asarray(x, dtype=xp.complex128), + n=n, norm=norm)[:(n//2 + 1)]) + xp_assert_close( + fft.rfft(x, n=n, norm="ortho"), + fft.rfft(x, n=n) / xp.sqrt(xp.asarray(n, dtype=xp.float64)) + ) + + def test_irfft(self, xp): + x = xp.asarray(random(30)) + xp_assert_close(fft.irfft(fft.rfft(x)), x) + for norm in ["backward", "ortho", "forward"]: + xp_assert_close(fft.irfft(fft.rfft(x, norm=norm), norm=norm), x) + + def test_rfft2(self, xp): + x = xp.asarray(random((30, 20)), dtype=xp.float64) + expect = fft.fft2(xp.asarray(x, dtype=xp.complex128))[:, :11] + xp_assert_close(fft.rfft2(x), expect) + xp_assert_close(fft.rfft2(x, norm="backward"), expect) + xp_assert_close(fft.rfft2(x, norm="ortho"), + expect / xp.sqrt(xp.asarray(30 * 20, dtype=xp.float64))) + xp_assert_close(fft.rfft2(x, norm="forward"), expect / (30 * 20)) + + def test_irfft2(self, xp): + x = xp.asarray(random((30, 20))) + xp_assert_close(fft.irfft2(fft.rfft2(x)), x) + for norm in ["backward", "ortho", "forward"]: + xp_assert_close(fft.irfft2(fft.rfft2(x, norm=norm), norm=norm), x) + + def test_rfftn(self, xp): + x = xp.asarray(random((30, 20, 10)), dtype=xp.float64) + expect = fft.fftn(xp.asarray(x, dtype=xp.complex128))[:, :, :6] + xp_assert_close(fft.rfftn(x), expect) + xp_assert_close(fft.rfftn(x, norm="backward"), expect) + xp_assert_close(fft.rfftn(x, norm="ortho"), + expect / xp.sqrt(xp.asarray(30 * 20 * 10, dtype=xp.float64))) + xp_assert_close(fft.rfftn(x, norm="forward"), expect / (30 * 20 * 10)) + + def test_irfftn(self, xp): + x = xp.asarray(random((30, 20, 10))) + xp_assert_close(fft.irfftn(fft.rfftn(x)), x) + for norm in ["backward", "ortho", "forward"]: + xp_assert_close(fft.irfftn(fft.rfftn(x, norm=norm), norm=norm), x) + + def test_hfft(self, xp): + x = random(14) + 1j*random(14) + x_herm = np.concatenate((random(1), x, random(1))) + x = np.concatenate((x_herm, x[::-1].conj())) + x = xp.asarray(x) + x_herm = xp.asarray(x_herm) + expect = xp.real(fft.fft(x)) + xp_assert_close(fft.hfft(x_herm), expect) + xp_assert_close(fft.hfft(x_herm, norm="backward"), expect) + xp_assert_close(fft.hfft(x_herm, norm="ortho"), + expect / xp.sqrt(xp.asarray(30, dtype=xp.float64))) + xp_assert_close(fft.hfft(x_herm, norm="forward"), expect / 30) + + def test_ihfft(self, xp): + x = random(14) + 1j*random(14) + x_herm = np.concatenate((random(1), x, random(1))) + x = np.concatenate((x_herm, x[::-1].conj())) + x = xp.asarray(x) + x_herm = xp.asarray(x_herm) + xp_assert_close(fft.ihfft(fft.hfft(x_herm)), x_herm) + for norm in ["backward", "ortho", "forward"]: + xp_assert_close(fft.ihfft(fft.hfft(x_herm, norm=norm), norm=norm), x_herm) + + def test_hfft2(self, xp): + x = xp.asarray(random((30, 20))) + xp_assert_close(fft.hfft2(fft.ihfft2(x)), x) + for norm in ["backward", "ortho", "forward"]: + xp_assert_close(fft.hfft2(fft.ihfft2(x, norm=norm), norm=norm), x) + + def test_ihfft2(self, xp): + x = xp.asarray(random((30, 20)), dtype=xp.float64) + expect = fft.ifft2(xp.asarray(x, dtype=xp.complex128))[:, :11] + xp_assert_close(fft.ihfft2(x), expect) + xp_assert_close(fft.ihfft2(x, norm="backward"), expect) + xp_assert_close( + fft.ihfft2(x, norm="ortho"), + expect * xp.sqrt(xp.asarray(30 * 20, dtype=xp.float64)) + ) + xp_assert_close(fft.ihfft2(x, norm="forward"), expect * (30 * 20)) + + def test_hfftn(self, xp): + x = xp.asarray(random((30, 20, 10))) + xp_assert_close(fft.hfftn(fft.ihfftn(x)), x) + for norm in ["backward", "ortho", "forward"]: + xp_assert_close(fft.hfftn(fft.ihfftn(x, norm=norm), norm=norm), x) + + def test_ihfftn(self, xp): + x = xp.asarray(random((30, 20, 10)), dtype=xp.float64) + expect = fft.ifftn(xp.asarray(x, dtype=xp.complex128))[:, :, :6] + xp_assert_close(expect, fft.ihfftn(x)) + xp_assert_close(expect, fft.ihfftn(x, norm="backward")) + xp_assert_close( + fft.ihfftn(x, norm="ortho"), + expect * xp.sqrt(xp.asarray(30 * 20 * 10, dtype=xp.float64)) + ) + xp_assert_close(fft.ihfftn(x, norm="forward"), expect * (30 * 20 * 10)) + + def _check_axes(self, op, xp): + dtype = get_expected_input_dtype(op, xp) + x = xp.asarray(random((30, 20, 10)), dtype=dtype) + axes = [(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0)] + xp_test = array_namespace(x) + for a in axes: + op_tr = op(xp_test.permute_dims(x, axes=a)) + tr_op = xp_test.permute_dims(op(x, axes=a), axes=a) + xp_assert_close(op_tr, tr_op) + + @pytest.mark.parametrize("op", [fft.fftn, fft.ifftn, fft.rfftn, fft.irfftn]) + def test_axes_standard(self, op, xp): + self._check_axes(op, xp) + + @pytest.mark.parametrize("op", [fft.hfftn, fft.ihfftn]) + def test_axes_non_standard(self, op, xp): + self._check_axes(op, xp) + + @pytest.mark.parametrize("op", [fft.fftn, fft.ifftn, + fft.rfftn, fft.irfftn]) + def test_axes_subset_with_shape_standard(self, op, xp): + dtype = get_expected_input_dtype(op, xp) + x = xp.asarray(random((16, 8, 4)), dtype=dtype) + axes = [(0, 1, 2), (0, 2, 1), (1, 2, 0)] + xp_test = array_namespace(x) + for a in axes: + # different shape on the first two axes + shape = tuple([2*x.shape[ax] if ax in a[:2] else x.shape[ax] + for ax in range(x.ndim)]) + # transform only the first two axes + op_tr = op(xp_test.permute_dims(x, axes=a), + s=shape[:2], axes=(0, 1)) + tr_op = xp_test.permute_dims(op(x, s=shape[:2], axes=a[:2]), + axes=a) + xp_assert_close(op_tr, tr_op) + + @pytest.mark.parametrize("op", [fft.fft2, fft.ifft2, + fft.rfft2, fft.irfft2, + fft.hfft2, fft.ihfft2, + fft.hfftn, fft.ihfftn]) + def test_axes_subset_with_shape_non_standard(self, op, xp): + dtype = get_expected_input_dtype(op, xp) + x = xp.asarray(random((16, 8, 4)), dtype=dtype) + axes = [(0, 1, 2), (0, 2, 1), (1, 2, 0)] + xp_test = array_namespace(x) + for a in axes: + # different shape on the first two axes + shape = tuple([2*x.shape[ax] if ax in a[:2] else x.shape[ax] + for ax in range(x.ndim)]) + # transform only the first two axes + op_tr = op(xp_test.permute_dims(x, axes=a), s=shape[:2], axes=(0, 1)) + tr_op = xp_test.permute_dims(op(x, s=shape[:2], axes=a[:2]), axes=a) + xp_assert_close(op_tr, tr_op) + + def test_all_1d_norm_preserving(self, xp): + # verify that round-trip transforms are norm-preserving + x = xp.asarray(random(30), dtype=xp.float64) + xp_test = array_namespace(x) + x_norm = xp_test.linalg.vector_norm(x) + n = xp_size(x) * 2 + func_pairs = [(fft.rfft, fft.irfft), + # hfft: order so the first function takes x.size samples + # (necessary for comparison to x_norm above) + (fft.ihfft, fft.hfft), + # functions that expect complex dtypes at the end + (fft.fft, fft.ifft), + ] + for forw, back in func_pairs: + if forw == fft.fft: + x = xp.asarray(x, dtype=xp.complex128) + x_norm = xp_test.linalg.vector_norm(x) + for n in [xp_size(x), 2*xp_size(x)]: + for norm in ['backward', 'ortho', 'forward']: + tmp = forw(x, n=n, norm=norm) + tmp = back(tmp, n=n, norm=norm) + xp_assert_close(xp_test.linalg.vector_norm(tmp), x_norm) + + @skip_xp_backends(np_only=True) + @pytest.mark.parametrize("dtype", [np.float16, np.longdouble]) + def test_dtypes_nonstandard(self, dtype): + x = random(30).astype(dtype) + out_dtypes = {np.float16: np.complex64, np.longdouble: np.clongdouble} + x_complex = x.astype(out_dtypes[dtype]) + + res_fft = fft.ifft(fft.fft(x)) + res_rfft = fft.irfft(fft.rfft(x)) + res_hfft = fft.hfft(fft.ihfft(x), x.shape[0]) + # Check both numerical results and exact dtype matches + assert_array_almost_equal(res_fft, x_complex) + assert_array_almost_equal(res_rfft, x) + assert_array_almost_equal(res_hfft, x) + assert res_fft.dtype == x_complex.dtype + assert res_rfft.dtype == np.result_type(np.float32, x.dtype) + assert res_hfft.dtype == np.result_type(np.float32, x.dtype) + + @pytest.mark.parametrize("dtype", ["float32", "float64"]) + def test_dtypes_real(self, dtype, xp): + x = xp.asarray(random(30), dtype=getattr(xp, dtype)) + + res_rfft = fft.irfft(fft.rfft(x)) + res_hfft = fft.hfft(fft.ihfft(x), x.shape[0]) + # Check both numerical results and exact dtype matches + xp_assert_close(res_rfft, x) + xp_assert_close(res_hfft, x) + + @pytest.mark.parametrize("dtype", ["complex64", "complex128"]) + def test_dtypes_complex(self, dtype, xp): + rng = np.random.default_rng(1234) + x = xp.asarray(rng.random(30), dtype=getattr(xp, dtype)) + + res_fft = fft.ifft(fft.fft(x)) + # Check both numerical results and exact dtype matches + xp_assert_close(res_fft, x) + + @skip_xp_backends(np_only=True, + reason='array-likes only supported for NumPy backend') + @pytest.mark.parametrize("op", [fft.fft, fft.ifft, + fft.fft2, fft.ifft2, + fft.fftn, fft.ifftn, + fft.rfft, fft.irfft, + fft.rfft2, fft.irfft2, + fft.rfftn, fft.irfftn, + fft.hfft, fft.ihfft, + fft.hfft2, fft.ihfft2, + fft.hfftn, fft.ihfftn,]) + def test_array_like(self, xp, op): + x = [[[1.0, 1.0], [1.0, 1.0]], + [[1.0, 1.0], [1.0, 1.0]], + [[1.0, 1.0], [1.0, 1.0]]] + xp_assert_close(op(x), op(xp.asarray(x))) + + +@skip_xp_backends(np_only=True) +@pytest.mark.parametrize( + "dtype", + [np.float32, np.float64, np.longdouble, + np.complex64, np.complex128, np.clongdouble]) +@pytest.mark.parametrize("order", ["F", 'non-contiguous']) +@pytest.mark.parametrize( + "fft", + [fft.fft, fft.fft2, fft.fftn, + fft.ifft, fft.ifft2, fft.ifftn]) +def test_fft_with_order(dtype, order, fft): + # Check that FFT/IFFT produces identical results for C, Fortran and + # non contiguous arrays + rng = np.random.RandomState(42) + X = rng.rand(8, 7, 13).astype(dtype, copy=False) + if order == 'F': + Y = np.asfortranarray(X) + else: + # Make a non contiguous array + Y = X[::-1] + X = np.ascontiguousarray(X[::-1]) + + if fft.__name__.endswith('fft'): + for axis in range(3): + X_res = fft(X, axis=axis) + Y_res = fft(Y, axis=axis) + assert_array_almost_equal(X_res, Y_res) + elif fft.__name__.endswith(('fft2', 'fftn')): + axes = [(0, 1), (1, 2), (0, 2)] + if fft.__name__.endswith('fftn'): + axes.extend([(0,), (1,), (2,), None]) + for ax in axes: + X_res = fft(X, axes=ax) + Y_res = fft(Y, axes=ax) + assert_array_almost_equal(X_res, Y_res) + else: + raise ValueError + + +@skip_xp_backends(cpu_only=True) +class TestFFTThreadSafe: + threads = 16 + input_shape = (800, 200) + + def _test_mtsame(self, func, *args, xp=None): + def worker(args, q): + q.put(func(*args)) + + q = queue.Queue() + expected = func(*args) + + # Spin off a bunch of threads to call the same function simultaneously + t = [threading.Thread(target=worker, args=(args, q)) + for i in range(self.threads)] + [x.start() for x in t] + + [x.join() for x in t] + + # Make sure all threads returned the correct value + for i in range(self.threads): + xp_assert_equal( + q.get(timeout=5), expected, + err_msg='Function returned wrong value in multithreaded context' + ) + + def test_fft(self, xp): + a = xp.ones(self.input_shape, dtype=xp.complex128) + self._test_mtsame(fft.fft, a, xp=xp) + + def test_ifft(self, xp): + a = xp.full(self.input_shape, 1+0j) + self._test_mtsame(fft.ifft, a, xp=xp) + + def test_rfft(self, xp): + a = xp.ones(self.input_shape) + self._test_mtsame(fft.rfft, a, xp=xp) + + def test_irfft(self, xp): + a = xp.full(self.input_shape, 1+0j) + self._test_mtsame(fft.irfft, a, xp=xp) + + def test_hfft(self, xp): + a = xp.ones(self.input_shape, dtype=xp.complex64) + self._test_mtsame(fft.hfft, a, xp=xp) + + def test_ihfft(self, xp): + a = xp.ones(self.input_shape) + self._test_mtsame(fft.ihfft, a, xp=xp) + + +@skip_xp_backends(np_only=True) +@pytest.mark.parametrize("func", [fft.fft, fft.ifft, fft.rfft, fft.irfft]) +def test_multiprocess(func): + # Test that fft still works after fork (gh-10422) + + with multiprocessing.Pool(2) as p: + res = p.map(func, [np.ones(100) for _ in range(4)]) + + expect = func(np.ones(100)) + for x in res: + assert_allclose(x, expect) + + +class TestIRFFTN: + + def test_not_last_axis_success(self, xp): + ar, ai = np.random.random((2, 16, 8, 32)) + a = ar + 1j*ai + a = xp.asarray(a) + + axes = (-2,) + + # Should not raise error + fft.irfftn(a, axes=axes) + + +@pytest.mark.parametrize("func", [fft.fft, fft.ifft, fft.rfft, fft.irfft, + fft.fftn, fft.ifftn, + fft.rfftn, fft.irfftn, fft.hfft, fft.ihfft]) +def test_non_standard_params(func, xp): + if func in [fft.rfft, fft.rfftn, fft.ihfft]: + dtype = xp.float64 + else: + dtype = xp.complex128 + + if xp.__name__ != 'numpy': + x = xp.asarray([1, 2, 3], dtype=dtype) + # func(x) should not raise an exception + func(x) + assert_raises(ValueError, func, x, workers=2) + # `plan` param is not tested since SciPy does not use it currently + # but should be tested if it comes into use + + +@pytest.mark.parametrize("dtype", ['float32', 'float64']) +@pytest.mark.parametrize("func", [fft.fft, fft.ifft, fft.irfft, + fft.fftn, fft.ifftn, + fft.irfftn, fft.hfft,]) +def test_real_input(func, dtype, xp): + x = xp.asarray([1, 2, 3], dtype=getattr(xp, dtype)) + # func(x) should not raise an exception + func(x) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_fftlog.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_fftlog.py new file mode 100644 index 0000000000000000000000000000000000000000..3480e165180baf3866ca7c99a313996a2dbbca49 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_fftlog.py @@ -0,0 +1,206 @@ +import warnings +import math + +import numpy as np +import pytest + +from scipy.fft._fftlog import fht, ifht, fhtoffset +from scipy.special import poch + +from scipy.conftest import array_api_compatible +from scipy._lib._array_api import xp_assert_close, xp_assert_less, array_namespace + +pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends"),] +skip_xp_backends = pytest.mark.skip_xp_backends + + +def test_fht_agrees_with_fftlog(xp): + # check that fht numerically agrees with the output from Fortran FFTLog, + # the results were generated with the provided `fftlogtest` program, + # after fixing how the k array is generated (divide range by n-1, not n) + + # test function, analytical Hankel transform is of the same form + def f(r, mu): + return r**(mu+1)*np.exp(-r**2/2) + + r = np.logspace(-4, 4, 16) + + dln = np.log(r[1]/r[0]) + mu = 0.3 + offset = 0.0 + bias = 0.0 + + a = xp.asarray(f(r, mu)) + + # test 1: compute as given + ours = fht(a, dln, mu, offset=offset, bias=bias) + theirs = [-0.1159922613593045E-02, +0.1625822618458832E-02, + -0.1949518286432330E-02, +0.3789220182554077E-02, + +0.5093959119952945E-03, +0.2785387803618774E-01, + +0.9944952700848897E-01, +0.4599202164586588E+00, + +0.3157462160881342E+00, -0.8201236844404755E-03, + -0.7834031308271878E-03, +0.3931444945110708E-03, + -0.2697710625194777E-03, +0.3568398050238820E-03, + -0.5554454827797206E-03, +0.8286331026468585E-03] + theirs = xp.asarray(theirs, dtype=xp.float64) + xp_assert_close(ours, theirs) + + # test 2: change to optimal offset + offset = fhtoffset(dln, mu, bias=bias) + ours = fht(a, dln, mu, offset=offset, bias=bias) + theirs = [+0.4353768523152057E-04, -0.9197045663594285E-05, + +0.3150140927838524E-03, +0.9149121960963704E-03, + +0.5808089753959363E-02, +0.2548065256377240E-01, + +0.1339477692089897E+00, +0.4821530509479356E+00, + +0.2659899781579785E+00, -0.1116475278448113E-01, + +0.1791441617592385E-02, -0.4181810476548056E-03, + +0.1314963536765343E-03, -0.5422057743066297E-04, + +0.3208681804170443E-04, -0.2696849476008234E-04] + theirs = xp.asarray(theirs, dtype=xp.float64) + xp_assert_close(ours, theirs) + + # test 3: positive bias + bias = 0.8 + offset = fhtoffset(dln, mu, bias=bias) + ours = fht(a, dln, mu, offset=offset, bias=bias) + theirs = [-7.3436673558316850E+00, +0.1710271207817100E+00, + +0.1065374386206564E+00, -0.5121739602708132E-01, + +0.2636649319269470E-01, +0.1697209218849693E-01, + +0.1250215614723183E+00, +0.4739583261486729E+00, + +0.2841149874912028E+00, -0.8312764741645729E-02, + +0.1024233505508988E-02, -0.1644902767389120E-03, + +0.3305775476926270E-04, -0.7786993194882709E-05, + +0.1962258449520547E-05, -0.8977895734909250E-06] + theirs = xp.asarray(theirs, dtype=xp.float64) + xp_assert_close(ours, theirs) + + # test 4: negative bias + bias = -0.8 + offset = fhtoffset(dln, mu, bias=bias) + ours = fht(a, dln, mu, offset=offset, bias=bias) + theirs = [+0.8985777068568745E-05, +0.4074898209936099E-04, + +0.2123969254700955E-03, +0.1009558244834628E-02, + +0.5131386375222176E-02, +0.2461678673516286E-01, + +0.1235812845384476E+00, +0.4719570096404403E+00, + +0.2893487490631317E+00, -0.1686570611318716E-01, + +0.2231398155172505E-01, -0.1480742256379873E-01, + +0.1692387813500801E+00, +0.3097490354365797E+00, + +2.7593607182401860E+00, 10.5251075070045800E+00] + theirs = xp.asarray(theirs, dtype=xp.float64) + xp_assert_close(ours, theirs) + + +@pytest.mark.parametrize('optimal', [True, False]) +@pytest.mark.parametrize('offset', [0.0, 1.0, -1.0]) +@pytest.mark.parametrize('bias', [0, 0.1, -0.1]) +@pytest.mark.parametrize('n', [64, 63]) +def test_fht_identity(n, bias, offset, optimal, xp): + rng = np.random.RandomState(3491349965) + + a = xp.asarray(rng.standard_normal(n)) + dln = rng.uniform(-1, 1) + mu = rng.uniform(-2, 2) + + if optimal: + offset = fhtoffset(dln, mu, initial=offset, bias=bias) + + A = fht(a, dln, mu, offset=offset, bias=bias) + a_ = ifht(A, dln, mu, offset=offset, bias=bias) + + xp_assert_close(a_, a, rtol=1.5e-7) + + + + +@pytest.mark.thread_unsafe +def test_fht_special_cases(xp): + rng = np.random.RandomState(3491349965) + + a = xp.asarray(rng.standard_normal(64)) + dln = rng.uniform(-1, 1) + + # let x = (mu+1+q)/2, y = (mu+1-q)/2, M = {0, -1, -2, ...} + + # case 1: x in M, y in M => well-defined transform + mu, bias = -4.0, 1.0 + with warnings.catch_warnings(record=True) as record: + fht(a, dln, mu, bias=bias) + assert not record, 'fht warned about a well-defined transform' + + # case 2: x not in M, y in M => well-defined transform + mu, bias = -2.5, 0.5 + with warnings.catch_warnings(record=True) as record: + fht(a, dln, mu, bias=bias) + assert not record, 'fht warned about a well-defined transform' + + # with fht_lock: + # case 3: x in M, y not in M => singular transform + mu, bias = -3.5, 0.5 + with pytest.warns(Warning) as record: + fht(a, dln, mu, bias=bias) + assert record, 'fht did not warn about a singular transform' + + # with fht_lock: + # case 4: x not in M, y in M => singular inverse transform + mu, bias = -2.5, 0.5 + with pytest.warns(Warning) as record: + ifht(a, dln, mu, bias=bias) + assert record, 'ifht did not warn about a singular transform' + + +@pytest.mark.parametrize('n', [64, 63]) +def test_fht_exact(n, xp): + rng = np.random.RandomState(3491349965) + + # for a(r) a power law r^\gamma, the fast Hankel transform produces the + # exact continuous Hankel transform if biased with q = \gamma + + mu = rng.uniform(0, 3) + + # convergence of HT: -1-mu < gamma < 1/2 + gamma = rng.uniform(-1-mu, 1/2) + + r = np.logspace(-2, 2, n) + a = xp.asarray(r**gamma) + + dln = np.log(r[1]/r[0]) + + offset = fhtoffset(dln, mu, initial=0.0, bias=gamma) + + A = fht(a, dln, mu, offset=offset, bias=gamma) + + k = np.exp(offset)/r[::-1] + + # analytical result + At = xp.asarray((2/k)**gamma * poch((mu+1-gamma)/2, gamma)) + + xp_assert_close(A, At) + +@skip_xp_backends(np_only=True, + reason='array-likes only supported for NumPy backend') +@pytest.mark.parametrize("op", [fht, ifht]) +def test_array_like(xp, op): + x = [[[1.0, 1.0], [1.0, 1.0]], + [[1.0, 1.0], [1.0, 1.0]], + [[1.0, 1.0], [1.0, 1.0]]] + xp_assert_close(op(x, 1.0, 2.0), op(xp.asarray(x), 1.0, 2.0)) + +@pytest.mark.parametrize('n', [128, 129]) +def test_gh_21661(xp, n): + one = xp.asarray(1.0) + xp_test = array_namespace(one) + mu = 0.0 + r = np.logspace(-7, 1, n) + dln = math.log(r[1] / r[0]) + offset = fhtoffset(dln, initial=-6 * np.log(10), mu=mu) + r = xp.asarray(r, dtype=one.dtype) + k = math.exp(offset) / xp_test.flip(r, axis=-1) + + def f(x, mu): + return x**(mu + 1)*xp.exp(-x**2/2) + + a_r = f(r, mu) + fht_val = fht(a_r, dln, mu=mu, offset=offset) + a_k = f(k, mu) + rel_err = xp.max(xp.abs((fht_val - a_k) / a_k)) + xp_assert_less(rel_err, xp.asarray(7.28e+16)[()]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_helper.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_helper.py new file mode 100644 index 0000000000000000000000000000000000000000..4333886555ffbbb0831f830456a3290f32e317fa --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_helper.py @@ -0,0 +1,574 @@ +"""Includes test functions for fftpack.helper module + +Copied from fftpack.helper by Pearu Peterson, October 2005 +Modified for Array API, 2023 + +""" +from scipy.fft._helper import next_fast_len, prev_fast_len, _init_nd_shape_and_axes +from numpy.testing import assert_equal +from pytest import raises as assert_raises +import pytest +import numpy as np +import sys +from scipy.conftest import array_api_compatible +from scipy._lib._array_api import ( + xp_assert_close, get_xp_devices, xp_device, array_namespace +) +from scipy import fft + +pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends")] +skip_xp_backends = pytest.mark.skip_xp_backends + +_5_smooth_numbers = [ + 2, 3, 4, 5, 6, 8, 9, 10, + 2 * 3 * 5, + 2**3 * 3**5, + 2**3 * 3**3 * 5**2, +] + +def test_next_fast_len(): + for n in _5_smooth_numbers: + assert_equal(next_fast_len(n), n) + + +def _assert_n_smooth(x, n): + x_orig = x + if n < 2: + assert False + + while True: + q, r = divmod(x, 2) + if r != 0: + break + x = q + + for d in range(3, n+1, 2): + while True: + q, r = divmod(x, d) + if r != 0: + break + x = q + + assert x == 1, \ + f'x={x_orig} is not {n}-smooth, remainder={x}' + + +@skip_xp_backends(np_only=True) +class TestNextFastLen: + + def test_next_fast_len(self): + np.random.seed(1234) + + def nums(): + yield from range(1, 1000) + yield 2**5 * 3**5 * 4**5 + 1 + + for n in nums(): + m = next_fast_len(n) + _assert_n_smooth(m, 11) + assert m == next_fast_len(n, False) + + m = next_fast_len(n, True) + _assert_n_smooth(m, 5) + + def test_np_integers(self): + ITYPES = [np.int16, np.int32, np.int64, np.uint16, np.uint32, np.uint64] + for ityp in ITYPES: + x = ityp(12345) + testN = next_fast_len(x) + assert_equal(testN, next_fast_len(int(x))) + + def testnext_fast_len_small(self): + hams = { + 1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 7: 8, 8: 8, 14: 15, 15: 15, + 16: 16, 17: 18, 1021: 1024, 1536: 1536, 51200000: 51200000 + } + for x, y in hams.items(): + assert_equal(next_fast_len(x, True), y) + + @pytest.mark.xfail(sys.maxsize < 2**32, + reason="Hamming Numbers too large for 32-bit", + raises=ValueError, strict=True) + def testnext_fast_len_big(self): + hams = { + 510183360: 510183360, 510183360 + 1: 512000000, + 511000000: 512000000, + 854296875: 854296875, 854296875 + 1: 859963392, + 196608000000: 196608000000, 196608000000 + 1: 196830000000, + 8789062500000: 8789062500000, 8789062500000 + 1: 8796093022208, + 206391214080000: 206391214080000, + 206391214080000 + 1: 206624260800000, + 470184984576000: 470184984576000, + 470184984576000 + 1: 470715894135000, + 7222041363087360: 7222041363087360, + 7222041363087360 + 1: 7230196133913600, + # power of 5 5**23 + 11920928955078125: 11920928955078125, + 11920928955078125 - 1: 11920928955078125, + # power of 3 3**34 + 16677181699666569: 16677181699666569, + 16677181699666569 - 1: 16677181699666569, + # power of 2 2**54 + 18014398509481984: 18014398509481984, + 18014398509481984 - 1: 18014398509481984, + # above this, int(ceil(n)) == int(ceil(n+1)) + 19200000000000000: 19200000000000000, + 19200000000000000 + 1: 19221679687500000, + 288230376151711744: 288230376151711744, + 288230376151711744 + 1: 288325195312500000, + 288325195312500000 - 1: 288325195312500000, + 288325195312500000: 288325195312500000, + 288325195312500000 + 1: 288555831593533440, + } + for x, y in hams.items(): + assert_equal(next_fast_len(x, True), y) + + def test_keyword_args(self): + assert next_fast_len(11, real=True) == 12 + assert next_fast_len(target=7, real=False) == 7 + +@skip_xp_backends(np_only=True) +class TestPrevFastLen: + + def test_prev_fast_len(self): + np.random.seed(1234) + + def nums(): + yield from range(1, 1000) + yield 2**5 * 3**5 * 4**5 + 1 + + for n in nums(): + m = prev_fast_len(n) + _assert_n_smooth(m, 11) + assert m == prev_fast_len(n, False) + + m = prev_fast_len(n, True) + _assert_n_smooth(m, 5) + + def test_np_integers(self): + ITYPES = [np.int16, np.int32, np.int64, np.uint16, np.uint32, + np.uint64] + for ityp in ITYPES: + x = ityp(12345) + testN = prev_fast_len(x) + assert_equal(testN, prev_fast_len(int(x))) + + testN = prev_fast_len(x, real=True) + assert_equal(testN, prev_fast_len(int(x), real=True)) + + def testprev_fast_len_small(self): + hams = { + 1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 7: 6, 8: 8, 14: 12, 15: 15, + 16: 16, 17: 16, 1021: 1000, 1536: 1536, 51200000: 51200000 + } + for x, y in hams.items(): + assert_equal(prev_fast_len(x, True), y) + + hams = { + 1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 7: 7, 8: 8, 9: 9, 10: 10, + 11: 11, 12: 12, 13: 12, 14: 14, 15: 15, 16: 16, 17: 16, 18: 18, + 19: 18, 20: 20, 21: 21, 22: 22, 120: 120, 121: 121, 122: 121, + 1021: 1008, 1536: 1536, 51200000: 51200000 + } + for x, y in hams.items(): + assert_equal(prev_fast_len(x, False), y) + + @pytest.mark.xfail(sys.maxsize < 2**32, + reason="Hamming Numbers too large for 32-bit", + raises=ValueError, strict=True) + def testprev_fast_len_big(self): + hams = { + # 2**6 * 3**13 * 5**1 + 510183360: 510183360, + 510183360 + 1: 510183360, + 510183360 - 1: 509607936, # 2**21 * 3**5 + # 2**6 * 5**6 * 7**1 * 73**1 + 511000000: 510183360, + 511000000 + 1: 510183360, + 511000000 - 1: 510183360, # 2**6 * 3**13 * 5**1 + # 3**7 * 5**8 + 854296875: 854296875, + 854296875 + 1: 854296875, + 854296875 - 1: 850305600, # 2**6 * 3**12 * 5**2 + # 2**22 * 3**1 * 5**6 + 196608000000: 196608000000, + 196608000000 + 1: 196608000000, + 196608000000 - 1: 195910410240, # 2**13 * 3**14 * 5**1 + # 2**5 * 3**2 * 5**15 + 8789062500000: 8789062500000, + 8789062500000 + 1: 8789062500000, + 8789062500000 - 1: 8748000000000, # 2**11 * 3**7 * 5**9 + # 2**24 * 3**9 * 5**4 + 206391214080000: 206391214080000, + 206391214080000 + 1: 206391214080000, + 206391214080000 - 1: 206158430208000, # 2**39 * 3**1 * 5**3 + # 2**18 * 3**15 * 5**3 + 470184984576000: 470184984576000, + 470184984576000 + 1: 470184984576000, + 470184984576000 - 1: 469654673817600, # 2**33 * 3**7 **5**2 + # 2**25 * 3**16 * 5**1 + 7222041363087360: 7222041363087360, + 7222041363087360 + 1: 7222041363087360, + 7222041363087360 - 1: 7213895789838336, # 2**40 * 3**8 + # power of 5 5**23 + 11920928955078125: 11920928955078125, + 11920928955078125 + 1: 11920928955078125, + 11920928955078125 - 1: 11901557422080000, # 2**14 * 3**19 * 5**4 + # power of 3 3**34 + 16677181699666569: 16677181699666569, + 16677181699666569 + 1: 16677181699666569, + 16677181699666569 - 1: 16607531250000000, # 2**7 * 3**12 * 5**12 + # power of 2 2**54 + 18014398509481984: 18014398509481984, + 18014398509481984 + 1: 18014398509481984, + 18014398509481984 - 1: 18000000000000000, # 2**16 * 3**2 * 5**15 + # 2**20 * 3**1 * 5**14 + 19200000000000000: 19200000000000000, + 19200000000000000 + 1: 19200000000000000, + 19200000000000000 - 1: 19131876000000000, # 2**11 * 3**14 * 5**9 + # 2**58 + 288230376151711744: 288230376151711744, + 288230376151711744 + 1: 288230376151711744, + 288230376151711744 - 1: 288000000000000000, # 2**20 * 3**2 * 5**15 + # 2**5 * 3**10 * 5**16 + 288325195312500000: 288325195312500000, + 288325195312500000 + 1: 288325195312500000, + 288325195312500000 - 1: 288230376151711744, # 2**58 + } + for x, y in hams.items(): + assert_equal(prev_fast_len(x, True), y) + + def test_keyword_args(self): + assert prev_fast_len(11, real=True) == 10 + assert prev_fast_len(target=7, real=False) == 7 + + +@skip_xp_backends(cpu_only=True) +class Test_init_nd_shape_and_axes: + + def test_py_0d_defaults(self, xp): + x = xp.asarray(4) + shape = None + axes = None + + shape_expected = () + axes_expected = [] + + shape_res, axes_res = _init_nd_shape_and_axes(x, shape, axes) + + assert shape_res == shape_expected + assert axes_res == axes_expected + + def test_xp_0d_defaults(self, xp): + x = xp.asarray(7.) + shape = None + axes = None + + shape_expected = () + axes_expected = [] + + shape_res, axes_res = _init_nd_shape_and_axes(x, shape, axes) + + assert shape_res == shape_expected + assert axes_res == axes_expected + + def test_py_1d_defaults(self, xp): + x = xp.asarray([1, 2, 3]) + shape = None + axes = None + + shape_expected = (3,) + axes_expected = [0] + + shape_res, axes_res = _init_nd_shape_and_axes(x, shape, axes) + + assert shape_res == shape_expected + assert axes_res == axes_expected + + def test_xp_1d_defaults(self, xp): + x = xp.arange(0, 1, .1) + shape = None + axes = None + + shape_expected = (10,) + axes_expected = [0] + + shape_res, axes_res = _init_nd_shape_and_axes(x, shape, axes) + + assert shape_res == shape_expected + assert axes_res == axes_expected + + def test_py_2d_defaults(self, xp): + x = xp.asarray([[1, 2, 3, 4], + [5, 6, 7, 8]]) + shape = None + axes = None + + shape_expected = (2, 4) + axes_expected = [0, 1] + + shape_res, axes_res = _init_nd_shape_and_axes(x, shape, axes) + + assert shape_res == shape_expected + assert axes_res == axes_expected + + def test_xp_2d_defaults(self, xp): + x = xp.arange(0, 1, .1) + x = xp.reshape(x, (5, 2)) + shape = None + axes = None + + shape_expected = (5, 2) + axes_expected = [0, 1] + + shape_res, axes_res = _init_nd_shape_and_axes(x, shape, axes) + + assert shape_res == shape_expected + assert axes_res == axes_expected + + def test_xp_5d_defaults(self, xp): + x = xp.zeros([6, 2, 5, 3, 4]) + shape = None + axes = None + + shape_expected = (6, 2, 5, 3, 4) + axes_expected = [0, 1, 2, 3, 4] + + shape_res, axes_res = _init_nd_shape_and_axes(x, shape, axes) + + assert shape_res == shape_expected + assert axes_res == axes_expected + + def test_xp_5d_set_shape(self, xp): + x = xp.zeros([6, 2, 5, 3, 4]) + shape = [10, -1, -1, 1, 4] + axes = None + + shape_expected = (10, 2, 5, 1, 4) + axes_expected = [0, 1, 2, 3, 4] + + shape_res, axes_res = _init_nd_shape_and_axes(x, shape, axes) + + assert shape_res == shape_expected + assert axes_res == axes_expected + + def test_xp_5d_set_axes(self, xp): + x = xp.zeros([6, 2, 5, 3, 4]) + shape = None + axes = [4, 1, 2] + + shape_expected = (4, 2, 5) + axes_expected = [4, 1, 2] + + shape_res, axes_res = _init_nd_shape_and_axes(x, shape, axes) + + assert shape_res == shape_expected + assert axes_res == axes_expected + + def test_xp_5d_set_shape_axes(self, xp): + x = xp.zeros([6, 2, 5, 3, 4]) + shape = [10, -1, 2] + axes = [1, 0, 3] + + shape_expected = (10, 6, 2) + axes_expected = [1, 0, 3] + + shape_res, axes_res = _init_nd_shape_and_axes(x, shape, axes) + + assert shape_res == shape_expected + assert axes_res == axes_expected + + def test_shape_axes_subset(self, xp): + x = xp.zeros((2, 3, 4, 5)) + shape, axes = _init_nd_shape_and_axes(x, shape=(5, 5, 5), axes=None) + + assert shape == (5, 5, 5) + assert axes == [1, 2, 3] + + def test_errors(self, xp): + x = xp.zeros(1) + with assert_raises(ValueError, match="axes must be a scalar or " + "iterable of integers"): + _init_nd_shape_and_axes(x, shape=None, axes=[[1, 2], [3, 4]]) + + with assert_raises(ValueError, match="axes must be a scalar or " + "iterable of integers"): + _init_nd_shape_and_axes(x, shape=None, axes=[1., 2., 3., 4.]) + + with assert_raises(ValueError, + match="axes exceeds dimensionality of input"): + _init_nd_shape_and_axes(x, shape=None, axes=[1]) + + with assert_raises(ValueError, + match="axes exceeds dimensionality of input"): + _init_nd_shape_and_axes(x, shape=None, axes=[-2]) + + with assert_raises(ValueError, + match="all axes must be unique"): + _init_nd_shape_and_axes(x, shape=None, axes=[0, 0]) + + with assert_raises(ValueError, match="shape must be a scalar or " + "iterable of integers"): + _init_nd_shape_and_axes(x, shape=[[1, 2], [3, 4]], axes=None) + + with assert_raises(ValueError, match="shape must be a scalar or " + "iterable of integers"): + _init_nd_shape_and_axes(x, shape=[1., 2., 3., 4.], axes=None) + + with assert_raises(ValueError, + match="when given, axes and shape arguments" + " have to be of the same length"): + _init_nd_shape_and_axes(xp.zeros([1, 1, 1, 1]), + shape=[1, 2, 3], axes=[1]) + + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[0\]\) specified"): + _init_nd_shape_and_axes(x, shape=[0], axes=None) + + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[-2\]\) specified"): + _init_nd_shape_and_axes(x, shape=-2, axes=None) + + +class TestFFTShift: + + def test_definition(self, xp): + x = xp.asarray([0., 1, 2, 3, 4, -4, -3, -2, -1]) + y = xp.asarray([-4., -3, -2, -1, 0, 1, 2, 3, 4]) + xp_assert_close(fft.fftshift(x), y) + xp_assert_close(fft.ifftshift(y), x) + x = xp.asarray([0., 1, 2, 3, 4, -5, -4, -3, -2, -1]) + y = xp.asarray([-5., -4, -3, -2, -1, 0, 1, 2, 3, 4]) + xp_assert_close(fft.fftshift(x), y) + xp_assert_close(fft.ifftshift(y), x) + + def test_inverse(self, xp): + for n in [1, 4, 9, 100, 211]: + x = xp.asarray(np.random.random((n,))) + xp_assert_close(fft.ifftshift(fft.fftshift(x)), x) + + @skip_xp_backends('cupy', reason='cupy/cupy#8393') + def test_axes_keyword(self, xp): + freqs = xp.asarray([[0., 1, 2], [3, 4, -4], [-3, -2, -1]]) + shifted = xp.asarray([[-1., -3, -2], [2, 0, 1], [-4, 3, 4]]) + xp_assert_close(fft.fftshift(freqs, axes=(0, 1)), shifted) + xp_assert_close(fft.fftshift(freqs, axes=0), fft.fftshift(freqs, axes=(0,))) + xp_assert_close(fft.ifftshift(shifted, axes=(0, 1)), freqs) + xp_assert_close(fft.ifftshift(shifted, axes=0), + fft.ifftshift(shifted, axes=(0,))) + xp_assert_close(fft.fftshift(freqs), shifted) + xp_assert_close(fft.ifftshift(shifted), freqs) + + @skip_xp_backends('cupy', reason='cupy/cupy#8393') + def test_uneven_dims(self, xp): + """ Test 2D input, which has uneven dimension sizes """ + freqs = xp.asarray([ + [0, 1], + [2, 3], + [4, 5] + ], dtype=xp.float64) + + # shift in dimension 0 + shift_dim0 = xp.asarray([ + [4, 5], + [0, 1], + [2, 3] + ], dtype=xp.float64) + xp_assert_close(fft.fftshift(freqs, axes=0), shift_dim0) + xp_assert_close(fft.ifftshift(shift_dim0, axes=0), freqs) + xp_assert_close(fft.fftshift(freqs, axes=(0,)), shift_dim0) + xp_assert_close(fft.ifftshift(shift_dim0, axes=[0]), freqs) + + # shift in dimension 1 + shift_dim1 = xp.asarray([ + [1, 0], + [3, 2], + [5, 4] + ], dtype=xp.float64) + xp_assert_close(fft.fftshift(freqs, axes=1), shift_dim1) + xp_assert_close(fft.ifftshift(shift_dim1, axes=1), freqs) + + # shift in both dimensions + shift_dim_both = xp.asarray([ + [5, 4], + [1, 0], + [3, 2] + ], dtype=xp.float64) + xp_assert_close(fft.fftshift(freqs, axes=(0, 1)), shift_dim_both) + xp_assert_close(fft.ifftshift(shift_dim_both, axes=(0, 1)), freqs) + xp_assert_close(fft.fftshift(freqs, axes=[0, 1]), shift_dim_both) + xp_assert_close(fft.ifftshift(shift_dim_both, axes=[0, 1]), freqs) + + # axes=None (default) shift in all dimensions + xp_assert_close(fft.fftshift(freqs, axes=None), shift_dim_both) + xp_assert_close(fft.ifftshift(shift_dim_both, axes=None), freqs) + xp_assert_close(fft.fftshift(freqs), shift_dim_both) + xp_assert_close(fft.ifftshift(shift_dim_both), freqs) + + +@skip_xp_backends("cupy", + reason="CuPy has not implemented the `device` param") +@skip_xp_backends("jax.numpy", + reason="JAX has not implemented the `device` param") +class TestFFTFreq: + + def test_definition(self, xp): + x = xp.asarray([0, 1, 2, 3, 4, -4, -3, -2, -1], dtype=xp.float64) + x2 = xp.asarray([0, 1, 2, 3, 4, -5, -4, -3, -2, -1], dtype=xp.float64) + + # default dtype varies across backends + + y = 9 * fft.fftfreq(9, xp=xp) + xp_assert_close(y, x, check_dtype=False, check_namespace=True) + + y = 9 * xp.pi * fft.fftfreq(9, xp.pi, xp=xp) + xp_assert_close(y, x, check_dtype=False) + + y = 10 * fft.fftfreq(10, xp=xp) + xp_assert_close(y, x2, check_dtype=False) + + y = 10 * xp.pi * fft.fftfreq(10, xp.pi, xp=xp) + xp_assert_close(y, x2, check_dtype=False) + + def test_device(self, xp): + xp_test = array_namespace(xp.empty(0)) + devices = get_xp_devices(xp) + for d in devices: + y = fft.fftfreq(9, xp=xp, device=d) + x = xp_test.empty(0, device=d) + assert xp_device(y) == xp_device(x) + + +@skip_xp_backends("cupy", + reason="CuPy has not implemented the `device` param") +@skip_xp_backends("jax.numpy", + reason="JAX has not implemented the `device` param") +class TestRFFTFreq: + + def test_definition(self, xp): + x = xp.asarray([0, 1, 2, 3, 4], dtype=xp.float64) + x2 = xp.asarray([0, 1, 2, 3, 4, 5], dtype=xp.float64) + + # default dtype varies across backends + + y = 9 * fft.rfftfreq(9, xp=xp) + xp_assert_close(y, x, check_dtype=False, check_namespace=True) + + y = 9 * xp.pi * fft.rfftfreq(9, xp.pi, xp=xp) + xp_assert_close(y, x, check_dtype=False) + + y = 10 * fft.rfftfreq(10, xp=xp) + xp_assert_close(y, x2, check_dtype=False) + + y = 10 * xp.pi * fft.rfftfreq(10, xp.pi, xp=xp) + xp_assert_close(y, x2, check_dtype=False) + + def test_device(self, xp): + xp_test = array_namespace(xp.empty(0)) + devices = get_xp_devices(xp) + for d in devices: + y = fft.rfftfreq(9, xp=xp, device=d) + x = xp_test.empty(0, device=d) + assert xp_device(y) == xp_device(x) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_multithreading.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_multithreading.py new file mode 100644 index 0000000000000000000000000000000000000000..1a6b71b830211f8bcbe56e97ff71098be75021c8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_multithreading.py @@ -0,0 +1,84 @@ +from scipy import fft +import numpy as np +import pytest +from numpy.testing import assert_allclose +import multiprocessing +import os + + +@pytest.fixture(scope='module') +def x(): + return np.random.randn(512, 128) # Must be large enough to qualify for mt + + +@pytest.mark.parametrize("func", [ + fft.fft, fft.ifft, fft.fft2, fft.ifft2, fft.fftn, fft.ifftn, + fft.rfft, fft.irfft, fft.rfft2, fft.irfft2, fft.rfftn, fft.irfftn, + fft.hfft, fft.ihfft, fft.hfft2, fft.ihfft2, fft.hfftn, fft.ihfftn, + fft.dct, fft.idct, fft.dctn, fft.idctn, + fft.dst, fft.idst, fft.dstn, fft.idstn, +]) +@pytest.mark.parametrize("workers", [2, -1]) +def test_threaded_same(x, func, workers): + expected = func(x, workers=1) + actual = func(x, workers=workers) + assert_allclose(actual, expected) + + +def _mt_fft(x): + return fft.fft(x, workers=2) + + +@pytest.mark.slow +def test_mixed_threads_processes(x): + # Test that the fft threadpool is safe to use before & after fork + + expect = fft.fft(x, workers=2) + + with multiprocessing.Pool(2) as p: + res = p.map(_mt_fft, [x for _ in range(4)]) + + for r in res: + assert_allclose(r, expect) + + fft.fft(x, workers=2) + + +def test_invalid_workers(x): + cpus = os.cpu_count() + + fft.ifft([1], workers=-cpus) + + with pytest.raises(ValueError, match='workers must not be zero'): + fft.fft(x, workers=0) + + with pytest.raises(ValueError, match='workers value out of range'): + fft.ifft(x, workers=-cpus-1) + + +def test_set_get_workers(): + cpus = os.cpu_count() + assert fft.get_workers() == 1 + with fft.set_workers(4): + assert fft.get_workers() == 4 + + with fft.set_workers(-1): + assert fft.get_workers() == cpus + + assert fft.get_workers() == 4 + + assert fft.get_workers() == 1 + + with fft.set_workers(-cpus): + assert fft.get_workers() == 1 + + +def test_set_workers_invalid(): + + with pytest.raises(ValueError, match='workers must not be zero'): + with fft.set_workers(0): + pass + + with pytest.raises(ValueError, match='workers value out of range'): + with fft.set_workers(-os.cpu_count()-1): + pass diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_real_transforms.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_real_transforms.py new file mode 100644 index 0000000000000000000000000000000000000000..890dc79640aff571262f5a12f721f62f5c907069 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fft/tests/test_real_transforms.py @@ -0,0 +1,249 @@ +import numpy as np +from numpy.testing import assert_allclose, assert_array_equal +import pytest +import math + +from scipy.fft import dct, idct, dctn, idctn, dst, idst, dstn, idstn +import scipy.fft as fft +from scipy import fftpack +from scipy.conftest import array_api_compatible +from scipy._lib._array_api import xp_copy, xp_assert_close + +pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends")] +skip_xp_backends = pytest.mark.skip_xp_backends + +SQRT_2 = math.sqrt(2) + +# scipy.fft wraps the fftpack versions but with normalized inverse transforms. +# So, the forward transforms and definitions are already thoroughly tested in +# fftpack/test_real_transforms.py + + +@skip_xp_backends(cpu_only=True) +@pytest.mark.parametrize("forward, backward", [(dct, idct), (dst, idst)]) +@pytest.mark.parametrize("type", [1, 2, 3, 4]) +@pytest.mark.parametrize("n", [2, 3, 4, 5, 10, 16]) +@pytest.mark.parametrize("axis", [0, 1]) +@pytest.mark.parametrize("norm", [None, 'backward', 'ortho', 'forward']) +@pytest.mark.parametrize("orthogonalize", [False, True]) +def test_identity_1d(forward, backward, type, n, axis, norm, orthogonalize, xp): + # Test the identity f^-1(f(x)) == x + x = xp.asarray(np.random.rand(n, n)) + + y = forward(x, type, axis=axis, norm=norm, orthogonalize=orthogonalize) + z = backward(y, type, axis=axis, norm=norm, orthogonalize=orthogonalize) + xp_assert_close(z, x) + + pad = [(0, 0)] * 2 + pad[axis] = (0, 4) + + y2 = xp.asarray(np.pad(np.asarray(y), pad, mode='edge')) + z2 = backward(y2, type, n, axis, norm, orthogonalize=orthogonalize) + xp_assert_close(z2, x) + + +@skip_xp_backends(np_only=True, + reason='`overwrite_x` only supported for NumPy backend.') +@pytest.mark.parametrize("forward, backward", [(dct, idct), (dst, idst)]) +@pytest.mark.parametrize("type", [1, 2, 3, 4]) +@pytest.mark.parametrize("dtype", [np.float16, np.float32, np.float64, + np.complex64, np.complex128]) +@pytest.mark.parametrize("axis", [0, 1]) +@pytest.mark.parametrize("norm", [None, 'backward', 'ortho', 'forward']) +@pytest.mark.parametrize("overwrite_x", [True, False]) +def test_identity_1d_overwrite(forward, backward, type, dtype, axis, norm, + overwrite_x): + # Test the identity f^-1(f(x)) == x + x = np.random.rand(7, 8).astype(dtype) + x_orig = x.copy() + + y = forward(x, type, axis=axis, norm=norm, overwrite_x=overwrite_x) + y_orig = y.copy() + z = backward(y, type, axis=axis, norm=norm, overwrite_x=overwrite_x) + if not overwrite_x: + assert_allclose(z, x, rtol=1e-6, atol=1e-6) + assert_array_equal(x, x_orig) + assert_array_equal(y, y_orig) + else: + assert_allclose(z, x_orig, rtol=1e-6, atol=1e-6) + + +@skip_xp_backends(cpu_only=True) +@pytest.mark.parametrize("forward, backward", [(dctn, idctn), (dstn, idstn)]) +@pytest.mark.parametrize("type", [1, 2, 3, 4]) +@pytest.mark.parametrize("shape, axes", + [ + ((4, 4), 0), + ((4, 4), 1), + ((4, 4), None), + ((4, 4), (0, 1)), + ((10, 12), None), + ((10, 12), (0, 1)), + ((4, 5, 6), None), + ((4, 5, 6), 1), + ((4, 5, 6), (0, 2)), + ]) +@pytest.mark.parametrize("norm", [None, 'backward', 'ortho', 'forward']) +@pytest.mark.parametrize("orthogonalize", [False, True]) +def test_identity_nd(forward, backward, type, shape, axes, norm, + orthogonalize, xp): + # Test the identity f^-1(f(x)) == x + + x = xp.asarray(np.random.random(shape)) + + if axes is not None: + shape = np.take(shape, axes) + + y = forward(x, type, axes=axes, norm=norm, orthogonalize=orthogonalize) + z = backward(y, type, axes=axes, norm=norm, orthogonalize=orthogonalize) + xp_assert_close(z, x) + + if axes is None: + pad = [(0, 4)] * x.ndim + elif isinstance(axes, int): + pad = [(0, 0)] * x.ndim + pad[axes] = (0, 4) + else: + pad = [(0, 0)] * x.ndim + + for a in axes: + pad[a] = (0, 4) + + # TODO write an array-agnostic pad + y2 = xp.asarray(np.pad(np.asarray(y), pad, mode='edge')) + z2 = backward(y2, type, shape, axes, norm, orthogonalize=orthogonalize) + xp_assert_close(z2, x) + + +@skip_xp_backends(np_only=True, + reason='`overwrite_x` only supported for NumPy backend.') +@pytest.mark.parametrize("forward, backward", [(dctn, idctn), (dstn, idstn)]) +@pytest.mark.parametrize("type", [1, 2, 3, 4]) +@pytest.mark.parametrize("shape, axes", + [ + ((4, 5), 0), + ((4, 5), 1), + ((4, 5), None), + ]) +@pytest.mark.parametrize("dtype", [np.float16, np.float32, np.float64, + np.complex64, np.complex128]) +@pytest.mark.parametrize("norm", [None, 'backward', 'ortho', 'forward']) +@pytest.mark.parametrize("overwrite_x", [False, True]) +def test_identity_nd_overwrite(forward, backward, type, shape, axes, dtype, + norm, overwrite_x): + # Test the identity f^-1(f(x)) == x + + x = np.random.random(shape).astype(dtype) + x_orig = x.copy() + + if axes is not None: + shape = np.take(shape, axes) + + y = forward(x, type, axes=axes, norm=norm) + y_orig = y.copy() + z = backward(y, type, axes=axes, norm=norm) + if overwrite_x: + assert_allclose(z, x_orig, rtol=1e-6, atol=1e-6) + else: + assert_allclose(z, x, rtol=1e-6, atol=1e-6) + assert_array_equal(x, x_orig) + assert_array_equal(y, y_orig) + + +@skip_xp_backends(cpu_only=True) +@pytest.mark.parametrize("func", ['dct', 'dst', 'dctn', 'dstn']) +@pytest.mark.parametrize("type", [1, 2, 3, 4]) +@pytest.mark.parametrize("norm", [None, 'backward', 'ortho', 'forward']) +def test_fftpack_equivalience(func, type, norm, xp): + x = np.random.rand(8, 16) + fftpack_res = xp.asarray(getattr(fftpack, func)(x, type, norm=norm)) + x = xp.asarray(x) + fft_res = getattr(fft, func)(x, type, norm=norm) + + xp_assert_close(fft_res, fftpack_res) + + +@skip_xp_backends(cpu_only=True) +@pytest.mark.parametrize("func", [dct, dst, dctn, dstn]) +@pytest.mark.parametrize("type", [1, 2, 3, 4]) +def test_orthogonalize_default(func, type, xp): + # Test orthogonalize is the default when norm="ortho", but not otherwise + x = xp.asarray(np.random.rand(100)) + + for norm, ortho in [ + ("forward", False), + ("backward", False), + ("ortho", True), + ]: + a = func(x, type=type, norm=norm, orthogonalize=ortho) + b = func(x, type=type, norm=norm) + xp_assert_close(a, b) + + +@skip_xp_backends(cpu_only=True) +@pytest.mark.parametrize("norm", ["backward", "ortho", "forward"]) +@pytest.mark.parametrize("func, type", [ + (dct, 4), (dst, 1), (dst, 4)]) +def test_orthogonalize_noop(func, type, norm, xp): + # Transforms where orthogonalize is a no-op + x = xp.asarray(np.random.rand(100)) + y1 = func(x, type=type, norm=norm, orthogonalize=True) + y2 = func(x, type=type, norm=norm, orthogonalize=False) + xp_assert_close(y1, y2) + + +@skip_xp_backends('jax.numpy', + reason='jax arrays do not support item assignment', + cpu_only=True) +@pytest.mark.parametrize("norm", ["backward", "ortho", "forward"]) +def test_orthogonalize_dct1(norm, xp): + x = xp.asarray(np.random.rand(100)) + + x2 = xp_copy(x, xp=xp) + x2[0] *= SQRT_2 + x2[-1] *= SQRT_2 + + y1 = dct(x, type=1, norm=norm, orthogonalize=True) + y2 = dct(x2, type=1, norm=norm, orthogonalize=False) + + y2[0] /= SQRT_2 + y2[-1] /= SQRT_2 + xp_assert_close(y1, y2) + + +@skip_xp_backends('jax.numpy', + reason='jax arrays do not support item assignment', + cpu_only=True) +@pytest.mark.parametrize("norm", ["backward", "ortho", "forward"]) +@pytest.mark.parametrize("func", [dct, dst]) +def test_orthogonalize_dcst2(func, norm, xp): + x = xp.asarray(np.random.rand(100)) + y1 = func(x, type=2, norm=norm, orthogonalize=True) + y2 = func(x, type=2, norm=norm, orthogonalize=False) + + y2[0 if func == dct else -1] /= SQRT_2 + xp_assert_close(y1, y2) + + +@skip_xp_backends('jax.numpy', + reason='jax arrays do not support item assignment', + cpu_only=True) +@pytest.mark.parametrize("norm", ["backward", "ortho", "forward"]) +@pytest.mark.parametrize("func", [dct, dst]) +def test_orthogonalize_dcst3(func, norm, xp): + x = xp.asarray(np.random.rand(100)) + x2 = xp_copy(x, xp=xp) + x2[0 if func == dct else -1] *= SQRT_2 + + y1 = func(x, type=3, norm=norm, orthogonalize=True) + y2 = func(x2, type=3, norm=norm, orthogonalize=False) + xp_assert_close(y1, y2) + +@skip_xp_backends(np_only=True, + reason='array-likes only supported for NumPy backend') +@pytest.mark.parametrize("func", [dct, idct, dctn, idctn, dst, idst, dstn, idstn]) +def test_array_like(xp, func): + x = [[[1.0, 1.0], [1.0, 1.0]], + [[1.0, 1.0], [1.0, 1.0]], + [[1.0, 1.0], [1.0, 1.0]]] + xp_assert_close(func(x), func(xp.asarray(x))) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..10f4b39e48e2d6c0b042582ca65f572bde6ba575 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/__init__.py @@ -0,0 +1,103 @@ +""" +========================================================= +Legacy discrete Fourier transforms (:mod:`scipy.fftpack`) +========================================================= + +.. legacy:: + + New code should use :mod:`scipy.fft`. + +Fast Fourier Transforms (FFTs) +============================== + +.. autosummary:: + :toctree: generated/ + + fft - Fast (discrete) Fourier Transform (FFT) + ifft - Inverse FFT + fft2 - 2-D FFT + ifft2 - 2-D inverse FFT + fftn - N-D FFT + ifftn - N-D inverse FFT + rfft - FFT of strictly real-valued sequence + irfft - Inverse of rfft + dct - Discrete cosine transform + idct - Inverse discrete cosine transform + dctn - N-D Discrete cosine transform + idctn - N-D Inverse discrete cosine transform + dst - Discrete sine transform + idst - Inverse discrete sine transform + dstn - N-D Discrete sine transform + idstn - N-D Inverse discrete sine transform + +Differential and pseudo-differential operators +============================================== + +.. autosummary:: + :toctree: generated/ + + diff - Differentiation and integration of periodic sequences + tilbert - Tilbert transform: cs_diff(x,h,h) + itilbert - Inverse Tilbert transform: sc_diff(x,h,h) + hilbert - Hilbert transform: cs_diff(x,inf,inf) + ihilbert - Inverse Hilbert transform: sc_diff(x,inf,inf) + cs_diff - cosh/sinh pseudo-derivative of periodic sequences + sc_diff - sinh/cosh pseudo-derivative of periodic sequences + ss_diff - sinh/sinh pseudo-derivative of periodic sequences + cc_diff - cosh/cosh pseudo-derivative of periodic sequences + shift - Shift periodic sequences + +Helper functions +================ + +.. autosummary:: + :toctree: generated/ + + fftshift - Shift the zero-frequency component to the center of the spectrum + ifftshift - The inverse of `fftshift` + fftfreq - Return the Discrete Fourier Transform sample frequencies + rfftfreq - DFT sample frequencies (for usage with rfft, irfft) + next_fast_len - Find the optimal length to zero-pad an FFT for speed + +Note that ``fftshift``, ``ifftshift`` and ``fftfreq`` are numpy functions +exposed by ``fftpack``; importing them from ``numpy`` should be preferred. + +Convolutions (:mod:`scipy.fftpack.convolve`) +============================================ + +.. module:: scipy.fftpack.convolve + +.. autosummary:: + :toctree: generated/ + + convolve + convolve_z + init_convolution_kernel + destroy_convolve_cache + +""" + + +__all__ = ['fft','ifft','fftn','ifftn','rfft','irfft', + 'fft2','ifft2', + 'diff', + 'tilbert','itilbert','hilbert','ihilbert', + 'sc_diff','cs_diff','cc_diff','ss_diff', + 'shift', + 'fftfreq', 'rfftfreq', + 'fftshift', 'ifftshift', + 'next_fast_len', + 'dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn' + ] + +from ._basic import * +from ._pseudo_diffs import * +from ._helper import * +from ._realtransforms import * + +# Deprecated namespaces, to be removed in v2.0.0 +from . import basic, helper, pseudo_diffs, realtransforms + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_basic.py new file mode 100644 index 0000000000000000000000000000000000000000..59c85ae4b364464a66489ef221f7f7ac45624694 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_basic.py @@ -0,0 +1,428 @@ +""" +Discrete Fourier Transforms - _basic.py +""" +# Created by Pearu Peterson, August,September 2002 +__all__ = ['fft','ifft','fftn','ifftn','rfft','irfft', + 'fft2','ifft2'] + +from scipy.fft import _pocketfft +from ._helper import _good_shape + + +def fft(x, n=None, axis=-1, overwrite_x=False): + """ + Return discrete Fourier transform of real or complex sequence. + + The returned complex array contains ``y(0), y(1),..., y(n-1)``, where + + ``y(j) = (x * exp(-2*pi*sqrt(-1)*j*np.arange(n)/n)).sum()``. + + Parameters + ---------- + x : array_like + Array to Fourier transform. + n : int, optional + Length of the Fourier transform. If ``n < x.shape[axis]``, `x` is + truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The + default results in ``n = x.shape[axis]``. + axis : int, optional + Axis along which the fft's are computed; the default is over the + last axis (i.e., ``axis=-1``). + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + z : complex ndarray + with the elements:: + + [y(0),y(1),..,y(n/2),y(1-n/2),...,y(-1)] if n is even + [y(0),y(1),..,y((n-1)/2),y(-(n-1)/2),...,y(-1)] if n is odd + + where:: + + y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n), j = 0..n-1 + + See Also + -------- + ifft : Inverse FFT + rfft : FFT of a real sequence + + Notes + ----- + The packing of the result is "standard": If ``A = fft(a, n)``, then + ``A[0]`` contains the zero-frequency term, ``A[1:n/2]`` contains the + positive-frequency terms, and ``A[n/2:]`` contains the negative-frequency + terms, in order of decreasingly negative frequency. So ,for an 8-point + transform, the frequencies of the result are [0, 1, 2, 3, -4, -3, -2, -1]. + To rearrange the fft output so that the zero-frequency component is + centered, like [-4, -3, -2, -1, 0, 1, 2, 3], use `fftshift`. + + Both single and double precision routines are implemented. Half precision + inputs will be converted to single precision. Non-floating-point inputs + will be converted to double precision. Long-double precision inputs are + not supported. + + This function is most efficient when `n` is a power of two, and least + efficient when `n` is prime. + + Note that if ``x`` is real-valued, then ``A[j] == A[n-j].conjugate()``. + If ``x`` is real-valued and ``n`` is even, then ``A[n/2]`` is real. + + If the data type of `x` is real, a "real FFT" algorithm is automatically + used, which roughly halves the computation time. To increase efficiency + a little further, use `rfft`, which does the same calculation, but only + outputs half of the symmetrical spectrum. If the data is both real and + symmetrical, the `dct` can again double the efficiency by generating + half of the spectrum from half of the signal. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fftpack import fft, ifft + >>> x = np.arange(5) + >>> np.allclose(fft(ifft(x)), x, atol=1e-15) # within numerical accuracy. + True + + """ + return _pocketfft.fft(x, n, axis, None, overwrite_x) + + +def ifft(x, n=None, axis=-1, overwrite_x=False): + """ + Return discrete inverse Fourier transform of real or complex sequence. + + The returned complex array contains ``y(0), y(1),..., y(n-1)``, where + + ``y(j) = (x * exp(2*pi*sqrt(-1)*j*np.arange(n)/n)).mean()``. + + Parameters + ---------- + x : array_like + Transformed data to invert. + n : int, optional + Length of the inverse Fourier transform. If ``n < x.shape[axis]``, + `x` is truncated. If ``n > x.shape[axis]``, `x` is zero-padded. + The default results in ``n = x.shape[axis]``. + axis : int, optional + Axis along which the ifft's are computed; the default is over the + last axis (i.e., ``axis=-1``). + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + ifft : ndarray of floats + The inverse discrete Fourier transform. + + See Also + -------- + fft : Forward FFT + + Notes + ----- + Both single and double precision routines are implemented. Half precision + inputs will be converted to single precision. Non-floating-point inputs + will be converted to double precision. Long-double precision inputs are + not supported. + + This function is most efficient when `n` is a power of two, and least + efficient when `n` is prime. + + If the data type of `x` is real, a "real IFFT" algorithm is automatically + used, which roughly halves the computation time. + + Examples + -------- + >>> from scipy.fftpack import fft, ifft + >>> import numpy as np + >>> x = np.arange(5) + >>> np.allclose(ifft(fft(x)), x, atol=1e-15) # within numerical accuracy. + True + + """ + return _pocketfft.ifft(x, n, axis, None, overwrite_x) + + +def rfft(x, n=None, axis=-1, overwrite_x=False): + """ + Discrete Fourier transform of a real sequence. + + Parameters + ---------- + x : array_like, real-valued + The data to transform. + n : int, optional + Defines the length of the Fourier transform. If `n` is not specified + (the default) then ``n = x.shape[axis]``. If ``n < x.shape[axis]``, + `x` is truncated, if ``n > x.shape[axis]``, `x` is zero-padded. + axis : int, optional + The axis along which the transform is applied. The default is the + last axis. + overwrite_x : bool, optional + If set to true, the contents of `x` can be overwritten. Default is + False. + + Returns + ------- + z : real ndarray + The returned real array contains:: + + [y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2))] if n is even + [y(0),Re(y(1)),Im(y(1)),...,Re(y(n/2)),Im(y(n/2))] if n is odd + + where:: + + y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k*2*pi/n) + j = 0..n-1 + + See Also + -------- + fft, irfft, scipy.fft.rfft + + Notes + ----- + Within numerical accuracy, ``y == rfft(irfft(y))``. + + Both single and double precision routines are implemented. Half precision + inputs will be converted to single precision. Non-floating-point inputs + will be converted to double precision. Long-double precision inputs are + not supported. + + To get an output with a complex datatype, consider using the newer + function `scipy.fft.rfft`. + + Examples + -------- + >>> from scipy.fftpack import fft, rfft + >>> a = [9, -9, 1, 3] + >>> fft(a) + array([ 4. +0.j, 8.+12.j, 16. +0.j, 8.-12.j]) + >>> rfft(a) + array([ 4., 8., 12., 16.]) + + """ + return _pocketfft.rfft_fftpack(x, n, axis, None, overwrite_x) + + +def irfft(x, n=None, axis=-1, overwrite_x=False): + """ + Return inverse discrete Fourier transform of real sequence x. + + The contents of `x` are interpreted as the output of the `rfft` + function. + + Parameters + ---------- + x : array_like + Transformed data to invert. + n : int, optional + Length of the inverse Fourier transform. + If n < x.shape[axis], x is truncated. + If n > x.shape[axis], x is zero-padded. + The default results in n = x.shape[axis]. + axis : int, optional + Axis along which the ifft's are computed; the default is over + the last axis (i.e., axis=-1). + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + irfft : ndarray of floats + The inverse discrete Fourier transform. + + See Also + -------- + rfft, ifft, scipy.fft.irfft + + Notes + ----- + The returned real array contains:: + + [y(0),y(1),...,y(n-1)] + + where for n is even:: + + y(j) = 1/n (sum[k=1..n/2-1] (x[2*k-1]+sqrt(-1)*x[2*k]) + * exp(sqrt(-1)*j*k* 2*pi/n) + + c.c. + x[0] + (-1)**(j) x[n-1]) + + and for n is odd:: + + y(j) = 1/n (sum[k=1..(n-1)/2] (x[2*k-1]+sqrt(-1)*x[2*k]) + * exp(sqrt(-1)*j*k* 2*pi/n) + + c.c. + x[0]) + + c.c. denotes complex conjugate of preceding expression. + + For details on input parameters, see `rfft`. + + To process (conjugate-symmetric) frequency-domain data with a complex + datatype, consider using the newer function `scipy.fft.irfft`. + + Examples + -------- + >>> from scipy.fftpack import rfft, irfft + >>> a = [1.0, 2.0, 3.0, 4.0, 5.0] + >>> irfft(a) + array([ 2.6 , -3.16405192, 1.24398433, -1.14955713, 1.46962473]) + >>> irfft(rfft(a)) + array([1., 2., 3., 4., 5.]) + + """ + return _pocketfft.irfft_fftpack(x, n, axis, None, overwrite_x) + + +def fftn(x, shape=None, axes=None, overwrite_x=False): + """ + Return multidimensional discrete Fourier transform. + + The returned array contains:: + + y[j_1,..,j_d] = sum[k_1=0..n_1-1, ..., k_d=0..n_d-1] + x[k_1,..,k_d] * prod[i=1..d] exp(-sqrt(-1)*2*pi/n_i * j_i * k_i) + + where d = len(x.shape) and n = x.shape. + + Parameters + ---------- + x : array_like + The (N-D) array to transform. + shape : int or array_like of ints or None, optional + The shape of the result. If both `shape` and `axes` (see below) are + None, `shape` is ``x.shape``; if `shape` is None but `axes` is + not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``. + If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. + If ``shape[i] < x.shape[i]``, the ith dimension is truncated to + length ``shape[i]``. + If any element of `shape` is -1, the size of the corresponding + dimension of `x` is used. + axes : int or array_like of ints or None, optional + The axes of `x` (`y` if `shape` is not None) along which the + transform is applied. + The default is over all axes. + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed. Default is False. + + Returns + ------- + y : complex-valued N-D NumPy array + The (N-D) DFT of the input array. + + See Also + -------- + ifftn + + Notes + ----- + If ``x`` is real-valued, then + ``y[..., j_i, ...] == y[..., n_i-j_i, ...].conjugate()``. + + Both single and double precision routines are implemented. Half precision + inputs will be converted to single precision. Non-floating-point inputs + will be converted to double precision. Long-double precision inputs are + not supported. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fftpack import fftn, ifftn + >>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16)) + >>> np.allclose(y, fftn(ifftn(y))) + True + + """ + shape = _good_shape(x, shape, axes) + return _pocketfft.fftn(x, shape, axes, None, overwrite_x) + + +def ifftn(x, shape=None, axes=None, overwrite_x=False): + """ + Return inverse multidimensional discrete Fourier transform. + + The sequence can be of an arbitrary type. + + The returned array contains:: + + y[j_1,..,j_d] = 1/p * sum[k_1=0..n_1-1, ..., k_d=0..n_d-1] + x[k_1,..,k_d] * prod[i=1..d] exp(sqrt(-1)*2*pi/n_i * j_i * k_i) + + where ``d = len(x.shape)``, ``n = x.shape``, and ``p = prod[i=1..d] n_i``. + + For description of parameters see `fftn`. + + See Also + -------- + fftn : for detailed information. + + Examples + -------- + >>> from scipy.fftpack import fftn, ifftn + >>> import numpy as np + >>> y = (-np.arange(16), 8 - np.arange(16), np.arange(16)) + >>> np.allclose(y, ifftn(fftn(y))) + True + + """ + shape = _good_shape(x, shape, axes) + return _pocketfft.ifftn(x, shape, axes, None, overwrite_x) + + +def fft2(x, shape=None, axes=(-2,-1), overwrite_x=False): + """ + 2-D discrete Fourier transform. + + Return the 2-D discrete Fourier transform of the 2-D argument + `x`. + + See Also + -------- + fftn : for detailed information. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fftpack import fft2, ifft2 + >>> y = np.mgrid[:5, :5][0] + >>> y + array([[0, 0, 0, 0, 0], + [1, 1, 1, 1, 1], + [2, 2, 2, 2, 2], + [3, 3, 3, 3, 3], + [4, 4, 4, 4, 4]]) + >>> np.allclose(y, ifft2(fft2(y))) + True + """ + return fftn(x,shape,axes,overwrite_x) + + +def ifft2(x, shape=None, axes=(-2,-1), overwrite_x=False): + """ + 2-D discrete inverse Fourier transform of real or complex sequence. + + Return inverse 2-D discrete Fourier transform of + arbitrary type sequence x. + + See `ifft` for more information. + + See Also + -------- + fft2, ifft + + Examples + -------- + >>> import numpy as np + >>> from scipy.fftpack import fft2, ifft2 + >>> y = np.mgrid[:5, :5][0] + >>> y + array([[0, 0, 0, 0, 0], + [1, 1, 1, 1, 1], + [2, 2, 2, 2, 2], + [3, 3, 3, 3, 3], + [4, 4, 4, 4, 4]]) + >>> np.allclose(y, fft2(ifft2(y))) + True + + """ + return ifftn(x,shape,axes,overwrite_x) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_helper.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_helper.py new file mode 100644 index 0000000000000000000000000000000000000000..7892543732906dcd86b4c1aa9c1f249af701c137 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_helper.py @@ -0,0 +1,115 @@ +import operator + +import numpy as np +from numpy.fft import fftshift, ifftshift, fftfreq + +import scipy.fft._pocketfft.helper as _helper + +__all__ = ['fftshift', 'ifftshift', 'fftfreq', 'rfftfreq', 'next_fast_len'] + + +def rfftfreq(n, d=1.0): + """DFT sample frequencies (for usage with rfft, irfft). + + The returned float array contains the frequency bins in + cycles/unit (with zero at the start) given a window length `n` and a + sample spacing `d`:: + + f = [0,1,1,2,2,...,n/2-1,n/2-1,n/2]/(d*n) if n is even + f = [0,1,1,2,2,...,n/2-1,n/2-1,n/2,n/2]/(d*n) if n is odd + + Parameters + ---------- + n : int + Window length. + d : scalar, optional + Sample spacing. Default is 1. + + Returns + ------- + out : ndarray + The array of length `n`, containing the sample frequencies. + + Examples + -------- + >>> import numpy as np + >>> from scipy import fftpack + >>> sig = np.array([-2, 8, 6, 4, 1, 0, 3, 5], dtype=float) + >>> sig_fft = fftpack.rfft(sig) + >>> n = sig_fft.size + >>> timestep = 0.1 + >>> freq = fftpack.rfftfreq(n, d=timestep) + >>> freq + array([ 0. , 1.25, 1.25, 2.5 , 2.5 , 3.75, 3.75, 5. ]) + + """ + n = operator.index(n) + if n < 0: + raise ValueError(f"n = {n} is not valid. " + "n must be a nonnegative integer.") + + return (np.arange(1, n + 1, dtype=int) // 2) / float(n * d) + + +def next_fast_len(target): + """ + Find the next fast size of input data to `fft`, for zero-padding, etc. + + SciPy's FFTPACK has efficient functions for radix {2, 3, 4, 5}, so this + returns the next composite of the prime factors 2, 3, and 5 which is + greater than or equal to `target`. (These are also known as 5-smooth + numbers, regular numbers, or Hamming numbers.) + + Parameters + ---------- + target : int + Length to start searching from. Must be a positive integer. + + Returns + ------- + out : int + The first 5-smooth number greater than or equal to `target`. + + Notes + ----- + .. versionadded:: 0.18.0 + + Examples + -------- + On a particular machine, an FFT of prime length takes 133 ms: + + >>> from scipy import fftpack + >>> import numpy as np + >>> rng = np.random.default_rng() + >>> min_len = 10007 # prime length is worst case for speed + >>> a = rng.standard_normal(min_len) + >>> b = fftpack.fft(a) + + Zero-padding to the next 5-smooth length reduces computation time to + 211 us, a speedup of 630 times: + + >>> fftpack.next_fast_len(min_len) + 10125 + >>> b = fftpack.fft(a, 10125) + + Rounding up to the next power of 2 is not optimal, taking 367 us to + compute, 1.7 times as long as the 5-smooth size: + + >>> b = fftpack.fft(a, 16384) + + """ + # Real transforms use regular sizes so this is backwards compatible + return _helper.good_size(target, True) + + +def _good_shape(x, shape, axes): + """Ensure that shape argument is valid for scipy.fftpack + + scipy.fftpack does not support len(shape) < x.ndim when axes is not given. + """ + if shape is not None and axes is None: + shape = _helper._iterable_of_int(shape, 'shape') + if len(shape) != np.ndim(x): + raise ValueError("when given, axes and shape arguments" + " have to be of the same length") + return shape diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_pseudo_diffs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_pseudo_diffs.py new file mode 100644 index 0000000000000000000000000000000000000000..6dbcc8d3979b35c1497266fea34cf565cd3d11d7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_pseudo_diffs.py @@ -0,0 +1,554 @@ +""" +Differential and pseudo-differential operators. +""" +# Created by Pearu Peterson, September 2002 + +__all__ = ['diff', + 'tilbert','itilbert','hilbert','ihilbert', + 'cs_diff','cc_diff','sc_diff','ss_diff', + 'shift'] + +import threading + +from numpy import pi, asarray, sin, cos, sinh, cosh, tanh, iscomplexobj +from . import convolve + +from scipy.fft._pocketfft.helper import _datacopied + + +_cache = threading.local() + + +def diff(x,order=1,period=None, _cache=_cache): + """ + Return kth derivative (or integral) of a periodic sequence x. + + If x_j and y_j are Fourier coefficients of periodic functions x + and y, respectively, then:: + + y_j = pow(sqrt(-1)*j*2*pi/period, order) * x_j + y_0 = 0 if order is not 0. + + Parameters + ---------- + x : array_like + Input array. + order : int, optional + The order of differentiation. Default order is 1. If order is + negative, then integration is carried out under the assumption + that ``x_0 == 0``. + period : float, optional + The assumed period of the sequence. Default is ``2*pi``. + + Notes + ----- + If ``sum(x, axis=0) = 0`` then ``diff(diff(x, k), -k) == x`` (within + numerical accuracy). + + For odd order and even ``len(x)``, the Nyquist mode is taken zero. + + """ + if isinstance(_cache, threading.local): + if not hasattr(_cache, 'diff_cache'): + _cache.diff_cache = {} + _cache = _cache.diff_cache + + tmp = asarray(x) + if order == 0: + return tmp + if iscomplexobj(tmp): + return diff(tmp.real, order, period, _cache)+1j*diff( + tmp.imag, order, period, _cache) + if period is not None: + c = 2*pi/period + else: + c = 1.0 + n = len(x) + omega = _cache.get((n,order,c)) + if omega is None: + if len(_cache) > 20: + while _cache: + _cache.popitem() + + def kernel(k,order=order,c=c): + if k: + return pow(c*k,order) + return 0 + omega = convolve.init_convolution_kernel(n,kernel,d=order, + zero_nyquist=1) + _cache[(n,order,c)] = omega + overwrite_x = _datacopied(tmp, x) + return convolve.convolve(tmp,omega,swap_real_imag=order % 2, + overwrite_x=overwrite_x) + + +def tilbert(x, h, period=None, _cache=_cache): + """ + Return h-Tilbert transform of a periodic sequence x. + + If x_j and y_j are Fourier coefficients of periodic functions x + and y, respectively, then:: + + y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j + y_0 = 0 + + Parameters + ---------- + x : array_like + The input array to transform. + h : float + Defines the parameter of the Tilbert transform. + period : float, optional + The assumed period of the sequence. Default period is ``2*pi``. + + Returns + ------- + tilbert : ndarray + The result of the transform. + + Notes + ----- + If ``sum(x, axis=0) == 0`` and ``n = len(x)`` is odd, then + ``tilbert(itilbert(x)) == x``. + + If ``2 * pi * h / period`` is approximately 10 or larger, then + numerically ``tilbert == hilbert`` + (theoretically oo-Tilbert == Hilbert). + + For even ``len(x)``, the Nyquist mode of ``x`` is taken zero. + + """ + if isinstance(_cache, threading.local): + if not hasattr(_cache, 'tilbert_cache'): + _cache.tilbert_cache = {} + _cache = _cache.tilbert_cache + + tmp = asarray(x) + if iscomplexobj(tmp): + return tilbert(tmp.real, h, period, _cache) + \ + 1j * tilbert(tmp.imag, h, period, _cache) + + if period is not None: + h = h * 2 * pi / period + + n = len(x) + omega = _cache.get((n, h)) + if omega is None: + if len(_cache) > 20: + while _cache: + _cache.popitem() + + def kernel(k, h=h): + if k: + return 1.0/tanh(h*k) + + return 0 + + omega = convolve.init_convolution_kernel(n, kernel, d=1) + _cache[(n,h)] = omega + + overwrite_x = _datacopied(tmp, x) + return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x) + + +def itilbert(x,h,period=None, _cache=_cache): + """ + Return inverse h-Tilbert transform of a periodic sequence x. + + If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x + and y, respectively, then:: + + y_j = -sqrt(-1)*tanh(j*h*2*pi/period) * x_j + y_0 = 0 + + For more details, see `tilbert`. + + """ + if isinstance(_cache, threading.local): + if not hasattr(_cache, 'itilbert_cache'): + _cache.itilbert_cache = {} + _cache = _cache.itilbert_cache + + tmp = asarray(x) + if iscomplexobj(tmp): + return itilbert(tmp.real, h, period, _cache) + \ + 1j*itilbert(tmp.imag, h, period, _cache) + if period is not None: + h = h*2*pi/period + n = len(x) + omega = _cache.get((n,h)) + if omega is None: + if len(_cache) > 20: + while _cache: + _cache.popitem() + + def kernel(k,h=h): + if k: + return -tanh(h*k) + return 0 + omega = convolve.init_convolution_kernel(n,kernel,d=1) + _cache[(n,h)] = omega + overwrite_x = _datacopied(tmp, x) + return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x) + + +def hilbert(x, _cache=_cache): + """ + Return Hilbert transform of a periodic sequence x. + + If x_j and y_j are Fourier coefficients of periodic functions x + and y, respectively, then:: + + y_j = sqrt(-1)*sign(j) * x_j + y_0 = 0 + + Parameters + ---------- + x : array_like + The input array, should be periodic. + _cache : dict, optional + Dictionary that contains the kernel used to do a convolution with. + + Returns + ------- + y : ndarray + The transformed input. + + See Also + -------- + scipy.signal.hilbert : Compute the analytic signal, using the Hilbert + transform. + + Notes + ----- + If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``. + + For even len(x), the Nyquist mode of x is taken zero. + + The sign of the returned transform does not have a factor -1 that is more + often than not found in the definition of the Hilbert transform. Note also + that `scipy.signal.hilbert` does have an extra -1 factor compared to this + function. + + """ + if isinstance(_cache, threading.local): + if not hasattr(_cache, 'hilbert_cache'): + _cache.hilbert_cache = {} + _cache = _cache.hilbert_cache + + tmp = asarray(x) + if iscomplexobj(tmp): + return hilbert(tmp.real, _cache) + 1j * hilbert(tmp.imag, _cache) + n = len(x) + omega = _cache.get(n) + if omega is None: + if len(_cache) > 20: + while _cache: + _cache.popitem() + + def kernel(k): + if k > 0: + return 1.0 + elif k < 0: + return -1.0 + return 0.0 + omega = convolve.init_convolution_kernel(n,kernel,d=1) + _cache[n] = omega + overwrite_x = _datacopied(tmp, x) + return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x) + + +def ihilbert(x, _cache=_cache): + """ + Return inverse Hilbert transform of a periodic sequence x. + + If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x + and y, respectively, then:: + + y_j = -sqrt(-1)*sign(j) * x_j + y_0 = 0 + + """ + if isinstance(_cache, threading.local): + if not hasattr(_cache, 'ihilbert_cache'): + _cache.ihilbert_cache = {} + _cache = _cache.ihilbert_cache + return -hilbert(x, _cache) + + +def cs_diff(x, a, b, period=None, _cache=_cache): + """ + Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence. + + If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x + and y, respectively, then:: + + y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j + y_0 = 0 + + Parameters + ---------- + x : array_like + The array to take the pseudo-derivative from. + a, b : float + Defines the parameters of the cosh/sinh pseudo-differential + operator. + period : float, optional + The period of the sequence. Default period is ``2*pi``. + + Returns + ------- + cs_diff : ndarray + Pseudo-derivative of periodic sequence `x`. + + Notes + ----- + For even len(`x`), the Nyquist mode of `x` is taken as zero. + + """ + if isinstance(_cache, threading.local): + if not hasattr(_cache, 'cs_diff_cache'): + _cache.cs_diff_cache = {} + _cache = _cache.cs_diff_cache + + tmp = asarray(x) + if iscomplexobj(tmp): + return cs_diff(tmp.real, a, b, period, _cache) + \ + 1j*cs_diff(tmp.imag, a, b, period, _cache) + if period is not None: + a = a*2*pi/period + b = b*2*pi/period + n = len(x) + omega = _cache.get((n,a,b)) + if omega is None: + if len(_cache) > 20: + while _cache: + _cache.popitem() + + def kernel(k,a=a,b=b): + if k: + return -cosh(a*k)/sinh(b*k) + return 0 + omega = convolve.init_convolution_kernel(n,kernel,d=1) + _cache[(n,a,b)] = omega + overwrite_x = _datacopied(tmp, x) + return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x) + + +def sc_diff(x, a, b, period=None, _cache=_cache): + """ + Return (a,b)-sinh/cosh pseudo-derivative of a periodic sequence x. + + If x_j and y_j are Fourier coefficients of periodic functions x + and y, respectively, then:: + + y_j = sqrt(-1)*sinh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j + y_0 = 0 + + Parameters + ---------- + x : array_like + Input array. + a,b : float + Defines the parameters of the sinh/cosh pseudo-differential + operator. + period : float, optional + The period of the sequence x. Default is 2*pi. + + Notes + ----- + ``sc_diff(cs_diff(x,a,b),b,a) == x`` + For even ``len(x)``, the Nyquist mode of x is taken as zero. + + """ + if isinstance(_cache, threading.local): + if not hasattr(_cache, 'sc_diff_cache'): + _cache.sc_diff_cache = {} + _cache = _cache.sc_diff_cache + + tmp = asarray(x) + if iscomplexobj(tmp): + return sc_diff(tmp.real, a, b, period, _cache) + \ + 1j * sc_diff(tmp.imag, a, b, period, _cache) + if period is not None: + a = a*2*pi/period + b = b*2*pi/period + n = len(x) + omega = _cache.get((n,a,b)) + if omega is None: + if len(_cache) > 20: + while _cache: + _cache.popitem() + + def kernel(k,a=a,b=b): + if k: + return sinh(a*k)/cosh(b*k) + return 0 + omega = convolve.init_convolution_kernel(n,kernel,d=1) + _cache[(n,a,b)] = omega + overwrite_x = _datacopied(tmp, x) + return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x) + + +def ss_diff(x, a, b, period=None, _cache=_cache): + """ + Return (a,b)-sinh/sinh pseudo-derivative of a periodic sequence x. + + If x_j and y_j are Fourier coefficients of periodic functions x + and y, respectively, then:: + + y_j = sinh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j + y_0 = a/b * x_0 + + Parameters + ---------- + x : array_like + The array to take the pseudo-derivative from. + a,b + Defines the parameters of the sinh/sinh pseudo-differential + operator. + period : float, optional + The period of the sequence x. Default is ``2*pi``. + + Notes + ----- + ``ss_diff(ss_diff(x,a,b),b,a) == x`` + + """ + if isinstance(_cache, threading.local): + if not hasattr(_cache, 'ss_diff_cache'): + _cache.ss_diff_cache = {} + _cache = _cache.ss_diff_cache + + tmp = asarray(x) + if iscomplexobj(tmp): + return ss_diff(tmp.real, a, b, period, _cache) + \ + 1j*ss_diff(tmp.imag, a, b, period, _cache) + if period is not None: + a = a*2*pi/period + b = b*2*pi/period + n = len(x) + omega = _cache.get((n,a,b)) + if omega is None: + if len(_cache) > 20: + while _cache: + _cache.popitem() + + def kernel(k,a=a,b=b): + if k: + return sinh(a*k)/sinh(b*k) + return float(a)/b + omega = convolve.init_convolution_kernel(n,kernel) + _cache[(n,a,b)] = omega + overwrite_x = _datacopied(tmp, x) + return convolve.convolve(tmp,omega,overwrite_x=overwrite_x) + + +def cc_diff(x, a, b, period=None, _cache=_cache): + """ + Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence. + + If x_j and y_j are Fourier coefficients of periodic functions x + and y, respectively, then:: + + y_j = cosh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j + + Parameters + ---------- + x : array_like + The array to take the pseudo-derivative from. + a,b : float + Defines the parameters of the sinh/sinh pseudo-differential + operator. + period : float, optional + The period of the sequence x. Default is ``2*pi``. + + Returns + ------- + cc_diff : ndarray + Pseudo-derivative of periodic sequence `x`. + + Notes + ----- + ``cc_diff(cc_diff(x,a,b),b,a) == x`` + + """ + if isinstance(_cache, threading.local): + if not hasattr(_cache, 'cc_diff_cache'): + _cache.cc_diff_cache = {} + _cache = _cache.cc_diff_cache + + tmp = asarray(x) + if iscomplexobj(tmp): + return cc_diff(tmp.real, a, b, period, _cache) + \ + 1j * cc_diff(tmp.imag, a, b, period, _cache) + if period is not None: + a = a*2*pi/period + b = b*2*pi/period + n = len(x) + omega = _cache.get((n,a,b)) + if omega is None: + if len(_cache) > 20: + while _cache: + _cache.popitem() + + def kernel(k,a=a,b=b): + return cosh(a*k)/cosh(b*k) + omega = convolve.init_convolution_kernel(n,kernel) + _cache[(n,a,b)] = omega + overwrite_x = _datacopied(tmp, x) + return convolve.convolve(tmp,omega,overwrite_x=overwrite_x) + + +def shift(x, a, period=None, _cache=_cache): + """ + Shift periodic sequence x by a: y(u) = x(u+a). + + If x_j and y_j are Fourier coefficients of periodic functions x + and y, respectively, then:: + + y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f + + Parameters + ---------- + x : array_like + The array to take the pseudo-derivative from. + a : float + Defines the parameters of the sinh/sinh pseudo-differential + period : float, optional + The period of the sequences x and y. Default period is ``2*pi``. + """ + if isinstance(_cache, threading.local): + if not hasattr(_cache, 'shift_cache'): + _cache.shift_cache = {} + _cache = _cache.shift_cache + + tmp = asarray(x) + if iscomplexobj(tmp): + return shift(tmp.real, a, period, _cache) + 1j * shift( + tmp.imag, a, period, _cache) + if period is not None: + a = a*2*pi/period + n = len(x) + omega = _cache.get((n,a)) + if omega is None: + if len(_cache) > 20: + while _cache: + _cache.popitem() + + def kernel_real(k,a=a): + return cos(a*k) + + def kernel_imag(k,a=a): + return sin(a*k) + omega_real = convolve.init_convolution_kernel(n,kernel_real,d=0, + zero_nyquist=0) + omega_imag = convolve.init_convolution_kernel(n,kernel_imag,d=1, + zero_nyquist=0) + _cache[(n,a)] = omega_real,omega_imag + else: + omega_real,omega_imag = omega + overwrite_x = _datacopied(tmp, x) + return convolve.convolve_z(tmp,omega_real,omega_imag, + overwrite_x=overwrite_x) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_realtransforms.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_realtransforms.py new file mode 100644 index 0000000000000000000000000000000000000000..ad71d517b0ac829ab71850bf67f7dc38636161f2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/_realtransforms.py @@ -0,0 +1,598 @@ +""" +Real spectrum transforms (DCT, DST, MDCT) +""" + +__all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn'] + +from scipy.fft import _pocketfft +from ._helper import _good_shape + +_inverse_typemap = {1: 1, 2: 3, 3: 2, 4: 4} + + +def dctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False): + """ + Return multidimensional Discrete Cosine Transform along the specified axes. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DCT (see Notes). Default type is 2. + shape : int or array_like of ints or None, optional + The shape of the result. If both `shape` and `axes` (see below) are + None, `shape` is ``x.shape``; if `shape` is None but `axes` is + not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``. + If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. + If ``shape[i] < x.shape[i]``, the ith dimension is truncated to + length ``shape[i]``. + If any element of `shape` is -1, the size of the corresponding + dimension of `x` is used. + axes : int or array_like of ints or None, optional + Axes along which the DCT is computed. + The default is over all axes. + norm : {None, 'ortho'}, optional + Normalization mode (see Notes). Default is None. + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + y : ndarray of real + The transformed input array. + + See Also + -------- + idctn : Inverse multidimensional DCT + + Notes + ----- + For full details of the DCT types and normalization modes, as well as + references, see `dct`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fftpack import dctn, idctn + >>> rng = np.random.default_rng() + >>> y = rng.standard_normal((16, 16)) + >>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho')) + True + + """ + shape = _good_shape(x, shape, axes) + return _pocketfft.dctn(x, type, shape, axes, norm, overwrite_x) + + +def idctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False): + """ + Return multidimensional Discrete Cosine Transform along the specified axes. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DCT (see Notes). Default type is 2. + shape : int or array_like of ints or None, optional + The shape of the result. If both `shape` and `axes` (see below) are + None, `shape` is ``x.shape``; if `shape` is None but `axes` is + not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``. + If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. + If ``shape[i] < x.shape[i]``, the ith dimension is truncated to + length ``shape[i]``. + If any element of `shape` is -1, the size of the corresponding + dimension of `x` is used. + axes : int or array_like of ints or None, optional + Axes along which the IDCT is computed. + The default is over all axes. + norm : {None, 'ortho'}, optional + Normalization mode (see Notes). Default is None. + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + y : ndarray of real + The transformed input array. + + See Also + -------- + dctn : multidimensional DCT + + Notes + ----- + For full details of the IDCT types and normalization modes, as well as + references, see `idct`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fftpack import dctn, idctn + >>> rng = np.random.default_rng() + >>> y = rng.standard_normal((16, 16)) + >>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho')) + True + + """ + type = _inverse_typemap[type] + shape = _good_shape(x, shape, axes) + return _pocketfft.dctn(x, type, shape, axes, norm, overwrite_x) + + +def dstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False): + """ + Return multidimensional Discrete Sine Transform along the specified axes. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DST (see Notes). Default type is 2. + shape : int or array_like of ints or None, optional + The shape of the result. If both `shape` and `axes` (see below) are + None, `shape` is ``x.shape``; if `shape` is None but `axes` is + not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``. + If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. + If ``shape[i] < x.shape[i]``, the ith dimension is truncated to + length ``shape[i]``. + If any element of `shape` is -1, the size of the corresponding + dimension of `x` is used. + axes : int or array_like of ints or None, optional + Axes along which the DCT is computed. + The default is over all axes. + norm : {None, 'ortho'}, optional + Normalization mode (see Notes). Default is None. + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + y : ndarray of real + The transformed input array. + + See Also + -------- + idstn : Inverse multidimensional DST + + Notes + ----- + For full details of the DST types and normalization modes, as well as + references, see `dst`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fftpack import dstn, idstn + >>> rng = np.random.default_rng() + >>> y = rng.standard_normal((16, 16)) + >>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho')) + True + + """ + shape = _good_shape(x, shape, axes) + return _pocketfft.dstn(x, type, shape, axes, norm, overwrite_x) + + +def idstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False): + """ + Return multidimensional Discrete Sine Transform along the specified axes. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DST (see Notes). Default type is 2. + shape : int or array_like of ints or None, optional + The shape of the result. If both `shape` and `axes` (see below) are + None, `shape` is ``x.shape``; if `shape` is None but `axes` is + not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``. + If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. + If ``shape[i] < x.shape[i]``, the ith dimension is truncated to + length ``shape[i]``. + If any element of `shape` is -1, the size of the corresponding + dimension of `x` is used. + axes : int or array_like of ints or None, optional + Axes along which the IDST is computed. + The default is over all axes. + norm : {None, 'ortho'}, optional + Normalization mode (see Notes). Default is None. + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + y : ndarray of real + The transformed input array. + + See Also + -------- + dstn : multidimensional DST + + Notes + ----- + For full details of the IDST types and normalization modes, as well as + references, see `idst`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.fftpack import dstn, idstn + >>> rng = np.random.default_rng() + >>> y = rng.standard_normal((16, 16)) + >>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho')) + True + + """ + type = _inverse_typemap[type] + shape = _good_shape(x, shape, axes) + return _pocketfft.dstn(x, type, shape, axes, norm, overwrite_x) + + +def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False): + r""" + Return the Discrete Cosine Transform of arbitrary type sequence x. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DCT (see Notes). Default type is 2. + n : int, optional + Length of the transform. If ``n < x.shape[axis]``, `x` is + truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The + default results in ``n = x.shape[axis]``. + axis : int, optional + Axis along which the dct is computed; the default is over the + last axis (i.e., ``axis=-1``). + norm : {None, 'ortho'}, optional + Normalization mode (see Notes). Default is None. + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + y : ndarray of real + The transformed input array. + + See Also + -------- + idct : Inverse DCT + + Notes + ----- + For a single dimension array ``x``, ``dct(x, norm='ortho')`` is equal to + MATLAB ``dct(x)``. + + There are, theoretically, 8 types of the DCT, only the first 4 types are + implemented in scipy. 'The' DCT generally refers to DCT type 2, and 'the' + Inverse DCT generally refers to DCT type 3. + + **Type I** + + There are several definitions of the DCT-I; we use the following + (for ``norm=None``) + + .. math:: + + y_k = x_0 + (-1)^k x_{N-1} + 2 \sum_{n=1}^{N-2} x_n \cos\left( + \frac{\pi k n}{N-1} \right) + + If ``norm='ortho'``, ``x[0]`` and ``x[N-1]`` are multiplied by a scaling + factor of :math:`\sqrt{2}`, and ``y[k]`` is multiplied by a scaling factor + ``f`` + + .. math:: + + f = \begin{cases} + \frac{1}{2}\sqrt{\frac{1}{N-1}} & \text{if }k=0\text{ or }N-1, \\ + \frac{1}{2}\sqrt{\frac{2}{N-1}} & \text{otherwise} \end{cases} + + .. versionadded:: 1.2.0 + Orthonormalization in DCT-I. + + .. note:: + The DCT-I is only supported for input size > 1. + + **Type II** + + There are several definitions of the DCT-II; we use the following + (for ``norm=None``) + + .. math:: + + y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right) + + If ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor ``f`` + + .. math:: + f = \begin{cases} + \sqrt{\frac{1}{4N}} & \text{if }k=0, \\ + \sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases} + + which makes the corresponding matrix of coefficients orthonormal + (``O @ O.T = np.eye(N)``). + + **Type III** + + There are several definitions, we use the following (for ``norm=None``) + + .. math:: + + y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right) + + or, for ``norm='ortho'`` + + .. math:: + + y_k = \frac{x_0}{\sqrt{N}} + \sqrt{\frac{2}{N}} \sum_{n=1}^{N-1} x_n + \cos\left(\frac{\pi(2k+1)n}{2N}\right) + + The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up + to a factor ``2N``. The orthonormalized DCT-III is exactly the inverse of + the orthonormalized DCT-II. + + **Type IV** + + There are several definitions of the DCT-IV; we use the following + (for ``norm=None``) + + .. math:: + + y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right) + + If ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor ``f`` + + .. math:: + + f = \frac{1}{\sqrt{2N}} + + .. versionadded:: 1.2.0 + Support for DCT-IV. + + References + ---------- + .. [1] 'A Fast Cosine Transform in One and Two Dimensions', by J. + Makhoul, `IEEE Transactions on acoustics, speech and signal + processing` vol. 28(1), pp. 27-34, + :doi:`10.1109/TASSP.1980.1163351` (1980). + .. [2] Wikipedia, "Discrete cosine transform", + https://en.wikipedia.org/wiki/Discrete_cosine_transform + + Examples + -------- + The Type 1 DCT is equivalent to the FFT (though faster) for real, + even-symmetrical inputs. The output is also real and even-symmetrical. + Half of the FFT input is used to generate half of the FFT output: + + >>> from scipy.fftpack import fft, dct + >>> import numpy as np + >>> fft(np.array([4., 3., 5., 10., 5., 3.])).real + array([ 30., -8., 6., -2., 6., -8.]) + >>> dct(np.array([4., 3., 5., 10.]), 1) + array([ 30., -8., 6., -2.]) + + """ + return _pocketfft.dct(x, type, n, axis, norm, overwrite_x) + + +def idct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False): + """ + Return the Inverse Discrete Cosine Transform of an arbitrary type sequence. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DCT (see Notes). Default type is 2. + n : int, optional + Length of the transform. If ``n < x.shape[axis]``, `x` is + truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The + default results in ``n = x.shape[axis]``. + axis : int, optional + Axis along which the idct is computed; the default is over the + last axis (i.e., ``axis=-1``). + norm : {None, 'ortho'}, optional + Normalization mode (see Notes). Default is None. + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + idct : ndarray of real + The transformed input array. + + See Also + -------- + dct : Forward DCT + + Notes + ----- + For a single dimension array `x`, ``idct(x, norm='ortho')`` is equal to + MATLAB ``idct(x)``. + + 'The' IDCT is the IDCT of type 2, which is the same as DCT of type 3. + + IDCT of type 1 is the DCT of type 1, IDCT of type 2 is the DCT of type + 3, and IDCT of type 3 is the DCT of type 2. IDCT of type 4 is the DCT + of type 4. For the definition of these types, see `dct`. + + Examples + -------- + The Type 1 DCT is equivalent to the DFT for real, even-symmetrical + inputs. The output is also real and even-symmetrical. Half of the IFFT + input is used to generate half of the IFFT output: + + >>> from scipy.fftpack import ifft, idct + >>> import numpy as np + >>> ifft(np.array([ 30., -8., 6., -2., 6., -8.])).real + array([ 4., 3., 5., 10., 5., 3.]) + >>> idct(np.array([ 30., -8., 6., -2.]), 1) / 6 + array([ 4., 3., 5., 10.]) + + """ + type = _inverse_typemap[type] + return _pocketfft.dct(x, type, n, axis, norm, overwrite_x) + + +def dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False): + r""" + Return the Discrete Sine Transform of arbitrary type sequence x. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DST (see Notes). Default type is 2. + n : int, optional + Length of the transform. If ``n < x.shape[axis]``, `x` is + truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The + default results in ``n = x.shape[axis]``. + axis : int, optional + Axis along which the dst is computed; the default is over the + last axis (i.e., ``axis=-1``). + norm : {None, 'ortho'}, optional + Normalization mode (see Notes). Default is None. + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + dst : ndarray of reals + The transformed input array. + + See Also + -------- + idst : Inverse DST + + Notes + ----- + For a single dimension array ``x``. + + There are, theoretically, 8 types of the DST for different combinations of + even/odd boundary conditions and boundary off sets [1]_, only the first + 4 types are implemented in scipy. + + **Type I** + + There are several definitions of the DST-I; we use the following + for ``norm=None``. DST-I assumes the input is odd around `n=-1` and `n=N`. + + .. math:: + + y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(n+1)}{N+1}\right) + + Note that the DST-I is only supported for input size > 1. + The (unnormalized) DST-I is its own inverse, up to a factor ``2(N+1)``. + The orthonormalized DST-I is exactly its own inverse. + + **Type II** + + There are several definitions of the DST-II; we use the following for + ``norm=None``. DST-II assumes the input is odd around `n=-1/2` and + `n=N-1/2`; the output is odd around :math:`k=-1` and even around `k=N-1` + + .. math:: + + y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(2n+1)}{2N}\right) + + if ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor ``f`` + + .. math:: + + f = \begin{cases} + \sqrt{\frac{1}{4N}} & \text{if }k = 0, \\ + \sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases} + + **Type III** + + There are several definitions of the DST-III, we use the following (for + ``norm=None``). DST-III assumes the input is odd around `n=-1` and even + around `n=N-1` + + .. math:: + + y_k = (-1)^k x_{N-1} + 2 \sum_{n=0}^{N-2} x_n \sin\left( + \frac{\pi(2k+1)(n+1)}{2N}\right) + + The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up + to a factor ``2N``. The orthonormalized DST-III is exactly the inverse of the + orthonormalized DST-II. + + .. versionadded:: 0.11.0 + + **Type IV** + + There are several definitions of the DST-IV, we use the following (for + ``norm=None``). DST-IV assumes the input is odd around `n=-0.5` and even + around `n=N-0.5` + + .. math:: + + y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right) + + The (unnormalized) DST-IV is its own inverse, up to a factor ``2N``. The + orthonormalized DST-IV is exactly its own inverse. + + .. versionadded:: 1.2.0 + Support for DST-IV. + + References + ---------- + .. [1] Wikipedia, "Discrete sine transform", + https://en.wikipedia.org/wiki/Discrete_sine_transform + + """ + return _pocketfft.dst(x, type, n, axis, norm, overwrite_x) + + +def idst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False): + """ + Return the Inverse Discrete Sine Transform of an arbitrary type sequence. + + Parameters + ---------- + x : array_like + The input array. + type : {1, 2, 3, 4}, optional + Type of the DST (see Notes). Default type is 2. + n : int, optional + Length of the transform. If ``n < x.shape[axis]``, `x` is + truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The + default results in ``n = x.shape[axis]``. + axis : int, optional + Axis along which the idst is computed; the default is over the + last axis (i.e., ``axis=-1``). + norm : {None, 'ortho'}, optional + Normalization mode (see Notes). Default is None. + overwrite_x : bool, optional + If True, the contents of `x` can be destroyed; the default is False. + + Returns + ------- + idst : ndarray of real + The transformed input array. + + See Also + -------- + dst : Forward DST + + Notes + ----- + 'The' IDST is the IDST of type 2, which is the same as DST of type 3. + + IDST of type 1 is the DST of type 1, IDST of type 2 is the DST of type + 3, and IDST of type 3 is the DST of type 2. For the definition of these + types, see `dst`. + + .. versionadded:: 0.11.0 + + """ + type = _inverse_typemap[type] + return _pocketfft.dst(x, type, n, axis, norm, overwrite_x) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/basic.py new file mode 100644 index 0000000000000000000000000000000000000000..553f456fe1561c28928ecc4ebe2238459cc60443 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/basic.py @@ -0,0 +1,20 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.fftpack` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'fft','ifft','fftn','ifftn','rfft','irfft', + 'fft2','ifft2' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="fftpack", module="basic", + private_modules=["_basic"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/helper.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/helper.py new file mode 100644 index 0000000000000000000000000000000000000000..fcc7000c215f8a7605a2a59b5767b27b2fcd969d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/helper.py @@ -0,0 +1,19 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.fftpack` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'fftshift', 'ifftshift', 'fftfreq', 'rfftfreq', 'next_fast_len' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="fftpack", module="helper", + private_modules=["_helper"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/pseudo_diffs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/pseudo_diffs.py new file mode 100644 index 0000000000000000000000000000000000000000..ecf71ad3256d48d2131c8058072da724cb001af9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/pseudo_diffs.py @@ -0,0 +1,22 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.fftpack` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'diff', + 'tilbert', 'itilbert', 'hilbert', 'ihilbert', + 'cs_diff', 'cc_diff', 'sc_diff', 'ss_diff', + 'shift', 'convolve' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="fftpack", module="pseudo_diffs", + private_modules=["_pseudo_diffs"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/realtransforms.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/realtransforms.py new file mode 100644 index 0000000000000000000000000000000000000000..9a392198fccf213bc988a79058bd69515e39f510 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/realtransforms.py @@ -0,0 +1,19 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.fftpack` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="fftpack", module="realtransforms", + private_modules=["_realtransforms"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_basic.py new file mode 100644 index 0000000000000000000000000000000000000000..2951471d2abb5c4a88ae0b44c172d3ecc0862d4d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_basic.py @@ -0,0 +1,879 @@ +# Created by Pearu Peterson, September 2002 + +from numpy.testing import (assert_, assert_equal, assert_array_almost_equal, + assert_array_almost_equal_nulp, assert_array_less) +import pytest +from pytest import raises as assert_raises +from scipy.fftpack import ifft, fft, fftn, ifftn, rfft, irfft, fft2 + +from numpy import (arange, array, asarray, zeros, dot, exp, pi, + swapaxes, double, cdouble) +import numpy as np +import numpy.fft +from numpy.random import rand + +# "large" composite numbers supported by FFTPACK +LARGE_COMPOSITE_SIZES = [ + 2**13, + 2**5 * 3**5, + 2**3 * 3**3 * 5**2, +] +SMALL_COMPOSITE_SIZES = [ + 2, + 2*3*5, + 2*2*3*3, +] +# prime +LARGE_PRIME_SIZES = [ + 2011 +] +SMALL_PRIME_SIZES = [ + 29 +] + + +def _assert_close_in_norm(x, y, rtol, size, rdt): + # helper function for testing + err_msg = f"size: {size} rdt: {rdt}" + assert_array_less(np.linalg.norm(x - y), rtol*np.linalg.norm(x), err_msg) + + +def random(size): + return rand(*size) + + +def direct_dft(x): + x = asarray(x) + n = len(x) + y = zeros(n, dtype=cdouble) + w = -arange(n)*(2j*pi/n) + for i in range(n): + y[i] = dot(exp(i*w), x) + return y + + +def direct_idft(x): + x = asarray(x) + n = len(x) + y = zeros(n, dtype=cdouble) + w = arange(n)*(2j*pi/n) + for i in range(n): + y[i] = dot(exp(i*w), x)/n + return y + + +def direct_dftn(x): + x = asarray(x) + for axis in range(len(x.shape)): + x = fft(x, axis=axis) + return x + + +def direct_idftn(x): + x = asarray(x) + for axis in range(len(x.shape)): + x = ifft(x, axis=axis) + return x + + +def direct_rdft(x): + x = asarray(x) + n = len(x) + w = -arange(n)*(2j*pi/n) + r = zeros(n, dtype=double) + for i in range(n//2+1): + y = dot(exp(i*w), x) + if i: + r[2*i-1] = y.real + if 2*i < n: + r[2*i] = y.imag + else: + r[0] = y.real + return r + + +def direct_irdft(x): + x = asarray(x) + n = len(x) + x1 = zeros(n, dtype=cdouble) + for i in range(n//2+1): + if i: + if 2*i < n: + x1[i] = x[2*i-1] + 1j*x[2*i] + x1[n-i] = x[2*i-1] - 1j*x[2*i] + else: + x1[i] = x[2*i-1] + else: + x1[0] = x[0] + return direct_idft(x1).real + + +class _TestFFTBase: + def setup_method(self): + self.cdt = None + self.rdt = None + np.random.seed(1234) + + def test_definition(self): + x = np.array([1,2,3,4+1j,1,2,3,4+2j], dtype=self.cdt) + y = fft(x) + assert_equal(y.dtype, self.cdt) + y1 = direct_dft(x) + assert_array_almost_equal(y,y1) + x = np.array([1,2,3,4+0j,5], dtype=self.cdt) + assert_array_almost_equal(fft(x),direct_dft(x)) + + def test_n_argument_real(self): + x1 = np.array([1,2,3,4], dtype=self.rdt) + x2 = np.array([1,2,3,4], dtype=self.rdt) + y = fft([x1,x2],n=4) + assert_equal(y.dtype, self.cdt) + assert_equal(y.shape,(2,4)) + assert_array_almost_equal(y[0],direct_dft(x1)) + assert_array_almost_equal(y[1],direct_dft(x2)) + + def _test_n_argument_complex(self): + x1 = np.array([1,2,3,4+1j], dtype=self.cdt) + x2 = np.array([1,2,3,4+1j], dtype=self.cdt) + y = fft([x1,x2],n=4) + assert_equal(y.dtype, self.cdt) + assert_equal(y.shape,(2,4)) + assert_array_almost_equal(y[0],direct_dft(x1)) + assert_array_almost_equal(y[1],direct_dft(x2)) + + def test_invalid_sizes(self): + assert_raises(ValueError, fft, []) + assert_raises(ValueError, fft, [[1,1],[2,2]], -5) + + +class TestDoubleFFT(_TestFFTBase): + def setup_method(self): + self.cdt = np.complex128 + self.rdt = np.float64 + + +class TestSingleFFT(_TestFFTBase): + def setup_method(self): + self.cdt = np.complex64 + self.rdt = np.float32 + + reason = ("single-precision FFT implementation is partially disabled, " + "until accuracy issues with large prime powers are resolved") + + @pytest.mark.xfail(run=False, reason=reason) + def test_notice(self): + pass + + +class TestFloat16FFT: + + def test_1_argument_real(self): + x1 = np.array([1, 2, 3, 4], dtype=np.float16) + y = fft(x1, n=4) + assert_equal(y.dtype, np.complex64) + assert_equal(y.shape, (4, )) + assert_array_almost_equal(y, direct_dft(x1.astype(np.float32))) + + def test_n_argument_real(self): + x1 = np.array([1, 2, 3, 4], dtype=np.float16) + x2 = np.array([1, 2, 3, 4], dtype=np.float16) + y = fft([x1, x2], n=4) + assert_equal(y.dtype, np.complex64) + assert_equal(y.shape, (2, 4)) + assert_array_almost_equal(y[0], direct_dft(x1.astype(np.float32))) + assert_array_almost_equal(y[1], direct_dft(x2.astype(np.float32))) + + +class _TestIFFTBase: + def setup_method(self): + np.random.seed(1234) + + def test_definition(self): + x = np.array([1,2,3,4+1j,1,2,3,4+2j], self.cdt) + y = ifft(x) + y1 = direct_idft(x) + assert_equal(y.dtype, self.cdt) + assert_array_almost_equal(y,y1) + + x = np.array([1,2,3,4+0j,5], self.cdt) + assert_array_almost_equal(ifft(x),direct_idft(x)) + + def test_definition_real(self): + x = np.array([1,2,3,4,1,2,3,4], self.rdt) + y = ifft(x) + assert_equal(y.dtype, self.cdt) + y1 = direct_idft(x) + assert_array_almost_equal(y,y1) + + x = np.array([1,2,3,4,5], dtype=self.rdt) + assert_equal(y.dtype, self.cdt) + assert_array_almost_equal(ifft(x),direct_idft(x)) + + def test_random_complex(self): + for size in [1,51,111,100,200,64,128,256,1024]: + x = random([size]).astype(self.cdt) + x = random([size]).astype(self.cdt) + 1j*x + y1 = ifft(fft(x)) + y2 = fft(ifft(x)) + assert_equal(y1.dtype, self.cdt) + assert_equal(y2.dtype, self.cdt) + assert_array_almost_equal(y1, x) + assert_array_almost_equal(y2, x) + + def test_random_real(self): + for size in [1,51,111,100,200,64,128,256,1024]: + x = random([size]).astype(self.rdt) + y1 = ifft(fft(x)) + y2 = fft(ifft(x)) + assert_equal(y1.dtype, self.cdt) + assert_equal(y2.dtype, self.cdt) + assert_array_almost_equal(y1, x) + assert_array_almost_equal(y2, x) + + def test_size_accuracy(self): + # Sanity check for the accuracy for prime and non-prime sized inputs + if self.rdt == np.float32: + rtol = 1e-5 + elif self.rdt == np.float64: + rtol = 1e-10 + + for size in LARGE_COMPOSITE_SIZES + LARGE_PRIME_SIZES: + np.random.seed(1234) + x = np.random.rand(size).astype(self.rdt) + y = ifft(fft(x)) + _assert_close_in_norm(x, y, rtol, size, self.rdt) + y = fft(ifft(x)) + _assert_close_in_norm(x, y, rtol, size, self.rdt) + + x = (x + 1j*np.random.rand(size)).astype(self.cdt) + y = ifft(fft(x)) + _assert_close_in_norm(x, y, rtol, size, self.rdt) + y = fft(ifft(x)) + _assert_close_in_norm(x, y, rtol, size, self.rdt) + + def test_invalid_sizes(self): + assert_raises(ValueError, ifft, []) + assert_raises(ValueError, ifft, [[1,1],[2,2]], -5) + + +class TestDoubleIFFT(_TestIFFTBase): + def setup_method(self): + self.cdt = np.complex128 + self.rdt = np.float64 + + +class TestSingleIFFT(_TestIFFTBase): + def setup_method(self): + self.cdt = np.complex64 + self.rdt = np.float32 + + +class _TestRFFTBase: + def setup_method(self): + np.random.seed(1234) + + def test_definition(self): + for t in [[1, 2, 3, 4, 1, 2, 3, 4], [1, 2, 3, 4, 1, 2, 3, 4, 5]]: + x = np.array(t, dtype=self.rdt) + y = rfft(x) + y1 = direct_rdft(x) + assert_array_almost_equal(y,y1) + assert_equal(y.dtype, self.rdt) + + def test_invalid_sizes(self): + assert_raises(ValueError, rfft, []) + assert_raises(ValueError, rfft, [[1,1],[2,2]], -5) + + # See gh-5790 + class MockSeries: + def __init__(self, data): + self.data = np.asarray(data) + + def __getattr__(self, item): + try: + return getattr(self.data, item) + except AttributeError as e: + raise AttributeError("'MockSeries' object " + f"has no attribute '{item}'") from e + + def test_non_ndarray_with_dtype(self): + x = np.array([1., 2., 3., 4., 5.]) + xs = _TestRFFTBase.MockSeries(x) + + expected = [1, 2, 3, 4, 5] + rfft(xs) + + # Data should not have been overwritten + assert_equal(x, expected) + assert_equal(xs.data, expected) + + def test_complex_input(self): + assert_raises(TypeError, rfft, np.arange(4, dtype=np.complex64)) + + +class TestRFFTDouble(_TestRFFTBase): + def setup_method(self): + self.cdt = np.complex128 + self.rdt = np.float64 + + +class TestRFFTSingle(_TestRFFTBase): + def setup_method(self): + self.cdt = np.complex64 + self.rdt = np.float32 + + +class _TestIRFFTBase: + def setup_method(self): + np.random.seed(1234) + + def test_definition(self): + x1 = [1,2,3,4,1,2,3,4] + x1_1 = [1,2+3j,4+1j,2+3j,4,2-3j,4-1j,2-3j] + x2 = [1,2,3,4,1,2,3,4,5] + x2_1 = [1,2+3j,4+1j,2+3j,4+5j,4-5j,2-3j,4-1j,2-3j] + + def _test(x, xr): + y = irfft(np.array(x, dtype=self.rdt)) + y1 = direct_irdft(x) + assert_equal(y.dtype, self.rdt) + assert_array_almost_equal(y,y1, decimal=self.ndec) + assert_array_almost_equal(y,ifft(xr), decimal=self.ndec) + + _test(x1, x1_1) + _test(x2, x2_1) + + def test_random_real(self): + for size in [1,51,111,100,200,64,128,256,1024]: + x = random([size]).astype(self.rdt) + y1 = irfft(rfft(x)) + y2 = rfft(irfft(x)) + assert_equal(y1.dtype, self.rdt) + assert_equal(y2.dtype, self.rdt) + assert_array_almost_equal(y1, x, decimal=self.ndec, + err_msg="size=%d" % size) + assert_array_almost_equal(y2, x, decimal=self.ndec, + err_msg="size=%d" % size) + + def test_size_accuracy(self): + # Sanity check for the accuracy for prime and non-prime sized inputs + if self.rdt == np.float32: + rtol = 1e-5 + elif self.rdt == np.float64: + rtol = 1e-10 + + for size in LARGE_COMPOSITE_SIZES + LARGE_PRIME_SIZES: + np.random.seed(1234) + x = np.random.rand(size).astype(self.rdt) + y = irfft(rfft(x)) + _assert_close_in_norm(x, y, rtol, size, self.rdt) + y = rfft(irfft(x)) + _assert_close_in_norm(x, y, rtol, size, self.rdt) + + def test_invalid_sizes(self): + assert_raises(ValueError, irfft, []) + assert_raises(ValueError, irfft, [[1,1],[2,2]], -5) + + def test_complex_input(self): + assert_raises(TypeError, irfft, np.arange(4, dtype=np.complex64)) + + +# self.ndec is bogus; we should have a assert_array_approx_equal for number of +# significant digits + +class TestIRFFTDouble(_TestIRFFTBase): + def setup_method(self): + self.cdt = np.complex128 + self.rdt = np.float64 + self.ndec = 14 + + +class TestIRFFTSingle(_TestIRFFTBase): + def setup_method(self): + self.cdt = np.complex64 + self.rdt = np.float32 + self.ndec = 5 + + +class Testfft2: + def setup_method(self): + np.random.seed(1234) + + def test_regression_244(self): + """FFT returns wrong result with axes parameter.""" + # fftn (and hence fft2) used to break when both axes and shape were + # used + x = numpy.ones((4, 4, 2)) + y = fft2(x, shape=(8, 8), axes=(-3, -2)) + y_r = numpy.fft.fftn(x, s=(8, 8), axes=(-3, -2)) + assert_array_almost_equal(y, y_r) + + def test_invalid_sizes(self): + assert_raises(ValueError, fft2, [[]]) + assert_raises(ValueError, fft2, [[1, 1], [2, 2]], (4, -3)) + + +class TestFftnSingle: + def setup_method(self): + np.random.seed(1234) + + def test_definition(self): + x = [[1, 2, 3], + [4, 5, 6], + [7, 8, 9]] + y = fftn(np.array(x, np.float32)) + assert_(y.dtype == np.complex64, + msg="double precision output with single precision") + + y_r = np.array(fftn(x), np.complex64) + assert_array_almost_equal_nulp(y, y_r) + + @pytest.mark.parametrize('size', SMALL_COMPOSITE_SIZES + SMALL_PRIME_SIZES) + def test_size_accuracy_small(self, size): + rng = np.random.default_rng(1234) + x = rng.random((size, size)) + 1j*rng.random((size, size)) + y1 = fftn(x.real.astype(np.float32)) + y2 = fftn(x.real.astype(np.float64)).astype(np.complex64) + + assert_equal(y1.dtype, np.complex64) + assert_array_almost_equal_nulp(y1, y2, 2000) + + @pytest.mark.parametrize('size', LARGE_COMPOSITE_SIZES + LARGE_PRIME_SIZES) + def test_size_accuracy_large(self, size): + rand = np.random.default_rng(1234) + x = rand.random((size, 3)) + 1j*rand.random((size, 3)) + y1 = fftn(x.real.astype(np.float32)) + y2 = fftn(x.real.astype(np.float64)).astype(np.complex64) + + assert_equal(y1.dtype, np.complex64) + assert_array_almost_equal_nulp(y1, y2, 2000) + + def test_definition_float16(self): + x = [[1, 2, 3], + [4, 5, 6], + [7, 8, 9]] + y = fftn(np.array(x, np.float16)) + assert_equal(y.dtype, np.complex64) + y_r = np.array(fftn(x), np.complex64) + assert_array_almost_equal_nulp(y, y_r) + + @pytest.mark.parametrize('size', SMALL_COMPOSITE_SIZES + SMALL_PRIME_SIZES) + def test_float16_input_small(self, size): + rng = np.random.default_rng(1234) + x = rng.random((size, size)) + 1j * rng.random((size, size)) + y1 = fftn(x.real.astype(np.float16)) + y2 = fftn(x.real.astype(np.float64)).astype(np.complex64) + + assert_equal(y1.dtype, np.complex64) + assert_array_almost_equal_nulp(y1, y2, 5e5) + + @pytest.mark.parametrize('size', LARGE_COMPOSITE_SIZES + LARGE_PRIME_SIZES) + def test_float16_input_large(self, size): + rng = np.random.default_rng(1234) + x = rng.random((size, 3)) + 1j*rng.random((size, 3)) + y1 = fftn(x.real.astype(np.float16)) + y2 = fftn(x.real.astype(np.float64)).astype(np.complex64) + + assert_equal(y1.dtype, np.complex64) + assert_array_almost_equal_nulp(y1, y2, 2e6) + + +class TestFftn: + def setup_method(self): + np.random.seed(1234) + + def test_definition(self): + x = [[1, 2, 3], + [4, 5, 6], + [7, 8, 9]] + y = fftn(x) + assert_array_almost_equal(y, direct_dftn(x)) + + x = random((20, 26)) + assert_array_almost_equal(fftn(x), direct_dftn(x)) + + x = random((5, 4, 3, 20)) + assert_array_almost_equal(fftn(x), direct_dftn(x)) + + def test_axes_argument(self): + # plane == ji_plane, x== kji_space + plane1 = [[1, 2, 3], + [4, 5, 6], + [7, 8, 9]] + plane2 = [[10, 11, 12], + [13, 14, 15], + [16, 17, 18]] + plane3 = [[19, 20, 21], + [22, 23, 24], + [25, 26, 27]] + ki_plane1 = [[1, 2, 3], + [10, 11, 12], + [19, 20, 21]] + ki_plane2 = [[4, 5, 6], + [13, 14, 15], + [22, 23, 24]] + ki_plane3 = [[7, 8, 9], + [16, 17, 18], + [25, 26, 27]] + jk_plane1 = [[1, 10, 19], + [4, 13, 22], + [7, 16, 25]] + jk_plane2 = [[2, 11, 20], + [5, 14, 23], + [8, 17, 26]] + jk_plane3 = [[3, 12, 21], + [6, 15, 24], + [9, 18, 27]] + kj_plane1 = [[1, 4, 7], + [10, 13, 16], [19, 22, 25]] + kj_plane2 = [[2, 5, 8], + [11, 14, 17], [20, 23, 26]] + kj_plane3 = [[3, 6, 9], + [12, 15, 18], [21, 24, 27]] + ij_plane1 = [[1, 4, 7], + [2, 5, 8], + [3, 6, 9]] + ij_plane2 = [[10, 13, 16], + [11, 14, 17], + [12, 15, 18]] + ij_plane3 = [[19, 22, 25], + [20, 23, 26], + [21, 24, 27]] + ik_plane1 = [[1, 10, 19], + [2, 11, 20], + [3, 12, 21]] + ik_plane2 = [[4, 13, 22], + [5, 14, 23], + [6, 15, 24]] + ik_plane3 = [[7, 16, 25], + [8, 17, 26], + [9, 18, 27]] + ijk_space = [jk_plane1, jk_plane2, jk_plane3] + ikj_space = [kj_plane1, kj_plane2, kj_plane3] + jik_space = [ik_plane1, ik_plane2, ik_plane3] + jki_space = [ki_plane1, ki_plane2, ki_plane3] + kij_space = [ij_plane1, ij_plane2, ij_plane3] + x = array([plane1, plane2, plane3]) + + assert_array_almost_equal(fftn(x), + fftn(x, axes=(-3, -2, -1))) # kji_space + assert_array_almost_equal(fftn(x), fftn(x, axes=(0, 1, 2))) + assert_array_almost_equal(fftn(x, axes=(0, 2)), fftn(x, axes=(0, -1))) + y = fftn(x, axes=(2, 1, 0)) # ijk_space + assert_array_almost_equal(swapaxes(y, -1, -3), fftn(ijk_space)) + y = fftn(x, axes=(2, 0, 1)) # ikj_space + assert_array_almost_equal(swapaxes(swapaxes(y, -1, -3), -1, -2), + fftn(ikj_space)) + y = fftn(x, axes=(1, 2, 0)) # jik_space + assert_array_almost_equal(swapaxes(swapaxes(y, -1, -3), -3, -2), + fftn(jik_space)) + y = fftn(x, axes=(1, 0, 2)) # jki_space + assert_array_almost_equal(swapaxes(y, -2, -3), fftn(jki_space)) + y = fftn(x, axes=(0, 2, 1)) # kij_space + assert_array_almost_equal(swapaxes(y, -2, -1), fftn(kij_space)) + + y = fftn(x, axes=(-2, -1)) # ji_plane + assert_array_almost_equal(fftn(plane1), y[0]) + assert_array_almost_equal(fftn(plane2), y[1]) + assert_array_almost_equal(fftn(plane3), y[2]) + + y = fftn(x, axes=(1, 2)) # ji_plane + assert_array_almost_equal(fftn(plane1), y[0]) + assert_array_almost_equal(fftn(plane2), y[1]) + assert_array_almost_equal(fftn(plane3), y[2]) + + y = fftn(x, axes=(-3, -2)) # kj_plane + assert_array_almost_equal(fftn(x[:, :, 0]), y[:, :, 0]) + assert_array_almost_equal(fftn(x[:, :, 1]), y[:, :, 1]) + assert_array_almost_equal(fftn(x[:, :, 2]), y[:, :, 2]) + + y = fftn(x, axes=(-3, -1)) # ki_plane + assert_array_almost_equal(fftn(x[:, 0, :]), y[:, 0, :]) + assert_array_almost_equal(fftn(x[:, 1, :]), y[:, 1, :]) + assert_array_almost_equal(fftn(x[:, 2, :]), y[:, 2, :]) + + y = fftn(x, axes=(-1, -2)) # ij_plane + assert_array_almost_equal(fftn(ij_plane1), swapaxes(y[0], -2, -1)) + assert_array_almost_equal(fftn(ij_plane2), swapaxes(y[1], -2, -1)) + assert_array_almost_equal(fftn(ij_plane3), swapaxes(y[2], -2, -1)) + + y = fftn(x, axes=(-1, -3)) # ik_plane + assert_array_almost_equal(fftn(ik_plane1), + swapaxes(y[:, 0, :], -1, -2)) + assert_array_almost_equal(fftn(ik_plane2), + swapaxes(y[:, 1, :], -1, -2)) + assert_array_almost_equal(fftn(ik_plane3), + swapaxes(y[:, 2, :], -1, -2)) + + y = fftn(x, axes=(-2, -3)) # jk_plane + assert_array_almost_equal(fftn(jk_plane1), + swapaxes(y[:, :, 0], -1, -2)) + assert_array_almost_equal(fftn(jk_plane2), + swapaxes(y[:, :, 1], -1, -2)) + assert_array_almost_equal(fftn(jk_plane3), + swapaxes(y[:, :, 2], -1, -2)) + + y = fftn(x, axes=(-1,)) # i_line + for i in range(3): + for j in range(3): + assert_array_almost_equal(fft(x[i, j, :]), y[i, j, :]) + y = fftn(x, axes=(-2,)) # j_line + for i in range(3): + for j in range(3): + assert_array_almost_equal(fft(x[i, :, j]), y[i, :, j]) + y = fftn(x, axes=(0,)) # k_line + for i in range(3): + for j in range(3): + assert_array_almost_equal(fft(x[:, i, j]), y[:, i, j]) + + y = fftn(x, axes=()) # point + assert_array_almost_equal(y, x) + + def test_shape_argument(self): + small_x = [[1, 2, 3], + [4, 5, 6]] + large_x1 = [[1, 2, 3, 0], + [4, 5, 6, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]] + + y = fftn(small_x, shape=(4, 4)) + assert_array_almost_equal(y, fftn(large_x1)) + + y = fftn(small_x, shape=(3, 4)) + assert_array_almost_equal(y, fftn(large_x1[:-1])) + + def test_shape_axes_argument(self): + small_x = [[1, 2, 3], + [4, 5, 6], + [7, 8, 9]] + large_x1 = array([[1, 2, 3, 0], + [4, 5, 6, 0], + [7, 8, 9, 0], + [0, 0, 0, 0]]) + y = fftn(small_x, shape=(4, 4), axes=(-2, -1)) + assert_array_almost_equal(y, fftn(large_x1)) + y = fftn(small_x, shape=(4, 4), axes=(-1, -2)) + + assert_array_almost_equal(y, swapaxes( + fftn(swapaxes(large_x1, -1, -2)), -1, -2)) + + def test_shape_axes_argument2(self): + # Change shape of the last axis + x = numpy.random.random((10, 5, 3, 7)) + y = fftn(x, axes=(-1,), shape=(8,)) + assert_array_almost_equal(y, fft(x, axis=-1, n=8)) + + # Change shape of an arbitrary axis which is not the last one + x = numpy.random.random((10, 5, 3, 7)) + y = fftn(x, axes=(-2,), shape=(8,)) + assert_array_almost_equal(y, fft(x, axis=-2, n=8)) + + # Change shape of axes: cf #244, where shape and axes were mixed up + x = numpy.random.random((4, 4, 2)) + y = fftn(x, axes=(-3, -2), shape=(8, 8)) + assert_array_almost_equal(y, + numpy.fft.fftn(x, axes=(-3, -2), s=(8, 8))) + + def test_shape_argument_more(self): + x = zeros((4, 4, 2)) + with assert_raises(ValueError, + match="when given, axes and shape arguments" + " have to be of the same length"): + fftn(x, shape=(8, 8, 2, 1)) + + def test_invalid_sizes(self): + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[1, 0\]\) specified"): + fftn([[]]) + + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[4, -3\]\) specified"): + fftn([[1, 1], [2, 2]], (4, -3)) + + +class TestIfftn: + dtype = None + cdtype = None + + def setup_method(self): + np.random.seed(1234) + + @pytest.mark.parametrize('dtype,cdtype,maxnlp', + [(np.float64, np.complex128, 2000), + (np.float32, np.complex64, 3500)]) + def test_definition(self, dtype, cdtype, maxnlp): + rng = np.random.default_rng(1234) + x = np.array([[1, 2, 3], + [4, 5, 6], + [7, 8, 9]], dtype=dtype) + y = ifftn(x) + assert_equal(y.dtype, cdtype) + assert_array_almost_equal_nulp(y, direct_idftn(x), maxnlp) + + x = rng.random((20, 26)) + assert_array_almost_equal_nulp(ifftn(x), direct_idftn(x), maxnlp) + + x = rng.random((5, 4, 3, 20)) + assert_array_almost_equal_nulp(ifftn(x), direct_idftn(x), maxnlp) + + @pytest.mark.parametrize('maxnlp', [2000, 3500]) + @pytest.mark.parametrize('size', [1, 2, 51, 32, 64, 92]) + def test_random_complex(self, maxnlp, size): + rng = np.random.default_rng(1234) + x = rng.random([size, size]) + 1j * rng.random([size, size]) + assert_array_almost_equal_nulp(ifftn(fftn(x)), x, maxnlp) + assert_array_almost_equal_nulp(fftn(ifftn(x)), x, maxnlp) + + def test_invalid_sizes(self): + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[1, 0\]\) specified"): + ifftn([[]]) + + with assert_raises(ValueError, + match="invalid number of data points" + r" \(\[4, -3\]\) specified"): + ifftn([[1, 1], [2, 2]], (4, -3)) + + +class FakeArray: + def __init__(self, data): + self._data = data + self.__array_interface__ = data.__array_interface__ + + +class FakeArray2: + def __init__(self, data): + self._data = data + + def __array__(self, dtype=None, copy=None): + return self._data + + +class TestOverwrite: + """Check input overwrite behavior of the FFT functions.""" + + real_dtypes = (np.float32, np.float64) + dtypes = real_dtypes + (np.complex64, np.complex128) + fftsizes = [8, 16, 32] + + def _check(self, x, routine, fftsize, axis, overwrite_x): + x2 = x.copy() + for fake in [lambda x: x, FakeArray, FakeArray2]: + routine(fake(x2), fftsize, axis, overwrite_x=overwrite_x) + + sig = (f"{routine.__name__}({x.dtype}{x.shape!r}, {fftsize!r}, " + f"axis={axis!r}, overwrite_x={overwrite_x!r})") + if not overwrite_x: + assert_equal(x2, x, err_msg=f"spurious overwrite in {sig}") + + def _check_1d(self, routine, dtype, shape, axis, overwritable_dtypes, + fftsize, overwrite_x): + np.random.seed(1234) + if np.issubdtype(dtype, np.complexfloating): + data = np.random.randn(*shape) + 1j*np.random.randn(*shape) + else: + data = np.random.randn(*shape) + data = data.astype(dtype) + + self._check(data, routine, fftsize, axis, + overwrite_x=overwrite_x) + + @pytest.mark.parametrize('dtype', dtypes) + @pytest.mark.parametrize('fftsize', fftsizes) + @pytest.mark.parametrize('overwrite_x', [True, False]) + @pytest.mark.parametrize('shape,axes', [((16,), -1), + ((16, 2), 0), + ((2, 16), 1)]) + def test_fft_ifft(self, dtype, fftsize, overwrite_x, shape, axes): + overwritable = (np.complex128, np.complex64) + self._check_1d(fft, dtype, shape, axes, overwritable, + fftsize, overwrite_x) + self._check_1d(ifft, dtype, shape, axes, overwritable, + fftsize, overwrite_x) + + @pytest.mark.parametrize('dtype', real_dtypes) + @pytest.mark.parametrize('fftsize', fftsizes) + @pytest.mark.parametrize('overwrite_x', [True, False]) + @pytest.mark.parametrize('shape,axes', [((16,), -1), + ((16, 2), 0), + ((2, 16), 1)]) + def test_rfft_irfft(self, dtype, fftsize, overwrite_x, shape, axes): + overwritable = self.real_dtypes + self._check_1d(irfft, dtype, shape, axes, overwritable, + fftsize, overwrite_x) + self._check_1d(rfft, dtype, shape, axes, overwritable, + fftsize, overwrite_x) + + def _check_nd_one(self, routine, dtype, shape, axes, overwritable_dtypes, + overwrite_x): + np.random.seed(1234) + if np.issubdtype(dtype, np.complexfloating): + data = np.random.randn(*shape) + 1j*np.random.randn(*shape) + else: + data = np.random.randn(*shape) + data = data.astype(dtype) + + def fftshape_iter(shp): + if len(shp) <= 0: + yield () + else: + for j in (shp[0]//2, shp[0], shp[0]*2): + for rest in fftshape_iter(shp[1:]): + yield (j,) + rest + + if axes is None: + part_shape = shape + else: + part_shape = tuple(np.take(shape, axes)) + + for fftshape in fftshape_iter(part_shape): + self._check(data, routine, fftshape, axes, + overwrite_x=overwrite_x) + if data.ndim > 1: + self._check(data.T, routine, fftshape, axes, + overwrite_x=overwrite_x) + + @pytest.mark.parametrize('dtype', dtypes) + @pytest.mark.parametrize('overwrite_x', [True, False]) + @pytest.mark.parametrize('shape,axes', [((16,), None), + ((16,), (0,)), + ((16, 2), (0,)), + ((2, 16), (1,)), + ((8, 16), None), + ((8, 16), (0, 1)), + ((8, 16, 2), (0, 1)), + ((8, 16, 2), (1, 2)), + ((8, 16, 2), (0,)), + ((8, 16, 2), (1,)), + ((8, 16, 2), (2,)), + ((8, 16, 2), None), + ((8, 16, 2), (0, 1, 2))]) + def test_fftn_ifftn(self, dtype, overwrite_x, shape, axes): + overwritable = (np.complex128, np.complex64) + self._check_nd_one(fftn, dtype, shape, axes, overwritable, + overwrite_x) + self._check_nd_one(ifftn, dtype, shape, axes, overwritable, + overwrite_x) + + +@pytest.mark.parametrize('func', [fftn, ifftn, fft2]) +def test_shape_axes_ndarray(func): + # Test fftn and ifftn work with NumPy arrays for shape and axes arguments + # Regression test for gh-13342 + a = np.random.rand(10, 10) + + expect = func(a, shape=(5, 5)) + actual = func(a, shape=np.array([5, 5])) + assert_equal(expect, actual) + + expect = func(a, axes=(-1,)) + actual = func(a, axes=np.array([-1,])) + assert_equal(expect, actual) + + expect = func(a, shape=(4, 7), axes=(1, 0)) + actual = func(a, shape=np.array([4, 7]), axes=np.array([1, 0])) + assert_equal(expect, actual) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_helper.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_helper.py new file mode 100644 index 0000000000000000000000000000000000000000..5e7be04f3c0291502b50b101db82d299aadc7772 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_helper.py @@ -0,0 +1,54 @@ +# Created by Pearu Peterson, September 2002 + +__usage__ = """ +Build fftpack: + python setup_fftpack.py build +Run tests if scipy is installed: + python -c 'import scipy;scipy.fftpack.test()' +Run tests if fftpack is not installed: + python tests/test_helper.py [] +""" + +from numpy.testing import assert_array_almost_equal +from scipy.fftpack import fftshift, ifftshift, fftfreq, rfftfreq + +from numpy import pi, random + +class TestFFTShift: + + def test_definition(self): + x = [0,1,2,3,4,-4,-3,-2,-1] + y = [-4,-3,-2,-1,0,1,2,3,4] + assert_array_almost_equal(fftshift(x),y) + assert_array_almost_equal(ifftshift(y),x) + x = [0,1,2,3,4,-5,-4,-3,-2,-1] + y = [-5,-4,-3,-2,-1,0,1,2,3,4] + assert_array_almost_equal(fftshift(x),y) + assert_array_almost_equal(ifftshift(y),x) + + def test_inverse(self): + for n in [1,4,9,100,211]: + x = random.random((n,)) + assert_array_almost_equal(ifftshift(fftshift(x)),x) + + +class TestFFTFreq: + + def test_definition(self): + x = [0,1,2,3,4,-4,-3,-2,-1] + assert_array_almost_equal(9*fftfreq(9),x) + assert_array_almost_equal(9*pi*fftfreq(9,pi),x) + x = [0,1,2,3,4,-5,-4,-3,-2,-1] + assert_array_almost_equal(10*fftfreq(10),x) + assert_array_almost_equal(10*pi*fftfreq(10,pi),x) + + +class TestRFFTFreq: + + def test_definition(self): + x = [0,1,1,2,2,3,3,4,4] + assert_array_almost_equal(9*rfftfreq(9),x) + assert_array_almost_equal(9*pi*rfftfreq(9,pi),x) + x = [0,1,1,2,2,3,3,4,4,5] + assert_array_almost_equal(10*rfftfreq(10),x) + assert_array_almost_equal(10*pi*rfftfreq(10,pi),x) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_import.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_import.py new file mode 100644 index 0000000000000000000000000000000000000000..e71aec9bd07cd4ef486b7e74b9589b6f1634d629 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_import.py @@ -0,0 +1,33 @@ +"""Test possibility of patching fftpack with pyfftw. + +No module source outside of scipy.fftpack should contain an import of +the form `from scipy.fftpack import ...`, so that a simple replacement +of scipy.fftpack by the corresponding fftw interface completely swaps +the two FFT implementations. + +Because this simply inspects source files, we only need to run the test +on one version of Python. +""" + + +from pathlib import Path +import re +import tokenize +import pytest +from numpy.testing import assert_ +import scipy + +class TestFFTPackImport: + @pytest.mark.slow + def test_fftpack_import(self): + base = Path(scipy.__file__).parent + regexp = r"\s*from.+\.fftpack import .*\n" + for path in base.rglob("*.py"): + if base / "fftpack" in path.parents: + continue + # use tokenize to auto-detect encoding on systems where no + # default encoding is defined (e.g., LANG='C') + with tokenize.open(str(path)) as file: + assert_(all(not re.fullmatch(regexp, line) + for line in file), + f"{path} contains an import from fftpack") diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_pseudo_diffs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_pseudo_diffs.py new file mode 100644 index 0000000000000000000000000000000000000000..0a92729626a280c12aa3197e99b0d58ce00812d9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_pseudo_diffs.py @@ -0,0 +1,388 @@ +# Created by Pearu Peterson, September 2002 + +__usage__ = """ +Build fftpack: + python setup_fftpack.py build +Run tests if scipy is installed: + python -c 'import scipy;scipy.fftpack.test()' +Run tests if fftpack is not installed: + python tests/test_pseudo_diffs.py [] +""" + +from numpy.testing import (assert_equal, assert_almost_equal, + assert_array_almost_equal) +from scipy.fftpack import (diff, fft, ifft, tilbert, itilbert, hilbert, + ihilbert, shift, fftfreq, cs_diff, sc_diff, + ss_diff, cc_diff) + +import numpy as np +from numpy import arange, sin, cos, pi, exp, tanh, sum, sign +from numpy.random import random + + +def direct_diff(x,k=1,period=None): + fx = fft(x) + n = len(fx) + if period is None: + period = 2*pi + w = fftfreq(n)*2j*pi/period*n + if k < 0: + w = 1 / w**k + w[0] = 0.0 + else: + w = w**k + if n > 2000: + w[250:n-250] = 0.0 + return ifft(w*fx).real + + +def direct_tilbert(x,h=1,period=None): + fx = fft(x) + n = len(fx) + if period is None: + period = 2*pi + w = fftfreq(n)*h*2*pi/period*n + w[0] = 1 + w = 1j/tanh(w) + w[0] = 0j + return ifft(w*fx) + + +def direct_itilbert(x,h=1,period=None): + fx = fft(x) + n = len(fx) + if period is None: + period = 2*pi + w = fftfreq(n)*h*2*pi/period*n + w = -1j*tanh(w) + return ifft(w*fx) + + +def direct_hilbert(x): + fx = fft(x) + n = len(fx) + w = fftfreq(n)*n + w = 1j*sign(w) + return ifft(w*fx) + + +def direct_ihilbert(x): + return -direct_hilbert(x) + + +def direct_shift(x,a,period=None): + n = len(x) + if period is None: + k = fftfreq(n)*1j*n + else: + k = fftfreq(n)*2j*pi/period*n + return ifft(fft(x)*exp(k*a)).real + + +class TestDiff: + + def test_definition(self): + for n in [16,17,64,127,32]: + x = arange(n)*2*pi/n + assert_array_almost_equal(diff(sin(x)),direct_diff(sin(x))) + assert_array_almost_equal(diff(sin(x),2),direct_diff(sin(x),2)) + assert_array_almost_equal(diff(sin(x),3),direct_diff(sin(x),3)) + assert_array_almost_equal(diff(sin(x),4),direct_diff(sin(x),4)) + assert_array_almost_equal(diff(sin(x),5),direct_diff(sin(x),5)) + assert_array_almost_equal(diff(sin(2*x),3),direct_diff(sin(2*x),3)) + assert_array_almost_equal(diff(sin(2*x),4),direct_diff(sin(2*x),4)) + assert_array_almost_equal(diff(cos(x)),direct_diff(cos(x))) + assert_array_almost_equal(diff(cos(x),2),direct_diff(cos(x),2)) + assert_array_almost_equal(diff(cos(x),3),direct_diff(cos(x),3)) + assert_array_almost_equal(diff(cos(x),4),direct_diff(cos(x),4)) + assert_array_almost_equal(diff(cos(2*x)),direct_diff(cos(2*x))) + assert_array_almost_equal(diff(sin(x*n/8)),direct_diff(sin(x*n/8))) + assert_array_almost_equal(diff(cos(x*n/8)),direct_diff(cos(x*n/8))) + for k in range(5): + assert_array_almost_equal(diff(sin(4*x),k),direct_diff(sin(4*x),k)) + assert_array_almost_equal(diff(cos(4*x),k),direct_diff(cos(4*x),k)) + + def test_period(self): + for n in [17,64]: + x = arange(n)/float(n) + assert_array_almost_equal(diff(sin(2*pi*x),period=1), + 2*pi*cos(2*pi*x)) + assert_array_almost_equal(diff(sin(2*pi*x),3,period=1), + -(2*pi)**3*cos(2*pi*x)) + + def test_sin(self): + for n in [32,64,77]: + x = arange(n)*2*pi/n + assert_array_almost_equal(diff(sin(x)),cos(x)) + assert_array_almost_equal(diff(cos(x)),-sin(x)) + assert_array_almost_equal(diff(sin(x),2),-sin(x)) + assert_array_almost_equal(diff(sin(x),4),sin(x)) + assert_array_almost_equal(diff(sin(4*x)),4*cos(4*x)) + assert_array_almost_equal(diff(sin(sin(x))),cos(x)*cos(sin(x))) + + def test_expr(self): + for n in [64,77,100,128,256,512,1024,2048,4096,8192][:5]: + x = arange(n)*2*pi/n + f = sin(x)*cos(4*x)+exp(sin(3*x)) + df = cos(x)*cos(4*x)-4*sin(x)*sin(4*x)+3*cos(3*x)*exp(sin(3*x)) + ddf = -17*sin(x)*cos(4*x)-8*cos(x)*sin(4*x)\ + - 9*sin(3*x)*exp(sin(3*x))+9*cos(3*x)**2*exp(sin(3*x)) + d1 = diff(f) + assert_array_almost_equal(d1,df) + assert_array_almost_equal(diff(df),ddf) + assert_array_almost_equal(diff(f,2),ddf) + assert_array_almost_equal(diff(ddf,-1),df) + + def test_expr_large(self): + for n in [2048,4096]: + x = arange(n)*2*pi/n + f = sin(x)*cos(4*x)+exp(sin(3*x)) + df = cos(x)*cos(4*x)-4*sin(x)*sin(4*x)+3*cos(3*x)*exp(sin(3*x)) + ddf = -17*sin(x)*cos(4*x)-8*cos(x)*sin(4*x)\ + - 9*sin(3*x)*exp(sin(3*x))+9*cos(3*x)**2*exp(sin(3*x)) + assert_array_almost_equal(diff(f),df) + assert_array_almost_equal(diff(df),ddf) + assert_array_almost_equal(diff(ddf,-1),df) + assert_array_almost_equal(diff(f,2),ddf) + + def test_int(self): + n = 64 + x = arange(n)*2*pi/n + assert_array_almost_equal(diff(sin(x),-1),-cos(x)) + assert_array_almost_equal(diff(sin(x),-2),-sin(x)) + assert_array_almost_equal(diff(sin(x),-4),sin(x)) + assert_array_almost_equal(diff(2*cos(2*x),-1),sin(2*x)) + + def test_random_even(self): + rng = np.random.default_rng(1234) + for k in [0,2,4,6]: + for n in [60,32,64,56,55]: + f = rng.random((n,)) + af = sum(f,axis=0)/n + f = f-af + # zeroing Nyquist mode: + f = diff(diff(f,1),-1) + assert_almost_equal(sum(f,axis=0),0.0) + assert_array_almost_equal(diff(diff(f,k),-k),f) + assert_array_almost_equal(diff(diff(f,-k),k),f) + + def test_random_odd(self): + rng = np.random.default_rng(1234) + for k in [0,1,2,3,4,5,6]: + for n in [33,65,55]: + f = rng.random((n,)) + af = sum(f,axis=0)/n + f = f-af + assert_almost_equal(sum(f,axis=0),0.0) + assert_array_almost_equal(diff(diff(f,k),-k),f) + assert_array_almost_equal(diff(diff(f,-k),k),f) + + def test_zero_nyquist(self): + rng = np.random.default_rng(1234) + for k in [0,1,2,3,4,5,6]: + for n in [32,33,64,56,55]: + f = rng.random((n,)) + af = sum(f,axis=0)/n + f = f-af + # zeroing Nyquist mode: + f = diff(diff(f,1),-1) + assert_almost_equal(sum(f,axis=0),0.0) + assert_array_almost_equal(diff(diff(f,k),-k),f) + assert_array_almost_equal(diff(diff(f,-k),k),f) + + +class TestTilbert: + + def test_definition(self): + for h in [0.1,0.5,1,5.5,10]: + for n in [16,17,64,127]: + x = arange(n)*2*pi/n + y = tilbert(sin(x),h) + y1 = direct_tilbert(sin(x),h) + assert_array_almost_equal(y,y1) + assert_array_almost_equal(tilbert(sin(x),h), + direct_tilbert(sin(x),h)) + assert_array_almost_equal(tilbert(sin(2*x),h), + direct_tilbert(sin(2*x),h)) + + def test_random_even(self): + for h in [0.1,0.5,1,5.5,10]: + for n in [32,64,56]: + f = random((n,)) + af = sum(f,axis=0)/n + f = f-af + assert_almost_equal(sum(f,axis=0),0.0) + assert_array_almost_equal(direct_tilbert(direct_itilbert(f,h),h),f) + + def test_random_odd(self): + rng = np.random.default_rng(1234) + for h in [0.1,0.5,1,5.5,10]: + for n in [33,65,55]: + f = rng.random((n,)) + af = sum(f,axis=0)/n + f = f-af + assert_almost_equal(sum(f,axis=0),0.0) + assert_array_almost_equal(itilbert(tilbert(f,h),h),f) + assert_array_almost_equal(tilbert(itilbert(f,h),h),f) + + +class TestITilbert: + + def test_definition(self): + for h in [0.1,0.5,1,5.5,10]: + for n in [16,17,64,127]: + x = arange(n)*2*pi/n + y = itilbert(sin(x),h) + y1 = direct_itilbert(sin(x),h) + assert_array_almost_equal(y,y1) + assert_array_almost_equal(itilbert(sin(x),h), + direct_itilbert(sin(x),h)) + assert_array_almost_equal(itilbert(sin(2*x),h), + direct_itilbert(sin(2*x),h)) + + +class TestHilbert: + + def test_definition(self): + for n in [16,17,64,127]: + x = arange(n)*2*pi/n + y = hilbert(sin(x)) + y1 = direct_hilbert(sin(x)) + assert_array_almost_equal(y,y1) + assert_array_almost_equal(hilbert(sin(2*x)), + direct_hilbert(sin(2*x))) + + def test_tilbert_relation(self): + for n in [16,17,64,127]: + x = arange(n)*2*pi/n + f = sin(x)+cos(2*x)*sin(x) + y = hilbert(f) + y1 = direct_hilbert(f) + assert_array_almost_equal(y,y1) + y2 = tilbert(f,h=10) + assert_array_almost_equal(y,y2) + + def test_random_odd(self): + rng = np.random.default_rng(1234) + for n in [33,65,55]: + f = rng.random((n,)) + af = sum(f,axis=0)/n + f = f-af + assert_almost_equal(sum(f,axis=0),0.0) + assert_array_almost_equal(ihilbert(hilbert(f)),f) + assert_array_almost_equal(hilbert(ihilbert(f)),f) + + def test_random_even(self): + rng = np.random.default_rng(1234) + for n in [32,64,56]: + f = rng.random((n,)) + af = sum(f,axis=0)/n + f = f-af + # zeroing Nyquist mode: + f = diff(diff(f,1),-1) + assert_almost_equal(sum(f,axis=0),0.0) + assert_array_almost_equal(direct_hilbert(direct_ihilbert(f)),f) + assert_array_almost_equal(hilbert(ihilbert(f)),f) + + +class TestIHilbert: + + def test_definition(self): + for n in [16,17,64,127]: + x = arange(n)*2*pi/n + y = ihilbert(sin(x)) + y1 = direct_ihilbert(sin(x)) + assert_array_almost_equal(y,y1) + assert_array_almost_equal(ihilbert(sin(2*x)), + direct_ihilbert(sin(2*x))) + + def test_itilbert_relation(self): + for n in [16,17,64,127]: + x = arange(n)*2*pi/n + f = sin(x)+cos(2*x)*sin(x) + y = ihilbert(f) + y1 = direct_ihilbert(f) + assert_array_almost_equal(y,y1) + y2 = itilbert(f,h=10) + assert_array_almost_equal(y,y2) + + +class TestShift: + + def test_definition(self): + for n in [18,17,64,127,32,2048,256]: + x = arange(n)*2*pi/n + for a in [0.1,3]: + assert_array_almost_equal(shift(sin(x),a),direct_shift(sin(x),a)) + assert_array_almost_equal(shift(sin(x),a),sin(x+a)) + assert_array_almost_equal(shift(cos(x),a),cos(x+a)) + assert_array_almost_equal(shift(cos(2*x)+sin(x),a), + cos(2*(x+a))+sin(x+a)) + assert_array_almost_equal(shift(exp(sin(x)),a),exp(sin(x+a))) + assert_array_almost_equal(shift(sin(x),2*pi),sin(x)) + assert_array_almost_equal(shift(sin(x),pi),-sin(x)) + assert_array_almost_equal(shift(sin(x),pi/2),cos(x)) + + +class TestOverwrite: + """Check input overwrite behavior """ + + real_dtypes = (np.float32, np.float64) + dtypes = real_dtypes + (np.complex64, np.complex128) + + def _check(self, x, routine, *args, **kwargs): + x2 = x.copy() + routine(x2, *args, **kwargs) + sig = routine.__name__ + if args: + sig += repr(args) + if kwargs: + sig += repr(kwargs) + assert_equal(x2, x, err_msg=f"spurious overwrite in {sig}") + + def _check_1d(self, routine, dtype, shape, *args, **kwargs): + # rng = np.random.default_rng(1234) + rng = np.random.RandomState(1234) + # np.random.seed(1234) + if np.issubdtype(dtype, np.complexfloating): + data = rng.randn(*shape) + 1j*rng.randn(*shape) + else: + data = rng.randn(*shape) + data = data.astype(dtype) + self._check(data, routine, *args, **kwargs) + + def test_diff(self): + for dtype in self.dtypes: + self._check_1d(diff, dtype, (16,)) + + def test_tilbert(self): + for dtype in self.dtypes: + self._check_1d(tilbert, dtype, (16,), 1.6) + + def test_itilbert(self): + for dtype in self.dtypes: + self._check_1d(itilbert, dtype, (16,), 1.6) + + def test_hilbert(self): + for dtype in self.dtypes: + self._check_1d(hilbert, dtype, (16,)) + + def test_cs_diff(self): + for dtype in self.dtypes: + self._check_1d(cs_diff, dtype, (16,), 1.0, 4.0) + + def test_sc_diff(self): + for dtype in self.dtypes: + self._check_1d(sc_diff, dtype, (16,), 1.0, 4.0) + + def test_ss_diff(self): + for dtype in self.dtypes: + self._check_1d(ss_diff, dtype, (16,), 1.0, 4.0) + + def test_cc_diff(self): + for dtype in self.dtypes: + self._check_1d(cc_diff, dtype, (16,), 1.0, 4.0) + + def test_shift(self): + for dtype in self.dtypes: + self._check_1d(shift, dtype, (16,), 1.0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_real_transforms.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_real_transforms.py new file mode 100644 index 0000000000000000000000000000000000000000..876af8f18a312f88c508c6b2ae96f1e5c1dad96d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/fftpack/tests/test_real_transforms.py @@ -0,0 +1,837 @@ +from os.path import join, dirname +import threading + +import numpy as np +from numpy.testing import assert_array_almost_equal, assert_equal +import pytest +from pytest import raises as assert_raises + +from scipy.fftpack._realtransforms import ( + dct, idct, dst, idst, dctn, idctn, dstn, idstn) + +# Matlab reference data +MDATA = np.load(join(dirname(__file__), 'test.npz')) +X = [MDATA['x%d' % i] for i in range(8)] +Y = [MDATA['y%d' % i] for i in range(8)] + +# FFTW reference data: the data are organized as follows: +# * SIZES is an array containing all available sizes +# * for every type (1, 2, 3, 4) and every size, the array dct_type_size +# contains the output of the DCT applied to the input np.linspace(0, size-1, +# size) +FFTWDATA_DOUBLE = np.load(join(dirname(__file__), 'fftw_double_ref.npz')) +FFTWDATA_SINGLE = np.load(join(dirname(__file__), 'fftw_single_ref.npz')) +FFTWDATA_SIZES = FFTWDATA_DOUBLE['sizes'] + + +def fftw_dct_ref(type, size, dt): + x = np.linspace(0, size-1, size).astype(dt) + dt = np.result_type(np.float32, dt) + if dt == np.float64: + data = FFTWDATA_DOUBLE + elif dt == np.float32: + data = FFTWDATA_SINGLE + else: + raise ValueError() + y = (data['dct_%d_%d' % (type, size)]).astype(dt) + return x, y, dt + + +def fftw_dst_ref(type, size, dt): + x = np.linspace(0, size-1, size).astype(dt) + dt = np.result_type(np.float32, dt) + if dt == np.float64: + data = FFTWDATA_DOUBLE + elif dt == np.float32: + data = FFTWDATA_SINGLE + else: + raise ValueError() + y = (data['dst_%d_%d' % (type, size)]).astype(dt) + return x, y, dt + + +def dct_2d_ref(x, **kwargs): + """Calculate reference values for testing dct2.""" + x = np.array(x, copy=True) + for row in range(x.shape[0]): + x[row, :] = dct(x[row, :], **kwargs) + for col in range(x.shape[1]): + x[:, col] = dct(x[:, col], **kwargs) + return x + + +def idct_2d_ref(x, **kwargs): + """Calculate reference values for testing idct2.""" + x = np.array(x, copy=True) + for row in range(x.shape[0]): + x[row, :] = idct(x[row, :], **kwargs) + for col in range(x.shape[1]): + x[:, col] = idct(x[:, col], **kwargs) + return x + + +def dst_2d_ref(x, **kwargs): + """Calculate reference values for testing dst2.""" + x = np.array(x, copy=True) + for row in range(x.shape[0]): + x[row, :] = dst(x[row, :], **kwargs) + for col in range(x.shape[1]): + x[:, col] = dst(x[:, col], **kwargs) + return x + + +def idst_2d_ref(x, **kwargs): + """Calculate reference values for testing idst2.""" + x = np.array(x, copy=True) + for row in range(x.shape[0]): + x[row, :] = idst(x[row, :], **kwargs) + for col in range(x.shape[1]): + x[:, col] = idst(x[:, col], **kwargs) + return x + + +def naive_dct1(x, norm=None): + """Calculate textbook definition version of DCT-I.""" + x = np.array(x, copy=True) + N = len(x) + M = N-1 + y = np.zeros(N) + m0, m = 1, 2 + if norm == 'ortho': + m0 = np.sqrt(1.0/M) + m = np.sqrt(2.0/M) + for k in range(N): + for n in range(1, N-1): + y[k] += m*x[n]*np.cos(np.pi*n*k/M) + y[k] += m0 * x[0] + y[k] += m0 * x[N-1] * (1 if k % 2 == 0 else -1) + if norm == 'ortho': + y[0] *= 1/np.sqrt(2) + y[N-1] *= 1/np.sqrt(2) + return y + + +def naive_dst1(x, norm=None): + """Calculate textbook definition version of DST-I.""" + x = np.array(x, copy=True) + N = len(x) + M = N+1 + y = np.zeros(N) + for k in range(N): + for n in range(N): + y[k] += 2*x[n]*np.sin(np.pi*(n+1.0)*(k+1.0)/M) + if norm == 'ortho': + y *= np.sqrt(0.5/M) + return y + + +def naive_dct4(x, norm=None): + """Calculate textbook definition version of DCT-IV.""" + x = np.array(x, copy=True) + N = len(x) + y = np.zeros(N) + for k in range(N): + for n in range(N): + y[k] += x[n]*np.cos(np.pi*(n+0.5)*(k+0.5)/(N)) + if norm == 'ortho': + y *= np.sqrt(2.0/N) + else: + y *= 2 + return y + + +def naive_dst4(x, norm=None): + """Calculate textbook definition version of DST-IV.""" + x = np.array(x, copy=True) + N = len(x) + y = np.zeros(N) + for k in range(N): + for n in range(N): + y[k] += x[n]*np.sin(np.pi*(n+0.5)*(k+0.5)/(N)) + if norm == 'ortho': + y *= np.sqrt(2.0/N) + else: + y *= 2 + return y + + +class TestComplex: + def test_dct_complex64(self): + y = dct(1j*np.arange(5, dtype=np.complex64)) + x = 1j*dct(np.arange(5)) + assert_array_almost_equal(x, y) + + def test_dct_complex(self): + y = dct(np.arange(5)*1j) + x = 1j*dct(np.arange(5)) + assert_array_almost_equal(x, y) + + def test_idct_complex(self): + y = idct(np.arange(5)*1j) + x = 1j*idct(np.arange(5)) + assert_array_almost_equal(x, y) + + def test_dst_complex64(self): + y = dst(np.arange(5, dtype=np.complex64)*1j) + x = 1j*dst(np.arange(5)) + assert_array_almost_equal(x, y) + + def test_dst_complex(self): + y = dst(np.arange(5)*1j) + x = 1j*dst(np.arange(5)) + assert_array_almost_equal(x, y) + + def test_idst_complex(self): + y = idst(np.arange(5)*1j) + x = 1j*idst(np.arange(5)) + assert_array_almost_equal(x, y) + + +class _TestDCTBase: + def setup_method(self): + self.rdt = None + self.dec = 14 + self.type = None + + @pytest.fixture + def dct_lock(self): + return threading.Lock() + + def test_definition(self, dct_lock): + for i in FFTWDATA_SIZES: + with dct_lock: + x, yr, dt = fftw_dct_ref(self.type, i, self.rdt) + y = dct(x, type=self.type) + assert_equal(y.dtype, dt) + # XXX: we divide by np.max(y) because the tests fail otherwise. We + # should really use something like assert_array_approx_equal. The + # difference is due to fftw using a better algorithm w.r.t error + # propagation compared to the ones from fftpack. + assert_array_almost_equal(y / np.max(y), yr / np.max(y), decimal=self.dec, + err_msg="Size %d failed" % i) + + def test_axis(self): + nt = 2 + rng = np.random.RandomState(1234) + for i in [7, 8, 9, 16, 32, 64]: + x = rng.randn(nt, i) + y = dct(x, type=self.type) + for j in range(nt): + assert_array_almost_equal(y[j], dct(x[j], type=self.type), + decimal=self.dec) + + x = x.T + y = dct(x, axis=0, type=self.type) + for j in range(nt): + assert_array_almost_equal(y[:,j], dct(x[:,j], type=self.type), + decimal=self.dec) + + +class _TestDCTIBase(_TestDCTBase): + def test_definition_ortho(self): + # Test orthornomal mode. + dt = np.result_type(np.float32, self.rdt) + for xr in X: + x = np.array(xr, dtype=self.rdt) + y = dct(x, norm='ortho', type=1) + y2 = naive_dct1(x, norm='ortho') + assert_equal(y.dtype, dt) + assert_array_almost_equal(y / np.max(y), y2 / np.max(y), decimal=self.dec) + +class _TestDCTIIBase(_TestDCTBase): + def test_definition_matlab(self): + # Test correspondence with MATLAB (orthornomal mode). + dt = np.result_type(np.float32, self.rdt) + for xr, yr in zip(X, Y): + x = np.array(xr, dtype=dt) + y = dct(x, norm="ortho", type=2) + assert_equal(y.dtype, dt) + assert_array_almost_equal(y, yr, decimal=self.dec) + + +class _TestDCTIIIBase(_TestDCTBase): + def test_definition_ortho(self): + # Test orthornomal mode. + dt = np.result_type(np.float32, self.rdt) + for xr in X: + x = np.array(xr, dtype=self.rdt) + y = dct(x, norm='ortho', type=2) + xi = dct(y, norm="ortho", type=3) + assert_equal(xi.dtype, dt) + assert_array_almost_equal(xi, x, decimal=self.dec) + +class _TestDCTIVBase(_TestDCTBase): + def test_definition_ortho(self): + # Test orthornomal mode. + dt = np.result_type(np.float32, self.rdt) + for xr in X: + x = np.array(xr, dtype=self.rdt) + y = dct(x, norm='ortho', type=4) + y2 = naive_dct4(x, norm='ortho') + assert_equal(y.dtype, dt) + assert_array_almost_equal(y / np.max(y), y2 / np.max(y), decimal=self.dec) + + +class TestDCTIDouble(_TestDCTIBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 10 + self.type = 1 + + +class TestDCTIFloat(_TestDCTIBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 4 + self.type = 1 + + +class TestDCTIInt(_TestDCTIBase): + def setup_method(self): + self.rdt = int + self.dec = 5 + self.type = 1 + + +class TestDCTIIDouble(_TestDCTIIBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 10 + self.type = 2 + + +class TestDCTIIFloat(_TestDCTIIBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 5 + self.type = 2 + + +class TestDCTIIInt(_TestDCTIIBase): + def setup_method(self): + self.rdt = int + self.dec = 5 + self.type = 2 + + +class TestDCTIIIDouble(_TestDCTIIIBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 14 + self.type = 3 + + +class TestDCTIIIFloat(_TestDCTIIIBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 5 + self.type = 3 + + +class TestDCTIIIInt(_TestDCTIIIBase): + def setup_method(self): + self.rdt = int + self.dec = 5 + self.type = 3 + + +class TestDCTIVDouble(_TestDCTIVBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 12 + self.type = 3 + + +class TestDCTIVFloat(_TestDCTIVBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 5 + self.type = 3 + + +class TestDCTIVInt(_TestDCTIVBase): + def setup_method(self): + self.rdt = int + self.dec = 5 + self.type = 3 + + +class _TestIDCTBase: + def setup_method(self): + self.rdt = None + self.dec = 14 + self.type = None + + @pytest.fixture + def idct_lock(self): + return threading.Lock() + + def test_definition(self, idct_lock): + for i in FFTWDATA_SIZES: + with idct_lock: + xr, yr, dt = fftw_dct_ref(self.type, i, self.rdt) + x = idct(yr, type=self.type) + if self.type == 1: + x /= 2 * (i-1) + else: + x /= 2 * i + assert_equal(x.dtype, dt) + # XXX: we divide by np.max(y) because the tests fail otherwise. We + # should really use something like assert_array_approx_equal. The + # difference is due to fftw using a better algorithm w.r.t error + # propagation compared to the ones from fftpack. + assert_array_almost_equal(x / np.max(x), xr / np.max(x), decimal=self.dec, + err_msg="Size %d failed" % i) + + +class TestIDCTIDouble(_TestIDCTBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 10 + self.type = 1 + + +class TestIDCTIFloat(_TestIDCTBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 4 + self.type = 1 + + +class TestIDCTIInt(_TestIDCTBase): + def setup_method(self): + self.rdt = int + self.dec = 4 + self.type = 1 + + +class TestIDCTIIDouble(_TestIDCTBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 10 + self.type = 2 + + +class TestIDCTIIFloat(_TestIDCTBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 5 + self.type = 2 + + +class TestIDCTIIInt(_TestIDCTBase): + def setup_method(self): + self.rdt = int + self.dec = 5 + self.type = 2 + + +class TestIDCTIIIDouble(_TestIDCTBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 14 + self.type = 3 + + +class TestIDCTIIIFloat(_TestIDCTBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 5 + self.type = 3 + + +class TestIDCTIIIInt(_TestIDCTBase): + def setup_method(self): + self.rdt = int + self.dec = 5 + self.type = 3 + +class TestIDCTIVDouble(_TestIDCTBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 12 + self.type = 4 + + +class TestIDCTIVFloat(_TestIDCTBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 5 + self.type = 4 + + +class TestIDCTIVInt(_TestIDCTBase): + def setup_method(self): + self.rdt = int + self.dec = 5 + self.type = 4 + +class _TestDSTBase: + def setup_method(self): + self.rdt = None # dtype + self.dec = None # number of decimals to match + self.type = None # dst type + + @pytest.fixture + def dst_lock(self): + return threading.Lock() + + def test_definition(self, dst_lock): + for i in FFTWDATA_SIZES: + with dst_lock: + xr, yr, dt = fftw_dst_ref(self.type, i, self.rdt) + y = dst(xr, type=self.type) + assert_equal(y.dtype, dt) + # XXX: we divide by np.max(y) because the tests fail otherwise. We + # should really use something like assert_array_approx_equal. The + # difference is due to fftw using a better algorithm w.r.t error + # propagation compared to the ones from fftpack. + assert_array_almost_equal(y / np.max(y), yr / np.max(y), decimal=self.dec, + err_msg="Size %d failed" % i) + + +class _TestDSTIBase(_TestDSTBase): + def test_definition_ortho(self): + # Test orthornomal mode. + dt = np.result_type(np.float32, self.rdt) + for xr in X: + x = np.array(xr, dtype=self.rdt) + y = dst(x, norm='ortho', type=1) + y2 = naive_dst1(x, norm='ortho') + assert_equal(y.dtype, dt) + assert_array_almost_equal(y / np.max(y), y2 / np.max(y), decimal=self.dec) + +class _TestDSTIVBase(_TestDSTBase): + def test_definition_ortho(self): + # Test orthornomal mode. + dt = np.result_type(np.float32, self.rdt) + for xr in X: + x = np.array(xr, dtype=self.rdt) + y = dst(x, norm='ortho', type=4) + y2 = naive_dst4(x, norm='ortho') + assert_equal(y.dtype, dt) + assert_array_almost_equal(y, y2, decimal=self.dec) + +class TestDSTIDouble(_TestDSTIBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 12 + self.type = 1 + + +class TestDSTIFloat(_TestDSTIBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 4 + self.type = 1 + + +class TestDSTIInt(_TestDSTIBase): + def setup_method(self): + self.rdt = int + self.dec = 5 + self.type = 1 + + +class TestDSTIIDouble(_TestDSTBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 14 + self.type = 2 + + +class TestDSTIIFloat(_TestDSTBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 6 + self.type = 2 + + +class TestDSTIIInt(_TestDSTBase): + def setup_method(self): + self.rdt = int + self.dec = 6 + self.type = 2 + + +class TestDSTIIIDouble(_TestDSTBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 14 + self.type = 3 + + +class TestDSTIIIFloat(_TestDSTBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 7 + self.type = 3 + + +class TestDSTIIIInt(_TestDSTBase): + def setup_method(self): + self.rdt = int + self.dec = 7 + self.type = 3 + + +class TestDSTIVDouble(_TestDSTIVBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 12 + self.type = 4 + + +class TestDSTIVFloat(_TestDSTIVBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 4 + self.type = 4 + + +class TestDSTIVInt(_TestDSTIVBase): + def setup_method(self): + self.rdt = int + self.dec = 5 + self.type = 4 + + +class _TestIDSTBase: + def setup_method(self): + self.rdt = None + self.dec = None + self.type = None + + @pytest.fixture + def idst_lock(self): + return threading.Lock() + + def test_definition(self, idst_lock): + for i in FFTWDATA_SIZES: + with idst_lock: + xr, yr, dt = fftw_dst_ref(self.type, i, self.rdt) + x = idst(yr, type=self.type) + if self.type == 1: + x /= 2 * (i+1) + else: + x /= 2 * i + assert_equal(x.dtype, dt) + # XXX: we divide by np.max(x) because the tests fail otherwise. We + # should really use something like assert_array_approx_equal. The + # difference is due to fftw using a better algorithm w.r.t error + # propagation compared to the ones from fftpack. + assert_array_almost_equal(x / np.max(x), xr / np.max(x), decimal=self.dec, + err_msg="Size %d failed" % i) + + +class TestIDSTIDouble(_TestIDSTBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 12 + self.type = 1 + + +class TestIDSTIFloat(_TestIDSTBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 4 + self.type = 1 + + +class TestIDSTIInt(_TestIDSTBase): + def setup_method(self): + self.rdt = int + self.dec = 4 + self.type = 1 + + +class TestIDSTIIDouble(_TestIDSTBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 14 + self.type = 2 + + +class TestIDSTIIFloat(_TestIDSTBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 6 + self.type = 2 + + +class TestIDSTIIInt(_TestIDSTBase): + def setup_method(self): + self.rdt = int + self.dec = 6 + self.type = 2 + + +class TestIDSTIIIDouble(_TestIDSTBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 14 + self.type = 3 + + +class TestIDSTIIIFloat(_TestIDSTBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 6 + self.type = 3 + + +class TestIDSTIIIInt(_TestIDSTBase): + def setup_method(self): + self.rdt = int + self.dec = 6 + self.type = 3 + + +class TestIDSTIVDouble(_TestIDSTBase): + def setup_method(self): + self.rdt = np.float64 + self.dec = 12 + self.type = 4 + + +class TestIDSTIVFloat(_TestIDSTBase): + def setup_method(self): + self.rdt = np.float32 + self.dec = 6 + self.type = 4 + + +class TestIDSTIVnt(_TestIDSTBase): + def setup_method(self): + self.rdt = int + self.dec = 6 + self.type = 4 + + +class TestOverwrite: + """Check input overwrite behavior.""" + + real_dtypes = [np.float32, np.float64] + + def _check(self, x, routine, type, fftsize, axis, norm, overwrite_x, **kw): + x2 = x.copy() + routine(x2, type, fftsize, axis, norm, overwrite_x=overwrite_x) + + sig = (f"{routine.__name__}({x.dtype}{x.shape!r}, {fftsize!r}, " + f"axis={axis!r}, overwrite_x={overwrite_x!r})") + if not overwrite_x: + assert_equal(x2, x, err_msg=f"spurious overwrite in {sig}") + + def _check_1d(self, routine, dtype, shape, axis): + rng = np.random.RandomState(1234) + if np.issubdtype(dtype, np.complexfloating): + data = rng.randn(*shape) + 1j*rng.randn(*shape) + else: + data = rng.randn(*shape) + data = data.astype(dtype) + + for type in [1, 2, 3, 4]: + for overwrite_x in [True, False]: + for norm in [None, 'ortho']: + self._check(data, routine, type, None, axis, norm, + overwrite_x) + + def test_dct(self): + for dtype in self.real_dtypes: + self._check_1d(dct, dtype, (16,), -1) + self._check_1d(dct, dtype, (16, 2), 0) + self._check_1d(dct, dtype, (2, 16), 1) + + def test_idct(self): + for dtype in self.real_dtypes: + self._check_1d(idct, dtype, (16,), -1) + self._check_1d(idct, dtype, (16, 2), 0) + self._check_1d(idct, dtype, (2, 16), 1) + + def test_dst(self): + for dtype in self.real_dtypes: + self._check_1d(dst, dtype, (16,), -1) + self._check_1d(dst, dtype, (16, 2), 0) + self._check_1d(dst, dtype, (2, 16), 1) + + def test_idst(self): + for dtype in self.real_dtypes: + self._check_1d(idst, dtype, (16,), -1) + self._check_1d(idst, dtype, (16, 2), 0) + self._check_1d(idst, dtype, (2, 16), 1) + + +class Test_DCTN_IDCTN: + dec = 14 + dct_type = [1, 2, 3, 4] + norms = [None, 'ortho'] + rstate = np.random.RandomState(1234) + shape = (32, 16) + data = rstate.randn(*shape) + + @pytest.mark.parametrize('fforward,finverse', [(dctn, idctn), + (dstn, idstn)]) + @pytest.mark.parametrize('axes', [None, + 1, (1,), [1], + 0, (0,), [0], + (0, 1), [0, 1], + (-2, -1), [-2, -1]]) + @pytest.mark.parametrize('dct_type', dct_type) + @pytest.mark.parametrize('norm', ['ortho']) + def test_axes_round_trip(self, fforward, finverse, axes, dct_type, norm): + tmp = fforward(self.data, type=dct_type, axes=axes, norm=norm) + tmp = finverse(tmp, type=dct_type, axes=axes, norm=norm) + assert_array_almost_equal(self.data, tmp, decimal=12) + + @pytest.mark.parametrize('fforward,fforward_ref', [(dctn, dct_2d_ref), + (dstn, dst_2d_ref)]) + @pytest.mark.parametrize('dct_type', dct_type) + @pytest.mark.parametrize('norm', norms) + def test_dctn_vs_2d_reference(self, fforward, fforward_ref, + dct_type, norm): + y1 = fforward(self.data, type=dct_type, axes=None, norm=norm) + y2 = fforward_ref(self.data, type=dct_type, norm=norm) + assert_array_almost_equal(y1, y2, decimal=11) + + @pytest.mark.parametrize('finverse,finverse_ref', [(idctn, idct_2d_ref), + (idstn, idst_2d_ref)]) + @pytest.mark.parametrize('dct_type', dct_type) + @pytest.mark.parametrize('norm', [None, 'ortho']) + def test_idctn_vs_2d_reference(self, finverse, finverse_ref, + dct_type, norm): + fdata = dctn(self.data, type=dct_type, norm=norm) + y1 = finverse(fdata, type=dct_type, norm=norm) + y2 = finverse_ref(fdata, type=dct_type, norm=norm) + assert_array_almost_equal(y1, y2, decimal=11) + + @pytest.mark.parametrize('fforward,finverse', [(dctn, idctn), + (dstn, idstn)]) + def test_axes_and_shape(self, fforward, finverse): + with assert_raises(ValueError, + match="when given, axes and shape arguments" + " have to be of the same length"): + fforward(self.data, shape=self.data.shape[0], axes=(0, 1)) + + with assert_raises(ValueError, + match="when given, axes and shape arguments" + " have to be of the same length"): + fforward(self.data, shape=self.data.shape[0], axes=None) + + with assert_raises(ValueError, + match="when given, axes and shape arguments" + " have to be of the same length"): + fforward(self.data, shape=self.data.shape, axes=0) + + @pytest.mark.parametrize('fforward', [dctn, dstn]) + def test_shape(self, fforward): + tmp = fforward(self.data, shape=(128, 128), axes=None) + assert_equal(tmp.shape, (128, 128)) + + @pytest.mark.parametrize('fforward,finverse', [(dctn, idctn), + (dstn, idstn)]) + @pytest.mark.parametrize('axes', [1, (1,), [1], + 0, (0,), [0]]) + def test_shape_is_none_with_axes(self, fforward, finverse, axes): + tmp = fforward(self.data, shape=None, axes=axes, norm='ortho') + tmp = finverse(tmp, shape=None, axes=axes, norm='ortho') + assert_array_almost_equal(self.data, tmp, decimal=self.dec) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..1533b5c60b695fce0abf08e2163dfba3bdd4fb17 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/__init__.py @@ -0,0 +1,122 @@ +""" +============================================= +Integration and ODEs (:mod:`scipy.integrate`) +============================================= + +.. currentmodule:: scipy.integrate + +Integrating functions, given function object +============================================ + +.. autosummary:: + :toctree: generated/ + + quad -- General purpose integration + quad_vec -- General purpose integration of vector-valued functions + cubature -- General purpose multi-dimensional integration of array-valued functions + dblquad -- General purpose double integration + tplquad -- General purpose triple integration + nquad -- General purpose N-D integration + tanhsinh -- General purpose elementwise integration + fixed_quad -- Integrate func(x) using Gaussian quadrature of order n + newton_cotes -- Weights and error coefficient for Newton-Cotes integration + lebedev_rule + qmc_quad -- N-D integration using Quasi-Monte Carlo quadrature + IntegrationWarning -- Warning on issues during integration + + +Integrating functions, given fixed samples +========================================== + +.. autosummary:: + :toctree: generated/ + + trapezoid -- Use trapezoidal rule to compute integral. + cumulative_trapezoid -- Use trapezoidal rule to cumulatively compute integral. + simpson -- Use Simpson's rule to compute integral from samples. + cumulative_simpson -- Use Simpson's rule to cumulatively compute integral from samples. + romb -- Use Romberg Integration to compute integral from + -- (2**k + 1) evenly-spaced samples. + +.. seealso:: + + :mod:`scipy.special` for orthogonal polynomials (special) for Gaussian + quadrature roots and weights for other weighting factors and regions. + +Summation +========= + +.. autosummary:: + :toctree: generated/ + + nsum + +Solving initial value problems for ODE systems +============================================== + +The solvers are implemented as individual classes, which can be used directly +(low-level usage) or through a convenience function. + +.. autosummary:: + :toctree: generated/ + + solve_ivp -- Convenient function for ODE integration. + RK23 -- Explicit Runge-Kutta solver of order 3(2). + RK45 -- Explicit Runge-Kutta solver of order 5(4). + DOP853 -- Explicit Runge-Kutta solver of order 8. + Radau -- Implicit Runge-Kutta solver of order 5. + BDF -- Implicit multi-step variable order (1 to 5) solver. + LSODA -- LSODA solver from ODEPACK Fortran package. + OdeSolver -- Base class for ODE solvers. + DenseOutput -- Local interpolant for computing a dense output. + OdeSolution -- Class which represents a continuous ODE solution. + + +Old API +------- + +These are the routines developed earlier for SciPy. They wrap older solvers +implemented in Fortran (mostly ODEPACK). While the interface to them is not +particularly convenient and certain features are missing compared to the new +API, the solvers themselves are of good quality and work fast as compiled +Fortran code. In some cases, it might be worth using this old API. + +.. autosummary:: + :toctree: generated/ + + odeint -- General integration of ordinary differential equations. + ode -- Integrate ODE using VODE and ZVODE routines. + complex_ode -- Convert a complex-valued ODE to real-valued and integrate. + ODEintWarning -- Warning raised during the execution of `odeint`. + + +Solving boundary value problems for ODE systems +=============================================== + +.. autosummary:: + :toctree: generated/ + + solve_bvp -- Solve a boundary value problem for a system of ODEs. +""" # noqa: E501 + + +from ._quadrature import * +from ._odepack_py import * +from ._quadpack_py import * +from ._ode import * +from ._bvp import solve_bvp +from ._ivp import (solve_ivp, OdeSolution, DenseOutput, + OdeSolver, RK23, RK45, DOP853, Radau, BDF, LSODA) +from ._quad_vec import quad_vec +from ._tanhsinh import nsum, tanhsinh +from ._cubature import cubature +from ._lebedev import lebedev_rule + +# Deprecated namespaces, to be removed in v2.0.0 +from . import dop, lsoda, vode, odepack, quadpack + +__all__ = [s for s in dir() if not s.startswith('_')] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_bvp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_bvp.py new file mode 100644 index 0000000000000000000000000000000000000000..74406c89a689edc3de21fcb7274c90d41b8d2dcc --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_bvp.py @@ -0,0 +1,1154 @@ +"""Boundary value problem solver.""" +from warnings import warn + +import numpy as np +from numpy.linalg import pinv + +from scipy.sparse import coo_matrix, csc_matrix +from scipy.sparse.linalg import splu +from scipy.optimize import OptimizeResult + + +EPS = np.finfo(float).eps + + +def estimate_fun_jac(fun, x, y, p, f0=None): + """Estimate derivatives of an ODE system rhs with forward differences. + + Returns + ------- + df_dy : ndarray, shape (n, n, m) + Derivatives with respect to y. An element (i, j, q) corresponds to + d f_i(x_q, y_q) / d (y_q)_j. + df_dp : ndarray with shape (n, k, m) or None + Derivatives with respect to p. An element (i, j, q) corresponds to + d f_i(x_q, y_q, p) / d p_j. If `p` is empty, None is returned. + """ + n, m = y.shape + if f0 is None: + f0 = fun(x, y, p) + + dtype = y.dtype + + df_dy = np.empty((n, n, m), dtype=dtype) + h = EPS**0.5 * (1 + np.abs(y)) + for i in range(n): + y_new = y.copy() + y_new[i] += h[i] + hi = y_new[i] - y[i] + f_new = fun(x, y_new, p) + df_dy[:, i, :] = (f_new - f0) / hi + + k = p.shape[0] + if k == 0: + df_dp = None + else: + df_dp = np.empty((n, k, m), dtype=dtype) + h = EPS**0.5 * (1 + np.abs(p)) + for i in range(k): + p_new = p.copy() + p_new[i] += h[i] + hi = p_new[i] - p[i] + f_new = fun(x, y, p_new) + df_dp[:, i, :] = (f_new - f0) / hi + + return df_dy, df_dp + + +def estimate_bc_jac(bc, ya, yb, p, bc0=None): + """Estimate derivatives of boundary conditions with forward differences. + + Returns + ------- + dbc_dya : ndarray, shape (n + k, n) + Derivatives with respect to ya. An element (i, j) corresponds to + d bc_i / d ya_j. + dbc_dyb : ndarray, shape (n + k, n) + Derivatives with respect to yb. An element (i, j) corresponds to + d bc_i / d ya_j. + dbc_dp : ndarray with shape (n + k, k) or None + Derivatives with respect to p. An element (i, j) corresponds to + d bc_i / d p_j. If `p` is empty, None is returned. + """ + n = ya.shape[0] + k = p.shape[0] + + if bc0 is None: + bc0 = bc(ya, yb, p) + + dtype = ya.dtype + + dbc_dya = np.empty((n, n + k), dtype=dtype) + h = EPS**0.5 * (1 + np.abs(ya)) + for i in range(n): + ya_new = ya.copy() + ya_new[i] += h[i] + hi = ya_new[i] - ya[i] + bc_new = bc(ya_new, yb, p) + dbc_dya[i] = (bc_new - bc0) / hi + dbc_dya = dbc_dya.T + + h = EPS**0.5 * (1 + np.abs(yb)) + dbc_dyb = np.empty((n, n + k), dtype=dtype) + for i in range(n): + yb_new = yb.copy() + yb_new[i] += h[i] + hi = yb_new[i] - yb[i] + bc_new = bc(ya, yb_new, p) + dbc_dyb[i] = (bc_new - bc0) / hi + dbc_dyb = dbc_dyb.T + + if k == 0: + dbc_dp = None + else: + h = EPS**0.5 * (1 + np.abs(p)) + dbc_dp = np.empty((k, n + k), dtype=dtype) + for i in range(k): + p_new = p.copy() + p_new[i] += h[i] + hi = p_new[i] - p[i] + bc_new = bc(ya, yb, p_new) + dbc_dp[i] = (bc_new - bc0) / hi + dbc_dp = dbc_dp.T + + return dbc_dya, dbc_dyb, dbc_dp + + +def compute_jac_indices(n, m, k): + """Compute indices for the collocation system Jacobian construction. + + See `construct_global_jac` for the explanation. + """ + i_col = np.repeat(np.arange((m - 1) * n), n) + j_col = (np.tile(np.arange(n), n * (m - 1)) + + np.repeat(np.arange(m - 1) * n, n**2)) + + i_bc = np.repeat(np.arange((m - 1) * n, m * n + k), n) + j_bc = np.tile(np.arange(n), n + k) + + i_p_col = np.repeat(np.arange((m - 1) * n), k) + j_p_col = np.tile(np.arange(m * n, m * n + k), (m - 1) * n) + + i_p_bc = np.repeat(np.arange((m - 1) * n, m * n + k), k) + j_p_bc = np.tile(np.arange(m * n, m * n + k), n + k) + + i = np.hstack((i_col, i_col, i_bc, i_bc, i_p_col, i_p_bc)) + j = np.hstack((j_col, j_col + n, + j_bc, j_bc + (m - 1) * n, + j_p_col, j_p_bc)) + + return i, j + + +def stacked_matmul(a, b): + """Stacked matrix multiply: out[i,:,:] = np.dot(a[i,:,:], b[i,:,:]). + + Empirical optimization. Use outer Python loop and BLAS for large + matrices, otherwise use a single einsum call. + """ + if a.shape[1] > 50: + out = np.empty((a.shape[0], a.shape[1], b.shape[2])) + for i in range(a.shape[0]): + out[i] = np.dot(a[i], b[i]) + return out + else: + return np.einsum('...ij,...jk->...ik', a, b) + + +def construct_global_jac(n, m, k, i_jac, j_jac, h, df_dy, df_dy_middle, df_dp, + df_dp_middle, dbc_dya, dbc_dyb, dbc_dp): + """Construct the Jacobian of the collocation system. + + There are n * m + k functions: m - 1 collocations residuals, each + containing n components, followed by n + k boundary condition residuals. + + There are n * m + k variables: m vectors of y, each containing n + components, followed by k values of vector p. + + For example, let m = 4, n = 2 and k = 1, then the Jacobian will have + the following sparsity structure: + + 1 1 2 2 0 0 0 0 5 + 1 1 2 2 0 0 0 0 5 + 0 0 1 1 2 2 0 0 5 + 0 0 1 1 2 2 0 0 5 + 0 0 0 0 1 1 2 2 5 + 0 0 0 0 1 1 2 2 5 + + 3 3 0 0 0 0 4 4 6 + 3 3 0 0 0 0 4 4 6 + 3 3 0 0 0 0 4 4 6 + + Zeros denote identically zero values, other values denote different kinds + of blocks in the matrix (see below). The blank row indicates the separation + of collocation residuals from boundary conditions. And the blank column + indicates the separation of y values from p values. + + Refer to [1]_ (p. 306) for the formula of n x n blocks for derivatives + of collocation residuals with respect to y. + + Parameters + ---------- + n : int + Number of equations in the ODE system. + m : int + Number of nodes in the mesh. + k : int + Number of the unknown parameters. + i_jac, j_jac : ndarray + Row and column indices returned by `compute_jac_indices`. They + represent different blocks in the Jacobian matrix in the following + order (see the scheme above): + + * 1: m - 1 diagonal n x n blocks for the collocation residuals. + * 2: m - 1 off-diagonal n x n blocks for the collocation residuals. + * 3 : (n + k) x n block for the dependency of the boundary + conditions on ya. + * 4: (n + k) x n block for the dependency of the boundary + conditions on yb. + * 5: (m - 1) * n x k block for the dependency of the collocation + residuals on p. + * 6: (n + k) x k block for the dependency of the boundary + conditions on p. + + df_dy : ndarray, shape (n, n, m) + Jacobian of f with respect to y computed at the mesh nodes. + df_dy_middle : ndarray, shape (n, n, m - 1) + Jacobian of f with respect to y computed at the middle between the + mesh nodes. + df_dp : ndarray with shape (n, k, m) or None + Jacobian of f with respect to p computed at the mesh nodes. + df_dp_middle : ndarray with shape (n, k, m - 1) or None + Jacobian of f with respect to p computed at the middle between the + mesh nodes. + dbc_dya, dbc_dyb : ndarray, shape (n, n) + Jacobian of bc with respect to ya and yb. + dbc_dp : ndarray with shape (n, k) or None + Jacobian of bc with respect to p. + + Returns + ------- + J : csc_matrix, shape (n * m + k, n * m + k) + Jacobian of the collocation system in a sparse form. + + References + ---------- + .. [1] J. Kierzenka, L. F. Shampine, "A BVP Solver Based on Residual + Control and the Maltab PSE", ACM Trans. Math. Softw., Vol. 27, + Number 3, pp. 299-316, 2001. + """ + df_dy = np.transpose(df_dy, (2, 0, 1)) + df_dy_middle = np.transpose(df_dy_middle, (2, 0, 1)) + + h = h[:, np.newaxis, np.newaxis] + + dtype = df_dy.dtype + + # Computing diagonal n x n blocks. + dPhi_dy_0 = np.empty((m - 1, n, n), dtype=dtype) + dPhi_dy_0[:] = -np.identity(n) + dPhi_dy_0 -= h / 6 * (df_dy[:-1] + 2 * df_dy_middle) + T = stacked_matmul(df_dy_middle, df_dy[:-1]) + dPhi_dy_0 -= h**2 / 12 * T + + # Computing off-diagonal n x n blocks. + dPhi_dy_1 = np.empty((m - 1, n, n), dtype=dtype) + dPhi_dy_1[:] = np.identity(n) + dPhi_dy_1 -= h / 6 * (df_dy[1:] + 2 * df_dy_middle) + T = stacked_matmul(df_dy_middle, df_dy[1:]) + dPhi_dy_1 += h**2 / 12 * T + + values = np.hstack((dPhi_dy_0.ravel(), dPhi_dy_1.ravel(), dbc_dya.ravel(), + dbc_dyb.ravel())) + + if k > 0: + df_dp = np.transpose(df_dp, (2, 0, 1)) + df_dp_middle = np.transpose(df_dp_middle, (2, 0, 1)) + T = stacked_matmul(df_dy_middle, df_dp[:-1] - df_dp[1:]) + df_dp_middle += 0.125 * h * T + dPhi_dp = -h/6 * (df_dp[:-1] + df_dp[1:] + 4 * df_dp_middle) + values = np.hstack((values, dPhi_dp.ravel(), dbc_dp.ravel())) + + J = coo_matrix((values, (i_jac, j_jac))) + return csc_matrix(J) + + +def collocation_fun(fun, y, p, x, h): + """Evaluate collocation residuals. + + This function lies in the core of the method. The solution is sought + as a cubic C1 continuous spline with derivatives matching the ODE rhs + at given nodes `x`. Collocation conditions are formed from the equality + of the spline derivatives and rhs of the ODE system in the middle points + between nodes. + + Such method is classified to Lobbato IIIA family in ODE literature. + Refer to [1]_ for the formula and some discussion. + + Returns + ------- + col_res : ndarray, shape (n, m - 1) + Collocation residuals at the middle points of the mesh intervals. + y_middle : ndarray, shape (n, m - 1) + Values of the cubic spline evaluated at the middle points of the mesh + intervals. + f : ndarray, shape (n, m) + RHS of the ODE system evaluated at the mesh nodes. + f_middle : ndarray, shape (n, m - 1) + RHS of the ODE system evaluated at the middle points of the mesh + intervals (and using `y_middle`). + + References + ---------- + .. [1] J. Kierzenka, L. F. Shampine, "A BVP Solver Based on Residual + Control and the Maltab PSE", ACM Trans. Math. Softw., Vol. 27, + Number 3, pp. 299-316, 2001. + """ + f = fun(x, y, p) + y_middle = (0.5 * (y[:, 1:] + y[:, :-1]) - + 0.125 * h * (f[:, 1:] - f[:, :-1])) + f_middle = fun(x[:-1] + 0.5 * h, y_middle, p) + col_res = y[:, 1:] - y[:, :-1] - h / 6 * (f[:, :-1] + f[:, 1:] + + 4 * f_middle) + + return col_res, y_middle, f, f_middle + + +def prepare_sys(n, m, k, fun, bc, fun_jac, bc_jac, x, h): + """Create the function and the Jacobian for the collocation system.""" + x_middle = x[:-1] + 0.5 * h + i_jac, j_jac = compute_jac_indices(n, m, k) + + def col_fun(y, p): + return collocation_fun(fun, y, p, x, h) + + def sys_jac(y, p, y_middle, f, f_middle, bc0): + if fun_jac is None: + df_dy, df_dp = estimate_fun_jac(fun, x, y, p, f) + df_dy_middle, df_dp_middle = estimate_fun_jac( + fun, x_middle, y_middle, p, f_middle) + else: + df_dy, df_dp = fun_jac(x, y, p) + df_dy_middle, df_dp_middle = fun_jac(x_middle, y_middle, p) + + if bc_jac is None: + dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac(bc, y[:, 0], y[:, -1], + p, bc0) + else: + dbc_dya, dbc_dyb, dbc_dp = bc_jac(y[:, 0], y[:, -1], p) + + return construct_global_jac(n, m, k, i_jac, j_jac, h, df_dy, + df_dy_middle, df_dp, df_dp_middle, dbc_dya, + dbc_dyb, dbc_dp) + + return col_fun, sys_jac + + +def solve_newton(n, m, h, col_fun, bc, jac, y, p, B, bvp_tol, bc_tol): + """Solve the nonlinear collocation system by a Newton method. + + This is a simple Newton method with a backtracking line search. As + advised in [1]_, an affine-invariant criterion function F = ||J^-1 r||^2 + is used, where J is the Jacobian matrix at the current iteration and r is + the vector or collocation residuals (values of the system lhs). + + The method alters between full Newton iterations and the fixed-Jacobian + iterations based + + There are other tricks proposed in [1]_, but they are not used as they + don't seem to improve anything significantly, and even break the + convergence on some test problems I tried. + + All important parameters of the algorithm are defined inside the function. + + Parameters + ---------- + n : int + Number of equations in the ODE system. + m : int + Number of nodes in the mesh. + h : ndarray, shape (m-1,) + Mesh intervals. + col_fun : callable + Function computing collocation residuals. + bc : callable + Function computing boundary condition residuals. + jac : callable + Function computing the Jacobian of the whole system (including + collocation and boundary condition residuals). It is supposed to + return csc_matrix. + y : ndarray, shape (n, m) + Initial guess for the function values at the mesh nodes. + p : ndarray, shape (k,) + Initial guess for the unknown parameters. + B : ndarray with shape (n, n) or None + Matrix to force the S y(a) = 0 condition for a problems with the + singular term. If None, the singular term is assumed to be absent. + bvp_tol : float + Tolerance to which we want to solve a BVP. + bc_tol : float + Tolerance to which we want to satisfy the boundary conditions. + + Returns + ------- + y : ndarray, shape (n, m) + Final iterate for the function values at the mesh nodes. + p : ndarray, shape (k,) + Final iterate for the unknown parameters. + singular : bool + True, if the LU decomposition failed because Jacobian turned out + to be singular. + + References + ---------- + .. [1] U. Ascher, R. Mattheij and R. Russell "Numerical Solution of + Boundary Value Problems for Ordinary Differential Equations" + """ + # We know that the solution residuals at the middle points of the mesh + # are connected with collocation residuals r_middle = 1.5 * col_res / h. + # As our BVP solver tries to decrease relative residuals below a certain + # tolerance, it seems reasonable to terminated Newton iterations by + # comparison of r_middle / (1 + np.abs(f_middle)) with a certain threshold, + # which we choose to be 1.5 orders lower than the BVP tolerance. We rewrite + # the condition as col_res < tol_r * (1 + np.abs(f_middle)), then tol_r + # should be computed as follows: + tol_r = 2/3 * h * 5e-2 * bvp_tol + + # Maximum allowed number of Jacobian evaluation and factorization, in + # other words, the maximum number of full Newton iterations. A small value + # is recommended in the literature. + max_njev = 4 + + # Maximum number of iterations, considering that some of them can be + # performed with the fixed Jacobian. In theory, such iterations are cheap, + # but it's not that simple in Python. + max_iter = 8 + + # Minimum relative improvement of the criterion function to accept the + # step (Armijo constant). + sigma = 0.2 + + # Step size decrease factor for backtracking. + tau = 0.5 + + # Maximum number of backtracking steps, the minimum step is then + # tau ** n_trial. + n_trial = 4 + + col_res, y_middle, f, f_middle = col_fun(y, p) + bc_res = bc(y[:, 0], y[:, -1], p) + res = np.hstack((col_res.ravel(order='F'), bc_res)) + + njev = 0 + singular = False + recompute_jac = True + for iteration in range(max_iter): + if recompute_jac: + J = jac(y, p, y_middle, f, f_middle, bc_res) + njev += 1 + try: + LU = splu(J) + except RuntimeError: + singular = True + break + + step = LU.solve(res) + cost = np.dot(step, step) + + y_step = step[:m * n].reshape((n, m), order='F') + p_step = step[m * n:] + + alpha = 1 + for trial in range(n_trial + 1): + y_new = y - alpha * y_step + if B is not None: + y_new[:, 0] = np.dot(B, y_new[:, 0]) + p_new = p - alpha * p_step + + col_res, y_middle, f, f_middle = col_fun(y_new, p_new) + bc_res = bc(y_new[:, 0], y_new[:, -1], p_new) + res = np.hstack((col_res.ravel(order='F'), bc_res)) + + step_new = LU.solve(res) + cost_new = np.dot(step_new, step_new) + if cost_new < (1 - 2 * alpha * sigma) * cost: + break + + if trial < n_trial: + alpha *= tau + + y = y_new + p = p_new + + if njev == max_njev: + break + + if (np.all(np.abs(col_res) < tol_r * (1 + np.abs(f_middle))) and + np.all(np.abs(bc_res) < bc_tol)): + break + + # If the full step was taken, then we are going to continue with + # the same Jacobian. This is the approach of BVP_SOLVER. + if alpha == 1: + step = step_new + cost = cost_new + recompute_jac = False + else: + recompute_jac = True + + return y, p, singular + + +def print_iteration_header(): + print(f"{'Iteration':^15}{'Max residual':^15}{'Max BC residual':^15}" + f"{'Total nodes':^15}{'Nodes added':^15}") + + +def print_iteration_progress(iteration, residual, bc_residual, total_nodes, + nodes_added): + print(f"{iteration:^15}{residual:^15.2e}{bc_residual:^15.2e}" + f"{total_nodes:^15}{nodes_added:^15}") + + +class BVPResult(OptimizeResult): + pass + + +TERMINATION_MESSAGES = { + 0: "The algorithm converged to the desired accuracy.", + 1: "The maximum number of mesh nodes is exceeded.", + 2: "A singular Jacobian encountered when solving the collocation system.", + 3: "The solver was unable to satisfy boundary conditions tolerance on iteration 10." +} + + +def estimate_rms_residuals(fun, sol, x, h, p, r_middle, f_middle): + """Estimate rms values of collocation residuals using Lobatto quadrature. + + The residuals are defined as the difference between the derivatives of + our solution and rhs of the ODE system. We use relative residuals, i.e., + normalized by 1 + np.abs(f). RMS values are computed as sqrt from the + normalized integrals of the squared relative residuals over each interval. + Integrals are estimated using 5-point Lobatto quadrature [1]_, we use the + fact that residuals at the mesh nodes are identically zero. + + In [2] they don't normalize integrals by interval lengths, which gives + a higher rate of convergence of the residuals by the factor of h**0.5. + I chose to do such normalization for an ease of interpretation of return + values as RMS estimates. + + Returns + ------- + rms_res : ndarray, shape (m - 1,) + Estimated rms values of the relative residuals over each interval. + + References + ---------- + .. [1] http://mathworld.wolfram.com/LobattoQuadrature.html + .. [2] J. Kierzenka, L. F. Shampine, "A BVP Solver Based on Residual + Control and the Maltab PSE", ACM Trans. Math. Softw., Vol. 27, + Number 3, pp. 299-316, 2001. + """ + x_middle = x[:-1] + 0.5 * h + s = 0.5 * h * (3/7)**0.5 + x1 = x_middle + s + x2 = x_middle - s + y1 = sol(x1) + y2 = sol(x2) + y1_prime = sol(x1, 1) + y2_prime = sol(x2, 1) + f1 = fun(x1, y1, p) + f2 = fun(x2, y2, p) + r1 = y1_prime - f1 + r2 = y2_prime - f2 + + r_middle /= 1 + np.abs(f_middle) + r1 /= 1 + np.abs(f1) + r2 /= 1 + np.abs(f2) + + r1 = np.sum(np.real(r1 * np.conj(r1)), axis=0) + r2 = np.sum(np.real(r2 * np.conj(r2)), axis=0) + r_middle = np.sum(np.real(r_middle * np.conj(r_middle)), axis=0) + + return (0.5 * (32 / 45 * r_middle + 49 / 90 * (r1 + r2))) ** 0.5 + + +def create_spline(y, yp, x, h): + """Create a cubic spline given values and derivatives. + + Formulas for the coefficients are taken from interpolate.CubicSpline. + + Returns + ------- + sol : PPoly + Constructed spline as a PPoly instance. + """ + from scipy.interpolate import PPoly + + n, m = y.shape + c = np.empty((4, n, m - 1), dtype=y.dtype) + slope = (y[:, 1:] - y[:, :-1]) / h + t = (yp[:, :-1] + yp[:, 1:] - 2 * slope) / h + c[0] = t / h + c[1] = (slope - yp[:, :-1]) / h - t + c[2] = yp[:, :-1] + c[3] = y[:, :-1] + c = np.moveaxis(c, 1, 0) + + return PPoly(c, x, extrapolate=True, axis=1) + + +def modify_mesh(x, insert_1, insert_2): + """Insert nodes into a mesh. + + Nodes removal logic is not established, its impact on the solver is + presumably negligible. So, only insertion is done in this function. + + Parameters + ---------- + x : ndarray, shape (m,) + Mesh nodes. + insert_1 : ndarray + Intervals to each insert 1 new node in the middle. + insert_2 : ndarray + Intervals to each insert 2 new nodes, such that divide an interval + into 3 equal parts. + + Returns + ------- + x_new : ndarray + New mesh nodes. + + Notes + ----- + `insert_1` and `insert_2` should not have common values. + """ + # Because np.insert implementation apparently varies with a version of + # NumPy, we use a simple and reliable approach with sorting. + return np.sort(np.hstack(( + x, + 0.5 * (x[insert_1] + x[insert_1 + 1]), + (2 * x[insert_2] + x[insert_2 + 1]) / 3, + (x[insert_2] + 2 * x[insert_2 + 1]) / 3 + ))) + + +def wrap_functions(fun, bc, fun_jac, bc_jac, k, a, S, D, dtype): + """Wrap functions for unified usage in the solver.""" + if fun_jac is None: + fun_jac_wrapped = None + + if bc_jac is None: + bc_jac_wrapped = None + + if k == 0: + def fun_p(x, y, _): + return np.asarray(fun(x, y), dtype) + + def bc_wrapped(ya, yb, _): + return np.asarray(bc(ya, yb), dtype) + + if fun_jac is not None: + def fun_jac_p(x, y, _): + return np.asarray(fun_jac(x, y), dtype), None + + if bc_jac is not None: + def bc_jac_wrapped(ya, yb, _): + dbc_dya, dbc_dyb = bc_jac(ya, yb) + return (np.asarray(dbc_dya, dtype), + np.asarray(dbc_dyb, dtype), None) + else: + def fun_p(x, y, p): + return np.asarray(fun(x, y, p), dtype) + + def bc_wrapped(x, y, p): + return np.asarray(bc(x, y, p), dtype) + + if fun_jac is not None: + def fun_jac_p(x, y, p): + df_dy, df_dp = fun_jac(x, y, p) + return np.asarray(df_dy, dtype), np.asarray(df_dp, dtype) + + if bc_jac is not None: + def bc_jac_wrapped(ya, yb, p): + dbc_dya, dbc_dyb, dbc_dp = bc_jac(ya, yb, p) + return (np.asarray(dbc_dya, dtype), np.asarray(dbc_dyb, dtype), + np.asarray(dbc_dp, dtype)) + + if S is None: + fun_wrapped = fun_p + else: + def fun_wrapped(x, y, p): + f = fun_p(x, y, p) + if x[0] == a: + f[:, 0] = np.dot(D, f[:, 0]) + f[:, 1:] += np.dot(S, y[:, 1:]) / (x[1:] - a) + else: + f += np.dot(S, y) / (x - a) + return f + + if fun_jac is not None: + if S is None: + fun_jac_wrapped = fun_jac_p + else: + Sr = S[:, :, np.newaxis] + + def fun_jac_wrapped(x, y, p): + df_dy, df_dp = fun_jac_p(x, y, p) + if x[0] == a: + df_dy[:, :, 0] = np.dot(D, df_dy[:, :, 0]) + df_dy[:, :, 1:] += Sr / (x[1:] - a) + else: + df_dy += Sr / (x - a) + + return df_dy, df_dp + + return fun_wrapped, bc_wrapped, fun_jac_wrapped, bc_jac_wrapped + + +def solve_bvp(fun, bc, x, y, p=None, S=None, fun_jac=None, bc_jac=None, + tol=1e-3, max_nodes=1000, verbose=0, bc_tol=None): + """Solve a boundary value problem for a system of ODEs. + + This function numerically solves a first order system of ODEs subject to + two-point boundary conditions:: + + dy / dx = f(x, y, p) + S * y / (x - a), a <= x <= b + bc(y(a), y(b), p) = 0 + + Here x is a 1-D independent variable, y(x) is an n-D + vector-valued function and p is a k-D vector of unknown + parameters which is to be found along with y(x). For the problem to be + determined, there must be n + k boundary conditions, i.e., bc must be an + (n + k)-D function. + + The last singular term on the right-hand side of the system is optional. + It is defined by an n-by-n matrix S, such that the solution must satisfy + S y(a) = 0. This condition will be forced during iterations, so it must not + contradict boundary conditions. See [2]_ for the explanation how this term + is handled when solving BVPs numerically. + + Problems in a complex domain can be solved as well. In this case, y and p + are considered to be complex, and f and bc are assumed to be complex-valued + functions, but x stays real. Note that f and bc must be complex + differentiable (satisfy Cauchy-Riemann equations [4]_), otherwise you + should rewrite your problem for real and imaginary parts separately. To + solve a problem in a complex domain, pass an initial guess for y with a + complex data type (see below). + + Parameters + ---------- + fun : callable + Right-hand side of the system. The calling signature is ``fun(x, y)``, + or ``fun(x, y, p)`` if parameters are present. All arguments are + ndarray: ``x`` with shape (m,), ``y`` with shape (n, m), meaning that + ``y[:, i]`` corresponds to ``x[i]``, and ``p`` with shape (k,). The + return value must be an array with shape (n, m) and with the same + layout as ``y``. + bc : callable + Function evaluating residuals of the boundary conditions. The calling + signature is ``bc(ya, yb)``, or ``bc(ya, yb, p)`` if parameters are + present. All arguments are ndarray: ``ya`` and ``yb`` with shape (n,), + and ``p`` with shape (k,). The return value must be an array with + shape (n + k,). + x : array_like, shape (m,) + Initial mesh. Must be a strictly increasing sequence of real numbers + with ``x[0]=a`` and ``x[-1]=b``. + y : array_like, shape (n, m) + Initial guess for the function values at the mesh nodes, ith column + corresponds to ``x[i]``. For problems in a complex domain pass `y` + with a complex data type (even if the initial guess is purely real). + p : array_like with shape (k,) or None, optional + Initial guess for the unknown parameters. If None (default), it is + assumed that the problem doesn't depend on any parameters. + S : array_like with shape (n, n) or None + Matrix defining the singular term. If None (default), the problem is + solved without the singular term. + fun_jac : callable or None, optional + Function computing derivatives of f with respect to y and p. The + calling signature is ``fun_jac(x, y)``, or ``fun_jac(x, y, p)`` if + parameters are present. The return must contain 1 or 2 elements in the + following order: + + * df_dy : array_like with shape (n, n, m), where an element + (i, j, q) equals to d f_i(x_q, y_q, p) / d (y_q)_j. + * df_dp : array_like with shape (n, k, m), where an element + (i, j, q) equals to d f_i(x_q, y_q, p) / d p_j. + + Here q numbers nodes at which x and y are defined, whereas i and j + number vector components. If the problem is solved without unknown + parameters, df_dp should not be returned. + + If `fun_jac` is None (default), the derivatives will be estimated + by the forward finite differences. + bc_jac : callable or None, optional + Function computing derivatives of bc with respect to ya, yb, and p. + The calling signature is ``bc_jac(ya, yb)``, or ``bc_jac(ya, yb, p)`` + if parameters are present. The return must contain 2 or 3 elements in + the following order: + + * dbc_dya : array_like with shape (n, n), where an element (i, j) + equals to d bc_i(ya, yb, p) / d ya_j. + * dbc_dyb : array_like with shape (n, n), where an element (i, j) + equals to d bc_i(ya, yb, p) / d yb_j. + * dbc_dp : array_like with shape (n, k), where an element (i, j) + equals to d bc_i(ya, yb, p) / d p_j. + + If the problem is solved without unknown parameters, dbc_dp should not + be returned. + + If `bc_jac` is None (default), the derivatives will be estimated by + the forward finite differences. + tol : float, optional + Desired tolerance of the solution. If we define ``r = y' - f(x, y)``, + where y is the found solution, then the solver tries to achieve on each + mesh interval ``norm(r / (1 + abs(f)) < tol``, where ``norm`` is + estimated in a root mean squared sense (using a numerical quadrature + formula). Default is 1e-3. + max_nodes : int, optional + Maximum allowed number of the mesh nodes. If exceeded, the algorithm + terminates. Default is 1000. + verbose : {0, 1, 2}, optional + Level of algorithm's verbosity: + + * 0 (default) : work silently. + * 1 : display a termination report. + * 2 : display progress during iterations. + bc_tol : float, optional + Desired absolute tolerance for the boundary condition residuals: `bc` + value should satisfy ``abs(bc) < bc_tol`` component-wise. + Equals to `tol` by default. Up to 10 iterations are allowed to achieve this + tolerance. + + Returns + ------- + Bunch object with the following fields defined: + sol : PPoly + Found solution for y as `scipy.interpolate.PPoly` instance, a C1 + continuous cubic spline. + p : ndarray or None, shape (k,) + Found parameters. None, if the parameters were not present in the + problem. + x : ndarray, shape (m,) + Nodes of the final mesh. + y : ndarray, shape (n, m) + Solution values at the mesh nodes. + yp : ndarray, shape (n, m) + Solution derivatives at the mesh nodes. + rms_residuals : ndarray, shape (m - 1,) + RMS values of the relative residuals over each mesh interval (see the + description of `tol` parameter). + niter : int + Number of completed iterations. + status : int + Reason for algorithm termination: + + * 0: The algorithm converged to the desired accuracy. + * 1: The maximum number of mesh nodes is exceeded. + * 2: A singular Jacobian encountered when solving the collocation + system. + + message : string + Verbal description of the termination reason. + success : bool + True if the algorithm converged to the desired accuracy (``status=0``). + + Notes + ----- + This function implements a 4th order collocation algorithm with the + control of residuals similar to [1]_. A collocation system is solved + by a damped Newton method with an affine-invariant criterion function as + described in [3]_. + + Note that in [1]_ integral residuals are defined without normalization + by interval lengths. So, their definition is different by a multiplier of + h**0.5 (h is an interval length) from the definition used here. + + .. versionadded:: 0.18.0 + + References + ---------- + .. [1] J. Kierzenka, L. F. Shampine, "A BVP Solver Based on Residual + Control and the Maltab PSE", ACM Trans. Math. Softw., Vol. 27, + Number 3, pp. 299-316, 2001. + .. [2] L.F. Shampine, P. H. Muir and H. Xu, "A User-Friendly Fortran BVP + Solver". + .. [3] U. Ascher, R. Mattheij and R. Russell "Numerical Solution of + Boundary Value Problems for Ordinary Differential Equations". + .. [4] `Cauchy-Riemann equations + `_ on + Wikipedia. + + Examples + -------- + In the first example, we solve Bratu's problem:: + + y'' + k * exp(y) = 0 + y(0) = y(1) = 0 + + for k = 1. + + We rewrite the equation as a first-order system and implement its + right-hand side evaluation:: + + y1' = y2 + y2' = -exp(y1) + + >>> import numpy as np + >>> def fun(x, y): + ... return np.vstack((y[1], -np.exp(y[0]))) + + Implement evaluation of the boundary condition residuals: + + >>> def bc(ya, yb): + ... return np.array([ya[0], yb[0]]) + + Define the initial mesh with 5 nodes: + + >>> x = np.linspace(0, 1, 5) + + This problem is known to have two solutions. To obtain both of them, we + use two different initial guesses for y. We denote them by subscripts + a and b. + + >>> y_a = np.zeros((2, x.size)) + >>> y_b = np.zeros((2, x.size)) + >>> y_b[0] = 3 + + Now we are ready to run the solver. + + >>> from scipy.integrate import solve_bvp + >>> res_a = solve_bvp(fun, bc, x, y_a) + >>> res_b = solve_bvp(fun, bc, x, y_b) + + Let's plot the two found solutions. We take an advantage of having the + solution in a spline form to produce a smooth plot. + + >>> x_plot = np.linspace(0, 1, 100) + >>> y_plot_a = res_a.sol(x_plot)[0] + >>> y_plot_b = res_b.sol(x_plot)[0] + >>> import matplotlib.pyplot as plt + >>> plt.plot(x_plot, y_plot_a, label='y_a') + >>> plt.plot(x_plot, y_plot_b, label='y_b') + >>> plt.legend() + >>> plt.xlabel("x") + >>> plt.ylabel("y") + >>> plt.show() + + We see that the two solutions have similar shape, but differ in scale + significantly. + + In the second example, we solve a simple Sturm-Liouville problem:: + + y'' + k**2 * y = 0 + y(0) = y(1) = 0 + + It is known that a non-trivial solution y = A * sin(k * x) is possible for + k = pi * n, where n is an integer. To establish the normalization constant + A = 1 we add a boundary condition:: + + y'(0) = k + + Again, we rewrite our equation as a first-order system and implement its + right-hand side evaluation:: + + y1' = y2 + y2' = -k**2 * y1 + + >>> def fun(x, y, p): + ... k = p[0] + ... return np.vstack((y[1], -k**2 * y[0])) + + Note that parameters p are passed as a vector (with one element in our + case). + + Implement the boundary conditions: + + >>> def bc(ya, yb, p): + ... k = p[0] + ... return np.array([ya[0], yb[0], ya[1] - k]) + + Set up the initial mesh and guess for y. We aim to find the solution for + k = 2 * pi, to achieve that we set values of y to approximately follow + sin(2 * pi * x): + + >>> x = np.linspace(0, 1, 5) + >>> y = np.zeros((2, x.size)) + >>> y[0, 1] = 1 + >>> y[0, 3] = -1 + + Run the solver with 6 as an initial guess for k. + + >>> sol = solve_bvp(fun, bc, x, y, p=[6]) + + We see that the found k is approximately correct: + + >>> sol.p[0] + 6.28329460046 + + And, finally, plot the solution to see the anticipated sinusoid: + + >>> x_plot = np.linspace(0, 1, 100) + >>> y_plot = sol.sol(x_plot)[0] + >>> plt.plot(x_plot, y_plot) + >>> plt.xlabel("x") + >>> plt.ylabel("y") + >>> plt.show() + """ + x = np.asarray(x, dtype=float) + if x.ndim != 1: + raise ValueError("`x` must be 1 dimensional.") + h = np.diff(x) + if np.any(h <= 0): + raise ValueError("`x` must be strictly increasing.") + a = x[0] + + y = np.asarray(y) + if np.issubdtype(y.dtype, np.complexfloating): + dtype = complex + else: + dtype = float + y = y.astype(dtype, copy=False) + + if y.ndim != 2: + raise ValueError("`y` must be 2 dimensional.") + if y.shape[1] != x.shape[0]: + raise ValueError(f"`y` is expected to have {x.shape[0]} columns, but actually " + f"has {y.shape[1]}.") + + if p is None: + p = np.array([]) + else: + p = np.asarray(p, dtype=dtype) + if p.ndim != 1: + raise ValueError("`p` must be 1 dimensional.") + + if tol < 100 * EPS: + warn(f"`tol` is too low, setting to {100 * EPS:.2e}", stacklevel=2) + tol = 100 * EPS + + if verbose not in [0, 1, 2]: + raise ValueError("`verbose` must be in [0, 1, 2].") + + n = y.shape[0] + k = p.shape[0] + + if S is not None: + S = np.asarray(S, dtype=dtype) + if S.shape != (n, n): + raise ValueError(f"`S` is expected to have shape {(n, n)}, " + f"but actually has {S.shape}") + + # Compute I - S^+ S to impose necessary boundary conditions. + B = np.identity(n) - np.dot(pinv(S), S) + + y[:, 0] = np.dot(B, y[:, 0]) + + # Compute (I - S)^+ to correct derivatives at x=a. + D = pinv(np.identity(n) - S) + else: + B = None + D = None + + if bc_tol is None: + bc_tol = tol + + # Maximum number of iterations + max_iteration = 10 + + fun_wrapped, bc_wrapped, fun_jac_wrapped, bc_jac_wrapped = wrap_functions( + fun, bc, fun_jac, bc_jac, k, a, S, D, dtype) + + f = fun_wrapped(x, y, p) + if f.shape != y.shape: + raise ValueError(f"`fun` return is expected to have shape {y.shape}, " + f"but actually has {f.shape}.") + + bc_res = bc_wrapped(y[:, 0], y[:, -1], p) + if bc_res.shape != (n + k,): + raise ValueError(f"`bc` return is expected to have shape {(n + k,)}, " + f"but actually has {bc_res.shape}.") + + status = 0 + iteration = 0 + if verbose == 2: + print_iteration_header() + + while True: + m = x.shape[0] + + col_fun, jac_sys = prepare_sys(n, m, k, fun_wrapped, bc_wrapped, + fun_jac_wrapped, bc_jac_wrapped, x, h) + y, p, singular = solve_newton(n, m, h, col_fun, bc_wrapped, jac_sys, + y, p, B, tol, bc_tol) + iteration += 1 + + col_res, y_middle, f, f_middle = collocation_fun(fun_wrapped, y, + p, x, h) + bc_res = bc_wrapped(y[:, 0], y[:, -1], p) + max_bc_res = np.max(abs(bc_res)) + + # This relation is not trivial, but can be verified. + r_middle = 1.5 * col_res / h + sol = create_spline(y, f, x, h) + rms_res = estimate_rms_residuals(fun_wrapped, sol, x, h, p, + r_middle, f_middle) + max_rms_res = np.max(rms_res) + + if singular: + status = 2 + break + + insert_1, = np.nonzero((rms_res > tol) & (rms_res < 100 * tol)) + insert_2, = np.nonzero(rms_res >= 100 * tol) + nodes_added = insert_1.shape[0] + 2 * insert_2.shape[0] + + if m + nodes_added > max_nodes: + status = 1 + if verbose == 2: + nodes_added = f"({nodes_added})" + print_iteration_progress(iteration, max_rms_res, max_bc_res, + m, nodes_added) + break + + if verbose == 2: + print_iteration_progress(iteration, max_rms_res, max_bc_res, m, + nodes_added) + + if nodes_added > 0: + x = modify_mesh(x, insert_1, insert_2) + h = np.diff(x) + y = sol(x) + elif max_bc_res <= bc_tol: + status = 0 + break + elif iteration >= max_iteration: + status = 3 + break + + if verbose > 0: + if status == 0: + print(f"Solved in {iteration} iterations, number of nodes {x.shape[0]}. \n" + f"Maximum relative residual: {max_rms_res:.2e} \n" + f"Maximum boundary residual: {max_bc_res:.2e}") + elif status == 1: + print(f"Number of nodes is exceeded after iteration {iteration}. \n" + f"Maximum relative residual: {max_rms_res:.2e} \n" + f"Maximum boundary residual: {max_bc_res:.2e}") + elif status == 2: + print("Singular Jacobian encountered when solving the collocation " + f"system on iteration {iteration}. \n" + f"Maximum relative residual: {max_rms_res:.2e} \n" + f"Maximum boundary residual: {max_bc_res:.2e}") + elif status == 3: + print("The solver was unable to satisfy boundary conditions " + f"tolerance on iteration {iteration}. \n" + f"Maximum relative residual: {max_rms_res:.2e} \n" + f"Maximum boundary residual: {max_bc_res:.2e}") + + if p.size == 0: + p = None + + return BVPResult(sol=sol, p=p, x=x, y=y, yp=f, rms_residuals=rms_res, + niter=iteration, status=status, + message=TERMINATION_MESSAGES[status], success=status == 0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_cubature.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_cubature.py new file mode 100644 index 0000000000000000000000000000000000000000..3e6d8911d13eeaa2420ef65a12e9b4ba34400ca0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_cubature.py @@ -0,0 +1,728 @@ +import math +import heapq +import itertools + +from dataclasses import dataclass, field +from types import ModuleType +from typing import Any, TypeAlias + +from scipy._lib._array_api import ( + array_namespace, + xp_size, + xp_copy, + xp_broadcast_promote +) +from scipy._lib._util import MapWrapper + +from scipy.integrate._rules import ( + ProductNestedFixed, + GaussKronrodQuadrature, + GenzMalikCubature, +) +from scipy.integrate._rules._base import _split_subregion + +__all__ = ['cubature'] + +Array: TypeAlias = Any # To be changed to an array-api-typing Protocol later + + +@dataclass +class CubatureRegion: + estimate: Array + error: Array + a: Array + b: Array + _xp: ModuleType = field(repr=False) + + def __lt__(self, other): + # Consider regions with higher error estimates as being "less than" regions with + # lower order estimates, so that regions with high error estimates are placed at + # the top of the heap. + + this_err = self._xp.max(self._xp.abs(self.error)) + other_err = self._xp.max(self._xp.abs(other.error)) + + return this_err > other_err + + +@dataclass +class CubatureResult: + estimate: Array + error: Array + status: str + regions: list[CubatureRegion] + subdivisions: int + atol: float + rtol: float + + +def cubature(f, a, b, *, rule="gk21", rtol=1e-8, atol=0, max_subdivisions=10000, + args=(), workers=1, points=None): + r""" + Adaptive cubature of multidimensional array-valued function. + + Given an arbitrary integration rule, this function returns an estimate of the + integral to the requested tolerance over the region defined by the arrays `a` and + `b` specifying the corners of a hypercube. + + Convergence is not guaranteed for all integrals. + + Parameters + ---------- + f : callable + Function to integrate. `f` must have the signature:: + + f(x : ndarray, *args) -> ndarray + + `f` should accept arrays ``x`` of shape:: + + (npoints, ndim) + + and output arrays of shape:: + + (npoints, output_dim_1, ..., output_dim_n) + + In this case, `cubature` will return arrays of shape:: + + (output_dim_1, ..., output_dim_n) + a, b : array_like + Lower and upper limits of integration as 1D arrays specifying the left and right + endpoints of the intervals being integrated over. Limits can be infinite. + rule : str, optional + Rule used to estimate the integral. If passing a string, the options are + "gauss-kronrod" (21 node), or "genz-malik" (degree 7). If a rule like + "gauss-kronrod" is specified for an ``n``-dim integrand, the corresponding + Cartesian product rule is used. "gk21", "gk15" are also supported for + compatibility with `quad_vec`. See Notes. + rtol, atol : float, optional + Relative and absolute tolerances. Iterations are performed until the error is + estimated to be less than ``atol + rtol * abs(est)``. Here `rtol` controls + relative accuracy (number of correct digits), while `atol` controls absolute + accuracy (number of correct decimal places). To achieve the desired `rtol`, set + `atol` to be smaller than the smallest value that can be expected from + ``rtol * abs(y)`` so that `rtol` dominates the allowable error. If `atol` is + larger than ``rtol * abs(y)`` the number of correct digits is not guaranteed. + Conversely, to achieve the desired `atol`, set `rtol` such that + ``rtol * abs(y)`` is always smaller than `atol`. Default values are 1e-8 for + `rtol` and 0 for `atol`. + max_subdivisions : int, optional + Upper bound on the number of subdivisions to perform. Default is 10,000. + args : tuple, optional + Additional positional args passed to `f`, if any. + workers : int or map-like callable, optional + If `workers` is an integer, part of the computation is done in parallel + subdivided to this many tasks (using :class:`python:multiprocessing.pool.Pool`). + Supply `-1` to use all cores available to the Process. Alternatively, supply a + map-like callable, such as :meth:`python:multiprocessing.pool.Pool.map` for + evaluating the population in parallel. This evaluation is carried out as + ``workers(func, iterable)``. + points : list of array_like, optional + List of points to avoid evaluating `f` at, under the condition that the rule + being used does not evaluate `f` on the boundary of a region (which is the + case for all Genz-Malik and Gauss-Kronrod rules). This can be useful if `f` has + a singularity at the specified point. This should be a list of array-likes where + each element has length ``ndim``. Default is empty. See Examples. + + Returns + ------- + res : object + Object containing the results of the estimation. It has the following + attributes: + + estimate : ndarray + Estimate of the value of the integral over the overall region specified. + error : ndarray + Estimate of the error of the approximation over the overall region + specified. + status : str + Whether the estimation was successful. Can be either: "converged", + "not_converged". + subdivisions : int + Number of subdivisions performed. + atol, rtol : float + Requested tolerances for the approximation. + regions: list of object + List of objects containing the estimates of the integral over smaller + regions of the domain. + + Each object in ``regions`` has the following attributes: + + a, b : ndarray + Points describing the corners of the region. If the original integral + contained infinite limits or was over a region described by `region`, + then `a` and `b` are in the transformed coordinates. + estimate : ndarray + Estimate of the value of the integral over this region. + error : ndarray + Estimate of the error of the approximation over this region. + + Notes + ----- + The algorithm uses a similar algorithm to `quad_vec`, which itself is based on the + implementation of QUADPACK's DQAG* algorithms, implementing global error control and + adaptive subdivision. + + The source of the nodes and weights used for Gauss-Kronrod quadrature can be found + in [1]_, and the algorithm for calculating the nodes and weights in Genz-Malik + cubature can be found in [2]_. + + The rules currently supported via the `rule` argument are: + + - ``"gauss-kronrod"``, 21-node Gauss-Kronrod + - ``"genz-malik"``, n-node Genz-Malik + + If using Gauss-Kronrod for an ``n``-dim integrand where ``n > 2``, then the + corresponding Cartesian product rule will be found by taking the Cartesian product + of the nodes in the 1D case. This means that the number of nodes scales + exponentially as ``21^n`` in the Gauss-Kronrod case, which may be problematic in a + moderate number of dimensions. + + Genz-Malik is typically less accurate than Gauss-Kronrod but has much fewer nodes, + so in this situation using "genz-malik" might be preferable. + + Infinite limits are handled with an appropriate variable transformation. Assuming + ``a = [a_1, ..., a_n]`` and ``b = [b_1, ..., b_n]``: + + If :math:`a_i = -\infty` and :math:`b_i = \infty`, the i-th integration variable + will use the transformation :math:`x = \frac{1-|t|}{t}` and :math:`t \in (-1, 1)`. + + If :math:`a_i \ne \pm\infty` and :math:`b_i = \infty`, the i-th integration variable + will use the transformation :math:`x = a_i + \frac{1-t}{t}` and + :math:`t \in (0, 1)`. + + If :math:`a_i = -\infty` and :math:`b_i \ne \pm\infty`, the i-th integration + variable will use the transformation :math:`x = b_i - \frac{1-t}{t}` and + :math:`t \in (0, 1)`. + + References + ---------- + .. [1] R. Piessens, E. de Doncker, Quadpack: A Subroutine Package for Automatic + Integration, files: dqk21.f, dqk15.f (1983). + + .. [2] A.C. Genz, A.A. Malik, Remarks on algorithm 006: An adaptive algorithm for + numerical integration over an N-dimensional rectangular region, Journal of + Computational and Applied Mathematics, Volume 6, Issue 4, 1980, Pages 295-302, + ISSN 0377-0427 + :doi:`10.1016/0771-050X(80)90039-X` + + Examples + -------- + **1D integral with vector output**: + + .. math:: + + \int^1_0 \mathbf f(x) \text dx + + Where ``f(x) = x^n`` and ``n = np.arange(10)`` is a vector. Since no rule is + specified, the default "gk21" is used, which corresponds to Gauss-Kronrod + integration with 21 nodes. + + >>> import numpy as np + >>> from scipy.integrate import cubature + >>> def f(x, n): + ... # Make sure x and n are broadcastable + ... return x[:, np.newaxis]**n[np.newaxis, :] + >>> res = cubature( + ... f, + ... a=[0], + ... b=[1], + ... args=(np.arange(10),), + ... ) + >>> res.estimate + array([1. , 0.5 , 0.33333333, 0.25 , 0.2 , + 0.16666667, 0.14285714, 0.125 , 0.11111111, 0.1 ]) + + **7D integral with arbitrary-shaped array output**:: + + f(x) = cos(2*pi*r + alphas @ x) + + for some ``r`` and ``alphas``, and the integral is performed over the unit + hybercube, :math:`[0, 1]^7`. Since the integral is in a moderate number of + dimensions, "genz-malik" is used rather than the default "gauss-kronrod" to + avoid constructing a product rule with :math:`21^7 \approx 2 \times 10^9` nodes. + + >>> import numpy as np + >>> from scipy.integrate import cubature + >>> def f(x, r, alphas): + ... # f(x) = cos(2*pi*r + alphas @ x) + ... # Need to allow r and alphas to be arbitrary shape + ... npoints, ndim = x.shape[0], x.shape[-1] + ... alphas = alphas[np.newaxis, ...] + ... x = x.reshape(npoints, *([1]*(len(alphas.shape) - 1)), ndim) + ... return np.cos(2*np.pi*r + np.sum(alphas * x, axis=-1)) + >>> rng = np.random.default_rng() + >>> r, alphas = rng.random((2, 3)), rng.random((2, 3, 7)) + >>> res = cubature( + ... f=f, + ... a=np.array([0, 0, 0, 0, 0, 0, 0]), + ... b=np.array([1, 1, 1, 1, 1, 1, 1]), + ... rtol=1e-5, + ... rule="genz-malik", + ... args=(r, alphas), + ... ) + >>> res.estimate + array([[-0.79812452, 0.35246913, -0.52273628], + [ 0.88392779, 0.59139899, 0.41895111]]) + + **Parallel computation with** `workers`: + + >>> from concurrent.futures import ThreadPoolExecutor + >>> with ThreadPoolExecutor() as executor: + ... res = cubature( + ... f=f, + ... a=np.array([0, 0, 0, 0, 0, 0, 0]), + ... b=np.array([1, 1, 1, 1, 1, 1, 1]), + ... rtol=1e-5, + ... rule="genz-malik", + ... args=(r, alphas), + ... workers=executor.map, + ... ) + >>> res.estimate + array([[-0.79812452, 0.35246913, -0.52273628], + [ 0.88392779, 0.59139899, 0.41895111]]) + + **2D integral with infinite limits**: + + .. math:: + + \int^{ \infty }_{ -\infty } + \int^{ \infty }_{ -\infty } + e^{-x^2-y^2} + \text dy + \text dx + + >>> def gaussian(x): + ... return np.exp(-np.sum(x**2, axis=-1)) + >>> res = cubature(gaussian, [-np.inf, -np.inf], [np.inf, np.inf]) + >>> res.estimate + 3.1415926 + + **1D integral with singularities avoided using** `points`: + + .. math:: + + \int^{ 1 }_{ -1 } + \frac{\sin(x)}{x} + \text dx + + It is necessary to use the `points` parameter to avoid evaluating `f` at the origin. + + >>> def sinc(x): + ... return np.sin(x)/x + >>> res = cubature(sinc, [-1], [1], points=[[0]]) + >>> res.estimate + 1.8921661 + """ + + # It is also possible to use a custom rule, but this is not yet part of the public + # API. An example of this can be found in the class scipy.integrate._rules.Rule. + + xp = array_namespace(a, b) + max_subdivisions = float("inf") if max_subdivisions is None else max_subdivisions + points = [] if points is None else points + + # Convert a and b to arrays and convert each point in points to an array, promoting + # each to a common floating dtype. + a, b, *points = xp_broadcast_promote(a, b, *points, force_floating=True) + result_dtype = a.dtype + + if xp_size(a) == 0 or xp_size(b) == 0: + raise ValueError("`a` and `b` must be nonempty") + + if a.ndim != 1 or b.ndim != 1: + raise ValueError("`a` and `b` must be 1D arrays") + + # If the rule is a string, convert to a corresponding product rule + if isinstance(rule, str): + ndim = xp_size(a) + + if rule == "genz-malik": + rule = GenzMalikCubature(ndim, xp=xp) + else: + quadratues = { + "gauss-kronrod": GaussKronrodQuadrature(21, xp=xp), + + # Also allow names quad_vec uses: + "gk21": GaussKronrodQuadrature(21, xp=xp), + "gk15": GaussKronrodQuadrature(15, xp=xp), + } + + base_rule = quadratues.get(rule) + + if base_rule is None: + raise ValueError(f"unknown rule {rule}") + + rule = ProductNestedFixed([base_rule] * ndim) + + # If any of limits are the wrong way around (a > b), flip them and keep track of + # the sign. + sign = (-1) ** xp.sum(xp.astype(a > b, xp.int8), dtype=result_dtype) + + a_flipped = xp.min(xp.stack([a, b]), axis=0) + b_flipped = xp.max(xp.stack([a, b]), axis=0) + + a, b = a_flipped, b_flipped + + # If any of the limits are infinite, apply a transformation + if xp.any(xp.isinf(a)) or xp.any(xp.isinf(b)): + f = _InfiniteLimitsTransform(f, a, b, xp=xp) + a, b = f.transformed_limits + + # Map points from the original coordinates to the new transformed coordinates. + # + # `points` is a list of arrays of shape (ndim,), but transformations are applied + # to arrays of shape (npoints, ndim). + # + # It is not possible to combine all the points into one array and then apply + # f.inv to all of them at once since `points` needs to remain iterable. + # Instead, each point is reshaped to an array of shape (1, ndim), `f.inv` is + # applied, and then each is reshaped back to (ndim,). + points = [xp.reshape(point, (1, -1)) for point in points] + points = [f.inv(point) for point in points] + points = [xp.reshape(point, (-1,)) for point in points] + + # Include any problematic points introduced by the transformation + points.extend(f.points) + + # If any problematic points are specified, divide the initial region so that these + # points lie on the edge of a subregion. + # + # This means ``f`` won't be evaluated there if the rule being used has no evaluation + # points on the boundary. + if len(points) == 0: + initial_regions = [(a, b)] + else: + initial_regions = _split_region_at_points(a, b, points, xp) + + regions = [] + est = 0.0 + err = 0.0 + + for a_k, b_k in initial_regions: + est_k = rule.estimate(f, a_k, b_k, args) + err_k = rule.estimate_error(f, a_k, b_k, args) + regions.append(CubatureRegion(est_k, err_k, a_k, b_k, xp)) + + est += est_k + err += err_k + + subdivisions = 0 + success = True + + with MapWrapper(workers) as mapwrapper: + while xp.any(err > atol + rtol * xp.abs(est)): + # region_k is the region with highest estimated error + region_k = heapq.heappop(regions) + + est_k = region_k.estimate + err_k = region_k.error + + a_k, b_k = region_k.a, region_k.b + + # Subtract the estimate of the integral and its error over this region from + # the current global estimates, since these will be refined in the loop over + # all subregions. + est -= est_k + err -= err_k + + # Find all 2^ndim subregions formed by splitting region_k along each axis, + # e.g. for 1D integrals this splits an estimate over an interval into an + # estimate over two subintervals, for 3D integrals this splits an estimate + # over a cube into 8 subcubes. + # + # For each of the new subregions, calculate an estimate for the integral and + # the error there, and push these regions onto the heap for potential + # further subdividing. + + executor_args = zip( + itertools.repeat(f), + itertools.repeat(rule), + itertools.repeat(args), + _split_subregion(a_k, b_k, xp), + ) + + for subdivision_result in mapwrapper(_process_subregion, executor_args): + a_k_sub, b_k_sub, est_sub, err_sub = subdivision_result + + est += est_sub + err += err_sub + + new_region = CubatureRegion(est_sub, err_sub, a_k_sub, b_k_sub, xp) + + heapq.heappush(regions, new_region) + + subdivisions += 1 + + if subdivisions >= max_subdivisions: + success = False + break + + status = "converged" if success else "not_converged" + + # Apply sign change to handle any limits which were initially flipped. + est = sign * est + + return CubatureResult( + estimate=est, + error=err, + status=status, + subdivisions=subdivisions, + regions=regions, + atol=atol, + rtol=rtol, + ) + + +def _process_subregion(data): + f, rule, args, coord = data + a_k_sub, b_k_sub = coord + + est_sub = rule.estimate(f, a_k_sub, b_k_sub, args) + err_sub = rule.estimate_error(f, a_k_sub, b_k_sub, args) + + return a_k_sub, b_k_sub, est_sub, err_sub + + +def _is_strictly_in_region(a, b, point, xp): + if xp.all(point == a) or xp.all(point == b): + return False + + return xp.all(a <= point) and xp.all(point <= b) + + +def _split_region_at_points(a, b, points, xp): + """ + Given the integration limits `a` and `b` describing a rectangular region and a list + of `points`, find the list of ``[(a_1, b_1), ..., (a_l, b_l)]`` which breaks up the + initial region into smaller subregion such that no `points` lie strictly inside + any of the subregions. + """ + + regions = [(a, b)] + + for point in points: + if xp.any(xp.isinf(point)): + # If a point is specified at infinity, ignore. + # + # This case occurs when points are given by the user to avoid, but after + # applying a transformation, they are removed. + continue + + new_subregions = [] + + for a_k, b_k in regions: + if _is_strictly_in_region(a_k, b_k, point, xp): + subregions = _split_subregion(a_k, b_k, xp, point) + + for left, right in subregions: + # Skip any zero-width regions. + if xp.any(left == right): + continue + else: + new_subregions.append((left, right)) + + new_subregions.extend(subregions) + + else: + new_subregions.append((a_k, b_k)) + + regions = new_subregions + + return regions + + +class _VariableTransform: + """ + A transformation that can be applied to an integral. + """ + + @property + def transformed_limits(self): + """ + New limits of integration after applying the transformation. + """ + + raise NotImplementedError + + @property + def points(self): + """ + Any problematic points introduced by the transformation. + + These should be specified as points where ``_VariableTransform(f)(self, point)`` + would be problematic. + + For example, if the transformation ``x = 1/((1-t)(1+t))`` is applied to a + univariate integral, then points should return ``[ [1], [-1] ]``. + """ + + return [] + + def inv(self, x): + """ + Map points ``x`` to ``t`` such that if ``f`` is the original function and ``g`` + is the function after the transformation is applied, then:: + + f(x) = g(self.inv(x)) + """ + + raise NotImplementedError + + def __call__(self, t, *args, **kwargs): + """ + Apply the transformation to ``f`` and multiply by the Jacobian determinant. + This should be the new integrand after the transformation has been applied so + that the following is satisfied:: + + f_transformed = _VariableTransform(f) + + cubature(f, a, b) == cubature( + f_transformed, + *f_transformed.transformed_limits(a, b), + ) + """ + + raise NotImplementedError + + +class _InfiniteLimitsTransform(_VariableTransform): + r""" + Transformation for handling infinite limits. + + Assuming ``a = [a_1, ..., a_n]`` and ``b = [b_1, ..., b_n]``: + + If :math:`a_i = -\infty` and :math:`b_i = \infty`, the i-th integration variable + will use the transformation :math:`x = \frac{1-|t|}{t}` and :math:`t \in (-1, 1)`. + + If :math:`a_i \ne \pm\infty` and :math:`b_i = \infty`, the i-th integration variable + will use the transformation :math:`x = a_i + \frac{1-t}{t}` and + :math:`t \in (0, 1)`. + + If :math:`a_i = -\infty` and :math:`b_i \ne \pm\infty`, the i-th integration + variable will use the transformation :math:`x = b_i - \frac{1-t}{t}` and + :math:`t \in (0, 1)`. + """ + + def __init__(self, f, a, b, xp): + self._xp = xp + + self._f = f + self._orig_a = a + self._orig_b = b + + # (-oo, oo) will be mapped to (-1, 1). + self._double_inf_pos = (a == -math.inf) & (b == math.inf) + + # (start, oo) will be mapped to (0, 1). + start_inf_mask = (a != -math.inf) & (b == math.inf) + + # (-oo, end) will be mapped to (0, 1). + inf_end_mask = (a == -math.inf) & (b != math.inf) + + # This is handled by making the transformation t = -x and reducing it to + # the other semi-infinite case. + self._semi_inf_pos = start_inf_mask | inf_end_mask + + # Since we flip the limits, we don't need to separately multiply the + # integrand by -1. + self._orig_a[inf_end_mask] = -b[inf_end_mask] + self._orig_b[inf_end_mask] = -a[inf_end_mask] + + self._num_inf = self._xp.sum( + self._xp.astype(self._double_inf_pos | self._semi_inf_pos, self._xp.int64), + ).__int__() + + @property + def transformed_limits(self): + a = xp_copy(self._orig_a) + b = xp_copy(self._orig_b) + + a[self._double_inf_pos] = -1 + b[self._double_inf_pos] = 1 + + a[self._semi_inf_pos] = 0 + b[self._semi_inf_pos] = 1 + + return a, b + + @property + def points(self): + # If there are infinite limits, then the origin becomes a problematic point + # due to a division by zero there. + + # If the function using this class only wraps f when a and b contain infinite + # limits, this condition will always be met (as is the case with cubature). + # + # If a and b do not contain infinite limits but f is still wrapped with this + # class, then without this condition the initial region of integration will + # be split around the origin unnecessarily. + if self._num_inf != 0: + return [self._xp.zeros(self._orig_a.shape)] + else: + return [] + + def inv(self, x): + t = xp_copy(x) + npoints = x.shape[0] + + double_inf_mask = self._xp.tile( + self._double_inf_pos[self._xp.newaxis, :], + (npoints, 1), + ) + + semi_inf_mask = self._xp.tile( + self._semi_inf_pos[self._xp.newaxis, :], + (npoints, 1), + ) + + # If any components of x are 0, then this component will be mapped to infinity + # under the transformation used for doubly-infinite limits. + # + # Handle the zero values and non-zero values separately to avoid division by + # zero. + zero_mask = x[double_inf_mask] == 0 + non_zero_mask = double_inf_mask & ~zero_mask + t[zero_mask] = math.inf + t[non_zero_mask] = 1/(x[non_zero_mask] + self._xp.sign(x[non_zero_mask])) + + start = self._xp.tile(self._orig_a[self._semi_inf_pos], (npoints,)) + t[semi_inf_mask] = 1/(x[semi_inf_mask] - start + 1) + + return t + + def __call__(self, t, *args, **kwargs): + x = xp_copy(t) + npoints = t.shape[0] + + double_inf_mask = self._xp.tile( + self._double_inf_pos[self._xp.newaxis, :], + (npoints, 1), + ) + + semi_inf_mask = self._xp.tile( + self._semi_inf_pos[self._xp.newaxis, :], + (npoints, 1), + ) + + # For (-oo, oo) -> (-1, 1), use the transformation x = (1-|t|)/t. + x[double_inf_mask] = ( + (1 - self._xp.abs(t[double_inf_mask])) / t[double_inf_mask] + ) + + start = self._xp.tile(self._orig_a[self._semi_inf_pos], (npoints,)) + + # For (start, oo) -> (0, 1), use the transformation x = start + (1-t)/t. + x[semi_inf_mask] = start + (1 - t[semi_inf_mask]) / t[semi_inf_mask] + + jacobian_det = 1/self._xp.prod( + self._xp.reshape( + t[semi_inf_mask | double_inf_mask]**2, + (-1, self._num_inf), + ), + axis=-1, + ) + + f_x = self._f(x, *args, **kwargs) + jacobian_det = self._xp.reshape(jacobian_det, (-1, *([1]*(len(f_x.shape) - 1)))) + + return f_x * jacobian_det diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..f3c8aaa36588651ae5e48b58fbb1d443bc71fc77 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/__init__.py @@ -0,0 +1,8 @@ +"""Suite of ODE solvers implemented in Python.""" +from .ivp import solve_ivp +from .rk import RK23, RK45, DOP853 +from .radau import Radau +from .bdf import BDF +from .lsoda import LSODA +from .common import OdeSolution +from .base import DenseOutput, OdeSolver diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/base.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/base.py new file mode 100644 index 0000000000000000000000000000000000000000..46db9a69dfb3e7aee5c150ac6795234cd455dfe5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/base.py @@ -0,0 +1,290 @@ +import numpy as np + + +def check_arguments(fun, y0, support_complex): + """Helper function for checking arguments common to all solvers.""" + y0 = np.asarray(y0) + if np.issubdtype(y0.dtype, np.complexfloating): + if not support_complex: + raise ValueError("`y0` is complex, but the chosen solver does " + "not support integration in a complex domain.") + dtype = complex + else: + dtype = float + y0 = y0.astype(dtype, copy=False) + + if y0.ndim != 1: + raise ValueError("`y0` must be 1-dimensional.") + + if not np.isfinite(y0).all(): + raise ValueError("All components of the initial state `y0` must be finite.") + + def fun_wrapped(t, y): + return np.asarray(fun(t, y), dtype=dtype) + + return fun_wrapped, y0 + + +class OdeSolver: + """Base class for ODE solvers. + + In order to implement a new solver you need to follow the guidelines: + + 1. A constructor must accept parameters presented in the base class + (listed below) along with any other parameters specific to a solver. + 2. A constructor must accept arbitrary extraneous arguments + ``**extraneous``, but warn that these arguments are irrelevant + using `common.warn_extraneous` function. Do not pass these + arguments to the base class. + 3. A solver must implement a private method `_step_impl(self)` which + propagates a solver one step further. It must return tuple + ``(success, message)``, where ``success`` is a boolean indicating + whether a step was successful, and ``message`` is a string + containing description of a failure if a step failed or None + otherwise. + 4. A solver must implement a private method `_dense_output_impl(self)`, + which returns a `DenseOutput` object covering the last successful + step. + 5. A solver must have attributes listed below in Attributes section. + Note that ``t_old`` and ``step_size`` are updated automatically. + 6. Use `fun(self, t, y)` method for the system rhs evaluation, this + way the number of function evaluations (`nfev`) will be tracked + automatically. + 7. For convenience, a base class provides `fun_single(self, t, y)` and + `fun_vectorized(self, t, y)` for evaluating the rhs in + non-vectorized and vectorized fashions respectively (regardless of + how `fun` from the constructor is implemented). These calls don't + increment `nfev`. + 8. If a solver uses a Jacobian matrix and LU decompositions, it should + track the number of Jacobian evaluations (`njev`) and the number of + LU decompositions (`nlu`). + 9. By convention, the function evaluations used to compute a finite + difference approximation of the Jacobian should not be counted in + `nfev`, thus use `fun_single(self, t, y)` or + `fun_vectorized(self, t, y)` when computing a finite difference + approximation of the Jacobian. + + Parameters + ---------- + fun : callable + Right-hand side of the system: the time derivative of the state ``y`` + at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a + scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must + return an array of the same shape as ``y``. See `vectorized` for more + information. + t0 : float + Initial time. + y0 : array_like, shape (n,) + Initial state. + t_bound : float + Boundary time --- the integration won't continue beyond it. It also + determines the direction of the integration. + vectorized : bool + Whether `fun` can be called in a vectorized fashion. Default is False. + + If ``vectorized`` is False, `fun` will always be called with ``y`` of + shape ``(n,)``, where ``n = len(y0)``. + + If ``vectorized`` is True, `fun` may be called with ``y`` of shape + ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave + such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of + the returned array is the time derivative of the state corresponding + with a column of ``y``). + + Setting ``vectorized=True`` allows for faster finite difference + approximation of the Jacobian by methods 'Radau' and 'BDF', but + will result in slower execution for other methods. It can also + result in slower overall execution for 'Radau' and 'BDF' in some + circumstances (e.g. small ``len(y0)``). + support_complex : bool, optional + Whether integration in a complex domain should be supported. + Generally determined by a derived solver class capabilities. + Default is False. + + Attributes + ---------- + n : int + Number of equations. + status : string + Current status of the solver: 'running', 'finished' or 'failed'. + t_bound : float + Boundary time. + direction : float + Integration direction: +1 or -1. + t : float + Current time. + y : ndarray + Current state. + t_old : float + Previous time. None if no steps were made yet. + step_size : float + Size of the last successful step. None if no steps were made yet. + nfev : int + Number of the system's rhs evaluations. + njev : int + Number of the Jacobian evaluations. + nlu : int + Number of LU decompositions. + """ + TOO_SMALL_STEP = "Required step size is less than spacing between numbers." + + def __init__(self, fun, t0, y0, t_bound, vectorized, + support_complex=False): + self.t_old = None + self.t = t0 + self._fun, self.y = check_arguments(fun, y0, support_complex) + self.t_bound = t_bound + self.vectorized = vectorized + + if vectorized: + def fun_single(t, y): + return self._fun(t, y[:, None]).ravel() + fun_vectorized = self._fun + else: + fun_single = self._fun + + def fun_vectorized(t, y): + f = np.empty_like(y) + for i, yi in enumerate(y.T): + f[:, i] = self._fun(t, yi) + return f + + def fun(t, y): + self.nfev += 1 + return self.fun_single(t, y) + + self.fun = fun + self.fun_single = fun_single + self.fun_vectorized = fun_vectorized + + self.direction = np.sign(t_bound - t0) if t_bound != t0 else 1 + self.n = self.y.size + self.status = 'running' + + self.nfev = 0 + self.njev = 0 + self.nlu = 0 + + @property + def step_size(self): + if self.t_old is None: + return None + else: + return np.abs(self.t - self.t_old) + + def step(self): + """Perform one integration step. + + Returns + ------- + message : string or None + Report from the solver. Typically a reason for a failure if + `self.status` is 'failed' after the step was taken or None + otherwise. + """ + if self.status != 'running': + raise RuntimeError("Attempt to step on a failed or finished " + "solver.") + + if self.n == 0 or self.t == self.t_bound: + # Handle corner cases of empty solver or no integration. + self.t_old = self.t + self.t = self.t_bound + message = None + self.status = 'finished' + else: + t = self.t + success, message = self._step_impl() + + if not success: + self.status = 'failed' + else: + self.t_old = t + if self.direction * (self.t - self.t_bound) >= 0: + self.status = 'finished' + + return message + + def dense_output(self): + """Compute a local interpolant over the last successful step. + + Returns + ------- + sol : `DenseOutput` + Local interpolant over the last successful step. + """ + if self.t_old is None: + raise RuntimeError("Dense output is available after a successful " + "step was made.") + + if self.n == 0 or self.t == self.t_old: + # Handle corner cases of empty solver and no integration. + return ConstantDenseOutput(self.t_old, self.t, self.y) + else: + return self._dense_output_impl() + + def _step_impl(self): + raise NotImplementedError + + def _dense_output_impl(self): + raise NotImplementedError + + +class DenseOutput: + """Base class for local interpolant over step made by an ODE solver. + + It interpolates between `t_min` and `t_max` (see Attributes below). + Evaluation outside this interval is not forbidden, but the accuracy is not + guaranteed. + + Attributes + ---------- + t_min, t_max : float + Time range of the interpolation. + """ + def __init__(self, t_old, t): + self.t_old = t_old + self.t = t + self.t_min = min(t, t_old) + self.t_max = max(t, t_old) + + def __call__(self, t): + """Evaluate the interpolant. + + Parameters + ---------- + t : float or array_like with shape (n_points,) + Points to evaluate the solution at. + + Returns + ------- + y : ndarray, shape (n,) or (n, n_points) + Computed values. Shape depends on whether `t` was a scalar or a + 1-D array. + """ + t = np.asarray(t) + if t.ndim > 1: + raise ValueError("`t` must be a float or a 1-D array.") + return self._call_impl(t) + + def _call_impl(self, t): + raise NotImplementedError + + +class ConstantDenseOutput(DenseOutput): + """Constant value interpolator. + + This class used for degenerate integration cases: equal integration limits + or a system with 0 equations. + """ + def __init__(self, t_old, t, value): + super().__init__(t_old, t) + self.value = value + + def _call_impl(self, t): + if t.ndim == 0: + return self.value + else: + ret = np.empty((self.value.shape[0], t.shape[0])) + ret[:] = self.value[:, None] + return ret diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/bdf.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/bdf.py new file mode 100644 index 0000000000000000000000000000000000000000..33b47a642b976e623edc9047f6465e328095dcd2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/bdf.py @@ -0,0 +1,478 @@ +import numpy as np +from scipy.linalg import lu_factor, lu_solve +from scipy.sparse import issparse, csc_matrix, eye +from scipy.sparse.linalg import splu +from scipy.optimize._numdiff import group_columns +from .common import (validate_max_step, validate_tol, select_initial_step, + norm, EPS, num_jac, validate_first_step, + warn_extraneous) +from .base import OdeSolver, DenseOutput + + +MAX_ORDER = 5 +NEWTON_MAXITER = 4 +MIN_FACTOR = 0.2 +MAX_FACTOR = 10 + + +def compute_R(order, factor): + """Compute the matrix for changing the differences array.""" + I = np.arange(1, order + 1)[:, None] + J = np.arange(1, order + 1) + M = np.zeros((order + 1, order + 1)) + M[1:, 1:] = (I - 1 - factor * J) / I + M[0] = 1 + return np.cumprod(M, axis=0) + + +def change_D(D, order, factor): + """Change differences array in-place when step size is changed.""" + R = compute_R(order, factor) + U = compute_R(order, 1) + RU = R.dot(U) + D[:order + 1] = np.dot(RU.T, D[:order + 1]) + + +def solve_bdf_system(fun, t_new, y_predict, c, psi, LU, solve_lu, scale, tol): + """Solve the algebraic system resulting from BDF method.""" + d = 0 + y = y_predict.copy() + dy_norm_old = None + converged = False + for k in range(NEWTON_MAXITER): + f = fun(t_new, y) + if not np.all(np.isfinite(f)): + break + + dy = solve_lu(LU, c * f - psi - d) + dy_norm = norm(dy / scale) + + if dy_norm_old is None: + rate = None + else: + rate = dy_norm / dy_norm_old + + if (rate is not None and (rate >= 1 or + rate ** (NEWTON_MAXITER - k) / (1 - rate) * dy_norm > tol)): + break + + y += dy + d += dy + + if (dy_norm == 0 or + rate is not None and rate / (1 - rate) * dy_norm < tol): + converged = True + break + + dy_norm_old = dy_norm + + return converged, k + 1, y, d + + +class BDF(OdeSolver): + """Implicit method based on backward-differentiation formulas. + + This is a variable order method with the order varying automatically from + 1 to 5. The general framework of the BDF algorithm is described in [1]_. + This class implements a quasi-constant step size as explained in [2]_. + The error estimation strategy for the constant-step BDF is derived in [3]_. + An accuracy enhancement using modified formulas (NDF) [2]_ is also implemented. + + Can be applied in the complex domain. + + Parameters + ---------- + fun : callable + Right-hand side of the system: the time derivative of the state ``y`` + at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a + scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must + return an array of the same shape as ``y``. See `vectorized` for more + information. + t0 : float + Initial time. + y0 : array_like, shape (n,) + Initial state. + t_bound : float + Boundary time - the integration won't continue beyond it. It also + determines the direction of the integration. + first_step : float or None, optional + Initial step size. Default is ``None`` which means that the algorithm + should choose. + max_step : float, optional + Maximum allowed step size. Default is np.inf, i.e., the step size is not + bounded and determined solely by the solver. + rtol, atol : float and array_like, optional + Relative and absolute tolerances. The solver keeps the local error + estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a + relative accuracy (number of correct digits), while `atol` controls + absolute accuracy (number of correct decimal places). To achieve the + desired `rtol`, set `atol` to be smaller than the smallest value that + can be expected from ``rtol * abs(y)`` so that `rtol` dominates the + allowable error. If `atol` is larger than ``rtol * abs(y)`` the + number of correct digits is not guaranteed. Conversely, to achieve the + desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller + than `atol`. If components of y have different scales, it might be + beneficial to set different `atol` values for different components by + passing array_like with shape (n,) for `atol`. Default values are + 1e-3 for `rtol` and 1e-6 for `atol`. + jac : {None, array_like, sparse_matrix, callable}, optional + Jacobian matrix of the right-hand side of the system with respect to y, + required by this method. The Jacobian matrix has shape (n, n) and its + element (i, j) is equal to ``d f_i / d y_j``. + There are three ways to define the Jacobian: + + * If array_like or sparse_matrix, the Jacobian is assumed to + be constant. + * If callable, the Jacobian is assumed to depend on both + t and y; it will be called as ``jac(t, y)`` as necessary. + For the 'Radau' and 'BDF' methods, the return value might be a + sparse matrix. + * If None (default), the Jacobian will be approximated by + finite differences. + + It is generally recommended to provide the Jacobian rather than + relying on a finite-difference approximation. + jac_sparsity : {None, array_like, sparse matrix}, optional + Defines a sparsity structure of the Jacobian matrix for a + finite-difference approximation. Its shape must be (n, n). This argument + is ignored if `jac` is not `None`. If the Jacobian has only few non-zero + elements in *each* row, providing the sparsity structure will greatly + speed up the computations [4]_. A zero entry means that a corresponding + element in the Jacobian is always zero. If None (default), the Jacobian + is assumed to be dense. + vectorized : bool, optional + Whether `fun` can be called in a vectorized fashion. Default is False. + + If ``vectorized`` is False, `fun` will always be called with ``y`` of + shape ``(n,)``, where ``n = len(y0)``. + + If ``vectorized`` is True, `fun` may be called with ``y`` of shape + ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave + such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of + the returned array is the time derivative of the state corresponding + with a column of ``y``). + + Setting ``vectorized=True`` allows for faster finite difference + approximation of the Jacobian by this method, but may result in slower + execution overall in some circumstances (e.g. small ``len(y0)``). + + Attributes + ---------- + n : int + Number of equations. + status : string + Current status of the solver: 'running', 'finished' or 'failed'. + t_bound : float + Boundary time. + direction : float + Integration direction: +1 or -1. + t : float + Current time. + y : ndarray + Current state. + t_old : float + Previous time. None if no steps were made yet. + step_size : float + Size of the last successful step. None if no steps were made yet. + nfev : int + Number of evaluations of the right-hand side. + njev : int + Number of evaluations of the Jacobian. + nlu : int + Number of LU decompositions. + + References + ---------- + .. [1] G. D. Byrne, A. C. Hindmarsh, "A Polyalgorithm for the Numerical + Solution of Ordinary Differential Equations", ACM Transactions on + Mathematical Software, Vol. 1, No. 1, pp. 71-96, March 1975. + .. [2] L. F. Shampine, M. W. Reichelt, "THE MATLAB ODE SUITE", SIAM J. SCI. + COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997. + .. [3] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations I: + Nonstiff Problems", Sec. III.2. + .. [4] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of + sparse Jacobian matrices", Journal of the Institute of Mathematics + and its Applications, 13, pp. 117-120, 1974. + """ + def __init__(self, fun, t0, y0, t_bound, max_step=np.inf, + rtol=1e-3, atol=1e-6, jac=None, jac_sparsity=None, + vectorized=False, first_step=None, **extraneous): + warn_extraneous(extraneous) + super().__init__(fun, t0, y0, t_bound, vectorized, + support_complex=True) + self.max_step = validate_max_step(max_step) + self.rtol, self.atol = validate_tol(rtol, atol, self.n) + f = self.fun(self.t, self.y) + if first_step is None: + self.h_abs = select_initial_step(self.fun, self.t, self.y, + t_bound, max_step, f, + self.direction, 1, + self.rtol, self.atol) + else: + self.h_abs = validate_first_step(first_step, t0, t_bound) + self.h_abs_old = None + self.error_norm_old = None + + self.newton_tol = max(10 * EPS / rtol, min(0.03, rtol ** 0.5)) + + self.jac_factor = None + self.jac, self.J = self._validate_jac(jac, jac_sparsity) + if issparse(self.J): + def lu(A): + self.nlu += 1 + return splu(A) + + def solve_lu(LU, b): + return LU.solve(b) + + I = eye(self.n, format='csc', dtype=self.y.dtype) + else: + def lu(A): + self.nlu += 1 + return lu_factor(A, overwrite_a=True) + + def solve_lu(LU, b): + return lu_solve(LU, b, overwrite_b=True) + + I = np.identity(self.n, dtype=self.y.dtype) + + self.lu = lu + self.solve_lu = solve_lu + self.I = I + + kappa = np.array([0, -0.1850, -1/9, -0.0823, -0.0415, 0]) + self.gamma = np.hstack((0, np.cumsum(1 / np.arange(1, MAX_ORDER + 1)))) + self.alpha = (1 - kappa) * self.gamma + self.error_const = kappa * self.gamma + 1 / np.arange(1, MAX_ORDER + 2) + + D = np.empty((MAX_ORDER + 3, self.n), dtype=self.y.dtype) + D[0] = self.y + D[1] = f * self.h_abs * self.direction + self.D = D + + self.order = 1 + self.n_equal_steps = 0 + self.LU = None + + def _validate_jac(self, jac, sparsity): + t0 = self.t + y0 = self.y + + if jac is None: + if sparsity is not None: + if issparse(sparsity): + sparsity = csc_matrix(sparsity) + groups = group_columns(sparsity) + sparsity = (sparsity, groups) + + def jac_wrapped(t, y): + self.njev += 1 + f = self.fun_single(t, y) + J, self.jac_factor = num_jac(self.fun_vectorized, t, y, f, + self.atol, self.jac_factor, + sparsity) + return J + J = jac_wrapped(t0, y0) + elif callable(jac): + J = jac(t0, y0) + self.njev += 1 + if issparse(J): + J = csc_matrix(J, dtype=y0.dtype) + + def jac_wrapped(t, y): + self.njev += 1 + return csc_matrix(jac(t, y), dtype=y0.dtype) + else: + J = np.asarray(J, dtype=y0.dtype) + + def jac_wrapped(t, y): + self.njev += 1 + return np.asarray(jac(t, y), dtype=y0.dtype) + + if J.shape != (self.n, self.n): + raise ValueError(f"`jac` is expected to have shape {(self.n, self.n)}," + f" but actually has {J.shape}.") + else: + if issparse(jac): + J = csc_matrix(jac, dtype=y0.dtype) + else: + J = np.asarray(jac, dtype=y0.dtype) + + if J.shape != (self.n, self.n): + raise ValueError(f"`jac` is expected to have shape {(self.n, self.n)}," + f" but actually has {J.shape}.") + jac_wrapped = None + + return jac_wrapped, J + + def _step_impl(self): + t = self.t + D = self.D + + max_step = self.max_step + min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t) + if self.h_abs > max_step: + h_abs = max_step + change_D(D, self.order, max_step / self.h_abs) + self.n_equal_steps = 0 + elif self.h_abs < min_step: + h_abs = min_step + change_D(D, self.order, min_step / self.h_abs) + self.n_equal_steps = 0 + else: + h_abs = self.h_abs + + atol = self.atol + rtol = self.rtol + order = self.order + + alpha = self.alpha + gamma = self.gamma + error_const = self.error_const + + J = self.J + LU = self.LU + current_jac = self.jac is None + + step_accepted = False + while not step_accepted: + if h_abs < min_step: + return False, self.TOO_SMALL_STEP + + h = h_abs * self.direction + t_new = t + h + + if self.direction * (t_new - self.t_bound) > 0: + t_new = self.t_bound + change_D(D, order, np.abs(t_new - t) / h_abs) + self.n_equal_steps = 0 + LU = None + + h = t_new - t + h_abs = np.abs(h) + + y_predict = np.sum(D[:order + 1], axis=0) + + scale = atol + rtol * np.abs(y_predict) + psi = np.dot(D[1: order + 1].T, gamma[1: order + 1]) / alpha[order] + + converged = False + c = h / alpha[order] + while not converged: + if LU is None: + LU = self.lu(self.I - c * J) + + converged, n_iter, y_new, d = solve_bdf_system( + self.fun, t_new, y_predict, c, psi, LU, self.solve_lu, + scale, self.newton_tol) + + if not converged: + if current_jac: + break + J = self.jac(t_new, y_predict) + LU = None + current_jac = True + + if not converged: + factor = 0.5 + h_abs *= factor + change_D(D, order, factor) + self.n_equal_steps = 0 + LU = None + continue + + safety = 0.9 * (2 * NEWTON_MAXITER + 1) / (2 * NEWTON_MAXITER + + n_iter) + + scale = atol + rtol * np.abs(y_new) + error = error_const[order] * d + error_norm = norm(error / scale) + + if error_norm > 1: + factor = max(MIN_FACTOR, + safety * error_norm ** (-1 / (order + 1))) + h_abs *= factor + change_D(D, order, factor) + self.n_equal_steps = 0 + # As we didn't have problems with convergence, we don't + # reset LU here. + else: + step_accepted = True + + self.n_equal_steps += 1 + + self.t = t_new + self.y = y_new + + self.h_abs = h_abs + self.J = J + self.LU = LU + + # Update differences. The principal relation here is + # D^{j + 1} y_n = D^{j} y_n - D^{j} y_{n - 1}. Keep in mind that D + # contained difference for previous interpolating polynomial and + # d = D^{k + 1} y_n. Thus this elegant code follows. + D[order + 2] = d - D[order + 1] + D[order + 1] = d + for i in reversed(range(order + 1)): + D[i] += D[i + 1] + + if self.n_equal_steps < order + 1: + return True, None + + if order > 1: + error_m = error_const[order - 1] * D[order] + error_m_norm = norm(error_m / scale) + else: + error_m_norm = np.inf + + if order < MAX_ORDER: + error_p = error_const[order + 1] * D[order + 2] + error_p_norm = norm(error_p / scale) + else: + error_p_norm = np.inf + + error_norms = np.array([error_m_norm, error_norm, error_p_norm]) + with np.errstate(divide='ignore'): + factors = error_norms ** (-1 / np.arange(order, order + 3)) + + delta_order = np.argmax(factors) - 1 + order += delta_order + self.order = order + + factor = min(MAX_FACTOR, safety * np.max(factors)) + self.h_abs *= factor + change_D(D, order, factor) + self.n_equal_steps = 0 + self.LU = None + + return True, None + + def _dense_output_impl(self): + return BdfDenseOutput(self.t_old, self.t, self.h_abs * self.direction, + self.order, self.D[:self.order + 1].copy()) + + +class BdfDenseOutput(DenseOutput): + def __init__(self, t_old, t, h, order, D): + super().__init__(t_old, t) + self.order = order + self.t_shift = self.t - h * np.arange(self.order) + self.denom = h * (1 + np.arange(self.order)) + self.D = D + + def _call_impl(self, t): + if t.ndim == 0: + x = (t - self.t_shift) / self.denom + p = np.cumprod(x) + else: + x = (t - self.t_shift[:, None]) / self.denom[:, None] + p = np.cumprod(x, axis=0) + + y = np.dot(self.D[1:].T, p) + if y.ndim == 1: + y += self.D[0] + else: + y += self.D[0, :, None] + + return y diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/common.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/common.py new file mode 100644 index 0000000000000000000000000000000000000000..0c820ad97f5a26955e20f98d80b71168dac54b0a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/common.py @@ -0,0 +1,451 @@ +from itertools import groupby +from warnings import warn +import numpy as np +from scipy.sparse import find, coo_matrix + + +EPS = np.finfo(float).eps + + +def validate_first_step(first_step, t0, t_bound): + """Assert that first_step is valid and return it.""" + if first_step <= 0: + raise ValueError("`first_step` must be positive.") + if first_step > np.abs(t_bound - t0): + raise ValueError("`first_step` exceeds bounds.") + return first_step + + +def validate_max_step(max_step): + """Assert that max_Step is valid and return it.""" + if max_step <= 0: + raise ValueError("`max_step` must be positive.") + return max_step + + +def warn_extraneous(extraneous): + """Display a warning for extraneous keyword arguments. + + The initializer of each solver class is expected to collect keyword + arguments that it doesn't understand and warn about them. This function + prints a warning for each key in the supplied dictionary. + + Parameters + ---------- + extraneous : dict + Extraneous keyword arguments + """ + if extraneous: + warn("The following arguments have no effect for a chosen solver: " + f"{', '.join(f'`{x}`' for x in extraneous)}.", + stacklevel=3) + + +def validate_tol(rtol, atol, n): + """Validate tolerance values.""" + + if np.any(rtol < 100 * EPS): + warn("At least one element of `rtol` is too small. " + f"Setting `rtol = np.maximum(rtol, {100 * EPS})`.", + stacklevel=3) + rtol = np.maximum(rtol, 100 * EPS) + + atol = np.asarray(atol) + if atol.ndim > 0 and atol.shape != (n,): + raise ValueError("`atol` has wrong shape.") + + if np.any(atol < 0): + raise ValueError("`atol` must be positive.") + + return rtol, atol + + +def norm(x): + """Compute RMS norm.""" + return np.linalg.norm(x) / x.size ** 0.5 + + +def select_initial_step(fun, t0, y0, t_bound, + max_step, f0, direction, order, rtol, atol): + """Empirically select a good initial step. + + The algorithm is described in [1]_. + + Parameters + ---------- + fun : callable + Right-hand side of the system. + t0 : float + Initial value of the independent variable. + y0 : ndarray, shape (n,) + Initial value of the dependent variable. + t_bound : float + End-point of integration interval; used to ensure that t0+step<=tbound + and that fun is only evaluated in the interval [t0,tbound] + max_step : float + Maximum allowable step size. + f0 : ndarray, shape (n,) + Initial value of the derivative, i.e., ``fun(t0, y0)``. + direction : float + Integration direction. + order : float + Error estimator order. It means that the error controlled by the + algorithm is proportional to ``step_size ** (order + 1)`. + rtol : float + Desired relative tolerance. + atol : float + Desired absolute tolerance. + + Returns + ------- + h_abs : float + Absolute value of the suggested initial step. + + References + ---------- + .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential + Equations I: Nonstiff Problems", Sec. II.4. + """ + if y0.size == 0: + return np.inf + + interval_length = abs(t_bound - t0) + if interval_length == 0.0: + return 0.0 + + scale = atol + np.abs(y0) * rtol + d0 = norm(y0 / scale) + d1 = norm(f0 / scale) + if d0 < 1e-5 or d1 < 1e-5: + h0 = 1e-6 + else: + h0 = 0.01 * d0 / d1 + # Check t0+h0*direction doesn't take us beyond t_bound + h0 = min(h0, interval_length) + y1 = y0 + h0 * direction * f0 + f1 = fun(t0 + h0 * direction, y1) + d2 = norm((f1 - f0) / scale) / h0 + + if d1 <= 1e-15 and d2 <= 1e-15: + h1 = max(1e-6, h0 * 1e-3) + else: + h1 = (0.01 / max(d1, d2)) ** (1 / (order + 1)) + + return min(100 * h0, h1, interval_length, max_step) + + +class OdeSolution: + """Continuous ODE solution. + + It is organized as a collection of `DenseOutput` objects which represent + local interpolants. It provides an algorithm to select a right interpolant + for each given point. + + The interpolants cover the range between `t_min` and `t_max` (see + Attributes below). Evaluation outside this interval is not forbidden, but + the accuracy is not guaranteed. + + When evaluating at a breakpoint (one of the values in `ts`) a segment with + the lower index is selected. + + Parameters + ---------- + ts : array_like, shape (n_segments + 1,) + Time instants between which local interpolants are defined. Must + be strictly increasing or decreasing (zero segment with two points is + also allowed). + interpolants : list of DenseOutput with n_segments elements + Local interpolants. An i-th interpolant is assumed to be defined + between ``ts[i]`` and ``ts[i + 1]``. + alt_segment : boolean + Requests the alternative interpolant segment selection scheme. At each + solver integration point, two interpolant segments are available. The + default (False) and alternative (True) behaviours select the segment + for which the requested time corresponded to ``t`` and ``t_old``, + respectively. This functionality is only relevant for testing the + interpolants' accuracy: different integrators use different + construction strategies. + + Attributes + ---------- + t_min, t_max : float + Time range of the interpolation. + """ + def __init__(self, ts, interpolants, alt_segment=False): + ts = np.asarray(ts) + d = np.diff(ts) + # The first case covers integration on zero segment. + if not ((ts.size == 2 and ts[0] == ts[-1]) + or np.all(d > 0) or np.all(d < 0)): + raise ValueError("`ts` must be strictly increasing or decreasing.") + + self.n_segments = len(interpolants) + if ts.shape != (self.n_segments + 1,): + raise ValueError("Numbers of time stamps and interpolants " + "don't match.") + + self.ts = ts + self.interpolants = interpolants + if ts[-1] >= ts[0]: + self.t_min = ts[0] + self.t_max = ts[-1] + self.ascending = True + self.side = "right" if alt_segment else "left" + self.ts_sorted = ts + else: + self.t_min = ts[-1] + self.t_max = ts[0] + self.ascending = False + self.side = "left" if alt_segment else "right" + self.ts_sorted = ts[::-1] + + def _call_single(self, t): + # Here we preserve a certain symmetry that when t is in self.ts, + # if alt_segment=False, then we prioritize a segment with a lower + # index. + ind = np.searchsorted(self.ts_sorted, t, side=self.side) + + segment = min(max(ind - 1, 0), self.n_segments - 1) + if not self.ascending: + segment = self.n_segments - 1 - segment + + return self.interpolants[segment](t) + + def __call__(self, t): + """Evaluate the solution. + + Parameters + ---------- + t : float or array_like with shape (n_points,) + Points to evaluate at. + + Returns + ------- + y : ndarray, shape (n_states,) or (n_states, n_points) + Computed values. Shape depends on whether `t` is a scalar or a + 1-D array. + """ + t = np.asarray(t) + + if t.ndim == 0: + return self._call_single(t) + + order = np.argsort(t) + reverse = np.empty_like(order) + reverse[order] = np.arange(order.shape[0]) + t_sorted = t[order] + + # See comment in self._call_single. + segments = np.searchsorted(self.ts_sorted, t_sorted, side=self.side) + segments -= 1 + segments[segments < 0] = 0 + segments[segments > self.n_segments - 1] = self.n_segments - 1 + if not self.ascending: + segments = self.n_segments - 1 - segments + + ys = [] + group_start = 0 + for segment, group in groupby(segments): + group_end = group_start + len(list(group)) + y = self.interpolants[segment](t_sorted[group_start:group_end]) + ys.append(y) + group_start = group_end + + ys = np.hstack(ys) + ys = ys[:, reverse] + + return ys + + +NUM_JAC_DIFF_REJECT = EPS ** 0.875 +NUM_JAC_DIFF_SMALL = EPS ** 0.75 +NUM_JAC_DIFF_BIG = EPS ** 0.25 +NUM_JAC_MIN_FACTOR = 1e3 * EPS +NUM_JAC_FACTOR_INCREASE = 10 +NUM_JAC_FACTOR_DECREASE = 0.1 + + +def num_jac(fun, t, y, f, threshold, factor, sparsity=None): + """Finite differences Jacobian approximation tailored for ODE solvers. + + This function computes finite difference approximation to the Jacobian + matrix of `fun` with respect to `y` using forward differences. + The Jacobian matrix has shape (n, n) and its element (i, j) is equal to + ``d f_i / d y_j``. + + A special feature of this function is the ability to correct the step + size from iteration to iteration. The main idea is to keep the finite + difference significantly separated from its round-off error which + approximately equals ``EPS * np.abs(f)``. It reduces a possibility of a + huge error and assures that the estimated derivative are reasonably close + to the true values (i.e., the finite difference approximation is at least + qualitatively reflects the structure of the true Jacobian). + + Parameters + ---------- + fun : callable + Right-hand side of the system implemented in a vectorized fashion. + t : float + Current time. + y : ndarray, shape (n,) + Current state. + f : ndarray, shape (n,) + Value of the right hand side at (t, y). + threshold : float + Threshold for `y` value used for computing the step size as + ``factor * np.maximum(np.abs(y), threshold)``. Typically, the value of + absolute tolerance (atol) for a solver should be passed as `threshold`. + factor : ndarray with shape (n,) or None + Factor to use for computing the step size. Pass None for the very + evaluation, then use the value returned from this function. + sparsity : tuple (structure, groups) or None + Sparsity structure of the Jacobian, `structure` must be csc_matrix. + + Returns + ------- + J : ndarray or csc_matrix, shape (n, n) + Jacobian matrix. + factor : ndarray, shape (n,) + Suggested `factor` for the next evaluation. + """ + y = np.asarray(y) + n = y.shape[0] + if n == 0: + return np.empty((0, 0)), factor + + if factor is None: + factor = np.full(n, EPS ** 0.5) + else: + factor = factor.copy() + + # Direct the step as ODE dictates, hoping that such a step won't lead to + # a problematic region. For complex ODEs it makes sense to use the real + # part of f as we use steps along real axis. + f_sign = 2 * (np.real(f) >= 0).astype(float) - 1 + y_scale = f_sign * np.maximum(threshold, np.abs(y)) + h = (y + factor * y_scale) - y + + # Make sure that the step is not 0 to start with. Not likely it will be + # executed often. + for i in np.nonzero(h == 0)[0]: + while h[i] == 0: + factor[i] *= 10 + h[i] = (y[i] + factor[i] * y_scale[i]) - y[i] + + if sparsity is None: + return _dense_num_jac(fun, t, y, f, h, factor, y_scale) + else: + structure, groups = sparsity + return _sparse_num_jac(fun, t, y, f, h, factor, y_scale, + structure, groups) + + +def _dense_num_jac(fun, t, y, f, h, factor, y_scale): + n = y.shape[0] + h_vecs = np.diag(h) + f_new = fun(t, y[:, None] + h_vecs) + diff = f_new - f[:, None] + max_ind = np.argmax(np.abs(diff), axis=0) + r = np.arange(n) + max_diff = np.abs(diff[max_ind, r]) + scale = np.maximum(np.abs(f[max_ind]), np.abs(f_new[max_ind, r])) + + diff_too_small = max_diff < NUM_JAC_DIFF_REJECT * scale + if np.any(diff_too_small): + ind, = np.nonzero(diff_too_small) + new_factor = NUM_JAC_FACTOR_INCREASE * factor[ind] + h_new = (y[ind] + new_factor * y_scale[ind]) - y[ind] + h_vecs[ind, ind] = h_new + f_new = fun(t, y[:, None] + h_vecs[:, ind]) + diff_new = f_new - f[:, None] + max_ind = np.argmax(np.abs(diff_new), axis=0) + r = np.arange(ind.shape[0]) + max_diff_new = np.abs(diff_new[max_ind, r]) + scale_new = np.maximum(np.abs(f[max_ind]), np.abs(f_new[max_ind, r])) + + update = max_diff[ind] * scale_new < max_diff_new * scale[ind] + if np.any(update): + update, = np.nonzero(update) + update_ind = ind[update] + factor[update_ind] = new_factor[update] + h[update_ind] = h_new[update] + diff[:, update_ind] = diff_new[:, update] + scale[update_ind] = scale_new[update] + max_diff[update_ind] = max_diff_new[update] + + diff /= h + + factor[max_diff < NUM_JAC_DIFF_SMALL * scale] *= NUM_JAC_FACTOR_INCREASE + factor[max_diff > NUM_JAC_DIFF_BIG * scale] *= NUM_JAC_FACTOR_DECREASE + factor = np.maximum(factor, NUM_JAC_MIN_FACTOR) + + return diff, factor + + +def _sparse_num_jac(fun, t, y, f, h, factor, y_scale, structure, groups): + n = y.shape[0] + n_groups = np.max(groups) + 1 + h_vecs = np.empty((n_groups, n)) + for group in range(n_groups): + e = np.equal(group, groups) + h_vecs[group] = h * e + h_vecs = h_vecs.T + + f_new = fun(t, y[:, None] + h_vecs) + df = f_new - f[:, None] + + i, j, _ = find(structure) + diff = coo_matrix((df[i, groups[j]], (i, j)), shape=(n, n)).tocsc() + max_ind = np.array(abs(diff).argmax(axis=0)).ravel() + r = np.arange(n) + max_diff = np.asarray(np.abs(diff[max_ind, r])).ravel() + scale = np.maximum(np.abs(f[max_ind]), + np.abs(f_new[max_ind, groups[r]])) + + diff_too_small = max_diff < NUM_JAC_DIFF_REJECT * scale + if np.any(diff_too_small): + ind, = np.nonzero(diff_too_small) + new_factor = NUM_JAC_FACTOR_INCREASE * factor[ind] + h_new = (y[ind] + new_factor * y_scale[ind]) - y[ind] + h_new_all = np.zeros(n) + h_new_all[ind] = h_new + + groups_unique = np.unique(groups[ind]) + groups_map = np.empty(n_groups, dtype=int) + h_vecs = np.empty((groups_unique.shape[0], n)) + for k, group in enumerate(groups_unique): + e = np.equal(group, groups) + h_vecs[k] = h_new_all * e + groups_map[group] = k + h_vecs = h_vecs.T + + f_new = fun(t, y[:, None] + h_vecs) + df = f_new - f[:, None] + i, j, _ = find(structure[:, ind]) + diff_new = coo_matrix((df[i, groups_map[groups[ind[j]]]], + (i, j)), shape=(n, ind.shape[0])).tocsc() + + max_ind_new = np.array(abs(diff_new).argmax(axis=0)).ravel() + r = np.arange(ind.shape[0]) + max_diff_new = np.asarray(np.abs(diff_new[max_ind_new, r])).ravel() + scale_new = np.maximum( + np.abs(f[max_ind_new]), + np.abs(f_new[max_ind_new, groups_map[groups[ind]]])) + + update = max_diff[ind] * scale_new < max_diff_new * scale[ind] + if np.any(update): + update, = np.nonzero(update) + update_ind = ind[update] + factor[update_ind] = new_factor[update] + h[update_ind] = h_new[update] + diff[:, update_ind] = diff_new[:, update] + scale[update_ind] = scale_new[update] + max_diff[update_ind] = max_diff_new[update] + + diff.data /= np.repeat(h, np.diff(diff.indptr)) + + factor[max_diff < NUM_JAC_DIFF_SMALL * scale] *= NUM_JAC_FACTOR_INCREASE + factor[max_diff > NUM_JAC_DIFF_BIG * scale] *= NUM_JAC_FACTOR_DECREASE + factor = np.maximum(factor, NUM_JAC_MIN_FACTOR) + + return diff, factor diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/dop853_coefficients.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/dop853_coefficients.py new file mode 100644 index 0000000000000000000000000000000000000000..f39f2f3650d321e2c475d4e220f9769139118a5e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/dop853_coefficients.py @@ -0,0 +1,193 @@ +import numpy as np + +N_STAGES = 12 +N_STAGES_EXTENDED = 16 +INTERPOLATOR_POWER = 7 + +C = np.array([0.0, + 0.526001519587677318785587544488e-01, + 0.789002279381515978178381316732e-01, + 0.118350341907227396726757197510, + 0.281649658092772603273242802490, + 0.333333333333333333333333333333, + 0.25, + 0.307692307692307692307692307692, + 0.651282051282051282051282051282, + 0.6, + 0.857142857142857142857142857142, + 1.0, + 1.0, + 0.1, + 0.2, + 0.777777777777777777777777777778]) + +A = np.zeros((N_STAGES_EXTENDED, N_STAGES_EXTENDED)) +A[1, 0] = 5.26001519587677318785587544488e-2 + +A[2, 0] = 1.97250569845378994544595329183e-2 +A[2, 1] = 5.91751709536136983633785987549e-2 + +A[3, 0] = 2.95875854768068491816892993775e-2 +A[3, 2] = 8.87627564304205475450678981324e-2 + +A[4, 0] = 2.41365134159266685502369798665e-1 +A[4, 2] = -8.84549479328286085344864962717e-1 +A[4, 3] = 9.24834003261792003115737966543e-1 + +A[5, 0] = 3.7037037037037037037037037037e-2 +A[5, 3] = 1.70828608729473871279604482173e-1 +A[5, 4] = 1.25467687566822425016691814123e-1 + +A[6, 0] = 3.7109375e-2 +A[6, 3] = 1.70252211019544039314978060272e-1 +A[6, 4] = 6.02165389804559606850219397283e-2 +A[6, 5] = -1.7578125e-2 + +A[7, 0] = 3.70920001185047927108779319836e-2 +A[7, 3] = 1.70383925712239993810214054705e-1 +A[7, 4] = 1.07262030446373284651809199168e-1 +A[7, 5] = -1.53194377486244017527936158236e-2 +A[7, 6] = 8.27378916381402288758473766002e-3 + +A[8, 0] = 6.24110958716075717114429577812e-1 +A[8, 3] = -3.36089262944694129406857109825 +A[8, 4] = -8.68219346841726006818189891453e-1 +A[8, 5] = 2.75920996994467083049415600797e1 +A[8, 6] = 2.01540675504778934086186788979e1 +A[8, 7] = -4.34898841810699588477366255144e1 + +A[9, 0] = 4.77662536438264365890433908527e-1 +A[9, 3] = -2.48811461997166764192642586468 +A[9, 4] = -5.90290826836842996371446475743e-1 +A[9, 5] = 2.12300514481811942347288949897e1 +A[9, 6] = 1.52792336328824235832596922938e1 +A[9, 7] = -3.32882109689848629194453265587e1 +A[9, 8] = -2.03312017085086261358222928593e-2 + +A[10, 0] = -9.3714243008598732571704021658e-1 +A[10, 3] = 5.18637242884406370830023853209 +A[10, 4] = 1.09143734899672957818500254654 +A[10, 5] = -8.14978701074692612513997267357 +A[10, 6] = -1.85200656599969598641566180701e1 +A[10, 7] = 2.27394870993505042818970056734e1 +A[10, 8] = 2.49360555267965238987089396762 +A[10, 9] = -3.0467644718982195003823669022 + +A[11, 0] = 2.27331014751653820792359768449 +A[11, 3] = -1.05344954667372501984066689879e1 +A[11, 4] = -2.00087205822486249909675718444 +A[11, 5] = -1.79589318631187989172765950534e1 +A[11, 6] = 2.79488845294199600508499808837e1 +A[11, 7] = -2.85899827713502369474065508674 +A[11, 8] = -8.87285693353062954433549289258 +A[11, 9] = 1.23605671757943030647266201528e1 +A[11, 10] = 6.43392746015763530355970484046e-1 + +A[12, 0] = 5.42937341165687622380535766363e-2 +A[12, 5] = 4.45031289275240888144113950566 +A[12, 6] = 1.89151789931450038304281599044 +A[12, 7] = -5.8012039600105847814672114227 +A[12, 8] = 3.1116436695781989440891606237e-1 +A[12, 9] = -1.52160949662516078556178806805e-1 +A[12, 10] = 2.01365400804030348374776537501e-1 +A[12, 11] = 4.47106157277725905176885569043e-2 + +A[13, 0] = 5.61675022830479523392909219681e-2 +A[13, 6] = 2.53500210216624811088794765333e-1 +A[13, 7] = -2.46239037470802489917441475441e-1 +A[13, 8] = -1.24191423263816360469010140626e-1 +A[13, 9] = 1.5329179827876569731206322685e-1 +A[13, 10] = 8.20105229563468988491666602057e-3 +A[13, 11] = 7.56789766054569976138603589584e-3 +A[13, 12] = -8.298e-3 + +A[14, 0] = 3.18346481635021405060768473261e-2 +A[14, 5] = 2.83009096723667755288322961402e-2 +A[14, 6] = 5.35419883074385676223797384372e-2 +A[14, 7] = -5.49237485713909884646569340306e-2 +A[14, 10] = -1.08347328697249322858509316994e-4 +A[14, 11] = 3.82571090835658412954920192323e-4 +A[14, 12] = -3.40465008687404560802977114492e-4 +A[14, 13] = 1.41312443674632500278074618366e-1 + +A[15, 0] = -4.28896301583791923408573538692e-1 +A[15, 5] = -4.69762141536116384314449447206 +A[15, 6] = 7.68342119606259904184240953878 +A[15, 7] = 4.06898981839711007970213554331 +A[15, 8] = 3.56727187455281109270669543021e-1 +A[15, 12] = -1.39902416515901462129418009734e-3 +A[15, 13] = 2.9475147891527723389556272149 +A[15, 14] = -9.15095847217987001081870187138 + + +B = A[N_STAGES, :N_STAGES] + +E3 = np.zeros(N_STAGES + 1) +E3[:-1] = B.copy() +E3[0] -= 0.244094488188976377952755905512 +E3[8] -= 0.733846688281611857341361741547 +E3[11] -= 0.220588235294117647058823529412e-1 + +E5 = np.zeros(N_STAGES + 1) +E5[0] = 0.1312004499419488073250102996e-1 +E5[5] = -0.1225156446376204440720569753e+1 +E5[6] = -0.4957589496572501915214079952 +E5[7] = 0.1664377182454986536961530415e+1 +E5[8] = -0.3503288487499736816886487290 +E5[9] = 0.3341791187130174790297318841 +E5[10] = 0.8192320648511571246570742613e-1 +E5[11] = -0.2235530786388629525884427845e-1 + +# First 3 coefficients are computed separately. +D = np.zeros((INTERPOLATOR_POWER - 3, N_STAGES_EXTENDED)) +D[0, 0] = -0.84289382761090128651353491142e+1 +D[0, 5] = 0.56671495351937776962531783590 +D[0, 6] = -0.30689499459498916912797304727e+1 +D[0, 7] = 0.23846676565120698287728149680e+1 +D[0, 8] = 0.21170345824450282767155149946e+1 +D[0, 9] = -0.87139158377797299206789907490 +D[0, 10] = 0.22404374302607882758541771650e+1 +D[0, 11] = 0.63157877876946881815570249290 +D[0, 12] = -0.88990336451333310820698117400e-1 +D[0, 13] = 0.18148505520854727256656404962e+2 +D[0, 14] = -0.91946323924783554000451984436e+1 +D[0, 15] = -0.44360363875948939664310572000e+1 + +D[1, 0] = 0.10427508642579134603413151009e+2 +D[1, 5] = 0.24228349177525818288430175319e+3 +D[1, 6] = 0.16520045171727028198505394887e+3 +D[1, 7] = -0.37454675472269020279518312152e+3 +D[1, 8] = -0.22113666853125306036270938578e+2 +D[1, 9] = 0.77334326684722638389603898808e+1 +D[1, 10] = -0.30674084731089398182061213626e+2 +D[1, 11] = -0.93321305264302278729567221706e+1 +D[1, 12] = 0.15697238121770843886131091075e+2 +D[1, 13] = -0.31139403219565177677282850411e+2 +D[1, 14] = -0.93529243588444783865713862664e+1 +D[1, 15] = 0.35816841486394083752465898540e+2 + +D[2, 0] = 0.19985053242002433820987653617e+2 +D[2, 5] = -0.38703730874935176555105901742e+3 +D[2, 6] = -0.18917813819516756882830838328e+3 +D[2, 7] = 0.52780815920542364900561016686e+3 +D[2, 8] = -0.11573902539959630126141871134e+2 +D[2, 9] = 0.68812326946963000169666922661e+1 +D[2, 10] = -0.10006050966910838403183860980e+1 +D[2, 11] = 0.77771377980534432092869265740 +D[2, 12] = -0.27782057523535084065932004339e+1 +D[2, 13] = -0.60196695231264120758267380846e+2 +D[2, 14] = 0.84320405506677161018159903784e+2 +D[2, 15] = 0.11992291136182789328035130030e+2 + +D[3, 0] = -0.25693933462703749003312586129e+2 +D[3, 5] = -0.15418974869023643374053993627e+3 +D[3, 6] = -0.23152937917604549567536039109e+3 +D[3, 7] = 0.35763911791061412378285349910e+3 +D[3, 8] = 0.93405324183624310003907691704e+2 +D[3, 9] = -0.37458323136451633156875139351e+2 +D[3, 10] = 0.10409964950896230045147246184e+3 +D[3, 11] = 0.29840293426660503123344363579e+2 +D[3, 12] = -0.43533456590011143754432175058e+2 +D[3, 13] = 0.96324553959188282948394950600e+2 +D[3, 14] = -0.39177261675615439165231486172e+2 +D[3, 15] = -0.14972683625798562581422125276e+3 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/ivp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/ivp.py new file mode 100644 index 0000000000000000000000000000000000000000..8186982e4fddbd8c1058b59c745ca66ab3a9c224 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/ivp.py @@ -0,0 +1,755 @@ +import inspect +import numpy as np +from .bdf import BDF +from .radau import Radau +from .rk import RK23, RK45, DOP853 +from .lsoda import LSODA +from scipy.optimize import OptimizeResult +from .common import EPS, OdeSolution +from .base import OdeSolver + + +METHODS = {'RK23': RK23, + 'RK45': RK45, + 'DOP853': DOP853, + 'Radau': Radau, + 'BDF': BDF, + 'LSODA': LSODA} + + +MESSAGES = {0: "The solver successfully reached the end of the integration interval.", + 1: "A termination event occurred."} + + +class OdeResult(OptimizeResult): + pass + + +def prepare_events(events): + """Standardize event functions and extract attributes.""" + if callable(events): + events = (events,) + + max_events = np.empty(len(events)) + direction = np.empty(len(events)) + for i, event in enumerate(events): + terminal = getattr(event, 'terminal', None) + direction[i] = getattr(event, 'direction', 0) + + message = ('The `terminal` attribute of each event ' + 'must be a boolean or positive integer.') + if terminal is None or terminal == 0: + max_events[i] = np.inf + elif int(terminal) == terminal and terminal > 0: + max_events[i] = terminal + else: + raise ValueError(message) + + return events, max_events, direction + + +def solve_event_equation(event, sol, t_old, t): + """Solve an equation corresponding to an ODE event. + + The equation is ``event(t, y(t)) = 0``, here ``y(t)`` is known from an + ODE solver using some sort of interpolation. It is solved by + `scipy.optimize.brentq` with xtol=atol=4*EPS. + + Parameters + ---------- + event : callable + Function ``event(t, y)``. + sol : callable + Function ``sol(t)`` which evaluates an ODE solution between `t_old` + and `t`. + t_old, t : float + Previous and new values of time. They will be used as a bracketing + interval. + + Returns + ------- + root : float + Found solution. + """ + from scipy.optimize import brentq + return brentq(lambda t: event(t, sol(t)), t_old, t, + xtol=4 * EPS, rtol=4 * EPS) + + +def handle_events(sol, events, active_events, event_count, max_events, + t_old, t): + """Helper function to handle events. + + Parameters + ---------- + sol : DenseOutput + Function ``sol(t)`` which evaluates an ODE solution between `t_old` + and `t`. + events : list of callables, length n_events + Event functions with signatures ``event(t, y)``. + active_events : ndarray + Indices of events which occurred. + event_count : ndarray + Current number of occurrences for each event. + max_events : ndarray, shape (n_events,) + Number of occurrences allowed for each event before integration + termination is issued. + t_old, t : float + Previous and new values of time. + + Returns + ------- + root_indices : ndarray + Indices of events which take zero between `t_old` and `t` and before + a possible termination. + roots : ndarray + Values of t at which events occurred. + terminate : bool + Whether a terminal event occurred. + """ + roots = [solve_event_equation(events[event_index], sol, t_old, t) + for event_index in active_events] + + roots = np.asarray(roots) + + if np.any(event_count[active_events] >= max_events[active_events]): + if t > t_old: + order = np.argsort(roots) + else: + order = np.argsort(-roots) + active_events = active_events[order] + roots = roots[order] + t = np.nonzero(event_count[active_events] + >= max_events[active_events])[0][0] + active_events = active_events[:t + 1] + roots = roots[:t + 1] + terminate = True + else: + terminate = False + + return active_events, roots, terminate + + +def find_active_events(g, g_new, direction): + """Find which event occurred during an integration step. + + Parameters + ---------- + g, g_new : array_like, shape (n_events,) + Values of event functions at a current and next points. + direction : ndarray, shape (n_events,) + Event "direction" according to the definition in `solve_ivp`. + + Returns + ------- + active_events : ndarray + Indices of events which occurred during the step. + """ + g, g_new = np.asarray(g), np.asarray(g_new) + up = (g <= 0) & (g_new >= 0) + down = (g >= 0) & (g_new <= 0) + either = up | down + mask = (up & (direction > 0) | + down & (direction < 0) | + either & (direction == 0)) + + return np.nonzero(mask)[0] + + +def solve_ivp(fun, t_span, y0, method='RK45', t_eval=None, dense_output=False, + events=None, vectorized=False, args=None, **options): + """Solve an initial value problem for a system of ODEs. + + This function numerically integrates a system of ordinary differential + equations given an initial value:: + + dy / dt = f(t, y) + y(t0) = y0 + + Here t is a 1-D independent variable (time), y(t) is an + N-D vector-valued function (state), and an N-D + vector-valued function f(t, y) determines the differential equations. + The goal is to find y(t) approximately satisfying the differential + equations, given an initial value y(t0)=y0. + + Some of the solvers support integration in the complex domain, but note + that for stiff ODE solvers, the right-hand side must be + complex-differentiable (satisfy Cauchy-Riemann equations [11]_). + To solve a problem in the complex domain, pass y0 with a complex data type. + Another option always available is to rewrite your problem for real and + imaginary parts separately. + + Parameters + ---------- + fun : callable + Right-hand side of the system: the time derivative of the state ``y`` + at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a + scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. Additional + arguments need to be passed if ``args`` is used (see documentation of + ``args`` argument). ``fun`` must return an array of the same shape as + ``y``. See `vectorized` for more information. + t_span : 2-member sequence + Interval of integration (t0, tf). The solver starts with t=t0 and + integrates until it reaches t=tf. Both t0 and tf must be floats + or values interpretable by the float conversion function. + y0 : array_like, shape (n,) + Initial state. For problems in the complex domain, pass `y0` with a + complex data type (even if the initial value is purely real). + method : string or `OdeSolver`, optional + Integration method to use: + + * 'RK45' (default): Explicit Runge-Kutta method of order 5(4) [1]_. + The error is controlled assuming accuracy of the fourth-order + method, but steps are taken using the fifth-order accurate + formula (local extrapolation is done). A quartic interpolation + polynomial is used for the dense output [2]_. Can be applied in + the complex domain. + * 'RK23': Explicit Runge-Kutta method of order 3(2) [3]_. The error + is controlled assuming accuracy of the second-order method, but + steps are taken using the third-order accurate formula (local + extrapolation is done). A cubic Hermite polynomial is used for the + dense output. Can be applied in the complex domain. + * 'DOP853': Explicit Runge-Kutta method of order 8 [13]_. + Python implementation of the "DOP853" algorithm originally + written in Fortran [14]_. A 7-th order interpolation polynomial + accurate to 7-th order is used for the dense output. + Can be applied in the complex domain. + * 'Radau': Implicit Runge-Kutta method of the Radau IIA family of + order 5 [4]_. The error is controlled with a third-order accurate + embedded formula. A cubic polynomial which satisfies the + collocation conditions is used for the dense output. + * 'BDF': Implicit multi-step variable-order (1 to 5) method based + on a backward differentiation formula for the derivative + approximation [5]_. The implementation follows the one described + in [6]_. A quasi-constant step scheme is used and accuracy is + enhanced using the NDF modification. Can be applied in the + complex domain. + * 'LSODA': Adams/BDF method with automatic stiffness detection and + switching [7]_, [8]_. This is a wrapper of the Fortran solver + from ODEPACK. + + Explicit Runge-Kutta methods ('RK23', 'RK45', 'DOP853') should be used + for non-stiff problems and implicit methods ('Radau', 'BDF') for + stiff problems [9]_. Among Runge-Kutta methods, 'DOP853' is recommended + for solving with high precision (low values of `rtol` and `atol`). + + If not sure, first try to run 'RK45'. If it makes unusually many + iterations, diverges, or fails, your problem is likely to be stiff and + you should use 'Radau' or 'BDF'. 'LSODA' can also be a good universal + choice, but it might be somewhat less convenient to work with as it + wraps old Fortran code. + + You can also pass an arbitrary class derived from `OdeSolver` which + implements the solver. + t_eval : array_like or None, optional + Times at which to store the computed solution, must be sorted and lie + within `t_span`. If None (default), use points selected by the solver. + dense_output : bool, optional + Whether to compute a continuous solution. Default is False. + events : callable, or list of callables, optional + Events to track. If None (default), no events will be tracked. + Each event occurs at the zeros of a continuous function of time and + state. Each function must have the signature ``event(t, y)`` where + additional argument have to be passed if ``args`` is used (see + documentation of ``args`` argument). Each function must return a + float. The solver will find an accurate value of `t` at which + ``event(t, y(t)) = 0`` using a root-finding algorithm. By default, + all zeros will be found. The solver looks for a sign change over + each step, so if multiple zero crossings occur within one step, + events may be missed. Additionally each `event` function might + have the following attributes: + + terminal: bool or int, optional + When boolean, whether to terminate integration if this event occurs. + When integral, termination occurs after the specified the number of + occurrences of this event. + Implicitly False if not assigned. + direction: float, optional + Direction of a zero crossing. If `direction` is positive, + `event` will only trigger when going from negative to positive, + and vice versa if `direction` is negative. If 0, then either + direction will trigger event. Implicitly 0 if not assigned. + + You can assign attributes like ``event.terminal = True`` to any + function in Python. + vectorized : bool, optional + Whether `fun` can be called in a vectorized fashion. Default is False. + + If ``vectorized`` is False, `fun` will always be called with ``y`` of + shape ``(n,)``, where ``n = len(y0)``. + + If ``vectorized`` is True, `fun` may be called with ``y`` of shape + ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave + such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of + the returned array is the time derivative of the state corresponding + with a column of ``y``). + + Setting ``vectorized=True`` allows for faster finite difference + approximation of the Jacobian by methods 'Radau' and 'BDF', but + will result in slower execution for other methods and for 'Radau' and + 'BDF' in some circumstances (e.g. small ``len(y0)``). + args : tuple, optional + Additional arguments to pass to the user-defined functions. If given, + the additional arguments are passed to all user-defined functions. + So if, for example, `fun` has the signature ``fun(t, y, a, b, c)``, + then `jac` (if given) and any event functions must have the same + signature, and `args` must be a tuple of length 3. + **options + Options passed to a chosen solver. All options available for already + implemented solvers are listed below. + first_step : float or None, optional + Initial step size. Default is `None` which means that the algorithm + should choose. + max_step : float, optional + Maximum allowed step size. Default is np.inf, i.e., the step size is not + bounded and determined solely by the solver. + rtol, atol : float or array_like, optional + Relative and absolute tolerances. The solver keeps the local error + estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a + relative accuracy (number of correct digits), while `atol` controls + absolute accuracy (number of correct decimal places). To achieve the + desired `rtol`, set `atol` to be smaller than the smallest value that + can be expected from ``rtol * abs(y)`` so that `rtol` dominates the + allowable error. If `atol` is larger than ``rtol * abs(y)`` the + number of correct digits is not guaranteed. Conversely, to achieve the + desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller + than `atol`. If components of y have different scales, it might be + beneficial to set different `atol` values for different components by + passing array_like with shape (n,) for `atol`. Default values are + 1e-3 for `rtol` and 1e-6 for `atol`. + jac : array_like, sparse_matrix, callable or None, optional + Jacobian matrix of the right-hand side of the system with respect + to y, required by the 'Radau', 'BDF' and 'LSODA' method. The + Jacobian matrix has shape (n, n) and its element (i, j) is equal to + ``d f_i / d y_j``. There are three ways to define the Jacobian: + + * If array_like or sparse_matrix, the Jacobian is assumed to + be constant. Not supported by 'LSODA'. + * If callable, the Jacobian is assumed to depend on both + t and y; it will be called as ``jac(t, y)``, as necessary. + Additional arguments have to be passed if ``args`` is + used (see documentation of ``args`` argument). + For 'Radau' and 'BDF' methods, the return value might be a + sparse matrix. + * If None (default), the Jacobian will be approximated by + finite differences. + + It is generally recommended to provide the Jacobian rather than + relying on a finite-difference approximation. + jac_sparsity : array_like, sparse matrix or None, optional + Defines a sparsity structure of the Jacobian matrix for a finite- + difference approximation. Its shape must be (n, n). This argument + is ignored if `jac` is not `None`. If the Jacobian has only few + non-zero elements in *each* row, providing the sparsity structure + will greatly speed up the computations [10]_. A zero entry means that + a corresponding element in the Jacobian is always zero. If None + (default), the Jacobian is assumed to be dense. + Not supported by 'LSODA', see `lband` and `uband` instead. + lband, uband : int or None, optional + Parameters defining the bandwidth of the Jacobian for the 'LSODA' + method, i.e., ``jac[i, j] != 0 only for i - lband <= j <= i + uband``. + Default is None. Setting these requires your jac routine to return the + Jacobian in the packed format: the returned array must have ``n`` + columns and ``uband + lband + 1`` rows in which Jacobian diagonals are + written. Specifically ``jac_packed[uband + i - j , j] = jac[i, j]``. + The same format is used in `scipy.linalg.solve_banded` (check for an + illustration). These parameters can be also used with ``jac=None`` to + reduce the number of Jacobian elements estimated by finite differences. + min_step : float, optional + The minimum allowed step size for 'LSODA' method. + By default `min_step` is zero. + + Returns + ------- + Bunch object with the following fields defined: + t : ndarray, shape (n_points,) + Time points. + y : ndarray, shape (n, n_points) + Values of the solution at `t`. + sol : `OdeSolution` or None + Found solution as `OdeSolution` instance; None if `dense_output` was + set to False. + t_events : list of ndarray or None + Contains for each event type a list of arrays at which an event of + that type event was detected. None if `events` was None. + y_events : list of ndarray or None + For each value of `t_events`, the corresponding value of the solution. + None if `events` was None. + nfev : int + Number of evaluations of the right-hand side. + njev : int + Number of evaluations of the Jacobian. + nlu : int + Number of LU decompositions. + status : int + Reason for algorithm termination: + + * -1: Integration step failed. + * 0: The solver successfully reached the end of `tspan`. + * 1: A termination event occurred. + + message : string + Human-readable description of the termination reason. + success : bool + True if the solver reached the interval end or a termination event + occurred (``status >= 0``). + + References + ---------- + .. [1] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta + formulae", Journal of Computational and Applied Mathematics, Vol. 6, + No. 1, pp. 19-26, 1980. + .. [2] L. W. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics + of Computation,, Vol. 46, No. 173, pp. 135-150, 1986. + .. [3] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas", + Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989. + .. [4] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II: + Stiff and Differential-Algebraic Problems", Sec. IV.8. + .. [5] `Backward Differentiation Formula + `_ + on Wikipedia. + .. [6] L. F. Shampine, M. W. Reichelt, "THE MATLAB ODE SUITE", SIAM J. SCI. + COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997. + .. [7] A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE + Solvers," IMACS Transactions on Scientific Computation, Vol 1., + pp. 55-64, 1983. + .. [8] L. Petzold, "Automatic selection of methods for solving stiff and + nonstiff systems of ordinary differential equations", SIAM Journal + on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136-148, + 1983. + .. [9] `Stiff equation `_ on + Wikipedia. + .. [10] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of + sparse Jacobian matrices", Journal of the Institute of Mathematics + and its Applications, 13, pp. 117-120, 1974. + .. [11] `Cauchy-Riemann equations + `_ on + Wikipedia. + .. [12] `Lotka-Volterra equations + `_ + on Wikipedia. + .. [13] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential + Equations I: Nonstiff Problems", Sec. II. + .. [14] `Page with original Fortran code of DOP853 + `_. + + Examples + -------- + Basic exponential decay showing automatically chosen time points. + + >>> import numpy as np + >>> from scipy.integrate import solve_ivp + >>> def exponential_decay(t, y): return -0.5 * y + >>> sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8]) + >>> print(sol.t) + [ 0. 0.11487653 1.26364188 3.06061781 4.81611105 6.57445806 + 8.33328988 10. ] + >>> print(sol.y) + [[2. 1.88836035 1.06327177 0.43319312 0.18017253 0.07483045 + 0.03107158 0.01350781] + [4. 3.7767207 2.12654355 0.86638624 0.36034507 0.14966091 + 0.06214316 0.02701561] + [8. 7.5534414 4.25308709 1.73277247 0.72069014 0.29932181 + 0.12428631 0.05403123]] + + Specifying points where the solution is desired. + + >>> sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8], + ... t_eval=[0, 1, 2, 4, 10]) + >>> print(sol.t) + [ 0 1 2 4 10] + >>> print(sol.y) + [[2. 1.21305369 0.73534021 0.27066736 0.01350938] + [4. 2.42610739 1.47068043 0.54133472 0.02701876] + [8. 4.85221478 2.94136085 1.08266944 0.05403753]] + + Cannon fired upward with terminal event upon impact. The ``terminal`` and + ``direction`` fields of an event are applied by monkey patching a function. + Here ``y[0]`` is position and ``y[1]`` is velocity. The projectile starts + at position 0 with velocity +10. Note that the integration never reaches + t=100 because the event is terminal. + + >>> def upward_cannon(t, y): return [y[1], -0.5] + >>> def hit_ground(t, y): return y[0] + >>> hit_ground.terminal = True + >>> hit_ground.direction = -1 + >>> sol = solve_ivp(upward_cannon, [0, 100], [0, 10], events=hit_ground) + >>> print(sol.t_events) + [array([40.])] + >>> print(sol.t) + [0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02 + 1.11088891e-01 1.11098890e+00 1.11099890e+01 4.00000000e+01] + + Use `dense_output` and `events` to find position, which is 100, at the apex + of the cannonball's trajectory. Apex is not defined as terminal, so both + apex and hit_ground are found. There is no information at t=20, so the sol + attribute is used to evaluate the solution. The sol attribute is returned + by setting ``dense_output=True``. Alternatively, the `y_events` attribute + can be used to access the solution at the time of the event. + + >>> def apex(t, y): return y[1] + >>> sol = solve_ivp(upward_cannon, [0, 100], [0, 10], + ... events=(hit_ground, apex), dense_output=True) + >>> print(sol.t_events) + [array([40.]), array([20.])] + >>> print(sol.t) + [0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02 + 1.11088891e-01 1.11098890e+00 1.11099890e+01 4.00000000e+01] + >>> print(sol.sol(sol.t_events[1][0])) + [100. 0.] + >>> print(sol.y_events) + [array([[-5.68434189e-14, -1.00000000e+01]]), + array([[1.00000000e+02, 1.77635684e-15]])] + + As an example of a system with additional parameters, we'll implement + the Lotka-Volterra equations [12]_. + + >>> def lotkavolterra(t, z, a, b, c, d): + ... x, y = z + ... return [a*x - b*x*y, -c*y + d*x*y] + ... + + We pass in the parameter values a=1.5, b=1, c=3 and d=1 with the `args` + argument. + + >>> sol = solve_ivp(lotkavolterra, [0, 15], [10, 5], args=(1.5, 1, 3, 1), + ... dense_output=True) + + Compute a dense solution and plot it. + + >>> t = np.linspace(0, 15, 300) + >>> z = sol.sol(t) + >>> import matplotlib.pyplot as plt + >>> plt.plot(t, z.T) + >>> plt.xlabel('t') + >>> plt.legend(['x', 'y'], shadow=True) + >>> plt.title('Lotka-Volterra System') + >>> plt.show() + + A couple examples of using solve_ivp to solve the differential + equation ``y' = Ay`` with complex matrix ``A``. + + >>> A = np.array([[-0.25 + 0.14j, 0, 0.33 + 0.44j], + ... [0.25 + 0.58j, -0.2 + 0.14j, 0], + ... [0, 0.2 + 0.4j, -0.1 + 0.97j]]) + + Solving an IVP with ``A`` from above and ``y`` as 3x1 vector: + + >>> def deriv_vec(t, y): + ... return A @ y + >>> result = solve_ivp(deriv_vec, [0, 25], + ... np.array([10 + 0j, 20 + 0j, 30 + 0j]), + ... t_eval=np.linspace(0, 25, 101)) + >>> print(result.y[:, 0]) + [10.+0.j 20.+0.j 30.+0.j] + >>> print(result.y[:, -1]) + [18.46291039+45.25653651j 10.01569306+36.23293216j + -4.98662741+80.07360388j] + + Solving an IVP with ``A`` from above with ``y`` as 3x3 matrix : + + >>> def deriv_mat(t, y): + ... return (A @ y.reshape(3, 3)).flatten() + >>> y0 = np.array([[2 + 0j, 3 + 0j, 4 + 0j], + ... [5 + 0j, 6 + 0j, 7 + 0j], + ... [9 + 0j, 34 + 0j, 78 + 0j]]) + + >>> result = solve_ivp(deriv_mat, [0, 25], y0.flatten(), + ... t_eval=np.linspace(0, 25, 101)) + >>> print(result.y[:, 0].reshape(3, 3)) + [[ 2.+0.j 3.+0.j 4.+0.j] + [ 5.+0.j 6.+0.j 7.+0.j] + [ 9.+0.j 34.+0.j 78.+0.j]] + >>> print(result.y[:, -1].reshape(3, 3)) + [[ 5.67451179 +12.07938445j 17.2888073 +31.03278837j + 37.83405768 +63.25138759j] + [ 3.39949503 +11.82123994j 21.32530996 +44.88668871j + 53.17531184+103.80400411j] + [ -2.26105874 +22.19277664j -15.1255713 +70.19616341j + -38.34616845+153.29039931j]] + + + """ + if method not in METHODS and not ( + inspect.isclass(method) and issubclass(method, OdeSolver)): + raise ValueError(f"`method` must be one of {METHODS} or OdeSolver class.") + + t0, tf = map(float, t_span) + + if args is not None: + # Wrap the user's fun (and jac, if given) in lambdas to hide the + # additional parameters. Pass in the original fun as a keyword + # argument to keep it in the scope of the lambda. + try: + _ = [*(args)] + except TypeError as exp: + suggestion_tuple = ( + "Supplied 'args' cannot be unpacked. Please supply `args`" + f" as a tuple (e.g. `args=({args},)`)" + ) + raise TypeError(suggestion_tuple) from exp + + def fun(t, x, fun=fun): + return fun(t, x, *args) + jac = options.get('jac') + if callable(jac): + options['jac'] = lambda t, x: jac(t, x, *args) + + if t_eval is not None: + t_eval = np.asarray(t_eval) + if t_eval.ndim != 1: + raise ValueError("`t_eval` must be 1-dimensional.") + + if np.any(t_eval < min(t0, tf)) or np.any(t_eval > max(t0, tf)): + raise ValueError("Values in `t_eval` are not within `t_span`.") + + d = np.diff(t_eval) + if tf > t0 and np.any(d <= 0) or tf < t0 and np.any(d >= 0): + raise ValueError("Values in `t_eval` are not properly sorted.") + + if tf > t0: + t_eval_i = 0 + else: + # Make order of t_eval decreasing to use np.searchsorted. + t_eval = t_eval[::-1] + # This will be an upper bound for slices. + t_eval_i = t_eval.shape[0] + + if method in METHODS: + method = METHODS[method] + + solver = method(fun, t0, y0, tf, vectorized=vectorized, **options) + + if t_eval is None: + ts = [t0] + ys = [y0] + elif t_eval is not None and dense_output: + ts = [] + ti = [t0] + ys = [] + else: + ts = [] + ys = [] + + interpolants = [] + + if events is not None: + events, max_events, event_dir = prepare_events(events) + event_count = np.zeros(len(events)) + if args is not None: + # Wrap user functions in lambdas to hide the additional parameters. + # The original event function is passed as a keyword argument to the + # lambda to keep the original function in scope (i.e., avoid the + # late binding closure "gotcha"). + events = [lambda t, x, event=event: event(t, x, *args) + for event in events] + g = [event(t0, y0) for event in events] + t_events = [[] for _ in range(len(events))] + y_events = [[] for _ in range(len(events))] + else: + t_events = None + y_events = None + + status = None + while status is None: + message = solver.step() + + if solver.status == 'finished': + status = 0 + elif solver.status == 'failed': + status = -1 + break + + t_old = solver.t_old + t = solver.t + y = solver.y + + if dense_output: + sol = solver.dense_output() + interpolants.append(sol) + else: + sol = None + + if events is not None: + g_new = [event(t, y) for event in events] + active_events = find_active_events(g, g_new, event_dir) + if active_events.size > 0: + if sol is None: + sol = solver.dense_output() + + event_count[active_events] += 1 + root_indices, roots, terminate = handle_events( + sol, events, active_events, event_count, max_events, + t_old, t) + + for e, te in zip(root_indices, roots): + t_events[e].append(te) + y_events[e].append(sol(te)) + + if terminate: + status = 1 + t = roots[-1] + y = sol(t) + + g = g_new + + if t_eval is None: + donot_append = (len(ts) > 1 and + ts[-1] == t and + dense_output) + if not donot_append: + ts.append(t) + ys.append(y) + else: + if len(interpolants) > 0: + interpolants.pop() + else: + # The value in t_eval equal to t will be included. + if solver.direction > 0: + t_eval_i_new = np.searchsorted(t_eval, t, side='right') + t_eval_step = t_eval[t_eval_i:t_eval_i_new] + else: + t_eval_i_new = np.searchsorted(t_eval, t, side='left') + # It has to be done with two slice operations, because + # you can't slice to 0th element inclusive using backward + # slicing. + t_eval_step = t_eval[t_eval_i_new:t_eval_i][::-1] + + if t_eval_step.size > 0: + if sol is None: + sol = solver.dense_output() + ts.append(t_eval_step) + ys.append(sol(t_eval_step)) + t_eval_i = t_eval_i_new + + if t_eval is not None and dense_output: + ti.append(t) + + message = MESSAGES.get(status, message) + + if t_events is not None: + t_events = [np.asarray(te) for te in t_events] + y_events = [np.asarray(ye) for ye in y_events] + + if t_eval is None: + ts = np.array(ts) + ys = np.vstack(ys).T + elif ts: + ts = np.hstack(ts) + ys = np.hstack(ys) + + if dense_output: + if t_eval is None: + sol = OdeSolution( + ts, interpolants, alt_segment=True if method in [BDF, LSODA] else False + ) + else: + sol = OdeSolution( + ti, interpolants, alt_segment=True if method in [BDF, LSODA] else False + ) + else: + sol = None + + return OdeResult(t=ts, y=ys, sol=sol, t_events=t_events, y_events=y_events, + nfev=solver.nfev, njev=solver.njev, nlu=solver.nlu, + status=status, message=message, success=status >= 0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/lsoda.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/lsoda.py new file mode 100644 index 0000000000000000000000000000000000000000..2a5a7c530c04eddc9beff44e2d4f6df439d5ef01 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/lsoda.py @@ -0,0 +1,224 @@ +import numpy as np +from scipy.integrate import ode +from .common import validate_tol, validate_first_step, warn_extraneous +from .base import OdeSolver, DenseOutput + + +class LSODA(OdeSolver): + """Adams/BDF method with automatic stiffness detection and switching. + + This is a wrapper to the Fortran solver from ODEPACK [1]_. It switches + automatically between the nonstiff Adams method and the stiff BDF method. + The method was originally detailed in [2]_. + + Parameters + ---------- + fun : callable + Right-hand side of the system: the time derivative of the state ``y`` + at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a + scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must + return an array of the same shape as ``y``. See `vectorized` for more + information. + t0 : float + Initial time. + y0 : array_like, shape (n,) + Initial state. + t_bound : float + Boundary time - the integration won't continue beyond it. It also + determines the direction of the integration. + first_step : float or None, optional + Initial step size. Default is ``None`` which means that the algorithm + should choose. + min_step : float, optional + Minimum allowed step size. Default is 0.0, i.e., the step size is not + bounded and determined solely by the solver. + max_step : float, optional + Maximum allowed step size. Default is np.inf, i.e., the step size is not + bounded and determined solely by the solver. + rtol, atol : float and array_like, optional + Relative and absolute tolerances. The solver keeps the local error + estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a + relative accuracy (number of correct digits), while `atol` controls + absolute accuracy (number of correct decimal places). To achieve the + desired `rtol`, set `atol` to be smaller than the smallest value that + can be expected from ``rtol * abs(y)`` so that `rtol` dominates the + allowable error. If `atol` is larger than ``rtol * abs(y)`` the + number of correct digits is not guaranteed. Conversely, to achieve the + desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller + than `atol`. If components of y have different scales, it might be + beneficial to set different `atol` values for different components by + passing array_like with shape (n,) for `atol`. Default values are + 1e-3 for `rtol` and 1e-6 for `atol`. + jac : None or callable, optional + Jacobian matrix of the right-hand side of the system with respect to + ``y``. The Jacobian matrix has shape (n, n) and its element (i, j) is + equal to ``d f_i / d y_j``. The function will be called as + ``jac(t, y)``. If None (default), the Jacobian will be + approximated by finite differences. It is generally recommended to + provide the Jacobian rather than relying on a finite-difference + approximation. + lband, uband : int or None + Parameters defining the bandwidth of the Jacobian, + i.e., ``jac[i, j] != 0 only for i - lband <= j <= i + uband``. Setting + these requires your jac routine to return the Jacobian in the packed format: + the returned array must have ``n`` columns and ``uband + lband + 1`` + rows in which Jacobian diagonals are written. Specifically + ``jac_packed[uband + i - j , j] = jac[i, j]``. The same format is used + in `scipy.linalg.solve_banded` (check for an illustration). + These parameters can be also used with ``jac=None`` to reduce the + number of Jacobian elements estimated by finite differences. + vectorized : bool, optional + Whether `fun` may be called in a vectorized fashion. False (default) + is recommended for this solver. + + If ``vectorized`` is False, `fun` will always be called with ``y`` of + shape ``(n,)``, where ``n = len(y0)``. + + If ``vectorized`` is True, `fun` may be called with ``y`` of shape + ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave + such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of + the returned array is the time derivative of the state corresponding + with a column of ``y``). + + Setting ``vectorized=True`` allows for faster finite difference + approximation of the Jacobian by methods 'Radau' and 'BDF', but + will result in slower execution for this solver. + + Attributes + ---------- + n : int + Number of equations. + status : string + Current status of the solver: 'running', 'finished' or 'failed'. + t_bound : float + Boundary time. + direction : float + Integration direction: +1 or -1. + t : float + Current time. + y : ndarray + Current state. + t_old : float + Previous time. None if no steps were made yet. + nfev : int + Number of evaluations of the right-hand side. + njev : int + Number of evaluations of the Jacobian. + + References + ---------- + .. [1] A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE + Solvers," IMACS Transactions on Scientific Computation, Vol 1., + pp. 55-64, 1983. + .. [2] L. Petzold, "Automatic selection of methods for solving stiff and + nonstiff systems of ordinary differential equations", SIAM Journal + on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136-148, + 1983. + """ + def __init__(self, fun, t0, y0, t_bound, first_step=None, min_step=0.0, + max_step=np.inf, rtol=1e-3, atol=1e-6, jac=None, lband=None, + uband=None, vectorized=False, **extraneous): + warn_extraneous(extraneous) + super().__init__(fun, t0, y0, t_bound, vectorized) + + if first_step is None: + first_step = 0 # LSODA value for automatic selection. + else: + first_step = validate_first_step(first_step, t0, t_bound) + + first_step *= self.direction + + if max_step == np.inf: + max_step = 0 # LSODA value for infinity. + elif max_step <= 0: + raise ValueError("`max_step` must be positive.") + + if min_step < 0: + raise ValueError("`min_step` must be nonnegative.") + + rtol, atol = validate_tol(rtol, atol, self.n) + + solver = ode(self.fun, jac) + solver.set_integrator('lsoda', rtol=rtol, atol=atol, max_step=max_step, + min_step=min_step, first_step=first_step, + lband=lband, uband=uband) + solver.set_initial_value(y0, t0) + + # Inject t_bound into rwork array as needed for itask=5. + solver._integrator.rwork[0] = self.t_bound + solver._integrator.call_args[4] = solver._integrator.rwork + + self._lsoda_solver = solver + + def _step_impl(self): + solver = self._lsoda_solver + integrator = solver._integrator + + # From lsoda.step and lsoda.integrate itask=5 means take a single + # step and do not go past t_bound. + itask = integrator.call_args[2] + integrator.call_args[2] = 5 + solver._y, solver.t = integrator.run( + solver.f, solver.jac or (lambda: None), solver._y, solver.t, + self.t_bound, solver.f_params, solver.jac_params) + integrator.call_args[2] = itask + + if solver.successful(): + self.t = solver.t + self.y = solver._y + # From LSODA Fortran source njev is equal to nlu. + self.njev = integrator.iwork[12] + self.nlu = integrator.iwork[12] + return True, None + else: + return False, 'Unexpected istate in LSODA.' + + def _dense_output_impl(self): + iwork = self._lsoda_solver._integrator.iwork + rwork = self._lsoda_solver._integrator.rwork + + # We want to produce the Nordsieck history array, yh, up to the order + # used in the last successful iteration. The step size is unimportant + # because it will be scaled out in LsodaDenseOutput. Some additional + # work may be required because ODEPACK's LSODA implementation produces + # the Nordsieck history in the state needed for the next iteration. + + # iwork[13] contains order from last successful iteration, while + # iwork[14] contains order to be attempted next. + order = iwork[13] + + # rwork[11] contains the step size to be attempted next, while + # rwork[10] contains step size from last successful iteration. + h = rwork[11] + + # rwork[20:20 + (iwork[14] + 1) * self.n] contains entries of the + # Nordsieck array in state needed for next iteration. We want + # the entries up to order for the last successful step so use the + # following. + yh = np.reshape(rwork[20:20 + (order + 1) * self.n], + (self.n, order + 1), order='F').copy() + if iwork[14] < order: + # If the order is set to decrease then the final column of yh + # has not been updated within ODEPACK's LSODA + # implementation because this column will not be used in the + # next iteration. We must rescale this column to make the + # associated step size consistent with the other columns. + yh[:, -1] *= (h / rwork[10]) ** order + + return LsodaDenseOutput(self.t_old, self.t, h, order, yh) + + +class LsodaDenseOutput(DenseOutput): + def __init__(self, t_old, t, h, order, yh): + super().__init__(t_old, t) + self.h = h + self.yh = yh + self.p = np.arange(order + 1) + + def _call_impl(self, t): + if t.ndim == 0: + x = ((t - self.t) / self.h) ** self.p + else: + x = ((t - self.t) / self.h) ** self.p[:, None] + + return np.dot(self.yh, x) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/radau.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/radau.py new file mode 100644 index 0000000000000000000000000000000000000000..0d572b48de51ebc7e8f8fd278ce1000bdef581b5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/radau.py @@ -0,0 +1,572 @@ +import numpy as np +from scipy.linalg import lu_factor, lu_solve +from scipy.sparse import csc_matrix, issparse, eye +from scipy.sparse.linalg import splu +from scipy.optimize._numdiff import group_columns +from .common import (validate_max_step, validate_tol, select_initial_step, + norm, num_jac, EPS, warn_extraneous, + validate_first_step) +from .base import OdeSolver, DenseOutput + +S6 = 6 ** 0.5 + +# Butcher tableau. A is not used directly, see below. +C = np.array([(4 - S6) / 10, (4 + S6) / 10, 1]) +E = np.array([-13 - 7 * S6, -13 + 7 * S6, -1]) / 3 + +# Eigendecomposition of A is done: A = T L T**-1. There is 1 real eigenvalue +# and a complex conjugate pair. They are written below. +MU_REAL = 3 + 3 ** (2 / 3) - 3 ** (1 / 3) +MU_COMPLEX = (3 + 0.5 * (3 ** (1 / 3) - 3 ** (2 / 3)) + - 0.5j * (3 ** (5 / 6) + 3 ** (7 / 6))) + +# These are transformation matrices. +T = np.array([ + [0.09443876248897524, -0.14125529502095421, 0.03002919410514742], + [0.25021312296533332, 0.20412935229379994, -0.38294211275726192], + [1, 1, 0]]) +TI = np.array([ + [4.17871859155190428, 0.32768282076106237, 0.52337644549944951], + [-4.17871859155190428, -0.32768282076106237, 0.47662355450055044], + [0.50287263494578682, -2.57192694985560522, 0.59603920482822492]]) +# These linear combinations are used in the algorithm. +TI_REAL = TI[0] +TI_COMPLEX = TI[1] + 1j * TI[2] + +# Interpolator coefficients. +P = np.array([ + [13/3 + 7*S6/3, -23/3 - 22*S6/3, 10/3 + 5 * S6], + [13/3 - 7*S6/3, -23/3 + 22*S6/3, 10/3 - 5 * S6], + [1/3, -8/3, 10/3]]) + + +NEWTON_MAXITER = 6 # Maximum number of Newton iterations. +MIN_FACTOR = 0.2 # Minimum allowed decrease in a step size. +MAX_FACTOR = 10 # Maximum allowed increase in a step size. + + +def solve_collocation_system(fun, t, y, h, Z0, scale, tol, + LU_real, LU_complex, solve_lu): + """Solve the collocation system. + + Parameters + ---------- + fun : callable + Right-hand side of the system. + t : float + Current time. + y : ndarray, shape (n,) + Current state. + h : float + Step to try. + Z0 : ndarray, shape (3, n) + Initial guess for the solution. It determines new values of `y` at + ``t + h * C`` as ``y + Z0``, where ``C`` is the Radau method constants. + scale : ndarray, shape (n) + Problem tolerance scale, i.e. ``rtol * abs(y) + atol``. + tol : float + Tolerance to which solve the system. This value is compared with + the normalized by `scale` error. + LU_real, LU_complex + LU decompositions of the system Jacobians. + solve_lu : callable + Callable which solves a linear system given a LU decomposition. The + signature is ``solve_lu(LU, b)``. + + Returns + ------- + converged : bool + Whether iterations converged. + n_iter : int + Number of completed iterations. + Z : ndarray, shape (3, n) + Found solution. + rate : float + The rate of convergence. + """ + n = y.shape[0] + M_real = MU_REAL / h + M_complex = MU_COMPLEX / h + + W = TI.dot(Z0) + Z = Z0 + + F = np.empty((3, n)) + ch = h * C + + dW_norm_old = None + dW = np.empty_like(W) + converged = False + rate = None + for k in range(NEWTON_MAXITER): + for i in range(3): + F[i] = fun(t + ch[i], y + Z[i]) + + if not np.all(np.isfinite(F)): + break + + f_real = F.T.dot(TI_REAL) - M_real * W[0] + f_complex = F.T.dot(TI_COMPLEX) - M_complex * (W[1] + 1j * W[2]) + + dW_real = solve_lu(LU_real, f_real) + dW_complex = solve_lu(LU_complex, f_complex) + + dW[0] = dW_real + dW[1] = dW_complex.real + dW[2] = dW_complex.imag + + dW_norm = norm(dW / scale) + if dW_norm_old is not None: + rate = dW_norm / dW_norm_old + + if (rate is not None and (rate >= 1 or + rate ** (NEWTON_MAXITER - k) / (1 - rate) * dW_norm > tol)): + break + + W += dW + Z = T.dot(W) + + if (dW_norm == 0 or + rate is not None and rate / (1 - rate) * dW_norm < tol): + converged = True + break + + dW_norm_old = dW_norm + + return converged, k + 1, Z, rate + + +def predict_factor(h_abs, h_abs_old, error_norm, error_norm_old): + """Predict by which factor to increase/decrease the step size. + + The algorithm is described in [1]_. + + Parameters + ---------- + h_abs, h_abs_old : float + Current and previous values of the step size, `h_abs_old` can be None + (see Notes). + error_norm, error_norm_old : float + Current and previous values of the error norm, `error_norm_old` can + be None (see Notes). + + Returns + ------- + factor : float + Predicted factor. + + Notes + ----- + If `h_abs_old` and `error_norm_old` are both not None then a two-step + algorithm is used, otherwise a one-step algorithm is used. + + References + ---------- + .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential + Equations II: Stiff and Differential-Algebraic Problems", Sec. IV.8. + """ + if error_norm_old is None or h_abs_old is None or error_norm == 0: + multiplier = 1 + else: + multiplier = h_abs / h_abs_old * (error_norm_old / error_norm) ** 0.25 + + with np.errstate(divide='ignore'): + factor = min(1, multiplier) * error_norm ** -0.25 + + return factor + + +class Radau(OdeSolver): + """Implicit Runge-Kutta method of Radau IIA family of order 5. + + The implementation follows [1]_. The error is controlled with a + third-order accurate embedded formula. A cubic polynomial which satisfies + the collocation conditions is used for the dense output. + + Parameters + ---------- + fun : callable + Right-hand side of the system: the time derivative of the state ``y`` + at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a + scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must + return an array of the same shape as ``y``. See `vectorized` for more + information. + t0 : float + Initial time. + y0 : array_like, shape (n,) + Initial state. + t_bound : float + Boundary time - the integration won't continue beyond it. It also + determines the direction of the integration. + first_step : float or None, optional + Initial step size. Default is ``None`` which means that the algorithm + should choose. + max_step : float, optional + Maximum allowed step size. Default is np.inf, i.e., the step size is not + bounded and determined solely by the solver. + rtol, atol : float and array_like, optional + Relative and absolute tolerances. The solver keeps the local error + estimates less than ``atol + rtol * abs(y)``. HHere `rtol` controls a + relative accuracy (number of correct digits), while `atol` controls + absolute accuracy (number of correct decimal places). To achieve the + desired `rtol`, set `atol` to be smaller than the smallest value that + can be expected from ``rtol * abs(y)`` so that `rtol` dominates the + allowable error. If `atol` is larger than ``rtol * abs(y)`` the + number of correct digits is not guaranteed. Conversely, to achieve the + desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller + than `atol`. If components of y have different scales, it might be + beneficial to set different `atol` values for different components by + passing array_like with shape (n,) for `atol`. Default values are + 1e-3 for `rtol` and 1e-6 for `atol`. + jac : {None, array_like, sparse_matrix, callable}, optional + Jacobian matrix of the right-hand side of the system with respect to + y, required by this method. The Jacobian matrix has shape (n, n) and + its element (i, j) is equal to ``d f_i / d y_j``. + There are three ways to define the Jacobian: + + * If array_like or sparse_matrix, the Jacobian is assumed to + be constant. + * If callable, the Jacobian is assumed to depend on both + t and y; it will be called as ``jac(t, y)`` as necessary. + For the 'Radau' and 'BDF' methods, the return value might be a + sparse matrix. + * If None (default), the Jacobian will be approximated by + finite differences. + + It is generally recommended to provide the Jacobian rather than + relying on a finite-difference approximation. + jac_sparsity : {None, array_like, sparse matrix}, optional + Defines a sparsity structure of the Jacobian matrix for a + finite-difference approximation. Its shape must be (n, n). This argument + is ignored if `jac` is not `None`. If the Jacobian has only few non-zero + elements in *each* row, providing the sparsity structure will greatly + speed up the computations [2]_. A zero entry means that a corresponding + element in the Jacobian is always zero. If None (default), the Jacobian + is assumed to be dense. + vectorized : bool, optional + Whether `fun` can be called in a vectorized fashion. Default is False. + + If ``vectorized`` is False, `fun` will always be called with ``y`` of + shape ``(n,)``, where ``n = len(y0)``. + + If ``vectorized`` is True, `fun` may be called with ``y`` of shape + ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave + such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of + the returned array is the time derivative of the state corresponding + with a column of ``y``). + + Setting ``vectorized=True`` allows for faster finite difference + approximation of the Jacobian by this method, but may result in slower + execution overall in some circumstances (e.g. small ``len(y0)``). + + Attributes + ---------- + n : int + Number of equations. + status : string + Current status of the solver: 'running', 'finished' or 'failed'. + t_bound : float + Boundary time. + direction : float + Integration direction: +1 or -1. + t : float + Current time. + y : ndarray + Current state. + t_old : float + Previous time. None if no steps were made yet. + step_size : float + Size of the last successful step. None if no steps were made yet. + nfev : int + Number of evaluations of the right-hand side. + njev : int + Number of evaluations of the Jacobian. + nlu : int + Number of LU decompositions. + + References + ---------- + .. [1] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II: + Stiff and Differential-Algebraic Problems", Sec. IV.8. + .. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of + sparse Jacobian matrices", Journal of the Institute of Mathematics + and its Applications, 13, pp. 117-120, 1974. + """ + def __init__(self, fun, t0, y0, t_bound, max_step=np.inf, + rtol=1e-3, atol=1e-6, jac=None, jac_sparsity=None, + vectorized=False, first_step=None, **extraneous): + warn_extraneous(extraneous) + super().__init__(fun, t0, y0, t_bound, vectorized) + self.y_old = None + self.max_step = validate_max_step(max_step) + self.rtol, self.atol = validate_tol(rtol, atol, self.n) + self.f = self.fun(self.t, self.y) + # Select initial step assuming the same order which is used to control + # the error. + if first_step is None: + self.h_abs = select_initial_step( + self.fun, self.t, self.y, t_bound, max_step, self.f, self.direction, + 3, self.rtol, self.atol) + else: + self.h_abs = validate_first_step(first_step, t0, t_bound) + self.h_abs_old = None + self.error_norm_old = None + + self.newton_tol = max(10 * EPS / rtol, min(0.03, rtol ** 0.5)) + self.sol = None + + self.jac_factor = None + self.jac, self.J = self._validate_jac(jac, jac_sparsity) + if issparse(self.J): + def lu(A): + self.nlu += 1 + return splu(A) + + def solve_lu(LU, b): + return LU.solve(b) + + I = eye(self.n, format='csc') + else: + def lu(A): + self.nlu += 1 + return lu_factor(A, overwrite_a=True) + + def solve_lu(LU, b): + return lu_solve(LU, b, overwrite_b=True) + + I = np.identity(self.n) + + self.lu = lu + self.solve_lu = solve_lu + self.I = I + + self.current_jac = True + self.LU_real = None + self.LU_complex = None + self.Z = None + + def _validate_jac(self, jac, sparsity): + t0 = self.t + y0 = self.y + + if jac is None: + if sparsity is not None: + if issparse(sparsity): + sparsity = csc_matrix(sparsity) + groups = group_columns(sparsity) + sparsity = (sparsity, groups) + + def jac_wrapped(t, y, f): + self.njev += 1 + J, self.jac_factor = num_jac(self.fun_vectorized, t, y, f, + self.atol, self.jac_factor, + sparsity) + return J + J = jac_wrapped(t0, y0, self.f) + elif callable(jac): + J = jac(t0, y0) + self.njev = 1 + if issparse(J): + J = csc_matrix(J) + + def jac_wrapped(t, y, _=None): + self.njev += 1 + return csc_matrix(jac(t, y), dtype=float) + + else: + J = np.asarray(J, dtype=float) + + def jac_wrapped(t, y, _=None): + self.njev += 1 + return np.asarray(jac(t, y), dtype=float) + + if J.shape != (self.n, self.n): + raise ValueError(f"`jac` is expected to have shape {(self.n, self.n)}," + f" but actually has {J.shape}.") + else: + if issparse(jac): + J = csc_matrix(jac) + else: + J = np.asarray(jac, dtype=float) + + if J.shape != (self.n, self.n): + raise ValueError(f"`jac` is expected to have shape {(self.n, self.n)}," + f" but actually has {J.shape}.") + jac_wrapped = None + + return jac_wrapped, J + + def _step_impl(self): + t = self.t + y = self.y + f = self.f + + max_step = self.max_step + atol = self.atol + rtol = self.rtol + + min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t) + if self.h_abs > max_step: + h_abs = max_step + h_abs_old = None + error_norm_old = None + elif self.h_abs < min_step: + h_abs = min_step + h_abs_old = None + error_norm_old = None + else: + h_abs = self.h_abs + h_abs_old = self.h_abs_old + error_norm_old = self.error_norm_old + + J = self.J + LU_real = self.LU_real + LU_complex = self.LU_complex + + current_jac = self.current_jac + jac = self.jac + + rejected = False + step_accepted = False + message = None + while not step_accepted: + if h_abs < min_step: + return False, self.TOO_SMALL_STEP + + h = h_abs * self.direction + t_new = t + h + + if self.direction * (t_new - self.t_bound) > 0: + t_new = self.t_bound + + h = t_new - t + h_abs = np.abs(h) + + if self.sol is None: + Z0 = np.zeros((3, y.shape[0])) + else: + Z0 = self.sol(t + h * C).T - y + + scale = atol + np.abs(y) * rtol + + converged = False + while not converged: + if LU_real is None or LU_complex is None: + LU_real = self.lu(MU_REAL / h * self.I - J) + LU_complex = self.lu(MU_COMPLEX / h * self.I - J) + + converged, n_iter, Z, rate = solve_collocation_system( + self.fun, t, y, h, Z0, scale, self.newton_tol, + LU_real, LU_complex, self.solve_lu) + + if not converged: + if current_jac: + break + + J = self.jac(t, y, f) + current_jac = True + LU_real = None + LU_complex = None + + if not converged: + h_abs *= 0.5 + LU_real = None + LU_complex = None + continue + + y_new = y + Z[-1] + ZE = Z.T.dot(E) / h + error = self.solve_lu(LU_real, f + ZE) + scale = atol + np.maximum(np.abs(y), np.abs(y_new)) * rtol + error_norm = norm(error / scale) + safety = 0.9 * (2 * NEWTON_MAXITER + 1) / (2 * NEWTON_MAXITER + + n_iter) + + if rejected and error_norm > 1: + error = self.solve_lu(LU_real, self.fun(t, y + error) + ZE) + error_norm = norm(error / scale) + + if error_norm > 1: + factor = predict_factor(h_abs, h_abs_old, + error_norm, error_norm_old) + h_abs *= max(MIN_FACTOR, safety * factor) + + LU_real = None + LU_complex = None + rejected = True + else: + step_accepted = True + + recompute_jac = jac is not None and n_iter > 2 and rate > 1e-3 + + factor = predict_factor(h_abs, h_abs_old, error_norm, error_norm_old) + factor = min(MAX_FACTOR, safety * factor) + + if not recompute_jac and factor < 1.2: + factor = 1 + else: + LU_real = None + LU_complex = None + + f_new = self.fun(t_new, y_new) + if recompute_jac: + J = jac(t_new, y_new, f_new) + current_jac = True + elif jac is not None: + current_jac = False + + self.h_abs_old = self.h_abs + self.error_norm_old = error_norm + + self.h_abs = h_abs * factor + + self.y_old = y + + self.t = t_new + self.y = y_new + self.f = f_new + + self.Z = Z + + self.LU_real = LU_real + self.LU_complex = LU_complex + self.current_jac = current_jac + self.J = J + + self.t_old = t + self.sol = self._compute_dense_output() + + return step_accepted, message + + def _compute_dense_output(self): + Q = np.dot(self.Z.T, P) + return RadauDenseOutput(self.t_old, self.t, self.y_old, Q) + + def _dense_output_impl(self): + return self.sol + + +class RadauDenseOutput(DenseOutput): + def __init__(self, t_old, t, y_old, Q): + super().__init__(t_old, t) + self.h = t - t_old + self.Q = Q + self.order = Q.shape[1] - 1 + self.y_old = y_old + + def _call_impl(self, t): + x = (t - self.t_old) / self.h + if t.ndim == 0: + p = np.tile(x, self.order + 1) + p = np.cumprod(p) + else: + p = np.tile(x, (self.order + 1, 1)) + p = np.cumprod(p, axis=0) + # Here we don't multiply by h, not a mistake. + y = np.dot(self.Q, p) + if y.ndim == 2: + y += self.y_old[:, None] + else: + y += self.y_old + + return y diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/rk.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/rk.py new file mode 100644 index 0000000000000000000000000000000000000000..62a5347ffe91afc754e9b818d0b34c010d0c4d12 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/rk.py @@ -0,0 +1,601 @@ +import numpy as np +from .base import OdeSolver, DenseOutput +from .common import (validate_max_step, validate_tol, select_initial_step, + norm, warn_extraneous, validate_first_step) +from . import dop853_coefficients + +# Multiply steps computed from asymptotic behaviour of errors by this. +SAFETY = 0.9 + +MIN_FACTOR = 0.2 # Minimum allowed decrease in a step size. +MAX_FACTOR = 10 # Maximum allowed increase in a step size. + + +def rk_step(fun, t, y, f, h, A, B, C, K): + """Perform a single Runge-Kutta step. + + This function computes a prediction of an explicit Runge-Kutta method and + also estimates the error of a less accurate method. + + Notation for Butcher tableau is as in [1]_. + + Parameters + ---------- + fun : callable + Right-hand side of the system. + t : float + Current time. + y : ndarray, shape (n,) + Current state. + f : ndarray, shape (n,) + Current value of the derivative, i.e., ``fun(x, y)``. + h : float + Step to use. + A : ndarray, shape (n_stages, n_stages) + Coefficients for combining previous RK stages to compute the next + stage. For explicit methods the coefficients at and above the main + diagonal are zeros. + B : ndarray, shape (n_stages,) + Coefficients for combining RK stages for computing the final + prediction. + C : ndarray, shape (n_stages,) + Coefficients for incrementing time for consecutive RK stages. + The value for the first stage is always zero. + K : ndarray, shape (n_stages + 1, n) + Storage array for putting RK stages here. Stages are stored in rows. + The last row is a linear combination of the previous rows with + coefficients + + Returns + ------- + y_new : ndarray, shape (n,) + Solution at t + h computed with a higher accuracy. + f_new : ndarray, shape (n,) + Derivative ``fun(t + h, y_new)``. + + References + ---------- + .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential + Equations I: Nonstiff Problems", Sec. II.4. + """ + K[0] = f + for s, (a, c) in enumerate(zip(A[1:], C[1:]), start=1): + dy = np.dot(K[:s].T, a[:s]) * h + K[s] = fun(t + c * h, y + dy) + + y_new = y + h * np.dot(K[:-1].T, B) + f_new = fun(t + h, y_new) + + K[-1] = f_new + + return y_new, f_new + + +class RungeKutta(OdeSolver): + """Base class for explicit Runge-Kutta methods.""" + C: np.ndarray = NotImplemented + A: np.ndarray = NotImplemented + B: np.ndarray = NotImplemented + E: np.ndarray = NotImplemented + P: np.ndarray = NotImplemented + order: int = NotImplemented + error_estimator_order: int = NotImplemented + n_stages: int = NotImplemented + + def __init__(self, fun, t0, y0, t_bound, max_step=np.inf, + rtol=1e-3, atol=1e-6, vectorized=False, + first_step=None, **extraneous): + warn_extraneous(extraneous) + super().__init__(fun, t0, y0, t_bound, vectorized, + support_complex=True) + self.y_old = None + self.max_step = validate_max_step(max_step) + self.rtol, self.atol = validate_tol(rtol, atol, self.n) + self.f = self.fun(self.t, self.y) + if first_step is None: + self.h_abs = select_initial_step( + self.fun, self.t, self.y, t_bound, max_step, self.f, self.direction, + self.error_estimator_order, self.rtol, self.atol) + else: + self.h_abs = validate_first_step(first_step, t0, t_bound) + self.K = np.empty((self.n_stages + 1, self.n), dtype=self.y.dtype) + self.error_exponent = -1 / (self.error_estimator_order + 1) + self.h_previous = None + + def _estimate_error(self, K, h): + return np.dot(K.T, self.E) * h + + def _estimate_error_norm(self, K, h, scale): + return norm(self._estimate_error(K, h) / scale) + + def _step_impl(self): + t = self.t + y = self.y + + max_step = self.max_step + rtol = self.rtol + atol = self.atol + + min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t) + + if self.h_abs > max_step: + h_abs = max_step + elif self.h_abs < min_step: + h_abs = min_step + else: + h_abs = self.h_abs + + step_accepted = False + step_rejected = False + + while not step_accepted: + if h_abs < min_step: + return False, self.TOO_SMALL_STEP + + h = h_abs * self.direction + t_new = t + h + + if self.direction * (t_new - self.t_bound) > 0: + t_new = self.t_bound + + h = t_new - t + h_abs = np.abs(h) + + y_new, f_new = rk_step(self.fun, t, y, self.f, h, self.A, + self.B, self.C, self.K) + scale = atol + np.maximum(np.abs(y), np.abs(y_new)) * rtol + error_norm = self._estimate_error_norm(self.K, h, scale) + + if error_norm < 1: + if error_norm == 0: + factor = MAX_FACTOR + else: + factor = min(MAX_FACTOR, + SAFETY * error_norm ** self.error_exponent) + + if step_rejected: + factor = min(1, factor) + + h_abs *= factor + + step_accepted = True + else: + h_abs *= max(MIN_FACTOR, + SAFETY * error_norm ** self.error_exponent) + step_rejected = True + + self.h_previous = h + self.y_old = y + + self.t = t_new + self.y = y_new + + self.h_abs = h_abs + self.f = f_new + + return True, None + + def _dense_output_impl(self): + Q = self.K.T.dot(self.P) + return RkDenseOutput(self.t_old, self.t, self.y_old, Q) + + +class RK23(RungeKutta): + """Explicit Runge-Kutta method of order 3(2). + + This uses the Bogacki-Shampine pair of formulas [1]_. The error is controlled + assuming accuracy of the second-order method, but steps are taken using the + third-order accurate formula (local extrapolation is done). A cubic Hermite + polynomial is used for the dense output. + + Can be applied in the complex domain. + + Parameters + ---------- + fun : callable + Right-hand side of the system: the time derivative of the state ``y`` + at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a + scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must + return an array of the same shape as ``y``. See `vectorized` for more + information. + t0 : float + Initial time. + y0 : array_like, shape (n,) + Initial state. + t_bound : float + Boundary time - the integration won't continue beyond it. It also + determines the direction of the integration. + first_step : float or None, optional + Initial step size. Default is ``None`` which means that the algorithm + should choose. + max_step : float, optional + Maximum allowed step size. Default is np.inf, i.e., the step size is not + bounded and determined solely by the solver. + rtol, atol : float and array_like, optional + Relative and absolute tolerances. The solver keeps the local error + estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a + relative accuracy (number of correct digits), while `atol` controls + absolute accuracy (number of correct decimal places). To achieve the + desired `rtol`, set `atol` to be smaller than the smallest value that + can be expected from ``rtol * abs(y)`` so that `rtol` dominates the + allowable error. If `atol` is larger than ``rtol * abs(y)`` the + number of correct digits is not guaranteed. Conversely, to achieve the + desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller + than `atol`. If components of y have different scales, it might be + beneficial to set different `atol` values for different components by + passing array_like with shape (n,) for `atol`. Default values are + 1e-3 for `rtol` and 1e-6 for `atol`. + vectorized : bool, optional + Whether `fun` may be called in a vectorized fashion. False (default) + is recommended for this solver. + + If ``vectorized`` is False, `fun` will always be called with ``y`` of + shape ``(n,)``, where ``n = len(y0)``. + + If ``vectorized`` is True, `fun` may be called with ``y`` of shape + ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave + such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of + the returned array is the time derivative of the state corresponding + with a column of ``y``). + + Setting ``vectorized=True`` allows for faster finite difference + approximation of the Jacobian by methods 'Radau' and 'BDF', but + will result in slower execution for this solver. + + Attributes + ---------- + n : int + Number of equations. + status : string + Current status of the solver: 'running', 'finished' or 'failed'. + t_bound : float + Boundary time. + direction : float + Integration direction: +1 or -1. + t : float + Current time. + y : ndarray + Current state. + t_old : float + Previous time. None if no steps were made yet. + step_size : float + Size of the last successful step. None if no steps were made yet. + nfev : int + Number evaluations of the system's right-hand side. + njev : int + Number of evaluations of the Jacobian. + Is always 0 for this solver as it does not use the Jacobian. + nlu : int + Number of LU decompositions. Is always 0 for this solver. + + References + ---------- + .. [1] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas", + Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989. + """ + order = 3 + error_estimator_order = 2 + n_stages = 3 + C = np.array([0, 1/2, 3/4]) + A = np.array([ + [0, 0, 0], + [1/2, 0, 0], + [0, 3/4, 0] + ]) + B = np.array([2/9, 1/3, 4/9]) + E = np.array([5/72, -1/12, -1/9, 1/8]) + P = np.array([[1, -4 / 3, 5 / 9], + [0, 1, -2/3], + [0, 4/3, -8/9], + [0, -1, 1]]) + + +class RK45(RungeKutta): + """Explicit Runge-Kutta method of order 5(4). + + This uses the Dormand-Prince pair of formulas [1]_. The error is controlled + assuming accuracy of the fourth-order method accuracy, but steps are taken + using the fifth-order accurate formula (local extrapolation is done). + A quartic interpolation polynomial is used for the dense output [2]_. + + Can be applied in the complex domain. + + Parameters + ---------- + fun : callable + Right-hand side of the system. The calling signature is ``fun(t, y)``. + Here ``t`` is a scalar, and there are two options for the ndarray ``y``: + It can either have shape (n,); then ``fun`` must return array_like with + shape (n,). Alternatively it can have shape (n, k); then ``fun`` + must return an array_like with shape (n, k), i.e., each column + corresponds to a single column in ``y``. The choice between the two + options is determined by `vectorized` argument (see below). + t0 : float + Initial time. + y0 : array_like, shape (n,) + Initial state. + t_bound : float + Boundary time - the integration won't continue beyond it. It also + determines the direction of the integration. + first_step : float or None, optional + Initial step size. Default is ``None`` which means that the algorithm + should choose. + max_step : float, optional + Maximum allowed step size. Default is np.inf, i.e., the step size is not + bounded and determined solely by the solver. + rtol, atol : float and array_like, optional + Relative and absolute tolerances. The solver keeps the local error + estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a + relative accuracy (number of correct digits), while `atol` controls + absolute accuracy (number of correct decimal places). To achieve the + desired `rtol`, set `atol` to be smaller than the smallest value that + can be expected from ``rtol * abs(y)`` so that `rtol` dominates the + allowable error. If `atol` is larger than ``rtol * abs(y)`` the + number of correct digits is not guaranteed. Conversely, to achieve the + desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller + than `atol`. If components of y have different scales, it might be + beneficial to set different `atol` values for different components by + passing array_like with shape (n,) for `atol`. Default values are + 1e-3 for `rtol` and 1e-6 for `atol`. + vectorized : bool, optional + Whether `fun` is implemented in a vectorized fashion. Default is False. + + Attributes + ---------- + n : int + Number of equations. + status : string + Current status of the solver: 'running', 'finished' or 'failed'. + t_bound : float + Boundary time. + direction : float + Integration direction: +1 or -1. + t : float + Current time. + y : ndarray + Current state. + t_old : float + Previous time. None if no steps were made yet. + step_size : float + Size of the last successful step. None if no steps were made yet. + nfev : int + Number evaluations of the system's right-hand side. + njev : int + Number of evaluations of the Jacobian. + Is always 0 for this solver as it does not use the Jacobian. + nlu : int + Number of LU decompositions. Is always 0 for this solver. + + References + ---------- + .. [1] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta + formulae", Journal of Computational and Applied Mathematics, Vol. 6, + No. 1, pp. 19-26, 1980. + .. [2] L. W. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics + of Computation,, Vol. 46, No. 173, pp. 135-150, 1986. + """ + order = 5 + error_estimator_order = 4 + n_stages = 6 + C = np.array([0, 1/5, 3/10, 4/5, 8/9, 1]) + A = np.array([ + [0, 0, 0, 0, 0], + [1/5, 0, 0, 0, 0], + [3/40, 9/40, 0, 0, 0], + [44/45, -56/15, 32/9, 0, 0], + [19372/6561, -25360/2187, 64448/6561, -212/729, 0], + [9017/3168, -355/33, 46732/5247, 49/176, -5103/18656] + ]) + B = np.array([35/384, 0, 500/1113, 125/192, -2187/6784, 11/84]) + E = np.array([-71/57600, 0, 71/16695, -71/1920, 17253/339200, -22/525, + 1/40]) + # Corresponds to the optimum value of c_6 from [2]_. + P = np.array([ + [1, -8048581381/2820520608, 8663915743/2820520608, + -12715105075/11282082432], + [0, 0, 0, 0], + [0, 131558114200/32700410799, -68118460800/10900136933, + 87487479700/32700410799], + [0, -1754552775/470086768, 14199869525/1410260304, + -10690763975/1880347072], + [0, 127303824393/49829197408, -318862633887/49829197408, + 701980252875 / 199316789632], + [0, -282668133/205662961, 2019193451/616988883, -1453857185/822651844], + [0, 40617522/29380423, -110615467/29380423, 69997945/29380423]]) + + +class DOP853(RungeKutta): + """Explicit Runge-Kutta method of order 8. + + This is a Python implementation of "DOP853" algorithm originally written + in Fortran [1]_, [2]_. Note that this is not a literal translation, but + the algorithmic core and coefficients are the same. + + Can be applied in the complex domain. + + Parameters + ---------- + fun : callable + Right-hand side of the system. The calling signature is ``fun(t, y)``. + Here, ``t`` is a scalar, and there are two options for the ndarray ``y``: + It can either have shape (n,); then ``fun`` must return array_like with + shape (n,). Alternatively it can have shape (n, k); then ``fun`` + must return an array_like with shape (n, k), i.e. each column + corresponds to a single column in ``y``. The choice between the two + options is determined by `vectorized` argument (see below). + t0 : float + Initial time. + y0 : array_like, shape (n,) + Initial state. + t_bound : float + Boundary time - the integration won't continue beyond it. It also + determines the direction of the integration. + first_step : float or None, optional + Initial step size. Default is ``None`` which means that the algorithm + should choose. + max_step : float, optional + Maximum allowed step size. Default is np.inf, i.e. the step size is not + bounded and determined solely by the solver. + rtol, atol : float and array_like, optional + Relative and absolute tolerances. The solver keeps the local error + estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a + relative accuracy (number of correct digits), while `atol` controls + absolute accuracy (number of correct decimal places). To achieve the + desired `rtol`, set `atol` to be smaller than the smallest value that + can be expected from ``rtol * abs(y)`` so that `rtol` dominates the + allowable error. If `atol` is larger than ``rtol * abs(y)`` the + number of correct digits is not guaranteed. Conversely, to achieve the + desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller + than `atol`. If components of y have different scales, it might be + beneficial to set different `atol` values for different components by + passing array_like with shape (n,) for `atol`. Default values are + 1e-3 for `rtol` and 1e-6 for `atol`. + vectorized : bool, optional + Whether `fun` is implemented in a vectorized fashion. Default is False. + + Attributes + ---------- + n : int + Number of equations. + status : string + Current status of the solver: 'running', 'finished' or 'failed'. + t_bound : float + Boundary time. + direction : float + Integration direction: +1 or -1. + t : float + Current time. + y : ndarray + Current state. + t_old : float + Previous time. None if no steps were made yet. + step_size : float + Size of the last successful step. None if no steps were made yet. + nfev : int + Number evaluations of the system's right-hand side. + njev : int + Number of evaluations of the Jacobian. Is always 0 for this solver + as it does not use the Jacobian. + nlu : int + Number of LU decompositions. Is always 0 for this solver. + + References + ---------- + .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential + Equations I: Nonstiff Problems", Sec. II. + .. [2] `Page with original Fortran code of DOP853 + `_. + """ + n_stages = dop853_coefficients.N_STAGES + order = 8 + error_estimator_order = 7 + A = dop853_coefficients.A[:n_stages, :n_stages] + B = dop853_coefficients.B + C = dop853_coefficients.C[:n_stages] + E3 = dop853_coefficients.E3 + E5 = dop853_coefficients.E5 + D = dop853_coefficients.D + + A_EXTRA = dop853_coefficients.A[n_stages + 1:] + C_EXTRA = dop853_coefficients.C[n_stages + 1:] + + def __init__(self, fun, t0, y0, t_bound, max_step=np.inf, + rtol=1e-3, atol=1e-6, vectorized=False, + first_step=None, **extraneous): + super().__init__(fun, t0, y0, t_bound, max_step, rtol, atol, + vectorized, first_step, **extraneous) + self.K_extended = np.empty((dop853_coefficients.N_STAGES_EXTENDED, + self.n), dtype=self.y.dtype) + self.K = self.K_extended[:self.n_stages + 1] + + def _estimate_error(self, K, h): # Left for testing purposes. + err5 = np.dot(K.T, self.E5) + err3 = np.dot(K.T, self.E3) + denom = np.hypot(np.abs(err5), 0.1 * np.abs(err3)) + correction_factor = np.ones_like(err5) + mask = denom > 0 + correction_factor[mask] = np.abs(err5[mask]) / denom[mask] + return h * err5 * correction_factor + + def _estimate_error_norm(self, K, h, scale): + err5 = np.dot(K.T, self.E5) / scale + err3 = np.dot(K.T, self.E3) / scale + err5_norm_2 = np.linalg.norm(err5)**2 + err3_norm_2 = np.linalg.norm(err3)**2 + if err5_norm_2 == 0 and err3_norm_2 == 0: + return 0.0 + denom = err5_norm_2 + 0.01 * err3_norm_2 + return np.abs(h) * err5_norm_2 / np.sqrt(denom * len(scale)) + + def _dense_output_impl(self): + K = self.K_extended + h = self.h_previous + for s, (a, c) in enumerate(zip(self.A_EXTRA, self.C_EXTRA), + start=self.n_stages + 1): + dy = np.dot(K[:s].T, a[:s]) * h + K[s] = self.fun(self.t_old + c * h, self.y_old + dy) + + F = np.empty((dop853_coefficients.INTERPOLATOR_POWER, self.n), + dtype=self.y_old.dtype) + + f_old = K[0] + delta_y = self.y - self.y_old + + F[0] = delta_y + F[1] = h * f_old - delta_y + F[2] = 2 * delta_y - h * (self.f + f_old) + F[3:] = h * np.dot(self.D, K) + + return Dop853DenseOutput(self.t_old, self.t, self.y_old, F) + + +class RkDenseOutput(DenseOutput): + def __init__(self, t_old, t, y_old, Q): + super().__init__(t_old, t) + self.h = t - t_old + self.Q = Q + self.order = Q.shape[1] - 1 + self.y_old = y_old + + def _call_impl(self, t): + x = (t - self.t_old) / self.h + if t.ndim == 0: + p = np.tile(x, self.order + 1) + p = np.cumprod(p) + else: + p = np.tile(x, (self.order + 1, 1)) + p = np.cumprod(p, axis=0) + y = self.h * np.dot(self.Q, p) + if y.ndim == 2: + y += self.y_old[:, None] + else: + y += self.y_old + + return y + + +class Dop853DenseOutput(DenseOutput): + def __init__(self, t_old, t, y_old, F): + super().__init__(t_old, t) + self.h = t - t_old + self.F = F + self.y_old = y_old + + def _call_impl(self, t): + x = (t - self.t_old) / self.h + + if t.ndim == 0: + y = np.zeros_like(self.y_old) + else: + x = x[:, None] + y = np.zeros((len(x), len(self.y_old)), dtype=self.y_old.dtype) + + for i, f in enumerate(reversed(self.F)): + y += f + if i % 2 == 0: + y *= x + else: + y *= 1 - x + y += self.y_old + + return y.T diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/tests/test_ivp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/tests/test_ivp.py new file mode 100644 index 0000000000000000000000000000000000000000..cd318b9a165051293ac13b9b0e63be2df322963b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/tests/test_ivp.py @@ -0,0 +1,1287 @@ +from itertools import product +from numpy.testing import (assert_, assert_allclose, assert_array_less, + assert_equal, assert_no_warnings, suppress_warnings) +import pytest +from pytest import raises as assert_raises +import numpy as np +from scipy.optimize._numdiff import group_columns +from scipy.integrate import solve_ivp, RK23, RK45, DOP853, Radau, BDF, LSODA +from scipy.integrate import OdeSolution +from scipy.integrate._ivp.common import num_jac, select_initial_step +from scipy.integrate._ivp.base import ConstantDenseOutput +from scipy.sparse import coo_matrix, csc_matrix + + +def fun_zero(t, y): + return np.zeros_like(y) + + +def fun_linear(t, y): + return np.array([-y[0] - 5 * y[1], y[0] + y[1]]) + + +def jac_linear(): + return np.array([[-1, -5], [1, 1]]) + + +def sol_linear(t): + return np.vstack((-5 * np.sin(2 * t), + 2 * np.cos(2 * t) + np.sin(2 * t))) + + +def fun_rational(t, y): + return np.array([y[1] / t, + y[1] * (y[0] + 2 * y[1] - 1) / (t * (y[0] - 1))]) + + +def fun_rational_vectorized(t, y): + return np.vstack((y[1] / t, + y[1] * (y[0] + 2 * y[1] - 1) / (t * (y[0] - 1)))) + + +def jac_rational(t, y): + return np.array([ + [0, 1 / t], + [-2 * y[1] ** 2 / (t * (y[0] - 1) ** 2), + (y[0] + 4 * y[1] - 1) / (t * (y[0] - 1))] + ]) + + +def jac_rational_sparse(t, y): + return csc_matrix([ + [0, 1 / t], + [-2 * y[1] ** 2 / (t * (y[0] - 1) ** 2), + (y[0] + 4 * y[1] - 1) / (t * (y[0] - 1))] + ]) + + +def sol_rational(t): + return np.asarray((t / (t + 10), 10 * t / (t + 10) ** 2)) + + +def fun_medazko(t, y): + n = y.shape[0] // 2 + k = 100 + c = 4 + + phi = 2 if t <= 5 else 0 + y = np.hstack((phi, 0, y, y[-2])) + + d = 1 / n + j = np.arange(n) + 1 + alpha = 2 * (j * d - 1) ** 3 / c ** 2 + beta = (j * d - 1) ** 4 / c ** 2 + + j_2_p1 = 2 * j + 2 + j_2_m3 = 2 * j - 2 + j_2_m1 = 2 * j + j_2 = 2 * j + 1 + + f = np.empty(2 * n) + f[::2] = (alpha * (y[j_2_p1] - y[j_2_m3]) / (2 * d) + + beta * (y[j_2_m3] - 2 * y[j_2_m1] + y[j_2_p1]) / d ** 2 - + k * y[j_2_m1] * y[j_2]) + f[1::2] = -k * y[j_2] * y[j_2_m1] + + return f + + +def medazko_sparsity(n): + cols = [] + rows = [] + + i = np.arange(n) * 2 + + cols.append(i[1:]) + rows.append(i[1:] - 2) + + cols.append(i) + rows.append(i) + + cols.append(i) + rows.append(i + 1) + + cols.append(i[:-1]) + rows.append(i[:-1] + 2) + + i = np.arange(n) * 2 + 1 + + cols.append(i) + rows.append(i) + + cols.append(i) + rows.append(i - 1) + + cols = np.hstack(cols) + rows = np.hstack(rows) + + return coo_matrix((np.ones_like(cols), (cols, rows))) + + +def fun_complex(t, y): + return -y + + +def jac_complex(t, y): + return -np.eye(y.shape[0]) + + +def jac_complex_sparse(t, y): + return csc_matrix(jac_complex(t, y)) + + +def sol_complex(t): + y = (0.5 + 1j) * np.exp(-t) + return y.reshape((1, -1)) + + +def fun_event_dense_output_LSODA(t, y): + return y * (t - 2) + + +def jac_event_dense_output_LSODA(t, y): + return t - 2 + + +def sol_event_dense_output_LSODA(t): + return np.exp(t ** 2 / 2 - 2 * t + np.log(0.05) - 6) + + +def compute_error(y, y_true, rtol, atol): + e = (y - y_true) / (atol + rtol * np.abs(y_true)) + return np.linalg.norm(e, axis=0) / np.sqrt(e.shape[0]) + +def test_duplicate_timestamps(): + def upward_cannon(t, y): + return [y[1], -9.80665] + + def hit_ground(t, y): + return y[0] + + hit_ground.terminal = True + hit_ground.direction = -1 + + sol = solve_ivp(upward_cannon, [0, np.inf], [0, 0.01], + max_step=0.05 * 0.001 / 9.80665, + events=hit_ground, dense_output=True) + assert_allclose(sol.sol(0.01), np.asarray([-0.00039033, -0.08806632]), + rtol=1e-5, atol=1e-8) + assert_allclose(sol.t_events, np.asarray([[0.00203943]]), rtol=1e-5, atol=1e-8) + assert_allclose(sol.y_events, [np.asarray([[ 0.0, -0.01 ]])], atol=1e-9) + assert sol.success + assert_equal(sol.status, 1) + +@pytest.mark.thread_unsafe +def test_integration(): + rtol = 1e-3 + atol = 1e-6 + y0 = [1/3, 2/9] + + for vectorized, method, t_span, jac in product( + [False, True], + ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA'], + [[5, 9], [5, 1]], + [None, jac_rational, jac_rational_sparse]): + + if vectorized: + fun = fun_rational_vectorized + else: + fun = fun_rational + + with suppress_warnings() as sup: + sup.filter(UserWarning, + "The following arguments have no effect for a chosen " + "solver: `jac`") + res = solve_ivp(fun, t_span, y0, rtol=rtol, + atol=atol, method=method, dense_output=True, + jac=jac, vectorized=vectorized) + assert_equal(res.t[0], t_span[0]) + assert_(res.t_events is None) + assert_(res.y_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + if method == 'DOP853': + # DOP853 spends more functions evaluation because it doesn't + # have enough time to develop big enough step size. + assert_(res.nfev < 50) + else: + assert_(res.nfev < 40) + + if method in ['RK23', 'RK45', 'DOP853', 'LSODA']: + assert_equal(res.njev, 0) + assert_equal(res.nlu, 0) + else: + assert_(0 < res.njev < 3) + assert_(0 < res.nlu < 10) + + y_true = sol_rational(res.t) + e = compute_error(res.y, y_true, rtol, atol) + assert_(np.all(e < 5)) + + tc = np.linspace(*t_span) + yc_true = sol_rational(tc) + yc = res.sol(tc) + + e = compute_error(yc, yc_true, rtol, atol) + assert_(np.all(e < 5)) + + tc = (t_span[0] + t_span[-1]) / 2 + yc_true = sol_rational(tc) + yc = res.sol(tc) + + e = compute_error(yc, yc_true, rtol, atol) + assert_(np.all(e < 5)) + + assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15) + + +@pytest.mark.thread_unsafe +def test_integration_complex(): + rtol = 1e-3 + atol = 1e-6 + y0 = [0.5 + 1j] + t_span = [0, 1] + tc = np.linspace(t_span[0], t_span[1]) + for method, jac in product(['RK23', 'RK45', 'DOP853', 'BDF'], + [None, jac_complex, jac_complex_sparse]): + with suppress_warnings() as sup: + sup.filter(UserWarning, + "The following arguments have no effect for a chosen " + "solver: `jac`") + res = solve_ivp(fun_complex, t_span, y0, method=method, + dense_output=True, rtol=rtol, atol=atol, jac=jac) + + assert_equal(res.t[0], t_span[0]) + assert_(res.t_events is None) + assert_(res.y_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + if method == 'DOP853': + assert res.nfev < 35 + else: + assert res.nfev < 25 + + if method == 'BDF': + assert_equal(res.njev, 1) + assert res.nlu < 6 + else: + assert res.njev == 0 + assert res.nlu == 0 + + y_true = sol_complex(res.t) + e = compute_error(res.y, y_true, rtol, atol) + assert np.all(e < 5) + + yc_true = sol_complex(tc) + yc = res.sol(tc) + e = compute_error(yc, yc_true, rtol, atol) + + assert np.all(e < 5) + + +@pytest.mark.fail_slow(5) +def test_integration_sparse_difference(): + n = 200 + t_span = [0, 20] + y0 = np.zeros(2 * n) + y0[1::2] = 1 + sparsity = medazko_sparsity(n) + + for method in ['BDF', 'Radau']: + res = solve_ivp(fun_medazko, t_span, y0, method=method, + jac_sparsity=sparsity) + + assert_equal(res.t[0], t_span[0]) + assert_(res.t_events is None) + assert_(res.y_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + assert_allclose(res.y[78, -1], 0.233994e-3, rtol=1e-2) + assert_allclose(res.y[79, -1], 0, atol=1e-3) + assert_allclose(res.y[148, -1], 0.359561e-3, rtol=1e-2) + assert_allclose(res.y[149, -1], 0, atol=1e-3) + assert_allclose(res.y[198, -1], 0.117374129e-3, rtol=1e-2) + assert_allclose(res.y[199, -1], 0.6190807e-5, atol=1e-3) + assert_allclose(res.y[238, -1], 0, atol=1e-3) + assert_allclose(res.y[239, -1], 0.9999997, rtol=1e-2) + + +def test_integration_const_jac(): + rtol = 1e-3 + atol = 1e-6 + y0 = [0, 2] + t_span = [0, 2] + J = jac_linear() + J_sparse = csc_matrix(J) + + for method, jac in product(['Radau', 'BDF'], [J, J_sparse]): + res = solve_ivp(fun_linear, t_span, y0, rtol=rtol, atol=atol, + method=method, dense_output=True, jac=jac) + assert_equal(res.t[0], t_span[0]) + assert_(res.t_events is None) + assert_(res.y_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + assert_(res.nfev < 100) + assert_equal(res.njev, 0) + assert_(0 < res.nlu < 15) + + y_true = sol_linear(res.t) + e = compute_error(res.y, y_true, rtol, atol) + assert_(np.all(e < 10)) + + tc = np.linspace(*t_span) + yc_true = sol_linear(tc) + yc = res.sol(tc) + + e = compute_error(yc, yc_true, rtol, atol) + assert_(np.all(e < 15)) + + assert_allclose(res.sol(res.t), res.y, rtol=1e-14, atol=1e-14) + + +@pytest.mark.slow +@pytest.mark.parametrize('method', ['Radau', 'BDF', 'LSODA']) +def test_integration_stiff(method, num_parallel_threads): + rtol = 1e-6 + atol = 1e-6 + y0 = [1e4, 0, 0] + tspan = [0, 1e8] + + if method == 'LSODA' and num_parallel_threads > 1: + pytest.skip(reason='LSODA does not allow for concurrent calls') + + def fun_robertson(t, state): + x, y, z = state + return [ + -0.04 * x + 1e4 * y * z, + 0.04 * x - 1e4 * y * z - 3e7 * y * y, + 3e7 * y * y, + ] + + res = solve_ivp(fun_robertson, tspan, y0, rtol=rtol, + atol=atol, method=method) + + # If the stiff mode is not activated correctly, these numbers will be much bigger + assert res.nfev < 5000 + assert res.njev < 200 + + +def test_events(num_parallel_threads): + def event_rational_1(t, y): + return y[0] - y[1] ** 0.7 + + def event_rational_2(t, y): + return y[1] ** 0.6 - y[0] + + def event_rational_3(t, y): + return t - 7.4 + + event_rational_3.terminal = True + + for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']: + if method == 'LSODA' and num_parallel_threads > 1: + continue + + res = solve_ivp(fun_rational, [5, 8], [1/3, 2/9], method=method, + events=(event_rational_1, event_rational_2)) + assert_equal(res.status, 0) + assert_equal(res.t_events[0].size, 1) + assert_equal(res.t_events[1].size, 1) + assert_(5.3 < res.t_events[0][0] < 5.7) + assert_(7.3 < res.t_events[1][0] < 7.7) + + assert_equal(res.y_events[0].shape, (1, 2)) + assert_equal(res.y_events[1].shape, (1, 2)) + assert np.isclose( + event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0) + assert np.isclose( + event_rational_2(res.t_events[1][0], res.y_events[1][0]), 0) + + event_rational_1.direction = 1 + event_rational_2.direction = 1 + res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method, + events=(event_rational_1, event_rational_2)) + assert_equal(res.status, 0) + assert_equal(res.t_events[0].size, 1) + assert_equal(res.t_events[1].size, 0) + assert_(5.3 < res.t_events[0][0] < 5.7) + assert_equal(res.y_events[0].shape, (1, 2)) + assert_equal(res.y_events[1].shape, (0,)) + assert np.isclose( + event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0) + + event_rational_1.direction = -1 + event_rational_2.direction = -1 + res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method, + events=(event_rational_1, event_rational_2)) + assert_equal(res.status, 0) + assert_equal(res.t_events[0].size, 0) + assert_equal(res.t_events[1].size, 1) + assert_(7.3 < res.t_events[1][0] < 7.7) + assert_equal(res.y_events[0].shape, (0,)) + assert_equal(res.y_events[1].shape, (1, 2)) + assert np.isclose( + event_rational_2(res.t_events[1][0], res.y_events[1][0]), 0) + + event_rational_1.direction = 0 + event_rational_2.direction = 0 + + res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method, + events=(event_rational_1, event_rational_2, + event_rational_3), dense_output=True) + assert_equal(res.status, 1) + assert_equal(res.t_events[0].size, 1) + assert_equal(res.t_events[1].size, 0) + assert_equal(res.t_events[2].size, 1) + assert_(5.3 < res.t_events[0][0] < 5.7) + assert_(7.3 < res.t_events[2][0] < 7.5) + assert_equal(res.y_events[0].shape, (1, 2)) + assert_equal(res.y_events[1].shape, (0,)) + assert_equal(res.y_events[2].shape, (1, 2)) + assert np.isclose( + event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0) + assert np.isclose( + event_rational_3(res.t_events[2][0], res.y_events[2][0]), 0) + + res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method, + events=event_rational_1, dense_output=True) + assert_equal(res.status, 0) + assert_equal(res.t_events[0].size, 1) + assert_(5.3 < res.t_events[0][0] < 5.7) + + assert_equal(res.y_events[0].shape, (1, 2)) + assert np.isclose( + event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0) + + # Also test that termination by event doesn't break interpolants. + tc = np.linspace(res.t[0], res.t[-1]) + yc_true = sol_rational(tc) + yc = res.sol(tc) + e = compute_error(yc, yc_true, 1e-3, 1e-6) + assert_(np.all(e < 5)) + + # Test that the y_event matches solution + assert np.allclose(sol_rational(res.t_events[0][0]), res.y_events[0][0], + rtol=1e-3, atol=1e-6) + + # Test in backward direction. + event_rational_1.direction = 0 + event_rational_2.direction = 0 + for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']: + if method == 'LSODA' and num_parallel_threads > 1: + continue + + res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method, + events=(event_rational_1, event_rational_2)) + assert_equal(res.status, 0) + assert_equal(res.t_events[0].size, 1) + assert_equal(res.t_events[1].size, 1) + assert_(5.3 < res.t_events[0][0] < 5.7) + assert_(7.3 < res.t_events[1][0] < 7.7) + + assert_equal(res.y_events[0].shape, (1, 2)) + assert_equal(res.y_events[1].shape, (1, 2)) + assert np.isclose( + event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0) + assert np.isclose( + event_rational_2(res.t_events[1][0], res.y_events[1][0]), 0) + + event_rational_1.direction = -1 + event_rational_2.direction = -1 + res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method, + events=(event_rational_1, event_rational_2)) + assert_equal(res.status, 0) + assert_equal(res.t_events[0].size, 1) + assert_equal(res.t_events[1].size, 0) + assert_(5.3 < res.t_events[0][0] < 5.7) + + assert_equal(res.y_events[0].shape, (1, 2)) + assert_equal(res.y_events[1].shape, (0,)) + assert np.isclose( + event_rational_1(res.t_events[0][0], res.y_events[0][0]), 0) + + event_rational_1.direction = 1 + event_rational_2.direction = 1 + res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method, + events=(event_rational_1, event_rational_2)) + assert_equal(res.status, 0) + assert_equal(res.t_events[0].size, 0) + assert_equal(res.t_events[1].size, 1) + assert_(7.3 < res.t_events[1][0] < 7.7) + + assert_equal(res.y_events[0].shape, (0,)) + assert_equal(res.y_events[1].shape, (1, 2)) + assert np.isclose( + event_rational_2(res.t_events[1][0], res.y_events[1][0]), 0) + + event_rational_1.direction = 0 + event_rational_2.direction = 0 + + res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method, + events=(event_rational_1, event_rational_2, + event_rational_3), dense_output=True) + assert_equal(res.status, 1) + assert_equal(res.t_events[0].size, 0) + assert_equal(res.t_events[1].size, 1) + assert_equal(res.t_events[2].size, 1) + assert_(7.3 < res.t_events[1][0] < 7.7) + assert_(7.3 < res.t_events[2][0] < 7.5) + + assert_equal(res.y_events[0].shape, (0,)) + assert_equal(res.y_events[1].shape, (1, 2)) + assert_equal(res.y_events[2].shape, (1, 2)) + assert np.isclose( + event_rational_2(res.t_events[1][0], res.y_events[1][0]), 0) + assert np.isclose( + event_rational_3(res.t_events[2][0], res.y_events[2][0]), 0) + + # Also test that termination by event doesn't break interpolants. + tc = np.linspace(res.t[-1], res.t[0]) + yc_true = sol_rational(tc) + yc = res.sol(tc) + e = compute_error(yc, yc_true, 1e-3, 1e-6) + assert_(np.all(e < 5)) + + assert np.allclose(sol_rational(res.t_events[1][0]), res.y_events[1][0], + rtol=1e-3, atol=1e-6) + assert np.allclose(sol_rational(res.t_events[2][0]), res.y_events[2][0], + rtol=1e-3, atol=1e-6) + + +def _get_harmonic_oscillator(): + def f(t, y): + return [y[1], -y[0]] + + def event(t, y): + return y[0] + + return f, event + + +@pytest.mark.parametrize('n_events', [3, 4]) +def test_event_terminal_integer(n_events): + f, event = _get_harmonic_oscillator() + event.terminal = n_events + res = solve_ivp(f, (0, 100), [1, 0], events=event) + assert len(res.t_events[0]) == n_events + assert len(res.y_events[0]) == n_events + assert_allclose(res.y_events[0][:, 0], 0, atol=1e-14) + + +def test_event_terminal_iv(): + f, event = _get_harmonic_oscillator() + args = (f, (0, 100), [1, 0]) + + event.terminal = None + res = solve_ivp(*args, events=event) + event.terminal = 0 + ref = solve_ivp(*args, events=event) + assert_allclose(res.t_events, ref.t_events) + + message = "The `terminal` attribute..." + event.terminal = -1 + with pytest.raises(ValueError, match=message): + solve_ivp(*args, events=event) + event.terminal = 3.5 + with pytest.raises(ValueError, match=message): + solve_ivp(*args, events=event) + + +def test_max_step(num_parallel_threads): + rtol = 1e-3 + atol = 1e-6 + y0 = [1/3, 2/9] + for method in [RK23, RK45, DOP853, Radau, BDF, LSODA]: + if method is LSODA and num_parallel_threads > 1: + continue + for t_span in ([5, 9], [5, 1]): + res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, + max_step=0.5, atol=atol, method=method, + dense_output=True) + assert_equal(res.t[0], t_span[0]) + assert_equal(res.t[-1], t_span[-1]) + assert_(np.all(np.abs(np.diff(res.t)) <= 0.5 + 1e-15)) + assert_(res.t_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + y_true = sol_rational(res.t) + e = compute_error(res.y, y_true, rtol, atol) + assert_(np.all(e < 5)) + + tc = np.linspace(*t_span) + yc_true = sol_rational(tc) + yc = res.sol(tc) + + e = compute_error(yc, yc_true, rtol, atol) + assert_(np.all(e < 5)) + + assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15) + + assert_raises(ValueError, method, fun_rational, t_span[0], y0, + t_span[1], max_step=-1) + + if method is not LSODA: + solver = method(fun_rational, t_span[0], y0, t_span[1], + rtol=rtol, atol=atol, max_step=1e-20) + message = solver.step() + message = solver.step() # First step succeeds but second step fails. + assert_equal(solver.status, 'failed') + assert_("step size is less" in message) + assert_raises(RuntimeError, solver.step) + + +def test_first_step(num_parallel_threads): + rtol = 1e-3 + atol = 1e-6 + y0 = [1/3, 2/9] + first_step = 0.1 + for method in [RK23, RK45, DOP853, Radau, BDF, LSODA]: + if method is LSODA and num_parallel_threads > 1: + continue + for t_span in ([5, 9], [5, 1]): + res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, + max_step=0.5, atol=atol, method=method, + dense_output=True, first_step=first_step) + + assert_equal(res.t[0], t_span[0]) + assert_equal(res.t[-1], t_span[-1]) + assert_allclose(first_step, np.abs(res.t[1] - 5)) + assert_(res.t_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + y_true = sol_rational(res.t) + e = compute_error(res.y, y_true, rtol, atol) + assert_(np.all(e < 5)) + + tc = np.linspace(*t_span) + yc_true = sol_rational(tc) + yc = res.sol(tc) + + e = compute_error(yc, yc_true, rtol, atol) + assert_(np.all(e < 5)) + + assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15) + + assert_raises(ValueError, method, fun_rational, t_span[0], y0, + t_span[1], first_step=-1) + assert_raises(ValueError, method, fun_rational, t_span[0], y0, + t_span[1], first_step=5) + + +def test_t_eval(): + rtol = 1e-3 + atol = 1e-6 + y0 = [1/3, 2/9] + for t_span in ([5, 9], [5, 1]): + t_eval = np.linspace(t_span[0], t_span[1], 10) + res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol, + t_eval=t_eval) + assert_equal(res.t, t_eval) + assert_(res.t_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + y_true = sol_rational(res.t) + e = compute_error(res.y, y_true, rtol, atol) + assert_(np.all(e < 5)) + + t_eval = [5, 5.01, 7, 8, 8.01, 9] + res = solve_ivp(fun_rational, [5, 9], y0, rtol=rtol, atol=atol, + t_eval=t_eval) + assert_equal(res.t, t_eval) + assert_(res.t_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + y_true = sol_rational(res.t) + e = compute_error(res.y, y_true, rtol, atol) + assert_(np.all(e < 5)) + + t_eval = [5, 4.99, 3, 1.5, 1.1, 1.01, 1] + res = solve_ivp(fun_rational, [5, 1], y0, rtol=rtol, atol=atol, + t_eval=t_eval) + assert_equal(res.t, t_eval) + assert_(res.t_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + t_eval = [5.01, 7, 8, 8.01] + res = solve_ivp(fun_rational, [5, 9], y0, rtol=rtol, atol=atol, + t_eval=t_eval) + assert_equal(res.t, t_eval) + assert_(res.t_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + y_true = sol_rational(res.t) + e = compute_error(res.y, y_true, rtol, atol) + assert_(np.all(e < 5)) + + t_eval = [4.99, 3, 1.5, 1.1, 1.01] + res = solve_ivp(fun_rational, [5, 1], y0, rtol=rtol, atol=atol, + t_eval=t_eval) + assert_equal(res.t, t_eval) + assert_(res.t_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + t_eval = [4, 6] + assert_raises(ValueError, solve_ivp, fun_rational, [5, 9], y0, + rtol=rtol, atol=atol, t_eval=t_eval) + + +def test_t_eval_dense_output(): + rtol = 1e-3 + atol = 1e-6 + y0 = [1/3, 2/9] + t_span = [5, 9] + t_eval = np.linspace(t_span[0], t_span[1], 10) + res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol, + t_eval=t_eval) + res_d = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol, + t_eval=t_eval, dense_output=True) + assert_equal(res.t, t_eval) + assert_(res.t_events is None) + assert_(res.success) + assert_equal(res.status, 0) + + assert_equal(res.t, res_d.t) + assert_equal(res.y, res_d.y) + assert_(res_d.t_events is None) + assert_(res_d.success) + assert_equal(res_d.status, 0) + + # if t and y are equal only test values for one case + y_true = sol_rational(res.t) + e = compute_error(res.y, y_true, rtol, atol) + assert_(np.all(e < 5)) + + +@pytest.mark.thread_unsafe +def test_t_eval_early_event(): + def early_event(t, y): + return t - 7 + + early_event.terminal = True + + rtol = 1e-3 + atol = 1e-6 + y0 = [1/3, 2/9] + t_span = [5, 9] + t_eval = np.linspace(7.5, 9, 16) + for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']: + with suppress_warnings() as sup: + sup.filter(UserWarning, + "The following arguments have no effect for a chosen " + "solver: `jac`") + res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol, + method=method, t_eval=t_eval, events=early_event, + jac=jac_rational) + assert res.success + assert res.message == 'A termination event occurred.' + assert res.status == 1 + assert not res.t and not res.y + assert len(res.t_events) == 1 + assert res.t_events[0].size == 1 + assert res.t_events[0][0] == 7 + + +def test_event_dense_output_LSODA(num_parallel_threads): + if num_parallel_threads > 1: + pytest.skip('LSODA does not allow for concurrent execution') + + def event_lsoda(t, y): + return y[0] - 2.02e-5 + + rtol = 1e-3 + atol = 1e-6 + y0 = [0.05] + t_span = [-2, 2] + first_step = 1e-3 + res = solve_ivp( + fun_event_dense_output_LSODA, + t_span, + y0, + method="LSODA", + dense_output=True, + events=event_lsoda, + first_step=first_step, + max_step=1, + rtol=rtol, + atol=atol, + jac=jac_event_dense_output_LSODA, + ) + + assert_equal(res.t[0], t_span[0]) + assert_equal(res.t[-1], t_span[-1]) + assert_allclose(first_step, np.abs(res.t[1] - t_span[0])) + assert res.success + assert_equal(res.status, 0) + + y_true = sol_event_dense_output_LSODA(res.t) + e = compute_error(res.y, y_true, rtol, atol) + assert_array_less(e, 5) + + tc = np.linspace(*t_span) + yc_true = sol_event_dense_output_LSODA(tc) + yc = res.sol(tc) + e = compute_error(yc, yc_true, rtol, atol) + assert_array_less(e, 5) + + assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15) + + +def test_no_integration(): + for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']: + sol = solve_ivp(lambda t, y: -y, [4, 4], [2, 3], + method=method, dense_output=True) + assert_equal(sol.sol(4), [2, 3]) + assert_equal(sol.sol([4, 5, 6]), [[2, 2, 2], [3, 3, 3]]) + + +def test_no_integration_class(): + for method in [RK23, RK45, DOP853, Radau, BDF, LSODA]: + solver = method(lambda t, y: -y, 0.0, [10.0, 0.0], 0.0) + solver.step() + assert_equal(solver.status, 'finished') + sol = solver.dense_output() + assert_equal(sol(0.0), [10.0, 0.0]) + assert_equal(sol([0, 1, 2]), [[10, 10, 10], [0, 0, 0]]) + + solver = method(lambda t, y: -y, 0.0, [], np.inf) + solver.step() + assert_equal(solver.status, 'finished') + sol = solver.dense_output() + assert_equal(sol(100.0), []) + assert_equal(sol([0, 1, 2]), np.empty((0, 3))) + + +def test_empty(): + def fun(t, y): + return np.zeros((0,)) + + y0 = np.zeros((0,)) + + for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']: + sol = assert_no_warnings(solve_ivp, fun, [0, 10], y0, + method=method, dense_output=True) + assert_equal(sol.sol(10), np.zeros((0,))) + assert_equal(sol.sol([1, 2, 3]), np.zeros((0, 3))) + + for method in ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']: + sol = assert_no_warnings(solve_ivp, fun, [0, np.inf], y0, + method=method, dense_output=True) + assert_equal(sol.sol(10), np.zeros((0,))) + assert_equal(sol.sol([1, 2, 3]), np.zeros((0, 3))) + + +def test_ConstantDenseOutput(): + sol = ConstantDenseOutput(0, 1, np.array([1, 2])) + assert_allclose(sol(1.5), [1, 2]) + assert_allclose(sol([1, 1.5, 2]), [[1, 1, 1], [2, 2, 2]]) + + sol = ConstantDenseOutput(0, 1, np.array([])) + assert_allclose(sol(1.5), np.empty(0)) + assert_allclose(sol([1, 1.5, 2]), np.empty((0, 3))) + + +def test_classes(): + y0 = [1 / 3, 2 / 9] + for cls in [RK23, RK45, DOP853, Radau, BDF, LSODA]: + solver = cls(fun_rational, 5, y0, np.inf) + assert_equal(solver.n, 2) + assert_equal(solver.status, 'running') + assert_equal(solver.t_bound, np.inf) + assert_equal(solver.direction, 1) + assert_equal(solver.t, 5) + assert_equal(solver.y, y0) + assert_(solver.step_size is None) + if cls is not LSODA: + assert_(solver.nfev > 0) + assert_(solver.njev >= 0) + assert_equal(solver.nlu, 0) + else: + assert_equal(solver.nfev, 0) + assert_equal(solver.njev, 0) + assert_equal(solver.nlu, 0) + + assert_raises(RuntimeError, solver.dense_output) + + message = solver.step() + assert_equal(solver.status, 'running') + assert_equal(message, None) + assert_equal(solver.n, 2) + assert_equal(solver.t_bound, np.inf) + assert_equal(solver.direction, 1) + assert_(solver.t > 5) + assert_(not np.all(np.equal(solver.y, y0))) + assert_(solver.step_size > 0) + assert_(solver.nfev > 0) + assert_(solver.njev >= 0) + assert_(solver.nlu >= 0) + sol = solver.dense_output() + assert_allclose(sol(5), y0, rtol=1e-15, atol=0) + + +def test_OdeSolution(): + ts = np.array([0, 2, 5], dtype=float) + s1 = ConstantDenseOutput(ts[0], ts[1], np.array([-1])) + s2 = ConstantDenseOutput(ts[1], ts[2], np.array([1])) + + sol = OdeSolution(ts, [s1, s2]) + + assert_equal(sol(-1), [-1]) + assert_equal(sol(1), [-1]) + assert_equal(sol(2), [-1]) + assert_equal(sol(3), [1]) + assert_equal(sol(5), [1]) + assert_equal(sol(6), [1]) + + assert_equal(sol([0, 6, -2, 1.5, 4.5, 2.5, 5, 5.5, 2]), + np.array([[-1, 1, -1, -1, 1, 1, 1, 1, -1]])) + + ts = np.array([10, 4, -3]) + s1 = ConstantDenseOutput(ts[0], ts[1], np.array([-1])) + s2 = ConstantDenseOutput(ts[1], ts[2], np.array([1])) + + sol = OdeSolution(ts, [s1, s2]) + assert_equal(sol(11), [-1]) + assert_equal(sol(10), [-1]) + assert_equal(sol(5), [-1]) + assert_equal(sol(4), [-1]) + assert_equal(sol(0), [1]) + assert_equal(sol(-3), [1]) + assert_equal(sol(-4), [1]) + + assert_equal(sol([12, -5, 10, -3, 6, 1, 4]), + np.array([[-1, 1, -1, 1, -1, 1, -1]])) + + ts = np.array([1, 1]) + s = ConstantDenseOutput(1, 1, np.array([10])) + sol = OdeSolution(ts, [s]) + assert_equal(sol(0), [10]) + assert_equal(sol(1), [10]) + assert_equal(sol(2), [10]) + + assert_equal(sol([2, 1, 0]), np.array([[10, 10, 10]])) + + +def test_num_jac(): + def fun(t, y): + return np.vstack([ + -0.04 * y[0] + 1e4 * y[1] * y[2], + 0.04 * y[0] - 1e4 * y[1] * y[2] - 3e7 * y[1] ** 2, + 3e7 * y[1] ** 2 + ]) + + def jac(t, y): + return np.array([ + [-0.04, 1e4 * y[2], 1e4 * y[1]], + [0.04, -1e4 * y[2] - 6e7 * y[1], -1e4 * y[1]], + [0, 6e7 * y[1], 0] + ]) + + t = 1 + y = np.array([1, 0, 0]) + J_true = jac(t, y) + threshold = 1e-5 + f = fun(t, y).ravel() + + J_num, factor = num_jac(fun, t, y, f, threshold, None) + assert_allclose(J_num, J_true, rtol=1e-5, atol=1e-5) + + J_num, factor = num_jac(fun, t, y, f, threshold, factor) + assert_allclose(J_num, J_true, rtol=1e-5, atol=1e-5) + + +def test_num_jac_sparse(): + def fun(t, y): + e = y[1:]**3 - y[:-1]**2 + z = np.zeros(y.shape[1]) + return np.vstack((z, 3 * e)) + np.vstack((2 * e, z)) + + def structure(n): + A = np.zeros((n, n), dtype=int) + A[0, 0] = 1 + A[0, 1] = 1 + for i in range(1, n - 1): + A[i, i - 1: i + 2] = 1 + A[-1, -1] = 1 + A[-1, -2] = 1 + + return A + + np.random.seed(0) + n = 20 + y = np.random.randn(n) + A = structure(n) + groups = group_columns(A) + + f = fun(0, y[:, None]).ravel() + + # Compare dense and sparse results, assuming that dense implementation + # is correct (as it is straightforward). + J_num_sparse, factor_sparse = num_jac(fun, 0, y.ravel(), f, 1e-8, None, + sparsity=(A, groups)) + J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, None) + assert_allclose(J_num_dense, J_num_sparse.toarray(), + rtol=1e-12, atol=1e-14) + assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14) + + # Take small factors to trigger their recomputing inside. + factor = np.random.uniform(0, 1e-12, size=n) + J_num_sparse, factor_sparse = num_jac(fun, 0, y.ravel(), f, 1e-8, factor, + sparsity=(A, groups)) + J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, factor) + + assert_allclose(J_num_dense, J_num_sparse.toarray(), + rtol=1e-12, atol=1e-14) + assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14) + + +def test_args(): + + # sys3 is actually two decoupled systems. (x, y) form a + # linear oscillator, while z is a nonlinear first order + # system with equilibria at z=0 and z=1. If k > 0, z=1 + # is stable and z=0 is unstable. + + def sys3(t, w, omega, k, zfinal): + x, y, z = w + return [-omega*y, omega*x, k*z*(1 - z)] + + def sys3_jac(t, w, omega, k, zfinal): + x, y, z = w + J = np.array([[0, -omega, 0], + [omega, 0, 0], + [0, 0, k*(1 - 2*z)]]) + return J + + def sys3_x0decreasing(t, w, omega, k, zfinal): + x, y, z = w + return x + + def sys3_y0increasing(t, w, omega, k, zfinal): + x, y, z = w + return y + + def sys3_zfinal(t, w, omega, k, zfinal): + x, y, z = w + return z - zfinal + + # Set the event flags for the event functions. + sys3_x0decreasing.direction = -1 + sys3_y0increasing.direction = 1 + sys3_zfinal.terminal = True + + omega = 2 + k = 4 + + tfinal = 5 + zfinal = 0.99 + # Find z0 such that when z(0) = z0, z(tfinal) = zfinal. + # The condition z(tfinal) = zfinal is the terminal event. + z0 = np.exp(-k*tfinal)/((1 - zfinal)/zfinal + np.exp(-k*tfinal)) + + w0 = [0, -1, z0] + + # Provide the jac argument and use the Radau method to ensure that the use + # of the Jacobian function is exercised. + # If event handling is working, the solution will stop at tfinal, not tend. + tend = 2*tfinal + sol = solve_ivp(sys3, [0, tend], w0, + events=[sys3_x0decreasing, sys3_y0increasing, sys3_zfinal], + dense_output=True, args=(omega, k, zfinal), + method='Radau', jac=sys3_jac, + rtol=1e-10, atol=1e-13) + + # Check that we got the expected events at the expected times. + x0events_t = sol.t_events[0] + y0events_t = sol.t_events[1] + zfinalevents_t = sol.t_events[2] + assert_allclose(x0events_t, [0.5*np.pi, 1.5*np.pi]) + assert_allclose(y0events_t, [0.25*np.pi, 1.25*np.pi]) + assert_allclose(zfinalevents_t, [tfinal]) + + # Check that the solution agrees with the known exact solution. + t = np.linspace(0, zfinalevents_t[0], 250) + w = sol.sol(t) + assert_allclose(w[0], np.sin(omega*t), rtol=1e-9, atol=1e-12) + assert_allclose(w[1], -np.cos(omega*t), rtol=1e-9, atol=1e-12) + assert_allclose(w[2], 1/(((1 - z0)/z0)*np.exp(-k*t) + 1), + rtol=1e-9, atol=1e-12) + + # Check that the state variables have the expected values at the events. + x0events = sol.sol(x0events_t) + y0events = sol.sol(y0events_t) + zfinalevents = sol.sol(zfinalevents_t) + assert_allclose(x0events[0], np.zeros_like(x0events[0]), atol=5e-14) + assert_allclose(x0events[1], np.ones_like(x0events[1])) + assert_allclose(y0events[0], np.ones_like(y0events[0])) + assert_allclose(y0events[1], np.zeros_like(y0events[1]), atol=5e-14) + assert_allclose(zfinalevents[2], [zfinal]) + + +@pytest.mark.thread_unsafe +def test_array_rtol(): + # solve_ivp had a bug with array_like `rtol`; see gh-15482 + # check that it's fixed + def f(t, y): + return y[0], y[1] + + # no warning (or error) when `rtol` is array_like + sol = solve_ivp(f, (0, 1), [1., 1.], rtol=[1e-1, 1e-1]) + err1 = np.abs(np.linalg.norm(sol.y[:, -1] - np.exp(1))) + + # warning when an element of `rtol` is too small + with pytest.warns(UserWarning, match="At least one element..."): + sol = solve_ivp(f, (0, 1), [1., 1.], rtol=[1e-1, 1e-16]) + err2 = np.abs(np.linalg.norm(sol.y[:, -1] - np.exp(1))) + + # tighter rtol improves the error + assert err2 < err1 + + +@pytest.mark.parametrize('method', ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF', 'LSODA']) +def test_integration_zero_rhs(method, num_parallel_threads): + if method == 'LSODA' and num_parallel_threads > 1: + pytest.skip(reason='LSODA does not allow for concurrent execution') + + result = solve_ivp(fun_zero, [0, 10], np.ones(3), method=method) + assert_(result.success) + assert_equal(result.status, 0) + assert_allclose(result.y, 1.0, rtol=1e-15) + + +def test_args_single_value(): + def fun_with_arg(t, y, a): + return a*y + + message = "Supplied 'args' cannot be unpacked." + with pytest.raises(TypeError, match=message): + solve_ivp(fun_with_arg, (0, 0.1), [1], args=-1) + + sol = solve_ivp(fun_with_arg, (0, 0.1), [1], args=(-1,)) + assert_allclose(sol.y[0, -1], np.exp(-0.1)) + + +@pytest.mark.parametrize("f0_fill", [np.nan, np.inf]) +def test_initial_state_finiteness(f0_fill): + # regression test for gh-17846 + msg = "All components of the initial state `y0` must be finite." + with pytest.raises(ValueError, match=msg): + solve_ivp(fun_zero, [0, 10], np.full(3, f0_fill)) + + +@pytest.mark.parametrize('method', ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF']) +def test_zero_interval(method): + # Case where upper and lower limits of integration are the same + # Result of integration should match initial state. + # f[y(t)] = 2y(t) + def f(t, y): + return 2 * y + res = solve_ivp(f, (0.0, 0.0), np.array([1.0]), method=method) + assert res.success + assert_allclose(res.y[0, -1], 1.0) + + +@pytest.mark.parametrize('method', ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF']) +def test_tbound_respected_small_interval(method): + """Regression test for gh-17341""" + SMALL = 1e-4 + + # f[y(t)] = 2y(t) on t in [0,SMALL] + # undefined otherwise + def f(t, y): + if t > SMALL: + raise ValueError("Function was evaluated outside interval") + return 2 * y + res = solve_ivp(f, (0.0, SMALL), np.array([1]), method=method) + assert res.success + + +@pytest.mark.parametrize('method', ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF']) +def test_tbound_respected_larger_interval(method): + """Regression test for gh-8848""" + def V(r): + return -11/r + 10 * r / (0.05 + r**2) + + def func(t, p): + if t < -17 or t > 2: + raise ValueError("Function was evaluated outside interval") + P = p[0] + Q = p[1] + r = np.exp(t) + dPdr = r * Q + dQdr = -2.0 * r * ((-0.2 - V(r)) * P + 1 / r * Q) + return np.array([dPdr, dQdr]) + + result = solve_ivp(func, + (-17, 2), + y0=np.array([1, -11]), + max_step=0.03, + vectorized=False, + t_eval=None, + atol=1e-8, + rtol=1e-5) + assert result.success + + +@pytest.mark.parametrize('method', ['RK23', 'RK45', 'DOP853', 'Radau', 'BDF']) +def test_tbound_respected_oscillator(method): + "Regression test for gh-9198" + def reactions_func(t, y): + if (t > 205): + raise ValueError("Called outside interval") + yprime = np.array([1.73307544e-02, + 6.49376470e-06, + 0.00000000e+00, + 0.00000000e+00]) + return yprime + + def run_sim2(t_end, n_timepoints=10, shortest_delay_line=10000000): + init_state = np.array([134.08298555, 138.82348612, 100., 0.]) + t0 = 100.0 + t1 = 200.0 + return solve_ivp(reactions_func, + (t0, t1), + init_state.copy(), + dense_output=True, + max_step=t1 - t0) + result = run_sim2(1000, 100, 100) + assert result.success + + +def test_inital_maxstep(): + """Verify that select_inital_step respects max_step""" + rtol = 1e-3 + atol = 1e-6 + y0 = np.array([1/3, 2/9]) + for (t0, t_bound) in ((5, 9), (5, 1)): + for method_order in [RK23.error_estimator_order, + RK45.error_estimator_order, + DOP853.error_estimator_order, + 3, #RADAU + 1 #BDF + ]: + step_no_max = select_initial_step(fun_rational, t0, y0, t_bound, + np.inf, + fun_rational(t0,y0), + np.sign(t_bound - t0), + method_order, + rtol, atol) + max_step = step_no_max/2 + step_with_max = select_initial_step(fun_rational, t0, y0, t_bound, + max_step, + fun_rational(t0, y0), + np.sign(t_bound - t0), + method_order, + rtol, atol) + assert_equal(max_step, step_with_max) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/tests/test_rk.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/tests/test_rk.py new file mode 100644 index 0000000000000000000000000000000000000000..33cb27d0323d037c0937ab94b4de8f63b46be3d7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ivp/tests/test_rk.py @@ -0,0 +1,37 @@ +import pytest +from numpy.testing import assert_allclose, assert_ +import numpy as np +from scipy.integrate import RK23, RK45, DOP853 +from scipy.integrate._ivp import dop853_coefficients + + +@pytest.mark.parametrize("solver", [RK23, RK45, DOP853]) +def test_coefficient_properties(solver): + assert_allclose(np.sum(solver.B), 1, rtol=1e-15) + assert_allclose(np.sum(solver.A, axis=1), solver.C, rtol=1e-14) + + +def test_coefficient_properties_dop853(): + assert_allclose(np.sum(dop853_coefficients.B), 1, rtol=1e-15) + assert_allclose(np.sum(dop853_coefficients.A, axis=1), + dop853_coefficients.C, + rtol=1e-14) + + +@pytest.mark.parametrize("solver_class", [RK23, RK45, DOP853]) +def test_error_estimation(solver_class): + step = 0.2 + solver = solver_class(lambda t, y: y, 0, [1], 1, first_step=step) + solver.step() + error_estimate = solver._estimate_error(solver.K, step) + error = solver.y - np.exp([step]) + assert_(np.abs(error) < np.abs(error_estimate)) + + +@pytest.mark.parametrize("solver_class", [RK23, RK45, DOP853]) +def test_error_estimation_complex(solver_class): + h = 0.2 + solver = solver_class(lambda t, y: 1j * y, 0, [1j], 1, first_step=h) + solver.step() + err_norm = solver._estimate_error_norm(solver.K, h, scale=[1]) + assert np.isrealobj(err_norm) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_lebedev.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_lebedev.py new file mode 100644 index 0000000000000000000000000000000000000000..da200972f9d475162f84294ed335149dc86fe94b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_lebedev.py @@ -0,0 +1,5450 @@ +# getLebedevSphere +# Copyright (c) 2010, Robert Parrish +# All rights reserved. +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions are +# met: +# +# * Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# * Redistributions in binary form must reproduce the above copyright +# notice, this list of conditions and the following disclaimer in +# the documentation and/or other materials provided with the distribution +# +# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" +# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE +# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE +# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR +# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF +# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN +# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) +# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE +# POSSIBILITY OF SUCH DAMAGE. +# +# Brainlessly translated to Python + +import numpy as np +from numpy import pi, zeros, sqrt + + +__all__ = ['lebedev_rule'] + + +def get_lebedev_sphere(degree): + # getLebedevSphere + # @author Rob Parrish, The Sherrill Group, CCMST Georgia Tech + # @email robparrish@gmail.com + # @date 03/24/2010 + # + # @description - function to compute normalized points and weights + # for Lebedev quadratures on the surface of the unit sphere at double precision. + # **********Relative error is generally expected to be ~2.0E-14 [1]******** + # Lebedev quadratures are superbly accurate and efficient quadrature rules for + # approximating integrals of the form $v = \iint_{4\pi} f(\Omega) \ \ud + # \Omega$, where $\Omega is the solid angle on the surface of the unit + # sphere. Lebedev quadratures integrate all spherical harmonics up to $l = + # order$, where $degree \approx order(order+1)/3$. These grids may be easily + # combined with radial quadratures to provide robust cubature formulae. For + # example, see 'A. Becke, 1988c, J. Chem. Phys., 88(4), pp. 2547' (The first + # paper on tractable molecular Density Functional Theory methods, of which + # Lebedev grids and numerical cubature are an intrinsic part). + # + # @param degree - positive integer specifying number of points in the + # requested quadrature. Allowed values are (degree -> order): + # degree: { 6, 14, 26, 38, 50, 74, 86, 110, 146, 170, 194, 230, 266, 302, + # 350, 434, 590, 770, 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, + # 3470, 3890, 4334, 4802, 5294, 5810 } + # order: {3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,35,41,47,53,59,65,71,77, + # 83,89,95,101,107,113,119,125,131} + # + # + # @return leb_tmp - struct containing fields: + # x - x values of quadrature, constrained to unit sphere + # y - y values of quadrature, constrained to unit sphere + # z - z values of quadrature, constrained to unit sphere + # w - quadrature weights, normalized to $4\pi$. + # + # @example: $\int_S x^2+y^2-z^2 \ud \Omega = 4.188790204786399$ + # f = @(x,y,z) x.^2+y.^2-z.^2 + # leb = getLebedevSphere(590) + # v = f(leb.x,leb.y,leb.z) + # int = sum(v.*leb.w) + # + # @citation - Translated from a Fortran code kindly provided by Christoph van + # Wuellen (Ruhr-Universitaet, Bochum, Germany), which in turn came from the + # original C routines coded by Dmitri Laikov (Moscow State University, + # Moscow, Russia). The MATLAB implementation of this code is designed for + # benchmarking of new DFT integration techniques to be implemented in the + # open source Psi4 ab initio quantum chemistry program. + # + # As per Professor Wuellen's request, any papers published using this code + # or its derivatives are requested to include the following citation: + # + # [1] V.I. Lebedev, and D.N. Laikov + # "A quadrature formula for the sphere of the 131st + # algebraic order of accuracy" + # Doklady Mathematics, Vol. 59, No. 3, 1999, pp. 477-481. + + class Leb: + x, y, z, w = None, None, None, None + + leb_tmp = Leb() + + leb_tmp.x = zeros(degree) + leb_tmp.y = zeros(degree) + leb_tmp.z = zeros(degree) + leb_tmp.w = zeros(degree) + + start = 0 + a = 0.0 + b = 0.0 + + match degree: + + case 6: + + v = 0.1666666666666667E+0 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + + case 14: + + v = 0.6666666666666667E-1 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.7500000000000000E-1 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + + case 26: + + v = 0.4761904761904762E-1 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.3809523809523810E-1 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.3214285714285714E-1 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + + case 38: + + v = 0.9523809523809524E-2 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.3214285714285714E-1 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.4597008433809831E+0 + v = 0.2857142857142857E-1 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + + case 50: + + v = 0.1269841269841270E-1 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.2257495590828924E-1 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.2109375000000000E-1 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.3015113445777636E+0 + v = 0.2017333553791887E-1 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + + case 74: + + v = 0.5130671797338464E-3 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.1660406956574204E-1 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = -0.2958603896103896E-1 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.4803844614152614E+0 + v = 0.2657620708215946E-1 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3207726489807764E+0 + v = 0.1652217099371571E-1 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + + case 86: + + v = 0.1154401154401154E-1 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.1194390908585628E-1 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.3696028464541502E+0 + v = 0.1111055571060340E-1 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6943540066026664E+0 + v = 0.1187650129453714E-1 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3742430390903412E+0 + v = 0.1181230374690448E-1 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + + case 110: + + v = 0.3828270494937162E-2 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.9793737512487512E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.1851156353447362E+0 + v = 0.8211737283191111E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6904210483822922E+0 + v = 0.9942814891178103E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3956894730559419E+0 + v = 0.9595471336070963E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4783690288121502E+0 + v = 0.9694996361663028E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + + case 146: + + v = 0.5996313688621381E-3 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.7372999718620756E-2 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.7210515360144488E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.6764410400114264E+0 + v = 0.7116355493117555E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4174961227965453E+0 + v = 0.6753829486314477E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1574676672039082E+0 + v = 0.7574394159054034E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1403553811713183E+0 + b = 0.4493328323269557E+0 + v = 0.6991087353303262E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 170: + + v = 0.5544842902037365E-2 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.6071332770670752E-2 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.6383674773515093E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.2551252621114134E+0 + v = 0.5183387587747790E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6743601460362766E+0 + v = 0.6317929009813725E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4318910696719410E+0 + v = 0.6201670006589077E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2613931360335988E+0 + v = 0.5477143385137348E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4990453161796037E+0 + b = 0.1446630744325115E+0 + v = 0.5968383987681156E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 194: + + v = 0.1782340447244611E-2 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.5716905949977102E-2 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.5573383178848738E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.6712973442695226E+0 + v = 0.5608704082587997E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2892465627575439E+0 + v = 0.5158237711805383E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4446933178717437E+0 + v = 0.5518771467273614E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1299335447650067E+0 + v = 0.4106777028169394E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3457702197611283E+0 + v = 0.5051846064614808E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1590417105383530E+0 + b = 0.8360360154824589E+0 + v = 0.5530248916233094E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 230: + + v = -0.5522639919727325E-1 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.4450274607445226E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.4492044687397611E+0 + v = 0.4496841067921404E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2520419490210201E+0 + v = 0.5049153450478750E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6981906658447242E+0 + v = 0.3976408018051883E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6587405243460960E+0 + v = 0.4401400650381014E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4038544050097660E-1 + v = 0.1724544350544401E-1 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5823842309715585E+0 + v = 0.4231083095357343E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3545877390518688E+0 + v = 0.5198069864064399E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2272181808998187E+0 + b = 0.4864661535886647E+0 + v = 0.4695720972568883E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 266: + + v = -0.1313769127326952E-2 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = -0.2522728704859336E-2 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.4186853881700583E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.7039373391585475E+0 + v = 0.5315167977810885E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1012526248572414E+0 + v = 0.4047142377086219E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4647448726420539E+0 + v = 0.4112482394406990E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3277420654971629E+0 + v = 0.3595584899758782E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6620338663699974E+0 + v = 0.4256131351428158E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.8506508083520399E+0 + v = 0.4229582700647240E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3233484542692899E+0 + b = 0.1153112011009701E+0 + v = 0.4080914225780505E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2314790158712601E+0 + b = 0.5244939240922365E+0 + v = 0.4071467593830964E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 302: + + v = 0.8545911725128148E-3 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.3599119285025571E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.3515640345570105E+0 + v = 0.3449788424305883E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6566329410219612E+0 + v = 0.3604822601419882E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4729054132581005E+0 + v = 0.3576729661743367E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.9618308522614784E-1 + v = 0.2352101413689164E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2219645236294178E+0 + v = 0.3108953122413675E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7011766416089545E+0 + v = 0.3650045807677255E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2644152887060663E+0 + v = 0.2982344963171804E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5718955891878961E+0 + v = 0.3600820932216460E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2510034751770465E+0 + b = 0.8000727494073952E+0 + v = 0.3571540554273387E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1233548532583327E+0 + b = 0.4127724083168531E+0 + v = 0.3392312205006170E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 350: + + v = 0.3006796749453936E-2 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.3050627745650771E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.7068965463912316E+0 + v = 0.1621104600288991E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4794682625712025E+0 + v = 0.3005701484901752E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1927533154878019E+0 + v = 0.2990992529653774E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6930357961327123E+0 + v = 0.2982170644107595E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3608302115520091E+0 + v = 0.2721564237310992E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6498486161496169E+0 + v = 0.3033513795811141E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1932945013230339E+0 + v = 0.3007949555218533E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3800494919899303E+0 + v = 0.2881964603055307E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2899558825499574E+0 + b = 0.7934537856582316E+0 + v = 0.2958357626535696E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.9684121455103957E-1 + b = 0.8280801506686862E+0 + v = 0.3036020026407088E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1833434647041659E+0 + b = 0.9074658265305127E+0 + v = 0.2832187403926303E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 434: + + v = 0.5265897968224436E-3 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.2548219972002607E-2 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.2512317418927307E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.6909346307509111E+0 + v = 0.2530403801186355E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1774836054609158E+0 + v = 0.2014279020918528E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4914342637784746E+0 + v = 0.2501725168402936E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6456664707424256E+0 + v = 0.2513267174597564E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2861289010307638E+0 + v = 0.2302694782227416E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7568084367178018E-1 + v = 0.1462495621594614E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3927259763368002E+0 + v = 0.2445373437312980E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.8818132877794288E+0 + v = 0.2417442375638981E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.9776428111182649E+0 + v = 0.1910951282179532E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2054823696403044E+0 + b = 0.8689460322872412E+0 + v = 0.2416930044324775E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5905157048925271E+0 + b = 0.7999278543857286E+0 + v = 0.2512236854563495E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5550152361076807E+0 + b = 0.7717462626915901E+0 + v = 0.2496644054553086E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.9371809858553722E+0 + b = 0.3344363145343455E+0 + v = 0.2236607760437849E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 590: + + v = 0.3095121295306187E-3 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.1852379698597489E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.7040954938227469E+0 + v = 0.1871790639277744E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6807744066455243E+0 + v = 0.1858812585438317E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6372546939258752E+0 + v = 0.1852028828296213E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5044419707800358E+0 + v = 0.1846715956151242E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4215761784010967E+0 + v = 0.1818471778162769E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3317920736472123E+0 + v = 0.1749564657281154E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2384736701421887E+0 + v = 0.1617210647254411E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1459036449157763E+0 + v = 0.1384737234851692E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6095034115507196E-1 + v = 0.9764331165051050E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6116843442009876E+0 + v = 0.1857161196774078E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3964755348199858E+0 + v = 0.1705153996395864E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1724782009907724E+0 + v = 0.1300321685886048E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5610263808622060E+0 + b = 0.3518280927733519E+0 + v = 0.1842866472905286E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4742392842551980E+0 + b = 0.2634716655937950E+0 + v = 0.1802658934377451E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5984126497885380E+0 + b = 0.1816640840360209E+0 + v = 0.1849830560443660E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3791035407695563E+0 + b = 0.1720795225656878E+0 + v = 0.1713904507106709E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2778673190586244E+0 + b = 0.8213021581932511E-1 + v = 0.1555213603396808E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5033564271075117E+0 + b = 0.8999205842074875E-1 + v = 0.1802239128008525E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 770: + + v = 0.2192942088181184E-3 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.1436433617319080E-2 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.1421940344335877E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.5087204410502360E-1 + v = 0.6798123511050502E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1228198790178831E+0 + v = 0.9913184235294912E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2026890814408786E+0 + v = 0.1180207833238949E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2847745156464294E+0 + v = 0.1296599602080921E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3656719078978026E+0 + v = 0.1365871427428316E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4428264886713469E+0 + v = 0.1402988604775325E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5140619627249735E+0 + v = 0.1418645563595609E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6306401219166803E+0 + v = 0.1421376741851662E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6716883332022612E+0 + v = 0.1423996475490962E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6979792685336881E+0 + v = 0.1431554042178567E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1446865674195309E+0 + v = 0.9254401499865368E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3390263475411216E+0 + v = 0.1250239995053509E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5335804651263506E+0 + v = 0.1394365843329230E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6944024393349413E-1 + b = 0.2355187894242326E+0 + v = 0.1127089094671749E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2269004109529460E+0 + b = 0.4102182474045730E+0 + v = 0.1345753760910670E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.8025574607775339E-1 + b = 0.6214302417481605E+0 + v = 0.1424957283316783E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1467999527896572E+0 + b = 0.3245284345717394E+0 + v = 0.1261523341237750E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1571507769824727E+0 + b = 0.5224482189696630E+0 + v = 0.1392547106052696E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2365702993157246E+0 + b = 0.6017546634089558E+0 + v = 0.1418761677877656E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.7714815866765732E-1 + b = 0.4346575516141163E+0 + v = 0.1338366684479554E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3062936666210730E+0 + b = 0.4908826589037616E+0 + v = 0.1393700862676131E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3822477379524787E+0 + b = 0.5648768149099500E+0 + v = 0.1415914757466932E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 974: + + v = 0.1438294190527431E-3 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.1125772288287004E-2 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.4292963545341347E-1 + v = 0.4948029341949241E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1051426854086404E+0 + v = 0.7357990109125470E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1750024867623087E+0 + v = 0.8889132771304384E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2477653379650257E+0 + v = 0.9888347838921435E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3206567123955957E+0 + v = 0.1053299681709471E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3916520749849983E+0 + v = 0.1092778807014578E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4590825874187624E+0 + v = 0.1114389394063227E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5214563888415861E+0 + v = 0.1123724788051555E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6253170244654199E+0 + v = 0.1125239325243814E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6637926744523170E+0 + v = 0.1126153271815905E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6910410398498301E+0 + v = 0.1130286931123841E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7052907007457760E+0 + v = 0.1134986534363955E-2 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1236686762657990E+0 + v = 0.6823367927109931E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2940777114468387E+0 + v = 0.9454158160447096E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4697753849207649E+0 + v = 0.1074429975385679E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6334563241139567E+0 + v = 0.1129300086569132E-2 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5974048614181342E-1 + b = 0.2029128752777523E+0 + v = 0.8436884500901954E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1375760408473636E+0 + b = 0.4602621942484054E+0 + v = 0.1075255720448885E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3391016526336286E+0 + b = 0.5030673999662036E+0 + v = 0.1108577236864462E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1271675191439820E+0 + b = 0.2817606422442134E+0 + v = 0.9566475323783357E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2693120740413512E+0 + b = 0.4331561291720157E+0 + v = 0.1080663250717391E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1419786452601918E+0 + b = 0.6256167358580814E+0 + v = 0.1126797131196295E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6709284600738255E-1 + b = 0.3798395216859157E+0 + v = 0.1022568715358061E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.7057738183256172E-1 + b = 0.5517505421423520E+0 + v = 0.1108960267713108E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2783888477882155E+0 + b = 0.6029619156159187E+0 + v = 0.1122790653435766E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1979578938917407E+0 + b = 0.3589606329589096E+0 + v = 0.1032401847117460E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2087307061103274E+0 + b = 0.5348666438135476E+0 + v = 0.1107249382283854E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4055122137872836E+0 + b = 0.5674997546074373E+0 + v = 0.1121780048519972E-2 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 1202: + + v = 0.1105189233267572E-3 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.9205232738090741E-3 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.9133159786443561E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.3712636449657089E-1 + v = 0.3690421898017899E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.9140060412262223E-1 + v = 0.5603990928680660E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1531077852469906E+0 + v = 0.6865297629282609E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2180928891660612E+0 + v = 0.7720338551145630E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2839874532200175E+0 + v = 0.8301545958894795E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3491177600963764E+0 + v = 0.8686692550179628E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4121431461444309E+0 + v = 0.8927076285846890E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4718993627149127E+0 + v = 0.9060820238568219E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5273145452842337E+0 + v = 0.9119777254940867E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6209475332444019E+0 + v = 0.9128720138604181E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6569722711857291E+0 + v = 0.9130714935691735E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6841788309070143E+0 + v = 0.9152873784554116E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7012604330123631E+0 + v = 0.9187436274321654E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1072382215478166E+0 + v = 0.5176977312965694E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2582068959496968E+0 + v = 0.7331143682101417E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4172752955306717E+0 + v = 0.8463232836379928E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5700366911792503E+0 + v = 0.9031122694253992E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.9827986018263947E+0 + b = 0.1771774022615325E+0 + v = 0.6485778453163257E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.9624249230326228E+0 + b = 0.2475716463426288E+0 + v = 0.7435030910982369E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.9402007994128811E+0 + b = 0.3354616289066489E+0 + v = 0.7998527891839054E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.9320822040143202E+0 + b = 0.3173615246611977E+0 + v = 0.8101731497468018E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.9043674199393299E+0 + b = 0.4090268427085357E+0 + v = 0.8483389574594331E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.8912407560074747E+0 + b = 0.3854291150669224E+0 + v = 0.8556299257311812E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.8676435628462708E+0 + b = 0.4932221184851285E+0 + v = 0.8803208679738260E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.8581979986041619E+0 + b = 0.4785320675922435E+0 + v = 0.8811048182425720E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.8396753624049856E+0 + b = 0.4507422593157064E+0 + v = 0.8850282341265444E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.8165288564022188E+0 + b = 0.5632123020762100E+0 + v = 0.9021342299040653E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.8015469370783529E+0 + b = 0.5434303569693900E+0 + v = 0.9010091677105086E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.7773563069070351E+0 + b = 0.5123518486419871E+0 + v = 0.9022692938426915E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.7661621213900394E+0 + b = 0.6394279634749102E+0 + v = 0.9158016174693465E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.7553584143533510E+0 + b = 0.6269805509024392E+0 + v = 0.9131578003189435E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.7344305757559503E+0 + b = 0.6031161693096310E+0 + v = 0.9107813579482705E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.7043837184021765E+0 + b = 0.5693702498468441E+0 + v = 0.9105760258970126E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 1454: + + v = 0.7777160743261247E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.7557646413004701E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.3229290663413854E-1 + v = 0.2841633806090617E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.8036733271462222E-1 + v = 0.4374419127053555E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1354289960531653E+0 + v = 0.5417174740872172E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1938963861114426E+0 + v = 0.6148000891358593E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2537343715011275E+0 + v = 0.6664394485800705E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3135251434752570E+0 + v = 0.7025039356923220E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3721558339375338E+0 + v = 0.7268511789249627E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4286809575195696E+0 + v = 0.7422637534208629E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4822510128282994E+0 + v = 0.7509545035841214E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5320679333566263E+0 + v = 0.7548535057718401E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6172998195394274E+0 + v = 0.7554088969774001E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6510679849127481E+0 + v = 0.7553147174442808E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6777315251687360E+0 + v = 0.7564767653292297E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6963109410648741E+0 + v = 0.7587991808518730E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7058935009831749E+0 + v = 0.7608261832033027E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.9955546194091857E+0 + v = 0.4021680447874916E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.9734115901794209E+0 + v = 0.5804871793945964E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.9275693732388626E+0 + v = 0.6792151955945159E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.8568022422795103E+0 + v = 0.7336741211286294E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.7623495553719372E+0 + v = 0.7581866300989608E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5707522908892223E+0 + b = 0.4387028039889501E+0 + v = 0.7538257859800743E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5196463388403083E+0 + b = 0.3858908414762617E+0 + v = 0.7483517247053123E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4646337531215351E+0 + b = 0.3301937372343854E+0 + v = 0.7371763661112059E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4063901697557691E+0 + b = 0.2725423573563777E+0 + v = 0.7183448895756934E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3456329466643087E+0 + b = 0.2139510237495250E+0 + v = 0.6895815529822191E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2831395121050332E+0 + b = 0.1555922309786647E+0 + v = 0.6480105801792886E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2197682022925330E+0 + b = 0.9892878979686097E-1 + v = 0.5897558896594636E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1564696098650355E+0 + b = 0.4598642910675510E-1 + v = 0.5095708849247346E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6027356673721295E+0 + b = 0.3376625140173426E+0 + v = 0.7536906428909755E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5496032320255096E+0 + b = 0.2822301309727988E+0 + v = 0.7472505965575118E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4921707755234567E+0 + b = 0.2248632342592540E+0 + v = 0.7343017132279698E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4309422998598483E+0 + b = 0.1666224723456479E+0 + v = 0.7130871582177445E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3664108182313672E+0 + b = 0.1086964901822169E+0 + v = 0.6817022032112776E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2990189057758436E+0 + b = 0.5251989784120085E-1 + v = 0.6380941145604121E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6268724013144998E+0 + b = 0.2297523657550023E+0 + v = 0.7550381377920310E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5707324144834607E+0 + b = 0.1723080607093800E+0 + v = 0.7478646640144802E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5096360901960365E+0 + b = 0.1140238465390513E+0 + v = 0.7335918720601220E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4438729938312456E+0 + b = 0.5611522095882537E-1 + v = 0.7110120527658118E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6419978471082389E+0 + b = 0.1164174423140873E+0 + v = 0.7571363978689501E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5817218061802611E+0 + b = 0.5797589531445219E-1 + v = 0.7489908329079234E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 1730: + + v = 0.6309049437420976E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.6398287705571748E-3 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.6357185073530720E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.2860923126194662E-1 + v = 0.2221207162188168E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7142556767711522E-1 + v = 0.3475784022286848E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1209199540995559E+0 + v = 0.4350742443589804E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1738673106594379E+0 + v = 0.4978569136522127E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2284645438467734E+0 + v = 0.5435036221998053E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2834807671701512E+0 + v = 0.5765913388219542E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3379680145467339E+0 + v = 0.6001200359226003E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3911355454819537E+0 + v = 0.6162178172717512E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4422860353001403E+0 + v = 0.6265218152438485E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4907781568726057E+0 + v = 0.6323987160974212E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5360006153211468E+0 + v = 0.6350767851540569E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6142105973596603E+0 + v = 0.6354362775297107E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6459300387977504E+0 + v = 0.6352302462706235E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6718056125089225E+0 + v = 0.6358117881417972E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6910888533186254E+0 + v = 0.6373101590310117E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7030467416823252E+0 + v = 0.6390428961368665E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.8354951166354646E-1 + v = 0.3186913449946576E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2050143009099486E+0 + v = 0.4678028558591711E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3370208290706637E+0 + v = 0.5538829697598626E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4689051484233963E+0 + v = 0.6044475907190476E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5939400424557334E+0 + v = 0.6313575103509012E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1394983311832261E+0 + b = 0.4097581162050343E-1 + v = 0.4078626431855630E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1967999180485014E+0 + b = 0.8851987391293348E-1 + v = 0.4759933057812725E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2546183732548967E+0 + b = 0.1397680182969819E+0 + v = 0.5268151186413440E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3121281074713875E+0 + b = 0.1929452542226526E+0 + v = 0.5643048560507316E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3685981078502492E+0 + b = 0.2467898337061562E+0 + v = 0.5914501076613073E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4233760321547856E+0 + b = 0.3003104124785409E+0 + v = 0.6104561257874195E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4758671236059246E+0 + b = 0.3526684328175033E+0 + v = 0.6230252860707806E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5255178579796463E+0 + b = 0.4031134861145713E+0 + v = 0.6305618761760796E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5718025633734589E+0 + b = 0.4509426448342351E+0 + v = 0.6343092767597889E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2686927772723415E+0 + b = 0.4711322502423248E-1 + v = 0.5176268945737826E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3306006819904809E+0 + b = 0.9784487303942695E-1 + v = 0.5564840313313692E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3904906850594983E+0 + b = 0.1505395810025273E+0 + v = 0.5856426671038980E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4479957951904390E+0 + b = 0.2039728156296050E+0 + v = 0.6066386925777091E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5027076848919780E+0 + b = 0.2571529941121107E+0 + v = 0.6208824962234458E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5542087392260217E+0 + b = 0.3092191375815670E+0 + v = 0.6296314297822907E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6020850887375187E+0 + b = 0.3593807506130276E+0 + v = 0.6340423756791859E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4019851409179594E+0 + b = 0.5063389934378671E-1 + v = 0.5829627677107342E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4635614567449800E+0 + b = 0.1032422269160612E+0 + v = 0.6048693376081110E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5215860931591575E+0 + b = 0.1566322094006254E+0 + v = 0.6202362317732461E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5758202499099271E+0 + b = 0.2098082827491099E+0 + v = 0.6299005328403779E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6259893683876795E+0 + b = 0.2618824114553391E+0 + v = 0.6347722390609353E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5313795124811891E+0 + b = 0.5263245019338556E-1 + v = 0.6203778981238834E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5893317955931995E+0 + b = 0.1061059730982005E+0 + v = 0.6308414671239979E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6426246321215801E+0 + b = 0.1594171564034221E+0 + v = 0.6362706466959498E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6511904367376113E+0 + b = 0.5354789536565540E-1 + v = 0.6375414170333233E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 2030: + + v = 0.4656031899197431E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.5421549195295507E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.2540835336814348E-1 + v = 0.1778522133346553E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6399322800504915E-1 + v = 0.2811325405682796E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1088269469804125E+0 + v = 0.3548896312631459E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1570670798818287E+0 + v = 0.4090310897173364E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2071163932282514E+0 + v = 0.4493286134169965E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2578914044450844E+0 + v = 0.4793728447962723E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3085687558169623E+0 + v = 0.5015415319164265E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3584719706267024E+0 + v = 0.5175127372677937E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4070135594428709E+0 + v = 0.5285522262081019E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4536618626222638E+0 + v = 0.5356832703713962E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4979195686463577E+0 + v = 0.5397914736175170E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5393075111126999E+0 + v = 0.5416899441599930E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6115617676843916E+0 + v = 0.5419308476889938E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6414308435160159E+0 + v = 0.5416936902030596E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6664099412721607E+0 + v = 0.5419544338703164E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6859161771214913E+0 + v = 0.5428983656630975E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6993625593503890E+0 + v = 0.5442286500098193E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7062393387719380E+0 + v = 0.5452250345057301E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7479028168349763E-1 + v = 0.2568002497728530E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1848951153969366E+0 + v = 0.3827211700292145E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3059529066581305E+0 + v = 0.4579491561917824E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4285556101021362E+0 + v = 0.5042003969083574E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5468758653496526E+0 + v = 0.5312708889976025E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6565821978343439E+0 + v = 0.5438401790747117E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1253901572367117E+0 + b = 0.3681917226439641E-1 + v = 0.3316041873197344E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1775721510383941E+0 + b = 0.7982487607213301E-1 + v = 0.3899113567153771E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2305693358216114E+0 + b = 0.1264640966592335E+0 + v = 0.4343343327201309E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2836502845992063E+0 + b = 0.1751585683418957E+0 + v = 0.4679415262318919E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3361794746232590E+0 + b = 0.2247995907632670E+0 + v = 0.4930847981631031E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3875979172264824E+0 + b = 0.2745299257422246E+0 + v = 0.5115031867540091E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4374019316999074E+0 + b = 0.3236373482441118E+0 + v = 0.5245217148457367E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4851275843340022E+0 + b = 0.3714967859436741E+0 + v = 0.5332041499895321E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5303391803806868E+0 + b = 0.4175353646321745E+0 + v = 0.5384583126021542E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5726197380596287E+0 + b = 0.4612084406355461E+0 + v = 0.5411067210798852E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2431520732564863E+0 + b = 0.4258040133043952E-1 + v = 0.4259797391468714E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3002096800895869E+0 + b = 0.8869424306722721E-1 + v = 0.4604931368460021E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3558554457457432E+0 + b = 0.1368811706510655E+0 + v = 0.4871814878255202E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4097782537048887E+0 + b = 0.1860739985015033E+0 + v = 0.5072242910074885E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4616337666067458E+0 + b = 0.2354235077395853E+0 + v = 0.5217069845235350E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5110707008417874E+0 + b = 0.2842074921347011E+0 + v = 0.5315785966280310E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5577415286163795E+0 + b = 0.3317784414984102E+0 + v = 0.5376833708758905E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6013060431366950E+0 + b = 0.3775299002040700E+0 + v = 0.5408032092069521E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3661596767261781E+0 + b = 0.4599367887164592E-1 + v = 0.4842744917904866E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4237633153506581E+0 + b = 0.9404893773654421E-1 + v = 0.5048926076188130E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4786328454658452E+0 + b = 0.1431377109091971E+0 + v = 0.5202607980478373E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5305702076789774E+0 + b = 0.1924186388843570E+0 + v = 0.5309932388325743E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5793436224231788E+0 + b = 0.2411590944775190E+0 + v = 0.5377419770895208E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6247069017094747E+0 + b = 0.2886871491583605E+0 + v = 0.5411696331677717E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4874315552535204E+0 + b = 0.4804978774953206E-1 + v = 0.5197996293282420E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5427337322059053E+0 + b = 0.9716857199366665E-1 + v = 0.5311120836622945E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5943493747246700E+0 + b = 0.1465205839795055E+0 + v = 0.5384309319956951E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6421314033564943E+0 + b = 0.1953579449803574E+0 + v = 0.5421859504051886E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6020628374713980E+0 + b = 0.4916375015738108E-1 + v = 0.5390948355046314E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6529222529856881E+0 + b = 0.9861621540127005E-1 + v = 0.5433312705027845E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 2354: + + v = 0.3922616270665292E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.4703831750854424E-3 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.4678202801282136E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.2290024646530589E-1 + v = 0.1437832228979900E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5779086652271284E-1 + v = 0.2303572493577644E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.9863103576375984E-1 + v = 0.2933110752447454E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1428155792982185E+0 + v = 0.3402905998359838E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1888978116601463E+0 + v = 0.3759138466870372E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2359091682970210E+0 + v = 0.4030638447899798E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2831228833706171E+0 + v = 0.4236591432242211E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3299495857966693E+0 + v = 0.4390522656946746E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3758840802660796E+0 + v = 0.4502523466626247E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4204751831009480E+0 + v = 0.4580577727783541E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4633068518751051E+0 + v = 0.4631391616615899E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5039849474507313E+0 + v = 0.4660928953698676E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5421265793440747E+0 + v = 0.4674751807936953E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6092660230557310E+0 + v = 0.4676414903932920E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6374654204984869E+0 + v = 0.4674086492347870E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6615136472609892E+0 + v = 0.4674928539483207E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6809487285958127E+0 + v = 0.4680748979686447E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6952980021665196E+0 + v = 0.4690449806389040E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7041245497695400E+0 + v = 0.4699877075860818E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6744033088306065E-1 + v = 0.2099942281069176E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1678684485334166E+0 + v = 0.3172269150712804E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2793559049539613E+0 + v = 0.3832051358546523E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3935264218057639E+0 + v = 0.4252193818146985E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5052629268232558E+0 + v = 0.4513807963755000E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6107905315437531E+0 + v = 0.4657797469114178E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1135081039843524E+0 + b = 0.3331954884662588E-1 + v = 0.2733362800522836E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1612866626099378E+0 + b = 0.7247167465436538E-1 + v = 0.3235485368463559E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2100786550168205E+0 + b = 0.1151539110849745E+0 + v = 0.3624908726013453E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2592282009459942E+0 + b = 0.1599491097143677E+0 + v = 0.3925540070712828E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3081740561320203E+0 + b = 0.2058699956028027E+0 + v = 0.4156129781116235E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3564289781578164E+0 + b = 0.2521624953502911E+0 + v = 0.4330644984623263E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4035587288240703E+0 + b = 0.2982090785797674E+0 + v = 0.4459677725921312E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4491671196373903E+0 + b = 0.3434762087235733E+0 + v = 0.4551593004456795E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4928854782917489E+0 + b = 0.3874831357203437E+0 + v = 0.4613341462749918E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5343646791958988E+0 + b = 0.4297814821746926E+0 + v = 0.4651019618269806E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5732683216530990E+0 + b = 0.4699402260943537E+0 + v = 0.4670249536100625E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2214131583218986E+0 + b = 0.3873602040643895E-1 + v = 0.3549555576441708E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2741796504750071E+0 + b = 0.8089496256902013E-1 + v = 0.3856108245249010E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3259797439149485E+0 + b = 0.1251732177620872E+0 + v = 0.4098622845756882E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3765441148826891E+0 + b = 0.1706260286403185E+0 + v = 0.4286328604268950E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4255773574530558E+0 + b = 0.2165115147300408E+0 + v = 0.4427802198993945E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4727795117058430E+0 + b = 0.2622089812225259E+0 + v = 0.4530473511488561E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5178546895819012E+0 + b = 0.3071721431296201E+0 + v = 0.4600805475703138E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5605141192097460E+0 + b = 0.3508998998801138E+0 + v = 0.4644599059958017E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6004763319352512E+0 + b = 0.3929160876166931E+0 + v = 0.4667274455712508E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3352842634946949E+0 + b = 0.4202563457288019E-1 + v = 0.4069360518020356E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3891971629814670E+0 + b = 0.8614309758870850E-1 + v = 0.4260442819919195E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4409875565542281E+0 + b = 0.1314500879380001E+0 + v = 0.4408678508029063E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4904893058592484E+0 + b = 0.1772189657383859E+0 + v = 0.4518748115548597E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5375056138769549E+0 + b = 0.2228277110050294E+0 + v = 0.4595564875375116E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5818255708669969E+0 + b = 0.2677179935014386E+0 + v = 0.4643988774315846E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6232334858144959E+0 + b = 0.3113675035544165E+0 + v = 0.4668827491646946E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4489485354492058E+0 + b = 0.4409162378368174E-1 + v = 0.4400541823741973E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5015136875933150E+0 + b = 0.8939009917748489E-1 + v = 0.4514512890193797E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5511300550512623E+0 + b = 0.1351806029383365E+0 + v = 0.4596198627347549E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5976720409858000E+0 + b = 0.1808370355053196E+0 + v = 0.4648659016801781E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6409956378989354E+0 + b = 0.2257852192301602E+0 + v = 0.4675502017157673E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5581222330827514E+0 + b = 0.4532173421637160E-1 + v = 0.4598494476455523E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6074705984161695E+0 + b = 0.9117488031840314E-1 + v = 0.4654916955152048E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6532272537379033E+0 + b = 0.1369294213140155E+0 + v = 0.4684709779505137E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6594761494500487E+0 + b = 0.4589901487275583E-1 + v = 0.4691445539106986E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 2702: + + v = 0.2998675149888161E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.4077860529495355E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.2065562538818703E-1 + v = 0.1185349192520667E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5250918173022379E-1 + v = 0.1913408643425751E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.8993480082038376E-1 + v = 0.2452886577209897E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1306023924436019E+0 + v = 0.2862408183288702E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1732060388531418E+0 + v = 0.3178032258257357E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2168727084820249E+0 + v = 0.3422945667633690E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2609528309173586E+0 + v = 0.3612790520235922E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3049252927938952E+0 + v = 0.3758638229818521E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3483484138084404E+0 + v = 0.3868711798859953E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3908321549106406E+0 + v = 0.3949429933189938E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4320210071894814E+0 + v = 0.4006068107541156E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4715824795890053E+0 + v = 0.4043192149672723E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5091984794078453E+0 + v = 0.4064947495808078E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5445580145650803E+0 + v = 0.4075245619813152E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6072575796841768E+0 + v = 0.4076423540893566E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6339484505755803E+0 + v = 0.4074280862251555E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6570718257486958E+0 + v = 0.4074163756012244E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6762557330090709E+0 + v = 0.4077647795071246E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6911161696923790E+0 + v = 0.4084517552782530E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7012841911659961E+0 + v = 0.4092468459224052E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7064559272410020E+0 + v = 0.4097872687240906E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6123554989894765E-1 + v = 0.1738986811745028E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1533070348312393E+0 + v = 0.2659616045280191E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2563902605244206E+0 + v = 0.3240596008171533E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3629346991663361E+0 + v = 0.3621195964432943E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4683949968987538E+0 + v = 0.3868838330760539E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5694479240657952E+0 + v = 0.4018911532693111E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6634465430993955E+0 + v = 0.4089929432983252E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1033958573552305E+0 + b = 0.3034544009063584E-1 + v = 0.2279907527706409E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1473521412414395E+0 + b = 0.6618803044247135E-1 + v = 0.2715205490578897E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1924552158705967E+0 + b = 0.1054431128987715E+0 + v = 0.3057917896703976E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2381094362890328E+0 + b = 0.1468263551238858E+0 + v = 0.3326913052452555E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2838121707936760E+0 + b = 0.1894486108187886E+0 + v = 0.3537334711890037E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3291323133373415E+0 + b = 0.2326374238761579E+0 + v = 0.3700567500783129E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3736896978741460E+0 + b = 0.2758485808485768E+0 + v = 0.3825245372589122E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4171406040760013E+0 + b = 0.3186179331996921E+0 + v = 0.3918125171518296E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4591677985256915E+0 + b = 0.3605329796303794E+0 + v = 0.3984720419937579E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4994733831718418E+0 + b = 0.4012147253586509E+0 + v = 0.4029746003338211E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5377731830445096E+0 + b = 0.4403050025570692E+0 + v = 0.4057428632156627E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5737917830001331E+0 + b = 0.4774565904277483E+0 + v = 0.4071719274114857E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2027323586271389E+0 + b = 0.3544122504976147E-1 + v = 0.2990236950664119E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2516942375187273E+0 + b = 0.7418304388646328E-1 + v = 0.3262951734212878E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3000227995257181E+0 + b = 0.1150502745727186E+0 + v = 0.3482634608242413E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3474806691046342E+0 + b = 0.1571963371209364E+0 + v = 0.3656596681700892E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3938103180359209E+0 + b = 0.1999631877247100E+0 + v = 0.3791740467794218E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4387519590455703E+0 + b = 0.2428073457846535E+0 + v = 0.3894034450156905E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4820503960077787E+0 + b = 0.2852575132906155E+0 + v = 0.3968600245508371E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5234573778475101E+0 + b = 0.3268884208674639E+0 + v = 0.4019931351420050E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5627318647235282E+0 + b = 0.3673033321675939E+0 + v = 0.4052108801278599E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5996390607156954E+0 + b = 0.4061211551830290E+0 + v = 0.4068978613940934E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3084780753791947E+0 + b = 0.3860125523100059E-1 + v = 0.3454275351319704E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3589988275920223E+0 + b = 0.7928938987104867E-1 + v = 0.3629963537007920E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4078628415881973E+0 + b = 0.1212614643030087E+0 + v = 0.3770187233889873E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4549287258889735E+0 + b = 0.1638770827382693E+0 + v = 0.3878608613694378E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5000278512957279E+0 + b = 0.2065965798260176E+0 + v = 0.3959065270221274E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5429785044928199E+0 + b = 0.2489436378852235E+0 + v = 0.4015286975463570E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5835939850491711E+0 + b = 0.2904811368946891E+0 + v = 0.4050866785614717E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6216870353444856E+0 + b = 0.3307941957666609E+0 + v = 0.4069320185051913E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4151104662709091E+0 + b = 0.4064829146052554E-1 + v = 0.3760120964062763E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4649804275009218E+0 + b = 0.8258424547294755E-1 + v = 0.3870969564418064E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5124695757009662E+0 + b = 0.1251841962027289E+0 + v = 0.3955287790534055E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5574711100606224E+0 + b = 0.1679107505976331E+0 + v = 0.4015361911302668E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5998597333287227E+0 + b = 0.2102805057358715E+0 + v = 0.4053836986719548E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6395007148516600E+0 + b = 0.2518418087774107E+0 + v = 0.4073578673299117E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5188456224746252E+0 + b = 0.4194321676077518E-1 + v = 0.3954628379231406E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5664190707942778E+0 + b = 0.8457661551921499E-1 + v = 0.4017645508847530E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6110464353283153E+0 + b = 0.1273652932519396E+0 + v = 0.4059030348651293E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6526430302051563E+0 + b = 0.1698173239076354E+0 + v = 0.4080565809484880E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6167551880377548E+0 + b = 0.4266398851548864E-1 + v = 0.4063018753664651E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6607195418355383E+0 + b = 0.8551925814238349E-1 + v = 0.4087191292799671E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 3074: + + v = 0.2599095953754734E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.3603134089687541E-3 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.3586067974412447E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.1886108518723392E-1 + v = 0.9831528474385880E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4800217244625303E-1 + v = 0.1605023107954450E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.8244922058397242E-1 + v = 0.2072200131464099E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1200408362484023E+0 + v = 0.2431297618814187E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1595773530809965E+0 + v = 0.2711819064496707E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2002635973434064E+0 + v = 0.2932762038321116E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2415127590139982E+0 + v = 0.3107032514197368E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2828584158458477E+0 + v = 0.3243808058921213E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3239091015338138E+0 + v = 0.3349899091374030E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3643225097962194E+0 + v = 0.3430580688505218E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4037897083691802E+0 + v = 0.3490124109290343E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4420247515194127E+0 + v = 0.3532148948561955E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4787572538464938E+0 + v = 0.3559862669062833E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5137265251275234E+0 + v = 0.3576224317551411E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5466764056654611E+0 + v = 0.3584050533086076E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6054859420813535E+0 + v = 0.3584903581373224E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6308106701764562E+0 + v = 0.3582991879040586E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6530369230179584E+0 + v = 0.3582371187963125E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6718609524611158E+0 + v = 0.3584353631122350E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6869676499894013E+0 + v = 0.3589120166517785E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6980467077240748E+0 + v = 0.3595445704531601E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7048241721250522E+0 + v = 0.3600943557111074E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5591105222058232E-1 + v = 0.1456447096742039E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1407384078513916E+0 + v = 0.2252370188283782E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2364035438976309E+0 + v = 0.2766135443474897E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3360602737818170E+0 + v = 0.3110729491500851E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4356292630054665E+0 + v = 0.3342506712303391E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5321569415256174E+0 + v = 0.3491981834026860E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6232956305040554E+0 + v = 0.3576003604348932E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.9469870086838469E-1 + b = 0.2778748387309470E-1 + v = 0.1921921305788564E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1353170300568141E+0 + b = 0.6076569878628364E-1 + v = 0.2301458216495632E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1771679481726077E+0 + b = 0.9703072762711040E-1 + v = 0.2604248549522893E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2197066664231751E+0 + b = 0.1354112458524762E+0 + v = 0.2845275425870697E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2624783557374927E+0 + b = 0.1750996479744100E+0 + v = 0.3036870897974840E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3050969521214442E+0 + b = 0.2154896907449802E+0 + v = 0.3188414832298066E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3472252637196021E+0 + b = 0.2560954625740152E+0 + v = 0.3307046414722089E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3885610219026360E+0 + b = 0.2965070050624096E+0 + v = 0.3398330969031360E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4288273776062765E+0 + b = 0.3363641488734497E+0 + v = 0.3466757899705373E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4677662471302948E+0 + b = 0.3753400029836788E+0 + v = 0.3516095923230054E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5051333589553359E+0 + b = 0.4131297522144286E+0 + v = 0.3549645184048486E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5406942145810492E+0 + b = 0.4494423776081795E+0 + v = 0.3570415969441392E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5742204122576457E+0 + b = 0.4839938958841502E+0 + v = 0.3581251798496118E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1865407027225188E+0 + b = 0.3259144851070796E-1 + v = 0.2543491329913348E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2321186453689432E+0 + b = 0.6835679505297343E-1 + v = 0.2786711051330776E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2773159142523882E+0 + b = 0.1062284864451989E+0 + v = 0.2985552361083679E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3219200192237254E+0 + b = 0.1454404409323047E+0 + v = 0.3145867929154039E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3657032593944029E+0 + b = 0.1854018282582510E+0 + v = 0.3273290662067609E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4084376778363622E+0 + b = 0.2256297412014750E+0 + v = 0.3372705511943501E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4499004945751427E+0 + b = 0.2657104425000896E+0 + v = 0.3448274437851510E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4898758141326335E+0 + b = 0.3052755487631557E+0 + v = 0.3503592783048583E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5281547442266309E+0 + b = 0.3439863920645423E+0 + v = 0.3541854792663162E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5645346989813992E+0 + b = 0.3815229456121914E+0 + v = 0.3565995517909428E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5988181252159848E+0 + b = 0.4175752420966734E+0 + v = 0.3578802078302898E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2850425424471603E+0 + b = 0.3562149509862536E-1 + v = 0.2958644592860982E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3324619433027876E+0 + b = 0.7330318886871096E-1 + v = 0.3119548129116835E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3785848333076282E+0 + b = 0.1123226296008472E+0 + v = 0.3250745225005984E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4232891028562115E+0 + b = 0.1521084193337708E+0 + v = 0.3355153415935208E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4664287050829722E+0 + b = 0.1921844459223610E+0 + v = 0.3435847568549328E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5078458493735726E+0 + b = 0.2321360989678303E+0 + v = 0.3495786831622488E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5473779816204180E+0 + b = 0.2715886486360520E+0 + v = 0.3537767805534621E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5848617133811376E+0 + b = 0.3101924707571355E+0 + v = 0.3564459815421428E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6201348281584888E+0 + b = 0.3476121052890973E+0 + v = 0.3578464061225468E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3852191185387871E+0 + b = 0.3763224880035108E-1 + v = 0.3239748762836212E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4325025061073423E+0 + b = 0.7659581935637135E-1 + v = 0.3345491784174287E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4778486229734490E+0 + b = 0.1163381306083900E+0 + v = 0.3429126177301782E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5211663693009000E+0 + b = 0.1563890598752899E+0 + v = 0.3492420343097421E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5623469504853703E+0 + b = 0.1963320810149200E+0 + v = 0.3537399050235257E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6012718188659246E+0 + b = 0.2357847407258738E+0 + v = 0.3566209152659172E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6378179206390117E+0 + b = 0.2743846121244060E+0 + v = 0.3581084321919782E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4836936460214534E+0 + b = 0.3895902610739024E-1 + v = 0.3426522117591512E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5293792562683797E+0 + b = 0.7871246819312640E-1 + v = 0.3491848770121379E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5726281253100033E+0 + b = 0.1187963808202981E+0 + v = 0.3539318235231476E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6133658776169068E+0 + b = 0.1587914708061787E+0 + v = 0.3570231438458694E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6515085491865307E+0 + b = 0.1983058575227646E+0 + v = 0.3586207335051714E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5778692716064976E+0 + b = 0.3977209689791542E-1 + v = 0.3541196205164025E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6207904288086192E+0 + b = 0.7990157592981152E-1 + v = 0.3574296911573953E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6608688171046802E+0 + b = 0.1199671308754309E+0 + v = 0.3591993279818963E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6656263089489130E+0 + b = 0.4015955957805969E-1 + v = 0.3595855034661997E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 3470: + + v = 0.2040382730826330E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.3178149703889544E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.1721420832906233E-1 + v = 0.8288115128076110E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4408875374981770E-1 + v = 0.1360883192522954E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7594680813878681E-1 + v = 0.1766854454542662E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1108335359204799E+0 + v = 0.2083153161230153E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1476517054388567E+0 + v = 0.2333279544657158E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1856731870860615E+0 + v = 0.2532809539930247E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2243634099428821E+0 + v = 0.2692472184211158E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2633006881662727E+0 + v = 0.2819949946811885E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3021340904916283E+0 + v = 0.2920953593973030E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3405594048030089E+0 + v = 0.2999889782948352E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3783044434007372E+0 + v = 0.3060292120496902E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4151194767407910E+0 + v = 0.3105109167522192E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4507705766443257E+0 + v = 0.3136902387550312E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4850346056573187E+0 + v = 0.3157984652454632E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5176950817792470E+0 + v = 0.3170516518425422E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5485384240820989E+0 + v = 0.3176568425633755E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6039117238943308E+0 + v = 0.3177198411207062E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6279956655573113E+0 + v = 0.3175519492394733E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6493636169568952E+0 + v = 0.3174654952634756E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6677644117704504E+0 + v = 0.3175676415467654E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6829368572115624E+0 + v = 0.3178923417835410E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6946195818184121E+0 + v = 0.3183788287531909E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7025711542057026E+0 + v = 0.3188755151918807E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7066004767140119E+0 + v = 0.3191916889313849E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5132537689946062E-1 + v = 0.1231779611744508E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1297994661331225E+0 + v = 0.1924661373839880E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2188852049401307E+0 + v = 0.2380881867403424E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3123174824903457E+0 + v = 0.2693100663037885E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4064037620738195E+0 + v = 0.2908673382834366E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4984958396944782E+0 + v = 0.3053914619381535E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5864975046021365E+0 + v = 0.3143916684147777E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6686711634580175E+0 + v = 0.3187042244055363E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.8715738780835950E-1 + b = 0.2557175233367578E-1 + v = 0.1635219535869790E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1248383123134007E+0 + b = 0.5604823383376681E-1 + v = 0.1968109917696070E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1638062693383378E+0 + b = 0.8968568601900765E-1 + v = 0.2236754342249974E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2035586203373176E+0 + b = 0.1254086651976279E+0 + v = 0.2453186687017181E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2436798975293774E+0 + b = 0.1624780150162012E+0 + v = 0.2627551791580541E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2838207507773806E+0 + b = 0.2003422342683208E+0 + v = 0.2767654860152220E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3236787502217692E+0 + b = 0.2385628026255263E+0 + v = 0.2879467027765895E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3629849554840691E+0 + b = 0.2767731148783578E+0 + v = 0.2967639918918702E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4014948081992087E+0 + b = 0.3146542308245309E+0 + v = 0.3035900684660351E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4389818379260225E+0 + b = 0.3519196415895088E+0 + v = 0.3087338237298308E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4752331143674377E+0 + b = 0.3883050984023654E+0 + v = 0.3124608838860167E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5100457318374018E+0 + b = 0.4235613423908649E+0 + v = 0.3150084294226743E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5432238388954868E+0 + b = 0.4574484717196220E+0 + v = 0.3165958398598402E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5745758685072442E+0 + b = 0.4897311639255524E+0 + v = 0.3174320440957372E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1723981437592809E+0 + b = 0.3010630597881105E-1 + v = 0.2182188909812599E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2149553257844597E+0 + b = 0.6326031554204694E-1 + v = 0.2399727933921445E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2573256081247422E+0 + b = 0.9848566980258631E-1 + v = 0.2579796133514652E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2993163751238106E+0 + b = 0.1350835952384266E+0 + v = 0.2727114052623535E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3407238005148000E+0 + b = 0.1725184055442181E+0 + v = 0.2846327656281355E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3813454978483264E+0 + b = 0.2103559279730725E+0 + v = 0.2941491102051334E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4209848104423343E+0 + b = 0.2482278774554860E+0 + v = 0.3016049492136107E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4594519699996300E+0 + b = 0.2858099509982883E+0 + v = 0.3072949726175648E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4965640166185930E+0 + b = 0.3228075659915428E+0 + v = 0.3114768142886460E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5321441655571562E+0 + b = 0.3589459907204151E+0 + v = 0.3143823673666223E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5660208438582166E+0 + b = 0.3939630088864310E+0 + v = 0.3162269764661535E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5980264315964364E+0 + b = 0.4276029922949089E+0 + v = 0.3172164663759821E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2644215852350733E+0 + b = 0.3300939429072552E-1 + v = 0.2554575398967435E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3090113743443063E+0 + b = 0.6803887650078501E-1 + v = 0.2701704069135677E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3525871079197808E+0 + b = 0.1044326136206709E+0 + v = 0.2823693413468940E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3950418005354029E+0 + b = 0.1416751597517679E+0 + v = 0.2922898463214289E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4362475663430163E+0 + b = 0.1793408610504821E+0 + v = 0.3001829062162428E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4760661812145854E+0 + b = 0.2170630750175722E+0 + v = 0.3062890864542953E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5143551042512103E+0 + b = 0.2545145157815807E+0 + v = 0.3108328279264746E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5509709026935597E+0 + b = 0.2913940101706601E+0 + v = 0.3140243146201245E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5857711030329428E+0 + b = 0.3274169910910705E+0 + v = 0.3160638030977130E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6186149917404392E+0 + b = 0.3623081329317265E+0 + v = 0.3171462882206275E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3586894569557064E+0 + b = 0.3497354386450040E-1 + v = 0.2812388416031796E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4035266610019441E+0 + b = 0.7129736739757095E-1 + v = 0.2912137500288045E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4467775312332510E+0 + b = 0.1084758620193165E+0 + v = 0.2993241256502206E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4883638346608543E+0 + b = 0.1460915689241772E+0 + v = 0.3057101738983822E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5281908348434601E+0 + b = 0.1837790832369980E+0 + v = 0.3105319326251432E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5661542687149311E+0 + b = 0.2212075390874021E+0 + v = 0.3139565514428167E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6021450102031452E+0 + b = 0.2580682841160985E+0 + v = 0.3161543006806366E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6360520783610050E+0 + b = 0.2940656362094121E+0 + v = 0.3172985960613294E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4521611065087196E+0 + b = 0.3631055365867002E-1 + v = 0.2989400336901431E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4959365651560963E+0 + b = 0.7348318468484350E-1 + v = 0.3054555883947677E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5376815804038283E+0 + b = 0.1111087643812648E+0 + v = 0.3104764960807702E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5773314480243768E+0 + b = 0.1488226085145408E+0 + v = 0.3141015825977616E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6148113245575056E+0 + b = 0.1862892274135151E+0 + v = 0.3164520621159896E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6500407462842380E+0 + b = 0.2231909701714456E+0 + v = 0.3176652305912204E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5425151448707213E+0 + b = 0.3718201306118944E-1 + v = 0.3105097161023939E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5841860556907931E+0 + b = 0.7483616335067346E-1 + v = 0.3143014117890550E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6234632186851500E+0 + b = 0.1125990834266120E+0 + v = 0.3168172866287200E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6602934551848843E+0 + b = 0.1501303813157619E+0 + v = 0.3181401865570968E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6278573968375105E+0 + b = 0.3767559930245720E-1 + v = 0.3170663659156037E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6665611711264577E+0 + b = 0.7548443301360158E-1 + v = 0.3185447944625510E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 3890: + + v = 0.1807395252196920E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.2848008782238827E-3 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.2836065837530581E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.1587876419858352E-1 + v = 0.7013149266673816E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4069193593751206E-1 + v = 0.1162798021956766E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7025888115257997E-1 + v = 0.1518728583972105E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1027495450028704E+0 + v = 0.1798796108216934E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1371457730893426E+0 + v = 0.2022593385972785E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1727758532671953E+0 + v = 0.2203093105575464E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2091492038929037E+0 + v = 0.2349294234299855E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2458813281751915E+0 + v = 0.2467682058747003E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2826545859450066E+0 + v = 0.2563092683572224E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3191957291799622E+0 + v = 0.2639253896763318E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3552621469299578E+0 + v = 0.2699137479265108E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3906329503406230E+0 + v = 0.2745196420166739E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4251028614093031E+0 + v = 0.2779529197397593E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4584777520111870E+0 + v = 0.2803996086684265E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4905711358710193E+0 + v = 0.2820302356715842E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5212011669847385E+0 + v = 0.2830056747491068E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5501878488737995E+0 + v = 0.2834808950776839E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6025037877479342E+0 + v = 0.2835282339078929E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6254572689549016E+0 + v = 0.2833819267065800E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6460107179528248E+0 + v = 0.2832858336906784E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6639541138154251E+0 + v = 0.2833268235451244E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6790688515667495E+0 + v = 0.2835432677029253E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6911338580371512E+0 + v = 0.2839091722743049E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6999385956126490E+0 + v = 0.2843308178875841E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7053037748656896E+0 + v = 0.2846703550533846E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4732224387180115E-1 + v = 0.1051193406971900E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1202100529326803E+0 + v = 0.1657871838796974E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2034304820664855E+0 + v = 0.2064648113714232E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2912285643573002E+0 + v = 0.2347942745819741E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3802361792726768E+0 + v = 0.2547775326597726E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4680598511056146E+0 + v = 0.2686876684847025E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5528151052155599E+0 + v = 0.2778665755515867E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6329386307803041E+0 + v = 0.2830996616782929E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.8056516651369069E-1 + b = 0.2363454684003124E-1 + v = 0.1403063340168372E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1156476077139389E+0 + b = 0.5191291632545936E-1 + v = 0.1696504125939477E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1520473382760421E+0 + b = 0.8322715736994519E-1 + v = 0.1935787242745390E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1892986699745931E+0 + b = 0.1165855667993712E+0 + v = 0.2130614510521968E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2270194446777792E+0 + b = 0.1513077167409504E+0 + v = 0.2289381265931048E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2648908185093273E+0 + b = 0.1868882025807859E+0 + v = 0.2418630292816186E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3026389259574136E+0 + b = 0.2229277629776224E+0 + v = 0.2523400495631193E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3400220296151384E+0 + b = 0.2590951840746235E+0 + v = 0.2607623973449605E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3768217953335510E+0 + b = 0.2951047291750847E+0 + v = 0.2674441032689209E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4128372900921884E+0 + b = 0.3307019714169930E+0 + v = 0.2726432360343356E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4478807131815630E+0 + b = 0.3656544101087634E+0 + v = 0.2765787685924545E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4817742034089257E+0 + b = 0.3997448951939695E+0 + v = 0.2794428690642224E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5143472814653344E+0 + b = 0.4327667110812024E+0 + v = 0.2814099002062895E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5454346213905650E+0 + b = 0.4645196123532293E+0 + v = 0.2826429531578994E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5748739313170252E+0 + b = 0.4948063555703345E+0 + v = 0.2832983542550884E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1599598738286342E+0 + b = 0.2792357590048985E-1 + v = 0.1886695565284976E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1998097412500951E+0 + b = 0.5877141038139065E-1 + v = 0.2081867882748234E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2396228952566202E+0 + b = 0.9164573914691377E-1 + v = 0.2245148680600796E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2792228341097746E+0 + b = 0.1259049641962687E+0 + v = 0.2380370491511872E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3184251107546741E+0 + b = 0.1610594823400863E+0 + v = 0.2491398041852455E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3570481164426244E+0 + b = 0.1967151653460898E+0 + v = 0.2581632405881230E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3949164710492144E+0 + b = 0.2325404606175168E+0 + v = 0.2653965506227417E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4318617293970503E+0 + b = 0.2682461141151439E+0 + v = 0.2710857216747087E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4677221009931678E+0 + b = 0.3035720116011973E+0 + v = 0.2754434093903659E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5023417939270955E+0 + b = 0.3382781859197439E+0 + v = 0.2786579932519380E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5355701836636128E+0 + b = 0.3721383065625942E+0 + v = 0.2809011080679474E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5672608451328771E+0 + b = 0.4049346360466055E+0 + v = 0.2823336184560987E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5972704202540162E+0 + b = 0.4364538098633802E+0 + v = 0.2831101175806309E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2461687022333596E+0 + b = 0.3070423166833368E-1 + v = 0.2221679970354546E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2881774566286831E+0 + b = 0.6338034669281885E-1 + v = 0.2356185734270703E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3293963604116978E+0 + b = 0.9742862487067941E-1 + v = 0.2469228344805590E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3697303822241377E+0 + b = 0.1323799532282290E+0 + v = 0.2562726348642046E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4090663023135127E+0 + b = 0.1678497018129336E+0 + v = 0.2638756726753028E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4472819355411712E+0 + b = 0.2035095105326114E+0 + v = 0.2699311157390862E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4842513377231437E+0 + b = 0.2390692566672091E+0 + v = 0.2746233268403837E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5198477629962928E+0 + b = 0.2742649818076149E+0 + v = 0.2781225674454771E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5539453011883145E+0 + b = 0.3088503806580094E+0 + v = 0.2805881254045684E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5864196762401251E+0 + b = 0.3425904245906614E+0 + v = 0.2821719877004913E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6171484466668390E+0 + b = 0.3752562294789468E+0 + v = 0.2830222502333124E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3350337830565727E+0 + b = 0.3261589934634747E-1 + v = 0.2457995956744870E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3775773224758284E+0 + b = 0.6658438928081572E-1 + v = 0.2551474407503706E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4188155229848973E+0 + b = 0.1014565797157954E+0 + v = 0.2629065335195311E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4586805892009344E+0 + b = 0.1368573320843822E+0 + v = 0.2691900449925075E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4970895714224235E+0 + b = 0.1724614851951608E+0 + v = 0.2741275485754276E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5339505133960747E+0 + b = 0.2079779381416412E+0 + v = 0.2778530970122595E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5691665792531440E+0 + b = 0.2431385788322288E+0 + v = 0.2805010567646741E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6026387682680377E+0 + b = 0.2776901883049853E+0 + v = 0.2822055834031040E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6342676150163307E+0 + b = 0.3113881356386632E+0 + v = 0.2831016901243473E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4237951119537067E+0 + b = 0.3394877848664351E-1 + v = 0.2624474901131803E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4656918683234929E+0 + b = 0.6880219556291447E-1 + v = 0.2688034163039377E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5058857069185980E+0 + b = 0.1041946859721635E+0 + v = 0.2738932751287636E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5443204666713996E+0 + b = 0.1398039738736393E+0 + v = 0.2777944791242523E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5809298813759742E+0 + b = 0.1753373381196155E+0 + v = 0.2806011661660987E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6156416039447128E+0 + b = 0.2105215793514010E+0 + v = 0.2824181456597460E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6483801351066604E+0 + b = 0.2450953312157051E+0 + v = 0.2833585216577828E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5103616577251688E+0 + b = 0.3485560643800719E-1 + v = 0.2738165236962878E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5506738792580681E+0 + b = 0.7026308631512033E-1 + v = 0.2778365208203180E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5889573040995292E+0 + b = 0.1059035061296403E+0 + v = 0.2807852940418966E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6251641589516930E+0 + b = 0.1414823925236026E+0 + v = 0.2827245949674705E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6592414921570178E+0 + b = 0.1767207908214530E+0 + v = 0.2837342344829828E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5930314017533384E+0 + b = 0.3542189339561672E-1 + v = 0.2809233907610981E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6309812253390175E+0 + b = 0.7109574040369549E-1 + v = 0.2829930809742694E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6666296011353230E+0 + b = 0.1067259792282730E+0 + v = 0.2841097874111479E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6703715271049922E+0 + b = 0.3569455268820809E-1 + v = 0.2843455206008783E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 4334: + + v = 0.1449063022537883E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.2546377329828424E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.1462896151831013E-1 + v = 0.6018432961087496E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3769840812493139E-1 + v = 0.1002286583263673E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6524701904096891E-1 + v = 0.1315222931028093E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.9560543416134648E-1 + v = 0.1564213746876724E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1278335898929198E+0 + v = 0.1765118841507736E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1613096104466031E+0 + v = 0.1928737099311080E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1955806225745371E+0 + v = 0.2062658534263270E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2302935218498028E+0 + v = 0.2172395445953787E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2651584344113027E+0 + v = 0.2262076188876047E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2999276825183209E+0 + v = 0.2334885699462397E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3343828669718798E+0 + v = 0.2393355273179203E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3683265013750518E+0 + v = 0.2439559200468863E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4015763206518108E+0 + v = 0.2475251866060002E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4339612026399770E+0 + v = 0.2501965558158773E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4653180651114582E+0 + v = 0.2521081407925925E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4954893331080803E+0 + v = 0.2533881002388081E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5243207068924930E+0 + v = 0.2541582900848261E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5516590479041704E+0 + v = 0.2545365737525860E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6012371927804176E+0 + v = 0.2545726993066799E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6231574466449819E+0 + v = 0.2544456197465555E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6429416514181271E+0 + v = 0.2543481596881064E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6604124272943595E+0 + v = 0.2543506451429194E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6753851470408250E+0 + v = 0.2544905675493763E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6876717970626160E+0 + v = 0.2547611407344429E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6970895061319234E+0 + v = 0.2551060375448869E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7034746912553310E+0 + v = 0.2554291933816039E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7067017217542295E+0 + v = 0.2556255710686343E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4382223501131123E-1 + v = 0.9041339695118195E-4 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1117474077400006E+0 + v = 0.1438426330079022E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1897153252911440E+0 + v = 0.1802523089820518E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2724023009910331E+0 + v = 0.2060052290565496E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3567163308709902E+0 + v = 0.2245002248967466E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4404784483028087E+0 + v = 0.2377059847731150E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5219833154161411E+0 + v = 0.2468118955882525E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5998179868977553E+0 + v = 0.2525410872966528E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6727803154548222E+0 + v = 0.2553101409933397E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.7476563943166086E-1 + b = 0.2193168509461185E-1 + v = 0.1212879733668632E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1075341482001416E+0 + b = 0.4826419281533887E-1 + v = 0.1472872881270931E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1416344885203259E+0 + b = 0.7751191883575742E-1 + v = 0.1686846601010828E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1766325315388586E+0 + b = 0.1087558139247680E+0 + v = 0.1862698414660208E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2121744174481514E+0 + b = 0.1413661374253096E+0 + v = 0.2007430956991861E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2479669443408145E+0 + b = 0.1748768214258880E+0 + v = 0.2126568125394796E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2837600452294113E+0 + b = 0.2089216406612073E+0 + v = 0.2224394603372113E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3193344933193984E+0 + b = 0.2431987685545972E+0 + v = 0.2304264522673135E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3544935442438745E+0 + b = 0.2774497054377770E+0 + v = 0.2368854288424087E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3890571932288154E+0 + b = 0.3114460356156915E+0 + v = 0.2420352089461772E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4228581214259090E+0 + b = 0.3449806851913012E+0 + v = 0.2460597113081295E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4557387211304052E+0 + b = 0.3778618641248256E+0 + v = 0.2491181912257687E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4875487950541643E+0 + b = 0.4099086391698978E+0 + v = 0.2513528194205857E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5181436529962997E+0 + b = 0.4409474925853973E+0 + v = 0.2528943096693220E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5473824095600661E+0 + b = 0.4708094517711291E+0 + v = 0.2538660368488136E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5751263398976174E+0 + b = 0.4993275140354637E+0 + v = 0.2543868648299022E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1489515746840028E+0 + b = 0.2599381993267017E-1 + v = 0.1642595537825183E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1863656444351767E+0 + b = 0.5479286532462190E-1 + v = 0.1818246659849308E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2238602880356348E+0 + b = 0.8556763251425254E-1 + v = 0.1966565649492420E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2612723375728160E+0 + b = 0.1177257802267011E+0 + v = 0.2090677905657991E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2984332990206190E+0 + b = 0.1508168456192700E+0 + v = 0.2193820409510504E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3351786584663333E+0 + b = 0.1844801892177727E+0 + v = 0.2278870827661928E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3713505522209120E+0 + b = 0.2184145236087598E+0 + v = 0.2348283192282090E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4067981098954663E+0 + b = 0.2523590641486229E+0 + v = 0.2404139755581477E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4413769993687534E+0 + b = 0.2860812976901373E+0 + v = 0.2448227407760734E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4749487182516394E+0 + b = 0.3193686757808996E+0 + v = 0.2482110455592573E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5073798105075426E+0 + b = 0.3520226949547602E+0 + v = 0.2507192397774103E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5385410448878654E+0 + b = 0.3838544395667890E+0 + v = 0.2524765968534880E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5683065353670530E+0 + b = 0.4146810037640963E+0 + v = 0.2536052388539425E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5965527620663510E+0 + b = 0.4443224094681121E+0 + v = 0.2542230588033068E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2299227700856157E+0 + b = 0.2865757664057584E-1 + v = 0.1944817013047896E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2695752998553267E+0 + b = 0.5923421684485993E-1 + v = 0.2067862362746635E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3086178716611389E+0 + b = 0.9117817776057715E-1 + v = 0.2172440734649114E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3469649871659077E+0 + b = 0.1240593814082605E+0 + v = 0.2260125991723423E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3845153566319655E+0 + b = 0.1575272058259175E+0 + v = 0.2332655008689523E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4211600033403215E+0 + b = 0.1912845163525413E+0 + v = 0.2391699681532458E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4567867834329882E+0 + b = 0.2250710177858171E+0 + v = 0.2438801528273928E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4912829319232061E+0 + b = 0.2586521303440910E+0 + v = 0.2475370504260665E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5245364793303812E+0 + b = 0.2918112242865407E+0 + v = 0.2502707235640574E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5564369788915756E+0 + b = 0.3243439239067890E+0 + v = 0.2522031701054241E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5868757697775287E+0 + b = 0.3560536787835351E+0 + v = 0.2534511269978784E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6157458853519617E+0 + b = 0.3867480821242581E+0 + v = 0.2541284914955151E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3138461110672113E+0 + b = 0.3051374637507278E-1 + v = 0.2161509250688394E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3542495872050569E+0 + b = 0.6237111233730755E-1 + v = 0.2248778513437852E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3935751553120181E+0 + b = 0.9516223952401907E-1 + v = 0.2322388803404617E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4317634668111147E+0 + b = 0.1285467341508517E+0 + v = 0.2383265471001355E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4687413842250821E+0 + b = 0.1622318931656033E+0 + v = 0.2432476675019525E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5044274237060283E+0 + b = 0.1959581153836453E+0 + v = 0.2471122223750674E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5387354077925727E+0 + b = 0.2294888081183837E+0 + v = 0.2500291752486870E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5715768898356105E+0 + b = 0.2626031152713945E+0 + v = 0.2521055942764682E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6028627200136111E+0 + b = 0.2950904075286713E+0 + v = 0.2534472785575503E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6325039812653463E+0 + b = 0.3267458451113286E+0 + v = 0.2541599713080121E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3981986708423407E+0 + b = 0.3183291458749821E-1 + v = 0.2317380975862936E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4382791182133300E+0 + b = 0.6459548193880908E-1 + v = 0.2378550733719775E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4769233057218166E+0 + b = 0.9795757037087952E-1 + v = 0.2428884456739118E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5140823911194238E+0 + b = 0.1316307235126655E+0 + v = 0.2469002655757292E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5496977833862983E+0 + b = 0.1653556486358704E+0 + v = 0.2499657574265851E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5837047306512727E+0 + b = 0.1988931724126510E+0 + v = 0.2521676168486082E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6160349566926879E+0 + b = 0.2320174581438950E+0 + v = 0.2535935662645334E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6466185353209440E+0 + b = 0.2645106562168662E+0 + v = 0.2543356743363214E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4810835158795404E+0 + b = 0.3275917807743992E-1 + v = 0.2427353285201535E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5199925041324341E+0 + b = 0.6612546183967181E-1 + v = 0.2468258039744386E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5571717692207494E+0 + b = 0.9981498331474143E-1 + v = 0.2500060956440310E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5925789250836378E+0 + b = 0.1335687001410374E+0 + v = 0.2523238365420979E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6261658523859670E+0 + b = 0.1671444402896463E+0 + v = 0.2538399260252846E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6578811126669331E+0 + b = 0.2003106382156076E+0 + v = 0.2546255927268069E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5609624612998100E+0 + b = 0.3337500940231335E-1 + v = 0.2500583360048449E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5979959659984670E+0 + b = 0.6708750335901803E-1 + v = 0.2524777638260203E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6330523711054002E+0 + b = 0.1008792126424850E+0 + v = 0.2540951193860656E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6660960998103972E+0 + b = 0.1345050343171794E+0 + v = 0.2549524085027472E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6365384364585819E+0 + b = 0.3372799460737052E-1 + v = 0.2542569507009158E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6710994302899275E+0 + b = 0.6755249309678028E-1 + v = 0.2552114127580376E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 4802: + + v = 0.9687521879420705E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.2307897895367918E-3 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.2297310852498558E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.2335728608887064E-1 + v = 0.7386265944001919E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4352987836550653E-1 + v = 0.8257977698542210E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6439200521088801E-1 + v = 0.9706044762057630E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.9003943631993181E-1 + v = 0.1302393847117003E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1196706615548473E+0 + v = 0.1541957004600968E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1511715412838134E+0 + v = 0.1704459770092199E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1835982828503801E+0 + v = 0.1827374890942906E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2165081259155405E+0 + v = 0.1926360817436107E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2496208720417563E+0 + v = 0.2008010239494833E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2827200673567900E+0 + v = 0.2075635983209175E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3156190823994346E+0 + v = 0.2131306638690909E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3481476793749115E+0 + v = 0.2176562329937335E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3801466086947226E+0 + v = 0.2212682262991018E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4114652119634011E+0 + v = 0.2240799515668565E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4419598786519751E+0 + v = 0.2261959816187525E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4714925949329543E+0 + v = 0.2277156368808855E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4999293972879466E+0 + v = 0.2287351772128336E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5271387221431248E+0 + v = 0.2293490814084085E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5529896780837761E+0 + v = 0.2296505312376273E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6000856099481712E+0 + v = 0.2296793832318756E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6210562192785175E+0 + v = 0.2295785443842974E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6401165879934240E+0 + v = 0.2295017931529102E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6571144029244334E+0 + v = 0.2295059638184868E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6718910821718863E+0 + v = 0.2296232343237362E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6842845591099010E+0 + v = 0.2298530178740771E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6941353476269816E+0 + v = 0.2301579790280501E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7012965242212991E+0 + v = 0.2304690404996513E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7056471428242644E+0 + v = 0.2307027995907102E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4595557643585895E-1 + v = 0.9312274696671092E-4 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1049316742435023E+0 + v = 0.1199919385876926E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1773548879549274E+0 + v = 0.1598039138877690E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2559071411236127E+0 + v = 0.1822253763574900E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3358156837985898E+0 + v = 0.1988579593655040E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4155835743763893E+0 + v = 0.2112620102533307E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4937894296167472E+0 + v = 0.2201594887699007E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5691569694793316E+0 + v = 0.2261622590895036E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6405840854894251E+0 + v = 0.2296458453435705E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.7345133894143348E-1 + b = 0.2177844081486067E-1 + v = 0.1006006990267000E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1009859834044931E+0 + b = 0.4590362185775188E-1 + v = 0.1227676689635876E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1324289619748758E+0 + b = 0.7255063095690877E-1 + v = 0.1467864280270117E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1654272109607127E+0 + b = 0.1017825451960684E+0 + v = 0.1644178912101232E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1990767186776461E+0 + b = 0.1325652320980364E+0 + v = 0.1777664890718961E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2330125945523278E+0 + b = 0.1642765374496765E+0 + v = 0.1884825664516690E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2670080611108287E+0 + b = 0.1965360374337889E+0 + v = 0.1973269246453848E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3008753376294316E+0 + b = 0.2290726770542238E+0 + v = 0.2046767775855328E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3344475596167860E+0 + b = 0.2616645495370823E+0 + v = 0.2107600125918040E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3675709724070786E+0 + b = 0.2941150728843141E+0 + v = 0.2157416362266829E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4001000887587812E+0 + b = 0.3262440400919066E+0 + v = 0.2197557816920721E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4318956350436028E+0 + b = 0.3578835350611916E+0 + v = 0.2229192611835437E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4628239056795531E+0 + b = 0.3888751854043678E+0 + v = 0.2253385110212775E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4927563229773636E+0 + b = 0.4190678003222840E+0 + v = 0.2271137107548774E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5215687136707969E+0 + b = 0.4483151836883852E+0 + v = 0.2283414092917525E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5491402346984905E+0 + b = 0.4764740676087880E+0 + v = 0.2291161673130077E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5753520160126075E+0 + b = 0.5034021310998277E+0 + v = 0.2295313908576598E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1388326356417754E+0 + b = 0.2435436510372806E-1 + v = 0.1438204721359031E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1743686900537244E+0 + b = 0.5118897057342652E-1 + v = 0.1607738025495257E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2099737037950268E+0 + b = 0.8014695048539634E-1 + v = 0.1741483853528379E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2454492590908548E+0 + b = 0.1105117874155699E+0 + v = 0.1851918467519151E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2807219257864278E+0 + b = 0.1417950531570966E+0 + v = 0.1944628638070613E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3156842271975842E+0 + b = 0.1736604945719597E+0 + v = 0.2022495446275152E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3502090945177752E+0 + b = 0.2058466324693981E+0 + v = 0.2087462382438514E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3841684849519686E+0 + b = 0.2381284261195919E+0 + v = 0.2141074754818308E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4174372367906016E+0 + b = 0.2703031270422569E+0 + v = 0.2184640913748162E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4498926465011892E+0 + b = 0.3021845683091309E+0 + v = 0.2219309165220329E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4814146229807701E+0 + b = 0.3335993355165720E+0 + v = 0.2246123118340624E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5118863625734701E+0 + b = 0.3643833735518232E+0 + v = 0.2266062766915125E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5411947455119144E+0 + b = 0.3943789541958179E+0 + v = 0.2280072952230796E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5692301500357246E+0 + b = 0.4234320144403542E+0 + v = 0.2289082025202583E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5958857204139576E+0 + b = 0.4513897947419260E+0 + v = 0.2294012695120025E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2156270284785766E+0 + b = 0.2681225755444491E-1 + v = 0.1722434488736947E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2532385054909710E+0 + b = 0.5557495747805614E-1 + v = 0.1830237421455091E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2902564617771537E+0 + b = 0.8569368062950249E-1 + v = 0.1923855349997633E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3266979823143256E+0 + b = 0.1167367450324135E+0 + v = 0.2004067861936271E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3625039627493614E+0 + b = 0.1483861994003304E+0 + v = 0.2071817297354263E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3975838937548699E+0 + b = 0.1803821503011405E+0 + v = 0.2128250834102103E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4318396099009774E+0 + b = 0.2124962965666424E+0 + v = 0.2174513719440102E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4651706555732742E+0 + b = 0.2445221837805913E+0 + v = 0.2211661839150214E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4974752649620969E+0 + b = 0.2762701224322987E+0 + v = 0.2240665257813102E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5286517579627517E+0 + b = 0.3075627775211328E+0 + v = 0.2262439516632620E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5586001195731895E+0 + b = 0.3382311089826877E+0 + v = 0.2277874557231869E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5872229902021319E+0 + b = 0.3681108834741399E+0 + v = 0.2287854314454994E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6144258616235123E+0 + b = 0.3970397446872839E+0 + v = 0.2293268499615575E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2951676508064861E+0 + b = 0.2867499538750441E-1 + v = 0.1912628201529828E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3335085485472725E+0 + b = 0.5867879341903510E-1 + v = 0.1992499672238701E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3709561760636381E+0 + b = 0.8961099205022284E-1 + v = 0.2061275533454027E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4074722861667498E+0 + b = 0.1211627927626297E+0 + v = 0.2119318215968572E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4429923648839117E+0 + b = 0.1530748903554898E+0 + v = 0.2167416581882652E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4774428052721736E+0 + b = 0.1851176436721877E+0 + v = 0.2206430730516600E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5107446539535904E+0 + b = 0.2170829107658179E+0 + v = 0.2237186938699523E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5428151370542935E+0 + b = 0.2487786689026271E+0 + v = 0.2260480075032884E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5735699292556964E+0 + b = 0.2800239952795016E+0 + v = 0.2277098884558542E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6029253794562866E+0 + b = 0.3106445702878119E+0 + v = 0.2287845715109671E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6307998987073145E+0 + b = 0.3404689500841194E+0 + v = 0.2293547268236294E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3752652273692719E+0 + b = 0.2997145098184479E-1 + v = 0.2056073839852528E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4135383879344028E+0 + b = 0.6086725898678011E-1 + v = 0.2114235865831876E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4506113885153907E+0 + b = 0.9238849548435643E-1 + v = 0.2163175629770551E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4864401554606072E+0 + b = 0.1242786603851851E+0 + v = 0.2203392158111650E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5209708076611709E+0 + b = 0.1563086731483386E+0 + v = 0.2235473176847839E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5541422135830122E+0 + b = 0.1882696509388506E+0 + v = 0.2260024141501235E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5858880915113817E+0 + b = 0.2199672979126059E+0 + v = 0.2277675929329182E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6161399390603444E+0 + b = 0.2512165482924867E+0 + v = 0.2289102112284834E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6448296482255090E+0 + b = 0.2818368701871888E+0 + v = 0.2295027954625118E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4544796274917948E+0 + b = 0.3088970405060312E-1 + v = 0.2161281589879992E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4919389072146628E+0 + b = 0.6240947677636835E-1 + v = 0.2201980477395102E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5279313026985183E+0 + b = 0.9430706144280313E-1 + v = 0.2234952066593166E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5624169925571135E+0 + b = 0.1263547818770374E+0 + v = 0.2260540098520838E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5953484627093287E+0 + b = 0.1583430788822594E+0 + v = 0.2279157981899988E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6266730715339185E+0 + b = 0.1900748462555988E+0 + v = 0.2291296918565571E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6563363204278871E+0 + b = 0.2213599519592567E+0 + v = 0.2297533752536649E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5314574716585696E+0 + b = 0.3152508811515374E-1 + v = 0.2234927356465995E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5674614932298185E+0 + b = 0.6343865291465561E-1 + v = 0.2261288012985219E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6017706004970264E+0 + b = 0.9551503504223951E-1 + v = 0.2280818160923688E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6343471270264178E+0 + b = 0.1275440099801196E+0 + v = 0.2293773295180159E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6651494599127802E+0 + b = 0.1593252037671960E+0 + v = 0.2300528767338634E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6050184986005704E+0 + b = 0.3192538338496105E-1 + v = 0.2281893855065666E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6390163550880400E+0 + b = 0.6402824353962306E-1 + v = 0.2295720444840727E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6711199107088448E+0 + b = 0.9609805077002909E-1 + v = 0.2303227649026753E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6741354429572275E+0 + b = 0.3211853196273233E-1 + v = 0.2304831913227114E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 5294: + + v = 0.9080510764308163E-4 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.2084824361987793E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.2303261686261450E-1 + v = 0.5011105657239616E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3757208620162394E-1 + v = 0.5942520409683854E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5821912033821852E-1 + v = 0.9564394826109721E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.8403127529194872E-1 + v = 0.1185530657126338E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1122927798060578E+0 + v = 0.1364510114230331E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1420125319192987E+0 + v = 0.1505828825605415E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1726396437341978E+0 + v = 0.1619298749867023E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2038170058115696E+0 + v = 0.1712450504267789E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2352849892876508E+0 + v = 0.1789891098164999E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2668363354312461E+0 + v = 0.1854474955629795E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2982941279900452E+0 + v = 0.1908148636673661E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3295002922087076E+0 + v = 0.1952377405281833E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3603094918363593E+0 + v = 0.1988349254282232E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3905857895173920E+0 + v = 0.2017079807160050E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4202005758160837E+0 + v = 0.2039473082709094E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4490310061597227E+0 + v = 0.2056360279288953E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4769586160311491E+0 + v = 0.2068525823066865E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5038679887049750E+0 + v = 0.2076724877534488E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5296454286519961E+0 + v = 0.2081694278237885E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5541776207164850E+0 + v = 0.2084157631219326E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5990467321921213E+0 + v = 0.2084381531128593E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6191467096294587E+0 + v = 0.2083476277129307E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6375251212901849E+0 + v = 0.2082686194459732E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6540514381131168E+0 + v = 0.2082475686112415E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6685899064391510E+0 + v = 0.2083139860289915E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6810013009681648E+0 + v = 0.2084745561831237E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6911469578730340E+0 + v = 0.2087091313375890E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6988956915141736E+0 + v = 0.2089718413297697E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7041335794868720E+0 + v = 0.2092003303479793E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7067754398018567E+0 + v = 0.2093336148263241E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3840368707853623E-1 + v = 0.7591708117365267E-4 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.9835485954117399E-1 + v = 0.1083383968169186E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1665774947612998E+0 + v = 0.1403019395292510E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2405702335362910E+0 + v = 0.1615970179286436E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3165270770189046E+0 + v = 0.1771144187504911E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3927386145645443E+0 + v = 0.1887760022988168E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4678825918374656E+0 + v = 0.1973474670768214E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5408022024266935E+0 + v = 0.2033787661234659E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6104967445752438E+0 + v = 0.2072343626517331E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6760910702685738E+0 + v = 0.2091177834226918E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6655644120217392E-1 + b = 0.1936508874588424E-1 + v = 0.9316684484675566E-4 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.9446246161270182E-1 + b = 0.4252442002115869E-1 + v = 0.1116193688682976E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1242651925452509E+0 + b = 0.6806529315354374E-1 + v = 0.1298623551559414E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1553438064846751E+0 + b = 0.9560957491205369E-1 + v = 0.1450236832456426E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1871137110542670E+0 + b = 0.1245931657452888E+0 + v = 0.1572719958149914E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2192612628836257E+0 + b = 0.1545385828778978E+0 + v = 0.1673234785867195E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2515682807206955E+0 + b = 0.1851004249723368E+0 + v = 0.1756860118725188E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2838535866287290E+0 + b = 0.2160182608272384E+0 + v = 0.1826776290439367E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3159578817528521E+0 + b = 0.2470799012277111E+0 + v = 0.1885116347992865E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3477370882791392E+0 + b = 0.2781014208986402E+0 + v = 0.1933457860170574E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3790576960890540E+0 + b = 0.3089172523515731E+0 + v = 0.1973060671902064E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4097938317810200E+0 + b = 0.3393750055472244E+0 + v = 0.2004987099616311E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4398256572859637E+0 + b = 0.3693322470987730E+0 + v = 0.2030170909281499E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4690384114718480E+0 + b = 0.3986541005609877E+0 + v = 0.2049461460119080E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4973216048301053E+0 + b = 0.4272112491408562E+0 + v = 0.2063653565200186E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5245681526132446E+0 + b = 0.4548781735309936E+0 + v = 0.2073507927381027E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5506733911803888E+0 + b = 0.4815315355023251E+0 + v = 0.2079764593256122E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5755339829522475E+0 + b = 0.5070486445801855E+0 + v = 0.2083150534968778E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1305472386056362E+0 + b = 0.2284970375722366E-1 + v = 0.1262715121590664E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1637327908216477E+0 + b = 0.4812254338288384E-1 + v = 0.1414386128545972E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1972734634149637E+0 + b = 0.7531734457511935E-1 + v = 0.1538740401313898E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2308694653110130E+0 + b = 0.1039043639882017E+0 + v = 0.1642434942331432E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2643899218338160E+0 + b = 0.1334526587117626E+0 + v = 0.1729790609237496E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2977171599622171E+0 + b = 0.1636414868936382E+0 + v = 0.1803505190260828E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3307293903032310E+0 + b = 0.1942195406166568E+0 + v = 0.1865475350079657E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3633069198219073E+0 + b = 0.2249752879943753E+0 + v = 0.1917182669679069E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3953346955922727E+0 + b = 0.2557218821820032E+0 + v = 0.1959851709034382E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4267018394184914E+0 + b = 0.2862897925213193E+0 + v = 0.1994529548117882E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4573009622571704E+0 + b = 0.3165224536636518E+0 + v = 0.2022138911146548E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4870279559856109E+0 + b = 0.3462730221636496E+0 + v = 0.2043518024208592E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5157819581450322E+0 + b = 0.3754016870282835E+0 + v = 0.2059450313018110E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5434651666465393E+0 + b = 0.4037733784993613E+0 + v = 0.2070685715318472E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5699823887764627E+0 + b = 0.4312557784139123E+0 + v = 0.2077955310694373E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5952403350947741E+0 + b = 0.4577175367122110E+0 + v = 0.2081980387824712E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2025152599210369E+0 + b = 0.2520253617719557E-1 + v = 0.1521318610377956E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2381066653274425E+0 + b = 0.5223254506119000E-1 + v = 0.1622772720185755E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2732823383651612E+0 + b = 0.8060669688588620E-1 + v = 0.1710498139420709E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3080137692611118E+0 + b = 0.1099335754081255E+0 + v = 0.1785911149448736E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3422405614587601E+0 + b = 0.1399120955959857E+0 + v = 0.1850125313687736E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3758808773890420E+0 + b = 0.1702977801651705E+0 + v = 0.1904229703933298E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4088458383438932E+0 + b = 0.2008799256601680E+0 + v = 0.1949259956121987E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4410450550841152E+0 + b = 0.2314703052180836E+0 + v = 0.1986161545363960E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4723879420561312E+0 + b = 0.2618972111375892E+0 + v = 0.2015790585641370E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5027843561874343E+0 + b = 0.2920013195600270E+0 + v = 0.2038934198707418E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5321453674452458E+0 + b = 0.3216322555190551E+0 + v = 0.2056334060538251E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5603839113834030E+0 + b = 0.3506456615934198E+0 + v = 0.2068705959462289E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5874150706875146E+0 + b = 0.3789007181306267E+0 + v = 0.2076753906106002E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6131559381660038E+0 + b = 0.4062580170572782E+0 + v = 0.2081179391734803E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2778497016394506E+0 + b = 0.2696271276876226E-1 + v = 0.1700345216228943E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3143733562261912E+0 + b = 0.5523469316960465E-1 + v = 0.1774906779990410E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3501485810261827E+0 + b = 0.8445193201626464E-1 + v = 0.1839659377002642E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3851430322303653E+0 + b = 0.1143263119336083E+0 + v = 0.1894987462975169E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4193013979470415E+0 + b = 0.1446177898344475E+0 + v = 0.1941548809452595E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4525585960458567E+0 + b = 0.1751165438438091E+0 + v = 0.1980078427252384E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4848447779622947E+0 + b = 0.2056338306745660E+0 + v = 0.2011296284744488E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5160871208276894E+0 + b = 0.2359965487229226E+0 + v = 0.2035888456966776E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5462112185696926E+0 + b = 0.2660430223139146E+0 + v = 0.2054516325352142E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5751425068101757E+0 + b = 0.2956193664498032E+0 + v = 0.2067831033092635E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6028073872853596E+0 + b = 0.3245763905312779E+0 + v = 0.2076485320284876E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6291338275278409E+0 + b = 0.3527670026206972E+0 + v = 0.2081141439525255E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3541797528439391E+0 + b = 0.2823853479435550E-1 + v = 0.1834383015469222E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3908234972074657E+0 + b = 0.5741296374713106E-1 + v = 0.1889540591777677E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4264408450107590E+0 + b = 0.8724646633650199E-1 + v = 0.1936677023597375E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4609949666553286E+0 + b = 0.1175034422915616E+0 + v = 0.1976176495066504E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4944389496536006E+0 + b = 0.1479755652628428E+0 + v = 0.2008536004560983E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5267194884346086E+0 + b = 0.1784740659484352E+0 + v = 0.2034280351712291E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5577787810220990E+0 + b = 0.2088245700431244E+0 + v = 0.2053944466027758E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5875563763536670E+0 + b = 0.2388628136570763E+0 + v = 0.2068077642882360E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6159910016391269E+0 + b = 0.2684308928769185E+0 + v = 0.2077250949661599E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6430219602956268E+0 + b = 0.2973740761960252E+0 + v = 0.2082062440705320E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4300647036213646E+0 + b = 0.2916399920493977E-1 + v = 0.1934374486546626E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4661486308935531E+0 + b = 0.5898803024755659E-1 + v = 0.1974107010484300E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5009658555287261E+0 + b = 0.8924162698525409E-1 + v = 0.2007129290388658E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5344824270447704E+0 + b = 0.1197185199637321E+0 + v = 0.2033736947471293E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5666575997416371E+0 + b = 0.1502300756161382E+0 + v = 0.2054287125902493E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5974457471404752E+0 + b = 0.1806004191913564E+0 + v = 0.2069184936818894E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6267984444116886E+0 + b = 0.2106621764786252E+0 + v = 0.2078883689808782E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6546664713575417E+0 + b = 0.2402526932671914E+0 + v = 0.2083886366116359E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5042711004437253E+0 + b = 0.2982529203607657E-1 + v = 0.2006593275470817E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5392127456774380E+0 + b = 0.6008728062339922E-1 + v = 0.2033728426135397E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5726819437668618E+0 + b = 0.9058227674571398E-1 + v = 0.2055008781377608E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6046469254207278E+0 + b = 0.1211219235803400E+0 + v = 0.2070651783518502E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6350716157434952E+0 + b = 0.1515286404791580E+0 + v = 0.2080953335094320E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6639177679185454E+0 + b = 0.1816314681255552E+0 + v = 0.2086284998988521E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5757276040972253E+0 + b = 0.3026991752575440E-1 + v = 0.2055549387644668E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6090265823139755E+0 + b = 0.6078402297870770E-1 + v = 0.2071871850267654E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6406735344387661E+0 + b = 0.9135459984176636E-1 + v = 0.2082856600431965E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6706397927793709E+0 + b = 0.1218024155966590E+0 + v = 0.2088705858819358E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6435019674426665E+0 + b = 0.3052608357660639E-1 + v = 0.2083995867536322E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6747218676375681E+0 + b = 0.6112185773983089E-1 + v = 0.2090509712889637E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case 5810: + + v = 0.9735347946175486E-5 + leb_tmp, start = get_lebedev_recurrence_points(1, start, a, b, v, leb_tmp) + v = 0.1907581241803167E-3 + leb_tmp, start = get_lebedev_recurrence_points(2, start, a, b, v, leb_tmp) + v = 0.1901059546737578E-3 + leb_tmp, start = get_lebedev_recurrence_points(3, start, a, b, v, leb_tmp) + a = 0.1182361662400277E-1 + v = 0.3926424538919212E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3062145009138958E-1 + v = 0.6667905467294382E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5329794036834243E-1 + v = 0.8868891315019135E-4 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7848165532862220E-1 + v = 0.1066306000958872E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1054038157636201E+0 + v = 0.1214506743336128E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1335577797766211E+0 + v = 0.1338054681640871E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1625769955502252E+0 + v = 0.1441677023628504E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.1921787193412792E+0 + v = 0.1528880200826557E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2221340534690548E+0 + v = 0.1602330623773609E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2522504912791132E+0 + v = 0.1664102653445244E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.2823610860679697E+0 + v = 0.1715845854011323E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3123173966267560E+0 + v = 0.1758901000133069E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3419847036953789E+0 + v = 0.1794382485256736E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3712386456999758E+0 + v = 0.1823238106757407E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3999627649876828E+0 + v = 0.1846293252959976E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4280466458648093E+0 + v = 0.1864284079323098E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4553844360185711E+0 + v = 0.1877882694626914E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.4818736094437834E+0 + v = 0.1887716321852025E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5074138709260629E+0 + v = 0.1894381638175673E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5319061304570707E+0 + v = 0.1898454899533629E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5552514978677286E+0 + v = 0.1900497929577815E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.5981009025246183E+0 + v = 0.1900671501924092E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6173990192228116E+0 + v = 0.1899837555533510E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6351365239411131E+0 + v = 0.1899014113156229E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6512010228227200E+0 + v = 0.1898581257705106E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6654758363948120E+0 + v = 0.1898804756095753E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6778410414853370E+0 + v = 0.1899793610426402E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6881760887484110E+0 + v = 0.1901464554844117E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.6963645267094598E+0 + v = 0.1903533246259542E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7023010617153579E+0 + v = 0.1905556158463228E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.7059004636628753E+0 + v = 0.1907037155663528E-3 + leb_tmp, start = get_lebedev_recurrence_points(4, start, a, b, v, leb_tmp) + a = 0.3552470312472575E-1 + v = 0.5992997844249967E-4 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.9151176620841283E-1 + v = 0.9749059382456978E-4 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.1566197930068980E+0 + v = 0.1241680804599158E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2265467599271907E+0 + v = 0.1437626154299360E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.2988242318581361E+0 + v = 0.1584200054793902E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.3717482419703886E+0 + v = 0.1694436550982744E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.4440094491758889E+0 + v = 0.1776617014018108E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5145337096756642E+0 + v = 0.1836132434440077E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.5824053672860230E+0 + v = 0.1876494727075983E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6468283961043370E+0 + v = 0.1899906535336482E-3 + leb_tmp, start = get_lebedev_recurrence_points(5, start, a, b, v, leb_tmp) + a = 0.6095964259104373E-1 + b = 0.1787828275342931E-1 + v = 0.8143252820767350E-4 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.8811962270959388E-1 + b = 0.3953888740792096E-1 + v = 0.9998859890887728E-4 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1165936722428831E+0 + b = 0.6378121797722990E-1 + v = 0.1156199403068359E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1460232857031785E+0 + b = 0.8985890813745037E-1 + v = 0.1287632092635513E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1761197110181755E+0 + b = 0.1172606510576162E+0 + v = 0.1398378643365139E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2066471190463718E+0 + b = 0.1456102876970995E+0 + v = 0.1491876468417391E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2374076026328152E+0 + b = 0.1746153823011775E+0 + v = 0.1570855679175456E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2682305474337051E+0 + b = 0.2040383070295584E+0 + v = 0.1637483948103775E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2989653312142369E+0 + b = 0.2336788634003698E+0 + v = 0.1693500566632843E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3294762752772209E+0 + b = 0.2633632752654219E+0 + v = 0.1740322769393633E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3596390887276086E+0 + b = 0.2929369098051601E+0 + v = 0.1779126637278296E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3893383046398812E+0 + b = 0.3222592785275512E+0 + v = 0.1810908108835412E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4184653789358347E+0 + b = 0.3512004791195743E+0 + v = 0.1836529132600190E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4469172319076166E+0 + b = 0.3796385677684537E+0 + v = 0.1856752841777379E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4745950813276976E+0 + b = 0.4074575378263879E+0 + v = 0.1872270566606832E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5014034601410262E+0 + b = 0.4345456906027828E+0 + v = 0.1883722645591307E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5272493404551239E+0 + b = 0.4607942515205134E+0 + v = 0.1891714324525297E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5520413051846366E+0 + b = 0.4860961284181720E+0 + v = 0.1896827480450146E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5756887237503077E+0 + b = 0.5103447395342790E+0 + v = 0.1899628417059528E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1225039430588352E+0 + b = 0.2136455922655793E-1 + v = 0.1123301829001669E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1539113217321372E+0 + b = 0.4520926166137188E-1 + v = 0.1253698826711277E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1856213098637712E+0 + b = 0.7086468177864818E-1 + v = 0.1366266117678531E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2174998728035131E+0 + b = 0.9785239488772918E-1 + v = 0.1462736856106918E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2494128336938330E+0 + b = 0.1258106396267210E+0 + v = 0.1545076466685412E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2812321562143480E+0 + b = 0.1544529125047001E+0 + v = 0.1615096280814007E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3128372276456111E+0 + b = 0.1835433512202753E+0 + v = 0.1674366639741759E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3441145160177973E+0 + b = 0.2128813258619585E+0 + v = 0.1724225002437900E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3749567714853510E+0 + b = 0.2422913734880829E+0 + v = 0.1765810822987288E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4052621732015610E+0 + b = 0.2716163748391453E+0 + v = 0.1800104126010751E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4349335453522385E+0 + b = 0.3007127671240280E+0 + v = 0.1827960437331284E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4638776641524965E+0 + b = 0.3294470677216479E+0 + v = 0.1850140300716308E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4920046410462687E+0 + b = 0.3576932543699155E+0 + v = 0.1867333507394938E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5192273554861704E+0 + b = 0.3853307059757764E+0 + v = 0.1880178688638289E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5454609081136522E+0 + b = 0.4122425044452694E+0 + v = 0.1889278925654758E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5706220661424140E+0 + b = 0.4383139587781027E+0 + v = 0.1895213832507346E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5946286755181518E+0 + b = 0.4634312536300553E+0 + v = 0.1898548277397420E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.1905370790924295E+0 + b = 0.2371311537781979E-1 + v = 0.1349105935937341E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2242518717748009E+0 + b = 0.4917878059254806E-1 + v = 0.1444060068369326E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2577190808025936E+0 + b = 0.7595498960495142E-1 + v = 0.1526797390930008E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2908724534927187E+0 + b = 0.1036991083191100E+0 + v = 0.1598208771406474E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3236354020056219E+0 + b = 0.1321348584450234E+0 + v = 0.1659354368615331E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3559267359304543E+0 + b = 0.1610316571314789E+0 + v = 0.1711279910946440E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3876637123676956E+0 + b = 0.1901912080395707E+0 + v = 0.1754952725601440E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4187636705218842E+0 + b = 0.2194384950137950E+0 + v = 0.1791247850802529E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4491449019883107E+0 + b = 0.2486155334763858E+0 + v = 0.1820954300877716E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4787270932425445E+0 + b = 0.2775768931812335E+0 + v = 0.1844788524548449E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5074315153055574E+0 + b = 0.3061863786591120E+0 + v = 0.1863409481706220E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5351810507738336E+0 + b = 0.3343144718152556E+0 + v = 0.1877433008795068E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5619001025975381E+0 + b = 0.3618362729028427E+0 + v = 0.1887444543705232E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5875144035268046E+0 + b = 0.3886297583620408E+0 + v = 0.1894009829375006E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6119507308734495E+0 + b = 0.4145742277792031E+0 + v = 0.1897683345035198E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2619733870119463E+0 + b = 0.2540047186389353E-1 + v = 0.1517327037467653E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.2968149743237949E+0 + b = 0.5208107018543989E-1 + v = 0.1587740557483543E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3310451504860488E+0 + b = 0.7971828470885599E-1 + v = 0.1649093382274097E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3646215567376676E+0 + b = 0.1080465999177927E+0 + v = 0.1701915216193265E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3974916785279360E+0 + b = 0.1368413849366629E+0 + v = 0.1746847753144065E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4295967403772029E+0 + b = 0.1659073184763559E+0 + v = 0.1784555512007570E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4608742854473447E+0 + b = 0.1950703730454614E+0 + v = 0.1815687562112174E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4912598858949903E+0 + b = 0.2241721144376724E+0 + v = 0.1840864370663302E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5206882758945558E+0 + b = 0.2530655255406489E+0 + v = 0.1860676785390006E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5490940914019819E+0 + b = 0.2816118409731066E+0 + v = 0.1875690583743703E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5764123302025542E+0 + b = 0.3096780504593238E+0 + v = 0.1886453236347225E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6025786004213506E+0 + b = 0.3371348366394987E+0 + v = 0.1893501123329645E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6275291964794956E+0 + b = 0.3638547827694396E+0 + v = 0.1897366184519868E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3348189479861771E+0 + b = 0.2664841935537443E-1 + v = 0.1643908815152736E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.3699515545855295E+0 + b = 0.5424000066843495E-1 + v = 0.1696300350907768E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4042003071474669E+0 + b = 0.8251992715430854E-1 + v = 0.1741553103844483E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4375320100182624E+0 + b = 0.1112695182483710E+0 + v = 0.1780015282386092E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4699054490335947E+0 + b = 0.1402964116467816E+0 + v = 0.1812116787077125E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5012739879431952E+0 + b = 0.1694275117584291E+0 + v = 0.1838323158085421E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5315874883754966E+0 + b = 0.1985038235312689E+0 + v = 0.1859113119837737E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5607937109622117E+0 + b = 0.2273765660020893E+0 + v = 0.1874969220221698E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5888393223495521E+0 + b = 0.2559041492849764E+0 + v = 0.1886375612681076E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6156705979160163E+0 + b = 0.2839497251976899E+0 + v = 0.1893819575809276E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6412338809078123E+0 + b = 0.3113791060500690E+0 + v = 0.1897794748256767E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4076051259257167E+0 + b = 0.2757792290858463E-1 + v = 0.1738963926584846E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4423788125791520E+0 + b = 0.5584136834984293E-1 + v = 0.1777442359873466E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4760480917328258E+0 + b = 0.8457772087727143E-1 + v = 0.1810010815068719E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5085838725946297E+0 + b = 0.1135975846359248E+0 + v = 0.1836920318248129E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5399513637391218E+0 + b = 0.1427286904765053E+0 + v = 0.1858489473214328E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5701118433636380E+0 + b = 0.1718112740057635E+0 + v = 0.1875079342496592E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5990240530606021E+0 + b = 0.2006944855985351E+0 + v = 0.1887080239102310E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6266452685139695E+0 + b = 0.2292335090598907E+0 + v = 0.1894905752176822E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6529320971415942E+0 + b = 0.2572871512353714E+0 + v = 0.1898991061200695E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.4791583834610126E+0 + b = 0.2826094197735932E-1 + v = 0.1809065016458791E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5130373952796940E+0 + b = 0.5699871359683649E-1 + v = 0.1836297121596799E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5456252429628476E+0 + b = 0.8602712528554394E-1 + v = 0.1858426916241869E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5768956329682385E+0 + b = 0.1151748137221281E+0 + v = 0.1875654101134641E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6068186944699046E+0 + b = 0.1442811654136362E+0 + v = 0.1888240751833503E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6353622248024907E+0 + b = 0.1731930321657680E+0 + v = 0.1896497383866979E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6624927035731797E+0 + b = 0.2017619958756061E+0 + v = 0.1900775530219121E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5484933508028488E+0 + b = 0.2874219755907391E-1 + v = 0.1858525041478814E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.5810207682142106E+0 + b = 0.5778312123713695E-1 + v = 0.1876248690077947E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6120955197181352E+0 + b = 0.8695262371439526E-1 + v = 0.1889404439064607E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6416944284294319E+0 + b = 0.1160893767057166E+0 + v = 0.1898168539265290E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6697926391731260E+0 + b = 0.1450378826743251E+0 + v = 0.1902779940661772E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6147594390585488E+0 + b = 0.2904957622341456E-1 + v = 0.1890125641731815E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6455390026356783E+0 + b = 0.5823809152617197E-1 + v = 0.1899434637795751E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6747258588365477E+0 + b = 0.8740384899884715E-1 + v = 0.1904520856831751E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + a = 0.6772135750395347E+0 + b = 0.2919946135808105E-1 + v = 0.1905534498734563E-3 + leb_tmp, start = get_lebedev_recurrence_points(6, start, a, b, v, leb_tmp) + + case _: + raise Exception('Angular grid unrecognized, choices are 6, 14, 26, 38, 50, 74, 86, 110, 146, 170, 194, 230, 266, 302, 350, 434, 590, 770, 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470, 3890, 4334, 4802, 5294, 5810') # noqa: E501 + + leb_tmp.n = degree + return leb_tmp + + +def get_lebedev_recurrence_points(type_, start, a, b, v, leb): + c = 0.0 + + match type_: + + case 1: + a = 1.0 + + leb.x[start] = a + leb.y[start] = 0.0 + leb.z[start] = 0.0 + leb.w[start] = 4.0 * pi * v + + leb.x[start + 1] = -a + leb.y[start + 1] = 0.0 + leb.z[start + 1] = 0.0 + leb.w[start + 1] = 4.0 * pi * v + + leb.x[start + 2] = 0.0 + leb.y[start + 2] = a + leb.z[start + 2] = 0.0 + leb.w[start + 2] = 4.0 * pi * v + + leb.x[start + 3] = 0.0 + leb.y[start + 3] = -a + leb.z[start + 3] = 0.0 + leb.w[start + 3] = 4.0 * pi * v + + leb.x[start + 4] = 0.0 + leb.y[start + 4] = 0.0 + leb.z[start + 4] = a + leb.w[start + 4] = 4.0 * pi * v + + leb.x[start + 5] = 0.0 + leb.y[start + 5] = 0.0 + leb.z[start + 5] = -a + leb.w[start + 5] = 4.0 * pi * v + start = start + 6 + + case 2: + a = sqrt(0.5) + leb.x[start] = 0.0 + leb.y[start] = a + leb.z[start] = a + leb.w[start] = 4.0 * pi * v + + leb.x[start + 1] = 0.0 + leb.y[start + 1] = -a + leb.z[start + 1] = a + leb.w[start + 1] = 4.0 * pi * v + + leb.x[start + 2] = 0.0 + leb.y[start + 2] = a + leb.z[start + 2] = -a + leb.w[start + 2] = 4.0 * pi * v + + leb.x[start + 3] = 0.0 + leb.y[start + 3] = -a + leb.z[start + 3] = -a + leb.w[start + 3] = 4.0 * pi * v + + leb.x[start + 4] = a + leb.y[start + 4] = 0.0 + leb.z[start + 4] = a + leb.w[start + 4] = 4.0 * pi * v + + leb.x[start + 5] = a + leb.y[start + 5] = 0.0 + leb.z[start + 5] = -a + leb.w[start + 5] = 4.0 * pi * v + + leb.x[start + 6] = -a + leb.y[start + 6] = 0.0 + leb.z[start + 6] = a + leb.w[start + 6] = 4.0 * pi * v + + leb.x[start + 7] = -a + leb.y[start + 7] = 0.0 + leb.z[start + 7] = -a + leb.w[start + 7] = 4.0 * pi * v + + leb.x[start + 8] = a + leb.y[start + 8] = a + leb.z[start + 8] = 0.0 + leb.w[start + 8] = 4.0 * pi * v + + leb.x[start + 9] = -a + leb.y[start + 9] = a + leb.z[start + 9] = 0.0 + leb.w[start + 9] = 4.0 * pi * v + + leb.x[start + 10] = a + leb.y[start + 10] = -a + leb.z[start + 10] = 0.0 + leb.w[start + 10] = 4.0 * pi * v + + leb.x[start + 11] = -a + leb.y[start + 11] = -a + leb.z[start + 11] = 0.0 + leb.w[start + 11] = 4.0 * pi * v + start = start + 12 + + case 3: + a = sqrt(1.0 / 3.0) + leb.x[start] = a + leb.y[start] = a + leb.z[start] = a + leb.w[start] = 4.0 * pi * v + + leb.x[start + 1] = -a + leb.y[start + 1] = a + leb.z[start + 1] = a + leb.w[start + 1] = 4.0 * pi * v + + leb.x[start + 2] = a + leb.y[start + 2] = -a + leb.z[start + 2] = a + leb.w[start + 2] = 4.0 * pi * v + + leb.x[start + 3] = a + leb.y[start + 3] = a + leb.z[start + 3] = -a + leb.w[start + 3] = 4.0 * pi * v + + leb.x[start + 4] = -a + leb.y[start + 4] = -a + leb.z[start + 4] = a + leb.w[start + 4] = 4.0 * pi * v + + leb.x[start + 5] = a + leb.y[start + 5] = -a + leb.z[start + 5] = -a + leb.w[start + 5] = 4.0 * pi * v + + leb.x[start + 6] = -a + leb.y[start + 6] = a + leb.z[start + 6] = -a + leb.w[start + 6] = 4.0 * pi * v + + leb.x[start + 7] = -a + leb.y[start + 7] = -a + leb.z[start + 7] = -a + leb.w[start + 7] = 4.0 * pi * v + start = start + 8 + + case 4: + # /* In this case A is inputed */ + b = sqrt(1.0 - 2.0 * a * a) + leb.x[start] = a + leb.y[start] = a + leb.z[start] = b + leb.w[start] = 4.0 * pi * v + + leb.x[start + 1] = -a + leb.y[start + 1] = a + leb.z[start + 1] = b + leb.w[start + 1] = 4.0 * pi * v + + leb.x[start + 2] = a + leb.y[start + 2] = -a + leb.z[start + 2] = b + leb.w[start + 2] = 4.0 * pi * v + + leb.x[start + 3] = a + leb.y[start + 3] = a + leb.z[start + 3] = -b + leb.w[start + 3] = 4.0 * pi * v + + leb.x[start + 4] = -a + leb.y[start + 4] = -a + leb.z[start + 4] = b + leb.w[start + 4] = 4.0 * pi * v + + leb.x[start + 5] = -a + leb.y[start + 5] = a + leb.z[start + 5] = -b + leb.w[start + 5] = 4.0 * pi * v + + leb.x[start + 6] = a + leb.y[start + 6] = -a + leb.z[start + 6] = -b + leb.w[start + 6] = 4.0 * pi * v + + leb.x[start + 7] = -a + leb.y[start + 7] = -a + leb.z[start + 7] = -b + leb.w[start + 7] = 4.0 * pi * v + + leb.x[start + 8] = -a + leb.y[start + 8] = b + leb.z[start + 8] = a + leb.w[start + 8] = 4.0 * pi * v + + leb.x[start + 9] = a + leb.y[start + 9] = -b + leb.z[start + 9] = a + leb.w[start + 9] = 4.0 * pi * v + + leb.x[start + 10] = a + leb.y[start + 10] = b + leb.z[start + 10] = -a + leb.w[start + 10] = 4.0 * pi * v + + leb.x[start + 11] = -a + leb.y[start + 11] = -b + leb.z[start + 11] = a + leb.w[start + 11] = 4.0 * pi * v + + leb.x[start + 12] = -a + leb.y[start + 12] = b + leb.z[start + 12] = -a + leb.w[start + 12] = 4.0 * pi * v + + leb.x[start + 13] = a + leb.y[start + 13] = -b + leb.z[start + 13] = -a + leb.w[start + 13] = 4.0 * pi * v + + leb.x[start + 14] = -a + leb.y[start + 14] = -b + leb.z[start + 14] = -a + leb.w[start + 14] = 4.0 * pi * v + + leb.x[start + 15] = a + leb.y[start + 15] = b + leb.z[start + 15] = a + leb.w[start + 15] = 4.0 * pi * v + + leb.x[start + 16] = b + leb.y[start + 16] = a + leb.z[start + 16] = a + leb.w[start + 16] = 4.0 * pi * v + + leb.x[start + 17] = -b + leb.y[start + 17] = a + leb.z[start + 17] = a + leb.w[start + 17] = 4.0 * pi * v + + leb.x[start + 18] = b + leb.y[start + 18] = -a + leb.z[start + 18] = a + leb.w[start + 18] = 4.0 * pi * v + + leb.x[start + 19] = b + leb.y[start + 19] = a + leb.z[start + 19] = -a + leb.w[start + 19] = 4.0 * pi * v + + leb.x[start + 20] = -b + leb.y[start + 20] = -a + leb.z[start + 20] = a + leb.w[start + 20] = 4.0 * pi * v + + leb.x[start + 21] = -b + leb.y[start + 21] = a + leb.z[start + 21] = -a + leb.w[start + 21] = 4.0 * pi * v + + leb.x[start + 22] = b + leb.y[start + 22] = -a + leb.z[start + 22] = -a + leb.w[start + 22] = 4.0 * pi * v + + leb.x[start + 23] = -b + leb.y[start + 23] = -a + leb.z[start + 23] = -a + leb.w[start + 23] = 4.0 * pi * v + start = start + 24 + + case 5: + # /* A is inputed in this case as well*/ + b = sqrt(1 - a * a) + leb.x[start] = a + leb.y[start] = b + leb.z[start] = 0.0 + leb.w[start] = 4.0 * pi * v + + leb.x[start + 1] = -a + leb.y[start + 1] = b + leb.z[start + 1] = 0.0 + leb.w[start + 1] = 4.0 * pi * v + + leb.x[start + 2] = a + leb.y[start + 2] = -b + leb.z[start + 2] = 0.0 + leb.w[start + 2] = 4.0 * pi * v + + leb.x[start + 3] = -a + leb.y[start + 3] = -b + leb.z[start + 3] = 0.0 + leb.w[start + 3] = 4.0 * pi * v + + leb.x[start + 4] = b + leb.y[start + 4] = a + leb.z[start + 4] = 0.0 + leb.w[start + 4] = 4.0 * pi * v + + leb.x[start + 5] = -b + leb.y[start + 5] = a + leb.z[start + 5] = 0.0 + leb.w[start + 5] = 4.0 * pi * v + + leb.x[start + 6] = b + leb.y[start + 6] = -a + leb.z[start + 6] = 0.0 + leb.w[start + 6] = 4.0 * pi * v + + leb.x[start + 7] = -b + leb.y[start + 7] = -a + leb.z[start + 7] = 0.0 + leb.w[start + 7] = 4.0 * pi * v + + leb.x[start + 8] = a + leb.y[start + 8] = 0.0 + leb.z[start + 8] = b + leb.w[start + 8] = 4.0 * pi * v + + leb.x[start + 9] = -a + leb.y[start + 9] = 0.0 + leb.z[start + 9] = b + leb.w[start + 9] = 4.0 * pi * v + + leb.x[start + 10] = a + leb.y[start + 10] = 0.0 + leb.z[start + 10] = -b + leb.w[start + 10] = 4.0 * pi * v + + leb.x[start + 11] = -a + leb.y[start + 11] = 0.0 + leb.z[start + 11] = -b + leb.w[start + 11] = 4.0 * pi * v + + leb.x[start + 12] = b + leb.y[start + 12] = 0.0 + leb.z[start + 12] = a + leb.w[start + 12] = 4.0 * pi * v + + leb.x[start + 13] = -b + leb.y[start + 13] = 0.0 + leb.z[start + 13] = a + leb.w[start + 13] = 4.0 * pi * v + + leb.x[start + 14] = b + leb.y[start + 14] = 0.0 + leb.z[start + 14] = -a + leb.w[start + 14] = 4.0 * pi * v + + leb.x[start + 15] = -b + leb.y[start + 15] = 0.0 + leb.z[start + 15] = -a + leb.w[start + 15] = 4.0 * pi * v + + leb.x[start + 16] = 0.0 + leb.y[start + 16] = a + leb.z[start + 16] = b + leb.w[start + 16] = 4.0 * pi * v + + leb.x[start + 17] = 0.0 + leb.y[start + 17] = -a + leb.z[start + 17] = b + leb.w[start + 17] = 4.0 * pi * v + + leb.x[start + 18] = 0.0 + leb.y[start + 18] = a + leb.z[start + 18] = -b + leb.w[start + 18] = 4.0 * pi * v + + leb.x[start + 19] = 0.0 + leb.y[start + 19] = -a + leb.z[start + 19] = -b + leb.w[start + 19] = 4.0 * pi * v + + leb.x[start + 20] = 0.0 + leb.y[start + 20] = b + leb.z[start + 20] = a + leb.w[start + 20] = 4.0 * pi * v + + leb.x[start + 21] = 0.0 + leb.y[start + 21] = -b + leb.z[start + 21] = a + leb.w[start + 21] = 4.0 * pi * v + + leb.x[start + 22] = 0.0 + leb.y[start + 22] = b + leb.z[start + 22] = -a + leb.w[start + 22] = 4.0 * pi * v + + leb.x[start + 23] = 0.0 + leb.y[start + 23] = -b + leb.z[start + 23] = -a + leb.w[start + 23] = 4.0 * pi * v + start = start + 24 + + case 6: + # /* both A and B are inputed in this case */ + c = sqrt(1.0 - a * a - b * b) + leb.x[start] = a + leb.y[start] = b + leb.z[start] = c + leb.w[start] = 4.0 * pi * v + + leb.x[start + 1] = -a + leb.y[start + 1] = b + leb.z[start + 1] = c + leb.w[start + 1] = 4.0 * pi * v + + leb.x[start + 2] = a + leb.y[start + 2] = -b + leb.z[start + 2] = c + leb.w[start + 2] = 4.0 * pi * v + + leb.x[start + 3] = a + leb.y[start + 3] = b + leb.z[start + 3] = -c + leb.w[start + 3] = 4.0 * pi * v + + leb.x[start + 4] = -a + leb.y[start + 4] = -b + leb.z[start + 4] = c + leb.w[start + 4] = 4.0 * pi * v + + leb.x[start + 5] = a + leb.y[start + 5] = -b + leb.z[start + 5] = -c + leb.w[start + 5] = 4.0 * pi * v + + leb.x[start + 6] = -a + leb.y[start + 6] = b + leb.z[start + 6] = -c + leb.w[start + 6] = 4.0 * pi * v + + leb.x[start + 7] = -a + leb.y[start + 7] = -b + leb.z[start + 7] = -c + leb.w[start + 7] = 4.0 * pi * v + + leb.x[start + 8] = b + leb.y[start + 8] = a + leb.z[start + 8] = c + leb.w[start + 8] = 4.0 * pi * v + + leb.x[start + 9] = -b + leb.y[start + 9] = a + leb.z[start + 9] = c + leb.w[start + 9] = 4.0 * pi * v + + leb.x[start + 10] = b + leb.y[start + 10] = -a + leb.z[start + 10] = c + leb.w[start + 10] = 4.0 * pi * v + + leb.x[start + 11] = b + leb.y[start + 11] = a + leb.z[start + 11] = -c + leb.w[start + 11] = 4.0 * pi * v + + leb.x[start + 12] = -b + leb.y[start + 12] = -a + leb.z[start + 12] = c + leb.w[start + 12] = 4.0 * pi * v + + leb.x[start + 13] = b + leb.y[start + 13] = -a + leb.z[start + 13] = -c + leb.w[start + 13] = 4.0 * pi * v + + leb.x[start + 14] = -b + leb.y[start + 14] = a + leb.z[start + 14] = -c + leb.w[start + 14] = 4.0 * pi * v + + leb.x[start + 15] = -b + leb.y[start + 15] = -a + leb.z[start + 15] = -c + leb.w[start + 15] = 4.0 * pi * v + + leb.x[start + 16] = c + leb.y[start + 16] = a + leb.z[start + 16] = b + leb.w[start + 16] = 4.0 * pi * v + + leb.x[start + 17] = -c + leb.y[start + 17] = a + leb.z[start + 17] = b + leb.w[start + 17] = 4.0 * pi * v + + leb.x[start + 18] = c + leb.y[start + 18] = -a + leb.z[start + 18] = b + leb.w[start + 18] = 4.0 * pi * v + + leb.x[start + 19] = c + leb.y[start + 19] = a + leb.z[start + 19] = -b + leb.w[start + 19] = 4.0 * pi * v + + leb.x[start + 20] = -c + leb.y[start + 20] = -a + leb.z[start + 20] = b + leb.w[start + 20] = 4.0 * pi * v + + leb.x[start + 21] = c + leb.y[start + 21] = -a + leb.z[start + 21] = -b + leb.w[start + 21] = 4.0 * pi * v + + leb.x[start + 22] = -c + leb.y[start + 22] = a + leb.z[start + 22] = -b + leb.w[start + 22] = 4.0 * pi * v + + leb.x[start + 23] = -c + leb.y[start + 23] = -a + leb.z[start + 23] = -b + leb.w[start + 23] = 4.0 * pi * v + + leb.x[start + 24] = c + leb.y[start + 24] = b + leb.z[start + 24] = a + leb.w[start + 24] = 4.0 * pi * v + + leb.x[start + 25] = -c + leb.y[start + 25] = b + leb.z[start + 25] = a + leb.w[start + 25] = 4.0 * pi * v + + leb.x[start + 26] = c + leb.y[start + 26] = -b + leb.z[start + 26] = a + leb.w[start + 26] = 4.0 * pi * v + + leb.x[start + 27] = c + leb.y[start + 27] = b + leb.z[start + 27] = -a + leb.w[start + 27] = 4.0 * pi * v + + leb.x[start + 28] = -c + leb.y[start + 28] = -b + leb.z[start + 28] = a + leb.w[start + 28] = 4.0 * pi * v + + leb.x[start + 29] = c + leb.y[start + 29] = -b + leb.z[start + 29] = -a + leb.w[start + 29] = 4.0 * pi * v + + leb.x[start + 30] = -c + leb.y[start + 30] = b + leb.z[start + 30] = -a + leb.w[start + 30] = 4.0 * pi * v + + leb.x[start + 31] = -c + leb.y[start + 31] = -b + leb.z[start + 31] = -a + leb.w[start + 31] = 4.0 * pi * v + + leb.x[start + 32] = a + leb.y[start + 32] = c + leb.z[start + 32] = b + leb.w[start + 32] = 4.0 * pi * v + + leb.x[start + 33] = -a + leb.y[start + 33] = c + leb.z[start + 33] = b + leb.w[start + 33] = 4.0 * pi * v + + leb.x[start + 34] = a + leb.y[start + 34] = -c + leb.z[start + 34] = b + leb.w[start + 34] = 4.0 * pi * v + + leb.x[start + 35] = a + leb.y[start + 35] = c + leb.z[start + 35] = -b + leb.w[start + 35] = 4.0 * pi * v + + leb.x[start + 36] = -a + leb.y[start + 36] = -c + leb.z[start + 36] = b + leb.w[start + 36] = 4.0 * pi * v + + leb.x[start + 37] = a + leb.y[start + 37] = -c + leb.z[start + 37] = -b + leb.w[start + 37] = 4.0 * pi * v + + leb.x[start + 38] = -a + leb.y[start + 38] = c + leb.z[start + 38] = -b + leb.w[start + 38] = 4.0 * pi * v + + leb.x[start + 39] = -a + leb.y[start + 39] = -c + leb.z[start + 39] = -b + leb.w[start + 39] = 4.0 * pi * v + + leb.x[start + 40] = b + leb.y[start + 40] = c + leb.z[start + 40] = a + leb.w[start + 40] = 4.0 * pi * v + + leb.x[start + 41] = -b + leb.y[start + 41] = c + leb.z[start + 41] = a + leb.w[start + 41] = 4.0 * pi * v + + leb.x[start + 42] = b + leb.y[start + 42] = -c + leb.z[start + 42] = a + leb.w[start + 42] = 4.0 * pi * v + + leb.x[start + 43] = b + leb.y[start + 43] = c + leb.z[start + 43] = -a + leb.w[start + 43] = 4.0 * pi * v + + leb.x[start + 44] = -b + leb.y[start + 44] = -c + leb.z[start + 44] = a + leb.w[start + 44] = 4.0 * pi * v + + leb.x[start + 45] = b + leb.y[start + 45] = -c + leb.z[start + 45] = -a + leb.w[start + 45] = 4.0 * pi * v + + leb.x[start + 46] = -b + leb.y[start + 46] = c + leb.z[start + 46] = -a + leb.w[start + 46] = 4.0 * pi * v + + leb.x[start + 47] = -b + leb.y[start + 47] = -c + leb.z[start + 47] = -a + leb.w[start + 47] = 4.0 * pi * v + start = start + 48 + + case _: + raise Exception('Bad grid order') + + return leb, start + + +def lebedev_rule(n): + r"""Lebedev quadrature. + + Compute the sample points and weights for Lebedev quadrature [1]_ + for integration of a function over the surface of a unit sphere. + + Parameters + ---------- + n : int + Quadrature order. See Notes for supported values. + + Returns + ------- + x : ndarray of shape ``(3, m)`` + Sample points on the unit sphere in Cartesian coordinates. + ``m`` is the "degree" corresponding with the specified order; see Notes. + w : ndarray of shape ``(m,)`` + Weights + + Notes + ----- + Implemented by translating the Matlab code of [2]_ to Python. + + The available orders (argument `n`) are:: + + 3, 5, 7, 9, 11, 13, 15, 17, + 19, 21, 23, 25, 27, 29, 31, 35, + 41, 47, 53, 59, 65, 71, 77, 83, + 89, 95, 101, 107, 113, 119, 125, 131 + + The corresponding degrees ``m`` are:: + + 6, 14, 26, 38, 50, 74, 86, 110, + 146, 170, 194, 230, 266, 302, 350, 434, + 590, 770, 974, 1202, 1454, 1730, 2030, 2354, + 2702, 3074, 3470, 3890, 4334, 4802, 5294, 5810 + + References + ---------- + .. [1] V.I. Lebedev, and D.N. Laikov. "A quadrature formula for the sphere of + the 131st algebraic order of accuracy". Doklady Mathematics, Vol. 59, + No. 3, 1999, pp. 477-481. + .. [2] R. Parrish. ``getLebedevSphere``. Matlab Central File Exchange. + https://www.mathworks.com/matlabcentral/fileexchange/27097-getlebedevsphere. + .. [3] Bellet, Jean-Baptiste, Matthieu Brachet, and Jean-Pierre Croisille. + "Quadrature and symmetry on the Cubed Sphere." Journal of Computational and + Applied Mathematics 409 (2022): 114142. :doi:`10.1016/j.cam.2022.114142`. + + Examples + -------- + An example given in [3]_ is integration of :math:`f(x, y, z) = \exp(x)` over a + sphere of radius :math:`1`; the reference there is ``14.7680137457653``. + Show the convergence to the expected result as the order increases: + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> from scipy.integrate import lebedev_rule + >>> + >>> def f(x): + ... return np.exp(x[0]) + >>> + >>> res = [] + >>> orders = np.arange(3, 20, 2) + >>> for n in orders: + ... x, w = lebedev_rule(n) + ... res.append(w @ f(x)) + >>> + >>> ref = np.full_like(res, 14.7680137457653) + >>> err = abs(res - ref)/abs(ref) + >>> plt.semilogy(orders, err) + >>> plt.xlabel('order $n$') + >>> plt.ylabel('relative error') + >>> plt.title(r'Convergence for $f(x, y, z) = \exp(x)$') + >>> plt.show() + + """ + degree = [6, 14, 26, 38, 50, 74, 86, 110, 146, 170, 194, 230, 266, 302, 350, + 434, 590, 770, 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470, + 3890, 4334, 4802, 5294, 5810] + order = [3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 41, 47, + 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131] + order_degree_map = dict(zip(order, degree)) + + if n not in order_degree_map: + message = f"Order {n=} not available. Available orders are {order}." + raise NotImplementedError(message) + + degree = order_degree_map[n] + res = get_lebedev_sphere(degree) + x = np.stack((res.x, res.y, res.z)) + w = res.w + + return x, w diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ode.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ode.py new file mode 100644 index 0000000000000000000000000000000000000000..72d9da2495da768753f45796e8df1996cd70d382 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_ode.py @@ -0,0 +1,1388 @@ +# Authors: Pearu Peterson, Pauli Virtanen, John Travers +""" +First-order ODE integrators. + +User-friendly interface to various numerical integrators for solving a +system of first order ODEs with prescribed initial conditions:: + + d y(t)[i] + --------- = f(t,y(t))[i], + d t + + y(t=0)[i] = y0[i], + +where:: + + i = 0, ..., len(y0) - 1 + +class ode +--------- + +A generic interface class to numeric integrators. It has the following +methods:: + + integrator = ode(f, jac=None) + integrator = integrator.set_integrator(name, **params) + integrator = integrator.set_initial_value(y0, t0=0.0) + integrator = integrator.set_f_params(*args) + integrator = integrator.set_jac_params(*args) + y1 = integrator.integrate(t1, step=False, relax=False) + flag = integrator.successful() + +class complex_ode +----------------- + +This class has the same generic interface as ode, except it can handle complex +f, y and Jacobians by transparently translating them into the equivalent +real-valued system. It supports the real-valued solvers (i.e., not zvode) and is +an alternative to ode with the zvode solver, sometimes performing better. +""" +# XXX: Integrators must have: +# =========================== +# cvode - C version of vode and vodpk with many improvements. +# Get it from http://www.netlib.org/ode/cvode.tar.gz. +# To wrap cvode to Python, one must write the extension module by +# hand. Its interface is too much 'advanced C' that using f2py +# would be too complicated (or impossible). +# +# How to define a new integrator: +# =============================== +# +# class myodeint(IntegratorBase): +# +# runner = or None +# +# def __init__(self,...): # required +# +# +# def reset(self,n,has_jac): # optional +# # n - the size of the problem (number of equations) +# # has_jac - whether user has supplied its own routine for Jacobian +# +# +# def run(self,f,jac,y0,t0,t1,f_params,jac_params): # required +# # this method is called to integrate from t=t0 to t=t1 +# # with initial condition y0. f and jac are user-supplied functions +# # that define the problem. f_params,jac_params are additional +# # arguments +# # to these functions. +# +# if : +# self.success = 0 +# return t1,y1 +# +# # In addition, one can define step() and run_relax() methods (they +# # take the same arguments as run()) if the integrator can support +# # these features (see IntegratorBase doc strings). +# +# if myodeint.runner: +# IntegratorBase.integrator_classes.append(myodeint) + +__all__ = ['ode', 'complex_ode'] + +import re +import threading +import warnings + +from numpy import asarray, array, zeros, isscalar, real, imag, vstack + +from . import _vode +from . import _dop +from . import _lsoda + + +_dop_int_dtype = _dop.types.intvar.dtype +_vode_int_dtype = _vode.types.intvar.dtype +_lsoda_int_dtype = _lsoda.types.intvar.dtype + + +# lsoda, vode and zvode are not thread-safe. VODE_LOCK protects both vode and +# zvode; they share the `def run` implementation +LSODA_LOCK = threading.Lock() +VODE_LOCK = threading.Lock() + + +# ------------------------------------------------------------------------------ +# User interface +# ------------------------------------------------------------------------------ + + +class ode: + """ + A generic interface class to numeric integrators. + + Solve an equation system :math:`y'(t) = f(t,y)` with (optional) ``jac = df/dy``. + + *Note*: The first two arguments of ``f(t, y, ...)`` are in the + opposite order of the arguments in the system definition function used + by `scipy.integrate.odeint`. + + Parameters + ---------- + f : callable ``f(t, y, *f_args)`` + Right-hand side of the differential equation. t is a scalar, + ``y.shape == (n,)``. + ``f_args`` is set by calling ``set_f_params(*args)``. + `f` should return a scalar, array or list (not a tuple). + jac : callable ``jac(t, y, *jac_args)``, optional + Jacobian of the right-hand side, ``jac[i,j] = d f[i] / d y[j]``. + ``jac_args`` is set by calling ``set_jac_params(*args)``. + + Attributes + ---------- + t : float + Current time. + y : ndarray + Current variable values. + + See also + -------- + odeint : an integrator with a simpler interface based on lsoda from ODEPACK + quad : for finding the area under a curve + + Notes + ----- + Available integrators are listed below. They can be selected using + the `set_integrator` method. + + "vode" + + Real-valued Variable-coefficient Ordinary Differential Equation + solver, with fixed-leading-coefficient implementation. It provides + implicit Adams method (for non-stiff problems) and a method based on + backward differentiation formulas (BDF) (for stiff problems). + + Source: http://www.netlib.org/ode/vode.f + + .. warning:: + + This integrator is not re-entrant. You cannot have two `ode` + instances using the "vode" integrator at the same time. + + This integrator accepts the following parameters in `set_integrator` + method of the `ode` class: + + - atol : float or sequence + absolute tolerance for solution + - rtol : float or sequence + relative tolerance for solution + - lband : None or int + - uband : None or int + Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+uband. + Setting these requires your jac routine to return the jacobian + in packed format, jac_packed[i-j+uband, j] = jac[i,j]. The + dimension of the matrix must be (lband+uband+1, len(y)). + - method: 'adams' or 'bdf' + Which solver to use, Adams (non-stiff) or BDF (stiff) + - with_jacobian : bool + This option is only considered when the user has not supplied a + Jacobian function and has not indicated (by setting either band) + that the Jacobian is banded. In this case, `with_jacobian` specifies + whether the iteration method of the ODE solver's correction step is + chord iteration with an internally generated full Jacobian or + functional iteration with no Jacobian. + - nsteps : int + Maximum number of (internally defined) steps allowed during one + call to the solver. + - first_step : float + - min_step : float + - max_step : float + Limits for the step sizes used by the integrator. + - order : int + Maximum order used by the integrator, + order <= 12 for Adams, <= 5 for BDF. + + "zvode" + + Complex-valued Variable-coefficient Ordinary Differential Equation + solver, with fixed-leading-coefficient implementation. It provides + implicit Adams method (for non-stiff problems) and a method based on + backward differentiation formulas (BDF) (for stiff problems). + + Source: http://www.netlib.org/ode/zvode.f + + .. warning:: + + This integrator is not re-entrant. You cannot have two `ode` + instances using the "zvode" integrator at the same time. + + This integrator accepts the same parameters in `set_integrator` + as the "vode" solver. + + .. note:: + + When using ZVODE for a stiff system, it should only be used for + the case in which the function f is analytic, that is, when each f(i) + is an analytic function of each y(j). Analyticity means that the + partial derivative df(i)/dy(j) is a unique complex number, and this + fact is critical in the way ZVODE solves the dense or banded linear + systems that arise in the stiff case. For a complex stiff ODE system + in which f is not analytic, ZVODE is likely to have convergence + failures, and for this problem one should instead use DVODE on the + equivalent real system (in the real and imaginary parts of y). + + "lsoda" + + Real-valued Variable-coefficient Ordinary Differential Equation + solver, with fixed-leading-coefficient implementation. It provides + automatic method switching between implicit Adams method (for non-stiff + problems) and a method based on backward differentiation formulas (BDF) + (for stiff problems). + + Source: http://www.netlib.org/odepack + + .. warning:: + + This integrator is not re-entrant. You cannot have two `ode` + instances using the "lsoda" integrator at the same time. + + This integrator accepts the following parameters in `set_integrator` + method of the `ode` class: + + - atol : float or sequence + absolute tolerance for solution + - rtol : float or sequence + relative tolerance for solution + - lband : None or int + - uband : None or int + Jacobian band width, jac[i,j] != 0 for i-lband <= j <= i+uband. + Setting these requires your jac routine to return the jacobian + in packed format, jac_packed[i-j+uband, j] = jac[i,j]. + - with_jacobian : bool + *Not used.* + - nsteps : int + Maximum number of (internally defined) steps allowed during one + call to the solver. + - first_step : float + - min_step : float + - max_step : float + Limits for the step sizes used by the integrator. + - max_order_ns : int + Maximum order used in the nonstiff case (default 12). + - max_order_s : int + Maximum order used in the stiff case (default 5). + - max_hnil : int + Maximum number of messages reporting too small step size (t + h = t) + (default 0) + - ixpr : int + Whether to generate extra printing at method switches (default False). + + "dopri5" + + This is an explicit runge-kutta method of order (4)5 due to Dormand & + Prince (with stepsize control and dense output). + + Authors: + + E. Hairer and G. Wanner + Universite de Geneve, Dept. de Mathematiques + CH-1211 Geneve 24, Switzerland + e-mail: ernst.hairer@math.unige.ch, gerhard.wanner@math.unige.ch + + This code is described in [HNW93]_. + + This integrator accepts the following parameters in set_integrator() + method of the ode class: + + - atol : float or sequence + absolute tolerance for solution + - rtol : float or sequence + relative tolerance for solution + - nsteps : int + Maximum number of (internally defined) steps allowed during one + call to the solver. + - first_step : float + - max_step : float + - safety : float + Safety factor on new step selection (default 0.9) + - ifactor : float + - dfactor : float + Maximum factor to increase/decrease step size by in one step + - beta : float + Beta parameter for stabilised step size control. + - verbosity : int + Switch for printing messages (< 0 for no messages). + + "dop853" + + This is an explicit runge-kutta method of order 8(5,3) due to Dormand + & Prince (with stepsize control and dense output). + + Options and references the same as "dopri5". + + Examples + -------- + + A problem to integrate and the corresponding jacobian: + + >>> from scipy.integrate import ode + >>> + >>> y0, t0 = [1.0j, 2.0], 0 + >>> + >>> def f(t, y, arg1): + ... return [1j*arg1*y[0] + y[1], -arg1*y[1]**2] + >>> def jac(t, y, arg1): + ... return [[1j*arg1, 1], [0, -arg1*2*y[1]]] + + The integration: + + >>> r = ode(f, jac).set_integrator('zvode', method='bdf') + >>> r.set_initial_value(y0, t0).set_f_params(2.0).set_jac_params(2.0) + >>> t1 = 10 + >>> dt = 1 + >>> while r.successful() and r.t < t1: + ... print(r.t+dt, r.integrate(r.t+dt)) + 1 [-0.71038232+0.23749653j 0.40000271+0.j ] + 2.0 [0.19098503-0.52359246j 0.22222356+0.j ] + 3.0 [0.47153208+0.52701229j 0.15384681+0.j ] + 4.0 [-0.61905937+0.30726255j 0.11764744+0.j ] + 5.0 [0.02340997-0.61418799j 0.09523835+0.j ] + 6.0 [0.58643071+0.339819j 0.08000018+0.j ] + 7.0 [-0.52070105+0.44525141j 0.06896565+0.j ] + 8.0 [-0.15986733-0.61234476j 0.06060616+0.j ] + 9.0 [0.64850462+0.15048982j 0.05405414+0.j ] + 10.0 [-0.38404699+0.56382299j 0.04878055+0.j ] + + References + ---------- + .. [HNW93] E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary + Differential Equations i. Nonstiff Problems. 2nd edition. + Springer Series in Computational Mathematics, + Springer-Verlag (1993) + + """ + + def __init__(self, f, jac=None): + self.stiff = 0 + self.f = f + self.jac = jac + self.f_params = () + self.jac_params = () + self._y = [] + + @property + def y(self): + return self._y + + def set_initial_value(self, y, t=0.0): + """Set initial conditions y(t) = y.""" + if isscalar(y): + y = [y] + n_prev = len(self._y) + if not n_prev: + self.set_integrator('') # find first available integrator + self._y = asarray(y, self._integrator.scalar) + self.t = t + self._integrator.reset(len(self._y), self.jac is not None) + return self + + def set_integrator(self, name, **integrator_params): + """ + Set integrator by name. + + Parameters + ---------- + name : str + Name of the integrator. + **integrator_params + Additional parameters for the integrator. + """ + integrator = find_integrator(name) + if integrator is None: + # FIXME: this really should be raise an exception. Will that break + # any code? + message = f'No integrator name match with {name!r} or is not available.' + warnings.warn(message, stacklevel=2) + else: + self._integrator = integrator(**integrator_params) + if not len(self._y): + self.t = 0.0 + self._y = array([0.0], self._integrator.scalar) + self._integrator.reset(len(self._y), self.jac is not None) + return self + + def integrate(self, t, step=False, relax=False): + """Find y=y(t), set y as an initial condition, and return y. + + Parameters + ---------- + t : float + The endpoint of the integration step. + step : bool + If True, and if the integrator supports the step method, + then perform a single integration step and return. + This parameter is provided in order to expose internals of + the implementation, and should not be changed from its default + value in most cases. + relax : bool + If True and if the integrator supports the run_relax method, + then integrate until t_1 >= t and return. ``relax`` is not + referenced if ``step=True``. + This parameter is provided in order to expose internals of + the implementation, and should not be changed from its default + value in most cases. + + Returns + ------- + y : float + The integrated value at t + """ + if step and self._integrator.supports_step: + mth = self._integrator.step + elif relax and self._integrator.supports_run_relax: + mth = self._integrator.run_relax + else: + mth = self._integrator.run + + try: + self._y, self.t = mth(self.f, self.jac or (lambda: None), + self._y, self.t, t, + self.f_params, self.jac_params) + except SystemError as e: + # f2py issue with tuple returns, see ticket 1187. + raise ValueError( + 'Function to integrate must not return a tuple.' + ) from e + + return self._y + + def successful(self): + """Check if integration was successful.""" + try: + self._integrator + except AttributeError: + self.set_integrator('') + return self._integrator.success == 1 + + def get_return_code(self): + """Extracts the return code for the integration to enable better control + if the integration fails. + + In general, a return code > 0 implies success, while a return code < 0 + implies failure. + + Notes + ----- + This section describes possible return codes and their meaning, for available + integrators that can be selected by `set_integrator` method. + + "vode" + + =========== ======= + Return Code Message + =========== ======= + 2 Integration successful. + -1 Excess work done on this call. (Perhaps wrong MF.) + -2 Excess accuracy requested. (Tolerances too small.) + -3 Illegal input detected. (See printed message.) + -4 Repeated error test failures. (Check all input.) + -5 Repeated convergence failures. (Perhaps bad Jacobian + supplied or wrong choice of MF or tolerances.) + -6 Error weight became zero during problem. (Solution + component i vanished, and ATOL or ATOL(i) = 0.) + =========== ======= + + "zvode" + + =========== ======= + Return Code Message + =========== ======= + 2 Integration successful. + -1 Excess work done on this call. (Perhaps wrong MF.) + -2 Excess accuracy requested. (Tolerances too small.) + -3 Illegal input detected. (See printed message.) + -4 Repeated error test failures. (Check all input.) + -5 Repeated convergence failures. (Perhaps bad Jacobian + supplied or wrong choice of MF or tolerances.) + -6 Error weight became zero during problem. (Solution + component i vanished, and ATOL or ATOL(i) = 0.) + =========== ======= + + "dopri5" + + =========== ======= + Return Code Message + =========== ======= + 1 Integration successful. + 2 Integration successful (interrupted by solout). + -1 Input is not consistent. + -2 Larger nsteps is needed. + -3 Step size becomes too small. + -4 Problem is probably stiff (interrupted). + =========== ======= + + "dop853" + + =========== ======= + Return Code Message + =========== ======= + 1 Integration successful. + 2 Integration successful (interrupted by solout). + -1 Input is not consistent. + -2 Larger nsteps is needed. + -3 Step size becomes too small. + -4 Problem is probably stiff (interrupted). + =========== ======= + + "lsoda" + + =========== ======= + Return Code Message + =========== ======= + 2 Integration successful. + -1 Excess work done on this call (perhaps wrong Dfun type). + -2 Excess accuracy requested (tolerances too small). + -3 Illegal input detected (internal error). + -4 Repeated error test failures (internal error). + -5 Repeated convergence failures (perhaps bad Jacobian or tolerances). + -6 Error weight became zero during problem. + -7 Internal workspace insufficient to finish (internal error). + =========== ======= + """ + try: + self._integrator + except AttributeError: + self.set_integrator('') + return self._integrator.istate + + def set_f_params(self, *args): + """Set extra parameters for user-supplied function f.""" + self.f_params = args + return self + + def set_jac_params(self, *args): + """Set extra parameters for user-supplied function jac.""" + self.jac_params = args + return self + + def set_solout(self, solout): + """ + Set callable to be called at every successful integration step. + + Parameters + ---------- + solout : callable + ``solout(t, y)`` is called at each internal integrator step, + t is a scalar providing the current independent position + y is the current solution ``y.shape == (n,)`` + solout should return -1 to stop integration + otherwise it should return None or 0 + + """ + if self._integrator.supports_solout: + self._integrator.set_solout(solout) + if self._y is not None: + self._integrator.reset(len(self._y), self.jac is not None) + else: + raise ValueError("selected integrator does not support solout," + " choose another one") + + +def _transform_banded_jac(bjac): + """ + Convert a real matrix of the form (for example) + + [0 0 A B] [0 0 0 B] + [0 0 C D] [0 0 A D] + [E F G H] to [0 F C H] + [I J K L] [E J G L] + [I 0 K 0] + + That is, every other column is shifted up one. + """ + # Shift every other column. + newjac = zeros((bjac.shape[0] + 1, bjac.shape[1])) + newjac[1:, ::2] = bjac[:, ::2] + newjac[:-1, 1::2] = bjac[:, 1::2] + return newjac + + +class complex_ode(ode): + """ + A wrapper of ode for complex systems. + + This functions similarly as `ode`, but re-maps a complex-valued + equation system to a real-valued one before using the integrators. + + Parameters + ---------- + f : callable ``f(t, y, *f_args)`` + Rhs of the equation. t is a scalar, ``y.shape == (n,)``. + ``f_args`` is set by calling ``set_f_params(*args)``. + jac : callable ``jac(t, y, *jac_args)`` + Jacobian of the rhs, ``jac[i,j] = d f[i] / d y[j]``. + ``jac_args`` is set by calling ``set_f_params(*args)``. + + Attributes + ---------- + t : float + Current time. + y : ndarray + Current variable values. + + Examples + -------- + For usage examples, see `ode`. + + """ + + def __init__(self, f, jac=None): + self.cf = f + self.cjac = jac + if jac is None: + ode.__init__(self, self._wrap, None) + else: + ode.__init__(self, self._wrap, self._wrap_jac) + + def _wrap(self, t, y, *f_args): + f = self.cf(*((t, y[::2] + 1j * y[1::2]) + f_args)) + # self.tmp is a real-valued array containing the interleaved + # real and imaginary parts of f. + self.tmp[::2] = real(f) + self.tmp[1::2] = imag(f) + return self.tmp + + def _wrap_jac(self, t, y, *jac_args): + # jac is the complex Jacobian computed by the user-defined function. + jac = self.cjac(*((t, y[::2] + 1j * y[1::2]) + jac_args)) + + # jac_tmp is the real version of the complex Jacobian. Each complex + # entry in jac, say 2+3j, becomes a 2x2 block of the form + # [2 -3] + # [3 2] + jac_tmp = zeros((2 * jac.shape[0], 2 * jac.shape[1])) + jac_tmp[1::2, 1::2] = jac_tmp[::2, ::2] = real(jac) + jac_tmp[1::2, ::2] = imag(jac) + jac_tmp[::2, 1::2] = -jac_tmp[1::2, ::2] + + ml = getattr(self._integrator, 'ml', None) + mu = getattr(self._integrator, 'mu', None) + if ml is not None or mu is not None: + # Jacobian is banded. The user's Jacobian function has computed + # the complex Jacobian in packed format. The corresponding + # real-valued version has every other column shifted up. + jac_tmp = _transform_banded_jac(jac_tmp) + + return jac_tmp + + @property + def y(self): + return self._y[::2] + 1j * self._y[1::2] + + def set_integrator(self, name, **integrator_params): + """ + Set integrator by name. + + Parameters + ---------- + name : str + Name of the integrator + **integrator_params + Additional parameters for the integrator. + """ + if name == 'zvode': + raise ValueError("zvode must be used with ode, not complex_ode") + + lband = integrator_params.get('lband') + uband = integrator_params.get('uband') + if lband is not None or uband is not None: + # The Jacobian is banded. Override the user-supplied bandwidths + # (which are for the complex Jacobian) with the bandwidths of + # the corresponding real-valued Jacobian wrapper of the complex + # Jacobian. + integrator_params['lband'] = 2 * (lband or 0) + 1 + integrator_params['uband'] = 2 * (uband or 0) + 1 + + return ode.set_integrator(self, name, **integrator_params) + + def set_initial_value(self, y, t=0.0): + """Set initial conditions y(t) = y.""" + y = asarray(y) + self.tmp = zeros(y.size * 2, 'float') + self.tmp[::2] = real(y) + self.tmp[1::2] = imag(y) + return ode.set_initial_value(self, self.tmp, t) + + def integrate(self, t, step=False, relax=False): + """Find y=y(t), set y as an initial condition, and return y. + + Parameters + ---------- + t : float + The endpoint of the integration step. + step : bool + If True, and if the integrator supports the step method, + then perform a single integration step and return. + This parameter is provided in order to expose internals of + the implementation, and should not be changed from its default + value in most cases. + relax : bool + If True and if the integrator supports the run_relax method, + then integrate until t_1 >= t and return. ``relax`` is not + referenced if ``step=True``. + This parameter is provided in order to expose internals of + the implementation, and should not be changed from its default + value in most cases. + + Returns + ------- + y : float + The integrated value at t + """ + y = ode.integrate(self, t, step, relax) + return y[::2] + 1j * y[1::2] + + def set_solout(self, solout): + """ + Set callable to be called at every successful integration step. + + Parameters + ---------- + solout : callable + ``solout(t, y)`` is called at each internal integrator step, + t is a scalar providing the current independent position + y is the current solution ``y.shape == (n,)`` + solout should return -1 to stop integration + otherwise it should return None or 0 + + """ + if self._integrator.supports_solout: + self._integrator.set_solout(solout, complex=True) + else: + raise TypeError("selected integrator does not support solouta, " + "choose another one") + + +# ------------------------------------------------------------------------------ +# ODE integrators +# ------------------------------------------------------------------------------ + +def find_integrator(name): + for cl in IntegratorBase.integrator_classes: + if re.match(name, cl.__name__, re.I): + return cl + return None + + +class IntegratorConcurrencyError(RuntimeError): + """ + Failure due to concurrent usage of an integrator that can be used + only for a single problem at a time. + + """ + + def __init__(self, name): + msg = (f"Integrator `{name}` can be used to solve only a single problem " + "at a time. If you want to integrate multiple problems, " + "consider using a different integrator (see `ode.set_integrator`)") + RuntimeError.__init__(self, msg) + + +class IntegratorBase: + runner = None # runner is None => integrator is not available + success = None # success==1 if integrator was called successfully + istate = None # istate > 0 means success, istate < 0 means failure + supports_run_relax = None + supports_step = None + supports_solout = False + integrator_classes = [] + scalar = float + + def acquire_new_handle(self): + # Some of the integrators have internal state (ancient + # Fortran...), and so only one instance can use them at a time. + # We keep track of this, and fail when concurrent usage is tried. + self.__class__.active_global_handle += 1 + self.handle = self.__class__.active_global_handle + + def check_handle(self): + if self.handle is not self.__class__.active_global_handle: + raise IntegratorConcurrencyError(self.__class__.__name__) + + def reset(self, n, has_jac): + """Prepare integrator for call: allocate memory, set flags, etc. + n - number of equations. + has_jac - if user has supplied function for evaluating Jacobian. + """ + + def run(self, f, jac, y0, t0, t1, f_params, jac_params): + """Integrate from t=t0 to t=t1 using y0 as an initial condition. + Return 2-tuple (y1,t1) where y1 is the result and t=t1 + defines the stoppage coordinate of the result. + """ + raise NotImplementedError('all integrators must define ' + 'run(f, jac, t0, t1, y0, f_params, jac_params)') + + def step(self, f, jac, y0, t0, t1, f_params, jac_params): + """Make one integration step and return (y1,t1).""" + raise NotImplementedError(f'{self.__class__.__name__} ' + 'does not support step() method') + + def run_relax(self, f, jac, y0, t0, t1, f_params, jac_params): + """Integrate from t=t0 to t>=t1 and return (y1,t).""" + raise NotImplementedError(f'{self.__class__.__name__} ' + 'does not support run_relax() method') + + # XXX: __str__ method for getting visual state of the integrator + + +def _vode_banded_jac_wrapper(jacfunc, ml, jac_params): + """ + Wrap a banded Jacobian function with a function that pads + the Jacobian with `ml` rows of zeros. + """ + + def jac_wrapper(t, y): + jac = asarray(jacfunc(t, y, *jac_params)) + padded_jac = vstack((jac, zeros((ml, jac.shape[1])))) + return padded_jac + + return jac_wrapper + + +class vode(IntegratorBase): + runner = getattr(_vode, 'dvode', None) + + messages = {-1: 'Excess work done on this call. (Perhaps wrong MF.)', + -2: 'Excess accuracy requested. (Tolerances too small.)', + -3: 'Illegal input detected. (See printed message.)', + -4: 'Repeated error test failures. (Check all input.)', + -5: 'Repeated convergence failures. (Perhaps bad' + ' Jacobian supplied or wrong choice of MF or tolerances.)', + -6: 'Error weight became zero during problem. (Solution' + ' component i vanished, and ATOL or ATOL(i) = 0.)' + } + supports_run_relax = 1 + supports_step = 1 + active_global_handle = 0 + + def __init__(self, + method='adams', + with_jacobian=False, + rtol=1e-6, atol=1e-12, + lband=None, uband=None, + order=12, + nsteps=500, + max_step=0.0, # corresponds to infinite + min_step=0.0, + first_step=0.0, # determined by solver + ): + + if re.match(method, r'adams', re.I): + self.meth = 1 + elif re.match(method, r'bdf', re.I): + self.meth = 2 + else: + raise ValueError(f'Unknown integration method {method}') + self.with_jacobian = with_jacobian + self.rtol = rtol + self.atol = atol + self.mu = uband + self.ml = lband + + self.order = order + self.nsteps = nsteps + self.max_step = max_step + self.min_step = min_step + self.first_step = first_step + self.success = 1 + + self.initialized = False + + def _determine_mf_and_set_bands(self, has_jac): + """ + Determine the `MF` parameter (Method Flag) for the Fortran subroutine `dvode`. + + In the Fortran code, the legal values of `MF` are: + 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, + -11, -12, -14, -15, -21, -22, -24, -25 + but this Python wrapper does not use negative values. + + Returns + + mf = 10*self.meth + miter + + self.meth is the linear multistep method: + self.meth == 1: method="adams" + self.meth == 2: method="bdf" + + miter is the correction iteration method: + miter == 0: Functional iteration; no Jacobian involved. + miter == 1: Chord iteration with user-supplied full Jacobian. + miter == 2: Chord iteration with internally computed full Jacobian. + miter == 3: Chord iteration with internally computed diagonal Jacobian. + miter == 4: Chord iteration with user-supplied banded Jacobian. + miter == 5: Chord iteration with internally computed banded Jacobian. + + Side effects: If either self.mu or self.ml is not None and the other is None, + then the one that is None is set to 0. + """ + + jac_is_banded = self.mu is not None or self.ml is not None + if jac_is_banded: + if self.mu is None: + self.mu = 0 + if self.ml is None: + self.ml = 0 + + # has_jac is True if the user provided a Jacobian function. + if has_jac: + if jac_is_banded: + miter = 4 + else: + miter = 1 + else: + if jac_is_banded: + if self.ml == self.mu == 0: + miter = 3 # Chord iteration with internal diagonal Jacobian. + else: + miter = 5 # Chord iteration with internal banded Jacobian. + else: + # self.with_jacobian is set by the user in + # the call to ode.set_integrator. + if self.with_jacobian: + miter = 2 # Chord iteration with internal full Jacobian. + else: + miter = 0 # Functional iteration; no Jacobian involved. + + mf = 10 * self.meth + miter + return mf + + def reset(self, n, has_jac): + mf = self._determine_mf_and_set_bands(has_jac) + + if mf == 10: + lrw = 20 + 16 * n + elif mf in [11, 12]: + lrw = 22 + 16 * n + 2 * n * n + elif mf == 13: + lrw = 22 + 17 * n + elif mf in [14, 15]: + lrw = 22 + 18 * n + (3 * self.ml + 2 * self.mu) * n + elif mf == 20: + lrw = 20 + 9 * n + elif mf in [21, 22]: + lrw = 22 + 9 * n + 2 * n * n + elif mf == 23: + lrw = 22 + 10 * n + elif mf in [24, 25]: + lrw = 22 + 11 * n + (3 * self.ml + 2 * self.mu) * n + else: + raise ValueError(f'Unexpected mf={mf}') + + if mf % 10 in [0, 3]: + liw = 30 + else: + liw = 30 + n + + rwork = zeros((lrw,), float) + rwork[4] = self.first_step + rwork[5] = self.max_step + rwork[6] = self.min_step + self.rwork = rwork + + iwork = zeros((liw,), _vode_int_dtype) + if self.ml is not None: + iwork[0] = self.ml + if self.mu is not None: + iwork[1] = self.mu + iwork[4] = self.order + iwork[5] = self.nsteps + iwork[6] = 2 # mxhnil + self.iwork = iwork + + self.call_args = [self.rtol, self.atol, 1, 1, + self.rwork, self.iwork, mf] + self.success = 1 + self.initialized = False + + def run(self, f, jac, y0, t0, t1, f_params, jac_params): + if self.initialized: + self.check_handle() + else: + self.initialized = True + self.acquire_new_handle() + + if self.ml is not None and self.ml > 0: + # Banded Jacobian. Wrap the user-provided function with one + # that pads the Jacobian array with the extra `self.ml` rows + # required by the f2py-generated wrapper. + jac = _vode_banded_jac_wrapper(jac, self.ml, jac_params) + + args = ((f, jac, y0, t0, t1) + tuple(self.call_args) + + (f_params, jac_params)) + + with VODE_LOCK: + y1, t, istate = self.runner(*args) + + self.istate = istate + if istate < 0: + unexpected_istate_msg = f'Unexpected istate={istate:d}' + warnings.warn(f'{self.__class__.__name__:s}: ' + f'{self.messages.get(istate, unexpected_istate_msg):s}', + stacklevel=2) + self.success = 0 + else: + self.call_args[3] = 2 # upgrade istate from 1 to 2 + self.istate = 2 + return y1, t + + def step(self, *args): + itask = self.call_args[2] + self.call_args[2] = 2 + r = self.run(*args) + self.call_args[2] = itask + return r + + def run_relax(self, *args): + itask = self.call_args[2] + self.call_args[2] = 3 + r = self.run(*args) + self.call_args[2] = itask + return r + + +if vode.runner is not None: + IntegratorBase.integrator_classes.append(vode) + + +class zvode(vode): + runner = getattr(_vode, 'zvode', None) + + supports_run_relax = 1 + supports_step = 1 + scalar = complex + active_global_handle = 0 + + def reset(self, n, has_jac): + mf = self._determine_mf_and_set_bands(has_jac) + + if mf in (10,): + lzw = 15 * n + elif mf in (11, 12): + lzw = 15 * n + 2 * n ** 2 + elif mf in (-11, -12): + lzw = 15 * n + n ** 2 + elif mf in (13,): + lzw = 16 * n + elif mf in (14, 15): + lzw = 17 * n + (3 * self.ml + 2 * self.mu) * n + elif mf in (-14, -15): + lzw = 16 * n + (2 * self.ml + self.mu) * n + elif mf in (20,): + lzw = 8 * n + elif mf in (21, 22): + lzw = 8 * n + 2 * n ** 2 + elif mf in (-21, -22): + lzw = 8 * n + n ** 2 + elif mf in (23,): + lzw = 9 * n + elif mf in (24, 25): + lzw = 10 * n + (3 * self.ml + 2 * self.mu) * n + elif mf in (-24, -25): + lzw = 9 * n + (2 * self.ml + self.mu) * n + + lrw = 20 + n + + if mf % 10 in (0, 3): + liw = 30 + else: + liw = 30 + n + + zwork = zeros((lzw,), complex) + self.zwork = zwork + + rwork = zeros((lrw,), float) + rwork[4] = self.first_step + rwork[5] = self.max_step + rwork[6] = self.min_step + self.rwork = rwork + + iwork = zeros((liw,), _vode_int_dtype) + if self.ml is not None: + iwork[0] = self.ml + if self.mu is not None: + iwork[1] = self.mu + iwork[4] = self.order + iwork[5] = self.nsteps + iwork[6] = 2 # mxhnil + self.iwork = iwork + + self.call_args = [self.rtol, self.atol, 1, 1, + self.zwork, self.rwork, self.iwork, mf] + self.success = 1 + self.initialized = False + + +if zvode.runner is not None: + IntegratorBase.integrator_classes.append(zvode) + + +class dopri5(IntegratorBase): + runner = getattr(_dop, 'dopri5', None) + name = 'dopri5' + supports_solout = True + + messages = {1: 'computation successful', + 2: 'computation successful (interrupted by solout)', + -1: 'input is not consistent', + -2: 'larger nsteps is needed', + -3: 'step size becomes too small', + -4: 'problem is probably stiff (interrupted)', + } + + def __init__(self, + rtol=1e-6, atol=1e-12, + nsteps=500, + max_step=0.0, + first_step=0.0, # determined by solver + safety=0.9, + ifactor=10.0, + dfactor=0.2, + beta=0.0, + method=None, + verbosity=-1, # no messages if negative + ): + self.rtol = rtol + self.atol = atol + self.nsteps = nsteps + self.max_step = max_step + self.first_step = first_step + self.safety = safety + self.ifactor = ifactor + self.dfactor = dfactor + self.beta = beta + self.verbosity = verbosity + self.success = 1 + self.set_solout(None) + + def set_solout(self, solout, complex=False): + self.solout = solout + self.solout_cmplx = complex + if solout is None: + self.iout = 0 + else: + self.iout = 1 + + def reset(self, n, has_jac): + work = zeros((8 * n + 21,), float) + work[1] = self.safety + work[2] = self.dfactor + work[3] = self.ifactor + work[4] = self.beta + work[5] = self.max_step + work[6] = self.first_step + self.work = work + iwork = zeros((21,), _dop_int_dtype) + iwork[0] = self.nsteps + iwork[2] = self.verbosity + self.iwork = iwork + self.call_args = [self.rtol, self.atol, self._solout, + self.iout, self.work, self.iwork] + self.success = 1 + + def run(self, f, jac, y0, t0, t1, f_params, jac_params): + x, y, iwork, istate = self.runner(*((f, t0, y0, t1) + + tuple(self.call_args) + (f_params,))) + self.istate = istate + if istate < 0: + unexpected_istate_msg = f'Unexpected istate={istate:d}' + warnings.warn(f'{self.__class__.__name__:s}: ' + f'{self.messages.get(istate, unexpected_istate_msg):s}', + stacklevel=2) + self.success = 0 + return y, x + + def _solout(self, nr, xold, x, y, nd, icomp, con): + if self.solout is not None: + if self.solout_cmplx: + y = y[::2] + 1j * y[1::2] + return self.solout(x, y) + else: + return 1 + + +if dopri5.runner is not None: + IntegratorBase.integrator_classes.append(dopri5) + + +class dop853(dopri5): + runner = getattr(_dop, 'dop853', None) + name = 'dop853' + + def __init__(self, + rtol=1e-6, atol=1e-12, + nsteps=500, + max_step=0.0, + first_step=0.0, # determined by solver + safety=0.9, + ifactor=6.0, + dfactor=0.3, + beta=0.0, + method=None, + verbosity=-1, # no messages if negative + ): + super().__init__(rtol, atol, nsteps, max_step, first_step, safety, + ifactor, dfactor, beta, method, verbosity) + + def reset(self, n, has_jac): + work = zeros((11 * n + 21,), float) + work[1] = self.safety + work[2] = self.dfactor + work[3] = self.ifactor + work[4] = self.beta + work[5] = self.max_step + work[6] = self.first_step + self.work = work + iwork = zeros((21,), _dop_int_dtype) + iwork[0] = self.nsteps + iwork[2] = self.verbosity + self.iwork = iwork + self.call_args = [self.rtol, self.atol, self._solout, + self.iout, self.work, self.iwork] + self.success = 1 + + +if dop853.runner is not None: + IntegratorBase.integrator_classes.append(dop853) + + +class lsoda(IntegratorBase): + runner = getattr(_lsoda, 'lsoda', None) + active_global_handle = 0 + + messages = { + 2: "Integration successful.", + -1: "Excess work done on this call (perhaps wrong Dfun type).", + -2: "Excess accuracy requested (tolerances too small).", + -3: "Illegal input detected (internal error).", + -4: "Repeated error test failures (internal error).", + -5: "Repeated convergence failures (perhaps bad Jacobian or tolerances).", + -6: "Error weight became zero during problem.", + -7: "Internal workspace insufficient to finish (internal error)." + } + + def __init__(self, + with_jacobian=False, + rtol=1e-6, atol=1e-12, + lband=None, uband=None, + nsteps=500, + max_step=0.0, # corresponds to infinite + min_step=0.0, + first_step=0.0, # determined by solver + ixpr=0, + max_hnil=0, + max_order_ns=12, + max_order_s=5, + method=None + ): + + self.with_jacobian = with_jacobian + self.rtol = rtol + self.atol = atol + self.mu = uband + self.ml = lband + + self.max_order_ns = max_order_ns + self.max_order_s = max_order_s + self.nsteps = nsteps + self.max_step = max_step + self.min_step = min_step + self.first_step = first_step + self.ixpr = ixpr + self.max_hnil = max_hnil + self.success = 1 + + self.initialized = False + + def reset(self, n, has_jac): + # Calculate parameters for Fortran subroutine dvode. + if has_jac: + if self.mu is None and self.ml is None: + jt = 1 + else: + if self.mu is None: + self.mu = 0 + if self.ml is None: + self.ml = 0 + jt = 4 + else: + if self.mu is None and self.ml is None: + jt = 2 + else: + if self.mu is None: + self.mu = 0 + if self.ml is None: + self.ml = 0 + jt = 5 + lrn = 20 + (self.max_order_ns + 4) * n + if jt in [1, 2]: + lrs = 22 + (self.max_order_s + 4) * n + n * n + elif jt in [4, 5]: + lrs = 22 + (self.max_order_s + 5 + 2 * self.ml + self.mu) * n + else: + raise ValueError(f'Unexpected jt={jt}') + lrw = max(lrn, lrs) + liw = 20 + n + rwork = zeros((lrw,), float) + rwork[4] = self.first_step + rwork[5] = self.max_step + rwork[6] = self.min_step + self.rwork = rwork + iwork = zeros((liw,), _lsoda_int_dtype) + if self.ml is not None: + iwork[0] = self.ml + if self.mu is not None: + iwork[1] = self.mu + iwork[4] = self.ixpr + iwork[5] = self.nsteps + iwork[6] = self.max_hnil + iwork[7] = self.max_order_ns + iwork[8] = self.max_order_s + self.iwork = iwork + self.call_args = [self.rtol, self.atol, 1, 1, + self.rwork, self.iwork, jt] + self.success = 1 + self.initialized = False + + def run(self, f, jac, y0, t0, t1, f_params, jac_params): + if self.initialized: + self.check_handle() + else: + self.initialized = True + self.acquire_new_handle() + args = [f, y0, t0, t1] + self.call_args[:-1] + \ + [jac, self.call_args[-1], f_params, 0, jac_params] + + with LSODA_LOCK: + y1, t, istate = self.runner(*args) + + self.istate = istate + if istate < 0: + unexpected_istate_msg = f'Unexpected istate={istate:d}' + warnings.warn(f'{self.__class__.__name__:s}: ' + f'{self.messages.get(istate, unexpected_istate_msg):s}', + stacklevel=2) + self.success = 0 + else: + self.call_args[3] = 2 # upgrade istate from 1 to 2 + self.istate = 2 + return y1, t + + def step(self, *args): + itask = self.call_args[2] + self.call_args[2] = 2 + r = self.run(*args) + self.call_args[2] = itask + return r + + def run_relax(self, *args): + itask = self.call_args[2] + self.call_args[2] = 3 + r = self.run(*args) + self.call_args[2] = itask + return r + + +if lsoda.runner: + IntegratorBase.integrator_classes.append(lsoda) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_odepack_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_odepack_py.py new file mode 100644 index 0000000000000000000000000000000000000000..75dfe925b312ae609d19ccbec27927c6c945176f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_odepack_py.py @@ -0,0 +1,273 @@ +# Author: Travis Oliphant + +__all__ = ['odeint', 'ODEintWarning'] + +import numpy as np +from . import _odepack +from copy import copy +import warnings + +from threading import Lock + + +ODE_LOCK = Lock() + + +class ODEintWarning(Warning): + """Warning raised during the execution of `odeint`.""" + pass + + +_msgs = {2: "Integration successful.", + 1: "Nothing was done; the integration time was 0.", + -1: "Excess work done on this call (perhaps wrong Dfun type).", + -2: "Excess accuracy requested (tolerances too small).", + -3: "Illegal input detected (internal error).", + -4: "Repeated error test failures (internal error).", + -5: "Repeated convergence failures (perhaps bad Jacobian or tolerances).", + -6: "Error weight became zero during problem.", + -7: "Internal workspace insufficient to finish (internal error).", + -8: "Run terminated (internal error)." + } + + +def odeint(func, y0, t, args=(), Dfun=None, col_deriv=0, full_output=0, + ml=None, mu=None, rtol=None, atol=None, tcrit=None, h0=0.0, + hmax=0.0, hmin=0.0, ixpr=0, mxstep=0, mxhnil=0, mxordn=12, + mxords=5, printmessg=0, tfirst=False): + """ + Integrate a system of ordinary differential equations. + + .. note:: For new code, use `scipy.integrate.solve_ivp` to solve a + differential equation. + + Solve a system of ordinary differential equations using lsoda from the + FORTRAN library odepack. + + Solves the initial value problem for stiff or non-stiff systems + of first order ode-s:: + + dy/dt = func(y, t, ...) [or func(t, y, ...)] + + where y can be a vector. + + .. note:: By default, the required order of the first two arguments of + `func` are in the opposite order of the arguments in the system + definition function used by the `scipy.integrate.ode` class and + the function `scipy.integrate.solve_ivp`. To use a function with + the signature ``func(t, y, ...)``, the argument `tfirst` must be + set to ``True``. + + Parameters + ---------- + func : callable(y, t, ...) or callable(t, y, ...) + Computes the derivative of y at t. + If the signature is ``callable(t, y, ...)``, then the argument + `tfirst` must be set ``True``. + `func` must not modify the data in `y`, as it is a + view of the data used internally by the ODE solver. + y0 : array + Initial condition on y (can be a vector). + t : array + A sequence of time points for which to solve for y. The initial + value point should be the first element of this sequence. + This sequence must be monotonically increasing or monotonically + decreasing; repeated values are allowed. + args : tuple, optional + Extra arguments to pass to function. + Dfun : callable(y, t, ...) or callable(t, y, ...) + Gradient (Jacobian) of `func`. + If the signature is ``callable(t, y, ...)``, then the argument + `tfirst` must be set ``True``. + `Dfun` must not modify the data in `y`, as it is a + view of the data used internally by the ODE solver. + col_deriv : bool, optional + True if `Dfun` defines derivatives down columns (faster), + otherwise `Dfun` should define derivatives across rows. + full_output : bool, optional + True if to return a dictionary of optional outputs as the second output + printmessg : bool, optional + Whether to print the convergence message + tfirst : bool, optional + If True, the first two arguments of `func` (and `Dfun`, if given) + must ``t, y`` instead of the default ``y, t``. + + .. versionadded:: 1.1.0 + + Returns + ------- + y : array, shape (len(t), len(y0)) + Array containing the value of y for each desired time in t, + with the initial value `y0` in the first row. + infodict : dict, only returned if full_output == True + Dictionary containing additional output information + + ======= ============================================================ + key meaning + ======= ============================================================ + 'hu' vector of step sizes successfully used for each time step + 'tcur' vector with the value of t reached for each time step + (will always be at least as large as the input times) + 'tolsf' vector of tolerance scale factors, greater than 1.0, + computed when a request for too much accuracy was detected + 'tsw' value of t at the time of the last method switch + (given for each time step) + 'nst' cumulative number of time steps + 'nfe' cumulative number of function evaluations for each time step + 'nje' cumulative number of jacobian evaluations for each time step + 'nqu' a vector of method orders for each successful step + 'imxer' index of the component of largest magnitude in the + weighted local error vector (e / ewt) on an error return, -1 + otherwise + 'lenrw' the length of the double work array required + 'leniw' the length of integer work array required + 'mused' a vector of method indicators for each successful time step: + 1: adams (nonstiff), 2: bdf (stiff) + ======= ============================================================ + + Other Parameters + ---------------- + ml, mu : int, optional + If either of these are not None or non-negative, then the + Jacobian is assumed to be banded. These give the number of + lower and upper non-zero diagonals in this banded matrix. + For the banded case, `Dfun` should return a matrix whose + rows contain the non-zero bands (starting with the lowest diagonal). + Thus, the return matrix `jac` from `Dfun` should have shape + ``(ml + mu + 1, len(y0))`` when ``ml >=0`` or ``mu >=0``. + The data in `jac` must be stored such that ``jac[i - j + mu, j]`` + holds the derivative of the ``i``\\ th equation with respect to the + ``j``\\ th state variable. If `col_deriv` is True, the transpose of + this `jac` must be returned. + rtol, atol : float, optional + The input parameters `rtol` and `atol` determine the error + control performed by the solver. The solver will control the + vector, e, of estimated local errors in y, according to an + inequality of the form ``max-norm of (e / ewt) <= 1``, + where ewt is a vector of positive error weights computed as + ``ewt = rtol * abs(y) + atol``. + rtol and atol can be either vectors the same length as y or scalars. + Defaults to 1.49012e-8. + tcrit : ndarray, optional + Vector of critical points (e.g., singularities) where integration + care should be taken. + h0 : float, (0: solver-determined), optional + The step size to be attempted on the first step. + hmax : float, (0: solver-determined), optional + The maximum absolute step size allowed. + hmin : float, (0: solver-determined), optional + The minimum absolute step size allowed. + ixpr : bool, optional + Whether to generate extra printing at method switches. + mxstep : int, (0: solver-determined), optional + Maximum number of (internally defined) steps allowed for each + integration point in t. + mxhnil : int, (0: solver-determined), optional + Maximum number of messages printed. + mxordn : int, (0: solver-determined), optional + Maximum order to be allowed for the non-stiff (Adams) method. + mxords : int, (0: solver-determined), optional + Maximum order to be allowed for the stiff (BDF) method. + + See Also + -------- + solve_ivp : solve an initial value problem for a system of ODEs + ode : a more object-oriented integrator based on VODE + quad : for finding the area under a curve + + Examples + -------- + The second order differential equation for the angle `theta` of a + pendulum acted on by gravity with friction can be written:: + + theta''(t) + b*theta'(t) + c*sin(theta(t)) = 0 + + where `b` and `c` are positive constants, and a prime (') denotes a + derivative. To solve this equation with `odeint`, we must first convert + it to a system of first order equations. By defining the angular + velocity ``omega(t) = theta'(t)``, we obtain the system:: + + theta'(t) = omega(t) + omega'(t) = -b*omega(t) - c*sin(theta(t)) + + Let `y` be the vector [`theta`, `omega`]. We implement this system + in Python as: + + >>> import numpy as np + >>> def pend(y, t, b, c): + ... theta, omega = y + ... dydt = [omega, -b*omega - c*np.sin(theta)] + ... return dydt + ... + + We assume the constants are `b` = 0.25 and `c` = 5.0: + + >>> b = 0.25 + >>> c = 5.0 + + For initial conditions, we assume the pendulum is nearly vertical + with `theta(0)` = `pi` - 0.1, and is initially at rest, so + `omega(0)` = 0. Then the vector of initial conditions is + + >>> y0 = [np.pi - 0.1, 0.0] + + We will generate a solution at 101 evenly spaced samples in the interval + 0 <= `t` <= 10. So our array of times is: + + >>> t = np.linspace(0, 10, 101) + + Call `odeint` to generate the solution. To pass the parameters + `b` and `c` to `pend`, we give them to `odeint` using the `args` + argument. + + >>> from scipy.integrate import odeint + >>> sol = odeint(pend, y0, t, args=(b, c)) + + The solution is an array with shape (101, 2). The first column + is `theta(t)`, and the second is `omega(t)`. The following code + plots both components. + + >>> import matplotlib.pyplot as plt + >>> plt.plot(t, sol[:, 0], 'b', label='theta(t)') + >>> plt.plot(t, sol[:, 1], 'g', label='omega(t)') + >>> plt.legend(loc='best') + >>> plt.xlabel('t') + >>> plt.grid() + >>> plt.show() + """ + + if ml is None: + ml = -1 # changed to zero inside function call + if mu is None: + mu = -1 # changed to zero inside function call + + dt = np.diff(t) + if not ((dt >= 0).all() or (dt <= 0).all()): + raise ValueError("The values in t must be monotonically increasing " + "or monotonically decreasing; repeated values are " + "allowed.") + + t = copy(t) + y0 = copy(y0) + + with ODE_LOCK: + output = _odepack.odeint(func, y0, t, args, Dfun, col_deriv, ml, mu, + full_output, rtol, atol, tcrit, h0, hmax, hmin, + ixpr, mxstep, mxhnil, mxordn, mxords, + int(bool(tfirst))) + if output[-1] < 0: + warning_msg = (f"{_msgs[output[-1]]} Run with full_output = 1 to " + f"get quantitative information.") + warnings.warn(warning_msg, ODEintWarning, stacklevel=2) + elif printmessg: + warning_msg = _msgs[output[-1]] + warnings.warn(warning_msg, ODEintWarning, stacklevel=2) + + if full_output: + output[1]['message'] = _msgs[output[-1]] + + output = output[:-1] + if len(output) == 1: + return output[0] + else: + return output diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_quad_vec.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_quad_vec.py new file mode 100644 index 0000000000000000000000000000000000000000..758bac5138777dbe152c2b455b5160196d2282ca --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_quad_vec.py @@ -0,0 +1,682 @@ +import sys +import copy +import heapq +import collections +import functools +import warnings + +import numpy as np + +from scipy._lib._util import MapWrapper, _FunctionWrapper + + +class LRUDict(collections.OrderedDict): + def __init__(self, max_size): + self.__max_size = max_size + + def __setitem__(self, key, value): + existing_key = (key in self) + super().__setitem__(key, value) + if existing_key: + self.move_to_end(key) + elif len(self) > self.__max_size: + self.popitem(last=False) + + def update(self, other): + # Not needed below + raise NotImplementedError() + + +class SemiInfiniteFunc: + """ + Argument transform from (start, +-oo) to (0, 1) + """ + def __init__(self, func, start, infty): + self._func = func + self._start = start + self._sgn = -1 if infty < 0 else 1 + + # Overflow threshold for the 1/t**2 factor + self._tmin = sys.float_info.min**0.5 + + def get_t(self, x): + z = self._sgn * (x - self._start) + 1 + if z == 0: + # Can happen only if point not in range + return np.inf + return 1 / z + + def __call__(self, t): + if t < self._tmin: + return 0.0 + else: + x = self._start + self._sgn * (1 - t) / t + f = self._func(x) + return self._sgn * (f / t) / t + + +class DoubleInfiniteFunc: + """ + Argument transform from (-oo, oo) to (-1, 1) + """ + def __init__(self, func): + self._func = func + + # Overflow threshold for the 1/t**2 factor + self._tmin = sys.float_info.min**0.5 + + def get_t(self, x): + s = -1 if x < 0 else 1 + return s / (abs(x) + 1) + + def __call__(self, t): + if abs(t) < self._tmin: + return 0.0 + else: + x = (1 - abs(t)) / t + f = self._func(x) + return (f / t) / t + + +def _max_norm(x): + return np.amax(abs(x)) + + +def _get_sizeof(obj): + try: + return sys.getsizeof(obj) + except TypeError: + # occurs on pypy + if hasattr(obj, '__sizeof__'): + return int(obj.__sizeof__()) + return 64 + + +class _Bunch: + def __init__(self, **kwargs): + self.__keys = kwargs.keys() + self.__dict__.update(**kwargs) + + def __repr__(self): + key_value_pairs = ', '.join( + f'{k}={repr(self.__dict__[k])}' for k in self.__keys + ) + return f"_Bunch({key_value_pairs})" + + +def quad_vec(f, a, b, epsabs=1e-200, epsrel=1e-8, norm='2', cache_size=100e6, + limit=10000, workers=1, points=None, quadrature=None, full_output=False, + *, args=()): + r"""Adaptive integration of a vector-valued function. + + Parameters + ---------- + f : callable + Vector-valued function f(x) to integrate. + a : float + Initial point. + b : float + Final point. + epsabs : float, optional + Absolute tolerance. + epsrel : float, optional + Relative tolerance. + norm : {'max', '2'}, optional + Vector norm to use for error estimation. + cache_size : int, optional + Number of bytes to use for memoization. + limit : float or int, optional + An upper bound on the number of subintervals used in the adaptive + algorithm. + workers : int or map-like callable, optional + If `workers` is an integer, part of the computation is done in + parallel subdivided to this many tasks (using + :class:`python:multiprocessing.pool.Pool`). + Supply `-1` to use all cores available to the Process. + Alternatively, supply a map-like callable, such as + :meth:`python:multiprocessing.pool.Pool.map` for evaluating the + population in parallel. + This evaluation is carried out as ``workers(func, iterable)``. + points : list, optional + List of additional breakpoints. + quadrature : {'gk21', 'gk15', 'trapezoid'}, optional + Quadrature rule to use on subintervals. + Options: 'gk21' (Gauss-Kronrod 21-point rule), + 'gk15' (Gauss-Kronrod 15-point rule), + 'trapezoid' (composite trapezoid rule). + Default: 'gk21' for finite intervals and 'gk15' for (semi-)infinite + full_output : bool, optional + Return an additional ``info`` dictionary. + args : tuple, optional + Extra arguments to pass to function, if any. + + .. versionadded:: 1.8.0 + + Returns + ------- + res : {float, array-like} + Estimate for the result + err : float + Error estimate for the result in the given norm + info : dict + Returned only when ``full_output=True``. + Info dictionary. Is an object with the attributes: + + success : bool + Whether integration reached target precision. + status : int + Indicator for convergence, success (0), + failure (1), and failure due to rounding error (2). + neval : int + Number of function evaluations. + intervals : ndarray, shape (num_intervals, 2) + Start and end points of subdivision intervals. + integrals : ndarray, shape (num_intervals, ...) + Integral for each interval. + Note that at most ``cache_size`` values are recorded, + and the array may contains *nan* for missing items. + errors : ndarray, shape (num_intervals,) + Estimated integration error for each interval. + + Notes + ----- + The algorithm mainly follows the implementation of QUADPACK's + DQAG* algorithms, implementing global error control and adaptive + subdivision. + + The algorithm here has some differences to the QUADPACK approach: + + Instead of subdividing one interval at a time, the algorithm + subdivides N intervals with largest errors at once. This enables + (partial) parallelization of the integration. + + The logic of subdividing "next largest" intervals first is then + not implemented, and we rely on the above extension to avoid + concentrating on "small" intervals only. + + The Wynn epsilon table extrapolation is not used (QUADPACK uses it + for infinite intervals). This is because the algorithm here is + supposed to work on vector-valued functions, in an user-specified + norm, and the extension of the epsilon algorithm to this case does + not appear to be widely agreed. For max-norm, using elementwise + Wynn epsilon could be possible, but we do not do this here with + the hope that the epsilon extrapolation is mainly useful in + special cases. + + References + ---------- + [1] R. Piessens, E. de Doncker, QUADPACK (1983). + + Examples + -------- + We can compute integrations of a vector-valued function: + + >>> from scipy.integrate import quad_vec + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> alpha = np.linspace(0.0, 2.0, num=30) + >>> f = lambda x: x**alpha + >>> x0, x1 = 0, 2 + >>> y, err = quad_vec(f, x0, x1) + >>> plt.plot(alpha, y) + >>> plt.xlabel(r"$\alpha$") + >>> plt.ylabel(r"$\int_{0}^{2} x^\alpha dx$") + >>> plt.show() + + When using the argument `workers`, one should ensure + that the main module is import-safe, for instance + by rewriting the example above as: + + .. code-block:: python + + from scipy.integrate import quad_vec + import numpy as np + import matplotlib.pyplot as plt + + alpha = np.linspace(0.0, 2.0, num=30) + x0, x1 = 0, 2 + def f(x): + return x**alpha + + if __name__ == "__main__": + y, err = quad_vec(f, x0, x1, workers=2) + """ + a = float(a) + b = float(b) + + if args: + if not isinstance(args, tuple): + args = (args,) + + # create a wrapped function to allow the use of map and Pool.map + f = _FunctionWrapper(f, args) + + # Use simple transformations to deal with integrals over infinite + # intervals. + kwargs = dict(epsabs=epsabs, + epsrel=epsrel, + norm=norm, + cache_size=cache_size, + limit=limit, + workers=workers, + points=points, + quadrature='gk15' if quadrature is None else quadrature, + full_output=full_output) + if np.isfinite(a) and np.isinf(b): + f2 = SemiInfiniteFunc(f, start=a, infty=b) + if points is not None: + kwargs['points'] = tuple(f2.get_t(xp) for xp in points) + return quad_vec(f2, 0, 1, **kwargs) + elif np.isfinite(b) and np.isinf(a): + f2 = SemiInfiniteFunc(f, start=b, infty=a) + if points is not None: + kwargs['points'] = tuple(f2.get_t(xp) for xp in points) + res = quad_vec(f2, 0, 1, **kwargs) + return (-res[0],) + res[1:] + elif np.isinf(a) and np.isinf(b): + sgn = -1 if b < a else 1 + + # NB. explicitly split integral at t=0, which separates + # the positive and negative sides + f2 = DoubleInfiniteFunc(f) + if points is not None: + kwargs['points'] = (0,) + tuple(f2.get_t(xp) for xp in points) + else: + kwargs['points'] = (0,) + + if a != b: + res = quad_vec(f2, -1, 1, **kwargs) + else: + res = quad_vec(f2, 1, 1, **kwargs) + + return (res[0]*sgn,) + res[1:] + elif not (np.isfinite(a) and np.isfinite(b)): + raise ValueError(f"invalid integration bounds a={a}, b={b}") + + norm_funcs = { + None: _max_norm, + 'max': _max_norm, + '2': np.linalg.norm + } + if callable(norm): + norm_func = norm + else: + norm_func = norm_funcs[norm] + + parallel_count = 128 + min_intervals = 2 + + try: + _quadrature = {None: _quadrature_gk21, + 'gk21': _quadrature_gk21, + 'gk15': _quadrature_gk15, + 'trapz': _quadrature_trapezoid, # alias for backcompat + 'trapezoid': _quadrature_trapezoid}[quadrature] + except KeyError as e: + raise ValueError(f"unknown quadrature {quadrature!r}") from e + + if quadrature == "trapz": + msg = ("`quadrature='trapz'` is deprecated in favour of " + "`quadrature='trapezoid' and will raise an error from SciPy 1.16.0 " + "onwards.") + warnings.warn(msg, DeprecationWarning, stacklevel=2) + + # Initial interval set + if points is None: + initial_intervals = [(a, b)] + else: + prev = a + initial_intervals = [] + for p in sorted(points): + p = float(p) + if not (a < p < b) or p == prev: + continue + initial_intervals.append((prev, p)) + prev = p + initial_intervals.append((prev, b)) + + global_integral = None + global_error = None + rounding_error = None + interval_cache = None + intervals = [] + neval = 0 + + for x1, x2 in initial_intervals: + ig, err, rnd = _quadrature(x1, x2, f, norm_func) + neval += _quadrature.num_eval + + if global_integral is None: + if isinstance(ig, (float, complex)): + # Specialize for scalars + if norm_func in (_max_norm, np.linalg.norm): + norm_func = abs + + global_integral = ig + global_error = float(err) + rounding_error = float(rnd) + + cache_count = cache_size // _get_sizeof(ig) + interval_cache = LRUDict(cache_count) + else: + global_integral += ig + global_error += err + rounding_error += rnd + + interval_cache[(x1, x2)] = copy.copy(ig) + intervals.append((-err, x1, x2)) + + heapq.heapify(intervals) + + CONVERGED = 0 + NOT_CONVERGED = 1 + ROUNDING_ERROR = 2 + NOT_A_NUMBER = 3 + + status_msg = { + CONVERGED: "Target precision reached.", + NOT_CONVERGED: "Target precision not reached.", + ROUNDING_ERROR: "Target precision could not be reached due to rounding error.", + NOT_A_NUMBER: "Non-finite values encountered." + } + + # Process intervals + with MapWrapper(workers) as mapwrapper: + ier = NOT_CONVERGED + + while intervals and len(intervals) < limit: + # Select intervals with largest errors for subdivision + tol = max(epsabs, epsrel*norm_func(global_integral)) + + to_process = [] + err_sum = 0 + + for j in range(parallel_count): + if not intervals: + break + + if j > 0 and err_sum > global_error - tol/8: + # avoid unnecessary parallel splitting + break + + interval = heapq.heappop(intervals) + + neg_old_err, a, b = interval + old_int = interval_cache.pop((a, b), None) + to_process.append( + ((-neg_old_err, a, b, old_int), f, norm_func, _quadrature) + ) + err_sum += -neg_old_err + + # Subdivide intervals + for parts in mapwrapper(_subdivide_interval, to_process): + dint, derr, dround_err, subint, dneval = parts + neval += dneval + global_integral += dint + global_error += derr + rounding_error += dround_err + for x in subint: + x1, x2, ig, err = x + interval_cache[(x1, x2)] = ig + heapq.heappush(intervals, (-err, x1, x2)) + + # Termination check + if len(intervals) >= min_intervals: + tol = max(epsabs, epsrel*norm_func(global_integral)) + if global_error < tol/8: + ier = CONVERGED + break + if global_error < rounding_error: + ier = ROUNDING_ERROR + break + + if not (np.isfinite(global_error) and np.isfinite(rounding_error)): + ier = NOT_A_NUMBER + break + + res = global_integral + err = global_error + rounding_error + + if full_output: + res_arr = np.asarray(res) + dummy = np.full(res_arr.shape, np.nan, dtype=res_arr.dtype) + integrals = np.array([interval_cache.get((z[1], z[2]), dummy) + for z in intervals], dtype=res_arr.dtype) + errors = np.array([-z[0] for z in intervals]) + intervals = np.array([[z[1], z[2]] for z in intervals]) + + info = _Bunch(neval=neval, + success=(ier == CONVERGED), + status=ier, + message=status_msg[ier], + intervals=intervals, + integrals=integrals, + errors=errors) + return (res, err, info) + else: + return (res, err) + + +def _subdivide_interval(args): + interval, f, norm_func, _quadrature = args + old_err, a, b, old_int = interval + + c = 0.5 * (a + b) + + # Left-hand side + if getattr(_quadrature, 'cache_size', 0) > 0: + f = functools.lru_cache(_quadrature.cache_size)(f) + + s1, err1, round1 = _quadrature(a, c, f, norm_func) + dneval = _quadrature.num_eval + s2, err2, round2 = _quadrature(c, b, f, norm_func) + dneval += _quadrature.num_eval + if old_int is None: + old_int, _, _ = _quadrature(a, b, f, norm_func) + dneval += _quadrature.num_eval + + if getattr(_quadrature, 'cache_size', 0) > 0: + dneval = f.cache_info().misses + + dint = s1 + s2 - old_int + derr = err1 + err2 - old_err + dround_err = round1 + round2 + + subintervals = ((a, c, s1, err1), (c, b, s2, err2)) + return dint, derr, dround_err, subintervals, dneval + + +def _quadrature_trapezoid(x1, x2, f, norm_func): + """ + Composite trapezoid quadrature + """ + x3 = 0.5*(x1 + x2) + f1 = f(x1) + f2 = f(x2) + f3 = f(x3) + + s2 = 0.25 * (x2 - x1) * (f1 + 2*f3 + f2) + + round_err = 0.25 * abs(x2 - x1) * (float(norm_func(f1)) + + 2*float(norm_func(f3)) + + float(norm_func(f2))) * 2e-16 + + s1 = 0.5 * (x2 - x1) * (f1 + f2) + err = 1/3 * float(norm_func(s1 - s2)) + return s2, err, round_err + + +_quadrature_trapezoid.cache_size = 3 * 3 +_quadrature_trapezoid.num_eval = 3 + + +def _quadrature_gk(a, b, f, norm_func, x, w, v): + """ + Generic Gauss-Kronrod quadrature + """ + + fv = [0.0]*len(x) + + c = 0.5 * (a + b) + h = 0.5 * (b - a) + + # Gauss-Kronrod + s_k = 0.0 + s_k_abs = 0.0 + for i in range(len(x)): + ff = f(c + h*x[i]) + fv[i] = ff + + vv = v[i] + + # \int f(x) + s_k += vv * ff + # \int |f(x)| + s_k_abs += vv * abs(ff) + + # Gauss + s_g = 0.0 + for i in range(len(w)): + s_g += w[i] * fv[2*i + 1] + + # Quadrature of abs-deviation from average + s_k_dabs = 0.0 + y0 = s_k / 2.0 + for i in range(len(x)): + # \int |f(x) - y0| + s_k_dabs += v[i] * abs(fv[i] - y0) + + # Use similar error estimation as quadpack + err = float(norm_func((s_k - s_g) * h)) + dabs = float(norm_func(s_k_dabs * h)) + if dabs != 0 and err != 0: + err = dabs * min(1.0, (200 * err / dabs)**1.5) + + eps = sys.float_info.epsilon + round_err = float(norm_func(50 * eps * h * s_k_abs)) + + if round_err > sys.float_info.min: + err = max(err, round_err) + + return h * s_k, err, round_err + + +def _quadrature_gk21(a, b, f, norm_func): + """ + Gauss-Kronrod 21 quadrature with error estimate + """ + # Gauss-Kronrod points + x = (0.995657163025808080735527280689003, + 0.973906528517171720077964012084452, + 0.930157491355708226001207180059508, + 0.865063366688984510732096688423493, + 0.780817726586416897063717578345042, + 0.679409568299024406234327365114874, + 0.562757134668604683339000099272694, + 0.433395394129247190799265943165784, + 0.294392862701460198131126603103866, + 0.148874338981631210884826001129720, + 0, + -0.148874338981631210884826001129720, + -0.294392862701460198131126603103866, + -0.433395394129247190799265943165784, + -0.562757134668604683339000099272694, + -0.679409568299024406234327365114874, + -0.780817726586416897063717578345042, + -0.865063366688984510732096688423493, + -0.930157491355708226001207180059508, + -0.973906528517171720077964012084452, + -0.995657163025808080735527280689003) + + # 10-point weights + w = (0.066671344308688137593568809893332, + 0.149451349150580593145776339657697, + 0.219086362515982043995534934228163, + 0.269266719309996355091226921569469, + 0.295524224714752870173892994651338, + 0.295524224714752870173892994651338, + 0.269266719309996355091226921569469, + 0.219086362515982043995534934228163, + 0.149451349150580593145776339657697, + 0.066671344308688137593568809893332) + + # 21-point weights + v = (0.011694638867371874278064396062192, + 0.032558162307964727478818972459390, + 0.054755896574351996031381300244580, + 0.075039674810919952767043140916190, + 0.093125454583697605535065465083366, + 0.109387158802297641899210590325805, + 0.123491976262065851077958109831074, + 0.134709217311473325928054001771707, + 0.142775938577060080797094273138717, + 0.147739104901338491374841515972068, + 0.149445554002916905664936468389821, + 0.147739104901338491374841515972068, + 0.142775938577060080797094273138717, + 0.134709217311473325928054001771707, + 0.123491976262065851077958109831074, + 0.109387158802297641899210590325805, + 0.093125454583697605535065465083366, + 0.075039674810919952767043140916190, + 0.054755896574351996031381300244580, + 0.032558162307964727478818972459390, + 0.011694638867371874278064396062192) + + return _quadrature_gk(a, b, f, norm_func, x, w, v) + + +_quadrature_gk21.num_eval = 21 + + +def _quadrature_gk15(a, b, f, norm_func): + """ + Gauss-Kronrod 15 quadrature with error estimate + """ + # Gauss-Kronrod points + x = (0.991455371120812639206854697526329, + 0.949107912342758524526189684047851, + 0.864864423359769072789712788640926, + 0.741531185599394439863864773280788, + 0.586087235467691130294144838258730, + 0.405845151377397166906606412076961, + 0.207784955007898467600689403773245, + 0.000000000000000000000000000000000, + -0.207784955007898467600689403773245, + -0.405845151377397166906606412076961, + -0.586087235467691130294144838258730, + -0.741531185599394439863864773280788, + -0.864864423359769072789712788640926, + -0.949107912342758524526189684047851, + -0.991455371120812639206854697526329) + + # 7-point weights + w = (0.129484966168869693270611432679082, + 0.279705391489276667901467771423780, + 0.381830050505118944950369775488975, + 0.417959183673469387755102040816327, + 0.381830050505118944950369775488975, + 0.279705391489276667901467771423780, + 0.129484966168869693270611432679082) + + # 15-point weights + v = (0.022935322010529224963732008058970, + 0.063092092629978553290700663189204, + 0.104790010322250183839876322541518, + 0.140653259715525918745189590510238, + 0.169004726639267902826583426598550, + 0.190350578064785409913256402421014, + 0.204432940075298892414161999234649, + 0.209482141084727828012999174891714, + 0.204432940075298892414161999234649, + 0.190350578064785409913256402421014, + 0.169004726639267902826583426598550, + 0.140653259715525918745189590510238, + 0.104790010322250183839876322541518, + 0.063092092629978553290700663189204, + 0.022935322010529224963732008058970) + + return _quadrature_gk(a, b, f, norm_func, x, w, v) + + +_quadrature_gk15.num_eval = 15 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_quadpack_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_quadpack_py.py new file mode 100644 index 0000000000000000000000000000000000000000..0d273f6d2c9943f4a35f9b0c761a944b7be84cfe --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_quadpack_py.py @@ -0,0 +1,1279 @@ +# Author: Travis Oliphant 2001 +# Author: Nathan Woods 2013 (nquad &c) +import sys +import warnings +from functools import partial + +from . import _quadpack +import numpy as np + +__all__ = ["quad", "dblquad", "tplquad", "nquad", "IntegrationWarning"] + + +class IntegrationWarning(UserWarning): + """ + Warning on issues during integration. + """ + pass + + +def quad(func, a, b, args=(), full_output=0, epsabs=1.49e-8, epsrel=1.49e-8, + limit=50, points=None, weight=None, wvar=None, wopts=None, maxp1=50, + limlst=50, complex_func=False): + """ + Compute a definite integral. + + Integrate func from `a` to `b` (possibly infinite interval) using a + technique from the Fortran library QUADPACK. + + Parameters + ---------- + func : {function, scipy.LowLevelCallable} + A Python function or method to integrate. If `func` takes many + arguments, it is integrated along the axis corresponding to the + first argument. + + If the user desires improved integration performance, then `f` may + be a `scipy.LowLevelCallable` with one of the signatures:: + + double func(double x) + double func(double x, void *user_data) + double func(int n, double *xx) + double func(int n, double *xx, void *user_data) + + The ``user_data`` is the data contained in the `scipy.LowLevelCallable`. + In the call forms with ``xx``, ``n`` is the length of the ``xx`` + array which contains ``xx[0] == x`` and the rest of the items are + numbers contained in the ``args`` argument of quad. + + In addition, certain ctypes call signatures are supported for + backward compatibility, but those should not be used in new code. + a : float + Lower limit of integration (use -numpy.inf for -infinity). + b : float + Upper limit of integration (use numpy.inf for +infinity). + args : tuple, optional + Extra arguments to pass to `func`. + full_output : int, optional + Non-zero to return a dictionary of integration information. + If non-zero, warning messages are also suppressed and the + message is appended to the output tuple. + complex_func : bool, optional + Indicate if the function's (`func`) return type is real + (``complex_func=False``: default) or complex (``complex_func=True``). + In both cases, the function's argument is real. + If full_output is also non-zero, the `infodict`, `message`, and + `explain` for the real and complex components are returned in + a dictionary with keys "real output" and "imag output". + + Returns + ------- + y : float + The integral of func from `a` to `b`. + abserr : float + An estimate of the absolute error in the result. + infodict : dict + A dictionary containing additional information. + message + A convergence message. + explain + Appended only with 'cos' or 'sin' weighting and infinite + integration limits, it contains an explanation of the codes in + infodict['ierlst'] + + Other Parameters + ---------------- + epsabs : float or int, optional + Absolute error tolerance. Default is 1.49e-8. `quad` tries to obtain + an accuracy of ``abs(i-result) <= max(epsabs, epsrel*abs(i))`` + where ``i`` = integral of `func` from `a` to `b`, and ``result`` is the + numerical approximation. See `epsrel` below. + epsrel : float or int, optional + Relative error tolerance. Default is 1.49e-8. + If ``epsabs <= 0``, `epsrel` must be greater than both 5e-29 + and ``50 * (machine epsilon)``. See `epsabs` above. + limit : float or int, optional + An upper bound on the number of subintervals used in the adaptive + algorithm. + points : (sequence of floats,ints), optional + A sequence of break points in the bounded integration interval + where local difficulties of the integrand may occur (e.g., + singularities, discontinuities). The sequence does not have + to be sorted. Note that this option cannot be used in conjunction + with ``weight``. + weight : float or int, optional + String indicating weighting function. Full explanation for this + and the remaining arguments can be found below. + wvar : optional + Variables for use with weighting functions. + wopts : optional + Optional input for reusing Chebyshev moments. + maxp1 : float or int, optional + An upper bound on the number of Chebyshev moments. + limlst : int, optional + Upper bound on the number of cycles (>=3) for use with a sinusoidal + weighting and an infinite end-point. + + See Also + -------- + dblquad : double integral + tplquad : triple integral + nquad : n-dimensional integrals (uses `quad` recursively) + fixed_quad : fixed-order Gaussian quadrature + simpson : integrator for sampled data + romb : integrator for sampled data + scipy.special : for coefficients and roots of orthogonal polynomials + + Notes + ----- + For valid results, the integral must converge; behavior for divergent + integrals is not guaranteed. + + **Extra information for quad() inputs and outputs** + + If full_output is non-zero, then the third output argument + (infodict) is a dictionary with entries as tabulated below. For + infinite limits, the range is transformed to (0,1) and the + optional outputs are given with respect to this transformed range. + Let M be the input argument limit and let K be infodict['last']. + The entries are: + + 'neval' + The number of function evaluations. + 'last' + The number, K, of subintervals produced in the subdivision process. + 'alist' + A rank-1 array of length M, the first K elements of which are the + left end points of the subintervals in the partition of the + integration range. + 'blist' + A rank-1 array of length M, the first K elements of which are the + right end points of the subintervals. + 'rlist' + A rank-1 array of length M, the first K elements of which are the + integral approximations on the subintervals. + 'elist' + A rank-1 array of length M, the first K elements of which are the + moduli of the absolute error estimates on the subintervals. + 'iord' + A rank-1 integer array of length M, the first L elements of + which are pointers to the error estimates over the subintervals + with ``L=K`` if ``K<=M/2+2`` or ``L=M+1-K`` otherwise. Let I be the + sequence ``infodict['iord']`` and let E be the sequence + ``infodict['elist']``. Then ``E[I[1]], ..., E[I[L]]`` forms a + decreasing sequence. + + If the input argument points is provided (i.e., it is not None), + the following additional outputs are placed in the output + dictionary. Assume the points sequence is of length P. + + 'pts' + A rank-1 array of length P+2 containing the integration limits + and the break points of the intervals in ascending order. + This is an array giving the subintervals over which integration + will occur. + 'level' + A rank-1 integer array of length M (=limit), containing the + subdivision levels of the subintervals, i.e., if (aa,bb) is a + subinterval of ``(pts[1], pts[2])`` where ``pts[0]`` and ``pts[2]`` + are adjacent elements of ``infodict['pts']``, then (aa,bb) has level l + if ``|bb-aa| = |pts[2]-pts[1]| * 2**(-l)``. + 'ndin' + A rank-1 integer array of length P+2. After the first integration + over the intervals (pts[1], pts[2]), the error estimates over some + of the intervals may have been increased artificially in order to + put their subdivision forward. This array has ones in slots + corresponding to the subintervals for which this happens. + + **Weighting the integrand** + + The input variables, *weight* and *wvar*, are used to weight the + integrand by a select list of functions. Different integration + methods are used to compute the integral with these weighting + functions, and these do not support specifying break points. The + possible values of weight and the corresponding weighting functions are. + + ========== =================================== ===================== + ``weight`` Weight function used ``wvar`` + ========== =================================== ===================== + 'cos' cos(w*x) wvar = w + 'sin' sin(w*x) wvar = w + 'alg' g(x) = ((x-a)**alpha)*((b-x)**beta) wvar = (alpha, beta) + 'alg-loga' g(x)*log(x-a) wvar = (alpha, beta) + 'alg-logb' g(x)*log(b-x) wvar = (alpha, beta) + 'alg-log' g(x)*log(x-a)*log(b-x) wvar = (alpha, beta) + 'cauchy' 1/(x-c) wvar = c + ========== =================================== ===================== + + wvar holds the parameter w, (alpha, beta), or c depending on the weight + selected. In these expressions, a and b are the integration limits. + + For the 'cos' and 'sin' weighting, additional inputs and outputs are + available. + + For finite integration limits, the integration is performed using a + Clenshaw-Curtis method which uses Chebyshev moments. For repeated + calculations, these moments are saved in the output dictionary: + + 'momcom' + The maximum level of Chebyshev moments that have been computed, + i.e., if ``M_c`` is ``infodict['momcom']`` then the moments have been + computed for intervals of length ``|b-a| * 2**(-l)``, + ``l=0,1,...,M_c``. + 'nnlog' + A rank-1 integer array of length M(=limit), containing the + subdivision levels of the subintervals, i.e., an element of this + array is equal to l if the corresponding subinterval is + ``|b-a|* 2**(-l)``. + 'chebmo' + A rank-2 array of shape (25, maxp1) containing the computed + Chebyshev moments. These can be passed on to an integration + over the same interval by passing this array as the second + element of the sequence wopts and passing infodict['momcom'] as + the first element. + + If one of the integration limits is infinite, then a Fourier integral is + computed (assuming w neq 0). If full_output is 1 and a numerical error + is encountered, besides the error message attached to the output tuple, + a dictionary is also appended to the output tuple which translates the + error codes in the array ``info['ierlst']`` to English messages. The + output information dictionary contains the following entries instead of + 'last', 'alist', 'blist', 'rlist', and 'elist': + + 'lst' + The number of subintervals needed for the integration (call it ``K_f``). + 'rslst' + A rank-1 array of length M_f=limlst, whose first ``K_f`` elements + contain the integral contribution over the interval + ``(a+(k-1)c, a+kc)`` where ``c = (2*floor(|w|) + 1) * pi / |w|`` + and ``k=1,2,...,K_f``. + 'erlst' + A rank-1 array of length ``M_f`` containing the error estimate + corresponding to the interval in the same position in + ``infodict['rslist']``. + 'ierlst' + A rank-1 integer array of length ``M_f`` containing an error flag + corresponding to the interval in the same position in + ``infodict['rslist']``. See the explanation dictionary (last entry + in the output tuple) for the meaning of the codes. + + + **Details of QUADPACK level routines** + + `quad` calls routines from the FORTRAN library QUADPACK. This section + provides details on the conditions for each routine to be called and a + short description of each routine. The routine called depends on + `weight`, `points` and the integration limits `a` and `b`. + + ================ ============== ========== ===================== + QUADPACK routine `weight` `points` infinite bounds + ================ ============== ========== ===================== + qagse None No No + qagie None No Yes + qagpe None Yes No + qawoe 'sin', 'cos' No No + qawfe 'sin', 'cos' No either `a` or `b` + qawse 'alg*' No No + qawce 'cauchy' No No + ================ ============== ========== ===================== + + The following provides a short description from [1]_ for each + routine. + + qagse + is an integrator based on globally adaptive interval + subdivision in connection with extrapolation, which will + eliminate the effects of integrand singularities of + several types. + qagie + handles integration over infinite intervals. The infinite range is + mapped onto a finite interval and subsequently the same strategy as + in ``QAGS`` is applied. + qagpe + serves the same purposes as QAGS, but also allows the + user to provide explicit information about the location + and type of trouble-spots i.e. the abscissae of internal + singularities, discontinuities and other difficulties of + the integrand function. + qawoe + is an integrator for the evaluation of + :math:`\\int^b_a \\cos(\\omega x)f(x)dx` or + :math:`\\int^b_a \\sin(\\omega x)f(x)dx` + over a finite interval [a,b], where :math:`\\omega` and :math:`f` + are specified by the user. The rule evaluation component is based + on the modified Clenshaw-Curtis technique + + An adaptive subdivision scheme is used in connection + with an extrapolation procedure, which is a modification + of that in ``QAGS`` and allows the algorithm to deal with + singularities in :math:`f(x)`. + qawfe + calculates the Fourier transform + :math:`\\int^\\infty_a \\cos(\\omega x)f(x)dx` or + :math:`\\int^\\infty_a \\sin(\\omega x)f(x)dx` + for user-provided :math:`\\omega` and :math:`f`. The procedure of + ``QAWO`` is applied on successive finite intervals, and convergence + acceleration by means of the :math:`\\varepsilon`-algorithm is applied + to the series of integral approximations. + qawse + approximate :math:`\\int^b_a w(x)f(x)dx`, with :math:`a < b` where + :math:`w(x) = (x-a)^{\\alpha}(b-x)^{\\beta}v(x)` with + :math:`\\alpha,\\beta > -1`, where :math:`v(x)` may be one of the + following functions: :math:`1`, :math:`\\log(x-a)`, :math:`\\log(b-x)`, + :math:`\\log(x-a)\\log(b-x)`. + + The user specifies :math:`\\alpha`, :math:`\\beta` and the type of the + function :math:`v`. A globally adaptive subdivision strategy is + applied, with modified Clenshaw-Curtis integration on those + subintervals which contain `a` or `b`. + qawce + compute :math:`\\int^b_a f(x) / (x-c)dx` where the integral must be + interpreted as a Cauchy principal value integral, for user specified + :math:`c` and :math:`f`. The strategy is globally adaptive. Modified + Clenshaw-Curtis integration is used on those intervals containing the + point :math:`x = c`. + + **Integration of Complex Function of a Real Variable** + + A complex valued function, :math:`f`, of a real variable can be written as + :math:`f = g + ih`. Similarly, the integral of :math:`f` can be + written as + + .. math:: + \\int_a^b f(x) dx = \\int_a^b g(x) dx + i\\int_a^b h(x) dx + + assuming that the integrals of :math:`g` and :math:`h` exist + over the interval :math:`[a,b]` [2]_. Therefore, ``quad`` integrates + complex-valued functions by integrating the real and imaginary components + separately. + + + References + ---------- + + .. [1] Piessens, Robert; de Doncker-Kapenga, Elise; + Überhuber, Christoph W.; Kahaner, David (1983). + QUADPACK: A subroutine package for automatic integration. + Springer-Verlag. + ISBN 978-3-540-12553-2. + + .. [2] McCullough, Thomas; Phillips, Keith (1973). + Foundations of Analysis in the Complex Plane. + Holt Rinehart Winston. + ISBN 0-03-086370-8 + + Examples + -------- + Calculate :math:`\\int^4_0 x^2 dx` and compare with an analytic result + + >>> from scipy import integrate + >>> import numpy as np + >>> x2 = lambda x: x**2 + >>> integrate.quad(x2, 0, 4) + (21.333333333333332, 2.3684757858670003e-13) + >>> print(4**3 / 3.) # analytical result + 21.3333333333 + + Calculate :math:`\\int^\\infty_0 e^{-x} dx` + + >>> invexp = lambda x: np.exp(-x) + >>> integrate.quad(invexp, 0, np.inf) + (1.0, 5.842605999138044e-11) + + Calculate :math:`\\int^1_0 a x \\,dx` for :math:`a = 1, 3` + + >>> f = lambda x, a: a*x + >>> y, err = integrate.quad(f, 0, 1, args=(1,)) + >>> y + 0.5 + >>> y, err = integrate.quad(f, 0, 1, args=(3,)) + >>> y + 1.5 + + Calculate :math:`\\int^1_0 x^2 + y^2 dx` with ctypes, holding + y parameter as 1:: + + testlib.c => + double func(int n, double args[n]){ + return args[0]*args[0] + args[1]*args[1];} + compile to library testlib.* + + :: + + from scipy import integrate + import ctypes + lib = ctypes.CDLL('/home/.../testlib.*') #use absolute path + lib.func.restype = ctypes.c_double + lib.func.argtypes = (ctypes.c_int,ctypes.c_double) + integrate.quad(lib.func,0,1,(1)) + #(1.3333333333333333, 1.4802973661668752e-14) + print((1.0**3/3.0 + 1.0) - (0.0**3/3.0 + 0.0)) #Analytic result + # 1.3333333333333333 + + Be aware that pulse shapes and other sharp features as compared to the + size of the integration interval may not be integrated correctly using + this method. A simplified example of this limitation is integrating a + y-axis reflected step function with many zero values within the integrals + bounds. + + >>> y = lambda x: 1 if x<=0 else 0 + >>> integrate.quad(y, -1, 1) + (1.0, 1.1102230246251565e-14) + >>> integrate.quad(y, -1, 100) + (1.0000000002199108, 1.0189464580163188e-08) + >>> integrate.quad(y, -1, 10000) + (0.0, 0.0) + + """ + if not isinstance(args, tuple): + args = (args,) + + # check the limits of integration: \int_a^b, expect a < b + flip, a, b = b < a, min(a, b), max(a, b) + + if complex_func: + def imfunc(x, *args): + return func(x, *args).imag + + def refunc(x, *args): + return func(x, *args).real + + re_retval = quad(refunc, a, b, args, full_output, epsabs, + epsrel, limit, points, weight, wvar, wopts, + maxp1, limlst, complex_func=False) + im_retval = quad(imfunc, a, b, args, full_output, epsabs, + epsrel, limit, points, weight, wvar, wopts, + maxp1, limlst, complex_func=False) + integral = re_retval[0] + 1j*im_retval[0] + error_estimate = re_retval[1] + 1j*im_retval[1] + retval = integral, error_estimate + if full_output: + msgexp = {} + msgexp["real"] = re_retval[2:] + msgexp["imag"] = im_retval[2:] + retval = retval + (msgexp,) + + return retval + + if weight is None: + retval = _quad(func, a, b, args, full_output, epsabs, epsrel, limit, + points) + else: + if points is not None: + msg = ("Break points cannot be specified when using weighted integrand.\n" + "Continuing, ignoring specified points.") + warnings.warn(msg, IntegrationWarning, stacklevel=2) + retval = _quad_weight(func, a, b, args, full_output, epsabs, epsrel, + limlst, limit, maxp1, weight, wvar, wopts) + + if flip: + retval = (-retval[0],) + retval[1:] + + ier = retval[-1] + if ier == 0: + return retval[:-1] + + msgs = {80: "A Python error occurred possibly while calling the function.", + 1: f"The maximum number of subdivisions ({limit}) has been achieved.\n " + f"If increasing the limit yields no improvement it is advised to " + f"analyze \n the integrand in order to determine the difficulties. " + f"If the position of a \n local difficulty can be determined " + f"(singularity, discontinuity) one will \n probably gain from " + f"splitting up the interval and calling the integrator \n on the " + f"subranges. Perhaps a special-purpose integrator should be used.", + 2: "The occurrence of roundoff error is detected, which prevents \n " + "the requested tolerance from being achieved. " + "The error may be \n underestimated.", + 3: "Extremely bad integrand behavior occurs at some points of the\n " + "integration interval.", + 4: "The algorithm does not converge. Roundoff error is detected\n " + "in the extrapolation table. It is assumed that the requested " + "tolerance\n cannot be achieved, and that the returned result " + "(if full_output = 1) is \n the best which can be obtained.", + 5: "The integral is probably divergent, or slowly convergent.", + 6: "The input is invalid.", + 7: "Abnormal termination of the routine. The estimates for result\n " + "and error are less reliable. It is assumed that the requested " + "accuracy\n has not been achieved.", + 'unknown': "Unknown error."} + + if weight in ['cos','sin'] and (b == np.inf or a == -np.inf): + msgs[1] = ( + "The maximum number of cycles allowed has been achieved., e.e.\n of " + "subintervals (a+(k-1)c, a+kc) where c = (2*int(abs(omega)+1))\n " + "*pi/abs(omega), for k = 1, 2, ..., lst. " + "One can allow more cycles by increasing the value of limlst. " + "Look at info['ierlst'] with full_output=1." + ) + msgs[4] = ( + "The extrapolation table constructed for convergence acceleration\n of " + "the series formed by the integral contributions over the cycles, \n does " + "not converge to within the requested accuracy. " + "Look at \n info['ierlst'] with full_output=1." + ) + msgs[7] = ( + "Bad integrand behavior occurs within one or more of the cycles.\n " + "Location and type of the difficulty involved can be determined from \n " + "the vector info['ierlist'] obtained with full_output=1." + ) + explain = {1: "The maximum number of subdivisions (= limit) has been \n " + "achieved on this cycle.", + 2: "The occurrence of roundoff error is detected and prevents\n " + "the tolerance imposed on this cycle from being achieved.", + 3: "Extremely bad integrand behavior occurs at some points of\n " + "this cycle.", + 4: "The integral over this cycle does not converge (to within the " + "required accuracy) due to roundoff in the extrapolation " + "procedure invoked on this cycle. It is assumed that the result " + "on this interval is the best which can be obtained.", + 5: "The integral over this cycle is probably divergent or " + "slowly convergent."} + + try: + msg = msgs[ier] + except KeyError: + msg = msgs['unknown'] + + if ier in [1,2,3,4,5,7]: + if full_output: + if weight in ['cos', 'sin'] and (b == np.inf or a == -np.inf): + return retval[:-1] + (msg, explain) + else: + return retval[:-1] + (msg,) + else: + warnings.warn(msg, IntegrationWarning, stacklevel=2) + return retval[:-1] + + elif ier == 6: # Forensic decision tree when QUADPACK throws ier=6 + if epsabs <= 0: # Small error tolerance - applies to all methods + if epsrel < max(50 * sys.float_info.epsilon, 5e-29): + msg = ("If 'epsabs'<=0, 'epsrel' must be greater than both" + " 5e-29 and 50*(machine epsilon).") + elif weight in ['sin', 'cos'] and (abs(a) + abs(b) == np.inf): + msg = ("Sine or cosine weighted integrals with infinite domain" + " must have 'epsabs'>0.") + + elif weight is None: + if points is None: # QAGSE/QAGIE + msg = ("Invalid 'limit' argument. There must be" + " at least one subinterval") + else: # QAGPE + if not (min(a, b) <= min(points) <= max(points) <= max(a, b)): + msg = ("All break points in 'points' must lie within the" + " integration limits.") + elif len(points) >= limit: + msg = (f"Number of break points ({len(points):d}) " + f"must be less than subinterval limit ({limit:d})") + + else: + if maxp1 < 1: + msg = "Chebyshev moment limit maxp1 must be >=1." + + elif weight in ('cos', 'sin') and abs(a+b) == np.inf: # QAWFE + msg = "Cycle limit limlst must be >=3." + + elif weight.startswith('alg'): # QAWSE + if min(wvar) < -1: + msg = "wvar parameters (alpha, beta) must both be >= -1." + if b < a: + msg = "Integration limits a, b must satistfy a>> import numpy as np + >>> from scipy import integrate + >>> f = lambda y, x: x*y**2 + >>> integrate.dblquad(f, 0, 2, 0, 1) + (0.6666666666666667, 7.401486830834377e-15) + + Calculate :math:`\\int^{x=\\pi/4}_{x=0} \\int^{y=\\cos(x)}_{y=\\sin(x)} 1 + \\,dy \\,dx`. + + >>> f = lambda y, x: 1 + >>> integrate.dblquad(f, 0, np.pi/4, np.sin, np.cos) + (0.41421356237309503, 1.1083280054755938e-14) + + Calculate :math:`\\int^{x=1}_{x=0} \\int^{y=2-x}_{y=x} a x y \\,dy \\,dx` + for :math:`a=1, 3`. + + >>> f = lambda y, x, a: a*x*y + >>> integrate.dblquad(f, 0, 1, lambda x: x, lambda x: 2-x, args=(1,)) + (0.33333333333333337, 5.551115123125783e-15) + >>> integrate.dblquad(f, 0, 1, lambda x: x, lambda x: 2-x, args=(3,)) + (0.9999999999999999, 1.6653345369377348e-14) + + Compute the two-dimensional Gaussian Integral, which is the integral of the + Gaussian function :math:`f(x,y) = e^{-(x^{2} + y^{2})}`, over + :math:`(-\\infty,+\\infty)`. That is, compute the integral + :math:`\\iint^{+\\infty}_{-\\infty} e^{-(x^{2} + y^{2})} \\,dy\\,dx`. + + >>> f = lambda x, y: np.exp(-(x ** 2 + y ** 2)) + >>> integrate.dblquad(f, -np.inf, np.inf, -np.inf, np.inf) + (3.141592653589777, 2.5173086737433208e-08) + + """ + + def temp_ranges(*args): + return [gfun(args[0]) if callable(gfun) else gfun, + hfun(args[0]) if callable(hfun) else hfun] + + return nquad(func, [temp_ranges, [a, b]], args=args, + opts={"epsabs": epsabs, "epsrel": epsrel}) + + +def tplquad(func, a, b, gfun, hfun, qfun, rfun, args=(), epsabs=1.49e-8, + epsrel=1.49e-8): + """ + Compute a triple (definite) integral. + + Return the triple integral of ``func(z, y, x)`` from ``x = a..b``, + ``y = gfun(x)..hfun(x)``, and ``z = qfun(x,y)..rfun(x,y)``. + + Parameters + ---------- + func : function + A Python function or method of at least three variables in the + order (z, y, x). + a, b : float + The limits of integration in x: `a` < `b` + gfun : function or float + The lower boundary curve in y which is a function taking a single + floating point argument (x) and returning a floating point result + or a float indicating a constant boundary curve. + hfun : function or float + The upper boundary curve in y (same requirements as `gfun`). + qfun : function or float + The lower boundary surface in z. It must be a function that takes + two floats in the order (x, y) and returns a float or a float + indicating a constant boundary surface. + rfun : function or float + The upper boundary surface in z. (Same requirements as `qfun`.) + args : tuple, optional + Extra arguments to pass to `func`. + epsabs : float, optional + Absolute tolerance passed directly to the innermost 1-D quadrature + integration. Default is 1.49e-8. + epsrel : float, optional + Relative tolerance of the innermost 1-D integrals. Default is 1.49e-8. + + Returns + ------- + y : float + The resultant integral. + abserr : float + An estimate of the error. + + See Also + -------- + quad : Adaptive quadrature using QUADPACK + fixed_quad : Fixed-order Gaussian quadrature + dblquad : Double integrals + nquad : N-dimensional integrals + romb : Integrators for sampled data + simpson : Integrators for sampled data + scipy.special : For coefficients and roots of orthogonal polynomials + + Notes + ----- + For valid results, the integral must converge; behavior for divergent + integrals is not guaranteed. + + **Details of QUADPACK level routines** + + `quad` calls routines from the FORTRAN library QUADPACK. This section + provides details on the conditions for each routine to be called and a + short description of each routine. For each level of integration, ``qagse`` + is used for finite limits or ``qagie`` is used, if either limit (or both!) + are infinite. The following provides a short description from [1]_ for each + routine. + + qagse + is an integrator based on globally adaptive interval + subdivision in connection with extrapolation, which will + eliminate the effects of integrand singularities of + several types. + qagie + handles integration over infinite intervals. The infinite range is + mapped onto a finite interval and subsequently the same strategy as + in ``QAGS`` is applied. + + References + ---------- + + .. [1] Piessens, Robert; de Doncker-Kapenga, Elise; + Überhuber, Christoph W.; Kahaner, David (1983). + QUADPACK: A subroutine package for automatic integration. + Springer-Verlag. + ISBN 978-3-540-12553-2. + + Examples + -------- + Compute the triple integral of ``x * y * z``, over ``x`` ranging + from 1 to 2, ``y`` ranging from 2 to 3, ``z`` ranging from 0 to 1. + That is, :math:`\\int^{x=2}_{x=1} \\int^{y=3}_{y=2} \\int^{z=1}_{z=0} x y z + \\,dz \\,dy \\,dx`. + + >>> import numpy as np + >>> from scipy import integrate + >>> f = lambda z, y, x: x*y*z + >>> integrate.tplquad(f, 1, 2, 2, 3, 0, 1) + (1.8749999999999998, 3.3246447942574074e-14) + + Calculate :math:`\\int^{x=1}_{x=0} \\int^{y=1-2x}_{y=0} + \\int^{z=1-x-2y}_{z=0} x y z \\,dz \\,dy \\,dx`. + Note: `qfun`/`rfun` takes arguments in the order (x, y), even though ``f`` + takes arguments in the order (z, y, x). + + >>> f = lambda z, y, x: x*y*z + >>> integrate.tplquad(f, 0, 1, 0, lambda x: 1-2*x, 0, lambda x, y: 1-x-2*y) + (0.05416666666666668, 2.1774196738157757e-14) + + Calculate :math:`\\int^{x=1}_{x=0} \\int^{y=1}_{y=0} \\int^{z=1}_{z=0} + a x y z \\,dz \\,dy \\,dx` for :math:`a=1, 3`. + + >>> f = lambda z, y, x, a: a*x*y*z + >>> integrate.tplquad(f, 0, 1, 0, 1, 0, 1, args=(1,)) + (0.125, 5.527033708952211e-15) + >>> integrate.tplquad(f, 0, 1, 0, 1, 0, 1, args=(3,)) + (0.375, 1.6581101126856635e-14) + + Compute the three-dimensional Gaussian Integral, which is the integral of + the Gaussian function :math:`f(x,y,z) = e^{-(x^{2} + y^{2} + z^{2})}`, over + :math:`(-\\infty,+\\infty)`. That is, compute the integral + :math:`\\iiint^{+\\infty}_{-\\infty} e^{-(x^{2} + y^{2} + z^{2})} \\,dz + \\,dy\\,dx`. + + >>> f = lambda x, y, z: np.exp(-(x ** 2 + y ** 2 + z ** 2)) + >>> integrate.tplquad(f, -np.inf, np.inf, -np.inf, np.inf, -np.inf, np.inf) + (5.568327996830833, 4.4619078828029765e-08) + + """ + # f(z, y, x) + # qfun/rfun(x, y) + # gfun/hfun(x) + # nquad will hand (y, x, t0, ...) to ranges0 + # nquad will hand (x, t0, ...) to ranges1 + # Only qfun / rfun is different API... + + def ranges0(*args): + return [qfun(args[1], args[0]) if callable(qfun) else qfun, + rfun(args[1], args[0]) if callable(rfun) else rfun] + + def ranges1(*args): + return [gfun(args[0]) if callable(gfun) else gfun, + hfun(args[0]) if callable(hfun) else hfun] + + ranges = [ranges0, ranges1, [a, b]] + return nquad(func, ranges, args=args, + opts={"epsabs": epsabs, "epsrel": epsrel}) + + +def nquad(func, ranges, args=None, opts=None, full_output=False): + r""" + Integration over multiple variables. + + Wraps `quad` to enable integration over multiple variables. + Various options allow improved integration of discontinuous functions, as + well as the use of weighted integration, and generally finer control of the + integration process. + + Parameters + ---------- + func : {callable, scipy.LowLevelCallable} + The function to be integrated. Has arguments of ``x0, ... xn``, + ``t0, ... tm``, where integration is carried out over ``x0, ... xn``, + which must be floats. Where ``t0, ... tm`` are extra arguments + passed in args. + Function signature should be ``func(x0, x1, ..., xn, t0, t1, ..., tm)``. + Integration is carried out in order. That is, integration over ``x0`` + is the innermost integral, and ``xn`` is the outermost. + + If the user desires improved integration performance, then `f` may + be a `scipy.LowLevelCallable` with one of the signatures:: + + double func(int n, double *xx) + double func(int n, double *xx, void *user_data) + + where ``n`` is the number of variables and args. The ``xx`` array + contains the coordinates and extra arguments. ``user_data`` is the data + contained in the `scipy.LowLevelCallable`. + ranges : iterable object + Each element of ranges may be either a sequence of 2 numbers, or else + a callable that returns such a sequence. ``ranges[0]`` corresponds to + integration over x0, and so on. If an element of ranges is a callable, + then it will be called with all of the integration arguments available, + as well as any parametric arguments. e.g., if + ``func = f(x0, x1, x2, t0, t1)``, then ``ranges[0]`` may be defined as + either ``(a, b)`` or else as ``(a, b) = range0(x1, x2, t0, t1)``. + args : iterable object, optional + Additional arguments ``t0, ... tn``, required by ``func``, ``ranges``, + and ``opts``. + opts : iterable object or dict, optional + Options to be passed to `quad`. May be empty, a dict, or + a sequence of dicts or functions that return a dict. If empty, the + default options from scipy.integrate.quad are used. If a dict, the same + options are used for all levels of integraion. If a sequence, then each + element of the sequence corresponds to a particular integration. e.g., + ``opts[0]`` corresponds to integration over ``x0``, and so on. If a + callable, the signature must be the same as for ``ranges``. The + available options together with their default values are: + + - epsabs = 1.49e-08 + - epsrel = 1.49e-08 + - limit = 50 + - points = None + - weight = None + - wvar = None + - wopts = None + + For more information on these options, see `quad`. + + full_output : bool, optional + Partial implementation of ``full_output`` from scipy.integrate.quad. + The number of integrand function evaluations ``neval`` can be obtained + by setting ``full_output=True`` when calling nquad. + + Returns + ------- + result : float + The result of the integration. + abserr : float + The maximum of the estimates of the absolute error in the various + integration results. + out_dict : dict, optional + A dict containing additional information on the integration. + + See Also + -------- + quad : 1-D numerical integration + dblquad, tplquad : double and triple integrals + fixed_quad : fixed-order Gaussian quadrature + + Notes + ----- + For valid results, the integral must converge; behavior for divergent + integrals is not guaranteed. + + **Details of QUADPACK level routines** + + `nquad` calls routines from the FORTRAN library QUADPACK. This section + provides details on the conditions for each routine to be called and a + short description of each routine. The routine called depends on + `weight`, `points` and the integration limits `a` and `b`. + + ================ ============== ========== ===================== + QUADPACK routine `weight` `points` infinite bounds + ================ ============== ========== ===================== + qagse None No No + qagie None No Yes + qagpe None Yes No + qawoe 'sin', 'cos' No No + qawfe 'sin', 'cos' No either `a` or `b` + qawse 'alg*' No No + qawce 'cauchy' No No + ================ ============== ========== ===================== + + The following provides a short description from [1]_ for each + routine. + + qagse + is an integrator based on globally adaptive interval + subdivision in connection with extrapolation, which will + eliminate the effects of integrand singularities of + several types. + qagie + handles integration over infinite intervals. The infinite range is + mapped onto a finite interval and subsequently the same strategy as + in ``QAGS`` is applied. + qagpe + serves the same purposes as QAGS, but also allows the + user to provide explicit information about the location + and type of trouble-spots i.e. the abscissae of internal + singularities, discontinuities and other difficulties of + the integrand function. + qawoe + is an integrator for the evaluation of + :math:`\int^b_a \cos(\omega x)f(x)dx` or + :math:`\int^b_a \sin(\omega x)f(x)dx` + over a finite interval [a,b], where :math:`\omega` and :math:`f` + are specified by the user. The rule evaluation component is based + on the modified Clenshaw-Curtis technique + + An adaptive subdivision scheme is used in connection + with an extrapolation procedure, which is a modification + of that in ``QAGS`` and allows the algorithm to deal with + singularities in :math:`f(x)`. + qawfe + calculates the Fourier transform + :math:`\int^\infty_a \cos(\omega x)f(x)dx` or + :math:`\int^\infty_a \sin(\omega x)f(x)dx` + for user-provided :math:`\omega` and :math:`f`. The procedure of + ``QAWO`` is applied on successive finite intervals, and convergence + acceleration by means of the :math:`\varepsilon`-algorithm is applied + to the series of integral approximations. + qawse + approximate :math:`\int^b_a w(x)f(x)dx`, with :math:`a < b` where + :math:`w(x) = (x-a)^{\alpha}(b-x)^{\beta}v(x)` with + :math:`\alpha,\beta > -1`, where :math:`v(x)` may be one of the + following functions: :math:`1`, :math:`\log(x-a)`, :math:`\log(b-x)`, + :math:`\log(x-a)\log(b-x)`. + + The user specifies :math:`\alpha`, :math:`\beta` and the type of the + function :math:`v`. A globally adaptive subdivision strategy is + applied, with modified Clenshaw-Curtis integration on those + subintervals which contain `a` or `b`. + qawce + compute :math:`\int^b_a f(x) / (x-c)dx` where the integral must be + interpreted as a Cauchy principal value integral, for user specified + :math:`c` and :math:`f`. The strategy is globally adaptive. Modified + Clenshaw-Curtis integration is used on those intervals containing the + point :math:`x = c`. + + References + ---------- + + .. [1] Piessens, Robert; de Doncker-Kapenga, Elise; + Überhuber, Christoph W.; Kahaner, David (1983). + QUADPACK: A subroutine package for automatic integration. + Springer-Verlag. + ISBN 978-3-540-12553-2. + + Examples + -------- + Calculate + + .. math:: + + \int^{1}_{-0.15} \int^{0.8}_{0.13} \int^{1}_{-1} \int^{1}_{0} + f(x_0, x_1, x_2, x_3) \,dx_0 \,dx_1 \,dx_2 \,dx_3 , + + where + + .. math:: + + f(x_0, x_1, x_2, x_3) = \begin{cases} + x_0^2+x_1 x_2-x_3^3+ \sin{x_0}+1 & (x_0-0.2 x_3-0.5-0.25 x_1 > 0) \\ + x_0^2+x_1 x_2-x_3^3+ \sin{x_0}+0 & (x_0-0.2 x_3-0.5-0.25 x_1 \leq 0) + \end{cases} . + + >>> import numpy as np + >>> from scipy import integrate + >>> func = lambda x0,x1,x2,x3 : x0**2 + x1*x2 - x3**3 + np.sin(x0) + ( + ... 1 if (x0-.2*x3-.5-.25*x1>0) else 0) + >>> def opts0(*args, **kwargs): + ... return {'points':[0.2*args[2] + 0.5 + 0.25*args[0]]} + >>> integrate.nquad(func, [[0,1], [-1,1], [.13,.8], [-.15,1]], + ... opts=[opts0,{},{},{}], full_output=True) + (1.5267454070738633, 2.9437360001402324e-14, {'neval': 388962}) + + Calculate + + .. math:: + + \int^{t_0+t_1+1}_{t_0+t_1-1} + \int^{x_2+t_0^2 t_1^3+1}_{x_2+t_0^2 t_1^3-1} + \int^{t_0 x_1+t_1 x_2+1}_{t_0 x_1+t_1 x_2-1} + f(x_0,x_1, x_2,t_0,t_1) + \,dx_0 \,dx_1 \,dx_2, + + where + + .. math:: + + f(x_0, x_1, x_2, t_0, t_1) = \begin{cases} + x_0 x_2^2 + \sin{x_1}+2 & (x_0+t_1 x_1-t_0 > 0) \\ + x_0 x_2^2 +\sin{x_1}+1 & (x_0+t_1 x_1-t_0 \leq 0) + \end{cases} + + and :math:`(t_0, t_1) = (0, 1)` . + + >>> def func2(x0, x1, x2, t0, t1): + ... return x0*x2**2 + np.sin(x1) + 1 + (1 if x0+t1*x1-t0>0 else 0) + >>> def lim0(x1, x2, t0, t1): + ... return [t0*x1 + t1*x2 - 1, t0*x1 + t1*x2 + 1] + >>> def lim1(x2, t0, t1): + ... return [x2 + t0**2*t1**3 - 1, x2 + t0**2*t1**3 + 1] + >>> def lim2(t0, t1): + ... return [t0 + t1 - 1, t0 + t1 + 1] + >>> def opts0(x1, x2, t0, t1): + ... return {'points' : [t0 - t1*x1]} + >>> def opts1(x2, t0, t1): + ... return {} + >>> def opts2(t0, t1): + ... return {} + >>> integrate.nquad(func2, [lim0, lim1, lim2], args=(0,1), + ... opts=[opts0, opts1, opts2]) + (36.099919226771625, 1.8546948553373528e-07) + + """ + depth = len(ranges) + ranges = [rng if callable(rng) else _RangeFunc(rng) for rng in ranges] + if args is None: + args = () + if opts is None: + opts = [dict([])] * depth + + if isinstance(opts, dict): + opts = [_OptFunc(opts)] * depth + else: + opts = [opt if callable(opt) else _OptFunc(opt) for opt in opts] + return _NQuad(func, ranges, opts, full_output).integrate(*args) + + +class _RangeFunc: + def __init__(self, range_): + self.range_ = range_ + + def __call__(self, *args): + """Return stored value. + + *args needed because range_ can be float or func, and is called with + variable number of parameters. + """ + return self.range_ + + +class _OptFunc: + def __init__(self, opt): + self.opt = opt + + def __call__(self, *args): + """Return stored dict.""" + return self.opt + + +class _NQuad: + def __init__(self, func, ranges, opts, full_output): + self.abserr = 0 + self.func = func + self.ranges = ranges + self.opts = opts + self.maxdepth = len(ranges) + self.full_output = full_output + if self.full_output: + self.out_dict = {'neval': 0} + + def integrate(self, *args, **kwargs): + depth = kwargs.pop('depth', 0) + if kwargs: + raise ValueError('unexpected kwargs') + + # Get the integration range and options for this depth. + ind = -(depth + 1) + fn_range = self.ranges[ind] + low, high = fn_range(*args) + fn_opt = self.opts[ind] + opt = dict(fn_opt(*args)) + + if 'points' in opt: + opt['points'] = [x for x in opt['points'] if low <= x <= high] + if depth + 1 == self.maxdepth: + f = self.func + else: + f = partial(self.integrate, depth=depth+1) + quad_r = quad(f, low, high, args=args, full_output=self.full_output, + **opt) + value = quad_r[0] + abserr = quad_r[1] + if self.full_output: + infodict = quad_r[2] + # The 'neval' parameter in full_output returns the total + # number of times the integrand function was evaluated. + # Therefore, only the innermost integration loop counts. + if depth + 1 == self.maxdepth: + self.out_dict['neval'] += infodict['neval'] + self.abserr = max(self.abserr, abserr) + if depth > 0: + return value + else: + # Final result of N-D integration with error + if self.full_output: + return value, self.abserr, self.out_dict + else: + return value, self.abserr diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_quadrature.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_quadrature.py new file mode 100644 index 0000000000000000000000000000000000000000..44cf10b32335014cd7cb26459c8dc89ac8f851ff --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_quadrature.py @@ -0,0 +1,1336 @@ +import numpy as np +import numpy.typing as npt +import math +import warnings +from collections import namedtuple +from collections.abc import Callable + +from scipy.special import roots_legendre +from scipy.special import gammaln, logsumexp +from scipy._lib._util import _rng_spawn +from scipy._lib._array_api import _asarray, array_namespace, xp_broadcast_promote + + +__all__ = ['fixed_quad', 'romb', + 'trapezoid', 'simpson', + 'cumulative_trapezoid', 'newton_cotes', + 'qmc_quad', 'cumulative_simpson'] + + +def trapezoid(y, x=None, dx=1.0, axis=-1): + r""" + Integrate along the given axis using the composite trapezoidal rule. + + If `x` is provided, the integration happens in sequence along its + elements - they are not sorted. + + Integrate `y` (`x`) along each 1d slice on the given axis, compute + :math:`\int y(x) dx`. + When `x` is specified, this integrates along the parametric curve, + computing :math:`\int_t y(t) dt = + \int_t y(t) \left.\frac{dx}{dt}\right|_{x=x(t)} dt`. + + Parameters + ---------- + y : array_like + Input array to integrate. + x : array_like, optional + The sample points corresponding to the `y` values. If `x` is None, + the sample points are assumed to be evenly spaced `dx` apart. The + default is None. + dx : scalar, optional + The spacing between sample points when `x` is None. The default is 1. + axis : int, optional + The axis along which to integrate. The default is the last axis. + + Returns + ------- + trapezoid : float or ndarray + Definite integral of `y` = n-dimensional array as approximated along + a single axis by the trapezoidal rule. If `y` is a 1-dimensional array, + then the result is a float. If `n` is greater than 1, then the result + is an `n`-1 dimensional array. + + See Also + -------- + cumulative_trapezoid, simpson, romb + + Notes + ----- + Image [2]_ illustrates trapezoidal rule -- y-axis locations of points + will be taken from `y` array, by default x-axis distances between + points will be 1.0, alternatively they can be provided with `x` array + or with `dx` scalar. Return value will be equal to combined area under + the red lines. + + References + ---------- + .. [1] Wikipedia page: https://en.wikipedia.org/wiki/Trapezoidal_rule + + .. [2] Illustration image: + https://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png + + Examples + -------- + Use the trapezoidal rule on evenly spaced points: + + >>> import numpy as np + >>> from scipy import integrate + >>> integrate.trapezoid([1, 2, 3]) + 4.0 + + The spacing between sample points can be selected by either the + ``x`` or ``dx`` arguments: + + >>> integrate.trapezoid([1, 2, 3], x=[4, 6, 8]) + 8.0 + >>> integrate.trapezoid([1, 2, 3], dx=2) + 8.0 + + Using a decreasing ``x`` corresponds to integrating in reverse: + + >>> integrate.trapezoid([1, 2, 3], x=[8, 6, 4]) + -8.0 + + More generally ``x`` is used to integrate along a parametric curve. We can + estimate the integral :math:`\int_0^1 x^2 = 1/3` using: + + >>> x = np.linspace(0, 1, num=50) + >>> y = x**2 + >>> integrate.trapezoid(y, x) + 0.33340274885464394 + + Or estimate the area of a circle, noting we repeat the sample which closes + the curve: + + >>> theta = np.linspace(0, 2 * np.pi, num=1000, endpoint=True) + >>> integrate.trapezoid(np.cos(theta), x=np.sin(theta)) + 3.141571941375841 + + ``trapezoid`` can be applied along a specified axis to do multiple + computations in one call: + + >>> a = np.arange(6).reshape(2, 3) + >>> a + array([[0, 1, 2], + [3, 4, 5]]) + >>> integrate.trapezoid(a, axis=0) + array([1.5, 2.5, 3.5]) + >>> integrate.trapezoid(a, axis=1) + array([2., 8.]) + """ + xp = array_namespace(y) + y = _asarray(y, xp=xp, subok=True) + # Cannot just use the broadcasted arrays that are returned + # because trapezoid does not follow normal broadcasting rules + # cf. https://github.com/scipy/scipy/pull/21524#issuecomment-2354105942 + result_dtype = xp_broadcast_promote(y, force_floating=True, xp=xp)[0].dtype + nd = y.ndim + slice1 = [slice(None)]*nd + slice2 = [slice(None)]*nd + slice1[axis] = slice(1, None) + slice2[axis] = slice(None, -1) + if x is None: + d = dx + else: + x = _asarray(x, xp=xp, subok=True) + if x.ndim == 1: + d = x[1:] - x[:-1] + # make d broadcastable to y + slice3 = [None] * nd + slice3[axis] = slice(None) + d = d[tuple(slice3)] + else: + # if x is n-D it should be broadcastable to y + x = xp.broadcast_to(x, y.shape) + d = x[tuple(slice1)] - x[tuple(slice2)] + try: + ret = xp.sum( + d * (y[tuple(slice1)] + y[tuple(slice2)]) / 2.0, + axis=axis, dtype=result_dtype + ) + except ValueError: + # Operations didn't work, cast to ndarray + d = xp.asarray(d) + y = xp.asarray(y) + ret = xp.sum( + d * (y[tuple(slice1)] + y[tuple(slice2)]) / 2.0, + axis=axis, dtype=result_dtype + ) + return ret + + +def _cached_roots_legendre(n): + """ + Cache roots_legendre results to speed up calls of the fixed_quad + function. + """ + if n in _cached_roots_legendre.cache: + return _cached_roots_legendre.cache[n] + + _cached_roots_legendre.cache[n] = roots_legendre(n) + return _cached_roots_legendre.cache[n] + + +_cached_roots_legendre.cache = dict() + + +def fixed_quad(func, a, b, args=(), n=5): + """ + Compute a definite integral using fixed-order Gaussian quadrature. + + Integrate `func` from `a` to `b` using Gaussian quadrature of + order `n`. + + Parameters + ---------- + func : callable + A Python function or method to integrate (must accept vector inputs). + If integrating a vector-valued function, the returned array must have + shape ``(..., len(x))``. + a : float + Lower limit of integration. + b : float + Upper limit of integration. + args : tuple, optional + Extra arguments to pass to function, if any. + n : int, optional + Order of quadrature integration. Default is 5. + + Returns + ------- + val : float + Gaussian quadrature approximation to the integral + none : None + Statically returned value of None + + See Also + -------- + quad : adaptive quadrature using QUADPACK + dblquad : double integrals + tplquad : triple integrals + romb : integrators for sampled data + simpson : integrators for sampled data + cumulative_trapezoid : cumulative integration for sampled data + + Examples + -------- + >>> from scipy import integrate + >>> import numpy as np + >>> f = lambda x: x**8 + >>> integrate.fixed_quad(f, 0.0, 1.0, n=4) + (0.1110884353741496, None) + >>> integrate.fixed_quad(f, 0.0, 1.0, n=5) + (0.11111111111111102, None) + >>> print(1/9.0) # analytical result + 0.1111111111111111 + + >>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=4) + (0.9999999771971152, None) + >>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=5) + (1.000000000039565, None) + >>> np.sin(np.pi/2)-np.sin(0) # analytical result + 1.0 + + """ + x, w = _cached_roots_legendre(n) + x = np.real(x) + if np.isinf(a) or np.isinf(b): + raise ValueError("Gaussian quadrature is only available for " + "finite limits.") + y = (b-a)*(x+1)/2.0 + a + return (b-a)/2.0 * np.sum(w*func(y, *args), axis=-1), None + + +def tupleset(t, i, value): + l = list(t) + l[i] = value + return tuple(l) + + +def cumulative_trapezoid(y, x=None, dx=1.0, axis=-1, initial=None): + """ + Cumulatively integrate y(x) using the composite trapezoidal rule. + + Parameters + ---------- + y : array_like + Values to integrate. + x : array_like, optional + The coordinate to integrate along. If None (default), use spacing `dx` + between consecutive elements in `y`. + dx : float, optional + Spacing between elements of `y`. Only used if `x` is None. + axis : int, optional + Specifies the axis to cumulate. Default is -1 (last axis). + initial : scalar, optional + If given, insert this value at the beginning of the returned result. + 0 or None are the only values accepted. Default is None, which means + `res` has one element less than `y` along the axis of integration. + + Returns + ------- + res : ndarray + The result of cumulative integration of `y` along `axis`. + If `initial` is None, the shape is such that the axis of integration + has one less value than `y`. If `initial` is given, the shape is equal + to that of `y`. + + See Also + -------- + numpy.cumsum, numpy.cumprod + cumulative_simpson : cumulative integration using Simpson's 1/3 rule + quad : adaptive quadrature using QUADPACK + fixed_quad : fixed-order Gaussian quadrature + dblquad : double integrals + tplquad : triple integrals + romb : integrators for sampled data + + Examples + -------- + >>> from scipy import integrate + >>> import numpy as np + >>> import matplotlib.pyplot as plt + + >>> x = np.linspace(-2, 2, num=20) + >>> y = x + >>> y_int = integrate.cumulative_trapezoid(y, x, initial=0) + >>> plt.plot(x, y_int, 'ro', x, y[0] + 0.5 * x**2, 'b-') + >>> plt.show() + + """ + y = np.asarray(y) + if y.shape[axis] == 0: + raise ValueError("At least one point is required along `axis`.") + if x is None: + d = dx + else: + x = np.asarray(x) + if x.ndim == 1: + d = np.diff(x) + # reshape to correct shape + shape = [1] * y.ndim + shape[axis] = -1 + d = d.reshape(shape) + elif len(x.shape) != len(y.shape): + raise ValueError("If given, shape of x must be 1-D or the " + "same as y.") + else: + d = np.diff(x, axis=axis) + + if d.shape[axis] != y.shape[axis] - 1: + raise ValueError("If given, length of x along axis must be the " + "same as y.") + + nd = len(y.shape) + slice1 = tupleset((slice(None),)*nd, axis, slice(1, None)) + slice2 = tupleset((slice(None),)*nd, axis, slice(None, -1)) + res = np.cumsum(d * (y[slice1] + y[slice2]) / 2.0, axis=axis) + + if initial is not None: + if initial != 0: + raise ValueError("`initial` must be `None` or `0`.") + if not np.isscalar(initial): + raise ValueError("`initial` parameter should be a scalar.") + + shape = list(res.shape) + shape[axis] = 1 + res = np.concatenate([np.full(shape, initial, dtype=res.dtype), res], + axis=axis) + + return res + + +def _basic_simpson(y, start, stop, x, dx, axis): + nd = len(y.shape) + if start is None: + start = 0 + step = 2 + slice_all = (slice(None),)*nd + slice0 = tupleset(slice_all, axis, slice(start, stop, step)) + slice1 = tupleset(slice_all, axis, slice(start+1, stop+1, step)) + slice2 = tupleset(slice_all, axis, slice(start+2, stop+2, step)) + + if x is None: # Even-spaced Simpson's rule. + result = np.sum(y[slice0] + 4.0*y[slice1] + y[slice2], axis=axis) + result *= dx / 3.0 + else: + # Account for possibly different spacings. + # Simpson's rule changes a bit. + h = np.diff(x, axis=axis) + sl0 = tupleset(slice_all, axis, slice(start, stop, step)) + sl1 = tupleset(slice_all, axis, slice(start+1, stop+1, step)) + h0 = h[sl0].astype(float, copy=False) + h1 = h[sl1].astype(float, copy=False) + hsum = h0 + h1 + hprod = h0 * h1 + h0divh1 = np.true_divide(h0, h1, out=np.zeros_like(h0), where=h1 != 0) + tmp = hsum/6.0 * (y[slice0] * + (2.0 - np.true_divide(1.0, h0divh1, + out=np.zeros_like(h0divh1), + where=h0divh1 != 0)) + + y[slice1] * (hsum * + np.true_divide(hsum, hprod, + out=np.zeros_like(hsum), + where=hprod != 0)) + + y[slice2] * (2.0 - h0divh1)) + result = np.sum(tmp, axis=axis) + return result + + +def simpson(y, x=None, *, dx=1.0, axis=-1): + """ + Integrate y(x) using samples along the given axis and the composite + Simpson's rule. If x is None, spacing of dx is assumed. + + Parameters + ---------- + y : array_like + Array to be integrated. + x : array_like, optional + If given, the points at which `y` is sampled. + dx : float, optional + Spacing of integration points along axis of `x`. Only used when + `x` is None. Default is 1. + axis : int, optional + Axis along which to integrate. Default is the last axis. + + Returns + ------- + float + The estimated integral computed with the composite Simpson's rule. + + See Also + -------- + quad : adaptive quadrature using QUADPACK + fixed_quad : fixed-order Gaussian quadrature + dblquad : double integrals + tplquad : triple integrals + romb : integrators for sampled data + cumulative_trapezoid : cumulative integration for sampled data + cumulative_simpson : cumulative integration using Simpson's 1/3 rule + + Notes + ----- + For an odd number of samples that are equally spaced the result is + exact if the function is a polynomial of order 3 or less. If + the samples are not equally spaced, then the result is exact only + if the function is a polynomial of order 2 or less. + + References + ---------- + .. [1] Cartwright, Kenneth V. Simpson's Rule Cumulative Integration with + MS Excel and Irregularly-spaced Data. Journal of Mathematical + Sciences and Mathematics Education. 12 (2): 1-9 + + Examples + -------- + >>> from scipy import integrate + >>> import numpy as np + >>> x = np.arange(0, 10) + >>> y = np.arange(0, 10) + + >>> integrate.simpson(y, x=x) + 40.5 + + >>> y = np.power(x, 3) + >>> integrate.simpson(y, x=x) + 1640.5 + >>> integrate.quad(lambda x: x**3, 0, 9)[0] + 1640.25 + + """ + y = np.asarray(y) + nd = len(y.shape) + N = y.shape[axis] + last_dx = dx + returnshape = 0 + if x is not None: + x = np.asarray(x) + if len(x.shape) == 1: + shapex = [1] * nd + shapex[axis] = x.shape[0] + saveshape = x.shape + returnshape = 1 + x = x.reshape(tuple(shapex)) + elif len(x.shape) != len(y.shape): + raise ValueError("If given, shape of x must be 1-D or the " + "same as y.") + if x.shape[axis] != N: + raise ValueError("If given, length of x along axis must be the " + "same as y.") + + if N % 2 == 0: + val = 0.0 + result = 0.0 + slice_all = (slice(None),) * nd + + if N == 2: + # need at least 3 points in integration axis to form parabolic + # segment. If there are two points then any of 'avg', 'first', + # 'last' should give the same result. + slice1 = tupleset(slice_all, axis, -1) + slice2 = tupleset(slice_all, axis, -2) + if x is not None: + last_dx = x[slice1] - x[slice2] + val += 0.5 * last_dx * (y[slice1] + y[slice2]) + else: + # use Simpson's rule on first intervals + result = _basic_simpson(y, 0, N-3, x, dx, axis) + + slice1 = tupleset(slice_all, axis, -1) + slice2 = tupleset(slice_all, axis, -2) + slice3 = tupleset(slice_all, axis, -3) + + h = np.asarray([dx, dx], dtype=np.float64) + if x is not None: + # grab the last two spacings from the appropriate axis + hm2 = tupleset(slice_all, axis, slice(-2, -1, 1)) + hm1 = tupleset(slice_all, axis, slice(-1, None, 1)) + + diffs = np.float64(np.diff(x, axis=axis)) + h = [np.squeeze(diffs[hm2], axis=axis), + np.squeeze(diffs[hm1], axis=axis)] + + # This is the correction for the last interval according to + # Cartwright. + # However, I used the equations given at + # https://en.wikipedia.org/wiki/Simpson%27s_rule#Composite_Simpson's_rule_for_irregularly_spaced_data + # A footnote on Wikipedia says: + # Cartwright 2017, Equation 8. The equation in Cartwright is + # calculating the first interval whereas the equations in the + # Wikipedia article are adjusting for the last integral. If the + # proper algebraic substitutions are made, the equation results in + # the values shown. + num = 2 * h[1] ** 2 + 3 * h[0] * h[1] + den = 6 * (h[1] + h[0]) + alpha = np.true_divide( + num, + den, + out=np.zeros_like(den), + where=den != 0 + ) + + num = h[1] ** 2 + 3.0 * h[0] * h[1] + den = 6 * h[0] + beta = np.true_divide( + num, + den, + out=np.zeros_like(den), + where=den != 0 + ) + + num = 1 * h[1] ** 3 + den = 6 * h[0] * (h[0] + h[1]) + eta = np.true_divide( + num, + den, + out=np.zeros_like(den), + where=den != 0 + ) + + result += alpha*y[slice1] + beta*y[slice2] - eta*y[slice3] + + result += val + else: + result = _basic_simpson(y, 0, N-2, x, dx, axis) + if returnshape: + x = x.reshape(saveshape) + return result + + +def _cumulatively_sum_simpson_integrals( + y: np.ndarray, + dx: np.ndarray, + integration_func: Callable[[np.ndarray, np.ndarray], np.ndarray], +) -> np.ndarray: + """Calculate cumulative sum of Simpson integrals. + Takes as input the integration function to be used. + The integration_func is assumed to return the cumulative sum using + composite Simpson's rule. Assumes the axis of summation is -1. + """ + sub_integrals_h1 = integration_func(y, dx) + sub_integrals_h2 = integration_func(y[..., ::-1], dx[..., ::-1])[..., ::-1] + + shape = list(sub_integrals_h1.shape) + shape[-1] += 1 + sub_integrals = np.empty(shape) + sub_integrals[..., :-1:2] = sub_integrals_h1[..., ::2] + sub_integrals[..., 1::2] = sub_integrals_h2[..., ::2] + # Integral over last subinterval can only be calculated from + # formula for h2 + sub_integrals[..., -1] = sub_integrals_h2[..., -1] + res = np.cumsum(sub_integrals, axis=-1) + return res + + +def _cumulative_simpson_equal_intervals(y: np.ndarray, dx: np.ndarray) -> np.ndarray: + """Calculate the Simpson integrals for all h1 intervals assuming equal interval + widths. The function can also be used to calculate the integral for all + h2 intervals by reversing the inputs, `y` and `dx`. + """ + d = dx[..., :-1] + f1 = y[..., :-2] + f2 = y[..., 1:-1] + f3 = y[..., 2:] + + # Calculate integral over the subintervals (eqn (10) of Reference [2]) + return d / 3 * (5 * f1 / 4 + 2 * f2 - f3 / 4) + + +def _cumulative_simpson_unequal_intervals(y: np.ndarray, dx: np.ndarray) -> np.ndarray: + """Calculate the Simpson integrals for all h1 intervals assuming unequal interval + widths. The function can also be used to calculate the integral for all + h2 intervals by reversing the inputs, `y` and `dx`. + """ + x21 = dx[..., :-1] + x32 = dx[..., 1:] + f1 = y[..., :-2] + f2 = y[..., 1:-1] + f3 = y[..., 2:] + + x31 = x21 + x32 + x21_x31 = x21/x31 + x21_x32 = x21/x32 + x21x21_x31x32 = x21_x31 * x21_x32 + + # Calculate integral over the subintervals (eqn (8) of Reference [2]) + coeff1 = 3 - x21_x31 + coeff2 = 3 + x21x21_x31x32 + x21_x31 + coeff3 = -x21x21_x31x32 + + return x21/6 * (coeff1*f1 + coeff2*f2 + coeff3*f3) + + +def _ensure_float_array(arr: npt.ArrayLike) -> np.ndarray: + arr = np.asarray(arr) + if np.issubdtype(arr.dtype, np.integer): + arr = arr.astype(float, copy=False) + return arr + + +def cumulative_simpson(y, *, x=None, dx=1.0, axis=-1, initial=None): + r""" + Cumulatively integrate y(x) using the composite Simpson's 1/3 rule. + The integral of the samples at every point is calculated by assuming a + quadratic relationship between each point and the two adjacent points. + + Parameters + ---------- + y : array_like + Values to integrate. Requires at least one point along `axis`. If two or fewer + points are provided along `axis`, Simpson's integration is not possible and the + result is calculated with `cumulative_trapezoid`. + x : array_like, optional + The coordinate to integrate along. Must have the same shape as `y` or + must be 1D with the same length as `y` along `axis`. `x` must also be + strictly increasing along `axis`. + If `x` is None (default), integration is performed using spacing `dx` + between consecutive elements in `y`. + dx : scalar or array_like, optional + Spacing between elements of `y`. Only used if `x` is None. Can either + be a float, or an array with the same shape as `y`, but of length one along + `axis`. Default is 1.0. + axis : int, optional + Specifies the axis to integrate along. Default is -1 (last axis). + initial : scalar or array_like, optional + If given, insert this value at the beginning of the returned result, + and add it to the rest of the result. Default is None, which means no + value at ``x[0]`` is returned and `res` has one element less than `y` + along the axis of integration. Can either be a float, or an array with + the same shape as `y`, but of length one along `axis`. + + Returns + ------- + res : ndarray + The result of cumulative integration of `y` along `axis`. + If `initial` is None, the shape is such that the axis of integration + has one less value than `y`. If `initial` is given, the shape is equal + to that of `y`. + + See Also + -------- + numpy.cumsum + cumulative_trapezoid : cumulative integration using the composite + trapezoidal rule + simpson : integrator for sampled data using the Composite Simpson's Rule + + Notes + ----- + + .. versionadded:: 1.12.0 + + The composite Simpson's 1/3 method can be used to approximate the definite + integral of a sampled input function :math:`y(x)` [1]_. The method assumes + a quadratic relationship over the interval containing any three consecutive + sampled points. + + Consider three consecutive points: + :math:`(x_1, y_1), (x_2, y_2), (x_3, y_3)`. + + Assuming a quadratic relationship over the three points, the integral over + the subinterval between :math:`x_1` and :math:`x_2` is given by formula + (8) of [2]_: + + .. math:: + \int_{x_1}^{x_2} y(x) dx\ &= \frac{x_2-x_1}{6}\left[\ + \left\{3-\frac{x_2-x_1}{x_3-x_1}\right\} y_1 + \ + \left\{3 + \frac{(x_2-x_1)^2}{(x_3-x_2)(x_3-x_1)} + \ + \frac{x_2-x_1}{x_3-x_1}\right\} y_2\\ + - \frac{(x_2-x_1)^2}{(x_3-x_2)(x_3-x_1)} y_3\right] + + The integral between :math:`x_2` and :math:`x_3` is given by swapping + appearances of :math:`x_1` and :math:`x_3`. The integral is estimated + separately for each subinterval and then cumulatively summed to obtain + the final result. + + For samples that are equally spaced, the result is exact if the function + is a polynomial of order three or less [1]_ and the number of subintervals + is even. Otherwise, the integral is exact for polynomials of order two or + less. + + References + ---------- + .. [1] Wikipedia page: https://en.wikipedia.org/wiki/Simpson's_rule + .. [2] Cartwright, Kenneth V. Simpson's Rule Cumulative Integration with + MS Excel and Irregularly-spaced Data. Journal of Mathematical + Sciences and Mathematics Education. 12 (2): 1-9 + + Examples + -------- + >>> from scipy import integrate + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(-2, 2, num=20) + >>> y = x**2 + >>> y_int = integrate.cumulative_simpson(y, x=x, initial=0) + >>> fig, ax = plt.subplots() + >>> ax.plot(x, y_int, 'ro', x, x**3/3 - (x[0])**3/3, 'b-') + >>> ax.grid() + >>> plt.show() + + The output of `cumulative_simpson` is similar to that of iteratively + calling `simpson` with successively higher upper limits of integration, but + not identical. + + >>> def cumulative_simpson_reference(y, x): + ... return np.asarray([integrate.simpson(y[:i], x=x[:i]) + ... for i in range(2, len(y) + 1)]) + >>> + >>> rng = np.random.default_rng(354673834679465) + >>> x, y = rng.random(size=(2, 10)) + >>> x.sort() + >>> + >>> res = integrate.cumulative_simpson(y, x=x) + >>> ref = cumulative_simpson_reference(y, x) + >>> equal = np.abs(res - ref) < 1e-15 + >>> equal # not equal when `simpson` has even number of subintervals + array([False, True, False, True, False, True, False, True, True]) + + This is expected: because `cumulative_simpson` has access to more + information than `simpson`, it can typically produce more accurate + estimates of the underlying integral over subintervals. + + """ + y = _ensure_float_array(y) + + # validate `axis` and standardize to work along the last axis + original_y = y + original_shape = y.shape + try: + y = np.swapaxes(y, axis, -1) + except IndexError as e: + message = f"`axis={axis}` is not valid for `y` with `y.ndim={y.ndim}`." + raise ValueError(message) from e + if y.shape[-1] < 3: + res = cumulative_trapezoid(original_y, x, dx=dx, axis=axis, initial=None) + res = np.swapaxes(res, axis, -1) + + elif x is not None: + x = _ensure_float_array(x) + message = ("If given, shape of `x` must be the same as `y` or 1-D with " + "the same length as `y` along `axis`.") + if not (x.shape == original_shape + or (x.ndim == 1 and len(x) == original_shape[axis])): + raise ValueError(message) + + x = np.broadcast_to(x, y.shape) if x.ndim == 1 else np.swapaxes(x, axis, -1) + dx = np.diff(x, axis=-1) + if np.any(dx <= 0): + raise ValueError("Input x must be strictly increasing.") + res = _cumulatively_sum_simpson_integrals( + y, dx, _cumulative_simpson_unequal_intervals + ) + + else: + dx = _ensure_float_array(dx) + final_dx_shape = tupleset(original_shape, axis, original_shape[axis] - 1) + alt_input_dx_shape = tupleset(original_shape, axis, 1) + message = ("If provided, `dx` must either be a scalar or have the same " + "shape as `y` but with only 1 point along `axis`.") + if not (dx.ndim == 0 or dx.shape == alt_input_dx_shape): + raise ValueError(message) + dx = np.broadcast_to(dx, final_dx_shape) + dx = np.swapaxes(dx, axis, -1) + res = _cumulatively_sum_simpson_integrals( + y, dx, _cumulative_simpson_equal_intervals + ) + + if initial is not None: + initial = _ensure_float_array(initial) + alt_initial_input_shape = tupleset(original_shape, axis, 1) + message = ("If provided, `initial` must either be a scalar or have the " + "same shape as `y` but with only 1 point along `axis`.") + if not (initial.ndim == 0 or initial.shape == alt_initial_input_shape): + raise ValueError(message) + initial = np.broadcast_to(initial, alt_initial_input_shape) + initial = np.swapaxes(initial, axis, -1) + + res += initial + res = np.concatenate((initial, res), axis=-1) + + res = np.swapaxes(res, -1, axis) + return res + + +def romb(y, dx=1.0, axis=-1, show=False): + """ + Romberg integration using samples of a function. + + Parameters + ---------- + y : array_like + A vector of ``2**k + 1`` equally-spaced samples of a function. + dx : float, optional + The sample spacing. Default is 1. + axis : int, optional + The axis along which to integrate. Default is -1 (last axis). + show : bool, optional + When `y` is a single 1-D array, then if this argument is True + print the table showing Richardson extrapolation from the + samples. Default is False. + + Returns + ------- + romb : ndarray + The integrated result for `axis`. + + See Also + -------- + quad : adaptive quadrature using QUADPACK + fixed_quad : fixed-order Gaussian quadrature + dblquad : double integrals + tplquad : triple integrals + simpson : integrators for sampled data + cumulative_trapezoid : cumulative integration for sampled data + + Examples + -------- + >>> from scipy import integrate + >>> import numpy as np + >>> x = np.arange(10, 14.25, 0.25) + >>> y = np.arange(3, 12) + + >>> integrate.romb(y) + 56.0 + + >>> y = np.sin(np.power(x, 2.5)) + >>> integrate.romb(y) + -0.742561336672229 + + >>> integrate.romb(y, show=True) + Richardson Extrapolation Table for Romberg Integration + ====================================================== + -0.81576 + 4.63862 6.45674 + -1.10581 -3.02062 -3.65245 + -2.57379 -3.06311 -3.06595 -3.05664 + -1.34093 -0.92997 -0.78776 -0.75160 -0.74256 + ====================================================== + -0.742561336672229 # may vary + + """ + y = np.asarray(y) + nd = len(y.shape) + Nsamps = y.shape[axis] + Ninterv = Nsamps-1 + n = 1 + k = 0 + while n < Ninterv: + n <<= 1 + k += 1 + if n != Ninterv: + raise ValueError("Number of samples must be one plus a " + "non-negative power of 2.") + + R = {} + slice_all = (slice(None),) * nd + slice0 = tupleset(slice_all, axis, 0) + slicem1 = tupleset(slice_all, axis, -1) + h = Ninterv * np.asarray(dx, dtype=float) + R[(0, 0)] = (y[slice0] + y[slicem1])/2.0*h + slice_R = slice_all + start = stop = step = Ninterv + for i in range(1, k+1): + start >>= 1 + slice_R = tupleset(slice_R, axis, slice(start, stop, step)) + step >>= 1 + R[(i, 0)] = 0.5*(R[(i-1, 0)] + h*y[slice_R].sum(axis=axis)) + for j in range(1, i+1): + prev = R[(i, j-1)] + R[(i, j)] = prev + (prev-R[(i-1, j-1)]) / ((1 << (2*j))-1) + h /= 2.0 + + if show: + if not np.isscalar(R[(0, 0)]): + print("*** Printing table only supported for integrals" + + " of a single data set.") + else: + try: + precis = show[0] + except (TypeError, IndexError): + precis = 5 + try: + width = show[1] + except (TypeError, IndexError): + width = 8 + formstr = "%%%d.%df" % (width, precis) + + title = "Richardson Extrapolation Table for Romberg Integration" + print(title, "=" * len(title), sep="\n", end="\n") + for i in range(k+1): + for j in range(i+1): + print(formstr % R[(i, j)], end=" ") + print() + print("=" * len(title)) + + return R[(k, k)] + + +# Coefficients for Newton-Cotes quadrature +# +# These are the points being used +# to construct the local interpolating polynomial +# a are the weights for Newton-Cotes integration +# B is the error coefficient. +# error in these coefficients grows as N gets larger. +# or as samples are closer and closer together + +# You can use maxima to find these rational coefficients +# for equally spaced data using the commands +# a(i,N) := (integrate(product(r-j,j,0,i-1) * product(r-j,j,i+1,N),r,0,N) +# / ((N-i)! * i!) * (-1)^(N-i)); +# Be(N) := N^(N+2)/(N+2)! * (N/(N+3) - sum((i/N)^(N+2)*a(i,N),i,0,N)); +# Bo(N) := N^(N+1)/(N+1)! * (N/(N+2) - sum((i/N)^(N+1)*a(i,N),i,0,N)); +# B(N) := (if (mod(N,2)=0) then Be(N) else Bo(N)); +# +# pre-computed for equally-spaced weights +# +# num_a, den_a, int_a, num_B, den_B = _builtincoeffs[N] +# +# a = num_a*array(int_a)/den_a +# B = num_B*1.0 / den_B +# +# integrate(f(x),x,x_0,x_N) = dx*sum(a*f(x_i)) + B*(dx)^(2k+3) f^(2k+2)(x*) +# where k = N // 2 +# +_builtincoeffs = { + 1: (1,2,[1,1],-1,12), + 2: (1,3,[1,4,1],-1,90), + 3: (3,8,[1,3,3,1],-3,80), + 4: (2,45,[7,32,12,32,7],-8,945), + 5: (5,288,[19,75,50,50,75,19],-275,12096), + 6: (1,140,[41,216,27,272,27,216,41],-9,1400), + 7: (7,17280,[751,3577,1323,2989,2989,1323,3577,751],-8183,518400), + 8: (4,14175,[989,5888,-928,10496,-4540,10496,-928,5888,989], + -2368,467775), + 9: (9,89600,[2857,15741,1080,19344,5778,5778,19344,1080, + 15741,2857], -4671, 394240), + 10: (5,299376,[16067,106300,-48525,272400,-260550,427368, + -260550,272400,-48525,106300,16067], + -673175, 163459296), + 11: (11,87091200,[2171465,13486539,-3237113, 25226685,-9595542, + 15493566,15493566,-9595542,25226685,-3237113, + 13486539,2171465], -2224234463, 237758976000), + 12: (1, 5255250, [1364651,9903168,-7587864,35725120,-51491295, + 87516288,-87797136,87516288,-51491295,35725120, + -7587864,9903168,1364651], -3012, 875875), + 13: (13, 402361344000,[8181904909, 56280729661, -31268252574, + 156074417954,-151659573325,206683437987, + -43111992612,-43111992612,206683437987, + -151659573325,156074417954,-31268252574, + 56280729661,8181904909], -2639651053, + 344881152000), + 14: (7, 2501928000, [90241897,710986864,-770720657,3501442784, + -6625093363,12630121616,-16802270373,19534438464, + -16802270373,12630121616,-6625093363,3501442784, + -770720657,710986864,90241897], -3740727473, + 1275983280000) + } + + +def newton_cotes(rn, equal=0): + r""" + Return weights and error coefficient for Newton-Cotes integration. + + Suppose we have (N+1) samples of f at the positions + x_0, x_1, ..., x_N. Then an N-point Newton-Cotes formula for the + integral between x_0 and x_N is: + + :math:`\int_{x_0}^{x_N} f(x)dx = \Delta x \sum_{i=0}^{N} a_i f(x_i) + + B_N (\Delta x)^{N+2} f^{N+1} (\xi)` + + where :math:`\xi \in [x_0,x_N]` + and :math:`\Delta x = \frac{x_N-x_0}{N}` is the average samples spacing. + + If the samples are equally-spaced and N is even, then the error + term is :math:`B_N (\Delta x)^{N+3} f^{N+2}(\xi)`. + + Parameters + ---------- + rn : int + The integer order for equally-spaced data or the relative positions of + the samples with the first sample at 0 and the last at N, where N+1 is + the length of `rn`. N is the order of the Newton-Cotes integration. + equal : int, optional + Set to 1 to enforce equally spaced data. + + Returns + ------- + an : ndarray + 1-D array of weights to apply to the function at the provided sample + positions. + B : float + Error coefficient. + + Notes + ----- + Normally, the Newton-Cotes rules are used on smaller integration + regions and a composite rule is used to return the total integral. + + Examples + -------- + Compute the integral of sin(x) in [0, :math:`\pi`]: + + >>> from scipy.integrate import newton_cotes + >>> import numpy as np + >>> def f(x): + ... return np.sin(x) + >>> a = 0 + >>> b = np.pi + >>> exact = 2 + >>> for N in [2, 4, 6, 8, 10]: + ... x = np.linspace(a, b, N + 1) + ... an, B = newton_cotes(N, 1) + ... dx = (b - a) / N + ... quad = dx * np.sum(an * f(x)) + ... error = abs(quad - exact) + ... print('{:2d} {:10.9f} {:.5e}'.format(N, quad, error)) + ... + 2 2.094395102 9.43951e-02 + 4 1.998570732 1.42927e-03 + 6 2.000017814 1.78136e-05 + 8 1.999999835 1.64725e-07 + 10 2.000000001 1.14677e-09 + + """ + try: + N = len(rn)-1 + if equal: + rn = np.arange(N+1) + elif np.all(np.diff(rn) == 1): + equal = 1 + except Exception: + N = rn + rn = np.arange(N+1) + equal = 1 + + if equal and N in _builtincoeffs: + na, da, vi, nb, db = _builtincoeffs[N] + an = na * np.array(vi, dtype=float) / da + return an, float(nb)/db + + if (rn[0] != 0) or (rn[-1] != N): + raise ValueError("The sample positions must start at 0" + " and end at N") + yi = rn / float(N) + ti = 2 * yi - 1 + nvec = np.arange(N+1) + C = ti ** nvec[:, np.newaxis] + Cinv = np.linalg.inv(C) + # improve precision of result + for i in range(2): + Cinv = 2*Cinv - Cinv.dot(C).dot(Cinv) + vec = 2.0 / (nvec[::2]+1) + ai = Cinv[:, ::2].dot(vec) * (N / 2.) + + if (N % 2 == 0) and equal: + BN = N/(N+3.) + power = N+2 + else: + BN = N/(N+2.) + power = N+1 + + BN = BN - np.dot(yi**power, ai) + p1 = power+1 + fac = power*math.log(N) - gammaln(p1) + fac = math.exp(fac) + return ai, BN*fac + + +def _qmc_quad_iv(func, a, b, n_points, n_estimates, qrng, log): + + # lazy import to avoid issues with partially-initialized submodule + if not hasattr(qmc_quad, 'qmc'): + from scipy import stats + qmc_quad.stats = stats + else: + stats = qmc_quad.stats + + if not callable(func): + message = "`func` must be callable." + raise TypeError(message) + + # a, b will be modified, so copy. Oh well if it's copied twice. + a = np.atleast_1d(a).copy() + b = np.atleast_1d(b).copy() + a, b = np.broadcast_arrays(a, b) + dim = a.shape[0] + + try: + func((a + b) / 2) + except Exception as e: + message = ("`func` must evaluate the integrand at points within " + "the integration range; e.g. `func( (a + b) / 2)` " + "must return the integrand at the centroid of the " + "integration volume.") + raise ValueError(message) from e + + try: + func(np.array([a, b]).T) + vfunc = func + except Exception as e: + message = ("Exception encountered when attempting vectorized call to " + f"`func`: {e}. For better performance, `func` should " + "accept two-dimensional array `x` with shape `(len(a), " + "n_points)` and return an array of the integrand value at " + "each of the `n_points.") + warnings.warn(message, stacklevel=3) + + def vfunc(x): + return np.apply_along_axis(func, axis=-1, arr=x) + + n_points_int = np.int64(n_points) + if n_points != n_points_int: + message = "`n_points` must be an integer." + raise TypeError(message) + + n_estimates_int = np.int64(n_estimates) + if n_estimates != n_estimates_int: + message = "`n_estimates` must be an integer." + raise TypeError(message) + + if qrng is None: + qrng = stats.qmc.Halton(dim) + elif not isinstance(qrng, stats.qmc.QMCEngine): + message = "`qrng` must be an instance of scipy.stats.qmc.QMCEngine." + raise TypeError(message) + + if qrng.d != a.shape[0]: + message = ("`qrng` must be initialized with dimensionality equal to " + "the number of variables in `a`, i.e., " + "`qrng.random().shape[-1]` must equal `a.shape[0]`.") + raise ValueError(message) + + rng_seed = getattr(qrng, 'rng_seed', None) + rng = stats._qmc.check_random_state(rng_seed) + + if log not in {True, False}: + message = "`log` must be boolean (`True` or `False`)." + raise TypeError(message) + + return (vfunc, a, b, n_points_int, n_estimates_int, qrng, rng, log, stats) + + +QMCQuadResult = namedtuple('QMCQuadResult', ['integral', 'standard_error']) + + +def qmc_quad(func, a, b, *, n_estimates=8, n_points=1024, qrng=None, + log=False): + """ + Compute an integral in N-dimensions using Quasi-Monte Carlo quadrature. + + Parameters + ---------- + func : callable + The integrand. Must accept a single argument ``x``, an array which + specifies the point(s) at which to evaluate the scalar-valued + integrand, and return the value(s) of the integrand. + For efficiency, the function should be vectorized to accept an array of + shape ``(d, n_points)``, where ``d`` is the number of variables (i.e. + the dimensionality of the function domain) and `n_points` is the number + of quadrature points, and return an array of shape ``(n_points,)``, + the integrand at each quadrature point. + a, b : array-like + One-dimensional arrays specifying the lower and upper integration + limits, respectively, of each of the ``d`` variables. + n_estimates, n_points : int, optional + `n_estimates` (default: 8) statistically independent QMC samples, each + of `n_points` (default: 1024) points, will be generated by `qrng`. + The total number of points at which the integrand `func` will be + evaluated is ``n_points * n_estimates``. See Notes for details. + qrng : `~scipy.stats.qmc.QMCEngine`, optional + An instance of the QMCEngine from which to sample QMC points. + The QMCEngine must be initialized to a number of dimensions ``d`` + corresponding with the number of variables ``x1, ..., xd`` passed to + `func`. + The provided QMCEngine is used to produce the first integral estimate. + If `n_estimates` is greater than one, additional QMCEngines are + spawned from the first (with scrambling enabled, if it is an option.) + If a QMCEngine is not provided, the default `scipy.stats.qmc.Halton` + will be initialized with the number of dimensions determine from + the length of `a`. + log : boolean, default: False + When set to True, `func` returns the log of the integrand, and + the result object contains the log of the integral. + + Returns + ------- + result : object + A result object with attributes: + + integral : float + The estimate of the integral. + standard_error : + The error estimate. See Notes for interpretation. + + Notes + ----- + Values of the integrand at each of the `n_points` points of a QMC sample + are used to produce an estimate of the integral. This estimate is drawn + from a population of possible estimates of the integral, the value of + which we obtain depends on the particular points at which the integral + was evaluated. We perform this process `n_estimates` times, each time + evaluating the integrand at different scrambled QMC points, effectively + drawing i.i.d. random samples from the population of integral estimates. + The sample mean :math:`m` of these integral estimates is an + unbiased estimator of the true value of the integral, and the standard + error of the mean :math:`s` of these estimates may be used to generate + confidence intervals using the t distribution with ``n_estimates - 1`` + degrees of freedom. Perhaps counter-intuitively, increasing `n_points` + while keeping the total number of function evaluation points + ``n_points * n_estimates`` fixed tends to reduce the actual error, whereas + increasing `n_estimates` tends to decrease the error estimate. + + Examples + -------- + QMC quadrature is particularly useful for computing integrals in higher + dimensions. An example integrand is the probability density function + of a multivariate normal distribution. + + >>> import numpy as np + >>> from scipy import stats + >>> dim = 8 + >>> mean = np.zeros(dim) + >>> cov = np.eye(dim) + >>> def func(x): + ... # `multivariate_normal` expects the _last_ axis to correspond with + ... # the dimensionality of the space, so `x` must be transposed + ... return stats.multivariate_normal.pdf(x.T, mean, cov) + + To compute the integral over the unit hypercube: + + >>> from scipy.integrate import qmc_quad + >>> a = np.zeros(dim) + >>> b = np.ones(dim) + >>> rng = np.random.default_rng() + >>> qrng = stats.qmc.Halton(d=dim, seed=rng) + >>> n_estimates = 8 + >>> res = qmc_quad(func, a, b, n_estimates=n_estimates, qrng=qrng) + >>> res.integral, res.standard_error + (0.00018429555666024108, 1.0389431116001344e-07) + + A two-sided, 99% confidence interval for the integral may be estimated + as: + + >>> t = stats.t(df=n_estimates-1, loc=res.integral, + ... scale=res.standard_error) + >>> t.interval(0.99) + (0.0001839319802536469, 0.00018465913306683527) + + Indeed, the value reported by `scipy.stats.multivariate_normal` is + within this range. + + >>> stats.multivariate_normal.cdf(b, mean, cov, lower_limit=a) + 0.00018430867675187443 + + """ + args = _qmc_quad_iv(func, a, b, n_points, n_estimates, qrng, log) + func, a, b, n_points, n_estimates, qrng, rng, log, stats = args + + def sum_product(integrands, dA, log=False): + if log: + return logsumexp(integrands) + np.log(dA) + else: + return np.sum(integrands * dA) + + def mean(estimates, log=False): + if log: + return logsumexp(estimates) - np.log(n_estimates) + else: + return np.mean(estimates) + + def std(estimates, m=None, ddof=0, log=False): + m = m or mean(estimates, log) + if log: + estimates, m = np.broadcast_arrays(estimates, m) + temp = np.vstack((estimates, m + np.pi * 1j)) + diff = logsumexp(temp, axis=0) + return np.real(0.5 * (logsumexp(2 * diff) + - np.log(n_estimates - ddof))) + else: + return np.std(estimates, ddof=ddof) + + def sem(estimates, m=None, s=None, log=False): + m = m or mean(estimates, log) + s = s or std(estimates, m, ddof=1, log=log) + if log: + return s - 0.5*np.log(n_estimates) + else: + return s / np.sqrt(n_estimates) + + # The sign of the integral depends on the order of the limits. Fix this by + # ensuring that lower bounds are indeed lower and setting sign of resulting + # integral manually + if np.any(a == b): + message = ("A lower limit was equal to an upper limit, so the value " + "of the integral is zero by definition.") + warnings.warn(message, stacklevel=2) + return QMCQuadResult(-np.inf if log else 0, 0) + + i_swap = b < a + sign = (-1)**(i_swap.sum(axis=-1)) # odd # of swaps -> negative + a[i_swap], b[i_swap] = b[i_swap], a[i_swap] + + A = np.prod(b - a) + dA = A / n_points + + estimates = np.zeros(n_estimates) + rngs = _rng_spawn(qrng.rng, n_estimates) + for i in range(n_estimates): + # Generate integral estimate + sample = qrng.random(n_points) + # The rationale for transposing is that this allows users to easily + # unpack `x` into separate variables, if desired. This is consistent + # with the `xx` array passed into the `scipy.integrate.nquad` `func`. + x = stats.qmc.scale(sample, a, b).T # (n_dim, n_points) + integrands = func(x) + estimates[i] = sum_product(integrands, dA, log) + + # Get a new, independently-scrambled QRNG for next time + qrng = type(qrng)(seed=rngs[i], **qrng._init_quad) + + integral = mean(estimates, log) + standard_error = sem(estimates, m=integral, log=log) + integral = integral + np.pi*1j if (log and sign < 0) else integral*sign + return QMCQuadResult(integral, standard_error) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..4c91aa324478d49a8723f05618801f9b256d07af --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/__init__.py @@ -0,0 +1,12 @@ +"""Numerical cubature algorithms""" + +from ._base import ( + Rule, FixedRule, + NestedFixedRule, + ProductNestedFixed, +) +from ._genz_malik import GenzMalikCubature +from ._gauss_kronrod import GaussKronrodQuadrature +from ._gauss_legendre import GaussLegendreQuadrature + +__all__ = [s for s in dir() if not s.startswith('_')] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_base.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_base.py new file mode 100644 index 0000000000000000000000000000000000000000..3a3ae5f506505c9c03b2ac8be33d301d60074681 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_base.py @@ -0,0 +1,518 @@ +from scipy._lib._array_api import array_namespace, xp_size + +from functools import cached_property + + +class Rule: + """ + Base class for numerical integration algorithms (cubatures). + + Finds an estimate for the integral of ``f`` over the region described by two arrays + ``a`` and ``b`` via `estimate`, and find an estimate for the error of this + approximation via `estimate_error`. + + If a subclass does not implement its own `estimate_error`, then it will use a + default error estimate based on the difference between the estimate over the whole + region and the sum of estimates over that region divided into ``2^ndim`` subregions. + + See Also + -------- + FixedRule + + Examples + -------- + In the following, a custom rule is created which uses 3D Genz-Malik cubature for + the estimate of the integral, and the difference between this estimate and a less + accurate estimate using 5-node Gauss-Legendre quadrature as an estimate for the + error. + + >>> import numpy as np + >>> from scipy.integrate import cubature + >>> from scipy.integrate._rules import ( + ... Rule, ProductNestedFixed, GenzMalikCubature, GaussLegendreQuadrature + ... ) + >>> def f(x, r, alphas): + ... # f(x) = cos(2*pi*r + alpha @ x) + ... # Need to allow r and alphas to be arbitrary shape + ... npoints, ndim = x.shape[0], x.shape[-1] + ... alphas_reshaped = alphas[np.newaxis, :] + ... x_reshaped = x.reshape(npoints, *([1]*(len(alphas.shape) - 1)), ndim) + ... return np.cos(2*np.pi*r + np.sum(alphas_reshaped * x_reshaped, axis=-1)) + >>> genz = GenzMalikCubature(ndim=3) + >>> gauss = GaussKronrodQuadrature(npoints=21) + >>> # Gauss-Kronrod is 1D, so we find the 3D product rule: + >>> gauss_3d = ProductNestedFixed([gauss, gauss, gauss]) + >>> class CustomRule(Rule): + ... def estimate(self, f, a, b, args=()): + ... return genz.estimate(f, a, b, args) + ... def estimate_error(self, f, a, b, args=()): + ... return np.abs( + ... genz.estimate(f, a, b, args) + ... - gauss_3d.estimate(f, a, b, args) + ... ) + >>> rng = np.random.default_rng() + >>> res = cubature( + ... f=f, + ... a=np.array([0, 0, 0]), + ... b=np.array([1, 1, 1]), + ... rule=CustomRule(), + ... args=(rng.random((2,)), rng.random((3, 2, 3))) + ... ) + >>> res.estimate + array([[-0.95179502, 0.12444608], + [-0.96247411, 0.60866385], + [-0.97360014, 0.25515587]]) + """ + + def estimate(self, f, a, b, args=()): + r""" + Calculate estimate of integral of `f` in rectangular region described by + corners `a` and ``b``. + + Parameters + ---------- + f : callable + Function to integrate. `f` must have the signature:: + f(x : ndarray, \*args) -> ndarray + + `f` should accept arrays ``x`` of shape:: + (npoints, ndim) + + and output arrays of shape:: + (npoints, output_dim_1, ..., output_dim_n) + + In this case, `estimate` will return arrays of shape:: + (output_dim_1, ..., output_dim_n) + a, b : ndarray + Lower and upper limits of integration as rank-1 arrays specifying the left + and right endpoints of the intervals being integrated over. Infinite limits + are currently not supported. + args : tuple, optional + Additional positional args passed to ``f``, if any. + + Returns + ------- + est : ndarray + Result of estimation. If `f` returns arrays of shape ``(npoints, + output_dim_1, ..., output_dim_n)``, then `est` will be of shape + ``(output_dim_1, ..., output_dim_n)``. + """ + raise NotImplementedError + + def estimate_error(self, f, a, b, args=()): + r""" + Estimate the error of the approximation for the integral of `f` in rectangular + region described by corners `a` and `b`. + + If a subclass does not override this method, then a default error estimator is + used. This estimates the error as ``|est - refined_est|`` where ``est`` is + ``estimate(f, a, b)`` and ``refined_est`` is the sum of + ``estimate(f, a_k, b_k)`` where ``a_k, b_k`` are the coordinates of each + subregion of the region described by ``a`` and ``b``. In the 1D case, this + is equivalent to comparing the integral over an entire interval ``[a, b]`` to + the sum of the integrals over the left and right subintervals, ``[a, (a+b)/2]`` + and ``[(a+b)/2, b]``. + + Parameters + ---------- + f : callable + Function to estimate error for. `f` must have the signature:: + f(x : ndarray, \*args) -> ndarray + + `f` should accept arrays `x` of shape:: + (npoints, ndim) + + and output arrays of shape:: + (npoints, output_dim_1, ..., output_dim_n) + + In this case, `estimate` will return arrays of shape:: + (output_dim_1, ..., output_dim_n) + a, b : ndarray + Lower and upper limits of integration as rank-1 arrays specifying the left + and right endpoints of the intervals being integrated over. Infinite limits + are currently not supported. + args : tuple, optional + Additional positional args passed to `f`, if any. + + Returns + ------- + err_est : ndarray + Result of error estimation. If `f` returns arrays of shape + ``(npoints, output_dim_1, ..., output_dim_n)``, then `est` will be + of shape ``(output_dim_1, ..., output_dim_n)``. + """ + + est = self.estimate(f, a, b, args) + refined_est = 0 + + for a_k, b_k in _split_subregion(a, b): + refined_est += self.estimate(f, a_k, b_k, args) + + return self.xp.abs(est - refined_est) + + +class FixedRule(Rule): + """ + A rule implemented as the weighted sum of function evaluations at fixed nodes. + + Attributes + ---------- + nodes_and_weights : (ndarray, ndarray) + A tuple ``(nodes, weights)`` of nodes at which to evaluate ``f`` and the + corresponding weights. ``nodes`` should be of shape ``(num_nodes,)`` for 1D + cubature rules (quadratures) and more generally for N-D cubature rules, it + should be of shape ``(num_nodes, ndim)``. ``weights`` should be of shape + ``(num_nodes,)``. The nodes and weights should be for integrals over + :math:`[-1, 1]^n`. + + See Also + -------- + GaussLegendreQuadrature, GaussKronrodQuadrature, GenzMalikCubature + + Examples + -------- + + Implementing Simpson's 1/3 rule: + + >>> import numpy as np + >>> from scipy.integrate._rules import FixedRule + >>> class SimpsonsQuad(FixedRule): + ... @property + ... def nodes_and_weights(self): + ... nodes = np.array([-1, 0, 1]) + ... weights = np.array([1/3, 4/3, 1/3]) + ... return (nodes, weights) + >>> rule = SimpsonsQuad() + >>> rule.estimate( + ... f=lambda x: x**2, + ... a=np.array([0]), + ... b=np.array([1]), + ... ) + [0.3333333] + """ + + def __init__(self): + self.xp = None + + @property + def nodes_and_weights(self): + raise NotImplementedError + + def estimate(self, f, a, b, args=()): + r""" + Calculate estimate of integral of `f` in rectangular region described by + corners `a` and `b` as ``sum(weights * f(nodes))``. + + Nodes and weights will automatically be adjusted from calculating integrals over + :math:`[-1, 1]^n` to :math:`[a, b]^n`. + + Parameters + ---------- + f : callable + Function to integrate. `f` must have the signature:: + f(x : ndarray, \*args) -> ndarray + + `f` should accept arrays `x` of shape:: + (npoints, ndim) + + and output arrays of shape:: + (npoints, output_dim_1, ..., output_dim_n) + + In this case, `estimate` will return arrays of shape:: + (output_dim_1, ..., output_dim_n) + a, b : ndarray + Lower and upper limits of integration as rank-1 arrays specifying the left + and right endpoints of the intervals being integrated over. Infinite limits + are currently not supported. + args : tuple, optional + Additional positional args passed to `f`, if any. + + Returns + ------- + est : ndarray + Result of estimation. If `f` returns arrays of shape ``(npoints, + output_dim_1, ..., output_dim_n)``, then `est` will be of shape + ``(output_dim_1, ..., output_dim_n)``. + """ + nodes, weights = self.nodes_and_weights + + if self.xp is None: + self.xp = array_namespace(nodes) + + return _apply_fixed_rule(f, a, b, nodes, weights, args, self.xp) + + +class NestedFixedRule(FixedRule): + r""" + A cubature rule with error estimate given by the difference between two underlying + fixed rules. + + If constructed as ``NestedFixedRule(higher, lower)``, this will use:: + + estimate(f, a, b) := higher.estimate(f, a, b) + estimate_error(f, a, b) := \|higher.estimate(f, a, b) - lower.estimate(f, a, b)| + + (where the absolute value is taken elementwise). + + Attributes + ---------- + higher : Rule + Higher accuracy rule. + + lower : Rule + Lower accuracy rule. + + See Also + -------- + GaussKronrodQuadrature + + Examples + -------- + + >>> from scipy.integrate import cubature + >>> from scipy.integrate._rules import ( + ... GaussLegendreQuadrature, NestedFixedRule, ProductNestedFixed + ... ) + >>> higher = GaussLegendreQuadrature(10) + >>> lower = GaussLegendreQuadrature(5) + >>> rule = NestedFixedRule( + ... higher, + ... lower + ... ) + >>> rule_2d = ProductNestedFixed([rule, rule]) + """ + + def __init__(self, higher, lower): + self.higher = higher + self.lower = lower + self.xp = None + + @property + def nodes_and_weights(self): + if self.higher is not None: + return self.higher.nodes_and_weights + else: + raise NotImplementedError + + @property + def lower_nodes_and_weights(self): + if self.lower is not None: + return self.lower.nodes_and_weights + else: + raise NotImplementedError + + def estimate_error(self, f, a, b, args=()): + r""" + Estimate the error of the approximation for the integral of `f` in rectangular + region described by corners `a` and `b`. + + Parameters + ---------- + f : callable + Function to estimate error for. `f` must have the signature:: + f(x : ndarray, \*args) -> ndarray + + `f` should accept arrays `x` of shape:: + (npoints, ndim) + + and output arrays of shape:: + (npoints, output_dim_1, ..., output_dim_n) + + In this case, `estimate` will return arrays of shape:: + (output_dim_1, ..., output_dim_n) + a, b : ndarray + Lower and upper limits of integration as rank-1 arrays specifying the left + and right endpoints of the intervals being integrated over. Infinite limits + are currently not supported. + args : tuple, optional + Additional positional args passed to `f`, if any. + + Returns + ------- + err_est : ndarray + Result of error estimation. If `f` returns arrays of shape + ``(npoints, output_dim_1, ..., output_dim_n)``, then `est` will be + of shape ``(output_dim_1, ..., output_dim_n)``. + """ + + nodes, weights = self.nodes_and_weights + lower_nodes, lower_weights = self.lower_nodes_and_weights + + if self.xp is None: + self.xp = array_namespace(nodes) + + error_nodes = self.xp.concat([nodes, lower_nodes], axis=0) + error_weights = self.xp.concat([weights, -lower_weights], axis=0) + + return self.xp.abs( + _apply_fixed_rule(f, a, b, error_nodes, error_weights, args, self.xp) + ) + + +class ProductNestedFixed(NestedFixedRule): + """ + Find the n-dimensional cubature rule constructed from the Cartesian product of 1-D + `NestedFixedRule` quadrature rules. + + Given a list of N 1-dimensional quadrature rules which support error estimation + using NestedFixedRule, this will find the N-dimensional cubature rule obtained by + taking the Cartesian product of their nodes, and estimating the error by taking the + difference with a lower-accuracy N-dimensional cubature rule obtained using the + ``.lower_nodes_and_weights`` rule in each of the base 1-dimensional rules. + + Parameters + ---------- + base_rules : list of NestedFixedRule + List of base 1-dimensional `NestedFixedRule` quadrature rules. + + Attributes + ---------- + base_rules : list of NestedFixedRule + List of base 1-dimensional `NestedFixedRule` qudarature rules. + + Examples + -------- + + Evaluate a 2D integral by taking the product of two 1D rules: + + >>> import numpy as np + >>> from scipy.integrate import cubature + >>> from scipy.integrate._rules import ( + ... ProductNestedFixed, GaussKronrodQuadrature + ... ) + >>> def f(x): + ... # f(x) = cos(x_1) + cos(x_2) + ... return np.sum(np.cos(x), axis=-1) + >>> rule = ProductNestedFixed( + ... [GaussKronrodQuadrature(15), GaussKronrodQuadrature(15)] + ... ) # Use 15-point Gauss-Kronrod, which implements NestedFixedRule + >>> a, b = np.array([0, 0]), np.array([1, 1]) + >>> rule.estimate(f, a, b) # True value 2*sin(1), approximately 1.6829 + np.float64(1.682941969615793) + >>> rule.estimate_error(f, a, b) + np.float64(2.220446049250313e-16) + """ + + def __init__(self, base_rules): + for rule in base_rules: + if not isinstance(rule, NestedFixedRule): + raise ValueError("base rules for product need to be instance of" + "NestedFixedRule") + + self.base_rules = base_rules + self.xp = None + + @cached_property + def nodes_and_weights(self): + nodes = _cartesian_product( + [rule.nodes_and_weights[0] for rule in self.base_rules] + ) + + if self.xp is None: + self.xp = array_namespace(nodes) + + weights = self.xp.prod( + _cartesian_product( + [rule.nodes_and_weights[1] for rule in self.base_rules] + ), + axis=-1, + ) + + return nodes, weights + + @cached_property + def lower_nodes_and_weights(self): + nodes = _cartesian_product( + [cubature.lower_nodes_and_weights[0] for cubature in self.base_rules] + ) + + if self.xp is None: + self.xp = array_namespace(nodes) + + weights = self.xp.prod( + _cartesian_product( + [cubature.lower_nodes_and_weights[1] for cubature in self.base_rules] + ), + axis=-1, + ) + + return nodes, weights + + +def _cartesian_product(arrays): + xp = array_namespace(*arrays) + + arrays_ix = xp.meshgrid(*arrays, indexing='ij') + result = xp.reshape(xp.stack(arrays_ix, axis=-1), (-1, len(arrays))) + + return result + + +def _split_subregion(a, b, xp, split_at=None): + """ + Given the coordinates of a region like a=[0, 0] and b=[1, 1], yield the coordinates + of all subregions, which in this case would be:: + + ([0, 0], [1/2, 1/2]), + ([0, 1/2], [1/2, 1]), + ([1/2, 0], [1, 1/2]), + ([1/2, 1/2], [1, 1]) + """ + xp = array_namespace(a, b) + + if split_at is None: + split_at = (a + b) / 2 + + left = [xp.asarray([a[i], split_at[i]]) for i in range(a.shape[0])] + right = [xp.asarray([split_at[i], b[i]]) for i in range(b.shape[0])] + + a_sub = _cartesian_product(left) + b_sub = _cartesian_product(right) + + for i in range(a_sub.shape[0]): + yield a_sub[i, ...], b_sub[i, ...] + + +def _apply_fixed_rule(f, a, b, orig_nodes, orig_weights, args, xp): + # Downcast nodes and weights to common dtype of a and b + result_dtype = a.dtype + orig_nodes = xp.astype(orig_nodes, result_dtype) + orig_weights = xp.astype(orig_weights, result_dtype) + + # Ensure orig_nodes are at least 2D, since 1D cubature methods can return arrays of + # shape (npoints,) rather than (npoints, 1) + if orig_nodes.ndim == 1: + orig_nodes = orig_nodes[:, None] + + rule_ndim = orig_nodes.shape[-1] + + a_ndim = xp_size(a) + b_ndim = xp_size(b) + + if rule_ndim != a_ndim or rule_ndim != b_ndim: + raise ValueError(f"rule and function are of incompatible dimension, nodes have" + f"ndim {rule_ndim}, while limit of integration has ndim" + f"a_ndim={a_ndim}, b_ndim={b_ndim}") + + lengths = b - a + + # The underlying rule is for the hypercube [-1, 1]^n. + # + # To handle arbitrary regions of integration, it's necessary to apply a linear + # change of coordinates to map each interval [a[i], b[i]] to [-1, 1]. + nodes = (orig_nodes + 1) * (lengths * 0.5) + a + + # Also need to multiply the weights by a scale factor equal to the determinant + # of the Jacobian for this coordinate change. + weight_scale_factor = xp.prod(lengths, dtype=result_dtype) / 2**rule_ndim + weights = orig_weights * weight_scale_factor + + f_nodes = f(nodes, *args) + weights_reshaped = xp.reshape(weights, (-1, *([1] * (f_nodes.ndim - 1)))) + + # f(nodes) will have shape (num_nodes, output_dim_1, ..., output_dim_n) + # Summing along the first axis means estimate will shape (output_dim_1, ..., + # output_dim_n) + est = xp.sum(weights_reshaped * f_nodes, axis=0, dtype=result_dtype) + + return est diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_gauss_kronrod.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_gauss_kronrod.py new file mode 100644 index 0000000000000000000000000000000000000000..b2a3518c55cf49cd14c777d243ea7e93a489f86c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_gauss_kronrod.py @@ -0,0 +1,202 @@ +from scipy._lib._array_api import np_compat, array_namespace + +from functools import cached_property + +from ._base import NestedFixedRule +from ._gauss_legendre import GaussLegendreQuadrature + + +class GaussKronrodQuadrature(NestedFixedRule): + """ + Gauss-Kronrod quadrature. + + Gauss-Kronrod rules consist of two quadrature rules, one higher-order and one + lower-order. The higher-order rule is used as the estimate of the integral and the + difference between them is used as an estimate for the error. + + Gauss-Kronrod is a 1D rule. To use it for multidimensional integrals, it will be + necessary to use ProductNestedFixed and multiple Gauss-Kronrod rules. See Examples. + + For n-node Gauss-Kronrod, the lower-order rule has ``n//2`` nodes, which are the + ordinary Gauss-Legendre nodes with corresponding weights. The higher-order rule has + ``n`` nodes, ``n//2`` of which are the same as the lower-order rule and the + remaining nodes are the Kronrod extension of those nodes. + + Parameters + ---------- + npoints : int + Number of nodes for the higher-order rule. + + xp : array_namespace, optional + The namespace for the node and weight arrays. Default is None, where NumPy is + used. + + Attributes + ---------- + lower : Rule + Lower-order rule. + + References + ---------- + .. [1] R. Piessens, E. de Doncker, Quadpack: A Subroutine Package for Automatic + Integration, files: dqk21.f, dqk15.f (1983). + + Examples + -------- + Evaluate a 1D integral. Note in this example that ``f`` returns an array, so the + estimates will also be arrays, despite the fact that this is a 1D problem. + + >>> import numpy as np + >>> from scipy.integrate import cubature + >>> from scipy.integrate._rules import GaussKronrodQuadrature + >>> def f(x): + ... return np.cos(x) + >>> rule = GaussKronrodQuadrature(21) # Use 21-point GaussKronrod + >>> a, b = np.array([0]), np.array([1]) + >>> rule.estimate(f, a, b) # True value sin(1), approximately 0.84147 + array([0.84147098]) + >>> rule.estimate_error(f, a, b) + array([1.11022302e-16]) + + Evaluate a 2D integral. Note that in this example ``f`` returns a float, so the + estimates will also be floats. + + >>> import numpy as np + >>> from scipy.integrate import cubature + >>> from scipy.integrate._rules import ( + ... ProductNestedFixed, GaussKronrodQuadrature + ... ) + >>> def f(x): + ... # f(x) = cos(x_1) + cos(x_2) + ... return np.sum(np.cos(x), axis=-1) + >>> rule = ProductNestedFixed( + ... [GaussKronrodQuadrature(15), GaussKronrodQuadrature(15)] + ... ) # Use 15-point Gauss-Kronrod + >>> a, b = np.array([0, 0]), np.array([1, 1]) + >>> rule.estimate(f, a, b) # True value 2*sin(1), approximately 1.6829 + np.float64(1.682941969615793) + >>> rule.estimate_error(f, a, b) + np.float64(2.220446049250313e-16) + """ + + def __init__(self, npoints, xp=None): + # TODO: nodes and weights are currently hard-coded for values 15 and 21, but in + # the future it would be best to compute the Kronrod extension of the lower rule + if npoints != 15 and npoints != 21: + raise NotImplementedError("Gauss-Kronrod quadrature is currently only" + "supported for 15 or 21 nodes") + + self.npoints = npoints + + if xp is None: + xp = np_compat + + self.xp = array_namespace(xp.empty(0)) + + self.gauss = GaussLegendreQuadrature(npoints//2, xp=self.xp) + + @cached_property + def nodes_and_weights(self): + # These values are from QUADPACK's `dqk21.f` and `dqk15.f` (1983). + if self.npoints == 21: + nodes = self.xp.asarray( + [ + 0.995657163025808080735527280689003, + 0.973906528517171720077964012084452, + 0.930157491355708226001207180059508, + 0.865063366688984510732096688423493, + 0.780817726586416897063717578345042, + 0.679409568299024406234327365114874, + 0.562757134668604683339000099272694, + 0.433395394129247190799265943165784, + 0.294392862701460198131126603103866, + 0.148874338981631210884826001129720, + 0, + -0.148874338981631210884826001129720, + -0.294392862701460198131126603103866, + -0.433395394129247190799265943165784, + -0.562757134668604683339000099272694, + -0.679409568299024406234327365114874, + -0.780817726586416897063717578345042, + -0.865063366688984510732096688423493, + -0.930157491355708226001207180059508, + -0.973906528517171720077964012084452, + -0.995657163025808080735527280689003, + ], + dtype=self.xp.float64, + ) + + weights = self.xp.asarray( + [ + 0.011694638867371874278064396062192, + 0.032558162307964727478818972459390, + 0.054755896574351996031381300244580, + 0.075039674810919952767043140916190, + 0.093125454583697605535065465083366, + 0.109387158802297641899210590325805, + 0.123491976262065851077958109831074, + 0.134709217311473325928054001771707, + 0.142775938577060080797094273138717, + 0.147739104901338491374841515972068, + 0.149445554002916905664936468389821, + 0.147739104901338491374841515972068, + 0.142775938577060080797094273138717, + 0.134709217311473325928054001771707, + 0.123491976262065851077958109831074, + 0.109387158802297641899210590325805, + 0.093125454583697605535065465083366, + 0.075039674810919952767043140916190, + 0.054755896574351996031381300244580, + 0.032558162307964727478818972459390, + 0.011694638867371874278064396062192, + ], + dtype=self.xp.float64, + ) + elif self.npoints == 15: + nodes = self.xp.asarray( + [ + 0.991455371120812639206854697526329, + 0.949107912342758524526189684047851, + 0.864864423359769072789712788640926, + 0.741531185599394439863864773280788, + 0.586087235467691130294144838258730, + 0.405845151377397166906606412076961, + 0.207784955007898467600689403773245, + 0.000000000000000000000000000000000, + -0.207784955007898467600689403773245, + -0.405845151377397166906606412076961, + -0.586087235467691130294144838258730, + -0.741531185599394439863864773280788, + -0.864864423359769072789712788640926, + -0.949107912342758524526189684047851, + -0.991455371120812639206854697526329, + ], + dtype=self.xp.float64, + ) + + weights = self.xp.asarray( + [ + 0.022935322010529224963732008058970, + 0.063092092629978553290700663189204, + 0.104790010322250183839876322541518, + 0.140653259715525918745189590510238, + 0.169004726639267902826583426598550, + 0.190350578064785409913256402421014, + 0.204432940075298892414161999234649, + 0.209482141084727828012999174891714, + 0.204432940075298892414161999234649, + 0.190350578064785409913256402421014, + 0.169004726639267902826583426598550, + 0.140653259715525918745189590510238, + 0.104790010322250183839876322541518, + 0.063092092629978553290700663189204, + 0.022935322010529224963732008058970, + ], + dtype=self.xp.float64, + ) + + return nodes, weights + + @property + def lower_nodes_and_weights(self): + return self.gauss.nodes_and_weights diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_gauss_legendre.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_gauss_legendre.py new file mode 100644 index 0000000000000000000000000000000000000000..1163aec5370fb93951402ab99ee2ae4b79158d52 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_gauss_legendre.py @@ -0,0 +1,62 @@ +from scipy._lib._array_api import array_namespace, np_compat + +from functools import cached_property + +from scipy.special import roots_legendre + +from ._base import FixedRule + + +class GaussLegendreQuadrature(FixedRule): + """ + Gauss-Legendre quadrature. + + Parameters + ---------- + npoints : int + Number of nodes for the higher-order rule. + + xp : array_namespace, optional + The namespace for the node and weight arrays. Default is None, where NumPy is + used. + + Examples + -------- + Evaluate a 1D integral. Note in this example that ``f`` returns an array, so the + estimates will also be arrays. + + >>> import numpy as np + >>> from scipy.integrate import cubature + >>> from scipy.integrate._rules import GaussLegendreQuadrature + >>> def f(x): + ... return np.cos(x) + >>> rule = GaussLegendreQuadrature(21) # Use 21-point GaussLegendre + >>> a, b = np.array([0]), np.array([1]) + >>> rule.estimate(f, a, b) # True value sin(1), approximately 0.84147 + array([0.84147098]) + >>> rule.estimate_error(f, a, b) + array([1.11022302e-16]) + """ + + def __init__(self, npoints, xp=None): + if npoints < 2: + raise ValueError( + "At least 2 nodes required for Gauss-Legendre cubature" + ) + + self.npoints = npoints + + if xp is None: + xp = np_compat + + self.xp = array_namespace(xp.empty(0)) + + @cached_property + def nodes_and_weights(self): + # TODO: current converting to/from numpy + nodes, weights = roots_legendre(self.npoints) + + return ( + self.xp.asarray(nodes, dtype=self.xp.float64), + self.xp.asarray(weights, dtype=self.xp.float64) + ) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_genz_malik.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_genz_malik.py new file mode 100644 index 0000000000000000000000000000000000000000..4873805e3364b10a3366de47c15fe3c4b306e5d6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_rules/_genz_malik.py @@ -0,0 +1,210 @@ +import math +import itertools + +from functools import cached_property + +from scipy._lib._array_api import array_namespace, np_compat + +from scipy.integrate._rules import NestedFixedRule + + +class GenzMalikCubature(NestedFixedRule): + """ + Genz-Malik cubature. + + Genz-Malik is only defined for integrals of dimension >= 2. + + Parameters + ---------- + ndim : int + The spatial dimension of the integrand. + + xp : array_namespace, optional + The namespace for the node and weight arrays. Default is None, where NumPy is + used. + + Attributes + ---------- + higher : Cubature + Higher-order rule. + + lower : Cubature + Lower-order rule. + + References + ---------- + .. [1] A.C. Genz, A.A. Malik, Remarks on algorithm 006: An adaptive algorithm for + numerical integration over an N-dimensional rectangular region, Journal of + Computational and Applied Mathematics, Volume 6, Issue 4, 1980, Pages 295-302, + ISSN 0377-0427, https://doi.org/10.1016/0771-050X(80)90039-X. + + Examples + -------- + Evaluate a 3D integral: + + >>> import numpy as np + >>> from scipy.integrate import cubature + >>> from scipy.integrate._rules import GenzMalikCubature + >>> def f(x): + ... # f(x) = cos(x_1) + cos(x_2) + cos(x_3) + ... return np.sum(np.cos(x), axis=-1) + >>> rule = GenzMalikCubature(3) # Use 3D Genz-Malik + >>> a, b = np.array([0, 0, 0]), np.array([1, 1, 1]) + >>> rule.estimate(f, a, b) # True value 3*sin(1), approximately 2.5244 + np.float64(2.5244129547230862) + >>> rule.estimate_error(f, a, b) + np.float64(1.378269656626685e-06) + """ + + def __init__(self, ndim, degree=7, lower_degree=5, xp=None): + if ndim < 2: + raise ValueError("Genz-Malik cubature is only defined for ndim >= 2") + + if degree != 7 or lower_degree != 5: + raise NotImplementedError("Genz-Malik cubature is currently only supported" + "for degree=7, lower_degree=5") + + self.ndim = ndim + self.degree = degree + self.lower_degree = lower_degree + + if xp is None: + xp = np_compat + + self.xp = array_namespace(xp.empty(0)) + + @cached_property + def nodes_and_weights(self): + # TODO: Currently only support for degree 7 Genz-Malik cubature, should aim to + # support arbitrary degree + l_2 = math.sqrt(9/70) + l_3 = math.sqrt(9/10) + l_4 = math.sqrt(9/10) + l_5 = math.sqrt(9/19) + + its = itertools.chain( + [(0,) * self.ndim], + _distinct_permutations((l_2,) + (0,) * (self.ndim - 1)), + _distinct_permutations((-l_2,) + (0,) * (self.ndim - 1)), + _distinct_permutations((l_3,) + (0,) * (self.ndim - 1)), + _distinct_permutations((-l_3,) + (0,) * (self.ndim - 1)), + _distinct_permutations((l_4, l_4) + (0,) * (self.ndim - 2)), + _distinct_permutations((l_4, -l_4) + (0,) * (self.ndim - 2)), + _distinct_permutations((-l_4, -l_4) + (0,) * (self.ndim - 2)), + itertools.product((l_5, -l_5), repeat=self.ndim), + ) + + nodes_size = 1 + (2 * (self.ndim + 1) * self.ndim) + 2**self.ndim + + nodes = self.xp.asarray( + list(zip(*its)), + dtype=self.xp.float64, + ) + + nodes = self.xp.reshape(nodes, (self.ndim, nodes_size)) + + # It's convenient to generate the nodes as a sequence of evaluation points + # as an array of shape (npoints, ndim), but nodes needs to have shape + # (ndim, npoints) + nodes = nodes.T + + w_1 = ( + (2**self.ndim) * (12824 - 9120*self.ndim + (400 * self.ndim**2)) / 19683 + ) + w_2 = (2**self.ndim) * 980/6561 + w_3 = (2**self.ndim) * (1820 - 400 * self.ndim) / 19683 + w_4 = (2**self.ndim) * (200 / 19683) + w_5 = 6859 / 19683 + + weights = self.xp.concat([ + self.xp.asarray([w_1] * 1, dtype=self.xp.float64), + self.xp.asarray([w_2] * (2 * self.ndim), dtype=self.xp.float64), + self.xp.asarray([w_3] * (2 * self.ndim), dtype=self.xp.float64), + self.xp.asarray( + [w_4] * (2 * (self.ndim - 1) * self.ndim), + dtype=self.xp.float64, + ), + self.xp.asarray([w_5] * (2**self.ndim), dtype=self.xp.float64), + ]) + + return nodes, weights + + @cached_property + def lower_nodes_and_weights(self): + # TODO: Currently only support for the degree 5 lower rule, in the future it + # would be worth supporting arbitrary degree + + # Nodes are almost the same as the full rule, but there are no nodes + # corresponding to l_5. + l_2 = math.sqrt(9/70) + l_3 = math.sqrt(9/10) + l_4 = math.sqrt(9/10) + + its = itertools.chain( + [(0,) * self.ndim], + _distinct_permutations((l_2,) + (0,) * (self.ndim - 1)), + _distinct_permutations((-l_2,) + (0,) * (self.ndim - 1)), + _distinct_permutations((l_3,) + (0,) * (self.ndim - 1)), + _distinct_permutations((-l_3,) + (0,) * (self.ndim - 1)), + _distinct_permutations((l_4, l_4) + (0,) * (self.ndim - 2)), + _distinct_permutations((l_4, -l_4) + (0,) * (self.ndim - 2)), + _distinct_permutations((-l_4, -l_4) + (0,) * (self.ndim - 2)), + ) + + nodes_size = 1 + (2 * (self.ndim + 1) * self.ndim) + + nodes = self.xp.asarray(list(zip(*its)), dtype=self.xp.float64) + nodes = self.xp.reshape(nodes, (self.ndim, nodes_size)) + nodes = nodes.T + + # Weights are different from those in the full rule. + w_1 = (2**self.ndim) * (729 - 950*self.ndim + 50*self.ndim**2) / 729 + w_2 = (2**self.ndim) * (245 / 486) + w_3 = (2**self.ndim) * (265 - 100*self.ndim) / 1458 + w_4 = (2**self.ndim) * (25 / 729) + + weights = self.xp.concat([ + self.xp.asarray([w_1] * 1, dtype=self.xp.float64), + self.xp.asarray([w_2] * (2 * self.ndim), dtype=self.xp.float64), + self.xp.asarray([w_3] * (2 * self.ndim), dtype=self.xp.float64), + self.xp.asarray( + [w_4] * (2 * (self.ndim - 1) * self.ndim), + dtype=self.xp.float64, + ), + ]) + + return nodes, weights + + +def _distinct_permutations(iterable): + """ + Find the number of distinct permutations of elements of `iterable`. + """ + + # Algorithm: https://w.wiki/Qai + + items = sorted(iterable) + size = len(items) + + while True: + # Yield the permutation we have + yield tuple(items) + + # Find the largest index i such that A[i] < A[i + 1] + for i in range(size - 2, -1, -1): + if items[i] < items[i + 1]: + break + + # If no such index exists, this permutation is the last one + else: + return + + # Find the largest index j greater than j such that A[i] < A[j] + for j in range(size - 1, i, -1): + if items[i] < items[j]: + break + + # Swap the value of A[i] with that of A[j], then reverse the + # sequence from A[i + 1] to form the new permutation + items[i], items[j] = items[j], items[i] + items[i+1:] = items[:i-size:-1] # A[i + 1:][::-1] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_tanhsinh.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_tanhsinh.py new file mode 100644 index 0000000000000000000000000000000000000000..de1d844f88f999d96d4616d8060b0b47de1d8dbe --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_tanhsinh.py @@ -0,0 +1,1384 @@ +# mypy: disable-error-code="attr-defined" +import math +import numpy as np +from scipy import special +import scipy._lib._elementwise_iterative_method as eim +from scipy._lib._util import _RichResult +from scipy._lib._array_api import (array_namespace, xp_copy, xp_ravel, + xp_real, xp_take_along_axis) + + +__all__ = ['nsum'] + + +# todo: +# figure out warning situation +# address https://github.com/scipy/scipy/pull/18650#discussion_r1233032521 +# without `minweight`, we are also suppressing infinities within the interval. +# Is that OK? If so, we can probably get rid of `status=3`. +# Add heuristic to stop when improvement is too slow / antithrashing +# support singularities? interval subdivision? this feature will be added +# eventually, but do we adjust the interface now? +# When doing log-integration, should the tolerances control the error of the +# log-integral or the error of the integral? The trouble is that `log` +# inherently looses some precision so it may not be possible to refine +# the integral further. Example: 7th moment of stats.f(15, 20) +# respect function evaluation limit? +# make public? + + +def tanhsinh(f, a, b, *, args=(), log=False, maxlevel=None, minlevel=2, + atol=None, rtol=None, preserve_shape=False, callback=None): + """Evaluate a convergent integral numerically using tanh-sinh quadrature. + + In practice, tanh-sinh quadrature achieves quadratic convergence for + many integrands: the number of accurate *digits* scales roughly linearly + with the number of function evaluations [1]_. + + Either or both of the limits of integration may be infinite, and + singularities at the endpoints are acceptable. Divergent integrals and + integrands with non-finite derivatives or singularities within an interval + are out of scope, but the latter may be evaluated be calling `tanhsinh` on + each sub-interval separately. + + Parameters + ---------- + f : callable + The function to be integrated. The signature must be:: + + f(xi: ndarray, *argsi) -> ndarray + + where each element of ``xi`` is a finite real number and ``argsi`` is a tuple, + which may contain an arbitrary number of arrays that are broadcastable + with ``xi``. `f` must be an elementwise function: see documentation of parameter + `preserve_shape` for details. It must not mutate the array ``xi`` or the arrays + in ``argsi``. + If ``f`` returns a value with complex dtype when evaluated at + either endpoint, subsequent arguments ``x`` will have complex dtype + (but zero imaginary part). + a, b : float array_like + Real lower and upper limits of integration. Must be broadcastable with one + another and with arrays in `args`. Elements may be infinite. + args : tuple of array_like, optional + Additional positional array arguments to be passed to `f`. Arrays + must be broadcastable with one another and the arrays of `a` and `b`. + If the callable for which the root is desired requires arguments that are + not broadcastable with `x`, wrap that callable with `f` such that `f` + accepts only `x` and broadcastable ``*args``. + log : bool, default: False + Setting to True indicates that `f` returns the log of the integrand + and that `atol` and `rtol` are expressed as the logs of the absolute + and relative errors. In this case, the result object will contain the + log of the integral and error. This is useful for integrands for which + numerical underflow or overflow would lead to inaccuracies. + When ``log=True``, the integrand (the exponential of `f`) must be real, + but it may be negative, in which case the log of the integrand is a + complex number with an imaginary part that is an odd multiple of π. + maxlevel : int, default: 10 + The maximum refinement level of the algorithm. + + At the zeroth level, `f` is called once, performing 16 function + evaluations. At each subsequent level, `f` is called once more, + approximately doubling the number of function evaluations that have + been performed. Accordingly, for many integrands, each successive level + will double the number of accurate digits in the result (up to the + limits of floating point precision). + + The algorithm will terminate after completing level `maxlevel` or after + another termination condition is satisfied, whichever comes first. + minlevel : int, default: 2 + The level at which to begin iteration (default: 2). This does not + change the total number of function evaluations or the abscissae at + which the function is evaluated; it changes only the *number of times* + `f` is called. If ``minlevel=k``, then the integrand is evaluated at + all abscissae from levels ``0`` through ``k`` in a single call. + Note that if `minlevel` exceeds `maxlevel`, the provided `minlevel` is + ignored, and `minlevel` is set equal to `maxlevel`. + atol, rtol : float, optional + Absolute termination tolerance (default: 0) and relative termination + tolerance (default: ``eps**0.75``, where ``eps`` is the precision of + the result dtype), respectively. Iteration will stop when + ``res.error < atol`` or ``res.error < res.integral * rtol``. The error + estimate is as described in [1]_ Section 5 but with a lower bound of + ``eps * res.integral``. While not theoretically rigorous or + conservative, it is said to work well in practice. Must be non-negative + and finite if `log` is False, and must be expressed as the log of a + non-negative and finite number if `log` is True. + preserve_shape : bool, default: False + In the following, "arguments of `f`" refers to the array ``xi`` and + any arrays within ``argsi``. Let ``shape`` be the broadcasted shape + of `a`, `b`, and all elements of `args` (which is conceptually + distinct from ``xi` and ``argsi`` passed into `f`). + + - When ``preserve_shape=False`` (default), `f` must accept arguments + of *any* broadcastable shapes. + + - When ``preserve_shape=True``, `f` must accept arguments of shape + ``shape`` *or* ``shape + (n,)``, where ``(n,)`` is the number of + abscissae at which the function is being evaluated. + + In either case, for each scalar element ``xi[j]`` within ``xi``, the array + returned by `f` must include the scalar ``f(xi[j])`` at the same index. + Consequently, the shape of the output is always the shape of the input + ``xi``. + + See Examples. + + callback : callable, optional + An optional user-supplied function to be called before the first + iteration and after each iteration. + Called as ``callback(res)``, where ``res`` is a ``_RichResult`` + similar to that returned by `_differentiate` (but containing the + current iterate's values of all variables). If `callback` raises a + ``StopIteration``, the algorithm will terminate immediately and + `tanhsinh` will return a result object. `callback` must not mutate + `res` or its attributes. + + Returns + ------- + res : _RichResult + An object similar to an instance of `scipy.optimize.OptimizeResult` with the + following attributes. (The descriptions are written as though the values will + be scalars; however, if `f` returns an array, the outputs will be + arrays of the same shape.) + + success : bool array + ``True`` when the algorithm terminated successfully (status ``0``). + ``False`` otherwise. + status : int array + An integer representing the exit status of the algorithm. + + ``0`` : The algorithm converged to the specified tolerances. + ``-1`` : (unused) + ``-2`` : The maximum number of iterations was reached. + ``-3`` : A non-finite value was encountered. + ``-4`` : Iteration was terminated by `callback`. + ``1`` : The algorithm is proceeding normally (in `callback` only). + + integral : float array + An estimate of the integral. + error : float array + An estimate of the error. Only available if level two or higher + has been completed; otherwise NaN. + maxlevel : int array + The maximum refinement level used. + nfev : int array + The number of points at which `f` was evaluated. + + See Also + -------- + quad + + Notes + ----- + Implements the algorithm as described in [1]_ with minor adaptations for + finite-precision arithmetic, including some described by [2]_ and [3]_. The + tanh-sinh scheme was originally introduced in [4]_. + + Due to floating-point error in the abscissae, the function may be evaluated + at the endpoints of the interval during iterations, but the values returned by + the function at the endpoints will be ignored. + + References + ---------- + .. [1] Bailey, David H., Karthik Jeyabalan, and Xiaoye S. Li. "A comparison of + three high-precision quadrature schemes." Experimental Mathematics 14.3 + (2005): 317-329. + .. [2] Vanherck, Joren, Bart Sorée, and Wim Magnus. "Tanh-sinh quadrature for + single and multiple integration using floating-point arithmetic." + arXiv preprint arXiv:2007.15057 (2020). + .. [3] van Engelen, Robert A. "Improving the Double Exponential Quadrature + Tanh-Sinh, Sinh-Sinh and Exp-Sinh Formulas." + https://www.genivia.com/files/qthsh.pdf + .. [4] Takahasi, Hidetosi, and Masatake Mori. "Double exponential formulas for + numerical integration." Publications of the Research Institute for + Mathematical Sciences 9.3 (1974): 721-741. + + Examples + -------- + Evaluate the Gaussian integral: + + >>> import numpy as np + >>> from scipy.integrate import tanhsinh + >>> def f(x): + ... return np.exp(-x**2) + >>> res = tanhsinh(f, -np.inf, np.inf) + >>> res.integral # true value is np.sqrt(np.pi), 1.7724538509055159 + 1.7724538509055159 + >>> res.error # actual error is 0 + 4.0007963937534104e-16 + + The value of the Gaussian function (bell curve) is nearly zero for + arguments sufficiently far from zero, so the value of the integral + over a finite interval is nearly the same. + + >>> tanhsinh(f, -20, 20).integral + 1.772453850905518 + + However, with unfavorable integration limits, the integration scheme + may not be able to find the important region. + + >>> tanhsinh(f, -np.inf, 1000).integral + 4.500490856616431 + + In such cases, or when there are singularities within the interval, + break the integral into parts with endpoints at the important points. + + >>> tanhsinh(f, -np.inf, 0).integral + tanhsinh(f, 0, 1000).integral + 1.772453850905404 + + For integration involving very large or very small magnitudes, use + log-integration. (For illustrative purposes, the following example shows a + case in which both regular and log-integration work, but for more extreme + limits of integration, log-integration would avoid the underflow + experienced when evaluating the integral normally.) + + >>> res = tanhsinh(f, 20, 30, rtol=1e-10) + >>> res.integral, res.error + (4.7819613911309014e-176, 4.670364401645202e-187) + >>> def log_f(x): + ... return -x**2 + >>> res = tanhsinh(log_f, 20, 30, log=True, rtol=np.log(1e-10)) + >>> np.exp(res.integral), np.exp(res.error) + (4.7819613911306924e-176, 4.670364401645093e-187) + + The limits of integration and elements of `args` may be broadcastable + arrays, and integration is performed elementwise. + + >>> from scipy import stats + >>> dist = stats.gausshyper(13.8, 3.12, 2.51, 5.18) + >>> a, b = dist.support() + >>> x = np.linspace(a, b, 100) + >>> res = tanhsinh(dist.pdf, a, x) + >>> ref = dist.cdf(x) + >>> np.allclose(res.integral, ref) + True + + By default, `preserve_shape` is False, and therefore the callable + `f` may be called with arrays of any broadcastable shapes. + For example: + + >>> shapes = [] + >>> def f(x, c): + ... shape = np.broadcast_shapes(x.shape, c.shape) + ... shapes.append(shape) + ... return np.sin(c*x) + >>> + >>> c = [1, 10, 30, 100] + >>> res = tanhsinh(f, 0, 1, args=(c,), minlevel=1) + >>> shapes + [(4,), (4, 34), (4, 32), (3, 64), (2, 128), (1, 256)] + + To understand where these shapes are coming from - and to better + understand how `tanhsinh` computes accurate results - note that + higher values of ``c`` correspond with higher frequency sinusoids. + The higher frequency sinusoids make the integrand more complicated, + so more function evaluations are required to achieve the target + accuracy: + + >>> res.nfev + array([ 67, 131, 259, 515], dtype=int32) + + The initial ``shape``, ``(4,)``, corresponds with evaluating the + integrand at a single abscissa and all four frequencies; this is used + for input validation and to determine the size and dtype of the arrays + that store results. The next shape corresponds with evaluating the + integrand at an initial grid of abscissae and all four frequencies. + Successive calls to the function double the total number of abscissae at + which the function has been evaluated. However, in later function + evaluations, the integrand is evaluated at fewer frequencies because + the corresponding integral has already converged to the required + tolerance. This saves function evaluations to improve performance, but + it requires the function to accept arguments of any shape. + + "Vector-valued" integrands, such as those written for use with + `scipy.integrate.quad_vec`, are unlikely to satisfy this requirement. + For example, consider + + >>> def f(x): + ... return [x, np.sin(10*x), np.cos(30*x), x*np.sin(100*x)**2] + + This integrand is not compatible with `tanhsinh` as written; for instance, + the shape of the output will not be the same as the shape of ``x``. Such a + function *could* be converted to a compatible form with the introduction of + additional parameters, but this would be inconvenient. In such cases, + a simpler solution would be to use `preserve_shape`. + + >>> shapes = [] + >>> def f(x): + ... shapes.append(x.shape) + ... x0, x1, x2, x3 = x + ... return [x0, np.sin(10*x1), np.cos(30*x2), x3*np.sin(100*x3)] + >>> + >>> a = np.zeros(4) + >>> res = tanhsinh(f, a, 1, preserve_shape=True) + >>> shapes + [(4,), (4, 66), (4, 64), (4, 128), (4, 256)] + + Here, the broadcasted shape of `a` and `b` is ``(4,)``. With + ``preserve_shape=True``, the function may be called with argument + ``x`` of shape ``(4,)`` or ``(4, n)``, and this is what we observe. + + """ + maxfun = None # unused right now + (f, a, b, log, maxfun, maxlevel, minlevel, + atol, rtol, args, preserve_shape, callback, xp) = _tanhsinh_iv( + f, a, b, log, maxfun, maxlevel, minlevel, atol, + rtol, args, preserve_shape, callback) + + # Initialization + # `eim._initialize` does several important jobs, including + # ensuring that limits, each of the `args`, and the output of `f` + # broadcast correctly and are of consistent types. To save a function + # evaluation, I pass the midpoint of the integration interval. This comes + # at a cost of some gymnastics to ensure that the midpoint has the right + # shape and dtype. Did you know that 0d and >0d arrays follow different + # type promotion rules? + with np.errstate(over='ignore', invalid='ignore', divide='ignore'): + c = xp.reshape((xp_ravel(a) + xp_ravel(b))/2, a.shape) + inf_a, inf_b = xp.isinf(a), xp.isinf(b) + c[inf_a] = b[inf_a] - 1. # takes care of infinite a + c[inf_b] = a[inf_b] + 1. # takes care of infinite b + c[inf_a & inf_b] = 0. # takes care of infinite a and b + temp = eim._initialize(f, (c,), args, complex_ok=True, + preserve_shape=preserve_shape, xp=xp) + f, xs, fs, args, shape, dtype, xp = temp + a = xp_ravel(xp.astype(xp.broadcast_to(a, shape), dtype)) + b = xp_ravel(xp.astype(xp.broadcast_to(b, shape), dtype)) + + # Transform improper integrals + a, b, a0, negative, abinf, ainf, binf = _transform_integrals(a, b, xp) + + # Define variables we'll need + nit, nfev = 0, 1 # one function evaluation performed above + zero = -xp.inf if log else 0 + pi = xp.asarray(xp.pi, dtype=dtype)[()] + maxiter = maxlevel - minlevel + 1 + eps = xp.finfo(dtype).eps + if rtol is None: + rtol = 0.75*math.log(eps) if log else eps**0.75 + + Sn = xp_ravel(xp.full(shape, zero, dtype=dtype)) # latest integral estimate + Sn[xp.isnan(a) | xp.isnan(b) | xp.isnan(fs[0])] = xp.nan + Sk = xp.reshape(xp.empty_like(Sn), (-1, 1))[:, 0:0] # all integral estimates + aerr = xp_ravel(xp.full(shape, xp.nan, dtype=dtype)) # absolute error + status = xp_ravel(xp.full(shape, eim._EINPROGRESS, dtype=xp.int32)) + h0 = _get_base_step(dtype, xp) + h0 = xp_real(h0) # base step + + # For term `d4` of error estimate ([1] Section 5), we need to keep the + # most extreme abscissae and corresponding `fj`s, `wj`s in Euler-Maclaurin + # sum. Here, we initialize these variables. + xr0 = xp_ravel(xp.full(shape, -xp.inf, dtype=dtype)) + fr0 = xp_ravel(xp.full(shape, xp.nan, dtype=dtype)) + wr0 = xp_ravel(xp.zeros(shape, dtype=dtype)) + xl0 = xp_ravel(xp.full(shape, xp.inf, dtype=dtype)) + fl0 = xp_ravel(xp.full(shape, xp.nan, dtype=dtype)) + wl0 = xp_ravel(xp.zeros(shape, dtype=dtype)) + d4 = xp_ravel(xp.zeros(shape, dtype=dtype)) + + work = _RichResult( + Sn=Sn, Sk=Sk, aerr=aerr, h=h0, log=log, dtype=dtype, pi=pi, eps=eps, + a=xp.reshape(a, (-1, 1)), b=xp.reshape(b, (-1, 1)), # integration limits + n=minlevel, nit=nit, nfev=nfev, status=status, # iter/eval counts + xr0=xr0, fr0=fr0, wr0=wr0, xl0=xl0, fl0=fl0, wl0=wl0, d4=d4, # err est + ainf=ainf, binf=binf, abinf=abinf, a0=xp.reshape(a0, (-1, 1)), # transforms + # Store the xjc/wj pair cache in an object so they can't get compressed + # Using RichResult to allow dot notation, but a dictionary would suffice + pair_cache=_RichResult(xjc=None, wj=None, indices=[0], h0=None)) # pair cache + + # Constant scalars don't need to be put in `work` unless they need to be + # passed outside `tanhsinh`. Examples: atol, rtol, h0, minlevel. + + # Correspondence between terms in the `work` object and the result + res_work_pairs = [('status', 'status'), ('integral', 'Sn'), + ('error', 'aerr'), ('nit', 'nit'), ('nfev', 'nfev')] + + def pre_func_eval(work): + # Determine abscissae at which to evaluate `f` + work.h = h0 / 2**work.n + xjc, wj = _get_pairs(work.n, h0, dtype=work.dtype, + inclusive=(work.n == minlevel), xp=xp, work=work) + work.xj, work.wj = _transform_to_limits(xjc, wj, work.a, work.b, xp) + + # Perform abscissae substitutions for infinite limits of integration + xj = xp_copy(work.xj) + # use xp_real here to avoid cupy/cupy#8434 + xj[work.abinf] = xj[work.abinf] / (1 - xp_real(xj[work.abinf])**2) + xj[work.binf] = 1/xj[work.binf] - 1 + work.a0[work.binf] + xj[work.ainf] *= -1 + return xj + + def post_func_eval(x, fj, work): + # Weight integrand as required by substitutions for infinite limits + if work.log: + fj[work.abinf] += (xp.log(1 + work.xj[work.abinf]**2) + - 2*xp.log(1 - work.xj[work.abinf]**2)) + fj[work.binf] -= 2 * xp.log(work.xj[work.binf]) + else: + fj[work.abinf] *= ((1 + work.xj[work.abinf]**2) / + (1 - work.xj[work.abinf]**2)**2) + fj[work.binf] *= work.xj[work.binf]**-2. + + # Estimate integral with Euler-Maclaurin Sum + fjwj, Sn = _euler_maclaurin_sum(fj, work, xp) + if work.Sk.shape[-1]: + Snm1 = work.Sk[:, -1] + Sn = (special.logsumexp(xp.stack([Snm1 - math.log(2), Sn]), axis=0) if log + else Snm1 / 2 + Sn) + + work.fjwj = fjwj + work.Sn = Sn + + def check_termination(work): + """Terminate due to convergence or encountering non-finite values""" + stop = xp.zeros(work.Sn.shape, dtype=bool) + + # Terminate before first iteration if integration limits are equal + if work.nit == 0: + i = xp_ravel(work.a == work.b) # ravel singleton dimension + zero = xp.asarray(-xp.inf if log else 0.) + zero = xp.full(work.Sn.shape, zero, dtype=Sn.dtype) + zero[xp.isnan(Sn)] = xp.nan + work.Sn[i] = zero[i] + work.aerr[i] = zero[i] + work.status[i] = eim._ECONVERGED + stop[i] = True + else: + # Terminate if convergence criterion is met + rerr, aerr = _estimate_error(work, xp) + i = (rerr < rtol) | (aerr < atol) + work.aerr = xp.reshape(xp.astype(aerr, work.dtype), work.Sn.shape) + work.status[i] = eim._ECONVERGED + stop[i] = True + + # Terminate if integral estimate becomes invalid + if log: + Sn_real = xp_real(work.Sn) + Sn_pos_inf = xp.isinf(Sn_real) & (Sn_real > 0) + i = (Sn_pos_inf | xp.isnan(work.Sn)) & ~stop + else: + i = ~xp.isfinite(work.Sn) & ~stop + work.status[i] = eim._EVALUEERR + stop[i] = True + + return stop + + def post_termination_check(work): + work.n += 1 + work.Sk = xp.concat((work.Sk, work.Sn[:, xp.newaxis]), axis=-1) + return + + def customize_result(res, shape): + # If the integration limits were such that b < a, we reversed them + # to perform the calculation, and the final result needs to be negated. + if log and xp.any(negative): + dtype = res['integral'].dtype + pi = xp.asarray(xp.pi, dtype=dtype)[()] + j = xp.asarray(1j, dtype=xp.complex64)[()] # minimum complex type + res['integral'] = res['integral'] + negative*pi*j + else: + res['integral'][negative] *= -1 + + # For this algorithm, it seems more appropriate to report the maximum + # level rather than the number of iterations in which it was performed. + res['maxlevel'] = minlevel + res['nit'] - 1 + res['maxlevel'][res['nit'] == 0] = -1 + del res['nit'] + return shape + + # Suppress all warnings initially, since there are many places in the code + # for which this is expected behavior. + with np.errstate(over='ignore', invalid='ignore', divide='ignore'): + res = eim._loop(work, callback, shape, maxiter, f, args, dtype, pre_func_eval, + post_func_eval, check_termination, post_termination_check, + customize_result, res_work_pairs, xp, preserve_shape) + return res + + +def _get_base_step(dtype, xp): + # Compute the base step length for the provided dtype. Theoretically, the + # Euler-Maclaurin sum is infinite, but it gets cut off when either the + # weights underflow or the abscissae cannot be distinguished from the + # limits of integration. The latter happens to occur first for float32 and + # float64, and it occurs when `xjc` (the abscissa complement) + # in `_compute_pair` underflows. We can solve for the argument `tmax` at + # which it will underflow using [2] Eq. 13. + fmin = 4*xp.finfo(dtype).smallest_normal # stay a little away from the limit + tmax = math.asinh(math.log(2/fmin - 1) / xp.pi) + + # Based on this, we can choose a base step size `h` for level 0. + # The number of function evaluations will be `2 + m*2^(k+1)`, where `k` is + # the level and `m` is an integer we get to choose. I choose + # m = _N_BASE_STEPS = `8` somewhat arbitrarily, but a rationale is that a + # power of 2 makes floating point arithmetic more predictable. It also + # results in a base step size close to `1`, which is what [1] uses (and I + # used here until I found [2] and these ideas settled). + h0 = tmax / _N_BASE_STEPS + return xp.asarray(h0, dtype=dtype)[()] + + +_N_BASE_STEPS = 8 + + +def _compute_pair(k, h0, xp): + # Compute the abscissa-weight pairs for each level k. See [1] page 9. + + # For now, we compute and store in 64-bit precision. If higher-precision + # data types become better supported, it would be good to compute these + # using the highest precision available. Or, once there is an Array API- + # compatible arbitrary precision array, we can compute at the required + # precision. + + # "....each level k of abscissa-weight pairs uses h = 2 **-k" + # We adapt to floating point arithmetic using ideas of [2]. + h = h0 / 2**k + max = _N_BASE_STEPS * 2**k + + # For iterations after the first, "....the integrand function needs to be + # evaluated only at the odd-indexed abscissas at each level." + j = xp.arange(max+1) if k == 0 else xp.arange(1, max+1, 2) + jh = j * h + + # "In this case... the weights wj = u1/cosh(u2)^2, where..." + pi_2 = xp.pi / 2 + u1 = pi_2*xp.cosh(jh) + u2 = pi_2*xp.sinh(jh) + # Denominators get big here. Overflow then underflow doesn't need warning. + # with np.errstate(under='ignore', over='ignore'): + wj = u1 / xp.cosh(u2)**2 + # "We actually store 1-xj = 1/(...)." + xjc = 1 / (xp.exp(u2) * xp.cosh(u2)) # complement of xj = xp.tanh(u2) + + # When level k == 0, the zeroth xj corresponds with xj = 0. To simplify + # code, the function will be evaluated there twice; each gets half weight. + wj[0] = wj[0] / 2 if k == 0 else wj[0] + + return xjc, wj # store at full precision + + +def _pair_cache(k, h0, xp, work): + # Cache the abscissa-weight pairs up to a specified level. + # Abscissae and weights of consecutive levels are concatenated. + # `index` records the indices that correspond with each level: + # `xjc[index[k]:index[k+1]` extracts the level `k` abscissae. + if not isinstance(h0, type(work.pair_cache.h0)) or h0 != work.pair_cache.h0: + work.pair_cache.xjc = xp.empty(0) + work.pair_cache.wj = xp.empty(0) + work.pair_cache.indices = [0] + + xjcs = [work.pair_cache.xjc] + wjs = [work.pair_cache.wj] + + for i in range(len(work.pair_cache.indices)-1, k + 1): + xjc, wj = _compute_pair(i, h0, xp) + xjcs.append(xjc) + wjs.append(wj) + work.pair_cache.indices.append(work.pair_cache.indices[-1] + xjc.shape[0]) + + work.pair_cache.xjc = xp.concat(xjcs) + work.pair_cache.wj = xp.concat(wjs) + work.pair_cache.h0 = h0 + + +def _get_pairs(k, h0, inclusive, dtype, xp, work): + # Retrieve the specified abscissa-weight pairs from the cache + # If `inclusive`, return all up to and including the specified level + if (len(work.pair_cache.indices) <= k+2 + or not isinstance (h0, type(work.pair_cache.h0)) + or h0 != work.pair_cache.h0): + _pair_cache(k, h0, xp, work) + + xjc = work.pair_cache.xjc + wj = work.pair_cache.wj + indices = work.pair_cache.indices + + start = 0 if inclusive else indices[k] + end = indices[k+1] + + return xp.astype(xjc[start:end], dtype), xp.astype(wj[start:end], dtype) + + +def _transform_to_limits(xjc, wj, a, b, xp): + # Transform integral according to user-specified limits. This is just + # math that follows from the fact that the standard limits are (-1, 1). + # Note: If we had stored xj instead of xjc, we would have + # xj = alpha * xj + beta, where beta = (a + b)/2 + alpha = (b - a) / 2 + xj = xp.concat((-alpha * xjc + b, alpha * xjc + a), axis=-1) + wj = wj*alpha # arguments get broadcasted, so we can't use *= + wj = xp.concat((wj, wj), axis=-1) + + # Points at the boundaries can be generated due to finite precision + # arithmetic, but these function values aren't supposed to be included in + # the Euler-Maclaurin sum. Ideally we wouldn't evaluate the function at + # these points; however, we can't easily filter out points since this + # function is vectorized. Instead, zero the weights. + # Note: values may have complex dtype, but have zero imaginary part + xj_real, a_real, b_real = xp_real(xj), xp_real(a), xp_real(b) + invalid = (xj_real <= a_real) | (xj_real >= b_real) + wj[invalid] = 0 + return xj, wj + + +def _euler_maclaurin_sum(fj, work, xp): + # Perform the Euler-Maclaurin Sum, [1] Section 4 + + # The error estimate needs to know the magnitude of the last term + # omitted from the Euler-Maclaurin sum. This is a bit involved because + # it may have been computed at a previous level. I sure hope it's worth + # all the trouble. + xr0, fr0, wr0 = work.xr0, work.fr0, work.wr0 + xl0, fl0, wl0 = work.xl0, work.fl0, work.wl0 + + # It is much more convenient to work with the transposes of our work + # variables here. + xj, fj, wj = work.xj.T, fj.T, work.wj.T + n_x, n_active = xj.shape # number of abscissae, number of active elements + + # We'll work with the left and right sides separately + xr, xl = xp_copy(xp.reshape(xj, (2, n_x // 2, n_active))) # this gets modified + fr, fl = xp.reshape(fj, (2, n_x // 2, n_active)) + wr, wl = xp.reshape(wj, (2, n_x // 2, n_active)) + + invalid_r = ~xp.isfinite(fr) | (wr == 0) + invalid_l = ~xp.isfinite(fl) | (wl == 0) + + # integer index of the maximum abscissa at this level + xr[invalid_r] = -xp.inf + ir = xp.argmax(xp_real(xr), axis=0, keepdims=True) + # abscissa, function value, and weight at this index + ### Not Array API Compatible... yet ### + xr_max = xp_take_along_axis(xr, ir, axis=0)[0] + fr_max = xp_take_along_axis(fr, ir, axis=0)[0] + wr_max = xp_take_along_axis(wr, ir, axis=0)[0] + # boolean indices at which maximum abscissa at this level exceeds + # the incumbent maximum abscissa (from all previous levels) + # note: abscissa may have complex dtype, but will have zero imaginary part + j = xp_real(xr_max) > xp_real(xr0) + # Update record of the incumbent abscissa, function value, and weight + xr0[j] = xr_max[j] + fr0[j] = fr_max[j] + wr0[j] = wr_max[j] + + # integer index of the minimum abscissa at this level + xl[invalid_l] = xp.inf + il = xp.argmin(xp_real(xl), axis=0, keepdims=True) + # abscissa, function value, and weight at this index + xl_min = xp_take_along_axis(xl, il, axis=0)[0] + fl_min = xp_take_along_axis(fl, il, axis=0)[0] + wl_min = xp_take_along_axis(wl, il, axis=0)[0] + # boolean indices at which minimum abscissa at this level is less than + # the incumbent minimum abscissa (from all previous levels) + # note: abscissa may have complex dtype, but will have zero imaginary part + j = xp_real(xl_min) < xp_real(xl0) + # Update record of the incumbent abscissa, function value, and weight + xl0[j] = xl_min[j] + fl0[j] = fl_min[j] + wl0[j] = wl_min[j] + fj = fj.T + + # Compute the error estimate `d4` - the magnitude of the leftmost or + # rightmost term, whichever is greater. + flwl0 = fl0 + xp.log(wl0) if work.log else fl0 * wl0 # leftmost term + frwr0 = fr0 + xp.log(wr0) if work.log else fr0 * wr0 # rightmost term + magnitude = xp_real if work.log else xp.abs + work.d4 = xp.maximum(magnitude(flwl0), magnitude(frwr0)) + + # There are two approaches to dealing with function values that are + # numerically infinite due to approaching a singularity - zero them, or + # replace them with the function value at the nearest non-infinite point. + # [3] pg. 22 suggests the latter, so let's do that given that we have the + # information. + fr0b = xp.broadcast_to(fr0[xp.newaxis, :], fr.shape) + fl0b = xp.broadcast_to(fl0[xp.newaxis, :], fl.shape) + fr[invalid_r] = fr0b[invalid_r] + fl[invalid_l] = fl0b[invalid_l] + + # When wj is zero, log emits a warning + # with np.errstate(divide='ignore'): + fjwj = fj + xp.log(work.wj) if work.log else fj * work.wj + + # update integral estimate + Sn = (special.logsumexp(fjwj + xp.log(work.h), axis=-1) if work.log + else xp.sum(fjwj, axis=-1) * work.h) + + work.xr0, work.fr0, work.wr0 = xr0, fr0, wr0 + work.xl0, work.fl0, work.wl0 = xl0, fl0, wl0 + + return fjwj, Sn + + +def _estimate_error(work, xp): + # Estimate the error according to [1] Section 5 + + if work.n == 0 or work.nit == 0: + # The paper says to use "one" as the error before it can be calculated. + # NaN seems to be more appropriate. + nan = xp.full_like(work.Sn, xp.nan) + return nan, nan + + indices = work.pair_cache.indices + + n_active = work.Sn.shape[0] # number of active elements + axis_kwargs = dict(axis=-1, keepdims=True) + + # With a jump start (starting at level higher than 0), we haven't + # explicitly calculated the integral estimate at lower levels. But we have + # all the function value-weight products, so we can compute the + # lower-level estimates. + if work.Sk.shape[-1] == 0: + h = 2 * work.h # step size at this level + n_x = indices[work.n] # number of abscissa up to this level + # The right and left fjwj terms from all levels are concatenated along + # the last axis. Get out only the terms up to this level. + fjwj_rl = xp.reshape(work.fjwj, (n_active, 2, -1)) + fjwj = xp.reshape(fjwj_rl[:, :, :n_x], (n_active, 2*n_x)) + # Compute the Euler-Maclaurin sum at this level + Snm1 = (special.logsumexp(fjwj, **axis_kwargs) + xp.log(h) if work.log + else xp.sum(fjwj, **axis_kwargs) * h) + work.Sk = xp.concat((Snm1, work.Sk), axis=-1) + + if work.n == 1: + nan = xp.full_like(work.Sn, xp.nan) + return nan, nan + + # The paper says not to calculate the error for n<=2, but it's not clear + # about whether it starts at level 0 or level 1. We start at level 0, so + # why not compute the error beginning in level 2? + if work.Sk.shape[-1] < 2: + h = 4 * work.h # step size at this level + n_x = indices[work.n-1] # number of abscissa up to this level + # The right and left fjwj terms from all levels are concatenated along + # the last axis. Get out only the terms up to this level. + fjwj_rl = xp.reshape(work.fjwj, (work.Sn.shape[0], 2, -1)) + fjwj = xp.reshape(fjwj_rl[..., :n_x], (n_active, 2*n_x)) + # Compute the Euler-Maclaurin sum at this level + Snm2 = (special.logsumexp(fjwj, **axis_kwargs) + xp.log(h) if work.log + else xp.sum(fjwj, **axis_kwargs) * h) + work.Sk = xp.concat((Snm2, work.Sk), axis=-1) + + Snm2 = work.Sk[..., -2] + Snm1 = work.Sk[..., -1] + + e1 = xp.asarray(work.eps)[()] + + if work.log: + log_e1 = xp.log(e1) + # Currently, only real integrals are supported in log-scale. All + # complex values have imaginary part in increments of pi*j, which just + # carries sign information of the original integral, so use of + # `xp.real` here is equivalent to absolute value in real scale. + d1 = xp_real(special.logsumexp(xp.stack([work.Sn, Snm1 + work.pi*1j]), axis=0)) + d2 = xp_real(special.logsumexp(xp.stack([work.Sn, Snm2 + work.pi*1j]), axis=0)) + d3 = log_e1 + xp.max(xp_real(work.fjwj), axis=-1) + d4 = work.d4 + d5 = log_e1 + xp.real(work.Sn) + temp = xp.where(d1 > -xp.inf, d1 ** 2 / d2, -xp.inf) + ds = xp.stack([temp, 2 * d1, d3, d4, d5]) + aerr = xp.max(ds, axis=0) + rerr = aerr - xp.real(work.Sn) + else: + # Note: explicit computation of log10 of each of these is unnecessary. + d1 = xp.abs(work.Sn - Snm1) + d2 = xp.abs(work.Sn - Snm2) + d3 = e1 * xp.max(xp.abs(work.fjwj), axis=-1) + d4 = work.d4 + d5 = e1 * xp.abs(work.Sn) + temp = xp.where(d1 > 0, d1**(xp.log(d1)/xp.log(d2)), 0) + ds = xp.stack([temp, d1**2, d3, d4, d5]) + aerr = xp.max(ds, axis=0) + rerr = aerr/xp.abs(work.Sn) + + return rerr, aerr + + +def _transform_integrals(a, b, xp): + # Transform integrals to a form with finite a <= b + # For b == a (even infinite), we ensure that the limits remain equal + # For b < a, we reverse the limits and will multiply the final result by -1 + # For infinite limit on the right, we use the substitution x = 1/t - 1 + a + # For infinite limit on the left, we substitute x = -x and treat as above + # For infinite limits, we substitute x = t / (1-t**2) + ab_same = (a == b) + a[ab_same], b[ab_same] = 1, 1 + + # `a, b` may have complex dtype but have zero imaginary part + negative = xp_real(b) < xp_real(a) + a[negative], b[negative] = b[negative], a[negative] + + abinf = xp.isinf(a) & xp.isinf(b) + a[abinf], b[abinf] = -1, 1 + + ainf = xp.isinf(a) + a[ainf], b[ainf] = -b[ainf], -a[ainf] + + binf = xp.isinf(b) + a0 = xp_copy(a) + a[binf], b[binf] = 0, 1 + + return a, b, a0, negative, abinf, ainf, binf + + +def _tanhsinh_iv(f, a, b, log, maxfun, maxlevel, minlevel, + atol, rtol, args, preserve_shape, callback): + # Input validation and standardization + + xp = array_namespace(a, b) + + message = '`f` must be callable.' + if not callable(f): + raise ValueError(message) + + message = 'All elements of `a` and `b` must be real numbers.' + a, b = xp.asarray(a), xp.asarray(b) + a, b = xp.broadcast_arrays(a, b) + if (xp.isdtype(a.dtype, 'complex floating') + or xp.isdtype(b.dtype, 'complex floating')): + raise ValueError(message) + + message = '`log` must be True or False.' + if log not in {True, False}: + raise ValueError(message) + log = bool(log) + + if atol is None: + atol = -xp.inf if log else 0 + + rtol_temp = rtol if rtol is not None else 0. + + # using NumPy for convenience here; these are just floats, not arrays + params = np.asarray([atol, rtol_temp, 0.]) + message = "`atol` and `rtol` must be real numbers." + if not np.issubdtype(params.dtype, np.floating): + raise ValueError(message) + + if log: + message = '`atol` and `rtol` may not be positive infinity.' + if np.any(np.isposinf(params)): + raise ValueError(message) + else: + message = '`atol` and `rtol` must be non-negative and finite.' + if np.any(params < 0) or np.any(np.isinf(params)): + raise ValueError(message) + atol = params[0] + rtol = rtol if rtol is None else params[1] + + BIGINT = float(2**62) + if maxfun is None and maxlevel is None: + maxlevel = 10 + + maxfun = BIGINT if maxfun is None else maxfun + maxlevel = BIGINT if maxlevel is None else maxlevel + + message = '`maxfun`, `maxlevel`, and `minlevel` must be integers.' + params = np.asarray([maxfun, maxlevel, minlevel]) + if not (np.issubdtype(params.dtype, np.number) + and np.all(np.isreal(params)) + and np.all(params.astype(np.int64) == params)): + raise ValueError(message) + message = '`maxfun`, `maxlevel`, and `minlevel` must be non-negative.' + if np.any(params < 0): + raise ValueError(message) + maxfun, maxlevel, minlevel = params.astype(np.int64) + minlevel = min(minlevel, maxlevel) + + if not np.iterable(args): + args = (args,) + args = (xp.asarray(arg) for arg in args) + + message = '`preserve_shape` must be True or False.' + if preserve_shape not in {True, False}: + raise ValueError(message) + + if callback is not None and not callable(callback): + raise ValueError('`callback` must be callable.') + + return (f, a, b, log, maxfun, maxlevel, minlevel, + atol, rtol, args, preserve_shape, callback, xp) + + +def _nsum_iv(f, a, b, step, args, log, maxterms, tolerances): + # Input validation and standardization + + xp = array_namespace(a, b) + + message = '`f` must be callable.' + if not callable(f): + raise ValueError(message) + + message = 'All elements of `a`, `b`, and `step` must be real numbers.' + a, b, step = xp.broadcast_arrays(xp.asarray(a), xp.asarray(b), xp.asarray(step)) + dtype = xp.result_type(a.dtype, b.dtype, step.dtype) + if not xp.isdtype(dtype, 'numeric') or xp.isdtype(dtype, 'complex floating'): + raise ValueError(message) + + valid_b = b >= a # NaNs will be False + valid_step = xp.isfinite(step) & (step > 0) + valid_abstep = valid_b & valid_step + + message = '`log` must be True or False.' + if log not in {True, False}: + raise ValueError(message) + + tolerances = {} if tolerances is None else tolerances + + atol = tolerances.get('atol', None) + if atol is None: + atol = -xp.inf if log else 0 + + rtol = tolerances.get('rtol', None) + rtol_temp = rtol if rtol is not None else 0. + + # using NumPy for convenience here; these are just floats, not arrays + params = np.asarray([atol, rtol_temp, 0.]) + message = "`atol` and `rtol` must be real numbers." + if not np.issubdtype(params.dtype, np.floating): + raise ValueError(message) + + if log: + message = '`atol`, `rtol` may not be positive infinity or NaN.' + if np.any(np.isposinf(params) | np.isnan(params)): + raise ValueError(message) + else: + message = '`atol`, and `rtol` must be non-negative and finite.' + if np.any((params < 0) | (~np.isfinite(params))): + raise ValueError(message) + atol = params[0] + rtol = rtol if rtol is None else params[1] + + maxterms_int = int(maxterms) + if maxterms_int != maxterms or maxterms < 0: + message = "`maxterms` must be a non-negative integer." + raise ValueError(message) + + if not np.iterable(args): + args = (args,) + + return f, a, b, step, valid_abstep, args, log, maxterms_int, atol, rtol, xp + + +def nsum(f, a, b, *, step=1, args=(), log=False, maxterms=int(2**20), tolerances=None): + r"""Evaluate a convergent finite or infinite series. + + For finite `a` and `b`, this evaluates:: + + f(a + np.arange(n)*step).sum() + + where ``n = int((b - a) / step) + 1``, where `f` is smooth, positive, and + unimodal. The number of terms in the sum may be very large or infinite, + in which case a partial sum is evaluated directly and the remainder is + approximated using integration. + + Parameters + ---------- + f : callable + The function that evaluates terms to be summed. The signature must be:: + + f(x: ndarray, *args) -> ndarray + + where each element of ``x`` is a finite real and ``args`` is a tuple, + which may contain an arbitrary number of arrays that are broadcastable + with ``x``. + + `f` must be an elementwise function: each element ``f(x)[i]`` + must equal ``f(x[i])`` for all indices ``i``. It must not mutate the + array ``x`` or the arrays in ``args``, and it must return NaN where + the argument is NaN. + + `f` must represent a smooth, positive, unimodal function of `x` defined at + *all reals* between `a` and `b`. + a, b : float array_like + Real lower and upper limits of summed terms. Must be broadcastable. + Each element of `a` must be less than the corresponding element in `b`. + step : float array_like + Finite, positive, real step between summed terms. Must be broadcastable + with `a` and `b`. Note that the number of terms included in the sum will + be ``floor((b - a) / step)`` + 1; adjust `b` accordingly to ensure + that ``f(b)`` is included if intended. + args : tuple of array_like, optional + Additional positional arguments to be passed to `f`. Must be arrays + broadcastable with `a`, `b`, and `step`. If the callable to be summed + requires arguments that are not broadcastable with `a`, `b`, and `step`, + wrap that callable with `f` such that `f` accepts only `x` and + broadcastable ``*args``. See Examples. + log : bool, default: False + Setting to True indicates that `f` returns the log of the terms + and that `atol` and `rtol` are expressed as the logs of the absolute + and relative errors. In this case, the result object will contain the + log of the sum and error. This is useful for summands for which + numerical underflow or overflow would lead to inaccuracies. + maxterms : int, default: 2**20 + The maximum number of terms to evaluate for direct summation. + Additional function evaluations may be performed for input + validation and integral evaluation. + atol, rtol : float, optional + Absolute termination tolerance (default: 0) and relative termination + tolerance (default: ``eps**0.5``, where ``eps`` is the precision of + the result dtype), respectively. Must be non-negative + and finite if `log` is False, and must be expressed as the log of a + non-negative and finite number if `log` is True. + + Returns + ------- + res : _RichResult + An object similar to an instance of `scipy.optimize.OptimizeResult` with the + following attributes. (The descriptions are written as though the values will + be scalars; however, if `f` returns an array, the outputs will be + arrays of the same shape.) + + success : bool + ``True`` when the algorithm terminated successfully (status ``0``); + ``False`` otherwise. + status : int array + An integer representing the exit status of the algorithm. + + - ``0`` : The algorithm converged to the specified tolerances. + - ``-1`` : Element(s) of `a`, `b`, or `step` are invalid + - ``-2`` : Numerical integration reached its iteration limit; + the sum may be divergent. + - ``-3`` : A non-finite value was encountered. + - ``-4`` : The magnitude of the last term of the partial sum exceeds + the tolerances, so the error estimate exceeds the tolerances. + Consider increasing `maxterms` or loosening `tolerances`. + Alternatively, the callable may not be unimodal, or the limits of + summation may be too far from the function maximum. Consider + increasing `maxterms` or breaking the sum into pieces. + + sum : float array + An estimate of the sum. + error : float array + An estimate of the absolute error, assuming all terms are non-negative, + the function is computed exactly, and direct summation is accurate to + the precision of the result dtype. + nfev : int array + The number of points at which `f` was evaluated. + + See Also + -------- + mpmath.nsum + + Notes + ----- + The method implemented for infinite summation is related to the integral + test for convergence of an infinite series: assuming `step` size 1 for + simplicity of exposition, the sum of a monotone decreasing function is bounded by + + .. math:: + + \int_u^\infty f(x) dx \leq \sum_{k=u}^\infty f(k) \leq \int_u^\infty f(x) dx + f(u) + + Let :math:`a` represent `a`, :math:`n` represent `maxterms`, :math:`\epsilon_a` + represent `atol`, and :math:`\epsilon_r` represent `rtol`. + The implementation first evaluates the integral :math:`S_l=\int_a^\infty f(x) dx` + as a lower bound of the infinite sum. Then, it seeks a value :math:`c > a` such + that :math:`f(c) < \epsilon_a + S_l \epsilon_r`, if it exists; otherwise, + let :math:`c = a + n`. Then the infinite sum is approximated as + + .. math:: + + \sum_{k=a}^{c-1} f(k) + \int_c^\infty f(x) dx + f(c)/2 + + and the reported error is :math:`f(c)/2` plus the error estimate of + numerical integration. Note that the integral approximations may require + evaluation of the function at points besides those that appear in the sum, + so `f` must be a continuous and monotonically decreasing function defined + for all reals within the integration interval. However, due to the nature + of the integral approximation, the shape of the function between points + that appear in the sum has little effect. If there is not a natural + extension of the function to all reals, consider using linear interpolation, + which is easy to evaluate and preserves monotonicity. + + The approach described above is generalized for non-unit + `step` and finite `b` that is too large for direct evaluation of the sum, + i.e. ``b - a + 1 > maxterms``. It is further generalized to unimodal + functions by directly summing terms surrounding the maximum. + This strategy may fail: + + - If the left limit is finite and the maximum is far from it. + - If the right limit is finite and the maximum is far from it. + - If both limits are finite and the maximum is far from the origin. + + In these cases, accuracy may be poor, and `nsum` may return status code ``4``. + + Although the callable `f` must be non-negative and unimodal, + `nsum` can be used to evaluate more general forms of series. For instance, to + evaluate an alternating series, pass a callable that returns the difference + between pairs of adjacent terms, and adjust `step` accordingly. See Examples. + + References + ---------- + .. [1] Wikipedia. "Integral test for convergence." + https://en.wikipedia.org/wiki/Integral_test_for_convergence + + Examples + -------- + Compute the infinite sum of the reciprocals of squared integers. + + >>> import numpy as np + >>> from scipy.integrate import nsum + >>> res = nsum(lambda k: 1/k**2, 1, np.inf) + >>> ref = np.pi**2/6 # true value + >>> res.error # estimated error + np.float64(7.448762306416137e-09) + >>> (res.sum - ref)/ref # true error + np.float64(-1.839871898894426e-13) + >>> res.nfev # number of points at which callable was evaluated + np.int32(8561) + + Compute the infinite sums of the reciprocals of integers raised to powers ``p``, + where ``p`` is an array. + + >>> from scipy import special + >>> p = np.arange(3, 10) + >>> res = nsum(lambda k, p: 1/k**p, 1, np.inf, maxterms=1e3, args=(p,)) + >>> ref = special.zeta(p, 1) + >>> np.allclose(res.sum, ref) + True + + Evaluate the alternating harmonic series. + + >>> res = nsum(lambda x: 1/x - 1/(x+1), 1, np.inf, step=2) + >>> res.sum, res.sum - np.log(2) # result, difference vs analytical sum + (np.float64(0.6931471805598691), np.float64(-7.616129948928574e-14)) + + """ # noqa: E501 + # Potential future work: + # - improve error estimate of `_direct` sum + # - add other methods for convergence acceleration (Richardson, epsilon) + # - support negative monotone increasing functions? + # - b < a / negative step? + # - complex-valued function? + # - check for violations of monotonicity? + + # Function-specific input validation / standardization + tmp = _nsum_iv(f, a, b, step, args, log, maxterms, tolerances) + f, a, b, step, valid_abstep, args, log, maxterms, atol, rtol, xp = tmp + + # Additional elementwise algorithm input validation / standardization + tmp = eim._initialize(f, (a,), args, complex_ok=False, xp=xp) + f, xs, fs, args, shape, dtype, xp = tmp + + # Finish preparing `a`, `b`, and `step` arrays + a = xs[0] + b = xp.astype(xp_ravel(xp.broadcast_to(b, shape)), dtype) + step = xp.astype(xp_ravel(xp.broadcast_to(step, shape)), dtype) + valid_abstep = xp_ravel(xp.broadcast_to(valid_abstep, shape)) + nterms = xp.floor((b - a) / step) + finite_terms = xp.isfinite(nterms) + b[finite_terms] = a[finite_terms] + nterms[finite_terms]*step[finite_terms] + + # Define constants + eps = xp.finfo(dtype).eps + zero = xp.asarray(-xp.inf if log else 0, dtype=dtype)[()] + if rtol is None: + rtol = 0.5*math.log(eps) if log else eps**0.5 + constants = (dtype, log, eps, zero, rtol, atol, maxterms) + + # Prepare result arrays + S = xp.empty_like(a) + E = xp.empty_like(a) + status = xp.zeros(len(a), dtype=xp.int32) + nfev = xp.ones(len(a), dtype=xp.int32) # one function evaluation above + + # Branch for direct sum evaluation / integral approximation / invalid input + i0 = ~valid_abstep # invalid + i1 = (nterms + 1 <= maxterms) & ~i0 # direct sum evaluation + i2 = xp.isfinite(a) & ~i1 & ~i0 # infinite sum to the right + i3 = xp.isfinite(b) & ~i2 & ~i1 & ~i0 # infinite sum to the left + i4 = ~i3 & ~i2 & ~i1 & ~i0 # infinite sum on both sides + + if xp.any(i0): + S[i0], E[i0] = xp.nan, xp.nan + status[i0] = -1 + + if xp.any(i1): + args_direct = [arg[i1] for arg in args] + tmp = _direct(f, a[i1], b[i1], step[i1], args_direct, constants, xp) + S[i1], E[i1] = tmp[:-1] + nfev[i1] += tmp[-1] + status[i1] = -3 * xp.asarray(~xp.isfinite(S[i1]), dtype=xp.int32) + + if xp.any(i2): + args_indirect = [arg[i2] for arg in args] + tmp = _integral_bound(f, a[i2], b[i2], step[i2], + args_indirect, constants, xp) + S[i2], E[i2], status[i2] = tmp[:-1] + nfev[i2] += tmp[-1] + + if xp.any(i3): + args_indirect = [arg[i3] for arg in args] + def _f(x, *args): return f(-x, *args) + tmp = _integral_bound(_f, -b[i3], -a[i3], step[i3], + args_indirect, constants, xp) + S[i3], E[i3], status[i3] = tmp[:-1] + nfev[i3] += tmp[-1] + + if xp.any(i4): + args_indirect = [arg[i4] for arg in args] + + # There are two obvious high-level strategies: + # - Do two separate half-infinite sums (e.g. from -inf to 0 and 1 to inf) + # - Make a callable that returns f(x) + f(-x) and do a single half-infinite sum + # I thought the latter would have about half the overhead, so I went that way. + # Then there are two ways of ensuring that f(0) doesn't get counted twice. + # - Evaluate the sum from 1 to inf and add f(0) + # - Evaluate the sum from 0 to inf and subtract f(0) + # - Evaluate the sum from 0 to inf, but apply a weight of 0.5 when `x = 0` + # The last option has more overhead, but is simpler to implement correctly + # (especially getting the status message right) + if log: + def _f(x, *args): + log_factor = xp.where(x==0, math.log(0.5), 0) + out = xp.stack([f(x, *args), f(-x, *args)], axis=0) + return special.logsumexp(out, axis=0) + log_factor + + else: + def _f(x, *args): + factor = xp.where(x==0, 0.5, 1) + return (f(x, *args) + f(-x, *args)) * factor + + zero = xp.zeros_like(a[i4]) + tmp = _integral_bound(_f, zero, b[i4], step[i4], args_indirect, constants, xp) + S[i4], E[i4], status[i4] = tmp[:-1] + nfev[i4] += 2*tmp[-1] + + # Return results + S, E = S.reshape(shape)[()], E.reshape(shape)[()] + status, nfev = status.reshape(shape)[()], nfev.reshape(shape)[()] + return _RichResult(sum=S, error=E, status=status, success=status == 0, + nfev=nfev) + + +def _direct(f, a, b, step, args, constants, xp, inclusive=True): + # Directly evaluate the sum. + + # When used in the context of distributions, `args` would contain the + # distribution parameters. We have broadcasted for simplicity, but we could + # reduce function evaluations when distribution parameters are the same but + # sum limits differ. Roughly: + # - compute the function at all points between min(a) and max(b), + # - compute the cumulative sum, + # - take the difference between elements of the cumulative sum + # corresponding with b and a. + # This is left to future enhancement + + dtype, log, eps, zero, _, _, _ = constants + + # To allow computation in a single vectorized call, find the maximum number + # of points (over all slices) at which the function needs to be evaluated. + # Note: if `inclusive` is `True`, then we want `1` more term in the sum. + # I didn't think it was great style to use `True` as `1` in Python, so I + # explicitly converted it to an `int` before using it. + inclusive_adjustment = int(inclusive) + steps = xp.round((b - a) / step) + inclusive_adjustment + # Equivalently, steps = xp.round((b - a) / step) + inclusive + max_steps = int(xp.max(steps)) + + # In each slice, the function will be evaluated at the same number of points, + # but excessive points (those beyond the right sum limit `b`) are replaced + # with NaN to (potentially) reduce the time of these unnecessary calculations. + # Use a new last axis for these calculations for consistency with other + # elementwise algorithms. + a2, b2, step2 = a[:, xp.newaxis], b[:, xp.newaxis], step[:, xp.newaxis] + args2 = [arg[:, xp.newaxis] for arg in args] + ks = a2 + xp.arange(max_steps, dtype=dtype) * step2 + i_nan = ks >= (b2 + inclusive_adjustment*step2/2) + ks[i_nan] = xp.nan + fs = f(ks, *args2) + + # The function evaluated at NaN is NaN, and NaNs are zeroed in the sum. + # In some cases it may be faster to loop over slices than to vectorize + # like this. This is an optimization that can be added later. + fs[i_nan] = zero + nfev = max_steps - i_nan.sum(axis=-1) + S = special.logsumexp(fs, axis=-1) if log else xp.sum(fs, axis=-1) + # Rough, non-conservative error estimate. See gh-19667 for improvement ideas. + E = xp_real(S) + math.log(eps) if log else eps * abs(S) + return S, E, nfev + + +def _integral_bound(f, a, b, step, args, constants, xp): + # Estimate the sum with integral approximation + dtype, log, _, _, rtol, atol, maxterms = constants + log2 = xp.asarray(math.log(2), dtype=dtype) + + # Get a lower bound on the sum and compute effective absolute tolerance + lb = tanhsinh(f, a, b, args=args, atol=atol, rtol=rtol, log=log) + tol = xp.broadcast_to(xp.asarray(atol), lb.integral.shape) + if log: + tol = special.logsumexp(xp.stack((tol, rtol + lb.integral)), axis=0) + else: + tol = tol + rtol*lb.integral + i_skip = lb.status < 0 # avoid unnecessary f_evals if integral is divergent + tol[i_skip] = xp.nan + status = lb.status + + # As in `_direct`, we'll need a temporary new axis for points + # at which to evaluate the function. Append axis at the end for + # consistency with other elementwise algorithms. + a2 = a[..., xp.newaxis] + step2 = step[..., xp.newaxis] + args2 = [arg[..., xp.newaxis] for arg in args] + + # Find the location of a term that is less than the tolerance (if possible) + log2maxterms = math.floor(math.log2(maxterms)) if maxterms else 0 + n_steps = xp.concat((2**xp.arange(0, log2maxterms), xp.asarray([maxterms]))) + n_steps = xp.astype(n_steps, dtype) + nfev = len(n_steps) * 2 + ks = a2 + n_steps * step2 + fks = f(ks, *args2) + fksp1 = f(ks + step2, *args2) # check that the function is decreasing + fk_insufficient = (fks > tol[:, xp.newaxis]) | (fksp1 > fks) + n_fk_insufficient = xp.sum(fk_insufficient, axis=-1) + nt = xp.minimum(n_fk_insufficient, xp.asarray(n_steps.shape[-1]-1)) + n_steps = n_steps[nt] + + # If `maxterms` is insufficient (i.e. either the magnitude of the last term of the + # partial sum exceeds the tolerance or the function is not decreasing), finish the + # calculation, but report nonzero status. (Improvement: separate the status codes + # for these two cases.) + i_fk_insufficient = (n_fk_insufficient == nfev//2) + + # Directly evaluate the sum up to this term + k = a + n_steps * step + left, left_error, left_nfev = _direct(f, a, k, step, args, + constants, xp, inclusive=False) + left_is_pos_inf = xp.isinf(left) & (left > 0) + i_skip |= left_is_pos_inf # if sum is infinite, no sense in continuing + status[left_is_pos_inf] = -3 + k[i_skip] = xp.nan + + # Use integration to estimate the remaining sum + # Possible optimization for future work: if there were no terms less than + # the tolerance, there is no need to compute the integral to better accuracy. + # Something like: + # atol = xp.maximum(atol, xp.minimum(fk/2 - fb/2)) + # rtol = xp.maximum(rtol, xp.minimum((fk/2 - fb/2)/left)) + # where `fk`/`fb` are currently calculated below. + right = tanhsinh(f, k, b, args=args, atol=atol, rtol=rtol, log=log) + + # Calculate the full estimate and error from the pieces + fk = fks[xp.arange(len(fks)), nt] + + # fb = f(b, *args), but some functions return NaN at infinity. + # instead of 0 like they must (for the sum to be convergent). + fb = xp.full_like(fk, -xp.inf) if log else xp.zeros_like(fk) + i = xp.isfinite(b) + if xp.any(i): # better not call `f` with empty arrays + fb[i] = f(b[i], *[arg[i] for arg in args]) + nfev = nfev + xp.asarray(i, dtype=left_nfev.dtype) + + if log: + log_step = xp.log(step) + S_terms = (left, right.integral - log_step, fk - log2, fb - log2) + S = special.logsumexp(xp.stack(S_terms), axis=0) + E_terms = (left_error, right.error - log_step, fk-log2, fb-log2+xp.pi*1j) + E = xp_real(special.logsumexp(xp.stack(E_terms), axis=0)) + else: + S = left + right.integral/step + fk/2 + fb/2 + E = left_error + right.error/step + fk/2 - fb/2 + status[~i_skip] = right.status[~i_skip] + + status[(status == 0) & i_fk_insufficient] = -4 + return S, E, status, left_nfev + right.nfev + nfev + lb.nfev diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_test_multivariate.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_test_multivariate.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..fbe799fa8bfe4c5f1b2d2ed5edc07fe91db628ef Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/_test_multivariate.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/dop.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/dop.py new file mode 100644 index 0000000000000000000000000000000000000000..bf67a9a35b7d2959c2617aadc5638b577a45b9b5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/dop.py @@ -0,0 +1,15 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="integrate", module="dop", + private_modules=["_dop"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/lsoda.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/lsoda.py new file mode 100644 index 0000000000000000000000000000000000000000..1bc1f1da3c4f0aefad9da73b6405b957ce9335b4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/lsoda.py @@ -0,0 +1,15 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = ['lsoda'] # noqa: F822 + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="integrate", module="lsoda", + private_modules=["_lsoda"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/odepack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/odepack.py new file mode 100644 index 0000000000000000000000000000000000000000..7bb4c1a8c9be375df855abe6e1b30ca9711f2607 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/odepack.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.integrate` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = ['odeint', 'ODEintWarning'] # noqa: F822 + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="integrate", module="odepack", + private_modules=["_odepack_py"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/quadpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/quadpack.py new file mode 100644 index 0000000000000000000000000000000000000000..144584988095c8855da8c34253c045f1a3940572 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/quadpack.py @@ -0,0 +1,23 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.integrate` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + "quad", + "dblquad", + "tplquad", + "nquad", + "IntegrationWarning", +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="integrate", module="quadpack", + private_modules=["_quadpack_py"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test__quad_vec.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test__quad_vec.py new file mode 100644 index 0000000000000000000000000000000000000000..851d28f5671c3eb5821a7379547c1ba66a7e1340 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test__quad_vec.py @@ -0,0 +1,217 @@ +import pytest + +import numpy as np +from numpy.testing import assert_allclose + +from scipy.integrate import quad_vec + +from multiprocessing.dummy import Pool + + +quadrature_params = pytest.mark.parametrize( + 'quadrature', [None, "gk15", "gk21", "trapezoid"]) + + +@quadrature_params +def test_quad_vec_simple(quadrature): + n = np.arange(10) + def f(x): + return x ** n + for epsabs in [0.1, 1e-3, 1e-6]: + if quadrature == 'trapezoid' and epsabs < 1e-4: + # slow: skip + continue + + kwargs = dict(epsabs=epsabs, quadrature=quadrature) + + exact = 2**(n+1)/(n + 1) + + res, err = quad_vec(f, 0, 2, norm='max', **kwargs) + assert_allclose(res, exact, rtol=0, atol=epsabs) + + res, err = quad_vec(f, 0, 2, norm='2', **kwargs) + assert np.linalg.norm(res - exact) < epsabs + + res, err = quad_vec(f, 0, 2, norm='max', points=(0.5, 1.0), **kwargs) + assert_allclose(res, exact, rtol=0, atol=epsabs) + + res, err, *rest = quad_vec(f, 0, 2, norm='max', + epsrel=1e-8, + full_output=True, + limit=10000, + **kwargs) + assert_allclose(res, exact, rtol=0, atol=epsabs) + + +@quadrature_params +def test_quad_vec_simple_inf(quadrature): + def f(x): + return 1 / (1 + np.float64(x) ** 2) + + for epsabs in [0.1, 1e-3, 1e-6]: + if quadrature == 'trapezoid' and epsabs < 1e-4: + # slow: skip + continue + + kwargs = dict(norm='max', epsabs=epsabs, quadrature=quadrature) + + res, err = quad_vec(f, 0, np.inf, **kwargs) + assert_allclose(res, np.pi/2, rtol=0, atol=max(epsabs, err)) + + res, err = quad_vec(f, 0, -np.inf, **kwargs) + assert_allclose(res, -np.pi/2, rtol=0, atol=max(epsabs, err)) + + res, err = quad_vec(f, -np.inf, 0, **kwargs) + assert_allclose(res, np.pi/2, rtol=0, atol=max(epsabs, err)) + + res, err = quad_vec(f, np.inf, 0, **kwargs) + assert_allclose(res, -np.pi/2, rtol=0, atol=max(epsabs, err)) + + res, err = quad_vec(f, -np.inf, np.inf, **kwargs) + assert_allclose(res, np.pi, rtol=0, atol=max(epsabs, err)) + + res, err = quad_vec(f, np.inf, -np.inf, **kwargs) + assert_allclose(res, -np.pi, rtol=0, atol=max(epsabs, err)) + + res, err = quad_vec(f, np.inf, np.inf, **kwargs) + assert_allclose(res, 0, rtol=0, atol=max(epsabs, err)) + + res, err = quad_vec(f, -np.inf, -np.inf, **kwargs) + assert_allclose(res, 0, rtol=0, atol=max(epsabs, err)) + + res, err = quad_vec(f, 0, np.inf, points=(1.0, 2.0), **kwargs) + assert_allclose(res, np.pi/2, rtol=0, atol=max(epsabs, err)) + + def f(x): + return np.sin(x + 2) / (1 + x ** 2) + exact = np.pi / np.e * np.sin(2) + epsabs = 1e-5 + + res, err, info = quad_vec(f, -np.inf, np.inf, limit=1000, norm='max', epsabs=epsabs, + quadrature=quadrature, full_output=True) + assert info.status == 1 + assert_allclose(res, exact, rtol=0, atol=max(epsabs, 1.5 * err)) + + +def test_quad_vec_args(): + def f(x, a): + return x * (x + a) * np.arange(3) + a = 2 + exact = np.array([0, 4/3, 8/3]) + + res, err = quad_vec(f, 0, 1, args=(a,)) + assert_allclose(res, exact, rtol=0, atol=1e-4) + + +def _lorenzian(x): + return 1 / (1 + x**2) + + +@pytest.mark.fail_slow(10) +def test_quad_vec_pool(): + f = _lorenzian + res, err = quad_vec(f, -np.inf, np.inf, norm='max', epsabs=1e-4, workers=4) + assert_allclose(res, np.pi, rtol=0, atol=1e-4) + + with Pool(10) as pool: + def f(x): + return 1 / (1 + x ** 2) + res, _ = quad_vec(f, -np.inf, np.inf, norm='max', epsabs=1e-4, workers=pool.map) + assert_allclose(res, np.pi, rtol=0, atol=1e-4) + + +def _func_with_args(x, a): + return x * (x + a) * np.arange(3) + + +@pytest.mark.fail_slow(10) +@pytest.mark.parametrize('extra_args', [2, (2,)]) +@pytest.mark.parametrize('workers', [1, 10]) +def test_quad_vec_pool_args(extra_args, workers): + f = _func_with_args + exact = np.array([0, 4/3, 8/3]) + + res, err = quad_vec(f, 0, 1, args=extra_args, workers=workers) + assert_allclose(res, exact, rtol=0, atol=1e-4) + + with Pool(workers) as pool: + res, err = quad_vec(f, 0, 1, args=extra_args, workers=pool.map) + assert_allclose(res, exact, rtol=0, atol=1e-4) + + +@quadrature_params +def test_num_eval(quadrature): + def f(x): + count[0] += 1 + return x**5 + + count = [0] + res = quad_vec(f, 0, 1, norm='max', full_output=True, quadrature=quadrature) + assert res[2].neval == count[0] + + +def test_info(): + def f(x): + return np.ones((3, 2, 1)) + + res, err, info = quad_vec(f, 0, 1, norm='max', full_output=True) + + assert info.success is True + assert info.status == 0 + assert info.message == 'Target precision reached.' + assert info.neval > 0 + assert info.intervals.shape[1] == 2 + assert info.integrals.shape == (info.intervals.shape[0], 3, 2, 1) + assert info.errors.shape == (info.intervals.shape[0],) + + +def test_nan_inf(): + def f_nan(x): + return np.nan + + def f_inf(x): + return np.inf if x < 0.1 else 1/x + + res, err, info = quad_vec(f_nan, 0, 1, full_output=True) + assert info.status == 3 + + res, err, info = quad_vec(f_inf, 0, 1, full_output=True) + assert info.status == 3 + + +@pytest.mark.parametrize('a,b', [(0, 1), (0, np.inf), (np.inf, 0), + (-np.inf, np.inf), (np.inf, -np.inf)]) +def test_points(a, b): + # Check that initial interval splitting is done according to + # `points`, by checking that consecutive sets of 15 point (for + # gk15) function evaluations lie between `points` + + points = (0, 0.25, 0.5, 0.75, 1.0) + points += tuple(-x for x in points) + + quadrature_points = 15 + interval_sets = [] + count = 0 + + def f(x): + nonlocal count + + if count % quadrature_points == 0: + interval_sets.append(set()) + + count += 1 + interval_sets[-1].add(float(x)) + return 0.0 + + quad_vec(f, a, b, points=points, quadrature='gk15', limit=0) + + # Check that all point sets lie in a single `points` interval + for p in interval_sets: + j = np.searchsorted(sorted(points), tuple(p)) + assert np.all(j == j[0]) + + +@pytest.mark.thread_unsafe +def test_trapz_deprecation(): + with pytest.deprecated_call(match="`quadrature='trapz'`"): + quad_vec(lambda x: x, 0, 1, quadrature="trapz") diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_banded_ode_solvers.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_banded_ode_solvers.py new file mode 100644 index 0000000000000000000000000000000000000000..358c5e3d1fcfe7ccd7e3691bd9af2f47656f4e2b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_banded_ode_solvers.py @@ -0,0 +1,220 @@ +import itertools +import pytest +import numpy as np +from numpy.testing import assert_allclose +from scipy.integrate import ode + + +def _band_count(a): + """Returns ml and mu, the lower and upper band sizes of a.""" + nrows, ncols = a.shape + ml = 0 + for k in range(-nrows+1, 0): + if np.diag(a, k).any(): + ml = -k + break + mu = 0 + for k in range(nrows-1, 0, -1): + if np.diag(a, k).any(): + mu = k + break + return ml, mu + + +def _linear_func(t, y, a): + """Linear system dy/dt = a * y""" + return a.dot(y) + + +def _linear_jac(t, y, a): + """Jacobian of a * y is a.""" + return a + + +def _linear_banded_jac(t, y, a): + """Banded Jacobian.""" + ml, mu = _band_count(a) + bjac = [np.r_[[0] * k, np.diag(a, k)] for k in range(mu, 0, -1)] + bjac.append(np.diag(a)) + for k in range(-1, -ml-1, -1): + bjac.append(np.r_[np.diag(a, k), [0] * (-k)]) + return bjac + + +def _solve_linear_sys(a, y0, tend=1, dt=0.1, + solver=None, method='bdf', use_jac=True, + with_jacobian=False, banded=False): + """Use scipy.integrate.ode to solve a linear system of ODEs. + + a : square ndarray + Matrix of the linear system to be solved. + y0 : ndarray + Initial condition + tend : float + Stop time. + dt : float + Step size of the output. + solver : str + If not None, this must be "vode", "lsoda" or "zvode". + method : str + Either "bdf" or "adams". + use_jac : bool + Determines if the jacobian function is passed to ode(). + with_jacobian : bool + Passed to ode.set_integrator(). + banded : bool + Determines whether a banded or full jacobian is used. + If `banded` is True, `lband` and `uband` are determined by the + values in `a`. + """ + if banded: + lband, uband = _band_count(a) + else: + lband = None + uband = None + + if use_jac: + if banded: + r = ode(_linear_func, _linear_banded_jac) + else: + r = ode(_linear_func, _linear_jac) + else: + r = ode(_linear_func) + + if solver is None: + if np.iscomplexobj(a): + solver = "zvode" + else: + solver = "vode" + + r.set_integrator(solver, + with_jacobian=with_jacobian, + method=method, + lband=lband, uband=uband, + rtol=1e-9, atol=1e-10, + ) + t0 = 0 + r.set_initial_value(y0, t0) + r.set_f_params(a) + r.set_jac_params(a) + + t = [t0] + y = [y0] + while r.successful() and r.t < tend: + r.integrate(r.t + dt) + t.append(r.t) + y.append(r.y) + + t = np.array(t) + y = np.array(y) + return t, y + + +def _analytical_solution(a, y0, t): + """ + Analytical solution to the linear differential equations dy/dt = a*y. + + The solution is only valid if `a` is diagonalizable. + + Returns a 2-D array with shape (len(t), len(y0)). + """ + lam, v = np.linalg.eig(a) + c = np.linalg.solve(v, y0) + e = c * np.exp(lam * t.reshape(-1, 1)) + sol = e.dot(v.T) + return sol + + +@pytest.mark.thread_unsafe +def test_banded_ode_solvers(): + # Test the "lsoda", "vode" and "zvode" solvers of the `ode` class + # with a system that has a banded Jacobian matrix. + + t_exact = np.linspace(0, 1.0, 5) + + # --- Real arrays for testing the "lsoda" and "vode" solvers --- + + # lband = 2, uband = 1: + a_real = np.array([[-0.6, 0.1, 0.0, 0.0, 0.0], + [0.2, -0.5, 0.9, 0.0, 0.0], + [0.1, 0.1, -0.4, 0.1, 0.0], + [0.0, 0.3, -0.1, -0.9, -0.3], + [0.0, 0.0, 0.1, 0.1, -0.7]]) + + # lband = 0, uband = 1: + a_real_upper = np.triu(a_real) + + # lband = 2, uband = 0: + a_real_lower = np.tril(a_real) + + # lband = 0, uband = 0: + a_real_diag = np.triu(a_real_lower) + + real_matrices = [a_real, a_real_upper, a_real_lower, a_real_diag] + real_solutions = [] + + for a in real_matrices: + y0 = np.arange(1, a.shape[0] + 1) + y_exact = _analytical_solution(a, y0, t_exact) + real_solutions.append((y0, t_exact, y_exact)) + + def check_real(idx, solver, meth, use_jac, with_jac, banded): + a = real_matrices[idx] + y0, t_exact, y_exact = real_solutions[idx] + t, y = _solve_linear_sys(a, y0, + tend=t_exact[-1], + dt=t_exact[1] - t_exact[0], + solver=solver, + method=meth, + use_jac=use_jac, + with_jacobian=with_jac, + banded=banded) + assert_allclose(t, t_exact) + assert_allclose(y, y_exact) + + for idx in range(len(real_matrices)): + p = [['vode', 'lsoda'], # solver + ['bdf', 'adams'], # method + [False, True], # use_jac + [False, True], # with_jacobian + [False, True]] # banded + for solver, meth, use_jac, with_jac, banded in itertools.product(*p): + check_real(idx, solver, meth, use_jac, with_jac, banded) + + # --- Complex arrays for testing the "zvode" solver --- + + # complex, lband = 2, uband = 1: + a_complex = a_real - 0.5j * a_real + + # complex, lband = 0, uband = 0: + a_complex_diag = np.diag(np.diag(a_complex)) + + complex_matrices = [a_complex, a_complex_diag] + complex_solutions = [] + + for a in complex_matrices: + y0 = np.arange(1, a.shape[0] + 1) + 1j + y_exact = _analytical_solution(a, y0, t_exact) + complex_solutions.append((y0, t_exact, y_exact)) + + def check_complex(idx, solver, meth, use_jac, with_jac, banded): + a = complex_matrices[idx] + y0, t_exact, y_exact = complex_solutions[idx] + t, y = _solve_linear_sys(a, y0, + tend=t_exact[-1], + dt=t_exact[1] - t_exact[0], + solver=solver, + method=meth, + use_jac=use_jac, + with_jacobian=with_jac, + banded=banded) + assert_allclose(t, t_exact) + assert_allclose(y, y_exact) + + for idx in range(len(complex_matrices)): + p = [['bdf', 'adams'], # method + [False, True], # use_jac + [False, True], # with_jacobian + [False, True]] # banded + for meth, use_jac, with_jac, banded in itertools.product(*p): + check_complex(idx, "zvode", meth, use_jac, with_jac, banded) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_bvp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_bvp.py new file mode 100644 index 0000000000000000000000000000000000000000..4ef9eb6ff0502e1113d6bea7ad1e0088633d3151 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_bvp.py @@ -0,0 +1,714 @@ +import sys + +try: + from StringIO import StringIO +except ImportError: + from io import StringIO + +import numpy as np +from numpy.testing import (assert_, assert_array_equal, assert_allclose, + assert_equal) +from pytest import raises as assert_raises + +from scipy.sparse import coo_matrix +from scipy.special import erf +from scipy.integrate._bvp import (modify_mesh, estimate_fun_jac, + estimate_bc_jac, compute_jac_indices, + construct_global_jac, solve_bvp) + +import pytest + + +def exp_fun(x, y): + return np.vstack((y[1], y[0])) + + +def exp_fun_jac(x, y): + df_dy = np.empty((2, 2, x.shape[0])) + df_dy[0, 0] = 0 + df_dy[0, 1] = 1 + df_dy[1, 0] = 1 + df_dy[1, 1] = 0 + return df_dy + + +def exp_bc(ya, yb): + return np.hstack((ya[0] - 1, yb[0])) + + +def exp_bc_complex(ya, yb): + return np.hstack((ya[0] - 1 - 1j, yb[0])) + + +def exp_bc_jac(ya, yb): + dbc_dya = np.array([ + [1, 0], + [0, 0] + ]) + dbc_dyb = np.array([ + [0, 0], + [1, 0] + ]) + return dbc_dya, dbc_dyb + + +def exp_sol(x): + return (np.exp(-x) - np.exp(x - 2)) / (1 - np.exp(-2)) + + +def sl_fun(x, y, p): + return np.vstack((y[1], -p[0]**2 * y[0])) + + +def sl_fun_jac(x, y, p): + n, m = y.shape + df_dy = np.empty((n, 2, m)) + df_dy[0, 0] = 0 + df_dy[0, 1] = 1 + df_dy[1, 0] = -p[0]**2 + df_dy[1, 1] = 0 + + df_dp = np.empty((n, 1, m)) + df_dp[0, 0] = 0 + df_dp[1, 0] = -2 * p[0] * y[0] + + return df_dy, df_dp + + +def sl_bc(ya, yb, p): + return np.hstack((ya[0], yb[0], ya[1] - p[0])) + + +def sl_bc_jac(ya, yb, p): + dbc_dya = np.zeros((3, 2)) + dbc_dya[0, 0] = 1 + dbc_dya[2, 1] = 1 + + dbc_dyb = np.zeros((3, 2)) + dbc_dyb[1, 0] = 1 + + dbc_dp = np.zeros((3, 1)) + dbc_dp[2, 0] = -1 + + return dbc_dya, dbc_dyb, dbc_dp + + +def sl_sol(x, p): + return np.sin(p[0] * x) + + +def emden_fun(x, y): + return np.vstack((y[1], -y[0]**5)) + + +def emden_fun_jac(x, y): + df_dy = np.empty((2, 2, x.shape[0])) + df_dy[0, 0] = 0 + df_dy[0, 1] = 1 + df_dy[1, 0] = -5 * y[0]**4 + df_dy[1, 1] = 0 + return df_dy + + +def emden_bc(ya, yb): + return np.array([ya[1], yb[0] - (3/4)**0.5]) + + +def emden_bc_jac(ya, yb): + dbc_dya = np.array([ + [0, 1], + [0, 0] + ]) + dbc_dyb = np.array([ + [0, 0], + [1, 0] + ]) + return dbc_dya, dbc_dyb + + +def emden_sol(x): + return (1 + x**2/3)**-0.5 + + +def undefined_fun(x, y): + return np.zeros_like(y) + + +def undefined_bc(ya, yb): + return np.array([ya[0], yb[0] - 1]) + + +def big_fun(x, y): + f = np.zeros_like(y) + f[::2] = y[1::2] + return f + + +def big_bc(ya, yb): + return np.hstack((ya[::2], yb[::2] - 1)) + + +def big_sol(x, n): + y = np.ones((2 * n, x.size)) + y[::2] = x + return x + + +def big_fun_with_parameters(x, y, p): + """ Big version of sl_fun, with two parameters. + + The two differential equations represented by sl_fun are broadcast to the + number of rows of y, rotating between the parameters p[0] and p[1]. + Here are the differential equations: + + dy[0]/dt = y[1] + dy[1]/dt = -p[0]**2 * y[0] + dy[2]/dt = y[3] + dy[3]/dt = -p[1]**2 * y[2] + dy[4]/dt = y[5] + dy[5]/dt = -p[0]**2 * y[4] + dy[6]/dt = y[7] + dy[7]/dt = -p[1]**2 * y[6] + . + . + . + + """ + f = np.zeros_like(y) + f[::2] = y[1::2] + f[1::4] = -p[0]**2 * y[::4] + f[3::4] = -p[1]**2 * y[2::4] + return f + + +def big_fun_with_parameters_jac(x, y, p): + # big version of sl_fun_jac, with two parameters + n, m = y.shape + df_dy = np.zeros((n, n, m)) + df_dy[range(0, n, 2), range(1, n, 2)] = 1 + df_dy[range(1, n, 4), range(0, n, 4)] = -p[0]**2 + df_dy[range(3, n, 4), range(2, n, 4)] = -p[1]**2 + + df_dp = np.zeros((n, 2, m)) + df_dp[range(1, n, 4), 0] = -2 * p[0] * y[range(0, n, 4)] + df_dp[range(3, n, 4), 1] = -2 * p[1] * y[range(2, n, 4)] + + return df_dy, df_dp + + +def big_bc_with_parameters(ya, yb, p): + # big version of sl_bc, with two parameters + return np.hstack((ya[::2], yb[::2], ya[1] - p[0], ya[3] - p[1])) + + +def big_bc_with_parameters_jac(ya, yb, p): + # big version of sl_bc_jac, with two parameters + n = ya.shape[0] + dbc_dya = np.zeros((n + 2, n)) + dbc_dyb = np.zeros((n + 2, n)) + + dbc_dya[range(n // 2), range(0, n, 2)] = 1 + dbc_dyb[range(n // 2, n), range(0, n, 2)] = 1 + + dbc_dp = np.zeros((n + 2, 2)) + dbc_dp[n, 0] = -1 + dbc_dya[n, 1] = 1 + dbc_dp[n + 1, 1] = -1 + dbc_dya[n + 1, 3] = 1 + + return dbc_dya, dbc_dyb, dbc_dp + + +def big_sol_with_parameters(x, p): + # big version of sl_sol, with two parameters + return np.vstack((np.sin(p[0] * x), np.sin(p[1] * x))) + + +def shock_fun(x, y): + eps = 1e-3 + return np.vstack(( + y[1], + -(x * y[1] + eps * np.pi**2 * np.cos(np.pi * x) + + np.pi * x * np.sin(np.pi * x)) / eps + )) + + +def shock_bc(ya, yb): + return np.array([ya[0] + 2, yb[0]]) + + +def shock_sol(x): + eps = 1e-3 + k = np.sqrt(2 * eps) + return np.cos(np.pi * x) + erf(x / k) / erf(1 / k) + + +def nonlin_bc_fun(x, y): + # laplace eq. + return np.stack([y[1], np.zeros_like(x)]) + + +def nonlin_bc_bc(ya, yb): + phiA, phipA = ya + phiC, phipC = yb + + kappa, ioA, ioC, V, f = 1.64, 0.01, 1.0e-4, 0.5, 38.9 + + # Butler-Volmer Kinetics at Anode + hA = 0.0-phiA-0.0 + iA = ioA * (np.exp(f*hA) - np.exp(-f*hA)) + res0 = iA + kappa * phipA + + # Butler-Volmer Kinetics at Cathode + hC = V - phiC - 1.0 + iC = ioC * (np.exp(f*hC) - np.exp(-f*hC)) + res1 = iC - kappa*phipC + + return np.array([res0, res1]) + + +def nonlin_bc_sol(x): + return -0.13426436116763119 - 1.1308709 * x + + +def test_modify_mesh(): + x = np.array([0, 1, 3, 9], dtype=float) + x_new = modify_mesh(x, np.array([0]), np.array([2])) + assert_array_equal(x_new, np.array([0, 0.5, 1, 3, 5, 7, 9])) + + x = np.array([-6, -3, 0, 3, 6], dtype=float) + x_new = modify_mesh(x, np.array([1], dtype=int), np.array([0, 2, 3])) + assert_array_equal(x_new, [-6, -5, -4, -3, -1.5, 0, 1, 2, 3, 4, 5, 6]) + + +def test_compute_fun_jac(): + x = np.linspace(0, 1, 5) + y = np.empty((2, x.shape[0])) + y[0] = 0.01 + y[1] = 0.02 + p = np.array([]) + df_dy, df_dp = estimate_fun_jac(lambda x, y, p: exp_fun(x, y), x, y, p) + df_dy_an = exp_fun_jac(x, y) + assert_allclose(df_dy, df_dy_an) + assert_(df_dp is None) + + x = np.linspace(0, np.pi, 5) + y = np.empty((2, x.shape[0])) + y[0] = np.sin(x) + y[1] = np.cos(x) + p = np.array([1.0]) + df_dy, df_dp = estimate_fun_jac(sl_fun, x, y, p) + df_dy_an, df_dp_an = sl_fun_jac(x, y, p) + assert_allclose(df_dy, df_dy_an) + assert_allclose(df_dp, df_dp_an) + + x = np.linspace(0, 1, 10) + y = np.empty((2, x.shape[0])) + y[0] = (3/4)**0.5 + y[1] = 1e-4 + p = np.array([]) + df_dy, df_dp = estimate_fun_jac(lambda x, y, p: emden_fun(x, y), x, y, p) + df_dy_an = emden_fun_jac(x, y) + assert_allclose(df_dy, df_dy_an) + assert_(df_dp is None) + + +def test_compute_bc_jac(): + ya = np.array([-1.0, 2]) + yb = np.array([0.5, 3]) + p = np.array([]) + dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac( + lambda ya, yb, p: exp_bc(ya, yb), ya, yb, p) + dbc_dya_an, dbc_dyb_an = exp_bc_jac(ya, yb) + assert_allclose(dbc_dya, dbc_dya_an) + assert_allclose(dbc_dyb, dbc_dyb_an) + assert_(dbc_dp is None) + + ya = np.array([0.0, 1]) + yb = np.array([0.0, -1]) + p = np.array([0.5]) + dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac(sl_bc, ya, yb, p) + dbc_dya_an, dbc_dyb_an, dbc_dp_an = sl_bc_jac(ya, yb, p) + assert_allclose(dbc_dya, dbc_dya_an) + assert_allclose(dbc_dyb, dbc_dyb_an) + assert_allclose(dbc_dp, dbc_dp_an) + + ya = np.array([0.5, 100]) + yb = np.array([-1000, 10.5]) + p = np.array([]) + dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac( + lambda ya, yb, p: emden_bc(ya, yb), ya, yb, p) + dbc_dya_an, dbc_dyb_an = emden_bc_jac(ya, yb) + assert_allclose(dbc_dya, dbc_dya_an) + assert_allclose(dbc_dyb, dbc_dyb_an) + assert_(dbc_dp is None) + + +def test_compute_jac_indices(): + n = 2 + m = 4 + k = 2 + i, j = compute_jac_indices(n, m, k) + s = coo_matrix((np.ones_like(i), (i, j))).toarray() + s_true = np.array([ + [1, 1, 1, 1, 0, 0, 0, 0, 1, 1], + [1, 1, 1, 1, 0, 0, 0, 0, 1, 1], + [0, 0, 1, 1, 1, 1, 0, 0, 1, 1], + [0, 0, 1, 1, 1, 1, 0, 0, 1, 1], + [0, 0, 0, 0, 1, 1, 1, 1, 1, 1], + [0, 0, 0, 0, 1, 1, 1, 1, 1, 1], + [1, 1, 0, 0, 0, 0, 1, 1, 1, 1], + [1, 1, 0, 0, 0, 0, 1, 1, 1, 1], + [1, 1, 0, 0, 0, 0, 1, 1, 1, 1], + [1, 1, 0, 0, 0, 0, 1, 1, 1, 1], + ]) + assert_array_equal(s, s_true) + + +def test_compute_global_jac(): + n = 2 + m = 5 + k = 1 + i_jac, j_jac = compute_jac_indices(2, 5, 1) + x = np.linspace(0, 1, 5) + h = np.diff(x) + y = np.vstack((np.sin(np.pi * x), np.pi * np.cos(np.pi * x))) + p = np.array([3.0]) + + f = sl_fun(x, y, p) + + x_middle = x[:-1] + 0.5 * h + y_middle = 0.5 * (y[:, :-1] + y[:, 1:]) - h/8 * (f[:, 1:] - f[:, :-1]) + + df_dy, df_dp = sl_fun_jac(x, y, p) + df_dy_middle, df_dp_middle = sl_fun_jac(x_middle, y_middle, p) + dbc_dya, dbc_dyb, dbc_dp = sl_bc_jac(y[:, 0], y[:, -1], p) + + J = construct_global_jac(n, m, k, i_jac, j_jac, h, df_dy, df_dy_middle, + df_dp, df_dp_middle, dbc_dya, dbc_dyb, dbc_dp) + J = J.toarray() + + def J_block(h, p): + return np.array([ + [h**2*p**2/12 - 1, -0.5*h, -h**2*p**2/12 + 1, -0.5*h], + [0.5*h*p**2, h**2*p**2/12 - 1, 0.5*h*p**2, 1 - h**2*p**2/12] + ]) + + J_true = np.zeros((m * n + k, m * n + k)) + for i in range(m - 1): + J_true[i * n: (i + 1) * n, i * n: (i + 2) * n] = J_block(h[i], p[0]) + + J_true[:(m - 1) * n:2, -1] = p * h**2/6 * (y[0, :-1] - y[0, 1:]) + J_true[1:(m - 1) * n:2, -1] = p * (h * (y[0, :-1] + y[0, 1:]) + + h**2/6 * (y[1, :-1] - y[1, 1:])) + + J_true[8, 0] = 1 + J_true[9, 8] = 1 + J_true[10, 1] = 1 + J_true[10, 10] = -1 + + assert_allclose(J, J_true, rtol=1e-10) + + df_dy, df_dp = estimate_fun_jac(sl_fun, x, y, p) + df_dy_middle, df_dp_middle = estimate_fun_jac(sl_fun, x_middle, y_middle, p) + dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac(sl_bc, y[:, 0], y[:, -1], p) + J = construct_global_jac(n, m, k, i_jac, j_jac, h, df_dy, df_dy_middle, + df_dp, df_dp_middle, dbc_dya, dbc_dyb, dbc_dp) + J = J.toarray() + assert_allclose(J, J_true, rtol=2e-8, atol=2e-8) + + +def test_parameter_validation(): + x = [0, 1, 0.5] + y = np.zeros((2, 3)) + assert_raises(ValueError, solve_bvp, exp_fun, exp_bc, x, y) + + x = np.linspace(0, 1, 5) + y = np.zeros((2, 4)) + assert_raises(ValueError, solve_bvp, exp_fun, exp_bc, x, y) + + def fun(x, y, p): + return exp_fun(x, y) + def bc(ya, yb, p): + return exp_bc(ya, yb) + + y = np.zeros((2, x.shape[0])) + assert_raises(ValueError, solve_bvp, fun, bc, x, y, p=[1]) + + def wrong_shape_fun(x, y): + return np.zeros(3) + + assert_raises(ValueError, solve_bvp, wrong_shape_fun, bc, x, y) + + S = np.array([[0, 0]]) + assert_raises(ValueError, solve_bvp, exp_fun, exp_bc, x, y, S=S) + + +def test_no_params(): + x = np.linspace(0, 1, 5) + x_test = np.linspace(0, 1, 100) + y = np.zeros((2, x.shape[0])) + for fun_jac in [None, exp_fun_jac]: + for bc_jac in [None, exp_bc_jac]: + sol = solve_bvp(exp_fun, exp_bc, x, y, fun_jac=fun_jac, + bc_jac=bc_jac) + + assert_equal(sol.status, 0) + assert_(sol.success) + + assert_equal(sol.x.size, 5) + + sol_test = sol.sol(x_test) + + assert_allclose(sol_test[0], exp_sol(x_test), atol=1e-5) + + f_test = exp_fun(x_test, sol_test) + r = sol.sol(x_test, 1) - f_test + rel_res = r / (1 + np.abs(f_test)) + norm_res = np.sum(rel_res**2, axis=0)**0.5 + assert_(np.all(norm_res < 1e-3)) + + assert_(np.all(sol.rms_residuals < 1e-3)) + assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10) + assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10) + + +def test_with_params(): + x = np.linspace(0, np.pi, 5) + x_test = np.linspace(0, np.pi, 100) + y = np.ones((2, x.shape[0])) + + for fun_jac in [None, sl_fun_jac]: + for bc_jac in [None, sl_bc_jac]: + sol = solve_bvp(sl_fun, sl_bc, x, y, p=[0.5], fun_jac=fun_jac, + bc_jac=bc_jac) + + assert_equal(sol.status, 0) + assert_(sol.success) + + assert_(sol.x.size < 10) + + assert_allclose(sol.p, [1], rtol=1e-4) + + sol_test = sol.sol(x_test) + + assert_allclose(sol_test[0], sl_sol(x_test, [1]), + rtol=1e-4, atol=1e-4) + + f_test = sl_fun(x_test, sol_test, [1]) + r = sol.sol(x_test, 1) - f_test + rel_res = r / (1 + np.abs(f_test)) + norm_res = np.sum(rel_res ** 2, axis=0) ** 0.5 + assert_(np.all(norm_res < 1e-3)) + + assert_(np.all(sol.rms_residuals < 1e-3)) + assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10) + assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10) + + +def test_singular_term(): + x = np.linspace(0, 1, 10) + x_test = np.linspace(0.05, 1, 100) + y = np.empty((2, 10)) + y[0] = (3/4)**0.5 + y[1] = 1e-4 + S = np.array([[0, 0], [0, -2]]) + + for fun_jac in [None, emden_fun_jac]: + for bc_jac in [None, emden_bc_jac]: + sol = solve_bvp(emden_fun, emden_bc, x, y, S=S, fun_jac=fun_jac, + bc_jac=bc_jac) + + assert_equal(sol.status, 0) + assert_(sol.success) + + assert_equal(sol.x.size, 10) + + sol_test = sol.sol(x_test) + assert_allclose(sol_test[0], emden_sol(x_test), atol=1e-5) + + f_test = emden_fun(x_test, sol_test) + S.dot(sol_test) / x_test + r = sol.sol(x_test, 1) - f_test + rel_res = r / (1 + np.abs(f_test)) + norm_res = np.sum(rel_res ** 2, axis=0) ** 0.5 + + assert_(np.all(norm_res < 1e-3)) + assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10) + assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10) + + +def test_complex(): + # The test is essentially the same as test_no_params, but boundary + # conditions are turned into complex. + x = np.linspace(0, 1, 5) + x_test = np.linspace(0, 1, 100) + y = np.zeros((2, x.shape[0]), dtype=complex) + for fun_jac in [None, exp_fun_jac]: + for bc_jac in [None, exp_bc_jac]: + sol = solve_bvp(exp_fun, exp_bc_complex, x, y, fun_jac=fun_jac, + bc_jac=bc_jac) + + assert_equal(sol.status, 0) + assert_(sol.success) + + sol_test = sol.sol(x_test) + + assert_allclose(sol_test[0].real, exp_sol(x_test), atol=1e-5) + assert_allclose(sol_test[0].imag, exp_sol(x_test), atol=1e-5) + + f_test = exp_fun(x_test, sol_test) + r = sol.sol(x_test, 1) - f_test + rel_res = r / (1 + np.abs(f_test)) + norm_res = np.sum(np.real(rel_res * np.conj(rel_res)), + axis=0) ** 0.5 + assert_(np.all(norm_res < 1e-3)) + + assert_(np.all(sol.rms_residuals < 1e-3)) + assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10) + assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10) + + +def test_failures(): + x = np.linspace(0, 1, 2) + y = np.zeros((2, x.size)) + res = solve_bvp(exp_fun, exp_bc, x, y, tol=1e-5, max_nodes=5) + assert_equal(res.status, 1) + assert_(not res.success) + + x = np.linspace(0, 1, 5) + y = np.zeros((2, x.size)) + res = solve_bvp(undefined_fun, undefined_bc, x, y) + assert_equal(res.status, 2) + assert_(not res.success) + + +def test_big_problem(): + n = 30 + x = np.linspace(0, 1, 5) + y = np.zeros((2 * n, x.size)) + sol = solve_bvp(big_fun, big_bc, x, y) + + assert_equal(sol.status, 0) + assert_(sol.success) + + sol_test = sol.sol(x) + + assert_allclose(sol_test[0], big_sol(x, n)) + + f_test = big_fun(x, sol_test) + r = sol.sol(x, 1) - f_test + rel_res = r / (1 + np.abs(f_test)) + norm_res = np.sum(np.real(rel_res * np.conj(rel_res)), axis=0) ** 0.5 + assert_(np.all(norm_res < 1e-3)) + + assert_(np.all(sol.rms_residuals < 1e-3)) + assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10) + assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10) + + +def test_big_problem_with_parameters(): + n = 30 + x = np.linspace(0, np.pi, 5) + x_test = np.linspace(0, np.pi, 100) + y = np.ones((2 * n, x.size)) + + for fun_jac in [None, big_fun_with_parameters_jac]: + for bc_jac in [None, big_bc_with_parameters_jac]: + sol = solve_bvp(big_fun_with_parameters, big_bc_with_parameters, x, + y, p=[0.5, 0.5], fun_jac=fun_jac, bc_jac=bc_jac) + + assert_equal(sol.status, 0) + assert_(sol.success) + + assert_allclose(sol.p, [1, 1], rtol=1e-4) + + sol_test = sol.sol(x_test) + + for isol in range(0, n, 4): + assert_allclose(sol_test[isol], + big_sol_with_parameters(x_test, [1, 1])[0], + rtol=1e-4, atol=1e-4) + assert_allclose(sol_test[isol + 2], + big_sol_with_parameters(x_test, [1, 1])[1], + rtol=1e-4, atol=1e-4) + + f_test = big_fun_with_parameters(x_test, sol_test, [1, 1]) + r = sol.sol(x_test, 1) - f_test + rel_res = r / (1 + np.abs(f_test)) + norm_res = np.sum(rel_res ** 2, axis=0) ** 0.5 + assert_(np.all(norm_res < 1e-3)) + + assert_(np.all(sol.rms_residuals < 1e-3)) + assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10) + assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10) + + +def test_shock_layer(): + x = np.linspace(-1, 1, 5) + x_test = np.linspace(-1, 1, 100) + y = np.zeros((2, x.size)) + sol = solve_bvp(shock_fun, shock_bc, x, y) + + assert_equal(sol.status, 0) + assert_(sol.success) + + assert_(sol.x.size < 110) + + sol_test = sol.sol(x_test) + assert_allclose(sol_test[0], shock_sol(x_test), rtol=1e-5, atol=1e-5) + + f_test = shock_fun(x_test, sol_test) + r = sol.sol(x_test, 1) - f_test + rel_res = r / (1 + np.abs(f_test)) + norm_res = np.sum(rel_res ** 2, axis=0) ** 0.5 + + assert_(np.all(norm_res < 1e-3)) + assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10) + assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10) + + +def test_nonlin_bc(): + x = np.linspace(0, 0.1, 5) + x_test = x + y = np.zeros([2, x.size]) + sol = solve_bvp(nonlin_bc_fun, nonlin_bc_bc, x, y) + + assert_equal(sol.status, 0) + assert_(sol.success) + + assert_(sol.x.size < 8) + + sol_test = sol.sol(x_test) + assert_allclose(sol_test[0], nonlin_bc_sol(x_test), rtol=1e-5, atol=1e-5) + + f_test = nonlin_bc_fun(x_test, sol_test) + r = sol.sol(x_test, 1) - f_test + rel_res = r / (1 + np.abs(f_test)) + norm_res = np.sum(rel_res ** 2, axis=0) ** 0.5 + + assert_(np.all(norm_res < 1e-3)) + assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10) + assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10) + + +@pytest.mark.thread_unsafe +def test_verbose(): + # Smoke test that checks the printing does something and does not crash + x = np.linspace(0, 1, 5) + y = np.zeros((2, x.shape[0])) + for verbose in [0, 1, 2]: + old_stdout = sys.stdout + sys.stdout = StringIO() + try: + sol = solve_bvp(exp_fun, exp_bc, x, y, verbose=verbose) + text = sys.stdout.getvalue() + finally: + sys.stdout = old_stdout + + assert_(sol.success) + if verbose == 0: + assert_(not text, text) + if verbose >= 1: + assert_("Solved in" in text, text) + if verbose >= 2: + assert_("Max residual" in text, text) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_cubature.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_cubature.py new file mode 100644 index 0000000000000000000000000000000000000000..899655c7631fbc86d06eb97c514761d4c882a632 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_cubature.py @@ -0,0 +1,1389 @@ +import math +import scipy +import itertools + +import pytest + +from scipy._lib._array_api import ( + array_namespace, + xp_assert_close, + xp_size, + np_compat, + is_array_api_strict, +) +from scipy.conftest import array_api_compatible + +from scipy.integrate import cubature + +from scipy.integrate._rules import ( + Rule, FixedRule, + NestedFixedRule, + GaussLegendreQuadrature, GaussKronrodQuadrature, + GenzMalikCubature, +) + +from scipy.integrate._cubature import _InfiniteLimitsTransform + +pytestmark = [pytest.mark.usefixtures("skip_xp_backends"),] +skip_xp_backends = pytest.mark.skip_xp_backends + +# The integrands ``genz_malik_1980_*`` come from the paper: +# A.C. Genz, A.A. Malik, Remarks on algorithm 006: An adaptive algorithm for +# numerical integration over an N-dimensional rectangular region, Journal of +# Computational and Applied Mathematics, Volume 6, Issue 4, 1980, Pages 295-302, +# ISSN 0377-0427, https://doi.org/10.1016/0771-050X(80)90039-X. + + +def basic_1d_integrand(x, n, xp): + x_reshaped = xp.reshape(x, (-1, 1, 1)) + n_reshaped = xp.reshape(n, (1, -1, 1)) + + return x_reshaped**n_reshaped + + +def basic_1d_integrand_exact(n, xp): + # Exact only for integration over interval [0, 2]. + return xp.reshape(2**(n+1)/(n+1), (-1, 1)) + + +def basic_nd_integrand(x, n, xp): + return xp.reshape(xp.sum(x, axis=-1), (-1, 1))**xp.reshape(n, (1, -1)) + + +def basic_nd_integrand_exact(n, xp): + # Exact only for integration over interval [0, 2]. + return (-2**(3+n) + 4**(2+n))/((1+n)*(2+n)) + + +def genz_malik_1980_f_1(x, r, alphas, xp): + r""" + .. math:: f_1(\mathbf x) = \cos\left(2\pi r + \sum^n_{i = 1}\alpha_i x_i\right) + + .. code-block:: mathematica + + genzMalik1980f1[x_List, r_, alphas_List] := Cos[2*Pi*r + Total[x*alphas]] + """ + + npoints, ndim = x.shape[0], x.shape[-1] + + alphas_reshaped = alphas[None, ...] + x_reshaped = xp.reshape(x, (npoints, *([1]*(len(alphas.shape) - 1)), ndim)) + + return xp.cos(2*math.pi*r + xp.sum(alphas_reshaped * x_reshaped, axis=-1)) + + +def genz_malik_1980_f_1_exact(a, b, r, alphas, xp): + ndim = xp_size(a) + a = xp.reshape(a, (*([1]*(len(alphas.shape) - 1)), ndim)) + b = xp.reshape(b, (*([1]*(len(alphas.shape) - 1)), ndim)) + + return ( + (-2)**ndim + * 1/xp.prod(alphas, axis=-1) + * xp.cos(2*math.pi*r + xp.sum(alphas * (a+b) * 0.5, axis=-1)) + * xp.prod(xp.sin(alphas * (a-b)/2), axis=-1) + ) + + +def genz_malik_1980_f_1_random_args(rng, shape, xp): + r = xp.asarray(rng.random(shape[:-1])) + alphas = xp.asarray(rng.random(shape)) + + difficulty = 9 + normalisation_factors = xp.sum(alphas, axis=-1)[..., None] + alphas = difficulty * alphas / normalisation_factors + + return (r, alphas) + + +def genz_malik_1980_f_2(x, alphas, betas, xp): + r""" + .. math:: f_2(\mathbf x) = \prod^n_{i = 1} (\alpha_i^2 + (x_i - \beta_i)^2)^{-1} + + .. code-block:: mathematica + + genzMalik1980f2[x_List, alphas_List, betas_List] := + 1/Times @@ ((alphas^2 + (x - betas)^2)) + """ + npoints, ndim = x.shape[0], x.shape[-1] + + alphas_reshaped = alphas[None, ...] + betas_reshaped = betas[None, ...] + + x_reshaped = xp.reshape(x, (npoints, *([1]*(len(alphas.shape) - 1)), ndim)) + + return 1/xp.prod(alphas_reshaped**2 + (x_reshaped-betas_reshaped)**2, axis=-1) + + +def genz_malik_1980_f_2_exact(a, b, alphas, betas, xp): + ndim = xp_size(a) + a = xp.reshape(a, (*([1]*(len(alphas.shape) - 1)), ndim)) + b = xp.reshape(b, (*([1]*(len(alphas.shape) - 1)), ndim)) + + # `xp` is the unwrapped namespace, so `.atan` won't work for `xp = np` and np<2. + xp_test = array_namespace(a) + + return ( + (-1)**ndim * 1/xp.prod(alphas, axis=-1) + * xp.prod( + xp_test.atan((a - betas)/alphas) - xp_test.atan((b - betas)/alphas), + axis=-1, + ) + ) + + +def genz_malik_1980_f_2_random_args(rng, shape, xp): + ndim = shape[-1] + alphas = xp.asarray(rng.random(shape)) + betas = xp.asarray(rng.random(shape)) + + difficulty = 25.0 + products = xp.prod(alphas**xp.asarray(-2.0), axis=-1) + normalisation_factors = (products**xp.asarray(1 / (2*ndim)))[..., None] + alphas = alphas * normalisation_factors * math.pow(difficulty, 1 / (2*ndim)) + + # Adjust alphas from distribution used in Genz and Malik 1980 since denominator + # is very small for high dimensions. + alphas *= 10 + + return alphas, betas + + +def genz_malik_1980_f_3(x, alphas, xp): + r""" + .. math:: f_3(\mathbf x) = \exp\left(\sum^n_{i = 1} \alpha_i x_i\right) + + .. code-block:: mathematica + + genzMalik1980f3[x_List, alphas_List] := Exp[Dot[x, alphas]] + """ + + npoints, ndim = x.shape[0], x.shape[-1] + + alphas_reshaped = alphas[None, ...] + x_reshaped = xp.reshape(x, (npoints, *([1]*(len(alphas.shape) - 1)), ndim)) + + return xp.exp(xp.sum(alphas_reshaped * x_reshaped, axis=-1)) + + +def genz_malik_1980_f_3_exact(a, b, alphas, xp): + ndim = xp_size(a) + a = xp.reshape(a, (*([1]*(len(alphas.shape) - 1)), ndim)) + b = xp.reshape(b, (*([1]*(len(alphas.shape) - 1)), ndim)) + + return ( + (-1)**ndim * 1/xp.prod(alphas, axis=-1) + * xp.prod(xp.exp(alphas * a) - xp.exp(alphas * b), axis=-1) + ) + + +def genz_malik_1980_f_3_random_args(rng, shape, xp): + alphas = xp.asarray(rng.random(shape)) + normalisation_factors = xp.sum(alphas, axis=-1)[..., None] + difficulty = 12.0 + alphas = difficulty * alphas / normalisation_factors + + return (alphas,) + + +def genz_malik_1980_f_4(x, alphas, xp): + r""" + .. math:: f_4(\mathbf x) = \left(1 + \sum^n_{i = 1} \alpha_i x_i\right)^{-n-1} + + .. code-block:: mathematica + genzMalik1980f4[x_List, alphas_List] := + (1 + Dot[x, alphas])^(-Length[alphas] - 1) + """ + + npoints, ndim = x.shape[0], x.shape[-1] + + alphas_reshaped = alphas[None, ...] + x_reshaped = xp.reshape(x, (npoints, *([1]*(len(alphas.shape) - 1)), ndim)) + + return (1 + xp.sum(alphas_reshaped * x_reshaped, axis=-1))**(-ndim-1) + + +def genz_malik_1980_f_4_exact(a, b, alphas, xp): + ndim = xp_size(a) + + def F(x): + x_reshaped = xp.reshape(x, (*([1]*(len(alphas.shape) - 1)), ndim)) + + return ( + (-1)**ndim/xp.prod(alphas, axis=-1) + / math.factorial(ndim) + / (1 + xp.sum(alphas * x_reshaped, axis=-1)) + ) + + return _eval_indefinite_integral(F, a, b, xp) + + +def _eval_indefinite_integral(F, a, b, xp): + """ + Calculates a definite integral from points `a` to `b` by summing up over the corners + of the corresponding hyperrectangle. + """ + + ndim = xp_size(a) + points = xp.stack([a, b], axis=0) + + out = 0 + for ind in itertools.product(range(2), repeat=ndim): + selected_points = xp.asarray([points[i, j] for i, j in zip(ind, range(ndim))]) + out += pow(-1, sum(ind) + ndim) * F(selected_points) + + return out + + +def genz_malik_1980_f_4_random_args(rng, shape, xp): + ndim = shape[-1] + + alphas = xp.asarray(rng.random(shape)) + normalisation_factors = xp.sum(alphas, axis=-1)[..., None] + difficulty = 14.0 + alphas = (difficulty / ndim) * alphas / normalisation_factors + + return (alphas,) + + +def genz_malik_1980_f_5(x, alphas, betas, xp): + r""" + .. math:: + + f_5(\mathbf x) = \exp\left(-\sum^n_{i = 1} \alpha^2_i (x_i - \beta_i)^2\right) + + .. code-block:: mathematica + + genzMalik1980f5[x_List, alphas_List, betas_List] := + Exp[-Total[alphas^2 * (x - betas)^2]] + """ + + npoints, ndim = x.shape[0], x.shape[-1] + + alphas_reshaped = alphas[None, ...] + betas_reshaped = betas[None, ...] + + x_reshaped = xp.reshape(x, (npoints, *([1]*(len(alphas.shape) - 1)), ndim)) + + return xp.exp( + -xp.sum(alphas_reshaped**2 * (x_reshaped - betas_reshaped)**2, axis=-1) + ) + + +def genz_malik_1980_f_5_exact(a, b, alphas, betas, xp): + ndim = xp_size(a) + a = xp.reshape(a, (*([1]*(len(alphas.shape) - 1)), ndim)) + b = xp.reshape(b, (*([1]*(len(alphas.shape) - 1)), ndim)) + + return ( + (1/2)**ndim + * 1/xp.prod(alphas, axis=-1) + * (math.pi**(ndim/2)) + * xp.prod( + scipy.special.erf(alphas * (betas - a)) + + scipy.special.erf(alphas * (b - betas)), + axis=-1, + ) + ) + + +def genz_malik_1980_f_5_random_args(rng, shape, xp): + alphas = xp.asarray(rng.random(shape)) + betas = xp.asarray(rng.random(shape)) + + difficulty = 21.0 + normalisation_factors = xp.sqrt(xp.sum(alphas**xp.asarray(2.0), axis=-1))[..., None] + alphas = alphas / normalisation_factors * math.sqrt(difficulty) + + return alphas, betas + + +def f_gaussian(x, alphas, xp): + r""" + .. math:: + + f(\mathbf x) = \exp\left(-\sum^n_{i = 1} (\alpha_i x_i)^2 \right) + """ + npoints, ndim = x.shape[0], x.shape[-1] + alphas_reshaped = alphas[None, ...] + x_reshaped = xp.reshape(x, (npoints, *([1]*(len(alphas.shape) - 1)), ndim)) + + return xp.exp(-xp.sum((alphas_reshaped * x_reshaped)**2, axis=-1)) + + +def f_gaussian_exact(a, b, alphas, xp): + # Exact only when `a` and `b` are one of: + # (-oo, oo), or + # (0, oo), or + # (-oo, 0) + # `alphas` can be arbitrary. + + ndim = xp_size(a) + double_infinite_count = 0 + semi_infinite_count = 0 + + for i in range(ndim): + if xp.isinf(a[i]) and xp.isinf(b[i]): # doubly-infinite + double_infinite_count += 1 + elif xp.isinf(a[i]) != xp.isinf(b[i]): # exclusive or, so semi-infinite + semi_infinite_count += 1 + + return (math.sqrt(math.pi) ** ndim) / ( + 2**semi_infinite_count * xp.prod(alphas, axis=-1) + ) + + +def f_gaussian_random_args(rng, shape, xp): + alphas = xp.asarray(rng.random(shape)) + + # If alphas are very close to 0 this makes the problem very difficult due to large + # values of ``f``. + alphas *= 100 + + return (alphas,) + + +def f_modified_gaussian(x_arr, n, xp): + r""" + .. math:: + + f(x, y, z, w) = x^n \sqrt{y} \exp(-y-z^2-w^2) + """ + x, y, z, w = x_arr[:, 0], x_arr[:, 1], x_arr[:, 2], x_arr[:, 3] + res = (x ** n[:, None]) * xp.sqrt(y) * xp.exp(-y-z**2-w**2) + + return res.T + + +def f_modified_gaussian_exact(a, b, n, xp): + # Exact only for the limits + # a = (0, 0, -oo, -oo) + # b = (1, oo, oo, oo) + # but defined here as a function to match the format of the other integrands. + return 1/(2 + 2*n) * math.pi ** (3/2) + + +def f_with_problematic_points(x_arr, points, xp): + """ + This emulates a function with a list of singularities given by `points`. + + If no `x_arr` are one of the `points`, then this function returns 1. + """ + + for point in points: + if xp.any(x_arr == point): + raise ValueError("called with a problematic point") + + return xp.ones(x_arr.shape[0]) + + +@array_api_compatible +class TestCubature: + """ + Tests related to the interface of `cubature`. + """ + + @pytest.mark.parametrize("rule_str", [ + "gauss-kronrod", + "genz-malik", + "gk21", + "gk15", + ]) + def test_pass_str(self, rule_str, xp): + n = xp.arange(5, dtype=xp.float64) + a = xp.asarray([0, 0], dtype=xp.float64) + b = xp.asarray([2, 2], dtype=xp.float64) + + res = cubature(basic_nd_integrand, a, b, rule=rule_str, args=(n, xp)) + + xp_assert_close( + res.estimate, + basic_nd_integrand_exact(n, xp), + rtol=1e-8, + atol=0, + ) + + @skip_xp_backends(np_only=True, + reason='array-likes only supported for NumPy backend') + def test_pass_array_like_not_array(self, xp): + n = np_compat.arange(5, dtype=np_compat.float64) + a = [0] + b = [2] + + res = cubature( + basic_1d_integrand, + a, + b, + args=(n, xp) + ) + + xp_assert_close( + res.estimate, + basic_1d_integrand_exact(n, xp), + rtol=1e-8, + atol=0, + ) + + def test_stops_after_max_subdivisions(self, xp): + a = xp.asarray([0]) + b = xp.asarray([1]) + rule = BadErrorRule() + + res = cubature( + basic_1d_integrand, # Any function would suffice + a, + b, + rule=rule, + max_subdivisions=10, + args=(xp.arange(5, dtype=xp.float64), xp), + ) + + assert res.subdivisions == 10 + assert res.status == "not_converged" + + def test_a_and_b_must_be_1d(self, xp): + a = xp.asarray([[0]], dtype=xp.float64) + b = xp.asarray([[1]], dtype=xp.float64) + + with pytest.raises(Exception, match="`a` and `b` must be 1D arrays"): + cubature(basic_1d_integrand, a, b, args=(xp,)) + + def test_a_and_b_must_be_nonempty(self, xp): + a = xp.asarray([]) + b = xp.asarray([]) + + with pytest.raises(Exception, match="`a` and `b` must be nonempty"): + cubature(basic_1d_integrand, a, b, args=(xp,)) + + def test_zero_width_limits(self, xp): + n = xp.arange(5, dtype=xp.float64) + + a = xp.asarray([0], dtype=xp.float64) + b = xp.asarray([0], dtype=xp.float64) + + res = cubature( + basic_1d_integrand, + a, + b, + args=(n, xp), + ) + + xp_assert_close( + res.estimate, + xp.asarray([[0], [0], [0], [0], [0]], dtype=xp.float64), + rtol=1e-8, + atol=0, + ) + + def test_limits_other_way_around(self, xp): + n = xp.arange(5, dtype=xp.float64) + + a = xp.asarray([2], dtype=xp.float64) + b = xp.asarray([0], dtype=xp.float64) + + res = cubature( + basic_1d_integrand, + a, + b, + args=(n, xp), + ) + + xp_assert_close( + res.estimate, + -basic_1d_integrand_exact(n, xp), + rtol=1e-8, + atol=0, + ) + + def test_result_dtype_promoted_correctly(self, xp): + result_dtype = cubature( + basic_1d_integrand, + xp.asarray([0], dtype=xp.float64), + xp.asarray([1], dtype=xp.float64), + points=[], + args=(xp.asarray([1], dtype=xp.float64), xp), + ).estimate.dtype + + assert result_dtype == xp.float64 + + result_dtype = cubature( + basic_1d_integrand, + xp.asarray([0], dtype=xp.float32), + xp.asarray([1], dtype=xp.float32), + points=[], + args=(xp.asarray([1], dtype=xp.float32), xp), + ).estimate.dtype + + assert result_dtype == xp.float32 + + result_dtype = cubature( + basic_1d_integrand, + xp.asarray([0], dtype=xp.float32), + xp.asarray([1], dtype=xp.float64), + points=[], + args=(xp.asarray([1], dtype=xp.float32), xp), + ).estimate.dtype + + assert result_dtype == xp.float64 + + +@pytest.mark.parametrize("rtol", [1e-4]) +@pytest.mark.parametrize("atol", [1e-5]) +@pytest.mark.parametrize("rule", [ + "gk15", + "gk21", + "genz-malik", +]) +@array_api_compatible +class TestCubatureProblems: + """ + Tests that `cubature` gives the correct answer. + """ + + @pytest.mark.parametrize("problem", [ + # -- f1 -- + ( + # Function to integrate, like `f(x, *args)` + genz_malik_1980_f_1, + + # Exact solution, like `exact(a, b, *args)` + genz_malik_1980_f_1_exact, + + # Coordinates of `a` + [0], + + # Coordinates of `b` + [10], + + # Arguments to pass to `f` and `exact` + ( + 1/4, + [5], + ) + ), + ( + genz_malik_1980_f_1, + genz_malik_1980_f_1_exact, + [0, 0], + [1, 1], + ( + 1/4, + [2, 4], + ), + ), + ( + genz_malik_1980_f_1, + genz_malik_1980_f_1_exact, + [0, 0], + [5, 5], + ( + 1/2, + [2, 4], + ) + ), + ( + genz_malik_1980_f_1, + genz_malik_1980_f_1_exact, + [0, 0, 0], + [5, 5, 5], + ( + 1/2, + [1, 1, 1], + ) + ), + + # -- f2 -- + ( + genz_malik_1980_f_2, + genz_malik_1980_f_2_exact, + [-1], + [1], + ( + [5], + [4], + ) + ), + ( + genz_malik_1980_f_2, + genz_malik_1980_f_2_exact, + + [0, 0], + [10, 50], + ( + [-3, 3], + [-2, 2], + ), + ), + ( + genz_malik_1980_f_2, + genz_malik_1980_f_2_exact, + [0, 0, 0], + [1, 1, 1], + ( + [1, 1, 1], + [1, 1, 1], + ) + ), + ( + genz_malik_1980_f_2, + genz_malik_1980_f_2_exact, + [0, 0, 0], + [1, 1, 1], + ( + [2, 3, 4], + [2, 3, 4], + ) + ), + ( + genz_malik_1980_f_2, + genz_malik_1980_f_2_exact, + [-1, -1, -1], + [1, 1, 1], + ( + [1, 1, 1], + [2, 2, 2], + ) + ), + ( + genz_malik_1980_f_2, + genz_malik_1980_f_2_exact, + [-1, -1, -1, -1], + [1, 1, 1, 1], + ( + [1, 1, 1, 1], + [1, 1, 1, 1], + ) + ), + + # -- f3 -- + ( + genz_malik_1980_f_3, + genz_malik_1980_f_3_exact, + [-1], + [1], + ( + [1/2], + ), + ), + ( + genz_malik_1980_f_3, + genz_malik_1980_f_3_exact, + [0, -1], + [1, 1], + ( + [5, 5], + ), + ), + ( + genz_malik_1980_f_3, + genz_malik_1980_f_3_exact, + [-1, -1, -1], + [1, 1, 1], + ( + [1, 1, 1], + ), + ), + + # -- f4 -- + ( + genz_malik_1980_f_4, + genz_malik_1980_f_4_exact, + [0], + [2], + ( + [1], + ), + ), + ( + genz_malik_1980_f_4, + genz_malik_1980_f_4_exact, + [0, 0], + [2, 1], + ([1, 1],), + ), + ( + genz_malik_1980_f_4, + genz_malik_1980_f_4_exact, + [0, 0, 0], + [1, 1, 1], + ([1, 1, 1],), + ), + + # -- f5 -- + ( + genz_malik_1980_f_5, + genz_malik_1980_f_5_exact, + [-1], + [1], + ( + [-2], + [2], + ), + ), + ( + genz_malik_1980_f_5, + genz_malik_1980_f_5_exact, + [-1, -1], + [1, 1], + ( + [2, 3], + [4, 5], + ), + ), + ( + genz_malik_1980_f_5, + genz_malik_1980_f_5_exact, + [-1, -1], + [1, 1], + ( + [-1, 1], + [0, 0], + ), + ), + ( + genz_malik_1980_f_5, + genz_malik_1980_f_5_exact, + [-1, -1, -1], + [1, 1, 1], + ( + [1, 1, 1], + [1, 1, 1], + ), + ), + ]) + def test_scalar_output(self, problem, rule, rtol, atol, xp): + f, exact, a, b, args = problem + + a = xp.asarray(a, dtype=xp.float64) + b = xp.asarray(b, dtype=xp.float64) + args = tuple(xp.asarray(arg, dtype=xp.float64) for arg in args) + + ndim = xp_size(a) + + if rule == "genz-malik" and ndim < 2: + pytest.skip("Genz-Malik cubature does not support 1D integrals") + + res = cubature( + f, + a, + b, + rule=rule, + rtol=rtol, + atol=atol, + args=(*args, xp), + ) + + assert res.status == "converged" + + est = res.estimate + exact_sol = exact(a, b, *args, xp) + + xp_assert_close( + est, + exact_sol, + rtol=rtol, + atol=atol, + err_msg=f"estimate_error={res.error}, subdivisions={res.subdivisions}", + ) + + @pytest.mark.parametrize("problem", [ + ( + # Function to integrate, like `f(x, *args)` + genz_malik_1980_f_1, + + # Exact solution, like `exact(a, b, *args)` + genz_malik_1980_f_1_exact, + + # Function that generates random args of a certain shape. + genz_malik_1980_f_1_random_args, + ), + ( + genz_malik_1980_f_2, + genz_malik_1980_f_2_exact, + genz_malik_1980_f_2_random_args, + ), + ( + genz_malik_1980_f_3, + genz_malik_1980_f_3_exact, + genz_malik_1980_f_3_random_args + ), + ( + genz_malik_1980_f_4, + genz_malik_1980_f_4_exact, + genz_malik_1980_f_4_random_args + ), + ( + genz_malik_1980_f_5, + genz_malik_1980_f_5_exact, + genz_malik_1980_f_5_random_args, + ), + ]) + @pytest.mark.parametrize("shape", [ + (2,), + (3,), + (4,), + (1, 2), + (1, 3), + (1, 4), + (3, 2), + (3, 4, 2), + (2, 1, 3), + ]) + def test_array_output(self, problem, rule, shape, rtol, atol, xp): + rng = np_compat.random.default_rng(1) + ndim = shape[-1] + + if rule == "genz-malik" and ndim < 2: + pytest.skip("Genz-Malik cubature does not support 1D integrals") + + if rule == "genz-malik" and ndim >= 5: + pytest.mark.slow("Gauss-Kronrod is slow in >= 5 dim") + + f, exact, random_args = problem + args = random_args(rng, shape, xp) + + a = xp.asarray([0] * ndim, dtype=xp.float64) + b = xp.asarray([1] * ndim, dtype=xp.float64) + + res = cubature( + f, + a, + b, + rule=rule, + rtol=rtol, + atol=atol, + args=(*args, xp), + ) + + est = res.estimate + exact_sol = exact(a, b, *args, xp) + + xp_assert_close( + est, + exact_sol, + rtol=rtol, + atol=atol, + err_msg=f"estimate_error={res.error}, subdivisions={res.subdivisions}", + ) + + err_msg = (f"estimate_error={res.error}, " + f"subdivisions= {res.subdivisions}, " + f"true_error={xp.abs(res.estimate - exact_sol)}") + assert res.status == "converged", err_msg + + assert res.estimate.shape == shape[:-1] + + @pytest.mark.parametrize("problem", [ + ( + # Function to integrate + lambda x, xp: x, + + # Exact value + [50.0], + + # Coordinates of `a` + [0], + + # Coordinates of `b` + [10], + + # Points by which to split up the initial region + None, + ), + ( + lambda x, xp: xp.sin(x)/x, + [2.551496047169878], # si(1) + si(2), + [-1], + [2], + [ + [0.0], + ], + ), + ( + lambda x, xp: xp.ones((x.shape[0], 1)), + [1.0], + [0, 0, 0], + [1, 1, 1], + [ + [0.5, 0.5, 0.5], + ], + ), + ( + lambda x, xp: xp.ones((x.shape[0], 1)), + [1.0], + [0, 0, 0], + [1, 1, 1], + [ + [0.25, 0.25, 0.25], + [0.5, 0.5, 0.5], + ], + ), + ( + lambda x, xp: xp.ones((x.shape[0], 1)), + [1.0], + [0, 0, 0], + [1, 1, 1], + [ + [0.1, 0.25, 0.5], + [0.25, 0.25, 0.25], + [0.5, 0.5, 0.5], + ], + ) + ]) + def test_break_points(self, problem, rule, rtol, atol, xp): + f, exact, a, b, points = problem + + a = xp.asarray(a, dtype=xp.float64) + b = xp.asarray(b, dtype=xp.float64) + exact = xp.asarray(exact, dtype=xp.float64) + + if points is not None: + points = [xp.asarray(point, dtype=xp.float64) for point in points] + + ndim = xp_size(a) + + if rule == "genz-malik" and ndim < 2: + pytest.skip("Genz-Malik cubature does not support 1D integrals") + + if rule == "genz-malik" and ndim >= 5: + pytest.mark.slow("Gauss-Kronrod is slow in >= 5 dim") + + res = cubature( + f, + a, + b, + rule=rule, + rtol=rtol, + atol=atol, + points=points, + args=(xp,), + ) + + xp_assert_close( + res.estimate, + exact, + rtol=rtol, + atol=atol, + err_msg=f"estimate_error={res.error}, subdivisions={res.subdivisions}", + check_dtype=False, + ) + + err_msg = (f"estimate_error={res.error}, " + f"subdivisions= {res.subdivisions}, " + f"true_error={xp.abs(res.estimate - exact)}") + assert res.status == "converged", err_msg + + @skip_xp_backends( + "jax.numpy", + reasons=["transforms make use of indexing assignment"], + ) + @pytest.mark.parametrize("problem", [ + ( + # Function to integrate + f_gaussian, + + # Exact solution + f_gaussian_exact, + + # Arguments passed to f + f_gaussian_random_args, + (1, 1), + + # Limits, have to match the shape of the arguments + [-math.inf], # a + [math.inf], # b + ), + ( + f_gaussian, + f_gaussian_exact, + f_gaussian_random_args, + (2, 2), + [-math.inf, -math.inf], + [math.inf, math.inf], + ), + ( + f_gaussian, + f_gaussian_exact, + f_gaussian_random_args, + (1, 1), + [0], + [math.inf], + ), + ( + f_gaussian, + f_gaussian_exact, + f_gaussian_random_args, + (1, 1), + [-math.inf], + [0], + ), + ( + f_gaussian, + f_gaussian_exact, + f_gaussian_random_args, + (2, 2), + [0, 0], + [math.inf, math.inf], + ), + ( + f_gaussian, + f_gaussian_exact, + f_gaussian_random_args, + (2, 2), + [0, -math.inf], + [math.inf, math.inf], + ), + ( + f_gaussian, + f_gaussian_exact, + f_gaussian_random_args, + (1, 4), + [0, 0, -math.inf, -math.inf], + [math.inf, math.inf, math.inf, math.inf], + ), + ( + f_gaussian, + f_gaussian_exact, + f_gaussian_random_args, + (1, 4), + [-math.inf, -math.inf, -math.inf, -math.inf], + [0, 0, math.inf, math.inf], + ), + ( + lambda x, xp: 1/xp.prod(x, axis=-1)**2, + + # Exact only for the below limits, not for general `a` and `b`. + lambda a, b, xp: xp.asarray(1/6, dtype=xp.float64), + + # Arguments + lambda rng, shape, xp: tuple(), + tuple(), + + [1, -math.inf, 3], + [math.inf, -2, math.inf], + ), + + # This particular problem can be slow + pytest.param( + ( + # f(x, y, z, w) = x^n * sqrt(y) * exp(-y-z**2-w**2) for n in [0,1,2,3] + f_modified_gaussian, + + # This exact solution is for the below limits, not in general + f_modified_gaussian_exact, + + # Constant arguments + lambda rng, shape, xp: (xp.asarray([0, 1, 2, 3, 4], dtype=xp.float64),), + tuple(), + + [0, 0, -math.inf, -math.inf], + [1, math.inf, math.inf, math.inf] + ), + + marks=pytest.mark.xslow, + ), + ]) + def test_infinite_limits(self, problem, rule, rtol, atol, xp): + rng = np_compat.random.default_rng(1) + f, exact, random_args_func, random_args_shape, a, b = problem + + a = xp.asarray(a, dtype=xp.float64) + b = xp.asarray(b, dtype=xp.float64) + args = random_args_func(rng, random_args_shape, xp) + + ndim = xp_size(a) + + if rule == "genz-malik" and ndim < 2: + pytest.skip("Genz-Malik cubature does not support 1D integrals") + + if rule == "genz-malik" and ndim >= 4: + pytest.mark.slow("Genz-Malik is slow in >= 5 dim") + + if rule == "genz-malik" and ndim >= 4 and is_array_api_strict(xp): + pytest.mark.xslow("Genz-Malik very slow for array_api_strict in >= 4 dim") + + res = cubature( + f, + a, + b, + rule=rule, + rtol=rtol, + atol=atol, + args=(*args, xp), + ) + + assert res.status == "converged" + + xp_assert_close( + res.estimate, + exact(a, b, *args, xp), + rtol=rtol, + atol=atol, + err_msg=f"error_estimate={res.error}, subdivisions={res.subdivisions}", + check_0d=False, + ) + + @skip_xp_backends( + "jax.numpy", + reasons=["transforms make use of indexing assignment"], + ) + @pytest.mark.parametrize("problem", [ + ( + # Function to integrate + lambda x, xp: (xp.sin(x) / x)**8, + + # Exact value + [151/315 * math.pi], + + # Limits + [-math.inf], + [math.inf], + + # Breakpoints + [[0]], + + ), + ( + # Function to integrate + lambda x, xp: (xp.sin(x[:, 0]) / x[:, 0])**8, + + # Exact value + 151/315 * math.pi, + + # Limits + [-math.inf, 0], + [math.inf, 1], + + # Breakpoints + [[0, 0.5]], + + ) + ]) + def test_infinite_limits_and_break_points(self, problem, rule, rtol, atol, xp): + f, exact, a, b, points = problem + + a = xp.asarray(a, dtype=xp.float64) + b = xp.asarray(b, dtype=xp.float64) + exact = xp.asarray(exact, dtype=xp.float64) + + ndim = xp_size(a) + + if rule == "genz-malik" and ndim < 2: + pytest.skip("Genz-Malik cubature does not support 1D integrals") + + if points is not None: + points = [xp.asarray(point, dtype=xp.float64) for point in points] + + res = cubature( + f, + a, + b, + rule=rule, + rtol=rtol, + atol=atol, + points=points, + args=(xp,), + ) + + assert res.status == "converged" + + xp_assert_close( + res.estimate, + exact, + rtol=rtol, + atol=atol, + err_msg=f"error_estimate={res.error}, subdivisions={res.subdivisions}", + check_0d=False, + ) + + +@array_api_compatible +class TestRules: + """ + Tests related to the general Rule interface (currently private). + """ + + @pytest.mark.parametrize("problem", [ + ( + # 2D problem, 1D rule + [0, 0], + [1, 1], + GaussKronrodQuadrature, + (21,), + ), + ( + # 1D problem, 2D rule + [0], + [1], + GenzMalikCubature, + (2,), + ) + ]) + def test_incompatible_dimension_raises_error(self, problem, xp): + a, b, quadrature, quadrature_args = problem + rule = quadrature(*quadrature_args, xp=xp) + + a = xp.asarray(a, dtype=xp.float64) + b = xp.asarray(b, dtype=xp.float64) + + with pytest.raises(Exception, match="incompatible dimension"): + rule.estimate(basic_1d_integrand, a, b, args=(xp,)) + + def test_estimate_with_base_classes_raise_error(self, xp): + a = xp.asarray([0]) + b = xp.asarray([1]) + + for base_class in [Rule(), FixedRule()]: + with pytest.raises(Exception): + base_class.estimate(basic_1d_integrand, a, b, args=(xp,)) + + +@array_api_compatible +class TestRulesQuadrature: + """ + Tests underlying quadrature rules (ndim == 1). + """ + + @pytest.mark.parametrize(("rule", "rule_args"), [ + (GaussLegendreQuadrature, (3,)), + (GaussLegendreQuadrature, (5,)), + (GaussLegendreQuadrature, (10,)), + (GaussKronrodQuadrature, (15,)), + (GaussKronrodQuadrature, (21,)), + ]) + def test_base_1d_quadratures_simple(self, rule, rule_args, xp): + quadrature = rule(*rule_args, xp=xp) + + n = xp.arange(5, dtype=xp.float64) + + def f(x): + x_reshaped = xp.reshape(x, (-1, 1, 1)) + n_reshaped = xp.reshape(n, (1, -1, 1)) + + return x_reshaped**n_reshaped + + a = xp.asarray([0], dtype=xp.float64) + b = xp.asarray([2], dtype=xp.float64) + + exact = xp.reshape(2**(n+1)/(n+1), (-1, 1)) + estimate = quadrature.estimate(f, a, b) + + xp_assert_close( + estimate, + exact, + rtol=1e-8, + atol=0, + ) + + @pytest.mark.parametrize(("rule_pair", "rule_pair_args"), [ + ((GaussLegendreQuadrature, GaussLegendreQuadrature), (10, 5)), + ]) + def test_base_1d_quadratures_error_from_difference(self, rule_pair, rule_pair_args, + xp): + n = xp.arange(5, dtype=xp.float64) + a = xp.asarray([0], dtype=xp.float64) + b = xp.asarray([2], dtype=xp.float64) + + higher = rule_pair[0](rule_pair_args[0], xp=xp) + lower = rule_pair[1](rule_pair_args[1], xp=xp) + + rule = NestedFixedRule(higher, lower) + res = cubature( + basic_1d_integrand, + a, b, + rule=rule, + rtol=1e-8, + args=(n, xp), + ) + + xp_assert_close( + res.estimate, + basic_1d_integrand_exact(n, xp), + rtol=1e-8, + atol=0, + ) + + @pytest.mark.parametrize("quadrature", [ + GaussLegendreQuadrature + ]) + def test_one_point_fixed_quad_impossible(self, quadrature, xp): + with pytest.raises(Exception): + quadrature(1, xp=xp) + + +@array_api_compatible +class TestRulesCubature: + """ + Tests underlying cubature rules (ndim >= 2). + """ + + @pytest.mark.parametrize("ndim", range(2, 11)) + def test_genz_malik_func_evaluations(self, ndim, xp): + """ + Tests that the number of function evaluations required for Genz-Malik cubature + matches the number in Genz and Malik 1980. + """ + + nodes, _ = GenzMalikCubature(ndim, xp=xp).nodes_and_weights + + assert nodes.shape[0] == (2**ndim) + 2*ndim**2 + 2*ndim + 1 + + def test_genz_malik_1d_raises_error(self, xp): + with pytest.raises(Exception, match="only defined for ndim >= 2"): + GenzMalikCubature(1, xp=xp) + + +@array_api_compatible +@skip_xp_backends( + "jax.numpy", + reasons=["transforms make use of indexing assignment"], +) +class TestTransformations: + @pytest.mark.parametrize(("a", "b", "points"), [ + ( + [0, 1, -math.inf], + [1, math.inf, math.inf], + [ + [1, 1, 1], + [0.5, 10, 10], + ] + ) + ]) + def test_infinite_limits_maintains_points(self, a, b, points, xp): + """ + Test that break points are correctly mapped under the _InfiniteLimitsTransform + transformation. + """ + + xp_compat = array_namespace(xp.empty(0)) + points = [xp.asarray(p, dtype=xp.float64) for p in points] + + f_transformed = _InfiniteLimitsTransform( + # Bind `points` and `xp` argument in f + lambda x: f_with_problematic_points(x, points, xp_compat), + xp.asarray(a, dtype=xp_compat.float64), + xp.asarray(b, dtype=xp_compat.float64), + xp=xp_compat, + ) + + for point in points: + transformed_point = f_transformed.inv(xp_compat.reshape(point, (1, -1))) + + with pytest.raises(Exception, match="called with a problematic point"): + f_transformed(transformed_point) + + +class BadErrorRule(Rule): + """ + A rule with fake high error so that cubature will keep on subdividing. + """ + + def estimate(self, f, a, b, args=()): + xp = array_namespace(a, b) + underlying = GaussLegendreQuadrature(10, xp=xp) + + return underlying.estimate(f, a, b, args) + + def estimate_error(self, f, a, b, args=()): + xp = array_namespace(a, b) + return xp.asarray(1e6, dtype=xp.float64) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_integrate.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_integrate.py new file mode 100644 index 0000000000000000000000000000000000000000..44bfecdaac0f00b413538510c61dd1317a076261 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_integrate.py @@ -0,0 +1,840 @@ +# Authors: Nils Wagner, Ed Schofield, Pauli Virtanen, John Travers +""" +Tests for numerical integration. +""" +import numpy as np +from numpy import (arange, zeros, array, dot, sqrt, cos, sin, eye, pi, exp, + allclose) + +from numpy.testing import ( + assert_, assert_array_almost_equal, + assert_allclose, assert_array_equal, assert_equal, assert_warns) +import pytest +from pytest import raises as assert_raises +from scipy.integrate import odeint, ode, complex_ode + +#------------------------------------------------------------------------------ +# Test ODE integrators +#------------------------------------------------------------------------------ + + +class TestOdeint: + # Check integrate.odeint + + def _do_problem(self, problem): + t = arange(0.0, problem.stop_t, 0.05) + + # Basic case + z, infodict = odeint(problem.f, problem.z0, t, full_output=True) + assert_(problem.verify(z, t)) + + # Use tfirst=True + z, infodict = odeint(lambda t, y: problem.f(y, t), problem.z0, t, + full_output=True, tfirst=True) + assert_(problem.verify(z, t)) + + if hasattr(problem, 'jac'): + # Use Dfun + z, infodict = odeint(problem.f, problem.z0, t, Dfun=problem.jac, + full_output=True) + assert_(problem.verify(z, t)) + + # Use Dfun and tfirst=True + z, infodict = odeint(lambda t, y: problem.f(y, t), problem.z0, t, + Dfun=lambda t, y: problem.jac(y, t), + full_output=True, tfirst=True) + assert_(problem.verify(z, t)) + + def test_odeint(self): + for problem_cls in PROBLEMS: + problem = problem_cls() + if problem.cmplx: + continue + self._do_problem(problem) + + +class TestODEClass: + + ode_class = None # Set in subclass. + + def _do_problem(self, problem, integrator, method='adams'): + + # ode has callback arguments in different order than odeint + def f(t, z): + return problem.f(z, t) + jac = None + if hasattr(problem, 'jac'): + def jac(t, z): + return problem.jac(z, t) + + integrator_params = {} + if problem.lband is not None or problem.uband is not None: + integrator_params['uband'] = problem.uband + integrator_params['lband'] = problem.lband + + ig = self.ode_class(f, jac) + ig.set_integrator(integrator, + atol=problem.atol/10, + rtol=problem.rtol/10, + method=method, + **integrator_params) + + ig.set_initial_value(problem.z0, t=0.0) + z = ig.integrate(problem.stop_t) + + assert_array_equal(z, ig.y) + assert_(ig.successful(), (problem, method)) + assert_(ig.get_return_code() > 0, (problem, method)) + assert_(problem.verify(array([z]), problem.stop_t), (problem, method)) + + +class TestOde(TestODEClass): + + ode_class = ode + + def test_vode(self): + # Check the vode solver + for problem_cls in PROBLEMS: + problem = problem_cls() + if problem.cmplx: + continue + if not problem.stiff: + self._do_problem(problem, 'vode', 'adams') + self._do_problem(problem, 'vode', 'bdf') + + def test_zvode(self): + # Check the zvode solver + for problem_cls in PROBLEMS: + problem = problem_cls() + if not problem.stiff: + self._do_problem(problem, 'zvode', 'adams') + self._do_problem(problem, 'zvode', 'bdf') + + def test_lsoda(self): + # Check the lsoda solver + for problem_cls in PROBLEMS: + problem = problem_cls() + if problem.cmplx: + continue + self._do_problem(problem, 'lsoda') + + def test_dopri5(self): + # Check the dopri5 solver + for problem_cls in PROBLEMS: + problem = problem_cls() + if problem.cmplx: + continue + if problem.stiff: + continue + if hasattr(problem, 'jac'): + continue + self._do_problem(problem, 'dopri5') + + def test_dop853(self): + # Check the dop853 solver + for problem_cls in PROBLEMS: + problem = problem_cls() + if problem.cmplx: + continue + if problem.stiff: + continue + if hasattr(problem, 'jac'): + continue + self._do_problem(problem, 'dop853') + + @pytest.mark.thread_unsafe + def test_concurrent_fail(self): + for sol in ('vode', 'zvode', 'lsoda'): + def f(t, y): + return 1.0 + + r = ode(f).set_integrator(sol) + r.set_initial_value(0, 0) + + r2 = ode(f).set_integrator(sol) + r2.set_initial_value(0, 0) + + r.integrate(r.t + 0.1) + r2.integrate(r2.t + 0.1) + + assert_raises(RuntimeError, r.integrate, r.t + 0.1) + + def test_concurrent_ok(self, num_parallel_threads): + def f(t, y): + return 1.0 + + for k in range(3): + for sol in ('vode', 'zvode', 'lsoda', 'dopri5', 'dop853'): + if sol in {'vode', 'zvode', 'lsoda'} and num_parallel_threads > 1: + continue + r = ode(f).set_integrator(sol) + r.set_initial_value(0, 0) + + r2 = ode(f).set_integrator(sol) + r2.set_initial_value(0, 0) + + r.integrate(r.t + 0.1) + r2.integrate(r2.t + 0.1) + r2.integrate(r2.t + 0.1) + + assert_allclose(r.y, 0.1) + assert_allclose(r2.y, 0.2) + + for sol in ('dopri5', 'dop853'): + r = ode(f).set_integrator(sol) + r.set_initial_value(0, 0) + + r2 = ode(f).set_integrator(sol) + r2.set_initial_value(0, 0) + + r.integrate(r.t + 0.1) + r.integrate(r.t + 0.1) + r2.integrate(r2.t + 0.1) + r.integrate(r.t + 0.1) + r2.integrate(r2.t + 0.1) + + assert_allclose(r.y, 0.3) + assert_allclose(r2.y, 0.2) + + +class TestComplexOde(TestODEClass): + + ode_class = complex_ode + + def test_vode(self): + # Check the vode solver + for problem_cls in PROBLEMS: + problem = problem_cls() + if not problem.stiff: + self._do_problem(problem, 'vode', 'adams') + else: + self._do_problem(problem, 'vode', 'bdf') + + def test_lsoda(self): + + # Check the lsoda solver + for problem_cls in PROBLEMS: + problem = problem_cls() + self._do_problem(problem, 'lsoda') + + def test_dopri5(self): + # Check the dopri5 solver + for problem_cls in PROBLEMS: + problem = problem_cls() + if problem.stiff: + continue + if hasattr(problem, 'jac'): + continue + self._do_problem(problem, 'dopri5') + + def test_dop853(self): + # Check the dop853 solver + for problem_cls in PROBLEMS: + problem = problem_cls() + if problem.stiff: + continue + if hasattr(problem, 'jac'): + continue + self._do_problem(problem, 'dop853') + + +class TestSolout: + # Check integrate.ode correctly handles solout for dopri5 and dop853 + def _run_solout_test(self, integrator): + # Check correct usage of solout + ts = [] + ys = [] + t0 = 0.0 + tend = 10.0 + y0 = [1.0, 2.0] + + def solout(t, y): + ts.append(t) + ys.append(y.copy()) + + def rhs(t, y): + return [y[0] + y[1], -y[1]**2] + + ig = ode(rhs).set_integrator(integrator) + ig.set_solout(solout) + ig.set_initial_value(y0, t0) + ret = ig.integrate(tend) + assert_array_equal(ys[0], y0) + assert_array_equal(ys[-1], ret) + assert_equal(ts[0], t0) + assert_equal(ts[-1], tend) + + def test_solout(self): + for integrator in ('dopri5', 'dop853'): + self._run_solout_test(integrator) + + def _run_solout_after_initial_test(self, integrator): + # Check if solout works even if it is set after the initial value. + ts = [] + ys = [] + t0 = 0.0 + tend = 10.0 + y0 = [1.0, 2.0] + + def solout(t, y): + ts.append(t) + ys.append(y.copy()) + + def rhs(t, y): + return [y[0] + y[1], -y[1]**2] + + ig = ode(rhs).set_integrator(integrator) + ig.set_initial_value(y0, t0) + ig.set_solout(solout) + ret = ig.integrate(tend) + assert_array_equal(ys[0], y0) + assert_array_equal(ys[-1], ret) + assert_equal(ts[0], t0) + assert_equal(ts[-1], tend) + + def test_solout_after_initial(self): + for integrator in ('dopri5', 'dop853'): + self._run_solout_after_initial_test(integrator) + + def _run_solout_break_test(self, integrator): + # Check correct usage of stopping via solout + ts = [] + ys = [] + t0 = 0.0 + tend = 10.0 + y0 = [1.0, 2.0] + + def solout(t, y): + ts.append(t) + ys.append(y.copy()) + if t > tend/2.0: + return -1 + + def rhs(t, y): + return [y[0] + y[1], -y[1]**2] + + ig = ode(rhs).set_integrator(integrator) + ig.set_solout(solout) + ig.set_initial_value(y0, t0) + ret = ig.integrate(tend) + assert_array_equal(ys[0], y0) + assert_array_equal(ys[-1], ret) + assert_equal(ts[0], t0) + assert_(ts[-1] > tend/2.0) + assert_(ts[-1] < tend) + + def test_solout_break(self): + for integrator in ('dopri5', 'dop853'): + self._run_solout_break_test(integrator) + + +class TestComplexSolout: + # Check integrate.ode correctly handles solout for dopri5 and dop853 + def _run_solout_test(self, integrator): + # Check correct usage of solout + ts = [] + ys = [] + t0 = 0.0 + tend = 20.0 + y0 = [0.0] + + def solout(t, y): + ts.append(t) + ys.append(y.copy()) + + def rhs(t, y): + return [1.0/(t - 10.0 - 1j)] + + ig = complex_ode(rhs).set_integrator(integrator) + ig.set_solout(solout) + ig.set_initial_value(y0, t0) + ret = ig.integrate(tend) + assert_array_equal(ys[0], y0) + assert_array_equal(ys[-1], ret) + assert_equal(ts[0], t0) + assert_equal(ts[-1], tend) + + def test_solout(self): + for integrator in ('dopri5', 'dop853'): + self._run_solout_test(integrator) + + def _run_solout_break_test(self, integrator): + # Check correct usage of stopping via solout + ts = [] + ys = [] + t0 = 0.0 + tend = 20.0 + y0 = [0.0] + + def solout(t, y): + ts.append(t) + ys.append(y.copy()) + if t > tend/2.0: + return -1 + + def rhs(t, y): + return [1.0/(t - 10.0 - 1j)] + + ig = complex_ode(rhs).set_integrator(integrator) + ig.set_solout(solout) + ig.set_initial_value(y0, t0) + ret = ig.integrate(tend) + assert_array_equal(ys[0], y0) + assert_array_equal(ys[-1], ret) + assert_equal(ts[0], t0) + assert_(ts[-1] > tend/2.0) + assert_(ts[-1] < tend) + + def test_solout_break(self): + for integrator in ('dopri5', 'dop853'): + self._run_solout_break_test(integrator) + + +#------------------------------------------------------------------------------ +# Test problems +#------------------------------------------------------------------------------ + + +class ODE: + """ + ODE problem + """ + stiff = False + cmplx = False + stop_t = 1 + z0 = [] + + lband = None + uband = None + + atol = 1e-6 + rtol = 1e-5 + + +class SimpleOscillator(ODE): + r""" + Free vibration of a simple oscillator:: + m \ddot{u} + k u = 0, u(0) = u_0 \dot{u}(0) \dot{u}_0 + Solution:: + u(t) = u_0*cos(sqrt(k/m)*t)+\dot{u}_0*sin(sqrt(k/m)*t)/sqrt(k/m) + """ + stop_t = 1 + 0.09 + z0 = array([1.0, 0.1], float) + + k = 4.0 + m = 1.0 + + def f(self, z, t): + tmp = zeros((2, 2), float) + tmp[0, 1] = 1.0 + tmp[1, 0] = -self.k / self.m + return dot(tmp, z) + + def verify(self, zs, t): + omega = sqrt(self.k / self.m) + u = self.z0[0]*cos(omega*t) + self.z0[1]*sin(omega*t)/omega + return allclose(u, zs[:, 0], atol=self.atol, rtol=self.rtol) + + +class ComplexExp(ODE): + r"""The equation :lm:`\dot u = i u`""" + stop_t = 1.23*pi + z0 = exp([1j, 2j, 3j, 4j, 5j]) + cmplx = True + + def f(self, z, t): + return 1j*z + + def jac(self, z, t): + return 1j*eye(5) + + def verify(self, zs, t): + u = self.z0 * exp(1j*t) + return allclose(u, zs, atol=self.atol, rtol=self.rtol) + + +class Pi(ODE): + r"""Integrate 1/(t + 1j) from t=-10 to t=10""" + stop_t = 20 + z0 = [0] + cmplx = True + + def f(self, z, t): + return array([1./(t - 10 + 1j)]) + + def verify(self, zs, t): + u = -2j * np.arctan(10) + return allclose(u, zs[-1, :], atol=self.atol, rtol=self.rtol) + + +class CoupledDecay(ODE): + r""" + 3 coupled decays suited for banded treatment + (banded mode makes it necessary when N>>3) + """ + + stiff = True + stop_t = 0.5 + z0 = [5.0, 7.0, 13.0] + lband = 1 + uband = 0 + + lmbd = [0.17, 0.23, 0.29] # fictitious decay constants + + def f(self, z, t): + lmbd = self.lmbd + return np.array([-lmbd[0]*z[0], + -lmbd[1]*z[1] + lmbd[0]*z[0], + -lmbd[2]*z[2] + lmbd[1]*z[1]]) + + def jac(self, z, t): + # The full Jacobian is + # + # [-lmbd[0] 0 0 ] + # [ lmbd[0] -lmbd[1] 0 ] + # [ 0 lmbd[1] -lmbd[2]] + # + # The lower and upper bandwidths are lband=1 and uband=0, resp. + # The representation of this array in packed format is + # + # [-lmbd[0] -lmbd[1] -lmbd[2]] + # [ lmbd[0] lmbd[1] 0 ] + + lmbd = self.lmbd + j = np.zeros((self.lband + self.uband + 1, 3), order='F') + + def set_j(ri, ci, val): + j[self.uband + ri - ci, ci] = val + set_j(0, 0, -lmbd[0]) + set_j(1, 0, lmbd[0]) + set_j(1, 1, -lmbd[1]) + set_j(2, 1, lmbd[1]) + set_j(2, 2, -lmbd[2]) + return j + + def verify(self, zs, t): + # Formulae derived by hand + lmbd = np.array(self.lmbd) + d10 = lmbd[1] - lmbd[0] + d21 = lmbd[2] - lmbd[1] + d20 = lmbd[2] - lmbd[0] + e0 = np.exp(-lmbd[0] * t) + e1 = np.exp(-lmbd[1] * t) + e2 = np.exp(-lmbd[2] * t) + u = np.vstack(( + self.z0[0] * e0, + self.z0[1] * e1 + self.z0[0] * lmbd[0] / d10 * (e0 - e1), + self.z0[2] * e2 + self.z0[1] * lmbd[1] / d21 * (e1 - e2) + + lmbd[1] * lmbd[0] * self.z0[0] / d10 * + (1 / d20 * (e0 - e2) - 1 / d21 * (e1 - e2)))).transpose() + return allclose(u, zs, atol=self.atol, rtol=self.rtol) + + +PROBLEMS = [SimpleOscillator, ComplexExp, Pi, CoupledDecay] + +#------------------------------------------------------------------------------ + + +def f(t, x): + dxdt = [x[1], -x[0]] + return dxdt + + +def jac(t, x): + j = array([[0.0, 1.0], + [-1.0, 0.0]]) + return j + + +def f1(t, x, omega): + dxdt = [omega*x[1], -omega*x[0]] + return dxdt + + +def jac1(t, x, omega): + j = array([[0.0, omega], + [-omega, 0.0]]) + return j + + +def f2(t, x, omega1, omega2): + dxdt = [omega1*x[1], -omega2*x[0]] + return dxdt + + +def jac2(t, x, omega1, omega2): + j = array([[0.0, omega1], + [-omega2, 0.0]]) + return j + + +def fv(t, x, omega): + dxdt = [omega[0]*x[1], -omega[1]*x[0]] + return dxdt + + +def jacv(t, x, omega): + j = array([[0.0, omega[0]], + [-omega[1], 0.0]]) + return j + + +class ODECheckParameterUse: + """Call an ode-class solver with several cases of parameter use.""" + + # solver_name must be set before tests can be run with this class. + + # Set these in subclasses. + solver_name = '' + solver_uses_jac = False + + def _get_solver(self, f, jac): + solver = ode(f, jac) + if self.solver_uses_jac: + solver.set_integrator(self.solver_name, atol=1e-9, rtol=1e-7, + with_jacobian=self.solver_uses_jac) + else: + # XXX Shouldn't set_integrator *always* accept the keyword arg + # 'with_jacobian', and perhaps raise an exception if it is set + # to True if the solver can't actually use it? + solver.set_integrator(self.solver_name, atol=1e-9, rtol=1e-7) + return solver + + def _check_solver(self, solver): + ic = [1.0, 0.0] + solver.set_initial_value(ic, 0.0) + solver.integrate(pi) + assert_array_almost_equal(solver.y, [-1.0, 0.0]) + + def test_no_params(self): + solver = self._get_solver(f, jac) + self._check_solver(solver) + + def test_one_scalar_param(self): + solver = self._get_solver(f1, jac1) + omega = 1.0 + solver.set_f_params(omega) + if self.solver_uses_jac: + solver.set_jac_params(omega) + self._check_solver(solver) + + def test_two_scalar_params(self): + solver = self._get_solver(f2, jac2) + omega1 = 1.0 + omega2 = 1.0 + solver.set_f_params(omega1, omega2) + if self.solver_uses_jac: + solver.set_jac_params(omega1, omega2) + self._check_solver(solver) + + def test_vector_param(self): + solver = self._get_solver(fv, jacv) + omega = [1.0, 1.0] + solver.set_f_params(omega) + if self.solver_uses_jac: + solver.set_jac_params(omega) + self._check_solver(solver) + + @pytest.mark.thread_unsafe + def test_warns_on_failure(self): + # Set nsteps small to ensure failure + solver = self._get_solver(f, jac) + solver.set_integrator(self.solver_name, nsteps=1) + ic = [1.0, 0.0] + solver.set_initial_value(ic, 0.0) + assert_warns(UserWarning, solver.integrate, pi) + + +class TestDOPRI5CheckParameterUse(ODECheckParameterUse): + solver_name = 'dopri5' + solver_uses_jac = False + + +class TestDOP853CheckParameterUse(ODECheckParameterUse): + solver_name = 'dop853' + solver_uses_jac = False + + +class TestVODECheckParameterUse(ODECheckParameterUse): + solver_name = 'vode' + solver_uses_jac = True + + +class TestZVODECheckParameterUse(ODECheckParameterUse): + solver_name = 'zvode' + solver_uses_jac = True + + +class TestLSODACheckParameterUse(ODECheckParameterUse): + solver_name = 'lsoda' + solver_uses_jac = True + + +def test_odeint_trivial_time(): + # Test that odeint succeeds when given a single time point + # and full_output=True. This is a regression test for gh-4282. + y0 = 1 + t = [0] + y, info = odeint(lambda y, t: -y, y0, t, full_output=True) + assert_array_equal(y, np.array([[y0]])) + + +def test_odeint_banded_jacobian(): + # Test the use of the `Dfun`, `ml` and `mu` options of odeint. + + def func(y, t, c): + return c.dot(y) + + def jac(y, t, c): + return c + + def jac_transpose(y, t, c): + return c.T.copy(order='C') + + def bjac_rows(y, t, c): + jac = np.vstack((np.r_[0, np.diag(c, 1)], + np.diag(c), + np.r_[np.diag(c, -1), 0], + np.r_[np.diag(c, -2), 0, 0])) + return jac + + def bjac_cols(y, t, c): + return bjac_rows(y, t, c).T.copy(order='C') + + c = array([[-205, 0.01, 0.00, 0.0], + [0.1, -2.50, 0.02, 0.0], + [1e-3, 0.01, -2.0, 0.01], + [0.00, 0.00, 0.1, -1.0]]) + + y0 = np.ones(4) + t = np.array([0, 5, 10, 100]) + + # Use the full Jacobian. + sol1, info1 = odeint(func, y0, t, args=(c,), full_output=True, + atol=1e-13, rtol=1e-11, mxstep=10000, + Dfun=jac) + + # Use the transposed full Jacobian, with col_deriv=True. + sol2, info2 = odeint(func, y0, t, args=(c,), full_output=True, + atol=1e-13, rtol=1e-11, mxstep=10000, + Dfun=jac_transpose, col_deriv=True) + + # Use the banded Jacobian. + sol3, info3 = odeint(func, y0, t, args=(c,), full_output=True, + atol=1e-13, rtol=1e-11, mxstep=10000, + Dfun=bjac_rows, ml=2, mu=1) + + # Use the transposed banded Jacobian, with col_deriv=True. + sol4, info4 = odeint(func, y0, t, args=(c,), full_output=True, + atol=1e-13, rtol=1e-11, mxstep=10000, + Dfun=bjac_cols, ml=2, mu=1, col_deriv=True) + + assert_allclose(sol1, sol2, err_msg="sol1 != sol2") + assert_allclose(sol1, sol3, atol=1e-12, err_msg="sol1 != sol3") + assert_allclose(sol3, sol4, err_msg="sol3 != sol4") + + # Verify that the number of jacobian evaluations was the same for the + # calls of odeint with a full jacobian and with a banded jacobian. This is + # a regression test--there was a bug in the handling of banded jacobians + # that resulted in an incorrect jacobian matrix being passed to the LSODA + # code. That would cause errors or excessive jacobian evaluations. + assert_array_equal(info1['nje'], info2['nje']) + assert_array_equal(info3['nje'], info4['nje']) + + # Test the use of tfirst + sol1ty, info1ty = odeint(lambda t, y, c: func(y, t, c), y0, t, args=(c,), + full_output=True, atol=1e-13, rtol=1e-11, + mxstep=10000, + Dfun=lambda t, y, c: jac(y, t, c), tfirst=True) + # The code should execute the exact same sequence of floating point + # calculations, so these should be exactly equal. We'll be safe and use + # a small tolerance. + assert_allclose(sol1, sol1ty, rtol=1e-12, err_msg="sol1 != sol1ty") + + +def test_odeint_errors(): + def sys1d(x, t): + return -100*x + + def bad1(x, t): + return 1.0/0 + + def bad2(x, t): + return "foo" + + def bad_jac1(x, t): + return 1.0/0 + + def bad_jac2(x, t): + return [["foo"]] + + def sys2d(x, t): + return [-100*x[0], -0.1*x[1]] + + def sys2d_bad_jac(x, t): + return [[1.0/0, 0], [0, -0.1]] + + assert_raises(ZeroDivisionError, odeint, bad1, 1.0, [0, 1]) + assert_raises(ValueError, odeint, bad2, 1.0, [0, 1]) + + assert_raises(ZeroDivisionError, odeint, sys1d, 1.0, [0, 1], Dfun=bad_jac1) + assert_raises(ValueError, odeint, sys1d, 1.0, [0, 1], Dfun=bad_jac2) + + assert_raises(ZeroDivisionError, odeint, sys2d, [1.0, 1.0], [0, 1], + Dfun=sys2d_bad_jac) + + +def test_odeint_bad_shapes(): + # Tests of some errors that can occur with odeint. + + def badrhs(x, t): + return [1, -1] + + def sys1(x, t): + return -100*x + + def badjac(x, t): + return [[0, 0, 0]] + + # y0 must be at most 1-d. + bad_y0 = [[0, 0], [0, 0]] + assert_raises(ValueError, odeint, sys1, bad_y0, [0, 1]) + + # t must be at most 1-d. + bad_t = [[0, 1], [2, 3]] + assert_raises(ValueError, odeint, sys1, [10.0], bad_t) + + # y0 is 10, but badrhs(x, t) returns [1, -1]. + assert_raises(RuntimeError, odeint, badrhs, 10, [0, 1]) + + # shape of array returned by badjac(x, t) is not correct. + assert_raises(RuntimeError, odeint, sys1, [10, 10], [0, 1], Dfun=badjac) + + +def test_repeated_t_values(): + """Regression test for gh-8217.""" + + def func(x, t): + return -0.25*x + + t = np.zeros(10) + sol = odeint(func, [1.], t) + assert_array_equal(sol, np.ones((len(t), 1))) + + tau = 4*np.log(2) + t = [0]*9 + [tau, 2*tau, 2*tau, 3*tau] + sol = odeint(func, [1, 2], t, rtol=1e-12, atol=1e-12) + expected_sol = np.array([[1.0, 2.0]]*9 + + [[0.5, 1.0], + [0.25, 0.5], + [0.25, 0.5], + [0.125, 0.25]]) + assert_allclose(sol, expected_sol) + + # Edge case: empty t sequence. + sol = odeint(func, [1.], []) + assert_array_equal(sol, np.array([], dtype=np.float64).reshape((0, 1))) + + # t values are not monotonic. + assert_raises(ValueError, odeint, func, [1.], [0, 1, 0.5, 0]) + assert_raises(ValueError, odeint, func, [1, 2, 3], [0, -1, -2, 3]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_odeint_jac.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_odeint_jac.py new file mode 100644 index 0000000000000000000000000000000000000000..7d28ccc93f4444f3f2e0b71da01c573d4f903dbc --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_odeint_jac.py @@ -0,0 +1,74 @@ +import numpy as np +from numpy.testing import assert_equal, assert_allclose +from scipy.integrate import odeint +import scipy.integrate._test_odeint_banded as banded5x5 + + +def rhs(y, t): + dydt = np.zeros_like(y) + banded5x5.banded5x5(t, y, dydt) + return dydt + + +def jac(y, t): + n = len(y) + jac = np.zeros((n, n), order='F') + banded5x5.banded5x5_jac(t, y, 1, 1, jac) + return jac + + +def bjac(y, t): + n = len(y) + bjac = np.zeros((4, n), order='F') + banded5x5.banded5x5_bjac(t, y, 1, 1, bjac) + return bjac + + +JACTYPE_FULL = 1 +JACTYPE_BANDED = 4 + + +def check_odeint(jactype): + if jactype == JACTYPE_FULL: + ml = None + mu = None + jacobian = jac + elif jactype == JACTYPE_BANDED: + ml = 2 + mu = 1 + jacobian = bjac + else: + raise ValueError(f"invalid jactype: {jactype!r}") + + y0 = np.arange(1.0, 6.0) + # These tolerances must match the tolerances used in banded5x5.f. + rtol = 1e-11 + atol = 1e-13 + dt = 0.125 + nsteps = 64 + t = dt * np.arange(nsteps+1) + + sol, info = odeint(rhs, y0, t, + Dfun=jacobian, ml=ml, mu=mu, + atol=atol, rtol=rtol, full_output=True) + yfinal = sol[-1] + odeint_nst = info['nst'][-1] + odeint_nfe = info['nfe'][-1] + odeint_nje = info['nje'][-1] + + y1 = y0.copy() + # Pure Fortran solution. y1 is modified in-place. + nst, nfe, nje = banded5x5.banded5x5_solve(y1, nsteps, dt, jactype) + + # It is likely that yfinal and y1 are *exactly* the same, but + # we'll be cautious and use assert_allclose. + assert_allclose(yfinal, y1, rtol=1e-12) + assert_equal((odeint_nst, odeint_nfe, odeint_nje), (nst, nfe, nje)) + + +def test_odeint_full_jac(): + check_odeint(JACTYPE_FULL) + + +def test_odeint_banded_jac(): + check_odeint(JACTYPE_BANDED) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_quadpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_quadpack.py new file mode 100644 index 0000000000000000000000000000000000000000..e61a69df40f9b5975a6f02f40e6f72e34dbbf297 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_quadpack.py @@ -0,0 +1,680 @@ +import sys +import math +import numpy as np +from numpy import sqrt, cos, sin, arctan, exp, log, pi +from numpy.testing import (assert_, + assert_allclose, assert_array_less, assert_almost_equal) +import pytest + +from scipy.integrate import quad, dblquad, tplquad, nquad +from scipy.special import erf, erfc +from scipy._lib._ccallback import LowLevelCallable + +import ctypes +import ctypes.util +from scipy._lib._ccallback_c import sine_ctypes + +import scipy.integrate._test_multivariate as clib_test + + +def assert_quad(value_and_err, tabled_value, error_tolerance=1.5e-8): + value, err = value_and_err + assert_allclose(value, tabled_value, atol=err, rtol=0) + if error_tolerance is not None: + assert_array_less(err, error_tolerance) + + +def get_clib_test_routine(name, restype, *argtypes): + ptr = getattr(clib_test, name) + return ctypes.cast(ptr, ctypes.CFUNCTYPE(restype, *argtypes)) + + +class TestCtypesQuad: + def setup_method(self): + if sys.platform == 'win32': + files = ['api-ms-win-crt-math-l1-1-0.dll'] + elif sys.platform == 'darwin': + files = ['libm.dylib'] + else: + files = ['libm.so', 'libm.so.6'] + + for file in files: + try: + self.lib = ctypes.CDLL(file) + break + except OSError: + pass + else: + # This test doesn't work on some Linux platforms (Fedora for + # example) that put an ld script in libm.so - see gh-5370 + pytest.skip("Ctypes can't import libm.so") + + restype = ctypes.c_double + argtypes = (ctypes.c_double,) + for name in ['sin', 'cos', 'tan']: + func = getattr(self.lib, name) + func.restype = restype + func.argtypes = argtypes + + def test_typical(self): + assert_quad(quad(self.lib.sin, 0, 5), quad(math.sin, 0, 5)[0]) + assert_quad(quad(self.lib.cos, 0, 5), quad(math.cos, 0, 5)[0]) + assert_quad(quad(self.lib.tan, 0, 1), quad(math.tan, 0, 1)[0]) + + def test_ctypes_sine(self): + quad(LowLevelCallable(sine_ctypes), 0, 1) + + def test_ctypes_variants(self): + sin_0 = get_clib_test_routine('_sin_0', ctypes.c_double, + ctypes.c_double, ctypes.c_void_p) + + sin_1 = get_clib_test_routine('_sin_1', ctypes.c_double, + ctypes.c_int, ctypes.POINTER(ctypes.c_double), + ctypes.c_void_p) + + sin_2 = get_clib_test_routine('_sin_2', ctypes.c_double, + ctypes.c_double) + + sin_3 = get_clib_test_routine('_sin_3', ctypes.c_double, + ctypes.c_int, ctypes.POINTER(ctypes.c_double)) + + sin_4 = get_clib_test_routine('_sin_3', ctypes.c_double, + ctypes.c_int, ctypes.c_double) + + all_sigs = [sin_0, sin_1, sin_2, sin_3, sin_4] + legacy_sigs = [sin_2, sin_4] + legacy_only_sigs = [sin_4] + + # LowLevelCallables work for new signatures + for j, func in enumerate(all_sigs): + callback = LowLevelCallable(func) + if func in legacy_only_sigs: + pytest.raises(ValueError, quad, callback, 0, pi) + else: + assert_allclose(quad(callback, 0, pi)[0], 2.0) + + # Plain ctypes items work only for legacy signatures + for j, func in enumerate(legacy_sigs): + if func in legacy_sigs: + assert_allclose(quad(func, 0, pi)[0], 2.0) + else: + pytest.raises(ValueError, quad, func, 0, pi) + + +class TestMultivariateCtypesQuad: + def setup_method(self): + restype = ctypes.c_double + argtypes = (ctypes.c_int, ctypes.c_double) + for name in ['_multivariate_typical', '_multivariate_indefinite', + '_multivariate_sin']: + func = get_clib_test_routine(name, restype, *argtypes) + setattr(self, name, func) + + def test_typical(self): + # 1) Typical function with two extra arguments: + assert_quad(quad(self._multivariate_typical, 0, pi, (2, 1.8)), + 0.30614353532540296487) + + def test_indefinite(self): + # 2) Infinite integration limits --- Euler's constant + assert_quad(quad(self._multivariate_indefinite, 0, np.inf), + 0.577215664901532860606512) + + def test_threadsafety(self): + # Ensure multivariate ctypes are threadsafe + def threadsafety(y): + return y + quad(self._multivariate_sin, 0, 1)[0] + assert_quad(quad(threadsafety, 0, 1), 0.9596976941318602) + + +class TestQuad: + def test_typical(self): + # 1) Typical function with two extra arguments: + def myfunc(x, n, z): # Bessel function integrand + return cos(n*x-z*sin(x))/pi + assert_quad(quad(myfunc, 0, pi, (2, 1.8)), 0.30614353532540296487) + + def test_indefinite(self): + # 2) Infinite integration limits --- Euler's constant + def myfunc(x): # Euler's constant integrand + return -exp(-x)*log(x) + assert_quad(quad(myfunc, 0, np.inf), 0.577215664901532860606512) + + def test_singular(self): + # 3) Singular points in region of integration. + def myfunc(x): + if 0 < x < 2.5: + return sin(x) + elif 2.5 <= x <= 5.0: + return exp(-x) + else: + return 0.0 + + assert_quad(quad(myfunc, 0, 10, points=[2.5, 5.0]), + 1 - cos(2.5) + exp(-2.5) - exp(-5.0)) + + def test_sine_weighted_finite(self): + # 4) Sine weighted integral (finite limits) + def myfunc(x, a): + return exp(a*(x-1)) + + ome = 2.0**3.4 + assert_quad(quad(myfunc, 0, 1, args=20, weight='sin', wvar=ome), + (20*sin(ome)-ome*cos(ome)+ome*exp(-20))/(20**2 + ome**2)) + + def test_sine_weighted_infinite(self): + # 5) Sine weighted integral (infinite limits) + def myfunc(x, a): + return exp(-x*a) + + a = 4.0 + ome = 3.0 + assert_quad(quad(myfunc, 0, np.inf, args=a, weight='sin', wvar=ome), + ome/(a**2 + ome**2)) + + def test_cosine_weighted_infinite(self): + # 6) Cosine weighted integral (negative infinite limits) + def myfunc(x, a): + return exp(x*a) + + a = 2.5 + ome = 2.3 + assert_quad(quad(myfunc, -np.inf, 0, args=a, weight='cos', wvar=ome), + a/(a**2 + ome**2)) + + def test_algebraic_log_weight(self): + # 6) Algebraic-logarithmic weight. + def myfunc(x, a): + return 1/(1+x+2**(-a)) + + a = 1.5 + assert_quad(quad(myfunc, -1, 1, args=a, weight='alg', + wvar=(-0.5, -0.5)), + pi/sqrt((1+2**(-a))**2 - 1)) + + def test_cauchypv_weight(self): + # 7) Cauchy prinicpal value weighting w(x) = 1/(x-c) + def myfunc(x, a): + return 2.0**(-a)/((x-1)**2+4.0**(-a)) + + a = 0.4 + tabledValue = ((2.0**(-0.4)*log(1.5) - + 2.0**(-1.4)*log((4.0**(-a)+16) / (4.0**(-a)+1)) - + arctan(2.0**(a+2)) - + arctan(2.0**a)) / + (4.0**(-a) + 1)) + assert_quad(quad(myfunc, 0, 5, args=0.4, weight='cauchy', wvar=2.0), + tabledValue, error_tolerance=1.9e-8) + + def test_b_less_than_a(self): + def f(x, p, q): + return p * np.exp(-q*x) + + val_1, err_1 = quad(f, 0, np.inf, args=(2, 3)) + val_2, err_2 = quad(f, np.inf, 0, args=(2, 3)) + assert_allclose(val_1, -val_2, atol=max(err_1, err_2)) + + def test_b_less_than_a_2(self): + def f(x, s): + return np.exp(-x**2 / 2 / s) / np.sqrt(2.*s) + + val_1, err_1 = quad(f, -np.inf, np.inf, args=(2,)) + val_2, err_2 = quad(f, np.inf, -np.inf, args=(2,)) + assert_allclose(val_1, -val_2, atol=max(err_1, err_2)) + + def test_b_less_than_a_3(self): + def f(x): + return 1.0 + + val_1, err_1 = quad(f, 0, 1, weight='alg', wvar=(0, 0)) + val_2, err_2 = quad(f, 1, 0, weight='alg', wvar=(0, 0)) + assert_allclose(val_1, -val_2, atol=max(err_1, err_2)) + + def test_b_less_than_a_full_output(self): + def f(x): + return 1.0 + + res_1 = quad(f, 0, 1, weight='alg', wvar=(0, 0), full_output=True) + res_2 = quad(f, 1, 0, weight='alg', wvar=(0, 0), full_output=True) + err = max(res_1[1], res_2[1]) + assert_allclose(res_1[0], -res_2[0], atol=err) + + def test_double_integral(self): + # 8) Double Integral test + def simpfunc(y, x): # Note order of arguments. + return x+y + + a, b = 1.0, 2.0 + assert_quad(dblquad(simpfunc, a, b, lambda x: x, lambda x: 2*x), + 5/6.0 * (b**3.0-a**3.0)) + + def test_double_integral2(self): + def func(x0, x1, t0, t1): + return x0 + x1 + t0 + t1 + def g(x): + return x + def h(x): + return 2 * x + args = 1, 2 + assert_quad(dblquad(func, 1, 2, g, h, args=args),35./6 + 9*.5) + + def test_double_integral3(self): + def func(x0, x1): + return x0 + x1 + 1 + 2 + assert_quad(dblquad(func, 1, 2, 1, 2),6.) + + @pytest.mark.parametrize( + "x_lower, x_upper, y_lower, y_upper, expected", + [ + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [-inf, 0] for all n. + (-np.inf, 0, -np.inf, 0, np.pi / 4), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [-inf, -1] for each n (one at a time). + (-np.inf, -1, -np.inf, 0, np.pi / 4 * erfc(1)), + (-np.inf, 0, -np.inf, -1, np.pi / 4 * erfc(1)), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [-inf, -1] for all n. + (-np.inf, -1, -np.inf, -1, np.pi / 4 * (erfc(1) ** 2)), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [-inf, 1] for each n (one at a time). + (-np.inf, 1, -np.inf, 0, np.pi / 4 * (erf(1) + 1)), + (-np.inf, 0, -np.inf, 1, np.pi / 4 * (erf(1) + 1)), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [-inf, 1] for all n. + (-np.inf, 1, -np.inf, 1, np.pi / 4 * ((erf(1) + 1) ** 2)), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain Dx = [-inf, -1] and Dy = [-inf, 1]. + (-np.inf, -1, -np.inf, 1, np.pi / 4 * ((erf(1) + 1) * erfc(1))), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain Dx = [-inf, 1] and Dy = [-inf, -1]. + (-np.inf, 1, -np.inf, -1, np.pi / 4 * ((erf(1) + 1) * erfc(1))), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [0, inf] for all n. + (0, np.inf, 0, np.inf, np.pi / 4), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [1, inf] for each n (one at a time). + (1, np.inf, 0, np.inf, np.pi / 4 * erfc(1)), + (0, np.inf, 1, np.inf, np.pi / 4 * erfc(1)), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [1, inf] for all n. + (1, np.inf, 1, np.inf, np.pi / 4 * (erfc(1) ** 2)), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [-1, inf] for each n (one at a time). + (-1, np.inf, 0, np.inf, np.pi / 4 * (erf(1) + 1)), + (0, np.inf, -1, np.inf, np.pi / 4 * (erf(1) + 1)), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [-1, inf] for all n. + (-1, np.inf, -1, np.inf, np.pi / 4 * ((erf(1) + 1) ** 2)), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain Dx = [-1, inf] and Dy = [1, inf]. + (-1, np.inf, 1, np.inf, np.pi / 4 * ((erf(1) + 1) * erfc(1))), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain Dx = [1, inf] and Dy = [-1, inf]. + (1, np.inf, -1, np.inf, np.pi / 4 * ((erf(1) + 1) * erfc(1))), + # Multiple integration of a function in n = 2 variables: f(x, y, z) + # over domain D = [-inf, inf] for all n. + (-np.inf, np.inf, -np.inf, np.inf, np.pi) + ] + ) + def test_double_integral_improper( + self, x_lower, x_upper, y_lower, y_upper, expected + ): + # The Gaussian Integral. + def f(x, y): + return np.exp(-x ** 2 - y ** 2) + + assert_quad( + dblquad(f, x_lower, x_upper, y_lower, y_upper), + expected, + error_tolerance=3e-8 + ) + + def test_triple_integral(self): + # 9) Triple Integral test + def simpfunc(z, y, x, t): # Note order of arguments. + return (x+y+z)*t + + a, b = 1.0, 2.0 + assert_quad(tplquad(simpfunc, a, b, + lambda x: x, lambda x: 2*x, + lambda x, y: x - y, lambda x, y: x + y, + (2.,)), + 2*8/3.0 * (b**4.0 - a**4.0)) + + @pytest.mark.xslow + @pytest.mark.parametrize( + "x_lower, x_upper, y_lower, y_upper, z_lower, z_upper, expected", + [ + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-inf, 0] for all n. + (-np.inf, 0, -np.inf, 0, -np.inf, 0, (np.pi ** (3 / 2)) / 8), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-inf, -1] for each n (one at a time). + (-np.inf, -1, -np.inf, 0, -np.inf, 0, + (np.pi ** (3 / 2)) / 8 * erfc(1)), + (-np.inf, 0, -np.inf, -1, -np.inf, 0, + (np.pi ** (3 / 2)) / 8 * erfc(1)), + (-np.inf, 0, -np.inf, 0, -np.inf, -1, + (np.pi ** (3 / 2)) / 8 * erfc(1)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-inf, -1] for each n (two at a time). + (-np.inf, -1, -np.inf, -1, -np.inf, 0, + (np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)), + (-np.inf, -1, -np.inf, 0, -np.inf, -1, + (np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)), + (-np.inf, 0, -np.inf, -1, -np.inf, -1, + (np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-inf, -1] for all n. + (-np.inf, -1, -np.inf, -1, -np.inf, -1, + (np.pi ** (3 / 2)) / 8 * (erfc(1) ** 3)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = [-inf, -1] and Dy = Dz = [-inf, 1]. + (-np.inf, -1, -np.inf, 1, -np.inf, 1, + (np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = Dy = [-inf, -1] and Dz = [-inf, 1]. + (-np.inf, -1, -np.inf, -1, -np.inf, 1, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = Dz = [-inf, -1] and Dy = [-inf, 1]. + (-np.inf, -1, -np.inf, 1, -np.inf, -1, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = [-inf, 1] and Dy = Dz = [-inf, -1]. + (-np.inf, 1, -np.inf, -1, -np.inf, -1, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = Dy = [-inf, 1] and Dz = [-inf, -1]. + (-np.inf, 1, -np.inf, 1, -np.inf, -1, + (np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = Dz = [-inf, 1] and Dy = [-inf, -1]. + (-np.inf, 1, -np.inf, -1, -np.inf, 1, + (np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-inf, 1] for each n (one at a time). + (-np.inf, 1, -np.inf, 0, -np.inf, 0, + (np.pi ** (3 / 2)) / 8 * (erf(1) + 1)), + (-np.inf, 0, -np.inf, 1, -np.inf, 0, + (np.pi ** (3 / 2)) / 8 * (erf(1) + 1)), + (-np.inf, 0, -np.inf, 0, -np.inf, 1, + (np.pi ** (3 / 2)) / 8 * (erf(1) + 1)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-inf, 1] for each n (two at a time). + (-np.inf, 1, -np.inf, 1, -np.inf, 0, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)), + (-np.inf, 1, -np.inf, 0, -np.inf, 1, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)), + (-np.inf, 0, -np.inf, 1, -np.inf, 1, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-inf, 1] for all n. + (-np.inf, 1, -np.inf, 1, -np.inf, 1, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 3)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [0, inf] for all n. + (0, np.inf, 0, np.inf, 0, np.inf, (np.pi ** (3 / 2)) / 8), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [1, inf] for each n (one at a time). + (1, np.inf, 0, np.inf, 0, np.inf, + (np.pi ** (3 / 2)) / 8 * erfc(1)), + (0, np.inf, 1, np.inf, 0, np.inf, + (np.pi ** (3 / 2)) / 8 * erfc(1)), + (0, np.inf, 0, np.inf, 1, np.inf, + (np.pi ** (3 / 2)) / 8 * erfc(1)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [1, inf] for each n (two at a time). + (1, np.inf, 1, np.inf, 0, np.inf, + (np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)), + (1, np.inf, 0, np.inf, 1, np.inf, + (np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)), + (0, np.inf, 1, np.inf, 1, np.inf, + (np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [1, inf] for all n. + (1, np.inf, 1, np.inf, 1, np.inf, + (np.pi ** (3 / 2)) / 8 * (erfc(1) ** 3)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-1, inf] for each n (one at a time). + (-1, np.inf, 0, np.inf, 0, np.inf, + (np.pi ** (3 / 2)) / 8 * (erf(1) + 1)), + (0, np.inf, -1, np.inf, 0, np.inf, + (np.pi ** (3 / 2)) / 8 * (erf(1) + 1)), + (0, np.inf, 0, np.inf, -1, np.inf, + (np.pi ** (3 / 2)) / 8 * (erf(1) + 1)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-1, inf] for each n (two at a time). + (-1, np.inf, -1, np.inf, 0, np.inf, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)), + (-1, np.inf, 0, np.inf, -1, np.inf, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)), + (0, np.inf, -1, np.inf, -1, np.inf, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-1, inf] for all n. + (-1, np.inf, -1, np.inf, -1, np.inf, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 3)), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = [1, inf] and Dy = Dz = [-1, inf]. + (1, np.inf, -1, np.inf, -1, np.inf, + (np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = Dy = [1, inf] and Dz = [-1, inf]. + (1, np.inf, 1, np.inf, -1, np.inf, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = Dz = [1, inf] and Dy = [-1, inf]. + (1, np.inf, -1, np.inf, 1, np.inf, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = [-1, inf] and Dy = Dz = [1, inf]. + (-1, np.inf, 1, np.inf, 1, np.inf, + (np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = Dy = [-1, inf] and Dz = [1, inf]. + (-1, np.inf, -1, np.inf, 1, np.inf, + (np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain Dx = Dz = [-1, inf] and Dy = [1, inf]. + (-1, np.inf, 1, np.inf, -1, np.inf, + (np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))), + # Multiple integration of a function in n = 3 variables: f(x, y, z) + # over domain D = [-inf, inf] for all n. + (-np.inf, np.inf, -np.inf, np.inf, -np.inf, np.inf, + np.pi ** (3 / 2)), + ], + ) + def test_triple_integral_improper( + self, + x_lower, + x_upper, + y_lower, + y_upper, + z_lower, + z_upper, + expected + ): + # The Gaussian Integral. + def f(x, y, z): + return np.exp(-x ** 2 - y ** 2 - z ** 2) + + assert_quad( + tplquad(f, x_lower, x_upper, y_lower, y_upper, z_lower, z_upper), + expected, + error_tolerance=6e-8 + ) + + def test_complex(self): + def tfunc(x): + return np.exp(1j*x) + + assert np.allclose( + quad(tfunc, 0, np.pi/2, complex_func=True)[0], + 1+1j) + + # We consider a divergent case in order to force quadpack + # to return an error message. The output is compared + # against what is returned by explicit integration + # of the parts. + kwargs = {'a': 0, 'b': np.inf, 'full_output': True, + 'weight': 'cos', 'wvar': 1} + res_c = quad(tfunc, complex_func=True, **kwargs) + res_r = quad(lambda x: np.real(np.exp(1j*x)), + complex_func=False, + **kwargs) + res_i = quad(lambda x: np.imag(np.exp(1j*x)), + complex_func=False, + **kwargs) + + np.testing.assert_equal(res_c[0], res_r[0] + 1j*res_i[0]) + np.testing.assert_equal(res_c[1], res_r[1] + 1j*res_i[1]) + + assert len(res_c[2]['real']) == len(res_r[2:]) == 3 + assert res_c[2]['real'][2] == res_r[4] + assert res_c[2]['real'][1] == res_r[3] + assert res_c[2]['real'][0]['lst'] == res_r[2]['lst'] + + assert len(res_c[2]['imag']) == len(res_i[2:]) == 1 + assert res_c[2]['imag'][0]['lst'] == res_i[2]['lst'] + + +class TestNQuad: + @pytest.mark.fail_slow(5) + def test_fixed_limits(self): + def func1(x0, x1, x2, x3): + val = (x0**2 + x1*x2 - x3**3 + np.sin(x0) + + (1 if (x0 - 0.2*x3 - 0.5 - 0.25*x1 > 0) else 0)) + return val + + def opts_basic(*args): + return {'points': [0.2*args[2] + 0.5 + 0.25*args[0]]} + + res = nquad(func1, [[0, 1], [-1, 1], [.13, .8], [-.15, 1]], + opts=[opts_basic, {}, {}, {}], full_output=True) + assert_quad(res[:-1], 1.5267454070738635) + assert_(res[-1]['neval'] > 0 and res[-1]['neval'] < 4e5) + + @pytest.mark.fail_slow(5) + def test_variable_limits(self): + scale = .1 + + def func2(x0, x1, x2, x3, t0, t1): + val = (x0*x1*x3**2 + np.sin(x2) + 1 + + (1 if x0 + t1*x1 - t0 > 0 else 0)) + return val + + def lim0(x1, x2, x3, t0, t1): + return [scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) - 1, + scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) + 1] + + def lim1(x2, x3, t0, t1): + return [scale * (t0*x2 + t1*x3) - 1, + scale * (t0*x2 + t1*x3) + 1] + + def lim2(x3, t0, t1): + return [scale * (x3 + t0**2*t1**3) - 1, + scale * (x3 + t0**2*t1**3) + 1] + + def lim3(t0, t1): + return [scale * (t0 + t1) - 1, scale * (t0 + t1) + 1] + + def opts0(x1, x2, x3, t0, t1): + return {'points': [t0 - t1*x1]} + + def opts1(x2, x3, t0, t1): + return {} + + def opts2(x3, t0, t1): + return {} + + def opts3(t0, t1): + return {} + + res = nquad(func2, [lim0, lim1, lim2, lim3], args=(0, 0), + opts=[opts0, opts1, opts2, opts3]) + assert_quad(res, 25.066666666666663) + + def test_square_separate_ranges_and_opts(self): + def f(y, x): + return 1.0 + + assert_quad(nquad(f, [[-1, 1], [-1, 1]], opts=[{}, {}]), 4.0) + + def test_square_aliased_ranges_and_opts(self): + def f(y, x): + return 1.0 + + r = [-1, 1] + opt = {} + assert_quad(nquad(f, [r, r], opts=[opt, opt]), 4.0) + + def test_square_separate_fn_ranges_and_opts(self): + def f(y, x): + return 1.0 + + def fn_range0(*args): + return (-1, 1) + + def fn_range1(*args): + return (-1, 1) + + def fn_opt0(*args): + return {} + + def fn_opt1(*args): + return {} + + ranges = [fn_range0, fn_range1] + opts = [fn_opt0, fn_opt1] + assert_quad(nquad(f, ranges, opts=opts), 4.0) + + def test_square_aliased_fn_ranges_and_opts(self): + def f(y, x): + return 1.0 + + def fn_range(*args): + return (-1, 1) + + def fn_opt(*args): + return {} + + ranges = [fn_range, fn_range] + opts = [fn_opt, fn_opt] + assert_quad(nquad(f, ranges, opts=opts), 4.0) + + def test_matching_quad(self): + def func(x): + return x**2 + 1 + + res, reserr = quad(func, 0, 4) + res2, reserr2 = nquad(func, ranges=[[0, 4]]) + assert_almost_equal(res, res2) + assert_almost_equal(reserr, reserr2) + + def test_matching_dblquad(self): + def func2d(x0, x1): + return x0**2 + x1**3 - x0 * x1 + 1 + + res, reserr = dblquad(func2d, -2, 2, lambda x: -3, lambda x: 3) + res2, reserr2 = nquad(func2d, [[-3, 3], (-2, 2)]) + assert_almost_equal(res, res2) + assert_almost_equal(reserr, reserr2) + + def test_matching_tplquad(self): + def func3d(x0, x1, x2, c0, c1): + return x0**2 + c0 * x1**3 - x0 * x1 + 1 + c1 * np.sin(x2) + + res = tplquad(func3d, -1, 2, lambda x: -2, lambda x: 2, + lambda x, y: -np.pi, lambda x, y: np.pi, + args=(2, 3)) + res2 = nquad(func3d, [[-np.pi, np.pi], [-2, 2], (-1, 2)], args=(2, 3)) + assert_almost_equal(res, res2) + + def test_dict_as_opts(self): + try: + nquad(lambda x, y: x * y, [[0, 1], [0, 1]], opts={'epsrel': 0.0001}) + except TypeError: + assert False + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_quadrature.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_quadrature.py new file mode 100644 index 0000000000000000000000000000000000000000..0198b53093a79c15d2fd644956cb0d2862ca92a2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_quadrature.py @@ -0,0 +1,732 @@ +# mypy: disable-error-code="attr-defined" +import pytest +import numpy as np +from numpy.testing import assert_equal, assert_almost_equal, assert_allclose +from hypothesis import given +import hypothesis.strategies as st +import hypothesis.extra.numpy as hyp_num + +from scipy.integrate import (romb, newton_cotes, + cumulative_trapezoid, trapezoid, + quad, simpson, fixed_quad, + qmc_quad, cumulative_simpson) +from scipy.integrate._quadrature import _cumulative_simpson_unequal_intervals + +from scipy import stats, special, integrate +from scipy.conftest import array_api_compatible, skip_xp_invalid_arg +from scipy._lib._array_api_no_0d import xp_assert_close + +skip_xp_backends = pytest.mark.skip_xp_backends + + +class TestFixedQuad: + def test_scalar(self): + n = 4 + expected = 1/(2*n) + got, _ = fixed_quad(lambda x: x**(2*n - 1), 0, 1, n=n) + # quadrature exact for this input + assert_allclose(got, expected, rtol=1e-12) + + def test_vector(self): + n = 4 + p = np.arange(1, 2*n) + expected = 1/(p + 1) + got, _ = fixed_quad(lambda x: x**p[:, None], 0, 1, n=n) + assert_allclose(got, expected, rtol=1e-12) + + +class TestQuadrature: + def quad(self, x, a, b, args): + raise NotImplementedError + + def test_romb(self): + assert_equal(romb(np.arange(17)), 128) + + def test_romb_gh_3731(self): + # Check that romb makes maximal use of data points + x = np.arange(2**4+1) + y = np.cos(0.2*x) + val = romb(y) + val2, err = quad(lambda x: np.cos(0.2*x), x.min(), x.max()) + assert_allclose(val, val2, rtol=1e-8, atol=0) + + def test_newton_cotes(self): + """Test the first few degrees, for evenly spaced points.""" + n = 1 + wts, errcoff = newton_cotes(n, 1) + assert_equal(wts, n*np.array([0.5, 0.5])) + assert_almost_equal(errcoff, -n**3/12.0) + + n = 2 + wts, errcoff = newton_cotes(n, 1) + assert_almost_equal(wts, n*np.array([1.0, 4.0, 1.0])/6.0) + assert_almost_equal(errcoff, -n**5/2880.0) + + n = 3 + wts, errcoff = newton_cotes(n, 1) + assert_almost_equal(wts, n*np.array([1.0, 3.0, 3.0, 1.0])/8.0) + assert_almost_equal(errcoff, -n**5/6480.0) + + n = 4 + wts, errcoff = newton_cotes(n, 1) + assert_almost_equal(wts, n*np.array([7.0, 32.0, 12.0, 32.0, 7.0])/90.0) + assert_almost_equal(errcoff, -n**7/1935360.0) + + def test_newton_cotes2(self): + """Test newton_cotes with points that are not evenly spaced.""" + + x = np.array([0.0, 1.5, 2.0]) + y = x**2 + wts, errcoff = newton_cotes(x) + exact_integral = 8.0/3 + numeric_integral = np.dot(wts, y) + assert_almost_equal(numeric_integral, exact_integral) + + x = np.array([0.0, 1.4, 2.1, 3.0]) + y = x**2 + wts, errcoff = newton_cotes(x) + exact_integral = 9.0 + numeric_integral = np.dot(wts, y) + assert_almost_equal(numeric_integral, exact_integral) + + def test_simpson(self): + y = np.arange(17) + assert_equal(simpson(y), 128) + assert_equal(simpson(y, dx=0.5), 64) + assert_equal(simpson(y, x=np.linspace(0, 4, 17)), 32) + + # integral should be exactly 21 + x = np.linspace(1, 4, 4) + def f(x): + return x**2 + + assert_allclose(simpson(f(x), x=x), 21.0) + + # integral should be exactly 114 + x = np.linspace(1, 7, 4) + assert_allclose(simpson(f(x), dx=2.0), 114) + + # test multi-axis behaviour + a = np.arange(16).reshape(4, 4) + x = np.arange(64.).reshape(4, 4, 4) + y = f(x) + for i in range(3): + r = simpson(y, x=x, axis=i) + it = np.nditer(a, flags=['multi_index']) + for _ in it: + idx = list(it.multi_index) + idx.insert(i, slice(None)) + integral = x[tuple(idx)][-1]**3 / 3 - x[tuple(idx)][0]**3 / 3 + assert_allclose(r[it.multi_index], integral) + + # test when integration axis only has two points + x = np.arange(16).reshape(8, 2) + y = f(x) + r = simpson(y, x=x, axis=-1) + + integral = 0.5 * (y[:, 1] + y[:, 0]) * (x[:, 1] - x[:, 0]) + assert_allclose(r, integral) + + # odd points, test multi-axis behaviour + a = np.arange(25).reshape(5, 5) + x = np.arange(125).reshape(5, 5, 5) + y = f(x) + for i in range(3): + r = simpson(y, x=x, axis=i) + it = np.nditer(a, flags=['multi_index']) + for _ in it: + idx = list(it.multi_index) + idx.insert(i, slice(None)) + integral = x[tuple(idx)][-1]**3 / 3 - x[tuple(idx)][0]**3 / 3 + assert_allclose(r[it.multi_index], integral) + + # Tests for checking base case + x = np.array([3]) + y = np.power(x, 2) + assert_allclose(simpson(y, x=x, axis=0), 0.0) + assert_allclose(simpson(y, x=x, axis=-1), 0.0) + + x = np.array([3, 3, 3, 3]) + y = np.power(x, 2) + assert_allclose(simpson(y, x=x, axis=0), 0.0) + assert_allclose(simpson(y, x=x, axis=-1), 0.0) + + x = np.array([[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]]) + y = np.power(x, 2) + zero_axis = [0.0, 0.0, 0.0, 0.0] + default_axis = [170 + 1/3] * 3 # 8**3 / 3 - 1/3 + assert_allclose(simpson(y, x=x, axis=0), zero_axis) + # the following should be exact + assert_allclose(simpson(y, x=x, axis=-1), default_axis) + + x = np.array([[1, 2, 4, 8], [1, 2, 4, 8], [1, 8, 16, 32]]) + y = np.power(x, 2) + zero_axis = [0.0, 136.0, 1088.0, 8704.0] + default_axis = [170 + 1/3, 170 + 1/3, 32**3 / 3 - 1/3] + assert_allclose(simpson(y, x=x, axis=0), zero_axis) + assert_allclose(simpson(y, x=x, axis=-1), default_axis) + + + @pytest.mark.parametrize('droplast', [False, True]) + def test_simpson_2d_integer_no_x(self, droplast): + # The inputs are 2d integer arrays. The results should be + # identical to the results when the inputs are floating point. + y = np.array([[2, 2, 4, 4, 8, 8, -4, 5], + [4, 4, 2, -4, 10, 22, -2, 10]]) + if droplast: + y = y[:, :-1] + result = simpson(y, axis=-1) + expected = simpson(np.array(y, dtype=np.float64), axis=-1) + assert_equal(result, expected) + + +class TestCumulative_trapezoid: + def test_1d(self): + x = np.linspace(-2, 2, num=5) + y = x + y_int = cumulative_trapezoid(y, x, initial=0) + y_expected = [0., -1.5, -2., -1.5, 0.] + assert_allclose(y_int, y_expected) + + y_int = cumulative_trapezoid(y, x, initial=None) + assert_allclose(y_int, y_expected[1:]) + + def test_y_nd_x_nd(self): + x = np.arange(3 * 2 * 4).reshape(3, 2, 4) + y = x + y_int = cumulative_trapezoid(y, x, initial=0) + y_expected = np.array([[[0., 0.5, 2., 4.5], + [0., 4.5, 10., 16.5]], + [[0., 8.5, 18., 28.5], + [0., 12.5, 26., 40.5]], + [[0., 16.5, 34., 52.5], + [0., 20.5, 42., 64.5]]]) + + assert_allclose(y_int, y_expected) + + # Try with all axes + shapes = [(2, 2, 4), (3, 1, 4), (3, 2, 3)] + for axis, shape in zip([0, 1, 2], shapes): + y_int = cumulative_trapezoid(y, x, initial=0, axis=axis) + assert_equal(y_int.shape, (3, 2, 4)) + y_int = cumulative_trapezoid(y, x, initial=None, axis=axis) + assert_equal(y_int.shape, shape) + + def test_y_nd_x_1d(self): + y = np.arange(3 * 2 * 4).reshape(3, 2, 4) + x = np.arange(4)**2 + # Try with all axes + ys_expected = ( + np.array([[[4., 5., 6., 7.], + [8., 9., 10., 11.]], + [[40., 44., 48., 52.], + [56., 60., 64., 68.]]]), + np.array([[[2., 3., 4., 5.]], + [[10., 11., 12., 13.]], + [[18., 19., 20., 21.]]]), + np.array([[[0.5, 5., 17.5], + [4.5, 21., 53.5]], + [[8.5, 37., 89.5], + [12.5, 53., 125.5]], + [[16.5, 69., 161.5], + [20.5, 85., 197.5]]])) + + for axis, y_expected in zip([0, 1, 2], ys_expected): + y_int = cumulative_trapezoid(y, x=x[:y.shape[axis]], axis=axis, + initial=None) + assert_allclose(y_int, y_expected) + + def test_x_none(self): + y = np.linspace(-2, 2, num=5) + + y_int = cumulative_trapezoid(y) + y_expected = [-1.5, -2., -1.5, 0.] + assert_allclose(y_int, y_expected) + + y_int = cumulative_trapezoid(y, initial=0) + y_expected = [0, -1.5, -2., -1.5, 0.] + assert_allclose(y_int, y_expected) + + y_int = cumulative_trapezoid(y, dx=3) + y_expected = [-4.5, -6., -4.5, 0.] + assert_allclose(y_int, y_expected) + + y_int = cumulative_trapezoid(y, dx=3, initial=0) + y_expected = [0, -4.5, -6., -4.5, 0.] + assert_allclose(y_int, y_expected) + + @pytest.mark.parametrize( + "initial", [1, 0.5] + ) + def test_initial_error(self, initial): + """If initial is not None or 0, a ValueError is raised.""" + y = np.linspace(0, 10, num=10) + with pytest.raises(ValueError, match="`initial`"): + cumulative_trapezoid(y, initial=initial) + + def test_zero_len_y(self): + with pytest.raises(ValueError, match="At least one point is required"): + cumulative_trapezoid(y=[]) + + +@array_api_compatible +class TestTrapezoid: + def test_simple(self, xp): + x = xp.arange(-10, 10, .1) + r = trapezoid(xp.exp(-.5 * x ** 2) / xp.sqrt(2 * xp.asarray(xp.pi)), dx=0.1) + # check integral of normal equals 1 + xp_assert_close(r, xp.asarray(1.0)) + + @skip_xp_backends('jax.numpy', + reasons=["JAX arrays do not support item assignment"]) + @pytest.mark.usefixtures("skip_xp_backends") + def test_ndim(self, xp): + x = xp.linspace(0, 1, 3) + y = xp.linspace(0, 2, 8) + z = xp.linspace(0, 3, 13) + + wx = xp.ones_like(x) * (x[1] - x[0]) + wx[0] /= 2 + wx[-1] /= 2 + wy = xp.ones_like(y) * (y[1] - y[0]) + wy[0] /= 2 + wy[-1] /= 2 + wz = xp.ones_like(z) * (z[1] - z[0]) + wz[0] /= 2 + wz[-1] /= 2 + + q = x[:, None, None] + y[None,:, None] + z[None, None,:] + + qx = xp.sum(q * wx[:, None, None], axis=0) + qy = xp.sum(q * wy[None, :, None], axis=1) + qz = xp.sum(q * wz[None, None, :], axis=2) + + # n-d `x` + r = trapezoid(q, x=x[:, None, None], axis=0) + xp_assert_close(r, qx) + r = trapezoid(q, x=y[None,:, None], axis=1) + xp_assert_close(r, qy) + r = trapezoid(q, x=z[None, None,:], axis=2) + xp_assert_close(r, qz) + + # 1-d `x` + r = trapezoid(q, x=x, axis=0) + xp_assert_close(r, qx) + r = trapezoid(q, x=y, axis=1) + xp_assert_close(r, qy) + r = trapezoid(q, x=z, axis=2) + xp_assert_close(r, qz) + + @skip_xp_backends('jax.numpy', + reasons=["JAX arrays do not support item assignment"]) + @pytest.mark.usefixtures("skip_xp_backends") + def test_gh21908(self, xp): + # extended testing for n-dim arrays + x = xp.reshape(xp.linspace(0, 29, 30), (3, 10)) + y = xp.reshape(xp.linspace(0, 29, 30), (3, 10)) + + out0 = xp.linspace(200, 380, 10) + xp_assert_close(trapezoid(y, x=x, axis=0), out0) + xp_assert_close(trapezoid(y, x=xp.asarray([0, 10., 20.]), axis=0), out0) + # x needs to be broadcastable against y + xp_assert_close( + trapezoid(y, x=xp.asarray([0, 10., 20.])[:, None], axis=0), + out0 + ) + with pytest.raises(Exception): + # x is not broadcastable against y + trapezoid(y, x=xp.asarray([0, 10., 20.])[None, :], axis=0) + + out1 = xp.asarray([ 40.5, 130.5, 220.5]) + xp_assert_close(trapezoid(y, x=x, axis=1), out1) + xp_assert_close( + trapezoid(y, x=xp.linspace(0, 9, 10), axis=1), + out1 + ) + + @skip_xp_invalid_arg + def test_masked(self, xp): + # Testing that masked arrays behave as if the function is 0 where + # masked + x = np.arange(5) + y = x * x + mask = x == 2 + ym = np.ma.array(y, mask=mask) + r = 13.0 # sum(0.5 * (0 + 1) * 1.0 + 0.5 * (9 + 16)) + assert_allclose(trapezoid(ym, x), r) + + xm = np.ma.array(x, mask=mask) + assert_allclose(trapezoid(ym, xm), r) + + xm = np.ma.array(x, mask=mask) + assert_allclose(trapezoid(y, xm), r) + + @skip_xp_backends(np_only=True, + reasons=['array-likes only supported for NumPy backend']) + @pytest.mark.usefixtures("skip_xp_backends") + def test_array_like(self, xp): + x = list(range(5)) + y = [t * t for t in x] + xarr = xp.asarray(x, dtype=xp.float64) + yarr = xp.asarray(y, dtype=xp.float64) + res = trapezoid(y, x) + resarr = trapezoid(yarr, xarr) + xp_assert_close(res, resarr) + + +class TestQMCQuad: + @pytest.mark.thread_unsafe + def test_input_validation(self): + message = "`func` must be callable." + with pytest.raises(TypeError, match=message): + qmc_quad("a duck", [0, 0], [1, 1]) + + message = "`func` must evaluate the integrand at points..." + with pytest.raises(ValueError, match=message): + qmc_quad(lambda: 1, [0, 0], [1, 1]) + + def func(x): + assert x.ndim == 1 + return np.sum(x) + message = "Exception encountered when attempting vectorized call..." + with pytest.warns(UserWarning, match=message): + qmc_quad(func, [0, 0], [1, 1]) + + message = "`n_points` must be an integer." + with pytest.raises(TypeError, match=message): + qmc_quad(lambda x: 1, [0, 0], [1, 1], n_points=1024.5) + + message = "`n_estimates` must be an integer." + with pytest.raises(TypeError, match=message): + qmc_quad(lambda x: 1, [0, 0], [1, 1], n_estimates=8.5) + + message = "`qrng` must be an instance of scipy.stats.qmc.QMCEngine." + with pytest.raises(TypeError, match=message): + qmc_quad(lambda x: 1, [0, 0], [1, 1], qrng="a duck") + + message = "`qrng` must be initialized with dimensionality equal to " + with pytest.raises(ValueError, match=message): + qmc_quad(lambda x: 1, [0, 0], [1, 1], qrng=stats.qmc.Sobol(1)) + + message = r"`log` must be boolean \(`True` or `False`\)." + with pytest.raises(TypeError, match=message): + qmc_quad(lambda x: 1, [0, 0], [1, 1], log=10) + + def basic_test(self, n_points=2**8, n_estimates=8, signs=None): + if signs is None: + signs = np.ones(2) + ndim = 2 + mean = np.zeros(ndim) + cov = np.eye(ndim) + + def func(x): + return stats.multivariate_normal.pdf(x.T, mean, cov) + + rng = np.random.default_rng(2879434385674690281) + qrng = stats.qmc.Sobol(ndim, seed=rng) + a = np.zeros(ndim) + b = np.ones(ndim) * signs + res = qmc_quad(func, a, b, n_points=n_points, + n_estimates=n_estimates, qrng=qrng) + ref = stats.multivariate_normal.cdf(b, mean, cov, lower_limit=a) + atol = special.stdtrit(n_estimates-1, 0.995) * res.standard_error # 99% CI + assert_allclose(res.integral, ref, atol=atol) + assert np.prod(signs)*res.integral > 0 + + rng = np.random.default_rng(2879434385674690281) + qrng = stats.qmc.Sobol(ndim, seed=rng) + logres = qmc_quad(lambda *args: np.log(func(*args)), a, b, + n_points=n_points, n_estimates=n_estimates, + log=True, qrng=qrng) + assert_allclose(np.exp(logres.integral), res.integral, rtol=1e-14) + assert np.imag(logres.integral) == (np.pi if np.prod(signs) < 0 else 0) + assert_allclose(np.exp(logres.standard_error), + res.standard_error, rtol=1e-14, atol=1e-16) + + @pytest.mark.parametrize("n_points", [2**8, 2**12]) + @pytest.mark.parametrize("n_estimates", [8, 16]) + def test_basic(self, n_points, n_estimates): + self.basic_test(n_points, n_estimates) + + @pytest.mark.parametrize("signs", [[1, 1], [-1, -1], [-1, 1], [1, -1]]) + def test_sign(self, signs): + self.basic_test(signs=signs) + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize("log", [False, True]) + def test_zero(self, log): + message = "A lower limit was equal to an upper limit, so" + with pytest.warns(UserWarning, match=message): + res = qmc_quad(lambda x: 1, [0, 0], [0, 1], log=log) + assert res.integral == (-np.inf if log else 0) + assert res.standard_error == 0 + + def test_flexible_input(self): + # check that qrng is not required + # also checks that for 1d problems, a and b can be scalars + def func(x): + return stats.norm.pdf(x, scale=2) + + res = qmc_quad(func, 0, 1) + ref = stats.norm.cdf(1, scale=2) - stats.norm.cdf(0, scale=2) + assert_allclose(res.integral, ref, 1e-2) + + +def cumulative_simpson_nd_reference(y, *, x=None, dx=None, initial=None, axis=-1): + # Use cumulative_trapezoid if length of y < 3 + if y.shape[axis] < 3: + if initial is None: + return cumulative_trapezoid(y, x=x, dx=dx, axis=axis, initial=None) + else: + return initial + cumulative_trapezoid(y, x=x, dx=dx, axis=axis, initial=0) + + # Ensure that working axis is last axis + y = np.moveaxis(y, axis, -1) + x = np.moveaxis(x, axis, -1) if np.ndim(x) > 1 else x + dx = np.moveaxis(dx, axis, -1) if np.ndim(dx) > 1 else dx + initial = np.moveaxis(initial, axis, -1) if np.ndim(initial) > 1 else initial + + # If `x` is not present, create it from `dx` + n = y.shape[-1] + x = dx * np.arange(n) if dx is not None else x + # Similarly, if `initial` is not present, set it to 0 + initial_was_none = initial is None + initial = 0 if initial_was_none else initial + + # `np.apply_along_axis` accepts only one array, so concatenate arguments + x = np.broadcast_to(x, y.shape) + initial = np.broadcast_to(initial, y.shape[:-1] + (1,)) + z = np.concatenate((y, x, initial), axis=-1) + + # Use `np.apply_along_axis` to compute result + def f(z): + return cumulative_simpson(z[:n], x=z[n:2*n], initial=z[2*n:]) + res = np.apply_along_axis(f, -1, z) + + # Remove `initial` and undo axis move as needed + res = res[..., 1:] if initial_was_none else res + res = np.moveaxis(res, -1, axis) + return res + + +class TestCumulativeSimpson: + x0 = np.arange(4) + y0 = x0**2 + + @pytest.mark.parametrize('use_dx', (False, True)) + @pytest.mark.parametrize('use_initial', (False, True)) + def test_1d(self, use_dx, use_initial): + # Test for exact agreement with polynomial of highest + # possible order (3 if `dx` is constant, 2 otherwise). + rng = np.random.default_rng(82456839535679456794) + n = 10 + + # Generate random polynomials and ground truth + # integral of appropriate order + order = 3 if use_dx else 2 + dx = rng.random() + x = (np.sort(rng.random(n)) if order == 2 + else np.arange(n)*dx + rng.random()) + i = np.arange(order + 1)[:, np.newaxis] + c = rng.random(order + 1)[:, np.newaxis] + y = np.sum(c*x**i, axis=0) + Y = np.sum(c*x**(i + 1)/(i + 1), axis=0) + ref = Y if use_initial else (Y-Y[0])[1:] + + # Integrate with `cumulative_simpson` + initial = Y[0] if use_initial else None + kwarg = {'dx': dx} if use_dx else {'x': x} + res = cumulative_simpson(y, **kwarg, initial=initial) + + # Compare result against reference + if not use_dx: + assert_allclose(res, ref, rtol=2e-15) + else: + i0 = 0 if use_initial else 1 + # all terms are "close" + assert_allclose(res, ref, rtol=0.0025) + # only even-interval terms are "exact" + assert_allclose(res[i0::2], ref[i0::2], rtol=2e-15) + + @pytest.mark.parametrize('axis', np.arange(-3, 3)) + @pytest.mark.parametrize('x_ndim', (1, 3)) + @pytest.mark.parametrize('x_len', (1, 2, 7)) + @pytest.mark.parametrize('i_ndim', (None, 0, 3,)) + @pytest.mark.parametrize('dx', (None, True)) + def test_nd(self, axis, x_ndim, x_len, i_ndim, dx): + # Test behavior of `cumulative_simpson` with N-D `y` + rng = np.random.default_rng(82456839535679456794) + + # determine shapes + shape = [5, 6, x_len] + shape[axis], shape[-1] = shape[-1], shape[axis] + shape_len_1 = shape.copy() + shape_len_1[axis] = 1 + i_shape = shape_len_1 if i_ndim == 3 else () + + # initialize arguments + y = rng.random(size=shape) + x, dx = None, None + if dx: + dx = rng.random(size=shape_len_1) if x_ndim > 1 else rng.random() + else: + x = (np.sort(rng.random(size=shape), axis=axis) if x_ndim > 1 + else np.sort(rng.random(size=shape[axis]))) + initial = None if i_ndim is None else rng.random(size=i_shape) + + # compare results + res = cumulative_simpson(y, x=x, dx=dx, initial=initial, axis=axis) + ref = cumulative_simpson_nd_reference(y, x=x, dx=dx, initial=initial, axis=axis) + np.testing.assert_allclose(res, ref, rtol=1e-15) + + @pytest.mark.parametrize(('message', 'kwarg_update'), [ + ("x must be strictly increasing", dict(x=[2, 2, 3, 4])), + ("x must be strictly increasing", dict(x=[x0, [2, 2, 4, 8]], y=[y0, y0])), + ("x must be strictly increasing", dict(x=[x0, x0, x0], y=[y0, y0, y0], axis=0)), + ("At least one point is required", dict(x=[], y=[])), + ("`axis=4` is not valid for `y` with `y.ndim=1`", dict(axis=4)), + ("shape of `x` must be the same as `y` or 1-D", dict(x=np.arange(5))), + ("`initial` must either be a scalar or...", dict(initial=np.arange(5))), + ("`dx` must either be a scalar or...", dict(x=None, dx=np.arange(5))), + ]) + def test_simpson_exceptions(self, message, kwarg_update): + kwargs0 = dict(y=self.y0, x=self.x0, dx=None, initial=None, axis=-1) + with pytest.raises(ValueError, match=message): + cumulative_simpson(**dict(kwargs0, **kwarg_update)) + + def test_special_cases(self): + # Test special cases not checked elsewhere + rng = np.random.default_rng(82456839535679456794) + y = rng.random(size=10) + res = cumulative_simpson(y, dx=0) + assert_equal(res, 0) + + # Should add tests of: + # - all elements of `x` identical + # These should work as they do for `simpson` + + def _get_theoretical_diff_between_simps_and_cum_simps(self, y, x): + """`cumulative_simpson` and `simpson` can be tested against other to verify + they give consistent results. `simpson` will iteratively be called with + successively higher upper limits of integration. This function calculates + the theoretical correction required to `simpson` at even intervals to match + with `cumulative_simpson`. + """ + d = np.diff(x, axis=-1) + sub_integrals_h1 = _cumulative_simpson_unequal_intervals(y, d) + sub_integrals_h2 = _cumulative_simpson_unequal_intervals( + y[..., ::-1], d[..., ::-1] + )[..., ::-1] + + # Concatenate to build difference array + zeros_shape = (*y.shape[:-1], 1) + theoretical_difference = np.concatenate( + [ + np.zeros(zeros_shape), + (sub_integrals_h1[..., 1:] - sub_integrals_h2[..., :-1]), + np.zeros(zeros_shape), + ], + axis=-1, + ) + # Differences only expected at even intervals. Odd intervals will + # match exactly so there is no correction + theoretical_difference[..., 1::2] = 0.0 + # Note: the first interval will not match from this correction as + # `simpson` uses the trapezoidal rule + return theoretical_difference + + @pytest.mark.thread_unsafe + @pytest.mark.slow + @given( + y=hyp_num.arrays( + np.float64, + hyp_num.array_shapes(max_dims=4, min_side=3, max_side=10), + elements=st.floats(-10, 10, allow_nan=False).filter(lambda x: abs(x) > 1e-7) + ) + ) + def test_cumulative_simpson_against_simpson_with_default_dx( + self, y + ): + """Theoretically, the output of `cumulative_simpson` will be identical + to `simpson` at all even indices and in the last index. The first index + will not match as `simpson` uses the trapezoidal rule when there are only two + data points. Odd indices after the first index are shown to match with + a mathematically-derived correction.""" + def simpson_reference(y): + return np.stack( + [simpson(y[..., :i], dx=1.0) for i in range(2, y.shape[-1]+1)], axis=-1, + ) + + res = cumulative_simpson(y, dx=1.0) + ref = simpson_reference(y) + theoretical_difference = self._get_theoretical_diff_between_simps_and_cum_simps( + y, x=np.arange(y.shape[-1]) + ) + np.testing.assert_allclose( + res[..., 1:], ref[..., 1:] + theoretical_difference[..., 1:], atol=1e-16 + ) + + @pytest.mark.thread_unsafe + @pytest.mark.slow + @given( + y=hyp_num.arrays( + np.float64, + hyp_num.array_shapes(max_dims=4, min_side=3, max_side=10), + elements=st.floats(-10, 10, allow_nan=False).filter(lambda x: abs(x) > 1e-7) + ) + ) + def test_cumulative_simpson_against_simpson( + self, y + ): + """Theoretically, the output of `cumulative_simpson` will be identical + to `simpson` at all even indices and in the last index. The first index + will not match as `simpson` uses the trapezoidal rule when there are only two + data points. Odd indices after the first index are shown to match with + a mathematically-derived correction.""" + interval = 10/(y.shape[-1] - 1) + x = np.linspace(0, 10, num=y.shape[-1]) + x[1:] = x[1:] + 0.2*interval*np.random.uniform(-1, 1, len(x) - 1) + + def simpson_reference(y, x): + return np.stack( + [simpson(y[..., :i], x=x[..., :i]) for i in range(2, y.shape[-1]+1)], + axis=-1, + ) + + res = cumulative_simpson(y, x=x) + ref = simpson_reference(y, x) + theoretical_difference = self._get_theoretical_diff_between_simps_and_cum_simps( + y, x + ) + np.testing.assert_allclose( + res[..., 1:], ref[..., 1:] + theoretical_difference[..., 1:] + ) + +class TestLebedev: + def test_input_validation(self): + # only certain rules are available + message = "Order n=-1 not available..." + with pytest.raises(NotImplementedError, match=message): + integrate.lebedev_rule(-1) + + def test_quadrature(self): + # Test points/weights to integrate an example function + + def f(x): + return np.exp(x[0]) + + x, w = integrate.lebedev_rule(15) + res = w @ f(x) + ref = 14.7680137457653 # lebedev_rule reference [3] + assert_allclose(res, ref, rtol=1e-14) + assert_allclose(np.sum(w), 4 * np.pi) + + @pytest.mark.parametrize('order', list(range(3, 32, 2)) + list(range(35, 132, 6))) + def test_properties(self, order): + x, w = integrate.lebedev_rule(order) + # dispersion should be maximal; no clear spherical mean + with np.errstate(divide='ignore', invalid='ignore'): + res = stats.directional_stats(x.T, axis=0) + assert_allclose(res.mean_resultant_length, 0, atol=1e-15) + # weights should sum to 4*pi (surface area of unit sphere) + assert_allclose(np.sum(w), 4*np.pi) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_tanhsinh.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_tanhsinh.py new file mode 100644 index 0000000000000000000000000000000000000000..15782ba13efcb16cf8982adf94b8b2f74be63a18 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/tests/test_tanhsinh.py @@ -0,0 +1,1163 @@ +# mypy: disable-error-code="attr-defined" +import os +import pytest +import math + +import numpy as np +from numpy.testing import assert_allclose + +from scipy.conftest import array_api_compatible +import scipy._lib._elementwise_iterative_method as eim +from scipy._lib._array_api_no_0d import xp_assert_close, xp_assert_equal +from scipy._lib._array_api import array_namespace, xp_size, xp_ravel, xp_copy, is_numpy +from scipy import special, stats +from scipy.integrate import quad_vec, nsum, tanhsinh as _tanhsinh +from scipy.integrate._tanhsinh import _pair_cache +from scipy.stats._discrete_distns import _gen_harmonic_gt1 + + +def norm_pdf(x, xp=None): + xp = array_namespace(x) if xp is None else xp + return 1/(2*xp.pi)**0.5 * xp.exp(-x**2/2) + +def norm_logpdf(x, xp=None): + xp = array_namespace(x) if xp is None else xp + return -0.5*math.log(2*xp.pi) - x**2/2 + + +def _vectorize(xp): + # xp-compatible version of np.vectorize + # assumes arguments are all arrays of the same shape + def decorator(f): + def wrapped(*arg_arrays): + shape = arg_arrays[0].shape + arg_arrays = [xp_ravel(arg_array) for arg_array in arg_arrays] + res = [] + for i in range(math.prod(shape)): + arg_scalars = [arg_array[i] for arg_array in arg_arrays] + res.append(f(*arg_scalars)) + return res + + return wrapped + + return decorator + + +@array_api_compatible +@pytest.mark.usefixtures("skip_xp_backends") +@pytest.mark.skip_xp_backends( + 'array_api_strict', reason='Currently uses fancy indexing assignment.' +) +@pytest.mark.skip_xp_backends( + 'jax.numpy', reason='JAX arrays do not support item assignment.' +) +class TestTanhSinh: + + # Test problems from [1] Section 6 + def f1(self, t): + return t * np.log(1 + t) + + f1.ref = 0.25 + f1.b = 1 + + def f2(self, t): + return t ** 2 * np.arctan(t) + + f2.ref = (np.pi - 2 + 2 * np.log(2)) / 12 + f2.b = 1 + + def f3(self, t): + return np.exp(t) * np.cos(t) + + f3.ref = (np.exp(np.pi / 2) - 1) / 2 + f3.b = np.pi / 2 + + def f4(self, t): + a = np.sqrt(2 + t ** 2) + return np.arctan(a) / ((1 + t ** 2) * a) + + f4.ref = 5 * np.pi ** 2 / 96 + f4.b = 1 + + def f5(self, t): + return np.sqrt(t) * np.log(t) + + f5.ref = -4 / 9 + f5.b = 1 + + def f6(self, t): + return np.sqrt(1 - t ** 2) + + f6.ref = np.pi / 4 + f6.b = 1 + + def f7(self, t): + return np.sqrt(t) / np.sqrt(1 - t ** 2) + + f7.ref = 2 * np.sqrt(np.pi) * special.gamma(3 / 4) / special.gamma(1 / 4) + f7.b = 1 + + def f8(self, t): + return np.log(t) ** 2 + + f8.ref = 2 + f8.b = 1 + + def f9(self, t): + return np.log(np.cos(t)) + + f9.ref = -np.pi * np.log(2) / 2 + f9.b = np.pi / 2 + + def f10(self, t): + return np.sqrt(np.tan(t)) + + f10.ref = np.pi * np.sqrt(2) / 2 + f10.b = np.pi / 2 + + def f11(self, t): + return 1 / (1 + t ** 2) + + f11.ref = np.pi / 2 + f11.b = np.inf + + def f12(self, t): + return np.exp(-t) / np.sqrt(t) + + f12.ref = np.sqrt(np.pi) + f12.b = np.inf + + def f13(self, t): + return np.exp(-t ** 2 / 2) + + f13.ref = np.sqrt(np.pi / 2) + f13.b = np.inf + + def f14(self, t): + return np.exp(-t) * np.cos(t) + + f14.ref = 0.5 + f14.b = np.inf + + def f15(self, t): + return np.sin(t) / t + + f15.ref = np.pi / 2 + f15.b = np.inf + + def error(self, res, ref, log=False, xp=None): + xp = array_namespace(res, ref) if xp is None else xp + err = abs(res - ref) + + if not log: + return err + + with np.errstate(divide='ignore'): + return xp.log10(err) + + def test_input_validation(self, xp): + f = self.f1 + + zero = xp.asarray(0) + f_b = xp.asarray(f.b) + + message = '`f` must be callable.' + with pytest.raises(ValueError, match=message): + _tanhsinh(42, zero, f_b) + + message = '...must be True or False.' + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, log=2) + + message = '...must be real numbers.' + with pytest.raises(ValueError, match=message): + _tanhsinh(f, xp.asarray(1+1j), f_b) + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, atol='ekki') + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, rtol=pytest) + + message = '...must be non-negative and finite.' + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, rtol=-1) + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, atol=xp.inf) + + message = '...may not be positive infinity.' + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, rtol=xp.inf, log=True) + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, atol=xp.inf, log=True) + + message = '...must be integers.' + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, maxlevel=object()) + # with pytest.raises(ValueError, match=message): # unused for now + # _tanhsinh(f, zero, f_b, maxfun=1+1j) + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, minlevel="migratory coconut") + + message = '...must be non-negative.' + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, maxlevel=-1) + # with pytest.raises(ValueError, match=message): # unused for now + # _tanhsinh(f, zero, f_b, maxfun=-1) + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, minlevel=-1) + + message = '...must be True or False.' + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, preserve_shape=2) + + message = '...must be callable.' + with pytest.raises(ValueError, match=message): + _tanhsinh(f, zero, f_b, callback='elderberry') + + @pytest.mark.parametrize("limits, ref", [ + [(0, math.inf), 0.5], # b infinite + [(-math.inf, 0), 0.5], # a infinite + [(-math.inf, math.inf), 1.], # a and b infinite + [(math.inf, -math.inf), -1.], # flipped limits + [(1, -1), stats.norm.cdf(-1.) - stats.norm.cdf(1.)], # flipped limits + ]) + def test_integral_transforms(self, limits, ref, xp): + # Check that the integral transforms are behaving for both normal and + # log integration + limits = [xp.asarray(limit) for limit in limits] + dtype = xp.asarray(float(limits[0])).dtype + ref = xp.asarray(ref, dtype=dtype) + + res = _tanhsinh(norm_pdf, *limits) + xp_assert_close(res.integral, ref) + + logres = _tanhsinh(norm_logpdf, *limits, log=True) + xp_assert_close(xp.exp(logres.integral), ref, check_dtype=False) + # Transformation should not make the result complex unnecessarily + xp_test = array_namespace(*limits) # we need xp.isdtype + assert (xp_test.isdtype(logres.integral.dtype, "real floating") if ref > 0 + else xp_test.isdtype(logres.integral.dtype, "complex floating")) + + xp_assert_close(xp.exp(logres.error), res.error, atol=1e-16, check_dtype=False) + + # 15 skipped intentionally; it's very difficult numerically + @pytest.mark.skip_xp_backends(np_only=True, + reason='Cumbersome to convert everything.') + @pytest.mark.parametrize('f_number', range(1, 15)) + def test_basic(self, f_number, xp): + f = getattr(self, f"f{f_number}") + rtol = 2e-8 + res = _tanhsinh(f, 0, f.b, rtol=rtol) + assert_allclose(res.integral, f.ref, rtol=rtol) + if f_number not in {14}: # mildly underestimates error here + true_error = abs(self.error(res.integral, f.ref)/res.integral) + assert true_error < res.error + + if f_number in {7, 10, 12}: # succeeds, but doesn't know it + return + + assert res.success + assert res.status == 0 + + @pytest.mark.skip_xp_backends(np_only=True, + reason="Distributions aren't xp-compatible.") + @pytest.mark.parametrize('ref', (0.5, [0.4, 0.6])) + @pytest.mark.parametrize('case', stats._distr_params.distcont) + def test_accuracy(self, ref, case, xp): + distname, params = case + if distname in {'dgamma', 'dweibull', 'laplace', 'kstwo'}: + # should split up interval at first-derivative discontinuity + pytest.skip('tanh-sinh is not great for non-smooth integrands') + if (distname in {'studentized_range', 'levy_stable'} + and not int(os.getenv('SCIPY_XSLOW', 0))): + pytest.skip('This case passes, but it is too slow.') + dist = getattr(stats, distname)(*params) + x = dist.interval(ref) + res = _tanhsinh(dist.pdf, *x) + assert_allclose(res.integral, ref) + + @pytest.mark.parametrize('shape', [tuple(), (12,), (3, 4), (3, 2, 2)]) + def test_vectorization(self, shape, xp): + # Test for correct functionality, output shapes, and dtypes for various + # input shapes. + rng = np.random.default_rng(82456839535679456794) + a = xp.asarray(rng.random(shape)) + b = xp.asarray(rng.random(shape)) + p = xp.asarray(rng.random(shape)) + n = math.prod(shape) + + def f(x, p): + f.ncall += 1 + f.feval += 1 if (xp_size(x) == n or x.ndim <= 1) else x.shape[-1] + return x**p + f.ncall = 0 + f.feval = 0 + + @_vectorize(xp) + def _tanhsinh_single(a, b, p): + return _tanhsinh(lambda x: x**p, a, b) + + res = _tanhsinh(f, a, b, args=(p,)) + refs = _tanhsinh_single(a, b, p) + + xp_test = array_namespace(a) # need xp.stack, isdtype + attrs = ['integral', 'error', 'success', 'status', 'nfev', 'maxlevel'] + for attr in attrs: + ref_attr = xp_test.stack([getattr(ref, attr) for ref in refs]) + res_attr = xp_ravel(getattr(res, attr)) + xp_assert_close(res_attr, ref_attr, rtol=1e-15) + assert getattr(res, attr).shape == shape + + assert xp_test.isdtype(res.success.dtype, 'bool') + assert xp_test.isdtype(res.status.dtype, 'integral') + assert xp_test.isdtype(res.nfev.dtype, 'integral') + assert xp_test.isdtype(res.maxlevel.dtype, 'integral') + assert xp.max(res.nfev) == f.feval + # maxlevel = 2 -> 3 function calls (2 initialization, 1 work) + assert xp.max(res.maxlevel) >= 2 + assert xp.max(res.maxlevel) == f.ncall + + def test_flags(self, xp): + # Test cases that should produce different status flags; show that all + # can be produced simultaneously. + def f(xs, js): + f.nit += 1 + funcs = [lambda x: xp.exp(-x**2), # converges + lambda x: xp.exp(x), # reaches maxiter due to order=2 + lambda x: xp.full_like(x, xp.nan)] # stops due to NaN + res = [] + for i in range(xp_size(js)): + x = xs[i, ...] + j = int(xp_ravel(js)[i]) + res.append(funcs[j](x)) + return xp.stack(res) + f.nit = 0 + + args = (xp.arange(3, dtype=xp.int64),) + a = xp.asarray([xp.inf]*3) + b = xp.asarray([-xp.inf] * 3) + res = _tanhsinh(f, a, b, maxlevel=5, args=args) + ref_flags = xp.asarray([0, -2, -3], dtype=xp.int32) + xp_assert_equal(res.status, ref_flags) + + def test_flags_preserve_shape(self, xp): + # Same test as above but using `preserve_shape` option to simplify. + def f(x): + res = [xp.exp(-x[0]**2), # converges + xp.exp(x[1]), # reaches maxiter due to order=2 + xp.full_like(x[2], xp.nan)] # stops due to NaN + return xp.stack(res) + + a = xp.asarray([xp.inf] * 3) + b = xp.asarray([-xp.inf] * 3) + res = _tanhsinh(f, a, b, maxlevel=5, preserve_shape=True) + ref_flags = xp.asarray([0, -2, -3], dtype=xp.int32) + xp_assert_equal(res.status, ref_flags) + + def test_preserve_shape(self, xp): + # Test `preserve_shape` option + def f(x, xp): + return xp.stack([xp.stack([x, xp.sin(10 * x)]), + xp.stack([xp.cos(30 * x), x * xp.sin(100 * x)])]) + + ref = quad_vec(lambda x: f(x, np), 0, 1) + res = _tanhsinh(lambda x: f(x, xp), xp.asarray(0), xp.asarray(1), + preserve_shape=True) + dtype = xp.asarray(0.).dtype + xp_assert_close(res.integral, xp.asarray(ref[0], dtype=dtype)) + + def test_convergence(self, xp): + # demonstrate that number of accurate digits doubles each iteration + dtype = xp.float64 # this only works with good precision + def f(t): + return t * xp.log(1 + t) + ref = xp.asarray(0.25, dtype=dtype) + a, b = xp.asarray(0., dtype=dtype), xp.asarray(1., dtype=dtype) + + last_logerr = 0 + for i in range(4): + res = _tanhsinh(f, a, b, minlevel=0, maxlevel=i) + logerr = self.error(res.integral, ref, log=True, xp=xp) + assert (logerr < last_logerr * 2 or logerr < -15.5) + last_logerr = logerr + + def test_options_and_result_attributes(self, xp): + # demonstrate that options are behaving as advertised and status + # messages are as intended + xp_test = array_namespace(xp.asarray(1.)) # need xp.atan + + def f(x): + f.calls += 1 + f.feval += xp_size(xp.asarray(x)) + return x**2 * xp_test.atan(x) + + f.ref = xp.asarray((math.pi - 2 + 2 * math.log(2)) / 12, dtype=xp.float64) + + default_rtol = 1e-12 + default_atol = f.ref * default_rtol # effective default absolute tol + + # Keep things simpler by leaving tolerances fixed rather than + # having to make them dtype-dependent + a = xp.asarray(0., dtype=xp.float64) + b = xp.asarray(1., dtype=xp.float64) + + # Test default options + f.feval, f.calls = 0, 0 + ref = _tanhsinh(f, a, b) + assert self.error(ref.integral, f.ref) < ref.error < default_atol + assert ref.nfev == f.feval + ref.calls = f.calls # reference number of function calls + assert ref.success + assert ref.status == 0 + + # Test `maxlevel` equal to required max level + # We should get all the same results + f.feval, f.calls = 0, 0 + maxlevel = int(ref.maxlevel) + res = _tanhsinh(f, a, b, maxlevel=maxlevel) + res.calls = f.calls + assert res == ref + + # Now reduce the maximum level. We won't meet tolerances. + f.feval, f.calls = 0, 0 + maxlevel -= 1 + assert maxlevel >= 2 # can't compare errors otherwise + res = _tanhsinh(f, a, b, maxlevel=maxlevel) + assert self.error(res.integral, f.ref) < res.error > default_atol + assert res.nfev == f.feval < ref.nfev + assert f.calls == ref.calls - 1 + assert not res.success + assert res.status == eim._ECONVERR + + # `maxfun` is currently not enforced + + # # Test `maxfun` equal to required number of function evaluations + # # We should get all the same results + # f.feval, f.calls = 0, 0 + # maxfun = ref.nfev + # res = _tanhsinh(f, 0, f.b, maxfun = maxfun) + # assert res == ref + # + # # Now reduce `maxfun`. We won't meet tolerances. + # f.feval, f.calls = 0, 0 + # maxfun -= 1 + # res = _tanhsinh(f, 0, f.b, maxfun=maxfun) + # assert self.error(res.integral, f.ref) < res.error > default_atol + # assert res.nfev == f.feval < ref.nfev + # assert f.calls == ref.calls - 1 + # assert not res.success + # assert res.status == 2 + + # Take this result to be the new reference + ref = res + ref.calls = f.calls + + # Test `atol` + f.feval, f.calls = 0, 0 + # With this tolerance, we should get the exact same result as ref + atol = np.nextafter(float(ref.error), np.inf) + res = _tanhsinh(f, a, b, rtol=0, atol=atol) + assert res.integral == ref.integral + assert res.error == ref.error + assert res.nfev == f.feval == ref.nfev + assert f.calls == ref.calls + # Except the result is considered to be successful + assert res.success + assert res.status == 0 + + f.feval, f.calls = 0, 0 + # With a tighter tolerance, we should get a more accurate result + atol = np.nextafter(float(ref.error), -np.inf) + res = _tanhsinh(f, a, b, rtol=0, atol=atol) + assert self.error(res.integral, f.ref) < res.error < atol + assert res.nfev == f.feval > ref.nfev + assert f.calls > ref.calls + assert res.success + assert res.status == 0 + + # Test `rtol` + f.feval, f.calls = 0, 0 + # With this tolerance, we should get the exact same result as ref + rtol = np.nextafter(float(ref.error/ref.integral), np.inf) + res = _tanhsinh(f, a, b, rtol=rtol) + assert res.integral == ref.integral + assert res.error == ref.error + assert res.nfev == f.feval == ref.nfev + assert f.calls == ref.calls + # Except the result is considered to be successful + assert res.success + assert res.status == 0 + + f.feval, f.calls = 0, 0 + # With a tighter tolerance, we should get a more accurate result + rtol = np.nextafter(float(ref.error/ref.integral), -np.inf) + res = _tanhsinh(f, a, b, rtol=rtol) + assert self.error(res.integral, f.ref)/f.ref < res.error/res.integral < rtol + assert res.nfev == f.feval > ref.nfev + assert f.calls > ref.calls + assert res.success + assert res.status == 0 + + @pytest.mark.skip_xp_backends('torch', reason= + 'https://github.com/scipy/scipy/pull/21149#issuecomment-2330477359', + ) + @pytest.mark.parametrize('rtol', [1e-4, 1e-14]) + def test_log(self, rtol, xp): + # Test equivalence of log-integration and regular integration + test_tols = dict(atol=1e-18, rtol=1e-15) + + # Positive integrand (real log-integrand) + a = xp.asarray(-1., dtype=xp.float64) + b = xp.asarray(2., dtype=xp.float64) + res = _tanhsinh(norm_logpdf, a, b, log=True, rtol=math.log(rtol)) + ref = _tanhsinh(norm_pdf, a, b, rtol=rtol) + xp_assert_close(xp.exp(res.integral), ref.integral, **test_tols) + xp_assert_close(xp.exp(res.error), ref.error, **test_tols) + assert res.nfev == ref.nfev + + # Real integrand (complex log-integrand) + def f(x): + return -norm_logpdf(x)*norm_pdf(x) + + def logf(x): + return xp.log(norm_logpdf(x) + 0j) + norm_logpdf(x) + xp.pi * 1j + + a = xp.asarray(-xp.inf, dtype=xp.float64) + b = xp.asarray(xp.inf, dtype=xp.float64) + res = _tanhsinh(logf, a, b, log=True) + ref = _tanhsinh(f, a, b) + # In gh-19173, we saw `invalid` warnings on one CI platform. + # Silencing `all` because I can't reproduce locally and don't want + # to risk the need to run CI again. + with np.errstate(all='ignore'): + xp_assert_close(xp.exp(res.integral), ref.integral, **test_tols, + check_dtype=False) + xp_assert_close(xp.exp(res.error), ref.error, **test_tols, + check_dtype=False) + assert res.nfev == ref.nfev + + def test_complex(self, xp): + # Test integration of complex integrand + # Finite limits + def f(x): + return xp.exp(1j * x) + + a, b = xp.asarray(0.), xp.asarray(xp.pi/4) + res = _tanhsinh(f, a, b) + ref = math.sqrt(2)/2 + (1-math.sqrt(2)/2)*1j + xp_assert_close(res.integral, xp.asarray(ref)) + + # Infinite limits + def f(x): + return norm_pdf(x) + 1j/2*norm_pdf(x/2) + + a, b = xp.asarray(xp.inf), xp.asarray(-xp.inf) + res = _tanhsinh(f, a, b) + xp_assert_close(res.integral, xp.asarray(-(1+1j))) + + @pytest.mark.parametrize("maxlevel", range(4)) + def test_minlevel(self, maxlevel, xp): + # Verify that minlevel does not change the values at which the + # integrand is evaluated or the integral/error estimates, only the + # number of function calls + + # need `xp.concat`, `xp.atan`, and `xp.sort` + xp_test = array_namespace(xp.asarray(1.)) + + def f(x): + f.calls += 1 + f.feval += xp_size(xp.asarray(x)) + f.x = xp_test.concat((f.x, xp_ravel(x))) + return x**2 * xp_test.atan(x) + + f.feval, f.calls, f.x = 0, 0, xp.asarray([]) + + a = xp.asarray(0, dtype=xp.float64) + b = xp.asarray(1, dtype=xp.float64) + ref = _tanhsinh(f, a, b, minlevel=0, maxlevel=maxlevel) + ref_x = xp_test.sort(f.x) + + for minlevel in range(0, maxlevel + 1): + f.feval, f.calls, f.x = 0, 0, xp.asarray([]) + options = dict(minlevel=minlevel, maxlevel=maxlevel) + res = _tanhsinh(f, a, b, **options) + # Should be very close; all that has changed is the order of values + xp_assert_close(res.integral, ref.integral, rtol=4e-16) + # Difference in absolute errors << magnitude of integral + xp_assert_close(res.error, ref.error, atol=4e-16 * ref.integral) + assert res.nfev == f.feval == f.x.shape[0] + assert f.calls == maxlevel - minlevel + 1 + 1 # 1 validation call + assert res.status == ref.status + xp_assert_equal(ref_x, xp_test.sort(f.x)) + + def test_improper_integrals(self, xp): + # Test handling of infinite limits of integration (mixed with finite limits) + def f(x): + x[xp.isinf(x)] = xp.nan + return xp.exp(-x**2) + a = xp.asarray([-xp.inf, 0, -xp.inf, xp.inf, -20, -xp.inf, -20]) + b = xp.asarray([xp.inf, xp.inf, 0, -xp.inf, 20, 20, xp.inf]) + ref = math.sqrt(math.pi) + ref = xp.asarray([ref, ref/2, ref/2, -ref, ref, ref, ref]) + res = _tanhsinh(f, a, b) + xp_assert_close(res.integral, ref) + + @pytest.mark.parametrize("limits", ((0, 3), ([-math.inf, 0], [3, 3]))) + @pytest.mark.parametrize("dtype", ('float32', 'float64')) + def test_dtype(self, limits, dtype, xp): + # Test that dtypes are preserved + dtype = getattr(xp, dtype) + a, b = xp.asarray(limits, dtype=dtype) + + def f(x): + assert x.dtype == dtype + return xp.exp(x) + + rtol = 1e-12 if dtype == xp.float64 else 1e-5 + res = _tanhsinh(f, a, b, rtol=rtol) + assert res.integral.dtype == dtype + assert res.error.dtype == dtype + assert xp.all(res.success) + xp_assert_close(res.integral, xp.exp(b)-xp.exp(a)) + + def test_maxiter_callback(self, xp): + # Test behavior of `maxiter` parameter and `callback` interface + a, b = xp.asarray(-xp.inf), xp.asarray(xp.inf) + def f(x): + return xp.exp(-x*x) + + minlevel, maxlevel = 0, 2 + maxiter = maxlevel - minlevel + 1 + kwargs = dict(minlevel=minlevel, maxlevel=maxlevel, rtol=1e-15) + res = _tanhsinh(f, a, b, **kwargs) + assert not res.success + assert res.maxlevel == maxlevel + + def callback(res): + callback.iter += 1 + callback.res = res + assert hasattr(res, 'integral') + assert res.status == 1 + if callback.iter == maxiter: + raise StopIteration + callback.iter = -1 # callback called once before first iteration + callback.res = None + + del kwargs['maxlevel'] + res2 = _tanhsinh(f, a, b, **kwargs, callback=callback) + # terminating with callback is identical to terminating due to maxiter + # (except for `status`) + for key in res.keys(): + if key == 'status': + assert res[key] == -2 + assert res2[key] == -4 + else: + assert res2[key] == callback.res[key] == res[key] + + def test_jumpstart(self, xp): + # The intermediate results at each level i should be the same as the + # final results when jumpstarting at level i; i.e. minlevel=maxlevel=i + a = xp.asarray(-xp.inf, dtype=xp.float64) + b = xp.asarray(xp.inf, dtype=xp.float64) + + def f(x): + return xp.exp(-x*x) + + def callback(res): + callback.integrals.append(xp_copy(res.integral)[()]) + callback.errors.append(xp_copy(res.error)[()]) + callback.integrals = [] + callback.errors = [] + + maxlevel = 4 + _tanhsinh(f, a, b, minlevel=0, maxlevel=maxlevel, callback=callback) + + for i in range(maxlevel + 1): + res = _tanhsinh(f, a, b, minlevel=i, maxlevel=i) + xp_assert_close(callback.integrals[1+i], res.integral, rtol=1e-15) + xp_assert_close(callback.errors[1+i], res.error, rtol=1e-15, atol=1e-16) + + def test_special_cases(self, xp): + # Test edge cases and other special cases + a, b = xp.asarray(0), xp.asarray(1) + xp_test = array_namespace(a, b) # need `xp.isdtype` + + def f(x): + assert xp_test.isdtype(x.dtype, "real floating") + return x + + res = _tanhsinh(f, a, b) + assert res.success + xp_assert_close(res.integral, xp.asarray(0.5)) + + # Test levels 0 and 1; error is NaN + res = _tanhsinh(f, a, b, maxlevel=0) + assert res.integral > 0 + xp_assert_equal(res.error, xp.asarray(xp.nan)) + res = _tanhsinh(f, a, b, maxlevel=1) + assert res.integral > 0 + xp_assert_equal(res.error, xp.asarray(xp.nan)) + + # Test equal left and right integration limits + res = _tanhsinh(f, b, b) + assert res.success + assert res.maxlevel == -1 + xp_assert_close(res.integral, xp.asarray(0.)) + + # Test scalar `args` (not in tuple) + def f(x, c): + return x**c + + res = _tanhsinh(f, a, b, args=29) + xp_assert_close(res.integral, xp.asarray(1/30)) + + # Test NaNs + a = xp.asarray([xp.nan, 0, 0, 0]) + b = xp.asarray([1, xp.nan, 1, 1]) + c = xp.asarray([1, 1, xp.nan, 1]) + res = _tanhsinh(f, a, b, args=(c,)) + xp_assert_close(res.integral, xp.asarray([xp.nan, xp.nan, xp.nan, 0.5])) + xp_assert_equal(res.error[:3], xp.full((3,), xp.nan)) + xp_assert_equal(res.status, xp.asarray([-3, -3, -3, 0], dtype=xp.int32)) + xp_assert_equal(res.success, xp.asarray([False, False, False, True])) + xp_assert_equal(res.nfev[:3], xp.full((3,), 1, dtype=xp.int32)) + + # Test complex integral followed by real integral + # Previously, h0 was of the result dtype. If the `dtype` were complex, + # this could lead to complex cached abscissae/weights. If these get + # cast to real dtype for a subsequent real integral, we would get a + # ComplexWarning. Check that this is avoided. + _pair_cache.xjc = xp.empty(0) + _pair_cache.wj = xp.empty(0) + _pair_cache.indices = [0] + _pair_cache.h0 = None + a, b = xp.asarray(0), xp.asarray(1) + res = _tanhsinh(lambda x: xp.asarray(x*1j), a, b) + xp_assert_close(res.integral, xp.asarray(0.5*1j)) + res = _tanhsinh(lambda x: x, a, b) + xp_assert_close(res.integral, xp.asarray(0.5)) + + # Test zero-size + shape = (0, 3) + res = _tanhsinh(lambda x: x, xp.asarray(0), xp.zeros(shape)) + attrs = ['integral', 'error', 'success', 'status', 'nfev', 'maxlevel'] + for attr in attrs: + assert res[attr].shape == shape + + @pytest.mark.skip_xp_backends(np_only=True) + def test_compress_nodes_weights_gh21496(self, xp): + # See discussion in: + # https://github.com/scipy/scipy/pull/21496#discussion_r1878681049 + # This would cause "ValueError: attempt to get argmax of an empty sequence" + # Check that this has been resolved. + x = np.full(65, 3) + x[-1] = 1000 + _tanhsinh(np.sin, 1, x) + + def test_gh_22681_finite_error(self, xp): + # gh-22681 noted a case in which the error was NaN on some platforms; + # check that this does in fact fail in CI. + a = complex(12, -10) + b = complex(12, 39) + def f(t): + return xp.sin(a * (1 - t) + b * t) + res = _tanhsinh(f, xp.asarray(0.), xp.asarray(1.), atol=0, rtol=0, maxlevel=10) + assert xp.isfinite(res.error) + + +@array_api_compatible +@pytest.mark.usefixtures("skip_xp_backends") +@pytest.mark.skip_xp_backends('array_api_strict', reason='No fancy indexing.') +@pytest.mark.skip_xp_backends('jax.numpy', reason='No mutation.') +class TestNSum: + rng = np.random.default_rng(5895448232066142650) + p = rng.uniform(1, 10, size=10).tolist() + + def f1(self, k): + # Integers are never passed to `f1`; if they were, we'd get + # integer to negative integer power error + return k**(-2) + + f1.ref = np.pi**2/6 + f1.a = 1 + f1.b = np.inf + f1.args = tuple() + + def f2(self, k, p): + return 1 / k**p + + f2.ref = special.zeta(p, 1) + f2.a = 1. + f2.b = np.inf + f2.args = (p,) + + def f3(self, k, p): + return 1 / k**p + + f3.a = 1 + f3.b = rng.integers(5, 15, size=(3, 1)) + f3.ref = _gen_harmonic_gt1(f3.b, p) + f3.args = (p,) + + def test_input_validation(self, xp): + f = self.f1 + a, b = xp.asarray(f.a), xp.asarray(f.b) + + message = '`f` must be callable.' + with pytest.raises(ValueError, match=message): + nsum(42, a, b) + + message = '...must be True or False.' + with pytest.raises(ValueError, match=message): + nsum(f, a, b, log=2) + + message = '...must be real numbers.' + with pytest.raises(ValueError, match=message): + nsum(f, xp.asarray(1+1j), b) + with pytest.raises(ValueError, match=message): + nsum(f, a, xp.asarray(1+1j)) + with pytest.raises(ValueError, match=message): + nsum(f, a, b, step=xp.asarray(1+1j)) + with pytest.raises(ValueError, match=message): + nsum(f, a, b, tolerances=dict(atol='ekki')) + with pytest.raises(ValueError, match=message): + nsum(f, a, b, tolerances=dict(rtol=pytest)) + + with np.errstate(all='ignore'): + res = nsum(f, xp.asarray([np.nan, np.inf]), xp.asarray(1.)) + assert xp.all((res.status == -1) & xp.isnan(res.sum) + & xp.isnan(res.error) & ~res.success & res.nfev == 1) + res = nsum(f, xp.asarray(10.), xp.asarray([np.nan, 1])) + assert xp.all((res.status == -1) & xp.isnan(res.sum) + & xp.isnan(res.error) & ~res.success & res.nfev == 1) + res = nsum(f, xp.asarray(1.), xp.asarray(10.), + step=xp.asarray([xp.nan, -xp.inf, xp.inf, -1, 0])) + assert xp.all((res.status == -1) & xp.isnan(res.sum) + & xp.isnan(res.error) & ~res.success & res.nfev == 1) + + message = '...must be non-negative and finite.' + with pytest.raises(ValueError, match=message): + nsum(f, a, b, tolerances=dict(rtol=-1)) + with pytest.raises(ValueError, match=message): + nsum(f, a, b, tolerances=dict(atol=np.inf)) + + message = '...may not be positive infinity.' + with pytest.raises(ValueError, match=message): + nsum(f, a, b, tolerances=dict(rtol=np.inf), log=True) + with pytest.raises(ValueError, match=message): + nsum(f, a, b, tolerances=dict(atol=np.inf), log=True) + + message = '...must be a non-negative integer.' + with pytest.raises(ValueError, match=message): + nsum(f, a, b, maxterms=3.5) + with pytest.raises(ValueError, match=message): + nsum(f, a, b, maxterms=-2) + + @pytest.mark.parametrize('f_number', range(1, 4)) + def test_basic(self, f_number, xp): + dtype = xp.asarray(1.).dtype + f = getattr(self, f"f{f_number}") + a, b = xp.asarray(f.a), xp.asarray(f.b), + args = tuple(xp.asarray(arg) for arg in f.args) + ref = xp.asarray(f.ref, dtype=dtype) + res = nsum(f, a, b, args=args) + xp_assert_close(res.sum, ref) + xp_assert_equal(res.status, xp.zeros(ref.shape, dtype=xp.int32)) + xp_test = array_namespace(a) # CuPy doesn't have `bool` + xp_assert_equal(res.success, xp.ones(ref.shape, dtype=xp_test.bool)) + + with np.errstate(divide='ignore'): + logres = nsum(lambda *args: xp.log(f(*args)), + a, b, log=True, args=args) + xp_assert_close(xp.exp(logres.sum), res.sum) + xp_assert_close(xp.exp(logres.error), res.error, atol=1e-15) + xp_assert_equal(logres.status, res.status) + xp_assert_equal(logres.success, res.success) + + @pytest.mark.parametrize('maxterms', [0, 1, 10, 20, 100]) + def test_integral(self, maxterms, xp): + # test precise behavior of integral approximation + f = self.f1 + + def logf(x): + return -2*xp.log(x) + + def F(x): + return -1 / x + + a = xp.asarray([1, 5], dtype=xp.float64)[:, xp.newaxis] + b = xp.asarray([20, 100, xp.inf], dtype=xp.float64)[:, xp.newaxis, xp.newaxis] + step = xp.asarray([0.5, 1, 2], dtype=xp.float64).reshape((-1, 1, 1, 1)) + nsteps = xp.floor((b - a)/step) + b_original = b + b = a + nsteps*step + + k = a + maxterms*step + # partial sum + direct = xp.sum(f(a + xp.arange(maxterms)*step), axis=-1, keepdims=True) + integral = (F(b) - F(k))/step # integral approximation of remainder + low = direct + integral + f(b) # theoretical lower bound + high = direct + integral + f(k) # theoretical upper bound + ref_sum = (low + high)/2 # nsum uses average of the two + ref_err = (high - low)/2 # error (assuming perfect quadrature) + + # correct reference values where number of terms < maxterms + xp_test = array_namespace(a) # torch needs broadcast_arrays + a, b, step = xp_test.broadcast_arrays(a, b, step) + for i in np.ndindex(a.shape): + ai, bi, stepi = float(a[i]), float(b[i]), float(step[i]) + if (bi - ai)/stepi + 1 <= maxterms: + direct = xp.sum(f(xp.arange(ai, bi+stepi, stepi, dtype=xp.float64))) + ref_sum[i] = direct + ref_err[i] = direct * xp.finfo(direct.dtype).eps + + rtol = 1e-12 + res = nsum(f, a, b_original, step=step, maxterms=maxterms, + tolerances=dict(rtol=rtol)) + xp_assert_close(res.sum, ref_sum, rtol=10*rtol) + xp_assert_close(res.error, ref_err, rtol=100*rtol) + + i = ((b_original - a)/step + 1 <= maxterms) + xp_assert_close(res.sum[i], ref_sum[i], rtol=1e-15) + xp_assert_close(res.error[i], ref_err[i], rtol=1e-15) + + logres = nsum(logf, a, b_original, step=step, log=True, + tolerances=dict(rtol=math.log(rtol)), maxterms=maxterms) + xp_assert_close(xp.exp(logres.sum), res.sum) + xp_assert_close(xp.exp(logres.error), res.error) + + @pytest.mark.parametrize('shape', [tuple(), (12,), (3, 4), (3, 2, 2)]) + def test_vectorization(self, shape, xp): + # Test for correct functionality, output shapes, and dtypes for various + # input shapes. + rng = np.random.default_rng(82456839535679456794) + a = rng.integers(1, 10, size=shape) + # when the sum can be computed directly or `maxterms` is large enough + # to meet `atol`, there are slight differences (for good reason) + # between vectorized call and looping. + b = np.inf + p = rng.random(shape) + 1 + n = math.prod(shape) + + def f(x, p): + f.feval += 1 if (x.size == n or x.ndim <= 1) else x.shape[-1] + return 1 / x ** p + + f.feval = 0 + + @np.vectorize + def nsum_single(a, b, p, maxterms): + return nsum(lambda x: 1 / x**p, a, b, maxterms=maxterms) + + res = nsum(f, xp.asarray(a), xp.asarray(b), maxterms=1000, + args=(xp.asarray(p),)) + refs = nsum_single(a, b, p, maxterms=1000).ravel() + + attrs = ['sum', 'error', 'success', 'status', 'nfev'] + for attr in attrs: + ref_attr = [xp.asarray(getattr(ref, attr)) for ref in refs] + res_attr = getattr(res, attr) + xp_assert_close(xp_ravel(res_attr), xp.asarray(ref_attr), rtol=1e-15) + assert res_attr.shape == shape + + xp_test = array_namespace(xp.asarray(1.)) + assert xp_test.isdtype(res.success.dtype, 'bool') + assert xp_test.isdtype(res.status.dtype, 'integral') + assert xp_test.isdtype(res.nfev.dtype, 'integral') + if is_numpy(xp): # other libraries might have different number + assert int(xp.max(res.nfev)) == f.feval + + def test_status(self, xp): + f = self.f2 + + p = [2, 2, 0.9, 1.1, 2, 2] + a = xp.asarray([0, 0, 1, 1, 1, np.nan], dtype=xp.float64) + b = xp.asarray([10, np.inf, np.inf, np.inf, np.inf, np.inf], dtype=xp.float64) + ref = special.zeta(p, 1) + p = xp.asarray(p, dtype=xp.float64) + + with np.errstate(divide='ignore'): # intentionally dividing by zero + res = nsum(f, a, b, args=(p,)) + + ref_success = xp.asarray([False, False, False, False, True, False]) + ref_status = xp.asarray([-3, -3, -2, -4, 0, -1], dtype=xp.int32) + xp_assert_equal(res.success, ref_success) + xp_assert_equal(res.status, ref_status) + xp_assert_close(res.sum[res.success], xp.asarray(ref)[res.success]) + + def test_nfev(self, xp): + def f(x): + f.nfev += xp_size(x) + return 1 / x**2 + + f.nfev = 0 + res = nsum(f, xp.asarray(1), xp.asarray(10)) + assert res.nfev == f.nfev + + f.nfev = 0 + res = nsum(f, xp.asarray(1), xp.asarray(xp.inf), tolerances=dict(atol=1e-6)) + assert res.nfev == f.nfev + + def test_inclusive(self, xp): + # There was an edge case off-by one bug when `_direct` was called with + # `inclusive=True`. Check that this is resolved. + a = xp.asarray([1, 4]) + b = xp.asarray(xp.inf) + res = nsum(lambda k: 1 / k ** 2, a, b, + maxterms=500, tolerances=dict(atol=0.1)) + ref = nsum(lambda k: 1 / k ** 2, a, b) + assert xp.all(res.sum > (ref.sum - res.error)) + assert xp.all(res.sum < (ref.sum + res.error)) + + @pytest.mark.parametrize('log', [True, False]) + def test_infinite_bounds(self, log, xp): + a = xp.asarray([1, -np.inf, -np.inf]) + b = xp.asarray([np.inf, -1, np.inf]) + c = xp.asarray([1, 2, 3]) + + def f(x, a): + return (xp.log(xp.tanh(a / 2)) - a*xp.abs(x) if log + else xp.tanh(a/2) * xp.exp(-a*xp.abs(x))) + + res = nsum(f, a, b, args=(c,), log=log) + ref = xp.asarray([stats.dlaplace.sf(0, 1), stats.dlaplace.sf(0, 2), 1]) + ref = xp.log(ref) if log else ref + atol = (1e-10 if a.dtype==xp.float64 else 1e-5) if log else 0 + xp_assert_close(res.sum, xp.asarray(ref, dtype=a.dtype), atol=atol) + + # # Make sure the sign of `x` passed into `f` is correct. + def f(x, c): + return -3*xp.log(c*x) if log else 1 / (c*x)**3 + + a = xp.asarray([1, -np.inf]) + b = xp.asarray([np.inf, -1]) + arg = xp.asarray([1, -1]) + res = nsum(f, a, b, args=(arg,), log=log) + ref = np.log(special.zeta(3)) if log else special.zeta(3) + xp_assert_close(res.sum, xp.full(a.shape, ref, dtype=a.dtype)) + + def test_decreasing_check(self, xp): + # Test accuracy when we start sum on an uphill slope. + # Without the decreasing check, the terms would look small enough to + # use the integral approximation. Because the function is not decreasing, + # the error is not bounded by the magnitude of the last term of the + # partial sum. In this case, the error would be ~1e-4, causing the test + # to fail. + def f(x): + return xp.exp(-x ** 2) + + a, b = xp.asarray(-25, dtype=xp.float64), xp.asarray(np.inf, dtype=xp.float64) + res = nsum(f, a, b) + + # Reference computed with mpmath: + # from mpmath import mp + # mp.dps = 50 + # def fmp(x): return mp.exp(-x**2) + # ref = mp.nsum(fmp, (-25, 0)) + mp.nsum(fmp, (1, mp.inf)) + ref = xp.asarray(1.772637204826652, dtype=xp.float64) + + xp_assert_close(res.sum, ref, rtol=1e-15) + + def test_special_case(self, xp): + # test equal lower/upper limit + f = self.f1 + a = b = xp.asarray(2) + res = nsum(f, a, b) + xp_assert_equal(res.sum, xp.asarray(f(2))) + + # Test scalar `args` (not in tuple) + res = nsum(self.f2, xp.asarray(1), xp.asarray(np.inf), args=xp.asarray(2)) + xp_assert_close(res.sum, xp.asarray(self.f1.ref)) # f1.ref is correct w/ args=2 + + # Test 0 size input + a = xp.empty((3, 1, 1)) # arbitrary broadcastable shapes + b = xp.empty((0, 1)) # could use Hypothesis + p = xp.empty(4) # but it's overkill + shape = np.broadcast_shapes(a.shape, b.shape, p.shape) + res = nsum(self.f2, a, b, args=(p,)) + assert res.sum.shape == shape + assert res.status.shape == shape + assert res.nfev.shape == shape + + # Test maxterms=0 + def f(x): + with np.errstate(divide='ignore'): + return 1 / x + + res = nsum(f, xp.asarray(0), xp.asarray(10), maxterms=0) + assert xp.isnan(res.sum) + assert xp.isnan(res.error) + assert res.status == -2 + + res = nsum(f, xp.asarray(0), xp.asarray(10), maxterms=1) + assert xp.isnan(res.sum) + assert xp.isnan(res.error) + assert res.status == -3 + + # Test NaNs + # should skip both direct and integral methods if there are NaNs + a = xp.asarray([xp.nan, 1, 1, 1]) + b = xp.asarray([xp.inf, xp.nan, xp.inf, xp.inf]) + p = xp.asarray([2, 2, xp.nan, 2]) + res = nsum(self.f2, a, b, args=(p,)) + xp_assert_close(res.sum, xp.asarray([xp.nan, xp.nan, xp.nan, self.f1.ref])) + xp_assert_close(res.error[:3], xp.full((3,), xp.nan)) + xp_assert_equal(res.status, xp.asarray([-1, -1, -3, 0], dtype=xp.int32)) + xp_assert_equal(res.success, xp.asarray([False, False, False, True])) + # Ideally res.nfev[2] would be 1, but `tanhsinh` has some function evals + xp_assert_equal(res.nfev[:2], xp.full((2,), 1, dtype=xp.int32)) + + @pytest.mark.parametrize('dtype', ['float32', 'float64']) + def test_dtype(self, dtype, xp): + dtype = getattr(xp, dtype) + + def f(k): + assert k.dtype == dtype + return 1 / k ** xp.asarray(2, dtype=dtype) + + a = xp.asarray(1, dtype=dtype) + b = xp.asarray([10, xp.inf], dtype=dtype) + res = nsum(f, a, b) + assert res.sum.dtype == dtype + assert res.error.dtype == dtype + + rtol = 1e-12 if dtype == xp.float64 else 1e-6 + ref = _gen_harmonic_gt1(np.asarray([10, xp.inf]), 2) + xp_assert_close(res.sum, xp.asarray(ref, dtype=dtype), rtol=rtol) + + @pytest.mark.parametrize('case', [(10, 100), (100, 10)]) + def test_nondivisible_interval(self, case, xp): + # When the limits of the sum are such that (b - a)/step + # is not exactly integral, check that only floor((b - a)/step) + # terms are included. + n, maxterms = case + + def f(k): + return 1 / k ** 2 + + a = np.e + step = 1 / 3 + b0 = a + n * step + i = np.arange(-2, 3) + b = b0 + i * np.spacing(b0) + ns = np.floor((b - a) / step) + assert len(set(ns)) == 2 + + a, b = xp.asarray(a, dtype=xp.float64), xp.asarray(b, dtype=xp.float64) + step, ns = xp.asarray(step, dtype=xp.float64), xp.asarray(ns, dtype=xp.float64) + res = nsum(f, a, b, step=step, maxterms=maxterms) + xp_assert_equal(xp.diff(ns) > 0, xp.diff(res.sum) > 0) + xp_assert_close(res.sum[-1], res.sum[0] + f(b0)) + + @pytest.mark.skip_xp_backends(np_only=True, reason='Needs beta function.') + def test_logser_kurtosis_gh20648(self, xp): + # Some functions return NaN at infinity rather than 0 like they should. + # Check that this is accounted for. + ref = stats.yulesimon.moment(4, 5) + def f(x): + return stats.yulesimon._pmf(x, 5) * x**4 + + with np.errstate(invalid='ignore'): + assert np.isnan(f(np.inf)) + + res = nsum(f, 1, np.inf) + assert_allclose(res.sum, ref) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/vode.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/vode.py new file mode 100644 index 0000000000000000000000000000000000000000..f92927901084ce33cdeb006057d85dd501b13aae --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/integrate/vode.py @@ -0,0 +1,15 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="integrate", module="vode", + private_modules=["_vode"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..1c4f97134d20b8d3acb1bea54c8384c510314aaa --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/__init__.py @@ -0,0 +1,216 @@ +""" +======================================== +Interpolation (:mod:`scipy.interpolate`) +======================================== + +.. currentmodule:: scipy.interpolate + +Sub-package for objects used in interpolation. + +As listed below, this sub-package contains spline functions and classes, +1-D and multidimensional (univariate and multivariate) +interpolation classes, Lagrange and Taylor polynomial interpolators, and +wrappers for `FITPACK `__ +and DFITPACK functions. + +Univariate interpolation +======================== + +.. autosummary:: + :toctree: generated/ + + interp1d + BarycentricInterpolator + KroghInterpolator + barycentric_interpolate + krogh_interpolate + pchip_interpolate + CubicHermiteSpline + PchipInterpolator + Akima1DInterpolator + CubicSpline + PPoly + BPoly + FloaterHormannInterpolator + + +Multivariate interpolation +========================== + +Unstructured data: + +.. autosummary:: + :toctree: generated/ + + griddata + LinearNDInterpolator + NearestNDInterpolator + CloughTocher2DInterpolator + RBFInterpolator + Rbf + interp2d + +For data on a grid: + +.. autosummary:: + :toctree: generated/ + + interpn + RegularGridInterpolator + RectBivariateSpline + +.. seealso:: + + `scipy.ndimage.map_coordinates` + +Tensor product polynomials: + +.. autosummary:: + :toctree: generated/ + + NdPPoly + NdBSpline + +1-D Splines +=========== + +.. autosummary:: + :toctree: generated/ + + BSpline + make_interp_spline + make_lsq_spline + make_smoothing_spline + generate_knots + make_splrep + make_splprep + +Functional interface to FITPACK routines: + +.. autosummary:: + :toctree: generated/ + + splrep + splprep + splev + splint + sproot + spalde + splder + splantider + insert + +Object-oriented FITPACK interface: + +.. autosummary:: + :toctree: generated/ + + UnivariateSpline + InterpolatedUnivariateSpline + LSQUnivariateSpline + + + +2-D Splines +=========== + +For data on a grid: + +.. autosummary:: + :toctree: generated/ + + RectBivariateSpline + RectSphereBivariateSpline + +For unstructured data: + +.. autosummary:: + :toctree: generated/ + + BivariateSpline + SmoothBivariateSpline + SmoothSphereBivariateSpline + LSQBivariateSpline + LSQSphereBivariateSpline + +Low-level interface to FITPACK functions: + +.. autosummary:: + :toctree: generated/ + + bisplrep + bisplev + +Rational Approximation +====================== + +.. autosummary:: + :toctree: generated/ + + pade + AAA + +Additional tools +================ + +.. autosummary:: + :toctree: generated/ + + lagrange + approximate_taylor_polynomial + +.. seealso:: + + `scipy.ndimage.map_coordinates`, + `scipy.ndimage.spline_filter`, + `scipy.signal.resample`, + `scipy.signal.bspline`, + `scipy.signal.gauss_spline`, + `scipy.signal.qspline1d`, + `scipy.signal.cspline1d`, + `scipy.signal.qspline1d_eval`, + `scipy.signal.cspline1d_eval`, + `scipy.signal.qspline2d`, + `scipy.signal.cspline2d`. + +``pchip`` is an alias of `PchipInterpolator` for backward compatibility +(should not be used in new code). +""" +from ._interpolate import * +from ._fitpack_py import * + +# New interface to fitpack library: +from ._fitpack2 import * + +from ._rbf import Rbf + +from ._rbfinterp import * + +from ._polyint import * + +from ._cubic import * + +from ._ndgriddata import * + +from ._bsplines import * +from ._fitpack_repro import generate_knots, make_splrep, make_splprep + +from ._pade import * + +from ._rgi import * + +from ._ndbspline import NdBSpline + +from ._bary_rational import * + +# Deprecated namespaces, to be removed in v2.0.0 +from . import fitpack, fitpack2, interpolate, ndgriddata, polyint, rbf, interpnd + +__all__ = [s for s in dir() if not s.startswith('_')] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester + +# Backward compatibility +pchip = PchipInterpolator diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_bary_rational.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_bary_rational.py new file mode 100644 index 0000000000000000000000000000000000000000..be13c06e27cb8df7ec4c55993dd3937e867429e5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_bary_rational.py @@ -0,0 +1,715 @@ +# Copyright (c) 2017, The Chancellor, Masters and Scholars of the University +# of Oxford, and the Chebfun Developers. All rights reserved. +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions are met: +# * Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# * Redistributions in binary form must reproduce the above copyright +# notice, this list of conditions and the following disclaimer in the +# documentation and/or other materials provided with the distribution. +# * Neither the name of the University of Oxford nor the names of its +# contributors may be used to endorse or promote products derived from +# this software without specific prior written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND +# ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE +# DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR +# ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES +# (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; +# LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND +# ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT +# (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +import warnings +import operator + +import numpy as np +import scipy + + +__all__ = ["AAA", "FloaterHormannInterpolator"] + + +class _BarycentricRational: + """Base class for Barycentric representation of a rational function.""" + def __init__(self, x, y, **kwargs): + # input validation + z = np.asarray(x) + f = np.asarray(y) + + self._input_validation(z, f, **kwargs) + + # Remove infinite or NaN function values and repeated entries + to_keep = np.logical_and.reduce( + ((np.isfinite(f)) & (~np.isnan(f))).reshape(f.shape[0], -1), + axis=-1 + ) + f = f[to_keep, ...] + z = z[to_keep] + z, uni = np.unique(z, return_index=True) + f = f[uni, ...] + + self._shape = f.shape[1:] + self._support_points, self._support_values, self.weights = ( + self._compute_weights(z, f, **kwargs) + ) + + # only compute once + self._poles = None + self._residues = None + self._roots = None + + def _input_validation(self, x, y, **kwargs): + if x.ndim != 1: + raise ValueError("`x` must be 1-D.") + + if not y.ndim >= 1: + raise ValueError("`y` must be at least 1-D.") + + if x.size != y.shape[0]: + raise ValueError("`x` be the same size as the first dimension of `y`.") + + if not np.all(np.isfinite(x)): + raise ValueError("`x` must be finite.") + + def _compute_weights(z, f, **kwargs): + raise NotImplementedError + + def __call__(self, z): + """Evaluate the rational approximation at given values. + + Parameters + ---------- + z : array_like + Input values. + """ + # evaluate rational function in barycentric form. + z = np.asarray(z) + zv = np.ravel(z) + + support_values = self._support_values.reshape( + (self._support_values.shape[0], -1) + ) + weights = self.weights[..., np.newaxis] + + # Cauchy matrix + # Ignore errors due to inf/inf at support points, these will be fixed later + with np.errstate(invalid="ignore", divide="ignore"): + CC = 1 / np.subtract.outer(zv, self._support_points) + # Vector of values + r = CC @ (weights * support_values) / (CC @ weights) + + # Deal with input inf: `r(inf) = lim r(z) = sum(w*f) / sum(w)` + if np.any(np.isinf(zv)): + r[np.isinf(zv)] = (np.sum(weights * support_values) + / np.sum(weights)) + + # Deal with NaN + ii = np.nonzero(np.isnan(r))[0] + for jj in ii: + if np.isnan(zv[jj]) or not np.any(zv[jj] == self._support_points): + # r(NaN) = NaN is fine. + # The second case may happen if `r(zv[ii]) = 0/0` at some point. + pass + else: + # Clean up values `NaN = inf/inf` at support points. + # Find the corresponding node and set entry to correct value: + r[jj] = support_values[zv[jj] == self._support_points].squeeze() + + return np.reshape(r, z.shape + self._shape) + + def poles(self): + """Compute the poles of the rational approximation. + + Returns + ------- + poles : array + Poles of the AAA approximation, repeated according to their multiplicity + but not in any specific order. + """ + if self._poles is None: + # Compute poles via generalized eigenvalue problem + m = self.weights.size + B = np.eye(m + 1, dtype=self.weights.dtype) + B[0, 0] = 0 + + E = np.zeros_like(B, dtype=np.result_type(self.weights, + self._support_points)) + E[0, 1:] = self.weights + E[1:, 0] = 1 + np.fill_diagonal(E[1:, 1:], self._support_points) + + pol = scipy.linalg.eigvals(E, B) + self._poles = pol[np.isfinite(pol)] + return self._poles + + def residues(self): + """Compute the residues of the poles of the approximation. + + Returns + ------- + residues : array + Residues associated with the `poles` of the approximation + """ + if self._residues is None: + # Compute residues via formula for res of quotient of analytic functions + with np.errstate(divide="ignore", invalid="ignore"): + N = (1/(np.subtract.outer(self.poles(), self._support_points))) @ ( + self._support_values * self.weights + ) + Ddiff = ( + -((1/np.subtract.outer(self.poles(), self._support_points))**2) + @ self.weights + ) + self._residues = N / Ddiff + return self._residues + + def roots(self): + """Compute the zeros of the rational approximation. + + Returns + ------- + zeros : array + Zeros of the AAA approximation, repeated according to their multiplicity + but not in any specific order. + """ + if self._roots is None: + # Compute zeros via generalized eigenvalue problem + m = self.weights.size + B = np.eye(m + 1, dtype=self.weights.dtype) + B[0, 0] = 0 + E = np.zeros_like(B, dtype=np.result_type(self.weights, + self._support_values, + self._support_points)) + E[0, 1:] = self.weights * self._support_values + E[1:, 0] = 1 + np.fill_diagonal(E[1:, 1:], self._support_points) + + zer = scipy.linalg.eigvals(E, B) + self._roots = zer[np.isfinite(zer)] + return self._roots + + +class AAA(_BarycentricRational): + r""" + AAA real or complex rational approximation. + + As described in [1]_, the AAA algorithm is a greedy algorithm for approximation by + rational functions on a real or complex set of points. The rational approximation is + represented in a barycentric form from which the roots (zeros), poles, and residues + can be computed. + + Parameters + ---------- + x : 1D array_like, shape (n,) + 1-D array containing values of the independent variable. Values may be real or + complex but must be finite. + y : 1D array_like, shape (n,) + Function values ``f(x)``. Infinite and NaN values of `values` and + corresponding values of `points` will be discarded. + rtol : float, optional + Relative tolerance, defaults to ``eps**0.75``. If a small subset of the entries + in `values` are much larger than the rest the default tolerance may be too + loose. If the tolerance is too tight then the approximation may contain + Froissart doublets or the algorithm may fail to converge entirely. + max_terms : int, optional + Maximum number of terms in the barycentric representation, defaults to ``100``. + Must be greater than or equal to one. + clean_up : bool, optional + Automatic removal of Froissart doublets, defaults to ``True``. See notes for + more details. + clean_up_tol : float, optional + Poles with residues less than this number times the geometric mean + of `values` times the minimum distance to `points` are deemed spurious by the + cleanup procedure, defaults to 1e-13. See notes for more details. + + Attributes + ---------- + support_points : array + Support points of the approximation. These are a subset of the provided `x` at + which the approximation strictly interpolates `y`. + See notes for more details. + support_values : array + Value of the approximation at the `support_points`. + weights : array + Weights of the barycentric approximation. + errors : array + Error :math:`|f(z) - r(z)|_\infty` over `points` in the successive iterations + of AAA. + + Warns + ----- + RuntimeWarning + If `rtol` is not achieved in `max_terms` iterations. + + See Also + -------- + FloaterHormannInterpolator : Floater-Hormann barycentric rational interpolation. + pade : Padé approximation. + + Notes + ----- + At iteration :math:`m` (at which point there are :math:`m` terms in the both the + numerator and denominator of the approximation), the + rational approximation in the AAA algorithm takes the barycentric form + + .. math:: + + r(z) = n(z)/d(z) = + \frac{\sum_{j=1}^m\ w_j f_j / (z - z_j)}{\sum_{j=1}^m w_j / (z - z_j)}, + + where :math:`z_1,\dots,z_m` are real or complex support points selected from + `x`, :math:`f_1,\dots,f_m` are the corresponding real or complex data values + from `y`, and :math:`w_1,\dots,w_m` are real or complex weights. + + Each iteration of the algorithm has two parts: the greedy selection the next support + point and the computation of the weights. The first part of each iteration is to + select the next support point to be added :math:`z_{m+1}` from the remaining + unselected `x`, such that the nonlinear residual + :math:`|f(z_{m+1}) - n(z_{m+1})/d(z_{m+1})|` is maximised. The algorithm terminates + when this maximum is less than ``rtol * np.linalg.norm(f, ord=np.inf)``. This means + the interpolation property is only satisfied up to a tolerance, except at the + support points where approximation exactly interpolates the supplied data. + + In the second part of each iteration, the weights :math:`w_j` are selected to solve + the least-squares problem + + .. math:: + + \text{minimise}_{w_j}|fd - n| \quad \text{subject to} \quad + \sum_{j=1}^{m+1} w_j = 1, + + over the unselected elements of `x`. + + One of the challenges with working with rational approximations is the presence of + Froissart doublets, which are either poles with vanishingly small residues or + pole-zero pairs that are close enough together to nearly cancel, see [2]_. The + greedy nature of the AAA algorithm means Froissart doublets are rare. However, if + `rtol` is set too tight then the approximation will stagnate and many Froissart + doublets will appear. Froissart doublets can usually be removed by removing support + points and then resolving the least squares problem. The support point :math:`z_j`, + which is the closest support point to the pole :math:`a` with residue + :math:`\alpha`, is removed if the following is satisfied + + .. math:: + + |\alpha| / |z_j - a| < \verb|clean_up_tol| \cdot \tilde{f}, + + where :math:`\tilde{f}` is the geometric mean of `support_values`. + + + References + ---------- + .. [1] Y. Nakatsukasa, O. Sete, and L. N. Trefethen, "The AAA algorithm for + rational approximation", SIAM J. Sci. Comp. 40 (2018), A1494-A1522. + :doi:`10.1137/16M1106122` + .. [2] J. Gilewicz and M. Pindor, Pade approximants and noise: rational functions, + J. Comp. Appl. Math. 105 (1999), pp. 285-297. + :doi:`10.1016/S0377-0427(02)00674-X` + + Examples + -------- + + Here we reproduce a number of the numerical examples from [1]_ as a demonstration + of the functionality offered by this method. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import AAA + >>> import warnings + + For the first example we approximate the gamma function on ``[-3.5, 4.5]`` by + extrapolating from 100 samples in ``[-1.5, 1.5]``. + + >>> from scipy.special import gamma + >>> sample_points = np.linspace(-1.5, 1.5, num=100) + >>> r = AAA(sample_points, gamma(sample_points)) + >>> z = np.linspace(-3.5, 4.5, num=1000) + >>> fig, ax = plt.subplots() + >>> ax.plot(z, gamma(z), label="Gamma") + >>> ax.plot(sample_points, gamma(sample_points), label="Sample points") + >>> ax.plot(z, r(z).real, '--', label="AAA approximation") + >>> ax.set(xlabel="z", ylabel="r(z)", ylim=[-8, 8], xlim=[-3.5, 4.5]) + >>> ax.legend() + >>> plt.show() + + We can also view the poles of the rational approximation and their residues: + + >>> order = np.argsort(r.poles()) + >>> r.poles()[order] + array([-3.81591039e+00+0.j , -3.00269049e+00+0.j , + -1.99999988e+00+0.j , -1.00000000e+00+0.j , + 5.85842812e-17+0.j , 4.77485458e+00-3.06919376j, + 4.77485458e+00+3.06919376j, 5.29095868e+00-0.97373072j, + 5.29095868e+00+0.97373072j]) + >>> r.residues()[order] + array([ 0.03658074 +0.j , -0.16915426 -0.j , + 0.49999915 +0.j , -1. +0.j , + 1. +0.j , -0.81132013 -2.30193429j, + -0.81132013 +2.30193429j, 0.87326839+10.70148546j, + 0.87326839-10.70148546j]) + + For the second example, we call `AAA` with a spiral of 1000 points that wind 7.5 + times around the origin in the complex plane. + + >>> z = np.exp(np.linspace(-0.5, 0.5 + 15j*np.pi, 1000)) + >>> r = AAA(z, np.tan(np.pi*z/2), rtol=1e-13) + + We see that AAA takes 12 steps to converge with the following errors: + + >>> r.errors.size + 12 + >>> r.errors + array([2.49261500e+01, 4.28045609e+01, 1.71346935e+01, 8.65055336e-02, + 1.27106444e-02, 9.90889874e-04, 5.86910543e-05, 1.28735561e-06, + 3.57007424e-08, 6.37007837e-10, 1.67103357e-11, 1.17112299e-13]) + + We can also plot the computed poles: + + >>> fig, ax = plt.subplots() + >>> ax.plot(z.real, z.imag, '.', markersize=2, label="Sample points") + >>> ax.plot(r.poles().real, r.poles().imag, '.', markersize=5, + ... label="Computed poles") + >>> ax.set(xlim=[-3.5, 3.5], ylim=[-3.5, 3.5], aspect="equal") + >>> ax.legend() + >>> plt.show() + + We now demonstrate the removal of Froissart doublets using the `clean_up` method + using an example from [1]_. Here we approximate the function + :math:`f(z)=\log(2 + z^4)/(1 + 16z^4)` by sampling it at 1000 roots of unity. The + algorithm is run with ``rtol=0`` and ``clean_up=False`` to deliberately cause + Froissart doublets to appear. + + >>> z = np.exp(1j*2*np.pi*np.linspace(0,1, num=1000)) + >>> def f(z): + ... return np.log(2 + z**4)/(1 - 16*z**4) + >>> with warnings.catch_warnings(): # filter convergence warning due to rtol=0 + ... warnings.simplefilter('ignore', RuntimeWarning) + ... r = AAA(z, f(z), rtol=0, max_terms=50, clean_up=False) + >>> mask = np.abs(r.residues()) < 1e-13 + >>> fig, axs = plt.subplots(ncols=2) + >>> axs[0].plot(r.poles().real[~mask], r.poles().imag[~mask], '.') + >>> axs[0].plot(r.poles().real[mask], r.poles().imag[mask], 'r.') + + Now we call the `clean_up` method to remove Froissart doublets. + + >>> with warnings.catch_warnings(): + ... warnings.simplefilter('ignore', RuntimeWarning) + ... r.clean_up() + 4 + >>> mask = np.abs(r.residues()) < 1e-13 + >>> axs[1].plot(r.poles().real[~mask], r.poles().imag[~mask], '.') + >>> axs[1].plot(r.poles().real[mask], r.poles().imag[mask], 'r.') + >>> plt.show() + + The left image shows the poles prior of the approximation ``clean_up=False`` with + poles with residue less than ``10^-13`` in absolute value shown in red. The right + image then shows the poles after the `clean_up` method has been called. + """ + def __init__(self, x, y, *, rtol=None, max_terms=100, clean_up=True, + clean_up_tol=1e-13): + super().__init__(x, y, rtol=rtol, max_terms=max_terms) + + if clean_up: + self.clean_up(clean_up_tol) + + def _input_validation(self, x, y, rtol=None, max_terms=100, clean_up=True, + clean_up_tol=1e-13): + max_terms = operator.index(max_terms) + if max_terms < 1: + raise ValueError("`max_terms` must be an integer value greater than or " + "equal to one.") + + if y.ndim != 1: + raise ValueError("`y` must be 1-D.") + + super()._input_validation(x, y) + + @property + def support_points(self): + return self._support_points + + @property + def support_values(self): + return self._support_values + + def _compute_weights(self, z, f, rtol, max_terms): + # Initialization for AAA iteration + M = np.size(z) + mask = np.ones(M, dtype=np.bool_) + dtype = np.result_type(z, f, 1.0) + rtol = np.finfo(dtype).eps**0.75 if rtol is None else rtol + atol = rtol * np.linalg.norm(f, ord=np.inf) + zj = np.empty(max_terms, dtype=dtype) + fj = np.empty(max_terms, dtype=dtype) + # Cauchy matrix + C = np.empty((M, max_terms), dtype=dtype) + # Loewner matrix + A = np.empty((M, max_terms), dtype=dtype) + errors = np.empty(max_terms, dtype=A.real.dtype) + R = np.repeat(np.mean(f), M) + + # AAA iteration + for m in range(max_terms): + # Introduce next support point + # Select next support point + jj = np.argmax(np.abs(f[mask] - R[mask])) + # Update support points + zj[m] = z[mask][jj] + # Update data values + fj[m] = f[mask][jj] + # Next column of Cauchy matrix + # Ignore errors as we manually interpolate at support points + with np.errstate(divide="ignore", invalid="ignore"): + C[:, m] = 1 / (z - z[mask][jj]) + # Update mask + mask[np.nonzero(mask)[0][jj]] = False + # Update Loewner matrix + # Ignore errors as inf values will be masked out in SVD call + with np.errstate(invalid="ignore"): + A[:, m] = (f - fj[m]) * C[:, m] + + # Compute weights + rows = mask.sum() + if rows >= m + 1: + # The usual tall-skinny case + _, s, V = scipy.linalg.svd( + A[mask, : m + 1], full_matrices=False, check_finite=False, + ) + # Treat case of multiple min singular values + mm = s == np.min(s) + # Aim for non-sparse weight vector + wj = (V.conj()[mm, :].sum(axis=0) / np.sqrt(mm.sum())).astype(dtype) + else: + # Fewer rows than columns + V = scipy.linalg.null_space(A[mask, : m + 1], check_finite=False) + nm = V.shape[-1] + # Aim for non-sparse wt vector + wj = V.sum(axis=-1) / np.sqrt(nm) + + # Compute rational approximant + # Omit columns with `wj == 0` + i0 = wj != 0 + # Ignore errors as we manually interpolate at support points + with np.errstate(invalid="ignore"): + # Numerator + N = C[:, : m + 1][:, i0] @ (wj[i0] * fj[: m + 1][i0]) + # Denominator + D = C[:, : m + 1][:, i0] @ wj[i0] + # Interpolate at support points with `wj !=0` + D_inf = np.isinf(D) | np.isnan(D) + D[D_inf] = 1 + N[D_inf] = f[D_inf] + R = N / D + + # Check if converged + max_error = np.linalg.norm(f - R, ord=np.inf) + errors[m] = max_error + if max_error <= atol: + break + + if m == max_terms - 1: + warnings.warn(f"AAA failed to converge within {max_terms} iterations.", + RuntimeWarning, stacklevel=2) + + # Trim off unused array allocation + zj = zj[: m + 1] + fj = fj[: m + 1] + + # Remove support points with zero weight + i_non_zero = wj != 0 + self.errors = errors[: m + 1] + self._points = z + self._values = f + return zj[i_non_zero], fj[i_non_zero], wj[i_non_zero] + + def clean_up(self, cleanup_tol=1e-13): + """Automatic removal of Froissart doublets. + + Parameters + ---------- + cleanup_tol : float, optional + Poles with residues less than this number times the geometric mean + of `values` times the minimum distance to `points` are deemed spurious by + the cleanup procedure, defaults to 1e-13. + + Returns + ------- + int + Number of Froissart doublets detected + """ + # Find negligible residues + geom_mean_abs_f = scipy.stats.gmean(np.abs(self._values)) + + Z_distances = np.min( + np.abs(np.subtract.outer(self.poles(), self._points)), axis=1 + ) + + with np.errstate(divide="ignore", invalid="ignore"): + ii = np.nonzero( + np.abs(self.residues()) / Z_distances < cleanup_tol * geom_mean_abs_f + ) + + ni = ii[0].size + if ni == 0: + return ni + + warnings.warn(f"{ni} Froissart doublets detected.", RuntimeWarning, + stacklevel=2) + + # For each spurious pole find and remove closest support point + closest_spt_point = np.argmin( + np.abs(np.subtract.outer(self._support_points, self.poles()[ii])), axis=0 + ) + self._support_points = np.delete(self._support_points, closest_spt_point) + self._support_values = np.delete(self._support_values, closest_spt_point) + + # Remove support points z from sample set + mask = np.logical_and.reduce( + np.not_equal.outer(self._points, self._support_points), axis=1 + ) + f = self._values[mask] + z = self._points[mask] + + # recompute weights, we resolve the least squares problem for the remaining + # support points + + m = self._support_points.size + + # Cauchy matrix + C = 1 / np.subtract.outer(z, self._support_points) + # Loewner matrix + A = f[:, np.newaxis] * C - C * self._support_values + + # Solve least-squares problem to obtain weights + _, _, V = scipy.linalg.svd(A, check_finite=False) + self.weights = np.conj(V[m - 1,:]) + + # reset roots, poles, residues as cached values will be wrong with new weights + self._poles = None + self._residues = None + self._roots = None + + return ni + + +class FloaterHormannInterpolator(_BarycentricRational): + r""" + Floater-Hormann barycentric rational interpolation. + + As described in [1]_, the method of Floater and Hormann computes weights for a + Barycentric rational interpolant with no poles on the real axis. + + Parameters + ---------- + x : 1D array_like, shape (n,) + 1-D array containing values of the independent variable. Values may be real or + complex but must be finite. + y : array_like, shape (n, ...) + Array containing values of the dependent variable. Infinite and NaN values + of `values` and corresponding values of `x` will be discarded. + d : int, optional + Blends ``n - d`` degree `d` polynomials together. For ``d = n - 1`` it is + equivalent to polynomial interpolation. Must satisfy ``0 <= d < n``, + defaults to 3. + + Attributes + ---------- + weights : array + Weights of the barycentric approximation. + + See Also + -------- + AAA : Barycentric rational approximation of real and complex functions. + pade : Padé approximation. + + Notes + ----- + The Floater-Hormann interpolant is a rational function that interpolates the data + with approximation order :math:`O(h^{d+1})`. The rational function blends ``n - d`` + polynomials of degree `d` together to produce a rational interpolant that contains + no poles on the real axis, unlike `AAA`. The interpolant is given + by + + .. math:: + + r(x) = \frac{\sum_{i=0}^{n-d} \lambda_i(x) p_i(x)} + {\sum_{i=0}^{n-d} \lambda_i(x)}, + + where :math:`p_i(x)` is an interpolating polynomials of at most degree `d` through + the points :math:`(x_i,y_i),\dots,(x_{i+d},y_{i+d}), and :math:`\lambda_i(z)` are + blending functions defined by + + .. math:: + + \lambda_i(x) = \frac{(-1)^i}{(x - x_i)\cdots(x - x_{i+d})}. + + When ``d = n - 1`` this reduces to polynomial interpolation. + + Due to its stability following barycentric representation of the above equation + is used instead for computation + + .. math:: + + r(z) = \frac{\sum_{k=1}^m\ w_k f_k / (x - x_k)}{\sum_{k=1}^m w_k / (x - x_k)}, + + where the weights :math:`w_j` are computed as + + .. math:: + + w_k &= (-1)^{k - d} \sum_{i \in J_k} \prod_{j = i, j \neq k}^{i + d} + 1/|x_k - x_j|, \\ + J_k &= \{ i \in I: k - d \leq i \leq k\},\\ + I &= \{0, 1, \dots, n - d\}. + + References + ---------- + .. [1] M.S. Floater and K. Hormann, "Barycentric rational interpolation with no + poles and high rates of approximation", Numer. Math. 107, 315 (2007). + :doi:`10.1007/s00211-007-0093-y` + + Examples + -------- + + Here we compare the method against polynomial interpolation for an example where + the polynomial interpolation fails due to Runge's phenomenon. + + >>> import numpy as np + >>> from scipy.interpolate import (FloaterHormannInterpolator, + ... BarycentricInterpolator) + >>> def f(z): + ... return 1/(1 + z**2) + >>> z = np.linspace(-5, 5, num=15) + >>> r = FloaterHormannInterpolator(z, f(z)) + >>> p = BarycentricInterpolator(z, f(z)) + >>> zz = np.linspace(-5, 5, num=1000) + >>> import matplotlib.pyplot as plt + >>> fig, ax = plt.subplots() + >>> ax.plot(zz, r(zz), label="Floater=Hormann") + >>> ax.plot(zz, p(zz), label="Polynomial") + >>> ax.legend() + >>> plt.show() + """ + def __init__(self, points, values, *, d=3): + super().__init__(points, values, d=d) + + def _input_validation(self, x, y, d): + d = operator.index(d) + if not (0 <= d < len(x)): + raise ValueError("`d` must satisfy 0 <= d < n") + + super()._input_validation(x, y) + + def _compute_weights(self, z, f, d): + # Floater and Hormann 2007 Eqn. (18) 3 equations later + w = np.zeros_like(z, dtype=np.result_type(z, 1.0)) + n = w.size + for k in range(n): + for i in range(max(k-d, 0), min(k+1, n-d)): + w[k] += 1/np.prod(np.abs(np.delete(z[k] - z[i : i + d + 1], k - i))) + w *= (-1.)**(np.arange(n) - d) + + return z, f, w diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_bsplines.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_bsplines.py new file mode 100644 index 0000000000000000000000000000000000000000..3d68e8d532100f4926328d68cb68d1048f4290e8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_bsplines.py @@ -0,0 +1,2416 @@ +import operator +from math import prod + +import numpy as np +from scipy._lib._util import normalize_axis_index +from scipy.linalg import (get_lapack_funcs, LinAlgError, + cholesky_banded, cho_solve_banded, + solve, solve_banded) +from scipy.optimize import minimize_scalar +from . import _dierckx +from . import _fitpack_impl +from scipy.sparse import csr_array +from scipy.special import poch +from itertools import combinations + + +__all__ = ["BSpline", "make_interp_spline", "make_lsq_spline", + "make_smoothing_spline"] + + +def _get_dtype(dtype): + """Return np.complex128 for complex dtypes, np.float64 otherwise.""" + if np.issubdtype(dtype, np.complexfloating): + return np.complex128 + else: + return np.float64 + + +def _as_float_array(x, check_finite=False): + """Convert the input into a C contiguous float array. + + NB: Upcasts half- and single-precision floats to double precision. + """ + x = np.ascontiguousarray(x) + dtyp = _get_dtype(x.dtype) + x = x.astype(dtyp, copy=False) + if check_finite and not np.isfinite(x).all(): + raise ValueError("Array must not contain infs or nans.") + return x + + +def _dual_poly(j, k, t, y): + """ + Dual polynomial of the B-spline B_{j,k,t} - + polynomial which is associated with B_{j,k,t}: + $p_{j,k}(y) = (y - t_{j+1})(y - t_{j+2})...(y - t_{j+k})$ + """ + if k == 0: + return 1 + return np.prod([(y - t[j + i]) for i in range(1, k + 1)]) + + +def _diff_dual_poly(j, k, y, d, t): + """ + d-th derivative of the dual polynomial $p_{j,k}(y)$ + """ + if d == 0: + return _dual_poly(j, k, t, y) + if d == k: + return poch(1, k) + comb = list(combinations(range(j + 1, j + k + 1), d)) + res = 0 + for i in range(len(comb) * len(comb[0])): + res += np.prod([(y - t[j + p]) for p in range(1, k + 1) + if (j + p) not in comb[i//d]]) + return res + + +class BSpline: + r"""Univariate spline in the B-spline basis. + + .. math:: + + S(x) = \sum_{j=0}^{n-1} c_j B_{j, k; t}(x) + + where :math:`B_{j, k; t}` are B-spline basis functions of degree `k` + and knots `t`. + + Parameters + ---------- + t : ndarray, shape (n+k+1,) + knots + c : ndarray, shape (>=n, ...) + spline coefficients + k : int + B-spline degree + extrapolate : bool or 'periodic', optional + whether to extrapolate beyond the base interval, ``t[k] .. t[n]``, + or to return nans. + If True, extrapolates the first and last polynomial pieces of b-spline + functions active on the base interval. + If 'periodic', periodic extrapolation is used. + Default is True. + axis : int, optional + Interpolation axis. Default is zero. + + Attributes + ---------- + t : ndarray + knot vector + c : ndarray + spline coefficients + k : int + spline degree + extrapolate : bool + If True, extrapolates the first and last polynomial pieces of b-spline + functions active on the base interval. + axis : int + Interpolation axis. + tck : tuple + A read-only equivalent of ``(self.t, self.c, self.k)`` + + Methods + ------- + __call__ + basis_element + derivative + antiderivative + integrate + insert_knot + construct_fast + design_matrix + from_power_basis + + Notes + ----- + B-spline basis elements are defined via + + .. math:: + + B_{i, 0}(x) = 1, \textrm{if $t_i \le x < t_{i+1}$, otherwise $0$,} + + B_{i, k}(x) = \frac{x - t_i}{t_{i+k} - t_i} B_{i, k-1}(x) + + \frac{t_{i+k+1} - x}{t_{i+k+1} - t_{i+1}} B_{i+1, k-1}(x) + + **Implementation details** + + - At least ``k+1`` coefficients are required for a spline of degree `k`, + so that ``n >= k+1``. Additional coefficients, ``c[j]`` with + ``j > n``, are ignored. + + - B-spline basis elements of degree `k` form a partition of unity on the + *base interval*, ``t[k] <= x <= t[n]``. + + + Examples + -------- + + Translating the recursive definition of B-splines into Python code, we have: + + >>> def B(x, k, i, t): + ... if k == 0: + ... return 1.0 if t[i] <= x < t[i+1] else 0.0 + ... if t[i+k] == t[i]: + ... c1 = 0.0 + ... else: + ... c1 = (x - t[i])/(t[i+k] - t[i]) * B(x, k-1, i, t) + ... if t[i+k+1] == t[i+1]: + ... c2 = 0.0 + ... else: + ... c2 = (t[i+k+1] - x)/(t[i+k+1] - t[i+1]) * B(x, k-1, i+1, t) + ... return c1 + c2 + + >>> def bspline(x, t, c, k): + ... n = len(t) - k - 1 + ... assert (n >= k+1) and (len(c) >= n) + ... return sum(c[i] * B(x, k, i, t) for i in range(n)) + + Note that this is an inefficient (if straightforward) way to + evaluate B-splines --- this spline class does it in an equivalent, + but much more efficient way. + + Here we construct a quadratic spline function on the base interval + ``2 <= x <= 4`` and compare with the naive way of evaluating the spline: + + >>> from scipy.interpolate import BSpline + >>> k = 2 + >>> t = [0, 1, 2, 3, 4, 5, 6] + >>> c = [-1, 2, 0, -1] + >>> spl = BSpline(t, c, k) + >>> spl(2.5) + array(1.375) + >>> bspline(2.5, t, c, k) + 1.375 + + Note that outside of the base interval results differ. This is because + `BSpline` extrapolates the first and last polynomial pieces of B-spline + functions active on the base interval. + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> fig, ax = plt.subplots() + >>> xx = np.linspace(1.5, 4.5, 50) + >>> ax.plot(xx, [bspline(x, t, c ,k) for x in xx], 'r-', lw=3, label='naive') + >>> ax.plot(xx, spl(xx), 'b-', lw=4, alpha=0.7, label='BSpline') + >>> ax.grid(True) + >>> ax.legend(loc='best') + >>> plt.show() + + + References + ---------- + .. [1] Tom Lyche and Knut Morken, Spline methods, + http://www.uio.no/studier/emner/matnat/ifi/INF-MAT5340/v05/undervisningsmateriale/ + .. [2] Carl de Boor, A practical guide to splines, Springer, 2001. + + """ + + def __init__(self, t, c, k, extrapolate=True, axis=0): + super().__init__() + + self.k = operator.index(k) + self.c = np.asarray(c) + self.t = np.ascontiguousarray(t, dtype=np.float64) + + if extrapolate == 'periodic': + self.extrapolate = extrapolate + else: + self.extrapolate = bool(extrapolate) + + n = self.t.shape[0] - self.k - 1 + + axis = normalize_axis_index(axis, self.c.ndim) + + # Note that the normalized axis is stored in the object. + self.axis = axis + if axis != 0: + # roll the interpolation axis to be the first one in self.c + # More specifically, the target shape for self.c is (n, ...), + # and axis !=0 means that we have c.shape (..., n, ...) + # ^ + # axis + self.c = np.moveaxis(self.c, axis, 0) + + if k < 0: + raise ValueError("Spline order cannot be negative.") + if self.t.ndim != 1: + raise ValueError("Knot vector must be one-dimensional.") + if n < self.k + 1: + raise ValueError("Need at least %d knots for degree %d" % + (2*k + 2, k)) + if (np.diff(self.t) < 0).any(): + raise ValueError("Knots must be in a non-decreasing order.") + if len(np.unique(self.t[k:n+1])) < 2: + raise ValueError("Need at least two internal knots.") + if not np.isfinite(self.t).all(): + raise ValueError("Knots should not have nans or infs.") + if self.c.ndim < 1: + raise ValueError("Coefficients must be at least 1-dimensional.") + if self.c.shape[0] < n: + raise ValueError("Knots, coefficients and degree are inconsistent.") + + dt = _get_dtype(self.c.dtype) + self.c = np.ascontiguousarray(self.c, dtype=dt) + + @classmethod + def construct_fast(cls, t, c, k, extrapolate=True, axis=0): + """Construct a spline without making checks. + + Accepts same parameters as the regular constructor. Input arrays + `t` and `c` must of correct shape and dtype. + """ + self = object.__new__(cls) + self.t, self.c, self.k = t, c, k + self.extrapolate = extrapolate + self.axis = axis + return self + + @property + def tck(self): + """Equivalent to ``(self.t, self.c, self.k)`` (read-only). + """ + return self.t, self.c, self.k + + @classmethod + def basis_element(cls, t, extrapolate=True): + """Return a B-spline basis element ``B(x | t[0], ..., t[k+1])``. + + Parameters + ---------- + t : ndarray, shape (k+2,) + internal knots + extrapolate : bool or 'periodic', optional + whether to extrapolate beyond the base interval, ``t[0] .. t[k+1]``, + or to return nans. + If 'periodic', periodic extrapolation is used. + Default is True. + + Returns + ------- + basis_element : callable + A callable representing a B-spline basis element for the knot + vector `t`. + + Notes + ----- + The degree of the B-spline, `k`, is inferred from the length of `t` as + ``len(t)-2``. The knot vector is constructed by appending and prepending + ``k+1`` elements to internal knots `t`. + + Examples + -------- + + Construct a cubic B-spline: + + >>> import numpy as np + >>> from scipy.interpolate import BSpline + >>> b = BSpline.basis_element([0, 1, 2, 3, 4]) + >>> k = b.k + >>> b.t[k:-k] + array([ 0., 1., 2., 3., 4.]) + >>> k + 3 + + Construct a quadratic B-spline on ``[0, 1, 1, 2]``, and compare + to its explicit form: + + >>> t = [0, 1, 1, 2] + >>> b = BSpline.basis_element(t) + >>> def f(x): + ... return np.where(x < 1, x*x, (2. - x)**2) + + >>> import matplotlib.pyplot as plt + >>> fig, ax = plt.subplots() + >>> x = np.linspace(0, 2, 51) + >>> ax.plot(x, b(x), 'g', lw=3) + >>> ax.plot(x, f(x), 'r', lw=8, alpha=0.4) + >>> ax.grid(True) + >>> plt.show() + + """ + k = len(t) - 2 + t = _as_float_array(t) + t = np.r_[(t[0]-1,) * k, t, (t[-1]+1,) * k] + c = np.zeros_like(t) + c[k] = 1. + return cls.construct_fast(t, c, k, extrapolate) + + @classmethod + def design_matrix(cls, x, t, k, extrapolate=False): + """ + Returns a design matrix as a CSR format sparse array. + + Parameters + ---------- + x : array_like, shape (n,) + Points to evaluate the spline at. + t : array_like, shape (nt,) + Sorted 1D array of knots. + k : int + B-spline degree. + extrapolate : bool or 'periodic', optional + Whether to extrapolate based on the first and last intervals + or raise an error. If 'periodic', periodic extrapolation is used. + Default is False. + + .. versionadded:: 1.10.0 + + Returns + ------- + design_matrix : `csr_array` object + Sparse matrix in CSR format where each row contains all the basis + elements of the input row (first row = basis elements of x[0], + ..., last row = basis elements x[-1]). + + Examples + -------- + Construct a design matrix for a B-spline + + >>> from scipy.interpolate import make_interp_spline, BSpline + >>> import numpy as np + >>> x = np.linspace(0, np.pi * 2, 4) + >>> y = np.sin(x) + >>> k = 3 + >>> bspl = make_interp_spline(x, y, k=k) + >>> design_matrix = bspl.design_matrix(x, bspl.t, k) + >>> design_matrix.toarray() + [[1. , 0. , 0. , 0. ], + [0.2962963 , 0.44444444, 0.22222222, 0.03703704], + [0.03703704, 0.22222222, 0.44444444, 0.2962963 ], + [0. , 0. , 0. , 1. ]] + + Construct a design matrix for some vector of knots + + >>> k = 2 + >>> t = [-1, 0, 1, 2, 3, 4, 5, 6] + >>> x = [1, 2, 3, 4] + >>> design_matrix = BSpline.design_matrix(x, t, k).toarray() + >>> design_matrix + [[0.5, 0.5, 0. , 0. , 0. ], + [0. , 0.5, 0.5, 0. , 0. ], + [0. , 0. , 0.5, 0.5, 0. ], + [0. , 0. , 0. , 0.5, 0.5]] + + This result is equivalent to the one created in the sparse format + + >>> c = np.eye(len(t) - k - 1) + >>> design_matrix_gh = BSpline(t, c, k)(x) + >>> np.allclose(design_matrix, design_matrix_gh, atol=1e-14) + True + + Notes + ----- + .. versionadded:: 1.8.0 + + In each row of the design matrix all the basis elements are evaluated + at the certain point (first row - x[0], ..., last row - x[-1]). + + `nt` is a length of the vector of knots: as far as there are + `nt - k - 1` basis elements, `nt` should be not less than `2 * k + 2` + to have at least `k + 1` basis element. + + Out of bounds `x` raises a ValueError. + """ + x = _as_float_array(x, True) + t = _as_float_array(t, True) + + if extrapolate != 'periodic': + extrapolate = bool(extrapolate) + + if k < 0: + raise ValueError("Spline order cannot be negative.") + if t.ndim != 1 or np.any(t[1:] < t[:-1]): + raise ValueError(f"Expect t to be a 1-D sorted array_like, but " + f"got t={t}.") + # There are `nt - k - 1` basis elements in a BSpline built on the + # vector of knots with length `nt`, so to have at least `k + 1` basis + # elements we need to have at least `2 * k + 2` elements in the vector + # of knots. + if len(t) < 2 * k + 2: + raise ValueError(f"Length t is not enough for k={k}.") + + if extrapolate == 'periodic': + # With periodic extrapolation we map x to the segment + # [t[k], t[n]]. + n = t.size - k - 1 + x = t[k] + (x - t[k]) % (t[n] - t[k]) + extrapolate = False + elif not extrapolate and ( + (min(x) < t[k]) or (max(x) > t[t.shape[0] - k - 1]) + ): + # Checks from `find_interval` function + raise ValueError(f'Out of bounds w/ x = {x}.') + + # Compute number of non-zeros of final CSR array in order to determine + # the dtype of indices and indptr of the CSR array. + n = x.shape[0] + nnz = n * (k + 1) + if nnz < np.iinfo(np.int32).max: + int_dtype = np.int32 + else: + int_dtype = np.int64 + + # Get the non-zero elements of the design matrix and per-row `offsets`: + # In row `i`, k+1 nonzero elements are consecutive, and start from `offset[i]` + data, offsets, _ = _dierckx.data_matrix(x, t, k, np.ones_like(x), extrapolate) + data = data.ravel() + + if offsets.dtype != int_dtype: + offsets = offsets.astype(int_dtype) + + # Convert from per-row offsets to the CSR indices/indptr format + indices = np.repeat(offsets, k+1).reshape(-1, k+1) + indices = indices + np.arange(k+1, dtype=int_dtype) + indices = indices.ravel() + + indptr = np.arange(0, (n + 1) * (k + 1), k + 1, dtype=int_dtype) + + return csr_array( + (data, indices, indptr), + shape=(x.shape[0], t.shape[0] - k - 1) + ) + + def __call__(self, x, nu=0, extrapolate=None): + """ + Evaluate a spline function. + + Parameters + ---------- + x : array_like + points to evaluate the spline at. + nu : int, optional + derivative to evaluate (default is 0). + extrapolate : bool or 'periodic', optional + whether to extrapolate based on the first and last intervals + or return nans. If 'periodic', periodic extrapolation is used. + Default is `self.extrapolate`. + + Returns + ------- + y : array_like + Shape is determined by replacing the interpolation axis + in the coefficient array with the shape of `x`. + + """ + if extrapolate is None: + extrapolate = self.extrapolate + x = np.asarray(x) + x_shape, x_ndim = x.shape, x.ndim + x = np.ascontiguousarray(x.ravel(), dtype=np.float64) + + # With periodic extrapolation we map x to the segment + # [self.t[k], self.t[n]]. + if extrapolate == 'periodic': + n = self.t.size - self.k - 1 + x = self.t[self.k] + (x - self.t[self.k]) % (self.t[n] - + self.t[self.k]) + extrapolate = False + + out = np.empty((len(x), prod(self.c.shape[1:])), dtype=self.c.dtype) + self._ensure_c_contiguous() + + # if self.c is complex, so is `out`; cython code in _bspl.pyx expectes + # floats though, so make a view---this expands the last axis, and + # the view is C contiguous if the original is. + # if c.dtype is complex of shape (n,), c.view(float).shape == (2*n,) + # if c.dtype is complex of shape (n, m), c.view(float).shape == (n, 2*m) + + cc = self.c.view(float) + if self.c.ndim == 1 and self.c.dtype.kind == 'c': + cc = cc.reshape(self.c.shape[0], 2) + + _dierckx.evaluate_spline(self.t, cc.reshape(cc.shape[0], -1), + self.k, x, nu, extrapolate, out.view(float)) + + out = out.reshape(x_shape + self.c.shape[1:]) + if self.axis != 0: + # transpose to move the calculated values to the interpolation axis + l = list(range(out.ndim)) + l = l[x_ndim:x_ndim+self.axis] + l[:x_ndim] + l[x_ndim+self.axis:] + out = out.transpose(l) + return out + + def _ensure_c_contiguous(self): + """ + c and t may be modified by the user. The Cython code expects + that they are C contiguous. + + """ + if not self.t.flags.c_contiguous: + self.t = self.t.copy() + if not self.c.flags.c_contiguous: + self.c = self.c.copy() + + def derivative(self, nu=1): + """Return a B-spline representing the derivative. + + Parameters + ---------- + nu : int, optional + Derivative order. + Default is 1. + + Returns + ------- + b : BSpline object + A new instance representing the derivative. + + See Also + -------- + splder, splantider + + """ + c = self.c.copy() + # pad the c array if needed + ct = len(self.t) - len(c) + if ct > 0: + c = np.r_[c, np.zeros((ct,) + c.shape[1:])] + tck = _fitpack_impl.splder((self.t, c, self.k), nu) + return self.construct_fast(*tck, extrapolate=self.extrapolate, + axis=self.axis) + + def antiderivative(self, nu=1): + """Return a B-spline representing the antiderivative. + + Parameters + ---------- + nu : int, optional + Antiderivative order. Default is 1. + + Returns + ------- + b : BSpline object + A new instance representing the antiderivative. + + Notes + ----- + If antiderivative is computed and ``self.extrapolate='periodic'``, + it will be set to False for the returned instance. This is done because + the antiderivative is no longer periodic and its correct evaluation + outside of the initially given x interval is difficult. + + See Also + -------- + splder, splantider + + """ + c = self.c.copy() + # pad the c array if needed + ct = len(self.t) - len(c) + if ct > 0: + c = np.r_[c, np.zeros((ct,) + c.shape[1:])] + tck = _fitpack_impl.splantider((self.t, c, self.k), nu) + + if self.extrapolate == 'periodic': + extrapolate = False + else: + extrapolate = self.extrapolate + + return self.construct_fast(*tck, extrapolate=extrapolate, + axis=self.axis) + + def integrate(self, a, b, extrapolate=None): + """Compute a definite integral of the spline. + + Parameters + ---------- + a : float + Lower limit of integration. + b : float + Upper limit of integration. + extrapolate : bool or 'periodic', optional + whether to extrapolate beyond the base interval, + ``t[k] .. t[-k-1]``, or take the spline to be zero outside of the + base interval. If 'periodic', periodic extrapolation is used. + If None (default), use `self.extrapolate`. + + Returns + ------- + I : array_like + Definite integral of the spline over the interval ``[a, b]``. + + Examples + -------- + Construct the linear spline ``x if x < 1 else 2 - x`` on the base + interval :math:`[0, 2]`, and integrate it + + >>> from scipy.interpolate import BSpline + >>> b = BSpline.basis_element([0, 1, 2]) + >>> b.integrate(0, 1) + array(0.5) + + If the integration limits are outside of the base interval, the result + is controlled by the `extrapolate` parameter + + >>> b.integrate(-1, 1) + array(0.0) + >>> b.integrate(-1, 1, extrapolate=False) + array(0.5) + + >>> import matplotlib.pyplot as plt + >>> fig, ax = plt.subplots() + >>> ax.grid(True) + >>> ax.axvline(0, c='r', lw=5, alpha=0.5) # base interval + >>> ax.axvline(2, c='r', lw=5, alpha=0.5) + >>> xx = [-1, 1, 2] + >>> ax.plot(xx, b(xx)) + >>> plt.show() + + """ + if extrapolate is None: + extrapolate = self.extrapolate + + # Prepare self.t and self.c. + self._ensure_c_contiguous() + + # Swap integration bounds if needed. + sign = 1 + if b < a: + a, b = b, a + sign = -1 + n = self.t.size - self.k - 1 + + if extrapolate != "periodic" and not extrapolate: + # Shrink the integration interval, if needed. + a = max(a, self.t[self.k]) + b = min(b, self.t[n]) + + if self.c.ndim == 1: + # Fast path: use FITPACK's routine + # (cf _fitpack_impl.splint). + integral = _fitpack_impl.splint(a, b, self.tck) + return np.asarray(integral * sign) + + out = np.empty((2, prod(self.c.shape[1:])), dtype=self.c.dtype) + + # Compute the antiderivative. + c = self.c + ct = len(self.t) - len(c) + if ct > 0: + c = np.r_[c, np.zeros((ct,) + c.shape[1:])] + ta, ca, ka = _fitpack_impl.splantider((self.t, c, self.k), 1) + + if extrapolate == 'periodic': + # Split the integral into the part over period (can be several + # of them) and the remaining part. + + ts, te = self.t[self.k], self.t[n] + period = te - ts + interval = b - a + n_periods, left = divmod(interval, period) + + if n_periods > 0: + # Evaluate the difference of antiderivatives. + x = np.asarray([ts, te], dtype=np.float64) + _dierckx.evaluate_spline(ta, ca.reshape(ca.shape[0], -1), + ka, x, 0, False, out) + integral = out[1] - out[0] + integral *= n_periods + else: + integral = np.zeros((1, prod(self.c.shape[1:])), + dtype=self.c.dtype) + + # Map a to [ts, te], b is always a + left. + a = ts + (a - ts) % period + b = a + left + + # If b <= te then we need to integrate over [a, b], otherwise + # over [a, te] and from xs to what is remained. + if b <= te: + x = np.asarray([a, b], dtype=np.float64) + _dierckx.evaluate_spline(ta, ca.reshape(ca.shape[0], -1), + ka, x, 0, False, out) + integral += out[1] - out[0] + else: + x = np.asarray([a, te], dtype=np.float64) + _dierckx.evaluate_spline(ta, ca.reshape(ca.shape[0], -1), + ka, x, 0, False, out) + integral += out[1] - out[0] + + x = np.asarray([ts, ts + b - te], dtype=np.float64) + _dierckx.evaluate_spline(ta, ca.reshape(ca.shape[0], -1), + ka, x, 0, False, out) + integral += out[1] - out[0] + else: + # Evaluate the difference of antiderivatives. + x = np.asarray([a, b], dtype=np.float64) + _dierckx.evaluate_spline(ta, ca.reshape(ca.shape[0], -1), + ka, x, 0, extrapolate, out) + integral = out[1] - out[0] + + integral *= sign + return integral.reshape(ca.shape[1:]) + + @classmethod + def from_power_basis(cls, pp, bc_type='not-a-knot'): + r""" + Construct a polynomial in the B-spline basis + from a piecewise polynomial in the power basis. + + For now, accepts ``CubicSpline`` instances only. + + Parameters + ---------- + pp : CubicSpline + A piecewise polynomial in the power basis, as created + by ``CubicSpline`` + bc_type : string, optional + Boundary condition type as in ``CubicSpline``: one of the + ``not-a-knot``, ``natural``, ``clamped``, or ``periodic``. + Necessary for construction an instance of ``BSpline`` class. + Default is ``not-a-knot``. + + Returns + ------- + b : BSpline object + A new instance representing the initial polynomial + in the B-spline basis. + + Notes + ----- + .. versionadded:: 1.8.0 + + Accepts only ``CubicSpline`` instances for now. + + The algorithm follows from differentiation + the Marsden's identity [1]: each of coefficients of spline + interpolation function in the B-spline basis is computed as follows: + + .. math:: + + c_j = \sum_{m=0}^{k} \frac{(k-m)!}{k!} + c_{m,i} (-1)^{k-m} D^m p_{j,k}(x_i) + + :math:`c_{m, i}` - a coefficient of CubicSpline, + :math:`D^m p_{j, k}(x_i)` - an m-th defivative of a dual polynomial + in :math:`x_i`. + + ``k`` always equals 3 for now. + + First ``n - 2`` coefficients are computed in :math:`x_i = x_j`, e.g. + + .. math:: + + c_1 = \sum_{m=0}^{k} \frac{(k-1)!}{k!} c_{m,1} D^m p_{j,3}(x_1) + + Last ``nod + 2`` coefficients are computed in ``x[-2]``, + ``nod`` - number of derivatives at the ends. + + For example, consider :math:`x = [0, 1, 2, 3, 4]`, + :math:`y = [1, 1, 1, 1, 1]` and bc_type = ``natural`` + + The coefficients of CubicSpline in the power basis: + + :math:`[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], [1, 1, 1, 1, 1]]` + + The knot vector: :math:`t = [0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4]` + + In this case + + .. math:: + + c_j = \frac{0!}{k!} c_{3, i} k! = c_{3, i} = 1,~j = 0, ..., 6 + + References + ---------- + .. [1] Tom Lyche and Knut Morken, Spline Methods, 2005, Section 3.1.2 + + """ + from ._cubic import CubicSpline + if not isinstance(pp, CubicSpline): + raise NotImplementedError(f"Only CubicSpline objects are accepted " + f"for now. Got {type(pp)} instead.") + x = pp.x + coef = pp.c + k = pp.c.shape[0] - 1 + n = x.shape[0] + + if bc_type == 'not-a-knot': + t = _not_a_knot(x, k) + elif bc_type == 'natural' or bc_type == 'clamped': + t = _augknt(x, k) + elif bc_type == 'periodic': + t = _periodic_knots(x, k) + else: + raise TypeError(f'Unknown boundary condition: {bc_type}') + + nod = t.shape[0] - (n + k + 1) # number of derivatives at the ends + c = np.zeros(n + nod, dtype=pp.c.dtype) + for m in range(k + 1): + for i in range(n - 2): + c[i] += poch(k + 1, -m) * coef[m, i]\ + * np.power(-1, k - m)\ + * _diff_dual_poly(i, k, x[i], m, t) + for j in range(n - 2, n + nod): + c[j] += poch(k + 1, -m) * coef[m, n - 2]\ + * np.power(-1, k - m)\ + * _diff_dual_poly(j, k, x[n - 2], m, t) + return cls.construct_fast(t, c, k, pp.extrapolate, pp.axis) + + def insert_knot(self, x, m=1): + """Insert a new knot at `x` of multiplicity `m`. + + Given the knots and coefficients of a B-spline representation, create a + new B-spline with a knot inserted `m` times at point `x`. + + Parameters + ---------- + x : float + The position of the new knot + m : int, optional + The number of times to insert the given knot (its multiplicity). + Default is 1. + + Returns + ------- + spl : BSpline object + A new BSpline object with the new knot inserted. + + Notes + ----- + Based on algorithms from [1]_ and [2]_. + + In case of a periodic spline (``self.extrapolate == "periodic"``) + there must be either at least k interior knots t(j) satisfying + ``t(k+1)>> import numpy as np + >>> from scipy.interpolate import BSpline, make_interp_spline + >>> x = np.linspace(0, 10, 5) + >>> y = np.sin(x) + >>> spl = make_interp_spline(x, y, k=3) + >>> spl.t + array([ 0., 0., 0., 0., 5., 10., 10., 10., 10.]) + + Insert a single knot + + >>> spl_1 = spl.insert_knot(3) + >>> spl_1.t + array([ 0., 0., 0., 0., 3., 5., 10., 10., 10., 10.]) + + Insert a multiple knot + + >>> spl_2 = spl.insert_knot(8, m=3) + >>> spl_2.t + array([ 0., 0., 0., 0., 5., 8., 8., 8., 10., 10., 10., 10.]) + + """ + if x < self.t[self.k] or x > self.t[-self.k-1]: + raise ValueError(f"Cannot insert a knot at {x}.") + if m <= 0: + raise ValueError(f"`m` must be positive, got {m = }.") + + tt = self.t.copy() + cc = self.c.copy() + + for _ in range(m): + tt, cc = _insert(x, tt, cc, self.k, self.extrapolate == "periodic") + return self.construct_fast(tt, cc, self.k, self.extrapolate, self.axis) + + +def _insert(xval, t, c, k, periodic=False): + """Insert a single knot at `xval`.""" + # + # This is a port of the FORTRAN `insert` routine by P. Dierckx, + # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/insert.f + # which carries the following comment: + # + # subroutine insert inserts a new knot x into a spline function s(x) + # of degree k and calculates the b-spline representation of s(x) with + # respect to the new set of knots. in addition, if iopt.ne.0, s(x) + # will be considered as a periodic spline with period per=t(n-k)-t(k+1) + # satisfying the boundary constraints + # t(i+n-2*k-1) = t(i)+per ,i=1,2,...,2*k+1 + # c(i+n-2*k-1) = c(i) ,i=1,2,...,k + # in that case, the knots and b-spline coefficients returned will also + # satisfy these boundary constraints, i.e. + # tt(i+nn-2*k-1) = tt(i)+per ,i=1,2,...,2*k+1 + # cc(i+nn-2*k-1) = cc(i) ,i=1,2,...,k + interval = _dierckx.find_interval(t, k, float(xval), k, False) + if interval < 0: + # extrapolated values are guarded for in BSpline.insert_knot + raise ValueError(f"Cannot insert the knot at {xval}.") + + # super edge case: a knot with multiplicity > k+1 + # see https://github.com/scipy/scipy/commit/037204c3e91 + if t[interval] == t[interval + k + 1]: + interval -= 1 + + if periodic: + if (interval + 1 <= 2*k) and (interval + 1 >= t.shape[0] - 2*k): + # in case of a periodic spline (iopt.ne.0) there must be + # either at least k interior knots t(j) satisfying t(k+1)= nk - k: + # adjust the left-hand boundary knots & coefs + tt[:k] = tt[nk - k:nk] - T + cc[:k, ...] = cc[n2k:n2k + k, ...] + + if interval <= 2*k-1: + # adjust the right-hand boundary knots & coefs + tt[n-k:] = tt[k+1:k+1+k] + T + cc[n2k:n2k + k, ...] = cc[:k, ...] + + return tt, cc + + +################################# +# Interpolating spline helpers # +################################# + +def _not_a_knot(x, k): + """Given data x, construct the knot vector w/ not-a-knot BC. + cf de Boor, XIII(12). + + For even k, it's a bit ad hoc: Greville sites + omit 2nd and 2nd-to-last + data points, a la not-a-knot. + This seems to match what Dierckx does, too: + https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L63-L80 + """ + x = np.asarray(x) + if k % 2 == 1: + k2 = (k + 1) // 2 + t = x.copy() + else: + k2 = k // 2 + t = (x[1:] + x[:-1]) / 2 + + t = t[k2:-k2] + t = np.r_[(x[0],)*(k+1), t, (x[-1],)*(k+1)] + return t + + +def _augknt(x, k): + """Construct a knot vector appropriate for the order-k interpolation.""" + return np.r_[(x[0],)*k, x, (x[-1],)*k] + + +def _convert_string_aliases(deriv, target_shape): + if isinstance(deriv, str): + if deriv == "clamped": + deriv = [(1, np.zeros(target_shape))] + elif deriv == "natural": + deriv = [(2, np.zeros(target_shape))] + else: + raise ValueError(f"Unknown boundary condition : {deriv}") + return deriv + + +def _process_deriv_spec(deriv): + if deriv is not None: + try: + ords, vals = zip(*deriv) + except TypeError as e: + msg = ("Derivatives, `bc_type`, should be specified as a pair of " + "iterables of pairs of (order, value).") + raise ValueError(msg) from e + else: + ords, vals = [], [] + return np.atleast_1d(ords, vals) + + +def _woodbury_algorithm(A, ur, ll, b, k): + ''' + Solve a cyclic banded linear system with upper right + and lower blocks of size ``(k-1) / 2`` using + the Woodbury formula + + Parameters + ---------- + A : 2-D array, shape(k, n) + Matrix of diagonals of original matrix (see + ``solve_banded`` documentation). + ur : 2-D array, shape(bs, bs) + Upper right block matrix. + ll : 2-D array, shape(bs, bs) + Lower left block matrix. + b : 1-D array, shape(n,) + Vector of constant terms of the system of linear equations. + k : int + B-spline degree. + + Returns + ------- + c : 1-D array, shape(n,) + Solution of the original system of linear equations. + + Notes + ----- + This algorithm works only for systems with banded matrix A plus + a correction term U @ V.T, where the matrix U @ V.T gives upper right + and lower left block of A + The system is solved with the following steps: + 1. New systems of linear equations are constructed: + A @ z_i = u_i, + u_i - column vector of U, + i = 1, ..., k - 1 + 2. Matrix Z is formed from vectors z_i: + Z = [ z_1 | z_2 | ... | z_{k - 1} ] + 3. Matrix H = (1 + V.T @ Z)^{-1} + 4. The system A' @ y = b is solved + 5. x = y - Z @ (H @ V.T @ y) + Also, ``n`` should be greater than ``k``, otherwise corner block + elements will intersect with diagonals. + + Examples + -------- + Consider the case of n = 8, k = 5 (size of blocks - 2 x 2). + The matrix of a system: U: V: + x x x * * a b a b 0 0 0 0 1 0 + x x x x * * c 0 c 0 0 0 0 0 1 + x x x x x * * 0 0 0 0 0 0 0 0 + * x x x x x * 0 0 0 0 0 0 0 0 + * * x x x x x 0 0 0 0 0 0 0 0 + d * * x x x x 0 0 d 0 1 0 0 0 + e f * * x x x 0 0 e f 0 1 0 0 + + References + ---------- + .. [1] William H. Press, Saul A. Teukolsky, William T. Vetterling + and Brian P. Flannery, Numerical Recipes, 2007, Section 2.7.3 + + ''' + k_mod = k - k % 2 + bs = int((k - 1) / 2) + (k + 1) % 2 + + n = A.shape[1] + 1 + U = np.zeros((n - 1, k_mod)) + VT = np.zeros((k_mod, n - 1)) # V transpose + + # upper right block + U[:bs, :bs] = ur + VT[np.arange(bs), np.arange(bs) - bs] = 1 + + # lower left block + U[-bs:, -bs:] = ll + VT[np.arange(bs) - bs, np.arange(bs)] = 1 + + Z = solve_banded((bs, bs), A, U) + + H = solve(np.identity(k_mod) + VT @ Z, np.identity(k_mod)) + + y = solve_banded((bs, bs), A, b) + c = y - Z @ (H @ (VT @ y)) + + return c + + +def _periodic_knots(x, k): + ''' + returns vector of nodes on circle + ''' + xc = np.copy(x) + n = len(xc) + if k % 2 == 0: + dx = np.diff(xc) + xc[1: -1] -= dx[:-1] / 2 + dx = np.diff(xc) + t = np.zeros(n + 2 * k) + t[k: -k] = xc + for i in range(0, k): + # filling first `k` elements in descending order + t[k - i - 1] = t[k - i] - dx[-(i % (n - 1)) - 1] + # filling last `k` elements in ascending order + t[-k + i] = t[-k + i - 1] + dx[i % (n - 1)] + return t + + +def _make_interp_per_full_matr(x, y, t, k): + ''' + Returns a solution of a system for B-spline interpolation with periodic + boundary conditions. First ``k - 1`` rows of matrix are conditions of + periodicity (continuity of ``k - 1`` derivatives at the boundary points). + Last ``n`` rows are interpolation conditions. + RHS is ``k - 1`` zeros and ``n`` ordinates in this case. + + Parameters + ---------- + x : 1-D array, shape (n,) + Values of x - coordinate of a given set of points. + y : 1-D array, shape (n,) + Values of y - coordinate of a given set of points. + t : 1-D array, shape(n+2*k,) + Vector of knots. + k : int + The maximum degree of spline + + Returns + ------- + c : 1-D array, shape (n+k-1,) + B-spline coefficients + + Notes + ----- + ``t`` is supposed to be taken on circle. + + ''' + + x, y, t = map(np.asarray, (x, y, t)) + + n = x.size + # LHS: the colocation matrix + derivatives at edges + matr = np.zeros((n + k - 1, n + k - 1)) + + # derivatives at x[0] and x[-1]: + for i in range(k - 1): + bb = _dierckx.evaluate_all_bspl(t, k, x[0], k, i + 1) + matr[i, : k + 1] += bb + bb = _dierckx.evaluate_all_bspl(t, k, x[-1], n + k - 1, i + 1)[:-1] + matr[i, -k:] -= bb + + # colocation matrix + for i in range(n): + xval = x[i] + # find interval + if xval == t[k]: + left = k + else: + left = np.searchsorted(t, xval) - 1 + + # fill a row + bb = _dierckx.evaluate_all_bspl(t, k, xval, left) + matr[i + k - 1, left-k:left+1] = bb + + # RHS + b = np.r_[[0] * (k - 1), y] + + c = solve(matr, b) + return c + + +def _handle_lhs_derivatives(t, k, xval, ab, kl, ku, deriv_ords, offset=0): + """ Fill in the entries of the colocation matrix corresponding to known + derivatives at `xval`. + + The colocation matrix is in the banded storage, as prepared by _coloc. + No error checking. + + Parameters + ---------- + t : ndarray, shape (nt + k + 1,) + knots + k : integer + B-spline order + xval : float + The value at which to evaluate the derivatives at. + ab : ndarray, shape(2*kl + ku + 1, nt), Fortran order + B-spline colocation matrix. + This argument is modified *in-place*. + kl : integer + Number of lower diagonals of ab. + ku : integer + Number of upper diagonals of ab. + deriv_ords : 1D ndarray + Orders of derivatives known at xval + offset : integer, optional + Skip this many rows of the matrix ab. + + """ + # find where `xval` is in the knot vector, `t` + left = _dierckx.find_interval(t, k, float(xval), k, False) + + # compute and fill in the derivatives @ xval + for row in range(deriv_ords.shape[0]): + nu = deriv_ords[row] + wrk = _dierckx.evaluate_all_bspl(t, k, xval, left, nu) + + # if A were a full matrix, it would be just + # ``A[row + offset, left-k:left+1] = bb``. + for a in range(k+1): + clmn = left - k + a + ab[kl + ku + offset + row - clmn, clmn] = wrk[a] + + +def _make_periodic_spline(x, y, t, k, axis): + ''' + Compute the (coefficients of) interpolating B-spline with periodic + boundary conditions. + + Parameters + ---------- + x : array_like, shape (n,) + Abscissas. + y : array_like, shape (n,) + Ordinates. + k : int + B-spline degree. + t : array_like, shape (n + 2 * k,). + Knots taken on a circle, ``k`` on the left and ``k`` on the right + of the vector ``x``. + + Returns + ------- + b : a BSpline object of the degree ``k`` and with knots ``t``. + + Notes + ----- + The original system is formed by ``n + k - 1`` equations where the first + ``k - 1`` of them stand for the ``k - 1`` derivatives continuity on the + edges while the other equations correspond to an interpolating case + (matching all the input points). Due to a special form of knot vector, it + can be proved that in the original system the first and last ``k`` + coefficients of a spline function are the same, respectively. It follows + from the fact that all ``k - 1`` derivatives are equal term by term at ends + and that the matrix of the original system of linear equations is + non-degenerate. So, we can reduce the number of equations to ``n - 1`` + (first ``k - 1`` equations could be reduced). Another trick of this + implementation is cyclic shift of values of B-splines due to equality of + ``k`` unknown coefficients. With this we can receive matrix of the system + with upper right and lower left blocks, and ``k`` diagonals. It allows + to use Woodbury formula to optimize the computations. + + ''' + n = y.shape[0] + + extradim = prod(y.shape[1:]) + y_new = y.reshape(n, extradim) + c = np.zeros((n + k - 1, extradim)) + + # n <= k case is solved with full matrix + if n <= k: + for i in range(extradim): + c[:, i] = _make_interp_per_full_matr(x, y_new[:, i], t, k) + c = np.ascontiguousarray(c.reshape((n + k - 1,) + y.shape[1:])) + return BSpline.construct_fast(t, c, k, extrapolate='periodic', axis=axis) + + nt = len(t) - k - 1 + + # size of block elements + kul = int(k / 2) + + # kl = ku = k + ab = np.zeros((3 * k + 1, nt), dtype=np.float64, order='F') + + # upper right and lower left blocks + ur = np.zeros((kul, kul)) + ll = np.zeros_like(ur) + + # `offset` is made to shift all the non-zero elements to the end of the + # matrix + # NB: 1. drop the last element of `x` because `x[0] = x[-1] + T` & `y[0] == y[-1]` + # 2. pass ab.T to _coloc to make it C-ordered; below it'll be fed to banded + # LAPACK, which needs F-ordered arrays + _dierckx._coloc(x[:-1], t, k, ab.T, k) + + # remove zeros before the matrix + ab = ab[-k - (k + 1) % 2:, :] + + # The least elements in rows (except repetitions) are diagonals + # of block matrices. Upper right matrix is an upper triangular + # matrix while lower left is a lower triangular one. + for i in range(kul): + ur += np.diag(ab[-i - 1, i: kul], k=i) + ll += np.diag(ab[i, -kul - (k % 2): n - 1 + 2 * kul - i], k=-i) + + # remove elements that occur in the last point + # (first and last points are equivalent) + A = ab[:, kul: -k + kul] + + for i in range(extradim): + cc = _woodbury_algorithm(A, ur, ll, y_new[:, i][:-1], k) + c[:, i] = np.concatenate((cc[-kul:], cc, cc[:kul + k % 2])) + c = np.ascontiguousarray(c.reshape((n + k - 1,) + y.shape[1:])) + return BSpline.construct_fast(t, c, k, extrapolate='periodic', axis=axis) + + +def make_interp_spline(x, y, k=3, t=None, bc_type=None, axis=0, + check_finite=True): + """Compute the (coefficients of) interpolating B-spline. + + Parameters + ---------- + x : array_like, shape (n,) + Abscissas. + y : array_like, shape (n, ...) + Ordinates. + k : int, optional + B-spline degree. Default is cubic, ``k = 3``. + t : array_like, shape (nt + k + 1,), optional. + Knots. + The number of knots needs to agree with the number of data points and + the number of derivatives at the edges. Specifically, ``nt - n`` must + equal ``len(deriv_l) + len(deriv_r)``. + bc_type : 2-tuple or None + Boundary conditions. + Default is None, which means choosing the boundary conditions + automatically. Otherwise, it must be a length-two tuple where the first + element (``deriv_l``) sets the boundary conditions at ``x[0]`` and + the second element (``deriv_r``) sets the boundary conditions at + ``x[-1]``. Each of these must be an iterable of pairs + ``(order, value)`` which gives the values of derivatives of specified + orders at the given edge of the interpolation interval. + Alternatively, the following string aliases are recognized: + + * ``"clamped"``: The first derivatives at the ends are zero. This is + equivalent to ``bc_type=([(1, 0.0)], [(1, 0.0)])``. + * ``"natural"``: The second derivatives at ends are zero. This is + equivalent to ``bc_type=([(2, 0.0)], [(2, 0.0)])``. + * ``"not-a-knot"`` (default): The first and second segments are the + same polynomial. This is equivalent to having ``bc_type=None``. + * ``"periodic"``: The values and the first ``k-1`` derivatives at the + ends are equivalent. + + axis : int, optional + Interpolation axis. Default is 0. + check_finite : bool, optional + Whether to check that the input arrays contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + Default is True. + + Returns + ------- + b : a BSpline object of the degree ``k`` and with knots ``t``. + + See Also + -------- + BSpline : base class representing the B-spline objects + CubicSpline : a cubic spline in the polynomial basis + make_lsq_spline : a similar factory function for spline fitting + UnivariateSpline : a wrapper over FITPACK spline fitting routines + splrep : a wrapper over FITPACK spline fitting routines + + Examples + -------- + + Use cubic interpolation on Chebyshev nodes: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> def cheb_nodes(N): + ... jj = 2.*np.arange(N) + 1 + ... x = np.cos(np.pi * jj / 2 / N)[::-1] + ... return x + + >>> x = cheb_nodes(20) + >>> y = np.sqrt(1 - x**2) + + >>> from scipy.interpolate import BSpline, make_interp_spline + >>> b = make_interp_spline(x, y) + >>> np.allclose(b(x), y) + True + + Note that the default is a cubic spline with a not-a-knot boundary condition + + >>> b.k + 3 + + Here we use a 'natural' spline, with zero 2nd derivatives at edges: + + >>> l, r = [(2, 0.0)], [(2, 0.0)] + >>> b_n = make_interp_spline(x, y, bc_type=(l, r)) # or, bc_type="natural" + >>> np.allclose(b_n(x), y) + True + >>> x0, x1 = x[0], x[-1] + >>> np.allclose([b_n(x0, 2), b_n(x1, 2)], [0, 0]) + True + + Interpolation of parametric curves is also supported. As an example, we + compute a discretization of a snail curve in polar coordinates + + >>> phi = np.linspace(0, 2.*np.pi, 40) + >>> r = 0.3 + np.cos(phi) + >>> x, y = r*np.cos(phi), r*np.sin(phi) # convert to Cartesian coordinates + + Build an interpolating curve, parameterizing it by the angle + + >>> spl = make_interp_spline(phi, np.c_[x, y]) + + Evaluate the interpolant on a finer grid (note that we transpose the result + to unpack it into a pair of x- and y-arrays) + + >>> phi_new = np.linspace(0, 2.*np.pi, 100) + >>> x_new, y_new = spl(phi_new).T + + Plot the result + + >>> plt.plot(x, y, 'o') + >>> plt.plot(x_new, y_new, '-') + >>> plt.show() + + Build a B-spline curve with 2 dimensional y + + >>> x = np.linspace(0, 2*np.pi, 10) + >>> y = np.array([np.sin(x), np.cos(x)]) + + Periodic condition is satisfied because y coordinates of points on the ends + are equivalent + + >>> ax = plt.axes(projection='3d') + >>> xx = np.linspace(0, 2*np.pi, 100) + >>> bspl = make_interp_spline(x, y, k=5, bc_type='periodic', axis=1) + >>> ax.plot3D(xx, *bspl(xx)) + >>> ax.scatter3D(x, *y, color='red') + >>> plt.show() + + """ + # convert string aliases for the boundary conditions + if bc_type is None or bc_type == 'not-a-knot' or bc_type == 'periodic': + deriv_l, deriv_r = None, None + elif isinstance(bc_type, str): + deriv_l, deriv_r = bc_type, bc_type + else: + try: + deriv_l, deriv_r = bc_type + except TypeError as e: + raise ValueError(f"Unknown boundary condition: {bc_type}") from e + + y = np.asarray(y) + + axis = normalize_axis_index(axis, y.ndim) + + x = _as_float_array(x, check_finite) + y = _as_float_array(y, check_finite) + + y = np.moveaxis(y, axis, 0) # now internally interp axis is zero + + # sanity check the input + if bc_type == 'periodic' and not np.allclose(y[0], y[-1], atol=1e-15): + raise ValueError("First and last points does not match while " + "periodic case expected") + if x.size != y.shape[0]: + raise ValueError(f'Shapes of x {x.shape} and y {y.shape} are incompatible') + if np.any(x[1:] == x[:-1]): + raise ValueError("Expect x to not have duplicates") + if x.ndim != 1 or np.any(x[1:] < x[:-1]): + raise ValueError("Expect x to be a 1D strictly increasing sequence.") + + # special-case k=0 right away + if k == 0: + if any(_ is not None for _ in (t, deriv_l, deriv_r)): + raise ValueError("Too much info for k=0: t and bc_type can only " + "be None.") + t = np.r_[x, x[-1]] + c = np.asarray(y) + c = np.ascontiguousarray(c, dtype=_get_dtype(c.dtype)) + return BSpline.construct_fast(t, c, k, axis=axis) + + # special-case k=1 (e.g., Lyche and Morken, Eq.(2.16)) + if k == 1 and t is None: + if not (deriv_l is None and deriv_r is None): + raise ValueError("Too much info for k=1: bc_type can only be None.") + t = np.r_[x[0], x, x[-1]] + c = np.asarray(y) + c = np.ascontiguousarray(c, dtype=_get_dtype(c.dtype)) + return BSpline.construct_fast(t, c, k, axis=axis) + + k = operator.index(k) + + if bc_type == 'periodic' and t is not None: + raise NotImplementedError("For periodic case t is constructed " + "automatically and can not be passed " + "manually") + + # come up with a sensible knot vector, if needed + if t is None: + if deriv_l is None and deriv_r is None: + if bc_type == 'periodic': + t = _periodic_knots(x, k) + else: + t = _not_a_knot(x, k) + else: + t = _augknt(x, k) + + t = _as_float_array(t, check_finite) + + if k < 0: + raise ValueError("Expect non-negative k.") + if t.ndim != 1 or np.any(t[1:] < t[:-1]): + raise ValueError("Expect t to be a 1-D sorted array_like.") + if t.size < x.size + k + 1: + raise ValueError('Got %d knots, need at least %d.' % + (t.size, x.size + k + 1)) + if (x[0] < t[k]) or (x[-1] > t[-k]): + raise ValueError(f'Out of bounds w/ x = {x}.') + + if bc_type == 'periodic': + return _make_periodic_spline(x, y, t, k, axis) + + # Here : deriv_l, r = [(nu, value), ...] + deriv_l = _convert_string_aliases(deriv_l, y.shape[1:]) + deriv_l_ords, deriv_l_vals = _process_deriv_spec(deriv_l) + nleft = deriv_l_ords.shape[0] + + deriv_r = _convert_string_aliases(deriv_r, y.shape[1:]) + deriv_r_ords, deriv_r_vals = _process_deriv_spec(deriv_r) + nright = deriv_r_ords.shape[0] + + if not all(0 <= i <= k for i in deriv_l_ords): + raise ValueError(f"Bad boundary conditions at {x[0]}.") + + if not all(0 <= i <= k for i in deriv_r_ords): + raise ValueError(f"Bad boundary conditions at {x[-1]}.") + + # have `n` conditions for `nt` coefficients; need nt-n derivatives + n = x.size + nt = t.size - k - 1 + + if nt - n != nleft + nright: + raise ValueError("The number of derivatives at boundaries does not " + f"match: expected {nt-n}, got {nleft}+{nright}") + + # bail out if the `y` array is zero-sized + if y.size == 0: + c = np.zeros((nt,) + y.shape[1:], dtype=float) + return BSpline.construct_fast(t, c, k, axis=axis) + + # set up the LHS: the colocation matrix + derivatives at boundaries + # NB: ab is in F order for banded LAPACK; _coloc needs C-ordered arrays, + # this pass ab.T into _coloc + kl = ku = k + ab = np.zeros((2*kl + ku + 1, nt), dtype=np.float64, order='F') + _dierckx._coloc(x, t, k, ab.T, nleft) + if nleft > 0: + _handle_lhs_derivatives(t, k, x[0], ab, kl, ku, deriv_l_ords) + if nright > 0: + _handle_lhs_derivatives(t, k, x[-1], ab, kl, ku, deriv_r_ords, + offset=nt-nright) + + # set up the RHS: values to interpolate (+ derivative values, if any) + extradim = prod(y.shape[1:]) + rhs = np.empty((nt, extradim), dtype=y.dtype) + if nleft > 0: + rhs[:nleft] = deriv_l_vals.reshape(-1, extradim) + rhs[nleft:nt - nright] = y.reshape(-1, extradim) + if nright > 0: + rhs[nt - nright:] = deriv_r_vals.reshape(-1, extradim) + + # solve Ab @ x = rhs; this is the relevant part of linalg.solve_banded + if check_finite: + ab, rhs = map(np.asarray_chkfinite, (ab, rhs)) + gbsv, = get_lapack_funcs(('gbsv',), (ab, rhs)) + lu, piv, c, info = gbsv(kl, ku, ab, rhs, + overwrite_ab=True, overwrite_b=True) + + if info > 0: + raise LinAlgError("Colocation matrix is singular.") + elif info < 0: + raise ValueError('illegal value in %d-th argument of internal gbsv' % -info) + + c = np.ascontiguousarray(c.reshape((nt,) + y.shape[1:])) + return BSpline.construct_fast(t, c, k, axis=axis) + + +def make_lsq_spline(x, y, t, k=3, w=None, axis=0, check_finite=True, *, method="qr"): + r"""Compute the (coefficients of) an LSQ (Least SQuared) based + fitting B-spline. + + The result is a linear combination + + .. math:: + + S(x) = \sum_j c_j B_j(x; t) + + of the B-spline basis elements, :math:`B_j(x; t)`, which minimizes + + .. math:: + + \sum_{j} \left( w_j \times (S(x_j) - y_j) \right)^2 + + Parameters + ---------- + x : array_like, shape (m,) + Abscissas. + y : array_like, shape (m, ...) + Ordinates. + t : array_like, shape (n + k + 1,). + Knots. + Knots and data points must satisfy Schoenberg-Whitney conditions. + k : int, optional + B-spline degree. Default is cubic, ``k = 3``. + w : array_like, shape (m,), optional + Weights for spline fitting. Must be positive. If ``None``, + then weights are all equal. + Default is ``None``. + axis : int, optional + Interpolation axis. Default is zero. + check_finite : bool, optional + Whether to check that the input arrays contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + Default is True. + method : str, optional + Method for solving the linear LSQ problem. Allowed values are "norm-eq" + (Explicitly construct and solve the normal system of equations), and + "qr" (Use the QR factorization of the design matrix). + Default is "qr". + + Returns + ------- + b : a BSpline object of the degree ``k`` with knots ``t``. + + See Also + -------- + BSpline : base class representing the B-spline objects + make_interp_spline : a similar factory function for interpolating splines + LSQUnivariateSpline : a FITPACK-based spline fitting routine + splrep : a FITPACK-based fitting routine + + Notes + ----- + The number of data points must be larger than the spline degree ``k``. + + Knots ``t`` must satisfy the Schoenberg-Whitney conditions, + i.e., there must be a subset of data points ``x[j]`` such that + ``t[j] < x[j] < t[j+k+1]``, for ``j=0, 1,...,n-k-2``. + + Examples + -------- + Generate some noisy data: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + >>> x = np.linspace(-3, 3, 50) + >>> y = np.exp(-x**2) + 0.1 * rng.standard_normal(50) + + Now fit a smoothing cubic spline with a pre-defined internal knots. + Here we make the knot vector (k+1)-regular by adding boundary knots: + + >>> from scipy.interpolate import make_lsq_spline, BSpline + >>> t = [-1, 0, 1] + >>> k = 3 + >>> t = np.r_[(x[0],)*(k+1), + ... t, + ... (x[-1],)*(k+1)] + >>> spl = make_lsq_spline(x, y, t, k) + + For comparison, we also construct an interpolating spline for the same + set of data: + + >>> from scipy.interpolate import make_interp_spline + >>> spl_i = make_interp_spline(x, y) + + Plot both: + + >>> xs = np.linspace(-3, 3, 100) + >>> plt.plot(x, y, 'ro', ms=5) + >>> plt.plot(xs, spl(xs), 'g-', lw=3, label='LSQ spline') + >>> plt.plot(xs, spl_i(xs), 'b-', lw=3, alpha=0.7, label='interp spline') + >>> plt.legend(loc='best') + >>> plt.show() + + **NaN handling**: If the input arrays contain ``nan`` values, the result is + not useful since the underlying spline fitting routines cannot deal with + ``nan``. A workaround is to use zero weights for not-a-number data points: + + >>> y[8] = np.nan + >>> w = np.isnan(y) + >>> y[w] = 0. + >>> tck = make_lsq_spline(x, y, t, w=~w) + + Notice the need to replace a ``nan`` by a numerical value (precise value + does not matter as long as the corresponding weight is zero.) + + """ + x = _as_float_array(x, check_finite) + y = _as_float_array(y, check_finite) + t = _as_float_array(t, check_finite) + if w is not None: + w = _as_float_array(w, check_finite) + else: + w = np.ones_like(x) + k = operator.index(k) + + axis = normalize_axis_index(axis, y.ndim) + + y = np.moveaxis(y, axis, 0) # now internally interp axis is zero + + if x.ndim != 1: + raise ValueError("Expect x to be a 1-D sequence.") + if x.shape[0] < k+1: + raise ValueError("Need more x points.") + if k < 0: + raise ValueError("Expect non-negative k.") + if t.ndim != 1 or np.any(t[1:] - t[:-1] < 0): + raise ValueError("Expect t to be a 1D strictly increasing sequence.") + if x.size != y.shape[0]: + raise ValueError(f'Shapes of x {x.shape} and y {y.shape} are incompatible') + if k > 0 and np.any((x < t[k]) | (x > t[-k])): + raise ValueError(f'Out of bounds w/ x = {x}.') + if x.size != w.size: + raise ValueError(f'Shapes of x {x.shape} and w {w.shape} are incompatible') + if method == "norm-eq" and np.any(x[1:] - x[:-1] <= 0): + raise ValueError("Expect x to be a 1D strictly increasing sequence.") + if method == "qr" and any(x[1:] - x[:-1] < 0): + raise ValueError("Expect x to be a 1D non-decreasing sequence.") + + # number of coefficients + n = t.size - k - 1 + + # complex y: view as float, preserve the length + was_complex = y.dtype.kind == 'c' + yy = y.view(float) + if was_complex and y.ndim == 1: + yy = yy.reshape(y.shape[0], 2) + + # multiple r.h.s + extradim = prod(yy.shape[1:]) + yy = yy.reshape(-1, extradim) + + # complex y: view as float, preserve the length + was_complex = y.dtype.kind == 'c' + yy = y.view(float) + if was_complex and y.ndim == 1: + yy = yy.reshape(y.shape[0], 2) + + # multiple r.h.s + extradim = prod(yy.shape[1:]) + yy = yy.reshape(-1, extradim) + + if method == "norm-eq": + # construct A.T @ A and rhs with A the colocation matrix, and + # rhs = A.T @ y for solving the LSQ problem ``A.T @ A @ c = A.T @ y`` + lower = True + ab = np.zeros((k+1, n), dtype=np.float64, order='F') + rhs = np.zeros((n, extradim), dtype=np.float64) + _dierckx._norm_eq_lsq(x, t, k, + yy, + w, + ab.T, rhs) + + # undo complex -> float and flattening the trailing dims + if was_complex: + rhs = rhs.view(complex) + + rhs = rhs.reshape((n,) + y.shape[1:]) + + # have observation matrix & rhs, can solve the LSQ problem + cho_decomp = cholesky_banded(ab, overwrite_ab=True, lower=lower, + check_finite=check_finite) + c = cho_solve_banded((cho_decomp, lower), rhs, overwrite_b=True, + check_finite=check_finite) + elif method == "qr": + _, _, c = _lsq_solve_qr(x, yy, t, k, w) + + if was_complex: + c = c.view(complex) + + else: + raise ValueError(f"Unknown {method =}.") + + + # restore the shape of `c` for both single and multiple r.h.s. + c = c.reshape((n,) + y.shape[1:]) + c = np.ascontiguousarray(c) + return BSpline.construct_fast(t, c, k, axis=axis) + + +###################### +# LSQ spline helpers # +###################### + +def _lsq_solve_qr(x, y, t, k, w): + """Solve for the LSQ spline coeffs given x, y and knots. + + `y` is always 2D: for 1D data, the shape is ``(m, 1)``. + `w` is always 1D: one weight value per `x` value. + + """ + assert y.ndim == 2 + + y_w = y * w[:, None] + A, offset, nc = _dierckx.data_matrix(x, t, k, w) + _dierckx.qr_reduce(A, offset, nc, y_w) # modifies arguments in-place + c = _dierckx.fpback(A, nc, y_w) + + return A, y_w, c + + +############################# +# Smoothing spline helpers # +############################# + +def _compute_optimal_gcv_parameter(X, wE, y, w): + """ + Returns an optimal regularization parameter from the GCV criteria [1]. + + Parameters + ---------- + X : array, shape (5, n) + 5 bands of the design matrix ``X`` stored in LAPACK banded storage. + wE : array, shape (5, n) + 5 bands of the penalty matrix :math:`W^{-1} E` stored in LAPACK banded + storage. + y : array, shape (n,) + Ordinates. + w : array, shape (n,) + Vector of weights. + + Returns + ------- + lam : float + An optimal from the GCV criteria point of view regularization + parameter. + + Notes + ----- + No checks are performed. + + References + ---------- + .. [1] G. Wahba, "Estimating the smoothing parameter" in Spline models + for observational data, Philadelphia, Pennsylvania: Society for + Industrial and Applied Mathematics, 1990, pp. 45-65. + :doi:`10.1137/1.9781611970128` + + """ + + def compute_banded_symmetric_XT_W_Y(X, w, Y): + """ + Assuming that the product :math:`X^T W Y` is symmetric and both ``X`` + and ``Y`` are 5-banded, compute the unique bands of the product. + + Parameters + ---------- + X : array, shape (5, n) + 5 bands of the matrix ``X`` stored in LAPACK banded storage. + w : array, shape (n,) + Array of weights + Y : array, shape (5, n) + 5 bands of the matrix ``Y`` stored in LAPACK banded storage. + + Returns + ------- + res : array, shape (4, n) + The result of the product :math:`X^T Y` stored in the banded way. + + Notes + ----- + As far as the matrices ``X`` and ``Y`` are 5-banded, their product + :math:`X^T W Y` is 7-banded. It is also symmetric, so we can store only + unique diagonals. + + """ + # compute W Y + W_Y = np.copy(Y) + + W_Y[2] *= w + for i in range(2): + W_Y[i, 2 - i:] *= w[:-2 + i] + W_Y[3 + i, :-1 - i] *= w[1 + i:] + + n = X.shape[1] + res = np.zeros((4, n)) + for i in range(n): + for j in range(min(n-i, 4)): + res[-j-1, i + j] = sum(X[j:, i] * W_Y[:5-j, i + j]) + return res + + def compute_b_inv(A): + """ + Inverse 3 central bands of matrix :math:`A=U^T D^{-1} U` assuming that + ``U`` is a unit upper triangular banded matrix using an algorithm + proposed in [1]. + + Parameters + ---------- + A : array, shape (4, n) + Matrix to inverse, stored in LAPACK banded storage. + + Returns + ------- + B : array, shape (4, n) + 3 unique bands of the symmetric matrix that is an inverse to ``A``. + The first row is filled with zeros. + + Notes + ----- + The algorithm is based on the cholesky decomposition and, therefore, + in case matrix ``A`` is close to not positive defined, the function + raises LinalgError. + + Both matrices ``A`` and ``B`` are stored in LAPACK banded storage. + + References + ---------- + .. [1] M. F. Hutchinson and F. R. de Hoog, "Smoothing noisy data with + spline functions," Numerische Mathematik, vol. 47, no. 1, + pp. 99-106, 1985. + :doi:`10.1007/BF01389878` + + """ + + def find_b_inv_elem(i, j, U, D, B): + rng = min(3, n - i - 1) + rng_sum = 0. + if j == 0: + # use 2-nd formula from [1] + for k in range(1, rng + 1): + rng_sum -= U[-k - 1, i + k] * B[-k - 1, i + k] + rng_sum += D[i] + B[-1, i] = rng_sum + else: + # use 1-st formula from [1] + for k in range(1, rng + 1): + diag = abs(k - j) + ind = i + min(k, j) + rng_sum -= U[-k - 1, i + k] * B[-diag - 1, ind + diag] + B[-j - 1, i + j] = rng_sum + + U = cholesky_banded(A) + for i in range(2, 5): + U[-i, i-1:] /= U[-1, :-i+1] + D = 1. / (U[-1])**2 + U[-1] /= U[-1] + + n = U.shape[1] + + B = np.zeros(shape=(4, n)) + for i in range(n - 1, -1, -1): + for j in range(min(3, n - i - 1), -1, -1): + find_b_inv_elem(i, j, U, D, B) + # the first row contains garbage and should be removed + B[0] = [0.] * n + return B + + def _gcv(lam, X, XtWX, wE, XtE): + r""" + Computes the generalized cross-validation criteria [1]. + + Parameters + ---------- + lam : float, (:math:`\lambda \geq 0`) + Regularization parameter. + X : array, shape (5, n) + Matrix is stored in LAPACK banded storage. + XtWX : array, shape (4, n) + Product :math:`X^T W X` stored in LAPACK banded storage. + wE : array, shape (5, n) + Matrix :math:`W^{-1} E` stored in LAPACK banded storage. + XtE : array, shape (4, n) + Product :math:`X^T E` stored in LAPACK banded storage. + + Returns + ------- + res : float + Value of the GCV criteria with the regularization parameter + :math:`\lambda`. + + Notes + ----- + Criteria is computed from the formula (1.3.2) [3]: + + .. math: + + GCV(\lambda) = \dfrac{1}{n} \sum\limits_{k = 1}^{n} \dfrac{ \left( + y_k - f_{\lambda}(x_k) \right)^2}{\left( 1 - \Tr{A}/n\right)^2}$. + The criteria is discussed in section 1.3 [3]. + + The numerator is computed using (2.2.4) [3] and the denominator is + computed using an algorithm from [2] (see in the ``compute_b_inv`` + function). + + References + ---------- + .. [1] G. Wahba, "Estimating the smoothing parameter" in Spline models + for observational data, Philadelphia, Pennsylvania: Society for + Industrial and Applied Mathematics, 1990, pp. 45-65. + :doi:`10.1137/1.9781611970128` + .. [2] M. F. Hutchinson and F. R. de Hoog, "Smoothing noisy data with + spline functions," Numerische Mathematik, vol. 47, no. 1, + pp. 99-106, 1985. + :doi:`10.1007/BF01389878` + .. [3] E. Zemlyanoy, "Generalized cross-validation smoothing splines", + BSc thesis, 2022. Might be available (in Russian) + `here `_ + + """ + # Compute the numerator from (2.2.4) [3] + n = X.shape[1] + c = solve_banded((2, 2), X + lam * wE, y) + res = np.zeros(n) + # compute ``W^{-1} E c`` with respect to banded-storage of ``E`` + tmp = wE * c + for i in range(n): + for j in range(max(0, i - n + 3), min(5, i + 3)): + res[i] += tmp[j, i + 2 - j] + numer = np.linalg.norm(lam * res)**2 / n + + # compute the denominator + lhs = XtWX + lam * XtE + try: + b_banded = compute_b_inv(lhs) + # compute the trace of the product b_banded @ XtX + tr = b_banded * XtWX + tr[:-1] *= 2 + # find the denominator + denom = (1 - sum(sum(tr)) / n)**2 + except LinAlgError: + # cholesky decomposition cannot be performed + raise ValueError('Seems like the problem is ill-posed') + + res = numer / denom + + return res + + n = X.shape[1] + + XtWX = compute_banded_symmetric_XT_W_Y(X, w, X) + XtE = compute_banded_symmetric_XT_W_Y(X, w, wE) + + def fun(lam): + return _gcv(lam, X, XtWX, wE, XtE) + + gcv_est = minimize_scalar(fun, bounds=(0, n), method='Bounded') + if gcv_est.success: + return gcv_est.x + raise ValueError(f"Unable to find minimum of the GCV " + f"function: {gcv_est.message}") + + +def _coeff_of_divided_diff(x): + """ + Returns the coefficients of the divided difference. + + Parameters + ---------- + x : array, shape (n,) + Array which is used for the computation of divided difference. + + Returns + ------- + res : array_like, shape (n,) + Coefficients of the divided difference. + + Notes + ----- + Vector ``x`` should have unique elements, otherwise an error division by + zero might be raised. + + No checks are performed. + + """ + n = x.shape[0] + res = np.zeros(n) + for i in range(n): + pp = 1. + for k in range(n): + if k != i: + pp *= (x[i] - x[k]) + res[i] = 1. / pp + return res + + +def make_smoothing_spline(x, y, w=None, lam=None): + r""" + Compute the (coefficients of) smoothing cubic spline function using + ``lam`` to control the tradeoff between the amount of smoothness of the + curve and its proximity to the data. In case ``lam`` is None, using the + GCV criteria [1] to find it. + + A smoothing spline is found as a solution to the regularized weighted + linear regression problem: + + .. math:: + + \sum\limits_{i=1}^n w_i\lvert y_i - f(x_i) \rvert^2 + + \lambda\int\limits_{x_1}^{x_n} (f^{(2)}(u))^2 d u + + where :math:`f` is a spline function, :math:`w` is a vector of weights and + :math:`\lambda` is a regularization parameter. + + If ``lam`` is None, we use the GCV criteria to find an optimal + regularization parameter, otherwise we solve the regularized weighted + linear regression problem with given parameter. The parameter controls + the tradeoff in the following way: the larger the parameter becomes, the + smoother the function gets. + + Parameters + ---------- + x : array_like, shape (n,) + Abscissas. `n` must be at least 5. + y : array_like, shape (n,) + Ordinates. `n` must be at least 5. + w : array_like, shape (n,), optional + Vector of weights. Default is ``np.ones_like(x)``. + lam : float, (:math:`\lambda \geq 0`), optional + Regularization parameter. If ``lam`` is None, then it is found from + the GCV criteria. Default is None. + + Returns + ------- + func : a BSpline object. + A callable representing a spline in the B-spline basis + as a solution of the problem of smoothing splines using + the GCV criteria [1] in case ``lam`` is None, otherwise using the + given parameter ``lam``. + + Notes + ----- + This algorithm is a clean room reimplementation of the algorithm + introduced by Woltring in FORTRAN [2]. The original version cannot be used + in SciPy source code because of the license issues. The details of the + reimplementation are discussed here (available only in Russian) [4]. + + If the vector of weights ``w`` is None, we assume that all the points are + equal in terms of weights, and vector of weights is vector of ones. + + Note that in weighted residual sum of squares, weights are not squared: + :math:`\sum\limits_{i=1}^n w_i\lvert y_i - f(x_i) \rvert^2` while in + ``splrep`` the sum is built from the squared weights. + + In cases when the initial problem is ill-posed (for example, the product + :math:`X^T W X` where :math:`X` is a design matrix is not a positive + defined matrix) a ValueError is raised. + + References + ---------- + .. [1] G. Wahba, "Estimating the smoothing parameter" in Spline models for + observational data, Philadelphia, Pennsylvania: Society for Industrial + and Applied Mathematics, 1990, pp. 45-65. + :doi:`10.1137/1.9781611970128` + .. [2] H. J. Woltring, A Fortran package for generalized, cross-validatory + spline smoothing and differentiation, Advances in Engineering + Software, vol. 8, no. 2, pp. 104-113, 1986. + :doi:`10.1016/0141-1195(86)90098-7` + .. [3] T. Hastie, J. Friedman, and R. Tisbshirani, "Smoothing Splines" in + The elements of Statistical Learning: Data Mining, Inference, and + prediction, New York: Springer, 2017, pp. 241-249. + :doi:`10.1007/978-0-387-84858-7` + .. [4] E. Zemlyanoy, "Generalized cross-validation smoothing splines", + BSc thesis, 2022. + ``_ (in + Russian) + + Examples + -------- + Generate some noisy data + + >>> import numpy as np + >>> np.random.seed(1234) + >>> n = 200 + >>> def func(x): + ... return x**3 + x**2 * np.sin(4 * x) + >>> x = np.sort(np.random.random_sample(n) * 4 - 2) + >>> y = func(x) + np.random.normal(scale=1.5, size=n) + + Make a smoothing spline function + + >>> from scipy.interpolate import make_smoothing_spline + >>> spl = make_smoothing_spline(x, y) + + Plot both + + >>> import matplotlib.pyplot as plt + >>> grid = np.linspace(x[0], x[-1], 400) + >>> plt.plot(grid, spl(grid), label='Spline') + >>> plt.plot(grid, func(grid), label='Original function') + >>> plt.scatter(x, y, marker='.') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + + x = np.ascontiguousarray(x, dtype=float) + y = np.ascontiguousarray(y, dtype=float) + + if any(x[1:] - x[:-1] <= 0): + raise ValueError('``x`` should be an ascending array') + + if x.ndim != 1 or y.ndim != 1 or x.shape[0] != y.shape[0]: + raise ValueError('``x`` and ``y`` should be one dimensional and the' + ' same size') + + if w is None: + w = np.ones(len(x)) + else: + w = np.ascontiguousarray(w) + if any(w <= 0): + raise ValueError('Invalid vector of weights') + + t = np.r_[[x[0]] * 3, x, [x[-1]] * 3] + n = x.shape[0] + + if n <= 4: + raise ValueError('``x`` and ``y`` length must be at least 5') + + # It is known that the solution to the stated minimization problem exists + # and is a natural cubic spline with vector of knots equal to the unique + # elements of ``x`` [3], so we will solve the problem in the basis of + # natural splines. + + # create design matrix in the B-spline basis + X_bspl = BSpline.design_matrix(x, t, 3) + # move from B-spline basis to the basis of natural splines using equations + # (2.1.7) [4] + # central elements + X = np.zeros((5, n)) + for i in range(1, 4): + X[i, 2: -2] = X_bspl[i: i - 4, 3: -3][np.diag_indices(n - 4)] + + # first elements + X[1, 1] = X_bspl[0, 0] + X[2, :2] = ((x[2] + x[1] - 2 * x[0]) * X_bspl[0, 0], + X_bspl[1, 1] + X_bspl[1, 2]) + X[3, :2] = ((x[2] - x[0]) * X_bspl[1, 1], X_bspl[2, 2]) + + # last elements + X[1, -2:] = (X_bspl[-3, -3], (x[-1] - x[-3]) * X_bspl[-2, -2]) + X[2, -2:] = (X_bspl[-2, -3] + X_bspl[-2, -2], + (2 * x[-1] - x[-2] - x[-3]) * X_bspl[-1, -1]) + X[3, -2] = X_bspl[-1, -1] + + # create penalty matrix and divide it by vector of weights: W^{-1} E + wE = np.zeros((5, n)) + wE[2:, 0] = _coeff_of_divided_diff(x[:3]) / w[:3] + wE[1:, 1] = _coeff_of_divided_diff(x[:4]) / w[:4] + for j in range(2, n - 2): + wE[:, j] = (x[j+2] - x[j-2]) * _coeff_of_divided_diff(x[j-2:j+3])\ + / w[j-2: j+3] + + wE[:-1, -2] = -_coeff_of_divided_diff(x[-4:]) / w[-4:] + wE[:-2, -1] = _coeff_of_divided_diff(x[-3:]) / w[-3:] + wE *= 6 + + if lam is None: + lam = _compute_optimal_gcv_parameter(X, wE, y, w) + elif lam < 0.: + raise ValueError('Regularization parameter should be non-negative') + + # solve the initial problem in the basis of natural splines + c = solve_banded((2, 2), X + lam * wE, y) + # move back to B-spline basis using equations (2.2.10) [4] + c_ = np.r_[c[0] * (t[5] + t[4] - 2 * t[3]) + c[1], + c[0] * (t[5] - t[3]) + c[1], + c[1: -1], + c[-1] * (t[-4] - t[-6]) + c[-2], + c[-1] * (2 * t[-4] - t[-5] - t[-6]) + c[-2]] + + return BSpline.construct_fast(t, c_, 3) + + +######################## +# FITPACK look-alikes # +######################## + +def fpcheck(x, t, k): + """ Check consistency of the data vector `x` and the knot vector `t`. + + Return None if inputs are consistent, raises a ValueError otherwise. + """ + # This routine is a clone of the `fpchec` Fortran routine, + # https://github.com/scipy/scipy/blob/main/scipy/interpolate/fitpack/fpchec.f + # which carries the following comment: + # + # subroutine fpchec verifies the number and the position of the knots + # t(j),j=1,2,...,n of a spline of degree k, in relation to the number + # and the position of the data points x(i),i=1,2,...,m. if all of the + # following conditions are fulfilled, the error parameter ier is set + # to zero. if one of the conditions is violated ier is set to ten. + # 1) k+1 <= n-k-1 <= m + # 2) t(1) <= t(2) <= ... <= t(k+1) + # t(n-k) <= t(n-k+1) <= ... <= t(n) + # 3) t(k+1) < t(k+2) < ... < t(n-k) + # 4) t(k+1) <= x(i) <= t(n-k) + # 5) the conditions specified by schoenberg and whitney must hold + # for at least one subset of data points, i.e. there must be a + # subset of data points y(j) such that + # t(j) < y(j) < t(j+k+1), j=1,2,...,n-k-1 + x = np.asarray(x) + t = np.asarray(t) + + if x.ndim != 1 or t.ndim != 1: + raise ValueError(f"Expect `x` and `t` be 1D sequences. Got {x = } and {t = }") + + m = x.shape[0] + n = t.shape[0] + nk1 = n - k - 1 + + # check condition no 1 + # c 1) k+1 <= n-k-1 <= m + if not (k + 1 <= nk1 <= m): + raise ValueError(f"Need k+1 <= n-k-1 <= m. Got {m = }, {n = } and {k = }.") + + # check condition no 2 + # c 2) t(1) <= t(2) <= ... <= t(k+1) + # c t(n-k) <= t(n-k+1) <= ... <= t(n) + if (t[:k+1] > t[1:k+2]).any(): + raise ValueError(f"First k knots must be ordered; got {t = }.") + + if (t[nk1:] < t[nk1-1:-1]).any(): + raise ValueError(f"Last k knots must be ordered; got {t = }.") + + # c check condition no 3 + # c 3) t(k+1) < t(k+2) < ... < t(n-k) + if (t[k+1:n-k] <= t[k:n-k-1]).any(): + raise ValueError(f"Internal knots must be distinct. Got {t = }.") + + # c check condition no 4 + # c 4) t(k+1) <= x(i) <= t(n-k) + # NB: FITPACK's fpchec only checks x[0] & x[-1], so we follow. + if (x[0] < t[k]) or (x[-1] > t[n-k-1]): + raise ValueError(f"Out of bounds: {x = } and {t = }.") + + # c check condition no 5 + # c 5) the conditions specified by schoenberg and whitney must hold + # c for at least one subset of data points, i.e. there must be a + # c subset of data points y(j) such that + # c t(j) < y(j) < t(j+k+1), j=1,2,...,n-k-1 + mesg = f"Schoenberg-Whitney condition is violated with {t = } and {x =}." + + if (x[0] >= t[k+1]) or (x[-1] <= t[n-k-2]): + raise ValueError(mesg) + + m = x.shape[0] + l = k+1 + nk3 = n - k - 3 + if nk3 < 2: + return + for j in range(1, nk3+1): + tj = t[j] + l += 1 + tl = t[l] + i = np.argmax(x > tj) + if i >= m-1: + raise ValueError(mesg) + if x[i] >= tl: + raise ValueError(mesg) + return diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_cubic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_cubic.py new file mode 100644 index 0000000000000000000000000000000000000000..3139e145916fa0637552331974ce531da625836f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_cubic.py @@ -0,0 +1,958 @@ +"""Interpolation algorithms using piecewise cubic polynomials.""" + +from typing import Literal + +import numpy as np + +from scipy.linalg import solve, solve_banded + +from . import PPoly +from ._polyint import _isscalar + +__all__ = ["CubicHermiteSpline", "PchipInterpolator", "pchip_interpolate", + "Akima1DInterpolator", "CubicSpline"] + + +def prepare_input(x, y, axis, dydx=None): + """Prepare input for cubic spline interpolators. + + All data are converted to numpy arrays and checked for correctness. + Axes equal to `axis` of arrays `y` and `dydx` are moved to be the 0th + axis. The value of `axis` is converted to lie in + [0, number of dimensions of `y`). + """ + + x, y = map(np.asarray, (x, y)) + if np.issubdtype(x.dtype, np.complexfloating): + raise ValueError("`x` must contain real values.") + x = x.astype(float) + + if np.issubdtype(y.dtype, np.complexfloating): + dtype = complex + else: + dtype = float + + if dydx is not None: + dydx = np.asarray(dydx) + if y.shape != dydx.shape: + raise ValueError("The shapes of `y` and `dydx` must be identical.") + if np.issubdtype(dydx.dtype, np.complexfloating): + dtype = complex + dydx = dydx.astype(dtype, copy=False) + + y = y.astype(dtype, copy=False) + axis = axis % y.ndim + if x.ndim != 1: + raise ValueError("`x` must be 1-dimensional.") + if x.shape[0] < 2: + raise ValueError("`x` must contain at least 2 elements.") + if x.shape[0] != y.shape[axis]: + raise ValueError(f"The length of `y` along `axis`={axis} doesn't " + "match the length of `x`") + + if not np.all(np.isfinite(x)): + raise ValueError("`x` must contain only finite values.") + if not np.all(np.isfinite(y)): + raise ValueError("`y` must contain only finite values.") + + if dydx is not None and not np.all(np.isfinite(dydx)): + raise ValueError("`dydx` must contain only finite values.") + + dx = np.diff(x) + if np.any(dx <= 0): + raise ValueError("`x` must be strictly increasing sequence.") + + y = np.moveaxis(y, axis, 0) + if dydx is not None: + dydx = np.moveaxis(dydx, axis, 0) + + return x, dx, y, axis, dydx + + +class CubicHermiteSpline(PPoly): + """Piecewise-cubic interpolator matching values and first derivatives. + + The result is represented as a `PPoly` instance. + + Parameters + ---------- + x : array_like, shape (n,) + 1-D array containing values of the independent variable. + Values must be real, finite and in strictly increasing order. + y : array_like + Array containing values of the dependent variable. It can have + arbitrary number of dimensions, but the length along ``axis`` + (see below) must match the length of ``x``. Values must be finite. + dydx : array_like + Array containing derivatives of the dependent variable. It can have + arbitrary number of dimensions, but the length along ``axis`` + (see below) must match the length of ``x``. Values must be finite. + axis : int, optional + Axis along which `y` is assumed to be varying. Meaning that for + ``x[i]`` the corresponding values are ``np.take(y, i, axis=axis)``. + Default is 0. + extrapolate : {bool, 'periodic', None}, optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. If 'periodic', + periodic extrapolation is used. If None (default), it is set to True. + + Attributes + ---------- + x : ndarray, shape (n,) + Breakpoints. The same ``x`` which was passed to the constructor. + c : ndarray, shape (4, n-1, ...) + Coefficients of the polynomials on each segment. The trailing + dimensions match the dimensions of `y`, excluding ``axis``. + For example, if `y` is 1-D, then ``c[k, i]`` is a coefficient for + ``(x-x[i])**(3-k)`` on the segment between ``x[i]`` and ``x[i+1]``. + axis : int + Interpolation axis. The same axis which was passed to the + constructor. + + Methods + ------- + __call__ + derivative + antiderivative + integrate + roots + + See Also + -------- + Akima1DInterpolator : Akima 1D interpolator. + PchipInterpolator : PCHIP 1-D monotonic cubic interpolator. + CubicSpline : Cubic spline data interpolator. + PPoly : Piecewise polynomial in terms of coefficients and breakpoints + + Notes + ----- + If you want to create a higher-order spline matching higher-order + derivatives, use `BPoly.from_derivatives`. + + References + ---------- + .. [1] `Cubic Hermite spline + `_ + on Wikipedia. + """ + + def __init__(self, x, y, dydx, axis=0, extrapolate=None): + if extrapolate is None: + extrapolate = True + + x, dx, y, axis, dydx = prepare_input(x, y, axis, dydx) + + dxr = dx.reshape([dx.shape[0]] + [1] * (y.ndim - 1)) + slope = np.diff(y, axis=0) / dxr + t = (dydx[:-1] + dydx[1:] - 2 * slope) / dxr + + c = np.empty((4, len(x) - 1) + y.shape[1:], dtype=t.dtype) + c[0] = t / dxr + c[1] = (slope - dydx[:-1]) / dxr - t + c[2] = dydx[:-1] + c[3] = y[:-1] + + super().__init__(c, x, extrapolate=extrapolate) + self.axis = axis + + +class PchipInterpolator(CubicHermiteSpline): + r"""PCHIP 1-D monotonic cubic interpolation. + + ``x`` and ``y`` are arrays of values used to approximate some function f, + with ``y = f(x)``. The interpolant uses monotonic cubic splines + to find the value of new points. (PCHIP stands for Piecewise Cubic + Hermite Interpolating Polynomial). + + Parameters + ---------- + x : ndarray, shape (npoints, ) + A 1-D array of monotonically increasing real values. ``x`` cannot + include duplicate values (otherwise f is overspecified) + y : ndarray, shape (..., npoints, ...) + A N-D array of real values. ``y``'s length along the interpolation + axis must be equal to the length of ``x``. Use the ``axis`` + parameter to select the interpolation axis. + axis : int, optional + Axis in the ``y`` array corresponding to the x-coordinate values. Defaults + to ``axis=0``. + extrapolate : bool, optional + Whether to extrapolate to out-of-bounds points based on first + and last intervals, or to return NaNs. + + Methods + ------- + __call__ + derivative + antiderivative + roots + + See Also + -------- + CubicHermiteSpline : Piecewise-cubic interpolator. + Akima1DInterpolator : Akima 1D interpolator. + CubicSpline : Cubic spline data interpolator. + PPoly : Piecewise polynomial in terms of coefficients and breakpoints. + + Notes + ----- + The interpolator preserves monotonicity in the interpolation data and does + not overshoot if the data is not smooth. + + The first derivatives are guaranteed to be continuous, but the second + derivatives may jump at :math:`x_k`. + + Determines the derivatives at the points :math:`x_k`, :math:`f'_k`, + by using PCHIP algorithm [1]_. + + Let :math:`h_k = x_{k+1} - x_k`, and :math:`d_k = (y_{k+1} - y_k) / h_k` + are the slopes at internal points :math:`x_k`. + If the signs of :math:`d_k` and :math:`d_{k-1}` are different or either of + them equals zero, then :math:`f'_k = 0`. Otherwise, it is given by the + weighted harmonic mean + + .. math:: + + \frac{w_1 + w_2}{f'_k} = \frac{w_1}{d_{k-1}} + \frac{w_2}{d_k} + + where :math:`w_1 = 2 h_k + h_{k-1}` and :math:`w_2 = h_k + 2 h_{k-1}`. + + The end slopes are set using a one-sided scheme [2]_. + + + References + ---------- + .. [1] F. N. Fritsch and J. Butland, + A method for constructing local + monotone piecewise cubic interpolants, + SIAM J. Sci. Comput., 5(2), 300-304 (1984). + :doi:`10.1137/0905021`. + .. [2] see, e.g., C. Moler, Numerical Computing with Matlab, 2004. + :doi:`10.1137/1.9780898717952` + + """ + + def __init__(self, x, y, axis=0, extrapolate=None): + x, _, y, axis, _ = prepare_input(x, y, axis) + if np.iscomplexobj(y): + msg = ("`PchipInterpolator` only works with real values for `y`. " + "If you are trying to use the real components of the passed array, " + "use `np.real` on the array before passing to `PchipInterpolator`.") + raise ValueError(msg) + xp = x.reshape((x.shape[0],) + (1,)*(y.ndim-1)) + dk = self._find_derivatives(xp, y) + super().__init__(x, y, dk, axis=0, extrapolate=extrapolate) + self.axis = axis + + @staticmethod + def _edge_case(h0, h1, m0, m1): + # one-sided three-point estimate for the derivative + d = ((2*h0 + h1)*m0 - h0*m1) / (h0 + h1) + + # try to preserve shape + mask = np.sign(d) != np.sign(m0) + mask2 = (np.sign(m0) != np.sign(m1)) & (np.abs(d) > 3.*np.abs(m0)) + mmm = (~mask) & mask2 + + d[mask] = 0. + d[mmm] = 3.*m0[mmm] + + return d + + @staticmethod + def _find_derivatives(x, y): + # Determine the derivatives at the points y_k, d_k, by using + # PCHIP algorithm is: + # We choose the derivatives at the point x_k by + # Let m_k be the slope of the kth segment (between k and k+1) + # If m_k=0 or m_{k-1}=0 or sgn(m_k) != sgn(m_{k-1}) then d_k == 0 + # else use weighted harmonic mean: + # w_1 = 2h_k + h_{k-1}, w_2 = h_k + 2h_{k-1} + # 1/d_k = 1/(w_1 + w_2)*(w_1 / m_k + w_2 / m_{k-1}) + # where h_k is the spacing between x_k and x_{k+1} + y_shape = y.shape + if y.ndim == 1: + # So that _edge_case doesn't end up assigning to scalars + x = x[:, None] + y = y[:, None] + + hk = x[1:] - x[:-1] + mk = (y[1:] - y[:-1]) / hk + + if y.shape[0] == 2: + # edge case: only have two points, use linear interpolation + dk = np.zeros_like(y) + dk[0] = mk + dk[1] = mk + return dk.reshape(y_shape) + + smk = np.sign(mk) + condition = (smk[1:] != smk[:-1]) | (mk[1:] == 0) | (mk[:-1] == 0) + + w1 = 2*hk[1:] + hk[:-1] + w2 = hk[1:] + 2*hk[:-1] + + # values where division by zero occurs will be excluded + # by 'condition' afterwards + with np.errstate(divide='ignore', invalid='ignore'): + whmean = (w1/mk[:-1] + w2/mk[1:]) / (w1 + w2) + + dk = np.zeros_like(y) + dk[1:-1][condition] = 0.0 + dk[1:-1][~condition] = 1.0 / whmean[~condition] + + # special case endpoints, as suggested in + # Cleve Moler, Numerical Computing with MATLAB, Chap 3.6 (pchiptx.m) + dk[0] = PchipInterpolator._edge_case(hk[0], hk[1], mk[0], mk[1]) + dk[-1] = PchipInterpolator._edge_case(hk[-1], hk[-2], mk[-1], mk[-2]) + + return dk.reshape(y_shape) + + +def pchip_interpolate(xi, yi, x, der=0, axis=0): + """ + Convenience function for pchip interpolation. + + xi and yi are arrays of values used to approximate some function f, + with ``yi = f(xi)``. The interpolant uses monotonic cubic splines + to find the value of new points x and the derivatives there. + + See `scipy.interpolate.PchipInterpolator` for details. + + Parameters + ---------- + xi : array_like + A sorted list of x-coordinates, of length N. + yi : array_like + A 1-D array of real values. `yi`'s length along the interpolation + axis must be equal to the length of `xi`. If N-D array, use axis + parameter to select correct axis. + + .. deprecated:: 1.13.0 + Complex data is deprecated and will raise an error in + SciPy 1.15.0. If you are trying to use the real components of + the passed array, use ``np.real`` on `yi`. + + x : scalar or array_like + Of length M. + der : int or list, optional + Derivatives to extract. The 0th derivative can be included to + return the function value. + axis : int, optional + Axis in the yi array corresponding to the x-coordinate values. + + Returns + ------- + y : scalar or array_like + The result, of length R or length M or M by R. + + See Also + -------- + PchipInterpolator : PCHIP 1-D monotonic cubic interpolator. + + Examples + -------- + We can interpolate 2D observed data using pchip interpolation: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import pchip_interpolate + >>> x_observed = np.linspace(0.0, 10.0, 11) + >>> y_observed = np.sin(x_observed) + >>> x = np.linspace(min(x_observed), max(x_observed), num=100) + >>> y = pchip_interpolate(x_observed, y_observed, x) + >>> plt.plot(x_observed, y_observed, "o", label="observation") + >>> plt.plot(x, y, label="pchip interpolation") + >>> plt.legend() + >>> plt.show() + + """ + P = PchipInterpolator(xi, yi, axis=axis) + + if der == 0: + return P(x) + elif _isscalar(der): + return P.derivative(der)(x) + else: + return [P.derivative(nu)(x) for nu in der] + + +class Akima1DInterpolator(CubicHermiteSpline): + r""" + Akima interpolator + + Fit piecewise cubic polynomials, given vectors x and y. The interpolation + method by Akima uses a continuously differentiable sub-spline built from + piecewise cubic polynomials. The resultant curve passes through the given + data points and will appear smooth and natural. + + Parameters + ---------- + x : ndarray, shape (npoints, ) + 1-D array of monotonically increasing real values. + y : ndarray, shape (..., npoints, ...) + N-D array of real values. The length of ``y`` along the interpolation axis + must be equal to the length of ``x``. Use the ``axis`` parameter to + select the interpolation axis. + axis : int, optional + Axis in the ``y`` array corresponding to the x-coordinate values. Defaults + to ``axis=0``. + method : {'akima', 'makima'}, optional + If ``"makima"``, use the modified Akima interpolation [2]_. + Defaults to ``"akima"``, use the Akima interpolation [1]_. + + .. versionadded:: 1.13.0 + + extrapolate : {bool, None}, optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. If None, + ``extrapolate`` is set to False. + + Methods + ------- + __call__ + derivative + antiderivative + roots + + See Also + -------- + PchipInterpolator : PCHIP 1-D monotonic cubic interpolator. + CubicSpline : Cubic spline data interpolator. + PPoly : Piecewise polynomial in terms of coefficients and breakpoints + + Notes + ----- + .. versionadded:: 0.14 + + Use only for precise data, as the fitted curve passes through the given + points exactly. This routine is useful for plotting a pleasingly smooth + curve through a few given points for purposes of plotting. + + Let :math:`\delta_i = (y_{i+1} - y_i) / (x_{i+1} - x_i)` be the slopes of + the interval :math:`\left[x_i, x_{i+1}\right)`. Akima's derivative at + :math:`x_i` is defined as: + + .. math:: + + d_i = \frac{w_1}{w_1 + w_2}\delta_{i-1} + \frac{w_2}{w_1 + w_2}\delta_i + + In the Akima interpolation [1]_ (``method="akima"``), the weights are: + + .. math:: + + \begin{aligned} + w_1 &= |\delta_{i+1} - \delta_i| \\ + w_2 &= |\delta_{i-1} - \delta_{i-2}| + \end{aligned} + + In the modified Akima interpolation [2]_ (``method="makima"``), + to eliminate overshoot and avoid edge cases of both numerator and + denominator being equal to 0, the weights are modified as follows: + + .. math:: + + \begin{align*} + w_1 &= |\delta_{i+1} - \delta_i| + |\delta_{i+1} + \delta_i| / 2 \\ + w_2 &= |\delta_{i-1} - \delta_{i-2}| + |\delta_{i-1} + \delta_{i-2}| / 2 + \end{align*} + + Examples + -------- + Comparison of ``method="akima"`` and ``method="makima"``: + + >>> import numpy as np + >>> from scipy.interpolate import Akima1DInterpolator + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(1, 7, 7) + >>> y = np.array([-1, -1, -1, 0, 1, 1, 1]) + >>> xs = np.linspace(min(x), max(x), num=100) + >>> y_akima = Akima1DInterpolator(x, y, method="akima")(xs) + >>> y_makima = Akima1DInterpolator(x, y, method="makima")(xs) + + >>> fig, ax = plt.subplots() + >>> ax.plot(x, y, "o", label="data") + >>> ax.plot(xs, y_akima, label="akima") + >>> ax.plot(xs, y_makima, label="makima") + >>> ax.legend() + >>> fig.show() + + The overshoot that occurred in ``"akima"`` has been avoided in ``"makima"``. + + References + ---------- + .. [1] A new method of interpolation and smooth curve fitting based + on local procedures. Hiroshi Akima, J. ACM, October 1970, 17(4), + 589-602. :doi:`10.1145/321607.321609` + .. [2] Makima Piecewise Cubic Interpolation. Cleve Moler and Cosmin Ionita, 2019. + https://blogs.mathworks.com/cleve/2019/04/29/makima-piecewise-cubic-interpolation/ + + """ + + def __init__(self, x, y, axis=0, *, method: Literal["akima", "makima"]="akima", + extrapolate:bool | None = None): + if method not in {"akima", "makima"}: + raise NotImplementedError(f"`method`={method} is unsupported.") + # Original implementation in MATLAB by N. Shamsundar (BSD licensed), see + # https://www.mathworks.com/matlabcentral/fileexchange/1814-akima-interpolation + x, dx, y, axis, _ = prepare_input(x, y, axis) + + if np.iscomplexobj(y): + msg = ("`Akima1DInterpolator` only works with real values for `y`. " + "If you are trying to use the real components of the passed array, " + "use `np.real` on the array before passing to " + "`Akima1DInterpolator`.") + raise ValueError(msg) + + # Akima extrapolation historically False; parent class defaults to True. + extrapolate = False if extrapolate is None else extrapolate + + # determine slopes between breakpoints + m = np.empty((x.size + 3, ) + y.shape[1:]) + dx = dx[(slice(None), ) + (None, ) * (y.ndim - 1)] + m[2:-2] = np.diff(y, axis=0) / dx + + # add two additional points on the left ... + m[1] = 2. * m[2] - m[3] + m[0] = 2. * m[1] - m[2] + # ... and on the right + m[-2] = 2. * m[-3] - m[-4] + m[-1] = 2. * m[-2] - m[-3] + + # if m1 == m2 != m3 == m4, the slope at the breakpoint is not + # defined. This is the fill value: + t = .5 * (m[3:] + m[:-3]) + # get the denominator of the slope t + dm = np.abs(np.diff(m, axis=0)) + if method == "makima": + pm = np.abs(m[1:] + m[:-1]) + f1 = dm[2:] + 0.5 * pm[2:] + f2 = dm[:-2] + 0.5 * pm[:-2] + else: + f1 = dm[2:] + f2 = dm[:-2] + f12 = f1 + f2 + # These are the mask of where the slope at breakpoint is defined: + ind = np.nonzero(f12 > 1e-9 * np.max(f12, initial=-np.inf)) + x_ind, y_ind = ind[0], ind[1:] + # Set the slope at breakpoint + t[ind] = (f1[ind] * m[(x_ind + 1,) + y_ind] + + f2[ind] * m[(x_ind + 2,) + y_ind]) / f12[ind] + + super().__init__(x, y, t, axis=0, extrapolate=extrapolate) + self.axis = axis + + def extend(self, c, x, right=True): + raise NotImplementedError("Extending a 1-D Akima interpolator is not " + "yet implemented") + + # These are inherited from PPoly, but they do not produce an Akima + # interpolator. Hence stub them out. + @classmethod + def from_spline(cls, tck, extrapolate=None): + raise NotImplementedError("This method does not make sense for " + "an Akima interpolator.") + + @classmethod + def from_bernstein_basis(cls, bp, extrapolate=None): + raise NotImplementedError("This method does not make sense for " + "an Akima interpolator.") + + +class CubicSpline(CubicHermiteSpline): + """Cubic spline data interpolator. + + Interpolate data with a piecewise cubic polynomial which is twice + continuously differentiable [1]_. The result is represented as a `PPoly` + instance with breakpoints matching the given data. + + Parameters + ---------- + x : array_like, shape (n,) + 1-D array containing values of the independent variable. + Values must be real, finite and in strictly increasing order. + y : array_like + Array containing values of the dependent variable. It can have + arbitrary number of dimensions, but the length along ``axis`` + (see below) must match the length of ``x``. Values must be finite. + axis : int, optional + Axis along which `y` is assumed to be varying. Meaning that for + ``x[i]`` the corresponding values are ``np.take(y, i, axis=axis)``. + Default is 0. + bc_type : string or 2-tuple, optional + Boundary condition type. Two additional equations, given by the + boundary conditions, are required to determine all coefficients of + polynomials on each segment [2]_. + + If `bc_type` is a string, then the specified condition will be applied + at both ends of a spline. Available conditions are: + + * 'not-a-knot' (default): The first and second segment at a curve end + are the same polynomial. It is a good default when there is no + information on boundary conditions. + * 'periodic': The interpolated functions is assumed to be periodic + of period ``x[-1] - x[0]``. The first and last value of `y` must be + identical: ``y[0] == y[-1]``. This boundary condition will result in + ``y'[0] == y'[-1]`` and ``y''[0] == y''[-1]``. + * 'clamped': The first derivative at curves ends are zero. Assuming + a 1D `y`, ``bc_type=((1, 0.0), (1, 0.0))`` is the same condition. + * 'natural': The second derivative at curve ends are zero. Assuming + a 1D `y`, ``bc_type=((2, 0.0), (2, 0.0))`` is the same condition. + + If `bc_type` is a 2-tuple, the first and the second value will be + applied at the curve start and end respectively. The tuple values can + be one of the previously mentioned strings (except 'periodic') or a + tuple ``(order, deriv_values)`` allowing to specify arbitrary + derivatives at curve ends: + + * `order`: the derivative order, 1 or 2. + * `deriv_value`: array_like containing derivative values, shape must + be the same as `y`, excluding ``axis`` dimension. For example, if + `y` is 1-D, then `deriv_value` must be a scalar. If `y` is 3-D with + the shape (n0, n1, n2) and axis=2, then `deriv_value` must be 2-D + and have the shape (n0, n1). + extrapolate : {bool, 'periodic', None}, optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. If 'periodic', + periodic extrapolation is used. If None (default), ``extrapolate`` is + set to 'periodic' for ``bc_type='periodic'`` and to True otherwise. + + Attributes + ---------- + x : ndarray, shape (n,) + Breakpoints. The same ``x`` which was passed to the constructor. + c : ndarray, shape (4, n-1, ...) + Coefficients of the polynomials on each segment. The trailing + dimensions match the dimensions of `y`, excluding ``axis``. + For example, if `y` is 1-d, then ``c[k, i]`` is a coefficient for + ``(x-x[i])**(3-k)`` on the segment between ``x[i]`` and ``x[i+1]``. + axis : int + Interpolation axis. The same axis which was passed to the + constructor. + + Methods + ------- + __call__ + derivative + antiderivative + integrate + roots + + See Also + -------- + Akima1DInterpolator : Akima 1D interpolator. + PchipInterpolator : PCHIP 1-D monotonic cubic interpolator. + PPoly : Piecewise polynomial in terms of coefficients and breakpoints. + + Notes + ----- + Parameters `bc_type` and ``extrapolate`` work independently, i.e. the + former controls only construction of a spline, and the latter only + evaluation. + + When a boundary condition is 'not-a-knot' and n = 2, it is replaced by + a condition that the first derivative is equal to the linear interpolant + slope. When both boundary conditions are 'not-a-knot' and n = 3, the + solution is sought as a parabola passing through given points. + + When 'not-a-knot' boundary conditions is applied to both ends, the + resulting spline will be the same as returned by `splrep` (with ``s=0``) + and `InterpolatedUnivariateSpline`, but these two methods use a + representation in B-spline basis. + + .. versionadded:: 0.18.0 + + Examples + -------- + In this example the cubic spline is used to interpolate a sampled sinusoid. + You can see that the spline continuity property holds for the first and + second derivatives and violates only for the third derivative. + + >>> import numpy as np + >>> from scipy.interpolate import CubicSpline + >>> import matplotlib.pyplot as plt + >>> x = np.arange(10) + >>> y = np.sin(x) + >>> cs = CubicSpline(x, y) + >>> xs = np.arange(-0.5, 9.6, 0.1) + >>> fig, ax = plt.subplots(figsize=(6.5, 4)) + >>> ax.plot(x, y, 'o', label='data') + >>> ax.plot(xs, np.sin(xs), label='true') + >>> ax.plot(xs, cs(xs), label="S") + >>> ax.plot(xs, cs(xs, 1), label="S'") + >>> ax.plot(xs, cs(xs, 2), label="S''") + >>> ax.plot(xs, cs(xs, 3), label="S'''") + >>> ax.set_xlim(-0.5, 9.5) + >>> ax.legend(loc='lower left', ncol=2) + >>> plt.show() + + In the second example, the unit circle is interpolated with a spline. A + periodic boundary condition is used. You can see that the first derivative + values, ds/dx=0, ds/dy=1 at the periodic point (1, 0) are correctly + computed. Note that a circle cannot be exactly represented by a cubic + spline. To increase precision, more breakpoints would be required. + + >>> theta = 2 * np.pi * np.linspace(0, 1, 5) + >>> y = np.c_[np.cos(theta), np.sin(theta)] + >>> cs = CubicSpline(theta, y, bc_type='periodic') + >>> print("ds/dx={:.1f} ds/dy={:.1f}".format(cs(0, 1)[0], cs(0, 1)[1])) + ds/dx=0.0 ds/dy=1.0 + >>> xs = 2 * np.pi * np.linspace(0, 1, 100) + >>> fig, ax = plt.subplots(figsize=(6.5, 4)) + >>> ax.plot(y[:, 0], y[:, 1], 'o', label='data') + >>> ax.plot(np.cos(xs), np.sin(xs), label='true') + >>> ax.plot(cs(xs)[:, 0], cs(xs)[:, 1], label='spline') + >>> ax.axes.set_aspect('equal') + >>> ax.legend(loc='center') + >>> plt.show() + + The third example is the interpolation of a polynomial y = x**3 on the + interval 0 <= x<= 1. A cubic spline can represent this function exactly. + To achieve that we need to specify values and first derivatives at + endpoints of the interval. Note that y' = 3 * x**2 and thus y'(0) = 0 and + y'(1) = 3. + + >>> cs = CubicSpline([0, 1], [0, 1], bc_type=((1, 0), (1, 3))) + >>> x = np.linspace(0, 1) + >>> np.allclose(x**3, cs(x)) + True + + References + ---------- + .. [1] `Cubic Spline Interpolation + `_ + on Wikiversity. + .. [2] Carl de Boor, "A Practical Guide to Splines", Springer-Verlag, 1978. + """ + + def __init__(self, x, y, axis=0, bc_type='not-a-knot', extrapolate=None): + x, dx, y, axis, _ = prepare_input(x, y, axis) + n = len(x) + + bc, y = self._validate_bc(bc_type, y, y.shape[1:], axis) + + if extrapolate is None: + if bc[0] == 'periodic': + extrapolate = 'periodic' + else: + extrapolate = True + + if y.size == 0: + # bail out early for zero-sized arrays + s = np.zeros_like(y) + else: + dxr = dx.reshape([dx.shape[0]] + [1] * (y.ndim - 1)) + slope = np.diff(y, axis=0) / dxr + + # If bc is 'not-a-knot' this change is just a convention. + # If bc is 'periodic' then we already checked that y[0] == y[-1], + # and the spline is just a constant, we handle this case in the + # same way by setting the first derivatives to slope, which is 0. + if n == 2: + if bc[0] in ['not-a-knot', 'periodic']: + bc[0] = (1, slope[0]) + if bc[1] in ['not-a-knot', 'periodic']: + bc[1] = (1, slope[0]) + + # This is a special case, when both conditions are 'not-a-knot' + # and n == 3. In this case 'not-a-knot' can't be handled regularly + # as the both conditions are identical. We handle this case by + # constructing a parabola passing through given points. + if n == 3 and bc[0] == 'not-a-knot' and bc[1] == 'not-a-knot': + A = np.zeros((3, 3)) # This is a standard matrix. + b = np.empty((3,) + y.shape[1:], dtype=y.dtype) + + A[0, 0] = 1 + A[0, 1] = 1 + A[1, 0] = dx[1] + A[1, 1] = 2 * (dx[0] + dx[1]) + A[1, 2] = dx[0] + A[2, 1] = 1 + A[2, 2] = 1 + + b[0] = 2 * slope[0] + b[1] = 3 * (dxr[0] * slope[1] + dxr[1] * slope[0]) + b[2] = 2 * slope[1] + + s = solve(A, b, overwrite_a=True, overwrite_b=True, + check_finite=False) + elif n == 3 and bc[0] == 'periodic': + # In case when number of points is 3 we compute the derivatives + # manually + t = (slope / dxr).sum(0) / (1. / dxr).sum(0) + s = np.broadcast_to(t, (n,) + y.shape[1:]) + else: + # Find derivative values at each x[i] by solving a tridiagonal + # system. + A = np.zeros((3, n)) # This is a banded matrix representation. + b = np.empty((n,) + y.shape[1:], dtype=y.dtype) + + # Filling the system for i=1..n-2 + # (x[i-1] - x[i]) * s[i-1] +\ + # 2 * ((x[i] - x[i-1]) + (x[i+1] - x[i])) * s[i] +\ + # (x[i] - x[i-1]) * s[i+1] =\ + # 3 * ((x[i+1] - x[i])*(y[i] - y[i-1])/(x[i] - x[i-1]) +\ + # (x[i] - x[i-1])*(y[i+1] - y[i])/(x[i+1] - x[i])) + + A[1, 1:-1] = 2 * (dx[:-1] + dx[1:]) # The diagonal + A[0, 2:] = dx[:-1] # The upper diagonal + A[-1, :-2] = dx[1:] # The lower diagonal + + b[1:-1] = 3 * (dxr[1:] * slope[:-1] + dxr[:-1] * slope[1:]) + + bc_start, bc_end = bc + + if bc_start == 'periodic': + # Due to the periodicity, and because y[-1] = y[0], the + # linear system has (n-1) unknowns/equations instead of n: + A = A[:, 0:-1] + A[1, 0] = 2 * (dx[-1] + dx[0]) + A[0, 1] = dx[-1] + + b = b[:-1] + + # Also, due to the periodicity, the system is not tri-diagonal. + # We need to compute a "condensed" matrix of shape (n-2, n-2). + # See https://web.archive.org/web/20151220180652/http://www.cfm.brown.edu/people/gk/chap6/node14.html + # for more explanations. + # The condensed matrix is obtained by removing the last column + # and last row of the (n-1, n-1) system matrix. The removed + # values are saved in scalar variables with the (n-1, n-1) + # system matrix indices forming their names: + a_m1_0 = dx[-2] # lower left corner value: A[-1, 0] + a_m1_m2 = dx[-1] + a_m1_m1 = 2 * (dx[-1] + dx[-2]) + a_m2_m1 = dx[-3] + a_0_m1 = dx[0] + + b[0] = 3 * (dxr[0] * slope[-1] + dxr[-1] * slope[0]) + b[-1] = 3 * (dxr[-1] * slope[-2] + dxr[-2] * slope[-1]) + + Ac = A[:, :-1] + b1 = b[:-1] + b2 = np.zeros_like(b1) + b2[0] = -a_0_m1 + b2[-1] = -a_m2_m1 + + # s1 and s2 are the solutions of (n-2, n-2) system + s1 = solve_banded((1, 1), Ac, b1, overwrite_ab=False, + overwrite_b=False, check_finite=False) + + s2 = solve_banded((1, 1), Ac, b2, overwrite_ab=False, + overwrite_b=False, check_finite=False) + + # computing the s[n-2] solution: + s_m1 = ((b[-1] - a_m1_0 * s1[0] - a_m1_m2 * s1[-1]) / + (a_m1_m1 + a_m1_0 * s2[0] + a_m1_m2 * s2[-1])) + + # s is the solution of the (n, n) system: + s = np.empty((n,) + y.shape[1:], dtype=y.dtype) + s[:-2] = s1 + s_m1 * s2 + s[-2] = s_m1 + s[-1] = s[0] + else: + if bc_start == 'not-a-knot': + A[1, 0] = dx[1] + A[0, 1] = x[2] - x[0] + d = x[2] - x[0] + b[0] = ((dxr[0] + 2*d) * dxr[1] * slope[0] + + dxr[0]**2 * slope[1]) / d + elif bc_start[0] == 1: + A[1, 0] = 1 + A[0, 1] = 0 + b[0] = bc_start[1] + elif bc_start[0] == 2: + A[1, 0] = 2 * dx[0] + A[0, 1] = dx[0] + b[0] = -0.5 * bc_start[1] * dx[0]**2 + 3 * (y[1] - y[0]) + + if bc_end == 'not-a-knot': + A[1, -1] = dx[-2] + A[-1, -2] = x[-1] - x[-3] + d = x[-1] - x[-3] + b[-1] = ((dxr[-1]**2*slope[-2] + + (2*d + dxr[-1])*dxr[-2]*slope[-1]) / d) + elif bc_end[0] == 1: + A[1, -1] = 1 + A[-1, -2] = 0 + b[-1] = bc_end[1] + elif bc_end[0] == 2: + A[1, -1] = 2 * dx[-1] + A[-1, -2] = dx[-1] + b[-1] = 0.5 * bc_end[1] * dx[-1]**2 + 3 * (y[-1] - y[-2]) + + s = solve_banded((1, 1), A, b, overwrite_ab=True, + overwrite_b=True, check_finite=False) + + super().__init__(x, y, s, axis=0, extrapolate=extrapolate) + self.axis = axis + + @staticmethod + def _validate_bc(bc_type, y, expected_deriv_shape, axis): + """Validate and prepare boundary conditions. + + Returns + ------- + validated_bc : 2-tuple + Boundary conditions for a curve start and end. + y : ndarray + y casted to complex dtype if one of the boundary conditions has + complex dtype. + """ + if isinstance(bc_type, str): + if bc_type == 'periodic': + if not np.allclose(y[0], y[-1], rtol=1e-15, atol=1e-15): + raise ValueError( + f"The first and last `y` point along axis {axis} must " + "be identical (within machine precision) when " + "bc_type='periodic'.") + + bc_type = (bc_type, bc_type) + + else: + if len(bc_type) != 2: + raise ValueError("`bc_type` must contain 2 elements to " + "specify start and end conditions.") + + if 'periodic' in bc_type: + raise ValueError("'periodic' `bc_type` is defined for both " + "curve ends and cannot be used with other " + "boundary conditions.") + + validated_bc = [] + for bc in bc_type: + if isinstance(bc, str): + if bc == 'clamped': + validated_bc.append((1, np.zeros(expected_deriv_shape))) + elif bc == 'natural': + validated_bc.append((2, np.zeros(expected_deriv_shape))) + elif bc in ['not-a-knot', 'periodic']: + validated_bc.append(bc) + else: + raise ValueError(f"bc_type={bc} is not allowed.") + else: + try: + deriv_order, deriv_value = bc + except Exception as e: + raise ValueError( + "A specified derivative value must be " + "given in the form (order, value)." + ) from e + + if deriv_order not in [1, 2]: + raise ValueError("The specified derivative order must " + "be 1 or 2.") + + deriv_value = np.asarray(deriv_value) + if deriv_value.shape != expected_deriv_shape: + raise ValueError( + f"`deriv_value` shape {deriv_value.shape} is not " + f"the expected one {expected_deriv_shape}." + ) + + if np.issubdtype(deriv_value.dtype, np.complexfloating): + y = y.astype(complex, copy=False) + + validated_bc.append((deriv_order, deriv_value)) + + return validated_bc, y diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..5f1de8a506d90631543927894cd35196744a879a Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack2.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack2.py new file mode 100644 index 0000000000000000000000000000000000000000..daa7773a0f3be6590ea9c0ab328d610efa745859 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack2.py @@ -0,0 +1,2394 @@ +""" +fitpack --- curve and surface fitting with splines + +fitpack is based on a collection of Fortran routines DIERCKX +by P. Dierckx (see http://www.netlib.org/dierckx/) transformed +to double routines by Pearu Peterson. +""" +# Created by Pearu Peterson, June,August 2003 +__all__ = [ + 'UnivariateSpline', + 'InterpolatedUnivariateSpline', + 'LSQUnivariateSpline', + 'BivariateSpline', + 'LSQBivariateSpline', + 'SmoothBivariateSpline', + 'LSQSphereBivariateSpline', + 'SmoothSphereBivariateSpline', + 'RectBivariateSpline', + 'RectSphereBivariateSpline'] + + +import warnings +from threading import Lock + +from numpy import zeros, concatenate, ravel, diff, array +import numpy as np + +from . import _fitpack_impl +from . import _dfitpack as dfitpack + + +dfitpack_int = dfitpack.types.intvar.dtype +FITPACK_LOCK = Lock() + + +# ############### Univariate spline #################### + +_curfit_messages = {1: """ +The required storage space exceeds the available storage space, as +specified by the parameter nest: nest too small. If nest is already +large (say nest > m/2), it may also indicate that s is too small. +The approximation returned is the weighted least-squares spline +according to the knots t[0],t[1],...,t[n-1]. (n=nest) the parameter fp +gives the corresponding weighted sum of squared residuals (fp>s). +""", + 2: """ +A theoretically impossible result was found during the iteration +process for finding a smoothing spline with fp = s: s too small. +There is an approximation returned but the corresponding weighted sum +of squared residuals does not satisfy the condition abs(fp-s)/s < tol.""", + 3: """ +The maximal number of iterations maxit (set to 20 by the program) +allowed for finding a smoothing spline with fp=s has been reached: s +too small. +There is an approximation returned but the corresponding weighted sum +of squared residuals does not satisfy the condition abs(fp-s)/s < tol.""", + 10: """ +Error on entry, no approximation returned. The following conditions +must hold: +xb<=x[0]0, i=0..m-1 +if iopt=-1: + xb>> import numpy as np + >>> from scipy.interpolate import UnivariateSpline + >>> x, y = np.array([1, 2, 3, 4]), np.array([1, np.nan, 3, 4]) + >>> w = np.isnan(y) + >>> y[w] = 0. + >>> spl = UnivariateSpline(x, y, w=~w) + + Notice the need to replace a ``nan`` by a numerical value (precise value + does not matter as long as the corresponding weight is zero.) + + References + ---------- + Based on algorithms described in [1]_, [2]_, [3]_, and [4]_: + + .. [1] P. Dierckx, "An algorithm for smoothing, differentiation and + integration of experimental data using spline functions", + J.Comp.Appl.Maths 1 (1975) 165-184. + .. [2] P. Dierckx, "A fast algorithm for smoothing data on a rectangular + grid while using spline functions", SIAM J.Numer.Anal. 19 (1982) + 1286-1304. + .. [3] P. Dierckx, "An improved algorithm for curve fitting with spline + functions", report tw54, Dept. Computer Science,K.U. Leuven, 1981. + .. [4] P. Dierckx, "Curve and surface fitting with splines", Monographs on + Numerical Analysis, Oxford University Press, 1993. + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import UnivariateSpline + >>> rng = np.random.default_rng() + >>> x = np.linspace(-3, 3, 50) + >>> y = np.exp(-x**2) + 0.1 * rng.standard_normal(50) + >>> plt.plot(x, y, 'ro', ms=5) + + Use the default value for the smoothing parameter: + + >>> spl = UnivariateSpline(x, y) + >>> xs = np.linspace(-3, 3, 1000) + >>> plt.plot(xs, spl(xs), 'g', lw=3) + + Manually change the amount of smoothing: + + >>> spl.set_smoothing_factor(0.5) + >>> plt.plot(xs, spl(xs), 'b', lw=3) + >>> plt.show() + + """ + + def __init__(self, x, y, w=None, bbox=[None]*2, k=3, s=None, + ext=0, check_finite=False): + + x, y, w, bbox, self.ext = self.validate_input(x, y, w, bbox, k, s, ext, + check_finite) + + # _data == x,y,w,xb,xe,k,s,n,t,c,fp,fpint,nrdata,ier + with FITPACK_LOCK: + data = dfitpack.fpcurf0(x, y, k, w=w, xb=bbox[0], + xe=bbox[1], s=s) + if data[-1] == 1: + # nest too small, setting to maximum bound + data = self._reset_nest(data) + self._data = data + self._reset_class() + + @staticmethod + def validate_input(x, y, w, bbox, k, s, ext, check_finite): + x, y, bbox = np.asarray(x), np.asarray(y), np.asarray(bbox) + if w is not None: + w = np.asarray(w) + if check_finite: + w_finite = np.isfinite(w).all() if w is not None else True + if (not np.isfinite(x).all() or not np.isfinite(y).all() or + not w_finite): + raise ValueError("x and y array must not contain " + "NaNs or infs.") + if s is None or s > 0: + if not np.all(diff(x) >= 0.0): + raise ValueError("x must be increasing if s > 0") + else: + if not np.all(diff(x) > 0.0): + raise ValueError("x must be strictly increasing if s = 0") + if x.size != y.size: + raise ValueError("x and y should have a same length") + elif w is not None and not x.size == y.size == w.size: + raise ValueError("x, y, and w should have a same length") + elif bbox.shape != (2,): + raise ValueError("bbox shape should be (2,)") + elif not (1 <= k <= 5): + raise ValueError("k should be 1 <= k <= 5") + elif s is not None and not s >= 0.0: + raise ValueError("s should be s >= 0.0") + + try: + ext = _extrap_modes[ext] + except KeyError as e: + raise ValueError(f"Unknown extrapolation mode {ext}.") from e + + return x, y, w, bbox, ext + + @classmethod + def _from_tck(cls, tck, ext=0): + """Construct a spline object from given tck""" + self = cls.__new__(cls) + t, c, k = tck + self._eval_args = tck + # _data == x,y,w,xb,xe,k,s,n,t,c,fp,fpint,nrdata,ier + self._data = (None, None, None, None, None, k, None, len(t), t, + c, None, None, None, None) + self.ext = ext + return self + + def _reset_class(self): + data = self._data + n, t, c, k, ier = data[7], data[8], data[9], data[5], data[-1] + self._eval_args = t[:n], c[:n], k + if ier == 0: + # the spline returned has a residual sum of squares fp + # such that abs(fp-s)/s <= tol with tol a relative + # tolerance set to 0.001 by the program + pass + elif ier == -1: + # the spline returned is an interpolating spline + self._set_class(InterpolatedUnivariateSpline) + elif ier == -2: + # the spline returned is the weighted least-squares + # polynomial of degree k. In this extreme case fp gives + # the upper bound fp0 for the smoothing factor s. + self._set_class(LSQUnivariateSpline) + else: + # error + if ier == 1: + self._set_class(LSQUnivariateSpline) + message = _curfit_messages.get(ier, f'ier={ier}') + warnings.warn(message, stacklevel=3) + + def _set_class(self, cls): + self._spline_class = cls + if self.__class__ in (UnivariateSpline, InterpolatedUnivariateSpline, + LSQUnivariateSpline): + self.__class__ = cls + else: + # It's an unknown subclass -- don't change class. cf. #731 + pass + + def _reset_nest(self, data, nest=None): + n = data[10] + if nest is None: + k, m = data[5], len(data[0]) + nest = m+k+1 # this is the maximum bound for nest + else: + if not n <= nest: + raise ValueError("`nest` can only be increased") + t, c, fpint, nrdata = (np.resize(data[j], nest) for j in + [8, 9, 11, 12]) + + args = data[:8] + (t, c, n, fpint, nrdata, data[13]) + with FITPACK_LOCK: + data = dfitpack.fpcurf1(*args) + return data + + def set_smoothing_factor(self, s): + """ Continue spline computation with the given smoothing + factor s and with the knots found at the last call. + + This routine modifies the spline in place. + + """ + data = self._data + if data[6] == -1: + warnings.warn('smoothing factor unchanged for' + 'LSQ spline with fixed knots', + stacklevel=2) + return + args = data[:6] + (s,) + data[7:] + with FITPACK_LOCK: + data = dfitpack.fpcurf1(*args) + if data[-1] == 1: + # nest too small, setting to maximum bound + data = self._reset_nest(data) + self._data = data + self._reset_class() + + def __call__(self, x, nu=0, ext=None): + """ + Evaluate spline (or its nu-th derivative) at positions x. + + Parameters + ---------- + x : array_like + A 1-D array of points at which to return the value of the smoothed + spline or its derivatives. Note: `x` can be unordered but the + evaluation is more efficient if `x` is (partially) ordered. + nu : int + The order of derivative of the spline to compute. + ext : int + Controls the value returned for elements of `x` not in the + interval defined by the knot sequence. + + * if ext=0 or 'extrapolate', return the extrapolated value. + * if ext=1 or 'zeros', return 0 + * if ext=2 or 'raise', raise a ValueError + * if ext=3 or 'const', return the boundary value. + + The default value is 0, passed from the initialization of + UnivariateSpline. + + """ + x = np.asarray(x) + # empty input yields empty output + if x.size == 0: + return array([]) + if ext is None: + ext = self.ext + else: + try: + ext = _extrap_modes[ext] + except KeyError as e: + raise ValueError(f"Unknown extrapolation mode {ext}.") from e + with FITPACK_LOCK: + return _fitpack_impl.splev(x, self._eval_args, der=nu, ext=ext) + + def get_knots(self): + """ Return positions of interior knots of the spline. + + Internally, the knot vector contains ``2*k`` additional boundary knots. + """ + data = self._data + k, n = data[5], data[7] + return data[8][k:n-k] + + def get_coeffs(self): + """Return spline coefficients.""" + data = self._data + k, n = data[5], data[7] + return data[9][:n-k-1] + + def get_residual(self): + """Return weighted sum of squared residuals of the spline approximation. + + This is equivalent to:: + + sum((w[i] * (y[i]-spl(x[i])))**2, axis=0) + + """ + return self._data[10] + + def integral(self, a, b): + """ Return definite integral of the spline between two given points. + + Parameters + ---------- + a : float + Lower limit of integration. + b : float + Upper limit of integration. + + Returns + ------- + integral : float + The value of the definite integral of the spline between limits. + + Examples + -------- + >>> import numpy as np + >>> from scipy.interpolate import UnivariateSpline + >>> x = np.linspace(0, 3, 11) + >>> y = x**2 + >>> spl = UnivariateSpline(x, y) + >>> spl.integral(0, 3) + 9.0 + + which agrees with :math:`\\int x^2 dx = x^3 / 3` between the limits + of 0 and 3. + + A caveat is that this routine assumes the spline to be zero outside of + the data limits: + + >>> spl.integral(-1, 4) + 9.0 + >>> spl.integral(-1, 0) + 0.0 + + """ + with FITPACK_LOCK: + return _fitpack_impl.splint(a, b, self._eval_args) + + def derivatives(self, x): + """ Return all derivatives of the spline at the point x. + + Parameters + ---------- + x : float + The point to evaluate the derivatives at. + + Returns + ------- + der : ndarray, shape(k+1,) + Derivatives of the orders 0 to k. + + Examples + -------- + >>> import numpy as np + >>> from scipy.interpolate import UnivariateSpline + >>> x = np.linspace(0, 3, 11) + >>> y = x**2 + >>> spl = UnivariateSpline(x, y) + >>> spl.derivatives(1.5) + array([2.25, 3.0, 2.0, 0]) + + """ + with FITPACK_LOCK: + return _fitpack_impl.spalde(x, self._eval_args) + + def roots(self): + """ Return the zeros of the spline. + + Notes + ----- + Restriction: only cubic splines are supported by FITPACK. For non-cubic + splines, use `PPoly.root` (see below for an example). + + Examples + -------- + + For some data, this method may miss a root. This happens when one of + the spline knots (which FITPACK places automatically) happens to + coincide with the true root. A workaround is to convert to `PPoly`, + which uses a different root-finding algorithm. + + For example, + + >>> x = [1.96, 1.97, 1.98, 1.99, 2.00, 2.01, 2.02, 2.03, 2.04, 2.05] + >>> y = [-6.365470e-03, -4.790580e-03, -3.204320e-03, -1.607270e-03, + ... 4.440892e-16, 1.616930e-03, 3.243000e-03, 4.877670e-03, + ... 6.520430e-03, 8.170770e-03] + >>> from scipy.interpolate import UnivariateSpline + >>> spl = UnivariateSpline(x, y, s=0) + >>> spl.roots() + array([], dtype=float64) + + Converting to a PPoly object does find the roots at `x=2`: + + >>> from scipy.interpolate import splrep, PPoly + >>> tck = splrep(x, y, s=0) + >>> ppoly = PPoly.from_spline(tck) + >>> ppoly.roots(extrapolate=False) + array([2.]) + + See Also + -------- + sproot + PPoly.roots + + """ + k = self._data[5] + if k == 3: + t = self._eval_args[0] + mest = 3 * (len(t) - 7) + with FITPACK_LOCK: + return _fitpack_impl.sproot(self._eval_args, mest=mest) + raise NotImplementedError('finding roots unsupported for ' + 'non-cubic splines') + + def derivative(self, n=1): + """ + Construct a new spline representing the derivative of this spline. + + Parameters + ---------- + n : int, optional + Order of derivative to evaluate. Default: 1 + + Returns + ------- + spline : UnivariateSpline + Spline of order k2=k-n representing the derivative of this + spline. + + See Also + -------- + splder, antiderivative + + Notes + ----- + + .. versionadded:: 0.13.0 + + Examples + -------- + This can be used for finding maxima of a curve: + + >>> import numpy as np + >>> from scipy.interpolate import UnivariateSpline + >>> x = np.linspace(0, 10, 70) + >>> y = np.sin(x) + >>> spl = UnivariateSpline(x, y, k=4, s=0) + + Now, differentiate the spline and find the zeros of the + derivative. (NB: `sproot` only works for order 3 splines, so we + fit an order 4 spline): + + >>> spl.derivative().roots() / np.pi + array([ 0.50000001, 1.5 , 2.49999998]) + + This agrees well with roots :math:`\\pi/2 + n\\pi` of + :math:`\\cos(x) = \\sin'(x)`. + + """ + with FITPACK_LOCK: + tck = _fitpack_impl.splder(self._eval_args, n) + # if self.ext is 'const', derivative.ext will be 'zeros' + ext = 1 if self.ext == 3 else self.ext + return UnivariateSpline._from_tck(tck, ext=ext) + + def antiderivative(self, n=1): + """ + Construct a new spline representing the antiderivative of this spline. + + Parameters + ---------- + n : int, optional + Order of antiderivative to evaluate. Default: 1 + + Returns + ------- + spline : UnivariateSpline + Spline of order k2=k+n representing the antiderivative of this + spline. + + Notes + ----- + + .. versionadded:: 0.13.0 + + See Also + -------- + splantider, derivative + + Examples + -------- + >>> import numpy as np + >>> from scipy.interpolate import UnivariateSpline + >>> x = np.linspace(0, np.pi/2, 70) + >>> y = 1 / np.sqrt(1 - 0.8*np.sin(x)**2) + >>> spl = UnivariateSpline(x, y, s=0) + + The derivative is the inverse operation of the antiderivative, + although some floating point error accumulates: + + >>> spl(1.7), spl.antiderivative().derivative()(1.7) + (array(2.1565429877197317), array(2.1565429877201865)) + + Antiderivative can be used to evaluate definite integrals: + + >>> ispl = spl.antiderivative() + >>> ispl(np.pi/2) - ispl(0) + 2.2572053588768486 + + This is indeed an approximation to the complete elliptic integral + :math:`K(m) = \\int_0^{\\pi/2} [1 - m\\sin^2 x]^{-1/2} dx`: + + >>> from scipy.special import ellipk + >>> ellipk(0.8) + 2.2572053268208538 + + """ + with FITPACK_LOCK: + tck = _fitpack_impl.splantider(self._eval_args, n) + return UnivariateSpline._from_tck(tck, self.ext) + + +class InterpolatedUnivariateSpline(UnivariateSpline): + """ + 1-D interpolating spline for a given set of data points. + + .. legacy:: class + + Specifically, we recommend using `make_interp_spline` instead. + + Fits a spline y = spl(x) of degree `k` to the provided `x`, `y` data. + Spline function passes through all provided points. Equivalent to + `UnivariateSpline` with `s` = 0. + + Parameters + ---------- + x : (N,) array_like + Input dimension of data points -- must be strictly increasing + y : (N,) array_like + input dimension of data points + w : (N,) array_like, optional + Weights for spline fitting. Must be positive. If None (default), + weights are all 1. + bbox : (2,) array_like, optional + 2-sequence specifying the boundary of the approximation interval. If + None (default), ``bbox=[x[0], x[-1]]``. + k : int, optional + Degree of the smoothing spline. Must be ``1 <= k <= 5``. Default is + ``k = 3``, a cubic spline. + ext : int or str, optional + Controls the extrapolation mode for elements + not in the interval defined by the knot sequence. + + * if ext=0 or 'extrapolate', return the extrapolated value. + * if ext=1 or 'zeros', return 0 + * if ext=2 or 'raise', raise a ValueError + * if ext=3 of 'const', return the boundary value. + + The default value is 0. + + check_finite : bool, optional + Whether to check that the input arrays contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination or non-sensical results) if the inputs + do contain infinities or NaNs. + Default is False. + + See Also + -------- + UnivariateSpline : + a smooth univariate spline to fit a given set of data points. + LSQUnivariateSpline : + a spline for which knots are user-selected + SmoothBivariateSpline : + a smoothing bivariate spline through the given points + LSQBivariateSpline : + a bivariate spline using weighted least-squares fitting + splrep : + a function to find the B-spline representation of a 1-D curve + splev : + a function to evaluate a B-spline or its derivatives + sproot : + a function to find the roots of a cubic B-spline + splint : + a function to evaluate the definite integral of a B-spline between two + given points + spalde : + a function to evaluate all derivatives of a B-spline + + Notes + ----- + The number of data points must be larger than the spline degree `k`. + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import InterpolatedUnivariateSpline + >>> rng = np.random.default_rng() + >>> x = np.linspace(-3, 3, 50) + >>> y = np.exp(-x**2) + 0.1 * rng.standard_normal(50) + >>> spl = InterpolatedUnivariateSpline(x, y) + >>> plt.plot(x, y, 'ro', ms=5) + >>> xs = np.linspace(-3, 3, 1000) + >>> plt.plot(xs, spl(xs), 'g', lw=3, alpha=0.7) + >>> plt.show() + + Notice that the ``spl(x)`` interpolates `y`: + + >>> spl.get_residual() + 0.0 + + """ + + def __init__(self, x, y, w=None, bbox=[None]*2, k=3, + ext=0, check_finite=False): + + x, y, w, bbox, self.ext = self.validate_input(x, y, w, bbox, k, None, + ext, check_finite) + if not np.all(diff(x) > 0.0): + raise ValueError('x must be strictly increasing') + + # _data == x,y,w,xb,xe,k,s,n,t,c,fp,fpint,nrdata,ier + with FITPACK_LOCK: + self._data = dfitpack.fpcurf0(x, y, k, w=w, xb=bbox[0], + xe=bbox[1], s=0) + self._reset_class() + + +_fpchec_error_string = """The input parameters have been rejected by fpchec. \ +This means that at least one of the following conditions is violated: + +1) k+1 <= n-k-1 <= m +2) t(1) <= t(2) <= ... <= t(k+1) + t(n-k) <= t(n-k+1) <= ... <= t(n) +3) t(k+1) < t(k+2) < ... < t(n-k) +4) t(k+1) <= x(i) <= t(n-k) +5) The conditions specified by Schoenberg and Whitney must hold + for at least one subset of data points, i.e., there must be a + subset of data points y(j) such that + t(j) < y(j) < t(j+k+1), j=1,2,...,n-k-1 +""" + + +class LSQUnivariateSpline(UnivariateSpline): + """ + 1-D spline with explicit internal knots. + + .. legacy:: class + + Specifically, we recommend using `make_lsq_spline` instead. + + + Fits a spline y = spl(x) of degree `k` to the provided `x`, `y` data. `t` + specifies the internal knots of the spline + + Parameters + ---------- + x : (N,) array_like + Input dimension of data points -- must be increasing + y : (N,) array_like + Input dimension of data points + t : (M,) array_like + interior knots of the spline. Must be in ascending order and:: + + bbox[0] < t[0] < ... < t[-1] < bbox[-1] + + w : (N,) array_like, optional + weights for spline fitting. Must be positive. If None (default), + weights are all 1. + bbox : (2,) array_like, optional + 2-sequence specifying the boundary of the approximation interval. If + None (default), ``bbox = [x[0], x[-1]]``. + k : int, optional + Degree of the smoothing spline. Must be 1 <= `k` <= 5. + Default is `k` = 3, a cubic spline. + ext : int or str, optional + Controls the extrapolation mode for elements + not in the interval defined by the knot sequence. + + * if ext=0 or 'extrapolate', return the extrapolated value. + * if ext=1 or 'zeros', return 0 + * if ext=2 or 'raise', raise a ValueError + * if ext=3 of 'const', return the boundary value. + + The default value is 0. + + check_finite : bool, optional + Whether to check that the input arrays contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination or non-sensical results) if the inputs + do contain infinities or NaNs. + Default is False. + + Raises + ------ + ValueError + If the interior knots do not satisfy the Schoenberg-Whitney conditions + + See Also + -------- + UnivariateSpline : + a smooth univariate spline to fit a given set of data points. + InterpolatedUnivariateSpline : + a interpolating univariate spline for a given set of data points. + splrep : + a function to find the B-spline representation of a 1-D curve + splev : + a function to evaluate a B-spline or its derivatives + sproot : + a function to find the roots of a cubic B-spline + splint : + a function to evaluate the definite integral of a B-spline between two + given points + spalde : + a function to evaluate all derivatives of a B-spline + + Notes + ----- + The number of data points must be larger than the spline degree `k`. + + Knots `t` must satisfy the Schoenberg-Whitney conditions, + i.e., there must be a subset of data points ``x[j]`` such that + ``t[j] < x[j] < t[j+k+1]``, for ``j=0, 1,...,n-k-2``. + + Examples + -------- + >>> import numpy as np + >>> from scipy.interpolate import LSQUnivariateSpline, UnivariateSpline + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + >>> x = np.linspace(-3, 3, 50) + >>> y = np.exp(-x**2) + 0.1 * rng.standard_normal(50) + + Fit a smoothing spline with a pre-defined internal knots: + + >>> t = [-1, 0, 1] + >>> spl = LSQUnivariateSpline(x, y, t) + + >>> xs = np.linspace(-3, 3, 1000) + >>> plt.plot(x, y, 'ro', ms=5) + >>> plt.plot(xs, spl(xs), 'g-', lw=3) + >>> plt.show() + + Check the knot vector: + + >>> spl.get_knots() + array([-3., -1., 0., 1., 3.]) + + Constructing lsq spline using the knots from another spline: + + >>> x = np.arange(10) + >>> s = UnivariateSpline(x, x, s=0) + >>> s.get_knots() + array([ 0., 2., 3., 4., 5., 6., 7., 9.]) + >>> knt = s.get_knots() + >>> s1 = LSQUnivariateSpline(x, x, knt[1:-1]) # Chop 1st and last knot + >>> s1.get_knots() + array([ 0., 2., 3., 4., 5., 6., 7., 9.]) + + """ + + def __init__(self, x, y, t, w=None, bbox=[None]*2, k=3, + ext=0, check_finite=False): + + x, y, w, bbox, self.ext = self.validate_input(x, y, w, bbox, k, None, + ext, check_finite) + if not np.all(diff(x) >= 0.0): + raise ValueError('x must be increasing') + + # _data == x,y,w,xb,xe,k,s,n,t,c,fp,fpint,nrdata,ier + xb = bbox[0] + xe = bbox[1] + if xb is None: + xb = x[0] + if xe is None: + xe = x[-1] + t = concatenate(([xb]*(k+1), t, [xe]*(k+1))) + n = len(t) + if not np.all(t[k+1:n-k]-t[k:n-k-1] > 0, axis=0): + raise ValueError('Interior knots t must satisfy ' + 'Schoenberg-Whitney conditions') + with FITPACK_LOCK: + if not dfitpack.fpchec(x, t, k) == 0: + raise ValueError(_fpchec_error_string) + data = dfitpack.fpcurfm1(x, y, k, t, w=w, xb=xb, xe=xe) + self._data = data[:-3] + (None, None, data[-1]) + self._reset_class() + + +# ############### Bivariate spline #################### + +class _BivariateSplineBase: + """ Base class for Bivariate spline s(x,y) interpolation on the rectangle + [xb,xe] x [yb, ye] calculated from a given set of data points + (x,y,z). + + See Also + -------- + bisplrep : + a function to find a bivariate B-spline representation of a surface + bisplev : + a function to evaluate a bivariate B-spline and its derivatives + BivariateSpline : + a base class for bivariate splines. + SphereBivariateSpline : + a bivariate spline on a spherical grid + """ + + @classmethod + def _from_tck(cls, tck): + """Construct a spline object from given tck and degree""" + self = cls.__new__(cls) + if len(tck) != 5: + raise ValueError("tck should be a 5 element tuple of tx," + " ty, c, kx, ky") + self.tck = tck[:3] + self.degrees = tck[3:] + return self + + def get_residual(self): + """ Return weighted sum of squared residuals of the spline + approximation: sum ((w[i]*(z[i]-s(x[i],y[i])))**2,axis=0) + """ + return self.fp + + def get_knots(self): + """ Return a tuple (tx,ty) where tx,ty contain knots positions + of the spline with respect to x-, y-variable, respectively. + The position of interior and additional knots are given as + t[k+1:-k-1] and t[:k+1]=b, t[-k-1:]=e, respectively. + """ + return self.tck[:2] + + def get_coeffs(self): + """ Return spline coefficients.""" + return self.tck[2] + + def __call__(self, x, y, dx=0, dy=0, grid=True): + """ + Evaluate the spline or its derivatives at given positions. + + Parameters + ---------- + x, y : array_like + Input coordinates. + + If `grid` is False, evaluate the spline at points ``(x[i], + y[i]), i=0, ..., len(x)-1``. Standard Numpy broadcasting + is obeyed. + + If `grid` is True: evaluate spline at the grid points + defined by the coordinate arrays x, y. The arrays must be + sorted to increasing order. + + The ordering of axes is consistent with + ``np.meshgrid(..., indexing="ij")`` and inconsistent with the + default ordering ``np.meshgrid(..., indexing="xy")``. + dx : int + Order of x-derivative + + .. versionadded:: 0.14.0 + dy : int + Order of y-derivative + + .. versionadded:: 0.14.0 + grid : bool + Whether to evaluate the results on a grid spanned by the + input arrays, or at points specified by the input arrays. + + .. versionadded:: 0.14.0 + + Examples + -------- + Suppose that we want to bilinearly interpolate an exponentially decaying + function in 2 dimensions. + + >>> import numpy as np + >>> from scipy.interpolate import RectBivariateSpline + + We sample the function on a coarse grid. Note that the default indexing="xy" + of meshgrid would result in an unexpected (transposed) result after + interpolation. + + >>> xarr = np.linspace(-3, 3, 100) + >>> yarr = np.linspace(-3, 3, 100) + >>> xgrid, ygrid = np.meshgrid(xarr, yarr, indexing="ij") + + The function to interpolate decays faster along one axis than the other. + + >>> zdata = np.exp(-np.sqrt((xgrid / 2) ** 2 + ygrid**2)) + + Next we sample on a finer grid using interpolation (kx=ky=1 for bilinear). + + >>> rbs = RectBivariateSpline(xarr, yarr, zdata, kx=1, ky=1) + >>> xarr_fine = np.linspace(-3, 3, 200) + >>> yarr_fine = np.linspace(-3, 3, 200) + >>> xgrid_fine, ygrid_fine = np.meshgrid(xarr_fine, yarr_fine, indexing="ij") + >>> zdata_interp = rbs(xgrid_fine, ygrid_fine, grid=False) + + And check that the result agrees with the input by plotting both. + + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> ax1 = fig.add_subplot(1, 2, 1, aspect="equal") + >>> ax2 = fig.add_subplot(1, 2, 2, aspect="equal") + >>> ax1.imshow(zdata) + >>> ax2.imshow(zdata_interp) + >>> plt.show() + """ + x = np.asarray(x) + y = np.asarray(y) + + tx, ty, c = self.tck[:3] + kx, ky = self.degrees + if grid: + if x.size == 0 or y.size == 0: + return np.zeros((x.size, y.size), dtype=self.tck[2].dtype) + + if (x.size >= 2) and (not np.all(np.diff(x) >= 0.0)): + raise ValueError("x must be strictly increasing when `grid` is True") + if (y.size >= 2) and (not np.all(np.diff(y) >= 0.0)): + raise ValueError("y must be strictly increasing when `grid` is True") + + if dx or dy: + with FITPACK_LOCK: + z, ier = dfitpack.parder(tx, ty, c, kx, ky, dx, dy, x, y) + if not ier == 0: + raise ValueError(f"Error code returned by parder: {ier}") + else: + with FITPACK_LOCK: + z, ier = dfitpack.bispev(tx, ty, c, kx, ky, x, y) + if not ier == 0: + raise ValueError(f"Error code returned by bispev: {ier}") + else: + # standard Numpy broadcasting + if x.shape != y.shape: + x, y = np.broadcast_arrays(x, y) + + shape = x.shape + x = x.ravel() + y = y.ravel() + + if x.size == 0 or y.size == 0: + return np.zeros(shape, dtype=self.tck[2].dtype) + + if dx or dy: + with FITPACK_LOCK: + z, ier = dfitpack.pardeu(tx, ty, c, kx, ky, dx, dy, x, y) + if not ier == 0: + raise ValueError(f"Error code returned by pardeu: {ier}") + else: + with FITPACK_LOCK: + z, ier = dfitpack.bispeu(tx, ty, c, kx, ky, x, y) + if not ier == 0: + raise ValueError(f"Error code returned by bispeu: {ier}") + + z = z.reshape(shape) + return z + + def partial_derivative(self, dx, dy): + """Construct a new spline representing a partial derivative of this + spline. + + Parameters + ---------- + dx, dy : int + Orders of the derivative in x and y respectively. They must be + non-negative integers and less than the respective degree of the + original spline (self) in that direction (``kx``, ``ky``). + + Returns + ------- + spline : + A new spline of degrees (``kx - dx``, ``ky - dy``) representing the + derivative of this spline. + + Notes + ----- + + .. versionadded:: 1.9.0 + + """ + if dx == 0 and dy == 0: + return self + else: + kx, ky = self.degrees + if not (dx >= 0 and dy >= 0): + raise ValueError("order of derivative must be positive or" + " zero") + if not (dx < kx and dy < ky): + raise ValueError("order of derivative must be less than" + " degree of spline") + tx, ty, c = self.tck[:3] + with FITPACK_LOCK: + newc, ier = dfitpack.pardtc(tx, ty, c, kx, ky, dx, dy) + if ier != 0: + # This should not happen under normal conditions. + raise ValueError("Unexpected error code returned by" + " pardtc: %d" % ier) + nx = len(tx) + ny = len(ty) + newtx = tx[dx:nx - dx] + newty = ty[dy:ny - dy] + newkx, newky = kx - dx, ky - dy + newclen = (nx - dx - kx - 1) * (ny - dy - ky - 1) + return _DerivedBivariateSpline._from_tck((newtx, newty, + newc[:newclen], + newkx, newky)) + + +_surfit_messages = {1: """ +The required storage space exceeds the available storage space: nxest +or nyest too small, or s too small. +The weighted least-squares spline corresponds to the current set of +knots.""", + 2: """ +A theoretically impossible result was found during the iteration +process for finding a smoothing spline with fp = s: s too small or +badly chosen eps. +Weighted sum of squared residuals does not satisfy abs(fp-s)/s < tol.""", + 3: """ +the maximal number of iterations maxit (set to 20 by the program) +allowed for finding a smoothing spline with fp=s has been reached: +s too small. +Weighted sum of squared residuals does not satisfy abs(fp-s)/s < tol.""", + 4: """ +No more knots can be added because the number of b-spline coefficients +(nx-kx-1)*(ny-ky-1) already exceeds the number of data points m: +either s or m too small. +The weighted least-squares spline corresponds to the current set of +knots.""", + 5: """ +No more knots can be added because the additional knot would (quasi) +coincide with an old one: s too small or too large a weight to an +inaccurate data point. +The weighted least-squares spline corresponds to the current set of +knots.""", + 10: """ +Error on entry, no approximation returned. The following conditions +must hold: +xb<=x[i]<=xe, yb<=y[i]<=ye, w[i]>0, i=0..m-1 +If iopt==-1, then + xb>> import numpy as np + >>> from scipy.interpolate import RectBivariateSpline + >>> def f(x, y): + ... return np.exp(-np.sqrt((x / 2) ** 2 + y**2)) + + We sample the function on a coarse grid and set up the interpolator. Note that + the default ``indexing="xy"`` of meshgrid would result in an unexpected + (transposed) result after interpolation. + + >>> xarr = np.linspace(-3, 3, 21) + >>> yarr = np.linspace(-3, 3, 21) + >>> xgrid, ygrid = np.meshgrid(xarr, yarr, indexing="ij") + >>> zdata = f(xgrid, ygrid) + >>> rbs = RectBivariateSpline(xarr, yarr, zdata, kx=1, ky=1) + + Next we sample the function along a diagonal slice through the coordinate space + on a finer grid using interpolation. + + >>> xinterp = np.linspace(-3, 3, 201) + >>> yinterp = np.linspace(3, -3, 201) + >>> zinterp = rbs.ev(xinterp, yinterp) + + And check that the interpolation passes through the function evaluations as a + function of the distance from the origin along the slice. + + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> ax1 = fig.add_subplot(1, 1, 1) + >>> ax1.plot(np.sqrt(xarr**2 + yarr**2), np.diag(zdata), "or") + >>> ax1.plot(np.sqrt(xinterp**2 + yinterp**2), zinterp, "-b") + >>> plt.show() + """ + return self.__call__(xi, yi, dx=dx, dy=dy, grid=False) + + def integral(self, xa, xb, ya, yb): + """ + Evaluate the integral of the spline over area [xa,xb] x [ya,yb]. + + Parameters + ---------- + xa, xb : float + The end-points of the x integration interval. + ya, yb : float + The end-points of the y integration interval. + + Returns + ------- + integ : float + The value of the resulting integral. + + """ + tx, ty, c = self.tck[:3] + kx, ky = self.degrees + with FITPACK_LOCK: + return dfitpack.dblint(tx, ty, c, kx, ky, xa, xb, ya, yb) + + @staticmethod + def _validate_input(x, y, z, w, kx, ky, eps): + x, y, z = np.asarray(x), np.asarray(y), np.asarray(z) + if not x.size == y.size == z.size: + raise ValueError('x, y, and z should have a same length') + + if w is not None: + w = np.asarray(w) + if x.size != w.size: + raise ValueError('x, y, z, and w should have a same length') + elif not np.all(w >= 0.0): + raise ValueError('w should be positive') + if (eps is not None) and (not 0.0 < eps < 1.0): + raise ValueError('eps should be between (0, 1)') + if not x.size >= (kx + 1) * (ky + 1): + raise ValueError('The length of x, y and z should be at least' + ' (kx+1) * (ky+1)') + return x, y, z, w + + +class _DerivedBivariateSpline(_BivariateSplineBase): + """Bivariate spline constructed from the coefficients and knots of another + spline. + + Notes + ----- + The class is not meant to be instantiated directly from the data to be + interpolated or smoothed. As a result, its ``fp`` attribute and + ``get_residual`` method are inherited but overridden; ``AttributeError`` is + raised when they are accessed. + + The other inherited attributes can be used as usual. + """ + _invalid_why = ("is unavailable, because _DerivedBivariateSpline" + " instance is not constructed from data that are to be" + " interpolated or smoothed, but derived from the" + " underlying knots and coefficients of another spline" + " object") + + @property + def fp(self): + raise AttributeError(f"attribute \"fp\" {self._invalid_why}") + + def get_residual(self): + raise AttributeError(f"method \"get_residual\" {self._invalid_why}") + + +class SmoothBivariateSpline(BivariateSpline): + """ + Smooth bivariate spline approximation. + + Parameters + ---------- + x, y, z : array_like + 1-D sequences of data points (order is not important). + w : array_like, optional + Positive 1-D sequence of weights, of same length as `x`, `y` and `z`. + bbox : array_like, optional + Sequence of length 4 specifying the boundary of the rectangular + approximation domain. By default, + ``bbox=[min(x), max(x), min(y), max(y)]``. + kx, ky : ints, optional + Degrees of the bivariate spline. Default is 3. + s : float, optional + Positive smoothing factor defined for estimation condition: + ``sum((w[i]*(z[i]-s(x[i], y[i])))**2, axis=0) <= s`` + Default ``s=len(w)`` which should be a good value if ``1/w[i]`` is an + estimate of the standard deviation of ``z[i]``. + eps : float, optional + A threshold for determining the effective rank of an over-determined + linear system of equations. `eps` should have a value within the open + interval ``(0, 1)``, the default is 1e-16. + + See Also + -------- + BivariateSpline : + a base class for bivariate splines. + UnivariateSpline : + a smooth univariate spline to fit a given set of data points. + LSQBivariateSpline : + a bivariate spline using weighted least-squares fitting + RectSphereBivariateSpline : + a bivariate spline over a rectangular mesh on a sphere + SmoothSphereBivariateSpline : + a smoothing bivariate spline in spherical coordinates + LSQSphereBivariateSpline : + a bivariate spline in spherical coordinates using weighted + least-squares fitting + RectBivariateSpline : + a bivariate spline over a rectangular mesh + bisplrep : + a function to find a bivariate B-spline representation of a surface + bisplev : + a function to evaluate a bivariate B-spline and its derivatives + + Notes + ----- + The length of `x`, `y` and `z` should be at least ``(kx+1) * (ky+1)``. + + If the input data is such that input dimensions have incommensurate + units and differ by many orders of magnitude, the interpolant may have + numerical artifacts. Consider rescaling the data before interpolating. + + This routine constructs spline knot vectors automatically via the FITPACK + algorithm. The spline knots may be placed away from the data points. For + some data sets, this routine may fail to construct an interpolating spline, + even if one is requested via ``s=0`` parameter. In such situations, it is + recommended to use `bisplrep` / `bisplev` directly instead of this routine + and, if needed, increase the values of ``nxest`` and ``nyest`` parameters + of `bisplrep`. + + For linear interpolation, prefer `LinearNDInterpolator`. + See ``https://gist.github.com/ev-br/8544371b40f414b7eaf3fe6217209bff`` + for discussion. + + """ + + def __init__(self, x, y, z, w=None, bbox=[None] * 4, kx=3, ky=3, s=None, + eps=1e-16): + + x, y, z, w = self._validate_input(x, y, z, w, kx, ky, eps) + bbox = ravel(bbox) + if not bbox.shape == (4,): + raise ValueError('bbox shape should be (4,)') + if s is not None and not s >= 0.0: + raise ValueError("s should be s >= 0.0") + + xb, xe, yb, ye = bbox + with FITPACK_LOCK: + nx, tx, ny, ty, c, fp, wrk1, ier = dfitpack.surfit_smth( + x, y, z, w, xb, xe, yb, ye, kx, ky, s=s, eps=eps, lwrk2=1) + if ier > 10: # lwrk2 was to small, re-run + nx, tx, ny, ty, c, fp, wrk1, ier = dfitpack.surfit_smth( + x, y, z, w, xb, xe, yb, ye, kx, ky, s=s, eps=eps, + lwrk2=ier) + if ier in [0, -1, -2]: # normal return + pass + else: + message = _surfit_messages.get(ier, f'ier={ier}') + warnings.warn(message, stacklevel=2) + + self.fp = fp + self.tck = tx[:nx], ty[:ny], c[:(nx-kx-1)*(ny-ky-1)] + self.degrees = kx, ky + + +class LSQBivariateSpline(BivariateSpline): + """ + Weighted least-squares bivariate spline approximation. + + Parameters + ---------- + x, y, z : array_like + 1-D sequences of data points (order is not important). + tx, ty : array_like + Strictly ordered 1-D sequences of knots coordinates. + w : array_like, optional + Positive 1-D array of weights, of the same length as `x`, `y` and `z`. + bbox : (4,) array_like, optional + Sequence of length 4 specifying the boundary of the rectangular + approximation domain. By default, + ``bbox=[min(x,tx),max(x,tx), min(y,ty),max(y,ty)]``. + kx, ky : ints, optional + Degrees of the bivariate spline. Default is 3. + eps : float, optional + A threshold for determining the effective rank of an over-determined + linear system of equations. `eps` should have a value within the open + interval ``(0, 1)``, the default is 1e-16. + + See Also + -------- + BivariateSpline : + a base class for bivariate splines. + UnivariateSpline : + a smooth univariate spline to fit a given set of data points. + SmoothBivariateSpline : + a smoothing bivariate spline through the given points + RectSphereBivariateSpline : + a bivariate spline over a rectangular mesh on a sphere + SmoothSphereBivariateSpline : + a smoothing bivariate spline in spherical coordinates + LSQSphereBivariateSpline : + a bivariate spline in spherical coordinates using weighted + least-squares fitting + RectBivariateSpline : + a bivariate spline over a rectangular mesh. + bisplrep : + a function to find a bivariate B-spline representation of a surface + bisplev : + a function to evaluate a bivariate B-spline and its derivatives + + Notes + ----- + The length of `x`, `y` and `z` should be at least ``(kx+1) * (ky+1)``. + + If the input data is such that input dimensions have incommensurate + units and differ by many orders of magnitude, the interpolant may have + numerical artifacts. Consider rescaling the data before interpolating. + + """ + + def __init__(self, x, y, z, tx, ty, w=None, bbox=[None]*4, kx=3, ky=3, + eps=None): + + x, y, z, w = self._validate_input(x, y, z, w, kx, ky, eps) + bbox = ravel(bbox) + if not bbox.shape == (4,): + raise ValueError('bbox shape should be (4,)') + + nx = 2*kx+2+len(tx) + ny = 2*ky+2+len(ty) + # The Fortran subroutine "surfit" (called as dfitpack.surfit_lsq) + # requires that the knot arrays passed as input should be "real + # array(s) of dimension nmax" where "nmax" refers to the greater of nx + # and ny. We pad the tx1/ty1 arrays here so that this is satisfied, and + # slice them to the desired sizes upon return. + nmax = max(nx, ny) + tx1 = zeros((nmax,), float) + ty1 = zeros((nmax,), float) + tx1[kx+1:nx-kx-1] = tx + ty1[ky+1:ny-ky-1] = ty + + xb, xe, yb, ye = bbox + with FITPACK_LOCK: + tx1, ty1, c, fp, ier = dfitpack.surfit_lsq(x, y, z, nx, tx1, ny, ty1, + w, xb, xe, yb, ye, + kx, ky, eps, lwrk2=1) + if ier > 10: + tx1, ty1, c, fp, ier = dfitpack.surfit_lsq(x, y, z, + nx, tx1, ny, ty1, w, + xb, xe, yb, ye, + kx, ky, eps, lwrk2=ier) + if ier in [0, -1, -2]: # normal return + pass + else: + if ier < -2: + deficiency = (nx-kx-1)*(ny-ky-1)+ier + message = _surfit_messages.get(-3) % (deficiency) + else: + message = _surfit_messages.get(ier, f'ier={ier}') + warnings.warn(message, stacklevel=2) + self.fp = fp + self.tck = tx1[:nx], ty1[:ny], c + self.degrees = kx, ky + + +class RectBivariateSpline(BivariateSpline): + """ + Bivariate spline approximation over a rectangular mesh. + + Can be used for both smoothing and interpolating data. + + Parameters + ---------- + x,y : array_like + 1-D arrays of coordinates in strictly ascending order. + Evaluated points outside the data range will be extrapolated. + z : array_like + 2-D array of data with shape (x.size,y.size). + bbox : array_like, optional + Sequence of length 4 specifying the boundary of the rectangular + approximation domain, which means the start and end spline knots of + each dimension are set by these values. By default, + ``bbox=[min(x), max(x), min(y), max(y)]``. + kx, ky : ints, optional + Degrees of the bivariate spline. Default is 3. + s : float, optional + Positive smoothing factor defined for estimation condition: + ``sum((z[i]-f(x[i], y[i]))**2, axis=0) <= s`` where f is a spline + function. Default is ``s=0``, which is for interpolation. + + See Also + -------- + BivariateSpline : + a base class for bivariate splines. + UnivariateSpline : + a smooth univariate spline to fit a given set of data points. + SmoothBivariateSpline : + a smoothing bivariate spline through the given points + LSQBivariateSpline : + a bivariate spline using weighted least-squares fitting + RectSphereBivariateSpline : + a bivariate spline over a rectangular mesh on a sphere + SmoothSphereBivariateSpline : + a smoothing bivariate spline in spherical coordinates + LSQSphereBivariateSpline : + a bivariate spline in spherical coordinates using weighted + least-squares fitting + bisplrep : + a function to find a bivariate B-spline representation of a surface + bisplev : + a function to evaluate a bivariate B-spline and its derivatives + + Notes + ----- + + If the input data is such that input dimensions have incommensurate + units and differ by many orders of magnitude, the interpolant may have + numerical artifacts. Consider rescaling the data before interpolating. + + """ + + def __init__(self, x, y, z, bbox=[None] * 4, kx=3, ky=3, s=0): + x, y, bbox = ravel(x), ravel(y), ravel(bbox) + z = np.asarray(z) + if not np.all(diff(x) > 0.0): + raise ValueError('x must be strictly increasing') + if not np.all(diff(y) > 0.0): + raise ValueError('y must be strictly increasing') + if not x.size == z.shape[0]: + raise ValueError('x dimension of z must have same number of ' + 'elements as x') + if not y.size == z.shape[1]: + raise ValueError('y dimension of z must have same number of ' + 'elements as y') + if not bbox.shape == (4,): + raise ValueError('bbox shape should be (4,)') + if s is not None and not s >= 0.0: + raise ValueError("s should be s >= 0.0") + + z = ravel(z) + xb, xe, yb, ye = bbox + with FITPACK_LOCK: + nx, tx, ny, ty, c, fp, ier = dfitpack.regrid_smth(x, y, z, xb, xe, yb, + ye, kx, ky, s) + + if ier not in [0, -1, -2]: + msg = _surfit_messages.get(ier, f'ier={ier}') + raise ValueError(msg) + + self.fp = fp + self.tck = tx[:nx], ty[:ny], c[:(nx - kx - 1) * (ny - ky - 1)] + self.degrees = kx, ky + + +_spherefit_messages = _surfit_messages.copy() +_spherefit_messages[10] = """ +ERROR. On entry, the input data are controlled on validity. The following + restrictions must be satisfied: + -1<=iopt<=1, m>=2, ntest>=8 ,npest >=8, 00, i=1,...,m + lwrk1 >= 185+52*v+10*u+14*u*v+8*(u-1)*v**2+8*m + kwrk >= m+(ntest-7)*(npest-7) + if iopt=-1: 8<=nt<=ntest , 9<=np<=npest + 0=0: s>=0 + if one of these conditions is found to be violated,control + is immediately repassed to the calling program. in that + case there is no approximation returned.""" +_spherefit_messages[-3] = """ +WARNING. The coefficients of the spline returned have been computed as the + minimal norm least-squares solution of a (numerically) rank + deficient system (deficiency=%i, rank=%i). Especially if the rank + deficiency, which is computed by 6+(nt-8)*(np-7)+ier, is large, + the results may be inaccurate. They could also seriously depend on + the value of eps.""" + + +class SphereBivariateSpline(_BivariateSplineBase): + """ + Bivariate spline s(x,y) of degrees 3 on a sphere, calculated from a + given set of data points (theta,phi,r). + + .. versionadded:: 0.11.0 + + See Also + -------- + bisplrep : + a function to find a bivariate B-spline representation of a surface + bisplev : + a function to evaluate a bivariate B-spline and its derivatives + UnivariateSpline : + a smooth univariate spline to fit a given set of data points. + SmoothBivariateSpline : + a smoothing bivariate spline through the given points + LSQUnivariateSpline : + a univariate spline using weighted least-squares fitting + """ + + def __call__(self, theta, phi, dtheta=0, dphi=0, grid=True): + """ + Evaluate the spline or its derivatives at given positions. + + Parameters + ---------- + theta, phi : array_like + Input coordinates. + + If `grid` is False, evaluate the spline at points + ``(theta[i], phi[i]), i=0, ..., len(x)-1``. Standard + Numpy broadcasting is obeyed. + + If `grid` is True: evaluate spline at the grid points + defined by the coordinate arrays theta, phi. The arrays + must be sorted to increasing order. + The ordering of axes is consistent with + ``np.meshgrid(..., indexing="ij")`` and inconsistent with the + default ordering ``np.meshgrid(..., indexing="xy")``. + dtheta : int, optional + Order of theta-derivative + + .. versionadded:: 0.14.0 + dphi : int + Order of phi-derivative + + .. versionadded:: 0.14.0 + grid : bool + Whether to evaluate the results on a grid spanned by the + input arrays, or at points specified by the input arrays. + + .. versionadded:: 0.14.0 + + Examples + -------- + + Suppose that we want to use splines to interpolate a bivariate function on a + sphere. The value of the function is known on a grid of longitudes and + colatitudes. + + >>> import numpy as np + >>> from scipy.interpolate import RectSphereBivariateSpline + >>> def f(theta, phi): + ... return np.sin(theta) * np.cos(phi) + + We evaluate the function on the grid. Note that the default indexing="xy" + of meshgrid would result in an unexpected (transposed) result after + interpolation. + + >>> thetaarr = np.linspace(0, np.pi, 22)[1:-1] + >>> phiarr = np.linspace(0, 2 * np.pi, 21)[:-1] + >>> thetagrid, phigrid = np.meshgrid(thetaarr, phiarr, indexing="ij") + >>> zdata = f(thetagrid, phigrid) + + We next set up the interpolator and use it to evaluate the function + on a finer grid. + + >>> rsbs = RectSphereBivariateSpline(thetaarr, phiarr, zdata) + >>> thetaarr_fine = np.linspace(0, np.pi, 200) + >>> phiarr_fine = np.linspace(0, 2 * np.pi, 200) + >>> zdata_fine = rsbs(thetaarr_fine, phiarr_fine) + + Finally we plot the coarsly-sampled input data alongside the + finely-sampled interpolated data to check that they agree. + + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> ax1 = fig.add_subplot(1, 2, 1) + >>> ax2 = fig.add_subplot(1, 2, 2) + >>> ax1.imshow(zdata) + >>> ax2.imshow(zdata_fine) + >>> plt.show() + """ + theta = np.asarray(theta) + phi = np.asarray(phi) + + if theta.size > 0 and (theta.min() < 0. or theta.max() > np.pi): + raise ValueError("requested theta out of bounds.") + + return _BivariateSplineBase.__call__(self, theta, phi, + dx=dtheta, dy=dphi, grid=grid) + + def ev(self, theta, phi, dtheta=0, dphi=0): + """ + Evaluate the spline at points + + Returns the interpolated value at ``(theta[i], phi[i]), + i=0,...,len(theta)-1``. + + Parameters + ---------- + theta, phi : array_like + Input coordinates. Standard Numpy broadcasting is obeyed. + The ordering of axes is consistent with + np.meshgrid(..., indexing="ij") and inconsistent with the + default ordering np.meshgrid(..., indexing="xy"). + dtheta : int, optional + Order of theta-derivative + + .. versionadded:: 0.14.0 + dphi : int, optional + Order of phi-derivative + + .. versionadded:: 0.14.0 + + Examples + -------- + Suppose that we want to use splines to interpolate a bivariate function on a + sphere. The value of the function is known on a grid of longitudes and + colatitudes. + + >>> import numpy as np + >>> from scipy.interpolate import RectSphereBivariateSpline + >>> def f(theta, phi): + ... return np.sin(theta) * np.cos(phi) + + We evaluate the function on the grid. Note that the default indexing="xy" + of meshgrid would result in an unexpected (transposed) result after + interpolation. + + >>> thetaarr = np.linspace(0, np.pi, 22)[1:-1] + >>> phiarr = np.linspace(0, 2 * np.pi, 21)[:-1] + >>> thetagrid, phigrid = np.meshgrid(thetaarr, phiarr, indexing="ij") + >>> zdata = f(thetagrid, phigrid) + + We next set up the interpolator and use it to evaluate the function + at points not on the original grid. + + >>> rsbs = RectSphereBivariateSpline(thetaarr, phiarr, zdata) + >>> thetainterp = np.linspace(thetaarr[0], thetaarr[-1], 200) + >>> phiinterp = np.linspace(phiarr[0], phiarr[-1], 200) + >>> zinterp = rsbs.ev(thetainterp, phiinterp) + + Finally we plot the original data for a diagonal slice through the + initial grid, and the spline approximation along the same slice. + + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> ax1 = fig.add_subplot(1, 1, 1) + >>> ax1.plot(np.sin(thetaarr) * np.sin(phiarr), np.diag(zdata), "or") + >>> ax1.plot(np.sin(thetainterp) * np.sin(phiinterp), zinterp, "-b") + >>> plt.show() + """ + return self.__call__(theta, phi, dtheta=dtheta, dphi=dphi, grid=False) + + +class SmoothSphereBivariateSpline(SphereBivariateSpline): + """ + Smooth bivariate spline approximation in spherical coordinates. + + .. versionadded:: 0.11.0 + + Parameters + ---------- + theta, phi, r : array_like + 1-D sequences of data points (order is not important). Coordinates + must be given in radians. Theta must lie within the interval + ``[0, pi]``, and phi must lie within the interval ``[0, 2pi]``. + w : array_like, optional + Positive 1-D sequence of weights. + s : float, optional + Positive smoothing factor defined for estimation condition: + ``sum((w(i)*(r(i) - s(theta(i), phi(i))))**2, axis=0) <= s`` + Default ``s=len(w)`` which should be a good value if ``1/w[i]`` is an + estimate of the standard deviation of ``r[i]``. + eps : float, optional + A threshold for determining the effective rank of an over-determined + linear system of equations. `eps` should have a value within the open + interval ``(0, 1)``, the default is 1e-16. + + See Also + -------- + BivariateSpline : + a base class for bivariate splines. + UnivariateSpline : + a smooth univariate spline to fit a given set of data points. + SmoothBivariateSpline : + a smoothing bivariate spline through the given points + LSQBivariateSpline : + a bivariate spline using weighted least-squares fitting + RectSphereBivariateSpline : + a bivariate spline over a rectangular mesh on a sphere + LSQSphereBivariateSpline : + a bivariate spline in spherical coordinates using weighted + least-squares fitting + RectBivariateSpline : + a bivariate spline over a rectangular mesh. + bisplrep : + a function to find a bivariate B-spline representation of a surface + bisplev : + a function to evaluate a bivariate B-spline and its derivatives + + Notes + ----- + For more information, see the FITPACK_ site about this function. + + .. _FITPACK: http://www.netlib.org/dierckx/sphere.f + + Examples + -------- + Suppose we have global data on a coarse grid (the input data does not + have to be on a grid): + + >>> import numpy as np + >>> theta = np.linspace(0., np.pi, 7) + >>> phi = np.linspace(0., 2*np.pi, 9) + >>> data = np.empty((theta.shape[0], phi.shape[0])) + >>> data[:,0], data[0,:], data[-1,:] = 0., 0., 0. + >>> data[1:-1,1], data[1:-1,-1] = 1., 1. + >>> data[1,1:-1], data[-2,1:-1] = 1., 1. + >>> data[2:-2,2], data[2:-2,-2] = 2., 2. + >>> data[2,2:-2], data[-3,2:-2] = 2., 2. + >>> data[3,3:-2] = 3. + >>> data = np.roll(data, 4, 1) + + We need to set up the interpolator object + + >>> lats, lons = np.meshgrid(theta, phi) + >>> from scipy.interpolate import SmoothSphereBivariateSpline + >>> lut = SmoothSphereBivariateSpline(lats.ravel(), lons.ravel(), + ... data.T.ravel(), s=3.5) + + As a first test, we'll see what the algorithm returns when run on the + input coordinates + + >>> data_orig = lut(theta, phi) + + Finally we interpolate the data to a finer grid + + >>> fine_lats = np.linspace(0., np.pi, 70) + >>> fine_lons = np.linspace(0., 2 * np.pi, 90) + + >>> data_smth = lut(fine_lats, fine_lons) + + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> ax1 = fig.add_subplot(131) + >>> ax1.imshow(data, interpolation='nearest') + >>> ax2 = fig.add_subplot(132) + >>> ax2.imshow(data_orig, interpolation='nearest') + >>> ax3 = fig.add_subplot(133) + >>> ax3.imshow(data_smth, interpolation='nearest') + >>> plt.show() + + """ + + def __init__(self, theta, phi, r, w=None, s=0., eps=1E-16): + + theta, phi, r = np.asarray(theta), np.asarray(phi), np.asarray(r) + + # input validation + if not ((0.0 <= theta).all() and (theta <= np.pi).all()): + raise ValueError('theta should be between [0, pi]') + if not ((0.0 <= phi).all() and (phi <= 2.0 * np.pi).all()): + raise ValueError('phi should be between [0, 2pi]') + if w is not None: + w = np.asarray(w) + if not (w >= 0.0).all(): + raise ValueError('w should be positive') + if not s >= 0.0: + raise ValueError('s should be positive') + if not 0.0 < eps < 1.0: + raise ValueError('eps should be between (0, 1)') + + with FITPACK_LOCK: + nt_, tt_, np_, tp_, c, fp, ier = dfitpack.spherfit_smth(theta, phi, + r, w=w, s=s, + eps=eps) + if ier not in [0, -1, -2]: + message = _spherefit_messages.get(ier, f'ier={ier}') + raise ValueError(message) + + self.fp = fp + self.tck = tt_[:nt_], tp_[:np_], c[:(nt_ - 4) * (np_ - 4)] + self.degrees = (3, 3) + + def __call__(self, theta, phi, dtheta=0, dphi=0, grid=True): + + theta = np.asarray(theta) + phi = np.asarray(phi) + + if phi.size > 0 and (phi.min() < 0. or phi.max() > 2. * np.pi): + raise ValueError("requested phi out of bounds.") + + return SphereBivariateSpline.__call__(self, theta, phi, dtheta=dtheta, + dphi=dphi, grid=grid) + + +class LSQSphereBivariateSpline(SphereBivariateSpline): + """ + Weighted least-squares bivariate spline approximation in spherical + coordinates. + + Determines a smoothing bicubic spline according to a given + set of knots in the `theta` and `phi` directions. + + .. versionadded:: 0.11.0 + + Parameters + ---------- + theta, phi, r : array_like + 1-D sequences of data points (order is not important). Coordinates + must be given in radians. Theta must lie within the interval + ``[0, pi]``, and phi must lie within the interval ``[0, 2pi]``. + tt, tp : array_like + Strictly ordered 1-D sequences of knots coordinates. + Coordinates must satisfy ``0 < tt[i] < pi``, ``0 < tp[i] < 2*pi``. + w : array_like, optional + Positive 1-D sequence of weights, of the same length as `theta`, `phi` + and `r`. + eps : float, optional + A threshold for determining the effective rank of an over-determined + linear system of equations. `eps` should have a value within the + open interval ``(0, 1)``, the default is 1e-16. + + See Also + -------- + BivariateSpline : + a base class for bivariate splines. + UnivariateSpline : + a smooth univariate spline to fit a given set of data points. + SmoothBivariateSpline : + a smoothing bivariate spline through the given points + LSQBivariateSpline : + a bivariate spline using weighted least-squares fitting + RectSphereBivariateSpline : + a bivariate spline over a rectangular mesh on a sphere + SmoothSphereBivariateSpline : + a smoothing bivariate spline in spherical coordinates + RectBivariateSpline : + a bivariate spline over a rectangular mesh. + bisplrep : + a function to find a bivariate B-spline representation of a surface + bisplev : + a function to evaluate a bivariate B-spline and its derivatives + + Notes + ----- + For more information, see the FITPACK_ site about this function. + + .. _FITPACK: http://www.netlib.org/dierckx/sphere.f + + Examples + -------- + Suppose we have global data on a coarse grid (the input data does not + have to be on a grid): + + >>> from scipy.interpolate import LSQSphereBivariateSpline + >>> import numpy as np + >>> import matplotlib.pyplot as plt + + >>> theta = np.linspace(0, np.pi, num=7) + >>> phi = np.linspace(0, 2*np.pi, num=9) + >>> data = np.empty((theta.shape[0], phi.shape[0])) + >>> data[:,0], data[0,:], data[-1,:] = 0., 0., 0. + >>> data[1:-1,1], data[1:-1,-1] = 1., 1. + >>> data[1,1:-1], data[-2,1:-1] = 1., 1. + >>> data[2:-2,2], data[2:-2,-2] = 2., 2. + >>> data[2,2:-2], data[-3,2:-2] = 2., 2. + >>> data[3,3:-2] = 3. + >>> data = np.roll(data, 4, 1) + + We need to set up the interpolator object. Here, we must also specify the + coordinates of the knots to use. + + >>> lats, lons = np.meshgrid(theta, phi) + >>> knotst, knotsp = theta.copy(), phi.copy() + >>> knotst[0] += .0001 + >>> knotst[-1] -= .0001 + >>> knotsp[0] += .0001 + >>> knotsp[-1] -= .0001 + >>> lut = LSQSphereBivariateSpline(lats.ravel(), lons.ravel(), + ... data.T.ravel(), knotst, knotsp) + + As a first test, we'll see what the algorithm returns when run on the + input coordinates + + >>> data_orig = lut(theta, phi) + + Finally we interpolate the data to a finer grid + + >>> fine_lats = np.linspace(0., np.pi, 70) + >>> fine_lons = np.linspace(0., 2*np.pi, 90) + >>> data_lsq = lut(fine_lats, fine_lons) + + >>> fig = plt.figure() + >>> ax1 = fig.add_subplot(131) + >>> ax1.imshow(data, interpolation='nearest') + >>> ax2 = fig.add_subplot(132) + >>> ax2.imshow(data_orig, interpolation='nearest') + >>> ax3 = fig.add_subplot(133) + >>> ax3.imshow(data_lsq, interpolation='nearest') + >>> plt.show() + + """ + + def __init__(self, theta, phi, r, tt, tp, w=None, eps=1E-16): + + theta, phi, r = np.asarray(theta), np.asarray(phi), np.asarray(r) + tt, tp = np.asarray(tt), np.asarray(tp) + + if not ((0.0 <= theta).all() and (theta <= np.pi).all()): + raise ValueError('theta should be between [0, pi]') + if not ((0.0 <= phi).all() and (phi <= 2*np.pi).all()): + raise ValueError('phi should be between [0, 2pi]') + if not ((0.0 < tt).all() and (tt < np.pi).all()): + raise ValueError('tt should be between (0, pi)') + if not ((0.0 < tp).all() and (tp < 2*np.pi).all()): + raise ValueError('tp should be between (0, 2pi)') + if w is not None: + w = np.asarray(w) + if not (w >= 0.0).all(): + raise ValueError('w should be positive') + if not 0.0 < eps < 1.0: + raise ValueError('eps should be between (0, 1)') + + nt_, np_ = 8 + len(tt), 8 + len(tp) + tt_, tp_ = zeros((nt_,), float), zeros((np_,), float) + tt_[4:-4], tp_[4:-4] = tt, tp + tt_[-4:], tp_[-4:] = np.pi, 2. * np.pi + with FITPACK_LOCK: + tt_, tp_, c, fp, ier = dfitpack.spherfit_lsq(theta, phi, r, tt_, tp_, + w=w, eps=eps) + if ier > 0: + message = _spherefit_messages.get(ier, f'ier={ier}') + raise ValueError(message) + + self.fp = fp + self.tck = tt_, tp_, c + self.degrees = (3, 3) + + def __call__(self, theta, phi, dtheta=0, dphi=0, grid=True): + + theta = np.asarray(theta) + phi = np.asarray(phi) + + if phi.size > 0 and (phi.min() < 0. or phi.max() > 2. * np.pi): + raise ValueError("requested phi out of bounds.") + + return SphereBivariateSpline.__call__(self, theta, phi, dtheta=dtheta, + dphi=dphi, grid=grid) + + +_spfit_messages = _surfit_messages.copy() +_spfit_messages[10] = """ +ERROR: on entry, the input data are controlled on validity + the following restrictions must be satisfied. + -1<=iopt(1)<=1, 0<=iopt(2)<=1, 0<=iopt(3)<=1, + -1<=ider(1)<=1, 0<=ider(2)<=1, ider(2)=0 if iopt(2)=0. + -1<=ider(3)<=1, 0<=ider(4)<=1, ider(4)=0 if iopt(3)=0. + mu >= mumin (see above), mv >= 4, nuest >=8, nvest >= 8, + kwrk>=5+mu+mv+nuest+nvest, + lwrk >= 12+nuest*(mv+nvest+3)+nvest*24+4*mu+8*mv+max(nuest,mv+nvest) + 0< u(i-1)=0: s>=0 + if s=0: nuest>=mu+6+iopt(2)+iopt(3), nvest>=mv+7 + if one of these conditions is found to be violated,control is + immediately repassed to the calling program. in that case there is no + approximation returned.""" + + +class RectSphereBivariateSpline(SphereBivariateSpline): + """ + Bivariate spline approximation over a rectangular mesh on a sphere. + + Can be used for smoothing data. + + .. versionadded:: 0.11.0 + + Parameters + ---------- + u : array_like + 1-D array of colatitude coordinates in strictly ascending order. + Coordinates must be given in radians and lie within the open interval + ``(0, pi)``. + v : array_like + 1-D array of longitude coordinates in strictly ascending order. + Coordinates must be given in radians. First element (``v[0]``) must lie + within the interval ``[-pi, pi)``. Last element (``v[-1]``) must satisfy + ``v[-1] <= v[0] + 2*pi``. + r : array_like + 2-D array of data with shape ``(u.size, v.size)``. + s : float, optional + Positive smoothing factor defined for estimation condition + (``s=0`` is for interpolation). + pole_continuity : bool or (bool, bool), optional + Order of continuity at the poles ``u=0`` (``pole_continuity[0]``) and + ``u=pi`` (``pole_continuity[1]``). The order of continuity at the pole + will be 1 or 0 when this is True or False, respectively. + Defaults to False. + pole_values : float or (float, float), optional + Data values at the poles ``u=0`` and ``u=pi``. Either the whole + parameter or each individual element can be None. Defaults to None. + pole_exact : bool or (bool, bool), optional + Data value exactness at the poles ``u=0`` and ``u=pi``. If True, the + value is considered to be the right function value, and it will be + fitted exactly. If False, the value will be considered to be a data + value just like the other data values. Defaults to False. + pole_flat : bool or (bool, bool), optional + For the poles at ``u=0`` and ``u=pi``, specify whether or not the + approximation has vanishing derivatives. Defaults to False. + + See Also + -------- + BivariateSpline : + a base class for bivariate splines. + UnivariateSpline : + a smooth univariate spline to fit a given set of data points. + SmoothBivariateSpline : + a smoothing bivariate spline through the given points + LSQBivariateSpline : + a bivariate spline using weighted least-squares fitting + SmoothSphereBivariateSpline : + a smoothing bivariate spline in spherical coordinates + LSQSphereBivariateSpline : + a bivariate spline in spherical coordinates using weighted + least-squares fitting + RectBivariateSpline : + a bivariate spline over a rectangular mesh. + bisplrep : + a function to find a bivariate B-spline representation of a surface + bisplev : + a function to evaluate a bivariate B-spline and its derivatives + + Notes + ----- + Currently, only the smoothing spline approximation (``iopt[0] = 0`` and + ``iopt[0] = 1`` in the FITPACK routine) is supported. The exact + least-squares spline approximation is not implemented yet. + + When actually performing the interpolation, the requested `v` values must + lie within the same length 2pi interval that the original `v` values were + chosen from. + + For more information, see the FITPACK_ site about this function. + + .. _FITPACK: http://www.netlib.org/dierckx/spgrid.f + + Examples + -------- + Suppose we have global data on a coarse grid + + >>> import numpy as np + >>> lats = np.linspace(10, 170, 9) * np.pi / 180. + >>> lons = np.linspace(0, 350, 18) * np.pi / 180. + >>> data = np.dot(np.atleast_2d(90. - np.linspace(-80., 80., 18)).T, + ... np.atleast_2d(180. - np.abs(np.linspace(0., 350., 9)))).T + + We want to interpolate it to a global one-degree grid + + >>> new_lats = np.linspace(1, 180, 180) * np.pi / 180 + >>> new_lons = np.linspace(1, 360, 360) * np.pi / 180 + >>> new_lats, new_lons = np.meshgrid(new_lats, new_lons) + + We need to set up the interpolator object + + >>> from scipy.interpolate import RectSphereBivariateSpline + >>> lut = RectSphereBivariateSpline(lats, lons, data) + + Finally we interpolate the data. The `RectSphereBivariateSpline` object + only takes 1-D arrays as input, therefore we need to do some reshaping. + + >>> data_interp = lut.ev(new_lats.ravel(), + ... new_lons.ravel()).reshape((360, 180)).T + + Looking at the original and the interpolated data, one can see that the + interpolant reproduces the original data very well: + + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> ax1 = fig.add_subplot(211) + >>> ax1.imshow(data, interpolation='nearest') + >>> ax2 = fig.add_subplot(212) + >>> ax2.imshow(data_interp, interpolation='nearest') + >>> plt.show() + + Choosing the optimal value of ``s`` can be a delicate task. Recommended + values for ``s`` depend on the accuracy of the data values. If the user + has an idea of the statistical errors on the data, she can also find a + proper estimate for ``s``. By assuming that, if she specifies the + right ``s``, the interpolator will use a spline ``f(u,v)`` which exactly + reproduces the function underlying the data, she can evaluate + ``sum((r(i,j)-s(u(i),v(j)))**2)`` to find a good estimate for this ``s``. + For example, if she knows that the statistical errors on her + ``r(i,j)``-values are not greater than 0.1, she may expect that a good + ``s`` should have a value not larger than ``u.size * v.size * (0.1)**2``. + + If nothing is known about the statistical error in ``r(i,j)``, ``s`` must + be determined by trial and error. The best is then to start with a very + large value of ``s`` (to determine the least-squares polynomial and the + corresponding upper bound ``fp0`` for ``s``) and then to progressively + decrease the value of ``s`` (say by a factor 10 in the beginning, i.e. + ``s = fp0 / 10, fp0 / 100, ...`` and more carefully as the approximation + shows more detail) to obtain closer fits. + + The interpolation results for different values of ``s`` give some insight + into this process: + + >>> fig2 = plt.figure() + >>> s = [3e9, 2e9, 1e9, 1e8] + >>> for idx, sval in enumerate(s, 1): + ... lut = RectSphereBivariateSpline(lats, lons, data, s=sval) + ... data_interp = lut.ev(new_lats.ravel(), + ... new_lons.ravel()).reshape((360, 180)).T + ... ax = fig2.add_subplot(2, 2, idx) + ... ax.imshow(data_interp, interpolation='nearest') + ... ax.set_title(f"s = {sval:g}") + >>> plt.show() + + """ + + def __init__(self, u, v, r, s=0., pole_continuity=False, pole_values=None, + pole_exact=False, pole_flat=False): + iopt = np.array([0, 0, 0], dtype=dfitpack_int) + ider = np.array([-1, 0, -1, 0], dtype=dfitpack_int) + if pole_values is None: + pole_values = (None, None) + elif isinstance(pole_values, (float, np.float32, np.float64)): + pole_values = (pole_values, pole_values) + if isinstance(pole_continuity, bool): + pole_continuity = (pole_continuity, pole_continuity) + if isinstance(pole_exact, bool): + pole_exact = (pole_exact, pole_exact) + if isinstance(pole_flat, bool): + pole_flat = (pole_flat, pole_flat) + + r0, r1 = pole_values + iopt[1:] = pole_continuity + if r0 is None: + ider[0] = -1 + else: + ider[0] = pole_exact[0] + + if r1 is None: + ider[2] = -1 + else: + ider[2] = pole_exact[1] + + ider[1], ider[3] = pole_flat + + u, v = np.ravel(u), np.ravel(v) + r = np.asarray(r) + + if not (0.0 < u[0] and u[-1] < np.pi): + raise ValueError('u should be between (0, pi)') + if not -np.pi <= v[0] < np.pi: + raise ValueError('v[0] should be between [-pi, pi)') + if not v[-1] <= v[0] + 2*np.pi: + raise ValueError('v[-1] should be v[0] + 2pi or less ') + + if not np.all(np.diff(u) > 0.0): + raise ValueError('u must be strictly increasing') + if not np.all(np.diff(v) > 0.0): + raise ValueError('v must be strictly increasing') + + if not u.size == r.shape[0]: + raise ValueError('u dimension of r must have same number of ' + 'elements as u') + if not v.size == r.shape[1]: + raise ValueError('v dimension of r must have same number of ' + 'elements as v') + + if pole_continuity[1] is False and pole_flat[1] is True: + raise ValueError('if pole_continuity is False, so must be ' + 'pole_flat') + if pole_continuity[0] is False and pole_flat[0] is True: + raise ValueError('if pole_continuity is False, so must be ' + 'pole_flat') + + if not s >= 0.0: + raise ValueError('s should be positive') + + r = np.ravel(r) + with FITPACK_LOCK: + nu, tu, nv, tv, c, fp, ier = dfitpack.regrid_smth_spher(iopt, ider, + u.copy(), + v.copy(), + r.copy(), + r0, r1, s) + + if ier not in [0, -1, -2]: + msg = _spfit_messages.get(ier, f'ier={ier}') + raise ValueError(msg) + + self.fp = fp + self.tck = tu[:nu], tv[:nv], c[:(nu - 4) * (nv-4)] + self.degrees = (3, 3) + self.v0 = v[0] + + def __call__(self, theta, phi, dtheta=0, dphi=0, grid=True): + + theta = np.asarray(theta) + phi = np.asarray(phi) + + return SphereBivariateSpline.__call__(self, theta, phi, dtheta=dtheta, + dphi=dphi, grid=grid) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack_impl.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack_impl.py new file mode 100644 index 0000000000000000000000000000000000000000..a00ca101b591dd69b6b590278083f0bdafbd3a01 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack_impl.py @@ -0,0 +1,805 @@ +""" +fitpack (dierckx in netlib) --- A Python-C wrapper to FITPACK (by P. Dierckx). + FITPACK is a collection of FORTRAN programs for curve and surface + fitting with splines and tensor product splines. + +See + https://web.archive.org/web/20010524124604/http://www.cs.kuleuven.ac.be:80/cwis/research/nalag/research/topics/fitpack.html +or + http://www.netlib.org/dierckx/ + +Copyright 2002 Pearu Peterson all rights reserved, +Pearu Peterson +Permission to use, modify, and distribute this software is given under the +terms of the SciPy (BSD style) license. See LICENSE.txt that came with +this distribution for specifics. + +NO WARRANTY IS EXPRESSED OR IMPLIED. USE AT YOUR OWN RISK. + +TODO: Make interfaces to the following fitpack functions: + For univariate splines: cocosp, concon, fourco, insert + For bivariate splines: profil, regrid, parsur, surev +""" + +__all__ = ['splrep', 'splprep', 'splev', 'splint', 'sproot', 'spalde', + 'bisplrep', 'bisplev', 'insert', 'splder', 'splantider'] + +import warnings +import numpy as np +from . import _fitpack +from numpy import (atleast_1d, array, ones, zeros, sqrt, ravel, transpose, + empty, iinfo, asarray) + +# Try to replace _fitpack interface with +# f2py-generated version +from . import _dfitpack as dfitpack + + +dfitpack_int = dfitpack.types.intvar.dtype + + +def _int_overflow(x, exception, msg=None): + """Cast the value to an dfitpack_int and raise an OverflowError if the value + cannot fit. + """ + if x > iinfo(dfitpack_int).max: + if msg is None: + msg = f'{x!r} cannot fit into an {dfitpack_int!r}' + raise exception(msg) + return dfitpack_int.type(x) + + +_iermess = { + 0: ["The spline has a residual sum of squares fp such that " + "abs(fp-s)/s<=0.001", None], + -1: ["The spline is an interpolating spline (fp=0)", None], + -2: ["The spline is weighted least-squares polynomial of degree k.\n" + "fp gives the upper bound fp0 for the smoothing factor s", None], + 1: ["The required storage space exceeds the available storage space.\n" + "Probable causes: data (x,y) size is too small or smoothing parameter" + "\ns is too small (fp>s).", ValueError], + 2: ["A theoretically impossible result when finding a smoothing spline\n" + "with fp = s. Probable cause: s too small. (abs(fp-s)/s>0.001)", + ValueError], + 3: ["The maximal number of iterations (20) allowed for finding smoothing\n" + "spline with fp=s has been reached. Probable cause: s too small.\n" + "(abs(fp-s)/s>0.001)", ValueError], + 10: ["Error on input data", ValueError], + 'unknown': ["An error occurred", TypeError] +} + +_iermess2 = { + 0: ["The spline has a residual sum of squares fp such that " + "abs(fp-s)/s<=0.001", None], + -1: ["The spline is an interpolating spline (fp=0)", None], + -2: ["The spline is weighted least-squares polynomial of degree kx and ky." + "\nfp gives the upper bound fp0 for the smoothing factor s", None], + -3: ["Warning. The coefficients of the spline have been computed as the\n" + "minimal norm least-squares solution of a rank deficient system.", + None], + 1: ["The required storage space exceeds the available storage space.\n" + "Probable causes: nxest or nyest too small or s is too small. (fp>s)", + ValueError], + 2: ["A theoretically impossible result when finding a smoothing spline\n" + "with fp = s. Probable causes: s too small or badly chosen eps.\n" + "(abs(fp-s)/s>0.001)", ValueError], + 3: ["The maximal number of iterations (20) allowed for finding smoothing\n" + "spline with fp=s has been reached. Probable cause: s too small.\n" + "(abs(fp-s)/s>0.001)", ValueError], + 4: ["No more knots can be added because the number of B-spline\n" + "coefficients already exceeds the number of data points m.\n" + "Probable causes: either s or m too small. (fp>s)", ValueError], + 5: ["No more knots can be added because the additional knot would\n" + "coincide with an old one. Probable cause: s too small or too large\n" + "a weight to an inaccurate data point. (fp>s)", ValueError], + 10: ["Error on input data", ValueError], + 11: ["rwrk2 too small, i.e., there is not enough workspace for computing\n" + "the minimal least-squares solution of a rank deficient system of\n" + "linear equations.", ValueError], + 'unknown': ["An error occurred", TypeError] +} + +_parcur_cache = {'t': array([], float), 'wrk': array([], float), + 'iwrk': array([], dfitpack_int), 'u': array([], float), + 'ub': 0, 'ue': 1} + + +def splprep(x, w=None, u=None, ub=None, ue=None, k=3, task=0, s=None, t=None, + full_output=0, nest=None, per=0, quiet=1): + # see the docstring of `_fitpack_py/splprep` + if task <= 0: + _parcur_cache = {'t': array([], float), 'wrk': array([], float), + 'iwrk': array([], dfitpack_int), 'u': array([], float), + 'ub': 0, 'ue': 1} + x = atleast_1d(x) + idim, m = x.shape + if per: + for i in range(idim): + if x[i][0] != x[i][-1]: + if not quiet: + warnings.warn(RuntimeWarning('Setting x[%d][%d]=x[%d][0]' % + (i, m, i)), + stacklevel=2) + x[i][-1] = x[i][0] + if not 0 < idim < 11: + raise TypeError('0 < idim < 11 must hold') + if w is None: + w = ones(m, float) + else: + w = atleast_1d(w) + ipar = (u is not None) + if ipar: + _parcur_cache['u'] = u + if ub is None: + _parcur_cache['ub'] = u[0] + else: + _parcur_cache['ub'] = ub + if ue is None: + _parcur_cache['ue'] = u[-1] + else: + _parcur_cache['ue'] = ue + else: + _parcur_cache['u'] = zeros(m, float) + if not (1 <= k <= 5): + raise TypeError('1 <= k= %d <=5 must hold' % k) + if not (-1 <= task <= 1): + raise TypeError('task must be -1, 0 or 1') + if (not len(w) == m) or (ipar == 1 and (not len(u) == m)): + raise TypeError('Mismatch of input dimensions') + if s is None: + s = m - sqrt(2*m) + if t is None and task == -1: + raise TypeError('Knots must be given for task=-1') + if t is not None: + _parcur_cache['t'] = atleast_1d(t) + n = len(_parcur_cache['t']) + if task == -1 and n < 2*k + 2: + raise TypeError('There must be at least 2*k+2 knots for task=-1') + if m <= k: + raise TypeError('m > k must hold') + if nest is None: + nest = m + 2*k + + if (task >= 0 and s == 0) or (nest < 0): + if per: + nest = m + 2*k + else: + nest = m + k + 1 + nest = max(nest, 2*k + 3) + u = _parcur_cache['u'] + ub = _parcur_cache['ub'] + ue = _parcur_cache['ue'] + t = _parcur_cache['t'] + wrk = _parcur_cache['wrk'] + iwrk = _parcur_cache['iwrk'] + t, c, o = _fitpack._parcur(ravel(transpose(x)), w, u, ub, ue, k, + task, ipar, s, t, nest, wrk, iwrk, per) + _parcur_cache['u'] = o['u'] + _parcur_cache['ub'] = o['ub'] + _parcur_cache['ue'] = o['ue'] + _parcur_cache['t'] = t + _parcur_cache['wrk'] = o['wrk'] + _parcur_cache['iwrk'] = o['iwrk'] + ier = o['ier'] + fp = o['fp'] + n = len(t) + u = o['u'] + c.shape = idim, n - k - 1 + tcku = [t, list(c), k], u + if ier <= 0 and not quiet: + warnings.warn(RuntimeWarning(_iermess[ier][0] + + "\tk=%d n=%d m=%d fp=%f s=%f" % + (k, len(t), m, fp, s)), + stacklevel=2) + if ier > 0 and not full_output: + if ier in [1, 2, 3]: + warnings.warn(RuntimeWarning(_iermess[ier][0]), stacklevel=2) + else: + try: + raise _iermess[ier][1](_iermess[ier][0]) + except KeyError as e: + raise _iermess['unknown'][1](_iermess['unknown'][0]) from e + if full_output: + try: + return tcku, fp, ier, _iermess[ier][0] + except KeyError: + return tcku, fp, ier, _iermess['unknown'][0] + else: + return tcku + + +_curfit_cache = {'t': array([], float), 'wrk': array([], float), + 'iwrk': array([], dfitpack_int)} + + +def splrep(x, y, w=None, xb=None, xe=None, k=3, task=0, s=None, t=None, + full_output=0, per=0, quiet=1): + # see the docstring of `_fitpack_py/splrep` + if task <= 0: + _curfit_cache = {} + x, y = map(atleast_1d, [x, y]) + m = len(x) + if w is None: + w = ones(m, float) + if s is None: + s = 0.0 + else: + w = atleast_1d(w) + if s is None: + s = m - sqrt(2*m) + if not len(w) == m: + raise TypeError('len(w)=%d is not equal to m=%d' % (len(w), m)) + if (m != len(y)) or (m != len(w)): + raise TypeError('Lengths of the first three arguments (x,y,w) must ' + 'be equal') + if not (1 <= k <= 5): + raise TypeError('Given degree of the spline (k=%d) is not supported. ' + '(1<=k<=5)' % k) + if m <= k: + raise TypeError('m > k must hold') + if xb is None: + xb = x[0] + if xe is None: + xe = x[-1] + if not (-1 <= task <= 1): + raise TypeError('task must be -1, 0 or 1') + if t is not None: + task = -1 + if task == -1: + if t is None: + raise TypeError('Knots must be given for task=-1') + numknots = len(t) + _curfit_cache['t'] = empty((numknots + 2*k + 2,), float) + _curfit_cache['t'][k+1:-k-1] = t + nest = len(_curfit_cache['t']) + elif task == 0: + if per: + nest = max(m + 2*k, 2*k + 3) + else: + nest = max(m + k + 1, 2*k + 3) + t = empty((nest,), float) + _curfit_cache['t'] = t + if task <= 0: + if per: + _curfit_cache['wrk'] = empty((m*(k + 1) + nest*(8 + 5*k),), float) + else: + _curfit_cache['wrk'] = empty((m*(k + 1) + nest*(7 + 3*k),), float) + _curfit_cache['iwrk'] = empty((nest,), dfitpack_int) + try: + t = _curfit_cache['t'] + wrk = _curfit_cache['wrk'] + iwrk = _curfit_cache['iwrk'] + except KeyError as e: + raise TypeError("must call with task=1 only after" + " call with task=0,-1") from e + if not per: + n, c, fp, ier = dfitpack.curfit(task, x, y, w, t, wrk, iwrk, + xb, xe, k, s) + else: + n, c, fp, ier = dfitpack.percur(task, x, y, w, t, wrk, iwrk, k, s) + tck = (t[:n], c[:n], k) + if ier <= 0 and not quiet: + _mess = (_iermess[ier][0] + "\tk=%d n=%d m=%d fp=%f s=%f" % + (k, len(t), m, fp, s)) + warnings.warn(RuntimeWarning(_mess), stacklevel=2) + if ier > 0 and not full_output: + if ier in [1, 2, 3]: + warnings.warn(RuntimeWarning(_iermess[ier][0]), stacklevel=2) + else: + try: + raise _iermess[ier][1](_iermess[ier][0]) + except KeyError as e: + raise _iermess['unknown'][1](_iermess['unknown'][0]) from e + if full_output: + try: + return tck, fp, ier, _iermess[ier][0] + except KeyError: + return tck, fp, ier, _iermess['unknown'][0] + else: + return tck + + +def splev(x, tck, der=0, ext=0): + # see the docstring of `_fitpack_py/splev` + t, c, k = tck + try: + c[0][0] + parametric = True + except Exception: + parametric = False + if parametric: + return list(map(lambda c, x=x, t=t, k=k, der=der: + splev(x, [t, c, k], der, ext), c)) + else: + if not (0 <= der <= k): + raise ValueError("0<=der=%d<=k=%d must hold" % (der, k)) + if ext not in (0, 1, 2, 3): + raise ValueError(f"ext = {ext} not in (0, 1, 2, 3) ") + + x = asarray(x) + shape = x.shape + x = atleast_1d(x).ravel() + if der == 0: + y, ier = dfitpack.splev(t, c, k, x, ext) + else: + y, ier = dfitpack.splder(t, c, k, x, der, ext) + + if ier == 10: + raise ValueError("Invalid input data") + if ier == 1: + raise ValueError("Found x value not in the domain") + if ier: + raise TypeError("An error occurred") + + return y.reshape(shape) + + +def splint(a, b, tck, full_output=0): + # see the docstring of `_fitpack_py/splint` + t, c, k = tck + try: + c[0][0] + parametric = True + except Exception: + parametric = False + if parametric: + return list(map(lambda c, a=a, b=b, t=t, k=k: + splint(a, b, [t, c, k]), c)) + else: + aint, wrk = dfitpack.splint(t, c, k, a, b) + if full_output: + return aint, wrk + else: + return aint + + +def sproot(tck, mest=10): + # see the docstring of `_fitpack_py/sproot` + t, c, k = tck + if k != 3: + raise ValueError("sproot works only for cubic (k=3) splines") + try: + c[0][0] + parametric = True + except Exception: + parametric = False + if parametric: + return list(map(lambda c, t=t, k=k, mest=mest: + sproot([t, c, k], mest), c)) + else: + if len(t) < 8: + raise TypeError(f"The number of knots {len(t)}>=8") + z, m, ier = dfitpack.sproot(t, c, mest) + if ier == 10: + raise TypeError("Invalid input data. " + "t1<=..<=t4 1: + return list(map(lambda x, tck=tck: spalde(x, tck), x)) + d, ier = dfitpack.spalde(t, c, k+1, x[0]) + if ier == 0: + return d + if ier == 10: + raise TypeError("Invalid input data. t(k)<=x<=t(n-k+1) must hold.") + raise TypeError("Unknown error") + +# def _curfit(x,y,w=None,xb=None,xe=None,k=3,task=0,s=None,t=None, +# full_output=0,nest=None,per=0,quiet=1): + + +_surfit_cache = {'tx': array([], float), 'ty': array([], float), + 'wrk': array([], float), 'iwrk': array([], dfitpack_int)} + + +def bisplrep(x, y, z, w=None, xb=None, xe=None, yb=None, ye=None, + kx=3, ky=3, task=0, s=None, eps=1e-16, tx=None, ty=None, + full_output=0, nxest=None, nyest=None, quiet=1): + """ + Find a bivariate B-spline representation of a surface. + + Given a set of data points (x[i], y[i], z[i]) representing a surface + z=f(x,y), compute a B-spline representation of the surface. Based on + the routine SURFIT from FITPACK. + + Parameters + ---------- + x, y, z : ndarray + Rank-1 arrays of data points. + w : ndarray, optional + Rank-1 array of weights. By default ``w=np.ones(len(x))``. + xb, xe : float, optional + End points of approximation interval in `x`. + By default ``xb = x.min(), xe=x.max()``. + yb, ye : float, optional + End points of approximation interval in `y`. + By default ``yb=y.min(), ye = y.max()``. + kx, ky : int, optional + The degrees of the spline (1 <= kx, ky <= 5). + Third order (kx=ky=3) is recommended. + task : int, optional + If task=0, find knots in x and y and coefficients for a given + smoothing factor, s. + If task=1, find knots and coefficients for another value of the + smoothing factor, s. bisplrep must have been previously called + with task=0 or task=1. + If task=-1, find coefficients for a given set of knots tx, ty. + s : float, optional + A non-negative smoothing factor. If weights correspond + to the inverse of the standard-deviation of the errors in z, + then a good s-value should be found in the range + ``(m-sqrt(2*m),m+sqrt(2*m))`` where m=len(x). + eps : float, optional + A threshold for determining the effective rank of an + over-determined linear system of equations (0 < eps < 1). + `eps` is not likely to need changing. + tx, ty : ndarray, optional + Rank-1 arrays of the knots of the spline for task=-1 + full_output : int, optional + Non-zero to return optional outputs. + nxest, nyest : int, optional + Over-estimates of the total number of knots. If None then + ``nxest = max(kx+sqrt(m/2),2*kx+3)``, + ``nyest = max(ky+sqrt(m/2),2*ky+3)``. + quiet : int, optional + Non-zero to suppress printing of messages. + + Returns + ------- + tck : array_like + A list [tx, ty, c, kx, ky] containing the knots (tx, ty) and + coefficients (c) of the bivariate B-spline representation of the + surface along with the degree of the spline. + fp : ndarray + The weighted sum of squared residuals of the spline approximation. + ier : int + An integer flag about splrep success. Success is indicated if + ier<=0. If ier in [1,2,3] an error occurred but was not raised. + Otherwise an error is raised. + msg : str + A message corresponding to the integer flag, ier. + + See Also + -------- + splprep, splrep, splint, sproot, splev + UnivariateSpline, BivariateSpline + + Notes + ----- + See `bisplev` to evaluate the value of the B-spline given its tck + representation. + + If the input data is such that input dimensions have incommensurate + units and differ by many orders of magnitude, the interpolant may have + numerical artifacts. Consider rescaling the data before interpolation. + + References + ---------- + .. [1] Dierckx P.:An algorithm for surface fitting with spline functions + Ima J. Numer. Anal. 1 (1981) 267-283. + .. [2] Dierckx P.:An algorithm for surface fitting with spline functions + report tw50, Dept. Computer Science,K.U.Leuven, 1980. + .. [3] Dierckx P.:Curve and surface fitting with splines, Monographs on + Numerical Analysis, Oxford University Press, 1993. + + Examples + -------- + Examples are given :ref:`in the tutorial `. + + """ + x, y, z = map(ravel, [x, y, z]) # ensure 1-d arrays. + m = len(x) + if not (m == len(y) == len(z)): + raise TypeError('len(x)==len(y)==len(z) must hold.') + if w is None: + w = ones(m, float) + else: + w = atleast_1d(w) + if not len(w) == m: + raise TypeError('len(w)=%d is not equal to m=%d' % (len(w), m)) + if xb is None: + xb = x.min() + if xe is None: + xe = x.max() + if yb is None: + yb = y.min() + if ye is None: + ye = y.max() + if not (-1 <= task <= 1): + raise TypeError('task must be -1, 0 or 1') + if s is None: + s = m - sqrt(2*m) + if tx is None and task == -1: + raise TypeError('Knots_x must be given for task=-1') + if tx is not None: + _surfit_cache['tx'] = atleast_1d(tx) + nx = len(_surfit_cache['tx']) + if ty is None and task == -1: + raise TypeError('Knots_y must be given for task=-1') + if ty is not None: + _surfit_cache['ty'] = atleast_1d(ty) + ny = len(_surfit_cache['ty']) + if task == -1 and nx < 2*kx+2: + raise TypeError('There must be at least 2*kx+2 knots_x for task=-1') + if task == -1 and ny < 2*ky+2: + raise TypeError('There must be at least 2*ky+2 knots_x for task=-1') + if not ((1 <= kx <= 5) and (1 <= ky <= 5)): + raise TypeError('Given degree of the spline (kx,ky=%d,%d) is not ' + 'supported. (1<=k<=5)' % (kx, ky)) + if m < (kx + 1)*(ky + 1): + raise TypeError('m >= (kx+1)(ky+1) must hold') + if nxest is None: + nxest = int(kx + sqrt(m/2)) + if nyest is None: + nyest = int(ky + sqrt(m/2)) + nxest, nyest = max(nxest, 2*kx + 3), max(nyest, 2*ky + 3) + if task >= 0 and s == 0: + nxest = int(kx + sqrt(3*m)) + nyest = int(ky + sqrt(3*m)) + if task == -1: + _surfit_cache['tx'] = atleast_1d(tx) + _surfit_cache['ty'] = atleast_1d(ty) + tx, ty = _surfit_cache['tx'], _surfit_cache['ty'] + wrk = _surfit_cache['wrk'] + u = nxest - kx - 1 + v = nyest - ky - 1 + km = max(kx, ky) + 1 + ne = max(nxest, nyest) + bx, by = kx*v + ky + 1, ky*u + kx + 1 + b1, b2 = bx, bx + v - ky + if bx > by: + b1, b2 = by, by + u - kx + msg = "Too many data points to interpolate" + lwrk1 = _int_overflow(u*v*(2 + b1 + b2) + + 2*(u + v + km*(m + ne) + ne - kx - ky) + b2 + 1, + OverflowError, + msg=msg) + lwrk2 = _int_overflow(u*v*(b2 + 1) + b2, OverflowError, msg=msg) + tx, ty, c, o = _fitpack._surfit(x, y, z, w, xb, xe, yb, ye, kx, ky, + task, s, eps, tx, ty, nxest, nyest, + wrk, lwrk1, lwrk2) + _curfit_cache['tx'] = tx + _curfit_cache['ty'] = ty + _curfit_cache['wrk'] = o['wrk'] + ier, fp = o['ier'], o['fp'] + tck = [tx, ty, c, kx, ky] + + ierm = min(11, max(-3, ier)) + if ierm <= 0 and not quiet: + _mess = (_iermess2[ierm][0] + + "\tkx,ky=%d,%d nx,ny=%d,%d m=%d fp=%f s=%f" % + (kx, ky, len(tx), len(ty), m, fp, s)) + warnings.warn(RuntimeWarning(_mess), stacklevel=2) + if ierm > 0 and not full_output: + if ier in [1, 2, 3, 4, 5]: + _mess = ("\n\tkx,ky=%d,%d nx,ny=%d,%d m=%d fp=%f s=%f" % + (kx, ky, len(tx), len(ty), m, fp, s)) + warnings.warn(RuntimeWarning(_iermess2[ierm][0] + _mess), stacklevel=2) + else: + try: + raise _iermess2[ierm][1](_iermess2[ierm][0]) + except KeyError as e: + raise _iermess2['unknown'][1](_iermess2['unknown'][0]) from e + if full_output: + try: + return tck, fp, ier, _iermess2[ierm][0] + except KeyError: + return tck, fp, ier, _iermess2['unknown'][0] + else: + return tck + + +def bisplev(x, y, tck, dx=0, dy=0): + """ + Evaluate a bivariate B-spline and its derivatives. + + Return a rank-2 array of spline function values (or spline derivative + values) at points given by the cross-product of the rank-1 arrays `x` and + `y`. In special cases, return an array or just a float if either `x` or + `y` or both are floats. Based on BISPEV and PARDER from FITPACK. + + Parameters + ---------- + x, y : ndarray + Rank-1 arrays specifying the domain over which to evaluate the + spline or its derivative. + tck : tuple + A sequence of length 5 returned by `bisplrep` containing the knot + locations, the coefficients, and the degree of the spline: + [tx, ty, c, kx, ky]. + dx, dy : int, optional + The orders of the partial derivatives in `x` and `y` respectively. + + Returns + ------- + vals : ndarray + The B-spline or its derivative evaluated over the set formed by + the cross-product of `x` and `y`. + + See Also + -------- + splprep, splrep, splint, sproot, splev + UnivariateSpline, BivariateSpline + + Notes + ----- + See `bisplrep` to generate the `tck` representation. + + References + ---------- + .. [1] Dierckx P. : An algorithm for surface fitting + with spline functions + Ima J. Numer. Anal. 1 (1981) 267-283. + .. [2] Dierckx P. : An algorithm for surface fitting + with spline functions + report tw50, Dept. Computer Science,K.U.Leuven, 1980. + .. [3] Dierckx P. : Curve and surface fitting with splines, + Monographs on Numerical Analysis, Oxford University Press, 1993. + + Examples + -------- + Examples are given :ref:`in the tutorial `. + + """ + tx, ty, c, kx, ky = tck + if not (0 <= dx < kx): + raise ValueError("0 <= dx = %d < kx = %d must hold" % (dx, kx)) + if not (0 <= dy < ky): + raise ValueError("0 <= dy = %d < ky = %d must hold" % (dy, ky)) + x, y = map(atleast_1d, [x, y]) + if (len(x.shape) != 1) or (len(y.shape) != 1): + raise ValueError("First two entries should be rank-1 arrays.") + + msg = "Too many data points to interpolate." + + _int_overflow(x.size * y.size, MemoryError, msg=msg) + + if dx != 0 or dy != 0: + _int_overflow((tx.size - kx - 1)*(ty.size - ky - 1), + MemoryError, msg=msg) + z, ier = dfitpack.parder(tx, ty, c, kx, ky, dx, dy, x, y) + else: + z, ier = dfitpack.bispev(tx, ty, c, kx, ky, x, y) + + if ier == 10: + raise ValueError("Invalid input data") + if ier: + raise TypeError("An error occurred") + z.shape = len(x), len(y) + if len(z) > 1: + return z + if len(z[0]) > 1: + return z[0] + return z[0][0] + + +def dblint(xa, xb, ya, yb, tck): + """Evaluate the integral of a spline over area [xa,xb] x [ya,yb]. + + Parameters + ---------- + xa, xb : float + The end-points of the x integration interval. + ya, yb : float + The end-points of the y integration interval. + tck : list [tx, ty, c, kx, ky] + A sequence of length 5 returned by bisplrep containing the knot + locations tx, ty, the coefficients c, and the degrees kx, ky + of the spline. + + Returns + ------- + integ : float + The value of the resulting integral. + """ + tx, ty, c, kx, ky = tck + return dfitpack.dblint(tx, ty, c, kx, ky, xa, xb, ya, yb) + + +def insert(x, tck, m=1, per=0): + # see the docstring of `_fitpack_py/insert` + t, c, k = tck + try: + c[0][0] + parametric = True + except Exception: + parametric = False + if parametric: + cc = [] + for c_vals in c: + tt, cc_val, kk = insert(x, [t, c_vals, k], m) + cc.append(cc_val) + return (tt, cc, kk) + else: + tt, cc, ier = _fitpack._insert(per, t, c, k, x, m) + if ier == 10: + raise ValueError("Invalid input data") + if ier: + raise TypeError("An error occurred") + return (tt, cc, k) + + +def splder(tck, n=1): + # see the docstring of `_fitpack_py/splder` + if n < 0: + return splantider(tck, -n) + + t, c, k = tck + + if n > k: + raise ValueError(f"Order of derivative (n = {n!r}) must be <= " + f"order of spline (k = {tck[2]!r})") + + # Extra axes for the trailing dims of the `c` array: + sh = (slice(None),) + ((None,)*len(c.shape[1:])) + + with np.errstate(invalid='raise', divide='raise'): + try: + for j in range(n): + # See e.g. Schumaker, Spline Functions: Basic Theory, Chapter 5 + + # Compute the denominator in the differentiation formula. + # (and append trailing dims, if necessary) + dt = t[k+1:-1] - t[1:-k-1] + dt = dt[sh] + # Compute the new coefficients + c = (c[1:-1-k] - c[:-2-k]) * k / dt + # Pad coefficient array to same size as knots (FITPACK + # convention) + c = np.r_[c, np.zeros((k,) + c.shape[1:])] + # Adjust knots + t = t[1:-1] + k -= 1 + except FloatingPointError as e: + raise ValueError(("The spline has internal repeated knots " + "and is not differentiable %d times") % n) from e + + return t, c, k + + +def splantider(tck, n=1): + # see the docstring of `_fitpack_py/splantider` + if n < 0: + return splder(tck, -n) + + t, c, k = tck + + # Extra axes for the trailing dims of the `c` array: + sh = (slice(None),) + (None,)*len(c.shape[1:]) + + for j in range(n): + # This is the inverse set of operations to splder. + + # Compute the multiplier in the antiderivative formula. + dt = t[k+1:] - t[:-k-1] + dt = dt[sh] + # Compute the new coefficients + c = np.cumsum(c[:-k-1] * dt, axis=0) / (k + 1) + c = np.r_[np.zeros((1,) + c.shape[1:]), + c, + [c[-1]] * (k+2)] + # New knots + t = np.r_[t[0], t, t[-1]] + k += 1 + + return t, c, k diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack_py.py new file mode 100644 index 0000000000000000000000000000000000000000..9f7a2ded7e46885b4e0e0e4ccdb8065c25742e6a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack_py.py @@ -0,0 +1,898 @@ +__all__ = ['splrep', 'splprep', 'splev', 'splint', 'sproot', 'spalde', + 'bisplrep', 'bisplev', 'insert', 'splder', 'splantider'] + + +import numpy as np + +# These are in the API for fitpack even if not used in fitpack.py itself. +from ._fitpack_impl import bisplrep, bisplev, dblint # noqa: F401 +from . import _fitpack_impl as _impl +from ._bsplines import BSpline + + +def splprep(x, w=None, u=None, ub=None, ue=None, k=3, task=0, s=None, t=None, + full_output=0, nest=None, per=0, quiet=1): + """ + Find the B-spline representation of an N-D curve. + + .. legacy:: function + + Specifically, we recommend using `make_splprep` in new code. + + Given a list of N rank-1 arrays, `x`, which represent a curve in + N-dimensional space parametrized by `u`, find a smooth approximating + spline curve g(`u`). Uses the FORTRAN routine parcur from FITPACK. + + Parameters + ---------- + x : array_like + A list of sample vector arrays representing the curve. + w : array_like, optional + Strictly positive rank-1 array of weights the same length as `x[0]`. + The weights are used in computing the weighted least-squares spline + fit. If the errors in the `x` values have standard-deviation given by + the vector d, then `w` should be 1/d. Default is ``ones(len(x[0]))``. + u : array_like, optional + An array of parameter values. If not given, these values are + calculated automatically as ``M = len(x[0])``, where + + v[0] = 0 + + v[i] = v[i-1] + distance(`x[i]`, `x[i-1]`) + + u[i] = v[i] / v[M-1] + + ub, ue : int, optional + The end-points of the parameters interval. Defaults to + u[0] and u[-1]. + k : int, optional + Degree of the spline. Cubic splines are recommended. + Even values of `k` should be avoided especially with a small s-value. + ``1 <= k <= 5``, default is 3. + task : int, optional + If task==0 (default), find t and c for a given smoothing factor, s. + If task==1, find t and c for another value of the smoothing factor, s. + There must have been a previous call with task=0 or task=1 + for the same set of data. + If task=-1 find the weighted least square spline for a given set of + knots, t. + s : float, optional + A smoothing condition. The amount of smoothness is determined by + satisfying the conditions: ``sum((w * (y - g))**2,axis=0) <= s``, + where g(x) is the smoothed interpolation of (x,y). The user can + use `s` to control the trade-off between closeness and smoothness + of fit. Larger `s` means more smoothing while smaller values of `s` + indicate less smoothing. Recommended values of `s` depend on the + weights, w. If the weights represent the inverse of the + standard-deviation of y, then a good `s` value should be found in + the range ``(m-sqrt(2*m),m+sqrt(2*m))``, where m is the number of + data points in x, y, and w. + t : array, optional + The knots needed for ``task=-1``. + There must be at least ``2*k+2`` knots. + full_output : int, optional + If non-zero, then return optional outputs. + nest : int, optional + An over-estimate of the total number of knots of the spline to + help in determining the storage space. By default nest=m/2. + Always large enough is nest=m+k+1. + per : int, optional + If non-zero, data points are considered periodic with period + ``x[m-1] - x[0]`` and a smooth periodic spline approximation is + returned. Values of ``y[m-1]`` and ``w[m-1]`` are not used. + quiet : int, optional + Non-zero to suppress messages. + + Returns + ------- + tck : tuple + A tuple, ``(t,c,k)`` containing the vector of knots, the B-spline + coefficients, and the degree of the spline. + u : array + An array of the values of the parameter. + fp : float + The weighted sum of squared residuals of the spline approximation. + ier : int + An integer flag about splrep success. Success is indicated + if ier<=0. If ier in [1,2,3] an error occurred but was not raised. + Otherwise an error is raised. + msg : str + A message corresponding to the integer flag, ier. + + See Also + -------- + splrep, splev, sproot, spalde, splint, + bisplrep, bisplev + UnivariateSpline, BivariateSpline + BSpline + make_interp_spline + + Notes + ----- + See `splev` for evaluation of the spline and its derivatives. + The number of dimensions N must be smaller than 11. + + The number of coefficients in the `c` array is ``k+1`` less than the number + of knots, ``len(t)``. This is in contrast with `splrep`, which zero-pads + the array of coefficients to have the same length as the array of knots. + These additional coefficients are ignored by evaluation routines, `splev` + and `BSpline`. + + References + ---------- + .. [1] P. Dierckx, "Algorithms for smoothing data with periodic and + parametric splines, Computer Graphics and Image Processing", + 20 (1982) 171-184. + .. [2] P. Dierckx, "Algorithms for smoothing data with periodic and + parametric splines", report tw55, Dept. Computer Science, + K.U.Leuven, 1981. + .. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs on + Numerical Analysis, Oxford University Press, 1993. + + Examples + -------- + Generate a discretization of a limacon curve in the polar coordinates: + + >>> import numpy as np + >>> phi = np.linspace(0, 2.*np.pi, 40) + >>> r = 0.5 + np.cos(phi) # polar coords + >>> x, y = r * np.cos(phi), r * np.sin(phi) # convert to cartesian + + And interpolate: + + >>> from scipy.interpolate import splprep, splev + >>> tck, u = splprep([x, y], s=0) + >>> new_points = splev(u, tck) + + Notice that (i) we force interpolation by using ``s=0``, + (ii) the parameterization, ``u``, is generated automatically. + Now plot the result: + + >>> import matplotlib.pyplot as plt + >>> fig, ax = plt.subplots() + >>> ax.plot(x, y, 'ro') + >>> ax.plot(new_points[0], new_points[1], 'r-') + >>> plt.show() + + """ + + res = _impl.splprep(x, w, u, ub, ue, k, task, s, t, full_output, nest, per, + quiet) + return res + + +def splrep(x, y, w=None, xb=None, xe=None, k=3, task=0, s=None, t=None, + full_output=0, per=0, quiet=1): + """ + Find the B-spline representation of a 1-D curve. + + .. legacy:: function + + Specifically, we recommend using `make_splrep` in new code. + + + Given the set of data points ``(x[i], y[i])`` determine a smooth spline + approximation of degree k on the interval ``xb <= x <= xe``. + + Parameters + ---------- + x, y : array_like + The data points defining a curve ``y = f(x)``. + w : array_like, optional + Strictly positive rank-1 array of weights the same length as `x` and `y`. + The weights are used in computing the weighted least-squares spline + fit. If the errors in the `y` values have standard-deviation given by the + vector ``d``, then `w` should be ``1/d``. Default is ``ones(len(x))``. + xb, xe : float, optional + The interval to fit. If None, these default to ``x[0]`` and ``x[-1]`` + respectively. + k : int, optional + The degree of the spline fit. It is recommended to use cubic splines. + Even values of `k` should be avoided especially with small `s` values. + ``1 <= k <= 5``. + task : {1, 0, -1}, optional + If ``task==0``, find ``t`` and ``c`` for a given smoothing factor, `s`. + + If ``task==1`` find ``t`` and ``c`` for another value of the smoothing factor, + `s`. There must have been a previous call with ``task=0`` or ``task=1`` for + the same set of data (``t`` will be stored an used internally) + + If ``task=-1`` find the weighted least square spline for a given set of + knots, ``t``. These should be interior knots as knots on the ends will be + added automatically. + s : float, optional + A smoothing condition. The amount of smoothness is determined by + satisfying the conditions: ``sum((w * (y - g))**2,axis=0) <= s`` where ``g(x)`` + is the smoothed interpolation of ``(x,y)``. The user can use `s` to control + the tradeoff between closeness and smoothness of fit. Larger `s` means + more smoothing while smaller values of `s` indicate less smoothing. + Recommended values of `s` depend on the weights, `w`. If the weights + represent the inverse of the standard-deviation of `y`, then a good `s` + value should be found in the range ``(m-sqrt(2*m),m+sqrt(2*m))`` where ``m`` is + the number of datapoints in `x`, `y`, and `w`. default : ``s=m-sqrt(2*m)`` if + weights are supplied. ``s = 0.0`` (interpolating) if no weights are + supplied. + t : array_like, optional + The knots needed for ``task=-1``. If given then task is automatically set + to ``-1``. + full_output : bool, optional + If non-zero, then return optional outputs. + per : bool, optional + If non-zero, data points are considered periodic with period ``x[m-1]`` - + ``x[0]`` and a smooth periodic spline approximation is returned. Values of + ``y[m-1]`` and ``w[m-1]`` are not used. + The default is zero, corresponding to boundary condition 'not-a-knot'. + quiet : bool, optional + Non-zero to suppress messages. + + Returns + ------- + tck : tuple + A tuple ``(t,c,k)`` containing the vector of knots, the B-spline + coefficients, and the degree of the spline. + fp : array, optional + The weighted sum of squared residuals of the spline approximation. + ier : int, optional + An integer flag about splrep success. Success is indicated if ``ier<=0``. + If ``ier in [1,2,3]``, an error occurred but was not raised. Otherwise an + error is raised. + msg : str, optional + A message corresponding to the integer flag, `ier`. + + See Also + -------- + UnivariateSpline, BivariateSpline + splprep, splev, sproot, spalde, splint + bisplrep, bisplev + BSpline + make_interp_spline + + Notes + ----- + See `splev` for evaluation of the spline and its derivatives. Uses the + FORTRAN routine ``curfit`` from FITPACK. + + The user is responsible for assuring that the values of `x` are unique. + Otherwise, `splrep` will not return sensible results. + + If provided, knots `t` must satisfy the Schoenberg-Whitney conditions, + i.e., there must be a subset of data points ``x[j]`` such that + ``t[j] < x[j] < t[j+k+1]``, for ``j=0, 1,...,n-k-2``. + + This routine zero-pads the coefficients array ``c`` to have the same length + as the array of knots ``t`` (the trailing ``k + 1`` coefficients are ignored + by the evaluation routines, `splev` and `BSpline`.) This is in contrast with + `splprep`, which does not zero-pad the coefficients. + + The default boundary condition is 'not-a-knot', i.e. the first and second + segment at a curve end are the same polynomial. More boundary conditions are + available in `CubicSpline`. + + References + ---------- + Based on algorithms described in [1]_, [2]_, [3]_, and [4]_: + + .. [1] P. Dierckx, "An algorithm for smoothing, differentiation and + integration of experimental data using spline functions", + J.Comp.Appl.Maths 1 (1975) 165-184. + .. [2] P. Dierckx, "A fast algorithm for smoothing data on a rectangular + grid while using spline functions", SIAM J.Numer.Anal. 19 (1982) + 1286-1304. + .. [3] P. Dierckx, "An improved algorithm for curve fitting with spline + functions", report tw54, Dept. Computer Science,K.U. Leuven, 1981. + .. [4] P. Dierckx, "Curve and surface fitting with splines", Monographs on + Numerical Analysis, Oxford University Press, 1993. + + Examples + -------- + You can interpolate 1-D points with a B-spline curve. + Further examples are given in + :ref:`in the tutorial `. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import splev, splrep + >>> x = np.linspace(0, 10, 10) + >>> y = np.sin(x) + >>> spl = splrep(x, y) + >>> x2 = np.linspace(0, 10, 200) + >>> y2 = splev(x2, spl) + >>> plt.plot(x, y, 'o', x2, y2) + >>> plt.show() + + """ + res = _impl.splrep(x, y, w, xb, xe, k, task, s, t, full_output, per, quiet) + return res + + +def splev(x, tck, der=0, ext=0): + """ + Evaluate a B-spline or its derivatives. + + .. legacy:: function + + Specifically, we recommend constructing a `BSpline` object and using + its ``__call__`` method. + + Given the knots and coefficients of a B-spline representation, evaluate + the value of the smoothing polynomial and its derivatives. This is a + wrapper around the FORTRAN routines splev and splder of FITPACK. + + Parameters + ---------- + x : array_like + An array of points at which to return the value of the smoothed + spline or its derivatives. If `tck` was returned from `splprep`, + then the parameter values, u should be given. + tck : BSpline instance or tuple + If a tuple, then it should be a sequence of length 3 returned by + `splrep` or `splprep` containing the knots, coefficients, and degree + of the spline. (Also see Notes.) + der : int, optional + The order of derivative of the spline to compute (must be less than + or equal to k, the degree of the spline). + ext : int, optional + Controls the value returned for elements of ``x`` not in the + interval defined by the knot sequence. + + * if ext=0, return the extrapolated value. + * if ext=1, return 0 + * if ext=2, raise a ValueError + * if ext=3, return the boundary value. + + The default value is 0. + + Returns + ------- + y : ndarray or list of ndarrays + An array of values representing the spline function evaluated at + the points in `x`. If `tck` was returned from `splprep`, then this + is a list of arrays representing the curve in an N-D space. + + See Also + -------- + splprep, splrep, sproot, spalde, splint + bisplrep, bisplev + BSpline + + Notes + ----- + Manipulating the tck-tuples directly is not recommended. In new code, + prefer using `BSpline` objects. + + References + ---------- + .. [1] C. de Boor, "On calculating with b-splines", J. Approximation + Theory, 6, p.50-62, 1972. + .. [2] M. G. Cox, "The numerical evaluation of b-splines", J. Inst. Maths + Applics, 10, p.134-149, 1972. + .. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs + on Numerical Analysis, Oxford University Press, 1993. + + Examples + -------- + Examples are given :ref:`in the tutorial `. + + A comparison between `splev`, `splder` and `spalde` to compute the derivatives of a + B-spline can be found in the `spalde` examples section. + + """ + if isinstance(tck, BSpline): + if tck.c.ndim > 1: + mesg = ("Calling splev() with BSpline objects with c.ndim > 1 is " + "not allowed. Use BSpline.__call__(x) instead.") + raise ValueError(mesg) + + # remap the out-of-bounds behavior + try: + extrapolate = {0: True, }[ext] + except KeyError as e: + raise ValueError(f"Extrapolation mode {ext} is not supported " + "by BSpline.") from e + + return tck(x, der, extrapolate=extrapolate) + else: + return _impl.splev(x, tck, der, ext) + + +def splint(a, b, tck, full_output=0): + """ + Evaluate the definite integral of a B-spline between two given points. + + .. legacy:: function + + Specifically, we recommend constructing a `BSpline` object and using its + ``integrate`` method. + + Parameters + ---------- + a, b : float + The end-points of the integration interval. + tck : tuple or a BSpline instance + If a tuple, then it should be a sequence of length 3, containing the + vector of knots, the B-spline coefficients, and the degree of the + spline (see `splev`). + full_output : int, optional + Non-zero to return optional output. + + Returns + ------- + integral : float + The resulting integral. + wrk : ndarray + An array containing the integrals of the normalized B-splines + defined on the set of knots. + (Only returned if `full_output` is non-zero) + + See Also + -------- + splprep, splrep, sproot, spalde, splev + bisplrep, bisplev + BSpline + + Notes + ----- + `splint` silently assumes that the spline function is zero outside the data + interval (`a`, `b`). + + Manipulating the tck-tuples directly is not recommended. In new code, + prefer using the `BSpline` objects. + + References + ---------- + .. [1] P.W. Gaffney, The calculation of indefinite integrals of b-splines", + J. Inst. Maths Applics, 17, p.37-41, 1976. + .. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs + on Numerical Analysis, Oxford University Press, 1993. + + Examples + -------- + Examples are given :ref:`in the tutorial `. + + """ + if isinstance(tck, BSpline): + if tck.c.ndim > 1: + mesg = ("Calling splint() with BSpline objects with c.ndim > 1 is " + "not allowed. Use BSpline.integrate() instead.") + raise ValueError(mesg) + + if full_output != 0: + mesg = (f"full_output = {full_output} is not supported. Proceeding as if " + "full_output = 0") + + return tck.integrate(a, b, extrapolate=False) + else: + return _impl.splint(a, b, tck, full_output) + + +def sproot(tck, mest=10): + """ + Find the roots of a cubic B-spline. + + .. legacy:: function + + Specifically, we recommend constructing a `BSpline` object and using the + following pattern: `PPoly.from_spline(spl).roots()`. + + Given the knots (>=8) and coefficients of a cubic B-spline return the + roots of the spline. + + Parameters + ---------- + tck : tuple or a BSpline object + If a tuple, then it should be a sequence of length 3, containing the + vector of knots, the B-spline coefficients, and the degree of the + spline. + The number of knots must be >= 8, and the degree must be 3. + The knots must be a montonically increasing sequence. + mest : int, optional + An estimate of the number of zeros (Default is 10). + + Returns + ------- + zeros : ndarray + An array giving the roots of the spline. + + See Also + -------- + splprep, splrep, splint, spalde, splev + bisplrep, bisplev + BSpline + + Notes + ----- + Manipulating the tck-tuples directly is not recommended. In new code, + prefer using the `BSpline` objects. + + References + ---------- + .. [1] C. de Boor, "On calculating with b-splines", J. Approximation + Theory, 6, p.50-62, 1972. + .. [2] M. G. Cox, "The numerical evaluation of b-splines", J. Inst. Maths + Applics, 10, p.134-149, 1972. + .. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs + on Numerical Analysis, Oxford University Press, 1993. + + Examples + -------- + + For some data, this method may miss a root. This happens when one of + the spline knots (which FITPACK places automatically) happens to + coincide with the true root. A workaround is to convert to `PPoly`, + which uses a different root-finding algorithm. + + For example, + + >>> x = [1.96, 1.97, 1.98, 1.99, 2.00, 2.01, 2.02, 2.03, 2.04, 2.05] + >>> y = [-6.365470e-03, -4.790580e-03, -3.204320e-03, -1.607270e-03, + ... 4.440892e-16, 1.616930e-03, 3.243000e-03, 4.877670e-03, + ... 6.520430e-03, 8.170770e-03] + >>> from scipy.interpolate import splrep, sproot, PPoly + >>> tck = splrep(x, y, s=0) + >>> sproot(tck) + array([], dtype=float64) + + Converting to a PPoly object does find the roots at ``x=2``: + + >>> ppoly = PPoly.from_spline(tck) + >>> ppoly.roots(extrapolate=False) + array([2.]) + + + Further examples are given :ref:`in the tutorial + `. + + """ + if isinstance(tck, BSpline): + if tck.c.ndim > 1: + mesg = ("Calling sproot() with BSpline objects with c.ndim > 1 is " + "not allowed.") + raise ValueError(mesg) + + t, c, k = tck.tck + + # _impl.sproot expects the interpolation axis to be last, so roll it. + # NB: This transpose is a no-op if c is 1D. + sh = tuple(range(c.ndim)) + c = c.transpose(sh[1:] + (0,)) + return _impl.sproot((t, c, k), mest) + else: + return _impl.sproot(tck, mest) + + +def spalde(x, tck): + """ + Evaluate a B-spline and all its derivatives at one point (or set of points) up + to order k (the degree of the spline), being 0 the spline itself. + + .. legacy:: function + + Specifically, we recommend constructing a `BSpline` object and evaluate + its derivative in a loop or a list comprehension. + + Parameters + ---------- + x : array_like + A point or a set of points at which to evaluate the derivatives. + Note that ``t(k) <= x <= t(n-k+1)`` must hold for each `x`. + tck : tuple + A tuple (t,c,k) containing the vector of knots, + the B-spline coefficients, and the degree of the spline whose + derivatives to compute. + + Returns + ------- + results : {ndarray, list of ndarrays} + An array (or a list of arrays) containing all derivatives + up to order k inclusive for each point `x`, being the first element the + spline itself. + + See Also + -------- + splprep, splrep, splint, sproot, splev, bisplrep, bisplev, + UnivariateSpline, BivariateSpline + + References + ---------- + .. [1] de Boor C : On calculating with b-splines, J. Approximation Theory + 6 (1972) 50-62. + .. [2] Cox M.G. : The numerical evaluation of b-splines, J. Inst. Maths + applics 10 (1972) 134-149. + .. [3] Dierckx P. : Curve and surface fitting with splines, Monographs on + Numerical Analysis, Oxford University Press, 1993. + + Examples + -------- + To calculate the derivatives of a B-spline there are several aproaches. + In this example, we will demonstrate that `spalde` is equivalent to + calling `splev` and `splder`. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import BSpline, spalde, splder, splev + + >>> # Store characteristic parameters of a B-spline + >>> tck = ((-2, -2, -2, -2, -1, 0, 1, 2, 2, 2, 2), # knots + ... (0, 0, 0, 6, 0, 0, 0), # coefficients + ... 3) # degree (cubic) + >>> # Instance a B-spline object + >>> # `BSpline` objects are preferred, except for spalde() + >>> bspl = BSpline(tck[0], tck[1], tck[2]) + >>> # Generate extra points to get a smooth curve + >>> x = np.linspace(min(tck[0]), max(tck[0]), 100) + + Evaluate the curve and all derivatives + + >>> # The order of derivative must be less or equal to k, the degree of the spline + >>> # Method 1: spalde() + >>> f1_y_bsplin = [spalde(i, tck)[0] for i in x ] # The B-spline itself + >>> f1_y_deriv1 = [spalde(i, tck)[1] for i in x ] # 1st derivative + >>> f1_y_deriv2 = [spalde(i, tck)[2] for i in x ] # 2nd derivative + >>> f1_y_deriv3 = [spalde(i, tck)[3] for i in x ] # 3rd derivative + >>> # You can reach the same result by using `splev`and `splder` + >>> f2_y_deriv3 = splev(x, bspl, der=3) + >>> f3_y_deriv3 = splder(bspl, n=3)(x) + + >>> # Generate a figure with three axes for graphic comparison + >>> fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(16, 5)) + >>> suptitle = fig.suptitle(f'Evaluate a B-spline and all derivatives') + >>> # Plot B-spline and all derivatives using the three methods + >>> orders = range(4) + >>> linetypes = ['-', '--', '-.', ':'] + >>> labels = ['B-Spline', '1st deriv.', '2nd deriv.', '3rd deriv.'] + >>> functions = ['splev()', 'splder()', 'spalde()'] + >>> for order, linetype, label in zip(orders, linetypes, labels): + ... ax1.plot(x, splev(x, bspl, der=order), linetype, label=label) + ... ax2.plot(x, splder(bspl, n=order)(x), linetype, label=label) + ... ax3.plot(x, [spalde(i, tck)[order] for i in x], linetype, label=label) + >>> for ax, function in zip((ax1, ax2, ax3), functions): + ... ax.set_title(function) + ... ax.legend() + >>> plt.tight_layout() + >>> plt.show() + + """ + if isinstance(tck, BSpline): + raise TypeError("spalde does not accept BSpline instances.") + else: + return _impl.spalde(x, tck) + + +def insert(x, tck, m=1, per=0): + """ + Insert knots into a B-spline. + + .. legacy:: function + + Specifically, we recommend constructing a `BSpline` object and using + its ``insert_knot`` method. + + Given the knots and coefficients of a B-spline representation, create a + new B-spline with a knot inserted `m` times at point `x`. + This is a wrapper around the FORTRAN routine insert of FITPACK. + + Parameters + ---------- + x (u) : float + A knot value at which to insert a new knot. If `tck` was returned + from ``splprep``, then the parameter values, u should be given. + tck : a `BSpline` instance or a tuple + If tuple, then it is expected to be a tuple (t,c,k) containing + the vector of knots, the B-spline coefficients, and the degree of + the spline. + m : int, optional + The number of times to insert the given knot (its multiplicity). + Default is 1. + per : int, optional + If non-zero, the input spline is considered periodic. + + Returns + ------- + BSpline instance or a tuple + A new B-spline with knots t, coefficients c, and degree k. + ``t(k+1) <= x <= t(n-k)``, where k is the degree of the spline. + In case of a periodic spline (``per != 0``) there must be + either at least k interior knots t(j) satisfying ``t(k+1)>> from scipy.interpolate import splrep, insert + >>> import numpy as np + >>> x = np.linspace(0, 10, 5) + >>> y = np.sin(x) + >>> tck = splrep(x, y) + >>> tck[0] + array([ 0., 0., 0., 0., 5., 10., 10., 10., 10.]) + + A knot is inserted: + + >>> tck_inserted = insert(3, tck) + >>> tck_inserted[0] + array([ 0., 0., 0., 0., 3., 5., 10., 10., 10., 10.]) + + Some knots are inserted: + + >>> tck_inserted2 = insert(8, tck, m=3) + >>> tck_inserted2[0] + array([ 0., 0., 0., 0., 5., 8., 8., 8., 10., 10., 10., 10.]) + + """ + if isinstance(tck, BSpline): + + t, c, k = tck.tck + + # FITPACK expects the interpolation axis to be last, so roll it over + # NB: if c array is 1D, transposes are no-ops + sh = tuple(range(c.ndim)) + c = c.transpose(sh[1:] + (0,)) + t_, c_, k_ = _impl.insert(x, (t, c, k), m, per) + + # and roll the last axis back + c_ = np.asarray(c_) + c_ = c_.transpose((sh[-1],) + sh[:-1]) + return BSpline(t_, c_, k_) + else: + return _impl.insert(x, tck, m, per) + + +def splder(tck, n=1): + """ + Compute the spline representation of the derivative of a given spline + + .. legacy:: function + + Specifically, we recommend constructing a `BSpline` object and using its + ``derivative`` method. + + Parameters + ---------- + tck : BSpline instance or tuple + BSpline instance or a tuple (t,c,k) containing the vector of knots, + the B-spline coefficients, and the degree of the spline whose + derivative to compute + n : int, optional + Order of derivative to evaluate. Default: 1 + + Returns + ------- + `BSpline` instance or tuple + Spline of order k2=k-n representing the derivative + of the input spline. + A tuple is returned if the input argument `tck` is a tuple, otherwise + a BSpline object is constructed and returned. + + See Also + -------- + splantider, splev, spalde + BSpline + + Notes + ----- + + .. versionadded:: 0.13.0 + + Examples + -------- + This can be used for finding maxima of a curve: + + >>> from scipy.interpolate import splrep, splder, sproot + >>> import numpy as np + >>> x = np.linspace(0, 10, 70) + >>> y = np.sin(x) + >>> spl = splrep(x, y, k=4) + + Now, differentiate the spline and find the zeros of the + derivative. (NB: `sproot` only works for order 3 splines, so we + fit an order 4 spline): + + >>> dspl = splder(spl) + >>> sproot(dspl) / np.pi + array([ 0.50000001, 1.5 , 2.49999998]) + + This agrees well with roots :math:`\\pi/2 + n\\pi` of + :math:`\\cos(x) = \\sin'(x)`. + + A comparison between `splev`, `splder` and `spalde` to compute the derivatives of a + B-spline can be found in the `spalde` examples section. + + """ + if isinstance(tck, BSpline): + return tck.derivative(n) + else: + return _impl.splder(tck, n) + + +def splantider(tck, n=1): + """ + Compute the spline for the antiderivative (integral) of a given spline. + + .. legacy:: function + + Specifically, we recommend constructing a `BSpline` object and using its + ``antiderivative`` method. + + Parameters + ---------- + tck : BSpline instance or a tuple of (t, c, k) + Spline whose antiderivative to compute + n : int, optional + Order of antiderivative to evaluate. Default: 1 + + Returns + ------- + BSpline instance or a tuple of (t2, c2, k2) + Spline of order k2=k+n representing the antiderivative of the input + spline. + A tuple is returned iff the input argument `tck` is a tuple, otherwise + a BSpline object is constructed and returned. + + See Also + -------- + splder, splev, spalde + BSpline + + Notes + ----- + The `splder` function is the inverse operation of this function. + Namely, ``splder(splantider(tck))`` is identical to `tck`, modulo + rounding error. + + .. versionadded:: 0.13.0 + + Examples + -------- + >>> from scipy.interpolate import splrep, splder, splantider, splev + >>> import numpy as np + >>> x = np.linspace(0, np.pi/2, 70) + >>> y = 1 / np.sqrt(1 - 0.8*np.sin(x)**2) + >>> spl = splrep(x, y) + + The derivative is the inverse operation of the antiderivative, + although some floating point error accumulates: + + >>> splev(1.7, spl), splev(1.7, splder(splantider(spl))) + (array(2.1565429877197317), array(2.1565429877201865)) + + Antiderivative can be used to evaluate definite integrals: + + >>> ispl = splantider(spl) + >>> splev(np.pi/2, ispl) - splev(0, ispl) + 2.2572053588768486 + + This is indeed an approximation to the complete elliptic integral + :math:`K(m) = \\int_0^{\\pi/2} [1 - m\\sin^2 x]^{-1/2} dx`: + + >>> from scipy.special import ellipk + >>> ellipk(0.8) + 2.2572053268208538 + + """ + if isinstance(tck, BSpline): + return tck.antiderivative(n) + else: + return _impl.splantider(tck, n) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack_repro.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack_repro.py new file mode 100644 index 0000000000000000000000000000000000000000..f5697f3ad716500f6557175e88adead6b3b4caac --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_fitpack_repro.py @@ -0,0 +1,992 @@ +""" Replicate FITPACK's logic for constructing smoothing spline functions and curves. + + Currently provides analogs of splrep and splprep python routines, i.e. + curfit.f and parcur.f routines (the drivers are fpcurf.f and fppara.f, respectively) + + The Fortran sources are from + https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/ + + .. [1] P. Dierckx, "Algorithms for smoothing data with periodic and + parametric splines, Computer Graphics and Image Processing", + 20 (1982) 171-184. + :doi:`10.1016/0146-664X(82)90043-0`. + .. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs on + Numerical Analysis, Oxford University Press, 1993. + .. [3] P. Dierckx, "An algorithm for smoothing, differentiation and integration + of experimental data using spline functions", + Journal of Computational and Applied Mathematics, vol. I, no 3, p. 165 (1975). + https://doi.org/10.1016/0771-050X(75)90034-0 +""" +import warnings +import operator +import numpy as np + +from ._bsplines import ( + _not_a_knot, make_interp_spline, BSpline, fpcheck, _lsq_solve_qr +) +from . import _dierckx # type: ignore[attr-defined] + + +# cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc +# c part 1: determination of the number of knots and their position c +# c ************************************************************** c +# +# https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L31 + + +# Hardcoded in curfit.f +TOL = 0.001 +MAXIT = 20 + + +def _get_residuals(x, y, t, k, w): + # FITPACK has (w*(spl(x)-y))**2; make_lsq_spline has w*(spl(x)-y)**2 + w2 = w**2 + + # inline the relevant part of + # >>> spl = make_lsq_spline(x, y, w=w2, t=t, k=k) + # NB: + # 1. y is assumed to be 2D here. For 1D case (parametric=False), + # the call must have been preceded by y = y[:, None] (cf _validate_inputs) + # 2. We always sum the squares across axis=1: + # * For 1D (parametric=False), the last dimension has size one, + # so the summation is a no-op. + # * For 2D (parametric=True), the summation is actually how the + # 'residuals' are defined, see Eq. (42) in Dierckx1982 + # (the reference is in the docstring of `class F`) below. + _, _, c = _lsq_solve_qr(x, y, t, k, w) + c = np.ascontiguousarray(c) + spl = BSpline(t, c, k) + return _compute_residuals(w2, spl(x), y) + + +def _compute_residuals(w2, splx, y): + delta = ((splx - y)**2).sum(axis=1) + return w2 * delta + + +def add_knot(x, t, k, residuals): + """Add a new knot. + + (Approximately) replicate FITPACK's logic: + 1. split the `x` array into knot intervals, ``t(j+k) <= x(i) <= t(j+k+1)`` + 2. find the interval with the maximum sum of residuals + 3. insert a new knot into the middle of that interval. + + NB: a new knot is in fact an `x` value at the middle of the interval. + So *the knots are a subset of `x`*. + + This routine is an analog of + https://github.com/scipy/scipy/blob/v1.11.4/scipy/interpolate/fitpack/fpcurf.f#L190-L215 + (cf _split function) + + and https://github.com/scipy/scipy/blob/v1.11.4/scipy/interpolate/fitpack/fpknot.f + """ + new_knot = _dierckx.fpknot(x, t, k, residuals) + + idx_t = np.searchsorted(t, new_knot) + t_new = np.r_[t[:idx_t], new_knot, t[idx_t:]] + return t_new + + +def _validate_inputs(x, y, w, k, s, xb, xe, parametric): + """Common input validations for generate_knots and make_splrep. + """ + x = np.asarray(x, dtype=float) + y = np.asarray(y, dtype=float) + + if w is None: + w = np.ones_like(x, dtype=float) + else: + w = np.asarray(w, dtype=float) + if w.ndim != 1: + raise ValueError(f"{w.ndim = } not implemented yet.") + if (w < 0).any(): + raise ValueError("Weights must be non-negative") + + if y.ndim == 0 or y.ndim > 2: + raise ValueError(f"{y.ndim = } not supported (must be 1 or 2.)") + + parametric = bool(parametric) + if parametric: + if y.ndim != 2: + raise ValueError(f"{y.ndim = } != 2 not supported with {parametric =}.") + else: + if y.ndim != 1: + raise ValueError(f"{y.ndim = } != 1 not supported with {parametric =}.") + # all _impl functions expect y.ndim = 2 + y = y[:, None] + + if w.shape[0] != x.shape[0]: + raise ValueError(f"Weights is incompatible: {w.shape =} != {x.shape}.") + + if x.shape[0] != y.shape[0]: + raise ValueError(f"Data is incompatible: {x.shape = } and {y.shape = }.") + if x.ndim != 1 or (x[1:] < x[:-1]).any(): + raise ValueError("Expect `x` to be an ordered 1D sequence.") + + k = operator.index(k) + + if s < 0: + raise ValueError(f"`s` must be non-negative. Got {s = }") + + if xb is None: + xb = min(x) + if xe is None: + xe = max(x) + + return x, y, w, k, s, xb, xe + + +def generate_knots(x, y, *, w=None, xb=None, xe=None, k=3, s=0, nest=None): + """Replicate FITPACK's constructing the knot vector. + + Parameters + ---------- + x, y : array_like + The data points defining the curve ``y = f(x)``. + w : array_like, optional + Weights. + xb : float, optional + The boundary of the approximation interval. If None (default), + is set to ``x[0]``. + xe : float, optional + The boundary of the approximation interval. If None (default), + is set to ``x[-1]``. + k : int, optional + The spline degree. Default is cubic, ``k = 3``. + s : float, optional + The smoothing factor. Default is ``s = 0``. + nest : int, optional + Stop when at least this many knots are placed. + + Yields + ------ + t : ndarray + Knot vectors with an increasing number of knots. + The generator is finite: it stops when the smoothing critetion is + satisfied, or when then number of knots exceeds the maximum value: + the user-provided `nest` or `x.size + k + 1` --- which is the knot vector + for the interpolating spline. + + Examples + -------- + Generate some noisy data and fit a sequence of LSQ splines: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import make_lsq_spline, generate_knots + >>> rng = np.random.default_rng(12345) + >>> x = np.linspace(-3, 3, 50) + >>> y = np.exp(-x**2) + 0.1 * rng.standard_normal(size=50) + + >>> knots = list(generate_knots(x, y, s=1e-10)) + >>> for t in knots[::3]: + ... spl = make_lsq_spline(x, y, t) + ... xs = xs = np.linspace(-3, 3, 201) + ... plt.plot(xs, spl(xs), '-', label=f'n = {len(t)}', lw=3, alpha=0.7) + >>> plt.plot(x, y, 'o', label='data') + >>> plt.plot(xs, np.exp(-xs**2), '--') + >>> plt.legend() + + Note that increasing the number of knots make the result follow the data + more and more closely. + + Also note that a step of the generator may add multiple knots: + + >>> [len(t) for t in knots] + [8, 9, 10, 12, 16, 24, 40, 48, 52, 54] + + Notes + ----- + The routine generates successive knots vectors of increasing length, starting + from ``2*(k+1)`` to ``len(x) + k + 1``, trying to make knots more dense + in the regions where the deviation of the LSQ spline from data is large. + + When the maximum number of knots, ``len(x) + k + 1`` is reached + (this happens when ``s`` is small and ``nest`` is large), the generator + stops, and the last output is the knots for the interpolation with the + not-a-knot boundary condition. + + Knots are located at data sites, unless ``k`` is even and the number of knots + is ``len(x) + k + 1``. In that case, the last output of the generator + has internal knots at Greville sites, ``(x[1:] + x[:-1]) / 2``. + + .. versionadded:: 1.15.0 + + """ + if s == 0: + if nest is not None or w is not None: + raise ValueError("s == 0 is interpolation only") + t = _not_a_knot(x, k) + yield t + return + + x, y, w, k, s, xb, xe = _validate_inputs( + x, y, w, k, s, xb, xe, parametric=np.ndim(y) == 2 + ) + + yield from _generate_knots_impl(x, y, w=w, xb=xb, xe=xe, k=k, s=s, nest=nest) + + +def _generate_knots_impl(x, y, *, w=None, xb=None, xe=None, k=3, s=0, nest=None): + + acc = s * TOL + m = x.size # the number of data points + + if nest is None: + # the max number of knots. This is set in _fitpack_impl.py line 274 + # and fitpack.pyf line 198 + nest = max(m + k + 1, 2*k + 3) + else: + if nest < 2*(k + 1): + raise ValueError(f"`nest` too small: {nest = } < 2*(k+1) = {2*(k+1)}.") + + nmin = 2*(k + 1) # the number of knots for an LSQ polynomial approximation + nmax = m + k + 1 # the number of knots for the spline interpolation + + # start from no internal knots + t = np.asarray([xb]*(k+1) + [xe]*(k+1), dtype=float) + n = t.shape[0] + fp = 0.0 + fpold = 0.0 + + # c main loop for the different sets of knots. m is a safe upper bound + # c for the number of trials. + for _ in range(m): + yield t + + # construct the LSQ spline with this set of knots + fpold = fp + residuals = _get_residuals(x, y, t, k, w=w) + fp = residuals.sum() + fpms = fp - s + + # c test whether the approximation sinf(x) is an acceptable solution. + # c if f(p=inf) < s accept the choice of knots. + if (abs(fpms) < acc) or (fpms < 0): + return + + # ### c increase the number of knots. ### + + # c determine the number of knots nplus we are going to add. + if n == nmin: + # the first iteration + nplus = 1 + else: + delta = fpold - fp + npl1 = int(nplus * fpms / delta) if delta > acc else nplus*2 + nplus = min(nplus*2, max(npl1, nplus//2, 1)) + + # actually add knots + for j in range(nplus): + t = add_knot(x, t, k, residuals) + + # check if we have enough knots already + + n = t.shape[0] + # c if n = nmax, sinf(x) is an interpolating spline. + # c if n=nmax we locate the knots as for interpolation. + if n >= nmax: + t = _not_a_knot(x, k) + yield t + return + + # c if n=nest we cannot increase the number of knots because of + # c the storage capacity limitation. + if n >= nest: + yield t + return + + # recompute if needed + if j < nplus - 1: + residuals = _get_residuals(x, y, t, k, w=w) + + # this should never be reached + return + + +# cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc +# c part 2: determination of the smoothing spline sp(x). c +# c *************************************************** c +# c we have determined the number of knots and their position. c +# c we now compute the b-spline coefficients of the smoothing spline c +# c sp(x). the observation matrix a is extended by the rows of matrix c +# c b expressing that the kth derivative discontinuities of sp(x) at c +# c the interior knots t(k+2),...t(n-k-1) must be zero. the corres- c +# c ponding weights of these additional rows are set to 1/p. c +# c iteratively we then have to determine the value of p such that c +# c f(p)=sum((w(i)*(y(i)-sp(x(i))))**2) be = s. we already know that c +# c the least-squares kth degree polynomial corresponds to p=0, and c +# c that the least-squares spline corresponds to p=infinity. the c +# c iteration process which is proposed here, makes use of rational c +# c interpolation. since f(p) is a convex and strictly decreasing c +# c function of p, it can be approximated by a rational function c +# c r(p) = (u*p+v)/(p+w). three values of p(p1,p2,p3) with correspond- c +# c ing values of f(p) (f1=f(p1)-s,f2=f(p2)-s,f3=f(p3)-s) are used c +# c to calculate the new value of p such that r(p)=s. convergence is c +# c guaranteed by taking f1>0 and f3<0. c +# cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc + + +def prodd(t, i, j, k): + res = 1.0 + for s in range(k+2): + if i + s != j: + res *= (t[j] - t[i+s]) + return res + + +def disc(t, k): + """Discontinuity matrix: jumps of k-th derivatives of b-splines at internal knots. + + See Eqs. (9)-(10) of Ref. [1], or, equivalently, Eq. (3.43) of Ref. [2]. + + This routine assumes internal knots are all simple (have multiplicity =1). + + Parameters + ---------- + t : ndarray, 1D, shape(n,) + Knots. + k : int + The spline degree + + Returns + ------- + disc : ndarray, shape(n-2*k-1, k+2) + The jumps of the k-th derivatives of b-splines at internal knots, + ``t[k+1], ...., t[n-k-1]``. + offset : ndarray, shape(2-2*k-1,) + Offsets + nc : int + + Notes + ----- + + The normalization here follows FITPACK: + (https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpdisc.f#L36) + + The k-th derivative jumps are multiplied by a factor:: + + (delta / nrint)**k / k! + + where ``delta`` is the length of the interval spanned by internal knots, and + ``nrint`` is one less the number of internal knots (i.e., the number of + subintervals between them). + + References + ---------- + .. [1] Paul Dierckx, Algorithms for smoothing data with periodic and parametric + splines, Computer Graphics and Image Processing, vol. 20, p. 171 (1982). + :doi:`10.1016/0146-664X(82)90043-0` + + .. [2] Tom Lyche and Knut Morken, Spline methods, + http://www.uio.no/studier/emner/matnat/ifi/INF-MAT5340/v05/undervisningsmateriale/ + + """ + n = t.shape[0] + + # the length of the base interval spanned by internal knots & the number + # of subintervas between these internal knots + delta = t[n - k - 1] - t[k] + nrint = n - 2*k - 1 + + matr = np.empty((nrint - 1, k + 2), dtype=float) + for jj in range(nrint - 1): + j = jj + k + 1 + for ii in range(k + 2): + i = jj + ii + matr[jj, ii] = (t[i + k + 1] - t[i]) / prodd(t, i, j, k) + # NB: equivalent to + # row = [(t[i + k + 1] - t[i]) / prodd(t, i, j, k) for i in range(j-k-1, j+1)] + # assert (matr[j-k-1, :] == row).all() + + # follow FITPACK + matr *= (delta/ nrint)**k + + # make it packed + offset = np.array([i for i in range(nrint-1)], dtype=np.int64) + nc = n - k - 1 + return matr, offset, nc + + +class F: + """ The r.h.s. of ``f(p) = s``. + + Given scalar `p`, we solve the system of equations in the LSQ sense: + + | A | @ | c | = | y | + | B / p | | 0 | | 0 | + + where `A` is the matrix of b-splines and `b` is the discontinuity matrix + (the jumps of the k-th derivatives of b-spline basis elements at knots). + + Since we do that repeatedly while minimizing over `p`, we QR-factorize + `A` only once and update the QR factorization only of the `B` rows of the + augmented matrix |A, B/p|. + + The system of equations is Eq. (15) Ref. [1]_, the strategy and implementation + follows that of FITPACK, see specific links below. + + References + ---------- + [1] P. Dierckx, Algorithms for Smoothing Data with Periodic and Parametric Splines, + COMPUTER GRAPHICS AND IMAGE PROCESSING vol. 20, pp 171-184 (1982.) + https://doi.org/10.1016/0146-664X(82)90043-0 + + """ + def __init__(self, x, y, t, k, s, w=None, *, R=None, Y=None): + self.x = x + self.y = y + self.t = t + self.k = k + w = np.ones_like(x, dtype=float) if w is None else w + if w.ndim != 1: + raise ValueError(f"{w.ndim = } != 1.") + self.w = w + self.s = s + + if y.ndim != 2: + raise ValueError(f"F: expected y.ndim == 2, got {y.ndim = } instead.") + + # ### precompute what we can ### + + # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L250 + # c evaluate the discontinuity jump of the kth derivative of the + # c b-splines at the knots t(l),l=k+2,...n-k-1 and store in b. + b, b_offset, b_nc = disc(t, k) + + # the QR factorization of the data matrix, if not provided + # NB: otherwise, must be consistent with x,y & s, but this is not checked + if R is None and Y is None: + R, Y, _ = _lsq_solve_qr(x, y, t, k, w) + + # prepare to combine R and the discontinuity matrix (AB); also r.h.s. (YY) + # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L269 + # c the rows of matrix b with weight 1/p are rotated into the + # c triangularised observation matrix a which is stored in g. + nc = t.shape[0] - k - 1 + nz = k + 1 + if R.shape[1] != nz: + raise ValueError(f"Internal error: {R.shape[1] =} != {k+1 =}.") + + # r.h.s. of the augmented system + z = np.zeros((b.shape[0], Y.shape[1]), dtype=float) + self.YY = np.r_[Y[:nc], z] + + # l.h.s. of the augmented system + AA = np.zeros((nc + b.shape[0], self.k+2), dtype=float) + AA[:nc, :nz] = R[:nc, :] + # AA[nc:, :] = b.a / p # done in __call__(self, p) + self.AA = AA + self.offset = np.r_[np.arange(nc, dtype=np.int64), b_offset] + + self.nc = nc + self.b = b + + def __call__(self, p): + # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L279 + # c the row of matrix b is rotated into triangle by givens transformation + + # copy the precomputed matrices over for in-place work + # R = PackedMatrix(self.AB.a.copy(), self.AB.offset.copy(), nc) + AB = self.AA.copy() + offset = self.offset.copy() + nc = self.nc + + AB[nc:, :] = self.b / p + QY = self.YY.copy() + + # heavy lifting happens here, in-place + _dierckx.qr_reduce(AB, offset, nc, QY, startrow=nc) + + # solve for the coefficients + c = _dierckx.fpback(AB, nc, QY) + + spl = BSpline(self.t, c, self.k) + residuals = _compute_residuals(self.w**2, spl(self.x), self.y) + fp = residuals.sum() + + self.spl = spl # store it + + return fp - self.s + + +def fprati(p1, f1, p2, f2, p3, f3): + """The root of r(p) = (u*p + v) / (p + w) given three points and values, + (p1, f2), (p2, f2) and (p3, f3). + + The FITPACK analog adjusts the bounds, and we do not + https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fprati.f + + NB: FITPACK uses p < 0 to encode p=infinity. We just use the infinity itself. + Since the bracket is ``p1 <= p2 <= p3``, ``p3`` can be infinite (in fact, + this is what the minimizer starts with, ``p3=inf``). + """ + h1 = f1 * (f2 - f3) + h2 = f2 * (f3 - f1) + h3 = f3 * (f1 - f2) + if p3 == np.inf: + return -(p2*h1 + p1*h2) / h3 + return -(p1*p2*h3 + p2*p3*h1 + p1*p3*h2) / (p1*h1 + p2*h2 + p3*h3) + + +class Bunch: + def __init__(self, **kwargs): + self.__dict__.update(**kwargs) + + +_iermesg = { +2: """error. a theoretically impossible result was found during +the iteration process for finding a smoothing spline with +fp = s. probably causes : s too small. +there is an approximation returned but the corresponding +weighted sum of squared residuals does not satisfy the +condition abs(fp-s)/s < tol. +""", +3: """error. the maximal number of iterations maxit (set to 20 +by the program) allowed for finding a smoothing spline +with fp=s has been reached. probably causes : s too small +there is an approximation returned but the corresponding +weighted sum of squared residuals does not satisfy the +condition abs(fp-s)/s < tol. +""" +} + + +def root_rati(f, p0, bracket, acc): + """Solve `f(p) = 0` using a rational function approximation. + + In a nutshell, since the function f(p) is known to be monotonically decreasing, we + - maintain the bracket (p1, f1), (p2, f2) and (p3, f3) + - at each iteration step, approximate f(p) by a rational function + r(p) = (u*p + v) / (p + w) + and make a step to p_new to the root of f(p): r(p_new) = 0. + The coefficients u, v and w are found from the bracket values p1..3 and f1...3 + + The algorithm and implementation follows + https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L229 + and + https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fppara.f#L290 + + Note that the latter is for parametric splines and the former is for 1D spline + functions. The minimization is indentical though [modulo a summation over the + dimensions in the computation of f(p)], so we reuse the minimizer for both + d=1 and d>1. + """ + # Magic values from + # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L27 + con1 = 0.1 + con9 = 0.9 + con4 = 0.04 + + # bracketing flags (follow FITPACK) + # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fppara.f#L365 + ich1, ich3 = 0, 0 + + (p1, f1), (p3, f3) = bracket + p = p0 + + for it in range(MAXIT): + p2, f2 = p, f(p) + + # c test whether the approximation sp(x) is an acceptable solution. + if abs(f2) < acc: + ier, converged = 0, True + break + + # c carry out one more step of the iteration process. + if ich3 == 0: + if f2 - f3 <= acc: + # c our initial choice of p is too large. + p3 = p2 + f3 = f2 + p = p*con4 + if p <= p1: + p = p1*con9 + p2*con1 + continue + else: + if f2 < 0: + ich3 = 1 + + if ich1 == 0: + if f1 - f2 <= acc: + # c our initial choice of p is too small + p1 = p2 + f1 = f2 + p = p/con4 + if p3 != np.inf and p <= p3: + p = p2*con1 + p3*con9 + continue + else: + if f2 > 0: + ich1 = 1 + + # c test whether the iteration process proceeds as theoretically expected. + # [f(p) should be monotonically decreasing] + if f1 <= f2 or f2 <= f3: + ier, converged = 2, False + break + + # actually make the iteration step + p = fprati(p1, f1, p2, f2, p3, f3) + + # c adjust the value of p1,f1,p3 and f3 such that f1 > 0 and f3 < 0. + if f2 < 0: + p3, f3 = p2, f2 + else: + p1, f1 = p2, f2 + + else: + # not converged in MAXIT iterations + ier, converged = 3, False + + if ier != 0: + warnings.warn(RuntimeWarning(_iermesg[ier]), stacklevel=2) + + return Bunch(converged=converged, root=p, iterations=it, ier=ier) + + +def _make_splrep_impl(x, y, *, w=None, xb=None, xe=None, k=3, s=0, t=None, nest=None): + """Shared infra for make_splrep and make_splprep. + """ + acc = s * TOL + m = x.size # the number of data points + + if nest is None: + # the max number of knots. This is set in _fitpack_impl.py line 274 + # and fitpack.pyf line 198 + nest = max(m + k + 1, 2*k + 3) + else: + if nest < 2*(k + 1): + raise ValueError(f"`nest` too small: {nest = } < 2*(k+1) = {2*(k+1)}.") + if t is not None: + raise ValueError("Either supply `t` or `nest`.") + + if t is None: + gen = _generate_knots_impl(x, y, w=w, k=k, s=s, xb=xb, xe=xe, nest=nest) + t = list(gen)[-1] + else: + fpcheck(x, t, k) + + if t.shape[0] == 2 * (k + 1): + # nothing to optimize + _, _, c = _lsq_solve_qr(x, y, t, k, w) + return BSpline(t, c, k) + + ### solve ### + + # c initial value for p. + # https://github.com/scipy/scipy/blob/maintenance/1.11.x/scipy/interpolate/fitpack/fpcurf.f#L253 + R, Y, _ = _lsq_solve_qr(x, y, t, k, w) + nc = t.shape[0] -k -1 + p = nc / R[:, 0].sum() + + # ### bespoke solver #### + # initial conditions + # f(p=inf) : LSQ spline with knots t (XXX: reuse R, c) + residuals = _get_residuals(x, y, t, k, w=w) + fp = residuals.sum() + fpinf = fp - s + + # f(p=0): LSQ spline without internal knots + residuals = _get_residuals(x, y, np.array([xb]*(k+1) + [xe]*(k+1)), k, w) + fp0 = residuals.sum() + fp0 = fp0 - s + + # solve + bracket = (0, fp0), (np.inf, fpinf) + f = F(x, y, t, k=k, s=s, w=w, R=R, Y=Y) + _ = root_rati(f, p, bracket, acc) + + # solve ALTERNATIVE: is roughly equivalent, gives slightly different results + # starting from scratch, that would have probably been tolerable; + # backwards compatibility dictates that we replicate the FITPACK minimizer though. + # f = F(x, y, t, k=k, s=s, w=w, R=R, Y=Y) + # from scipy.optimize import root_scalar + # res_ = root_scalar(f, x0=p, rtol=acc) + # assert res_.converged + + # f.spl is the spline corresponding to the found `p` value + return f.spl + + +def make_splrep(x, y, *, w=None, xb=None, xe=None, k=3, s=0, t=None, nest=None): + r"""Find the B-spline representation of a 1D function. + + Given the set of data points ``(x[i], y[i])``, determine a smooth spline + approximation of degree ``k`` on the interval ``xb <= x <= xe``. + + Parameters + ---------- + x, y : array_like, shape (m,) + The data points defining a curve ``y = f(x)``. + w : array_like, shape (m,), optional + Strictly positive 1D array of weights, of the same length as `x` and `y`. + The weights are used in computing the weighted least-squares spline + fit. If the errors in the y values have standard-deviation given by the + vector ``d``, then `w` should be ``1/d``. + Default is ``np.ones(m)``. + xb, xe : float, optional + The interval to fit. If None, these default to ``x[0]`` and ``x[-1]``, + respectively. + k : int, optional + The degree of the spline fit. It is recommended to use cubic splines, + ``k=3``, which is the default. Even values of `k` should be avoided, + especially with small `s` values. + s : float, optional + The smoothing condition. The amount of smoothness is determined by + satisfying the conditions:: + + sum((w * (g(x) - y))**2 ) <= s + + where ``g(x)`` is the smoothed fit to ``(x, y)``. The user can use `s` + to control the tradeoff between closeness to data and smoothness of fit. + Larger `s` means more smoothing while smaller values of `s` indicate less + smoothing. + Recommended values of `s` depend on the weights, `w`. If the weights + represent the inverse of the standard deviation of `y`, then a good `s` + value should be found in the range ``(m-sqrt(2*m), m+sqrt(2*m))`` where + ``m`` is the number of datapoints in `x`, `y`, and `w`. + Default is ``s = 0.0``, i.e. interpolation. + t : array_like, optional + The spline knots. If None (default), the knots will be constructed + automatically. + There must be at least ``2*k + 2`` and at most ``m + k + 1`` knots. + nest : int, optional + The target length of the knot vector. Should be between ``2*(k + 1)`` + (the minimum number of knots for a degree-``k`` spline), and + ``m + k + 1`` (the number of knots of the interpolating spline). + The actual number of knots returned by this routine may be slightly + larger than `nest`. + Default is None (no limit, add up to ``m + k + 1`` knots). + + Returns + ------- + spl : a `BSpline` instance + For `s=0`, ``spl(x) == y``. + For non-zero values of `s` the `spl` represents the smoothed approximation + to `(x, y)`, generally with fewer knots. + + See Also + -------- + generate_knots : is used under the hood for generating the knots + make_splprep : the analog of this routine for parametric curves + make_interp_spline : construct an interpolating spline (``s = 0``) + make_lsq_spline : construct the least-squares spline given the knot vector + splrep : a FITPACK analog of this routine + + References + ---------- + .. [1] P. Dierckx, "Algorithms for smoothing data with periodic and + parametric splines, Computer Graphics and Image Processing", + 20 (1982) 171-184. + .. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs on + Numerical Analysis, Oxford University Press, 1993. + + Notes + ----- + This routine constructs the smoothing spline function, :math:`g(x)`, to + minimize the sum of jumps, :math:`D_j`, of the ``k``-th derivative at the + internal knots (:math:`x_b < t_i < x_e`), where + + .. math:: + + D_i = g^{(k)}(t_i + 0) - g^{(k)}(t_i - 0) + + Specifically, the routine constructs the spline function :math:`g(x)` which + minimizes + + .. math:: + + \sum_i | D_i |^2 \to \mathrm{min} + + provided that + + .. math:: + + \sum_{j=1}^m (w_j \times (g(x_j) - y_j))^2 \leqslant s , + + where :math:`s > 0` is the input parameter. + + In other words, we balance maximizing the smoothness (measured as the jumps + of the derivative, the first criterion), and the deviation of :math:`g(x_j)` + from the data :math:`y_j` (the second criterion). + + Note that the summation in the second criterion is over all data points, + and in the first criterion it is over the internal spline knots (i.e. + those with ``xb < t[i] < xe``). The spline knots are in general a subset + of data, see `generate_knots` for details. + + Also note the difference of this routine to `make_lsq_spline`: the latter + routine does not consider smoothness and simply solves a least-squares + problem + + .. math:: + + \sum w_j \times (g(x_j) - y_j)^2 \to \mathrm{min} + + for a spline function :math:`g(x)` with a _fixed_ knot vector ``t``. + + .. versionadded:: 1.15.0 + """ + if s == 0: + if t is not None or w is not None or nest is not None: + raise ValueError("s==0 is for interpolation only") + return make_interp_spline(x, y, k=k) + + x, y, w, k, s, xb, xe = _validate_inputs(x, y, w, k, s, xb, xe, parametric=False) + + spl = _make_splrep_impl(x, y, w=w, xb=xb, xe=xe, k=k, s=s, t=t, nest=nest) + + # postprocess: squeeze out the last dimension: was added to simplify the internals. + spl.c = spl.c[:, 0] + return spl + + +def make_splprep(x, *, w=None, u=None, ub=None, ue=None, k=3, s=0, t=None, nest=None): + r""" + Find a smoothed B-spline representation of a parametric N-D curve. + + Given a list of N 1D arrays, `x`, which represent a curve in + N-dimensional space parametrized by `u`, find a smooth approximating + spline curve ``g(u)``. + + Parameters + ---------- + x : array_like, shape (m, ndim) + Sampled data points representing the curve in ``ndim`` dimensions. + The typical use is a list of 1D arrays, each of length ``m``. + w : array_like, shape(m,), optional + Strictly positive 1D array of weights. + The weights are used in computing the weighted least-squares spline + fit. If the errors in the `x` values have standard deviation given by + the vector d, then `w` should be 1/d. Default is ``np.ones(m)``. + u : array_like, optional + An array of parameter values for the curve in the parametric form. + If not given, these values are calculated automatically, according to:: + + v[0] = 0 + v[i] = v[i-1] + distance(x[i], x[i-1]) + u[i] = v[i] / v[-1] + + ub, ue : float, optional + The end-points of the parameters interval. Default to ``u[0]`` and ``u[-1]``. + k : int, optional + Degree of the spline. Cubic splines, ``k=3``, are recommended. + Even values of `k` should be avoided especially with a small ``s`` value. + Default is ``k=3`` + s : float, optional + A smoothing condition. The amount of smoothness is determined by + satisfying the conditions:: + + sum((w * (g(u) - x))**2) <= s, + + where ``g(u)`` is the smoothed approximation to ``x``. The user can + use `s` to control the trade-off between closeness and smoothness + of fit. Larger ``s`` means more smoothing while smaller values of ``s`` + indicate less smoothing. + Recommended values of ``s`` depend on the weights, ``w``. If the weights + represent the inverse of the standard deviation of ``x``, then a good + ``s`` value should be found in the range ``(m - sqrt(2*m), m + sqrt(2*m))``, + where ``m`` is the number of data points in ``x`` and ``w``. + t : array_like, optional + The spline knots. If None (default), the knots will be constructed + automatically. + There must be at least ``2*k + 2`` and at most ``m + k + 1`` knots. + nest : int, optional + The target length of the knot vector. Should be between ``2*(k + 1)`` + (the minimum number of knots for a degree-``k`` spline), and + ``m + k + 1`` (the number of knots of the interpolating spline). + The actual number of knots returned by this routine may be slightly + larger than `nest`. + Default is None (no limit, add up to ``m + k + 1`` knots). + + Returns + ------- + spl : a `BSpline` instance + For `s=0`, ``spl(u) == x``. + For non-zero values of ``s``, `spl` represents the smoothed approximation + to ``x``, generally with fewer knots. + u : ndarray + The values of the parameters + + See Also + -------- + generate_knots : is used under the hood for generating the knots + make_splrep : the analog of this routine 1D functions + make_interp_spline : construct an interpolating spline (``s = 0``) + make_lsq_spline : construct the least-squares spline given the knot vector + splprep : a FITPACK analog of this routine + + Notes + ----- + Given a set of :math:`m` data points in :math:`D` dimensions, :math:`\vec{x}_j`, + with :math:`j=1, ..., m` and :math:`\vec{x}_j = (x_{j; 1}, ..., x_{j; D})`, + this routine constructs the parametric spline curve :math:`g_a(u)` with + :math:`a=1, ..., D`, to minimize the sum of jumps, :math:`D_{i; a}`, of the + ``k``-th derivative at the internal knots (:math:`u_b < t_i < u_e`), where + + .. math:: + + D_{i; a} = g_a^{(k)}(t_i + 0) - g_a^{(k)}(t_i - 0) + + Specifically, the routine constructs the spline function :math:`g(u)` which + minimizes + + .. math:: + + \sum_i \sum_{a=1}^D | D_{i; a} |^2 \to \mathrm{min} + + provided that + + .. math:: + + \sum_{j=1}^m \sum_{a=1}^D (w_j \times (g_a(u_j) - x_{j; a}))^2 \leqslant s + + where :math:`u_j` is the value of the parameter corresponding to the data point + :math:`(x_{j; 1}, ..., x_{j; D})`, and :math:`s > 0` is the input parameter. + + In other words, we balance maximizing the smoothness (measured as the jumps + of the derivative, the first criterion), and the deviation of :math:`g(u_j)` + from the data :math:`x_j` (the second criterion). + + Note that the summation in the second criterion is over all data points, + and in the first criterion it is over the internal spline knots (i.e. + those with ``ub < t[i] < ue``). The spline knots are in general a subset + of data, see `generate_knots` for details. + + .. versionadded:: 1.15.0 + + References + ---------- + .. [1] P. Dierckx, "Algorithms for smoothing data with periodic and + parametric splines, Computer Graphics and Image Processing", + 20 (1982) 171-184. + .. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs on + Numerical Analysis, Oxford University Press, 1993. + """ + x = np.stack(x, axis=1) + + # construct the default parametrization of the curve + if u is None: + dp = (x[1:, :] - x[:-1, :])**2 + u = np.sqrt((dp).sum(axis=1)).cumsum() + u = np.r_[0, u / u[-1]] + + if s == 0: + if t is not None or w is not None or nest is not None: + raise ValueError("s==0 is for interpolation only") + return make_interp_spline(u, x.T, k=k, axis=1), u + + u, x, w, k, s, ub, ue = _validate_inputs(u, x, w, k, s, ub, ue, parametric=True) + + spl = _make_splrep_impl(u, x, w=w, xb=ub, xe=ue, k=k, s=s, t=t, nest=nest) + + # posprocess: `axis=1` so that spl(u).shape == np.shape(x) + # when `x` is a list of 1D arrays (cf original splPrep) + cc = spl.c.T + spl1 = BSpline(spl.t, cc, spl.k, axis=1) + + return spl1, u + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_interpolate.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_interpolate.py new file mode 100644 index 0000000000000000000000000000000000000000..7558bd7db25cbd60206f908aabbcb6dc9c567fa4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_interpolate.py @@ -0,0 +1,2248 @@ +__all__ = ['interp1d', 'interp2d', 'lagrange', 'PPoly', 'BPoly', 'NdPPoly'] + +from math import prod + +import numpy as np +from numpy import array, asarray, intp, poly1d, searchsorted + +import scipy.special as spec +from scipy._lib._util import copy_if_needed +from scipy.special import comb + +from . import _fitpack_py +from ._polyint import _Interpolator1D +from . import _ppoly +from ._interpnd import _ndim_coords_from_arrays +from ._bsplines import make_interp_spline, BSpline + + +def lagrange(x, w): + r""" + Return a Lagrange interpolating polynomial. + + Given two 1-D arrays `x` and `w,` returns the Lagrange interpolating + polynomial through the points ``(x, w)``. + + Warning: This implementation is numerically unstable. Do not expect to + be able to use more than about 20 points even if they are chosen optimally. + + Parameters + ---------- + x : array_like + `x` represents the x-coordinates of a set of datapoints. + w : array_like + `w` represents the y-coordinates of a set of datapoints, i.e., f(`x`). + + Returns + ------- + lagrange : `numpy.poly1d` instance + The Lagrange interpolating polynomial. + + Examples + -------- + Interpolate :math:`f(x) = x^3` by 3 points. + + >>> import numpy as np + >>> from scipy.interpolate import lagrange + >>> x = np.array([0, 1, 2]) + >>> y = x**3 + >>> poly = lagrange(x, y) + + Since there are only 3 points, Lagrange polynomial has degree 2. Explicitly, + it is given by + + .. math:: + + \begin{aligned} + L(x) &= 1\times \frac{x (x - 2)}{-1} + 8\times \frac{x (x-1)}{2} \\ + &= x (-2 + 3x) + \end{aligned} + + >>> from numpy.polynomial.polynomial import Polynomial + >>> Polynomial(poly.coef[::-1]).coef + array([ 0., -2., 3.]) + + >>> import matplotlib.pyplot as plt + >>> x_new = np.arange(0, 2.1, 0.1) + >>> plt.scatter(x, y, label='data') + >>> plt.plot(x_new, Polynomial(poly.coef[::-1])(x_new), label='Polynomial') + >>> plt.plot(x_new, 3*x_new**2 - 2*x_new + 0*x_new, + ... label=r"$3 x^2 - 2 x$", linestyle='-.') + >>> plt.legend() + >>> plt.show() + + """ + + M = len(x) + p = poly1d(0.0) + for j in range(M): + pt = poly1d(w[j]) + for k in range(M): + if k == j: + continue + fac = x[j]-x[k] + pt *= poly1d([1.0, -x[k]])/fac + p += pt + return p + + +# !! Need to find argument for keeping initialize. If it isn't +# !! found, get rid of it! + + +err_mesg = """\ +`interp2d` has been removed in SciPy 1.14.0. + +For legacy code, nearly bug-for-bug compatible replacements are +`RectBivariateSpline` on regular grids, and `bisplrep`/`bisplev` for +scattered 2D data. + +In new code, for regular grids use `RegularGridInterpolator` instead. +For scattered data, prefer `LinearNDInterpolator` or +`CloughTocher2DInterpolator`. + +For more details see +https://scipy.github.io/devdocs/tutorial/interpolate/interp_transition_guide.html +""" + +class interp2d: + """ + interp2d(x, y, z, kind='linear', copy=True, bounds_error=False, + fill_value=None) + + .. versionremoved:: 1.14.0 + + `interp2d` has been removed in SciPy 1.14.0. + + For legacy code, nearly bug-for-bug compatible replacements are + `RectBivariateSpline` on regular grids, and `bisplrep`/`bisplev` for + scattered 2D data. + + In new code, for regular grids use `RegularGridInterpolator` instead. + For scattered data, prefer `LinearNDInterpolator` or + `CloughTocher2DInterpolator`. + + For more details see :ref:`interp-transition-guide`. + """ + def __init__(self, x, y, z, kind='linear', copy=True, bounds_error=False, + fill_value=None): + raise NotImplementedError(err_mesg) + + +def _check_broadcast_up_to(arr_from, shape_to, name): + """Helper to check that arr_from broadcasts up to shape_to""" + shape_from = arr_from.shape + if len(shape_to) >= len(shape_from): + for t, f in zip(shape_to[::-1], shape_from[::-1]): + if f != 1 and f != t: + break + else: # all checks pass, do the upcasting that we need later + if arr_from.size != 1 and arr_from.shape != shape_to: + arr_from = np.ones(shape_to, arr_from.dtype) * arr_from + return arr_from.ravel() + # at least one check failed + raise ValueError(f'{name} argument must be able to broadcast up ' + f'to shape {shape_to} but had shape {shape_from}') + + +def _do_extrapolate(fill_value): + """Helper to check if fill_value == "extrapolate" without warnings""" + return (isinstance(fill_value, str) and + fill_value == 'extrapolate') + + +class interp1d(_Interpolator1D): + """ + Interpolate a 1-D function. + + .. legacy:: class + + For a guide to the intended replacements for `interp1d` see + :ref:`tutorial-interpolate_1Dsection`. + + `x` and `y` are arrays of values used to approximate some function f: + ``y = f(x)``. This class returns a function whose call method uses + interpolation to find the value of new points. + + Parameters + ---------- + x : (npoints, ) array_like + A 1-D array of real values. + y : (..., npoints, ...) array_like + A N-D array of real values. The length of `y` along the interpolation + axis must be equal to the length of `x`. Use the ``axis`` parameter + to select correct axis. Unlike other interpolators, the default + interpolation axis is the last axis of `y`. + kind : str or int, optional + Specifies the kind of interpolation as a string or as an integer + specifying the order of the spline interpolator to use. + The string has to be one of 'linear', 'nearest', 'nearest-up', 'zero', + 'slinear', 'quadratic', 'cubic', 'previous', or 'next'. 'zero', + 'slinear', 'quadratic' and 'cubic' refer to a spline interpolation of + zeroth, first, second or third order; 'previous' and 'next' simply + return the previous or next value of the point; 'nearest-up' and + 'nearest' differ when interpolating half-integers (e.g. 0.5, 1.5) + in that 'nearest-up' rounds up and 'nearest' rounds down. Default + is 'linear'. + axis : int, optional + Axis in the ``y`` array corresponding to the x-coordinate values. Unlike + other interpolators, defaults to ``axis=-1``. + copy : bool, optional + If ``True``, the class makes internal copies of x and y. If ``False``, + references to ``x`` and ``y`` are used if possible. The default is to copy. + bounds_error : bool, optional + If True, a ValueError is raised any time interpolation is attempted on + a value outside of the range of x (where extrapolation is + necessary). If False, out of bounds values are assigned `fill_value`. + By default, an error is raised unless ``fill_value="extrapolate"``. + fill_value : array-like or (array-like, array_like) or "extrapolate", optional + - if a ndarray (or float), this value will be used to fill in for + requested points outside of the data range. If not provided, then + the default is NaN. The array-like must broadcast properly to the + dimensions of the non-interpolation axes. + - If a two-element tuple, then the first element is used as a + fill value for ``x_new < x[0]`` and the second element is used for + ``x_new > x[-1]``. Anything that is not a 2-element tuple (e.g., + list or ndarray, regardless of shape) is taken to be a single + array-like argument meant to be used for both bounds as + ``below, above = fill_value, fill_value``. Using a two-element tuple + or ndarray requires ``bounds_error=False``. + + .. versionadded:: 0.17.0 + - If "extrapolate", then points outside the data range will be + extrapolated. + + .. versionadded:: 0.17.0 + assume_sorted : bool, optional + If False, values of `x` can be in any order and they are sorted first. + If True, `x` has to be an array of monotonically increasing values. + + Attributes + ---------- + fill_value + + Methods + ------- + __call__ + + See Also + -------- + splrep, splev + Spline interpolation/smoothing based on FITPACK. + UnivariateSpline : An object-oriented wrapper of the FITPACK routines. + interp2d : 2-D interpolation + + Notes + ----- + Calling `interp1d` with NaNs present in input values results in + undefined behaviour. + + Input values `x` and `y` must be convertible to `float` values like + `int` or `float`. + + If the values in `x` are not unique, the resulting behavior is + undefined and specific to the choice of `kind`, i.e., changing + `kind` will change the behavior for duplicates. + + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy import interpolate + >>> x = np.arange(0, 10) + >>> y = np.exp(-x/3.0) + >>> f = interpolate.interp1d(x, y) + + >>> xnew = np.arange(0, 9, 0.1) + >>> ynew = f(xnew) # use interpolation function returned by `interp1d` + >>> plt.plot(x, y, 'o', xnew, ynew, '-') + >>> plt.show() + """ + + def __init__(self, x, y, kind='linear', axis=-1, + copy=True, bounds_error=None, fill_value=np.nan, + assume_sorted=False): + """ Initialize a 1-D linear interpolation class.""" + _Interpolator1D.__init__(self, x, y, axis=axis) + + self.bounds_error = bounds_error # used by fill_value setter + + # `copy` keyword semantics changed in NumPy 2.0, once that is + # the minimum version this can use `copy=None`. + self.copy = copy + if not copy: + self.copy = copy_if_needed + + if kind in ['zero', 'slinear', 'quadratic', 'cubic']: + order = {'zero': 0, 'slinear': 1, + 'quadratic': 2, 'cubic': 3}[kind] + kind = 'spline' + elif isinstance(kind, int): + order = kind + kind = 'spline' + elif kind not in ('linear', 'nearest', 'nearest-up', 'previous', + 'next'): + raise NotImplementedError(f"{kind} is unsupported: Use fitpack " + "routines for other types.") + x = array(x, copy=self.copy) + y = array(y, copy=self.copy) + + if not assume_sorted: + ind = np.argsort(x, kind="mergesort") + x = x[ind] + y = np.take(y, ind, axis=axis) + + if x.ndim != 1: + raise ValueError("the x array must have exactly one dimension.") + if y.ndim == 0: + raise ValueError("the y array must have at least one dimension.") + + # Force-cast y to a floating-point type, if it's not yet one + if not issubclass(y.dtype.type, np.inexact): + y = y.astype(np.float64) + + # Backward compatibility + self.axis = axis % y.ndim + + # Interpolation goes internally along the first axis + self.y = y + self._y = self._reshape_yi(self.y) + self.x = x + del y, x # clean up namespace to prevent misuse; use attributes + self._kind = kind + + # Adjust to interpolation kind; store reference to *unbound* + # interpolation methods, in order to avoid circular references to self + # stored in the bound instance methods, and therefore delayed garbage + # collection. See: https://docs.python.org/reference/datamodel.html + if kind in ('linear', 'nearest', 'nearest-up', 'previous', 'next'): + # Make a "view" of the y array that is rotated to the interpolation + # axis. + minval = 1 + if kind == 'nearest': + # Do division before addition to prevent possible integer + # overflow + self._side = 'left' + self.x_bds = self.x / 2.0 + self.x_bds = self.x_bds[1:] + self.x_bds[:-1] + + self._call = self.__class__._call_nearest + elif kind == 'nearest-up': + # Do division before addition to prevent possible integer + # overflow + self._side = 'right' + self.x_bds = self.x / 2.0 + self.x_bds = self.x_bds[1:] + self.x_bds[:-1] + + self._call = self.__class__._call_nearest + elif kind == 'previous': + # Side for np.searchsorted and index for clipping + self._side = 'left' + self._ind = 0 + # Move x by one floating point value to the left + self._x_shift = np.nextafter(self.x, -np.inf) + self._call = self.__class__._call_previousnext + if _do_extrapolate(fill_value): + self._check_and_update_bounds_error_for_extrapolation() + # assume y is sorted by x ascending order here. + fill_value = (np.nan, np.take(self.y, -1, axis)) + elif kind == 'next': + self._side = 'right' + self._ind = 1 + # Move x by one floating point value to the right + self._x_shift = np.nextafter(self.x, np.inf) + self._call = self.__class__._call_previousnext + if _do_extrapolate(fill_value): + self._check_and_update_bounds_error_for_extrapolation() + # assume y is sorted by x ascending order here. + fill_value = (np.take(self.y, 0, axis), np.nan) + else: + # Check if we can delegate to numpy.interp (2x-10x faster). + np_dtypes = (np.dtype(np.float64), np.dtype(int)) + cond = self.x.dtype in np_dtypes and self.y.dtype in np_dtypes + cond = cond and self.y.ndim == 1 + cond = cond and not _do_extrapolate(fill_value) + + if cond: + self._call = self.__class__._call_linear_np + else: + self._call = self.__class__._call_linear + else: + minval = order + 1 + + rewrite_nan = False + xx, yy = self.x, self._y + if order > 1: + # Quadratic or cubic spline. If input contains even a single + # nan, then the output is all nans. We cannot just feed data + # with nans to make_interp_spline because it calls LAPACK. + # So, we make up a bogus x and y with no nans and use it + # to get the correct shape of the output, which we then fill + # with nans. + # For slinear or zero order spline, we just pass nans through. + mask = np.isnan(self.x) + if mask.any(): + sx = self.x[~mask] + if sx.size == 0: + raise ValueError("`x` array is all-nan") + xx = np.linspace(np.nanmin(self.x), + np.nanmax(self.x), + len(self.x)) + rewrite_nan = True + if np.isnan(self._y).any(): + yy = np.ones_like(self._y) + rewrite_nan = True + + self._spline = make_interp_spline(xx, yy, k=order, + check_finite=False) + if rewrite_nan: + self._call = self.__class__._call_nan_spline + else: + self._call = self.__class__._call_spline + + if len(self.x) < minval: + raise ValueError("x and y arrays must have at " + "least %d entries" % minval) + + self.fill_value = fill_value # calls the setter, can modify bounds_err + + @property + def fill_value(self): + """The fill value.""" + # backwards compat: mimic a public attribute + return self._fill_value_orig + + @fill_value.setter + def fill_value(self, fill_value): + # extrapolation only works for nearest neighbor and linear methods + if _do_extrapolate(fill_value): + self._check_and_update_bounds_error_for_extrapolation() + self._extrapolate = True + else: + broadcast_shape = (self.y.shape[:self.axis] + + self.y.shape[self.axis + 1:]) + if len(broadcast_shape) == 0: + broadcast_shape = (1,) + # it's either a pair (_below_range, _above_range) or a single value + # for both above and below range + if isinstance(fill_value, tuple) and len(fill_value) == 2: + below_above = [np.asarray(fill_value[0]), + np.asarray(fill_value[1])] + names = ('fill_value (below)', 'fill_value (above)') + for ii in range(2): + below_above[ii] = _check_broadcast_up_to( + below_above[ii], broadcast_shape, names[ii]) + else: + fill_value = np.asarray(fill_value) + below_above = [_check_broadcast_up_to( + fill_value, broadcast_shape, 'fill_value')] * 2 + self._fill_value_below, self._fill_value_above = below_above + self._extrapolate = False + if self.bounds_error is None: + self.bounds_error = True + # backwards compat: fill_value was a public attr; make it writeable + self._fill_value_orig = fill_value + + def _check_and_update_bounds_error_for_extrapolation(self): + if self.bounds_error: + raise ValueError("Cannot extrapolate and raise " + "at the same time.") + self.bounds_error = False + + def _call_linear_np(self, x_new): + # Note that out-of-bounds values are taken care of in self._evaluate + return np.interp(x_new, self.x, self.y) + + def _call_linear(self, x_new): + # 2. Find where in the original data, the values to interpolate + # would be inserted. + # Note: If x_new[n] == x[m], then m is returned by searchsorted. + x_new_indices = searchsorted(self.x, x_new) + + # 3. Clip x_new_indices so that they are within the range of + # self.x indices and at least 1. Removes mis-interpolation + # of x_new[n] = x[0] + x_new_indices = x_new_indices.clip(1, len(self.x)-1).astype(int) + + # 4. Calculate the slope of regions that each x_new value falls in. + lo = x_new_indices - 1 + hi = x_new_indices + + x_lo = self.x[lo] + x_hi = self.x[hi] + y_lo = self._y[lo] + y_hi = self._y[hi] + + # Note that the following two expressions rely on the specifics of the + # broadcasting semantics. + slope = (y_hi - y_lo) / (x_hi - x_lo)[:, None] + + # 5. Calculate the actual value for each entry in x_new. + y_new = slope*(x_new - x_lo)[:, None] + y_lo + + return y_new + + def _call_nearest(self, x_new): + """ Find nearest neighbor interpolated y_new = f(x_new).""" + + # 2. Find where in the averaged data the values to interpolate + # would be inserted. + # Note: use side='left' (right) to searchsorted() to define the + # halfway point to be nearest to the left (right) neighbor + x_new_indices = searchsorted(self.x_bds, x_new, side=self._side) + + # 3. Clip x_new_indices so that they are within the range of x indices. + x_new_indices = x_new_indices.clip(0, len(self.x)-1).astype(intp) + + # 4. Calculate the actual value for each entry in x_new. + y_new = self._y[x_new_indices] + + return y_new + + def _call_previousnext(self, x_new): + """Use previous/next neighbor of x_new, y_new = f(x_new).""" + + # 1. Get index of left/right value + x_new_indices = searchsorted(self._x_shift, x_new, side=self._side) + + # 2. Clip x_new_indices so that they are within the range of x indices. + x_new_indices = x_new_indices.clip(1-self._ind, + len(self.x)-self._ind).astype(intp) + + # 3. Calculate the actual value for each entry in x_new. + y_new = self._y[x_new_indices+self._ind-1] + + return y_new + + def _call_spline(self, x_new): + return self._spline(x_new) + + def _call_nan_spline(self, x_new): + out = self._spline(x_new) + out[...] = np.nan + return out + + def _evaluate(self, x_new): + # 1. Handle values in x_new that are outside of x. Throw error, + # or return a list of mask array indicating the outofbounds values. + # The behavior is set by the bounds_error variable. + x_new = asarray(x_new) + y_new = self._call(self, x_new) + if not self._extrapolate: + below_bounds, above_bounds = self._check_bounds(x_new) + if len(y_new) > 0: + # Note fill_value must be broadcast up to the proper size + # and flattened to work here + y_new[below_bounds] = self._fill_value_below + y_new[above_bounds] = self._fill_value_above + return y_new + + def _check_bounds(self, x_new): + """Check the inputs for being in the bounds of the interpolated data. + + Parameters + ---------- + x_new : array + + Returns + ------- + out_of_bounds : bool array + The mask on x_new of values that are out of the bounds. + """ + + # If self.bounds_error is True, we raise an error if any x_new values + # fall outside the range of x. Otherwise, we return an array indicating + # which values are outside the boundary region. + below_bounds = x_new < self.x[0] + above_bounds = x_new > self.x[-1] + + if self.bounds_error and below_bounds.any(): + below_bounds_value = x_new[np.argmax(below_bounds)] + raise ValueError(f"A value ({below_bounds_value}) in x_new is below " + f"the interpolation range's minimum value ({self.x[0]}).") + if self.bounds_error and above_bounds.any(): + above_bounds_value = x_new[np.argmax(above_bounds)] + raise ValueError(f"A value ({above_bounds_value}) in x_new is above " + f"the interpolation range's maximum value ({self.x[-1]}).") + + # !! Should we emit a warning if some values are out of bounds? + # !! matlab does not. + return below_bounds, above_bounds + + +class _PPolyBase: + """Base class for piecewise polynomials.""" + __slots__ = ('c', 'x', 'extrapolate', 'axis') + + def __init__(self, c, x, extrapolate=None, axis=0): + self.c = np.asarray(c) + self.x = np.ascontiguousarray(x, dtype=np.float64) + + if extrapolate is None: + extrapolate = True + elif extrapolate != 'periodic': + extrapolate = bool(extrapolate) + self.extrapolate = extrapolate + + if self.c.ndim < 2: + raise ValueError("Coefficients array must be at least " + "2-dimensional.") + + if not (0 <= axis < self.c.ndim - 1): + raise ValueError(f"axis={axis} must be between 0 and {self.c.ndim-1}") + + self.axis = axis + if axis != 0: + # move the interpolation axis to be the first one in self.c + # More specifically, the target shape for self.c is (k, m, ...), + # and axis !=0 means that we have c.shape (..., k, m, ...) + # ^ + # axis + # So we roll two of them. + self.c = np.moveaxis(self.c, axis+1, 0) + self.c = np.moveaxis(self.c, axis+1, 0) + + if self.x.ndim != 1: + raise ValueError("x must be 1-dimensional") + if self.x.size < 2: + raise ValueError("at least 2 breakpoints are needed") + if self.c.ndim < 2: + raise ValueError("c must have at least 2 dimensions") + if self.c.shape[0] == 0: + raise ValueError("polynomial must be at least of order 0") + if self.c.shape[1] != self.x.size-1: + raise ValueError("number of coefficients != len(x)-1") + dx = np.diff(self.x) + if not (np.all(dx >= 0) or np.all(dx <= 0)): + raise ValueError("`x` must be strictly increasing or decreasing.") + + dtype = self._get_dtype(self.c.dtype) + self.c = np.ascontiguousarray(self.c, dtype=dtype) + + def _get_dtype(self, dtype): + if np.issubdtype(dtype, np.complexfloating) \ + or np.issubdtype(self.c.dtype, np.complexfloating): + return np.complex128 + else: + return np.float64 + + @classmethod + def construct_fast(cls, c, x, extrapolate=None, axis=0): + """ + Construct the piecewise polynomial without making checks. + + Takes the same parameters as the constructor. Input arguments + ``c`` and ``x`` must be arrays of the correct shape and type. The + ``c`` array can only be of dtypes float and complex, and ``x`` + array must have dtype float. + """ + self = object.__new__(cls) + self.c = c + self.x = x + self.axis = axis + if extrapolate is None: + extrapolate = True + self.extrapolate = extrapolate + return self + + def _ensure_c_contiguous(self): + """ + c and x may be modified by the user. The Cython code expects + that they are C contiguous. + """ + if not self.x.flags.c_contiguous: + self.x = self.x.copy() + if not self.c.flags.c_contiguous: + self.c = self.c.copy() + + def extend(self, c, x): + """ + Add additional breakpoints and coefficients to the polynomial. + + Parameters + ---------- + c : ndarray, size (k, m, ...) + Additional coefficients for polynomials in intervals. Note that + the first additional interval will be formed using one of the + ``self.x`` end points. + x : ndarray, size (m,) + Additional breakpoints. Must be sorted in the same order as + ``self.x`` and either to the right or to the left of the current + breakpoints. + + Notes + ----- + This method is not thread safe and must not be executed concurrently + with other methods available in this class. Doing so may cause + unexpected errors or numerical output mismatches. + """ + + c = np.asarray(c) + x = np.asarray(x) + + if c.ndim < 2: + raise ValueError("invalid dimensions for c") + if x.ndim != 1: + raise ValueError("invalid dimensions for x") + if x.shape[0] != c.shape[1]: + raise ValueError(f"Shapes of x {x.shape} and c {c.shape} are incompatible") + if c.shape[2:] != self.c.shape[2:] or c.ndim != self.c.ndim: + raise ValueError( + f"Shapes of c {c.shape} and self.c {self.c.shape} are incompatible" + ) + + if c.size == 0: + return + + dx = np.diff(x) + if not (np.all(dx >= 0) or np.all(dx <= 0)): + raise ValueError("`x` is not sorted.") + + if self.x[-1] >= self.x[0]: + if not x[-1] >= x[0]: + raise ValueError("`x` is in the different order " + "than `self.x`.") + + if x[0] >= self.x[-1]: + action = 'append' + elif x[-1] <= self.x[0]: + action = 'prepend' + else: + raise ValueError("`x` is neither on the left or on the right " + "from `self.x`.") + else: + if not x[-1] <= x[0]: + raise ValueError("`x` is in the different order " + "than `self.x`.") + + if x[0] <= self.x[-1]: + action = 'append' + elif x[-1] >= self.x[0]: + action = 'prepend' + else: + raise ValueError("`x` is neither on the left or on the right " + "from `self.x`.") + + dtype = self._get_dtype(c.dtype) + + k2 = max(c.shape[0], self.c.shape[0]) + c2 = np.zeros((k2, self.c.shape[1] + c.shape[1]) + self.c.shape[2:], + dtype=dtype) + + if action == 'append': + c2[k2-self.c.shape[0]:, :self.c.shape[1]] = self.c + c2[k2-c.shape[0]:, self.c.shape[1]:] = c + self.x = np.r_[self.x, x] + elif action == 'prepend': + c2[k2-self.c.shape[0]:, :c.shape[1]] = c + c2[k2-c.shape[0]:, c.shape[1]:] = self.c + self.x = np.r_[x, self.x] + + self.c = c2 + + def __call__(self, x, nu=0, extrapolate=None): + """ + Evaluate the piecewise polynomial or its derivative. + + Parameters + ---------- + x : array_like + Points to evaluate the interpolant at. + nu : int, optional + Order of derivative to evaluate. Must be non-negative. + extrapolate : {bool, 'periodic', None}, optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. + If 'periodic', periodic extrapolation is used. + If None (default), use `self.extrapolate`. + + Returns + ------- + y : array_like + Interpolated values. Shape is determined by replacing + the interpolation axis in the original array with the shape of x. + + Notes + ----- + Derivatives are evaluated piecewise for each polynomial + segment, even if the polynomial is not differentiable at the + breakpoints. The polynomial intervals are considered half-open, + ``[a, b)``, except for the last interval which is closed + ``[a, b]``. + """ + if extrapolate is None: + extrapolate = self.extrapolate + x = np.asarray(x) + x_shape, x_ndim = x.shape, x.ndim + x = np.ascontiguousarray(x.ravel(), dtype=np.float64) + + # With periodic extrapolation we map x to the segment + # [self.x[0], self.x[-1]]. + if extrapolate == 'periodic': + x = self.x[0] + (x - self.x[0]) % (self.x[-1] - self.x[0]) + extrapolate = False + + out = np.empty((len(x), prod(self.c.shape[2:])), dtype=self.c.dtype) + self._ensure_c_contiguous() + self._evaluate(x, nu, extrapolate, out) + out = out.reshape(x_shape + self.c.shape[2:]) + if self.axis != 0: + # transpose to move the calculated values to the interpolation axis + l = list(range(out.ndim)) + l = l[x_ndim:x_ndim+self.axis] + l[:x_ndim] + l[x_ndim+self.axis:] + out = out.transpose(l) + return out + + +class PPoly(_PPolyBase): + """ + Piecewise polynomial in terms of coefficients and breakpoints + + The polynomial between ``x[i]`` and ``x[i + 1]`` is written in the + local power basis:: + + S = sum(c[m, i] * (xp - x[i])**(k-m) for m in range(k+1)) + + where ``k`` is the degree of the polynomial. + + Parameters + ---------- + c : ndarray, shape (k, m, ...) + Polynomial coefficients, order `k` and `m` intervals. + x : ndarray, shape (m+1,) + Polynomial breakpoints. Must be sorted in either increasing or + decreasing order. + extrapolate : bool or 'periodic', optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. If 'periodic', + periodic extrapolation is used. Default is True. + axis : int, optional + Interpolation axis. Default is zero. + + Attributes + ---------- + x : ndarray + Breakpoints. + c : ndarray + Coefficients of the polynomials. They are reshaped + to a 3-D array with the last dimension representing + the trailing dimensions of the original coefficient array. + axis : int + Interpolation axis. + + Methods + ------- + __call__ + derivative + antiderivative + integrate + solve + roots + extend + from_spline + from_bernstein_basis + construct_fast + + See also + -------- + BPoly : piecewise polynomials in the Bernstein basis + + Notes + ----- + High-order polynomials in the power basis can be numerically + unstable. Precision problems can start to appear for orders + larger than 20-30. + """ + + def _evaluate(self, x, nu, extrapolate, out): + _ppoly.evaluate(self.c.reshape(self.c.shape[0], self.c.shape[1], -1), + self.x, x, nu, bool(extrapolate), out) + + def derivative(self, nu=1): + """ + Construct a new piecewise polynomial representing the derivative. + + Parameters + ---------- + nu : int, optional + Order of derivative to evaluate. Default is 1, i.e., compute the + first derivative. If negative, the antiderivative is returned. + + Returns + ------- + pp : PPoly + Piecewise polynomial of order k2 = k - n representing the derivative + of this polynomial. + + Notes + ----- + Derivatives are evaluated piecewise for each polynomial + segment, even if the polynomial is not differentiable at the + breakpoints. The polynomial intervals are considered half-open, + ``[a, b)``, except for the last interval which is closed + ``[a, b]``. + """ + if nu < 0: + return self.antiderivative(-nu) + + # reduce order + if nu == 0: + c2 = self.c.copy() + else: + c2 = self.c[:-nu, :].copy() + + if c2.shape[0] == 0: + # derivative of order 0 is zero + c2 = np.zeros((1,) + c2.shape[1:], dtype=c2.dtype) + + # multiply by the correct rising factorials + factor = spec.poch(np.arange(c2.shape[0], 0, -1), nu) + c2 *= factor[(slice(None),) + (None,)*(c2.ndim-1)] + + # construct a compatible polynomial + return self.construct_fast(c2, self.x, self.extrapolate, self.axis) + + def antiderivative(self, nu=1): + """ + Construct a new piecewise polynomial representing the antiderivative. + + Antiderivative is also the indefinite integral of the function, + and derivative is its inverse operation. + + Parameters + ---------- + nu : int, optional + Order of antiderivative to evaluate. Default is 1, i.e., compute + the first integral. If negative, the derivative is returned. + + Returns + ------- + pp : PPoly + Piecewise polynomial of order k2 = k + n representing + the antiderivative of this polynomial. + + Notes + ----- + The antiderivative returned by this function is continuous and + continuously differentiable to order n-1, up to floating point + rounding error. + + If antiderivative is computed and ``self.extrapolate='periodic'``, + it will be set to False for the returned instance. This is done because + the antiderivative is no longer periodic and its correct evaluation + outside of the initially given x interval is difficult. + """ + if nu <= 0: + return self.derivative(-nu) + + c = np.zeros((self.c.shape[0] + nu, self.c.shape[1]) + self.c.shape[2:], + dtype=self.c.dtype) + c[:-nu] = self.c + + # divide by the correct rising factorials + factor = spec.poch(np.arange(self.c.shape[0], 0, -1), nu) + c[:-nu] /= factor[(slice(None),) + (None,)*(c.ndim-1)] + + # fix continuity of added degrees of freedom + self._ensure_c_contiguous() + _ppoly.fix_continuity(c.reshape(c.shape[0], c.shape[1], -1), + self.x, nu - 1) + + if self.extrapolate == 'periodic': + extrapolate = False + else: + extrapolate = self.extrapolate + + # construct a compatible polynomial + return self.construct_fast(c, self.x, extrapolate, self.axis) + + def integrate(self, a, b, extrapolate=None): + """ + Compute a definite integral over a piecewise polynomial. + + Parameters + ---------- + a : float + Lower integration bound + b : float + Upper integration bound + extrapolate : {bool, 'periodic', None}, optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. + If 'periodic', periodic extrapolation is used. + If None (default), use `self.extrapolate`. + + Returns + ------- + ig : array_like + Definite integral of the piecewise polynomial over [a, b] + """ + if extrapolate is None: + extrapolate = self.extrapolate + + # Swap integration bounds if needed + sign = 1 + if b < a: + a, b = b, a + sign = -1 + + range_int = np.empty((prod(self.c.shape[2:]),), dtype=self.c.dtype) + self._ensure_c_contiguous() + + # Compute the integral. + if extrapolate == 'periodic': + # Split the integral into the part over period (can be several + # of them) and the remaining part. + + xs, xe = self.x[0], self.x[-1] + period = xe - xs + interval = b - a + n_periods, left = divmod(interval, period) + + if n_periods > 0: + _ppoly.integrate( + self.c.reshape(self.c.shape[0], self.c.shape[1], -1), + self.x, xs, xe, False, out=range_int) + range_int *= n_periods + else: + range_int.fill(0) + + # Map a to [xs, xe], b is always a + left. + a = xs + (a - xs) % period + b = a + left + + # If b <= xe then we need to integrate over [a, b], otherwise + # over [a, xe] and from xs to what is remained. + remainder_int = np.empty_like(range_int) + if b <= xe: + _ppoly.integrate( + self.c.reshape(self.c.shape[0], self.c.shape[1], -1), + self.x, a, b, False, out=remainder_int) + range_int += remainder_int + else: + _ppoly.integrate( + self.c.reshape(self.c.shape[0], self.c.shape[1], -1), + self.x, a, xe, False, out=remainder_int) + range_int += remainder_int + + _ppoly.integrate( + self.c.reshape(self.c.shape[0], self.c.shape[1], -1), + self.x, xs, xs + left + a - xe, False, out=remainder_int) + range_int += remainder_int + else: + _ppoly.integrate( + self.c.reshape(self.c.shape[0], self.c.shape[1], -1), + self.x, a, b, bool(extrapolate), out=range_int) + + # Return + range_int *= sign + return range_int.reshape(self.c.shape[2:]) + + def solve(self, y=0., discontinuity=True, extrapolate=None): + """ + Find real solutions of the equation ``pp(x) == y``. + + Parameters + ---------- + y : float, optional + Right-hand side. Default is zero. + discontinuity : bool, optional + Whether to report sign changes across discontinuities at + breakpoints as roots. + extrapolate : {bool, 'periodic', None}, optional + If bool, determines whether to return roots from the polynomial + extrapolated based on first and last intervals, 'periodic' works + the same as False. If None (default), use `self.extrapolate`. + + Returns + ------- + roots : ndarray + Roots of the polynomial(s). + + If the PPoly object describes multiple polynomials, the + return value is an object array whose each element is an + ndarray containing the roots. + + Notes + ----- + This routine works only on real-valued polynomials. + + If the piecewise polynomial contains sections that are + identically zero, the root list will contain the start point + of the corresponding interval, followed by a ``nan`` value. + + If the polynomial is discontinuous across a breakpoint, and + there is a sign change across the breakpoint, this is reported + if the `discont` parameter is True. + + Examples + -------- + + Finding roots of ``[x**2 - 1, (x - 1)**2]`` defined on intervals + ``[-2, 1], [1, 2]``: + + >>> import numpy as np + >>> from scipy.interpolate import PPoly + >>> pp = PPoly(np.array([[1, -4, 3], [1, 0, 0]]).T, [-2, 1, 2]) + >>> pp.solve() + array([-1., 1.]) + """ + if extrapolate is None: + extrapolate = self.extrapolate + + self._ensure_c_contiguous() + + if np.issubdtype(self.c.dtype, np.complexfloating): + raise ValueError("Root finding is only for " + "real-valued polynomials") + + y = float(y) + r = _ppoly.real_roots(self.c.reshape(self.c.shape[0], self.c.shape[1], -1), + self.x, y, bool(discontinuity), + bool(extrapolate)) + if self.c.ndim == 2: + return r[0] + else: + r2 = np.empty(prod(self.c.shape[2:]), dtype=object) + # this for-loop is equivalent to ``r2[...] = r``, but that's broken + # in NumPy 1.6.0 + for ii, root in enumerate(r): + r2[ii] = root + + return r2.reshape(self.c.shape[2:]) + + def roots(self, discontinuity=True, extrapolate=None): + """ + Find real roots of the piecewise polynomial. + + Parameters + ---------- + discontinuity : bool, optional + Whether to report sign changes across discontinuities at + breakpoints as roots. + extrapolate : {bool, 'periodic', None}, optional + If bool, determines whether to return roots from the polynomial + extrapolated based on first and last intervals, 'periodic' works + the same as False. If None (default), use `self.extrapolate`. + + Returns + ------- + roots : ndarray + Roots of the polynomial(s). + + If the PPoly object describes multiple polynomials, the + return value is an object array whose each element is an + ndarray containing the roots. + + See Also + -------- + PPoly.solve + """ + return self.solve(0, discontinuity, extrapolate) + + @classmethod + def from_spline(cls, tck, extrapolate=None): + """ + Construct a piecewise polynomial from a spline + + Parameters + ---------- + tck + A spline, as returned by `splrep` or a BSpline object. + extrapolate : bool or 'periodic', optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. + If 'periodic', periodic extrapolation is used. Default is True. + + Examples + -------- + Construct an interpolating spline and convert it to a `PPoly` instance + + >>> import numpy as np + >>> from scipy.interpolate import splrep, PPoly + >>> x = np.linspace(0, 1, 11) + >>> y = np.sin(2*np.pi*x) + >>> tck = splrep(x, y, s=0) + >>> p = PPoly.from_spline(tck) + >>> isinstance(p, PPoly) + True + + Note that this function only supports 1D splines out of the box. + + If the ``tck`` object represents a parametric spline (e.g. constructed + by `splprep` or a `BSpline` with ``c.ndim > 1``), you will need to loop + over the dimensions manually. + + >>> from scipy.interpolate import splprep, splev + >>> t = np.linspace(0, 1, 11) + >>> x = np.sin(2*np.pi*t) + >>> y = np.cos(2*np.pi*t) + >>> (t, c, k), u = splprep([x, y], s=0) + + Note that ``c`` is a list of two arrays of length 11. + + >>> unew = np.arange(0, 1.01, 0.01) + >>> out = splev(unew, (t, c, k)) + + To convert this spline to the power basis, we convert each + component of the list of b-spline coefficients, ``c``, into the + corresponding cubic polynomial. + + >>> polys = [PPoly.from_spline((t, cj, k)) for cj in c] + >>> polys[0].c.shape + (4, 14) + + Note that the coefficients of the polynomials `polys` are in the + power basis and their dimensions reflect just that: here 4 is the order + (degree+1), and 14 is the number of intervals---which is nothing but + the length of the knot array of the original `tck` minus one. + + Optionally, we can stack the components into a single `PPoly` along + the third dimension: + + >>> cc = np.dstack([p.c for p in polys]) # has shape = (4, 14, 2) + >>> poly = PPoly(cc, polys[0].x) + >>> np.allclose(poly(unew).T, # note the transpose to match `splev` + ... out, atol=1e-15) + True + + """ + if isinstance(tck, BSpline): + t, c, k = tck.tck + if extrapolate is None: + extrapolate = tck.extrapolate + else: + t, c, k = tck + + cvals = np.empty((k + 1, len(t)-1), dtype=c.dtype) + for m in range(k, -1, -1): + y = _fitpack_py.splev(t[:-1], tck, der=m) + cvals[k - m, :] = y/spec.gamma(m+1) + + return cls.construct_fast(cvals, t, extrapolate) + + @classmethod + def from_bernstein_basis(cls, bp, extrapolate=None): + """ + Construct a piecewise polynomial in the power basis + from a polynomial in Bernstein basis. + + Parameters + ---------- + bp : BPoly + A Bernstein basis polynomial, as created by BPoly + extrapolate : bool or 'periodic', optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. + If 'periodic', periodic extrapolation is used. Default is True. + """ + if not isinstance(bp, BPoly): + raise TypeError(f".from_bernstein_basis only accepts BPoly instances. " + f"Got {type(bp)} instead.") + + dx = np.diff(bp.x) + k = bp.c.shape[0] - 1 # polynomial order + + rest = (None,)*(bp.c.ndim-2) + + c = np.zeros_like(bp.c) + for a in range(k+1): + factor = (-1)**a * comb(k, a) * bp.c[a] + for s in range(a, k+1): + val = comb(k-a, s-a) * (-1)**s + c[k-s] += factor * val / dx[(slice(None),)+rest]**s + + if extrapolate is None: + extrapolate = bp.extrapolate + + return cls.construct_fast(c, bp.x, extrapolate, bp.axis) + + +class BPoly(_PPolyBase): + """Piecewise polynomial in terms of coefficients and breakpoints. + + The polynomial between ``x[i]`` and ``x[i + 1]`` is written in the + Bernstein polynomial basis:: + + S = sum(c[a, i] * b(a, k; x) for a in range(k+1)), + + where ``k`` is the degree of the polynomial, and:: + + b(a, k; x) = binom(k, a) * t**a * (1 - t)**(k - a), + + with ``t = (x - x[i]) / (x[i+1] - x[i])`` and ``binom`` is the binomial + coefficient. + + Parameters + ---------- + c : ndarray, shape (k, m, ...) + Polynomial coefficients, order `k` and `m` intervals + x : ndarray, shape (m+1,) + Polynomial breakpoints. Must be sorted in either increasing or + decreasing order. + extrapolate : bool, optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. If 'periodic', + periodic extrapolation is used. Default is True. + axis : int, optional + Interpolation axis. Default is zero. + + Attributes + ---------- + x : ndarray + Breakpoints. + c : ndarray + Coefficients of the polynomials. They are reshaped + to a 3-D array with the last dimension representing + the trailing dimensions of the original coefficient array. + axis : int + Interpolation axis. + + Methods + ------- + __call__ + extend + derivative + antiderivative + integrate + construct_fast + from_power_basis + from_derivatives + + See also + -------- + PPoly : piecewise polynomials in the power basis + + Notes + ----- + Properties of Bernstein polynomials are well documented in the literature, + see for example [1]_ [2]_ [3]_. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Bernstein_polynomial + + .. [2] Kenneth I. Joy, Bernstein polynomials, + http://www.idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials.pdf + + .. [3] E. H. Doha, A. H. Bhrawy, and M. A. Saker, Boundary Value Problems, + vol 2011, article ID 829546, :doi:`10.1155/2011/829543`. + + Examples + -------- + >>> from scipy.interpolate import BPoly + >>> x = [0, 1] + >>> c = [[1], [2], [3]] + >>> bp = BPoly(c, x) + + This creates a 2nd order polynomial + + .. math:: + + B(x) = 1 \\times b_{0, 2}(x) + 2 \\times b_{1, 2}(x) + 3 + \\times b_{2, 2}(x) \\\\ + = 1 \\times (1-x)^2 + 2 \\times 2 x (1 - x) + 3 \\times x^2 + + """ # noqa: E501 + + def _evaluate(self, x, nu, extrapolate, out): + _ppoly.evaluate_bernstein( + self.c.reshape(self.c.shape[0], self.c.shape[1], -1), + self.x, x, nu, bool(extrapolate), out) + + def derivative(self, nu=1): + """ + Construct a new piecewise polynomial representing the derivative. + + Parameters + ---------- + nu : int, optional + Order of derivative to evaluate. Default is 1, i.e., compute the + first derivative. If negative, the antiderivative is returned. + + Returns + ------- + bp : BPoly + Piecewise polynomial of order k - nu representing the derivative of + this polynomial. + + """ + if nu < 0: + return self.antiderivative(-nu) + + if nu > 1: + bp = self + for k in range(nu): + bp = bp.derivative() + return bp + + # reduce order + if nu == 0: + c2 = self.c.copy() + else: + # For a polynomial + # B(x) = \sum_{a=0}^{k} c_a b_{a, k}(x), + # we use the fact that + # b'_{a, k} = k ( b_{a-1, k-1} - b_{a, k-1} ), + # which leads to + # B'(x) = \sum_{a=0}^{k-1} (c_{a+1} - c_a) b_{a, k-1} + # + # finally, for an interval [y, y + dy] with dy != 1, + # we need to correct for an extra power of dy + + rest = (None,)*(self.c.ndim-2) + + k = self.c.shape[0] - 1 + dx = np.diff(self.x)[(None, slice(None))+rest] + c2 = k * np.diff(self.c, axis=0) / dx + + if c2.shape[0] == 0: + # derivative of order 0 is zero + c2 = np.zeros((1,) + c2.shape[1:], dtype=c2.dtype) + + # construct a compatible polynomial + return self.construct_fast(c2, self.x, self.extrapolate, self.axis) + + def antiderivative(self, nu=1): + """ + Construct a new piecewise polynomial representing the antiderivative. + + Parameters + ---------- + nu : int, optional + Order of antiderivative to evaluate. Default is 1, i.e., compute + the first integral. If negative, the derivative is returned. + + Returns + ------- + bp : BPoly + Piecewise polynomial of order k + nu representing the + antiderivative of this polynomial. + + Notes + ----- + If antiderivative is computed and ``self.extrapolate='periodic'``, + it will be set to False for the returned instance. This is done because + the antiderivative is no longer periodic and its correct evaluation + outside of the initially given x interval is difficult. + """ + if nu <= 0: + return self.derivative(-nu) + + if nu > 1: + bp = self + for k in range(nu): + bp = bp.antiderivative() + return bp + + # Construct the indefinite integrals on individual intervals + c, x = self.c, self.x + k = c.shape[0] + c2 = np.zeros((k+1,) + c.shape[1:], dtype=c.dtype) + + c2[1:, ...] = np.cumsum(c, axis=0) / k + delta = x[1:] - x[:-1] + c2 *= delta[(None, slice(None)) + (None,)*(c.ndim-2)] + + # Now fix continuity: on the very first interval, take the integration + # constant to be zero; on an interval [x_j, x_{j+1}) with j>0, + # the integration constant is then equal to the jump of the `bp` at x_j. + # The latter is given by the coefficient of B_{n+1, n+1} + # *on the previous interval* (other B. polynomials are zero at the + # breakpoint). Finally, use the fact that BPs form a partition of unity. + c2[:,1:] += np.cumsum(c2[k, :], axis=0)[:-1] + + if self.extrapolate == 'periodic': + extrapolate = False + else: + extrapolate = self.extrapolate + + return self.construct_fast(c2, x, extrapolate, axis=self.axis) + + def integrate(self, a, b, extrapolate=None): + """ + Compute a definite integral over a piecewise polynomial. + + Parameters + ---------- + a : float + Lower integration bound + b : float + Upper integration bound + extrapolate : {bool, 'periodic', None}, optional + Whether to extrapolate to out-of-bounds points based on first + and last intervals, or to return NaNs. If 'periodic', periodic + extrapolation is used. If None (default), use `self.extrapolate`. + + Returns + ------- + array_like + Definite integral of the piecewise polynomial over [a, b] + + """ + # XXX: can probably use instead the fact that + # \int_0^{1} B_{j, n}(x) \dx = 1/(n+1) + ib = self.antiderivative() + if extrapolate is None: + extrapolate = self.extrapolate + + # ib.extrapolate shouldn't be 'periodic', it is converted to + # False for 'periodic. in antiderivative() call. + if extrapolate != 'periodic': + ib.extrapolate = extrapolate + + if extrapolate == 'periodic': + # Split the integral into the part over period (can be several + # of them) and the remaining part. + + # For simplicity and clarity convert to a <= b case. + if a <= b: + sign = 1 + else: + a, b = b, a + sign = -1 + + xs, xe = self.x[0], self.x[-1] + period = xe - xs + interval = b - a + n_periods, left = divmod(interval, period) + res = n_periods * (ib(xe) - ib(xs)) + + # Map a and b to [xs, xe]. + a = xs + (a - xs) % period + b = a + left + + # If b <= xe then we need to integrate over [a, b], otherwise + # over [a, xe] and from xs to what is remained. + if b <= xe: + res += ib(b) - ib(a) + else: + res += ib(xe) - ib(a) + ib(xs + left + a - xe) - ib(xs) + + return sign * res + else: + return ib(b) - ib(a) + + def extend(self, c, x): + k = max(self.c.shape[0], c.shape[0]) + self.c = self._raise_degree(self.c, k - self.c.shape[0]) + c = self._raise_degree(c, k - c.shape[0]) + return _PPolyBase.extend(self, c, x) + extend.__doc__ = _PPolyBase.extend.__doc__ + + @classmethod + def from_power_basis(cls, pp, extrapolate=None): + """ + Construct a piecewise polynomial in Bernstein basis + from a power basis polynomial. + + Parameters + ---------- + pp : PPoly + A piecewise polynomial in the power basis + extrapolate : bool or 'periodic', optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. + If 'periodic', periodic extrapolation is used. Default is True. + """ + if not isinstance(pp, PPoly): + raise TypeError(f".from_power_basis only accepts PPoly instances. " + f"Got {type(pp)} instead.") + + dx = np.diff(pp.x) + k = pp.c.shape[0] - 1 # polynomial order + + rest = (None,)*(pp.c.ndim-2) + + c = np.zeros_like(pp.c) + for a in range(k+1): + factor = pp.c[a] / comb(k, k-a) * dx[(slice(None),)+rest]**(k-a) + for j in range(k-a, k+1): + c[j] += factor * comb(j, k-a) + + if extrapolate is None: + extrapolate = pp.extrapolate + + return cls.construct_fast(c, pp.x, extrapolate, pp.axis) + + @classmethod + def from_derivatives(cls, xi, yi, orders=None, extrapolate=None): + """Construct a piecewise polynomial in the Bernstein basis, + compatible with the specified values and derivatives at breakpoints. + + Parameters + ---------- + xi : array_like + sorted 1-D array of x-coordinates + yi : array_like or list of array_likes + ``yi[i][j]`` is the ``j``\\ th derivative known at ``xi[i]`` + orders : None or int or array_like of ints. Default: None. + Specifies the degree of local polynomials. If not None, some + derivatives are ignored. + extrapolate : bool or 'periodic', optional + If bool, determines whether to extrapolate to out-of-bounds points + based on first and last intervals, or to return NaNs. + If 'periodic', periodic extrapolation is used. Default is True. + + Notes + ----- + If ``k`` derivatives are specified at a breakpoint ``x``, the + constructed polynomial is exactly ``k`` times continuously + differentiable at ``x``, unless the ``order`` is provided explicitly. + In the latter case, the smoothness of the polynomial at + the breakpoint is controlled by the ``order``. + + Deduces the number of derivatives to match at each end + from ``order`` and the number of derivatives available. If + possible it uses the same number of derivatives from + each end; if the number is odd it tries to take the + extra one from y2. In any case if not enough derivatives + are available at one end or another it draws enough to + make up the total from the other end. + + If the order is too high and not enough derivatives are available, + an exception is raised. + + Examples + -------- + + >>> from scipy.interpolate import BPoly + >>> BPoly.from_derivatives([0, 1], [[1, 2], [3, 4]]) + + Creates a polynomial `f(x)` of degree 3, defined on ``[0, 1]`` + such that `f(0) = 1, df/dx(0) = 2, f(1) = 3, df/dx(1) = 4` + + >>> BPoly.from_derivatives([0, 1, 2], [[0, 1], [0], [2]]) + + Creates a piecewise polynomial `f(x)`, such that + `f(0) = f(1) = 0`, `f(2) = 2`, and `df/dx(0) = 1`. + Based on the number of derivatives provided, the order of the + local polynomials is 2 on ``[0, 1]`` and 1 on ``[1, 2]``. + Notice that no restriction is imposed on the derivatives at + ``x = 1`` and ``x = 2``. + + Indeed, the explicit form of the polynomial is:: + + f(x) = | x * (1 - x), 0 <= x < 1 + | 2 * (x - 1), 1 <= x <= 2 + + So that f'(1-0) = -1 and f'(1+0) = 2 + + """ + xi = np.asarray(xi) + if len(xi) != len(yi): + raise ValueError("xi and yi need to have the same length") + if np.any(xi[1:] - xi[:1] <= 0): + raise ValueError("x coordinates are not in increasing order") + + # number of intervals + m = len(xi) - 1 + + # global poly order is k-1, local orders are <=k and can vary + try: + k = max(len(yi[i]) + len(yi[i+1]) for i in range(m)) + except TypeError as e: + raise ValueError( + "Using a 1-D array for y? Please .reshape(-1, 1)." + ) from e + + if orders is None: + orders = [None] * m + else: + if isinstance(orders, (int, np.integer)): + orders = [orders] * m + k = max(k, max(orders)) + + if any(o <= 0 for o in orders): + raise ValueError("Orders must be positive.") + + c = [] + for i in range(m): + y1, y2 = yi[i], yi[i+1] + if orders[i] is None: + n1, n2 = len(y1), len(y2) + else: + n = orders[i]+1 + n1 = min(n//2, len(y1)) + n2 = min(n - n1, len(y2)) + n1 = min(n - n2, len(y2)) + if n1+n2 != n: + mesg = ("Point %g has %d derivatives, point %g" + " has %d derivatives, but order %d requested" % ( + xi[i], len(y1), xi[i+1], len(y2), orders[i])) + raise ValueError(mesg) + + if not (n1 <= len(y1) and n2 <= len(y2)): + raise ValueError("`order` input incompatible with" + " length y1 or y2.") + + b = BPoly._construct_from_derivatives(xi[i], xi[i+1], + y1[:n1], y2[:n2]) + if len(b) < k: + b = BPoly._raise_degree(b, k - len(b)) + c.append(b) + + c = np.asarray(c) + return cls(c.swapaxes(0, 1), xi, extrapolate) + + @staticmethod + def _construct_from_derivatives(xa, xb, ya, yb): + r"""Compute the coefficients of a polynomial in the Bernstein basis + given the values and derivatives at the edges. + + Return the coefficients of a polynomial in the Bernstein basis + defined on ``[xa, xb]`` and having the values and derivatives at the + endpoints `xa` and `xb` as specified by `ya` and `yb`. + The polynomial constructed is of the minimal possible degree, i.e., + if the lengths of `ya` and `yb` are `na` and `nb`, the degree + of the polynomial is ``na + nb - 1``. + + Parameters + ---------- + xa : float + Left-hand end point of the interval + xb : float + Right-hand end point of the interval + ya : array_like + Derivatives at `xa`. ``ya[0]`` is the value of the function, and + ``ya[i]`` for ``i > 0`` is the value of the ``i``\ th derivative. + yb : array_like + Derivatives at `xb`. + + Returns + ------- + array + coefficient array of a polynomial having specified derivatives + + Notes + ----- + This uses several facts from life of Bernstein basis functions. + First of all, + + .. math:: b'_{a, n} = n (b_{a-1, n-1} - b_{a, n-1}) + + If B(x) is a linear combination of the form + + .. math:: B(x) = \sum_{a=0}^{n} c_a b_{a, n}, + + then :math: B'(x) = n \sum_{a=0}^{n-1} (c_{a+1} - c_{a}) b_{a, n-1}. + Iterating the latter one, one finds for the q-th derivative + + .. math:: B^{q}(x) = n!/(n-q)! \sum_{a=0}^{n-q} Q_a b_{a, n-q}, + + with + + .. math:: Q_a = \sum_{j=0}^{q} (-)^{j+q} comb(q, j) c_{j+a} + + This way, only `a=0` contributes to :math: `B^{q}(x = xa)`, and + `c_q` are found one by one by iterating `q = 0, ..., na`. + + At ``x = xb`` it's the same with ``a = n - q``. + + """ + ya, yb = np.asarray(ya), np.asarray(yb) + if ya.shape[1:] != yb.shape[1:]: + raise ValueError( + f"Shapes of ya {ya.shape} and yb {yb.shape} are incompatible" + ) + + dta, dtb = ya.dtype, yb.dtype + if (np.issubdtype(dta, np.complexfloating) or + np.issubdtype(dtb, np.complexfloating)): + dt = np.complex128 + else: + dt = np.float64 + + na, nb = len(ya), len(yb) + n = na + nb + + c = np.empty((na+nb,) + ya.shape[1:], dtype=dt) + + # compute coefficients of a polynomial degree na+nb-1 + # walk left-to-right + for q in range(0, na): + c[q] = ya[q] / spec.poch(n - q, q) * (xb - xa)**q + for j in range(0, q): + c[q] -= (-1)**(j+q) * comb(q, j) * c[j] + + # now walk right-to-left + for q in range(0, nb): + c[-q-1] = yb[q] / spec.poch(n - q, q) * (-1)**q * (xb - xa)**q + for j in range(0, q): + c[-q-1] -= (-1)**(j+1) * comb(q, j+1) * c[-q+j] + + return c + + @staticmethod + def _raise_degree(c, d): + r"""Raise a degree of a polynomial in the Bernstein basis. + + Given the coefficients of a polynomial degree `k`, return (the + coefficients of) the equivalent polynomial of degree `k+d`. + + Parameters + ---------- + c : array_like + coefficient array, 1-D + d : integer + + Returns + ------- + array + coefficient array, 1-D array of length `c.shape[0] + d` + + Notes + ----- + This uses the fact that a Bernstein polynomial `b_{a, k}` can be + identically represented as a linear combination of polynomials of + a higher degree `k+d`: + + .. math:: b_{a, k} = comb(k, a) \sum_{j=0}^{d} b_{a+j, k+d} \ + comb(d, j) / comb(k+d, a+j) + + """ + if d == 0: + return c + + k = c.shape[0] - 1 + out = np.zeros((c.shape[0] + d,) + c.shape[1:], dtype=c.dtype) + + for a in range(c.shape[0]): + f = c[a] * comb(k, a) + for j in range(d+1): + out[a+j] += f * comb(d, j) / comb(k+d, a+j) + return out + + +class NdPPoly: + """ + Piecewise tensor product polynomial + + The value at point ``xp = (x', y', z', ...)`` is evaluated by first + computing the interval indices `i` such that:: + + x[0][i[0]] <= x' < x[0][i[0]+1] + x[1][i[1]] <= y' < x[1][i[1]+1] + ... + + and then computing:: + + S = sum(c[k0-m0-1,...,kn-mn-1,i[0],...,i[n]] + * (xp[0] - x[0][i[0]])**m0 + * ... + * (xp[n] - x[n][i[n]])**mn + for m0 in range(k[0]+1) + ... + for mn in range(k[n]+1)) + + where ``k[j]`` is the degree of the polynomial in dimension j. This + representation is the piecewise multivariate power basis. + + Parameters + ---------- + c : ndarray, shape (k0, ..., kn, m0, ..., mn, ...) + Polynomial coefficients, with polynomial order `kj` and + `mj+1` intervals for each dimension `j`. + x : ndim-tuple of ndarrays, shapes (mj+1,) + Polynomial breakpoints for each dimension. These must be + sorted in increasing order. + extrapolate : bool, optional + Whether to extrapolate to out-of-bounds points based on first + and last intervals, or to return NaNs. Default: True. + + Attributes + ---------- + x : tuple of ndarrays + Breakpoints. + c : ndarray + Coefficients of the polynomials. + + Methods + ------- + __call__ + derivative + antiderivative + integrate + integrate_1d + construct_fast + + See also + -------- + PPoly : piecewise polynomials in 1D + + Notes + ----- + High-order polynomials in the power basis can be numerically + unstable. + + """ + + def __init__(self, c, x, extrapolate=None): + self.x = tuple(np.ascontiguousarray(v, dtype=np.float64) for v in x) + self.c = np.asarray(c) + if extrapolate is None: + extrapolate = True + self.extrapolate = bool(extrapolate) + + ndim = len(self.x) + if any(v.ndim != 1 for v in self.x): + raise ValueError("x arrays must all be 1-dimensional") + if any(v.size < 2 for v in self.x): + raise ValueError("x arrays must all contain at least 2 points") + if c.ndim < 2*ndim: + raise ValueError("c must have at least 2*len(x) dimensions") + if any(np.any(v[1:] - v[:-1] < 0) for v in self.x): + raise ValueError("x-coordinates are not in increasing order") + if any(a != b.size - 1 for a, b in zip(c.shape[ndim:2*ndim], self.x)): + raise ValueError("x and c do not agree on the number of intervals") + + dtype = self._get_dtype(self.c.dtype) + self.c = np.ascontiguousarray(self.c, dtype=dtype) + + @classmethod + def construct_fast(cls, c, x, extrapolate=None): + """ + Construct the piecewise polynomial without making checks. + + Takes the same parameters as the constructor. Input arguments + ``c`` and ``x`` must be arrays of the correct shape and type. The + ``c`` array can only be of dtypes float and complex, and ``x`` + array must have dtype float. + + """ + self = object.__new__(cls) + self.c = c + self.x = x + if extrapolate is None: + extrapolate = True + self.extrapolate = extrapolate + return self + + def _get_dtype(self, dtype): + if np.issubdtype(dtype, np.complexfloating) \ + or np.issubdtype(self.c.dtype, np.complexfloating): + return np.complex128 + else: + return np.float64 + + def _ensure_c_contiguous(self): + if not self.c.flags.c_contiguous: + self.c = self.c.copy() + if not isinstance(self.x, tuple): + self.x = tuple(self.x) + + def __call__(self, x, nu=None, extrapolate=None): + """ + Evaluate the piecewise polynomial or its derivative + + Parameters + ---------- + x : array-like + Points to evaluate the interpolant at. + nu : tuple, optional + Orders of derivatives to evaluate. Each must be non-negative. + extrapolate : bool, optional + Whether to extrapolate to out-of-bounds points based on first + and last intervals, or to return NaNs. + + Returns + ------- + y : array-like + Interpolated values. Shape is determined by replacing + the interpolation axis in the original array with the shape of x. + + Notes + ----- + Derivatives are evaluated piecewise for each polynomial + segment, even if the polynomial is not differentiable at the + breakpoints. The polynomial intervals are considered half-open, + ``[a, b)``, except for the last interval which is closed + ``[a, b]``. + + """ + if extrapolate is None: + extrapolate = self.extrapolate + else: + extrapolate = bool(extrapolate) + + ndim = len(self.x) + + x = _ndim_coords_from_arrays(x) + x_shape = x.shape + x = np.ascontiguousarray(x.reshape(-1, x.shape[-1]), dtype=np.float64) + + if nu is None: + nu = np.zeros((ndim,), dtype=np.intc) + else: + nu = np.asarray(nu, dtype=np.intc) + if nu.ndim != 1 or nu.shape[0] != ndim: + raise ValueError("invalid number of derivative orders nu") + + dim1 = prod(self.c.shape[:ndim]) + dim2 = prod(self.c.shape[ndim:2*ndim]) + dim3 = prod(self.c.shape[2*ndim:]) + ks = np.array(self.c.shape[:ndim], dtype=np.intc) + + out = np.empty((x.shape[0], dim3), dtype=self.c.dtype) + self._ensure_c_contiguous() + + _ppoly.evaluate_nd(self.c.reshape(dim1, dim2, dim3), + self.x, + ks, + x, + nu, + bool(extrapolate), + out) + + return out.reshape(x_shape[:-1] + self.c.shape[2*ndim:]) + + def _derivative_inplace(self, nu, axis): + """ + Compute 1-D derivative along a selected dimension in-place + May result to non-contiguous c array. + """ + if nu < 0: + return self._antiderivative_inplace(-nu, axis) + + ndim = len(self.x) + axis = axis % ndim + + # reduce order + if nu == 0: + # noop + return + else: + sl = [slice(None)]*ndim + sl[axis] = slice(None, -nu, None) + c2 = self.c[tuple(sl)] + + if c2.shape[axis] == 0: + # derivative of order 0 is zero + shp = list(c2.shape) + shp[axis] = 1 + c2 = np.zeros(shp, dtype=c2.dtype) + + # multiply by the correct rising factorials + factor = spec.poch(np.arange(c2.shape[axis], 0, -1), nu) + sl = [None]*c2.ndim + sl[axis] = slice(None) + c2 *= factor[tuple(sl)] + + self.c = c2 + + def _antiderivative_inplace(self, nu, axis): + """ + Compute 1-D antiderivative along a selected dimension + May result to non-contiguous c array. + """ + if nu <= 0: + return self._derivative_inplace(-nu, axis) + + ndim = len(self.x) + axis = axis % ndim + + perm = list(range(ndim)) + perm[0], perm[axis] = perm[axis], perm[0] + perm = perm + list(range(ndim, self.c.ndim)) + + c = self.c.transpose(perm) + + c2 = np.zeros((c.shape[0] + nu,) + c.shape[1:], + dtype=c.dtype) + c2[:-nu] = c + + # divide by the correct rising factorials + factor = spec.poch(np.arange(c.shape[0], 0, -1), nu) + c2[:-nu] /= factor[(slice(None),) + (None,)*(c.ndim-1)] + + # fix continuity of added degrees of freedom + perm2 = list(range(c2.ndim)) + perm2[1], perm2[ndim+axis] = perm2[ndim+axis], perm2[1] + + c2 = c2.transpose(perm2) + c2 = c2.copy() + _ppoly.fix_continuity(c2.reshape(c2.shape[0], c2.shape[1], -1), + self.x[axis], nu-1) + + c2 = c2.transpose(perm2) + c2 = c2.transpose(perm) + + # Done + self.c = c2 + + def derivative(self, nu): + """ + Construct a new piecewise polynomial representing the derivative. + + Parameters + ---------- + nu : ndim-tuple of int + Order of derivatives to evaluate for each dimension. + If negative, the antiderivative is returned. + + Returns + ------- + pp : NdPPoly + Piecewise polynomial of orders (k[0] - nu[0], ..., k[n] - nu[n]) + representing the derivative of this polynomial. + + Notes + ----- + Derivatives are evaluated piecewise for each polynomial + segment, even if the polynomial is not differentiable at the + breakpoints. The polynomial intervals in each dimension are + considered half-open, ``[a, b)``, except for the last interval + which is closed ``[a, b]``. + + """ + p = self.construct_fast(self.c.copy(), self.x, self.extrapolate) + + for axis, n in enumerate(nu): + p._derivative_inplace(n, axis) + + p._ensure_c_contiguous() + return p + + def antiderivative(self, nu): + """ + Construct a new piecewise polynomial representing the antiderivative. + + Antiderivative is also the indefinite integral of the function, + and derivative is its inverse operation. + + Parameters + ---------- + nu : ndim-tuple of int + Order of derivatives to evaluate for each dimension. + If negative, the derivative is returned. + + Returns + ------- + pp : PPoly + Piecewise polynomial of order k2 = k + n representing + the antiderivative of this polynomial. + + Notes + ----- + The antiderivative returned by this function is continuous and + continuously differentiable to order n-1, up to floating point + rounding error. + + """ + p = self.construct_fast(self.c.copy(), self.x, self.extrapolate) + + for axis, n in enumerate(nu): + p._antiderivative_inplace(n, axis) + + p._ensure_c_contiguous() + return p + + def integrate_1d(self, a, b, axis, extrapolate=None): + r""" + Compute NdPPoly representation for one dimensional definite integral + + The result is a piecewise polynomial representing the integral: + + .. math:: + + p(y, z, ...) = \int_a^b dx\, p(x, y, z, ...) + + where the dimension integrated over is specified with the + `axis` parameter. + + Parameters + ---------- + a, b : float + Lower and upper bound for integration. + axis : int + Dimension over which to compute the 1-D integrals + extrapolate : bool, optional + Whether to extrapolate to out-of-bounds points based on first + and last intervals, or to return NaNs. + + Returns + ------- + ig : NdPPoly or array-like + Definite integral of the piecewise polynomial over [a, b]. + If the polynomial was 1D, an array is returned, + otherwise, an NdPPoly object. + + """ + if extrapolate is None: + extrapolate = self.extrapolate + else: + extrapolate = bool(extrapolate) + + ndim = len(self.x) + axis = int(axis) % ndim + + # reuse 1-D integration routines + c = self.c + swap = list(range(c.ndim)) + swap.insert(0, swap[axis]) + del swap[axis + 1] + swap.insert(1, swap[ndim + axis]) + del swap[ndim + axis + 1] + + c = c.transpose(swap) + p = PPoly.construct_fast(c.reshape(c.shape[0], c.shape[1], -1), + self.x[axis], + extrapolate=extrapolate) + out = p.integrate(a, b, extrapolate=extrapolate) + + # Construct result + if ndim == 1: + return out.reshape(c.shape[2:]) + else: + c = out.reshape(c.shape[2:]) + x = self.x[:axis] + self.x[axis+1:] + return self.construct_fast(c, x, extrapolate=extrapolate) + + def integrate(self, ranges, extrapolate=None): + """ + Compute a definite integral over a piecewise polynomial. + + Parameters + ---------- + ranges : ndim-tuple of 2-tuples float + Sequence of lower and upper bounds for each dimension, + ``[(a[0], b[0]), ..., (a[ndim-1], b[ndim-1])]`` + extrapolate : bool, optional + Whether to extrapolate to out-of-bounds points based on first + and last intervals, or to return NaNs. + + Returns + ------- + ig : array_like + Definite integral of the piecewise polynomial over + [a[0], b[0]] x ... x [a[ndim-1], b[ndim-1]] + + """ + + ndim = len(self.x) + + if extrapolate is None: + extrapolate = self.extrapolate + else: + extrapolate = bool(extrapolate) + + if not hasattr(ranges, '__len__') or len(ranges) != ndim: + raise ValueError("Range not a sequence of correct length") + + self._ensure_c_contiguous() + + # Reuse 1D integration routine + c = self.c + for n, (a, b) in enumerate(ranges): + swap = list(range(c.ndim)) + swap.insert(1, swap[ndim - n]) + del swap[ndim - n + 1] + + c = c.transpose(swap) + + p = PPoly.construct_fast(c, self.x[n], extrapolate=extrapolate) + out = p.integrate(a, b, extrapolate=extrapolate) + c = out.reshape(c.shape[2:]) + + return c diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_ndbspline.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_ndbspline.py new file mode 100644 index 0000000000000000000000000000000000000000..51ac566ed5ff1271a46ffafcc04c0e180f2ec3f1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_ndbspline.py @@ -0,0 +1,420 @@ +import itertools +import functools +import operator +import numpy as np + +from math import prod + +from . import _bspl # type: ignore[attr-defined] + +import scipy.sparse.linalg as ssl +from scipy.sparse import csr_array + +from ._bsplines import _not_a_knot + +__all__ = ["NdBSpline"] + + +def _get_dtype(dtype): + """Return np.complex128 for complex dtypes, np.float64 otherwise.""" + if np.issubdtype(dtype, np.complexfloating): + return np.complex128 + else: + return np.float64 + + +class NdBSpline: + """Tensor product spline object. + + The value at point ``xp = (x1, x2, ..., xN)`` is evaluated as a linear + combination of products of one-dimensional b-splines in each of the ``N`` + dimensions:: + + c[i1, i2, ..., iN] * B(x1; i1, t1) * B(x2; i2, t2) * ... * B(xN; iN, tN) + + + Here ``B(x; i, t)`` is the ``i``-th b-spline defined by the knot vector + ``t`` evaluated at ``x``. + + Parameters + ---------- + t : tuple of 1D ndarrays + knot vectors in directions 1, 2, ... N, + ``len(t[i]) == n[i] + k + 1`` + c : ndarray, shape (n1, n2, ..., nN, ...) + b-spline coefficients + k : int or length-d tuple of integers + spline degrees. + A single integer is interpreted as having this degree for + all dimensions. + extrapolate : bool, optional + Whether to extrapolate out-of-bounds inputs, or return `nan`. + Default is to extrapolate. + + Attributes + ---------- + t : tuple of ndarrays + Knots vectors. + c : ndarray + Coefficients of the tensor-product spline. + k : tuple of integers + Degrees for each dimension. + extrapolate : bool, optional + Whether to extrapolate or return nans for out-of-bounds inputs. + Defaults to true. + + Methods + ------- + __call__ + design_matrix + + See Also + -------- + BSpline : a one-dimensional B-spline object + NdPPoly : an N-dimensional piecewise tensor product polynomial + + """ + def __init__(self, t, c, k, *, extrapolate=None): + self._k, self._indices_k1d, (self._t, self._len_t) = _preprocess_inputs(k, t) + + if extrapolate is None: + extrapolate = True + self.extrapolate = bool(extrapolate) + + self.c = np.asarray(c) + + ndim = self._t.shape[0] # == len(self.t) + if self.c.ndim < ndim: + raise ValueError(f"Coefficients must be at least {ndim}-dimensional.") + + for d in range(ndim): + td = self.t[d] + kd = self.k[d] + n = td.shape[0] - kd - 1 + + if self.c.shape[d] != n: + raise ValueError(f"Knots, coefficients and degree in dimension" + f" {d} are inconsistent:" + f" got {self.c.shape[d]} coefficients for" + f" {len(td)} knots, need at least {n} for" + f" k={k}.") + + dt = _get_dtype(self.c.dtype) + self.c = np.ascontiguousarray(self.c, dtype=dt) + + @property + def k(self): + return tuple(self._k) + + @property + def t(self): + # repack the knots into a tuple + return tuple(self._t[d, :self._len_t[d]] for d in range(self._t.shape[0])) + + def __call__(self, xi, *, nu=None, extrapolate=None): + """Evaluate the tensor product b-spline at ``xi``. + + Parameters + ---------- + xi : array_like, shape(..., ndim) + The coordinates to evaluate the interpolator at. + This can be a list or tuple of ndim-dimensional points + or an array with the shape (num_points, ndim). + nu : array_like, optional, shape (ndim,) + Orders of derivatives to evaluate. Each must be non-negative. + Defaults to the zeroth derivivative. + extrapolate : bool, optional + Whether to exrapolate based on first and last intervals in each + dimension, or return `nan`. Default is to ``self.extrapolate``. + + Returns + ------- + values : ndarray, shape ``xi.shape[:-1] + self.c.shape[ndim:]`` + Interpolated values at ``xi`` + """ + ndim = self._t.shape[0] # == len(self.t) + + if extrapolate is None: + extrapolate = self.extrapolate + extrapolate = bool(extrapolate) + + if nu is None: + nu = np.zeros((ndim,), dtype=np.intc) + else: + nu = np.asarray(nu, dtype=np.intc) + if nu.ndim != 1 or nu.shape[0] != ndim: + raise ValueError( + f"invalid number of derivative orders {nu = } for " + f"ndim = {len(self.t)}.") + if any(nu < 0): + raise ValueError(f"derivatives must be positive, got {nu = }") + + # prepare xi : shape (..., m1, ..., md) -> (1, m1, ..., md) + xi = np.asarray(xi, dtype=float) + xi_shape = xi.shape + xi = xi.reshape(-1, xi_shape[-1]) + xi = np.ascontiguousarray(xi) + + if xi_shape[-1] != ndim: + raise ValueError(f"Shapes: xi.shape={xi_shape} and ndim={ndim}") + + # complex -> double + was_complex = self.c.dtype.kind == 'c' + cc = self.c + if was_complex and self.c.ndim == ndim: + # make sure that core dimensions are intact, and complex->float + # size doubling only adds a trailing dimension + cc = self.c[..., None] + cc = cc.view(float) + + # prepare the coefficients: flatten the trailing dimensions + c1 = cc.reshape(cc.shape[:ndim] + (-1,)) + c1r = c1.ravel() + + # replacement for np.ravel_multi_index for indexing of `c1`: + _strides_c1 = np.asarray([s // c1.dtype.itemsize + for s in c1.strides], dtype=np.intp) + + num_c_tr = c1.shape[-1] # # of trailing coefficients + out = np.empty(xi.shape[:-1] + (num_c_tr,), dtype=c1.dtype) + + _bspl.evaluate_ndbspline(xi, + self._t, + self._len_t, + self._k, + nu, + extrapolate, + c1r, + num_c_tr, + _strides_c1, + self._indices_k1d, + out,) + out = out.view(self.c.dtype) + return out.reshape(xi_shape[:-1] + self.c.shape[ndim:]) + + @classmethod + def design_matrix(cls, xvals, t, k, extrapolate=True): + """Construct the design matrix as a CSR format sparse array. + + Parameters + ---------- + xvals : ndarray, shape(npts, ndim) + Data points. ``xvals[j, :]`` gives the ``j``-th data point as an + ``ndim``-dimensional array. + t : tuple of 1D ndarrays, length-ndim + Knot vectors in directions 1, 2, ... ndim, + k : int + B-spline degree. + extrapolate : bool, optional + Whether to extrapolate out-of-bounds values of raise a `ValueError` + + Returns + ------- + design_matrix : a CSR array + Each row of the design matrix corresponds to a value in `xvals` and + contains values of b-spline basis elements which are non-zero + at this value. + + """ + xvals = np.asarray(xvals, dtype=float) + ndim = xvals.shape[-1] + if len(t) != ndim: + raise ValueError( + f"Data and knots are inconsistent: len(t) = {len(t)} for " + f" {ndim = }." + ) + + # tabulate the flat indices for iterating over the (k+1)**ndim subarray + k, _indices_k1d, (_t, len_t) = _preprocess_inputs(k, t) + + # Precompute the shape and strides of the 'coefficients array'. + # This would have been the NdBSpline coefficients; in the present context + # this is a helper to compute the indices into the colocation matrix. + c_shape = tuple(len_t[d] - k[d] - 1 for d in range(ndim)) + + # The strides of the coeffs array: the computation is equivalent to + # >>> cstrides = [s // 8 for s in np.empty(c_shape).strides] + cs = c_shape[1:] + (1,) + cstrides = np.cumprod(cs[::-1], dtype=np.intp)[::-1].copy() + + # heavy lifting happens here + data, indices, indptr = _bspl._colloc_nd(xvals, + _t, + len_t, + k, + _indices_k1d, + cstrides) + return csr_array((data, indices, indptr)) + + +def _preprocess_inputs(k, t_tpl): + """Helpers: validate and preprocess NdBSpline inputs. + + Parameters + ---------- + k : int or tuple + Spline orders + t_tpl : tuple or array-likes + Knots. + """ + # 1. Make sure t_tpl is a tuple + if not isinstance(t_tpl, tuple): + raise ValueError(f"Expect `t` to be a tuple of array-likes. " + f"Got {t_tpl} instead." + ) + + # 2. Make ``k`` a tuple of integers + ndim = len(t_tpl) + try: + len(k) + except TypeError: + # make k a tuple + k = (k,)*ndim + + k = np.asarray([operator.index(ki) for ki in k], dtype=np.int32) + + if len(k) != ndim: + raise ValueError(f"len(t) = {len(t_tpl)} != {len(k) = }.") + + # 3. Validate inputs + ndim = len(t_tpl) + for d in range(ndim): + td = np.asarray(t_tpl[d]) + kd = k[d] + n = td.shape[0] - kd - 1 + if kd < 0: + raise ValueError(f"Spline degree in dimension {d} cannot be" + f" negative.") + if td.ndim != 1: + raise ValueError(f"Knot vector in dimension {d} must be" + f" one-dimensional.") + if n < kd + 1: + raise ValueError(f"Need at least {2*kd + 2} knots for degree" + f" {kd} in dimension {d}.") + if (np.diff(td) < 0).any(): + raise ValueError(f"Knots in dimension {d} must be in a" + f" non-decreasing order.") + if len(np.unique(td[kd:n + 1])) < 2: + raise ValueError(f"Need at least two internal knots in" + f" dimension {d}.") + if not np.isfinite(td).all(): + raise ValueError(f"Knots in dimension {d} should not have" + f" nans or infs.") + + # 4. tabulate the flat indices for iterating over the (k+1)**ndim subarray + # non-zero b-spline elements + shape = tuple(kd + 1 for kd in k) + indices = np.unravel_index(np.arange(prod(shape)), shape) + _indices_k1d = np.asarray(indices, dtype=np.intp).T.copy() + + # 5. pack the knots into a single array: + # ([1, 2, 3, 4], [5, 6], (7, 8, 9)) --> + # array([[1, 2, 3, 4], + # [5, 6, nan, nan], + # [7, 8, 9, nan]]) + ndim = len(t_tpl) + len_t = [len(ti) for ti in t_tpl] + _t = np.empty((ndim, max(len_t)), dtype=float) + _t.fill(np.nan) + for d in range(ndim): + _t[d, :len(t_tpl[d])] = t_tpl[d] + len_t = np.asarray(len_t, dtype=np.int32) + + return k, _indices_k1d, (_t, len_t) + + +def _iter_solve(a, b, solver=ssl.gcrotmk, **solver_args): + # work around iterative solvers not accepting multiple r.h.s. + + # also work around a.dtype == float64 and b.dtype == complex128 + # cf https://github.com/scipy/scipy/issues/19644 + if np.issubdtype(b.dtype, np.complexfloating): + real = _iter_solve(a, b.real, solver, **solver_args) + imag = _iter_solve(a, b.imag, solver, **solver_args) + return real + 1j*imag + + if b.ndim == 2 and b.shape[1] !=1: + res = np.empty_like(b) + for j in range(b.shape[1]): + res[:, j], info = solver(a, b[:, j], **solver_args) + if info != 0: + raise ValueError(f"{solver = } returns {info =} for column {j}.") + return res + else: + res, info = solver(a, b, **solver_args) + if info != 0: + raise ValueError(f"{solver = } returns {info = }.") + return res + + +def make_ndbspl(points, values, k=3, *, solver=ssl.gcrotmk, **solver_args): + """Construct an interpolating NdBspline. + + Parameters + ---------- + points : tuple of ndarrays of float, with shapes (m1,), ... (mN,) + The points defining the regular grid in N dimensions. The points in + each dimension (i.e. every element of the `points` tuple) must be + strictly ascending or descending. + values : ndarray of float, shape (m1, ..., mN, ...) + The data on the regular grid in n dimensions. + k : int, optional + The spline degree. Must be odd. Default is cubic, k=3 + solver : a `scipy.sparse.linalg` solver (iterative or direct), optional. + An iterative solver from `scipy.sparse.linalg` or a direct one, + `sparse.sparse.linalg.spsolve`. + Used to solve the sparse linear system + ``design_matrix @ coefficients = rhs`` for the coefficients. + Default is `scipy.sparse.linalg.gcrotmk` + solver_args : dict, optional + Additional arguments for the solver. The call signature is + ``solver(csr_array, rhs_vector, **solver_args)`` + + Returns + ------- + spl : NdBSpline object + + Notes + ----- + Boundary conditions are not-a-knot in all dimensions. + """ + ndim = len(points) + xi_shape = tuple(len(x) for x in points) + + try: + len(k) + except TypeError: + # make k a tuple + k = (k,)*ndim + + for d, point in enumerate(points): + numpts = len(np.atleast_1d(point)) + if numpts <= k[d]: + raise ValueError(f"There are {numpts} points in dimension {d}," + f" but order {k[d]} requires at least " + f" {k[d]+1} points per dimension.") + + t = tuple(_not_a_knot(np.asarray(points[d], dtype=float), k[d]) + for d in range(ndim)) + xvals = np.asarray([xv for xv in itertools.product(*points)], dtype=float) + + # construct the colocation matrix + matr = NdBSpline.design_matrix(xvals, t, k) + + # Solve for the coefficients given `values`. + # Trailing dimensions: first ndim dimensions are data, the rest are batch + # dimensions, so stack `values` into a 2D array for `spsolve` to undestand. + v_shape = values.shape + vals_shape = (prod(v_shape[:ndim]), prod(v_shape[ndim:])) + vals = values.reshape(vals_shape) + + if solver != ssl.spsolve: + solver = functools.partial(_iter_solve, solver=solver) + if "atol" not in solver_args: + # avoid a DeprecationWarning, grumble grumble + solver_args["atol"] = 1e-6 + + coef = solver(matr, vals, **solver_args) + coef = coef.reshape(xi_shape + v_shape[ndim:]) + return NdBSpline(t, coef, k) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_ndgriddata.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_ndgriddata.py new file mode 100644 index 0000000000000000000000000000000000000000..78fe9d6995141ad238002e0b48feb94017dc272a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_ndgriddata.py @@ -0,0 +1,332 @@ +""" +Convenience interface to N-D interpolation + +.. versionadded:: 0.9 + +""" +import numpy as np +from ._interpnd import (LinearNDInterpolator, NDInterpolatorBase, + CloughTocher2DInterpolator, _ndim_coords_from_arrays) +from scipy.spatial import cKDTree + +__all__ = ['griddata', 'NearestNDInterpolator', 'LinearNDInterpolator', + 'CloughTocher2DInterpolator'] + +#------------------------------------------------------------------------------ +# Nearest-neighbor interpolation +#------------------------------------------------------------------------------ + + +class NearestNDInterpolator(NDInterpolatorBase): + """NearestNDInterpolator(x, y). + + Nearest-neighbor interpolator in N > 1 dimensions. + + .. versionadded:: 0.9 + + Methods + ------- + __call__ + + Parameters + ---------- + x : (npoints, ndims) 2-D ndarray of floats + Data point coordinates. + y : (npoints, ) 1-D ndarray of float or complex + Data values. + rescale : boolean, optional + Rescale points to unit cube before performing interpolation. + This is useful if some of the input dimensions have + incommensurable units and differ by many orders of magnitude. + + .. versionadded:: 0.14.0 + tree_options : dict, optional + Options passed to the underlying ``cKDTree``. + + .. versionadded:: 0.17.0 + + See Also + -------- + griddata : + Interpolate unstructured D-D data. + LinearNDInterpolator : + Piecewise linear interpolator in N dimensions. + CloughTocher2DInterpolator : + Piecewise cubic, C1 smooth, curvature-minimizing interpolator in 2D. + interpn : Interpolation on a regular grid or rectilinear grid. + RegularGridInterpolator : Interpolator on a regular or rectilinear grid + in arbitrary dimensions (`interpn` wraps this + class). + + Notes + ----- + Uses ``scipy.spatial.cKDTree`` + + .. note:: For data on a regular grid use `interpn` instead. + + Examples + -------- + We can interpolate values on a 2D plane: + + >>> from scipy.interpolate import NearestNDInterpolator + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + >>> x = rng.random(10) - 0.5 + >>> y = rng.random(10) - 0.5 + >>> z = np.hypot(x, y) + >>> X = np.linspace(min(x), max(x)) + >>> Y = np.linspace(min(y), max(y)) + >>> X, Y = np.meshgrid(X, Y) # 2D grid for interpolation + >>> interp = NearestNDInterpolator(list(zip(x, y)), z) + >>> Z = interp(X, Y) + >>> plt.pcolormesh(X, Y, Z, shading='auto') + >>> plt.plot(x, y, "ok", label="input point") + >>> plt.legend() + >>> plt.colorbar() + >>> plt.axis("equal") + >>> plt.show() + + """ + + def __init__(self, x, y, rescale=False, tree_options=None): + NDInterpolatorBase.__init__(self, x, y, rescale=rescale, + need_contiguous=False, + need_values=False) + if tree_options is None: + tree_options = dict() + self.tree = cKDTree(self.points, **tree_options) + self.values = np.asarray(y) + + def __call__(self, *args, **query_options): + """ + Evaluate interpolator at given points. + + Parameters + ---------- + x1, x2, ... xn : array-like of float + Points where to interpolate data at. + x1, x2, ... xn can be array-like of float with broadcastable shape. + or x1 can be array-like of float with shape ``(..., ndim)`` + **query_options + This allows ``eps``, ``p``, ``distance_upper_bound``, and ``workers`` + being passed to the cKDTree's query function to be explicitly set. + See `scipy.spatial.cKDTree.query` for an overview of the different options. + + .. versionadded:: 1.12.0 + + """ + # For the sake of enabling subclassing, NDInterpolatorBase._set_xi performs + # some operations which are not required by NearestNDInterpolator.__call__, + # hence here we operate on xi directly, without calling a parent class function. + xi = _ndim_coords_from_arrays(args, ndim=self.points.shape[1]) + xi = self._check_call_shape(xi) + xi = self._scale_x(xi) + + # We need to handle two important cases: + # (1) the case where xi has trailing dimensions (..., ndim), and + # (2) the case where y has trailing dimensions + # We will first flatten xi to deal with case (1), + # do the computation in flattened array while retaining y's dimensionality, + # and then reshape the interpolated values back to match xi's shape. + + # Flatten xi for the query + xi_flat = xi.reshape(-1, xi.shape[-1]) + original_shape = xi.shape + flattened_shape = xi_flat.shape + + # if distance_upper_bound is set to not be infinite, + # then we need to consider the case where cKDtree + # does not find any points within distance_upper_bound to return. + # It marks those points as having infinte distance, which is what will be used + # below to mask the array and return only the points that were deemed + # to have a close enough neighbor to return something useful. + dist, i = self.tree.query(xi_flat, **query_options) + valid_mask = np.isfinite(dist) + + # create a holder interp_values array and fill with nans. + if self.values.ndim > 1: + interp_shape = flattened_shape[:-1] + self.values.shape[1:] + else: + interp_shape = flattened_shape[:-1] + + if np.issubdtype(self.values.dtype, np.complexfloating): + interp_values = np.full(interp_shape, np.nan, dtype=self.values.dtype) + else: + interp_values = np.full(interp_shape, np.nan) + + interp_values[valid_mask] = self.values[i[valid_mask], ...] + + if self.values.ndim > 1: + new_shape = original_shape[:-1] + self.values.shape[1:] + else: + new_shape = original_shape[:-1] + interp_values = interp_values.reshape(new_shape) + + return interp_values + + +#------------------------------------------------------------------------------ +# Convenience interface function +#------------------------------------------------------------------------------ + + +def griddata(points, values, xi, method='linear', fill_value=np.nan, + rescale=False): + """ + Interpolate unstructured D-D data. + + Parameters + ---------- + points : 2-D ndarray of floats with shape (n, D), or length D tuple of 1-D ndarrays with shape (n,). + Data point coordinates. + values : ndarray of float or complex, shape (n,) + Data values. + xi : 2-D ndarray of floats with shape (m, D), or length D tuple of ndarrays broadcastable to the same shape. + Points at which to interpolate data. + method : {'linear', 'nearest', 'cubic'}, optional + Method of interpolation. One of + + ``nearest`` + return the value at the data point closest to + the point of interpolation. See `NearestNDInterpolator` for + more details. + + ``linear`` + tessellate the input point set to N-D + simplices, and interpolate linearly on each simplex. See + `LinearNDInterpolator` for more details. + + ``cubic`` (1-D) + return the value determined from a cubic + spline. + + ``cubic`` (2-D) + return the value determined from a + piecewise cubic, continuously differentiable (C1), and + approximately curvature-minimizing polynomial surface. See + `CloughTocher2DInterpolator` for more details. + fill_value : float, optional + Value used to fill in for requested points outside of the + convex hull of the input points. If not provided, then the + default is ``nan``. This option has no effect for the + 'nearest' method. + rescale : bool, optional + Rescale points to unit cube before performing interpolation. + This is useful if some of the input dimensions have + incommensurable units and differ by many orders of magnitude. + + .. versionadded:: 0.14.0 + + Returns + ------- + ndarray + Array of interpolated values. + + See Also + -------- + LinearNDInterpolator : + Piecewise linear interpolator in N dimensions. + NearestNDInterpolator : + Nearest-neighbor interpolator in N dimensions. + CloughTocher2DInterpolator : + Piecewise cubic, C1 smooth, curvature-minimizing interpolator in 2D. + interpn : Interpolation on a regular grid or rectilinear grid. + RegularGridInterpolator : Interpolator on a regular or rectilinear grid + in arbitrary dimensions (`interpn` wraps this + class). + + Notes + ----- + + .. versionadded:: 0.9 + + .. note:: For data on a regular grid use `interpn` instead. + + Examples + -------- + + Suppose we want to interpolate the 2-D function + + >>> import numpy as np + >>> def func(x, y): + ... return x*(1-x)*np.cos(4*np.pi*x) * np.sin(4*np.pi*y**2)**2 + + on a grid in [0, 1]x[0, 1] + + >>> grid_x, grid_y = np.mgrid[0:1:100j, 0:1:200j] + + but we only know its values at 1000 data points: + + >>> rng = np.random.default_rng() + >>> points = rng.random((1000, 2)) + >>> values = func(points[:,0], points[:,1]) + + This can be done with `griddata` -- below we try out all of the + interpolation methods: + + >>> from scipy.interpolate import griddata + >>> grid_z0 = griddata(points, values, (grid_x, grid_y), method='nearest') + >>> grid_z1 = griddata(points, values, (grid_x, grid_y), method='linear') + >>> grid_z2 = griddata(points, values, (grid_x, grid_y), method='cubic') + + One can see that the exact result is reproduced by all of the + methods to some degree, but for this smooth function the piecewise + cubic interpolant gives the best results: + + >>> import matplotlib.pyplot as plt + >>> plt.subplot(221) + >>> plt.imshow(func(grid_x, grid_y).T, extent=(0,1,0,1), origin='lower') + >>> plt.plot(points[:,0], points[:,1], 'k.', ms=1) + >>> plt.title('Original') + >>> plt.subplot(222) + >>> plt.imshow(grid_z0.T, extent=(0,1,0,1), origin='lower') + >>> plt.title('Nearest') + >>> plt.subplot(223) + >>> plt.imshow(grid_z1.T, extent=(0,1,0,1), origin='lower') + >>> plt.title('Linear') + >>> plt.subplot(224) + >>> plt.imshow(grid_z2.T, extent=(0,1,0,1), origin='lower') + >>> plt.title('Cubic') + >>> plt.gcf().set_size_inches(6, 6) + >>> plt.show() + + """ # numpy/numpydoc#87 # noqa: E501 + + points = _ndim_coords_from_arrays(points) + + if points.ndim < 2: + ndim = points.ndim + else: + ndim = points.shape[-1] + + if ndim == 1 and method in ('nearest', 'linear', 'cubic'): + from ._interpolate import interp1d + points = points.ravel() + if isinstance(xi, tuple): + if len(xi) != 1: + raise ValueError("invalid number of dimensions in xi") + xi, = xi + # Sort points/values together, necessary as input for interp1d + idx = np.argsort(points) + points = points[idx] + values = values[idx] + if method == 'nearest': + fill_value = 'extrapolate' + ip = interp1d(points, values, kind=method, axis=0, bounds_error=False, + fill_value=fill_value) + return ip(xi) + elif method == 'nearest': + ip = NearestNDInterpolator(points, values, rescale=rescale) + return ip(xi) + elif method == 'linear': + ip = LinearNDInterpolator(points, values, fill_value=fill_value, + rescale=rescale) + return ip(xi) + elif method == 'cubic' and ndim == 2: + ip = CloughTocher2DInterpolator(points, values, fill_value=fill_value, + rescale=rescale) + return ip(xi) + else: + raise ValueError("Unknown interpolation method %r for " + "%d dimensional data" % (method, ndim)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_pade.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_pade.py new file mode 100644 index 0000000000000000000000000000000000000000..387ef11dde5d3ace8a15324058c10fa31899c92c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_pade.py @@ -0,0 +1,67 @@ +from numpy import zeros, asarray, eye, poly1d, hstack, r_ +from scipy import linalg + +__all__ = ["pade"] + +def pade(an, m, n=None): + """ + Return Pade approximation to a polynomial as the ratio of two polynomials. + + Parameters + ---------- + an : (N,) array_like + Taylor series coefficients. + m : int + The order of the returned approximating polynomial `q`. + n : int, optional + The order of the returned approximating polynomial `p`. By default, + the order is ``len(an)-1-m``. + + Returns + ------- + p, q : Polynomial class + The Pade approximation of the polynomial defined by `an` is + ``p(x)/q(x)``. + + Examples + -------- + >>> import numpy as np + >>> from scipy.interpolate import pade + >>> e_exp = [1.0, 1.0, 1.0/2.0, 1.0/6.0, 1.0/24.0, 1.0/120.0] + >>> p, q = pade(e_exp, 2) + + >>> e_exp.reverse() + >>> e_poly = np.poly1d(e_exp) + + Compare ``e_poly(x)`` and the Pade approximation ``p(x)/q(x)`` + + >>> e_poly(1) + 2.7166666666666668 + + >>> p(1)/q(1) + 2.7179487179487181 + + """ + an = asarray(an) + if n is None: + n = len(an) - 1 - m + if n < 0: + raise ValueError("Order of q must be smaller than len(an)-1.") + if n < 0: + raise ValueError("Order of p must be greater than 0.") + N = m + n + if N > len(an)-1: + raise ValueError("Order of q+p must be smaller than len(an).") + an = an[:N+1] + Akj = eye(N+1, n+1, dtype=an.dtype) + Bkj = zeros((N+1, m), dtype=an.dtype) + for row in range(1, m+1): + Bkj[row,:row] = -(an[:row])[::-1] + for row in range(m+1, N+1): + Bkj[row,:] = -(an[row-m:row])[::-1] + C = hstack((Akj, Bkj)) + pq = linalg.solve(C, an) + p = pq[:n+1] + q = r_[1.0, pq[n+1:]] + return poly1d(p[::-1]), poly1d(q[::-1]) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_polyint.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_polyint.py new file mode 100644 index 0000000000000000000000000000000000000000..9cec3eb9939abba36505010334911c167797b750 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_polyint.py @@ -0,0 +1,961 @@ +import warnings + +import numpy as np +from scipy.special import factorial +from scipy._lib._util import (_asarray_validated, float_factorial, check_random_state, + _transition_to_rng) + + +__all__ = ["KroghInterpolator", "krogh_interpolate", + "BarycentricInterpolator", "barycentric_interpolate", + "approximate_taylor_polynomial"] + + +def _isscalar(x): + """Check whether x is if a scalar type, or 0-dim""" + return np.isscalar(x) or hasattr(x, 'shape') and x.shape == () + + +class _Interpolator1D: + """ + Common features in univariate interpolation + + Deal with input data type and interpolation axis rolling. The + actual interpolator can assume the y-data is of shape (n, r) where + `n` is the number of x-points, and `r` the number of variables, + and use self.dtype as the y-data type. + + Attributes + ---------- + _y_axis + Axis along which the interpolation goes in the original array + _y_extra_shape + Additional trailing shape of the input arrays, excluding + the interpolation axis. + dtype + Dtype of the y-data arrays. Can be set via _set_dtype, which + forces it to be float or complex. + + Methods + ------- + __call__ + _prepare_x + _finish_y + _reshape_yi + _set_yi + _set_dtype + _evaluate + + """ + + __slots__ = ('_y_axis', '_y_extra_shape', 'dtype') + + def __init__(self, xi=None, yi=None, axis=None): + self._y_axis = axis + self._y_extra_shape = None + self.dtype = None + if yi is not None: + self._set_yi(yi, xi=xi, axis=axis) + + def __call__(self, x): + """ + Evaluate the interpolant + + Parameters + ---------- + x : array_like + Point or points at which to evaluate the interpolant. + + Returns + ------- + y : array_like + Interpolated values. Shape is determined by replacing + the interpolation axis in the original array with the shape of `x`. + + Notes + ----- + Input values `x` must be convertible to `float` values like `int` + or `float`. + + """ + x, x_shape = self._prepare_x(x) + y = self._evaluate(x) + return self._finish_y(y, x_shape) + + def _evaluate(self, x): + """ + Actually evaluate the value of the interpolator. + """ + raise NotImplementedError() + + def _prepare_x(self, x): + """Reshape input x array to 1-D""" + x = _asarray_validated(x, check_finite=False, as_inexact=True) + x_shape = x.shape + return x.ravel(), x_shape + + def _finish_y(self, y, x_shape): + """Reshape interpolated y back to an N-D array similar to initial y""" + y = y.reshape(x_shape + self._y_extra_shape) + if self._y_axis != 0 and x_shape != (): + nx = len(x_shape) + ny = len(self._y_extra_shape) + s = (list(range(nx, nx + self._y_axis)) + + list(range(nx)) + list(range(nx+self._y_axis, nx+ny))) + y = y.transpose(s) + return y + + def _reshape_yi(self, yi, check=False): + yi = np.moveaxis(np.asarray(yi), self._y_axis, 0) + if check and yi.shape[1:] != self._y_extra_shape: + ok_shape = (f"{self._y_extra_shape[-self._y_axis:]!r} + (N,) + " + f"{self._y_extra_shape[:-self._y_axis]!r}") + raise ValueError(f"Data must be of shape {ok_shape}") + return yi.reshape((yi.shape[0], -1)) + + def _set_yi(self, yi, xi=None, axis=None): + if axis is None: + axis = self._y_axis + if axis is None: + raise ValueError("no interpolation axis specified") + + yi = np.asarray(yi) + + shape = yi.shape + if shape == (): + shape = (1,) + if xi is not None and shape[axis] != len(xi): + raise ValueError("x and y arrays must be equal in length along " + "interpolation axis.") + + self._y_axis = (axis % yi.ndim) + self._y_extra_shape = yi.shape[:self._y_axis] + yi.shape[self._y_axis+1:] + self.dtype = None + self._set_dtype(yi.dtype) + + def _set_dtype(self, dtype, union=False): + if np.issubdtype(dtype, np.complexfloating) \ + or np.issubdtype(self.dtype, np.complexfloating): + self.dtype = np.complex128 + else: + if not union or self.dtype != np.complex128: + self.dtype = np.float64 + + +class _Interpolator1DWithDerivatives(_Interpolator1D): + def derivatives(self, x, der=None): + """ + Evaluate several derivatives of the polynomial at the point `x` + + Produce an array of derivatives evaluated at the point `x`. + + Parameters + ---------- + x : array_like + Point or points at which to evaluate the derivatives + der : int or list or None, optional + How many derivatives to evaluate, or None for all potentially + nonzero derivatives (that is, a number equal to the number + of points), or a list of derivatives to evaluate. This number + includes the function value as the '0th' derivative. + + Returns + ------- + d : ndarray + Array with derivatives; ``d[j]`` contains the jth derivative. + Shape of ``d[j]`` is determined by replacing the interpolation + axis in the original array with the shape of `x`. + + Examples + -------- + >>> from scipy.interpolate import KroghInterpolator + >>> KroghInterpolator([0,0,0],[1,2,3]).derivatives(0) + array([1.0,2.0,3.0]) + >>> KroghInterpolator([0,0,0],[1,2,3]).derivatives([0,0]) + array([[1.0,1.0], + [2.0,2.0], + [3.0,3.0]]) + + """ + x, x_shape = self._prepare_x(x) + y = self._evaluate_derivatives(x, der) + + y = y.reshape((y.shape[0],) + x_shape + self._y_extra_shape) + if self._y_axis != 0 and x_shape != (): + nx = len(x_shape) + ny = len(self._y_extra_shape) + s = ([0] + list(range(nx+1, nx + self._y_axis+1)) + + list(range(1, nx+1)) + + list(range(nx+1+self._y_axis, nx+ny+1))) + y = y.transpose(s) + return y + + def derivative(self, x, der=1): + """ + Evaluate a single derivative of the polynomial at the point `x`. + + Parameters + ---------- + x : array_like + Point or points at which to evaluate the derivatives + + der : integer, optional + Which derivative to evaluate (default: first derivative). + This number includes the function value as 0th derivative. + + Returns + ------- + d : ndarray + Derivative interpolated at the x-points. Shape of `d` is + determined by replacing the interpolation axis in the + original array with the shape of `x`. + + Notes + ----- + This may be computed by evaluating all derivatives up to the desired + one (using self.derivatives()) and then discarding the rest. + + """ + x, x_shape = self._prepare_x(x) + y = self._evaluate_derivatives(x, der+1) + return self._finish_y(y[der], x_shape) + + def _evaluate_derivatives(self, x, der=None): + """ + Actually evaluate the derivatives. + + Parameters + ---------- + x : array_like + 1D array of points at which to evaluate the derivatives + der : integer, optional + The number of derivatives to evaluate, from 'order 0' (der=1) + to order der-1. If omitted, return all possibly-non-zero + derivatives, ie 0 to order n-1. + + Returns + ------- + d : ndarray + Array of shape ``(der, x.size, self.yi.shape[1])`` containing + the derivatives from 0 to der-1 + """ + raise NotImplementedError() + + +class KroghInterpolator(_Interpolator1DWithDerivatives): + """ + Interpolating polynomial for a set of points. + + The polynomial passes through all the pairs ``(xi, yi)``. One may + additionally specify a number of derivatives at each point `xi`; + this is done by repeating the value `xi` and specifying the + derivatives as successive `yi` values. + + Allows evaluation of the polynomial and all its derivatives. + For reasons of numerical stability, this function does not compute + the coefficients of the polynomial, although they can be obtained + by evaluating all the derivatives. + + Parameters + ---------- + xi : array_like, shape (npoints, ) + Known x-coordinates. Must be sorted in increasing order. + yi : array_like, shape (..., npoints, ...) + Known y-coordinates. When an xi occurs two or more times in + a row, the corresponding yi's represent derivative values. The length of `yi` + along the interpolation axis must be equal to the length of `xi`. Use the + `axis` parameter to select the correct axis. + axis : int, optional + Axis in the `yi` array corresponding to the x-coordinate values. Defaults to + ``axis=0``. + + Notes + ----- + Be aware that the algorithms implemented here are not necessarily + the most numerically stable known. Moreover, even in a world of + exact computation, unless the x coordinates are chosen very + carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice - + polynomial interpolation itself is a very ill-conditioned process + due to the Runge phenomenon. In general, even with well-chosen + x values, degrees higher than about thirty cause problems with + numerical instability in this code. + + Based on [1]_. + + References + ---------- + .. [1] Krogh, "Efficient Algorithms for Polynomial Interpolation + and Numerical Differentiation", 1970. + + Examples + -------- + To produce a polynomial that is zero at 0 and 1 and has + derivative 2 at 0, call + + >>> from scipy.interpolate import KroghInterpolator + >>> KroghInterpolator([0,0,1],[0,2,0]) + + This constructs the quadratic :math:`2x^2-2x`. The derivative condition + is indicated by the repeated zero in the `xi` array; the corresponding + yi values are 0, the function value, and 2, the derivative value. + + For another example, given `xi`, `yi`, and a derivative `ypi` for each + point, appropriate arrays can be constructed as: + + >>> import numpy as np + >>> rng = np.random.default_rng() + >>> xi = np.linspace(0, 1, 5) + >>> yi, ypi = rng.random((2, 5)) + >>> xi_k, yi_k = np.repeat(xi, 2), np.ravel(np.dstack((yi,ypi))) + >>> KroghInterpolator(xi_k, yi_k) + + To produce a vector-valued polynomial, supply a higher-dimensional + array for `yi`: + + >>> KroghInterpolator([0,1],[[2,3],[4,5]]) + + This constructs a linear polynomial giving (2,3) at 0 and (4,5) at 1. + + """ + + def __init__(self, xi, yi, axis=0): + super().__init__(xi, yi, axis) + + self.xi = np.asarray(xi) + self.yi = self._reshape_yi(yi) + self.n, self.r = self.yi.shape + + if (deg := self.xi.size) > 30: + warnings.warn(f"{deg} degrees provided, degrees higher than about" + " thirty cause problems with numerical instability " + "with 'KroghInterpolator'", stacklevel=2) + + c = np.zeros((self.n+1, self.r), dtype=self.dtype) + c[0] = self.yi[0] + Vk = np.zeros((self.n, self.r), dtype=self.dtype) + for k in range(1, self.n): + s = 0 + while s <= k and xi[k-s] == xi[k]: + s += 1 + s -= 1 + Vk[0] = self.yi[k]/float_factorial(s) + for i in range(k-s): + if xi[i] == xi[k]: + raise ValueError("Elements of `xi` can't be equal.") + if s == 0: + Vk[i+1] = (c[i]-Vk[i])/(xi[i]-xi[k]) + else: + Vk[i+1] = (Vk[i+1]-Vk[i])/(xi[i]-xi[k]) + c[k] = Vk[k-s] + self.c = c + + def _evaluate(self, x): + pi = 1 + p = np.zeros((len(x), self.r), dtype=self.dtype) + p += self.c[0,np.newaxis,:] + for k in range(1, self.n): + w = x - self.xi[k-1] + pi = w*pi + p += pi[:,np.newaxis] * self.c[k] + return p + + def _evaluate_derivatives(self, x, der=None): + n = self.n + r = self.r + + if der is None: + der = self.n + + pi = np.zeros((n, len(x))) + w = np.zeros((n, len(x))) + pi[0] = 1 + p = np.zeros((len(x), self.r), dtype=self.dtype) + p += self.c[0, np.newaxis, :] + + for k in range(1, n): + w[k-1] = x - self.xi[k-1] + pi[k] = w[k-1] * pi[k-1] + p += pi[k, :, np.newaxis] * self.c[k] + + cn = np.zeros((max(der, n+1), len(x), r), dtype=self.dtype) + cn[:n+1, :, :] += self.c[:n+1, np.newaxis, :] + cn[0] = p + for k in range(1, n): + for i in range(1, n-k+1): + pi[i] = w[k+i-1]*pi[i-1] + pi[i] + cn[k] = cn[k] + pi[i, :, np.newaxis]*cn[k+i] + cn[k] *= float_factorial(k) + + cn[n, :, :] = 0 + return cn[:der] + + +def krogh_interpolate(xi, yi, x, der=0, axis=0): + """ + Convenience function for polynomial interpolation. + + See `KroghInterpolator` for more details. + + Parameters + ---------- + xi : array_like + Interpolation points (known x-coordinates). + yi : array_like + Known y-coordinates, of shape ``(xi.size, R)``. Interpreted as + vectors of length R, or scalars if R=1. + x : array_like + Point or points at which to evaluate the derivatives. + der : int or list or None, optional + How many derivatives to evaluate, or None for all potentially + nonzero derivatives (that is, a number equal to the number + of points), or a list of derivatives to evaluate. This number + includes the function value as the '0th' derivative. + axis : int, optional + Axis in the `yi` array corresponding to the x-coordinate values. + + Returns + ------- + d : ndarray + If the interpolator's values are R-D then the + returned array will be the number of derivatives by N by R. + If `x` is a scalar, the middle dimension will be dropped; if + the `yi` are scalars then the last dimension will be dropped. + + See Also + -------- + KroghInterpolator : Krogh interpolator + + Notes + ----- + Construction of the interpolating polynomial is a relatively expensive + process. If you want to evaluate it repeatedly consider using the class + KroghInterpolator (which is what this function uses). + + Examples + -------- + We can interpolate 2D observed data using Krogh interpolation: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import krogh_interpolate + >>> x_observed = np.linspace(0.0, 10.0, 11) + >>> y_observed = np.sin(x_observed) + >>> x = np.linspace(min(x_observed), max(x_observed), num=100) + >>> y = krogh_interpolate(x_observed, y_observed, x) + >>> plt.plot(x_observed, y_observed, "o", label="observation") + >>> plt.plot(x, y, label="krogh interpolation") + >>> plt.legend() + >>> plt.show() + """ + + P = KroghInterpolator(xi, yi, axis=axis) + if der == 0: + return P(x) + elif _isscalar(der): + return P.derivative(x, der=der) + else: + return P.derivatives(x, der=np.amax(der)+1)[der] + + +def approximate_taylor_polynomial(f,x,degree,scale,order=None): + """ + Estimate the Taylor polynomial of f at x by polynomial fitting. + + Parameters + ---------- + f : callable + The function whose Taylor polynomial is sought. Should accept + a vector of `x` values. + x : scalar + The point at which the polynomial is to be evaluated. + degree : int + The degree of the Taylor polynomial + scale : scalar + The width of the interval to use to evaluate the Taylor polynomial. + Function values spread over a range this wide are used to fit the + polynomial. Must be chosen carefully. + order : int or None, optional + The order of the polynomial to be used in the fitting; `f` will be + evaluated ``order+1`` times. If None, use `degree`. + + Returns + ------- + p : poly1d instance + The Taylor polynomial (translated to the origin, so that + for example p(0)=f(x)). + + Notes + ----- + The appropriate choice of "scale" is a trade-off; too large and the + function differs from its Taylor polynomial too much to get a good + answer, too small and round-off errors overwhelm the higher-order terms. + The algorithm used becomes numerically unstable around order 30 even + under ideal circumstances. + + Choosing order somewhat larger than degree may improve the higher-order + terms. + + Examples + -------- + We can calculate Taylor approximation polynomials of sin function with + various degrees: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import approximate_taylor_polynomial + >>> x = np.linspace(-10.0, 10.0, num=100) + >>> plt.plot(x, np.sin(x), label="sin curve") + >>> for degree in np.arange(1, 15, step=2): + ... sin_taylor = approximate_taylor_polynomial(np.sin, 0, degree, 1, + ... order=degree + 2) + ... plt.plot(x, sin_taylor(x), label=f"degree={degree}") + >>> plt.legend(bbox_to_anchor=(1.05, 1), loc='upper left', + ... borderaxespad=0.0, shadow=True) + >>> plt.tight_layout() + >>> plt.axis([-10, 10, -10, 10]) + >>> plt.show() + + """ + if order is None: + order = degree + + n = order+1 + # Choose n points that cluster near the endpoints of the interval in + # a way that avoids the Runge phenomenon. Ensure, by including the + # endpoint or not as appropriate, that one point always falls at x + # exactly. + xs = scale*np.cos(np.linspace(0,np.pi,n,endpoint=n % 1)) + x + + P = KroghInterpolator(xs, f(xs)) + d = P.derivatives(x,der=degree+1) + + return np.poly1d((d/factorial(np.arange(degree+1)))[::-1]) + + +class BarycentricInterpolator(_Interpolator1DWithDerivatives): + r"""Interpolating polynomial for a set of points. + + Constructs a polynomial that passes through a given set of points. + Allows evaluation of the polynomial and all its derivatives, + efficient changing of the y-values to be interpolated, + and updating by adding more x- and y-values. + + For reasons of numerical stability, this function does not compute + the coefficients of the polynomial. + + The values `yi` need to be provided before the function is + evaluated, but none of the preprocessing depends on them, so rapid + updates are possible. + + Parameters + ---------- + xi : array_like, shape (npoints, ) + 1-D array of x coordinates of the points the polynomial + should pass through + yi : array_like, shape (..., npoints, ...), optional + N-D array of y coordinates of the points the polynomial should pass through. + If None, the y values will be supplied later via the `set_y` method. + The length of `yi` along the interpolation axis must be equal to the length + of `xi`. Use the ``axis`` parameter to select correct axis. + axis : int, optional + Axis in the yi array corresponding to the x-coordinate values. Defaults + to ``axis=0``. + wi : array_like, optional + The barycentric weights for the chosen interpolation points `xi`. + If absent or None, the weights will be computed from `xi` (default). + This allows for the reuse of the weights `wi` if several interpolants + are being calculated using the same nodes `xi`, without re-computation. + rng : {None, int, `numpy.random.Generator`}, optional + If `rng` is passed by keyword, types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + If `rng` is already a ``Generator`` instance, then the provided instance is + used. Specify `rng` for repeatable interpolation. + + If this argument `random_state` is passed by keyword, + legacy behavior for the argument `random_state` applies: + + - If `random_state` is None (or `numpy.random`), the `numpy.random.RandomState` + singleton is used. + - If `random_state` is an int, a new ``RandomState`` instance is used, + seeded with `random_state`. + - If `random_state` is already a ``Generator`` or ``RandomState`` instance then + that instance is used. + + .. versionchanged:: 1.15.0 + As part of the `SPEC-007 `_ + transition from use of `numpy.random.RandomState` to + `numpy.random.Generator` this keyword was changed from `random_state` to `rng`. + For an interim period, both keywords will continue to work (only specify + one of them). After the interim period using the `random_state` keyword will emit + warnings. The behavior of the `random_state` and `rng` keywords is outlined above. + + Notes + ----- + This class uses a "barycentric interpolation" method that treats + the problem as a special case of rational function interpolation. + This algorithm is quite stable, numerically, but even in a world of + exact computation, unless the x coordinates are chosen very + carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice - + polynomial interpolation itself is a very ill-conditioned process + due to the Runge phenomenon. + + Based on Berrut and Trefethen 2004, "Barycentric Lagrange Interpolation". + + Examples + -------- + To produce a quintic barycentric interpolant approximating the function + :math:`\sin x`, and its first four derivatives, using six randomly-spaced + nodes in :math:`(0, \frac{\pi}{2})`: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import BarycentricInterpolator + >>> rng = np.random.default_rng() + >>> xi = rng.random(6) * np.pi/2 + >>> f, f_d1, f_d2, f_d3, f_d4 = np.sin, np.cos, lambda x: -np.sin(x), lambda x: -np.cos(x), np.sin + >>> P = BarycentricInterpolator(xi, f(xi), random_state=rng) + >>> fig, axs = plt.subplots(5, 1, sharex=True, layout='constrained', figsize=(7,10)) + >>> x = np.linspace(0, np.pi, 100) + >>> axs[0].plot(x, P(x), 'r:', x, f(x), 'k--', xi, f(xi), 'xk') + >>> axs[1].plot(x, P.derivative(x), 'r:', x, f_d1(x), 'k--', xi, f_d1(xi), 'xk') + >>> axs[2].plot(x, P.derivative(x, 2), 'r:', x, f_d2(x), 'k--', xi, f_d2(xi), 'xk') + >>> axs[3].plot(x, P.derivative(x, 3), 'r:', x, f_d3(x), 'k--', xi, f_d3(xi), 'xk') + >>> axs[4].plot(x, P.derivative(x, 4), 'r:', x, f_d4(x), 'k--', xi, f_d4(xi), 'xk') + >>> axs[0].set_xlim(0, np.pi) + >>> axs[4].set_xlabel(r"$x$") + >>> axs[4].set_xticks([i * np.pi / 4 for i in range(5)], + ... ["0", r"$\frac{\pi}{4}$", r"$\frac{\pi}{2}$", r"$\frac{3\pi}{4}$", r"$\pi$"]) + >>> axs[0].set_ylabel("$f(x)$") + >>> axs[1].set_ylabel("$f'(x)$") + >>> axs[2].set_ylabel("$f''(x)$") + >>> axs[3].set_ylabel("$f^{(3)}(x)$") + >>> axs[4].set_ylabel("$f^{(4)}(x)$") + >>> labels = ['Interpolation nodes', 'True function $f$', 'Barycentric interpolation'] + >>> axs[0].legend(axs[0].get_lines()[::-1], labels, bbox_to_anchor=(0., 1.02, 1., .102), + ... loc='lower left', ncols=3, mode="expand", borderaxespad=0., frameon=False) + >>> plt.show() + """ # numpy/numpydoc#87 # noqa: E501 + + @_transition_to_rng("random_state", replace_doc=False) + def __init__(self, xi, yi=None, axis=0, *, wi=None, rng=None): + super().__init__(xi, yi, axis) + + rng = check_random_state(rng) + + self.xi = np.asarray(xi, dtype=np.float64) + self.set_yi(yi) + self.n = len(self.xi) + + # cache derivative object to avoid re-computing the weights with every call. + self._diff_cij = None + + if wi is not None: + self.wi = wi + else: + # See page 510 of Berrut and Trefethen 2004 for an explanation of the + # capacity scaling and the suggestion of using a random permutation of + # the input factors. + # At the moment, the permutation is not performed for xi that are + # appended later through the add_xi interface. It's not clear to me how + # to implement that and it seems that most situations that require + # these numerical stability improvements will be able to provide all + # the points to the constructor. + self._inv_capacity = 4.0 / (np.max(self.xi) - np.min(self.xi)) + permute = rng.permutation(self.n, ) + inv_permute = np.zeros(self.n, dtype=np.int32) + inv_permute[permute] = np.arange(self.n) + self.wi = np.zeros(self.n) + + for i in range(self.n): + dist = self._inv_capacity * (self.xi[i] - self.xi[permute]) + dist[inv_permute[i]] = 1.0 + prod = np.prod(dist) + if prod == 0.0: + raise ValueError("Interpolation points xi must be" + " distinct.") + self.wi[i] = 1.0 / prod + + def set_yi(self, yi, axis=None): + """ + Update the y values to be interpolated + + The barycentric interpolation algorithm requires the calculation + of weights, but these depend only on the `xi`. The `yi` can be changed + at any time. + + Parameters + ---------- + yi : array_like + The y-coordinates of the points the polynomial will pass through. + If None, the y values must be supplied later. + axis : int, optional + Axis in the `yi` array corresponding to the x-coordinate values. + + """ + if yi is None: + self.yi = None + return + self._set_yi(yi, xi=self.xi, axis=axis) + self.yi = self._reshape_yi(yi) + self.n, self.r = self.yi.shape + self._diff_baryint = None + + def add_xi(self, xi, yi=None): + """ + Add more x values to the set to be interpolated + + The barycentric interpolation algorithm allows easy updating by + adding more points for the polynomial to pass through. + + Parameters + ---------- + xi : array_like + The x coordinates of the points that the polynomial should pass + through. + yi : array_like, optional + The y coordinates of the points the polynomial should pass through. + Should have shape ``(xi.size, R)``; if R > 1 then the polynomial is + vector-valued. + If `yi` is not given, the y values will be supplied later. `yi` + should be given if and only if the interpolator has y values + specified. + + Notes + ----- + The new points added by `add_xi` are not randomly permuted + so there is potential for numerical instability, + especially for a large number of points. If this + happens, please reconstruct interpolation from scratch instead. + """ + if yi is not None: + if self.yi is None: + raise ValueError("No previous yi value to update!") + yi = self._reshape_yi(yi, check=True) + self.yi = np.vstack((self.yi,yi)) + else: + if self.yi is not None: + raise ValueError("No update to yi provided!") + old_n = self.n + self.xi = np.concatenate((self.xi,xi)) + self.n = len(self.xi) + self.wi **= -1 + old_wi = self.wi + self.wi = np.zeros(self.n) + self.wi[:old_n] = old_wi + for j in range(old_n, self.n): + self.wi[:j] *= self._inv_capacity * (self.xi[j]-self.xi[:j]) + self.wi[j] = np.multiply.reduce( + self._inv_capacity * (self.xi[:j]-self.xi[j]) + ) + self.wi **= -1 + self._diff_cij = None + self._diff_baryint = None + + def __call__(self, x): + """Evaluate the interpolating polynomial at the points x + + Parameters + ---------- + x : array_like + Point or points at which to evaluate the interpolant. + + Returns + ------- + y : array_like + Interpolated values. Shape is determined by replacing + the interpolation axis in the original array with the shape of `x`. + + Notes + ----- + Currently the code computes an outer product between `x` and the + weights, that is, it constructs an intermediate array of size + ``(N, len(x))``, where N is the degree of the polynomial. + """ + return _Interpolator1D.__call__(self, x) + + def _evaluate(self, x): + if x.size == 0: + p = np.zeros((0, self.r), dtype=self.dtype) + else: + c = x[..., np.newaxis] - self.xi + z = c == 0 + c[z] = 1 + c = self.wi / c + with np.errstate(divide='ignore'): + p = np.dot(c, self.yi) / np.sum(c, axis=-1)[..., np.newaxis] + # Now fix where x==some xi + r = np.nonzero(z) + if len(r) == 1: # evaluation at a scalar + if len(r[0]) > 0: # equals one of the points + p = self.yi[r[0][0]] + else: + p[r[:-1]] = self.yi[r[-1]] + return p + + def derivative(self, x, der=1): + """ + Evaluate a single derivative of the polynomial at the point x. + + Parameters + ---------- + x : array_like + Point or points at which to evaluate the derivatives + der : integer, optional + Which derivative to evaluate (default: first derivative). + This number includes the function value as 0th derivative. + + Returns + ------- + d : ndarray + Derivative interpolated at the x-points. Shape of `d` is + determined by replacing the interpolation axis in the + original array with the shape of `x`. + """ + x, x_shape = self._prepare_x(x) + y = self._evaluate_derivatives(x, der+1, all_lower=False) + return self._finish_y(y, x_shape) + + def _evaluate_derivatives(self, x, der=None, all_lower=True): + # NB: der here is not the order of the highest derivative; + # instead, it is the size of the derivatives matrix that + # would be returned with all_lower=True, including the + # '0th' derivative (the undifferentiated function). + # E.g. to evaluate the 5th derivative alone, call + # _evaluate_derivatives(x, der=6, all_lower=False). + + if (not all_lower) and (x.size == 0 or self.r == 0): + return np.zeros((0, self.r), dtype=self.dtype) + + if (not all_lower) and der == 1: + return self._evaluate(x) + + if (not all_lower) and (der > self.n): + return np.zeros((len(x), self.r), dtype=self.dtype) + + if der is None: + der = self.n + + if all_lower and (x.size == 0 or self.r == 0): + return np.zeros((der, len(x), self.r), dtype=self.dtype) + + if self._diff_cij is None: + # c[i,j] = xi[i] - xi[j] + c = self.xi[:, np.newaxis] - self.xi + + # avoid division by 0 (diagonal entries are so far zero by construction) + np.fill_diagonal(c, 1) + + # c[i,j] = (w[j] / w[i]) / (xi[i] - xi[j]) (equation 9.4) + c = self.wi/ (c * self.wi[..., np.newaxis]) + + # fill in correct diagonal entries: each column sums to 0 + np.fill_diagonal(c, 0) + + # calculate diagonal + # c[j,j] = -sum_{i != j} c[i,j] (equation 9.5) + d = -c.sum(axis=1) + # c[i,j] = l_j(x_i) + np.fill_diagonal(c, d) + + self._diff_cij = c + + if self._diff_baryint is None: + # initialise and cache derivative interpolator and cijs; + # reuse weights wi (which depend only on interpolation points xi), + # to avoid unnecessary re-computation + self._diff_baryint = BarycentricInterpolator(xi=self.xi, + yi=self._diff_cij @ self.yi, + wi=self.wi) + self._diff_baryint._diff_cij = self._diff_cij + + if all_lower: + # assemble matrix of derivatives from order 0 to order der-1, + # in the format required by _Interpolator1DWithDerivatives. + cn = np.zeros((der, len(x), self.r), dtype=self.dtype) + for d in range(der): + cn[d, :, :] = self._evaluate_derivatives(x, d+1, all_lower=False) + return cn + + # recursively evaluate only the derivative requested + return self._diff_baryint._evaluate_derivatives(x, der-1, all_lower=False) + + +def barycentric_interpolate(xi, yi, x, axis=0, *, der=0, rng=None): + """ + Convenience function for polynomial interpolation. + + Constructs a polynomial that passes through a given set of points, + then evaluates the polynomial. For reasons of numerical stability, + this function does not compute the coefficients of the polynomial. + + This function uses a "barycentric interpolation" method that treats + the problem as a special case of rational function interpolation. + This algorithm is quite stable, numerically, but even in a world of + exact computation, unless the `x` coordinates are chosen very + carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice - + polynomial interpolation itself is a very ill-conditioned process + due to the Runge phenomenon. + + Parameters + ---------- + xi : array_like + 1-D array of x coordinates of the points the polynomial should + pass through + yi : array_like + The y coordinates of the points the polynomial should pass through. + x : scalar or array_like + Point or points at which to evaluate the interpolant. + axis : int, optional + Axis in the `yi` array corresponding to the x-coordinate values. + der : int or list or None, optional + How many derivatives to evaluate, or None for all potentially + nonzero derivatives (that is, a number equal to the number + of points), or a list of derivatives to evaluate. This number + includes the function value as the '0th' derivative. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + + Returns + ------- + y : scalar or array_like + Interpolated values. Shape is determined by replacing + the interpolation axis in the original array with the shape of `x`. + + See Also + -------- + BarycentricInterpolator : Barycentric interpolator + + Notes + ----- + Construction of the interpolation weights is a relatively slow process. + If you want to call this many times with the same xi (but possibly + varying yi or x) you should use the class `BarycentricInterpolator`. + This is what this function uses internally. + + Examples + -------- + We can interpolate 2D observed data using barycentric interpolation: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import barycentric_interpolate + >>> x_observed = np.linspace(0.0, 10.0, 11) + >>> y_observed = np.sin(x_observed) + >>> x = np.linspace(min(x_observed), max(x_observed), num=100) + >>> y = barycentric_interpolate(x_observed, y_observed, x) + >>> plt.plot(x_observed, y_observed, "o", label="observation") + >>> plt.plot(x, y, label="barycentric interpolation") + >>> plt.legend() + >>> plt.show() + + """ + P = BarycentricInterpolator(xi, yi, axis=axis, rng=rng) + if der == 0: + return P(x) + elif _isscalar(der): + return P.derivative(x, der=der) + else: + return P.derivatives(x, der=np.amax(der)+1)[der] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_rbf.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_rbf.py new file mode 100644 index 0000000000000000000000000000000000000000..ed52230dd1cce678e56ca4427e10bafd07e501c0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_rbf.py @@ -0,0 +1,290 @@ +"""rbf - Radial basis functions for interpolation/smoothing scattered N-D data. + +Written by John Travers , February 2007 +Based closely on Matlab code by Alex Chirokov +Additional, large, improvements by Robert Hetland +Some additional alterations by Travis Oliphant +Interpolation with multi-dimensional target domain by Josua Sassen + +Permission to use, modify, and distribute this software is given under the +terms of the SciPy (BSD style) license. See LICENSE.txt that came with +this distribution for specifics. + +NO WARRANTY IS EXPRESSED OR IMPLIED. USE AT YOUR OWN RISK. + +Copyright (c) 2006-2007, Robert Hetland +Copyright (c) 2007, John Travers + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + * Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + * Redistributions in binary form must reproduce the above + copyright notice, this list of conditions and the following + disclaimer in the documentation and/or other materials provided + with the distribution. + + * Neither the name of Robert Hetland nor the names of any + contributors may be used to endorse or promote products derived + from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR +A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT +OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, +SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT +LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, +DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY +THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT +(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE +OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. +""" +import numpy as np + +from scipy import linalg +from scipy.special import xlogy +from scipy.spatial.distance import cdist, pdist, squareform + +__all__ = ['Rbf'] + + +class Rbf: + """ + Rbf(*args, **kwargs) + + A class for radial basis function interpolation of functions from + N-D scattered data to an M-D domain. + + .. legacy:: class + + `Rbf` is legacy code, for new usage please use `RBFInterpolator` + instead. + + Parameters + ---------- + *args : arrays + x, y, z, ..., d, where x, y, z, ... are the coordinates of the nodes + and d is the array of values at the nodes + function : str or callable, optional + The radial basis function, based on the radius, r, given by the norm + (default is Euclidean distance); the default is 'multiquadric':: + + 'multiquadric': sqrt((r/self.epsilon)**2 + 1) + 'inverse': 1.0/sqrt((r/self.epsilon)**2 + 1) + 'gaussian': exp(-(r/self.epsilon)**2) + 'linear': r + 'cubic': r**3 + 'quintic': r**5 + 'thin_plate': r**2 * log(r) + + If callable, then it must take 2 arguments (self, r). The epsilon + parameter will be available as self.epsilon. Other keyword + arguments passed in will be available as well. + + epsilon : float, optional + Adjustable constant for gaussian or multiquadrics functions + - defaults to approximate average distance between nodes (which is + a good start). + smooth : float, optional + Values greater than zero increase the smoothness of the + approximation. 0 is for interpolation (default), the function will + always go through the nodal points in this case. + norm : str, callable, optional + A function that returns the 'distance' between two points, with + inputs as arrays of positions (x, y, z, ...), and an output as an + array of distance. E.g., the default: 'euclidean', such that the result + is a matrix of the distances from each point in ``x1`` to each point in + ``x2``. For more options, see documentation of + `scipy.spatial.distances.cdist`. + mode : str, optional + Mode of the interpolation, can be '1-D' (default) or 'N-D'. When it is + '1-D' the data `d` will be considered as 1-D and flattened + internally. When it is 'N-D' the data `d` is assumed to be an array of + shape (n_samples, m), where m is the dimension of the target domain. + + + Attributes + ---------- + N : int + The number of data points (as determined by the input arrays). + di : ndarray + The 1-D array of data values at each of the data coordinates `xi`. + xi : ndarray + The 2-D array of data coordinates. + function : str or callable + The radial basis function. See description under Parameters. + epsilon : float + Parameter used by gaussian or multiquadrics functions. See Parameters. + smooth : float + Smoothing parameter. See description under Parameters. + norm : str or callable + The distance function. See description under Parameters. + mode : str + Mode of the interpolation. See description under Parameters. + nodes : ndarray + A 1-D array of node values for the interpolation. + A : internal property, do not use + + See Also + -------- + RBFInterpolator + + Examples + -------- + >>> import numpy as np + >>> from scipy.interpolate import Rbf + >>> rng = np.random.default_rng() + >>> x, y, z, d = rng.random((4, 50)) + >>> rbfi = Rbf(x, y, z, d) # radial basis function interpolator instance + >>> xi = yi = zi = np.linspace(0, 1, 20) + >>> di = rbfi(xi, yi, zi) # interpolated values + >>> di.shape + (20,) + + """ + # Available radial basis functions that can be selected as strings; + # they all start with _h_ (self._init_function relies on that) + def _h_multiquadric(self, r): + return np.sqrt((1.0/self.epsilon*r)**2 + 1) + + def _h_inverse_multiquadric(self, r): + return 1.0/np.sqrt((1.0/self.epsilon*r)**2 + 1) + + def _h_gaussian(self, r): + return np.exp(-(1.0/self.epsilon*r)**2) + + def _h_linear(self, r): + return r + + def _h_cubic(self, r): + return r**3 + + def _h_quintic(self, r): + return r**5 + + def _h_thin_plate(self, r): + return xlogy(r**2, r) + + # Setup self._function and do smoke test on initial r + def _init_function(self, r): + if isinstance(self.function, str): + self.function = self.function.lower() + _mapped = {'inverse': 'inverse_multiquadric', + 'inverse multiquadric': 'inverse_multiquadric', + 'thin-plate': 'thin_plate'} + if self.function in _mapped: + self.function = _mapped[self.function] + + func_name = "_h_" + self.function + if hasattr(self, func_name): + self._function = getattr(self, func_name) + else: + functionlist = [x[3:] for x in dir(self) + if x.startswith('_h_')] + raise ValueError("function must be a callable or one of " + + ", ".join(functionlist)) + self._function = getattr(self, "_h_"+self.function) + elif callable(self.function): + allow_one = False + if hasattr(self.function, 'func_code') or \ + hasattr(self.function, '__code__'): + val = self.function + allow_one = True + elif hasattr(self.function, "__call__"): + val = self.function.__call__.__func__ + else: + raise ValueError("Cannot determine number of arguments to " + "function") + + argcount = val.__code__.co_argcount + if allow_one and argcount == 1: + self._function = self.function + elif argcount == 2: + self._function = self.function.__get__(self, Rbf) + else: + raise ValueError("Function argument must take 1 or 2 " + "arguments.") + + a0 = self._function(r) + if a0.shape != r.shape: + raise ValueError("Callable must take array and return array of " + "the same shape") + return a0 + + def __init__(self, *args, **kwargs): + # `args` can be a variable number of arrays; we flatten them and store + # them as a single 2-D array `xi` of shape (n_args-1, array_size), + # plus a 1-D array `di` for the values. + # All arrays must have the same number of elements + self.xi = np.asarray([np.asarray(a, dtype=np.float64).flatten() + for a in args[:-1]]) + self.N = self.xi.shape[-1] + + self.mode = kwargs.pop('mode', '1-D') + + if self.mode == '1-D': + self.di = np.asarray(args[-1]).flatten() + self._target_dim = 1 + elif self.mode == 'N-D': + self.di = np.asarray(args[-1]) + self._target_dim = self.di.shape[-1] + else: + raise ValueError("Mode has to be 1-D or N-D.") + + if not all([x.size == self.di.shape[0] for x in self.xi]): + raise ValueError("All arrays must be equal length.") + + self.norm = kwargs.pop('norm', 'euclidean') + self.epsilon = kwargs.pop('epsilon', None) + if self.epsilon is None: + # default epsilon is the "the average distance between nodes" based + # on a bounding hypercube + ximax = np.amax(self.xi, axis=1) + ximin = np.amin(self.xi, axis=1) + edges = ximax - ximin + edges = edges[np.nonzero(edges)] + self.epsilon = np.power(np.prod(edges)/self.N, 1.0/edges.size) + + self.smooth = kwargs.pop('smooth', 0.0) + self.function = kwargs.pop('function', 'multiquadric') + + # attach anything left in kwargs to self for use by any user-callable + # function or to save on the object returned. + for item, value in kwargs.items(): + setattr(self, item, value) + + # Compute weights + if self._target_dim > 1: # If we have more than one target dimension, + # we first factorize the matrix + self.nodes = np.zeros((self.N, self._target_dim), dtype=self.di.dtype) + lu, piv = linalg.lu_factor(self.A) + for i in range(self._target_dim): + self.nodes[:, i] = linalg.lu_solve((lu, piv), self.di[:, i]) + else: + self.nodes = linalg.solve(self.A, self.di) + + @property + def A(self): + # this only exists for backwards compatibility: self.A was available + # and, at least technically, public. + r = squareform(pdist(self.xi.T, self.norm)) # Pairwise norm + return self._init_function(r) - np.eye(self.N)*self.smooth + + def _call_norm(self, x1, x2): + return cdist(x1.T, x2.T, self.norm) + + def __call__(self, *args): + args = [np.asarray(x) for x in args] + if not all([x.shape == y.shape for x in args for y in args]): + raise ValueError("Array lengths must be equal") + if self._target_dim > 1: + shp = args[0].shape + (self._target_dim,) + else: + shp = args[0].shape + xa = np.asarray([a.flatten() for a in args], dtype=np.float64) + r = self._call_norm(xa, self.xi) + return np.dot(self._function(r), self.nodes).reshape(shp) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_rbfinterp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_rbfinterp.py new file mode 100644 index 0000000000000000000000000000000000000000..6690e6ccf7d5499db10efffb0ef1c0139a90d2ba --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_rbfinterp.py @@ -0,0 +1,550 @@ +"""Module for RBF interpolation.""" +import warnings +from itertools import combinations_with_replacement + +import numpy as np +from numpy.linalg import LinAlgError +from scipy.spatial import KDTree +from scipy.special import comb +from scipy.linalg.lapack import dgesv # type: ignore[attr-defined] + +from ._rbfinterp_pythran import (_build_system, + _build_evaluation_coefficients, + _polynomial_matrix) + + +__all__ = ["RBFInterpolator"] + + +# These RBFs are implemented. +_AVAILABLE = { + "linear", + "thin_plate_spline", + "cubic", + "quintic", + "multiquadric", + "inverse_multiquadric", + "inverse_quadratic", + "gaussian" + } + + +# The shape parameter does not need to be specified when using these RBFs. +_SCALE_INVARIANT = {"linear", "thin_plate_spline", "cubic", "quintic"} + + +# For RBFs that are conditionally positive definite of order m, the interpolant +# should include polynomial terms with degree >= m - 1. Define the minimum +# degrees here. These values are from Chapter 8 of Fasshauer's "Meshfree +# Approximation Methods with MATLAB". The RBFs that are not in this dictionary +# are positive definite and do not need polynomial terms. +_NAME_TO_MIN_DEGREE = { + "multiquadric": 0, + "linear": 0, + "thin_plate_spline": 1, + "cubic": 1, + "quintic": 2 + } + + +def _monomial_powers(ndim, degree): + """Return the powers for each monomial in a polynomial. + + Parameters + ---------- + ndim : int + Number of variables in the polynomial. + degree : int + Degree of the polynomial. + + Returns + ------- + (nmonos, ndim) int ndarray + Array where each row contains the powers for each variable in a + monomial. + + """ + nmonos = comb(degree + ndim, ndim, exact=True) + out = np.zeros((nmonos, ndim), dtype=np.dtype("long")) + count = 0 + for deg in range(degree + 1): + for mono in combinations_with_replacement(range(ndim), deg): + # `mono` is a tuple of variables in the current monomial with + # multiplicity indicating power (e.g., (0, 1, 1) represents x*y**2) + for var in mono: + out[count, var] += 1 + + count += 1 + + return out + + +def _build_and_solve_system(y, d, smoothing, kernel, epsilon, powers): + """Build and solve the RBF interpolation system of equations. + + Parameters + ---------- + y : (P, N) float ndarray + Data point coordinates. + d : (P, S) float ndarray + Data values at `y`. + smoothing : (P,) float ndarray + Smoothing parameter for each data point. + kernel : str + Name of the RBF. + epsilon : float + Shape parameter. + powers : (R, N) int ndarray + The exponents for each monomial in the polynomial. + + Returns + ------- + coeffs : (P + R, S) float ndarray + Coefficients for each RBF and monomial. + shift : (N,) float ndarray + Domain shift used to create the polynomial matrix. + scale : (N,) float ndarray + Domain scaling used to create the polynomial matrix. + + """ + lhs, rhs, shift, scale = _build_system( + y, d, smoothing, kernel, epsilon, powers + ) + _, _, coeffs, info = dgesv(lhs, rhs, overwrite_a=True, overwrite_b=True) + if info < 0: + raise ValueError(f"The {-info}-th argument had an illegal value.") + elif info > 0: + msg = "Singular matrix." + nmonos = powers.shape[0] + if nmonos > 0: + pmat = _polynomial_matrix((y - shift)/scale, powers) + rank = np.linalg.matrix_rank(pmat) + if rank < nmonos: + msg = ( + "Singular matrix. The matrix of monomials evaluated at " + "the data point coordinates does not have full column " + f"rank ({rank}/{nmonos})." + ) + + raise LinAlgError(msg) + + return shift, scale, coeffs + + +class RBFInterpolator: + """Radial basis function (RBF) interpolation in N dimensions. + + Parameters + ---------- + y : (npoints, ndims) array_like + 2-D array of data point coordinates. + d : (npoints, ...) array_like + N-D array of data values at `y`. The length of `d` along the first + axis must be equal to the length of `y`. Unlike some interpolators, the + interpolation axis cannot be changed. + neighbors : int, optional + If specified, the value of the interpolant at each evaluation point + will be computed using only this many nearest data points. All the data + points are used by default. + smoothing : float or (npoints, ) array_like, optional + Smoothing parameter. The interpolant perfectly fits the data when this + is set to 0. For large values, the interpolant approaches a least + squares fit of a polynomial with the specified degree. Default is 0. + kernel : str, optional + Type of RBF. This should be one of + + - 'linear' : ``-r`` + - 'thin_plate_spline' : ``r**2 * log(r)`` + - 'cubic' : ``r**3`` + - 'quintic' : ``-r**5`` + - 'multiquadric' : ``-sqrt(1 + r**2)`` + - 'inverse_multiquadric' : ``1/sqrt(1 + r**2)`` + - 'inverse_quadratic' : ``1/(1 + r**2)`` + - 'gaussian' : ``exp(-r**2)`` + + Default is 'thin_plate_spline'. + epsilon : float, optional + Shape parameter that scales the input to the RBF. If `kernel` is + 'linear', 'thin_plate_spline', 'cubic', or 'quintic', this defaults to + 1 and can be ignored because it has the same effect as scaling the + smoothing parameter. Otherwise, this must be specified. + degree : int, optional + Degree of the added polynomial. For some RBFs the interpolant may not + be well-posed if the polynomial degree is too small. Those RBFs and + their corresponding minimum degrees are + + - 'multiquadric' : 0 + - 'linear' : 0 + - 'thin_plate_spline' : 1 + - 'cubic' : 1 + - 'quintic' : 2 + + The default value is the minimum degree for `kernel` or 0 if there is + no minimum degree. Set this to -1 for no added polynomial. + + Notes + ----- + An RBF is a scalar valued function in N-dimensional space whose value at + :math:`x` can be expressed in terms of :math:`r=||x - c||`, where :math:`c` + is the center of the RBF. + + An RBF interpolant for the vector of data values :math:`d`, which are from + locations :math:`y`, is a linear combination of RBFs centered at :math:`y` + plus a polynomial with a specified degree. The RBF interpolant is written + as + + .. math:: + f(x) = K(x, y) a + P(x) b, + + where :math:`K(x, y)` is a matrix of RBFs with centers at :math:`y` + evaluated at the points :math:`x`, and :math:`P(x)` is a matrix of + monomials, which span polynomials with the specified degree, evaluated at + :math:`x`. The coefficients :math:`a` and :math:`b` are the solution to the + linear equations + + .. math:: + (K(y, y) + \\lambda I) a + P(y) b = d + + and + + .. math:: + P(y)^T a = 0, + + where :math:`\\lambda` is a non-negative smoothing parameter that controls + how well we want to fit the data. The data are fit exactly when the + smoothing parameter is 0. + + The above system is uniquely solvable if the following requirements are + met: + + - :math:`P(y)` must have full column rank. :math:`P(y)` always has full + column rank when `degree` is -1 or 0. When `degree` is 1, + :math:`P(y)` has full column rank if the data point locations are not + all collinear (N=2), coplanar (N=3), etc. + - If `kernel` is 'multiquadric', 'linear', 'thin_plate_spline', + 'cubic', or 'quintic', then `degree` must not be lower than the + minimum value listed above. + - If `smoothing` is 0, then each data point location must be distinct. + + When using an RBF that is not scale invariant ('multiquadric', + 'inverse_multiquadric', 'inverse_quadratic', or 'gaussian'), an appropriate + shape parameter must be chosen (e.g., through cross validation). Smaller + values for the shape parameter correspond to wider RBFs. The problem can + become ill-conditioned or singular when the shape parameter is too small. + + The memory required to solve for the RBF interpolation coefficients + increases quadratically with the number of data points, which can become + impractical when interpolating more than about a thousand data points. + To overcome memory limitations for large interpolation problems, the + `neighbors` argument can be specified to compute an RBF interpolant for + each evaluation point using only the nearest data points. + + .. versionadded:: 1.7.0 + + See Also + -------- + NearestNDInterpolator + LinearNDInterpolator + CloughTocher2DInterpolator + + References + ---------- + .. [1] Fasshauer, G., 2007. Meshfree Approximation Methods with Matlab. + World Scientific Publishing Co. + + .. [2] http://amadeus.math.iit.edu/~fass/603_ch3.pdf + + .. [3] Wahba, G., 1990. Spline Models for Observational Data. SIAM. + + .. [4] http://pages.stat.wisc.edu/~wahba/stat860public/lect/lect8/lect8.pdf + + Examples + -------- + Demonstrate interpolating scattered data to a grid in 2-D. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.interpolate import RBFInterpolator + >>> from scipy.stats.qmc import Halton + + >>> rng = np.random.default_rng() + >>> xobs = 2*Halton(2, seed=rng).random(100) - 1 + >>> yobs = np.sum(xobs, axis=1)*np.exp(-6*np.sum(xobs**2, axis=1)) + + >>> xgrid = np.mgrid[-1:1:50j, -1:1:50j] + >>> xflat = xgrid.reshape(2, -1).T + >>> yflat = RBFInterpolator(xobs, yobs)(xflat) + >>> ygrid = yflat.reshape(50, 50) + + >>> fig, ax = plt.subplots() + >>> ax.pcolormesh(*xgrid, ygrid, vmin=-0.25, vmax=0.25, shading='gouraud') + >>> p = ax.scatter(*xobs.T, c=yobs, s=50, ec='k', vmin=-0.25, vmax=0.25) + >>> fig.colorbar(p) + >>> plt.show() + + """ + + def __init__(self, y, d, + neighbors=None, + smoothing=0.0, + kernel="thin_plate_spline", + epsilon=None, + degree=None): + y = np.asarray(y, dtype=float, order="C") + if y.ndim != 2: + raise ValueError("`y` must be a 2-dimensional array.") + + ny, ndim = y.shape + + d_dtype = complex if np.iscomplexobj(d) else float + d = np.asarray(d, dtype=d_dtype, order="C") + if d.shape[0] != ny: + raise ValueError( + f"Expected the first axis of `d` to have length {ny}." + ) + + d_shape = d.shape[1:] + d = d.reshape((ny, -1)) + # If `d` is complex, convert it to a float array with twice as many + # columns. Otherwise, the LHS matrix would need to be converted to + # complex and take up 2x more memory than necessary. + d = d.view(float) + + if np.isscalar(smoothing): + smoothing = np.full(ny, smoothing, dtype=float) + else: + smoothing = np.asarray(smoothing, dtype=float, order="C") + if smoothing.shape != (ny,): + raise ValueError( + "Expected `smoothing` to be a scalar or have shape " + f"({ny},)." + ) + + kernel = kernel.lower() + if kernel not in _AVAILABLE: + raise ValueError(f"`kernel` must be one of {_AVAILABLE}.") + + if epsilon is None: + if kernel in _SCALE_INVARIANT: + epsilon = 1.0 + else: + raise ValueError( + "`epsilon` must be specified if `kernel` is not one of " + f"{_SCALE_INVARIANT}." + ) + else: + epsilon = float(epsilon) + + min_degree = _NAME_TO_MIN_DEGREE.get(kernel, -1) + if degree is None: + degree = max(min_degree, 0) + else: + degree = int(degree) + if degree < -1: + raise ValueError("`degree` must be at least -1.") + elif -1 < degree < min_degree: + warnings.warn( + f"`degree` should not be below {min_degree} except -1 " + f"when `kernel` is '{kernel}'." + f"The interpolant may not be uniquely " + f"solvable, and the smoothing parameter may have an " + f"unintuitive effect.", + UserWarning, stacklevel=2 + ) + + if neighbors is None: + nobs = ny + else: + # Make sure the number of nearest neighbors used for interpolation + # does not exceed the number of observations. + neighbors = int(min(neighbors, ny)) + nobs = neighbors + + powers = _monomial_powers(ndim, degree) + # The polynomial matrix must have full column rank in order for the + # interpolant to be well-posed, which is not possible if there are + # fewer observations than monomials. + if powers.shape[0] > nobs: + raise ValueError( + f"At least {powers.shape[0]} data points are required when " + f"`degree` is {degree} and the number of dimensions is {ndim}." + ) + + if neighbors is None: + shift, scale, coeffs = _build_and_solve_system( + y, d, smoothing, kernel, epsilon, powers + ) + + # Make these attributes private since they do not always exist. + self._shift = shift + self._scale = scale + self._coeffs = coeffs + + else: + self._tree = KDTree(y) + + self.y = y + self.d = d + self.d_shape = d_shape + self.d_dtype = d_dtype + self.neighbors = neighbors + self.smoothing = smoothing + self.kernel = kernel + self.epsilon = epsilon + self.powers = powers + + def _chunk_evaluator( + self, + x, + y, + shift, + scale, + coeffs, + memory_budget=1000000 + ): + """ + Evaluate the interpolation while controlling memory consumption. + We chunk the input if we need more memory than specified. + + Parameters + ---------- + x : (Q, N) float ndarray + array of points on which to evaluate + y: (P, N) float ndarray + array of points on which we know function values + shift: (N, ) ndarray + Domain shift used to create the polynomial matrix. + scale : (N,) float ndarray + Domain scaling used to create the polynomial matrix. + coeffs: (P+R, S) float ndarray + Coefficients in front of basis functions + memory_budget: int + Total amount of memory (in units of sizeof(float)) we wish + to devote for storing the array of coefficients for + interpolated points. If we need more memory than that, we + chunk the input. + + Returns + ------- + (Q, S) float ndarray + Interpolated array + """ + nx, ndim = x.shape + if self.neighbors is None: + nnei = len(y) + else: + nnei = self.neighbors + # in each chunk we consume the same space we already occupy + chunksize = memory_budget // (self.powers.shape[0] + nnei) + 1 + if chunksize <= nx: + out = np.empty((nx, self.d.shape[1]), dtype=float) + for i in range(0, nx, chunksize): + vec = _build_evaluation_coefficients( + x[i:i + chunksize, :], + y, + self.kernel, + self.epsilon, + self.powers, + shift, + scale) + out[i:i + chunksize, :] = np.dot(vec, coeffs) + else: + vec = _build_evaluation_coefficients( + x, + y, + self.kernel, + self.epsilon, + self.powers, + shift, + scale) + out = np.dot(vec, coeffs) + return out + + def __call__(self, x): + """Evaluate the interpolant at `x`. + + Parameters + ---------- + x : (Q, N) array_like + Evaluation point coordinates. + + Returns + ------- + (Q, ...) ndarray + Values of the interpolant at `x`. + + """ + x = np.asarray(x, dtype=float, order="C") + if x.ndim != 2: + raise ValueError("`x` must be a 2-dimensional array.") + + nx, ndim = x.shape + if ndim != self.y.shape[1]: + raise ValueError("Expected the second axis of `x` to have length " + f"{self.y.shape[1]}.") + + # Our memory budget for storing RBF coefficients is + # based on how many floats in memory we already occupy + # If this number is below 1e6 we just use 1e6 + # This memory budget is used to decide how we chunk + # the inputs + memory_budget = max(x.size + self.y.size + self.d.size, 1000000) + + if self.neighbors is None: + out = self._chunk_evaluator( + x, + self.y, + self._shift, + self._scale, + self._coeffs, + memory_budget=memory_budget) + else: + # Get the indices of the k nearest observation points to each + # evaluation point. + _, yindices = self._tree.query(x, self.neighbors) + if self.neighbors == 1: + # `KDTree` squeezes the output when neighbors=1. + yindices = yindices[:, None] + + # Multiple evaluation points may have the same neighborhood of + # observation points. Make the neighborhoods unique so that we only + # compute the interpolation coefficients once for each + # neighborhood. + yindices = np.sort(yindices, axis=1) + yindices, inv = np.unique(yindices, return_inverse=True, axis=0) + inv = np.reshape(inv, (-1,)) # flatten, we need 1-D indices + # `inv` tells us which neighborhood will be used by each evaluation + # point. Now we find which evaluation points will be using each + # neighborhood. + xindices = [[] for _ in range(len(yindices))] + for i, j in enumerate(inv): + xindices[j].append(i) + + out = np.empty((nx, self.d.shape[1]), dtype=float) + for xidx, yidx in zip(xindices, yindices): + # `yidx` are the indices of the observations in this + # neighborhood. `xidx` are the indices of the evaluation points + # that are using this neighborhood. + xnbr = x[xidx] + ynbr = self.y[yidx] + dnbr = self.d[yidx] + snbr = self.smoothing[yidx] + shift, scale, coeffs = _build_and_solve_system( + ynbr, + dnbr, + snbr, + self.kernel, + self.epsilon, + self.powers, + ) + out[xidx] = self._chunk_evaluator( + xnbr, + ynbr, + shift, + scale, + coeffs, + memory_budget=memory_budget) + + out = out.view(self.d_dtype) + out = out.reshape((nx, ) + self.d_shape) + return out diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_rgi.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_rgi.py new file mode 100644 index 0000000000000000000000000000000000000000..8e20200568ed961849b5510e8626cdbe6e5b9643 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/_rgi.py @@ -0,0 +1,759 @@ +__all__ = ['RegularGridInterpolator', 'interpn'] + +import itertools + +import numpy as np + +import scipy.sparse.linalg as ssl + +from ._interpnd import _ndim_coords_from_arrays +from ._cubic import PchipInterpolator +from ._rgi_cython import evaluate_linear_2d, find_indices +from ._bsplines import make_interp_spline +from ._fitpack2 import RectBivariateSpline +from ._ndbspline import make_ndbspl + + +def _check_points(points): + descending_dimensions = [] + grid = [] + for i, p in enumerate(points): + # early make points float + # see https://github.com/scipy/scipy/pull/17230 + p = np.asarray(p, dtype=float) + if not np.all(p[1:] > p[:-1]): + if np.all(p[1:] < p[:-1]): + # input is descending, so make it ascending + descending_dimensions.append(i) + p = np.flip(p) + else: + raise ValueError( + "The points in dimension %d must be strictly " + "ascending or descending" % i) + # see https://github.com/scipy/scipy/issues/17716 + p = np.ascontiguousarray(p) + grid.append(p) + return tuple(grid), tuple(descending_dimensions) + + +def _check_dimensionality(points, values): + if len(points) > values.ndim: + raise ValueError("There are %d point arrays, but values has %d " + "dimensions" % (len(points), values.ndim)) + for i, p in enumerate(points): + if not np.asarray(p).ndim == 1: + raise ValueError("The points in dimension %d must be " + "1-dimensional" % i) + if not values.shape[i] == len(p): + raise ValueError("There are %d points and %d values in " + "dimension %d" % (len(p), values.shape[i], i)) + + +class RegularGridInterpolator: + """ + Interpolator on a regular or rectilinear grid in arbitrary dimensions. + + The data must be defined on a rectilinear grid; that is, a rectangular + grid with even or uneven spacing. Linear, nearest-neighbor, spline + interpolations are supported. After setting up the interpolator object, + the interpolation method may be chosen at each evaluation. + + Parameters + ---------- + points : tuple of ndarray of float, with shapes (m1, ), ..., (mn, ) + The points defining the regular grid in n dimensions. The points in + each dimension (i.e. every elements of the points tuple) must be + strictly ascending or descending. + + values : array_like, shape (m1, ..., mn, ...) + The data on the regular grid in n dimensions. Complex data is + accepted. + + method : str, optional + The method of interpolation to perform. Supported are "linear", + "nearest", "slinear", "cubic", "quintic" and "pchip". This + parameter will become the default for the object's ``__call__`` + method. Default is "linear". + + bounds_error : bool, optional + If True, when interpolated values are requested outside of the + domain of the input data, a ValueError is raised. + If False, then `fill_value` is used. + Default is True. + + fill_value : float or None, optional + The value to use for points outside of the interpolation domain. + If None, values outside the domain are extrapolated. + Default is ``np.nan``. + + solver : callable, optional + Only used for methods "slinear", "cubic" and "quintic". + Sparse linear algebra solver for construction of the NdBSpline instance. + Default is the iterative solver `scipy.sparse.linalg.gcrotmk`. + + .. versionadded:: 1.13 + + solver_args: dict, optional + Additional arguments to pass to `solver`, if any. + + .. versionadded:: 1.13 + + Methods + ------- + __call__ + + Attributes + ---------- + grid : tuple of ndarrays + The points defining the regular grid in n dimensions. + This tuple defines the full grid via + ``np.meshgrid(*grid, indexing='ij')`` + values : ndarray + Data values at the grid. + method : str + Interpolation method. + fill_value : float or ``None`` + Use this value for out-of-bounds arguments to `__call__`. + bounds_error : bool + If ``True``, out-of-bounds argument raise a ``ValueError``. + + Notes + ----- + Contrary to `LinearNDInterpolator` and `NearestNDInterpolator`, this class + avoids expensive triangulation of the input data by taking advantage of the + regular grid structure. + + In other words, this class assumes that the data is defined on a + *rectilinear* grid. + + .. versionadded:: 0.14 + + The 'slinear'(k=1), 'cubic'(k=3), and 'quintic'(k=5) methods are + tensor-product spline interpolators, where `k` is the spline degree, + If any dimension has fewer points than `k` + 1, an error will be raised. + + .. versionadded:: 1.9 + + If the input data is such that dimensions have incommensurate + units and differ by many orders of magnitude, the interpolant may have + numerical artifacts. Consider rescaling the data before interpolating. + + **Choosing a solver for spline methods** + + Spline methods, "slinear", "cubic" and "quintic" involve solving a + large sparse linear system at instantiation time. Depending on data, + the default solver may or may not be adequate. When it is not, you may + need to experiment with an optional `solver` argument, where you may + choose between the direct solver (`scipy.sparse.linalg.spsolve`) or + iterative solvers from `scipy.sparse.linalg`. You may need to supply + additional parameters via the optional `solver_args` parameter (for instance, + you may supply the starting value or target tolerance). See the + `scipy.sparse.linalg` documentation for the full list of available options. + + Alternatively, you may instead use the legacy methods, "slinear_legacy", + "cubic_legacy" and "quintic_legacy". These methods allow faster construction + but evaluations will be much slower. + + Examples + -------- + **Evaluate a function on the points of a 3-D grid** + + As a first example, we evaluate a simple example function on the points of + a 3-D grid: + + >>> from scipy.interpolate import RegularGridInterpolator + >>> import numpy as np + >>> def f(x, y, z): + ... return 2 * x**3 + 3 * y**2 - z + >>> x = np.linspace(1, 4, 11) + >>> y = np.linspace(4, 7, 22) + >>> z = np.linspace(7, 9, 33) + >>> xg, yg ,zg = np.meshgrid(x, y, z, indexing='ij', sparse=True) + >>> data = f(xg, yg, zg) + + ``data`` is now a 3-D array with ``data[i, j, k] = f(x[i], y[j], z[k])``. + Next, define an interpolating function from this data: + + >>> interp = RegularGridInterpolator((x, y, z), data) + + Evaluate the interpolating function at the two points + ``(x,y,z) = (2.1, 6.2, 8.3)`` and ``(3.3, 5.2, 7.1)``: + + >>> pts = np.array([[2.1, 6.2, 8.3], + ... [3.3, 5.2, 7.1]]) + >>> interp(pts) + array([ 125.80469388, 146.30069388]) + + which is indeed a close approximation to + + >>> f(2.1, 6.2, 8.3), f(3.3, 5.2, 7.1) + (125.54200000000002, 145.894) + + **Interpolate and extrapolate a 2D dataset** + + As a second example, we interpolate and extrapolate a 2D data set: + + >>> x, y = np.array([-2, 0, 4]), np.array([-2, 0, 2, 5]) + >>> def ff(x, y): + ... return x**2 + y**2 + + >>> xg, yg = np.meshgrid(x, y, indexing='ij') + >>> data = ff(xg, yg) + >>> interp = RegularGridInterpolator((x, y), data, + ... bounds_error=False, fill_value=None) + + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> ax = fig.add_subplot(projection='3d') + >>> ax.scatter(xg.ravel(), yg.ravel(), data.ravel(), + ... s=60, c='k', label='data') + + Evaluate and plot the interpolator on a finer grid + + >>> xx = np.linspace(-4, 9, 31) + >>> yy = np.linspace(-4, 9, 31) + >>> X, Y = np.meshgrid(xx, yy, indexing='ij') + + >>> # interpolator + >>> ax.plot_wireframe(X, Y, interp((X, Y)), rstride=3, cstride=3, + ... alpha=0.4, color='m', label='linear interp') + + >>> # ground truth + >>> ax.plot_wireframe(X, Y, ff(X, Y), rstride=3, cstride=3, + ... alpha=0.4, label='ground truth') + >>> plt.legend() + >>> plt.show() + + Other examples are given + :ref:`in the tutorial `. + + See Also + -------- + NearestNDInterpolator : Nearest neighbor interpolator on *unstructured* + data in N dimensions + + LinearNDInterpolator : Piecewise linear interpolator on *unstructured* data + in N dimensions + + interpn : a convenience function which wraps `RegularGridInterpolator` + + scipy.ndimage.map_coordinates : interpolation on grids with equal spacing + (suitable for e.g., N-D image resampling) + + References + ---------- + .. [1] Python package *regulargrid* by Johannes Buchner, see + https://pypi.python.org/pypi/regulargrid/ + .. [2] Wikipedia, "Trilinear interpolation", + https://en.wikipedia.org/wiki/Trilinear_interpolation + .. [3] Weiser, Alan, and Sergio E. Zarantonello. "A note on piecewise linear + and multilinear table interpolation in many dimensions." MATH. + COMPUT. 50.181 (1988): 189-196. + https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917826-0/S0025-5718-1988-0917826-0.pdf + :doi:`10.1090/S0025-5718-1988-0917826-0` + + """ + # this class is based on code originally programmed by Johannes Buchner, + # see https://github.com/JohannesBuchner/regulargrid + + _SPLINE_DEGREE_MAP = {"slinear": 1, "cubic": 3, "quintic": 5, 'pchip': 3, + "slinear_legacy": 1, "cubic_legacy": 3, "quintic_legacy": 5,} + _SPLINE_METHODS_recursive = {"slinear_legacy", "cubic_legacy", + "quintic_legacy", "pchip"} + _SPLINE_METHODS_ndbspl = {"slinear", "cubic", "quintic"} + _SPLINE_METHODS = list(_SPLINE_DEGREE_MAP.keys()) + _ALL_METHODS = ["linear", "nearest"] + _SPLINE_METHODS + + def __init__(self, points, values, method="linear", bounds_error=True, + fill_value=np.nan, *, solver=None, solver_args=None): + if method not in self._ALL_METHODS: + raise ValueError(f"Method '{method}' is not defined") + elif method in self._SPLINE_METHODS: + self._validate_grid_dimensions(points, method) + self.method = method + self._spline = None + self.bounds_error = bounds_error + self.grid, self._descending_dimensions = _check_points(points) + self.values = self._check_values(values) + self._check_dimensionality(self.grid, self.values) + self.fill_value = self._check_fill_value(self.values, fill_value) + if self._descending_dimensions: + self.values = np.flip(values, axis=self._descending_dimensions) + if self.method == "pchip" and np.iscomplexobj(self.values): + msg = ("`PchipInterpolator` only works with real values. If you are trying " + "to use the real components of the passed array, use `np.real` on " + "the array before passing to `RegularGridInterpolator`.") + raise ValueError(msg) + if method in self._SPLINE_METHODS_ndbspl: + if solver_args is None: + solver_args = {} + self._spline = self._construct_spline(method, solver, **solver_args) + else: + if solver is not None or solver_args: + raise ValueError( + f"{method =} does not accept the 'solver' argument. Got " + f" {solver = } and with arguments {solver_args}." + ) + + def _construct_spline(self, method, solver=None, **solver_args): + if solver is None: + solver = ssl.gcrotmk + spl = make_ndbspl( + self.grid, self.values, self._SPLINE_DEGREE_MAP[method], + solver=solver, **solver_args + ) + return spl + + def _check_dimensionality(self, grid, values): + _check_dimensionality(grid, values) + + def _check_points(self, points): + return _check_points(points) + + def _check_values(self, values): + if not hasattr(values, 'ndim'): + # allow reasonable duck-typed values + values = np.asarray(values) + + if hasattr(values, 'dtype') and hasattr(values, 'astype'): + if not np.issubdtype(values.dtype, np.inexact): + values = values.astype(float) + + return values + + def _check_fill_value(self, values, fill_value): + if fill_value is not None: + fill_value_dtype = np.asarray(fill_value).dtype + if (hasattr(values, 'dtype') and not + np.can_cast(fill_value_dtype, values.dtype, + casting='same_kind')): + raise ValueError("fill_value must be either 'None' or " + "of a type compatible with values") + return fill_value + + def __call__(self, xi, method=None, *, nu=None): + """ + Interpolation at coordinates. + + Parameters + ---------- + xi : ndarray of shape (..., ndim) + The coordinates to evaluate the interpolator at. + + method : str, optional + The method of interpolation to perform. Supported are "linear", + "nearest", "slinear", "cubic", "quintic" and "pchip". Default is + the method chosen when the interpolator was created. + + nu : sequence of ints, length ndim, optional + If not None, the orders of the derivatives to evaluate. + Each entry must be non-negative. + Only allowed for methods "slinear", "cubic" and "quintic". + + .. versionadded:: 1.13 + + Returns + ------- + values_x : ndarray, shape xi.shape[:-1] + values.shape[ndim:] + Interpolated values at `xi`. See notes for behaviour when + ``xi.ndim == 1``. + + Notes + ----- + In the case that ``xi.ndim == 1`` a new axis is inserted into + the 0 position of the returned array, values_x, so its shape is + instead ``(1,) + values.shape[ndim:]``. + + Examples + -------- + Here we define a nearest-neighbor interpolator of a simple function + + >>> import numpy as np + >>> x, y = np.array([0, 1, 2]), np.array([1, 3, 7]) + >>> def f(x, y): + ... return x**2 + y**2 + >>> data = f(*np.meshgrid(x, y, indexing='ij', sparse=True)) + >>> from scipy.interpolate import RegularGridInterpolator + >>> interp = RegularGridInterpolator((x, y), data, method='nearest') + + By construction, the interpolator uses the nearest-neighbor + interpolation + + >>> interp([[1.5, 1.3], [0.3, 4.5]]) + array([2., 9.]) + + We can however evaluate the linear interpolant by overriding the + `method` parameter + + >>> interp([[1.5, 1.3], [0.3, 4.5]], method='linear') + array([ 4.7, 24.3]) + """ + _spline = self._spline + method = self.method if method is None else method + is_method_changed = self.method != method + if method not in self._ALL_METHODS: + raise ValueError(f"Method '{method}' is not defined") + if is_method_changed and method in self._SPLINE_METHODS_ndbspl: + _spline = self._construct_spline(method) + + if nu is not None and method not in self._SPLINE_METHODS_ndbspl: + raise ValueError( + f"Can only compute derivatives for methods " + f"{self._SPLINE_METHODS_ndbspl}, got {method =}." + ) + + xi, xi_shape, ndim, nans, out_of_bounds = self._prepare_xi(xi) + + if method == "linear": + indices, norm_distances = self._find_indices(xi.T) + if (ndim == 2 and hasattr(self.values, 'dtype') and + self.values.ndim == 2 and self.values.flags.writeable and + self.values.dtype in (np.float64, np.complex128) and + self.values.dtype.byteorder == '='): + # until cython supports const fused types, the fast path + # cannot support non-writeable values + # a fast path + out = np.empty(indices.shape[1], dtype=self.values.dtype) + result = evaluate_linear_2d(self.values, + indices, + norm_distances, + self.grid, + out) + else: + result = self._evaluate_linear(indices, norm_distances) + elif method == "nearest": + indices, norm_distances = self._find_indices(xi.T) + result = self._evaluate_nearest(indices, norm_distances) + elif method in self._SPLINE_METHODS: + if is_method_changed: + self._validate_grid_dimensions(self.grid, method) + if method in self._SPLINE_METHODS_recursive: + result = self._evaluate_spline(xi, method) + else: + result = _spline(xi, nu=nu) + + if not self.bounds_error and self.fill_value is not None: + result[out_of_bounds] = self.fill_value + + # f(nan) = nan, if any + if np.any(nans): + result[nans] = np.nan + return result.reshape(xi_shape[:-1] + self.values.shape[ndim:]) + + def _prepare_xi(self, xi): + ndim = len(self.grid) + xi = _ndim_coords_from_arrays(xi, ndim=ndim) + if xi.shape[-1] != len(self.grid): + raise ValueError("The requested sample points xi have dimension " + f"{xi.shape[-1]} but this " + f"RegularGridInterpolator has dimension {ndim}") + + xi_shape = xi.shape + xi = xi.reshape(-1, xi_shape[-1]) + xi = np.asarray(xi, dtype=float) + + # find nans in input + nans = np.any(np.isnan(xi), axis=-1) + + if self.bounds_error: + for i, p in enumerate(xi.T): + if not np.logical_and(np.all(self.grid[i][0] <= p), + np.all(p <= self.grid[i][-1])): + raise ValueError("One of the requested xi is out of bounds " + "in dimension %d" % i) + out_of_bounds = None + else: + out_of_bounds = self._find_out_of_bounds(xi.T) + + return xi, xi_shape, ndim, nans, out_of_bounds + + def _evaluate_linear(self, indices, norm_distances): + # slice for broadcasting over trailing dimensions in self.values + vslice = (slice(None),) + (None,)*(self.values.ndim - len(indices)) + + # Compute shifting up front before zipping everything together + shift_norm_distances = [1 - yi for yi in norm_distances] + shift_indices = [i + 1 for i in indices] + + # The formula for linear interpolation in 2d takes the form: + # values = self.values[(i0, i1)] * (1 - y0) * (1 - y1) + \ + # self.values[(i0, i1 + 1)] * (1 - y0) * y1 + \ + # self.values[(i0 + 1, i1)] * y0 * (1 - y1) + \ + # self.values[(i0 + 1, i1 + 1)] * y0 * y1 + # We pair i with 1 - yi (zipped1) and i + 1 with yi (zipped2) + zipped1 = zip(indices, shift_norm_distances) + zipped2 = zip(shift_indices, norm_distances) + + # Take all products of zipped1 and zipped2 and iterate over them + # to get the terms in the above formula. This corresponds to iterating + # over the vertices of a hypercube. + hypercube = itertools.product(*zip(zipped1, zipped2)) + value = np.array([0.]) + for h in hypercube: + edge_indices, weights = zip(*h) + weight = np.array([1.]) + for w in weights: + weight = weight * w + term = np.asarray(self.values[edge_indices]) * weight[vslice] + value = value + term # cannot use += because broadcasting + return value + + def _evaluate_nearest(self, indices, norm_distances): + idx_res = [np.where(yi <= .5, i, i + 1) + for i, yi in zip(indices, norm_distances)] + return self.values[tuple(idx_res)] + + def _validate_grid_dimensions(self, points, method): + k = self._SPLINE_DEGREE_MAP[method] + for i, point in enumerate(points): + ndim = len(np.atleast_1d(point)) + if ndim <= k: + raise ValueError(f"There are {ndim} points in dimension {i}," + f" but method {method} requires at least " + f" {k+1} points per dimension.") + + def _evaluate_spline(self, xi, method): + # ensure xi is 2D list of points to evaluate (`m` is the number of + # points and `n` is the number of interpolation dimensions, + # ``n == len(self.grid)``.) + if xi.ndim == 1: + xi = xi.reshape((1, xi.size)) + m, n = xi.shape + + # Reorder the axes: n-dimensional process iterates over the + # interpolation axes from the last axis downwards: E.g. for a 4D grid + # the order of axes is 3, 2, 1, 0. Each 1D interpolation works along + # the 0th axis of its argument array (for 1D routine it's its ``y`` + # array). Thus permute the interpolation axes of `values` *and keep + # trailing dimensions trailing*. + axes = tuple(range(self.values.ndim)) + axx = axes[:n][::-1] + axes[n:] + values = self.values.transpose(axx) + + if method == 'pchip': + _eval_func = self._do_pchip + else: + _eval_func = self._do_spline_fit + k = self._SPLINE_DEGREE_MAP[method] + + # Non-stationary procedure: difficult to vectorize this part entirely + # into numpy-level operations. Unfortunately this requires explicit + # looping over each point in xi. + + # can at least vectorize the first pass across all points in the + # last variable of xi. + last_dim = n - 1 + first_values = _eval_func(self.grid[last_dim], + values, + xi[:, last_dim], + k) + + # the rest of the dimensions have to be on a per point-in-xi basis + shape = (m, *self.values.shape[n:]) + result = np.empty(shape, dtype=self.values.dtype) + for j in range(m): + # Main process: Apply 1D interpolate in each dimension + # sequentially, starting with the last dimension. + # These are then "folded" into the next dimension in-place. + folded_values = first_values[j, ...] + for i in range(last_dim-1, -1, -1): + # Interpolate for each 1D from the last dimensions. + # This collapses each 1D sequence into a scalar. + folded_values = _eval_func(self.grid[i], + folded_values, + xi[j, i], + k) + result[j, ...] = folded_values + + return result + + @staticmethod + def _do_spline_fit(x, y, pt, k): + local_interp = make_interp_spline(x, y, k=k, axis=0) + values = local_interp(pt) + return values + + @staticmethod + def _do_pchip(x, y, pt, k): + local_interp = PchipInterpolator(x, y, axis=0) + values = local_interp(pt) + return values + + def _find_indices(self, xi): + return find_indices(self.grid, xi) + + def _find_out_of_bounds(self, xi): + # check for out of bounds xi + out_of_bounds = np.zeros((xi.shape[1]), dtype=bool) + # iterate through dimensions + for x, grid in zip(xi, self.grid): + out_of_bounds += x < grid[0] + out_of_bounds += x > grid[-1] + return out_of_bounds + + +def interpn(points, values, xi, method="linear", bounds_error=True, + fill_value=np.nan): + """ + Multidimensional interpolation on regular or rectilinear grids. + + Strictly speaking, not all regular grids are supported - this function + works on *rectilinear* grids, that is, a rectangular grid with even or + uneven spacing. + + Parameters + ---------- + points : tuple of ndarray of float, with shapes (m1, ), ..., (mn, ) + The points defining the regular grid in n dimensions. The points in + each dimension (i.e. every elements of the points tuple) must be + strictly ascending or descending. + + values : array_like, shape (m1, ..., mn, ...) + The data on the regular grid in n dimensions. Complex data is + accepted. + + .. deprecated:: 1.13.0 + Complex data is deprecated with ``method="pchip"`` and will raise an + error in SciPy 1.15.0. This is because ``PchipInterpolator`` only + works with real values. If you are trying to use the real components of + the passed array, use ``np.real`` on ``values``. + + xi : ndarray of shape (..., ndim) + The coordinates to sample the gridded data at + + method : str, optional + The method of interpolation to perform. Supported are "linear", + "nearest", "slinear", "cubic", "quintic", "pchip", and "splinef2d". + "splinef2d" is only supported for 2-dimensional data. + + bounds_error : bool, optional + If True, when interpolated values are requested outside of the + domain of the input data, a ValueError is raised. + If False, then `fill_value` is used. + + fill_value : number, optional + If provided, the value to use for points outside of the + interpolation domain. If None, values outside + the domain are extrapolated. Extrapolation is not supported by method + "splinef2d". + + Returns + ------- + values_x : ndarray, shape xi.shape[:-1] + values.shape[ndim:] + Interpolated values at `xi`. See notes for behaviour when + ``xi.ndim == 1``. + + See Also + -------- + NearestNDInterpolator : Nearest neighbor interpolation on unstructured + data in N dimensions + LinearNDInterpolator : Piecewise linear interpolant on unstructured data + in N dimensions + RegularGridInterpolator : interpolation on a regular or rectilinear grid + in arbitrary dimensions (`interpn` wraps this + class). + RectBivariateSpline : Bivariate spline approximation over a rectangular mesh + scipy.ndimage.map_coordinates : interpolation on grids with equal spacing + (suitable for e.g., N-D image resampling) + + Notes + ----- + + .. versionadded:: 0.14 + + In the case that ``xi.ndim == 1`` a new axis is inserted into + the 0 position of the returned array, values_x, so its shape is + instead ``(1,) + values.shape[ndim:]``. + + If the input data is such that input dimensions have incommensurate + units and differ by many orders of magnitude, the interpolant may have + numerical artifacts. Consider rescaling the data before interpolation. + + Examples + -------- + Evaluate a simple example function on the points of a regular 3-D grid: + + >>> import numpy as np + >>> from scipy.interpolate import interpn + >>> def value_func_3d(x, y, z): + ... return 2 * x + 3 * y - z + >>> x = np.linspace(0, 4, 5) + >>> y = np.linspace(0, 5, 6) + >>> z = np.linspace(0, 6, 7) + >>> points = (x, y, z) + >>> values = value_func_3d(*np.meshgrid(*points, indexing='ij')) + + Evaluate the interpolating function at a point + + >>> point = np.array([2.21, 3.12, 1.15]) + >>> print(interpn(points, values, point)) + [12.63] + + """ + # sanity check 'method' kwarg + if method not in ["linear", "nearest", "cubic", "quintic", "pchip", + "splinef2d", "slinear", + "slinear_legacy", "cubic_legacy", "quintic_legacy"]: + raise ValueError("interpn only understands the methods 'linear', " + "'nearest', 'slinear', 'cubic', 'quintic', 'pchip', " + f"and 'splinef2d'. You provided {method}.") + + if not hasattr(values, 'ndim'): + values = np.asarray(values) + + ndim = values.ndim + if ndim > 2 and method == "splinef2d": + raise ValueError("The method splinef2d can only be used for " + "2-dimensional input data") + if not bounds_error and fill_value is None and method == "splinef2d": + raise ValueError("The method splinef2d does not support extrapolation.") + + # sanity check consistency of input dimensions + if len(points) > ndim: + raise ValueError("There are %d point arrays, but values has %d " + "dimensions" % (len(points), ndim)) + if len(points) != ndim and method == 'splinef2d': + raise ValueError("The method splinef2d can only be used for " + "scalar data with one point per coordinate") + + grid, descending_dimensions = _check_points(points) + _check_dimensionality(grid, values) + + # sanity check requested xi + xi = _ndim_coords_from_arrays(xi, ndim=len(grid)) + if xi.shape[-1] != len(grid): + raise ValueError("The requested sample points xi have dimension " + "%d, but this RegularGridInterpolator has " + "dimension %d" % (xi.shape[-1], len(grid))) + + if bounds_error: + for i, p in enumerate(xi.T): + if not np.logical_and(np.all(grid[i][0] <= p), + np.all(p <= grid[i][-1])): + raise ValueError("One of the requested xi is out of bounds " + "in dimension %d" % i) + + # perform interpolation + if method in RegularGridInterpolator._ALL_METHODS: + interp = RegularGridInterpolator(points, values, method=method, + bounds_error=bounds_error, + fill_value=fill_value) + return interp(xi) + elif method == "splinef2d": + xi_shape = xi.shape + xi = xi.reshape(-1, xi.shape[-1]) + + # RectBivariateSpline doesn't support fill_value; we need to wrap here + idx_valid = np.all((grid[0][0] <= xi[:, 0], xi[:, 0] <= grid[0][-1], + grid[1][0] <= xi[:, 1], xi[:, 1] <= grid[1][-1]), + axis=0) + result = np.empty_like(xi[:, 0]) + + # make a copy of values for RectBivariateSpline + interp = RectBivariateSpline(points[0], points[1], values[:]) + result[idx_valid] = interp.ev(xi[idx_valid, 0], xi[idx_valid, 1]) + result[np.logical_not(idx_valid)] = fill_value + + return result.reshape(xi_shape[:-1]) + else: + raise ValueError(f"unknown {method = }") diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/dfitpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/dfitpack.py new file mode 100644 index 0000000000000000000000000000000000000000..e10da3b3fd0c69dede4767dc17b62c327818ecce --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/dfitpack.py @@ -0,0 +1,44 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.interpolate` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'bispeu', + 'bispev', + 'curfit', + 'dblint', + 'fpchec', + 'fpcurf0', + 'fpcurf1', + 'fpcurfm1', + 'parcur', + 'parder', + 'pardeu', + 'pardtc', + 'percur', + 'regrid_smth', + 'regrid_smth_spher', + 'spalde', + 'spherfit_lsq', + 'spherfit_smth', + 'splder', + 'splev', + 'splint', + 'sproot', + 'surfit_lsq', + 'surfit_smth', + 'types', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="interpolate", module="dfitpack", + private_modules=["_dfitpack"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/fitpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/fitpack.py new file mode 100644 index 0000000000000000000000000000000000000000..6490c93fe02b4c665b032d09e2ad3c269e1f7970 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/fitpack.py @@ -0,0 +1,31 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.interpolate` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'BSpline', + 'bisplev', + 'bisplrep', + 'insert', + 'spalde', + 'splantider', + 'splder', + 'splev', + 'splint', + 'splprep', + 'splrep', + 'sproot', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="interpolate", module="fitpack", + private_modules=["_fitpack_py"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/fitpack2.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/fitpack2.py new file mode 100644 index 0000000000000000000000000000000000000000..f993961f94d913d632aa3d2cc7b1348659a6a613 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/fitpack2.py @@ -0,0 +1,29 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.interpolate` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'BivariateSpline', + 'InterpolatedUnivariateSpline', + 'LSQBivariateSpline', + 'LSQSphereBivariateSpline', + 'LSQUnivariateSpline', + 'RectBivariateSpline', + 'RectSphereBivariateSpline', + 'SmoothBivariateSpline', + 'SmoothSphereBivariateSpline', + 'UnivariateSpline', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="interpolate", module="fitpack2", + private_modules=["_fitpack2"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/interpnd.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/interpnd.py new file mode 100644 index 0000000000000000000000000000000000000000..4288ac233fdde98dbb19aed84b916cfd15302f4c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/interpnd.py @@ -0,0 +1,25 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.interpolate` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'CloughTocher2DInterpolator', + 'GradientEstimationWarning', + 'LinearNDInterpolator', + 'NDInterpolatorBase', + 'estimate_gradients_2d_global', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="interpolate", module="interpnd", + private_modules=["_interpnd"], all=__all__, + attribute=name) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/interpolate.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/interpolate.py new file mode 100644 index 0000000000000000000000000000000000000000..341d13954c81130cceb8afe070db023a82550e7a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/interpolate.py @@ -0,0 +1,30 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.interpolate` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'BPoly', + 'BSpline', + 'NdPPoly', + 'PPoly', + 'RectBivariateSpline', + 'RegularGridInterpolator', + 'interp1d', + 'interp2d', + 'interpn', + 'lagrange', + 'make_interp_spline', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="interpolate", module="interpolate", + private_modules=["_interpolate", "fitpack2", "_rgi"], + all=__all__, attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/ndgriddata.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/ndgriddata.py new file mode 100644 index 0000000000000000000000000000000000000000..20373eaaedaa1cdec6c7a4bc12639d9658bfa85b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/ndgriddata.py @@ -0,0 +1,23 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.interpolate` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'CloughTocher2DInterpolator', + 'LinearNDInterpolator', + 'NearestNDInterpolator', + 'griddata', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="interpolate", module="ndgriddata", + private_modules=["_ndgriddata"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/polyint.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/polyint.py new file mode 100644 index 0000000000000000000000000000000000000000..e81306304abffb313ab5abe09116a162642a9d67 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/polyint.py @@ -0,0 +1,24 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.interpolate` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'BarycentricInterpolator', + 'KroghInterpolator', + 'approximate_taylor_polynomial', + 'barycentric_interpolate', + 'krogh_interpolate', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="interpolate", module="polyint", + private_modules=["_polyint"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/rbf.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/rbf.py new file mode 100644 index 0000000000000000000000000000000000000000..772752ef536f4a3b47fb6f9b5d250c5d7f198d85 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/rbf.py @@ -0,0 +1,18 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.interpolate` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = ["Rbf"] # noqa: F822 + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="interpolate", module="rbf", + private_modules=["_rbf"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_bary_rational.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_bary_rational.py new file mode 100644 index 0000000000000000000000000000000000000000..fbeea868ea24292693ced4cfb230badc3a551a89 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_bary_rational.py @@ -0,0 +1,368 @@ +# Copyright (c) 2017, The Chancellor, Masters and Scholars of the University +# of Oxford, and the Chebfun Developers. All rights reserved. +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions are met: +# * Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# * Redistributions in binary form must reproduce the above copyright +# notice, this list of conditions and the following disclaimer in the +# documentation and/or other materials provided with the distribution. +# * Neither the name of the University of Oxford nor the names of its +# contributors may be used to endorse or promote products derived from +# this software without specific prior written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND +# ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE +# DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR +# ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES +# (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; +# LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND +# ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT +# (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +from math import factorial + +import numpy as np +from numpy.testing import assert_allclose, assert_equal, assert_array_less +import pytest +import scipy +from scipy.interpolate import AAA, FloaterHormannInterpolator, BarycentricInterpolator + +TOL = 1e4 * np.finfo(np.float64).eps +UNIT_INTERVAL = np.linspace(-1, 1, num=1000) +PTS = np.logspace(-15, 0, base=10, num=500) +PTS = np.concatenate([-PTS[::-1], [0], PTS]) + + +@pytest.mark.parametrize("method", [AAA, FloaterHormannInterpolator]) +@pytest.mark.parametrize("dtype", [np.float32, np.float64, np.complex64, np.complex128]) +def test_dtype_preservation(method, dtype): + rtol = np.finfo(dtype).eps ** 0.75 * 100 + if method is FloaterHormannInterpolator: + rtol *= 100 + rng = np.random.default_rng(59846294526092468) + + z = np.linspace(-1, 1, dtype=dtype) + r = method(z, np.sin(z)) + + z2 = rng.uniform(-1, 1, size=100).astype(dtype) + assert_allclose(r(z2), np.sin(z2), rtol=rtol) + assert r(z2).dtype == dtype + + if method is AAA: + assert r.support_points.dtype == dtype + assert r.support_values.dtype == dtype + assert r.errors.dtype == z.real.dtype + assert r.weights.dtype == dtype + assert r.poles().dtype == np.result_type(dtype, 1j) + assert r.residues().dtype == np.result_type(dtype, 1j) + assert r.roots().dtype == np.result_type(dtype, 1j) + + +@pytest.mark.parametrize("method", [AAA, FloaterHormannInterpolator]) +@pytest.mark.parametrize("dtype", [np.int16, np.int32, np.int64]) +def test_integer_promotion(method, dtype): + z = np.arange(10, dtype=dtype) + r = method(z, z) + assert r.weights.dtype == np.result_type(dtype, 1.0) + if method is AAA: + assert r.support_points.dtype == np.result_type(dtype, 1.0) + assert r.support_values.dtype == np.result_type(dtype, 1.0) + assert r.errors.dtype == np.result_type(dtype, 1.0) + assert r.poles().dtype == np.result_type(dtype, 1j) + assert r.residues().dtype == np.result_type(dtype, 1j) + assert r.roots().dtype == np.result_type(dtype, 1j) + + assert r(z).dtype == np.result_type(dtype, 1.0) + + +class TestAAA: + def test_input_validation(self): + with pytest.raises(ValueError, match="same size"): + AAA([0], [1, 1]) + with pytest.raises(ValueError, match="1-D"): + AAA([[0], [0]], [[1], [1]]) + with pytest.raises(ValueError, match="finite"): + AAA([np.inf], [1]) + with pytest.raises(TypeError): + AAA([1], [1], max_terms=1.0) + with pytest.raises(ValueError, match="greater"): + AAA([1], [1], max_terms=-1) + + @pytest.mark.thread_unsafe + def test_convergence_error(self): + with pytest.warns(RuntimeWarning, match="AAA failed"): + AAA(UNIT_INTERVAL, np.exp(UNIT_INTERVAL), max_terms=1) + + # The following tests are based on: + # https://github.com/chebfun/chebfun/blob/master/tests/chebfun/test_aaa.m + def test_exp(self): + f = np.exp(UNIT_INTERVAL) + r = AAA(UNIT_INTERVAL, f) + + assert_allclose(r(UNIT_INTERVAL), f, atol=TOL) + assert_equal(r(np.nan), np.nan) + assert np.isfinite(r(np.inf)) + + m1 = r.support_points.size + r = AAA(UNIT_INTERVAL, f, rtol=1e-3) + assert r.support_points.size < m1 + + def test_tan(self): + f = np.tan(np.pi * UNIT_INTERVAL) + r = AAA(UNIT_INTERVAL, f) + + assert_allclose(r(UNIT_INTERVAL), f, atol=10 * TOL, rtol=1.4e-7) + assert_allclose(np.min(np.abs(r.roots())), 0, atol=3e-10) + assert_allclose(np.min(np.abs(r.poles() - 0.5)), 0, atol=TOL) + # Test for spurious poles (poles with tiny residue are likely spurious) + assert np.min(np.abs(r.residues())) > 1e-13 + + def test_short_cases(self): + # Computed using Chebfun: + # >> format long + # >> [r, pol, res, zer, zj, fj, wj, errvec] = aaa([1 2], [0 1]) + z = np.array([0, 1]) + f = np.array([1, 2]) + r = AAA(z, f, rtol=1e-13) + assert_allclose(r(z), f, atol=TOL) + assert_allclose(r.poles(), 0.5) + assert_allclose(r.residues(), 0.25) + assert_allclose(r.roots(), 1/3) + assert_equal(r.support_points, z) + assert_equal(r.support_values, f) + assert_allclose(r.weights, [0.707106781186547, 0.707106781186547]) + assert_equal(r.errors, [1, 0]) + + # >> format long + # >> [r, pol, res, zer, zj, fj, wj, errvec] = aaa([1 0 0], [0 1 2]) + z = np.array([0, 1, 2]) + f = np.array([1, 0, 0]) + r = AAA(z, f, rtol=1e-13) + assert_allclose(r(z), f, atol=TOL) + assert_allclose(np.sort(r.poles()), + np.sort([1.577350269189626, 0.422649730810374])) + assert_allclose(np.sort(r.residues()), + np.sort([-0.070441621801729, -0.262891711531604])) + assert_allclose(np.sort(r.roots()), np.sort([2, 1])) + assert_equal(r.support_points, z) + assert_equal(r.support_values, f) + assert_allclose(r.weights, [0.577350269189626, 0.577350269189626, + 0.577350269189626]) + assert_equal(r.errors, [1, 1, 0]) + + def test_scale_invariance(self): + z = np.linspace(0.3, 1.5) + f = np.exp(z) / (1 + 1j) + r1 = AAA(z, f) + r2 = AAA(z, (2**311 * f).astype(np.complex128)) + r3 = AAA(z, (2**-311 * f).astype(np.complex128)) + assert_equal(r1(0.2j), 2**-311 * r2(0.2j)) + assert_equal(r1(1.4), 2**311 * r3(1.4)) + + def test_log_func(self): + rng = np.random.default_rng(1749382759832758297) + z = rng.standard_normal(10000) + 3j * rng.standard_normal(10000) + + def f(z): + return np.log(5 - z) / (1 + z**2) + + r = AAA(z, f(z)) + assert_allclose(r(0), f(0), atol=TOL) + + def test_infinite_data(self): + z = np.linspace(-1, 1) + r = AAA(z, scipy.special.gamma(z)) + assert_allclose(r(0.63), scipy.special.gamma(0.63), atol=1e-15) + + def test_nan(self): + x = np.linspace(0, 20) + with np.errstate(invalid="ignore"): + f = np.sin(x) / x + r = AAA(x, f) + assert_allclose(r(2), np.sin(2) / 2, atol=1e-15) + + def test_residues(self): + x = np.linspace(-1.337, 2, num=537) + r = AAA(x, np.exp(x) / x) + ii = np.flatnonzero(np.abs(r.poles()) < 1e-8) + assert_allclose(r.residues()[ii], 1, atol=1e-15) + + r = AAA(x, (1 + 1j) * scipy.special.gamma(x)) + ii = np.flatnonzero(abs(r.poles() - (-1)) < 1e-8) + assert_allclose(r.residues()[ii], -1 - 1j, atol=1e-15) + + # The following tests are based on: + # https://github.com/complexvariables/RationalFunctionApproximation.jl/blob/main/test/interval.jl + @pytest.mark.parametrize("func,atol,rtol", + [(lambda x: np.abs(x + 0.5 + 0.01j), 5e-13, 1e-7), + (lambda x: np.sin(1/(1.05 - x)), 2e-13, 1e-7), + (lambda x: np.exp(-1/(x**2)), 3.5e-13, 0), + (lambda x: np.exp(-100*x**2), 8e-13, 0), + (lambda x: np.exp(-10/(1.2 - x)), 1e-14, 0), + (lambda x: 1/(1+np.exp(100*(x + 0.5))), 2e-13, 1e-7), + (lambda x: np.abs(x - 0.95), 1e-6, 1e-7)]) + def test_basic_functions(self, func, atol, rtol): + with np.errstate(divide="ignore"): + f = func(PTS) + assert_allclose(AAA(UNIT_INTERVAL, func(UNIT_INTERVAL))(PTS), + f, atol=atol, rtol=rtol) + + def test_poles_zeros_residues(self): + def f(z): + return (z+1) * (z+2) / ((z+3) * (z+4)) + r = AAA(UNIT_INTERVAL, f(UNIT_INTERVAL)) + assert_allclose(np.sum(r.poles() + r.roots()), -10, atol=1e-12) + + def f(z): + return 2/(3 + z) + 5/(z - 2j) + r = AAA(UNIT_INTERVAL, f(UNIT_INTERVAL)) + assert_allclose(r.residues().prod(), 10, atol=1e-8) + + r = AAA(UNIT_INTERVAL, np.sin(10*np.pi*UNIT_INTERVAL)) + assert_allclose(np.sort(np.abs(r.roots()))[18], 0.9, atol=1e-12) + + def f(z): + return (z - (3 + 3j))/(z + 2) + r = AAA(UNIT_INTERVAL, f(UNIT_INTERVAL)) + assert_allclose(r.poles()[0]*r.roots()[0], -6-6j, atol=1e-12) + + @pytest.mark.parametrize("func", + [lambda z: np.zeros_like(z), lambda z: z, lambda z: 1j*z, + lambda z: z**2 + z, lambda z: z**3 + z, + lambda z: 1/(1.1 + z), lambda z: 1/(1 + 1j*z), + lambda z: 1/(3 + z + z**2), lambda z: 1/(1.01 + z**3)]) + def test_polynomials_and_reciprocals(self, func): + assert_allclose(AAA(UNIT_INTERVAL, func(UNIT_INTERVAL))(PTS), + func(PTS), atol=2e-13) + + # The following tests are taken from: + # https://github.com/macd/BaryRational.jl/blob/main/test/test_aaa.jl + def test_spiral(self): + z = np.exp(np.linspace(-0.5, 0.5 + 15j*np.pi, num=1000)) + r = AAA(z, np.tan(np.pi*z/2)) + assert_allclose(np.sort(np.abs(r.poles()))[:4], [1, 1, 3, 3], rtol=9e-7) + + @pytest.mark.thread_unsafe + def test_spiral_cleanup(self): + z = np.exp(np.linspace(-0.5, 0.5 + 15j*np.pi, num=1000)) + # here we set `rtol=0` to force froissart doublets, without cleanup there + # are many spurious poles + with pytest.warns(RuntimeWarning): + r = AAA(z, np.tan(np.pi*z/2), rtol=0, max_terms=60, clean_up=False) + n_spurious = np.sum(np.abs(r.residues()) < 1e-14) + with pytest.warns(RuntimeWarning): + assert r.clean_up() >= 1 + # check there are less potentially spurious poles than before + assert np.sum(np.abs(r.residues()) < 1e-14) < n_spurious + # check accuracy + assert_allclose(r(z), np.tan(np.pi*z/2), atol=6e-12, rtol=3e-12) + + +class TestFloaterHormann: + def runge(self, z): + return 1/(1 + z**2) + + def scale(self, n, d): + return (-1)**(np.arange(n) + d) * factorial(d) + + def test_iv(self): + with pytest.raises(ValueError, match="`x`"): + FloaterHormannInterpolator([[0]], [0], d=0) + with pytest.raises(ValueError, match="`y`"): + FloaterHormannInterpolator([0], 0, d=0) + with pytest.raises(ValueError, match="dimension"): + FloaterHormannInterpolator([0], [[1, 1], [1, 1]], d=0) + with pytest.raises(ValueError, match="finite"): + FloaterHormannInterpolator([np.inf], [1], d=0) + with pytest.raises(ValueError, match="`d`"): + FloaterHormannInterpolator([0], [0], d=-1) + with pytest.raises(ValueError, match="`d`"): + FloaterHormannInterpolator([0], [0], d=10) + with pytest.raises(TypeError): + FloaterHormannInterpolator([0], [0], d=0.0) + + # reference values from Floater and Hormann 2007 page 8. + @pytest.mark.parametrize("d,expected", [ + (0, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), + (1, [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1]), + (2, [1, 3, 4, 4, 4, 4, 4, 4, 4, 3, 1]), + (3, [1, 4, 7, 8, 8, 8, 8, 8, 7, 4, 1]), + (4, [1, 5, 11, 15, 16, 16, 16, 15, 11, 5, 1]) + ]) + def test_uniform_grid(self, d, expected): + # Check against explicit results on an uniform grid + x = np.arange(11) + r = FloaterHormannInterpolator(x, 0.0*x, d=d) + assert_allclose(r.weights.ravel()*self.scale(x.size, d), expected, + rtol=1e-15, atol=1e-15) + + @pytest.mark.parametrize("d", range(10)) + def test_runge(self, d): + x = np.linspace(0, 1, 51) + rng = np.random.default_rng(802754237598370893) + xx = rng.uniform(0, 1, size=1000) + y = self.runge(x) + h = x[1] - x[0] + + r = FloaterHormannInterpolator(x, y, d=d) + + tol = 10*h**(d+1) + assert_allclose(r(xx), self.runge(xx), atol=1e-10, rtol=tol) + # check interpolation property + assert_equal(r(x), self.runge(x)) + + def test_complex(self): + x = np.linspace(-1, 1) + z = x + x*1j + r = FloaterHormannInterpolator(z, np.sin(z), d=12) + xx = np.linspace(-1, 1, num=1000) + zz = xx + xx*1j + assert_allclose(r(zz), np.sin(zz), rtol=1e-12) + + def test_polyinterp(self): + # check that when d=n-1 FH gives a polynomial interpolant + x = np.linspace(0, 1, 11) + xx = np.linspace(0, 1, 1001) + y = np.sin(x) + r = FloaterHormannInterpolator(x, y, d=x.size-1) + p = BarycentricInterpolator(x, y) + assert_allclose(r(xx), p(xx), rtol=1e-12, atol=1e-12) + + @pytest.mark.parametrize("y_shape", [(2,), (2, 3, 1), (1, 5, 6, 4)]) + @pytest.mark.parametrize("xx_shape", [(100), (10, 10)]) + def test_trailing_dim(self, y_shape, xx_shape): + x = np.linspace(0, 1) + y = np.broadcast_to( + np.expand_dims(np.sin(x), tuple(range(1, len(y_shape) + 1))), + x.shape + y_shape + ) + + r = FloaterHormannInterpolator(x, y) + + rng = np.random.default_rng(897138947238097528091759187597) + xx = rng.random(xx_shape) + yy = np.broadcast_to( + np.expand_dims(np.sin(xx), tuple(range(xx.ndim, len(y_shape) + xx.ndim))), + xx.shape + y_shape + ) + rr = r(xx) + assert rr.shape == xx.shape + y_shape + assert_allclose(rr, yy, rtol=1e-6) + + def test_zeros(self): + x = np.linspace(0, 10, num=100) + r = FloaterHormannInterpolator(x, np.sin(np.pi*x)) + + err = np.abs(np.subtract.outer(r.roots(), np.arange(11))).min(axis=0) + assert_array_less(err, 1e-5) + + def test_no_poles(self): + x = np.linspace(-1, 1) + r = FloaterHormannInterpolator(x, 1/x**2) + p = r.poles() + mask = (p.real >= -1) & (p.real <= 1) & (np.abs(p.imag) < 1.e-12) + assert np.sum(mask) == 0 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_bsplines.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_bsplines.py new file mode 100644 index 0000000000000000000000000000000000000000..e8b1b2b58afcf5ad5a34babe65fff96bf268df03 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_bsplines.py @@ -0,0 +1,3658 @@ +import os +import operator +import itertools +import math +import threading + +import numpy as np +from numpy.testing import suppress_warnings +from scipy._lib._array_api import xp_assert_equal, xp_assert_close +from pytest import raises as assert_raises +import pytest + +from scipy.interpolate import ( + BSpline, BPoly, PPoly, make_interp_spline, make_lsq_spline, + splev, splrep, splprep, splder, splantider, sproot, splint, insert, + CubicSpline, NdBSpline, make_smoothing_spline, RegularGridInterpolator, +) +import scipy.linalg as sl +import scipy.sparse.linalg as ssl + +from scipy.interpolate._bsplines import (_not_a_knot, _augknt, + _woodbury_algorithm, _periodic_knots, + _make_interp_per_full_matr) + +from scipy.interpolate import generate_knots, make_splrep, make_splprep + +import scipy.interpolate._fitpack_impl as _impl +from scipy._lib._util import AxisError +from scipy._lib._testutils import _run_concurrent_barrier + +# XXX: move to the interpolate namespace +from scipy.interpolate._ndbspline import make_ndbspl + +from scipy.interpolate import _dfitpack as dfitpack +from scipy.interpolate import _bsplines as _b +from scipy.interpolate import _dierckx + + +class TestBSpline: + + def test_ctor(self): + # knots should be an ordered 1-D array of finite real numbers + assert_raises((TypeError, ValueError), BSpline, + **dict(t=[1, 1.j], c=[1.], k=0)) + with np.errstate(invalid='ignore'): + assert_raises(ValueError, BSpline, **dict(t=[1, np.nan], c=[1.], k=0)) + assert_raises(ValueError, BSpline, **dict(t=[1, np.inf], c=[1.], k=0)) + assert_raises(ValueError, BSpline, **dict(t=[1, -1], c=[1.], k=0)) + assert_raises(ValueError, BSpline, **dict(t=[[1], [1]], c=[1.], k=0)) + + # for n+k+1 knots and degree k need at least n coefficients + assert_raises(ValueError, BSpline, **dict(t=[0, 1, 2], c=[1], k=0)) + assert_raises(ValueError, BSpline, + **dict(t=[0, 1, 2, 3, 4], c=[1., 1.], k=2)) + + # non-integer orders + assert_raises(TypeError, BSpline, + **dict(t=[0., 0., 1., 2., 3., 4.], c=[1., 1., 1.], k="cubic")) + assert_raises(TypeError, BSpline, + **dict(t=[0., 0., 1., 2., 3., 4.], c=[1., 1., 1.], k=2.5)) + + # basic interval cannot have measure zero (here: [1..1]) + assert_raises(ValueError, BSpline, + **dict(t=[0., 0, 1, 1, 2, 3], c=[1., 1, 1], k=2)) + + # tck vs self.tck + n, k = 11, 3 + t = np.arange(n+k+1, dtype=np.float64) + c = np.random.random(n) + b = BSpline(t, c, k) + + xp_assert_close(t, b.t) + xp_assert_close(c, b.c) + assert k == b.k + + def test_tck(self): + b = _make_random_spline() + tck = b.tck + + xp_assert_close(b.t, tck[0], atol=1e-15, rtol=1e-15) + xp_assert_close(b.c, tck[1], atol=1e-15, rtol=1e-15) + assert b.k == tck[2] + + # b.tck is read-only + with pytest.raises(AttributeError): + b.tck = 'foo' + + def test_degree_0(self): + xx = np.linspace(0, 1, 10) + + b = BSpline(t=[0, 1], c=[3.], k=0) + xp_assert_close(b(xx), np.ones_like(xx) * 3.0) + + b = BSpline(t=[0, 0.35, 1], c=[3, 4], k=0) + xp_assert_close(b(xx), np.where(xx < 0.35, 3.0, 4.0)) + + def test_degree_1(self): + t = [0, 1, 2, 3, 4] + c = [1, 2, 3] + k = 1 + b = BSpline(t, c, k) + + x = np.linspace(1, 3, 50) + xp_assert_close(c[0]*B_012(x) + c[1]*B_012(x-1) + c[2]*B_012(x-2), + b(x), atol=1e-14) + xp_assert_close(splev(x, (t, c, k)), b(x), atol=1e-14) + + def test_bernstein(self): + # a special knot vector: Bernstein polynomials + k = 3 + t = np.asarray([0]*(k+1) + [1]*(k+1)) + c = np.asarray([1., 2., 3., 4.]) + bp = BPoly(c.reshape(-1, 1), [0, 1]) + bspl = BSpline(t, c, k) + + xx = np.linspace(-1., 2., 10) + xp_assert_close(bp(xx, extrapolate=True), + bspl(xx, extrapolate=True), atol=1e-14) + xp_assert_close(splev(xx, (t, c, k)), + bspl(xx), atol=1e-14) + + def test_rndm_naive_eval(self): + # test random coefficient spline *on the base interval*, + # t[k] <= x < t[-k-1] + b = _make_random_spline() + t, c, k = b.tck + xx = np.linspace(t[k], t[-k-1], 50) + y_b = b(xx) + + y_n = [_naive_eval(x, t, c, k) for x in xx] + xp_assert_close(y_b, y_n, atol=1e-14) + + y_n2 = [_naive_eval_2(x, t, c, k) for x in xx] + xp_assert_close(y_b, y_n2, atol=1e-14) + + def test_rndm_splev(self): + b = _make_random_spline() + t, c, k = b.tck + xx = np.linspace(t[k], t[-k-1], 50) + xp_assert_close(b(xx), splev(xx, (t, c, k)), atol=1e-14) + + def test_rndm_splrep(self): + rng = np.random.RandomState(1234) + x = np.sort(rng.random(20)) + y = rng.random(20) + + tck = splrep(x, y) + b = BSpline(*tck) + + t, k = b.t, b.k + xx = np.linspace(t[k], t[-k-1], 80) + xp_assert_close(b(xx), splev(xx, tck), atol=1e-14) + + def test_rndm_unity(self): + b = _make_random_spline() + b.c = np.ones_like(b.c) + xx = np.linspace(b.t[b.k], b.t[-b.k-1], 100) + xp_assert_close(b(xx), np.ones_like(xx)) + + def test_vectorization(self): + rng = np.random.RandomState(1234) + n, k = 22, 3 + t = np.sort(rng.random(n)) + c = rng.random(size=(n, 6, 7)) + b = BSpline(t, c, k) + tm, tp = t[k], t[-k-1] + xx = tm + (tp - tm) * rng.random((3, 4, 5)) + assert b(xx).shape == (3, 4, 5, 6, 7) + + def test_len_c(self): + # for n+k+1 knots, only first n coefs are used. + # and BTW this is consistent with FITPACK + rng = np.random.RandomState(1234) + n, k = 33, 3 + t = np.sort(rng.random(n+k+1)) + c = rng.random(n) + + # pad coefficients with random garbage + c_pad = np.r_[c, rng.random(k+1)] + + b, b_pad = BSpline(t, c, k), BSpline(t, c_pad, k) + + dt = t[-1] - t[0] + xx = np.linspace(t[0] - dt, t[-1] + dt, 50) + xp_assert_close(b(xx), b_pad(xx), atol=1e-14) + xp_assert_close(b(xx), splev(xx, (t, c, k)), atol=1e-14) + xp_assert_close(b(xx), splev(xx, (t, c_pad, k)), atol=1e-14) + + def test_endpoints(self, num_parallel_threads): + # base interval is closed + b = _make_random_spline() + t, _, k = b.tck + tm, tp = t[k], t[-k-1] + # atol = 1e-9 if num_parallel_threads == 1 else 1e-7 + for extrap in (True, False): + xp_assert_close(b([tm, tp], extrap), + b([tm + 1e-10, tp - 1e-10], extrap), atol=1e-9, rtol=1e-7) + + def test_continuity(self, num_parallel_threads): + # assert continuity at internal knots + b = _make_random_spline() + t, _, k = b.tck + xp_assert_close(b(t[k+1:-k-1] - 1e-10), b(t[k+1:-k-1] + 1e-10), + atol=1e-9) + + def test_extrap(self): + b = _make_random_spline() + t, c, k = b.tck + dt = t[-1] - t[0] + xx = np.linspace(t[k] - dt, t[-k-1] + dt, 50) + mask = (t[k] < xx) & (xx < t[-k-1]) + + # extrap has no effect within the base interval + xp_assert_close(b(xx[mask], extrapolate=True), + b(xx[mask], extrapolate=False)) + + # extrapolated values agree with FITPACK + xp_assert_close(b(xx, extrapolate=True), + splev(xx, (t, c, k), ext=0)) + + def test_default_extrap(self): + # BSpline defaults to extrapolate=True + b = _make_random_spline() + t, _, k = b.tck + xx = [t[0] - 1, t[-1] + 1] + yy = b(xx) + assert not np.all(np.isnan(yy)) + + def test_periodic_extrap(self): + rng = np.random.RandomState(1234) + t = np.sort(rng.random(8)) + c = rng.random(4) + k = 3 + b = BSpline(t, c, k, extrapolate='periodic') + n = t.size - (k + 1) + + dt = t[-1] - t[0] + xx = np.linspace(t[k] - dt, t[n] + dt, 50) + xy = t[k] + (xx - t[k]) % (t[n] - t[k]) + xp_assert_close(b(xx), splev(xy, (t, c, k))) + + # Direct check + xx = [-1, 0, 0.5, 1] + xy = t[k] + (xx - t[k]) % (t[n] - t[k]) + xp_assert_equal(b(xx, extrapolate='periodic'), b(xy, extrapolate=True)) + + def test_ppoly(self): + b = _make_random_spline() + t, c, k = b.tck + pp = PPoly.from_spline((t, c, k)) + + xx = np.linspace(t[k], t[-k], 100) + xp_assert_close(b(xx), pp(xx), atol=1e-14, rtol=1e-14) + + def test_derivative_rndm(self): + b = _make_random_spline() + t, c, k = b.tck + xx = np.linspace(t[0], t[-1], 50) + xx = np.r_[xx, t] + + for der in range(1, k+1): + yd = splev(xx, (t, c, k), der=der) + xp_assert_close(yd, b(xx, nu=der), atol=1e-14) + + # higher derivatives all vanish + xp_assert_close(b(xx, nu=k+1), np.zeros_like(xx), atol=1e-14) + + def test_derivative_jumps(self): + # example from de Boor, Chap IX, example (24) + # NB: knots augmented & corresp coefs are zeroed out + # in agreement with the convention (29) + k = 2 + t = [-1, -1, 0, 1, 1, 3, 4, 6, 6, 6, 7, 7] + rng = np.random.RandomState(1234) + c = np.r_[0, 0, rng.random(5), 0, 0] + b = BSpline(t, c, k) + + # b is continuous at x != 6 (triple knot) + x = np.asarray([1, 3, 4, 6]) + xp_assert_close(b(x[x != 6] - 1e-10), + b(x[x != 6] + 1e-10)) + assert not np.allclose(b(6.-1e-10), b(6+1e-10)) + + # 1st derivative jumps at double knots, 1 & 6: + x0 = np.asarray([3, 4]) + xp_assert_close(b(x0 - 1e-10, nu=1), + b(x0 + 1e-10, nu=1)) + x1 = np.asarray([1, 6]) + assert not np.allclose(b(x1 - 1e-10, nu=1), b(x1 + 1e-10, nu=1)) + + # 2nd derivative is not guaranteed to be continuous either + assert not np.allclose(b(x - 1e-10, nu=2), b(x + 1e-10, nu=2)) + + def test_basis_element_quadratic(self): + xx = np.linspace(-1, 4, 20) + b = BSpline.basis_element(t=[0, 1, 2, 3]) + xp_assert_close(b(xx), + splev(xx, (b.t, b.c, b.k)), atol=1e-14) + xp_assert_close(b(xx), + B_0123(xx), atol=1e-14) + + b = BSpline.basis_element(t=[0, 1, 1, 2]) + xx = np.linspace(0, 2, 10) + xp_assert_close(b(xx), + np.where(xx < 1, xx*xx, (2.-xx)**2), atol=1e-14) + + def test_basis_element_rndm(self): + b = _make_random_spline() + t, c, k = b.tck + xx = np.linspace(t[k], t[-k-1], 20) + xp_assert_close(b(xx), _sum_basis_elements(xx, t, c, k), atol=1e-14) + + def test_cmplx(self): + b = _make_random_spline() + t, c, k = b.tck + cc = c * (1. + 3.j) + + b = BSpline(t, cc, k) + b_re = BSpline(t, b.c.real, k) + b_im = BSpline(t, b.c.imag, k) + + xx = np.linspace(t[k], t[-k-1], 20) + xp_assert_close(b(xx).real, b_re(xx), atol=1e-14) + xp_assert_close(b(xx).imag, b_im(xx), atol=1e-14) + + def test_nan(self): + # nan in, nan out. + b = BSpline.basis_element([0, 1, 1, 2]) + assert np.isnan(b(np.nan)) + + def test_derivative_method(self): + b = _make_random_spline(k=5) + t, c, k = b.tck + b0 = BSpline(t, c, k) + xx = np.linspace(t[k], t[-k-1], 20) + for j in range(1, k): + b = b.derivative() + xp_assert_close(b0(xx, j), b(xx), atol=1e-12, rtol=1e-12) + + def test_antiderivative_method(self): + b = _make_random_spline() + t, c, k = b.tck + xx = np.linspace(t[k], t[-k-1], 20) + xp_assert_close(b.antiderivative().derivative()(xx), + b(xx), atol=1e-14, rtol=1e-14) + + # repeat with N-D array for c + c = np.c_[c, c, c] + c = np.dstack((c, c)) + b = BSpline(t, c, k) + xp_assert_close(b.antiderivative().derivative()(xx), + b(xx), atol=1e-14, rtol=1e-14) + + def test_integral(self): + b = BSpline.basis_element([0, 1, 2]) # x for x < 1 else 2 - x + xp_assert_close(b.integrate(0, 1), np.asarray(0.5)) + xp_assert_close(b.integrate(1, 0), np.asarray(-1 * 0.5)) + xp_assert_close(b.integrate(1, 0), np.asarray(-0.5)) + + # extrapolate or zeros outside of [0, 2]; default is yes + xp_assert_close(b.integrate(-1, 1), np.asarray(0.0)) + xp_assert_close(b.integrate(-1, 1, extrapolate=True), np.asarray(0.0)) + xp_assert_close(b.integrate(-1, 1, extrapolate=False), np.asarray(0.5)) + xp_assert_close(b.integrate(1, -1, extrapolate=False), np.asarray(-1 * 0.5)) + + # Test ``_fitpack._splint()`` + xp_assert_close(b.integrate(1, -1, extrapolate=False), + np.asarray(_impl.splint(1, -1, b.tck))) + + # Test ``extrapolate='periodic'``. + b.extrapolate = 'periodic' + i = b.antiderivative() + period_int = np.asarray(i(2) - i(0)) + + xp_assert_close(b.integrate(0, 2), period_int) + xp_assert_close(b.integrate(2, 0), np.asarray(-1 * period_int)) + xp_assert_close(b.integrate(-9, -7), period_int) + xp_assert_close(b.integrate(-8, -4), np.asarray(2 * period_int)) + + xp_assert_close(b.integrate(0.5, 1.5), + np.asarray(i(1.5) - i(0.5))) + xp_assert_close(b.integrate(1.5, 3), + np.asarray(i(1) - i(0) + i(2) - i(1.5))) + xp_assert_close(b.integrate(1.5 + 12, 3 + 12), + np.asarray(i(1) - i(0) + i(2) - i(1.5))) + xp_assert_close(b.integrate(1.5, 3 + 12), + np.asarray(i(1) - i(0) + i(2) - i(1.5) + 6 * period_int)) + + xp_assert_close(b.integrate(0, -1), np.asarray(i(0) - i(1))) + xp_assert_close(b.integrate(-9, -10), np.asarray(i(0) - i(1))) + xp_assert_close(b.integrate(0, -9), + np.asarray(i(1) - i(2) - 4 * period_int)) + + def test_integrate_ppoly(self): + # test .integrate method to be consistent with PPoly.integrate + x = [0, 1, 2, 3, 4] + b = make_interp_spline(x, x) + b.extrapolate = 'periodic' + p = PPoly.from_spline(b) + + for x0, x1 in [(-5, 0.5), (0.5, 5), (-4, 13)]: + xp_assert_close(b.integrate(x0, x1), + p.integrate(x0, x1)) + + def test_integrate_0D_always(self): + # make sure the result is always a 0D array (not a python scalar) + b = BSpline.basis_element([0, 1, 2]) + for extrapolate in (True, False): + res = b.integrate(0, 1, extrapolate=extrapolate) + assert isinstance(res, np.ndarray) + assert res.ndim == 0 + + def test_subclassing(self): + # classmethods should not decay to the base class + class B(BSpline): + pass + + b = B.basis_element([0, 1, 2, 2]) + assert b.__class__ == B + assert b.derivative().__class__ == B + assert b.antiderivative().__class__ == B + + @pytest.mark.parametrize('axis', range(-4, 4)) + def test_axis(self, axis): + n, k = 22, 3 + t = np.linspace(0, 1, n + k + 1) + sh = [6, 7, 8] + # We need the positive axis for some of the indexing and slices used + # in this test. + pos_axis = axis % 4 + sh.insert(pos_axis, n) # [22, 6, 7, 8] etc + sh = tuple(sh) + rng = np.random.RandomState(1234) + c = rng.random(size=sh) + b = BSpline(t, c, k, axis=axis) + assert b.c.shape == (sh[pos_axis],) + sh[:pos_axis] + sh[pos_axis+1:] + + xp = rng.random((3, 4, 5)) + assert b(xp).shape == sh[:pos_axis] + xp.shape + sh[pos_axis+1:] + + # -c.ndim <= axis < c.ndim + for ax in [-c.ndim - 1, c.ndim]: + assert_raises(AxisError, BSpline, + **dict(t=t, c=c, k=k, axis=ax)) + + # derivative, antiderivative keeps the axis + for b1 in [BSpline(t, c, k, axis=axis).derivative(), + BSpline(t, c, k, axis=axis).derivative(2), + BSpline(t, c, k, axis=axis).antiderivative(), + BSpline(t, c, k, axis=axis).antiderivative(2)]: + assert b1.axis == b.axis + + def test_neg_axis(self): + k = 2 + t = [0, 1, 2, 3, 4, 5, 6] + c = np.array([[-1, 2, 0, -1], [2, 0, -3, 1]]) + + spl = BSpline(t, c, k, axis=-1) + spl0 = BSpline(t, c[0], k) + spl1 = BSpline(t, c[1], k) + xp_assert_equal(spl(2.5), [spl0(2.5), spl1(2.5)]) + + @pytest.mark.thread_unsafe + def test_design_matrix_bc_types(self): + ''' + Splines with different boundary conditions are built on different + types of vectors of knots. As far as design matrix depends only on + vector of knots, `k` and `x` it is useful to make tests for different + boundary conditions (and as following different vectors of knots). + ''' + def run_design_matrix_tests(n, k, bc_type): + ''' + To avoid repetition of code the following function is provided. + ''' + rng = np.random.RandomState(1234) + x = np.sort(rng.random_sample(n) * 40 - 20) + y = rng.random_sample(n) * 40 - 20 + if bc_type == "periodic": + y[0] = y[-1] + + bspl = make_interp_spline(x, y, k=k, bc_type=bc_type) + + c = np.eye(len(bspl.t) - k - 1) + des_matr_def = BSpline(bspl.t, c, k)(x) + des_matr_csr = BSpline.design_matrix(x, + bspl.t, + k).toarray() + xp_assert_close(des_matr_csr @ bspl.c, y, atol=1e-14) + xp_assert_close(des_matr_def, des_matr_csr, atol=1e-14) + + # "clamped" and "natural" work only with `k = 3` + n = 11 + k = 3 + for bc in ["clamped", "natural"]: + run_design_matrix_tests(n, k, bc) + + # "not-a-knot" works with odd `k` + for k in range(3, 8, 2): + run_design_matrix_tests(n, k, "not-a-knot") + + # "periodic" works with any `k` (even more than `n`) + n = 5 # smaller `n` to test `k > n` case + for k in range(2, 7): + run_design_matrix_tests(n, k, "periodic") + + @pytest.mark.parametrize('extrapolate', [False, True, 'periodic']) + @pytest.mark.parametrize('degree', range(5)) + def test_design_matrix_same_as_BSpline_call(self, extrapolate, degree): + """Test that design_matrix(x) is equivalent to BSpline(..)(x).""" + rng = np.random.RandomState(1234) + x = rng.random_sample(10 * (degree + 1)) + xmin, xmax = np.amin(x), np.amax(x) + k = degree + t = np.r_[np.linspace(xmin - 2, xmin - 1, degree), + np.linspace(xmin, xmax, 2 * (degree + 1)), + np.linspace(xmax + 1, xmax + 2, degree)] + c = np.eye(len(t) - k - 1) + bspline = BSpline(t, c, k, extrapolate) + xp_assert_close( + bspline(x), BSpline.design_matrix(x, t, k, extrapolate).toarray() + ) + + # extrapolation regime + x = np.array([xmin - 10, xmin - 1, xmax + 1.5, xmax + 10]) + if not extrapolate: + with pytest.raises(ValueError): + BSpline.design_matrix(x, t, k, extrapolate) + else: + xp_assert_close( + bspline(x), + BSpline.design_matrix(x, t, k, extrapolate).toarray() + ) + + def test_design_matrix_x_shapes(self): + # test for different `x` shapes + rng = np.random.RandomState(1234) + n = 10 + k = 3 + x = np.sort(rng.random_sample(n) * 40 - 20) + y = rng.random_sample(n) * 40 - 20 + + bspl = make_interp_spline(x, y, k=k) + for i in range(1, 4): + xc = x[:i] + yc = y[:i] + des_matr_csr = BSpline.design_matrix(xc, + bspl.t, + k).toarray() + xp_assert_close(des_matr_csr @ bspl.c, yc, atol=1e-14) + + def test_design_matrix_t_shapes(self): + # test for minimal possible `t` shape + t = [1., 1., 1., 2., 3., 4., 4., 4.] + des_matr = BSpline.design_matrix(2., t, 3).toarray() + xp_assert_close(des_matr, + [[0.25, 0.58333333, 0.16666667, 0.]], + atol=1e-14) + + def test_design_matrix_asserts(self): + rng = np.random.RandomState(1234) + n = 10 + k = 3 + x = np.sort(rng.random_sample(n) * 40 - 20) + y = rng.random_sample(n) * 40 - 20 + bspl = make_interp_spline(x, y, k=k) + # invalid vector of knots (should be a 1D non-descending array) + # here the actual vector of knots is reversed, so it is invalid + with assert_raises(ValueError): + BSpline.design_matrix(x, bspl.t[::-1], k) + k = 2 + t = [0., 1., 2., 3., 4., 5.] + x = [1., 2., 3., 4.] + # out of bounds + with assert_raises(ValueError): + BSpline.design_matrix(x, t, k) + + @pytest.mark.parametrize('bc_type', ['natural', 'clamped', + 'periodic', 'not-a-knot']) + def test_from_power_basis(self, bc_type): + rng = np.random.RandomState(1234) + x = np.sort(rng.random(20)) + y = rng.random(20) + if bc_type == 'periodic': + y[-1] = y[0] + cb = CubicSpline(x, y, bc_type=bc_type) + bspl = BSpline.from_power_basis(cb, bc_type=bc_type) + xx = np.linspace(0, 1, 20) + xp_assert_close(cb(xx), bspl(xx), atol=1e-15) + bspl_new = make_interp_spline(x, y, bc_type=bc_type) + xp_assert_close(bspl.c, bspl_new.c, atol=1e-15) + + @pytest.mark.parametrize('bc_type', ['natural', 'clamped', + 'periodic', 'not-a-knot']) + def test_from_power_basis_complex(self, bc_type): + rng = np.random.RandomState(1234) + x = np.sort(rng.random(20)) + y = rng.random(20) + rng.random(20) * 1j + if bc_type == 'periodic': + y[-1] = y[0] + cb = CubicSpline(x, y, bc_type=bc_type) + bspl = BSpline.from_power_basis(cb, bc_type=bc_type) + bspl_new_real = make_interp_spline(x, y.real, bc_type=bc_type) + bspl_new_imag = make_interp_spline(x, y.imag, bc_type=bc_type) + xp_assert_close(bspl.c, bspl_new_real.c + 1j * bspl_new_imag.c, atol=1e-15) + + def test_from_power_basis_exmp(self): + ''' + For x = [0, 1, 2, 3, 4] and y = [1, 1, 1, 1, 1] + the coefficients of Cubic Spline in the power basis: + + $[[0, 0, 0, 0, 0],\\$ + $[0, 0, 0, 0, 0],\\$ + $[0, 0, 0, 0, 0],\\$ + $[1, 1, 1, 1, 1]]$ + + It could be shown explicitly that coefficients of the interpolating + function in B-spline basis are c = [1, 1, 1, 1, 1, 1, 1] + ''' + x = np.array([0, 1, 2, 3, 4]) + y = np.array([1, 1, 1, 1, 1]) + bspl = BSpline.from_power_basis(CubicSpline(x, y, bc_type='natural'), + bc_type='natural') + xp_assert_close(bspl.c, [1.0, 1, 1, 1, 1, 1, 1], atol=1e-15) + + def test_read_only(self): + # BSpline must work on read-only knots and coefficients. + t = np.array([0, 1]) + c = np.array([3.0]) + t.setflags(write=False) + c.setflags(write=False) + + xx = np.linspace(0, 1, 10) + xx.setflags(write=False) + + b = BSpline(t=t, c=c, k=0) + xp_assert_close(b(xx), np.ones_like(xx) * 3.0) + + @pytest.mark.thread_unsafe + def test_concurrency(self): + # Check that no segfaults appear with concurrent access to BSpline + b = _make_random_spline() + + def worker_fn(_, b): + t, _, k = b.tck + xx = np.linspace(t[k], t[-k-1], 10000) + b(xx) + + _run_concurrent_barrier(10, worker_fn, b) + + + def test_memmap(self, tmpdir): + # Make sure that memmaps can be used as t and c atrributes after the + # spline has been constructed. This is similar to what happens in a + # scikit-learn context, where joblib can create read-only memmap to + # share objects between workers. For more details, see + # https://github.com/scipy/scipy/issues/22143 + b = _make_random_spline() + xx = np.linspace(0, 1, 10) + + expected = b(xx) + + tid = threading.get_native_id() + t_mm = np.memmap(str(tmpdir.join(f't{tid}.dat')), mode='w+', + dtype=b.t.dtype, shape=b.t.shape) + t_mm[:] = b.t + c_mm = np.memmap(str(tmpdir.join(f'c{tid}.dat')), mode='w+', + dtype=b.c.dtype, shape=b.c.shape) + c_mm[:] = b.c + b.t = t_mm + b.c = c_mm + + xp_assert_close(b(xx), expected) + +class TestInsert: + + @pytest.mark.parametrize('xval', [0.0, 1.0, 2.5, 4, 6.5, 7.0]) + def test_insert(self, xval): + # insert a knot, incl edges (0.0, 7.0) and exactly at an existing knot (4.0) + x = np.arange(8) + y = np.sin(x)**3 + spl = make_interp_spline(x, y, k=3) + + spl_1f = insert(xval, spl) # FITPACK + spl_1 = spl.insert_knot(xval) + + xp_assert_close(spl_1.t, spl_1f.t, atol=1e-15) + xp_assert_close(spl_1.c, spl_1f.c[:-spl.k-1], atol=1e-15) + + # knot insertion preserves values, unless multiplicity >= k+1 + xx = x if xval != x[-1] else x[:-1] + xx = np.r_[xx, 0.5*(x[1:] + x[:-1])] + xp_assert_close(spl(xx), spl_1(xx), atol=1e-15) + + # ... repeat with ndim > 1 + y1 = np.cos(x)**3 + spl_y1 = make_interp_spline(x, y1, k=3) + spl_yy = make_interp_spline(x, np.c_[y, y1], k=3) + spl_yy1 = spl_yy.insert_knot(xval) + + xp_assert_close(spl_yy1.t, spl_1.t, atol=1e-15) + xp_assert_close(spl_yy1.c, np.c_[spl.insert_knot(xval).c, + spl_y1.insert_knot(xval).c], atol=1e-15) + + xx = x if xval != x[-1] else x[:-1] + xx = np.r_[xx, 0.5*(x[1:] + x[:-1])] + xp_assert_close(spl_yy(xx), spl_yy1(xx), atol=1e-15) + + + @pytest.mark.parametrize( + 'xval, m', [(0.0, 2), (1.0, 3), (1.5, 5), (4, 2), (7.0, 2)] + ) + def test_insert_multi(self, xval, m): + x = np.arange(8) + y = np.sin(x)**3 + spl = make_interp_spline(x, y, k=3) + + spl_1f = insert(xval, spl, m=m) + spl_1 = spl.insert_knot(xval, m) + + xp_assert_close(spl_1.t, spl_1f.t, atol=1e-15) + xp_assert_close(spl_1.c, spl_1f.c[:-spl.k-1], atol=1e-15) + + xx = x if xval != x[-1] else x[:-1] + xx = np.r_[xx, 0.5*(x[1:] + x[:-1])] + xp_assert_close(spl(xx), spl_1(xx), atol=1e-15) + + def test_insert_random(self): + rng = np.random.default_rng(12345) + n, k = 11, 3 + + t = np.sort(rng.uniform(size=n+k+1)) + c = rng.uniform(size=(n, 3, 2)) + spl = BSpline(t, c, k) + + xv = rng.uniform(low=t[k+1], high=t[-k-1]) + spl_1 = spl.insert_knot(xv) + + xx = rng.uniform(low=t[k+1], high=t[-k-1], size=33) + xp_assert_close(spl(xx), spl_1(xx), atol=1e-15) + + @pytest.mark.parametrize('xv', [0, 0.1, 2.0, 4.0, 4.5, # l.h. edge + 5.5, 6.0, 6.1, 7.0] # r.h. edge + ) + def test_insert_periodic(self, xv): + x = np.arange(8) + y = np.sin(x)**3 + tck = splrep(x, y, k=3) + spl = BSpline(*tck, extrapolate="periodic") + + spl_1 = spl.insert_knot(xv) + tf, cf, k = insert(xv, spl.tck, per=True) + + xp_assert_close(spl_1.t, tf, atol=1e-15) + xp_assert_close(spl_1.c[:-k-1], cf[:-k-1], atol=1e-15) + + xx = np.random.default_rng(1234).uniform(low=0, high=7, size=41) + xp_assert_close(spl_1(xx), splev(xx, (tf, cf, k)), atol=1e-15) + + @pytest.mark.parametrize('extrapolate', [None, 'periodic']) + def test_complex(self, extrapolate): + x = np.arange(8)*2*np.pi + y_re, y_im = np.sin(x), np.cos(x) + + spl = make_interp_spline(x, y_re + 1j*y_im, k=3) + spl.extrapolate = extrapolate + + spl_re = make_interp_spline(x, y_re, k=3) + spl_re.extrapolate = extrapolate + + spl_im = make_interp_spline(x, y_im, k=3) + spl_im.extrapolate = extrapolate + + xv = 3.5 + spl_1 = spl.insert_knot(xv) + spl_1re = spl_re.insert_knot(xv) + spl_1im = spl_im.insert_knot(xv) + + xp_assert_close(spl_1.t, spl_1re.t, atol=1e-15) + xp_assert_close(spl_1.t, spl_1im.t, atol=1e-15) + xp_assert_close(spl_1.c, spl_1re.c + 1j*spl_1im.c, atol=1e-15) + + def test_insert_periodic_too_few_internal_knots(self): + # both FITPACK and spl.insert_knot raise when there's not enough + # internal knots to make a periodic extension. + # Below the internal knots are 2, 3, , 4, 5 + # ^ + # 2, 3, 3.5, 4, 5 + # so two knots from each side from the new one, while need at least + # from either left or right. + xv = 3.5 + k = 3 + t = np.array([0]*(k+1) + [2, 3, 4, 5] + [7]*(k+1)) + c = np.ones(len(t) - k - 1) + spl = BSpline(t, c, k, extrapolate="periodic") + + with assert_raises(ValueError): + insert(xv, (t, c, k), per=True) + + with assert_raises(ValueError): + spl.insert_knot(xv) + + def test_insert_no_extrap(self): + k = 3 + t = np.array([0]*(k+1) + [2, 3, 4, 5] + [7]*(k+1)) + c = np.ones(len(t) - k - 1) + spl = BSpline(t, c, k) + + with assert_raises(ValueError): + spl.insert_knot(-1) + + with assert_raises(ValueError): + spl.insert_knot(8) + + with assert_raises(ValueError): + spl.insert_knot(3, m=0) + + +def test_knots_multiplicity(): + # Take a spline w/ random coefficients, throw in knots of varying + # multiplicity. + + def check_splev(b, j, der=0, atol=1e-14, rtol=1e-14): + # check evaluations against FITPACK, incl extrapolations + t, c, k = b.tck + x = np.unique(t) + x = np.r_[t[0]-0.1, 0.5*(x[1:] + x[:1]), t[-1]+0.1] + xp_assert_close(splev(x, (t, c, k), der), b(x, der), + atol=atol, rtol=rtol, err_msg=f'der = {der} k = {b.k}') + + # test loop itself + # [the index `j` is for interpreting the traceback in case of a failure] + for k in [1, 2, 3, 4, 5]: + b = _make_random_spline(k=k) + for j, b1 in enumerate(_make_multiples(b)): + check_splev(b1, j) + for der in range(1, k+1): + check_splev(b1, j, der, 1e-12, 1e-12) + + +### stolen from @pv, verbatim +def _naive_B(x, k, i, t): + """ + Naive way to compute B-spline basis functions. Useful only for testing! + computes B(x; t[i],..., t[i+k+1]) + """ + if k == 0: + return 1.0 if t[i] <= x < t[i+1] else 0.0 + if t[i+k] == t[i]: + c1 = 0.0 + else: + c1 = (x - t[i])/(t[i+k] - t[i]) * _naive_B(x, k-1, i, t) + if t[i+k+1] == t[i+1]: + c2 = 0.0 + else: + c2 = (t[i+k+1] - x)/(t[i+k+1] - t[i+1]) * _naive_B(x, k-1, i+1, t) + return (c1 + c2) + + +### stolen from @pv, verbatim +def _naive_eval(x, t, c, k): + """ + Naive B-spline evaluation. Useful only for testing! + """ + if x == t[k]: + i = k + else: + i = np.searchsorted(t, x) - 1 + assert t[i] <= x <= t[i+1] + assert i >= k and i < len(t) - k + return sum(c[i-j] * _naive_B(x, k, i-j, t) for j in range(0, k+1)) + + +def _naive_eval_2(x, t, c, k): + """Naive B-spline evaluation, another way.""" + n = len(t) - (k+1) + assert n >= k+1 + assert len(c) >= n + assert t[k] <= x <= t[n] + return sum(c[i] * _naive_B(x, k, i, t) for i in range(n)) + + +def _sum_basis_elements(x, t, c, k): + n = len(t) - (k+1) + assert n >= k+1 + assert len(c) >= n + s = 0. + for i in range(n): + b = BSpline.basis_element(t[i:i+k+2], extrapolate=False)(x) + s += c[i] * np.nan_to_num(b) # zero out out-of-bounds elements + return s + + +def B_012(x): + """ A linear B-spline function B(x | 0, 1, 2).""" + x = np.atleast_1d(x) + return np.piecewise(x, [(x < 0) | (x > 2), + (x >= 0) & (x < 1), + (x >= 1) & (x <= 2)], + [lambda x: 0., lambda x: x, lambda x: 2.-x]) + + +def B_0123(x, der=0): + """A quadratic B-spline function B(x | 0, 1, 2, 3).""" + x = np.atleast_1d(x) + conds = [x < 1, (x > 1) & (x < 2), x > 2] + if der == 0: + funcs = [lambda x: x*x/2., + lambda x: 3./4 - (x-3./2)**2, + lambda x: (3.-x)**2 / 2] + elif der == 2: + funcs = [lambda x: 1., + lambda x: -2., + lambda x: 1.] + else: + raise ValueError(f'never be here: der={der}') + pieces = np.piecewise(x, conds, funcs) + return pieces + + +def _make_random_spline(n=35, k=3): + rng = np.random.RandomState(123) + t = np.sort(rng.random(n+k+1)) + c = rng.random(n) + return BSpline.construct_fast(t, c, k) + + +def _make_multiples(b): + """Increase knot multiplicity.""" + c, k = b.c, b.k + + t1 = b.t.copy() + t1[17:19] = t1[17] + t1[22] = t1[21] + yield BSpline(t1, c, k) + + t1 = b.t.copy() + t1[:k+1] = t1[0] + yield BSpline(t1, c, k) + + t1 = b.t.copy() + t1[-k-1:] = t1[-1] + yield BSpline(t1, c, k) + + +class TestInterop: + # + # Test that FITPACK-based spl* functions can deal with BSpline objects + # + def setup_method(self): + xx = np.linspace(0, 4.*np.pi, 41) + yy = np.cos(xx) + b = make_interp_spline(xx, yy) + self.tck = (b.t, b.c, b.k) + self.xx, self.yy, self.b = xx, yy, b + + self.xnew = np.linspace(0, 4.*np.pi, 21) + + c2 = np.c_[b.c, b.c, b.c] + self.c2 = np.dstack((c2, c2)) + self.b2 = BSpline(b.t, self.c2, b.k) + + def test_splev(self): + xnew, b, b2 = self.xnew, self.b, self.b2 + + # check that splev works with 1-D array of coefficients + # for array and scalar `x` + xp_assert_close(splev(xnew, b), + b(xnew), atol=1e-15, rtol=1e-15) + xp_assert_close(splev(xnew, b.tck), + b(xnew), atol=1e-15, rtol=1e-15) + xp_assert_close(np.asarray([splev(x, b) for x in xnew]), + b(xnew), atol=1e-15, rtol=1e-15) + + # With N-D coefficients, there's a quirck: + # splev(x, BSpline) is equivalent to BSpline(x) + with assert_raises(ValueError, match="Calling splev.. with BSpline"): + splev(xnew, b2) + + # However, splev(x, BSpline.tck) needs some transposes. This is because + # BSpline interpolates along the first axis, while the legacy FITPACK + # wrapper does list(map(...)) which effectively interpolates along the + # last axis. Like so: + sh = tuple(range(1, b2.c.ndim)) + (0,) # sh = (1, 2, 0) + cc = b2.c.transpose(sh) + tck = (b2.t, cc, b2.k) + xp_assert_close(np.asarray(splev(xnew, tck)), + b2(xnew).transpose(sh), atol=1e-15, rtol=1e-15) + + def test_splrep(self): + x, y = self.xx, self.yy + # test that "new" splrep is equivalent to _impl.splrep + tck = splrep(x, y) + t, c, k = _impl.splrep(x, y) + xp_assert_close(tck[0], t, atol=1e-15) + xp_assert_close(tck[1], c, atol=1e-15) + assert tck[2] == k + + # also cover the `full_output=True` branch + tck_f, _, _, _ = splrep(x, y, full_output=True) + xp_assert_close(tck_f[0], t, atol=1e-15) + xp_assert_close(tck_f[1], c, atol=1e-15) + assert tck_f[2] == k + + # test that the result of splrep roundtrips with splev: + # evaluate the spline on the original `x` points + yy = splev(x, tck) + xp_assert_close(y, yy, atol=1e-15) + + # ... and also it roundtrips if wrapped in a BSpline + b = BSpline(*tck) + xp_assert_close(y, b(x), atol=1e-15) + + def test_splrep_errors(self): + # test that both "old" and "new" splrep raise for an N-D ``y`` array + # with n > 1 + x, y = self.xx, self.yy + y2 = np.c_[y, y] + with assert_raises(ValueError): + splrep(x, y2) + with assert_raises(ValueError): + _impl.splrep(x, y2) + + # input below minimum size + with assert_raises(TypeError, match="m > k must hold"): + splrep(x[:3], y[:3]) + with assert_raises(TypeError, match="m > k must hold"): + _impl.splrep(x[:3], y[:3]) + + def test_splprep(self): + x = np.arange(15, dtype=np.float64).reshape((3, 5)) + b, u = splprep(x) + tck, u1 = _impl.splprep(x) + + # test the roundtrip with splev for both "old" and "new" output + xp_assert_close(u, u1, atol=1e-15) + xp_assert_close(np.asarray(splev(u, b)), x, atol=1e-15) + xp_assert_close(np.asarray(splev(u, tck)), x, atol=1e-15) + + # cover the ``full_output=True`` branch + (b_f, u_f), _, _, _ = splprep(x, s=0, full_output=True) + xp_assert_close(u, u_f, atol=1e-15) + xp_assert_close(np.asarray(splev(u_f, b_f)), x, atol=1e-15) + + def test_splprep_errors(self): + # test that both "old" and "new" code paths raise for x.ndim > 2 + x = np.arange(3*4*5).reshape((3, 4, 5)) + with assert_raises(ValueError, match="too many values to unpack"): + splprep(x) + with assert_raises(ValueError, match="too many values to unpack"): + _impl.splprep(x) + + # input below minimum size + x = np.linspace(0, 40, num=3) + with assert_raises(TypeError, match="m > k must hold"): + splprep([x]) + with assert_raises(TypeError, match="m > k must hold"): + _impl.splprep([x]) + + # automatically calculated parameters are non-increasing + # see gh-7589 + x = [-50.49072266, -50.49072266, -54.49072266, -54.49072266] + with assert_raises(ValueError, match="Invalid inputs"): + splprep([x]) + with assert_raises(ValueError, match="Invalid inputs"): + _impl.splprep([x]) + + # given non-increasing parameter values u + x = [1, 3, 2, 4] + u = [0, 0.3, 0.2, 1] + with assert_raises(ValueError, match="Invalid inputs"): + splprep(*[[x], None, u]) + + def test_sproot(self): + b, b2 = self.b, self.b2 + roots = np.array([0.5, 1.5, 2.5, 3.5])*np.pi + # sproot accepts a BSpline obj w/ 1-D coef array + xp_assert_close(sproot(b), roots, atol=1e-7, rtol=1e-7) + xp_assert_close(sproot((b.t, b.c, b.k)), roots, atol=1e-7, rtol=1e-7) + + # ... and deals with trailing dimensions if coef array is N-D + with assert_raises(ValueError, match="Calling sproot.. with BSpline"): + sproot(b2, mest=50) + + # and legacy behavior is preserved for a tck tuple w/ N-D coef + c2r = b2.c.transpose(1, 2, 0) + rr = np.asarray(sproot((b2.t, c2r, b2.k), mest=50)) + assert rr.shape == (3, 2, 4) + xp_assert_close(rr - roots, np.zeros_like(rr), atol=1e-12) + + def test_splint(self): + # test that splint accepts BSpline objects + b, b2 = self.b, self.b2 + + xp_assert_close(splint(0, 1, b), + splint(0, 1, b.tck), atol=1e-14, check_0d=False) + xp_assert_close(splint(0, 1, b), + b.integrate(0, 1), atol=1e-14, check_0d=False) + + # ... and deals with N-D arrays of coefficients + with assert_raises(ValueError, match="Calling splint.. with BSpline"): + splint(0, 1, b2) + + # and the legacy behavior is preserved for a tck tuple w/ N-D coef + c2r = b2.c.transpose(1, 2, 0) + integr = np.asarray(splint(0, 1, (b2.t, c2r, b2.k))) + assert integr.shape == (3, 2) + xp_assert_close(integr, + splint(0, 1, b), atol=1e-14, check_shape=False) + + def test_splder(self): + for b in [self.b, self.b2]: + # pad the c array (FITPACK convention) + ct = len(b.t) - len(b.c) + b_c = b.c.copy() + if ct > 0: + b_c = np.r_[b_c, np.zeros((ct,) + b_c.shape[1:])] + + for n in [1, 2, 3]: + bd = splder(b) + tck_d = _impl.splder((b.t.copy(), b_c, b.k)) + xp_assert_close(bd.t, tck_d[0], atol=1e-15) + xp_assert_close(bd.c, tck_d[1], atol=1e-15) + assert bd.k == tck_d[2] + assert isinstance(bd, BSpline) + assert isinstance(tck_d, tuple) # back-compat: tck in and out + + def test_splantider(self): + for b in [self.b, self.b2]: + # pad the c array (FITPACK convention) + ct = len(b.t) - len(b.c) + b_c = b.c.copy() + if ct > 0: + b_c = np.r_[b_c, np.zeros((ct,) + b_c.shape[1:])] + + for n in [1, 2, 3]: + bd = splantider(b) + tck_d = _impl.splantider((b.t.copy(), b_c, b.k)) + xp_assert_close(bd.t, tck_d[0], atol=1e-15) + xp_assert_close(bd.c, tck_d[1], atol=1e-15) + assert bd.k == tck_d[2] + assert isinstance(bd, BSpline) + assert isinstance(tck_d, tuple) # back-compat: tck in and out + + def test_insert(self): + b, b2, xx = self.b, self.b2, self.xx + + j = b.t.size // 2 + tn = 0.5*(b.t[j] + b.t[j+1]) + + bn, tck_n = insert(tn, b), insert(tn, (b.t, b.c, b.k)) + xp_assert_close(splev(xx, bn), + splev(xx, tck_n), atol=1e-15) + assert isinstance(bn, BSpline) + assert isinstance(tck_n, tuple) # back-compat: tck in, tck out + + # for N-D array of coefficients, BSpline.c needs to be transposed + # after that, the results are equivalent. + sh = tuple(range(b2.c.ndim)) + c_ = b2.c.transpose(sh[1:] + (0,)) + tck_n2 = insert(tn, (b2.t, c_, b2.k)) + + bn2 = insert(tn, b2) + + # need a transpose for comparing the results, cf test_splev + xp_assert_close(np.asarray(splev(xx, tck_n2)).transpose(2, 0, 1), + bn2(xx), atol=1e-15) + assert isinstance(bn2, BSpline) + assert isinstance(tck_n2, tuple) # back-compat: tck in, tck out + + +class TestInterp: + # + # Test basic ways of constructing interpolating splines. + # + xx = np.linspace(0., 2.*np.pi) + yy = np.sin(xx) + + def test_non_int_order(self): + with assert_raises(TypeError): + make_interp_spline(self.xx, self.yy, k=2.5) + + def test_order_0(self): + b = make_interp_spline(self.xx, self.yy, k=0) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + b = make_interp_spline(self.xx, self.yy, k=0, axis=-1) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + + def test_linear(self): + b = make_interp_spline(self.xx, self.yy, k=1) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + b = make_interp_spline(self.xx, self.yy, k=1, axis=-1) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + + @pytest.mark.parametrize('k', [0, 1, 2, 3]) + def test_incompatible_x_y(self, k): + x = [0, 1, 2, 3, 4, 5] + y = [0, 1, 2, 3, 4, 5, 6, 7] + with assert_raises(ValueError, match="Shapes of x"): + make_interp_spline(x, y, k=k) + + @pytest.mark.parametrize('k', [0, 1, 2, 3]) + def test_broken_x(self, k): + x = [0, 1, 1, 2, 3, 4] # duplicates + y = [0, 1, 2, 3, 4, 5] + with assert_raises(ValueError, match="x to not have duplicates"): + make_interp_spline(x, y, k=k) + + x = [0, 2, 1, 3, 4, 5] # unsorted + with assert_raises(ValueError, match="Expect x to be a 1D strictly"): + make_interp_spline(x, y, k=k) + + x = [0, 1, 2, 3, 4, 5] + x = np.asarray(x).reshape((1, -1)) # 1D + with assert_raises(ValueError, match="Expect x to be a 1D strictly"): + make_interp_spline(x, y, k=k) + + def test_not_a_knot(self): + for k in [2, 3, 4, 5, 6, 7]: + b = make_interp_spline(self.xx, self.yy, k) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + + def test_periodic(self): + # k = 5 here for more derivatives + b = make_interp_spline(self.xx, self.yy, k=5, bc_type='periodic') + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + # in periodic case it is expected equality of k-1 first + # derivatives at the boundaries + for i in range(1, 5): + xp_assert_close(b(self.xx[0], nu=i), b(self.xx[-1], nu=i), atol=1e-11) + # tests for axis=-1 + b = make_interp_spline(self.xx, self.yy, k=5, bc_type='periodic', axis=-1) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + for i in range(1, 5): + xp_assert_close(b(self.xx[0], nu=i), b(self.xx[-1], nu=i), atol=1e-11) + + @pytest.mark.parametrize('k', [2, 3, 4, 5, 6, 7]) + def test_periodic_random(self, k): + # tests for both cases (k > n and k <= n) + n = 5 + rng = np.random.RandomState(1234) + x = np.sort(rng.random_sample(n) * 10) + y = rng.random_sample(n) * 100 + y[0] = y[-1] + b = make_interp_spline(x, y, k=k, bc_type='periodic') + xp_assert_close(b(x), y, atol=1e-14) + + def test_periodic_axis(self): + n = self.xx.shape[0] + rng = np.random.RandomState(1234) + x = rng.random_sample(n) * 2 * np.pi + x = np.sort(x) + x[0] = 0. + x[-1] = 2 * np.pi + y = np.zeros((2, n)) + y[0] = np.sin(x) + y[1] = np.cos(x) + b = make_interp_spline(x, y, k=5, bc_type='periodic', axis=1) + for i in range(n): + xp_assert_close(b(x[i]), y[:, i], atol=1e-14) + xp_assert_close(b(x[0]), b(x[-1]), atol=1e-14) + + def test_periodic_points_exception(self): + # first and last points should match when periodic case expected + rng = np.random.RandomState(1234) + k = 5 + n = 8 + x = np.sort(rng.random_sample(n)) + y = rng.random_sample(n) + y[0] = y[-1] - 1 # to be sure that they are not equal + with assert_raises(ValueError): + make_interp_spline(x, y, k=k, bc_type='periodic') + + def test_periodic_knots_exception(self): + # `periodic` case does not work with passed vector of knots + rng = np.random.RandomState(1234) + k = 3 + n = 7 + x = np.sort(rng.random_sample(n)) + y = rng.random_sample(n) + t = np.zeros(n + 2 * k) + with assert_raises(ValueError): + make_interp_spline(x, y, k, t, 'periodic') + + @pytest.mark.parametrize('k', [2, 3, 4, 5]) + def test_periodic_splev(self, k): + # comparison values of periodic b-spline with splev + b = make_interp_spline(self.xx, self.yy, k=k, bc_type='periodic') + tck = splrep(self.xx, self.yy, per=True, k=k) + spl = splev(self.xx, tck) + xp_assert_close(spl, b(self.xx), atol=1e-14) + + # comparison derivatives of periodic b-spline with splev + for i in range(1, k): + spl = splev(self.xx, tck, der=i) + xp_assert_close(spl, b(self.xx, nu=i), atol=1e-10) + + def test_periodic_cubic(self): + # comparison values of cubic periodic b-spline with CubicSpline + b = make_interp_spline(self.xx, self.yy, k=3, bc_type='periodic') + cub = CubicSpline(self.xx, self.yy, bc_type='periodic') + xp_assert_close(b(self.xx), cub(self.xx), atol=1e-14) + + # edge case: Cubic interpolation on 3 points + rng = np.random.RandomState(1234) + n = 3 + x = np.sort(rng.random_sample(n) * 10) + y = rng.random_sample(n) * 100 + y[0] = y[-1] + b = make_interp_spline(x, y, k=3, bc_type='periodic') + cub = CubicSpline(x, y, bc_type='periodic') + xp_assert_close(b(x), cub(x), atol=1e-14) + + def test_periodic_full_matrix(self): + # comparison values of cubic periodic b-spline with + # solution of the system with full matrix + k = 3 + b = make_interp_spline(self.xx, self.yy, k=k, bc_type='periodic') + t = _periodic_knots(self.xx, k) + c = _make_interp_per_full_matr(self.xx, self.yy, t, k) + b1 = np.vectorize(lambda x: _naive_eval(x, t, c, k)) + xp_assert_close(b(self.xx), b1(self.xx), atol=1e-14) + + def test_quadratic_deriv(self): + der = [(1, 8.)] # order, value: f'(x) = 8. + + # derivative at right-hand edge + b = make_interp_spline(self.xx, self.yy, k=2, bc_type=(None, der)) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + xp_assert_close( + b(self.xx[-1], 1), der[0][1], atol=1e-14, rtol=1e-14, check_0d=False + ) + + # derivative at left-hand edge + b = make_interp_spline(self.xx, self.yy, k=2, bc_type=(der, None)) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + xp_assert_close( + b(self.xx[0], 1), der[0][1], atol=1e-14, rtol=1e-14, check_0d=False + ) + + def test_cubic_deriv(self): + k = 3 + + # first derivatives at left & right edges: + der_l, der_r = [(1, 3.)], [(1, 4.)] + b = make_interp_spline(self.xx, self.yy, k, bc_type=(der_l, der_r)) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + xp_assert_close(np.asarray([b(self.xx[0], 1), b(self.xx[-1], 1)]), + np.asarray([der_l[0][1], der_r[0][1]]), atol=1e-14, rtol=1e-14) + + # 'natural' cubic spline, zero out 2nd derivatives at the boundaries + der_l, der_r = [(2, 0)], [(2, 0)] + b = make_interp_spline(self.xx, self.yy, k, bc_type=(der_l, der_r)) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + + def test_quintic_derivs(self): + k, n = 5, 7 + x = np.arange(n).astype(np.float64) + y = np.sin(x) + der_l = [(1, -12.), (2, 1)] + der_r = [(1, 8.), (2, 3.)] + b = make_interp_spline(x, y, k=k, bc_type=(der_l, der_r)) + xp_assert_close(b(x), y, atol=1e-14, rtol=1e-14) + xp_assert_close(np.asarray([b(x[0], 1), b(x[0], 2)]), + np.asarray([val for (nu, val) in der_l])) + xp_assert_close(np.asarray([b(x[-1], 1), b(x[-1], 2)]), + np.asarray([val for (nu, val) in der_r])) + + @pytest.mark.xfail(reason='unstable') + def test_cubic_deriv_unstable(self): + # 1st and 2nd derivative at x[0], no derivative information at x[-1] + # The problem is not that it fails [who would use this anyway], + # the problem is that it fails *silently*, and I've no idea + # how to detect this sort of instability. + # In this particular case: it's OK for len(t) < 20, goes haywire + # at larger `len(t)`. + k = 3 + t = _augknt(self.xx, k) + + der_l = [(1, 3.), (2, 4.)] + b = make_interp_spline(self.xx, self.yy, k, t, bc_type=(der_l, None)) + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + + def test_knots_not_data_sites(self): + # Knots need not coincide with the data sites. + # use a quadratic spline, knots are at data averages, + # two additional constraints are zero 2nd derivatives at edges + k = 2 + t = np.r_[(self.xx[0],)*(k+1), + (self.xx[1:] + self.xx[:-1]) / 2., + (self.xx[-1],)*(k+1)] + b = make_interp_spline(self.xx, self.yy, k, t, + bc_type=([(2, 0)], [(2, 0)])) + + xp_assert_close(b(self.xx), self.yy, atol=1e-14, rtol=1e-14) + xp_assert_close(b(self.xx[0], 2), np.asarray(0.0), atol=1e-14) + xp_assert_close(b(self.xx[-1], 2), np.asarray(0.0), atol=1e-14) + + def test_minimum_points_and_deriv(self): + # interpolation of f(x) = x**3 between 0 and 1. f'(x) = 3 * xx**2 and + # f'(0) = 0, f'(1) = 3. + k = 3 + x = [0., 1.] + y = [0., 1.] + b = make_interp_spline(x, y, k, bc_type=([(1, 0.)], [(1, 3.)])) + + xx = np.linspace(0., 1.) + yy = xx**3 + xp_assert_close(b(xx), yy, atol=1e-14, rtol=1e-14) + + def test_deriv_spec(self): + # If one of the derivatives is omitted, the spline definition is + # incomplete. + x = y = [1.0, 2, 3, 4, 5, 6] + + with assert_raises(ValueError): + make_interp_spline(x, y, bc_type=([(1, 0.)], None)) + + with assert_raises(ValueError): + make_interp_spline(x, y, bc_type=(1, 0.)) + + with assert_raises(ValueError): + make_interp_spline(x, y, bc_type=[(1, 0.)]) + + with assert_raises(ValueError): + make_interp_spline(x, y, bc_type=42) + + # CubicSpline expects`bc_type=(left_pair, right_pair)`, while + # here we expect `bc_type=(iterable, iterable)`. + l, r = (1, 0.0), (1, 0.0) + with assert_raises(ValueError): + make_interp_spline(x, y, bc_type=(l, r)) + + def test_deriv_order_too_large(self): + x = np.arange(7) + y = x**2 + l, r = [(6, 0)], [(1, 0)] # 6th derivative = 0 at x[0] for k=3 + with assert_raises(ValueError, match="Bad boundary conditions at 0."): + # cannot fix 6th derivative at x[0]: does not segfault + make_interp_spline(x, y, bc_type=(l, r)) + + l, r = [(1, 0)], [(-6, 0)] # derivative order < 0 at x[-1] + with assert_raises(ValueError, match="Bad boundary conditions at 6."): + # does not segfault + make_interp_spline(x, y, bc_type=(l, r)) + + def test_complex(self): + k = 3 + xx = self.xx + yy = self.yy + 1.j*self.yy + + # first derivatives at left & right edges: + der_l, der_r = [(1, 3.j)], [(1, 4.+2.j)] + b = make_interp_spline(xx, yy, k, bc_type=(der_l, der_r)) + xp_assert_close(b(xx), yy, atol=1e-14, rtol=1e-14) + xp_assert_close( + b(xx[0], 1), der_l[0][1], atol=1e-14, rtol=1e-14, check_0d=False + ) + xp_assert_close( + b(xx[-1], 1), der_r[0][1], atol=1e-14, rtol=1e-14, check_0d=False + ) + + # also test zero and first order + for k in (0, 1): + b = make_interp_spline(xx, yy, k=k) + xp_assert_close(b(xx), yy, atol=1e-14, rtol=1e-14) + + def test_int_xy(self): + x = np.arange(10).astype(int) + y = np.arange(10).astype(int) + + # Cython chokes on "buffer type mismatch" (construction) or + # "no matching signature found" (evaluation) + for k in (0, 1, 2, 3): + b = make_interp_spline(x, y, k=k) + b(x) + + def test_sliced_input(self): + # Cython code chokes on non C contiguous arrays + xx = np.linspace(-1, 1, 100) + + x = xx[::5] + y = xx[::5] + + for k in (0, 1, 2, 3): + make_interp_spline(x, y, k=k) + + def test_check_finite(self): + # check_finite defaults to True; nans and such trigger a ValueError + x = np.arange(10).astype(float) + y = x**2 + + for z in [np.nan, np.inf, -np.inf]: + y[-1] = z + assert_raises(ValueError, make_interp_spline, x, y) + + @pytest.mark.parametrize('k', [1, 2, 3, 5]) + def test_list_input(self, k): + # regression test for gh-8714: TypeError for x, y being lists and k=2 + x = list(range(10)) + y = [a**2 for a in x] + make_interp_spline(x, y, k=k) + + def test_multiple_rhs(self): + yy = np.c_[np.sin(self.xx), np.cos(self.xx)] + der_l = [(1, [1., 2.])] + der_r = [(1, [3., 4.])] + + b = make_interp_spline(self.xx, yy, k=3, bc_type=(der_l, der_r)) + xp_assert_close(b(self.xx), yy, atol=1e-14, rtol=1e-14) + xp_assert_close(b(self.xx[0], 1), der_l[0][1], atol=1e-14, rtol=1e-14) + xp_assert_close(b(self.xx[-1], 1), der_r[0][1], atol=1e-14, rtol=1e-14) + + def test_shapes(self): + rng = np.random.RandomState(1234) + k, n = 3, 22 + x = np.sort(rng.random(size=n)) + y = rng.random(size=(n, 5, 6, 7)) + + b = make_interp_spline(x, y, k) + assert b.c.shape == (n, 5, 6, 7) + + # now throw in some derivatives + d_l = [(1, rng.random((5, 6, 7)))] + d_r = [(1, rng.random((5, 6, 7)))] + b = make_interp_spline(x, y, k, bc_type=(d_l, d_r)) + assert b.c.shape == (n + k - 1, 5, 6, 7) + + def test_string_aliases(self): + yy = np.sin(self.xx) + + # a single string is duplicated + b1 = make_interp_spline(self.xx, yy, k=3, bc_type='natural') + b2 = make_interp_spline(self.xx, yy, k=3, bc_type=([(2, 0)], [(2, 0)])) + xp_assert_close(b1.c, b2.c, atol=1e-15) + + # two strings are handled + b1 = make_interp_spline(self.xx, yy, k=3, + bc_type=('natural', 'clamped')) + b2 = make_interp_spline(self.xx, yy, k=3, + bc_type=([(2, 0)], [(1, 0)])) + xp_assert_close(b1.c, b2.c, atol=1e-15) + + # one-sided BCs are OK + b1 = make_interp_spline(self.xx, yy, k=2, bc_type=(None, 'clamped')) + b2 = make_interp_spline(self.xx, yy, k=2, bc_type=(None, [(1, 0.0)])) + xp_assert_close(b1.c, b2.c, atol=1e-15) + + # 'not-a-knot' is equivalent to None + b1 = make_interp_spline(self.xx, yy, k=3, bc_type='not-a-knot') + b2 = make_interp_spline(self.xx, yy, k=3, bc_type=None) + xp_assert_close(b1.c, b2.c, atol=1e-15) + + # unknown strings do not pass + with assert_raises(ValueError): + make_interp_spline(self.xx, yy, k=3, bc_type='typo') + + # string aliases are handled for 2D values + yy = np.c_[np.sin(self.xx), np.cos(self.xx)] + der_l = [(1, [0., 0.])] + der_r = [(2, [0., 0.])] + b2 = make_interp_spline(self.xx, yy, k=3, bc_type=(der_l, der_r)) + b1 = make_interp_spline(self.xx, yy, k=3, + bc_type=('clamped', 'natural')) + xp_assert_close(b1.c, b2.c, atol=1e-15) + + # ... and for N-D values: + rng = np.random.RandomState(1234) + k, n = 3, 22 + x = np.sort(rng.random(size=n)) + y = rng.random(size=(n, 5, 6, 7)) + + # now throw in some derivatives + d_l = [(1, np.zeros((5, 6, 7)))] + d_r = [(1, np.zeros((5, 6, 7)))] + b1 = make_interp_spline(x, y, k, bc_type=(d_l, d_r)) + b2 = make_interp_spline(x, y, k, bc_type='clamped') + xp_assert_close(b1.c, b2.c, atol=1e-15) + + def test_full_matrix(self): + rng = np.random.RandomState(1234) + k, n = 3, 7 + x = np.sort(rng.random(size=n)) + y = rng.random(size=n) + t = _not_a_knot(x, k) + + b = make_interp_spline(x, y, k, t) + cf = make_interp_full_matr(x, y, t, k) + xp_assert_close(b.c, cf, atol=1e-14, rtol=1e-14) + + def test_woodbury(self): + ''' + Random elements in diagonal matrix with blocks in the + left lower and right upper corners checking the + implementation of Woodbury algorithm. + ''' + rng = np.random.RandomState(1234) + n = 201 + for k in range(3, 32, 2): + offset = int((k - 1) / 2) + a = np.diagflat(rng.random((1, n))) + for i in range(1, offset + 1): + a[:-i, i:] += np.diagflat(rng.random((1, n - i))) + a[i:, :-i] += np.diagflat(rng.random((1, n - i))) + ur = rng.random((offset, offset)) + a[:offset, -offset:] = ur + ll = rng.random((offset, offset)) + a[-offset:, :offset] = ll + d = np.zeros((k, n)) + for i, j in enumerate(range(offset, -offset - 1, -1)): + if j < 0: + d[i, :j] = np.diagonal(a, offset=j) + else: + d[i, j:] = np.diagonal(a, offset=j) + b = rng.random(n) + xp_assert_close(_woodbury_algorithm(d, ur, ll, b, k), + np.linalg.solve(a, b), atol=1e-14) + + +def make_interp_full_matr(x, y, t, k): + """Assemble an spline order k with knots t to interpolate + y(x) using full matrices. + Not-a-knot BC only. + + This routine is here for testing only (even though it's functional). + """ + assert x.size == y.size + assert t.size == x.size + k + 1 + n = x.size + + A = np.zeros((n, n), dtype=np.float64) + + for j in range(n): + xval = x[j] + if xval == t[k]: + left = k + else: + left = np.searchsorted(t, xval) - 1 + + # fill a row + bb = _dierckx.evaluate_all_bspl(t, k, xval, left) + A[j, left-k:left+1] = bb + + c = sl.solve(A, y) + return c + + +def make_lsq_full_matrix(x, y, t, k=3): + """Make the least-square spline, full matrices.""" + x, y, t = map(np.asarray, (x, y, t)) + m = x.size + n = t.size - k - 1 + + A = np.zeros((m, n), dtype=np.float64) + + for j in range(m): + xval = x[j] + # find interval + if xval == t[k]: + left = k + else: + left = np.searchsorted(t, xval) - 1 + + # fill a row + bb = _dierckx.evaluate_all_bspl(t, k, xval, left) + A[j, left-k:left+1] = bb + + # have observation matrix, can solve the LSQ problem + B = np.dot(A.T, A) + Y = np.dot(A.T, y) + c = sl.solve(B, Y) + + return c, (A, Y) + + +parametrize_lsq_methods = pytest.mark.parametrize("method", ["norm-eq", "qr"]) + +class TestLSQ: + # + # Test make_lsq_spline + # + rng = np.random.RandomState(1234) + n, k = 13, 3 + x = np.sort(rng.random(n)) + y = rng.random(n) + t = _augknt(np.linspace(x[0], x[-1], 7), k) + + @parametrize_lsq_methods + def test_lstsq(self, method): + # check LSQ construction vs a full matrix version + x, y, t, k = self.x, self.y, self.t, self.k + + c0, AY = make_lsq_full_matrix(x, y, t, k) + b = make_lsq_spline(x, y, t, k, method=method) + + xp_assert_close(b.c, c0) + assert b.c.shape == (t.size - k - 1,) + + # also check against numpy.lstsq + aa, yy = AY + c1, _, _, _ = np.linalg.lstsq(aa, y, rcond=-1) + xp_assert_close(b.c, c1) + + @parametrize_lsq_methods + def test_weights(self, method): + # weights = 1 is same as None + x, y, t, k = self.x, self.y, self.t, self.k + w = np.ones_like(x) + + b = make_lsq_spline(x, y, t, k, method=method) + b_w = make_lsq_spline(x, y, t, k, w=w, method=method) + + xp_assert_close(b.t, b_w.t, atol=1e-14) + xp_assert_close(b.c, b_w.c, atol=1e-14) + assert b.k == b_w.k + + def test_weights_same(self): + # both methods treat weights + x, y, t, k = self.x, self.y, self.t, self.k + w = np.random.default_rng(1234).uniform(size=x.shape[0]) + + b_ne = make_lsq_spline(x, y, t, k, w=w, method="norm-eq") + b_qr = make_lsq_spline(x, y, t, k, w=w, method="qr") + b_no_w = make_lsq_spline(x, y, t, k, method="qr") + + xp_assert_close(b_ne.c, b_qr.c, atol=1e-14) + assert not np.allclose(b_no_w.c, b_qr.c, atol=1e-14) + + @parametrize_lsq_methods + def test_multiple_rhs(self, method): + x, t, k, n = self.x, self.t, self.k, self.n + rng = np.random.RandomState(1234) + y = rng.random(size=(n, 5, 6, 7)) + b = make_lsq_spline(x, y, t, k, method=method) + assert b.c.shape == (t.size-k-1, 5, 6, 7) + + @parametrize_lsq_methods + def test_multiple_rhs_2(self, method): + x, t, k, n = self.x, self.t, self.k, self.n + nrhs = 3 + rng = np.random.RandomState(1234) + y = rng.random(size=(n, nrhs)) + b = make_lsq_spline(x, y, t, k, method=method) + + bb = [make_lsq_spline(x, y[:, i], t, k, method=method) + for i in range(nrhs)] + coefs = np.vstack([bb[i].c for i in range(nrhs)]).T + + xp_assert_close(coefs, b.c, atol=1e-15) + + def test_multiple_rhs_3(self): + x, t, k, n = self.x, self.t, self.k, self.n + nrhs = 3 + y = np.random.random(size=(n, nrhs)) + b_qr = make_lsq_spline(x, y, t, k, method="qr") + b_neq = make_lsq_spline(x, y, t, k, method="norm-eq") + xp_assert_close(b_qr.c, b_neq.c, atol=1e-15) + + @parametrize_lsq_methods + def test_complex(self, method): + # cmplx-valued `y` + x, t, k = self.x, self.t, self.k + yc = self.y * (1. + 2.j) + + b = make_lsq_spline(x, yc, t, k, method=method) + b_re = make_lsq_spline(x, yc.real, t, k, method=method) + b_im = make_lsq_spline(x, yc.imag, t, k, method=method) + + xp_assert_close(b(x), b_re(x) + 1.j*b_im(x), atol=1e-15, rtol=1e-15) + + def test_complex_2(self): + # test complex-valued y with y.ndim > 1 + + x, t, k = self.x, self.t, self.k + yc = self.y * (1. + 2.j) + yc = np.stack((yc, yc), axis=1) + + b = make_lsq_spline(x, yc, t, k) + b_re = make_lsq_spline(x, yc.real, t, k) + b_im = make_lsq_spline(x, yc.imag, t, k) + + xp_assert_close(b(x), b_re(x) + 1.j*b_im(x), atol=1e-15, rtol=1e-15) + + # repeat with num_trailing_dims > 1 : yc.shape[1:] = (2, 2) + yc = np.stack((yc, yc), axis=1) + + b = make_lsq_spline(x, yc, t, k) + b_re = make_lsq_spline(x, yc.real, t, k) + b_im = make_lsq_spline(x, yc.imag, t, k) + + xp_assert_close(b(x), b_re(x) + 1.j*b_im(x), atol=1e-15, rtol=1e-15) + + @parametrize_lsq_methods + def test_int_xy(self, method): + x = np.arange(10).astype(int) + y = np.arange(10).astype(int) + t = _augknt(x, k=1) + # Cython chokes on "buffer type mismatch" + make_lsq_spline(x, y, t, k=1, method=method) + + @parametrize_lsq_methods + def test_f32_xy(self, method): + x = np.arange(10, dtype=np.float32) + y = np.arange(10, dtype=np.float32) + t = _augknt(x, k=1) + spl_f32 = make_lsq_spline(x, y, t, k=1, method=method) + spl_f64 = make_lsq_spline( + x.astype(float), y.astype(float), t.astype(float), k=1, method=method + ) + + x2 = (x[1:] + x[:-1]) / 2.0 + xp_assert_close(spl_f32(x2), spl_f64(x2), atol=1e-15) + + @parametrize_lsq_methods + def test_sliced_input(self, method): + # Cython code chokes on non C contiguous arrays + xx = np.linspace(-1, 1, 100) + + x = xx[::3] + y = xx[::3] + t = _augknt(x, 1) + make_lsq_spline(x, y, t, k=1, method=method) + + @parametrize_lsq_methods + def test_checkfinite(self, method): + # check_finite defaults to True; nans and such trigger a ValueError + x = np.arange(12).astype(float) + y = x**2 + t = _augknt(x, 3) + + for z in [np.nan, np.inf, -np.inf]: + y[-1] = z + assert_raises(ValueError, make_lsq_spline, x, y, t, method=method) + + @parametrize_lsq_methods + def test_read_only(self, method): + # Check that make_lsq_spline works with read only arrays + x, y, t = self.x, self.y, self.t + x.setflags(write=False) + y.setflags(write=False) + t.setflags(write=False) + make_lsq_spline(x=x, y=y, t=t, method=method) + + @pytest.mark.parametrize('k', list(range(1, 7))) + def test_qr_vs_norm_eq(self, k): + # check that QR and normal eq solutions match + x, y = self.x, self.y + t = _augknt(np.linspace(x[0], x[-1], 7), k) + spl_norm_eq = make_lsq_spline(x, y, t, k=k, method='norm-eq') + spl_qr = make_lsq_spline(x, y, t, k=k, method='qr') + + xx = (x[1:] + x[:-1]) / 2.0 + xp_assert_close(spl_norm_eq(xx), spl_qr(xx), atol=1e-15) + + def test_duplicates(self): + # method="qr" can handle duplicated data points + x = np.repeat(self.x, 2) + y = np.repeat(self.y, 2) + spl_1 = make_lsq_spline(self.x, self.y, self.t, k=3, method='qr') + spl_2 = make_lsq_spline(x, y, self.t, k=3, method='qr') + + xx = (x[1:] + x[:-1]) / 2.0 + xp_assert_close(spl_1(xx), spl_2(xx), atol=1e-15) + + +class PackedMatrix: + """A simplified CSR format for when non-zeros in each row are consecutive. + + Assuming that each row of an `(m, nc)` matrix 1) only has `nz` non-zeros, and + 2) these non-zeros are consecutive, we only store an `(m, nz)` matrix of + non-zeros and a 1D array of row offsets. This way, a row `i` of the original + matrix A is ``A[i, offset[i]: offset[i] + nz]``. + + """ + def __init__(self, a, offset, nc): + self.a = a + self.offset = offset + self.nc = nc + + assert a.ndim == 2 + assert offset.ndim == 1 + assert a.shape[0] == offset.shape[0] + + @property + def shape(self): + return self.a.shape[0], self.nc + + def todense(self): + out = np.zeros(self.shape) + nelem = self.a.shape[1] + for i in range(out.shape[0]): + nel = min(self.nc - self.offset[i], nelem) + out[i, self.offset[i]:self.offset[i] + nel] = self.a[i, :nel] + return out + + +def _qr_reduce_py(a_p, y, startrow=1): + """This is a python counterpart of the `_qr_reduce` routine, + declared in interpolate/src/__fitpack.h + """ + from scipy.linalg.lapack import dlartg + + # unpack the packed format + a = a_p.a + offset = a_p.offset + nc = a_p.nc + + m, nz = a.shape + + assert y.shape[0] == m + R = a.copy() + y1 = y.copy() + + for i in range(startrow, m): + oi = offset[i] + for j in range(oi, nc): + # rotate only the lower diagonal + if j >= min(i, nc): + break + + # In dense format: diag a1[j, j] vs a1[i, j] + c, s, r = dlartg(R[j, 0], R[i, 0]) + + # rotate l.h.s. + R[j, 0] = r + for l in range(1, nz): + R[j, l], R[i, l-1] = fprota(c, s, R[j, l], R[i, l]) + R[i, -1] = 0.0 + + # rotate r.h.s. + for l in range(y1.shape[1]): + y1[j, l], y1[i, l] = fprota(c, s, y1[j, l], y1[i, l]) + + # convert to packed + offs = list(range(R.shape[0])) + R_p = PackedMatrix(R, np.array(offs, dtype=np.int64), nc) + + return R_p, y1 + + +def fprota(c, s, a, b): + """Givens rotate [a, b]. + + [aa] = [ c s] @ [a] + [bb] [-s c] [b] + + """ + aa = c*a + s*b + bb = -s*a + c*b + return aa, bb + + +def fpback(R_p, y): + """Backsubsitution solve upper triangular banded `R @ c = y.` + + `R` is in the "packed" format: `R[i, :]` is `a[i, i:i+k+1]` + """ + R = R_p.a + _, nz = R.shape + nc = R_p.nc + assert y.shape[0] == R.shape[0] + + c = np.zeros_like(y[:nc]) + c[nc-1, ...] = y[nc-1] / R[nc-1, 0] + for i in range(nc-2, -1, -1): + nel = min(nz, nc-i) + # NB: broadcast R across trailing dimensions of `c`. + summ = (R[i, 1:nel, None] * c[i+1:i+nel, ...]).sum(axis=0) + c[i, ...] = ( y[i] - summ ) / R[i, 0] + return c + + +class TestGivensQR: + # Test row-by-row QR factorization, used for the LSQ spline construction. + # This is implementation detail; still test it separately. + def _get_xyt(self, n): + k = 3 + x = np.arange(n, dtype=float) + y = x**3 + 1/(1+x) + t = _not_a_knot(x, k) + return x, y, t, k + + def test_vs_full(self): + n = 10 + x, y, t, k = self._get_xyt(n) + + # design matrix + a_csr = BSpline.design_matrix(x, t, k) + + # dense QR + q, r = sl.qr(a_csr.todense()) + qTy = q.T @ y + + # prepare the PackedMatrix to factorize + # convert to "packed" format + m, nc = a_csr.shape + assert nc == t.shape[0] - k - 1 + + offset = a_csr.indices[::(k+1)] + offset = np.ascontiguousarray(offset, dtype=np.int64) + A = a_csr.data.reshape(m, k+1) + + R = PackedMatrix(A, offset, nc) + y_ = y[:, None] # _qr_reduce requires `y` a 2D array + _dierckx.qr_reduce(A, offset, nc, y_) # modifies arguments in-place + + # signs may differ + xp_assert_close(np.minimum(R.todense() + r, + R.todense() - r), np.zeros_like(r), atol=1e-15) + xp_assert_close(np.minimum(abs(qTy - y_[:, 0]), + abs(qTy + y_[:, 0])), np.zeros_like(qTy), atol=2e-13) + + # sign changes are consistent between Q and R: + c_full = sl.solve(r, qTy) + c_banded = _dierckx.fpback(R.a, R.nc, y_) + xp_assert_close(c_full, c_banded[:, 0], atol=5e-13) + + def test_py_vs_compiled(self): + # test _qr_reduce vs a python implementation + n = 10 + x, y, t, k = self._get_xyt(n) + + # design matrix + a_csr = BSpline.design_matrix(x, t, k) + m, nc = a_csr.shape + assert nc == t.shape[0] - k - 1 + + offset = a_csr.indices[::(k+1)] + offset = np.ascontiguousarray(offset, dtype=np.int64) + A = a_csr.data.reshape(m, k+1) + + R = PackedMatrix(A, offset, nc) + y_ = y[:, None] + + RR, yy = _qr_reduce_py(R, y_) + _dierckx.qr_reduce(A, offset, nc , y_) # in-place + + xp_assert_close(RR.a, R.a, atol=1e-15) + xp_assert_equal(RR.offset, R.offset, check_dtype=False) + assert RR.nc == R.nc + xp_assert_close(yy, y_, atol=1e-15) + + # Test C-level construction of the design matrix + + def test_data_matrix(self): + n = 10 + x, y, t, k = self._get_xyt(n) + w = np.arange(1, n+1, dtype=float) + + A, offset, nc = _dierckx.data_matrix(x, t, k, w) + + m = x.shape[0] + a_csr = BSpline.design_matrix(x, t, k) + a_w = (a_csr * w[:, None]).tocsr() + A_ = a_w.data.reshape((m, k+1)) + offset_ = a_w.indices[::(k+1)].astype(np.int64) + + xp_assert_close(A, A_, atol=1e-15) + xp_assert_equal(offset, offset_) + assert nc == t.shape[0] - k - 1 + + def test_fpback(self): + n = 10 + x, y, t, k = self._get_xyt(n) + y = np.c_[y, y**2] + A, offset, nc = _dierckx.data_matrix(x, t, k, np.ones_like(x)) + R = PackedMatrix(A, offset, nc) + _dierckx.qr_reduce(A, offset, nc, y) + + c = fpback(R, y) + cc = _dierckx.fpback(A, nc, y) + + xp_assert_close(cc, c, atol=1e-14) + + +def data_file(basename): + return os.path.join(os.path.abspath(os.path.dirname(__file__)), + 'data', basename) + + +class TestSmoothingSpline: + # + # test make_smoothing_spline + # + def test_invalid_input(self): + rng = np.random.RandomState(1234) + n = 100 + x = np.sort(rng.random_sample(n) * 4 - 2) + y = x**2 * np.sin(4 * x) + x**3 + rng.normal(0., 1.5, n) + + # ``x`` and ``y`` should have same shapes (1-D array) + with assert_raises(ValueError): + make_smoothing_spline(x, y[1:]) + with assert_raises(ValueError): + make_smoothing_spline(x[1:], y) + with assert_raises(ValueError): + make_smoothing_spline(x.reshape(1, n), y) + + # ``x`` should be an ascending array + with assert_raises(ValueError): + make_smoothing_spline(x[::-1], y) + + x_dupl = np.copy(x) + x_dupl[0] = x_dupl[1] + + with assert_raises(ValueError): + make_smoothing_spline(x_dupl, y) + + # x and y length must be >= 5 + x = np.arange(4) + y = np.ones(4) + exception_message = "``x`` and ``y`` length must be at least 5" + with pytest.raises(ValueError, match=exception_message): + make_smoothing_spline(x, y) + + def test_compare_with_GCVSPL(self): + """ + Data is generated in the following way: + >>> np.random.seed(1234) + >>> n = 100 + >>> x = np.sort(np.random.random_sample(n) * 4 - 2) + >>> y = np.sin(x) + np.random.normal(scale=.5, size=n) + >>> np.savetxt('x.csv', x) + >>> np.savetxt('y.csv', y) + + We obtain the result of performing the GCV smoothing splines + package (by Woltring, gcvspl) on the sample data points + using its version for Octave (https://github.com/srkuberski/gcvspl). + In order to use this implementation, one should clone the repository + and open the folder in Octave. + In Octave, we load up ``x`` and ``y`` (generated from Python code + above): + + >>> x = csvread('x.csv'); + >>> y = csvread('y.csv'); + + Then, in order to access the implementation, we compile gcvspl files in + Octave: + + >>> mex gcvsplmex.c gcvspl.c + >>> mex spldermex.c gcvspl.c + + The first function computes the vector of unknowns from the dataset + (x, y) while the second one evaluates the spline in certain points + with known vector of coefficients. + + >>> c = gcvsplmex( x, y, 2 ); + >>> y0 = spldermex( x, c, 2, x, 0 ); + + If we want to compare the results of the gcvspl code, we can save + ``y0`` in csv file: + + >>> csvwrite('y0.csv', y0); + + """ + # load the data sample + with np.load(data_file('gcvspl.npz')) as data: + # data points + x = data['x'] + y = data['y'] + + y_GCVSPL = data['y_GCVSPL'] + y_compr = make_smoothing_spline(x, y)(x) + + # such tolerance is explained by the fact that the spline is built + # using an iterative algorithm for minimizing the GCV criteria. These + # algorithms may vary, so the tolerance should be rather low. + # Not checking dtypes as gcvspl.npz stores little endian arrays, which + # result in conflicting dtypes on big endian systems. + xp_assert_close(y_compr, y_GCVSPL, atol=1e-4, rtol=1e-4, check_dtype=False) + + def test_non_regularized_case(self): + """ + In case the regularization parameter is 0, the resulting spline + is an interpolation spline with natural boundary conditions. + """ + # create data sample + rng = np.random.RandomState(1234) + n = 100 + x = np.sort(rng.random_sample(n) * 4 - 2) + y = x**2 * np.sin(4 * x) + x**3 + rng.normal(0., 1.5, n) + + spline_GCV = make_smoothing_spline(x, y, lam=0.) + spline_interp = make_interp_spline(x, y, 3, bc_type='natural') + + grid = np.linspace(x[0], x[-1], 2 * n) + xp_assert_close(spline_GCV(grid), + spline_interp(grid), + atol=1e-15) + + @pytest.mark.fail_slow(2) + def test_weighted_smoothing_spline(self): + # create data sample + rng = np.random.RandomState(1234) + n = 100 + x = np.sort(rng.random_sample(n) * 4 - 2) + y = x**2 * np.sin(4 * x) + x**3 + rng.normal(0., 1.5, n) + + spl = make_smoothing_spline(x, y) + + # in order not to iterate over all of the indices, we select 10 of + # them randomly + for ind in rng.choice(range(100), size=10): + w = np.ones(n) + w[ind] = 30. + spl_w = make_smoothing_spline(x, y, w) + # check that spline with weight in a certain point is closer to the + # original point than the one without weights + orig = abs(spl(x[ind]) - y[ind]) + weighted = abs(spl_w(x[ind]) - y[ind]) + + if orig < weighted: + raise ValueError(f'Spline with weights should be closer to the' + f' points than the original one: {orig:.4} < ' + f'{weighted:.4}') + + +################################ +# NdBSpline tests +def bspline2(xy, t, c, k): + """A naive 2D tensort product spline evaluation.""" + x, y = xy + tx, ty = t + nx = len(tx) - k - 1 + assert (nx >= k+1) + ny = len(ty) - k - 1 + assert (ny >= k+1) + res = sum(c[ix, iy] * B(x, k, ix, tx) * B(y, k, iy, ty) + for ix in range(nx) for iy in range(ny)) + return np.asarray(res) + + +def B(x, k, i, t): + if k == 0: + return 1.0 if t[i] <= x < t[i+1] else 0.0 + if t[i+k] == t[i]: + c1 = 0.0 + else: + c1 = (x - t[i])/(t[i+k] - t[i]) * B(x, k-1, i, t) + if t[i+k+1] == t[i+1]: + c2 = 0.0 + else: + c2 = (t[i+k+1] - x)/(t[i+k+1] - t[i+1]) * B(x, k-1, i+1, t) + return c1 + c2 + + +def bspline(x, t, c, k): + n = len(t) - k - 1 + assert (n >= k+1) and (len(c) >= n) + return sum(c[i] * B(x, k, i, t) for i in range(n)) + + +class NdBSpline0: + def __init__(self, t, c, k=3): + """Tensor product spline object. + + c[i1, i2, ..., id] * B(x1, i1) * B(x2, i2) * ... * B(xd, id) + + Parameters + ---------- + c : ndarray, shape (n1, n2, ..., nd, ...) + b-spline coefficients + t : tuple of 1D ndarrays + knot vectors in directions 1, 2, ... d + ``len(t[i]) == n[i] + k + 1`` + k : int or length-d tuple of integers + spline degrees. + """ + ndim = len(t) + assert ndim <= len(c.shape) + + try: + len(k) + except TypeError: + # make k a tuple + k = (k,)*ndim + + self.k = tuple(operator.index(ki) for ki in k) + self.t = tuple(np.asarray(ti, dtype=float) for ti in t) + self.c = c + + def __call__(self, x): + ndim = len(self.t) + # a single evaluation point: `x` is a 1D array_like, shape (ndim,) + assert len(x) == ndim + + # get the indices in an ndim-dimensional vector + i = ['none', ]*ndim + for d in range(ndim): + td, xd = self.t[d], x[d] + k = self.k[d] + + # find the index for x[d] + if xd == td[k]: + i[d] = k + else: + i[d] = np.searchsorted(td, xd) - 1 + assert td[i[d]] <= xd <= td[i[d]+1] + assert i[d] >= k and i[d] < len(td) - k + i = tuple(i) + + # iterate over the dimensions, form linear combinations of + # products B(x_1) * B(x_2) * ... B(x_N) of (k+1)**N b-splines + # which are non-zero at `i = (i_1, i_2, ..., i_N)`. + result = 0 + iters = [range(i[d] - self.k[d], i[d] + 1) for d in range(ndim)] + for idx in itertools.product(*iters): + term = self.c[idx] * np.prod([B(x[d], self.k[d], idx[d], self.t[d]) + for d in range(ndim)]) + result += term + return np.asarray(result) + + +class TestNdBSpline: + + def test_1D(self): + # test ndim=1 agrees with BSpline + rng = np.random.default_rng(12345) + n, k = 11, 3 + n_tr = 7 + t = np.sort(rng.uniform(size=n + k + 1)) + c = rng.uniform(size=(n, n_tr)) + + b = BSpline(t, c, k) + nb = NdBSpline((t,), c, k) + + xi = rng.uniform(size=21) + # NdBSpline expects xi.shape=(npts, ndim) + xp_assert_close(nb(xi[:, None]), + b(xi), atol=1e-14) + assert nb(xi[:, None]).shape == (xi.shape[0], c.shape[1]) + + def make_2d_case(self): + # make a 2D separable spline + x = np.arange(6) + y = x**3 + spl = make_interp_spline(x, y, k=3) + + y_1 = x**3 + 2*x + spl_1 = make_interp_spline(x, y_1, k=3) + + t2 = (spl.t, spl_1.t) + c2 = spl.c[:, None] * spl_1.c[None, :] + + return t2, c2, 3 + + def make_2d_mixed(self): + # make a 2D separable spline w/ kx=3, ky=2 + x = np.arange(6) + y = x**3 + spl = make_interp_spline(x, y, k=3) + + x = np.arange(5) + 1.5 + y_1 = x**2 + 2*x + spl_1 = make_interp_spline(x, y_1, k=2) + + t2 = (spl.t, spl_1.t) + c2 = spl.c[:, None] * spl_1.c[None, :] + + return t2, c2, spl.k, spl_1.k + + def test_2D_separable(self): + xi = [(1.5, 2.5), (2.5, 1), (0.5, 1.5)] + t2, c2, k = self.make_2d_case() + target = [x**3 * (y**3 + 2*y) for (x, y) in xi] + + # sanity check: bspline2 gives the product as constructed + xp_assert_close(np.asarray([bspline2(xy, t2, c2, k) for xy in xi]), + np.asarray(target), + check_shape=False, + atol=1e-14) + + # check evaluation on a 2D array: the 1D array of 2D points + bspl2 = NdBSpline(t2, c2, k=3) + assert bspl2(xi).shape == (len(xi), ) + xp_assert_close(bspl2(xi), + target, atol=1e-14) + + # now check on a multidim xi + rng = np.random.default_rng(12345) + xi = rng.uniform(size=(4, 3, 2)) * 5 + result = bspl2(xi) + assert result.shape == (4, 3) + + # also check the values + x, y = xi.reshape((-1, 2)).T + xp_assert_close(result.ravel(), + x**3 * (y**3 + 2*y), atol=1e-14) + + def test_2D_separable_2(self): + # test `c` with trailing dimensions, i.e. c.ndim > ndim + ndim = 2 + xi = [(1.5, 2.5), (2.5, 1), (0.5, 1.5)] + target = [x**3 * (y**3 + 2*y) for (x, y) in xi] + + t2, c2, k = self.make_2d_case() + c2_4 = np.dstack((c2, c2, c2, c2)) # c22.shape = (6, 6, 4) + + xy = (1.5, 2.5) + bspl2_4 = NdBSpline(t2, c2_4, k=3) + result = bspl2_4(xy) + val_single = NdBSpline(t2, c2, k)(xy) + assert result.shape == (4,) + xp_assert_close(result, + [val_single, ]*4, atol=1e-14) + + # now try the array xi : the output.shape is (3, 4) where 3 + # is the number of points in xi and 4 is the trailing dimension of c + assert bspl2_4(xi).shape == np.shape(xi)[:-1] + bspl2_4.c.shape[ndim:] + xp_assert_close(bspl2_4(xi), np.asarray(target)[:, None], + check_shape=False, + atol=5e-14) + + # two trailing dimensions + c2_22 = c2_4.reshape((6, 6, 2, 2)) + bspl2_22 = NdBSpline(t2, c2_22, k=3) + + result = bspl2_22(xy) + assert result.shape == (2, 2) + xp_assert_close(result, + [[val_single, val_single], + [val_single, val_single]], atol=1e-14) + + # now try the array xi : the output shape is (3, 2, 2) + # for 3 points in xi and c trailing dimensions being (2, 2) + assert (bspl2_22(xi).shape == + np.shape(xi)[:-1] + bspl2_22.c.shape[ndim:]) + xp_assert_close(bspl2_22(xi), np.asarray(target)[:, None, None], + check_shape=False, + atol=5e-14) + + + def test_2D_separable_2_complex(self): + # test `c` with c.dtype == complex, with and w/o trailing dims + xi = [(1.5, 2.5), (2.5, 1), (0.5, 1.5)] + target = [x**3 * (y**3 + 2*y) for (x, y) in xi] + + target = [t + 2j*t for t in target] + + t2, c2, k = self.make_2d_case() + c2 = c2 * (1 + 2j) + c2_4 = np.dstack((c2, c2, c2, c2)) # c2_4.shape = (6, 6, 4) + + xy = (1.5, 2.5) + bspl2_4 = NdBSpline(t2, c2_4, k=3) + result = bspl2_4(xy) + val_single = NdBSpline(t2, c2, k)(xy) + assert result.shape == (4,) + xp_assert_close(result, + [val_single, ]*4, atol=1e-14) + + def test_2D_random(self): + rng = np.random.default_rng(12345) + k = 3 + tx = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=7)) * 3, 3, 3, 3, 3] + ty = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=8)) * 4, 4, 4, 4, 4] + c = rng.uniform(size=(tx.size-k-1, ty.size-k-1)) + + spl = NdBSpline((tx, ty), c, k=k) + + xi = (1., 1.) + xp_assert_close(spl(xi), + bspline2(xi, (tx, ty), c, k), atol=1e-14) + + xi = np.c_[[1, 1.5, 2], + [1.1, 1.6, 2.1]] + xp_assert_close(spl(xi), + [bspline2(xy, (tx, ty), c, k) for xy in xi], + atol=1e-14) + + def test_2D_mixed(self): + t2, c2, kx, ky = self.make_2d_mixed() + xi = [(1.4, 4.5), (2.5, 2.4), (4.5, 3.5)] + target = [x**3 * (y**2 + 2*y) for (x, y) in xi] + bspl2 = NdBSpline(t2, c2, k=(kx, ky)) + assert bspl2(xi).shape == (len(xi), ) + xp_assert_close(bspl2(xi), + target, atol=1e-14) + + def test_2D_derivative(self): + t2, c2, kx, ky = self.make_2d_mixed() + xi = [(1.4, 4.5), (2.5, 2.4), (4.5, 3.5)] + bspl2 = NdBSpline(t2, c2, k=(kx, ky)) + + der = bspl2(xi, nu=(1, 0)) + xp_assert_close(der, + [3*x**2 * (y**2 + 2*y) for x, y in xi], atol=1e-14) + + der = bspl2(xi, nu=(1, 1)) + xp_assert_close(der, + [3*x**2 * (2*y + 2) for x, y in xi], atol=1e-14) + + der = bspl2(xi, nu=(0, 0)) + xp_assert_close(der, + [x**3 * (y**2 + 2*y) for x, y in xi], atol=1e-14) + + with assert_raises(ValueError): + # all(nu >= 0) + der = bspl2(xi, nu=(-1, 0)) + + with assert_raises(ValueError): + # len(nu) == ndim + der = bspl2(xi, nu=(-1, 0, 1)) + + def test_2D_mixed_random(self): + rng = np.random.default_rng(12345) + kx, ky = 2, 3 + tx = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=7)) * 3, 3, 3, 3, 3] + ty = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=8)) * 4, 4, 4, 4, 4] + c = rng.uniform(size=(tx.size - kx - 1, ty.size - ky - 1)) + + xi = np.c_[[1, 1.5, 2], + [1.1, 1.6, 2.1]] + + bspl2 = NdBSpline((tx, ty), c, k=(kx, ky)) + bspl2_0 = NdBSpline0((tx, ty), c, k=(kx, ky)) + + xp_assert_close(bspl2(xi), + [bspl2_0(xp) for xp in xi], atol=1e-14) + + def test_tx_neq_ty(self): + # 2D separable spline w/ len(tx) != len(ty) + x = np.arange(6) + y = np.arange(7) + 1.5 + + spl_x = make_interp_spline(x, x**3, k=3) + spl_y = make_interp_spline(y, y**2 + 2*y, k=3) + cc = spl_x.c[:, None] * spl_y.c[None, :] + bspl = NdBSpline((spl_x.t, spl_y.t), cc, (spl_x.k, spl_y.k)) + + values = (x**3)[:, None] * (y**2 + 2*y)[None, :] + rgi = RegularGridInterpolator((x, y), values) + + xi = [(a, b) for a, b in itertools.product(x, y)] + bxi = bspl(xi) + + assert not np.isnan(bxi).any() + xp_assert_close(bxi, rgi(xi), atol=1e-14) + xp_assert_close(bxi.reshape(values.shape), values, atol=1e-14) + + def make_3d_case(self): + # make a 3D separable spline + x = np.arange(6) + y = x**3 + spl = make_interp_spline(x, y, k=3) + + y_1 = x**3 + 2*x + spl_1 = make_interp_spline(x, y_1, k=3) + + y_2 = x**3 + 3*x + 1 + spl_2 = make_interp_spline(x, y_2, k=3) + + t2 = (spl.t, spl_1.t, spl_2.t) + c2 = (spl.c[:, None, None] * + spl_1.c[None, :, None] * + spl_2.c[None, None, :]) + + return t2, c2, 3 + + def test_3D_separable(self): + rng = np.random.default_rng(12345) + x, y, z = rng.uniform(size=(3, 11)) * 5 + target = x**3 * (y**3 + 2*y) * (z**3 + 3*z + 1) + + t3, c3, k = self.make_3d_case() + bspl3 = NdBSpline(t3, c3, k=3) + + xi = [_ for _ in zip(x, y, z)] + result = bspl3(xi) + assert result.shape == (11,) + xp_assert_close(result, target, atol=1e-14) + + def test_3D_derivative(self): + t3, c3, k = self.make_3d_case() + bspl3 = NdBSpline(t3, c3, k=3) + rng = np.random.default_rng(12345) + x, y, z = rng.uniform(size=(3, 11)) * 5 + xi = [_ for _ in zip(x, y, z)] + + xp_assert_close(bspl3(xi, nu=(1, 0, 0)), + 3*x**2 * (y**3 + 2*y) * (z**3 + 3*z + 1), atol=1e-14) + + xp_assert_close(bspl3(xi, nu=(2, 0, 0)), + 6*x * (y**3 + 2*y) * (z**3 + 3*z + 1), atol=1e-14) + + xp_assert_close(bspl3(xi, nu=(2, 1, 0)), + 6*x * (3*y**2 + 2) * (z**3 + 3*z + 1), atol=1e-14) + + xp_assert_close(bspl3(xi, nu=(2, 1, 3)), + 6*x * (3*y**2 + 2) * (6), atol=1e-14) + + xp_assert_close(bspl3(xi, nu=(2, 1, 4)), + np.zeros(len(xi)), atol=1e-14) + + def test_3D_random(self): + rng = np.random.default_rng(12345) + k = 3 + tx = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=7)) * 3, 3, 3, 3, 3] + ty = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=8)) * 4, 4, 4, 4, 4] + tz = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=8)) * 4, 4, 4, 4, 4] + c = rng.uniform(size=(tx.size-k-1, ty.size-k-1, tz.size-k-1)) + + spl = NdBSpline((tx, ty, tz), c, k=k) + spl_0 = NdBSpline0((tx, ty, tz), c, k=k) + + xi = (1., 1., 1) + xp_assert_close(spl(xi), spl_0(xi), atol=1e-14) + + xi = np.c_[[1, 1.5, 2], + [1.1, 1.6, 2.1], + [0.9, 1.4, 1.9]] + xp_assert_close(spl(xi), [spl_0(xp) for xp in xi], atol=1e-14) + + def test_3D_random_complex(self): + rng = np.random.default_rng(12345) + k = 3 + tx = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=7)) * 3, 3, 3, 3, 3] + ty = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=8)) * 4, 4, 4, 4, 4] + tz = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=8)) * 4, 4, 4, 4, 4] + c = (rng.uniform(size=(tx.size-k-1, ty.size-k-1, tz.size-k-1)) + + rng.uniform(size=(tx.size-k-1, ty.size-k-1, tz.size-k-1))*1j) + + spl = NdBSpline((tx, ty, tz), c, k=k) + spl_re = NdBSpline((tx, ty, tz), c.real, k=k) + spl_im = NdBSpline((tx, ty, tz), c.imag, k=k) + + xi = np.c_[[1, 1.5, 2], + [1.1, 1.6, 2.1], + [0.9, 1.4, 1.9]] + xp_assert_close(spl(xi), + spl_re(xi) + 1j*spl_im(xi), atol=1e-14) + + @pytest.mark.parametrize('cls_extrap', [None, True]) + @pytest.mark.parametrize('call_extrap', [None, True]) + def test_extrapolate_3D_separable(self, cls_extrap, call_extrap): + # test that extrapolate=True does extrapolate + t3, c3, k = self.make_3d_case() + bspl3 = NdBSpline(t3, c3, k=3, extrapolate=cls_extrap) + + # evaluate out of bounds + x, y, z = [-2, -1, 7], [-3, -0.5, 6.5], [-1, -1.5, 7.5] + x, y, z = map(np.asarray, (x, y, z)) + xi = [_ for _ in zip(x, y, z)] + target = x**3 * (y**3 + 2*y) * (z**3 + 3*z + 1) + + result = bspl3(xi, extrapolate=call_extrap) + xp_assert_close(result, target, atol=1e-14) + + @pytest.mark.parametrize('extrap', [(False, True), (True, None)]) + def test_extrapolate_3D_separable_2(self, extrap): + # test that call(..., extrapolate=None) defers to self.extrapolate, + # otherwise supersedes self.extrapolate + t3, c3, k = self.make_3d_case() + cls_extrap, call_extrap = extrap + bspl3 = NdBSpline(t3, c3, k=3, extrapolate=cls_extrap) + + # evaluate out of bounds + x, y, z = [-2, -1, 7], [-3, -0.5, 6.5], [-1, -1.5, 7.5] + x, y, z = map(np.asarray, (x, y, z)) + xi = [_ for _ in zip(x, y, z)] + target = x**3 * (y**3 + 2*y) * (z**3 + 3*z + 1) + + result = bspl3(xi, extrapolate=call_extrap) + xp_assert_close(result, target, atol=1e-14) + + def test_extrapolate_false_3D_separable(self): + # test that extrapolate=False produces nans for out-of-bounds values + t3, c3, k = self.make_3d_case() + bspl3 = NdBSpline(t3, c3, k=3) + + # evaluate out of bounds and inside + x, y, z = [-2, 1, 7], [-3, 0.5, 6.5], [-1, 1.5, 7.5] + x, y, z = map(np.asarray, (x, y, z)) + xi = [_ for _ in zip(x, y, z)] + target = x**3 * (y**3 + 2*y) * (z**3 + 3*z + 1) + + result = bspl3(xi, extrapolate=False) + assert np.isnan(result[0]) + assert np.isnan(result[-1]) + xp_assert_close(result[1:-1], target[1:-1], atol=1e-14) + + def test_x_nan_3D(self): + # test that spline(nan) is nan + t3, c3, k = self.make_3d_case() + bspl3 = NdBSpline(t3, c3, k=3) + + # evaluate out of bounds and inside + x = np.asarray([-2, 3, np.nan, 1, 2, 7, np.nan]) + y = np.asarray([-3, 3.5, 1, np.nan, 3, 6.5, 6.5]) + z = np.asarray([-1, 3.5, 2, 3, np.nan, 7.5, 7.5]) + xi = [_ for _ in zip(x, y, z)] + target = x**3 * (y**3 + 2*y) * (z**3 + 3*z + 1) + mask = np.isnan(x) | np.isnan(y) | np.isnan(z) + target[mask] = np.nan + + result = bspl3(xi) + assert np.isnan(result[mask]).all() + xp_assert_close(result, target, atol=1e-14) + + def test_non_c_contiguous(self): + # check that non C-contiguous inputs are OK + rng = np.random.default_rng(12345) + kx, ky = 3, 3 + tx = np.sort(rng.uniform(low=0, high=4, size=16)) + tx = np.r_[(tx[0],)*kx, tx, (tx[-1],)*kx] + ty = np.sort(rng.uniform(low=0, high=4, size=16)) + ty = np.r_[(ty[0],)*ky, ty, (ty[-1],)*ky] + + assert not tx[::2].flags.c_contiguous + assert not ty[::2].flags.c_contiguous + + c = rng.uniform(size=(tx.size//2 - kx - 1, ty.size//2 - ky - 1)) + c = c.T + assert not c.flags.c_contiguous + + xi = np.c_[[1, 1.5, 2], + [1.1, 1.6, 2.1]] + + bspl2 = NdBSpline((tx[::2], ty[::2]), c, k=(kx, ky)) + bspl2_0 = NdBSpline0((tx[::2], ty[::2]), c, k=(kx, ky)) + + xp_assert_close(bspl2(xi), + [bspl2_0(xp) for xp in xi], atol=1e-14) + + def test_readonly(self): + t3, c3, k = self.make_3d_case() + bspl3 = NdBSpline(t3, c3, k=3) + + for i in range(3): + t3[i].flags.writeable = False + c3.flags.writeable = False + + bspl3_ = NdBSpline(t3, c3, k=3) + + assert bspl3((1, 2, 3)) == bspl3_((1, 2, 3)) + + def test_design_matrix(self): + t3, c3, k = self.make_3d_case() + + xi = np.asarray([[1, 2, 3], [4, 5, 6]]) + dm = NdBSpline(t3, c3, k).design_matrix(xi, t3, k) + dm1 = NdBSpline.design_matrix(xi, t3, [k, k, k]) + assert dm.shape[0] == xi.shape[0] + xp_assert_close(dm.todense(), dm1.todense(), atol=1e-16) + + with assert_raises(ValueError): + NdBSpline.design_matrix([1, 2, 3], t3, [k]*3) + + with assert_raises(ValueError, match="Data and knots*"): + NdBSpline.design_matrix([[1, 2]], t3, [k]*3) + + @pytest.mark.thread_unsafe + def test_concurrency(self): + rng = np.random.default_rng(12345) + k = 3 + tx = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=7)) * 3, 3, 3, 3, 3] + ty = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=8)) * 4, 4, 4, 4, 4] + tz = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=8)) * 4, 4, 4, 4, 4] + c = rng.uniform(size=(tx.size-k-1, ty.size-k-1, tz.size-k-1)) + + spl = NdBSpline((tx, ty, tz), c, k=k) + + def worker_fn(_, spl): + xi = np.c_[[1, 1.5, 2], + [1.1, 1.6, 2.1], + [0.9, 1.4, 1.9]] + spl(xi) + + _run_concurrent_barrier(10, worker_fn, spl) + + +class TestMakeND: + def test_2D_separable_simple(self): + x = np.arange(6) + y = np.arange(6) + 0.5 + values = x[:, None]**3 * (y**3 + 2*y)[None, :] + xi = [(a, b) for a, b in itertools.product(x, y)] + + bspl = make_ndbspl((x, y), values, k=1) + xp_assert_close(bspl(xi), values.ravel(), atol=1e-15) + + # test the coefficients vs outer product of 1D coefficients + spl_x = make_interp_spline(x, x**3, k=1) + spl_y = make_interp_spline(y, y**3 + 2*y, k=1) + cc = spl_x.c[:, None] * spl_y.c[None, :] + xp_assert_close(cc, bspl.c, atol=1e-11, rtol=0) + + # test against RGI + from scipy.interpolate import RegularGridInterpolator as RGI + rgi = RGI((x, y), values, method='linear') + xp_assert_close(rgi(xi), bspl(xi), atol=1e-14) + + def test_2D_separable_trailing_dims(self): + # test `c` with trailing dimensions, i.e. c.ndim > ndim + x = np.arange(6) + y = np.arange(6) + xi = [(a, b) for a, b in itertools.product(x, y)] + + # make values4.shape = (6, 6, 4) + values = x[:, None]**3 * (y**3 + 2*y)[None, :] + values4 = np.dstack((values, values, values, values)) + bspl = make_ndbspl((x, y), values4, k=3, solver=ssl.spsolve) + + result = bspl(xi) + target = np.dstack((values, values, values, values)).astype(float) + assert result.shape == (36, 4) + xp_assert_close(result.reshape(6, 6, 4), + target, atol=1e-14) + + # now two trailing dimensions + values22 = values4.reshape((6, 6, 2, 2)) + bspl = make_ndbspl((x, y), values22, k=3, solver=ssl.spsolve) + + result = bspl(xi) + assert result.shape == (36, 2, 2) + xp_assert_close(result.reshape(6, 6, 2, 2), + target.reshape((6, 6, 2, 2)), atol=1e-14) + + @pytest.mark.parametrize('k', [(3, 3), (1, 1), (3, 1), (1, 3), (3, 5)]) + def test_2D_mixed(self, k): + # make a 2D separable spline w/ len(tx) != len(ty) + x = np.arange(6) + y = np.arange(7) + 1.5 + xi = [(a, b) for a, b in itertools.product(x, y)] + + values = (x**3)[:, None] * (y**2 + 2*y)[None, :] + bspl = make_ndbspl((x, y), values, k=k, solver=ssl.spsolve) + xp_assert_close(bspl(xi), values.ravel(), atol=1e-15) + + def _get_sample_2d_data(self): + # from test_rgi.py::TestIntepN + x = np.array([.5, 2., 3., 4., 5.5, 6.]) + y = np.array([.5, 2., 3., 4., 5.5, 6.]) + z = np.array( + [ + [1, 2, 1, 2, 1, 1], + [1, 2, 1, 2, 1, 1], + [1, 2, 3, 2, 1, 1], + [1, 2, 2, 2, 1, 1], + [1, 2, 1, 2, 1, 1], + [1, 2, 2, 2, 1, 1], + ] + ) + return x, y, z + + def test_2D_vs_RGI_linear(self): + x, y, z = self._get_sample_2d_data() + bspl = make_ndbspl((x, y), z, k=1) + rgi = RegularGridInterpolator((x, y), z, method='linear') + + xi = np.array([[1, 2.3, 5.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, 4.0, 5.0, 1.0, 3]]).T + + xp_assert_close(bspl(xi), rgi(xi), atol=1e-14) + + def test_2D_vs_RGI_cubic(self): + x, y, z = self._get_sample_2d_data() + bspl = make_ndbspl((x, y), z, k=3, solver=ssl.spsolve) + rgi = RegularGridInterpolator((x, y), z, method='cubic_legacy') + + xi = np.array([[1, 2.3, 5.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, 4.0, 5.0, 1.0, 3]]).T + + xp_assert_close(bspl(xi), rgi(xi), atol=1e-14) + + @pytest.mark.parametrize('solver', [ssl.gmres, ssl.gcrotmk]) + def test_2D_vs_RGI_cubic_iterative(self, solver): + # same as `test_2D_vs_RGI_cubic`, only with an iterative solver. + # Note the need to add an explicit `rtol` solver_arg to achieve the + # target accuracy of 1e-14. (the relation between solver atol/rtol + # and the accuracy of the final result is not direct and needs experimenting) + x, y, z = self._get_sample_2d_data() + bspl = make_ndbspl((x, y), z, k=3, solver=solver, rtol=1e-6) + rgi = RegularGridInterpolator((x, y), z, method='cubic_legacy') + + xi = np.array([[1, 2.3, 5.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, 4.0, 5.0, 1.0, 3]]).T + + xp_assert_close(bspl(xi), rgi(xi), atol=1e-14, rtol=1e-7) + + def test_2D_vs_RGI_quintic(self): + x, y, z = self._get_sample_2d_data() + bspl = make_ndbspl((x, y), z, k=5, solver=ssl.spsolve) + rgi = RegularGridInterpolator((x, y), z, method='quintic_legacy') + + xi = np.array([[1, 2.3, 5.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, 4.0, 5.0, 1.0, 3]]).T + + xp_assert_close(bspl(xi), rgi(xi), atol=1e-14) + + @pytest.mark.parametrize( + 'k, meth', [(1, 'linear'), (3, 'cubic_legacy'), (5, 'quintic_legacy')] + ) + def test_3D_random_vs_RGI(self, k, meth): + rndm = np.random.default_rng(123456) + x = np.cumsum(rndm.uniform(size=6)) + y = np.cumsum(rndm.uniform(size=7)) + z = np.cumsum(rndm.uniform(size=8)) + values = rndm.uniform(size=(6, 7, 8)) + + bspl = make_ndbspl((x, y, z), values, k=k, solver=ssl.spsolve) + rgi = RegularGridInterpolator((x, y, z), values, method=meth) + + xi = np.random.uniform(low=0.7, high=2.1, size=(11, 3)) + xp_assert_close(bspl(xi), rgi(xi), atol=1e-14) + + def test_solver_err_not_converged(self): + x, y, z = self._get_sample_2d_data() + solver_args = {'maxiter': 1} + with assert_raises(ValueError, match='solver'): + make_ndbspl((x, y), z, k=3, **solver_args) + + with assert_raises(ValueError, match='solver'): + make_ndbspl((x, y), np.dstack((z, z)), k=3, **solver_args) + + +class TestFpchec: + # https://github.com/scipy/scipy/blob/main/scipy/interpolate/fitpack/fpchec.f + + def test_1D_x_t(self): + k = 1 + t = np.arange(12).reshape(2, 6) + x = np.arange(12) + + with pytest.raises(ValueError, match="1D sequence"): + _b.fpcheck(x, t, k) + + with pytest.raises(ValueError, match="1D sequence"): + _b.fpcheck(t, x, k) + + def test_condition_1(self): + # c 1) k+1 <= n-k-1 <= m + k = 3 + n = 2*(k + 1) - 1 # not OK + m = n + 11 # OK + t = np.arange(n) + x = np.arange(m) + + assert dfitpack.fpchec(x, t, k) == 10 + with pytest.raises(ValueError, match="Need k+1*"): + _b.fpcheck(x, t, k) + + n = 2*(k+1) + 1 # OK + m = n - k - 2 # not OK + t = np.arange(n) + x = np.arange(m) + + assert dfitpack.fpchec(x, t, k) == 10 + with pytest.raises(ValueError, match="Need k+1*"): + _b.fpcheck(x, t, k) + + def test_condition_2(self): + # c 2) t(1) <= t(2) <= ... <= t(k+1) + # c t(n-k) <= t(n-k+1) <= ... <= t(n) + k = 3 + t = [0]*(k+1) + [2] + [5]*(k+1) # this is OK + x = [1, 2, 3, 4, 4.5] + + assert dfitpack.fpchec(x, t, k) == 0 + assert _b.fpcheck(x, t, k) is None # does not raise + + tt = t.copy() + tt[-1] = tt[0] # not OK + assert dfitpack.fpchec(x, tt, k) == 20 + with pytest.raises(ValueError, match="Last k knots*"): + _b.fpcheck(x, tt, k) + + tt = t.copy() + tt[0] = tt[-1] # not OK + assert dfitpack.fpchec(x, tt, k) == 20 + with pytest.raises(ValueError, match="First k knots*"): + _b.fpcheck(x, tt, k) + + def test_condition_3(self): + # c 3) t(k+1) < t(k+2) < ... < t(n-k) + k = 3 + t = [0]*(k+1) + [2, 3] + [5]*(k+1) # this is OK + x = [1, 2, 3, 3.5, 4, 4.5] + assert dfitpack.fpchec(x, t, k) == 0 + assert _b.fpcheck(x, t, k) is None + + t = [0]*(k+1) + [2, 2] + [5]*(k+1) # this is not OK + assert dfitpack.fpchec(x, t, k) == 30 + with pytest.raises(ValueError, match="Internal knots*"): + _b.fpcheck(x, t, k) + + def test_condition_4(self): + # c 4) t(k+1) <= x(i) <= t(n-k) + # NB: FITPACK's fpchec only checks x[0] & x[-1], so we follow. + k = 3 + t = [0]*(k+1) + [5]*(k+1) + x = [1, 2, 3, 3.5, 4, 4.5] # this is OK + assert dfitpack.fpchec(x, t, k) == 0 + assert _b.fpcheck(x, t, k) is None + + xx = x.copy() + xx[0] = t[0] # still OK + assert dfitpack.fpchec(xx, t, k) == 0 + assert _b.fpcheck(x, t, k) is None + + xx = x.copy() + xx[0] = t[0] - 1 # not OK + assert dfitpack.fpchec(xx, t, k) == 40 + with pytest.raises(ValueError, match="Out of bounds*"): + _b.fpcheck(xx, t, k) + + xx = x.copy() + xx[-1] = t[-1] + 1 # not OK + assert dfitpack.fpchec(xx, t, k) == 40 + with pytest.raises(ValueError, match="Out of bounds*"): + _b.fpcheck(xx, t, k) + + # ### Test the S-W condition (no 5) + # c 5) the conditions specified by schoenberg and whitney must hold + # c for at least one subset of data points, i.e. there must be a + # c subset of data points y(j) such that + # c t(j) < y(j) < t(j+k+1), j=1,2,...,n-k-1 + def test_condition_5_x1xm(self): + # x(1).ge.t(k2) .or. x(m).le.t(nk1) + k = 1 + t = [0, 0, 1, 2, 2] + x = [1.1, 1.1, 1.1] + assert dfitpack.fpchec(x, t, k) == 50 + with pytest.raises(ValueError, match="Schoenberg-Whitney*"): + _b.fpcheck(x, t, k) + + x = [0.5, 0.5, 0.5] + assert dfitpack.fpchec(x, t, k) == 50 + with pytest.raises(ValueError, match="Schoenberg-Whitney*"): + _b.fpcheck(x, t, k) + + def test_condition_5_k1(self): + # special case nk3 (== n - k - 2) < 2 + k = 1 + t = [0, 0, 1, 1] + x = [0.5, 0.6] + assert dfitpack.fpchec(x, t, k) == 0 + assert _b.fpcheck(x, t, k) is None + + def test_condition_5_1(self): + # basically, there can't be an interval of t[j]..t[j+k+1] with no x + k = 3 + t = [0]*(k+1) + [2] + [5]*(k+1) + x = [3]*5 + assert dfitpack.fpchec(x, t, k) == 50 + with pytest.raises(ValueError, match="Schoenberg-Whitney*"): + _b.fpcheck(x, t, k) + + t = [0]*(k+1) + [2] + [5]*(k+1) + x = [1]*5 + assert dfitpack.fpchec(x, t, k) == 50 + with pytest.raises(ValueError, match="Schoenberg-Whitney*"): + _b.fpcheck(x, t, k) + + def test_condition_5_2(self): + # same as _5_1, only the empty interval is in the middle + k = 3 + t = [0]*(k+1) + [2, 3] + [5]*(k+1) + x = [1.1]*5 + [4] + + assert dfitpack.fpchec(x, t, k) == 50 + with pytest.raises(ValueError, match="Schoenberg-Whitney*"): + _b.fpcheck(x, t, k) + + # and this one is OK + x = [1.1]*4 + [4, 4] + assert dfitpack.fpchec(x, t, k) == 0 + assert _b.fpcheck(x, t, k) is None + + def test_condition_5_3(self): + # similar to _5_2, covers a different failure branch + k = 1 + t = [0, 0, 2, 3, 4, 5, 6, 7, 7] + x = [1, 1, 1, 5.2, 5.2, 5.2, 6.5] + + assert dfitpack.fpchec(x, t, k) == 50 + with pytest.raises(ValueError, match="Schoenberg-Whitney*"): + _b.fpcheck(x, t, k) + + +# ### python replicas of generate_knots(...) implementation details, for testing. +# ### see TestGenerateKnots::test_split_and_add_knot +def _split(x, t, k, residuals): + """Split the knot interval into "runs". + """ + ix = np.searchsorted(x, t[k:-k]) + # sum half-open intervals + fparts = [residuals[ix[i]:ix[i+1]].sum() for i in range(len(ix)-1)] + carries = residuals[ix[1:-1]] + + for i in range(len(carries)): # split residuals at internal knots + carry = carries[i] / 2 + fparts[i] += carry + fparts[i+1] -= carry + + fparts[-1] += residuals[-1] # add the contribution of the last knot + + xp_assert_close(sum(fparts), sum(residuals), atol=1e-15) + + return fparts, ix + + +def _add_knot(x, t, k, residuals): + """Insert a new knot given reduals.""" + fparts, ix = _split(x, t, k, residuals) + + # find the interval with max fparts and non-zero number of x values inside + idx_max = -101 + fpart_max = -1e100 + for i in range(len(fparts)): + if ix[i+1] - ix[i] > 1 and fparts[i] > fpart_max: + idx_max = i + fpart_max = fparts[i] + + if idx_max == -101: + raise ValueError("Internal error, please report it to SciPy developers.") + + # round up, like Dierckx does? This is really arbitrary though. + idx_newknot = (ix[idx_max] + ix[idx_max+1] + 1) // 2 + new_knot = x[idx_newknot] + idx_t = np.searchsorted(t, new_knot) + t_new = np.r_[t[:idx_t], new_knot, t[idx_t:]] + return t_new + + +class TestGenerateKnots: + def test_split_add_knot(self): + # smoke test implementation details: insert a new knot given residuals + x = np.arange(8, dtype=float) + y = x**3 + 1./(1 + x) + k = 3 + t = np.array([0.]*(k+1) + [7.]*(k+1)) + spl = make_lsq_spline(x, y, k=k, t=t) + residuals = (spl(x) - y)**2 + + from scipy.interpolate import _fitpack_repro as _fr + new_t = _fr.add_knot(x, t, k, residuals) + new_t_py = _add_knot(x, t, k, residuals) + + xp_assert_close(new_t, new_t_py, atol=1e-15) + + # redo with new knots + spl2 = make_lsq_spline(x, y, k=k, t=new_t) + residuals2 = (spl2(x) - y)**2 + + new_t2 = _fr.add_knot(x, new_t, k, residuals2) + new_t2_py = _add_knot(x, new_t, k, residuals2) + + xp_assert_close(new_t2, new_t2_py, atol=1e-15) + + @pytest.mark.parametrize('k', [1, 2, 3, 4, 5]) + def test_s0(self, k): + x = np.arange(8, dtype=np.float64) + y = np.sin(x*np.pi/8) + t = list(generate_knots(x, y, k=k, s=0))[-1] + + tt = splrep(x, y, k=k, s=0)[0] + xp_assert_close(t, tt, atol=1e-15) + + def test_s0_1(self): + # with these data, naive algorithm tries to insert >= nmax knots + n = 10 + x = np.arange(n) + y = x**3 + knots = list(generate_knots(x, y, k=3, s=0)) # does not error out + xp_assert_close(knots[-1], _not_a_knot(x, 3), atol=1e-15) + + def test_s0_n20(self): + n = 20 + x = np.arange(n) + y = x**3 + knots = list(generate_knots(x, y, k=3, s=0)) + xp_assert_close(knots[-1], _not_a_knot(x, 3), atol=1e-15) + + def test_s0_nest(self): + # s=0 and non-default nest: not implemented, errors out + x = np.arange(10) + y = x**3 + with assert_raises(ValueError): + list(generate_knots(x, y, k=3, s=0, nest=10)) + + def test_s_switch(self): + # test the process switching to interpolating knots when len(t) == m + k + 1 + """ + To generate the `wanted` list below apply the following diff and rerun + the test. The stdout will contain successive iterations of the `t` + array. + +$ git diff scipy/interpolate/fitpack/fpcurf.f +diff --git a/scipy/interpolate/fitpack/fpcurf.f b/scipy/interpolate/fitpack/fpcurf.f +index 1afb1900f1..d817e51ad8 100644 +--- a/scipy/interpolate/fitpack/fpcurf.f ++++ b/scipy/interpolate/fitpack/fpcurf.f +@@ -216,6 +216,9 @@ c t(j+k) <= x(i) <= t(j+k+1) and store it in fpint(j),j=1,2,...nrint. + do 190 l=1,nplus + c add a new knot. + call fpknot(x,m,t,n,fpint,nrdata,nrint,nest,1) ++ print*, l, nest, ': ', t ++ print*, "n, nmax = ", n, nmax ++ + c if n=nmax we locate the knots as for interpolation. + if(n.eq.nmax) go to 10 + c test whether we cannot further increase the number of knots. + """ # NOQA: E501 + x = np.arange(8) + y = np.sin(x*np.pi/8) + k = 3 + + knots = list(generate_knots(x, y, k=k, s=1e-7)) + wanted = [[0., 0., 0., 0., 7., 7., 7., 7.], + [0., 0., 0., 0., 4., 7., 7., 7., 7.], + [0., 0., 0., 0., 2., 4., 7., 7., 7., 7.], + [0., 0., 0., 0., 2., 4., 6., 7., 7., 7., 7.], + [0., 0., 0., 0., 2., 3., 4., 5., 7, 7., 7., 7.] + ] + + assert len(knots) == len(wanted) + for t, tt in zip(knots, wanted): + xp_assert_close(t, tt, atol=1e-15) + + # also check that the last knot vector matches FITPACK + t, _, _ = splrep(x, y, k=k, s=1e-7) + xp_assert_close(knots[-1], t, atol=1e-15) + + def test_list_input(self): + # test that list inputs are accepted + x = list(range(8)) + gen = generate_knots(x, x, s=0.1, k=1) + next(gen) + + def test_nest(self): + # test that nest < nmax stops the process early (and we get 10 knots not 12) + x = np.arange(8) + y = np.sin(x*np.pi/8) + s = 1e-7 + + knots = list(generate_knots(x, y, k=3, s=s, nest=10)) + xp_assert_close(knots[-1], + [0., 0., 0., 0., 2., 4., 7., 7., 7., 7.], atol=1e-15) + + with assert_raises(ValueError): + # nest < 2*(k+1) + list(generate_knots(x, y, k=3, nest=4)) + + def test_weights(self): + x = np.arange(8) + y = np.sin(x*np.pi/8) + + with assert_raises(ValueError): + list(generate_knots(x, y, w=np.arange(11))) # len(w) != len(x) + + with assert_raises(ValueError): + list(generate_knots(x, y, w=-np.ones(8))) # w < 0 + + @pytest.mark.parametrize("npts", [30, 50, 100]) + @pytest.mark.parametrize("s", [0.1, 1e-2, 0]) + def test_vs_splrep(self, s, npts): + # XXX this test is brittle: differences start apearing for k=3 and s=1e-6, + # also for k != 3. Might be worth investigating at some point. + # I think we do not really guarantee exact agreement with splrep. Instead, + # we guarantee it is the same *in most cases*; otherwise slight differences + # are allowed. There is no theorem, it is al heuristics by P. Dierckx. + # The best we can do it to best-effort reproduce it. + rndm = np.random.RandomState(12345) + x = 10*np.sort(rndm.uniform(size=npts)) + y = np.sin(x*np.pi/10) + np.exp(-(x-6)**2) + + k = 3 + t = splrep(x, y, k=k, s=s)[0] + tt = list(generate_knots(x, y, k=k, s=s))[-1] + + xp_assert_close(tt, t, atol=1e-15) + + @pytest.mark.thread_unsafe + def test_s_too_small(self): + n = 14 + x = np.arange(n) + y = x**3 + + # XXX splrep warns that "s too small": ier=2 + knots = list(generate_knots(x, y, k=3, s=1e-50)) + + with suppress_warnings() as sup: + r = sup.record(RuntimeWarning) + tck = splrep(x, y, k=3, s=1e-50) + assert len(r) == 1 + xp_assert_equal(knots[-1], tck[0]) + + +def disc_naive(t, k): + """Straitforward way to compute the discontinuity matrix. For testing ONLY. + + This routine returns a dense matrix, while `_fitpack_repro.disc` returns + a packed one. + """ + n = t.shape[0] + + delta = t[n - k - 1] - t[k] + nrint = n - 2*k - 1 + + ti = t[k+1:n-k-1] # internal knots + tii = np.repeat(ti, 2) + tii[::2] += 1e-10 + tii[1::2] -= 1e-10 + m = BSpline(t, np.eye(n - k - 1), k)(tii, nu=k) + + matr = np.empty((nrint-1, m.shape[1]), dtype=float) + for i in range(0, m.shape[0], 2): + matr[i//2, :] = m[i, :] - m[i+1, :] + + matr *= (delta/nrint)**k / math.factorial(k) + return matr + + +class F_dense: + """ The r.h.s. of ``f(p) = s``, an analog of _fitpack_repro.F + Uses full matrices, so is for tests only. + """ + def __init__(self, x, y, t, k, s, w=None): + self.x = x + self.y = y + self.t = t + self.k = k + self.w = np.ones_like(x, dtype=float) if w is None else w + assert self.w.ndim == 1 + + # lhs + a_dense = BSpline(t, np.eye(t.shape[0] - k - 1), k)(x) + self.a_dense = a_dense * self.w[:, None] + + from scipy.interpolate import _fitpack_repro as _fr + self.b_dense = PackedMatrix(*_fr.disc(t, k)).todense() + + # rhs + assert y.ndim == 1 + yy = y * self.w + self.yy = np.r_[yy, np.zeros(self.b_dense.shape[0])] + + self.s = s + + def __call__(self, p): + ab = np.vstack((self.a_dense, self.b_dense / p)) + + # LSQ solution of ab @ c = yy + from scipy.linalg import qr, solve + q, r = qr(ab, mode='economic') + + qy = q.T @ self.yy + + nc = r.shape[1] + c = solve(r[:nc, :nc], qy[:nc]) + + spl = BSpline(self.t, c, self.k) + fp = np.sum(self.w**2 * (spl(self.x) - self.y)**2) + + self.spl = spl # store it + + return fp - self.s + + +class TestMakeSplrep: + def test_input_errors(self): + x = np.linspace(0, 10, 11) + y = np.linspace(0, 10, 12) + with assert_raises(ValueError): + # len(x) != len(y) + make_splrep(x, y) + + with assert_raises(ValueError): + # 0D inputs + make_splrep(1, 2, s=0.1) + + with assert_raises(ValueError): + # y.ndim > 2 + y = np.ones((x.size, 2, 2, 2)) + make_splrep(x, y, s=0.1) + + w = np.ones(12) + with assert_raises(ValueError): + # len(weights) != len(x) + make_splrep(x, x**3, w=w, s=0.1) + + w = -np.ones(12) + with assert_raises(ValueError): + # w < 0 + make_splrep(x, x**3, w=w, s=0.1) + + w = np.ones((x.shape[0], 2)) + with assert_raises(ValueError): + # w.ndim != 1 + make_splrep(x, x**3, w=w, s=0.1) + + with assert_raises(ValueError): + # x not ordered + make_splrep(x[::-1], x**3, s=0.1) + + with assert_raises(TypeError): + # k != int(k) + make_splrep(x, x**3, k=2.5, s=0.1) + + with assert_raises(ValueError): + # s < 0 + make_splrep(x, x**3, s=-1) + + with assert_raises(ValueError): + # nest < 2*k + 2 + make_splrep(x, x**3, k=3, nest=2, s=0.1) + + with assert_raises(ValueError): + # nest not None and s==0 + make_splrep(x, x**3, s=0, nest=11) + + with assert_raises(ValueError): + # len(x) != len(y) + make_splrep(np.arange(8), np.arange(9), s=0.1) + + def _get_xykt(self): + x = np.linspace(0, 5, 11) + y = np.sin(x*3.14 / 5)**2 + k = 3 + s = 1.7e-4 + tt = np.array([0]*(k+1) + [2.5, 4.0] + [5]*(k+1)) + + return x, y, k, s, tt + + def test_fitpack_F(self): + # test an implementation detail: banded/packed linalg vs full matrices + from scipy.interpolate._fitpack_repro import F + + x, y, k, s, t = self._get_xykt() + f = F(x, y[:, None], t, k, s) # F expects y to be 2D + f_d = F_dense(x, y, t, k, s) + for p in [1, 10, 100]: + xp_assert_close(f(p), f_d(p), atol=1e-15) + + def test_fitpack_F_with_weights(self): + # repeat test_fitpack_F, with weights + from scipy.interpolate._fitpack_repro import F + + x, y, k, s, t = self._get_xykt() + w = np.arange(x.shape[0], dtype=float) + fw = F(x, y[:, None], t, k, s, w=w) # F expects y to be 2D + fw_d = F_dense(x, y, t, k, s, w=w) + + f_d = F_dense(x, y, t, k, s) # no weights + + for p in [1, 10, 100]: + xp_assert_close(fw(p), fw_d(p), atol=1e-15) + assert not np.allclose(f_d(p), fw_d(p), atol=1e-15) + + def test_disc_matrix(self): + # test an implementation detail: discontinuity matrix + # (jumps of k-th derivative at knots) + import scipy.interpolate._fitpack_repro as _fr + + rng = np.random.default_rng(12345) + t = np.r_[0, 0, 0, 0, np.sort(rng.uniform(size=7))*5, 5, 5, 5, 5] + + n, k = len(t), 3 + D = PackedMatrix(*_fr.disc(t, k)).todense() + D_dense = disc_naive(t, k) + assert D.shape[0] == n - 2*k - 2 # number of internal knots + xp_assert_close(D, D_dense, atol=1e-15) + + def test_simple_vs_splrep(self): + x, y, k, s, tt = self._get_xykt() + tt = np.array([0]*(k+1) + [2.5, 4.0] + [5]*(k+1)) + + t,c,k = splrep(x, y, k=k, s=s) + assert all(t == tt) + + spl = make_splrep(x, y, k=k, s=s) + xp_assert_close(c[:spl.c.size], spl.c, atol=1e-15) + + def test_with_knots(self): + x, y, k, s, _ = self._get_xykt() + + t = list(generate_knots(x, y, k=k, s=s))[-1] + + spl_auto = make_splrep(x, y, k=k, s=s) + spl_t = make_splrep(x, y, t=t, k=k, s=s) + + xp_assert_close(spl_auto.t, spl_t.t, atol=1e-15) + xp_assert_close(spl_auto.c, spl_t.c, atol=1e-15) + assert spl_auto.k == spl_t.k + + def test_no_internal_knots(self): + # should not fail if there are no internal knots + n = 10 + x = np.arange(n) + y = x**3 + k = 3 + spl = make_splrep(x, y, k=k, s=1) + assert spl.t.shape[0] == 2*(k+1) + + def test_default_s(self): + n = 10 + x = np.arange(n) + y = x**3 + spl = make_splrep(x, y, k=3) + spl_i = make_interp_spline(x, y, k=3) + + xp_assert_close(spl.c, spl_i.c, atol=1e-15) + + @pytest.mark.thread_unsafe + def test_s_too_small(self): + # both splrep and make_splrep warn that "s too small": ier=2 + n = 14 + x = np.arange(n) + y = x**3 + + with suppress_warnings() as sup: + r = sup.record(RuntimeWarning) + tck = splrep(x, y, k=3, s=1e-50) + spl = make_splrep(x, y, k=3, s=1e-50) + assert len(r) == 2 + xp_assert_equal(spl.t, tck[0]) + xp_assert_close(np.r_[spl.c, [0]*(spl.k+1)], + tck[1], atol=5e-13) + + def test_shape(self): + # make sure coefficients have the right shape (not extra dims) + n, k = 10, 3 + x = np.arange(n) + y = x**3 + + spl = make_splrep(x, y, k=k) + spl_1 = make_splrep(x, y, k=k, s=1e-5) + + assert spl.c.ndim == 1 + assert spl_1.c.ndim == 1 + + # force the general code path, not shortcuts + spl_2 = make_splrep(x, y + 1/(1+y), k=k, s=1e-5) + assert spl_2.c.ndim == 1 + + def test_s0_vs_not(self): + # check that the shapes are consistent + n, k = 10, 3 + x = np.arange(n) + y = x**3 + + spl_0 = make_splrep(x, y, k=3, s=0) + spl_1 = make_splrep(x, y, k=3, s=1) + + assert spl_0.c.ndim == 1 + assert spl_1.c.ndim == 1 + + assert spl_0.t.shape[0] == n + k + 1 + assert spl_1.t.shape[0] == 2 * (k + 1) + + +class TestMakeSplprep: + def _get_xyk(self, m=10, k=3): + x = np.arange(m) * np.pi / m + y = [np.sin(x), np.cos(x)] + return x, y, k + + @pytest.mark.parametrize('s', [0, 0.1, 1e-3, 1e-5]) + def test_simple_vs_splprep(self, s): + # Check/document the interface vs splPrep + # The four values of `s` are to probe all code paths and shortcuts + m, k = 10, 3 + x = np.arange(m) * np.pi / m + y = [np.sin(x), np.cos(x)] + + # the number of knots depends on `s` (this is by construction) + num_knots = {0: 14, 0.1: 8, 1e-3: 8 + 1, 1e-5: 8 + 2} + + # construct the splines + (t, c, k), u_ = splprep(y, s=s) + spl, u = make_splprep(y, s=s) + + # parameters + xp_assert_close(u, u_, atol=1e-15) + + # knots + xp_assert_close(spl.t, t, atol=1e-15) + assert len(t) == num_knots[s] + + # coefficients: note the transpose + cc = np.asarray(c).T + xp_assert_close(spl.c, cc, atol=1e-15) + + # values: note axis=1 + xp_assert_close(spl(u), + BSpline(t, c, k, axis=1)(u), atol=1e-15) + + @pytest.mark.parametrize('s', [0, 0.1, 1e-3, 1e-5]) + def test_array_not_list(self, s): + # the argument of splPrep is either a list of arrays or a 2D array (sigh) + _, y, _ = self._get_xyk() + assert isinstance(y, list) + assert np.shape(y)[0] == 2 + + # assert the behavior of FITPACK's splrep + tck, u = splprep(y, s=s) + tck_a, u_a = splprep(np.asarray(y), s=s) + xp_assert_close(u, u_a, atol=s) + xp_assert_close(tck[0], tck_a[0], atol=1e-15) + assert len(tck[1]) == len(tck_a[1]) + for c1, c2 in zip(tck[1], tck_a[1]): + xp_assert_close(c1, c2, atol=1e-15) + assert tck[2] == tck_a[2] + assert np.shape(splev(u, tck)) == np.shape(y) + + spl, u = make_splprep(y, s=s) + xp_assert_close(u, u_a, atol=1e-15) + xp_assert_close(spl.t, tck_a[0], atol=1e-15) + xp_assert_close(spl.c.T, tck_a[1], atol=1e-15) + assert spl.k == tck_a[2] + assert spl(u).shape == np.shape(y) + + spl, u = make_splprep(np.asarray(y), s=s) + xp_assert_close(u, u_a, atol=1e-15) + xp_assert_close(spl.t, tck_a[0], atol=1e-15) + xp_assert_close(spl.c.T, tck_a[1], atol=1e-15) + assert spl.k == tck_a[2] + assert spl(u).shape == np.shape(y) + + with assert_raises(ValueError): + make_splprep(np.asarray(y).T, s=s) + + def test_default_s_is_zero(self): + x, y, k = self._get_xyk(m=10) + + spl, u = make_splprep(y) + xp_assert_close(spl(u), y, atol=1e-15) + + def test_s_zero_vs_near_zero(self): + # s=0 and s \approx 0 are consistent + x, y, k = self._get_xyk(m=10) + + spl_i, u_i = make_splprep(y, s=0) + spl_n, u_n = make_splprep(y, s=1e-15) + + xp_assert_close(u_i, u_n, atol=1e-15) + xp_assert_close(spl_i(u_i), y, atol=1e-15) + xp_assert_close(spl_n(u_n), y, atol=1e-7) + assert spl_i.axis == spl_n.axis + assert spl_i.c.shape == spl_n.c.shape + + def test_1D(self): + x = np.arange(8, dtype=float) + with assert_raises(ValueError): + splprep(x) + + with assert_raises(ValueError): + make_splprep(x, s=0) + + with assert_raises(ValueError): + make_splprep(x, s=0.1) + + tck, u_ = splprep([x], s=1e-5) + spl, u = make_splprep([x], s=1e-5) + + assert spl(u).shape == (1, 8) + xp_assert_close(spl(u), [x], atol=1e-15) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_fitpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_fitpack.py new file mode 100644 index 0000000000000000000000000000000000000000..d798f0eda4eb0c099bdf46cb4b3468628013c9d3 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_fitpack.py @@ -0,0 +1,519 @@ +import itertools +import os + +import numpy as np +from scipy._lib._array_api import ( + xp_assert_equal, xp_assert_close, assert_almost_equal, assert_array_almost_equal +) +from pytest import raises as assert_raises +import pytest +from scipy._lib._testutils import check_free_memory + +from scipy.interpolate import RectBivariateSpline +from scipy.interpolate import make_splrep + +from scipy.interpolate._fitpack_py import (splrep, splev, bisplrep, bisplev, + sproot, splprep, splint, spalde, splder, splantider, insert, dblint) +from scipy.interpolate._dfitpack import regrid_smth +from scipy.interpolate._fitpack2 import dfitpack_int + + +def data_file(basename): + return os.path.join(os.path.abspath(os.path.dirname(__file__)), + 'data', basename) + + +def norm2(x): + return np.sqrt(np.dot(x.T, x)) + + +def f1(x, d=0): + """Derivatives of sin->cos->-sin->-cos.""" + if d % 4 == 0: + return np.sin(x) + if d % 4 == 1: + return np.cos(x) + if d % 4 == 2: + return -np.sin(x) + if d % 4 == 3: + return -np.cos(x) + + +def makepairs(x, y): + """Helper function to create an array of pairs of x and y.""" + xy = np.array(list(itertools.product(np.asarray(x), np.asarray(y)))) + return xy.T + + +class TestSmokeTests: + """ + Smoke tests (with a few asserts) for fitpack routines -- mostly + check that they are runnable + """ + def check_1(self, per=0, s=0, a=0, b=2*np.pi, at_nodes=False, + xb=None, xe=None): + if xb is None: + xb = a + if xe is None: + xe = b + + N = 20 + # nodes and middle points of the nodes + x = np.linspace(a, b, N + 1) + x1 = a + (b - a) * np.arange(1, N, dtype=float) / float(N - 1) + v = f1(x) + + def err_est(k, d): + # Assume f has all derivatives < 1 + h = 1.0 / N + tol = 5 * h**(.75*(k-d)) + if s > 0: + tol += 1e5*s + return tol + + for k in range(1, 6): + tck = splrep(x, v, s=s, per=per, k=k, xe=xe) + tt = tck[0][k:-k] if at_nodes else x1 + + for d in range(k+1): + tol = err_est(k, d) + err = norm2(f1(tt, d) - splev(tt, tck, d)) / norm2(f1(tt, d)) + assert err < tol + + # smoke test make_splrep + if not per: + spl = make_splrep(x, v, k=k, s=s, xb=xb, xe=xe) + if len(spl.t) == len(tck[0]): + xp_assert_close(spl.t, tck[0], atol=1e-15) + xp_assert_close(spl.c, tck[1][:spl.c.size], atol=1e-13) + else: + assert k == 5 # knot length differ in some k=5 cases + + def check_2(self, per=0, N=20, ia=0, ib=2*np.pi): + a, b, dx = 0, 2*np.pi, 0.2*np.pi + x = np.linspace(a, b, N+1) # nodes + v = np.sin(x) + + def err_est(k, d): + # Assume f has all derivatives < 1 + h = 1.0 / N + tol = 5 * h**(.75*(k-d)) + return tol + + nk = [] + for k in range(1, 6): + tck = splrep(x, v, s=0, per=per, k=k, xe=b) + nk.append([splint(ia, ib, tck), spalde(dx, tck)]) + + k = 1 + for r in nk: + d = 0 + for dr in r[1]: + tol = err_est(k, d) + xp_assert_close(dr, f1(dx, d), atol=0, rtol=tol) + d = d+1 + k = k+1 + + def test_smoke_splrep_splev(self): + self.check_1(s=1e-6) + self.check_1(b=1.5*np.pi) + self.check_1(b=1.5*np.pi, xe=2*np.pi, per=1, s=1e-1) + + @pytest.mark.parametrize('per', [0, 1]) + @pytest.mark.parametrize('at_nodes', [True, False]) + def test_smoke_splrep_splev_2(self, per, at_nodes): + self.check_1(per=per, at_nodes=at_nodes) + + @pytest.mark.parametrize('N', [20, 50]) + @pytest.mark.parametrize('per', [0, 1]) + def test_smoke_splint_spalde(self, N, per): + self.check_2(per=per, N=N) + + @pytest.mark.parametrize('N', [20, 50]) + @pytest.mark.parametrize('per', [0, 1]) + def test_smoke_splint_spalde_iaib(self, N, per): + self.check_2(ia=0.2*np.pi, ib=np.pi, N=N, per=per) + + def test_smoke_sproot(self): + # sproot is only implemented for k=3 + a, b = 0.1, 15 + x = np.linspace(a, b, 20) + v = np.sin(x) + + for k in [1, 2, 4, 5]: + tck = splrep(x, v, s=0, per=0, k=k, xe=b) + with assert_raises(ValueError): + sproot(tck) + + k = 3 + tck = splrep(x, v, s=0, k=3) + roots = sproot(tck) + xp_assert_close(splev(roots, tck), np.zeros(len(roots)), atol=1e-10, rtol=1e-10) + xp_assert_close(roots, np.pi * np.array([1, 2, 3, 4]), rtol=1e-3) + + @pytest.mark.parametrize('N', [20, 50]) + @pytest.mark.parametrize('k', [1, 2, 3, 4, 5]) + def test_smoke_splprep_splrep_splev(self, N, k): + a, b, dx = 0, 2.*np.pi, 0.2*np.pi + x = np.linspace(a, b, N+1) # nodes + v = np.sin(x) + + tckp, u = splprep([x, v], s=0, per=0, k=k, nest=-1) + uv = splev(dx, tckp) + err1 = abs(uv[1] - np.sin(uv[0])) + assert err1 < 1e-2 + + tck = splrep(x, v, s=0, per=0, k=k) + err2 = abs(splev(uv[0], tck) - np.sin(uv[0])) + assert err2 < 1e-2 + + # Derivatives of parametric cubic spline at u (first function) + if k == 3: + tckp, u = splprep([x, v], s=0, per=0, k=k, nest=-1) + for d in range(1, k+1): + uv = splev(dx, tckp, d) + + def test_smoke_bisplrep_bisplev(self): + xb, xe = 0, 2.*np.pi + yb, ye = 0, 2.*np.pi + kx, ky = 3, 3 + Nx, Ny = 20, 20 + + def f2(x, y): + return np.sin(x+y) + + x = np.linspace(xb, xe, Nx + 1) + y = np.linspace(yb, ye, Ny + 1) + xy = makepairs(x, y) + tck = bisplrep(xy[0], xy[1], f2(xy[0], xy[1]), s=0, kx=kx, ky=ky) + + tt = [tck[0][kx:-kx], tck[1][ky:-ky]] + t2 = makepairs(tt[0], tt[1]) + v1 = bisplev(tt[0], tt[1], tck) + v2 = f2(t2[0], t2[1]) + v2.shape = len(tt[0]), len(tt[1]) + + assert norm2(np.ravel(v1 - v2)) < 1e-2 + + +class TestSplev: + def test_1d_shape(self): + x = [1,2,3,4,5] + y = [4,5,6,7,8] + tck = splrep(x, y) + z = splev([1], tck) + assert z.shape == (1,) + z = splev(1, tck) + assert z.shape == () + + def test_2d_shape(self): + x = [1, 2, 3, 4, 5] + y = [4, 5, 6, 7, 8] + tck = splrep(x, y) + t = np.array([[1.0, 1.5, 2.0, 2.5], + [3.0, 3.5, 4.0, 4.5]]) + z = splev(t, tck) + z0 = splev(t[0], tck) + z1 = splev(t[1], tck) + xp_assert_equal(z, np.vstack((z0, z1))) + + def test_extrapolation_modes(self): + # test extrapolation modes + # * if ext=0, return the extrapolated value. + # * if ext=1, return 0 + # * if ext=2, raise a ValueError + # * if ext=3, return the boundary value. + x = [1,2,3] + y = [0,2,4] + tck = splrep(x, y, k=1) + + rstl = [[-2, 6], [0, 0], None, [0, 4]] + for ext in (0, 1, 3): + assert_array_almost_equal(splev([0, 4], tck, ext=ext), rstl[ext]) + + assert_raises(ValueError, splev, [0, 4], tck, ext=2) + + +class TestSplder: + def setup_method(self): + # non-uniform grid, just to make it sure + x = np.linspace(0, 1, 100)**3 + y = np.sin(20 * x) + self.spl = splrep(x, y) + + # double check that knots are non-uniform + assert np.ptp(np.diff(self.spl[0])) > 0 + + def test_inverse(self): + # Check that antiderivative + derivative is identity. + for n in range(5): + spl2 = splantider(self.spl, n) + spl3 = splder(spl2, n) + xp_assert_close(self.spl[0], spl3[0]) + xp_assert_close(self.spl[1], spl3[1]) + assert self.spl[2] == spl3[2] + + def test_splder_vs_splev(self): + # Check derivative vs. FITPACK + + for n in range(3+1): + # Also extrapolation! + xx = np.linspace(-1, 2, 2000) + if n == 3: + # ... except that FITPACK extrapolates strangely for + # order 0, so let's not check that. + xx = xx[(xx >= 0) & (xx <= 1)] + + dy = splev(xx, self.spl, n) + spl2 = splder(self.spl, n) + dy2 = splev(xx, spl2) + if n == 1: + xp_assert_close(dy, dy2, rtol=2e-6) + else: + xp_assert_close(dy, dy2) + + def test_splantider_vs_splint(self): + # Check antiderivative vs. FITPACK + spl2 = splantider(self.spl) + + # no extrapolation, splint assumes function is zero outside + # range + xx = np.linspace(0, 1, 20) + + for x1 in xx: + for x2 in xx: + y1 = splint(x1, x2, self.spl) + y2 = splev(x2, spl2) - splev(x1, spl2) + xp_assert_close(np.asarray(y1), np.asarray(y2)) + + def test_order0_diff(self): + assert_raises(ValueError, splder, self.spl, 4) + + def test_kink(self): + # Should refuse to differentiate splines with kinks + + spl2 = insert(0.5, self.spl, m=2) + splder(spl2, 2) # Should work + assert_raises(ValueError, splder, spl2, 3) + + spl2 = insert(0.5, self.spl, m=3) + splder(spl2, 1) # Should work + assert_raises(ValueError, splder, spl2, 2) + + spl2 = insert(0.5, self.spl, m=4) + assert_raises(ValueError, splder, spl2, 1) + + def test_multidim(self): + # c can have trailing dims + for n in range(3): + t, c, k = self.spl + c2 = np.c_[c, c, c] + c2 = np.dstack((c2, c2)) + + spl2 = splantider((t, c2, k), n) + spl3 = splder(spl2, n) + + xp_assert_close(t, spl3[0]) + xp_assert_close(c2, spl3[1]) + assert k == spl3[2] + + +class TestSplint: + def test_len_c(self): + n, k = 7, 3 + x = np.arange(n) + y = x**3 + t, c, k = splrep(x, y, s=0) + + # note that len(c) == len(t) == 11 (== len(x) + 2*(k-1)) + assert len(t) == len(c) == n + 2*(k-1) + + # integrate directly: $\int_0^6 x^3 dx = 6^4 / 4$ + res = splint(0, 6, (t, c, k)) + expected = 6**4 / 4 + assert abs(res - expected) < 1e-13 + + # check that the coefficients past len(t) - k - 1 are ignored + c0 = c.copy() + c0[len(t) - k - 1:] = np.nan + res0 = splint(0, 6, (t, c0, k)) + assert abs(res0 - expected) < 1e-13 + + # however, all other coefficients *are* used + c0[6] = np.nan + assert np.isnan(splint(0, 6, (t, c0, k))) + + # check that the coefficient array can have length `len(t) - k - 1` + c1 = c[:len(t) - k - 1] + res1 = splint(0, 6, (t, c1, k)) + assert (res1 - expected) < 1e-13 + + + # however shorter c arrays raise. The error from f2py is a + # `dftipack.error`, which is an Exception but not ValueError etc. + with assert_raises(Exception, match=r">=n-k-1"): + splint(0, 1, (np.ones(10), np.ones(5), 3)) + + +class TestBisplrep: + def test_overflow(self): + from numpy.lib.stride_tricks import as_strided + if dfitpack_int.itemsize == 8: + size = 1500000**2 + else: + size = 400**2 + # Don't allocate a real array, as it's very big, but rely + # on that it's not referenced + x = as_strided(np.zeros(()), shape=(size,)) + assert_raises(OverflowError, bisplrep, x, x, x, w=x, + xb=0, xe=1, yb=0, ye=1, s=0) + + def test_regression_1310(self): + # Regression test for gh-1310 + with np.load(data_file('bug-1310.npz')) as loaded_data: + data = loaded_data['data'] + + # Shouldn't crash -- the input data triggers work array sizes + # that caused previously some data to not be aligned on + # sizeof(double) boundaries in memory, which made the Fortran + # code to crash when compiled with -O3 + bisplrep(data[:,0], data[:,1], data[:,2], kx=3, ky=3, s=0, + full_output=True) + + @pytest.mark.skipif(dfitpack_int != np.int64, reason="needs ilp64 fitpack") + def test_ilp64_bisplrep(self): + check_free_memory(28000) # VM size, doesn't actually use the pages + x = np.linspace(0, 1, 400) + y = np.linspace(0, 1, 400) + x, y = np.meshgrid(x, y) + z = np.zeros_like(x) + tck = bisplrep(x, y, z, kx=3, ky=3, s=0) + xp_assert_close(bisplev(0.5, 0.5, tck), 0.0) + + +def test_dblint(): + # Basic test to see it runs and gives the correct result on a trivial + # problem. Note that `dblint` is not exposed in the interpolate namespace. + x = np.linspace(0, 1) + y = np.linspace(0, 1) + xx, yy = np.meshgrid(x, y) + rect = RectBivariateSpline(x, y, 4 * xx * yy) + tck = list(rect.tck) + tck.extend(rect.degrees) + + assert abs(dblint(0, 1, 0, 1, tck) - 1) < 1e-10 + assert abs(dblint(0, 0.5, 0, 1, tck) - 0.25) < 1e-10 + assert abs(dblint(0.5, 1, 0, 1, tck) - 0.75) < 1e-10 + assert abs(dblint(-100, 100, -100, 100, tck) - 1) < 1e-10 + + +def test_splev_der_k(): + # regression test for gh-2188: splev(x, tck, der=k) gives garbage or crashes + # for x outside of knot range + + # test case from gh-2188 + tck = (np.array([0., 0., 2.5, 2.5]), + np.array([-1.56679978, 2.43995873, 0., 0.]), + 1) + t, c, k = tck + x = np.array([-3, 0, 2.5, 3]) + + # an explicit form of the linear spline + xp_assert_close(splev(x, tck), c[0] + (c[1] - c[0]) * x/t[2]) + xp_assert_close(splev(x, tck, 1), + np.ones_like(x) * (c[1] - c[0]) / t[2] + ) + + # now check a random spline vs splder + np.random.seed(1234) + x = np.sort(np.random.random(30)) + y = np.random.random(30) + t, c, k = splrep(x, y) + + x = [t[0] - 1., t[-1] + 1.] + tck2 = splder((t, c, k), k) + xp_assert_close(splev(x, (t, c, k), k), splev(x, tck2)) + + +def test_splprep_segfault(): + # regression test for gh-3847: splprep segfaults if knots are specified + # for task=-1 + t = np.arange(0, 1.1, 0.1) + x = np.sin(2*np.pi*t) + y = np.cos(2*np.pi*t) + tck, u = splprep([x, y], s=0) + np.arange(0, 1.01, 0.01) + + uknots = tck[0] # using the knots from the previous fitting + tck, u = splprep([x, y], task=-1, t=uknots) # here is the crash + + +def test_bisplev_integer_overflow(): + np.random.seed(1) + + x = np.linspace(0, 1, 11) + y = x + z = np.random.randn(11, 11).ravel() + kx = 1 + ky = 1 + + nx, tx, ny, ty, c, fp, ier = regrid_smth( + x, y, z, None, None, None, None, kx=kx, ky=ky, s=0.0) + tck = (tx[:nx], ty[:ny], c[:(nx - kx - 1) * (ny - ky - 1)], kx, ky) + + xp = np.zeros([2621440]) + yp = np.zeros([2621440]) + + assert_raises((RuntimeError, MemoryError), bisplev, xp, yp, tck) + + +@pytest.mark.xslow +def test_gh_1766(): + # this should fail gracefully instead of segfaulting (int overflow) + size = 22 + kx, ky = 3, 3 + def f2(x, y): + return np.sin(x+y) + + x = np.linspace(0, 10, size) + y = np.linspace(50, 700, size) + xy = makepairs(x, y) + tck = bisplrep(xy[0], xy[1], f2(xy[0], xy[1]), s=0, kx=kx, ky=ky) + # the size value here can either segfault + # or produce a MemoryError on main + tx_ty_size = 500000 + tck[0] = np.arange(tx_ty_size) + tck[1] = np.arange(tx_ty_size) * 4 + tt_0 = np.arange(50) + tt_1 = np.arange(50) * 3 + with pytest.raises(MemoryError): + bisplev(tt_0, tt_1, tck, 1, 1) + + +def test_spalde_scalar_input(): + # Ticket #629 + x = np.linspace(0, 10) + y = x**3 + tck = splrep(x, y, k=3, t=[5]) + res = spalde(np.float64(1), tck) + des = np.array([1., 3., 6., 6.]) + assert_almost_equal(res, des) + + +def test_spalde_nc(): + # regression test for https://github.com/scipy/scipy/issues/19002 + # here len(t) = 29 and len(c) = 25 (== len(t) - k - 1) + x = np.asarray([-10., -9., -8., -7., -6., -5., -4., -3., -2.5, -2., -1.5, + -1., -0.5, 0., 0.5, 1., 1.5, 2., 2.5, 3., 4., 5., 6.], + dtype="float") + t = [-10.0, -10.0, -10.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, + -2.5, -2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, + 5.0, 6.0, 6.0, 6.0, 6.0] + c = np.asarray([1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., + 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]) + k = 3 + + res = spalde(x, (t, c, k)) + res = np.vstack(res) + res_splev = np.asarray([splev(x, (t, c, k), nu) for nu in range(4)]) + xp_assert_close(res, res_splev.T, atol=1e-15) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_fitpack2.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_fitpack2.py new file mode 100644 index 0000000000000000000000000000000000000000..044ace830bc6af26ee12edbcc6488e46bdfbcce6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_fitpack2.py @@ -0,0 +1,1393 @@ +# Created by Pearu Peterson, June 2003 +import itertools +from threading import Lock +import numpy as np +from numpy.testing import suppress_warnings +import pytest +from pytest import raises as assert_raises +from scipy._lib._array_api import ( + xp_assert_equal, xp_assert_close, assert_almost_equal, assert_array_almost_equal +) + +from numpy import array, diff, linspace, meshgrid, ones, pi, shape +from scipy.interpolate._fitpack_py import bisplrep, bisplev, splrep, spalde +from scipy.interpolate._fitpack2 import (UnivariateSpline, + LSQUnivariateSpline, InterpolatedUnivariateSpline, + LSQBivariateSpline, SmoothBivariateSpline, RectBivariateSpline, + LSQSphereBivariateSpline, SmoothSphereBivariateSpline, + RectSphereBivariateSpline) + +from scipy._lib._testutils import _run_concurrent_barrier + +from scipy.interpolate import make_splrep + +class TestUnivariateSpline: + def test_linear_constant(self): + x = [1,2,3] + y = [3,3,3] + lut = UnivariateSpline(x,y,k=1) + assert_array_almost_equal(lut.get_knots(), [1, 3]) + assert_array_almost_equal(lut.get_coeffs(), [3, 3]) + assert abs(lut.get_residual()) < 1e-10 + assert_array_almost_equal(lut([1, 1.5, 2]), [3, 3, 3]) + + spl = make_splrep(x, y, k=1, s=len(x)) + xp_assert_close(spl.t[1:-1], lut.get_knots(), atol=1e-15) + xp_assert_close(spl.c, lut.get_coeffs(), atol=1e-15) + + def test_preserve_shape(self): + x = [1, 2, 3] + y = [0, 2, 4] + lut = UnivariateSpline(x, y, k=1) + arg = 2 + assert shape(arg) == shape(lut(arg)) + assert shape(arg) == shape(lut(arg, nu=1)) + arg = [1.5, 2, 2.5] + assert shape(arg) == shape(lut(arg)) + assert shape(arg) == shape(lut(arg, nu=1)) + + def test_linear_1d(self): + x = [1,2,3] + y = [0,2,4] + lut = UnivariateSpline(x,y,k=1) + assert_array_almost_equal(lut.get_knots(),[1,3]) + assert_array_almost_equal(lut.get_coeffs(),[0,4]) + assert abs(lut.get_residual()) < 1e-15 + assert_array_almost_equal(lut([1,1.5,2]),[0,1,2]) + + def test_subclassing(self): + # See #731 + + class ZeroSpline(UnivariateSpline): + def __call__(self, x): + return 0*array(x) + + sp = ZeroSpline([1,2,3,4,5], [3,2,3,2,3], k=2) + xp_assert_equal(sp([1.5, 2.5]), [0., 0.]) + + def test_empty_input(self): + # Test whether empty input returns an empty output. Ticket 1014 + x = [1,3,5,7,9] + y = [0,4,9,12,21] + spl = UnivariateSpline(x, y, k=3) + xp_assert_equal(spl([]), array([])) + + def test_roots(self): + x = [1, 3, 5, 7, 9] + y = [0, 4, 9, 12, 21] + spl = UnivariateSpline(x, y, k=3) + assert_almost_equal(spl.roots()[0], 1.050290639101332) + + def test_roots_length(self): # for gh18335 + x = np.linspace(0, 50 * np.pi, 1000) + y = np.cos(x) + spl = UnivariateSpline(x, y, s=0) + assert len(spl.roots()) == 50 + + def test_derivatives(self): + x = [1, 3, 5, 7, 9] + y = [0, 4, 9, 12, 21] + spl = UnivariateSpline(x, y, k=3) + assert_almost_equal(spl.derivatives(3.5), + [5.5152902, 1.7146577, -0.1830357, 0.3125]) + + def test_derivatives_2(self): + x = np.arange(8) + y = x**3 + 2.*x**2 + + tck = splrep(x, y, s=0) + ders = spalde(3, tck) + xp_assert_close(ders, [45., # 3**3 + 2*(3)**2 + 39., # 3*(3)**2 + 4*(3) + 22., # 6*(3) + 4 + 6.], # 6*3**0 + atol=1e-15) + spl = UnivariateSpline(x, y, s=0, k=3) + xp_assert_close(spl.derivatives(3), + ders, + atol=1e-15) + + def test_resize_regression(self): + """Regression test for #1375.""" + x = [-1., -0.65016502, -0.58856235, -0.26903553, -0.17370892, + -0.10011001, 0., 0.10011001, 0.17370892, 0.26903553, 0.58856235, + 0.65016502, 1.] + y = [1.,0.62928599, 0.5797223, 0.39965815, 0.36322694, 0.3508061, + 0.35214793, 0.3508061, 0.36322694, 0.39965815, 0.5797223, + 0.62928599, 1.] + w = [1.00000000e+12, 6.88875973e+02, 4.89314737e+02, 4.26864807e+02, + 6.07746770e+02, 4.51341444e+02, 3.17480210e+02, 4.51341444e+02, + 6.07746770e+02, 4.26864807e+02, 4.89314737e+02, 6.88875973e+02, + 1.00000000e+12] + spl = UnivariateSpline(x=x, y=y, w=w, s=None) + desired = array([0.35100374, 0.51715855, 0.87789547, 0.98719344]) + xp_assert_close(spl([0.1, 0.5, 0.9, 0.99]), desired, atol=5e-4) + + def test_out_of_range_regression(self): + # Test different extrapolation modes. See ticket 3557 + x = np.arange(5, dtype=float) + y = x**3 + + xp = linspace(-8, 13, 100) + xp_zeros = xp.copy() + xp_zeros[np.logical_or(xp_zeros < 0., xp_zeros > 4.)] = 0 + xp_clip = xp.copy() + xp_clip[xp_clip < x[0]] = x[0] + xp_clip[xp_clip > x[-1]] = x[-1] + + for cls in [UnivariateSpline, InterpolatedUnivariateSpline]: + spl = cls(x=x, y=y) + for ext in [0, 'extrapolate']: + xp_assert_close(spl(xp, ext=ext), xp**3, atol=1e-16) + xp_assert_close(cls(x, y, ext=ext)(xp), xp**3, atol=1e-16) + for ext in [1, 'zeros']: + xp_assert_close(spl(xp, ext=ext), xp_zeros**3, atol=1e-16) + xp_assert_close(cls(x, y, ext=ext)(xp), xp_zeros**3, atol=1e-16) + for ext in [2, 'raise']: + assert_raises(ValueError, spl, xp, **dict(ext=ext)) + for ext in [3, 'const']: + xp_assert_close(spl(xp, ext=ext), xp_clip**3, atol=2e-16) + xp_assert_close(cls(x, y, ext=ext)(xp), xp_clip**3, atol=2e-16) + + # also test LSQUnivariateSpline [which needs explicit knots] + t = spl.get_knots()[3:4] # interior knots w/ default k=3 + spl = LSQUnivariateSpline(x, y, t) + xp_assert_close(spl(xp, ext=0), xp**3, atol=1e-16) + xp_assert_close(spl(xp, ext=1), xp_zeros**3, atol=1e-16) + assert_raises(ValueError, spl, xp, **dict(ext=2)) + xp_assert_close(spl(xp, ext=3), xp_clip**3, atol=1e-16) + + # also make sure that unknown values for `ext` are caught early + for ext in [-1, 'unknown']: + spl = UnivariateSpline(x, y) + assert_raises(ValueError, spl, xp, **dict(ext=ext)) + assert_raises(ValueError, UnivariateSpline, + **dict(x=x, y=y, ext=ext)) + + def test_lsq_fpchec(self): + xs = np.arange(100) * 1. + ys = np.arange(100) * 1. + knots = np.linspace(0, 99, 10) + bbox = (-1, 101) + assert_raises(ValueError, LSQUnivariateSpline, xs, ys, knots, + bbox=bbox) + + def test_derivative_and_antiderivative(self): + # Thin wrappers to splder/splantider, so light smoke test only. + x = np.linspace(0, 1, 70)**3 + y = np.cos(x) + + spl = UnivariateSpline(x, y, s=0) + spl2 = spl.antiderivative(2).derivative(2) + xp_assert_close(spl(0.3), spl2(0.3)) + + spl2 = spl.antiderivative(1) + xp_assert_close(spl2(0.6) - spl2(0.2), + spl.integral(0.2, 0.6)) + + def test_derivative_extrapolation(self): + # Regression test for gh-10195: for a const-extrapolation spline + # its derivative evaluates to zero for extrapolation + x_values = [1, 2, 4, 6, 8.5] + y_values = [0.5, 0.8, 1.3, 2.5, 5] + f = UnivariateSpline(x_values, y_values, ext='const', k=3) + + x = [-1, 0, -0.5, 9, 9.5, 10] + xp_assert_close(f.derivative()(x), np.zeros_like(x), atol=1e-15) + + def test_integral_out_of_bounds(self): + # Regression test for gh-7906: .integral(a, b) is wrong if both + # a and b are out-of-bounds + x = np.linspace(0., 1., 7) + for ext in range(4): + f = UnivariateSpline(x, x, s=0, ext=ext) + for (a, b) in [(1, 1), (1, 5), (2, 5), + (0, 0), (-2, 0), (-2, -1)]: + assert abs(f.integral(a, b)) < 1e-15 + + def test_nan(self): + # bail out early if the input data contains nans + x = np.arange(10, dtype=float) + y = x**3 + w = np.ones_like(x) + # also test LSQUnivariateSpline [which needs explicit knots] + spl = UnivariateSpline(x, y, check_finite=True) + t = spl.get_knots()[3:4] # interior knots w/ default k=3 + y_end = y[-1] + for z in [np.nan, np.inf, -np.inf]: + y[-1] = z + assert_raises(ValueError, UnivariateSpline, + **dict(x=x, y=y, check_finite=True)) + assert_raises(ValueError, InterpolatedUnivariateSpline, + **dict(x=x, y=y, check_finite=True)) + assert_raises(ValueError, LSQUnivariateSpline, + **dict(x=x, y=y, t=t, check_finite=True)) + y[-1] = y_end # check valid y but invalid w + w[-1] = z + assert_raises(ValueError, UnivariateSpline, + **dict(x=x, y=y, w=w, check_finite=True)) + assert_raises(ValueError, InterpolatedUnivariateSpline, + **dict(x=x, y=y, w=w, check_finite=True)) + assert_raises(ValueError, LSQUnivariateSpline, + **dict(x=x, y=y, t=t, w=w, check_finite=True)) + + def test_strictly_increasing_x(self): + # Test the x is required to be strictly increasing for + # UnivariateSpline if s=0 and for InterpolatedUnivariateSpline, + # but merely increasing for UnivariateSpline if s>0 + # and for LSQUnivariateSpline; see gh-8535 + xx = np.arange(10, dtype=float) + yy = xx**3 + x = np.arange(10, dtype=float) + x[1] = x[0] + y = x**3 + w = np.ones_like(x) + # also test LSQUnivariateSpline [which needs explicit knots] + spl = UnivariateSpline(xx, yy, check_finite=True) + t = spl.get_knots()[3:4] # interior knots w/ default k=3 + UnivariateSpline(x=x, y=y, w=w, s=1, check_finite=True) + LSQUnivariateSpline(x=x, y=y, t=t, w=w, check_finite=True) + assert_raises(ValueError, UnivariateSpline, + **dict(x=x, y=y, s=0, check_finite=True)) + assert_raises(ValueError, InterpolatedUnivariateSpline, + **dict(x=x, y=y, check_finite=True)) + + def test_increasing_x(self): + # Test that x is required to be increasing, see gh-8535 + xx = np.arange(10, dtype=float) + yy = xx**3 + x = np.arange(10, dtype=float) + x[1] = x[0] - 1.0 + y = x**3 + w = np.ones_like(x) + # also test LSQUnivariateSpline [which needs explicit knots] + spl = UnivariateSpline(xx, yy, check_finite=True) + t = spl.get_knots()[3:4] # interior knots w/ default k=3 + assert_raises(ValueError, UnivariateSpline, + **dict(x=x, y=y, check_finite=True)) + assert_raises(ValueError, InterpolatedUnivariateSpline, + **dict(x=x, y=y, check_finite=True)) + assert_raises(ValueError, LSQUnivariateSpline, + **dict(x=x, y=y, t=t, w=w, check_finite=True)) + + def test_invalid_input_for_univariate_spline(self): + + with assert_raises(ValueError) as info: + x_values = [1, 2, 4, 6, 8.5] + y_values = [0.5, 0.8, 1.3, 2.5] + UnivariateSpline(x_values, y_values) + assert "x and y should have a same length" in str(info.value) + + with assert_raises(ValueError) as info: + x_values = [1, 2, 4, 6, 8.5] + y_values = [0.5, 0.8, 1.3, 2.5, 2.8] + w_values = [-1.0, 1.0, 1.0, 1.0] + UnivariateSpline(x_values, y_values, w=w_values) + assert "x, y, and w should have a same length" in str(info.value) + + with assert_raises(ValueError) as info: + bbox = (-1) + UnivariateSpline(x_values, y_values, bbox=bbox) + assert "bbox shape should be (2,)" in str(info.value) + + with assert_raises(ValueError) as info: + UnivariateSpline(x_values, y_values, k=6) + assert "k should be 1 <= k <= 5" in str(info.value) + + with assert_raises(ValueError) as info: + UnivariateSpline(x_values, y_values, s=-1.0) + assert "s should be s >= 0.0" in str(info.value) + + def test_invalid_input_for_interpolated_univariate_spline(self): + + with assert_raises(ValueError) as info: + x_values = [1, 2, 4, 6, 8.5] + y_values = [0.5, 0.8, 1.3, 2.5] + InterpolatedUnivariateSpline(x_values, y_values) + assert "x and y should have a same length" in str(info.value) + + with assert_raises(ValueError) as info: + x_values = [1, 2, 4, 6, 8.5] + y_values = [0.5, 0.8, 1.3, 2.5, 2.8] + w_values = [-1.0, 1.0, 1.0, 1.0] + InterpolatedUnivariateSpline(x_values, y_values, w=w_values) + assert "x, y, and w should have a same length" in str(info.value) + + with assert_raises(ValueError) as info: + bbox = (-1) + InterpolatedUnivariateSpline(x_values, y_values, bbox=bbox) + assert "bbox shape should be (2,)" in str(info.value) + + with assert_raises(ValueError) as info: + InterpolatedUnivariateSpline(x_values, y_values, k=6) + assert "k should be 1 <= k <= 5" in str(info.value) + + def test_invalid_input_for_lsq_univariate_spline(self): + + x_values = [1, 2, 4, 6, 8.5] + y_values = [0.5, 0.8, 1.3, 2.5, 2.8] + spl = UnivariateSpline(x_values, y_values, check_finite=True) + t_values = spl.get_knots()[3:4] # interior knots w/ default k=3 + + with assert_raises(ValueError) as info: + x_values = [1, 2, 4, 6, 8.5] + y_values = [0.5, 0.8, 1.3, 2.5] + LSQUnivariateSpline(x_values, y_values, t_values) + assert "x and y should have a same length" in str(info.value) + + with assert_raises(ValueError) as info: + x_values = [1, 2, 4, 6, 8.5] + y_values = [0.5, 0.8, 1.3, 2.5, 2.8] + w_values = [1.0, 1.0, 1.0, 1.0] + LSQUnivariateSpline(x_values, y_values, t_values, w=w_values) + assert "x, y, and w should have a same length" in str(info.value) + + message = "Interior knots t must satisfy Schoenberg-Whitney conditions" + with assert_raises(ValueError, match=message) as info: + bbox = (100, -100) + LSQUnivariateSpline(x_values, y_values, t_values, bbox=bbox) + + with assert_raises(ValueError) as info: + bbox = (-1) + LSQUnivariateSpline(x_values, y_values, t_values, bbox=bbox) + assert "bbox shape should be (2,)" in str(info.value) + + with assert_raises(ValueError) as info: + LSQUnivariateSpline(x_values, y_values, t_values, k=6) + assert "k should be 1 <= k <= 5" in str(info.value) + + def test_array_like_input(self): + x_values = np.array([1, 2, 4, 6, 8.5]) + y_values = np.array([0.5, 0.8, 1.3, 2.5, 2.8]) + w_values = np.array([1.0, 1.0, 1.0, 1.0, 1.0]) + bbox = np.array([-100, 100]) + # np.array input + spl1 = UnivariateSpline(x=x_values, y=y_values, w=w_values, + bbox=bbox) + # list input + spl2 = UnivariateSpline(x=x_values.tolist(), y=y_values.tolist(), + w=w_values.tolist(), bbox=bbox.tolist()) + + xp_assert_close(spl1([0.1, 0.5, 0.9, 0.99]), + spl2([0.1, 0.5, 0.9, 0.99])) + + @pytest.mark.thread_unsafe + def test_fpknot_oob_crash(self): + # https://github.com/scipy/scipy/issues/3691 + x = range(109) + y = [0., 0., 0., 0., 0., 10.9, 0., 11., 0., + 0., 0., 10.9, 0., 0., 0., 0., 0., 0., + 10.9, 0., 0., 0., 11., 0., 0., 0., 10.9, + 0., 0., 0., 10.5, 0., 0., 0., 10.7, 0., + 0., 0., 11., 0., 0., 0., 0., 0., 0., + 10.9, 0., 0., 10.7, 0., 0., 0., 10.6, 0., + 0., 0., 10.5, 0., 0., 10.7, 0., 0., 10.5, + 0., 0., 11.5, 0., 0., 0., 10.7, 0., 0., + 10.7, 0., 0., 10.9, 0., 0., 10.8, 0., 0., + 0., 10.7, 0., 0., 10.6, 0., 0., 0., 10.4, + 0., 0., 10.6, 0., 0., 10.5, 0., 0., 0., + 10.7, 0., 0., 0., 10.4, 0., 0., 0., 10.8, 0.] + with suppress_warnings() as sup: + r = sup.record( + UserWarning, + r""" +The maximal number of iterations maxit \(set to 20 by the program\) +allowed for finding a smoothing spline with fp=s has been reached: s +too small. +There is an approximation returned but the corresponding weighted sum +of squared residuals does not satisfy the condition abs\(fp-s\)/s < tol.""") + UnivariateSpline(x, y, k=1) + assert len(r) == 1 + + def test_concurrency(self): + # Check that no segfaults appear with concurrent access to + # UnivariateSpline + xx = np.arange(100, dtype=float) + yy = xx**3 + x = np.arange(100, dtype=float) + x[1] = x[0] + spl = UnivariateSpline(xx, yy, check_finite=True) + + def worker_fn(_, interp, x): + interp(x) + + _run_concurrent_barrier(10, worker_fn, spl, x) + + +class TestLSQBivariateSpline: + # NOTE: The systems in this test class are rank-deficient + @pytest.mark.thread_unsafe + def test_linear_constant(self): + x = [1,1,1,2,2,2,3,3,3] + y = [1,2,3,1,2,3,1,2,3] + z = [3,3,3,3,3,3,3,3,3] + s = 0.1 + tx = [1+s,3-s] + ty = [1+s,3-s] + with suppress_warnings() as sup: + r = sup.record(UserWarning, "\nThe coefficients of the spline") + lut = LSQBivariateSpline(x,y,z,tx,ty,kx=1,ky=1) + assert len(r) == 1 + + assert_almost_equal(lut(2, 2), np.asarray(3.)) + + def test_bilinearity(self): + x = [1,1,1,2,2,2,3,3,3] + y = [1,2,3,1,2,3,1,2,3] + z = [0,7,8,3,4,7,1,3,4] + s = 0.1 + tx = [1+s,3-s] + ty = [1+s,3-s] + with suppress_warnings() as sup: + # This seems to fail (ier=1, see ticket 1642). + sup.filter(UserWarning, "\nThe coefficients of the spline") + lut = LSQBivariateSpline(x,y,z,tx,ty,kx=1,ky=1) + + tx, ty = lut.get_knots() + for xa, xb in zip(tx[:-1], tx[1:]): + for ya, yb in zip(ty[:-1], ty[1:]): + for t in [0.1, 0.5, 0.9]: + for s in [0.3, 0.4, 0.7]: + xp = xa*(1-t) + xb*t + yp = ya*(1-s) + yb*s + zp = (+ lut(xa, ya)*(1-t)*(1-s) + + lut(xb, ya)*t*(1-s) + + lut(xa, yb)*(1-t)*s + + lut(xb, yb)*t*s) + assert_almost_equal(lut(xp,yp), zp) + + @pytest.mark.thread_unsafe + def test_integral(self): + x = [1,1,1,2,2,2,8,8,8] + y = [1,2,3,1,2,3,1,2,3] + z = array([0,7,8,3,4,7,1,3,4]) + + s = 0.1 + tx = [1+s,3-s] + ty = [1+s,3-s] + with suppress_warnings() as sup: + r = sup.record(UserWarning, "\nThe coefficients of the spline") + lut = LSQBivariateSpline(x, y, z, tx, ty, kx=1, ky=1) + assert len(r) == 1 + tx, ty = lut.get_knots() + tz = lut(tx, ty) + trpz = .25*(diff(tx)[:,None]*diff(ty)[None,:] + * (tz[:-1,:-1]+tz[1:,:-1]+tz[:-1,1:]+tz[1:,1:])).sum() + + assert_almost_equal(np.asarray(lut.integral(tx[0], tx[-1], ty[0], ty[-1])), + np.asarray(trpz)) + + @pytest.mark.thread_unsafe + def test_empty_input(self): + # Test whether empty inputs returns an empty output. Ticket 1014 + x = [1,1,1,2,2,2,3,3,3] + y = [1,2,3,1,2,3,1,2,3] + z = [3,3,3,3,3,3,3,3,3] + s = 0.1 + tx = [1+s,3-s] + ty = [1+s,3-s] + with suppress_warnings() as sup: + r = sup.record(UserWarning, "\nThe coefficients of the spline") + lut = LSQBivariateSpline(x, y, z, tx, ty, kx=1, ky=1) + assert len(r) == 1 + + xp_assert_equal(lut([], []), np.zeros((0,0))) + xp_assert_equal(lut([], [], grid=False), np.zeros((0,))) + + def test_invalid_input(self): + s = 0.1 + tx = [1 + s, 3 - s] + ty = [1 + s, 3 - s] + + with assert_raises(ValueError) as info: + x = np.linspace(1.0, 10.0) + y = np.linspace(1.0, 10.0) + z = np.linspace(1.0, 10.0, num=10) + LSQBivariateSpline(x, y, z, tx, ty) + assert "x, y, and z should have a same length" in str(info.value) + + with assert_raises(ValueError) as info: + x = np.linspace(1.0, 10.0) + y = np.linspace(1.0, 10.0) + z = np.linspace(1.0, 10.0) + w = np.linspace(1.0, 10.0, num=20) + LSQBivariateSpline(x, y, z, tx, ty, w=w) + assert "x, y, z, and w should have a same length" in str(info.value) + + with assert_raises(ValueError) as info: + w = np.linspace(-1.0, 10.0) + LSQBivariateSpline(x, y, z, tx, ty, w=w) + assert "w should be positive" in str(info.value) + + with assert_raises(ValueError) as info: + bbox = (-100, 100, -100) + LSQBivariateSpline(x, y, z, tx, ty, bbox=bbox) + assert "bbox shape should be (4,)" in str(info.value) + + with assert_raises(ValueError) as info: + LSQBivariateSpline(x, y, z, tx, ty, kx=10, ky=10) + assert "The length of x, y and z should be at least (kx+1) * (ky+1)" in \ + str(info.value) + + with assert_raises(ValueError) as exc_info: + LSQBivariateSpline(x, y, z, tx, ty, eps=0.0) + assert "eps should be between (0, 1)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + LSQBivariateSpline(x, y, z, tx, ty, eps=1.0) + assert "eps should be between (0, 1)" in str(exc_info.value) + + @pytest.mark.thread_unsafe + def test_array_like_input(self): + s = 0.1 + tx = np.array([1 + s, 3 - s]) + ty = np.array([1 + s, 3 - s]) + x = np.linspace(1.0, 10.0) + y = np.linspace(1.0, 10.0) + z = np.linspace(1.0, 10.0) + w = np.linspace(1.0, 10.0) + bbox = np.array([1.0, 10.0, 1.0, 10.0]) + + with suppress_warnings() as sup: + r = sup.record(UserWarning, "\nThe coefficients of the spline") + # np.array input + spl1 = LSQBivariateSpline(x, y, z, tx, ty, w=w, bbox=bbox) + # list input + spl2 = LSQBivariateSpline(x.tolist(), y.tolist(), z.tolist(), + tx.tolist(), ty.tolist(), w=w.tolist(), + bbox=bbox) + xp_assert_close(spl1(2.0, 2.0), spl2(2.0, 2.0)) + assert len(r) == 2 + + @pytest.mark.thread_unsafe + def test_unequal_length_of_knots(self): + """Test for the case when the input knot-location arrays in x and y are + of different lengths. + """ + x, y = np.mgrid[0:100, 0:100] + x = x.ravel() + y = y.ravel() + z = 3.0 * np.ones_like(x) + tx = np.linspace(0.1, 98.0, 29) + ty = np.linspace(0.1, 98.0, 33) + with suppress_warnings() as sup: + r = sup.record(UserWarning, "\nThe coefficients of the spline") + lut = LSQBivariateSpline(x,y,z,tx,ty) + assert len(r) == 1 + + assert_almost_equal(lut(x, y, grid=False), z) + + +class TestSmoothBivariateSpline: + def test_linear_constant(self): + x = [1,1,1,2,2,2,3,3,3] + y = [1,2,3,1,2,3,1,2,3] + z = [3,3,3,3,3,3,3,3,3] + lut = SmoothBivariateSpline(x,y,z,kx=1,ky=1) + for t in lut.get_knots(): + assert_array_almost_equal(t, [1, 1, 3, 3]) + + assert_array_almost_equal(lut.get_coeffs(), [3, 3, 3, 3]) + assert abs(lut.get_residual()) < 1e-15 + assert_array_almost_equal(lut([1, 1.5, 2], [1, 1.5]), [[3, 3], [3, 3], [3, 3]]) + + def test_linear_1d(self): + x = [1,1,1,2,2,2,3,3,3] + y = [1,2,3,1,2,3,1,2,3] + z = [0,0,0,2,2,2,4,4,4] + lut = SmoothBivariateSpline(x,y,z,kx=1,ky=1) + for t in lut.get_knots(): + xp_assert_close(t, np.asarray([1.0, 1, 3, 3])) + assert_array_almost_equal(lut.get_coeffs(), [0, 0, 4, 4]) + assert abs(lut.get_residual()) < 1e-15 + assert_array_almost_equal(lut([1,1.5,2],[1,1.5]),[[0,0],[1,1],[2,2]]) + + @pytest.mark.thread_unsafe + def test_integral(self): + x = [1,1,1,2,2,2,4,4,4] + y = [1,2,3,1,2,3,1,2,3] + z = array([0,7,8,3,4,7,1,3,4]) + + with suppress_warnings() as sup: + # This seems to fail (ier=1, see ticket 1642). + sup.filter(UserWarning, "\nThe required storage space") + lut = SmoothBivariateSpline(x, y, z, kx=1, ky=1, s=0) + + tx = [1,2,4] + ty = [1,2,3] + + tz = lut(tx, ty) + trpz = .25*(diff(tx)[:,None]*diff(ty)[None,:] + * (tz[:-1,:-1]+tz[1:,:-1]+tz[:-1,1:]+tz[1:,1:])).sum() + assert_almost_equal(np.asarray(lut.integral(tx[0], tx[-1], ty[0], ty[-1])), + np.asarray(trpz)) + + lut2 = SmoothBivariateSpline(x, y, z, kx=2, ky=2, s=0) + assert_almost_equal(np.asarray(lut2.integral(tx[0], tx[-1], ty[0], ty[-1])), + np.asarray(trpz), + decimal=0) # the quadratures give 23.75 and 23.85 + + tz = lut(tx[:-1], ty[:-1]) + trpz = .25*(diff(tx[:-1])[:,None]*diff(ty[:-1])[None,:] + * (tz[:-1,:-1]+tz[1:,:-1]+tz[:-1,1:]+tz[1:,1:])).sum() + assert_almost_equal(np.asarray(lut.integral(tx[0], tx[-2], ty[0], ty[-2])), + np.asarray(trpz)) + + def test_rerun_lwrk2_too_small(self): + # in this setting, lwrk2 is too small in the default run. Here we + # check for equality with the bisplrep/bisplev output because there, + # an automatic re-run of the spline representation is done if ier>10. + x = np.linspace(-2, 2, 80) + y = np.linspace(-2, 2, 80) + z = x + y + xi = np.linspace(-1, 1, 100) + yi = np.linspace(-2, 2, 100) + tck = bisplrep(x, y, z) + res1 = bisplev(xi, yi, tck) + interp_ = SmoothBivariateSpline(x, y, z) + res2 = interp_(xi, yi) + assert_almost_equal(res1, res2) + + def test_invalid_input(self): + + with assert_raises(ValueError) as info: + x = np.linspace(1.0, 10.0) + y = np.linspace(1.0, 10.0) + z = np.linspace(1.0, 10.0, num=10) + SmoothBivariateSpline(x, y, z) + assert "x, y, and z should have a same length" in str(info.value) + + with assert_raises(ValueError) as info: + x = np.linspace(1.0, 10.0) + y = np.linspace(1.0, 10.0) + z = np.linspace(1.0, 10.0) + w = np.linspace(1.0, 10.0, num=20) + SmoothBivariateSpline(x, y, z, w=w) + assert "x, y, z, and w should have a same length" in str(info.value) + + with assert_raises(ValueError) as info: + w = np.linspace(-1.0, 10.0) + SmoothBivariateSpline(x, y, z, w=w) + assert "w should be positive" in str(info.value) + + with assert_raises(ValueError) as info: + bbox = (-100, 100, -100) + SmoothBivariateSpline(x, y, z, bbox=bbox) + assert "bbox shape should be (4,)" in str(info.value) + + with assert_raises(ValueError) as info: + SmoothBivariateSpline(x, y, z, kx=10, ky=10) + assert "The length of x, y and z should be at least (kx+1) * (ky+1)" in\ + str(info.value) + + with assert_raises(ValueError) as info: + SmoothBivariateSpline(x, y, z, s=-1.0) + assert "s should be s >= 0.0" in str(info.value) + + with assert_raises(ValueError) as exc_info: + SmoothBivariateSpline(x, y, z, eps=0.0) + assert "eps should be between (0, 1)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + SmoothBivariateSpline(x, y, z, eps=1.0) + assert "eps should be between (0, 1)" in str(exc_info.value) + + def test_array_like_input(self): + x = np.array([1, 1, 1, 2, 2, 2, 3, 3, 3]) + y = np.array([1, 2, 3, 1, 2, 3, 1, 2, 3]) + z = np.array([3, 3, 3, 3, 3, 3, 3, 3, 3]) + w = np.array([1, 1, 1, 1, 1, 1, 1, 1, 1]) + bbox = np.array([1.0, 3.0, 1.0, 3.0]) + # np.array input + spl1 = SmoothBivariateSpline(x, y, z, w=w, bbox=bbox, kx=1, ky=1) + # list input + spl2 = SmoothBivariateSpline(x.tolist(), y.tolist(), z.tolist(), + bbox=bbox.tolist(), w=w.tolist(), + kx=1, ky=1) + xp_assert_close(spl1(0.1, 0.5), spl2(0.1, 0.5)) + + +class TestLSQSphereBivariateSpline: + def setup_method(self): + # define the input data and coordinates + ntheta, nphi = 70, 90 + theta = linspace(0.5/(ntheta - 1), 1 - 0.5/(ntheta - 1), ntheta) * pi + phi = linspace(0.5/(nphi - 1), 1 - 0.5/(nphi - 1), nphi) * 2. * pi + data = ones((theta.shape[0], phi.shape[0])) + # define knots and extract data values at the knots + knotst = theta[::5] + knotsp = phi[::5] + knotdata = data[::5, ::5] + # calculate spline coefficients + lats, lons = meshgrid(theta, phi) + lut_lsq = LSQSphereBivariateSpline(lats.ravel(), lons.ravel(), + data.T.ravel(), knotst, knotsp) + self.lut_lsq = lut_lsq + self.data = knotdata + self.new_lons, self.new_lats = knotsp, knotst + + def test_linear_constant(self): + assert abs(self.lut_lsq.get_residual()) < 1e-15 + assert_array_almost_equal(self.lut_lsq(self.new_lats, self.new_lons), + self.data) + + def test_empty_input(self): + assert_array_almost_equal(self.lut_lsq([], []), np.zeros((0,0))) + assert_array_almost_equal(self.lut_lsq([], [], grid=False), np.zeros((0,))) + + def test_invalid_input(self): + ntheta, nphi = 70, 90 + theta = linspace(0.5 / (ntheta - 1), 1 - 0.5 / (ntheta - 1), + ntheta) * pi + phi = linspace(0.5 / (nphi - 1), 1 - 0.5 / (nphi - 1), nphi) * 2. * pi + data = ones((theta.shape[0], phi.shape[0])) + # define knots and extract data values at the knots + knotst = theta[::5] + knotsp = phi[::5] + + with assert_raises(ValueError) as exc_info: + invalid_theta = linspace(-0.1, 1.0, num=ntheta) * pi + invalid_lats, lons = meshgrid(invalid_theta, phi) + LSQSphereBivariateSpline(invalid_lats.ravel(), lons.ravel(), + data.T.ravel(), knotst, knotsp) + assert "theta should be between [0, pi]" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_theta = linspace(0.1, 1.1, num=ntheta) * pi + invalid_lats, lons = meshgrid(invalid_theta, phi) + LSQSphereBivariateSpline(invalid_lats.ravel(), lons.ravel(), + data.T.ravel(), knotst, knotsp) + assert "theta should be between [0, pi]" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_phi = linspace(-0.1, 1.0, num=ntheta) * 2.0 * pi + lats, invalid_lons = meshgrid(theta, invalid_phi) + LSQSphereBivariateSpline(lats.ravel(), invalid_lons.ravel(), + data.T.ravel(), knotst, knotsp) + assert "phi should be between [0, 2pi]" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_phi = linspace(0.0, 1.1, num=ntheta) * 2.0 * pi + lats, invalid_lons = meshgrid(theta, invalid_phi) + LSQSphereBivariateSpline(lats.ravel(), invalid_lons.ravel(), + data.T.ravel(), knotst, knotsp) + assert "phi should be between [0, 2pi]" in str(exc_info.value) + + lats, lons = meshgrid(theta, phi) + + with assert_raises(ValueError) as exc_info: + invalid_knotst = np.copy(knotst) + invalid_knotst[0] = -0.1 + LSQSphereBivariateSpline(lats.ravel(), lons.ravel(), + data.T.ravel(), invalid_knotst, knotsp) + assert "tt should be between (0, pi)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_knotst = np.copy(knotst) + invalid_knotst[0] = pi + LSQSphereBivariateSpline(lats.ravel(), lons.ravel(), + data.T.ravel(), invalid_knotst, knotsp) + assert "tt should be between (0, pi)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_knotsp = np.copy(knotsp) + invalid_knotsp[0] = -0.1 + LSQSphereBivariateSpline(lats.ravel(), lons.ravel(), + data.T.ravel(), knotst, invalid_knotsp) + assert "tp should be between (0, 2pi)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_knotsp = np.copy(knotsp) + invalid_knotsp[0] = 2 * pi + LSQSphereBivariateSpline(lats.ravel(), lons.ravel(), + data.T.ravel(), knotst, invalid_knotsp) + assert "tp should be between (0, 2pi)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_w = array([-1.0, 1.0, 1.5, 0.5, 1.0, 1.5, 0.5, 1.0, 1.0]) + LSQSphereBivariateSpline(lats.ravel(), lons.ravel(), data.T.ravel(), + knotst, knotsp, w=invalid_w) + assert "w should be positive" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + LSQSphereBivariateSpline(lats.ravel(), lons.ravel(), data.T.ravel(), + knotst, knotsp, eps=0.0) + assert "eps should be between (0, 1)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + LSQSphereBivariateSpline(lats.ravel(), lons.ravel(), data.T.ravel(), + knotst, knotsp, eps=1.0) + assert "eps should be between (0, 1)" in str(exc_info.value) + + def test_array_like_input(self): + ntheta, nphi = 70, 90 + theta = linspace(0.5 / (ntheta - 1), 1 - 0.5 / (ntheta - 1), + ntheta) * pi + phi = linspace(0.5 / (nphi - 1), 1 - 0.5 / (nphi - 1), + nphi) * 2. * pi + lats, lons = meshgrid(theta, phi) + data = ones((theta.shape[0], phi.shape[0])) + # define knots and extract data values at the knots + knotst = theta[::5] + knotsp = phi[::5] + w = ones(lats.ravel().shape[0]) + + # np.array input + spl1 = LSQSphereBivariateSpline(lats.ravel(), lons.ravel(), + data.T.ravel(), knotst, knotsp, w=w) + # list input + spl2 = LSQSphereBivariateSpline(lats.ravel().tolist(), + lons.ravel().tolist(), + data.T.ravel().tolist(), + knotst.tolist(), + knotsp.tolist(), w=w.tolist()) + assert_array_almost_equal(spl1(1.0, 1.0), spl2(1.0, 1.0)) + + +class TestSmoothSphereBivariateSpline: + def setup_method(self): + theta = array([.25*pi, .25*pi, .25*pi, .5*pi, .5*pi, .5*pi, .75*pi, + .75*pi, .75*pi]) + phi = array([.5 * pi, pi, 1.5 * pi, .5 * pi, pi, 1.5 * pi, .5 * pi, pi, + 1.5 * pi]) + r = array([3, 3, 3, 3, 3, 3, 3, 3, 3]) + self.lut = SmoothSphereBivariateSpline(theta, phi, r, s=1E10) + + def test_linear_constant(self): + assert abs(self.lut.get_residual()) < 1e-15 + assert_array_almost_equal(self.lut([1, 1.5, 2],[1, 1.5]), + [[3, 3], [3, 3], [3, 3]]) + + def test_empty_input(self): + assert_array_almost_equal(self.lut([], []), np.zeros((0,0))) + assert_array_almost_equal(self.lut([], [], grid=False), np.zeros((0,))) + + def test_invalid_input(self): + theta = array([.25 * pi, .25 * pi, .25 * pi, .5 * pi, .5 * pi, .5 * pi, + .75 * pi, .75 * pi, .75 * pi]) + phi = array([.5 * pi, pi, 1.5 * pi, .5 * pi, pi, 1.5 * pi, .5 * pi, pi, + 1.5 * pi]) + r = array([3, 3, 3, 3, 3, 3, 3, 3, 3]) + + with assert_raises(ValueError) as exc_info: + invalid_theta = array([-0.1 * pi, .25 * pi, .25 * pi, .5 * pi, + .5 * pi, .5 * pi, .75 * pi, .75 * pi, + .75 * pi]) + SmoothSphereBivariateSpline(invalid_theta, phi, r, s=1E10) + assert "theta should be between [0, pi]" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_theta = array([.25 * pi, .25 * pi, .25 * pi, .5 * pi, + .5 * pi, .5 * pi, .75 * pi, .75 * pi, + 1.1 * pi]) + SmoothSphereBivariateSpline(invalid_theta, phi, r, s=1E10) + assert "theta should be between [0, pi]" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_phi = array([-.1 * pi, pi, 1.5 * pi, .5 * pi, pi, 1.5 * pi, + .5 * pi, pi, 1.5 * pi]) + SmoothSphereBivariateSpline(theta, invalid_phi, r, s=1E10) + assert "phi should be between [0, 2pi]" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_phi = array([1.0 * pi, pi, 1.5 * pi, .5 * pi, pi, 1.5 * pi, + .5 * pi, pi, 2.1 * pi]) + SmoothSphereBivariateSpline(theta, invalid_phi, r, s=1E10) + assert "phi should be between [0, 2pi]" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + invalid_w = array([-1.0, 1.0, 1.5, 0.5, 1.0, 1.5, 0.5, 1.0, 1.0]) + SmoothSphereBivariateSpline(theta, phi, r, w=invalid_w, s=1E10) + assert "w should be positive" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + SmoothSphereBivariateSpline(theta, phi, r, s=-1.0) + assert "s should be positive" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + SmoothSphereBivariateSpline(theta, phi, r, eps=-1.0) + assert "eps should be between (0, 1)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + SmoothSphereBivariateSpline(theta, phi, r, eps=1.0) + assert "eps should be between (0, 1)" in str(exc_info.value) + + def test_array_like_input(self): + theta = np.array([.25 * pi, .25 * pi, .25 * pi, .5 * pi, .5 * pi, + .5 * pi, .75 * pi, .75 * pi, .75 * pi]) + phi = np.array([.5 * pi, pi, 1.5 * pi, .5 * pi, pi, 1.5 * pi, .5 * pi, + pi, 1.5 * pi]) + r = np.array([3, 3, 3, 3, 3, 3, 3, 3, 3]) + w = np.array([1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]) + + # np.array input + spl1 = SmoothSphereBivariateSpline(theta, phi, r, w=w, s=1E10) + + # list input + spl2 = SmoothSphereBivariateSpline(theta.tolist(), phi.tolist(), + r.tolist(), w=w.tolist(), s=1E10) + assert_array_almost_equal(spl1(1.0, 1.0), spl2(1.0, 1.0)) + + +class TestRectBivariateSpline: + def test_defaults(self): + x = array([1,2,3,4,5]) + y = array([1,2,3,4,5]) + z = array([[1,2,1,2,1],[1,2,1,2,1],[1,2,3,2,1],[1,2,2,2,1],[1,2,1,2,1]]) + lut = RectBivariateSpline(x,y,z) + assert_array_almost_equal(lut(x,y),z) + + def test_evaluate(self): + x = array([1,2,3,4,5]) + y = array([1,2,3,4,5]) + z = array([[1,2,1,2,1],[1,2,1,2,1],[1,2,3,2,1],[1,2,2,2,1],[1,2,1,2,1]]) + lut = RectBivariateSpline(x,y,z) + + xi = [1, 2.3, 5.3, 0.5, 3.3, 1.2, 3] + yi = [1, 3.3, 1.2, 4.0, 5.0, 1.0, 3] + zi = lut.ev(xi, yi) + zi2 = array([lut(xp, yp)[0,0] for xp, yp in zip(xi, yi)]) + + assert_almost_equal(zi, zi2) + + def test_derivatives_grid(self): + x = array([1,2,3,4,5]) + y = array([1,2,3,4,5]) + z = array([[1,2,1,2,1],[1,2,1,2,1],[1,2,3,2,1],[1,2,2,2,1],[1,2,1,2,1]]) + dx = array([[0,0,-20,0,0],[0,0,13,0,0],[0,0,4,0,0], + [0,0,-11,0,0],[0,0,4,0,0]])/6. + dy = array([[4,-1,0,1,-4],[4,-1,0,1,-4],[0,1.5,0,-1.5,0], + [2,.25,0,-.25,-2],[4,-1,0,1,-4]]) + dxdy = array([[40,-25,0,25,-40],[-26,16.25,0,-16.25,26], + [-8,5,0,-5,8],[22,-13.75,0,13.75,-22],[-8,5,0,-5,8]])/6. + lut = RectBivariateSpline(x,y,z) + assert_array_almost_equal(lut(x,y,dx=1),dx) + assert_array_almost_equal(lut(x,y,dy=1),dy) + assert_array_almost_equal(lut(x,y,dx=1,dy=1),dxdy) + + def test_derivatives(self): + x = array([1,2,3,4,5]) + y = array([1,2,3,4,5]) + z = array([[1,2,1,2,1],[1,2,1,2,1],[1,2,3,2,1],[1,2,2,2,1],[1,2,1,2,1]]) + dx = array([0,0,2./3,0,0]) + dy = array([4,-1,0,-.25,-4]) + dxdy = array([160,65,0,55,32])/24. + lut = RectBivariateSpline(x,y,z) + assert_array_almost_equal(lut(x,y,dx=1,grid=False),dx) + assert_array_almost_equal(lut(x,y,dy=1,grid=False),dy) + assert_array_almost_equal(lut(x,y,dx=1,dy=1,grid=False),dxdy) + + def test_partial_derivative_method_grid(self): + x = array([1, 2, 3, 4, 5]) + y = array([1, 2, 3, 4, 5]) + z = array([[1, 2, 1, 2, 1], + [1, 2, 1, 2, 1], + [1, 2, 3, 2, 1], + [1, 2, 2, 2, 1], + [1, 2, 1, 2, 1]]) + dx = array([[0, 0, -20, 0, 0], + [0, 0, 13, 0, 0], + [0, 0, 4, 0, 0], + [0, 0, -11, 0, 0], + [0, 0, 4, 0, 0]]) / 6. + dy = array([[4, -1, 0, 1, -4], + [4, -1, 0, 1, -4], + [0, 1.5, 0, -1.5, 0], + [2, .25, 0, -.25, -2], + [4, -1, 0, 1, -4]]) + dxdy = array([[40, -25, 0, 25, -40], + [-26, 16.25, 0, -16.25, 26], + [-8, 5, 0, -5, 8], + [22, -13.75, 0, 13.75, -22], + [-8, 5, 0, -5, 8]]) / 6. + lut = RectBivariateSpline(x, y, z) + assert_array_almost_equal(lut.partial_derivative(1, 0)(x, y), dx) + assert_array_almost_equal(lut.partial_derivative(0, 1)(x, y), dy) + assert_array_almost_equal(lut.partial_derivative(1, 1)(x, y), dxdy) + + def test_partial_derivative_method(self): + x = array([1, 2, 3, 4, 5]) + y = array([1, 2, 3, 4, 5]) + z = array([[1, 2, 1, 2, 1], + [1, 2, 1, 2, 1], + [1, 2, 3, 2, 1], + [1, 2, 2, 2, 1], + [1, 2, 1, 2, 1]]) + dx = array([0, 0, 2./3, 0, 0]) + dy = array([4, -1, 0, -.25, -4]) + dxdy = array([160, 65, 0, 55, 32]) / 24. + lut = RectBivariateSpline(x, y, z) + assert_array_almost_equal(lut.partial_derivative(1, 0)(x, y, + grid=False), + dx) + assert_array_almost_equal(lut.partial_derivative(0, 1)(x, y, + grid=False), + dy) + assert_array_almost_equal(lut.partial_derivative(1, 1)(x, y, + grid=False), + dxdy) + + def test_partial_derivative_order_too_large(self): + x = array([0, 1, 2, 3, 4], dtype=float) + y = x.copy() + z = ones((x.size, y.size)) + lut = RectBivariateSpline(x, y, z) + with assert_raises(ValueError): + lut.partial_derivative(4, 1) + + def test_broadcast(self): + x = array([1,2,3,4,5]) + y = array([1,2,3,4,5]) + z = array([[1,2,1,2,1],[1,2,1,2,1],[1,2,3,2,1],[1,2,2,2,1],[1,2,1,2,1]]) + lut = RectBivariateSpline(x,y,z) + xp_assert_close(lut(x, y), lut(x[:,None], y[None,:], grid=False)) + + def test_invalid_input(self): + + with assert_raises(ValueError) as info: + x = array([6, 2, 3, 4, 5]) + y = array([1, 2, 3, 4, 5]) + z = array([[1, 2, 1, 2, 1], [1, 2, 1, 2, 1], [1, 2, 3, 2, 1], + [1, 2, 2, 2, 1], [1, 2, 1, 2, 1]]) + RectBivariateSpline(x, y, z) + assert "x must be strictly increasing" in str(info.value) + + with assert_raises(ValueError) as info: + x = array([1, 2, 3, 4, 5]) + y = array([2, 2, 3, 4, 5]) + z = array([[1, 2, 1, 2, 1], [1, 2, 1, 2, 1], [1, 2, 3, 2, 1], + [1, 2, 2, 2, 1], [1, 2, 1, 2, 1]]) + RectBivariateSpline(x, y, z) + assert "y must be strictly increasing" in str(info.value) + + with assert_raises(ValueError) as info: + x = array([1, 2, 3, 4, 5]) + y = array([1, 2, 3, 4, 5]) + z = array([[1, 2, 1, 2, 1], [1, 2, 1, 2, 1], [1, 2, 3, 2, 1], + [1, 2, 2, 2, 1]]) + RectBivariateSpline(x, y, z) + assert "x dimension of z must have same number of elements as x"\ + in str(info.value) + + with assert_raises(ValueError) as info: + x = array([1, 2, 3, 4, 5]) + y = array([1, 2, 3, 4, 5]) + z = array([[1, 2, 1, 2], [1, 2, 1, 2], [1, 2, 3, 2], + [1, 2, 2, 2], [1, 2, 1, 2]]) + RectBivariateSpline(x, y, z) + assert "y dimension of z must have same number of elements as y"\ + in str(info.value) + + with assert_raises(ValueError) as info: + x = array([1, 2, 3, 4, 5]) + y = array([1, 2, 3, 4, 5]) + z = array([[1, 2, 1, 2, 1], [1, 2, 1, 2, 1], [1, 2, 3, 2, 1], + [1, 2, 2, 2, 1], [1, 2, 1, 2, 1]]) + bbox = (-100, 100, -100) + RectBivariateSpline(x, y, z, bbox=bbox) + assert "bbox shape should be (4,)" in str(info.value) + + with assert_raises(ValueError) as info: + RectBivariateSpline(x, y, z, s=-1.0) + assert "s should be s >= 0.0" in str(info.value) + + def test_array_like_input(self): + x = array([1, 2, 3, 4, 5]) + y = array([1, 2, 3, 4, 5]) + z = array([[1, 2, 1, 2, 1], [1, 2, 1, 2, 1], [1, 2, 3, 2, 1], + [1, 2, 2, 2, 1], [1, 2, 1, 2, 1]]) + bbox = array([1, 5, 1, 5]) + + spl1 = RectBivariateSpline(x, y, z, bbox=bbox) + spl2 = RectBivariateSpline(x.tolist(), y.tolist(), z.tolist(), + bbox=bbox.tolist()) + assert_array_almost_equal(spl1(1.0, 1.0), spl2(1.0, 1.0)) + + def test_not_increasing_input(self): + # gh-8565 + NSamp = 20 + Theta = np.random.uniform(0, np.pi, NSamp) + Phi = np.random.uniform(0, 2 * np.pi, NSamp) + Data = np.ones(NSamp) + + Interpolator = SmoothSphereBivariateSpline(Theta, Phi, Data, s=3.5) + + NLon = 6 + NLat = 3 + GridPosLats = np.arange(NLat) / NLat * np.pi + GridPosLons = np.arange(NLon) / NLon * 2 * np.pi + + # No error + Interpolator(GridPosLats, GridPosLons) + + nonGridPosLats = GridPosLats.copy() + nonGridPosLats[2] = 0.001 + with assert_raises(ValueError) as exc_info: + Interpolator(nonGridPosLats, GridPosLons) + assert "x must be strictly increasing" in str(exc_info.value) + + nonGridPosLons = GridPosLons.copy() + nonGridPosLons[2] = 0.001 + with assert_raises(ValueError) as exc_info: + Interpolator(GridPosLats, nonGridPosLons) + assert "y must be strictly increasing" in str(exc_info.value) + + +class TestRectSphereBivariateSpline: + def test_defaults(self): + y = linspace(0.01, 2*pi-0.01, 7) + x = linspace(0.01, pi-0.01, 7) + z = array([[1,2,1,2,1,2,1],[1,2,1,2,1,2,1],[1,2,3,2,1,2,1], + [1,2,2,2,1,2,1],[1,2,1,2,1,2,1],[1,2,2,2,1,2,1], + [1,2,1,2,1,2,1]]) + lut = RectSphereBivariateSpline(x,y,z) + assert_array_almost_equal(lut(x,y),z) + + def test_evaluate(self): + y = linspace(0.01, 2*pi-0.01, 7) + x = linspace(0.01, pi-0.01, 7) + z = array([[1,2,1,2,1,2,1],[1,2,1,2,1,2,1],[1,2,3,2,1,2,1], + [1,2,2,2,1,2,1],[1,2,1,2,1,2,1],[1,2,2,2,1,2,1], + [1,2,1,2,1,2,1]]) + lut = RectSphereBivariateSpline(x,y,z) + yi = [0.2, 1, 2.3, 2.35, 3.0, 3.99, 5.25] + xi = [1.5, 0.4, 1.1, 0.45, 0.2345, 1., 0.0001] + zi = lut.ev(xi, yi) + zi2 = array([lut(xp, yp)[0,0] for xp, yp in zip(xi, yi)]) + assert_almost_equal(zi, zi2) + + def test_invalid_input(self): + data = np.dot(np.atleast_2d(90. - np.linspace(-80., 80., 18)).T, + np.atleast_2d(180. - np.abs(np.linspace(0., 350., 9)))).T + + with assert_raises(ValueError) as exc_info: + lats = np.linspace(-1, 170, 9) * np.pi / 180. + lons = np.linspace(0, 350, 18) * np.pi / 180. + RectSphereBivariateSpline(lats, lons, data) + assert "u should be between (0, pi)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + lats = np.linspace(10, 181, 9) * np.pi / 180. + lons = np.linspace(0, 350, 18) * np.pi / 180. + RectSphereBivariateSpline(lats, lons, data) + assert "u should be between (0, pi)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + lats = np.linspace(10, 170, 9) * np.pi / 180. + lons = np.linspace(-181, 10, 18) * np.pi / 180. + RectSphereBivariateSpline(lats, lons, data) + assert "v[0] should be between [-pi, pi)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + lats = np.linspace(10, 170, 9) * np.pi / 180. + lons = np.linspace(-10, 360, 18) * np.pi / 180. + RectSphereBivariateSpline(lats, lons, data) + assert "v[-1] should be v[0] + 2pi or less" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + lats = np.linspace(10, 170, 9) * np.pi / 180. + lons = np.linspace(10, 350, 18) * np.pi / 180. + RectSphereBivariateSpline(lats, lons, data, s=-1) + assert "s should be positive" in str(exc_info.value) + + def test_derivatives_grid(self): + y = linspace(0.01, 2*pi-0.01, 7) + x = linspace(0.01, pi-0.01, 7) + z = array([[1,2,1,2,1,2,1],[1,2,1,2,1,2,1],[1,2,3,2,1,2,1], + [1,2,2,2,1,2,1],[1,2,1,2,1,2,1],[1,2,2,2,1,2,1], + [1,2,1,2,1,2,1]]) + + lut = RectSphereBivariateSpline(x,y,z) + + y = linspace(0.02, 2*pi-0.02, 7) + x = linspace(0.02, pi-0.02, 7) + + xp_assert_close(lut(x, y, dtheta=1), _numdiff_2d(lut, x, y, dx=1), + rtol=1e-4, atol=1e-4) + xp_assert_close(lut(x, y, dphi=1), _numdiff_2d(lut, x, y, dy=1), + rtol=1e-4, atol=1e-4) + xp_assert_close(lut(x, y, dtheta=1, dphi=1), + _numdiff_2d(lut, x, y, dx=1, dy=1, eps=1e-6), + rtol=1e-3, atol=1e-3) + + xp_assert_equal(lut(x, y, dtheta=1), + lut.partial_derivative(1, 0)(x, y)) + xp_assert_equal(lut(x, y, dphi=1), + lut.partial_derivative(0, 1)(x, y)) + xp_assert_equal(lut(x, y, dtheta=1, dphi=1), + lut.partial_derivative(1, 1)(x, y)) + + xp_assert_equal(lut(x, y, dtheta=1, grid=False), + lut.partial_derivative(1, 0)(x, y, grid=False)) + xp_assert_equal(lut(x, y, dphi=1, grid=False), + lut.partial_derivative(0, 1)(x, y, grid=False)) + xp_assert_equal(lut(x, y, dtheta=1, dphi=1, grid=False), + lut.partial_derivative(1, 1)(x, y, grid=False)) + + def test_derivatives(self): + y = linspace(0.01, 2*pi-0.01, 7) + x = linspace(0.01, pi-0.01, 7) + z = array([[1,2,1,2,1,2,1],[1,2,1,2,1,2,1],[1,2,3,2,1,2,1], + [1,2,2,2,1,2,1],[1,2,1,2,1,2,1],[1,2,2,2,1,2,1], + [1,2,1,2,1,2,1]]) + + lut = RectSphereBivariateSpline(x,y,z) + + y = linspace(0.02, 2*pi-0.02, 7) + x = linspace(0.02, pi-0.02, 7) + + assert lut(x, y, dtheta=1, grid=False).shape == x.shape + xp_assert_close(lut(x, y, dtheta=1, grid=False), + _numdiff_2d(lambda x,y: lut(x,y,grid=False), x, y, dx=1), + rtol=1e-4, atol=1e-4) + xp_assert_close(lut(x, y, dphi=1, grid=False), + _numdiff_2d(lambda x,y: lut(x,y,grid=False), x, y, dy=1), + rtol=1e-4, atol=1e-4) + xp_assert_close(lut(x, y, dtheta=1, dphi=1, grid=False), + _numdiff_2d(lambda x,y: lut(x,y,grid=False), + x, y, dx=1, dy=1, eps=1e-6), + rtol=1e-3, atol=1e-3) + + def test_invalid_input_2(self): + data = np.dot(np.atleast_2d(90. - np.linspace(-80., 80., 18)).T, + np.atleast_2d(180. - np.abs(np.linspace(0., 350., 9)))).T + + with assert_raises(ValueError) as exc_info: + lats = np.linspace(0, 170, 9) * np.pi / 180. + lons = np.linspace(0, 350, 18) * np.pi / 180. + RectSphereBivariateSpline(lats, lons, data) + assert "u should be between (0, pi)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + lats = np.linspace(10, 180, 9) * np.pi / 180. + lons = np.linspace(0, 350, 18) * np.pi / 180. + RectSphereBivariateSpline(lats, lons, data) + assert "u should be between (0, pi)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + lats = np.linspace(10, 170, 9) * np.pi / 180. + lons = np.linspace(-181, 10, 18) * np.pi / 180. + RectSphereBivariateSpline(lats, lons, data) + assert "v[0] should be between [-pi, pi)" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + lats = np.linspace(10, 170, 9) * np.pi / 180. + lons = np.linspace(-10, 360, 18) * np.pi / 180. + RectSphereBivariateSpline(lats, lons, data) + assert "v[-1] should be v[0] + 2pi or less" in str(exc_info.value) + + with assert_raises(ValueError) as exc_info: + lats = np.linspace(10, 170, 9) * np.pi / 180. + lons = np.linspace(10, 350, 18) * np.pi / 180. + RectSphereBivariateSpline(lats, lons, data, s=-1) + assert "s should be positive" in str(exc_info.value) + + def test_array_like_input(self): + y = linspace(0.01, 2 * pi - 0.01, 7) + x = linspace(0.01, pi - 0.01, 7) + z = array([[1, 2, 1, 2, 1, 2, 1], [1, 2, 1, 2, 1, 2, 1], + [1, 2, 3, 2, 1, 2, 1], + [1, 2, 2, 2, 1, 2, 1], [1, 2, 1, 2, 1, 2, 1], + [1, 2, 2, 2, 1, 2, 1], + [1, 2, 1, 2, 1, 2, 1]]) + # np.array input + spl1 = RectSphereBivariateSpline(x, y, z) + # list input + spl2 = RectSphereBivariateSpline(x.tolist(), y.tolist(), z.tolist()) + assert_array_almost_equal(spl1(x, y), spl2(x, y)) + + def test_negative_evaluation(self): + lats = np.array([25, 30, 35, 40, 45]) + lons = np.array([-90, -85, -80, -75, 70]) + mesh = np.meshgrid(lats, lons) + data = mesh[0] + mesh[1] # lon + lat value + lat_r = np.radians(lats) + lon_r = np.radians(lons) + interpolator = RectSphereBivariateSpline(lat_r, lon_r, data) + query_lat = np.radians(np.array([35, 37.5])) + query_lon = np.radians(np.array([-80, -77.5])) + data_interp = interpolator(query_lat, query_lon) + ans = np.array([[-45.0, -42.480862], + [-49.0625, -46.54315]]) + assert_array_almost_equal(data_interp, ans) + + def test_pole_continuity_gh_14591(self): + # regression test for https://github.com/scipy/scipy/issues/14591 + # with pole_continuty=(True, True), the internal work array size + # was too small, leading to a FITPACK data validation error. + + # The reproducer in gh-14591 was using a NetCDF4 file with + # 361x507 arrays, so here we trivialize array sizes to a minimum + # which still demonstrates the issue. + u = np.arange(1, 10) * np.pi / 10 + v = np.arange(1, 10) * np.pi / 10 + r = np.zeros((9, 9)) + for p in [(True, True), (True, False), (False, False)]: + RectSphereBivariateSpline(u, v, r, s=0, pole_continuity=p) + + +def _numdiff_2d(func, x, y, dx=0, dy=0, eps=1e-8): + if dx == 0 and dy == 0: + return func(x, y) + elif dx == 1 and dy == 0: + return (func(x + eps, y) - func(x - eps, y)) / (2*eps) + elif dx == 0 and dy == 1: + return (func(x, y + eps) - func(x, y - eps)) / (2*eps) + elif dx == 1 and dy == 1: + return (func(x + eps, y + eps) - func(x - eps, y + eps) + - func(x + eps, y - eps) + func(x - eps, y - eps)) / (2*eps)**2 + else: + raise ValueError("invalid derivative order") + + +class Test_DerivedBivariateSpline: + """Test the creation, usage, and attribute access of the (private) + _DerivedBivariateSpline class. + """ + def setup_method(self): + x = np.concatenate(list(zip(range(10), range(10)))) + y = np.concatenate(list(zip(range(10), range(1, 11)))) + z = np.concatenate((np.linspace(3, 1, 10), np.linspace(1, 3, 10))) + with suppress_warnings() as sup: + sup.record(UserWarning, "\nThe coefficients of the spline") + self.lut_lsq = LSQBivariateSpline(x, y, z, + linspace(0.5, 19.5, 4), + linspace(1.5, 20.5, 4), + eps=1e-2) + self.lut_smooth = SmoothBivariateSpline(x, y, z) + xx = linspace(0, 1, 20) + yy = xx + 1.0 + zz = array([np.roll(z, i) for i in range(z.size)]) + self.lut_rect = RectBivariateSpline(xx, yy, zz) + self.orders = list(itertools.product(range(3), range(3))) + + def test_creation_from_LSQ(self): + for nux, nuy in self.orders: + lut_der = self.lut_lsq.partial_derivative(nux, nuy) + a = lut_der(3.5, 3.5, grid=False) + b = self.lut_lsq(3.5, 3.5, dx=nux, dy=nuy, grid=False) + assert a == b + + def test_creation_from_Smooth(self): + for nux, nuy in self.orders: + lut_der = self.lut_smooth.partial_derivative(nux, nuy) + a = lut_der(5.5, 5.5, grid=False) + b = self.lut_smooth(5.5, 5.5, dx=nux, dy=nuy, grid=False) + assert a == b + + def test_creation_from_Rect(self): + for nux, nuy in self.orders: + lut_der = self.lut_rect.partial_derivative(nux, nuy) + a = lut_der(0.5, 1.5, grid=False) + b = self.lut_rect(0.5, 1.5, dx=nux, dy=nuy, grid=False) + assert a == b + + def test_invalid_attribute_fp(self): + der = self.lut_rect.partial_derivative(1, 1) + with assert_raises(AttributeError): + der.fp + + def test_invalid_attribute_get_residual(self): + der = self.lut_smooth.partial_derivative(1, 1) + with assert_raises(AttributeError): + der.get_residual() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_gil.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_gil.py new file mode 100644 index 0000000000000000000000000000000000000000..48197062e0b83a9ef54e45089d9089d49b8ad367 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_gil.py @@ -0,0 +1,64 @@ +import itertools +import threading +import time + +import numpy as np +import pytest +import scipy.interpolate + + +class TestGIL: + """Check if the GIL is properly released by scipy.interpolate functions.""" + + def setup_method(self): + self.messages = [] + + def log(self, message): + self.messages.append(message) + + def make_worker_thread(self, target, args): + log = self.log + + class WorkerThread(threading.Thread): + def run(self): + log('interpolation started') + target(*args) + log('interpolation complete') + + return WorkerThread() + + @pytest.mark.xslow + @pytest.mark.xfail(reason='race conditions, may depend on system load') + def test_rectbivariatespline(self): + def generate_params(n_points): + x = y = np.linspace(0, 1000, n_points) + x_grid, y_grid = np.meshgrid(x, y) + z = x_grid * y_grid + return x, y, z + + def calibrate_delay(requested_time): + for n_points in itertools.count(5000, 1000): + args = generate_params(n_points) + time_started = time.time() + interpolate(*args) + if time.time() - time_started > requested_time: + return args + + def interpolate(x, y, z): + scipy.interpolate.RectBivariateSpline(x, y, z) + + args = calibrate_delay(requested_time=3) + worker_thread = self.make_worker_thread(interpolate, args) + worker_thread.start() + for i in range(3): + time.sleep(0.5) + self.log('working') + worker_thread.join() + assert self.messages == [ + 'interpolation started', + 'working', + 'working', + 'working', + 'interpolation complete', + ] + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_interpnd.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_interpnd.py new file mode 100644 index 0000000000000000000000000000000000000000..981cd99d9d56e11e8d8ce0635e7b7240c19eef1f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_interpnd.py @@ -0,0 +1,440 @@ +import os +import sys + +import numpy as np +from numpy.testing import suppress_warnings +from pytest import raises as assert_raises +import pytest +from scipy._lib._array_api import xp_assert_close, assert_almost_equal + +from scipy._lib._testutils import check_free_memory +import scipy.interpolate._interpnd as interpnd +import scipy.spatial._qhull as qhull + +import pickle +import threading + +_IS_32BIT = (sys.maxsize < 2**32) + + +def data_file(basename): + return os.path.join(os.path.abspath(os.path.dirname(__file__)), + 'data', basename) + + +class TestLinearNDInterpolation: + def test_smoketest(self): + # Test at single points + x = np.array([(0,0), (-0.5,-0.5), (-0.5,0.5), (0.5, 0.5), (0.25, 0.3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + + yi = interpnd.LinearNDInterpolator(x, y)(x) + assert_almost_equal(y, yi) + + def test_smoketest_alternate(self): + # Test at single points, alternate calling convention + x = np.array([(0,0), (-0.5,-0.5), (-0.5,0.5), (0.5, 0.5), (0.25, 0.3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + + yi = interpnd.LinearNDInterpolator((x[:,0], x[:,1]), y)(x[:,0], x[:,1]) + assert_almost_equal(y, yi) + + def test_complex_smoketest(self): + # Test at single points + x = np.array([(0,0), (-0.5,-0.5), (-0.5,0.5), (0.5, 0.5), (0.25, 0.3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + y = y - 3j*y + + yi = interpnd.LinearNDInterpolator(x, y)(x) + assert_almost_equal(y, yi) + + def test_tri_input(self): + # Test at single points + x = np.array([(0,0), (-0.5,-0.5), (-0.5,0.5), (0.5, 0.5), (0.25, 0.3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + y = y - 3j*y + + tri = qhull.Delaunay(x) + interpolator = interpnd.LinearNDInterpolator(tri, y) + yi = interpolator(x) + assert_almost_equal(y, yi) + assert interpolator.tri is tri + + def test_square(self): + # Test barycentric interpolation on a square against a manual + # implementation + + points = np.array([(0,0), (0,1), (1,1), (1,0)], dtype=np.float64) + values = np.array([1., 2., -3., 5.], dtype=np.float64) + + # NB: assume triangles (0, 1, 3) and (1, 2, 3) + # + # 1----2 + # | \ | + # | \ | + # 0----3 + + def ip(x, y): + t1 = (x + y <= 1) + t2 = ~t1 + + x1 = x[t1] + y1 = y[t1] + + x2 = x[t2] + y2 = y[t2] + + z = 0*x + + z[t1] = (values[0]*(1 - x1 - y1) + + values[1]*y1 + + values[3]*x1) + + z[t2] = (values[2]*(x2 + y2 - 1) + + values[1]*(1 - x2) + + values[3]*(1 - y2)) + return z + + xx, yy = np.broadcast_arrays(np.linspace(0, 1, 14)[:,None], + np.linspace(0, 1, 14)[None,:]) + xx = xx.ravel() + yy = yy.ravel() + + xi = np.array([xx, yy]).T.copy() + zi = interpnd.LinearNDInterpolator(points, values)(xi) + + assert_almost_equal(zi, ip(xx, yy)) + + def test_smoketest_rescale(self): + # Test at single points + x = np.array([(0, 0), (-5, -5), (-5, 5), (5, 5), (2.5, 3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + + yi = interpnd.LinearNDInterpolator(x, y, rescale=True)(x) + assert_almost_equal(y, yi) + + def test_square_rescale(self): + # Test barycentric interpolation on a rectangle with rescaling + # agaings the same implementation without rescaling + + points = np.array([(0,0), (0,100), (10,100), (10,0)], dtype=np.float64) + values = np.array([1., 2., -3., 5.], dtype=np.float64) + + xx, yy = np.broadcast_arrays(np.linspace(0, 10, 14)[:,None], + np.linspace(0, 100, 14)[None,:]) + xx = xx.ravel() + yy = yy.ravel() + xi = np.array([xx, yy]).T.copy() + zi = interpnd.LinearNDInterpolator(points, values)(xi) + zi_rescaled = interpnd.LinearNDInterpolator(points, values, + rescale=True)(xi) + + assert_almost_equal(zi, zi_rescaled) + + def test_tripoints_input_rescale(self): + # Test at single points + x = np.array([(0,0), (-5,-5), (-5,5), (5, 5), (2.5, 3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + y = y - 3j*y + + tri = qhull.Delaunay(x) + yi = interpnd.LinearNDInterpolator(tri.points, y)(x) + yi_rescale = interpnd.LinearNDInterpolator(tri.points, y, + rescale=True)(x) + assert_almost_equal(yi, yi_rescale) + + def test_tri_input_rescale(self): + # Test at single points + x = np.array([(0,0), (-5,-5), (-5,5), (5, 5), (2.5, 3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + y = y - 3j*y + + tri = qhull.Delaunay(x) + match = ("Rescaling is not supported when passing a " + "Delaunay triangulation as ``points``.") + with pytest.raises(ValueError, match=match): + interpnd.LinearNDInterpolator(tri, y, rescale=True)(x) + + def test_pickle(self): + # Test at single points + np.random.seed(1234) + x = np.random.rand(30, 2) + y = np.random.rand(30) + 1j*np.random.rand(30) + + ip = interpnd.LinearNDInterpolator(x, y) + ip2 = pickle.loads(pickle.dumps(ip)) + + assert_almost_equal(ip(0.5, 0.5), ip2(0.5, 0.5)) + + @pytest.mark.slow + @pytest.mark.thread_unsafe + @pytest.mark.skipif(_IS_32BIT, reason='it fails on 32-bit') + def test_threading(self): + # This test was taken from issue 8856 + # https://github.com/scipy/scipy/issues/8856 + check_free_memory(10000) + + r_ticks = np.arange(0, 4200, 10) + phi_ticks = np.arange(0, 4200, 10) + r_grid, phi_grid = np.meshgrid(r_ticks, phi_ticks) + + def do_interp(interpolator, slice_rows, slice_cols): + grid_x, grid_y = np.mgrid[slice_rows, slice_cols] + res = interpolator((grid_x, grid_y)) + return res + + points = np.vstack((r_grid.ravel(), phi_grid.ravel())).T + values = (r_grid * phi_grid).ravel() + interpolator = interpnd.LinearNDInterpolator(points, values) + + worker_thread_1 = threading.Thread( + target=do_interp, + args=(interpolator, slice(0, 2100), slice(0, 2100))) + worker_thread_2 = threading.Thread( + target=do_interp, + args=(interpolator, slice(2100, 4200), slice(0, 2100))) + worker_thread_3 = threading.Thread( + target=do_interp, + args=(interpolator, slice(0, 2100), slice(2100, 4200))) + worker_thread_4 = threading.Thread( + target=do_interp, + args=(interpolator, slice(2100, 4200), slice(2100, 4200))) + + worker_thread_1.start() + worker_thread_2.start() + worker_thread_3.start() + worker_thread_4.start() + + worker_thread_1.join() + worker_thread_2.join() + worker_thread_3.join() + worker_thread_4.join() + + +class TestEstimateGradients2DGlobal: + def test_smoketest(self): + x = np.array([(0, 0), (0, 2), + (1, 0), (1, 2), (0.25, 0.75), (0.6, 0.8)], dtype=float) + tri = qhull.Delaunay(x) + + # Should be exact for linear functions, independent of triangulation + + funcs = [ + (lambda x, y: 0*x + 1, (0, 0)), + (lambda x, y: 0 + x, (1, 0)), + (lambda x, y: -2 + y, (0, 1)), + (lambda x, y: 3 + 3*x + 14.15*y, (3, 14.15)) + ] + + for j, (func, grad) in enumerate(funcs): + z = func(x[:,0], x[:,1]) + dz = interpnd.estimate_gradients_2d_global(tri, z, tol=1e-6) + + assert dz.shape == (6, 2) + xp_assert_close(dz, np.array(grad)[None,:] + 0*dz, + rtol=1e-5, atol=1e-5, err_msg="item %d" % j) + + def test_regression_2359(self): + # Check regression --- for certain point sets, gradient + # estimation could end up in an infinite loop + points = np.load(data_file('estimate_gradients_hang.npy')) + values = np.random.rand(points.shape[0]) + tri = qhull.Delaunay(points) + + # This should not hang + with suppress_warnings() as sup: + sup.filter(interpnd.GradientEstimationWarning, + "Gradient estimation did not converge") + interpnd.estimate_gradients_2d_global(tri, values, maxiter=1) + + +class TestCloughTocher2DInterpolator: + + def _check_accuracy(self, func, x=None, tol=1e-6, alternate=False, + rescale=False, **kw): + rng = np.random.RandomState(1234) + # np.random.seed(1234) + if x is None: + x = np.array([(0, 0), (0, 1), + (1, 0), (1, 1), (0.25, 0.75), (0.6, 0.8), + (0.5, 0.2)], + dtype=float) + + if not alternate: + ip = interpnd.CloughTocher2DInterpolator(x, func(x[:,0], x[:,1]), + tol=1e-6, rescale=rescale) + else: + ip = interpnd.CloughTocher2DInterpolator((x[:,0], x[:,1]), + func(x[:,0], x[:,1]), + tol=1e-6, rescale=rescale) + + p = rng.rand(50, 2) + + if not alternate: + a = ip(p) + else: + a = ip(p[:,0], p[:,1]) + b = func(p[:,0], p[:,1]) + + try: + xp_assert_close(a, b, **kw) + except AssertionError: + print("_check_accuracy: abs(a-b):", abs(a - b)) + print("ip.grad:", ip.grad) + raise + + def test_linear_smoketest(self): + # Should be exact for linear functions, independent of triangulation + funcs = [ + lambda x, y: 0*x + 1, + lambda x, y: 0 + x, + lambda x, y: -2 + y, + lambda x, y: 3 + 3*x + 14.15*y, + ] + + for j, func in enumerate(funcs): + self._check_accuracy(func, tol=1e-13, atol=1e-7, rtol=1e-7, + err_msg="Function %d" % j) + self._check_accuracy(func, tol=1e-13, atol=1e-7, rtol=1e-7, + alternate=True, + err_msg="Function (alternate) %d" % j) + # check rescaling + self._check_accuracy(func, tol=1e-13, atol=1e-7, rtol=1e-7, + err_msg="Function (rescaled) %d" % j, rescale=True) + self._check_accuracy(func, tol=1e-13, atol=1e-7, rtol=1e-7, + alternate=True, rescale=True, + err_msg="Function (alternate, rescaled) %d" % j) + + def test_quadratic_smoketest(self): + # Should be reasonably accurate for quadratic functions + funcs = [ + lambda x, y: x**2, + lambda x, y: y**2, + lambda x, y: x**2 - y**2, + lambda x, y: x*y, + ] + + for j, func in enumerate(funcs): + self._check_accuracy(func, tol=1e-9, atol=0.22, rtol=0, + err_msg="Function %d" % j) + self._check_accuracy(func, tol=1e-9, atol=0.22, rtol=0, + err_msg="Function %d" % j, rescale=True) + + def test_tri_input(self): + # Test at single points + x = np.array([(0,0), (-0.5,-0.5), (-0.5,0.5), (0.5, 0.5), (0.25, 0.3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + y = y - 3j*y + + tri = qhull.Delaunay(x) + yi = interpnd.CloughTocher2DInterpolator(tri, y)(x) + assert_almost_equal(y, yi) + + def test_tri_input_rescale(self): + # Test at single points + x = np.array([(0,0), (-5,-5), (-5,5), (5, 5), (2.5, 3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + y = y - 3j*y + + tri = qhull.Delaunay(x) + match = ("Rescaling is not supported when passing a " + "Delaunay triangulation as ``points``.") + with pytest.raises(ValueError, match=match): + interpnd.CloughTocher2DInterpolator(tri, y, rescale=True)(x) + + def test_tripoints_input_rescale(self): + # Test at single points + x = np.array([(0,0), (-5,-5), (-5,5), (5, 5), (2.5, 3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + y = y - 3j*y + + tri = qhull.Delaunay(x) + yi = interpnd.CloughTocher2DInterpolator(tri.points, y)(x) + yi_rescale = interpnd.CloughTocher2DInterpolator(tri.points, y, rescale=True)(x) + assert_almost_equal(yi, yi_rescale) + + @pytest.mark.fail_slow(5) + def test_dense(self): + # Should be more accurate for dense meshes + funcs = [ + lambda x, y: x**2, + lambda x, y: y**2, + lambda x, y: x**2 - y**2, + lambda x, y: x*y, + lambda x, y: np.cos(2*np.pi*x)*np.sin(2*np.pi*y) + ] + + rng = np.random.RandomState(4321) # use a different seed than the check! + grid = np.r_[np.array([(0,0), (0,1), (1,0), (1,1)], dtype=float), + rng.rand(30*30, 2)] + + for j, func in enumerate(funcs): + self._check_accuracy(func, x=grid, tol=1e-9, atol=5e-3, rtol=1e-2, + err_msg="Function %d" % j) + self._check_accuracy(func, x=grid, tol=1e-9, atol=5e-3, rtol=1e-2, + err_msg="Function %d" % j, rescale=True) + + def test_wrong_ndim(self): + x = np.random.randn(30, 3) + y = np.random.randn(30) + assert_raises(ValueError, interpnd.CloughTocher2DInterpolator, x, y) + + def test_pickle(self): + # Test at single points + rng = np.random.RandomState(1234) + x = rng.rand(30, 2) + y = rng.rand(30) + 1j*rng.rand(30) + + ip = interpnd.CloughTocher2DInterpolator(x, y) + ip2 = pickle.loads(pickle.dumps(ip)) + + assert_almost_equal(ip(0.5, 0.5), ip2(0.5, 0.5)) + + def test_boundary_tri_symmetry(self): + # Interpolation at neighbourless triangles should retain + # symmetry with mirroring the triangle. + + # Equilateral triangle + points = np.array([(0, 0), (1, 0), (0.5, np.sqrt(3)/2)]) + values = np.array([1, 0, 0]) + + ip = interpnd.CloughTocher2DInterpolator(points, values) + + # Set gradient to zero at vertices + ip.grad[...] = 0 + + # Interpolation should be symmetric vs. bisector + alpha = 0.3 + p1 = np.array([0.5 * np.cos(alpha), 0.5 * np.sin(alpha)]) + p2 = np.array([0.5 * np.cos(np.pi/3 - alpha), 0.5 * np.sin(np.pi/3 - alpha)]) + + v1 = ip(p1) + v2 = ip(p2) + xp_assert_close(v1, v2) + + # ... and affine invariant + rng = np.random.RandomState(1) + A = rng.randn(2, 2) + b = rng.randn(2) + + points = A.dot(points.T).T + b[None,:] + p1 = A.dot(p1) + b + p2 = A.dot(p2) + b + + ip = interpnd.CloughTocher2DInterpolator(points, values) + ip.grad[...] = 0 + + w1 = ip(p1) + w2 = ip(p2) + xp_assert_close(w1, v1) + xp_assert_close(w2, v2) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_interpolate.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_interpolate.py new file mode 100644 index 0000000000000000000000000000000000000000..24a6907b7b050ddb0686cf8b1761bfd50eaf692a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_interpolate.py @@ -0,0 +1,2586 @@ +from scipy._lib._array_api import ( + xp_assert_equal, xp_assert_close, assert_almost_equal, assert_array_almost_equal +) +from pytest import raises as assert_raises +import pytest + +from numpy import mgrid, pi, sin, poly1d +import numpy as np + +from scipy.interpolate import (interp1d, interp2d, lagrange, PPoly, BPoly, + splrep, splev, splantider, splint, sproot, Akima1DInterpolator, + NdPPoly, BSpline, PchipInterpolator) + +from scipy.special import poch, gamma + +from scipy.interpolate import _ppoly + +from scipy._lib._gcutils import assert_deallocated, IS_PYPY +from scipy._lib._testutils import _run_concurrent_barrier + +from scipy.integrate import nquad + +from scipy.special import binom + + +class TestInterp2D: + def test_interp2d(self): + y, x = mgrid[0:2:20j, 0:pi:21j] + z = sin(x+0.5*y) + with assert_raises(NotImplementedError): + interp2d(x, y, z) + + +class TestInterp1D: + + def setup_method(self): + self.x5 = np.arange(5.) + self.x10 = np.arange(10.) + self.y10 = np.arange(10.) + self.x25 = self.x10.reshape((2,5)) + self.x2 = np.arange(2.) + self.y2 = np.arange(2.) + self.x1 = np.array([0.]) + self.y1 = np.array([0.]) + + self.y210 = np.arange(20.).reshape((2, 10)) + self.y102 = np.arange(20.).reshape((10, 2)) + self.y225 = np.arange(20.).reshape((2, 2, 5)) + self.y25 = np.arange(10.).reshape((2, 5)) + self.y235 = np.arange(30.).reshape((2, 3, 5)) + self.y325 = np.arange(30.).reshape((3, 2, 5)) + + # Edge updated test matrix 1 + # array([[ 30, 1, 2, 3, 4, 5, 6, 7, 8, -30], + # [ 30, 11, 12, 13, 14, 15, 16, 17, 18, -30]]) + self.y210_edge_updated = np.arange(20.).reshape((2, 10)) + self.y210_edge_updated[:, 0] = 30 + self.y210_edge_updated[:, -1] = -30 + + # Edge updated test matrix 2 + # array([[ 30, 30], + # [ 2, 3], + # [ 4, 5], + # [ 6, 7], + # [ 8, 9], + # [ 10, 11], + # [ 12, 13], + # [ 14, 15], + # [ 16, 17], + # [-30, -30]]) + self.y102_edge_updated = np.arange(20.).reshape((10, 2)) + self.y102_edge_updated[0, :] = 30 + self.y102_edge_updated[-1, :] = -30 + + self.fill_value = -100.0 + + def test_validation(self): + # Make sure that appropriate exceptions are raised when invalid values + # are given to the constructor. + + # These should all work. + for kind in ('nearest', 'nearest-up', 'zero', 'linear', 'slinear', + 'quadratic', 'cubic', 'previous', 'next'): + interp1d(self.x10, self.y10, kind=kind) + interp1d(self.x10, self.y10, kind=kind, fill_value="extrapolate") + interp1d(self.x10, self.y10, kind='linear', fill_value=(-1, 1)) + interp1d(self.x10, self.y10, kind='linear', + fill_value=np.array([-1])) + interp1d(self.x10, self.y10, kind='linear', + fill_value=(-1,)) + interp1d(self.x10, self.y10, kind='linear', + fill_value=-1) + interp1d(self.x10, self.y10, kind='linear', + fill_value=(-1, -1)) + interp1d(self.x10, self.y10, kind=0) + interp1d(self.x10, self.y10, kind=1) + interp1d(self.x10, self.y10, kind=2) + interp1d(self.x10, self.y10, kind=3) + interp1d(self.x10, self.y210, kind='linear', axis=-1, + fill_value=(-1, -1)) + interp1d(self.x2, self.y210, kind='linear', axis=0, + fill_value=np.ones(10)) + interp1d(self.x2, self.y210, kind='linear', axis=0, + fill_value=(np.ones(10), np.ones(10))) + interp1d(self.x2, self.y210, kind='linear', axis=0, + fill_value=(np.ones(10), -1)) + + # x array must be 1D. + assert_raises(ValueError, interp1d, self.x25, self.y10) + + # y array cannot be a scalar. + assert_raises(ValueError, interp1d, self.x10, np.array(0)) + + # Check for x and y arrays having the same length. + assert_raises(ValueError, interp1d, self.x10, self.y2) + assert_raises(ValueError, interp1d, self.x2, self.y10) + assert_raises(ValueError, interp1d, self.x10, self.y102) + interp1d(self.x10, self.y210) + interp1d(self.x10, self.y102, axis=0) + + # Check for x and y having at least 1 element. + assert_raises(ValueError, interp1d, self.x1, self.y10) + assert_raises(ValueError, interp1d, self.x10, self.y1) + + # Bad fill values + assert_raises(ValueError, interp1d, self.x10, self.y10, kind='linear', + fill_value=(-1, -1, -1)) # doesn't broadcast + assert_raises(ValueError, interp1d, self.x10, self.y10, kind='linear', + fill_value=[-1, -1, -1]) # doesn't broadcast + assert_raises(ValueError, interp1d, self.x10, self.y10, kind='linear', + fill_value=np.array((-1, -1, -1))) # doesn't broadcast + assert_raises(ValueError, interp1d, self.x10, self.y10, kind='linear', + fill_value=[[-1]]) # doesn't broadcast + assert_raises(ValueError, interp1d, self.x10, self.y10, kind='linear', + fill_value=[-1, -1]) # doesn't broadcast + assert_raises(ValueError, interp1d, self.x10, self.y10, kind='linear', + fill_value=np.array([])) # doesn't broadcast + assert_raises(ValueError, interp1d, self.x10, self.y10, kind='linear', + fill_value=()) # doesn't broadcast + assert_raises(ValueError, interp1d, self.x2, self.y210, kind='linear', + axis=0, fill_value=[-1, -1]) # doesn't broadcast + assert_raises(ValueError, interp1d, self.x2, self.y210, kind='linear', + axis=0, fill_value=(0., [-1, -1])) # above doesn't bc + + def test_init(self): + # Check that the attributes are initialized appropriately by the + # constructor. + assert interp1d(self.x10, self.y10).copy + assert not interp1d(self.x10, self.y10, copy=False).copy + assert interp1d(self.x10, self.y10).bounds_error + assert not interp1d(self.x10, self.y10, bounds_error=False).bounds_error + assert np.isnan(interp1d(self.x10, self.y10).fill_value) + assert interp1d(self.x10, self.y10, fill_value=3.0).fill_value == 3.0 + assert (interp1d(self.x10, self.y10, fill_value=(1.0, 2.0)).fill_value == + (1.0, 2.0) + ) + assert interp1d(self.x10, self.y10).axis == 0 + assert interp1d(self.x10, self.y210).axis == 1 + assert interp1d(self.x10, self.y102, axis=0).axis == 0 + xp_assert_equal(interp1d(self.x10, self.y10).x, self.x10) + xp_assert_equal(interp1d(self.x10, self.y10).y, self.y10) + xp_assert_equal(interp1d(self.x10, self.y210).y, self.y210) + + def test_assume_sorted(self): + # Check for unsorted arrays + interp10 = interp1d(self.x10, self.y10) + interp10_unsorted = interp1d(self.x10[::-1], self.y10[::-1]) + + assert_array_almost_equal(interp10_unsorted(self.x10), self.y10) + assert_array_almost_equal(interp10_unsorted(1.2), np.array(1.2)) + assert_array_almost_equal(interp10_unsorted([2.4, 5.6, 6.0]), + interp10([2.4, 5.6, 6.0])) + + # Check assume_sorted keyword (defaults to False) + interp10_assume_kw = interp1d(self.x10[::-1], self.y10[::-1], + assume_sorted=False) + assert_array_almost_equal(interp10_assume_kw(self.x10), self.y10) + + interp10_assume_kw2 = interp1d(self.x10[::-1], self.y10[::-1], + assume_sorted=True) + # Should raise an error for unsorted input if assume_sorted=True + assert_raises(ValueError, interp10_assume_kw2, self.x10) + + # Check that if y is a 2-D array, things are still consistent + interp10_y_2d = interp1d(self.x10, self.y210) + interp10_y_2d_unsorted = interp1d(self.x10[::-1], self.y210[:, ::-1]) + assert_array_almost_equal(interp10_y_2d(self.x10), + interp10_y_2d_unsorted(self.x10)) + + def test_linear(self): + for kind in ['linear', 'slinear']: + self._check_linear(kind) + + def _check_linear(self, kind): + # Check the actual implementation of linear interpolation. + interp10 = interp1d(self.x10, self.y10, kind=kind) + assert_array_almost_equal(interp10(self.x10), self.y10) + assert_array_almost_equal(interp10(1.2), np.array(1.2)) + assert_array_almost_equal(interp10([2.4, 5.6, 6.0]), + np.array([2.4, 5.6, 6.0])) + + # test fill_value="extrapolate" + extrapolator = interp1d(self.x10, self.y10, kind=kind, + fill_value='extrapolate') + xp_assert_close(extrapolator([-1., 0, 9, 11]), + np.asarray([-1.0, 0, 9, 11]), rtol=1e-14) + + opts = dict(kind=kind, + fill_value='extrapolate', + bounds_error=True) + assert_raises(ValueError, interp1d, self.x10, self.y10, **opts) + + def test_linear_dtypes(self): + # regression test for gh-5898, where 1D linear interpolation has been + # delegated to numpy.interp for all float dtypes, and the latter was + # not handling e.g. np.float128. + for dtyp in [np.float16, + np.float32, + np.float64, + np.longdouble]: + x = np.arange(8, dtype=dtyp) + y = x + yp = interp1d(x, y, kind='linear')(x) + assert yp.dtype == dtyp + xp_assert_close(yp, y, atol=1e-15) + + # regression test for gh-14531, where 1D linear interpolation has been + # has been extended to delegate to numpy.interp for integer dtypes + x = [0, 1, 2] + y = [np.nan, 0, 1] + yp = interp1d(x, y)(x) + xp_assert_close(yp, y, atol=1e-15) + + def test_slinear_dtypes(self): + # regression test for gh-7273: 1D slinear interpolation fails with + # float32 inputs + dt_r = [np.float16, np.float32, np.float64] + dt_rc = dt_r + [np.complex64, np.complex128] + spline_kinds = ['slinear', 'zero', 'quadratic', 'cubic'] + for dtx in dt_r: + x = np.arange(0, 10, dtype=dtx) + for dty in dt_rc: + y = np.exp(-x/3.0).astype(dty) + for dtn in dt_r: + xnew = x.astype(dtn) + for kind in spline_kinds: + f = interp1d(x, y, kind=kind, bounds_error=False) + xp_assert_close(f(xnew), y, atol=1e-7, + check_dtype=False, + err_msg=f"{dtx}, {dty} {dtn}") + + def test_cubic(self): + # Check the actual implementation of spline interpolation. + interp10 = interp1d(self.x10, self.y10, kind='cubic') + assert_array_almost_equal(interp10(self.x10), self.y10) + assert_array_almost_equal(interp10(1.2), np.array(1.2)) + assert_array_almost_equal(interp10(1.5), np.array(1.5)) + assert_array_almost_equal(interp10([2.4, 5.6, 6.0]), + np.array([2.4, 5.6, 6.0]),) + + def test_nearest(self): + # Check the actual implementation of nearest-neighbour interpolation. + # Nearest asserts that half-integer case (1.5) rounds down to 1 + interp10 = interp1d(self.x10, self.y10, kind='nearest') + assert_array_almost_equal(interp10(self.x10), self.y10) + assert_array_almost_equal(interp10(1.2), np.array(1.)) + assert_array_almost_equal(interp10(1.5), np.array(1.)) + assert_array_almost_equal(interp10([2.4, 5.6, 6.0]), + np.array([2., 6., 6.]),) + + # test fill_value="extrapolate" + extrapolator = interp1d(self.x10, self.y10, kind='nearest', + fill_value='extrapolate') + xp_assert_close(extrapolator([-1., 0, 9, 11]), + [0.0, 0, 9, 9], rtol=1e-14) + + opts = dict(kind='nearest', + fill_value='extrapolate', + bounds_error=True) + assert_raises(ValueError, interp1d, self.x10, self.y10, **opts) + + def test_nearest_up(self): + # Check the actual implementation of nearest-neighbour interpolation. + # Nearest-up asserts that half-integer case (1.5) rounds up to 2 + interp10 = interp1d(self.x10, self.y10, kind='nearest-up') + assert_array_almost_equal(interp10(self.x10), self.y10) + assert_array_almost_equal(interp10(1.2), np.array(1.)) + assert_array_almost_equal(interp10(1.5), np.array(2.)) + assert_array_almost_equal(interp10([2.4, 5.6, 6.0]), + np.array([2., 6., 6.]),) + + # test fill_value="extrapolate" + extrapolator = interp1d(self.x10, self.y10, kind='nearest-up', + fill_value='extrapolate') + xp_assert_close(extrapolator([-1., 0, 9, 11]), + [0.0, 0, 9, 9], rtol=1e-14) + + opts = dict(kind='nearest-up', + fill_value='extrapolate', + bounds_error=True) + assert_raises(ValueError, interp1d, self.x10, self.y10, **opts) + + def test_previous(self): + # Check the actual implementation of previous interpolation. + interp10 = interp1d(self.x10, self.y10, kind='previous') + assert_array_almost_equal(interp10(self.x10), self.y10) + assert_array_almost_equal(interp10(1.2), np.array(1.)) + assert_array_almost_equal(interp10(1.5), np.array(1.)) + assert_array_almost_equal(interp10([2.4, 5.6, 6.0]), + np.array([2., 5., 6.]),) + + # test fill_value="extrapolate" + extrapolator = interp1d(self.x10, self.y10, kind='previous', + fill_value='extrapolate') + xp_assert_close(extrapolator([-1., 0, 9, 11]), + [np.nan, 0, 9, 9], rtol=1e-14) + + # Tests for gh-9591 + interpolator1D = interp1d(self.x10, self.y10, kind="previous", + fill_value='extrapolate') + xp_assert_close(interpolator1D([-1, -2, 5, 8, 12, 25]), + [np.nan, np.nan, 5, 8, 9, 9]) + + interpolator2D = interp1d(self.x10, self.y210, kind="previous", + fill_value='extrapolate') + xp_assert_close(interpolator2D([-1, -2, 5, 8, 12, 25]), + [[np.nan, np.nan, 5, 8, 9, 9], + [np.nan, np.nan, 15, 18, 19, 19]]) + + interpolator2DAxis0 = interp1d(self.x10, self.y102, kind="previous", + axis=0, fill_value='extrapolate') + xp_assert_close(interpolator2DAxis0([-2, 5, 12]), + [[np.nan, np.nan], + [10, 11], + [18, 19]]) + + opts = dict(kind='previous', + fill_value='extrapolate', + bounds_error=True) + assert_raises(ValueError, interp1d, self.x10, self.y10, **opts) + + # Tests for gh-16813 + interpolator1D = interp1d([0, 1, 2], + [0, 1, -1], kind="previous", + fill_value='extrapolate', + assume_sorted=True) + xp_assert_close(interpolator1D([-2, -1, 0, 1, 2, 3, 5]), + [np.nan, np.nan, 0, 1, -1, -1, -1]) + + interpolator1D = interp1d([2, 0, 1], # x is not ascending + [-1, 0, 1], kind="previous", + fill_value='extrapolate', + assume_sorted=False) + xp_assert_close(interpolator1D([-2, -1, 0, 1, 2, 3, 5]), + [np.nan, np.nan, 0, 1, -1, -1, -1]) + + interpolator2D = interp1d(self.x10, self.y210_edge_updated, + kind="previous", + fill_value='extrapolate') + xp_assert_close(interpolator2D([-1, -2, 5, 8, 12, 25]), + [[np.nan, np.nan, 5, 8, -30, -30], + [np.nan, np.nan, 15, 18, -30, -30]]) + + interpolator2DAxis0 = interp1d(self.x10, self.y102_edge_updated, + kind="previous", + axis=0, fill_value='extrapolate') + xp_assert_close(interpolator2DAxis0([-2, 5, 12]), + [[np.nan, np.nan], + [10, 11], + [-30, -30]]) + + def test_next(self): + # Check the actual implementation of next interpolation. + interp10 = interp1d(self.x10, self.y10, kind='next') + assert_array_almost_equal(interp10(self.x10), self.y10) + assert_array_almost_equal(interp10(1.2), np.array(2.)) + assert_array_almost_equal(interp10(1.5), np.array(2.)) + assert_array_almost_equal(interp10([2.4, 5.6, 6.0]), + np.array([3., 6., 6.]),) + + # test fill_value="extrapolate" + extrapolator = interp1d(self.x10, self.y10, kind='next', + fill_value='extrapolate') + xp_assert_close(extrapolator([-1., 0, 9, 11]), + [0, 0, 9, np.nan], rtol=1e-14) + + # Tests for gh-9591 + interpolator1D = interp1d(self.x10, self.y10, kind="next", + fill_value='extrapolate') + xp_assert_close(interpolator1D([-1, -2, 5, 8, 12, 25]), + [0, 0, 5, 8, np.nan, np.nan]) + + interpolator2D = interp1d(self.x10, self.y210, kind="next", + fill_value='extrapolate') + xp_assert_close(interpolator2D([-1, -2, 5, 8, 12, 25]), + [[0, 0, 5, 8, np.nan, np.nan], + [10, 10, 15, 18, np.nan, np.nan]]) + + interpolator2DAxis0 = interp1d(self.x10, self.y102, kind="next", + axis=0, fill_value='extrapolate') + xp_assert_close(interpolator2DAxis0([-2, 5, 12]), + [[0, 1], + [10, 11], + [np.nan, np.nan]]) + + opts = dict(kind='next', + fill_value='extrapolate', + bounds_error=True) + assert_raises(ValueError, interp1d, self.x10, self.y10, **opts) + + # Tests for gh-16813 + interpolator1D = interp1d([0, 1, 2], + [0, 1, -1], kind="next", + fill_value='extrapolate', + assume_sorted=True) + xp_assert_close(interpolator1D([-2, -1, 0, 1, 2, 3, 5]), + [0, 0, 0, 1, -1, np.nan, np.nan]) + + interpolator1D = interp1d([2, 0, 1], # x is not ascending + [-1, 0, 1], kind="next", + fill_value='extrapolate', + assume_sorted=False) + xp_assert_close(interpolator1D([-2, -1, 0, 1, 2, 3, 5]), + [0, 0, 0, 1, -1, np.nan, np.nan]) + + interpolator2D = interp1d(self.x10, self.y210_edge_updated, + kind="next", + fill_value='extrapolate') + xp_assert_close(interpolator2D([-1, -2, 5, 8, 12, 25]), + [[30, 30, 5, 8, np.nan, np.nan], + [30, 30, 15, 18, np.nan, np.nan]]) + + interpolator2DAxis0 = interp1d(self.x10, self.y102_edge_updated, + kind="next", + axis=0, fill_value='extrapolate') + xp_assert_close(interpolator2DAxis0([-2, 5, 12]), + [[30, 30], + [10, 11], + [np.nan, np.nan]]) + + def test_zero(self): + # Check the actual implementation of zero-order spline interpolation. + interp10 = interp1d(self.x10, self.y10, kind='zero') + assert_array_almost_equal(interp10(self.x10), self.y10) + assert_array_almost_equal(interp10(1.2), np.array(1.)) + assert_array_almost_equal(interp10(1.5), np.array(1.)) + assert_array_almost_equal(interp10([2.4, 5.6, 6.0]), + np.array([2., 5., 6.])) + + def bounds_check_helper(self, interpolant, test_array, fail_value): + # Asserts that a ValueError is raised and that the error message + # contains the value causing this exception. + assert_raises(ValueError, interpolant, test_array) + try: + interpolant(test_array) + except ValueError as err: + assert (f"{fail_value}" in str(err)) + + def _bounds_check(self, kind='linear'): + # Test that our handling of out-of-bounds input is correct. + extrap10 = interp1d(self.x10, self.y10, fill_value=self.fill_value, + bounds_error=False, kind=kind) + + xp_assert_equal(extrap10(11.2), np.array(self.fill_value)) + xp_assert_equal(extrap10(-3.4), np.array(self.fill_value)) + xp_assert_equal(extrap10([[[11.2], [-3.4], [12.6], [19.3]]]), + np.array(self.fill_value), check_shape=False) + xp_assert_equal(extrap10._check_bounds( + np.array([-1.0, 0.0, 5.0, 9.0, 11.0])), + np.array([[True, False, False, False, False], + [False, False, False, False, True]])) + + raises_bounds_error = interp1d(self.x10, self.y10, bounds_error=True, + kind=kind) + + self.bounds_check_helper(raises_bounds_error, -1.0, -1.0) + self.bounds_check_helper(raises_bounds_error, 11.0, 11.0) + self.bounds_check_helper(raises_bounds_error, [0.0, -1.0, 0.0], -1.0) + self.bounds_check_helper(raises_bounds_error, [0.0, 1.0, 21.0], 21.0) + + raises_bounds_error([0.0, 5.0, 9.0]) + + def _bounds_check_int_nan_fill(self, kind='linear'): + x = np.arange(10).astype(int) + y = np.arange(10).astype(int) + c = interp1d(x, y, kind=kind, fill_value=np.nan, bounds_error=False) + yi = c(x - 1) + assert np.isnan(yi[0]) + assert_array_almost_equal(yi, np.r_[np.nan, y[:-1]]) + + def test_bounds(self): + for kind in ('linear', 'cubic', 'nearest', 'previous', 'next', + 'slinear', 'zero', 'quadratic'): + self._bounds_check(kind) + self._bounds_check_int_nan_fill(kind) + + def _check_fill_value(self, kind): + interp = interp1d(self.x10, self.y10, kind=kind, + fill_value=(-100, 100), bounds_error=False) + assert_array_almost_equal(interp(10), np.asarray(100.)) + assert_array_almost_equal(interp(-10), np.asarray(-100.)) + assert_array_almost_equal(interp([-10, 10]), [-100, 100]) + + # Proper broadcasting: + # interp along axis of length 5 + # other dim=(2, 3), (3, 2), (2, 2), or (2,) + + # one singleton fill_value (works for all) + for y in (self.y235, self.y325, self.y225, self.y25): + interp = interp1d(self.x5, y, kind=kind, axis=-1, + fill_value=100, bounds_error=False) + assert_array_almost_equal(interp(10), np.asarray(100.)) + assert_array_almost_equal(interp(-10), np.asarray(100.)) + assert_array_almost_equal(interp([-10, 10]), np.asarray(100.)) + + # singleton lower, singleton upper + interp = interp1d(self.x5, y, kind=kind, axis=-1, + fill_value=(-100, 100), bounds_error=False) + assert_array_almost_equal(interp(10), np.asarray(100.)) + assert_array_almost_equal(interp(-10), np.asarray(-100.)) + if y.ndim == 3: + result = [[[-100, 100]] * y.shape[1]] * y.shape[0] + else: + result = [[-100, 100]] * y.shape[0] + assert_array_almost_equal(interp([-10, 10]), result) + + # one broadcastable (3,) fill_value + fill_value = [100, 200, 300] + for y in (self.y325, self.y225): + assert_raises(ValueError, interp1d, self.x5, y, kind=kind, + axis=-1, fill_value=fill_value, bounds_error=False) + interp = interp1d(self.x5, self.y235, kind=kind, axis=-1, + fill_value=fill_value, bounds_error=False) + assert_array_almost_equal(interp(10), [[100, 200, 300]] * 2) + assert_array_almost_equal(interp(-10), [[100, 200, 300]] * 2) + assert_array_almost_equal(interp([-10, 10]), [[[100, 100], + [200, 200], + [300, 300]]] * 2) + + # one broadcastable (2,) fill_value + fill_value = [100, 200] + assert_raises(ValueError, interp1d, self.x5, self.y235, kind=kind, + axis=-1, fill_value=fill_value, bounds_error=False) + for y in (self.y225, self.y325, self.y25): + interp = interp1d(self.x5, y, kind=kind, axis=-1, + fill_value=fill_value, bounds_error=False) + result = [100, 200] + if y.ndim == 3: + result = [result] * y.shape[0] + assert_array_almost_equal(interp(10), result) + assert_array_almost_equal(interp(-10), result) + result = [[100, 100], [200, 200]] + if y.ndim == 3: + result = [result] * y.shape[0] + assert_array_almost_equal(interp([-10, 10]), result) + + # broadcastable (3,) lower, singleton upper + fill_value = (np.array([-100, -200, -300]), 100) + for y in (self.y325, self.y225): + assert_raises(ValueError, interp1d, self.x5, y, kind=kind, + axis=-1, fill_value=fill_value, bounds_error=False) + interp = interp1d(self.x5, self.y235, kind=kind, axis=-1, + fill_value=fill_value, bounds_error=False) + assert_array_almost_equal(interp(10), np.asarray(100.)) + assert_array_almost_equal(interp(-10), [[-100, -200, -300]] * 2) + assert_array_almost_equal(interp([-10, 10]), [[[-100, 100], + [-200, 100], + [-300, 100]]] * 2) + + # broadcastable (2,) lower, singleton upper + fill_value = (np.array([-100, -200]), 100) + assert_raises(ValueError, interp1d, self.x5, self.y235, kind=kind, + axis=-1, fill_value=fill_value, bounds_error=False) + for y in (self.y225, self.y325, self.y25): + interp = interp1d(self.x5, y, kind=kind, axis=-1, + fill_value=fill_value, bounds_error=False) + assert_array_almost_equal(interp(10), np.asarray(100)) + result = [-100, -200] + if y.ndim == 3: + result = [result] * y.shape[0] + assert_array_almost_equal(interp(-10), result) + result = [[-100, 100], [-200, 100]] + if y.ndim == 3: + result = [result] * y.shape[0] + assert_array_almost_equal(interp([-10, 10]), result) + + # broadcastable (3,) lower, broadcastable (3,) upper + fill_value = ([-100, -200, -300], [100, 200, 300]) + for y in (self.y325, self.y225): + assert_raises(ValueError, interp1d, self.x5, y, kind=kind, + axis=-1, fill_value=fill_value, bounds_error=False) + for ii in range(2): # check ndarray as well as list here + if ii == 1: + fill_value = tuple(np.array(f) for f in fill_value) + interp = interp1d(self.x5, self.y235, kind=kind, axis=-1, + fill_value=fill_value, bounds_error=False) + assert_array_almost_equal(interp(10), [[100, 200, 300]] * 2) + assert_array_almost_equal(interp(-10), [[-100, -200, -300]] * 2) + assert_array_almost_equal(interp([-10, 10]), [[[-100, 100], + [-200, 200], + [-300, 300]]] * 2) + # broadcastable (2,) lower, broadcastable (2,) upper + fill_value = ([-100, -200], [100, 200]) + assert_raises(ValueError, interp1d, self.x5, self.y235, kind=kind, + axis=-1, fill_value=fill_value, bounds_error=False) + for y in (self.y325, self.y225, self.y25): + interp = interp1d(self.x5, y, kind=kind, axis=-1, + fill_value=fill_value, bounds_error=False) + result = [100, 200] + if y.ndim == 3: + result = [result] * y.shape[0] + assert_array_almost_equal(interp(10), result) + result = [-100, -200] + if y.ndim == 3: + result = [result] * y.shape[0] + assert_array_almost_equal(interp(-10), result) + result = [[-100, 100], [-200, 200]] + if y.ndim == 3: + result = [result] * y.shape[0] + assert_array_almost_equal(interp([-10, 10]), result) + + # one broadcastable (2, 2) array-like + fill_value = [[100, 200], [1000, 2000]] + for y in (self.y235, self.y325, self.y25): + assert_raises(ValueError, interp1d, self.x5, y, kind=kind, + axis=-1, fill_value=fill_value, bounds_error=False) + for ii in range(2): + if ii == 1: + fill_value = np.array(fill_value) + interp = interp1d(self.x5, self.y225, kind=kind, axis=-1, + fill_value=fill_value, bounds_error=False) + assert_array_almost_equal(interp(10), [[100, 200], [1000, 2000]]) + assert_array_almost_equal(interp(-10), [[100, 200], [1000, 2000]]) + assert_array_almost_equal(interp([-10, 10]), [[[100, 100], + [200, 200]], + [[1000, 1000], + [2000, 2000]]]) + + # broadcastable (2, 2) lower, broadcastable (2, 2) upper + fill_value = ([[-100, -200], [-1000, -2000]], + [[100, 200], [1000, 2000]]) + for y in (self.y235, self.y325, self.y25): + assert_raises(ValueError, interp1d, self.x5, y, kind=kind, + axis=-1, fill_value=fill_value, bounds_error=False) + for ii in range(2): + if ii == 1: + fill_value = (np.array(fill_value[0]), np.array(fill_value[1])) + interp = interp1d(self.x5, self.y225, kind=kind, axis=-1, + fill_value=fill_value, bounds_error=False) + assert_array_almost_equal(interp(10), [[100, 200], [1000, 2000]]) + assert_array_almost_equal(interp(-10), [[-100, -200], + [-1000, -2000]]) + assert_array_almost_equal(interp([-10, 10]), [[[-100, 100], + [-200, 200]], + [[-1000, 1000], + [-2000, 2000]]]) + + def test_fill_value(self): + # test that two-element fill value works + for kind in ('linear', 'nearest', 'cubic', 'slinear', 'quadratic', + 'zero', 'previous', 'next'): + self._check_fill_value(kind) + + def test_fill_value_writeable(self): + # backwards compat: fill_value is a public writeable attribute + interp = interp1d(self.x10, self.y10, fill_value=123.0) + assert interp.fill_value == 123.0 + interp.fill_value = 321.0 + assert interp.fill_value == 321.0 + + def _nd_check_interp(self, kind='linear'): + # Check the behavior when the inputs and outputs are multidimensional. + + # Multidimensional input. + interp10 = interp1d(self.x10, self.y10, kind=kind) + assert_array_almost_equal(interp10(np.array([[3., 5.], [2., 7.]])), + np.array([[3., 5.], [2., 7.]])) + + # Scalar input -> 0-dim scalar array output + assert isinstance(interp10(1.2), np.ndarray) + assert interp10(1.2).shape == () + + # Multidimensional outputs. + interp210 = interp1d(self.x10, self.y210, kind=kind) + assert_array_almost_equal(interp210(1.), np.array([1., 11.])) + assert_array_almost_equal(interp210(np.array([1., 2.])), + np.array([[1., 2.], [11., 12.]])) + + interp102 = interp1d(self.x10, self.y102, axis=0, kind=kind) + assert_array_almost_equal(interp102(1.), np.array([2.0, 3.0])) + assert_array_almost_equal(interp102(np.array([1., 3.])), + np.array([[2., 3.], [6., 7.]])) + + # Both at the same time! + x_new = np.array([[3., 5.], [2., 7.]]) + assert_array_almost_equal(interp210(x_new), + np.array([[[3., 5.], [2., 7.]], + [[13., 15.], [12., 17.]]])) + assert_array_almost_equal(interp102(x_new), + np.array([[[6., 7.], [10., 11.]], + [[4., 5.], [14., 15.]]])) + + def _nd_check_shape(self, kind='linear'): + # Check large N-D output shape + a = [4, 5, 6, 7] + y = np.arange(np.prod(a)).reshape(*a) + for n, s in enumerate(a): + x = np.arange(s) + z = interp1d(x, y, axis=n, kind=kind) + assert_array_almost_equal(z(x), y, err_msg=kind) + + x2 = np.arange(2*3*1).reshape((2,3,1)) / 12. + b = list(a) + b[n:n+1] = [2, 3, 1] + assert z(x2).shape == tuple(b), kind + + def test_nd(self): + for kind in ('linear', 'cubic', 'slinear', 'quadratic', 'nearest', + 'zero', 'previous', 'next'): + self._nd_check_interp(kind) + self._nd_check_shape(kind) + + def _check_complex(self, dtype=np.complex128, kind='linear'): + x = np.array([1, 2.5, 3, 3.1, 4, 6.4, 7.9, 8.0, 9.5, 10]) + y = x * x ** (1 + 2j) + y = y.astype(dtype) + + # simple test + c = interp1d(x, y, kind=kind) + assert_array_almost_equal(y[:-1], c(x)[:-1]) + + # check against interpolating real+imag separately + xi = np.linspace(1, 10, 31) + cr = interp1d(x, y.real, kind=kind) + ci = interp1d(x, y.imag, kind=kind) + assert_array_almost_equal(c(xi).real, cr(xi)) + assert_array_almost_equal(c(xi).imag, ci(xi)) + + def test_complex(self): + for kind in ('linear', 'nearest', 'cubic', 'slinear', 'quadratic', + 'zero', 'previous', 'next'): + self._check_complex(np.complex64, kind) + self._check_complex(np.complex128, kind) + + @pytest.mark.skipif(IS_PYPY, reason="Test not meaningful on PyPy") + def test_circular_refs(self): + # Test interp1d can be automatically garbage collected + x = np.linspace(0, 1) + y = np.linspace(0, 1) + # Confirm interp can be released from memory after use + with assert_deallocated(interp1d, x, y) as interp: + interp([0.1, 0.2]) + del interp + + def test_overflow_nearest(self): + # Test that the x range doesn't overflow when given integers as input + for kind in ('nearest', 'previous', 'next'): + x = np.array([0, 50, 127], dtype=np.int8) + ii = interp1d(x, x, kind=kind) + assert_array_almost_equal(ii(x), x) + + def test_local_nans(self): + # check that for local interpolation kinds (slinear, zero) a single nan + # only affects its local neighborhood + x = np.arange(10).astype(float) + y = x.copy() + y[6] = np.nan + for kind in ('zero', 'slinear'): + ir = interp1d(x, y, kind=kind) + vals = ir([4.9, 7.0]) + assert np.isfinite(vals).all() + + def test_spline_nans(self): + # Backwards compat: a single nan makes the whole spline interpolation + # return nans in an array of the correct shape. And it doesn't raise, + # just quiet nans because of backcompat. + x = np.arange(8).astype(float) + y = x.copy() + yn = y.copy() + yn[3] = np.nan + + for kind in ['quadratic', 'cubic']: + ir = interp1d(x, y, kind=kind) + irn = interp1d(x, yn, kind=kind) + for xnew in (6, [1, 6], [[1, 6], [3, 5]]): + xnew = np.asarray(xnew) + out, outn = ir(x), irn(x) + assert np.isnan(outn).all() + assert out.shape == outn.shape + + def test_all_nans(self): + # regression test for gh-11637: interp1d core dumps with all-nan `x` + x = np.ones(10) * np.nan + y = np.arange(10) + with assert_raises(ValueError): + interp1d(x, y, kind='cubic') + + def test_read_only(self): + x = np.arange(0, 10) + y = np.exp(-x / 3.0) + xnew = np.arange(0, 9, 0.1) + # Check both read-only and not read-only: + for xnew_writeable in (True, False): + xnew.flags.writeable = xnew_writeable + x.flags.writeable = False + for kind in ('linear', 'nearest', 'zero', 'slinear', 'quadratic', + 'cubic'): + f = interp1d(x, y, kind=kind) + vals = f(xnew) + assert np.isfinite(vals).all() + + @pytest.mark.parametrize( + "kind", ("linear", "nearest", "nearest-up", "previous", "next") + ) + def test_single_value(self, kind): + # https://github.com/scipy/scipy/issues/4043 + f = interp1d([1.5], [6], kind=kind, bounds_error=False, + fill_value=(2, 10)) + xp_assert_equal(f([1, 1.5, 2]), np.asarray([2.0, 6, 10])) + # check still error if bounds_error=True + f = interp1d([1.5], [6], kind=kind, bounds_error=True) + with assert_raises(ValueError, match="x_new is above"): + f(2.0) + + +class TestLagrange: + + def test_lagrange(self): + p = poly1d([5,2,1,4,3]) + xs = np.arange(len(p.coeffs)) + ys = p(xs) + pl = lagrange(xs,ys) + assert_array_almost_equal(p.coeffs,pl.coeffs) + + +class TestAkima1DInterpolator: + def test_eval(self): + x = np.arange(0., 11.) + y = np.array([0., 2., 1., 3., 2., 6., 5.5, 5.5, 2.7, 5.1, 3.]) + ak = Akima1DInterpolator(x, y) + xi = np.array([0., 0.5, 1., 1.5, 2.5, 3.5, 4.5, 5.1, 6.5, 7.2, + 8.6, 9.9, 10.]) + yi = np.array([0., 1.375, 2., 1.5, 1.953125, 2.484375, + 4.1363636363636366866103344, 5.9803623910336236590978842, + 5.5067291516462386624652936, 5.2031367459745245795943447, + 4.1796554159017080820603951, 3.4110386597938129327189927, + 3.]) + xp_assert_close(ak(xi), yi) + + def test_eval_mod(self): + # Reference values generated with the following MATLAB code: + # format longG + # x = 0:10; y = [0. 2. 1. 3. 2. 6. 5.5 5.5 2.7 5.1 3.]; + # xi = [0. 0.5 1. 1.5 2.5 3.5 4.5 5.1 6.5 7.2 8.6 9.9 10.]; + # makima(x, y, xi) + x = np.arange(0., 11.) + y = np.array([0., 2., 1., 3., 2., 6., 5.5, 5.5, 2.7, 5.1, 3.]) + ak = Akima1DInterpolator(x, y, method="makima") + xi = np.array([0., 0.5, 1., 1.5, 2.5, 3.5, 4.5, 5.1, 6.5, 7.2, + 8.6, 9.9, 10.]) + yi = np.array([ + 0.0, 1.34471153846154, 2.0, 1.44375, 1.94375, 2.51939102564103, + 4.10366931918656, 5.98501550899192, 5.51756330960439, 5.1757231914014, + 4.12326636931311, 3.32931513157895, 3.0]) + xp_assert_close(ak(xi), yi) + + def test_eval_2d(self): + x = np.arange(0., 11.) + y = np.array([0., 2., 1., 3., 2., 6., 5.5, 5.5, 2.7, 5.1, 3.]) + y = np.column_stack((y, 2. * y)) + ak = Akima1DInterpolator(x, y) + xi = np.array([0., 0.5, 1., 1.5, 2.5, 3.5, 4.5, 5.1, 6.5, 7.2, + 8.6, 9.9, 10.]) + yi = np.array([0., 1.375, 2., 1.5, 1.953125, 2.484375, + 4.1363636363636366866103344, + 5.9803623910336236590978842, + 5.5067291516462386624652936, + 5.2031367459745245795943447, + 4.1796554159017080820603951, + 3.4110386597938129327189927, 3.]) + yi = np.column_stack((yi, 2. * yi)) + xp_assert_close(ak(xi), yi) + + def test_eval_3d(self): + x = np.arange(0., 11.) + y_ = np.array([0., 2., 1., 3., 2., 6., 5.5, 5.5, 2.7, 5.1, 3.]) + y = np.empty((11, 2, 2)) + y[:, 0, 0] = y_ + y[:, 1, 0] = 2. * y_ + y[:, 0, 1] = 3. * y_ + y[:, 1, 1] = 4. * y_ + ak = Akima1DInterpolator(x, y) + xi = np.array([0., 0.5, 1., 1.5, 2.5, 3.5, 4.5, 5.1, 6.5, 7.2, + 8.6, 9.9, 10.]) + yi = np.empty((13, 2, 2)) + yi_ = np.array([0., 1.375, 2., 1.5, 1.953125, 2.484375, + 4.1363636363636366866103344, + 5.9803623910336236590978842, + 5.5067291516462386624652936, + 5.2031367459745245795943447, + 4.1796554159017080820603951, + 3.4110386597938129327189927, 3.]) + yi[:, 0, 0] = yi_ + yi[:, 1, 0] = 2. * yi_ + yi[:, 0, 1] = 3. * yi_ + yi[:, 1, 1] = 4. * yi_ + xp_assert_close(ak(xi), yi) + + def test_degenerate_case_multidimensional(self): + # This test is for issue #5683. + x = np.array([0, 1, 2]) + y = np.vstack((x, x**2)).T + ak = Akima1DInterpolator(x, y) + x_eval = np.array([0.5, 1.5]) + y_eval = ak(x_eval) + xp_assert_close(y_eval, np.vstack((x_eval, x_eval**2)).T) + + def test_extend(self): + x = np.arange(0., 11.) + y = np.array([0., 2., 1., 3., 2., 6., 5.5, 5.5, 2.7, 5.1, 3.]) + ak = Akima1DInterpolator(x, y) + match = "Extending a 1-D Akima interpolator is not yet implemented" + with pytest.raises(NotImplementedError, match=match): + ak.extend(None, None) + + def test_mod_invalid_method(self): + x = np.arange(0., 11.) + y = np.array([0., 2., 1., 3., 2., 6., 5.5, 5.5, 2.7, 5.1, 3.]) + match = "`method`=invalid is unsupported." + with pytest.raises(NotImplementedError, match=match): + Akima1DInterpolator(x, y, method="invalid") # type: ignore + + def test_extrapolate_attr(self): + # + x = np.linspace(-5, 5, 11) + y = x**2 + x_ext = np.linspace(-10, 10, 17) + y_ext = x_ext**2 + # Testing all extrapolate cases. + ak_true = Akima1DInterpolator(x, y, extrapolate=True) + ak_false = Akima1DInterpolator(x, y, extrapolate=False) + ak_none = Akima1DInterpolator(x, y, extrapolate=None) + # None should default to False; extrapolated points are NaN. + xp_assert_close(ak_false(x_ext), ak_none(x_ext), atol=1e-15) + xp_assert_equal(ak_false(x_ext)[0:4], np.full(4, np.nan)) + xp_assert_equal(ak_false(x_ext)[-4:-1], np.full(3, np.nan)) + # Extrapolation on call and attribute should be equal. + xp_assert_close(ak_false(x_ext, extrapolate=True), ak_true(x_ext), atol=1e-15) + # Testing extrapoation to actual function. + xp_assert_close(y_ext, ak_true(x_ext), atol=1e-15) + + +@pytest.mark.parametrize("method", [Akima1DInterpolator, PchipInterpolator]) +def test_complex(method): + # Complex-valued data deprecated + x = np.arange(0., 11.) + y = np.array([0., 2., 1., 3., 2., 6., 5.5, 5.5, 2.7, 5.1, 3.]) + y = y - 2j*y + msg = "real values" + with pytest.raises(ValueError, match=msg): + method(x, y) + + def test_concurrency(self): + # Check that no segfaults appear with concurrent access to Akima1D + x = np.linspace(-5, 5, 11) + y = x**2 + x_ext = np.linspace(-10, 10, 17) + ak = Akima1DInterpolator(x, y, extrapolate=True) + + def worker_fn(_, ak, x_ext): + ak(x_ext) + + _run_concurrent_barrier(10, worker_fn, ak, x_ext) + + +class TestPPolyCommon: + # test basic functionality for PPoly and BPoly + def test_sort_check(self): + c = np.array([[1, 4], [2, 5], [3, 6]]) + x = np.array([0, 1, 0.5]) + assert_raises(ValueError, PPoly, c, x) + assert_raises(ValueError, BPoly, c, x) + + def test_ctor_c(self): + # wrong shape: `c` must be at least 2D + with assert_raises(ValueError): + PPoly([1, 2], [0, 1]) + + def test_extend(self): + # Test adding new points to the piecewise polynomial + np.random.seed(1234) + + order = 3 + x = np.unique(np.r_[0, 10 * np.random.rand(30), 10]) + c = 2*np.random.rand(order+1, len(x)-1, 2, 3) - 1 + + for cls in (PPoly, BPoly): + pp = cls(c[:,:9], x[:10]) + pp.extend(c[:,9:], x[10:]) + + pp2 = cls(c[:, 10:], x[10:]) + pp2.extend(c[:, :10], x[:10]) + + pp3 = cls(c, x) + + xp_assert_equal(pp.c, pp3.c) + xp_assert_equal(pp.x, pp3.x) + xp_assert_equal(pp2.c, pp3.c) + xp_assert_equal(pp2.x, pp3.x) + + def test_extend_diff_orders(self): + # Test extending polynomial with different order one + np.random.seed(1234) + + x = np.linspace(0, 1, 6) + c = np.random.rand(2, 5) + + x2 = np.linspace(1, 2, 6) + c2 = np.random.rand(4, 5) + + for cls in (PPoly, BPoly): + pp1 = cls(c, x) + pp2 = cls(c2, x2) + + pp_comb = cls(c, x) + pp_comb.extend(c2, x2[1:]) + + # NB. doesn't match to pp1 at the endpoint, because pp1 is not + # continuous with pp2 as we took random coefs. + xi1 = np.linspace(0, 1, 300, endpoint=False) + xi2 = np.linspace(1, 2, 300) + + xp_assert_close(pp1(xi1), pp_comb(xi1)) + xp_assert_close(pp2(xi2), pp_comb(xi2)) + + def test_extend_descending(self): + np.random.seed(0) + + order = 3 + x = np.sort(np.random.uniform(0, 10, 20)) + c = np.random.rand(order + 1, x.shape[0] - 1, 2, 3) + + for cls in (PPoly, BPoly): + p = cls(c, x) + + p1 = cls(c[:, :9], x[:10]) + p1.extend(c[:, 9:], x[10:]) + + p2 = cls(c[:, 10:], x[10:]) + p2.extend(c[:, :10], x[:10]) + + xp_assert_equal(p1.c, p.c) + xp_assert_equal(p1.x, p.x) + xp_assert_equal(p2.c, p.c) + xp_assert_equal(p2.x, p.x) + + def test_shape(self): + np.random.seed(1234) + c = np.random.rand(8, 12, 5, 6, 7) + x = np.sort(np.random.rand(13)) + xp = np.random.rand(3, 4) + for cls in (PPoly, BPoly): + p = cls(c, x) + assert p(xp).shape == (3, 4, 5, 6, 7) + + # 'scalars' + for cls in (PPoly, BPoly): + p = cls(c[..., 0, 0, 0], x) + + assert np.shape(p(0.5)) == () + assert np.shape(p(np.array(0.5))) == () + + assert_raises(ValueError, p, np.array([[0.1, 0.2], [0.4]], dtype=object)) + + def test_concurrency(self): + # Check that no segfaults appear with concurrent access to BPoly, PPoly + c = np.random.rand(8, 12, 5, 6, 7) + x = np.sort(np.random.rand(13)) + xp = np.random.rand(3, 4) + + for cls in (PPoly, BPoly): + interp = cls(c, x) + + def worker_fn(_, interp, xp): + interp(xp) + + _run_concurrent_barrier(10, worker_fn, interp, xp) + + + def test_complex_coef(self): + np.random.seed(12345) + x = np.sort(np.random.random(13)) + c = np.random.random((8, 12)) * (1. + 0.3j) + c_re, c_im = c.real, c.imag + xp = np.random.random(5) + for cls in (PPoly, BPoly): + p, p_re, p_im = cls(c, x), cls(c_re, x), cls(c_im, x) + for nu in [0, 1, 2]: + xp_assert_close(p(xp, nu).real, p_re(xp, nu)) + xp_assert_close(p(xp, nu).imag, p_im(xp, nu)) + + def test_axis(self): + np.random.seed(12345) + c = np.random.rand(3, 4, 5, 6, 7, 8) + c_s = c.shape + xp = np.random.random((1, 2)) + for axis in (0, 1, 2, 3): + m = c.shape[axis+1] + x = np.sort(np.random.rand(m+1)) + for cls in (PPoly, BPoly): + p = cls(c, x, axis=axis) + assert p.c.shape == c_s[axis:axis+2] + c_s[:axis] + c_s[axis+2:] + res = p(xp) + targ_shape = c_s[:axis] + xp.shape + c_s[2+axis:] + assert res.shape == targ_shape + + # deriv/antideriv does not drop the axis + for p1 in [cls(c, x, axis=axis).derivative(), + cls(c, x, axis=axis).derivative(2), + cls(c, x, axis=axis).antiderivative(), + cls(c, x, axis=axis).antiderivative(2)]: + assert p1.axis == p.axis + + # c array needs two axes for the coefficients and intervals, so + # 0 <= axis < c.ndim-1; raise otherwise + for axis in (-1, 4, 5, 6): + for cls in (BPoly, PPoly): + assert_raises(ValueError, cls, **dict(c=c, x=x, axis=axis)) + + +class TestPolySubclassing: + class P(PPoly): + pass + + class B(BPoly): + pass + + def _make_polynomials(self): + np.random.seed(1234) + x = np.sort(np.random.random(3)) + c = np.random.random((4, 2)) + return self.P(c, x), self.B(c, x) + + def test_derivative(self): + pp, bp = self._make_polynomials() + for p in (pp, bp): + pd = p.derivative() + assert p.__class__ == pd.__class__ + + ppa = pp.antiderivative() + assert pp.__class__ == ppa.__class__ + + def test_from_spline(self): + np.random.seed(1234) + x = np.sort(np.r_[0, np.random.rand(11), 1]) + y = np.random.rand(len(x)) + + spl = splrep(x, y, s=0) + pp = self.P.from_spline(spl) + assert pp.__class__ == self.P + + def test_conversions(self): + pp, bp = self._make_polynomials() + + pp1 = self.P.from_bernstein_basis(bp) + assert pp1.__class__ == self.P + + bp1 = self.B.from_power_basis(pp) + assert bp1.__class__ == self.B + + def test_from_derivatives(self): + x = [0, 1, 2] + y = [[1], [2], [3]] + bp = self.B.from_derivatives(x, y) + assert bp.__class__ == self.B + + +class TestPPoly: + def test_simple(self): + c = np.array([[1, 4], [2, 5], [3, 6]]) + x = np.array([0, 0.5, 1]) + p = PPoly(c, x) + xp_assert_close(p(0.3), np.asarray(1*0.3**2 + 2*0.3 + 3)) + xp_assert_close(p(0.7), np.asarray(4*(0.7-0.5)**2 + 5*(0.7-0.5) + 6)) + + def test_periodic(self): + c = np.array([[1, 4], [2, 5], [3, 6]]) + x = np.array([0, 0.5, 1]) + p = PPoly(c, x, extrapolate='periodic') + + xp_assert_close(p(1.3), + np.asarray(1 * 0.3 ** 2 + 2 * 0.3 + 3)) + xp_assert_close(p(-0.3), + np.asarray(4 * (0.7 - 0.5) ** 2 + 5 * (0.7 - 0.5) + 6)) + + xp_assert_close(p(1.3, 1), np.asarray(2 * 0.3 + 2)) + xp_assert_close(p(-0.3, 1), np.asarray(8 * (0.7 - 0.5) + 5)) + + def test_read_only(self): + c = np.array([[1, 4], [2, 5], [3, 6]]) + x = np.array([0, 0.5, 1]) + xnew = np.array([0, 0.1, 0.2]) + PPoly(c, x, extrapolate='periodic') + + for writeable in (True, False): + x.flags.writeable = writeable + c.flags.writeable = writeable + f = PPoly(c, x) + vals = f(xnew) + assert np.isfinite(vals).all() + + def test_descending(self): + def binom_matrix(power): + n = np.arange(power + 1).reshape(-1, 1) + k = np.arange(power + 1) + B = binom(n, k) + return B[::-1, ::-1] + + rng = np.random.RandomState(0) + + power = 3 + for m in [10, 20, 30]: + x = np.sort(rng.uniform(0, 10, m + 1)) + ca = rng.uniform(-2, 2, size=(power + 1, m)) + + h = np.diff(x) + h_powers = h[None, :] ** np.arange(power + 1)[::-1, None] + B = binom_matrix(power) + cap = ca * h_powers + cdp = np.dot(B.T, cap) + cd = cdp / h_powers + + pa = PPoly(ca, x, extrapolate=True) + pd = PPoly(cd[:, ::-1], x[::-1], extrapolate=True) + + x_test = rng.uniform(-10, 20, 100) + xp_assert_close(pa(x_test), pd(x_test), rtol=1e-13) + xp_assert_close(pa(x_test, 1), pd(x_test, 1), rtol=1e-13) + + pa_d = pa.derivative() + pd_d = pd.derivative() + + xp_assert_close(pa_d(x_test), pd_d(x_test), rtol=1e-13) + + # Antiderivatives won't be equal because fixing continuity is + # done in the reverse order, but surely the differences should be + # equal. + pa_i = pa.antiderivative() + pd_i = pd.antiderivative() + for a, b in rng.uniform(-10, 20, (5, 2)): + int_a = pa.integrate(a, b) + int_d = pd.integrate(a, b) + xp_assert_close(int_a, int_d, rtol=1e-13) + xp_assert_close(pa_i(b) - pa_i(a), pd_i(b) - pd_i(a), + rtol=1e-13) + + roots_d = pd.roots() + roots_a = pa.roots() + xp_assert_close(roots_a, np.sort(roots_d), rtol=1e-12) + + def test_multi_shape(self): + c = np.random.rand(6, 2, 1, 2, 3) + x = np.array([0, 0.5, 1]) + p = PPoly(c, x) + assert p.x.shape == x.shape + assert p.c.shape == c.shape + assert p(0.3).shape == c.shape[2:] + + assert p(np.random.rand(5, 6)).shape == (5, 6) + c.shape[2:] + + dp = p.derivative() + assert dp.c.shape == (5, 2, 1, 2, 3) + ip = p.antiderivative() + assert ip.c.shape == (7, 2, 1, 2, 3) + + def test_construct_fast(self): + np.random.seed(1234) + c = np.array([[1, 4], [2, 5], [3, 6]], dtype=float) + x = np.array([0, 0.5, 1]) + p = PPoly.construct_fast(c, x) + xp_assert_close(p(0.3), np.asarray(1*0.3**2 + 2*0.3 + 3)) + xp_assert_close(p(0.7), np.asarray(4*(0.7-0.5)**2 + 5*(0.7-0.5) + 6)) + + def test_vs_alternative_implementations(self): + rng = np.random.RandomState(1234) + c = rng.rand(3, 12, 22) + x = np.sort(np.r_[0, rng.rand(11), 1]) + + p = PPoly(c, x) + + xp = np.r_[0.3, 0.5, 0.33, 0.6] + expected = _ppoly_eval_1(c, x, xp) + xp_assert_close(p(xp), expected) + + expected = _ppoly_eval_2(c[:,:,0], x, xp) + xp_assert_close(p(xp)[:, 0], expected) + + def test_from_spline(self): + rng = np.random.RandomState(1234) + x = np.sort(np.r_[0, rng.rand(11), 1]) + y = rng.rand(len(x)) + + spl = splrep(x, y, s=0) + pp = PPoly.from_spline(spl) + + xi = np.linspace(0, 1, 200) + xp_assert_close(pp(xi), splev(xi, spl)) + + # make sure .from_spline accepts BSpline objects + b = BSpline(*spl) + ppp = PPoly.from_spline(b) + xp_assert_close(ppp(xi), b(xi)) + + # BSpline's extrapolate attribute propagates unless overridden + t, c, k = spl + for extrap in (None, True, False): + b = BSpline(t, c, k, extrapolate=extrap) + p = PPoly.from_spline(b) + assert p.extrapolate == b.extrapolate + + def test_derivative_simple(self): + np.random.seed(1234) + c = np.array([[4, 3, 2, 1]]).T + dc = np.array([[3*4, 2*3, 2]]).T + ddc = np.array([[2*3*4, 1*2*3]]).T + x = np.array([0, 1]) + + pp = PPoly(c, x) + dpp = PPoly(dc, x) + ddpp = PPoly(ddc, x) + + xp_assert_close(pp.derivative().c, dpp.c) + xp_assert_close(pp.derivative(2).c, ddpp.c) + + def test_derivative_eval(self): + rng = np.random.RandomState(1234) + x = np.sort(np.r_[0, rng.rand(11), 1]) + y = rng.rand(len(x)) + + spl = splrep(x, y, s=0) + pp = PPoly.from_spline(spl) + + xi = np.linspace(0, 1, 200) + for dx in range(0, 3): + xp_assert_close(pp(xi, dx), splev(xi, spl, dx)) + + def test_derivative(self): + rng = np.random.RandomState(1234) + x = np.sort(np.r_[0, rng.rand(11), 1]) + y = rng.rand(len(x)) + + spl = splrep(x, y, s=0, k=5) + pp = PPoly.from_spline(spl) + + xi = np.linspace(0, 1, 200) + for dx in range(0, 10): + xp_assert_close(pp(xi, dx), pp.derivative(dx)(xi), + err_msg="dx=%d" % (dx,)) + + def test_antiderivative_of_constant(self): + # https://github.com/scipy/scipy/issues/4216 + p = PPoly([[1.]], [0, 1]) + xp_assert_equal(p.antiderivative().c, PPoly([[1], [0]], [0, 1]).c) + xp_assert_equal(p.antiderivative().x, PPoly([[1], [0]], [0, 1]).x) + + def test_antiderivative_regression_4355(self): + # https://github.com/scipy/scipy/issues/4355 + p = PPoly([[1., 0.5]], [0, 1, 2]) + q = p.antiderivative() + xp_assert_equal(q.c, [[1, 0.5], [0, 1]]) + xp_assert_equal(q.x, [0.0, 1, 2]) + xp_assert_close(p.integrate(0, 2), np.asarray(1.5)) + xp_assert_close(np.asarray(q(2) - q(0)), + np.asarray(1.5)) + + def test_antiderivative_simple(self): + np.random.seed(1234) + # [ p1(x) = 3*x**2 + 2*x + 1, + # p2(x) = 1.6875] + c = np.array([[3, 2, 1], [0, 0, 1.6875]]).T + # [ pp1(x) = x**3 + x**2 + x, + # pp2(x) = 1.6875*(x - 0.25) + pp1(0.25)] + ic = np.array([[1, 1, 1, 0], [0, 0, 1.6875, 0.328125]]).T + # [ ppp1(x) = (1/4)*x**4 + (1/3)*x**3 + (1/2)*x**2, + # ppp2(x) = (1.6875/2)*(x - 0.25)**2 + pp1(0.25)*x + ppp1(0.25)] + iic = np.array([[1/4, 1/3, 1/2, 0, 0], + [0, 0, 1.6875/2, 0.328125, 0.037434895833333336]]).T + x = np.array([0, 0.25, 1]) + + pp = PPoly(c, x) + ipp = pp.antiderivative() + iipp = pp.antiderivative(2) + iipp2 = ipp.antiderivative() + + xp_assert_close(ipp.x, x) + xp_assert_close(ipp.c.T, ic.T) + xp_assert_close(iipp.c.T, iic.T) + xp_assert_close(iipp2.c.T, iic.T) + + def test_antiderivative_vs_derivative(self): + rng = np.random.RandomState(1234) + x = np.linspace(0, 1, 30)**2 + y = rng.rand(len(x)) + spl = splrep(x, y, s=0, k=5) + pp = PPoly.from_spline(spl) + + for dx in range(0, 10): + ipp = pp.antiderivative(dx) + + # check that derivative is inverse op + pp2 = ipp.derivative(dx) + xp_assert_close(pp.c, pp2.c) + + # check continuity + for k in range(dx): + pp2 = ipp.derivative(k) + + r = 1e-13 + endpoint = r*pp2.x[:-1] + (1 - r)*pp2.x[1:] + + xp_assert_close(pp2(pp2.x[1:]), pp2(endpoint), + rtol=1e-7, err_msg="dx=%d k=%d" % (dx, k)) + + def test_antiderivative_vs_spline(self): + rng = np.random.RandomState(1234) + x = np.sort(np.r_[0, rng.rand(11), 1]) + y = rng.rand(len(x)) + + spl = splrep(x, y, s=0, k=5) + pp = PPoly.from_spline(spl) + + for dx in range(0, 10): + pp2 = pp.antiderivative(dx) + spl2 = splantider(spl, dx) + + xi = np.linspace(0, 1, 200) + xp_assert_close(pp2(xi), splev(xi, spl2), + rtol=1e-7) + + def test_antiderivative_continuity(self): + c = np.array([[2, 1, 2, 2], [2, 1, 3, 3]]).T + x = np.array([0, 0.5, 1]) + + p = PPoly(c, x) + ip = p.antiderivative() + + # check continuity + xp_assert_close(ip(0.5 - 1e-9), ip(0.5 + 1e-9), rtol=1e-8) + + # check that only lowest order coefficients were changed + p2 = ip.derivative() + xp_assert_close(p2.c, p.c) + + def test_integrate(self): + rng = np.random.RandomState(1234) + x = np.sort(np.r_[0, rng.rand(11), 1]) + y = rng.rand(len(x)) + + spl = splrep(x, y, s=0, k=5) + pp = PPoly.from_spline(spl) + + a, b = 0.3, 0.9 + ig = pp.integrate(a, b) + + ipp = pp.antiderivative() + xp_assert_close(ig, ipp(b) - ipp(a), check_0d=False) + xp_assert_close(ig, splint(a, b, spl), check_0d=False) + + a, b = -0.3, 0.9 + ig = pp.integrate(a, b, extrapolate=True) + xp_assert_close(ig, ipp(b) - ipp(a), check_0d=False) + + assert np.isnan(pp.integrate(a, b, extrapolate=False)).all() + + def test_integrate_readonly(self): + x = np.array([1, 2, 4]) + c = np.array([[0., 0.], [-1., -1.], [2., -0.], [1., 2.]]) + + for writeable in (True, False): + x.flags.writeable = writeable + + P = PPoly(c, x) + vals = P.integrate(1, 4) + + assert np.isfinite(vals).all() + + def test_integrate_periodic(self): + x = np.array([1, 2, 4]) + c = np.array([[0., 0.], [-1., -1.], [2., -0.], [1., 2.]]) + + P = PPoly(c, x, extrapolate='periodic') + I = P.antiderivative() + + period_int = np.asarray(I(4) - I(1)) + + xp_assert_close(P.integrate(1, 4), period_int) + xp_assert_close(P.integrate(-10, -7), period_int) + xp_assert_close(P.integrate(-10, -4), np.asarray(2 * period_int)) + + xp_assert_close(P.integrate(1.5, 2.5), + np.asarray(I(2.5) - I(1.5))) + xp_assert_close(P.integrate(3.5, 5), + np.asarray(I(2) - I(1) + I(4) - I(3.5))) + xp_assert_close(P.integrate(3.5 + 12, 5 + 12), + np.asarray(I(2) - I(1) + I(4) - I(3.5))) + xp_assert_close(P.integrate(3.5, 5 + 12), + np.asarray(I(2) - I(1) + I(4) - I(3.5) + 4 * period_int)) + xp_assert_close(P.integrate(0, -1), + np.asarray(I(2) - I(3))) + xp_assert_close(P.integrate(-9, -10), + np.asarray(I(2) - I(3))) + xp_assert_close(P.integrate(0, -10), + np.asarray(I(2) - I(3) - 3 * period_int)) + + def test_roots(self): + x = np.linspace(0, 1, 31)**2 + y = np.sin(30*x) + + spl = splrep(x, y, s=0, k=3) + pp = PPoly.from_spline(spl) + + r = pp.roots() + r = r[(r >= 0 - 1e-15) & (r <= 1 + 1e-15)] + xp_assert_close(r, sproot(spl), atol=1e-15) + + def test_roots_idzero(self): + # Roots for piecewise polynomials with identically zero + # sections. + c = np.array([[-1, 0.25], [0, 0], [-1, 0.25]]).T + x = np.array([0, 0.4, 0.6, 1.0]) + + pp = PPoly(c, x) + xp_assert_equal(pp.roots(), + [0.25, 0.4, np.nan, 0.6 + 0.25]) + + # ditto for p.solve(const) with sections identically equal const + const = 2. + c1 = c.copy() + c1[1, :] += const + pp1 = PPoly(c1, x) + + xp_assert_equal(pp1.solve(const), + [0.25, 0.4, np.nan, 0.6 + 0.25]) + + def test_roots_all_zero(self): + # test the code path for the polynomial being identically zero everywhere + c = [[0], [0]] + x = [0, 1] + p = PPoly(c, x) + xp_assert_equal(p.roots(), [0, np.nan]) + xp_assert_equal(p.solve(0), [0, np.nan]) + xp_assert_equal(p.solve(1), []) + + c = [[0, 0], [0, 0]] + x = [0, 1, 2] + p = PPoly(c, x) + xp_assert_equal(p.roots(), [0, np.nan, 1, np.nan]) + xp_assert_equal(p.solve(0), [0, np.nan, 1, np.nan]) + xp_assert_equal(p.solve(1), []) + + def test_roots_repeated(self): + # Check roots repeated in multiple sections are reported only + # once. + + # [(x + 1)**2 - 1, -x**2] ; x == 0 is a repeated root + c = np.array([[1, 0, -1], [-1, 0, 0]]).T + x = np.array([-1, 0, 1]) + + pp = PPoly(c, x) + xp_assert_equal(pp.roots(), np.asarray([-2.0, 0.0])) + xp_assert_equal(pp.roots(extrapolate=False), np.asarray([0.0])) + + def test_roots_discont(self): + # Check that a discontinuity across zero is reported as root + c = np.array([[1], [-1]]).T + x = np.array([0, 0.5, 1]) + pp = PPoly(c, x) + xp_assert_equal(pp.roots(), np.asarray([0.5])) + xp_assert_equal(pp.roots(discontinuity=False), np.asarray([])) + + # ditto for a discontinuity across y: + xp_assert_equal(pp.solve(0.5), np.asarray([0.5])) + xp_assert_equal(pp.solve(0.5, discontinuity=False), np.asarray([])) + + xp_assert_equal(pp.solve(1.5), np.asarray([])) + xp_assert_equal(pp.solve(1.5, discontinuity=False), np.asarray([])) + + def test_roots_random(self): + # Check high-order polynomials with random coefficients + rng = np.random.RandomState(1234) + + num = 0 + + for extrapolate in (True, False): + for order in range(0, 20): + x = np.unique(np.r_[0, 10 * rng.rand(30), 10]) + c = 2*rng.rand(order+1, len(x)-1, 2, 3) - 1 + + pp = PPoly(c, x) + for y in [0, rng.random()]: + r = pp.solve(y, discontinuity=False, extrapolate=extrapolate) + + for i in range(2): + for j in range(3): + rr = r[i,j] + if rr.size > 0: + # Check that the reported roots indeed are roots + num += rr.size + val = pp(rr, extrapolate=extrapolate)[:,i,j] + cmpval = pp(rr, nu=1, + extrapolate=extrapolate)[:,i,j] + msg = f"({extrapolate!r}) r = {repr(rr)}" + xp_assert_close((val-y) / cmpval, np.asarray(0.0), + atol=1e-7, + err_msg=msg, check_shape=False) + + # Check that we checked a number of roots + assert num > 100, repr(num) + + def test_roots_croots(self): + # Test the complex root finding algorithm + rng = np.random.RandomState(1234) + + for k in range(1, 15): + c = rng.rand(k, 1, 130) + + if k == 3: + # add a case with zero discriminant + c[:,0,0] = 1, 2, 1 + + for y in [0, rng.random()]: + w = np.empty(c.shape, dtype=complex) + _ppoly._croots_poly1(c, w, y) + + if k == 1: + assert np.isnan(w).all() + continue + + res = -y + cres = 0 + for i in range(k): + res += c[i,None] * w**(k-1-i) + cres += abs(c[i,None] * w**(k-1-i)) + with np.errstate(invalid='ignore'): + res /= cres + res = res.ravel() + res = res[~np.isnan(res)] + xp_assert_close(res, np.zeros_like(res), atol=1e-10) + + def test_extrapolate_attr(self): + # [ 1 - x**2 ] + c = np.array([[-1, 0, 1]]).T + x = np.array([0, 1]) + + for extrapolate in [True, False, None]: + pp = PPoly(c, x, extrapolate=extrapolate) + pp_d = pp.derivative() + pp_i = pp.antiderivative() + + if extrapolate is False: + assert np.isnan(pp([-0.1, 1.1])).all() + assert np.isnan(pp_i([-0.1, 1.1])).all() + assert np.isnan(pp_d([-0.1, 1.1])).all() + assert pp.roots() == [1] + else: + xp_assert_close(pp([-0.1, 1.1]), [1-0.1**2, 1-1.1**2]) + assert not np.isnan(pp_i([-0.1, 1.1])).any() + assert not np.isnan(pp_d([-0.1, 1.1])).any() + xp_assert_close(pp.roots(), np.asarray([1.0, -1.0])) + + +class TestBPoly: + def test_simple(self): + x = [0, 1] + c = [[3]] + bp = BPoly(c, x) + xp_assert_close(bp(0.1), np.asarray(3.)) + + def test_simple2(self): + x = [0, 1] + c = [[3], [1]] + bp = BPoly(c, x) # 3*(1-x) + 1*x + xp_assert_close(bp(0.1), np.asarray(3*0.9 + 1.*0.1)) + + def test_simple3(self): + x = [0, 1] + c = [[3], [1], [4]] + bp = BPoly(c, x) # 3 * (1-x)**2 + 2 * x (1-x) + 4 * x**2 + xp_assert_close(bp(0.2), + np.asarray(3 * 0.8*0.8 + 1 * 2*0.2*0.8 + 4 * 0.2*0.2)) + + def test_simple4(self): + x = [0, 1] + c = [[1], [1], [1], [2]] + bp = BPoly(c, x) + xp_assert_close(bp(0.3), + np.asarray( 0.7**3 + + 3 * 0.7**2 * 0.3 + + 3 * 0.7 * 0.3**2 + + 2 * 0.3**3) + ) + + def test_simple5(self): + x = [0, 1] + c = [[1], [1], [8], [2], [1]] + bp = BPoly(c, x) + xp_assert_close(bp(0.3), + np.asarray( 0.7**4 + + 4 * 0.7**3 * 0.3 + + 8 * 6 * 0.7**2 * 0.3**2 + + 2 * 4 * 0.7 * 0.3**3 + + 0.3**4) + ) + + def test_periodic(self): + x = [0, 1, 3] + c = [[3, 0], [0, 0], [0, 2]] + # [3*(1-x)**2, 2*((x-1)/2)**2] + bp = BPoly(c, x, extrapolate='periodic') + + xp_assert_close(bp(3.4), np.asarray(3 * 0.6**2)) + xp_assert_close(bp(-1.3), np.asarray(2 * (0.7/2)**2)) + + xp_assert_close(bp(3.4, 1), np.asarray(-6 * 0.6)) + xp_assert_close(bp(-1.3, 1), np.asarray(2 * (0.7/2))) + + def test_descending(self): + rng = np.random.RandomState(0) + + power = 3 + for m in [10, 20, 30]: + x = np.sort(rng.uniform(0, 10, m + 1)) + ca = rng.uniform(-0.1, 0.1, size=(power + 1, m)) + # We need only to flip coefficients to get it right! + cd = ca[::-1].copy() + + pa = BPoly(ca, x, extrapolate=True) + pd = BPoly(cd[:, ::-1], x[::-1], extrapolate=True) + + x_test = rng.uniform(-10, 20, 100) + xp_assert_close(pa(x_test), pd(x_test), rtol=1e-13) + xp_assert_close(pa(x_test, 1), pd(x_test, 1), rtol=1e-13) + + pa_d = pa.derivative() + pd_d = pd.derivative() + + xp_assert_close(pa_d(x_test), pd_d(x_test), rtol=1e-13) + + # Antiderivatives won't be equal because fixing continuity is + # done in the reverse order, but surely the differences should be + # equal. + pa_i = pa.antiderivative() + pd_i = pd.antiderivative() + for a, b in rng.uniform(-10, 20, (5, 2)): + int_a = pa.integrate(a, b) + int_d = pd.integrate(a, b) + xp_assert_close(int_a, int_d, rtol=1e-12) + xp_assert_close(pa_i(b) - pa_i(a), pd_i(b) - pd_i(a), + rtol=1e-12) + + def test_multi_shape(self): + rng = np.random.RandomState(1234) + c = rng.rand(6, 2, 1, 2, 3) + x = np.array([0, 0.5, 1]) + p = BPoly(c, x) + assert p.x.shape == x.shape + assert p.c.shape == c.shape + assert p(0.3).shape == c.shape[2:] + assert p(rng.rand(5, 6)).shape == (5, 6) + c.shape[2:] + + dp = p.derivative() + assert dp.c.shape == (5, 2, 1, 2, 3) + + def test_interval_length(self): + x = [0, 2] + c = [[3], [1], [4]] + bp = BPoly(c, x) + xval = 0.1 + s = xval / 2 # s = (x - xa) / (xb - xa) + xp_assert_close(bp(xval), + np.asarray(3 * (1-s)*(1-s) + 1 * 2*s*(1-s) + 4 * s*s) + ) + + def test_two_intervals(self): + x = [0, 1, 3] + c = [[3, 0], [0, 0], [0, 2]] + bp = BPoly(c, x) # [3*(1-x)**2, 2*((x-1)/2)**2] + + xp_assert_close(bp(0.4), np.asarray(3 * 0.6*0.6)) + xp_assert_close(bp(1.7), np.asarray(2 * (0.7/2)**2)) + + def test_extrapolate_attr(self): + x = [0, 2] + c = [[3], [1], [4]] + bp = BPoly(c, x) + + for extrapolate in (True, False, None): + bp = BPoly(c, x, extrapolate=extrapolate) + bp_d = bp.derivative() + if extrapolate is False: + assert np.isnan(bp([-0.1, 2.1])).all() + assert np.isnan(bp_d([-0.1, 2.1])).all() + else: + assert not np.isnan(bp([-0.1, 2.1])).any() + assert not np.isnan(bp_d([-0.1, 2.1])).any() + + +class TestBPolyCalculus: + def test_derivative(self): + x = [0, 1, 3] + c = [[3, 0], [0, 0], [0, 2]] + bp = BPoly(c, x) # [3*(1-x)**2, 2*((x-1)/2)**2] + bp_der = bp.derivative() + xp_assert_close(bp_der(0.4), np.asarray(-6*(0.6))) + xp_assert_close(bp_der(1.7), np.asarray(0.7)) + + # derivatives in-place + xp_assert_close(np.asarray([bp(0.4, nu) for nu in [1, 2, 3]]), + np.asarray([-6*(1-0.4), 6., 0.]) + ) + xp_assert_close(np.asarray([bp(1.7, nu) for nu in [1, 2, 3]]), + np.asarray([0.7, 1., 0]) + ) + + def test_derivative_ppoly(self): + # make sure it's consistent w/ power basis + rng = np.random.RandomState(1234) + m, k = 5, 8 # number of intervals, order + x = np.sort(rng.random(m)) + c = rng.random((k, m-1)) + bp = BPoly(c, x) + pp = PPoly.from_bernstein_basis(bp) + + for d in range(k): + bp = bp.derivative() + pp = pp.derivative() + xp = np.linspace(x[0], x[-1], 21) + xp_assert_close(bp(xp), pp(xp)) + + def test_deriv_inplace(self): + rng = np.random.RandomState(1234) + m, k = 5, 8 # number of intervals, order + x = np.sort(rng.random(m)) + c = rng.random((k, m-1)) + + # test both real and complex coefficients + for cc in [c.copy(), c*(1. + 2.j)]: + bp = BPoly(cc, x) + xp = np.linspace(x[0], x[-1], 21) + for i in range(k): + xp_assert_close(bp(xp, i), bp.derivative(i)(xp)) + + def test_antiderivative_simple(self): + # f(x) = x for x \in [0, 1), + # (x-1)/2 for x \in [1, 3] + # + # antiderivative is then + # F(x) = x**2 / 2 for x \in [0, 1), + # 0.5*x*(x/2 - 1) + A for x \in [1, 3] + # where A = 3/4 for continuity at x = 1. + x = [0, 1, 3] + c = [[0, 0], [1, 1]] + + bp = BPoly(c, x) + bi = bp.antiderivative() + + xx = np.linspace(0, 3, 11) + xp_assert_close(bi(xx), + np.where(xx < 1, xx**2 / 2., + 0.5 * xx * (xx/2. - 1) + 3./4), + atol=1e-12, rtol=1e-12) + + def test_der_antider(self): + rng = np.random.RandomState(1234) + x = np.sort(rng.random(11)) + c = rng.random((4, 10, 2, 3)) + bp = BPoly(c, x) + + xx = np.linspace(x[0], x[-1], 100) + xp_assert_close(bp.antiderivative().derivative()(xx), + bp(xx), atol=1e-12, rtol=1e-12) + + def test_antider_ppoly(self): + rng = np.random.RandomState(1234) + x = np.sort(rng.random(11)) + c = rng.random((4, 10, 2, 3)) + bp = BPoly(c, x) + pp = PPoly.from_bernstein_basis(bp) + + xx = np.linspace(x[0], x[-1], 10) + + xp_assert_close(bp.antiderivative(2)(xx), + pp.antiderivative(2)(xx), atol=1e-12, rtol=1e-12) + + def test_antider_continuous(self): + rng = np.random.RandomState(1234) + x = np.sort(rng.random(11)) + c = rng.random((4, 10)) + bp = BPoly(c, x).antiderivative() + + xx = bp.x[1:-1] + xp_assert_close(bp(xx - 1e-14), + bp(xx + 1e-14), atol=1e-12, rtol=1e-12) + + def test_integrate(self): + rng = np.random.RandomState(1234) + x = np.sort(rng.random(11)) + c = rng.random((4, 10)) + bp = BPoly(c, x) + pp = PPoly.from_bernstein_basis(bp) + xp_assert_close(bp.integrate(0, 1), + pp.integrate(0, 1), atol=1e-12, rtol=1e-12, check_0d=False) + + def test_integrate_extrap(self): + c = [[1]] + x = [0, 1] + b = BPoly(c, x) + + # default is extrapolate=True + xp_assert_close(b.integrate(0, 2), np.asarray(2.), + atol=1e-14, check_0d=False) + + # .integrate argument overrides self.extrapolate + b1 = BPoly(c, x, extrapolate=False) + assert np.isnan(b1.integrate(0, 2)) + xp_assert_close(b1.integrate(0, 2, extrapolate=True), + np.asarray(2.), atol=1e-14, check_0d=False) + + def test_integrate_periodic(self): + x = np.array([1, 2, 4]) + c = np.array([[0., 0.], [-1., -1.], [2., -0.], [1., 2.]]) + + P = BPoly.from_power_basis(PPoly(c, x), extrapolate='periodic') + I = P.antiderivative() + + period_int = I(4) - I(1) + + xp_assert_close(P.integrate(1, 4), period_int) #, check_0d=False) + xp_assert_close(P.integrate(-10, -7), period_int) + xp_assert_close(P.integrate(-10, -4), 2 * period_int) + + xp_assert_close(P.integrate(1.5, 2.5), I(2.5) - I(1.5)) + xp_assert_close(P.integrate(3.5, 5), I(2) - I(1) + I(4) - I(3.5)) + xp_assert_close(P.integrate(3.5 + 12, 5 + 12), + I(2) - I(1) + I(4) - I(3.5)) + xp_assert_close(P.integrate(3.5, 5 + 12), + I(2) - I(1) + I(4) - I(3.5) + 4 * period_int) + + xp_assert_close(P.integrate(0, -1), I(2) - I(3)) + xp_assert_close(P.integrate(-9, -10), I(2) - I(3)) + xp_assert_close(P.integrate(0, -10), I(2) - I(3) - 3 * period_int) + + def test_antider_neg(self): + # .derivative(-nu) ==> .andiderivative(nu) and vice versa + c = [[1]] + x = [0, 1] + b = BPoly(c, x) + + xx = np.linspace(0, 1, 21) + + xp_assert_close(b.derivative(-1)(xx), b.antiderivative()(xx), + atol=1e-12, rtol=1e-12) + xp_assert_close(b.derivative(1)(xx), b.antiderivative(-1)(xx), + atol=1e-12, rtol=1e-12) + + +class TestPolyConversions: + def test_bp_from_pp(self): + x = [0, 1, 3] + c = [[3, 2], [1, 8], [4, 3]] + pp = PPoly(c, x) + bp = BPoly.from_power_basis(pp) + pp1 = PPoly.from_bernstein_basis(bp) + + xp = [0.1, 1.4] + xp_assert_close(pp(xp), bp(xp)) + xp_assert_close(pp(xp), pp1(xp)) + + def test_bp_from_pp_random(self): + rng = np.random.RandomState(1234) + m, k = 5, 8 # number of intervals, order + x = np.sort(rng.random(m)) + c = rng.random((k, m-1)) + pp = PPoly(c, x) + bp = BPoly.from_power_basis(pp) + pp1 = PPoly.from_bernstein_basis(bp) + + xp = np.linspace(x[0], x[-1], 21) + xp_assert_close(pp(xp), bp(xp)) + xp_assert_close(pp(xp), pp1(xp)) + + def test_pp_from_bp(self): + x = [0, 1, 3] + c = [[3, 3], [1, 1], [4, 2]] + bp = BPoly(c, x) + pp = PPoly.from_bernstein_basis(bp) + bp1 = BPoly.from_power_basis(pp) + + xp = [0.1, 1.4] + xp_assert_close(bp(xp), pp(xp)) + xp_assert_close(bp(xp), bp1(xp)) + + def test_broken_conversions(self): + # regression test for gh-10597: from_power_basis only accepts PPoly etc. + x = [0, 1, 3] + c = [[3, 3], [1, 1], [4, 2]] + pp = PPoly(c, x) + with assert_raises(TypeError): + PPoly.from_bernstein_basis(pp) + + bp = BPoly(c, x) + with assert_raises(TypeError): + BPoly.from_power_basis(bp) + + +class TestBPolyFromDerivatives: + def test_make_poly_1(self): + c1 = BPoly._construct_from_derivatives(0, 1, [2], [3]) + xp_assert_close(c1, [2., 3.]) + + def test_make_poly_2(self): + c1 = BPoly._construct_from_derivatives(0, 1, [1, 0], [1]) + xp_assert_close(c1, [1., 1., 1.]) + + # f'(0) = 3 + c2 = BPoly._construct_from_derivatives(0, 1, [2, 3], [1]) + xp_assert_close(c2, [2., 7./2, 1.]) + + # f'(1) = 3 + c3 = BPoly._construct_from_derivatives(0, 1, [2], [1, 3]) + xp_assert_close(c3, [2., -0.5, 1.]) + + def test_make_poly_3(self): + # f'(0)=2, f''(0)=3 + c1 = BPoly._construct_from_derivatives(0, 1, [1, 2, 3], [4]) + xp_assert_close(c1, [1., 5./3, 17./6, 4.]) + + # f'(1)=2, f''(1)=3 + c2 = BPoly._construct_from_derivatives(0, 1, [1], [4, 2, 3]) + xp_assert_close(c2, [1., 19./6, 10./3, 4.]) + + # f'(0)=2, f'(1)=3 + c3 = BPoly._construct_from_derivatives(0, 1, [1, 2], [4, 3]) + xp_assert_close(c3, [1., 5./3, 3., 4.]) + + def test_make_poly_12(self): + rng = np.random.RandomState(12345) + ya = np.r_[0, rng.random(5)] + yb = np.r_[0, rng.random(5)] + + c = BPoly._construct_from_derivatives(0, 1, ya, yb) + pp = BPoly(c[:, None], [0, 1]) + for j in range(6): + xp_assert_close(pp(0.), ya[j], check_0d=False) + xp_assert_close(pp(1.), yb[j], check_0d=False) + pp = pp.derivative() + + def test_raise_degree(self): + rng = np.random.RandomState(12345) + x = [0, 1] + k, d = 8, 5 + c = rng.random((k, 1, 2, 3, 4)) + bp = BPoly(c, x) + + c1 = BPoly._raise_degree(c, d) + bp1 = BPoly(c1, x) + + xp = np.linspace(0, 1, 11) + xp_assert_close(bp(xp), bp1(xp)) + + def test_xi_yi(self): + assert_raises(ValueError, BPoly.from_derivatives, [0, 1], [0]) + + def test_coords_order(self): + xi = [0, 0, 1] + yi = [[0], [0], [0]] + assert_raises(ValueError, BPoly.from_derivatives, xi, yi) + + def test_zeros(self): + xi = [0, 1, 2, 3] + yi = [[0, 0], [0], [0, 0], [0, 0]] # NB: will have to raise the degree + pp = BPoly.from_derivatives(xi, yi) + assert pp.c.shape == (4, 3) + + ppd = pp.derivative() + for xp in [0., 0.1, 1., 1.1, 1.9, 2., 2.5]: + xp_assert_close(pp(xp), np.asarray(0.0)) + xp_assert_close(ppd(xp), np.asarray(0.0)) + + + def _make_random_mk(self, m, k): + # k derivatives at each breakpoint + rng = np.random.RandomState(1234) + xi = np.asarray([1. * j**2 for j in range(m+1)]) + yi = [rng.random(k) for j in range(m+1)] + return xi, yi + + def test_random_12(self): + m, k = 5, 12 + xi, yi = self._make_random_mk(m, k) + pp = BPoly.from_derivatives(xi, yi) + + for order in range(k//2): + xp_assert_close(pp(xi), [yy[order] for yy in yi]) + pp = pp.derivative() + + def test_order_zero(self): + m, k = 5, 12 + xi, yi = self._make_random_mk(m, k) + assert_raises(ValueError, BPoly.from_derivatives, + **dict(xi=xi, yi=yi, orders=0)) + + def test_orders_too_high(self): + m, k = 5, 12 + xi, yi = self._make_random_mk(m, k) + + BPoly.from_derivatives(xi, yi, orders=2*k-1) # this is still ok + assert_raises(ValueError, BPoly.from_derivatives, # but this is not + **dict(xi=xi, yi=yi, orders=2*k)) + + def test_orders_global(self): + m, k = 5, 12 + xi, yi = self._make_random_mk(m, k) + + # ok, this is confusing. Local polynomials will be of the order 5 + # which means that up to the 2nd derivatives will be used at each point + order = 5 + pp = BPoly.from_derivatives(xi, yi, orders=order) + + for j in range(order//2+1): + xp_assert_close(pp(xi[1:-1] - 1e-12), pp(xi[1:-1] + 1e-12)) + pp = pp.derivative() + assert not np.allclose(pp(xi[1:-1] - 1e-12), pp(xi[1:-1] + 1e-12)) + + # now repeat with `order` being even: on each interval, it uses + # order//2 'derivatives' @ the right-hand endpoint and + # order//2+1 @ 'derivatives' the left-hand endpoint + order = 6 + pp = BPoly.from_derivatives(xi, yi, orders=order) + for j in range(order//2): + xp_assert_close(pp(xi[1:-1] - 1e-12), pp(xi[1:-1] + 1e-12)) + pp = pp.derivative() + assert not np.allclose(pp(xi[1:-1] - 1e-12), pp(xi[1:-1] + 1e-12)) + + def test_orders_local(self): + m, k = 7, 12 + xi, yi = self._make_random_mk(m, k) + + orders = [o + 1 for o in range(m)] + for i, x in enumerate(xi[1:-1]): + pp = BPoly.from_derivatives(xi, yi, orders=orders) + for j in range(orders[i] // 2 + 1): + xp_assert_close(pp(x - 1e-12), pp(x + 1e-12)) + pp = pp.derivative() + assert not np.allclose(pp(x - 1e-12), pp(x + 1e-12)) + + def test_yi_trailing_dims(self): + rng = np.random.RandomState(1234) + m, k = 7, 5 + xi = np.sort(rng.random(m+1)) + yi = rng.random((m+1, k, 6, 7, 8)) + pp = BPoly.from_derivatives(xi, yi) + assert pp.c.shape == (2*k, m, 6, 7, 8) + + def test_gh_5430(self): + # At least one of these raises an error unless gh-5430 is + # fixed. In py2k an int is implemented using a C long, so + # which one fails depends on your system. In py3k there is only + # one arbitrary precision integer type, so both should fail. + orders = np.int32(1) + p = BPoly.from_derivatives([0, 1], [[0], [0]], orders=orders) + assert_almost_equal(p(0), np.asarray(0)) + orders = np.int64(1) + p = BPoly.from_derivatives([0, 1], [[0], [0]], orders=orders) + assert_almost_equal(p(0), np.asarray(0)) + orders = 1 + # This worked before; make sure it still works + p = BPoly.from_derivatives([0, 1], [[0], [0]], orders=orders) + assert_almost_equal(p(0), np.asarray(0)) + orders = 1 + + +class TestNdPPoly: + def test_simple_1d(self): + rng = np.random.RandomState(1234) + + c = rng.rand(4, 5) + x = np.linspace(0, 1, 5+1) + + xi = rng.rand(200) + + p = NdPPoly(c, (x,)) + v1 = p((xi,)) + + v2 = _ppoly_eval_1(c[:,:,None], x, xi).ravel() + xp_assert_close(v1, v2) + + def test_simple_2d(self): + rng = np.random.RandomState(1234) + + c = rng.rand(4, 5, 6, 7) + x = np.linspace(0, 1, 6+1) + y = np.linspace(0, 1, 7+1)**2 + + xi = rng.rand(200) + yi = rng.rand(200) + + v1 = np.empty([len(xi), 1], dtype=c.dtype) + v1.fill(np.nan) + _ppoly.evaluate_nd(c.reshape(4*5, 6*7, 1), + (x, y), + np.array([4, 5], dtype=np.intc), + np.c_[xi, yi], + np.array([0, 0], dtype=np.intc), + 1, + v1) + v1 = v1.ravel() + v2 = _ppoly2d_eval(c, (x, y), xi, yi) + xp_assert_close(v1, v2) + + p = NdPPoly(c, (x, y)) + for nu in (None, (0, 0), (0, 1), (1, 0), (2, 3), (9, 2)): + v1 = p(np.c_[xi, yi], nu=nu) + v2 = _ppoly2d_eval(c, (x, y), xi, yi, nu=nu) + xp_assert_close(v1, v2, err_msg=repr(nu)) + + def test_simple_3d(self): + rng = np.random.RandomState(1234) + + c = rng.rand(4, 5, 6, 7, 8, 9) + x = np.linspace(0, 1, 7+1) + y = np.linspace(0, 1, 8+1)**2 + z = np.linspace(0, 1, 9+1)**3 + + xi = rng.rand(40) + yi = rng.rand(40) + zi = rng.rand(40) + + p = NdPPoly(c, (x, y, z)) + + for nu in (None, (0, 0, 0), (0, 1, 0), (1, 0, 0), (2, 3, 0), + (6, 0, 2)): + v1 = p((xi, yi, zi), nu=nu) + v2 = _ppoly3d_eval(c, (x, y, z), xi, yi, zi, nu=nu) + xp_assert_close(v1, v2, err_msg=repr(nu)) + + def test_simple_4d(self): + rng = np.random.RandomState(1234) + + c = rng.rand(4, 5, 6, 7, 8, 9, 10, 11) + x = np.linspace(0, 1, 8+1) + y = np.linspace(0, 1, 9+1)**2 + z = np.linspace(0, 1, 10+1)**3 + u = np.linspace(0, 1, 11+1)**4 + + xi = rng.rand(20) + yi = rng.rand(20) + zi = rng.rand(20) + ui = rng.rand(20) + + p = NdPPoly(c, (x, y, z, u)) + v1 = p((xi, yi, zi, ui)) + + v2 = _ppoly4d_eval(c, (x, y, z, u), xi, yi, zi, ui) + xp_assert_close(v1, v2) + + def test_deriv_1d(self): + rng = np.random.RandomState(1234) + + c = rng.rand(4, 5) + x = np.linspace(0, 1, 5+1) + + p = NdPPoly(c, (x,)) + + # derivative + dp = p.derivative(nu=[1]) + p1 = PPoly(c, x) + dp1 = p1.derivative() + xp_assert_close(dp.c, dp1.c) + + # antiderivative + dp = p.antiderivative(nu=[2]) + p1 = PPoly(c, x) + dp1 = p1.antiderivative(2) + xp_assert_close(dp.c, dp1.c) + + def test_deriv_3d(self): + rng = np.random.RandomState(1234) + + c = rng.rand(4, 5, 6, 7, 8, 9) + x = np.linspace(0, 1, 7+1) + y = np.linspace(0, 1, 8+1)**2 + z = np.linspace(0, 1, 9+1)**3 + + p = NdPPoly(c, (x, y, z)) + + # differentiate vs x + p1 = PPoly(c.transpose(0, 3, 1, 2, 4, 5), x) + dp = p.derivative(nu=[2]) + dp1 = p1.derivative(2) + xp_assert_close(dp.c, + dp1.c.transpose(0, 2, 3, 1, 4, 5)) + + # antidifferentiate vs y + p1 = PPoly(c.transpose(1, 4, 0, 2, 3, 5), y) + dp = p.antiderivative(nu=[0, 1, 0]) + dp1 = p1.antiderivative(1) + xp_assert_close(dp.c, + dp1.c.transpose(2, 0, 3, 4, 1, 5)) + + # differentiate vs z + p1 = PPoly(c.transpose(2, 5, 0, 1, 3, 4), z) + dp = p.derivative(nu=[0, 0, 3]) + dp1 = p1.derivative(3) + xp_assert_close(dp.c, + dp1.c.transpose(2, 3, 0, 4, 5, 1)) + + def test_deriv_3d_simple(self): + # Integrate to obtain function x y**2 z**4 / (2! 4!) + rng = np.random.RandomState(1234) + + c = np.ones((1, 1, 1, 3, 4, 5)) + x = np.linspace(0, 1, 3+1)**1 + y = np.linspace(0, 1, 4+1)**2 + z = np.linspace(0, 1, 5+1)**3 + + p = NdPPoly(c, (x, y, z)) + ip = p.antiderivative((1, 0, 4)) + ip = ip.antiderivative((0, 2, 0)) + + xi = rng.rand(20) + yi = rng.rand(20) + zi = rng.rand(20) + + xp_assert_close(ip((xi, yi, zi)), + xi * yi**2 * zi**4 / (gamma(3)*gamma(5))) + + def test_integrate_2d(self): + rng = np.random.RandomState(1234) + c = rng.rand(4, 5, 16, 17) + x = np.linspace(0, 1, 16+1)**1 + y = np.linspace(0, 1, 17+1)**2 + + # make continuously differentiable so that nquad() has an + # easier time + c = c.transpose(0, 2, 1, 3) + cx = c.reshape(c.shape[0], c.shape[1], -1).copy() + _ppoly.fix_continuity(cx, x, 2) + c = cx.reshape(c.shape) + c = c.transpose(0, 2, 1, 3) + c = c.transpose(1, 3, 0, 2) + cx = c.reshape(c.shape[0], c.shape[1], -1).copy() + _ppoly.fix_continuity(cx, y, 2) + c = cx.reshape(c.shape) + c = c.transpose(2, 0, 3, 1).copy() + + # Check integration + p = NdPPoly(c, (x, y)) + + for ranges in [[(0, 1), (0, 1)], + [(0, 0.5), (0, 1)], + [(0, 1), (0, 0.5)], + [(0.3, 0.7), (0.6, 0.2)]]: + + ig = p.integrate(ranges) + ig2, err2 = nquad(lambda x, y: p((x, y)), ranges, + opts=[dict(epsrel=1e-5, epsabs=1e-5)]*2) + xp_assert_close(ig, ig2, rtol=1e-5, atol=1e-5, check_0d=False, + err_msg=repr(ranges)) + + def test_integrate_1d(self): + rng = np.random.RandomState(1234) + c = rng.rand(4, 5, 6, 16, 17, 18) + x = np.linspace(0, 1, 16+1)**1 + y = np.linspace(0, 1, 17+1)**2 + z = np.linspace(0, 1, 18+1)**3 + + # Check 1-D integration + p = NdPPoly(c, (x, y, z)) + + u = rng.rand(200) + v = rng.rand(200) + a, b = 0.2, 0.7 + + px = p.integrate_1d(a, b, axis=0) + pax = p.antiderivative((1, 0, 0)) + xp_assert_close(px((u, v)), pax((b, u, v)) - pax((a, u, v))) + + py = p.integrate_1d(a, b, axis=1) + pay = p.antiderivative((0, 1, 0)) + xp_assert_close(py((u, v)), pay((u, b, v)) - pay((u, a, v))) + + pz = p.integrate_1d(a, b, axis=2) + paz = p.antiderivative((0, 0, 1)) + xp_assert_close(pz((u, v)), paz((u, v, b)) - paz((u, v, a))) + + @pytest.mark.thread_unsafe + def test_concurrency(self): + rng = np.random.default_rng(12345) + + c = rng.uniform(size=(4, 5, 6, 7, 8, 9)) + x = np.linspace(0, 1, 7+1) + y = np.linspace(0, 1, 8+1)**2 + z = np.linspace(0, 1, 9+1)**3 + + p = NdPPoly(c, (x, y, z)) + + def worker_fn(_, spl): + xi = rng.uniform(size=40) + yi = rng.uniform(size=40) + zi = rng.uniform(size=40) + spl((xi, yi, zi)) + + _run_concurrent_barrier(10, worker_fn, p) + + +def _ppoly_eval_1(c, x, xps): + """Evaluate piecewise polynomial manually""" + out = np.zeros((len(xps), c.shape[2])) + for i, xp in enumerate(xps): + if xp < 0 or xp > 1: + out[i,:] = np.nan + continue + j = np.searchsorted(x, xp) - 1 + d = xp - x[j] + assert x[j] <= xp < x[j+1] + r = sum(c[k,j] * d**(c.shape[0]-k-1) + for k in range(c.shape[0])) + out[i,:] = r + return out + + +def _ppoly_eval_2(coeffs, breaks, xnew, fill=np.nan): + """Evaluate piecewise polynomial manually (another way)""" + a = breaks[0] + b = breaks[-1] + K = coeffs.shape[0] + + saveshape = np.shape(xnew) + xnew = np.ravel(xnew) + res = np.empty_like(xnew) + mask = (xnew >= a) & (xnew <= b) + res[~mask] = fill + xx = xnew.compress(mask) + indxs = np.searchsorted(breaks, xx)-1 + indxs = indxs.clip(0, len(breaks)) + pp = coeffs + diff = xx - breaks.take(indxs) + V = np.vander(diff, N=K) + values = np.array([np.dot(V[k, :], pp[:, indxs[k]]) for k in range(len(xx))]) + res[mask] = values + res.shape = saveshape + return res + + +def _dpow(x, y, n): + """ + d^n (x**y) / dx^n + """ + if n < 0: + raise ValueError("invalid derivative order") + elif n > y: + return 0 + else: + return poch(y - n + 1, n) * x**(y - n) + + +def _ppoly2d_eval(c, xs, xnew, ynew, nu=None): + """ + Straightforward evaluation of 2-D piecewise polynomial + """ + if nu is None: + nu = (0, 0) + + out = np.empty((len(xnew),), dtype=c.dtype) + + nx, ny = c.shape[:2] + + for jout, (x, y) in enumerate(zip(xnew, ynew)): + if not ((xs[0][0] <= x <= xs[0][-1]) and + (xs[1][0] <= y <= xs[1][-1])): + out[jout] = np.nan + continue + + j1 = np.searchsorted(xs[0], x) - 1 + j2 = np.searchsorted(xs[1], y) - 1 + + s1 = x - xs[0][j1] + s2 = y - xs[1][j2] + + val = 0 + + for k1 in range(c.shape[0]): + for k2 in range(c.shape[1]): + val += (c[nx-k1-1,ny-k2-1,j1,j2] + * _dpow(s1, k1, nu[0]) + * _dpow(s2, k2, nu[1])) + + out[jout] = val + + return out + + +def _ppoly3d_eval(c, xs, xnew, ynew, znew, nu=None): + """ + Straightforward evaluation of 3-D piecewise polynomial + """ + if nu is None: + nu = (0, 0, 0) + + out = np.empty((len(xnew),), dtype=c.dtype) + + nx, ny, nz = c.shape[:3] + + for jout, (x, y, z) in enumerate(zip(xnew, ynew, znew)): + if not ((xs[0][0] <= x <= xs[0][-1]) and + (xs[1][0] <= y <= xs[1][-1]) and + (xs[2][0] <= z <= xs[2][-1])): + out[jout] = np.nan + continue + + j1 = np.searchsorted(xs[0], x) - 1 + j2 = np.searchsorted(xs[1], y) - 1 + j3 = np.searchsorted(xs[2], z) - 1 + + s1 = x - xs[0][j1] + s2 = y - xs[1][j2] + s3 = z - xs[2][j3] + + val = 0 + for k1 in range(c.shape[0]): + for k2 in range(c.shape[1]): + for k3 in range(c.shape[2]): + val += (c[nx-k1-1,ny-k2-1,nz-k3-1,j1,j2,j3] + * _dpow(s1, k1, nu[0]) + * _dpow(s2, k2, nu[1]) + * _dpow(s3, k3, nu[2])) + + out[jout] = val + + return out + + +def _ppoly4d_eval(c, xs, xnew, ynew, znew, unew, nu=None): + """ + Straightforward evaluation of 4-D piecewise polynomial + """ + if nu is None: + nu = (0, 0, 0, 0) + + out = np.empty((len(xnew),), dtype=c.dtype) + + mx, my, mz, mu = c.shape[:4] + + for jout, (x, y, z, u) in enumerate(zip(xnew, ynew, znew, unew)): + if not ((xs[0][0] <= x <= xs[0][-1]) and + (xs[1][0] <= y <= xs[1][-1]) and + (xs[2][0] <= z <= xs[2][-1]) and + (xs[3][0] <= u <= xs[3][-1])): + out[jout] = np.nan + continue + + j1 = np.searchsorted(xs[0], x) - 1 + j2 = np.searchsorted(xs[1], y) - 1 + j3 = np.searchsorted(xs[2], z) - 1 + j4 = np.searchsorted(xs[3], u) - 1 + + s1 = x - xs[0][j1] + s2 = y - xs[1][j2] + s3 = z - xs[2][j3] + s4 = u - xs[3][j4] + + val = 0 + for k1 in range(c.shape[0]): + for k2 in range(c.shape[1]): + for k3 in range(c.shape[2]): + for k4 in range(c.shape[3]): + val += (c[mx-k1-1,my-k2-1,mz-k3-1,mu-k4-1,j1,j2,j3,j4] + * _dpow(s1, k1, nu[0]) + * _dpow(s2, k2, nu[1]) + * _dpow(s3, k3, nu[2]) + * _dpow(s4, k4, nu[3])) + + out[jout] = val + + return out diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_ndgriddata.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_ndgriddata.py new file mode 100644 index 0000000000000000000000000000000000000000..047a940b3efcb24ec85a94a87dd1050baa01f165 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_ndgriddata.py @@ -0,0 +1,308 @@ +import numpy as np +from scipy._lib._array_api import ( + xp_assert_equal, xp_assert_close +) +import pytest +from pytest import raises as assert_raises + +from scipy.interpolate import (griddata, NearestNDInterpolator, + LinearNDInterpolator, + CloughTocher2DInterpolator) +from scipy._lib._testutils import _run_concurrent_barrier + + +parametrize_interpolators = pytest.mark.parametrize( + "interpolator", [NearestNDInterpolator, LinearNDInterpolator, + CloughTocher2DInterpolator] +) +parametrize_methods = pytest.mark.parametrize( + 'method', + ('nearest', 'linear', 'cubic'), +) +parametrize_rescale = pytest.mark.parametrize( + 'rescale', + (True, False), +) + + +class TestGriddata: + def test_fill_value(self): + x = [(0,0), (0,1), (1,0)] + y = [1, 2, 3] + + yi = griddata(x, y, [(1,1), (1,2), (0,0)], fill_value=-1) + xp_assert_equal(yi, [-1., -1, 1]) + + yi = griddata(x, y, [(1,1), (1,2), (0,0)]) + xp_assert_equal(yi, [np.nan, np.nan, 1]) + + @parametrize_methods + @parametrize_rescale + def test_alternative_call(self, method, rescale): + x = np.array([(0,0), (-0.5,-0.5), (-0.5,0.5), (0.5, 0.5), (0.25, 0.3)], + dtype=np.float64) + y = (np.arange(x.shape[0], dtype=np.float64)[:,None] + + np.array([0,1])[None,:]) + + msg = repr((method, rescale)) + yi = griddata((x[:,0], x[:,1]), y, (x[:,0], x[:,1]), method=method, + rescale=rescale) + xp_assert_close(y, yi, atol=1e-14, err_msg=msg) + + @parametrize_methods + @parametrize_rescale + def test_multivalue_2d(self, method, rescale): + x = np.array([(0,0), (-0.5,-0.5), (-0.5,0.5), (0.5, 0.5), (0.25, 0.3)], + dtype=np.float64) + y = (np.arange(x.shape[0], dtype=np.float64)[:,None] + + np.array([0,1])[None,:]) + + msg = repr((method, rescale)) + yi = griddata(x, y, x, method=method, rescale=rescale) + xp_assert_close(y, yi, atol=1e-14, err_msg=msg) + + @parametrize_methods + @parametrize_rescale + def test_multipoint_2d(self, method, rescale): + x = np.array([(0,0), (-0.5,-0.5), (-0.5,0.5), (0.5, 0.5), (0.25, 0.3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + + xi = x[:,None,:] + np.array([0,0,0])[None,:,None] + + msg = repr((method, rescale)) + yi = griddata(x, y, xi, method=method, rescale=rescale) + + assert yi.shape == (5, 3), msg + xp_assert_close(yi, np.tile(y[:,None], (1, 3)), + atol=1e-14, err_msg=msg) + + @parametrize_methods + @parametrize_rescale + def test_complex_2d(self, method, rescale): + x = np.array([(0,0), (-0.5,-0.5), (-0.5,0.5), (0.5, 0.5), (0.25, 0.3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + y = y - 2j*y[::-1] + + xi = x[:,None,:] + np.array([0,0,0])[None,:,None] + + msg = repr((method, rescale)) + yi = griddata(x, y, xi, method=method, rescale=rescale) + + assert yi.shape == (5, 3) + xp_assert_close(yi, np.tile(y[:,None], (1, 3)), + atol=1e-14, err_msg=msg) + + @parametrize_methods + def test_1d(self, method): + x = np.array([1, 2.5, 3, 4.5, 5, 6]) + y = np.array([1, 2, 0, 3.9, 2, 1]) + + xp_assert_close(griddata(x, y, x, method=method), y, + err_msg=method, atol=1e-14) + xp_assert_close(griddata(x.reshape(6, 1), y, x, method=method), y, + err_msg=method, atol=1e-14) + xp_assert_close(griddata((x,), y, (x,), method=method), y, + err_msg=method, atol=1e-14) + + def test_1d_borders(self): + # Test for nearest neighbor case with xi outside + # the range of the values. + x = np.array([1, 2.5, 3, 4.5, 5, 6]) + y = np.array([1, 2, 0, 3.9, 2, 1]) + xi = np.array([0.9, 6.5]) + yi_should = np.array([1.0, 1.0]) + + method = 'nearest' + xp_assert_close(griddata(x, y, xi, + method=method), yi_should, + err_msg=method, + atol=1e-14) + xp_assert_close(griddata(x.reshape(6, 1), y, xi, + method=method), yi_should, + err_msg=method, + atol=1e-14) + xp_assert_close(griddata((x, ), y, (xi, ), + method=method), yi_should, + err_msg=method, + atol=1e-14) + + @parametrize_methods + def test_1d_unsorted(self, method): + x = np.array([2.5, 1, 4.5, 5, 6, 3]) + y = np.array([1, 2, 0, 3.9, 2, 1]) + + xp_assert_close(griddata(x, y, x, method=method), y, + err_msg=method, atol=1e-10) + xp_assert_close(griddata(x.reshape(6, 1), y, x, method=method), y, + err_msg=method, atol=1e-10) + xp_assert_close(griddata((x,), y, (x,), method=method), y, + err_msg=method, atol=1e-10) + + @parametrize_methods + def test_square_rescale_manual(self, method): + points = np.array([(0,0), (0,100), (10,100), (10,0), (1, 5)], dtype=np.float64) + points_rescaled = np.array([(0,0), (0,1), (1,1), (1,0), (0.1, 0.05)], + dtype=np.float64) + values = np.array([1., 2., -3., 5., 9.], dtype=np.float64) + + xx, yy = np.broadcast_arrays(np.linspace(0, 10, 14)[:,None], + np.linspace(0, 100, 14)[None,:]) + xx = xx.ravel() + yy = yy.ravel() + xi = np.array([xx, yy]).T.copy() + + msg = method + zi = griddata(points_rescaled, values, xi/np.array([10, 100.]), + method=method) + zi_rescaled = griddata(points, values, xi, method=method, + rescale=True) + xp_assert_close(zi, zi_rescaled, err_msg=msg, + atol=1e-12) + + @parametrize_methods + def test_xi_1d(self, method): + # Check that 1-D xi is interpreted as a coordinate + x = np.array([(0,0), (-0.5,-0.5), (-0.5,0.5), (0.5, 0.5), (0.25, 0.3)], + dtype=np.float64) + y = np.arange(x.shape[0], dtype=np.float64) + y = y - 2j*y[::-1] + + xi = np.array([0.5, 0.5]) + + p1 = griddata(x, y, xi, method=method) + p2 = griddata(x, y, xi[None,:], method=method) + xp_assert_close(p1, p2, err_msg=method) + + xi1 = np.array([0.5]) + xi3 = np.array([0.5, 0.5, 0.5]) + assert_raises(ValueError, griddata, x, y, xi1, + method=method) + assert_raises(ValueError, griddata, x, y, xi3, + method=method) + + +class TestNearestNDInterpolator: + def test_nearest_options(self): + # smoke test that NearestNDInterpolator accept cKDTree options + npts, nd = 4, 3 + x = np.arange(npts*nd).reshape((npts, nd)) + y = np.arange(npts) + nndi = NearestNDInterpolator(x, y) + + opts = {'balanced_tree': False, 'compact_nodes': False} + nndi_o = NearestNDInterpolator(x, y, tree_options=opts) + xp_assert_close(nndi(x), nndi_o(x), atol=1e-14) + + def test_nearest_list_argument(self): + nd = np.array([[0, 0, 0, 0, 1, 0, 1], + [0, 0, 0, 0, 0, 1, 1], + [0, 0, 0, 0, 1, 1, 2]]) + d = nd[:, 3:] + + # z is np.array + NI = NearestNDInterpolator((d[0], d[1]), d[2]) + xp_assert_equal(NI([0.1, 0.9], [0.1, 0.9]), [0.0, 2.0]) + + # z is list + NI = NearestNDInterpolator((d[0], d[1]), list(d[2])) + xp_assert_equal(NI([0.1, 0.9], [0.1, 0.9]), [0.0, 2.0]) + + def test_nearest_query_options(self): + nd = np.array([[0, 0.5, 0, 1], + [0, 0, 0.5, 1], + [0, 1, 1, 2]]) + delta = 0.1 + query_points = [0 + delta, 1 + delta], [0 + delta, 1 + delta] + + # case 1 - query max_dist is smaller than + # the query points' nearest distance to nd. + NI = NearestNDInterpolator((nd[0], nd[1]), nd[2]) + distance_upper_bound = np.sqrt(delta ** 2 + delta ** 2) - 1e-7 + xp_assert_equal(NI(query_points, distance_upper_bound=distance_upper_bound), + [np.nan, np.nan]) + + # case 2 - query p is inf, will return [0, 2] + distance_upper_bound = np.sqrt(delta ** 2 + delta ** 2) - 1e-7 + p = np.inf + xp_assert_equal( + NI(query_points, distance_upper_bound=distance_upper_bound, p=p), + [0.0, 2.0] + ) + + # case 3 - query max_dist is larger, so should return non np.nan + distance_upper_bound = np.sqrt(delta ** 2 + delta ** 2) + 1e-7 + xp_assert_equal( + NI(query_points, distance_upper_bound=distance_upper_bound), + [0.0, 2.0] + ) + + def test_nearest_query_valid_inputs(self): + nd = np.array([[0, 1, 0, 1], + [0, 0, 1, 1], + [0, 1, 1, 2]]) + NI = NearestNDInterpolator((nd[0], nd[1]), nd[2]) + with assert_raises(TypeError): + NI([0.5, 0.5], query_options="not a dictionary") + + @pytest.mark.thread_unsafe + def test_concurrency(self): + npts, nd = 50, 3 + x = np.arange(npts * nd).reshape((npts, nd)) + y = np.arange(npts) + nndi = NearestNDInterpolator(x, y) + + def worker_fn(_, spl): + spl(x) + + _run_concurrent_barrier(10, worker_fn, nndi) + + +class TestNDInterpolators: + @parametrize_interpolators + def test_broadcastable_input(self, interpolator): + # input data + rng = np.random.RandomState(0) + x = rng.random(10) + y = rng.random(10) + z = np.hypot(x, y) + + # x-y grid for interpolation + X = np.linspace(min(x), max(x)) + Y = np.linspace(min(y), max(y)) + X, Y = np.meshgrid(X, Y) + XY = np.vstack((X.ravel(), Y.ravel())).T + interp = interpolator(list(zip(x, y)), z) + # single array input + interp_points0 = interp(XY) + # tuple input + interp_points1 = interp((X, Y)) + interp_points2 = interp((X, 0.0)) + # broadcastable input + interp_points3 = interp(X, Y) + interp_points4 = interp(X, 0.0) + + assert (interp_points0.size == + interp_points1.size == + interp_points2.size == + interp_points3.size == + interp_points4.size) + + @parametrize_interpolators + def test_read_only(self, interpolator): + # input data + rng = np.random.RandomState(0) + xy = rng.random((10, 2)) + x, y = xy[:, 0], xy[:, 1] + z = np.hypot(x, y) + + # interpolation points + XY = rng.random((50, 2)) + + xy.setflags(write=False) + z.setflags(write=False) + XY.setflags(write=False) + + interp = interpolator(xy, z) + interp(XY) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_pade.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_pade.py new file mode 100644 index 0000000000000000000000000000000000000000..119b7d1c5667368b284fbf6458174ea14e71957a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_pade.py @@ -0,0 +1,107 @@ +import numpy as np +from scipy.interpolate import pade +from scipy._lib._array_api import ( + xp_assert_equal, assert_array_almost_equal +) + +def test_pade_trivial(): + nump, denomp = pade([1.0], 0) + xp_assert_equal(nump.c, np.asarray([1.0])) + xp_assert_equal(denomp.c, np.asarray([1.0])) + + nump, denomp = pade([1.0], 0, 0) + xp_assert_equal(nump.c, np.asarray([1.0])) + xp_assert_equal(denomp.c, np.asarray([1.0])) + + +def test_pade_4term_exp(): + # First four Taylor coefficients of exp(x). + # Unlike poly1d, the first array element is the zero-order term. + an = [1.0, 1.0, 0.5, 1.0/6] + + nump, denomp = pade(an, 0) + assert_array_almost_equal(nump.c, [1.0/6, 0.5, 1.0, 1.0]) + assert_array_almost_equal(denomp.c, [1.0]) + + nump, denomp = pade(an, 1) + assert_array_almost_equal(nump.c, [1.0/6, 2.0/3, 1.0]) + assert_array_almost_equal(denomp.c, [-1.0/3, 1.0]) + + nump, denomp = pade(an, 2) + assert_array_almost_equal(nump.c, [1.0/3, 1.0]) + assert_array_almost_equal(denomp.c, [1.0/6, -2.0/3, 1.0]) + + nump, denomp = pade(an, 3) + assert_array_almost_equal(nump.c, [1.0]) + assert_array_almost_equal(denomp.c, [-1.0/6, 0.5, -1.0, 1.0]) + + # Testing inclusion of optional parameter + nump, denomp = pade(an, 0, 3) + assert_array_almost_equal(nump.c, [1.0/6, 0.5, 1.0, 1.0]) + assert_array_almost_equal(denomp.c, [1.0]) + + nump, denomp = pade(an, 1, 2) + assert_array_almost_equal(nump.c, [1.0/6, 2.0/3, 1.0]) + assert_array_almost_equal(denomp.c, [-1.0/3, 1.0]) + + nump, denomp = pade(an, 2, 1) + assert_array_almost_equal(nump.c, [1.0/3, 1.0]) + assert_array_almost_equal(denomp.c, [1.0/6, -2.0/3, 1.0]) + + nump, denomp = pade(an, 3, 0) + assert_array_almost_equal(nump.c, [1.0]) + assert_array_almost_equal(denomp.c, [-1.0/6, 0.5, -1.0, 1.0]) + + # Testing reducing array. + nump, denomp = pade(an, 0, 2) + assert_array_almost_equal(nump.c, [0.5, 1.0, 1.0]) + assert_array_almost_equal(denomp.c, [1.0]) + + nump, denomp = pade(an, 1, 1) + assert_array_almost_equal(nump.c, [1.0/2, 1.0]) + assert_array_almost_equal(denomp.c, [-1.0/2, 1.0]) + + nump, denomp = pade(an, 2, 0) + assert_array_almost_equal(nump.c, [1.0]) + assert_array_almost_equal(denomp.c, [1.0/2, -1.0, 1.0]) + + +def test_pade_ints(): + # Simple test sequences (one of ints, one of floats). + an_int = [1, 2, 3, 4] + an_flt = [1.0, 2.0, 3.0, 4.0] + + # Make sure integer arrays give the same result as float arrays with same values. + for i in range(0, len(an_int)): + for j in range(0, len(an_int) - i): + + # Create float and int pade approximation for given order. + nump_int, denomp_int = pade(an_int, i, j) + nump_flt, denomp_flt = pade(an_flt, i, j) + + # Check that they are the same. + xp_assert_equal(nump_int.c, nump_flt.c) + xp_assert_equal(denomp_int.c, denomp_flt.c) + + +def test_pade_complex(): + # Test sequence with known solutions - see page 6 of 10.1109/PESGM.2012.6344759. + # Variable x is parameter - these tests will work with any complex number. + x = 0.2 + 0.6j + an = [1.0, x, -x*x.conjugate(), x.conjugate()*(x**2) + x*(x.conjugate()**2), + -(x**3)*x.conjugate() - 3*(x*x.conjugate())**2 - x*(x.conjugate()**3)] + + nump, denomp = pade(an, 1, 1) + assert_array_almost_equal(nump.c, [x + x.conjugate(), 1.0]) + assert_array_almost_equal(denomp.c, [x.conjugate(), 1.0]) + + nump, denomp = pade(an, 1, 2) + assert_array_almost_equal(nump.c, [x**2, 2*x + x.conjugate(), 1.0]) + assert_array_almost_equal(denomp.c, [x + x.conjugate(), 1.0]) + + nump, denomp = pade(an, 2, 2) + assert_array_almost_equal( + nump.c, + [x**2 + x*x.conjugate() + x.conjugate()**2, 2*(x + x.conjugate()), 1.0] + ) + assert_array_almost_equal(denomp.c, [x.conjugate()**2, x + 2*x.conjugate(), 1.0]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_polyint.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_polyint.py new file mode 100644 index 0000000000000000000000000000000000000000..e3e6cb7894ea344289c12b522b45c5e0f22748e6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_polyint.py @@ -0,0 +1,972 @@ +import warnings +import io +import numpy as np + +from scipy._lib._array_api import ( + xp_assert_equal, xp_assert_close, assert_array_almost_equal, assert_almost_equal +) +from pytest import raises as assert_raises +import pytest + +from scipy.interpolate import ( + KroghInterpolator, krogh_interpolate, + BarycentricInterpolator, barycentric_interpolate, + approximate_taylor_polynomial, CubicHermiteSpline, pchip, + PchipInterpolator, pchip_interpolate, Akima1DInterpolator, CubicSpline, + make_interp_spline) +from scipy._lib._testutils import _run_concurrent_barrier + + +def check_shape(interpolator_cls, x_shape, y_shape, deriv_shape=None, axis=0, + extra_args=None): + if extra_args is None: + extra_args = {} + rng = np.random.RandomState(1234) + + x = [-1, 0, 1, 2, 3, 4] + s = list(range(1, len(y_shape)+1)) + s.insert(axis % (len(y_shape)+1), 0) + y = rng.rand(*((6,) + y_shape)).transpose(s) + + xi = np.zeros(x_shape) + if interpolator_cls is CubicHermiteSpline: + dydx = rng.rand(*((6,) + y_shape)).transpose(s) + yi = interpolator_cls(x, y, dydx, axis=axis, **extra_args)(xi) + else: + yi = interpolator_cls(x, y, axis=axis, **extra_args)(xi) + + target_shape = ((deriv_shape or ()) + y.shape[:axis] + + x_shape + y.shape[axis:][1:]) + assert yi.shape == target_shape + + # check it works also with lists + if x_shape and y.size > 0: + if interpolator_cls is CubicHermiteSpline: + interpolator_cls(list(x), list(y), list(dydx), axis=axis, + **extra_args)(list(xi)) + else: + interpolator_cls(list(x), list(y), axis=axis, + **extra_args)(list(xi)) + + # check also values + if xi.size > 0 and deriv_shape is None: + bs_shape = y.shape[:axis] + (1,)*len(x_shape) + y.shape[axis:][1:] + yv = y[((slice(None,),)*(axis % y.ndim)) + (1,)] + yv = yv.reshape(bs_shape) + + yi, y = np.broadcast_arrays(yi, yv) + xp_assert_close(yi, y) + + +SHAPES = [(), (0,), (1,), (6, 2, 5)] + + +def test_shapes(): + + def spl_interp(x, y, axis): + return make_interp_spline(x, y, axis=axis) + + for ip in [KroghInterpolator, BarycentricInterpolator, CubicHermiteSpline, + pchip, Akima1DInterpolator, CubicSpline, spl_interp]: + for s1 in SHAPES: + for s2 in SHAPES: + for axis in range(-len(s2), len(s2)): + if ip != CubicSpline: + check_shape(ip, s1, s2, None, axis) + else: + for bc in ['natural', 'clamped']: + extra = {'bc_type': bc} + check_shape(ip, s1, s2, None, axis, extra) + +def test_derivs_shapes(): + for ip in [KroghInterpolator, BarycentricInterpolator]: + def interpolator_derivs(x, y, axis=0): + return ip(x, y, axis).derivatives + + for s1 in SHAPES: + for s2 in SHAPES: + for axis in range(-len(s2), len(s2)): + check_shape(interpolator_derivs, s1, s2, (6,), axis) + + +def test_deriv_shapes(): + def krogh_deriv(x, y, axis=0): + return KroghInterpolator(x, y, axis).derivative + + def bary_deriv(x, y, axis=0): + return BarycentricInterpolator(x, y, axis).derivative + + def pchip_deriv(x, y, axis=0): + return pchip(x, y, axis).derivative() + + def pchip_deriv2(x, y, axis=0): + return pchip(x, y, axis).derivative(2) + + def pchip_antideriv(x, y, axis=0): + return pchip(x, y, axis).antiderivative() + + def pchip_antideriv2(x, y, axis=0): + return pchip(x, y, axis).antiderivative(2) + + def pchip_deriv_inplace(x, y, axis=0): + class P(PchipInterpolator): + def __call__(self, x): + return PchipInterpolator.__call__(self, x, 1) + pass + return P(x, y, axis) + + def akima_deriv(x, y, axis=0): + return Akima1DInterpolator(x, y, axis).derivative() + + def akima_antideriv(x, y, axis=0): + return Akima1DInterpolator(x, y, axis).antiderivative() + + def cspline_deriv(x, y, axis=0): + return CubicSpline(x, y, axis).derivative() + + def cspline_antideriv(x, y, axis=0): + return CubicSpline(x, y, axis).antiderivative() + + def bspl_deriv(x, y, axis=0): + return make_interp_spline(x, y, axis=axis).derivative() + + def bspl_antideriv(x, y, axis=0): + return make_interp_spline(x, y, axis=axis).antiderivative() + + for ip in [krogh_deriv, bary_deriv, pchip_deriv, pchip_deriv2, pchip_deriv_inplace, + pchip_antideriv, pchip_antideriv2, akima_deriv, akima_antideriv, + cspline_deriv, cspline_antideriv, bspl_deriv, bspl_antideriv]: + for s1 in SHAPES: + for s2 in SHAPES: + for axis in range(-len(s2), len(s2)): + check_shape(ip, s1, s2, (), axis) + + +def test_complex(): + x = [1, 2, 3, 4] + y = [1, 2, 1j, 3] + + for ip in [KroghInterpolator, BarycentricInterpolator, CubicSpline]: + p = ip(x, y) + xp_assert_close(p(x), np.asarray(y)) + + dydx = [0, -1j, 2, 3j] + p = CubicHermiteSpline(x, y, dydx) + xp_assert_close(p(x), np.asarray(y)) + xp_assert_close(p(x, 1), np.asarray(dydx)) + + +class TestKrogh: + def setup_method(self): + self.true_poly = np.polynomial.Polynomial([-4, 5, 1, 3, -2]) + self.test_xs = np.linspace(-1,1,100) + self.xs = np.linspace(-1,1,5) + self.ys = self.true_poly(self.xs) + + def test_lagrange(self): + P = KroghInterpolator(self.xs,self.ys) + assert_almost_equal(self.true_poly(self.test_xs),P(self.test_xs)) + + def test_scalar(self): + P = KroghInterpolator(self.xs,self.ys) + assert_almost_equal(self.true_poly(7), P(7), check_0d=False) + assert_almost_equal(self.true_poly(np.array(7)), P(np.array(7)), check_0d=False) + + def test_derivatives(self): + P = KroghInterpolator(self.xs,self.ys) + D = P.derivatives(self.test_xs) + for i in range(D.shape[0]): + assert_almost_equal(self.true_poly.deriv(i)(self.test_xs), + D[i]) + + def test_low_derivatives(self): + P = KroghInterpolator(self.xs,self.ys) + D = P.derivatives(self.test_xs,len(self.xs)+2) + for i in range(D.shape[0]): + assert_almost_equal(self.true_poly.deriv(i)(self.test_xs), + D[i]) + + def test_derivative(self): + P = KroghInterpolator(self.xs,self.ys) + m = 10 + r = P.derivatives(self.test_xs,m) + for i in range(m): + assert_almost_equal(P.derivative(self.test_xs,i),r[i]) + + def test_high_derivative(self): + P = KroghInterpolator(self.xs,self.ys) + for i in range(len(self.xs), 2*len(self.xs)): + assert_almost_equal(P.derivative(self.test_xs,i), + np.zeros(len(self.test_xs))) + + def test_ndim_derivatives(self): + poly1 = self.true_poly + poly2 = np.polynomial.Polynomial([-2, 5, 3, -1]) + poly3 = np.polynomial.Polynomial([12, -3, 4, -5, 6]) + ys = np.stack((poly1(self.xs), poly2(self.xs), poly3(self.xs)), axis=-1) + + P = KroghInterpolator(self.xs, ys, axis=0) + D = P.derivatives(self.test_xs) + for i in range(D.shape[0]): + xp_assert_close(D[i], + np.stack((poly1.deriv(i)(self.test_xs), + poly2.deriv(i)(self.test_xs), + poly3.deriv(i)(self.test_xs)), + axis=-1)) + + def test_ndim_derivative(self): + poly1 = self.true_poly + poly2 = np.polynomial.Polynomial([-2, 5, 3, -1]) + poly3 = np.polynomial.Polynomial([12, -3, 4, -5, 6]) + ys = np.stack((poly1(self.xs), poly2(self.xs), poly3(self.xs)), axis=-1) + + P = KroghInterpolator(self.xs, ys, axis=0) + for i in range(P.n): + xp_assert_close(P.derivative(self.test_xs, i), + np.stack((poly1.deriv(i)(self.test_xs), + poly2.deriv(i)(self.test_xs), + poly3.deriv(i)(self.test_xs)), + axis=-1)) + + def test_hermite(self): + P = KroghInterpolator(self.xs,self.ys) + assert_almost_equal(self.true_poly(self.test_xs),P(self.test_xs)) + + def test_vector(self): + xs = [0, 1, 2] + ys = np.array([[0,1],[1,0],[2,1]]) + P = KroghInterpolator(xs,ys) + Pi = [KroghInterpolator(xs,ys[:,i]) for i in range(ys.shape[1])] + test_xs = np.linspace(-1,3,100) + assert_almost_equal(P(test_xs), + np.asarray([p(test_xs) for p in Pi]).T) + assert_almost_equal(P.derivatives(test_xs), + np.transpose(np.asarray([p.derivatives(test_xs) for p in Pi]), + (1,2,0))) + + def test_empty(self): + P = KroghInterpolator(self.xs,self.ys) + xp_assert_equal(P([]), np.asarray([])) + + def test_shapes_scalarvalue(self): + P = KroghInterpolator(self.xs,self.ys) + assert np.shape(P(0)) == () + assert np.shape(P(np.array(0))) == () + assert np.shape(P([0])) == (1,) + assert np.shape(P([0,1])) == (2,) + + def test_shapes_scalarvalue_derivative(self): + P = KroghInterpolator(self.xs,self.ys) + n = P.n + assert np.shape(P.derivatives(0)) == (n,) + assert np.shape(P.derivatives(np.array(0))) == (n,) + assert np.shape(P.derivatives([0])) == (n, 1) + assert np.shape(P.derivatives([0, 1])) == (n, 2) + + def test_shapes_vectorvalue(self): + P = KroghInterpolator(self.xs,np.outer(self.ys,np.arange(3))) + assert np.shape(P(0)) == (3,) + assert np.shape(P([0])) == (1, 3) + assert np.shape(P([0, 1])) == (2, 3) + + def test_shapes_1d_vectorvalue(self): + P = KroghInterpolator(self.xs,np.outer(self.ys,[1])) + assert np.shape(P(0)) == (1,) + assert np.shape(P([0])) == (1, 1) + assert np.shape(P([0,1])) == (2, 1) + + def test_shapes_vectorvalue_derivative(self): + P = KroghInterpolator(self.xs,np.outer(self.ys,np.arange(3))) + n = P.n + assert np.shape(P.derivatives(0)) == (n, 3) + assert np.shape(P.derivatives([0])) == (n, 1, 3) + assert np.shape(P.derivatives([0,1])) == (n, 2, 3) + + def test_wrapper(self): + P = KroghInterpolator(self.xs, self.ys) + ki = krogh_interpolate + assert_almost_equal(P(self.test_xs), ki(self.xs, self.ys, self.test_xs)) + assert_almost_equal(P.derivative(self.test_xs, 2), + ki(self.xs, self.ys, self.test_xs, der=2)) + assert_almost_equal(P.derivatives(self.test_xs, 2), + ki(self.xs, self.ys, self.test_xs, der=[0, 1])) + + def test_int_inputs(self): + # Check input args are cast correctly to floats, gh-3669 + x = [0, 234, 468, 702, 936, 1170, 1404, 2340, 3744, 6084, 8424, + 13104, 60000] + offset_cdf = np.array([-0.95, -0.86114777, -0.8147762, -0.64072425, + -0.48002351, -0.34925329, -0.26503107, + -0.13148093, -0.12988833, -0.12979296, + -0.12973574, -0.08582937, 0.05]) + f = KroghInterpolator(x, offset_cdf) + + xp_assert_close(abs((f(x) - offset_cdf) / f.derivative(x, 1)), + np.zeros_like(offset_cdf), atol=1e-10) + + def test_derivatives_complex(self): + # regression test for gh-7381: krogh.derivatives(0) fails complex y + x, y = np.array([-1, -1, 0, 1, 1]), np.array([1, 1.0j, 0, -1, 1.0j]) + func = KroghInterpolator(x, y) + cmplx = func.derivatives(0) + + cmplx2 = (KroghInterpolator(x, y.real).derivatives(0) + + 1j*KroghInterpolator(x, y.imag).derivatives(0)) + xp_assert_close(cmplx, cmplx2, atol=1e-15) + + @pytest.mark.thread_unsafe + def test_high_degree_warning(self): + with pytest.warns(UserWarning, match="40 degrees provided,"): + KroghInterpolator(np.arange(40), np.ones(40)) + + @pytest.mark.thread_unsafe + def test_concurrency(self): + P = KroghInterpolator(self.xs, self.ys) + + def worker_fn(_, interp): + interp(self.xs) + + _run_concurrent_barrier(10, worker_fn, P) + + +class TestTaylor: + def test_exponential(self): + degree = 5 + p = approximate_taylor_polynomial(np.exp, 0, degree, 1, 15) + for i in range(degree+1): + assert_almost_equal(p(0),1) + p = p.deriv() + assert_almost_equal(p(0),0) + + +class TestBarycentric: + def setup_method(self): + self.true_poly = np.polynomial.Polynomial([-4, 5, 1, 3, -2]) + self.test_xs = np.linspace(-1, 1, 100) + self.xs = np.linspace(-1, 1, 5) + self.ys = self.true_poly(self.xs) + + def test_lagrange(self): + # Ensure backwards compatible post SPEC7 + P = BarycentricInterpolator(self.xs, self.ys, random_state=1) + xp_assert_close(P(self.test_xs), self.true_poly(self.test_xs)) + + def test_scalar(self): + P = BarycentricInterpolator(self.xs, self.ys, rng=1) + xp_assert_close(P(7), self.true_poly(7), check_0d=False) + xp_assert_close(P(np.array(7)), self.true_poly(np.array(7)), check_0d=False) + + def test_derivatives(self): + P = BarycentricInterpolator(self.xs, self.ys) + D = P.derivatives(self.test_xs) + for i in range(D.shape[0]): + xp_assert_close(self.true_poly.deriv(i)(self.test_xs), D[i]) + + def test_low_derivatives(self): + P = BarycentricInterpolator(self.xs, self.ys) + D = P.derivatives(self.test_xs, len(self.xs)+2) + for i in range(D.shape[0]): + xp_assert_close(self.true_poly.deriv(i)(self.test_xs), + D[i], + atol=1e-12) + + def test_derivative(self): + P = BarycentricInterpolator(self.xs, self.ys) + m = 10 + r = P.derivatives(self.test_xs, m) + for i in range(m): + xp_assert_close(P.derivative(self.test_xs, i), r[i]) + + def test_high_derivative(self): + P = BarycentricInterpolator(self.xs, self.ys) + for i in range(len(self.xs), 5*len(self.xs)): + xp_assert_close(P.derivative(self.test_xs, i), + np.zeros(len(self.test_xs))) + + def test_ndim_derivatives(self): + poly1 = self.true_poly + poly2 = np.polynomial.Polynomial([-2, 5, 3, -1]) + poly3 = np.polynomial.Polynomial([12, -3, 4, -5, 6]) + ys = np.stack((poly1(self.xs), poly2(self.xs), poly3(self.xs)), axis=-1) + + P = BarycentricInterpolator(self.xs, ys, axis=0) + D = P.derivatives(self.test_xs) + for i in range(D.shape[0]): + xp_assert_close(D[i], + np.stack((poly1.deriv(i)(self.test_xs), + poly2.deriv(i)(self.test_xs), + poly3.deriv(i)(self.test_xs)), + axis=-1), + atol=1e-12) + + def test_ndim_derivative(self): + poly1 = self.true_poly + poly2 = np.polynomial.Polynomial([-2, 5, 3, -1]) + poly3 = np.polynomial.Polynomial([12, -3, 4, -5, 6]) + ys = np.stack((poly1(self.xs), poly2(self.xs), poly3(self.xs)), axis=-1) + + P = BarycentricInterpolator(self.xs, ys, axis=0) + for i in range(P.n): + xp_assert_close(P.derivative(self.test_xs, i), + np.stack((poly1.deriv(i)(self.test_xs), + poly2.deriv(i)(self.test_xs), + poly3.deriv(i)(self.test_xs)), + axis=-1), + atol=1e-12) + + def test_delayed(self): + P = BarycentricInterpolator(self.xs) + P.set_yi(self.ys) + assert_almost_equal(self.true_poly(self.test_xs), P(self.test_xs)) + + def test_append(self): + P = BarycentricInterpolator(self.xs[:3], self.ys[:3]) + P.add_xi(self.xs[3:], self.ys[3:]) + assert_almost_equal(self.true_poly(self.test_xs), P(self.test_xs)) + + def test_vector(self): + xs = [0, 1, 2] + ys = np.array([[0, 1], [1, 0], [2, 1]]) + BI = BarycentricInterpolator + P = BI(xs, ys) + Pi = [BI(xs, ys[:, i]) for i in range(ys.shape[1])] + test_xs = np.linspace(-1, 3, 100) + assert_almost_equal(P(test_xs), + np.asarray([p(test_xs) for p in Pi]).T) + + def test_shapes_scalarvalue(self): + P = BarycentricInterpolator(self.xs, self.ys) + assert np.shape(P(0)) == () + assert np.shape(P(np.array(0))) == () + assert np.shape(P([0])) == (1,) + assert np.shape(P([0, 1])) == (2,) + + def test_shapes_scalarvalue_derivative(self): + P = BarycentricInterpolator(self.xs,self.ys) + n = P.n + assert np.shape(P.derivatives(0)) == (n,) + assert np.shape(P.derivatives(np.array(0))) == (n,) + assert np.shape(P.derivatives([0])) == (n,1) + assert np.shape(P.derivatives([0,1])) == (n,2) + + def test_shapes_vectorvalue(self): + P = BarycentricInterpolator(self.xs, np.outer(self.ys, np.arange(3))) + assert np.shape(P(0)) == (3,) + assert np.shape(P([0])) == (1, 3) + assert np.shape(P([0, 1])) == (2, 3) + + def test_shapes_1d_vectorvalue(self): + P = BarycentricInterpolator(self.xs, np.outer(self.ys, [1])) + assert np.shape(P(0)) == (1,) + assert np.shape(P([0])) == (1, 1) + assert np.shape(P([0, 1])) == (2, 1) + + def test_shapes_vectorvalue_derivative(self): + P = BarycentricInterpolator(self.xs,np.outer(self.ys,np.arange(3))) + n = P.n + assert np.shape(P.derivatives(0)) == (n, 3) + assert np.shape(P.derivatives([0])) == (n, 1, 3) + assert np.shape(P.derivatives([0, 1])) == (n, 2, 3) + + def test_wrapper(self): + P = BarycentricInterpolator(self.xs, self.ys, rng=1) + bi = barycentric_interpolate + xp_assert_close(P(self.test_xs), bi(self.xs, self.ys, self.test_xs, rng=1)) + xp_assert_close(P.derivative(self.test_xs, 2), + bi(self.xs, self.ys, self.test_xs, der=2, rng=1)) + xp_assert_close(P.derivatives(self.test_xs, 2), + bi(self.xs, self.ys, self.test_xs, der=[0, 1], rng=1)) + + def test_int_input(self): + x = 1000 * np.arange(1, 11) # np.prod(x[-1] - x[:-1]) overflows + y = np.arange(1, 11) + value = barycentric_interpolate(x, y, 1000 * 9.5) + assert_almost_equal(value, np.asarray(9.5)) + + def test_large_chebyshev(self): + # The weights for Chebyshev points of the second kind have analytically + # solvable weights. Naive calculation of barycentric weights will fail + # for large N because of numerical underflow and overflow. We test + # correctness for large N against analytical Chebyshev weights. + + # Without capacity scaling or permutation, n=800 fails, + # With just capacity scaling, n=1097 fails + # With both capacity scaling and random permutation, n=30000 succeeds + n = 1100 + j = np.arange(n + 1).astype(np.float64) + x = np.cos(j * np.pi / n) + + # See page 506 of Berrut and Trefethen 2004 for this formula + w = (-1) ** j + w[0] *= 0.5 + w[-1] *= 0.5 + + P = BarycentricInterpolator(x) + + # It's okay to have a constant scaling factor in the weights because it + # cancels out in the evaluation of the polynomial. + factor = P.wi[0] + assert_almost_equal(P.wi / (2 * factor), w) + + def test_warning(self): + # Test if the divide-by-zero warning is properly ignored when computing + # interpolated values equals to interpolation points + P = BarycentricInterpolator([0, 1], [1, 2]) + with np.errstate(divide='raise'): + yi = P(P.xi) + + # Check if the interpolated values match the input values + # at the nodes + assert_almost_equal(yi, P.yi.ravel()) + + @pytest.mark.thread_unsafe + def test_repeated_node(self): + # check that a repeated node raises a ValueError + # (computing the weights requires division by xi[i] - xi[j]) + xis = np.array([0.1, 0.5, 0.9, 0.5]) + ys = np.array([1, 2, 3, 4]) + with pytest.raises(ValueError, + match="Interpolation points xi must be distinct."): + BarycentricInterpolator(xis, ys) + + @pytest.mark.thread_unsafe + def test_concurrency(self): + P = BarycentricInterpolator(self.xs, self.ys) + + def worker_fn(_, interp): + interp(self.xs) + + _run_concurrent_barrier(10, worker_fn, P) + + +class TestPCHIP: + def _make_random(self, npts=20): + rng = np.random.RandomState(1234) + xi = np.sort(rng.random(npts)) + yi = rng.random(npts) + return pchip(xi, yi), xi, yi + + def test_overshoot(self): + # PCHIP should not overshoot + p, xi, yi = self._make_random() + for i in range(len(xi)-1): + x1, x2 = xi[i], xi[i+1] + y1, y2 = yi[i], yi[i+1] + if y1 > y2: + y1, y2 = y2, y1 + xp = np.linspace(x1, x2, 10) + yp = p(xp) + assert ((y1 <= yp + 1e-15) & (yp <= y2 + 1e-15)).all() + + def test_monotone(self): + # PCHIP should preserve monotonicty + p, xi, yi = self._make_random() + for i in range(len(xi)-1): + x1, x2 = xi[i], xi[i+1] + y1, y2 = yi[i], yi[i+1] + xp = np.linspace(x1, x2, 10) + yp = p(xp) + assert ((y2-y1) * (yp[1:] - yp[:1]) > 0).all() + + def test_cast(self): + # regression test for integer input data, see gh-3453 + data = np.array([[0, 4, 12, 27, 47, 60, 79, 87, 99, 100], + [-33, -33, -19, -2, 12, 26, 38, 45, 53, 55]]) + xx = np.arange(100) + curve = pchip(data[0], data[1])(xx) + + data1 = data * 1.0 + curve1 = pchip(data1[0], data1[1])(xx) + + xp_assert_close(curve, curve1, atol=1e-14, rtol=1e-14) + + def test_nag(self): + # Example from NAG C implementation, + # http://nag.com/numeric/cl/nagdoc_cl25/html/e01/e01bec.html + # suggested in gh-5326 as a smoke test for the way the derivatives + # are computed (see also gh-3453) + dataStr = ''' + 7.99 0.00000E+0 + 8.09 0.27643E-4 + 8.19 0.43750E-1 + 8.70 0.16918E+0 + 9.20 0.46943E+0 + 10.00 0.94374E+0 + 12.00 0.99864E+0 + 15.00 0.99992E+0 + 20.00 0.99999E+0 + ''' + data = np.loadtxt(io.StringIO(dataStr)) + pch = pchip(data[:,0], data[:,1]) + + resultStr = ''' + 7.9900 0.0000 + 9.1910 0.4640 + 10.3920 0.9645 + 11.5930 0.9965 + 12.7940 0.9992 + 13.9950 0.9998 + 15.1960 0.9999 + 16.3970 1.0000 + 17.5980 1.0000 + 18.7990 1.0000 + 20.0000 1.0000 + ''' + result = np.loadtxt(io.StringIO(resultStr)) + xp_assert_close(result[:,1], pch(result[:,0]), rtol=0., atol=5e-5) + + def test_endslopes(self): + # this is a smoke test for gh-3453: PCHIP interpolator should not + # set edge slopes to zero if the data do not suggest zero edge derivatives + x = np.array([0.0, 0.1, 0.25, 0.35]) + y1 = np.array([279.35, 0.5e3, 1.0e3, 2.5e3]) + y2 = np.array([279.35, 2.5e3, 1.50e3, 1.0e3]) + for pp in (pchip(x, y1), pchip(x, y2)): + for t in (x[0], x[-1]): + assert pp(t, 1) != 0 + + @pytest.mark.thread_unsafe + def test_all_zeros(self): + x = np.arange(10) + y = np.zeros_like(x) + + # this should work and not generate any warnings + with warnings.catch_warnings(): + warnings.filterwarnings('error') + pch = pchip(x, y) + + xx = np.linspace(0, 9, 101) + assert all(pch(xx) == 0.) + + def test_two_points(self): + # regression test for gh-6222: pchip([0, 1], [0, 1]) fails because + # it tries to use a three-point scheme to estimate edge derivatives, + # while there are only two points available. + # Instead, it should construct a linear interpolator. + x = np.linspace(0, 1, 11) + p = pchip([0, 1], [0, 2]) + xp_assert_close(p(x), 2*x, atol=1e-15) + + def test_pchip_interpolate(self): + assert_array_almost_equal( + pchip_interpolate([1, 2, 3], [4, 5, 6], [0.5], der=1), + np.asarray([1.])) + + assert_array_almost_equal( + pchip_interpolate([1, 2, 3], [4, 5, 6], [0.5], der=0), + np.asarray([3.5])) + + assert_array_almost_equal( + np.asarray(pchip_interpolate([1, 2, 3], [4, 5, 6], [0.5], der=[0, 1])), + np.asarray([[3.5], [1]])) + + def test_roots(self): + # regression test for gh-6357: .roots method should work + p = pchip([0, 1], [-1, 1]) + r = p.roots() + xp_assert_close(r, np.asarray([0.5])) + + +class TestCubicSpline: + @staticmethod + def check_correctness(S, bc_start='not-a-knot', bc_end='not-a-knot', + tol=1e-14): + """Check that spline coefficients satisfy the continuity and boundary + conditions.""" + x = S.x + c = S.c + dx = np.diff(x) + dx = dx.reshape([dx.shape[0]] + [1] * (c.ndim - 2)) + dxi = dx[:-1] + + # Check C2 continuity. + xp_assert_close(c[3, 1:], c[0, :-1] * dxi**3 + c[1, :-1] * dxi**2 + + c[2, :-1] * dxi + c[3, :-1], rtol=tol, atol=tol) + xp_assert_close(c[2, 1:], 3 * c[0, :-1] * dxi**2 + + 2 * c[1, :-1] * dxi + c[2, :-1], rtol=tol, atol=tol) + xp_assert_close(c[1, 1:], 3 * c[0, :-1] * dxi + c[1, :-1], + rtol=tol, atol=tol) + + # Check that we found a parabola, the third derivative is 0. + if x.size == 3 and bc_start == 'not-a-knot' and bc_end == 'not-a-knot': + xp_assert_close(c[0], np.zeros_like(c[0]), rtol=tol, atol=tol) + return + + # Check periodic boundary conditions. + if bc_start == 'periodic': + xp_assert_close(S(x[0], 0), S(x[-1], 0), rtol=tol, atol=tol) + xp_assert_close(S(x[0], 1), S(x[-1], 1), rtol=tol, atol=tol) + xp_assert_close(S(x[0], 2), S(x[-1], 2), rtol=tol, atol=tol) + return + + # Check other boundary conditions. + if bc_start == 'not-a-knot': + if x.size == 2: + slope = (S(x[1]) - S(x[0])) / dx[0] + slope = np.asarray(slope) + xp_assert_close(S(x[0], 1), slope, rtol=tol, atol=tol) + else: + xp_assert_close(c[0, 0], c[0, 1], rtol=tol, atol=tol) + elif bc_start == 'clamped': + xp_assert_close( + S(x[0], 1), np.zeros_like(S(x[0], 1)), rtol=tol, atol=tol) + elif bc_start == 'natural': + xp_assert_close( + S(x[0], 2), np.zeros_like(S(x[0], 2)), rtol=tol, atol=tol) + else: + order, value = bc_start + xp_assert_close(S(x[0], order), np.asarray(value), rtol=tol, atol=tol) + + if bc_end == 'not-a-knot': + if x.size == 2: + slope = (S(x[1]) - S(x[0])) / dx[0] + slope = np.asarray(slope) + xp_assert_close(S(x[1], 1), slope, rtol=tol, atol=tol) + else: + xp_assert_close(c[0, -1], c[0, -2], rtol=tol, atol=tol) + elif bc_end == 'clamped': + xp_assert_close(S(x[-1], 1), np.zeros_like(S(x[-1], 1)), + rtol=tol, atol=tol) + elif bc_end == 'natural': + xp_assert_close(S(x[-1], 2), np.zeros_like(S(x[-1], 2)), + rtol=2*tol, atol=2*tol) + else: + order, value = bc_end + xp_assert_close(S(x[-1], order), np.asarray(value), rtol=tol, atol=tol) + + def check_all_bc(self, x, y, axis): + deriv_shape = list(y.shape) + del deriv_shape[axis] + first_deriv = np.empty(deriv_shape) + first_deriv.fill(2) + second_deriv = np.empty(deriv_shape) + second_deriv.fill(-1) + bc_all = [ + 'not-a-knot', + 'natural', + 'clamped', + (1, first_deriv), + (2, second_deriv) + ] + for bc in bc_all[:3]: + S = CubicSpline(x, y, axis=axis, bc_type=bc) + self.check_correctness(S, bc, bc) + + for bc_start in bc_all: + for bc_end in bc_all: + S = CubicSpline(x, y, axis=axis, bc_type=(bc_start, bc_end)) + self.check_correctness(S, bc_start, bc_end, tol=2e-14) + + def test_general(self): + x = np.array([-1, 0, 0.5, 2, 4, 4.5, 5.5, 9]) + y = np.array([0, -0.5, 2, 3, 2.5, 1, 1, 0.5]) + for n in [2, 3, x.size]: + self.check_all_bc(x[:n], y[:n], 0) + + Y = np.empty((2, n, 2)) + Y[0, :, 0] = y[:n] + Y[0, :, 1] = y[:n] - 1 + Y[1, :, 0] = y[:n] + 2 + Y[1, :, 1] = y[:n] + 3 + self.check_all_bc(x[:n], Y, 1) + + def test_periodic(self): + for n in [2, 3, 5]: + x = np.linspace(0, 2 * np.pi, n) + y = np.cos(x) + S = CubicSpline(x, y, bc_type='periodic') + self.check_correctness(S, 'periodic', 'periodic') + + Y = np.empty((2, n, 2)) + Y[0, :, 0] = y + Y[0, :, 1] = y + 2 + Y[1, :, 0] = y - 1 + Y[1, :, 1] = y + 5 + S = CubicSpline(x, Y, axis=1, bc_type='periodic') + self.check_correctness(S, 'periodic', 'periodic') + + def test_periodic_eval(self): + x = np.linspace(0, 2 * np.pi, 10) + y = np.cos(x) + S = CubicSpline(x, y, bc_type='periodic') + assert_almost_equal(S(1), S(1 + 2 * np.pi), decimal=15) + + def test_second_derivative_continuity_gh_11758(self): + # gh-11758: C2 continuity fail + x = np.array([0.9, 1.3, 1.9, 2.1, 2.6, 3.0, 3.9, 4.4, 4.7, 5.0, 6.0, + 7.0, 8.0, 9.2, 10.5, 11.3, 11.6, 12.0, 12.6, 13.0, 13.3]) + y = np.array([1.3, 1.5, 1.85, 2.1, 2.6, 2.7, 2.4, 2.15, 2.05, 2.1, + 2.25, 2.3, 2.25, 1.95, 1.4, 0.9, 0.7, 0.6, 0.5, 0.4, 1.3]) + S = CubicSpline(x, y, bc_type='periodic', extrapolate='periodic') + self.check_correctness(S, 'periodic', 'periodic') + + def test_three_points(self): + # gh-11758: Fails computing a_m2_m1 + # In this case, s (first derivatives) could be found manually by solving + # system of 2 linear equations. Due to solution of this system, + # s[i] = (h1m2 + h2m1) / (h1 + h2), where h1 = x[1] - x[0], h2 = x[2] - x[1], + # m1 = (y[1] - y[0]) / h1, m2 = (y[2] - y[1]) / h2 + x = np.array([1.0, 2.75, 3.0]) + y = np.array([1.0, 15.0, 1.0]) + S = CubicSpline(x, y, bc_type='periodic') + self.check_correctness(S, 'periodic', 'periodic') + xp_assert_close(S.derivative(1)(x), np.array([-48.0, -48.0, -48.0])) + + def test_periodic_three_points_multidim(self): + # make sure one multidimensional interpolator does the same as multiple + # one-dimensional interpolators + x = np.array([0.0, 1.0, 3.0]) + y = np.array([[0.0, 1.0], [1.0, 0.0], [0.0, 1.0]]) + S = CubicSpline(x, y, bc_type="periodic") + self.check_correctness(S, 'periodic', 'periodic') + S0 = CubicSpline(x, y[:, 0], bc_type="periodic") + S1 = CubicSpline(x, y[:, 1], bc_type="periodic") + q = np.linspace(0, 2, 5) + xp_assert_close(S(q)[:, 0], S0(q)) + xp_assert_close(S(q)[:, 1], S1(q)) + + def test_dtypes(self): + x = np.array([0, 1, 2, 3], dtype=int) + y = np.array([-5, 2, 3, 1], dtype=int) + S = CubicSpline(x, y) + self.check_correctness(S) + + y = np.array([-1+1j, 0.0, 1-1j, 0.5-1.5j]) + S = CubicSpline(x, y) + self.check_correctness(S) + + S = CubicSpline(x, x ** 3, bc_type=("natural", (1, 2j))) + self.check_correctness(S, "natural", (1, 2j)) + + y = np.array([-5, 2, 3, 1]) + S = CubicSpline(x, y, bc_type=[(1, 2 + 0.5j), (2, 0.5 - 1j)]) + self.check_correctness(S, (1, 2 + 0.5j), (2, 0.5 - 1j)) + + def test_small_dx(self): + rng = np.random.RandomState(0) + x = np.sort(rng.uniform(size=100)) + y = 1e4 + rng.uniform(size=100) + S = CubicSpline(x, y) + self.check_correctness(S, tol=1e-13) + + def test_incorrect_inputs(self): + x = np.array([1, 2, 3, 4]) + y = np.array([1, 2, 3, 4]) + xc = np.array([1 + 1j, 2, 3, 4]) + xn = np.array([np.nan, 2, 3, 4]) + xo = np.array([2, 1, 3, 4]) + yn = np.array([np.nan, 2, 3, 4]) + y3 = [1, 2, 3] + x1 = [1] + y1 = [1] + + assert_raises(ValueError, CubicSpline, xc, y) + assert_raises(ValueError, CubicSpline, xn, y) + assert_raises(ValueError, CubicSpline, x, yn) + assert_raises(ValueError, CubicSpline, xo, y) + assert_raises(ValueError, CubicSpline, x, y3) + assert_raises(ValueError, CubicSpline, x[:, np.newaxis], y) + assert_raises(ValueError, CubicSpline, x1, y1) + + wrong_bc = [('periodic', 'clamped'), + ((2, 0), (3, 10)), + ((1, 0), ), + (0., 0.), + 'not-a-typo'] + + for bc_type in wrong_bc: + assert_raises(ValueError, CubicSpline, x, y, 0, bc_type, True) + + # Shapes mismatch when giving arbitrary derivative values: + Y = np.c_[y, y] + bc1 = ('clamped', (1, 0)) + bc2 = ('clamped', (1, [0, 0, 0])) + bc3 = ('clamped', (1, [[0, 0]])) + assert_raises(ValueError, CubicSpline, x, Y, 0, bc1, True) + assert_raises(ValueError, CubicSpline, x, Y, 0, bc2, True) + assert_raises(ValueError, CubicSpline, x, Y, 0, bc3, True) + + # periodic condition, y[-1] must be equal to y[0]: + assert_raises(ValueError, CubicSpline, x, y, 0, 'periodic', True) + + +def test_CubicHermiteSpline_correctness(): + x = [0, 2, 7] + y = [-1, 2, 3] + dydx = [0, 3, 7] + s = CubicHermiteSpline(x, y, dydx) + xp_assert_close(s(x), y, check_shape=False, check_dtype=False, rtol=1e-15) + xp_assert_close(s(x, 1), dydx, check_shape=False, check_dtype=False, rtol=1e-15) + + +def test_CubicHermiteSpline_error_handling(): + x = [1, 2, 3] + y = [0, 3, 5] + dydx = [1, -1, 2, 3] + assert_raises(ValueError, CubicHermiteSpline, x, y, dydx) + + dydx_with_nan = [1, 0, np.nan] + assert_raises(ValueError, CubicHermiteSpline, x, y, dydx_with_nan) + + +def test_roots_extrapolate_gh_11185(): + x = np.array([0.001, 0.002]) + y = np.array([1.66066935e-06, 1.10410807e-06]) + dy = np.array([-1.60061854, -1.600619]) + p = CubicHermiteSpline(x, y, dy) + + # roots(extrapolate=True) for a polynomial with a single interval + # should return all three real roots + r = p.roots(extrapolate=True) + assert p.c.shape[1] == 1 + assert r.size == 3 + + +class TestZeroSizeArrays: + # regression tests for gh-17241 : CubicSpline et al must not segfault + # when y.size == 0 + # The two methods below are _almost_ the same, but not quite: + # one is for objects which have the `bc_type` argument (CubicSpline) + # and the other one is for those which do not (Pchip, Akima1D) + + @pytest.mark.parametrize('y', [np.zeros((10, 0, 5)), + np.zeros((10, 5, 0))]) + @pytest.mark.parametrize('bc_type', + ['not-a-knot', 'periodic', 'natural', 'clamped']) + @pytest.mark.parametrize('axis', [0, 1, 2]) + @pytest.mark.parametrize('cls', [make_interp_spline, CubicSpline]) + def test_zero_size(self, cls, y, bc_type, axis): + x = np.arange(10) + xval = np.arange(3) + + obj = cls(x, y, bc_type=bc_type) + assert obj(xval).size == 0 + assert obj(xval).shape == xval.shape + y.shape[1:] + + # Also check with an explicit non-default axis + yt = np.moveaxis(y, 0, axis) # (10, 0, 5) --> (0, 10, 5) if axis=1 etc + + obj = cls(x, yt, bc_type=bc_type, axis=axis) + sh = yt.shape[:axis] + (xval.size, ) + yt.shape[axis+1:] + assert obj(xval).size == 0 + assert obj(xval).shape == sh + + @pytest.mark.parametrize('y', [np.zeros((10, 0, 5)), + np.zeros((10, 5, 0))]) + @pytest.mark.parametrize('axis', [0, 1, 2]) + @pytest.mark.parametrize('cls', [PchipInterpolator, Akima1DInterpolator]) + def test_zero_size_2(self, cls, y, axis): + x = np.arange(10) + xval = np.arange(3) + + obj = cls(x, y) + assert obj(xval).size == 0 + assert obj(xval).shape == xval.shape + y.shape[1:] + + # Also check with an explicit non-default axis + yt = np.moveaxis(y, 0, axis) # (10, 0, 5) --> (0, 10, 5) if axis=1 etc + + obj = cls(x, yt, axis=axis) + sh = yt.shape[:axis] + (xval.size, ) + yt.shape[axis+1:] + assert obj(xval).size == 0 + assert obj(xval).shape == sh diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_rbf.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_rbf.py new file mode 100644 index 0000000000000000000000000000000000000000..d824a84a80eda316b680ba4e43d0f418c774af99 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_rbf.py @@ -0,0 +1,246 @@ +# Created by John Travers, Robert Hetland, 2007 +""" Test functions for rbf module """ + +import numpy as np + +from scipy._lib._array_api import assert_array_almost_equal, assert_almost_equal + +from numpy import linspace, sin, cos, exp, allclose +from scipy.interpolate._rbf import Rbf +from scipy._lib._testutils import _run_concurrent_barrier + +import pytest + + +FUNCTIONS = ('multiquadric', 'inverse multiquadric', 'gaussian', + 'cubic', 'quintic', 'thin-plate', 'linear') + + +def check_rbf1d_interpolation(function): + # Check that the Rbf function interpolates through the nodes (1D) + x = linspace(0,10,9) + y = sin(x) + rbf = Rbf(x, y, function=function) + yi = rbf(x) + assert_array_almost_equal(y, yi) + assert_almost_equal(rbf(float(x[0])), y[0], check_0d=False) + + +def check_rbf2d_interpolation(function): + # Check that the Rbf function interpolates through the nodes (2D). + rng = np.random.RandomState(1234) + x = rng.rand(50,1)*4-2 + y = rng.rand(50,1)*4-2 + z = x*exp(-x**2-1j*y**2) + rbf = Rbf(x, y, z, epsilon=2, function=function) + zi = rbf(x, y) + zi.shape = x.shape + assert_array_almost_equal(z, zi) + + +def check_rbf3d_interpolation(function): + # Check that the Rbf function interpolates through the nodes (3D). + rng = np.random.RandomState(1234) + x = rng.rand(50, 1)*4 - 2 + y = rng.rand(50, 1)*4 - 2 + z = rng.rand(50, 1)*4 - 2 + d = x*exp(-x**2 - y**2) + rbf = Rbf(x, y, z, d, epsilon=2, function=function) + di = rbf(x, y, z) + di.shape = x.shape + assert_array_almost_equal(di, d) + + +def test_rbf_interpolation(): + for function in FUNCTIONS: + check_rbf1d_interpolation(function) + check_rbf2d_interpolation(function) + check_rbf3d_interpolation(function) + + +def check_2drbf1d_interpolation(function): + # Check that the 2-D Rbf function interpolates through the nodes (1D) + x = linspace(0, 10, 9) + y0 = sin(x) + y1 = cos(x) + y = np.vstack([y0, y1]).T + rbf = Rbf(x, y, function=function, mode='N-D') + yi = rbf(x) + assert_array_almost_equal(y, yi) + assert_almost_equal(rbf(float(x[0])), y[0]) + + +def check_2drbf2d_interpolation(function): + # Check that the 2-D Rbf function interpolates through the nodes (2D). + rng = np.random.RandomState(1234) + x = rng.rand(50, ) * 4 - 2 + y = rng.rand(50, ) * 4 - 2 + z0 = x * exp(-x ** 2 - 1j * y ** 2) + z1 = y * exp(-y ** 2 - 1j * x ** 2) + z = np.vstack([z0, z1]).T + rbf = Rbf(x, y, z, epsilon=2, function=function, mode='N-D') + zi = rbf(x, y) + zi.shape = z.shape + assert_array_almost_equal(z, zi) + + +def check_2drbf3d_interpolation(function): + # Check that the 2-D Rbf function interpolates through the nodes (3D). + rng = np.random.RandomState(1234) + x = rng.rand(50, ) * 4 - 2 + y = rng.rand(50, ) * 4 - 2 + z = rng.rand(50, ) * 4 - 2 + d0 = x * exp(-x ** 2 - y ** 2) + d1 = y * exp(-y ** 2 - x ** 2) + d = np.vstack([d0, d1]).T + rbf = Rbf(x, y, z, d, epsilon=2, function=function, mode='N-D') + di = rbf(x, y, z) + di.shape = d.shape + assert_array_almost_equal(di, d) + + +def test_2drbf_interpolation(): + for function in FUNCTIONS: + check_2drbf1d_interpolation(function) + check_2drbf2d_interpolation(function) + check_2drbf3d_interpolation(function) + + +def check_rbf1d_regularity(function, atol): + # Check that the Rbf function approximates a smooth function well away + # from the nodes. + x = linspace(0, 10, 9) + y = sin(x) + rbf = Rbf(x, y, function=function) + xi = linspace(0, 10, 100) + yi = rbf(xi) + msg = f"abs-diff: {abs(yi - sin(xi)).max():f}" + assert allclose(yi, sin(xi), atol=atol), msg + + +def test_rbf_regularity(): + tolerances = { + 'multiquadric': 0.1, + 'inverse multiquadric': 0.15, + 'gaussian': 0.15, + 'cubic': 0.15, + 'quintic': 0.1, + 'thin-plate': 0.1, + 'linear': 0.2 + } + for function in FUNCTIONS: + check_rbf1d_regularity(function, tolerances.get(function, 1e-2)) + + +def check_2drbf1d_regularity(function, atol): + # Check that the 2-D Rbf function approximates a smooth function well away + # from the nodes. + x = linspace(0, 10, 9) + y0 = sin(x) + y1 = cos(x) + y = np.vstack([y0, y1]).T + rbf = Rbf(x, y, function=function, mode='N-D') + xi = linspace(0, 10, 100) + yi = rbf(xi) + msg = f"abs-diff: {abs(yi - np.vstack([sin(xi), cos(xi)]).T).max():f}" + assert allclose(yi, np.vstack([sin(xi), cos(xi)]).T, atol=atol), msg + + +def test_2drbf_regularity(): + tolerances = { + 'multiquadric': 0.1, + 'inverse multiquadric': 0.15, + 'gaussian': 0.15, + 'cubic': 0.15, + 'quintic': 0.1, + 'thin-plate': 0.15, + 'linear': 0.2 + } + for function in FUNCTIONS: + check_2drbf1d_regularity(function, tolerances.get(function, 1e-2)) + + +def check_rbf1d_stability(function): + # Check that the Rbf function with default epsilon is not subject + # to overshoot. Regression for issue #4523. + # + # Generate some data (fixed random seed hence deterministic) + rng = np.random.RandomState(1234) + x = np.linspace(0, 10, 50) + z = x + 4.0 * rng.randn(len(x)) + + rbf = Rbf(x, z, function=function) + xi = np.linspace(0, 10, 1000) + yi = rbf(xi) + + # subtract the linear trend and make sure there no spikes + assert np.abs(yi-xi).max() / np.abs(z-x).max() < 1.1 + +def test_rbf_stability(): + for function in FUNCTIONS: + check_rbf1d_stability(function) + + +def test_default_construction(): + # Check that the Rbf class can be constructed with the default + # multiquadric basis function. Regression test for ticket #1228. + x = linspace(0,10,9) + y = sin(x) + rbf = Rbf(x, y) + yi = rbf(x) + assert_array_almost_equal(y, yi) + + +def test_function_is_callable(): + # Check that the Rbf class can be constructed with function=callable. + x = linspace(0,10,9) + y = sin(x) + def linfunc(x): + return x + rbf = Rbf(x, y, function=linfunc) + yi = rbf(x) + assert_array_almost_equal(y, yi) + + +def test_two_arg_function_is_callable(): + # Check that the Rbf class can be constructed with a two argument + # function=callable. + def _func(self, r): + return self.epsilon + r + + x = linspace(0,10,9) + y = sin(x) + rbf = Rbf(x, y, function=_func) + yi = rbf(x) + assert_array_almost_equal(y, yi) + + +def test_rbf_epsilon_none(): + x = linspace(0, 10, 9) + y = sin(x) + Rbf(x, y, epsilon=None) + + +def test_rbf_epsilon_none_collinear(): + # Check that collinear points in one dimension doesn't cause an error + # due to epsilon = 0 + x = [1, 2, 3] + y = [4, 4, 4] + z = [5, 6, 7] + rbf = Rbf(x, y, z, epsilon=None) + assert rbf.epsilon > 0 + + +@pytest.mark.thread_unsafe +def test_rbf_concurrency(): + x = linspace(0, 10, 100) + y0 = sin(x) + y1 = cos(x) + y = np.vstack([y0, y1]).T + rbf = Rbf(x, y, mode='N-D') + + def worker_fn(_, interp, xp): + interp(xp) + + _run_concurrent_barrier(10, worker_fn, rbf, x) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_rbfinterp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_rbfinterp.py new file mode 100644 index 0000000000000000000000000000000000000000..3d2759fdee41fa64c09bb00979f10b70e49a2855 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_rbfinterp.py @@ -0,0 +1,534 @@ +import pickle +import pytest +import numpy as np +from numpy.linalg import LinAlgError +from scipy._lib._array_api import xp_assert_close +from scipy.stats.qmc import Halton +from scipy.spatial import cKDTree # type: ignore[attr-defined] +from scipy.interpolate._rbfinterp import ( + _AVAILABLE, _SCALE_INVARIANT, _NAME_TO_MIN_DEGREE, _monomial_powers, + RBFInterpolator + ) +from scipy.interpolate import _rbfinterp_pythran +from scipy._lib._testutils import _run_concurrent_barrier + + +def _vandermonde(x, degree): + # Returns a matrix of monomials that span polynomials with the specified + # degree evaluated at x. + powers = _monomial_powers(x.shape[1], degree) + return _rbfinterp_pythran._polynomial_matrix(x, powers) + + +def _1d_test_function(x): + # Test function used in Wahba's "Spline Models for Observational Data". + # domain ~= (0, 3), range ~= (-1.0, 0.2) + x = x[:, 0] + y = 4.26*(np.exp(-x) - 4*np.exp(-2*x) + 3*np.exp(-3*x)) + return y + + +def _2d_test_function(x): + # Franke's test function. + # domain ~= (0, 1) X (0, 1), range ~= (0.0, 1.2) + x1, x2 = x[:, 0], x[:, 1] + term1 = 0.75 * np.exp(-(9*x1-2)**2/4 - (9*x2-2)**2/4) + term2 = 0.75 * np.exp(-(9*x1+1)**2/49 - (9*x2+1)/10) + term3 = 0.5 * np.exp(-(9*x1-7)**2/4 - (9*x2-3)**2/4) + term4 = -0.2 * np.exp(-(9*x1-4)**2 - (9*x2-7)**2) + y = term1 + term2 + term3 + term4 + return y + + +def _is_conditionally_positive_definite(kernel, m): + # Tests whether the kernel is conditionally positive definite of order m. + # See chapter 7 of Fasshauer's "Meshfree Approximation Methods with + # MATLAB". + nx = 10 + ntests = 100 + for ndim in [1, 2, 3, 4, 5]: + # Generate sample points with a Halton sequence to avoid samples that + # are too close to each other, which can make the matrix singular. + seq = Halton(ndim, scramble=False, seed=np.random.RandomState()) + for _ in range(ntests): + x = 2*seq.random(nx) - 1 + A = _rbfinterp_pythran._kernel_matrix(x, kernel) + P = _vandermonde(x, m - 1) + Q, R = np.linalg.qr(P, mode='complete') + # Q2 forms a basis spanning the space where P.T.dot(x) = 0. Project + # A onto this space, and then see if it is positive definite using + # the Cholesky decomposition. If not, then the kernel is not c.p.d. + # of order m. + Q2 = Q[:, P.shape[1]:] + B = Q2.T.dot(A).dot(Q2) + try: + np.linalg.cholesky(B) + except np.linalg.LinAlgError: + return False + + return True + + +# Sorting the parametrize arguments is necessary to avoid a parallelization +# issue described here: https://github.com/pytest-dev/pytest-xdist/issues/432. +@pytest.mark.parametrize('kernel', sorted(_AVAILABLE)) +def test_conditionally_positive_definite(kernel): + # Test if each kernel in _AVAILABLE is conditionally positive definite of + # order m, where m comes from _NAME_TO_MIN_DEGREE. This is a necessary + # condition for the smoothed RBF interpolant to be well-posed in general. + m = _NAME_TO_MIN_DEGREE.get(kernel, -1) + 1 + assert _is_conditionally_positive_definite(kernel, m) + + +class _TestRBFInterpolator: + @pytest.mark.parametrize('kernel', sorted(_SCALE_INVARIANT)) + def test_scale_invariance_1d(self, kernel): + # Verify that the functions in _SCALE_INVARIANT are insensitive to the + # shape parameter (when smoothing == 0) in 1d. + seq = Halton(1, scramble=False, seed=np.random.RandomState()) + x = 3*seq.random(50) + y = _1d_test_function(x) + xitp = 3*seq.random(50) + yitp1 = self.build(x, y, epsilon=1.0, kernel=kernel)(xitp) + yitp2 = self.build(x, y, epsilon=2.0, kernel=kernel)(xitp) + xp_assert_close(yitp1, yitp2, atol=1e-8) + + @pytest.mark.parametrize('kernel', sorted(_SCALE_INVARIANT)) + def test_scale_invariance_2d(self, kernel): + # Verify that the functions in _SCALE_INVARIANT are insensitive to the + # shape parameter (when smoothing == 0) in 2d. + seq = Halton(2, scramble=False, seed=np.random.RandomState()) + x = seq.random(100) + y = _2d_test_function(x) + xitp = seq.random(100) + yitp1 = self.build(x, y, epsilon=1.0, kernel=kernel)(xitp) + yitp2 = self.build(x, y, epsilon=2.0, kernel=kernel)(xitp) + xp_assert_close(yitp1, yitp2, atol=1e-8) + + @pytest.mark.parametrize('kernel', sorted(_AVAILABLE)) + def test_extreme_domains(self, kernel): + # Make sure the interpolant remains numerically stable for very + # large/small domains. + seq = Halton(2, scramble=False, seed=np.random.RandomState()) + scale = 1e50 + shift = 1e55 + + x = seq.random(100) + y = _2d_test_function(x) + xitp = seq.random(100) + + if kernel in _SCALE_INVARIANT: + yitp1 = self.build(x, y, kernel=kernel)(xitp) + yitp2 = self.build( + x*scale + shift, y, + kernel=kernel + )(xitp*scale + shift) + else: + yitp1 = self.build(x, y, epsilon=5.0, kernel=kernel)(xitp) + yitp2 = self.build( + x*scale + shift, y, + epsilon=5.0/scale, + kernel=kernel + )(xitp*scale + shift) + + xp_assert_close(yitp1, yitp2, atol=1e-8) + + def test_polynomial_reproduction(self): + # If the observed data comes from a polynomial, then the interpolant + # should be able to reproduce the polynomial exactly, provided that + # `degree` is sufficiently high. + rng = np.random.RandomState(0) + seq = Halton(2, scramble=False, seed=rng) + degree = 3 + + x = seq.random(50) + xitp = seq.random(50) + + P = _vandermonde(x, degree) + Pitp = _vandermonde(xitp, degree) + + poly_coeffs = rng.normal(0.0, 1.0, P.shape[1]) + + y = P.dot(poly_coeffs) + yitp1 = Pitp.dot(poly_coeffs) + yitp2 = self.build(x, y, degree=degree)(xitp) + + xp_assert_close(yitp1, yitp2, atol=1e-8) + + @pytest.mark.slow + def test_chunking(self, monkeypatch): + # If the observed data comes from a polynomial, then the interpolant + # should be able to reproduce the polynomial exactly, provided that + # `degree` is sufficiently high. + rng = np.random.RandomState(0) + seq = Halton(2, scramble=False, seed=rng) + degree = 3 + + largeN = 1000 + 33 + # this is large to check that chunking of the RBFInterpolator is tested + x = seq.random(50) + xitp = seq.random(largeN) + + P = _vandermonde(x, degree) + Pitp = _vandermonde(xitp, degree) + + poly_coeffs = rng.normal(0.0, 1.0, P.shape[1]) + + y = P.dot(poly_coeffs) + yitp1 = Pitp.dot(poly_coeffs) + interp = self.build(x, y, degree=degree) + ce_real = interp._chunk_evaluator + + def _chunk_evaluator(*args, **kwargs): + kwargs.update(memory_budget=100) + return ce_real(*args, **kwargs) + + monkeypatch.setattr(interp, '_chunk_evaluator', _chunk_evaluator) + yitp2 = interp(xitp) + xp_assert_close(yitp1, yitp2, atol=1e-8) + + def test_vector_data(self): + # Make sure interpolating a vector field is the same as interpolating + # each component separately. + seq = Halton(2, scramble=False, seed=np.random.RandomState()) + + x = seq.random(100) + xitp = seq.random(100) + + y = np.array([_2d_test_function(x), + _2d_test_function(x[:, ::-1])]).T + + yitp1 = self.build(x, y)(xitp) + yitp2 = self.build(x, y[:, 0])(xitp) + yitp3 = self.build(x, y[:, 1])(xitp) + + xp_assert_close(yitp1[:, 0], yitp2) + xp_assert_close(yitp1[:, 1], yitp3) + + def test_complex_data(self): + # Interpolating complex input should be the same as interpolating the + # real and complex components. + seq = Halton(2, scramble=False, seed=np.random.RandomState()) + + x = seq.random(100) + xitp = seq.random(100) + + y = _2d_test_function(x) + 1j*_2d_test_function(x[:, ::-1]) + + yitp1 = self.build(x, y)(xitp) + yitp2 = self.build(x, y.real)(xitp) + yitp3 = self.build(x, y.imag)(xitp) + + xp_assert_close(yitp1.real, yitp2) + xp_assert_close(yitp1.imag, yitp3) + + @pytest.mark.parametrize('kernel', sorted(_AVAILABLE)) + def test_interpolation_misfit_1d(self, kernel): + # Make sure that each kernel, with its default `degree` and an + # appropriate `epsilon`, does a good job at interpolation in 1d. + seq = Halton(1, scramble=False, seed=np.random.RandomState()) + + x = 3*seq.random(50) + xitp = 3*seq.random(50) + + y = _1d_test_function(x) + ytrue = _1d_test_function(xitp) + yitp = self.build(x, y, epsilon=5.0, kernel=kernel)(xitp) + + mse = np.mean((yitp - ytrue)**2) + assert mse < 1.0e-4 + + @pytest.mark.parametrize('kernel', sorted(_AVAILABLE)) + def test_interpolation_misfit_2d(self, kernel): + # Make sure that each kernel, with its default `degree` and an + # appropriate `epsilon`, does a good job at interpolation in 2d. + seq = Halton(2, scramble=False, seed=np.random.RandomState()) + + x = seq.random(100) + xitp = seq.random(100) + + y = _2d_test_function(x) + ytrue = _2d_test_function(xitp) + yitp = self.build(x, y, epsilon=5.0, kernel=kernel)(xitp) + + mse = np.mean((yitp - ytrue)**2) + assert mse < 2.0e-4 + + @pytest.mark.parametrize('kernel', sorted(_AVAILABLE)) + def test_smoothing_misfit(self, kernel): + # Make sure we can find a smoothing parameter for each kernel that + # removes a sufficient amount of noise. + rng = np.random.RandomState(0) + seq = Halton(1, scramble=False, seed=rng) + + noise = 0.2 + rmse_tol = 0.1 + smoothing_range = 10**np.linspace(-4, 1, 20) + + x = 3*seq.random(100) + y = _1d_test_function(x) + rng.normal(0.0, noise, (100,)) + ytrue = _1d_test_function(x) + rmse_within_tol = False + for smoothing in smoothing_range: + ysmooth = self.build( + x, y, + epsilon=1.0, + smoothing=smoothing, + kernel=kernel)(x) + rmse = np.sqrt(np.mean((ysmooth - ytrue)**2)) + if rmse < rmse_tol: + rmse_within_tol = True + break + + assert rmse_within_tol + + def test_array_smoothing(self): + # Test using an array for `smoothing` to give less weight to a known + # outlier. + rng = np.random.RandomState(0) + seq = Halton(1, scramble=False, seed=rng) + degree = 2 + + x = seq.random(50) + P = _vandermonde(x, degree) + poly_coeffs = rng.normal(0.0, 1.0, P.shape[1]) + y = P.dot(poly_coeffs) + y_with_outlier = np.copy(y) + y_with_outlier[10] += 1.0 + smoothing = np.zeros((50,)) + smoothing[10] = 1000.0 + yitp = self.build(x, y_with_outlier, smoothing=smoothing)(x) + # Should be able to reproduce the uncorrupted data almost exactly. + xp_assert_close(yitp, y, atol=1e-4) + + def test_inconsistent_x_dimensions_error(self): + # ValueError should be raised if the observation points and evaluation + # points have a different number of dimensions. + y = Halton(2, scramble=False, seed=np.random.RandomState()).random(10) + d = _2d_test_function(y) + x = Halton(1, scramble=False, seed=np.random.RandomState()).random(10) + match = 'Expected the second axis of `x`' + with pytest.raises(ValueError, match=match): + self.build(y, d)(x) + + def test_inconsistent_d_length_error(self): + y = np.linspace(0, 1, 5)[:, None] + d = np.zeros(1) + match = 'Expected the first axis of `d`' + with pytest.raises(ValueError, match=match): + self.build(y, d) + + def test_y_not_2d_error(self): + y = np.linspace(0, 1, 5) + d = np.zeros(5) + match = '`y` must be a 2-dimensional array.' + with pytest.raises(ValueError, match=match): + self.build(y, d) + + def test_inconsistent_smoothing_length_error(self): + y = np.linspace(0, 1, 5)[:, None] + d = np.zeros(5) + smoothing = np.ones(1) + match = 'Expected `smoothing` to be' + with pytest.raises(ValueError, match=match): + self.build(y, d, smoothing=smoothing) + + def test_invalid_kernel_name_error(self): + y = np.linspace(0, 1, 5)[:, None] + d = np.zeros(5) + match = '`kernel` must be one of' + with pytest.raises(ValueError, match=match): + self.build(y, d, kernel='test') + + def test_epsilon_not_specified_error(self): + y = np.linspace(0, 1, 5)[:, None] + d = np.zeros(5) + for kernel in _AVAILABLE: + if kernel in _SCALE_INVARIANT: + continue + + match = '`epsilon` must be specified' + with pytest.raises(ValueError, match=match): + self.build(y, d, kernel=kernel) + + def test_x_not_2d_error(self): + y = np.linspace(0, 1, 5)[:, None] + x = np.linspace(0, 1, 5) + d = np.zeros(5) + match = '`x` must be a 2-dimensional array.' + with pytest.raises(ValueError, match=match): + self.build(y, d)(x) + + def test_not_enough_observations_error(self): + y = np.linspace(0, 1, 1)[:, None] + d = np.zeros(1) + match = 'At least 2 data points are required' + with pytest.raises(ValueError, match=match): + self.build(y, d, kernel='thin_plate_spline') + + @pytest.mark.thread_unsafe + def test_degree_warning(self): + y = np.linspace(0, 1, 5)[:, None] + d = np.zeros(5) + for kernel, deg in _NAME_TO_MIN_DEGREE.items(): + # Only test for kernels that its minimum degree is not 0. + if deg >= 1: + match = f'`degree` should not be below {deg}' + with pytest.warns(Warning, match=match): + self.build(y, d, epsilon=1.0, kernel=kernel, degree=deg-1) + + def test_minus_one_degree(self): + # Make sure a degree of -1 is accepted without any warning. + y = np.linspace(0, 1, 5)[:, None] + d = np.zeros(5) + for kernel, _ in _NAME_TO_MIN_DEGREE.items(): + self.build(y, d, epsilon=1.0, kernel=kernel, degree=-1) + + def test_rank_error(self): + # An error should be raised when `kernel` is "thin_plate_spline" and + # observations are 2-D and collinear. + y = np.array([[2.0, 0.0], [1.0, 0.0], [0.0, 0.0]]) + d = np.array([0.0, 0.0, 0.0]) + match = 'does not have full column rank' + with pytest.raises(LinAlgError, match=match): + self.build(y, d, kernel='thin_plate_spline')(y) + + def test_single_point(self): + # Make sure interpolation still works with only one point (in 1, 2, and + # 3 dimensions). + for dim in [1, 2, 3]: + y = np.zeros((1, dim)) + d = np.ones((1,)) + f = self.build(y, d, kernel='linear')(y) + xp_assert_close(d, f) + + def test_pickleable(self): + # Make sure we can pickle and unpickle the interpolant without any + # changes in the behavior. + seq = Halton(1, scramble=False, seed=np.random.RandomState(2305982309)) + + x = 3*seq.random(50) + xitp = 3*seq.random(50) + + y = _1d_test_function(x) + + interp = self.build(x, y) + + yitp1 = interp(xitp) + yitp2 = pickle.loads(pickle.dumps(interp))(xitp) + + xp_assert_close(yitp1, yitp2, atol=1e-16) + + +class TestRBFInterpolatorNeighborsNone(_TestRBFInterpolator): + def build(self, *args, **kwargs): + return RBFInterpolator(*args, **kwargs) + + def test_smoothing_limit_1d(self): + # For large smoothing parameters, the interpolant should approach a + # least squares fit of a polynomial with the specified degree. + seq = Halton(1, scramble=False, seed=np.random.RandomState()) + + degree = 3 + smoothing = 1e8 + + x = 3*seq.random(50) + xitp = 3*seq.random(50) + + y = _1d_test_function(x) + + yitp1 = self.build( + x, y, + degree=degree, + smoothing=smoothing + )(xitp) + + P = _vandermonde(x, degree) + Pitp = _vandermonde(xitp, degree) + yitp2 = Pitp.dot(np.linalg.lstsq(P, y, rcond=None)[0]) + + xp_assert_close(yitp1, yitp2, atol=1e-8) + + def test_smoothing_limit_2d(self): + # For large smoothing parameters, the interpolant should approach a + # least squares fit of a polynomial with the specified degree. + seq = Halton(2, scramble=False, seed=np.random.RandomState()) + + degree = 3 + smoothing = 1e8 + + x = seq.random(100) + xitp = seq.random(100) + + y = _2d_test_function(x) + + yitp1 = self.build( + x, y, + degree=degree, + smoothing=smoothing + )(xitp) + + P = _vandermonde(x, degree) + Pitp = _vandermonde(xitp, degree) + yitp2 = Pitp.dot(np.linalg.lstsq(P, y, rcond=None)[0]) + + xp_assert_close(yitp1, yitp2, atol=1e-8) + + +class TestRBFInterpolatorNeighbors20(_TestRBFInterpolator): + # RBFInterpolator using 20 nearest neighbors. + def build(self, *args, **kwargs): + return RBFInterpolator(*args, **kwargs, neighbors=20) + + def test_equivalent_to_rbf_interpolator(self): + seq = Halton(2, scramble=False, seed=np.random.RandomState()) + + x = seq.random(100) + xitp = seq.random(100) + + y = _2d_test_function(x) + + yitp1 = self.build(x, y)(xitp) + + yitp2 = [] + tree = cKDTree(x) + for xi in xitp: + _, nbr = tree.query(xi, 20) + yitp2.append(RBFInterpolator(x[nbr], y[nbr])(xi[None])[0]) + + xp_assert_close(yitp1, yitp2, atol=1e-8) + + def test_concurrency(self): + # Check that no segfaults appear with concurrent access to + # RbfInterpolator + seq = Halton(2, scramble=False, seed=np.random.RandomState(0)) + x = seq.random(100) + xitp = seq.random(100) + + y = _2d_test_function(x) + + interp = self.build(x, y) + + def worker_fn(_, interp, xp): + interp(xp) + + _run_concurrent_barrier(10, worker_fn, interp, xitp) + + +class TestRBFInterpolatorNeighborsInf(TestRBFInterpolatorNeighborsNone): + # RBFInterpolator using neighbors=np.inf. This should give exactly the same + # results as neighbors=None, but it will be slower. + def build(self, *args, **kwargs): + return RBFInterpolator(*args, **kwargs, neighbors=np.inf) + + def test_equivalent_to_rbf_interpolator(self): + seq = Halton(1, scramble=False, seed=np.random.RandomState()) + + x = 3*seq.random(50) + xitp = 3*seq.random(50) + + y = _1d_test_function(x) + yitp1 = self.build(x, y)(xitp) + yitp2 = RBFInterpolator(x, y)(xitp) + + xp_assert_close(yitp1, yitp2, atol=1e-8) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_rgi.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_rgi.py new file mode 100644 index 0000000000000000000000000000000000000000..54c4f380ad7d51f54a5947cfc7764a60c1fc235e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/interpolate/tests/test_rgi.py @@ -0,0 +1,1150 @@ +import itertools + +import pytest +import numpy as np + +from numpy.testing import assert_warns +from scipy._lib._array_api import ( + xp_assert_equal, xp_assert_close, assert_array_almost_equal +) +from scipy.conftest import skip_xp_invalid_arg + +from pytest import raises as assert_raises + +from scipy.interpolate import (RegularGridInterpolator, interpn, + RectBivariateSpline, + NearestNDInterpolator, LinearNDInterpolator) + +from scipy.sparse._sputils import matrix +from scipy._lib._util import ComplexWarning +from scipy._lib._testutils import _run_concurrent_barrier + + +parametrize_rgi_interp_methods = pytest.mark.parametrize( + "method", RegularGridInterpolator._ALL_METHODS +) + +class TestRegularGridInterpolator: + def _get_sample_4d(self): + # create a 4-D grid of 3 points in each dimension + points = [(0., .5, 1.)] * 4 + values = np.asarray([0., .5, 1.]) + values0 = values[:, np.newaxis, np.newaxis, np.newaxis] + values1 = values[np.newaxis, :, np.newaxis, np.newaxis] + values2 = values[np.newaxis, np.newaxis, :, np.newaxis] + values3 = values[np.newaxis, np.newaxis, np.newaxis, :] + values = (values0 + values1 * 10 + values2 * 100 + values3 * 1000) + return points, values + + def _get_sample_4d_2(self): + # create another 4-D grid of 3 points in each dimension + points = [(0., .5, 1.)] * 2 + [(0., 5., 10.)] * 2 + values = np.asarray([0., .5, 1.]) + values0 = values[:, np.newaxis, np.newaxis, np.newaxis] + values1 = values[np.newaxis, :, np.newaxis, np.newaxis] + values2 = values[np.newaxis, np.newaxis, :, np.newaxis] + values3 = values[np.newaxis, np.newaxis, np.newaxis, :] + values = (values0 + values1 * 10 + values2 * 100 + values3 * 1000) + return points, values + + def _get_sample_4d_3(self): + # create another 4-D grid of 7 points in each dimension + points = [(0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0)] * 4 + values = np.asarray([0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0]) + values0 = values[:, np.newaxis, np.newaxis, np.newaxis] + values1 = values[np.newaxis, :, np.newaxis, np.newaxis] + values2 = values[np.newaxis, np.newaxis, :, np.newaxis] + values3 = values[np.newaxis, np.newaxis, np.newaxis, :] + values = (values0 + values1 * 10 + values2 * 100 + values3 * 1000) + return points, values + + def _get_sample_4d_4(self): + # create another 4-D grid of 2 points in each dimension + points = [(0.0, 1.0)] * 4 + values = np.asarray([0.0, 1.0]) + values0 = values[:, np.newaxis, np.newaxis, np.newaxis] + values1 = values[np.newaxis, :, np.newaxis, np.newaxis] + values2 = values[np.newaxis, np.newaxis, :, np.newaxis] + values3 = values[np.newaxis, np.newaxis, np.newaxis, :] + values = (values0 + values1 * 10 + values2 * 100 + values3 * 1000) + return points, values + + @parametrize_rgi_interp_methods + def test_list_input(self, method): + points, values = self._get_sample_4d_3() + + sample = np.asarray([[0.1, 0.1, 1., .9], [0.2, 0.1, .45, .8], + [0.5, 0.5, .5, .5]]) + + interp = RegularGridInterpolator(points, + values.tolist(), + method=method) + v1 = interp(sample.tolist()) + interp = RegularGridInterpolator(points, + values, + method=method) + v2 = interp(sample) + xp_assert_close(v1, v2) + + @pytest.mark.parametrize('method', ['cubic', 'quintic', 'pchip']) + def test_spline_dim_error(self, method): + points, values = self._get_sample_4d_4() + match = "points in dimension" + + # Check error raise when creating interpolator + with pytest.raises(ValueError, match=match): + RegularGridInterpolator(points, values, method=method) + + # Check error raise when creating interpolator + interp = RegularGridInterpolator(points, values) + sample = np.asarray([[0.1, 0.1, 1., .9], [0.2, 0.1, .45, .8], + [0.5, 0.5, .5, .5]]) + with pytest.raises(ValueError, match=match): + interp(sample, method=method) + + @pytest.mark.parametrize( + "points_values, sample", + [ + ( + _get_sample_4d, + np.asarray( + [[0.1, 0.1, 1.0, 0.9], + [0.2, 0.1, 0.45, 0.8], + [0.5, 0.5, 0.5, 0.5]] + ), + ), + (_get_sample_4d_2, np.asarray([0.1, 0.1, 10.0, 9.0])), + ], + ) + def test_linear_and_slinear_close(self, points_values, sample): + points, values = points_values(self) + interp = RegularGridInterpolator(points, values, method="linear") + v1 = interp(sample) + interp = RegularGridInterpolator(points, values, method="slinear") + v2 = interp(sample) + xp_assert_close(v1, v2) + + def test_derivatives(self): + points, values = self._get_sample_4d() + sample = np.array([[0.1 , 0.1 , 1. , 0.9 ], + [0.2 , 0.1 , 0.45, 0.8 ], + [0.5 , 0.5 , 0.5 , 0.5 ]]) + interp = RegularGridInterpolator(points, values, method="slinear") + + with assert_raises(ValueError): + # wrong number of derivatives (need 4) + interp(sample, nu=1) + + xp_assert_close(interp(sample, nu=(1, 0, 0, 0)), + np.asarray([1.0, 1, 1]), atol=1e-15) + xp_assert_close(interp(sample, nu=(0, 1, 0, 0)), + np.asarray([10.0, 10, 10]), atol=1e-15) + + # 2nd derivatives of a linear function are zero + xp_assert_close(interp(sample, nu=(0, 1, 1, 0)), + np.asarray([0.0, 0, 0]), atol=2e-12) + + @parametrize_rgi_interp_methods + def test_complex(self, method): + if method == "pchip": + pytest.skip("pchip does not make sense for complex data") + points, values = self._get_sample_4d_3() + values = values - 2j*values + sample = np.asarray([[0.1, 0.1, 1., .9], [0.2, 0.1, .45, .8], + [0.5, 0.5, .5, .5]]) + + interp = RegularGridInterpolator(points, values, method=method) + rinterp = RegularGridInterpolator(points, values.real, method=method) + iinterp = RegularGridInterpolator(points, values.imag, method=method) + + v1 = interp(sample) + v2 = rinterp(sample) + 1j*iinterp(sample) + xp_assert_close(v1, v2) + + def test_cubic_vs_pchip(self): + x, y = [1, 2, 3, 4], [1, 2, 3, 4] + xg, yg = np.meshgrid(x, y, indexing='ij') + + values = (lambda x, y: x**4 * y**4)(xg, yg) + cubic = RegularGridInterpolator((x, y), values, method='cubic') + pchip = RegularGridInterpolator((x, y), values, method='pchip') + + vals_cubic = cubic([1.5, 2]) + vals_pchip = pchip([1.5, 2]) + assert not np.allclose(vals_cubic, vals_pchip, atol=1e-14, rtol=0) + + def test_linear_xi1d(self): + points, values = self._get_sample_4d_2() + interp = RegularGridInterpolator(points, values) + sample = np.asarray([0.1, 0.1, 10., 9.]) + wanted = np.asarray([1001.1]) + assert_array_almost_equal(interp(sample), wanted) + + def test_linear_xi3d(self): + points, values = self._get_sample_4d() + interp = RegularGridInterpolator(points, values) + sample = np.asarray([[0.1, 0.1, 1., .9], [0.2, 0.1, .45, .8], + [0.5, 0.5, .5, .5]]) + wanted = np.asarray([1001.1, 846.2, 555.5]) + assert_array_almost_equal(interp(sample), wanted) + + @pytest.mark.parametrize( + "sample, wanted", + [ + (np.asarray([0.1, 0.1, 0.9, 0.9]), 1100.0), + (np.asarray([0.1, 0.1, 0.1, 0.1]), 0.0), + (np.asarray([0.0, 0.0, 0.0, 0.0]), 0.0), + (np.asarray([1.0, 1.0, 1.0, 1.0]), 1111.0), + (np.asarray([0.1, 0.4, 0.6, 0.9]), 1055.0), + ], + ) + def test_nearest(self, sample, wanted): + points, values = self._get_sample_4d() + interp = RegularGridInterpolator(points, values, method="nearest") + wanted = np.asarray([wanted]) + assert_array_almost_equal(interp(sample), wanted) + + def test_linear_edges(self): + points, values = self._get_sample_4d() + interp = RegularGridInterpolator(points, values) + sample = np.asarray([[0., 0., 0., 0.], [1., 1., 1., 1.]]) + wanted = np.asarray([0., 1111.]) + assert_array_almost_equal(interp(sample), wanted) + + def test_valid_create(self): + # create a 2-D grid of 3 points in each dimension + points = [(0., .5, 1.), (0., 1., .5)] + values = np.asarray([0., .5, 1.]) + values0 = values[:, np.newaxis] + values1 = values[np.newaxis, :] + values = (values0 + values1 * 10) + assert_raises(ValueError, RegularGridInterpolator, points, values) + points = [((0., .5, 1.), ), (0., .5, 1.)] + assert_raises(ValueError, RegularGridInterpolator, points, values) + points = [(0., .5, .75, 1.), (0., .5, 1.)] + assert_raises(ValueError, RegularGridInterpolator, points, values) + points = [(0., .5, 1.), (0., .5, 1.), (0., .5, 1.)] + assert_raises(ValueError, RegularGridInterpolator, points, values) + points = [(0., .5, 1.), (0., .5, 1.)] + assert_raises(ValueError, RegularGridInterpolator, points, values, + method="undefmethod") + + def test_valid_call(self): + points, values = self._get_sample_4d() + interp = RegularGridInterpolator(points, values) + sample = np.asarray([[0., 0., 0., 0.], [1., 1., 1., 1.]]) + assert_raises(ValueError, interp, sample, "undefmethod") + sample = np.asarray([[0., 0., 0.], [1., 1., 1.]]) + assert_raises(ValueError, interp, sample) + sample = np.asarray([[0., 0., 0., 0.], [1., 1., 1., 1.1]]) + assert_raises(ValueError, interp, sample) + + def test_out_of_bounds_extrap(self): + points, values = self._get_sample_4d() + interp = RegularGridInterpolator(points, values, bounds_error=False, + fill_value=None) + sample = np.asarray([[-.1, -.1, -.1, -.1], [1.1, 1.1, 1.1, 1.1], + [21, 2.1, -1.1, -11], [2.1, 2.1, -1.1, -1.1]]) + wanted = np.asarray([0., 1111., 11., 11.]) + assert_array_almost_equal(interp(sample, method="nearest"), wanted) + wanted = np.asarray([-111.1, 1222.1, -11068., -1186.9]) + assert_array_almost_equal(interp(sample, method="linear"), wanted) + + def test_out_of_bounds_extrap2(self): + points, values = self._get_sample_4d_2() + interp = RegularGridInterpolator(points, values, bounds_error=False, + fill_value=None) + sample = np.asarray([[-.1, -.1, -.1, -.1], [1.1, 1.1, 1.1, 1.1], + [21, 2.1, -1.1, -11], [2.1, 2.1, -1.1, -1.1]]) + wanted = np.asarray([0., 11., 11., 11.]) + assert_array_almost_equal(interp(sample, method="nearest"), wanted) + wanted = np.asarray([-12.1, 133.1, -1069., -97.9]) + assert_array_almost_equal(interp(sample, method="linear"), wanted) + + def test_out_of_bounds_fill(self): + points, values = self._get_sample_4d() + interp = RegularGridInterpolator(points, values, bounds_error=False, + fill_value=np.nan) + sample = np.asarray([[-.1, -.1, -.1, -.1], [1.1, 1.1, 1.1, 1.1], + [2.1, 2.1, -1.1, -1.1]]) + wanted = np.asarray([np.nan, np.nan, np.nan]) + assert_array_almost_equal(interp(sample, method="nearest"), wanted) + assert_array_almost_equal(interp(sample, method="linear"), wanted) + sample = np.asarray([[0.1, 0.1, 1., .9], [0.2, 0.1, .45, .8], + [0.5, 0.5, .5, .5]]) + wanted = np.asarray([1001.1, 846.2, 555.5]) + assert_array_almost_equal(interp(sample), wanted) + + def test_nearest_compare_qhull(self): + points, values = self._get_sample_4d() + interp = RegularGridInterpolator(points, values, method="nearest") + points_qhull = itertools.product(*points) + points_qhull = [p for p in points_qhull] + points_qhull = np.asarray(points_qhull) + values_qhull = values.reshape(-1) + interp_qhull = NearestNDInterpolator(points_qhull, values_qhull) + sample = np.asarray([[0.1, 0.1, 1., .9], [0.2, 0.1, .45, .8], + [0.5, 0.5, .5, .5]]) + assert_array_almost_equal(interp(sample), interp_qhull(sample)) + + def test_linear_compare_qhull(self): + points, values = self._get_sample_4d() + interp = RegularGridInterpolator(points, values) + points_qhull = itertools.product(*points) + points_qhull = [p for p in points_qhull] + points_qhull = np.asarray(points_qhull) + values_qhull = values.reshape(-1) + interp_qhull = LinearNDInterpolator(points_qhull, values_qhull) + sample = np.asarray([[0.1, 0.1, 1., .9], [0.2, 0.1, .45, .8], + [0.5, 0.5, .5, .5]]) + assert_array_almost_equal(interp(sample), interp_qhull(sample)) + + @pytest.mark.parametrize("method", ["nearest", "linear"]) + def test_duck_typed_values(self, method): + x = np.linspace(0, 2, 5) + y = np.linspace(0, 1, 7) + + values = MyValue((5, 7)) + + interp = RegularGridInterpolator((x, y), values, method=method) + v1 = interp([0.4, 0.7]) + + interp = RegularGridInterpolator((x, y), values._v, method=method) + v2 = interp([0.4, 0.7]) + xp_assert_close(v1, v2, check_dtype=False) + + def test_invalid_fill_value(self): + np.random.seed(1234) + x = np.linspace(0, 2, 5) + y = np.linspace(0, 1, 7) + values = np.random.rand(5, 7) + + # integers can be cast to floats + RegularGridInterpolator((x, y), values, fill_value=1) + + # complex values cannot + assert_raises(ValueError, RegularGridInterpolator, + (x, y), values, fill_value=1+2j) + + def test_fillvalue_type(self): + # from #3703; test that interpolator object construction succeeds + values = np.ones((10, 20, 30), dtype='>f4') + points = [np.arange(n) for n in values.shape] + # xi = [(1, 1, 1)] + RegularGridInterpolator(points, values) + RegularGridInterpolator(points, values, fill_value=0.) + + def test_length_one_axis(self): + # gh-5890, gh-9524 : length-1 axis is legal for method='linear'. + # Along the axis it's linear interpolation; away from the length-1 + # axis, it's an extrapolation, so fill_value should be used. + def f(x, y): + return x + y + x = np.linspace(1, 1, 1) + y = np.linspace(1, 10, 10) + data = f(*np.meshgrid(x, y, indexing="ij", sparse=True)) + + interp = RegularGridInterpolator((x, y), data, method="linear", + bounds_error=False, fill_value=101) + + # check values at the grid + xp_assert_close(interp(np.array([[1, 1], [1, 5], [1, 10]])), + np.asarray([2.0, 6, 11]), + atol=1e-14) + + # check off-grid interpolation is indeed linear + xp_assert_close(interp(np.array([[1, 1.4], [1, 5.3], [1, 10]])), + [2.4, 6.3, 11], + atol=1e-14) + + # check exrapolation w/ fill_value + xp_assert_close(interp(np.array([1.1, 2.4])), + interp.fill_value, + check_dtype=False, check_shape=False, check_0d=False, + atol=1e-14) + + # check extrapolation: linear along the `y` axis, const along `x` + interp.fill_value = None + xp_assert_close(interp([[1, 0.3], [1, 11.5]]), + [1.3, 12.5], atol=1e-15) + + xp_assert_close(interp([[1.5, 0.3], [1.9, 11.5]]), + [1.3, 12.5], atol=1e-15) + + # extrapolation with method='nearest' + interp = RegularGridInterpolator((x, y), data, method="nearest", + bounds_error=False, fill_value=None) + xp_assert_close(interp([[1.5, 1.8], [-4, 5.1]]), + np.asarray([3.0, 6]), + atol=1e-15) + + @pytest.mark.parametrize("fill_value", [None, np.nan, np.pi]) + @pytest.mark.parametrize("method", ['linear', 'nearest']) + def test_length_one_axis2(self, fill_value, method): + options = {"fill_value": fill_value, "bounds_error": False, + "method": method} + + x = np.linspace(0, 2*np.pi, 20) + z = np.sin(x) + + fa = RegularGridInterpolator((x,), z[:], **options) + fb = RegularGridInterpolator((x, [0]), z[:, None], **options) + + x1a = np.linspace(-1, 2*np.pi+1, 100) + za = fa(x1a) + + # evaluated at provided y-value, fb should behave exactly as fa + y1b = np.zeros(100) + zb = fb(np.vstack([x1a, y1b]).T) + xp_assert_close(zb, za) + + # evaluated at a different y-value, fb should return fill value + y1b = np.ones(100) + zb = fb(np.vstack([x1a, y1b]).T) + if fill_value is None: + xp_assert_close(zb, za) + else: + xp_assert_close(zb, np.full_like(zb, fill_value)) + + @pytest.mark.parametrize("method", ['nearest', 'linear']) + def test_nan_x_1d(self, method): + # gh-6624 : if x is nan, result should be nan + f = RegularGridInterpolator(([1, 2, 3],), [10, 20, 30], fill_value=1, + bounds_error=False, method=method) + assert np.isnan(f([np.nan])) + + # test arbitrary nan pattern + rng = np.random.default_rng(8143215468) + x = rng.random(size=100)*4 + i = rng.random(size=100) > 0.5 + x[i] = np.nan + with np.errstate(invalid='ignore'): + # out-of-bounds comparisons, `out_of_bounds += x < grid[0]`, + # generate numpy warnings if `x` contains nans. + # These warnings should propagate to user (since `x` is user + # input) and we simply filter them out. + res = f(x) + + assert np.isnan(res[i]).all() + xp_assert_equal(res[~i], f(x[~i])) + + # also test the length-one axis f(nan) + x = [1, 2, 3] + y = [1, ] + data = np.ones((3, 1)) + f = RegularGridInterpolator((x, y), data, fill_value=1, + bounds_error=False, method=method) + assert np.all(np.isnan(f([np.nan, 1]))) + assert np.all(np.isnan(f([1, np.nan]))) + + @pytest.mark.parametrize("method", ['nearest', 'linear']) + def test_nan_x_2d(self, method): + x, y = np.array([0, 1, 2]), np.array([1, 3, 7]) + + def f(x, y): + return x**2 + y**2 + + xg, yg = np.meshgrid(x, y, indexing='ij', sparse=True) + data = f(xg, yg) + interp = RegularGridInterpolator((x, y), data, + method=method, bounds_error=False) + + with np.errstate(invalid='ignore'): + res = interp([[1.5, np.nan], [1, 1]]) + xp_assert_close(res[1], 2.0, atol=1e-14) + assert np.isnan(res[0]) + + # test arbitrary nan pattern + rng = np.random.default_rng(8143215468) + x = rng.random(size=100)*4-1 + y = rng.random(size=100)*8 + i1 = rng.random(size=100) > 0.5 + i2 = rng.random(size=100) > 0.5 + i = i1 | i2 + x[i1] = np.nan + y[i2] = np.nan + z = np.array([x, y]).T + with np.errstate(invalid='ignore'): + # out-of-bounds comparisons, `out_of_bounds += x < grid[0]`, + # generate numpy warnings if `x` contains nans. + # These warnings should propagate to user (since `x` is user + # input) and we simply filter them out. + res = interp(z) + + assert np.isnan(res[i]).all() + xp_assert_equal(res[~i], interp(z[~i]), check_dtype=False) + + @pytest.mark.fail_slow(10) + @parametrize_rgi_interp_methods + @pytest.mark.parametrize(("ndims", "func"), [ + (2, lambda x, y: 2 * x ** 3 + 3 * y ** 2), + (3, lambda x, y, z: 2 * x ** 3 + 3 * y ** 2 - z), + (4, lambda x, y, z, a: 2 * x ** 3 + 3 * y ** 2 - z + a), + (5, lambda x, y, z, a, b: 2 * x ** 3 + 3 * y ** 2 - z + a * b), + ]) + def test_descending_points_nd(self, method, ndims, func): + + if ndims >= 4 and method in {"cubic", "quintic"}: + pytest.skip("too slow; OOM (quintic); or nearly so (cubic)") + + rng = np.random.default_rng(42) + sample_low = 1 + sample_high = 5 + test_points = rng.uniform(sample_low, sample_high, size=(2, ndims)) + + ascending_points = [np.linspace(sample_low, sample_high, 12) + for _ in range(ndims)] + + ascending_values = func(*np.meshgrid(*ascending_points, + indexing="ij", + sparse=True)) + + ascending_interp = RegularGridInterpolator(ascending_points, + ascending_values, + method=method) + ascending_result = ascending_interp(test_points) + + descending_points = [xi[::-1] for xi in ascending_points] + descending_values = func(*np.meshgrid(*descending_points, + indexing="ij", + sparse=True)) + descending_interp = RegularGridInterpolator(descending_points, + descending_values, + method=method) + descending_result = descending_interp(test_points) + + xp_assert_equal(ascending_result, descending_result) + + def test_invalid_points_order(self): + def val_func_2d(x, y): + return 2 * x ** 3 + 3 * y ** 2 + + x = np.array([.5, 2., 0., 4., 5.5]) # not ascending or descending + y = np.array([.5, 2., 3., 4., 5.5]) + points = (x, y) + values = val_func_2d(*np.meshgrid(*points, indexing='ij', + sparse=True)) + match = "must be strictly ascending or descending" + with pytest.raises(ValueError, match=match): + RegularGridInterpolator(points, values) + + @parametrize_rgi_interp_methods + def test_fill_value(self, method): + interp = RegularGridInterpolator([np.arange(6)], np.ones(6), + method=method, bounds_error=False) + assert np.isnan(interp([10])) + + @pytest.mark.fail_slow(5) + @parametrize_rgi_interp_methods + def test_nonscalar_values(self, method): + + if method == "quintic": + pytest.skip("Way too slow.") + + # Verify that non-scalar valued values also works + points = [(0.0, 0.5, 1.0, 1.5, 2.0, 2.5)] * 2 + [ + (0.0, 5.0, 10.0, 15.0, 20, 25.0) + ] * 2 + + rng = np.random.default_rng(1234) + values = rng.random((6, 6, 6, 6, 8)) + sample = rng.random((7, 3, 4)) + + interp = RegularGridInterpolator(points, values, method=method, + bounds_error=False) + v = interp(sample) + assert v.shape == (7, 3, 8), method + + vs = [] + for j in range(8): + interp = RegularGridInterpolator(points, values[..., j], + method=method, + bounds_error=False) + vs.append(interp(sample)) + v2 = np.array(vs).transpose(1, 2, 0) + + xp_assert_close(v, v2, atol=1e-14, err_msg=method) + + @parametrize_rgi_interp_methods + @pytest.mark.parametrize("flip_points", [False, True]) + def test_nonscalar_values_2(self, method, flip_points): + + if method in {"cubic", "quintic"}: + pytest.skip("Way too slow.") + + # Verify that non-scalar valued values also work : use different + # lengths of axes to simplify tracing the internals + points = [(0.0, 0.5, 1.0, 1.5, 2.0, 2.5), + (0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0), + (0.0, 5.0, 10.0, 15.0, 20, 25.0, 35.0, 36.0), + (0.0, 5.0, 10.0, 15.0, 20, 25.0, 35.0, 36.0, 47)] + + # verify, that strictly decreasing dimensions work + if flip_points: + points = [tuple(reversed(p)) for p in points] + + rng = np.random.default_rng(1234) + + trailing_points = (3, 2) + # NB: values has a `num_trailing_dims` trailing dimension + values = rng.random((6, 7, 8, 9, *trailing_points)) + sample = rng.random(4) # a single sample point ! + + interp = RegularGridInterpolator(points, values, method=method, + bounds_error=False) + v = interp(sample) + + # v has a single sample point *per entry in the trailing dimensions* + assert v.shape == (1, *trailing_points) + + # check the values, too : manually loop over the trailing dimensions + vs = np.empty(values.shape[-2:]) + for i in range(values.shape[-2]): + for j in range(values.shape[-1]): + interp = RegularGridInterpolator(points, values[..., i, j], + method=method, + bounds_error=False) + vs[i, j] = interp(sample).item() + v2 = np.expand_dims(vs, axis=0) + xp_assert_close(v, v2, atol=1e-14, err_msg=method) + + def test_nonscalar_values_linear_2D(self): + # Verify that non-scalar values work in the 2D fast path + method = 'linear' + points = [(0.0, 0.5, 1.0, 1.5, 2.0, 2.5), + (0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0), ] + + rng = np.random.default_rng(1234) + + trailing_points = (3, 4) + # NB: values has a `num_trailing_dims` trailing dimension + values = rng.random((6, 7, *trailing_points)) + sample = rng.random(2) # a single sample point ! + + interp = RegularGridInterpolator(points, values, method=method, + bounds_error=False) + v = interp(sample) + + # v has a single sample point *per entry in the trailing dimensions* + assert v.shape == (1, *trailing_points) + + # check the values, too : manually loop over the trailing dimensions + vs = np.empty(values.shape[-2:]) + for i in range(values.shape[-2]): + for j in range(values.shape[-1]): + interp = RegularGridInterpolator(points, values[..., i, j], + method=method, + bounds_error=False) + vs[i, j] = interp(sample).item() + v2 = np.expand_dims(vs, axis=0) + xp_assert_close(v, v2, atol=1e-14, err_msg=method) + + @pytest.mark.parametrize( + "dtype", + [np.float32, np.float64, np.complex64, np.complex128] + ) + @pytest.mark.parametrize("xi_dtype", [np.float32, np.float64]) + def test_float32_values(self, dtype, xi_dtype): + # regression test for gh-17718: values.dtype=float32 fails + def f(x, y): + return 2 * x**3 + 3 * y**2 + + x = np.linspace(1, 4, 11) + y = np.linspace(4, 7, 22) + + xg, yg = np.meshgrid(x, y, indexing='ij', sparse=True) + data = f(xg, yg) + + data = data.astype(dtype) + + interp = RegularGridInterpolator((x, y), data) + + pts = np.array([[2.1, 6.2], + [3.3, 5.2]], dtype=xi_dtype) + + # the values here are just what the call returns; the test checks that + # that the call succeeds at all, instead of failing with cython not + # having a float32 kernel + xp_assert_close(interp(pts), [134.10469388, 153.40069388], + atol=1e-7, rtol=1e-7, check_dtype=False) + + def test_bad_solver(self): + x = np.linspace(0, 3, 7) + y = np.linspace(0, 3, 7) + xg, yg = np.meshgrid(x, y, indexing='ij', sparse=True) + data = xg + yg + + # default method 'linear' does not accept 'solver' + with assert_raises(ValueError): + RegularGridInterpolator((x, y), data, solver=lambda x: x) + + with assert_raises(TypeError): + # wrong solver interface + RegularGridInterpolator( + (x, y), data, method='slinear', solver=lambda x: x + ) + + with assert_raises(TypeError): + # unknown argument + RegularGridInterpolator( + (x, y), data, method='slinear', solver=lambda x: x, woof='woof' + ) + + with assert_raises(TypeError): + # unknown argument + RegularGridInterpolator( + (x, y), data, method='slinear', solver_args={'woof': 42} + ) + + @pytest.mark.thread_unsafe + def test_concurrency(self): + points, values = self._get_sample_4d() + sample = np.array([[0.1 , 0.1 , 1. , 0.9 ], + [0.2 , 0.1 , 0.45, 0.8 ], + [0.5 , 0.5 , 0.5 , 0.5 ], + [0.3 , 0.1 , 0.2 , 0.4 ]]) + interp = RegularGridInterpolator(points, values, method="slinear") + + # A call to RGI with a method different from the one specified on the + # constructor, should not mutate it. + methods = ['slinear', 'nearest'] + def worker_fn(tid, interp): + spline = interp._spline + method = methods[tid % 2] + interp(sample, method=method) + assert interp._spline is spline + + _run_concurrent_barrier(10, worker_fn, interp) + + +class MyValue: + """ + Minimal indexable object + """ + + def __init__(self, shape): + self.ndim = 2 + self.shape = shape + self._v = np.arange(np.prod(shape)).reshape(shape) + + def __getitem__(self, idx): + return self._v[idx] + + def __array_interface__(self): + return None + + def __array__(self, dtype=None, copy=None): + raise RuntimeError("No array representation") + + +class TestInterpN: + def _sample_2d_data(self): + x = np.array([.5, 2., 3., 4., 5.5, 6.]) + y = np.array([.5, 2., 3., 4., 5.5, 6.]) + z = np.array( + [ + [1, 2, 1, 2, 1, 1], + [1, 2, 1, 2, 1, 1], + [1, 2, 3, 2, 1, 1], + [1, 2, 2, 2, 1, 1], + [1, 2, 1, 2, 1, 1], + [1, 2, 2, 2, 1, 1], + ] + ) + return x, y, z + + def test_spline_2d(self): + x, y, z = self._sample_2d_data() + lut = RectBivariateSpline(x, y, z) + + xi = np.array([[1, 2.3, 5.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, 4.0, 5.0, 1.0, 3]]).T + assert_array_almost_equal(interpn((x, y), z, xi, method="splinef2d"), + lut.ev(xi[:, 0], xi[:, 1])) + + @parametrize_rgi_interp_methods + def test_list_input(self, method): + x, y, z = self._sample_2d_data() + xi = np.array([[1, 2.3, 5.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, 4.0, 5.0, 1.0, 3]]).T + v1 = interpn((x, y), z, xi, method=method) + v2 = interpn( + (x.tolist(), y.tolist()), z.tolist(), xi.tolist(), method=method + ) + xp_assert_close(v1, v2, err_msg=method) + + def test_spline_2d_outofbounds(self): + x = np.array([.5, 2., 3., 4., 5.5]) + y = np.array([.5, 2., 3., 4., 5.5]) + z = np.array([[1, 2, 1, 2, 1], [1, 2, 1, 2, 1], [1, 2, 3, 2, 1], + [1, 2, 2, 2, 1], [1, 2, 1, 2, 1]]) + lut = RectBivariateSpline(x, y, z) + + xi = np.array([[1, 2.3, 6.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, -4.0, 5.0, 1.0, 3]]).T + actual = interpn((x, y), z, xi, method="splinef2d", + bounds_error=False, fill_value=999.99) + expected = lut.ev(xi[:, 0], xi[:, 1]) + expected[2:4] = 999.99 + assert_array_almost_equal(actual, expected) + + # no extrapolation for splinef2d + assert_raises(ValueError, interpn, (x, y), z, xi, method="splinef2d", + bounds_error=False, fill_value=None) + + def _sample_4d_data(self): + points = [(0., .5, 1.)] * 2 + [(0., 5., 10.)] * 2 + values = np.asarray([0., .5, 1.]) + values0 = values[:, np.newaxis, np.newaxis, np.newaxis] + values1 = values[np.newaxis, :, np.newaxis, np.newaxis] + values2 = values[np.newaxis, np.newaxis, :, np.newaxis] + values3 = values[np.newaxis, np.newaxis, np.newaxis, :] + values = (values0 + values1 * 10 + values2 * 100 + values3 * 1000) + return points, values + + def test_linear_4d(self): + # create a 4-D grid of 3 points in each dimension + points, values = self._sample_4d_data() + interp_rg = RegularGridInterpolator(points, values) + sample = np.asarray([[0.1, 0.1, 10., 9.]]) + wanted = interpn(points, values, sample, method="linear") + assert_array_almost_equal(interp_rg(sample), wanted) + + def test_4d_linear_outofbounds(self): + # create a 4-D grid of 3 points in each dimension + points, values = self._sample_4d_data() + sample = np.asarray([[0.1, -0.1, 10.1, 9.]]) + wanted = np.asarray([999.99]) + actual = interpn(points, values, sample, method="linear", + bounds_error=False, fill_value=999.99) + assert_array_almost_equal(actual, wanted) + + def test_nearest_4d(self): + # create a 4-D grid of 3 points in each dimension + points, values = self._sample_4d_data() + interp_rg = RegularGridInterpolator(points, values, method="nearest") + sample = np.asarray([[0.1, 0.1, 10., 9.]]) + wanted = interpn(points, values, sample, method="nearest") + assert_array_almost_equal(interp_rg(sample), wanted) + + def test_4d_nearest_outofbounds(self): + # create a 4-D grid of 3 points in each dimension + points, values = self._sample_4d_data() + sample = np.asarray([[0.1, -0.1, 10.1, 9.]]) + wanted = np.asarray([999.99]) + actual = interpn(points, values, sample, method="nearest", + bounds_error=False, fill_value=999.99) + assert_array_almost_equal(actual, wanted) + + def test_xi_1d(self): + # verify that 1-D xi works as expected + points, values = self._sample_4d_data() + sample = np.asarray([0.1, 0.1, 10., 9.]) + v1 = interpn(points, values, sample, bounds_error=False) + v2 = interpn(points, values, sample[None,:], bounds_error=False) + xp_assert_close(v1, v2) + + def test_xi_nd(self): + # verify that higher-d xi works as expected + points, values = self._sample_4d_data() + + np.random.seed(1234) + sample = np.random.rand(2, 3, 4) + + v1 = interpn(points, values, sample, method='nearest', + bounds_error=False) + assert v1.shape == (2, 3) + + v2 = interpn(points, values, sample.reshape(-1, 4), + method='nearest', bounds_error=False) + xp_assert_close(v1, v2.reshape(v1.shape)) + + @parametrize_rgi_interp_methods + def test_xi_broadcast(self, method): + # verify that the interpolators broadcast xi + x, y, values = self._sample_2d_data() + points = (x, y) + + xi = np.linspace(0, 1, 2) + yi = np.linspace(0, 3, 3) + + sample = (xi[:, None], yi[None, :]) + v1 = interpn(points, values, sample, method=method, bounds_error=False) + assert v1.shape == (2, 3) + + xx, yy = np.meshgrid(xi, yi) + sample = np.c_[xx.T.ravel(), yy.T.ravel()] + + v2 = interpn(points, values, sample, + method=method, bounds_error=False) + xp_assert_close(v1, v2.reshape(v1.shape)) + + @pytest.mark.fail_slow(5) + @parametrize_rgi_interp_methods + def test_nonscalar_values(self, method): + + if method == "quintic": + pytest.skip("Way too slow.") + + # Verify that non-scalar valued values also works + points = [(0.0, 0.5, 1.0, 1.5, 2.0, 2.5)] * 2 + [ + (0.0, 5.0, 10.0, 15.0, 20, 25.0) + ] * 2 + + rng = np.random.default_rng(1234) + values = rng.random((6, 6, 6, 6, 8)) + sample = rng.random((7, 3, 4)) + + v = interpn(points, values, sample, method=method, + bounds_error=False) + assert v.shape == (7, 3, 8), method + + vs = [interpn(points, values[..., j], sample, method=method, + bounds_error=False) for j in range(8)] + v2 = np.array(vs).transpose(1, 2, 0) + + xp_assert_close(v, v2, atol=1e-14, err_msg=method) + + @parametrize_rgi_interp_methods + def test_nonscalar_values_2(self, method): + + if method in {"cubic", "quintic"}: + pytest.skip("Way too slow.") + + # Verify that non-scalar valued values also work : use different + # lengths of axes to simplify tracing the internals + points = [(0.0, 0.5, 1.0, 1.5, 2.0, 2.5), + (0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0), + (0.0, 5.0, 10.0, 15.0, 20, 25.0, 35.0, 36.0), + (0.0, 5.0, 10.0, 15.0, 20, 25.0, 35.0, 36.0, 47)] + + rng = np.random.default_rng(1234) + + trailing_points = (3, 2) + # NB: values has a `num_trailing_dims` trailing dimension + values = rng.random((6, 7, 8, 9, *trailing_points)) + sample = rng.random(4) # a single sample point ! + + v = interpn(points, values, sample, method=method, bounds_error=False) + + # v has a single sample point *per entry in the trailing dimensions* + assert v.shape == (1, *trailing_points) + + # check the values, too : manually loop over the trailing dimensions + vs = [[ + interpn(points, values[..., i, j], sample, method=method, + bounds_error=False) for i in range(values.shape[-2]) + ] for j in range(values.shape[-1])] + + xp_assert_close(v, np.asarray(vs).T, atol=1e-14, err_msg=method) + + def test_non_scalar_values_splinef2d(self): + # Vector-valued splines supported with fitpack + points, values = self._sample_4d_data() + + np.random.seed(1234) + values = np.random.rand(3, 3, 3, 3, 6) + sample = np.random.rand(7, 11, 4) + assert_raises(ValueError, interpn, points, values, sample, + method='splinef2d') + + @parametrize_rgi_interp_methods + def test_complex(self, method): + if method == "pchip": + pytest.skip("pchip does not make sense for complex data") + + x, y, values = self._sample_2d_data() + points = (x, y) + values = values - 2j*values + + sample = np.array([[1, 2.3, 5.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, 4.0, 5.0, 1.0, 3]]).T + + v1 = interpn(points, values, sample, method=method) + v2r = interpn(points, values.real, sample, method=method) + v2i = interpn(points, values.imag, sample, method=method) + v2 = v2r + 1j*v2i + + xp_assert_close(v1, v2) + + @pytest.mark.thread_unsafe + def test_complex_pchip(self): + # Complex-valued data deprecated for pchip + x, y, values = self._sample_2d_data() + points = (x, y) + values = values - 2j*values + + sample = np.array([[1, 2.3, 5.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, 4.0, 5.0, 1.0, 3]]).T + with pytest.raises(ValueError, match='real'): + interpn(points, values, sample, method='pchip') + + @pytest.mark.thread_unsafe + def test_complex_spline2fd(self): + # Complex-valued data not supported by spline2fd + x, y, values = self._sample_2d_data() + points = (x, y) + values = values - 2j*values + + sample = np.array([[1, 2.3, 5.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, 4.0, 5.0, 1.0, 3]]).T + with assert_warns(ComplexWarning): + interpn(points, values, sample, method='splinef2d') + + @pytest.mark.parametrize( + "method", + ["linear", "nearest"] + ) + def test_duck_typed_values(self, method): + x = np.linspace(0, 2, 5) + y = np.linspace(0, 1, 7) + + values = MyValue((5, 7)) + + v1 = interpn((x, y), values, [0.4, 0.7], method=method) + v2 = interpn((x, y), values._v, [0.4, 0.7], method=method) + xp_assert_close(v1, v2, check_dtype=False) + + @skip_xp_invalid_arg + @parametrize_rgi_interp_methods + def test_matrix_input(self, method): + """np.matrix inputs are allowed for backwards compatibility""" + x = np.linspace(0, 2, 6) + y = np.linspace(0, 1, 7) + + values = matrix(np.random.rand(6, 7)) + + sample = np.random.rand(3, 7, 2) + + v1 = interpn((x, y), values, sample, method=method) + v2 = interpn((x, y), np.asarray(values), sample, method=method) + if method == "quintic": + # https://github.com/scipy/scipy/issues/20472 + xp_assert_close(v1, v2, atol=5e-5, rtol=2e-6) + else: + xp_assert_close(v1, v2) + + def test_length_one_axis(self): + # gh-5890, gh-9524 : length-1 axis is legal for method='linear'. + # Along the axis it's linear interpolation; away from the length-1 + # axis, it's an extrapolation, so fill_value should be used. + + values = np.array([[0.1, 1, 10]]) + xi = np.array([[1, 2.2], [1, 3.2], [1, 3.8]]) + + res = interpn(([1], [2, 3, 4]), values, xi) + wanted = [0.9*0.2 + 0.1, # on [2, 3) it's 0.9*(x-2) + 0.1 + 9*0.2 + 1, # on [3, 4] it's 9*(x-3) + 1 + 9*0.8 + 1] + + xp_assert_close(res, wanted, atol=1e-15) + + # check extrapolation + xi = np.array([[1.1, 2.2], [1.5, 3.2], [-2.3, 3.8]]) + res = interpn(([1], [2, 3, 4]), values, xi, + bounds_error=False, fill_value=None) + + xp_assert_close(res, wanted, atol=1e-15) + + def test_descending_points(self): + def value_func_4d(x, y, z, a): + return 2 * x ** 3 + 3 * y ** 2 - z - a + + x1 = np.array([0, 1, 2, 3]) + x2 = np.array([0, 10, 20, 30]) + x3 = np.array([0, 10, 20, 30]) + x4 = np.array([0, .1, .2, .30]) + points = (x1, x2, x3, x4) + values = value_func_4d( + *np.meshgrid(*points, indexing='ij', sparse=True)) + pts = (0.1, 0.3, np.transpose(np.linspace(0, 30, 4)), + np.linspace(0, 0.3, 4)) + correct_result = interpn(points, values, pts) + + x1_descend = x1[::-1] + x2_descend = x2[::-1] + x3_descend = x3[::-1] + x4_descend = x4[::-1] + points_shuffled = (x1_descend, x2_descend, x3_descend, x4_descend) + values_shuffled = value_func_4d( + *np.meshgrid(*points_shuffled, indexing='ij', sparse=True)) + test_result = interpn(points_shuffled, values_shuffled, pts) + + xp_assert_equal(correct_result, test_result) + + def test_invalid_points_order(self): + x = np.array([.5, 2., 0., 4., 5.5]) # not ascending or descending + y = np.array([.5, 2., 3., 4., 5.5]) + z = np.array([[1, 2, 1, 2, 1], [1, 2, 1, 2, 1], [1, 2, 3, 2, 1], + [1, 2, 2, 2, 1], [1, 2, 1, 2, 1]]) + xi = np.array([[1, 2.3, 6.3, 0.5, 3.3, 1.2, 3], + [1, 3.3, 1.2, -4.0, 5.0, 1.0, 3]]).T + + match = "must be strictly ascending or descending" + with pytest.raises(ValueError, match=match): + interpn((x, y), z, xi) + + def test_invalid_xi_dimensions(self): + # https://github.com/scipy/scipy/issues/16519 + points = [(0, 1)] + values = [0, 1] + xi = np.ones((1, 1, 3)) + msg = ("The requested sample points xi have dimension 3, but this " + "RegularGridInterpolator has dimension 1") + with assert_raises(ValueError, match=msg): + interpn(points, values, xi) + + def test_readonly_grid(self): + # https://github.com/scipy/scipy/issues/17716 + x = np.linspace(0, 4, 5) + y = np.linspace(0, 5, 6) + z = np.linspace(0, 6, 7) + points = (x, y, z) + values = np.ones((5, 6, 7)) + point = np.array([2.21, 3.12, 1.15]) + for d in points: + d.flags.writeable = False + values.flags.writeable = False + point.flags.writeable = False + interpn(points, values, point) + RegularGridInterpolator(points, values)(point) + + def test_2d_readonly_grid(self): + # https://github.com/scipy/scipy/issues/17716 + # test special 2d case + x = np.linspace(0, 4, 5) + y = np.linspace(0, 5, 6) + points = (x, y) + values = np.ones((5, 6)) + point = np.array([2.21, 3.12]) + for d in points: + d.flags.writeable = False + values.flags.writeable = False + point.flags.writeable = False + interpn(points, values, point) + RegularGridInterpolator(points, values)(point) + + def test_non_c_contiguous_grid(self): + # https://github.com/scipy/scipy/issues/17716 + x = np.linspace(0, 4, 5) + x = np.vstack((x, np.empty_like(x))).T.copy()[:, 0] + assert not x.flags.c_contiguous + y = np.linspace(0, 5, 6) + z = np.linspace(0, 6, 7) + points = (x, y, z) + values = np.ones((5, 6, 7)) + point = np.array([2.21, 3.12, 1.15]) + interpn(points, values, point) + RegularGridInterpolator(points, values)(point) + + @pytest.mark.parametrize("dtype", ['>f8', '`__ + +MATLAB® files +============= + +.. autosummary:: + :toctree: generated/ + + loadmat - Read a MATLAB style mat file (version 4 through 7.1) + savemat - Write a MATLAB style mat file (version 4 through 7.1) + whosmat - List contents of a MATLAB style mat file (version 4 through 7.1) + +For low-level MATLAB reading and writing utilities, see `scipy.io.matlab`. + +IDL® files +========== + +.. autosummary:: + :toctree: generated/ + + readsav - Read an IDL 'save' file + +Matrix Market files +=================== + +.. autosummary:: + :toctree: generated/ + + mminfo - Query matrix info from Matrix Market formatted file + mmread - Read matrix from Matrix Market formatted file + mmwrite - Write matrix to Matrix Market formatted file + +Unformatted Fortran files +=============================== + +.. autosummary:: + :toctree: generated/ + + FortranFile - A file object for unformatted sequential Fortran files + FortranEOFError - Exception indicating the end of a well-formed file + FortranFormattingError - Exception indicating an inappropriate end + +Netcdf +====== + +.. autosummary:: + :toctree: generated/ + + netcdf_file - A file object for NetCDF data + netcdf_variable - A data object for the netcdf module + +Harwell-Boeing files +==================== + +.. autosummary:: + :toctree: generated/ + + hb_read -- read H-B file + hb_write -- write H-B file + +Wav sound files (:mod:`scipy.io.wavfile`) +========================================= + +.. module:: scipy.io.wavfile + +.. autosummary:: + :toctree: generated/ + + read + write + WavFileWarning + +Arff files (:mod:`scipy.io.arff`) +================================= + +.. module:: scipy.io.arff + +.. autosummary:: + :toctree: generated/ + + loadarff + MetaData + ArffError + ParseArffError +""" +# matfile read and write +from .matlab import loadmat, savemat, whosmat + +# netCDF file support +from ._netcdf import netcdf_file, netcdf_variable + +# Fortran file support +from ._fortran import FortranFile, FortranEOFError, FortranFormattingError + +from ._fast_matrix_market import mminfo, mmread, mmwrite +from ._idl import readsav +from ._harwell_boeing import hb_read, hb_write + +# Deprecated namespaces, to be removed in v2.0.0 +from . import arff, harwell_boeing, idl, mmio, netcdf, wavfile + +__all__ = [s for s in dir() if not s.startswith('_')] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_fast_matrix_market/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_fast_matrix_market/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..cfd6f8fb30ea8dcc2a9de7b1be23a9538c130718 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_fast_matrix_market/__init__.py @@ -0,0 +1,598 @@ +# Copyright (C) 2022-2023 Adam Lugowski. All rights reserved. +# Use of this source code is governed by the BSD 2-clause license found in +# the LICENSE.txt file. +# SPDX-License-Identifier: BSD-2-Clause +""" +Matrix Market I/O with a C++ backend. +See http://math.nist.gov/MatrixMarket/formats.html +for information about the Matrix Market format. + +.. versionadded:: 1.12.0 +""" +import io +import os + +import numpy as np +from scipy.sparse import coo_array, issparse, coo_matrix +from scipy.io import _mmio + +__all__ = ['mminfo', 'mmread', 'mmwrite'] + +PARALLELISM = 0 +""" +Number of threads that `mmread()` and `mmwrite()` use. +0 means number of CPUs in the system. +Use `threadpoolctl` to set this value. +""" + +ALWAYS_FIND_SYMMETRY = False +""" +Whether mmwrite() with symmetry='AUTO' will always search for symmetry +inside the matrix. This is scipy.io._mmio.mmwrite()'s default behavior, +but has a significant performance cost on large matrices. +""" + +_field_to_dtype = { + "integer": "int64", + "unsigned-integer": "uint64", + "real": "float64", + "complex": "complex", + "pattern": "float64", +} + + +def _fmm_version(): + from . import _fmm_core + return _fmm_core.__version__ + + +# Register with threadpoolctl, if available +try: + import threadpoolctl + + class _FMMThreadPoolCtlController(threadpoolctl.LibController): + user_api = "scipy" + internal_api = "scipy_mmio" + + filename_prefixes = ("_fmm_core",) + + def get_num_threads(self): + global PARALLELISM + return PARALLELISM + + def set_num_threads(self, num_threads): + global PARALLELISM + PARALLELISM = num_threads + + def get_version(self): + return _fmm_version + + def set_additional_attributes(self): + pass + + threadpoolctl.register(_FMMThreadPoolCtlController) +except (ImportError, AttributeError): + # threadpoolctl not installed or version too old + pass + + +class _TextToBytesWrapper(io.BufferedReader): + """ + Convert a TextIOBase string stream to a byte stream. + """ + + def __init__(self, text_io_buffer, encoding=None, errors=None, **kwargs): + super().__init__(text_io_buffer, **kwargs) + self.encoding = encoding or text_io_buffer.encoding or 'utf-8' + self.errors = errors or text_io_buffer.errors or 'strict' + + def __del__(self): + # do not close the wrapped stream + self.detach() + + def _encoding_call(self, method_name, *args, **kwargs): + raw_method = getattr(self.raw, method_name) + val = raw_method(*args, **kwargs) + return val.encode(self.encoding, errors=self.errors) + + def read(self, size=-1): + return self._encoding_call('read', size) + + def read1(self, size=-1): + return self._encoding_call('read1', size) + + def peek(self, size=-1): + return self._encoding_call('peek', size) + + def seek(self, offset, whence=0): + # Random seeks are not allowed because of non-trivial conversion + # between byte and character offsets, + # with the possibility of a byte offset landing within a character. + if offset == 0 and whence == 0 or \ + offset == 0 and whence == 2: + # seek to start or end is ok + super().seek(offset, whence) + else: + # Drop any other seek + # In this application this may happen when pystreambuf seeks during sync(), + # which can happen when closing a partially-read stream. + # Ex. when mminfo() only reads the header then exits. + pass + + +def _read_body_array(cursor): + """ + Read MatrixMarket array body + """ + from . import _fmm_core + + vals = np.zeros(cursor.header.shape, dtype=_field_to_dtype.get(cursor.header.field)) + _fmm_core.read_body_array(cursor, vals) + return vals + + +def _read_body_coo(cursor, generalize_symmetry=True): + """ + Read MatrixMarket coordinate body + """ + from . import _fmm_core + + index_dtype = "int32" + if cursor.header.nrows >= 2**31 or cursor.header.ncols >= 2**31: + # Dimensions are too large to fit in int32 + index_dtype = "int64" + + i = np.zeros(cursor.header.nnz, dtype=index_dtype) + j = np.zeros(cursor.header.nnz, dtype=index_dtype) + data = np.zeros(cursor.header.nnz, dtype=_field_to_dtype.get(cursor.header.field)) + + _fmm_core.read_body_coo(cursor, i, j, data) + + if generalize_symmetry and cursor.header.symmetry != "general": + off_diagonal_mask = (i != j) + off_diagonal_rows = i[off_diagonal_mask] + off_diagonal_cols = j[off_diagonal_mask] + off_diagonal_data = data[off_diagonal_mask] + + if cursor.header.symmetry == "skew-symmetric": + off_diagonal_data *= -1 + elif cursor.header.symmetry == "hermitian": + off_diagonal_data = off_diagonal_data.conjugate() + + i = np.concatenate((i, off_diagonal_cols)) + j = np.concatenate((j, off_diagonal_rows)) + data = np.concatenate((data, off_diagonal_data)) + + return (data, (i, j)), cursor.header.shape + + +def _get_read_cursor(source, parallelism=None): + """ + Open file for reading. + """ + from . import _fmm_core + + ret_stream_to_close = None + if parallelism is None: + parallelism = PARALLELISM + + try: + source = os.fspath(source) + # It's a file path + is_path = True + except TypeError: + is_path = False + + if is_path: + path = str(source) + if path.endswith('.gz'): + import gzip + source = gzip.GzipFile(path, 'r') + ret_stream_to_close = source + elif path.endswith('.bz2'): + import bz2 + source = bz2.BZ2File(path, 'rb') + ret_stream_to_close = source + else: + return _fmm_core.open_read_file(path, parallelism), ret_stream_to_close + + # Stream object. + if hasattr(source, "read"): + if isinstance(source, io.TextIOBase): + source = _TextToBytesWrapper(source) + return _fmm_core.open_read_stream(source, parallelism), ret_stream_to_close + else: + raise TypeError("Unknown source type") + + +def _get_write_cursor(target, h=None, comment=None, parallelism=None, + symmetry="general", precision=None): + """ + Open file for writing. + """ + from . import _fmm_core + + if parallelism is None: + parallelism = PARALLELISM + if comment is None: + comment = '' + if symmetry is None: + symmetry = "general" + if precision is None: + precision = -1 + + if not h: + h = _fmm_core.header(comment=comment, symmetry=symmetry) + + try: + target = os.fspath(target) + # It's a file path + if target[-4:] != '.mtx': + target += '.mtx' + return _fmm_core.open_write_file(str(target), h, parallelism, precision) + except TypeError: + pass + + if hasattr(target, "write"): + # Stream object. + if isinstance(target, io.TextIOBase): + raise TypeError("target stream must be open in binary mode.") + return _fmm_core.open_write_stream(target, h, parallelism, precision) + else: + raise TypeError("Unknown source object") + + +def _apply_field(data, field, no_pattern=False): + """ + Ensure that ``data.dtype`` is compatible with the specified MatrixMarket field type. + + Parameters + ---------- + data : ndarray + Input array. + + field : str + Matrix Market field, such as 'real', 'complex', 'integer', 'pattern'. + + no_pattern : bool, optional + Whether an empty array may be returned for a 'pattern' field. + + Returns + ------- + data : ndarray + Input data if no conversion necessary, or a converted version + """ + + if field is None: + return data + if field == "pattern": + if no_pattern: + return data + else: + return np.zeros(0) + + dtype = _field_to_dtype.get(field, None) + if dtype is None: + raise ValueError("Invalid field.") + + return np.asarray(data, dtype=dtype) + + +def _validate_symmetry(symmetry): + """ + Check that the symmetry parameter is one that MatrixMarket allows.. + """ + if symmetry is None: + return "general" + + symmetry = str(symmetry).lower() + symmetries = ["general", "symmetric", "skew-symmetric", "hermitian"] + if symmetry not in symmetries: + raise ValueError("Invalid symmetry. Must be one of: " + ", ".join(symmetries)) + + return symmetry + + +def mmread(source, *, spmatrix=True): + """ + Reads the contents of a Matrix Market file-like 'source' into a matrix. + + Parameters + ---------- + source : str or file-like + Matrix Market filename (extensions .mtx, .mtz.gz) + or open file-like object. + spmatrix : bool, optional (default: True) + If ``True``, return sparse ``coo_matrix``. Otherwise return ``coo_array``. + + Returns + ------- + a : ndarray or coo_array + Dense or sparse array depending on the matrix format in the + Matrix Market file. + + Notes + ----- + .. versionchanged:: 1.12.0 + C++ implementation. + + Examples + -------- + >>> from io import StringIO + >>> from scipy.io import mmread + + >>> text = '''%%MatrixMarket matrix coordinate real general + ... 5 5 7 + ... 2 3 1.0 + ... 3 4 2.0 + ... 3 5 3.0 + ... 4 1 4.0 + ... 4 2 5.0 + ... 4 3 6.0 + ... 4 4 7.0 + ... ''' + + ``mmread(source)`` returns the data as sparse array in COO format. + + >>> m = mmread(StringIO(text), spmatrix=False) + >>> m + + >>> m.toarray() + array([[0., 0., 0., 0., 0.], + [0., 0., 1., 0., 0.], + [0., 0., 0., 2., 3.], + [4., 5., 6., 7., 0.], + [0., 0., 0., 0., 0.]]) + + This method is threaded. + The default number of threads is equal to the number of CPUs in the system. + Use `threadpoolctl `_ to override: + + >>> import threadpoolctl + >>> + >>> with threadpoolctl.threadpool_limits(limits=2): + ... m = mmread(StringIO(text), spmatrix=False) + + """ + cursor, stream_to_close = _get_read_cursor(source) + + if cursor.header.format == "array": + mat = _read_body_array(cursor) + if stream_to_close: + stream_to_close.close() + return mat + else: + triplet, shape = _read_body_coo(cursor, generalize_symmetry=True) + if stream_to_close: + stream_to_close.close() + if spmatrix: + return coo_matrix(triplet, shape=shape) + return coo_array(triplet, shape=shape) + + +def mmwrite(target, a, comment=None, field=None, precision=None, symmetry="AUTO"): + r""" + Writes the sparse or dense array `a` to Matrix Market file-like `target`. + + Parameters + ---------- + target : str or file-like + Matrix Market filename (extension .mtx) or open file-like object. + a : array like + Sparse or dense 2-D array. + comment : str, optional + Comments to be prepended to the Matrix Market file. + field : None or str, optional + Either 'real', 'complex', 'pattern', or 'integer'. + precision : None or int, optional + Number of digits to display for real or complex values. + symmetry : None or str, optional + Either 'AUTO', 'general', 'symmetric', 'skew-symmetric', or 'hermitian'. + If symmetry is None the symmetry type of 'a' is determined by its + values. If symmetry is 'AUTO' the symmetry type of 'a' is either + determined or set to 'general', at mmwrite's discretion. + + Returns + ------- + None + + Notes + ----- + .. versionchanged:: 1.12.0 + C++ implementation. + + Examples + -------- + >>> from io import BytesIO + >>> import numpy as np + >>> from scipy.sparse import coo_array + >>> from scipy.io import mmwrite + + Write a small NumPy array to a matrix market file. The file will be + written in the ``'array'`` format. + + >>> a = np.array([[1.0, 0, 0, 0], [0, 2.5, 0, 6.25]]) + >>> target = BytesIO() + >>> mmwrite(target, a) + >>> print(target.getvalue().decode('latin1')) + %%MatrixMarket matrix array real general + % + 2 4 + 1 + 0 + 0 + 2.5 + 0 + 0 + 0 + 6.25 + + Add a comment to the output file, and set the precision to 3. + + >>> target = BytesIO() + >>> mmwrite(target, a, comment='\n Some test data.\n', precision=3) + >>> print(target.getvalue().decode('latin1')) + %%MatrixMarket matrix array real general + % + % Some test data. + % + 2 4 + 1.00e+00 + 0.00e+00 + 0.00e+00 + 2.50e+00 + 0.00e+00 + 0.00e+00 + 0.00e+00 + 6.25e+00 + + Convert to a sparse matrix before calling ``mmwrite``. This will + result in the output format being ``'coordinate'`` rather than + ``'array'``. + + >>> target = BytesIO() + >>> mmwrite(target, coo_array(a), precision=3) + >>> print(target.getvalue().decode('latin1')) + %%MatrixMarket matrix coordinate real general + % + 2 4 3 + 1 1 1.00e+00 + 2 2 2.50e+00 + 2 4 6.25e+00 + + Write a complex Hermitian array to a matrix market file. Note that + only six values are actually written to the file; the other values + are implied by the symmetry. + + >>> z = np.array([[3, 1+2j, 4-3j], [1-2j, 1, -5j], [4+3j, 5j, 2.5]]) + >>> z + array([[ 3. +0.j, 1. +2.j, 4. -3.j], + [ 1. -2.j, 1. +0.j, -0. -5.j], + [ 4. +3.j, 0. +5.j, 2.5+0.j]]) + + >>> target = BytesIO() + >>> mmwrite(target, z, precision=2) + >>> print(target.getvalue().decode('latin1')) + %%MatrixMarket matrix array complex hermitian + % + 3 3 + 3.0e+00 0.0e+00 + 1.0e+00 -2.0e+00 + 4.0e+00 3.0e+00 + 1.0e+00 0.0e+00 + 0.0e+00 5.0e+00 + 2.5e+00 0.0e+00 + + This method is threaded. + The default number of threads is equal to the number of CPUs in the system. + Use `threadpoolctl `_ to override: + + >>> import threadpoolctl + >>> + >>> target = BytesIO() + >>> with threadpoolctl.threadpool_limits(limits=2): + ... mmwrite(target, a) + + """ + from . import _fmm_core + + if isinstance(a, list) or isinstance(a, tuple) or hasattr(a, "__array__"): + a = np.asarray(a) + + if symmetry == "AUTO": + if ALWAYS_FIND_SYMMETRY or (hasattr(a, "shape") and max(a.shape) < 100): + symmetry = None + else: + symmetry = "general" + + if symmetry is None: + symmetry = _mmio.MMFile()._get_symmetry(a) + + symmetry = _validate_symmetry(symmetry) + cursor = _get_write_cursor(target, comment=comment, + precision=precision, symmetry=symmetry) + + if isinstance(a, np.ndarray): + # Write dense numpy arrays + a = _apply_field(a, field, no_pattern=True) + _fmm_core.write_body_array(cursor, a) + + elif issparse(a): + # Write sparse scipy matrices + a = a.tocoo() + + if symmetry is not None and symmetry != "general": + # A symmetric matrix only specifies the elements below the diagonal. + # Ensure that the matrix satisfies this requirement. + lower_triangle_mask = a.row >= a.col + a = coo_array((a.data[lower_triangle_mask], + (a.row[lower_triangle_mask], + a.col[lower_triangle_mask])), shape=a.shape) + + data = _apply_field(a.data, field) + _fmm_core.write_body_coo(cursor, a.shape, a.row, a.col, data) + + else: + raise ValueError(f"unknown matrix type: {type(a)}") + + +def mminfo(source): + """ + Return size and storage parameters from Matrix Market file-like 'source'. + + Parameters + ---------- + source : str or file-like + Matrix Market filename (extension .mtx) or open file-like object + + Returns + ------- + rows : int + Number of matrix rows. + cols : int + Number of matrix columns. + entries : int + Number of non-zero entries of a sparse matrix + or rows*cols for a dense matrix. + format : str + Either 'coordinate' or 'array'. + field : str + Either 'real', 'complex', 'pattern', or 'integer'. + symmetry : str + Either 'general', 'symmetric', 'skew-symmetric', or 'hermitian'. + + Notes + ----- + .. versionchanged:: 1.12.0 + C++ implementation. + + Examples + -------- + >>> from io import StringIO + >>> from scipy.io import mminfo + + >>> text = '''%%MatrixMarket matrix coordinate real general + ... 5 5 7 + ... 2 3 1.0 + ... 3 4 2.0 + ... 3 5 3.0 + ... 4 1 4.0 + ... 4 2 5.0 + ... 4 3 6.0 + ... 4 4 7.0 + ... ''' + + + ``mminfo(source)`` returns the number of rows, number of columns, + format, field type and symmetry attribute of the source file. + + >>> mminfo(StringIO(text)) + (5, 5, 7, 'coordinate', 'real', 'general') + """ + cursor, stream_to_close = _get_read_cursor(source, 1) + h = cursor.header + cursor.close() + if stream_to_close: + stream_to_close.close() + return h.nrows, h.ncols, h.nnz, h.format, h.field, h.symmetry diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_fortran.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_fortran.py new file mode 100644 index 0000000000000000000000000000000000000000..ac491dce68fe2f2f171dcee5a3097b0f4c4ea10c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_fortran.py @@ -0,0 +1,354 @@ +""" +Module to read / write Fortran unformatted sequential files. + +This is in the spirit of code written by Neil Martinsen-Burrell and Joe Zuntz. + +""" +import warnings +import numpy as np + +__all__ = ['FortranFile', 'FortranEOFError', 'FortranFormattingError'] + + +class FortranEOFError(TypeError, OSError): + """Indicates that the file ended properly. + + This error descends from TypeError because the code used to raise + TypeError (and this was the only way to know that the file had + ended) so users might have ``except TypeError:``. + + """ + pass + + +class FortranFormattingError(TypeError, OSError): + """Indicates that the file ended mid-record. + + Descends from TypeError for backward compatibility. + + """ + pass + + +class FortranFile: + """ + A file object for unformatted sequential files from Fortran code. + + Parameters + ---------- + filename : file or str + Open file object or filename. + mode : {'r', 'w'}, optional + Read-write mode, default is 'r'. + header_dtype : dtype, optional + Data type of the header. Size and endianness must match the input/output file. + + Notes + ----- + These files are broken up into records of unspecified types. The size of + each record is given at the start (although the size of this header is not + standard) and the data is written onto disk without any formatting. Fortran + compilers supporting the BACKSPACE statement will write a second copy of + the size to facilitate backwards seeking. + + This class only supports files written with both sizes for the record. + It also does not support the subrecords used in Intel and gfortran compilers + for records which are greater than 2GB with a 4-byte header. + + An example of an unformatted sequential file in Fortran would be written as:: + + OPEN(1, FILE=myfilename, FORM='unformatted') + + WRITE(1) myvariable + + Since this is a non-standard file format, whose contents depend on the + compiler and the endianness of the machine, caution is advised. Files from + gfortran 4.8.0 and gfortran 4.1.2 on x86_64 are known to work. + + Consider using Fortran direct-access files or files from the newer Stream + I/O, which can be easily read by `numpy.fromfile`. + + Examples + -------- + To create an unformatted sequential Fortran file: + + >>> from scipy.io import FortranFile + >>> import numpy as np + >>> f = FortranFile('test.unf', 'w') + >>> f.write_record(np.array([1,2,3,4,5], dtype=np.int32)) + >>> f.write_record(np.linspace(0,1,20).reshape((5,4)).T) + >>> f.close() + + To read this file: + + >>> f = FortranFile('test.unf', 'r') + >>> print(f.read_ints(np.int32)) + [1 2 3 4 5] + >>> print(f.read_reals(float).reshape((5,4), order="F")) + [[0. 0.05263158 0.10526316 0.15789474] + [0.21052632 0.26315789 0.31578947 0.36842105] + [0.42105263 0.47368421 0.52631579 0.57894737] + [0.63157895 0.68421053 0.73684211 0.78947368] + [0.84210526 0.89473684 0.94736842 1. ]] + >>> f.close() + + Or, in Fortran:: + + integer :: a(5), i + double precision :: b(5,4) + open(1, file='test.unf', form='unformatted') + read(1) a + read(1) b + close(1) + write(*,*) a + do i = 1, 5 + write(*,*) b(i,:) + end do + + """ + def __init__(self, filename, mode='r', header_dtype=np.uint32): + if header_dtype is None: + raise ValueError('Must specify dtype') + + header_dtype = np.dtype(header_dtype) + if header_dtype.kind != 'u': + warnings.warn("Given a dtype which is not unsigned.", stacklevel=2) + + if mode not in 'rw' or len(mode) != 1: + raise ValueError('mode must be either r or w') + + if hasattr(filename, 'seek'): + self._fp = filename + else: + self._fp = open(filename, f'{mode}b') + + self._header_dtype = header_dtype + + def _read_size(self, eof_ok=False): + n = self._header_dtype.itemsize + b = self._fp.read(n) + if (not b) and eof_ok: + raise FortranEOFError("End of file occurred at end of record") + elif len(b) < n: + raise FortranFormattingError( + "End of file in the middle of the record size") + return int(np.frombuffer(b, dtype=self._header_dtype, count=1)[0]) + + def write_record(self, *items): + """ + Write a record (including sizes) to the file. + + Parameters + ---------- + *items : array_like + The data arrays to write. + + Notes + ----- + Writes data items to a file:: + + write_record(a.T, b.T, c.T, ...) + + write(1) a, b, c, ... + + Note that data in multidimensional arrays is written in + row-major order --- to make them read correctly by Fortran + programs, you need to transpose the arrays yourself when + writing them. + + """ + items = tuple(np.asarray(item) for item in items) + total_size = sum(item.nbytes for item in items) + + nb = np.array([total_size], dtype=self._header_dtype) + + nb.tofile(self._fp) + for item in items: + item.tofile(self._fp) + nb.tofile(self._fp) + + def read_record(self, *dtypes, **kwargs): + """ + Reads a record of a given type from the file. + + Parameters + ---------- + *dtypes : dtypes, optional + Data type(s) specifying the size and endianness of the data. + + Returns + ------- + data : ndarray + A 1-D array object. + + Raises + ------ + FortranEOFError + To signal that no further records are available + FortranFormattingError + To signal that the end of the file was encountered + part-way through a record + + Notes + ----- + If the record contains a multidimensional array, you can specify + the size in the dtype. For example:: + + INTEGER var(5,4) + + can be read with:: + + read_record('(4,5)i4').T + + Note that this function does **not** assume the file data is in Fortran + column major order, so you need to (i) swap the order of dimensions + when reading and (ii) transpose the resulting array. + + Alternatively, you can read the data as a 1-D array and handle the + ordering yourself. For example:: + + read_record('i4').reshape(5, 4, order='F') + + For records that contain several variables or mixed types (as opposed + to single scalar or array types), give them as separate arguments:: + + double precision :: a + integer :: b + write(1) a, b + + record = f.read_record(' 0, -n and n if n < 0 + + Parameters + ---------- + n : int + max number one wants to be able to represent + min : int + minimum number of characters to use for the format + + Returns + ------- + res : IntFormat + IntFormat instance with reasonable (see Notes) computed width + + Notes + ----- + Reasonable should be understood as the minimal string length necessary + without losing precision. For example, IntFormat.from_number(1) will + return an IntFormat instance of width 2, so that any 0 and 1 may be + represented as 1-character strings without loss of information. + """ + width = number_digits(n) + 1 + if n < 0: + width += 1 + repeat = 80 // width + return cls(width, min, repeat=repeat) + + def __init__(self, width, min=None, repeat=None): + self.width = width + self.repeat = repeat + self.min = min + + def __repr__(self): + r = "IntFormat(" + if self.repeat: + r += "%d" % self.repeat + r += "I%d" % self.width + if self.min: + r += ".%d" % self.min + return r + ")" + + @property + def fortran_format(self): + r = "(" + if self.repeat: + r += "%d" % self.repeat + r += "I%d" % self.width + if self.min: + r += ".%d" % self.min + return r + ")" + + @property + def python_format(self): + return "%" + str(self.width) + "d" + + +class ExpFormat: + @classmethod + def from_number(cls, n, min=None): + """Given a float number, returns a "reasonable" ExpFormat instance to + represent any number between -n and n. + + Parameters + ---------- + n : float + max number one wants to be able to represent + min : int + minimum number of characters to use for the format + + Returns + ------- + res : ExpFormat + ExpFormat instance with reasonable (see Notes) computed width + + Notes + ----- + Reasonable should be understood as the minimal string length necessary + to avoid losing precision. + """ + # len of one number in exp format: sign + 1|0 + "." + + # number of digit for fractional part + 'E' + sign of exponent + + # len of exponent + finfo = np.finfo(n.dtype) + # Number of digits for fractional part + n_prec = finfo.precision + 1 + # Number of digits for exponential part + n_exp = number_digits(np.max(np.abs([finfo.maxexp, finfo.minexp]))) + width = 1 + 1 + n_prec + 1 + n_exp + 1 + if n < 0: + width += 1 + repeat = int(np.floor(80 / width)) + return cls(width, n_prec, min, repeat=repeat) + + def __init__(self, width, significand, min=None, repeat=None): + """\ + Parameters + ---------- + width : int + number of characters taken by the string (includes space). + """ + self.width = width + self.significand = significand + self.repeat = repeat + self.min = min + + def __repr__(self): + r = "ExpFormat(" + if self.repeat: + r += "%d" % self.repeat + r += "E%d.%d" % (self.width, self.significand) + if self.min: + r += "E%d" % self.min + return r + ")" + + @property + def fortran_format(self): + r = "(" + if self.repeat: + r += "%d" % self.repeat + r += "E%d.%d" % (self.width, self.significand) + if self.min: + r += "E%d" % self.min + return r + ")" + + @property + def python_format(self): + return "%" + str(self.width-1) + "." + str(self.significand) + "E" + + +class Token: + def __init__(self, type, value, pos): + self.type = type + self.value = value + self.pos = pos + + def __str__(self): + return f"""Token('{self.type}', "{self.value}")""" + + def __repr__(self): + return self.__str__() + + +class Tokenizer: + def __init__(self): + self.tokens = list(TOKENS.keys()) + self.res = [re.compile(TOKENS[i]) for i in self.tokens] + + def input(self, s): + self.data = s + self.curpos = 0 + self.len = len(s) + + def next_token(self): + curpos = self.curpos + + while curpos < self.len: + for i, r in enumerate(self.res): + m = r.match(self.data, curpos) + if m is None: + continue + else: + self.curpos = m.end() + return Token(self.tokens[i], m.group(), self.curpos) + raise SyntaxError("Unknown character at position %d (%s)" + % (self.curpos, self.data[curpos])) + + +# Grammar for fortran format: +# format : LPAR format_string RPAR +# format_string : repeated | simple +# repeated : repeat simple +# simple : int_fmt | exp_fmt +# int_fmt : INT_ID width +# exp_fmt : simple_exp_fmt +# simple_exp_fmt : EXP_ID width DOT significand +# extended_exp_fmt : EXP_ID width DOT significand EXP_ID ndigits +# repeat : INT +# width : INT +# significand : INT +# ndigits : INT + +# Naive fortran formatter - parser is hand-made +class FortranFormatParser: + """Parser for Fortran format strings. The parse method returns a *Format + instance. + + Notes + ----- + Only ExpFormat (exponential format for floating values) and IntFormat + (integer format) for now. + """ + def __init__(self): + self.tokenizer = threading.local() + + def parse(self, s): + if not hasattr(self.tokenizer, 't'): + self.tokenizer.t = Tokenizer() + + self.tokenizer.t.input(s) + + tokens = [] + + try: + while True: + t = self.tokenizer.t.next_token() + if t is None: + break + else: + tokens.append(t) + return self._parse_format(tokens) + except SyntaxError as e: + raise BadFortranFormat(str(e)) from e + + def _get_min(self, tokens): + next = tokens.pop(0) + if not next.type == "DOT": + raise SyntaxError() + next = tokens.pop(0) + return next.value + + def _expect(self, token, tp): + if not token.type == tp: + raise SyntaxError() + + def _parse_format(self, tokens): + if not tokens[0].type == "LPAR": + raise SyntaxError("Expected left parenthesis at position " + "%d (got '%s')" % (0, tokens[0].value)) + elif not tokens[-1].type == "RPAR": + raise SyntaxError("Expected right parenthesis at position " + f"{len(tokens)} (got '{tokens[-1].value}')") + + tokens = tokens[1:-1] + types = [t.type for t in tokens] + if types[0] == "INT": + repeat = int(tokens.pop(0).value) + else: + repeat = None + + next = tokens.pop(0) + if next.type == "INT_ID": + next = self._next(tokens, "INT") + width = int(next.value) + if tokens: + min = int(self._get_min(tokens)) + else: + min = None + return IntFormat(width, min, repeat) + elif next.type == "EXP_ID": + next = self._next(tokens, "INT") + width = int(next.value) + + next = self._next(tokens, "DOT") + + next = self._next(tokens, "INT") + significand = int(next.value) + + if tokens: + next = self._next(tokens, "EXP_ID") + + next = self._next(tokens, "INT") + min = int(next.value) + else: + min = None + return ExpFormat(width, significand, min, repeat) + else: + raise SyntaxError(f"Invalid formatter type {next.value}") + + def _next(self, tokens, tp): + if not len(tokens) > 0: + raise SyntaxError() + next = tokens.pop(0) + self._expect(next, tp) + return next diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/hb.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/hb.py new file mode 100644 index 0000000000000000000000000000000000000000..96fef89ac35a271f3a5501beaefd06da13ede685 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/hb.py @@ -0,0 +1,575 @@ +""" +Implementation of Harwell-Boeing read/write. + +At the moment not the full Harwell-Boeing format is supported. Supported +features are: + + - assembled, non-symmetric, real matrices + - integer for pointer/indices + - exponential format for float values, and int format + +""" +# TODO: +# - Add more support (symmetric/complex matrices, non-assembled matrices ?) + +# XXX: reading is reasonably efficient (>= 85 % is in numpy.fromstring), but +# takes a lot of memory. Being faster would require compiled code. +# write is not efficient. Although not a terribly exciting task, +# having reusable facilities to efficiently read/write fortran-formatted files +# would be useful outside this module. + +import warnings + +import numpy as np +from scipy.sparse import csc_array, csc_matrix +from ._fortran_format_parser import FortranFormatParser, IntFormat, ExpFormat + +__all__ = ["hb_read", "hb_write"] + + +class MalformedHeader(Exception): + pass + + +class LineOverflow(Warning): + pass + + +def _nbytes_full(fmt, nlines): + """Return the number of bytes to read to get every full lines for the + given parsed fortran format.""" + return (fmt.repeat * fmt.width + 1) * (nlines - 1) + + +class HBInfo: + @classmethod + def from_data(cls, m, title="Default title", key="0", mxtype=None, fmt=None): + """Create a HBInfo instance from an existing sparse matrix. + + Parameters + ---------- + m : sparse array or matrix + the HBInfo instance will derive its parameters from m + title : str + Title to put in the HB header + key : str + Key + mxtype : HBMatrixType + type of the input matrix + fmt : dict + not implemented + + Returns + ------- + hb_info : HBInfo instance + """ + m = m.tocsc(copy=False) + + pointer = m.indptr + indices = m.indices + values = m.data + + nrows, ncols = m.shape + nnon_zeros = m.nnz + + if fmt is None: + # +1 because HB use one-based indexing (Fortran), and we will write + # the indices /pointer as such + pointer_fmt = IntFormat.from_number(np.max(pointer+1)) + indices_fmt = IntFormat.from_number(np.max(indices+1)) + + if values.dtype.kind in np.typecodes["AllFloat"]: + values_fmt = ExpFormat.from_number(-np.max(np.abs(values))) + elif values.dtype.kind in np.typecodes["AllInteger"]: + values_fmt = IntFormat.from_number(-np.max(np.abs(values))) + else: + message = f"type {values.dtype.kind} not implemented yet" + raise NotImplementedError(message) + else: + raise NotImplementedError("fmt argument not supported yet.") + + if mxtype is None: + if not np.isrealobj(values): + raise ValueError("Complex values not supported yet") + if values.dtype.kind in np.typecodes["AllInteger"]: + tp = "integer" + elif values.dtype.kind in np.typecodes["AllFloat"]: + tp = "real" + else: + raise NotImplementedError( + f"type {values.dtype} for values not implemented") + mxtype = HBMatrixType(tp, "unsymmetric", "assembled") + else: + raise ValueError("mxtype argument not handled yet.") + + def _nlines(fmt, size): + nlines = size // fmt.repeat + if nlines * fmt.repeat != size: + nlines += 1 + return nlines + + pointer_nlines = _nlines(pointer_fmt, pointer.size) + indices_nlines = _nlines(indices_fmt, indices.size) + values_nlines = _nlines(values_fmt, values.size) + + total_nlines = pointer_nlines + indices_nlines + values_nlines + + return cls(title, key, + total_nlines, pointer_nlines, indices_nlines, values_nlines, + mxtype, nrows, ncols, nnon_zeros, + pointer_fmt.fortran_format, indices_fmt.fortran_format, + values_fmt.fortran_format) + + @classmethod + def from_file(cls, fid): + """Create a HBInfo instance from a file object containing a matrix in the + HB format. + + Parameters + ---------- + fid : file-like matrix + File or file-like object containing a matrix in the HB format. + + Returns + ------- + hb_info : HBInfo instance + """ + # First line + line = fid.readline().strip("\n") + if not len(line) > 72: + raise ValueError("Expected at least 72 characters for first line, " + f"got: \n{line}") + title = line[:72] + key = line[72:] + + # Second line + line = fid.readline().strip("\n") + if not len(line.rstrip()) >= 56: + raise ValueError("Expected at least 56 characters for second line, " + f"got: \n{line}") + total_nlines = _expect_int(line[:14]) + pointer_nlines = _expect_int(line[14:28]) + indices_nlines = _expect_int(line[28:42]) + values_nlines = _expect_int(line[42:56]) + + rhs_nlines = line[56:72].strip() + if rhs_nlines == '': + rhs_nlines = 0 + else: + rhs_nlines = _expect_int(rhs_nlines) + if not rhs_nlines == 0: + raise ValueError("Only files without right hand side supported for " + "now.") + + # Third line + line = fid.readline().strip("\n") + if not len(line) >= 70: + raise ValueError(f"Expected at least 72 character for third line, " + f"got:\n{line}") + + mxtype_s = line[:3].upper() + if not len(mxtype_s) == 3: + raise ValueError("mxtype expected to be 3 characters long") + + mxtype = HBMatrixType.from_fortran(mxtype_s) + if mxtype.value_type not in ["real", "integer"]: + raise ValueError("Only real or integer matrices supported for " + f"now (detected {mxtype})") + if not mxtype.structure == "unsymmetric": + raise ValueError("Only unsymmetric matrices supported for " + f"now (detected {mxtype})") + if not mxtype.storage == "assembled": + raise ValueError("Only assembled matrices supported for now") + + if not line[3:14] == " " * 11: + raise ValueError(f"Malformed data for third line: {line}") + + nrows = _expect_int(line[14:28]) + ncols = _expect_int(line[28:42]) + nnon_zeros = _expect_int(line[42:56]) + nelementals = _expect_int(line[56:70]) + if not nelementals == 0: + raise ValueError("Unexpected value %d for nltvl (last entry of line 3)" + % nelementals) + + # Fourth line + line = fid.readline().strip("\n") + + ct = line.split() + if not len(ct) == 3: + raise ValueError(f"Expected 3 formats, got {ct}") + + return cls(title, key, + total_nlines, pointer_nlines, indices_nlines, values_nlines, + mxtype, nrows, ncols, nnon_zeros, + ct[0], ct[1], ct[2], + rhs_nlines, nelementals) + + def __init__(self, title, key, + total_nlines, pointer_nlines, indices_nlines, values_nlines, + mxtype, nrows, ncols, nnon_zeros, + pointer_format_str, indices_format_str, values_format_str, + right_hand_sides_nlines=0, nelementals=0): + """Do not use this directly, but the class ctrs (from_* functions).""" + if title is None: + title = "No Title" + if len(title) > 72: + raise ValueError("title cannot be > 72 characters") + + if key is None: + key = "|No Key" + if len(key) > 8: + warnings.warn(f"key is > 8 characters (key is {key})", + LineOverflow, stacklevel=3) + self.title = title + self.key = key + + self.total_nlines = total_nlines + self.pointer_nlines = pointer_nlines + self.indices_nlines = indices_nlines + self.values_nlines = values_nlines + + parser = FortranFormatParser() + pointer_format = parser.parse(pointer_format_str) + if not isinstance(pointer_format, IntFormat): + raise ValueError("Expected int format for pointer format, got " + f"{pointer_format}") + + indices_format = parser.parse(indices_format_str) + if not isinstance(indices_format, IntFormat): + raise ValueError("Expected int format for indices format, got " + f"{indices_format}") + + values_format = parser.parse(values_format_str) + if isinstance(values_format, ExpFormat): + if mxtype.value_type not in ["real", "complex"]: + raise ValueError(f"Inconsistency between matrix type {mxtype} and " + f"value type {values_format}") + values_dtype = np.float64 + elif isinstance(values_format, IntFormat): + if mxtype.value_type not in ["integer"]: + raise ValueError(f"Inconsistency between matrix type {mxtype} and " + f"value type {values_format}") + # XXX: fortran int -> dtype association ? + values_dtype = int + else: + raise ValueError(f"Unsupported format for values {values_format!r}") + + self.pointer_format = pointer_format + self.indices_format = indices_format + self.values_format = values_format + + self.pointer_dtype = np.int32 + self.indices_dtype = np.int32 + self.values_dtype = values_dtype + + self.pointer_nlines = pointer_nlines + self.pointer_nbytes_full = _nbytes_full(pointer_format, pointer_nlines) + + self.indices_nlines = indices_nlines + self.indices_nbytes_full = _nbytes_full(indices_format, indices_nlines) + + self.values_nlines = values_nlines + self.values_nbytes_full = _nbytes_full(values_format, values_nlines) + + self.nrows = nrows + self.ncols = ncols + self.nnon_zeros = nnon_zeros + self.nelementals = nelementals + self.mxtype = mxtype + + def dump(self): + """Gives the header corresponding to this instance as a string.""" + header = [self.title.ljust(72) + self.key.ljust(8)] + + header.append("%14d%14d%14d%14d" % + (self.total_nlines, self.pointer_nlines, + self.indices_nlines, self.values_nlines)) + header.append("%14s%14d%14d%14d%14d" % + (self.mxtype.fortran_format.ljust(14), self.nrows, + self.ncols, self.nnon_zeros, 0)) + + pffmt = self.pointer_format.fortran_format + iffmt = self.indices_format.fortran_format + vffmt = self.values_format.fortran_format + header.append("%16s%16s%20s" % + (pffmt.ljust(16), iffmt.ljust(16), vffmt.ljust(20))) + return "\n".join(header) + + +def _expect_int(value, msg=None): + try: + return int(value) + except ValueError as e: + if msg is None: + msg = "Expected an int, got %s" + raise ValueError(msg % value) from e + + +def _read_hb_data(content, header): + # XXX: look at a way to reduce memory here (big string creation) + ptr_string = "".join([content.read(header.pointer_nbytes_full), + content.readline()]) + ptr = np.fromstring(ptr_string, + dtype=int, sep=' ') + + ind_string = "".join([content.read(header.indices_nbytes_full), + content.readline()]) + ind = np.fromstring(ind_string, + dtype=int, sep=' ') + + val_string = "".join([content.read(header.values_nbytes_full), + content.readline()]) + val = np.fromstring(val_string, + dtype=header.values_dtype, sep=' ') + + return csc_array((val, ind-1, ptr-1), shape=(header.nrows, header.ncols)) + + +def _write_data(m, fid, header): + m = m.tocsc(copy=False) + + def write_array(f, ar, nlines, fmt): + # ar_nlines is the number of full lines, n is the number of items per + # line, ffmt the fortran format + pyfmt = fmt.python_format + pyfmt_full = pyfmt * fmt.repeat + + # for each array to write, we first write the full lines, and special + # case for partial line + full = ar[:(nlines - 1) * fmt.repeat] + for row in full.reshape((nlines-1, fmt.repeat)): + f.write(pyfmt_full % tuple(row) + "\n") + nremain = ar.size - full.size + if nremain > 0: + f.write((pyfmt * nremain) % tuple(ar[ar.size - nremain:]) + "\n") + + fid.write(header.dump()) + fid.write("\n") + # +1 is for Fortran one-based indexing + write_array(fid, m.indptr+1, header.pointer_nlines, + header.pointer_format) + write_array(fid, m.indices+1, header.indices_nlines, + header.indices_format) + write_array(fid, m.data, header.values_nlines, + header.values_format) + + +class HBMatrixType: + """Class to hold the matrix type.""" + # q2f* translates qualified names to Fortran character + _q2f_type = { + "real": "R", + "complex": "C", + "pattern": "P", + "integer": "I", + } + _q2f_structure = { + "symmetric": "S", + "unsymmetric": "U", + "hermitian": "H", + "skewsymmetric": "Z", + "rectangular": "R" + } + _q2f_storage = { + "assembled": "A", + "elemental": "E", + } + + _f2q_type = {j: i for i, j in _q2f_type.items()} + _f2q_structure = {j: i for i, j in _q2f_structure.items()} + _f2q_storage = {j: i for i, j in _q2f_storage.items()} + + @classmethod + def from_fortran(cls, fmt): + if not len(fmt) == 3: + raise ValueError("Fortran format for matrix type should be 3 " + "characters long") + try: + value_type = cls._f2q_type[fmt[0]] + structure = cls._f2q_structure[fmt[1]] + storage = cls._f2q_storage[fmt[2]] + return cls(value_type, structure, storage) + except KeyError as e: + raise ValueError(f"Unrecognized format {fmt}") from e + + def __init__(self, value_type, structure, storage="assembled"): + self.value_type = value_type + self.structure = structure + self.storage = storage + + if value_type not in self._q2f_type: + raise ValueError(f"Unrecognized type {value_type}") + if structure not in self._q2f_structure: + raise ValueError(f"Unrecognized structure {structure}") + if storage not in self._q2f_storage: + raise ValueError(f"Unrecognized storage {storage}") + + @property + def fortran_format(self): + return self._q2f_type[self.value_type] + \ + self._q2f_structure[self.structure] + \ + self._q2f_storage[self.storage] + + def __repr__(self): + return f"HBMatrixType({self.value_type}, {self.structure}, {self.storage})" + + +class HBFile: + def __init__(self, file, hb_info=None): + """Create a HBFile instance. + + Parameters + ---------- + file : file-object + StringIO work as well + hb_info : HBInfo, optional + Should be given as an argument for writing, in which case the file + should be writable. + """ + self._fid = file + if hb_info is None: + self._hb_info = HBInfo.from_file(file) + else: + #raise OSError("file %s is not writable, and hb_info " + # "was given." % file) + self._hb_info = hb_info + + @property + def title(self): + return self._hb_info.title + + @property + def key(self): + return self._hb_info.key + + @property + def type(self): + return self._hb_info.mxtype.value_type + + @property + def structure(self): + return self._hb_info.mxtype.structure + + @property + def storage(self): + return self._hb_info.mxtype.storage + + def read_matrix(self): + return _read_hb_data(self._fid, self._hb_info) + + def write_matrix(self, m): + return _write_data(m, self._fid, self._hb_info) + + +def hb_read(path_or_open_file, *, spmatrix=True): + """Read HB-format file. + + Parameters + ---------- + path_or_open_file : path-like or file-like + If a file-like object, it is used as-is. Otherwise, it is opened + before reading. + spmatrix : bool, optional (default: True) + If ``True``, return sparse ``coo_matrix``. Otherwise return ``coo_array``. + + Returns + ------- + data : csc_array or csc_matrix + The data read from the HB file as a sparse array. + + Notes + ----- + At the moment not the full Harwell-Boeing format is supported. Supported + features are: + + - assembled, non-symmetric, real matrices + - integer for pointer/indices + - exponential format for float values, and int format + + Examples + -------- + We can read and write a harwell-boeing format file: + + >>> from scipy.io import hb_read, hb_write + >>> from scipy.sparse import csr_array, eye + >>> data = csr_array(eye(3)) # create a sparse array + >>> hb_write("data.hb", data) # write a hb file + >>> print(hb_read("data.hb", spmatrix=False)) # read a hb file + + Coords Values + (0, 0) 1.0 + (1, 1) 1.0 + (2, 2) 1.0 + """ + def _get_matrix(fid): + hb = HBFile(fid) + return hb.read_matrix() + + if hasattr(path_or_open_file, 'read'): + data = _get_matrix(path_or_open_file) + else: + with open(path_or_open_file) as f: + data = _get_matrix(f) + if spmatrix: + return csc_matrix(data) + return data + + +def hb_write(path_or_open_file, m, hb_info=None): + """Write HB-format file. + + Parameters + ---------- + path_or_open_file : path-like or file-like + If a file-like object, it is used as-is. Otherwise, it is opened + before writing. + m : sparse array or matrix + the sparse array to write + hb_info : HBInfo + contains the meta-data for write + + Returns + ------- + None + + Notes + ----- + At the moment not the full Harwell-Boeing format is supported. Supported + features are: + + - assembled, non-symmetric, real matrices + - integer for pointer/indices + - exponential format for float values, and int format + + Examples + -------- + We can read and write a harwell-boeing format file: + + >>> from scipy.io import hb_read, hb_write + >>> from scipy.sparse import csr_array, eye + >>> data = csr_array(eye(3)) # create a sparse array + >>> hb_write("data.hb", data) # write a hb file + >>> print(hb_read("data.hb", spmatrix=False)) # read a hb file + + Coords Values + (0, 0) 1.0 + (1, 1) 1.0 + (2, 2) 1.0 + """ + m = m.tocsc(copy=False) + + if hb_info is None: + hb_info = HBInfo.from_data(m) + + def _set_matrix(fid): + hb = HBFile(fid, hb_info) + return hb.write_matrix(m) + + if hasattr(path_or_open_file, 'write'): + return _set_matrix(path_or_open_file) + else: + with open(path_or_open_file, 'w') as f: + return _set_matrix(f) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/tests/test_fortran_format.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/tests/test_fortran_format.py new file mode 100644 index 0000000000000000000000000000000000000000..dae040c523d6a6d618e89402d39a0cb05bad927a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/tests/test_fortran_format.py @@ -0,0 +1,74 @@ +import numpy as np + +from numpy.testing import assert_equal +from pytest import raises as assert_raises + +from scipy.io._harwell_boeing._fortran_format_parser import ( + FortranFormatParser, IntFormat, ExpFormat, BadFortranFormat) + + +class TestFortranFormatParser: + def setup_method(self): + self.parser = FortranFormatParser() + + def _test_equal(self, format, ref): + ret = self.parser.parse(format) + assert_equal(ret.__dict__, ref.__dict__) + + def test_simple_int(self): + self._test_equal("(I4)", IntFormat(4)) + + def test_simple_repeated_int(self): + self._test_equal("(3I4)", IntFormat(4, repeat=3)) + + def test_simple_exp(self): + self._test_equal("(E4.3)", ExpFormat(4, 3)) + + def test_exp_exp(self): + self._test_equal("(E8.3E3)", ExpFormat(8, 3, 3)) + + def test_repeat_exp(self): + self._test_equal("(2E4.3)", ExpFormat(4, 3, repeat=2)) + + def test_repeat_exp_exp(self): + self._test_equal("(2E8.3E3)", ExpFormat(8, 3, 3, repeat=2)) + + def test_wrong_formats(self): + def _test_invalid(bad_format): + assert_raises(BadFortranFormat, lambda: self.parser.parse(bad_format)) + _test_invalid("I4") + _test_invalid("(E4)") + _test_invalid("(E4.)") + _test_invalid("(E4.E3)") + + +class TestIntFormat: + def test_to_fortran(self): + f = [IntFormat(10), IntFormat(12, 10), IntFormat(12, 10, 3)] + res = ["(I10)", "(I12.10)", "(3I12.10)"] + + for i, j in zip(f, res): + assert_equal(i.fortran_format, j) + + def test_from_number(self): + f = [10, -12, 123456789] + r_f = [IntFormat(3, repeat=26), IntFormat(4, repeat=20), + IntFormat(10, repeat=8)] + for i, j in zip(f, r_f): + assert_equal(IntFormat.from_number(i).__dict__, j.__dict__) + + +class TestExpFormat: + def test_to_fortran(self): + f = [ExpFormat(10, 5), ExpFormat(12, 10), ExpFormat(12, 10, min=3), + ExpFormat(10, 5, repeat=3)] + res = ["(E10.5)", "(E12.10)", "(E12.10E3)", "(3E10.5)"] + + for i, j in zip(f, res): + assert_equal(i.fortran_format, j) + + def test_from_number(self): + f = np.array([1.0, -1.2]) + r_f = [ExpFormat(24, 16, repeat=3), ExpFormat(25, 16, repeat=3)] + for i, j in zip(f, r_f): + assert_equal(ExpFormat.from_number(i).__dict__, j.__dict__) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/tests/test_hb.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/tests/test_hb.py new file mode 100644 index 0000000000000000000000000000000000000000..d0c9ee5635fdd45c0f6560e686c9e40176ef95e4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_harwell_boeing/tests/test_hb.py @@ -0,0 +1,70 @@ +from io import StringIO +import tempfile + +import numpy as np + +from numpy.testing import assert_equal, \ + assert_array_almost_equal_nulp + +from scipy.sparse import coo_array, csc_array, random_array, isspmatrix + +from scipy.io import hb_read, hb_write + + +SIMPLE = """\ +No Title |No Key + 9 4 1 4 +RUA 100 100 10 0 +(26I3) (26I3) (3E23.15) +1 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 +3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 +3 3 3 3 3 3 3 4 4 4 6 6 6 6 6 6 6 6 6 6 6 8 9 9 9 9 +9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 11 +37 71 89 18 30 45 70 19 25 52 +2.971243799687726e-01 3.662366682877375e-01 4.786962174699534e-01 +6.490068647991184e-01 6.617490424831662e-02 8.870370343191623e-01 +4.196478590163001e-01 5.649603072111251e-01 9.934423887087086e-01 +6.912334991524289e-01 +""" + +SIMPLE_MATRIX = coo_array( + ((0.297124379969, 0.366236668288, 0.47869621747, 0.649006864799, + 0.0661749042483, 0.887037034319, 0.419647859016, + 0.564960307211, 0.993442388709, 0.691233499152,), + (np.array([[36, 70, 88, 17, 29, 44, 69, 18, 24, 51], + [0, 4, 58, 61, 61, 72, 72, 73, 99, 99]])))) + + +def assert_csc_almost_equal(r, l): + r = csc_array(r) + l = csc_array(l) + assert_equal(r.indptr, l.indptr) + assert_equal(r.indices, l.indices) + assert_array_almost_equal_nulp(r.data, l.data, 10000) + + +class TestHBReader: + def test_simple(self): + m = hb_read(StringIO(SIMPLE), spmatrix=False) + assert_csc_almost_equal(m, SIMPLE_MATRIX) + assert not isspmatrix(m) + m = hb_read(StringIO(SIMPLE), spmatrix=True) + assert isspmatrix(m) + m = hb_read(StringIO(SIMPLE)) # default + assert isspmatrix(m) + + +class TestHBReadWrite: + + def check_save_load(self, value): + with tempfile.NamedTemporaryFile(mode='w+t') as file: + hb_write(file, value) + file.file.seek(0) + value_loaded = hb_read(file, spmatrix=False) + assert_csc_almost_equal(value, value_loaded) + + def test_simple(self): + random_arr = random_array((10, 100), density=0.1) + for format in ('coo', 'csc', 'csr', 'bsr', 'dia', 'dok', 'lil'): + arr = random_arr.asformat(format, copy=False) + self.check_save_load(arr) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_idl.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_idl.py new file mode 100644 index 0000000000000000000000000000000000000000..5730a9d4fe1beda5a71aa668bda13d17f2fc2436 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_idl.py @@ -0,0 +1,919 @@ +# IDLSave - a python module to read IDL 'save' files +# Copyright (c) 2010 Thomas P. Robitaille + +# Many thanks to Craig Markwardt for publishing the Unofficial Format +# Specification for IDL .sav files, without which this Python module would not +# exist (http://cow.physics.wisc.edu/~craigm/idl/savefmt). + +# This code was developed by with permission from ITT Visual Information +# Systems. IDL(r) is a registered trademark of ITT Visual Information Systems, +# Inc. for their Interactive Data Language software. + +# Permission is hereby granted, free of charge, to any person obtaining a +# copy of this software and associated documentation files (the "Software"), +# to deal in the Software without restriction, including without limitation +# the rights to use, copy, modify, merge, publish, distribute, sublicense, +# and/or sell copies of the Software, and to permit persons to whom the +# Software is furnished to do so, subject to the following conditions: + +# The above copyright notice and this permission notice shall be included in +# all copies or substantial portions of the Software. + +# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING +# FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER +# DEALINGS IN THE SOFTWARE. + +__all__ = ['readsav'] + +import struct +import numpy as np +import tempfile +import zlib +import warnings + +# Define the different data types that can be found in an IDL save file +DTYPE_DICT = {1: '>u1', + 2: '>i2', + 3: '>i4', + 4: '>f4', + 5: '>f8', + 6: '>c8', + 7: '|O', + 8: '|O', + 9: '>c16', + 10: '|O', + 11: '|O', + 12: '>u2', + 13: '>u4', + 14: '>i8', + 15: '>u8'} + +# Define the different record types that can be found in an IDL save file +RECTYPE_DICT = {0: "START_MARKER", + 1: "COMMON_VARIABLE", + 2: "VARIABLE", + 3: "SYSTEM_VARIABLE", + 6: "END_MARKER", + 10: "TIMESTAMP", + 12: "COMPILED", + 13: "IDENTIFICATION", + 14: "VERSION", + 15: "HEAP_HEADER", + 16: "HEAP_DATA", + 17: "PROMOTE64", + 19: "NOTICE", + 20: "DESCRIPTION"} + +# Define a dictionary to contain structure definitions +STRUCT_DICT = {} + + +def _align_32(f): + '''Align to the next 32-bit position in a file''' + + pos = f.tell() + if pos % 4 != 0: + f.seek(pos + 4 - pos % 4) + return + + +def _skip_bytes(f, n): + '''Skip `n` bytes''' + f.read(n) + return + + +def _read_bytes(f, n): + '''Read the next `n` bytes''' + return f.read(n) + + +def _read_byte(f): + '''Read a single byte''' + return np.uint8(struct.unpack('>B', f.read(4)[:1])[0]) + + +def _read_long(f): + '''Read a signed 32-bit integer''' + return np.int32(struct.unpack('>l', f.read(4))[0]) + + +def _read_int16(f): + '''Read a signed 16-bit integer''' + return np.int16(struct.unpack('>h', f.read(4)[2:4])[0]) + + +def _read_int32(f): + '''Read a signed 32-bit integer''' + return np.int32(struct.unpack('>i', f.read(4))[0]) + + +def _read_int64(f): + '''Read a signed 64-bit integer''' + return np.int64(struct.unpack('>q', f.read(8))[0]) + + +def _read_uint16(f): + '''Read an unsigned 16-bit integer''' + return np.uint16(struct.unpack('>H', f.read(4)[2:4])[0]) + + +def _read_uint32(f): + '''Read an unsigned 32-bit integer''' + return np.uint32(struct.unpack('>I', f.read(4))[0]) + + +def _read_uint64(f): + '''Read an unsigned 64-bit integer''' + return np.uint64(struct.unpack('>Q', f.read(8))[0]) + + +def _read_float32(f): + '''Read a 32-bit float''' + return np.float32(struct.unpack('>f', f.read(4))[0]) + + +def _read_float64(f): + '''Read a 64-bit float''' + return np.float64(struct.unpack('>d', f.read(8))[0]) + + +class Pointer: + '''Class used to define pointers''' + + def __init__(self, index): + self.index = index + return + + +class ObjectPointer(Pointer): + '''Class used to define object pointers''' + pass + + +def _read_string(f): + '''Read a string''' + length = _read_long(f) + if length > 0: + chars = _read_bytes(f, length).decode('latin1') + _align_32(f) + else: + chars = '' + return chars + + +def _read_string_data(f): + '''Read a data string (length is specified twice)''' + length = _read_long(f) + if length > 0: + length = _read_long(f) + string_data = _read_bytes(f, length) + _align_32(f) + else: + string_data = '' + return string_data + + +def _read_data(f, dtype): + '''Read a variable with a specified data type''' + if dtype == 1: + if _read_int32(f) != 1: + raise Exception("Error occurred while reading byte variable") + return _read_byte(f) + elif dtype == 2: + return _read_int16(f) + elif dtype == 3: + return _read_int32(f) + elif dtype == 4: + return _read_float32(f) + elif dtype == 5: + return _read_float64(f) + elif dtype == 6: + real = _read_float32(f) + imag = _read_float32(f) + return np.complex64(real + imag * 1j) + elif dtype == 7: + return _read_string_data(f) + elif dtype == 8: + raise Exception("Should not be here - please report this") + elif dtype == 9: + real = _read_float64(f) + imag = _read_float64(f) + return np.complex128(real + imag * 1j) + elif dtype == 10: + return Pointer(_read_int32(f)) + elif dtype == 11: + return ObjectPointer(_read_int32(f)) + elif dtype == 12: + return _read_uint16(f) + elif dtype == 13: + return _read_uint32(f) + elif dtype == 14: + return _read_int64(f) + elif dtype == 15: + return _read_uint64(f) + else: + raise Exception("Unknown IDL type: %i - please report this" % dtype) + + +def _read_structure(f, array_desc, struct_desc): + ''' + Read a structure, with the array and structure descriptors given as + `array_desc` and `structure_desc` respectively. + ''' + + nrows = array_desc['nelements'] + columns = struct_desc['tagtable'] + + dtype = [] + for col in columns: + if col['structure'] or col['array']: + dtype.append(((col['name'].lower(), col['name']), np.object_)) + else: + if col['typecode'] in DTYPE_DICT: + dtype.append(((col['name'].lower(), col['name']), + DTYPE_DICT[col['typecode']])) + else: + raise Exception("Variable type %i not implemented" % + col['typecode']) + + structure = np.rec.recarray((nrows, ), dtype=dtype) + + for i in range(nrows): + for col in columns: + dtype = col['typecode'] + if col['structure']: + structure[col['name']][i] = _read_structure(f, + struct_desc['arrtable'][col['name']], + struct_desc['structtable'][col['name']]) + elif col['array']: + structure[col['name']][i] = _read_array(f, dtype, + struct_desc['arrtable'][col['name']]) + else: + structure[col['name']][i] = _read_data(f, dtype) + + # Reshape structure if needed + if array_desc['ndims'] > 1: + dims = array_desc['dims'][:int(array_desc['ndims'])] + dims.reverse() + structure = structure.reshape(dims) + + return structure + + +def _read_array(f, typecode, array_desc): + ''' + Read an array of type `typecode`, with the array descriptor given as + `array_desc`. + ''' + + if typecode in [1, 3, 4, 5, 6, 9, 13, 14, 15]: + + if typecode == 1: + nbytes = _read_int32(f) + if nbytes != array_desc['nbytes']: + warnings.warn("Not able to verify number of bytes from header", + stacklevel=3) + + # Read bytes as numpy array + array = np.frombuffer(f.read(array_desc['nbytes']), + dtype=DTYPE_DICT[typecode]) + + elif typecode in [2, 12]: + + # These are 2 byte types, need to skip every two as they are not packed + + array = np.frombuffer(f.read(array_desc['nbytes']*2), + dtype=DTYPE_DICT[typecode])[1::2] + + else: + + # Read bytes into list + array = [] + for i in range(array_desc['nelements']): + dtype = typecode + data = _read_data(f, dtype) + array.append(data) + + array = np.array(array, dtype=np.object_) + + # Reshape array if needed + if array_desc['ndims'] > 1: + dims = array_desc['dims'][:int(array_desc['ndims'])] + dims.reverse() + array = array.reshape(dims) + + # Go to next alignment position + _align_32(f) + + return array + + +def _read_record(f): + '''Function to read in a full record''' + + record = {'rectype': _read_long(f)} + + nextrec = _read_uint32(f) + nextrec += _read_uint32(f).astype(np.int64) * 2**32 + + _skip_bytes(f, 4) + + if record['rectype'] not in RECTYPE_DICT: + raise Exception("Unknown RECTYPE: %i" % record['rectype']) + + record['rectype'] = RECTYPE_DICT[record['rectype']] + + if record['rectype'] in ["VARIABLE", "HEAP_DATA"]: + + if record['rectype'] == "VARIABLE": + record['varname'] = _read_string(f) + else: + record['heap_index'] = _read_long(f) + _skip_bytes(f, 4) + + rectypedesc = _read_typedesc(f) + + if rectypedesc['typecode'] == 0: + + if nextrec == f.tell(): + record['data'] = None # Indicates NULL value + else: + raise ValueError("Unexpected type code: 0") + + else: + + varstart = _read_long(f) + if varstart != 7: + raise Exception("VARSTART is not 7") + + if rectypedesc['structure']: + record['data'] = _read_structure(f, rectypedesc['array_desc'], + rectypedesc['struct_desc']) + elif rectypedesc['array']: + record['data'] = _read_array(f, rectypedesc['typecode'], + rectypedesc['array_desc']) + else: + dtype = rectypedesc['typecode'] + record['data'] = _read_data(f, dtype) + + elif record['rectype'] == "TIMESTAMP": + + _skip_bytes(f, 4*256) + record['date'] = _read_string(f) + record['user'] = _read_string(f) + record['host'] = _read_string(f) + + elif record['rectype'] == "VERSION": + + record['format'] = _read_long(f) + record['arch'] = _read_string(f) + record['os'] = _read_string(f) + record['release'] = _read_string(f) + + elif record['rectype'] == "IDENTIFICATON": + + record['author'] = _read_string(f) + record['title'] = _read_string(f) + record['idcode'] = _read_string(f) + + elif record['rectype'] == "NOTICE": + + record['notice'] = _read_string(f) + + elif record['rectype'] == "DESCRIPTION": + + record['description'] = _read_string_data(f) + + elif record['rectype'] == "HEAP_HEADER": + + record['nvalues'] = _read_long(f) + record['indices'] = [_read_long(f) for _ in range(record['nvalues'])] + + elif record['rectype'] == "COMMONBLOCK": + + record['nvars'] = _read_long(f) + record['name'] = _read_string(f) + record['varnames'] = [_read_string(f) for _ in range(record['nvars'])] + + elif record['rectype'] == "END_MARKER": + + record['end'] = True + + elif record['rectype'] == "UNKNOWN": + + warnings.warn("Skipping UNKNOWN record", stacklevel=3) + + elif record['rectype'] == "SYSTEM_VARIABLE": + + warnings.warn("Skipping SYSTEM_VARIABLE record", stacklevel=3) + + else: + + raise Exception(f"record['rectype']={record['rectype']} not implemented") + + f.seek(nextrec) + + return record + + +def _read_typedesc(f): + '''Function to read in a type descriptor''' + + typedesc = {'typecode': _read_long(f), 'varflags': _read_long(f)} + + if typedesc['varflags'] & 2 == 2: + raise Exception("System variables not implemented") + + typedesc['array'] = typedesc['varflags'] & 4 == 4 + typedesc['structure'] = typedesc['varflags'] & 32 == 32 + + if typedesc['structure']: + typedesc['array_desc'] = _read_arraydesc(f) + typedesc['struct_desc'] = _read_structdesc(f) + elif typedesc['array']: + typedesc['array_desc'] = _read_arraydesc(f) + + return typedesc + + +def _read_arraydesc(f): + '''Function to read in an array descriptor''' + + arraydesc = {'arrstart': _read_long(f)} + + if arraydesc['arrstart'] == 8: + + _skip_bytes(f, 4) + + arraydesc['nbytes'] = _read_long(f) + arraydesc['nelements'] = _read_long(f) + arraydesc['ndims'] = _read_long(f) + + _skip_bytes(f, 8) + + arraydesc['nmax'] = _read_long(f) + + arraydesc['dims'] = [_read_long(f) for _ in range(arraydesc['nmax'])] + + elif arraydesc['arrstart'] == 18: + + warnings.warn("Using experimental 64-bit array read", stacklevel=3) + + _skip_bytes(f, 8) + + arraydesc['nbytes'] = _read_uint64(f) + arraydesc['nelements'] = _read_uint64(f) + arraydesc['ndims'] = _read_long(f) + + _skip_bytes(f, 8) + + arraydesc['nmax'] = 8 + + arraydesc['dims'] = [] + for d in range(arraydesc['nmax']): + v = _read_long(f) + if v != 0: + raise Exception("Expected a zero in ARRAY_DESC") + arraydesc['dims'].append(_read_long(f)) + + else: + + raise Exception("Unknown ARRSTART: %i" % arraydesc['arrstart']) + + return arraydesc + + +def _read_structdesc(f): + '''Function to read in a structure descriptor''' + + structdesc = {} + + structstart = _read_long(f) + if structstart != 9: + raise Exception("STRUCTSTART should be 9") + + structdesc['name'] = _read_string(f) + predef = _read_long(f) + structdesc['ntags'] = _read_long(f) + structdesc['nbytes'] = _read_long(f) + + structdesc['predef'] = predef & 1 + structdesc['inherits'] = predef & 2 + structdesc['is_super'] = predef & 4 + + if not structdesc['predef']: + + structdesc['tagtable'] = [_read_tagdesc(f) + for _ in range(structdesc['ntags'])] + + for tag in structdesc['tagtable']: + tag['name'] = _read_string(f) + + structdesc['arrtable'] = {tag['name']: _read_arraydesc(f) + for tag in structdesc['tagtable'] + if tag['array']} + + structdesc['structtable'] = {tag['name']: _read_structdesc(f) + for tag in structdesc['tagtable'] + if tag['structure']} + + if structdesc['inherits'] or structdesc['is_super']: + structdesc['classname'] = _read_string(f) + structdesc['nsupclasses'] = _read_long(f) + structdesc['supclassnames'] = [ + _read_string(f) for _ in range(structdesc['nsupclasses'])] + structdesc['supclasstable'] = [ + _read_structdesc(f) for _ in range(structdesc['nsupclasses'])] + + STRUCT_DICT[structdesc['name']] = structdesc + + else: + + if structdesc['name'] not in STRUCT_DICT: + raise Exception("PREDEF=1 but can't find definition") + + structdesc = STRUCT_DICT[structdesc['name']] + + return structdesc + + +def _read_tagdesc(f): + '''Function to read in a tag descriptor''' + + tagdesc = {'offset': _read_long(f)} + + if tagdesc['offset'] == -1: + tagdesc['offset'] = _read_uint64(f) + + tagdesc['typecode'] = _read_long(f) + tagflags = _read_long(f) + + tagdesc['array'] = tagflags & 4 == 4 + tagdesc['structure'] = tagflags & 32 == 32 + tagdesc['scalar'] = tagdesc['typecode'] in DTYPE_DICT + # Assume '10'x is scalar + + return tagdesc + + +def _replace_heap(variable, heap): + + if isinstance(variable, Pointer): + + while isinstance(variable, Pointer): + + if variable.index == 0: + variable = None + else: + if variable.index in heap: + variable = heap[variable.index] + else: + warnings.warn("Variable referenced by pointer not found " + "in heap: variable will be set to None", + stacklevel=3) + variable = None + + replace, new = _replace_heap(variable, heap) + + if replace: + variable = new + + return True, variable + + elif isinstance(variable, np.rec.recarray): + + # Loop over records + for ir, record in enumerate(variable): + + replace, new = _replace_heap(record, heap) + + if replace: + variable[ir] = new + + return False, variable + + elif isinstance(variable, np.record): + + # Loop over values + for iv, value in enumerate(variable): + + replace, new = _replace_heap(value, heap) + + if replace: + variable[iv] = new + + return False, variable + + elif isinstance(variable, np.ndarray): + + # Loop over values if type is np.object_ + if variable.dtype.type is np.object_: + + for iv in range(variable.size): + + replace, new = _replace_heap(variable.item(iv), heap) + + if replace: + variable.reshape(-1)[iv] = new + + return False, variable + + else: + + return False, variable + + +class AttrDict(dict): + ''' + A case-insensitive dictionary with access via item, attribute, and call + notations: + + >>> from scipy.io._idl import AttrDict + >>> d = AttrDict() + >>> d['Variable'] = 123 + >>> d['Variable'] + 123 + >>> d.Variable + 123 + >>> d.variable + 123 + >>> d('VARIABLE') + 123 + >>> d['missing'] + Traceback (most recent error last): + ... + KeyError: 'missing' + >>> d.missing + Traceback (most recent error last): + ... + AttributeError: 'AttrDict' object has no attribute 'missing' + ''' + + def __init__(self, init=None): + if init is None: + init = {} + dict.__init__(self, init) + + def __getitem__(self, name): + return super().__getitem__(name.lower()) + + def __setitem__(self, key, value): + return super().__setitem__(key.lower(), value) + + def __getattr__(self, name): + try: + return self.__getitem__(name) + except KeyError: + raise AttributeError( + f"'{type(self)}' object has no attribute '{name}'") from None + + __setattr__ = __setitem__ + __call__ = __getitem__ + + +def readsav(file_name, idict=None, python_dict=False, + uncompressed_file_name=None, verbose=False): + """ + Read an IDL .sav file. + + Parameters + ---------- + file_name : str + Name of the IDL save file. + idict : dict, optional + Dictionary in which to insert .sav file variables. + python_dict : bool, optional + By default, the object return is not a Python dictionary, but a + case-insensitive dictionary with item, attribute, and call access + to variables. To get a standard Python dictionary, set this option + to True. + uncompressed_file_name : str, optional + This option only has an effect for .sav files written with the + /compress option. If a file name is specified, compressed .sav + files are uncompressed to this file. Otherwise, readsav will use + the `tempfile` module to determine a temporary filename + automatically, and will remove the temporary file upon successfully + reading it in. + verbose : bool, optional + Whether to print out information about the save file, including + the records read, and available variables. + + Returns + ------- + idl_dict : AttrDict or dict + If `python_dict` is set to False (default), this function returns a + case-insensitive dictionary with item, attribute, and call access + to variables. If `python_dict` is set to True, this function + returns a Python dictionary with all variable names in lowercase. + If `idict` was specified, then variables are written to the + dictionary specified, and the updated dictionary is returned. + + Examples + -------- + >>> from os.path import dirname, join as pjoin + >>> import scipy.io as sio + >>> from scipy.io import readsav + + Get the filename for an example .sav file from the tests/data directory. + + >>> data_dir = pjoin(dirname(sio.__file__), 'tests', 'data') + >>> sav_fname = pjoin(data_dir, 'array_float32_1d.sav') + + Load the .sav file contents. + + >>> sav_data = readsav(sav_fname) + + Get keys of the .sav file contents. + + >>> print(sav_data.keys()) + dict_keys(['array1d']) + + Access a content with a key. + + >>> print(sav_data['array1d']) + [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. + 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. + 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. + 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. + 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. + 0. 0. 0.] + + """ + + # Initialize record and variable holders + records = [] + if python_dict or idict: + variables = {} + else: + variables = AttrDict() + + # Open the IDL file + f = open(file_name, 'rb') + + # Read the signature, which should be 'SR' + signature = _read_bytes(f, 2) + if signature != b'SR': + raise Exception(f"Invalid SIGNATURE: {signature}") + + # Next, the record format, which is '\x00\x04' for normal .sav + # files, and '\x00\x06' for compressed .sav files. + recfmt = _read_bytes(f, 2) + + if recfmt == b'\x00\x04': + pass + + elif recfmt == b'\x00\x06': + + if verbose: + print("IDL Save file is compressed") + + if uncompressed_file_name: + fout = open(uncompressed_file_name, 'w+b') + else: + fout = tempfile.NamedTemporaryFile(suffix='.sav') + + if verbose: + print(f" -> expanding to {fout.name}") + + # Write header + fout.write(b'SR\x00\x04') + + # Cycle through records + while True: + + # Read record type + rectype = _read_long(f) + fout.write(struct.pack('>l', int(rectype))) + + # Read position of next record and return as int + nextrec = _read_uint32(f) + nextrec += _read_uint32(f).astype(np.int64) * 2**32 + + # Read the unknown 4 bytes + unknown = f.read(4) + + # Check if the end of the file has been reached + if RECTYPE_DICT[rectype] == 'END_MARKER': + modval = np.int64(2**32) + fout.write(struct.pack('>I', int(nextrec) % modval)) + fout.write( + struct.pack('>I', int((nextrec - (nextrec % modval)) / modval)) + ) + fout.write(unknown) + break + + # Find current position + pos = f.tell() + + # Decompress record + rec_string = zlib.decompress(f.read(nextrec-pos)) + + # Find new position of next record + nextrec = fout.tell() + len(rec_string) + 12 + + # Write out record + fout.write(struct.pack('>I', int(nextrec % 2**32))) + fout.write(struct.pack('>I', int((nextrec - (nextrec % 2**32)) / 2**32))) + fout.write(unknown) + fout.write(rec_string) + + # Close the original compressed file + f.close() + + # Set f to be the decompressed file, and skip the first four bytes + f = fout + f.seek(4) + + else: + raise Exception(f"Invalid RECFMT: {recfmt}") + + # Loop through records, and add them to the list + while True: + r = _read_record(f) + records.append(r) + if 'end' in r: + if r['end']: + break + + # Close the file + f.close() + + # Find heap data variables + heap = {} + for r in records: + if r['rectype'] == "HEAP_DATA": + heap[r['heap_index']] = r['data'] + + # Find all variables + for r in records: + if r['rectype'] == "VARIABLE": + replace, new = _replace_heap(r['data'], heap) + if replace: + r['data'] = new + variables[r['varname'].lower()] = r['data'] + + if verbose: + + # Print out timestamp info about the file + for record in records: + if record['rectype'] == "TIMESTAMP": + print("-"*50) + print(f"Date: {record['date']}") + print(f"User: {record['user']}") + print(f"Host: {record['host']}") + break + + # Print out version info about the file + for record in records: + if record['rectype'] == "VERSION": + print("-"*50) + print(f"Format: {record['format']}") + print(f"Architecture: {record['arch']}") + print(f"Operating System: {record['os']}") + print(f"IDL Version: {record['release']}") + break + + # Print out identification info about the file + for record in records: + if record['rectype'] == "IDENTIFICATON": + print("-"*50) + print(f"Author: {record['author']}") + print(f"Title: {record['title']}") + print(f"ID Code: {record['idcode']}") + break + + # Print out descriptions saved with the file + for record in records: + if record['rectype'] == "DESCRIPTION": + print("-"*50) + print(f"Description: {record['description']}") + break + + print("-"*50) + print(f"Successfully read {len(records)} records of which:") + + # Create convenience list of record types + rectypes = [r['rectype'] for r in records] + + for rt in set(rectypes): + if rt != 'END_MARKER': + print(" - %i are of type %s" % (rectypes.count(rt), rt)) + print("-"*50) + + if 'VARIABLE' in rectypes: + print("Available variables:") + for var in variables: + print(f" - {var} [{type(variables[var])}]") + print("-"*50) + + if idict: + for var in variables: + idict[var] = variables[var] + return idict + else: + return variables diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_mmio.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_mmio.py new file mode 100644 index 0000000000000000000000000000000000000000..32db20065d632d582f04addcf766daa4e6b5fd8e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_mmio.py @@ -0,0 +1,968 @@ +""" + Matrix Market I/O in Python. + See http://math.nist.gov/MatrixMarket/formats.html + for information about the Matrix Market format. +""" +# +# Author: Pearu Peterson +# Created: October, 2004 +# +# References: +# http://math.nist.gov/MatrixMarket/ +# +import os + +import numpy as np +from numpy import (asarray, real, imag, conj, zeros, ndarray, concatenate, + ones, can_cast) + +from scipy.sparse import coo_array, issparse, coo_matrix + +__all__ = ['mminfo', 'mmread', 'mmwrite', 'MMFile'] + + +# ----------------------------------------------------------------------------- +def asstr(s): + if isinstance(s, bytes): + return s.decode('latin1') + return str(s) + + +def mminfo(source): + """ + Return size and storage parameters from Matrix Market file-like 'source'. + + Parameters + ---------- + source : str or file-like + Matrix Market filename (extension .mtx) or open file-like object + + Returns + ------- + rows : int + Number of matrix rows. + cols : int + Number of matrix columns. + entries : int + Number of non-zero entries of a sparse matrix + or rows*cols for a dense matrix. + format : str + Either 'coordinate' or 'array'. + field : str + Either 'real', 'complex', 'pattern', or 'integer'. + symmetry : str + Either 'general', 'symmetric', 'skew-symmetric', or 'hermitian'. + + Examples + -------- + >>> from io import StringIO + >>> from scipy.io import mminfo + + >>> text = '''%%MatrixMarket matrix coordinate real general + ... 5 5 7 + ... 2 3 1.0 + ... 3 4 2.0 + ... 3 5 3.0 + ... 4 1 4.0 + ... 4 2 5.0 + ... 4 3 6.0 + ... 4 4 7.0 + ... ''' + + + ``mminfo(source)`` returns the number of rows, number of columns, + format, field type and symmetry attribute of the source file. + + >>> mminfo(StringIO(text)) + (5, 5, 7, 'coordinate', 'real', 'general') + """ + return MMFile.info(source) + +# ----------------------------------------------------------------------------- + + +def mmread(source, *, spmatrix=True): + """ + Reads the contents of a Matrix Market file-like 'source' into a matrix. + + Parameters + ---------- + source : str or file-like + Matrix Market filename (extensions .mtx, .mtz.gz) + or open file-like object. + spmatrix : bool, optional (default: True) + If ``True``, return sparse ``coo_matrix``. Otherwise return ``coo_array``. + + Returns + ------- + a : ndarray or coo_array or coo_matrix + Dense or sparse array depending on the matrix format in the + Matrix Market file. + + Examples + -------- + >>> from io import StringIO + >>> from scipy.io import mmread + + >>> text = '''%%MatrixMarket matrix coordinate real general + ... 5 5 7 + ... 2 3 1.0 + ... 3 4 2.0 + ... 3 5 3.0 + ... 4 1 4.0 + ... 4 2 5.0 + ... 4 3 6.0 + ... 4 4 7.0 + ... ''' + + ``mmread(source)`` returns the data as sparse matrix in COO format. + + >>> m = mmread(StringIO(text), spmatrix=False) + >>> m + + >>> m.toarray() + array([[0., 0., 0., 0., 0.], + [0., 0., 1., 0., 0.], + [0., 0., 0., 2., 3.], + [4., 5., 6., 7., 0.], + [0., 0., 0., 0., 0.]]) + """ + return MMFile().read(source, spmatrix=spmatrix) + +# ----------------------------------------------------------------------------- + + +def mmwrite(target, a, comment='', field=None, precision=None, symmetry=None): + r""" + Writes the sparse or dense array `a` to Matrix Market file-like `target`. + + Parameters + ---------- + target : str or file-like + Matrix Market filename (extension .mtx) or open file-like object. + a : array like + Sparse or dense 2-D array. + comment : str, optional + Comments to be prepended to the Matrix Market file. + field : None or str, optional + Either 'real', 'complex', 'pattern', or 'integer'. + precision : None or int, optional + Number of digits to display for real or complex values. + symmetry : None or str, optional + Either 'general', 'symmetric', 'skew-symmetric', or 'hermitian'. + If symmetry is None the symmetry type of 'a' is determined by its + values. + + Returns + ------- + None + + Examples + -------- + >>> from io import BytesIO + >>> import numpy as np + >>> from scipy.sparse import coo_array + >>> from scipy.io import mmwrite + + Write a small NumPy array to a matrix market file. The file will be + written in the ``'array'`` format. + + >>> a = np.array([[1.0, 0, 0, 0], [0, 2.5, 0, 6.25]]) + >>> target = BytesIO() + >>> mmwrite(target, a) + >>> print(target.getvalue().decode('latin1')) + %%MatrixMarket matrix array real general + % + 2 4 + 1 + 0 + 0 + 2.5 + 0 + 0 + 0 + 6.25 + + Add a comment to the output file, and set the precision to 3. + + >>> target = BytesIO() + >>> mmwrite(target, a, comment='\n Some test data.\n', precision=3) + >>> print(target.getvalue().decode('latin1')) + %%MatrixMarket matrix array real general + % + % Some test data. + % + 2 4 + 1.00e+00 + 0.00e+00 + 0.00e+00 + 2.50e+00 + 0.00e+00 + 0.00e+00 + 0.00e+00 + 6.25e+00 + + Convert to a sparse matrix before calling ``mmwrite``. This will + result in the output format being ``'coordinate'`` rather than + ``'array'``. + + >>> target = BytesIO() + >>> mmwrite(target, coo_array(a), precision=3) + >>> print(target.getvalue().decode('latin1')) + %%MatrixMarket matrix coordinate real general + % + 2 4 3 + 1 1 1.00e+00 + 2 2 2.50e+00 + 2 4 6.25e+00 + + Write a complex Hermitian array to a matrix market file. Note that + only six values are actually written to the file; the other values + are implied by the symmetry. + + >>> z = np.array([[3, 1+2j, 4-3j], [1-2j, 1, -5j], [4+3j, 5j, 2.5]]) + >>> z + array([[ 3. +0.j, 1. +2.j, 4. -3.j], + [ 1. -2.j, 1. +0.j, -0. -5.j], + [ 4. +3.j, 0. +5.j, 2.5+0.j]]) + + >>> target = BytesIO() + >>> mmwrite(target, z, precision=2) + >>> print(target.getvalue().decode('latin1')) + %%MatrixMarket matrix array complex hermitian + % + 3 3 + 3.0e+00 0.0e+00 + 1.0e+00 -2.0e+00 + 4.0e+00 3.0e+00 + 1.0e+00 0.0e+00 + 0.0e+00 5.0e+00 + 2.5e+00 0.0e+00 + + """ + MMFile().write(target, a, comment, field, precision, symmetry) + + +############################################################################### +class MMFile: + __slots__ = ('_rows', + '_cols', + '_entries', + '_format', + '_field', + '_symmetry') + + @property + def rows(self): + return self._rows + + @property + def cols(self): + return self._cols + + @property + def entries(self): + return self._entries + + @property + def format(self): + return self._format + + @property + def field(self): + return self._field + + @property + def symmetry(self): + return self._symmetry + + @property + def has_symmetry(self): + return self._symmetry in (self.SYMMETRY_SYMMETRIC, + self.SYMMETRY_SKEW_SYMMETRIC, + self.SYMMETRY_HERMITIAN) + + # format values + FORMAT_COORDINATE = 'coordinate' + FORMAT_ARRAY = 'array' + FORMAT_VALUES = (FORMAT_COORDINATE, FORMAT_ARRAY) + + @classmethod + def _validate_format(self, format): + if format not in self.FORMAT_VALUES: + msg = f'unknown format type {format}, must be one of {self.FORMAT_VALUES}' + raise ValueError(msg) + + # field values + FIELD_INTEGER = 'integer' + FIELD_UNSIGNED = 'unsigned-integer' + FIELD_REAL = 'real' + FIELD_COMPLEX = 'complex' + FIELD_PATTERN = 'pattern' + FIELD_VALUES = (FIELD_INTEGER, FIELD_UNSIGNED, FIELD_REAL, FIELD_COMPLEX, + FIELD_PATTERN) + + @classmethod + def _validate_field(self, field): + if field not in self.FIELD_VALUES: + msg = f'unknown field type {field}, must be one of {self.FIELD_VALUES}' + raise ValueError(msg) + + # symmetry values + SYMMETRY_GENERAL = 'general' + SYMMETRY_SYMMETRIC = 'symmetric' + SYMMETRY_SKEW_SYMMETRIC = 'skew-symmetric' + SYMMETRY_HERMITIAN = 'hermitian' + SYMMETRY_VALUES = (SYMMETRY_GENERAL, SYMMETRY_SYMMETRIC, + SYMMETRY_SKEW_SYMMETRIC, SYMMETRY_HERMITIAN) + + @classmethod + def _validate_symmetry(self, symmetry): + if symmetry not in self.SYMMETRY_VALUES: + raise ValueError(f'unknown symmetry type {symmetry}, ' + f'must be one of {self.SYMMETRY_VALUES}') + + DTYPES_BY_FIELD = {FIELD_INTEGER: 'intp', + FIELD_UNSIGNED: 'uint64', + FIELD_REAL: 'd', + FIELD_COMPLEX: 'D', + FIELD_PATTERN: 'd'} + + # ------------------------------------------------------------------------- + @staticmethod + def reader(): + pass + + # ------------------------------------------------------------------------- + @staticmethod + def writer(): + pass + + # ------------------------------------------------------------------------- + @classmethod + def info(self, source): + """ + Return size, storage parameters from Matrix Market file-like 'source'. + + Parameters + ---------- + source : str or file-like + Matrix Market filename (extension .mtx) or open file-like object + + Returns + ------- + rows : int + Number of matrix rows. + cols : int + Number of matrix columns. + entries : int + Number of non-zero entries of a sparse matrix + or rows*cols for a dense matrix. + format : str + Either 'coordinate' or 'array'. + field : str + Either 'real', 'complex', 'pattern', or 'integer'. + symmetry : str + Either 'general', 'symmetric', 'skew-symmetric', or 'hermitian'. + """ + + stream, close_it = self._open(source) + + try: + + # read and validate header line + line = stream.readline() + mmid, matrix, format, field, symmetry = \ + (asstr(part.strip()) for part in line.split()) + if not mmid.startswith('%%MatrixMarket'): + raise ValueError('source is not in Matrix Market format') + if not matrix.lower() == 'matrix': + raise ValueError("Problem reading file header: " + line) + + # http://math.nist.gov/MatrixMarket/formats.html + if format.lower() == 'array': + format = self.FORMAT_ARRAY + elif format.lower() == 'coordinate': + format = self.FORMAT_COORDINATE + + # skip comments + # line.startswith('%') + while line: + if line.lstrip() and line.lstrip()[0] in ['%', 37]: + line = stream.readline() + else: + break + + # skip empty lines + while not line.strip(): + line = stream.readline() + + split_line = line.split() + if format == self.FORMAT_ARRAY: + if not len(split_line) == 2: + raise ValueError("Header line not of length 2: " + + line.decode('ascii')) + rows, cols = map(int, split_line) + entries = rows * cols + else: + if not len(split_line) == 3: + raise ValueError("Header line not of length 3: " + + line.decode('ascii')) + rows, cols, entries = map(int, split_line) + + return (rows, cols, entries, format, field.lower(), + symmetry.lower()) + + finally: + if close_it: + stream.close() + + # ------------------------------------------------------------------------- + @staticmethod + def _open(filespec, mode='rb'): + """ Return an open file stream for reading based on source. + + If source is a file name, open it (after trying to find it with mtx and + gzipped mtx extensions). Otherwise, just return source. + + Parameters + ---------- + filespec : str or file-like + String giving file name or file-like object + mode : str, optional + Mode with which to open file, if `filespec` is a file name. + + Returns + ------- + fobj : file-like + Open file-like object. + close_it : bool + True if the calling function should close this file when done, + false otherwise. + """ + # If 'filespec' is path-like (str, pathlib.Path, os.DirEntry, other class + # implementing a '__fspath__' method), try to convert it to str. If this + # fails by throwing a 'TypeError', assume it's an open file handle and + # return it as-is. + try: + filespec = os.fspath(filespec) + except TypeError: + return filespec, False + + # 'filespec' is definitely a str now + + # open for reading + if mode[0] == 'r': + + # determine filename plus extension + if not os.path.isfile(filespec): + if os.path.isfile(filespec+'.mtx'): + filespec = filespec + '.mtx' + elif os.path.isfile(filespec+'.mtx.gz'): + filespec = filespec + '.mtx.gz' + elif os.path.isfile(filespec+'.mtx.bz2'): + filespec = filespec + '.mtx.bz2' + # open filename + if filespec.endswith('.gz'): + import gzip + stream = gzip.open(filespec, mode) + elif filespec.endswith('.bz2'): + import bz2 + stream = bz2.BZ2File(filespec, 'rb') + else: + stream = open(filespec, mode) + + # open for writing + else: + if filespec[-4:] != '.mtx': + filespec = filespec + '.mtx' + stream = open(filespec, mode) + + return stream, True + + # ------------------------------------------------------------------------- + @staticmethod + def _get_symmetry(a): + m, n = a.shape + if m != n: + return MMFile.SYMMETRY_GENERAL + issymm = True + isskew = True + isherm = a.dtype.char in 'FD' + + # sparse input + if issparse(a): + # check if number of nonzero entries of lower and upper triangle + # matrix are equal + a = a.tocoo() + (row, col) = a.nonzero() + if (row < col).sum() != (row > col).sum(): + return MMFile.SYMMETRY_GENERAL + + # define iterator over symmetric pair entries + a = a.todok() + + def symm_iterator(): + for ((i, j), aij) in a.items(): + if i > j: + aji = a[j, i] + yield (aij, aji, False) + elif i == j: + yield (aij, aij, True) + + # non-sparse input + else: + # define iterator over symmetric pair entries + def symm_iterator(): + for j in range(n): + for i in range(j, n): + aij, aji = a[i][j], a[j][i] + yield (aij, aji, i == j) + + # check for symmetry + # yields aij, aji, is_diagonal + for (aij, aji, is_diagonal) in symm_iterator(): + if isskew and is_diagonal and aij != 0: + isskew = False + else: + if issymm and aij != aji: + issymm = False + with np.errstate(over="ignore"): + # This can give a warning for uint dtypes, so silence that + if isskew and aij != -aji: + isskew = False + if isherm and aij != conj(aji): + isherm = False + if not (issymm or isskew or isherm): + break + + # return symmetry value + if issymm: + return MMFile.SYMMETRY_SYMMETRIC + if isskew: + return MMFile.SYMMETRY_SKEW_SYMMETRIC + if isherm: + return MMFile.SYMMETRY_HERMITIAN + return MMFile.SYMMETRY_GENERAL + + # ------------------------------------------------------------------------- + @staticmethod + def _field_template(field, precision): + return {MMFile.FIELD_REAL: '%%.%ie\n' % precision, + MMFile.FIELD_INTEGER: '%i\n', + MMFile.FIELD_UNSIGNED: '%u\n', + MMFile.FIELD_COMPLEX: '%%.%ie %%.%ie\n' % + (precision, precision) + }.get(field, None) + + # ------------------------------------------------------------------------- + def __init__(self, **kwargs): + self._init_attrs(**kwargs) + + # ------------------------------------------------------------------------- + def read(self, source, *, spmatrix=True): + """ + Reads the contents of a Matrix Market file-like 'source' into a matrix. + + Parameters + ---------- + source : str or file-like + Matrix Market filename (extensions .mtx, .mtz.gz) + or open file object. + spmatrix : bool, optional (default: True) + If ``True``, return sparse ``coo_matrix``. Otherwise return ``coo_array``. + + Returns + ------- + a : ndarray or coo_array or coo_matrix + Dense or sparse array depending on the matrix format in the + Matrix Market file. + """ + stream, close_it = self._open(source) + + try: + self._parse_header(stream) + data = self._parse_body(stream) + + finally: + if close_it: + stream.close() + if spmatrix and isinstance(data, coo_array): + data = coo_matrix(data) + return data + + + # ------------------------------------------------------------------------- + def write(self, target, a, comment='', field=None, precision=None, + symmetry=None): + """ + Writes sparse or dense array `a` to Matrix Market file-like `target`. + + Parameters + ---------- + target : str or file-like + Matrix Market filename (extension .mtx) or open file-like object. + a : array like + Sparse or dense 2-D array. + comment : str, optional + Comments to be prepended to the Matrix Market file. + field : None or str, optional + Either 'real', 'complex', 'pattern', or 'integer'. + precision : None or int, optional + Number of digits to display for real or complex values. + symmetry : None or str, optional + Either 'general', 'symmetric', 'skew-symmetric', or 'hermitian'. + If symmetry is None the symmetry type of 'a' is determined by its + values. + """ + + stream, close_it = self._open(target, 'wb') + + try: + self._write(stream, a, comment, field, precision, symmetry) + + finally: + if close_it: + stream.close() + else: + stream.flush() + + # ------------------------------------------------------------------------- + def _init_attrs(self, **kwargs): + """ + Initialize each attributes with the corresponding keyword arg value + or a default of None + """ + + attrs = self.__class__.__slots__ + public_attrs = [attr[1:] for attr in attrs] + invalid_keys = set(kwargs.keys()) - set(public_attrs) + + if invalid_keys: + raise ValueError(f"found {tuple(invalid_keys)} invalid keyword " + f"arguments, please only use {public_attrs}") + + for attr in attrs: + setattr(self, attr, kwargs.get(attr[1:], None)) + + # ------------------------------------------------------------------------- + def _parse_header(self, stream): + rows, cols, entries, format, field, symmetry = \ + self.__class__.info(stream) + self._init_attrs(rows=rows, cols=cols, entries=entries, format=format, + field=field, symmetry=symmetry) + + # ------------------------------------------------------------------------- + def _parse_body(self, stream): + rows, cols, entries, format, field, symm = (self.rows, self.cols, + self.entries, self.format, + self.field, self.symmetry) + + dtype = self.DTYPES_BY_FIELD.get(field, None) + + has_symmetry = self.has_symmetry + is_integer = field == self.FIELD_INTEGER + is_unsigned_integer = field == self.FIELD_UNSIGNED + is_complex = field == self.FIELD_COMPLEX + is_skew = symm == self.SYMMETRY_SKEW_SYMMETRIC + is_herm = symm == self.SYMMETRY_HERMITIAN + is_pattern = field == self.FIELD_PATTERN + + if format == self.FORMAT_ARRAY: + a = zeros((rows, cols), dtype=dtype) + line = 1 + i, j = 0, 0 + if is_skew: + a[i, j] = 0 + if i < rows - 1: + i += 1 + while line: + line = stream.readline() + # line.startswith('%') + if not line or line[0] in ['%', 37] or not line.strip(): + continue + if is_integer: + aij = int(line) + elif is_unsigned_integer: + aij = int(line) + elif is_complex: + aij = complex(*map(float, line.split())) + else: + aij = float(line) + a[i, j] = aij + if has_symmetry and i != j: + if is_skew: + a[j, i] = -aij + elif is_herm: + a[j, i] = conj(aij) + else: + a[j, i] = aij + if i < rows-1: + i = i + 1 + else: + j = j + 1 + if not has_symmetry: + i = 0 + else: + i = j + if is_skew: + a[i, j] = 0 + if i < rows-1: + i += 1 + + if is_skew: + if not (i in [0, j] and j == cols - 1): + raise ValueError("Parse error, did not read all lines.") + else: + if not (i in [0, j] and j == cols): + raise ValueError("Parse error, did not read all lines.") + + elif format == self.FORMAT_COORDINATE: + # Read sparse COOrdinate format + + if entries == 0: + # empty matrix + return coo_array((rows, cols), dtype=dtype) + + I = zeros(entries, dtype='intc') + J = zeros(entries, dtype='intc') + if is_pattern: + V = ones(entries, dtype='int8') + elif is_integer: + V = zeros(entries, dtype='intp') + elif is_unsigned_integer: + V = zeros(entries, dtype='uint64') + elif is_complex: + V = zeros(entries, dtype='complex') + else: + V = zeros(entries, dtype='float') + + entry_number = 0 + for line in stream: + # line.startswith('%') + if not line or line[0] in ['%', 37] or not line.strip(): + continue + + if entry_number+1 > entries: + raise ValueError("'entries' in header is smaller than " + "number of entries") + l = line.split() + I[entry_number], J[entry_number] = map(int, l[:2]) + + if not is_pattern: + if is_integer: + V[entry_number] = int(l[2]) + elif is_unsigned_integer: + V[entry_number] = int(l[2]) + elif is_complex: + V[entry_number] = complex(*map(float, l[2:])) + else: + V[entry_number] = float(l[2]) + entry_number += 1 + if entry_number < entries: + raise ValueError("'entries' in header is larger than " + "number of entries") + + I -= 1 # adjust indices (base 1 -> base 0) + J -= 1 + + if has_symmetry: + mask = (I != J) # off diagonal mask + od_I = I[mask] + od_J = J[mask] + od_V = V[mask] + + I = concatenate((I, od_J)) + J = concatenate((J, od_I)) + + if is_skew: + od_V *= -1 + elif is_herm: + od_V = od_V.conjugate() + + V = concatenate((V, od_V)) + + a = coo_array((V, (I, J)), shape=(rows, cols), dtype=dtype) + else: + raise NotImplementedError(format) + + return a + + # ------------------------------------------------------------------------ + def _write(self, stream, a, comment='', field=None, precision=None, + symmetry=None): + if isinstance(a, list) or isinstance(a, ndarray) or \ + isinstance(a, tuple) or hasattr(a, '__array__'): + rep = self.FORMAT_ARRAY + a = asarray(a) + if len(a.shape) != 2: + raise ValueError('Expected 2 dimensional array') + rows, cols = a.shape + + if field is not None: + + if field == self.FIELD_INTEGER: + if not can_cast(a.dtype, 'intp'): + raise OverflowError("mmwrite does not support integer " + "dtypes larger than native 'intp'.") + a = a.astype('intp') + elif field == self.FIELD_REAL: + if a.dtype.char not in 'fd': + a = a.astype('d') + elif field == self.FIELD_COMPLEX: + if a.dtype.char not in 'FD': + a = a.astype('D') + + else: + if not issparse(a): + raise ValueError(f'unknown matrix type: {type(a)}') + + rep = 'coordinate' + rows, cols = a.shape + + typecode = a.dtype.char + + if precision is None: + if typecode in 'fF': + precision = 8 + else: + precision = 16 + if field is None: + kind = a.dtype.kind + if kind == 'i': + if not can_cast(a.dtype, 'intp'): + raise OverflowError("mmwrite does not support integer " + "dtypes larger than native 'intp'.") + field = 'integer' + elif kind == 'f': + field = 'real' + elif kind == 'c': + field = 'complex' + elif kind == 'u': + field = 'unsigned-integer' + else: + raise TypeError('unexpected dtype kind ' + kind) + + if symmetry is None: + symmetry = self._get_symmetry(a) + + # validate rep, field, and symmetry + self.__class__._validate_format(rep) + self.__class__._validate_field(field) + self.__class__._validate_symmetry(symmetry) + + # write initial header line + data = f'%%MatrixMarket matrix {rep} {field} {symmetry}\n' + stream.write(data.encode('latin1')) + + # write comments + for line in comment.split('\n'): + data = f'%{line}\n' + stream.write(data.encode('latin1')) + + template = self._field_template(field, precision) + # write dense format + if rep == self.FORMAT_ARRAY: + # write shape spec + data = '%i %i\n' % (rows, cols) + stream.write(data.encode('latin1')) + + if field in (self.FIELD_INTEGER, self.FIELD_REAL, + self.FIELD_UNSIGNED): + if symmetry == self.SYMMETRY_GENERAL: + for j in range(cols): + for i in range(rows): + data = template % a[i, j] + stream.write(data.encode('latin1')) + + elif symmetry == self.SYMMETRY_SKEW_SYMMETRIC: + for j in range(cols): + for i in range(j + 1, rows): + data = template % a[i, j] + stream.write(data.encode('latin1')) + + else: + for j in range(cols): + for i in range(j, rows): + data = template % a[i, j] + stream.write(data.encode('latin1')) + + elif field == self.FIELD_COMPLEX: + + if symmetry == self.SYMMETRY_GENERAL: + for j in range(cols): + for i in range(rows): + aij = a[i, j] + data = template % (real(aij), imag(aij)) + stream.write(data.encode('latin1')) + else: + for j in range(cols): + for i in range(j, rows): + aij = a[i, j] + data = template % (real(aij), imag(aij)) + stream.write(data.encode('latin1')) + + elif field == self.FIELD_PATTERN: + raise ValueError('pattern type inconsisted with dense format') + + else: + raise TypeError(f'Unknown field type {field}') + + # write sparse format + else: + coo = a.tocoo() # convert to COOrdinate format + + # if symmetry format used, remove values above main diagonal + if symmetry != self.SYMMETRY_GENERAL: + lower_triangle_mask = coo.row >= coo.col + coo = coo_array((coo.data[lower_triangle_mask], + (coo.row[lower_triangle_mask], + coo.col[lower_triangle_mask])), + shape=coo.shape) + + # write shape spec + data = '%i %i %i\n' % (rows, cols, coo.nnz) + stream.write(data.encode('latin1')) + + template = self._field_template(field, precision-1) + + if field == self.FIELD_PATTERN: + for r, c in zip(coo.row+1, coo.col+1): + data = "%i %i\n" % (r, c) + stream.write(data.encode('latin1')) + elif field in (self.FIELD_INTEGER, self.FIELD_REAL, + self.FIELD_UNSIGNED): + for r, c, d in zip(coo.row+1, coo.col+1, coo.data): + data = ("%i %i " % (r, c)) + (template % d) + stream.write(data.encode('latin1')) + elif field == self.FIELD_COMPLEX: + for r, c, d in zip(coo.row+1, coo.col+1, coo.data): + data = ("%i %i " % (r, c)) + (template % (d.real, d.imag)) + stream.write(data.encode('latin1')) + else: + raise TypeError(f'Unknown field type {field}') + + +def _is_fromfile_compatible(stream): + """ + Check whether `stream` is compatible with numpy.fromfile. + + Passing a gzipped file object to ``fromfile/fromstring`` doesn't work with + Python 3. + """ + + bad_cls = [] + try: + import gzip + bad_cls.append(gzip.GzipFile) + except ImportError: + pass + try: + import bz2 + bad_cls.append(bz2.BZ2File) + except ImportError: + pass + + bad_cls = tuple(bad_cls) + return not isinstance(stream, bad_cls) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_netcdf.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_netcdf.py new file mode 100644 index 0000000000000000000000000000000000000000..3f4bddd0126facebd39c7ae996eb885a630cf550 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_netcdf.py @@ -0,0 +1,1094 @@ +""" +NetCDF reader/writer module. + +This module is used to read and create NetCDF files. NetCDF files are +accessed through the `netcdf_file` object. Data written to and from NetCDF +files are contained in `netcdf_variable` objects. Attributes are given +as member variables of the `netcdf_file` and `netcdf_variable` objects. + +This module implements the Scientific.IO.NetCDF API to read and create +NetCDF files. The same API is also used in the PyNIO and pynetcdf +modules, allowing these modules to be used interchangeably when working +with NetCDF files. + +Only NetCDF3 is supported here; for NetCDF4 see +`netCDF4-python `__, +which has a similar API. + +""" + +# TODO: +# * properly implement ``_FillValue``. +# * fix character variables. +# * implement PAGESIZE for Python 2.6? + +# The Scientific.IO.NetCDF API allows attributes to be added directly to +# instances of ``netcdf_file`` and ``netcdf_variable``. To differentiate +# between user-set attributes and instance attributes, user-set attributes +# are automatically stored in the ``_attributes`` attribute by overloading +#``__setattr__``. This is the reason why the code sometimes uses +#``obj.__dict__['key'] = value``, instead of simply ``obj.key = value``; +# otherwise the key would be inserted into userspace attributes. + + +__all__ = ['netcdf_file', 'netcdf_variable'] + + +import warnings +import weakref +from operator import mul +from platform import python_implementation + +import mmap as mm + +import numpy as np +from numpy import frombuffer, dtype, empty, array, asarray +from numpy import little_endian as LITTLE_ENDIAN +from functools import reduce + + +IS_PYPY = python_implementation() == 'PyPy' + +ABSENT = b'\x00\x00\x00\x00\x00\x00\x00\x00' +ZERO = b'\x00\x00\x00\x00' +NC_BYTE = b'\x00\x00\x00\x01' +NC_CHAR = b'\x00\x00\x00\x02' +NC_SHORT = b'\x00\x00\x00\x03' +NC_INT = b'\x00\x00\x00\x04' +NC_FLOAT = b'\x00\x00\x00\x05' +NC_DOUBLE = b'\x00\x00\x00\x06' +NC_DIMENSION = b'\x00\x00\x00\n' +NC_VARIABLE = b'\x00\x00\x00\x0b' +NC_ATTRIBUTE = b'\x00\x00\x00\x0c' +FILL_BYTE = b'\x81' +FILL_CHAR = b'\x00' +FILL_SHORT = b'\x80\x01' +FILL_INT = b'\x80\x00\x00\x01' +FILL_FLOAT = b'\x7C\xF0\x00\x00' +FILL_DOUBLE = b'\x47\x9E\x00\x00\x00\x00\x00\x00' + +TYPEMAP = {NC_BYTE: ('b', 1), + NC_CHAR: ('c', 1), + NC_SHORT: ('h', 2), + NC_INT: ('i', 4), + NC_FLOAT: ('f', 4), + NC_DOUBLE: ('d', 8)} + +FILLMAP = {NC_BYTE: FILL_BYTE, + NC_CHAR: FILL_CHAR, + NC_SHORT: FILL_SHORT, + NC_INT: FILL_INT, + NC_FLOAT: FILL_FLOAT, + NC_DOUBLE: FILL_DOUBLE} + +REVERSE = {('b', 1): NC_BYTE, + ('B', 1): NC_CHAR, + ('c', 1): NC_CHAR, + ('h', 2): NC_SHORT, + ('i', 4): NC_INT, + ('f', 4): NC_FLOAT, + ('d', 8): NC_DOUBLE, + + # these come from asarray(1).dtype.char and asarray('foo').dtype.char, + # used when getting the types from generic attributes. + ('l', 4): NC_INT, + ('S', 1): NC_CHAR} + + +class netcdf_file: + """ + A file object for NetCDF data. + + A `netcdf_file` object has two standard attributes: `dimensions` and + `variables`. The values of both are dictionaries, mapping dimension + names to their associated lengths and variable names to variables, + respectively. Application programs should never modify these + dictionaries. + + All other attributes correspond to global attributes defined in the + NetCDF file. Global file attributes are created by assigning to an + attribute of the `netcdf_file` object. + + Parameters + ---------- + filename : string or file-like + string -> filename + mode : {'r', 'w', 'a'}, optional + read-write-append mode, default is 'r' + mmap : None or bool, optional + Whether to mmap `filename` when reading. Default is True + when `filename` is a file name, False when `filename` is a + file-like object. Note that when mmap is in use, data arrays + returned refer directly to the mmapped data on disk, and the + file cannot be closed as long as references to it exist. + version : {1, 2}, optional + version of netcdf to read / write, where 1 means *Classic + format* and 2 means *64-bit offset format*. Default is 1. See + `here `__ + for more info. + maskandscale : bool, optional + Whether to automatically scale and/or mask data based on attributes. + Default is False. + + Notes + ----- + The major advantage of this module over other modules is that it doesn't + require the code to be linked to the NetCDF libraries. This module is + derived from `pupynere `_. + + NetCDF files are a self-describing binary data format. The file contains + metadata that describes the dimensions and variables in the file. More + details about NetCDF files can be found `here + `__. There + are three main sections to a NetCDF data structure: + + 1. Dimensions + 2. Variables + 3. Attributes + + The dimensions section records the name and length of each dimension used + by the variables. The variables would then indicate which dimensions it + uses and any attributes such as data units, along with containing the data + values for the variable. It is good practice to include a + variable that is the same name as a dimension to provide the values for + that axes. Lastly, the attributes section would contain additional + information such as the name of the file creator or the instrument used to + collect the data. + + When writing data to a NetCDF file, there is often the need to indicate the + 'record dimension'. A record dimension is the unbounded dimension for a + variable. For example, a temperature variable may have dimensions of + latitude, longitude and time. If one wants to add more temperature data to + the NetCDF file as time progresses, then the temperature variable should + have the time dimension flagged as the record dimension. + + In addition, the NetCDF file header contains the position of the data in + the file, so access can be done in an efficient manner without loading + unnecessary data into memory. It uses the ``mmap`` module to create + Numpy arrays mapped to the data on disk, for the same purpose. + + Note that when `netcdf_file` is used to open a file with mmap=True + (default for read-only), arrays returned by it refer to data + directly on the disk. The file should not be closed, and cannot be cleanly + closed when asked, if such arrays are alive. You may want to copy data arrays + obtained from mmapped Netcdf file if they are to be processed after the file + is closed, see the example below. + + Examples + -------- + To create a NetCDF file: + + >>> from scipy.io import netcdf_file + >>> import numpy as np + >>> f = netcdf_file('simple.nc', 'w') + >>> f.history = 'Created for a test' + >>> f.createDimension('time', 10) + >>> time = f.createVariable('time', 'i', ('time',)) + >>> time[:] = np.arange(10) + >>> time.units = 'days since 2008-01-01' + >>> f.close() + + Note the assignment of ``arange(10)`` to ``time[:]``. Exposing the slice + of the time variable allows for the data to be set in the object, rather + than letting ``arange(10)`` overwrite the ``time`` variable. + + To read the NetCDF file we just created: + + >>> from scipy.io import netcdf_file + >>> f = netcdf_file('simple.nc', 'r') + >>> print(f.history) + b'Created for a test' + >>> time = f.variables['time'] + >>> print(time.units) + b'days since 2008-01-01' + >>> print(time.shape) + (10,) + >>> print(time[-1]) + 9 + + NetCDF files, when opened read-only, return arrays that refer + directly to memory-mapped data on disk: + + >>> data = time[:] + + If the data is to be processed after the file is closed, it needs + to be copied to main memory: + + >>> data = time[:].copy() + >>> del time + >>> f.close() + >>> data.mean() + 4.5 + + A NetCDF file can also be used as context manager: + + >>> from scipy.io import netcdf_file + >>> with netcdf_file('simple.nc', 'r') as f: + ... print(f.history) + b'Created for a test' + + """ + def __init__(self, filename, mode='r', mmap=None, version=1, + maskandscale=False): + """Initialize netcdf_file from fileobj (str or file-like).""" + if mode not in 'rwa': + raise ValueError("Mode must be either 'r', 'w' or 'a'.") + + if hasattr(filename, 'seek'): # file-like + self.fp = filename + self.filename = 'None' + if mmap is None: + mmap = False + elif mmap and not hasattr(filename, 'fileno'): + raise ValueError('Cannot use file object for mmap') + else: # maybe it's a string + self.filename = filename + omode = 'r+' if mode == 'a' else mode + self.fp = open(self.filename, f'{omode}b') + if mmap is None: + # Mmapped files on PyPy cannot be usually closed + # before the GC runs, so it's better to use mmap=False + # as the default. + mmap = (not IS_PYPY) + + if mode != 'r': + # Cannot read write-only files + mmap = False + + self.use_mmap = mmap + self.mode = mode + self.version_byte = version + self.maskandscale = maskandscale + + self.dimensions = {} + self.variables = {} + + self._dims = [] + self._recs = 0 + self._recsize = 0 + + self._mm = None + self._mm_buf = None + if self.use_mmap: + self._mm = mm.mmap(self.fp.fileno(), 0, access=mm.ACCESS_READ) + self._mm_buf = np.frombuffer(self._mm, dtype=np.int8) + + self._attributes = {} + + if mode in 'ra': + self._read() + + def __setattr__(self, attr, value): + # Store user defined attributes in a separate dict, + # so we can save them to file later. + try: + self._attributes[attr] = value + except AttributeError: + pass + self.__dict__[attr] = value + + def close(self): + """Closes the NetCDF file.""" + if hasattr(self, 'fp') and not self.fp.closed: + try: + self.flush() + finally: + self.variables = {} + if self._mm_buf is not None: + ref = weakref.ref(self._mm_buf) + self._mm_buf = None + if ref() is None: + # self._mm_buf is gc'd, and we can close the mmap + self._mm.close() + else: + # we cannot close self._mm, since self._mm_buf is + # alive and there may still be arrays referring to it + warnings.warn( + "Cannot close a netcdf_file opened with mmap=True, when " + "netcdf_variables or arrays referring to its data still " + "exist. All data arrays obtained from such files refer " + "directly to data on disk, and must be copied before the " + "file can be cleanly closed. " + "(See netcdf_file docstring for more information on mmap.)", + category=RuntimeWarning, stacklevel=2, + ) + self._mm = None + self.fp.close() + __del__ = close + + def __enter__(self): + return self + + def __exit__(self, type, value, traceback): + self.close() + + def createDimension(self, name, length): + """ + Adds a dimension to the Dimension section of the NetCDF data structure. + + Note that this function merely adds a new dimension that the variables can + reference. The values for the dimension, if desired, should be added as + a variable using `createVariable`, referring to this dimension. + + Parameters + ---------- + name : str + Name of the dimension (Eg, 'lat' or 'time'). + length : int + Length of the dimension. + + See Also + -------- + createVariable + + """ + if length is None and self._dims: + raise ValueError("Only first dimension may be unlimited!") + + self.dimensions[name] = length + self._dims.append(name) + + def createVariable(self, name, type, dimensions): + """ + Create an empty variable for the `netcdf_file` object, specifying its data + type and the dimensions it uses. + + Parameters + ---------- + name : str + Name of the new variable. + type : dtype or str + Data type of the variable. + dimensions : sequence of str + List of the dimension names used by the variable, in the desired order. + + Returns + ------- + variable : netcdf_variable + The newly created ``netcdf_variable`` object. + This object has also been added to the `netcdf_file` object as well. + + See Also + -------- + createDimension + + Notes + ----- + Any dimensions to be used by the variable should already exist in the + NetCDF data structure or should be created by `createDimension` prior to + creating the NetCDF variable. + + """ + shape = tuple([self.dimensions[dim] for dim in dimensions]) + shape_ = tuple([dim or 0 for dim in shape]) # replace None with 0 for NumPy + + type = dtype(type) + typecode, size = type.char, type.itemsize + if (typecode, size) not in REVERSE: + raise ValueError(f"NetCDF 3 does not support type {type}") + + # convert to big endian always for NetCDF 3 + data = empty(shape_, dtype=type.newbyteorder("B")) + self.variables[name] = netcdf_variable( + data, typecode, size, shape, dimensions, + maskandscale=self.maskandscale) + return self.variables[name] + + def flush(self): + """ + Perform a sync-to-disk flush if the `netcdf_file` object is in write mode. + + See Also + -------- + sync : Identical function + + """ + if hasattr(self, 'mode') and self.mode in 'wa': + self._write() + sync = flush + + def _write(self): + self.fp.seek(0) + self.fp.write(b'CDF') + self.fp.write(array(self.version_byte, '>b').tobytes()) + + # Write headers and data. + self._write_numrecs() + self._write_dim_array() + self._write_gatt_array() + self._write_var_array() + + def _write_numrecs(self): + # Get highest record count from all record variables. + for var in self.variables.values(): + if var.isrec and len(var.data) > self._recs: + self.__dict__['_recs'] = len(var.data) + self._pack_int(self._recs) + + def _write_dim_array(self): + if self.dimensions: + self.fp.write(NC_DIMENSION) + self._pack_int(len(self.dimensions)) + for name in self._dims: + self._pack_string(name) + length = self.dimensions[name] + self._pack_int(length or 0) # replace None with 0 for record dimension + else: + self.fp.write(ABSENT) + + def _write_gatt_array(self): + self._write_att_array(self._attributes) + + def _write_att_array(self, attributes): + if attributes: + self.fp.write(NC_ATTRIBUTE) + self._pack_int(len(attributes)) + for name, values in attributes.items(): + self._pack_string(name) + self._write_att_values(values) + else: + self.fp.write(ABSENT) + + def _write_var_array(self): + if self.variables: + self.fp.write(NC_VARIABLE) + self._pack_int(len(self.variables)) + + # Sort variable names non-recs first, then recs. + def sortkey(n): + v = self.variables[n] + if v.isrec: + return (-1,) + return v._shape + variables = sorted(self.variables, key=sortkey, reverse=True) + + # Set the metadata for all variables. + for name in variables: + self._write_var_metadata(name) + # Now that we have the metadata, we know the vsize of + # each record variable, so we can calculate recsize. + self.__dict__['_recsize'] = sum([ + var._vsize for var in self.variables.values() + if var.isrec]) + # Set the data for all variables. + for name in variables: + self._write_var_data(name) + else: + self.fp.write(ABSENT) + + def _write_var_metadata(self, name): + var = self.variables[name] + + self._pack_string(name) + self._pack_int(len(var.dimensions)) + for dimname in var.dimensions: + dimid = self._dims.index(dimname) + self._pack_int(dimid) + + self._write_att_array(var._attributes) + + nc_type = REVERSE[var.typecode(), var.itemsize()] + self.fp.write(nc_type) + + if not var.isrec: + vsize = var.data.size * var.data.itemsize + vsize += -vsize % 4 + else: # record variable + try: + vsize = var.data[0].size * var.data.itemsize + except IndexError: + vsize = 0 + rec_vars = len([v for v in self.variables.values() + if v.isrec]) + if rec_vars > 1: + vsize += -vsize % 4 + self.variables[name].__dict__['_vsize'] = vsize + self._pack_int(vsize) + + # Pack a bogus begin, and set the real value later. + self.variables[name].__dict__['_begin'] = self.fp.tell() + self._pack_begin(0) + + def _write_var_data(self, name): + var = self.variables[name] + + # Set begin in file header. + the_beguine = self.fp.tell() + self.fp.seek(var._begin) + self._pack_begin(the_beguine) + self.fp.seek(the_beguine) + + # Write data. + if not var.isrec: + self.fp.write(var.data.tobytes()) + count = var.data.size * var.data.itemsize + self._write_var_padding(var, var._vsize - count) + else: # record variable + # Handle rec vars with shape[0] < nrecs. + if self._recs > len(var.data): + shape = (self._recs,) + var.data.shape[1:] + # Resize in-place does not always work since + # the array might not be single-segment + try: + var.data.resize(shape) + except ValueError: + dtype = var.data.dtype + var.__dict__['data'] = np.resize(var.data, shape).astype(dtype) + + pos0 = pos = self.fp.tell() + for rec in var.data: + # Apparently scalars cannot be converted to big endian. If we + # try to convert a ``=i4`` scalar to, say, '>i4' the dtype + # will remain as ``=i4``. + if not rec.shape and (rec.dtype.byteorder == '<' or + (rec.dtype.byteorder == '=' and LITTLE_ENDIAN)): + rec = rec.byteswap() + self.fp.write(rec.tobytes()) + # Padding + count = rec.size * rec.itemsize + self._write_var_padding(var, var._vsize - count) + pos += self._recsize + self.fp.seek(pos) + self.fp.seek(pos0 + var._vsize) + + def _write_var_padding(self, var, size): + encoded_fill_value = var._get_encoded_fill_value() + num_fills = size // len(encoded_fill_value) + self.fp.write(encoded_fill_value * num_fills) + + def _write_att_values(self, values): + if hasattr(values, 'dtype'): + nc_type = REVERSE[values.dtype.char, values.dtype.itemsize] + else: + types = [(int, NC_INT), (float, NC_FLOAT), (str, NC_CHAR)] + + # bytes index into scalars in py3k. Check for "string" types + if isinstance(values, (str, bytes)): + sample = values + else: + try: + sample = values[0] # subscriptable? + except TypeError: + sample = values # scalar + + for class_, nc_type in types: + if isinstance(sample, class_): + break + + typecode, size = TYPEMAP[nc_type] + dtype_ = f'>{typecode}' + # asarray() dies with bytes and '>c' in py3k. Change to 'S' + dtype_ = 'S' if dtype_ == '>c' else dtype_ + + values = asarray(values, dtype=dtype_) + + self.fp.write(nc_type) + + if values.dtype.char == 'S': + nelems = values.itemsize + else: + nelems = values.size + self._pack_int(nelems) + + if not values.shape and (values.dtype.byteorder == '<' or + (values.dtype.byteorder == '=' and LITTLE_ENDIAN)): + values = values.byteswap() + self.fp.write(values.tobytes()) + count = values.size * values.itemsize + self.fp.write(b'\x00' * (-count % 4)) # pad + + def _read(self): + # Check magic bytes and version + magic = self.fp.read(3) + if not magic == b'CDF': + raise TypeError(f"Error: {self.filename} is not a valid NetCDF 3 file") + self.__dict__['version_byte'] = frombuffer(self.fp.read(1), '>b')[0] + + # Read file headers and set data. + self._read_numrecs() + self._read_dim_array() + self._read_gatt_array() + self._read_var_array() + + def _read_numrecs(self): + self.__dict__['_recs'] = self._unpack_int() + + def _read_dim_array(self): + header = self.fp.read(4) + if header not in [ZERO, NC_DIMENSION]: + raise ValueError("Unexpected header.") + count = self._unpack_int() + + for dim in range(count): + name = self._unpack_string().decode('latin1') + length = self._unpack_int() or None # None for record dimension + self.dimensions[name] = length + self._dims.append(name) # preserve order + + def _read_gatt_array(self): + for k, v in self._read_att_array().items(): + self.__setattr__(k, v) + + def _read_att_array(self): + header = self.fp.read(4) + if header not in [ZERO, NC_ATTRIBUTE]: + raise ValueError("Unexpected header.") + count = self._unpack_int() + + attributes = {} + for attr in range(count): + name = self._unpack_string().decode('latin1') + attributes[name] = self._read_att_values() + return attributes + + def _read_var_array(self): + header = self.fp.read(4) + if header not in [ZERO, NC_VARIABLE]: + raise ValueError("Unexpected header.") + + begin = 0 + dtypes = {'names': [], 'formats': []} + rec_vars = [] + count = self._unpack_int() + for var in range(count): + (name, dimensions, shape, attributes, + typecode, size, dtype_, begin_, vsize) = self._read_var() + # https://www.unidata.ucar.edu/software/netcdf/guide_toc.html + # Note that vsize is the product of the dimension lengths + # (omitting the record dimension) and the number of bytes + # per value (determined from the type), increased to the + # next multiple of 4, for each variable. If a record + # variable, this is the amount of space per record. The + # netCDF "record size" is calculated as the sum of the + # vsize's of all the record variables. + # + # The vsize field is actually redundant, because its value + # may be computed from other information in the header. The + # 32-bit vsize field is not large enough to contain the size + # of variables that require more than 2^32 - 4 bytes, so + # 2^32 - 1 is used in the vsize field for such variables. + if shape and shape[0] is None: # record variable + rec_vars.append(name) + # The netCDF "record size" is calculated as the sum of + # the vsize's of all the record variables. + self.__dict__['_recsize'] += vsize + if begin == 0: + begin = begin_ + dtypes['names'].append(name) + dtypes['formats'].append(str(shape[1:]) + dtype_) + + # Handle padding with a virtual variable. + if typecode in 'bch': + actual_size = reduce(mul, (1,) + shape[1:]) * size + padding = -actual_size % 4 + if padding: + dtypes['names'].append('_padding_%d' % var) + dtypes['formats'].append('(%d,)>b' % padding) + + # Data will be set later. + data = None + else: # not a record variable + # Calculate size to avoid problems with vsize (above) + a_size = reduce(mul, shape, 1) * size + if self.use_mmap: + data = self._mm_buf[begin_:begin_+a_size].view(dtype=dtype_) + data.shape = shape + else: + pos = self.fp.tell() + self.fp.seek(begin_) + data = frombuffer(self.fp.read(a_size), dtype=dtype_ + ).copy() + data.shape = shape + self.fp.seek(pos) + + # Add variable. + self.variables[name] = netcdf_variable( + data, typecode, size, shape, dimensions, attributes, + maskandscale=self.maskandscale) + + if rec_vars: + # Remove padding when only one record variable. + if len(rec_vars) == 1: + dtypes['names'] = dtypes['names'][:1] + dtypes['formats'] = dtypes['formats'][:1] + + # Build rec array. + if self.use_mmap: + buf = self._mm_buf[begin:begin+self._recs*self._recsize] + rec_array = buf.view(dtype=dtypes) + rec_array.shape = (self._recs,) + else: + pos = self.fp.tell() + self.fp.seek(begin) + rec_array = frombuffer(self.fp.read(self._recs*self._recsize), + dtype=dtypes).copy() + rec_array.shape = (self._recs,) + self.fp.seek(pos) + + for var in rec_vars: + self.variables[var].__dict__['data'] = rec_array[var] + + def _read_var(self): + name = self._unpack_string().decode('latin1') + dimensions = [] + shape = [] + dims = self._unpack_int() + + for i in range(dims): + dimid = self._unpack_int() + dimname = self._dims[dimid] + dimensions.append(dimname) + dim = self.dimensions[dimname] + shape.append(dim) + dimensions = tuple(dimensions) + shape = tuple(shape) + + attributes = self._read_att_array() + nc_type = self.fp.read(4) + vsize = self._unpack_int() + begin = [self._unpack_int, self._unpack_int64][self.version_byte-1]() + + typecode, size = TYPEMAP[nc_type] + dtype_ = f'>{typecode}' + + return name, dimensions, shape, attributes, typecode, size, dtype_, begin, vsize + + def _read_att_values(self): + nc_type = self.fp.read(4) + n = self._unpack_int() + + typecode, size = TYPEMAP[nc_type] + + count = n*size + values = self.fp.read(int(count)) + self.fp.read(-count % 4) # read padding + + if typecode != 'c': + values = frombuffer(values, dtype=f'>{typecode}').copy() + if values.shape == (1,): + values = values[0] + else: + values = values.rstrip(b'\x00') + return values + + def _pack_begin(self, begin): + if self.version_byte == 1: + self._pack_int(begin) + elif self.version_byte == 2: + self._pack_int64(begin) + + def _pack_int(self, value): + self.fp.write(array(value, '>i').tobytes()) + _pack_int32 = _pack_int + + def _unpack_int(self): + return int(frombuffer(self.fp.read(4), '>i')[0]) + _unpack_int32 = _unpack_int + + def _pack_int64(self, value): + self.fp.write(array(value, '>q').tobytes()) + + def _unpack_int64(self): + return frombuffer(self.fp.read(8), '>q')[0] + + def _pack_string(self, s): + count = len(s) + self._pack_int(count) + self.fp.write(s.encode('latin1')) + self.fp.write(b'\x00' * (-count % 4)) # pad + + def _unpack_string(self): + count = self._unpack_int() + s = self.fp.read(count).rstrip(b'\x00') + self.fp.read(-count % 4) # read padding + return s + + +class netcdf_variable: + """ + A data object for netcdf files. + + `netcdf_variable` objects are constructed by calling the method + `netcdf_file.createVariable` on the `netcdf_file` object. `netcdf_variable` + objects behave much like array objects defined in numpy, except that their + data resides in a file. Data is read by indexing and written by assigning + to an indexed subset; the entire array can be accessed by the index ``[:]`` + or (for scalars) by using the methods `getValue` and `assignValue`. + `netcdf_variable` objects also have attribute `shape` with the same meaning + as for arrays, but the shape cannot be modified. There is another read-only + attribute `dimensions`, whose value is the tuple of dimension names. + + All other attributes correspond to variable attributes defined in + the NetCDF file. Variable attributes are created by assigning to an + attribute of the `netcdf_variable` object. + + Parameters + ---------- + data : array_like + The data array that holds the values for the variable. + Typically, this is initialized as empty, but with the proper shape. + typecode : dtype character code + Desired data-type for the data array. + size : int + Desired element size for the data array. + shape : sequence of ints + The shape of the array. This should match the lengths of the + variable's dimensions. + dimensions : sequence of strings + The names of the dimensions used by the variable. Must be in the + same order of the dimension lengths given by `shape`. + attributes : dict, optional + Attribute values (any type) keyed by string names. These attributes + become attributes for the netcdf_variable object. + maskandscale : bool, optional + Whether to automatically scale and/or mask data based on attributes. + Default is False. + + + Attributes + ---------- + dimensions : list of str + List of names of dimensions used by the variable object. + isrec, shape + Properties + + See also + -------- + isrec, shape + + """ + def __init__(self, data, typecode, size, shape, dimensions, + attributes=None, + maskandscale=False): + self.data = data + self._typecode = typecode + self._size = size + self._shape = shape + self.dimensions = dimensions + self.maskandscale = maskandscale + + self._attributes = attributes or {} + for k, v in self._attributes.items(): + self.__dict__[k] = v + + def __setattr__(self, attr, value): + # Store user defined attributes in a separate dict, + # so we can save them to file later. + try: + self._attributes[attr] = value + except AttributeError: + pass + self.__dict__[attr] = value + + def isrec(self): + """Returns whether the variable has a record dimension or not. + + A record dimension is a dimension along which additional data could be + easily appended in the netcdf data structure without much rewriting of + the data file. This attribute is a read-only property of the + `netcdf_variable`. + + """ + return bool(self.data.shape) and not self._shape[0] + isrec = property(isrec) + + def shape(self): + """Returns the shape tuple of the data variable. + + This is a read-only attribute and can not be modified in the + same manner of other numpy arrays. + """ + return self.data.shape + shape = property(shape) + + def getValue(self): + """ + Retrieve a scalar value from a `netcdf_variable` of length one. + + Raises + ------ + ValueError + If the netcdf variable is an array of length greater than one, + this exception will be raised. + + """ + return self.data.item() + + def assignValue(self, value): + """ + Assign a scalar value to a `netcdf_variable` of length one. + + Parameters + ---------- + value : scalar + Scalar value (of compatible type) to assign to a length-one netcdf + variable. This value will be written to file. + + Raises + ------ + ValueError + If the input is not a scalar, or if the destination is not a length-one + netcdf variable. + + """ + if not self.data.flags.writeable: + # Work-around for a bug in NumPy. Calling itemset() on a read-only + # memory-mapped array causes a seg. fault. + # See NumPy ticket #1622, and SciPy ticket #1202. + # This check for `writeable` can be removed when the oldest version + # of NumPy still supported by scipy contains the fix for #1622. + raise RuntimeError("variable is not writeable") + + self.data[:] = value + + def typecode(self): + """ + Return the typecode of the variable. + + Returns + ------- + typecode : char + The character typecode of the variable (e.g., 'i' for int). + + """ + return self._typecode + + def itemsize(self): + """ + Return the itemsize of the variable. + + Returns + ------- + itemsize : int + The element size of the variable (e.g., 8 for float64). + + """ + return self._size + + def __getitem__(self, index): + if not self.maskandscale: + return self.data[index] + + data = self.data[index].copy() + missing_value = self._get_missing_value() + data = self._apply_missing_value(data, missing_value) + scale_factor = self._attributes.get('scale_factor') + add_offset = self._attributes.get('add_offset') + if add_offset is not None or scale_factor is not None: + data = data.astype(np.float64) + if scale_factor is not None: + data = data * scale_factor + if add_offset is not None: + data += add_offset + + return data + + def __setitem__(self, index, data): + if self.maskandscale: + missing_value = ( + self._get_missing_value() or + getattr(data, 'fill_value', 999999)) + self._attributes.setdefault('missing_value', missing_value) + self._attributes.setdefault('_FillValue', missing_value) + data = ((data - self._attributes.get('add_offset', 0.0)) / + self._attributes.get('scale_factor', 1.0)) + data = np.ma.asarray(data).filled(missing_value) + if self._typecode not in 'fd' and data.dtype.kind == 'f': + data = np.round(data) + + # Expand data for record vars? + if self.isrec: + if isinstance(index, tuple): + rec_index = index[0] + else: + rec_index = index + if isinstance(rec_index, slice): + recs = (rec_index.start or 0) + len(data) + else: + recs = rec_index + 1 + if recs > len(self.data): + shape = (recs,) + self._shape[1:] + # Resize in-place does not always work since + # the array might not be single-segment + try: + self.data.resize(shape) + except ValueError: + dtype = self.data.dtype + self.__dict__['data'] = np.resize(self.data, shape).astype(dtype) + self.data[index] = data + + def _default_encoded_fill_value(self): + """ + The default encoded fill-value for this Variable's data type. + """ + nc_type = REVERSE[self.typecode(), self.itemsize()] + return FILLMAP[nc_type] + + def _get_encoded_fill_value(self): + """ + Returns the encoded fill value for this variable as bytes. + + This is taken from either the _FillValue attribute, or the default fill + value for this variable's data type. + """ + if '_FillValue' in self._attributes: + fill_value = np.array(self._attributes['_FillValue'], + dtype=self.data.dtype).tobytes() + if len(fill_value) == self.itemsize(): + return fill_value + else: + return self._default_encoded_fill_value() + else: + return self._default_encoded_fill_value() + + def _get_missing_value(self): + """ + Returns the value denoting "no data" for this variable. + + If this variable does not have a missing/fill value, returns None. + + If both _FillValue and missing_value are given, give precedence to + _FillValue. The netCDF standard gives special meaning to _FillValue; + missing_value is just used for compatibility with old datasets. + """ + + if '_FillValue' in self._attributes: + missing_value = self._attributes['_FillValue'] + elif 'missing_value' in self._attributes: + missing_value = self._attributes['missing_value'] + else: + missing_value = None + + return missing_value + + @staticmethod + def _apply_missing_value(data, missing_value): + """ + Applies the given missing value to the data array. + + Returns a numpy.ma array, with any value equal to missing_value masked + out (unless missing_value is None, in which case the original array is + returned). + """ + + if missing_value is None: + newdata = data + else: + try: + missing_value_isnan = np.isnan(missing_value) + except (TypeError, NotImplementedError): + # some data types (e.g., characters) cannot be tested for NaN + missing_value_isnan = False + + if missing_value_isnan: + mymask = np.isnan(data) + else: + mymask = (data == missing_value) + + newdata = np.ma.masked_where(mymask, data) + + return newdata + + +NetCDFFile = netcdf_file +NetCDFVariable = netcdf_variable diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_test_fortran.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_test_fortran.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..8a863a591370a3b1bd4afdb05e621c1af8eab3e2 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/_test_fortran.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..dcfe1c4237e69054b582a0fa52f710b25a1d7914 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/__init__.py @@ -0,0 +1,28 @@ +""" +Module to read ARFF files +========================= +ARFF is the standard data format for WEKA. +It is a text file format which support numerical, string and data values. +The format can also represent missing data and sparse data. + +Notes +----- +The ARFF support in ``scipy.io`` provides file reading functionality only. +For more extensive ARFF functionality, see `liac-arff +`_. + +See the `WEKA website `_ +for more details about the ARFF format and available datasets. + +""" +from ._arffread import * +from . import _arffread + +# Deprecated namespaces, to be removed in v2.0.0 +from .import arffread + +__all__ = _arffread.__all__ + ['arffread'] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/_arffread.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/_arffread.py new file mode 100644 index 0000000000000000000000000000000000000000..65495b8d98386492eecbccb8968715590c0faf7c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/_arffread.py @@ -0,0 +1,873 @@ +# Last Change: Mon Aug 20 08:00 PM 2007 J +import re +import datetime + +import numpy as np + +import csv +import ctypes + +"""A module to read arff files.""" + +__all__ = ['MetaData', 'loadarff', 'ArffError', 'ParseArffError'] + +# An Arff file is basically two parts: +# - header +# - data +# +# A header has each of its components starting by @META where META is one of +# the keyword (attribute of relation, for now). + +# TODO: +# - both integer and reals are treated as numeric -> the integer info +# is lost! +# - Replace ValueError by ParseError or something + +# We know can handle the following: +# - numeric and nominal attributes +# - missing values for numeric attributes + +r_meta = re.compile(r'^\s*@') +# Match a comment +r_comment = re.compile(r'^%') +# Match an empty line +r_empty = re.compile(r'^\s+$') +# Match a header line, that is a line which starts by @ + a word +r_headerline = re.compile(r'^\s*@\S*') +r_datameta = re.compile(r'^@[Dd][Aa][Tt][Aa]') +r_relation = re.compile(r'^@[Rr][Ee][Ll][Aa][Tt][Ii][Oo][Nn]\s*(\S*)') +r_attribute = re.compile(r'^\s*@[Aa][Tt][Tt][Rr][Ii][Bb][Uu][Tt][Ee]\s*(..*$)') + +r_nominal = re.compile(r'{(.+)}') +r_date = re.compile(r"[Dd][Aa][Tt][Ee]\s+[\"']?(.+?)[\"']?$") + +# To get attributes name enclosed with '' +r_comattrval = re.compile(r"'(..+)'\s+(..+$)") +# To get normal attributes +r_wcomattrval = re.compile(r"(\S+)\s+(..+$)") + +# ------------------------ +# Module defined exception +# ------------------------ + + +class ArffError(OSError): + pass + + +class ParseArffError(ArffError): + pass + + +# ---------- +# Attributes +# ---------- +class Attribute: + + type_name = None + + def __init__(self, name): + self.name = name + self.range = None + self.dtype = np.object_ + + @classmethod + def parse_attribute(cls, name, attr_string): + """ + Parse the attribute line if it knows how. Returns the parsed + attribute, or None. + """ + return None + + def parse_data(self, data_str): + """ + Parse a value of this type. + """ + return None + + def __str__(self): + """ + Parse a value of this type. + """ + return self.name + ',' + self.type_name + + +class NominalAttribute(Attribute): + + type_name = 'nominal' + + def __init__(self, name, values): + super().__init__(name) + self.values = values + self.range = values + self.dtype = (np.bytes_, max(len(i) for i in values)) + + @staticmethod + def _get_nom_val(atrv): + """Given a string containing a nominal type, returns a tuple of the + possible values. + + A nominal type is defined as something framed between braces ({}). + + Parameters + ---------- + atrv : str + Nominal type definition + + Returns + ------- + poss_vals : tuple + possible values + + Examples + -------- + >>> from scipy.io.arff._arffread import NominalAttribute + >>> NominalAttribute._get_nom_val("{floup, bouga, fl, ratata}") + ('floup', 'bouga', 'fl', 'ratata') + """ + m = r_nominal.match(atrv) + if m: + attrs, _ = split_data_line(m.group(1)) + return tuple(attrs) + else: + raise ValueError("This does not look like a nominal string") + + @classmethod + def parse_attribute(cls, name, attr_string): + """ + Parse the attribute line if it knows how. Returns the parsed + attribute, or None. + + For nominal attributes, the attribute string would be like '{, + , }'. + """ + if attr_string[0] == '{': + values = cls._get_nom_val(attr_string) + return cls(name, values) + else: + return None + + def parse_data(self, data_str): + """ + Parse a value of this type. + """ + if data_str in self.values: + return data_str + elif data_str == '?': + return data_str + else: + raise ValueError(f"{str(data_str)} value not in {str(self.values)}") + + def __str__(self): + msg = self.name + ",{" + for i in range(len(self.values)-1): + msg += self.values[i] + "," + msg += self.values[-1] + msg += "}" + return msg + + +class NumericAttribute(Attribute): + + def __init__(self, name): + super().__init__(name) + self.type_name = 'numeric' + self.dtype = np.float64 + + @classmethod + def parse_attribute(cls, name, attr_string): + """ + Parse the attribute line if it knows how. Returns the parsed + attribute, or None. + + For numeric attributes, the attribute string would be like + 'numeric' or 'int' or 'real'. + """ + + attr_string = attr_string.lower().strip() + + if (attr_string[:len('numeric')] == 'numeric' or + attr_string[:len('int')] == 'int' or + attr_string[:len('real')] == 'real'): + return cls(name) + else: + return None + + def parse_data(self, data_str): + """ + Parse a value of this type. + + Parameters + ---------- + data_str : str + string to convert + + Returns + ------- + f : float + where float can be nan + + Examples + -------- + >>> from scipy.io.arff._arffread import NumericAttribute + >>> atr = NumericAttribute('atr') + >>> atr.parse_data('1') + 1.0 + >>> atr.parse_data('1\\n') + 1.0 + >>> atr.parse_data('?\\n') + nan + """ + if '?' in data_str: + return np.nan + else: + return float(data_str) + + def _basic_stats(self, data): + nbfac = data.size * 1. / (data.size - 1) + return (np.nanmin(data), np.nanmax(data), + np.mean(data), np.std(data) * nbfac) + + +class StringAttribute(Attribute): + + def __init__(self, name): + super().__init__(name) + self.type_name = 'string' + + @classmethod + def parse_attribute(cls, name, attr_string): + """ + Parse the attribute line if it knows how. Returns the parsed + attribute, or None. + + For string attributes, the attribute string would be like + 'string'. + """ + + attr_string = attr_string.lower().strip() + + if attr_string[:len('string')] == 'string': + return cls(name) + else: + return None + + +class DateAttribute(Attribute): + + def __init__(self, name, date_format, datetime_unit): + super().__init__(name) + self.date_format = date_format + self.datetime_unit = datetime_unit + self.type_name = 'date' + self.range = date_format + self.dtype = np.datetime64(0, self.datetime_unit) + + @staticmethod + def _get_date_format(atrv): + m = r_date.match(atrv) + if m: + pattern = m.group(1).strip() + # convert time pattern from Java's SimpleDateFormat to C's format + datetime_unit = None + if "yyyy" in pattern: + pattern = pattern.replace("yyyy", "%Y") + datetime_unit = "Y" + elif "yy": + pattern = pattern.replace("yy", "%y") + datetime_unit = "Y" + if "MM" in pattern: + pattern = pattern.replace("MM", "%m") + datetime_unit = "M" + if "dd" in pattern: + pattern = pattern.replace("dd", "%d") + datetime_unit = "D" + if "HH" in pattern: + pattern = pattern.replace("HH", "%H") + datetime_unit = "h" + if "mm" in pattern: + pattern = pattern.replace("mm", "%M") + datetime_unit = "m" + if "ss" in pattern: + pattern = pattern.replace("ss", "%S") + datetime_unit = "s" + if "z" in pattern or "Z" in pattern: + raise ValueError("Date type attributes with time zone not " + "supported, yet") + + if datetime_unit is None: + raise ValueError("Invalid or unsupported date format") + + return pattern, datetime_unit + else: + raise ValueError("Invalid or no date format") + + @classmethod + def parse_attribute(cls, name, attr_string): + """ + Parse the attribute line if it knows how. Returns the parsed + attribute, or None. + + For date attributes, the attribute string would be like + 'date '. + """ + + attr_string_lower = attr_string.lower().strip() + + if attr_string_lower[:len('date')] == 'date': + date_format, datetime_unit = cls._get_date_format(attr_string) + return cls(name, date_format, datetime_unit) + else: + return None + + def parse_data(self, data_str): + """ + Parse a value of this type. + """ + date_str = data_str.strip().strip("'").strip('"') + if date_str == '?': + return np.datetime64('NaT', self.datetime_unit) + else: + dt = datetime.datetime.strptime(date_str, self.date_format) + return np.datetime64(dt).astype( + f"datetime64[{self.datetime_unit}]") + + def __str__(self): + return super().__str__() + ',' + self.date_format + + +class RelationalAttribute(Attribute): + + def __init__(self, name): + super().__init__(name) + self.type_name = 'relational' + self.dtype = np.object_ + self.attributes = [] + self.dialect = None + + @classmethod + def parse_attribute(cls, name, attr_string): + """ + Parse the attribute line if it knows how. Returns the parsed + attribute, or None. + + For date attributes, the attribute string would be like + 'date '. + """ + + attr_string_lower = attr_string.lower().strip() + + if attr_string_lower[:len('relational')] == 'relational': + return cls(name) + else: + return None + + def parse_data(self, data_str): + # Copy-pasted + elems = list(range(len(self.attributes))) + + escaped_string = data_str.encode().decode("unicode-escape") + + row_tuples = [] + + for raw in escaped_string.split("\n"): + row, self.dialect = split_data_line(raw, self.dialect) + + row_tuples.append(tuple( + [self.attributes[i].parse_data(row[i]) for i in elems])) + + return np.array(row_tuples, + [(a.name, a.dtype) for a in self.attributes]) + + def __str__(self): + return (super().__str__() + '\n\t' + + '\n\t'.join(str(a) for a in self.attributes)) + + +# ----------------- +# Various utilities +# ----------------- +def to_attribute(name, attr_string): + attr_classes = (NominalAttribute, NumericAttribute, DateAttribute, + StringAttribute, RelationalAttribute) + + for cls in attr_classes: + attr = cls.parse_attribute(name, attr_string) + if attr is not None: + return attr + + raise ParseArffError(f"unknown attribute {attr_string}") + + +def csv_sniffer_has_bug_last_field(): + """ + Checks if the bug https://bugs.python.org/issue30157 is unpatched. + """ + + # We only compute this once. + has_bug = getattr(csv_sniffer_has_bug_last_field, "has_bug", None) + + if has_bug is None: + dialect = csv.Sniffer().sniff("3, 'a'") + csv_sniffer_has_bug_last_field.has_bug = dialect.quotechar != "'" + has_bug = csv_sniffer_has_bug_last_field.has_bug + + return has_bug + + +def workaround_csv_sniffer_bug_last_field(sniff_line, dialect, delimiters): + """ + Workaround for the bug https://bugs.python.org/issue30157 if is unpatched. + """ + if csv_sniffer_has_bug_last_field(): + # Reuses code from the csv module + right_regex = r'(?P[^\w\n"\'])(?P ?)(?P["\']).*?(?P=quote)(?:$|\n)' # noqa: E501 + + for restr in (r'(?P[^\w\n"\'])(?P ?)(?P["\']).*?(?P=quote)(?P=delim)', # ,".*?", # noqa: E501 + r'(?:^|\n)(?P["\']).*?(?P=quote)(?P[^\w\n"\'])(?P ?)', # .*?", # noqa: E501 + right_regex, # ,".*?" + r'(?:^|\n)(?P["\']).*?(?P=quote)(?:$|\n)'): # ".*?" (no delim, no space) # noqa: E501 + regexp = re.compile(restr, re.DOTALL | re.MULTILINE) + matches = regexp.findall(sniff_line) + if matches: + break + + # If it does not match the expression that was bugged, + # then this bug does not apply + if restr != right_regex: + return + + groupindex = regexp.groupindex + + # There is only one end of the string + assert len(matches) == 1 + m = matches[0] + + n = groupindex['quote'] - 1 + quote = m[n] + + n = groupindex['delim'] - 1 + delim = m[n] + + n = groupindex['space'] - 1 + space = bool(m[n]) + + dq_regexp = re.compile( + rf"(({re.escape(delim)})|^)\W*{quote}[^{re.escape(delim)}\n]*{quote}[^{re.escape(delim)}\n]*{quote}\W*(({re.escape(delim)})|$)", re.MULTILINE # noqa: E501 + ) + + doublequote = bool(dq_regexp.search(sniff_line)) + + dialect.quotechar = quote + if delim in delimiters: + dialect.delimiter = delim + dialect.doublequote = doublequote + dialect.skipinitialspace = space + + +def split_data_line(line, dialect=None): + delimiters = ",\t" + + # This can not be done in a per reader basis, and relational fields + # can be HUGE + csv.field_size_limit(int(ctypes.c_ulong(-1).value // 2)) + + # Remove the line end if any + if line[-1] == '\n': + line = line[:-1] + + # Remove potential trailing whitespace + line = line.strip() + + sniff_line = line + + # Add a delimiter if none is present, so that the csv.Sniffer + # does not complain for a single-field CSV. + if not any(d in line for d in delimiters): + sniff_line += "," + + if dialect is None: + dialect = csv.Sniffer().sniff(sniff_line, delimiters=delimiters) + workaround_csv_sniffer_bug_last_field(sniff_line=sniff_line, + dialect=dialect, + delimiters=delimiters) + + row = next(csv.reader([line], dialect)) + + return row, dialect + + +# -------------- +# Parsing header +# -------------- +def tokenize_attribute(iterable, attribute): + """Parse a raw string in header (e.g., starts by @attribute). + + Given a raw string attribute, try to get the name and type of the + attribute. Constraints: + + * The first line must start with @attribute (case insensitive, and + space like characters before @attribute are allowed) + * Works also if the attribute is spread on multilines. + * Works if empty lines or comments are in between + + Parameters + ---------- + attribute : str + the attribute string. + + Returns + ------- + name : str + name of the attribute + value : str + value of the attribute + next : str + next line to be parsed + + Examples + -------- + If attribute is a string defined in python as r"floupi real", will + return floupi as name, and real as value. + + >>> from scipy.io.arff._arffread import tokenize_attribute + >>> iterable = iter([0] * 10) # dummy iterator + >>> tokenize_attribute(iterable, r"@attribute floupi real") + ('floupi', 'real', 0) + + If attribute is r"'floupi 2' real", will return 'floupi 2' as name, + and real as value. + + >>> tokenize_attribute(iterable, r" @attribute 'floupi 2' real ") + ('floupi 2', 'real', 0) + + """ + sattr = attribute.strip() + mattr = r_attribute.match(sattr) + if mattr: + # atrv is everything after @attribute + atrv = mattr.group(1) + if r_comattrval.match(atrv): + name, type = tokenize_single_comma(atrv) + next_item = next(iterable) + elif r_wcomattrval.match(atrv): + name, type = tokenize_single_wcomma(atrv) + next_item = next(iterable) + else: + # Not sure we should support this, as it does not seem supported by + # weka. + raise ValueError("multi line not supported yet") + else: + raise ValueError(f"First line unparsable: {sattr}") + + attribute = to_attribute(name, type) + + if type.lower() == 'relational': + next_item = read_relational_attribute(iterable, attribute, next_item) + # raise ValueError("relational attributes not supported yet") + + return attribute, next_item + + +def tokenize_single_comma(val): + # XXX we match twice the same string (here and at the caller level). It is + # stupid, but it is easier for now... + m = r_comattrval.match(val) + if m: + try: + name = m.group(1).strip() + type = m.group(2).strip() + except IndexError as e: + raise ValueError("Error while tokenizing attribute") from e + else: + raise ValueError(f"Error while tokenizing single {val}") + return name, type + + +def tokenize_single_wcomma(val): + # XXX we match twice the same string (here and at the caller level). It is + # stupid, but it is easier for now... + m = r_wcomattrval.match(val) + if m: + try: + name = m.group(1).strip() + type = m.group(2).strip() + except IndexError as e: + raise ValueError("Error while tokenizing attribute") from e + else: + raise ValueError(f"Error while tokenizing single {val}") + return name, type + + +def read_relational_attribute(ofile, relational_attribute, i): + """Read the nested attributes of a relational attribute""" + + r_end_relational = re.compile(r'^@[Ee][Nn][Dd]\s*' + + relational_attribute.name + r'\s*$') + + while not r_end_relational.match(i): + m = r_headerline.match(i) + if m: + isattr = r_attribute.match(i) + if isattr: + attr, i = tokenize_attribute(ofile, i) + relational_attribute.attributes.append(attr) + else: + raise ValueError(f"Error parsing line {i}") + else: + i = next(ofile) + + i = next(ofile) + return i + + +def read_header(ofile): + """Read the header of the iterable ofile.""" + i = next(ofile) + + # Pass first comments + while r_comment.match(i): + i = next(ofile) + + # Header is everything up to DATA attribute ? + relation = None + attributes = [] + while not r_datameta.match(i): + m = r_headerline.match(i) + if m: + isattr = r_attribute.match(i) + if isattr: + attr, i = tokenize_attribute(ofile, i) + attributes.append(attr) + else: + isrel = r_relation.match(i) + if isrel: + relation = isrel.group(1) + else: + raise ValueError(f"Error parsing line {i}") + i = next(ofile) + else: + i = next(ofile) + + return relation, attributes + + +class MetaData: + """Small container to keep useful information on a ARFF dataset. + + Knows about attributes names and types. + + Examples + -------- + :: + + data, meta = loadarff('iris.arff') + # This will print the attributes names of the iris.arff dataset + for i in meta: + print(i) + # This works too + meta.names() + # Getting attribute type + types = meta.types() + + Methods + ------- + names + types + + Notes + ----- + Also maintains the list of attributes in order, i.e., doing for i in + meta, where meta is an instance of MetaData, will return the + different attribute names in the order they were defined. + """ + def __init__(self, rel, attr): + self.name = rel + self._attributes = {a.name: a for a in attr} + + def __repr__(self): + msg = "" + msg += f"Dataset: {self.name}\n" + for i in self._attributes: + msg += f"\t{i}'s type is {self._attributes[i].type_name}" + if self._attributes[i].range: + msg += f", range is {str(self._attributes[i].range)}" + msg += '\n' + return msg + + def __iter__(self): + return iter(self._attributes) + + def __getitem__(self, key): + attr = self._attributes[key] + + return (attr.type_name, attr.range) + + def names(self): + """Return the list of attribute names. + + Returns + ------- + attrnames : list of str + The attribute names. + """ + return list(self._attributes) + + def types(self): + """Return the list of attribute types. + + Returns + ------- + attr_types : list of str + The attribute types. + """ + attr_types = [self._attributes[name].type_name + for name in self._attributes] + return attr_types + + +def loadarff(f): + """ + Read an arff file. + + The data is returned as a record array, which can be accessed much like + a dictionary of NumPy arrays. For example, if one of the attributes is + called 'pressure', then its first 10 data points can be accessed from the + ``data`` record array like so: ``data['pressure'][0:10]`` + + + Parameters + ---------- + f : file-like or str + File-like object to read from, or filename to open. + + Returns + ------- + data : record array + The data of the arff file, accessible by attribute names. + meta : `MetaData` + Contains information about the arff file such as name and + type of attributes, the relation (name of the dataset), etc. + + Raises + ------ + ParseArffError + This is raised if the given file is not ARFF-formatted. + NotImplementedError + The ARFF file has an attribute which is not supported yet. + + Notes + ----- + + This function should be able to read most arff files. Not + implemented functionality include: + + * date type attributes + * string type attributes + + It can read files with numeric and nominal attributes. It cannot read + files with sparse data ({} in the file). However, this function can + read files with missing data (? in the file), representing the data + points as NaNs. + + Examples + -------- + >>> from scipy.io import arff + >>> from io import StringIO + >>> content = \"\"\" + ... @relation foo + ... @attribute width numeric + ... @attribute height numeric + ... @attribute color {red,green,blue,yellow,black} + ... @data + ... 5.0,3.25,blue + ... 4.5,3.75,green + ... 3.0,4.00,red + ... \"\"\" + >>> f = StringIO(content) + >>> data, meta = arff.loadarff(f) + >>> data + array([(5.0, 3.25, 'blue'), (4.5, 3.75, 'green'), (3.0, 4.0, 'red')], + dtype=[('width', '>> meta + Dataset: foo + \twidth's type is numeric + \theight's type is numeric + \tcolor's type is nominal, range is ('red', 'green', 'blue', 'yellow', 'black') + + """ + if hasattr(f, 'read'): + ofile = f + else: + ofile = open(f) + try: + return _loadarff(ofile) + finally: + if ofile is not f: # only close what we opened + ofile.close() + + +def _loadarff(ofile): + # Parse the header file + try: + rel, attr = read_header(ofile) + except ValueError as e: + msg = "Error while parsing header, error was: " + str(e) + raise ParseArffError(msg) from e + + # Check whether we have a string attribute (not supported yet) + hasstr = False + for a in attr: + if isinstance(a, StringAttribute): + hasstr = True + + meta = MetaData(rel, attr) + + # XXX The following code is not great + # Build the type descriptor descr and the list of converters to convert + # each attribute to the suitable type (which should match the one in + # descr). + + # This can be used once we want to support integer as integer values and + # not as numeric anymore (using masked arrays ?). + + if hasstr: + # How to support string efficiently ? Ideally, we should know the max + # size of the string before allocating the numpy array. + raise NotImplementedError("String attributes not supported yet, sorry") + + ni = len(attr) + + def generator(row_iter, delim=','): + # TODO: this is where we are spending time (~80%). I think things + # could be made more efficiently: + # - We could for example "compile" the function, because some values + # do not change here. + # - The function to convert a line to dtyped values could also be + # generated on the fly from a string and be executed instead of + # looping. + # - The regex are overkill: for comments, checking that a line starts + # by % should be enough and faster, and for empty lines, same thing + # --> this does not seem to change anything. + + # 'compiling' the range since it does not change + # Note, I have already tried zipping the converters and + # row elements and got slightly worse performance. + elems = list(range(ni)) + + dialect = None + for raw in row_iter: + # We do not abstract skipping comments and empty lines for + # performance reasons. + if r_comment.match(raw) or r_empty.match(raw): + continue + + row, dialect = split_data_line(raw, dialect) + + yield tuple([attr[i].parse_data(row[i]) for i in elems]) + + a = list(generator(ofile)) + # No error should happen here: it is a bug otherwise + data = np.array(a, [(a.name, a.dtype) for a in attr]) + return data, meta + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/arffread.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/arffread.py new file mode 100644 index 0000000000000000000000000000000000000000..c42ae31db6bde3987bd059cc7451d1ae87f0073c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/arffread.py @@ -0,0 +1,19 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io.arff` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'MetaData', 'loadarff', 'ArffError', 'ParseArffError', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io.arff", module="arffread", + private_modules=["_arffread"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/iris.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/iris.arff new file mode 100644 index 0000000000000000000000000000000000000000..780480c7c6b9a68bf71aaf357c7d3f7a5b3b3f57 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/iris.arff @@ -0,0 +1,225 @@ +% 1. Title: Iris Plants Database +% +% 2. Sources: +% (a) Creator: R.A. Fisher +% (b) Donor: Michael Marshall (MARSHALL%PLU@io.arc.nasa.gov) +% (c) Date: July, 1988 +% +% 3. Past Usage: +% - Publications: too many to mention!!! Here are a few. +% 1. Fisher,R.A. "The use of multiple measurements in taxonomic problems" +% Annual Eugenics, 7, Part II, 179-188 (1936); also in "Contributions +% to Mathematical Statistics" (John Wiley, NY, 1950). +% 2. Duda,R.O., & Hart,P.E. (1973) Pattern Classification and Scene Analysis. +% (Q327.D83) John Wiley & Sons. ISBN 0-471-22361-1. See page 218. +% 3. Dasarathy, B.V. (1980) "Nosing Around the Neighborhood: A New System +% Structure and Classification Rule for Recognition in Partially Exposed +% Environments". IEEE Transactions on Pattern Analysis and Machine +% Intelligence, Vol. PAMI-2, No. 1, 67-71. +% -- Results: +% -- very low misclassification rates (0% for the setosa class) +% 4. Gates, G.W. (1972) "The Reduced Nearest Neighbor Rule". IEEE +% Transactions on Information Theory, May 1972, 431-433. +% -- Results: +% -- very low misclassification rates again +% 5. See also: 1988 MLC Proceedings, 54-64. Cheeseman et al's AUTOCLASS II +% conceptual clustering system finds 3 classes in the data. +% +% 4. Relevant Information: +% --- This is perhaps the best known database to be found in the pattern +% recognition literature. Fisher's paper is a classic in the field +% and is referenced frequently to this day. (See Duda & Hart, for +% example.) The data set contains 3 classes of 50 instances each, +% where each class refers to a type of iris plant. One class is +% linearly separable from the other 2; the latter are NOT linearly +% separable from each other. +% --- Predicted attribute: class of iris plant. +% --- This is an exceedingly simple domain. +% +% 5. Number of Instances: 150 (50 in each of three classes) +% +% 6. Number of Attributes: 4 numeric, predictive attributes and the class +% +% 7. Attribute Information: +% 1. sepal length in cm +% 2. sepal width in cm +% 3. petal length in cm +% 4. petal width in cm +% 5. class: +% -- Iris Setosa +% -- Iris Versicolour +% -- Iris Virginica +% +% 8. Missing Attribute Values: None +% +% Summary Statistics: +% Min Max Mean SD Class Correlation +% sepal length: 4.3 7.9 5.84 0.83 0.7826 +% sepal width: 2.0 4.4 3.05 0.43 -0.4194 +% petal length: 1.0 6.9 3.76 1.76 0.9490 (high!) +% petal width: 0.1 2.5 1.20 0.76 0.9565 (high!) +% +% 9. Class Distribution: 33.3% for each of 3 classes. + +@RELATION iris + +@ATTRIBUTE sepallength REAL +@ATTRIBUTE sepalwidth REAL +@ATTRIBUTE petallength REAL +@ATTRIBUTE petalwidth REAL +@ATTRIBUTE class {Iris-setosa,Iris-versicolor,Iris-virginica} + +@DATA +5.1,3.5,1.4,0.2,Iris-setosa +4.9,3.0,1.4,0.2,Iris-setosa +4.7,3.2,1.3,0.2,Iris-setosa +4.6,3.1,1.5,0.2,Iris-setosa +5.0,3.6,1.4,0.2,Iris-setosa +5.4,3.9,1.7,0.4,Iris-setosa +4.6,3.4,1.4,0.3,Iris-setosa +5.0,3.4,1.5,0.2,Iris-setosa +4.4,2.9,1.4,0.2,Iris-setosa +4.9,3.1,1.5,0.1,Iris-setosa +5.4,3.7,1.5,0.2,Iris-setosa +4.8,3.4,1.6,0.2,Iris-setosa +4.8,3.0,1.4,0.1,Iris-setosa +4.3,3.0,1.1,0.1,Iris-setosa +5.8,4.0,1.2,0.2,Iris-setosa +5.7,4.4,1.5,0.4,Iris-setosa +5.4,3.9,1.3,0.4,Iris-setosa +5.1,3.5,1.4,0.3,Iris-setosa +5.7,3.8,1.7,0.3,Iris-setosa +5.1,3.8,1.5,0.3,Iris-setosa +5.4,3.4,1.7,0.2,Iris-setosa +5.1,3.7,1.5,0.4,Iris-setosa +4.6,3.6,1.0,0.2,Iris-setosa +5.1,3.3,1.7,0.5,Iris-setosa +4.8,3.4,1.9,0.2,Iris-setosa 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+5.9,3.0,5.1,1.8,Iris-virginica +% +% +% diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/missing.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/missing.arff new file mode 100644 index 0000000000000000000000000000000000000000..dedc64c8fa2fcdc0081b30b7804be85114495ce2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/missing.arff @@ -0,0 +1,8 @@ +% This arff file contains some missing data +@relation missing +@attribute yop real +@attribute yap real +@data +1,5 +2,4 +?,? diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/nodata.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/nodata.arff new file mode 100644 index 0000000000000000000000000000000000000000..5766aeb229a1b31378026274c366e8e9e44fd487 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/nodata.arff @@ -0,0 +1,11 @@ +@RELATION iris + +@ATTRIBUTE sepallength REAL +@ATTRIBUTE sepalwidth REAL +@ATTRIBUTE petallength REAL +@ATTRIBUTE petalwidth REAL +@ATTRIBUTE class {Iris-setosa,Iris-versicolor,Iris-virginica} + +@DATA + +% This file has no data diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/quoted_nominal.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/quoted_nominal.arff new file mode 100644 index 0000000000000000000000000000000000000000..7cd16d1ef9b50cc1194d034ef4d458ef3cf0d417 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/quoted_nominal.arff @@ -0,0 +1,13 @@ +% Regression test for issue #10232 : Exception in loadarff with quoted nominal attributes +% Spaces between elements are stripped by the parser + +@relation SOME_DATA +@attribute age numeric +@attribute smoker {'yes', 'no'} +@data +18, 'no' +24, 'yes' +44, 'no' +56, 'no' +89,'yes' +11, 'no' diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/quoted_nominal_spaces.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/quoted_nominal_spaces.arff new file mode 100644 index 0000000000000000000000000000000000000000..c799127862b6060442b29c9a0382836cc9c55537 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/quoted_nominal_spaces.arff @@ -0,0 +1,13 @@ +% Regression test for issue #10232 : Exception in loadarff with quoted nominal attributes +% Spaces inside quotes are NOT stripped by the parser + +@relation SOME_DATA +@attribute age numeric +@attribute smoker {' yes', 'no '} +@data +18,'no ' +24,' yes' +44,'no ' +56,'no ' +89,' yes' +11,'no ' diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test1.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test1.arff new file mode 100644 index 0000000000000000000000000000000000000000..ccc8e0cc7c43dc66ad7b3a8e4738c3322d3f79d8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test1.arff @@ -0,0 +1,10 @@ +@RELATION test1 + +@ATTRIBUTE attr0 REAL +@ATTRIBUTE attr1 REAL +@ATTRIBUTE attr2 REAL +@ATTRIBUTE attr3 REAL +@ATTRIBUTE class {class0, class1, class2, class3} + +@DATA +0.1, 0.2, 0.3, 0.4,class1 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test10.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test10.arff new file mode 100644 index 0000000000000000000000000000000000000000..094ac5094a842866666726b358d2c66bf927c9d2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test10.arff @@ -0,0 +1,8 @@ +@relation test9 + +@attribute attr_relational relational + @attribute attr_number integer +@end attr_relational + +@data 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\ No newline at end of file diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test11.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test11.arff new file mode 100644 index 0000000000000000000000000000000000000000..fadfaee884e3e91cd59f691afd954a6a6d4042da --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test11.arff @@ -0,0 +1,11 @@ +@RELATION test11 + +@ATTRIBUTE attr0 REAL +@ATTRIBUTE attr1 REAL +@ATTRIBUTE attr2 REAL +@ATTRIBUTE attr3 REAL +@ATTRIBUTE class { class0, class1, class2, class3 } +@DATA +0.1, 0.2, 0.3, 0.4,class1 +-0.1, -0.2, -0.3, -0.4,class2 +1, 2, 3, 4,class3 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test2.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test2.arff new file mode 100644 index 0000000000000000000000000000000000000000..30f0dbf91b078ef670868d5e7321f956a6a7a506 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test2.arff @@ -0,0 +1,15 @@ +@RELATION test2 + +@ATTRIBUTE attr0 REAL +@ATTRIBUTE attr1 real +@ATTRIBUTE attr2 integer +@ATTRIBUTE attr3 Integer +@ATTRIBUTE attr4 Numeric +@ATTRIBUTE attr5 numeric +@ATTRIBUTE attr6 string +@ATTRIBUTE attr7 STRING +@ATTRIBUTE attr8 {bla} +@ATTRIBUTE attr9 {bla, bla} + +@DATA +0.1, 0.2, 0.3, 0.4,class1 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test3.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test3.arff new file mode 100644 index 0000000000000000000000000000000000000000..23da3b30967fcc95d70883f70be9ef6e39d577fa --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test3.arff @@ -0,0 +1,6 @@ +@RELATION test3 + +@ATTRIBUTE attr0 crap + +@DATA +0.1, 0.2, 0.3, 0.4,class1 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test4.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test4.arff new file mode 100644 index 0000000000000000000000000000000000000000..bf5f99ca89375fbd980185fd25711901f23ff844 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test4.arff @@ -0,0 +1,11 @@ +@RELATION test5 + +@ATTRIBUTE attr0 REAL +@ATTRIBUTE attr1 REAL +@ATTRIBUTE attr2 REAL +@ATTRIBUTE attr3 REAL +@ATTRIBUTE class {class0, class1, class2, class3} +@DATA +0.1, 0.2, 0.3, 0.4,class1 +-0.1, -0.2, -0.3, -0.4,class2 +1, 2, 3, 4,class3 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test5.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test5.arff new file mode 100644 index 0000000000000000000000000000000000000000..0075daf05e7792e80dcd565e791ce40e4dd49e85 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test5.arff @@ -0,0 +1,26 @@ +@RELATION test4 + +@ATTRIBUTE attr0 REAL +@ATTRIBUTE attr1 REAL +@ATTRIBUTE attr2 REAL +@ATTRIBUTE attr3 REAL +@ATTRIBUTE class {class0, class1, class2, class3} + +@DATA + +% lsdflkjhaksjdhf + +% lsdflkjhaksjdhf + +0.1, 0.2, 0.3, 0.4,class1 +% laksjdhf + +% lsdflkjhaksjdhf +-0.1, -0.2, -0.3, -0.4,class2 + +% lsdflkjhaksjdhf +% lsdflkjhaksjdhf + +% lsdflkjhaksjdhf + +1, 2, 3, 4,class3 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test6.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test6.arff new file mode 100644 index 0000000000000000000000000000000000000000..b63280b03aef8e0553a83fbf96692d280a3f86b7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test6.arff @@ -0,0 +1,12 @@ +@RELATION test6 + +@ATTRIBUTE attr0 REAL +@ATTRIBUTE attr1 REAL +@ATTRIBUTE attr2 REAL +@ATTRIBUTE attr3 REAL +@ATTRIBUTE class {C} + +@DATA +0.1, 0.2, 0.3, 0.4,C +-0.1, -0.2, -0.3, -0.4,C +1, 2, 3, 4,C diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test7.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test7.arff new file mode 100644 index 0000000000000000000000000000000000000000..38ef6c9a7a10afb10caa5913687ea3636ab1d38e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test7.arff @@ -0,0 +1,15 @@ +@RELATION test7 + +@ATTRIBUTE attr_year DATE yyyy +@ATTRIBUTE attr_month DATE yyyy-MM +@ATTRIBUTE attr_date DATE yyyy-MM-dd +@ATTRIBUTE attr_datetime_local DATE "yyyy-MM-dd HH:mm" +@ATTRIBUTE attr_datetime_missing DATE "yyyy-MM-dd HH:mm" + +@DATA +1999,1999-01,1999-01-31,"1999-01-31 00:01",? +2004,2004-12,2004-12-01,"2004-12-01 23:59","2004-12-01 23:59" +1817,1817-04,1817-04-28,"1817-04-28 13:00",? +2100,2100-09,2100-09-10,"2100-09-10 12:00",? +2013,2013-11,2013-11-30,"2013-11-30 04:55","2013-11-30 04:55" +1631,1631-10,1631-10-15,"1631-10-15 20:04","1631-10-15 20:04" \ No newline at end of file diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test8.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test8.arff new file mode 100644 index 0000000000000000000000000000000000000000..776deb4c9e7550eafdb26d16826f5651da37ef12 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test8.arff @@ -0,0 +1,12 @@ +@RELATION test8 + +@ATTRIBUTE attr_datetime_utc DATE "yyyy-MM-dd HH:mm Z" +@ATTRIBUTE attr_datetime_full DATE "yy-MM-dd HH:mm:ss z" + +@DATA +"1999-01-31 00:01 UTC","99-01-31 00:01:08 +0430" +"2004-12-01 23:59 UTC","04-12-01 23:59:59 -0800" +"1817-04-28 13:00 UTC","17-04-28 13:00:33 +1000" +"2100-09-10 12:00 UTC","21-09-10 12:00:21 -0300" +"2013-11-30 04:55 UTC","13-11-30 04:55:48 -1100" +"1631-10-15 20:04 UTC","31-10-15 20:04:10 +0000" \ No newline at end of file diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test9.arff b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test9.arff new file mode 100644 index 0000000000000000000000000000000000000000..b3f97e32a3fd4909a3f9cbf8d5d2e8d250f8dbad --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/data/test9.arff @@ -0,0 +1,14 @@ +@RELATION test9 + +@ATTRIBUTE attr_date_number RELATIONAL + @ATTRIBUTE attr_date DATE "yyyy-MM-dd" + @ATTRIBUTE attr_number INTEGER +@END attr_date_number + +@DATA +"1999-01-31 1\n1935-11-27 10" +"2004-12-01 2\n1942-08-13 20" +"1817-04-28 3" +"2100-09-10 4\n1957-04-17 40\n1721-01-14 400" +"2013-11-30 5" +"1631-10-15 6" \ No newline at end of file diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/test_arffread.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/test_arffread.py new file mode 100644 index 0000000000000000000000000000000000000000..d13ebe6dd1af3044794b28f5375d06ed60787966 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/arff/tests/test_arffread.py @@ -0,0 +1,421 @@ +import datetime +import os +import sys +from os.path import join as pjoin + +from io import StringIO + +import numpy as np + +from numpy.testing import (assert_array_almost_equal, + assert_array_equal, assert_equal, assert_) +from pytest import raises as assert_raises + +from scipy.io.arff import loadarff +from scipy.io.arff._arffread import read_header, ParseArffError + + +data_path = pjoin(os.path.dirname(__file__), 'data') + +test1 = pjoin(data_path, 'test1.arff') +test2 = pjoin(data_path, 'test2.arff') +test3 = pjoin(data_path, 'test3.arff') + +test4 = pjoin(data_path, 'test4.arff') +test5 = pjoin(data_path, 'test5.arff') +test6 = pjoin(data_path, 'test6.arff') +test7 = pjoin(data_path, 'test7.arff') +test8 = pjoin(data_path, 'test8.arff') +test9 = pjoin(data_path, 'test9.arff') +test10 = pjoin(data_path, 'test10.arff') +test11 = pjoin(data_path, 'test11.arff') +test_quoted_nominal = pjoin(data_path, 'quoted_nominal.arff') +test_quoted_nominal_spaces = pjoin(data_path, 'quoted_nominal_spaces.arff') + +expect4_data = [(0.1, 0.2, 0.3, 0.4, 'class1'), + (-0.1, -0.2, -0.3, -0.4, 'class2'), + (1, 2, 3, 4, 'class3')] +expected_types = ['numeric', 'numeric', 'numeric', 'numeric', 'nominal'] + +missing = pjoin(data_path, 'missing.arff') +expect_missing_raw = np.array([[1, 5], [2, 4], [np.nan, np.nan]]) +expect_missing = np.empty(3, [('yop', float), ('yap', float)]) +expect_missing['yop'] = expect_missing_raw[:, 0] # type: ignore[call-overload] +expect_missing['yap'] = expect_missing_raw[:, 1] # type: ignore[call-overload] + + +class TestData: + def test1(self): + # Parsing trivial file with nothing. + self._test(test4) + + def test2(self): + # Parsing trivial file with some comments in the data section. + self._test(test5) + + def test3(self): + # Parsing trivial file with nominal attribute of 1 character. + self._test(test6) + + def test4(self): + # Parsing trivial file with trailing spaces in attribute declaration. + self._test(test11) + + def _test(self, test_file): + data, meta = loadarff(test_file) + for i in range(len(data)): + for j in range(4): + assert_array_almost_equal(expect4_data[i][j], data[i][j]) + assert_equal(meta.types(), expected_types) + + def test_filelike(self): + # Test reading from file-like object (StringIO) + with open(test1) as f1: + data1, meta1 = loadarff(f1) + with open(test1) as f2: + data2, meta2 = loadarff(StringIO(f2.read())) + assert_(data1 == data2) + assert_(repr(meta1) == repr(meta2)) + + def test_path(self): + # Test reading from `pathlib.Path` object + from pathlib import Path + + with open(test1) as f1: + data1, meta1 = loadarff(f1) + + data2, meta2 = loadarff(Path(test1)) + + assert_(data1 == data2) + assert_(repr(meta1) == repr(meta2)) + + +class TestMissingData: + def test_missing(self): + data, meta = loadarff(missing) + for i in ['yop', 'yap']: + assert_array_almost_equal(data[i], expect_missing[i]) + + +class TestNoData: + def test_nodata(self): + # The file nodata.arff has no data in the @DATA section. + # Reading it should result in an array with length 0. + nodata_filename = os.path.join(data_path, 'nodata.arff') + data, meta = loadarff(nodata_filename) + if sys.byteorder == 'big': + end = '>' + else: + end = '<' + expected_dtype = np.dtype([('sepallength', f'{end}f8'), + ('sepalwidth', f'{end}f8'), + ('petallength', f'{end}f8'), + ('petalwidth', f'{end}f8'), + ('class', 'S15')]) + assert_equal(data.dtype, expected_dtype) + assert_equal(data.size, 0) + + +class TestHeader: + def test_type_parsing(self): + # Test parsing type of attribute from their value. + with open(test2) as ofile: + rel, attrs = read_header(ofile) + + expected = ['numeric', 'numeric', 'numeric', 'numeric', 'numeric', + 'numeric', 'string', 'string', 'nominal', 'nominal'] + + for i in range(len(attrs)): + assert_(attrs[i].type_name == expected[i]) + + def test_badtype_parsing(self): + # Test parsing wrong type of attribute from their value. + def badtype_read(): + with open(test3) as ofile: + _, _ = read_header(ofile) + + assert_raises(ParseArffError, badtype_read) + + def test_fullheader1(self): + # Parsing trivial header with nothing. + with open(test1) as ofile: + rel, attrs = read_header(ofile) + + # Test relation + assert_(rel == 'test1') + + # Test numerical attributes + assert_(len(attrs) == 5) + for i in range(4): + assert_(attrs[i].name == 'attr%d' % i) + assert_(attrs[i].type_name == 'numeric') + + # Test nominal attribute + assert_(attrs[4].name == 'class') + assert_(attrs[4].values == ('class0', 'class1', 'class2', 'class3')) + + def test_dateheader(self): + with open(test7) as ofile: + rel, attrs = read_header(ofile) + + assert_(rel == 'test7') + + assert_(len(attrs) == 5) + + assert_(attrs[0].name == 'attr_year') + assert_(attrs[0].date_format == '%Y') + + assert_(attrs[1].name == 'attr_month') + assert_(attrs[1].date_format == '%Y-%m') + + assert_(attrs[2].name == 'attr_date') + assert_(attrs[2].date_format == '%Y-%m-%d') + + assert_(attrs[3].name == 'attr_datetime_local') + assert_(attrs[3].date_format == '%Y-%m-%d %H:%M') + + assert_(attrs[4].name == 'attr_datetime_missing') + assert_(attrs[4].date_format == '%Y-%m-%d %H:%M') + + def test_dateheader_unsupported(self): + def read_dateheader_unsupported(): + with open(test8) as ofile: + _, _ = read_header(ofile) + + assert_raises(ValueError, read_dateheader_unsupported) + + +class TestDateAttribute: + def setup_method(self): + self.data, self.meta = loadarff(test7) + + def test_year_attribute(self): + expected = np.array([ + '1999', + '2004', + '1817', + '2100', + '2013', + '1631' + ], dtype='datetime64[Y]') + + assert_array_equal(self.data["attr_year"], expected) + + def test_month_attribute(self): + expected = np.array([ + '1999-01', + '2004-12', + '1817-04', + '2100-09', + '2013-11', + '1631-10' + ], dtype='datetime64[M]') + + assert_array_equal(self.data["attr_month"], expected) + + def test_date_attribute(self): + expected = np.array([ + '1999-01-31', + '2004-12-01', + '1817-04-28', + '2100-09-10', + '2013-11-30', + '1631-10-15' + ], dtype='datetime64[D]') + + assert_array_equal(self.data["attr_date"], expected) + + def test_datetime_local_attribute(self): + expected = np.array([ + datetime.datetime(year=1999, month=1, day=31, hour=0, minute=1), + datetime.datetime(year=2004, month=12, day=1, hour=23, minute=59), + datetime.datetime(year=1817, month=4, day=28, hour=13, minute=0), + datetime.datetime(year=2100, month=9, day=10, hour=12, minute=0), + datetime.datetime(year=2013, month=11, day=30, hour=4, minute=55), + datetime.datetime(year=1631, month=10, day=15, hour=20, minute=4) + ], dtype='datetime64[m]') + + assert_array_equal(self.data["attr_datetime_local"], expected) + + def test_datetime_missing(self): + expected = np.array([ + 'nat', + '2004-12-01T23:59', + 'nat', + 'nat', + '2013-11-30T04:55', + '1631-10-15T20:04' + ], dtype='datetime64[m]') + + assert_array_equal(self.data["attr_datetime_missing"], expected) + + def test_datetime_timezone(self): + assert_raises(ParseArffError, loadarff, test8) + + +class TestRelationalAttribute: + def setup_method(self): + self.data, self.meta = loadarff(test9) + + def test_attributes(self): + assert_equal(len(self.meta._attributes), 1) + + relational = list(self.meta._attributes.values())[0] + + assert_equal(relational.name, 'attr_date_number') + assert_equal(relational.type_name, 'relational') + assert_equal(len(relational.attributes), 2) + assert_equal(relational.attributes[0].name, + 'attr_date') + assert_equal(relational.attributes[0].type_name, + 'date') + assert_equal(relational.attributes[1].name, + 'attr_number') + assert_equal(relational.attributes[1].type_name, + 'numeric') + + def test_data(self): + dtype_instance = [('attr_date', 'datetime64[D]'), + ('attr_number', np.float64)] + + expected = [ + np.array([('1999-01-31', 1), ('1935-11-27', 10)], + dtype=dtype_instance), + np.array([('2004-12-01', 2), ('1942-08-13', 20)], + dtype=dtype_instance), + np.array([('1817-04-28', 3)], + dtype=dtype_instance), + np.array([('2100-09-10', 4), ('1957-04-17', 40), + ('1721-01-14', 400)], + dtype=dtype_instance), + np.array([('2013-11-30', 5)], + dtype=dtype_instance), + np.array([('1631-10-15', 6)], + dtype=dtype_instance) + ] + + for i in range(len(self.data["attr_date_number"])): + assert_array_equal(self.data["attr_date_number"][i], + expected[i]) + + +class TestRelationalAttributeLong: + def setup_method(self): + self.data, self.meta = loadarff(test10) + + def test_attributes(self): + assert_equal(len(self.meta._attributes), 1) + + relational = list(self.meta._attributes.values())[0] + + assert_equal(relational.name, 'attr_relational') + assert_equal(relational.type_name, 'relational') + assert_equal(len(relational.attributes), 1) + assert_equal(relational.attributes[0].name, + 'attr_number') + assert_equal(relational.attributes[0].type_name, 'numeric') + + def test_data(self): + dtype_instance = [('attr_number', np.float64)] + + expected = np.array([(n,) for n in range(30000)], + dtype=dtype_instance) + + assert_array_equal(self.data["attr_relational"][0], + expected) + + +class TestQuotedNominal: + """ + Regression test for issue #10232: + + Exception in loadarff with quoted nominal attributes. + """ + + def setup_method(self): + self.data, self.meta = loadarff(test_quoted_nominal) + + def test_attributes(self): + assert_equal(len(self.meta._attributes), 2) + + age, smoker = self.meta._attributes.values() + + assert_equal(age.name, 'age') + assert_equal(age.type_name, 'numeric') + assert_equal(smoker.name, 'smoker') + assert_equal(smoker.type_name, 'nominal') + assert_equal(smoker.values, ['yes', 'no']) + + def test_data(self): + + age_dtype_instance = np.float64 + smoker_dtype_instance = '' (big endian) + +''' +import sys + +__all__ = [ + 'aliases', 'native_code', 'swapped_code', + 'sys_is_le', 'to_numpy_code' +] + +sys_is_le = sys.byteorder == 'little' +native_code = sys_is_le and '<' or '>' +swapped_code = sys_is_le and '>' or '<' + +aliases = {'little': ('little', '<', 'l', 'le'), + 'big': ('big', '>', 'b', 'be'), + 'native': ('native', '='), + 'swapped': ('swapped', 'S')} + + +def to_numpy_code(code): + """ + Convert various order codings to NumPy format. + + Parameters + ---------- + code : str + The code to convert. It is converted to lower case before parsing. + Legal values are: + 'little', 'big', 'l', 'b', 'le', 'be', '<', '>', 'native', '=', + 'swapped', 's'. + + Returns + ------- + out_code : {'<', '>'} + Here '<' is the numpy dtype code for little endian, + and '>' is the code for big endian. + + Examples + -------- + >>> import sys + >>> from scipy.io.matlab._byteordercodes import to_numpy_code + >>> sys_is_le = (sys.byteorder == 'little') + >>> sys_is_le + True + >>> to_numpy_code('big') + '>' + >>> to_numpy_code('little') + '<' + >>> nc = to_numpy_code('native') + >>> nc == '<' if sys_is_le else nc == '>' + True + >>> sc = to_numpy_code('swapped') + >>> sc == '>' if sys_is_le else sc == '<' + True + + """ + code = code.lower() + if code is None: + return native_code + if code in aliases['little']: + return '<' + elif code in aliases['big']: + return '>' + elif code in aliases['native']: + return native_code + elif code in aliases['swapped']: + return swapped_code + else: + raise ValueError( + f'We cannot handle byte order {code}') diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio.py new file mode 100644 index 0000000000000000000000000000000000000000..4c86d873bd11fb45676d8db37a5d60b032276ecc --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio.py @@ -0,0 +1,372 @@ +""" +Module for reading and writing matlab (TM) .mat files +""" +# Authors: Travis Oliphant, Matthew Brett + +from contextlib import contextmanager + +from ._miobase import _get_matfile_version, docfiller +from ._mio4 import MatFile4Reader, MatFile4Writer +from ._mio5 import MatFile5Reader, MatFile5Writer + +__all__ = ['loadmat', 'savemat', 'whosmat'] + + +@contextmanager +def _open_file_context(file_like, appendmat, mode='rb'): + f, opened = _open_file(file_like, appendmat, mode) + try: + yield f + finally: + if opened: + f.close() + + +def _open_file(file_like, appendmat, mode='rb'): + """ + Open `file_like` and return as file-like object. First, check if object is + already file-like; if so, return it as-is. Otherwise, try to pass it + to open(). If that fails, and `file_like` is a string, and `appendmat` is true, + append '.mat' and try again. + """ + reqs = {'read'} if set(mode) & set('r+') else set() + if set(mode) & set('wax+'): + reqs.add('write') + if reqs.issubset(dir(file_like)): + return file_like, False + + try: + return open(file_like, mode), True + except OSError as e: + # Probably "not found" + if isinstance(file_like, str): + if appendmat and not file_like.endswith('.mat'): + file_like += '.mat' + return open(file_like, mode), True + else: + raise OSError( + 'Reader needs file name or open file-like object' + ) from e + + +@docfiller +def mat_reader_factory(file_name, appendmat=True, **kwargs): + """ + Create reader for matlab .mat format files. + + Parameters + ---------- + %(file_arg)s + %(append_arg)s + %(load_args)s + %(struct_arg)s + + Returns + ------- + matreader : MatFileReader object + Initialized instance of MatFileReader class matching the mat file + type detected in `filename`. + file_opened : bool + Whether the file was opened by this routine. + + """ + byte_stream, file_opened = _open_file(file_name, appendmat) + mjv, mnv = _get_matfile_version(byte_stream) + if mjv == 0: + return MatFile4Reader(byte_stream, **kwargs), file_opened + elif mjv == 1: + return MatFile5Reader(byte_stream, **kwargs), file_opened + elif mjv == 2: + raise NotImplementedError('Please use HDF reader for matlab v7.3 ' + 'files, e.g. h5py') + else: + raise TypeError(f'Did not recognize version {mjv}') + + +@docfiller +def loadmat(file_name, mdict=None, appendmat=True, *, spmatrix=True, **kwargs): + """ + Load MATLAB file. + + Parameters + ---------- + file_name : str + Name of the mat file (do not need .mat extension if + appendmat==True). Can also pass open file-like object. + mdict : dict, optional + Dictionary in which to insert matfile variables. + appendmat : bool, optional + True to append the .mat extension to the end of the given + filename, if not already present. Default is True. + spmatrix : bool, optional (default: True) + If ``True``, return sparse ``coo_matrix``. Otherwise return ``coo_array``. + Only relevant for sparse variables. + byte_order : str or None, optional + None by default, implying byte order guessed from mat + file. Otherwise can be one of ('native', '=', 'little', '<', + 'BIG', '>'). + mat_dtype : bool, optional + If True, return arrays in same dtype as would be loaded into + MATLAB (instead of the dtype with which they are saved). + squeeze_me : bool, optional + Whether to squeeze unit matrix dimensions or not. + chars_as_strings : bool, optional + Whether to convert char arrays to string arrays. + matlab_compatible : bool, optional + Returns matrices as would be loaded by MATLAB (implies + squeeze_me=False, chars_as_strings=False, mat_dtype=True, + struct_as_record=True). + struct_as_record : bool, optional + Whether to load MATLAB structs as NumPy record arrays, or as + old-style NumPy arrays with dtype=object. Setting this flag to + False replicates the behavior of scipy version 0.7.x (returning + NumPy object arrays). The default setting is True, because it + allows easier round-trip load and save of MATLAB files. + verify_compressed_data_integrity : bool, optional + Whether the length of compressed sequences in the MATLAB file + should be checked, to ensure that they are not longer than we expect. + It is advisable to enable this (the default) because overlong + compressed sequences in MATLAB files generally indicate that the + files have experienced some sort of corruption. + variable_names : None or sequence + If None (the default) - read all variables in file. Otherwise, + `variable_names` should be a sequence of strings, giving names of the + MATLAB variables to read from the file. The reader will skip any + variable with a name not in this sequence, possibly saving some read + processing. + simplify_cells : False, optional + If True, return a simplified dict structure (which is useful if the mat + file contains cell arrays). Note that this only affects the structure + of the result and not its contents (which is identical for both output + structures). If True, this automatically sets `struct_as_record` to + False and `squeeze_me` to True, which is required to simplify cells. + uint16_codec : str, optional + The codec to use for decoding characters, which are stored as uint16 + values. The default uses the system encoding, but this can be manually + set to other values such as 'ascii', 'latin1', and 'utf-8'. This + parameter is relevant only for files stored as v6 and above, and not + for files stored as v4. + + Returns + ------- + mat_dict : dict + dictionary with variable names as keys, and loaded matrices as values. + + Notes + ----- + v4 (Level 1.0), v6 and v7 to 7.2 matfiles are supported. + + You will need an HDF5 Python library to read MATLAB 7.3 format mat + files. Because SciPy does not supply one, we do not implement the + HDF5 / 7.3 interface here. + + Examples + -------- + >>> from os.path import dirname, join as pjoin + >>> import scipy.io as sio + + Get the filename for an example .mat file from the tests/data directory. + + >>> data_dir = pjoin(dirname(sio.__file__), 'matlab', 'tests', 'data') + >>> mat_fname = pjoin(data_dir, 'testdouble_7.4_GLNX86.mat') + + Load the .mat file contents. + + >>> mat_contents = sio.loadmat(mat_fname, spmatrix=False) + + The result is a dictionary, one key/value pair for each variable: + + >>> sorted(mat_contents.keys()) + ['__globals__', '__header__', '__version__', 'testdouble'] + >>> mat_contents['testdouble'] + array([[0. , 0.78539816, 1.57079633, 2.35619449, 3.14159265, + 3.92699082, 4.71238898, 5.49778714, 6.28318531]]) + + By default SciPy reads MATLAB structs as structured NumPy arrays where the + dtype fields are of type `object` and the names correspond to the MATLAB + struct field names. This can be disabled by setting the optional argument + `struct_as_record=False`. + + Get the filename for an example .mat file that contains a MATLAB struct + called `teststruct` and load the contents. + + >>> matstruct_fname = pjoin(data_dir, 'teststruct_7.4_GLNX86.mat') + >>> matstruct_contents = sio.loadmat(matstruct_fname) + >>> teststruct = matstruct_contents['teststruct'] + >>> teststruct.dtype + dtype([('stringfield', 'O'), ('doublefield', 'O'), ('complexfield', 'O')]) + + The size of the structured array is the size of the MATLAB struct, not the + number of elements in any particular field. The shape defaults to 2-D + unless the optional argument `squeeze_me=True`, in which case all length 1 + dimensions are removed. + + >>> teststruct.size + 1 + >>> teststruct.shape + (1, 1) + + Get the 'stringfield' of the first element in the MATLAB struct. + + >>> teststruct[0, 0]['stringfield'] + array(['Rats live on no evil star.'], + dtype='>> teststruct['doublefield'][0, 0] + array([[ 1.41421356, 2.71828183, 3.14159265]]) + + Load the MATLAB struct, squeezing out length 1 dimensions, and get the item + from the 'complexfield'. + + >>> matstruct_squeezed = sio.loadmat(matstruct_fname, squeeze_me=True) + >>> matstruct_squeezed['teststruct'].shape + () + >>> matstruct_squeezed['teststruct']['complexfield'].shape + () + >>> matstruct_squeezed['teststruct']['complexfield'].item() + array([ 1.41421356+1.41421356j, 2.71828183+2.71828183j, + 3.14159265+3.14159265j]) + """ + variable_names = kwargs.pop('variable_names', None) + with _open_file_context(file_name, appendmat) as f: + MR, _ = mat_reader_factory(f, **kwargs) + matfile_dict = MR.get_variables(variable_names) + if spmatrix: + from scipy.sparse import issparse, coo_matrix + for name, var in list(matfile_dict.items()): + if issparse(var): + matfile_dict[name] = coo_matrix(var) + + if mdict is not None: + mdict.update(matfile_dict) + else: + mdict = matfile_dict + + return mdict + + +@docfiller +def savemat(file_name, mdict, + appendmat=True, + format='5', + long_field_names=False, + do_compression=False, + oned_as='row'): + """ + Save a dictionary of names and arrays into a MATLAB-style .mat file. + + This saves the array objects in the given dictionary to a MATLAB- + style .mat file. + + Parameters + ---------- + file_name : str or file-like object + Name of the .mat file (.mat extension not needed if ``appendmat == + True``). + Can also pass open file_like object. + mdict : dict + Dictionary from which to save matfile variables. + appendmat : bool, optional + True (the default) to append the .mat extension to the end of the + given filename, if not already present. + format : {'5', '4'}, string, optional + '5' (the default) for MATLAB 5 and up (to 7.2), + '4' for MATLAB 4 .mat files. + long_field_names : bool, optional + False (the default) - maximum field name length in a structure is + 31 characters which is the documented maximum length. + True - maximum field name length in a structure is 63 characters + which works for MATLAB 7.6+. + do_compression : bool, optional + Whether or not to compress matrices on write. Default is False. + oned_as : {'row', 'column'}, optional + If 'column', write 1-D NumPy arrays as column vectors. + If 'row', write 1-D NumPy arrays as row vectors. + + Examples + -------- + >>> from scipy.io import savemat + >>> import numpy as np + >>> a = np.arange(20) + >>> mdic = {"a": a, "label": "experiment"} + >>> mdic + {'a': array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, + 17, 18, 19]), + 'label': 'experiment'} + >>> savemat("matlab_matrix.mat", mdic) + """ + with _open_file_context(file_name, appendmat, 'wb') as file_stream: + if format == '4': + if long_field_names: + message = "Long field names are not available for version 4 files" + raise ValueError(message) + MW = MatFile4Writer(file_stream, oned_as) + elif format == '5': + MW = MatFile5Writer(file_stream, + do_compression=do_compression, + unicode_strings=True, + long_field_names=long_field_names, + oned_as=oned_as) + else: + raise ValueError("Format should be '4' or '5'") + MW.put_variables(mdict) + + +@docfiller +def whosmat(file_name, appendmat=True, **kwargs): + """ + List variables inside a MATLAB file. + + Parameters + ---------- + %(file_arg)s + %(append_arg)s + %(load_args)s + %(struct_arg)s + + Returns + ------- + variables : list of tuples + A list of tuples, where each tuple holds the matrix name (a string), + its shape (tuple of ints), and its data class (a string). + Possible data classes are: int8, uint8, int16, uint16, int32, uint32, + int64, uint64, single, double, cell, struct, object, char, sparse, + function, opaque, logical, unknown. + + Notes + ----- + v4 (Level 1.0), v6 and v7 to 7.2 matfiles are supported. + + You will need an HDF5 python library to read matlab 7.3 format mat + files (e.g. h5py). Because SciPy does not supply one, we do not implement the + HDF5 / 7.3 interface here. + + .. versionadded:: 0.12.0 + + Examples + -------- + >>> from io import BytesIO + >>> import numpy as np + >>> from scipy.io import savemat, whosmat + + Create some arrays, and use `savemat` to write them to a ``BytesIO`` + instance. + + >>> a = np.array([[10, 20, 30], [11, 21, 31]], dtype=np.int32) + >>> b = np.geomspace(1, 10, 5) + >>> f = BytesIO() + >>> savemat(f, {'a': a, 'b': b}) + + Use `whosmat` to inspect ``f``. Each tuple in the output list gives + the name, shape and data type of the array in ``f``. + + >>> whosmat(f) + [('a', (2, 3), 'int32'), ('b', (1, 5), 'double')] + + """ + with _open_file_context(file_name, appendmat) as f: + ML, file_opened = mat_reader_factory(f, **kwargs) + variables = ML.list_variables() + return variables diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio4.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio4.py new file mode 100644 index 0000000000000000000000000000000000000000..b108386d110e6062d088498259e849603583eb94 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio4.py @@ -0,0 +1,632 @@ +''' Classes for read / write of matlab (TM) 4 files +''' +import sys +import warnings +import math +from operator import mul + +import numpy as np + +import scipy.sparse + +from ._miobase import (MatFileReader, docfiller, matdims, read_dtype, + convert_dtypes, arr_to_chars, arr_dtype_number) + +from ._mio_utils import squeeze_element, chars_to_strings +from functools import reduce + + +__all__ = [ + 'MatFile4Reader', 'MatFile4Writer', 'SYS_LITTLE_ENDIAN', + 'VarHeader4', 'VarReader4', 'VarWriter4', 'arr_to_2d', 'mclass_info', + 'mdtypes_template', 'miDOUBLE', 'miINT16', 'miINT32', 'miSINGLE', + 'miUINT16', 'miUINT8', 'mxCHAR_CLASS', 'mxFULL_CLASS', 'mxSPARSE_CLASS', + 'np_to_mtypes', 'order_codes' +] + + +SYS_LITTLE_ENDIAN = sys.byteorder == 'little' + +miDOUBLE = 0 +miSINGLE = 1 +miINT32 = 2 +miINT16 = 3 +miUINT16 = 4 +miUINT8 = 5 + +mdtypes_template = { + miDOUBLE: 'f8', + miSINGLE: 'f4', + miINT32: 'i4', + miINT16: 'i2', + miUINT16: 'u2', + miUINT8: 'u1', + 'header': [('mopt', 'i4'), + ('mrows', 'i4'), + ('ncols', 'i4'), + ('imagf', 'i4'), + ('namlen', 'i4')], + 'U1': 'U1', + } + +np_to_mtypes = { + 'f8': miDOUBLE, + 'c32': miDOUBLE, + 'c24': miDOUBLE, + 'c16': miDOUBLE, + 'f4': miSINGLE, + 'c8': miSINGLE, + 'i4': miINT32, + 'i2': miINT16, + 'u2': miUINT16, + 'u1': miUINT8, + 'S1': miUINT8, + } + +# matrix classes +mxFULL_CLASS = 0 +mxCHAR_CLASS = 1 +mxSPARSE_CLASS = 2 + +order_codes = { + 0: '<', + 1: '>', + 2: 'VAX D-float', # ! + 3: 'VAX G-float', + 4: 'Cray', # !! + } + +mclass_info = { + mxFULL_CLASS: 'double', + mxCHAR_CLASS: 'char', + mxSPARSE_CLASS: 'sparse', + } + + +_MAX_INTP = np.iinfo(np.intp).max + + +class VarHeader4: + # Mat4 variables never logical or global + is_logical = False + is_global = False + + def __init__(self, + name, + dtype, + mclass, + dims, + is_complex): + self.name = name + self.dtype = dtype + self.mclass = mclass + self.dims = dims + self.is_complex = is_complex + + +class VarReader4: + ''' Class to read matlab 4 variables ''' + + def __init__(self, file_reader): + self.file_reader = file_reader + self.mat_stream = file_reader.mat_stream + self.dtypes = file_reader.dtypes + self.chars_as_strings = file_reader.chars_as_strings + self.squeeze_me = file_reader.squeeze_me + + def read_header(self): + ''' Read and return header for variable ''' + data = read_dtype(self.mat_stream, self.dtypes['header']) + name = self.mat_stream.read(int(data['namlen'])).strip(b'\x00') + if data['mopt'] < 0 or data['mopt'] > 5000: + raise ValueError('Mat 4 mopt wrong format, byteswapping problem?') + M, rest = divmod(data['mopt'], 1000) # order code + if M not in (0, 1): + warnings.warn(f"We do not support byte ordering '{order_codes[M]}';" + " returned data may be corrupt", + UserWarning, stacklevel=3) + O, rest = divmod(rest, 100) # unused, should be 0 + if O != 0: + raise ValueError('O in MOPT integer should be 0, wrong format?') + P, rest = divmod(rest, 10) # data type code e.g miDOUBLE (see above) + T = rest # matrix type code e.g., mxFULL_CLASS (see above) + dims = (data['mrows'], data['ncols']) + is_complex = data['imagf'] == 1 + dtype = self.dtypes[P] + return VarHeader4( + name, + dtype, + T, + dims, + is_complex) + + def array_from_header(self, hdr, process=True): + mclass = hdr.mclass + if mclass == mxFULL_CLASS: + arr = self.read_full_array(hdr) + elif mclass == mxCHAR_CLASS: + arr = self.read_char_array(hdr) + if process and self.chars_as_strings: + arr = chars_to_strings(arr) + elif mclass == mxSPARSE_CLASS: + # no current processing (below) makes sense for sparse + return self.read_sparse_array(hdr) + else: + raise TypeError(f'No reader for class code {mclass}') + if process and self.squeeze_me: + return squeeze_element(arr) + return arr + + def read_sub_array(self, hdr, copy=True): + ''' Mat4 read using header `hdr` dtype and dims + + Parameters + ---------- + hdr : object + object with attributes ``dtype``, ``dims``. dtype is assumed to be + the correct endianness + copy : bool, optional + copies array before return if True (default True) + (buffer is usually read only) + + Returns + ------- + arr : ndarray + of dtype given by `hdr` ``dtype`` and shape given by `hdr` ``dims`` + ''' + dt = hdr.dtype + # Fast product for large (>2GB) arrays. + num_bytes = reduce(mul, hdr.dims, np.int64(dt.itemsize)) + if num_bytes > _MAX_INTP: + raise ValueError( + f"Variable '{hdr.name.decode('latin1')}' has byte length " + f"longer than largest possible NumPy array on this platform.") + buffer = self.mat_stream.read(num_bytes) + if len(buffer) != num_bytes: + raise ValueError( + f"Not enough bytes to read matrix " + f"'{hdr.name.decode('latin1')}'; is this a badly-formed file? " + f"Consider listing matrices with `whosmat` and loading named " + f"matrices with `variable_names` kwarg to `loadmat`") + arr = np.ndarray(shape=hdr.dims, + dtype=dt, + buffer=buffer, + order='F') + if copy: + arr = arr.copy() + return arr + + def read_full_array(self, hdr): + ''' Full (rather than sparse) matrix getter + + Read matrix (array) can be real or complex + + Parameters + ---------- + hdr : ``VarHeader4`` instance + + Returns + ------- + arr : ndarray + complex array if ``hdr.is_complex`` is True, otherwise a real + numeric array + ''' + if hdr.is_complex: + # avoid array copy to save memory + res = self.read_sub_array(hdr, copy=False) + res_j = self.read_sub_array(hdr, copy=False) + return res + (res_j * 1j) + return self.read_sub_array(hdr) + + def read_char_array(self, hdr): + ''' latin-1 text matrix (char matrix) reader + + Parameters + ---------- + hdr : ``VarHeader4`` instance + + Returns + ------- + arr : ndarray + with dtype 'U1', shape given by `hdr` ``dims`` + ''' + arr = self.read_sub_array(hdr).astype(np.uint8) + S = arr.tobytes().decode('latin-1') + return np.ndarray(shape=hdr.dims, + dtype=np.dtype('U1'), + buffer=np.array(S)).copy() + + def read_sparse_array(self, hdr): + ''' Read and return sparse matrix type + + Parameters + ---------- + hdr : ``VarHeader4`` instance + + Returns + ------- + arr : coo_array + with dtype ``float`` and shape read from the sparse array data + + Notes + ----- + MATLAB 4 real sparse arrays are saved in a N+1 by 3 array format, where + N is the number of non-zero values. Column 1 values [0:N] are the + (1-based) row indices of the each non-zero value, column 2 [0:N] are the + column indices, column 3 [0:N] are the (real) values. The last values + [-1,0:2] of the rows, column indices are shape[0] and shape[1] + respectively of the output matrix. The last value for the values column + is a padding 0. mrows and ncols values from the header give the shape of + the stored matrix, here [N+1, 3]. Complex data are saved as a 4 column + matrix, where the fourth column contains the imaginary component; the + last value is again 0. Complex sparse data do *not* have the header + ``imagf`` field set to True; the fact that the data are complex is only + detectable because there are 4 storage columns. + ''' + res = self.read_sub_array(hdr) + tmp = res[:-1,:] + # All numbers are float64 in Matlab, but SciPy sparse expects int shape + dims = (int(res[-1,0]), int(res[-1,1])) + I = np.ascontiguousarray(tmp[:,0],dtype='intc') # fixes byte order also + J = np.ascontiguousarray(tmp[:,1],dtype='intc') + I -= 1 # for 1-based indexing + J -= 1 + if res.shape[1] == 3: + V = np.ascontiguousarray(tmp[:,2],dtype='float') + else: + V = np.ascontiguousarray(tmp[:,2],dtype='complex') + V.imag = tmp[:,3] + return scipy.sparse.coo_array((V,(I,J)), dims) + + def shape_from_header(self, hdr): + '''Read the shape of the array described by the header. + The file position after this call is unspecified. + ''' + mclass = hdr.mclass + if mclass == mxFULL_CLASS: + shape = tuple(map(int, hdr.dims)) + elif mclass == mxCHAR_CLASS: + shape = tuple(map(int, hdr.dims)) + if self.chars_as_strings: + shape = shape[:-1] + elif mclass == mxSPARSE_CLASS: + dt = hdr.dtype + dims = hdr.dims + + if not (len(dims) == 2 and dims[0] >= 1 and dims[1] >= 1): + return () + + # Read only the row and column counts + self.mat_stream.seek(dt.itemsize * (dims[0] - 1), 1) + rows = np.ndarray(shape=(), dtype=dt, + buffer=self.mat_stream.read(dt.itemsize)) + self.mat_stream.seek(dt.itemsize * (dims[0] - 1), 1) + cols = np.ndarray(shape=(), dtype=dt, + buffer=self.mat_stream.read(dt.itemsize)) + + shape = (int(rows), int(cols)) + else: + raise TypeError(f'No reader for class code {mclass}') + + if self.squeeze_me: + shape = tuple([x for x in shape if x != 1]) + return shape + + +class MatFile4Reader(MatFileReader): + ''' Reader for Mat4 files ''' + @docfiller + def __init__(self, mat_stream, *args, **kwargs): + ''' Initialize matlab 4 file reader + + %(matstream_arg)s + %(load_args)s + ''' + super().__init__(mat_stream, *args, **kwargs) + self._matrix_reader = None + + def guess_byte_order(self): + self.mat_stream.seek(0) + mopt = read_dtype(self.mat_stream, np.dtype('i4')) + self.mat_stream.seek(0) + if mopt == 0: + return '<' + if mopt < 0 or mopt > 5000: + # Number must have been byteswapped + return SYS_LITTLE_ENDIAN and '>' or '<' + # Not byteswapped + return SYS_LITTLE_ENDIAN and '<' or '>' + + def initialize_read(self): + ''' Run when beginning read of variables + + Sets up readers from parameters in `self` + ''' + self.dtypes = convert_dtypes(mdtypes_template, self.byte_order) + self._matrix_reader = VarReader4(self) + + def read_var_header(self): + ''' Read and return header, next position + + Parameters + ---------- + None + + Returns + ------- + header : object + object that can be passed to self.read_var_array, and that + has attributes ``name`` and ``is_global`` + next_position : int + position in stream of next variable + ''' + hdr = self._matrix_reader.read_header() + # Fast product for large (>2GB) arrays. + remaining_bytes = reduce(mul, hdr.dims, np.int64(hdr.dtype.itemsize)) + if hdr.is_complex and not hdr.mclass == mxSPARSE_CLASS: + remaining_bytes *= 2 + next_position = self.mat_stream.tell() + remaining_bytes + return hdr, next_position + + def read_var_array(self, header, process=True): + ''' Read array, given `header` + + Parameters + ---------- + header : header object + object with fields defining variable header + process : {True, False}, optional + If True, apply recursive post-processing during loading of array. + + Returns + ------- + arr : array + array with post-processing applied or not according to + `process`. + ''' + return self._matrix_reader.array_from_header(header, process) + + def get_variables(self, variable_names=None): + ''' get variables from stream as dictionary + + Parameters + ---------- + variable_names : None or str or sequence of str, optional + variable name, or sequence of variable names to get from Mat file / + file stream. If None, then get all variables in file. + ''' + if isinstance(variable_names, str): + variable_names = [variable_names] + elif variable_names is not None: + variable_names = list(variable_names) + self.mat_stream.seek(0) + # set up variable reader + self.initialize_read() + mdict = {} + while not self.end_of_stream(): + hdr, next_position = self.read_var_header() + name = 'None' if hdr.name is None else hdr.name.decode('latin1') + if variable_names is not None and name not in variable_names: + self.mat_stream.seek(next_position) + continue + mdict[name] = self.read_var_array(hdr) + self.mat_stream.seek(next_position) + if variable_names is not None: + variable_names.remove(name) + if len(variable_names) == 0: + break + return mdict + + def list_variables(self): + ''' list variables from stream ''' + self.mat_stream.seek(0) + # set up variable reader + self.initialize_read() + vars = [] + while not self.end_of_stream(): + hdr, next_position = self.read_var_header() + name = 'None' if hdr.name is None else hdr.name.decode('latin1') + shape = self._matrix_reader.shape_from_header(hdr) + info = mclass_info.get(hdr.mclass, 'unknown') + vars.append((name, shape, info)) + + self.mat_stream.seek(next_position) + return vars + + +def arr_to_2d(arr, oned_as='row'): + ''' Make ``arr`` exactly two dimensional + + If `arr` has more than 2 dimensions, raise a ValueError + + Parameters + ---------- + arr : array + oned_as : {'row', 'column'}, optional + Whether to reshape 1-D vectors as row vectors or column vectors. + See documentation for ``matdims`` for more detail + + Returns + ------- + arr2d : array + 2-D version of the array + ''' + dims = matdims(arr, oned_as) + if len(dims) > 2: + raise ValueError('Matlab 4 files cannot save arrays with more than ' + '2 dimensions') + return arr.reshape(dims) + + +class VarWriter4: + def __init__(self, file_writer): + self.file_stream = file_writer.file_stream + self.oned_as = file_writer.oned_as + + def write_bytes(self, arr): + self.file_stream.write(arr.tobytes(order='F')) + + def write_string(self, s): + self.file_stream.write(s) + + def write_header(self, name, shape, P=miDOUBLE, T=mxFULL_CLASS, imagf=0): + ''' Write header for given data options + + Parameters + ---------- + name : str + name of variable + shape : sequence + Shape of array as it will be read in matlab + P : int, optional + code for mat4 data type, one of ``miDOUBLE, miSINGLE, miINT32, + miINT16, miUINT16, miUINT8`` + T : int, optional + code for mat4 matrix class, one of ``mxFULL_CLASS, mxCHAR_CLASS, + mxSPARSE_CLASS`` + imagf : int, optional + flag indicating complex + ''' + header = np.empty((), mdtypes_template['header']) + M = not SYS_LITTLE_ENDIAN + O = 0 + header['mopt'] = (M * 1000 + + O * 100 + + P * 10 + + T) + header['mrows'] = shape[0] + header['ncols'] = shape[1] + header['imagf'] = imagf + header['namlen'] = len(name) + 1 + self.write_bytes(header) + data = name + '\0' + self.write_string(data.encode('latin1')) + + def write(self, arr, name): + ''' Write matrix `arr`, with name `name` + + Parameters + ---------- + arr : array_like + array to write + name : str + name in matlab workspace + ''' + # we need to catch sparse first, because np.asarray returns an + # an object array for scipy.sparse + if scipy.sparse.issparse(arr): + self.write_sparse(arr, name) + return + arr = np.asarray(arr) + dt = arr.dtype + if not dt.isnative: + arr = arr.astype(dt.newbyteorder('=')) + dtt = dt.type + if dtt is np.object_: + raise TypeError('Cannot save object arrays in Mat4') + elif dtt is np.void: + raise TypeError('Cannot save void type arrays') + elif dtt in (np.str_, np.bytes_): + self.write_char(arr, name) + return + self.write_numeric(arr, name) + + def write_numeric(self, arr, name): + arr = arr_to_2d(arr, self.oned_as) + imagf = arr.dtype.kind == 'c' + try: + P = np_to_mtypes[arr.dtype.str[1:]] + except KeyError: + if imagf: + arr = arr.astype('c128') + else: + arr = arr.astype('f8') + P = miDOUBLE + self.write_header(name, + arr.shape, + P=P, + T=mxFULL_CLASS, + imagf=imagf) + if imagf: + self.write_bytes(arr.real) + self.write_bytes(arr.imag) + else: + self.write_bytes(arr) + + def write_char(self, arr, name): + if arr.dtype.type == np.str_ and arr.dtype.itemsize != np.dtype('U1').itemsize: + arr = arr_to_chars(arr) + arr = arr_to_2d(arr, self.oned_as) + dims = arr.shape + self.write_header( + name, + dims, + P=miUINT8, + T=mxCHAR_CLASS) + if arr.dtype.kind == 'U': + # Recode unicode to latin1 + n_chars = math.prod(dims) + st_arr = np.ndarray(shape=(), + dtype=arr_dtype_number(arr, n_chars), + buffer=arr) + st = st_arr.item().encode('latin-1') + arr = np.ndarray(shape=dims, dtype='S1', buffer=st) + self.write_bytes(arr) + + def write_sparse(self, arr, name): + ''' Sparse matrices are 2-D + + See docstring for VarReader4.read_sparse_array + ''' + A = arr.tocoo() # convert to sparse COO format (ijv) + imagf = A.dtype.kind == 'c' + ijv = np.zeros((A.nnz + 1, 3+imagf), dtype='f8') + ijv[:-1,0] = A.row + ijv[:-1,1] = A.col + ijv[:-1,0:2] += 1 # 1 based indexing + if imagf: + ijv[:-1,2] = A.data.real + ijv[:-1,3] = A.data.imag + else: + ijv[:-1,2] = A.data + ijv[-1,0:2] = A.shape + self.write_header( + name, + ijv.shape, + P=miDOUBLE, + T=mxSPARSE_CLASS) + self.write_bytes(ijv) + + +class MatFile4Writer: + ''' Class for writing matlab 4 format files ''' + def __init__(self, file_stream, oned_as=None): + self.file_stream = file_stream + if oned_as is None: + oned_as = 'row' + self.oned_as = oned_as + self._matrix_writer = None + + def put_variables(self, mdict, write_header=None): + ''' Write variables in `mdict` to stream + + Parameters + ---------- + mdict : mapping + mapping with method ``items`` return name, contents pairs + where ``name`` which will appeak in the matlab workspace in + file load, and ``contents`` is something writeable to a + matlab file, such as a NumPy array. + write_header : {None, True, False} + If True, then write the matlab file header before writing the + variables. If None (the default) then write the file header + if we are at position 0 in the stream. By setting False + here, and setting the stream position to the end of the file, + you can append variables to a matlab file + ''' + # there is no header for a matlab 4 mat file, so we ignore the + # ``write_header`` input argument. It's there for compatibility + # with the matlab 5 version of this method + self._matrix_writer = VarWriter4(self) + for name, var in mdict.items(): + self._matrix_writer.write(var, name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio5.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio5.py new file mode 100644 index 0000000000000000000000000000000000000000..5c4ed0361a603e10338a8d494838ebd861f56cb8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio5.py @@ -0,0 +1,895 @@ +''' Classes for read / write of matlab (TM) 5 files + +The matfile specification last found here: + +https://www.mathworks.com/access/helpdesk/help/pdf_doc/matlab/matfile_format.pdf + +(as of December 5 2008) + +================================= + Note on functions and mat files +================================= + +The document above does not give any hints as to the storage of matlab +function handles, or anonymous function handles. I had, therefore, to +guess the format of matlab arrays of ``mxFUNCTION_CLASS`` and +``mxOPAQUE_CLASS`` by looking at example mat files. + +``mxFUNCTION_CLASS`` stores all types of matlab functions. It seems to +contain a struct matrix with a set pattern of fields. For anonymous +functions, a sub-fields of one of these fields seems to contain the +well-named ``mxOPAQUE_CLASS``. This seems to contain: + +* array flags as for any matlab matrix +* 3 int8 strings +* a matrix + +It seems that whenever the mat file contains a ``mxOPAQUE_CLASS`` +instance, there is also an un-named matrix (name == '') at the end of +the mat file. I'll call this the ``__function_workspace__`` matrix. + +When I saved two anonymous functions in a mat file, or appended another +anonymous function to the mat file, there was still only one +``__function_workspace__`` un-named matrix at the end, but larger than +that for a mat file with a single anonymous function, suggesting that +the workspaces for the two functions had been merged. + +The ``__function_workspace__`` matrix appears to be of double class +(``mxCLASS_DOUBLE``), but stored as uint8, the memory for which is in +the format of a mini .mat file, without the first 124 bytes of the file +header (the description and the subsystem_offset), but with the version +U2 bytes, and the S2 endian test bytes. There follow 4 zero bytes, +presumably for 8 byte padding, and then a series of ``miMATRIX`` +entries, as in a standard mat file. The ``miMATRIX`` entries appear to +be series of un-named (name == '') matrices, and may also contain arrays +of this same mini-mat format. + +I guess that: + +* saving an anonymous function back to a mat file will need the + associated ``__function_workspace__`` matrix saved as well for the + anonymous function to work correctly. +* appending to a mat file that has a ``__function_workspace__`` would + involve first pulling off this workspace, appending, checking whether + there were any more anonymous functions appended, and then somehow + merging the relevant workspaces, and saving at the end of the mat + file. + +The mat files I was playing with are in ``tests/data``: + +* sqr.mat +* parabola.mat +* some_functions.mat + +See ``tests/test_mio.py:test_mio_funcs.py`` for the debugging +script I was working with. + +Small fragments of current code adapted from matfile.py by Heiko +Henkelmann; parts of the code for simplify_cells=True adapted from +http://blog.nephics.com/2019/08/28/better-loadmat-for-scipy/. +''' + +import math +import os +import time +import sys +import zlib + +from io import BytesIO + +import warnings + +import numpy as np + +import scipy.sparse + +from ._byteordercodes import native_code, swapped_code + +from ._miobase import (MatFileReader, docfiller, matdims, read_dtype, + arr_to_chars, arr_dtype_number, MatWriteError, + MatReadError, MatReadWarning) + +# Reader object for matlab 5 format variables +from ._mio5_utils import VarReader5 + +# Constants and helper objects +from ._mio5_params import (MatlabObject, MatlabFunction, MDTYPES, NP_TO_MTYPES, + NP_TO_MXTYPES, miCOMPRESSED, miMATRIX, miINT8, + miUTF8, miUINT32, mxCELL_CLASS, mxSTRUCT_CLASS, + mxOBJECT_CLASS, mxCHAR_CLASS, mxSPARSE_CLASS, + mxDOUBLE_CLASS, mclass_info, mat_struct) + +from ._streams import ZlibInputStream + + +def _has_struct(elem): + """Determine if elem is an array and if first array item is a struct.""" + return (isinstance(elem, np.ndarray) and (elem.size > 0) and (elem.ndim > 0) and + isinstance(elem[0], mat_struct)) + + +def _inspect_cell_array(ndarray): + """Construct lists from cell arrays (loaded as numpy ndarrays), recursing + into items if they contain mat_struct objects.""" + elem_list = [] + for sub_elem in ndarray: + if isinstance(sub_elem, mat_struct): + elem_list.append(_matstruct_to_dict(sub_elem)) + elif _has_struct(sub_elem): + elem_list.append(_inspect_cell_array(sub_elem)) + else: + elem_list.append(sub_elem) + return elem_list + + +def _matstruct_to_dict(matobj): + """Construct nested dicts from mat_struct objects.""" + d = {} + for f in matobj._fieldnames: + elem = matobj.__dict__[f] + if isinstance(elem, mat_struct): + d[f] = _matstruct_to_dict(elem) + elif _has_struct(elem): + d[f] = _inspect_cell_array(elem) + else: + d[f] = elem + return d + + +def _simplify_cells(d): + """Convert mat objects in dict to nested dicts.""" + for key in d: + if isinstance(d[key], mat_struct): + d[key] = _matstruct_to_dict(d[key]) + elif _has_struct(d[key]): + d[key] = _inspect_cell_array(d[key]) + return d + + +class MatFile5Reader(MatFileReader): + ''' Reader for Mat 5 mat files + Adds the following attribute to base class + + uint16_codec - char codec to use for uint16 char arrays + (defaults to system default codec) + + Uses variable reader that has the following standard interface (see + abstract class in ``miobase``:: + + __init__(self, file_reader) + read_header(self) + array_from_header(self) + + and added interface:: + + set_stream(self, stream) + read_full_tag(self) + + ''' + @docfiller + def __init__(self, + mat_stream, + byte_order=None, + mat_dtype=False, + squeeze_me=False, + chars_as_strings=True, + matlab_compatible=False, + struct_as_record=True, + verify_compressed_data_integrity=True, + uint16_codec=None, + simplify_cells=False): + '''Initializer for matlab 5 file format reader + + %(matstream_arg)s + %(load_args)s + %(struct_arg)s + uint16_codec : {None, string} + Set codec to use for uint16 char arrays (e.g., 'utf-8'). + Use system default codec if None + ''' + super().__init__( + mat_stream, + byte_order, + mat_dtype, + squeeze_me, + chars_as_strings, + matlab_compatible, + struct_as_record, + verify_compressed_data_integrity, + simplify_cells) + # Set uint16 codec + if not uint16_codec: + uint16_codec = sys.getdefaultencoding() + self.uint16_codec = uint16_codec + # placeholders for readers - see initialize_read method + self._file_reader = None + self._matrix_reader = None + + def guess_byte_order(self): + ''' Guess byte order. + Sets stream pointer to 0''' + self.mat_stream.seek(126) + mi = self.mat_stream.read(2) + self.mat_stream.seek(0) + return mi == b'IM' and '<' or '>' + + def read_file_header(self): + ''' Read in mat 5 file header ''' + hdict = {} + hdr_dtype = MDTYPES[self.byte_order]['dtypes']['file_header'] + hdr = read_dtype(self.mat_stream, hdr_dtype) + hdict['__header__'] = hdr['description'].item().strip(b' \t\n\000') + v_major = hdr['version'] >> 8 + v_minor = hdr['version'] & 0xFF + hdict['__version__'] = '%d.%d' % (v_major, v_minor) + return hdict + + def initialize_read(self): + ''' Run when beginning read of variables + + Sets up readers from parameters in `self` + ''' + # reader for top level stream. We need this extra top-level + # reader because we use the matrix_reader object to contain + # compressed matrices (so they have their own stream) + self._file_reader = VarReader5(self) + # reader for matrix streams + self._matrix_reader = VarReader5(self) + + def read_var_header(self): + ''' Read header, return header, next position + + Header has to define at least .name and .is_global + + Parameters + ---------- + None + + Returns + ------- + header : object + object that can be passed to self.read_var_array, and that + has attributes .name and .is_global + next_position : int + position in stream of next variable + ''' + mdtype, byte_count = self._file_reader.read_full_tag() + if not byte_count > 0: + raise ValueError("Did not read any bytes") + next_pos = self.mat_stream.tell() + byte_count + if mdtype == miCOMPRESSED: + # Make new stream from compressed data + stream = ZlibInputStream(self.mat_stream, byte_count) + self._matrix_reader.set_stream(stream) + check_stream_limit = self.verify_compressed_data_integrity + mdtype, byte_count = self._matrix_reader.read_full_tag() + else: + check_stream_limit = False + self._matrix_reader.set_stream(self.mat_stream) + if not mdtype == miMATRIX: + raise TypeError('Expecting miMATRIX type here, got %d' % mdtype) + header = self._matrix_reader.read_header(check_stream_limit) + return header, next_pos + + def read_var_array(self, header, process=True): + ''' Read array, given `header` + + Parameters + ---------- + header : header object + object with fields defining variable header + process : {True, False} bool, optional + If True, apply recursive post-processing during loading of + array. + + Returns + ------- + arr : array + array with post-processing applied or not according to + `process`. + ''' + return self._matrix_reader.array_from_header(header, process) + + def get_variables(self, variable_names=None): + ''' get variables from stream as dictionary + + variable_names - optional list of variable names to get + + If variable_names is None, then get all variables in file + ''' + if isinstance(variable_names, str): + variable_names = [variable_names] + elif variable_names is not None: + variable_names = list(variable_names) + + self.mat_stream.seek(0) + # Here we pass all the parameters in self to the reading objects + self.initialize_read() + mdict = self.read_file_header() + mdict['__globals__'] = [] + while not self.end_of_stream(): + hdr, next_position = self.read_var_header() + name = 'None' if hdr.name is None else hdr.name.decode('latin1') + if name in mdict: + msg = ( + f'Duplicate variable name "{name}" in stream' + " - replacing previous with new\nConsider" + "scipy.io.matlab.varmats_from_mat to split " + "file into single variable files" + ) + warnings.warn(msg, MatReadWarning, stacklevel=2) + if name == '': + # can only be a matlab 7 function workspace + name = '__function_workspace__' + # We want to keep this raw because mat_dtype processing + # will break the format (uint8 as mxDOUBLE_CLASS) + process = False + else: + process = True + if variable_names is not None and name not in variable_names: + self.mat_stream.seek(next_position) + continue + try: + res = self.read_var_array(hdr, process) + except MatReadError as err: + warnings.warn( + f'Unreadable variable "{name}", because "{err}"', + Warning, stacklevel=2) + res = f"Read error: {err}" + self.mat_stream.seek(next_position) + mdict[name] = res + if hdr.is_global: + mdict['__globals__'].append(name) + if variable_names is not None: + variable_names.remove(name) + if len(variable_names) == 0: + break + if self.simplify_cells: + return _simplify_cells(mdict) + else: + return mdict + + def list_variables(self): + ''' list variables from stream ''' + self.mat_stream.seek(0) + # Here we pass all the parameters in self to the reading objects + self.initialize_read() + self.read_file_header() + vars = [] + while not self.end_of_stream(): + hdr, next_position = self.read_var_header() + name = 'None' if hdr.name is None else hdr.name.decode('latin1') + if name == '': + # can only be a matlab 7 function workspace + name = '__function_workspace__' + + shape = self._matrix_reader.shape_from_header(hdr) + if hdr.is_logical: + info = 'logical' + else: + info = mclass_info.get(hdr.mclass, 'unknown') + vars.append((name, shape, info)) + + self.mat_stream.seek(next_position) + return vars + + +def varmats_from_mat(file_obj): + """ Pull variables out of mat 5 file as a sequence of mat file objects + + This can be useful with a difficult mat file, containing unreadable + variables. This routine pulls the variables out in raw form and puts them, + unread, back into a file stream for saving or reading. Another use is the + pathological case where there is more than one variable of the same name in + the file; this routine returns the duplicates, whereas the standard reader + will overwrite duplicates in the returned dictionary. + + The file pointer in `file_obj` will be undefined. File pointers for the + returned file-like objects are set at 0. + + Parameters + ---------- + file_obj : file-like + file object containing mat file + + Returns + ------- + named_mats : list + list contains tuples of (name, BytesIO) where BytesIO is a file-like + object containing mat file contents as for a single variable. The + BytesIO contains a string with the original header and a single var. If + ``var_file_obj`` is an individual BytesIO instance, then save as a mat + file with something like ``open('test.mat', + 'wb').write(var_file_obj.read())`` + + Examples + -------- + >>> import scipy.io + >>> import numpy as np + >>> from io import BytesIO + >>> from scipy.io.matlab._mio5 import varmats_from_mat + >>> mat_fileobj = BytesIO() + >>> scipy.io.savemat(mat_fileobj, {'b': np.arange(10), 'a': 'a string'}) + >>> varmats = varmats_from_mat(mat_fileobj) + >>> sorted([name for name, str_obj in varmats]) + ['a', 'b'] + """ + rdr = MatFile5Reader(file_obj) + file_obj.seek(0) + # Raw read of top-level file header + hdr_len = MDTYPES[native_code]['dtypes']['file_header'].itemsize + raw_hdr = file_obj.read(hdr_len) + # Initialize variable reading + file_obj.seek(0) + rdr.initialize_read() + rdr.read_file_header() + next_position = file_obj.tell() + named_mats = [] + while not rdr.end_of_stream(): + start_position = next_position + hdr, next_position = rdr.read_var_header() + name = 'None' if hdr.name is None else hdr.name.decode('latin1') + # Read raw variable string + file_obj.seek(start_position) + byte_count = next_position - start_position + var_str = file_obj.read(byte_count) + # write to stringio object + out_obj = BytesIO() + out_obj.write(raw_hdr) + out_obj.write(var_str) + out_obj.seek(0) + named_mats.append((name, out_obj)) + return named_mats + + +class EmptyStructMarker: + """ Class to indicate presence of empty matlab struct on output """ + + +def to_writeable(source): + ''' Convert input object ``source`` to something we can write + + Parameters + ---------- + source : object + + Returns + ------- + arr : None or ndarray or EmptyStructMarker + If `source` cannot be converted to something we can write to a matfile, + return None. If `source` is equivalent to an empty dictionary, return + ``EmptyStructMarker``. Otherwise return `source` converted to an + ndarray with contents for writing to matfile. + ''' + if isinstance(source, np.ndarray): + return source + if source is None: + return None + if hasattr(source, "__array__"): + return np.asarray(source) + # Objects that implement mappings + is_mapping = (hasattr(source, 'keys') and hasattr(source, 'values') and + hasattr(source, 'items')) + # Objects that don't implement mappings, but do have dicts + if isinstance(source, np.generic): + # NumPy scalars are never mappings (PyPy issue workaround) + pass + elif not is_mapping and hasattr(source, '__dict__'): + source = {key: value for key, value in source.__dict__.items() + if not key.startswith('_')} + is_mapping = True + if is_mapping: + dtype = [] + values = [] + for field, value in source.items(): + if (isinstance(field, str) and + field[0] not in '_0123456789'): + dtype.append((str(field), object)) + values.append(value) + if dtype: + return np.array([tuple(values)], dtype) + else: + return EmptyStructMarker + # Next try and convert to an array + try: + narr = np.asanyarray(source) + except ValueError: + narr = np.asanyarray(source, dtype=object) + if narr.dtype.type in (object, np.object_) and \ + narr.shape == () and narr == source: + # No interesting conversion possible + return None + return narr + + +# Native byte ordered dtypes for convenience for writers +NDT_FILE_HDR = MDTYPES[native_code]['dtypes']['file_header'] +NDT_TAG_FULL = MDTYPES[native_code]['dtypes']['tag_full'] +NDT_TAG_SMALL = MDTYPES[native_code]['dtypes']['tag_smalldata'] +NDT_ARRAY_FLAGS = MDTYPES[native_code]['dtypes']['array_flags'] + + +class VarWriter5: + ''' Generic matlab matrix writing class ''' + mat_tag = np.zeros((), NDT_TAG_FULL) + mat_tag['mdtype'] = miMATRIX # type: ignore[call-overload] + + def __init__(self, file_writer): + self.file_stream = file_writer.file_stream + self.unicode_strings = file_writer.unicode_strings + self.long_field_names = file_writer.long_field_names + self.oned_as = file_writer.oned_as + # These are used for top level writes, and unset after + self._var_name = None + self._var_is_global = False + + def write_bytes(self, arr): + self.file_stream.write(arr.tobytes(order='F')) + + def write_string(self, s): + self.file_stream.write(s) + + def write_element(self, arr, mdtype=None): + ''' write tag and data ''' + if mdtype is None: + mdtype = NP_TO_MTYPES[arr.dtype.str[1:]] + # Array needs to be in native byte order + if arr.dtype.byteorder == swapped_code: + arr = arr.byteswap().view(arr.dtype.newbyteorder()) + byte_count = arr.size*arr.itemsize + if byte_count <= 4: + self.write_smalldata_element(arr, mdtype, byte_count) + else: + self.write_regular_element(arr, mdtype, byte_count) + + def write_smalldata_element(self, arr, mdtype, byte_count): + # write tag with embedded data + tag = np.zeros((), NDT_TAG_SMALL) + tag['byte_count_mdtype'] = (byte_count << 16) + mdtype + # if arr.tobytes is < 4, the element will be zero-padded as needed. + tag['data'] = arr.tobytes(order='F') + self.write_bytes(tag) + + def write_regular_element(self, arr, mdtype, byte_count): + # write tag, data + tag = np.zeros((), NDT_TAG_FULL) + tag['mdtype'] = mdtype + tag['byte_count'] = byte_count + self.write_bytes(tag) + self.write_bytes(arr) + # pad to next 64-bit boundary + bc_mod_8 = byte_count % 8 + if bc_mod_8: + self.file_stream.write(b'\x00' * (8-bc_mod_8)) + + def write_header(self, + shape, + mclass, + is_complex=False, + is_logical=False, + nzmax=0): + ''' Write header for given data options + shape : sequence + array shape + mclass - mat5 matrix class + is_complex - True if matrix is complex + is_logical - True if matrix is logical + nzmax - max non zero elements for sparse arrays + + We get the name and the global flag from the object, and reset + them to defaults after we've used them + ''' + # get name and is_global from one-shot object store + name = self._var_name + is_global = self._var_is_global + # initialize the top-level matrix tag, store position + self._mat_tag_pos = self.file_stream.tell() + self.write_bytes(self.mat_tag) + # write array flags (complex, global, logical, class, nzmax) + af = np.zeros((), NDT_ARRAY_FLAGS) + af['data_type'] = miUINT32 + af['byte_count'] = 8 + flags = is_complex << 3 | is_global << 2 | is_logical << 1 + af['flags_class'] = mclass | flags << 8 + af['nzmax'] = nzmax + self.write_bytes(af) + # shape + self.write_element(np.array(shape, dtype='i4')) + # write name + name = np.asarray(name) + if name == '': # empty string zero-terminated + self.write_smalldata_element(name, miINT8, 0) + else: + self.write_element(name, miINT8) + # reset the one-shot store to defaults + self._var_name = '' + self._var_is_global = False + + def update_matrix_tag(self, start_pos): + curr_pos = self.file_stream.tell() + self.file_stream.seek(start_pos) + byte_count = curr_pos - start_pos - 8 + if byte_count >= 2**32: + raise MatWriteError("Matrix too large to save with Matlab " + "5 format") + self.mat_tag['byte_count'] = byte_count + self.write_bytes(self.mat_tag) + self.file_stream.seek(curr_pos) + + def write_top(self, arr, name, is_global): + """ Write variable at top level of mat file + + Parameters + ---------- + arr : array_like + array-like object to create writer for + name : str, optional + name as it will appear in matlab workspace + default is empty string + is_global : {False, True}, optional + whether variable will be global on load into matlab + """ + # these are set before the top-level header write, and unset at + # the end of the same write, because they do not apply for lower levels + self._var_is_global = is_global + self._var_name = name + # write the header and data + self.write(arr) + + def write(self, arr): + ''' Write `arr` to stream at top and sub levels + + Parameters + ---------- + arr : array_like + array-like object to create writer for + ''' + # store position, so we can update the matrix tag + mat_tag_pos = self.file_stream.tell() + # First check if these are sparse + if scipy.sparse.issparse(arr): + self.write_sparse(arr) + self.update_matrix_tag(mat_tag_pos) + return + # Try to convert things that aren't arrays + narr = to_writeable(arr) + if narr is None: + raise TypeError(f'Could not convert {arr} (type {type(arr)}) to array') + if isinstance(narr, MatlabObject): + self.write_object(narr) + elif isinstance(narr, MatlabFunction): + raise MatWriteError('Cannot write matlab functions') + elif narr is EmptyStructMarker: # empty struct array + self.write_empty_struct() + elif narr.dtype.fields: # struct array + self.write_struct(narr) + elif narr.dtype.hasobject: # cell array + self.write_cells(narr) + elif narr.dtype.kind in ('U', 'S'): + if self.unicode_strings: + codec = 'UTF8' + else: + codec = 'ascii' + self.write_char(narr, codec) + else: + self.write_numeric(narr) + self.update_matrix_tag(mat_tag_pos) + + def write_numeric(self, arr): + imagf = arr.dtype.kind == 'c' + logif = arr.dtype.kind == 'b' + try: + mclass = NP_TO_MXTYPES[arr.dtype.str[1:]] + except KeyError: + # No matching matlab type, probably complex256 / float128 / float96 + # Cast data to complex128 / float64. + if imagf: + arr = arr.astype('c128') + elif logif: + arr = arr.astype('i1') # Should only contain 0/1 + else: + arr = arr.astype('f8') + mclass = mxDOUBLE_CLASS + self.write_header(matdims(arr, self.oned_as), + mclass, + is_complex=imagf, + is_logical=logif) + if imagf: + self.write_element(arr.real) + self.write_element(arr.imag) + else: + self.write_element(arr) + + def write_char(self, arr, codec='ascii'): + ''' Write string array `arr` with given `codec` + ''' + if arr.size == 0 or np.all(arr == ''): + # This an empty string array or a string array containing + # only empty strings. Matlab cannot distinguish between a + # string array that is empty, and a string array containing + # only empty strings, because it stores strings as arrays of + # char. There is no way of having an array of char that is + # not empty, but contains an empty string. We have to + # special-case the array-with-empty-strings because even + # empty strings have zero padding, which would otherwise + # appear in matlab as a string with a space. + shape = (0,) * np.max([arr.ndim, 2]) + self.write_header(shape, mxCHAR_CLASS) + self.write_smalldata_element(arr, miUTF8, 0) + return + # non-empty string. + # + # Convert to char array + arr = arr_to_chars(arr) + # We have to write the shape directly, because we are going + # recode the characters, and the resulting stream of chars + # may have a different length + shape = arr.shape + self.write_header(shape, mxCHAR_CLASS) + if arr.dtype.kind == 'U' and arr.size: + # Make one long string from all the characters. We need to + # transpose here, because we're flattening the array, before + # we write the bytes. The bytes have to be written in + # Fortran order. + n_chars = math.prod(shape) + st_arr = np.ndarray(shape=(), + dtype=arr_dtype_number(arr, n_chars), + buffer=arr.T.copy()) # Fortran order + # Recode with codec to give byte string + st = st_arr.item().encode(codec) + # Reconstruct as 1-D byte array + arr = np.ndarray(shape=(len(st),), + dtype='S1', + buffer=st) + self.write_element(arr, mdtype=miUTF8) + + def write_sparse(self, arr): + ''' Sparse matrices are 2D + ''' + A = arr.tocsc() # convert to sparse CSC format + A.sort_indices() # MATLAB expects sorted row indices + is_complex = (A.dtype.kind == 'c') + is_logical = (A.dtype.kind == 'b') + nz = A.nnz + self.write_header(matdims(arr, self.oned_as), + mxSPARSE_CLASS, + is_complex=is_complex, + is_logical=is_logical, + # matlab won't load file with 0 nzmax + nzmax=1 if nz == 0 else nz) + self.write_element(A.indices.astype('i4')) + self.write_element(A.indptr.astype('i4')) + self.write_element(A.data.real) + if is_complex: + self.write_element(A.data.imag) + + def write_cells(self, arr): + self.write_header(matdims(arr, self.oned_as), + mxCELL_CLASS) + # loop over data, column major + A = np.atleast_2d(arr).flatten('F') + for el in A: + self.write(el) + + def write_empty_struct(self): + self.write_header((1, 1), mxSTRUCT_CLASS) + # max field name length set to 1 in an example matlab struct + self.write_element(np.array(1, dtype=np.int32)) + # Field names element is empty + self.write_element(np.array([], dtype=np.int8)) + + def write_struct(self, arr): + self.write_header(matdims(arr, self.oned_as), + mxSTRUCT_CLASS) + self._write_items(arr) + + def _write_items(self, arr): + # write fieldnames + fieldnames = [f[0] for f in arr.dtype.descr] + length = max([len(fieldname) for fieldname in fieldnames])+1 + max_length = (self.long_field_names and 64) or 32 + if length > max_length: + raise ValueError("Field names are restricted to %d characters" % + (max_length-1)) + self.write_element(np.array([length], dtype='i4')) + self.write_element( + np.array(fieldnames, dtype='S%d' % (length)), + mdtype=miINT8) + A = np.atleast_2d(arr).flatten('F') + for el in A: + for f in fieldnames: + self.write(el[f]) + + def write_object(self, arr): + '''Same as writing structs, except different mx class, and extra + classname element after header + ''' + self.write_header(matdims(arr, self.oned_as), + mxOBJECT_CLASS) + self.write_element(np.array(arr.classname, dtype='S'), + mdtype=miINT8) + self._write_items(arr) + + +class MatFile5Writer: + ''' Class for writing mat5 files ''' + + @docfiller + def __init__(self, file_stream, + do_compression=False, + unicode_strings=False, + global_vars=None, + long_field_names=False, + oned_as='row'): + ''' Initialize writer for matlab 5 format files + + Parameters + ---------- + %(do_compression)s + %(unicode_strings)s + global_vars : None or sequence of strings, optional + Names of variables to be marked as global for matlab + %(long_fields)s + %(oned_as)s + ''' + self.file_stream = file_stream + self.do_compression = do_compression + self.unicode_strings = unicode_strings + if global_vars: + self.global_vars = global_vars + else: + self.global_vars = [] + self.long_field_names = long_field_names + self.oned_as = oned_as + self._matrix_writer = None + + def write_file_header(self): + # write header + hdr = np.zeros((), NDT_FILE_HDR) + hdr['description'] = (f'MATLAB 5.0 MAT-file Platform: {os.name}, ' + f'Created on: {time.asctime()}') + hdr['version'] = 0x0100 + hdr['endian_test'] = np.ndarray(shape=(), + dtype='S2', + buffer=np.uint16(0x4d49)) + self.file_stream.write(hdr.tobytes()) + + def put_variables(self, mdict, write_header=None): + ''' Write variables in `mdict` to stream + + Parameters + ---------- + mdict : mapping + mapping with method ``items`` returns name, contents pairs where + ``name`` which will appear in the matlab workspace in file load, and + ``contents`` is something writeable to a matlab file, such as a NumPy + array. + write_header : {None, True, False}, optional + If True, then write the matlab file header before writing the + variables. If None (the default) then write the file header + if we are at position 0 in the stream. By setting False + here, and setting the stream position to the end of the file, + you can append variables to a matlab file + ''' + # write header if requested, or None and start of file + if write_header is None: + write_header = self.file_stream.tell() == 0 + if write_header: + self.write_file_header() + self._matrix_writer = VarWriter5(self) + for name, var in mdict.items(): + if name[0] == '_': + continue + is_global = name in self.global_vars + if self.do_compression: + stream = BytesIO() + self._matrix_writer.file_stream = stream + self._matrix_writer.write_top(var, name.encode('latin1'), is_global) + out_str = zlib.compress(stream.getvalue()) + tag = np.empty((), NDT_TAG_FULL) + tag['mdtype'] = miCOMPRESSED + tag['byte_count'] = len(out_str) + self.file_stream.write(tag.tobytes()) + self.file_stream.write(out_str) + else: # not compressing + self._matrix_writer.write_top(var, name.encode('latin1'), is_global) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio5_params.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio5_params.py new file mode 100644 index 0000000000000000000000000000000000000000..0d60b8e7a4a2dd1e6a336139f67ce984743e27bb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio5_params.py @@ -0,0 +1,281 @@ +''' Constants and classes for matlab 5 read and write + +See also mio5_utils.pyx where these same constants arise as c enums. + +If you make changes in this file, don't forget to change mio5_utils.pyx +''' +import numpy as np + +from ._miobase import convert_dtypes + + +__all__ = [ + 'MDTYPES', 'MatlabFunction', 'MatlabObject', 'MatlabOpaque', + 'NP_TO_MTYPES', 'NP_TO_MXTYPES', 'OPAQUE_DTYPE', 'codecs_template', + 'mat_struct', 'mclass_dtypes_template', 'mclass_info', 'mdtypes_template', + 'miCOMPRESSED', 'miDOUBLE', 'miINT16', 'miINT32', 'miINT64', 'miINT8', + 'miMATRIX', 'miSINGLE', 'miUINT16', 'miUINT32', 'miUINT64', 'miUINT8', + 'miUTF16', 'miUTF32', 'miUTF8', 'mxCELL_CLASS', 'mxCHAR_CLASS', + 'mxDOUBLE_CLASS', 'mxFUNCTION_CLASS', 'mxINT16_CLASS', 'mxINT32_CLASS', + 'mxINT64_CLASS', 'mxINT8_CLASS', 'mxOBJECT_CLASS', + 'mxOBJECT_CLASS_FROM_MATRIX_H', 'mxOPAQUE_CLASS', 'mxSINGLE_CLASS', + 'mxSPARSE_CLASS', 'mxSTRUCT_CLASS', 'mxUINT16_CLASS', 'mxUINT32_CLASS', + 'mxUINT64_CLASS', 'mxUINT8_CLASS' +] +miINT8 = 1 +miUINT8 = 2 +miINT16 = 3 +miUINT16 = 4 +miINT32 = 5 +miUINT32 = 6 +miSINGLE = 7 +miDOUBLE = 9 +miINT64 = 12 +miUINT64 = 13 +miMATRIX = 14 +miCOMPRESSED = 15 +miUTF8 = 16 +miUTF16 = 17 +miUTF32 = 18 + +mxCELL_CLASS = 1 +mxSTRUCT_CLASS = 2 +# The March 2008 edition of "Matlab 7 MAT-File Format" says that +# mxOBJECT_CLASS = 3, whereas matrix.h says that mxLOGICAL = 3. +# Matlab 2008a appears to save logicals as type 9, so we assume that +# the document is correct. See type 18, below. +mxOBJECT_CLASS = 3 +mxCHAR_CLASS = 4 +mxSPARSE_CLASS = 5 +mxDOUBLE_CLASS = 6 +mxSINGLE_CLASS = 7 +mxINT8_CLASS = 8 +mxUINT8_CLASS = 9 +mxINT16_CLASS = 10 +mxUINT16_CLASS = 11 +mxINT32_CLASS = 12 +mxUINT32_CLASS = 13 +# The following are not in the March 2008 edition of "Matlab 7 +# MAT-File Format," but were guessed from matrix.h. +mxINT64_CLASS = 14 +mxUINT64_CLASS = 15 +mxFUNCTION_CLASS = 16 +# Not doing anything with these at the moment. +mxOPAQUE_CLASS = 17 # This appears to be a function workspace +# Thread 'saving/loading symbol table of annymous functions', +# octave-maintainers, April-May 2007 +# https://lists.gnu.org/archive/html/octave-maintainers/2007-04/msg00031.html +# https://lists.gnu.org/archive/html/octave-maintainers/2007-05/msg00032.html +# (Was/Deprecated: https://www-old.cae.wisc.edu/pipermail/octave-maintainers/2007-May/002824.html) +mxOBJECT_CLASS_FROM_MATRIX_H = 18 + +mdtypes_template = { + miINT8: 'i1', + miUINT8: 'u1', + miINT16: 'i2', + miUINT16: 'u2', + miINT32: 'i4', + miUINT32: 'u4', + miSINGLE: 'f4', + miDOUBLE: 'f8', + miINT64: 'i8', + miUINT64: 'u8', + miUTF8: 'u1', + miUTF16: 'u2', + miUTF32: 'u4', + 'file_header': [('description', 'S116'), + ('subsystem_offset', 'i8'), + ('version', 'u2'), + ('endian_test', 'S2')], + 'tag_full': [('mdtype', 'u4'), ('byte_count', 'u4')], + 'tag_smalldata':[('byte_count_mdtype', 'u4'), ('data', 'S4')], + 'array_flags': [('data_type', 'u4'), + ('byte_count', 'u4'), + ('flags_class','u4'), + ('nzmax', 'u4')], + 'U1': 'U1', + } + +mclass_dtypes_template = { + mxINT8_CLASS: 'i1', + mxUINT8_CLASS: 'u1', + mxINT16_CLASS: 'i2', + mxUINT16_CLASS: 'u2', + mxINT32_CLASS: 'i4', + mxUINT32_CLASS: 'u4', + mxINT64_CLASS: 'i8', + mxUINT64_CLASS: 'u8', + mxSINGLE_CLASS: 'f4', + mxDOUBLE_CLASS: 'f8', + } + +mclass_info = { + mxINT8_CLASS: 'int8', + mxUINT8_CLASS: 'uint8', + mxINT16_CLASS: 'int16', + mxUINT16_CLASS: 'uint16', + mxINT32_CLASS: 'int32', + mxUINT32_CLASS: 'uint32', + mxINT64_CLASS: 'int64', + mxUINT64_CLASS: 'uint64', + mxSINGLE_CLASS: 'single', + mxDOUBLE_CLASS: 'double', + mxCELL_CLASS: 'cell', + mxSTRUCT_CLASS: 'struct', + mxOBJECT_CLASS: 'object', + mxCHAR_CLASS: 'char', + mxSPARSE_CLASS: 'sparse', + mxFUNCTION_CLASS: 'function', + mxOPAQUE_CLASS: 'opaque', + } + +NP_TO_MTYPES = { + 'f8': miDOUBLE, + 'c32': miDOUBLE, + 'c24': miDOUBLE, + 'c16': miDOUBLE, + 'f4': miSINGLE, + 'c8': miSINGLE, + 'i8': miINT64, + 'i4': miINT32, + 'i2': miINT16, + 'i1': miINT8, + 'u8': miUINT64, + 'u4': miUINT32, + 'u2': miUINT16, + 'u1': miUINT8, + 'S1': miUINT8, + 'U1': miUTF16, + 'b1': miUINT8, # not standard but seems MATLAB uses this (gh-4022) + } + + +NP_TO_MXTYPES = { + 'f8': mxDOUBLE_CLASS, + 'c32': mxDOUBLE_CLASS, + 'c24': mxDOUBLE_CLASS, + 'c16': mxDOUBLE_CLASS, + 'f4': mxSINGLE_CLASS, + 'c8': mxSINGLE_CLASS, + 'i8': mxINT64_CLASS, + 'i4': mxINT32_CLASS, + 'i2': mxINT16_CLASS, + 'i1': mxINT8_CLASS, + 'u8': mxUINT64_CLASS, + 'u4': mxUINT32_CLASS, + 'u2': mxUINT16_CLASS, + 'u1': mxUINT8_CLASS, + 'S1': mxUINT8_CLASS, + 'b1': mxUINT8_CLASS, # not standard but seems MATLAB uses this + } + +''' Before release v7.1 (release 14) matlab (TM) used the system +default character encoding scheme padded out to 16-bits. Release 14 +and later use Unicode. When saving character data, R14 checks if it +can be encoded in 7-bit ascii, and saves in that format if so.''' + +codecs_template = { + miUTF8: {'codec': 'utf_8', 'width': 1}, + miUTF16: {'codec': 'utf_16', 'width': 2}, + miUTF32: {'codec': 'utf_32','width': 4}, + } + + +def _convert_codecs(template, byte_order): + ''' Convert codec template mapping to byte order + + Set codecs not on this system to None + + Parameters + ---------- + template : mapping + key, value are respectively codec name, and root name for codec + (without byte order suffix) + byte_order : {'<', '>'} + code for little or big endian + + Returns + ------- + codecs : dict + key, value are name, codec (as in .encode(codec)) + ''' + codecs = {} + postfix = byte_order == '<' and '_le' or '_be' + for k, v in template.items(): + codec = v['codec'] + try: + " ".encode(codec) + except LookupError: + codecs[k] = None + continue + if v['width'] > 1: + codec += postfix + codecs[k] = codec + return codecs.copy() + + +MDTYPES = {} +for _bytecode in '<>': + _def = {'dtypes': convert_dtypes(mdtypes_template, _bytecode), + 'classes': convert_dtypes(mclass_dtypes_template, _bytecode), + 'codecs': _convert_codecs(codecs_template, _bytecode)} + MDTYPES[_bytecode] = _def + + +class mat_struct: + """Placeholder for holding read data from structs. + + We use instances of this class when the user passes False as a value to the + ``struct_as_record`` parameter of the :func:`scipy.io.loadmat` function. + """ + pass + + +class MatlabObject(np.ndarray): + """Subclass of ndarray to signal this is a matlab object. + + This is a simple subclass of :class:`numpy.ndarray` meant to be used + by :func:`scipy.io.loadmat` and should not be instantiated directly. + """ + + def __new__(cls, input_array, classname=None): + # Input array is an already formed ndarray instance + # We first cast to be our class type + obj = np.asarray(input_array).view(cls) + # add the new attribute to the created instance + obj.classname = classname + # Finally, we must return the newly created object: + return obj + + def __array_finalize__(self,obj): + # reset the attribute from passed original object + self.classname = getattr(obj, 'classname', None) + # We do not need to return anything + + +class MatlabFunction(np.ndarray): + """Subclass for a MATLAB function. + + This is a simple subclass of :class:`numpy.ndarray` meant to be used + by :func:`scipy.io.loadmat` and should not be directly instantiated. + """ + + def __new__(cls, input_array): + obj = np.asarray(input_array).view(cls) + return obj + + +class MatlabOpaque(np.ndarray): + """Subclass for a MATLAB opaque matrix. + + This is a simple subclass of :class:`numpy.ndarray` meant to be used + by :func:`scipy.io.loadmat` and should not be directly instantiated. + """ + + def __new__(cls, input_array): + obj = np.asarray(input_array).view(cls) + return obj + + +OPAQUE_DTYPE = np.dtype( + [('s0', 'O'), ('s1', 'O'), ('s2', 'O'), ('arr', 'O')]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio_utils.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio_utils.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..fc8a904d06026119fb14be7ced171f184d44e58b Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_mio_utils.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_miobase.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_miobase.py new file mode 100644 index 0000000000000000000000000000000000000000..1ad7fd7395bb67cc2926b03ed7a6002dc4f9e3f6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/_miobase.py @@ -0,0 +1,432 @@ +# Authors: Travis Oliphant, Matthew Brett + +""" +Base classes for MATLAB file stream reading. + +MATLAB is a registered trademark of the Mathworks inc. +""" + +from typing import Final + +import numpy as np +from scipy._lib import doccer + +from . import _byteordercodes as boc + +__all__ = [ + 'MatReadError', 'MatReadWarning', 'MatWriteError', +] + +class MatReadError(Exception): + """Exception indicating a read issue.""" + + +class MatWriteError(Exception): + """Exception indicating a write issue.""" + + +class MatReadWarning(UserWarning): + """Warning class for read issues.""" + + +doc_dict = \ + {'file_arg': + '''file_name : str + Name of the mat file (do not need .mat extension if + appendmat==True) Can also pass open file-like object.''', + 'append_arg': + '''appendmat : bool, optional + True to append the .mat extension to the end of the given + filename, if not already present. Default is True.''', + 'load_args': + '''byte_order : str or None, optional + None by default, implying byte order guessed from mat + file. Otherwise can be one of ('native', '=', 'little', '<', + 'BIG', '>'). +mat_dtype : bool, optional + If True, return arrays in same dtype as would be loaded into + MATLAB (instead of the dtype with which they are saved). +squeeze_me : bool, optional + Whether to squeeze unit matrix dimensions or not. +chars_as_strings : bool, optional + Whether to convert char arrays to string arrays. +matlab_compatible : bool, optional + Returns matrices as would be loaded by MATLAB (implies + squeeze_me=False, chars_as_strings=False, mat_dtype=True, + struct_as_record=True).''', + 'struct_arg': + '''struct_as_record : bool, optional + Whether to load MATLAB structs as NumPy record arrays, or as + old-style NumPy arrays with dtype=object. Setting this flag to + False replicates the behavior of SciPy version 0.7.x (returning + numpy object arrays). The default setting is True, because it + allows easier round-trip load and save of MATLAB files.''', + 'matstream_arg': + '''mat_stream : file-like + Object with file API, open for reading.''', + 'long_fields': + '''long_field_names : bool, optional + * False - maximum field name length in a structure is 31 characters + which is the documented maximum length. This is the default. + * True - maximum field name length in a structure is 63 characters + which works for MATLAB 7.6''', + 'do_compression': + '''do_compression : bool, optional + Whether to compress matrices on write. Default is False.''', + 'oned_as': + '''oned_as : {'row', 'column'}, optional + If 'column', write 1-D NumPy arrays as column vectors. + If 'row', write 1D NumPy arrays as row vectors.''', + 'unicode_strings': + '''unicode_strings : bool, optional + If True, write strings as Unicode, else MATLAB usual encoding.'''} + +docfiller: Final = doccer.filldoc(doc_dict) + +''' + + Note on architecture +====================== + +There are three sets of parameters relevant for reading files. The +first are *file read parameters* - containing options that are common +for reading the whole file, and therefore every variable within that +file. At the moment these are: + +* mat_stream +* dtypes (derived from byte code) +* byte_order +* chars_as_strings +* squeeze_me +* struct_as_record (MATLAB 5 files) +* class_dtypes (derived from order code, MATLAB 5 files) +* codecs (MATLAB 5 files) +* uint16_codec (MATLAB 5 files) + +Another set of parameters are those that apply only to the current +variable being read - the *header*: + +* header related variables (different for v4 and v5 mat files) +* is_complex +* mclass +* var_stream + +With the header, we need ``next_position`` to tell us where the next +variable in the stream is. + +Then, for each element in a matrix, there can be *element read +parameters*. An element is, for example, one element in a MATLAB cell +array. At the moment, these are: + +* mat_dtype + +The file-reading object contains the *file read parameters*. The +*header* is passed around as a data object, or may be read and discarded +in a single function. The *element read parameters* - the mat_dtype in +this instance, is passed into a general post-processing function - see +``mio_utils`` for details. +''' + + +def convert_dtypes(dtype_template, order_code): + ''' Convert dtypes in mapping to given order + + Parameters + ---------- + dtype_template : mapping + mapping with values returning numpy dtype from ``np.dtype(val)`` + order_code : str + an order code suitable for using in ``dtype.newbyteorder()`` + + Returns + ------- + dtypes : mapping + mapping where values have been replaced by + ``np.dtype(val).newbyteorder(order_code)`` + + ''' + dtypes = dtype_template.copy() + for k in dtypes: + dtypes[k] = np.dtype(dtypes[k]).newbyteorder(order_code) + return dtypes + + +def read_dtype(mat_stream, a_dtype): + """ + Generic get of byte stream data of known type + + Parameters + ---------- + mat_stream : file_like object + MATLAB (tm) mat file stream + a_dtype : dtype + dtype of array to read. `a_dtype` is assumed to be correct + endianness. + + Returns + ------- + arr : ndarray + Array of dtype `a_dtype` read from stream. + + """ + num_bytes = a_dtype.itemsize + arr = np.ndarray(shape=(), + dtype=a_dtype, + buffer=mat_stream.read(num_bytes), + order='F') + return arr + + +def matfile_version(file_name, *, appendmat=True): + """ + Return major, minor tuple depending on apparent mat file type + + Where: + + #. 0,x -> version 4 format mat files + #. 1,x -> version 5 format mat files + #. 2,x -> version 7.3 format mat files (HDF format) + + Parameters + ---------- + file_name : str + Name of the mat file (do not need .mat extension if + appendmat==True). Can also pass open file-like object. + appendmat : bool, optional + True to append the .mat extension to the end of the given + filename, if not already present. Default is True. + + Returns + ------- + major_version : {0, 1, 2} + major MATLAB File format version + minor_version : int + minor MATLAB file format version + + Raises + ------ + MatReadError + If the file is empty. + ValueError + The matfile version is unknown. + + Notes + ----- + Has the side effect of setting the file read pointer to 0 + """ + from ._mio import _open_file_context + with _open_file_context(file_name, appendmat=appendmat) as fileobj: + return _get_matfile_version(fileobj) + + +get_matfile_version = matfile_version + + +_HDR_N_BYTES = 20 + + +def _get_matfile_version(fileobj): + # Mat4 files have a zero somewhere in first 4 bytes + fileobj.seek(0) + hdr_bytes = fileobj.read(_HDR_N_BYTES) + if len(hdr_bytes) < _HDR_N_BYTES: + raise MatReadError("Mat file appears to be truncated") + if hdr_bytes.count(0) == _HDR_N_BYTES: + raise MatReadError("Mat file appears to be corrupt " + f"(first {_HDR_N_BYTES} bytes == 0)") + mopt_ints = np.ndarray(shape=(4,), dtype=np.uint8, buffer=hdr_bytes[:4]) + if 0 in mopt_ints: + fileobj.seek(0) + return (0,0) + # For 5 format or 7.3 format we need to read an integer in the + # header. Bytes 124 through 128 contain a version integer and an + # endian test string + fileobj.seek(124) + tst_str = fileobj.read(4) + fileobj.seek(0) + maj_ind = int(tst_str[2] == b'I'[0]) + maj_val = int(tst_str[maj_ind]) + min_val = int(tst_str[1 - maj_ind]) + ret = (maj_val, min_val) + if maj_val in (1, 2): + return ret + raise ValueError('Unknown mat file type, version {}, {}'.format(*ret)) + + +def matdims(arr, oned_as='column'): + """ + Determine equivalent MATLAB dimensions for given array + + Parameters + ---------- + arr : ndarray + Input array + oned_as : {'column', 'row'}, optional + Whether 1-D arrays are returned as MATLAB row or column matrices. + Default is 'column'. + + Returns + ------- + dims : tuple + Shape tuple, in the form MATLAB expects it. + + Notes + ----- + We had to decide what shape a 1 dimensional array would be by + default. ``np.atleast_2d`` thinks it is a row vector. The + default for a vector in MATLAB (e.g., ``>> 1:12``) is a row vector. + + Versions of scipy up to and including 0.11 resulted (accidentally) + in 1-D arrays being read as column vectors. For the moment, we + maintain the same tradition here. + + Examples + -------- + >>> import numpy as np + >>> from scipy.io.matlab._miobase import matdims + >>> matdims(np.array(1)) # NumPy scalar + (1, 1) + >>> matdims(np.array([1])) # 1-D array, 1 element + (1, 1) + >>> matdims(np.array([1,2])) # 1-D array, 2 elements + (2, 1) + >>> matdims(np.array([[2],[3]])) # 2-D array, column vector + (2, 1) + >>> matdims(np.array([[2,3]])) # 2-D array, row vector + (1, 2) + >>> matdims(np.array([[[2,3]]])) # 3-D array, rowish vector + (1, 1, 2) + >>> matdims(np.array([])) # empty 1-D array + (0, 0) + >>> matdims(np.array([[]])) # empty 2-D array + (0, 0) + >>> matdims(np.array([[[]]])) # empty 3-D array + (0, 0, 0) + + Optional argument flips 1-D shape behavior. + + >>> matdims(np.array([1,2]), 'row') # 1-D array, 2 elements + (1, 2) + + The argument has to make sense though + + >>> matdims(np.array([1,2]), 'bizarre') + Traceback (most recent call last): + ... + ValueError: 1-D option "bizarre" is strange + + """ + shape = arr.shape + if shape == (): # scalar + return (1, 1) + if len(shape) == 1: # 1D + if shape[0] == 0: + return (0, 0) + elif oned_as == 'column': + return shape + (1,) + elif oned_as == 'row': + return (1,) + shape + else: + raise ValueError(f'1-D option "{oned_as}" is strange') + return shape + + +class MatVarReader: + ''' Abstract class defining required interface for var readers''' + def __init__(self, file_reader): + pass + + def read_header(self): + ''' Returns header ''' + pass + + def array_from_header(self, header): + ''' Reads array given header ''' + pass + + +class MatFileReader: + """ Base object for reading mat files + + To make this class functional, you will need to override the + following methods: + + matrix_getter_factory - gives object to fetch next matrix from stream + guess_byte_order - guesses file byte order from file + """ + + @docfiller + def __init__(self, mat_stream, + byte_order=None, + mat_dtype=False, + squeeze_me=False, + chars_as_strings=True, + matlab_compatible=False, + struct_as_record=True, + verify_compressed_data_integrity=True, + simplify_cells=False): + ''' + Initializer for mat file reader + + mat_stream : file-like + object with file API, open for reading + %(load_args)s + ''' + # Initialize stream + self.mat_stream = mat_stream + self.dtypes = {} + if not byte_order: + byte_order = self.guess_byte_order() + else: + byte_order = boc.to_numpy_code(byte_order) + self.byte_order = byte_order + self.struct_as_record = struct_as_record + if matlab_compatible: + self.set_matlab_compatible() + else: + self.squeeze_me = squeeze_me + self.chars_as_strings = chars_as_strings + self.mat_dtype = mat_dtype + self.verify_compressed_data_integrity = verify_compressed_data_integrity + self.simplify_cells = simplify_cells + if simplify_cells: + self.squeeze_me = True + self.struct_as_record = False + + def set_matlab_compatible(self): + ''' Sets options to return arrays as MATLAB loads them ''' + self.mat_dtype = True + self.squeeze_me = False + self.chars_as_strings = False + + def guess_byte_order(self): + ''' As we do not know what file type we have, assume native ''' + return boc.native_code + + def end_of_stream(self): + b = self.mat_stream.read(1) + curpos = self.mat_stream.tell() + self.mat_stream.seek(curpos-1) + return len(b) == 0 + + +def arr_dtype_number(arr, num): + ''' Return dtype for given number of items per element''' + return np.dtype(arr.dtype.str[:2] + str(num)) + + +def arr_to_chars(arr): + ''' Convert string array to char array ''' + dims = list(arr.shape) + if not dims: + dims = [1] + dims.append(int(arr.dtype.str[2:])) + arr = np.ndarray(shape=dims, + dtype=arr_dtype_number(arr, 1), + buffer=arr) + empties = [arr == np.array('', dtype=arr.dtype)] + if not np.any(empties): + return arr + arr = arr.copy() + arr[tuple(empties)] = ' ' + return arr diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/byteordercodes.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/byteordercodes.py new file mode 100644 index 0000000000000000000000000000000000000000..0a1c5b0f5e77fdd461d6085037bfdf2850f40fa0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/byteordercodes.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io.matlab` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io.matlab", module="byteordercodes", + private_modules=["_byteordercodes"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio.py new file mode 100644 index 0000000000000000000000000000000000000000..65bb31e52dc719b485b12ba1294fc3d09806c9d0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio.py @@ -0,0 +1,16 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io.matlab` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = ["loadmat", "savemat", "whosmat"] # noqa: F822 + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io.matlab", module="mio", + private_modules=["_mio"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio4.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio4.py new file mode 100644 index 0000000000000000000000000000000000000000..d13b99a0bcedc9746f7681843989791e0918df2e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio4.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io.matlab` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io.matlab", module="mio4", + private_modules=["_mio4"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio5.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio5.py new file mode 100644 index 0000000000000000000000000000000000000000..b84ca19799b32999032833b4e1be1b21f6bc70da --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio5.py @@ -0,0 +1,19 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io.matlab` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'MatWriteError', 'MatReadError', 'MatReadWarning', 'MatlabObject', + 'MatlabFunction', 'mat_struct', 'varmats_from_mat', +] + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io.matlab", module="mio5", + private_modules=["_mio5"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio5_params.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio5_params.py new file mode 100644 index 0000000000000000000000000000000000000000..2dcc9a4f353794546f0d8c07f9afe369baa992f5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio5_params.py @@ -0,0 +1,18 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io.matlab` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'MatlabFunction', 'MatlabObject', 'MatlabOpaque', 'mat_struct', +] + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io.matlab", module="mio5_params", + private_modules=["_mio5_params"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio5_utils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio5_utils.py new file mode 100644 index 0000000000000000000000000000000000000000..37ad9e2dc2f50b85bf5aba517c4ac7d661b5039a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio5_utils.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io.matlab` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io.matlab", module="mio5_utils", + private_modules=["_mio5_utils"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio_utils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio_utils.py new file mode 100644 index 0000000000000000000000000000000000000000..6920511d2635b44acd33ce6f5e00247daf6578d9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/mio_utils.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io.matlab` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io.matlab", module="mio_utils", + private_modules=["_mio_utils"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/miobase.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/miobase.py new file mode 100644 index 0000000000000000000000000000000000000000..13e16848394471f9a1744a7b27fa4e6c86a9248b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/miobase.py @@ -0,0 +1,16 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io.matlab` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = ["MatReadError", "MatReadWarning", "MatWriteError"] # noqa: F822 + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io.matlab", module="miobase", + private_modules=["_miobase"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/streams.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/streams.py new file mode 100644 index 0000000000000000000000000000000000000000..8125271b06cc6f44cee19b2f6079d26b8f32e268 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/streams.py @@ -0,0 +1,16 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io.matlab` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io.matlab", module="streams", + private_modules=["_streams"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/data/bad_miuint32.mat b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/data/bad_miuint32.mat new file mode 100644 index 0000000000000000000000000000000000000000..c9ab357ec85972cf0014752a1e0ccb08ff284af9 Binary files /dev/null and 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byteorder module ''' + +import sys + +from numpy.testing import assert_ +from pytest import raises as assert_raises + +import scipy.io.matlab._byteordercodes as sibc + + +def test_native(): + native_is_le = sys.byteorder == 'little' + assert_(sibc.sys_is_le == native_is_le) + + +def test_to_numpy(): + if sys.byteorder == 'little': + assert_(sibc.to_numpy_code('native') == '<') + assert_(sibc.to_numpy_code('swapped') == '>') + else: + assert_(sibc.to_numpy_code('native') == '>') + assert_(sibc.to_numpy_code('swapped') == '<') + assert_(sibc.to_numpy_code('native') == sibc.to_numpy_code('=')) + assert_(sibc.to_numpy_code('big') == '>') + for code in ('little', '<', 'l', 'L', 'le'): + assert_(sibc.to_numpy_code(code) == '<') + for code in ('big', '>', 'b', 'B', 'be'): + assert_(sibc.to_numpy_code(code) == '>') + assert_raises(ValueError, sibc.to_numpy_code, 'silly string') diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_mio.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_mio.py new file mode 100644 index 0000000000000000000000000000000000000000..ef8b3e34ee666fa297d9e25eec1e409ef68edb5f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_mio.py @@ -0,0 +1,1371 @@ +import os +from collections import OrderedDict +from os.path import join as pjoin, dirname +from glob import glob +from io import BytesIO +import re +from tempfile import mkdtemp + +import warnings +import shutil +import gzip + +from numpy.testing import (assert_array_equal, assert_array_almost_equal, + assert_equal, assert_, assert_warns, assert_allclose) +import pytest +from pytest import raises as assert_raises + +import numpy as np +from numpy import array +from scipy.sparse import issparse, eye_array, coo_array, csc_array + +import scipy.io +from scipy.io.matlab import MatlabOpaque, MatlabFunction, MatlabObject +import scipy.io.matlab._byteordercodes as boc +from scipy.io.matlab._miobase import ( + matdims, MatWriteError, MatReadError, matfile_version) +from scipy.io.matlab._mio import mat_reader_factory, loadmat, savemat, whosmat +from scipy.io.matlab._mio5 import ( + MatFile5Writer, MatFile5Reader, varmats_from_mat, to_writeable, + EmptyStructMarker) +import scipy.io.matlab._mio5_params as mio5p +from scipy._lib._util import VisibleDeprecationWarning + + +test_data_path = pjoin(dirname(__file__), 'data') +pytestmark = pytest.mark.thread_unsafe + + +def mlarr(*args, **kwargs): + """Convenience function to return matlab-compatible 2-D array.""" + arr = np.array(*args, **kwargs) + arr.shape = matdims(arr) + return arr + + +# Define cases to test +theta = np.pi/4*np.arange(9,dtype=float).reshape(1,9) +case_table4 = [ + {'name': 'double', + 'classes': {'testdouble': 'double'}, + 'expected': {'testdouble': theta} + }] +case_table4.append( + {'name': 'string', + 'classes': {'teststring': 'char'}, + 'expected': {'teststring': + array(['"Do nine men interpret?" "Nine men," I nod.'])} + }) +case_table4.append( + {'name': 'complex', + 'classes': {'testcomplex': 'double'}, + 'expected': {'testcomplex': np.cos(theta) + 1j*np.sin(theta)} + }) +A = np.zeros((3,5)) +A[0] = list(range(1,6)) +A[:,0] = list(range(1,4)) +case_table4.append( + {'name': 'matrix', + 'classes': {'testmatrix': 'double'}, + 'expected': {'testmatrix': A}, + }) +case_table4.append( + {'name': 'sparse', + 'classes': {'testsparse': 'sparse'}, + 'expected': {'testsparse': coo_array(A)}, + }) +B = A.astype(complex) +B[0,0] += 1j +case_table4.append( + {'name': 'sparsecomplex', + 'classes': {'testsparsecomplex': 'sparse'}, + 'expected': {'testsparsecomplex': coo_array(B)}, + }) +case_table4.append( + {'name': 'multi', + 'classes': {'theta': 'double', 'a': 'double'}, + 'expected': {'theta': theta, 'a': A}, + }) +case_table4.append( + {'name': 'minus', + 'classes': {'testminus': 'double'}, + 'expected': {'testminus': mlarr(-1)}, + }) +case_table4.append( + {'name': 'onechar', + 'classes': {'testonechar': 'char'}, + 'expected': {'testonechar': array(['r'])}, + }) +# Cell arrays stored as object arrays +CA = mlarr(( # tuple for object array creation + [], + mlarr([1]), + mlarr([[1,2]]), + mlarr([[1,2,3]])), dtype=object).reshape(1,-1) +CA[0,0] = array( + ['This cell contains this string and 3 arrays of increasing length']) +case_table5 = [ + {'name': 'cell', + 'classes': {'testcell': 'cell'}, + 'expected': {'testcell': CA}}] +CAE = mlarr(( # tuple for object array creation + mlarr(1), + mlarr(2), + mlarr([]), + mlarr([]), + mlarr(3)), dtype=object).reshape(1,-1) +objarr = np.empty((1,1),dtype=object) +objarr[0,0] = mlarr(1) +case_table5.append( + {'name': 'scalarcell', + 'classes': {'testscalarcell': 'cell'}, + 'expected': {'testscalarcell': objarr} + }) +case_table5.append( + {'name': 'emptycell', + 'classes': {'testemptycell': 'cell'}, + 'expected': {'testemptycell': CAE}}) +case_table5.append( + {'name': 'stringarray', + 'classes': {'teststringarray': 'char'}, + 'expected': {'teststringarray': array( + ['one ', 'two ', 'three'])}, + }) +case_table5.append( + {'name': '3dmatrix', + 'classes': {'test3dmatrix': 'double'}, + 'expected': { + 'test3dmatrix': np.transpose(np.reshape(list(range(1,25)), (4,3,2)))} + }) +st_sub_arr = array([np.sqrt(2),np.exp(1),np.pi]).reshape(1,3) +dtype = [(n, object) for n in ['stringfield', 'doublefield', 'complexfield']] +st1 = np.zeros((1,1), dtype) +st1['stringfield'][0,0] = array(['Rats live on no evil star.']) +st1['doublefield'][0,0] = st_sub_arr +st1['complexfield'][0,0] = st_sub_arr * (1 + 1j) +case_table5.append( + {'name': 'struct', + 'classes': {'teststruct': 'struct'}, + 'expected': {'teststruct': st1} + }) +CN = np.zeros((1,2), dtype=object) +CN[0,0] = mlarr(1) +CN[0,1] = np.zeros((1,3), dtype=object) +CN[0,1][0,0] = mlarr(2, dtype=np.uint8) +CN[0,1][0,1] = mlarr([[3]], dtype=np.uint8) +CN[0,1][0,2] = np.zeros((1,2), dtype=object) +CN[0,1][0,2][0,0] = mlarr(4, dtype=np.uint8) +CN[0,1][0,2][0,1] = mlarr(5, dtype=np.uint8) +case_table5.append( + {'name': 'cellnest', + 'classes': {'testcellnest': 'cell'}, + 'expected': {'testcellnest': CN}, + }) +st2 = np.empty((1,1), dtype=[(n, object) for n in ['one', 'two']]) +st2[0,0]['one'] = mlarr(1) +st2[0,0]['two'] = np.empty((1,1), dtype=[('three', object)]) +st2[0,0]['two'][0,0]['three'] = array(['number 3']) +case_table5.append( + {'name': 'structnest', + 'classes': {'teststructnest': 'struct'}, + 'expected': {'teststructnest': st2} + }) +a = np.empty((1,2), dtype=[(n, object) for n in ['one', 'two']]) +a[0,0]['one'] = mlarr(1) +a[0,0]['two'] = mlarr(2) +a[0,1]['one'] = array(['number 1']) +a[0,1]['two'] = array(['number 2']) +case_table5.append( + {'name': 'structarr', + 'classes': {'teststructarr': 'struct'}, + 'expected': {'teststructarr': a} + }) +ODT = np.dtype([(n, object) for n in + ['expr', 'inputExpr', 'args', + 'isEmpty', 'numArgs', 'version']]) +MO = MatlabObject(np.zeros((1,1), dtype=ODT), 'inline') +m0 = MO[0,0] +m0['expr'] = array(['x']) +m0['inputExpr'] = array([' x = INLINE_INPUTS_{1};']) +m0['args'] = array(['x']) +m0['isEmpty'] = mlarr(0) +m0['numArgs'] = mlarr(1) +m0['version'] = mlarr(1) +case_table5.append( + {'name': 'object', + 'classes': {'testobject': 'object'}, + 'expected': {'testobject': MO} + }) +fp_u_str = open(pjoin(test_data_path, 'japanese_utf8.txt'), 'rb') +u_str = fp_u_str.read().decode('utf-8') +fp_u_str.close() +case_table5.append( + {'name': 'unicode', + 'classes': {'testunicode': 'char'}, + 'expected': {'testunicode': array([u_str])} + }) +case_table5.append( + {'name': 'sparse', + 'classes': {'testsparse': 'sparse'}, + 'expected': {'testsparse': coo_array(A)}, + }) +case_table5.append( + {'name': 'sparsecomplex', + 'classes': {'testsparsecomplex': 'sparse'}, + 'expected': {'testsparsecomplex': coo_array(B)}, + }) +case_table5.append( + {'name': 'bool', + 'classes': {'testbools': 'logical'}, + 'expected': {'testbools': + array([[True], [False]])}, + }) + +case_table5_rt = case_table5[:] +# Inline functions can't be concatenated in matlab, so RT only +case_table5_rt.append( + {'name': 'objectarray', + 'classes': {'testobjectarray': 'object'}, + 'expected': {'testobjectarray': np.repeat(MO, 2).reshape(1,2)}}) + + +def types_compatible(var1, var2): + """Check if types are same or compatible. + + 0-D numpy scalars are compatible with bare python scalars. + """ + type1 = type(var1) + type2 = type(var2) + if type1 is type2: + return True + if type1 is np.ndarray and var1.shape == (): + return type(var1.item()) is type2 + if type2 is np.ndarray and var2.shape == (): + return type(var2.item()) is type1 + return False + + +def _check_level(label, expected, actual): + """ Check one level of a potentially nested array """ + if issparse(expected): # allow different types of sparse matrices + assert_(issparse(actual)) + assert_array_almost_equal(actual.toarray(), + expected.toarray(), + err_msg=label, + decimal=5) + return + # Check types are as expected + assert_(types_compatible(expected, actual), + f"Expected type {type(expected)}, got {type(actual)} at {label}") + # A field in a record array may not be an ndarray + # A scalar from a record array will be type np.void + if not isinstance(expected, np.void | np.ndarray | MatlabObject): + assert_equal(expected, actual) + return + # This is an ndarray-like thing + assert_(expected.shape == actual.shape, + msg=f'Expected shape {expected.shape}, got {actual.shape} at {label}') + ex_dtype = expected.dtype + if ex_dtype.hasobject: # array of objects + if isinstance(expected, MatlabObject): + assert_equal(expected.classname, actual.classname) + for i, ev in enumerate(expected): + level_label = "%s, [%d], " % (label, i) + _check_level(level_label, ev, actual[i]) + return + if ex_dtype.fields: # probably recarray + for fn in ex_dtype.fields: + level_label = f"{label}, field {fn}, " + _check_level(level_label, + expected[fn], actual[fn]) + return + if ex_dtype.type in (str, # string or bool + np.str_, + np.bool_): + assert_equal(actual, expected, err_msg=label) + return + # Something numeric + assert_array_almost_equal(actual, expected, err_msg=label, decimal=5) + + +def _load_check_case(name, files, case): + for file_name in files: + matdict = loadmat(file_name, struct_as_record=True, spmatrix=False) + label = f"test {name}; file {file_name}" + for k, expected in case.items(): + k_label = f"{label}, variable {k}" + assert_(k in matdict, f"Missing key at {k_label}") + _check_level(k_label, expected, matdict[k]) + + +def _whos_check_case(name, files, case, classes): + for file_name in files: + label = f"test {name}; file {file_name}" + + whos = whosmat(file_name) + + expected_whos = [ + (k, expected.shape, classes[k]) for k, expected in case.items()] + + whos.sort() + expected_whos.sort() + assert_equal(whos, expected_whos, + f"{label}: {whos!r} != {expected_whos!r}" + ) + + +# Round trip tests +def _rt_check_case(name, expected, format): + mat_stream = BytesIO() + savemat(mat_stream, expected, format=format) + mat_stream.seek(0) + _load_check_case(name, [mat_stream], expected) + + +# generator for tests +def _cases(version, filt='test%(name)s_*.mat'): + if version == '4': + cases = case_table4 + elif version == '5': + cases = case_table5 + else: + assert version == '5_rt' + cases = case_table5_rt + for case in cases: + name = case['name'] + expected = case['expected'] + if filt is None: + files = None + else: + use_filt = pjoin(test_data_path, filt % dict(name=name)) + files = glob(use_filt) + assert len(files) > 0, \ + f"No files for test {name} using filter {filt}" + classes = case['classes'] + yield name, files, expected, classes + + +@pytest.mark.parametrize('version', ('4', '5')) +def test_load(version): + for case in _cases(version): + _load_check_case(*case[:3]) + + +@pytest.mark.parametrize('version', ('4', '5')) +def test_whos(version): + for case in _cases(version): + _whos_check_case(*case) + + +# generator for round trip tests +@pytest.mark.parametrize('version, fmts', [ + ('4', ['4', '5']), + ('5_rt', ['5']), +]) +def test_round_trip(version, fmts): + for case in _cases(version, filt=None): + for fmt in fmts: + _rt_check_case(case[0], case[2], fmt) + + +def test_gzip_simple(): + xdense = np.zeros((20,20)) + xdense[2,3] = 2.3 + xdense[4,5] = 4.5 + x = csc_array(xdense) + + name = 'gzip_test' + expected = {'x':x} + format = '4' + + tmpdir = mkdtemp() + try: + fname = pjoin(tmpdir,name) + mat_stream = gzip.open(fname, mode='wb') + savemat(mat_stream, expected, format=format) + mat_stream.close() + + mat_stream = gzip.open(fname, mode='rb') + actual = loadmat(mat_stream, struct_as_record=True, spmatrix=False) + mat_stream.close() + finally: + shutil.rmtree(tmpdir) + + assert_array_almost_equal(actual['x'].toarray(), + expected['x'].toarray(), + err_msg=repr(actual)) + + +def test_multiple_open(): + # Ticket #1039, on Windows: check that files are not left open + tmpdir = mkdtemp() + try: + x = dict(x=np.zeros((2, 2))) + + fname = pjoin(tmpdir, "a.mat") + + # Check that file is not left open + savemat(fname, x) + os.unlink(fname) + savemat(fname, x) + loadmat(fname) + os.unlink(fname) + + # Check that stream is left open + f = open(fname, 'wb') + savemat(f, x) + f.seek(0) + f.close() + + f = open(fname, 'rb') + loadmat(f) + f.seek(0) + f.close() + finally: + shutil.rmtree(tmpdir) + + +def test_mat73(): + # Check any hdf5 files raise an error + filenames = glob( + pjoin(test_data_path, 'testhdf5*.mat')) + assert_(len(filenames) > 0) + for filename in filenames: + fp = open(filename, 'rb') + assert_raises(NotImplementedError, + loadmat, + fp, + struct_as_record=True) + fp.close() + + +def test_warnings(): + # This test is an echo of the previous behavior, which was to raise a + # warning if the user triggered a search for mat files on the Python system + # path. We can remove the test in the next version after upcoming (0.13). + fname = pjoin(test_data_path, 'testdouble_7.1_GLNX86.mat') + with warnings.catch_warnings(): + warnings.simplefilter('error') + # This should not generate a warning + loadmat(fname, struct_as_record=True) + # This neither + loadmat(fname, struct_as_record=False) + + +def test_regression_653(): + # Saving a dictionary with only invalid keys used to raise an error. Now we + # save this as an empty struct in matlab space. + sio = BytesIO() + savemat(sio, {'d':{1:2}}, format='5') + back = loadmat(sio)['d'] + # Check we got an empty struct equivalent + assert_equal(back.shape, (1,1)) + assert_equal(back.dtype, np.dtype(object)) + assert_(back[0,0] is None) + + +def test_structname_len(): + # Test limit for length of field names in structs + lim = 31 + fldname = 'a' * lim + st1 = np.zeros((1,1), dtype=[(fldname, object)]) + savemat(BytesIO(), {'longstruct': st1}, format='5') + fldname = 'a' * (lim+1) + st1 = np.zeros((1,1), dtype=[(fldname, object)]) + assert_raises(ValueError, savemat, BytesIO(), + {'longstruct': st1}, format='5') + + +def test_4_and_long_field_names_incompatible(): + # Long field names option not supported in 4 + my_struct = np.zeros((1,1),dtype=[('my_fieldname',object)]) + assert_raises(ValueError, savemat, BytesIO(), + {'my_struct':my_struct}, format='4', long_field_names=True) + + +def test_long_field_names(): + # Test limit for length of field names in structs + lim = 63 + fldname = 'a' * lim + st1 = np.zeros((1,1), dtype=[(fldname, object)]) + savemat(BytesIO(), {'longstruct': st1}, format='5',long_field_names=True) + fldname = 'a' * (lim+1) + st1 = np.zeros((1,1), dtype=[(fldname, object)]) + assert_raises(ValueError, savemat, BytesIO(), + {'longstruct': st1}, format='5',long_field_names=True) + + +def test_long_field_names_in_struct(): + # Regression test - long_field_names was erased if you passed a struct + # within a struct + lim = 63 + fldname = 'a' * lim + cell = np.ndarray((1,2),dtype=object) + st1 = np.zeros((1,1), dtype=[(fldname, object)]) + cell[0,0] = st1 + cell[0,1] = st1 + savemat(BytesIO(), {'longstruct': cell}, format='5',long_field_names=True) + # + # Check to make sure it fails with long field names off + # + assert_raises(ValueError, savemat, BytesIO(), + {'longstruct': cell}, format='5', long_field_names=False) + + +def test_cell_with_one_thing_in_it(): + # Regression test - make a cell array that's 1 x 2 and put two + # strings in it. It works. Make a cell array that's 1 x 1 and put + # a string in it. It should work but, in the old days, it didn't. + cells = np.ndarray((1,2),dtype=object) + cells[0,0] = 'Hello' + cells[0,1] = 'World' + savemat(BytesIO(), {'x': cells}, format='5') + + cells = np.ndarray((1,1),dtype=object) + cells[0,0] = 'Hello, world' + savemat(BytesIO(), {'x': cells}, format='5') + + +def test_writer_properties(): + # Tests getting, setting of properties of matrix writer + mfw = MatFile5Writer(BytesIO()) + assert_equal(mfw.global_vars, []) + mfw.global_vars = ['avar'] + assert_equal(mfw.global_vars, ['avar']) + assert_equal(mfw.unicode_strings, False) + mfw.unicode_strings = True + assert_equal(mfw.unicode_strings, True) + assert_equal(mfw.long_field_names, False) + mfw.long_field_names = True + assert_equal(mfw.long_field_names, True) + + +def test_use_small_element(): + # Test whether we're using small data element or not + sio = BytesIO() + wtr = MatFile5Writer(sio) + # First check size for no sde for name + arr = np.zeros(10) + wtr.put_variables({'aaaaa': arr}) + w_sz = len(sio.getvalue()) + # Check small name results in largish difference in size + sio.truncate(0) + sio.seek(0) + wtr.put_variables({'aaaa': arr}) + assert_(w_sz - len(sio.getvalue()) > 4) + # Whereas increasing name size makes less difference + sio.truncate(0) + sio.seek(0) + wtr.put_variables({'aaaaaa': arr}) + assert_(len(sio.getvalue()) - w_sz < 4) + + +def test_save_dict(): + # Test that both dict and OrderedDict can be saved (as recarray), + # loaded as matstruct, and preserve order + ab_exp = np.array([[(1, 2)]], dtype=[('a', object), ('b', object)]) + for dict_type in (dict, OrderedDict): + # Initialize with tuples to keep order + d = dict_type([('a', 1), ('b', 2)]) + stream = BytesIO() + savemat(stream, {'dict': d}) + stream.seek(0) + vals = loadmat(stream)['dict'] + assert_equal(vals.dtype.names, ('a', 'b')) + assert_array_equal(vals, ab_exp) + + +def test_1d_shape(): + # New 5 behavior is 1D -> row vector + arr = np.arange(5) + for format in ('4', '5'): + # Column is the default + stream = BytesIO() + savemat(stream, {'oned': arr}, format=format) + vals = loadmat(stream) + assert_equal(vals['oned'].shape, (1, 5)) + # can be explicitly 'column' for oned_as + stream = BytesIO() + savemat(stream, {'oned':arr}, + format=format, + oned_as='column') + vals = loadmat(stream) + assert_equal(vals['oned'].shape, (5,1)) + # but different from 'row' + stream = BytesIO() + savemat(stream, {'oned':arr}, + format=format, + oned_as='row') + vals = loadmat(stream) + assert_equal(vals['oned'].shape, (1,5)) + + +def test_compression(): + arr = np.zeros(100).reshape((5,20)) + arr[2,10] = 1 + stream = BytesIO() + savemat(stream, {'arr':arr}) + raw_len = len(stream.getvalue()) + vals = loadmat(stream) + assert_array_equal(vals['arr'], arr) + stream = BytesIO() + savemat(stream, {'arr':arr}, do_compression=True) + compressed_len = len(stream.getvalue()) + vals = loadmat(stream) + assert_array_equal(vals['arr'], arr) + assert_(raw_len > compressed_len) + # Concatenate, test later + arr2 = arr.copy() + arr2[0,0] = 1 + stream = BytesIO() + savemat(stream, {'arr':arr, 'arr2':arr2}, do_compression=False) + vals = loadmat(stream) + assert_array_equal(vals['arr2'], arr2) + stream = BytesIO() + savemat(stream, {'arr':arr, 'arr2':arr2}, do_compression=True) + vals = loadmat(stream) + assert_array_equal(vals['arr2'], arr2) + + +def test_single_object(): + stream = BytesIO() + savemat(stream, {'A':np.array(1, dtype=object)}) + + +def test_skip_variable(): + # Test skipping over the first of two variables in a MAT file + # using mat_reader_factory and put_variables to read them in. + # + # This is a regression test of a problem that's caused by + # using the compressed file reader seek instead of the raw file + # I/O seek when skipping over a compressed chunk. + # + # The problem arises when the chunk is large: this file has + # a 256x256 array of random (uncompressible) doubles. + # + filename = pjoin(test_data_path,'test_skip_variable.mat') + # + # Prove that it loads with loadmat + # + d = loadmat(filename, struct_as_record=True) + assert_('first' in d) + assert_('second' in d) + # + # Make the factory + # + factory, file_opened = mat_reader_factory(filename, struct_as_record=True) + # + # This is where the factory breaks with an error in MatMatrixGetter.to_next + # + d = factory.get_variables('second') + assert_('second' in d) + factory.mat_stream.close() + + +def test_empty_struct(): + # ticket 885 + filename = pjoin(test_data_path,'test_empty_struct.mat') + # before ticket fix, this would crash with ValueError, empty data + # type + d = loadmat(filename, struct_as_record=True) + a = d['a'] + assert_equal(a.shape, (1,1)) + assert_equal(a.dtype, np.dtype(object)) + assert_(a[0,0] is None) + stream = BytesIO() + arr = np.array((), dtype='U') + # before ticket fix, this used to give data type not understood + savemat(stream, {'arr':arr}) + d = loadmat(stream) + a2 = d['arr'] + assert_array_equal(a2, arr) + + +def test_save_empty_dict(): + # saving empty dict also gives empty struct + stream = BytesIO() + savemat(stream, {'arr': {}}) + d = loadmat(stream) + a = d['arr'] + assert_equal(a.shape, (1,1)) + assert_equal(a.dtype, np.dtype(object)) + assert_(a[0,0] is None) + + +def assert_any_equal(output, alternatives): + """ Assert `output` is equal to at least one element in `alternatives` + """ + one_equal = False + for expected in alternatives: + if np.all(output == expected): + one_equal = True + break + assert_(one_equal) + + +def test_to_writeable(): + # Test to_writeable function + res = to_writeable(np.array([1])) # pass through ndarrays + assert_equal(res.shape, (1,)) + assert_array_equal(res, 1) + # Dict fields can be written in any order + expected1 = np.array([(1, 2)], dtype=[('a', '|O8'), ('b', '|O8')]) + expected2 = np.array([(2, 1)], dtype=[('b', '|O8'), ('a', '|O8')]) + alternatives = (expected1, expected2) + assert_any_equal(to_writeable({'a':1,'b':2}), alternatives) + # Fields with underscores discarded + assert_any_equal(to_writeable({'a':1,'b':2, '_c':3}), alternatives) + # Not-string fields discarded + assert_any_equal(to_writeable({'a':1,'b':2, 100:3}), alternatives) + # String fields that are valid Python identifiers discarded + assert_any_equal(to_writeable({'a':1,'b':2, '99':3}), alternatives) + # Object with field names is equivalent + + class klass: + pass + + c = klass + c.a = 1 + c.b = 2 + assert_any_equal(to_writeable(c), alternatives) + # empty list and tuple go to empty array + res = to_writeable([]) + assert_equal(res.shape, (0,)) + assert_equal(res.dtype.type, np.float64) + res = to_writeable(()) + assert_equal(res.shape, (0,)) + assert_equal(res.dtype.type, np.float64) + # None -> None + assert_(to_writeable(None) is None) + # String to strings + assert_equal(to_writeable('a string').dtype.type, np.str_) + # Scalars to numpy to NumPy scalars + res = to_writeable(1) + assert_equal(res.shape, ()) + assert_equal(res.dtype.type, np.array(1).dtype.type) + assert_array_equal(res, 1) + # Empty dict returns EmptyStructMarker + assert_(to_writeable({}) is EmptyStructMarker) + # Object does not have (even empty) __dict__ + assert_(to_writeable(object()) is None) + # Custom object does have empty __dict__, returns EmptyStructMarker + + class C: + pass + + assert_(to_writeable(c()) is EmptyStructMarker) + # dict keys with legal characters are convertible + res = to_writeable({'a': 1})['a'] + assert_equal(res.shape, (1,)) + assert_equal(res.dtype.type, np.object_) + # Only fields with illegal characters, falls back to EmptyStruct + assert_(to_writeable({'1':1}) is EmptyStructMarker) + assert_(to_writeable({'_a':1}) is EmptyStructMarker) + # Unless there are valid fields, in which case structured array + assert_equal(to_writeable({'1':1, 'f': 2}), + np.array([(2,)], dtype=[('f', '|O8')])) + + +def test_recarray(): + # check roundtrip of structured array + dt = [('f1', 'f8'), + ('f2', 'S10')] + arr = np.zeros((2,), dtype=dt) + arr[0]['f1'] = 0.5 + arr[0]['f2'] = 'python' + arr[1]['f1'] = 99 + arr[1]['f2'] = 'not perl' + stream = BytesIO() + savemat(stream, {'arr': arr}) + d = loadmat(stream, struct_as_record=False) + a20 = d['arr'][0,0] + assert_equal(a20.f1, 0.5) + assert_equal(a20.f2, 'python') + d = loadmat(stream, struct_as_record=True) + a20 = d['arr'][0,0] + assert_equal(a20['f1'], 0.5) + assert_equal(a20['f2'], 'python') + # structs always come back as object types + assert_equal(a20.dtype, np.dtype([('f1', 'O'), + ('f2', 'O')])) + a21 = d['arr'].flat[1] + assert_equal(a21['f1'], 99) + assert_equal(a21['f2'], 'not perl') + + +def test_save_object(): + class C: + pass + c = C() + c.field1 = 1 + c.field2 = 'a string' + stream = BytesIO() + savemat(stream, {'c': c}) + d = loadmat(stream, struct_as_record=False) + c2 = d['c'][0,0] + assert_equal(c2.field1, 1) + assert_equal(c2.field2, 'a string') + d = loadmat(stream, struct_as_record=True) + c2 = d['c'][0,0] + assert_equal(c2['field1'], 1) + assert_equal(c2['field2'], 'a string') + + +def test_read_opts(): + # tests if read is seeing option sets, at initialization and after + # initialization + arr = np.arange(6).reshape(1,6) + stream = BytesIO() + savemat(stream, {'a': arr}) + rdr = MatFile5Reader(stream) + back_dict = rdr.get_variables() + rarr = back_dict['a'] + assert_array_equal(rarr, arr) + rdr = MatFile5Reader(stream, squeeze_me=True) + assert_array_equal(rdr.get_variables()['a'], arr.reshape((6,))) + rdr.squeeze_me = False + assert_array_equal(rarr, arr) + rdr = MatFile5Reader(stream, byte_order=boc.native_code) + assert_array_equal(rdr.get_variables()['a'], arr) + # inverted byte code leads to error on read because of swapped + # header etc. + rdr = MatFile5Reader(stream, byte_order=boc.swapped_code) + assert_raises(Exception, rdr.get_variables) + rdr.byte_order = boc.native_code + assert_array_equal(rdr.get_variables()['a'], arr) + arr = np.array(['a string']) + stream.truncate(0) + stream.seek(0) + savemat(stream, {'a': arr}) + rdr = MatFile5Reader(stream) + assert_array_equal(rdr.get_variables()['a'], arr) + rdr = MatFile5Reader(stream, chars_as_strings=False) + carr = np.atleast_2d(np.array(list(arr.item()), dtype='U1')) + assert_array_equal(rdr.get_variables()['a'], carr) + rdr.chars_as_strings = True + assert_array_equal(rdr.get_variables()['a'], arr) + + +def test_empty_string(): + # make sure reading empty string does not raise error + estring_fname = pjoin(test_data_path, 'single_empty_string.mat') + fp = open(estring_fname, 'rb') + rdr = MatFile5Reader(fp) + d = rdr.get_variables() + fp.close() + assert_array_equal(d['a'], np.array([], dtype='U1')) + # Empty string round trip. Matlab cannot distinguish + # between a string array that is empty, and a string array + # containing a single empty string, because it stores strings as + # arrays of char. There is no way of having an array of char that + # is not empty, but contains an empty string. + stream = BytesIO() + savemat(stream, {'a': np.array([''])}) + rdr = MatFile5Reader(stream) + d = rdr.get_variables() + assert_array_equal(d['a'], np.array([], dtype='U1')) + stream.truncate(0) + stream.seek(0) + savemat(stream, {'a': np.array([], dtype='U1')}) + rdr = MatFile5Reader(stream) + d = rdr.get_variables() + assert_array_equal(d['a'], np.array([], dtype='U1')) + stream.close() + + +def test_corrupted_data(): + import zlib + for exc, fname in [(ValueError, 'corrupted_zlib_data.mat'), + (zlib.error, 'corrupted_zlib_checksum.mat')]: + with open(pjoin(test_data_path, fname), 'rb') as fp: + rdr = MatFile5Reader(fp) + assert_raises(exc, rdr.get_variables) + + +def test_corrupted_data_check_can_be_disabled(): + with open(pjoin(test_data_path, 'corrupted_zlib_data.mat'), 'rb') as fp: + rdr = MatFile5Reader(fp, verify_compressed_data_integrity=False) + rdr.get_variables() + + +def test_read_both_endian(): + # make sure big- and little- endian data is read correctly + for fname in ('big_endian.mat', 'little_endian.mat'): + fp = open(pjoin(test_data_path, fname), 'rb') + rdr = MatFile5Reader(fp) + d = rdr.get_variables() + fp.close() + assert_array_equal(d['strings'], + np.array([['hello'], + ['world']], dtype=object)) + assert_array_equal(d['floats'], + np.array([[2., 3.], + [3., 4.]], dtype=np.float32)) + + +def test_write_opposite_endian(): + # We don't support writing opposite endian .mat files, but we need to behave + # correctly if the user supplies an other-endian NumPy array to write out. + float_arr = np.array([[2., 3.], + [3., 4.]]) + int_arr = np.arange(6).reshape((2, 3)) + uni_arr = np.array(['hello', 'world'], dtype='U') + stream = BytesIO() + savemat(stream, { + 'floats': float_arr.byteswap().view(float_arr.dtype.newbyteorder()), + 'ints': int_arr.byteswap().view(int_arr.dtype.newbyteorder()), + 'uni_arr': uni_arr.byteswap().view(uni_arr.dtype.newbyteorder()), + }) + rdr = MatFile5Reader(stream) + d = rdr.get_variables() + assert_array_equal(d['floats'], float_arr) + assert_array_equal(d['ints'], int_arr) + assert_array_equal(d['uni_arr'], uni_arr) + stream.close() + + +def test_logical_array(): + # The roundtrip test doesn't verify that we load the data up with the + # correct (bool) dtype + with open(pjoin(test_data_path, 'testbool_8_WIN64.mat'), 'rb') as fobj: + rdr = MatFile5Reader(fobj, mat_dtype=True) + d = rdr.get_variables() + x = np.array([[True], [False]], dtype=np.bool_) + assert_array_equal(d['testbools'], x) + assert_equal(d['testbools'].dtype, x.dtype) + + +def test_logical_out_type(): + # Confirm that bool type written as uint8, uint8 class + # See gh-4022 + stream = BytesIO() + barr = np.array([False, True, False]) + savemat(stream, {'barray': barr}) + stream.seek(0) + reader = MatFile5Reader(stream) + reader.initialize_read() + reader.read_file_header() + hdr, _ = reader.read_var_header() + assert_equal(hdr.mclass, mio5p.mxUINT8_CLASS) + assert_equal(hdr.is_logical, True) + var = reader.read_var_array(hdr, False) + assert_equal(var.dtype.type, np.uint8) + + +def test_roundtrip_zero_dimensions(): + stream = BytesIO() + savemat(stream, {'d':np.empty((10, 0))}) + d = loadmat(stream) + assert d['d'].shape == (10, 0) + + +def test_mat4_3d(): + # test behavior when writing 3-D arrays to matlab 4 files + stream = BytesIO() + arr = np.arange(24).reshape((2,3,4)) + assert_raises(ValueError, savemat, stream, {'a': arr}, True, '4') + + +def test_func_read(): + func_eg = pjoin(test_data_path, 'testfunc_7.4_GLNX86.mat') + fp = open(func_eg, 'rb') + rdr = MatFile5Reader(fp) + d = rdr.get_variables() + fp.close() + assert isinstance(d['testfunc'], MatlabFunction) + stream = BytesIO() + wtr = MatFile5Writer(stream) + assert_raises(MatWriteError, wtr.put_variables, d) + + +def test_mat_dtype(): + double_eg = pjoin(test_data_path, 'testmatrix_6.1_SOL2.mat') + fp = open(double_eg, 'rb') + rdr = MatFile5Reader(fp, mat_dtype=False) + d = rdr.get_variables() + fp.close() + assert_equal(d['testmatrix'].dtype.kind, 'u') + + fp = open(double_eg, 'rb') + rdr = MatFile5Reader(fp, mat_dtype=True) + d = rdr.get_variables() + fp.close() + assert_equal(d['testmatrix'].dtype.kind, 'f') + + +def test_sparse_in_struct(): + # reproduces bug found by DC where Cython code was insisting on + # ndarray return type, but getting sparse matrix + st = {'sparsefield': eye_array(4)} + stream = BytesIO() + savemat(stream, {'a':st}) + d = loadmat(stream, struct_as_record=True) + assert_array_equal(d['a'][0, 0]['sparsefield'].toarray(), np.eye(4)) + + +def test_mat_struct_squeeze(): + stream = BytesIO() + in_d = {'st':{'one':1, 'two':2}} + savemat(stream, in_d) + # no error without squeeze + loadmat(stream, struct_as_record=False) + # previous error was with squeeze, with mat_struct + loadmat(stream, struct_as_record=False, squeeze_me=True) + + +def test_scalar_squeeze(): + stream = BytesIO() + in_d = {'scalar': [[0.1]], 'string': 'my name', 'st':{'one':1, 'two':2}} + savemat(stream, in_d) + out_d = loadmat(stream, squeeze_me=True) + assert_(isinstance(out_d['scalar'], float)) + assert_(isinstance(out_d['string'], str)) + assert_(isinstance(out_d['st'], np.ndarray)) + + +def test_str_round(): + # from report by Angus McMorland on mailing list 3 May 2010 + stream = BytesIO() + in_arr = np.array(['Hello', 'Foob']) + out_arr = np.array(['Hello', 'Foob ']) + savemat(stream, dict(a=in_arr)) + res = loadmat(stream) + # resulted in ['HloolFoa', 'elWrdobr'] + assert_array_equal(res['a'], out_arr) + stream.truncate(0) + stream.seek(0) + # Make Fortran ordered version of string + in_str = in_arr.tobytes(order='F') + in_from_str = np.ndarray(shape=a.shape, + dtype=in_arr.dtype, + order='F', + buffer=in_str) + savemat(stream, dict(a=in_from_str)) + assert_array_equal(res['a'], out_arr) + # unicode save did lead to buffer too small error + stream.truncate(0) + stream.seek(0) + in_arr_u = in_arr.astype('U') + out_arr_u = out_arr.astype('U') + savemat(stream, {'a': in_arr_u}) + res = loadmat(stream) + assert_array_equal(res['a'], out_arr_u) + + +def test_fieldnames(): + # Check that field names are as expected + stream = BytesIO() + savemat(stream, {'a': {'a':1, 'b':2}}) + res = loadmat(stream) + field_names = res['a'].dtype.names + assert_equal(set(field_names), {'a', 'b'}) + + +def test_loadmat_varnames(): + # Test that we can get just one variable from a mat file using loadmat + mat5_sys_names = ['__globals__', + '__header__', + '__version__'] + for eg_file, sys_v_names in ( + (pjoin(test_data_path, 'testmulti_4.2c_SOL2.mat'), []), (pjoin( + test_data_path, 'testmulti_7.4_GLNX86.mat'), mat5_sys_names)): + vars = loadmat(eg_file) + assert_equal(set(vars.keys()), set(['a', 'theta'] + sys_v_names)) + vars = loadmat(eg_file, variable_names='a') + assert_equal(set(vars.keys()), set(['a'] + sys_v_names)) + vars = loadmat(eg_file, variable_names=['a']) + assert_equal(set(vars.keys()), set(['a'] + sys_v_names)) + vars = loadmat(eg_file, variable_names=['theta']) + assert_equal(set(vars.keys()), set(['theta'] + sys_v_names)) + vars = loadmat(eg_file, variable_names=('theta',)) + assert_equal(set(vars.keys()), set(['theta'] + sys_v_names)) + vars = loadmat(eg_file, variable_names=[]) + assert_equal(set(vars.keys()), set(sys_v_names)) + vnames = ['theta'] + vars = loadmat(eg_file, variable_names=vnames) + assert_equal(vnames, ['theta']) + + +def test_round_types(): + # Check that saving, loading preserves dtype in most cases + arr = np.arange(10) + stream = BytesIO() + for dts in ('f8','f4','i8','i4','i2','i1', + 'u8','u4','u2','u1','c16','c8'): + stream.truncate(0) + stream.seek(0) # needed for BytesIO in Python 3 + savemat(stream, {'arr': arr.astype(dts)}) + vars = loadmat(stream) + assert_equal(np.dtype(dts), vars['arr'].dtype) + + +def test_varmats_from_mat(): + # Make a mat file with several variables, write it, read it back + names_vars = (('arr', mlarr(np.arange(10))), + ('mystr', mlarr('a string')), + ('mynum', mlarr(10))) + + # Dict like thing to give variables in defined order + class C: + def items(self): + return names_vars + stream = BytesIO() + savemat(stream, C()) + varmats = varmats_from_mat(stream) + assert_equal(len(varmats), 3) + for i in range(3): + name, var_stream = varmats[i] + exp_name, exp_res = names_vars[i] + assert_equal(name, exp_name) + res = loadmat(var_stream) + assert_array_equal(res[name], exp_res) + + +def test_one_by_zero(): + # Test 1x0 chars get read correctly + func_eg = pjoin(test_data_path, 'one_by_zero_char.mat') + fp = open(func_eg, 'rb') + rdr = MatFile5Reader(fp) + d = rdr.get_variables() + fp.close() + assert_equal(d['var'].shape, (0,)) + + +def test_load_mat4_le(): + # We were getting byte order wrong when reading little-endian floa64 dense + # matrices on big-endian platforms + mat4_fname = pjoin(test_data_path, 'test_mat4_le_floats.mat') + vars = loadmat(mat4_fname) + assert_array_equal(vars['a'], [[0.1, 1.2]]) + + +def test_unicode_mat4(): + # Mat4 should save unicode as latin1 + bio = BytesIO() + var = {'second_cat': 'Schrödinger'} + savemat(bio, var, format='4') + var_back = loadmat(bio) + assert_equal(var_back['second_cat'], var['second_cat']) + + +def test_logical_sparse(): + # Test we can read logical sparse stored in mat file as bytes. + # See https://github.com/scipy/scipy/issues/3539. + # In some files saved by MATLAB, the sparse data elements (Real Part + # Subelement in MATLAB speak) are stored with apparent type double + # (miDOUBLE) but are in fact single bytes. + filename = pjoin(test_data_path,'logical_sparse.mat') + # Before fix, this would crash with: + # ValueError: indices and data should have the same size + d = loadmat(filename, struct_as_record=True, spmatrix=False) + log_sp = d['sp_log_5_4'] + assert_(issparse(log_sp) and log_sp.format == "csc") + assert_equal(log_sp.dtype.type, np.bool_) + assert_array_equal(log_sp.toarray(), + [[True, True, True, False], + [False, False, True, False], + [False, False, True, False], + [False, False, False, False], + [False, False, False, False]]) + + +def test_empty_sparse(): + # Can we read empty sparse matrices? + sio = BytesIO() + import scipy.sparse + empty_sparse = scipy.sparse.csr_array([[0,0],[0,0]]) + savemat(sio, dict(x=empty_sparse)) + sio.seek(0) + + res = loadmat(sio, spmatrix=False) + assert not scipy.sparse.isspmatrix(res['x']) + res = loadmat(sio, spmatrix=True) + assert scipy.sparse.isspmatrix(res['x']) + res = loadmat(sio) # chk default + assert scipy.sparse.isspmatrix(res['x']) + + assert_array_equal(res['x'].shape, empty_sparse.shape) + assert_array_equal(res['x'].toarray(), 0) + # Do empty sparse matrices get written with max nnz 1? + # See https://github.com/scipy/scipy/issues/4208 + sio.seek(0) + reader = MatFile5Reader(sio) + reader.initialize_read() + reader.read_file_header() + hdr, _ = reader.read_var_header() + assert_equal(hdr.nzmax, 1) + + +def test_empty_mat_error(): + # Test we get a specific warning for an empty mat file + sio = BytesIO() + assert_raises(MatReadError, loadmat, sio) + + +def test_miuint32_compromise(): + # Reader should accept miUINT32 for miINT32, but check signs + # mat file with miUINT32 for miINT32, but OK values + filename = pjoin(test_data_path, 'miuint32_for_miint32.mat') + res = loadmat(filename) + assert_equal(res['an_array'], np.arange(10)[None, :]) + # mat file with miUINT32 for miINT32, with negative value + filename = pjoin(test_data_path, 'bad_miuint32.mat') + with assert_raises(ValueError): + loadmat(filename) + + +def test_miutf8_for_miint8_compromise(): + # Check reader accepts ascii as miUTF8 for array names + filename = pjoin(test_data_path, 'miutf8_array_name.mat') + res = loadmat(filename) + assert_equal(res['array_name'], [[1]]) + # mat file with non-ascii utf8 name raises error + filename = pjoin(test_data_path, 'bad_miutf8_array_name.mat') + with assert_raises(ValueError): + loadmat(filename) + + +def test_bad_utf8(): + # Check that reader reads bad UTF with 'replace' option + filename = pjoin(test_data_path,'broken_utf8.mat') + res = loadmat(filename) + assert_equal(res['bad_string'], + b'\x80 am broken'.decode('utf8', 'replace')) + + +def test_save_unicode_field(tmpdir): + filename = os.path.join(str(tmpdir), 'test.mat') + test_dict = {'a':{'b':1,'c':'test_str'}} + savemat(filename, test_dict) + + +def test_save_custom_array_type(tmpdir): + class CustomArray: + def __array__(self, dtype=None, copy=None): + return np.arange(6.0).reshape(2, 3) + a = CustomArray() + filename = os.path.join(str(tmpdir), 'test.mat') + savemat(filename, {'a': a}) + out = loadmat(filename) + assert_array_equal(out['a'], np.array(a)) + + +def test_filenotfound(): + # Check the correct error is thrown + assert_raises(OSError, loadmat, "NotExistentFile00.mat") + assert_raises(OSError, loadmat, "NotExistentFile00") + + +def test_simplify_cells(): + # Test output when simplify_cells=True + filename = pjoin(test_data_path, 'testsimplecell.mat') + res1 = loadmat(filename, simplify_cells=True) + res2 = loadmat(filename, simplify_cells=False) + assert_(isinstance(res1["s"], dict)) + assert_(isinstance(res2["s"], np.ndarray)) + assert_array_equal(res1["s"]["mycell"], np.array(["a", "b", "c"])) + + +@pytest.mark.parametrize('version, filt, regex', [ + (0, '_4*_*', None), + (1, '_5*_*', None), + (1, '_6*_*', None), + (1, '_7*_*', '^((?!hdf5).)*$'), # not containing hdf5 + (2, '_7*_*', '.*hdf5.*'), + (1, '8*_*', None), +]) +def test_matfile_version(version, filt, regex): + use_filt = pjoin(test_data_path, f'test*{filt}.mat') + files = glob(use_filt) + if regex is not None: + files = [file for file in files if re.match(regex, file) is not None] + assert len(files) > 0, \ + f"No files for version {version} using filter {filt}" + for file in files: + got_version = matfile_version(file) + assert got_version[0] == version + + +def test_opaque(): + """Test that we can read a MatlabOpaque object.""" + data = loadmat(pjoin(test_data_path, 'parabola.mat')) + assert isinstance(data['parabola'], MatlabFunction) + assert isinstance(data['parabola'].item()[3].item()[3], MatlabOpaque) + + +def test_opaque_simplify(): + """Test that we can read a MatlabOpaque object when simplify_cells=True.""" + data = loadmat(pjoin(test_data_path, 'parabola.mat'), simplify_cells=True) + assert isinstance(data['parabola'], MatlabFunction) + + +def test_deprecation(): + """Test that access to previous attributes still works.""" + # This should be accessible immediately from scipy.io import + with assert_warns(DeprecationWarning): + scipy.io.matlab.mio5_params.MatlabOpaque + + # These should be importable but warn as well + with assert_warns(DeprecationWarning): + from scipy.io.matlab.miobase import MatReadError # noqa: F401 + + +def test_gh_17992(tmp_path): + rng = np.random.default_rng(12345) + outfile = tmp_path / "lists.mat" + array_one = rng.random((5,3)) + array_two = rng.random((6,3)) + list_of_arrays = [array_one, array_two] + # warning suppression only needed for NumPy < 1.24.0 + with np.testing.suppress_warnings() as sup: + sup.filter(VisibleDeprecationWarning) + savemat(outfile, + {'data': list_of_arrays}, + long_field_names=True, + do_compression=True) + # round trip check + new_dict = {} + loadmat(outfile, + new_dict) + assert_allclose(new_dict["data"][0][0], array_one) + assert_allclose(new_dict["data"][0][1], array_two) + + +def test_gh_19659(tmp_path): + d = { + "char_array": np.array([list("char"), list("char")], dtype="U1"), + "string_array": np.array(["string", "string"]), + } + outfile = tmp_path / "tmp.mat" + # should not error: + savemat(outfile, d, format="4") + + +def test_large_m4(): + # Test we can read a Matlab 4 file with array > 2GB. + # (In fact, test we get the correct error from reading a truncated + # version). + # See https://github.com/scipy/scipy/issues/21256 + # Data file is first 1024 bytes of: + # >>> a = np.zeros((134217728, 3)) + # >>> siom.savemat('big_m4.mat', {'a': a}, format='4') + truncated_mat = pjoin(test_data_path, 'debigged_m4.mat') + match = ("Not enough bytes to read matrix 'a';" + if np.intp == np.int64 else + "Variable 'a' has byte length longer than largest possible") + with pytest.raises(ValueError, match=match): + loadmat(truncated_mat) + + +def test_gh_19223(): + from scipy.io.matlab import varmats_from_mat # noqa: F401 + +def test_corrupt_files(): + # Test we can detect truncated or corrupt (all zero) files. + for n in (2, 4, 10, 19): + with pytest.raises(MatReadError, + match="Mat file appears to be truncated"): + loadmat(BytesIO(b'\x00' * n)) + with pytest.raises(MatReadError, + match="Mat file appears to be corrupt"): + loadmat(BytesIO(b'\x00' * 20)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_mio5_utils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_mio5_utils.py new file mode 100644 index 0000000000000000000000000000000000000000..b3f27114c4a4ed10c1a2526058f4d0dbbd0e5638 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_mio5_utils.py @@ -0,0 +1,179 @@ +""" Testing mio5_utils Cython module + +""" +import sys + +from io import BytesIO + +import numpy as np + +from numpy.testing import assert_array_equal, assert_equal, assert_ +from pytest import raises as assert_raises + +import scipy.io.matlab._byteordercodes as boc +import scipy.io.matlab._streams as streams +import scipy.io.matlab._mio5_params as mio5p +import scipy.io.matlab._mio5_utils as m5u + + +def test_byteswap(): + for val in ( + 1, + 0x100, + 0x10000): + a = np.array(val, dtype=np.uint32) + b = a.byteswap() + c = m5u.byteswap_u4(a) + assert_equal(b.item(), c) + d = m5u.byteswap_u4(c) + assert_equal(a.item(), d) + + +def _make_tag(base_dt, val, mdtype, sde=False): + ''' Makes a simple matlab tag, full or sde ''' + base_dt = np.dtype(base_dt) + bo = boc.to_numpy_code(base_dt.byteorder) + byte_count = base_dt.itemsize + if not sde: + udt = bo + 'u4' + padding = 8 - (byte_count % 8) + all_dt = [('mdtype', udt), + ('byte_count', udt), + ('val', base_dt)] + if padding: + all_dt.append(('padding', 'u1', padding)) + else: # is sde + udt = bo + 'u2' + padding = 4-byte_count + if bo == '<': # little endian + all_dt = [('mdtype', udt), + ('byte_count', udt), + ('val', base_dt)] + else: # big endian + all_dt = [('byte_count', udt), + ('mdtype', udt), + ('val', base_dt)] + if padding: + all_dt.append(('padding', 'u1', padding)) + tag = np.zeros((1,), dtype=all_dt) + tag['mdtype'] = mdtype + tag['byte_count'] = byte_count + tag['val'] = val + return tag + + +def _write_stream(stream, *strings): + stream.truncate(0) + stream.seek(0) + for s in strings: + stream.write(s) + stream.seek(0) + + +def _make_readerlike(stream, byte_order=boc.native_code): + class R: + pass + r = R() + r.mat_stream = stream + r.byte_order = byte_order + r.struct_as_record = True + r.uint16_codec = sys.getdefaultencoding() + r.chars_as_strings = False + r.mat_dtype = False + r.squeeze_me = False + return r + + +def test_read_tag(): + # mainly to test errors + # make reader-like thing + str_io = BytesIO() + r = _make_readerlike(str_io) + c_reader = m5u.VarReader5(r) + # This works for StringIO but _not_ BytesIO + assert_raises(OSError, c_reader.read_tag) + # bad SDE + tag = _make_tag('i4', 1, mio5p.miINT32, sde=True) + tag['byte_count'] = 5 + _write_stream(str_io, tag.tobytes()) + assert_raises(ValueError, c_reader.read_tag) + + +def test_read_stream(): + tag = _make_tag('i4', 1, mio5p.miINT32, sde=True) + tag_str = tag.tobytes() + str_io = BytesIO(tag_str) + st = streams.make_stream(str_io) + s = streams._read_into(st, tag.itemsize) + assert_equal(s, tag.tobytes()) + + +def test_read_numeric(): + # make reader-like thing + str_io = BytesIO() + r = _make_readerlike(str_io) + # check simplest of tags + for base_dt, val, mdtype in (('u2', 30, mio5p.miUINT16), + ('i4', 1, mio5p.miINT32), + ('i2', -1, mio5p.miINT16)): + for byte_code in ('<', '>'): + r.byte_order = byte_code + c_reader = m5u.VarReader5(r) + assert_equal(c_reader.little_endian, byte_code == '<') + assert_equal(c_reader.is_swapped, byte_code != boc.native_code) + for sde_f in (False, True): + dt = np.dtype(base_dt).newbyteorder(byte_code) + a = _make_tag(dt, val, mdtype, sde_f) + a_str = a.tobytes() + _write_stream(str_io, a_str) + el = c_reader.read_numeric() + assert_equal(el, val) + # two sequential reads + _write_stream(str_io, a_str, a_str) + el = c_reader.read_numeric() + assert_equal(el, val) + el = c_reader.read_numeric() + assert_equal(el, val) + + +def test_read_numeric_writeable(): + # make reader-like thing + str_io = BytesIO() + r = _make_readerlike(str_io, '<') + c_reader = m5u.VarReader5(r) + dt = np.dtype('' + rdr.mat_stream.read(4) # presumably byte padding + mdict = read_minimat_vars(rdr) + fp.close() + return mdict + + +def test_jottings(): + # example + fname = os.path.join(test_data_path, 'parabola.mat') + read_workspace_vars(fname) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_mio_utils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_mio_utils.py new file mode 100644 index 0000000000000000000000000000000000000000..1d19a9797faa2221307a7330b69fffa26410f624 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_mio_utils.py @@ -0,0 +1,45 @@ +""" Testing + +""" + +import numpy as np + +from numpy.testing import assert_array_equal, assert_ + +from scipy.io.matlab._mio_utils import squeeze_element, chars_to_strings + + +def test_squeeze_element(): + a = np.zeros((1,3)) + assert_array_equal(np.squeeze(a), squeeze_element(a)) + # 0-D output from squeeze gives scalar + sq_int = squeeze_element(np.zeros((1,1), dtype=float)) + assert_(isinstance(sq_int, float)) + # Unless it's a structured array + sq_sa = squeeze_element(np.zeros((1,1),dtype=[('f1', 'f')])) + assert_(isinstance(sq_sa, np.ndarray)) + # Squeezing empty arrays maintain their dtypes. + sq_empty = squeeze_element(np.empty(0, np.uint8)) + assert sq_empty.dtype == np.uint8 + + +def test_chars_strings(): + # chars as strings + strings = ['learn ', 'python', 'fast ', 'here '] + str_arr = np.array(strings, dtype='U6') # shape (4,) + chars = [list(s) for s in strings] + char_arr = np.array(chars, dtype='U1') # shape (4,6) + assert_array_equal(chars_to_strings(char_arr), str_arr) + ca2d = char_arr.reshape((2,2,6)) + sa2d = str_arr.reshape((2,2)) + assert_array_equal(chars_to_strings(ca2d), sa2d) + ca3d = char_arr.reshape((1,2,2,6)) + sa3d = str_arr.reshape((1,2,2)) + assert_array_equal(chars_to_strings(ca3d), sa3d) + # Fortran ordered arrays + char_arrf = np.array(chars, dtype='U1', order='F') # shape (4,6) + assert_array_equal(chars_to_strings(char_arrf), str_arr) + # empty array + arr = np.array([['']], dtype='U1') + out_arr = np.array([''], dtype='U1') + assert_array_equal(chars_to_strings(arr), out_arr) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_miobase.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_miobase.py new file mode 100644 index 0000000000000000000000000000000000000000..d8c8eb2a56aaa1d1de77bfb90c859ed0af0b7bc4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_miobase.py @@ -0,0 +1,32 @@ +""" Testing miobase module +""" + +import numpy as np + +from numpy.testing import assert_equal +from pytest import raises as assert_raises + +from scipy.io.matlab._miobase import matdims + + +def test_matdims(): + # Test matdims dimension finder + assert_equal(matdims(np.array(1)), (1, 1)) # NumPy scalar + assert_equal(matdims(np.array([1])), (1, 1)) # 1-D array, 1 element + assert_equal(matdims(np.array([1,2])), (2, 1)) # 1-D array, 2 elements + assert_equal(matdims(np.array([[2],[3]])), (2, 1)) # 2-D array, column vector + assert_equal(matdims(np.array([[2,3]])), (1, 2)) # 2-D array, row vector + # 3d array, rowish vector + assert_equal(matdims(np.array([[[2,3]]])), (1, 1, 2)) + assert_equal(matdims(np.array([])), (0, 0)) # empty 1-D array + assert_equal(matdims(np.array([[]])), (1, 0)) # empty 2-D array + assert_equal(matdims(np.array([[[]]])), (1, 1, 0)) # empty 3-D array + assert_equal(matdims(np.empty((1, 0, 1))), (1, 0, 1)) # empty 3-D array + # Optional argument flips 1-D shape behavior. + assert_equal(matdims(np.array([1,2]), 'row'), (1, 2)) # 1-D array, 2 elements + # The argument has to make sense though + assert_raises(ValueError, matdims, np.array([1,2]), 'bizarre') + # Check empty sparse matrices get their own shape + from scipy.sparse import csr_array, csc_array + assert_equal(matdims(csr_array(np.zeros((3, 3)))), (3, 3)) + assert_equal(matdims(csc_array(np.zeros((2, 2)))), (2, 2)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_pathological.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_pathological.py new file mode 100644 index 0000000000000000000000000000000000000000..c5c86decb7e90f69f293e90eba74fb47dd4f1277 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_pathological.py @@ -0,0 +1,33 @@ +""" Test reading of files not conforming to matlab specification + +We try and read any file that matlab reads, these files included +""" +from os.path import dirname, join as pjoin + +from numpy.testing import assert_ +from pytest import raises as assert_raises + +from scipy.io.matlab._mio import loadmat + +TEST_DATA_PATH = pjoin(dirname(__file__), 'data') + + +def test_multiple_fieldnames(): + # Example provided by Dharhas Pothina + # Extracted using mio5.varmats_from_mat + multi_fname = pjoin(TEST_DATA_PATH, 'nasty_duplicate_fieldnames.mat') + vars = loadmat(multi_fname) + funny_names = vars['Summary'].dtype.names + assert_({'_1_Station_Q', '_2_Station_Q', + '_3_Station_Q'}.issubset(funny_names)) + + +def test_malformed1(): + # Example from gh-6072 + # Contains malformed header data, which previously resulted into a + # buffer overflow. + # + # Should raise an exception, not segfault + fname = pjoin(TEST_DATA_PATH, 'malformed1.mat') + with open(fname, 'rb') as f: + assert_raises(ValueError, loadmat, f) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_streams.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_streams.py new file mode 100644 index 0000000000000000000000000000000000000000..d8768d8e9251c6e47debeb65dff3ec056d38ee56 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/matlab/tests/test_streams.py @@ -0,0 +1,232 @@ +""" Testing + +""" + +import os +import zlib + +from io import BytesIO + + +from tempfile import mkstemp +from contextlib import contextmanager + +import numpy as np + +from numpy.testing import assert_, assert_equal +from pytest import raises as assert_raises + +from scipy.io.matlab._streams import (make_stream, + GenericStream, ZlibInputStream, + _read_into, _read_string, BLOCK_SIZE) + + +@contextmanager +def setup_test_file(): + val = b'a\x00string' + fd, fname = mkstemp() + + with os.fdopen(fd, 'wb') as fs: + fs.write(val) + with open(fname, 'rb') as fs: + gs = BytesIO(val) + cs = BytesIO(val) + yield fs, gs, cs + os.unlink(fname) + + +def test_make_stream(): + with setup_test_file() as (fs, gs, cs): + # test stream initialization + assert_(isinstance(make_stream(gs), GenericStream)) + + +def test_tell_seek(): + with setup_test_file() as (fs, gs, cs): + for s in (fs, gs, cs): + st = make_stream(s) + res = st.seek(0) + assert_equal(res, 0) + assert_equal(st.tell(), 0) + res = st.seek(5) + assert_equal(res, 0) + assert_equal(st.tell(), 5) + res = st.seek(2, 1) + assert_equal(res, 0) + assert_equal(st.tell(), 7) + res = st.seek(-2, 2) + assert_equal(res, 0) + assert_equal(st.tell(), 6) + + +def test_read(): + with setup_test_file() as (fs, gs, cs): + for s in (fs, gs, cs): + st = make_stream(s) + st.seek(0) + res = st.read(-1) + assert_equal(res, b'a\x00string') + st.seek(0) + res = st.read(4) + assert_equal(res, b'a\x00st') + # read into + st.seek(0) + res = _read_into(st, 4) + assert_equal(res, b'a\x00st') + res = _read_into(st, 4) + assert_equal(res, b'ring') + assert_raises(OSError, _read_into, st, 2) + # read alloc + st.seek(0) + res = _read_string(st, 4) + assert_equal(res, b'a\x00st') + res = _read_string(st, 4) + assert_equal(res, b'ring') + assert_raises(OSError, _read_string, st, 2) + + +class TestZlibInputStream: + def _get_data(self, size): + data = np.random.randint(0, 256, size).astype(np.uint8).tobytes() + compressed_data = zlib.compress(data) + stream = BytesIO(compressed_data) + return stream, len(compressed_data), data + + def test_read(self): + SIZES = [0, 1, 10, BLOCK_SIZE//2, BLOCK_SIZE-1, + BLOCK_SIZE, BLOCK_SIZE+1, 2*BLOCK_SIZE-1] + + READ_SIZES = [BLOCK_SIZE//2, BLOCK_SIZE-1, + BLOCK_SIZE, BLOCK_SIZE+1] + + def check(size, read_size): + compressed_stream, compressed_data_len, data = self._get_data(size) + stream = ZlibInputStream(compressed_stream, compressed_data_len) + data2 = b'' + so_far = 0 + while True: + block = stream.read(min(read_size, + size - so_far)) + if not block: + break + so_far += len(block) + data2 += block + assert_equal(data, data2) + + for size in SIZES: + for read_size in READ_SIZES: + check(size, read_size) + + def test_read_max_length(self): + size = 1234 + data = np.random.randint(0, 256, size).astype(np.uint8).tobytes() + compressed_data = zlib.compress(data) + compressed_stream = BytesIO(compressed_data + b"abbacaca") + stream = ZlibInputStream(compressed_stream, len(compressed_data)) + + stream.read(len(data)) + assert_equal(compressed_stream.tell(), len(compressed_data)) + + assert_raises(OSError, stream.read, 1) + + def test_read_bad_checksum(self): + data = np.random.randint(0, 256, 10).astype(np.uint8).tobytes() + compressed_data = zlib.compress(data) + + # break checksum + compressed_data = (compressed_data[:-1] + + bytes([(compressed_data[-1] + 1) & 255])) + + compressed_stream = BytesIO(compressed_data) + stream = ZlibInputStream(compressed_stream, len(compressed_data)) + + assert_raises(zlib.error, stream.read, len(data)) + + def test_seek(self): + compressed_stream, compressed_data_len, data = self._get_data(1024) + + stream = ZlibInputStream(compressed_stream, compressed_data_len) + + stream.seek(123) + p = 123 + assert_equal(stream.tell(), p) + d1 = stream.read(11) + assert_equal(d1, data[p:p+11]) + + stream.seek(321, 1) + p = 123+11+321 + assert_equal(stream.tell(), p) + d2 = stream.read(21) + assert_equal(d2, data[p:p+21]) + + stream.seek(641, 0) + p = 641 + assert_equal(stream.tell(), p) + d3 = stream.read(11) + assert_equal(d3, data[p:p+11]) + + assert_raises(OSError, stream.seek, 10, 2) + assert_raises(OSError, stream.seek, -1, 1) + assert_raises(ValueError, stream.seek, 1, 123) + + stream.seek(10000, 1) + assert_raises(OSError, stream.read, 12) + + def test_seek_bad_checksum(self): + data = np.random.randint(0, 256, 10).astype(np.uint8).tobytes() + compressed_data = zlib.compress(data) + + # break checksum + compressed_data = (compressed_data[:-1] + + bytes([(compressed_data[-1] + 1) & 255])) + + compressed_stream = BytesIO(compressed_data) + stream = ZlibInputStream(compressed_stream, len(compressed_data)) + + assert_raises(zlib.error, stream.seek, len(data)) + + def test_all_data_read(self): + compressed_stream, compressed_data_len, data = self._get_data(1024) + stream = ZlibInputStream(compressed_stream, compressed_data_len) + assert_(not stream.all_data_read()) + stream.seek(512) + assert_(not stream.all_data_read()) + stream.seek(1024) + assert_(stream.all_data_read()) + + def test_all_data_read_overlap(self): + COMPRESSION_LEVEL = 6 + + data = np.arange(33707000).astype(np.uint8).tobytes() + compressed_data = zlib.compress(data, COMPRESSION_LEVEL) + compressed_data_len = len(compressed_data) + + # check that part of the checksum overlaps + assert_(compressed_data_len == BLOCK_SIZE + 2) + + compressed_stream = BytesIO(compressed_data) + stream = ZlibInputStream(compressed_stream, compressed_data_len) + assert_(not stream.all_data_read()) + stream.seek(len(data)) + assert_(stream.all_data_read()) + + def test_all_data_read_bad_checksum(self): + COMPRESSION_LEVEL = 6 + + data = np.arange(33707000).astype(np.uint8).tobytes() + compressed_data = zlib.compress(data, COMPRESSION_LEVEL) + compressed_data_len = len(compressed_data) + + # check that part of the checksum overlaps + assert_(compressed_data_len == BLOCK_SIZE + 2) + + # break checksum + compressed_data = (compressed_data[:-1] + + bytes([(compressed_data[-1] + 1) & 255])) + + compressed_stream = BytesIO(compressed_data) + stream = ZlibInputStream(compressed_stream, compressed_data_len) + assert_(not stream.all_data_read()) + stream.seek(len(data)) + + assert_raises(zlib.error, stream.all_data_read) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/mmio.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/mmio.py new file mode 100644 index 0000000000000000000000000000000000000000..67cf0684cbf9468468027957a5b7f3da2c43c845 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/mmio.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = ["mminfo", "mmread", "mmwrite"] # noqa: F822 + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="io", module="mmio", + private_modules=["_mmio"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/netcdf.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/netcdf.py new file mode 100644 index 0000000000000000000000000000000000000000..c1f119dd2bad72d772c3d1db6ceec9fd3d91316d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/netcdf.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.io` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import 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io_lock(): + return threading.Lock() + + +def test_fortranfiles_read(io_lock): + for filename in iglob(path.join(DATA_PATH, "fortran-*-*x*x*.dat")): + m = re.search(r'fortran-([^-]+)-(\d+)x(\d+)x(\d+).dat', filename, re.I) + if not m: + raise RuntimeError(f"Couldn't match {filename} filename to regex") + + dims = (int(m.group(2)), int(m.group(3)), int(m.group(4))) + + dtype = m.group(1).replace('s', '<') + + with io_lock: + f = FortranFile(filename, 'r', ' 0] = 1 + info = (2, 2, 3, 'coordinate', 'pattern', 'general') + mmwrite(self.fn, a, field='pattern') + assert_equal(mminfo(self.fn), info) + b = mmread(self.fn, spmatrix=False) + assert_array_almost_equal(p, b.toarray()) + assert not scipy.sparse.isspmatrix(b) + + b = mmread(self.fn, spmatrix=True) + assert scipy.sparse.isspmatrix(b) + b = mmread(self.fn) # chk default + assert scipy.sparse.isspmatrix(b) + + def test_gh13634_non_skew_symmetric_int(self): + a = scipy.sparse.csr_array([[1, 2], [-2, 99]], dtype=np.int32) + self.check_exact(a, (2, 2, 4, 'coordinate', 'integer', 'general')) + + def test_gh13634_non_skew_symmetric_float(self): + a = scipy.sparse.csr_array([[1, 2], [-2, 99.]], dtype=np.float32) + self.check(a, (2, 2, 4, 'coordinate', 'real', 'general')) + + +_32bit_integer_dense_example = '''\ +%%MatrixMarket matrix array integer general +2 2 +2147483647 +2147483646 +2147483647 +2147483646 +''' + +_32bit_integer_sparse_example = '''\ +%%MatrixMarket matrix coordinate integer symmetric +2 2 2 +1 1 2147483647 +2 2 2147483646 +''' + +_64bit_integer_dense_example = '''\ +%%MatrixMarket matrix array integer general +2 2 + 2147483648 +-9223372036854775806 + -2147483648 + 9223372036854775807 +''' + +_64bit_integer_sparse_general_example = '''\ +%%MatrixMarket matrix coordinate integer general +2 2 3 +1 1 2147483648 +1 2 9223372036854775807 +2 2 9223372036854775807 +''' + +_64bit_integer_sparse_symmetric_example = '''\ +%%MatrixMarket matrix coordinate integer symmetric +2 2 3 +1 1 2147483648 +1 2 -9223372036854775807 +2 2 9223372036854775807 +''' + +_64bit_integer_sparse_skew_example = '''\ +%%MatrixMarket matrix coordinate integer skew-symmetric +2 2 3 +1 1 2147483648 +1 2 -9223372036854775807 +2 2 9223372036854775807 +''' + +_over64bit_integer_dense_example = '''\ +%%MatrixMarket matrix array integer general +2 2 + 2147483648 +9223372036854775807 + 2147483648 +9223372036854775808 +''' + +_over64bit_integer_sparse_example = '''\ +%%MatrixMarket matrix coordinate integer symmetric +2 2 2 +1 1 2147483648 +2 2 19223372036854775808 +''' + + +class TestMMIOReadLargeIntegers: + def setup_method(self): + self.tmpdir = mkdtemp(suffix=str(threading.get_native_id())) + self.fn = os.path.join(self.tmpdir, 'testfile.mtx') + + def teardown_method(self): + shutil.rmtree(self.tmpdir) + + def check_read(self, example, a, info, dense, over32, over64): + with open(self.fn, 'w') as f: + f.write(example) + assert_equal(mminfo(self.fn), info) + if ((over32 and (np.intp(0).itemsize < 8) and mmwrite == scipy.io._mmio.mmwrite) + or over64): + assert_raises(OverflowError, mmread, self.fn) + else: + b = mmread(self.fn, spmatrix=False) + if not dense: + b = b.toarray() + assert_equal(a, b) + + def test_read_32bit_integer_dense(self): + a = array([[2**31-1, 2**31-1], + [2**31-2, 2**31-2]], dtype=np.int64) + self.check_read(_32bit_integer_dense_example, + a, + (2, 2, 4, 'array', 'integer', 'general'), + dense=True, + over32=False, + over64=False) + + def test_read_32bit_integer_sparse(self): + a = array([[2**31-1, 0], + [0, 2**31-2]], dtype=np.int64) + self.check_read(_32bit_integer_sparse_example, + a, + (2, 2, 2, 'coordinate', 'integer', 'symmetric'), + dense=False, + over32=False, + over64=False) + + def test_read_64bit_integer_dense(self): + a = array([[2**31, -2**31], + [-2**63+2, 2**63-1]], dtype=np.int64) + self.check_read(_64bit_integer_dense_example, + a, + (2, 2, 4, 'array', 'integer', 'general'), + dense=True, + over32=True, + over64=False) + + def test_read_64bit_integer_sparse_general(self): + a = array([[2**31, 2**63-1], + [0, 2**63-1]], dtype=np.int64) + self.check_read(_64bit_integer_sparse_general_example, + a, + (2, 2, 3, 'coordinate', 'integer', 'general'), + dense=False, + over32=True, + over64=False) + + def test_read_64bit_integer_sparse_symmetric(self): + a = array([[2**31, -2**63+1], + [-2**63+1, 2**63-1]], dtype=np.int64) + self.check_read(_64bit_integer_sparse_symmetric_example, + a, + (2, 2, 3, 'coordinate', 'integer', 'symmetric'), + dense=False, + over32=True, + over64=False) + + def test_read_64bit_integer_sparse_skew(self): + a = array([[2**31, -2**63+1], + [2**63-1, 2**63-1]], dtype=np.int64) + self.check_read(_64bit_integer_sparse_skew_example, + a, + (2, 2, 3, 'coordinate', 'integer', 'skew-symmetric'), + dense=False, + over32=True, + over64=False) + + def test_read_over64bit_integer_dense(self): + self.check_read(_over64bit_integer_dense_example, + None, + (2, 2, 4, 'array', 'integer', 'general'), + dense=True, + over32=True, + over64=True) + + def test_read_over64bit_integer_sparse(self): + self.check_read(_over64bit_integer_sparse_example, + None, + (2, 2, 2, 'coordinate', 'integer', 'symmetric'), + dense=False, + over32=True, + over64=True) + + +_general_example = '''\ +%%MatrixMarket matrix coordinate real general +%================================================================================= +% +% This ASCII file represents a sparse MxN matrix with L +% nonzeros in the following Matrix Market format: +% +% +----------------------------------------------+ +% |%%MatrixMarket matrix coordinate real general | <--- header line +% |% | <--+ +% |% comments | |-- 0 or more comment lines +% |% | <--+ +% | M N L | <--- rows, columns, entries +% | I1 J1 A(I1, J1) | <--+ +% | I2 J2 A(I2, J2) | | +% | I3 J3 A(I3, J3) | |-- L lines +% | . . . | | +% | IL JL A(IL, JL) | <--+ +% +----------------------------------------------+ +% +% Indices are 1-based, i.e. A(1,1) is the first element. +% +%================================================================================= + 5 5 8 + 1 1 1.000e+00 + 2 2 1.050e+01 + 3 3 1.500e-02 + 1 4 6.000e+00 + 4 2 2.505e+02 + 4 4 -2.800e+02 + 4 5 3.332e+01 + 5 5 1.200e+01 +''' + +_hermitian_example = '''\ +%%MatrixMarket matrix coordinate complex hermitian + 5 5 7 + 1 1 1.0 0 + 2 2 10.5 0 + 4 2 250.5 22.22 + 3 3 1.5e-2 0 + 4 4 -2.8e2 0 + 5 5 12. 0 + 5 4 0 33.32 +''' + +_skew_example = '''\ +%%MatrixMarket matrix coordinate real skew-symmetric + 5 5 7 + 1 1 1.0 + 2 2 10.5 + 4 2 250.5 + 3 3 1.5e-2 + 4 4 -2.8e2 + 5 5 12. + 5 4 0 +''' + +_symmetric_example = '''\ +%%MatrixMarket matrix coordinate real symmetric + 5 5 7 + 1 1 1.0 + 2 2 10.5 + 4 2 250.5 + 3 3 1.5e-2 + 4 4 -2.8e2 + 5 5 12. + 5 4 8 +''' + +_symmetric_pattern_example = '''\ +%%MatrixMarket matrix coordinate pattern symmetric + 5 5 7 + 1 1 + 2 2 + 4 2 + 3 3 + 4 4 + 5 5 + 5 4 +''' + +# example (without comment lines) from Figure 1 in +# https://math.nist.gov/MatrixMarket/reports/MMformat.ps +_empty_lines_example = '''\ +%%MatrixMarket MATRIX Coordinate Real General + + 5 5 8 + +1 1 1.0 +2 2 10.5 +3 3 1.5e-2 +4 4 -2.8E2 +5 5 12. + 1 4 6 + 4 2 250.5 + 4 5 33.32 + +''' + + +class TestMMIOCoordinate: + def setup_method(self): + self.tmpdir = mkdtemp(suffix=str(threading.get_native_id())) + self.fn = os.path.join(self.tmpdir, 'testfile.mtx') + + def teardown_method(self): + shutil.rmtree(self.tmpdir) + + def check_read(self, example, a, info): + f = open(self.fn, 'w') + f.write(example) + f.close() + assert_equal(mminfo(self.fn), info) + b = mmread(self.fn, spmatrix=False).toarray() + assert_array_almost_equal(a, b) + + def test_read_general(self): + a = [[1, 0, 0, 6, 0], + [0, 10.5, 0, 0, 0], + [0, 0, .015, 0, 0], + [0, 250.5, 0, -280, 33.32], + [0, 0, 0, 0, 12]] + self.check_read(_general_example, a, + (5, 5, 8, 'coordinate', 'real', 'general')) + + def test_read_hermitian(self): + a = [[1, 0, 0, 0, 0], + [0, 10.5, 0, 250.5 - 22.22j, 0], + [0, 0, .015, 0, 0], + [0, 250.5 + 22.22j, 0, -280, -33.32j], + [0, 0, 0, 33.32j, 12]] + self.check_read(_hermitian_example, a, + (5, 5, 7, 'coordinate', 'complex', 'hermitian')) + + def test_read_skew(self): + a = [[1, 0, 0, 0, 0], + [0, 10.5, 0, -250.5, 0], + [0, 0, .015, 0, 0], + [0, 250.5, 0, -280, 0], + [0, 0, 0, 0, 12]] + self.check_read(_skew_example, a, + (5, 5, 7, 'coordinate', 'real', 'skew-symmetric')) + + def test_read_symmetric(self): + a = [[1, 0, 0, 0, 0], + [0, 10.5, 0, 250.5, 0], + [0, 0, .015, 0, 0], + [0, 250.5, 0, -280, 8], + [0, 0, 0, 8, 12]] + self.check_read(_symmetric_example, a, + (5, 5, 7, 'coordinate', 'real', 'symmetric')) + + def test_read_symmetric_pattern(self): + a = [[1, 0, 0, 0, 0], + [0, 1, 0, 1, 0], + [0, 0, 1, 0, 0], + [0, 1, 0, 1, 1], + [0, 0, 0, 1, 1]] + self.check_read(_symmetric_pattern_example, a, + (5, 5, 7, 'coordinate', 'pattern', 'symmetric')) + + def test_read_empty_lines(self): + a = [[1, 0, 0, 6, 0], + [0, 10.5, 0, 0, 0], + [0, 0, .015, 0, 0], + [0, 250.5, 0, -280, 33.32], + [0, 0, 0, 0, 12]] + self.check_read(_empty_lines_example, a, + (5, 5, 8, 'coordinate', 'real', 'general')) + + def test_empty_write_read(self): + # https://github.com/scipy/scipy/issues/1410 (Trac #883) + + b = scipy.sparse.coo_array((10, 10)) + mmwrite(self.fn, b) + + assert_equal(mminfo(self.fn), + (10, 10, 0, 'coordinate', 'real', 'symmetric')) + a = b.toarray() + b = mmread(self.fn, spmatrix=False).toarray() + assert_array_almost_equal(a, b) + + def test_bzip2_py3(self): + # test if fix for #2152 works + try: + # bz2 module isn't always built when building Python. + import bz2 + except ImportError: + return + I = array([0, 0, 1, 2, 3, 3, 3, 4]) + J = array([0, 3, 1, 2, 1, 3, 4, 4]) + V = array([1.0, 6.0, 10.5, 0.015, 250.5, -280.0, 33.32, 12.0]) + + b = scipy.sparse.coo_array((V, (I, J)), shape=(5, 5)) + + mmwrite(self.fn, b) + + fn_bzip2 = f"{self.fn}.bz2" + with open(self.fn, 'rb') as f_in: + f_out = bz2.BZ2File(fn_bzip2, 'wb') + f_out.write(f_in.read()) + f_out.close() + + a = mmread(fn_bzip2, spmatrix=False).toarray() + assert_array_almost_equal(a, b.toarray()) + + def test_gzip_py3(self): + # test if fix for #2152 works + try: + # gzip module can be missing from Python installation + import gzip + except ImportError: + return + I = array([0, 0, 1, 2, 3, 3, 3, 4]) + J = array([0, 3, 1, 2, 1, 3, 4, 4]) + V = array([1.0, 6.0, 10.5, 0.015, 250.5, -280.0, 33.32, 12.0]) + + b = scipy.sparse.coo_array((V, (I, J)), shape=(5, 5)) + + mmwrite(self.fn, b) + + fn_gzip = f"{self.fn}.gz" + with open(self.fn, 'rb') as f_in: + f_out = gzip.open(fn_gzip, 'wb') + f_out.write(f_in.read()) + f_out.close() + + a = mmread(fn_gzip, spmatrix=False).toarray() + assert_array_almost_equal(a, b.toarray()) + + def test_real_write_read(self): + I = array([0, 0, 1, 2, 3, 3, 3, 4]) + J = array([0, 3, 1, 2, 1, 3, 4, 4]) + V = array([1.0, 6.0, 10.5, 0.015, 250.5, -280.0, 33.32, 12.0]) + + b = scipy.sparse.coo_array((V, (I, J)), shape=(5, 5)) + + mmwrite(self.fn, b) + + assert_equal(mminfo(self.fn), + (5, 5, 8, 'coordinate', 'real', 'general')) + a = b.toarray() + b = mmread(self.fn, spmatrix=False).toarray() + assert_array_almost_equal(a, b) + + def test_complex_write_read(self): + I = array([0, 0, 1, 2, 3, 3, 3, 4]) + J = array([0, 3, 1, 2, 1, 3, 4, 4]) + V = array([1.0 + 3j, 6.0 + 2j, 10.50 + 0.9j, 0.015 + -4.4j, + 250.5 + 0j, -280.0 + 5j, 33.32 + 6.4j, 12.00 + 0.8j]) + + b = scipy.sparse.coo_array((V, (I, J)), shape=(5, 5)) + + mmwrite(self.fn, b) + + assert_equal(mminfo(self.fn), + (5, 5, 8, 'coordinate', 'complex', 'general')) + a = b.toarray() + b = mmread(self.fn, spmatrix=False).toarray() + assert_array_almost_equal(a, b) + + def test_sparse_formats(self, tmp_path): + # Note: `tmp_path` is a pytest fixture, it handles cleanup + tmpdir = tmp_path / 'sparse_formats' + tmpdir.mkdir() + + mats = [] + I = array([0, 0, 1, 2, 3, 3, 3, 4]) + J = array([0, 3, 1, 2, 1, 3, 4, 4]) + + V = array([1.0, 6.0, 10.5, 0.015, 250.5, -280.0, 33.32, 12.0]) + mats.append(scipy.sparse.coo_array((V, (I, J)), shape=(5, 5))) + + V = array([1.0 + 3j, 6.0 + 2j, 10.50 + 0.9j, 0.015 + -4.4j, + 250.5 + 0j, -280.0 + 5j, 33.32 + 6.4j, 12.00 + 0.8j]) + mats.append(scipy.sparse.coo_array((V, (I, J)), shape=(5, 5))) + + for mat in mats: + expected = mat.toarray() + for fmt in ['csr', 'csc', 'coo']: + fname = tmpdir / (fmt + '.mtx') + mmwrite(fname, mat.asformat(fmt)) + result = mmread(fname, spmatrix=False).toarray() + assert_array_almost_equal(result, expected) + + def test_precision(self): + test_values = [pi] + [10**(i) for i in range(0, -10, -1)] + test_precisions = range(1, 10) + for value in test_values: + for precision in test_precisions: + # construct sparse matrix with test value at last main diagonal + n = 10**precision + 1 + A = scipy.sparse.dok_array((n, n)) + A[n-1, n-1] = value + # write matrix with test precision and read again + mmwrite(self.fn, A, precision=precision) + A = scipy.io.mmread(self.fn, spmatrix=False) + # check for right entries in matrix + assert_array_equal(A.row, [n-1]) + assert_array_equal(A.col, [n-1]) + assert_allclose(A.data, [float('%%.%dg' % precision % value)]) + + def test_bad_number_of_coordinate_header_fields(self): + s = """\ + %%MatrixMarket matrix coordinate real general + 5 5 8 999 + 1 1 1.000e+00 + 2 2 1.050e+01 + 3 3 1.500e-02 + 1 4 6.000e+00 + 4 2 2.505e+02 + 4 4 -2.800e+02 + 4 5 3.332e+01 + 5 5 1.200e+01 + """ + text = textwrap.dedent(s).encode('ascii') + with pytest.raises(ValueError, match='not of length 3'): + scipy.io.mmread(io.BytesIO(text)) + + +def test_gh11389(): + mmread(io.StringIO("%%MatrixMarket matrix coordinate complex symmetric\n" + " 1 1 1\n" + "1 1 -2.1846000000000e+02 0.0000000000000e+00"), + spmatrix=False) + + +def test_gh18123(tmp_path): + lines = [" %%MatrixMarket matrix coordinate real general\n", + "5 5 3\n", + "2 3 1.0\n", + "3 4 2.0\n", + "3 5 3.0\n"] + test_file = tmp_path / "test.mtx" + with open(test_file, "w") as f: + f.writelines(lines) + mmread(test_file, spmatrix=False) + +def test_mtx_append(tmp_path): + a = mmread(io.StringIO("%%MatrixMarket matrix coordinate complex symmetric\n" + " 1 1 1\n" + "1 1 -2.1846000000000e+02 0.0000000000000e+00"), + spmatrix=False) + test_writefile = tmp_path / "test_mtx" + test_readfile = tmp_path / "test_mtx.mtx" + mmwrite(test_writefile, a) + mmread(test_readfile, spmatrix=False) + + +def test_threadpoolctl(): + try: + import threadpoolctl + if not hasattr(threadpoolctl, "register"): + pytest.skip("threadpoolctl too old") + return + except ImportError: + pytest.skip("no threadpoolctl") + return + + with threadpoolctl.threadpool_limits(limits=4): + assert_equal(fmm.PARALLELISM, 4) + + with threadpoolctl.threadpool_limits(limits=2, user_api='scipy'): + assert_equal(fmm.PARALLELISM, 2) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/tests/test_netcdf.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/tests/test_netcdf.py new file mode 100644 index 0000000000000000000000000000000000000000..161406076d0b5078e8e11aa5762b7715cd83c4a7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/tests/test_netcdf.py @@ -0,0 +1,550 @@ +''' Tests for netcdf ''' +import os +from os.path import join as pjoin, dirname +import shutil +import tempfile +import warnings +from io import BytesIO +from glob import glob +from contextlib import contextmanager + +import numpy as np +from numpy.testing import (assert_, assert_allclose, assert_equal, + break_cycles, suppress_warnings, IS_PYPY) +import pytest +from pytest import raises as assert_raises + +from scipy.io import netcdf_file +from scipy._lib._tmpdirs import in_tempdir + +TEST_DATA_PATH = pjoin(dirname(__file__), 'data') + +N_EG_ELS = 11 # number of elements for example variable +VARTYPE_EG = 'b' # var type for example variable + + +pytestmark = pytest.mark.thread_unsafe + + +@contextmanager +def make_simple(*args, **kwargs): + f = netcdf_file(*args, **kwargs) + f.history = 'Created for a test' + f.createDimension('time', N_EG_ELS) + time = f.createVariable('time', VARTYPE_EG, ('time',)) + time[:] = np.arange(N_EG_ELS) + time.units = 'days since 2008-01-01' + f.flush() + yield f + f.close() + + +def check_simple(ncfileobj): + '''Example fileobj tests ''' + assert_equal(ncfileobj.history, b'Created for a test') + time = ncfileobj.variables['time'] + assert_equal(time.units, b'days since 2008-01-01') + assert_equal(time.shape, (N_EG_ELS,)) + assert_equal(time[-1], N_EG_ELS-1) + +def assert_mask_matches(arr, expected_mask): + ''' + Asserts that the mask of arr is effectively the same as expected_mask. + + In contrast to numpy.ma.testutils.assert_mask_equal, this function allows + testing the 'mask' of a standard numpy array (the mask in this case is treated + as all False). + + Parameters + ---------- + arr : ndarray or MaskedArray + Array to test. + expected_mask : array_like of booleans + A list giving the expected mask. + ''' + + mask = np.ma.getmaskarray(arr) + assert_equal(mask, expected_mask) + + +def test_read_write_files(): + # test round trip for example file + cwd = os.getcwd() + try: + tmpdir = tempfile.mkdtemp() + os.chdir(tmpdir) + with make_simple('simple.nc', 'w') as f: + pass + # read the file we just created in 'a' mode + with netcdf_file('simple.nc', 'a') as f: + check_simple(f) + # add something + f._attributes['appendRan'] = 1 + + # To read the NetCDF file we just created:: + with netcdf_file('simple.nc') as f: + # Using mmap is the default (but not on pypy) + assert_equal(f.use_mmap, not IS_PYPY) + check_simple(f) + assert_equal(f._attributes['appendRan'], 1) + + # Read it in append (and check mmap is off) + with netcdf_file('simple.nc', 'a') as f: + assert_(not f.use_mmap) + check_simple(f) + assert_equal(f._attributes['appendRan'], 1) + + # Now without mmap + with netcdf_file('simple.nc', mmap=False) as f: + # Using mmap is the default + assert_(not f.use_mmap) + check_simple(f) + + # To read the NetCDF file we just created, as file object, no + # mmap. When n * n_bytes(var_type) is not divisible by 4, this + # raised an error in pupynere 1.0.12 and scipy rev 5893, because + # calculated vsize was rounding up in units of 4 - see + # https://www.unidata.ucar.edu/software/netcdf/guide_toc.html + with open('simple.nc', 'rb') as fobj: + with netcdf_file(fobj) as f: + # by default, don't use mmap for file-like + assert_(not f.use_mmap) + check_simple(f) + + # Read file from fileobj, with mmap + with suppress_warnings() as sup: + if IS_PYPY: + sup.filter(RuntimeWarning, + "Cannot close a netcdf_file opened with mmap=True.*") + with open('simple.nc', 'rb') as fobj: + with netcdf_file(fobj, mmap=True) as f: + assert_(f.use_mmap) + check_simple(f) + + # Again read it in append mode (adding another att) + with open('simple.nc', 'r+b') as fobj: + with netcdf_file(fobj, 'a') as f: + assert_(not f.use_mmap) + check_simple(f) + f.createDimension('app_dim', 1) + var = f.createVariable('app_var', 'i', ('app_dim',)) + var[:] = 42 + + # And... check that app_var made it in... + with netcdf_file('simple.nc') as f: + check_simple(f) + assert_equal(f.variables['app_var'][:], 42) + + finally: + if IS_PYPY: + # windows cannot remove a dead file held by a mmap + # that has not been collected in PyPy + break_cycles() + break_cycles() + os.chdir(cwd) + shutil.rmtree(tmpdir) + + +def test_read_write_sio(): + eg_sio1 = BytesIO() + with make_simple(eg_sio1, 'w'): + str_val = eg_sio1.getvalue() + + eg_sio2 = BytesIO(str_val) + with netcdf_file(eg_sio2) as f2: + check_simple(f2) + + # Test that error is raised if attempting mmap for sio + eg_sio3 = BytesIO(str_val) + assert_raises(ValueError, netcdf_file, eg_sio3, 'r', True) + # Test 64-bit offset write / read + eg_sio_64 = BytesIO() + with make_simple(eg_sio_64, 'w', version=2) as f_64: + str_val = eg_sio_64.getvalue() + + eg_sio_64 = BytesIO(str_val) + with netcdf_file(eg_sio_64) as f_64: + check_simple(f_64) + assert_equal(f_64.version_byte, 2) + # also when version 2 explicitly specified + eg_sio_64 = BytesIO(str_val) + with netcdf_file(eg_sio_64, version=2) as f_64: + check_simple(f_64) + assert_equal(f_64.version_byte, 2) + + +def test_bytes(): + raw_file = BytesIO() + f = netcdf_file(raw_file, mode='w') + # Dataset only has a single variable, dimension and attribute to avoid + # any ambiguity related to order. + f.a = 'b' + f.createDimension('dim', 1) + var = f.createVariable('var', np.int16, ('dim',)) + var[0] = -9999 + var.c = 'd' + f.sync() + + actual = raw_file.getvalue() + + expected = (b'CDF\x01' + b'\x00\x00\x00\x00' + b'\x00\x00\x00\x0a' + b'\x00\x00\x00\x01' + b'\x00\x00\x00\x03' + b'dim\x00' + b'\x00\x00\x00\x01' + b'\x00\x00\x00\x0c' + b'\x00\x00\x00\x01' + b'\x00\x00\x00\x01' + b'a\x00\x00\x00' + b'\x00\x00\x00\x02' + b'\x00\x00\x00\x01' + b'b\x00\x00\x00' + b'\x00\x00\x00\x0b' + b'\x00\x00\x00\x01' + b'\x00\x00\x00\x03' + b'var\x00' + b'\x00\x00\x00\x01' + b'\x00\x00\x00\x00' + b'\x00\x00\x00\x0c' + b'\x00\x00\x00\x01' + b'\x00\x00\x00\x01' + b'c\x00\x00\x00' + b'\x00\x00\x00\x02' + b'\x00\x00\x00\x01' + b'd\x00\x00\x00' + b'\x00\x00\x00\x03' + b'\x00\x00\x00\x04' + b'\x00\x00\x00\x78' + b'\xd8\xf1\x80\x01') + + assert_equal(actual, expected) + + +def test_encoded_fill_value(): + with netcdf_file(BytesIO(), mode='w') as f: + f.createDimension('x', 1) + var = f.createVariable('var', 'S1', ('x',)) + assert_equal(var._get_encoded_fill_value(), b'\x00') + var._FillValue = b'\x01' + assert_equal(var._get_encoded_fill_value(), b'\x01') + var._FillValue = b'\x00\x00' # invalid, wrong size + assert_equal(var._get_encoded_fill_value(), b'\x00') + + +def test_read_example_data(): + # read any example data files + for fname in glob(pjoin(TEST_DATA_PATH, '*.nc')): + with netcdf_file(fname, 'r'): + pass + with netcdf_file(fname, 'r', mmap=False): + pass + + +def test_itemset_no_segfault_on_readonly(): + # Regression test for ticket #1202. + # Open the test file in read-only mode. + + filename = pjoin(TEST_DATA_PATH, 'example_1.nc') + with suppress_warnings() as sup: + message = ("Cannot close a netcdf_file opened with mmap=True, when " + "netcdf_variables or arrays referring to its data still exist") + sup.filter(RuntimeWarning, message) + with netcdf_file(filename, 'r', mmap=True) as f: + time_var = f.variables['time'] + + # time_var.assignValue(42) should raise a RuntimeError--not seg. fault! + assert_raises(RuntimeError, time_var.assignValue, 42) + + +def test_appending_issue_gh_8625(): + stream = BytesIO() + + with make_simple(stream, mode='w') as f: + f.createDimension('x', 2) + f.createVariable('x', float, ('x',)) + f.variables['x'][...] = 1 + f.flush() + contents = stream.getvalue() + + stream = BytesIO(contents) + with netcdf_file(stream, mode='a') as f: + f.variables['x'][...] = 2 + + +def test_write_invalid_dtype(): + dtypes = ['int64', 'uint64'] + if np.dtype('int').itemsize == 8: # 64-bit machines + dtypes.append('int') + if np.dtype('uint').itemsize == 8: # 64-bit machines + dtypes.append('uint') + + with netcdf_file(BytesIO(), 'w') as f: + f.createDimension('time', N_EG_ELS) + for dt in dtypes: + assert_raises(ValueError, f.createVariable, 'time', dt, ('time',)) + + +def test_flush_rewind(): + stream = BytesIO() + with make_simple(stream, mode='w') as f: + f.createDimension('x',4) # x is used in createVariable + v = f.createVariable('v', 'i2', ['x']) + v[:] = 1 + f.flush() + len_single = len(stream.getvalue()) + f.flush() + len_double = len(stream.getvalue()) + + assert_(len_single == len_double) + + +def test_dtype_specifiers(): + # Numpy 1.7.0-dev had a bug where 'i2' wouldn't work. + # Specifying np.int16 or similar only works from the same commit as this + # comment was made. + with make_simple(BytesIO(), mode='w') as f: + f.createDimension('x',4) + f.createVariable('v1', 'i2', ['x']) + f.createVariable('v2', np.int16, ['x']) + f.createVariable('v3', np.dtype(np.int16), ['x']) + + +def test_ticket_1720(): + io = BytesIO() + + items = [0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9] + + with netcdf_file(io, 'w') as f: + f.history = 'Created for a test' + f.createDimension('float_var', 10) + float_var = f.createVariable('float_var', 'f', ('float_var',)) + float_var[:] = items + float_var.units = 'metres' + f.flush() + contents = io.getvalue() + + io = BytesIO(contents) + with netcdf_file(io, 'r') as f: + assert_equal(f.history, b'Created for a test') + float_var = f.variables['float_var'] + assert_equal(float_var.units, b'metres') + assert_equal(float_var.shape, (10,)) + assert_allclose(float_var[:], items) + + +def test_mmaps_segfault(): + filename = pjoin(TEST_DATA_PATH, 'example_1.nc') + + if not IS_PYPY: + with warnings.catch_warnings(): + warnings.simplefilter("error") + with netcdf_file(filename, mmap=True) as f: + x = f.variables['lat'][:] + # should not raise warnings + del x + + def doit(): + with netcdf_file(filename, mmap=True) as f: + return f.variables['lat'][:] + + # should not crash + with suppress_warnings() as sup: + message = ("Cannot close a netcdf_file opened with mmap=True, when " + "netcdf_variables or arrays referring to its data still exist") + sup.filter(RuntimeWarning, message) + x = doit() + x.sum() + + +def test_zero_dimensional_var(): + io = BytesIO() + with make_simple(io, 'w') as f: + v = f.createVariable('zerodim', 'i2', []) + # This is checking that .isrec returns a boolean - don't simplify it + # to 'assert not ...' + assert v.isrec is False, v.isrec + f.flush() + + +def test_byte_gatts(): + # Check that global "string" atts work like they did before py3k + # unicode and general bytes confusion + with in_tempdir(): + filename = 'g_byte_atts.nc' + f = netcdf_file(filename, 'w') + f._attributes['holy'] = b'grail' + f._attributes['witch'] = 'floats' + f.close() + f = netcdf_file(filename, 'r') + assert_equal(f._attributes['holy'], b'grail') + assert_equal(f._attributes['witch'], b'floats') + f.close() + + +def test_open_append(): + # open 'w' put one attr + with in_tempdir(): + filename = 'append_dat.nc' + f = netcdf_file(filename, 'w') + f._attributes['Kilroy'] = 'was here' + f.close() + + # open again in 'a', read the att and a new one + f = netcdf_file(filename, 'a') + assert_equal(f._attributes['Kilroy'], b'was here') + f._attributes['naughty'] = b'Zoot' + f.close() + + # open yet again in 'r' and check both atts + f = netcdf_file(filename, 'r') + assert_equal(f._attributes['Kilroy'], b'was here') + assert_equal(f._attributes['naughty'], b'Zoot') + f.close() + + +def test_append_recordDimension(): + dataSize = 100 + + with in_tempdir(): + # Create file with record time dimension + with netcdf_file('withRecordDimension.nc', 'w') as f: + f.createDimension('time', None) + f.createVariable('time', 'd', ('time',)) + f.createDimension('x', dataSize) + x = f.createVariable('x', 'd', ('x',)) + x[:] = np.array(range(dataSize)) + f.createDimension('y', dataSize) + y = f.createVariable('y', 'd', ('y',)) + y[:] = np.array(range(dataSize)) + f.createVariable('testData', 'i', ('time', 'x', 'y')) + f.flush() + f.close() + + for i in range(2): + # Open the file in append mode and add data + with netcdf_file('withRecordDimension.nc', 'a') as f: + f.variables['time'].data = np.append(f.variables["time"].data, i) + f.variables['testData'][i, :, :] = np.full((dataSize, dataSize), i) + f.flush() + + # Read the file and check that append worked + with netcdf_file('withRecordDimension.nc') as f: + assert_equal(f.variables['time'][-1], i) + assert_equal(f.variables['testData'][-1, :, :].copy(), + np.full((dataSize, dataSize), i)) + assert_equal(f.variables['time'].data.shape[0], i+1) + assert_equal(f.variables['testData'].data.shape[0], i+1) + + # Read the file and check that 'data' was not saved as user defined + # attribute of testData variable during append operation + with netcdf_file('withRecordDimension.nc') as f: + with assert_raises(KeyError) as ar: + f.variables['testData']._attributes['data'] + ex = ar.value + assert_equal(ex.args[0], 'data') + +def test_maskandscale(): + t = np.linspace(20, 30, 15) + t[3] = 100 + tm = np.ma.masked_greater(t, 99) + fname = pjoin(TEST_DATA_PATH, 'example_2.nc') + with netcdf_file(fname, maskandscale=True) as f: + Temp = f.variables['Temperature'] + assert_equal(Temp.missing_value, 9999) + assert_equal(Temp.add_offset, 20) + assert_equal(Temp.scale_factor, np.float32(0.01)) + found = Temp[:].compressed() + del Temp # Remove ref to mmap, so file can be closed. + expected = np.round(tm.compressed(), 2) + assert_allclose(found, expected) + + with in_tempdir(): + newfname = 'ms.nc' + f = netcdf_file(newfname, 'w', maskandscale=True) + f.createDimension('Temperature', len(tm)) + temp = f.createVariable('Temperature', 'i', ('Temperature',)) + temp.missing_value = 9999 + temp.scale_factor = 0.01 + temp.add_offset = 20 + temp[:] = tm + f.close() + + with netcdf_file(newfname, maskandscale=True) as f: + Temp = f.variables['Temperature'] + assert_equal(Temp.missing_value, 9999) + assert_equal(Temp.add_offset, 20) + assert_equal(Temp.scale_factor, np.float32(0.01)) + expected = np.round(tm.compressed(), 2) + found = Temp[:].compressed() + del Temp + assert_allclose(found, expected) + + +# ------------------------------------------------------------------------ +# Test reading with masked values (_FillValue / missing_value) +# ------------------------------------------------------------------------ + +def test_read_withValuesNearFillValue(): + # Regression test for ticket #5626 + fname = pjoin(TEST_DATA_PATH, 'example_3_maskedvals.nc') + with netcdf_file(fname, maskandscale=True) as f: + vardata = f.variables['var1_fillval0'][:] + assert_mask_matches(vardata, [False, True, False]) + +def test_read_withNoFillValue(): + # For a variable with no fill value, reading data with maskandscale=True + # should return unmasked data + fname = pjoin(TEST_DATA_PATH, 'example_3_maskedvals.nc') + with netcdf_file(fname, maskandscale=True) as f: + vardata = f.variables['var2_noFillval'][:] + assert_mask_matches(vardata, [False, False, False]) + assert_equal(vardata, [1,2,3]) + +def test_read_withFillValueAndMissingValue(): + # For a variable with both _FillValue and missing_value, the _FillValue + # should be used + IRRELEVANT_VALUE = 9999 + fname = pjoin(TEST_DATA_PATH, 'example_3_maskedvals.nc') + with netcdf_file(fname, maskandscale=True) as f: + vardata = f.variables['var3_fillvalAndMissingValue'][:] + assert_mask_matches(vardata, [True, False, False]) + assert_equal(vardata, [IRRELEVANT_VALUE, 2, 3]) + +def test_read_withMissingValue(): + # For a variable with missing_value but not _FillValue, the missing_value + # should be used + fname = pjoin(TEST_DATA_PATH, 'example_3_maskedvals.nc') + with netcdf_file(fname, maskandscale=True) as f: + vardata = f.variables['var4_missingValue'][:] + assert_mask_matches(vardata, [False, True, False]) + +def test_read_withFillValNaN(): + fname = pjoin(TEST_DATA_PATH, 'example_3_maskedvals.nc') + with netcdf_file(fname, maskandscale=True) as f: + vardata = f.variables['var5_fillvalNaN'][:] + assert_mask_matches(vardata, [False, True, False]) + +def test_read_withChar(): + fname = pjoin(TEST_DATA_PATH, 'example_3_maskedvals.nc') + with netcdf_file(fname, maskandscale=True) as f: + vardata = f.variables['var6_char'][:] + assert_mask_matches(vardata, [False, True, False]) + +def test_read_with2dVar(): + fname = pjoin(TEST_DATA_PATH, 'example_3_maskedvals.nc') + with netcdf_file(fname, maskandscale=True) as f: + vardata = f.variables['var7_2d'][:] + assert_mask_matches(vardata, [[True, False], [False, False], [False, True]]) + +def test_read_withMaskAndScaleFalse(): + # If a variable has a _FillValue (or missing_value) attribute, but is read + # with maskandscale set to False, the result should be unmasked + fname = pjoin(TEST_DATA_PATH, 'example_3_maskedvals.nc') + # Open file with mmap=False to avoid problems with closing a mmap'ed file + # when arrays referring to its data still exist: + with netcdf_file(fname, maskandscale=False, mmap=False) as f: + vardata = f.variables['var3_fillvalAndMissingValue'][:] + assert_mask_matches(vardata, [False, False, False]) + assert_equal(vardata, [1, 2, 3]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/tests/test_paths.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/tests/test_paths.py new file mode 100644 index 0000000000000000000000000000000000000000..1e7c4167ace335fb5fc86f6499ee54c3360ded6e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/tests/test_paths.py @@ -0,0 +1,93 @@ +""" +Ensure that we can use pathlib.Path objects in all relevant IO functions. +""" +from pathlib import Path + +import numpy as np + +import scipy.io +import scipy.io.wavfile +from scipy._lib._tmpdirs import tempdir +import scipy.sparse + + +class TestPaths: + data = np.arange(5).astype(np.int64) + + def test_savemat(self): + with tempdir() as temp_dir: + path = Path(temp_dir) / 'data.mat' + scipy.io.savemat(path, {'data': self.data}) + assert path.is_file() + + def test_loadmat(self): + # Save data with string path, load with pathlib.Path + with tempdir() as temp_dir: + path = Path(temp_dir) / 'data.mat' + scipy.io.savemat(str(path), {'data': self.data}) + + mat_contents = scipy.io.loadmat(path) + assert (mat_contents['data'] == self.data).all() + + def test_whosmat(self): + # Save data with string path, load with pathlib.Path + with tempdir() as temp_dir: + path = Path(temp_dir) / 'data.mat' + scipy.io.savemat(str(path), {'data': self.data}) + + contents = scipy.io.whosmat(path) + assert contents[0] == ('data', (1, 5), 'int64') + + def test_readsav(self): + path = Path(__file__).parent / 'data/scalar_string.sav' + scipy.io.readsav(path) + + def test_hb_read(self): + # Save data with string path, load with pathlib.Path + with tempdir() as temp_dir: + data = scipy.sparse.eye_array(3, format='csr') + path = Path(temp_dir) / 'data.hb' + scipy.io.hb_write(str(path), data) + + data_new = scipy.io.hb_read(path, spmatrix=False) + assert (data_new != data).nnz == 0 + + def test_hb_write(self): + with tempdir() as temp_dir: + data = scipy.sparse.eye_array(3, format='csr') + path = Path(temp_dir) / 'data.hb' + scipy.io.hb_write(path, data) + assert path.is_file() + + def test_mmio_read(self): + # Save data with string path, load with pathlib.Path + with tempdir() as temp_dir: + data = scipy.sparse.eye_array(3, format='csr') + path = Path(temp_dir) / 'data.mtx' + scipy.io.mmwrite(str(path), data) + + data_new = scipy.io.mmread(path, spmatrix=False) + assert (data_new != data).nnz == 0 + + def test_mmio_write(self): + with tempdir() as temp_dir: + data = scipy.sparse.eye_array(3, format='csr') + path = Path(temp_dir) / 'data.mtx' + scipy.io.mmwrite(path, data) + + def test_netcdf_file(self): + path = Path(__file__).parent / 'data/example_1.nc' + scipy.io.netcdf_file(path) + + def test_wavfile_read(self): + path = Path(__file__).parent / 'data/test-8000Hz-le-2ch-1byteu.wav' + scipy.io.wavfile.read(path) + + def test_wavfile_write(self): + # Read from str path, write to Path + input_path = Path(__file__).parent / 'data/test-8000Hz-le-2ch-1byteu.wav' + rate, data = scipy.io.wavfile.read(str(input_path)) + + with tempdir() as temp_dir: + output_path = Path(temp_dir) / input_path.name + scipy.io.wavfile.write(output_path, rate, data) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/tests/test_wavfile.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/tests/test_wavfile.py new file mode 100644 index 0000000000000000000000000000000000000000..8e0a545495a842916a3cddd48c0b1b3859ae4ca5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/tests/test_wavfile.py @@ -0,0 +1,460 @@ +import os +import sys +from io import BytesIO +import threading + +import numpy as np +from numpy.testing import (assert_equal, assert_, assert_array_equal, + break_cycles, suppress_warnings, IS_PYPY) +import pytest +from pytest import raises, warns + +from scipy.io import wavfile + + +def datafile(fn): + return os.path.join(os.path.dirname(__file__), 'data', fn) + + +def test_read_1(): + # 32-bit PCM (which uses extensible format) + for mmap in [False, True]: + filename = 'test-44100Hz-le-1ch-4bytes.wav' + rate, data = wavfile.read(datafile(filename), mmap=mmap) + + assert_equal(rate, 44100) + assert_(np.issubdtype(data.dtype, np.int32)) + assert_equal(data.shape, (4410,)) + + del data + + +def test_read_2(): + # 8-bit unsigned PCM + for mmap in [False, True]: + filename = 'test-8000Hz-le-2ch-1byteu.wav' + rate, data = wavfile.read(datafile(filename), mmap=mmap) + + assert_equal(rate, 8000) + assert_(np.issubdtype(data.dtype, np.uint8)) + assert_equal(data.shape, (800, 2)) + + del data + + +def test_read_3(): + # Little-endian float + for mmap in [False, True]: + filename = 'test-44100Hz-2ch-32bit-float-le.wav' + rate, data = wavfile.read(datafile(filename), mmap=mmap) + + assert_equal(rate, 44100) + assert_(np.issubdtype(data.dtype, np.float32)) + assert_equal(data.shape, (441, 2)) + + del data + + +def test_read_4(): + # Contains unsupported 'PEAK' chunk + for mmap in [False, True]: + with suppress_warnings() as sup: + sup.filter(wavfile.WavFileWarning, + "Chunk .non-data. not understood, skipping it") + filename = 'test-48000Hz-2ch-64bit-float-le-wavex.wav' + rate, data = wavfile.read(datafile(filename), mmap=mmap) + + assert_equal(rate, 48000) + assert_(np.issubdtype(data.dtype, np.float64)) + assert_equal(data.shape, (480, 2)) + + del data + + +def test_read_5(): + # Big-endian float + for mmap in [False, True]: + filename = 'test-44100Hz-2ch-32bit-float-be.wav' + rate, data = wavfile.read(datafile(filename), mmap=mmap) + + assert_equal(rate, 44100) + assert_(np.issubdtype(data.dtype, np.float32)) + assert_(data.dtype.byteorder == '>' or (sys.byteorder == 'big' and + data.dtype.byteorder == '=')) + assert_equal(data.shape, (441, 2)) + + del data + + +def test_5_bit_odd_size_no_pad(): + # 5-bit, 1 B container, 5 channels, 9 samples, 45 B data chunk + # Generated by LTspice, which incorrectly omits pad byte, but should be + # readable anyway + for mmap in [False, True]: + filename = 'test-8000Hz-le-5ch-9S-5bit.wav' + rate, data = wavfile.read(datafile(filename), mmap=mmap) + + assert_equal(rate, 8000) + assert_(np.issubdtype(data.dtype, np.uint8)) + assert_equal(data.shape, (9, 5)) + + # 8-5 = 3 LSBits should be 0 + assert_equal(data & 0b00000111, 0) + + # Unsigned + assert_equal(data.max(), 0b11111000) # Highest possible + assert_equal(data[0, 0], 128) # Midpoint is 128 for <= 8-bit + assert_equal(data.min(), 0) # Lowest possible + + del data + + +def test_12_bit_even_size(): + # 12-bit, 2 B container, 4 channels, 9 samples, 72 B data chunk + # Generated by LTspice from 1 Vpk sine waves + for mmap in [False, True]: + filename = 'test-8000Hz-le-4ch-9S-12bit.wav' + rate, data = wavfile.read(datafile(filename), mmap=mmap) + + assert_equal(rate, 8000) + assert_(np.issubdtype(data.dtype, np.int16)) + assert_equal(data.shape, (9, 4)) + + # 16-12 = 4 LSBits should be 0 + assert_equal(data & 0b00000000_00001111, 0) + + # Signed + assert_equal(data.max(), 0b01111111_11110000) # Highest possible + assert_equal(data[0, 0], 0) # Midpoint is 0 for >= 9-bit + assert_equal(data.min(), -0b10000000_00000000) # Lowest possible + + del data + + +def test_24_bit_odd_size_with_pad(): + # 24-bit, 3 B container, 3 channels, 5 samples, 45 B data chunk + # Should not raise any warnings about the data chunk pad byte + filename = 'test-8000Hz-le-3ch-5S-24bit.wav' + rate, data = wavfile.read(datafile(filename), mmap=False) + + assert_equal(rate, 8000) + assert_(np.issubdtype(data.dtype, np.int32)) + assert_equal(data.shape, (5, 3)) + + # All LSBytes should be 0 + assert_equal(data & 0xff, 0) + + # Hand-made max/min samples under different conventions: + # 2**(N-1) 2**(N-1)-1 LSB + assert_equal(data, [[-0x8000_0000, -0x7fff_ff00, -0x200], + [-0x4000_0000, -0x3fff_ff00, -0x100], + [+0x0000_0000, +0x0000_0000, +0x000], + [+0x4000_0000, +0x3fff_ff00, +0x100], + [+0x7fff_ff00, +0x7fff_ff00, +0x200]]) + # ^ clipped + + +def test_20_bit_extra_data(): + # 20-bit, 3 B container, 1 channel, 10 samples, 30 B data chunk + # with extra data filling container beyond the bit depth + filename = 'test-1234Hz-le-1ch-10S-20bit-extra.wav' + rate, data = wavfile.read(datafile(filename), mmap=False) + + assert_equal(rate, 1234) + assert_(np.issubdtype(data.dtype, np.int32)) + assert_equal(data.shape, (10,)) + + # All LSBytes should still be 0, because 3 B container in 4 B dtype + assert_equal(data & 0xff, 0) + + # But it should load the data beyond 20 bits + assert_((data & 0xf00).any()) + + # Full-scale positive/negative samples, then being halved each time + assert_equal(data, [+0x7ffff000, # +full-scale 20-bit + -0x7ffff000, # -full-scale 20-bit + +0x7ffff000 >> 1, # +1/2 + -0x7ffff000 >> 1, # -1/2 + +0x7ffff000 >> 2, # +1/4 + -0x7ffff000 >> 2, # -1/4 + +0x7ffff000 >> 3, # +1/8 + -0x7ffff000 >> 3, # -1/8 + +0x7ffff000 >> 4, # +1/16 + -0x7ffff000 >> 4, # -1/16 + ]) + + +def test_36_bit_odd_size(): + # 36-bit, 5 B container, 3 channels, 5 samples, 75 B data chunk + pad + filename = 'test-8000Hz-le-3ch-5S-36bit.wav' + rate, data = wavfile.read(datafile(filename), mmap=False) + + assert_equal(rate, 8000) + assert_(np.issubdtype(data.dtype, np.int64)) + assert_equal(data.shape, (5, 3)) + + # 28 LSBits should be 0 + assert_equal(data & 0xfffffff, 0) + + # Hand-made max/min samples under different conventions: + # Fixed-point 2**(N-1) Full-scale 2**(N-1)-1 LSB + correct = [[-0x8000_0000_0000_0000, -0x7fff_ffff_f000_0000, -0x2000_0000], + [-0x4000_0000_0000_0000, -0x3fff_ffff_f000_0000, -0x1000_0000], + [+0x0000_0000_0000_0000, +0x0000_0000_0000_0000, +0x0000_0000], + [+0x4000_0000_0000_0000, +0x3fff_ffff_f000_0000, +0x1000_0000], + [+0x7fff_ffff_f000_0000, +0x7fff_ffff_f000_0000, +0x2000_0000]] + # ^ clipped + + assert_equal(data, correct) + + +def test_45_bit_even_size(): + # 45-bit, 6 B container, 3 channels, 5 samples, 90 B data chunk + filename = 'test-8000Hz-le-3ch-5S-45bit.wav' + rate, data = wavfile.read(datafile(filename), mmap=False) + + assert_equal(rate, 8000) + assert_(np.issubdtype(data.dtype, np.int64)) + assert_equal(data.shape, (5, 3)) + + # 19 LSBits should be 0 + assert_equal(data & 0x7ffff, 0) + + # Hand-made max/min samples under different conventions: + # Fixed-point 2**(N-1) Full-scale 2**(N-1)-1 LSB + correct = [[-0x8000_0000_0000_0000, -0x7fff_ffff_fff8_0000, -0x10_0000], + [-0x4000_0000_0000_0000, -0x3fff_ffff_fff8_0000, -0x08_0000], + [+0x0000_0000_0000_0000, +0x0000_0000_0000_0000, +0x00_0000], + [+0x4000_0000_0000_0000, +0x3fff_ffff_fff8_0000, +0x08_0000], + [+0x7fff_ffff_fff8_0000, +0x7fff_ffff_fff8_0000, +0x10_0000]] + # ^ clipped + + assert_equal(data, correct) + + +def test_53_bit_odd_size(): + # 53-bit, 7 B container, 3 channels, 5 samples, 105 B data chunk + pad + filename = 'test-8000Hz-le-3ch-5S-53bit.wav' + rate, data = wavfile.read(datafile(filename), mmap=False) + + assert_equal(rate, 8000) + assert_(np.issubdtype(data.dtype, np.int64)) + assert_equal(data.shape, (5, 3)) + + # 11 LSBits should be 0 + assert_equal(data & 0x7ff, 0) + + # Hand-made max/min samples under different conventions: + # Fixed-point 2**(N-1) Full-scale 2**(N-1)-1 LSB + correct = [[-0x8000_0000_0000_0000, -0x7fff_ffff_ffff_f800, -0x1000], + [-0x4000_0000_0000_0000, -0x3fff_ffff_ffff_f800, -0x0800], + [+0x0000_0000_0000_0000, +0x0000_0000_0000_0000, +0x0000], + [+0x4000_0000_0000_0000, +0x3fff_ffff_ffff_f800, +0x0800], + [+0x7fff_ffff_ffff_f800, +0x7fff_ffff_ffff_f800, +0x1000]] + # ^ clipped + + assert_equal(data, correct) + + +def test_64_bit_even_size(): + # 64-bit, 8 B container, 3 channels, 5 samples, 120 B data chunk + for mmap in [False, True]: + filename = 'test-8000Hz-le-3ch-5S-64bit.wav' + rate, data = wavfile.read(datafile(filename), mmap=mmap) + + assert_equal(rate, 8000) + assert_(np.issubdtype(data.dtype, np.int64)) + assert_equal(data.shape, (5, 3)) + + # Hand-made max/min samples under different conventions: + # Fixed-point 2**(N-1) Full-scale 2**(N-1)-1 LSB + correct = [[-0x8000_0000_0000_0000, -0x7fff_ffff_ffff_ffff, -0x2], + [-0x4000_0000_0000_0000, -0x3fff_ffff_ffff_ffff, -0x1], + [+0x0000_0000_0000_0000, +0x0000_0000_0000_0000, +0x0], + [+0x4000_0000_0000_0000, +0x3fff_ffff_ffff_ffff, +0x1], + [+0x7fff_ffff_ffff_ffff, +0x7fff_ffff_ffff_ffff, +0x2]] + # ^ clipped + + assert_equal(data, correct) + + del data + + +def test_unsupported_mmap(): + # Test containers that cannot be mapped to numpy types + for filename in {'test-8000Hz-le-3ch-5S-24bit.wav', + 'test-8000Hz-le-3ch-5S-36bit.wav', + 'test-8000Hz-le-3ch-5S-45bit.wav', + 'test-8000Hz-le-3ch-5S-53bit.wav', + 'test-1234Hz-le-1ch-10S-20bit-extra.wav'}: + with raises(ValueError, match="mmap.*not compatible"): + rate, data = wavfile.read(datafile(filename), mmap=True) + + +def test_rifx(): + # Compare equivalent RIFX and RIFF files + for rifx, riff in {('test-44100Hz-be-1ch-4bytes.wav', + 'test-44100Hz-le-1ch-4bytes.wav'), + ('test-8000Hz-be-3ch-5S-24bit.wav', + 'test-8000Hz-le-3ch-5S-24bit.wav')}: + rate1, data1 = wavfile.read(datafile(rifx), mmap=False) + rate2, data2 = wavfile.read(datafile(riff), mmap=False) + assert_equal(rate1, rate2) + assert_equal(data1, data2) + + +def test_rf64(): + # Compare equivalent RF64 and RIFF files + for rf64, riff in {('test-44100Hz-le-1ch-4bytes-rf64.wav', + 'test-44100Hz-le-1ch-4bytes.wav'), + ('test-8000Hz-le-3ch-5S-24bit-rf64.wav', + 'test-8000Hz-le-3ch-5S-24bit.wav')}: + rate1, data1 = wavfile.read(datafile(rf64), mmap=False) + rate2, data2 = wavfile.read(datafile(riff), mmap=False) + assert_array_equal(rate1, rate2) + assert_array_equal(data1, data2) + + +@pytest.mark.xslow +def test_write_roundtrip_rf64(tmpdir): + dtype = np.dtype(" 0 + assert rate == 44100 + # also test writing (gh-12176) + data[0] = 0 + + +def test_read_early_eof(): + # File ends after 'fact' chunk at boundary, no data read + for mmap in [False, True]: + filename = 'test-44100Hz-le-1ch-4bytes-early-eof-no-data.wav' + with open(datafile(filename), 'rb') as fp: + with raises(ValueError, match="Unexpected end of file."): + wavfile.read(fp, mmap=mmap) + + +def test_read_incomplete_chunk(): + # File ends inside 'fmt ' chunk ID, no data read + for mmap in [False, True]: + filename = 'test-44100Hz-le-1ch-4bytes-incomplete-chunk.wav' + with open(datafile(filename), 'rb') as fp: + with raises(ValueError, match="Incomplete chunk ID.*b'f'"): + wavfile.read(fp, mmap=mmap) + + +def test_read_inconsistent_header(): + # File header's size fields contradict each other + for mmap in [False, True]: + filename = 'test-8000Hz-le-3ch-5S-24bit-inconsistent.wav' + with open(datafile(filename), 'rb') as fp: + with raises(ValueError, match="header is invalid"): + wavfile.read(fp, mmap=mmap) + + +# signed 8-bit integer PCM is not allowed +# unsigned > 8-bit integer PCM is not allowed +# 8- or 16-bit float PCM is not expected +# g and q are platform-dependent, so not included +@pytest.mark.parametrize("dt_str", ["i2", ">i4", ">i8", ">f4", ">f8", '|u1']) +@pytest.mark.parametrize("channels", [1, 2, 5]) +@pytest.mark.parametrize("rate", [8000, 32000]) +@pytest.mark.parametrize("mmap", [False, True]) +@pytest.mark.parametrize("realfile", [False, True]) +def test_write_roundtrip(realfile, mmap, rate, channels, dt_str, tmpdir): + dtype = np.dtype(dt_str) + if realfile: + tmpfile = str(tmpdir.join(str(threading.get_native_id()), 'temp.wav')) + os.makedirs(os.path.dirname(tmpfile), exist_ok=True) + else: + tmpfile = BytesIO() + data = np.random.rand(100, channels) + if channels == 1: + data = data[:, 0] + if dtype.kind == 'f': + # The range of the float type should be in [-1, 1] + data = data.astype(dtype) + else: + data = (data*128).astype(dtype) + + wavfile.write(tmpfile, rate, data) + + rate2, data2 = wavfile.read(tmpfile, mmap=mmap) + + assert_equal(rate, rate2) + assert_(data2.dtype.byteorder in ('<', '=', '|'), msg=data2.dtype) + assert_array_equal(data, data2) + # also test writing (gh-12176) + if realfile: + data2[0] = 0 + else: + with pytest.raises(ValueError, match='read-only'): + data2[0] = 0 + + if realfile and mmap and IS_PYPY and sys.platform == 'win32': + # windows cannot remove a dead file held by a mmap but not collected + # in PyPy; since the filename gets reused in this test, clean this up + break_cycles() + break_cycles() + + +@pytest.mark.parametrize("dtype", [np.float16]) +def test_wavfile_dtype_unsupported(tmpdir, dtype): + tmpfile = str(tmpdir.join('temp.wav')) + rng = np.random.default_rng(1234) + data = rng.random((100, 5)).astype(dtype) + rate = 8000 + with pytest.raises(ValueError, match="Unsupported"): + wavfile.write(tmpfile, rate, data) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/wavfile.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/wavfile.py new file mode 100644 index 0000000000000000000000000000000000000000..b6978a1c461c825e35b8a1f0d7de39fceba38bd6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/io/wavfile.py @@ -0,0 +1,891 @@ +""" +Module to read / write wav files using NumPy arrays + +Functions +--------- +`read`: Return the sample rate (in samples/sec) and data from a WAV file. + +`write`: Write a NumPy array as a WAV file. + +""" +import io +import sys +import numpy as np +import struct +import warnings +from enum import IntEnum + + +__all__ = [ + 'WavFileWarning', + 'read', + 'write' +] + + +class WavFileWarning(UserWarning): + pass + + +class WAVE_FORMAT(IntEnum): + """ + WAVE form wFormatTag IDs + + Complete list is in mmreg.h in Windows 10 SDK. ALAC and OPUS are the + newest additions, in v10.0.14393 2016-07 + """ + UNKNOWN = 0x0000 + PCM = 0x0001 + ADPCM = 0x0002 + IEEE_FLOAT = 0x0003 + VSELP = 0x0004 + IBM_CVSD = 0x0005 + ALAW = 0x0006 + MULAW = 0x0007 + DTS = 0x0008 + DRM = 0x0009 + WMAVOICE9 = 0x000A + WMAVOICE10 = 0x000B + OKI_ADPCM = 0x0010 + DVI_ADPCM = 0x0011 + IMA_ADPCM = 0x0011 # Duplicate + MEDIASPACE_ADPCM = 0x0012 + SIERRA_ADPCM = 0x0013 + G723_ADPCM = 0x0014 + DIGISTD = 0x0015 + DIGIFIX = 0x0016 + DIALOGIC_OKI_ADPCM = 0x0017 + MEDIAVISION_ADPCM = 0x0018 + CU_CODEC = 0x0019 + HP_DYN_VOICE = 0x001A + YAMAHA_ADPCM = 0x0020 + SONARC = 0x0021 + DSPGROUP_TRUESPEECH = 0x0022 + ECHOSC1 = 0x0023 + AUDIOFILE_AF36 = 0x0024 + APTX = 0x0025 + AUDIOFILE_AF10 = 0x0026 + PROSODY_1612 = 0x0027 + LRC = 0x0028 + DOLBY_AC2 = 0x0030 + GSM610 = 0x0031 + MSNAUDIO = 0x0032 + ANTEX_ADPCME = 0x0033 + CONTROL_RES_VQLPC = 0x0034 + DIGIREAL = 0x0035 + DIGIADPCM = 0x0036 + CONTROL_RES_CR10 = 0x0037 + NMS_VBXADPCM = 0x0038 + CS_IMAADPCM = 0x0039 + ECHOSC3 = 0x003A + ROCKWELL_ADPCM = 0x003B + ROCKWELL_DIGITALK = 0x003C + XEBEC = 0x003D + G721_ADPCM = 0x0040 + G728_CELP = 0x0041 + MSG723 = 0x0042 + INTEL_G723_1 = 0x0043 + INTEL_G729 = 0x0044 + SHARP_G726 = 0x0045 + MPEG = 0x0050 + RT24 = 0x0052 + PAC = 0x0053 + MPEGLAYER3 = 0x0055 + LUCENT_G723 = 0x0059 + CIRRUS = 0x0060 + ESPCM = 0x0061 + VOXWARE = 0x0062 + CANOPUS_ATRAC = 0x0063 + G726_ADPCM = 0x0064 + G722_ADPCM = 0x0065 + DSAT = 0x0066 + DSAT_DISPLAY = 0x0067 + VOXWARE_BYTE_ALIGNED = 0x0069 + VOXWARE_AC8 = 0x0070 + VOXWARE_AC10 = 0x0071 + VOXWARE_AC16 = 0x0072 + VOXWARE_AC20 = 0x0073 + VOXWARE_RT24 = 0x0074 + VOXWARE_RT29 = 0x0075 + VOXWARE_RT29HW = 0x0076 + VOXWARE_VR12 = 0x0077 + VOXWARE_VR18 = 0x0078 + VOXWARE_TQ40 = 0x0079 + VOXWARE_SC3 = 0x007A + VOXWARE_SC3_1 = 0x007B + SOFTSOUND = 0x0080 + VOXWARE_TQ60 = 0x0081 + MSRT24 = 0x0082 + G729A = 0x0083 + MVI_MVI2 = 0x0084 + DF_G726 = 0x0085 + DF_GSM610 = 0x0086 + ISIAUDIO = 0x0088 + ONLIVE = 0x0089 + MULTITUDE_FT_SX20 = 0x008A + INFOCOM_ITS_G721_ADPCM = 0x008B + CONVEDIA_G729 = 0x008C + CONGRUENCY = 0x008D + SBC24 = 0x0091 + DOLBY_AC3_SPDIF = 0x0092 + MEDIASONIC_G723 = 0x0093 + PROSODY_8KBPS = 0x0094 + ZYXEL_ADPCM = 0x0097 + PHILIPS_LPCBB = 0x0098 + PACKED = 0x0099 + MALDEN_PHONYTALK = 0x00A0 + RACAL_RECORDER_GSM = 0x00A1 + RACAL_RECORDER_G720_A = 0x00A2 + RACAL_RECORDER_G723_1 = 0x00A3 + RACAL_RECORDER_TETRA_ACELP = 0x00A4 + NEC_AAC = 0x00B0 + RAW_AAC1 = 0x00FF + RHETOREX_ADPCM = 0x0100 + IRAT = 0x0101 + VIVO_G723 = 0x0111 + VIVO_SIREN = 0x0112 + PHILIPS_CELP = 0x0120 + PHILIPS_GRUNDIG = 0x0121 + DIGITAL_G723 = 0x0123 + SANYO_LD_ADPCM = 0x0125 + SIPROLAB_ACEPLNET = 0x0130 + SIPROLAB_ACELP4800 = 0x0131 + SIPROLAB_ACELP8V3 = 0x0132 + SIPROLAB_G729 = 0x0133 + SIPROLAB_G729A = 0x0134 + SIPROLAB_KELVIN = 0x0135 + VOICEAGE_AMR = 0x0136 + G726ADPCM = 0x0140 + DICTAPHONE_CELP68 = 0x0141 + DICTAPHONE_CELP54 = 0x0142 + QUALCOMM_PUREVOICE = 0x0150 + QUALCOMM_HALFRATE = 0x0151 + TUBGSM = 0x0155 + MSAUDIO1 = 0x0160 + WMAUDIO2 = 0x0161 + WMAUDIO3 = 0x0162 + WMAUDIO_LOSSLESS = 0x0163 + WMASPDIF = 0x0164 + UNISYS_NAP_ADPCM = 0x0170 + UNISYS_NAP_ULAW = 0x0171 + UNISYS_NAP_ALAW = 0x0172 + UNISYS_NAP_16K = 0x0173 + SYCOM_ACM_SYC008 = 0x0174 + SYCOM_ACM_SYC701_G726L = 0x0175 + SYCOM_ACM_SYC701_CELP54 = 0x0176 + SYCOM_ACM_SYC701_CELP68 = 0x0177 + KNOWLEDGE_ADVENTURE_ADPCM = 0x0178 + FRAUNHOFER_IIS_MPEG2_AAC = 0x0180 + DTS_DS = 0x0190 + CREATIVE_ADPCM = 0x0200 + CREATIVE_FASTSPEECH8 = 0x0202 + CREATIVE_FASTSPEECH10 = 0x0203 + UHER_ADPCM = 0x0210 + ULEAD_DV_AUDIO = 0x0215 + ULEAD_DV_AUDIO_1 = 0x0216 + QUARTERDECK = 0x0220 + ILINK_VC = 0x0230 + RAW_SPORT = 0x0240 + ESST_AC3 = 0x0241 + GENERIC_PASSTHRU = 0x0249 + IPI_HSX = 0x0250 + IPI_RPELP = 0x0251 + CS2 = 0x0260 + SONY_SCX = 0x0270 + SONY_SCY = 0x0271 + SONY_ATRAC3 = 0x0272 + SONY_SPC = 0x0273 + TELUM_AUDIO = 0x0280 + TELUM_IA_AUDIO = 0x0281 + NORCOM_VOICE_SYSTEMS_ADPCM = 0x0285 + FM_TOWNS_SND = 0x0300 + MICRONAS = 0x0350 + MICRONAS_CELP833 = 0x0351 + BTV_DIGITAL = 0x0400 + INTEL_MUSIC_CODER = 0x0401 + INDEO_AUDIO = 0x0402 + QDESIGN_MUSIC = 0x0450 + ON2_VP7_AUDIO = 0x0500 + ON2_VP6_AUDIO = 0x0501 + VME_VMPCM = 0x0680 + TPC = 0x0681 + LIGHTWAVE_LOSSLESS = 0x08AE + OLIGSM = 0x1000 + OLIADPCM = 0x1001 + OLICELP = 0x1002 + OLISBC = 0x1003 + OLIOPR = 0x1004 + LH_CODEC = 0x1100 + LH_CODEC_CELP = 0x1101 + LH_CODEC_SBC8 = 0x1102 + LH_CODEC_SBC12 = 0x1103 + LH_CODEC_SBC16 = 0x1104 + NORRIS = 0x1400 + ISIAUDIO_2 = 0x1401 + SOUNDSPACE_MUSICOMPRESS = 0x1500 + MPEG_ADTS_AAC = 0x1600 + MPEG_RAW_AAC = 0x1601 + MPEG_LOAS = 0x1602 + NOKIA_MPEG_ADTS_AAC = 0x1608 + NOKIA_MPEG_RAW_AAC = 0x1609 + VODAFONE_MPEG_ADTS_AAC = 0x160A + VODAFONE_MPEG_RAW_AAC = 0x160B + MPEG_HEAAC = 0x1610 + VOXWARE_RT24_SPEECH = 0x181C + SONICFOUNDRY_LOSSLESS = 0x1971 + INNINGS_TELECOM_ADPCM = 0x1979 + LUCENT_SX8300P = 0x1C07 + LUCENT_SX5363S = 0x1C0C + CUSEEME = 0x1F03 + NTCSOFT_ALF2CM_ACM = 0x1FC4 + DVM = 0x2000 + DTS2 = 0x2001 + MAKEAVIS = 0x3313 + DIVIO_MPEG4_AAC = 0x4143 + NOKIA_ADAPTIVE_MULTIRATE = 0x4201 + DIVIO_G726 = 0x4243 + LEAD_SPEECH = 0x434C + LEAD_VORBIS = 0x564C + WAVPACK_AUDIO = 0x5756 + OGG_VORBIS_MODE_1 = 0x674F + OGG_VORBIS_MODE_2 = 0x6750 + OGG_VORBIS_MODE_3 = 0x6751 + OGG_VORBIS_MODE_1_PLUS = 0x676F + OGG_VORBIS_MODE_2_PLUS = 0x6770 + OGG_VORBIS_MODE_3_PLUS = 0x6771 + ALAC = 0x6C61 + _3COM_NBX = 0x7000 # Can't have leading digit + OPUS = 0x704F + FAAD_AAC = 0x706D + AMR_NB = 0x7361 + AMR_WB = 0x7362 + AMR_WP = 0x7363 + GSM_AMR_CBR = 0x7A21 + GSM_AMR_VBR_SID = 0x7A22 + COMVERSE_INFOSYS_G723_1 = 0xA100 + COMVERSE_INFOSYS_AVQSBC = 0xA101 + COMVERSE_INFOSYS_SBC = 0xA102 + SYMBOL_G729_A = 0xA103 + VOICEAGE_AMR_WB = 0xA104 + INGENIENT_G726 = 0xA105 + MPEG4_AAC = 0xA106 + ENCORE_G726 = 0xA107 + ZOLL_ASAO = 0xA108 + SPEEX_VOICE = 0xA109 + VIANIX_MASC = 0xA10A + WM9_SPECTRUM_ANALYZER = 0xA10B + WMF_SPECTRUM_ANAYZER = 0xA10C + GSM_610 = 0xA10D + GSM_620 = 0xA10E + GSM_660 = 0xA10F + GSM_690 = 0xA110 + GSM_ADAPTIVE_MULTIRATE_WB = 0xA111 + POLYCOM_G722 = 0xA112 + POLYCOM_G728 = 0xA113 + POLYCOM_G729_A = 0xA114 + POLYCOM_SIREN = 0xA115 + GLOBAL_IP_ILBC = 0xA116 + RADIOTIME_TIME_SHIFT_RADIO = 0xA117 + NICE_ACA = 0xA118 + NICE_ADPCM = 0xA119 + VOCORD_G721 = 0xA11A + VOCORD_G726 = 0xA11B + VOCORD_G722_1 = 0xA11C + VOCORD_G728 = 0xA11D + VOCORD_G729 = 0xA11E + VOCORD_G729_A = 0xA11F + VOCORD_G723_1 = 0xA120 + VOCORD_LBC = 0xA121 + NICE_G728 = 0xA122 + FRACE_TELECOM_G729 = 0xA123 + CODIAN = 0xA124 + FLAC = 0xF1AC + EXTENSIBLE = 0xFFFE + DEVELOPMENT = 0xFFFF + + +KNOWN_WAVE_FORMATS = {WAVE_FORMAT.PCM, WAVE_FORMAT.IEEE_FLOAT} + + +def _raise_bad_format(format_tag): + try: + format_name = WAVE_FORMAT(format_tag).name + except ValueError: + format_name = f'{format_tag:#06x}' + raise ValueError(f"Unknown wave file format: {format_name}. Supported " + "formats: " + + ', '.join(x.name for x in KNOWN_WAVE_FORMATS)) + + +def _read_fmt_chunk(fid, is_big_endian): + """ + Returns + ------- + size : int + size of format subchunk in bytes (minus 8 for "fmt " and itself) + format_tag : int + PCM, float, or compressed format + channels : int + number of channels + fs : int + sampling frequency in samples per second + bytes_per_second : int + overall byte rate for the file + block_align : int + bytes per sample, including all channels + bit_depth : int + bits per sample + + Notes + ----- + Assumes file pointer is immediately after the 'fmt ' id + """ + if is_big_endian: + fmt = '>' + else: + fmt = '<' + + size = struct.unpack(fmt+'I', fid.read(4))[0] + + if size < 16: + raise ValueError("Binary structure of wave file is not compliant") + + res = struct.unpack(fmt+'HHIIHH', fid.read(16)) + bytes_read = 16 + + format_tag, channels, fs, bytes_per_second, block_align, bit_depth = res + + if format_tag == WAVE_FORMAT.EXTENSIBLE and size >= (16+2): + ext_chunk_size = struct.unpack(fmt+'H', fid.read(2))[0] + bytes_read += 2 + if ext_chunk_size >= 22: + extensible_chunk_data = fid.read(22) + bytes_read += 22 + raw_guid = extensible_chunk_data[2+4:2+4+16] + # GUID template {XXXXXXXX-0000-0010-8000-00AA00389B71} (RFC-2361) + # MS GUID byte order: first three groups are native byte order, + # rest is Big Endian + if is_big_endian: + tail = b'\x00\x00\x00\x10\x80\x00\x00\xAA\x00\x38\x9B\x71' + else: + tail = b'\x00\x00\x10\x00\x80\x00\x00\xAA\x00\x38\x9B\x71' + if raw_guid.endswith(tail): + format_tag = struct.unpack(fmt+'I', raw_guid[:4])[0] + else: + raise ValueError("Binary structure of wave file is not compliant") + + if format_tag not in KNOWN_WAVE_FORMATS: + _raise_bad_format(format_tag) + + # move file pointer to next chunk + if size > bytes_read: + fid.read(size - bytes_read) + + # fmt should always be 16, 18 or 40, but handle it just in case + _handle_pad_byte(fid, size) + + if format_tag == WAVE_FORMAT.PCM: + if bytes_per_second != fs * block_align: + raise ValueError("WAV header is invalid: nAvgBytesPerSec must" + " equal product of nSamplesPerSec and" + " nBlockAlign, but file has nSamplesPerSec =" + f" {fs}, nBlockAlign = {block_align}, and" + f" nAvgBytesPerSec = {bytes_per_second}") + + return (size, format_tag, channels, fs, bytes_per_second, block_align, + bit_depth) + + +def _read_data_chunk(fid, format_tag, channels, bit_depth, is_big_endian, is_rf64, + block_align, mmap=False): + """ + Notes + ----- + Assumes file pointer is immediately after the 'data' id + + It's possible to not use all available bits in a container, or to store + samples in a container bigger than necessary, so bytes_per_sample uses + the actual reported container size (nBlockAlign / nChannels). Real-world + examples: + + Adobe Audition's "24-bit packed int (type 1, 20-bit)" + + nChannels = 2, nBlockAlign = 6, wBitsPerSample = 20 + + http://www-mmsp.ece.mcgill.ca/Documents/AudioFormats/WAVE/Samples/AFsp/M1F1-int12-AFsp.wav + is: + + nChannels = 2, nBlockAlign = 4, wBitsPerSample = 12 + + http://www-mmsp.ece.mcgill.ca/Documents/AudioFormats/WAVE/Docs/multichaudP.pdf + gives an example of: + + nChannels = 2, nBlockAlign = 8, wBitsPerSample = 20 + """ + if is_big_endian: + fmt = '>' + else: + fmt = '<' + + # Size of the data subchunk in bytes + if not is_rf64: + size = struct.unpack(fmt+'I', fid.read(4))[0] + else: + pos = fid.tell() + # chunk size is stored in global file header for RF64 + fid.seek(28) + size = struct.unpack(' 1: + data = data.reshape(-1, channels) + return data + + +def _skip_unknown_chunk(fid, is_big_endian): + if is_big_endian: + fmt = '>I' + else: + fmt = '>> from os.path import dirname, join as pjoin + >>> from scipy.io import wavfile + >>> import scipy.io + + Get the filename for an example .wav file from the tests/data directory. + + >>> data_dir = pjoin(dirname(scipy.io.__file__), 'tests', 'data') + >>> wav_fname = pjoin(data_dir, 'test-44100Hz-2ch-32bit-float-be.wav') + + Load the .wav file contents. + + >>> samplerate, data = wavfile.read(wav_fname) + >>> print(f"number of channels = {data.shape[1]}") + number of channels = 2 + >>> length = data.shape[0] / samplerate + >>> print(f"length = {length}s") + length = 0.01s + + Plot the waveform. + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> time = np.linspace(0., length, data.shape[0]) + >>> plt.plot(time, data[:, 0], label="Left channel") + >>> plt.plot(time, data[:, 1], label="Right channel") + >>> plt.legend() + >>> plt.xlabel("Time [s]") + >>> plt.ylabel("Amplitude") + >>> plt.show() + + """ + if hasattr(filename, 'read'): + fid = filename + mmap = False + else: + fid = open(filename, 'rb') + + try: + file_size, is_big_endian, is_rf64 = _read_riff_chunk(fid) + fmt_chunk_received = False + data_chunk_received = False + while fid.tell() < file_size: + # read the next chunk + chunk_id = fid.read(4) + + if not chunk_id: + if data_chunk_received: + # End of file but data successfully read + warnings.warn( + f"Reached EOF prematurely; finished at {fid.tell():d} bytes, " + f"expected {file_size:d} bytes from header.", + WavFileWarning, stacklevel=2) + break + else: + raise ValueError("Unexpected end of file.") + elif len(chunk_id) < 4: + msg = f"Incomplete chunk ID: {repr(chunk_id)}" + # If we have the data, ignore the broken chunk + if fmt_chunk_received and data_chunk_received: + warnings.warn(msg + ", ignoring it.", WavFileWarning, + stacklevel=2) + else: + raise ValueError(msg) + + if chunk_id == b'fmt ': + fmt_chunk_received = True + fmt_chunk = _read_fmt_chunk(fid, is_big_endian) + format_tag, channels, fs = fmt_chunk[1:4] + bit_depth = fmt_chunk[6] + block_align = fmt_chunk[5] + elif chunk_id == b'fact': + _skip_unknown_chunk(fid, is_big_endian) + elif chunk_id == b'data': + data_chunk_received = True + if not fmt_chunk_received: + raise ValueError("No fmt chunk before data") + data = _read_data_chunk(fid, format_tag, channels, bit_depth, + is_big_endian, is_rf64, block_align, mmap) + elif chunk_id == b'LIST': + # Someday this could be handled properly but for now skip it + _skip_unknown_chunk(fid, is_big_endian) + elif chunk_id in {b'JUNK', b'Fake'}: + # Skip alignment chunks without warning + _skip_unknown_chunk(fid, is_big_endian) + else: + warnings.warn("Chunk (non-data) not understood, skipping it.", + WavFileWarning, stacklevel=2) + _skip_unknown_chunk(fid, is_big_endian) + finally: + if not hasattr(filename, 'read'): + fid.close() + else: + fid.seek(0) + + return fs, data + + +def write(filename, rate, data): + """ + Write a NumPy array as a WAV file. + + Parameters + ---------- + filename : string or open file handle + Output wav file. + rate : int + The sample rate (in samples/sec). + data : ndarray + A 1-D or 2-D NumPy array of either integer or float data-type. + + Notes + ----- + * Writes a simple uncompressed WAV file. + * To write multiple-channels, use a 2-D array of shape + (Nsamples, Nchannels). + * The bits-per-sample and PCM/float will be determined by the data-type. + + Common data types: [1]_ + + ===================== =========== =========== ============= + WAV format Min Max NumPy dtype + ===================== =========== =========== ============= + 32-bit floating-point -1.0 +1.0 float32 + 32-bit PCM -2147483648 +2147483647 int32 + 16-bit PCM -32768 +32767 int16 + 8-bit PCM 0 255 uint8 + ===================== =========== =========== ============= + + Note that 8-bit PCM is unsigned. + + References + ---------- + .. [1] IBM Corporation and Microsoft Corporation, "Multimedia Programming + Interface and Data Specifications 1.0", section "Data Format of the + Samples", August 1991 + http://www.tactilemedia.com/info/MCI_Control_Info.html + + Examples + -------- + Create a 100Hz sine wave, sampled at 44100Hz. + Write to 16-bit PCM, Mono. + + >>> from scipy.io.wavfile import write + >>> import numpy as np + >>> samplerate = 44100; fs = 100 + >>> t = np.linspace(0., 1., samplerate) + >>> amplitude = np.iinfo(np.int16).max + >>> data = amplitude * np.sin(2. * np.pi * fs * t) + >>> write("example.wav", samplerate, data.astype(np.int16)) + + """ + if hasattr(filename, 'write'): + fid = filename + else: + fid = open(filename, 'wb') + + fs = rate + + try: + dkind = data.dtype.kind + allowed_dtypes = ['float32', 'float64', + 'uint8', 'int16', 'int32', 'int64'] + if data.dtype.name not in allowed_dtypes: + raise ValueError(f"Unsupported data type '{data.dtype}'") + + header_data = b'' + + header_data += b'RIFF' + header_data += b'\x00\x00\x00\x00' + header_data += b'WAVE' + + # fmt chunk + header_data += b'fmt ' + if dkind == 'f': + format_tag = WAVE_FORMAT.IEEE_FLOAT + else: + format_tag = WAVE_FORMAT.PCM + if data.ndim == 1: + channels = 1 + else: + channels = data.shape[1] + bit_depth = data.dtype.itemsize * 8 + bytes_per_second = fs*(bit_depth // 8)*channels + block_align = channels * (bit_depth // 8) + + fmt_chunk_data = struct.pack(' 0xFFFFFFFF + if is_rf64: + header_data = b'' + header_data += b'RF64' + header_data += b'\xFF\xFF\xFF\xFF' + header_data += b'WAVE' + header_data += b'ds64' + # size of ds64 chunk + header_data += struct.pack('' or (data.dtype.byteorder == '=' and + sys.byteorder == 'big'): + data = data.byteswap() + _array_tofile(fid, data) + + # Determine file size and place it in correct + # position at start of the file or the data chunk. + size = fid.tell() + if not is_rf64: + fid.seek(4) + fid.write(struct.pack('`__ + for more linear algebra functions. Note that + although `scipy.linalg` imports most of them, identically named + functions from `scipy.linalg` may offer more or slightly differing + functionality. + + +Basics +====== + +.. autosummary:: + :toctree: generated/ + + inv - Find the inverse of a square matrix + solve - Solve a linear system of equations + solve_banded - Solve a banded linear system + solveh_banded - Solve a Hermitian or symmetric banded system + solve_circulant - Solve a circulant system + solve_triangular - Solve a triangular matrix + solve_toeplitz - Solve a toeplitz matrix + matmul_toeplitz - Multiply a Toeplitz matrix with an array. + det - Find the determinant of a square matrix + norm - Matrix and vector norm + lstsq - Solve a linear least-squares problem + pinv - Pseudo-inverse (Moore-Penrose) using lstsq + pinvh - Pseudo-inverse of hermitian matrix + kron - Kronecker product of two arrays + khatri_rao - Khatri-Rao product of two arrays + orthogonal_procrustes - Solve an orthogonal Procrustes problem + matrix_balance - Balance matrix entries with a similarity transformation + subspace_angles - Compute the subspace angles between two matrices + bandwidth - Return the lower and upper bandwidth of an array + issymmetric - Check if a square 2D array is symmetric + ishermitian - Check if a square 2D array is Hermitian + LinAlgError + LinAlgWarning + +Eigenvalue Problems +=================== + +.. autosummary:: + :toctree: generated/ + + eig - Find the eigenvalues and eigenvectors of a square matrix + eigvals - Find just the eigenvalues of a square matrix + eigh - Find the e-vals and e-vectors of a Hermitian or symmetric matrix + eigvalsh - Find just the eigenvalues of a Hermitian or symmetric matrix + eig_banded - Find the eigenvalues and eigenvectors of a banded matrix + eigvals_banded - Find just the eigenvalues of a banded matrix + eigh_tridiagonal - Find the eigenvalues and eigenvectors of a tridiagonal matrix + eigvalsh_tridiagonal - Find just the eigenvalues of a tridiagonal matrix + +Decompositions +============== + +.. autosummary:: + :toctree: generated/ + + lu - LU decomposition of a matrix + lu_factor - LU decomposition returning unordered matrix and pivots + lu_solve - Solve Ax=b using back substitution with output of lu_factor + svd - Singular value decomposition of a matrix + svdvals - Singular values of a matrix + diagsvd - Construct matrix of singular values from output of svd + orth - Construct orthonormal basis for the range of A using svd + null_space - Construct orthonormal basis for the null space of A using svd + ldl - LDL.T decomposition of a Hermitian or a symmetric matrix. + cholesky - Cholesky decomposition of a matrix + cholesky_banded - Cholesky decomp. of a sym. or Hermitian banded matrix + cho_factor - Cholesky decomposition for use in solving a linear system + cho_solve - Solve previously factored linear system + cho_solve_banded - Solve previously factored banded linear system + polar - Compute the polar decomposition. + qr - QR decomposition of a matrix + qr_multiply - QR decomposition and multiplication by Q + qr_update - Rank k QR update + qr_delete - QR downdate on row or column deletion + qr_insert - QR update on row or column insertion + rq - RQ decomposition of a matrix + qz - QZ decomposition of a pair of matrices + ordqz - QZ decomposition of a pair of matrices with reordering + schur - Schur decomposition of a matrix + rsf2csf - Real to complex Schur form + hessenberg - Hessenberg form of a matrix + cdf2rdf - Complex diagonal form to real diagonal block form + cossin - Cosine sine decomposition of a unitary or orthogonal matrix + +.. seealso:: + + `scipy.linalg.interpolative` -- Interpolative matrix decompositions + + +Matrix Functions +================ + +.. autosummary:: + :toctree: generated/ + + expm - Matrix exponential + logm - Matrix logarithm + cosm - Matrix cosine + sinm - Matrix sine + tanm - Matrix tangent + coshm - Matrix hyperbolic cosine + sinhm - Matrix hyperbolic sine + tanhm - Matrix hyperbolic tangent + signm - Matrix sign + sqrtm - Matrix square root + funm - Evaluating an arbitrary matrix function + expm_frechet - Frechet derivative of the matrix exponential + expm_cond - Relative condition number of expm in the Frobenius norm + fractional_matrix_power - Fractional matrix power + + +Matrix Equation Solvers +======================= + +.. autosummary:: + :toctree: generated/ + + solve_sylvester - Solve the Sylvester matrix equation + solve_continuous_are - Solve the continuous-time algebraic Riccati equation + solve_discrete_are - Solve the discrete-time algebraic Riccati equation + solve_continuous_lyapunov - Solve the continuous-time Lyapunov equation + solve_discrete_lyapunov - Solve the discrete-time Lyapunov equation + + +Sketches and Random Projections +=============================== + +.. autosummary:: + :toctree: generated/ + + clarkson_woodruff_transform - Applies the Clarkson Woodruff Sketch (a.k.a CountMin Sketch) + +Special Matrices +================ + +.. autosummary:: + :toctree: generated/ + + block_diag - Construct a block diagonal matrix from submatrices + circulant - Circulant matrix + companion - Companion matrix + convolution_matrix - Convolution matrix + dft - Discrete Fourier transform matrix + fiedler - Fiedler matrix + fiedler_companion - Fiedler companion matrix + hadamard - Hadamard matrix of order 2**n + hankel - Hankel matrix + helmert - Helmert matrix + hilbert - Hilbert matrix + invhilbert - Inverse Hilbert matrix + leslie - Leslie matrix + pascal - Pascal matrix + invpascal - Inverse Pascal matrix + toeplitz - Toeplitz matrix + +Low-level routines +================== + +.. autosummary:: + :toctree: generated/ + + get_blas_funcs + get_lapack_funcs + find_best_blas_type + +.. seealso:: + + `scipy.linalg.blas` -- Low-level BLAS functions + + `scipy.linalg.lapack` -- Low-level LAPACK functions + + `scipy.linalg.cython_blas` -- Low-level BLAS functions for Cython + + `scipy.linalg.cython_lapack` -- Low-level LAPACK functions for Cython + +""" # noqa: E501 + +from ._misc import * +from ._cythonized_array_utils import * +from ._basic import * +from ._decomp import * +from ._decomp_lu import * +from ._decomp_ldl import * +from ._decomp_cholesky import * +from ._decomp_qr import * +from ._decomp_qz import * +from ._decomp_svd import * +from ._decomp_schur import * +from ._decomp_polar import * +from ._matfuncs import * +from .blas import * +from .lapack import * +from ._special_matrices import * +from ._solvers import * +from ._procrustes import * +from ._decomp_update import * +from ._sketches import * +from ._decomp_cossin import * + +# Deprecated namespaces, to be removed in v2.0.0 +from . import ( + decomp, decomp_cholesky, decomp_lu, decomp_qr, decomp_svd, decomp_schur, + basic, misc, special_matrices, matfuncs, +) + +__all__ = [s for s in dir() if not s.startswith('_')] + + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_basic.py new file mode 100644 index 0000000000000000000000000000000000000000..70841fa80976c62ed7d894d824fdfbbcb4673273 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_basic.py @@ -0,0 +1,2119 @@ +# +# Author: Pearu Peterson, March 2002 +# +# w/ additions by Travis Oliphant, March 2002 +# and Jake Vanderplas, August 2012 + +import warnings +from warnings import warn +from itertools import product +import numpy as np +from numpy import atleast_1d, atleast_2d +from .lapack import get_lapack_funcs, _compute_lwork +from ._misc import LinAlgError, _datacopied, LinAlgWarning +from ._decomp import _asarray_validated +from . import _decomp, _decomp_svd +from ._solve_toeplitz import levinson +from ._cythonized_array_utils import (find_det_from_lu, bandwidth, issymmetric, + ishermitian) + +__all__ = ['solve', 'solve_triangular', 'solveh_banded', 'solve_banded', + 'solve_toeplitz', 'solve_circulant', 'inv', 'det', 'lstsq', + 'pinv', 'pinvh', 'matrix_balance', 'matmul_toeplitz'] + + +# The numpy facilities for type-casting checks are too slow for small sized +# arrays and eat away the time budget for the checkups. Here we set a +# precomputed dict container of the numpy.can_cast() table. + +# It can be used to determine quickly what a dtype can be cast to LAPACK +# compatible types, i.e., 'float32, float64, complex64, complex128'. +# Then it can be checked via "casting_dict[arr.dtype.char]" +lapack_cast_dict = {x: ''.join([y for y in 'fdFD' if np.can_cast(x, y)]) + for x in np.typecodes['All']} + + +# Linear equations +def _solve_check(n, info, lamch=None, rcond=None): + """ Check arguments during the different steps of the solution phase """ + if info < 0: + raise ValueError(f'LAPACK reported an illegal value in {-info}-th argument.') + elif 0 < info: + raise LinAlgError('Matrix is singular.') + + if lamch is None: + return + E = lamch('E') + if rcond < E: + warn(f'Ill-conditioned matrix (rcond={rcond:.6g}): ' + 'result may not be accurate.', + LinAlgWarning, stacklevel=3) + + +def _find_matrix_structure(a): + n = a.shape[0] + n_below, n_above = bandwidth(a) + + if n_below == n_above == 0: + kind = 'diagonal' + elif n_above == 0: + kind = 'lower triangular' + elif n_below == 0: + kind = 'upper triangular' + elif n_above <= 1 and n_below <= 1 and n > 3: + kind = 'tridiagonal' + elif np.issubdtype(a.dtype, np.complexfloating) and ishermitian(a): + kind = 'hermitian' + elif issymmetric(a): + kind = 'symmetric' + else: + kind = 'general' + + return kind, n_below, n_above + + +def solve(a, b, lower=False, overwrite_a=False, + overwrite_b=False, check_finite=True, assume_a=None, + transposed=False): + """ + Solves the linear equation set ``a @ x == b`` for the unknown ``x`` + for square `a` matrix. + + If the data matrix is known to be a particular type then supplying the + corresponding string to ``assume_a`` key chooses the dedicated solver. + The available options are + + =================== ================================ + diagonal 'diagonal' + tridiagonal 'tridiagonal' + banded 'banded' + upper triangular 'upper triangular' + lower triangular 'lower triangular' + symmetric 'symmetric' (or 'sym') + hermitian 'hermitian' (or 'her') + positive definite 'positive definite' (or 'pos') + general 'general' (or 'gen') + =================== ================================ + + Parameters + ---------- + a : (N, N) array_like + Square input data + b : (N, NRHS) array_like + Input data for the right hand side. + lower : bool, default: False + Ignored unless ``assume_a`` is one of ``'sym'``, ``'her'``, or ``'pos'``. + If True, the calculation uses only the data in the lower triangle of `a`; + entries above the diagonal are ignored. If False (default), the + calculation uses only the data in the upper triangle of `a`; entries + below the diagonal are ignored. + overwrite_a : bool, default: False + Allow overwriting data in `a` (may enhance performance). + overwrite_b : bool, default: False + Allow overwriting data in `b` (may enhance performance). + check_finite : bool, default: True + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + assume_a : str, optional + Valid entries are described above. + If omitted or ``None``, checks are performed to identify structure so the + appropriate solver can be called. + transposed : bool, default: False + If True, solve ``a.T @ x == b``. Raises `NotImplementedError` + for complex `a`. + + Returns + ------- + x : (N, NRHS) ndarray + The solution array. + + Raises + ------ + ValueError + If size mismatches detected or input a is not square. + LinAlgError + If the matrix is singular. + LinAlgWarning + If an ill-conditioned input a is detected. + NotImplementedError + If transposed is True and input a is a complex matrix. + + Notes + ----- + If the input b matrix is a 1-D array with N elements, when supplied + together with an NxN input a, it is assumed as a valid column vector + despite the apparent size mismatch. This is compatible with the + numpy.dot() behavior and the returned result is still 1-D array. + + The general, symmetric, Hermitian and positive definite solutions are + obtained via calling ?GESV, ?SYSV, ?HESV, and ?POSV routines of + LAPACK respectively. + + The datatype of the arrays define which solver is called regardless + of the values. In other words, even when the complex array entries have + precisely zero imaginary parts, the complex solver will be called based + on the data type of the array. + + Examples + -------- + Given `a` and `b`, solve for `x`: + + >>> import numpy as np + >>> a = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]]) + >>> b = np.array([2, 4, -1]) + >>> from scipy import linalg + >>> x = linalg.solve(a, b) + >>> x + array([ 2., -2., 9.]) + >>> np.dot(a, x) == b + array([ True, True, True], dtype=bool) + + """ + # Flags for 1-D or N-D right-hand side + b_is_1D = False + + # check finite after determining structure + a1 = atleast_2d(_asarray_validated(a, check_finite=False)) + b1 = atleast_1d(_asarray_validated(b, check_finite=False)) + a1, b1 = _ensure_dtype_cdsz(a1, b1) + n = a1.shape[0] + + overwrite_a = overwrite_a or _datacopied(a1, a) + overwrite_b = overwrite_b or _datacopied(b1, b) + + if a1.shape[0] != a1.shape[1]: + raise ValueError('Input a needs to be a square matrix.') + + if n != b1.shape[0]: + # Last chance to catch 1x1 scalar a and 1-D b arrays + if not (n == 1 and b1.size != 0): + raise ValueError('Input b has to have same number of rows as ' + 'input a') + + # accommodate empty arrays + if b1.size == 0: + dt = solve(np.eye(2, dtype=a1.dtype), np.ones(2, dtype=b1.dtype)).dtype + return np.empty_like(b1, dtype=dt) + + # regularize 1-D b arrays to 2D + if b1.ndim == 1: + if n == 1: + b1 = b1[None, :] + else: + b1 = b1[:, None] + b_is_1D = True + + if assume_a not in {None, 'diagonal', 'tridiagonal', 'banded', 'lower triangular', + 'upper triangular', 'symmetric', 'hermitian', + 'positive definite', 'general', 'sym', 'her', 'pos', 'gen'}: + raise ValueError(f'{assume_a} is not a recognized matrix structure') + + # for a real matrix, describe it as "symmetric", not "hermitian" + # (lapack doesn't know what to do with real hermitian matrices) + if assume_a in {'hermitian', 'her'} and not np.iscomplexobj(a1): + assume_a = 'symmetric' + + n_below, n_above = None, None + if assume_a is None: + assume_a, n_below, n_above = _find_matrix_structure(a1) + + # Get the correct lamch function. + # The LAMCH functions only exists for S and D + # So for complex values we have to convert to real/double. + if a1.dtype.char in 'fF': # single precision + lamch = get_lapack_funcs('lamch', dtype='f') + else: + lamch = get_lapack_funcs('lamch', dtype='d') + + # Currently we do not have the other forms of the norm calculators + # lansy, lanpo, lanhe. + # However, in any case they only reduce computations slightly... + if assume_a == 'diagonal': + _matrix_norm = _matrix_norm_diagonal + elif assume_a == 'tridiagonal': + _matrix_norm = _matrix_norm_tridiagonal + elif assume_a in {'lower triangular', 'upper triangular'}: + _matrix_norm = _matrix_norm_triangular(assume_a) + else: + _matrix_norm = _matrix_norm_general + + # Since the I-norm and 1-norm are the same for symmetric matrices + # we can collect them all in this one call + # Note however, that when issuing 'gen' and form!='none', then + # the I-norm should be used + if transposed: + trans = 1 + norm = 'I' + if np.iscomplexobj(a1): + raise NotImplementedError('scipy.linalg.solve can currently ' + 'not solve a^T x = b or a^H x = b ' + 'for complex matrices.') + else: + trans = 0 + norm = '1' + + anorm = _matrix_norm(norm, a1, check_finite) + + info, rcond = 0, np.inf + + # Generalized case 'gesv' + if assume_a in {'general', 'gen'}: + gecon, getrf, getrs = get_lapack_funcs(('gecon', 'getrf', 'getrs'), + (a1, b1)) + lu, ipvt, info = getrf(a1, overwrite_a=overwrite_a) + _solve_check(n, info) + x, info = getrs(lu, ipvt, b1, + trans=trans, overwrite_b=overwrite_b) + _solve_check(n, info) + rcond, info = gecon(lu, anorm, norm=norm) + # Hermitian case 'hesv' + elif assume_a in {'hermitian', 'her'}: + hecon, hesv, hesv_lw = get_lapack_funcs(('hecon', 'hesv', + 'hesv_lwork'), (a1, b1)) + lwork = _compute_lwork(hesv_lw, n, lower) + lu, ipvt, x, info = hesv(a1, b1, lwork=lwork, + lower=lower, + overwrite_a=overwrite_a, + overwrite_b=overwrite_b) + _solve_check(n, info) + rcond, info = hecon(lu, ipvt, anorm) + # Symmetric case 'sysv' + elif assume_a in {'symmetric', 'sym'}: + sycon, sysv, sysv_lw = get_lapack_funcs(('sycon', 'sysv', + 'sysv_lwork'), (a1, b1)) + lwork = _compute_lwork(sysv_lw, n, lower) + lu, ipvt, x, info = sysv(a1, b1, lwork=lwork, + lower=lower, + overwrite_a=overwrite_a, + overwrite_b=overwrite_b) + _solve_check(n, info) + rcond, info = sycon(lu, ipvt, anorm) + # Diagonal case + elif assume_a == 'diagonal': + diag_a = np.diag(a1) + x = (b1.T / diag_a).T + abs_diag_a = np.abs(diag_a) + rcond = abs_diag_a.min() / abs_diag_a.max() + # Tri-diagonal case + elif assume_a == 'tridiagonal': + a1 = a1.T if transposed else a1 + dl, d, du = np.diag(a1, -1), np.diag(a1, 0), np.diag(a1, 1) + _gttrf, _gttrs, _gtcon = get_lapack_funcs(('gttrf', 'gttrs', 'gtcon'), (a1, b1)) + dl, d, du, du2, ipiv, info = _gttrf(dl, d, du) + _solve_check(n, info) + x, info = _gttrs(dl, d, du, du2, ipiv, b1, overwrite_b=overwrite_b) + _solve_check(n, info) + rcond, info = _gtcon(dl, d, du, du2, ipiv, anorm) + # Banded case + elif assume_a == 'banded': + a1, n_below, n_above = ((a1.T, n_above, n_below) if transposed + else (a1, n_below, n_above)) + n_below, n_above = bandwidth(a1) if n_below is None else (n_below, n_above) + ab = _to_banded(n_below, n_above, a1) + gbsv, = get_lapack_funcs(('gbsv',), (a1, b1)) + # Next two lines copied from `solve_banded` + a2 = np.zeros((2*n_below + n_above + 1, ab.shape[1]), dtype=gbsv.dtype) + a2[n_below:, :] = ab + _, _, x, info = gbsv(n_below, n_above, a2, b1, + overwrite_ab=True, overwrite_b=overwrite_b) + _solve_check(n, info) + # TODO: wrap gbcon and use to get rcond + # Triangular case + elif assume_a in {'lower triangular', 'upper triangular'}: + lower = assume_a == 'lower triangular' + x, info = _solve_triangular(a1, b1, lower=lower, overwrite_b=overwrite_b, + trans=transposed) + _solve_check(n, info) + _trcon = get_lapack_funcs(('trcon'), (a1, b1)) + rcond, info = _trcon(a1, uplo='L' if lower else 'U') + # Positive definite case 'posv' + else: + pocon, posv = get_lapack_funcs(('pocon', 'posv'), + (a1, b1)) + lu, x, info = posv(a1, b1, lower=lower, + overwrite_a=overwrite_a, + overwrite_b=overwrite_b) + _solve_check(n, info) + rcond, info = pocon(lu, anorm) + + _solve_check(n, info, lamch, rcond) + + if b_is_1D: + x = x.ravel() + + return x + + +def _matrix_norm_diagonal(_, a, check_finite): + # Equivalent of dlange for diagonal matrix, assuming + # norm is either 'I' or '1' (really just not the Frobenius norm) + d = np.diag(a) + d = np.asarray_chkfinite(d) if check_finite else d + return np.abs(d).max() + + +def _matrix_norm_tridiagonal(norm, a, check_finite): + # Equivalent of dlange for tridiagonal matrix, assuming + # norm is either 'I' or '1' + if norm == 'I': + a = a.T + # Context to avoid warning before error in cases like -inf + inf + with np.errstate(invalid='ignore'): + d = np.abs(np.diag(a)) + d[1:] += np.abs(np.diag(a, 1)) + d[:-1] += np.abs(np.diag(a, -1)) + d = np.asarray_chkfinite(d) if check_finite else d + return d.max() + + +def _matrix_norm_triangular(structure): + def fun(norm, a, check_finite): + a = np.asarray_chkfinite(a) if check_finite else a + lantr = get_lapack_funcs('lantr', (a,)) + return lantr(norm, a, 'L' if structure == 'lower triangular' else 'U' ) + return fun + + +def _matrix_norm_general(norm, a, check_finite): + a = np.asarray_chkfinite(a) if check_finite else a + lange = get_lapack_funcs('lange', (a,)) + return lange(norm, a) + + +def _to_banded(n_below, n_above, a): + n = a.shape[0] + rows = n_above + n_below + 1 + ab = np.zeros((rows, n), dtype=a.dtype) + ab[n_above] = np.diag(a) + for i in range(1, n_above + 1): + ab[n_above - i, i:] = np.diag(a, i) + for i in range(1, n_below + 1): + ab[n_above + i, :-i] = np.diag(a, -i) + return ab + + +def _ensure_dtype_cdsz(*arrays): + # Ensure that the dtype of arrays is one of the standard types + # compatible with LAPACK functions (single or double precision + # real or complex). + dtype = np.result_type(*arrays) + if not np.issubdtype(dtype, np.inexact): + return (array.astype(np.float64) for array in arrays) + complex = np.issubdtype(dtype, np.complexfloating) + if np.finfo(dtype).bits <= 32: + dtype = np.complex64 if complex else np.float32 + elif np.finfo(dtype).bits >= 64: + dtype = np.complex128 if complex else np.float64 + return (array.astype(dtype, copy=False) for array in arrays) + + +def solve_triangular(a, b, trans=0, lower=False, unit_diagonal=False, + overwrite_b=False, check_finite=True): + """ + Solve the equation ``a x = b`` for `x`, assuming a is a triangular matrix. + + Parameters + ---------- + a : (M, M) array_like + A triangular matrix + b : (M,) or (M, N) array_like + Right-hand side matrix in ``a x = b`` + lower : bool, optional + Use only data contained in the lower triangle of `a`. + Default is to use upper triangle. + trans : {0, 1, 2, 'N', 'T', 'C'}, optional + Type of system to solve: + + ======== ========= + trans system + ======== ========= + 0 or 'N' a x = b + 1 or 'T' a^T x = b + 2 or 'C' a^H x = b + ======== ========= + unit_diagonal : bool, optional + If True, diagonal elements of `a` are assumed to be 1 and + will not be referenced. + overwrite_b : bool, optional + Allow overwriting data in `b` (may enhance performance) + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + x : (M,) or (M, N) ndarray + Solution to the system ``a x = b``. Shape of return matches `b`. + + Raises + ------ + LinAlgError + If `a` is singular + + Notes + ----- + .. versionadded:: 0.9.0 + + Examples + -------- + Solve the lower triangular system a x = b, where:: + + [3 0 0 0] [4] + a = [2 1 0 0] b = [2] + [1 0 1 0] [4] + [1 1 1 1] [2] + + >>> import numpy as np + >>> from scipy.linalg import solve_triangular + >>> a = np.array([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 1, 1, 1]]) + >>> b = np.array([4, 2, 4, 2]) + >>> x = solve_triangular(a, b, lower=True) + >>> x + array([ 1.33333333, -0.66666667, 2.66666667, -1.33333333]) + >>> a.dot(x) # Check the result + array([ 4., 2., 4., 2.]) + + """ + + a1 = _asarray_validated(a, check_finite=check_finite) + b1 = _asarray_validated(b, check_finite=check_finite) + + if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]: + raise ValueError('expected square matrix') + + if a1.shape[0] != b1.shape[0]: + raise ValueError(f'shapes of a {a1.shape} and b {b1.shape} are incompatible') + + # accommodate empty arrays + if b1.size == 0: + dt_nonempty = solve_triangular( + np.eye(2, dtype=a1.dtype), np.ones(2, dtype=b1.dtype) + ).dtype + return np.empty_like(b1, dtype=dt_nonempty) + + overwrite_b = overwrite_b or _datacopied(b1, b) + + x, _ = _solve_triangular(a1, b1, trans, lower, unit_diagonal, overwrite_b) + return x + + +# solve_triangular without the input validation +def _solve_triangular(a1, b1, trans=0, lower=False, unit_diagonal=False, + overwrite_b=False): + + trans = {'N': 0, 'T': 1, 'C': 2}.get(trans, trans) + trtrs, = get_lapack_funcs(('trtrs',), (a1, b1)) + if a1.flags.f_contiguous or trans == 2: + x, info = trtrs(a1, b1, overwrite_b=overwrite_b, lower=lower, + trans=trans, unitdiag=unit_diagonal) + else: + # transposed system is solved since trtrs expects Fortran ordering + x, info = trtrs(a1.T, b1, overwrite_b=overwrite_b, lower=not lower, + trans=not trans, unitdiag=unit_diagonal) + + if info == 0: + return x, info + if info > 0: + raise LinAlgError("singular matrix: resolution failed at diagonal %d" % + (info-1)) + raise ValueError('illegal value in %dth argument of internal trtrs' % + (-info)) + + +def solve_banded(l_and_u, ab, b, overwrite_ab=False, overwrite_b=False, + check_finite=True): + """ + Solve the equation a x = b for x, assuming a is banded matrix. + + The matrix a is stored in `ab` using the matrix diagonal ordered form:: + + ab[u + i - j, j] == a[i,j] + + Example of `ab` (shape of a is (6,6), `u` =1, `l` =2):: + + * a01 a12 a23 a34 a45 + a00 a11 a22 a33 a44 a55 + a10 a21 a32 a43 a54 * + a20 a31 a42 a53 * * + + Parameters + ---------- + (l, u) : (integer, integer) + Number of non-zero lower and upper diagonals + ab : (`l` + `u` + 1, M) array_like + Banded matrix + b : (M,) or (M, K) array_like + Right-hand side + overwrite_ab : bool, optional + Discard data in `ab` (may enhance performance) + overwrite_b : bool, optional + Discard data in `b` (may enhance performance) + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + x : (M,) or (M, K) ndarray + The solution to the system a x = b. Returned shape depends on the + shape of `b`. + + Examples + -------- + Solve the banded system a x = b, where:: + + [5 2 -1 0 0] [0] + [1 4 2 -1 0] [1] + a = [0 1 3 2 -1] b = [2] + [0 0 1 2 2] [2] + [0 0 0 1 1] [3] + + There is one nonzero diagonal below the main diagonal (l = 1), and + two above (u = 2). The diagonal banded form of the matrix is:: + + [* * -1 -1 -1] + ab = [* 2 2 2 2] + [5 4 3 2 1] + [1 1 1 1 *] + + >>> import numpy as np + >>> from scipy.linalg import solve_banded + >>> ab = np.array([[0, 0, -1, -1, -1], + ... [0, 2, 2, 2, 2], + ... [5, 4, 3, 2, 1], + ... [1, 1, 1, 1, 0]]) + >>> b = np.array([0, 1, 2, 2, 3]) + >>> x = solve_banded((1, 2), ab, b) + >>> x + array([-2.37288136, 3.93220339, -4. , 4.3559322 , -1.3559322 ]) + + """ + + a1 = _asarray_validated(ab, check_finite=check_finite, as_inexact=True) + b1 = _asarray_validated(b, check_finite=check_finite, as_inexact=True) + + # Validate shapes. + if a1.shape[-1] != b1.shape[0]: + raise ValueError("shapes of ab and b are not compatible.") + + (nlower, nupper) = l_and_u + if nlower + nupper + 1 != a1.shape[0]: + raise ValueError("invalid values for the number of lower and upper " + "diagonals: l+u+1 (%d) does not equal ab.shape[0] " + "(%d)" % (nlower + nupper + 1, ab.shape[0])) + + # accommodate empty arrays + if b1.size == 0: + dt = solve(np.eye(1, dtype=a1.dtype), np.ones(1, dtype=b1.dtype)).dtype + return np.empty_like(b1, dtype=dt) + + overwrite_b = overwrite_b or _datacopied(b1, b) + if a1.shape[-1] == 1: + b2 = np.array(b1, copy=(not overwrite_b)) + # a1.shape[-1] == 1 -> original matrix is 1x1. Typically, the user + # will pass u = l = 0 and `a1` will be 1x1. However, the rest of the + # function works with unnecessary rows in `a1` as long as + # `a1[u + i - j, j] == a[i,j]`. In the 1x1 case, we want i = j = 0, + # so the diagonal is in row `u` of `a1`. See gh-8906. + b2 /= a1[nupper, 0] + return b2 + if nlower == nupper == 1: + overwrite_ab = overwrite_ab or _datacopied(a1, ab) + gtsv, = get_lapack_funcs(('gtsv',), (a1, b1)) + du = a1[0, 1:] + d = a1[1, :] + dl = a1[2, :-1] + du2, d, du, x, info = gtsv(dl, d, du, b1, overwrite_ab, overwrite_ab, + overwrite_ab, overwrite_b) + else: + gbsv, = get_lapack_funcs(('gbsv',), (a1, b1)) + a2 = np.zeros((2*nlower + nupper + 1, a1.shape[1]), dtype=gbsv.dtype) + a2[nlower:, :] = a1 + lu, piv, x, info = gbsv(nlower, nupper, a2, b1, overwrite_ab=True, + overwrite_b=overwrite_b) + if info == 0: + return x + if info > 0: + raise LinAlgError("singular matrix") + raise ValueError('illegal value in %d-th argument of internal ' + 'gbsv/gtsv' % -info) + + +def solveh_banded(ab, b, overwrite_ab=False, overwrite_b=False, lower=False, + check_finite=True): + """ + Solve equation a x = b. a is Hermitian positive-definite banded matrix. + + Uses Thomas' Algorithm, which is more efficient than standard LU + factorization, but should only be used for Hermitian positive-definite + matrices. + + The matrix ``a`` is stored in `ab` either in lower diagonal or upper + diagonal ordered form: + + ab[u + i - j, j] == a[i,j] (if upper form; i <= j) + ab[ i - j, j] == a[i,j] (if lower form; i >= j) + + Example of `ab` (shape of ``a`` is (6, 6), number of upper diagonals, + ``u`` =2):: + + upper form: + * * a02 a13 a24 a35 + * a01 a12 a23 a34 a45 + a00 a11 a22 a33 a44 a55 + + lower form: + a00 a11 a22 a33 a44 a55 + a10 a21 a32 a43 a54 * + a20 a31 a42 a53 * * + + Cells marked with * are not used. + + Parameters + ---------- + ab : (``u`` + 1, M) array_like + Banded matrix + b : (M,) or (M, K) array_like + Right-hand side + overwrite_ab : bool, optional + Discard data in `ab` (may enhance performance) + overwrite_b : bool, optional + Discard data in `b` (may enhance performance) + lower : bool, optional + Is the matrix in the lower form. (Default is upper form) + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + x : (M,) or (M, K) ndarray + The solution to the system ``a x = b``. Shape of return matches shape + of `b`. + + Notes + ----- + In the case of a non-positive definite matrix ``a``, the solver + `solve_banded` may be used. + + Examples + -------- + Solve the banded system ``A x = b``, where:: + + [ 4 2 -1 0 0 0] [1] + [ 2 5 2 -1 0 0] [2] + A = [-1 2 6 2 -1 0] b = [2] + [ 0 -1 2 7 2 -1] [3] + [ 0 0 -1 2 8 2] [3] + [ 0 0 0 -1 2 9] [3] + + >>> import numpy as np + >>> from scipy.linalg import solveh_banded + + ``ab`` contains the main diagonal and the nonzero diagonals below the + main diagonal. That is, we use the lower form: + + >>> ab = np.array([[ 4, 5, 6, 7, 8, 9], + ... [ 2, 2, 2, 2, 2, 0], + ... [-1, -1, -1, -1, 0, 0]]) + >>> b = np.array([1, 2, 2, 3, 3, 3]) + >>> x = solveh_banded(ab, b, lower=True) + >>> x + array([ 0.03431373, 0.45938375, 0.05602241, 0.47759104, 0.17577031, + 0.34733894]) + + + Solve the Hermitian banded system ``H x = b``, where:: + + [ 8 2-1j 0 0 ] [ 1 ] + H = [2+1j 5 1j 0 ] b = [1+1j] + [ 0 -1j 9 -2-1j] [1-2j] + [ 0 0 -2+1j 6 ] [ 0 ] + + In this example, we put the upper diagonals in the array ``hb``: + + >>> hb = np.array([[0, 2-1j, 1j, -2-1j], + ... [8, 5, 9, 6 ]]) + >>> b = np.array([1, 1+1j, 1-2j, 0]) + >>> x = solveh_banded(hb, b) + >>> x + array([ 0.07318536-0.02939412j, 0.11877624+0.17696461j, + 0.10077984-0.23035393j, -0.00479904-0.09358128j]) + + """ + a1 = _asarray_validated(ab, check_finite=check_finite) + b1 = _asarray_validated(b, check_finite=check_finite) + + # Validate shapes. + if a1.shape[-1] != b1.shape[0]: + raise ValueError("shapes of ab and b are not compatible.") + + # accommodate empty arrays + if b1.size == 0: + dt = solve(np.eye(1, dtype=a1.dtype), np.ones(1, dtype=b1.dtype)).dtype + return np.empty_like(b1, dtype=dt) + + overwrite_b = overwrite_b or _datacopied(b1, b) + overwrite_ab = overwrite_ab or _datacopied(a1, ab) + + if a1.shape[0] == 2: + ptsv, = get_lapack_funcs(('ptsv',), (a1, b1)) + if lower: + d = a1[0, :].real + e = a1[1, :-1] + else: + d = a1[1, :].real + e = a1[0, 1:].conj() + d, du, x, info = ptsv(d, e, b1, overwrite_ab, overwrite_ab, + overwrite_b) + else: + pbsv, = get_lapack_funcs(('pbsv',), (a1, b1)) + c, x, info = pbsv(a1, b1, lower=lower, overwrite_ab=overwrite_ab, + overwrite_b=overwrite_b) + if info > 0: + raise LinAlgError("%dth leading minor not positive definite" % info) + if info < 0: + raise ValueError('illegal value in %dth argument of internal ' + 'pbsv' % -info) + return x + + +def solve_toeplitz(c_or_cr, b, check_finite=True): + r"""Solve a Toeplitz system using Levinson Recursion + + The Toeplitz matrix has constant diagonals, with c as its first column + and r as its first row. If r is not given, ``r == conjugate(c)`` is + assumed. + + .. warning:: + + Beginning in SciPy 1.17, multidimensional input will be treated as a batch, + not ``ravel``\ ed. To preserve the existing behavior, ``ravel`` arguments + before passing them to `solve_toeplitz`. + + Parameters + ---------- + c_or_cr : array_like or tuple of (array_like, array_like) + The vector ``c``, or a tuple of arrays (``c``, ``r``). If not + supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is + real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row + of the Toeplitz matrix is ``[c[0], r[1:]]``. + b : (M,) or (M, K) array_like + Right-hand side in ``T x = b``. + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (result entirely NaNs) if the inputs do contain infinities or NaNs. + + Returns + ------- + x : (M,) or (M, K) ndarray + The solution to the system ``T x = b``. Shape of return matches shape + of `b`. + + See Also + -------- + toeplitz : Toeplitz matrix + + Notes + ----- + The solution is computed using Levinson-Durbin recursion, which is faster + than generic least-squares methods, but can be less numerically stable. + + Examples + -------- + Solve the Toeplitz system T x = b, where:: + + [ 1 -1 -2 -3] [1] + T = [ 3 1 -1 -2] b = [2] + [ 6 3 1 -1] [2] + [10 6 3 1] [5] + + To specify the Toeplitz matrix, only the first column and the first + row are needed. + + >>> import numpy as np + >>> c = np.array([1, 3, 6, 10]) # First column of T + >>> r = np.array([1, -1, -2, -3]) # First row of T + >>> b = np.array([1, 2, 2, 5]) + + >>> from scipy.linalg import solve_toeplitz, toeplitz + >>> x = solve_toeplitz((c, r), b) + >>> x + array([ 1.66666667, -1. , -2.66666667, 2.33333333]) + + Check the result by creating the full Toeplitz matrix and + multiplying it by `x`. We should get `b`. + + >>> T = toeplitz(c, r) + >>> T.dot(x) + array([ 1., 2., 2., 5.]) + + """ + # If numerical stability of this algorithm is a problem, a future + # developer might consider implementing other O(N^2) Toeplitz solvers, + # such as GKO (https://www.jstor.org/stable/2153371) or Bareiss. + + r, c, b, dtype, b_shape = _validate_args_for_toeplitz_ops( + c_or_cr, b, check_finite, keep_b_shape=True) + + # accommodate empty arrays + if b.size == 0: + return np.empty_like(b) + + # Form a 1-D array of values to be used in the matrix, containing a + # reversed copy of r[1:], followed by c. + vals = np.concatenate((r[-1:0:-1], c)) + if b is None: + raise ValueError('illegal value, `b` is a required argument') + + if b.ndim == 1: + x, _ = levinson(vals, np.ascontiguousarray(b)) + else: + x = np.column_stack([levinson(vals, np.ascontiguousarray(b[:, i]))[0] + for i in range(b.shape[1])]) + x = x.reshape(*b_shape) + + return x + + +def _get_axis_len(aname, a, axis): + ax = axis + if ax < 0: + ax += a.ndim + if 0 <= ax < a.ndim: + return a.shape[ax] + raise ValueError(f"'{aname}axis' entry is out of bounds") + + +def solve_circulant(c, b, singular='raise', tol=None, + caxis=-1, baxis=0, outaxis=0): + """Solve C x = b for x, where C is a circulant matrix. + + `C` is the circulant matrix associated with the vector `c`. + + The system is solved by doing division in Fourier space. The + calculation is:: + + x = ifft(fft(b) / fft(c)) + + where `fft` and `ifft` are the fast Fourier transform and its inverse, + respectively. For a large vector `c`, this is *much* faster than + solving the system with the full circulant matrix. + + Parameters + ---------- + c : array_like + The coefficients of the circulant matrix. + b : array_like + Right-hand side matrix in ``a x = b``. + singular : str, optional + This argument controls how a near singular circulant matrix is + handled. If `singular` is "raise" and the circulant matrix is + near singular, a `LinAlgError` is raised. If `singular` is + "lstsq", the least squares solution is returned. Default is "raise". + tol : float, optional + If any eigenvalue of the circulant matrix has an absolute value + that is less than or equal to `tol`, the matrix is considered to be + near singular. If not given, `tol` is set to:: + + tol = abs_eigs.max() * abs_eigs.size * np.finfo(np.float64).eps + + where `abs_eigs` is the array of absolute values of the eigenvalues + of the circulant matrix. + caxis : int + When `c` has dimension greater than 1, it is viewed as a collection + of circulant vectors. In this case, `caxis` is the axis of `c` that + holds the vectors of circulant coefficients. + baxis : int + When `b` has dimension greater than 1, it is viewed as a collection + of vectors. In this case, `baxis` is the axis of `b` that holds the + right-hand side vectors. + outaxis : int + When `c` or `b` are multidimensional, the value returned by + `solve_circulant` is multidimensional. In this case, `outaxis` is + the axis of the result that holds the solution vectors. + + Returns + ------- + x : ndarray + Solution to the system ``C x = b``. + + Raises + ------ + LinAlgError + If the circulant matrix associated with `c` is near singular. + + See Also + -------- + circulant : circulant matrix + + Notes + ----- + For a 1-D vector `c` with length `m`, and an array `b` + with shape ``(m, ...)``, + + solve_circulant(c, b) + + returns the same result as + + solve(circulant(c), b) + + where `solve` and `circulant` are from `scipy.linalg`. + + .. versionadded:: 0.16.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import solve_circulant, solve, circulant, lstsq + + >>> c = np.array([2, 2, 4]) + >>> b = np.array([1, 2, 3]) + >>> solve_circulant(c, b) + array([ 0.75, -0.25, 0.25]) + + Compare that result to solving the system with `scipy.linalg.solve`: + + >>> solve(circulant(c), b) + array([ 0.75, -0.25, 0.25]) + + A singular example: + + >>> c = np.array([1, 1, 0, 0]) + >>> b = np.array([1, 2, 3, 4]) + + Calling ``solve_circulant(c, b)`` will raise a `LinAlgError`. For the + least square solution, use the option ``singular='lstsq'``: + + >>> solve_circulant(c, b, singular='lstsq') + array([ 0.25, 1.25, 2.25, 1.25]) + + Compare to `scipy.linalg.lstsq`: + + >>> x, resid, rnk, s = lstsq(circulant(c), b) + >>> x + array([ 0.25, 1.25, 2.25, 1.25]) + + A broadcasting example: + + Suppose we have the vectors of two circulant matrices stored in an array + with shape (2, 5), and three `b` vectors stored in an array with shape + (3, 5). For example, + + >>> c = np.array([[1.5, 2, 3, 0, 0], [1, 1, 4, 3, 2]]) + >>> b = np.arange(15).reshape(-1, 5) + + We want to solve all combinations of circulant matrices and `b` vectors, + with the result stored in an array with shape (2, 3, 5). When we + disregard the axes of `c` and `b` that hold the vectors of coefficients, + the shapes of the collections are (2,) and (3,), respectively, which are + not compatible for broadcasting. To have a broadcast result with shape + (2, 3), we add a trivial dimension to `c`: ``c[:, np.newaxis, :]`` has + shape (2, 1, 5). The last dimension holds the coefficients of the + circulant matrices, so when we call `solve_circulant`, we can use the + default ``caxis=-1``. The coefficients of the `b` vectors are in the last + dimension of the array `b`, so we use ``baxis=-1``. If we use the + default `outaxis`, the result will have shape (5, 2, 3), so we'll use + ``outaxis=-1`` to put the solution vectors in the last dimension. + + >>> x = solve_circulant(c[:, np.newaxis, :], b, baxis=-1, outaxis=-1) + >>> x.shape + (2, 3, 5) + >>> np.set_printoptions(precision=3) # For compact output of numbers. + >>> x + array([[[-0.118, 0.22 , 1.277, -0.142, 0.302], + [ 0.651, 0.989, 2.046, 0.627, 1.072], + [ 1.42 , 1.758, 2.816, 1.396, 1.841]], + [[ 0.401, 0.304, 0.694, -0.867, 0.377], + [ 0.856, 0.758, 1.149, -0.412, 0.831], + [ 1.31 , 1.213, 1.603, 0.042, 1.286]]]) + + Check by solving one pair of `c` and `b` vectors (cf. ``x[1, 1, :]``): + + >>> solve_circulant(c[1], b[1, :]) + array([ 0.856, 0.758, 1.149, -0.412, 0.831]) + + """ + c = np.atleast_1d(c) + nc = _get_axis_len("c", c, caxis) + b = np.atleast_1d(b) + nb = _get_axis_len("b", b, baxis) + if nc != nb: + raise ValueError(f'Shapes of c {c.shape} and b {b.shape} are incompatible') + + # accommodate empty arrays + if b.size == 0: + dt = solve_circulant(np.arange(3, dtype=c.dtype), + np.ones(3, dtype=b.dtype)).dtype + return np.empty_like(b, dtype=dt) + + fc = np.fft.fft(np.moveaxis(c, caxis, -1), axis=-1) + abs_fc = np.abs(fc) + if tol is None: + # This is the same tolerance as used in np.linalg.matrix_rank. + tol = abs_fc.max(axis=-1) * nc * np.finfo(np.float64).eps + if tol.shape != (): + tol.shape = tol.shape + (1,) + else: + tol = np.atleast_1d(tol) + + near_zeros = abs_fc <= tol + is_near_singular = np.any(near_zeros) + if is_near_singular: + if singular == 'raise': + raise LinAlgError("near singular circulant matrix.") + else: + # Replace the small values with 1 to avoid errors in the + # division fb/fc below. + fc[near_zeros] = 1 + + fb = np.fft.fft(np.moveaxis(b, baxis, -1), axis=-1) + + q = fb / fc + + if is_near_singular: + # `near_zeros` is a boolean array, same shape as `c`, that is + # True where `fc` is (near) zero. `q` is the broadcasted result + # of fb / fc, so to set the values of `q` to 0 where `fc` is near + # zero, we use a mask that is the broadcast result of an array + # of True values shaped like `b` with `near_zeros`. + mask = np.ones_like(b, dtype=bool) & near_zeros + q[mask] = 0 + + x = np.fft.ifft(q, axis=-1) + if not (np.iscomplexobj(c) or np.iscomplexobj(b)): + x = x.real + if outaxis != -1: + x = np.moveaxis(x, -1, outaxis) + return x + + +# matrix inversion +def inv(a, overwrite_a=False, check_finite=True): + """ + Compute the inverse of a matrix. + + Parameters + ---------- + a : array_like + Square matrix to be inverted. + overwrite_a : bool, optional + Discard data in `a` (may improve performance). Default is False. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + ainv : ndarray + Inverse of the matrix `a`. + + Raises + ------ + LinAlgError + If `a` is singular. + ValueError + If `a` is not square, or not 2D. + + Examples + -------- + >>> import numpy as np + >>> from scipy import linalg + >>> a = np.array([[1., 2.], [3., 4.]]) + >>> linalg.inv(a) + array([[-2. , 1. ], + [ 1.5, -0.5]]) + >>> np.dot(a, linalg.inv(a)) + array([[ 1., 0.], + [ 0., 1.]]) + + """ + a1 = _asarray_validated(a, check_finite=check_finite) + if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]: + raise ValueError('expected square matrix') + + # accommodate empty square matrices + if a1.size == 0: + dt = inv(np.eye(2, dtype=a1.dtype)).dtype + return np.empty_like(a1, dtype=dt) + + overwrite_a = overwrite_a or _datacopied(a1, a) + getrf, getri, getri_lwork = get_lapack_funcs(('getrf', 'getri', + 'getri_lwork'), + (a1,)) + lu, piv, info = getrf(a1, overwrite_a=overwrite_a) + if info == 0: + lwork = _compute_lwork(getri_lwork, a1.shape[0]) + + # XXX: the following line fixes curious SEGFAULT when + # benchmarking 500x500 matrix inverse. This seems to + # be a bug in LAPACK ?getri routine because if lwork is + # minimal (when using lwork[0] instead of lwork[1]) then + # all tests pass. Further investigation is required if + # more such SEGFAULTs occur. + lwork = int(1.01 * lwork) + inv_a, info = getri(lu, piv, lwork=lwork, overwrite_lu=1) + if info > 0: + raise LinAlgError("singular matrix") + if info < 0: + raise ValueError('illegal value in %d-th argument of internal ' + 'getrf|getri' % -info) + return inv_a + + +# Determinant + +def det(a, overwrite_a=False, check_finite=True): + """ + Compute the determinant of a matrix + + The determinant is a scalar that is a function of the associated square + matrix coefficients. The determinant value is zero for singular matrices. + + Parameters + ---------- + a : (..., M, M) array_like + Input array to compute determinants for. + overwrite_a : bool, optional + Allow overwriting data in a (may enhance performance). + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + det : (...) float or complex + Determinant of `a`. For stacked arrays, a scalar is returned for each + (m, m) slice in the last two dimensions of the input. For example, an + input of shape (p, q, m, m) will produce a result of shape (p, q). If + all dimensions are 1 a scalar is returned regardless of ndim. + + Notes + ----- + The determinant is computed by performing an LU factorization of the + input with LAPACK routine 'getrf', and then calculating the product of + diagonal entries of the U factor. + + Even if the input array is single precision (float32 or complex64), the + result will be returned in double precision (float64 or complex128) to + prevent overflows. + + Examples + -------- + >>> import numpy as np + >>> from scipy import linalg + >>> a = np.array([[1,2,3], [4,5,6], [7,8,9]]) # A singular matrix + >>> linalg.det(a) + 0.0 + >>> b = np.array([[0,2,3], [4,5,6], [7,8,9]]) + >>> linalg.det(b) + 3.0 + >>> # An array with the shape (3, 2, 2, 2) + >>> c = np.array([[[[1., 2.], [3., 4.]], + ... [[5., 6.], [7., 8.]]], + ... [[[9., 10.], [11., 12.]], + ... [[13., 14.], [15., 16.]]], + ... [[[17., 18.], [19., 20.]], + ... [[21., 22.], [23., 24.]]]]) + >>> linalg.det(c) # The resulting shape is (3, 2) + array([[-2., -2.], + [-2., -2.], + [-2., -2.]]) + >>> linalg.det(c[0, 0]) # Confirm the (0, 0) slice, [[1, 2], [3, 4]] + -2.0 + """ + # The goal is to end up with a writable contiguous array to pass to Cython + + # First we check and make arrays. + a1 = np.asarray_chkfinite(a) if check_finite else np.asarray(a) + if a1.ndim < 2: + raise ValueError('The input array must be at least two-dimensional.') + if a1.shape[-1] != a1.shape[-2]: + raise ValueError('Last 2 dimensions of the array must be square' + f' but received shape {a1.shape}.') + + # Also check if dtype is LAPACK compatible + if a1.dtype.char not in 'fdFD': + dtype_char = lapack_cast_dict[a1.dtype.char] + if not dtype_char: # No casting possible + raise TypeError(f'The dtype "{a1.dtype.name}" cannot be cast ' + 'to float(32, 64) or complex(64, 128).') + + a1 = a1.astype(dtype_char[0]) # makes a copy, free to scratch + overwrite_a = True + + # Empty array has determinant 1 because math. + if min(*a1.shape) == 0: + dtyp = np.float64 if a1.dtype.char not in 'FD' else np.complex128 + if a1.ndim == 2: + return dtyp(1.0) + else: + return np.ones(shape=a1.shape[:-2], dtype=dtyp) + + # Scalar case + if a1.shape[-2:] == (1, 1): + a1 = a1[..., 0, 0] + if a1.ndim == 0: + a1 = a1[()] + # Convert float32 to float64, and complex64 to complex128. + if a1.dtype.char in 'dD': + return a1 + return a1.astype('d') if a1.dtype.char == 'f' else a1.astype('D') + + # Then check overwrite permission + if not _datacopied(a1, a): # "a" still alive through "a1" + if not overwrite_a: + # Data belongs to "a" so make a copy + a1 = a1.copy(order='C') + # else: Do nothing we'll use "a" if possible + # else: a1 has its own data thus free to scratch + + # Then layout checks, might happen that overwrite is allowed but original + # array was read-only or non-C-contiguous. + if not (a1.flags['C_CONTIGUOUS'] and a1.flags['WRITEABLE']): + a1 = a1.copy(order='C') + + if a1.ndim == 2: + det = find_det_from_lu(a1) + # Convert float, complex to NumPy scalars + return (np.float64(det) if np.isrealobj(det) else np.complex128(det)) + + # loop over the stacked array, and avoid overflows for single precision + # Cf. np.linalg.det(np.diag([1e+38, 1e+38]).astype(np.float32)) + dtype_char = a1.dtype.char + if dtype_char in 'fF': + dtype_char = 'd' if dtype_char.islower() else 'D' + + det = np.empty(a1.shape[:-2], dtype=dtype_char) + for ind in product(*[range(x) for x in a1.shape[:-2]]): + det[ind] = find_det_from_lu(a1[ind]) + return det + + +# Linear Least Squares +def lstsq(a, b, cond=None, overwrite_a=False, overwrite_b=False, + check_finite=True, lapack_driver=None): + """ + Compute least-squares solution to equation Ax = b. + + Compute a vector x such that the 2-norm ``|b - A x|`` is minimized. + + Parameters + ---------- + a : (M, N) array_like + Left-hand side array + b : (M,) or (M, K) array_like + Right hand side array + cond : float, optional + Cutoff for 'small' singular values; used to determine effective + rank of a. Singular values smaller than + ``cond * largest_singular_value`` are considered zero. + overwrite_a : bool, optional + Discard data in `a` (may enhance performance). Default is False. + overwrite_b : bool, optional + Discard data in `b` (may enhance performance). Default is False. + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + lapack_driver : str, optional + Which LAPACK driver is used to solve the least-squares problem. + Options are ``'gelsd'``, ``'gelsy'``, ``'gelss'``. Default + (``'gelsd'``) is a good choice. However, ``'gelsy'`` can be slightly + faster on many problems. ``'gelss'`` was used historically. It is + generally slow but uses less memory. + + .. versionadded:: 0.17.0 + + Returns + ------- + x : (N,) or (N, K) ndarray + Least-squares solution. + residues : (K,) ndarray or float + Square of the 2-norm for each column in ``b - a x``, if ``M > N`` and + ``rank(A) == n`` (returns a scalar if ``b`` is 1-D). Otherwise a + (0,)-shaped array is returned. + rank : int + Effective rank of `a`. + s : (min(M, N),) ndarray or None + Singular values of `a`. The condition number of ``a`` is + ``s[0] / s[-1]``. + + Raises + ------ + LinAlgError + If computation does not converge. + + ValueError + When parameters are not compatible. + + See Also + -------- + scipy.optimize.nnls : linear least squares with non-negativity constraint + + Notes + ----- + When ``'gelsy'`` is used as a driver, `residues` is set to a (0,)-shaped + array and `s` is always ``None``. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import lstsq + >>> import matplotlib.pyplot as plt + + Suppose we have the following data: + + >>> x = np.array([1, 2.5, 3.5, 4, 5, 7, 8.5]) + >>> y = np.array([0.3, 1.1, 1.5, 2.0, 3.2, 6.6, 8.6]) + + We want to fit a quadratic polynomial of the form ``y = a + b*x**2`` + to this data. We first form the "design matrix" M, with a constant + column of 1s and a column containing ``x**2``: + + >>> M = x[:, np.newaxis]**[0, 2] + >>> M + array([[ 1. , 1. ], + [ 1. , 6.25], + [ 1. , 12.25], + [ 1. , 16. ], + [ 1. , 25. ], + [ 1. , 49. ], + [ 1. , 72.25]]) + + We want to find the least-squares solution to ``M.dot(p) = y``, + where ``p`` is a vector with length 2 that holds the parameters + ``a`` and ``b``. + + >>> p, res, rnk, s = lstsq(M, y) + >>> p + array([ 0.20925829, 0.12013861]) + + Plot the data and the fitted curve. + + >>> plt.plot(x, y, 'o', label='data') + >>> xx = np.linspace(0, 9, 101) + >>> yy = p[0] + p[1]*xx**2 + >>> plt.plot(xx, yy, label='least squares fit, $y = a + bx^2$') + >>> plt.xlabel('x') + >>> plt.ylabel('y') + >>> plt.legend(framealpha=1, shadow=True) + >>> plt.grid(alpha=0.25) + >>> plt.show() + + """ + a1 = _asarray_validated(a, check_finite=check_finite) + b1 = _asarray_validated(b, check_finite=check_finite) + if len(a1.shape) != 2: + raise ValueError('Input array a should be 2D') + m, n = a1.shape + if len(b1.shape) == 2: + nrhs = b1.shape[1] + else: + nrhs = 1 + if m != b1.shape[0]: + raise ValueError('Shape mismatch: a and b should have the same number' + f' of rows ({m} != {b1.shape[0]}).') + if m == 0 or n == 0: # Zero-sized problem, confuses LAPACK + x = np.zeros((n,) + b1.shape[1:], dtype=np.common_type(a1, b1)) + if n == 0: + residues = np.linalg.norm(b1, axis=0)**2 + else: + residues = np.empty((0,)) + return x, residues, 0, np.empty((0,)) + + driver = lapack_driver + if driver is None: + driver = lstsq.default_lapack_driver + if driver not in ('gelsd', 'gelsy', 'gelss'): + raise ValueError(f'LAPACK driver "{driver}" is not found') + + lapack_func, lapack_lwork = get_lapack_funcs((driver, + f'{driver}_lwork'), + (a1, b1)) + real_data = True if (lapack_func.dtype.kind == 'f') else False + + if m < n: + # need to extend b matrix as it will be filled with + # a larger solution matrix + if len(b1.shape) == 2: + b2 = np.zeros((n, nrhs), dtype=lapack_func.dtype) + b2[:m, :] = b1 + else: + b2 = np.zeros(n, dtype=lapack_func.dtype) + b2[:m] = b1 + b1 = b2 + + overwrite_a = overwrite_a or _datacopied(a1, a) + overwrite_b = overwrite_b or _datacopied(b1, b) + + if cond is None: + cond = np.finfo(lapack_func.dtype).eps + + if driver in ('gelss', 'gelsd'): + if driver == 'gelss': + lwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond) + v, x, s, rank, work, info = lapack_func(a1, b1, cond, lwork, + overwrite_a=overwrite_a, + overwrite_b=overwrite_b) + + elif driver == 'gelsd': + if real_data: + lwork, iwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond) + x, s, rank, info = lapack_func(a1, b1, lwork, + iwork, cond, False, False) + else: # complex data + lwork, rwork, iwork = _compute_lwork(lapack_lwork, m, n, + nrhs, cond) + x, s, rank, info = lapack_func(a1, b1, lwork, rwork, iwork, + cond, False, False) + if info > 0: + raise LinAlgError("SVD did not converge in Linear Least Squares") + if info < 0: + raise ValueError('illegal value in %d-th argument of internal %s' + % (-info, lapack_driver)) + resids = np.asarray([], dtype=x.dtype) + if m > n: + x1 = x[:n] + if rank == n: + resids = np.sum(np.abs(x[n:])**2, axis=0) + x = x1 + return x, resids, rank, s + + elif driver == 'gelsy': + lwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond) + jptv = np.zeros((a1.shape[1], 1), dtype=np.int32) + v, x, j, rank, info = lapack_func(a1, b1, jptv, cond, + lwork, False, False) + if info < 0: + raise ValueError("illegal value in %d-th argument of internal " + "gelsy" % -info) + if m > n: + x1 = x[:n] + x = x1 + return x, np.array([], x.dtype), rank, None + + +lstsq.default_lapack_driver = 'gelsd' + + +def pinv(a, *, atol=None, rtol=None, return_rank=False, check_finite=True): + """ + Compute the (Moore-Penrose) pseudo-inverse of a matrix. + + Calculate a generalized inverse of a matrix using its + singular-value decomposition ``U @ S @ V`` in the economy mode and picking + up only the columns/rows that are associated with significant singular + values. + + If ``s`` is the maximum singular value of ``a``, then the + significance cut-off value is determined by ``atol + rtol * s``. Any + singular value below this value is assumed insignificant. + + Parameters + ---------- + a : (M, N) array_like + Matrix to be pseudo-inverted. + atol : float, optional + Absolute threshold term, default value is 0. + + .. versionadded:: 1.7.0 + + rtol : float, optional + Relative threshold term, default value is ``max(M, N) * eps`` where + ``eps`` is the machine precision value of the datatype of ``a``. + + .. versionadded:: 1.7.0 + + return_rank : bool, optional + If True, return the effective rank of the matrix. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + B : (N, M) ndarray + The pseudo-inverse of matrix `a`. + rank : int + The effective rank of the matrix. Returned if `return_rank` is True. + + Raises + ------ + LinAlgError + If SVD computation does not converge. + + See Also + -------- + pinvh : Moore-Penrose pseudoinverse of a hermitian matrix. + + Notes + ----- + If ``A`` is invertible then the Moore-Penrose pseudoinverse is exactly + the inverse of ``A`` [1]_. If ``A`` is not invertible then the + Moore-Penrose pseudoinverse computes the ``x`` solution to ``Ax = b`` such + that ``||Ax - b||`` is minimized [1]_. + + References + ---------- + .. [1] Penrose, R. (1956). On best approximate solutions of linear matrix + equations. Mathematical Proceedings of the Cambridge Philosophical + Society, 52(1), 17-19. doi:10.1017/S0305004100030929 + + Examples + -------- + + Given an ``m x n`` matrix ``A`` and an ``n x m`` matrix ``B`` the four + Moore-Penrose conditions are: + + 1. ``ABA = A`` (``B`` is a generalized inverse of ``A``), + 2. ``BAB = B`` (``A`` is a generalized inverse of ``B``), + 3. ``(AB)* = AB`` (``AB`` is hermitian), + 4. ``(BA)* = BA`` (``BA`` is hermitian) [1]_. + + Here, ``A*`` denotes the conjugate transpose. The Moore-Penrose + pseudoinverse is a unique ``B`` that satisfies all four of these + conditions and exists for any ``A``. Note that, unlike the standard + matrix inverse, ``A`` does not have to be a square matrix or have + linearly independent columns/rows. + + As an example, we can calculate the Moore-Penrose pseudoinverse of a + random non-square matrix and verify it satisfies the four conditions. + + >>> import numpy as np + >>> from scipy import linalg + >>> rng = np.random.default_rng() + >>> A = rng.standard_normal((9, 6)) + >>> B = linalg.pinv(A) + >>> np.allclose(A @ B @ A, A) # Condition 1 + True + >>> np.allclose(B @ A @ B, B) # Condition 2 + True + >>> np.allclose((A @ B).conj().T, A @ B) # Condition 3 + True + >>> np.allclose((B @ A).conj().T, B @ A) # Condition 4 + True + + """ + a = _asarray_validated(a, check_finite=check_finite) + u, s, vh = _decomp_svd.svd(a, full_matrices=False, check_finite=False) + t = u.dtype.char.lower() + maxS = np.max(s, initial=0.) + + atol = 0. if atol is None else atol + rtol = max(a.shape) * np.finfo(t).eps if (rtol is None) else rtol + + if (atol < 0.) or (rtol < 0.): + raise ValueError("atol and rtol values must be positive.") + + val = atol + maxS * rtol + rank = np.sum(s > val) + + u = u[:, :rank] + u /= s[:rank] + B = (u @ vh[:rank]).conj().T + + if return_rank: + return B, rank + else: + return B + + +def pinvh(a, atol=None, rtol=None, lower=True, return_rank=False, + check_finite=True): + """ + Compute the (Moore-Penrose) pseudo-inverse of a Hermitian matrix. + + Calculate a generalized inverse of a complex Hermitian/real symmetric + matrix using its eigenvalue decomposition and including all eigenvalues + with 'large' absolute value. + + Parameters + ---------- + a : (N, N) array_like + Real symmetric or complex hermetian matrix to be pseudo-inverted + + atol : float, optional + Absolute threshold term, default value is 0. + + .. versionadded:: 1.7.0 + + rtol : float, optional + Relative threshold term, default value is ``N * eps`` where + ``eps`` is the machine precision value of the datatype of ``a``. + + .. versionadded:: 1.7.0 + + lower : bool, optional + Whether the pertinent array data is taken from the lower or upper + triangle of `a`. (Default: lower) + return_rank : bool, optional + If True, return the effective rank of the matrix. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + B : (N, N) ndarray + The pseudo-inverse of matrix `a`. + rank : int + The effective rank of the matrix. Returned if `return_rank` is True. + + Raises + ------ + LinAlgError + If eigenvalue algorithm does not converge. + + See Also + -------- + pinv : Moore-Penrose pseudoinverse of a matrix. + + Examples + -------- + + For a more detailed example see `pinv`. + + >>> import numpy as np + >>> from scipy.linalg import pinvh + >>> rng = np.random.default_rng() + >>> a = rng.standard_normal((9, 6)) + >>> a = np.dot(a, a.T) + >>> B = pinvh(a) + >>> np.allclose(a, a @ B @ a) + True + >>> np.allclose(B, B @ a @ B) + True + + """ + a = _asarray_validated(a, check_finite=check_finite) + s, u = _decomp.eigh(a, lower=lower, check_finite=False, driver='ev') + t = u.dtype.char.lower() + maxS = np.max(np.abs(s), initial=0.) + + atol = 0. if atol is None else atol + rtol = max(a.shape) * np.finfo(t).eps if (rtol is None) else rtol + + if (atol < 0.) or (rtol < 0.): + raise ValueError("atol and rtol values must be positive.") + + val = atol + maxS * rtol + above_cutoff = (abs(s) > val) + + psigma_diag = 1.0 / s[above_cutoff] + u = u[:, above_cutoff] + + B = (u * psigma_diag) @ u.conj().T + + if return_rank: + return B, len(psigma_diag) + else: + return B + + +def matrix_balance(A, permute=True, scale=True, separate=False, + overwrite_a=False): + """ + Compute a diagonal similarity transformation for row/column balancing. + + The balancing tries to equalize the row and column 1-norms by applying + a similarity transformation such that the magnitude variation of the + matrix entries is reflected to the scaling matrices. + + Moreover, if enabled, the matrix is first permuted to isolate the upper + triangular parts of the matrix and, again if scaling is also enabled, + only the remaining subblocks are subjected to scaling. + + The balanced matrix satisfies the following equality + + .. math:: + + B = T^{-1} A T + + The scaling coefficients are approximated to the nearest power of 2 + to avoid round-off errors. + + Parameters + ---------- + A : (n, n) array_like + Square data matrix for the balancing. + permute : bool, optional + The selector to define whether permutation of A is also performed + prior to scaling. + scale : bool, optional + The selector to turn on and off the scaling. If False, the matrix + will not be scaled. + separate : bool, optional + This switches from returning a full matrix of the transformation + to a tuple of two separate 1-D permutation and scaling arrays. + overwrite_a : bool, optional + This is passed to xGEBAL directly. Essentially, overwrites the result + to the data. It might increase the space efficiency. See LAPACK manual + for details. This is False by default. + + Returns + ------- + B : (n, n) ndarray + Balanced matrix + T : (n, n) ndarray + A possibly permuted diagonal matrix whose nonzero entries are + integer powers of 2 to avoid numerical truncation errors. + scale, perm : (n,) ndarray + If ``separate`` keyword is set to True then instead of the array + ``T`` above, the scaling and the permutation vectors are given + separately as a tuple without allocating the full array ``T``. + + Notes + ----- + This algorithm is particularly useful for eigenvalue and matrix + decompositions and in many cases it is already called by various + LAPACK routines. + + The algorithm is based on the well-known technique of [1]_ and has + been modified to account for special cases. See [2]_ for details + which have been implemented since LAPACK v3.5.0. Before this version + there are corner cases where balancing can actually worsen the + conditioning. See [3]_ for such examples. + + The code is a wrapper around LAPACK's xGEBAL routine family for matrix + balancing. + + .. versionadded:: 0.19.0 + + References + ---------- + .. [1] B.N. Parlett and C. Reinsch, "Balancing a Matrix for + Calculation of Eigenvalues and Eigenvectors", Numerische Mathematik, + Vol.13(4), 1969, :doi:`10.1007/BF02165404` + .. [2] R. James, J. Langou, B.R. Lowery, "On matrix balancing and + eigenvector computation", 2014, :arxiv:`1401.5766` + .. [3] D.S. Watkins. A case where balancing is harmful. + Electron. Trans. Numer. Anal, Vol.23, 2006. + + Examples + -------- + >>> import numpy as np + >>> from scipy import linalg + >>> x = np.array([[1,2,0], [9,1,0.01], [1,2,10*np.pi]]) + + >>> y, permscale = linalg.matrix_balance(x) + >>> np.abs(x).sum(axis=0) / np.abs(x).sum(axis=1) + array([ 3.66666667, 0.4995005 , 0.91312162]) + + >>> np.abs(y).sum(axis=0) / np.abs(y).sum(axis=1) + array([ 1.2 , 1.27041742, 0.92658316]) # may vary + + >>> permscale # only powers of 2 (0.5 == 2^(-1)) + array([[ 0.5, 0. , 0. ], # may vary + [ 0. , 1. , 0. ], + [ 0. , 0. , 1. ]]) + + """ + + A = np.atleast_2d(_asarray_validated(A, check_finite=True)) + + if not np.equal(*A.shape): + raise ValueError('The data matrix for balancing should be square.') + + # accommodate empty arrays + if A.size == 0: + b_n, t_n = matrix_balance(np.eye(2, dtype=A.dtype)) + B = np.empty_like(A, dtype=b_n.dtype) + if separate: + scaling = np.ones_like(A, shape=len(A)) + perm = np.arange(len(A)) + return B, (scaling, perm) + return B, np.empty_like(A, dtype=t_n.dtype) + + gebal = get_lapack_funcs(('gebal'), (A,)) + B, lo, hi, ps, info = gebal(A, scale=scale, permute=permute, + overwrite_a=overwrite_a) + + if info < 0: + raise ValueError('xGEBAL exited with the internal error ' + f'"illegal value in argument number {-info}.". See ' + 'LAPACK documentation for the xGEBAL error codes.') + + # Separate the permutations from the scalings and then convert to int + scaling = np.ones_like(ps, dtype=float) + scaling[lo:hi+1] = ps[lo:hi+1] + + # gebal uses 1-indexing + ps = ps.astype(int, copy=False) - 1 + n = A.shape[0] + perm = np.arange(n) + + # LAPACK permutes with the ordering n --> hi, then 0--> lo + if hi < n: + for ind, x in enumerate(ps[hi+1:][::-1], 1): + if n-ind == x: + continue + perm[[x, n-ind]] = perm[[n-ind, x]] + + if lo > 0: + for ind, x in enumerate(ps[:lo]): + if ind == x: + continue + perm[[x, ind]] = perm[[ind, x]] + + if separate: + return B, (scaling, perm) + + # get the inverse permutation + iperm = np.empty_like(perm) + iperm[perm] = np.arange(n) + + return B, np.diag(scaling)[iperm, :] + + +def _validate_args_for_toeplitz_ops(c_or_cr, b, check_finite, keep_b_shape, + enforce_square=True): + """Validate arguments and format inputs for toeplitz functions + + Parameters + ---------- + c_or_cr : array_like or tuple of (array_like, array_like) + The vector ``c``, or a tuple of arrays (``c``, ``r``). Whatever the + actual shape of ``c``, it will be converted to a 1-D array. If not + supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is + real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row + of the Toeplitz matrix is ``[c[0], r[1:]]``. Whatever the actual shape + of ``r``, it will be converted to a 1-D array. + b : (M,) or (M, K) array_like + Right-hand side in ``T x = b``. + check_finite : bool + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (result entirely NaNs) if the inputs do contain infinities or NaNs. + keep_b_shape : bool + Whether to convert a (M,) dimensional b into a (M, 1) dimensional + matrix. + enforce_square : bool, optional + If True (default), this verifies that the Toeplitz matrix is square. + + Returns + ------- + r : array + 1d array corresponding to the first row of the Toeplitz matrix. + c: array + 1d array corresponding to the first column of the Toeplitz matrix. + b: array + (M,), (M, 1) or (M, K) dimensional array, post validation, + corresponding to ``b``. + dtype: numpy datatype + ``dtype`` stores the datatype of ``r``, ``c`` and ``b``. If any of + ``r``, ``c`` or ``b`` are complex, ``dtype`` is ``np.complex128``, + otherwise, it is ``np.float``. + b_shape: tuple + Shape of ``b`` after passing it through ``_asarray_validated``. + + """ + + if isinstance(c_or_cr, tuple): + c, r = c_or_cr + c = _asarray_validated(c, check_finite=check_finite) + r = _asarray_validated(r, check_finite=check_finite) + else: + c = _asarray_validated(c_or_cr, check_finite=check_finite) + r = c.conjugate() + + if c.ndim > 1 or r.ndim > 1: + msg = ("Beginning in SciPy 1.17, multidimensional input will be treated as a " + "batch, not `ravel`ed. To preserve the existing behavior and silence " + "this warning, `ravel` arguments before passing them to " + "`toeplitz`, `matmul_toeplitz`, and `solve_toeplitz`.") + warnings.warn(msg, FutureWarning, stacklevel=2) + c = c.ravel() + r = r.ravel() + + if b is None: + raise ValueError('`b` must be an array, not None.') + + b = _asarray_validated(b, check_finite=check_finite) + b_shape = b.shape + + is_not_square = r.shape[0] != c.shape[0] + if (enforce_square and is_not_square) or b.shape[0] != r.shape[0]: + raise ValueError('Incompatible dimensions.') + + is_cmplx = np.iscomplexobj(r) or np.iscomplexobj(c) or np.iscomplexobj(b) + dtype = np.complex128 if is_cmplx else np.float64 + r, c, b = (np.asarray(i, dtype=dtype) for i in (r, c, b)) + + if b.ndim == 1 and not keep_b_shape: + b = b.reshape(-1, 1) + elif b.ndim != 1: + b = b.reshape(b.shape[0], -1 if b.size > 0 else 0) + + return r, c, b, dtype, b_shape + + +def matmul_toeplitz(c_or_cr, x, check_finite=False, workers=None): + r"""Efficient Toeplitz Matrix-Matrix Multiplication using FFT + + This function returns the matrix multiplication between a Toeplitz + matrix and a dense matrix. + + The Toeplitz matrix has constant diagonals, with c as its first column + and r as its first row. If r is not given, ``r == conjugate(c)`` is + assumed. + + .. warning:: + + Beginning in SciPy 1.17, multidimensional input will be treated as a batch, + not ``ravel``\ ed. To preserve the existing behavior, ``ravel`` arguments + before passing them to `matmul_toeplitz`. + + Parameters + ---------- + c_or_cr : array_like or tuple of (array_like, array_like) + The vector ``c``, or a tuple of arrays (``c``, ``r``). If not + supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is + real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row + of the Toeplitz matrix is ``[c[0], r[1:]]``. + x : (M,) or (M, K) array_like + Matrix with which to multiply. + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (result entirely NaNs) if the inputs do contain infinities or NaNs. + workers : int, optional + To pass to scipy.fft.fft and ifft. Maximum number of workers to use + for parallel computation. If negative, the value wraps around from + ``os.cpu_count()``. See scipy.fft.fft for more details. + + Returns + ------- + T @ x : (M,) or (M, K) ndarray + The result of the matrix multiplication ``T @ x``. Shape of return + matches shape of `x`. + + See Also + -------- + toeplitz : Toeplitz matrix + solve_toeplitz : Solve a Toeplitz system using Levinson Recursion + + Notes + ----- + The Toeplitz matrix is embedded in a circulant matrix and the FFT is used + to efficiently calculate the matrix-matrix product. + + Because the computation is based on the FFT, integer inputs will + result in floating point outputs. This is unlike NumPy's `matmul`, + which preserves the data type of the input. + + This is partly based on the implementation that can be found in [1]_, + licensed under the MIT license. More information about the method can be + found in reference [2]_. References [3]_ and [4]_ have more reference + implementations in Python. + + .. versionadded:: 1.6.0 + + References + ---------- + .. [1] Jacob R Gardner, Geoff Pleiss, David Bindel, Kilian + Q Weinberger, Andrew Gordon Wilson, "GPyTorch: Blackbox Matrix-Matrix + Gaussian Process Inference with GPU Acceleration" with contributions + from Max Balandat and Ruihan Wu. Available online: + https://github.com/cornellius-gp/gpytorch + + .. [2] J. Demmel, P. Koev, and X. Li, "A Brief Survey of Direct Linear + Solvers". In Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der + Vorst, editors. Templates for the Solution of Algebraic Eigenvalue + Problems: A Practical Guide. SIAM, Philadelphia, 2000. Available at: + http://www.netlib.org/utk/people/JackDongarra/etemplates/node384.html + + .. [3] R. Scheibler, E. Bezzam, I. Dokmanic, Pyroomacoustics: A Python + package for audio room simulations and array processing algorithms, + Proc. IEEE ICASSP, Calgary, CA, 2018. + https://github.com/LCAV/pyroomacoustics/blob/pypi-release/ + pyroomacoustics/adaptive/util.py + + .. [4] Marano S, Edwards B, Ferrari G and Fah D (2017), "Fitting + Earthquake Spectra: Colored Noise and Incomplete Data", Bulletin of + the Seismological Society of America., January, 2017. Vol. 107(1), + pp. 276-291. + + Examples + -------- + Multiply the Toeplitz matrix T with matrix x:: + + [ 1 -1 -2 -3] [1 10] + T = [ 3 1 -1 -2] x = [2 11] + [ 6 3 1 -1] [2 11] + [10 6 3 1] [5 19] + + To specify the Toeplitz matrix, only the first column and the first + row are needed. + + >>> import numpy as np + >>> c = np.array([1, 3, 6, 10]) # First column of T + >>> r = np.array([1, -1, -2, -3]) # First row of T + >>> x = np.array([[1, 10], [2, 11], [2, 11], [5, 19]]) + + >>> from scipy.linalg import toeplitz, matmul_toeplitz + >>> matmul_toeplitz((c, r), x) + array([[-20., -80.], + [ -7., -8.], + [ 9., 85.], + [ 33., 218.]]) + + Check the result by creating the full Toeplitz matrix and + multiplying it by ``x``. + + >>> toeplitz(c, r) @ x + array([[-20, -80], + [ -7, -8], + [ 9, 85], + [ 33, 218]]) + + The full matrix is never formed explicitly, so this routine + is suitable for very large Toeplitz matrices. + + >>> n = 1000000 + >>> matmul_toeplitz([1] + [0]*(n-1), np.ones(n)) + array([1., 1., 1., ..., 1., 1., 1.], shape=(1000000,)) + + """ + + from ..fft import fft, ifft, rfft, irfft + + r, c, x, dtype, x_shape = _validate_args_for_toeplitz_ops( + c_or_cr, x, check_finite, keep_b_shape=False, enforce_square=False) + n, m = x.shape + + T_nrows = len(c) + T_ncols = len(r) + p = T_nrows + T_ncols - 1 # equivalent to len(embedded_col) + return_shape = (T_nrows,) if len(x_shape) == 1 else (T_nrows, m) + + # accommodate empty arrays + if x.size == 0: + return np.empty_like(x, shape=return_shape) + + embedded_col = np.concatenate((c, r[-1:0:-1])) + + if np.iscomplexobj(embedded_col) or np.iscomplexobj(x): + fft_mat = fft(embedded_col, axis=0, workers=workers).reshape(-1, 1) + fft_x = fft(x, n=p, axis=0, workers=workers) + + mat_times_x = ifft(fft_mat*fft_x, axis=0, + workers=workers)[:T_nrows, :] + else: + # Real inputs; using rfft is faster + fft_mat = rfft(embedded_col, axis=0, workers=workers).reshape(-1, 1) + fft_x = rfft(x, n=p, axis=0, workers=workers) + + mat_times_x = irfft(fft_mat*fft_x, axis=0, + workers=workers, n=p)[:T_nrows, :] + + return mat_times_x.reshape(*return_shape) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_blas_subroutines.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_blas_subroutines.h new file mode 100644 index 0000000000000000000000000000000000000000..a175ca15f4adbed6d5c576e9e5ee1117abbd31ec --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_blas_subroutines.h @@ -0,0 +1,164 @@ +/* +This file was generated by _generate_pyx.py. +Do not edit this file directly. +*/ + +#include "npy_cblas.h" +#include "fortran_defs.h" + +#ifdef __cplusplus +extern "C" { +#endif + +void BLAS_FUNC(caxpy)(int *n, npy_complex64 *ca, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy); +void BLAS_FUNC(ccopy)(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy); +void F_FUNC(cdotcwrp,CDOTCWRP)(npy_complex64 *out, int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy); +void F_FUNC(cdotuwrp,CDOTUWRP)(npy_complex64 *out, int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy); +void BLAS_FUNC(cgbmv)(char *trans, int *m, int *n, int *kl, int *ku, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy); +void BLAS_FUNC(cgemm)(char *transa, char *transb, int *m, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc); +void BLAS_FUNC(cgemv)(char *trans, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy); +void BLAS_FUNC(cgerc)(int *m, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *a, int *lda); +void BLAS_FUNC(cgeru)(int *m, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *a, int *lda); +void BLAS_FUNC(chbmv)(char *uplo, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy); +void BLAS_FUNC(chemm)(char *side, char *uplo, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc); +void BLAS_FUNC(chemv)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy); +void BLAS_FUNC(cher)(char *uplo, int *n, float *alpha, npy_complex64 *x, int *incx, npy_complex64 *a, int *lda); +void BLAS_FUNC(cher2)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *a, int *lda); +void BLAS_FUNC(cher2k)(char *uplo, char *trans, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, float *beta, npy_complex64 *c, int *ldc); +void BLAS_FUNC(cherk)(char *uplo, char *trans, int *n, int *k, float *alpha, npy_complex64 *a, int *lda, float *beta, npy_complex64 *c, int *ldc); +void BLAS_FUNC(chpmv)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *ap, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy); +void BLAS_FUNC(chpr)(char *uplo, int *n, float *alpha, npy_complex64 *x, int *incx, npy_complex64 *ap); +void BLAS_FUNC(chpr2)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *ap); +void BLAS_FUNC(crotg)(npy_complex64 *ca, npy_complex64 *cb, float *c, npy_complex64 *s); +void BLAS_FUNC(cscal)(int *n, npy_complex64 *ca, npy_complex64 *cx, int *incx); +void BLAS_FUNC(csrot)(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy, float *c, float *s); +void BLAS_FUNC(csscal)(int *n, float *sa, npy_complex64 *cx, int *incx); +void BLAS_FUNC(cswap)(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy); +void BLAS_FUNC(csymm)(char *side, char *uplo, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc); +void BLAS_FUNC(csyr2k)(char *uplo, char *trans, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc); +void BLAS_FUNC(csyrk)(char *uplo, char *trans, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *beta, npy_complex64 *c, int *ldc); +void BLAS_FUNC(ctbmv)(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx); +void BLAS_FUNC(ctbsv)(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx); +void BLAS_FUNC(ctpmv)(char *uplo, char *trans, char *diag, int *n, npy_complex64 *ap, npy_complex64 *x, int *incx); +void BLAS_FUNC(ctpsv)(char *uplo, char *trans, char *diag, int *n, npy_complex64 *ap, npy_complex64 *x, int *incx); +void BLAS_FUNC(ctrmm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb); +void BLAS_FUNC(ctrmv)(char *uplo, char *trans, char *diag, int *n, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx); +void BLAS_FUNC(ctrsm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb); +void BLAS_FUNC(ctrsv)(char *uplo, char *trans, char *diag, int *n, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx); +double BLAS_FUNC(dasum)(int *n, double *dx, int *incx); +void BLAS_FUNC(daxpy)(int *n, double *da, double *dx, int *incx, double *dy, int *incy); +double BLAS_FUNC(dcabs1)(npy_complex128 *z); +void BLAS_FUNC(dcopy)(int *n, double *dx, int *incx, double *dy, int *incy); +double BLAS_FUNC(ddot)(int *n, double *dx, int *incx, double *dy, int *incy); +void BLAS_FUNC(dgbmv)(char *trans, int *m, int *n, int *kl, int *ku, double *alpha, double *a, int *lda, double *x, int *incx, double *beta, double *y, int *incy); +void BLAS_FUNC(dgemm)(char *transa, char *transb, int *m, int *n, int *k, double *alpha, double *a, int *lda, double *b, int *ldb, double *beta, double *c, int *ldc); +void BLAS_FUNC(dgemv)(char *trans, int *m, int *n, double *alpha, double *a, int *lda, double *x, int *incx, double *beta, double *y, int *incy); +void BLAS_FUNC(dger)(int *m, int *n, double *alpha, double *x, int *incx, double *y, int *incy, double *a, int *lda); +double BLAS_FUNC(dnrm2)(int *n, double *x, int *incx); +void BLAS_FUNC(drot)(int *n, double *dx, int *incx, double *dy, int *incy, double *c, double *s); +void BLAS_FUNC(drotg)(double *da, double *db, double *c, double *s); +void BLAS_FUNC(drotm)(int *n, double *dx, int *incx, double *dy, int *incy, double *dparam); +void BLAS_FUNC(drotmg)(double *dd1, double *dd2, double *dx1, double *dy1, double *dparam); +void BLAS_FUNC(dsbmv)(char *uplo, int *n, int *k, double *alpha, double *a, int *lda, double *x, int *incx, double *beta, double *y, int *incy); +void BLAS_FUNC(dscal)(int *n, double *da, double *dx, int *incx); +double BLAS_FUNC(dsdot)(int *n, float *sx, int *incx, float *sy, int *incy); +void BLAS_FUNC(dspmv)(char *uplo, int *n, double *alpha, double *ap, double *x, int *incx, double *beta, double *y, int *incy); +void BLAS_FUNC(dspr)(char *uplo, int *n, double *alpha, double *x, int *incx, double *ap); +void BLAS_FUNC(dspr2)(char *uplo, int *n, double *alpha, double *x, int *incx, double *y, int *incy, double *ap); +void BLAS_FUNC(dswap)(int *n, double *dx, int *incx, double *dy, int *incy); +void BLAS_FUNC(dsymm)(char *side, char *uplo, int *m, int *n, double *alpha, double *a, int *lda, double *b, int *ldb, double *beta, double *c, int *ldc); +void BLAS_FUNC(dsymv)(char *uplo, int *n, double *alpha, double *a, int *lda, double *x, int *incx, double *beta, double *y, int *incy); +void BLAS_FUNC(dsyr)(char *uplo, int *n, double *alpha, double *x, int *incx, double *a, int *lda); +void BLAS_FUNC(dsyr2)(char *uplo, int *n, double *alpha, double *x, int *incx, double *y, int *incy, double *a, int *lda); +void BLAS_FUNC(dsyr2k)(char *uplo, char *trans, int *n, int *k, double *alpha, double *a, int *lda, double *b, int *ldb, double *beta, double *c, int *ldc); +void BLAS_FUNC(dsyrk)(char *uplo, char *trans, int *n, int *k, double *alpha, double *a, int *lda, double *beta, double *c, int *ldc); +void BLAS_FUNC(dtbmv)(char *uplo, char *trans, char *diag, int *n, int *k, double *a, int *lda, double *x, int *incx); +void BLAS_FUNC(dtbsv)(char *uplo, char *trans, char *diag, int *n, int *k, double *a, int *lda, double *x, int *incx); +void BLAS_FUNC(dtpmv)(char *uplo, char *trans, char *diag, int *n, double *ap, double *x, int *incx); +void BLAS_FUNC(dtpsv)(char *uplo, char *trans, char *diag, int *n, double *ap, double *x, int *incx); +void BLAS_FUNC(dtrmm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, double *alpha, double *a, int *lda, double *b, int *ldb); +void BLAS_FUNC(dtrmv)(char *uplo, char *trans, char *diag, int *n, double *a, int *lda, double *x, int *incx); +void BLAS_FUNC(dtrsm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, double *alpha, double *a, int *lda, double *b, int *ldb); +void BLAS_FUNC(dtrsv)(char *uplo, char *trans, char *diag, int *n, double *a, int *lda, double *x, int *incx); +double BLAS_FUNC(dzasum)(int *n, npy_complex128 *zx, int *incx); +double BLAS_FUNC(dznrm2)(int *n, npy_complex128 *x, int *incx); +int BLAS_FUNC(icamax)(int *n, npy_complex64 *cx, int *incx); +int BLAS_FUNC(idamax)(int *n, double *dx, int *incx); +int BLAS_FUNC(isamax)(int *n, float *sx, int *incx); +int BLAS_FUNC(izamax)(int *n, npy_complex128 *zx, int *incx); +int BLAS_FUNC(lsame)(char *ca, char *cb); +float BLAS_FUNC(sasum)(int *n, float *sx, int *incx); +void BLAS_FUNC(saxpy)(int *n, float *sa, float *sx, int *incx, float *sy, int *incy); +float BLAS_FUNC(scasum)(int *n, npy_complex64 *cx, int *incx); +float BLAS_FUNC(scnrm2)(int *n, npy_complex64 *x, int *incx); +void BLAS_FUNC(scopy)(int *n, float *sx, int *incx, float *sy, int *incy); +float BLAS_FUNC(sdot)(int *n, float *sx, int *incx, float *sy, int *incy); +float BLAS_FUNC(sdsdot)(int *n, float *sb, float *sx, int *incx, float *sy, int *incy); +void BLAS_FUNC(sgbmv)(char *trans, int *m, int *n, int *kl, int *ku, float *alpha, float *a, int *lda, float *x, int *incx, float *beta, float *y, int *incy); +void BLAS_FUNC(sgemm)(char *transa, char *transb, int *m, int *n, int *k, float *alpha, float *a, int *lda, float *b, int *ldb, float *beta, float *c, int *ldc); +void BLAS_FUNC(sgemv)(char *trans, int *m, int *n, float *alpha, float *a, int *lda, float *x, int *incx, float *beta, float *y, int *incy); +void BLAS_FUNC(sger)(int *m, int *n, float *alpha, float *x, int *incx, float *y, int *incy, float *a, int *lda); +float BLAS_FUNC(snrm2)(int *n, float *x, int *incx); +void BLAS_FUNC(srot)(int *n, float *sx, int *incx, float *sy, int *incy, float *c, float *s); +void BLAS_FUNC(srotg)(float *sa, float *sb, float *c, float *s); +void BLAS_FUNC(srotm)(int *n, float *sx, int *incx, float *sy, int *incy, float *sparam); +void BLAS_FUNC(srotmg)(float *sd1, float *sd2, float *sx1, float *sy1, float *sparam); +void BLAS_FUNC(ssbmv)(char *uplo, int *n, int *k, float *alpha, float *a, int *lda, float *x, int *incx, float *beta, float *y, int *incy); +void BLAS_FUNC(sscal)(int *n, float *sa, float *sx, int *incx); +void BLAS_FUNC(sspmv)(char *uplo, int *n, float *alpha, float *ap, float *x, int *incx, float *beta, float *y, int *incy); +void BLAS_FUNC(sspr)(char *uplo, int *n, float *alpha, float *x, int *incx, float *ap); +void BLAS_FUNC(sspr2)(char *uplo, int *n, float *alpha, float *x, int *incx, float *y, int *incy, float *ap); +void BLAS_FUNC(sswap)(int *n, float *sx, int *incx, float *sy, int *incy); +void BLAS_FUNC(ssymm)(char *side, char *uplo, int *m, int *n, float *alpha, float *a, int *lda, float *b, int *ldb, float *beta, float *c, int *ldc); +void BLAS_FUNC(ssymv)(char *uplo, int *n, float *alpha, float *a, int *lda, float *x, int *incx, float *beta, float *y, int *incy); +void BLAS_FUNC(ssyr)(char *uplo, int *n, float *alpha, float *x, int *incx, float *a, int *lda); +void BLAS_FUNC(ssyr2)(char *uplo, int *n, float *alpha, float *x, int *incx, float *y, int *incy, float *a, int *lda); +void BLAS_FUNC(ssyr2k)(char *uplo, char *trans, int *n, int *k, float *alpha, float *a, int *lda, float *b, int *ldb, float *beta, float *c, int *ldc); +void BLAS_FUNC(ssyrk)(char *uplo, char *trans, int *n, int *k, float *alpha, float *a, int *lda, float *beta, float *c, int *ldc); +void BLAS_FUNC(stbmv)(char *uplo, char *trans, char *diag, int *n, int *k, float *a, int *lda, float *x, int *incx); +void BLAS_FUNC(stbsv)(char *uplo, char *trans, char *diag, int *n, int *k, float *a, int *lda, float *x, int *incx); +void BLAS_FUNC(stpmv)(char *uplo, char *trans, char *diag, int *n, float *ap, float *x, int *incx); +void BLAS_FUNC(stpsv)(char *uplo, char *trans, char *diag, int *n, float *ap, float *x, int *incx); +void BLAS_FUNC(strmm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, float *alpha, float *a, int *lda, float *b, int *ldb); +void BLAS_FUNC(strmv)(char *uplo, char *trans, char *diag, int *n, float *a, int *lda, float *x, int *incx); +void BLAS_FUNC(strsm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, float *alpha, float *a, int *lda, float *b, int *ldb); +void BLAS_FUNC(strsv)(char *uplo, char *trans, char *diag, int *n, float *a, int *lda, float *x, int *incx); +void BLAS_FUNC(zaxpy)(int *n, npy_complex128 *za, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy); +void BLAS_FUNC(zcopy)(int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy); +void F_FUNC(zdotcwrp,ZDOTCWRP)(npy_complex128 *out, int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy); +void F_FUNC(zdotuwrp,ZDOTUWRP)(npy_complex128 *out, int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy); +void BLAS_FUNC(zdrot)(int *n, npy_complex128 *cx, int *incx, npy_complex128 *cy, int *incy, double *c, double *s); +void BLAS_FUNC(zdscal)(int *n, double *da, npy_complex128 *zx, int *incx); +void BLAS_FUNC(zgbmv)(char *trans, int *m, int *n, int *kl, int *ku, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy); +void BLAS_FUNC(zgemm)(char *transa, char *transb, int *m, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc); +void BLAS_FUNC(zgemv)(char *trans, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy); +void BLAS_FUNC(zgerc)(int *m, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *a, int *lda); +void BLAS_FUNC(zgeru)(int *m, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *a, int *lda); +void BLAS_FUNC(zhbmv)(char *uplo, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy); +void BLAS_FUNC(zhemm)(char *side, char *uplo, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc); +void BLAS_FUNC(zhemv)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy); +void BLAS_FUNC(zher)(char *uplo, int *n, double *alpha, npy_complex128 *x, int *incx, npy_complex128 *a, int *lda); +void BLAS_FUNC(zher2)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *a, int *lda); +void BLAS_FUNC(zher2k)(char *uplo, char *trans, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, double *beta, npy_complex128 *c, int *ldc); +void BLAS_FUNC(zherk)(char *uplo, char *trans, int *n, int *k, double *alpha, npy_complex128 *a, int *lda, double *beta, npy_complex128 *c, int *ldc); +void BLAS_FUNC(zhpmv)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *ap, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy); +void BLAS_FUNC(zhpr)(char *uplo, int *n, double *alpha, npy_complex128 *x, int *incx, npy_complex128 *ap); +void BLAS_FUNC(zhpr2)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *ap); +void BLAS_FUNC(zrotg)(npy_complex128 *ca, npy_complex128 *cb, double *c, npy_complex128 *s); +void BLAS_FUNC(zscal)(int *n, npy_complex128 *za, npy_complex128 *zx, int *incx); +void BLAS_FUNC(zswap)(int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy); +void BLAS_FUNC(zsymm)(char *side, char *uplo, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc); +void BLAS_FUNC(zsyr2k)(char *uplo, char *trans, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc); +void BLAS_FUNC(zsyrk)(char *uplo, char *trans, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *beta, npy_complex128 *c, int *ldc); +void BLAS_FUNC(ztbmv)(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx); +void BLAS_FUNC(ztbsv)(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx); +void BLAS_FUNC(ztpmv)(char *uplo, char *trans, char *diag, int *n, npy_complex128 *ap, npy_complex128 *x, int *incx); +void BLAS_FUNC(ztpsv)(char *uplo, char *trans, char *diag, int *n, npy_complex128 *ap, npy_complex128 *x, int *incx); +void BLAS_FUNC(ztrmm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb); +void BLAS_FUNC(ztrmv)(char *uplo, char *trans, char *diag, int *n, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx); +void BLAS_FUNC(ztrsm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb); +void BLAS_FUNC(ztrsv)(char *uplo, char *trans, char *diag, int *n, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx); + +#ifdef __cplusplus +} +#endif diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_cythonized_array_utils.pxd b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_cythonized_array_utils.pxd new file mode 100644 index 0000000000000000000000000000000000000000..ccec61c078e57ba7b6a310ec57189fcf236c972d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_cythonized_array_utils.pxd @@ -0,0 +1,40 @@ +cimport numpy as cnp + +ctypedef fused lapack_t: + float + double + (float complex) + (double complex) + +ctypedef fused lapack_cz_t: + (float complex) + (double complex) + +ctypedef fused lapack_sd_t: + float + double + +ctypedef fused np_numeric_t: + cnp.int8_t + cnp.int16_t + cnp.int32_t + cnp.int64_t + cnp.uint8_t + cnp.uint16_t + cnp.uint32_t + cnp.uint64_t + cnp.float32_t + cnp.float64_t + cnp.longdouble_t + cnp.complex64_t + cnp.complex128_t + +ctypedef fused np_complex_numeric_t: + cnp.complex64_t + cnp.complex128_t + + +cdef void swap_c_and_f_layout(lapack_t *a, lapack_t *b, int r, int c) noexcept nogil +cdef (int, int) band_check_internal_c(np_numeric_t[:, ::1]A) noexcept nogil +cdef bint is_sym_her_real_c_internal(np_numeric_t[:, ::1]A) noexcept nogil +cdef bint is_sym_her_complex_c_internal(np_complex_numeric_t[:, ::1]A) noexcept nogil diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_cythonized_array_utils.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_cythonized_array_utils.pyi new file mode 100644 index 0000000000000000000000000000000000000000..5633cb61ecf3a90eba901120f64fa6cc6634fa5a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_cythonized_array_utils.pyi @@ -0,0 +1,16 @@ +from numpy.typing import NDArray +from typing import Any + +def bandwidth(a: NDArray[Any]) -> tuple[int, int]: ... + +def issymmetric( + a: NDArray[Any], + atol: None | float = ..., + rtol: None | float = ..., +) -> bool: ... + +def ishermitian( + a: NDArray[Any], + atol: None | float = ..., + rtol: None | float = ..., +) -> bool: ... diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp.py new file mode 100644 index 0000000000000000000000000000000000000000..c520d6b04b6bf0a22c4fcad62d98a17faea7c9fd --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp.py @@ -0,0 +1,1632 @@ +# +# Author: Pearu Peterson, March 2002 +# +# additions by Travis Oliphant, March 2002 +# additions by Eric Jones, June 2002 +# additions by Johannes Loehnert, June 2006 +# additions by Bart Vandereycken, June 2006 +# additions by Andrew D Straw, May 2007 +# additions by Tiziano Zito, November 2008 +# +# April 2010: Functions for LU, QR, SVD, Schur, and Cholesky decompositions +# were moved to their own files. Still in this file are functions for +# eigenstuff and for the Hessenberg form. + +__all__ = ['eig', 'eigvals', 'eigh', 'eigvalsh', + 'eig_banded', 'eigvals_banded', + 'eigh_tridiagonal', 'eigvalsh_tridiagonal', 'hessenberg', 'cdf2rdf'] + +import numpy as np +from numpy import (array, isfinite, inexact, nonzero, iscomplexobj, + flatnonzero, conj, asarray, argsort, empty, + iscomplex, zeros, einsum, eye, inf) +# Local imports +from scipy._lib._util import _asarray_validated +from ._misc import LinAlgError, _datacopied, norm +from .lapack import get_lapack_funcs, _compute_lwork + + +_I = np.array(1j, dtype='F') + + +def _make_complex_eigvecs(w, vin, dtype): + """ + Produce complex-valued eigenvectors from LAPACK DGGEV real-valued output + """ + # - see LAPACK man page DGGEV at ALPHAI + v = np.array(vin, dtype=dtype) + m = (w.imag > 0) + m[:-1] |= (w.imag[1:] < 0) # workaround for LAPACK bug, cf. ticket #709 + for i in flatnonzero(m): + v.imag[:, i] = vin[:, i+1] + conj(v[:, i], v[:, i+1]) + return v + + +def _make_eigvals(alpha, beta, homogeneous_eigvals): + if homogeneous_eigvals: + if beta is None: + return np.vstack((alpha, np.ones_like(alpha))) + else: + return np.vstack((alpha, beta)) + else: + if beta is None: + return alpha + else: + w = np.empty_like(alpha) + alpha_zero = (alpha == 0) + beta_zero = (beta == 0) + beta_nonzero = ~beta_zero + w[beta_nonzero] = alpha[beta_nonzero]/beta[beta_nonzero] + # Use np.inf for complex values too since + # 1/np.inf = 0, i.e., it correctly behaves as projective + # infinity. + w[~alpha_zero & beta_zero] = np.inf + if np.all(alpha.imag == 0): + w[alpha_zero & beta_zero] = np.nan + else: + w[alpha_zero & beta_zero] = complex(np.nan, np.nan) + return w + + +def _geneig(a1, b1, left, right, overwrite_a, overwrite_b, + homogeneous_eigvals): + ggev, = get_lapack_funcs(('ggev',), (a1, b1)) + cvl, cvr = left, right + res = ggev(a1, b1, lwork=-1) + lwork = res[-2][0].real.astype(np.int_) + if ggev.typecode in 'cz': + alpha, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr, lwork, + overwrite_a, overwrite_b) + w = _make_eigvals(alpha, beta, homogeneous_eigvals) + else: + alphar, alphai, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr, + lwork, overwrite_a, + overwrite_b) + alpha = alphar + _I * alphai + w = _make_eigvals(alpha, beta, homogeneous_eigvals) + _check_info(info, 'generalized eig algorithm (ggev)') + + only_real = np.all(w.imag == 0.0) + if not (ggev.typecode in 'cz' or only_real): + t = w.dtype.char + if left: + vl = _make_complex_eigvecs(w, vl, t) + if right: + vr = _make_complex_eigvecs(w, vr, t) + + # the eigenvectors returned by the lapack function are NOT normalized + for i in range(vr.shape[0]): + if right: + vr[:, i] /= norm(vr[:, i]) + if left: + vl[:, i] /= norm(vl[:, i]) + + if not (left or right): + return w + if left: + if right: + return w, vl, vr + return w, vl + return w, vr + + +def eig(a, b=None, left=False, right=True, overwrite_a=False, + overwrite_b=False, check_finite=True, homogeneous_eigvals=False): + """ + Solve an ordinary or generalized eigenvalue problem of a square matrix. + + Find eigenvalues w and right or left eigenvectors of a general matrix:: + + a vr[:,i] = w[i] b vr[:,i] + a.H vl[:,i] = w[i].conj() b.H vl[:,i] + + where ``.H`` is the Hermitian conjugation. + + Parameters + ---------- + a : (M, M) array_like + A complex or real matrix whose eigenvalues and eigenvectors + will be computed. + b : (M, M) array_like, optional + Right-hand side matrix in a generalized eigenvalue problem. + Default is None, identity matrix is assumed. + left : bool, optional + Whether to calculate and return left eigenvectors. Default is False. + right : bool, optional + Whether to calculate and return right eigenvectors. Default is True. + overwrite_a : bool, optional + Whether to overwrite `a`; may improve performance. Default is False. + overwrite_b : bool, optional + Whether to overwrite `b`; may improve performance. Default is False. + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + homogeneous_eigvals : bool, optional + If True, return the eigenvalues in homogeneous coordinates. + In this case ``w`` is a (2, M) array so that:: + + w[1,i] a vr[:,i] = w[0,i] b vr[:,i] + + Default is False. + + Returns + ------- + w : (M,) or (2, M) double or complex ndarray + The eigenvalues, each repeated according to its + multiplicity. The shape is (M,) unless + ``homogeneous_eigvals=True``. + vl : (M, M) double or complex ndarray + The left eigenvector corresponding to the eigenvalue + ``w[i]`` is the column ``vl[:,i]``. Only returned if ``left=True``. + The left eigenvector is not normalized. + vr : (M, M) double or complex ndarray + The normalized right eigenvector corresponding to the eigenvalue + ``w[i]`` is the column ``vr[:,i]``. Only returned if ``right=True``. + + Raises + ------ + LinAlgError + If eigenvalue computation does not converge. + + See Also + -------- + eigvals : eigenvalues of general arrays + eigh : Eigenvalues and right eigenvectors for symmetric/Hermitian arrays. + eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian + band matrices + eigh_tridiagonal : eigenvalues and right eiegenvectors for + symmetric/Hermitian tridiagonal matrices + + Examples + -------- + >>> import numpy as np + >>> from scipy import linalg + >>> a = np.array([[0., -1.], [1., 0.]]) + >>> linalg.eigvals(a) + array([0.+1.j, 0.-1.j]) + + >>> b = np.array([[0., 1.], [1., 1.]]) + >>> linalg.eigvals(a, b) + array([ 1.+0.j, -1.+0.j]) + + >>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]]) + >>> linalg.eigvals(a, homogeneous_eigvals=True) + array([[3.+0.j, 8.+0.j, 7.+0.j], + [1.+0.j, 1.+0.j, 1.+0.j]]) + + >>> a = np.array([[0., -1.], [1., 0.]]) + >>> linalg.eigvals(a) == linalg.eig(a)[0] + array([ True, True]) + >>> linalg.eig(a, left=True, right=False)[1] # normalized left eigenvector + array([[-0.70710678+0.j , -0.70710678-0.j ], + [-0. +0.70710678j, -0. -0.70710678j]]) + >>> linalg.eig(a, left=False, right=True)[1] # normalized right eigenvector + array([[0.70710678+0.j , 0.70710678-0.j ], + [0. -0.70710678j, 0. +0.70710678j]]) + + + + """ + a1 = _asarray_validated(a, check_finite=check_finite) + if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]: + raise ValueError('expected square matrix') + + # accommodate square empty matrices + if a1.size == 0: + w_n, vr_n = eig(np.eye(2, dtype=a1.dtype)) + w = np.empty_like(a1, shape=(0,), dtype=w_n.dtype) + w = _make_eigvals(w, None, homogeneous_eigvals) + vl = np.empty_like(a1, shape=(0, 0), dtype=vr_n.dtype) + vr = np.empty_like(a1, shape=(0, 0), dtype=vr_n.dtype) + if not (left or right): + return w + if left: + if right: + return w, vl, vr + return w, vl + return w, vr + + overwrite_a = overwrite_a or (_datacopied(a1, a)) + if b is not None: + b1 = _asarray_validated(b, check_finite=check_finite) + overwrite_b = overwrite_b or _datacopied(b1, b) + if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]: + raise ValueError('expected square matrix') + if b1.shape != a1.shape: + raise ValueError('a and b must have the same shape') + return _geneig(a1, b1, left, right, overwrite_a, overwrite_b, + homogeneous_eigvals) + + geev, geev_lwork = get_lapack_funcs(('geev', 'geev_lwork'), (a1,)) + compute_vl, compute_vr = left, right + + lwork = _compute_lwork(geev_lwork, a1.shape[0], + compute_vl=compute_vl, + compute_vr=compute_vr) + + if geev.typecode in 'cz': + w, vl, vr, info = geev(a1, lwork=lwork, + compute_vl=compute_vl, + compute_vr=compute_vr, + overwrite_a=overwrite_a) + w = _make_eigvals(w, None, homogeneous_eigvals) + else: + wr, wi, vl, vr, info = geev(a1, lwork=lwork, + compute_vl=compute_vl, + compute_vr=compute_vr, + overwrite_a=overwrite_a) + w = wr + _I * wi + w = _make_eigvals(w, None, homogeneous_eigvals) + + _check_info(info, 'eig algorithm (geev)', + positive='did not converge (only eigenvalues ' + 'with order >= %d have converged)') + + only_real = np.all(w.imag == 0.0) + if not (geev.typecode in 'cz' or only_real): + t = w.dtype.char + if left: + vl = _make_complex_eigvecs(w, vl, t) + if right: + vr = _make_complex_eigvecs(w, vr, t) + if not (left or right): + return w + if left: + if right: + return w, vl, vr + return w, vl + return w, vr + + +def eigh(a, b=None, *, lower=True, eigvals_only=False, overwrite_a=False, + overwrite_b=False, type=1, check_finite=True, subset_by_index=None, + subset_by_value=None, driver=None): + """ + Solve a standard or generalized eigenvalue problem for a complex + Hermitian or real symmetric matrix. + + Find eigenvalues array ``w`` and optionally eigenvectors array ``v`` of + array ``a``, where ``b`` is positive definite such that for every + eigenvalue λ (i-th entry of w) and its eigenvector ``vi`` (i-th column of + ``v``) satisfies:: + + a @ vi = λ * b @ vi + vi.conj().T @ a @ vi = λ + vi.conj().T @ b @ vi = 1 + + In the standard problem, ``b`` is assumed to be the identity matrix. + + Parameters + ---------- + a : (M, M) array_like + A complex Hermitian or real symmetric matrix whose eigenvalues and + eigenvectors will be computed. + b : (M, M) array_like, optional + A complex Hermitian or real symmetric definite positive matrix in. + If omitted, identity matrix is assumed. + lower : bool, optional + Whether the pertinent array data is taken from the lower or upper + triangle of ``a`` and, if applicable, ``b``. (Default: lower) + eigvals_only : bool, optional + Whether to calculate only eigenvalues and no eigenvectors. + (Default: both are calculated) + subset_by_index : iterable, optional + If provided, this two-element iterable defines the start and the end + indices of the desired eigenvalues (ascending order and 0-indexed). + To return only the second smallest to fifth smallest eigenvalues, + ``[1, 4]`` is used. ``[n-3, n-1]`` returns the largest three. Only + available with "evr", "evx", and "gvx" drivers. The entries are + directly converted to integers via ``int()``. + subset_by_value : iterable, optional + If provided, this two-element iterable defines the half-open interval + ``(a, b]`` that, if any, only the eigenvalues between these values + are returned. Only available with "evr", "evx", and "gvx" drivers. Use + ``np.inf`` for the unconstrained ends. + driver : str, optional + Defines which LAPACK driver should be used. Valid options are "ev", + "evd", "evr", "evx" for standard problems and "gv", "gvd", "gvx" for + generalized (where b is not None) problems. See the Notes section. + The default for standard problems is "evr". For generalized problems, + "gvd" is used for full set, and "gvx" for subset requested cases. + type : int, optional + For the generalized problems, this keyword specifies the problem type + to be solved for ``w`` and ``v`` (only takes 1, 2, 3 as possible + inputs):: + + 1 => a @ v = w @ b @ v + 2 => a @ b @ v = w @ v + 3 => b @ a @ v = w @ v + + This keyword is ignored for standard problems. + overwrite_a : bool, optional + Whether to overwrite data in ``a`` (may improve performance). Default + is False. + overwrite_b : bool, optional + Whether to overwrite data in ``b`` (may improve performance). Default + is False. + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + w : (N,) ndarray + The N (N<=M) selected eigenvalues, in ascending order, each + repeated according to its multiplicity. + v : (M, N) ndarray + The normalized eigenvector corresponding to the eigenvalue ``w[i]`` is + the column ``v[:,i]``. Only returned if ``eigvals_only=False``. + + Raises + ------ + LinAlgError + If eigenvalue computation does not converge, an error occurred, or + b matrix is not definite positive. Note that if input matrices are + not symmetric or Hermitian, no error will be reported but results will + be wrong. + + See Also + -------- + eigvalsh : eigenvalues of symmetric or Hermitian arrays + eig : eigenvalues and right eigenvectors for non-symmetric arrays + eigh_tridiagonal : eigenvalues and right eiegenvectors for + symmetric/Hermitian tridiagonal matrices + + Notes + ----- + This function does not check the input array for being Hermitian/symmetric + in order to allow for representing arrays with only their upper/lower + triangular parts. Also, note that even though not taken into account, + finiteness check applies to the whole array and unaffected by "lower" + keyword. + + This function uses LAPACK drivers for computations in all possible keyword + combinations, prefixed with ``sy`` if arrays are real and ``he`` if + complex, e.g., a float array with "evr" driver is solved via + "syevr", complex arrays with "gvx" driver problem is solved via "hegvx" + etc. + + As a brief summary, the slowest and the most robust driver is the + classical ``ev`` which uses symmetric QR. ``evr`` is seen as + the optimal choice for the most general cases. However, there are certain + occasions that ``evd`` computes faster at the expense of more + memory usage. ``evx``, while still being faster than ``ev``, + often performs worse than the rest except when very few eigenvalues are + requested for large arrays though there is still no performance guarantee. + + Note that the underlying LAPACK algorithms are different depending on whether + `eigvals_only` is True or False --- thus the eigenvalues may differ + depending on whether eigenvectors are requested or not. The difference is + generally of the order of machine epsilon times the largest eigenvalue, + so is likely only visible for zero or nearly zero eigenvalues. + + For the generalized problem, normalization with respect to the given + type argument:: + + type 1 and 3 : v.conj().T @ a @ v = w + type 2 : inv(v).conj().T @ a @ inv(v) = w + + type 1 or 2 : v.conj().T @ b @ v = I + type 3 : v.conj().T @ inv(b) @ v = I + + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import eigh + >>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]]) + >>> w, v = eigh(A) + >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4))) + True + + Request only the eigenvalues + + >>> w = eigh(A, eigvals_only=True) + + Request eigenvalues that are less than 10. + + >>> A = np.array([[34, -4, -10, -7, 2], + ... [-4, 7, 2, 12, 0], + ... [-10, 2, 44, 2, -19], + ... [-7, 12, 2, 79, -34], + ... [2, 0, -19, -34, 29]]) + >>> eigh(A, eigvals_only=True, subset_by_value=[-np.inf, 10]) + array([6.69199443e-07, 9.11938152e+00]) + + Request the second smallest eigenvalue and its eigenvector + + >>> w, v = eigh(A, subset_by_index=[1, 1]) + >>> w + array([9.11938152]) + >>> v.shape # only a single column is returned + (5, 1) + + """ + # set lower + uplo = 'L' if lower else 'U' + # Set job for Fortran routines + _job = 'N' if eigvals_only else 'V' + + drv_str = [None, "ev", "evd", "evr", "evx", "gv", "gvd", "gvx"] + if driver not in drv_str: + raise ValueError('"{}" is unknown. Possible values are "None", "{}".' + ''.format(driver, '", "'.join(drv_str[1:]))) + + a1 = _asarray_validated(a, check_finite=check_finite) + if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]: + raise ValueError('expected square "a" matrix') + + # accommodate square empty matrices + if a1.size == 0: + w_n, v_n = eigh(np.eye(2, dtype=a1.dtype)) + + w = np.empty_like(a1, shape=(0,), dtype=w_n.dtype) + v = np.empty_like(a1, shape=(0, 0), dtype=v_n.dtype) + if eigvals_only: + return w + else: + return w, v + + overwrite_a = overwrite_a or (_datacopied(a1, a)) + cplx = True if iscomplexobj(a1) else False + n = a1.shape[0] + drv_args = {'overwrite_a': overwrite_a} + + if b is not None: + b1 = _asarray_validated(b, check_finite=check_finite) + overwrite_b = overwrite_b or _datacopied(b1, b) + if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]: + raise ValueError('expected square "b" matrix') + + if b1.shape != a1.shape: + raise ValueError(f"wrong b dimensions {b1.shape}, should be {a1.shape}") + + if type not in [1, 2, 3]: + raise ValueError('"type" keyword only accepts 1, 2, and 3.') + + cplx = True if iscomplexobj(b1) else (cplx or False) + drv_args.update({'overwrite_b': overwrite_b, 'itype': type}) + + subset = (subset_by_index is not None) or (subset_by_value is not None) + + # Both subsets can't be given + if subset_by_index and subset_by_value: + raise ValueError('Either index or value subset can be requested.') + + # Check indices if given + if subset_by_index: + lo, hi = (int(x) for x in subset_by_index) + if not (0 <= lo <= hi < n): + raise ValueError('Requested eigenvalue indices are not valid. ' + f'Valid range is [0, {n-1}] and start <= end, but ' + f'start={lo}, end={hi} is given') + # fortran is 1-indexed + drv_args.update({'range': 'I', 'il': lo + 1, 'iu': hi + 1}) + + if subset_by_value: + lo, hi = subset_by_value + if not (-inf <= lo < hi <= inf): + raise ValueError('Requested eigenvalue bounds are not valid. ' + 'Valid range is (-inf, inf) and low < high, but ' + f'low={lo}, high={hi} is given') + + drv_args.update({'range': 'V', 'vl': lo, 'vu': hi}) + + # fix prefix for lapack routines + pfx = 'he' if cplx else 'sy' + + # decide on the driver if not given + # first early exit on incompatible choice + if driver: + if b is None and (driver in ["gv", "gvd", "gvx"]): + raise ValueError(f'{driver} requires input b array to be supplied ' + 'for generalized eigenvalue problems.') + if (b is not None) and (driver in ['ev', 'evd', 'evr', 'evx']): + raise ValueError(f'"{driver}" does not accept input b array ' + 'for standard eigenvalue problems.') + if subset and (driver in ["ev", "evd", "gv", "gvd"]): + raise ValueError(f'"{driver}" cannot compute subsets of eigenvalues') + + # Default driver is evr and gvd + else: + driver = "evr" if b is None else ("gvx" if subset else "gvd") + + lwork_spec = { + 'syevd': ['lwork', 'liwork'], + 'syevr': ['lwork', 'liwork'], + 'heevd': ['lwork', 'liwork', 'lrwork'], + 'heevr': ['lwork', 'lrwork', 'liwork'], + } + + if b is None: # Standard problem + drv, drvlw = get_lapack_funcs((pfx + driver, pfx+driver+'_lwork'), + [a1]) + clw_args = {'n': n, 'lower': lower} + if driver == 'evd': + clw_args.update({'compute_v': 0 if _job == "N" else 1}) + + lw = _compute_lwork(drvlw, **clw_args) + # Multiple lwork vars + if isinstance(lw, tuple): + lwork_args = dict(zip(lwork_spec[pfx+driver], lw)) + else: + lwork_args = {'lwork': lw} + + drv_args.update({'lower': lower, 'compute_v': 0 if _job == "N" else 1}) + w, v, *other_args, info = drv(a=a1, **drv_args, **lwork_args) + + else: # Generalized problem + # 'gvd' doesn't have lwork query + if driver == "gvd": + drv = get_lapack_funcs(pfx + "gvd", [a1, b1]) + lwork_args = {} + else: + drv, drvlw = get_lapack_funcs((pfx + driver, pfx+driver+'_lwork'), + [a1, b1]) + # generalized drivers use uplo instead of lower + lw = _compute_lwork(drvlw, n, uplo=uplo) + lwork_args = {'lwork': lw} + + drv_args.update({'uplo': uplo, 'jobz': _job}) + + w, v, *other_args, info = drv(a=a1, b=b1, **drv_args, **lwork_args) + + # m is always the first extra argument + w = w[:other_args[0]] if subset else w + v = v[:, :other_args[0]] if (subset and not eigvals_only) else v + + # Check if we had a successful exit + if info == 0: + if eigvals_only: + return w + else: + return w, v + else: + if info < -1: + raise LinAlgError(f'Illegal value in argument {-info} of internal ' + f'{drv.typecode + pfx + driver}') + elif info > n: + raise LinAlgError(f'The leading minor of order {info-n} of B is not ' + 'positive definite. The factorization of B ' + 'could not be completed and no eigenvalues ' + 'or eigenvectors were computed.') + else: + drv_err = {'ev': 'The algorithm failed to converge; {} ' + 'off-diagonal elements of an intermediate ' + 'tridiagonal form did not converge to zero.', + 'evx': '{} eigenvectors failed to converge.', + 'evd': 'The algorithm failed to compute an eigenvalue ' + 'while working on the submatrix lying in rows ' + 'and columns {0}/{1} through mod({0},{1}).', + 'evr': 'Internal Error.' + } + if driver in ['ev', 'gv']: + msg = drv_err['ev'].format(info) + elif driver in ['evx', 'gvx']: + msg = drv_err['evx'].format(info) + elif driver in ['evd', 'gvd']: + if eigvals_only: + msg = drv_err['ev'].format(info) + else: + msg = drv_err['evd'].format(info, n+1) + else: + msg = drv_err['evr'] + + raise LinAlgError(msg) + + +_conv_dict = {0: 0, 1: 1, 2: 2, + 'all': 0, 'value': 1, 'index': 2, + 'a': 0, 'v': 1, 'i': 2} + + +def _check_select(select, select_range, max_ev, max_len): + """Check that select is valid, convert to Fortran style.""" + if isinstance(select, str): + select = select.lower() + try: + select = _conv_dict[select] + except KeyError as e: + raise ValueError('invalid argument for select') from e + vl, vu = 0., 1. + il = iu = 1 + if select != 0: # (non-all) + sr = asarray(select_range) + if sr.ndim != 1 or sr.size != 2 or sr[1] < sr[0]: + raise ValueError('select_range must be a 2-element array-like ' + 'in nondecreasing order') + if select == 1: # (value) + vl, vu = sr + if max_ev == 0: + max_ev = max_len + else: # 2 (index) + if sr.dtype.char.lower() not in 'hilqp': + raise ValueError( + f'when using select="i", select_range must ' + f'contain integers, got dtype {sr.dtype} ({sr.dtype.char})' + ) + # translate Python (0 ... N-1) into Fortran (1 ... N) with + 1 + il, iu = sr + 1 + if min(il, iu) < 1 or max(il, iu) > max_len: + raise ValueError('select_range out of bounds') + max_ev = iu - il + 1 + return select, vl, vu, il, iu, max_ev + + +def eig_banded(a_band, lower=False, eigvals_only=False, overwrite_a_band=False, + select='a', select_range=None, max_ev=0, check_finite=True): + """ + Solve real symmetric or complex Hermitian band matrix eigenvalue problem. + + Find eigenvalues w and optionally right eigenvectors v of a:: + + a v[:,i] = w[i] v[:,i] + v.H v = identity + + The matrix a is stored in a_band either in lower diagonal or upper + diagonal ordered form: + + a_band[u + i - j, j] == a[i,j] (if upper form; i <= j) + a_band[ i - j, j] == a[i,j] (if lower form; i >= j) + + where u is the number of bands above the diagonal. + + Example of a_band (shape of a is (6,6), u=2):: + + upper form: + * * a02 a13 a24 a35 + * a01 a12 a23 a34 a45 + a00 a11 a22 a33 a44 a55 + + lower form: + a00 a11 a22 a33 a44 a55 + a10 a21 a32 a43 a54 * + a20 a31 a42 a53 * * + + Cells marked with * are not used. + + Parameters + ---------- + a_band : (u+1, M) array_like + The bands of the M by M matrix a. + lower : bool, optional + Is the matrix in the lower form. (Default is upper form) + eigvals_only : bool, optional + Compute only the eigenvalues and no eigenvectors. + (Default: calculate also eigenvectors) + overwrite_a_band : bool, optional + Discard data in a_band (may enhance performance) + select : {'a', 'v', 'i'}, optional + Which eigenvalues to calculate + + ====== ======================================== + select calculated + ====== ======================================== + 'a' All eigenvalues + 'v' Eigenvalues in the interval (min, max] + 'i' Eigenvalues with indices min <= i <= max + ====== ======================================== + select_range : (min, max), optional + Range of selected eigenvalues + max_ev : int, optional + For select=='v', maximum number of eigenvalues expected. + For other values of select, has no meaning. + + In doubt, leave this parameter untouched. + + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + w : (M,) ndarray + The eigenvalues, in ascending order, each repeated according to its + multiplicity. + v : (M, M) float or complex ndarray + The normalized eigenvector corresponding to the eigenvalue w[i] is + the column v[:,i]. Only returned if ``eigvals_only=False``. + + Raises + ------ + LinAlgError + If eigenvalue computation does not converge. + + See Also + -------- + eigvals_banded : eigenvalues for symmetric/Hermitian band matrices + eig : eigenvalues and right eigenvectors of general arrays. + eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays + eigh_tridiagonal : eigenvalues and right eigenvectors for + symmetric/Hermitian tridiagonal matrices + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import eig_banded + >>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]]) + >>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]]) + >>> w, v = eig_banded(Ab, lower=True) + >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4))) + True + >>> w = eig_banded(Ab, lower=True, eigvals_only=True) + >>> w + array([-4.26200532, -2.22987175, 3.95222349, 12.53965359]) + + Request only the eigenvalues between ``[-3, 4]`` + + >>> w, v = eig_banded(Ab, lower=True, select='v', select_range=[-3, 4]) + >>> w + array([-2.22987175, 3.95222349]) + + """ + if eigvals_only or overwrite_a_band: + a1 = _asarray_validated(a_band, check_finite=check_finite) + overwrite_a_band = overwrite_a_band or (_datacopied(a1, a_band)) + else: + a1 = array(a_band) + if issubclass(a1.dtype.type, inexact) and not isfinite(a1).all(): + raise ValueError("array must not contain infs or NaNs") + overwrite_a_band = 1 + + if len(a1.shape) != 2: + raise ValueError('expected a 2-D array') + + # accommodate square empty matrices + if a1.size == 0: + w_n, v_n = eig_banded(np.array([[0, 0], [1, 1]], dtype=a1.dtype)) + + w = np.empty_like(a1, shape=(0,), dtype=w_n.dtype) + v = np.empty_like(a1, shape=(0, 0), dtype=v_n.dtype) + if eigvals_only: + return w + else: + return w, v + + select, vl, vu, il, iu, max_ev = _check_select( + select, select_range, max_ev, a1.shape[1]) + + del select_range + if select == 0: + if a1.dtype.char in 'GFD': + # FIXME: implement this somewhen, for now go with builtin values + # FIXME: calc optimal lwork by calling ?hbevd(lwork=-1) + # or by using calc_lwork.f ??? + # lwork = calc_lwork.hbevd(bevd.typecode, a1.shape[0], lower) + internal_name = 'hbevd' + else: # a1.dtype.char in 'fd': + # FIXME: implement this somewhen, for now go with builtin values + # see above + # lwork = calc_lwork.sbevd(bevd.typecode, a1.shape[0], lower) + internal_name = 'sbevd' + bevd, = get_lapack_funcs((internal_name,), (a1,)) + w, v, info = bevd(a1, compute_v=not eigvals_only, + lower=lower, overwrite_ab=overwrite_a_band) + else: # select in [1, 2] + if eigvals_only: + max_ev = 1 + # calculate optimal abstol for dsbevx (see manpage) + if a1.dtype.char in 'fF': # single precision + lamch, = get_lapack_funcs(('lamch',), (array(0, dtype='f'),)) + else: + lamch, = get_lapack_funcs(('lamch',), (array(0, dtype='d'),)) + abstol = 2 * lamch('s') + if a1.dtype.char in 'GFD': + internal_name = 'hbevx' + else: # a1.dtype.char in 'gfd' + internal_name = 'sbevx' + bevx, = get_lapack_funcs((internal_name,), (a1,)) + w, v, m, ifail, info = bevx( + a1, vl, vu, il, iu, compute_v=not eigvals_only, mmax=max_ev, + range=select, lower=lower, overwrite_ab=overwrite_a_band, + abstol=abstol) + # crop off w and v + w = w[:m] + if not eigvals_only: + v = v[:, :m] + _check_info(info, internal_name) + + if eigvals_only: + return w + return w, v + + +def eigvals(a, b=None, overwrite_a=False, check_finite=True, + homogeneous_eigvals=False): + """ + Compute eigenvalues from an ordinary or generalized eigenvalue problem. + + Find eigenvalues of a general matrix:: + + a vr[:,i] = w[i] b vr[:,i] + + Parameters + ---------- + a : (M, M) array_like + A complex or real matrix whose eigenvalues and eigenvectors + will be computed. + b : (M, M) array_like, optional + Right-hand side matrix in a generalized eigenvalue problem. + If omitted, identity matrix is assumed. + overwrite_a : bool, optional + Whether to overwrite data in a (may improve performance) + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities + or NaNs. + homogeneous_eigvals : bool, optional + If True, return the eigenvalues in homogeneous coordinates. + In this case ``w`` is a (2, M) array so that:: + + w[1,i] a vr[:,i] = w[0,i] b vr[:,i] + + Default is False. + + Returns + ------- + w : (M,) or (2, M) double or complex ndarray + The eigenvalues, each repeated according to its multiplicity + but not in any specific order. The shape is (M,) unless + ``homogeneous_eigvals=True``. + + Raises + ------ + LinAlgError + If eigenvalue computation does not converge + + See Also + -------- + eig : eigenvalues and right eigenvectors of general arrays. + eigvalsh : eigenvalues of symmetric or Hermitian arrays + eigvals_banded : eigenvalues for symmetric/Hermitian band matrices + eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal + matrices + + Examples + -------- + >>> import numpy as np + >>> from scipy import linalg + >>> a = np.array([[0., -1.], [1., 0.]]) + >>> linalg.eigvals(a) + array([0.+1.j, 0.-1.j]) + + >>> b = np.array([[0., 1.], [1., 1.]]) + >>> linalg.eigvals(a, b) + array([ 1.+0.j, -1.+0.j]) + + >>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]]) + >>> linalg.eigvals(a, homogeneous_eigvals=True) + array([[3.+0.j, 8.+0.j, 7.+0.j], + [1.+0.j, 1.+0.j, 1.+0.j]]) + + """ + return eig(a, b=b, left=0, right=0, overwrite_a=overwrite_a, + check_finite=check_finite, + homogeneous_eigvals=homogeneous_eigvals) + + +def eigvalsh(a, b=None, *, lower=True, overwrite_a=False, + overwrite_b=False, type=1, check_finite=True, subset_by_index=None, + subset_by_value=None, driver=None): + """ + Solves a standard or generalized eigenvalue problem for a complex + Hermitian or real symmetric matrix. + + Find eigenvalues array ``w`` of array ``a``, where ``b`` is positive + definite such that for every eigenvalue λ (i-th entry of w) and its + eigenvector vi (i-th column of v) satisfies:: + + a @ vi = λ * b @ vi + vi.conj().T @ a @ vi = λ + vi.conj().T @ b @ vi = 1 + + In the standard problem, b is assumed to be the identity matrix. + + Parameters + ---------- + a : (M, M) array_like + A complex Hermitian or real symmetric matrix whose eigenvalues will + be computed. + b : (M, M) array_like, optional + A complex Hermitian or real symmetric definite positive matrix in. + If omitted, identity matrix is assumed. + lower : bool, optional + Whether the pertinent array data is taken from the lower or upper + triangle of ``a`` and, if applicable, ``b``. (Default: lower) + overwrite_a : bool, optional + Whether to overwrite data in ``a`` (may improve performance). Default + is False. + overwrite_b : bool, optional + Whether to overwrite data in ``b`` (may improve performance). Default + is False. + type : int, optional + For the generalized problems, this keyword specifies the problem type + to be solved for ``w`` and ``v`` (only takes 1, 2, 3 as possible + inputs):: + + 1 => a @ v = w @ b @ v + 2 => a @ b @ v = w @ v + 3 => b @ a @ v = w @ v + + This keyword is ignored for standard problems. + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + subset_by_index : iterable, optional + If provided, this two-element iterable defines the start and the end + indices of the desired eigenvalues (ascending order and 0-indexed). + To return only the second smallest to fifth smallest eigenvalues, + ``[1, 4]`` is used. ``[n-3, n-1]`` returns the largest three. Only + available with "evr", "evx", and "gvx" drivers. The entries are + directly converted to integers via ``int()``. + subset_by_value : iterable, optional + If provided, this two-element iterable defines the half-open interval + ``(a, b]`` that, if any, only the eigenvalues between these values + are returned. Only available with "evr", "evx", and "gvx" drivers. Use + ``np.inf`` for the unconstrained ends. + driver : str, optional + Defines which LAPACK driver should be used. Valid options are "ev", + "evd", "evr", "evx" for standard problems and "gv", "gvd", "gvx" for + generalized (where b is not None) problems. See the Notes section of + `scipy.linalg.eigh`. + + Returns + ------- + w : (N,) ndarray + The N (N<=M) selected eigenvalues, in ascending order, each + repeated according to its multiplicity. + + Raises + ------ + LinAlgError + If eigenvalue computation does not converge, an error occurred, or + b matrix is not definite positive. Note that if input matrices are + not symmetric or Hermitian, no error will be reported but results will + be wrong. + + See Also + -------- + eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays + eigvals : eigenvalues of general arrays + eigvals_banded : eigenvalues for symmetric/Hermitian band matrices + eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal + matrices + + Notes + ----- + This function does not check the input array for being Hermitian/symmetric + in order to allow for representing arrays with only their upper/lower + triangular parts. + + This function serves as a one-liner shorthand for `scipy.linalg.eigh` with + the option ``eigvals_only=True`` to get the eigenvalues and not the + eigenvectors. Here it is kept as a legacy convenience. It might be + beneficial to use the main function to have full control and to be a bit + more pythonic. + + Examples + -------- + For more examples see `scipy.linalg.eigh`. + + >>> import numpy as np + >>> from scipy.linalg import eigvalsh + >>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]]) + >>> w = eigvalsh(A) + >>> w + array([-3.74637491, -0.76263923, 6.08502336, 12.42399079]) + + """ + return eigh(a, b=b, lower=lower, eigvals_only=True, overwrite_a=overwrite_a, + overwrite_b=overwrite_b, type=type, check_finite=check_finite, + subset_by_index=subset_by_index, subset_by_value=subset_by_value, + driver=driver) + + +def eigvals_banded(a_band, lower=False, overwrite_a_band=False, + select='a', select_range=None, check_finite=True): + """ + Solve real symmetric or complex Hermitian band matrix eigenvalue problem. + + Find eigenvalues w of a:: + + a v[:,i] = w[i] v[:,i] + v.H v = identity + + The matrix a is stored in a_band either in lower diagonal or upper + diagonal ordered form: + + a_band[u + i - j, j] == a[i,j] (if upper form; i <= j) + a_band[ i - j, j] == a[i,j] (if lower form; i >= j) + + where u is the number of bands above the diagonal. + + Example of a_band (shape of a is (6,6), u=2):: + + upper form: + * * a02 a13 a24 a35 + * a01 a12 a23 a34 a45 + a00 a11 a22 a33 a44 a55 + + lower form: + a00 a11 a22 a33 a44 a55 + a10 a21 a32 a43 a54 * + a20 a31 a42 a53 * * + + Cells marked with * are not used. + + Parameters + ---------- + a_band : (u+1, M) array_like + The bands of the M by M matrix a. + lower : bool, optional + Is the matrix in the lower form. (Default is upper form) + overwrite_a_band : bool, optional + Discard data in a_band (may enhance performance) + select : {'a', 'v', 'i'}, optional + Which eigenvalues to calculate + + ====== ======================================== + select calculated + ====== ======================================== + 'a' All eigenvalues + 'v' Eigenvalues in the interval (min, max] + 'i' Eigenvalues with indices min <= i <= max + ====== ======================================== + select_range : (min, max), optional + Range of selected eigenvalues + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + w : (M,) ndarray + The eigenvalues, in ascending order, each repeated according to its + multiplicity. + + Raises + ------ + LinAlgError + If eigenvalue computation does not converge. + + See Also + -------- + eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian + band matrices + eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal + matrices + eigvals : eigenvalues of general arrays + eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays + eig : eigenvalues and right eigenvectors for non-symmetric arrays + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import eigvals_banded + >>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]]) + >>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]]) + >>> w = eigvals_banded(Ab, lower=True) + >>> w + array([-4.26200532, -2.22987175, 3.95222349, 12.53965359]) + """ + return eig_banded(a_band, lower=lower, eigvals_only=1, + overwrite_a_band=overwrite_a_band, select=select, + select_range=select_range, check_finite=check_finite) + + +def eigvalsh_tridiagonal(d, e, select='a', select_range=None, + check_finite=True, tol=0., lapack_driver='auto'): + """ + Solve eigenvalue problem for a real symmetric tridiagonal matrix. + + Find eigenvalues `w` of ``a``:: + + a v[:,i] = w[i] v[:,i] + v.H v = identity + + For a real symmetric matrix ``a`` with diagonal elements `d` and + off-diagonal elements `e`. + + Parameters + ---------- + d : ndarray, shape (ndim,) + The diagonal elements of the array. + e : ndarray, shape (ndim-1,) + The off-diagonal elements of the array. + select : {'a', 'v', 'i'}, optional + Which eigenvalues to calculate + + ====== ======================================== + select calculated + ====== ======================================== + 'a' All eigenvalues + 'v' Eigenvalues in the interval (min, max] + 'i' Eigenvalues with indices min <= i <= max + ====== ======================================== + select_range : (min, max), optional + Range of selected eigenvalues + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + tol : float + The absolute tolerance to which each eigenvalue is required + (only used when ``lapack_driver='stebz'``). + An eigenvalue (or cluster) is considered to have converged if it + lies in an interval of this width. If <= 0. (default), + the value ``eps*|a|`` is used where eps is the machine precision, + and ``|a|`` is the 1-norm of the matrix ``a``. + lapack_driver : str + LAPACK function to use, can be 'auto', 'stemr', 'stebz', 'sterf', + or 'stev'. When 'auto' (default), it will use 'stemr' if ``select='a'`` + and 'stebz' otherwise. 'sterf' and 'stev' can only be used when + ``select='a'``. + + Returns + ------- + w : (M,) ndarray + The eigenvalues, in ascending order, each repeated according to its + multiplicity. + + Raises + ------ + LinAlgError + If eigenvalue computation does not converge. + + See Also + -------- + eigh_tridiagonal : eigenvalues and right eiegenvectors for + symmetric/Hermitian tridiagonal matrices + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import eigvalsh_tridiagonal, eigvalsh + >>> d = 3*np.ones(4) + >>> e = -1*np.ones(3) + >>> w = eigvalsh_tridiagonal(d, e) + >>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1) + >>> w2 = eigvalsh(A) # Verify with other eigenvalue routines + >>> np.allclose(w - w2, np.zeros(4)) + True + """ + return eigh_tridiagonal( + d, e, eigvals_only=True, select=select, select_range=select_range, + check_finite=check_finite, tol=tol, lapack_driver=lapack_driver) + + +def eigh_tridiagonal(d, e, eigvals_only=False, select='a', select_range=None, + check_finite=True, tol=0., lapack_driver='auto'): + """ + Solve eigenvalue problem for a real symmetric tridiagonal matrix. + + Find eigenvalues `w` and optionally right eigenvectors `v` of ``a``:: + + a v[:,i] = w[i] v[:,i] + v.H v = identity + + For a real symmetric matrix ``a`` with diagonal elements `d` and + off-diagonal elements `e`. + + Parameters + ---------- + d : ndarray, shape (ndim,) + The diagonal elements of the array. + e : ndarray, shape (ndim-1,) + The off-diagonal elements of the array. + eigvals_only : bool, optional + Compute only the eigenvalues and no eigenvectors. + (Default: calculate also eigenvectors) + select : {'a', 'v', 'i'}, optional + Which eigenvalues to calculate + + ====== ======================================== + select calculated + ====== ======================================== + 'a' All eigenvalues + 'v' Eigenvalues in the interval (min, max] + 'i' Eigenvalues with indices min <= i <= max + ====== ======================================== + select_range : (min, max), optional + Range of selected eigenvalues + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + tol : float + The absolute tolerance to which each eigenvalue is required + (only used when 'stebz' is the `lapack_driver`). + An eigenvalue (or cluster) is considered to have converged if it + lies in an interval of this width. If <= 0. (default), + the value ``eps*|a|`` is used where eps is the machine precision, + and ``|a|`` is the 1-norm of the matrix ``a``. + lapack_driver : str + LAPACK function to use, can be 'auto', 'stemr', 'stebz', 'sterf', + or 'stev'. When 'auto' (default), it will use 'stemr' if ``select='a'`` + and 'stebz' otherwise. When 'stebz' is used to find the eigenvalues and + ``eigvals_only=False``, then a second LAPACK call (to ``?STEIN``) is + used to find the corresponding eigenvectors. 'sterf' can only be + used when ``eigvals_only=True`` and ``select='a'``. 'stev' can only + be used when ``select='a'``. + + Returns + ------- + w : (M,) ndarray + The eigenvalues, in ascending order, each repeated according to its + multiplicity. + v : (M, M) ndarray + The normalized eigenvector corresponding to the eigenvalue ``w[i]`` is + the column ``v[:,i]``. Only returned if ``eigvals_only=False``. + + Raises + ------ + LinAlgError + If eigenvalue computation does not converge. + + See Also + -------- + eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal + matrices + eig : eigenvalues and right eigenvectors for non-symmetric arrays + eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays + eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian + band matrices + + Notes + ----- + This function makes use of LAPACK ``S/DSTEMR`` routines. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import eigh_tridiagonal + >>> d = 3*np.ones(4) + >>> e = -1*np.ones(3) + >>> w, v = eigh_tridiagonal(d, e) + >>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1) + >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4))) + True + """ + d = _asarray_validated(d, check_finite=check_finite) + e = _asarray_validated(e, check_finite=check_finite) + for check in (d, e): + if check.ndim != 1: + raise ValueError('expected a 1-D array') + if check.dtype.char in 'GFD': # complex + raise TypeError('Only real arrays currently supported') + if d.size != e.size + 1: + raise ValueError(f'd ({d.size}) must have one more element than e ({e.size})') + select, vl, vu, il, iu, _ = _check_select( + select, select_range, 0, d.size) + if not isinstance(lapack_driver, str): + raise TypeError('lapack_driver must be str') + drivers = ('auto', 'stemr', 'sterf', 'stebz', 'stev') + if lapack_driver not in drivers: + raise ValueError(f'lapack_driver must be one of {drivers}, ' + f'got {lapack_driver}') + if lapack_driver == 'auto': + lapack_driver = 'stemr' if select == 0 else 'stebz' + + # Quick exit for 1x1 case + if len(d) == 1: + if select == 1 and (not (vl < d[0] <= vu)): # request by value + w = array([]) + v = empty([1, 0], dtype=d.dtype) + else: # all and request by index + w = array([d[0]], dtype=d.dtype) + v = array([[1.]], dtype=d.dtype) + + if eigvals_only: + return w + else: + return w, v + + func, = get_lapack_funcs((lapack_driver,), (d, e)) + compute_v = not eigvals_only + if lapack_driver == 'sterf': + if select != 0: + raise ValueError('sterf can only be used when select == "a"') + if not eigvals_only: + raise ValueError('sterf can only be used when eigvals_only is ' + 'True') + w, info = func(d, e) + m = len(w) + elif lapack_driver == 'stev': + if select != 0: + raise ValueError('stev can only be used when select == "a"') + w, v, info = func(d, e, compute_v=compute_v) + m = len(w) + elif lapack_driver == 'stebz': + tol = float(tol) + internal_name = 'stebz' + stebz, = get_lapack_funcs((internal_name,), (d, e)) + # If getting eigenvectors, needs to be block-ordered (B) instead of + # matrix-ordered (E), and we will reorder later + order = 'E' if eigvals_only else 'B' + m, w, iblock, isplit, info = stebz(d, e, select, vl, vu, il, iu, tol, + order) + else: # 'stemr' + # ?STEMR annoyingly requires size N instead of N-1 + e_ = empty(e.size+1, e.dtype) + e_[:-1] = e + stemr_lwork, = get_lapack_funcs(('stemr_lwork',), (d, e)) + lwork, liwork, info = stemr_lwork(d, e_, select, vl, vu, il, iu, + compute_v=compute_v) + _check_info(info, 'stemr_lwork') + m, w, v, info = func(d, e_, select, vl, vu, il, iu, + compute_v=compute_v, lwork=lwork, liwork=liwork) + _check_info(info, lapack_driver + ' (eigh_tridiagonal)') + w = w[:m] + if eigvals_only: + return w + else: + # Do we still need to compute the eigenvalues? + if lapack_driver == 'stebz': + func, = get_lapack_funcs(('stein',), (d, e)) + v, info = func(d, e, w, iblock, isplit) + _check_info(info, 'stein (eigh_tridiagonal)', + positive='%d eigenvectors failed to converge') + # Convert block-order to matrix-order + order = argsort(w) + w, v = w[order], v[:, order] + else: + v = v[:, :m] + return w, v + + +def _check_info(info, driver, positive='did not converge (LAPACK info=%d)'): + """Check info return value.""" + if info < 0: + raise ValueError('illegal value in argument %d of internal %s' + % (-info, driver)) + if info > 0 and positive: + raise LinAlgError(("%s " + positive) % (driver, info,)) + + +def hessenberg(a, calc_q=False, overwrite_a=False, check_finite=True): + """ + Compute Hessenberg form of a matrix. + + The Hessenberg decomposition is:: + + A = Q H Q^H + + where `Q` is unitary/orthogonal and `H` has only zero elements below + the first sub-diagonal. + + Parameters + ---------- + a : (M, M) array_like + Matrix to bring into Hessenberg form. + calc_q : bool, optional + Whether to compute the transformation matrix. Default is False. + overwrite_a : bool, optional + Whether to overwrite `a`; may improve performance. + Default is False. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + H : (M, M) ndarray + Hessenberg form of `a`. + Q : (M, M) ndarray + Unitary/orthogonal similarity transformation matrix ``A = Q H Q^H``. + Only returned if ``calc_q=True``. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import hessenberg + >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]]) + >>> H, Q = hessenberg(A, calc_q=True) + >>> H + array([[ 2. , -11.65843866, 1.42005301, 0.25349066], + [ -9.94987437, 14.53535354, -5.31022304, 2.43081618], + [ 0. , -1.83299243, 0.38969961, -0.51527034], + [ 0. , 0. , -3.83189513, 1.07494686]]) + >>> np.allclose(Q @ H @ Q.conj().T - A, np.zeros((4, 4))) + True + """ + a1 = _asarray_validated(a, check_finite=check_finite) + if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]): + raise ValueError('expected square matrix') + overwrite_a = overwrite_a or (_datacopied(a1, a)) + + if a1.size == 0: + h3 = hessenberg(np.eye(3, dtype=a1.dtype)) + h = np.empty(a1.shape, dtype=h3.dtype) + if not calc_q: + return h + else: + h3, q3 = hessenberg(np.eye(3, dtype=a1.dtype), calc_q=True) + q = np.empty(a1.shape, dtype=q3.dtype) + h = np.empty(a1.shape, dtype=h3.dtype) + return h, q + + # if 2x2 or smaller: already in Hessenberg + if a1.shape[0] <= 2: + if calc_q: + return a1, eye(a1.shape[0]) + return a1 + + gehrd, gebal, gehrd_lwork = get_lapack_funcs(('gehrd', 'gebal', + 'gehrd_lwork'), (a1,)) + ba, lo, hi, pivscale, info = gebal(a1, permute=0, overwrite_a=overwrite_a) + _check_info(info, 'gebal (hessenberg)', positive=False) + n = len(a1) + + lwork = _compute_lwork(gehrd_lwork, ba.shape[0], lo=lo, hi=hi) + + hq, tau, info = gehrd(ba, lo=lo, hi=hi, lwork=lwork, overwrite_a=1) + _check_info(info, 'gehrd (hessenberg)', positive=False) + h = np.triu(hq, -1) + if not calc_q: + return h + + # use orghr/unghr to compute q + orghr, orghr_lwork = get_lapack_funcs(('orghr', 'orghr_lwork'), (a1,)) + lwork = _compute_lwork(orghr_lwork, n, lo=lo, hi=hi) + + q, info = orghr(a=hq, tau=tau, lo=lo, hi=hi, lwork=lwork, overwrite_a=1) + _check_info(info, 'orghr (hessenberg)', positive=False) + return h, q + + +def cdf2rdf(w, v): + """ + Converts complex eigenvalues ``w`` and eigenvectors ``v`` to real + eigenvalues in a block diagonal form ``wr`` and the associated real + eigenvectors ``vr``, such that:: + + vr @ wr = X @ vr + + continues to hold, where ``X`` is the original array for which ``w`` and + ``v`` are the eigenvalues and eigenvectors. + + .. versionadded:: 1.1.0 + + Parameters + ---------- + w : (..., M) array_like + Complex or real eigenvalues, an array or stack of arrays + + Conjugate pairs must not be interleaved, else the wrong result + will be produced. So ``[1+1j, 1, 1-1j]`` will give a correct result, + but ``[1+1j, 2+1j, 1-1j, 2-1j]`` will not. + + v : (..., M, M) array_like + Complex or real eigenvectors, a square array or stack of square arrays. + + Returns + ------- + wr : (..., M, M) ndarray + Real diagonal block form of eigenvalues + vr : (..., M, M) ndarray + Real eigenvectors associated with ``wr`` + + See Also + -------- + eig : Eigenvalues and right eigenvectors for non-symmetric arrays + rsf2csf : Convert real Schur form to complex Schur form + + Notes + ----- + ``w``, ``v`` must be the eigenstructure for some *real* matrix ``X``. + For example, obtained by ``w, v = scipy.linalg.eig(X)`` or + ``w, v = numpy.linalg.eig(X)`` in which case ``X`` can also represent + stacked arrays. + + .. versionadded:: 1.1.0 + + Examples + -------- + >>> import numpy as np + >>> X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]]) + >>> X + array([[ 1, 2, 3], + [ 0, 4, 5], + [ 0, -5, 4]]) + + >>> from scipy import linalg + >>> w, v = linalg.eig(X) + >>> w + array([ 1.+0.j, 4.+5.j, 4.-5.j]) + >>> v + array([[ 1.00000+0.j , -0.01906-0.40016j, -0.01906+0.40016j], + [ 0.00000+0.j , 0.00000-0.64788j, 0.00000+0.64788j], + [ 0.00000+0.j , 0.64788+0.j , 0.64788-0.j ]]) + + >>> wr, vr = linalg.cdf2rdf(w, v) + >>> wr + array([[ 1., 0., 0.], + [ 0., 4., 5.], + [ 0., -5., 4.]]) + >>> vr + array([[ 1. , 0.40016, -0.01906], + [ 0. , 0.64788, 0. ], + [ 0. , 0. , 0.64788]]) + + >>> vr @ wr + array([[ 1. , 1.69593, 1.9246 ], + [ 0. , 2.59153, 3.23942], + [ 0. , -3.23942, 2.59153]]) + >>> X @ vr + array([[ 1. , 1.69593, 1.9246 ], + [ 0. , 2.59153, 3.23942], + [ 0. , -3.23942, 2.59153]]) + """ + w, v = _asarray_validated(w), _asarray_validated(v) + + # check dimensions + if w.ndim < 1: + raise ValueError('expected w to be at least 1D') + if v.ndim < 2: + raise ValueError('expected v to be at least 2D') + if v.ndim != w.ndim + 1: + raise ValueError('expected eigenvectors array to have exactly one ' + 'dimension more than eigenvalues array') + + # check shapes + n = w.shape[-1] + M = w.shape[:-1] + if v.shape[-2] != v.shape[-1]: + raise ValueError('expected v to be a square matrix or stacked square ' + 'matrices: v.shape[-2] = v.shape[-1]') + if v.shape[-1] != n: + raise ValueError('expected the same number of eigenvalues as ' + 'eigenvectors') + + # get indices for each first pair of complex eigenvalues + complex_mask = iscomplex(w) + n_complex = complex_mask.sum(axis=-1) + + # check if all complex eigenvalues have conjugate pairs + if not (n_complex % 2 == 0).all(): + raise ValueError('expected complex-conjugate pairs of eigenvalues') + + # find complex indices + idx = nonzero(complex_mask) + idx_stack = idx[:-1] + idx_elem = idx[-1] + + # filter them to conjugate indices, assuming pairs are not interleaved + j = idx_elem[0::2] + k = idx_elem[1::2] + stack_ind = () + for i in idx_stack: + # should never happen, assuming nonzero orders by the last axis + assert (i[0::2] == i[1::2]).all(), \ + "Conjugate pair spanned different arrays!" + stack_ind += (i[0::2],) + + # all eigenvalues to diagonal form + wr = zeros(M + (n, n), dtype=w.real.dtype) + di = range(n) + wr[..., di, di] = w.real + + # complex eigenvalues to real block diagonal form + wr[stack_ind + (j, k)] = w[stack_ind + (j,)].imag + wr[stack_ind + (k, j)] = w[stack_ind + (k,)].imag + + # compute real eigenvectors associated with real block diagonal eigenvalues + u = zeros(M + (n, n), dtype=np.cdouble) + u[..., di, di] = 1.0 + u[stack_ind + (j, j)] = 0.5j + u[stack_ind + (j, k)] = 0.5 + u[stack_ind + (k, j)] = -0.5j + u[stack_ind + (k, k)] = 0.5 + + # multiply matrices v and u (equivalent to v @ u) + vr = einsum('...ij,...jk->...ik', v, u).real + + return wr, vr diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_cholesky.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_cholesky.py new file mode 100644 index 0000000000000000000000000000000000000000..c35b6a4920dea6bb638fe54a1fa719ebc74fb773 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_cholesky.py @@ -0,0 +1,398 @@ +"""Cholesky decomposition functions.""" + +import numpy as np +from numpy import asarray_chkfinite, asarray, atleast_2d, empty_like + +# Local imports +from ._misc import LinAlgError, _datacopied +from .lapack import get_lapack_funcs + +__all__ = ['cholesky', 'cho_factor', 'cho_solve', 'cholesky_banded', + 'cho_solve_banded'] + + +def _cholesky(a, lower=False, overwrite_a=False, clean=True, + check_finite=True): + """Common code for cholesky() and cho_factor().""" + + a1 = asarray_chkfinite(a) if check_finite else asarray(a) + a1 = atleast_2d(a1) + + # Dimension check + if a1.ndim != 2: + raise ValueError(f'Input array needs to be 2D but received a {a1.ndim}d-array.') + # Squareness check + if a1.shape[0] != a1.shape[1]: + raise ValueError('Input array is expected to be square but has ' + f'the shape: {a1.shape}.') + + # Quick return for square empty array + if a1.size == 0: + dt = cholesky(np.eye(1, dtype=a1.dtype)).dtype + return empty_like(a1, dtype=dt), lower + + overwrite_a = overwrite_a or _datacopied(a1, a) + potrf, = get_lapack_funcs(('potrf',), (a1,)) + c, info = potrf(a1, lower=lower, overwrite_a=overwrite_a, clean=clean) + if info > 0: + raise LinAlgError("%d-th leading minor of the array is not positive " + "definite" % info) + if info < 0: + raise ValueError(f'LAPACK reported an illegal value in {-info}-th argument' + 'on entry to "POTRF".') + return c, lower + + +def cholesky(a, lower=False, overwrite_a=False, check_finite=True): + """ + Compute the Cholesky decomposition of a matrix. + + Returns the Cholesky decomposition, :math:`A = L L^*` or + :math:`A = U^* U` of a Hermitian positive-definite matrix A. + + Parameters + ---------- + a : (M, M) array_like + Matrix to be decomposed + lower : bool, optional + Whether to compute the upper- or lower-triangular Cholesky + factorization. During decomposition, only the selected half of the + matrix is referenced. Default is upper-triangular. + overwrite_a : bool, optional + Whether to overwrite data in `a` (may improve performance). + check_finite : bool, optional + Whether to check that the entire input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + c : (M, M) ndarray + Upper- or lower-triangular Cholesky factor of `a`. + + Raises + ------ + LinAlgError : if decomposition fails. + + Notes + ----- + During the finiteness check (if selected), the entire matrix `a` is + checked. During decomposition, `a` is assumed to be symmetric or Hermitian + (as applicable), and only the half selected by option `lower` is referenced. + Consequently, if `a` is asymmetric/non-Hermitian, `cholesky` may still + succeed if the symmetric/Hermitian matrix represented by the selected half + is positive definite, yet it may fail if an element in the other half is + non-finite. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import cholesky + >>> a = np.array([[1,-2j],[2j,5]]) + >>> L = cholesky(a, lower=True) + >>> L + array([[ 1.+0.j, 0.+0.j], + [ 0.+2.j, 1.+0.j]]) + >>> L @ L.T.conj() + array([[ 1.+0.j, 0.-2.j], + [ 0.+2.j, 5.+0.j]]) + + """ + c, lower = _cholesky(a, lower=lower, overwrite_a=overwrite_a, clean=True, + check_finite=check_finite) + return c + + +def cho_factor(a, lower=False, overwrite_a=False, check_finite=True): + """ + Compute the Cholesky decomposition of a matrix, to use in cho_solve + + Returns a matrix containing the Cholesky decomposition, + ``A = L L*`` or ``A = U* U`` of a Hermitian positive-definite matrix `a`. + The return value can be directly used as the first parameter to cho_solve. + + .. warning:: + The returned matrix also contains random data in the entries not + used by the Cholesky decomposition. If you need to zero these + entries, use the function `cholesky` instead. + + Parameters + ---------- + a : (M, M) array_like + Matrix to be decomposed + lower : bool, optional + Whether to compute the upper or lower triangular Cholesky factorization. + During decomposition, only the selected half of the matrix is referenced. + (Default: upper-triangular) + overwrite_a : bool, optional + Whether to overwrite data in a (may improve performance) + check_finite : bool, optional + Whether to check that the entire input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + c : (M, M) ndarray + Matrix whose upper or lower triangle contains the Cholesky factor + of `a`. Other parts of the matrix contain random data. + lower : bool + Flag indicating whether the factor is in the lower or upper triangle + + Raises + ------ + LinAlgError + Raised if decomposition fails. + + See Also + -------- + cho_solve : Solve a linear set equations using the Cholesky factorization + of a matrix. + + Notes + ----- + During the finiteness check (if selected), the entire matrix `a` is + checked. During decomposition, `a` is assumed to be symmetric or Hermitian + (as applicable), and only the half selected by option `lower` is referenced. + Consequently, if `a` is asymmetric/non-Hermitian, `cholesky` may still + succeed if the symmetric/Hermitian matrix represented by the selected half + is positive definite, yet it may fail if an element in the other half is + non-finite. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import cho_factor + >>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]]) + >>> c, low = cho_factor(A) + >>> c + array([[3. , 1. , 0.33333333, 1.66666667], + [3. , 2.44948974, 1.90515869, -0.27216553], + [1. , 5. , 2.29330749, 0.8559528 ], + [5. , 1. , 2. , 1.55418563]]) + >>> np.allclose(np.triu(c).T @ np. triu(c) - A, np.zeros((4, 4))) + True + + """ + c, lower = _cholesky(a, lower=lower, overwrite_a=overwrite_a, clean=False, + check_finite=check_finite) + return c, lower + + +def cho_solve(c_and_lower, b, overwrite_b=False, check_finite=True): + """Solve the linear equations A x = b, given the Cholesky factorization of A. + + Parameters + ---------- + (c, lower) : tuple, (array, bool) + Cholesky factorization of a, as given by cho_factor + b : array + Right-hand side + overwrite_b : bool, optional + Whether to overwrite data in b (may improve performance) + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + x : array + The solution to the system A x = b + + See Also + -------- + cho_factor : Cholesky factorization of a matrix + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import cho_factor, cho_solve + >>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]]) + >>> c, low = cho_factor(A) + >>> x = cho_solve((c, low), [1, 1, 1, 1]) + >>> np.allclose(A @ x - [1, 1, 1, 1], np.zeros(4)) + True + + """ + (c, lower) = c_and_lower + if check_finite: + b1 = asarray_chkfinite(b) + c = asarray_chkfinite(c) + else: + b1 = asarray(b) + c = asarray(c) + + if c.ndim != 2 or c.shape[0] != c.shape[1]: + raise ValueError("The factored matrix c is not square.") + if c.shape[1] != b1.shape[0]: + raise ValueError(f"incompatible dimensions ({c.shape} and {b1.shape})") + + # accommodate empty arrays + if b1.size == 0: + dt = cho_solve((np.eye(2, dtype=b1.dtype), True), + np.ones(2, dtype=c.dtype)).dtype + return empty_like(b1, dtype=dt) + + overwrite_b = overwrite_b or _datacopied(b1, b) + + potrs, = get_lapack_funcs(('potrs',), (c, b1)) + x, info = potrs(c, b1, lower=lower, overwrite_b=overwrite_b) + if info != 0: + raise ValueError('illegal value in %dth argument of internal potrs' + % -info) + return x + + +def cholesky_banded(ab, overwrite_ab=False, lower=False, check_finite=True): + """ + Cholesky decompose a banded Hermitian positive-definite matrix + + The matrix a is stored in ab either in lower-diagonal or upper- + diagonal ordered form:: + + ab[u + i - j, j] == a[i,j] (if upper form; i <= j) + ab[ i - j, j] == a[i,j] (if lower form; i >= j) + + Example of ab (shape of a is (6,6), u=2):: + + upper form: + * * a02 a13 a24 a35 + * a01 a12 a23 a34 a45 + a00 a11 a22 a33 a44 a55 + + lower form: + a00 a11 a22 a33 a44 a55 + a10 a21 a32 a43 a54 * + a20 a31 a42 a53 * * + + Parameters + ---------- + ab : (u + 1, M) array_like + Banded matrix + overwrite_ab : bool, optional + Discard data in ab (may enhance performance) + lower : bool, optional + Is the matrix in the lower form. (Default is upper form) + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + c : (u + 1, M) ndarray + Cholesky factorization of a, in the same banded format as ab + + See Also + -------- + cho_solve_banded : + Solve a linear set equations, given the Cholesky factorization + of a banded Hermitian. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import cholesky_banded + >>> from numpy import allclose, zeros, diag + >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]]) + >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1) + >>> A = A + A.conj().T + np.diag(Ab[2, :]) + >>> c = cholesky_banded(Ab) + >>> C = np.diag(c[0, 2:], k=2) + np.diag(c[1, 1:], k=1) + np.diag(c[2, :]) + >>> np.allclose(C.conj().T @ C - A, np.zeros((5, 5))) + True + + """ + if check_finite: + ab = asarray_chkfinite(ab) + else: + ab = asarray(ab) + + # accommodate square empty matrices + if ab.size == 0: + dt = cholesky_banded(np.array([[0, 0], [1, 1]], dtype=ab.dtype)).dtype + return empty_like(ab, dtype=dt) + + pbtrf, = get_lapack_funcs(('pbtrf',), (ab,)) + c, info = pbtrf(ab, lower=lower, overwrite_ab=overwrite_ab) + if info > 0: + raise LinAlgError("%d-th leading minor not positive definite" % info) + if info < 0: + raise ValueError('illegal value in %d-th argument of internal pbtrf' + % -info) + return c + + +def cho_solve_banded(cb_and_lower, b, overwrite_b=False, check_finite=True): + """ + Solve the linear equations ``A x = b``, given the Cholesky factorization of + the banded Hermitian ``A``. + + Parameters + ---------- + (cb, lower) : tuple, (ndarray, bool) + `cb` is the Cholesky factorization of A, as given by cholesky_banded. + `lower` must be the same value that was given to cholesky_banded. + b : array_like + Right-hand side + overwrite_b : bool, optional + If True, the function will overwrite the values in `b`. + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + x : array + The solution to the system A x = b + + See Also + -------- + cholesky_banded : Cholesky factorization of a banded matrix + + Notes + ----- + + .. versionadded:: 0.8.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import cholesky_banded, cho_solve_banded + >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]]) + >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1) + >>> A = A + A.conj().T + np.diag(Ab[2, :]) + >>> c = cholesky_banded(Ab) + >>> x = cho_solve_banded((c, False), np.ones(5)) + >>> np.allclose(A @ x - np.ones(5), np.zeros(5)) + True + + """ + (cb, lower) = cb_and_lower + if check_finite: + cb = asarray_chkfinite(cb) + b = asarray_chkfinite(b) + else: + cb = asarray(cb) + b = asarray(b) + + # Validate shapes. + if cb.shape[-1] != b.shape[0]: + raise ValueError("shapes of cb and b are not compatible.") + + # accommodate empty arrays + if b.size == 0: + m = cholesky_banded(np.array([[0, 0], [1, 1]], dtype=cb.dtype)) + dt = cho_solve_banded((m, True), np.ones(2, dtype=b.dtype)).dtype + return empty_like(b, dtype=dt) + + pbtrs, = get_lapack_funcs(('pbtrs',), (cb, b)) + x, info = pbtrs(cb, b, lower=lower, overwrite_b=overwrite_b) + if info > 0: + raise LinAlgError("%dth leading minor not positive definite" % info) + if info < 0: + raise ValueError('illegal value in %dth argument of internal pbtrs' + % -info) + return x diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_cossin.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_cossin.py new file mode 100644 index 0000000000000000000000000000000000000000..e10c04fe5ebc196e1b84724b25f0fc20a5e46857 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_cossin.py @@ -0,0 +1,221 @@ +from collections.abc import Iterable +import numpy as np + +from scipy._lib._util import _asarray_validated +from scipy.linalg import block_diag, LinAlgError +from .lapack import _compute_lwork, get_lapack_funcs + +__all__ = ['cossin'] + + +def cossin(X, p=None, q=None, separate=False, + swap_sign=False, compute_u=True, compute_vh=True): + """ + Compute the cosine-sine (CS) decomposition of an orthogonal/unitary matrix. + + X is an ``(m, m)`` orthogonal/unitary matrix, partitioned as the following + where upper left block has the shape of ``(p, q)``:: + + ┌ ┐ + │ I 0 0 │ 0 0 0 │ + ┌ ┐ ┌ ┐│ 0 C 0 │ 0 -S 0 │┌ ┐* + │ X11 │ X12 │ │ U1 │ ││ 0 0 0 │ 0 0 -I ││ V1 │ │ + │ ────┼──── │ = │────┼────││─────────┼─────────││────┼────│ + │ X21 │ X22 │ │ │ U2 ││ 0 0 0 │ I 0 0 ││ │ V2 │ + └ ┘ └ ┘│ 0 S 0 │ 0 C 0 │└ ┘ + │ 0 0 I │ 0 0 0 │ + └ ┘ + + ``U1``, ``U2``, ``V1``, ``V2`` are square orthogonal/unitary matrices of + dimensions ``(p,p)``, ``(m-p,m-p)``, ``(q,q)``, and ``(m-q,m-q)`` + respectively, and ``C`` and ``S`` are ``(r, r)`` nonnegative diagonal + matrices satisfying ``C^2 + S^2 = I`` where ``r = min(p, m-p, q, m-q)``. + + Moreover, the rank of the identity matrices are ``min(p, q) - r``, + ``min(p, m - q) - r``, ``min(m - p, q) - r``, and ``min(m - p, m - q) - r`` + respectively. + + X can be supplied either by itself and block specifications p, q or its + subblocks in an iterable from which the shapes would be derived. See the + examples below. + + Parameters + ---------- + X : array_like, iterable + complex unitary or real orthogonal matrix to be decomposed, or iterable + of subblocks ``X11``, ``X12``, ``X21``, ``X22``, when ``p``, ``q`` are + omitted. + p : int, optional + Number of rows of the upper left block ``X11``, used only when X is + given as an array. + q : int, optional + Number of columns of the upper left block ``X11``, used only when X is + given as an array. + separate : bool, optional + if ``True``, the low level components are returned instead of the + matrix factors, i.e. ``(u1,u2)``, ``theta``, ``(v1h,v2h)`` instead of + ``u``, ``cs``, ``vh``. + swap_sign : bool, optional + if ``True``, the ``-S``, ``-I`` block will be the bottom left, + otherwise (by default) they will be in the upper right block. + compute_u : bool, optional + if ``False``, ``u`` won't be computed and an empty array is returned. + compute_vh : bool, optional + if ``False``, ``vh`` won't be computed and an empty array is returned. + + Returns + ------- + u : ndarray + When ``compute_u=True``, contains the block diagonal orthogonal/unitary + matrix consisting of the blocks ``U1`` (``p`` x ``p``) and ``U2`` + (``m-p`` x ``m-p``) orthogonal/unitary matrices. If ``separate=True``, + this contains the tuple of ``(U1, U2)``. + cs : ndarray + The cosine-sine factor with the structure described above. + If ``separate=True``, this contains the ``theta`` array containing the + angles in radians. + vh : ndarray + When ``compute_vh=True`, contains the block diagonal orthogonal/unitary + matrix consisting of the blocks ``V1H`` (``q`` x ``q``) and ``V2H`` + (``m-q`` x ``m-q``) orthogonal/unitary matrices. If ``separate=True``, + this contains the tuple of ``(V1H, V2H)``. + + References + ---------- + .. [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. + Algorithms, 50(1):33-65, 2009. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import cossin + >>> from scipy.stats import unitary_group + >>> x = unitary_group.rvs(4) + >>> u, cs, vdh = cossin(x, p=2, q=2) + >>> np.allclose(x, u @ cs @ vdh) + True + + Same can be entered via subblocks without the need of ``p`` and ``q``. Also + let's skip the computation of ``u`` + + >>> ue, cs, vdh = cossin((x[:2, :2], x[:2, 2:], x[2:, :2], x[2:, 2:]), + ... compute_u=False) + >>> print(ue) + [] + >>> np.allclose(x, u @ cs @ vdh) + True + + """ + + if p or q: + p = 1 if p is None else int(p) + q = 1 if q is None else int(q) + X = _asarray_validated(X, check_finite=True) + if not np.equal(*X.shape): + raise ValueError("Cosine Sine decomposition only supports square" + f" matrices, got {X.shape}") + m = X.shape[0] + if p >= m or p <= 0: + raise ValueError(f"invalid p={p}, 0= m or q <= 0: + raise ValueError(f"invalid q={q}, 0 0: + raise LinAlgError(f"{method_name} did not converge: {info}") + + if separate: + return (u1, u2), theta, (v1h, v2h) + + U = block_diag(u1, u2) + VDH = block_diag(v1h, v2h) + + # Construct the middle factor CS + c = np.diag(np.cos(theta)) + s = np.diag(np.sin(theta)) + r = min(p, q, m - p, m - q) + n11 = min(p, q) - r + n12 = min(p, m - q) - r + n21 = min(m - p, q) - r + n22 = min(m - p, m - q) - r + Id = np.eye(np.max([n11, n12, n21, n22, r]), dtype=theta.dtype) + CS = np.zeros((m, m), dtype=theta.dtype) + + CS[:n11, :n11] = Id[:n11, :n11] + + xs = n11 + r + xe = n11 + r + n12 + ys = n11 + n21 + n22 + 2 * r + ye = n11 + n21 + n22 + 2 * r + n12 + CS[xs: xe, ys:ye] = Id[:n12, :n12] if swap_sign else -Id[:n12, :n12] + + xs = p + n22 + r + xe = p + n22 + r + + n21 + ys = n11 + r + ye = n11 + r + n21 + CS[xs:xe, ys:ye] = -Id[:n21, :n21] if swap_sign else Id[:n21, :n21] + + CS[p:p + n22, q:q + n22] = Id[:n22, :n22] + CS[n11:n11 + r, n11:n11 + r] = c + CS[p + n22:p + n22 + r, n11 + r + n21 + n22:2 * r + n11 + n21 + n22] = c + + xs = n11 + xe = n11 + r + ys = n11 + n21 + n22 + r + ye = n11 + n21 + n22 + 2 * r + CS[xs:xe, ys:ye] = s if swap_sign else -s + + CS[p + n22:p + n22 + r, n11:n11 + r] = -s if swap_sign else s + + return U, CS, VDH diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_ldl.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_ldl.py new file mode 100644 index 0000000000000000000000000000000000000000..336df1d5fb416f635c91afe3cc2cfb3c340239fc --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_ldl.py @@ -0,0 +1,353 @@ +from warnings import warn + +import numpy as np +from numpy import (atleast_2d, arange, zeros_like, imag, diag, + iscomplexobj, tril, triu, argsort, empty_like) +from scipy._lib._util import ComplexWarning +from ._decomp import _asarray_validated +from .lapack import get_lapack_funcs, _compute_lwork + +__all__ = ['ldl'] + + +def ldl(A, lower=True, hermitian=True, overwrite_a=False, check_finite=True): + """ Computes the LDLt or Bunch-Kaufman factorization of a symmetric/ + hermitian matrix. + + This function returns a block diagonal matrix D consisting blocks of size + at most 2x2 and also a possibly permuted unit lower triangular matrix + ``L`` such that the factorization ``A = L D L^H`` or ``A = L D L^T`` + holds. If `lower` is False then (again possibly permuted) upper + triangular matrices are returned as outer factors. + + The permutation array can be used to triangularize the outer factors + simply by a row shuffle, i.e., ``lu[perm, :]`` is an upper/lower + triangular matrix. This is also equivalent to multiplication with a + permutation matrix ``P.dot(lu)``, where ``P`` is a column-permuted + identity matrix ``I[:, perm]``. + + Depending on the value of the boolean `lower`, only upper or lower + triangular part of the input array is referenced. Hence, a triangular + matrix on entry would give the same result as if the full matrix is + supplied. + + Parameters + ---------- + A : array_like + Square input array + lower : bool, optional + This switches between the lower and upper triangular outer factors of + the factorization. Lower triangular (``lower=True``) is the default. + hermitian : bool, optional + For complex-valued arrays, this defines whether ``A = A.conj().T`` or + ``A = A.T`` is assumed. For real-valued arrays, this switch has no + effect. + overwrite_a : bool, optional + Allow overwriting data in `A` (may enhance performance). The default + is False. + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + lu : ndarray + The (possibly) permuted upper/lower triangular outer factor of the + factorization. + d : ndarray + The block diagonal multiplier of the factorization. + perm : ndarray + The row-permutation index array that brings lu into triangular form. + + Raises + ------ + ValueError + If input array is not square. + ComplexWarning + If a complex-valued array with nonzero imaginary parts on the + diagonal is given and hermitian is set to True. + + See Also + -------- + cholesky, lu + + Notes + ----- + This function uses ``?SYTRF`` routines for symmetric matrices and + ``?HETRF`` routines for Hermitian matrices from LAPACK. See [1]_ for + the algorithm details. + + Depending on the `lower` keyword value, only lower or upper triangular + part of the input array is referenced. Moreover, this keyword also defines + the structure of the outer factors of the factorization. + + .. versionadded:: 1.1.0 + + References + ---------- + .. [1] J.R. Bunch, L. Kaufman, Some stable methods for calculating + inertia and solving symmetric linear systems, Math. Comput. Vol.31, + 1977. :doi:`10.2307/2005787` + + Examples + -------- + Given an upper triangular array ``a`` that represents the full symmetric + array with its entries, obtain ``l``, 'd' and the permutation vector `perm`: + + >>> import numpy as np + >>> from scipy.linalg import ldl + >>> a = np.array([[2, -1, 3], [0, 2, 0], [0, 0, 1]]) + >>> lu, d, perm = ldl(a, lower=0) # Use the upper part + >>> lu + array([[ 0. , 0. , 1. ], + [ 0. , 1. , -0.5], + [ 1. , 1. , 1.5]]) + >>> d + array([[-5. , 0. , 0. ], + [ 0. , 1.5, 0. ], + [ 0. , 0. , 2. ]]) + >>> perm + array([2, 1, 0]) + >>> lu[perm, :] + array([[ 1. , 1. , 1.5], + [ 0. , 1. , -0.5], + [ 0. , 0. , 1. ]]) + >>> lu.dot(d).dot(lu.T) + array([[ 2., -1., 3.], + [-1., 2., 0.], + [ 3., 0., 1.]]) + + """ + a = atleast_2d(_asarray_validated(A, check_finite=check_finite)) + if a.shape[0] != a.shape[1]: + raise ValueError('The input array "a" should be square.') + # Return empty arrays for empty square input + if a.size == 0: + return empty_like(a), empty_like(a), np.array([], dtype=int) + + n = a.shape[0] + r_or_c = complex if iscomplexobj(a) else float + + # Get the LAPACK routine + if r_or_c is complex and hermitian: + s, sl = 'hetrf', 'hetrf_lwork' + if np.any(imag(diag(a))): + warn('scipy.linalg.ldl():\nThe imaginary parts of the diagonal' + 'are ignored. Use "hermitian=False" for factorization of' + 'complex symmetric arrays.', ComplexWarning, stacklevel=2) + else: + s, sl = 'sytrf', 'sytrf_lwork' + + solver, solver_lwork = get_lapack_funcs((s, sl), (a,)) + lwork = _compute_lwork(solver_lwork, n, lower=lower) + ldu, piv, info = solver(a, lwork=lwork, lower=lower, + overwrite_a=overwrite_a) + if info < 0: + raise ValueError(f'{s.upper()} exited with the internal error "illegal value ' + f'in argument number {-info}". See LAPACK documentation ' + 'for the error codes.') + + swap_arr, pivot_arr = _ldl_sanitize_ipiv(piv, lower=lower) + d, lu = _ldl_get_d_and_l(ldu, pivot_arr, lower=lower, hermitian=hermitian) + lu, perm = _ldl_construct_tri_factor(lu, swap_arr, pivot_arr, lower=lower) + + return lu, d, perm + + +def _ldl_sanitize_ipiv(a, lower=True): + """ + This helper function takes the rather strangely encoded permutation array + returned by the LAPACK routines ?(HE/SY)TRF and converts it into + regularized permutation and diagonal pivot size format. + + Since FORTRAN uses 1-indexing and LAPACK uses different start points for + upper and lower formats there are certain offsets in the indices used + below. + + Let's assume a result where the matrix is 6x6 and there are two 2x2 + and two 1x1 blocks reported by the routine. To ease the coding efforts, + we still populate a 6-sized array and fill zeros as the following :: + + pivots = [2, 0, 2, 0, 1, 1] + + This denotes a diagonal matrix of the form :: + + [x x ] + [x x ] + [ x x ] + [ x x ] + [ x ] + [ x] + + In other words, we write 2 when the 2x2 block is first encountered and + automatically write 0 to the next entry and skip the next spin of the + loop. Thus, a separate counter or array appends to keep track of block + sizes are avoided. If needed, zeros can be filtered out later without + losing the block structure. + + Parameters + ---------- + a : ndarray + The permutation array ipiv returned by LAPACK + lower : bool, optional + The switch to select whether upper or lower triangle is chosen in + the LAPACK call. + + Returns + ------- + swap_ : ndarray + The array that defines the row/column swap operations. For example, + if row two is swapped with row four, the result is [0, 3, 2, 3]. + pivots : ndarray + The array that defines the block diagonal structure as given above. + + """ + n = a.size + swap_ = arange(n) + pivots = zeros_like(swap_, dtype=int) + skip_2x2 = False + + # Some upper/lower dependent offset values + # range (s)tart, r(e)nd, r(i)ncrement + x, y, rs, re, ri = (1, 0, 0, n, 1) if lower else (-1, -1, n-1, -1, -1) + + for ind in range(rs, re, ri): + # If previous spin belonged already to a 2x2 block + if skip_2x2: + skip_2x2 = False + continue + + cur_val = a[ind] + # do we have a 1x1 block or not? + if cur_val > 0: + if cur_val != ind+1: + # Index value != array value --> permutation required + swap_[ind] = swap_[cur_val-1] + pivots[ind] = 1 + # Not. + elif cur_val < 0 and cur_val == a[ind+x]: + # first neg entry of 2x2 block identifier + if -cur_val != ind+2: + # Index value != array value --> permutation required + swap_[ind+x] = swap_[-cur_val-1] + pivots[ind+y] = 2 + skip_2x2 = True + else: # Doesn't make sense, give up + raise ValueError('While parsing the permutation array ' + 'in "scipy.linalg.ldl", invalid entries ' + 'found. The array syntax is invalid.') + return swap_, pivots + + +def _ldl_get_d_and_l(ldu, pivs, lower=True, hermitian=True): + """ + Helper function to extract the diagonal and triangular matrices for + LDL.T factorization. + + Parameters + ---------- + ldu : ndarray + The compact output returned by the LAPACK routing + pivs : ndarray + The sanitized array of {0, 1, 2} denoting the sizes of the pivots. For + every 2 there is a succeeding 0. + lower : bool, optional + If set to False, upper triangular part is considered. + hermitian : bool, optional + If set to False a symmetric complex array is assumed. + + Returns + ------- + d : ndarray + The block diagonal matrix. + lu : ndarray + The upper/lower triangular matrix + """ + is_c = iscomplexobj(ldu) + d = diag(diag(ldu)) + n = d.shape[0] + blk_i = 0 # block index + + # row/column offsets for selecting sub-, super-diagonal + x, y = (1, 0) if lower else (0, 1) + + lu = tril(ldu, -1) if lower else triu(ldu, 1) + diag_inds = arange(n) + lu[diag_inds, diag_inds] = 1 + + for blk in pivs[pivs != 0]: + # increment the block index and check for 2s + # if 2 then copy the off diagonals depending on uplo + inc = blk_i + blk + + if blk == 2: + d[blk_i+x, blk_i+y] = ldu[blk_i+x, blk_i+y] + # If Hermitian matrix is factorized, the cross-offdiagonal element + # should be conjugated. + if is_c and hermitian: + d[blk_i+y, blk_i+x] = ldu[blk_i+x, blk_i+y].conj() + else: + d[blk_i+y, blk_i+x] = ldu[blk_i+x, blk_i+y] + + lu[blk_i+x, blk_i+y] = 0. + blk_i = inc + + return d, lu + + +def _ldl_construct_tri_factor(lu, swap_vec, pivs, lower=True): + """ + Helper function to construct explicit outer factors of LDL factorization. + + If lower is True the permuted factors are multiplied as L(1)*L(2)*...*L(k). + Otherwise, the permuted factors are multiplied as L(k)*...*L(2)*L(1). See + LAPACK documentation for more details. + + Parameters + ---------- + lu : ndarray + The triangular array that is extracted from LAPACK routine call with + ones on the diagonals. + swap_vec : ndarray + The array that defines the row swapping indices. If the kth entry is m + then rows k,m are swapped. Notice that the mth entry is not necessarily + k to avoid undoing the swapping. + pivs : ndarray + The array that defines the block diagonal structure returned by + _ldl_sanitize_ipiv(). + lower : bool, optional + The boolean to switch between lower and upper triangular structure. + + Returns + ------- + lu : ndarray + The square outer factor which satisfies the L * D * L.T = A + perm : ndarray + The permutation vector that brings the lu to the triangular form + + Notes + ----- + Note that the original argument "lu" is overwritten. + + """ + n = lu.shape[0] + perm = arange(n) + # Setup the reading order of the permutation matrix for upper/lower + rs, re, ri = (n-1, -1, -1) if lower else (0, n, 1) + + for ind in range(rs, re, ri): + s_ind = swap_vec[ind] + if s_ind != ind: + # Column start and end positions + col_s = ind if lower else 0 + col_e = n if lower else ind+1 + + # If we stumble upon a 2x2 block include both cols in the perm. + if pivs[ind] == (0 if lower else 2): + col_s += -1 if lower else 0 + col_e += 0 if lower else 1 + lu[[s_ind, ind], col_s:col_e] = lu[[ind, s_ind], col_s:col_e] + perm[[s_ind, ind]] = perm[[ind, s_ind]] + + return lu, argsort(perm) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_lu.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_lu.py new file mode 100644 index 0000000000000000000000000000000000000000..06562a4a490a4328c08d4de45a5463427e33562b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_lu.py @@ -0,0 +1,389 @@ +"""LU decomposition functions.""" + +from warnings import warn + +from numpy import asarray, asarray_chkfinite +import numpy as np +from itertools import product + +# Local imports +from ._misc import _datacopied, LinAlgWarning +from .lapack import get_lapack_funcs +from ._decomp_lu_cython import lu_dispatcher + +lapack_cast_dict = {x: ''.join([y for y in 'fdFD' if np.can_cast(x, y)]) + for x in np.typecodes['All']} + +__all__ = ['lu', 'lu_solve', 'lu_factor'] + + +def lu_factor(a, overwrite_a=False, check_finite=True): + """ + Compute pivoted LU decomposition of a matrix. + + The decomposition is:: + + A = P L U + + where P is a permutation matrix, L lower triangular with unit + diagonal elements, and U upper triangular. + + Parameters + ---------- + a : (M, N) array_like + Matrix to decompose + overwrite_a : bool, optional + Whether to overwrite data in A (may increase performance) + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + lu : (M, N) ndarray + Matrix containing U in its upper triangle, and L in its lower triangle. + The unit diagonal elements of L are not stored. + piv : (K,) ndarray + Pivot indices representing the permutation matrix P: + row i of matrix was interchanged with row piv[i]. + Of shape ``(K,)``, with ``K = min(M, N)``. + + See Also + -------- + lu : gives lu factorization in more user-friendly format + lu_solve : solve an equation system using the LU factorization of a matrix + + Notes + ----- + This is a wrapper to the ``*GETRF`` routines from LAPACK. Unlike + :func:`lu`, it outputs the L and U factors into a single array + and returns pivot indices instead of a permutation matrix. + + While the underlying ``*GETRF`` routines return 1-based pivot indices, the + ``piv`` array returned by ``lu_factor`` contains 0-based indices. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import lu_factor + >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]]) + >>> lu, piv = lu_factor(A) + >>> piv + array([2, 2, 3, 3], dtype=int32) + + Convert LAPACK's ``piv`` array to NumPy index and test the permutation + + >>> def pivot_to_permutation(piv): + ... perm = np.arange(len(piv)) + ... for i in range(len(piv)): + ... perm[i], perm[piv[i]] = perm[piv[i]], perm[i] + ... return perm + ... + >>> p_inv = pivot_to_permutation(piv) + >>> p_inv + array([2, 0, 3, 1]) + >>> L, U = np.tril(lu, k=-1) + np.eye(4), np.triu(lu) + >>> np.allclose(A[p_inv] - L @ U, np.zeros((4, 4))) + True + + The P matrix in P L U is defined by the inverse permutation and + can be recovered using argsort: + + >>> p = np.argsort(p_inv) + >>> p + array([1, 3, 0, 2]) + >>> np.allclose(A - L[p] @ U, np.zeros((4, 4))) + True + + or alternatively: + + >>> P = np.eye(4)[p] + >>> np.allclose(A - P @ L @ U, np.zeros((4, 4))) + True + """ + if check_finite: + a1 = asarray_chkfinite(a) + else: + a1 = asarray(a) + + # accommodate empty arrays + if a1.size == 0: + lu = np.empty_like(a1) + piv = np.arange(0, dtype=np.int32) + return lu, piv + + overwrite_a = overwrite_a or (_datacopied(a1, a)) + + getrf, = get_lapack_funcs(('getrf',), (a1,)) + lu, piv, info = getrf(a1, overwrite_a=overwrite_a) + if info < 0: + raise ValueError('illegal value in %dth argument of ' + 'internal getrf (lu_factor)' % -info) + if info > 0: + warn("Diagonal number %d is exactly zero. Singular matrix." % info, + LinAlgWarning, stacklevel=2) + return lu, piv + + +def lu_solve(lu_and_piv, b, trans=0, overwrite_b=False, check_finite=True): + """Solve an equation system, a x = b, given the LU factorization of a + + Parameters + ---------- + (lu, piv) + Factorization of the coefficient matrix a, as given by lu_factor. + In particular piv are 0-indexed pivot indices. + b : array + Right-hand side + trans : {0, 1, 2}, optional + Type of system to solve: + + ===== ========= + trans system + ===== ========= + 0 a x = b + 1 a^T x = b + 2 a^H x = b + ===== ========= + overwrite_b : bool, optional + Whether to overwrite data in b (may increase performance) + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + x : array + Solution to the system + + See Also + -------- + lu_factor : LU factorize a matrix + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import lu_factor, lu_solve + >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]]) + >>> b = np.array([1, 1, 1, 1]) + >>> lu, piv = lu_factor(A) + >>> x = lu_solve((lu, piv), b) + >>> np.allclose(A @ x - b, np.zeros((4,))) + True + + """ + (lu, piv) = lu_and_piv + if check_finite: + b1 = asarray_chkfinite(b) + else: + b1 = asarray(b) + + overwrite_b = overwrite_b or _datacopied(b1, b) + + if lu.shape[0] != b1.shape[0]: + raise ValueError(f"Shapes of lu {lu.shape} and b {b1.shape} are incompatible") + + # accommodate empty arrays + if b1.size == 0: + m = lu_solve((np.eye(2, dtype=lu.dtype), [0, 1]), np.ones(2, dtype=b.dtype)) + return np.empty_like(b1, dtype=m.dtype) + + getrs, = get_lapack_funcs(('getrs',), (lu, b1)) + x, info = getrs(lu, piv, b1, trans=trans, overwrite_b=overwrite_b) + if info == 0: + return x + raise ValueError('illegal value in %dth argument of internal gesv|posv' + % -info) + + +def lu(a, permute_l=False, overwrite_a=False, check_finite=True, + p_indices=False): + """ + Compute LU decomposition of a matrix with partial pivoting. + + The decomposition satisfies:: + + A = P @ L @ U + + where ``P`` is a permutation matrix, ``L`` lower triangular with unit + diagonal elements, and ``U`` upper triangular. If `permute_l` is set to + ``True`` then ``L`` is returned already permuted and hence satisfying + ``A = L @ U``. + + Parameters + ---------- + a : (M, N) array_like + Array to decompose + permute_l : bool, optional + Perform the multiplication P*L (Default: do not permute) + overwrite_a : bool, optional + Whether to overwrite data in a (may improve performance) + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + p_indices : bool, optional + If ``True`` the permutation information is returned as row indices. + The default is ``False`` for backwards-compatibility reasons. + + Returns + ------- + **(If `permute_l` is ``False``)** + + p : (..., M, M) ndarray + Permutation arrays or vectors depending on `p_indices` + l : (..., M, K) ndarray + Lower triangular or trapezoidal array with unit diagonal. + ``K = min(M, N)`` + u : (..., K, N) ndarray + Upper triangular or trapezoidal array + + **(If `permute_l` is ``True``)** + + pl : (..., M, K) ndarray + Permuted L matrix. + ``K = min(M, N)`` + u : (..., K, N) ndarray + Upper triangular or trapezoidal array + + Notes + ----- + Permutation matrices are costly since they are nothing but row reorder of + ``L`` and hence indices are strongly recommended to be used instead if the + permutation is required. The relation in the 2D case then becomes simply + ``A = L[P, :] @ U``. In higher dimensions, it is better to use `permute_l` + to avoid complicated indexing tricks. + + In 2D case, if one has the indices however, for some reason, the + permutation matrix is still needed then it can be constructed by + ``np.eye(M)[P, :]``. + + Examples + -------- + + >>> import numpy as np + >>> from scipy.linalg import lu + >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]]) + >>> p, l, u = lu(A) + >>> np.allclose(A, p @ l @ u) + True + >>> p # Permutation matrix + array([[0., 1., 0., 0.], # Row index 1 + [0., 0., 0., 1.], # Row index 3 + [1., 0., 0., 0.], # Row index 0 + [0., 0., 1., 0.]]) # Row index 2 + >>> p, _, _ = lu(A, p_indices=True) + >>> p + array([1, 3, 0, 2], dtype=int32) # as given by row indices above + >>> np.allclose(A, l[p, :] @ u) + True + + We can also use nd-arrays, for example, a demonstration with 4D array: + + >>> rng = np.random.default_rng() + >>> A = rng.uniform(low=-4, high=4, size=[3, 2, 4, 8]) + >>> p, l, u = lu(A) + >>> p.shape, l.shape, u.shape + ((3, 2, 4, 4), (3, 2, 4, 4), (3, 2, 4, 8)) + >>> np.allclose(A, p @ l @ u) + True + >>> PL, U = lu(A, permute_l=True) + >>> np.allclose(A, PL @ U) + True + + """ + a1 = np.asarray_chkfinite(a) if check_finite else np.asarray(a) + if a1.ndim < 2: + raise ValueError('The input array must be at least two-dimensional.') + + # Also check if dtype is LAPACK compatible + if a1.dtype.char not in 'fdFD': + dtype_char = lapack_cast_dict[a1.dtype.char] + if not dtype_char: # No casting possible + raise TypeError(f'The dtype {a1.dtype} cannot be cast ' + 'to float(32, 64) or complex(64, 128).') + + a1 = a1.astype(dtype_char[0]) # makes a copy, free to scratch + overwrite_a = True + + *nd, m, n = a1.shape + k = min(m, n) + real_dchar = 'f' if a1.dtype.char in 'fF' else 'd' + + # Empty input + if min(*a1.shape) == 0: + if permute_l: + PL = np.empty(shape=[*nd, m, k], dtype=a1.dtype) + U = np.empty(shape=[*nd, k, n], dtype=a1.dtype) + return PL, U + else: + P = (np.empty([*nd, 0], dtype=np.int32) if p_indices else + np.empty([*nd, 0, 0], dtype=real_dchar)) + L = np.empty(shape=[*nd, m, k], dtype=a1.dtype) + U = np.empty(shape=[*nd, k, n], dtype=a1.dtype) + return P, L, U + + # Scalar case + if a1.shape[-2:] == (1, 1): + if permute_l: + return np.ones_like(a1), (a1 if overwrite_a else a1.copy()) + else: + P = (np.zeros(shape=[*nd, m], dtype=int) if p_indices + else np.ones_like(a1)) + return P, np.ones_like(a1), (a1 if overwrite_a else a1.copy()) + + # Then check overwrite permission + if not _datacopied(a1, a): # "a" still alive through "a1" + if not overwrite_a: + # Data belongs to "a" so make a copy + a1 = a1.copy(order='C') + # else: Do nothing we'll use "a" if possible + # else: a1 has its own data thus free to scratch + + # Then layout checks, might happen that overwrite is allowed but original + # array was read-only or non-contiguous. + + if not (a1.flags['C_CONTIGUOUS'] and a1.flags['WRITEABLE']): + a1 = a1.copy(order='C') + + if not nd: # 2D array + + p = np.empty(m, dtype=np.int32) + u = np.zeros([k, k], dtype=a1.dtype) + lu_dispatcher(a1, u, p, permute_l) + P, L, U = (p, a1, u) if m > n else (p, u, a1) + + else: # Stacked array + + # Prepare the contiguous data holders + P = np.empty([*nd, m], dtype=np.int32) # perm vecs + + if m > n: # Tall arrays, U will be created + U = np.zeros([*nd, k, k], dtype=a1.dtype) + for ind in product(*[range(x) for x in a1.shape[:-2]]): + lu_dispatcher(a1[ind], U[ind], P[ind], permute_l) + L = a1 + + else: # Fat arrays, L will be created + L = np.zeros([*nd, k, k], dtype=a1.dtype) + for ind in product(*[range(x) for x in a1.shape[:-2]]): + lu_dispatcher(a1[ind], L[ind], P[ind], permute_l) + U = a1 + + # Convert permutation vecs to permutation arrays + # permute_l=False needed to enter here to avoid wasted efforts + if (not p_indices) and (not permute_l): + if nd: + Pa = np.zeros([*nd, m, m], dtype=real_dchar) + # An unreadable index hack - One-hot encoding for perm matrices + nd_ix = np.ix_(*([np.arange(x) for x in nd]+[np.arange(m)])) + Pa[(*nd_ix, P)] = 1 + P = Pa + else: # 2D case + Pa = np.zeros([m, m], dtype=real_dchar) + Pa[np.arange(m), P] = 1 + P = Pa + + return (L, U) if permute_l else (P, L, U) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_lu_cython.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_lu_cython.pyi new file mode 100644 index 0000000000000000000000000000000000000000..0a175b1de32806102318cf69f7c5b4c3deddb03c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_lu_cython.pyi @@ -0,0 +1,6 @@ +from numpy.typing import NDArray +from typing import Any + +def lu_decompose(a: NDArray[Any], lu: NDArray[Any], perm: NDArray[Any], permute_l: bool) -> None: ... # noqa: E501 + +def lu_dispatcher(a: NDArray[Any], lu: NDArray[Any], perm: NDArray[Any], permute_l: bool) -> None: ... # noqa: E501 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_polar.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_polar.py new file mode 100644 index 0000000000000000000000000000000000000000..2fc3652899bed607ab1dd5e3f1663345010e93c1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_polar.py @@ -0,0 +1,111 @@ +import numpy as np +from scipy.linalg import svd + + +__all__ = ['polar'] + + +def polar(a, side="right"): + """ + Compute the polar decomposition. + + Returns the factors of the polar decomposition [1]_ `u` and `p` such + that ``a = up`` (if `side` is "right") or ``a = pu`` (if `side` is + "left"), where `p` is positive semidefinite. Depending on the shape + of `a`, either the rows or columns of `u` are orthonormal. When `a` + is a square array, `u` is a square unitary array. When `a` is not + square, the "canonical polar decomposition" [2]_ is computed. + + Parameters + ---------- + a : (m, n) array_like + The array to be factored. + side : {'left', 'right'}, optional + Determines whether a right or left polar decomposition is computed. + If `side` is "right", then ``a = up``. If `side` is "left", then + ``a = pu``. The default is "right". + + Returns + ------- + u : (m, n) ndarray + If `a` is square, then `u` is unitary. If m > n, then the columns + of `a` are orthonormal, and if m < n, then the rows of `u` are + orthonormal. + p : ndarray + `p` is Hermitian positive semidefinite. If `a` is nonsingular, `p` + is positive definite. The shape of `p` is (n, n) or (m, m), depending + on whether `side` is "right" or "left", respectively. + + References + ---------- + .. [1] R. A. Horn and C. R. Johnson, "Matrix Analysis", Cambridge + University Press, 1985. + .. [2] N. J. Higham, "Functions of Matrices: Theory and Computation", + SIAM, 2008. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import polar + >>> a = np.array([[1, -1], [2, 4]]) + >>> u, p = polar(a) + >>> u + array([[ 0.85749293, -0.51449576], + [ 0.51449576, 0.85749293]]) + >>> p + array([[ 1.88648444, 1.2004901 ], + [ 1.2004901 , 3.94446746]]) + + A non-square example, with m < n: + + >>> b = np.array([[0.5, 1, 2], [1.5, 3, 4]]) + >>> u, p = polar(b) + >>> u + array([[-0.21196618, -0.42393237, 0.88054056], + [ 0.39378971, 0.78757942, 0.4739708 ]]) + >>> p + array([[ 0.48470147, 0.96940295, 1.15122648], + [ 0.96940295, 1.9388059 , 2.30245295], + [ 1.15122648, 2.30245295, 3.65696431]]) + >>> u.dot(p) # Verify the decomposition. + array([[ 0.5, 1. , 2. ], + [ 1.5, 3. , 4. ]]) + >>> u.dot(u.T) # The rows of u are orthonormal. + array([[ 1.00000000e+00, -2.07353665e-17], + [ -2.07353665e-17, 1.00000000e+00]]) + + Another non-square example, with m > n: + + >>> c = b.T + >>> u, p = polar(c) + >>> u + array([[-0.21196618, 0.39378971], + [-0.42393237, 0.78757942], + [ 0.88054056, 0.4739708 ]]) + >>> p + array([[ 1.23116567, 1.93241587], + [ 1.93241587, 4.84930602]]) + >>> u.dot(p) # Verify the decomposition. + array([[ 0.5, 1.5], + [ 1. , 3. ], + [ 2. , 4. ]]) + >>> u.T.dot(u) # The columns of u are orthonormal. + array([[ 1.00000000e+00, -1.26363763e-16], + [ -1.26363763e-16, 1.00000000e+00]]) + + """ + if side not in ['right', 'left']: + raise ValueError("`side` must be either 'right' or 'left'") + a = np.asarray(a) + if a.ndim != 2: + raise ValueError("`a` must be a 2-D array.") + + w, s, vh = svd(a, full_matrices=False) + u = w.dot(vh) + if side == 'right': + # a = up + p = (vh.T.conj() * s).dot(vh) + else: + # a = pu + p = (w * s).dot(w.T.conj()) + return u, p diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_qr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_qr.py new file mode 100644 index 0000000000000000000000000000000000000000..a41ad90770e3d53c741d85e6f46d4040bf203e7a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_qr.py @@ -0,0 +1,490 @@ +"""QR decomposition functions.""" +import numpy as np + +# Local imports +from .lapack import get_lapack_funcs +from ._misc import _datacopied + +__all__ = ['qr', 'qr_multiply', 'rq'] + + +def safecall(f, name, *args, **kwargs): + """Call a LAPACK routine, determining lwork automatically and handling + error return values""" + lwork = kwargs.get("lwork", None) + if lwork in (None, -1): + kwargs['lwork'] = -1 + ret = f(*args, **kwargs) + kwargs['lwork'] = ret[-2][0].real.astype(np.int_) + ret = f(*args, **kwargs) + if ret[-1] < 0: + raise ValueError("illegal value in %dth argument of internal %s" + % (-ret[-1], name)) + return ret[:-2] + + +def qr(a, overwrite_a=False, lwork=None, mode='full', pivoting=False, + check_finite=True): + """ + Compute QR decomposition of a matrix. + + Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal + and R upper triangular. + + Parameters + ---------- + a : (M, N) array_like + Matrix to be decomposed + overwrite_a : bool, optional + Whether data in `a` is overwritten (may improve performance if + `overwrite_a` is set to True by reusing the existing input data + structure rather than creating a new one.) + lwork : int, optional + Work array size, lwork >= a.shape[1]. If None or -1, an optimal size + is computed. + mode : {'full', 'r', 'economic', 'raw'}, optional + Determines what information is to be returned: either both Q and R + ('full', default), only R ('r') or both Q and R but computed in + economy-size ('economic', see Notes). The final option 'raw' + (added in SciPy 0.11) makes the function return two matrices + (Q, TAU) in the internal format used by LAPACK. + pivoting : bool, optional + Whether or not factorization should include pivoting for rank-revealing + qr decomposition. If pivoting, compute the decomposition + ``A[:, P] = Q @ R`` as above, but where P is chosen such that the + diagonal of R is non-increasing. Equivalently, albeit less efficiently, + an explicit P matrix may be formed explicitly by permuting the rows or columns + (depending on the side of the equation on which it is to be used) of + an identity matrix. See Examples. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + Q : float or complex ndarray + Of shape (M, M), or (M, K) for ``mode='economic'``. Not returned + if ``mode='r'``. Replaced by tuple ``(Q, TAU)`` if ``mode='raw'``. + R : float or complex ndarray + Of shape (M, N), or (K, N) for ``mode in ['economic', 'raw']``. + ``K = min(M, N)``. + P : int ndarray + Of shape (N,) for ``pivoting=True``. Not returned if + ``pivoting=False``. + + Raises + ------ + LinAlgError + Raised if decomposition fails + + Notes + ----- + This is an interface to the LAPACK routines dgeqrf, zgeqrf, + dorgqr, zungqr, dgeqp3, and zgeqp3. + + If ``mode=economic``, the shapes of Q and R are (M, K) and (K, N) instead + of (M,M) and (M,N), with ``K=min(M,N)``. + + Examples + -------- + >>> import numpy as np + >>> from scipy import linalg + >>> rng = np.random.default_rng() + >>> a = rng.standard_normal((9, 6)) + + >>> q, r = linalg.qr(a) + >>> np.allclose(a, np.dot(q, r)) + True + >>> q.shape, r.shape + ((9, 9), (9, 6)) + + >>> r2 = linalg.qr(a, mode='r') + >>> np.allclose(r, r2) + True + + >>> q3, r3 = linalg.qr(a, mode='economic') + >>> q3.shape, r3.shape + ((9, 6), (6, 6)) + + >>> q4, r4, p4 = linalg.qr(a, pivoting=True) + >>> d = np.abs(np.diag(r4)) + >>> np.all(d[1:] <= d[:-1]) + True + >>> np.allclose(a[:, p4], np.dot(q4, r4)) + True + >>> P = np.eye(p4.size)[p4] + >>> np.allclose(a, np.dot(q4, r4) @ P) + True + >>> np.allclose(a @ P.T, np.dot(q4, r4)) + True + >>> q4.shape, r4.shape, p4.shape + ((9, 9), (9, 6), (6,)) + + >>> q5, r5, p5 = linalg.qr(a, mode='economic', pivoting=True) + >>> q5.shape, r5.shape, p5.shape + ((9, 6), (6, 6), (6,)) + >>> P = np.eye(6)[:, p5] + >>> np.allclose(a @ P, np.dot(q5, r5)) + True + + """ + # 'qr' was the old default, equivalent to 'full'. Neither 'full' nor + # 'qr' are used below. + # 'raw' is used internally by qr_multiply + if mode not in ['full', 'qr', 'r', 'economic', 'raw']: + raise ValueError("Mode argument should be one of ['full', 'r', " + "'economic', 'raw']") + + if check_finite: + a1 = np.asarray_chkfinite(a) + else: + a1 = np.asarray(a) + if len(a1.shape) != 2: + raise ValueError("expected a 2-D array") + + M, N = a1.shape + + # accommodate empty arrays + if a1.size == 0: + K = min(M, N) + + if mode not in ['economic', 'raw']: + Q = np.empty_like(a1, shape=(M, M)) + Q[...] = np.identity(M) + R = np.empty_like(a1) + else: + Q = np.empty_like(a1, shape=(M, K)) + R = np.empty_like(a1, shape=(K, N)) + + if pivoting: + Rj = R, np.arange(N, dtype=np.int32) + else: + Rj = R, + + if mode == 'r': + return Rj + elif mode == 'raw': + qr = np.empty_like(a1, shape=(M, N)) + tau = np.zeros_like(a1, shape=(K,)) + return ((qr, tau),) + Rj + return (Q,) + Rj + + overwrite_a = overwrite_a or (_datacopied(a1, a)) + + if pivoting: + geqp3, = get_lapack_funcs(('geqp3',), (a1,)) + qr, jpvt, tau = safecall(geqp3, "geqp3", a1, overwrite_a=overwrite_a) + jpvt -= 1 # geqp3 returns a 1-based index array, so subtract 1 + else: + geqrf, = get_lapack_funcs(('geqrf',), (a1,)) + qr, tau = safecall(geqrf, "geqrf", a1, lwork=lwork, + overwrite_a=overwrite_a) + + if mode not in ['economic', 'raw'] or M < N: + R = np.triu(qr) + else: + R = np.triu(qr[:N, :]) + + if pivoting: + Rj = R, jpvt + else: + Rj = R, + + if mode == 'r': + return Rj + elif mode == 'raw': + return ((qr, tau),) + Rj + + gor_un_gqr, = get_lapack_funcs(('orgqr',), (qr,)) + + if M < N: + Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qr[:, :M], tau, + lwork=lwork, overwrite_a=1) + elif mode == 'economic': + Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qr, tau, lwork=lwork, + overwrite_a=1) + else: + t = qr.dtype.char + qqr = np.empty((M, M), dtype=t) + qqr[:, :N] = qr + Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qqr, tau, lwork=lwork, + overwrite_a=1) + + return (Q,) + Rj + + +def qr_multiply(a, c, mode='right', pivoting=False, conjugate=False, + overwrite_a=False, overwrite_c=False): + """ + Calculate the QR decomposition and multiply Q with a matrix. + + Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal + and R upper triangular. Multiply Q with a vector or a matrix c. + + Parameters + ---------- + a : (M, N), array_like + Input array + c : array_like + Input array to be multiplied by ``q``. + mode : {'left', 'right'}, optional + ``Q @ c`` is returned if mode is 'left', ``c @ Q`` is returned if + mode is 'right'. + The shape of c must be appropriate for the matrix multiplications, + if mode is 'left', ``min(a.shape) == c.shape[0]``, + if mode is 'right', ``a.shape[0] == c.shape[1]``. + pivoting : bool, optional + Whether or not factorization should include pivoting for rank-revealing + qr decomposition, see the documentation of qr. + conjugate : bool, optional + Whether Q should be complex-conjugated. This might be faster + than explicit conjugation. + overwrite_a : bool, optional + Whether data in a is overwritten (may improve performance) + overwrite_c : bool, optional + Whether data in c is overwritten (may improve performance). + If this is used, c must be big enough to keep the result, + i.e. ``c.shape[0]`` = ``a.shape[0]`` if mode is 'left'. + + Returns + ------- + CQ : ndarray + The product of ``Q`` and ``c``. + R : (K, N), ndarray + R array of the resulting QR factorization where ``K = min(M, N)``. + P : (N,) ndarray + Integer pivot array. Only returned when ``pivoting=True``. + + Raises + ------ + LinAlgError + Raised if QR decomposition fails. + + Notes + ----- + This is an interface to the LAPACK routines ``?GEQRF``, ``?ORMQR``, + ``?UNMQR``, and ``?GEQP3``. + + .. versionadded:: 0.11.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import qr_multiply, qr + >>> A = np.array([[1, 3, 3], [2, 3, 2], [2, 3, 3], [1, 3, 2]]) + >>> qc, r1, piv1 = qr_multiply(A, 2*np.eye(4), pivoting=1) + >>> qc + array([[-1., 1., -1.], + [-1., -1., 1.], + [-1., -1., -1.], + [-1., 1., 1.]]) + >>> r1 + array([[-6., -3., -5. ], + [ 0., -1., -1.11022302e-16], + [ 0., 0., -1. ]]) + >>> piv1 + array([1, 0, 2], dtype=int32) + >>> q2, r2, piv2 = qr(A, mode='economic', pivoting=1) + >>> np.allclose(2*q2 - qc, np.zeros((4, 3))) + True + + """ + if mode not in ['left', 'right']: + raise ValueError("Mode argument can only be 'left' or 'right' but " + f"not '{mode}'") + c = np.asarray_chkfinite(c) + if c.ndim < 2: + onedim = True + c = np.atleast_2d(c) + if mode == "left": + c = c.T + else: + onedim = False + + a = np.atleast_2d(np.asarray(a)) # chkfinite done in qr + M, N = a.shape + + if mode == 'left': + if c.shape[0] != min(M, N + overwrite_c*(M-N)): + raise ValueError('Array shapes are not compatible for Q @ c' + f' operation: {a.shape} vs {c.shape}') + else: + if M != c.shape[1]: + raise ValueError('Array shapes are not compatible for c @ Q' + f' operation: {c.shape} vs {a.shape}') + + raw = qr(a, overwrite_a, None, "raw", pivoting) + Q, tau = raw[0] + + # accommodate empty arrays + if c.size == 0: + return (np.empty_like(c),) + raw[1:] + + gor_un_mqr, = get_lapack_funcs(('ormqr',), (Q,)) + if gor_un_mqr.typecode in ('s', 'd'): + trans = "T" + else: + trans = "C" + + Q = Q[:, :min(M, N)] + if M > N and mode == "left" and not overwrite_c: + if conjugate: + cc = np.zeros((c.shape[1], M), dtype=c.dtype, order="F") + cc[:, :N] = c.T + else: + cc = np.zeros((M, c.shape[1]), dtype=c.dtype, order="F") + cc[:N, :] = c + trans = "N" + if conjugate: + lr = "R" + else: + lr = "L" + overwrite_c = True + elif c.flags["C_CONTIGUOUS"] and trans == "T" or conjugate: + cc = c.T + if mode == "left": + lr = "R" + else: + lr = "L" + else: + trans = "N" + cc = c + if mode == "left": + lr = "L" + else: + lr = "R" + cQ, = safecall(gor_un_mqr, "gormqr/gunmqr", lr, trans, Q, tau, cc, + overwrite_c=overwrite_c) + if trans != "N": + cQ = cQ.T + if mode == "right": + cQ = cQ[:, :min(M, N)] + if onedim: + cQ = cQ.ravel() + + return (cQ,) + raw[1:] + + +def rq(a, overwrite_a=False, lwork=None, mode='full', check_finite=True): + """ + Compute RQ decomposition of a matrix. + + Calculate the decomposition ``A = R Q`` where Q is unitary/orthogonal + and R upper triangular. + + Parameters + ---------- + a : (M, N) array_like + Matrix to be decomposed + overwrite_a : bool, optional + Whether data in a is overwritten (may improve performance) + lwork : int, optional + Work array size, lwork >= a.shape[1]. If None or -1, an optimal size + is computed. + mode : {'full', 'r', 'economic'}, optional + Determines what information is to be returned: either both Q and R + ('full', default), only R ('r') or both Q and R but computed in + economy-size ('economic', see Notes). + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + R : float or complex ndarray + Of shape (M, N) or (M, K) for ``mode='economic'``. ``K = min(M, N)``. + Q : float or complex ndarray + Of shape (N, N) or (K, N) for ``mode='economic'``. Not returned + if ``mode='r'``. + + Raises + ------ + LinAlgError + If decomposition fails. + + Notes + ----- + This is an interface to the LAPACK routines sgerqf, dgerqf, cgerqf, zgerqf, + sorgrq, dorgrq, cungrq and zungrq. + + If ``mode=economic``, the shapes of Q and R are (K, N) and (M, K) instead + of (N,N) and (M,N), with ``K=min(M,N)``. + + Examples + -------- + >>> import numpy as np + >>> from scipy import linalg + >>> rng = np.random.default_rng() + >>> a = rng.standard_normal((6, 9)) + >>> r, q = linalg.rq(a) + >>> np.allclose(a, r @ q) + True + >>> r.shape, q.shape + ((6, 9), (9, 9)) + >>> r2 = linalg.rq(a, mode='r') + >>> np.allclose(r, r2) + True + >>> r3, q3 = linalg.rq(a, mode='economic') + >>> r3.shape, q3.shape + ((6, 6), (6, 9)) + + """ + if mode not in ['full', 'r', 'economic']: + raise ValueError( + "Mode argument should be one of ['full', 'r', 'economic']") + + if check_finite: + a1 = np.asarray_chkfinite(a) + else: + a1 = np.asarray(a) + if len(a1.shape) != 2: + raise ValueError('expected matrix') + + M, N = a1.shape + + # accommodate empty arrays + if a1.size == 0: + K = min(M, N) + + if not mode == 'economic': + R = np.empty_like(a1) + Q = np.empty_like(a1, shape=(N, N)) + Q[...] = np.identity(N) + else: + R = np.empty_like(a1, shape=(M, K)) + Q = np.empty_like(a1, shape=(K, N)) + + if mode == 'r': + return R + return R, Q + + overwrite_a = overwrite_a or (_datacopied(a1, a)) + + gerqf, = get_lapack_funcs(('gerqf',), (a1,)) + rq, tau = safecall(gerqf, 'gerqf', a1, lwork=lwork, + overwrite_a=overwrite_a) + if not mode == 'economic' or N < M: + R = np.triu(rq, N-M) + else: + R = np.triu(rq[-M:, -M:]) + + if mode == 'r': + return R + + gor_un_grq, = get_lapack_funcs(('orgrq',), (rq,)) + + if N < M: + Q, = safecall(gor_un_grq, "gorgrq/gungrq", rq[-N:], tau, lwork=lwork, + overwrite_a=1) + elif mode == 'economic': + Q, = safecall(gor_un_grq, "gorgrq/gungrq", rq, tau, lwork=lwork, + overwrite_a=1) + else: + rq1 = np.empty((N, N), dtype=rq.dtype) + rq1[-M:] = rq + Q, = safecall(gor_un_grq, "gorgrq/gungrq", rq1, tau, lwork=lwork, + overwrite_a=1) + + return R, Q diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_qz.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_qz.py new file mode 100644 index 0000000000000000000000000000000000000000..39361f172df7f1985c7ed0fbc4d919b5c4545725 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_qz.py @@ -0,0 +1,449 @@ +import warnings + +import numpy as np +from numpy import asarray_chkfinite +from ._misc import LinAlgError, _datacopied, LinAlgWarning +from .lapack import get_lapack_funcs + + +__all__ = ['qz', 'ordqz'] + +_double_precision = ['i', 'l', 'd'] + + +def _select_function(sort): + if callable(sort): + # assume the user knows what they're doing + sfunction = sort + elif sort == 'lhp': + sfunction = _lhp + elif sort == 'rhp': + sfunction = _rhp + elif sort == 'iuc': + sfunction = _iuc + elif sort == 'ouc': + sfunction = _ouc + else: + raise ValueError("sort parameter must be None, a callable, or " + "one of ('lhp','rhp','iuc','ouc')") + + return sfunction + + +def _lhp(x, y): + out = np.empty_like(x, dtype=bool) + nonzero = (y != 0) + # handles (x, y) = (0, 0) too + out[~nonzero] = False + out[nonzero] = (np.real(x[nonzero]/y[nonzero]) < 0.0) + return out + + +def _rhp(x, y): + out = np.empty_like(x, dtype=bool) + nonzero = (y != 0) + # handles (x, y) = (0, 0) too + out[~nonzero] = False + out[nonzero] = (np.real(x[nonzero]/y[nonzero]) > 0.0) + return out + + +def _iuc(x, y): + out = np.empty_like(x, dtype=bool) + nonzero = (y != 0) + # handles (x, y) = (0, 0) too + out[~nonzero] = False + out[nonzero] = (abs(x[nonzero]/y[nonzero]) < 1.0) + return out + + +def _ouc(x, y): + out = np.empty_like(x, dtype=bool) + xzero = (x == 0) + yzero = (y == 0) + out[xzero & yzero] = False + out[~xzero & yzero] = True + out[~yzero] = (abs(x[~yzero]/y[~yzero]) > 1.0) + return out + + +def _qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False, + overwrite_b=False, check_finite=True): + if sort is not None: + # Disabled due to segfaults on win32, see ticket 1717. + raise ValueError("The 'sort' input of qz() has to be None and will be " + "removed in a future release. Use ordqz instead.") + + if output not in ['real', 'complex', 'r', 'c']: + raise ValueError("argument must be 'real', or 'complex'") + + if check_finite: + a1 = asarray_chkfinite(A) + b1 = asarray_chkfinite(B) + else: + a1 = np.asarray(A) + b1 = np.asarray(B) + + a_m, a_n = a1.shape + b_m, b_n = b1.shape + if not (a_m == a_n == b_m == b_n): + raise ValueError("Array dimensions must be square and agree") + + typa = a1.dtype.char + if output in ['complex', 'c'] and typa not in ['F', 'D']: + if typa in _double_precision: + a1 = a1.astype('D') + typa = 'D' + else: + a1 = a1.astype('F') + typa = 'F' + typb = b1.dtype.char + if output in ['complex', 'c'] and typb not in ['F', 'D']: + if typb in _double_precision: + b1 = b1.astype('D') + typb = 'D' + else: + b1 = b1.astype('F') + typb = 'F' + + overwrite_a = overwrite_a or (_datacopied(a1, A)) + overwrite_b = overwrite_b or (_datacopied(b1, B)) + + gges, = get_lapack_funcs(('gges',), (a1, b1)) + + if lwork is None or lwork == -1: + # get optimal work array size + result = gges(lambda x: None, a1, b1, lwork=-1) + lwork = result[-2][0].real.astype(int) + + def sfunction(x): + return None + result = gges(sfunction, a1, b1, lwork=lwork, overwrite_a=overwrite_a, + overwrite_b=overwrite_b, sort_t=0) + + info = result[-1] + if info < 0: + raise ValueError(f"Illegal value in argument {-info} of gges") + elif info > 0 and info <= a_n: + warnings.warn("The QZ iteration failed. (a,b) are not in Schur " + "form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be " + f"correct for J={info-1},...,N", LinAlgWarning, + stacklevel=3) + elif info == a_n+1: + raise LinAlgError("Something other than QZ iteration failed") + elif info == a_n+2: + raise LinAlgError("After reordering, roundoff changed values of some " + "complex eigenvalues so that leading eigenvalues " + "in the Generalized Schur form no longer satisfy " + "sort=True. This could also be due to scaling.") + elif info == a_n+3: + raise LinAlgError("Reordering failed in tgsen") + + return result, gges.typecode + + +def qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False, + overwrite_b=False, check_finite=True): + """ + QZ decomposition for generalized eigenvalues of a pair of matrices. + + The QZ, or generalized Schur, decomposition for a pair of n-by-n + matrices (A,B) is:: + + (A,B) = (Q @ AA @ Z*, Q @ BB @ Z*) + + where AA, BB is in generalized Schur form if BB is upper-triangular + with non-negative diagonal and AA is upper-triangular, or for real QZ + decomposition (``output='real'``) block upper triangular with 1x1 + and 2x2 blocks. In this case, the 1x1 blocks correspond to real + generalized eigenvalues and 2x2 blocks are 'standardized' by making + the corresponding elements of BB have the form:: + + [ a 0 ] + [ 0 b ] + + and the pair of corresponding 2x2 blocks in AA and BB will have a complex + conjugate pair of generalized eigenvalues. If (``output='complex'``) or + A and B are complex matrices, Z' denotes the conjugate-transpose of Z. + Q and Z are unitary matrices. + + Parameters + ---------- + A : (N, N) array_like + 2-D array to decompose + B : (N, N) array_like + 2-D array to decompose + output : {'real', 'complex'}, optional + Construct the real or complex QZ decomposition for real matrices. + Default is 'real'. + lwork : int, optional + Work array size. If None or -1, it is automatically computed. + sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional + NOTE: THIS INPUT IS DISABLED FOR NOW. Use ordqz instead. + + Specifies whether the upper eigenvalues should be sorted. A callable + may be passed that, given a eigenvalue, returns a boolean denoting + whether the eigenvalue should be sorted to the top-left (True). For + real matrix pairs, the sort function takes three real arguments + (alphar, alphai, beta). The eigenvalue + ``x = (alphar + alphai*1j)/beta``. For complex matrix pairs or + output='complex', the sort function takes two complex arguments + (alpha, beta). The eigenvalue ``x = (alpha/beta)``. Alternatively, + string parameters may be used: + + - 'lhp' Left-hand plane (x.real < 0.0) + - 'rhp' Right-hand plane (x.real > 0.0) + - 'iuc' Inside the unit circle (x*x.conjugate() < 1.0) + - 'ouc' Outside the unit circle (x*x.conjugate() > 1.0) + + Defaults to None (no sorting). + overwrite_a : bool, optional + Whether to overwrite data in a (may improve performance) + overwrite_b : bool, optional + Whether to overwrite data in b (may improve performance) + check_finite : bool, optional + If true checks the elements of `A` and `B` are finite numbers. If + false does no checking and passes matrix through to + underlying algorithm. + + Returns + ------- + AA : (N, N) ndarray + Generalized Schur form of A. + BB : (N, N) ndarray + Generalized Schur form of B. + Q : (N, N) ndarray + The left Schur vectors. + Z : (N, N) ndarray + The right Schur vectors. + + See Also + -------- + ordqz + + Notes + ----- + Q is transposed versus the equivalent function in Matlab. + + .. versionadded:: 0.11.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import qz + + >>> A = np.array([[1, 2, -1], [5, 5, 5], [2, 4, -8]]) + >>> B = np.array([[1, 1, -3], [3, 1, -1], [5, 6, -2]]) + + Compute the decomposition. The QZ decomposition is not unique, so + depending on the underlying library that is used, there may be + differences in the signs of coefficients in the following output. + + >>> AA, BB, Q, Z = qz(A, B) + >>> AA + array([[-1.36949157, -4.05459025, 7.44389431], + [ 0. , 7.65653432, 5.13476017], + [ 0. , -0.65978437, 2.4186015 ]]) # may vary + >>> BB + array([[ 1.71890633, -1.64723705, -0.72696385], + [ 0. , 8.6965692 , -0. ], + [ 0. , 0. , 2.27446233]]) # may vary + >>> Q + array([[-0.37048362, 0.1903278 , 0.90912992], + [-0.90073232, 0.16534124, -0.40167593], + [ 0.22676676, 0.96769706, -0.11017818]]) # may vary + >>> Z + array([[-0.67660785, 0.63528924, -0.37230283], + [ 0.70243299, 0.70853819, -0.06753907], + [ 0.22088393, -0.30721526, -0.92565062]]) # may vary + + Verify the QZ decomposition. With real output, we only need the + transpose of ``Z`` in the following expressions. + + >>> Q @ AA @ Z.T # Should be A + array([[ 1., 2., -1.], + [ 5., 5., 5.], + [ 2., 4., -8.]]) + >>> Q @ BB @ Z.T # Should be B + array([[ 1., 1., -3.], + [ 3., 1., -1.], + [ 5., 6., -2.]]) + + Repeat the decomposition, but with ``output='complex'``. + + >>> AA, BB, Q, Z = qz(A, B, output='complex') + + For conciseness in the output, we use ``np.set_printoptions()`` to set + the output precision of NumPy arrays to 3 and display tiny values as 0. + + >>> np.set_printoptions(precision=3, suppress=True) + >>> AA + array([[-1.369+0.j , 2.248+4.237j, 4.861-5.022j], + [ 0. +0.j , 7.037+2.922j, 0.794+4.932j], + [ 0. +0.j , 0. +0.j , 2.655-1.103j]]) # may vary + >>> BB + array([[ 1.719+0.j , -1.115+1.j , -0.763-0.646j], + [ 0. +0.j , 7.24 +0.j , -3.144+3.322j], + [ 0. +0.j , 0. +0.j , 2.732+0.j ]]) # may vary + >>> Q + array([[ 0.326+0.175j, -0.273-0.029j, -0.886-0.052j], + [ 0.794+0.426j, -0.093+0.134j, 0.402-0.02j ], + [-0.2 -0.107j, -0.816+0.482j, 0.151-0.167j]]) # may vary + >>> Z + array([[ 0.596+0.32j , -0.31 +0.414j, 0.393-0.347j], + [-0.619-0.332j, -0.479+0.314j, 0.154-0.393j], + [-0.195-0.104j, 0.576+0.27j , 0.715+0.187j]]) # may vary + + With complex arrays, we must use ``Z.conj().T`` in the following + expressions to verify the decomposition. + + >>> Q @ AA @ Z.conj().T # Should be A + array([[ 1.-0.j, 2.-0.j, -1.-0.j], + [ 5.+0.j, 5.+0.j, 5.-0.j], + [ 2.+0.j, 4.+0.j, -8.+0.j]]) + >>> Q @ BB @ Z.conj().T # Should be B + array([[ 1.+0.j, 1.+0.j, -3.+0.j], + [ 3.-0.j, 1.-0.j, -1.+0.j], + [ 5.+0.j, 6.+0.j, -2.+0.j]]) + + """ + # output for real + # AA, BB, sdim, alphar, alphai, beta, vsl, vsr, work, info + # output for complex + # AA, BB, sdim, alpha, beta, vsl, vsr, work, info + result, _ = _qz(A, B, output=output, lwork=lwork, sort=sort, + overwrite_a=overwrite_a, overwrite_b=overwrite_b, + check_finite=check_finite) + return result[0], result[1], result[-4], result[-3] + + +def ordqz(A, B, sort='lhp', output='real', overwrite_a=False, + overwrite_b=False, check_finite=True): + """QZ decomposition for a pair of matrices with reordering. + + Parameters + ---------- + A : (N, N) array_like + 2-D array to decompose + B : (N, N) array_like + 2-D array to decompose + sort : {callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional + Specifies whether the upper eigenvalues should be sorted. A + callable may be passed that, given an ordered pair ``(alpha, + beta)`` representing the eigenvalue ``x = (alpha/beta)``, + returns a boolean denoting whether the eigenvalue should be + sorted to the top-left (True). For the real matrix pairs + ``beta`` is real while ``alpha`` can be complex, and for + complex matrix pairs both ``alpha`` and ``beta`` can be + complex. The callable must be able to accept a NumPy + array. Alternatively, string parameters may be used: + + - 'lhp' Left-hand plane (x.real < 0.0) + - 'rhp' Right-hand plane (x.real > 0.0) + - 'iuc' Inside the unit circle (x*x.conjugate() < 1.0) + - 'ouc' Outside the unit circle (x*x.conjugate() > 1.0) + + With the predefined sorting functions, an infinite eigenvalue + (i.e., ``alpha != 0`` and ``beta = 0``) is considered to lie in + neither the left-hand nor the right-hand plane, but it is + considered to lie outside the unit circle. For the eigenvalue + ``(alpha, beta) = (0, 0)``, the predefined sorting functions + all return `False`. + output : str {'real','complex'}, optional + Construct the real or complex QZ decomposition for real matrices. + Default is 'real'. + overwrite_a : bool, optional + If True, the contents of A are overwritten. + overwrite_b : bool, optional + If True, the contents of B are overwritten. + check_finite : bool, optional + If true checks the elements of `A` and `B` are finite numbers. If + false does no checking and passes matrix through to + underlying algorithm. + + Returns + ------- + AA : (N, N) ndarray + Generalized Schur form of A. + BB : (N, N) ndarray + Generalized Schur form of B. + alpha : (N,) ndarray + alpha = alphar + alphai * 1j. See notes. + beta : (N,) ndarray + See notes. + Q : (N, N) ndarray + The left Schur vectors. + Z : (N, N) ndarray + The right Schur vectors. + + See Also + -------- + qz + + Notes + ----- + On exit, ``(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N``, will be the + generalized eigenvalues. ``ALPHAR(j) + ALPHAI(j)*i`` and + ``BETA(j),j=1,...,N`` are the diagonals of the complex Schur form (S,T) + that would result if the 2-by-2 diagonal blocks of the real generalized + Schur form of (A,B) were further reduced to triangular form using complex + unitary transformations. If ALPHAI(j) is zero, then the jth eigenvalue is + real; if positive, then the ``j``\\ th and ``(j+1)``\\ st eigenvalues are a + complex conjugate pair, with ``ALPHAI(j+1)`` negative. + + .. versionadded:: 0.17.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import ordqz + >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]]) + >>> B = np.array([[0, 6, 0, 0], [5, 0, 2, 1], [5, 2, 6, 6], [4, 7, 7, 7]]) + >>> AA, BB, alpha, beta, Q, Z = ordqz(A, B, sort='lhp') + + Since we have sorted for left half plane eigenvalues, negatives come first + + >>> (alpha/beta).real < 0 + array([ True, True, False, False], dtype=bool) + + """ + (AA, BB, _, *ab, Q, Z, _, _), typ = _qz(A, B, output=output, sort=None, + overwrite_a=overwrite_a, + overwrite_b=overwrite_b, + check_finite=check_finite) + + if typ == 's': + alpha, beta = ab[0] + ab[1]*np.complex64(1j), ab[2] + elif typ == 'd': + alpha, beta = ab[0] + ab[1]*1.j, ab[2] + else: + alpha, beta = ab + + sfunction = _select_function(sort) + select = sfunction(alpha, beta) + + tgsen = get_lapack_funcs('tgsen', (AA, BB)) + # the real case needs 4n + 16 lwork + lwork = 4*AA.shape[0] + 16 if typ in 'sd' else 1 + AAA, BBB, *ab, QQ, ZZ, _, _, _, _, info = tgsen(select, AA, BB, Q, Z, + ijob=0, + lwork=lwork, liwork=1) + + # Once more for tgsen output + if typ == 's': + alpha, beta = ab[0] + ab[1]*np.complex64(1j), ab[2] + elif typ == 'd': + alpha, beta = ab[0] + ab[1]*1.j, ab[2] + else: + alpha, beta = ab + + if info < 0: + raise ValueError(f"Illegal value in argument {-info} of tgsen") + elif info == 1: + raise ValueError("Reordering of (A, B) failed because the transformed" + " matrix pair (A, B) would be too far from " + "generalized Schur form; the problem is very " + "ill-conditioned. (A, B) may have been partially " + "reordered.") + + return AAA, BBB, alpha, beta, QQ, ZZ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_schur.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_schur.py new file mode 100644 index 0000000000000000000000000000000000000000..8609a175e16d663938386c6b45d190cd0e5dafd8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_schur.py @@ -0,0 +1,334 @@ +"""Schur decomposition functions.""" +import numpy as np +from numpy import asarray_chkfinite, single, asarray, array +from numpy.linalg import norm + + +# Local imports. +from ._misc import LinAlgError, _datacopied +from .lapack import get_lapack_funcs +from ._decomp import eigvals + +__all__ = ['schur', 'rsf2csf'] + +_double_precision = ['i', 'l', 'd'] + + +def schur(a, output='real', lwork=None, overwrite_a=False, sort=None, + check_finite=True): + """ + Compute Schur decomposition of a matrix. + + The Schur decomposition is:: + + A = Z T Z^H + + where Z is unitary and T is either upper-triangular, or for real + Schur decomposition (output='real'), quasi-upper triangular. In + the quasi-triangular form, 2x2 blocks describing complex-valued + eigenvalue pairs may extrude from the diagonal. + + Parameters + ---------- + a : (M, M) array_like + Matrix to decompose + output : {'real', 'complex'}, optional + When the dtype of `a` is real, this specifies whether to compute + the real or complex Schur decomposition. + When the dtype of `a` is complex, this argument is ignored, and the + complex Schur decomposition is computed. + lwork : int, optional + Work array size. If None or -1, it is automatically computed. + overwrite_a : bool, optional + Whether to overwrite data in a (may improve performance). + sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional + Specifies whether the upper eigenvalues should be sorted. A callable + may be passed that, given an eigenvalue, returns a boolean denoting + whether the eigenvalue should be sorted to the top-left (True). + + - If ``output='complex'`` OR the dtype of `a` is complex, the callable + should have one argument: the eigenvalue expressed as a complex number. + - If ``output='real'`` AND the dtype of `a` is real, the callable should have + two arguments: the real and imaginary parts of the eigenvalue, respectively. + + Alternatively, string parameters may be used:: + + 'lhp' Left-hand plane (real(eigenvalue) < 0.0) + 'rhp' Right-hand plane (real(eigenvalue) >= 0.0) + 'iuc' Inside the unit circle (abs(eigenvalue) <= 1.0) + 'ouc' Outside the unit circle (abs(eigenvalue) > 1.0) + + Defaults to None (no sorting). + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + T : (M, M) ndarray + Schur form of A. It is real-valued for the real Schur decomposition. + Z : (M, M) ndarray + An unitary Schur transformation matrix for A. + It is real-valued for the real Schur decomposition. + sdim : int + If and only if sorting was requested, a third return value will + contain the number of eigenvalues satisfying the sort condition. + Note that complex conjugate pairs for which the condition is true + for either eigenvalue count as 2. + + Raises + ------ + LinAlgError + Error raised under three conditions: + + 1. The algorithm failed due to a failure of the QR algorithm to + compute all eigenvalues. + 2. If eigenvalue sorting was requested, the eigenvalues could not be + reordered due to a failure to separate eigenvalues, usually because + of poor conditioning. + 3. If eigenvalue sorting was requested, roundoff errors caused the + leading eigenvalues to no longer satisfy the sorting condition. + + See Also + -------- + rsf2csf : Convert real Schur form to complex Schur form + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import schur, eigvals + >>> A = np.array([[0, 2, 2], [0, 1, 2], [1, 0, 1]]) + >>> T, Z = schur(A) + >>> T + array([[ 2.65896708, 1.42440458, -1.92933439], + [ 0. , -0.32948354, -0.49063704], + [ 0. , 1.31178921, -0.32948354]]) + >>> Z + array([[0.72711591, -0.60156188, 0.33079564], + [0.52839428, 0.79801892, 0.28976765], + [0.43829436, 0.03590414, -0.89811411]]) + + >>> T2, Z2 = schur(A, output='complex') + >>> T2 + array([[ 2.65896708, -1.22839825+1.32378589j, 0.42590089+1.51937378j], # may vary + [ 0. , -0.32948354+0.80225456j, -0.59877807+0.56192146j], + [ 0. , 0. , -0.32948354-0.80225456j]]) + >>> eigvals(T2) + array([2.65896708, -0.32948354+0.80225456j, -0.32948354-0.80225456j]) # may vary + + A custom eigenvalue-sorting condition that sorts by positive imaginary part + is satisfied by only one eigenvalue. + + >>> _, _, sdim = schur(A, output='complex', sort=lambda x: x.imag > 1e-15) + >>> sdim + 1 + + When ``output='real'`` and the array `a` is real, the `sort` callable must accept + the real and imaginary parts as separate arguments. Note that now the complex + eigenvalues ``-0.32948354+0.80225456j`` and ``-0.32948354-0.80225456j`` will be + treated as a complex conjugate pair, and according to the `sdim` documentation, + complex conjugate pairs for which the condition is True for *either* eigenvalue + increase `sdim` by *two*. + + >>> _, _, sdim = schur(A, output='real', sort=lambda x, y: y > 1e-15) + >>> sdim + 2 + + """ + if output not in ['real', 'complex', 'r', 'c']: + raise ValueError("argument must be 'real', or 'complex'") + if check_finite: + a1 = asarray_chkfinite(a) + else: + a1 = asarray(a) + if np.issubdtype(a1.dtype, np.integer): + a1 = asarray(a, dtype=np.dtype("long")) + if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]): + raise ValueError('expected square matrix') + + typ = a1.dtype.char + if output in ['complex', 'c'] and typ not in ['F', 'D']: + if typ in _double_precision: + a1 = a1.astype('D') + else: + a1 = a1.astype('F') + + # accommodate empty matrix + if a1.size == 0: + t0, z0 = schur(np.eye(2, dtype=a1.dtype)) + if sort is None: + return (np.empty_like(a1, dtype=t0.dtype), + np.empty_like(a1, dtype=z0.dtype)) + else: + return (np.empty_like(a1, dtype=t0.dtype), + np.empty_like(a1, dtype=z0.dtype), 0) + + overwrite_a = overwrite_a or (_datacopied(a1, a)) + gees, = get_lapack_funcs(('gees',), (a1,)) + if lwork is None or lwork == -1: + # get optimal work array + result = gees(lambda x: None, a1, lwork=-1) + lwork = result[-2][0].real.astype(np.int_) + + if sort is None: + sort_t = 0 + def sfunction(x, y=None): + return None + else: + sort_t = 1 + if callable(sort): + sfunction = sort + elif sort == 'lhp': + def sfunction(x, y=None): + return x.real < 0.0 + elif sort == 'rhp': + def sfunction(x, y=None): + return x.real >= 0.0 + elif sort == 'iuc': + def sfunction(x, y=None): + z = x if y is None else x + y*1j + return abs(z) <= 1.0 + elif sort == 'ouc': + def sfunction(x, y=None): + z = x if y is None else x + y*1j + return abs(z) > 1.0 + else: + raise ValueError("'sort' parameter must either be 'None', or a " + "callable, or one of ('lhp','rhp','iuc','ouc')") + + result = gees(sfunction, a1, lwork=lwork, overwrite_a=overwrite_a, + sort_t=sort_t) + + info = result[-1] + if info < 0: + raise ValueError(f'illegal value in {-info}-th argument of internal gees') + elif info == a1.shape[0] + 1: + raise LinAlgError('Eigenvalues could not be separated for reordering.') + elif info == a1.shape[0] + 2: + raise LinAlgError('Leading eigenvalues do not satisfy sort condition.') + elif info > 0: + raise LinAlgError("Schur form not found. Possibly ill-conditioned.") + + if sort is None: + return result[0], result[-3] + else: + return result[0], result[-3], result[1] + + +eps = np.finfo(float).eps +feps = np.finfo(single).eps + +_array_kind = {'b': 0, 'h': 0, 'B': 0, 'i': 0, 'l': 0, + 'f': 0, 'd': 0, 'F': 1, 'D': 1} +_array_precision = {'i': 1, 'l': 1, 'f': 0, 'd': 1, 'F': 0, 'D': 1} +_array_type = [['f', 'd'], ['F', 'D']] + + +def _commonType(*arrays): + kind = 0 + precision = 0 + for a in arrays: + t = a.dtype.char + kind = max(kind, _array_kind[t]) + precision = max(precision, _array_precision[t]) + return _array_type[kind][precision] + + +def _castCopy(type, *arrays): + cast_arrays = () + for a in arrays: + if a.dtype.char == type: + cast_arrays = cast_arrays + (a.copy(),) + else: + cast_arrays = cast_arrays + (a.astype(type),) + if len(cast_arrays) == 1: + return cast_arrays[0] + else: + return cast_arrays + + +def rsf2csf(T, Z, check_finite=True): + """ + Convert real Schur form to complex Schur form. + + Convert a quasi-diagonal real-valued Schur form to the upper-triangular + complex-valued Schur form. + + Parameters + ---------- + T : (M, M) array_like + Real Schur form of the original array + Z : (M, M) array_like + Schur transformation matrix + check_finite : bool, optional + Whether to check that the input arrays contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + T : (M, M) ndarray + Complex Schur form of the original array + Z : (M, M) ndarray + Schur transformation matrix corresponding to the complex form + + See Also + -------- + schur : Schur decomposition of an array + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import schur, rsf2csf + >>> A = np.array([[0, 2, 2], [0, 1, 2], [1, 0, 1]]) + >>> T, Z = schur(A) + >>> T + array([[ 2.65896708, 1.42440458, -1.92933439], + [ 0. , -0.32948354, -0.49063704], + [ 0. , 1.31178921, -0.32948354]]) + >>> Z + array([[0.72711591, -0.60156188, 0.33079564], + [0.52839428, 0.79801892, 0.28976765], + [0.43829436, 0.03590414, -0.89811411]]) + >>> T2 , Z2 = rsf2csf(T, Z) + >>> T2 + array([[2.65896708+0.j, -1.64592781+0.743164187j, -1.21516887+1.00660462j], + [0.+0.j , -0.32948354+8.02254558e-01j, -0.82115218-2.77555756e-17j], + [0.+0.j , 0.+0.j, -0.32948354-0.802254558j]]) + >>> Z2 + array([[0.72711591+0.j, 0.28220393-0.31385693j, 0.51319638-0.17258824j], + [0.52839428+0.j, 0.24720268+0.41635578j, -0.68079517-0.15118243j], + [0.43829436+0.j, -0.76618703+0.01873251j, -0.03063006+0.46857912j]]) + + """ + if check_finite: + Z, T = map(asarray_chkfinite, (Z, T)) + else: + Z, T = map(asarray, (Z, T)) + + for ind, X in enumerate([Z, T]): + if X.ndim != 2 or X.shape[0] != X.shape[1]: + raise ValueError(f"Input '{'ZT'[ind]}' must be square.") + + if T.shape[0] != Z.shape[0]: + message = f"Input array shapes must match: Z: {Z.shape} vs. T: {T.shape}" + raise ValueError(message) + N = T.shape[0] + t = _commonType(Z, T, array([3.0], 'F')) + Z, T = _castCopy(t, Z, T) + + for m in range(N-1, 0, -1): + if abs(T[m, m-1]) > eps*(abs(T[m-1, m-1]) + abs(T[m, m])): + mu = eigvals(T[m-1:m+1, m-1:m+1]) - T[m, m] + r = norm([mu[0], T[m, m-1]]) + c = mu[0] / r + s = T[m, m-1] / r + G = array([[c.conj(), s], [-s, c]], dtype=t) + + T[m-1:m+1, m-1:] = G.dot(T[m-1:m+1, m-1:]) + T[:m+1, m-1:m+1] = T[:m+1, m-1:m+1].dot(G.conj().T) + Z[:, m-1:m+1] = Z[:, m-1:m+1].dot(G.conj().T) + + T[m, m-1] = 0.0 + return T, Z diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_svd.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_svd.py new file mode 100644 index 0000000000000000000000000000000000000000..98425f6c11e727d582102dce72baeb9cbdb6c40b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_decomp_svd.py @@ -0,0 +1,534 @@ +"""SVD decomposition functions.""" +import numpy as np +from numpy import zeros, r_, diag, dot, arccos, arcsin, where, clip + +# Local imports. +from ._misc import LinAlgError, _datacopied +from .lapack import get_lapack_funcs, _compute_lwork +from ._decomp import _asarray_validated + +__all__ = ['svd', 'svdvals', 'diagsvd', 'orth', 'subspace_angles', 'null_space'] + + +def svd(a, full_matrices=True, compute_uv=True, overwrite_a=False, + check_finite=True, lapack_driver='gesdd'): + """ + Singular Value Decomposition. + + Factorizes the matrix `a` into two unitary matrices ``U`` and ``Vh``, and + a 1-D array ``s`` of singular values (real, non-negative) such that + ``a == U @ S @ Vh``, where ``S`` is a suitably shaped matrix of zeros with + main diagonal ``s``. + + Parameters + ---------- + a : (M, N) array_like + Matrix to decompose. + full_matrices : bool, optional + If True (default), `U` and `Vh` are of shape ``(M, M)``, ``(N, N)``. + If False, the shapes are ``(M, K)`` and ``(K, N)``, where + ``K = min(M, N)``. + compute_uv : bool, optional + Whether to compute also ``U`` and ``Vh`` in addition to ``s``. + Default is True. + overwrite_a : bool, optional + Whether to overwrite `a`; may improve performance. + Default is False. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + lapack_driver : {'gesdd', 'gesvd'}, optional + Whether to use the more efficient divide-and-conquer approach + (``'gesdd'``) or general rectangular approach (``'gesvd'``) + to compute the SVD. MATLAB and Octave use the ``'gesvd'`` approach. + Default is ``'gesdd'``. + + Returns + ------- + U : ndarray + Unitary matrix having left singular vectors as columns. + Of shape ``(M, M)`` or ``(M, K)``, depending on `full_matrices`. + s : ndarray + The singular values, sorted in non-increasing order. + Of shape (K,), with ``K = min(M, N)``. + Vh : ndarray + Unitary matrix having right singular vectors as rows. + Of shape ``(N, N)`` or ``(K, N)`` depending on `full_matrices`. + + For ``compute_uv=False``, only ``s`` is returned. + + Raises + ------ + LinAlgError + If SVD computation does not converge. + + See Also + -------- + svdvals : Compute singular values of a matrix. + diagsvd : Construct the Sigma matrix, given the vector s. + + Examples + -------- + >>> import numpy as np + >>> from scipy import linalg + >>> rng = np.random.default_rng() + >>> m, n = 9, 6 + >>> a = rng.standard_normal((m, n)) + 1.j*rng.standard_normal((m, n)) + >>> U, s, Vh = linalg.svd(a) + >>> U.shape, s.shape, Vh.shape + ((9, 9), (6,), (6, 6)) + + Reconstruct the original matrix from the decomposition: + + >>> sigma = np.zeros((m, n)) + >>> for i in range(min(m, n)): + ... sigma[i, i] = s[i] + >>> a1 = np.dot(U, np.dot(sigma, Vh)) + >>> np.allclose(a, a1) + True + + Alternatively, use ``full_matrices=False`` (notice that the shape of + ``U`` is then ``(m, n)`` instead of ``(m, m)``): + + >>> U, s, Vh = linalg.svd(a, full_matrices=False) + >>> U.shape, s.shape, Vh.shape + ((9, 6), (6,), (6, 6)) + >>> S = np.diag(s) + >>> np.allclose(a, np.dot(U, np.dot(S, Vh))) + True + + >>> s2 = linalg.svd(a, compute_uv=False) + >>> np.allclose(s, s2) + True + + """ + a1 = _asarray_validated(a, check_finite=check_finite) + if len(a1.shape) != 2: + raise ValueError('expected matrix') + m, n = a1.shape + + # accommodate empty matrix + if a1.size == 0: + u0, s0, v0 = svd(np.eye(2, dtype=a1.dtype)) + + s = np.empty_like(a1, shape=(0,), dtype=s0.dtype) + if full_matrices: + u = np.empty_like(a1, shape=(m, m), dtype=u0.dtype) + u[...] = np.identity(m) + v = np.empty_like(a1, shape=(n, n), dtype=v0.dtype) + v[...] = np.identity(n) + else: + u = np.empty_like(a1, shape=(m, 0), dtype=u0.dtype) + v = np.empty_like(a1, shape=(0, n), dtype=v0.dtype) + if compute_uv: + return u, s, v + else: + return s + + overwrite_a = overwrite_a or (_datacopied(a1, a)) + + if not isinstance(lapack_driver, str): + raise TypeError('lapack_driver must be a string') + if lapack_driver not in ('gesdd', 'gesvd'): + message = f'lapack_driver must be "gesdd" or "gesvd", not "{lapack_driver}"' + raise ValueError(message) + + if compute_uv: + # XXX: revisit int32 when ILP64 lapack becomes a thing + max_mn, min_mn = (m, n) if m > n else (n, m) + if full_matrices: + if max_mn*max_mn > np.iinfo(np.int32).max: + raise ValueError(f"Indexing a matrix size {max_mn} x {max_mn} " + "would incur integer overflow in LAPACK. " + "Try using numpy.linalg.svd instead.") + else: + sz = max(m * min_mn, n * min_mn) + if max(m * min_mn, n * min_mn) > np.iinfo(np.int32).max: + raise ValueError(f"Indexing a matrix of {sz} elements would " + "incur an in integer overflow in LAPACK. " + "Try using numpy.linalg.svd instead.") + + funcs = (lapack_driver, lapack_driver + '_lwork') + # XXX: As of 1.14.1 it isn't possible to build SciPy with ILP64, + # so the following line always yields a LP64 (32-bit pointer size) variant + gesXd, gesXd_lwork = get_lapack_funcs(funcs, (a1,), ilp64="preferred") + + # compute optimal lwork + lwork = _compute_lwork(gesXd_lwork, a1.shape[0], a1.shape[1], + compute_uv=compute_uv, full_matrices=full_matrices) + + # perform decomposition + u, s, v, info = gesXd(a1, compute_uv=compute_uv, lwork=lwork, + full_matrices=full_matrices, overwrite_a=overwrite_a) + + if info > 0: + raise LinAlgError("SVD did not converge") + if info < 0: + raise ValueError('illegal value in %dth argument of internal gesdd' + % -info) + if compute_uv: + return u, s, v + else: + return s + + +def svdvals(a, overwrite_a=False, check_finite=True): + """ + Compute singular values of a matrix. + + Parameters + ---------- + a : (M, N) array_like + Matrix to decompose. + overwrite_a : bool, optional + Whether to overwrite `a`; may improve performance. + Default is False. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + s : (min(M, N),) ndarray + The singular values, sorted in decreasing order. + + Raises + ------ + LinAlgError + If SVD computation does not converge. + + See Also + -------- + svd : Compute the full singular value decomposition of a matrix. + diagsvd : Construct the Sigma matrix, given the vector s. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import svdvals + >>> m = np.array([[1.0, 0.0], + ... [2.0, 3.0], + ... [1.0, 1.0], + ... [0.0, 2.0], + ... [1.0, 0.0]]) + >>> svdvals(m) + array([ 4.28091555, 1.63516424]) + + We can verify the maximum singular value of `m` by computing the maximum + length of `m.dot(u)` over all the unit vectors `u` in the (x,y) plane. + We approximate "all" the unit vectors with a large sample. Because + of linearity, we only need the unit vectors with angles in [0, pi]. + + >>> t = np.linspace(0, np.pi, 2000) + >>> u = np.array([np.cos(t), np.sin(t)]) + >>> np.linalg.norm(m.dot(u), axis=0).max() + 4.2809152422538475 + + `p` is a projection matrix with rank 1. With exact arithmetic, + its singular values would be [1, 0, 0, 0]. + + >>> v = np.array([0.1, 0.3, 0.9, 0.3]) + >>> p = np.outer(v, v) + >>> svdvals(p) + array([ 1.00000000e+00, 2.02021698e-17, 1.56692500e-17, + 8.15115104e-34]) + + The singular values of an orthogonal matrix are all 1. Here, we + create a random orthogonal matrix by using the `rvs()` method of + `scipy.stats.ortho_group`. + + >>> from scipy.stats import ortho_group + >>> orth = ortho_group.rvs(4) + >>> svdvals(orth) + array([ 1., 1., 1., 1.]) + + """ + return svd(a, compute_uv=0, overwrite_a=overwrite_a, + check_finite=check_finite) + + +def diagsvd(s, M, N): + """ + Construct the sigma matrix in SVD from singular values and size M, N. + + Parameters + ---------- + s : (M,) or (N,) array_like + Singular values + M : int + Size of the matrix whose singular values are `s`. + N : int + Size of the matrix whose singular values are `s`. + + Returns + ------- + S : (M, N) ndarray + The S-matrix in the singular value decomposition + + See Also + -------- + svd : Singular value decomposition of a matrix + svdvals : Compute singular values of a matrix. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import diagsvd + >>> vals = np.array([1, 2, 3]) # The array representing the computed svd + >>> diagsvd(vals, 3, 4) + array([[1, 0, 0, 0], + [0, 2, 0, 0], + [0, 0, 3, 0]]) + >>> diagsvd(vals, 4, 3) + array([[1, 0, 0], + [0, 2, 0], + [0, 0, 3], + [0, 0, 0]]) + + """ + part = diag(s) + typ = part.dtype.char + MorN = len(s) + if MorN == M: + return np.hstack((part, zeros((M, N - M), dtype=typ))) + elif MorN == N: + return r_[part, zeros((M - N, N), dtype=typ)] + else: + raise ValueError("Length of s must be M or N.") + + +# Orthonormal decomposition + +def orth(A, rcond=None): + """ + Construct an orthonormal basis for the range of A using SVD + + Parameters + ---------- + A : (M, N) array_like + Input array + rcond : float, optional + Relative condition number. Singular values ``s`` smaller than + ``rcond * max(s)`` are considered zero. + Default: floating point eps * max(M,N). + + Returns + ------- + Q : (M, K) ndarray + Orthonormal basis for the range of A. + K = effective rank of A, as determined by rcond + + See Also + -------- + svd : Singular value decomposition of a matrix + null_space : Matrix null space + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import orth + >>> A = np.array([[2, 0, 0], [0, 5, 0]]) # rank 2 array + >>> orth(A) + array([[0., 1.], + [1., 0.]]) + >>> orth(A.T) + array([[0., 1.], + [1., 0.], + [0., 0.]]) + + """ + u, s, vh = svd(A, full_matrices=False) + M, N = u.shape[0], vh.shape[1] + if rcond is None: + rcond = np.finfo(s.dtype).eps * max(M, N) + tol = np.amax(s, initial=0.) * rcond + num = np.sum(s > tol, dtype=int) + Q = u[:, :num] + return Q + + +def null_space(A, rcond=None, *, overwrite_a=False, check_finite=True, + lapack_driver='gesdd'): + """ + Construct an orthonormal basis for the null space of A using SVD + + Parameters + ---------- + A : (M, N) array_like + Input array + rcond : float, optional + Relative condition number. Singular values ``s`` smaller than + ``rcond * max(s)`` are considered zero. + Default: floating point eps * max(M,N). + overwrite_a : bool, optional + Whether to overwrite `a`; may improve performance. + Default is False. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + lapack_driver : {'gesdd', 'gesvd'}, optional + Whether to use the more efficient divide-and-conquer approach + (``'gesdd'``) or general rectangular approach (``'gesvd'``) + to compute the SVD. MATLAB and Octave use the ``'gesvd'`` approach. + Default is ``'gesdd'``. + + Returns + ------- + Z : (N, K) ndarray + Orthonormal basis for the null space of A. + K = dimension of effective null space, as determined by rcond + + See Also + -------- + svd : Singular value decomposition of a matrix + orth : Matrix range + + Examples + -------- + 1-D null space: + + >>> import numpy as np + >>> from scipy.linalg import null_space + >>> A = np.array([[1, 1], [1, 1]]) + >>> ns = null_space(A) + >>> ns * np.copysign(1, ns[0,0]) # Remove the sign ambiguity of the vector + array([[ 0.70710678], + [-0.70710678]]) + + 2-D null space: + + >>> from numpy.random import default_rng + >>> rng = default_rng() + >>> B = rng.random((3, 5)) + >>> Z = null_space(B) + >>> Z.shape + (5, 2) + >>> np.allclose(B.dot(Z), 0) + True + + The basis vectors are orthonormal (up to rounding error): + + >>> Z.T.dot(Z) + array([[ 1.00000000e+00, 6.92087741e-17], + [ 6.92087741e-17, 1.00000000e+00]]) + + """ + u, s, vh = svd(A, full_matrices=True, overwrite_a=overwrite_a, + check_finite=check_finite, lapack_driver=lapack_driver) + M, N = u.shape[0], vh.shape[1] + if rcond is None: + rcond = np.finfo(s.dtype).eps * max(M, N) + tol = np.amax(s, initial=0.) * rcond + num = np.sum(s > tol, dtype=int) + Q = vh[num:,:].T.conj() + return Q + + +def subspace_angles(A, B): + r""" + Compute the subspace angles between two matrices. + + Parameters + ---------- + A : (M, N) array_like + The first input array. + B : (M, K) array_like + The second input array. + + Returns + ------- + angles : ndarray, shape (min(N, K),) + The subspace angles between the column spaces of `A` and `B` in + descending order. + + See Also + -------- + orth + svd + + Notes + ----- + This computes the subspace angles according to the formula + provided in [1]_. For equivalence with MATLAB and Octave behavior, + use ``angles[0]``. + + .. versionadded:: 1.0 + + References + ---------- + .. [1] Knyazev A, Argentati M (2002) Principal Angles between Subspaces + in an A-Based Scalar Product: Algorithms and Perturbation + Estimates. SIAM J. Sci. Comput. 23:2008-2040. + + Examples + -------- + An Hadamard matrix, which has orthogonal columns, so we expect that + the suspace angle to be :math:`\frac{\pi}{2}`: + + >>> import numpy as np + >>> from scipy.linalg import hadamard, subspace_angles + >>> rng = np.random.default_rng() + >>> H = hadamard(4) + >>> print(H) + [[ 1 1 1 1] + [ 1 -1 1 -1] + [ 1 1 -1 -1] + [ 1 -1 -1 1]] + >>> np.rad2deg(subspace_angles(H[:, :2], H[:, 2:])) + array([ 90., 90.]) + + And the subspace angle of a matrix to itself should be zero: + + >>> subspace_angles(H[:, :2], H[:, :2]) <= 2 * np.finfo(float).eps + array([ True, True], dtype=bool) + + The angles between non-orthogonal subspaces are in between these extremes: + + >>> x = rng.standard_normal((4, 3)) + >>> np.rad2deg(subspace_angles(x[:, :2], x[:, [2]])) + array([ 55.832]) # random + """ + # Steps here omit the U and V calculation steps from the paper + + # 1. Compute orthonormal bases of column-spaces + A = _asarray_validated(A, check_finite=True) + if len(A.shape) != 2: + raise ValueError(f'expected 2D array, got shape {A.shape}') + QA = orth(A) + del A + + B = _asarray_validated(B, check_finite=True) + if len(B.shape) != 2: + raise ValueError(f'expected 2D array, got shape {B.shape}') + if len(B) != len(QA): + raise ValueError('A and B must have the same number of rows, got ' + f'{QA.shape[0]} and {B.shape[0]}') + QB = orth(B) + del B + + # 2. Compute SVD for cosine + QA_H_QB = dot(QA.T.conj(), QB) + sigma = svdvals(QA_H_QB) + + # 3. Compute matrix B + if QA.shape[1] >= QB.shape[1]: + B = QB - dot(QA, QA_H_QB) + else: + B = QA - dot(QB, QA_H_QB.T.conj()) + del QA, QB, QA_H_QB + + # 4. Compute SVD for sine + mask = sigma ** 2 >= 0.5 + if mask.any(): + mu_arcsin = arcsin(clip(svdvals(B, overwrite_a=True), -1., 1.)) + else: + mu_arcsin = 0. + + # 5. Compute the principal angles + # with reverse ordering of sigma because smallest sigma belongs to largest + # angle theta + theta = where(mask, mu_arcsin, arccos(clip(sigma[::-1], -1., 1.))) + return theta diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_expm_frechet.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_expm_frechet.py new file mode 100644 index 0000000000000000000000000000000000000000..56ddbc45c3bc47f6beb122e2acadd274ebd9be95 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_expm_frechet.py @@ -0,0 +1,413 @@ +"""Frechet derivative of the matrix exponential.""" +import numpy as np +import scipy.linalg + +__all__ = ['expm_frechet', 'expm_cond'] + + +def expm_frechet(A, E, method=None, compute_expm=True, check_finite=True): + """ + Frechet derivative of the matrix exponential of A in the direction E. + + Parameters + ---------- + A : (N, N) array_like + Matrix of which to take the matrix exponential. + E : (N, N) array_like + Matrix direction in which to take the Frechet derivative. + method : str, optional + Choice of algorithm. Should be one of + + - `SPS` (default) + - `blockEnlarge` + + compute_expm : bool, optional + Whether to compute also `expm_A` in addition to `expm_frechet_AE`. + Default is True. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + expm_A : ndarray + Matrix exponential of A. + expm_frechet_AE : ndarray + Frechet derivative of the matrix exponential of A in the direction E. + For ``compute_expm = False``, only `expm_frechet_AE` is returned. + + See Also + -------- + expm : Compute the exponential of a matrix. + + Notes + ----- + This section describes the available implementations that can be selected + by the `method` parameter. The default method is *SPS*. + + Method *blockEnlarge* is a naive algorithm. + + Method *SPS* is Scaling-Pade-Squaring [1]_. + It is a sophisticated implementation which should take + only about 3/8 as much time as the naive implementation. + The asymptotics are the same. + + .. versionadded:: 0.13.0 + + References + ---------- + .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2009) + Computing the Frechet Derivative of the Matrix Exponential, + with an application to Condition Number Estimation. + SIAM Journal On Matrix Analysis and Applications., + 30 (4). pp. 1639-1657. ISSN 1095-7162 + + Examples + -------- + >>> import numpy as np + >>> from scipy import linalg + >>> rng = np.random.default_rng() + + >>> A = rng.standard_normal((3, 3)) + >>> E = rng.standard_normal((3, 3)) + >>> expm_A, expm_frechet_AE = linalg.expm_frechet(A, E) + >>> expm_A.shape, expm_frechet_AE.shape + ((3, 3), (3, 3)) + + Create a 6x6 matrix containing [[A, E], [0, A]]: + + >>> M = np.zeros((6, 6)) + >>> M[:3, :3] = A + >>> M[:3, 3:] = E + >>> M[3:, 3:] = A + + >>> expm_M = linalg.expm(M) + >>> np.allclose(expm_A, expm_M[:3, :3]) + True + >>> np.allclose(expm_frechet_AE, expm_M[:3, 3:]) + True + + """ + if check_finite: + A = np.asarray_chkfinite(A) + E = np.asarray_chkfinite(E) + else: + A = np.asarray(A) + E = np.asarray(E) + if A.ndim != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected A to be a square matrix') + if E.ndim != 2 or E.shape[0] != E.shape[1]: + raise ValueError('expected E to be a square matrix') + if A.shape != E.shape: + raise ValueError('expected A and E to be the same shape') + if method is None: + method = 'SPS' + if method == 'SPS': + expm_A, expm_frechet_AE = expm_frechet_algo_64(A, E) + elif method == 'blockEnlarge': + expm_A, expm_frechet_AE = expm_frechet_block_enlarge(A, E) + else: + raise ValueError(f'Unknown implementation {method}') + if compute_expm: + return expm_A, expm_frechet_AE + else: + return expm_frechet_AE + + +def expm_frechet_block_enlarge(A, E): + """ + This is a helper function, mostly for testing and profiling. + Return expm(A), frechet(A, E) + """ + n = A.shape[0] + M = np.vstack([ + np.hstack([A, E]), + np.hstack([np.zeros_like(A), A])]) + expm_M = scipy.linalg.expm(M) + return expm_M[:n, :n], expm_M[:n, n:] + + +""" +Maximal values ell_m of ||2**-s A|| such that the backward error bound +does not exceed 2**-53. +""" +ell_table_61 = ( + None, + # 1 + 2.11e-8, + 3.56e-4, + 1.08e-2, + 6.49e-2, + 2.00e-1, + 4.37e-1, + 7.83e-1, + 1.23e0, + 1.78e0, + 2.42e0, + # 11 + 3.13e0, + 3.90e0, + 4.74e0, + 5.63e0, + 6.56e0, + 7.52e0, + 8.53e0, + 9.56e0, + 1.06e1, + 1.17e1, + ) + + +# The b vectors and U and V are copypasted +# from scipy.sparse.linalg.matfuncs.py. +# M, Lu, Lv follow (6.11), (6.12), (6.13), (3.3) + +def _diff_pade3(A, E, ident): + b = (120., 60., 12., 1.) + A2 = A.dot(A) + M2 = np.dot(A, E) + np.dot(E, A) + U = A.dot(b[3]*A2 + b[1]*ident) + V = b[2]*A2 + b[0]*ident + Lu = A.dot(b[3]*M2) + E.dot(b[3]*A2 + b[1]*ident) + Lv = b[2]*M2 + return U, V, Lu, Lv + + +def _diff_pade5(A, E, ident): + b = (30240., 15120., 3360., 420., 30., 1.) + A2 = A.dot(A) + M2 = np.dot(A, E) + np.dot(E, A) + A4 = np.dot(A2, A2) + M4 = np.dot(A2, M2) + np.dot(M2, A2) + U = A.dot(b[5]*A4 + b[3]*A2 + b[1]*ident) + V = b[4]*A4 + b[2]*A2 + b[0]*ident + Lu = (A.dot(b[5]*M4 + b[3]*M2) + + E.dot(b[5]*A4 + b[3]*A2 + b[1]*ident)) + Lv = b[4]*M4 + b[2]*M2 + return U, V, Lu, Lv + + +def _diff_pade7(A, E, ident): + b = (17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.) + A2 = A.dot(A) + M2 = np.dot(A, E) + np.dot(E, A) + A4 = np.dot(A2, A2) + M4 = np.dot(A2, M2) + np.dot(M2, A2) + A6 = np.dot(A2, A4) + M6 = np.dot(A4, M2) + np.dot(M4, A2) + U = A.dot(b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident) + V = b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*ident + Lu = (A.dot(b[7]*M6 + b[5]*M4 + b[3]*M2) + + E.dot(b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident)) + Lv = b[6]*M6 + b[4]*M4 + b[2]*M2 + return U, V, Lu, Lv + + +def _diff_pade9(A, E, ident): + b = (17643225600., 8821612800., 2075673600., 302702400., 30270240., + 2162160., 110880., 3960., 90., 1.) + A2 = A.dot(A) + M2 = np.dot(A, E) + np.dot(E, A) + A4 = np.dot(A2, A2) + M4 = np.dot(A2, M2) + np.dot(M2, A2) + A6 = np.dot(A2, A4) + M6 = np.dot(A4, M2) + np.dot(M4, A2) + A8 = np.dot(A4, A4) + M8 = np.dot(A4, M4) + np.dot(M4, A4) + U = A.dot(b[9]*A8 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident) + V = b[8]*A8 + b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*ident + Lu = (A.dot(b[9]*M8 + b[7]*M6 + b[5]*M4 + b[3]*M2) + + E.dot(b[9]*A8 + b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident)) + Lv = b[8]*M8 + b[6]*M6 + b[4]*M4 + b[2]*M2 + return U, V, Lu, Lv + + +def expm_frechet_algo_64(A, E): + n = A.shape[0] + s = None + ident = np.identity(n) + A_norm_1 = scipy.linalg.norm(A, 1) + m_pade_pairs = ( + (3, _diff_pade3), + (5, _diff_pade5), + (7, _diff_pade7), + (9, _diff_pade9)) + for m, pade in m_pade_pairs: + if A_norm_1 <= ell_table_61[m]: + U, V, Lu, Lv = pade(A, E, ident) + s = 0 + break + if s is None: + # scaling + s = max(0, int(np.ceil(np.log2(A_norm_1 / ell_table_61[13])))) + A = A * 2.0**-s + E = E * 2.0**-s + # pade order 13 + A2 = np.dot(A, A) + M2 = np.dot(A, E) + np.dot(E, A) + A4 = np.dot(A2, A2) + M4 = np.dot(A2, M2) + np.dot(M2, A2) + A6 = np.dot(A2, A4) + M6 = np.dot(A4, M2) + np.dot(M4, A2) + b = (64764752532480000., 32382376266240000., 7771770303897600., + 1187353796428800., 129060195264000., 10559470521600., + 670442572800., 33522128640., 1323241920., 40840800., 960960., + 16380., 182., 1.) + W1 = b[13]*A6 + b[11]*A4 + b[9]*A2 + W2 = b[7]*A6 + b[5]*A4 + b[3]*A2 + b[1]*ident + Z1 = b[12]*A6 + b[10]*A4 + b[8]*A2 + Z2 = b[6]*A6 + b[4]*A4 + b[2]*A2 + b[0]*ident + W = np.dot(A6, W1) + W2 + U = np.dot(A, W) + V = np.dot(A6, Z1) + Z2 + Lw1 = b[13]*M6 + b[11]*M4 + b[9]*M2 + Lw2 = b[7]*M6 + b[5]*M4 + b[3]*M2 + Lz1 = b[12]*M6 + b[10]*M4 + b[8]*M2 + Lz2 = b[6]*M6 + b[4]*M4 + b[2]*M2 + Lw = np.dot(A6, Lw1) + np.dot(M6, W1) + Lw2 + Lu = np.dot(A, Lw) + np.dot(E, W) + Lv = np.dot(A6, Lz1) + np.dot(M6, Z1) + Lz2 + # factor once and solve twice + lu_piv = scipy.linalg.lu_factor(-U + V) + R = scipy.linalg.lu_solve(lu_piv, U + V) + L = scipy.linalg.lu_solve(lu_piv, Lu + Lv + np.dot((Lu - Lv), R)) + # squaring + for k in range(s): + L = np.dot(R, L) + np.dot(L, R) + R = np.dot(R, R) + return R, L + + +def vec(M): + """ + Stack columns of M to construct a single vector. + + This is somewhat standard notation in linear algebra. + + Parameters + ---------- + M : 2-D array_like + Input matrix + + Returns + ------- + v : 1-D ndarray + Output vector + + """ + return M.T.ravel() + + +def expm_frechet_kronform(A, method=None, check_finite=True): + """ + Construct the Kronecker form of the Frechet derivative of expm. + + Parameters + ---------- + A : array_like with shape (N, N) + Matrix to be expm'd. + method : str, optional + Extra keyword to be passed to expm_frechet. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + K : 2-D ndarray with shape (N*N, N*N) + Kronecker form of the Frechet derivative of the matrix exponential. + + Notes + ----- + This function is used to help compute the condition number + of the matrix exponential. + + See Also + -------- + expm : Compute a matrix exponential. + expm_frechet : Compute the Frechet derivative of the matrix exponential. + expm_cond : Compute the relative condition number of the matrix exponential + in the Frobenius norm. + + """ + if check_finite: + A = np.asarray_chkfinite(A) + else: + A = np.asarray(A) + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected a square matrix') + + n = A.shape[0] + ident = np.identity(n) + cols = [] + for i in range(n): + for j in range(n): + E = np.outer(ident[i], ident[j]) + F = expm_frechet(A, E, + method=method, compute_expm=False, check_finite=False) + cols.append(vec(F)) + return np.vstack(cols).T + + +def expm_cond(A, check_finite=True): + """ + Relative condition number of the matrix exponential in the Frobenius norm. + + Parameters + ---------- + A : 2-D array_like + Square input matrix with shape (N, N). + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + kappa : float + The relative condition number of the matrix exponential + in the Frobenius norm + + See Also + -------- + expm : Compute the exponential of a matrix. + expm_frechet : Compute the Frechet derivative of the matrix exponential. + + Notes + ----- + A faster estimate for the condition number in the 1-norm + has been published but is not yet implemented in SciPy. + + .. versionadded:: 0.14.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import expm_cond + >>> A = np.array([[-0.3, 0.2, 0.6], [0.6, 0.3, -0.1], [-0.7, 1.2, 0.9]]) + >>> k = expm_cond(A) + >>> k + 1.7787805864469866 + + """ + if check_finite: + A = np.asarray_chkfinite(A) + else: + A = np.asarray(A) + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected a square matrix') + + X = scipy.linalg.expm(A) + K = expm_frechet_kronform(A, check_finite=False) + + # The following norm choices are deliberate. + # The norms of A and X are Frobenius norms, + # and the norm of K is the induced 2-norm. + A_norm = scipy.linalg.norm(A, 'fro') + X_norm = scipy.linalg.norm(X, 'fro') + K_norm = scipy.linalg.norm(K, 2) + + kappa = (K_norm * A_norm) / X_norm + return kappa diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_lapack_subroutines.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_lapack_subroutines.h new file mode 100644 index 0000000000000000000000000000000000000000..676658205e41bcde69e3899e8e065c90738af246 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_lapack_subroutines.h @@ -0,0 +1,1521 @@ +/* +This file was generated by _generate_pyx.py. +Do not edit this file directly. +*/ + +#include "npy_cblas.h" +#include "fortran_defs.h" + +typedef int (*_cselect1)(npy_complex64*); +typedef int (*_cselect2)(npy_complex64*, npy_complex64*); +typedef int (*_dselect2)(double*, double*); +typedef int (*_dselect3)(double*, double*, double*); +typedef int (*_sselect2)(float*, float*); +typedef int (*_sselect3)(float*, float*, float*); +typedef int (*_zselect1)(npy_complex128*); +typedef int (*_zselect2)(npy_complex128*, npy_complex128*); + +#ifdef __cplusplus +extern "C" { +#endif + +void BLAS_FUNC(cbbcsd)(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, float *theta, float *phi, npy_complex64 *u1, int *ldu1, npy_complex64 *u2, int *ldu2, npy_complex64 *v1t, int *ldv1t, npy_complex64 *v2t, int *ldv2t, float *b11d, float *b11e, float *b12d, float *b12e, float *b21d, float *b21e, float *b22d, float *b22e, float *rwork, int *lrwork, int *info); +void BLAS_FUNC(cbdsqr)(char *uplo, int *n, int *ncvt, int *nru, int *ncc, float *d, float *e, npy_complex64 *vt, int *ldvt, npy_complex64 *u, int *ldu, npy_complex64 *c, int *ldc, float *rwork, int *info); +void BLAS_FUNC(cgbbrd)(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, npy_complex64 *ab, int *ldab, float *d, float *e, npy_complex64 *q, int *ldq, npy_complex64 *pt, int *ldpt, npy_complex64 *c, int *ldc, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cgbcon)(char *norm, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, int *ipiv, float *anorm, float *rcond, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cgbequ)(int *m, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, float *r, float *c, float *rowcnd, float *colcnd, float *amax, int *info); +void BLAS_FUNC(cgbequb)(int *m, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, float *r, float *c, float *rowcnd, float *colcnd, float *amax, int *info); +void BLAS_FUNC(cgbrfs)(char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *afb, int *ldafb, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cgbsv)(int *n, int *kl, int *ku, int *nrhs, npy_complex64 *ab, int *ldab, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cgbsvx)(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *afb, int *ldafb, int *ipiv, char *equed, float *r, float *c, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cgbtf2)(int *m, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, int *ipiv, int *info); +void BLAS_FUNC(cgbtrf)(int *m, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, int *ipiv, int *info); +void BLAS_FUNC(cgbtrs)(char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex64 *ab, int *ldab, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cgebak)(char *job, char *side, int *n, int *ilo, int *ihi, float *scale, int *m, npy_complex64 *v, int *ldv, int *info); +void BLAS_FUNC(cgebal)(char *job, int *n, npy_complex64 *a, int *lda, int *ilo, int *ihi, float *scale, int *info); +void BLAS_FUNC(cgebd2)(int *m, int *n, npy_complex64 *a, int *lda, float *d, float *e, npy_complex64 *tauq, npy_complex64 *taup, npy_complex64 *work, int *info); +void BLAS_FUNC(cgebrd)(int *m, int *n, npy_complex64 *a, int *lda, float *d, float *e, npy_complex64 *tauq, npy_complex64 *taup, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgecon)(char *norm, int *n, npy_complex64 *a, int *lda, float *anorm, float *rcond, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cgeequ)(int *m, int *n, npy_complex64 *a, int *lda, float *r, float *c, float *rowcnd, float *colcnd, float *amax, int *info); +void BLAS_FUNC(cgeequb)(int *m, int *n, npy_complex64 *a, int *lda, float *r, float *c, float *rowcnd, float *colcnd, float *amax, int *info); +void BLAS_FUNC(cgees)(char *jobvs, char *sort, _cselect1 *select, int *n, npy_complex64 *a, int *lda, int *sdim, npy_complex64 *w, npy_complex64 *vs, int *ldvs, npy_complex64 *work, int *lwork, float *rwork, int *bwork, int *info); +void BLAS_FUNC(cgeesx)(char *jobvs, char *sort, _cselect1 *select, char *sense, int *n, npy_complex64 *a, int *lda, int *sdim, npy_complex64 *w, npy_complex64 *vs, int *ldvs, float *rconde, float *rcondv, npy_complex64 *work, int *lwork, float *rwork, int *bwork, int *info); +void BLAS_FUNC(cgeev)(char *jobvl, char *jobvr, int *n, npy_complex64 *a, int *lda, npy_complex64 *w, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(cgeevx)(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, npy_complex64 *a, int *lda, npy_complex64 *w, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, int *ilo, int *ihi, float *scale, float *abnrm, float *rconde, float *rcondv, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(cgehd2)(int *n, int *ilo, int *ihi, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info); +void BLAS_FUNC(cgehrd)(int *n, int *ilo, int *ihi, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgelq2)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info); +void BLAS_FUNC(cgelqf)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgels)(char *trans, int *m, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgelsd)(int *m, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, float *s, float *rcond, int *rank, npy_complex64 *work, int *lwork, float *rwork, int *iwork, int *info); +void BLAS_FUNC(cgelss)(int *m, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, float *s, float *rcond, int *rank, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(cgelsy)(int *m, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *jpvt, float *rcond, int *rank, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(cgemqrt)(char *side, char *trans, int *m, int *n, int *k, int *nb, npy_complex64 *v, int *ldv, npy_complex64 *t, int *ldt, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info); +void BLAS_FUNC(cgeql2)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info); +void BLAS_FUNC(cgeqlf)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgeqp3)(int *m, int *n, npy_complex64 *a, int *lda, int *jpvt, npy_complex64 *tau, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(cgeqr2)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info); +void BLAS_FUNC(cgeqr2p)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info); +void BLAS_FUNC(cgeqrf)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgeqrfp)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgeqrt)(int *m, int *n, int *nb, npy_complex64 *a, int *lda, npy_complex64 *t, int *ldt, npy_complex64 *work, int *info); +void BLAS_FUNC(cgeqrt2)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *t, int *ldt, int *info); +void BLAS_FUNC(cgeqrt3)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *t, int *ldt, int *info); +void BLAS_FUNC(cgerfs)(char *trans, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cgerq2)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info); +void BLAS_FUNC(cgerqf)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgesc2)(int *n, npy_complex64 *a, int *lda, npy_complex64 *rhs, int *ipiv, int *jpiv, float *scale); +void BLAS_FUNC(cgesdd)(char *jobz, int *m, int *n, npy_complex64 *a, int *lda, float *s, npy_complex64 *u, int *ldu, npy_complex64 *vt, int *ldvt, npy_complex64 *work, int *lwork, float *rwork, int *iwork, int *info); +void BLAS_FUNC(cgesv)(int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cgesvd)(char *jobu, char *jobvt, int *m, int *n, npy_complex64 *a, int *lda, float *s, npy_complex64 *u, int *ldu, npy_complex64 *vt, int *ldvt, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(cgesvx)(char *fact, char *trans, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, char *equed, float *r, float *c, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cgetc2)(int *n, npy_complex64 *a, int *lda, int *ipiv, int *jpiv, int *info); +void BLAS_FUNC(cgetf2)(int *m, int *n, npy_complex64 *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(cgetrf)(int *m, int *n, npy_complex64 *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(cgetri)(int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgetrs)(char *trans, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cggbak)(char *job, char *side, int *n, int *ilo, int *ihi, float *lscale, float *rscale, int *m, npy_complex64 *v, int *ldv, int *info); +void BLAS_FUNC(cggbal)(char *job, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *ilo, int *ihi, float *lscale, float *rscale, float *work, int *info); +void BLAS_FUNC(cgges)(char *jobvsl, char *jobvsr, char *sort, _cselect2 *selctg, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *sdim, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *vsl, int *ldvsl, npy_complex64 *vsr, int *ldvsr, npy_complex64 *work, int *lwork, float *rwork, int *bwork, int *info); +void BLAS_FUNC(cggesx)(char *jobvsl, char *jobvsr, char *sort, _cselect2 *selctg, char *sense, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *sdim, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *vsl, int *ldvsl, npy_complex64 *vsr, int *ldvsr, float *rconde, float *rcondv, npy_complex64 *work, int *lwork, float *rwork, int *iwork, int *liwork, int *bwork, int *info); +void BLAS_FUNC(cggev)(char *jobvl, char *jobvr, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(cggevx)(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, int *ilo, int *ihi, float *lscale, float *rscale, float *abnrm, float *bbnrm, float *rconde, float *rcondv, npy_complex64 *work, int *lwork, float *rwork, int *iwork, int *bwork, int *info); +void BLAS_FUNC(cggglm)(int *n, int *m, int *p, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *d, npy_complex64 *x, npy_complex64 *y, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgghrd)(char *compq, char *compz, int *n, int *ilo, int *ihi, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *q, int *ldq, npy_complex64 *z, int *ldz, int *info); +void BLAS_FUNC(cgglse)(int *m, int *n, int *p, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *c, npy_complex64 *d, npy_complex64 *x, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cggqrf)(int *n, int *m, int *p, npy_complex64 *a, int *lda, npy_complex64 *taua, npy_complex64 *b, int *ldb, npy_complex64 *taub, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cggrqf)(int *m, int *p, int *n, npy_complex64 *a, int *lda, npy_complex64 *taua, npy_complex64 *b, int *ldb, npy_complex64 *taub, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cgtcon)(char *norm, int *n, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *du2, int *ipiv, float *anorm, float *rcond, npy_complex64 *work, int *info); +void BLAS_FUNC(cgtrfs)(char *trans, int *n, int *nrhs, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *dlf, npy_complex64 *df, npy_complex64 *duf, npy_complex64 *du2, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cgtsv)(int *n, int *nrhs, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cgtsvx)(char *fact, char *trans, int *n, int *nrhs, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *dlf, npy_complex64 *df, npy_complex64 *duf, npy_complex64 *du2, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cgttrf)(int *n, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *du2, int *ipiv, int *info); +void BLAS_FUNC(cgttrs)(char *trans, int *n, int *nrhs, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *du2, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cgtts2)(int *itrans, int *n, int *nrhs, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *du2, int *ipiv, npy_complex64 *b, int *ldb); +void BLAS_FUNC(chbev)(char *jobz, char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(chbevd)(char *jobz, char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, float *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(chbevx)(char *jobz, char *range, char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, npy_complex64 *q, int *ldq, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, float *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(chbgst)(char *vect, char *uplo, int *n, int *ka, int *kb, npy_complex64 *ab, int *ldab, npy_complex64 *bb, int *ldbb, npy_complex64 *x, int *ldx, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(chbgv)(char *jobz, char *uplo, int *n, int *ka, int *kb, npy_complex64 *ab, int *ldab, npy_complex64 *bb, int *ldbb, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(chbgvd)(char *jobz, char *uplo, int *n, int *ka, int *kb, npy_complex64 *ab, int *ldab, npy_complex64 *bb, int *ldbb, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, float *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(chbgvx)(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, npy_complex64 *ab, int *ldab, npy_complex64 *bb, int *ldbb, npy_complex64 *q, int *ldq, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, float *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(chbtrd)(char *vect, char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, float *d, float *e, npy_complex64 *q, int *ldq, npy_complex64 *work, int *info); +void BLAS_FUNC(checon)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, float *anorm, float *rcond, npy_complex64 *work, int *info); +void BLAS_FUNC(cheequb)(char *uplo, int *n, npy_complex64 *a, int *lda, float *s, float *scond, float *amax, npy_complex64 *work, int *info); +void BLAS_FUNC(cheev)(char *jobz, char *uplo, int *n, npy_complex64 *a, int *lda, float *w, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(cheevd)(char *jobz, char *uplo, int *n, npy_complex64 *a, int *lda, float *w, npy_complex64 *work, int *lwork, float *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(cheevr)(char *jobz, char *range, char *uplo, int *n, npy_complex64 *a, int *lda, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, npy_complex64 *z, int *ldz, int *isuppz, npy_complex64 *work, int *lwork, float *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(cheevx)(char *jobz, char *range, char *uplo, int *n, npy_complex64 *a, int *lda, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, float *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(chegs2)(int *itype, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(chegst)(int *itype, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(chegv)(int *itype, char *jobz, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, float *w, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(chegvd)(int *itype, char *jobz, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, float *w, npy_complex64 *work, int *lwork, float *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(chegvx)(int *itype, char *jobz, char *range, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, float *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(cherfs)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(chesv)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(chesvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(cheswapr)(char *uplo, int *n, npy_complex64 *a, int *lda, int *i1, int *i2); +void BLAS_FUNC(chetd2)(char *uplo, int *n, npy_complex64 *a, int *lda, float *d, float *e, npy_complex64 *tau, int *info); +void BLAS_FUNC(chetf2)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(chetrd)(char *uplo, int *n, npy_complex64 *a, int *lda, float *d, float *e, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(chetrf)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(chetri)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *info); +void BLAS_FUNC(chetri2)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(chetri2x)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *nb, int *info); +void BLAS_FUNC(chetrs)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(chetrs2)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *work, int *info); +void BLAS_FUNC(chfrk)(char *transr, char *uplo, char *trans, int *n, int *k, float *alpha, npy_complex64 *a, int *lda, float *beta, npy_complex64 *c); +void BLAS_FUNC(chgeqz)(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, npy_complex64 *h, int *ldh, npy_complex64 *t, int *ldt, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *q, int *ldq, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, float *rwork, int *info); +char BLAS_FUNC(chla_transtype)(int *trans); +void BLAS_FUNC(chpcon)(char *uplo, int *n, npy_complex64 *ap, int *ipiv, float *anorm, float *rcond, npy_complex64 *work, int *info); +void BLAS_FUNC(chpev)(char *jobz, char *uplo, int *n, npy_complex64 *ap, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(chpevd)(char *jobz, char *uplo, int *n, npy_complex64 *ap, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, float *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(chpevx)(char *jobz, char *range, char *uplo, int *n, npy_complex64 *ap, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, float *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(chpgst)(int *itype, char *uplo, int *n, npy_complex64 *ap, npy_complex64 *bp, int *info); +void BLAS_FUNC(chpgv)(int *itype, char *jobz, char *uplo, int *n, npy_complex64 *ap, npy_complex64 *bp, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(chpgvd)(int *itype, char *jobz, char *uplo, int *n, npy_complex64 *ap, npy_complex64 *bp, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, float *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(chpgvx)(int *itype, char *jobz, char *range, char *uplo, int *n, npy_complex64 *ap, npy_complex64 *bp, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, npy_complex64 *z, int *ldz, npy_complex64 *work, float *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(chprfs)(char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(chpsv)(char *uplo, int *n, int *nrhs, npy_complex64 *ap, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(chpsvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(chptrd)(char *uplo, int *n, npy_complex64 *ap, float *d, float *e, npy_complex64 *tau, int *info); +void BLAS_FUNC(chptrf)(char *uplo, int *n, npy_complex64 *ap, int *ipiv, int *info); +void BLAS_FUNC(chptri)(char *uplo, int *n, npy_complex64 *ap, int *ipiv, npy_complex64 *work, int *info); +void BLAS_FUNC(chptrs)(char *uplo, int *n, int *nrhs, npy_complex64 *ap, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(chsein)(char *side, char *eigsrc, char *initv, int *select, int *n, npy_complex64 *h, int *ldh, npy_complex64 *w, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, int *mm, int *m, npy_complex64 *work, float *rwork, int *ifaill, int *ifailr, int *info); +void BLAS_FUNC(chseqr)(char *job, char *compz, int *n, int *ilo, int *ihi, npy_complex64 *h, int *ldh, npy_complex64 *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(clabrd)(int *m, int *n, int *nb, npy_complex64 *a, int *lda, float *d, float *e, npy_complex64 *tauq, npy_complex64 *taup, npy_complex64 *x, int *ldx, npy_complex64 *y, int *ldy); +void BLAS_FUNC(clacgv)(int *n, npy_complex64 *x, int *incx); +void BLAS_FUNC(clacn2)(int *n, npy_complex64 *v, npy_complex64 *x, float *est, int *kase, int *isave); +void BLAS_FUNC(clacon)(int *n, npy_complex64 *v, npy_complex64 *x, float *est, int *kase); +void BLAS_FUNC(clacp2)(char *uplo, int *m, int *n, float *a, int *lda, npy_complex64 *b, int *ldb); +void BLAS_FUNC(clacpy)(char *uplo, int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb); +void BLAS_FUNC(clacrm)(int *m, int *n, npy_complex64 *a, int *lda, float *b, int *ldb, npy_complex64 *c, int *ldc, float *rwork); +void BLAS_FUNC(clacrt)(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy, npy_complex64 *c, npy_complex64 *s); +void F_FUNC(cladivwrp,CLADIVWRP)(npy_complex64 *out, npy_complex64 *x, npy_complex64 *y); +void BLAS_FUNC(claed0)(int *qsiz, int *n, float *d, float *e, npy_complex64 *q, int *ldq, npy_complex64 *qstore, int *ldqs, float *rwork, int *iwork, int *info); +void BLAS_FUNC(claed7)(int *n, int *cutpnt, int *qsiz, int *tlvls, int *curlvl, int *curpbm, float *d, npy_complex64 *q, int *ldq, float *rho, int *indxq, float *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, float *givnum, npy_complex64 *work, float *rwork, int *iwork, int *info); +void BLAS_FUNC(claed8)(int *k, int *n, int *qsiz, npy_complex64 *q, int *ldq, float *d, float *rho, int *cutpnt, float *z, float *dlamda, npy_complex64 *q2, int *ldq2, float *w, int *indxp, int *indx, int *indxq, int *perm, int *givptr, int *givcol, float *givnum, int *info); +void BLAS_FUNC(claein)(int *rightv, int *noinit, int *n, npy_complex64 *h, int *ldh, npy_complex64 *w, npy_complex64 *v, npy_complex64 *b, int *ldb, float *rwork, float *eps3, float *smlnum, int *info); +void BLAS_FUNC(claesy)(npy_complex64 *a, npy_complex64 *b, npy_complex64 *c, npy_complex64 *rt1, npy_complex64 *rt2, npy_complex64 *evscal, npy_complex64 *cs1, npy_complex64 *sn1); +void BLAS_FUNC(claev2)(npy_complex64 *a, npy_complex64 *b, npy_complex64 *c, float *rt1, float *rt2, float *cs1, npy_complex64 *sn1); +void BLAS_FUNC(clag2z)(int *m, int *n, npy_complex64 *sa, int *ldsa, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(clags2)(int *upper, float *a1, npy_complex64 *a2, float *a3, float *b1, npy_complex64 *b2, float *b3, float *csu, npy_complex64 *snu, float *csv, npy_complex64 *snv, float *csq, npy_complex64 *snq); +void BLAS_FUNC(clagtm)(char *trans, int *n, int *nrhs, float *alpha, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *x, int *ldx, float *beta, npy_complex64 *b, int *ldb); +void BLAS_FUNC(clahef)(char *uplo, int *n, int *nb, int *kb, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *w, int *ldw, int *info); +void BLAS_FUNC(clahqr)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, npy_complex64 *h, int *ldh, npy_complex64 *w, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, int *info); +void BLAS_FUNC(clahr2)(int *n, int *k, int *nb, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *t, int *ldt, npy_complex64 *y, int *ldy); +void BLAS_FUNC(claic1)(int *job, int *j, npy_complex64 *x, float *sest, npy_complex64 *w, npy_complex64 *gamma, float *sestpr, npy_complex64 *s, npy_complex64 *c); +void BLAS_FUNC(clals0)(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, npy_complex64 *b, int *ldb, npy_complex64 *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, float *givnum, int *ldgnum, float *poles, float *difl, float *difr, float *z, int *k, float *c, float *s, float *rwork, int *info); +void BLAS_FUNC(clalsa)(int *icompq, int *smlsiz, int *n, int *nrhs, npy_complex64 *b, int *ldb, npy_complex64 *bx, int *ldbx, float *u, int *ldu, float *vt, int *k, float *difl, float *difr, float *z, float *poles, int *givptr, int *givcol, int *ldgcol, int *perm, float *givnum, float *c, float *s, float *rwork, int *iwork, int *info); +void BLAS_FUNC(clalsd)(char *uplo, int *smlsiz, int *n, int *nrhs, float *d, float *e, npy_complex64 *b, int *ldb, float *rcond, int *rank, npy_complex64 *work, float *rwork, int *iwork, int *info); +float BLAS_FUNC(clangb)(char *norm, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, float *work); +float BLAS_FUNC(clange)(char *norm, int *m, int *n, npy_complex64 *a, int *lda, float *work); +float BLAS_FUNC(clangt)(char *norm, int *n, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du); +float BLAS_FUNC(clanhb)(char *norm, char *uplo, int *n, int *k, npy_complex64 *ab, int *ldab, float *work); +float BLAS_FUNC(clanhe)(char *norm, char *uplo, int *n, npy_complex64 *a, int *lda, float *work); +float BLAS_FUNC(clanhf)(char *norm, char *transr, char *uplo, int *n, npy_complex64 *a, float *work); +float BLAS_FUNC(clanhp)(char *norm, char *uplo, int *n, npy_complex64 *ap, float *work); +float BLAS_FUNC(clanhs)(char *norm, int *n, npy_complex64 *a, int *lda, float *work); +float BLAS_FUNC(clanht)(char *norm, int *n, float *d, npy_complex64 *e); +float BLAS_FUNC(clansb)(char *norm, char *uplo, int *n, int *k, npy_complex64 *ab, int *ldab, float *work); +float BLAS_FUNC(clansp)(char *norm, char *uplo, int *n, npy_complex64 *ap, float *work); +float BLAS_FUNC(clansy)(char *norm, char *uplo, int *n, npy_complex64 *a, int *lda, float *work); +float BLAS_FUNC(clantb)(char *norm, char *uplo, char *diag, int *n, int *k, npy_complex64 *ab, int *ldab, float *work); +float BLAS_FUNC(clantp)(char *norm, char *uplo, char *diag, int *n, npy_complex64 *ap, float *work); +float BLAS_FUNC(clantr)(char *norm, char *uplo, char *diag, int *m, int *n, npy_complex64 *a, int *lda, float *work); +void BLAS_FUNC(clapll)(int *n, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, float *ssmin); +void BLAS_FUNC(clapmr)(int *forwrd, int *m, int *n, npy_complex64 *x, int *ldx, int *k); +void BLAS_FUNC(clapmt)(int *forwrd, int *m, int *n, npy_complex64 *x, int *ldx, int *k); +void BLAS_FUNC(claqgb)(int *m, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, float *r, float *c, float *rowcnd, float *colcnd, float *amax, char *equed); +void BLAS_FUNC(claqge)(int *m, int *n, npy_complex64 *a, int *lda, float *r, float *c, float *rowcnd, float *colcnd, float *amax, char *equed); +void BLAS_FUNC(claqhb)(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, float *s, float *scond, float *amax, char *equed); +void BLAS_FUNC(claqhe)(char *uplo, int *n, npy_complex64 *a, int *lda, float *s, float *scond, float *amax, char *equed); +void BLAS_FUNC(claqhp)(char *uplo, int *n, npy_complex64 *ap, float *s, float *scond, float *amax, char *equed); +void BLAS_FUNC(claqp2)(int *m, int *n, int *offset, npy_complex64 *a, int *lda, int *jpvt, npy_complex64 *tau, float *vn1, float *vn2, npy_complex64 *work); +void BLAS_FUNC(claqps)(int *m, int *n, int *offset, int *nb, int *kb, npy_complex64 *a, int *lda, int *jpvt, npy_complex64 *tau, float *vn1, float *vn2, npy_complex64 *auxv, npy_complex64 *f, int *ldf); +void BLAS_FUNC(claqr0)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, npy_complex64 *h, int *ldh, npy_complex64 *w, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(claqr1)(int *n, npy_complex64 *h, int *ldh, npy_complex64 *s1, npy_complex64 *s2, npy_complex64 *v); +void BLAS_FUNC(claqr2)(int *wantt, int *wantz, int *n, int *ktop, int *kbot, int *nw, npy_complex64 *h, int *ldh, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, int *ns, int *nd, npy_complex64 *sh, npy_complex64 *v, int *ldv, int *nh, npy_complex64 *t, int *ldt, int *nv, npy_complex64 *wv, int *ldwv, npy_complex64 *work, int *lwork); +void BLAS_FUNC(claqr3)(int *wantt, int *wantz, int *n, int *ktop, int *kbot, int *nw, npy_complex64 *h, int *ldh, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, int *ns, int *nd, npy_complex64 *sh, npy_complex64 *v, int *ldv, int *nh, npy_complex64 *t, int *ldt, int *nv, npy_complex64 *wv, int *ldwv, npy_complex64 *work, int *lwork); +void BLAS_FUNC(claqr4)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, npy_complex64 *h, int *ldh, npy_complex64 *w, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(claqr5)(int *wantt, int *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, npy_complex64 *s, npy_complex64 *h, int *ldh, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, npy_complex64 *v, int *ldv, npy_complex64 *u, int *ldu, int *nv, npy_complex64 *wv, int *ldwv, int *nh, npy_complex64 *wh, int *ldwh); +void BLAS_FUNC(claqsb)(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, float *s, float *scond, float *amax, char *equed); +void BLAS_FUNC(claqsp)(char *uplo, int *n, npy_complex64 *ap, float *s, float *scond, float *amax, char *equed); +void BLAS_FUNC(claqsy)(char *uplo, int *n, npy_complex64 *a, int *lda, float *s, float *scond, float *amax, char *equed); +void BLAS_FUNC(clar1v)(int *n, int *b1, int *bn, float *lambda_, float *d, float *l, float *ld, float *lld, float *pivmin, float *gaptol, npy_complex64 *z, int *wantnc, int *negcnt, float *ztz, float *mingma, int *r, int *isuppz, float *nrminv, float *resid, float *rqcorr, float *work); +void BLAS_FUNC(clar2v)(int *n, npy_complex64 *x, npy_complex64 *y, npy_complex64 *z, int *incx, float *c, npy_complex64 *s, int *incc); +void BLAS_FUNC(clarcm)(int *m, int *n, float *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *c, int *ldc, float *rwork); +void BLAS_FUNC(clarf)(char *side, int *m, int *n, npy_complex64 *v, int *incv, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work); +void BLAS_FUNC(clarfb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, npy_complex64 *v, int *ldv, npy_complex64 *t, int *ldt, npy_complex64 *c, int *ldc, npy_complex64 *work, int *ldwork); +void BLAS_FUNC(clarfg)(int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *tau); +void BLAS_FUNC(clarfgp)(int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *tau); +void BLAS_FUNC(clarft)(char *direct, char *storev, int *n, int *k, npy_complex64 *v, int *ldv, npy_complex64 *tau, npy_complex64 *t, int *ldt); +void BLAS_FUNC(clarfx)(char *side, int *m, int *n, npy_complex64 *v, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work); +void BLAS_FUNC(clargv)(int *n, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, float *c, int *incc); +void BLAS_FUNC(clarnv)(int *idist, int *iseed, int *n, npy_complex64 *x); +void BLAS_FUNC(clarrv)(int *n, float *vl, float *vu, float *d, float *l, float *pivmin, int *isplit, int *m, int *dol, int *dou, float *minrgp, float *rtol1, float *rtol2, float *w, float *werr, float *wgap, int *iblock, int *indexw, float *gers, npy_complex64 *z, int *ldz, int *isuppz, float *work, int *iwork, int *info); +void BLAS_FUNC(clartg)(npy_complex64 *f, npy_complex64 *g, float *cs, npy_complex64 *sn, npy_complex64 *r); +void BLAS_FUNC(clartv)(int *n, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, float *c, npy_complex64 *s, int *incc); +void BLAS_FUNC(clarz)(char *side, int *m, int *n, int *l, npy_complex64 *v, int *incv, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work); +void BLAS_FUNC(clarzb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, npy_complex64 *v, int *ldv, npy_complex64 *t, int *ldt, npy_complex64 *c, int *ldc, npy_complex64 *work, int *ldwork); +void BLAS_FUNC(clarzt)(char *direct, char *storev, int *n, int *k, npy_complex64 *v, int *ldv, npy_complex64 *tau, npy_complex64 *t, int *ldt); +void BLAS_FUNC(clascl)(char *type_bn, int *kl, int *ku, float *cfrom, float *cto, int *m, int *n, npy_complex64 *a, int *lda, int *info); +void BLAS_FUNC(claset)(char *uplo, int *m, int *n, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *a, int *lda); +void BLAS_FUNC(clasr)(char *side, char *pivot, char *direct, int *m, int *n, float *c, float *s, npy_complex64 *a, int *lda); +void BLAS_FUNC(classq)(int *n, npy_complex64 *x, int *incx, float *scale, float *sumsq); +void BLAS_FUNC(claswp)(int *n, npy_complex64 *a, int *lda, int *k1, int *k2, int *ipiv, int *incx); +void BLAS_FUNC(clasyf)(char *uplo, int *n, int *nb, int *kb, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *w, int *ldw, int *info); +void BLAS_FUNC(clatbs)(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, npy_complex64 *ab, int *ldab, npy_complex64 *x, float *scale, float *cnorm, int *info); +void BLAS_FUNC(clatdf)(int *ijob, int *n, npy_complex64 *z, int *ldz, npy_complex64 *rhs, float *rdsum, float *rdscal, int *ipiv, int *jpiv); +void BLAS_FUNC(clatps)(char *uplo, char *trans, char *diag, char *normin, int *n, npy_complex64 *ap, npy_complex64 *x, float *scale, float *cnorm, int *info); +void BLAS_FUNC(clatrd)(char *uplo, int *n, int *nb, npy_complex64 *a, int *lda, float *e, npy_complex64 *tau, npy_complex64 *w, int *ldw); +void BLAS_FUNC(clatrs)(char *uplo, char *trans, char *diag, char *normin, int *n, npy_complex64 *a, int *lda, npy_complex64 *x, float *scale, float *cnorm, int *info); +void BLAS_FUNC(clatrz)(int *m, int *n, int *l, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work); +void BLAS_FUNC(clauu2)(char *uplo, int *n, npy_complex64 *a, int *lda, int *info); +void BLAS_FUNC(clauum)(char *uplo, int *n, npy_complex64 *a, int *lda, int *info); +void BLAS_FUNC(cpbcon)(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, float *anorm, float *rcond, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cpbequ)(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, float *s, float *scond, float *amax, int *info); +void BLAS_FUNC(cpbrfs)(char *uplo, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *afb, int *ldafb, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cpbstf)(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, int *info); +void BLAS_FUNC(cpbsv)(char *uplo, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cpbsvx)(char *fact, char *uplo, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *afb, int *ldafb, char *equed, float *s, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cpbtf2)(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, int *info); +void BLAS_FUNC(cpbtrf)(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, int *info); +void BLAS_FUNC(cpbtrs)(char *uplo, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cpftrf)(char *transr, char *uplo, int *n, npy_complex64 *a, int *info); +void BLAS_FUNC(cpftri)(char *transr, char *uplo, int *n, npy_complex64 *a, int *info); +void BLAS_FUNC(cpftrs)(char *transr, char *uplo, int *n, int *nrhs, npy_complex64 *a, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cpocon)(char *uplo, int *n, npy_complex64 *a, int *lda, float *anorm, float *rcond, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cpoequ)(int *n, npy_complex64 *a, int *lda, float *s, float *scond, float *amax, int *info); +void BLAS_FUNC(cpoequb)(int *n, npy_complex64 *a, int *lda, float *s, float *scond, float *amax, int *info); +void BLAS_FUNC(cporfs)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cposv)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cposvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, char *equed, float *s, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cpotf2)(char *uplo, int *n, npy_complex64 *a, int *lda, int *info); +void BLAS_FUNC(cpotrf)(char *uplo, int *n, npy_complex64 *a, int *lda, int *info); +void BLAS_FUNC(cpotri)(char *uplo, int *n, npy_complex64 *a, int *lda, int *info); +void BLAS_FUNC(cpotrs)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cppcon)(char *uplo, int *n, npy_complex64 *ap, float *anorm, float *rcond, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cppequ)(char *uplo, int *n, npy_complex64 *ap, float *s, float *scond, float *amax, int *info); +void BLAS_FUNC(cpprfs)(char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cppsv)(char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cppsvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, char *equed, float *s, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cpptrf)(char *uplo, int *n, npy_complex64 *ap, int *info); +void BLAS_FUNC(cpptri)(char *uplo, int *n, npy_complex64 *ap, int *info); +void BLAS_FUNC(cpptrs)(char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cpstf2)(char *uplo, int *n, npy_complex64 *a, int *lda, int *piv, int *rank, float *tol, float *work, int *info); +void BLAS_FUNC(cpstrf)(char *uplo, int *n, npy_complex64 *a, int *lda, int *piv, int *rank, float *tol, float *work, int *info); +void BLAS_FUNC(cptcon)(int *n, float *d, npy_complex64 *e, float *anorm, float *rcond, float *rwork, int *info); +void BLAS_FUNC(cpteqr)(char *compz, int *n, float *d, float *e, npy_complex64 *z, int *ldz, float *work, int *info); +void BLAS_FUNC(cptrfs)(char *uplo, int *n, int *nrhs, float *d, npy_complex64 *e, float *df, npy_complex64 *ef, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cptsv)(int *n, int *nrhs, float *d, npy_complex64 *e, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cptsvx)(char *fact, int *n, int *nrhs, float *d, npy_complex64 *e, float *df, npy_complex64 *ef, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cpttrf)(int *n, float *d, npy_complex64 *e, int *info); +void BLAS_FUNC(cpttrs)(char *uplo, int *n, int *nrhs, float *d, npy_complex64 *e, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cptts2)(int *iuplo, int *n, int *nrhs, float *d, npy_complex64 *e, npy_complex64 *b, int *ldb); +void BLAS_FUNC(crot)(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy, float *c, npy_complex64 *s); +void BLAS_FUNC(cspcon)(char *uplo, int *n, npy_complex64 *ap, int *ipiv, float *anorm, float *rcond, npy_complex64 *work, int *info); +void BLAS_FUNC(cspmv)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *ap, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy); +void BLAS_FUNC(cspr)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *ap); +void BLAS_FUNC(csprfs)(char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(cspsv)(char *uplo, int *n, int *nrhs, npy_complex64 *ap, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(cspsvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(csptrf)(char *uplo, int *n, npy_complex64 *ap, int *ipiv, int *info); +void BLAS_FUNC(csptri)(char *uplo, int *n, npy_complex64 *ap, int *ipiv, npy_complex64 *work, int *info); +void BLAS_FUNC(csptrs)(char *uplo, int *n, int *nrhs, npy_complex64 *ap, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(csrscl)(int *n, float *sa, npy_complex64 *sx, int *incx); +void BLAS_FUNC(cstedc)(char *compz, int *n, float *d, float *e, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, float *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(cstegr)(char *jobz, char *range, int *n, float *d, float *e, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, npy_complex64 *z, int *ldz, int *isuppz, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(cstein)(int *n, float *d, float *e, int *m, float *w, int *iblock, int *isplit, npy_complex64 *z, int *ldz, float *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(cstemr)(char *jobz, char *range, int *n, float *d, float *e, float *vl, float *vu, int *il, int *iu, int *m, float *w, npy_complex64 *z, int *ldz, int *nzc, int *isuppz, int *tryrac, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(csteqr)(char *compz, int *n, float *d, float *e, npy_complex64 *z, int *ldz, float *work, int *info); +void BLAS_FUNC(csycon)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, float *anorm, float *rcond, npy_complex64 *work, int *info); +void BLAS_FUNC(csyconv)(char *uplo, char *way, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *info); +void BLAS_FUNC(csyequb)(char *uplo, int *n, npy_complex64 *a, int *lda, float *s, float *scond, float *amax, npy_complex64 *work, int *info); +void BLAS_FUNC(csymv)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy); +void BLAS_FUNC(csyr)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *a, int *lda); +void BLAS_FUNC(csyrfs)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(csysv)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(csysvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *rcond, float *ferr, float *berr, npy_complex64 *work, int *lwork, float *rwork, int *info); +void BLAS_FUNC(csyswapr)(char *uplo, int *n, npy_complex64 *a, int *lda, int *i1, int *i2); +void BLAS_FUNC(csytf2)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(csytrf)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(csytri)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *info); +void BLAS_FUNC(csytri2)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(csytri2x)(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *nb, int *info); +void BLAS_FUNC(csytrs)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(csytrs2)(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *work, int *info); +void BLAS_FUNC(ctbcon)(char *norm, char *uplo, char *diag, int *n, int *kd, npy_complex64 *ab, int *ldab, float *rcond, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(ctbrfs)(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(ctbtrs)(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(ctfsm)(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, npy_complex64 *b, int *ldb); +void BLAS_FUNC(ctftri)(char *transr, char *uplo, char *diag, int *n, npy_complex64 *a, int *info); +void BLAS_FUNC(ctfttp)(char *transr, char *uplo, int *n, npy_complex64 *arf, npy_complex64 *ap, int *info); +void BLAS_FUNC(ctfttr)(char *transr, char *uplo, int *n, npy_complex64 *arf, npy_complex64 *a, int *lda, int *info); +void BLAS_FUNC(ctgevc)(char *side, char *howmny, int *select, int *n, npy_complex64 *s, int *lds, npy_complex64 *p, int *ldp, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, int *mm, int *m, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(ctgex2)(int *wantq, int *wantz, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *q, int *ldq, npy_complex64 *z, int *ldz, int *j1, int *info); +void BLAS_FUNC(ctgexc)(int *wantq, int *wantz, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *q, int *ldq, npy_complex64 *z, int *ldz, int *ifst, int *ilst, int *info); +void BLAS_FUNC(ctgsen)(int *ijob, int *wantq, int *wantz, int *select, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *q, int *ldq, npy_complex64 *z, int *ldz, int *m, float *pl, float *pr, float *dif, npy_complex64 *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(ctgsja)(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, float *tola, float *tolb, float *alpha, float *beta, npy_complex64 *u, int *ldu, npy_complex64 *v, int *ldv, npy_complex64 *q, int *ldq, npy_complex64 *work, int *ncycle, int *info); +void BLAS_FUNC(ctgsna)(char *job, char *howmny, int *select, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, float *s, float *dif, int *mm, int *m, npy_complex64 *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(ctgsy2)(char *trans, int *ijob, int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *c, int *ldc, npy_complex64 *d, int *ldd, npy_complex64 *e, int *lde, npy_complex64 *f, int *ldf, float *scale, float *rdsum, float *rdscal, int *info); +void BLAS_FUNC(ctgsyl)(char *trans, int *ijob, int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *c, int *ldc, npy_complex64 *d, int *ldd, npy_complex64 *e, int *lde, npy_complex64 *f, int *ldf, float *scale, float *dif, npy_complex64 *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(ctpcon)(char *norm, char *uplo, char *diag, int *n, npy_complex64 *ap, float *rcond, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(ctpmqrt)(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, npy_complex64 *v, int *ldv, npy_complex64 *t, int *ldt, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *work, int *info); +void BLAS_FUNC(ctpqrt)(int *m, int *n, int *l, int *nb, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *t, int *ldt, npy_complex64 *work, int *info); +void BLAS_FUNC(ctpqrt2)(int *m, int *n, int *l, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *t, int *ldt, int *info); +void BLAS_FUNC(ctprfb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, npy_complex64 *v, int *ldv, npy_complex64 *t, int *ldt, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *work, int *ldwork); +void BLAS_FUNC(ctprfs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(ctptri)(char *uplo, char *diag, int *n, npy_complex64 *ap, int *info); +void BLAS_FUNC(ctptrs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(ctpttf)(char *transr, char *uplo, int *n, npy_complex64 *ap, npy_complex64 *arf, int *info); +void BLAS_FUNC(ctpttr)(char *uplo, int *n, npy_complex64 *ap, npy_complex64 *a, int *lda, int *info); +void BLAS_FUNC(ctrcon)(char *norm, char *uplo, char *diag, int *n, npy_complex64 *a, int *lda, float *rcond, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(ctrevc)(char *side, char *howmny, int *select, int *n, npy_complex64 *t, int *ldt, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, int *mm, int *m, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(ctrexc)(char *compq, int *n, npy_complex64 *t, int *ldt, npy_complex64 *q, int *ldq, int *ifst, int *ilst, int *info); +void BLAS_FUNC(ctrrfs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, float *ferr, float *berr, npy_complex64 *work, float *rwork, int *info); +void BLAS_FUNC(ctrsen)(char *job, char *compq, int *select, int *n, npy_complex64 *t, int *ldt, npy_complex64 *q, int *ldq, npy_complex64 *w, int *m, float *s, float *sep, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(ctrsna)(char *job, char *howmny, int *select, int *n, npy_complex64 *t, int *ldt, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, float *s, float *sep, int *mm, int *m, npy_complex64 *work, int *ldwork, float *rwork, int *info); +void BLAS_FUNC(ctrsyl)(char *trana, char *tranb, int *isgn, int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *c, int *ldc, float *scale, int *info); +void BLAS_FUNC(ctrti2)(char *uplo, char *diag, int *n, npy_complex64 *a, int *lda, int *info); +void BLAS_FUNC(ctrtri)(char *uplo, char *diag, int *n, npy_complex64 *a, int *lda, int *info); +void BLAS_FUNC(ctrtrs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *info); +void BLAS_FUNC(ctrttf)(char *transr, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *arf, int *info); +void BLAS_FUNC(ctrttp)(char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *ap, int *info); +void BLAS_FUNC(ctzrzf)(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cunbdb)(char *trans, char *signs, int *m, int *p, int *q, npy_complex64 *x11, int *ldx11, npy_complex64 *x12, int *ldx12, npy_complex64 *x21, int *ldx21, npy_complex64 *x22, int *ldx22, float *theta, float *phi, npy_complex64 *taup1, npy_complex64 *taup2, npy_complex64 *tauq1, npy_complex64 *tauq2, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cuncsd)(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, npy_complex64 *x11, int *ldx11, npy_complex64 *x12, int *ldx12, npy_complex64 *x21, int *ldx21, npy_complex64 *x22, int *ldx22, float *theta, npy_complex64 *u1, int *ldu1, npy_complex64 *u2, int *ldu2, npy_complex64 *v1t, int *ldv1t, npy_complex64 *v2t, int *ldv2t, npy_complex64 *work, int *lwork, float *rwork, int *lrwork, int *iwork, int *info); +void BLAS_FUNC(cung2l)(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info); +void BLAS_FUNC(cung2r)(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info); +void BLAS_FUNC(cungbr)(char *vect, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cunghr)(int *n, int *ilo, int *ihi, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cungl2)(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info); +void BLAS_FUNC(cunglq)(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cungql)(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cungqr)(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cungr2)(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info); +void BLAS_FUNC(cungrq)(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cungtr)(char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cunm2l)(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info); +void BLAS_FUNC(cunm2r)(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info); +void BLAS_FUNC(cunmbr)(char *vect, char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cunmhr)(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cunml2)(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info); +void BLAS_FUNC(cunmlq)(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cunmql)(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cunmqr)(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cunmr2)(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info); +void BLAS_FUNC(cunmr3)(char *side, char *trans, int *m, int *n, int *k, int *l, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info); +void BLAS_FUNC(cunmrq)(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cunmrz)(char *side, char *trans, int *m, int *n, int *k, int *l, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cunmtr)(char *side, char *uplo, char *trans, int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info); +void BLAS_FUNC(cupgtr)(char *uplo, int *n, npy_complex64 *ap, npy_complex64 *tau, npy_complex64 *q, int *ldq, npy_complex64 *work, int *info); +void BLAS_FUNC(cupmtr)(char *side, char *uplo, char *trans, int *m, int *n, npy_complex64 *ap, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info); +void BLAS_FUNC(dbbcsd)(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, double *theta, double *phi, double *u1, int *ldu1, double *u2, int *ldu2, double *v1t, int *ldv1t, double *v2t, int *ldv2t, double *b11d, double *b11e, double *b12d, double *b12e, double *b21d, double *b21e, double *b22d, double *b22e, double *work, int *lwork, int *info); +void BLAS_FUNC(dbdsdc)(char *uplo, char *compq, int *n, double *d, double *e, double *u, int *ldu, double *vt, int *ldvt, double *q, int *iq, double *work, int *iwork, int *info); +void BLAS_FUNC(dbdsqr)(char *uplo, int *n, int *ncvt, int *nru, int *ncc, double *d, double *e, double *vt, int *ldvt, double *u, int *ldu, double *c, int *ldc, double *work, int *info); +void BLAS_FUNC(ddisna)(char *job, int *m, int *n, double *d, double *sep, int *info); +void BLAS_FUNC(dgbbrd)(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, double *ab, int *ldab, double *d, double *e, double *q, int *ldq, double *pt, int *ldpt, double *c, int *ldc, double *work, int *info); +void BLAS_FUNC(dgbcon)(char *norm, int *n, int *kl, int *ku, double *ab, int *ldab, int *ipiv, double *anorm, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dgbequ)(int *m, int *n, int *kl, int *ku, double *ab, int *ldab, double *r, double *c, double *rowcnd, double *colcnd, double *amax, int *info); +void BLAS_FUNC(dgbequb)(int *m, int *n, int *kl, int *ku, double *ab, int *ldab, double *r, double *c, double *rowcnd, double *colcnd, double *amax, int *info); +void BLAS_FUNC(dgbrfs)(char *trans, int *n, int *kl, int *ku, int *nrhs, double *ab, int *ldab, double *afb, int *ldafb, int *ipiv, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dgbsv)(int *n, int *kl, int *ku, int *nrhs, double *ab, int *ldab, int *ipiv, double *b, int *ldb, int *info); +void BLAS_FUNC(dgbsvx)(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, double *ab, int *ldab, double *afb, int *ldafb, int *ipiv, char *equed, double *r, double *c, double *b, int *ldb, double *x, int *ldx, double *rcond, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dgbtf2)(int *m, int *n, int *kl, int *ku, double *ab, int *ldab, int *ipiv, int *info); +void BLAS_FUNC(dgbtrf)(int *m, int *n, int *kl, int *ku, double *ab, int *ldab, int *ipiv, int *info); +void BLAS_FUNC(dgbtrs)(char *trans, int *n, int *kl, int *ku, int *nrhs, double *ab, int *ldab, int *ipiv, double *b, int *ldb, int *info); +void BLAS_FUNC(dgebak)(char *job, char *side, int *n, int *ilo, int *ihi, double *scale, int *m, double *v, int *ldv, int *info); +void BLAS_FUNC(dgebal)(char *job, int *n, double *a, int *lda, int *ilo, int *ihi, double *scale, int *info); +void BLAS_FUNC(dgebd2)(int *m, int *n, double *a, int *lda, double *d, double *e, double *tauq, double *taup, double *work, int *info); +void BLAS_FUNC(dgebrd)(int *m, int *n, double *a, int *lda, double *d, double *e, double *tauq, double *taup, double *work, int *lwork, int *info); +void BLAS_FUNC(dgecon)(char *norm, int *n, double *a, int *lda, double *anorm, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dgeequ)(int *m, int *n, double *a, int *lda, double *r, double *c, double *rowcnd, double *colcnd, double *amax, int *info); +void BLAS_FUNC(dgeequb)(int *m, int *n, double *a, int *lda, double *r, double *c, double *rowcnd, double *colcnd, double *amax, int *info); +void BLAS_FUNC(dgees)(char *jobvs, char *sort, _dselect2 *select, int *n, double *a, int *lda, int *sdim, double *wr, double *wi, double *vs, int *ldvs, double *work, int *lwork, int *bwork, int *info); +void BLAS_FUNC(dgeesx)(char *jobvs, char *sort, _dselect2 *select, char *sense, int *n, double *a, int *lda, int *sdim, double *wr, double *wi, double *vs, int *ldvs, double *rconde, double *rcondv, double *work, int *lwork, int *iwork, int *liwork, int *bwork, int *info); +void BLAS_FUNC(dgeev)(char *jobvl, char *jobvr, int *n, double *a, int *lda, double *wr, double *wi, double *vl, int *ldvl, double *vr, int *ldvr, double *work, int *lwork, int *info); +void BLAS_FUNC(dgeevx)(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, double *a, int *lda, double *wr, double *wi, double *vl, int *ldvl, double *vr, int *ldvr, int *ilo, int *ihi, double *scale, double *abnrm, double *rconde, double *rcondv, double *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(dgehd2)(int *n, int *ilo, int *ihi, double *a, int *lda, double *tau, double *work, int *info); +void BLAS_FUNC(dgehrd)(int *n, int *ilo, int *ihi, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dgejsv)(char *joba, char *jobu, char *jobv, char *jobr, char *jobt, char *jobp, int *m, int *n, double *a, int *lda, double *sva, double *u, int *ldu, double *v, int *ldv, double *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(dgelq2)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *info); +void BLAS_FUNC(dgelqf)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dgels)(char *trans, int *m, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, double *work, int *lwork, int *info); +void BLAS_FUNC(dgelsd)(int *m, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, double *s, double *rcond, int *rank, double *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(dgelss)(int *m, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, double *s, double *rcond, int *rank, double *work, int *lwork, int *info); +void BLAS_FUNC(dgelsy)(int *m, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, int *jpvt, double *rcond, int *rank, double *work, int *lwork, int *info); +void BLAS_FUNC(dgemqrt)(char *side, char *trans, int *m, int *n, int *k, int *nb, double *v, int *ldv, double *t, int *ldt, double *c, int *ldc, double *work, int *info); +void BLAS_FUNC(dgeql2)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *info); +void BLAS_FUNC(dgeqlf)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dgeqp3)(int *m, int *n, double *a, int *lda, int *jpvt, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dgeqr2)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *info); +void BLAS_FUNC(dgeqr2p)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *info); +void BLAS_FUNC(dgeqrf)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dgeqrfp)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dgeqrt)(int *m, int *n, int *nb, double *a, int *lda, double *t, int *ldt, double *work, int *info); +void BLAS_FUNC(dgeqrt2)(int *m, int *n, double *a, int *lda, double *t, int *ldt, int *info); +void BLAS_FUNC(dgeqrt3)(int *m, int *n, double *a, int *lda, double *t, int *ldt, int *info); +void BLAS_FUNC(dgerfs)(char *trans, int *n, int *nrhs, double *a, int *lda, double *af, int *ldaf, int *ipiv, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dgerq2)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *info); +void BLAS_FUNC(dgerqf)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dgesc2)(int *n, double *a, int *lda, double *rhs, int *ipiv, int *jpiv, double *scale); +void BLAS_FUNC(dgesdd)(char *jobz, int *m, int *n, double *a, int *lda, double *s, double *u, int *ldu, double *vt, int *ldvt, double *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(dgesv)(int *n, int *nrhs, double *a, int *lda, int *ipiv, double *b, int *ldb, int *info); +void BLAS_FUNC(dgesvd)(char *jobu, char *jobvt, int *m, int *n, double *a, int *lda, double *s, double *u, int *ldu, double *vt, int *ldvt, double *work, int *lwork, int *info); +void BLAS_FUNC(dgesvj)(char *joba, char *jobu, char *jobv, int *m, int *n, double *a, int *lda, double *sva, int *mv, double *v, int *ldv, double *work, int *lwork, int *info); +void BLAS_FUNC(dgesvx)(char *fact, char *trans, int *n, int *nrhs, double *a, int *lda, double *af, int *ldaf, int *ipiv, char *equed, double *r, double *c, double *b, int *ldb, double *x, int *ldx, double *rcond, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dgetc2)(int *n, double *a, int *lda, int *ipiv, int *jpiv, int *info); +void BLAS_FUNC(dgetf2)(int *m, int *n, double *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(dgetrf)(int *m, int *n, double *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(dgetri)(int *n, double *a, int *lda, int *ipiv, double *work, int *lwork, int *info); +void BLAS_FUNC(dgetrs)(char *trans, int *n, int *nrhs, double *a, int *lda, int *ipiv, double *b, int *ldb, int *info); +void BLAS_FUNC(dggbak)(char *job, char *side, int *n, int *ilo, int *ihi, double *lscale, double *rscale, int *m, double *v, int *ldv, int *info); +void BLAS_FUNC(dggbal)(char *job, int *n, double *a, int *lda, double *b, int *ldb, int *ilo, int *ihi, double *lscale, double *rscale, double *work, int *info); +void BLAS_FUNC(dgges)(char *jobvsl, char *jobvsr, char *sort, _dselect3 *selctg, int *n, double *a, int *lda, double *b, int *ldb, int *sdim, double *alphar, double *alphai, double *beta, double *vsl, int *ldvsl, double *vsr, int *ldvsr, double *work, int *lwork, int *bwork, int *info); +void BLAS_FUNC(dggesx)(char *jobvsl, char *jobvsr, char *sort, _dselect3 *selctg, char *sense, int *n, double *a, int *lda, double *b, int *ldb, int *sdim, double *alphar, double *alphai, double *beta, double *vsl, int *ldvsl, double *vsr, int *ldvsr, double *rconde, double *rcondv, double *work, int *lwork, int *iwork, int *liwork, int *bwork, int *info); +void BLAS_FUNC(dggev)(char *jobvl, char *jobvr, int *n, double *a, int *lda, double *b, int *ldb, double *alphar, double *alphai, double *beta, double *vl, int *ldvl, double *vr, int *ldvr, double *work, int *lwork, int *info); +void BLAS_FUNC(dggevx)(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, double *a, int *lda, double *b, int *ldb, double *alphar, double *alphai, double *beta, double *vl, int *ldvl, double *vr, int *ldvr, int *ilo, int *ihi, double *lscale, double *rscale, double *abnrm, double *bbnrm, double *rconde, double *rcondv, double *work, int *lwork, int *iwork, int *bwork, int *info); +void BLAS_FUNC(dggglm)(int *n, int *m, int *p, double *a, int *lda, double *b, int *ldb, double *d, double *x, double *y, double *work, int *lwork, int *info); +void BLAS_FUNC(dgghrd)(char *compq, char *compz, int *n, int *ilo, int *ihi, double *a, int *lda, double *b, int *ldb, double *q, int *ldq, double *z, int *ldz, int *info); +void BLAS_FUNC(dgglse)(int *m, int *n, int *p, double *a, int *lda, double *b, int *ldb, double *c, double *d, double *x, double *work, int *lwork, int *info); +void BLAS_FUNC(dggqrf)(int *n, int *m, int *p, double *a, int *lda, double *taua, double *b, int *ldb, double *taub, double *work, int *lwork, int *info); +void BLAS_FUNC(dggrqf)(int *m, int *p, int *n, double *a, int *lda, double *taua, double *b, int *ldb, double *taub, double *work, int *lwork, int *info); +void BLAS_FUNC(dgsvj0)(char *jobv, int *m, int *n, double *a, int *lda, double *d, double *sva, int *mv, double *v, int *ldv, double *eps, double *sfmin, double *tol, int *nsweep, double *work, int *lwork, int *info); +void BLAS_FUNC(dgsvj1)(char *jobv, int *m, int *n, int *n1, double *a, int *lda, double *d, double *sva, int *mv, double *v, int *ldv, double *eps, double *sfmin, double *tol, int *nsweep, double *work, int *lwork, int *info); +void BLAS_FUNC(dgtcon)(char *norm, int *n, double *dl, double *d, double *du, double *du2, int *ipiv, double *anorm, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dgtrfs)(char *trans, int *n, int *nrhs, double *dl, double *d, double *du, double *dlf, double *df, double *duf, double *du2, int *ipiv, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dgtsv)(int *n, int *nrhs, double *dl, double *d, double *du, double *b, int *ldb, int *info); +void BLAS_FUNC(dgtsvx)(char *fact, char *trans, int *n, int *nrhs, double *dl, double *d, double *du, double *dlf, double *df, double *duf, double *du2, int *ipiv, double *b, int *ldb, double *x, int *ldx, double *rcond, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dgttrf)(int *n, double *dl, double *d, double *du, double *du2, int *ipiv, int *info); +void BLAS_FUNC(dgttrs)(char *trans, int *n, int *nrhs, double *dl, double *d, double *du, double *du2, int *ipiv, double *b, int *ldb, int *info); +void BLAS_FUNC(dgtts2)(int *itrans, int *n, int *nrhs, double *dl, double *d, double *du, double *du2, int *ipiv, double *b, int *ldb); +void BLAS_FUNC(dhgeqz)(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, double *h, int *ldh, double *t, int *ldt, double *alphar, double *alphai, double *beta, double *q, int *ldq, double *z, int *ldz, double *work, int *lwork, int *info); +void BLAS_FUNC(dhsein)(char *side, char *eigsrc, char *initv, int *select, int *n, double *h, int *ldh, double *wr, double *wi, double *vl, int *ldvl, double *vr, int *ldvr, int *mm, int *m, double *work, int *ifaill, int *ifailr, int *info); +void BLAS_FUNC(dhseqr)(char *job, char *compz, int *n, int *ilo, int *ihi, double *h, int *ldh, double *wr, double *wi, double *z, int *ldz, double *work, int *lwork, int *info); +int BLAS_FUNC(disnan)(double *din); +void BLAS_FUNC(dlabad)(double *small, double *large); +void BLAS_FUNC(dlabrd)(int *m, int *n, int *nb, double *a, int *lda, double *d, double *e, double *tauq, double *taup, double *x, int *ldx, double *y, int *ldy); +void BLAS_FUNC(dlacn2)(int *n, double *v, double *x, int *isgn, double *est, int *kase, int *isave); +void BLAS_FUNC(dlacon)(int *n, double *v, double *x, int *isgn, double *est, int *kase); +void BLAS_FUNC(dlacpy)(char *uplo, int *m, int *n, double *a, int *lda, double *b, int *ldb); +void BLAS_FUNC(dladiv)(double *a, double *b, double *c, double *d, double *p, double *q); +void BLAS_FUNC(dlae2)(double *a, double *b, double *c, double *rt1, double *rt2); +void BLAS_FUNC(dlaebz)(int *ijob, int *nitmax, int *n, int *mmax, int *minp, int *nbmin, double *abstol, double *reltol, double *pivmin, double *d, double *e, double *e2, int *nval, double *ab, double *c, int *mout, int *nab, double *work, int *iwork, int *info); +void BLAS_FUNC(dlaed0)(int *icompq, int *qsiz, int *n, double *d, double *e, double *q, int *ldq, double *qstore, int *ldqs, double *work, int *iwork, int *info); +void BLAS_FUNC(dlaed1)(int *n, double *d, double *q, int *ldq, int *indxq, double *rho, int *cutpnt, double *work, int *iwork, int *info); +void BLAS_FUNC(dlaed2)(int *k, int *n, int *n1, double *d, double *q, int *ldq, int *indxq, double *rho, double *z, double *dlamda, double *w, double *q2, int *indx, int *indxc, int *indxp, int *coltyp, int *info); +void BLAS_FUNC(dlaed3)(int *k, int *n, int *n1, double *d, double *q, int *ldq, double *rho, double *dlamda, double *q2, int *indx, int *ctot, double *w, double *s, int *info); +void BLAS_FUNC(dlaed4)(int *n, int *i, double *d, double *z, double *delta, double *rho, double *dlam, int *info); +void BLAS_FUNC(dlaed5)(int *i, double *d, double *z, double *delta, double *rho, double *dlam); +void BLAS_FUNC(dlaed6)(int *kniter, int *orgati, double *rho, double *d, double *z, double *finit, double *tau, int *info); +void BLAS_FUNC(dlaed7)(int *icompq, int *n, int *qsiz, int *tlvls, int *curlvl, int *curpbm, double *d, double *q, int *ldq, int *indxq, double *rho, int *cutpnt, double *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, double *givnum, double *work, int *iwork, int *info); +void BLAS_FUNC(dlaed8)(int *icompq, int *k, int *n, int *qsiz, double *d, double *q, int *ldq, int *indxq, double *rho, int *cutpnt, double *z, double *dlamda, double *q2, int *ldq2, double *w, int *perm, int *givptr, int *givcol, double *givnum, int *indxp, int *indx, int *info); +void BLAS_FUNC(dlaed9)(int *k, int *kstart, int *kstop, int *n, double *d, double *q, int *ldq, double *rho, double *dlamda, double *w, double *s, int *lds, int *info); +void BLAS_FUNC(dlaeda)(int *n, int *tlvls, int *curlvl, int *curpbm, int *prmptr, int *perm, int *givptr, int *givcol, double *givnum, double *q, int *qptr, double *z, double *ztemp, int *info); +void BLAS_FUNC(dlaein)(int *rightv, int *noinit, int *n, double *h, int *ldh, double *wr, double *wi, double *vr, double *vi, double *b, int *ldb, double *work, double *eps3, double *smlnum, double *bignum, int *info); +void BLAS_FUNC(dlaev2)(double *a, double *b, double *c, double *rt1, double *rt2, double *cs1, double *sn1); +void BLAS_FUNC(dlaexc)(int *wantq, int *n, double *t, int *ldt, double *q, int *ldq, int *j1, int *n1, int *n2, double *work, int *info); +void BLAS_FUNC(dlag2)(double *a, int *lda, double *b, int *ldb, double *safmin, double *scale1, double *scale2, double *wr1, double *wr2, double *wi); +void BLAS_FUNC(dlag2s)(int *m, int *n, double *a, int *lda, float *sa, int *ldsa, int *info); +void BLAS_FUNC(dlags2)(int *upper, double *a1, double *a2, double *a3, double *b1, double *b2, double *b3, double *csu, double *snu, double *csv, double *snv, double *csq, double *snq); +void BLAS_FUNC(dlagtf)(int *n, double *a, double *lambda_, double *b, double *c, double *tol, double *d, int *in_, int *info); +void BLAS_FUNC(dlagtm)(char *trans, int *n, int *nrhs, double *alpha, double *dl, double *d, double *du, double *x, int *ldx, double *beta, double *b, int *ldb); +void BLAS_FUNC(dlagts)(int *job, int *n, double *a, double *b, double *c, double *d, int *in_, double *y, double *tol, int *info); +void BLAS_FUNC(dlagv2)(double *a, int *lda, double *b, int *ldb, double *alphar, double *alphai, double *beta, double *csl, double *snl, double *csr, double *snr); +void BLAS_FUNC(dlahqr)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, double *h, int *ldh, double *wr, double *wi, int *iloz, int *ihiz, double *z, int *ldz, int *info); +void BLAS_FUNC(dlahr2)(int *n, int *k, int *nb, double *a, int *lda, double *tau, double *t, int *ldt, double *y, int *ldy); +void BLAS_FUNC(dlaic1)(int *job, int *j, double *x, double *sest, double *w, double *gamma, double *sestpr, double *s, double *c); +void BLAS_FUNC(dlaln2)(int *ltrans, int *na, int *nw, double *smin, double *ca, double *a, int *lda, double *d1, double *d2, double *b, int *ldb, double *wr, double *wi, double *x, int *ldx, double *scale, double *xnorm, int *info); +void BLAS_FUNC(dlals0)(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, double *b, int *ldb, double *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, double *givnum, int *ldgnum, double *poles, double *difl, double *difr, double *z, int *k, double *c, double *s, double *work, int *info); +void BLAS_FUNC(dlalsa)(int *icompq, int *smlsiz, int *n, int *nrhs, double *b, int *ldb, double *bx, int *ldbx, double *u, int *ldu, double *vt, int *k, double *difl, double *difr, double *z, double *poles, int *givptr, int *givcol, int *ldgcol, int *perm, double *givnum, double *c, double *s, double *work, int *iwork, int *info); +void BLAS_FUNC(dlalsd)(char *uplo, int *smlsiz, int *n, int *nrhs, double *d, double *e, double *b, int *ldb, double *rcond, int *rank, double *work, int *iwork, int *info); +double BLAS_FUNC(dlamch)(char *cmach); +void BLAS_FUNC(dlamrg)(int *n1, int *n2, double *a, int *dtrd1, int *dtrd2, int *index_bn); +int BLAS_FUNC(dlaneg)(int *n, double *d, double *lld, double *sigma, double *pivmin, int *r); +double BLAS_FUNC(dlangb)(char *norm, int *n, int *kl, int *ku, double *ab, int *ldab, double *work); +double BLAS_FUNC(dlange)(char *norm, int *m, int *n, double *a, int *lda, double *work); +double BLAS_FUNC(dlangt)(char *norm, int *n, double *dl, double *d_, double *du); +double BLAS_FUNC(dlanhs)(char *norm, int *n, double *a, int *lda, double *work); +double BLAS_FUNC(dlansb)(char *norm, char *uplo, int *n, int *k, double *ab, int *ldab, double *work); +double BLAS_FUNC(dlansf)(char *norm, char *transr, char *uplo, int *n, double *a, double *work); +double BLAS_FUNC(dlansp)(char *norm, char *uplo, int *n, double *ap, double *work); +double BLAS_FUNC(dlanst)(char *norm, int *n, double *d_, double *e); +double BLAS_FUNC(dlansy)(char *norm, char *uplo, int *n, double *a, int *lda, double *work); +double BLAS_FUNC(dlantb)(char *norm, char *uplo, char *diag, int *n, int *k, double *ab, int *ldab, double *work); +double BLAS_FUNC(dlantp)(char *norm, char *uplo, char *diag, int *n, double *ap, double *work); +double BLAS_FUNC(dlantr)(char *norm, char *uplo, char *diag, int *m, int *n, double *a, int *lda, double *work); +void BLAS_FUNC(dlanv2)(double *a, double *b, double *c, double *d, double *rt1r, double *rt1i, double *rt2r, double *rt2i, double *cs, double *sn); +void BLAS_FUNC(dlapll)(int *n, double *x, int *incx, double *y, int *incy, double *ssmin); +void BLAS_FUNC(dlapmr)(int *forwrd, int *m, int *n, double *x, int *ldx, int *k); +void BLAS_FUNC(dlapmt)(int *forwrd, int *m, int *n, double *x, int *ldx, int *k); +double BLAS_FUNC(dlapy2)(double *x, double *y); +double BLAS_FUNC(dlapy3)(double *x, double *y, double *z); +void BLAS_FUNC(dlaqgb)(int *m, int *n, int *kl, int *ku, double *ab, int *ldab, double *r, double *c, double *rowcnd, double *colcnd, double *amax, char *equed); +void BLAS_FUNC(dlaqge)(int *m, int *n, double *a, int *lda, double *r, double *c, double *rowcnd, double *colcnd, double *amax, char *equed); +void BLAS_FUNC(dlaqp2)(int *m, int *n, int *offset, double *a, int *lda, int *jpvt, double *tau, double *vn1, double *vn2, double *work); +void BLAS_FUNC(dlaqps)(int *m, int *n, int *offset, int *nb, int *kb, double *a, int *lda, int *jpvt, double *tau, double *vn1, double *vn2, double *auxv, double *f, int *ldf); +void BLAS_FUNC(dlaqr0)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, double *h, int *ldh, double *wr, double *wi, int *iloz, int *ihiz, double *z, int *ldz, double *work, int *lwork, int *info); +void BLAS_FUNC(dlaqr1)(int *n, double *h, int *ldh, double *sr1, double *si1, double *sr2, double *si2, double *v); +void BLAS_FUNC(dlaqr2)(int *wantt, int *wantz, int *n, int *ktop, int *kbot, int *nw, double *h, int *ldh, int *iloz, int *ihiz, double *z, int *ldz, int *ns, int *nd, double *sr, double *si, double *v, int *ldv, int *nh, double *t, int *ldt, int *nv, double *wv, int *ldwv, double *work, int *lwork); +void BLAS_FUNC(dlaqr3)(int *wantt, int *wantz, int *n, int *ktop, int *kbot, int *nw, double *h, int *ldh, int *iloz, int *ihiz, double *z, int *ldz, int *ns, int *nd, double *sr, double *si, double *v, int *ldv, int *nh, double *t, int *ldt, int *nv, double *wv, int *ldwv, double *work, int *lwork); +void BLAS_FUNC(dlaqr4)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, double *h, int *ldh, double *wr, double *wi, int *iloz, int *ihiz, double *z, int *ldz, double *work, int *lwork, int *info); +void BLAS_FUNC(dlaqr5)(int *wantt, int *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, double *sr, double *si, double *h, int *ldh, int *iloz, int *ihiz, double *z, int *ldz, double *v, int *ldv, double *u, int *ldu, int *nv, double *wv, int *ldwv, int *nh, double *wh, int *ldwh); +void BLAS_FUNC(dlaqsb)(char *uplo, int *n, int *kd, double *ab, int *ldab, double *s, double *scond, double *amax, char *equed); +void BLAS_FUNC(dlaqsp)(char *uplo, int *n, double *ap, double *s, double *scond, double *amax, char *equed); +void BLAS_FUNC(dlaqsy)(char *uplo, int *n, double *a, int *lda, double *s, double *scond, double *amax, char *equed); +void BLAS_FUNC(dlaqtr)(int *ltran, int *lreal, int *n, double *t, int *ldt, double *b, double *w, double *scale, double *x, double *work, int *info); +void BLAS_FUNC(dlar1v)(int *n, int *b1, int *bn, double *lambda_, double *d, double *l, double *ld, double *lld, double *pivmin, double *gaptol, double *z, int *wantnc, int *negcnt, double *ztz, double *mingma, int *r, int *isuppz, double *nrminv, double *resid, double *rqcorr, double *work); +void BLAS_FUNC(dlar2v)(int *n, double *x, double *y, double *z, int *incx, double *c, double *s, int *incc); +void BLAS_FUNC(dlarf)(char *side, int *m, int *n, double *v, int *incv, double *tau, double *c, int *ldc, double *work); +void BLAS_FUNC(dlarfb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, double *v, int *ldv, double *t, int *ldt, double *c, int *ldc, double *work, int *ldwork); +void BLAS_FUNC(dlarfg)(int *n, double *alpha, double *x, int *incx, double *tau); +void BLAS_FUNC(dlarfgp)(int *n, double *alpha, double *x, int *incx, double *tau); +void BLAS_FUNC(dlarft)(char *direct, char *storev, int *n, int *k, double *v, int *ldv, double *tau, double *t, int *ldt); +void BLAS_FUNC(dlarfx)(char *side, int *m, int *n, double *v, double *tau, double *c, int *ldc, double *work); +void BLAS_FUNC(dlargv)(int *n, double *x, int *incx, double *y, int *incy, double *c, int *incc); +void BLAS_FUNC(dlarnv)(int *idist, int *iseed, int *n, double *x); +void BLAS_FUNC(dlarra)(int *n, double *d, double *e, double *e2, double *spltol, double *tnrm, int *nsplit, int *isplit, int *info); +void BLAS_FUNC(dlarrb)(int *n, double *d, double *lld, int *ifirst, int *ilast, double *rtol1, double *rtol2, int *offset, double *w, double *wgap, double *werr, double *work, int *iwork, double *pivmin, double *spdiam, int *twist, int *info); +void BLAS_FUNC(dlarrc)(char *jobt, int *n, double *vl, double *vu, double *d, double *e, double *pivmin, int *eigcnt, int *lcnt, int *rcnt, int *info); +void BLAS_FUNC(dlarrd)(char *range, char *order, int *n, double *vl, double *vu, int *il, int *iu, double *gers, double *reltol, double *d, double *e, double *e2, double *pivmin, int *nsplit, int *isplit, int *m, double *w, double *werr, double *wl, double *wu, int *iblock, int *indexw, double *work, int *iwork, int *info); +void BLAS_FUNC(dlarre)(char *range, int *n, double *vl, double *vu, int *il, int *iu, double *d, double *e, double *e2, double *rtol1, double *rtol2, double *spltol, int *nsplit, int *isplit, int *m, double *w, double *werr, double *wgap, int *iblock, int *indexw, double *gers, double *pivmin, double *work, int *iwork, int *info); +void BLAS_FUNC(dlarrf)(int *n, double *d, double *l, double *ld, int *clstrt, int *clend, double *w, double *wgap, double *werr, double *spdiam, double *clgapl, double *clgapr, double *pivmin, double *sigma, double *dplus, double *lplus, double *work, int *info); +void BLAS_FUNC(dlarrj)(int *n, double *d, double *e2, int *ifirst, int *ilast, double *rtol, int *offset, double *w, double *werr, double *work, int *iwork, double *pivmin, double *spdiam, int *info); +void BLAS_FUNC(dlarrk)(int *n, int *iw, double *gl, double *gu, double *d, double *e2, double *pivmin, double *reltol, double *w, double *werr, int *info); +void BLAS_FUNC(dlarrr)(int *n, double *d, double *e, int *info); +void BLAS_FUNC(dlarrv)(int *n, double *vl, double *vu, double *d, double *l, double *pivmin, int *isplit, int *m, int *dol, int *dou, double *minrgp, double *rtol1, double *rtol2, double *w, double *werr, double *wgap, int *iblock, int *indexw, double *gers, double *z, int *ldz, int *isuppz, double *work, int *iwork, int *info); +void BLAS_FUNC(dlartg)(double *f, double *g, double *cs, double *sn, double *r); +void BLAS_FUNC(dlartgp)(double *f, double *g, double *cs, double *sn, double *r); +void BLAS_FUNC(dlartgs)(double *x, double *y, double *sigma, double *cs, double *sn); +void BLAS_FUNC(dlartv)(int *n, double *x, int *incx, double *y, int *incy, double *c, double *s, int *incc); +void BLAS_FUNC(dlaruv)(int *iseed, int *n, double *x); +void BLAS_FUNC(dlarz)(char *side, int *m, int *n, int *l, double *v, int *incv, double *tau, double *c, int *ldc, double *work); +void BLAS_FUNC(dlarzb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, double *v, int *ldv, double *t, int *ldt, double *c, int *ldc, double *work, int *ldwork); +void BLAS_FUNC(dlarzt)(char *direct, char *storev, int *n, int *k, double *v, int *ldv, double *tau, double *t, int *ldt); +void BLAS_FUNC(dlas2)(double *f, double *g, double *h, double *ssmin, double *ssmax); +void BLAS_FUNC(dlascl)(char *type_bn, int *kl, int *ku, double *cfrom, double *cto, int *m, int *n, double *a, int *lda, int *info); +void BLAS_FUNC(dlasd0)(int *n, int *sqre, double *d, double *e, double *u, int *ldu, double *vt, int *ldvt, int *smlsiz, int *iwork, double *work, int *info); +void BLAS_FUNC(dlasd1)(int *nl, int *nr, int *sqre, double *d, double *alpha, double *beta, double *u, int *ldu, double *vt, int *ldvt, int *idxq, int *iwork, double *work, int *info); +void BLAS_FUNC(dlasd2)(int *nl, int *nr, int *sqre, int *k, double *d, double *z, double *alpha, double *beta, double *u, int *ldu, double *vt, int *ldvt, double *dsigma, double *u2, int *ldu2, double *vt2, int *ldvt2, int *idxp, int *idx, int *idxc, int *idxq, int *coltyp, int *info); +void BLAS_FUNC(dlasd3)(int *nl, int *nr, int *sqre, int *k, double *d, double *q, int *ldq, double *dsigma, double *u, int *ldu, double *u2, int *ldu2, double *vt, int *ldvt, double *vt2, int *ldvt2, int *idxc, int *ctot, double *z, int *info); +void BLAS_FUNC(dlasd4)(int *n, int *i, double *d, double *z, double *delta, double *rho, double *sigma, double *work, int *info); +void BLAS_FUNC(dlasd5)(int *i, double *d, double *z, double *delta, double *rho, double *dsigma, double *work); +void BLAS_FUNC(dlasd6)(int *icompq, int *nl, int *nr, int *sqre, double *d, double *vf, double *vl, double *alpha, double *beta, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, double *givnum, int *ldgnum, double *poles, double *difl, double *difr, double *z, int *k, double *c, double *s, double *work, int *iwork, int *info); +void BLAS_FUNC(dlasd7)(int *icompq, int *nl, int *nr, int *sqre, int *k, double *d, double *z, double *zw, double *vf, double *vfw, double *vl, double *vlw, double *alpha, double *beta, double *dsigma, int *idx, int *idxp, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, double *givnum, int *ldgnum, double *c, double *s, int *info); +void BLAS_FUNC(dlasd8)(int *icompq, int *k, double *d, double *z, double *vf, double *vl, double *difl, double *difr, int *lddifr, double *dsigma, double *work, int *info); +void BLAS_FUNC(dlasda)(int *icompq, int *smlsiz, int *n, int *sqre, double *d, double *e, double *u, int *ldu, double *vt, int *k, double *difl, double *difr, double *z, double *poles, int *givptr, int *givcol, int *ldgcol, int *perm, double *givnum, double *c, double *s, double *work, int *iwork, int *info); +void BLAS_FUNC(dlasdq)(char *uplo, int *sqre, int *n, int *ncvt, int *nru, int *ncc, double *d, double *e, double *vt, int *ldvt, double *u, int *ldu, double *c, int *ldc, double *work, int *info); +void BLAS_FUNC(dlasdt)(int *n, int *lvl, int *nd, int *inode, int *ndiml, int *ndimr, int *msub); +void BLAS_FUNC(dlaset)(char *uplo, int *m, int *n, double *alpha, double *beta, double *a, int *lda); +void BLAS_FUNC(dlasq1)(int *n, double *d, double *e, double *work, int *info); +void BLAS_FUNC(dlasq2)(int *n, double *z, int *info); +void BLAS_FUNC(dlasq3)(int *i0, int *n0, double *z, int *pp, double *dmin, double *sigma, double *desig, double *qmax, int *nfail, int *iter, int *ndiv, int *ieee, int *ttype, double *dmin1, double *dmin2, double *dn, double *dn1, double *dn2, double *g, double *tau); +void BLAS_FUNC(dlasq4)(int *i0, int *n0, double *z, int *pp, int *n0in, double *dmin, double *dmin1, double *dmin2, double *dn, double *dn1, double *dn2, double *tau, int *ttype, double *g); +void BLAS_FUNC(dlasq6)(int *i0, int *n0, double *z, int *pp, double *dmin, double *dmin1, double *dmin2, double *dn, double *dnm1, double *dnm2); +void BLAS_FUNC(dlasr)(char *side, char *pivot, char *direct, int *m, int *n, double *c, double *s, double *a, int *lda); +void BLAS_FUNC(dlasrt)(char *id, int *n, double *d, int *info); +void BLAS_FUNC(dlassq)(int *n, double *x, int *incx, double *scale, double *sumsq); +void BLAS_FUNC(dlasv2)(double *f, double *g, double *h, double *ssmin, double *ssmax, double *snr, double *csr, double *snl, double *csl); +void BLAS_FUNC(dlaswp)(int *n, double *a, int *lda, int *k1, int *k2, int *ipiv, int *incx); +void BLAS_FUNC(dlasy2)(int *ltranl, int *ltranr, int *isgn, int *n1, int *n2, double *tl, int *ldtl, double *tr, int *ldtr, double *b, int *ldb, double *scale, double *x, int *ldx, double *xnorm, int *info); +void BLAS_FUNC(dlasyf)(char *uplo, int *n, int *nb, int *kb, double *a, int *lda, int *ipiv, double *w, int *ldw, int *info); +void BLAS_FUNC(dlat2s)(char *uplo, int *n, double *a, int *lda, float *sa, int *ldsa, int *info); +void BLAS_FUNC(dlatbs)(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, double *ab, int *ldab, double *x, double *scale, double *cnorm, int *info); +void BLAS_FUNC(dlatdf)(int *ijob, int *n, double *z, int *ldz, double *rhs, double *rdsum, double *rdscal, int *ipiv, int *jpiv); +void BLAS_FUNC(dlatps)(char *uplo, char *trans, char *diag, char *normin, int *n, double *ap, double *x, double *scale, double *cnorm, int *info); +void BLAS_FUNC(dlatrd)(char *uplo, int *n, int *nb, double *a, int *lda, double *e, double *tau, double *w, int *ldw); +void BLAS_FUNC(dlatrs)(char *uplo, char *trans, char *diag, char *normin, int *n, double *a, int *lda, double *x, double *scale, double *cnorm, int *info); +void BLAS_FUNC(dlatrz)(int *m, int *n, int *l, double *a, int *lda, double *tau, double *work); +void BLAS_FUNC(dlauu2)(char *uplo, int *n, double *a, int *lda, int *info); +void BLAS_FUNC(dlauum)(char *uplo, int *n, double *a, int *lda, int *info); +void BLAS_FUNC(dopgtr)(char *uplo, int *n, double *ap, double *tau, double *q, int *ldq, double *work, int *info); +void BLAS_FUNC(dopmtr)(char *side, char *uplo, char *trans, int *m, int *n, double *ap, double *tau, double *c, int *ldc, double *work, int *info); +void BLAS_FUNC(dorbdb)(char *trans, char *signs, int *m, int *p, int *q, double *x11, int *ldx11, double *x12, int *ldx12, double *x21, int *ldx21, double *x22, int *ldx22, double *theta, double *phi, double *taup1, double *taup2, double *tauq1, double *tauq2, double *work, int *lwork, int *info); +void BLAS_FUNC(dorcsd)(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, double *x11, int *ldx11, double *x12, int *ldx12, double *x21, int *ldx21, double *x22, int *ldx22, double *theta, double *u1, int *ldu1, double *u2, int *ldu2, double *v1t, int *ldv1t, double *v2t, int *ldv2t, double *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(dorg2l)(int *m, int *n, int *k, double *a, int *lda, double *tau, double *work, int *info); +void BLAS_FUNC(dorg2r)(int *m, int *n, int *k, double *a, int *lda, double *tau, double *work, int *info); +void BLAS_FUNC(dorgbr)(char *vect, int *m, int *n, int *k, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dorghr)(int *n, int *ilo, int *ihi, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dorgl2)(int *m, int *n, int *k, double *a, int *lda, double *tau, double *work, int *info); +void BLAS_FUNC(dorglq)(int *m, int *n, int *k, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dorgql)(int *m, int *n, int *k, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dorgqr)(int *m, int *n, int *k, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dorgr2)(int *m, int *n, int *k, double *a, int *lda, double *tau, double *work, int *info); +void BLAS_FUNC(dorgrq)(int *m, int *n, int *k, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dorgtr)(char *uplo, int *n, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dorm2l)(char *side, char *trans, int *m, int *n, int *k, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *info); +void BLAS_FUNC(dorm2r)(char *side, char *trans, int *m, int *n, int *k, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *info); +void BLAS_FUNC(dormbr)(char *vect, char *side, char *trans, int *m, int *n, int *k, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *lwork, int *info); +void BLAS_FUNC(dormhr)(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *lwork, int *info); +void BLAS_FUNC(dorml2)(char *side, char *trans, int *m, int *n, int *k, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *info); +void BLAS_FUNC(dormlq)(char *side, char *trans, int *m, int *n, int *k, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *lwork, int *info); +void BLAS_FUNC(dormql)(char *side, char *trans, int *m, int *n, int *k, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *lwork, int *info); +void BLAS_FUNC(dormqr)(char *side, char *trans, int *m, int *n, int *k, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *lwork, int *info); +void BLAS_FUNC(dormr2)(char *side, char *trans, int *m, int *n, int *k, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *info); +void BLAS_FUNC(dormr3)(char *side, char *trans, int *m, int *n, int *k, int *l, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *info); +void BLAS_FUNC(dormrq)(char *side, char *trans, int *m, int *n, int *k, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *lwork, int *info); +void BLAS_FUNC(dormrz)(char *side, char *trans, int *m, int *n, int *k, int *l, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *lwork, int *info); +void BLAS_FUNC(dormtr)(char *side, char *uplo, char *trans, int *m, int *n, double *a, int *lda, double *tau, double *c, int *ldc, double *work, int *lwork, int *info); +void BLAS_FUNC(dpbcon)(char *uplo, int *n, int *kd, double *ab, int *ldab, double *anorm, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dpbequ)(char *uplo, int *n, int *kd, double *ab, int *ldab, double *s, double *scond, double *amax, int *info); +void BLAS_FUNC(dpbrfs)(char *uplo, int *n, int *kd, int *nrhs, double *ab, int *ldab, double *afb, int *ldafb, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dpbstf)(char *uplo, int *n, int *kd, double *ab, int *ldab, int *info); +void BLAS_FUNC(dpbsv)(char *uplo, int *n, int *kd, int *nrhs, double *ab, int *ldab, double *b, int *ldb, int *info); +void BLAS_FUNC(dpbsvx)(char *fact, char *uplo, int *n, int *kd, int *nrhs, double *ab, int *ldab, double *afb, int *ldafb, char *equed, double *s, double *b, int *ldb, double *x, int *ldx, double *rcond, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dpbtf2)(char *uplo, int *n, int *kd, double *ab, int *ldab, int *info); +void BLAS_FUNC(dpbtrf)(char *uplo, int *n, int *kd, double *ab, int *ldab, int *info); +void BLAS_FUNC(dpbtrs)(char *uplo, int *n, int *kd, int *nrhs, double *ab, int *ldab, double *b, int *ldb, int *info); +void BLAS_FUNC(dpftrf)(char *transr, char *uplo, int *n, double *a, int *info); +void BLAS_FUNC(dpftri)(char *transr, char *uplo, int *n, double *a, int *info); +void BLAS_FUNC(dpftrs)(char *transr, char *uplo, int *n, int *nrhs, double *a, double *b, int *ldb, int *info); +void BLAS_FUNC(dpocon)(char *uplo, int *n, double *a, int *lda, double *anorm, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dpoequ)(int *n, double *a, int *lda, double *s, double *scond, double *amax, int *info); +void BLAS_FUNC(dpoequb)(int *n, double *a, int *lda, double *s, double *scond, double *amax, int *info); +void BLAS_FUNC(dporfs)(char *uplo, int *n, int *nrhs, double *a, int *lda, double *af, int *ldaf, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dposv)(char *uplo, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, int *info); +void BLAS_FUNC(dposvx)(char *fact, char *uplo, int *n, int *nrhs, double *a, int *lda, double *af, int *ldaf, char *equed, double *s, double *b, int *ldb, double *x, int *ldx, double *rcond, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dpotf2)(char *uplo, int *n, double *a, int *lda, int *info); +void BLAS_FUNC(dpotrf)(char *uplo, int *n, double *a, int *lda, int *info); +void BLAS_FUNC(dpotri)(char *uplo, int *n, double *a, int *lda, int *info); +void BLAS_FUNC(dpotrs)(char *uplo, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, int *info); +void BLAS_FUNC(dppcon)(char *uplo, int *n, double *ap, double *anorm, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dppequ)(char *uplo, int *n, double *ap, double *s, double *scond, double *amax, int *info); +void BLAS_FUNC(dpprfs)(char *uplo, int *n, int *nrhs, double *ap, double *afp, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dppsv)(char *uplo, int *n, int *nrhs, double *ap, double *b, int *ldb, int *info); +void BLAS_FUNC(dppsvx)(char *fact, char *uplo, int *n, int *nrhs, double *ap, double *afp, char *equed, double *s, double *b, int *ldb, double *x, int *ldx, double *rcond, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dpptrf)(char *uplo, int *n, double *ap, int *info); +void BLAS_FUNC(dpptri)(char *uplo, int *n, double *ap, int *info); +void BLAS_FUNC(dpptrs)(char *uplo, int *n, int *nrhs, double *ap, double *b, int *ldb, int *info); +void BLAS_FUNC(dpstf2)(char *uplo, int *n, double *a, int *lda, int *piv, int *rank, double *tol, double *work, int *info); +void BLAS_FUNC(dpstrf)(char *uplo, int *n, double *a, int *lda, int *piv, int *rank, double *tol, double *work, int *info); +void BLAS_FUNC(dptcon)(int *n, double *d, double *e, double *anorm, double *rcond, double *work, int *info); +void BLAS_FUNC(dpteqr)(char *compz, int *n, double *d, double *e, double *z, int *ldz, double *work, int *info); +void BLAS_FUNC(dptrfs)(int *n, int *nrhs, double *d, double *e, double *df, double *ef, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *info); +void BLAS_FUNC(dptsv)(int *n, int *nrhs, double *d, double *e, double *b, int *ldb, int *info); +void BLAS_FUNC(dptsvx)(char *fact, int *n, int *nrhs, double *d, double *e, double *df, double *ef, double *b, int *ldb, double *x, int *ldx, double *rcond, double *ferr, double *berr, double *work, int *info); +void BLAS_FUNC(dpttrf)(int *n, double *d, double *e, int *info); +void BLAS_FUNC(dpttrs)(int *n, int *nrhs, double *d, double *e, double *b, int *ldb, int *info); +void BLAS_FUNC(dptts2)(int *n, int *nrhs, double *d, double *e, double *b, int *ldb); +void BLAS_FUNC(drscl)(int *n, double *sa, double *sx, int *incx); +void BLAS_FUNC(dsbev)(char *jobz, char *uplo, int *n, int *kd, double *ab, int *ldab, double *w, double *z, int *ldz, double *work, int *info); +void BLAS_FUNC(dsbevd)(char *jobz, char *uplo, int *n, int *kd, double *ab, int *ldab, double *w, double *z, int *ldz, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dsbevx)(char *jobz, char *range, char *uplo, int *n, int *kd, double *ab, int *ldab, double *q, int *ldq, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, double *z, int *ldz, double *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(dsbgst)(char *vect, char *uplo, int *n, int *ka, int *kb, double *ab, int *ldab, double *bb, int *ldbb, double *x, int *ldx, double *work, int *info); +void BLAS_FUNC(dsbgv)(char *jobz, char *uplo, int *n, int *ka, int *kb, double *ab, int *ldab, double *bb, int *ldbb, double *w, double *z, int *ldz, double *work, int *info); +void BLAS_FUNC(dsbgvd)(char *jobz, char *uplo, int *n, int *ka, int *kb, double *ab, int *ldab, double *bb, int *ldbb, double *w, double *z, int *ldz, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dsbgvx)(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, double *ab, int *ldab, double *bb, int *ldbb, double *q, int *ldq, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, double *z, int *ldz, double *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(dsbtrd)(char *vect, char *uplo, int *n, int *kd, double *ab, int *ldab, double *d, double *e, double *q, int *ldq, double *work, int *info); +void BLAS_FUNC(dsfrk)(char *transr, char *uplo, char *trans, int *n, int *k, double *alpha, double *a, int *lda, double *beta, double *c); +void BLAS_FUNC(dsgesv)(int *n, int *nrhs, double *a, int *lda, int *ipiv, double *b, int *ldb, double *x, int *ldx, double *work, float *swork, int *iter, int *info); +void BLAS_FUNC(dspcon)(char *uplo, int *n, double *ap, int *ipiv, double *anorm, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dspev)(char *jobz, char *uplo, int *n, double *ap, double *w, double *z, int *ldz, double *work, int *info); +void BLAS_FUNC(dspevd)(char *jobz, char *uplo, int *n, double *ap, double *w, double *z, int *ldz, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dspevx)(char *jobz, char *range, char *uplo, int *n, double *ap, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, double *z, int *ldz, double *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(dspgst)(int *itype, char *uplo, int *n, double *ap, double *bp, int *info); +void BLAS_FUNC(dspgv)(int *itype, char *jobz, char *uplo, int *n, double *ap, double *bp, double *w, double *z, int *ldz, double *work, int *info); +void BLAS_FUNC(dspgvd)(int *itype, char *jobz, char *uplo, int *n, double *ap, double *bp, double *w, double *z, int *ldz, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dspgvx)(int *itype, char *jobz, char *range, char *uplo, int *n, double *ap, double *bp, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, double *z, int *ldz, double *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(dsposv)(char *uplo, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, double *x, int *ldx, double *work, float *swork, int *iter, int *info); +void BLAS_FUNC(dsprfs)(char *uplo, int *n, int *nrhs, double *ap, double *afp, int *ipiv, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dspsv)(char *uplo, int *n, int *nrhs, double *ap, int *ipiv, double *b, int *ldb, int *info); +void BLAS_FUNC(dspsvx)(char *fact, char *uplo, int *n, int *nrhs, double *ap, double *afp, int *ipiv, double *b, int *ldb, double *x, int *ldx, double *rcond, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dsptrd)(char *uplo, int *n, double *ap, double *d, double *e, double *tau, int *info); +void BLAS_FUNC(dsptrf)(char *uplo, int *n, double *ap, int *ipiv, int *info); +void BLAS_FUNC(dsptri)(char *uplo, int *n, double *ap, int *ipiv, double *work, int *info); +void BLAS_FUNC(dsptrs)(char *uplo, int *n, int *nrhs, double *ap, int *ipiv, double *b, int *ldb, int *info); +void BLAS_FUNC(dstebz)(char *range, char *order, int *n, double *vl, double *vu, int *il, int *iu, double *abstol, double *d, double *e, int *m, int *nsplit, double *w, int *iblock, int *isplit, double *work, int *iwork, int *info); +void BLAS_FUNC(dstedc)(char *compz, int *n, double *d, double *e, double *z, int *ldz, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dstegr)(char *jobz, char *range, int *n, double *d, double *e, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, double *z, int *ldz, int *isuppz, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dstein)(int *n, double *d, double *e, int *m, double *w, int *iblock, int *isplit, double *z, int *ldz, double *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(dstemr)(char *jobz, char *range, int *n, double *d, double *e, double *vl, double *vu, int *il, int *iu, int *m, double *w, double *z, int *ldz, int *nzc, int *isuppz, int *tryrac, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dsteqr)(char *compz, int *n, double *d, double *e, double *z, int *ldz, double *work, int *info); +void BLAS_FUNC(dsterf)(int *n, double *d, double *e, int *info); +void BLAS_FUNC(dstev)(char *jobz, int *n, double *d, double *e, double *z, int *ldz, double *work, int *info); +void BLAS_FUNC(dstevd)(char *jobz, int *n, double *d, double *e, double *z, int *ldz, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dstevr)(char *jobz, char *range, int *n, double *d, double *e, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, double *z, int *ldz, int *isuppz, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dstevx)(char *jobz, char *range, int *n, double *d, double *e, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, double *z, int *ldz, double *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(dsycon)(char *uplo, int *n, double *a, int *lda, int *ipiv, double *anorm, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dsyconv)(char *uplo, char *way, int *n, double *a, int *lda, int *ipiv, double *work, int *info); +void BLAS_FUNC(dsyequb)(char *uplo, int *n, double *a, int *lda, double *s, double *scond, double *amax, double *work, int *info); +void BLAS_FUNC(dsyev)(char *jobz, char *uplo, int *n, double *a, int *lda, double *w, double *work, int *lwork, int *info); +void BLAS_FUNC(dsyevd)(char *jobz, char *uplo, int *n, double *a, int *lda, double *w, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dsyevr)(char *jobz, char *range, char *uplo, int *n, double *a, int *lda, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, double *z, int *ldz, int *isuppz, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dsyevx)(char *jobz, char *range, char *uplo, int *n, double *a, int *lda, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, double *z, int *ldz, double *work, int *lwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(dsygs2)(int *itype, char *uplo, int *n, double *a, int *lda, double *b, int *ldb, int *info); +void BLAS_FUNC(dsygst)(int *itype, char *uplo, int *n, double *a, int *lda, double *b, int *ldb, int *info); +void BLAS_FUNC(dsygv)(int *itype, char *jobz, char *uplo, int *n, double *a, int *lda, double *b, int *ldb, double *w, double *work, int *lwork, int *info); +void BLAS_FUNC(dsygvd)(int *itype, char *jobz, char *uplo, int *n, double *a, int *lda, double *b, int *ldb, double *w, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dsygvx)(int *itype, char *jobz, char *range, char *uplo, int *n, double *a, int *lda, double *b, int *ldb, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, double *z, int *ldz, double *work, int *lwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(dsyrfs)(char *uplo, int *n, int *nrhs, double *a, int *lda, double *af, int *ldaf, int *ipiv, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dsysv)(char *uplo, int *n, int *nrhs, double *a, int *lda, int *ipiv, double *b, int *ldb, double *work, int *lwork, int *info); +void BLAS_FUNC(dsysvx)(char *fact, char *uplo, int *n, int *nrhs, double *a, int *lda, double *af, int *ldaf, int *ipiv, double *b, int *ldb, double *x, int *ldx, double *rcond, double *ferr, double *berr, double *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(dsyswapr)(char *uplo, int *n, double *a, int *lda, int *i1, int *i2); +void BLAS_FUNC(dsytd2)(char *uplo, int *n, double *a, int *lda, double *d, double *e, double *tau, int *info); +void BLAS_FUNC(dsytf2)(char *uplo, int *n, double *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(dsytrd)(char *uplo, int *n, double *a, int *lda, double *d, double *e, double *tau, double *work, int *lwork, int *info); +void BLAS_FUNC(dsytrf)(char *uplo, int *n, double *a, int *lda, int *ipiv, double *work, int *lwork, int *info); +void BLAS_FUNC(dsytri)(char *uplo, int *n, double *a, int *lda, int *ipiv, double *work, int *info); +void BLAS_FUNC(dsytri2)(char *uplo, int *n, double *a, int *lda, int *ipiv, double *work, int *lwork, int *info); +void BLAS_FUNC(dsytri2x)(char *uplo, int *n, double *a, int *lda, int *ipiv, double *work, int *nb, int *info); +void BLAS_FUNC(dsytrs)(char *uplo, int *n, int *nrhs, double *a, int *lda, int *ipiv, double *b, int *ldb, int *info); +void BLAS_FUNC(dsytrs2)(char *uplo, int *n, int *nrhs, double *a, int *lda, int *ipiv, double *b, int *ldb, double *work, int *info); +void BLAS_FUNC(dtbcon)(char *norm, char *uplo, char *diag, int *n, int *kd, double *ab, int *ldab, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dtbrfs)(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, double *ab, int *ldab, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dtbtrs)(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, double *ab, int *ldab, double *b, int *ldb, int *info); +void BLAS_FUNC(dtfsm)(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, double *alpha, double *a, double *b, int *ldb); +void BLAS_FUNC(dtftri)(char *transr, char *uplo, char *diag, int *n, double *a, int *info); +void BLAS_FUNC(dtfttp)(char *transr, char *uplo, int *n, double *arf, double *ap, int *info); +void BLAS_FUNC(dtfttr)(char *transr, char *uplo, int *n, double *arf, double *a, int *lda, int *info); +void BLAS_FUNC(dtgevc)(char *side, char *howmny, int *select, int *n, double *s, int *lds, double *p, int *ldp, double *vl, int *ldvl, double *vr, int *ldvr, int *mm, int *m, double *work, int *info); +void BLAS_FUNC(dtgex2)(int *wantq, int *wantz, int *n, double *a, int *lda, double *b, int *ldb, double *q, int *ldq, double *z, int *ldz, int *j1, int *n1, int *n2, double *work, int *lwork, int *info); +void BLAS_FUNC(dtgexc)(int *wantq, int *wantz, int *n, double *a, int *lda, double *b, int *ldb, double *q, int *ldq, double *z, int *ldz, int *ifst, int *ilst, double *work, int *lwork, int *info); +void BLAS_FUNC(dtgsen)(int *ijob, int *wantq, int *wantz, int *select, int *n, double *a, int *lda, double *b, int *ldb, double *alphar, double *alphai, double *beta, double *q, int *ldq, double *z, int *ldz, int *m, double *pl, double *pr, double *dif, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dtgsja)(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, double *a, int *lda, double *b, int *ldb, double *tola, double *tolb, double *alpha, double *beta, double *u, int *ldu, double *v, int *ldv, double *q, int *ldq, double *work, int *ncycle, int *info); +void BLAS_FUNC(dtgsna)(char *job, char *howmny, int *select, int *n, double *a, int *lda, double *b, int *ldb, double *vl, int *ldvl, double *vr, int *ldvr, double *s, double *dif, int *mm, int *m, double *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(dtgsy2)(char *trans, int *ijob, int *m, int *n, double *a, int *lda, double *b, int *ldb, double *c, int *ldc, double *d, int *ldd, double *e, int *lde, double *f, int *ldf, double *scale, double *rdsum, double *rdscal, int *iwork, int *pq, int *info); +void BLAS_FUNC(dtgsyl)(char *trans, int *ijob, int *m, int *n, double *a, int *lda, double *b, int *ldb, double *c, int *ldc, double *d, int *ldd, double *e, int *lde, double *f, int *ldf, double *scale, double *dif, double *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(dtpcon)(char *norm, char *uplo, char *diag, int *n, double *ap, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dtpmqrt)(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, double *v, int *ldv, double *t, int *ldt, double *a, int *lda, double *b, int *ldb, double *work, int *info); +void BLAS_FUNC(dtpqrt)(int *m, int *n, int *l, int *nb, double *a, int *lda, double *b, int *ldb, double *t, int *ldt, double *work, int *info); +void BLAS_FUNC(dtpqrt2)(int *m, int *n, int *l, double *a, int *lda, double *b, int *ldb, double *t, int *ldt, int *info); +void BLAS_FUNC(dtprfb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, double *v, int *ldv, double *t, int *ldt, double *a, int *lda, double *b, int *ldb, double *work, int *ldwork); +void BLAS_FUNC(dtprfs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, double *ap, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dtptri)(char *uplo, char *diag, int *n, double *ap, int *info); +void BLAS_FUNC(dtptrs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, double *ap, double *b, int *ldb, int *info); +void BLAS_FUNC(dtpttf)(char *transr, char *uplo, int *n, double *ap, double *arf, int *info); +void BLAS_FUNC(dtpttr)(char *uplo, int *n, double *ap, double *a, int *lda, int *info); +void BLAS_FUNC(dtrcon)(char *norm, char *uplo, char *diag, int *n, double *a, int *lda, double *rcond, double *work, int *iwork, int *info); +void BLAS_FUNC(dtrevc)(char *side, char *howmny, int *select, int *n, double *t, int *ldt, double *vl, int *ldvl, double *vr, int *ldvr, int *mm, int *m, double *work, int *info); +void BLAS_FUNC(dtrexc)(char *compq, int *n, double *t, int *ldt, double *q, int *ldq, int *ifst, int *ilst, double *work, int *info); +void BLAS_FUNC(dtrrfs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, double *x, int *ldx, double *ferr, double *berr, double *work, int *iwork, int *info); +void BLAS_FUNC(dtrsen)(char *job, char *compq, int *select, int *n, double *t, int *ldt, double *q, int *ldq, double *wr, double *wi, int *m, double *s, double *sep, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(dtrsna)(char *job, char *howmny, int *select, int *n, double *t, int *ldt, double *vl, int *ldvl, double *vr, int *ldvr, double *s, double *sep, int *mm, int *m, double *work, int *ldwork, int *iwork, int *info); +void BLAS_FUNC(dtrsyl)(char *trana, char *tranb, int *isgn, int *m, int *n, double *a, int *lda, double *b, int *ldb, double *c, int *ldc, double *scale, int *info); +void BLAS_FUNC(dtrti2)(char *uplo, char *diag, int *n, double *a, int *lda, int *info); +void BLAS_FUNC(dtrtri)(char *uplo, char *diag, int *n, double *a, int *lda, int *info); +void BLAS_FUNC(dtrtrs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, double *a, int *lda, double *b, int *ldb, int *info); +void BLAS_FUNC(dtrttf)(char *transr, char *uplo, int *n, double *a, int *lda, double *arf, int *info); +void BLAS_FUNC(dtrttp)(char *uplo, int *n, double *a, int *lda, double *ap, int *info); +void BLAS_FUNC(dtzrzf)(int *m, int *n, double *a, int *lda, double *tau, double *work, int *lwork, int *info); +double BLAS_FUNC(dzsum1)(int *n, npy_complex128 *cx, int *incx); +int BLAS_FUNC(icmax1)(int *n, npy_complex64 *cx, int *incx); +int BLAS_FUNC(ieeeck)(int *ispec, float *zero, float *one); +int BLAS_FUNC(ilaclc)(int *m, int *n, npy_complex64 *a, int *lda); +int BLAS_FUNC(ilaclr)(int *m, int *n, npy_complex64 *a, int *lda); +int BLAS_FUNC(iladiag)(char *diag); +int BLAS_FUNC(iladlc)(int *m, int *n, double *a, int *lda); +int BLAS_FUNC(iladlr)(int *m, int *n, double *a, int *lda); +int BLAS_FUNC(ilaprec)(char *prec); +int BLAS_FUNC(ilaslc)(int *m, int *n, float *a, int *lda); +int BLAS_FUNC(ilaslr)(int *m, int *n, float *a, int *lda); +int BLAS_FUNC(ilatrans)(char *trans); +int BLAS_FUNC(ilauplo)(char *uplo); +void BLAS_FUNC(ilaver)(int *vers_major, int *vers_minor, int *vers_patch); +int BLAS_FUNC(ilazlc)(int *m, int *n, npy_complex128 *a, int *lda); +int BLAS_FUNC(ilazlr)(int *m, int *n, npy_complex128 *a, int *lda); +int BLAS_FUNC(izmax1)(int *n, npy_complex128 *cx, int *incx); +void BLAS_FUNC(sbbcsd)(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, float *theta, float *phi, float *u1, int *ldu1, float *u2, int *ldu2, float *v1t, int *ldv1t, float *v2t, int *ldv2t, float *b11d, float *b11e, float *b12d, float *b12e, float *b21d, float *b21e, float *b22d, float *b22e, float *work, int *lwork, int *info); +void BLAS_FUNC(sbdsdc)(char *uplo, char *compq, int *n, float *d, float *e, float *u, int *ldu, float *vt, int *ldvt, float *q, int *iq, float *work, int *iwork, int *info); +void BLAS_FUNC(sbdsqr)(char *uplo, int *n, int *ncvt, int *nru, int *ncc, float *d, float *e, float *vt, int *ldvt, float *u, int *ldu, float *c, int *ldc, float *work, int *info); +float BLAS_FUNC(scsum1)(int *n, npy_complex64 *cx, int *incx); +void BLAS_FUNC(sdisna)(char *job, int *m, int *n, float *d, float *sep, int *info); +void BLAS_FUNC(sgbbrd)(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, float *ab, int *ldab, float *d, float *e, float *q, int *ldq, float *pt, int *ldpt, float *c, int *ldc, float *work, int *info); +void BLAS_FUNC(sgbcon)(char *norm, int *n, int *kl, int *ku, float *ab, int *ldab, int *ipiv, float *anorm, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(sgbequ)(int *m, int *n, int *kl, int *ku, float *ab, int *ldab, float *r, float *c, float *rowcnd, float *colcnd, float *amax, int *info); +void BLAS_FUNC(sgbequb)(int *m, int *n, int *kl, int *ku, float *ab, int *ldab, float *r, float *c, float *rowcnd, float *colcnd, float *amax, int *info); +void BLAS_FUNC(sgbrfs)(char *trans, int *n, int *kl, int *ku, int *nrhs, float *ab, int *ldab, float *afb, int *ldafb, int *ipiv, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(sgbsv)(int *n, int *kl, int *ku, int *nrhs, float *ab, int *ldab, int *ipiv, float *b, int *ldb, int *info); +void BLAS_FUNC(sgbsvx)(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, float *ab, int *ldab, float *afb, int *ldafb, int *ipiv, char *equed, float *r, float *c, float *b, int *ldb, float *x, int *ldx, float *rcond, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(sgbtf2)(int *m, int *n, int *kl, int *ku, float *ab, int *ldab, int *ipiv, int *info); +void BLAS_FUNC(sgbtrf)(int *m, int *n, int *kl, int *ku, float *ab, int *ldab, int *ipiv, int *info); +void BLAS_FUNC(sgbtrs)(char *trans, int *n, int *kl, int *ku, int *nrhs, float *ab, int *ldab, int *ipiv, float *b, int *ldb, int *info); +void BLAS_FUNC(sgebak)(char *job, char *side, int *n, int *ilo, int *ihi, float *scale, int *m, float *v, int *ldv, int *info); +void BLAS_FUNC(sgebal)(char *job, int *n, float *a, int *lda, int *ilo, int *ihi, float *scale, int *info); +void BLAS_FUNC(sgebd2)(int *m, int *n, float *a, int *lda, float *d, float *e, float *tauq, float *taup, float *work, int *info); +void BLAS_FUNC(sgebrd)(int *m, int *n, float *a, int *lda, float *d, float *e, float *tauq, float *taup, float *work, int *lwork, int *info); +void BLAS_FUNC(sgecon)(char *norm, int *n, float *a, int *lda, float *anorm, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(sgeequ)(int *m, int *n, float *a, int *lda, float *r, float *c, float *rowcnd, float *colcnd, float *amax, int *info); +void BLAS_FUNC(sgeequb)(int *m, int *n, float *a, int *lda, float *r, float *c, float *rowcnd, float *colcnd, float *amax, int *info); +void BLAS_FUNC(sgees)(char *jobvs, char *sort, _sselect2 *select, int *n, float *a, int *lda, int *sdim, float *wr, float *wi, float *vs, int *ldvs, float *work, int *lwork, int *bwork, int *info); +void BLAS_FUNC(sgeesx)(char *jobvs, char *sort, _sselect2 *select, char *sense, int *n, float *a, int *lda, int *sdim, float *wr, float *wi, float *vs, int *ldvs, float *rconde, float *rcondv, float *work, int *lwork, int *iwork, int *liwork, int *bwork, int *info); +void BLAS_FUNC(sgeev)(char *jobvl, char *jobvr, int *n, float *a, int *lda, float *wr, float *wi, float *vl, int *ldvl, float *vr, int *ldvr, float *work, int *lwork, int *info); +void BLAS_FUNC(sgeevx)(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, float *a, int *lda, float *wr, float *wi, float *vl, int *ldvl, float *vr, int *ldvr, int *ilo, int *ihi, float *scale, float *abnrm, float *rconde, float *rcondv, float *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(sgehd2)(int *n, int *ilo, int *ihi, float *a, int *lda, float *tau, float *work, int *info); +void BLAS_FUNC(sgehrd)(int *n, int *ilo, int *ihi, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sgejsv)(char *joba, char *jobu, char *jobv, char *jobr, char *jobt, char *jobp, int *m, int *n, float *a, int *lda, float *sva, float *u, int *ldu, float *v, int *ldv, float *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(sgelq2)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *info); +void BLAS_FUNC(sgelqf)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sgels)(char *trans, int *m, int *n, int *nrhs, float *a, int *lda, float *b, int *ldb, float *work, int *lwork, int *info); +void BLAS_FUNC(sgelsd)(int *m, int *n, int *nrhs, float *a, int *lda, float *b, int *ldb, float *s, float *rcond, int *rank, float *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(sgelss)(int *m, int *n, int *nrhs, float *a, int *lda, float *b, int *ldb, float *s, float *rcond, int *rank, float *work, int *lwork, int *info); +void BLAS_FUNC(sgelsy)(int *m, int *n, int *nrhs, float *a, int *lda, float *b, int *ldb, int *jpvt, float *rcond, int *rank, float *work, int *lwork, int *info); +void BLAS_FUNC(sgemqrt)(char *side, char *trans, int *m, int *n, int *k, int *nb, float *v, int *ldv, float *t, int *ldt, float *c, int *ldc, float *work, int *info); +void BLAS_FUNC(sgeql2)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *info); +void BLAS_FUNC(sgeqlf)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sgeqp3)(int *m, int *n, float *a, int *lda, int *jpvt, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sgeqr2)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *info); +void BLAS_FUNC(sgeqr2p)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *info); +void BLAS_FUNC(sgeqrf)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sgeqrfp)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sgeqrt)(int *m, int *n, int *nb, float *a, int *lda, float *t, int *ldt, float *work, int *info); +void BLAS_FUNC(sgeqrt2)(int *m, int *n, float *a, int *lda, float *t, int *ldt, int *info); +void BLAS_FUNC(sgeqrt3)(int *m, int *n, float *a, int *lda, float *t, int *ldt, int *info); +void BLAS_FUNC(sgerfs)(char *trans, int *n, int *nrhs, float *a, int *lda, float *af, int *ldaf, int *ipiv, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(sgerq2)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *info); +void BLAS_FUNC(sgerqf)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sgesc2)(int *n, float *a, int *lda, float *rhs, int *ipiv, int *jpiv, float *scale); +void BLAS_FUNC(sgesdd)(char *jobz, int *m, int *n, float *a, int *lda, float *s, float *u, int *ldu, float *vt, int *ldvt, float *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(sgesv)(int *n, int *nrhs, float *a, int *lda, int *ipiv, float *b, int *ldb, int *info); +void BLAS_FUNC(sgesvd)(char *jobu, char *jobvt, int *m, int *n, float *a, int *lda, float *s, float *u, int *ldu, float *vt, int *ldvt, float *work, int *lwork, int *info); +void BLAS_FUNC(sgesvj)(char *joba, char *jobu, char *jobv, int *m, int *n, float *a, int *lda, float *sva, int *mv, float *v, int *ldv, float *work, int *lwork, int *info); +void BLAS_FUNC(sgesvx)(char *fact, char *trans, int *n, int *nrhs, float *a, int *lda, float *af, int *ldaf, int *ipiv, char *equed, float *r, float *c, float *b, int *ldb, float *x, int *ldx, float *rcond, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(sgetc2)(int *n, float *a, int *lda, int *ipiv, int *jpiv, int *info); +void BLAS_FUNC(sgetf2)(int *m, int *n, float *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(sgetrf)(int *m, int *n, float *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(sgetri)(int *n, float *a, int *lda, int *ipiv, float *work, int *lwork, int *info); +void BLAS_FUNC(sgetrs)(char *trans, int *n, int *nrhs, float *a, int *lda, int *ipiv, float *b, int *ldb, int *info); +void BLAS_FUNC(sggbak)(char *job, char *side, int *n, int *ilo, int *ihi, float *lscale, float *rscale, int *m, float *v, int *ldv, int *info); +void BLAS_FUNC(sggbal)(char *job, int *n, float *a, int *lda, float *b, int *ldb, int *ilo, int *ihi, float *lscale, float *rscale, float *work, int *info); +void BLAS_FUNC(sgges)(char *jobvsl, char *jobvsr, char *sort, _sselect3 *selctg, int *n, float *a, int *lda, float *b, int *ldb, int *sdim, float *alphar, float *alphai, float *beta, float *vsl, int *ldvsl, float *vsr, int *ldvsr, float *work, int *lwork, int *bwork, int *info); +void BLAS_FUNC(sggesx)(char *jobvsl, char *jobvsr, char *sort, _sselect3 *selctg, char *sense, int *n, float *a, int *lda, float *b, int *ldb, int *sdim, float *alphar, float *alphai, float *beta, float *vsl, int *ldvsl, float *vsr, int *ldvsr, float *rconde, float *rcondv, float *work, int *lwork, int *iwork, int *liwork, int *bwork, int *info); +void BLAS_FUNC(sggev)(char *jobvl, char *jobvr, int *n, float *a, int *lda, float *b, int *ldb, float *alphar, float *alphai, float *beta, float *vl, int *ldvl, float *vr, int *ldvr, float *work, int *lwork, int *info); +void BLAS_FUNC(sggevx)(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, float *a, int *lda, float *b, int *ldb, float *alphar, float *alphai, float *beta, float *vl, int *ldvl, float *vr, int *ldvr, int *ilo, int *ihi, float *lscale, float *rscale, float *abnrm, float *bbnrm, float *rconde, float *rcondv, float *work, int *lwork, int *iwork, int *bwork, int *info); +void BLAS_FUNC(sggglm)(int *n, int *m, int *p, float *a, int *lda, float *b, int *ldb, float *d, float *x, float *y, float *work, int *lwork, int *info); +void BLAS_FUNC(sgghrd)(char *compq, char *compz, int *n, int *ilo, int *ihi, float *a, int *lda, float *b, int *ldb, float *q, int *ldq, float *z, int *ldz, int *info); +void BLAS_FUNC(sgglse)(int *m, int *n, int *p, float *a, int *lda, float *b, int *ldb, float *c, float *d, float *x, float *work, int *lwork, int *info); +void BLAS_FUNC(sggqrf)(int *n, int *m, int *p, float *a, int *lda, float *taua, float *b, int *ldb, float *taub, float *work, int *lwork, int *info); +void BLAS_FUNC(sggrqf)(int *m, int *p, int *n, float *a, int *lda, float *taua, float *b, int *ldb, float *taub, float *work, int *lwork, int *info); +void BLAS_FUNC(sgsvj0)(char *jobv, int *m, int *n, float *a, int *lda, float *d, float *sva, int *mv, float *v, int *ldv, float *eps, float *sfmin, float *tol, int *nsweep, float *work, int *lwork, int *info); +void BLAS_FUNC(sgsvj1)(char *jobv, int *m, int *n, int *n1, float *a, int *lda, float *d, float *sva, int *mv, float *v, int *ldv, float *eps, float *sfmin, float *tol, int *nsweep, float *work, int *lwork, int *info); +void BLAS_FUNC(sgtcon)(char *norm, int *n, float *dl, float *d, float *du, float *du2, int *ipiv, float *anorm, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(sgtrfs)(char *trans, int *n, int *nrhs, float *dl, float *d, float *du, float *dlf, float *df, float *duf, float *du2, int *ipiv, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(sgtsv)(int *n, int *nrhs, float *dl, float *d, float *du, float *b, int *ldb, int *info); +void BLAS_FUNC(sgtsvx)(char *fact, char *trans, int *n, int *nrhs, float *dl, float *d, float *du, float *dlf, float *df, float *duf, float *du2, int *ipiv, float *b, int *ldb, float *x, int *ldx, float *rcond, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(sgttrf)(int *n, float *dl, float *d, float *du, float *du2, int *ipiv, int *info); +void BLAS_FUNC(sgttrs)(char *trans, int *n, int *nrhs, float *dl, float *d, float *du, float *du2, int *ipiv, float *b, int *ldb, int *info); +void BLAS_FUNC(sgtts2)(int *itrans, int *n, int *nrhs, float *dl, float *d, float *du, float *du2, int *ipiv, float *b, int *ldb); +void BLAS_FUNC(shgeqz)(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, float *h, int *ldh, float *t, int *ldt, float *alphar, float *alphai, float *beta, float *q, int *ldq, float *z, int *ldz, float *work, int *lwork, int *info); +void BLAS_FUNC(shsein)(char *side, char *eigsrc, char *initv, int *select, int *n, float *h, int *ldh, float *wr, float *wi, float *vl, int *ldvl, float *vr, int *ldvr, int *mm, int *m, float *work, int *ifaill, int *ifailr, int *info); +void BLAS_FUNC(shseqr)(char *job, char *compz, int *n, int *ilo, int *ihi, float *h, int *ldh, float *wr, float *wi, float *z, int *ldz, float *work, int *lwork, int *info); +void BLAS_FUNC(slabad)(float *small, float *large); +void BLAS_FUNC(slabrd)(int *m, int *n, int *nb, float *a, int *lda, float *d, float *e, float *tauq, float *taup, float *x, int *ldx, float *y, int *ldy); +void BLAS_FUNC(slacn2)(int *n, float *v, float *x, int *isgn, float *est, int *kase, int *isave); +void BLAS_FUNC(slacon)(int *n, float *v, float *x, int *isgn, float *est, int *kase); +void BLAS_FUNC(slacpy)(char *uplo, int *m, int *n, float *a, int *lda, float *b, int *ldb); +void BLAS_FUNC(sladiv)(float *a, float *b, float *c, float *d, float *p, float *q); +void BLAS_FUNC(slae2)(float *a, float *b, float *c, float *rt1, float *rt2); +void BLAS_FUNC(slaebz)(int *ijob, int *nitmax, int *n, int *mmax, int *minp, int *nbmin, float *abstol, float *reltol, float *pivmin, float *d, float *e, float *e2, int *nval, float *ab, float *c, int *mout, int *nab, float *work, int *iwork, int *info); +void BLAS_FUNC(slaed0)(int *icompq, int *qsiz, int *n, float *d, float *e, float *q, int *ldq, float *qstore, int *ldqs, float *work, int *iwork, int *info); +void BLAS_FUNC(slaed1)(int *n, float *d, float *q, int *ldq, int *indxq, float *rho, int *cutpnt, float *work, int *iwork, int *info); +void BLAS_FUNC(slaed2)(int *k, int *n, int *n1, float *d, float *q, int *ldq, int *indxq, float *rho, float *z, float *dlamda, float *w, float *q2, int *indx, int *indxc, int *indxp, int *coltyp, int *info); +void BLAS_FUNC(slaed3)(int *k, int *n, int *n1, float *d, float *q, int *ldq, float *rho, float *dlamda, float *q2, int *indx, int *ctot, float *w, float *s, int *info); +void BLAS_FUNC(slaed4)(int *n, int *i, float *d, float *z, float *delta, float *rho, float *dlam, int *info); +void BLAS_FUNC(slaed5)(int *i, float *d, float *z, float *delta, float *rho, float *dlam); +void BLAS_FUNC(slaed6)(int *kniter, int *orgati, float *rho, float *d, float *z, float *finit, float *tau, int *info); +void BLAS_FUNC(slaed7)(int *icompq, int *n, int *qsiz, int *tlvls, int *curlvl, int *curpbm, float *d, float *q, int *ldq, int *indxq, float *rho, int *cutpnt, float *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, float *givnum, float *work, int *iwork, int *info); +void BLAS_FUNC(slaed8)(int *icompq, int *k, int *n, int *qsiz, float *d, float *q, int *ldq, int *indxq, float *rho, int *cutpnt, float *z, float *dlamda, float *q2, int *ldq2, float *w, int *perm, int *givptr, int *givcol, float *givnum, int *indxp, int *indx, int *info); +void BLAS_FUNC(slaed9)(int *k, int *kstart, int *kstop, int *n, float *d, float *q, int *ldq, float *rho, float *dlamda, float *w, float *s, int *lds, int *info); +void BLAS_FUNC(slaeda)(int *n, int *tlvls, int *curlvl, int *curpbm, int *prmptr, int *perm, int *givptr, int *givcol, float *givnum, float *q, int *qptr, float *z, float *ztemp, int *info); +void BLAS_FUNC(slaein)(int *rightv, int *noinit, int *n, float *h, int *ldh, float *wr, float *wi, float *vr, float *vi, float *b, int *ldb, float *work, float *eps3, float *smlnum, float *bignum, int *info); +void BLAS_FUNC(slaev2)(float *a, float *b, float *c, float *rt1, float *rt2, float *cs1, float *sn1); +void BLAS_FUNC(slaexc)(int *wantq, int *n, float *t, int *ldt, float *q, int *ldq, int *j1, int *n1, int *n2, float *work, int *info); +void BLAS_FUNC(slag2)(float *a, int *lda, float *b, int *ldb, float *safmin, float *scale1, float *scale2, float *wr1, float *wr2, float *wi); +void BLAS_FUNC(slag2d)(int *m, int *n, float *sa, int *ldsa, double *a, int *lda, int *info); +void BLAS_FUNC(slags2)(int *upper, float *a1, float *a2, float *a3, float *b1, float *b2, float *b3, float *csu, float *snu, float *csv, float *snv, float *csq, float *snq); +void BLAS_FUNC(slagtf)(int *n, float *a, float *lambda_, float *b, float *c, float *tol, float *d, int *in_, int *info); +void BLAS_FUNC(slagtm)(char *trans, int *n, int *nrhs, float *alpha, float *dl, float *d, float *du, float *x, int *ldx, float *beta, float *b, int *ldb); +void BLAS_FUNC(slagts)(int *job, int *n, float *a, float *b, float *c, float *d, int *in_, float *y, float *tol, int *info); +void BLAS_FUNC(slagv2)(float *a, int *lda, float *b, int *ldb, float *alphar, float *alphai, float *beta, float *csl, float *snl, float *csr, float *snr); +void BLAS_FUNC(slahqr)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, float *h, int *ldh, float *wr, float *wi, int *iloz, int *ihiz, float *z, int *ldz, int *info); +void BLAS_FUNC(slahr2)(int *n, int *k, int *nb, float *a, int *lda, float *tau, float *t, int *ldt, float *y, int *ldy); +void BLAS_FUNC(slaic1)(int *job, int *j, float *x, float *sest, float *w, float *gamma, float *sestpr, float *s, float *c); +void BLAS_FUNC(slaln2)(int *ltrans, int *na, int *nw, float *smin, float *ca, float *a, int *lda, float *d1, float *d2, float *b, int *ldb, float *wr, float *wi, float *x, int *ldx, float *scale, float *xnorm, int *info); +void BLAS_FUNC(slals0)(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, float *b, int *ldb, float *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, float *givnum, int *ldgnum, float *poles, float *difl, float *difr, float *z, int *k, float *c, float *s, float *work, int *info); +void BLAS_FUNC(slalsa)(int *icompq, int *smlsiz, int *n, int *nrhs, float *b, int *ldb, float *bx, int *ldbx, float *u, int *ldu, float *vt, int *k, float *difl, float *difr, float *z, float *poles, int *givptr, int *givcol, int *ldgcol, int *perm, float *givnum, float *c, float *s, float *work, int *iwork, int *info); +void BLAS_FUNC(slalsd)(char *uplo, int *smlsiz, int *n, int *nrhs, float *d, float *e, float *b, int *ldb, float *rcond, int *rank, float *work, int *iwork, int *info); +float BLAS_FUNC(slamch)(char *cmach); +void BLAS_FUNC(slamrg)(int *n1, int *n2, float *a, int *strd1, int *strd2, int *index_bn); +float BLAS_FUNC(slangb)(char *norm, int *n, int *kl, int *ku, float *ab, int *ldab, float *work); +float BLAS_FUNC(slange)(char *norm, int *m, int *n, float *a, int *lda, float *work); +float BLAS_FUNC(slangt)(char *norm, int *n, float *dl, float *d, float *du); +float BLAS_FUNC(slanhs)(char *norm, int *n, float *a, int *lda, float *work); +float BLAS_FUNC(slansb)(char *norm, char *uplo, int *n, int *k, float *ab, int *ldab, float *work); +float BLAS_FUNC(slansf)(char *norm, char *transr, char *uplo, int *n, float *a, float *work); +float BLAS_FUNC(slansp)(char *norm, char *uplo, int *n, float *ap, float *work); +float BLAS_FUNC(slanst)(char *norm, int *n, float *d, float *e); +float BLAS_FUNC(slansy)(char *norm, char *uplo, int *n, float *a, int *lda, float *work); +float BLAS_FUNC(slantb)(char *norm, char *uplo, char *diag, int *n, int *k, float *ab, int *ldab, float *work); +float BLAS_FUNC(slantp)(char *norm, char *uplo, char *diag, int *n, float *ap, float *work); +float BLAS_FUNC(slantr)(char *norm, char *uplo, char *diag, int *m, int *n, float *a, int *lda, float *work); +void BLAS_FUNC(slanv2)(float *a, float *b, float *c, float *d, float *rt1r, float *rt1i, float *rt2r, float *rt2i, float *cs, float *sn); +void BLAS_FUNC(slapll)(int *n, float *x, int *incx, float *y, int *incy, float *ssmin); +void BLAS_FUNC(slapmr)(int *forwrd, int *m, int *n, float *x, int *ldx, int *k); +void BLAS_FUNC(slapmt)(int *forwrd, int *m, int *n, float *x, int *ldx, int *k); +float BLAS_FUNC(slapy2)(float *x, float *y); +float BLAS_FUNC(slapy3)(float *x, float *y, float *z); +void BLAS_FUNC(slaqgb)(int *m, int *n, int *kl, int *ku, float *ab, int *ldab, float *r, float *c, float *rowcnd, float *colcnd, float *amax, char *equed); +void BLAS_FUNC(slaqge)(int *m, int *n, float *a, int *lda, float *r, float *c, float *rowcnd, float *colcnd, float *amax, char *equed); +void BLAS_FUNC(slaqp2)(int *m, int *n, int *offset, float *a, int *lda, int *jpvt, float *tau, float *vn1, float *vn2, float *work); +void BLAS_FUNC(slaqps)(int *m, int *n, int *offset, int *nb, int *kb, float *a, int *lda, int *jpvt, float *tau, float *vn1, float *vn2, float *auxv, float *f, int *ldf); +void BLAS_FUNC(slaqr0)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, float *h, int *ldh, float *wr, float *wi, int *iloz, int *ihiz, float *z, int *ldz, float *work, int *lwork, int *info); +void BLAS_FUNC(slaqr1)(int *n, float *h, int *ldh, float *sr1, float *si1, float *sr2, float *si2, float *v); +void BLAS_FUNC(slaqr2)(int *wantt, int *wantz, int *n, int *ktop, int *kbot, int *nw, float *h, int *ldh, int *iloz, int *ihiz, float *z, int *ldz, int *ns, int *nd, float *sr, float *si, float *v, int *ldv, int *nh, float *t, int *ldt, int *nv, float *wv, int *ldwv, float *work, int *lwork); +void BLAS_FUNC(slaqr3)(int *wantt, int *wantz, int *n, int *ktop, int *kbot, int *nw, float *h, int *ldh, int *iloz, int *ihiz, float *z, int *ldz, int *ns, int *nd, float *sr, float *si, float *v, int *ldv, int *nh, float *t, int *ldt, int *nv, float *wv, int *ldwv, float *work, int *lwork); +void BLAS_FUNC(slaqr4)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, float *h, int *ldh, float *wr, float *wi, int *iloz, int *ihiz, float *z, int *ldz, float *work, int *lwork, int *info); +void BLAS_FUNC(slaqr5)(int *wantt, int *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, float *sr, float *si, float *h, int *ldh, int *iloz, int *ihiz, float *z, int *ldz, float *v, int *ldv, float *u, int *ldu, int *nv, float *wv, int *ldwv, int *nh, float *wh, int *ldwh); +void BLAS_FUNC(slaqsb)(char *uplo, int *n, int *kd, float *ab, int *ldab, float *s, float *scond, float *amax, char *equed); +void BLAS_FUNC(slaqsp)(char *uplo, int *n, float *ap, float *s, float *scond, float *amax, char *equed); +void BLAS_FUNC(slaqsy)(char *uplo, int *n, float *a, int *lda, float *s, float *scond, float *amax, char *equed); +void BLAS_FUNC(slaqtr)(int *ltran, int *lreal, int *n, float *t, int *ldt, float *b, float *w, float *scale, float *x, float *work, int *info); +void BLAS_FUNC(slar1v)(int *n, int *b1, int *bn, float *lambda_, float *d, float *l, float *ld, float *lld, float *pivmin, float *gaptol, float *z, int *wantnc, int *negcnt, float *ztz, float *mingma, int *r, int *isuppz, float *nrminv, float *resid, float *rqcorr, float *work); +void BLAS_FUNC(slar2v)(int *n, float *x, float *y, float *z, int *incx, float *c, float *s, int *incc); +void BLAS_FUNC(slarf)(char *side, int *m, int *n, float *v, int *incv, float *tau, float *c, int *ldc, float *work); +void BLAS_FUNC(slarfb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, float *v, int *ldv, float *t, int *ldt, float *c, int *ldc, float *work, int *ldwork); +void BLAS_FUNC(slarfg)(int *n, float *alpha, float *x, int *incx, float *tau); +void BLAS_FUNC(slarfgp)(int *n, float *alpha, float *x, int *incx, float *tau); +void BLAS_FUNC(slarft)(char *direct, char *storev, int *n, int *k, float *v, int *ldv, float *tau, float *t, int *ldt); +void BLAS_FUNC(slarfx)(char *side, int *m, int *n, float *v, float *tau, float *c, int *ldc, float *work); +void BLAS_FUNC(slargv)(int *n, float *x, int *incx, float *y, int *incy, float *c, int *incc); +void BLAS_FUNC(slarnv)(int *idist, int *iseed, int *n, float *x); +void BLAS_FUNC(slarra)(int *n, float *d, float *e, float *e2, float *spltol, float *tnrm, int *nsplit, int *isplit, int *info); +void BLAS_FUNC(slarrb)(int *n, float *d, float *lld, int *ifirst, int *ilast, float *rtol1, float *rtol2, int *offset, float *w, float *wgap, float *werr, float *work, int *iwork, float *pivmin, float *spdiam, int *twist, int *info); +void BLAS_FUNC(slarrc)(char *jobt, int *n, float *vl, float *vu, float *d, float *e, float *pivmin, int *eigcnt, int *lcnt, int *rcnt, int *info); +void BLAS_FUNC(slarrd)(char *range, char *order, int *n, float *vl, float *vu, int *il, int *iu, float *gers, float *reltol, float *d, float *e, float *e2, float *pivmin, int *nsplit, int *isplit, int *m, float *w, float *werr, float *wl, float *wu, int *iblock, int *indexw, float *work, int *iwork, int *info); +void BLAS_FUNC(slarre)(char *range, int *n, float *vl, float *vu, int *il, int *iu, float *d, float *e, float *e2, float *rtol1, float *rtol2, float *spltol, int *nsplit, int *isplit, int *m, float *w, float *werr, float *wgap, int *iblock, int *indexw, float *gers, float *pivmin, float *work, int *iwork, int *info); +void BLAS_FUNC(slarrf)(int *n, float *d, float *l, float *ld, int *clstrt, int *clend, float *w, float *wgap, float *werr, float *spdiam, float *clgapl, float *clgapr, float *pivmin, float *sigma, float *dplus, float *lplus, float *work, int *info); +void BLAS_FUNC(slarrj)(int *n, float *d, float *e2, int *ifirst, int *ilast, float *rtol, int *offset, float *w, float *werr, float *work, int *iwork, float *pivmin, float *spdiam, int *info); +void BLAS_FUNC(slarrk)(int *n, int *iw, float *gl, float *gu, float *d, float *e2, float *pivmin, float *reltol, float *w, float *werr, int *info); +void BLAS_FUNC(slarrr)(int *n, float *d, float *e, int *info); +void BLAS_FUNC(slarrv)(int *n, float *vl, float *vu, float *d, float *l, float *pivmin, int *isplit, int *m, int *dol, int *dou, float *minrgp, float *rtol1, float *rtol2, float *w, float *werr, float *wgap, int *iblock, int *indexw, float *gers, float *z, int *ldz, int *isuppz, float *work, int *iwork, int *info); +void BLAS_FUNC(slartg)(float *f, float *g, float *cs, float *sn, float *r); +void BLAS_FUNC(slartgp)(float *f, float *g, float *cs, float *sn, float *r); +void BLAS_FUNC(slartgs)(float *x, float *y, float *sigma, float *cs, float *sn); +void BLAS_FUNC(slartv)(int *n, float *x, int *incx, float *y, int *incy, float *c, float *s, int *incc); +void BLAS_FUNC(slaruv)(int *iseed, int *n, float *x); +void BLAS_FUNC(slarz)(char *side, int *m, int *n, int *l, float *v, int *incv, float *tau, float *c, int *ldc, float *work); +void BLAS_FUNC(slarzb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, float *v, int *ldv, float *t, int *ldt, float *c, int *ldc, float *work, int *ldwork); +void BLAS_FUNC(slarzt)(char *direct, char *storev, int *n, int *k, float *v, int *ldv, float *tau, float *t, int *ldt); +void BLAS_FUNC(slas2)(float *f, float *g, float *h, float *ssmin, float *ssmax); +void BLAS_FUNC(slascl)(char *type_bn, int *kl, int *ku, float *cfrom, float *cto, int *m, int *n, float *a, int *lda, int *info); +void BLAS_FUNC(slasd0)(int *n, int *sqre, float *d, float *e, float *u, int *ldu, float *vt, int *ldvt, int *smlsiz, int *iwork, float *work, int *info); +void BLAS_FUNC(slasd1)(int *nl, int *nr, int *sqre, float *d, float *alpha, float *beta, float *u, int *ldu, float *vt, int *ldvt, int *idxq, int *iwork, float *work, int *info); +void BLAS_FUNC(slasd2)(int *nl, int *nr, int *sqre, int *k, float *d, float *z, float *alpha, float *beta, float *u, int *ldu, float *vt, int *ldvt, float *dsigma, float *u2, int *ldu2, float *vt2, int *ldvt2, int *idxp, int *idx, int *idxc, int *idxq, int *coltyp, int *info); +void BLAS_FUNC(slasd3)(int *nl, int *nr, int *sqre, int *k, float *d, float *q, int *ldq, float *dsigma, float *u, int *ldu, float *u2, int *ldu2, float *vt, int *ldvt, float *vt2, int *ldvt2, int *idxc, int *ctot, float *z, int *info); +void BLAS_FUNC(slasd4)(int *n, int *i, float *d, float *z, float *delta, float *rho, float *sigma, float *work, int *info); +void BLAS_FUNC(slasd5)(int *i, float *d, float *z, float *delta, float *rho, float *dsigma, float *work); +void BLAS_FUNC(slasd6)(int *icompq, int *nl, int *nr, int *sqre, float *d, float *vf, float *vl, float *alpha, float *beta, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, float *givnum, int *ldgnum, float *poles, float *difl, float *difr, float *z, int *k, float *c, float *s, float *work, int *iwork, int *info); +void BLAS_FUNC(slasd7)(int *icompq, int *nl, int *nr, int *sqre, int *k, float *d, float *z, float *zw, float *vf, float *vfw, float *vl, float *vlw, float *alpha, float *beta, float *dsigma, int *idx, int *idxp, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, float *givnum, int *ldgnum, float *c, float *s, int *info); +void BLAS_FUNC(slasd8)(int *icompq, int *k, float *d, float *z, float *vf, float *vl, float *difl, float *difr, int *lddifr, float *dsigma, float *work, int *info); +void BLAS_FUNC(slasda)(int *icompq, int *smlsiz, int *n, int *sqre, float *d, float *e, float *u, int *ldu, float *vt, int *k, float *difl, float *difr, float *z, float *poles, int *givptr, int *givcol, int *ldgcol, int *perm, float *givnum, float *c, float *s, float *work, int *iwork, int *info); +void BLAS_FUNC(slasdq)(char *uplo, int *sqre, int *n, int *ncvt, int *nru, int *ncc, float *d, float *e, float *vt, int *ldvt, float *u, int *ldu, float *c, int *ldc, float *work, int *info); +void BLAS_FUNC(slasdt)(int *n, int *lvl, int *nd, int *inode, int *ndiml, int *ndimr, int *msub); +void BLAS_FUNC(slaset)(char *uplo, int *m, int *n, float *alpha, float *beta, float *a, int *lda); +void BLAS_FUNC(slasq1)(int *n, float *d, float *e, float *work, int *info); +void BLAS_FUNC(slasq2)(int *n, float *z, int *info); +void BLAS_FUNC(slasq3)(int *i0, int *n0, float *z, int *pp, float *dmin, float *sigma, float *desig, float *qmax, int *nfail, int *iter, int *ndiv, int *ieee, int *ttype, float *dmin1, float *dmin2, float *dn, float *dn1, float *dn2, float *g, float *tau); +void BLAS_FUNC(slasq4)(int *i0, int *n0, float *z, int *pp, int *n0in, float *dmin, float *dmin1, float *dmin2, float *dn, float *dn1, float *dn2, float *tau, int *ttype, float *g); +void BLAS_FUNC(slasq6)(int *i0, int *n0, float *z, int *pp, float *dmin, float *dmin1, float *dmin2, float *dn, float *dnm1, float *dnm2); +void BLAS_FUNC(slasr)(char *side, char *pivot, char *direct, int *m, int *n, float *c, float *s, float *a, int *lda); +void BLAS_FUNC(slasrt)(char *id, int *n, float *d, int *info); +void BLAS_FUNC(slassq)(int *n, float *x, int *incx, float *scale, float *sumsq); +void BLAS_FUNC(slasv2)(float *f, float *g, float *h, float *ssmin, float *ssmax, float *snr, float *csr, float *snl, float *csl); +void BLAS_FUNC(slaswp)(int *n, float *a, int *lda, int *k1, int *k2, int *ipiv, int *incx); +void BLAS_FUNC(slasy2)(int *ltranl, int *ltranr, int *isgn, int *n1, int *n2, float *tl, int *ldtl, float *tr, int *ldtr, float *b, int *ldb, float *scale, float *x, int *ldx, float *xnorm, int *info); +void BLAS_FUNC(slasyf)(char *uplo, int *n, int *nb, int *kb, float *a, int *lda, int *ipiv, float *w, int *ldw, int *info); +void BLAS_FUNC(slatbs)(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, float *ab, int *ldab, float *x, float *scale, float *cnorm, int *info); +void BLAS_FUNC(slatdf)(int *ijob, int *n, float *z, int *ldz, float *rhs, float *rdsum, float *rdscal, int *ipiv, int *jpiv); +void BLAS_FUNC(slatps)(char *uplo, char *trans, char *diag, char *normin, int *n, float *ap, float *x, float *scale, float *cnorm, int *info); +void BLAS_FUNC(slatrd)(char *uplo, int *n, int *nb, float *a, int *lda, float *e, float *tau, float *w, int *ldw); +void BLAS_FUNC(slatrs)(char *uplo, char *trans, char *diag, char *normin, int *n, float *a, int *lda, float *x, float *scale, float *cnorm, int *info); +void BLAS_FUNC(slatrz)(int *m, int *n, int *l, float *a, int *lda, float *tau, float *work); +void BLAS_FUNC(slauu2)(char *uplo, int *n, float *a, int *lda, int *info); +void BLAS_FUNC(slauum)(char *uplo, int *n, float *a, int *lda, int *info); +void BLAS_FUNC(sopgtr)(char *uplo, int *n, float *ap, float *tau, float *q, int *ldq, float *work, int *info); +void BLAS_FUNC(sopmtr)(char *side, char *uplo, char *trans, int *m, int *n, float *ap, float *tau, float *c, int *ldc, float *work, int *info); +void BLAS_FUNC(sorbdb)(char *trans, char *signs, int *m, int *p, int *q, float *x11, int *ldx11, float *x12, int *ldx12, float *x21, int *ldx21, float *x22, int *ldx22, float *theta, float *phi, float *taup1, float *taup2, float *tauq1, float *tauq2, float *work, int *lwork, int *info); +void BLAS_FUNC(sorcsd)(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, float *x11, int *ldx11, float *x12, int *ldx12, float *x21, int *ldx21, float *x22, int *ldx22, float *theta, float *u1, int *ldu1, float *u2, int *ldu2, float *v1t, int *ldv1t, float *v2t, int *ldv2t, float *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(sorg2l)(int *m, int *n, int *k, float *a, int *lda, float *tau, float *work, int *info); +void BLAS_FUNC(sorg2r)(int *m, int *n, int *k, float *a, int *lda, float *tau, float *work, int *info); +void BLAS_FUNC(sorgbr)(char *vect, int *m, int *n, int *k, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sorghr)(int *n, int *ilo, int *ihi, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sorgl2)(int *m, int *n, int *k, float *a, int *lda, float *tau, float *work, int *info); +void BLAS_FUNC(sorglq)(int *m, int *n, int *k, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sorgql)(int *m, int *n, int *k, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sorgqr)(int *m, int *n, int *k, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sorgr2)(int *m, int *n, int *k, float *a, int *lda, float *tau, float *work, int *info); +void BLAS_FUNC(sorgrq)(int *m, int *n, int *k, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sorgtr)(char *uplo, int *n, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(sorm2l)(char *side, char *trans, int *m, int *n, int *k, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *info); +void BLAS_FUNC(sorm2r)(char *side, char *trans, int *m, int *n, int *k, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *info); +void BLAS_FUNC(sormbr)(char *vect, char *side, char *trans, int *m, int *n, int *k, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *lwork, int *info); +void BLAS_FUNC(sormhr)(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *lwork, int *info); +void BLAS_FUNC(sorml2)(char *side, char *trans, int *m, int *n, int *k, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *info); +void BLAS_FUNC(sormlq)(char *side, char *trans, int *m, int *n, int *k, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *lwork, int *info); +void BLAS_FUNC(sormql)(char *side, char *trans, int *m, int *n, int *k, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *lwork, int *info); +void BLAS_FUNC(sormqr)(char *side, char *trans, int *m, int *n, int *k, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *lwork, int *info); +void BLAS_FUNC(sormr2)(char *side, char *trans, int *m, int *n, int *k, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *info); +void BLAS_FUNC(sormr3)(char *side, char *trans, int *m, int *n, int *k, int *l, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *info); +void BLAS_FUNC(sormrq)(char *side, char *trans, int *m, int *n, int *k, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *lwork, int *info); +void BLAS_FUNC(sormrz)(char *side, char *trans, int *m, int *n, int *k, int *l, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *lwork, int *info); +void BLAS_FUNC(sormtr)(char *side, char *uplo, char *trans, int *m, int *n, float *a, int *lda, float *tau, float *c, int *ldc, float *work, int *lwork, int *info); +void BLAS_FUNC(spbcon)(char *uplo, int *n, int *kd, float *ab, int *ldab, float *anorm, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(spbequ)(char *uplo, int *n, int *kd, float *ab, int *ldab, float *s, float *scond, float *amax, int *info); +void BLAS_FUNC(spbrfs)(char *uplo, int *n, int *kd, int *nrhs, float *ab, int *ldab, float *afb, int *ldafb, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(spbstf)(char *uplo, int *n, int *kd, float *ab, int *ldab, int *info); +void BLAS_FUNC(spbsv)(char *uplo, int *n, int *kd, int *nrhs, float *ab, int *ldab, float *b, int *ldb, int *info); +void BLAS_FUNC(spbsvx)(char *fact, char *uplo, int *n, int *kd, int *nrhs, float *ab, int *ldab, float *afb, int *ldafb, char *equed, float *s, float *b, int *ldb, float *x, int *ldx, float *rcond, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(spbtf2)(char *uplo, int *n, int *kd, float *ab, int *ldab, int *info); +void BLAS_FUNC(spbtrf)(char *uplo, int *n, int *kd, float *ab, int *ldab, int *info); +void BLAS_FUNC(spbtrs)(char *uplo, int *n, int *kd, int *nrhs, float *ab, int *ldab, float *b, int *ldb, int *info); +void BLAS_FUNC(spftrf)(char *transr, char *uplo, int *n, float *a, int *info); +void BLAS_FUNC(spftri)(char *transr, char *uplo, int *n, float *a, int *info); +void BLAS_FUNC(spftrs)(char *transr, char *uplo, int *n, int *nrhs, float *a, float *b, int *ldb, int *info); +void BLAS_FUNC(spocon)(char *uplo, int *n, float *a, int *lda, float *anorm, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(spoequ)(int *n, float *a, int *lda, float *s, float *scond, float *amax, int *info); +void BLAS_FUNC(spoequb)(int *n, float *a, int *lda, float *s, float *scond, float *amax, int *info); +void BLAS_FUNC(sporfs)(char *uplo, int *n, int *nrhs, float *a, int *lda, float *af, int *ldaf, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(sposv)(char *uplo, int *n, int *nrhs, float *a, int *lda, float *b, int *ldb, int *info); +void BLAS_FUNC(sposvx)(char *fact, char *uplo, int *n, int *nrhs, float *a, int *lda, float *af, int *ldaf, char *equed, float *s, float *b, int *ldb, float *x, int *ldx, float *rcond, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(spotf2)(char *uplo, int *n, float *a, int *lda, int *info); +void BLAS_FUNC(spotrf)(char *uplo, int *n, float *a, int *lda, int *info); +void BLAS_FUNC(spotri)(char *uplo, int *n, float *a, int *lda, int *info); +void BLAS_FUNC(spotrs)(char *uplo, int *n, int *nrhs, float *a, int *lda, float *b, int *ldb, int *info); +void BLAS_FUNC(sppcon)(char *uplo, int *n, float *ap, float *anorm, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(sppequ)(char *uplo, int *n, float *ap, float *s, float *scond, float *amax, int *info); +void BLAS_FUNC(spprfs)(char *uplo, int *n, int *nrhs, float *ap, float *afp, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(sppsv)(char *uplo, int *n, int *nrhs, float *ap, float *b, int *ldb, int *info); +void BLAS_FUNC(sppsvx)(char *fact, char *uplo, int *n, int *nrhs, float *ap, float *afp, char *equed, float *s, float *b, int *ldb, float *x, int *ldx, float *rcond, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(spptrf)(char *uplo, int *n, float *ap, int *info); +void BLAS_FUNC(spptri)(char *uplo, int *n, float *ap, int *info); +void BLAS_FUNC(spptrs)(char *uplo, int *n, int *nrhs, float *ap, float *b, int *ldb, int *info); +void BLAS_FUNC(spstf2)(char *uplo, int *n, float *a, int *lda, int *piv, int *rank, float *tol, float *work, int *info); +void BLAS_FUNC(spstrf)(char *uplo, int *n, float *a, int *lda, int *piv, int *rank, float *tol, float *work, int *info); +void BLAS_FUNC(sptcon)(int *n, float *d, float *e, float *anorm, float *rcond, float *work, int *info); +void BLAS_FUNC(spteqr)(char *compz, int *n, float *d, float *e, float *z, int *ldz, float *work, int *info); +void BLAS_FUNC(sptrfs)(int *n, int *nrhs, float *d, float *e, float *df, float *ef, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *info); +void BLAS_FUNC(sptsv)(int *n, int *nrhs, float *d, float *e, float *b, int *ldb, int *info); +void BLAS_FUNC(sptsvx)(char *fact, int *n, int *nrhs, float *d, float *e, float *df, float *ef, float *b, int *ldb, float *x, int *ldx, float *rcond, float *ferr, float *berr, float *work, int *info); +void BLAS_FUNC(spttrf)(int *n, float *d, float *e, int *info); +void BLAS_FUNC(spttrs)(int *n, int *nrhs, float *d, float *e, float *b, int *ldb, int *info); +void BLAS_FUNC(sptts2)(int *n, int *nrhs, float *d, float *e, float *b, int *ldb); +void BLAS_FUNC(srscl)(int *n, float *sa, float *sx, int *incx); +void BLAS_FUNC(ssbev)(char *jobz, char *uplo, int *n, int *kd, float *ab, int *ldab, float *w, float *z, int *ldz, float *work, int *info); +void BLAS_FUNC(ssbevd)(char *jobz, char *uplo, int *n, int *kd, float *ab, int *ldab, float *w, float *z, int *ldz, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(ssbevx)(char *jobz, char *range, char *uplo, int *n, int *kd, float *ab, int *ldab, float *q, int *ldq, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, float *z, int *ldz, float *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(ssbgst)(char *vect, char *uplo, int *n, int *ka, int *kb, float *ab, int *ldab, float *bb, int *ldbb, float *x, int *ldx, float *work, int *info); +void BLAS_FUNC(ssbgv)(char *jobz, char *uplo, int *n, int *ka, int *kb, float *ab, int *ldab, float *bb, int *ldbb, float *w, float *z, int *ldz, float *work, int *info); +void BLAS_FUNC(ssbgvd)(char *jobz, char *uplo, int *n, int *ka, int *kb, float *ab, int *ldab, float *bb, int *ldbb, float *w, float *z, int *ldz, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(ssbgvx)(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, float *ab, int *ldab, float *bb, int *ldbb, float *q, int *ldq, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, float *z, int *ldz, float *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(ssbtrd)(char *vect, char *uplo, int *n, int *kd, float *ab, int *ldab, float *d, float *e, float *q, int *ldq, float *work, int *info); +void BLAS_FUNC(ssfrk)(char *transr, char *uplo, char *trans, int *n, int *k, float *alpha, float *a, int *lda, float *beta, float *c); +void BLAS_FUNC(sspcon)(char *uplo, int *n, float *ap, int *ipiv, float *anorm, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(sspev)(char *jobz, char *uplo, int *n, float *ap, float *w, float *z, int *ldz, float *work, int *info); +void BLAS_FUNC(sspevd)(char *jobz, char *uplo, int *n, float *ap, float *w, float *z, int *ldz, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(sspevx)(char *jobz, char *range, char *uplo, int *n, float *ap, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, float *z, int *ldz, float *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(sspgst)(int *itype, char *uplo, int *n, float *ap, float *bp, int *info); +void BLAS_FUNC(sspgv)(int *itype, char *jobz, char *uplo, int *n, float *ap, float *bp, float *w, float *z, int *ldz, float *work, int *info); +void BLAS_FUNC(sspgvd)(int *itype, char *jobz, char *uplo, int *n, float *ap, float *bp, float *w, float *z, int *ldz, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(sspgvx)(int *itype, char *jobz, char *range, char *uplo, int *n, float *ap, float *bp, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, float *z, int *ldz, float *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(ssprfs)(char *uplo, int *n, int *nrhs, float *ap, float *afp, int *ipiv, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(sspsv)(char *uplo, int *n, int *nrhs, float *ap, int *ipiv, float *b, int *ldb, int *info); +void BLAS_FUNC(sspsvx)(char *fact, char *uplo, int *n, int *nrhs, float *ap, float *afp, int *ipiv, float *b, int *ldb, float *x, int *ldx, float *rcond, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(ssptrd)(char *uplo, int *n, float *ap, float *d, float *e, float *tau, int *info); +void BLAS_FUNC(ssptrf)(char *uplo, int *n, float *ap, int *ipiv, int *info); +void BLAS_FUNC(ssptri)(char *uplo, int *n, float *ap, int *ipiv, float *work, int *info); +void BLAS_FUNC(ssptrs)(char *uplo, int *n, int *nrhs, float *ap, int *ipiv, float *b, int *ldb, int *info); +void BLAS_FUNC(sstebz)(char *range, char *order, int *n, float *vl, float *vu, int *il, int *iu, float *abstol, float *d, float *e, int *m, int *nsplit, float *w, int *iblock, int *isplit, float *work, int *iwork, int *info); +void BLAS_FUNC(sstedc)(char *compz, int *n, float *d, float *e, float *z, int *ldz, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(sstegr)(char *jobz, char *range, int *n, float *d, float *e, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, float *z, int *ldz, int *isuppz, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(sstein)(int *n, float *d, float *e, int *m, float *w, int *iblock, int *isplit, float *z, int *ldz, float *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(sstemr)(char *jobz, char *range, int *n, float *d, float *e, float *vl, float *vu, int *il, int *iu, int *m, float *w, float *z, int *ldz, int *nzc, int *isuppz, int *tryrac, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(ssteqr)(char *compz, int *n, float *d, float *e, float *z, int *ldz, float *work, int *info); +void BLAS_FUNC(ssterf)(int *n, float *d, float *e, int *info); +void BLAS_FUNC(sstev)(char *jobz, int *n, float *d, float *e, float *z, int *ldz, float *work, int *info); +void BLAS_FUNC(sstevd)(char *jobz, int *n, float *d, float *e, float *z, int *ldz, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(sstevr)(char *jobz, char *range, int *n, float *d, float *e, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, float *z, int *ldz, int *isuppz, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(sstevx)(char *jobz, char *range, int *n, float *d, float *e, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, float *z, int *ldz, float *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(ssycon)(char *uplo, int *n, float *a, int *lda, int *ipiv, float *anorm, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(ssyconv)(char *uplo, char *way, int *n, float *a, int *lda, int *ipiv, float *work, int *info); +void BLAS_FUNC(ssyequb)(char *uplo, int *n, float *a, int *lda, float *s, float *scond, float *amax, float *work, int *info); +void BLAS_FUNC(ssyev)(char *jobz, char *uplo, int *n, float *a, int *lda, float *w, float *work, int *lwork, int *info); +void BLAS_FUNC(ssyevd)(char *jobz, char *uplo, int *n, float *a, int *lda, float *w, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(ssyevr)(char *jobz, char *range, char *uplo, int *n, float *a, int *lda, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, float *z, int *ldz, int *isuppz, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(ssyevx)(char *jobz, char *range, char *uplo, int *n, float *a, int *lda, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, float *z, int *ldz, float *work, int *lwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(ssygs2)(int *itype, char *uplo, int *n, float *a, int *lda, float *b, int *ldb, int *info); +void BLAS_FUNC(ssygst)(int *itype, char *uplo, int *n, float *a, int *lda, float *b, int *ldb, int *info); +void BLAS_FUNC(ssygv)(int *itype, char *jobz, char *uplo, int *n, float *a, int *lda, float *b, int *ldb, float *w, float *work, int *lwork, int *info); +void BLAS_FUNC(ssygvd)(int *itype, char *jobz, char *uplo, int *n, float *a, int *lda, float *b, int *ldb, float *w, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(ssygvx)(int *itype, char *jobz, char *range, char *uplo, int *n, float *a, int *lda, float *b, int *ldb, float *vl, float *vu, int *il, int *iu, float *abstol, int *m, float *w, float *z, int *ldz, float *work, int *lwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(ssyrfs)(char *uplo, int *n, int *nrhs, float *a, int *lda, float *af, int *ldaf, int *ipiv, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(ssysv)(char *uplo, int *n, int *nrhs, float *a, int *lda, int *ipiv, float *b, int *ldb, float *work, int *lwork, int *info); +void BLAS_FUNC(ssysvx)(char *fact, char *uplo, int *n, int *nrhs, float *a, int *lda, float *af, int *ldaf, int *ipiv, float *b, int *ldb, float *x, int *ldx, float *rcond, float *ferr, float *berr, float *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(ssyswapr)(char *uplo, int *n, float *a, int *lda, int *i1, int *i2); +void BLAS_FUNC(ssytd2)(char *uplo, int *n, float *a, int *lda, float *d, float *e, float *tau, int *info); +void BLAS_FUNC(ssytf2)(char *uplo, int *n, float *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(ssytrd)(char *uplo, int *n, float *a, int *lda, float *d, float *e, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(ssytrf)(char *uplo, int *n, float *a, int *lda, int *ipiv, float *work, int *lwork, int *info); +void BLAS_FUNC(ssytri)(char *uplo, int *n, float *a, int *lda, int *ipiv, float *work, int *info); +void BLAS_FUNC(ssytri2)(char *uplo, int *n, float *a, int *lda, int *ipiv, float *work, int *lwork, int *info); +void BLAS_FUNC(ssytri2x)(char *uplo, int *n, float *a, int *lda, int *ipiv, float *work, int *nb, int *info); +void BLAS_FUNC(ssytrs)(char *uplo, int *n, int *nrhs, float *a, int *lda, int *ipiv, float *b, int *ldb, int *info); +void BLAS_FUNC(ssytrs2)(char *uplo, int *n, int *nrhs, float *a, int *lda, int *ipiv, float *b, int *ldb, float *work, int *info); +void BLAS_FUNC(stbcon)(char *norm, char *uplo, char *diag, int *n, int *kd, float *ab, int *ldab, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(stbrfs)(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, float *ab, int *ldab, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(stbtrs)(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, float *ab, int *ldab, float *b, int *ldb, int *info); +void BLAS_FUNC(stfsm)(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, float *alpha, float *a, float *b, int *ldb); +void BLAS_FUNC(stftri)(char *transr, char *uplo, char *diag, int *n, float *a, int *info); +void BLAS_FUNC(stfttp)(char *transr, char *uplo, int *n, float *arf, float *ap, int *info); +void BLAS_FUNC(stfttr)(char *transr, char *uplo, int *n, float *arf, float *a, int *lda, int *info); +void BLAS_FUNC(stgevc)(char *side, char *howmny, int *select, int *n, float *s, int *lds, float *p, int *ldp, float *vl, int *ldvl, float *vr, int *ldvr, int *mm, int *m, float *work, int *info); +void BLAS_FUNC(stgex2)(int *wantq, int *wantz, int *n, float *a, int *lda, float *b, int *ldb, float *q, int *ldq, float *z, int *ldz, int *j1, int *n1, int *n2, float *work, int *lwork, int *info); +void BLAS_FUNC(stgexc)(int *wantq, int *wantz, int *n, float *a, int *lda, float *b, int *ldb, float *q, int *ldq, float *z, int *ldz, int *ifst, int *ilst, float *work, int *lwork, int *info); +void BLAS_FUNC(stgsen)(int *ijob, int *wantq, int *wantz, int *select, int *n, float *a, int *lda, float *b, int *ldb, float *alphar, float *alphai, float *beta, float *q, int *ldq, float *z, int *ldz, int *m, float *pl, float *pr, float *dif, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(stgsja)(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, float *a, int *lda, float *b, int *ldb, float *tola, float *tolb, float *alpha, float *beta, float *u, int *ldu, float *v, int *ldv, float *q, int *ldq, float *work, int *ncycle, int *info); +void BLAS_FUNC(stgsna)(char *job, char *howmny, int *select, int *n, float *a, int *lda, float *b, int *ldb, float *vl, int *ldvl, float *vr, int *ldvr, float *s, float *dif, int *mm, int *m, float *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(stgsy2)(char *trans, int *ijob, int *m, int *n, float *a, int *lda, float *b, int *ldb, float *c, int *ldc, float *d, int *ldd, float *e, int *lde, float *f, int *ldf, float *scale, float *rdsum, float *rdscal, int *iwork, int *pq, int *info); +void BLAS_FUNC(stgsyl)(char *trans, int *ijob, int *m, int *n, float *a, int *lda, float *b, int *ldb, float *c, int *ldc, float *d, int *ldd, float *e, int *lde, float *f, int *ldf, float *scale, float *dif, float *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(stpcon)(char *norm, char *uplo, char *diag, int *n, float *ap, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(stpmqrt)(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, float *v, int *ldv, float *t, int *ldt, float *a, int *lda, float *b, int *ldb, float *work, int *info); +void BLAS_FUNC(stpqrt)(int *m, int *n, int *l, int *nb, float *a, int *lda, float *b, int *ldb, float *t, int *ldt, float *work, int *info); +void BLAS_FUNC(stpqrt2)(int *m, int *n, int *l, float *a, int *lda, float *b, int *ldb, float *t, int *ldt, int *info); +void BLAS_FUNC(stprfb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, float *v, int *ldv, float *t, int *ldt, float *a, int *lda, float *b, int *ldb, float *work, int *ldwork); +void BLAS_FUNC(stprfs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, float *ap, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(stptri)(char *uplo, char *diag, int *n, float *ap, int *info); +void BLAS_FUNC(stptrs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, float *ap, float *b, int *ldb, int *info); +void BLAS_FUNC(stpttf)(char *transr, char *uplo, int *n, float *ap, float *arf, int *info); +void BLAS_FUNC(stpttr)(char *uplo, int *n, float *ap, float *a, int *lda, int *info); +void BLAS_FUNC(strcon)(char *norm, char *uplo, char *diag, int *n, float *a, int *lda, float *rcond, float *work, int *iwork, int *info); +void BLAS_FUNC(strevc)(char *side, char *howmny, int *select, int *n, float *t, int *ldt, float *vl, int *ldvl, float *vr, int *ldvr, int *mm, int *m, float *work, int *info); +void BLAS_FUNC(strexc)(char *compq, int *n, float *t, int *ldt, float *q, int *ldq, int *ifst, int *ilst, float *work, int *info); +void BLAS_FUNC(strrfs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, float *a, int *lda, float *b, int *ldb, float *x, int *ldx, float *ferr, float *berr, float *work, int *iwork, int *info); +void BLAS_FUNC(strsen)(char *job, char *compq, int *select, int *n, float *t, int *ldt, float *q, int *ldq, float *wr, float *wi, int *m, float *s, float *sep, float *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(strsna)(char *job, char *howmny, int *select, int *n, float *t, int *ldt, float *vl, int *ldvl, float *vr, int *ldvr, float *s, float *sep, int *mm, int *m, float *work, int *ldwork, int *iwork, int *info); +void BLAS_FUNC(strsyl)(char *trana, char *tranb, int *isgn, int *m, int *n, float *a, int *lda, float *b, int *ldb, float *c, int *ldc, float *scale, int *info); +void BLAS_FUNC(strti2)(char *uplo, char *diag, int *n, float *a, int *lda, int *info); +void BLAS_FUNC(strtri)(char *uplo, char *diag, int *n, float *a, int *lda, int *info); +void BLAS_FUNC(strtrs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, float *a, int *lda, float *b, int *ldb, int *info); +void BLAS_FUNC(strttf)(char *transr, char *uplo, int *n, float *a, int *lda, float *arf, int *info); +void BLAS_FUNC(strttp)(char *uplo, int *n, float *a, int *lda, float *ap, int *info); +void BLAS_FUNC(stzrzf)(int *m, int *n, float *a, int *lda, float *tau, float *work, int *lwork, int *info); +void BLAS_FUNC(xerbla_array)(char *srname_array, int *srname_len, int *info); +void BLAS_FUNC(zbbcsd)(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, double *theta, double *phi, npy_complex128 *u1, int *ldu1, npy_complex128 *u2, int *ldu2, npy_complex128 *v1t, int *ldv1t, npy_complex128 *v2t, int *ldv2t, double *b11d, double *b11e, double *b12d, double *b12e, double *b21d, double *b21e, double *b22d, double *b22e, double *rwork, int *lrwork, int *info); +void BLAS_FUNC(zbdsqr)(char *uplo, int *n, int *ncvt, int *nru, int *ncc, double *d, double *e, npy_complex128 *vt, int *ldvt, npy_complex128 *u, int *ldu, npy_complex128 *c, int *ldc, double *rwork, int *info); +void BLAS_FUNC(zcgesv)(int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, npy_complex128 *work, npy_complex64 *swork, double *rwork, int *iter, int *info); +void BLAS_FUNC(zcposv)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, npy_complex128 *work, npy_complex64 *swork, double *rwork, int *iter, int *info); +void BLAS_FUNC(zdrscl)(int *n, double *sa, npy_complex128 *sx, int *incx); +void BLAS_FUNC(zgbbrd)(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, npy_complex128 *ab, int *ldab, double *d, double *e, npy_complex128 *q, int *ldq, npy_complex128 *pt, int *ldpt, npy_complex128 *c, int *ldc, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zgbcon)(char *norm, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, int *ipiv, double *anorm, double *rcond, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zgbequ)(int *m, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, double *r, double *c, double *rowcnd, double *colcnd, double *amax, int *info); +void BLAS_FUNC(zgbequb)(int *m, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, double *r, double *c, double *rowcnd, double *colcnd, double *amax, int *info); +void BLAS_FUNC(zgbrfs)(char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *afb, int *ldafb, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zgbsv)(int *n, int *kl, int *ku, int *nrhs, npy_complex128 *ab, int *ldab, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zgbsvx)(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *afb, int *ldafb, int *ipiv, char *equed, double *r, double *c, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zgbtf2)(int *m, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, int *ipiv, int *info); +void BLAS_FUNC(zgbtrf)(int *m, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, int *ipiv, int *info); +void BLAS_FUNC(zgbtrs)(char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex128 *ab, int *ldab, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zgebak)(char *job, char *side, int *n, int *ilo, int *ihi, double *scale, int *m, npy_complex128 *v, int *ldv, int *info); +void BLAS_FUNC(zgebal)(char *job, int *n, npy_complex128 *a, int *lda, int *ilo, int *ihi, double *scale, int *info); +void BLAS_FUNC(zgebd2)(int *m, int *n, npy_complex128 *a, int *lda, double *d, double *e, npy_complex128 *tauq, npy_complex128 *taup, npy_complex128 *work, int *info); +void BLAS_FUNC(zgebrd)(int *m, int *n, npy_complex128 *a, int *lda, double *d, double *e, npy_complex128 *tauq, npy_complex128 *taup, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgecon)(char *norm, int *n, npy_complex128 *a, int *lda, double *anorm, double *rcond, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zgeequ)(int *m, int *n, npy_complex128 *a, int *lda, double *r, double *c, double *rowcnd, double *colcnd, double *amax, int *info); +void BLAS_FUNC(zgeequb)(int *m, int *n, npy_complex128 *a, int *lda, double *r, double *c, double *rowcnd, double *colcnd, double *amax, int *info); +void BLAS_FUNC(zgees)(char *jobvs, char *sort, _zselect1 *select, int *n, npy_complex128 *a, int *lda, int *sdim, npy_complex128 *w, npy_complex128 *vs, int *ldvs, npy_complex128 *work, int *lwork, double *rwork, int *bwork, int *info); +void BLAS_FUNC(zgeesx)(char *jobvs, char *sort, _zselect1 *select, char *sense, int *n, npy_complex128 *a, int *lda, int *sdim, npy_complex128 *w, npy_complex128 *vs, int *ldvs, double *rconde, double *rcondv, npy_complex128 *work, int *lwork, double *rwork, int *bwork, int *info); +void BLAS_FUNC(zgeev)(char *jobvl, char *jobvr, int *n, npy_complex128 *a, int *lda, npy_complex128 *w, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zgeevx)(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, npy_complex128 *a, int *lda, npy_complex128 *w, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, int *ilo, int *ihi, double *scale, double *abnrm, double *rconde, double *rcondv, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zgehd2)(int *n, int *ilo, int *ihi, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info); +void BLAS_FUNC(zgehrd)(int *n, int *ilo, int *ihi, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgelq2)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info); +void BLAS_FUNC(zgelqf)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgels)(char *trans, int *m, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgelsd)(int *m, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, double *s, double *rcond, int *rank, npy_complex128 *work, int *lwork, double *rwork, int *iwork, int *info); +void BLAS_FUNC(zgelss)(int *m, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, double *s, double *rcond, int *rank, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zgelsy)(int *m, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *jpvt, double *rcond, int *rank, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zgemqrt)(char *side, char *trans, int *m, int *n, int *k, int *nb, npy_complex128 *v, int *ldv, npy_complex128 *t, int *ldt, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info); +void BLAS_FUNC(zgeql2)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info); +void BLAS_FUNC(zgeqlf)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgeqp3)(int *m, int *n, npy_complex128 *a, int *lda, int *jpvt, npy_complex128 *tau, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zgeqr2)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info); +void BLAS_FUNC(zgeqr2p)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info); +void BLAS_FUNC(zgeqrf)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgeqrfp)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgeqrt)(int *m, int *n, int *nb, npy_complex128 *a, int *lda, npy_complex128 *t, int *ldt, npy_complex128 *work, int *info); +void BLAS_FUNC(zgeqrt2)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *t, int *ldt, int *info); +void BLAS_FUNC(zgeqrt3)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *t, int *ldt, int *info); +void BLAS_FUNC(zgerfs)(char *trans, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zgerq2)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info); +void BLAS_FUNC(zgerqf)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgesc2)(int *n, npy_complex128 *a, int *lda, npy_complex128 *rhs, int *ipiv, int *jpiv, double *scale); +void BLAS_FUNC(zgesdd)(char *jobz, int *m, int *n, npy_complex128 *a, int *lda, double *s, npy_complex128 *u, int *ldu, npy_complex128 *vt, int *ldvt, npy_complex128 *work, int *lwork, double *rwork, int *iwork, int *info); +void BLAS_FUNC(zgesv)(int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zgesvd)(char *jobu, char *jobvt, int *m, int *n, npy_complex128 *a, int *lda, double *s, npy_complex128 *u, int *ldu, npy_complex128 *vt, int *ldvt, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zgesvx)(char *fact, char *trans, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, char *equed, double *r, double *c, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zgetc2)(int *n, npy_complex128 *a, int *lda, int *ipiv, int *jpiv, int *info); +void BLAS_FUNC(zgetf2)(int *m, int *n, npy_complex128 *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(zgetrf)(int *m, int *n, npy_complex128 *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(zgetri)(int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgetrs)(char *trans, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zggbak)(char *job, char *side, int *n, int *ilo, int *ihi, double *lscale, double *rscale, int *m, npy_complex128 *v, int *ldv, int *info); +void BLAS_FUNC(zggbal)(char *job, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *ilo, int *ihi, double *lscale, double *rscale, double *work, int *info); +void BLAS_FUNC(zgges)(char *jobvsl, char *jobvsr, char *sort, _zselect2 *selctg, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *sdim, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *vsl, int *ldvsl, npy_complex128 *vsr, int *ldvsr, npy_complex128 *work, int *lwork, double *rwork, int *bwork, int *info); +void BLAS_FUNC(zggesx)(char *jobvsl, char *jobvsr, char *sort, _zselect2 *selctg, char *sense, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *sdim, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *vsl, int *ldvsl, npy_complex128 *vsr, int *ldvsr, double *rconde, double *rcondv, npy_complex128 *work, int *lwork, double *rwork, int *iwork, int *liwork, int *bwork, int *info); +void BLAS_FUNC(zggev)(char *jobvl, char *jobvr, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zggevx)(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, int *ilo, int *ihi, double *lscale, double *rscale, double *abnrm, double *bbnrm, double *rconde, double *rcondv, npy_complex128 *work, int *lwork, double *rwork, int *iwork, int *bwork, int *info); +void BLAS_FUNC(zggglm)(int *n, int *m, int *p, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *d, npy_complex128 *x, npy_complex128 *y, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgghrd)(char *compq, char *compz, int *n, int *ilo, int *ihi, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *q, int *ldq, npy_complex128 *z, int *ldz, int *info); +void BLAS_FUNC(zgglse)(int *m, int *n, int *p, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *c, npy_complex128 *d, npy_complex128 *x, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zggqrf)(int *n, int *m, int *p, npy_complex128 *a, int *lda, npy_complex128 *taua, npy_complex128 *b, int *ldb, npy_complex128 *taub, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zggrqf)(int *m, int *p, int *n, npy_complex128 *a, int *lda, npy_complex128 *taua, npy_complex128 *b, int *ldb, npy_complex128 *taub, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zgtcon)(char *norm, int *n, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *du2, int *ipiv, double *anorm, double *rcond, npy_complex128 *work, int *info); +void BLAS_FUNC(zgtrfs)(char *trans, int *n, int *nrhs, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *dlf, npy_complex128 *df, npy_complex128 *duf, npy_complex128 *du2, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zgtsv)(int *n, int *nrhs, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zgtsvx)(char *fact, char *trans, int *n, int *nrhs, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *dlf, npy_complex128 *df, npy_complex128 *duf, npy_complex128 *du2, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zgttrf)(int *n, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *du2, int *ipiv, int *info); +void BLAS_FUNC(zgttrs)(char *trans, int *n, int *nrhs, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *du2, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zgtts2)(int *itrans, int *n, int *nrhs, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *du2, int *ipiv, npy_complex128 *b, int *ldb); +void BLAS_FUNC(zhbev)(char *jobz, char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zhbevd)(char *jobz, char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, double *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(zhbevx)(char *jobz, char *range, char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, npy_complex128 *q, int *ldq, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, double *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(zhbgst)(char *vect, char *uplo, int *n, int *ka, int *kb, npy_complex128 *ab, int *ldab, npy_complex128 *bb, int *ldbb, npy_complex128 *x, int *ldx, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zhbgv)(char *jobz, char *uplo, int *n, int *ka, int *kb, npy_complex128 *ab, int *ldab, npy_complex128 *bb, int *ldbb, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zhbgvd)(char *jobz, char *uplo, int *n, int *ka, int *kb, npy_complex128 *ab, int *ldab, npy_complex128 *bb, int *ldbb, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, double *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(zhbgvx)(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, npy_complex128 *ab, int *ldab, npy_complex128 *bb, int *ldbb, npy_complex128 *q, int *ldq, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, double *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(zhbtrd)(char *vect, char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, double *d, double *e, npy_complex128 *q, int *ldq, npy_complex128 *work, int *info); +void BLAS_FUNC(zhecon)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, double *anorm, double *rcond, npy_complex128 *work, int *info); +void BLAS_FUNC(zheequb)(char *uplo, int *n, npy_complex128 *a, int *lda, double *s, double *scond, double *amax, npy_complex128 *work, int *info); +void BLAS_FUNC(zheev)(char *jobz, char *uplo, int *n, npy_complex128 *a, int *lda, double *w, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zheevd)(char *jobz, char *uplo, int *n, npy_complex128 *a, int *lda, double *w, npy_complex128 *work, int *lwork, double *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(zheevr)(char *jobz, char *range, char *uplo, int *n, npy_complex128 *a, int *lda, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, npy_complex128 *z, int *ldz, int *isuppz, npy_complex128 *work, int *lwork, double *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(zheevx)(char *jobz, char *range, char *uplo, int *n, npy_complex128 *a, int *lda, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, double *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(zhegs2)(int *itype, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zhegst)(int *itype, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zhegv)(int *itype, char *jobz, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, double *w, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zhegvd)(int *itype, char *jobz, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, double *w, npy_complex128 *work, int *lwork, double *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(zhegvx)(int *itype, char *jobz, char *range, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, double *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(zherfs)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zhesv)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zhesvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zheswapr)(char *uplo, int *n, npy_complex128 *a, int *lda, int *i1, int *i2); +void BLAS_FUNC(zhetd2)(char *uplo, int *n, npy_complex128 *a, int *lda, double *d, double *e, npy_complex128 *tau, int *info); +void BLAS_FUNC(zhetf2)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(zhetrd)(char *uplo, int *n, npy_complex128 *a, int *lda, double *d, double *e, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zhetrf)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zhetri)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *info); +void BLAS_FUNC(zhetri2)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zhetri2x)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *nb, int *info); +void BLAS_FUNC(zhetrs)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zhetrs2)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *work, int *info); +void BLAS_FUNC(zhfrk)(char *transr, char *uplo, char *trans, int *n, int *k, double *alpha, npy_complex128 *a, int *lda, double *beta, npy_complex128 *c); +void BLAS_FUNC(zhgeqz)(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, npy_complex128 *h, int *ldh, npy_complex128 *t, int *ldt, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *q, int *ldq, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zhpcon)(char *uplo, int *n, npy_complex128 *ap, int *ipiv, double *anorm, double *rcond, npy_complex128 *work, int *info); +void BLAS_FUNC(zhpev)(char *jobz, char *uplo, int *n, npy_complex128 *ap, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zhpevd)(char *jobz, char *uplo, int *n, npy_complex128 *ap, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, double *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(zhpevx)(char *jobz, char *range, char *uplo, int *n, npy_complex128 *ap, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, double *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(zhpgst)(int *itype, char *uplo, int *n, npy_complex128 *ap, npy_complex128 *bp, int *info); +void BLAS_FUNC(zhpgv)(int *itype, char *jobz, char *uplo, int *n, npy_complex128 *ap, npy_complex128 *bp, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zhpgvd)(int *itype, char *jobz, char *uplo, int *n, npy_complex128 *ap, npy_complex128 *bp, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, double *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(zhpgvx)(int *itype, char *jobz, char *range, char *uplo, int *n, npy_complex128 *ap, npy_complex128 *bp, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, npy_complex128 *z, int *ldz, npy_complex128 *work, double *rwork, int *iwork, int *ifail, int *info); +void BLAS_FUNC(zhprfs)(char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zhpsv)(char *uplo, int *n, int *nrhs, npy_complex128 *ap, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zhpsvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zhptrd)(char *uplo, int *n, npy_complex128 *ap, double *d, double *e, npy_complex128 *tau, int *info); +void BLAS_FUNC(zhptrf)(char *uplo, int *n, npy_complex128 *ap, int *ipiv, int *info); +void BLAS_FUNC(zhptri)(char *uplo, int *n, npy_complex128 *ap, int *ipiv, npy_complex128 *work, int *info); +void BLAS_FUNC(zhptrs)(char *uplo, int *n, int *nrhs, npy_complex128 *ap, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zhsein)(char *side, char *eigsrc, char *initv, int *select, int *n, npy_complex128 *h, int *ldh, npy_complex128 *w, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, int *mm, int *m, npy_complex128 *work, double *rwork, int *ifaill, int *ifailr, int *info); +void BLAS_FUNC(zhseqr)(char *job, char *compz, int *n, int *ilo, int *ihi, npy_complex128 *h, int *ldh, npy_complex128 *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zlabrd)(int *m, int *n, int *nb, npy_complex128 *a, int *lda, double *d, double *e, npy_complex128 *tauq, npy_complex128 *taup, npy_complex128 *x, int *ldx, npy_complex128 *y, int *ldy); +void BLAS_FUNC(zlacgv)(int *n, npy_complex128 *x, int *incx); +void BLAS_FUNC(zlacn2)(int *n, npy_complex128 *v, npy_complex128 *x, double *est, int *kase, int *isave); +void BLAS_FUNC(zlacon)(int *n, npy_complex128 *v, npy_complex128 *x, double *est, int *kase); +void BLAS_FUNC(zlacp2)(char *uplo, int *m, int *n, double *a, int *lda, npy_complex128 *b, int *ldb); +void BLAS_FUNC(zlacpy)(char *uplo, int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb); +void BLAS_FUNC(zlacrm)(int *m, int *n, npy_complex128 *a, int *lda, double *b, int *ldb, npy_complex128 *c, int *ldc, double *rwork); +void BLAS_FUNC(zlacrt)(int *n, npy_complex128 *cx, int *incx, npy_complex128 *cy, int *incy, npy_complex128 *c, npy_complex128 *s); +void F_FUNC(zladivwrp,ZLADIVWRP)(npy_complex128 *out, npy_complex128 *x, npy_complex128 *y); +void BLAS_FUNC(zlaed0)(int *qsiz, int *n, double *d, double *e, npy_complex128 *q, int *ldq, npy_complex128 *qstore, int *ldqs, double *rwork, int *iwork, int *info); +void BLAS_FUNC(zlaed7)(int *n, int *cutpnt, int *qsiz, int *tlvls, int *curlvl, int *curpbm, double *d, npy_complex128 *q, int *ldq, double *rho, int *indxq, double *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, double *givnum, npy_complex128 *work, double *rwork, int *iwork, int *info); +void BLAS_FUNC(zlaed8)(int *k, int *n, int *qsiz, npy_complex128 *q, int *ldq, double *d, double *rho, int *cutpnt, double *z, double *dlamda, npy_complex128 *q2, int *ldq2, double *w, int *indxp, int *indx, int *indxq, int *perm, int *givptr, int *givcol, double *givnum, int *info); +void BLAS_FUNC(zlaein)(int *rightv, int *noinit, int *n, npy_complex128 *h, int *ldh, npy_complex128 *w, npy_complex128 *v, npy_complex128 *b, int *ldb, double *rwork, double *eps3, double *smlnum, int *info); +void BLAS_FUNC(zlaesy)(npy_complex128 *a, npy_complex128 *b, npy_complex128 *c, npy_complex128 *rt1, npy_complex128 *rt2, npy_complex128 *evscal, npy_complex128 *cs1, npy_complex128 *sn1); +void BLAS_FUNC(zlaev2)(npy_complex128 *a, npy_complex128 *b, npy_complex128 *c, double *rt1, double *rt2, double *cs1, npy_complex128 *sn1); +void BLAS_FUNC(zlag2c)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex64 *sa, int *ldsa, int *info); +void BLAS_FUNC(zlags2)(int *upper, double *a1, npy_complex128 *a2, double *a3, double *b1, npy_complex128 *b2, double *b3, double *csu, npy_complex128 *snu, double *csv, npy_complex128 *snv, double *csq, npy_complex128 *snq); +void BLAS_FUNC(zlagtm)(char *trans, int *n, int *nrhs, double *alpha, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *x, int *ldx, double *beta, npy_complex128 *b, int *ldb); +void BLAS_FUNC(zlahef)(char *uplo, int *n, int *nb, int *kb, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *w, int *ldw, int *info); +void BLAS_FUNC(zlahqr)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, npy_complex128 *h, int *ldh, npy_complex128 *w, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, int *info); +void BLAS_FUNC(zlahr2)(int *n, int *k, int *nb, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *t, int *ldt, npy_complex128 *y, int *ldy); +void BLAS_FUNC(zlaic1)(int *job, int *j, npy_complex128 *x, double *sest, npy_complex128 *w, npy_complex128 *gamma, double *sestpr, npy_complex128 *s, npy_complex128 *c); +void BLAS_FUNC(zlals0)(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, npy_complex128 *b, int *ldb, npy_complex128 *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, double *givnum, int *ldgnum, double *poles, double *difl, double *difr, double *z, int *k, double *c, double *s, double *rwork, int *info); +void BLAS_FUNC(zlalsa)(int *icompq, int *smlsiz, int *n, int *nrhs, npy_complex128 *b, int *ldb, npy_complex128 *bx, int *ldbx, double *u, int *ldu, double *vt, int *k, double *difl, double *difr, double *z, double *poles, int *givptr, int *givcol, int *ldgcol, int *perm, double *givnum, double *c, double *s, double *rwork, int *iwork, int *info); +void BLAS_FUNC(zlalsd)(char *uplo, int *smlsiz, int *n, int *nrhs, double *d, double *e, npy_complex128 *b, int *ldb, double *rcond, int *rank, npy_complex128 *work, double *rwork, int *iwork, int *info); +double BLAS_FUNC(zlangb)(char *norm, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, double *work); +double BLAS_FUNC(zlange)(char *norm, int *m, int *n, npy_complex128 *a, int *lda, double *work); +double BLAS_FUNC(zlangt)(char *norm, int *n, npy_complex128 *dl, npy_complex128 *d_, npy_complex128 *du); +double BLAS_FUNC(zlanhb)(char *norm, char *uplo, int *n, int *k, npy_complex128 *ab, int *ldab, double *work); +double BLAS_FUNC(zlanhe)(char *norm, char *uplo, int *n, npy_complex128 *a, int *lda, double *work); +double BLAS_FUNC(zlanhf)(char *norm, char *transr, char *uplo, int *n, npy_complex128 *a, double *work); +double BLAS_FUNC(zlanhp)(char *norm, char *uplo, int *n, npy_complex128 *ap, double *work); +double BLAS_FUNC(zlanhs)(char *norm, int *n, npy_complex128 *a, int *lda, double *work); +double BLAS_FUNC(zlanht)(char *norm, int *n, double *d_, npy_complex128 *e); +double BLAS_FUNC(zlansb)(char *norm, char *uplo, int *n, int *k, npy_complex128 *ab, int *ldab, double *work); +double BLAS_FUNC(zlansp)(char *norm, char *uplo, int *n, npy_complex128 *ap, double *work); +double BLAS_FUNC(zlansy)(char *norm, char *uplo, int *n, npy_complex128 *a, int *lda, double *work); +double BLAS_FUNC(zlantb)(char *norm, char *uplo, char *diag, int *n, int *k, npy_complex128 *ab, int *ldab, double *work); +double BLAS_FUNC(zlantp)(char *norm, char *uplo, char *diag, int *n, npy_complex128 *ap, double *work); +double BLAS_FUNC(zlantr)(char *norm, char *uplo, char *diag, int *m, int *n, npy_complex128 *a, int *lda, double *work); +void BLAS_FUNC(zlapll)(int *n, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, double *ssmin); +void BLAS_FUNC(zlapmr)(int *forwrd, int *m, int *n, npy_complex128 *x, int *ldx, int *k); +void BLAS_FUNC(zlapmt)(int *forwrd, int *m, int *n, npy_complex128 *x, int *ldx, int *k); +void BLAS_FUNC(zlaqgb)(int *m, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, double *r, double *c, double *rowcnd, double *colcnd, double *amax, char *equed); +void BLAS_FUNC(zlaqge)(int *m, int *n, npy_complex128 *a, int *lda, double *r, double *c, double *rowcnd, double *colcnd, double *amax, char *equed); +void BLAS_FUNC(zlaqhb)(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, double *s, double *scond, double *amax, char *equed); +void BLAS_FUNC(zlaqhe)(char *uplo, int *n, npy_complex128 *a, int *lda, double *s, double *scond, double *amax, char *equed); +void BLAS_FUNC(zlaqhp)(char *uplo, int *n, npy_complex128 *ap, double *s, double *scond, double *amax, char *equed); +void BLAS_FUNC(zlaqp2)(int *m, int *n, int *offset, npy_complex128 *a, int *lda, int *jpvt, npy_complex128 *tau, double *vn1, double *vn2, npy_complex128 *work); +void BLAS_FUNC(zlaqps)(int *m, int *n, int *offset, int *nb, int *kb, npy_complex128 *a, int *lda, int *jpvt, npy_complex128 *tau, double *vn1, double *vn2, npy_complex128 *auxv, npy_complex128 *f, int *ldf); +void BLAS_FUNC(zlaqr0)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, npy_complex128 *h, int *ldh, npy_complex128 *w, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zlaqr1)(int *n, npy_complex128 *h, int *ldh, npy_complex128 *s1, npy_complex128 *s2, npy_complex128 *v); +void BLAS_FUNC(zlaqr2)(int *wantt, int *wantz, int *n, int *ktop, int *kbot, int *nw, npy_complex128 *h, int *ldh, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, int *ns, int *nd, npy_complex128 *sh, npy_complex128 *v, int *ldv, int *nh, npy_complex128 *t, int *ldt, int *nv, npy_complex128 *wv, int *ldwv, npy_complex128 *work, int *lwork); +void BLAS_FUNC(zlaqr3)(int *wantt, int *wantz, int *n, int *ktop, int *kbot, int *nw, npy_complex128 *h, int *ldh, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, int *ns, int *nd, npy_complex128 *sh, npy_complex128 *v, int *ldv, int *nh, npy_complex128 *t, int *ldt, int *nv, npy_complex128 *wv, int *ldwv, npy_complex128 *work, int *lwork); +void BLAS_FUNC(zlaqr4)(int *wantt, int *wantz, int *n, int *ilo, int *ihi, npy_complex128 *h, int *ldh, npy_complex128 *w, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zlaqr5)(int *wantt, int *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, npy_complex128 *s, npy_complex128 *h, int *ldh, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, npy_complex128 *v, int *ldv, npy_complex128 *u, int *ldu, int *nv, npy_complex128 *wv, int *ldwv, int *nh, npy_complex128 *wh, int *ldwh); +void BLAS_FUNC(zlaqsb)(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, double *s, double *scond, double *amax, char *equed); +void BLAS_FUNC(zlaqsp)(char *uplo, int *n, npy_complex128 *ap, double *s, double *scond, double *amax, char *equed); +void BLAS_FUNC(zlaqsy)(char *uplo, int *n, npy_complex128 *a, int *lda, double *s, double *scond, double *amax, char *equed); +void BLAS_FUNC(zlar1v)(int *n, int *b1, int *bn, double *lambda_, double *d, double *l, double *ld, double *lld, double *pivmin, double *gaptol, npy_complex128 *z, int *wantnc, int *negcnt, double *ztz, double *mingma, int *r, int *isuppz, double *nrminv, double *resid, double *rqcorr, double *work); +void BLAS_FUNC(zlar2v)(int *n, npy_complex128 *x, npy_complex128 *y, npy_complex128 *z, int *incx, double *c, npy_complex128 *s, int *incc); +void BLAS_FUNC(zlarcm)(int *m, int *n, double *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *c, int *ldc, double *rwork); +void BLAS_FUNC(zlarf)(char *side, int *m, int *n, npy_complex128 *v, int *incv, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work); +void BLAS_FUNC(zlarfb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, npy_complex128 *v, int *ldv, npy_complex128 *t, int *ldt, npy_complex128 *c, int *ldc, npy_complex128 *work, int *ldwork); +void BLAS_FUNC(zlarfg)(int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *tau); +void BLAS_FUNC(zlarfgp)(int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *tau); +void BLAS_FUNC(zlarft)(char *direct, char *storev, int *n, int *k, npy_complex128 *v, int *ldv, npy_complex128 *tau, npy_complex128 *t, int *ldt); +void BLAS_FUNC(zlarfx)(char *side, int *m, int *n, npy_complex128 *v, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work); +void BLAS_FUNC(zlargv)(int *n, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, double *c, int *incc); +void BLAS_FUNC(zlarnv)(int *idist, int *iseed, int *n, npy_complex128 *x); +void BLAS_FUNC(zlarrv)(int *n, double *vl, double *vu, double *d, double *l, double *pivmin, int *isplit, int *m, int *dol, int *dou, double *minrgp, double *rtol1, double *rtol2, double *w, double *werr, double *wgap, int *iblock, int *indexw, double *gers, npy_complex128 *z, int *ldz, int *isuppz, double *work, int *iwork, int *info); +void BLAS_FUNC(zlartg)(npy_complex128 *f, npy_complex128 *g, double *cs, npy_complex128 *sn, npy_complex128 *r); +void BLAS_FUNC(zlartv)(int *n, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, double *c, npy_complex128 *s, int *incc); +void BLAS_FUNC(zlarz)(char *side, int *m, int *n, int *l, npy_complex128 *v, int *incv, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work); +void BLAS_FUNC(zlarzb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, npy_complex128 *v, int *ldv, npy_complex128 *t, int *ldt, npy_complex128 *c, int *ldc, npy_complex128 *work, int *ldwork); +void BLAS_FUNC(zlarzt)(char *direct, char *storev, int *n, int *k, npy_complex128 *v, int *ldv, npy_complex128 *tau, npy_complex128 *t, int *ldt); +void BLAS_FUNC(zlascl)(char *type_bn, int *kl, int *ku, double *cfrom, double *cto, int *m, int *n, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(zlaset)(char *uplo, int *m, int *n, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *a, int *lda); +void BLAS_FUNC(zlasr)(char *side, char *pivot, char *direct, int *m, int *n, double *c, double *s, npy_complex128 *a, int *lda); +void BLAS_FUNC(zlassq)(int *n, npy_complex128 *x, int *incx, double *scale, double *sumsq); +void BLAS_FUNC(zlaswp)(int *n, npy_complex128 *a, int *lda, int *k1, int *k2, int *ipiv, int *incx); +void BLAS_FUNC(zlasyf)(char *uplo, int *n, int *nb, int *kb, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *w, int *ldw, int *info); +void BLAS_FUNC(zlat2c)(char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex64 *sa, int *ldsa, int *info); +void BLAS_FUNC(zlatbs)(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, npy_complex128 *ab, int *ldab, npy_complex128 *x, double *scale, double *cnorm, int *info); +void BLAS_FUNC(zlatdf)(int *ijob, int *n, npy_complex128 *z, int *ldz, npy_complex128 *rhs, double *rdsum, double *rdscal, int *ipiv, int *jpiv); +void BLAS_FUNC(zlatps)(char *uplo, char *trans, char *diag, char *normin, int *n, npy_complex128 *ap, npy_complex128 *x, double *scale, double *cnorm, int *info); +void BLAS_FUNC(zlatrd)(char *uplo, int *n, int *nb, npy_complex128 *a, int *lda, double *e, npy_complex128 *tau, npy_complex128 *w, int *ldw); +void BLAS_FUNC(zlatrs)(char *uplo, char *trans, char *diag, char *normin, int *n, npy_complex128 *a, int *lda, npy_complex128 *x, double *scale, double *cnorm, int *info); +void BLAS_FUNC(zlatrz)(int *m, int *n, int *l, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work); +void BLAS_FUNC(zlauu2)(char *uplo, int *n, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(zlauum)(char *uplo, int *n, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(zpbcon)(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, double *anorm, double *rcond, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zpbequ)(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, double *s, double *scond, double *amax, int *info); +void BLAS_FUNC(zpbrfs)(char *uplo, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *afb, int *ldafb, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zpbstf)(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, int *info); +void BLAS_FUNC(zpbsv)(char *uplo, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zpbsvx)(char *fact, char *uplo, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *afb, int *ldafb, char *equed, double *s, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zpbtf2)(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, int *info); +void BLAS_FUNC(zpbtrf)(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, int *info); +void BLAS_FUNC(zpbtrs)(char *uplo, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zpftrf)(char *transr, char *uplo, int *n, npy_complex128 *a, int *info); +void BLAS_FUNC(zpftri)(char *transr, char *uplo, int *n, npy_complex128 *a, int *info); +void BLAS_FUNC(zpftrs)(char *transr, char *uplo, int *n, int *nrhs, npy_complex128 *a, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zpocon)(char *uplo, int *n, npy_complex128 *a, int *lda, double *anorm, double *rcond, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zpoequ)(int *n, npy_complex128 *a, int *lda, double *s, double *scond, double *amax, int *info); +void BLAS_FUNC(zpoequb)(int *n, npy_complex128 *a, int *lda, double *s, double *scond, double *amax, int *info); +void BLAS_FUNC(zporfs)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zposv)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zposvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, char *equed, double *s, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zpotf2)(char *uplo, int *n, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(zpotrf)(char *uplo, int *n, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(zpotri)(char *uplo, int *n, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(zpotrs)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zppcon)(char *uplo, int *n, npy_complex128 *ap, double *anorm, double *rcond, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zppequ)(char *uplo, int *n, npy_complex128 *ap, double *s, double *scond, double *amax, int *info); +void BLAS_FUNC(zpprfs)(char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zppsv)(char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zppsvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, char *equed, double *s, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zpptrf)(char *uplo, int *n, npy_complex128 *ap, int *info); +void BLAS_FUNC(zpptri)(char *uplo, int *n, npy_complex128 *ap, int *info); +void BLAS_FUNC(zpptrs)(char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zpstf2)(char *uplo, int *n, npy_complex128 *a, int *lda, int *piv, int *rank, double *tol, double *work, int *info); +void BLAS_FUNC(zpstrf)(char *uplo, int *n, npy_complex128 *a, int *lda, int *piv, int *rank, double *tol, double *work, int *info); +void BLAS_FUNC(zptcon)(int *n, double *d, npy_complex128 *e, double *anorm, double *rcond, double *rwork, int *info); +void BLAS_FUNC(zpteqr)(char *compz, int *n, double *d, double *e, npy_complex128 *z, int *ldz, double *work, int *info); +void BLAS_FUNC(zptrfs)(char *uplo, int *n, int *nrhs, double *d, npy_complex128 *e, double *df, npy_complex128 *ef, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zptsv)(int *n, int *nrhs, double *d, npy_complex128 *e, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zptsvx)(char *fact, int *n, int *nrhs, double *d, npy_complex128 *e, double *df, npy_complex128 *ef, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zpttrf)(int *n, double *d, npy_complex128 *e, int *info); +void BLAS_FUNC(zpttrs)(char *uplo, int *n, int *nrhs, double *d, npy_complex128 *e, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zptts2)(int *iuplo, int *n, int *nrhs, double *d, npy_complex128 *e, npy_complex128 *b, int *ldb); +void BLAS_FUNC(zrot)(int *n, npy_complex128 *cx, int *incx, npy_complex128 *cy, int *incy, double *c, npy_complex128 *s); +void BLAS_FUNC(zspcon)(char *uplo, int *n, npy_complex128 *ap, int *ipiv, double *anorm, double *rcond, npy_complex128 *work, int *info); +void BLAS_FUNC(zspmv)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *ap, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy); +void BLAS_FUNC(zspr)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *ap); +void BLAS_FUNC(zsprfs)(char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zspsv)(char *uplo, int *n, int *nrhs, npy_complex128 *ap, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zspsvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zsptrf)(char *uplo, int *n, npy_complex128 *ap, int *ipiv, int *info); +void BLAS_FUNC(zsptri)(char *uplo, int *n, npy_complex128 *ap, int *ipiv, npy_complex128 *work, int *info); +void BLAS_FUNC(zsptrs)(char *uplo, int *n, int *nrhs, npy_complex128 *ap, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zstedc)(char *compz, int *n, double *d, double *e, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, double *rwork, int *lrwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(zstegr)(char *jobz, char *range, int *n, double *d, double *e, double *vl, double *vu, int *il, int *iu, double *abstol, int *m, double *w, npy_complex128 *z, int *ldz, int *isuppz, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(zstein)(int *n, double *d, double *e, int *m, double *w, int *iblock, int *isplit, npy_complex128 *z, int *ldz, double *work, int *iwork, int *ifail, int *info); +void BLAS_FUNC(zstemr)(char *jobz, char *range, int *n, double *d, double *e, double *vl, double *vu, int *il, int *iu, int *m, double *w, npy_complex128 *z, int *ldz, int *nzc, int *isuppz, int *tryrac, double *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(zsteqr)(char *compz, int *n, double *d, double *e, npy_complex128 *z, int *ldz, double *work, int *info); +void BLAS_FUNC(zsycon)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, double *anorm, double *rcond, npy_complex128 *work, int *info); +void BLAS_FUNC(zsyconv)(char *uplo, char *way, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *info); +void BLAS_FUNC(zsyequb)(char *uplo, int *n, npy_complex128 *a, int *lda, double *s, double *scond, double *amax, npy_complex128 *work, int *info); +void BLAS_FUNC(zsymv)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy); +void BLAS_FUNC(zsyr)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *a, int *lda); +void BLAS_FUNC(zsyrfs)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(zsysv)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zsysvx)(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *rcond, double *ferr, double *berr, npy_complex128 *work, int *lwork, double *rwork, int *info); +void BLAS_FUNC(zsyswapr)(char *uplo, int *n, npy_complex128 *a, int *lda, int *i1, int *i2); +void BLAS_FUNC(zsytf2)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, int *info); +void BLAS_FUNC(zsytrf)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zsytri)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *info); +void BLAS_FUNC(zsytri2)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zsytri2x)(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *nb, int *info); +void BLAS_FUNC(zsytrs)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(zsytrs2)(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *work, int *info); +void BLAS_FUNC(ztbcon)(char *norm, char *uplo, char *diag, int *n, int *kd, npy_complex128 *ab, int *ldab, double *rcond, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(ztbrfs)(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(ztbtrs)(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(ztfsm)(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, npy_complex128 *b, int *ldb); +void BLAS_FUNC(ztftri)(char *transr, char *uplo, char *diag, int *n, npy_complex128 *a, int *info); +void BLAS_FUNC(ztfttp)(char *transr, char *uplo, int *n, npy_complex128 *arf, npy_complex128 *ap, int *info); +void BLAS_FUNC(ztfttr)(char *transr, char *uplo, int *n, npy_complex128 *arf, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(ztgevc)(char *side, char *howmny, int *select, int *n, npy_complex128 *s, int *lds, npy_complex128 *p, int *ldp, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, int *mm, int *m, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(ztgex2)(int *wantq, int *wantz, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *q, int *ldq, npy_complex128 *z, int *ldz, int *j1, int *info); +void BLAS_FUNC(ztgexc)(int *wantq, int *wantz, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *q, int *ldq, npy_complex128 *z, int *ldz, int *ifst, int *ilst, int *info); +void BLAS_FUNC(ztgsen)(int *ijob, int *wantq, int *wantz, int *select, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *q, int *ldq, npy_complex128 *z, int *ldz, int *m, double *pl, double *pr, double *dif, npy_complex128 *work, int *lwork, int *iwork, int *liwork, int *info); +void BLAS_FUNC(ztgsja)(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, double *tola, double *tolb, double *alpha, double *beta, npy_complex128 *u, int *ldu, npy_complex128 *v, int *ldv, npy_complex128 *q, int *ldq, npy_complex128 *work, int *ncycle, int *info); +void BLAS_FUNC(ztgsna)(char *job, char *howmny, int *select, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, double *s, double *dif, int *mm, int *m, npy_complex128 *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(ztgsy2)(char *trans, int *ijob, int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *c, int *ldc, npy_complex128 *d, int *ldd, npy_complex128 *e, int *lde, npy_complex128 *f, int *ldf, double *scale, double *rdsum, double *rdscal, int *info); +void BLAS_FUNC(ztgsyl)(char *trans, int *ijob, int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *c, int *ldc, npy_complex128 *d, int *ldd, npy_complex128 *e, int *lde, npy_complex128 *f, int *ldf, double *scale, double *dif, npy_complex128 *work, int *lwork, int *iwork, int *info); +void BLAS_FUNC(ztpcon)(char *norm, char *uplo, char *diag, int *n, npy_complex128 *ap, double *rcond, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(ztpmqrt)(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, npy_complex128 *v, int *ldv, npy_complex128 *t, int *ldt, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *work, int *info); +void BLAS_FUNC(ztpqrt)(int *m, int *n, int *l, int *nb, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *t, int *ldt, npy_complex128 *work, int *info); +void BLAS_FUNC(ztpqrt2)(int *m, int *n, int *l, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *t, int *ldt, int *info); +void BLAS_FUNC(ztprfb)(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, npy_complex128 *v, int *ldv, npy_complex128 *t, int *ldt, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *work, int *ldwork); +void BLAS_FUNC(ztprfs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(ztptri)(char *uplo, char *diag, int *n, npy_complex128 *ap, int *info); +void BLAS_FUNC(ztptrs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(ztpttf)(char *transr, char *uplo, int *n, npy_complex128 *ap, npy_complex128 *arf, int *info); +void BLAS_FUNC(ztpttr)(char *uplo, int *n, npy_complex128 *ap, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(ztrcon)(char *norm, char *uplo, char *diag, int *n, npy_complex128 *a, int *lda, double *rcond, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(ztrevc)(char *side, char *howmny, int *select, int *n, npy_complex128 *t, int *ldt, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, int *mm, int *m, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(ztrexc)(char *compq, int *n, npy_complex128 *t, int *ldt, npy_complex128 *q, int *ldq, int *ifst, int *ilst, int *info); +void BLAS_FUNC(ztrrfs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, double *ferr, double *berr, npy_complex128 *work, double *rwork, int *info); +void BLAS_FUNC(ztrsen)(char *job, char *compq, int *select, int *n, npy_complex128 *t, int *ldt, npy_complex128 *q, int *ldq, npy_complex128 *w, int *m, double *s, double *sep, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(ztrsna)(char *job, char *howmny, int *select, int *n, npy_complex128 *t, int *ldt, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, double *s, double *sep, int *mm, int *m, npy_complex128 *work, int *ldwork, double *rwork, int *info); +void BLAS_FUNC(ztrsyl)(char *trana, char *tranb, int *isgn, int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *c, int *ldc, double *scale, int *info); +void BLAS_FUNC(ztrti2)(char *uplo, char *diag, int *n, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(ztrtri)(char *uplo, char *diag, int *n, npy_complex128 *a, int *lda, int *info); +void BLAS_FUNC(ztrtrs)(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *info); +void BLAS_FUNC(ztrttf)(char *transr, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *arf, int *info); +void BLAS_FUNC(ztrttp)(char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *ap, int *info); +void BLAS_FUNC(ztzrzf)(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zunbdb)(char *trans, char *signs, int *m, int *p, int *q, npy_complex128 *x11, int *ldx11, npy_complex128 *x12, int *ldx12, npy_complex128 *x21, int *ldx21, npy_complex128 *x22, int *ldx22, double *theta, double *phi, npy_complex128 *taup1, npy_complex128 *taup2, npy_complex128 *tauq1, npy_complex128 *tauq2, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zuncsd)(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, npy_complex128 *x11, int *ldx11, npy_complex128 *x12, int *ldx12, npy_complex128 *x21, int *ldx21, npy_complex128 *x22, int *ldx22, double *theta, npy_complex128 *u1, int *ldu1, npy_complex128 *u2, int *ldu2, npy_complex128 *v1t, int *ldv1t, npy_complex128 *v2t, int *ldv2t, npy_complex128 *work, int *lwork, double *rwork, int *lrwork, int *iwork, int *info); +void BLAS_FUNC(zung2l)(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info); +void BLAS_FUNC(zung2r)(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info); +void BLAS_FUNC(zungbr)(char *vect, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zunghr)(int *n, int *ilo, int *ihi, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zungl2)(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info); +void BLAS_FUNC(zunglq)(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zungql)(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zungqr)(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zungr2)(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info); +void BLAS_FUNC(zungrq)(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zungtr)(char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zunm2l)(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info); +void BLAS_FUNC(zunm2r)(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info); +void BLAS_FUNC(zunmbr)(char *vect, char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zunmhr)(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zunml2)(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info); +void BLAS_FUNC(zunmlq)(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zunmql)(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zunmqr)(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zunmr2)(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info); +void BLAS_FUNC(zunmr3)(char *side, char *trans, int *m, int *n, int *k, int *l, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info); +void BLAS_FUNC(zunmrq)(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zunmrz)(char *side, char *trans, int *m, int *n, int *k, int *l, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zunmtr)(char *side, char *uplo, char *trans, int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info); +void BLAS_FUNC(zupgtr)(char *uplo, int *n, npy_complex128 *ap, npy_complex128 *tau, npy_complex128 *q, int *ldq, npy_complex128 *work, int *info); +void BLAS_FUNC(zupmtr)(char *side, char *uplo, char *trans, int *m, int *n, npy_complex128 *ap, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info); + +#ifdef __cplusplus +} +#endif diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..3a0ba92af71f3f3188cc73ca441187db9701b052 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs.py @@ -0,0 +1,867 @@ +# +# Author: Travis Oliphant, March 2002 +# +import warnings +from itertools import product + +import numpy as np +from numpy import (dot, diag, prod, logical_not, ravel, transpose, + conjugate, absolute, amax, sign, isfinite, triu) + +# Local imports +from scipy.linalg import LinAlgError, bandwidth +from ._misc import norm +from ._basic import solve, inv +from ._decomp_svd import svd +from ._decomp_schur import schur, rsf2csf +from ._expm_frechet import expm_frechet, expm_cond +from ._matfuncs_sqrtm import sqrtm +from ._matfuncs_expm import pick_pade_structure, pade_UV_calc +from ._linalg_pythran import _funm_loops # type: ignore[import-not-found] + +__all__ = ['expm', 'cosm', 'sinm', 'tanm', 'coshm', 'sinhm', 'tanhm', 'logm', + 'funm', 'signm', 'sqrtm', 'fractional_matrix_power', 'expm_frechet', + 'expm_cond', 'khatri_rao'] + +eps = np.finfo('d').eps +feps = np.finfo('f').eps + +_array_precision = {'i': 1, 'l': 1, 'f': 0, 'd': 1, 'F': 0, 'D': 1} + + +############################################################################### +# Utility functions. + + +def _asarray_square(A): + """ + Wraps asarray with the extra requirement that the input be a square matrix. + + The motivation is that the matfuncs module has real functions that have + been lifted to square matrix functions. + + Parameters + ---------- + A : array_like + A square matrix. + + Returns + ------- + out : ndarray + An ndarray copy or view or other representation of A. + + """ + A = np.asarray(A) + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected square array_like input') + return A + + +def _maybe_real(A, B, tol=None): + """ + Return either B or the real part of B, depending on properties of A and B. + + The motivation is that B has been computed as a complicated function of A, + and B may be perturbed by negligible imaginary components. + If A is real and B is complex with small imaginary components, + then return a real copy of B. The assumption in that case would be that + the imaginary components of B are numerical artifacts. + + Parameters + ---------- + A : ndarray + Input array whose type is to be checked as real vs. complex. + B : ndarray + Array to be returned, possibly without its imaginary part. + tol : float + Absolute tolerance. + + Returns + ------- + out : real or complex array + Either the input array B or only the real part of the input array B. + + """ + # Note that booleans and integers compare as real. + if np.isrealobj(A) and np.iscomplexobj(B): + if tol is None: + tol = {0: feps*1e3, 1: eps*1e6}[_array_precision[B.dtype.char]] + if np.allclose(B.imag, 0.0, atol=tol): + B = B.real + return B + + +############################################################################### +# Matrix functions. + + +def fractional_matrix_power(A, t): + """ + Compute the fractional power of a matrix. + + Proceeds according to the discussion in section (6) of [1]_. + + Parameters + ---------- + A : (N, N) array_like + Matrix whose fractional power to evaluate. + t : float + Fractional power. + + Returns + ------- + X : (N, N) array_like + The fractional power of the matrix. + + References + ---------- + .. [1] Nicholas J. Higham and Lijing lin (2011) + "A Schur-Pade Algorithm for Fractional Powers of a Matrix." + SIAM Journal on Matrix Analysis and Applications, + 32 (3). pp. 1056-1078. ISSN 0895-4798 + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import fractional_matrix_power + >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) + >>> b = fractional_matrix_power(a, 0.5) + >>> b + array([[ 0.75592895, 1.13389342], + [ 0.37796447, 1.88982237]]) + >>> np.dot(b, b) # Verify square root + array([[ 1., 3.], + [ 1., 4.]]) + + """ + # This fixes some issue with imports; + # this function calls onenormest which is in scipy.sparse. + A = _asarray_square(A) + import scipy.linalg._matfuncs_inv_ssq + return scipy.linalg._matfuncs_inv_ssq._fractional_matrix_power(A, t) + + +def logm(A, disp=True): + """ + Compute matrix logarithm. + + The matrix logarithm is the inverse of + expm: expm(logm(`A`)) == `A` + + Parameters + ---------- + A : (N, N) array_like + Matrix whose logarithm to evaluate + disp : bool, optional + Emit warning if error in the result is estimated large + instead of returning estimated error. (Default: True) + + Returns + ------- + logm : (N, N) ndarray + Matrix logarithm of `A` + errest : float + (if disp == False) + + 1-norm of the estimated error, ||err||_1 / ||A||_1 + + References + ---------- + .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012) + "Improved Inverse Scaling and Squaring Algorithms + for the Matrix Logarithm." + SIAM Journal on Scientific Computing, 34 (4). C152-C169. + ISSN 1095-7197 + + .. [2] Nicholas J. Higham (2008) + "Functions of Matrices: Theory and Computation" + ISBN 978-0-898716-46-7 + + .. [3] Nicholas J. Higham and Lijing lin (2011) + "A Schur-Pade Algorithm for Fractional Powers of a Matrix." + SIAM Journal on Matrix Analysis and Applications, + 32 (3). pp. 1056-1078. ISSN 0895-4798 + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import logm, expm + >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) + >>> b = logm(a) + >>> b + array([[-1.02571087, 2.05142174], + [ 0.68380725, 1.02571087]]) + >>> expm(b) # Verify expm(logm(a)) returns a + array([[ 1., 3.], + [ 1., 4.]]) + + """ + A = np.asarray(A) # squareness checked in `_logm` + # Avoid circular import ... this is OK, right? + import scipy.linalg._matfuncs_inv_ssq + F = scipy.linalg._matfuncs_inv_ssq._logm(A) + F = _maybe_real(A, F) + errtol = 1000*eps + # TODO use a better error approximation + with np.errstate(divide='ignore', invalid='ignore'): + errest = norm(expm(F)-A, 1) / np.asarray(norm(A, 1), dtype=A.dtype).real[()] + if disp: + if not isfinite(errest) or errest >= errtol: + message = f"logm result may be inaccurate, approximate err = {errest}" + warnings.warn(message, RuntimeWarning, stacklevel=2) + return F + else: + return F, errest + + +def expm(A): + """Compute the matrix exponential of an array. + + Parameters + ---------- + A : ndarray + Input with last two dimensions are square ``(..., n, n)``. + + Returns + ------- + eA : ndarray + The resulting matrix exponential with the same shape of ``A`` + + Notes + ----- + Implements the algorithm given in [1], which is essentially a Pade + approximation with a variable order that is decided based on the array + data. + + For input with size ``n``, the memory usage is in the worst case in the + order of ``8*(n**2)``. If the input data is not of single and double + precision of real and complex dtypes, it is copied to a new array. + + For cases ``n >= 400``, the exact 1-norm computation cost, breaks even with + 1-norm estimation and from that point on the estimation scheme given in + [2] is used to decide on the approximation order. + + References + ---------- + .. [1] Awad H. Al-Mohy and Nicholas J. Higham, (2009), "A New Scaling + and Squaring Algorithm for the Matrix Exponential", SIAM J. Matrix + Anal. Appl. 31(3):970-989, :doi:`10.1137/09074721X` + + .. [2] Nicholas J. Higham and Francoise Tisseur (2000), "A Block Algorithm + for Matrix 1-Norm Estimation, with an Application to 1-Norm + Pseudospectra." SIAM J. Matrix Anal. Appl. 21(4):1185-1201, + :doi:`10.1137/S0895479899356080` + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import expm, sinm, cosm + + Matrix version of the formula exp(0) = 1: + + >>> expm(np.zeros((3, 2, 2))) + array([[[1., 0.], + [0., 1.]], + + [[1., 0.], + [0., 1.]], + + [[1., 0.], + [0., 1.]]]) + + Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) + applied to a matrix: + + >>> a = np.array([[1.0, 2.0], [-1.0, 3.0]]) + >>> expm(1j*a) + array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], + [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) + >>> cosm(a) + 1j*sinm(a) + array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], + [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) + + """ + a = np.asarray(A) + if a.size == 1 and a.ndim < 2: + return np.array([[np.exp(a.item())]]) + + if a.ndim < 2: + raise LinAlgError('The input array must be at least two-dimensional') + if a.shape[-1] != a.shape[-2]: + raise LinAlgError('Last 2 dimensions of the array must be square') + + # Empty array + if min(*a.shape) == 0: + dtype = expm(np.eye(2, dtype=a.dtype)).dtype + return np.empty_like(a, dtype=dtype) + + # Scalar case + if a.shape[-2:] == (1, 1): + return np.exp(a) + + if not np.issubdtype(a.dtype, np.inexact): + a = a.astype(np.float64) + elif a.dtype == np.float16: + a = a.astype(np.float32) + + # An explicit formula for 2x2 case exists (formula (2.2) in [1]). However, without + # Kahan's method, numerical instabilities can occur (See gh-19584). Hence removed + # here until we have a more stable implementation. + + n = a.shape[-1] + eA = np.empty(a.shape, dtype=a.dtype) + # working memory to hold intermediate arrays + Am = np.empty((5, n, n), dtype=a.dtype) + + # Main loop to go through the slices of an ndarray and passing to expm + for ind in product(*[range(x) for x in a.shape[:-2]]): + aw = a[ind] + + lu = bandwidth(aw) + if not any(lu): # a is diagonal? + eA[ind] = np.diag(np.exp(np.diag(aw))) + continue + + # Generic/triangular case; copy the slice into scratch and send. + # Am will be mutated by pick_pade_structure + # If s != 0, scaled Am will be returned from pick_pade_structure. + Am[0, :, :] = aw + m, s = pick_pade_structure(Am) + if (m < 0): + raise MemoryError("scipy.linalg.expm could not allocate sufficient" + " memory while trying to compute the Pade " + f"structure (error code {m}).") + info = pade_UV_calc(Am, m) + if info != 0: + if info <= -11: + # We raise it from failed mallocs; negative LAPACK codes > -7 + raise MemoryError("scipy.linalg.expm could not allocate " + "sufficient memory while trying to compute the " + f"exponential (error code {info}).") + else: + # LAPACK wrong argument error or exact singularity. + # Neither should happen. + raise RuntimeError("scipy.linalg.expm got an internal LAPACK " + "error during the exponential computation " + f"(error code {info})") + eAw = Am[0] + + if s != 0: # squaring needed + + if (lu[1] == 0) or (lu[0] == 0): # lower/upper triangular + # This branch implements Code Fragment 2.1 of [1] + + diag_aw = np.diag(aw) + # einsum returns a writable view + np.einsum('ii->i', eAw)[:] = np.exp(diag_aw * 2**(-s)) + # super/sub diagonal + sd = np.diag(aw, k=-1 if lu[1] == 0 else 1) + + for i in range(s-1, -1, -1): + eAw = eAw @ eAw + + # diagonal + np.einsum('ii->i', eAw)[:] = np.exp(diag_aw * 2.**(-i)) + exp_sd = _exp_sinch(diag_aw * (2.**(-i))) * (sd * 2**(-i)) + if lu[1] == 0: # lower + np.einsum('ii->i', eAw[1:, :-1])[:] = exp_sd + else: # upper + np.einsum('ii->i', eAw[:-1, 1:])[:] = exp_sd + + else: # generic + for _ in range(s): + eAw = eAw @ eAw + + # Zero out the entries from np.empty in case of triangular input + if (lu[0] == 0) or (lu[1] == 0): + eA[ind] = np.triu(eAw) if lu[0] == 0 else np.tril(eAw) + else: + eA[ind] = eAw + + return eA + + +def _exp_sinch(x): + # Higham's formula (10.42), might overflow, see GH-11839 + lexp_diff = np.diff(np.exp(x)) + l_diff = np.diff(x) + mask_z = l_diff == 0. + lexp_diff[~mask_z] /= l_diff[~mask_z] + lexp_diff[mask_z] = np.exp(x[:-1][mask_z]) + return lexp_diff + + +def cosm(A): + """ + Compute the matrix cosine. + + This routine uses expm to compute the matrix exponentials. + + Parameters + ---------- + A : (N, N) array_like + Input array + + Returns + ------- + cosm : (N, N) ndarray + Matrix cosine of A + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import expm, sinm, cosm + + Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) + applied to a matrix: + + >>> a = np.array([[1.0, 2.0], [-1.0, 3.0]]) + >>> expm(1j*a) + array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], + [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) + >>> cosm(a) + 1j*sinm(a) + array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], + [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) + + """ + A = _asarray_square(A) + if np.iscomplexobj(A): + return 0.5*(expm(1j*A) + expm(-1j*A)) + else: + return expm(1j*A).real + + +def sinm(A): + """ + Compute the matrix sine. + + This routine uses expm to compute the matrix exponentials. + + Parameters + ---------- + A : (N, N) array_like + Input array. + + Returns + ------- + sinm : (N, N) ndarray + Matrix sine of `A` + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import expm, sinm, cosm + + Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) + applied to a matrix: + + >>> a = np.array([[1.0, 2.0], [-1.0, 3.0]]) + >>> expm(1j*a) + array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], + [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) + >>> cosm(a) + 1j*sinm(a) + array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], + [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) + + """ + A = _asarray_square(A) + if np.iscomplexobj(A): + return -0.5j*(expm(1j*A) - expm(-1j*A)) + else: + return expm(1j*A).imag + + +def tanm(A): + """ + Compute the matrix tangent. + + This routine uses expm to compute the matrix exponentials. + + Parameters + ---------- + A : (N, N) array_like + Input array. + + Returns + ------- + tanm : (N, N) ndarray + Matrix tangent of `A` + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import tanm, sinm, cosm + >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) + >>> t = tanm(a) + >>> t + array([[ -2.00876993, -8.41880636], + [ -2.80626879, -10.42757629]]) + + Verify tanm(a) = sinm(a).dot(inv(cosm(a))) + + >>> s = sinm(a) + >>> c = cosm(a) + >>> s.dot(np.linalg.inv(c)) + array([[ -2.00876993, -8.41880636], + [ -2.80626879, -10.42757629]]) + + """ + A = _asarray_square(A) + return _maybe_real(A, solve(cosm(A), sinm(A))) + + +def coshm(A): + """ + Compute the hyperbolic matrix cosine. + + This routine uses expm to compute the matrix exponentials. + + Parameters + ---------- + A : (N, N) array_like + Input array. + + Returns + ------- + coshm : (N, N) ndarray + Hyperbolic matrix cosine of `A` + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import tanhm, sinhm, coshm + >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) + >>> c = coshm(a) + >>> c + array([[ 11.24592233, 38.76236492], + [ 12.92078831, 50.00828725]]) + + Verify tanhm(a) = sinhm(a).dot(inv(coshm(a))) + + >>> t = tanhm(a) + >>> s = sinhm(a) + >>> t - s.dot(np.linalg.inv(c)) + array([[ 2.72004641e-15, 4.55191440e-15], + [ 0.00000000e+00, -5.55111512e-16]]) + + """ + A = _asarray_square(A) + return _maybe_real(A, 0.5 * (expm(A) + expm(-A))) + + +def sinhm(A): + """ + Compute the hyperbolic matrix sine. + + This routine uses expm to compute the matrix exponentials. + + Parameters + ---------- + A : (N, N) array_like + Input array. + + Returns + ------- + sinhm : (N, N) ndarray + Hyperbolic matrix sine of `A` + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import tanhm, sinhm, coshm + >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) + >>> s = sinhm(a) + >>> s + array([[ 10.57300653, 39.28826594], + [ 13.09608865, 49.86127247]]) + + Verify tanhm(a) = sinhm(a).dot(inv(coshm(a))) + + >>> t = tanhm(a) + >>> c = coshm(a) + >>> t - s.dot(np.linalg.inv(c)) + array([[ 2.72004641e-15, 4.55191440e-15], + [ 0.00000000e+00, -5.55111512e-16]]) + + """ + A = _asarray_square(A) + return _maybe_real(A, 0.5 * (expm(A) - expm(-A))) + + +def tanhm(A): + """ + Compute the hyperbolic matrix tangent. + + This routine uses expm to compute the matrix exponentials. + + Parameters + ---------- + A : (N, N) array_like + Input array + + Returns + ------- + tanhm : (N, N) ndarray + Hyperbolic matrix tangent of `A` + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import tanhm, sinhm, coshm + >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) + >>> t = tanhm(a) + >>> t + array([[ 0.3428582 , 0.51987926], + [ 0.17329309, 0.86273746]]) + + Verify tanhm(a) = sinhm(a).dot(inv(coshm(a))) + + >>> s = sinhm(a) + >>> c = coshm(a) + >>> t - s.dot(np.linalg.inv(c)) + array([[ 2.72004641e-15, 4.55191440e-15], + [ 0.00000000e+00, -5.55111512e-16]]) + + """ + A = _asarray_square(A) + return _maybe_real(A, solve(coshm(A), sinhm(A))) + + +def funm(A, func, disp=True): + """ + Evaluate a matrix function specified by a callable. + + Returns the value of matrix-valued function ``f`` at `A`. The + function ``f`` is an extension of the scalar-valued function `func` + to matrices. + + Parameters + ---------- + A : (N, N) array_like + Matrix at which to evaluate the function + func : callable + Callable object that evaluates a scalar function f. + Must be vectorized (eg. using vectorize). + disp : bool, optional + Print warning if error in the result is estimated large + instead of returning estimated error. (Default: True) + + Returns + ------- + funm : (N, N) ndarray + Value of the matrix function specified by func evaluated at `A` + errest : float + (if disp == False) + + 1-norm of the estimated error, ||err||_1 / ||A||_1 + + Notes + ----- + This function implements the general algorithm based on Schur decomposition + (Algorithm 9.1.1. in [1]_). + + If the input matrix is known to be diagonalizable, then relying on the + eigendecomposition is likely to be faster. For example, if your matrix is + Hermitian, you can do + + >>> from scipy.linalg import eigh + >>> def funm_herm(a, func, check_finite=False): + ... w, v = eigh(a, check_finite=check_finite) + ... ## if you further know that your matrix is positive semidefinite, + ... ## you can optionally guard against precision errors by doing + ... # w = np.maximum(w, 0) + ... w = func(w) + ... return (v * w).dot(v.conj().T) + + References + ---------- + .. [1] Gene H. Golub, Charles F. van Loan, Matrix Computations 4th ed. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import funm + >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) + >>> funm(a, lambda x: x*x) + array([[ 4., 15.], + [ 5., 19.]]) + >>> a.dot(a) + array([[ 4., 15.], + [ 5., 19.]]) + + """ + A = _asarray_square(A) + # Perform Shur decomposition (lapack ?gees) + T, Z = schur(A) + T, Z = rsf2csf(T, Z) + n, n = T.shape + F = diag(func(diag(T))) # apply function to diagonal elements + F = F.astype(T.dtype.char) # e.g., when F is real but T is complex + + minden = abs(T[0, 0]) + + # implement Algorithm 11.1.1 from Golub and Van Loan + # "matrix Computations." + F, minden = _funm_loops(F, T, n, minden) + + F = dot(dot(Z, F), transpose(conjugate(Z))) + F = _maybe_real(A, F) + + tol = {0: feps, 1: eps}[_array_precision[F.dtype.char]] + if minden == 0.0: + minden = tol + err = min(1, max(tol, (tol/minden)*norm(triu(T, 1), 1))) + if prod(ravel(logical_not(isfinite(F))), axis=0): + err = np.inf + if disp: + if err > 1000*tol: + print("funm result may be inaccurate, approximate err =", err) + return F + else: + return F, err + + +def signm(A, disp=True): + """ + Matrix sign function. + + Extension of the scalar sign(x) to matrices. + + Parameters + ---------- + A : (N, N) array_like + Matrix at which to evaluate the sign function + disp : bool, optional + Print warning if error in the result is estimated large + instead of returning estimated error. (Default: True) + + Returns + ------- + signm : (N, N) ndarray + Value of the sign function at `A` + errest : float + (if disp == False) + + 1-norm of the estimated error, ||err||_1 / ||A||_1 + + Examples + -------- + >>> from scipy.linalg import signm, eigvals + >>> a = [[1,2,3], [1,2,1], [1,1,1]] + >>> eigvals(a) + array([ 4.12488542+0.j, -0.76155718+0.j, 0.63667176+0.j]) + >>> eigvals(signm(a)) + array([-1.+0.j, 1.+0.j, 1.+0.j]) + + """ + A = _asarray_square(A) + + def rounded_sign(x): + rx = np.real(x) + if rx.dtype.char == 'f': + c = 1e3*feps*amax(x) + else: + c = 1e3*eps*amax(x) + return sign((absolute(rx) > c) * rx) + result, errest = funm(A, rounded_sign, disp=0) + errtol = {0: 1e3*feps, 1: 1e3*eps}[_array_precision[result.dtype.char]] + if errest < errtol: + return result + + # Handle signm of defective matrices: + + # See "E.D.Denman and J.Leyva-Ramos, Appl.Math.Comp., + # 8:237-250,1981" for how to improve the following (currently a + # rather naive) iteration process: + + # a = result # sometimes iteration converges faster but where?? + + # Shifting to avoid zero eigenvalues. How to ensure that shifting does + # not change the spectrum too much? + vals = svd(A, compute_uv=False) + max_sv = np.amax(vals) + # min_nonzero_sv = vals[(vals>max_sv*errtol).tolist().count(1)-1] + # c = 0.5/min_nonzero_sv + c = 0.5/max_sv + S0 = A + c*np.identity(A.shape[0]) + prev_errest = errest + for i in range(100): + iS0 = inv(S0) + S0 = 0.5*(S0 + iS0) + Pp = 0.5*(dot(S0, S0)+S0) + errest = norm(dot(Pp, Pp)-Pp, 1) + if errest < errtol or prev_errest == errest: + break + prev_errest = errest + if disp: + if not isfinite(errest) or errest >= errtol: + print("signm result may be inaccurate, approximate err =", errest) + return S0 + else: + return S0, errest + + +def khatri_rao(a, b): + r""" + Khatri-rao product + + A column-wise Kronecker product of two matrices + + Parameters + ---------- + a : (n, k) array_like + Input array + b : (m, k) array_like + Input array + + Returns + ------- + c: (n*m, k) ndarray + Khatri-rao product of `a` and `b`. + + Notes + ----- + The mathematical definition of the Khatri-Rao product is: + + .. math:: + + (A_{ij} \bigotimes B_{ij})_{ij} + + which is the Kronecker product of every column of A and B, e.g.:: + + c = np.vstack([np.kron(a[:, k], b[:, k]) for k in range(b.shape[1])]).T + + Examples + -------- + >>> import numpy as np + >>> from scipy import linalg + >>> a = np.array([[1, 2, 3], [4, 5, 6]]) + >>> b = np.array([[3, 4, 5], [6, 7, 8], [2, 3, 9]]) + >>> linalg.khatri_rao(a, b) + array([[ 3, 8, 15], + [ 6, 14, 24], + [ 2, 6, 27], + [12, 20, 30], + [24, 35, 48], + [ 8, 15, 54]]) + + """ + a = np.asarray(a) + b = np.asarray(b) + + if not (a.ndim == 2 and b.ndim == 2): + raise ValueError("The both arrays should be 2-dimensional.") + + if not a.shape[1] == b.shape[1]: + raise ValueError("The number of columns for both arrays " + "should be equal.") + + # accommodate empty arrays + if a.size == 0 or b.size == 0: + m = a.shape[0] * b.shape[0] + n = a.shape[1] + return np.empty_like(a, shape=(m, n)) + + # c = np.vstack([np.kron(a[:, k], b[:, k]) for k in range(b.shape[1])]).T + c = a[..., :, np.newaxis, :] * b[..., np.newaxis, :, :] + return c.reshape((-1,) + c.shape[2:]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs_expm.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs_expm.pyi new file mode 100644 index 0000000000000000000000000000000000000000..98ca455c6eb06c1e95e6e11d3db2dc346a295fde --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs_expm.pyi @@ -0,0 +1,6 @@ +from numpy.typing import NDArray +from typing import Any + +def pick_pade_structure(a: NDArray[Any]) -> tuple[int, int]: ... + +def pade_UV_calc(Am: NDArray[Any], m: int) -> int: ... diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs_inv_ssq.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs_inv_ssq.py new file mode 100644 index 0000000000000000000000000000000000000000..1decffae2e521f0a9325b873cc33b095a4e3c166 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs_inv_ssq.py @@ -0,0 +1,886 @@ +""" +Matrix functions that use Pade approximation with inverse scaling and squaring. + +""" +import warnings + +import numpy as np + +from scipy.linalg._matfuncs_sqrtm import SqrtmError, _sqrtm_triu +from scipy.linalg._decomp_schur import schur, rsf2csf +from scipy.linalg._matfuncs import funm +from scipy.linalg import svdvals, solve_triangular +from scipy.sparse.linalg._interface import LinearOperator +from scipy.sparse.linalg import onenormest +import scipy.special + + +class LogmRankWarning(UserWarning): + pass + + +class LogmExactlySingularWarning(LogmRankWarning): + pass + + +class LogmNearlySingularWarning(LogmRankWarning): + pass + + +class LogmError(np.linalg.LinAlgError): + pass + + +class FractionalMatrixPowerError(np.linalg.LinAlgError): + pass + + +#TODO renovate or move this class when scipy operators are more mature +class _MatrixM1PowerOperator(LinearOperator): + """ + A representation of the linear operator (A - I)^p. + """ + + def __init__(self, A, p): + if A.ndim != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected A to be like a square matrix') + if p < 0 or p != int(p): + raise ValueError('expected p to be a non-negative integer') + self._A = A + self._p = p + self.ndim = A.ndim + self.shape = A.shape + + def _matvec(self, x): + for i in range(self._p): + x = self._A.dot(x) - x + return x + + def _rmatvec(self, x): + for i in range(self._p): + x = x.dot(self._A) - x + return x + + def _matmat(self, X): + for i in range(self._p): + X = self._A.dot(X) - X + return X + + def _adjoint(self): + return _MatrixM1PowerOperator(self._A.T, self._p) + + +#TODO renovate or move this function when SciPy operators are more mature +def _onenormest_m1_power(A, p, + t=2, itmax=5, compute_v=False, compute_w=False): + """ + Efficiently estimate the 1-norm of (A - I)^p. + + Parameters + ---------- + A : ndarray + Matrix whose 1-norm of a power is to be computed. + p : int + Non-negative integer power. + t : int, optional + A positive parameter controlling the tradeoff between + accuracy versus time and memory usage. + Larger values take longer and use more memory + but give more accurate output. + itmax : int, optional + Use at most this many iterations. + compute_v : bool, optional + Request a norm-maximizing linear operator input vector if True. + compute_w : bool, optional + Request a norm-maximizing linear operator output vector if True. + + Returns + ------- + est : float + An underestimate of the 1-norm of the sparse matrix. + v : ndarray, optional + The vector such that ||Av||_1 == est*||v||_1. + It can be thought of as an input to the linear operator + that gives an output with particularly large norm. + w : ndarray, optional + The vector Av which has relatively large 1-norm. + It can be thought of as an output of the linear operator + that is relatively large in norm compared to the input. + + """ + return onenormest(_MatrixM1PowerOperator(A, p), + t=t, itmax=itmax, compute_v=compute_v, compute_w=compute_w) + + +def _unwindk(z): + """ + Compute the scalar unwinding number. + + Uses Eq. (5.3) in [1]_, and should be equal to (z - log(exp(z)) / (2 pi i). + Note that this definition differs in sign from the original definition + in equations (5, 6) in [2]_. The sign convention is justified in [3]_. + + Parameters + ---------- + z : complex + A complex number. + + Returns + ------- + unwinding_number : integer + The scalar unwinding number of z. + + References + ---------- + .. [1] Nicholas J. Higham and Lijing lin (2011) + "A Schur-Pade Algorithm for Fractional Powers of a Matrix." + SIAM Journal on Matrix Analysis and Applications, + 32 (3). pp. 1056-1078. ISSN 0895-4798 + + .. [2] Robert M. Corless and David J. Jeffrey, + "The unwinding number." Newsletter ACM SIGSAM Bulletin + Volume 30, Issue 2, June 1996, Pages 28-35. + + .. [3] Russell Bradford and Robert M. Corless and James H. Davenport and + David J. Jeffrey and Stephen M. Watt, + "Reasoning about the elementary functions of complex analysis" + Annals of Mathematics and Artificial Intelligence, + 36: 303-318, 2002. + + """ + return int(np.ceil((z.imag - np.pi) / (2*np.pi))) + + +def _briggs_helper_function(a, k): + """ + Computes r = a^(1 / (2^k)) - 1. + + This is algorithm (2) of [1]_. + The purpose is to avoid a danger of subtractive cancellation. + For more computational efficiency it should probably be cythonized. + + Parameters + ---------- + a : complex + A complex number. + k : integer + A nonnegative integer. + + Returns + ------- + r : complex + The value r = a^(1 / (2^k)) - 1 computed with less cancellation. + + Notes + ----- + The algorithm as formulated in the reference does not handle k=0 or k=1 + correctly, so these are special-cased in this implementation. + This function is intended to not allow `a` to belong to the closed + negative real axis, but this constraint is relaxed. + + References + ---------- + .. [1] Awad H. Al-Mohy (2012) + "A more accurate Briggs method for the logarithm", + Numerical Algorithms, 59 : 393--402. + + """ + if k < 0 or int(k) != k: + raise ValueError('expected a nonnegative integer k') + if k == 0: + return a - 1 + elif k == 1: + return np.sqrt(a) - 1 + else: + k_hat = k + if np.angle(a) >= np.pi / 2: + a = np.sqrt(a) + k_hat = k - 1 + z0 = a - 1 + a = np.sqrt(a) + r = 1 + a + for j in range(1, k_hat): + a = np.sqrt(a) + r = r * (1 + a) + r = z0 / r + return r + + +def _fractional_power_superdiag_entry(l1, l2, t12, p): + """ + Compute a superdiagonal entry of a fractional matrix power. + + This is Eq. (5.6) in [1]_. + + Parameters + ---------- + l1 : complex + A diagonal entry of the matrix. + l2 : complex + A diagonal entry of the matrix. + t12 : complex + A superdiagonal entry of the matrix. + p : float + A fractional power. + + Returns + ------- + f12 : complex + A superdiagonal entry of the fractional matrix power. + + Notes + ----- + Care has been taken to return a real number if possible when + all of the inputs are real numbers. + + References + ---------- + .. [1] Nicholas J. Higham and Lijing lin (2011) + "A Schur-Pade Algorithm for Fractional Powers of a Matrix." + SIAM Journal on Matrix Analysis and Applications, + 32 (3). pp. 1056-1078. ISSN 0895-4798 + + """ + if l1 == l2: + f12 = t12 * p * l1**(p-1) + elif abs(l2 - l1) > abs(l1 + l2) / 2: + f12 = t12 * ((l2**p) - (l1**p)) / (l2 - l1) + else: + # This is Eq. (5.5) in [1]. + z = (l2 - l1) / (l2 + l1) + log_l1 = np.log(l1) + log_l2 = np.log(l2) + arctanh_z = np.arctanh(z) + tmp_a = t12 * np.exp((p/2)*(log_l2 + log_l1)) + tmp_u = _unwindk(log_l2 - log_l1) + if tmp_u: + tmp_b = p * (arctanh_z + np.pi * 1j * tmp_u) + else: + tmp_b = p * arctanh_z + tmp_c = 2 * np.sinh(tmp_b) / (l2 - l1) + f12 = tmp_a * tmp_c + return f12 + + +def _logm_superdiag_entry(l1, l2, t12): + """ + Compute a superdiagonal entry of a matrix logarithm. + + This is like Eq. (11.28) in [1]_, except the determination of whether + l1 and l2 are sufficiently far apart has been modified. + + Parameters + ---------- + l1 : complex + A diagonal entry of the matrix. + l2 : complex + A diagonal entry of the matrix. + t12 : complex + A superdiagonal entry of the matrix. + + Returns + ------- + f12 : complex + A superdiagonal entry of the matrix logarithm. + + Notes + ----- + Care has been taken to return a real number if possible when + all of the inputs are real numbers. + + References + ---------- + .. [1] Nicholas J. Higham (2008) + "Functions of Matrices: Theory and Computation" + ISBN 978-0-898716-46-7 + + """ + if l1 == l2: + f12 = t12 / l1 + elif abs(l2 - l1) > abs(l1 + l2) / 2: + f12 = t12 * (np.log(l2) - np.log(l1)) / (l2 - l1) + else: + z = (l2 - l1) / (l2 + l1) + u = _unwindk(np.log(l2) - np.log(l1)) + if u: + f12 = t12 * 2 * (np.arctanh(z) + np.pi*1j*u) / (l2 - l1) + else: + f12 = t12 * 2 * np.arctanh(z) / (l2 - l1) + return f12 + + +def _inverse_squaring_helper(T0, theta): + """ + A helper function for inverse scaling and squaring for Pade approximation. + + Parameters + ---------- + T0 : (N, N) array_like upper triangular + Matrix involved in inverse scaling and squaring. + theta : indexable + The values theta[1] .. theta[7] must be available. + They represent bounds related to Pade approximation, and they depend + on the matrix function which is being computed. + For example, different values of theta are required for + matrix logarithm than for fractional matrix power. + + Returns + ------- + R : (N, N) array_like upper triangular + Composition of zero or more matrix square roots of T0, minus I. + s : non-negative integer + Number of square roots taken. + m : positive integer + The degree of the Pade approximation. + + Notes + ----- + This subroutine appears as a chunk of lines within + a couple of published algorithms; for example it appears + as lines 4--35 in algorithm (3.1) of [1]_, and + as lines 3--34 in algorithm (4.1) of [2]_. + The instances of 'goto line 38' in algorithm (3.1) of [1]_ + probably mean 'goto line 36' and have been interpreted accordingly. + + References + ---------- + .. [1] Nicholas J. Higham and Lijing Lin (2013) + "An Improved Schur-Pade Algorithm for Fractional Powers + of a Matrix and their Frechet Derivatives." + + .. [2] Awad H. Al-Mohy and Nicholas J. Higham (2012) + "Improved Inverse Scaling and Squaring Algorithms + for the Matrix Logarithm." + SIAM Journal on Scientific Computing, 34 (4). C152-C169. + ISSN 1095-7197 + + """ + if len(T0.shape) != 2 or T0.shape[0] != T0.shape[1]: + raise ValueError('expected an upper triangular square matrix') + n, n = T0.shape + T = T0 + + # Find s0, the smallest s such that the spectral radius + # of a certain diagonal matrix is at most theta[7]. + # Note that because theta[7] < 1, + # this search will not terminate if any diagonal entry of T is zero. + s0 = 0 + tmp_diag = np.diag(T) + if np.count_nonzero(tmp_diag) != n: + raise Exception('Diagonal entries of T must be nonzero') + while np.max(np.absolute(tmp_diag - 1), initial=0.) > theta[7]: + tmp_diag = np.sqrt(tmp_diag) + s0 += 1 + + # Take matrix square roots of T. + for i in range(s0): + T = _sqrtm_triu(T) + + # Flow control in this section is a little odd. + # This is because I am translating algorithm descriptions + # which have GOTOs in the publication. + s = s0 + k = 0 + d2 = _onenormest_m1_power(T, 2) ** (1/2) + d3 = _onenormest_m1_power(T, 3) ** (1/3) + a2 = max(d2, d3) + m = None + for i in (1, 2): + if a2 <= theta[i]: + m = i + break + while m is None: + if s > s0: + d3 = _onenormest_m1_power(T, 3) ** (1/3) + d4 = _onenormest_m1_power(T, 4) ** (1/4) + a3 = max(d3, d4) + if a3 <= theta[7]: + j1 = min(i for i in (3, 4, 5, 6, 7) if a3 <= theta[i]) + if j1 <= 6: + m = j1 + break + elif a3 / 2 <= theta[5] and k < 2: + k += 1 + T = _sqrtm_triu(T) + s += 1 + continue + d5 = _onenormest_m1_power(T, 5) ** (1/5) + a4 = max(d4, d5) + eta = min(a3, a4) + for i in (6, 7): + if eta <= theta[i]: + m = i + break + if m is not None: + break + T = _sqrtm_triu(T) + s += 1 + + # The subtraction of the identity is redundant here, + # because the diagonal will be replaced for improved numerical accuracy, + # but this formulation should help clarify the meaning of R. + R = T - np.identity(n) + + # Replace the diagonal and first superdiagonal of T0^(1/(2^s)) - I + # using formulas that have less subtractive cancellation. + # Skip this step if the principal branch + # does not exist at T0; this happens when a diagonal entry of T0 + # is negative with imaginary part 0. + has_principal_branch = all(x.real > 0 or x.imag != 0 for x in np.diag(T0)) + if has_principal_branch: + for j in range(n): + a = T0[j, j] + r = _briggs_helper_function(a, s) + R[j, j] = r + p = np.exp2(-s) + for j in range(n-1): + l1 = T0[j, j] + l2 = T0[j+1, j+1] + t12 = T0[j, j+1] + f12 = _fractional_power_superdiag_entry(l1, l2, t12, p) + R[j, j+1] = f12 + + # Return the T-I matrix, the number of square roots, and the Pade degree. + if not np.array_equal(R, np.triu(R)): + raise Exception('R is not upper triangular') + return R, s, m + + +def _fractional_power_pade_constant(i, t): + # A helper function for matrix fractional power. + if i < 1: + raise ValueError('expected a positive integer i') + if not (-1 < t < 1): + raise ValueError('expected -1 < t < 1') + if i == 1: + return -t + elif i % 2 == 0: + j = i // 2 + return (-j + t) / (2 * (2*j - 1)) + elif i % 2 == 1: + j = (i - 1) // 2 + return (-j - t) / (2 * (2*j + 1)) + else: + raise Exception(f'unnexpected value of i, i = {i}') + + +def _fractional_power_pade(R, t, m): + """ + Evaluate the Pade approximation of a fractional matrix power. + + Evaluate the degree-m Pade approximation of R + to the fractional matrix power t using the continued fraction + in bottom-up fashion using algorithm (4.1) in [1]_. + + Parameters + ---------- + R : (N, N) array_like + Upper triangular matrix whose fractional power to evaluate. + t : float + Fractional power between -1 and 1 exclusive. + m : positive integer + Degree of Pade approximation. + + Returns + ------- + U : (N, N) array_like + The degree-m Pade approximation of R to the fractional power t. + This matrix will be upper triangular. + + References + ---------- + .. [1] Nicholas J. Higham and Lijing lin (2011) + "A Schur-Pade Algorithm for Fractional Powers of a Matrix." + SIAM Journal on Matrix Analysis and Applications, + 32 (3). pp. 1056-1078. ISSN 0895-4798 + + """ + if m < 1 or int(m) != m: + raise ValueError('expected a positive integer m') + if not (-1 < t < 1): + raise ValueError('expected -1 < t < 1') + R = np.asarray(R) + if len(R.shape) != 2 or R.shape[0] != R.shape[1]: + raise ValueError('expected an upper triangular square matrix') + n, n = R.shape + ident = np.identity(n) + Y = R * _fractional_power_pade_constant(2*m, t) + for j in range(2*m - 1, 0, -1): + rhs = R * _fractional_power_pade_constant(j, t) + Y = solve_triangular(ident + Y, rhs) + U = ident + Y + if not np.array_equal(U, np.triu(U)): + raise Exception('U is not upper triangular') + return U + + +def _remainder_matrix_power_triu(T, t): + """ + Compute a fractional power of an upper triangular matrix. + + The fractional power is restricted to fractions -1 < t < 1. + This uses algorithm (3.1) of [1]_. + The Pade approximation itself uses algorithm (4.1) of [2]_. + + Parameters + ---------- + T : (N, N) array_like + Upper triangular matrix whose fractional power to evaluate. + t : float + Fractional power between -1 and 1 exclusive. + + Returns + ------- + X : (N, N) array_like + The fractional power of the matrix. + + References + ---------- + .. [1] Nicholas J. Higham and Lijing Lin (2013) + "An Improved Schur-Pade Algorithm for Fractional Powers + of a Matrix and their Frechet Derivatives." + + .. [2] Nicholas J. Higham and Lijing lin (2011) + "A Schur-Pade Algorithm for Fractional Powers of a Matrix." + SIAM Journal on Matrix Analysis and Applications, + 32 (3). pp. 1056-1078. ISSN 0895-4798 + + """ + m_to_theta = { + 1: 1.51e-5, + 2: 2.24e-3, + 3: 1.88e-2, + 4: 6.04e-2, + 5: 1.24e-1, + 6: 2.00e-1, + 7: 2.79e-1, + } + n, n = T.shape + T0 = T + T0_diag = np.diag(T0) + if np.array_equal(T0, np.diag(T0_diag)): + U = np.diag(T0_diag ** t) + else: + R, s, m = _inverse_squaring_helper(T0, m_to_theta) + + # Evaluate the Pade approximation. + # Note that this function expects the negative of the matrix + # returned by the inverse squaring helper. + U = _fractional_power_pade(-R, t, m) + + # Undo the inverse scaling and squaring. + # Be less clever about this + # if the principal branch does not exist at T0; + # this happens when a diagonal entry of T0 + # is negative with imaginary part 0. + eivals = np.diag(T0) + has_principal_branch = all(x.real > 0 or x.imag != 0 for x in eivals) + for i in range(s, -1, -1): + if i < s: + U = U.dot(U) + else: + if has_principal_branch: + p = t * np.exp2(-i) + U[np.diag_indices(n)] = T0_diag ** p + for j in range(n-1): + l1 = T0[j, j] + l2 = T0[j+1, j+1] + t12 = T0[j, j+1] + f12 = _fractional_power_superdiag_entry(l1, l2, t12, p) + U[j, j+1] = f12 + if not np.array_equal(U, np.triu(U)): + raise Exception('U is not upper triangular') + return U + + +def _remainder_matrix_power(A, t): + """ + Compute the fractional power of a matrix, for fractions -1 < t < 1. + + This uses algorithm (3.1) of [1]_. + The Pade approximation itself uses algorithm (4.1) of [2]_. + + Parameters + ---------- + A : (N, N) array_like + Matrix whose fractional power to evaluate. + t : float + Fractional power between -1 and 1 exclusive. + + Returns + ------- + X : (N, N) array_like + The fractional power of the matrix. + + References + ---------- + .. [1] Nicholas J. Higham and Lijing Lin (2013) + "An Improved Schur-Pade Algorithm for Fractional Powers + of a Matrix and their Frechet Derivatives." + + .. [2] Nicholas J. Higham and Lijing lin (2011) + "A Schur-Pade Algorithm for Fractional Powers of a Matrix." + SIAM Journal on Matrix Analysis and Applications, + 32 (3). pp. 1056-1078. ISSN 0895-4798 + + """ + # This code block is copied from numpy.matrix_power(). + A = np.asarray(A) + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('input must be a square array') + + # Get the number of rows and columns. + n, n = A.shape + + # Triangularize the matrix if necessary, + # attempting to preserve dtype if possible. + if np.array_equal(A, np.triu(A)): + Z = None + T = A + else: + if np.isrealobj(A): + T, Z = schur(A) + if not np.array_equal(T, np.triu(T)): + T, Z = rsf2csf(T, Z) + else: + T, Z = schur(A, output='complex') + + # Zeros on the diagonal of the triangular matrix are forbidden, + # because the inverse scaling and squaring cannot deal with it. + T_diag = np.diag(T) + if np.count_nonzero(T_diag) != n: + raise FractionalMatrixPowerError( + 'cannot use inverse scaling and squaring to find ' + 'the fractional matrix power of a singular matrix') + + # If the triangular matrix is real and has a negative + # entry on the diagonal, then force the matrix to be complex. + if np.isrealobj(T) and np.min(T_diag) < 0: + T = T.astype(complex) + + # Get the fractional power of the triangular matrix, + # and de-triangularize it if necessary. + U = _remainder_matrix_power_triu(T, t) + if Z is not None: + ZH = np.conjugate(Z).T + return Z.dot(U).dot(ZH) + else: + return U + + +def _fractional_matrix_power(A, p): + """ + Compute the fractional power of a matrix. + + See the fractional_matrix_power docstring in matfuncs.py for more info. + + """ + A = np.asarray(A) + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected a square matrix') + if p == int(p): + return np.linalg.matrix_power(A, int(p)) + # Compute singular values. + s = svdvals(A) + # Inverse scaling and squaring cannot deal with a singular matrix, + # because the process of repeatedly taking square roots + # would not converge to the identity matrix. + if s[-1]: + # Compute the condition number relative to matrix inversion, + # and use this to decide between floor(p) and ceil(p). + k2 = s[0] / s[-1] + p1 = p - np.floor(p) + p2 = p - np.ceil(p) + if p1 * k2 ** (1 - p1) <= -p2 * k2: + a = int(np.floor(p)) + b = p1 + else: + a = int(np.ceil(p)) + b = p2 + try: + R = _remainder_matrix_power(A, b) + Q = np.linalg.matrix_power(A, a) + return Q.dot(R) + except np.linalg.LinAlgError: + pass + # If p is negative then we are going to give up. + # If p is non-negative then we can fall back to generic funm. + if p < 0: + X = np.empty_like(A) + X.fill(np.nan) + return X + else: + p1 = p - np.floor(p) + a = int(np.floor(p)) + b = p1 + R, info = funm(A, lambda x: pow(x, b), disp=False) + Q = np.linalg.matrix_power(A, a) + return Q.dot(R) + + +def _logm_triu(T): + """ + Compute matrix logarithm of an upper triangular matrix. + + The matrix logarithm is the inverse of + expm: expm(logm(`T`)) == `T` + + Parameters + ---------- + T : (N, N) array_like + Upper triangular matrix whose logarithm to evaluate + + Returns + ------- + logm : (N, N) ndarray + Matrix logarithm of `T` + + References + ---------- + .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012) + "Improved Inverse Scaling and Squaring Algorithms + for the Matrix Logarithm." + SIAM Journal on Scientific Computing, 34 (4). C152-C169. + ISSN 1095-7197 + + .. [2] Nicholas J. Higham (2008) + "Functions of Matrices: Theory and Computation" + ISBN 978-0-898716-46-7 + + .. [3] Nicholas J. Higham and Lijing lin (2011) + "A Schur-Pade Algorithm for Fractional Powers of a Matrix." + SIAM Journal on Matrix Analysis and Applications, + 32 (3). pp. 1056-1078. ISSN 0895-4798 + + """ + T = np.asarray(T) + if len(T.shape) != 2 or T.shape[0] != T.shape[1]: + raise ValueError('expected an upper triangular square matrix') + n, n = T.shape + + # Construct T0 with the appropriate type, + # depending on the dtype and the spectrum of T. + T_diag = np.diag(T) + keep_it_real = np.isrealobj(T) and np.min(T_diag, initial=0.) >= 0 + if keep_it_real: + T0 = T + else: + T0 = T.astype(complex) + + # Define bounds given in Table (2.1). + theta = (None, + 1.59e-5, 2.31e-3, 1.94e-2, 6.21e-2, + 1.28e-1, 2.06e-1, 2.88e-1, 3.67e-1, + 4.39e-1, 5.03e-1, 5.60e-1, 6.09e-1, + 6.52e-1, 6.89e-1, 7.21e-1, 7.49e-1) + + R, s, m = _inverse_squaring_helper(T0, theta) + + # Evaluate U = 2**s r_m(T - I) using the partial fraction expansion (1.1). + # This requires the nodes and weights + # corresponding to degree-m Gauss-Legendre quadrature. + # These quadrature arrays need to be transformed from the [-1, 1] interval + # to the [0, 1] interval. + nodes, weights = scipy.special.p_roots(m) + nodes = nodes.real + if nodes.shape != (m,) or weights.shape != (m,): + raise Exception('internal error') + nodes = 0.5 + 0.5 * nodes + weights = 0.5 * weights + ident = np.identity(n) + U = np.zeros_like(R) + for alpha, beta in zip(weights, nodes): + U += solve_triangular(ident + beta*R, alpha*R) + U *= np.exp2(s) + + # Skip this step if the principal branch + # does not exist at T0; this happens when a diagonal entry of T0 + # is negative with imaginary part 0. + has_principal_branch = all(x.real > 0 or x.imag != 0 for x in np.diag(T0)) + if has_principal_branch: + + # Recompute diagonal entries of U. + U[np.diag_indices(n)] = np.log(np.diag(T0)) + + # Recompute superdiagonal entries of U. + # This indexing of this code should be renovated + # when newer np.diagonal() becomes available. + for i in range(n-1): + l1 = T0[i, i] + l2 = T0[i+1, i+1] + t12 = T0[i, i+1] + U[i, i+1] = _logm_superdiag_entry(l1, l2, t12) + + # Return the logm of the upper triangular matrix. + if not np.array_equal(U, np.triu(U)): + raise Exception('U is not upper triangular') + return U + + +def _logm_force_nonsingular_triangular_matrix(T, inplace=False): + # The input matrix should be upper triangular. + # The eps is ad hoc and is not meant to be machine precision. + tri_eps = 1e-20 + abs_diag = np.absolute(np.diag(T)) + if np.any(abs_diag == 0): + exact_singularity_msg = 'The logm input matrix is exactly singular.' + warnings.warn(exact_singularity_msg, LogmExactlySingularWarning, stacklevel=3) + if not inplace: + T = T.copy() + n = T.shape[0] + for i in range(n): + if not T[i, i]: + T[i, i] = tri_eps + elif np.any(abs_diag < tri_eps): + near_singularity_msg = 'The logm input matrix may be nearly singular.' + warnings.warn(near_singularity_msg, LogmNearlySingularWarning, stacklevel=3) + return T + + +def _logm(A): + """ + Compute the matrix logarithm. + + See the logm docstring in matfuncs.py for more info. + + Notes + ----- + In this function we look at triangular matrices that are similar + to the input matrix. If any diagonal entry of such a triangular matrix + is exactly zero then the original matrix is singular. + The matrix logarithm does not exist for such matrices, + but in such cases we will pretend that the diagonal entries that are zero + are actually slightly positive by an ad-hoc amount, in the interest + of returning something more useful than NaN. This will cause a warning. + + """ + A = np.asarray(A) + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected a square matrix') + + # If the input matrix dtype is integer then copy to a float dtype matrix. + if issubclass(A.dtype.type, np.integer): + A = np.asarray(A, dtype=float) + + keep_it_real = np.isrealobj(A) + try: + if np.array_equal(A, np.triu(A)): + A = _logm_force_nonsingular_triangular_matrix(A) + if np.min(np.diag(A), initial=0.) < 0: + A = A.astype(complex) + return _logm_triu(A) + else: + if keep_it_real: + T, Z = schur(A) + if not np.array_equal(T, np.triu(T)): + T, Z = rsf2csf(T, Z) + else: + T, Z = schur(A, output='complex') + T = _logm_force_nonsingular_triangular_matrix(T, inplace=True) + U = _logm_triu(T) + ZH = np.conjugate(Z).T + return Z.dot(U).dot(ZH) + except (SqrtmError, LogmError): + X = np.empty_like(A) + X.fill(np.nan) + return X diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs_sqrtm.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs_sqrtm.py new file mode 100644 index 0000000000000000000000000000000000000000..b7da6ced474ee3db548a24ecc08d7e2627f0d7a4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_matfuncs_sqrtm.py @@ -0,0 +1,205 @@ +""" +Matrix square root for general matrices and for upper triangular matrices. + +This module exists to avoid cyclic imports. + +""" +__all__ = ['sqrtm'] + +import numpy as np + +from scipy._lib._util import _asarray_validated + +# Local imports +from ._misc import norm +from .lapack import ztrsyl, dtrsyl +from ._decomp_schur import schur, rsf2csf +from ._basic import _ensure_dtype_cdsz + + + +class SqrtmError(np.linalg.LinAlgError): + pass + + +from ._matfuncs_sqrtm_triu import within_block_loop # noqa: E402 + + +def _sqrtm_triu(T, blocksize=64): + """ + Matrix square root of an upper triangular matrix. + + This is a helper function for `sqrtm` and `logm`. + + Parameters + ---------- + T : (N, N) array_like upper triangular + Matrix whose square root to evaluate + blocksize : int, optional + If the blocksize is not degenerate with respect to the + size of the input array, then use a blocked algorithm. (Default: 64) + + Returns + ------- + sqrtm : (N, N) ndarray + Value of the sqrt function at `T` + + References + ---------- + .. [1] Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013) + "Blocked Schur Algorithms for Computing the Matrix Square Root, + Lecture Notes in Computer Science, 7782. pp. 171-182. + + """ + T_diag = np.diag(T) + keep_it_real = np.isrealobj(T) and np.min(T_diag, initial=0.) >= 0 + + # Cast to complex as necessary + ensure double precision + if not keep_it_real: + T = np.asarray(T, dtype=np.complex128, order="C") + T_diag = np.asarray(T_diag, dtype=np.complex128) + else: + T = np.asarray(T, dtype=np.float64, order="C") + T_diag = np.asarray(T_diag, dtype=np.float64) + + R = np.diag(np.sqrt(T_diag)) + + # Compute the number of blocks to use; use at least one block. + n, n = T.shape + nblocks = max(n // blocksize, 1) + + # Compute the smaller of the two sizes of blocks that + # we will actually use, and compute the number of large blocks. + bsmall, nlarge = divmod(n, nblocks) + blarge = bsmall + 1 + nsmall = nblocks - nlarge + if nsmall * bsmall + nlarge * blarge != n: + raise Exception('internal inconsistency') + + # Define the index range covered by each block. + start_stop_pairs = [] + start = 0 + for count, size in ((nsmall, bsmall), (nlarge, blarge)): + for i in range(count): + start_stop_pairs.append((start, start + size)) + start += size + + # Within-block interactions (Cythonized) + try: + within_block_loop(R, T, start_stop_pairs, nblocks) + except RuntimeError as e: + raise SqrtmError(*e.args) from e + + # Between-block interactions (Cython would give no significant speedup) + for j in range(nblocks): + jstart, jstop = start_stop_pairs[j] + for i in range(j-1, -1, -1): + istart, istop = start_stop_pairs[i] + S = T[istart:istop, jstart:jstop] + if j - i > 1: + S = S - R[istart:istop, istop:jstart].dot(R[istop:jstart, + jstart:jstop]) + + # Invoke LAPACK. + # For more details, see the solve_sylvester implementation + # and the fortran dtrsyl and ztrsyl docs. + Rii = R[istart:istop, istart:istop] + Rjj = R[jstart:jstop, jstart:jstop] + if keep_it_real: + x, scale, info = dtrsyl(Rii, Rjj, S) + else: + x, scale, info = ztrsyl(Rii, Rjj, S) + R[istart:istop, jstart:jstop] = x * scale + + # Return the matrix square root. + return R + + +def sqrtm(A, disp=True, blocksize=64): + """ + Matrix square root. + + Parameters + ---------- + A : (N, N) array_like + Matrix whose square root to evaluate + disp : bool, optional + Print warning if error in the result is estimated large + instead of returning estimated error. (Default: True) + blocksize : integer, optional + If the blocksize is not degenerate with respect to the + size of the input array, then use a blocked algorithm. (Default: 64) + + Returns + ------- + sqrtm : (N, N) ndarray + Value of the sqrt function at `A`. The dtype is float or complex. + The precision (data size) is determined based on the precision of + input `A`. + + errest : float + (if disp == False) + + Frobenius norm of the estimated error, ||err||_F / ||A||_F + + References + ---------- + .. [1] Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013) + "Blocked Schur Algorithms for Computing the Matrix Square Root, + Lecture Notes in Computer Science, 7782. pp. 171-182. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import sqrtm + >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) + >>> r = sqrtm(a) + >>> r + array([[ 0.75592895, 1.13389342], + [ 0.37796447, 1.88982237]]) + >>> r.dot(r) + array([[ 1., 3.], + [ 1., 4.]]) + + """ + A = _asarray_validated(A, check_finite=True, as_inexact=True) + if len(A.shape) != 2: + raise ValueError("Non-matrix input to matrix function.") + if blocksize < 1: + raise ValueError("The blocksize should be at least 1.") + A, = _ensure_dtype_cdsz(A) + keep_it_real = np.isrealobj(A) + if keep_it_real: + T, Z = schur(A) + d0 = np.diagonal(T) + d1 = np.diagonal(T, -1) + eps = np.finfo(T.dtype).eps + needs_conversion = abs(d1) > eps * (abs(d0[1:]) + abs(d0[:-1])) + if needs_conversion.any(): + T, Z = rsf2csf(T, Z) + else: + T, Z = schur(A, output='complex') + failflag = False + try: + R = _sqrtm_triu(T, blocksize=blocksize) + ZH = np.conjugate(Z).T + X = Z.dot(R).dot(ZH) + dtype = np.result_type(A.dtype, 1j if np.iscomplexobj(X) else 1) + X = X.astype(dtype, copy=False) + except SqrtmError: + failflag = True + X = np.empty_like(A) + X.fill(np.nan) + + if disp: + if failflag: + print("Failed to find a square root.") + return X + else: + try: + arg2 = norm(X.dot(X) - A, 'fro')**2 / norm(A, 'fro') + except ValueError: + # NaNs in matrix + arg2 = np.inf + + return X, arg2 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_misc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_misc.py new file mode 100644 index 0000000000000000000000000000000000000000..27cd442080c8569417694a8a612fe0a461c1a2ca --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_misc.py @@ -0,0 +1,191 @@ +import numpy as np +from numpy.linalg import LinAlgError +from .blas import get_blas_funcs +from .lapack import get_lapack_funcs + +__all__ = ['LinAlgError', 'LinAlgWarning', 'norm'] + + +class LinAlgWarning(RuntimeWarning): + """ + The warning emitted when a linear algebra related operation is close + to fail conditions of the algorithm or loss of accuracy is expected. + """ + pass + + +def norm(a, ord=None, axis=None, keepdims=False, check_finite=True): + """ + Matrix or vector norm. + + This function is able to return one of eight different matrix norms, + or one of an infinite number of vector norms (described below), depending + on the value of the ``ord`` parameter. For tensors with rank different from + 1 or 2, only `ord=None` is supported. + + Parameters + ---------- + a : array_like + Input array. If `axis` is None, `a` must be 1-D or 2-D, unless `ord` + is None. If both `axis` and `ord` are None, the 2-norm of + ``a.ravel`` will be returned. + ord : {int, inf, -inf, 'fro', 'nuc', None}, optional + Order of the norm (see table under ``Notes``). inf means NumPy's + `inf` object. + axis : {int, 2-tuple of ints, None}, optional + If `axis` is an integer, it specifies the axis of `a` along which to + compute the vector norms. If `axis` is a 2-tuple, it specifies the + axes that hold 2-D matrices, and the matrix norms of these matrices + are computed. If `axis` is None then either a vector norm (when `a` + is 1-D) or a matrix norm (when `a` is 2-D) is returned. + keepdims : bool, optional + If this is set to True, the axes which are normed over are left in the + result as dimensions with size one. With this option the result will + broadcast correctly against the original `a`. + check_finite : bool, optional + Whether to check that the input matrix contains only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + n : float or ndarray + Norm of the matrix or vector(s). + + Notes + ----- + For values of ``ord <= 0``, the result is, strictly speaking, not a + mathematical 'norm', but it may still be useful for various numerical + purposes. + + The following norms can be calculated: + + ===== ============================ ========================== + ord norm for matrices norm for vectors + ===== ============================ ========================== + None Frobenius norm 2-norm + 'fro' Frobenius norm -- + 'nuc' nuclear norm -- + inf max(sum(abs(a), axis=1)) max(abs(a)) + -inf min(sum(abs(a), axis=1)) min(abs(a)) + 0 -- sum(a != 0) + 1 max(sum(abs(a), axis=0)) as below + -1 min(sum(abs(a), axis=0)) as below + 2 2-norm (largest sing. value) as below + -2 smallest singular value as below + other -- sum(abs(a)**ord)**(1./ord) + ===== ============================ ========================== + + The Frobenius norm is given by [1]_: + + :math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}` + + The nuclear norm is the sum of the singular values. + + Both the Frobenius and nuclear norm orders are only defined for + matrices. + + References + ---------- + .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*, + Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15 + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import norm + >>> a = np.arange(9) - 4.0 + >>> a + array([-4., -3., -2., -1., 0., 1., 2., 3., 4.]) + >>> b = a.reshape((3, 3)) + >>> b + array([[-4., -3., -2.], + [-1., 0., 1.], + [ 2., 3., 4.]]) + + >>> norm(a) + 7.745966692414834 + >>> norm(b) + 7.745966692414834 + >>> norm(b, 'fro') + 7.745966692414834 + >>> norm(a, np.inf) + 4.0 + >>> norm(b, np.inf) + 9.0 + >>> norm(a, -np.inf) + 0.0 + >>> norm(b, -np.inf) + 2.0 + + >>> norm(a, 1) + 20.0 + >>> norm(b, 1) + 7.0 + >>> norm(a, -1) + -4.6566128774142013e-010 + >>> norm(b, -1) + 6.0 + >>> norm(a, 2) + 7.745966692414834 + >>> norm(b, 2) + 7.3484692283495345 + + >>> norm(a, -2) + 0.0 + >>> norm(b, -2) + 1.8570331885190563e-016 + >>> norm(a, 3) + 5.8480354764257312 + >>> norm(a, -3) + 0.0 + + """ + # Differs from numpy only in non-finite handling and the use of blas. + if check_finite: + a = np.asarray_chkfinite(a) + else: + a = np.asarray(a) + + if a.size and a.dtype.char in 'fdFD' and axis is None and not keepdims: + + if ord in (None, 2) and (a.ndim == 1): + # use blas for fast and stable euclidean norm + nrm2 = get_blas_funcs('nrm2', dtype=a.dtype, ilp64='preferred') + return nrm2(a) + + if a.ndim == 2: + # Use lapack for a couple fast matrix norms. + # For some reason the *lange frobenius norm is slow. + lange_args = None + # Make sure this works if the user uses the axis keywords + # to apply the norm to the transpose. + if ord == 1: + if np.isfortran(a): + lange_args = '1', a + elif np.isfortran(a.T): + lange_args = 'i', a.T + elif ord == np.inf: + if np.isfortran(a): + lange_args = 'i', a + elif np.isfortran(a.T): + lange_args = '1', a.T + if lange_args: + lange = get_lapack_funcs('lange', dtype=a.dtype, ilp64='preferred') + return lange(*lange_args) + + # fall back to numpy in every other case + return np.linalg.norm(a, ord=ord, axis=axis, keepdims=keepdims) + + +def _datacopied(arr, original): + """ + Strict check for `arr` not sharing any data with `original`, + under the assumption that arr = asarray(original) + + """ + if arr is original: + return False + if not isinstance(original, np.ndarray) and hasattr(original, '__array__'): + return False + return arr.base is None diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_procrustes.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_procrustes.py new file mode 100644 index 0000000000000000000000000000000000000000..7d68f0b737ead5d581095ad32a34ae88d153264c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_procrustes.py @@ -0,0 +1,111 @@ +""" +Solve the orthogonal Procrustes problem. + +""" +import numpy as np +from ._decomp_svd import svd + + +__all__ = ['orthogonal_procrustes'] + + +def orthogonal_procrustes(A, B, check_finite=True): + """ + Compute the matrix solution of the orthogonal (or unitary) Procrustes problem. + + Given matrices `A` and `B` of the same shape, find an orthogonal (or unitary in + the case of complex input) matrix `R` that most closely maps `A` to `B` using the + algorithm given in [1]_. + + Parameters + ---------- + A : (M, N) array_like + Matrix to be mapped. + B : (M, N) array_like + Target matrix. + check_finite : bool, optional + Whether to check that the input matrices contain only finite numbers. + Disabling may give a performance gain, but may result in problems + (crashes, non-termination) if the inputs do contain infinities or NaNs. + + Returns + ------- + R : (N, N) ndarray + The matrix solution of the orthogonal Procrustes problem. + Minimizes the Frobenius norm of ``(A @ R) - B``, subject to + ``R.conj().T @ R = I``. + scale : float + Sum of the singular values of ``A.conj().T @ B``. + + Raises + ------ + ValueError + If the input array shapes don't match or if check_finite is True and + the arrays contain Inf or NaN. + + Notes + ----- + Note that unlike higher level Procrustes analyses of spatial data, this + function only uses orthogonal transformations like rotations and + reflections, and it does not use scaling or translation. + + .. versionadded:: 0.15.0 + + References + ---------- + .. [1] Peter H. Schonemann, "A generalized solution of the orthogonal + Procrustes problem", Psychometrica -- Vol. 31, No. 1, March, 1966. + :doi:`10.1007/BF02289451` + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import orthogonal_procrustes + >>> A = np.array([[ 2, 0, 1], [-2, 0, 0]]) + + Flip the order of columns and check for the anti-diagonal mapping + + >>> R, sca = orthogonal_procrustes(A, np.fliplr(A)) + >>> R + array([[-5.34384992e-17, 0.00000000e+00, 1.00000000e+00], + [ 0.00000000e+00, 1.00000000e+00, 0.00000000e+00], + [ 1.00000000e+00, 0.00000000e+00, -7.85941422e-17]]) + >>> sca + 9.0 + + As an example of the unitary Procrustes problem, generate a + random complex matrix ``A``, a random unitary matrix ``Q``, + and their product ``B``. + + >>> shape = (4, 4) + >>> rng = np.random.default_rng(589234981235) + >>> A = rng.random(shape) + rng.random(shape)*1j + >>> Q = rng.random(shape) + rng.random(shape)*1j + >>> Q, _ = np.linalg.qr(Q) + >>> B = A @ Q + + `orthogonal_procrustes` recovers the unitary matrix ``Q`` + from ``A`` and ``B``. + + >>> R, _ = orthogonal_procrustes(A, B) + >>> np.allclose(R, Q) + True + + """ + if check_finite: + A = np.asarray_chkfinite(A) + B = np.asarray_chkfinite(B) + else: + A = np.asanyarray(A) + B = np.asanyarray(B) + if A.ndim != 2: + raise ValueError(f'expected ndim to be 2, but observed {A.ndim}') + if A.shape != B.shape: + raise ValueError(f'the shapes of A and B differ ({A.shape} vs {B.shape})') + # Be clever with transposes, with the intention to save memory. + # The conjugate has no effect for real inputs, but gives the correct solution + # for complex inputs. + u, w, vt = svd((B.T @ np.conjugate(A)).T) + R = u @ vt + scale = w.sum() + return R, scale diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_sketches.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_sketches.py new file mode 100644 index 0000000000000000000000000000000000000000..589172827f528799203cb3e93a4a013e07dc5ff8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_sketches.py @@ -0,0 +1,178 @@ +""" Sketching-based Matrix Computations """ + +# Author: Jordi Montes +# August 28, 2017 + +import numpy as np + +from scipy._lib._util import (check_random_state, rng_integers, + _transition_to_rng) +from scipy.sparse import csc_matrix + +__all__ = ['clarkson_woodruff_transform'] + + +def cwt_matrix(n_rows, n_columns, rng=None): + r""" + Generate a matrix S which represents a Clarkson-Woodruff transform. + + Given the desired size of matrix, the method returns a matrix S of size + (n_rows, n_columns) where each column has all the entries set to 0 + except for one position which has been randomly set to +1 or -1 with + equal probability. + + Parameters + ---------- + n_rows : int + Number of rows of S + n_columns : int + Number of columns of S + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + + + Returns + ------- + S : (n_rows, n_columns) csc_matrix + The returned matrix has ``n_columns`` nonzero entries. + + Notes + ----- + Given a matrix A, with probability at least 9/10, + .. math:: \|SA\| = (1 \pm \epsilon)\|A\| + Where the error epsilon is related to the size of S. + """ + rng = check_random_state(rng) + rows = rng_integers(rng, 0, n_rows, n_columns) + cols = np.arange(n_columns+1) + signs = rng.choice([1, -1], n_columns) + S = csc_matrix((signs, rows, cols), shape=(n_rows, n_columns)) + return S + + +@_transition_to_rng("seed", position_num=2) +def clarkson_woodruff_transform(input_matrix, sketch_size, rng=None): + r""" + Applies a Clarkson-Woodruff Transform/sketch to the input matrix. + + Given an input_matrix ``A`` of size ``(n, d)``, compute a matrix ``A'`` of + size (sketch_size, d) so that + + .. math:: \|Ax\| \approx \|A'x\| + + with high probability via the Clarkson-Woodruff Transform, otherwise + known as the CountSketch matrix. + + Parameters + ---------- + input_matrix : array_like + Input matrix, of shape ``(n, d)``. + sketch_size : int + Number of rows for the sketch. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + + Returns + ------- + A' : array_like + Sketch of the input matrix ``A``, of size ``(sketch_size, d)``. + + Notes + ----- + To make the statement + + .. math:: \|Ax\| \approx \|A'x\| + + precise, observe the following result which is adapted from the + proof of Theorem 14 of [2]_ via Markov's Inequality. If we have + a sketch size ``sketch_size=k`` which is at least + + .. math:: k \geq \frac{2}{\epsilon^2\delta} + + Then for any fixed vector ``x``, + + .. math:: \|Ax\| = (1\pm\epsilon)\|A'x\| + + with probability at least one minus delta. + + This implementation takes advantage of sparsity: computing + a sketch takes time proportional to ``A.nnz``. Data ``A`` which + is in ``scipy.sparse.csc_matrix`` format gives the quickest + computation time for sparse input. + + >>> import numpy as np + >>> from scipy import linalg + >>> from scipy import sparse + >>> rng = np.random.default_rng() + >>> n_rows, n_columns, density, sketch_n_rows = 15000, 100, 0.01, 200 + >>> A = sparse.rand(n_rows, n_columns, density=density, format='csc') + >>> B = sparse.rand(n_rows, n_columns, density=density, format='csr') + >>> C = sparse.rand(n_rows, n_columns, density=density, format='coo') + >>> D = rng.standard_normal((n_rows, n_columns)) + >>> SA = linalg.clarkson_woodruff_transform(A, sketch_n_rows) # fastest + >>> SB = linalg.clarkson_woodruff_transform(B, sketch_n_rows) # fast + >>> SC = linalg.clarkson_woodruff_transform(C, sketch_n_rows) # slower + >>> SD = linalg.clarkson_woodruff_transform(D, sketch_n_rows) # slowest + + That said, this method does perform well on dense inputs, just slower + on a relative scale. + + References + ---------- + .. [1] Kenneth L. Clarkson and David P. Woodruff. Low rank approximation + and regression in input sparsity time. In STOC, 2013. + .. [2] David P. Woodruff. Sketching as a tool for numerical linear algebra. + In Foundations and Trends in Theoretical Computer Science, 2014. + + Examples + -------- + Create a big dense matrix ``A`` for the example: + + >>> import numpy as np + >>> from scipy import linalg + >>> n_rows, n_columns = 15000, 100 + >>> rng = np.random.default_rng() + >>> A = rng.standard_normal((n_rows, n_columns)) + + Apply the transform to create a new matrix with 200 rows: + + >>> sketch_n_rows = 200 + >>> sketch = linalg.clarkson_woodruff_transform(A, sketch_n_rows, seed=rng) + >>> sketch.shape + (200, 100) + + Now with high probability, the true norm is close to the sketched norm + in absolute value. + + >>> linalg.norm(A) + 1224.2812927123198 + >>> linalg.norm(sketch) + 1226.518328407333 + + Similarly, applying our sketch preserves the solution to a linear + regression of :math:`\min \|Ax - b\|`. + + >>> b = rng.standard_normal(n_rows) + >>> x = linalg.lstsq(A, b)[0] + >>> Ab = np.hstack((A, b.reshape(-1, 1))) + >>> SAb = linalg.clarkson_woodruff_transform(Ab, sketch_n_rows, seed=rng) + >>> SA, Sb = SAb[:, :-1], SAb[:, -1] + >>> x_sketched = linalg.lstsq(SA, Sb)[0] + + As with the matrix norm example, ``linalg.norm(A @ x - b)`` is close + to ``linalg.norm(A @ x_sketched - b)`` with high probability. + + >>> linalg.norm(A @ x - b) + 122.83242365433877 + >>> linalg.norm(A @ x_sketched - b) + 166.58473879945151 + + """ + S = cwt_matrix(sketch_size, input_matrix.shape[0], rng=rng) + return S.dot(input_matrix) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_solvers.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_solvers.py new file mode 100644 index 0000000000000000000000000000000000000000..60a6a73e7bf9cce36090137c0e884d3ed22c55a8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_solvers.py @@ -0,0 +1,857 @@ +"""Matrix equation solver routines""" +# Author: Jeffrey Armstrong +# February 24, 2012 + +# Modified: Chad Fulton +# June 19, 2014 + +# Modified: Ilhan Polat +# September 13, 2016 + +import warnings +import numpy as np +from numpy.linalg import inv, LinAlgError, norm, cond, svd + +from ._basic import solve, solve_triangular, matrix_balance +from .lapack import get_lapack_funcs +from ._decomp_schur import schur +from ._decomp_lu import lu +from ._decomp_qr import qr +from ._decomp_qz import ordqz +from ._decomp import _asarray_validated +from ._special_matrices import block_diag + +__all__ = ['solve_sylvester', + 'solve_continuous_lyapunov', 'solve_discrete_lyapunov', + 'solve_lyapunov', + 'solve_continuous_are', 'solve_discrete_are'] + + +def solve_sylvester(a, b, q): + """ + Computes a solution (X) to the Sylvester equation :math:`AX + XB = Q`. + + Parameters + ---------- + a : (M, M) array_like + Leading matrix of the Sylvester equation + b : (N, N) array_like + Trailing matrix of the Sylvester equation + q : (M, N) array_like + Right-hand side + + Returns + ------- + x : (M, N) ndarray + The solution to the Sylvester equation. + + Raises + ------ + LinAlgError + If solution was not found + + Notes + ----- + Computes a solution to the Sylvester matrix equation via the Bartels- + Stewart algorithm. The A and B matrices first undergo Schur + decompositions. The resulting matrices are used to construct an + alternative Sylvester equation (``RY + YS^T = F``) where the R and S + matrices are in quasi-triangular form (or, when R, S or F are complex, + triangular form). The simplified equation is then solved using + ``*TRSYL`` from LAPACK directly. + + .. versionadded:: 0.11.0 + + Examples + -------- + Given `a`, `b`, and `q` solve for `x`: + + >>> import numpy as np + >>> from scipy import linalg + >>> a = np.array([[-3, -2, 0], [-1, -1, 3], [3, -5, -1]]) + >>> b = np.array([[1]]) + >>> q = np.array([[1],[2],[3]]) + >>> x = linalg.solve_sylvester(a, b, q) + >>> x + array([[ 0.0625], + [-0.5625], + [ 0.6875]]) + >>> np.allclose(a.dot(x) + x.dot(b), q) + True + + """ + # Accommodate empty a + if a.size == 0 or b.size == 0: + tdict = {'s': np.float32, 'd': np.float64, + 'c': np.complex64, 'z': np.complex128} + func, = get_lapack_funcs(('trsyl',), arrays=(a, b, q)) + return np.empty(q.shape, dtype=tdict[func.typecode]) + + # Compute the Schur decomposition form of a + r, u = schur(a, output='real') + + # Compute the Schur decomposition of b + s, v = schur(b.conj().transpose(), output='real') + + # Construct f = u'*q*v + f = np.dot(np.dot(u.conj().transpose(), q), v) + + # Call the Sylvester equation solver + trsyl, = get_lapack_funcs(('trsyl',), (r, s, f)) + if trsyl is None: + raise RuntimeError('LAPACK implementation does not contain a proper ' + 'Sylvester equation solver (TRSYL)') + y, scale, info = trsyl(r, s, f, tranb='C') + + y = scale*y + + if info < 0: + raise LinAlgError("Illegal value encountered in " + "the %d term" % (-info,)) + + return np.dot(np.dot(u, y), v.conj().transpose()) + + +def solve_continuous_lyapunov(a, q): + """ + Solves the continuous Lyapunov equation :math:`AX + XA^H = Q`. + + Uses the Bartels-Stewart algorithm to find :math:`X`. + + Parameters + ---------- + a : array_like + A square matrix + + q : array_like + Right-hand side square matrix + + Returns + ------- + x : ndarray + Solution to the continuous Lyapunov equation + + See Also + -------- + solve_discrete_lyapunov : computes the solution to the discrete-time + Lyapunov equation + solve_sylvester : computes the solution to the Sylvester equation + + Notes + ----- + The continuous Lyapunov equation is a special form of the Sylvester + equation, hence this solver relies on LAPACK routine ?TRSYL. + + .. versionadded:: 0.11.0 + + Examples + -------- + Given `a` and `q` solve for `x`: + + >>> import numpy as np + >>> from scipy import linalg + >>> a = np.array([[-3, -2, 0], [-1, -1, 0], [0, -5, -1]]) + >>> b = np.array([2, 4, -1]) + >>> q = np.eye(3) + >>> x = linalg.solve_continuous_lyapunov(a, q) + >>> x + array([[ -0.75 , 0.875 , -3.75 ], + [ 0.875 , -1.375 , 5.3125], + [ -3.75 , 5.3125, -27.0625]]) + >>> np.allclose(a.dot(x) + x.dot(a.T), q) + True + """ + + a = np.atleast_2d(_asarray_validated(a, check_finite=True)) + q = np.atleast_2d(_asarray_validated(q, check_finite=True)) + + r_or_c = float + + for ind, _ in enumerate((a, q)): + if np.iscomplexobj(_): + r_or_c = complex + + if not np.equal(*_.shape): + raise ValueError(f"Matrix {'aq'[ind]} should be square.") + + # Shape consistency check + if a.shape != q.shape: + raise ValueError("Matrix a and q should have the same shape.") + + # Accommodate empty array + if a.size == 0: + tdict = {'s': np.float32, 'd': np.float64, + 'c': np.complex64, 'z': np.complex128} + func, = get_lapack_funcs(('trsyl',), arrays=(a, q)) + return np.empty(a.shape, dtype=tdict[func.typecode]) + + # Compute the Schur decomposition form of a + r, u = schur(a, output='real') + + # Construct f = u'*q*u + f = u.conj().T.dot(q.dot(u)) + + # Call the Sylvester equation solver + trsyl = get_lapack_funcs('trsyl', (r, f)) + + dtype_string = 'T' if r_or_c is float else 'C' + y, scale, info = trsyl(r, r, f, tranb=dtype_string) + + if info < 0: + raise ValueError('?TRSYL exited with the internal error ' + f'"illegal value in argument number {-info}.". See ' + 'LAPACK documentation for the ?TRSYL error codes.') + elif info == 1: + warnings.warn('Input "a" has an eigenvalue pair whose sum is ' + 'very close to or exactly zero. The solution is ' + 'obtained via perturbing the coefficients.', + RuntimeWarning, stacklevel=2) + y *= scale + + return u.dot(y).dot(u.conj().T) + + +# For backwards compatibility, keep the old name +solve_lyapunov = solve_continuous_lyapunov + + +def _solve_discrete_lyapunov_direct(a, q): + """ + Solves the discrete Lyapunov equation directly. + + This function is called by the `solve_discrete_lyapunov` function with + `method=direct`. It is not supposed to be called directly. + """ + + lhs = np.kron(a, a.conj()) + lhs = np.eye(lhs.shape[0]) - lhs + x = solve(lhs, q.flatten()) + + return np.reshape(x, q.shape) + + +def _solve_discrete_lyapunov_bilinear(a, q): + """ + Solves the discrete Lyapunov equation using a bilinear transformation. + + This function is called by the `solve_discrete_lyapunov` function with + `method=bilinear`. It is not supposed to be called directly. + """ + eye = np.eye(a.shape[0]) + aH = a.conj().transpose() + aHI_inv = inv(aH + eye) + b = np.dot(aH - eye, aHI_inv) + c = 2*np.dot(np.dot(inv(a + eye), q), aHI_inv) + return solve_lyapunov(b.conj().transpose(), -c) + + +def solve_discrete_lyapunov(a, q, method=None): + """ + Solves the discrete Lyapunov equation :math:`AXA^H - X + Q = 0`. + + Parameters + ---------- + a, q : (M, M) array_like + Square matrices corresponding to A and Q in the equation + above respectively. Must have the same shape. + + method : {'direct', 'bilinear'}, optional + Type of solver. + + If not given, chosen to be ``direct`` if ``M`` is less than 10 and + ``bilinear`` otherwise. + + Returns + ------- + x : ndarray + Solution to the discrete Lyapunov equation + + See Also + -------- + solve_continuous_lyapunov : computes the solution to the continuous-time + Lyapunov equation + + Notes + ----- + This section describes the available solvers that can be selected by the + 'method' parameter. The default method is *direct* if ``M`` is less than 10 + and ``bilinear`` otherwise. + + Method *direct* uses a direct analytical solution to the discrete Lyapunov + equation. The algorithm is given in, for example, [1]_. However, it requires + the linear solution of a system with dimension :math:`M^2` so that + performance degrades rapidly for even moderately sized matrices. + + Method *bilinear* uses a bilinear transformation to convert the discrete + Lyapunov equation to a continuous Lyapunov equation :math:`(BX+XB'=-C)` + where :math:`B=(A-I)(A+I)^{-1}` and + :math:`C=2(A' + I)^{-1} Q (A + I)^{-1}`. The continuous equation can be + efficiently solved since it is a special case of a Sylvester equation. + The transformation algorithm is from Popov (1964) as described in [2]_. + + .. versionadded:: 0.11.0 + + References + ---------- + .. [1] "Lyapunov equation", Wikipedia, + https://en.wikipedia.org/wiki/Lyapunov_equation#Discrete_time + .. [2] Gajic, Z., and M.T.J. Qureshi. 2008. + Lyapunov Matrix Equation in System Stability and Control. + Dover Books on Engineering Series. Dover Publications. + + Examples + -------- + Given `a` and `q` solve for `x`: + + >>> import numpy as np + >>> from scipy import linalg + >>> a = np.array([[0.2, 0.5],[0.7, -0.9]]) + >>> q = np.eye(2) + >>> x = linalg.solve_discrete_lyapunov(a, q) + >>> x + array([[ 0.70872893, 1.43518822], + [ 1.43518822, -2.4266315 ]]) + >>> np.allclose(a.dot(x).dot(a.T)-x, -q) + True + + """ + a = np.asarray(a) + q = np.asarray(q) + if method is None: + # Select automatically based on size of matrices + if a.shape[0] >= 10: + method = 'bilinear' + else: + method = 'direct' + + meth = method.lower() + + if meth == 'direct': + x = _solve_discrete_lyapunov_direct(a, q) + elif meth == 'bilinear': + x = _solve_discrete_lyapunov_bilinear(a, q) + else: + raise ValueError(f'Unknown solver {method}') + + return x + + +def solve_continuous_are(a, b, q, r, e=None, s=None, balanced=True): + r""" + Solves the continuous-time algebraic Riccati equation (CARE). + + The CARE is defined as + + .. math:: + + X A + A^H X - X B R^{-1} B^H X + Q = 0 + + The limitations for a solution to exist are : + + * All eigenvalues of :math:`A` on the right half plane, should be + controllable. + + * The associated hamiltonian pencil (See Notes), should have + eigenvalues sufficiently away from the imaginary axis. + + Moreover, if ``e`` or ``s`` is not precisely ``None``, then the + generalized version of CARE + + .. math:: + + E^HXA + A^HXE - (E^HXB + S) R^{-1} (B^HXE + S^H) + Q = 0 + + is solved. When omitted, ``e`` is assumed to be the identity and ``s`` + is assumed to be the zero matrix with sizes compatible with ``a`` and + ``b``, respectively. + + Parameters + ---------- + a : (M, M) array_like + Square matrix + b : (M, N) array_like + Input + q : (M, M) array_like + Input + r : (N, N) array_like + Nonsingular square matrix + e : (M, M) array_like, optional + Nonsingular square matrix + s : (M, N) array_like, optional + Input + balanced : bool, optional + The boolean that indicates whether a balancing step is performed + on the data. The default is set to True. + + Returns + ------- + x : (M, M) ndarray + Solution to the continuous-time algebraic Riccati equation. + + Raises + ------ + LinAlgError + For cases where the stable subspace of the pencil could not be + isolated. See Notes section and the references for details. + + See Also + -------- + solve_discrete_are : Solves the discrete-time algebraic Riccati equation + + Notes + ----- + The equation is solved by forming the extended hamiltonian matrix pencil, + as described in [1]_, :math:`H - \lambda J` given by the block matrices :: + + [ A 0 B ] [ E 0 0 ] + [-Q -A^H -S ] - \lambda * [ 0 E^H 0 ] + [ S^H B^H R ] [ 0 0 0 ] + + and using a QZ decomposition method. + + In this algorithm, the fail conditions are linked to the symmetry + of the product :math:`U_2 U_1^{-1}` and condition number of + :math:`U_1`. Here, :math:`U` is the 2m-by-m matrix that holds the + eigenvectors spanning the stable subspace with 2-m rows and partitioned + into two m-row matrices. See [1]_ and [2]_ for more details. + + In order to improve the QZ decomposition accuracy, the pencil goes + through a balancing step where the sum of absolute values of + :math:`H` and :math:`J` entries (after removing the diagonal entries of + the sum) is balanced following the recipe given in [3]_. + + .. versionadded:: 0.11.0 + + References + ---------- + .. [1] P. van Dooren , "A Generalized Eigenvalue Approach For Solving + Riccati Equations.", SIAM Journal on Scientific and Statistical + Computing, Vol.2(2), :doi:`10.1137/0902010` + + .. [2] A.J. Laub, "A Schur Method for Solving Algebraic Riccati + Equations.", Massachusetts Institute of Technology. Laboratory for + Information and Decision Systems. LIDS-R ; 859. Available online : + http://hdl.handle.net/1721.1/1301 + + .. [3] P. Benner, "Symplectic Balancing of Hamiltonian Matrices", 2001, + SIAM J. Sci. Comput., 2001, Vol.22(5), :doi:`10.1137/S1064827500367993` + + Examples + -------- + Given `a`, `b`, `q`, and `r` solve for `x`: + + >>> import numpy as np + >>> from scipy import linalg + >>> a = np.array([[4, 3], [-4.5, -3.5]]) + >>> b = np.array([[1], [-1]]) + >>> q = np.array([[9, 6], [6, 4.]]) + >>> r = 1 + >>> x = linalg.solve_continuous_are(a, b, q, r) + >>> x + array([[ 21.72792206, 14.48528137], + [ 14.48528137, 9.65685425]]) + >>> np.allclose(a.T.dot(x) + x.dot(a)-x.dot(b).dot(b.T).dot(x), -q) + True + + """ + + # Validate input arguments + a, b, q, r, e, s, m, n, r_or_c, gen_are = _are_validate_args( + a, b, q, r, e, s, 'care') + + H = np.empty((2*m+n, 2*m+n), dtype=r_or_c) + H[:m, :m] = a + H[:m, m:2*m] = 0. + H[:m, 2*m:] = b + H[m:2*m, :m] = -q + H[m:2*m, m:2*m] = -a.conj().T + H[m:2*m, 2*m:] = 0. if s is None else -s + H[2*m:, :m] = 0. if s is None else s.conj().T + H[2*m:, m:2*m] = b.conj().T + H[2*m:, 2*m:] = r + + if gen_are and e is not None: + J = block_diag(e, e.conj().T, np.zeros_like(r, dtype=r_or_c)) + else: + J = block_diag(np.eye(2*m), np.zeros_like(r, dtype=r_or_c)) + + if balanced: + # xGEBAL does not remove the diagonals before scaling. Also + # to avoid destroying the Symplectic structure, we follow Ref.3 + M = np.abs(H) + np.abs(J) + np.fill_diagonal(M, 0.) + _, (sca, _) = matrix_balance(M, separate=1, permute=0) + # do we need to bother? + if not np.allclose(sca, np.ones_like(sca)): + # Now impose diag(D,inv(D)) from Benner where D is + # square root of s_i/s_(n+i) for i=0,.... + sca = np.log2(sca) + # NOTE: Py3 uses "Bankers Rounding: round to the nearest even" !! + s = np.round((sca[m:2*m] - sca[:m])/2) + sca = 2 ** np.r_[s, -s, sca[2*m:]] + # Elementwise multiplication via broadcasting. + elwisescale = sca[:, None] * np.reciprocal(sca) + H *= elwisescale + J *= elwisescale + + # Deflate the pencil to 2m x 2m ala Ref.1, eq.(55) + q, r = qr(H[:, -n:]) + H = q[:, n:].conj().T.dot(H[:, :2*m]) + J = q[:2*m, n:].conj().T.dot(J[:2*m, :2*m]) + + # Decide on which output type is needed for QZ + out_str = 'real' if r_or_c is float else 'complex' + + _, _, _, _, _, u = ordqz(H, J, sort='lhp', overwrite_a=True, + overwrite_b=True, check_finite=False, + output=out_str) + + # Get the relevant parts of the stable subspace basis + if e is not None: + u, _ = qr(np.vstack((e.dot(u[:m, :m]), u[m:, :m]))) + u00 = u[:m, :m] + u10 = u[m:, :m] + + # Solve via back-substituion after checking the condition of u00 + up, ul, uu = lu(u00) + if 1/cond(uu) < np.spacing(1.): + raise LinAlgError('Failed to find a finite solution.') + + # Exploit the triangular structure + x = solve_triangular(ul.conj().T, + solve_triangular(uu.conj().T, + u10.conj().T, + lower=True), + unit_diagonal=True, + ).conj().T.dot(up.conj().T) + if balanced: + x *= sca[:m, None] * sca[:m] + + # Check the deviation from symmetry for lack of success + # See proof of Thm.5 item 3 in [2] + u_sym = u00.conj().T.dot(u10) + n_u_sym = norm(u_sym, 1) + u_sym = u_sym - u_sym.conj().T + sym_threshold = np.max([np.spacing(1000.), 0.1*n_u_sym]) + + if norm(u_sym, 1) > sym_threshold: + raise LinAlgError('The associated Hamiltonian pencil has eigenvalues ' + 'too close to the imaginary axis') + + return (x + x.conj().T)/2 + + +def solve_discrete_are(a, b, q, r, e=None, s=None, balanced=True): + r""" + Solves the discrete-time algebraic Riccati equation (DARE). + + The DARE is defined as + + .. math:: + + A^HXA - X - (A^HXB) (R + B^HXB)^{-1} (B^HXA) + Q = 0 + + The limitations for a solution to exist are : + + * All eigenvalues of :math:`A` outside the unit disc, should be + controllable. + + * The associated symplectic pencil (See Notes), should have + eigenvalues sufficiently away from the unit circle. + + Moreover, if ``e`` and ``s`` are not both precisely ``None``, then the + generalized version of DARE + + .. math:: + + A^HXA - E^HXE - (A^HXB+S) (R+B^HXB)^{-1} (B^HXA+S^H) + Q = 0 + + is solved. When omitted, ``e`` is assumed to be the identity and ``s`` + is assumed to be the zero matrix. + + Parameters + ---------- + a : (M, M) array_like + Square matrix + b : (M, N) array_like + Input + q : (M, M) array_like + Input + r : (N, N) array_like + Square matrix + e : (M, M) array_like, optional + Nonsingular square matrix + s : (M, N) array_like, optional + Input + balanced : bool + The boolean that indicates whether a balancing step is performed + on the data. The default is set to True. + + Returns + ------- + x : (M, M) ndarray + Solution to the discrete algebraic Riccati equation. + + Raises + ------ + LinAlgError + For cases where the stable subspace of the pencil could not be + isolated. See Notes section and the references for details. + + See Also + -------- + solve_continuous_are : Solves the continuous algebraic Riccati equation + + Notes + ----- + The equation is solved by forming the extended symplectic matrix pencil, + as described in [1]_, :math:`H - \lambda J` given by the block matrices :: + + [ A 0 B ] [ E 0 B ] + [ -Q E^H -S ] - \lambda * [ 0 A^H 0 ] + [ S^H 0 R ] [ 0 -B^H 0 ] + + and using a QZ decomposition method. + + In this algorithm, the fail conditions are linked to the symmetry + of the product :math:`U_2 U_1^{-1}` and condition number of + :math:`U_1`. Here, :math:`U` is the 2m-by-m matrix that holds the + eigenvectors spanning the stable subspace with 2-m rows and partitioned + into two m-row matrices. See [1]_ and [2]_ for more details. + + In order to improve the QZ decomposition accuracy, the pencil goes + through a balancing step where the sum of absolute values of + :math:`H` and :math:`J` rows/cols (after removing the diagonal entries) + is balanced following the recipe given in [3]_. If the data has small + numerical noise, balancing may amplify their effects and some clean up + is required. + + .. versionadded:: 0.11.0 + + References + ---------- + .. [1] P. van Dooren , "A Generalized Eigenvalue Approach For Solving + Riccati Equations.", SIAM Journal on Scientific and Statistical + Computing, Vol.2(2), :doi:`10.1137/0902010` + + .. [2] A.J. Laub, "A Schur Method for Solving Algebraic Riccati + Equations.", Massachusetts Institute of Technology. Laboratory for + Information and Decision Systems. LIDS-R ; 859. Available online : + http://hdl.handle.net/1721.1/1301 + + .. [3] P. Benner, "Symplectic Balancing of Hamiltonian Matrices", 2001, + SIAM J. Sci. Comput., 2001, Vol.22(5), :doi:`10.1137/S1064827500367993` + + Examples + -------- + Given `a`, `b`, `q`, and `r` solve for `x`: + + >>> import numpy as np + >>> from scipy import linalg as la + >>> a = np.array([[0, 1], [0, -1]]) + >>> b = np.array([[1, 0], [2, 1]]) + >>> q = np.array([[-4, -4], [-4, 7]]) + >>> r = np.array([[9, 3], [3, 1]]) + >>> x = la.solve_discrete_are(a, b, q, r) + >>> x + array([[-4., -4.], + [-4., 7.]]) + >>> R = la.solve(r + b.T.dot(x).dot(b), b.T.dot(x).dot(a)) + >>> np.allclose(a.T.dot(x).dot(a) - x - a.T.dot(x).dot(b).dot(R), -q) + True + + """ + + # Validate input arguments + a, b, q, r, e, s, m, n, r_or_c, gen_are = _are_validate_args( + a, b, q, r, e, s, 'dare') + + # Form the matrix pencil + H = np.zeros((2*m+n, 2*m+n), dtype=r_or_c) + H[:m, :m] = a + H[:m, 2*m:] = b + H[m:2*m, :m] = -q + H[m:2*m, m:2*m] = np.eye(m) if e is None else e.conj().T + H[m:2*m, 2*m:] = 0. if s is None else -s + H[2*m:, :m] = 0. if s is None else s.conj().T + H[2*m:, 2*m:] = r + + J = np.zeros_like(H, dtype=r_or_c) + J[:m, :m] = np.eye(m) if e is None else e + J[m:2*m, m:2*m] = a.conj().T + J[2*m:, m:2*m] = -b.conj().T + + if balanced: + # xGEBAL does not remove the diagonals before scaling. Also + # to avoid destroying the Symplectic structure, we follow Ref.3 + M = np.abs(H) + np.abs(J) + np.fill_diagonal(M, 0.) + _, (sca, _) = matrix_balance(M, separate=1, permute=0) + # do we need to bother? + if not np.allclose(sca, np.ones_like(sca)): + # Now impose diag(D,inv(D)) from Benner where D is + # square root of s_i/s_(n+i) for i=0,.... + sca = np.log2(sca) + # NOTE: Py3 uses "Bankers Rounding: round to the nearest even" !! + s = np.round((sca[m:2*m] - sca[:m])/2) + sca = 2 ** np.r_[s, -s, sca[2*m:]] + # Elementwise multiplication via broadcasting. + elwisescale = sca[:, None] * np.reciprocal(sca) + H *= elwisescale + J *= elwisescale + + # Deflate the pencil by the R column ala Ref.1 + q_of_qr, _ = qr(H[:, -n:]) + H = q_of_qr[:, n:].conj().T.dot(H[:, :2*m]) + J = q_of_qr[:, n:].conj().T.dot(J[:, :2*m]) + + # Decide on which output type is needed for QZ + out_str = 'real' if r_or_c is float else 'complex' + + _, _, _, _, _, u = ordqz(H, J, sort='iuc', + overwrite_a=True, + overwrite_b=True, + check_finite=False, + output=out_str) + + # Get the relevant parts of the stable subspace basis + if e is not None: + u, _ = qr(np.vstack((e.dot(u[:m, :m]), u[m:, :m]))) + u00 = u[:m, :m] + u10 = u[m:, :m] + + # Solve via back-substituion after checking the condition of u00 + up, ul, uu = lu(u00) + + if 1/cond(uu) < np.spacing(1.): + raise LinAlgError('Failed to find a finite solution.') + + # Exploit the triangular structure + x = solve_triangular(ul.conj().T, + solve_triangular(uu.conj().T, + u10.conj().T, + lower=True), + unit_diagonal=True, + ).conj().T.dot(up.conj().T) + if balanced: + x *= sca[:m, None] * sca[:m] + + # Check the deviation from symmetry for lack of success + # See proof of Thm.5 item 3 in [2] + u_sym = u00.conj().T.dot(u10) + n_u_sym = norm(u_sym, 1) + u_sym = u_sym - u_sym.conj().T + sym_threshold = np.max([np.spacing(1000.), 0.1*n_u_sym]) + + if norm(u_sym, 1) > sym_threshold: + raise LinAlgError('The associated symplectic pencil has eigenvalues ' + 'too close to the unit circle') + + return (x + x.conj().T)/2 + + +def _are_validate_args(a, b, q, r, e, s, eq_type='care'): + """ + A helper function to validate the arguments supplied to the + Riccati equation solvers. Any discrepancy found in the input + matrices leads to a ``ValueError`` exception. + + Essentially, it performs: + + - a check whether the input is free of NaN and Infs + - a pass for the data through ``numpy.atleast_2d()`` + - squareness check of the relevant arrays + - shape consistency check of the arrays + - singularity check of the relevant arrays + - symmetricity check of the relevant matrices + - a check whether the regular or the generalized version is asked. + + This function is used by ``solve_continuous_are`` and + ``solve_discrete_are``. + + Parameters + ---------- + a, b, q, r, e, s : array_like + Input data + eq_type : str + Accepted arguments are 'care' and 'dare'. + + Returns + ------- + a, b, q, r, e, s : ndarray + Regularized input data + m, n : int + shape of the problem + r_or_c : type + Data type of the problem, returns float or complex + gen_or_not : bool + Type of the equation, True for generalized and False for regular ARE. + + """ + + if eq_type.lower() not in ("dare", "care"): + raise ValueError("Equation type unknown. " + "Only 'care' and 'dare' is understood") + + a = np.atleast_2d(_asarray_validated(a, check_finite=True)) + b = np.atleast_2d(_asarray_validated(b, check_finite=True)) + q = np.atleast_2d(_asarray_validated(q, check_finite=True)) + r = np.atleast_2d(_asarray_validated(r, check_finite=True)) + + # Get the correct data types otherwise NumPy complains + # about pushing complex numbers into real arrays. + r_or_c = complex if np.iscomplexobj(b) else float + + for ind, mat in enumerate((a, q, r)): + if np.iscomplexobj(mat): + r_or_c = complex + + if not np.equal(*mat.shape): + raise ValueError(f"Matrix {'aqr'[ind]} should be square.") + + # Shape consistency checks + m, n = b.shape + if m != a.shape[0]: + raise ValueError("Matrix a and b should have the same number of rows.") + if m != q.shape[0]: + raise ValueError("Matrix a and q should have the same shape.") + if n != r.shape[0]: + raise ValueError("Matrix b and r should have the same number of cols.") + + # Check if the data matrices q, r are (sufficiently) hermitian + for ind, mat in enumerate((q, r)): + if norm(mat - mat.conj().T, 1) > np.spacing(norm(mat, 1))*100: + raise ValueError(f"Matrix {'qr'[ind]} should be symmetric/hermitian.") + + # Continuous time ARE should have a nonsingular r matrix. + if eq_type == 'care': + min_sv = svd(r, compute_uv=False)[-1] + if min_sv == 0. or min_sv < np.spacing(1.)*norm(r, 1): + raise ValueError('Matrix r is numerically singular.') + + # Check if the generalized case is required with omitted arguments + # perform late shape checking etc. + generalized_case = e is not None or s is not None + + if generalized_case: + if e is not None: + e = np.atleast_2d(_asarray_validated(e, check_finite=True)) + if not np.equal(*e.shape): + raise ValueError("Matrix e should be square.") + if m != e.shape[0]: + raise ValueError("Matrix a and e should have the same shape.") + # numpy.linalg.cond doesn't check for exact zeros and + # emits a runtime warning. Hence the following manual check. + min_sv = svd(e, compute_uv=False)[-1] + if min_sv == 0. or min_sv < np.spacing(1.) * norm(e, 1): + raise ValueError('Matrix e is numerically singular.') + if np.iscomplexobj(e): + r_or_c = complex + if s is not None: + s = np.atleast_2d(_asarray_validated(s, check_finite=True)) + if s.shape != b.shape: + raise ValueError("Matrix b and s should have the same shape.") + if np.iscomplexobj(s): + r_or_c = complex + + return a, b, q, r, e, s, m, n, r_or_c, generalized_case diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_special_matrices.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_special_matrices.py new file mode 100644 index 0000000000000000000000000000000000000000..7dca572f2d5a12a27ccb468425f03c75e86d5da7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_special_matrices.py @@ -0,0 +1,1332 @@ +import math +import warnings + +import numpy as np +from numpy.lib.stride_tricks import as_strided + + +__all__ = ['toeplitz', 'circulant', 'hankel', + 'hadamard', 'leslie', 'kron', 'block_diag', 'companion', + 'helmert', 'hilbert', 'invhilbert', 'pascal', 'invpascal', 'dft', + 'fiedler', 'fiedler_companion', 'convolution_matrix'] + + +# ----------------------------------------------------------------------------- +# matrix construction functions +# ----------------------------------------------------------------------------- + + +def toeplitz(c, r=None): + r""" + Construct a Toeplitz matrix. + + The Toeplitz matrix has constant diagonals, with c as its first column + and r as its first row. If r is not given, ``r == conjugate(c)`` is + assumed. + + Parameters + ---------- + c : array_like + First column of the matrix. + r : array_like, optional + First row of the matrix. If None, ``r = conjugate(c)`` is assumed; + in this case, if c[0] is real, the result is a Hermitian matrix. + r[0] is ignored; the first row of the returned matrix is + ``[c[0], r[1:]]``. + + .. warning:: + + Beginning in SciPy 1.17, multidimensional input will be treated as a batch, + not ``ravel``\ ed. To preserve the existing behavior, ``ravel`` arguments + before passing them to `toeplitz`. + + Returns + ------- + A : (len(c), len(r)) ndarray + The Toeplitz matrix. Dtype is the same as ``(c[0] + r[0]).dtype``. + + See Also + -------- + circulant : circulant matrix + hankel : Hankel matrix + solve_toeplitz : Solve a Toeplitz system. + + Notes + ----- + The behavior when `c` or `r` is a scalar, or when `c` is complex and + `r` is None, was changed in version 0.8.0. The behavior in previous + versions was undocumented and is no longer supported. + + Examples + -------- + >>> from scipy.linalg import toeplitz + >>> toeplitz([1,2,3], [1,4,5,6]) + array([[1, 4, 5, 6], + [2, 1, 4, 5], + [3, 2, 1, 4]]) + >>> toeplitz([1.0, 2+3j, 4-1j]) + array([[ 1.+0.j, 2.-3.j, 4.+1.j], + [ 2.+3.j, 1.+0.j, 2.-3.j], + [ 4.-1.j, 2.+3.j, 1.+0.j]]) + + """ + c = np.asarray(c) + if r is None: + r = c.conjugate() + else: + r = np.asarray(r) + + if c.ndim > 1 or r.ndim > 1: + msg = ("Beginning in SciPy 1.17, multidimensional input will be treated as a " + "batch, not `ravel`ed. To preserve the existing behavior and silence " + "this warning, `ravel` arguments before passing them to `toeplitz`.") + warnings.warn(msg, FutureWarning, stacklevel=2) + + c, r = c.ravel(), r.ravel() + # Form a 1-D array containing a reversed c followed by r[1:] that could be + # strided to give us toeplitz matrix. + vals = np.concatenate((c[::-1], r[1:])) + out_shp = len(c), len(r) + n = vals.strides[0] + return as_strided(vals[len(c)-1:], shape=out_shp, strides=(-n, n)).copy() + + +def circulant(c): + """ + Construct a circulant matrix. + + Parameters + ---------- + c : (..., N,) array_like + The first column(s) of the matrix. Multidimensional arrays are treated as a + batch: each slice along the last axis is the first column of an output matrix. + + Returns + ------- + A : (..., N, N) ndarray + A circulant matrix whose first column is given by `c`. For batch input, each + slice of shape ``(N, N)`` along the last two dimensions of the output + corresponds with a slice of shape ``(N,)`` along the last dimension of the + input. + + + See Also + -------- + toeplitz : Toeplitz matrix + hankel : Hankel matrix + solve_circulant : Solve a circulant system. + + Notes + ----- + .. versionadded:: 0.8.0 + + Examples + -------- + >>> from scipy.linalg import circulant + >>> circulant([1, 2, 3]) + array([[1, 3, 2], + [2, 1, 3], + [3, 2, 1]]) + + >>> circulant([[1, 2, 3], [4, 5, 6]]) + array([[[1, 3, 2], + [2, 1, 3], + [3, 2, 1]], + [[4, 6, 5], + [5, 4, 6], + [6, 5, 4]]]) + """ + c = np.atleast_1d(c) + batch_shape, N = c.shape[:-1], c.shape[-1] + # Need to use `prod(batch_shape)` instead of `-1` in case array has zero size + c = c.reshape(math.prod(batch_shape), N) if batch_shape else c + # Form an extended array that could be strided to give circulant version + c_ext = np.concatenate((c[..., ::-1], c[..., :0:-1]), axis=-1).ravel() + L = c.shape[-1] + n = c_ext.strides[-1] + if c.ndim == 1: + A = as_strided(c_ext[L-1:], shape=(L, L), strides=(-n, n)) + else: + m = c.shape[0] + A = as_strided(c_ext[L-1:], shape=(m, L, L), strides=(n*(2*L-1), -n, n)) + return A.reshape(batch_shape + (N, N)).copy() + + +def hankel(c, r=None): + """ + Construct a Hankel matrix. + + The Hankel matrix has constant anti-diagonals, with `c` as its + first column and `r` as its last row. If the first element of `r` + differs from the last element of `c`, the first element of `r` is + replaced by the last element of `c` to ensure that anti-diagonals + remain constant. If `r` is not given, then `r = zeros_like(c)` is + assumed. + + Parameters + ---------- + c : array_like + First column of the matrix. Whatever the actual shape of `c`, it + will be converted to a 1-D array. + r : array_like, optional + Last row of the matrix. If None, ``r = zeros_like(c)`` is assumed. + r[0] is ignored; the last row of the returned matrix is + ``[c[-1], r[1:]]``. Whatever the actual shape of `r`, it will be + converted to a 1-D array. + + Returns + ------- + A : (len(c), len(r)) ndarray + The Hankel matrix. Dtype is the same as ``(c[0] + r[0]).dtype``. + + See Also + -------- + toeplitz : Toeplitz matrix + circulant : circulant matrix + + Examples + -------- + >>> from scipy.linalg import hankel + >>> hankel([1, 17, 99]) + array([[ 1, 17, 99], + [17, 99, 0], + [99, 0, 0]]) + >>> hankel([1,2,3,4], [4,7,7,8,9]) + array([[1, 2, 3, 4, 7], + [2, 3, 4, 7, 7], + [3, 4, 7, 7, 8], + [4, 7, 7, 8, 9]]) + + """ + c = np.asarray(c).ravel() + if r is None: + r = np.zeros_like(c) + else: + r = np.asarray(r).ravel() + # Form a 1-D array of values to be used in the matrix, containing `c` + # followed by r[1:]. + vals = np.concatenate((c, r[1:])) + # Stride on concatenated array to get hankel matrix + out_shp = len(c), len(r) + n = vals.strides[0] + return as_strided(vals, shape=out_shp, strides=(n, n)).copy() + + +def hadamard(n, dtype=int): + """ + Construct an Hadamard matrix. + + Constructs an n-by-n Hadamard matrix, using Sylvester's + construction. `n` must be a power of 2. + + Parameters + ---------- + n : int + The order of the matrix. `n` must be a power of 2. + dtype : dtype, optional + The data type of the array to be constructed. + + Returns + ------- + H : (n, n) ndarray + The Hadamard matrix. + + Notes + ----- + .. versionadded:: 0.8.0 + + Examples + -------- + >>> from scipy.linalg import hadamard + >>> hadamard(2, dtype=complex) + array([[ 1.+0.j, 1.+0.j], + [ 1.+0.j, -1.-0.j]]) + >>> hadamard(4) + array([[ 1, 1, 1, 1], + [ 1, -1, 1, -1], + [ 1, 1, -1, -1], + [ 1, -1, -1, 1]]) + + """ + + # This function is a slightly modified version of the + # function contributed by Ivo in ticket #675. + + if n < 1: + lg2 = 0 + else: + lg2 = int(math.log(n, 2)) + if 2 ** lg2 != n: + raise ValueError("n must be an positive integer, and n must be " + "a power of 2") + + H = np.array([[1]], dtype=dtype) + + # Sylvester's construction + for i in range(0, lg2): + H = np.vstack((np.hstack((H, H)), np.hstack((H, -H)))) + + return H + + +def leslie(f, s): + """ + Create a Leslie matrix. + + Given the length n array of fecundity coefficients `f` and the length + n-1 array of survival coefficients `s`, return the associated Leslie + matrix. + + Parameters + ---------- + f : (..., N,) array_like + The "fecundity" coefficients. + s : (..., N-1,) array_like + The "survival" coefficients. The length of each slice of `s` (along the last + axis) must be one less than the length of `f`, and it must be at least 1. + + Returns + ------- + L : (..., N, N) ndarray + The array is zero except for the first row, + which is `f`, and the first sub-diagonal, which is `s`. + For 1-D input, the data-type of the array will be the data-type of + ``f[0]+s[0]``. + + Notes + ----- + .. versionadded:: 0.8.0 + + The Leslie matrix is used to model discrete-time, age-structured + population growth [1]_ [2]_. In a population with `n` age classes, two sets + of parameters define a Leslie matrix: the `n` "fecundity coefficients", + which give the number of offspring per-capita produced by each age + class, and the `n` - 1 "survival coefficients", which give the + per-capita survival rate of each age class. + + N-dimensional input are treated as a batches of coefficient arrays: each + slice along the last axis of the input arrays is a 1-D coefficient array, + and each slice along the last two dimensions of the output is the + corresponding Leslie matrix. + + References + ---------- + .. [1] P. H. Leslie, On the use of matrices in certain population + mathematics, Biometrika, Vol. 33, No. 3, 183--212 (Nov. 1945) + .. [2] P. H. Leslie, Some further notes on the use of matrices in + population mathematics, Biometrika, Vol. 35, No. 3/4, 213--245 + (Dec. 1948) + + Examples + -------- + >>> from scipy.linalg import leslie + >>> leslie([0.1, 2.0, 1.0, 0.1], [0.2, 0.8, 0.7]) + array([[ 0.1, 2. , 1. , 0.1], + [ 0.2, 0. , 0. , 0. ], + [ 0. , 0.8, 0. , 0. ], + [ 0. , 0. , 0.7, 0. ]]) + + """ + f = np.atleast_1d(f) + s = np.atleast_1d(s) + + if f.shape[-1] != s.shape[-1] + 1: + raise ValueError("Incorrect lengths for f and s. The length of s along " + "the last axis must be one less than the length of f.") + if s.shape[-1] == 0: + raise ValueError("The length of s must be at least 1.") + + n = f.shape[-1] + + if f.ndim > 1 or s.ndim > 1: + from scipy.stats._resampling import _vectorize_statistic + _leslie_nd = _vectorize_statistic(leslie) + return np.moveaxis(_leslie_nd(f, s, axis=-1), [0, 1], [-2, -1]) + + tmp = f[0] + s[0] + a = np.zeros((n, n), dtype=tmp.dtype) + a[0] = f + a[list(range(1, n)), list(range(0, n - 1))] = s + return a + + +def kron(a, b): + """ + Kronecker product. + + .. deprecated:: 1.15.0 + `kron` has been deprecated in favour of `numpy.kron` and will be + removed in SciPy 1.17.0. + + The result is the block matrix:: + + a[0,0]*b a[0,1]*b ... a[0,-1]*b + a[1,0]*b a[1,1]*b ... a[1,-1]*b + ... + a[-1,0]*b a[-1,1]*b ... a[-1,-1]*b + + Parameters + ---------- + a : (M, N) ndarray + Input array + b : (P, Q) ndarray + Input array + + Returns + ------- + A : (M*P, N*Q) ndarray + Kronecker product of `a` and `b`. + + Examples + -------- + >>> from numpy import array + >>> from scipy.linalg import kron + >>> kron(array([[1,2],[3,4]]), array([[1,1,1]])) + array([[1, 1, 1, 2, 2, 2], + [3, 3, 3, 4, 4, 4]]) + + """ + msg = ("`kron` has been deprecated in favour of `numpy.kron` in SciPy" + " 1.15.0 and will be removed in SciPy 1.17.0.") + warnings.warn(msg, DeprecationWarning, stacklevel=2) + # accommodate empty arrays + if a.size == 0 or b.size == 0: + m = a.shape[0] * b.shape[0] + n = a.shape[1] * b.shape[1] + return np.empty_like(a, shape=(m, n)) + + if not a.flags['CONTIGUOUS']: + a = np.reshape(a, a.shape) + if not b.flags['CONTIGUOUS']: + b = np.reshape(b, b.shape) + o = np.outer(a, b) + o = o.reshape(a.shape + b.shape) + return np.concatenate(np.concatenate(o, axis=1), axis=1) + + +def block_diag(*arrs): + """ + Create a block diagonal matrix from provided arrays. + + Given the inputs `A`, `B` and `C`, the output will have these + arrays arranged on the diagonal:: + + [[A, 0, 0], + [0, B, 0], + [0, 0, C]] + + Parameters + ---------- + A, B, C, ... : array_like, up to 2-D + Input arrays. A 1-D array or array_like sequence of length `n` is + treated as a 2-D array with shape ``(1,n)``. + + Returns + ------- + D : ndarray + Array with `A`, `B`, `C`, ... on the diagonal. `D` has the + same dtype as `A`. + + Notes + ----- + If all the input arrays are square, the output is known as a + block diagonal matrix. + + Empty sequences (i.e., array-likes of zero size) will not be ignored. + Noteworthy, both [] and [[]] are treated as matrices with shape ``(1,0)``. + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import block_diag + >>> A = [[1, 0], + ... [0, 1]] + >>> B = [[3, 4, 5], + ... [6, 7, 8]] + >>> C = [[7]] + >>> P = np.zeros((2, 0), dtype='int32') + >>> block_diag(A, B, C) + array([[1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0], + [0, 0, 3, 4, 5, 0], + [0, 0, 6, 7, 8, 0], + [0, 0, 0, 0, 0, 7]]) + >>> block_diag(A, P, B, C) + array([[1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 3, 4, 5, 0], + [0, 0, 6, 7, 8, 0], + [0, 0, 0, 0, 0, 7]]) + >>> block_diag(1.0, [2, 3], [[4, 5], [6, 7]]) + array([[ 1., 0., 0., 0., 0.], + [ 0., 2., 3., 0., 0.], + [ 0., 0., 0., 4., 5.], + [ 0., 0., 0., 6., 7.]]) + + """ + if arrs == (): + arrs = ([],) + arrs = [np.atleast_2d(a) for a in arrs] + + bad_args = [k for k in range(len(arrs)) if arrs[k].ndim > 2] + if bad_args: + raise ValueError("arguments in the following positions " + f"have dimension greater than 2: {bad_args}") + + shapes = np.array([a.shape for a in arrs]) + out_dtype = np.result_type(*[arr.dtype for arr in arrs]) + out = np.zeros(np.sum(shapes, axis=0), dtype=out_dtype) + + r, c = 0, 0 + for i, (rr, cc) in enumerate(shapes): + out[r:r + rr, c:c + cc] = arrs[i] + r += rr + c += cc + return out + + +def companion(a): + """ + Create a companion matrix. + + Create the companion matrix [1]_ associated with the polynomial whose + coefficients are given in `a`. + + Parameters + ---------- + a : (..., N) array_like + 1-D array of polynomial coefficients. The length of `a` must be + at least two, and ``a[0]`` must not be zero. + M-dimensional arrays are treated as a batch: each slice along the last + axis is a 1-D array of polynomial coefficients. + + Returns + ------- + c : (..., N-1, N-1) ndarray + For 1-D input, the first row of `c` is ``-a[1:]/a[0]``, and the first + sub-diagonal is all ones. The data-type of the array is the same + as the data-type of ``1.0*a[0]``. + For batch input, each slice of shape ``(N-1, N-1)`` along the last two + dimensions of the output corresponds with a slice of shape ``(N,)`` + along the last dimension of the input. + + Raises + ------ + ValueError + If any of the following are true: a) ``a.shape[-1] < 2``; b) ``a[..., 0] == 0``. + + Notes + ----- + .. versionadded:: 0.8.0 + + References + ---------- + .. [1] R. A. Horn & C. R. Johnson, *Matrix Analysis*. Cambridge, UK: + Cambridge University Press, 1999, pp. 146-7. + + Examples + -------- + >>> from scipy.linalg import companion + >>> companion([1, -10, 31, -30]) + array([[ 10., -31., 30.], + [ 1., 0., 0.], + [ 0., 1., 0.]]) + + """ + a = np.atleast_1d(a) + n = a.shape[-1] + + if n < 2: + raise ValueError("The length of `a` along the last axis must be at least 2.") + + if np.any(a[..., 0] == 0): + raise ValueError("The first coefficient(s) of `a` (i.e. elements " + "of `a[..., 0]`) must not be zero.") + + first_row = -a[..., 1:] / (1.0 * a[..., 0:1]) + c = np.zeros(a.shape[:-1] + (n - 1, n - 1), dtype=first_row.dtype) + c[..., 0, :] = first_row + c[..., np.arange(1, n - 1), np.arange(0, n - 2)] = 1 + return c + + +def helmert(n, full=False): + """ + Create an Helmert matrix of order `n`. + + This has applications in statistics, compositional or simplicial analysis, + and in Aitchison geometry. + + Parameters + ---------- + n : int + The size of the array to create. + full : bool, optional + If True the (n, n) ndarray will be returned. + Otherwise the submatrix that does not include the first + row will be returned. + Default: False. + + Returns + ------- + M : ndarray + The Helmert matrix. + The shape is (n, n) or (n-1, n) depending on the `full` argument. + + Examples + -------- + >>> from scipy.linalg import helmert + >>> helmert(5, full=True) + array([[ 0.4472136 , 0.4472136 , 0.4472136 , 0.4472136 , 0.4472136 ], + [ 0.70710678, -0.70710678, 0. , 0. , 0. ], + [ 0.40824829, 0.40824829, -0.81649658, 0. , 0. ], + [ 0.28867513, 0.28867513, 0.28867513, -0.8660254 , 0. ], + [ 0.2236068 , 0.2236068 , 0.2236068 , 0.2236068 , -0.89442719]]) + + """ + H = np.tril(np.ones((n, n)), -1) - np.diag(np.arange(n)) + d = np.arange(n) * np.arange(1, n+1) + H[0] = 1 + d[0] = n + H_full = H / np.sqrt(d)[:, np.newaxis] + if full: + return H_full + else: + return H_full[1:] + + +def hilbert(n): + """ + Create a Hilbert matrix of order `n`. + + Returns the `n` by `n` array with entries `h[i,j] = 1 / (i + j + 1)`. + + Parameters + ---------- + n : int + The size of the array to create. + + Returns + ------- + h : (n, n) ndarray + The Hilbert matrix. + + See Also + -------- + invhilbert : Compute the inverse of a Hilbert matrix. + + Notes + ----- + .. versionadded:: 0.10.0 + + Examples + -------- + >>> from scipy.linalg import hilbert + >>> hilbert(3) + array([[ 1. , 0.5 , 0.33333333], + [ 0.5 , 0.33333333, 0.25 ], + [ 0.33333333, 0.25 , 0.2 ]]) + + """ + values = 1.0 / (1.0 + np.arange(2 * n - 1)) + h = hankel(values[:n], r=values[n - 1:]) + return h + + +def invhilbert(n, exact=False): + """ + Compute the inverse of the Hilbert matrix of order `n`. + + The entries in the inverse of a Hilbert matrix are integers. When `n` + is greater than 14, some entries in the inverse exceed the upper limit + of 64 bit integers. The `exact` argument provides two options for + dealing with these large integers. + + Parameters + ---------- + n : int + The order of the Hilbert matrix. + exact : bool, optional + If False, the data type of the array that is returned is np.float64, + and the array is an approximation of the inverse. + If True, the array is the exact integer inverse array. To represent + the exact inverse when n > 14, the returned array is an object array + of long integers. For n <= 14, the exact inverse is returned as an + array with data type np.int64. + + Returns + ------- + invh : (n, n) ndarray + The data type of the array is np.float64 if `exact` is False. + If `exact` is True, the data type is either np.int64 (for n <= 14) + or object (for n > 14). In the latter case, the objects in the + array will be long integers. + + See Also + -------- + hilbert : Create a Hilbert matrix. + + Notes + ----- + .. versionadded:: 0.10.0 + + Examples + -------- + >>> from scipy.linalg import invhilbert + >>> invhilbert(4) + array([[ 16., -120., 240., -140.], + [ -120., 1200., -2700., 1680.], + [ 240., -2700., 6480., -4200.], + [ -140., 1680., -4200., 2800.]]) + >>> invhilbert(4, exact=True) + array([[ 16, -120, 240, -140], + [ -120, 1200, -2700, 1680], + [ 240, -2700, 6480, -4200], + [ -140, 1680, -4200, 2800]], dtype=int64) + >>> invhilbert(16)[7,7] + 4.2475099528537506e+19 + >>> invhilbert(16, exact=True)[7,7] + 42475099528537378560 + + """ + from scipy.special import comb + if exact: + if n > 14: + dtype = object + else: + dtype = np.int64 + else: + dtype = np.float64 + invh = np.empty((n, n), dtype=dtype) + for i in range(n): + for j in range(0, i + 1): + s = i + j + invh[i, j] = ((-1) ** s * (s + 1) * + comb(n + i, n - j - 1, exact=exact) * + comb(n + j, n - i - 1, exact=exact) * + comb(s, i, exact=exact) ** 2) + if i != j: + invh[j, i] = invh[i, j] + return invh + + +def pascal(n, kind='symmetric', exact=True): + """ + Returns the n x n Pascal matrix. + + The Pascal matrix is a matrix containing the binomial coefficients as + its elements. + + Parameters + ---------- + n : int + The size of the matrix to create; that is, the result is an n x n + matrix. + kind : str, optional + Must be one of 'symmetric', 'lower', or 'upper'. + Default is 'symmetric'. + exact : bool, optional + If `exact` is True, the result is either an array of type + numpy.uint64 (if n < 35) or an object array of Python long integers. + If `exact` is False, the coefficients in the matrix are computed using + `scipy.special.comb` with ``exact=False``. The result will be a floating + point array, and the values in the array will not be the exact + coefficients, but this version is much faster than ``exact=True``. + + Returns + ------- + p : (n, n) ndarray + The Pascal matrix. + + See Also + -------- + invpascal + + Notes + ----- + See https://en.wikipedia.org/wiki/Pascal_matrix for more information + about Pascal matrices. + + .. versionadded:: 0.11.0 + + Examples + -------- + >>> from scipy.linalg import pascal + >>> pascal(4) + array([[ 1, 1, 1, 1], + [ 1, 2, 3, 4], + [ 1, 3, 6, 10], + [ 1, 4, 10, 20]], dtype=uint64) + >>> pascal(4, kind='lower') + array([[1, 0, 0, 0], + [1, 1, 0, 0], + [1, 2, 1, 0], + [1, 3, 3, 1]], dtype=uint64) + >>> pascal(50)[-1, -1] + 25477612258980856902730428600 + >>> from scipy.special import comb + >>> comb(98, 49, exact=True) + 25477612258980856902730428600 + + """ + + from scipy.special import comb + if kind not in ['symmetric', 'lower', 'upper']: + raise ValueError("kind must be 'symmetric', 'lower', or 'upper'") + + if exact: + if n >= 35: + L_n = np.empty((n, n), dtype=object) + L_n.fill(0) + else: + L_n = np.zeros((n, n), dtype=np.uint64) + for i in range(n): + for j in range(i + 1): + L_n[i, j] = comb(i, j, exact=True) + else: + L_n = comb(*np.ogrid[:n, :n]) + + if kind == 'lower': + p = L_n + elif kind == 'upper': + p = L_n.T + else: + p = np.dot(L_n, L_n.T) + + return p + + +def invpascal(n, kind='symmetric', exact=True): + """ + Returns the inverse of the n x n Pascal matrix. + + The Pascal matrix is a matrix containing the binomial coefficients as + its elements. + + Parameters + ---------- + n : int + The size of the matrix to create; that is, the result is an n x n + matrix. + kind : str, optional + Must be one of 'symmetric', 'lower', or 'upper'. + Default is 'symmetric'. + exact : bool, optional + If `exact` is True, the result is either an array of type + ``numpy.int64`` (if `n` <= 35) or an object array of Python integers. + If `exact` is False, the coefficients in the matrix are computed using + `scipy.special.comb` with `exact=False`. The result will be a floating + point array, and for large `n`, the values in the array will not be the + exact coefficients. + + Returns + ------- + invp : (n, n) ndarray + The inverse of the Pascal matrix. + + See Also + -------- + pascal + + Notes + ----- + + .. versionadded:: 0.16.0 + + References + ---------- + .. [1] "Pascal matrix", https://en.wikipedia.org/wiki/Pascal_matrix + .. [2] Cohen, A. M., "The inverse of a Pascal matrix", Mathematical + Gazette, 59(408), pp. 111-112, 1975. + + Examples + -------- + >>> from scipy.linalg import invpascal, pascal + >>> invp = invpascal(5) + >>> invp + array([[ 5, -10, 10, -5, 1], + [-10, 30, -35, 19, -4], + [ 10, -35, 46, -27, 6], + [ -5, 19, -27, 17, -4], + [ 1, -4, 6, -4, 1]]) + + >>> p = pascal(5) + >>> p.dot(invp) + array([[ 1., 0., 0., 0., 0.], + [ 0., 1., 0., 0., 0.], + [ 0., 0., 1., 0., 0.], + [ 0., 0., 0., 1., 0.], + [ 0., 0., 0., 0., 1.]]) + + An example of the use of `kind` and `exact`: + + >>> invpascal(5, kind='lower', exact=False) + array([[ 1., -0., 0., -0., 0.], + [-1., 1., -0., 0., -0.], + [ 1., -2., 1., -0., 0.], + [-1., 3., -3., 1., -0.], + [ 1., -4., 6., -4., 1.]]) + + """ + from scipy.special import comb + + if kind not in ['symmetric', 'lower', 'upper']: + raise ValueError("'kind' must be 'symmetric', 'lower' or 'upper'.") + + if kind == 'symmetric': + if exact: + if n > 34: + dt = object + else: + dt = np.int64 + else: + dt = np.float64 + invp = np.empty((n, n), dtype=dt) + for i in range(n): + for j in range(0, i + 1): + v = 0 + for k in range(n - i): + v += comb(i + k, k, exact=exact) * comb(i + k, i + k - j, + exact=exact) + invp[i, j] = (-1)**(i - j) * v + if i != j: + invp[j, i] = invp[i, j] + else: + # For the 'lower' and 'upper' cases, we computer the inverse by + # changing the sign of every other diagonal of the pascal matrix. + invp = pascal(n, kind=kind, exact=exact) + if invp.dtype == np.uint64: + # This cast from np.uint64 to int64 OK, because if `kind` is not + # "symmetric", the values in invp are all much less than 2**63. + invp = invp.view(np.int64) + + # The toeplitz matrix has alternating bands of 1 and -1. + invp *= toeplitz((-1)**np.arange(n)).astype(invp.dtype) + + return invp + + +def dft(n, scale=None): + """ + Discrete Fourier transform matrix. + + Create the matrix that computes the discrete Fourier transform of a + sequence [1]_. The nth primitive root of unity used to generate the + matrix is exp(-2*pi*i/n), where i = sqrt(-1). + + Parameters + ---------- + n : int + Size the matrix to create. + scale : str, optional + Must be None, 'sqrtn', or 'n'. + If `scale` is 'sqrtn', the matrix is divided by `sqrt(n)`. + If `scale` is 'n', the matrix is divided by `n`. + If `scale` is None (the default), the matrix is not normalized, and the + return value is simply the Vandermonde matrix of the roots of unity. + + Returns + ------- + m : (n, n) ndarray + The DFT matrix. + + Notes + ----- + When `scale` is None, multiplying a vector by the matrix returned by + `dft` is mathematically equivalent to (but much less efficient than) + the calculation performed by `scipy.fft.fft`. + + .. versionadded:: 0.14.0 + + References + ---------- + .. [1] "DFT matrix", https://en.wikipedia.org/wiki/DFT_matrix + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import dft + >>> np.set_printoptions(precision=2, suppress=True) # for compact output + >>> m = dft(5) + >>> m + array([[ 1. +0.j , 1. +0.j , 1. +0.j , 1. +0.j , 1. +0.j ], + [ 1. +0.j , 0.31-0.95j, -0.81-0.59j, -0.81+0.59j, 0.31+0.95j], + [ 1. +0.j , -0.81-0.59j, 0.31+0.95j, 0.31-0.95j, -0.81+0.59j], + [ 1. +0.j , -0.81+0.59j, 0.31-0.95j, 0.31+0.95j, -0.81-0.59j], + [ 1. +0.j , 0.31+0.95j, -0.81+0.59j, -0.81-0.59j, 0.31-0.95j]]) + >>> x = np.array([1, 2, 3, 0, 3]) + >>> m @ x # Compute the DFT of x + array([ 9. +0.j , 0.12-0.81j, -2.12+3.44j, -2.12-3.44j, 0.12+0.81j]) + + Verify that ``m @ x`` is the same as ``fft(x)``. + + >>> from scipy.fft import fft + >>> fft(x) # Same result as m @ x + array([ 9. +0.j , 0.12-0.81j, -2.12+3.44j, -2.12-3.44j, 0.12+0.81j]) + """ + if scale not in [None, 'sqrtn', 'n']: + raise ValueError("scale must be None, 'sqrtn', or 'n'; " + f"{scale!r} is not valid.") + + omegas = np.exp(-2j * np.pi * np.arange(n) / n).reshape(-1, 1) + m = omegas ** np.arange(n) + if scale == 'sqrtn': + m /= math.sqrt(n) + elif scale == 'n': + m /= n + return m + + +def fiedler(a): + """Returns a symmetric Fiedler matrix + + Given an sequence of numbers `a`, Fiedler matrices have the structure + ``F[i, j] = np.abs(a[i] - a[j])``, and hence zero diagonals and nonnegative + entries. A Fiedler matrix has a dominant positive eigenvalue and other + eigenvalues are negative. Although not valid generally, for certain inputs, + the inverse and the determinant can be derived explicitly as given in [1]_. + + Parameters + ---------- + a : (..., n,) array_like + Coefficient array. N-dimensional arrays are treated as a batch: + each slice along the last axis is a 1-D coefficient array. + + Returns + ------- + F : (..., n, n) ndarray + Fiedler matrix. For batch input, each slice of shape ``(n, n)`` + along the last two dimensions of the output corresponds with a + slice of shape ``(n,)`` along the last dimension of the input. + + See Also + -------- + circulant, toeplitz + + Notes + ----- + + .. versionadded:: 1.3.0 + + References + ---------- + .. [1] J. Todd, "Basic Numerical Mathematics: Vol.2 : Numerical Algebra", + 1977, Birkhauser, :doi:`10.1007/978-3-0348-7286-7` + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import det, inv, fiedler + >>> a = [1, 4, 12, 45, 77] + >>> n = len(a) + >>> A = fiedler(a) + >>> A + array([[ 0, 3, 11, 44, 76], + [ 3, 0, 8, 41, 73], + [11, 8, 0, 33, 65], + [44, 41, 33, 0, 32], + [76, 73, 65, 32, 0]]) + + The explicit formulas for determinant and inverse seem to hold only for + monotonically increasing/decreasing arrays. Note the tridiagonal structure + and the corners. + + >>> Ai = inv(A) + >>> Ai[np.abs(Ai) < 1e-12] = 0. # cleanup the numerical noise for display + >>> Ai + array([[-0.16008772, 0.16666667, 0. , 0. , 0.00657895], + [ 0.16666667, -0.22916667, 0.0625 , 0. , 0. ], + [ 0. , 0.0625 , -0.07765152, 0.01515152, 0. ], + [ 0. , 0. , 0.01515152, -0.03077652, 0.015625 ], + [ 0.00657895, 0. , 0. , 0.015625 , -0.00904605]]) + >>> det(A) + 15409151.999999998 + >>> (-1)**(n-1) * 2**(n-2) * np.diff(a).prod() * (a[-1] - a[0]) + 15409152 + + """ + a = np.atleast_1d(a) + + if a.ndim > 1: + return np.apply_along_axis(fiedler, -1, a) + + if a.size == 0: + return np.array([], dtype=float) + elif a.size == 1: + return np.array([[0.]]) + else: + return np.abs(a[:, None] - a) + + +def fiedler_companion(a): + """ Returns a Fiedler companion matrix + + Given a polynomial coefficient array ``a``, this function forms a + pentadiagonal matrix with a special structure whose eigenvalues coincides + with the roots of ``a``. + + Parameters + ---------- + a : (..., N) array_like + 1-D array of polynomial coefficients in descending order with a nonzero + leading coefficient. For ``N < 2``, an empty array is returned. + N-dimensional arrays are treated as a batch: each slice along the last + axis is a 1-D array of polynomial coefficients. + + Returns + ------- + c : (..., N-1, N-1) ndarray + Resulting companion matrix. For batch input, each slice of shape + ``(N-1, N-1)`` along the last two dimensions of the output corresponds + with a slice of shape ``(N,)`` along the last dimension of the input. + + See Also + -------- + companion + + Notes + ----- + Similar to `companion`, each leading coefficient along the last axis of the + input should be nonzero. + If the leading coefficient is not 1, other coefficients are rescaled before + the array generation. To avoid numerical issues, it is best to provide a + monic polynomial. + + .. versionadded:: 1.3.0 + + References + ---------- + .. [1] M. Fiedler, " A note on companion matrices", Linear Algebra and its + Applications, 2003, :doi:`10.1016/S0024-3795(03)00548-2` + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import fiedler_companion, eigvals + >>> p = np.poly(np.arange(1, 9, 2)) # [1., -16., 86., -176., 105.] + >>> fc = fiedler_companion(p) + >>> fc + array([[ 16., -86., 1., 0.], + [ 1., 0., 0., 0.], + [ 0., 176., 0., -105.], + [ 0., 1., 0., 0.]]) + >>> eigvals(fc) + array([7.+0.j, 5.+0.j, 3.+0.j, 1.+0.j]) + + """ + a = np.atleast_1d(a) + + if a.ndim > 1: + return np.apply_along_axis(fiedler_companion, -1, a) + + if a.size <= 2: + if a.size == 2: + return np.array([[-(a/a[0])[-1]]]) + return np.array([], dtype=a.dtype) + + if a[0] == 0.: + raise ValueError('Leading coefficient is zero.') + + a = a/a[0] + n = a.size - 1 + c = np.zeros((n, n), dtype=a.dtype) + # subdiagonals + c[range(3, n, 2), range(1, n-2, 2)] = 1. + c[range(2, n, 2), range(1, n-1, 2)] = -a[3::2] + # superdiagonals + c[range(0, n-2, 2), range(2, n, 2)] = 1. + c[range(0, n-1, 2), range(1, n, 2)] = -a[2::2] + c[[0, 1], 0] = [-a[1], 1] + + return c + + +def convolution_matrix(a, n, mode='full'): + """ + Construct a convolution matrix. + + Constructs the Toeplitz matrix representing one-dimensional + convolution [1]_. See the notes below for details. + + Parameters + ---------- + a : (..., m) array_like + The 1-D array to convolve. N-dimensional arrays are treated as a + batch: each slice along the last axis is a 1-D array to convolve. + n : int + The number of columns in the resulting matrix. It gives the length + of the input to be convolved with `a`. This is analogous to the + length of `v` in ``numpy.convolve(a, v)``. + mode : str + This is analogous to `mode` in ``numpy.convolve(v, a, mode)``. + It must be one of ('full', 'valid', 'same'). + See below for how `mode` determines the shape of the result. + + Returns + ------- + A : (..., k, n) ndarray + The convolution matrix whose row count `k` depends on `mode`:: + + ======= ========================= + mode k + ======= ========================= + 'full' m + n -1 + 'same' max(m, n) + 'valid' max(m, n) - min(m, n) + 1 + ======= ========================= + + For batch input, each slice of shape ``(k, n)`` along the last two + dimensions of the output corresponds with a slice of shape ``(m,)`` + along the last dimension of the input. + + See Also + -------- + toeplitz : Toeplitz matrix + + Notes + ----- + The code:: + + A = convolution_matrix(a, n, mode) + + creates a Toeplitz matrix `A` such that ``A @ v`` is equivalent to + using ``convolve(a, v, mode)``. The returned array always has `n` + columns. The number of rows depends on the specified `mode`, as + explained above. + + In the default 'full' mode, the entries of `A` are given by:: + + A[i, j] == (a[i-j] if (0 <= (i-j) < m) else 0) + + where ``m = len(a)``. Suppose, for example, the input array is + ``[x, y, z]``. The convolution matrix has the form:: + + [x, 0, 0, ..., 0, 0] + [y, x, 0, ..., 0, 0] + [z, y, x, ..., 0, 0] + ... + [0, 0, 0, ..., x, 0] + [0, 0, 0, ..., y, x] + [0, 0, 0, ..., z, y] + [0, 0, 0, ..., 0, z] + + In 'valid' mode, the entries of `A` are given by:: + + A[i, j] == (a[i-j+m-1] if (0 <= (i-j+m-1) < m) else 0) + + This corresponds to a matrix whose rows are the subset of those from + the 'full' case where all the coefficients in `a` are contained in the + row. For input ``[x, y, z]``, this array looks like:: + + [z, y, x, 0, 0, ..., 0, 0, 0] + [0, z, y, x, 0, ..., 0, 0, 0] + [0, 0, z, y, x, ..., 0, 0, 0] + ... + [0, 0, 0, 0, 0, ..., x, 0, 0] + [0, 0, 0, 0, 0, ..., y, x, 0] + [0, 0, 0, 0, 0, ..., z, y, x] + + In the 'same' mode, the entries of `A` are given by:: + + d = (m - 1) // 2 + A[i, j] == (a[i-j+d] if (0 <= (i-j+d) < m) else 0) + + The typical application of the 'same' mode is when one has a signal of + length `n` (with `n` greater than ``len(a)``), and the desired output + is a filtered signal that is still of length `n`. + + For input ``[x, y, z]``, this array looks like:: + + [y, x, 0, 0, ..., 0, 0, 0] + [z, y, x, 0, ..., 0, 0, 0] + [0, z, y, x, ..., 0, 0, 0] + [0, 0, z, y, ..., 0, 0, 0] + ... + [0, 0, 0, 0, ..., y, x, 0] + [0, 0, 0, 0, ..., z, y, x] + [0, 0, 0, 0, ..., 0, z, y] + + .. versionadded:: 1.5.0 + + References + ---------- + .. [1] "Convolution", https://en.wikipedia.org/wiki/Convolution + + Examples + -------- + >>> import numpy as np + >>> from scipy.linalg import convolution_matrix + >>> A = convolution_matrix([-1, 4, -2], 5, mode='same') + >>> A + array([[ 4, -1, 0, 0, 0], + [-2, 4, -1, 0, 0], + [ 0, -2, 4, -1, 0], + [ 0, 0, -2, 4, -1], + [ 0, 0, 0, -2, 4]]) + + Compare multiplication by `A` with the use of `numpy.convolve`. + + >>> x = np.array([1, 2, 0, -3, 0.5]) + >>> A @ x + array([ 2. , 6. , -1. , -12.5, 8. ]) + + Verify that ``A @ x`` produced the same result as applying the + convolution function. + + >>> np.convolve([-1, 4, -2], x, mode='same') + array([ 2. , 6. , -1. , -12.5, 8. ]) + + For comparison to the case ``mode='same'`` shown above, here are the + matrices produced by ``mode='full'`` and ``mode='valid'`` for the + same coefficients and size. + + >>> convolution_matrix([-1, 4, -2], 5, mode='full') + array([[-1, 0, 0, 0, 0], + [ 4, -1, 0, 0, 0], + [-2, 4, -1, 0, 0], + [ 0, -2, 4, -1, 0], + [ 0, 0, -2, 4, -1], + [ 0, 0, 0, -2, 4], + [ 0, 0, 0, 0, -2]]) + + >>> convolution_matrix([-1, 4, -2], 5, mode='valid') + array([[-2, 4, -1, 0, 0], + [ 0, -2, 4, -1, 0], + [ 0, 0, -2, 4, -1]]) + """ + if n <= 0: + raise ValueError('n must be a positive integer.') + + a = np.asarray(a) + + if a.size == 0: + raise ValueError('len(a) must be at least 1.') + + if mode not in ('full', 'valid', 'same'): + raise ValueError( + "'mode' argument must be one of ('full', 'valid', 'same')") + + if a.ndim > 1: + return np.apply_along_axis(lambda a: convolution_matrix(a, n, mode), -1, a) + + # create zero padded versions of the array + az = np.pad(a, (0, n-1), 'constant') + raz = np.pad(a[::-1], (0, n-1), 'constant') + + if mode == 'same': + trim = min(n, len(a)) - 1 + tb = trim//2 + te = trim - tb + col0 = az[tb:len(az)-te] + row0 = raz[-n-tb:len(raz)-tb] + elif mode == 'valid': + tb = min(n, len(a)) - 1 + te = tb + col0 = az[tb:len(az)-te] + row0 = raz[-n-tb:len(raz)-tb] + else: # 'full' + col0 = az + row0 = raz[-n:] + return toeplitz(col0, row0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_testutils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_testutils.py new file mode 100644 index 0000000000000000000000000000000000000000..f6d01d2b6e59b040f39c0b53cc2788bbd3d0888f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/_testutils.py @@ -0,0 +1,65 @@ +import numpy as np + + +class _FakeMatrix: + def __init__(self, data): + self._data = data + self.__array_interface__ = data.__array_interface__ + + +class _FakeMatrix2: + def __init__(self, data): + self._data = data + + def __array__(self, dtype=None, copy=None): + if copy: + return self._data.copy() + return self._data + + +def _get_array(shape, dtype): + """ + Get a test array of given shape and data type. + Returned NxN matrices are posdef, and 2xN are banded-posdef. + + """ + if len(shape) == 2 and shape[0] == 2: + # yield a banded positive definite one + x = np.zeros(shape, dtype=dtype) + x[0, 1:] = -1 + x[1] = 2 + return x + elif len(shape) == 2 and shape[0] == shape[1]: + # always yield a positive definite matrix + x = np.zeros(shape, dtype=dtype) + j = np.arange(shape[0]) + x[j, j] = 2 + x[j[:-1], j[:-1]+1] = -1 + x[j[:-1]+1, j[:-1]] = -1 + return x + else: + np.random.seed(1234) + return np.random.randn(*shape).astype(dtype) + + +def _id(x): + return x + + +def assert_no_overwrite(call, shapes, dtypes=None): + """ + Test that a call does not overwrite its input arguments + """ + + if dtypes is None: + dtypes = [np.float32, np.float64, np.complex64, np.complex128] + + for dtype in dtypes: + for order in ["C", "F"]: + for faker in [_id, _FakeMatrix, _FakeMatrix2]: + orig_inputs = [_get_array(s, dtype) for s in shapes] + inputs = [faker(x.copy(order)) for x in orig_inputs] + call(*inputs) + msg = f"call modified inputs [{dtype!r}, {faker!r}]" + for a, b in zip(inputs, orig_inputs): + np.testing.assert_equal(a, b, err_msg=msg) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/basic.py new file mode 100644 index 0000000000000000000000000000000000000000..04ef3645a2ed6a22106ed8ca1acf9e9ac93df5cf --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/basic.py @@ -0,0 +1,23 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'solve', 'solve_triangular', 'solveh_banded', 'solve_banded', + 'solve_toeplitz', 'solve_circulant', 'inv', 'det', 'lstsq', + 'pinv', 'pinvh', 'matrix_balance', 'matmul_toeplitz', + 'get_lapack_funcs', 'LinAlgError', 'LinAlgWarning', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="linalg", module="basic", + private_modules=["_basic"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/blas.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/blas.py new file mode 100644 index 0000000000000000000000000000000000000000..c943460e6bafcd9a382586d5a5155357382ea596 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/blas.py @@ -0,0 +1,484 @@ +""" +Low-level BLAS functions (:mod:`scipy.linalg.blas`) +=================================================== + +This module contains low-level functions from the BLAS library. + +.. versionadded:: 0.12.0 + +.. note:: + + The common ``overwrite_<>`` option in many routines, allows the + input arrays to be overwritten to avoid extra memory allocation. + However this requires the array to satisfy two conditions + which are memory order and the data type to match exactly the + order and the type expected by the routine. + + As an example, if you pass a double precision float array to any + ``S....`` routine which expects single precision arguments, f2py + will create an intermediate array to match the argument types and + overwriting will be performed on that intermediate array. + + Similarly, if a C-contiguous array is passed, f2py will pass a + FORTRAN-contiguous array internally. Please make sure that these + details are satisfied. More information can be found in the f2py + documentation. + +.. warning:: + + These functions do little to no error checking. + It is possible to cause crashes by mis-using them, + so prefer using the higher-level routines in `scipy.linalg`. + +Finding functions +----------------- + +.. autosummary:: + :toctree: generated/ + + get_blas_funcs + find_best_blas_type + +BLAS Level 1 functions +---------------------- + +.. autosummary:: + :toctree: generated/ + + caxpy + ccopy + cdotc + cdotu + crotg + cscal + csrot + csscal + cswap + dasum + daxpy + dcopy + ddot + dnrm2 + drot + drotg + drotm + drotmg + dscal + dswap + dzasum + dznrm2 + icamax + idamax + isamax + izamax + sasum + saxpy + scasum + scnrm2 + scopy + sdot + snrm2 + srot + srotg + srotm + srotmg + sscal + sswap + zaxpy + zcopy + zdotc + zdotu + zdrot + zdscal + zrotg + zscal + zswap + +BLAS Level 2 functions +---------------------- + +.. autosummary:: + :toctree: generated/ + + sgbmv + sgemv + sger + ssbmv + sspr + sspr2 + ssymv + ssyr + ssyr2 + stbmv + stpsv + strmv + strsv + dgbmv + dgemv + dger + dsbmv + dspr + dspr2 + dsymv + dsyr + dsyr2 + dtbmv + dtpsv + dtrmv + dtrsv + cgbmv + cgemv + cgerc + cgeru + chbmv + chemv + cher + cher2 + chpmv + chpr + chpr2 + ctbmv + ctbsv + ctpmv + ctpsv + ctrmv + ctrsv + csyr + zgbmv + zgemv + zgerc + zgeru + zhbmv + zhemv + zher + zher2 + zhpmv + zhpr + zhpr2 + ztbmv + ztbsv + ztpmv + ztrmv + ztrsv + zsyr + +BLAS Level 3 functions +---------------------- + +.. autosummary:: + :toctree: generated/ + + sgemm + ssymm + ssyr2k + ssyrk + strmm + strsm + dgemm + dsymm + dsyr2k + dsyrk + dtrmm + dtrsm + cgemm + chemm + cher2k + cherk + csymm + csyr2k + csyrk + ctrmm + ctrsm + zgemm + zhemm + zher2k + zherk + zsymm + zsyr2k + zsyrk + ztrmm + ztrsm + +""" +# +# Author: Pearu Peterson, March 2002 +# refactoring by Fabian Pedregosa, March 2010 +# + +__all__ = ['get_blas_funcs', 'find_best_blas_type'] + +import numpy as np +import functools + +from scipy.linalg import _fblas +try: + from scipy.linalg import _cblas +except ImportError: + _cblas = None + +try: + from scipy.linalg import _fblas_64 + HAS_ILP64 = True +except ImportError: + HAS_ILP64 = False + _fblas_64 = None + +# Expose all functions (only fblas --- cblas is an implementation detail) +empty_module = None +from scipy.linalg._fblas import * # noqa: E402, F403 +del empty_module + +# all numeric dtypes '?bBhHiIlLqQefdgFDGO' that are safe to be converted to + +# single precision float : '?bBhH!!!!!!ef!!!!!!' +# double precision float : '?bBhHiIlLqQefdg!!!!' +# single precision complex : '?bBhH!!!!!!ef!!F!!!' +# double precision complex : '?bBhHiIlLqQefdgFDG!' + +_type_score = {x: 1 for x in '?bBhHef'} +_type_score.update({x: 2 for x in 'iIlLqQd'}) + +# Handle float128(g) and complex256(G) separately in case non-Windows systems. +# On Windows, the values will be rewritten to the same key with the same value. +_type_score.update({'F': 3, 'D': 4, 'g': 2, 'G': 4}) + +# Final mapping to the actual prefixes and dtypes +_type_conv = {1: ('s', np.dtype('float32')), + 2: ('d', np.dtype('float64')), + 3: ('c', np.dtype('complex64')), + 4: ('z', np.dtype('complex128'))} + +# some convenience alias for complex functions +_blas_alias = {'cnrm2': 'scnrm2', 'znrm2': 'dznrm2', + 'cdot': 'cdotc', 'zdot': 'zdotc', + 'cger': 'cgerc', 'zger': 'zgerc', + 'sdotc': 'sdot', 'sdotu': 'sdot', + 'ddotc': 'ddot', 'ddotu': 'ddot'} + + +def find_best_blas_type(arrays=(), dtype=None): + """Find best-matching BLAS/LAPACK type. + + Arrays are used to determine the optimal prefix of BLAS routines. + + Parameters + ---------- + arrays : sequence of ndarrays, optional + Arrays can be given to determine optimal prefix of BLAS + routines. If not given, double-precision routines will be + used, otherwise the most generic type in arrays will be used. + dtype : str or dtype, optional + Data-type specifier. Not used if `arrays` is non-empty. + + Returns + ------- + prefix : str + BLAS/LAPACK prefix character. + dtype : dtype + Inferred Numpy data type. + prefer_fortran : bool + Whether to prefer Fortran order routines over C order. + + Examples + -------- + >>> import numpy as np + >>> import scipy.linalg.blas as bla + >>> rng = np.random.default_rng() + >>> a = rng.random((10,15)) + >>> b = np.asfortranarray(a) # Change the memory layout order + >>> bla.find_best_blas_type((a,)) + ('d', dtype('float64'), False) + >>> bla.find_best_blas_type((a*1j,)) + ('z', dtype('complex128'), False) + >>> bla.find_best_blas_type((b,)) + ('d', dtype('float64'), True) + + """ + dtype = np.dtype(dtype) + max_score = _type_score.get(dtype.char, 5) + prefer_fortran = False + + if arrays: + # In most cases, single element is passed through, quicker route + if len(arrays) == 1: + max_score = _type_score.get(arrays[0].dtype.char, 5) + prefer_fortran = arrays[0].flags['FORTRAN'] + else: + # use the most generic type in arrays + scores = [_type_score.get(x.dtype.char, 5) for x in arrays] + max_score = max(scores) + ind_max_score = scores.index(max_score) + # safe upcasting for mix of float64 and complex64 --> prefix 'z' + if max_score == 3 and (2 in scores): + max_score = 4 + + if arrays[ind_max_score].flags['FORTRAN']: + # prefer Fortran for leading array with column major order + prefer_fortran = True + + # Get the LAPACK prefix and the corresponding dtype if not fall back + # to 'd' and double precision float. + prefix, dtype = _type_conv.get(max_score, ('d', np.dtype('float64'))) + + return prefix, dtype, prefer_fortran + + +def _get_funcs(names, arrays, dtype, + lib_name, fmodule, cmodule, + fmodule_name, cmodule_name, alias, + ilp64=False): + """ + Return available BLAS/LAPACK functions. + + Used also in lapack.py. See get_blas_funcs for docstring. + """ + + funcs = [] + unpack = False + dtype = np.dtype(dtype) + module1 = (cmodule, cmodule_name) + module2 = (fmodule, fmodule_name) + + if isinstance(names, str): + names = (names,) + unpack = True + + prefix, dtype, prefer_fortran = find_best_blas_type(arrays, dtype) + + if prefer_fortran: + module1, module2 = module2, module1 + + for name in names: + func_name = prefix + name + func_name = alias.get(func_name, func_name) + func = getattr(module1[0], func_name, None) + module_name = module1[1] + if func is None: + func = getattr(module2[0], func_name, None) + module_name = module2[1] + if func is None: + raise ValueError( + f'{lib_name} function {func_name} could not be found') + func.module_name, func.typecode = module_name, prefix + func.dtype = dtype + if not ilp64: + func.int_dtype = np.dtype(np.intc) + else: + func.int_dtype = np.dtype(np.int64) + func.prefix = prefix # Backward compatibility + funcs.append(func) + + if unpack: + return funcs[0] + else: + return funcs + + +def _memoize_get_funcs(func): + """ + Memoized fast path for _get_funcs instances + """ + memo = {} + func.memo = memo + + @functools.wraps(func) + def getter(names, arrays=(), dtype=None, ilp64=False): + key = (names, dtype, ilp64) + for array in arrays: + # cf. find_blas_funcs + key += (array.dtype.char, array.flags.fortran) + + try: + value = memo.get(key) + except TypeError: + # unhashable key etc. + key = None + value = None + + if value is not None: + return value + + value = func(names, arrays, dtype, ilp64) + + if key is not None: + memo[key] = value + + return value + + return getter + + +@_memoize_get_funcs +def get_blas_funcs(names, arrays=(), dtype=None, ilp64=False): + """Return available BLAS function objects from names. + + Arrays are used to determine the optimal prefix of BLAS routines. + + Parameters + ---------- + names : str or sequence of str + Name(s) of BLAS functions without type prefix. + + arrays : sequence of ndarrays, optional + Arrays can be given to determine optimal prefix of BLAS + routines. If not given, double-precision routines will be + used, otherwise the most generic type in arrays will be used. + + dtype : str or dtype, optional + Data-type specifier. Not used if `arrays` is non-empty. + + ilp64 : {True, False, 'preferred'}, optional + Whether to return ILP64 routine variant. + Choosing 'preferred' returns ILP64 routine if available, + and otherwise the 32-bit routine. Default: False + + Returns + ------- + funcs : list + List containing the found function(s). + + + Notes + ----- + This routine automatically chooses between Fortran/C + interfaces. Fortran code is used whenever possible for arrays with + column major order. In all other cases, C code is preferred. + + In BLAS, the naming convention is that all functions start with a + type prefix, which depends on the type of the principal + matrix. These can be one of {'s', 'd', 'c', 'z'} for the NumPy + types {float32, float64, complex64, complex128} respectively. + The code and the dtype are stored in attributes `typecode` and `dtype` + of the returned functions. + + Examples + -------- + >>> import numpy as np + >>> import scipy.linalg as LA + >>> rng = np.random.default_rng() + >>> a = rng.random((3,2)) + >>> x_gemv = LA.get_blas_funcs('gemv', (a,)) + >>> x_gemv.typecode + 'd' + >>> x_gemv = LA.get_blas_funcs('gemv',(a*1j,)) + >>> x_gemv.typecode + 'z' + + """ + if isinstance(ilp64, str): + if ilp64 == 'preferred': + ilp64 = HAS_ILP64 + else: + raise ValueError("Invalid value for 'ilp64'") + + if not ilp64: + return _get_funcs(names, arrays, dtype, + "BLAS", _fblas, _cblas, "fblas", "cblas", + _blas_alias, ilp64=False) + else: + if not HAS_ILP64: + raise RuntimeError("BLAS ILP64 routine requested, but Scipy " + "compiled only with 32-bit BLAS") + return _get_funcs(names, arrays, dtype, + "BLAS", _fblas_64, None, "fblas_64", None, + _blas_alias, ilp64=True) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_blas.pxd b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_blas.pxd new file mode 100644 index 0000000000000000000000000000000000000000..7ed44f6ea8611f926e3ea5fd2670446cdf9b398c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_blas.pxd @@ -0,0 +1,169 @@ +""" +This file was generated by _generate_pyx.py. +Do not edit this file directly. +""" + +# Within scipy, these wrappers can be used via relative or absolute cimport. +# Examples: +# from ..linalg cimport cython_blas +# from scipy.linalg cimport cython_blas +# cimport scipy.linalg.cython_blas as cython_blas +# cimport ..linalg.cython_blas as cython_blas + +# Within SciPy, if BLAS functions are needed in C/C++/Fortran, +# these wrappers should not be used. +# The original libraries should be linked directly. + +ctypedef float s +ctypedef double d +ctypedef float complex c +ctypedef double complex z + +cdef void caxpy(int *n, c *ca, c *cx, int *incx, c *cy, int *incy) noexcept nogil +cdef void ccopy(int *n, c *cx, int *incx, c *cy, int *incy) noexcept nogil +cdef c cdotc(int *n, c *cx, int *incx, c *cy, int *incy) noexcept nogil +cdef c cdotu(int *n, c *cx, int *incx, c *cy, int *incy) noexcept nogil +cdef void cgbmv(char *trans, int *m, int *n, int *kl, int *ku, c *alpha, c *a, int *lda, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil +cdef void cgemm(char *transa, char *transb, int *m, int *n, int *k, c *alpha, c *a, int *lda, c *b, int *ldb, c *beta, c *c, int *ldc) noexcept nogil +cdef void cgemv(char *trans, int *m, int *n, c *alpha, c *a, int *lda, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil +cdef void cgerc(int *m, int *n, c *alpha, c *x, int *incx, c *y, int *incy, c *a, int *lda) noexcept nogil +cdef void cgeru(int *m, int *n, c *alpha, c *x, int *incx, c *y, int *incy, c *a, int *lda) noexcept nogil +cdef void chbmv(char *uplo, int *n, int *k, c *alpha, c *a, int *lda, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil +cdef void chemm(char *side, char *uplo, int *m, int *n, c *alpha, c *a, int *lda, c *b, int *ldb, c *beta, c *c, int *ldc) noexcept nogil +cdef void chemv(char *uplo, int *n, c *alpha, c *a, int *lda, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil +cdef void cher(char *uplo, int *n, s *alpha, c *x, int *incx, c *a, int *lda) noexcept nogil +cdef void cher2(char *uplo, int *n, c *alpha, c *x, int *incx, c *y, int *incy, c *a, int *lda) noexcept nogil +cdef void cher2k(char *uplo, char *trans, int *n, int *k, c *alpha, c *a, int *lda, c *b, int *ldb, s *beta, c *c, int *ldc) noexcept nogil +cdef void cherk(char *uplo, char *trans, int *n, int *k, s *alpha, c *a, int *lda, s *beta, c *c, int *ldc) noexcept nogil +cdef void chpmv(char *uplo, int *n, c *alpha, c *ap, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil +cdef void chpr(char *uplo, int *n, s *alpha, c *x, int *incx, c *ap) noexcept nogil +cdef void chpr2(char *uplo, int *n, c *alpha, c *x, int *incx, c *y, int *incy, c *ap) noexcept nogil +cdef void crotg(c *ca, c *cb, s *c, c *s) noexcept nogil +cdef void cscal(int *n, c *ca, c *cx, int *incx) noexcept nogil +cdef void csrot(int *n, c *cx, int *incx, c *cy, int *incy, s *c, s *s) noexcept nogil +cdef void csscal(int *n, s *sa, c *cx, int *incx) noexcept nogil +cdef void cswap(int *n, c *cx, int *incx, c *cy, int *incy) noexcept nogil +cdef void csymm(char *side, char *uplo, int *m, int *n, c *alpha, c *a, int *lda, c *b, int *ldb, c *beta, c *c, int *ldc) noexcept nogil +cdef void csyr2k(char *uplo, char *trans, int *n, int *k, c *alpha, c *a, int *lda, c *b, int *ldb, c *beta, c *c, int *ldc) noexcept nogil +cdef void csyrk(char *uplo, char *trans, int *n, int *k, c *alpha, c *a, int *lda, c *beta, c *c, int *ldc) noexcept nogil +cdef void ctbmv(char *uplo, char *trans, char *diag, int *n, int *k, c *a, int *lda, c *x, int *incx) noexcept nogil +cdef void ctbsv(char *uplo, char *trans, char *diag, int *n, int *k, c *a, int *lda, c *x, int *incx) noexcept nogil +cdef void ctpmv(char *uplo, char *trans, char *diag, int *n, c *ap, c *x, int *incx) noexcept nogil +cdef void ctpsv(char *uplo, char *trans, char *diag, int *n, c *ap, c *x, int *incx) noexcept nogil +cdef void ctrmm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, c *alpha, c *a, int *lda, c *b, int *ldb) noexcept nogil +cdef void ctrmv(char *uplo, char *trans, char *diag, int *n, c *a, int *lda, c *x, int *incx) noexcept nogil +cdef void ctrsm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, c *alpha, c *a, int *lda, c *b, int *ldb) noexcept nogil +cdef void ctrsv(char *uplo, char *trans, char *diag, int *n, c *a, int *lda, c *x, int *incx) noexcept nogil +cdef d dasum(int *n, d *dx, int *incx) noexcept nogil +cdef void daxpy(int *n, d *da, d *dx, int *incx, d *dy, int *incy) noexcept nogil +cdef d dcabs1(z *z) noexcept nogil +cdef void dcopy(int *n, d *dx, int *incx, d *dy, int *incy) noexcept nogil +cdef d ddot(int *n, d *dx, int *incx, d *dy, int *incy) noexcept nogil +cdef void dgbmv(char *trans, int *m, int *n, int *kl, int *ku, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) noexcept nogil +cdef void dgemm(char *transa, char *transb, int *m, int *n, int *k, d *alpha, d *a, int *lda, d *b, int *ldb, d *beta, d *c, int *ldc) noexcept nogil +cdef void dgemv(char *trans, int *m, int *n, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) noexcept nogil +cdef void dger(int *m, int *n, d *alpha, d *x, int *incx, d *y, int *incy, d *a, int *lda) noexcept nogil +cdef d dnrm2(int *n, d *x, int *incx) noexcept nogil +cdef void drot(int *n, d *dx, int *incx, d *dy, int *incy, d *c, d *s) noexcept nogil +cdef void drotg(d *da, d *db, d *c, d *s) noexcept nogil +cdef void drotm(int *n, d *dx, int *incx, d *dy, int *incy, d *dparam) noexcept nogil +cdef void drotmg(d *dd1, d *dd2, d *dx1, d *dy1, d *dparam) noexcept nogil +cdef void dsbmv(char *uplo, int *n, int *k, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) noexcept nogil +cdef void dscal(int *n, d *da, d *dx, int *incx) noexcept nogil +cdef d dsdot(int *n, s *sx, int *incx, s *sy, int *incy) noexcept nogil +cdef void dspmv(char *uplo, int *n, d *alpha, d *ap, d *x, int *incx, d *beta, d *y, int *incy) noexcept nogil +cdef void dspr(char *uplo, int *n, d *alpha, d *x, int *incx, d *ap) noexcept nogil +cdef void dspr2(char *uplo, int *n, d *alpha, d *x, int *incx, d *y, int *incy, d *ap) noexcept nogil +cdef void dswap(int *n, d *dx, int *incx, d *dy, int *incy) noexcept nogil +cdef void dsymm(char *side, char *uplo, int *m, int *n, d *alpha, d *a, int *lda, d *b, int *ldb, d *beta, d *c, int *ldc) noexcept nogil +cdef void dsymv(char *uplo, int *n, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) noexcept nogil +cdef void dsyr(char *uplo, int *n, d *alpha, d *x, int *incx, d *a, int *lda) noexcept nogil +cdef void dsyr2(char *uplo, int *n, d *alpha, d *x, int *incx, d *y, int *incy, d *a, int *lda) noexcept nogil +cdef void dsyr2k(char *uplo, char *trans, int *n, int *k, d *alpha, d *a, int *lda, d *b, int *ldb, d *beta, d *c, int *ldc) noexcept nogil +cdef void dsyrk(char *uplo, char *trans, int *n, int *k, d *alpha, d *a, int *lda, d *beta, d *c, int *ldc) noexcept nogil +cdef void dtbmv(char *uplo, char *trans, char *diag, int *n, int *k, d *a, int *lda, d *x, int *incx) noexcept nogil +cdef void dtbsv(char *uplo, char *trans, char *diag, int *n, int *k, d *a, int *lda, d *x, int *incx) noexcept nogil +cdef void dtpmv(char *uplo, char *trans, char *diag, int *n, d *ap, d *x, int *incx) noexcept nogil +cdef void dtpsv(char *uplo, char *trans, char *diag, int *n, d *ap, d *x, int *incx) noexcept nogil +cdef void dtrmm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, d *alpha, d *a, int *lda, d *b, int *ldb) noexcept nogil +cdef void dtrmv(char *uplo, char *trans, char *diag, int *n, d *a, int *lda, d *x, int *incx) noexcept nogil +cdef void dtrsm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, d *alpha, d *a, int *lda, d *b, int *ldb) noexcept nogil +cdef void dtrsv(char *uplo, char *trans, char *diag, int *n, d *a, int *lda, d *x, int *incx) noexcept nogil +cdef d dzasum(int *n, z *zx, int *incx) noexcept nogil +cdef d dznrm2(int *n, z *x, int *incx) noexcept nogil +cdef int icamax(int *n, c *cx, int *incx) noexcept nogil +cdef int idamax(int *n, d *dx, int *incx) noexcept nogil +cdef int isamax(int *n, s *sx, int *incx) noexcept nogil +cdef int izamax(int *n, z *zx, int *incx) noexcept nogil +cdef bint lsame(char *ca, char *cb) noexcept nogil +cdef s sasum(int *n, s *sx, int *incx) noexcept nogil +cdef void saxpy(int *n, s *sa, s *sx, int *incx, s *sy, int *incy) noexcept nogil +cdef s scasum(int *n, c *cx, int *incx) noexcept nogil +cdef s scnrm2(int *n, c *x, int *incx) noexcept nogil +cdef void scopy(int *n, s *sx, int *incx, s *sy, int *incy) noexcept nogil +cdef s sdot(int *n, s *sx, int *incx, s *sy, int *incy) noexcept nogil +cdef s sdsdot(int *n, s *sb, s *sx, int *incx, s *sy, int *incy) noexcept nogil +cdef void sgbmv(char *trans, int *m, int *n, int *kl, int *ku, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) noexcept nogil +cdef void sgemm(char *transa, char *transb, int *m, int *n, int *k, s *alpha, s *a, int *lda, s *b, int *ldb, s *beta, s *c, int *ldc) noexcept nogil +cdef void sgemv(char *trans, int *m, int *n, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) noexcept nogil +cdef void sger(int *m, int *n, s *alpha, s *x, int *incx, s *y, int *incy, s *a, int *lda) noexcept nogil +cdef s snrm2(int *n, s *x, int *incx) noexcept nogil +cdef void srot(int *n, s *sx, int *incx, s *sy, int *incy, s *c, s *s) noexcept nogil +cdef void srotg(s *sa, s *sb, s *c, s *s) noexcept nogil +cdef void srotm(int *n, s *sx, int *incx, s *sy, int *incy, s *sparam) noexcept nogil +cdef void srotmg(s *sd1, s *sd2, s *sx1, s *sy1, s *sparam) noexcept nogil +cdef void ssbmv(char *uplo, int *n, int *k, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) noexcept nogil +cdef void sscal(int *n, s *sa, s *sx, int *incx) noexcept nogil +cdef void sspmv(char *uplo, int *n, s *alpha, s *ap, s *x, int *incx, s *beta, s *y, int *incy) noexcept nogil +cdef void sspr(char *uplo, int *n, s *alpha, s *x, int *incx, s *ap) noexcept nogil +cdef void sspr2(char *uplo, int *n, s *alpha, s *x, int *incx, s *y, int *incy, s *ap) noexcept nogil +cdef void sswap(int *n, s *sx, int *incx, s *sy, int *incy) noexcept nogil +cdef void ssymm(char *side, char *uplo, int *m, int *n, s *alpha, s *a, int *lda, s *b, int *ldb, s *beta, s *c, int *ldc) noexcept nogil +cdef void ssymv(char *uplo, int *n, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) noexcept nogil +cdef void ssyr(char *uplo, int *n, s *alpha, s *x, int *incx, s *a, int *lda) noexcept nogil +cdef void ssyr2(char *uplo, int *n, s *alpha, s *x, int *incx, s *y, int *incy, s *a, int *lda) noexcept nogil +cdef void ssyr2k(char *uplo, char *trans, int *n, int *k, s *alpha, s *a, int *lda, s *b, int *ldb, s *beta, s *c, int *ldc) noexcept nogil +cdef void ssyrk(char *uplo, char *trans, int *n, int *k, s *alpha, s *a, int *lda, s *beta, s *c, int *ldc) noexcept nogil +cdef void stbmv(char *uplo, char *trans, char *diag, int *n, int *k, s *a, int *lda, s *x, int *incx) noexcept nogil +cdef void stbsv(char *uplo, char *trans, char *diag, int *n, int *k, s *a, int *lda, s *x, int *incx) noexcept nogil +cdef void stpmv(char *uplo, char *trans, char *diag, int *n, s *ap, s *x, int *incx) noexcept nogil +cdef void stpsv(char *uplo, char *trans, char *diag, int *n, s *ap, s *x, int *incx) noexcept nogil +cdef void strmm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, s *alpha, s *a, int *lda, s *b, int *ldb) noexcept nogil +cdef void strmv(char *uplo, char *trans, char *diag, int *n, s *a, int *lda, s *x, int *incx) noexcept nogil +cdef void strsm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, s *alpha, s *a, int *lda, s *b, int *ldb) noexcept nogil +cdef void strsv(char *uplo, char *trans, char *diag, int *n, s *a, int *lda, s *x, int *incx) noexcept nogil +cdef void zaxpy(int *n, z *za, z *zx, int *incx, z *zy, int *incy) noexcept nogil +cdef void zcopy(int *n, z *zx, int *incx, z *zy, int *incy) noexcept nogil +cdef z zdotc(int *n, z *zx, int *incx, z *zy, int *incy) noexcept nogil +cdef z zdotu(int *n, z *zx, int *incx, z *zy, int *incy) noexcept nogil +cdef void zdrot(int *n, z *cx, int *incx, z *cy, int *incy, d *c, d *s) noexcept nogil +cdef void zdscal(int *n, d *da, z *zx, int *incx) noexcept nogil +cdef void zgbmv(char *trans, int *m, int *n, int *kl, int *ku, z *alpha, z *a, int *lda, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil +cdef void zgemm(char *transa, char *transb, int *m, int *n, int *k, z *alpha, z *a, int *lda, z *b, int *ldb, z *beta, z *c, int *ldc) noexcept nogil +cdef void zgemv(char *trans, int *m, int *n, z *alpha, z *a, int *lda, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil +cdef void zgerc(int *m, int *n, z *alpha, z *x, int *incx, z *y, int *incy, z *a, int *lda) noexcept nogil +cdef void zgeru(int *m, int *n, z *alpha, z *x, int *incx, z *y, int *incy, z *a, int *lda) noexcept nogil +cdef void zhbmv(char *uplo, int *n, int *k, z *alpha, z *a, int *lda, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil +cdef void zhemm(char *side, char *uplo, int *m, int *n, z *alpha, z *a, int *lda, z *b, int *ldb, z *beta, z *c, int *ldc) noexcept nogil +cdef void zhemv(char *uplo, int *n, z *alpha, z *a, int *lda, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil +cdef void zher(char *uplo, int *n, d *alpha, z *x, int *incx, z *a, int *lda) noexcept nogil +cdef void zher2(char *uplo, int *n, z *alpha, z *x, int *incx, z *y, int *incy, z *a, int *lda) noexcept nogil +cdef void zher2k(char *uplo, char *trans, int *n, int *k, z *alpha, z *a, int *lda, z *b, int *ldb, d *beta, z *c, int *ldc) noexcept nogil +cdef void zherk(char *uplo, char *trans, int *n, int *k, d *alpha, z *a, int *lda, d *beta, z *c, int *ldc) noexcept nogil +cdef void zhpmv(char *uplo, int *n, z *alpha, z *ap, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil +cdef void zhpr(char *uplo, int *n, d *alpha, z *x, int *incx, z *ap) noexcept nogil +cdef void zhpr2(char *uplo, int *n, z *alpha, z *x, int *incx, z *y, int *incy, z *ap) noexcept nogil +cdef void zrotg(z *ca, z *cb, d *c, z *s) noexcept nogil +cdef void zscal(int *n, z *za, z *zx, int *incx) noexcept nogil +cdef void zswap(int *n, z *zx, int *incx, z *zy, int *incy) noexcept nogil +cdef void zsymm(char *side, char *uplo, int *m, int *n, z *alpha, z *a, int *lda, z *b, int *ldb, z *beta, z *c, int *ldc) noexcept nogil +cdef void zsyr2k(char *uplo, char *trans, int *n, int *k, z *alpha, z *a, int *lda, z *b, int *ldb, z *beta, z *c, int *ldc) noexcept nogil +cdef void zsyrk(char *uplo, char *trans, int *n, int *k, z *alpha, z *a, int *lda, z *beta, z *c, int *ldc) noexcept nogil +cdef void ztbmv(char *uplo, char *trans, char *diag, int *n, int *k, z *a, int *lda, z *x, int *incx) noexcept nogil +cdef void ztbsv(char *uplo, char *trans, char *diag, int *n, int *k, z *a, int *lda, z *x, int *incx) noexcept nogil +cdef void ztpmv(char *uplo, char *trans, char *diag, int *n, z *ap, z *x, int *incx) noexcept nogil +cdef void ztpsv(char *uplo, char *trans, char *diag, int *n, z *ap, z *x, int *incx) noexcept nogil +cdef void ztrmm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, z *alpha, z *a, int *lda, z *b, int *ldb) noexcept nogil +cdef void ztrmv(char *uplo, char *trans, char *diag, int *n, z *a, int *lda, z *x, int *incx) noexcept nogil +cdef void ztrsm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, z *alpha, z *a, int *lda, z *b, int *ldb) noexcept nogil +cdef void ztrsv(char *uplo, char *trans, char *diag, int *n, z *a, int *lda, z *x, int *incx) noexcept nogil diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_blas.pyx b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_blas.pyx new file mode 100644 index 0000000000000000000000000000000000000000..35286fe11d72226269c0e459d9a3109151f74a4a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_blas.pyx @@ -0,0 +1,1432 @@ +# This file was generated by _generate_pyx.py. +# Do not edit this file directly. +# cython: boundscheck = False +# cython: wraparound = False +# cython: cdivision = True + +""" +BLAS Functions for Cython +========================= + +Usable from Cython via:: + + cimport scipy.linalg.cython_blas + +These wrappers do not check for alignment of arrays. +Alignment should be checked before these wrappers are used. + +If using ``cdotu``, ``cdotc``, ``zdotu``, ``zdotc``, ``sladiv``, or ``dladiv``, +the ``CYTHON_CCOMPLEX`` define must be set to 0 during compilation. For +example, in a ``meson.build`` file when using Meson:: + + py.extension_module('ext_module' + 'ext_module.pyx', + c_args: ['-DCYTHON_CCOMPLEX=0'], + ... + ) + +Raw function pointers (Fortran-style pointer arguments): + +- caxpy +- ccopy +- cdotc +- cdotu +- cgbmv +- cgemm +- cgemv +- cgerc +- cgeru +- chbmv +- chemm +- chemv +- cher +- cher2 +- cher2k +- cherk +- chpmv +- chpr +- chpr2 +- crotg +- cscal +- csrot +- csscal +- cswap +- csymm +- csyr2k +- csyrk +- ctbmv +- ctbsv +- ctpmv +- ctpsv +- ctrmm +- ctrmv +- ctrsm +- ctrsv +- dasum +- daxpy +- dcabs1 +- dcopy +- ddot +- dgbmv +- dgemm +- dgemv +- dger +- dnrm2 +- drot +- drotg +- drotm +- drotmg +- dsbmv +- dscal +- dsdot +- dspmv +- dspr +- dspr2 +- dswap +- dsymm +- dsymv +- dsyr +- dsyr2 +- dsyr2k +- dsyrk +- dtbmv +- dtbsv +- dtpmv +- dtpsv +- dtrmm +- dtrmv +- dtrsm +- dtrsv +- dzasum +- dznrm2 +- icamax +- idamax +- isamax +- izamax +- lsame +- sasum +- saxpy +- scasum +- scnrm2 +- scopy +- sdot +- sdsdot +- sgbmv +- sgemm +- sgemv +- sger +- snrm2 +- srot +- srotg +- srotm +- srotmg +- ssbmv +- sscal +- sspmv +- sspr +- sspr2 +- sswap +- ssymm +- ssymv +- ssyr +- ssyr2 +- ssyr2k +- ssyrk +- stbmv +- stbsv +- stpmv +- stpsv +- strmm +- strmv +- strsm +- strsv +- zaxpy +- zcopy +- zdotc +- zdotu +- zdrot +- zdscal +- zgbmv +- zgemm +- zgemv +- zgerc +- zgeru +- zhbmv +- zhemm +- zhemv +- zher +- zher2 +- zher2k +- zherk +- zhpmv +- zhpr +- zhpr2 +- zrotg +- zscal +- zswap +- zsymm +- zsyr2k +- zsyrk +- ztbmv +- ztbsv +- ztpmv +- ztpsv +- ztrmm +- ztrmv +- ztrsm +- ztrsv + + +""" + +# Within SciPy, these wrappers can be used via relative or absolute cimport. +# Examples: +# from ..linalg cimport cython_blas +# from scipy.linalg cimport cython_blas +# cimport scipy.linalg.cython_blas as cython_blas +# cimport ..linalg.cython_blas as cython_blas + +# Within SciPy, if BLAS functions are needed in C/C++/Fortran, +# these wrappers should not be used. +# The original libraries should be linked directly. + +cdef extern from "fortran_defs.h": + pass + +from numpy cimport npy_complex64, npy_complex128 + + +cdef extern from "_blas_subroutines.h": + void _fortran_caxpy "BLAS_FUNC(caxpy)"(int *n, npy_complex64 *ca, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy) nogil +cdef void caxpy(int *n, c *ca, c *cx, int *incx, c *cy, int *incy) noexcept nogil: + + _fortran_caxpy(n, ca, cx, incx, cy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ccopy "BLAS_FUNC(ccopy)"(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy) nogil +cdef void ccopy(int *n, c *cx, int *incx, c *cy, int *incy) noexcept nogil: + + _fortran_ccopy(n, cx, incx, cy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cdotc "F_FUNC(cdotcwrp,CDOTCWRP)"(npy_complex64 *out, int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy) nogil +cdef c cdotc(int *n, c *cx, int *incx, c *cy, int *incy) noexcept nogil: + cdef c out + _fortran_cdotc(&out, n, cx, incx, cy, incy) + return out + +cdef extern from "_blas_subroutines.h": + void _fortran_cdotu "F_FUNC(cdotuwrp,CDOTUWRP)"(npy_complex64 *out, int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy) nogil +cdef c cdotu(int *n, c *cx, int *incx, c *cy, int *incy) noexcept nogil: + cdef c out + _fortran_cdotu(&out, n, cx, incx, cy, incy) + return out + +cdef extern from "_blas_subroutines.h": + void _fortran_cgbmv "BLAS_FUNC(cgbmv)"(char *trans, int *m, int *n, int *kl, int *ku, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy) nogil +cdef void cgbmv(char *trans, int *m, int *n, int *kl, int *ku, c *alpha, c *a, int *lda, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil: + + _fortran_cgbmv(trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cgemm "BLAS_FUNC(cgemm)"(char *transa, char *transb, int *m, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc) nogil +cdef void cgemm(char *transa, char *transb, int *m, int *n, int *k, c *alpha, c *a, int *lda, c *b, int *ldb, c *beta, c *c, int *ldc) noexcept nogil: + + _fortran_cgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cgemv "BLAS_FUNC(cgemv)"(char *trans, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy) nogil +cdef void cgemv(char *trans, int *m, int *n, c *alpha, c *a, int *lda, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil: + + _fortran_cgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cgerc "BLAS_FUNC(cgerc)"(int *m, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *a, int *lda) nogil +cdef void cgerc(int *m, int *n, c *alpha, c *x, int *incx, c *y, int *incy, c *a, int *lda) noexcept nogil: + + _fortran_cgerc(m, n, alpha, x, incx, y, incy, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cgeru "BLAS_FUNC(cgeru)"(int *m, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *a, int *lda) nogil +cdef void cgeru(int *m, int *n, c *alpha, c *x, int *incx, c *y, int *incy, c *a, int *lda) noexcept nogil: + + _fortran_cgeru(m, n, alpha, x, incx, y, incy, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_chbmv "BLAS_FUNC(chbmv)"(char *uplo, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy) nogil +cdef void chbmv(char *uplo, int *n, int *k, c *alpha, c *a, int *lda, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil: + + _fortran_chbmv(uplo, n, k, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_chemm "BLAS_FUNC(chemm)"(char *side, char *uplo, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc) nogil +cdef void chemm(char *side, char *uplo, int *m, int *n, c *alpha, c *a, int *lda, c *b, int *ldb, c *beta, c *c, int *ldc) noexcept nogil: + + _fortran_chemm(side, uplo, m, n, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_chemv "BLAS_FUNC(chemv)"(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy) nogil +cdef void chemv(char *uplo, int *n, c *alpha, c *a, int *lda, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil: + + _fortran_chemv(uplo, n, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cher "BLAS_FUNC(cher)"(char *uplo, int *n, s *alpha, npy_complex64 *x, int *incx, npy_complex64 *a, int *lda) nogil +cdef void cher(char *uplo, int *n, s *alpha, c *x, int *incx, c *a, int *lda) noexcept nogil: + + _fortran_cher(uplo, n, alpha, x, incx, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cher2 "BLAS_FUNC(cher2)"(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *a, int *lda) nogil +cdef void cher2(char *uplo, int *n, c *alpha, c *x, int *incx, c *y, int *incy, c *a, int *lda) noexcept nogil: + + _fortran_cher2(uplo, n, alpha, x, incx, y, incy, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cher2k "BLAS_FUNC(cher2k)"(char *uplo, char *trans, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, s *beta, npy_complex64 *c, int *ldc) nogil +cdef void cher2k(char *uplo, char *trans, int *n, int *k, c *alpha, c *a, int *lda, c *b, int *ldb, s *beta, c *c, int *ldc) noexcept nogil: + + _fortran_cher2k(uplo, trans, n, k, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cherk "BLAS_FUNC(cherk)"(char *uplo, char *trans, int *n, int *k, s *alpha, npy_complex64 *a, int *lda, s *beta, npy_complex64 *c, int *ldc) nogil +cdef void cherk(char *uplo, char *trans, int *n, int *k, s *alpha, c *a, int *lda, s *beta, c *c, int *ldc) noexcept nogil: + + _fortran_cherk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_chpmv "BLAS_FUNC(chpmv)"(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *ap, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy) nogil +cdef void chpmv(char *uplo, int *n, c *alpha, c *ap, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil: + + _fortran_chpmv(uplo, n, alpha, ap, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_chpr "BLAS_FUNC(chpr)"(char *uplo, int *n, s *alpha, npy_complex64 *x, int *incx, npy_complex64 *ap) nogil +cdef void chpr(char *uplo, int *n, s *alpha, c *x, int *incx, c *ap) noexcept nogil: + + _fortran_chpr(uplo, n, alpha, x, incx, ap) + + +cdef extern from "_blas_subroutines.h": + void _fortran_chpr2 "BLAS_FUNC(chpr2)"(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *ap) nogil +cdef void chpr2(char *uplo, int *n, c *alpha, c *x, int *incx, c *y, int *incy, c *ap) noexcept nogil: + + _fortran_chpr2(uplo, n, alpha, x, incx, y, incy, ap) + + +cdef extern from "_blas_subroutines.h": + void _fortran_crotg "BLAS_FUNC(crotg)"(npy_complex64 *ca, npy_complex64 *cb, s *c, npy_complex64 *s) nogil +cdef void crotg(c *ca, c *cb, s *c, c *s) noexcept nogil: + + _fortran_crotg(ca, cb, c, s) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cscal "BLAS_FUNC(cscal)"(int *n, npy_complex64 *ca, npy_complex64 *cx, int *incx) nogil +cdef void cscal(int *n, c *ca, c *cx, int *incx) noexcept nogil: + + _fortran_cscal(n, ca, cx, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_csrot "BLAS_FUNC(csrot)"(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy, s *c, s *s) nogil +cdef void csrot(int *n, c *cx, int *incx, c *cy, int *incy, s *c, s *s) noexcept nogil: + + _fortran_csrot(n, cx, incx, cy, incy, c, s) + + +cdef extern from "_blas_subroutines.h": + void _fortran_csscal "BLAS_FUNC(csscal)"(int *n, s *sa, npy_complex64 *cx, int *incx) nogil +cdef void csscal(int *n, s *sa, c *cx, int *incx) noexcept nogil: + + _fortran_csscal(n, sa, cx, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_cswap "BLAS_FUNC(cswap)"(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy) nogil +cdef void cswap(int *n, c *cx, int *incx, c *cy, int *incy) noexcept nogil: + + _fortran_cswap(n, cx, incx, cy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_csymm "BLAS_FUNC(csymm)"(char *side, char *uplo, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc) nogil +cdef void csymm(char *side, char *uplo, int *m, int *n, c *alpha, c *a, int *lda, c *b, int *ldb, c *beta, c *c, int *ldc) noexcept nogil: + + _fortran_csymm(side, uplo, m, n, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_csyr2k "BLAS_FUNC(csyr2k)"(char *uplo, char *trans, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc) nogil +cdef void csyr2k(char *uplo, char *trans, int *n, int *k, c *alpha, c *a, int *lda, c *b, int *ldb, c *beta, c *c, int *ldc) noexcept nogil: + + _fortran_csyr2k(uplo, trans, n, k, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_csyrk "BLAS_FUNC(csyrk)"(char *uplo, char *trans, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *beta, npy_complex64 *c, int *ldc) nogil +cdef void csyrk(char *uplo, char *trans, int *n, int *k, c *alpha, c *a, int *lda, c *beta, c *c, int *ldc) noexcept nogil: + + _fortran_csyrk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ctbmv "BLAS_FUNC(ctbmv)"(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx) nogil +cdef void ctbmv(char *uplo, char *trans, char *diag, int *n, int *k, c *a, int *lda, c *x, int *incx) noexcept nogil: + + _fortran_ctbmv(uplo, trans, diag, n, k, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ctbsv "BLAS_FUNC(ctbsv)"(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx) nogil +cdef void ctbsv(char *uplo, char *trans, char *diag, int *n, int *k, c *a, int *lda, c *x, int *incx) noexcept nogil: + + _fortran_ctbsv(uplo, trans, diag, n, k, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ctpmv "BLAS_FUNC(ctpmv)"(char *uplo, char *trans, char *diag, int *n, npy_complex64 *ap, npy_complex64 *x, int *incx) nogil +cdef void ctpmv(char *uplo, char *trans, char *diag, int *n, c *ap, c *x, int *incx) noexcept nogil: + + _fortran_ctpmv(uplo, trans, diag, n, ap, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ctpsv "BLAS_FUNC(ctpsv)"(char *uplo, char *trans, char *diag, int *n, npy_complex64 *ap, npy_complex64 *x, int *incx) nogil +cdef void ctpsv(char *uplo, char *trans, char *diag, int *n, c *ap, c *x, int *incx) noexcept nogil: + + _fortran_ctpsv(uplo, trans, diag, n, ap, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ctrmm "BLAS_FUNC(ctrmm)"(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb) nogil +cdef void ctrmm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, c *alpha, c *a, int *lda, c *b, int *ldb) noexcept nogil: + + _fortran_ctrmm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ctrmv "BLAS_FUNC(ctrmv)"(char *uplo, char *trans, char *diag, int *n, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx) nogil +cdef void ctrmv(char *uplo, char *trans, char *diag, int *n, c *a, int *lda, c *x, int *incx) noexcept nogil: + + _fortran_ctrmv(uplo, trans, diag, n, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ctrsm "BLAS_FUNC(ctrsm)"(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb) nogil +cdef void ctrsm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, c *alpha, c *a, int *lda, c *b, int *ldb) noexcept nogil: + + _fortran_ctrsm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ctrsv "BLAS_FUNC(ctrsv)"(char *uplo, char *trans, char *diag, int *n, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx) nogil +cdef void ctrsv(char *uplo, char *trans, char *diag, int *n, c *a, int *lda, c *x, int *incx) noexcept nogil: + + _fortran_ctrsv(uplo, trans, diag, n, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + d _fortran_dasum "BLAS_FUNC(dasum)"(int *n, d *dx, int *incx) nogil +cdef d dasum(int *n, d *dx, int *incx) noexcept nogil: + + return _fortran_dasum(n, dx, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_daxpy "BLAS_FUNC(daxpy)"(int *n, d *da, d *dx, int *incx, d *dy, int *incy) nogil +cdef void daxpy(int *n, d *da, d *dx, int *incx, d *dy, int *incy) noexcept nogil: + + _fortran_daxpy(n, da, dx, incx, dy, incy) + + +cdef extern from "_blas_subroutines.h": + d _fortran_dcabs1 "BLAS_FUNC(dcabs1)"(npy_complex128 *z) nogil +cdef d dcabs1(z *z) noexcept nogil: + + return _fortran_dcabs1(z) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dcopy "BLAS_FUNC(dcopy)"(int *n, d *dx, int *incx, d *dy, int *incy) nogil +cdef void dcopy(int *n, d *dx, int *incx, d *dy, int *incy) noexcept nogil: + + _fortran_dcopy(n, dx, incx, dy, incy) + + +cdef extern from "_blas_subroutines.h": + d _fortran_ddot "BLAS_FUNC(ddot)"(int *n, d *dx, int *incx, d *dy, int *incy) nogil +cdef d ddot(int *n, d *dx, int *incx, d *dy, int *incy) noexcept nogil: + + return _fortran_ddot(n, dx, incx, dy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dgbmv "BLAS_FUNC(dgbmv)"(char *trans, int *m, int *n, int *kl, int *ku, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) nogil +cdef void dgbmv(char *trans, int *m, int *n, int *kl, int *ku, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) noexcept nogil: + + _fortran_dgbmv(trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dgemm "BLAS_FUNC(dgemm)"(char *transa, char *transb, int *m, int *n, int *k, d *alpha, d *a, int *lda, d *b, int *ldb, d *beta, d *c, int *ldc) nogil +cdef void dgemm(char *transa, char *transb, int *m, int *n, int *k, d *alpha, d *a, int *lda, d *b, int *ldb, d *beta, d *c, int *ldc) noexcept nogil: + + _fortran_dgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dgemv "BLAS_FUNC(dgemv)"(char *trans, int *m, int *n, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) nogil +cdef void dgemv(char *trans, int *m, int *n, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) noexcept nogil: + + _fortran_dgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dger "BLAS_FUNC(dger)"(int *m, int *n, d *alpha, d *x, int *incx, d *y, int *incy, d *a, int *lda) nogil +cdef void dger(int *m, int *n, d *alpha, d *x, int *incx, d *y, int *incy, d *a, int *lda) noexcept nogil: + + _fortran_dger(m, n, alpha, x, incx, y, incy, a, lda) + + +cdef extern from "_blas_subroutines.h": + d _fortran_dnrm2 "BLAS_FUNC(dnrm2)"(int *n, d *x, int *incx) nogil +cdef d dnrm2(int *n, d *x, int *incx) noexcept nogil: + + return _fortran_dnrm2(n, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_drot "BLAS_FUNC(drot)"(int *n, d *dx, int *incx, d *dy, int *incy, d *c, d *s) nogil +cdef void drot(int *n, d *dx, int *incx, d *dy, int *incy, d *c, d *s) noexcept nogil: + + _fortran_drot(n, dx, incx, dy, incy, c, s) + + +cdef extern from "_blas_subroutines.h": + void _fortran_drotg "BLAS_FUNC(drotg)"(d *da, d *db, d *c, d *s) nogil +cdef void drotg(d *da, d *db, d *c, d *s) noexcept nogil: + + _fortran_drotg(da, db, c, s) + + +cdef extern from "_blas_subroutines.h": + void _fortran_drotm "BLAS_FUNC(drotm)"(int *n, d *dx, int *incx, d *dy, int *incy, d *dparam) nogil +cdef void drotm(int *n, d *dx, int *incx, d *dy, int *incy, d *dparam) noexcept nogil: + + _fortran_drotm(n, dx, incx, dy, incy, dparam) + + +cdef extern from "_blas_subroutines.h": + void _fortran_drotmg "BLAS_FUNC(drotmg)"(d *dd1, d *dd2, d *dx1, d *dy1, d *dparam) nogil +cdef void drotmg(d *dd1, d *dd2, d *dx1, d *dy1, d *dparam) noexcept nogil: + + _fortran_drotmg(dd1, dd2, dx1, dy1, dparam) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dsbmv "BLAS_FUNC(dsbmv)"(char *uplo, int *n, int *k, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) nogil +cdef void dsbmv(char *uplo, int *n, int *k, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) noexcept nogil: + + _fortran_dsbmv(uplo, n, k, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dscal "BLAS_FUNC(dscal)"(int *n, d *da, d *dx, int *incx) nogil +cdef void dscal(int *n, d *da, d *dx, int *incx) noexcept nogil: + + _fortran_dscal(n, da, dx, incx) + + +cdef extern from "_blas_subroutines.h": + d _fortran_dsdot "BLAS_FUNC(dsdot)"(int *n, s *sx, int *incx, s *sy, int *incy) nogil +cdef d dsdot(int *n, s *sx, int *incx, s *sy, int *incy) noexcept nogil: + + return _fortran_dsdot(n, sx, incx, sy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dspmv "BLAS_FUNC(dspmv)"(char *uplo, int *n, d *alpha, d *ap, d *x, int *incx, d *beta, d *y, int *incy) nogil +cdef void dspmv(char *uplo, int *n, d *alpha, d *ap, d *x, int *incx, d *beta, d *y, int *incy) noexcept nogil: + + _fortran_dspmv(uplo, n, alpha, ap, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dspr "BLAS_FUNC(dspr)"(char *uplo, int *n, d *alpha, d *x, int *incx, d *ap) nogil +cdef void dspr(char *uplo, int *n, d *alpha, d *x, int *incx, d *ap) noexcept nogil: + + _fortran_dspr(uplo, n, alpha, x, incx, ap) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dspr2 "BLAS_FUNC(dspr2)"(char *uplo, int *n, d *alpha, d *x, int *incx, d *y, int *incy, d *ap) nogil +cdef void dspr2(char *uplo, int *n, d *alpha, d *x, int *incx, d *y, int *incy, d *ap) noexcept nogil: + + _fortran_dspr2(uplo, n, alpha, x, incx, y, incy, ap) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dswap "BLAS_FUNC(dswap)"(int *n, d *dx, int *incx, d *dy, int *incy) nogil +cdef void dswap(int *n, d *dx, int *incx, d *dy, int *incy) noexcept nogil: + + _fortran_dswap(n, dx, incx, dy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dsymm "BLAS_FUNC(dsymm)"(char *side, char *uplo, int *m, int *n, d *alpha, d *a, int *lda, d *b, int *ldb, d *beta, d *c, int *ldc) nogil +cdef void dsymm(char *side, char *uplo, int *m, int *n, d *alpha, d *a, int *lda, d *b, int *ldb, d *beta, d *c, int *ldc) noexcept nogil: + + _fortran_dsymm(side, uplo, m, n, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dsymv "BLAS_FUNC(dsymv)"(char *uplo, int *n, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) nogil +cdef void dsymv(char *uplo, int *n, d *alpha, d *a, int *lda, d *x, int *incx, d *beta, d *y, int *incy) noexcept nogil: + + _fortran_dsymv(uplo, n, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dsyr "BLAS_FUNC(dsyr)"(char *uplo, int *n, d *alpha, d *x, int *incx, d *a, int *lda) nogil +cdef void dsyr(char *uplo, int *n, d *alpha, d *x, int *incx, d *a, int *lda) noexcept nogil: + + _fortran_dsyr(uplo, n, alpha, x, incx, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dsyr2 "BLAS_FUNC(dsyr2)"(char *uplo, int *n, d *alpha, d *x, int *incx, d *y, int *incy, d *a, int *lda) nogil +cdef void dsyr2(char *uplo, int *n, d *alpha, d *x, int *incx, d *y, int *incy, d *a, int *lda) noexcept nogil: + + _fortran_dsyr2(uplo, n, alpha, x, incx, y, incy, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dsyr2k "BLAS_FUNC(dsyr2k)"(char *uplo, char *trans, int *n, int *k, d *alpha, d *a, int *lda, d *b, int *ldb, d *beta, d *c, int *ldc) nogil +cdef void dsyr2k(char *uplo, char *trans, int *n, int *k, d *alpha, d *a, int *lda, d *b, int *ldb, d *beta, d *c, int *ldc) noexcept nogil: + + _fortran_dsyr2k(uplo, trans, n, k, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dsyrk "BLAS_FUNC(dsyrk)"(char *uplo, char *trans, int *n, int *k, d *alpha, d *a, int *lda, d *beta, d *c, int *ldc) nogil +cdef void dsyrk(char *uplo, char *trans, int *n, int *k, d *alpha, d *a, int *lda, d *beta, d *c, int *ldc) noexcept nogil: + + _fortran_dsyrk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dtbmv "BLAS_FUNC(dtbmv)"(char *uplo, char *trans, char *diag, int *n, int *k, d *a, int *lda, d *x, int *incx) nogil +cdef void dtbmv(char *uplo, char *trans, char *diag, int *n, int *k, d *a, int *lda, d *x, int *incx) noexcept nogil: + + _fortran_dtbmv(uplo, trans, diag, n, k, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dtbsv "BLAS_FUNC(dtbsv)"(char *uplo, char *trans, char *diag, int *n, int *k, d *a, int *lda, d *x, int *incx) nogil +cdef void dtbsv(char *uplo, char *trans, char *diag, int *n, int *k, d *a, int *lda, d *x, int *incx) noexcept nogil: + + _fortran_dtbsv(uplo, trans, diag, n, k, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dtpmv "BLAS_FUNC(dtpmv)"(char *uplo, char *trans, char *diag, int *n, d *ap, d *x, int *incx) nogil +cdef void dtpmv(char *uplo, char *trans, char *diag, int *n, d *ap, d *x, int *incx) noexcept nogil: + + _fortran_dtpmv(uplo, trans, diag, n, ap, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dtpsv "BLAS_FUNC(dtpsv)"(char *uplo, char *trans, char *diag, int *n, d *ap, d *x, int *incx) nogil +cdef void dtpsv(char *uplo, char *trans, char *diag, int *n, d *ap, d *x, int *incx) noexcept nogil: + + _fortran_dtpsv(uplo, trans, diag, n, ap, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dtrmm "BLAS_FUNC(dtrmm)"(char *side, char *uplo, char *transa, char *diag, int *m, int *n, d *alpha, d *a, int *lda, d *b, int *ldb) nogil +cdef void dtrmm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, d *alpha, d *a, int *lda, d *b, int *ldb) noexcept nogil: + + _fortran_dtrmm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dtrmv "BLAS_FUNC(dtrmv)"(char *uplo, char *trans, char *diag, int *n, d *a, int *lda, d *x, int *incx) nogil +cdef void dtrmv(char *uplo, char *trans, char *diag, int *n, d *a, int *lda, d *x, int *incx) noexcept nogil: + + _fortran_dtrmv(uplo, trans, diag, n, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dtrsm "BLAS_FUNC(dtrsm)"(char *side, char *uplo, char *transa, char *diag, int *m, int *n, d *alpha, d *a, int *lda, d *b, int *ldb) nogil +cdef void dtrsm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, d *alpha, d *a, int *lda, d *b, int *ldb) noexcept nogil: + + _fortran_dtrsm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb) + + +cdef extern from "_blas_subroutines.h": + void _fortran_dtrsv "BLAS_FUNC(dtrsv)"(char *uplo, char *trans, char *diag, int *n, d *a, int *lda, d *x, int *incx) nogil +cdef void dtrsv(char *uplo, char *trans, char *diag, int *n, d *a, int *lda, d *x, int *incx) noexcept nogil: + + _fortran_dtrsv(uplo, trans, diag, n, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + d _fortran_dzasum "BLAS_FUNC(dzasum)"(int *n, npy_complex128 *zx, int *incx) nogil +cdef d dzasum(int *n, z *zx, int *incx) noexcept nogil: + + return _fortran_dzasum(n, zx, incx) + + +cdef extern from "_blas_subroutines.h": + d _fortran_dznrm2 "BLAS_FUNC(dznrm2)"(int *n, npy_complex128 *x, int *incx) nogil +cdef d dznrm2(int *n, z *x, int *incx) noexcept nogil: + + return _fortran_dznrm2(n, x, incx) + + +cdef extern from "_blas_subroutines.h": + int _fortran_icamax "BLAS_FUNC(icamax)"(int *n, npy_complex64 *cx, int *incx) nogil +cdef int icamax(int *n, c *cx, int *incx) noexcept nogil: + + return _fortran_icamax(n, cx, incx) + + +cdef extern from "_blas_subroutines.h": + int _fortran_idamax "BLAS_FUNC(idamax)"(int *n, d *dx, int *incx) nogil +cdef int idamax(int *n, d *dx, int *incx) noexcept nogil: + + return _fortran_idamax(n, dx, incx) + + +cdef extern from "_blas_subroutines.h": + int _fortran_isamax "BLAS_FUNC(isamax)"(int *n, s *sx, int *incx) nogil +cdef int isamax(int *n, s *sx, int *incx) noexcept nogil: + + return _fortran_isamax(n, sx, incx) + + +cdef extern from "_blas_subroutines.h": + int _fortran_izamax "BLAS_FUNC(izamax)"(int *n, npy_complex128 *zx, int *incx) nogil +cdef int izamax(int *n, z *zx, int *incx) noexcept nogil: + + return _fortran_izamax(n, zx, incx) + + +cdef extern from "_blas_subroutines.h": + bint _fortran_lsame "BLAS_FUNC(lsame)"(char *ca, char *cb) nogil +cdef bint lsame(char *ca, char *cb) noexcept nogil: + + return _fortran_lsame(ca, cb) + + +cdef extern from "_blas_subroutines.h": + s _fortran_sasum "BLAS_FUNC(sasum)"(int *n, s *sx, int *incx) nogil +cdef s sasum(int *n, s *sx, int *incx) noexcept nogil: + + return _fortran_sasum(n, sx, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_saxpy "BLAS_FUNC(saxpy)"(int *n, s *sa, s *sx, int *incx, s *sy, int *incy) nogil +cdef void saxpy(int *n, s *sa, s *sx, int *incx, s *sy, int *incy) noexcept nogil: + + _fortran_saxpy(n, sa, sx, incx, sy, incy) + + +cdef extern from "_blas_subroutines.h": + s _fortran_scasum "BLAS_FUNC(scasum)"(int *n, npy_complex64 *cx, int *incx) nogil +cdef s scasum(int *n, c *cx, int *incx) noexcept nogil: + + return _fortran_scasum(n, cx, incx) + + +cdef extern from "_blas_subroutines.h": + s _fortran_scnrm2 "BLAS_FUNC(scnrm2)"(int *n, npy_complex64 *x, int *incx) nogil +cdef s scnrm2(int *n, c *x, int *incx) noexcept nogil: + + return _fortran_scnrm2(n, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_scopy "BLAS_FUNC(scopy)"(int *n, s *sx, int *incx, s *sy, int *incy) nogil +cdef void scopy(int *n, s *sx, int *incx, s *sy, int *incy) noexcept nogil: + + _fortran_scopy(n, sx, incx, sy, incy) + + +cdef extern from "_blas_subroutines.h": + s _fortran_sdot "BLAS_FUNC(sdot)"(int *n, s *sx, int *incx, s *sy, int *incy) nogil +cdef s sdot(int *n, s *sx, int *incx, s *sy, int *incy) noexcept nogil: + + return _fortran_sdot(n, sx, incx, sy, incy) + + +cdef extern from "_blas_subroutines.h": + s _fortran_sdsdot "BLAS_FUNC(sdsdot)"(int *n, s *sb, s *sx, int *incx, s *sy, int *incy) nogil +cdef s sdsdot(int *n, s *sb, s *sx, int *incx, s *sy, int *incy) noexcept nogil: + + return _fortran_sdsdot(n, sb, sx, incx, sy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_sgbmv "BLAS_FUNC(sgbmv)"(char *trans, int *m, int *n, int *kl, int *ku, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) nogil +cdef void sgbmv(char *trans, int *m, int *n, int *kl, int *ku, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) noexcept nogil: + + _fortran_sgbmv(trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_sgemm "BLAS_FUNC(sgemm)"(char *transa, char *transb, int *m, int *n, int *k, s *alpha, s *a, int *lda, s *b, int *ldb, s *beta, s *c, int *ldc) nogil +cdef void sgemm(char *transa, char *transb, int *m, int *n, int *k, s *alpha, s *a, int *lda, s *b, int *ldb, s *beta, s *c, int *ldc) noexcept nogil: + + _fortran_sgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_sgemv "BLAS_FUNC(sgemv)"(char *trans, int *m, int *n, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) nogil +cdef void sgemv(char *trans, int *m, int *n, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) noexcept nogil: + + _fortran_sgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_sger "BLAS_FUNC(sger)"(int *m, int *n, s *alpha, s *x, int *incx, s *y, int *incy, s *a, int *lda) nogil +cdef void sger(int *m, int *n, s *alpha, s *x, int *incx, s *y, int *incy, s *a, int *lda) noexcept nogil: + + _fortran_sger(m, n, alpha, x, incx, y, incy, a, lda) + + +cdef extern from "_blas_subroutines.h": + s _fortran_snrm2 "BLAS_FUNC(snrm2)"(int *n, s *x, int *incx) nogil +cdef s snrm2(int *n, s *x, int *incx) noexcept nogil: + + return _fortran_snrm2(n, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_srot "BLAS_FUNC(srot)"(int *n, s *sx, int *incx, s *sy, int *incy, s *c, s *s) nogil +cdef void srot(int *n, s *sx, int *incx, s *sy, int *incy, s *c, s *s) noexcept nogil: + + _fortran_srot(n, sx, incx, sy, incy, c, s) + + +cdef extern from "_blas_subroutines.h": + void _fortran_srotg "BLAS_FUNC(srotg)"(s *sa, s *sb, s *c, s *s) nogil +cdef void srotg(s *sa, s *sb, s *c, s *s) noexcept nogil: + + _fortran_srotg(sa, sb, c, s) + + +cdef extern from "_blas_subroutines.h": + void _fortran_srotm "BLAS_FUNC(srotm)"(int *n, s *sx, int *incx, s *sy, int *incy, s *sparam) nogil +cdef void srotm(int *n, s *sx, int *incx, s *sy, int *incy, s *sparam) noexcept nogil: + + _fortran_srotm(n, sx, incx, sy, incy, sparam) + + +cdef extern from "_blas_subroutines.h": + void _fortran_srotmg "BLAS_FUNC(srotmg)"(s *sd1, s *sd2, s *sx1, s *sy1, s *sparam) nogil +cdef void srotmg(s *sd1, s *sd2, s *sx1, s *sy1, s *sparam) noexcept nogil: + + _fortran_srotmg(sd1, sd2, sx1, sy1, sparam) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ssbmv "BLAS_FUNC(ssbmv)"(char *uplo, int *n, int *k, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) nogil +cdef void ssbmv(char *uplo, int *n, int *k, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) noexcept nogil: + + _fortran_ssbmv(uplo, n, k, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_sscal "BLAS_FUNC(sscal)"(int *n, s *sa, s *sx, int *incx) nogil +cdef void sscal(int *n, s *sa, s *sx, int *incx) noexcept nogil: + + _fortran_sscal(n, sa, sx, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_sspmv "BLAS_FUNC(sspmv)"(char *uplo, int *n, s *alpha, s *ap, s *x, int *incx, s *beta, s *y, int *incy) nogil +cdef void sspmv(char *uplo, int *n, s *alpha, s *ap, s *x, int *incx, s *beta, s *y, int *incy) noexcept nogil: + + _fortran_sspmv(uplo, n, alpha, ap, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_sspr "BLAS_FUNC(sspr)"(char *uplo, int *n, s *alpha, s *x, int *incx, s *ap) nogil +cdef void sspr(char *uplo, int *n, s *alpha, s *x, int *incx, s *ap) noexcept nogil: + + _fortran_sspr(uplo, n, alpha, x, incx, ap) + + +cdef extern from "_blas_subroutines.h": + void _fortran_sspr2 "BLAS_FUNC(sspr2)"(char *uplo, int *n, s *alpha, s *x, int *incx, s *y, int *incy, s *ap) nogil +cdef void sspr2(char *uplo, int *n, s *alpha, s *x, int *incx, s *y, int *incy, s *ap) noexcept nogil: + + _fortran_sspr2(uplo, n, alpha, x, incx, y, incy, ap) + + +cdef extern from "_blas_subroutines.h": + void _fortran_sswap "BLAS_FUNC(sswap)"(int *n, s *sx, int *incx, s *sy, int *incy) nogil +cdef void sswap(int *n, s *sx, int *incx, s *sy, int *incy) noexcept nogil: + + _fortran_sswap(n, sx, incx, sy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ssymm "BLAS_FUNC(ssymm)"(char *side, char *uplo, int *m, int *n, s *alpha, s *a, int *lda, s *b, int *ldb, s *beta, s *c, int *ldc) nogil +cdef void ssymm(char *side, char *uplo, int *m, int *n, s *alpha, s *a, int *lda, s *b, int *ldb, s *beta, s *c, int *ldc) noexcept nogil: + + _fortran_ssymm(side, uplo, m, n, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ssymv "BLAS_FUNC(ssymv)"(char *uplo, int *n, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) nogil +cdef void ssymv(char *uplo, int *n, s *alpha, s *a, int *lda, s *x, int *incx, s *beta, s *y, int *incy) noexcept nogil: + + _fortran_ssymv(uplo, n, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ssyr "BLAS_FUNC(ssyr)"(char *uplo, int *n, s *alpha, s *x, int *incx, s *a, int *lda) nogil +cdef void ssyr(char *uplo, int *n, s *alpha, s *x, int *incx, s *a, int *lda) noexcept nogil: + + _fortran_ssyr(uplo, n, alpha, x, incx, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ssyr2 "BLAS_FUNC(ssyr2)"(char *uplo, int *n, s *alpha, s *x, int *incx, s *y, int *incy, s *a, int *lda) nogil +cdef void ssyr2(char *uplo, int *n, s *alpha, s *x, int *incx, s *y, int *incy, s *a, int *lda) noexcept nogil: + + _fortran_ssyr2(uplo, n, alpha, x, incx, y, incy, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ssyr2k "BLAS_FUNC(ssyr2k)"(char *uplo, char *trans, int *n, int *k, s *alpha, s *a, int *lda, s *b, int *ldb, s *beta, s *c, int *ldc) nogil +cdef void ssyr2k(char *uplo, char *trans, int *n, int *k, s *alpha, s *a, int *lda, s *b, int *ldb, s *beta, s *c, int *ldc) noexcept nogil: + + _fortran_ssyr2k(uplo, trans, n, k, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ssyrk "BLAS_FUNC(ssyrk)"(char *uplo, char *trans, int *n, int *k, s *alpha, s *a, int *lda, s *beta, s *c, int *ldc) nogil +cdef void ssyrk(char *uplo, char *trans, int *n, int *k, s *alpha, s *a, int *lda, s *beta, s *c, int *ldc) noexcept nogil: + + _fortran_ssyrk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_stbmv "BLAS_FUNC(stbmv)"(char *uplo, char *trans, char *diag, int *n, int *k, s *a, int *lda, s *x, int *incx) nogil +cdef void stbmv(char *uplo, char *trans, char *diag, int *n, int *k, s *a, int *lda, s *x, int *incx) noexcept nogil: + + _fortran_stbmv(uplo, trans, diag, n, k, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_stbsv "BLAS_FUNC(stbsv)"(char *uplo, char *trans, char *diag, int *n, int *k, s *a, int *lda, s *x, int *incx) nogil +cdef void stbsv(char *uplo, char *trans, char *diag, int *n, int *k, s *a, int *lda, s *x, int *incx) noexcept nogil: + + _fortran_stbsv(uplo, trans, diag, n, k, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_stpmv "BLAS_FUNC(stpmv)"(char *uplo, char *trans, char *diag, int *n, s *ap, s *x, int *incx) nogil +cdef void stpmv(char *uplo, char *trans, char *diag, int *n, s *ap, s *x, int *incx) noexcept nogil: + + _fortran_stpmv(uplo, trans, diag, n, ap, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_stpsv "BLAS_FUNC(stpsv)"(char *uplo, char *trans, char *diag, int *n, s *ap, s *x, int *incx) nogil +cdef void stpsv(char *uplo, char *trans, char *diag, int *n, s *ap, s *x, int *incx) noexcept nogil: + + _fortran_stpsv(uplo, trans, diag, n, ap, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_strmm "BLAS_FUNC(strmm)"(char *side, char *uplo, char *transa, char *diag, int *m, int *n, s *alpha, s *a, int *lda, s *b, int *ldb) nogil +cdef void strmm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, s *alpha, s *a, int *lda, s *b, int *ldb) noexcept nogil: + + _fortran_strmm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb) + + +cdef extern from "_blas_subroutines.h": + void _fortran_strmv "BLAS_FUNC(strmv)"(char *uplo, char *trans, char *diag, int *n, s *a, int *lda, s *x, int *incx) nogil +cdef void strmv(char *uplo, char *trans, char *diag, int *n, s *a, int *lda, s *x, int *incx) noexcept nogil: + + _fortran_strmv(uplo, trans, diag, n, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_strsm "BLAS_FUNC(strsm)"(char *side, char *uplo, char *transa, char *diag, int *m, int *n, s *alpha, s *a, int *lda, s *b, int *ldb) nogil +cdef void strsm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, s *alpha, s *a, int *lda, s *b, int *ldb) noexcept nogil: + + _fortran_strsm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb) + + +cdef extern from "_blas_subroutines.h": + void _fortran_strsv "BLAS_FUNC(strsv)"(char *uplo, char *trans, char *diag, int *n, s *a, int *lda, s *x, int *incx) nogil +cdef void strsv(char *uplo, char *trans, char *diag, int *n, s *a, int *lda, s *x, int *incx) noexcept nogil: + + _fortran_strsv(uplo, trans, diag, n, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zaxpy "BLAS_FUNC(zaxpy)"(int *n, npy_complex128 *za, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy) nogil +cdef void zaxpy(int *n, z *za, z *zx, int *incx, z *zy, int *incy) noexcept nogil: + + _fortran_zaxpy(n, za, zx, incx, zy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zcopy "BLAS_FUNC(zcopy)"(int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy) nogil +cdef void zcopy(int *n, z *zx, int *incx, z *zy, int *incy) noexcept nogil: + + _fortran_zcopy(n, zx, incx, zy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zdotc "F_FUNC(zdotcwrp,ZDOTCWRP)"(npy_complex128 *out, int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy) nogil +cdef z zdotc(int *n, z *zx, int *incx, z *zy, int *incy) noexcept nogil: + cdef z out + _fortran_zdotc(&out, n, zx, incx, zy, incy) + return out + +cdef extern from "_blas_subroutines.h": + void _fortran_zdotu "F_FUNC(zdotuwrp,ZDOTUWRP)"(npy_complex128 *out, int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy) nogil +cdef z zdotu(int *n, z *zx, int *incx, z *zy, int *incy) noexcept nogil: + cdef z out + _fortran_zdotu(&out, n, zx, incx, zy, incy) + return out + +cdef extern from "_blas_subroutines.h": + void _fortran_zdrot "BLAS_FUNC(zdrot)"(int *n, npy_complex128 *cx, int *incx, npy_complex128 *cy, int *incy, d *c, d *s) nogil +cdef void zdrot(int *n, z *cx, int *incx, z *cy, int *incy, d *c, d *s) noexcept nogil: + + _fortran_zdrot(n, cx, incx, cy, incy, c, s) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zdscal "BLAS_FUNC(zdscal)"(int *n, d *da, npy_complex128 *zx, int *incx) nogil +cdef void zdscal(int *n, d *da, z *zx, int *incx) noexcept nogil: + + _fortran_zdscal(n, da, zx, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zgbmv "BLAS_FUNC(zgbmv)"(char *trans, int *m, int *n, int *kl, int *ku, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy) nogil +cdef void zgbmv(char *trans, int *m, int *n, int *kl, int *ku, z *alpha, z *a, int *lda, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil: + + _fortran_zgbmv(trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zgemm "BLAS_FUNC(zgemm)"(char *transa, char *transb, int *m, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc) nogil +cdef void zgemm(char *transa, char *transb, int *m, int *n, int *k, z *alpha, z *a, int *lda, z *b, int *ldb, z *beta, z *c, int *ldc) noexcept nogil: + + _fortran_zgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zgemv "BLAS_FUNC(zgemv)"(char *trans, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy) nogil +cdef void zgemv(char *trans, int *m, int *n, z *alpha, z *a, int *lda, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil: + + _fortran_zgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zgerc "BLAS_FUNC(zgerc)"(int *m, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *a, int *lda) nogil +cdef void zgerc(int *m, int *n, z *alpha, z *x, int *incx, z *y, int *incy, z *a, int *lda) noexcept nogil: + + _fortran_zgerc(m, n, alpha, x, incx, y, incy, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zgeru "BLAS_FUNC(zgeru)"(int *m, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *a, int *lda) nogil +cdef void zgeru(int *m, int *n, z *alpha, z *x, int *incx, z *y, int *incy, z *a, int *lda) noexcept nogil: + + _fortran_zgeru(m, n, alpha, x, incx, y, incy, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zhbmv "BLAS_FUNC(zhbmv)"(char *uplo, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy) nogil +cdef void zhbmv(char *uplo, int *n, int *k, z *alpha, z *a, int *lda, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil: + + _fortran_zhbmv(uplo, n, k, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zhemm "BLAS_FUNC(zhemm)"(char *side, char *uplo, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc) nogil +cdef void zhemm(char *side, char *uplo, int *m, int *n, z *alpha, z *a, int *lda, z *b, int *ldb, z *beta, z *c, int *ldc) noexcept nogil: + + _fortran_zhemm(side, uplo, m, n, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zhemv "BLAS_FUNC(zhemv)"(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy) nogil +cdef void zhemv(char *uplo, int *n, z *alpha, z *a, int *lda, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil: + + _fortran_zhemv(uplo, n, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zher "BLAS_FUNC(zher)"(char *uplo, int *n, d *alpha, npy_complex128 *x, int *incx, npy_complex128 *a, int *lda) nogil +cdef void zher(char *uplo, int *n, d *alpha, z *x, int *incx, z *a, int *lda) noexcept nogil: + + _fortran_zher(uplo, n, alpha, x, incx, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zher2 "BLAS_FUNC(zher2)"(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *a, int *lda) nogil +cdef void zher2(char *uplo, int *n, z *alpha, z *x, int *incx, z *y, int *incy, z *a, int *lda) noexcept nogil: + + _fortran_zher2(uplo, n, alpha, x, incx, y, incy, a, lda) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zher2k "BLAS_FUNC(zher2k)"(char *uplo, char *trans, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, d *beta, npy_complex128 *c, int *ldc) nogil +cdef void zher2k(char *uplo, char *trans, int *n, int *k, z *alpha, z *a, int *lda, z *b, int *ldb, d *beta, z *c, int *ldc) noexcept nogil: + + _fortran_zher2k(uplo, trans, n, k, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zherk "BLAS_FUNC(zherk)"(char *uplo, char *trans, int *n, int *k, d *alpha, npy_complex128 *a, int *lda, d *beta, npy_complex128 *c, int *ldc) nogil +cdef void zherk(char *uplo, char *trans, int *n, int *k, d *alpha, z *a, int *lda, d *beta, z *c, int *ldc) noexcept nogil: + + _fortran_zherk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zhpmv "BLAS_FUNC(zhpmv)"(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *ap, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy) nogil +cdef void zhpmv(char *uplo, int *n, z *alpha, z *ap, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil: + + _fortran_zhpmv(uplo, n, alpha, ap, x, incx, beta, y, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zhpr "BLAS_FUNC(zhpr)"(char *uplo, int *n, d *alpha, npy_complex128 *x, int *incx, npy_complex128 *ap) nogil +cdef void zhpr(char *uplo, int *n, d *alpha, z *x, int *incx, z *ap) noexcept nogil: + + _fortran_zhpr(uplo, n, alpha, x, incx, ap) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zhpr2 "BLAS_FUNC(zhpr2)"(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *ap) nogil +cdef void zhpr2(char *uplo, int *n, z *alpha, z *x, int *incx, z *y, int *incy, z *ap) noexcept nogil: + + _fortran_zhpr2(uplo, n, alpha, x, incx, y, incy, ap) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zrotg "BLAS_FUNC(zrotg)"(npy_complex128 *ca, npy_complex128 *cb, d *c, npy_complex128 *s) nogil +cdef void zrotg(z *ca, z *cb, d *c, z *s) noexcept nogil: + + _fortran_zrotg(ca, cb, c, s) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zscal "BLAS_FUNC(zscal)"(int *n, npy_complex128 *za, npy_complex128 *zx, int *incx) nogil +cdef void zscal(int *n, z *za, z *zx, int *incx) noexcept nogil: + + _fortran_zscal(n, za, zx, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zswap "BLAS_FUNC(zswap)"(int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy) nogil +cdef void zswap(int *n, z *zx, int *incx, z *zy, int *incy) noexcept nogil: + + _fortran_zswap(n, zx, incx, zy, incy) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zsymm "BLAS_FUNC(zsymm)"(char *side, char *uplo, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc) nogil +cdef void zsymm(char *side, char *uplo, int *m, int *n, z *alpha, z *a, int *lda, z *b, int *ldb, z *beta, z *c, int *ldc) noexcept nogil: + + _fortran_zsymm(side, uplo, m, n, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zsyr2k "BLAS_FUNC(zsyr2k)"(char *uplo, char *trans, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc) nogil +cdef void zsyr2k(char *uplo, char *trans, int *n, int *k, z *alpha, z *a, int *lda, z *b, int *ldb, z *beta, z *c, int *ldc) noexcept nogil: + + _fortran_zsyr2k(uplo, trans, n, k, alpha, a, lda, b, ldb, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_zsyrk "BLAS_FUNC(zsyrk)"(char *uplo, char *trans, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *beta, npy_complex128 *c, int *ldc) nogil +cdef void zsyrk(char *uplo, char *trans, int *n, int *k, z *alpha, z *a, int *lda, z *beta, z *c, int *ldc) noexcept nogil: + + _fortran_zsyrk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ztbmv "BLAS_FUNC(ztbmv)"(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx) nogil +cdef void ztbmv(char *uplo, char *trans, char *diag, int *n, int *k, z *a, int *lda, z *x, int *incx) noexcept nogil: + + _fortran_ztbmv(uplo, trans, diag, n, k, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ztbsv "BLAS_FUNC(ztbsv)"(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx) nogil +cdef void ztbsv(char *uplo, char *trans, char *diag, int *n, int *k, z *a, int *lda, z *x, int *incx) noexcept nogil: + + _fortran_ztbsv(uplo, trans, diag, n, k, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ztpmv "BLAS_FUNC(ztpmv)"(char *uplo, char *trans, char *diag, int *n, npy_complex128 *ap, npy_complex128 *x, int *incx) nogil +cdef void ztpmv(char *uplo, char *trans, char *diag, int *n, z *ap, z *x, int *incx) noexcept nogil: + + _fortran_ztpmv(uplo, trans, diag, n, ap, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ztpsv "BLAS_FUNC(ztpsv)"(char *uplo, char *trans, char *diag, int *n, npy_complex128 *ap, npy_complex128 *x, int *incx) nogil +cdef void ztpsv(char *uplo, char *trans, char *diag, int *n, z *ap, z *x, int *incx) noexcept nogil: + + _fortran_ztpsv(uplo, trans, diag, n, ap, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ztrmm "BLAS_FUNC(ztrmm)"(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb) nogil +cdef void ztrmm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, z *alpha, z *a, int *lda, z *b, int *ldb) noexcept nogil: + + _fortran_ztrmm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ztrmv "BLAS_FUNC(ztrmv)"(char *uplo, char *trans, char *diag, int *n, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx) nogil +cdef void ztrmv(char *uplo, char *trans, char *diag, int *n, z *a, int *lda, z *x, int *incx) noexcept nogil: + + _fortran_ztrmv(uplo, trans, diag, n, a, lda, x, incx) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ztrsm "BLAS_FUNC(ztrsm)"(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb) nogil +cdef void ztrsm(char *side, char *uplo, char *transa, char *diag, int *m, int *n, z *alpha, z *a, int *lda, z *b, int *ldb) noexcept nogil: + + _fortran_ztrsm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb) + + +cdef extern from "_blas_subroutines.h": + void _fortran_ztrsv "BLAS_FUNC(ztrsv)"(char *uplo, char *trans, char *diag, int *n, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx) nogil +cdef void ztrsv(char *uplo, char *trans, char *diag, int *n, z *a, int *lda, z *x, int *incx) noexcept nogil: + + _fortran_ztrsv(uplo, trans, diag, n, a, lda, x, incx) + + + +# Python-accessible wrappers for testing: + +cdef inline bint _is_contiguous(double[:,:] a, int axis) noexcept nogil: + return (a.strides[axis] == sizeof(a[0,0]) or a.shape[axis] == 1) + +cpdef float complex _test_cdotc(float complex[:] cx, float complex[:] cy) noexcept nogil: + cdef: + int n = cx.shape[0] + int incx = cx.strides[0] // sizeof(cx[0]) + int incy = cy.strides[0] // sizeof(cy[0]) + return cdotc(&n, &cx[0], &incx, &cy[0], &incy) + +cpdef float complex _test_cdotu(float complex[:] cx, float complex[:] cy) noexcept nogil: + cdef: + int n = cx.shape[0] + int incx = cx.strides[0] // sizeof(cx[0]) + int incy = cy.strides[0] // sizeof(cy[0]) + return cdotu(&n, &cx[0], &incx, &cy[0], &incy) + +cpdef double _test_dasum(double[:] dx) noexcept nogil: + cdef: + int n = dx.shape[0] + int incx = dx.strides[0] // sizeof(dx[0]) + return dasum(&n, &dx[0], &incx) + +cpdef double _test_ddot(double[:] dx, double[:] dy) noexcept nogil: + cdef: + int n = dx.shape[0] + int incx = dx.strides[0] // sizeof(dx[0]) + int incy = dy.strides[0] // sizeof(dy[0]) + return ddot(&n, &dx[0], &incx, &dy[0], &incy) + +cpdef int _test_dgemm(double alpha, double[:,:] a, double[:,:] b, double beta, + double[:,:] c) except -1 nogil: + cdef: + char *transa + char *transb + int m, n, k, lda, ldb, ldc + double *a0=&a[0,0] + double *b0=&b[0,0] + double *c0=&c[0,0] + # In the case that c is C contiguous, swap a and b and + # swap whether or not each of them is transposed. + # This can be done because a.dot(b) = b.T.dot(a.T).T. + if _is_contiguous(c, 1): + if _is_contiguous(a, 1): + transb = 'n' + ldb = (&a[1,0]) - a0 if a.shape[0] > 1 else 1 + elif _is_contiguous(a, 0): + transb = 't' + ldb = (&a[0,1]) - a0 if a.shape[1] > 1 else 1 + else: + with gil: + raise ValueError("Input 'a' is neither C nor Fortran contiguous.") + if _is_contiguous(b, 1): + transa = 'n' + lda = (&b[1,0]) - b0 if b.shape[0] > 1 else 1 + elif _is_contiguous(b, 0): + transa = 't' + lda = (&b[0,1]) - b0 if b.shape[1] > 1 else 1 + else: + with gil: + raise ValueError("Input 'b' is neither C nor Fortran contiguous.") + k = b.shape[0] + if k != a.shape[1]: + with gil: + raise ValueError("Shape mismatch in input arrays.") + m = b.shape[1] + n = a.shape[0] + if n != c.shape[0] or m != c.shape[1]: + with gil: + raise ValueError("Output array does not have the correct shape.") + ldc = (&c[1,0]) - c0 if c.shape[0] > 1 else 1 + dgemm(transa, transb, &m, &n, &k, &alpha, b0, &lda, a0, + &ldb, &beta, c0, &ldc) + elif _is_contiguous(c, 0): + if _is_contiguous(a, 1): + transa = 't' + lda = (&a[1,0]) - a0 if a.shape[0] > 1 else 1 + elif _is_contiguous(a, 0): + transa = 'n' + lda = (&a[0,1]) - a0 if a.shape[1] > 1 else 1 + else: + with gil: + raise ValueError("Input 'a' is neither C nor Fortran contiguous.") + if _is_contiguous(b, 1): + transb = 't' + ldb = (&b[1,0]) - b0 if b.shape[0] > 1 else 1 + elif _is_contiguous(b, 0): + transb = 'n' + ldb = (&b[0,1]) - b0 if b.shape[1] > 1 else 1 + else: + with gil: + raise ValueError("Input 'b' is neither C nor Fortran contiguous.") + m = a.shape[0] + k = a.shape[1] + if k != b.shape[0]: + with gil: + raise ValueError("Shape mismatch in input arrays.") + n = b.shape[1] + if m != c.shape[0] or n != c.shape[1]: + with gil: + raise ValueError("Output array does not have the correct shape.") + ldc = (&c[0,1]) - c0 if c.shape[1] > 1 else 1 + dgemm(transa, transb, &m, &n, &k, &alpha, a0, &lda, b0, + &ldb, &beta, c0, &ldc) + else: + with gil: + raise ValueError("Input 'c' is neither C nor Fortran contiguous.") + return 0 + +cpdef double _test_dnrm2(double[:] x) noexcept nogil: + cdef: + int n = x.shape[0] + int incx = x.strides[0] // sizeof(x[0]) + return dnrm2(&n, &x[0], &incx) + +cpdef double _test_dzasum(double complex[:] zx) noexcept nogil: + cdef: + int n = zx.shape[0] + int incx = zx.strides[0] // sizeof(zx[0]) + return dzasum(&n, &zx[0], &incx) + +cpdef double _test_dznrm2(double complex[:] x) noexcept nogil: + cdef: + int n = x.shape[0] + int incx = x.strides[0] // sizeof(x[0]) + return dznrm2(&n, &x[0], &incx) + +cpdef int _test_icamax(float complex[:] cx) noexcept nogil: + cdef: + int n = cx.shape[0] + int incx = cx.strides[0] // sizeof(cx[0]) + return icamax(&n, &cx[0], &incx) + +cpdef int _test_idamax(double[:] dx) noexcept nogil: + cdef: + int n = dx.shape[0] + int incx = dx.strides[0] // sizeof(dx[0]) + return idamax(&n, &dx[0], &incx) + +cpdef int _test_isamax(float[:] sx) noexcept nogil: + cdef: + int n = sx.shape[0] + int incx = sx.strides[0] // sizeof(sx[0]) + return isamax(&n, &sx[0], &incx) + +cpdef int _test_izamax(double complex[:] zx) noexcept nogil: + cdef: + int n = zx.shape[0] + int incx = zx.strides[0] // sizeof(zx[0]) + return izamax(&n, &zx[0], &incx) + +cpdef float _test_sasum(float[:] sx) noexcept nogil: + cdef: + int n = sx.shape[0] + int incx = sx.strides[0] // sizeof(sx[0]) + return sasum(&n, &sx[0], &incx) + +cpdef float _test_scasum(float complex[:] cx) noexcept nogil: + cdef: + int n = cx.shape[0] + int incx = cx.strides[0] // sizeof(cx[0]) + return scasum(&n, &cx[0], &incx) + +cpdef float _test_scnrm2(float complex[:] x) noexcept nogil: + cdef: + int n = x.shape[0] + int incx = x.strides[0] // sizeof(x[0]) + return scnrm2(&n, &x[0], &incx) + +cpdef float _test_sdot(float[:] sx, float[:] sy) noexcept nogil: + cdef: + int n = sx.shape[0] + int incx = sx.strides[0] // sizeof(sx[0]) + int incy = sy.strides[0] // sizeof(sy[0]) + return sdot(&n, &sx[0], &incx, &sy[0], &incy) + +cpdef float _test_snrm2(float[:] x) noexcept nogil: + cdef: + int n = x.shape[0] + int incx = x.strides[0] // sizeof(x[0]) + return snrm2(&n, &x[0], &incx) + +cpdef double complex _test_zdotc(double complex[:] zx, double complex[:] zy) noexcept nogil: + cdef: + int n = zx.shape[0] + int incx = zx.strides[0] // sizeof(zx[0]) + int incy = zy.strides[0] // sizeof(zy[0]) + return zdotc(&n, &zx[0], &incx, &zy[0], &incy) + +cpdef double complex _test_zdotu(double complex[:] zx, double complex[:] zy) noexcept nogil: + cdef: + int n = zx.shape[0] + int incx = zx.strides[0] // sizeof(zx[0]) + int incy = zy.strides[0] // sizeof(zy[0]) + return zdotu(&n, &zx[0], &incx, &zy[0], &incy) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_lapack.pxd b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_lapack.pxd new file mode 100644 index 0000000000000000000000000000000000000000..7964c52d766cd1b08bda6411960a29dbeb6bfe2d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_lapack.pxd @@ -0,0 +1,1528 @@ +""" +This file was generated by _generate_pyx.py. +Do not edit this file directly. +""" + +# Within SciPy, these wrappers can be used via relative or absolute cimport. +# Examples: +# from ..linalg cimport cython_lapack +# from scipy.linalg cimport cython_lapack +# cimport scipy.linalg.cython_lapack as cython_lapack +# cimport ..linalg.cython_lapack as cython_lapack + +# Within SciPy, if LAPACK functions are needed in C/C++/Fortran, +# these wrappers should not be used. +# The original libraries should be linked directly. + +ctypedef float s +ctypedef double d +ctypedef float complex c +ctypedef double complex z + +# Function pointer type declarations for +# gees and gges families of functions. +ctypedef bint cselect1(c*) +ctypedef bint cselect2(c*, c*) +ctypedef bint dselect2(d*, d*) +ctypedef bint dselect3(d*, d*, d*) +ctypedef bint sselect2(s*, s*) +ctypedef bint sselect3(s*, s*, s*) +ctypedef bint zselect1(z*) +ctypedef bint zselect2(z*, z*) + +cdef void cbbcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, s *theta, s *phi, c *u1, int *ldu1, c *u2, int *ldu2, c *v1t, int *ldv1t, c *v2t, int *ldv2t, s *b11d, s *b11e, s *b12d, s *b12e, s *b21d, s *b21e, s *b22d, s *b22e, s *rwork, int *lrwork, int *info) noexcept nogil +cdef void cbdsqr(char *uplo, int *n, int *ncvt, int *nru, int *ncc, s *d, s *e, c *vt, int *ldvt, c *u, int *ldu, c *c, int *ldc, s *rwork, int *info) noexcept nogil +cdef void cgbbrd(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, c *ab, int *ldab, s *d, s *e, c *q, int *ldq, c *pt, int *ldpt, c *c, int *ldc, c *work, s *rwork, int *info) noexcept nogil +cdef void cgbcon(char *norm, int *n, int *kl, int *ku, c *ab, int *ldab, int *ipiv, s *anorm, s *rcond, c *work, s *rwork, int *info) noexcept nogil +cdef void cgbequ(int *m, int *n, int *kl, int *ku, c *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil +cdef void cgbequb(int *m, int *n, int *kl, int *ku, c *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil +cdef void cgbrfs(char *trans, int *n, int *kl, int *ku, int *nrhs, c *ab, int *ldab, c *afb, int *ldafb, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cgbsv(int *n, int *kl, int *ku, int *nrhs, c *ab, int *ldab, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void cgbsvx(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, c *ab, int *ldab, c *afb, int *ldafb, int *ipiv, char *equed, s *r, s *c, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cgbtf2(int *m, int *n, int *kl, int *ku, c *ab, int *ldab, int *ipiv, int *info) noexcept nogil +cdef void cgbtrf(int *m, int *n, int *kl, int *ku, c *ab, int *ldab, int *ipiv, int *info) noexcept nogil +cdef void cgbtrs(char *trans, int *n, int *kl, int *ku, int *nrhs, c *ab, int *ldab, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void cgebak(char *job, char *side, int *n, int *ilo, int *ihi, s *scale, int *m, c *v, int *ldv, int *info) noexcept nogil +cdef void cgebal(char *job, int *n, c *a, int *lda, int *ilo, int *ihi, s *scale, int *info) noexcept nogil +cdef void cgebd2(int *m, int *n, c *a, int *lda, s *d, s *e, c *tauq, c *taup, c *work, int *info) noexcept nogil +cdef void cgebrd(int *m, int *n, c *a, int *lda, s *d, s *e, c *tauq, c *taup, c *work, int *lwork, int *info) noexcept nogil +cdef void cgecon(char *norm, int *n, c *a, int *lda, s *anorm, s *rcond, c *work, s *rwork, int *info) noexcept nogil +cdef void cgeequ(int *m, int *n, c *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil +cdef void cgeequb(int *m, int *n, c *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil +cdef void cgees(char *jobvs, char *sort, cselect1 *select, int *n, c *a, int *lda, int *sdim, c *w, c *vs, int *ldvs, c *work, int *lwork, s *rwork, bint *bwork, int *info) noexcept nogil +cdef void cgeesx(char *jobvs, char *sort, cselect1 *select, char *sense, int *n, c *a, int *lda, int *sdim, c *w, c *vs, int *ldvs, s *rconde, s *rcondv, c *work, int *lwork, s *rwork, bint *bwork, int *info) noexcept nogil +cdef void cgeev(char *jobvl, char *jobvr, int *n, c *a, int *lda, c *w, c *vl, int *ldvl, c *vr, int *ldvr, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void cgeevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, c *a, int *lda, c *w, c *vl, int *ldvl, c *vr, int *ldvr, int *ilo, int *ihi, s *scale, s *abnrm, s *rconde, s *rcondv, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void cgehd2(int *n, int *ilo, int *ihi, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil +cdef void cgehrd(int *n, int *ilo, int *ihi, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cgelq2(int *m, int *n, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil +cdef void cgelqf(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cgels(char *trans, int *m, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, c *work, int *lwork, int *info) noexcept nogil +cdef void cgelsd(int *m, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, s *s, s *rcond, int *rank, c *work, int *lwork, s *rwork, int *iwork, int *info) noexcept nogil +cdef void cgelss(int *m, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, s *s, s *rcond, int *rank, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void cgelsy(int *m, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, int *jpvt, s *rcond, int *rank, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void cgemqrt(char *side, char *trans, int *m, int *n, int *k, int *nb, c *v, int *ldv, c *t, int *ldt, c *c, int *ldc, c *work, int *info) noexcept nogil +cdef void cgeql2(int *m, int *n, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil +cdef void cgeqlf(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cgeqp3(int *m, int *n, c *a, int *lda, int *jpvt, c *tau, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void cgeqr2(int *m, int *n, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil +cdef void cgeqr2p(int *m, int *n, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil +cdef void cgeqrf(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cgeqrfp(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cgeqrt(int *m, int *n, int *nb, c *a, int *lda, c *t, int *ldt, c *work, int *info) noexcept nogil +cdef void cgeqrt2(int *m, int *n, c *a, int *lda, c *t, int *ldt, int *info) noexcept nogil +cdef void cgeqrt3(int *m, int *n, c *a, int *lda, c *t, int *ldt, int *info) noexcept nogil +cdef void cgerfs(char *trans, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cgerq2(int *m, int *n, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil +cdef void cgerqf(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cgesc2(int *n, c *a, int *lda, c *rhs, int *ipiv, int *jpiv, s *scale) noexcept nogil +cdef void cgesdd(char *jobz, int *m, int *n, c *a, int *lda, s *s, c *u, int *ldu, c *vt, int *ldvt, c *work, int *lwork, s *rwork, int *iwork, int *info) noexcept nogil +cdef void cgesv(int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void cgesvd(char *jobu, char *jobvt, int *m, int *n, c *a, int *lda, s *s, c *u, int *ldu, c *vt, int *ldvt, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void cgesvx(char *fact, char *trans, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, char *equed, s *r, s *c, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cgetc2(int *n, c *a, int *lda, int *ipiv, int *jpiv, int *info) noexcept nogil +cdef void cgetf2(int *m, int *n, c *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void cgetrf(int *m, int *n, c *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void cgetri(int *n, c *a, int *lda, int *ipiv, c *work, int *lwork, int *info) noexcept nogil +cdef void cgetrs(char *trans, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void cggbak(char *job, char *side, int *n, int *ilo, int *ihi, s *lscale, s *rscale, int *m, c *v, int *ldv, int *info) noexcept nogil +cdef void cggbal(char *job, int *n, c *a, int *lda, c *b, int *ldb, int *ilo, int *ihi, s *lscale, s *rscale, s *work, int *info) noexcept nogil +cdef void cgges(char *jobvsl, char *jobvsr, char *sort, cselect2 *selctg, int *n, c *a, int *lda, c *b, int *ldb, int *sdim, c *alpha, c *beta, c *vsl, int *ldvsl, c *vsr, int *ldvsr, c *work, int *lwork, s *rwork, bint *bwork, int *info) noexcept nogil +cdef void cggesx(char *jobvsl, char *jobvsr, char *sort, cselect2 *selctg, char *sense, int *n, c *a, int *lda, c *b, int *ldb, int *sdim, c *alpha, c *beta, c *vsl, int *ldvsl, c *vsr, int *ldvsr, s *rconde, s *rcondv, c *work, int *lwork, s *rwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil +cdef void cggev(char *jobvl, char *jobvr, int *n, c *a, int *lda, c *b, int *ldb, c *alpha, c *beta, c *vl, int *ldvl, c *vr, int *ldvr, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void cggevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, c *a, int *lda, c *b, int *ldb, c *alpha, c *beta, c *vl, int *ldvl, c *vr, int *ldvr, int *ilo, int *ihi, s *lscale, s *rscale, s *abnrm, s *bbnrm, s *rconde, s *rcondv, c *work, int *lwork, s *rwork, int *iwork, bint *bwork, int *info) noexcept nogil +cdef void cggglm(int *n, int *m, int *p, c *a, int *lda, c *b, int *ldb, c *d, c *x, c *y, c *work, int *lwork, int *info) noexcept nogil +cdef void cgghrd(char *compq, char *compz, int *n, int *ilo, int *ihi, c *a, int *lda, c *b, int *ldb, c *q, int *ldq, c *z, int *ldz, int *info) noexcept nogil +cdef void cgglse(int *m, int *n, int *p, c *a, int *lda, c *b, int *ldb, c *c, c *d, c *x, c *work, int *lwork, int *info) noexcept nogil +cdef void cggqrf(int *n, int *m, int *p, c *a, int *lda, c *taua, c *b, int *ldb, c *taub, c *work, int *lwork, int *info) noexcept nogil +cdef void cggrqf(int *m, int *p, int *n, c *a, int *lda, c *taua, c *b, int *ldb, c *taub, c *work, int *lwork, int *info) noexcept nogil +cdef void cgtcon(char *norm, int *n, c *dl, c *d, c *du, c *du2, int *ipiv, s *anorm, s *rcond, c *work, int *info) noexcept nogil +cdef void cgtrfs(char *trans, int *n, int *nrhs, c *dl, c *d, c *du, c *dlf, c *df, c *duf, c *du2, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cgtsv(int *n, int *nrhs, c *dl, c *d, c *du, c *b, int *ldb, int *info) noexcept nogil +cdef void cgtsvx(char *fact, char *trans, int *n, int *nrhs, c *dl, c *d, c *du, c *dlf, c *df, c *duf, c *du2, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cgttrf(int *n, c *dl, c *d, c *du, c *du2, int *ipiv, int *info) noexcept nogil +cdef void cgttrs(char *trans, int *n, int *nrhs, c *dl, c *d, c *du, c *du2, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void cgtts2(int *itrans, int *n, int *nrhs, c *dl, c *d, c *du, c *du2, int *ipiv, c *b, int *ldb) noexcept nogil +cdef void chbev(char *jobz, char *uplo, int *n, int *kd, c *ab, int *ldab, s *w, c *z, int *ldz, c *work, s *rwork, int *info) noexcept nogil +cdef void chbevd(char *jobz, char *uplo, int *n, int *kd, c *ab, int *ldab, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void chbevx(char *jobz, char *range, char *uplo, int *n, int *kd, c *ab, int *ldab, c *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void chbgst(char *vect, char *uplo, int *n, int *ka, int *kb, c *ab, int *ldab, c *bb, int *ldbb, c *x, int *ldx, c *work, s *rwork, int *info) noexcept nogil +cdef void chbgv(char *jobz, char *uplo, int *n, int *ka, int *kb, c *ab, int *ldab, c *bb, int *ldbb, s *w, c *z, int *ldz, c *work, s *rwork, int *info) noexcept nogil +cdef void chbgvd(char *jobz, char *uplo, int *n, int *ka, int *kb, c *ab, int *ldab, c *bb, int *ldbb, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void chbgvx(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, c *ab, int *ldab, c *bb, int *ldbb, c *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void chbtrd(char *vect, char *uplo, int *n, int *kd, c *ab, int *ldab, s *d, s *e, c *q, int *ldq, c *work, int *info) noexcept nogil +cdef void checon(char *uplo, int *n, c *a, int *lda, int *ipiv, s *anorm, s *rcond, c *work, int *info) noexcept nogil +cdef void cheequb(char *uplo, int *n, c *a, int *lda, s *s, s *scond, s *amax, c *work, int *info) noexcept nogil +cdef void cheev(char *jobz, char *uplo, int *n, c *a, int *lda, s *w, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void cheevd(char *jobz, char *uplo, int *n, c *a, int *lda, s *w, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void cheevr(char *jobz, char *range, char *uplo, int *n, c *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, int *isuppz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void cheevx(char *jobz, char *range, char *uplo, int *n, c *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void chegs2(int *itype, char *uplo, int *n, c *a, int *lda, c *b, int *ldb, int *info) noexcept nogil +cdef void chegst(int *itype, char *uplo, int *n, c *a, int *lda, c *b, int *ldb, int *info) noexcept nogil +cdef void chegv(int *itype, char *jobz, char *uplo, int *n, c *a, int *lda, c *b, int *ldb, s *w, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void chegvd(int *itype, char *jobz, char *uplo, int *n, c *a, int *lda, c *b, int *ldb, s *w, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void chegvx(int *itype, char *jobz, char *range, char *uplo, int *n, c *a, int *lda, c *b, int *ldb, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void cherfs(char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void chesv(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, c *work, int *lwork, int *info) noexcept nogil +cdef void chesvx(char *fact, char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void cheswapr(char *uplo, int *n, c *a, int *lda, int *i1, int *i2) noexcept nogil +cdef void chetd2(char *uplo, int *n, c *a, int *lda, s *d, s *e, c *tau, int *info) noexcept nogil +cdef void chetf2(char *uplo, int *n, c *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void chetrd(char *uplo, int *n, c *a, int *lda, s *d, s *e, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void chetrf(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *lwork, int *info) noexcept nogil +cdef void chetri(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *info) noexcept nogil +cdef void chetri2(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *lwork, int *info) noexcept nogil +cdef void chetri2x(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *nb, int *info) noexcept nogil +cdef void chetrs(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void chetrs2(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, c *work, int *info) noexcept nogil +cdef void chfrk(char *transr, char *uplo, char *trans, int *n, int *k, s *alpha, c *a, int *lda, s *beta, c *c) noexcept nogil +cdef void chgeqz(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, c *h, int *ldh, c *t, int *ldt, c *alpha, c *beta, c *q, int *ldq, c *z, int *ldz, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef char chla_transtype(int *trans) noexcept nogil +cdef void chpcon(char *uplo, int *n, c *ap, int *ipiv, s *anorm, s *rcond, c *work, int *info) noexcept nogil +cdef void chpev(char *jobz, char *uplo, int *n, c *ap, s *w, c *z, int *ldz, c *work, s *rwork, int *info) noexcept nogil +cdef void chpevd(char *jobz, char *uplo, int *n, c *ap, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void chpevx(char *jobz, char *range, char *uplo, int *n, c *ap, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void chpgst(int *itype, char *uplo, int *n, c *ap, c *bp, int *info) noexcept nogil +cdef void chpgv(int *itype, char *jobz, char *uplo, int *n, c *ap, c *bp, s *w, c *z, int *ldz, c *work, s *rwork, int *info) noexcept nogil +cdef void chpgvd(int *itype, char *jobz, char *uplo, int *n, c *ap, c *bp, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void chpgvx(int *itype, char *jobz, char *range, char *uplo, int *n, c *ap, c *bp, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void chprfs(char *uplo, int *n, int *nrhs, c *ap, c *afp, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void chpsv(char *uplo, int *n, int *nrhs, c *ap, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void chpsvx(char *fact, char *uplo, int *n, int *nrhs, c *ap, c *afp, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void chptrd(char *uplo, int *n, c *ap, s *d, s *e, c *tau, int *info) noexcept nogil +cdef void chptrf(char *uplo, int *n, c *ap, int *ipiv, int *info) noexcept nogil +cdef void chptri(char *uplo, int *n, c *ap, int *ipiv, c *work, int *info) noexcept nogil +cdef void chptrs(char *uplo, int *n, int *nrhs, c *ap, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void chsein(char *side, char *eigsrc, char *initv, bint *select, int *n, c *h, int *ldh, c *w, c *vl, int *ldvl, c *vr, int *ldvr, int *mm, int *m, c *work, s *rwork, int *ifaill, int *ifailr, int *info) noexcept nogil +cdef void chseqr(char *job, char *compz, int *n, int *ilo, int *ihi, c *h, int *ldh, c *w, c *z, int *ldz, c *work, int *lwork, int *info) noexcept nogil +cdef void clabrd(int *m, int *n, int *nb, c *a, int *lda, s *d, s *e, c *tauq, c *taup, c *x, int *ldx, c *y, int *ldy) noexcept nogil +cdef void clacgv(int *n, c *x, int *incx) noexcept nogil +cdef void clacn2(int *n, c *v, c *x, s *est, int *kase, int *isave) noexcept nogil +cdef void clacon(int *n, c *v, c *x, s *est, int *kase) noexcept nogil +cdef void clacp2(char *uplo, int *m, int *n, s *a, int *lda, c *b, int *ldb) noexcept nogil +cdef void clacpy(char *uplo, int *m, int *n, c *a, int *lda, c *b, int *ldb) noexcept nogil +cdef void clacrm(int *m, int *n, c *a, int *lda, s *b, int *ldb, c *c, int *ldc, s *rwork) noexcept nogil +cdef void clacrt(int *n, c *cx, int *incx, c *cy, int *incy, c *c, c *s) noexcept nogil +cdef c cladiv(c *x, c *y) noexcept nogil +cdef void claed0(int *qsiz, int *n, s *d, s *e, c *q, int *ldq, c *qstore, int *ldqs, s *rwork, int *iwork, int *info) noexcept nogil +cdef void claed7(int *n, int *cutpnt, int *qsiz, int *tlvls, int *curlvl, int *curpbm, s *d, c *q, int *ldq, s *rho, int *indxq, s *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, s *givnum, c *work, s *rwork, int *iwork, int *info) noexcept nogil +cdef void claed8(int *k, int *n, int *qsiz, c *q, int *ldq, s *d, s *rho, int *cutpnt, s *z, s *dlamda, c *q2, int *ldq2, s *w, int *indxp, int *indx, int *indxq, int *perm, int *givptr, int *givcol, s *givnum, int *info) noexcept nogil +cdef void claein(bint *rightv, bint *noinit, int *n, c *h, int *ldh, c *w, c *v, c *b, int *ldb, s *rwork, s *eps3, s *smlnum, int *info) noexcept nogil +cdef void claesy(c *a, c *b, c *c, c *rt1, c *rt2, c *evscal, c *cs1, c *sn1) noexcept nogil +cdef void claev2(c *a, c *b, c *c, s *rt1, s *rt2, s *cs1, c *sn1) noexcept nogil +cdef void clag2z(int *m, int *n, c *sa, int *ldsa, z *a, int *lda, int *info) noexcept nogil +cdef void clags2(bint *upper, s *a1, c *a2, s *a3, s *b1, c *b2, s *b3, s *csu, c *snu, s *csv, c *snv, s *csq, c *snq) noexcept nogil +cdef void clagtm(char *trans, int *n, int *nrhs, s *alpha, c *dl, c *d, c *du, c *x, int *ldx, s *beta, c *b, int *ldb) noexcept nogil +cdef void clahef(char *uplo, int *n, int *nb, int *kb, c *a, int *lda, int *ipiv, c *w, int *ldw, int *info) noexcept nogil +cdef void clahqr(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, c *h, int *ldh, c *w, int *iloz, int *ihiz, c *z, int *ldz, int *info) noexcept nogil +cdef void clahr2(int *n, int *k, int *nb, c *a, int *lda, c *tau, c *t, int *ldt, c *y, int *ldy) noexcept nogil +cdef void claic1(int *job, int *j, c *x, s *sest, c *w, c *gamma, s *sestpr, c *s, c *c) noexcept nogil +cdef void clals0(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, c *b, int *ldb, c *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *poles, s *difl, s *difr, s *z, int *k, s *c, s *s, s *rwork, int *info) noexcept nogil +cdef void clalsa(int *icompq, int *smlsiz, int *n, int *nrhs, c *b, int *ldb, c *bx, int *ldbx, s *u, int *ldu, s *vt, int *k, s *difl, s *difr, s *z, s *poles, int *givptr, int *givcol, int *ldgcol, int *perm, s *givnum, s *c, s *s, s *rwork, int *iwork, int *info) noexcept nogil +cdef void clalsd(char *uplo, int *smlsiz, int *n, int *nrhs, s *d, s *e, c *b, int *ldb, s *rcond, int *rank, c *work, s *rwork, int *iwork, int *info) noexcept nogil +cdef s clangb(char *norm, int *n, int *kl, int *ku, c *ab, int *ldab, s *work) noexcept nogil +cdef s clange(char *norm, int *m, int *n, c *a, int *lda, s *work) noexcept nogil +cdef s clangt(char *norm, int *n, c *dl, c *d, c *du) noexcept nogil +cdef s clanhb(char *norm, char *uplo, int *n, int *k, c *ab, int *ldab, s *work) noexcept nogil +cdef s clanhe(char *norm, char *uplo, int *n, c *a, int *lda, s *work) noexcept nogil +cdef s clanhf(char *norm, char *transr, char *uplo, int *n, c *a, s *work) noexcept nogil +cdef s clanhp(char *norm, char *uplo, int *n, c *ap, s *work) noexcept nogil +cdef s clanhs(char *norm, int *n, c *a, int *lda, s *work) noexcept nogil +cdef s clanht(char *norm, int *n, s *d, c *e) noexcept nogil +cdef s clansb(char *norm, char *uplo, int *n, int *k, c *ab, int *ldab, s *work) noexcept nogil +cdef s clansp(char *norm, char *uplo, int *n, c *ap, s *work) noexcept nogil +cdef s clansy(char *norm, char *uplo, int *n, c *a, int *lda, s *work) noexcept nogil +cdef s clantb(char *norm, char *uplo, char *diag, int *n, int *k, c *ab, int *ldab, s *work) noexcept nogil +cdef s clantp(char *norm, char *uplo, char *diag, int *n, c *ap, s *work) noexcept nogil +cdef s clantr(char *norm, char *uplo, char *diag, int *m, int *n, c *a, int *lda, s *work) noexcept nogil +cdef void clapll(int *n, c *x, int *incx, c *y, int *incy, s *ssmin) noexcept nogil +cdef void clapmr(bint *forwrd, int *m, int *n, c *x, int *ldx, int *k) noexcept nogil +cdef void clapmt(bint *forwrd, int *m, int *n, c *x, int *ldx, int *k) noexcept nogil +cdef void claqgb(int *m, int *n, int *kl, int *ku, c *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) noexcept nogil +cdef void claqge(int *m, int *n, c *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) noexcept nogil +cdef void claqhb(char *uplo, int *n, int *kd, c *ab, int *ldab, s *s, s *scond, s *amax, char *equed) noexcept nogil +cdef void claqhe(char *uplo, int *n, c *a, int *lda, s *s, s *scond, s *amax, char *equed) noexcept nogil +cdef void claqhp(char *uplo, int *n, c *ap, s *s, s *scond, s *amax, char *equed) noexcept nogil +cdef void claqp2(int *m, int *n, int *offset, c *a, int *lda, int *jpvt, c *tau, s *vn1, s *vn2, c *work) noexcept nogil +cdef void claqps(int *m, int *n, int *offset, int *nb, int *kb, c *a, int *lda, int *jpvt, c *tau, s *vn1, s *vn2, c *auxv, c *f, int *ldf) noexcept nogil +cdef void claqr0(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, c *h, int *ldh, c *w, int *iloz, int *ihiz, c *z, int *ldz, c *work, int *lwork, int *info) noexcept nogil +cdef void claqr1(int *n, c *h, int *ldh, c *s1, c *s2, c *v) noexcept nogil +cdef void claqr2(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, c *h, int *ldh, int *iloz, int *ihiz, c *z, int *ldz, int *ns, int *nd, c *sh, c *v, int *ldv, int *nh, c *t, int *ldt, int *nv, c *wv, int *ldwv, c *work, int *lwork) noexcept nogil +cdef void claqr3(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, c *h, int *ldh, int *iloz, int *ihiz, c *z, int *ldz, int *ns, int *nd, c *sh, c *v, int *ldv, int *nh, c *t, int *ldt, int *nv, c *wv, int *ldwv, c *work, int *lwork) noexcept nogil +cdef void claqr4(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, c *h, int *ldh, c *w, int *iloz, int *ihiz, c *z, int *ldz, c *work, int *lwork, int *info) noexcept nogil +cdef void claqr5(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, c *s, c *h, int *ldh, int *iloz, int *ihiz, c *z, int *ldz, c *v, int *ldv, c *u, int *ldu, int *nv, c *wv, int *ldwv, int *nh, c *wh, int *ldwh) noexcept nogil +cdef void claqsb(char *uplo, int *n, int *kd, c *ab, int *ldab, s *s, s *scond, s *amax, char *equed) noexcept nogil +cdef void claqsp(char *uplo, int *n, c *ap, s *s, s *scond, s *amax, char *equed) noexcept nogil +cdef void claqsy(char *uplo, int *n, c *a, int *lda, s *s, s *scond, s *amax, char *equed) noexcept nogil +cdef void clar1v(int *n, int *b1, int *bn, s *lambda_, s *d, s *l, s *ld, s *lld, s *pivmin, s *gaptol, c *z, bint *wantnc, int *negcnt, s *ztz, s *mingma, int *r, int *isuppz, s *nrminv, s *resid, s *rqcorr, s *work) noexcept nogil +cdef void clar2v(int *n, c *x, c *y, c *z, int *incx, s *c, c *s, int *incc) noexcept nogil +cdef void clarcm(int *m, int *n, s *a, int *lda, c *b, int *ldb, c *c, int *ldc, s *rwork) noexcept nogil +cdef void clarf(char *side, int *m, int *n, c *v, int *incv, c *tau, c *c, int *ldc, c *work) noexcept nogil +cdef void clarfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, c *v, int *ldv, c *t, int *ldt, c *c, int *ldc, c *work, int *ldwork) noexcept nogil +cdef void clarfg(int *n, c *alpha, c *x, int *incx, c *tau) noexcept nogil +cdef void clarfgp(int *n, c *alpha, c *x, int *incx, c *tau) noexcept nogil +cdef void clarft(char *direct, char *storev, int *n, int *k, c *v, int *ldv, c *tau, c *t, int *ldt) noexcept nogil +cdef void clarfx(char *side, int *m, int *n, c *v, c *tau, c *c, int *ldc, c *work) noexcept nogil +cdef void clargv(int *n, c *x, int *incx, c *y, int *incy, s *c, int *incc) noexcept nogil +cdef void clarnv(int *idist, int *iseed, int *n, c *x) noexcept nogil +cdef void clarrv(int *n, s *vl, s *vu, s *d, s *l, s *pivmin, int *isplit, int *m, int *dol, int *dou, s *minrgp, s *rtol1, s *rtol2, s *w, s *werr, s *wgap, int *iblock, int *indexw, s *gers, c *z, int *ldz, int *isuppz, s *work, int *iwork, int *info) noexcept nogil +cdef void clartg(c *f, c *g, s *cs, c *sn, c *r) noexcept nogil +cdef void clartv(int *n, c *x, int *incx, c *y, int *incy, s *c, c *s, int *incc) noexcept nogil +cdef void clarz(char *side, int *m, int *n, int *l, c *v, int *incv, c *tau, c *c, int *ldc, c *work) noexcept nogil +cdef void clarzb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, c *v, int *ldv, c *t, int *ldt, c *c, int *ldc, c *work, int *ldwork) noexcept nogil +cdef void clarzt(char *direct, char *storev, int *n, int *k, c *v, int *ldv, c *tau, c *t, int *ldt) noexcept nogil +cdef void clascl(char *type_bn, int *kl, int *ku, s *cfrom, s *cto, int *m, int *n, c *a, int *lda, int *info) noexcept nogil +cdef void claset(char *uplo, int *m, int *n, c *alpha, c *beta, c *a, int *lda) noexcept nogil +cdef void clasr(char *side, char *pivot, char *direct, int *m, int *n, s *c, s *s, c *a, int *lda) noexcept nogil +cdef void classq(int *n, c *x, int *incx, s *scale, s *sumsq) noexcept nogil +cdef void claswp(int *n, c *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) noexcept nogil +cdef void clasyf(char *uplo, int *n, int *nb, int *kb, c *a, int *lda, int *ipiv, c *w, int *ldw, int *info) noexcept nogil +cdef void clatbs(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, c *ab, int *ldab, c *x, s *scale, s *cnorm, int *info) noexcept nogil +cdef void clatdf(int *ijob, int *n, c *z, int *ldz, c *rhs, s *rdsum, s *rdscal, int *ipiv, int *jpiv) noexcept nogil +cdef void clatps(char *uplo, char *trans, char *diag, char *normin, int *n, c *ap, c *x, s *scale, s *cnorm, int *info) noexcept nogil +cdef void clatrd(char *uplo, int *n, int *nb, c *a, int *lda, s *e, c *tau, c *w, int *ldw) noexcept nogil +cdef void clatrs(char *uplo, char *trans, char *diag, char *normin, int *n, c *a, int *lda, c *x, s *scale, s *cnorm, int *info) noexcept nogil +cdef void clatrz(int *m, int *n, int *l, c *a, int *lda, c *tau, c *work) noexcept nogil +cdef void clauu2(char *uplo, int *n, c *a, int *lda, int *info) noexcept nogil +cdef void clauum(char *uplo, int *n, c *a, int *lda, int *info) noexcept nogil +cdef void cpbcon(char *uplo, int *n, int *kd, c *ab, int *ldab, s *anorm, s *rcond, c *work, s *rwork, int *info) noexcept nogil +cdef void cpbequ(char *uplo, int *n, int *kd, c *ab, int *ldab, s *s, s *scond, s *amax, int *info) noexcept nogil +cdef void cpbrfs(char *uplo, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *afb, int *ldafb, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cpbstf(char *uplo, int *n, int *kd, c *ab, int *ldab, int *info) noexcept nogil +cdef void cpbsv(char *uplo, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *b, int *ldb, int *info) noexcept nogil +cdef void cpbsvx(char *fact, char *uplo, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *afb, int *ldafb, char *equed, s *s, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cpbtf2(char *uplo, int *n, int *kd, c *ab, int *ldab, int *info) noexcept nogil +cdef void cpbtrf(char *uplo, int *n, int *kd, c *ab, int *ldab, int *info) noexcept nogil +cdef void cpbtrs(char *uplo, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *b, int *ldb, int *info) noexcept nogil +cdef void cpftrf(char *transr, char *uplo, int *n, c *a, int *info) noexcept nogil +cdef void cpftri(char *transr, char *uplo, int *n, c *a, int *info) noexcept nogil +cdef void cpftrs(char *transr, char *uplo, int *n, int *nrhs, c *a, c *b, int *ldb, int *info) noexcept nogil +cdef void cpocon(char *uplo, int *n, c *a, int *lda, s *anorm, s *rcond, c *work, s *rwork, int *info) noexcept nogil +cdef void cpoequ(int *n, c *a, int *lda, s *s, s *scond, s *amax, int *info) noexcept nogil +cdef void cpoequb(int *n, c *a, int *lda, s *s, s *scond, s *amax, int *info) noexcept nogil +cdef void cporfs(char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cposv(char *uplo, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, int *info) noexcept nogil +cdef void cposvx(char *fact, char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, char *equed, s *s, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cpotf2(char *uplo, int *n, c *a, int *lda, int *info) noexcept nogil +cdef void cpotrf(char *uplo, int *n, c *a, int *lda, int *info) noexcept nogil +cdef void cpotri(char *uplo, int *n, c *a, int *lda, int *info) noexcept nogil +cdef void cpotrs(char *uplo, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, int *info) noexcept nogil +cdef void cppcon(char *uplo, int *n, c *ap, s *anorm, s *rcond, c *work, s *rwork, int *info) noexcept nogil +cdef void cppequ(char *uplo, int *n, c *ap, s *s, s *scond, s *amax, int *info) noexcept nogil +cdef void cpprfs(char *uplo, int *n, int *nrhs, c *ap, c *afp, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cppsv(char *uplo, int *n, int *nrhs, c *ap, c *b, int *ldb, int *info) noexcept nogil +cdef void cppsvx(char *fact, char *uplo, int *n, int *nrhs, c *ap, c *afp, char *equed, s *s, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cpptrf(char *uplo, int *n, c *ap, int *info) noexcept nogil +cdef void cpptri(char *uplo, int *n, c *ap, int *info) noexcept nogil +cdef void cpptrs(char *uplo, int *n, int *nrhs, c *ap, c *b, int *ldb, int *info) noexcept nogil +cdef void cpstf2(char *uplo, int *n, c *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) noexcept nogil +cdef void cpstrf(char *uplo, int *n, c *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) noexcept nogil +cdef void cptcon(int *n, s *d, c *e, s *anorm, s *rcond, s *rwork, int *info) noexcept nogil +cdef void cpteqr(char *compz, int *n, s *d, s *e, c *z, int *ldz, s *work, int *info) noexcept nogil +cdef void cptrfs(char *uplo, int *n, int *nrhs, s *d, c *e, s *df, c *ef, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cptsv(int *n, int *nrhs, s *d, c *e, c *b, int *ldb, int *info) noexcept nogil +cdef void cptsvx(char *fact, int *n, int *nrhs, s *d, c *e, s *df, c *ef, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cpttrf(int *n, s *d, c *e, int *info) noexcept nogil +cdef void cpttrs(char *uplo, int *n, int *nrhs, s *d, c *e, c *b, int *ldb, int *info) noexcept nogil +cdef void cptts2(int *iuplo, int *n, int *nrhs, s *d, c *e, c *b, int *ldb) noexcept nogil +cdef void crot(int *n, c *cx, int *incx, c *cy, int *incy, s *c, c *s) noexcept nogil +cdef void cspcon(char *uplo, int *n, c *ap, int *ipiv, s *anorm, s *rcond, c *work, int *info) noexcept nogil +cdef void cspmv(char *uplo, int *n, c *alpha, c *ap, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil +cdef void cspr(char *uplo, int *n, c *alpha, c *x, int *incx, c *ap) noexcept nogil +cdef void csprfs(char *uplo, int *n, int *nrhs, c *ap, c *afp, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void cspsv(char *uplo, int *n, int *nrhs, c *ap, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void cspsvx(char *fact, char *uplo, int *n, int *nrhs, c *ap, c *afp, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void csptrf(char *uplo, int *n, c *ap, int *ipiv, int *info) noexcept nogil +cdef void csptri(char *uplo, int *n, c *ap, int *ipiv, c *work, int *info) noexcept nogil +cdef void csptrs(char *uplo, int *n, int *nrhs, c *ap, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void csrscl(int *n, s *sa, c *sx, int *incx) noexcept nogil +cdef void cstedc(char *compz, int *n, s *d, s *e, c *z, int *ldz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void cstegr(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void cstein(int *n, s *d, s *e, int *m, s *w, int *iblock, int *isplit, c *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void cstemr(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, int *m, s *w, c *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void csteqr(char *compz, int *n, s *d, s *e, c *z, int *ldz, s *work, int *info) noexcept nogil +cdef void csycon(char *uplo, int *n, c *a, int *lda, int *ipiv, s *anorm, s *rcond, c *work, int *info) noexcept nogil +cdef void csyconv(char *uplo, char *way, int *n, c *a, int *lda, int *ipiv, c *work, int *info) noexcept nogil +cdef void csyequb(char *uplo, int *n, c *a, int *lda, s *s, s *scond, s *amax, c *work, int *info) noexcept nogil +cdef void csymv(char *uplo, int *n, c *alpha, c *a, int *lda, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil +cdef void csyr(char *uplo, int *n, c *alpha, c *x, int *incx, c *a, int *lda) noexcept nogil +cdef void csyrfs(char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void csysv(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, c *work, int *lwork, int *info) noexcept nogil +cdef void csysvx(char *fact, char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, int *lwork, s *rwork, int *info) noexcept nogil +cdef void csyswapr(char *uplo, int *n, c *a, int *lda, int *i1, int *i2) noexcept nogil +cdef void csytf2(char *uplo, int *n, c *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void csytrf(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *lwork, int *info) noexcept nogil +cdef void csytri(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *info) noexcept nogil +cdef void csytri2(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *lwork, int *info) noexcept nogil +cdef void csytri2x(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *nb, int *info) noexcept nogil +cdef void csytrs(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, int *info) noexcept nogil +cdef void csytrs2(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, c *work, int *info) noexcept nogil +cdef void ctbcon(char *norm, char *uplo, char *diag, int *n, int *kd, c *ab, int *ldab, s *rcond, c *work, s *rwork, int *info) noexcept nogil +cdef void ctbrfs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void ctbtrs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *b, int *ldb, int *info) noexcept nogil +cdef void ctfsm(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, c *alpha, c *a, c *b, int *ldb) noexcept nogil +cdef void ctftri(char *transr, char *uplo, char *diag, int *n, c *a, int *info) noexcept nogil +cdef void ctfttp(char *transr, char *uplo, int *n, c *arf, c *ap, int *info) noexcept nogil +cdef void ctfttr(char *transr, char *uplo, int *n, c *arf, c *a, int *lda, int *info) noexcept nogil +cdef void ctgevc(char *side, char *howmny, bint *select, int *n, c *s, int *lds, c *p, int *ldp, c *vl, int *ldvl, c *vr, int *ldvr, int *mm, int *m, c *work, s *rwork, int *info) noexcept nogil +cdef void ctgex2(bint *wantq, bint *wantz, int *n, c *a, int *lda, c *b, int *ldb, c *q, int *ldq, c *z, int *ldz, int *j1, int *info) noexcept nogil +cdef void ctgexc(bint *wantq, bint *wantz, int *n, c *a, int *lda, c *b, int *ldb, c *q, int *ldq, c *z, int *ldz, int *ifst, int *ilst, int *info) noexcept nogil +cdef void ctgsen(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, c *a, int *lda, c *b, int *ldb, c *alpha, c *beta, c *q, int *ldq, c *z, int *ldz, int *m, s *pl, s *pr, s *dif, c *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void ctgsja(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, c *a, int *lda, c *b, int *ldb, s *tola, s *tolb, s *alpha, s *beta, c *u, int *ldu, c *v, int *ldv, c *q, int *ldq, c *work, int *ncycle, int *info) noexcept nogil +cdef void ctgsna(char *job, char *howmny, bint *select, int *n, c *a, int *lda, c *b, int *ldb, c *vl, int *ldvl, c *vr, int *ldvr, s *s, s *dif, int *mm, int *m, c *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void ctgsy2(char *trans, int *ijob, int *m, int *n, c *a, int *lda, c *b, int *ldb, c *c, int *ldc, c *d, int *ldd, c *e, int *lde, c *f, int *ldf, s *scale, s *rdsum, s *rdscal, int *info) noexcept nogil +cdef void ctgsyl(char *trans, int *ijob, int *m, int *n, c *a, int *lda, c *b, int *ldb, c *c, int *ldc, c *d, int *ldd, c *e, int *lde, c *f, int *ldf, s *scale, s *dif, c *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void ctpcon(char *norm, char *uplo, char *diag, int *n, c *ap, s *rcond, c *work, s *rwork, int *info) noexcept nogil +cdef void ctpmqrt(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, c *v, int *ldv, c *t, int *ldt, c *a, int *lda, c *b, int *ldb, c *work, int *info) noexcept nogil +cdef void ctpqrt(int *m, int *n, int *l, int *nb, c *a, int *lda, c *b, int *ldb, c *t, int *ldt, c *work, int *info) noexcept nogil +cdef void ctpqrt2(int *m, int *n, int *l, c *a, int *lda, c *b, int *ldb, c *t, int *ldt, int *info) noexcept nogil +cdef void ctprfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, c *v, int *ldv, c *t, int *ldt, c *a, int *lda, c *b, int *ldb, c *work, int *ldwork) noexcept nogil +cdef void ctprfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, c *ap, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void ctptri(char *uplo, char *diag, int *n, c *ap, int *info) noexcept nogil +cdef void ctptrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, c *ap, c *b, int *ldb, int *info) noexcept nogil +cdef void ctpttf(char *transr, char *uplo, int *n, c *ap, c *arf, int *info) noexcept nogil +cdef void ctpttr(char *uplo, int *n, c *ap, c *a, int *lda, int *info) noexcept nogil +cdef void ctrcon(char *norm, char *uplo, char *diag, int *n, c *a, int *lda, s *rcond, c *work, s *rwork, int *info) noexcept nogil +cdef void ctrevc(char *side, char *howmny, bint *select, int *n, c *t, int *ldt, c *vl, int *ldvl, c *vr, int *ldvr, int *mm, int *m, c *work, s *rwork, int *info) noexcept nogil +cdef void ctrexc(char *compq, int *n, c *t, int *ldt, c *q, int *ldq, int *ifst, int *ilst, int *info) noexcept nogil +cdef void ctrrfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil +cdef void ctrsen(char *job, char *compq, bint *select, int *n, c *t, int *ldt, c *q, int *ldq, c *w, int *m, s *s, s *sep, c *work, int *lwork, int *info) noexcept nogil +cdef void ctrsna(char *job, char *howmny, bint *select, int *n, c *t, int *ldt, c *vl, int *ldvl, c *vr, int *ldvr, s *s, s *sep, int *mm, int *m, c *work, int *ldwork, s *rwork, int *info) noexcept nogil +cdef void ctrsyl(char *trana, char *tranb, int *isgn, int *m, int *n, c *a, int *lda, c *b, int *ldb, c *c, int *ldc, s *scale, int *info) noexcept nogil +cdef void ctrti2(char *uplo, char *diag, int *n, c *a, int *lda, int *info) noexcept nogil +cdef void ctrtri(char *uplo, char *diag, int *n, c *a, int *lda, int *info) noexcept nogil +cdef void ctrtrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, int *info) noexcept nogil +cdef void ctrttf(char *transr, char *uplo, int *n, c *a, int *lda, c *arf, int *info) noexcept nogil +cdef void ctrttp(char *uplo, int *n, c *a, int *lda, c *ap, int *info) noexcept nogil +cdef void ctzrzf(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cunbdb(char *trans, char *signs, int *m, int *p, int *q, c *x11, int *ldx11, c *x12, int *ldx12, c *x21, int *ldx21, c *x22, int *ldx22, s *theta, s *phi, c *taup1, c *taup2, c *tauq1, c *tauq2, c *work, int *lwork, int *info) noexcept nogil +cdef void cuncsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, c *x11, int *ldx11, c *x12, int *ldx12, c *x21, int *ldx21, c *x22, int *ldx22, s *theta, c *u1, int *ldu1, c *u2, int *ldu2, c *v1t, int *ldv1t, c *v2t, int *ldv2t, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *info) noexcept nogil +cdef void cung2l(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil +cdef void cung2r(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil +cdef void cungbr(char *vect, int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cunghr(int *n, int *ilo, int *ihi, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cungl2(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil +cdef void cunglq(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cungql(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cungqr(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cungr2(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil +cdef void cungrq(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cungtr(char *uplo, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil +cdef void cunm2l(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil +cdef void cunm2r(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil +cdef void cunmbr(char *vect, char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil +cdef void cunmhr(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil +cdef void cunml2(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil +cdef void cunmlq(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil +cdef void cunmql(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil +cdef void cunmqr(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil +cdef void cunmr2(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil +cdef void cunmr3(char *side, char *trans, int *m, int *n, int *k, int *l, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil +cdef void cunmrq(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil +cdef void cunmrz(char *side, char *trans, int *m, int *n, int *k, int *l, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil +cdef void cunmtr(char *side, char *uplo, char *trans, int *m, int *n, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil +cdef void cupgtr(char *uplo, int *n, c *ap, c *tau, c *q, int *ldq, c *work, int *info) noexcept nogil +cdef void cupmtr(char *side, char *uplo, char *trans, int *m, int *n, c *ap, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil +cdef void dbbcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, d *theta, d *phi, d *u1, int *ldu1, d *u2, int *ldu2, d *v1t, int *ldv1t, d *v2t, int *ldv2t, d *b11d, d *b11e, d *b12d, d *b12e, d *b21d, d *b21e, d *b22d, d *b22e, d *work, int *lwork, int *info) noexcept nogil +cdef void dbdsdc(char *uplo, char *compq, int *n, d *d, d *e, d *u, int *ldu, d *vt, int *ldvt, d *q, int *iq, d *work, int *iwork, int *info) noexcept nogil +cdef void dbdsqr(char *uplo, int *n, int *ncvt, int *nru, int *ncc, d *d, d *e, d *vt, int *ldvt, d *u, int *ldu, d *c, int *ldc, d *work, int *info) noexcept nogil +cdef void ddisna(char *job, int *m, int *n, d *d, d *sep, int *info) noexcept nogil +cdef void dgbbrd(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, d *ab, int *ldab, d *d, d *e, d *q, int *ldq, d *pt, int *ldpt, d *c, int *ldc, d *work, int *info) noexcept nogil +cdef void dgbcon(char *norm, int *n, int *kl, int *ku, d *ab, int *ldab, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dgbequ(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil +cdef void dgbequb(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil +cdef void dgbrfs(char *trans, int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dgbsv(int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, int *ipiv, d *b, int *ldb, int *info) noexcept nogil +cdef void dgbsvx(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, int *ipiv, char *equed, d *r, d *c, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dgbtf2(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, int *ipiv, int *info) noexcept nogil +cdef void dgbtrf(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, int *ipiv, int *info) noexcept nogil +cdef void dgbtrs(char *trans, int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, int *ipiv, d *b, int *ldb, int *info) noexcept nogil +cdef void dgebak(char *job, char *side, int *n, int *ilo, int *ihi, d *scale, int *m, d *v, int *ldv, int *info) noexcept nogil +cdef void dgebal(char *job, int *n, d *a, int *lda, int *ilo, int *ihi, d *scale, int *info) noexcept nogil +cdef void dgebd2(int *m, int *n, d *a, int *lda, d *d, d *e, d *tauq, d *taup, d *work, int *info) noexcept nogil +cdef void dgebrd(int *m, int *n, d *a, int *lda, d *d, d *e, d *tauq, d *taup, d *work, int *lwork, int *info) noexcept nogil +cdef void dgecon(char *norm, int *n, d *a, int *lda, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dgeequ(int *m, int *n, d *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil +cdef void dgeequb(int *m, int *n, d *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil +cdef void dgees(char *jobvs, char *sort, dselect2 *select, int *n, d *a, int *lda, int *sdim, d *wr, d *wi, d *vs, int *ldvs, d *work, int *lwork, bint *bwork, int *info) noexcept nogil +cdef void dgeesx(char *jobvs, char *sort, dselect2 *select, char *sense, int *n, d *a, int *lda, int *sdim, d *wr, d *wi, d *vs, int *ldvs, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil +cdef void dgeev(char *jobvl, char *jobvr, int *n, d *a, int *lda, d *wr, d *wi, d *vl, int *ldvl, d *vr, int *ldvr, d *work, int *lwork, int *info) noexcept nogil +cdef void dgeevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, d *a, int *lda, d *wr, d *wi, d *vl, int *ldvl, d *vr, int *ldvr, int *ilo, int *ihi, d *scale, d *abnrm, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void dgehd2(int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil +cdef void dgehrd(int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dgejsv(char *joba, char *jobu, char *jobv, char *jobr, char *jobt, char *jobp, int *m, int *n, d *a, int *lda, d *sva, d *u, int *ldu, d *v, int *ldv, d *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void dgelq2(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil +cdef void dgelqf(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dgels(char *trans, int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *work, int *lwork, int *info) noexcept nogil +cdef void dgelsd(int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *s, d *rcond, int *rank, d *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void dgelss(int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *s, d *rcond, int *rank, d *work, int *lwork, int *info) noexcept nogil +cdef void dgelsy(int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *jpvt, d *rcond, int *rank, d *work, int *lwork, int *info) noexcept nogil +cdef void dgemqrt(char *side, char *trans, int *m, int *n, int *k, int *nb, d *v, int *ldv, d *t, int *ldt, d *c, int *ldc, d *work, int *info) noexcept nogil +cdef void dgeql2(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil +cdef void dgeqlf(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dgeqp3(int *m, int *n, d *a, int *lda, int *jpvt, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dgeqr2(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil +cdef void dgeqr2p(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil +cdef void dgeqrf(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dgeqrfp(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dgeqrt(int *m, int *n, int *nb, d *a, int *lda, d *t, int *ldt, d *work, int *info) noexcept nogil +cdef void dgeqrt2(int *m, int *n, d *a, int *lda, d *t, int *ldt, int *info) noexcept nogil +cdef void dgeqrt3(int *m, int *n, d *a, int *lda, d *t, int *ldt, int *info) noexcept nogil +cdef void dgerfs(char *trans, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dgerq2(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil +cdef void dgerqf(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dgesc2(int *n, d *a, int *lda, d *rhs, int *ipiv, int *jpiv, d *scale) noexcept nogil +cdef void dgesdd(char *jobz, int *m, int *n, d *a, int *lda, d *s, d *u, int *ldu, d *vt, int *ldvt, d *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void dgesv(int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, int *info) noexcept nogil +cdef void dgesvd(char *jobu, char *jobvt, int *m, int *n, d *a, int *lda, d *s, d *u, int *ldu, d *vt, int *ldvt, d *work, int *lwork, int *info) noexcept nogil +cdef void dgesvj(char *joba, char *jobu, char *jobv, int *m, int *n, d *a, int *lda, d *sva, int *mv, d *v, int *ldv, d *work, int *lwork, int *info) noexcept nogil +cdef void dgesvx(char *fact, char *trans, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, char *equed, d *r, d *c, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dgetc2(int *n, d *a, int *lda, int *ipiv, int *jpiv, int *info) noexcept nogil +cdef void dgetf2(int *m, int *n, d *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void dgetrf(int *m, int *n, d *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void dgetri(int *n, d *a, int *lda, int *ipiv, d *work, int *lwork, int *info) noexcept nogil +cdef void dgetrs(char *trans, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, int *info) noexcept nogil +cdef void dggbak(char *job, char *side, int *n, int *ilo, int *ihi, d *lscale, d *rscale, int *m, d *v, int *ldv, int *info) noexcept nogil +cdef void dggbal(char *job, int *n, d *a, int *lda, d *b, int *ldb, int *ilo, int *ihi, d *lscale, d *rscale, d *work, int *info) noexcept nogil +cdef void dgges(char *jobvsl, char *jobvsr, char *sort, dselect3 *selctg, int *n, d *a, int *lda, d *b, int *ldb, int *sdim, d *alphar, d *alphai, d *beta, d *vsl, int *ldvsl, d *vsr, int *ldvsr, d *work, int *lwork, bint *bwork, int *info) noexcept nogil +cdef void dggesx(char *jobvsl, char *jobvsr, char *sort, dselect3 *selctg, char *sense, int *n, d *a, int *lda, d *b, int *ldb, int *sdim, d *alphar, d *alphai, d *beta, d *vsl, int *ldvsl, d *vsr, int *ldvsr, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil +cdef void dggev(char *jobvl, char *jobvr, int *n, d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *vl, int *ldvl, d *vr, int *ldvr, d *work, int *lwork, int *info) noexcept nogil +cdef void dggevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *vl, int *ldvl, d *vr, int *ldvr, int *ilo, int *ihi, d *lscale, d *rscale, d *abnrm, d *bbnrm, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, bint *bwork, int *info) noexcept nogil +cdef void dggglm(int *n, int *m, int *p, d *a, int *lda, d *b, int *ldb, d *d, d *x, d *y, d *work, int *lwork, int *info) noexcept nogil +cdef void dgghrd(char *compq, char *compz, int *n, int *ilo, int *ihi, d *a, int *lda, d *b, int *ldb, d *q, int *ldq, d *z, int *ldz, int *info) noexcept nogil +cdef void dgglse(int *m, int *n, int *p, d *a, int *lda, d *b, int *ldb, d *c, d *d, d *x, d *work, int *lwork, int *info) noexcept nogil +cdef void dggqrf(int *n, int *m, int *p, d *a, int *lda, d *taua, d *b, int *ldb, d *taub, d *work, int *lwork, int *info) noexcept nogil +cdef void dggrqf(int *m, int *p, int *n, d *a, int *lda, d *taua, d *b, int *ldb, d *taub, d *work, int *lwork, int *info) noexcept nogil +cdef void dgsvj0(char *jobv, int *m, int *n, d *a, int *lda, d *d, d *sva, int *mv, d *v, int *ldv, d *eps, d *sfmin, d *tol, int *nsweep, d *work, int *lwork, int *info) noexcept nogil +cdef void dgsvj1(char *jobv, int *m, int *n, int *n1, d *a, int *lda, d *d, d *sva, int *mv, d *v, int *ldv, d *eps, d *sfmin, d *tol, int *nsweep, d *work, int *lwork, int *info) noexcept nogil +cdef void dgtcon(char *norm, int *n, d *dl, d *d, d *du, d *du2, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dgtrfs(char *trans, int *n, int *nrhs, d *dl, d *d, d *du, d *dlf, d *df, d *duf, d *du2, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dgtsv(int *n, int *nrhs, d *dl, d *d, d *du, d *b, int *ldb, int *info) noexcept nogil +cdef void dgtsvx(char *fact, char *trans, int *n, int *nrhs, d *dl, d *d, d *du, d *dlf, d *df, d *duf, d *du2, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dgttrf(int *n, d *dl, d *d, d *du, d *du2, int *ipiv, int *info) noexcept nogil +cdef void dgttrs(char *trans, int *n, int *nrhs, d *dl, d *d, d *du, d *du2, int *ipiv, d *b, int *ldb, int *info) noexcept nogil +cdef void dgtts2(int *itrans, int *n, int *nrhs, d *dl, d *d, d *du, d *du2, int *ipiv, d *b, int *ldb) noexcept nogil +cdef void dhgeqz(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *t, int *ldt, d *alphar, d *alphai, d *beta, d *q, int *ldq, d *z, int *ldz, d *work, int *lwork, int *info) noexcept nogil +cdef void dhsein(char *side, char *eigsrc, char *initv, bint *select, int *n, d *h, int *ldh, d *wr, d *wi, d *vl, int *ldvl, d *vr, int *ldvr, int *mm, int *m, d *work, int *ifaill, int *ifailr, int *info) noexcept nogil +cdef void dhseqr(char *job, char *compz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, d *z, int *ldz, d *work, int *lwork, int *info) noexcept nogil +cdef bint disnan(d *din) noexcept nogil +cdef void dlabad(d *small, d *large) noexcept nogil +cdef void dlabrd(int *m, int *n, int *nb, d *a, int *lda, d *d, d *e, d *tauq, d *taup, d *x, int *ldx, d *y, int *ldy) noexcept nogil +cdef void dlacn2(int *n, d *v, d *x, int *isgn, d *est, int *kase, int *isave) noexcept nogil +cdef void dlacon(int *n, d *v, d *x, int *isgn, d *est, int *kase) noexcept nogil +cdef void dlacpy(char *uplo, int *m, int *n, d *a, int *lda, d *b, int *ldb) noexcept nogil +cdef void dladiv(d *a, d *b, d *c, d *d, d *p, d *q) noexcept nogil +cdef void dlae2(d *a, d *b, d *c, d *rt1, d *rt2) noexcept nogil +cdef void dlaebz(int *ijob, int *nitmax, int *n, int *mmax, int *minp, int *nbmin, d *abstol, d *reltol, d *pivmin, d *d, d *e, d *e2, int *nval, d *ab, d *c, int *mout, int *nab, d *work, int *iwork, int *info) noexcept nogil +cdef void dlaed0(int *icompq, int *qsiz, int *n, d *d, d *e, d *q, int *ldq, d *qstore, int *ldqs, d *work, int *iwork, int *info) noexcept nogil +cdef void dlaed1(int *n, d *d, d *q, int *ldq, int *indxq, d *rho, int *cutpnt, d *work, int *iwork, int *info) noexcept nogil +cdef void dlaed2(int *k, int *n, int *n1, d *d, d *q, int *ldq, int *indxq, d *rho, d *z, d *dlamda, d *w, d *q2, int *indx, int *indxc, int *indxp, int *coltyp, int *info) noexcept nogil +cdef void dlaed3(int *k, int *n, int *n1, d *d, d *q, int *ldq, d *rho, d *dlamda, d *q2, int *indx, int *ctot, d *w, d *s, int *info) noexcept nogil +cdef void dlaed4(int *n, int *i, d *d, d *z, d *delta, d *rho, d *dlam, int *info) noexcept nogil +cdef void dlaed5(int *i, d *d, d *z, d *delta, d *rho, d *dlam) noexcept nogil +cdef void dlaed6(int *kniter, bint *orgati, d *rho, d *d, d *z, d *finit, d *tau, int *info) noexcept nogil +cdef void dlaed7(int *icompq, int *n, int *qsiz, int *tlvls, int *curlvl, int *curpbm, d *d, d *q, int *ldq, int *indxq, d *rho, int *cutpnt, d *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, d *givnum, d *work, int *iwork, int *info) noexcept nogil +cdef void dlaed8(int *icompq, int *k, int *n, int *qsiz, d *d, d *q, int *ldq, int *indxq, d *rho, int *cutpnt, d *z, d *dlamda, d *q2, int *ldq2, d *w, int *perm, int *givptr, int *givcol, d *givnum, int *indxp, int *indx, int *info) noexcept nogil +cdef void dlaed9(int *k, int *kstart, int *kstop, int *n, d *d, d *q, int *ldq, d *rho, d *dlamda, d *w, d *s, int *lds, int *info) noexcept nogil +cdef void dlaeda(int *n, int *tlvls, int *curlvl, int *curpbm, int *prmptr, int *perm, int *givptr, int *givcol, d *givnum, d *q, int *qptr, d *z, d *ztemp, int *info) noexcept nogil +cdef void dlaein(bint *rightv, bint *noinit, int *n, d *h, int *ldh, d *wr, d *wi, d *vr, d *vi, d *b, int *ldb, d *work, d *eps3, d *smlnum, d *bignum, int *info) noexcept nogil +cdef void dlaev2(d *a, d *b, d *c, d *rt1, d *rt2, d *cs1, d *sn1) noexcept nogil +cdef void dlaexc(bint *wantq, int *n, d *t, int *ldt, d *q, int *ldq, int *j1, int *n1, int *n2, d *work, int *info) noexcept nogil +cdef void dlag2(d *a, int *lda, d *b, int *ldb, d *safmin, d *scale1, d *scale2, d *wr1, d *wr2, d *wi) noexcept nogil +cdef void dlag2s(int *m, int *n, d *a, int *lda, s *sa, int *ldsa, int *info) noexcept nogil +cdef void dlags2(bint *upper, d *a1, d *a2, d *a3, d *b1, d *b2, d *b3, d *csu, d *snu, d *csv, d *snv, d *csq, d *snq) noexcept nogil +cdef void dlagtf(int *n, d *a, d *lambda_, d *b, d *c, d *tol, d *d, int *in_, int *info) noexcept nogil +cdef void dlagtm(char *trans, int *n, int *nrhs, d *alpha, d *dl, d *d, d *du, d *x, int *ldx, d *beta, d *b, int *ldb) noexcept nogil +cdef void dlagts(int *job, int *n, d *a, d *b, d *c, d *d, int *in_, d *y, d *tol, int *info) noexcept nogil +cdef void dlagv2(d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *csl, d *snl, d *csr, d *snr) noexcept nogil +cdef void dlahqr(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, int *iloz, int *ihiz, d *z, int *ldz, int *info) noexcept nogil +cdef void dlahr2(int *n, int *k, int *nb, d *a, int *lda, d *tau, d *t, int *ldt, d *y, int *ldy) noexcept nogil +cdef void dlaic1(int *job, int *j, d *x, d *sest, d *w, d *gamma, d *sestpr, d *s, d *c) noexcept nogil +cdef void dlaln2(bint *ltrans, int *na, int *nw, d *smin, d *ca, d *a, int *lda, d *d1, d *d2, d *b, int *ldb, d *wr, d *wi, d *x, int *ldx, d *scale, d *xnorm, int *info) noexcept nogil +cdef void dlals0(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, d *b, int *ldb, d *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *poles, d *difl, d *difr, d *z, int *k, d *c, d *s, d *work, int *info) noexcept nogil +cdef void dlalsa(int *icompq, int *smlsiz, int *n, int *nrhs, d *b, int *ldb, d *bx, int *ldbx, d *u, int *ldu, d *vt, int *k, d *difl, d *difr, d *z, d *poles, int *givptr, int *givcol, int *ldgcol, int *perm, d *givnum, d *c, d *s, d *work, int *iwork, int *info) noexcept nogil +cdef void dlalsd(char *uplo, int *smlsiz, int *n, int *nrhs, d *d, d *e, d *b, int *ldb, d *rcond, int *rank, d *work, int *iwork, int *info) noexcept nogil +cdef d dlamch(char *cmach) noexcept nogil +cdef void dlamrg(int *n1, int *n2, d *a, int *dtrd1, int *dtrd2, int *index_bn) noexcept nogil +cdef int dlaneg(int *n, d *d, d *lld, d *sigma, d *pivmin, int *r) noexcept nogil +cdef d dlangb(char *norm, int *n, int *kl, int *ku, d *ab, int *ldab, d *work) noexcept nogil +cdef d dlange(char *norm, int *m, int *n, d *a, int *lda, d *work) noexcept nogil +cdef d dlangt(char *norm, int *n, d *dl, d *d_, d *du) noexcept nogil +cdef d dlanhs(char *norm, int *n, d *a, int *lda, d *work) noexcept nogil +cdef d dlansb(char *norm, char *uplo, int *n, int *k, d *ab, int *ldab, d *work) noexcept nogil +cdef d dlansf(char *norm, char *transr, char *uplo, int *n, d *a, d *work) noexcept nogil +cdef d dlansp(char *norm, char *uplo, int *n, d *ap, d *work) noexcept nogil +cdef d dlanst(char *norm, int *n, d *d_, d *e) noexcept nogil +cdef d dlansy(char *norm, char *uplo, int *n, d *a, int *lda, d *work) noexcept nogil +cdef d dlantb(char *norm, char *uplo, char *diag, int *n, int *k, d *ab, int *ldab, d *work) noexcept nogil +cdef d dlantp(char *norm, char *uplo, char *diag, int *n, d *ap, d *work) noexcept nogil +cdef d dlantr(char *norm, char *uplo, char *diag, int *m, int *n, d *a, int *lda, d *work) noexcept nogil +cdef void dlanv2(d *a, d *b, d *c, d *d, d *rt1r, d *rt1i, d *rt2r, d *rt2i, d *cs, d *sn) noexcept nogil +cdef void dlapll(int *n, d *x, int *incx, d *y, int *incy, d *ssmin) noexcept nogil +cdef void dlapmr(bint *forwrd, int *m, int *n, d *x, int *ldx, int *k) noexcept nogil +cdef void dlapmt(bint *forwrd, int *m, int *n, d *x, int *ldx, int *k) noexcept nogil +cdef d dlapy2(d *x, d *y) noexcept nogil +cdef d dlapy3(d *x, d *y, d *z) noexcept nogil +cdef void dlaqgb(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) noexcept nogil +cdef void dlaqge(int *m, int *n, d *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) noexcept nogil +cdef void dlaqp2(int *m, int *n, int *offset, d *a, int *lda, int *jpvt, d *tau, d *vn1, d *vn2, d *work) noexcept nogil +cdef void dlaqps(int *m, int *n, int *offset, int *nb, int *kb, d *a, int *lda, int *jpvt, d *tau, d *vn1, d *vn2, d *auxv, d *f, int *ldf) noexcept nogil +cdef void dlaqr0(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, int *iloz, int *ihiz, d *z, int *ldz, d *work, int *lwork, int *info) noexcept nogil +cdef void dlaqr1(int *n, d *h, int *ldh, d *sr1, d *si1, d *sr2, d *si2, d *v) noexcept nogil +cdef void dlaqr2(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, d *h, int *ldh, int *iloz, int *ihiz, d *z, int *ldz, int *ns, int *nd, d *sr, d *si, d *v, int *ldv, int *nh, d *t, int *ldt, int *nv, d *wv, int *ldwv, d *work, int *lwork) noexcept nogil +cdef void dlaqr3(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, d *h, int *ldh, int *iloz, int *ihiz, d *z, int *ldz, int *ns, int *nd, d *sr, d *si, d *v, int *ldv, int *nh, d *t, int *ldt, int *nv, d *wv, int *ldwv, d *work, int *lwork) noexcept nogil +cdef void dlaqr4(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, int *iloz, int *ihiz, d *z, int *ldz, d *work, int *lwork, int *info) noexcept nogil +cdef void dlaqr5(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, d *sr, d *si, d *h, int *ldh, int *iloz, int *ihiz, d *z, int *ldz, d *v, int *ldv, d *u, int *ldu, int *nv, d *wv, int *ldwv, int *nh, d *wh, int *ldwh) noexcept nogil +cdef void dlaqsb(char *uplo, int *n, int *kd, d *ab, int *ldab, d *s, d *scond, d *amax, char *equed) noexcept nogil +cdef void dlaqsp(char *uplo, int *n, d *ap, d *s, d *scond, d *amax, char *equed) noexcept nogil +cdef void dlaqsy(char *uplo, int *n, d *a, int *lda, d *s, d *scond, d *amax, char *equed) noexcept nogil +cdef void dlaqtr(bint *ltran, bint *lreal, int *n, d *t, int *ldt, d *b, d *w, d *scale, d *x, d *work, int *info) noexcept nogil +cdef void dlar1v(int *n, int *b1, int *bn, d *lambda_, d *d, d *l, d *ld, d *lld, d *pivmin, d *gaptol, d *z, bint *wantnc, int *negcnt, d *ztz, d *mingma, int *r, int *isuppz, d *nrminv, d *resid, d *rqcorr, d *work) noexcept nogil +cdef void dlar2v(int *n, d *x, d *y, d *z, int *incx, d *c, d *s, int *incc) noexcept nogil +cdef void dlarf(char *side, int *m, int *n, d *v, int *incv, d *tau, d *c, int *ldc, d *work) noexcept nogil +cdef void dlarfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, d *v, int *ldv, d *t, int *ldt, d *c, int *ldc, d *work, int *ldwork) noexcept nogil +cdef void dlarfg(int *n, d *alpha, d *x, int *incx, d *tau) noexcept nogil +cdef void dlarfgp(int *n, d *alpha, d *x, int *incx, d *tau) noexcept nogil +cdef void dlarft(char *direct, char *storev, int *n, int *k, d *v, int *ldv, d *tau, d *t, int *ldt) noexcept nogil +cdef void dlarfx(char *side, int *m, int *n, d *v, d *tau, d *c, int *ldc, d *work) noexcept nogil +cdef void dlargv(int *n, d *x, int *incx, d *y, int *incy, d *c, int *incc) noexcept nogil +cdef void dlarnv(int *idist, int *iseed, int *n, d *x) noexcept nogil +cdef void dlarra(int *n, d *d, d *e, d *e2, d *spltol, d *tnrm, int *nsplit, int *isplit, int *info) noexcept nogil +cdef void dlarrb(int *n, d *d, d *lld, int *ifirst, int *ilast, d *rtol1, d *rtol2, int *offset, d *w, d *wgap, d *werr, d *work, int *iwork, d *pivmin, d *spdiam, int *twist, int *info) noexcept nogil +cdef void dlarrc(char *jobt, int *n, d *vl, d *vu, d *d, d *e, d *pivmin, int *eigcnt, int *lcnt, int *rcnt, int *info) noexcept nogil +cdef void dlarrd(char *range, char *order, int *n, d *vl, d *vu, int *il, int *iu, d *gers, d *reltol, d *d, d *e, d *e2, d *pivmin, int *nsplit, int *isplit, int *m, d *w, d *werr, d *wl, d *wu, int *iblock, int *indexw, d *work, int *iwork, int *info) noexcept nogil +cdef void dlarre(char *range, int *n, d *vl, d *vu, int *il, int *iu, d *d, d *e, d *e2, d *rtol1, d *rtol2, d *spltol, int *nsplit, int *isplit, int *m, d *w, d *werr, d *wgap, int *iblock, int *indexw, d *gers, d *pivmin, d *work, int *iwork, int *info) noexcept nogil +cdef void dlarrf(int *n, d *d, d *l, d *ld, int *clstrt, int *clend, d *w, d *wgap, d *werr, d *spdiam, d *clgapl, d *clgapr, d *pivmin, d *sigma, d *dplus, d *lplus, d *work, int *info) noexcept nogil +cdef void dlarrj(int *n, d *d, d *e2, int *ifirst, int *ilast, d *rtol, int *offset, d *w, d *werr, d *work, int *iwork, d *pivmin, d *spdiam, int *info) noexcept nogil +cdef void dlarrk(int *n, int *iw, d *gl, d *gu, d *d, d *e2, d *pivmin, d *reltol, d *w, d *werr, int *info) noexcept nogil +cdef void dlarrr(int *n, d *d, d *e, int *info) noexcept nogil +cdef void dlarrv(int *n, d *vl, d *vu, d *d, d *l, d *pivmin, int *isplit, int *m, int *dol, int *dou, d *minrgp, d *rtol1, d *rtol2, d *w, d *werr, d *wgap, int *iblock, int *indexw, d *gers, d *z, int *ldz, int *isuppz, d *work, int *iwork, int *info) noexcept nogil +cdef void dlartg(d *f, d *g, d *cs, d *sn, d *r) noexcept nogil +cdef void dlartgp(d *f, d *g, d *cs, d *sn, d *r) noexcept nogil +cdef void dlartgs(d *x, d *y, d *sigma, d *cs, d *sn) noexcept nogil +cdef void dlartv(int *n, d *x, int *incx, d *y, int *incy, d *c, d *s, int *incc) noexcept nogil +cdef void dlaruv(int *iseed, int *n, d *x) noexcept nogil +cdef void dlarz(char *side, int *m, int *n, int *l, d *v, int *incv, d *tau, d *c, int *ldc, d *work) noexcept nogil +cdef void dlarzb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, d *v, int *ldv, d *t, int *ldt, d *c, int *ldc, d *work, int *ldwork) noexcept nogil +cdef void dlarzt(char *direct, char *storev, int *n, int *k, d *v, int *ldv, d *tau, d *t, int *ldt) noexcept nogil +cdef void dlas2(d *f, d *g, d *h, d *ssmin, d *ssmax) noexcept nogil +cdef void dlascl(char *type_bn, int *kl, int *ku, d *cfrom, d *cto, int *m, int *n, d *a, int *lda, int *info) noexcept nogil +cdef void dlasd0(int *n, int *sqre, d *d, d *e, d *u, int *ldu, d *vt, int *ldvt, int *smlsiz, int *iwork, d *work, int *info) noexcept nogil +cdef void dlasd1(int *nl, int *nr, int *sqre, d *d, d *alpha, d *beta, d *u, int *ldu, d *vt, int *ldvt, int *idxq, int *iwork, d *work, int *info) noexcept nogil +cdef void dlasd2(int *nl, int *nr, int *sqre, int *k, d *d, d *z, d *alpha, d *beta, d *u, int *ldu, d *vt, int *ldvt, d *dsigma, d *u2, int *ldu2, d *vt2, int *ldvt2, int *idxp, int *idx, int *idxc, int *idxq, int *coltyp, int *info) noexcept nogil +cdef void dlasd3(int *nl, int *nr, int *sqre, int *k, d *d, d *q, int *ldq, d *dsigma, d *u, int *ldu, d *u2, int *ldu2, d *vt, int *ldvt, d *vt2, int *ldvt2, int *idxc, int *ctot, d *z, int *info) noexcept nogil +cdef void dlasd4(int *n, int *i, d *d, d *z, d *delta, d *rho, d *sigma, d *work, int *info) noexcept nogil +cdef void dlasd5(int *i, d *d, d *z, d *delta, d *rho, d *dsigma, d *work) noexcept nogil +cdef void dlasd6(int *icompq, int *nl, int *nr, int *sqre, d *d, d *vf, d *vl, d *alpha, d *beta, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *poles, d *difl, d *difr, d *z, int *k, d *c, d *s, d *work, int *iwork, int *info) noexcept nogil +cdef void dlasd7(int *icompq, int *nl, int *nr, int *sqre, int *k, d *d, d *z, d *zw, d *vf, d *vfw, d *vl, d *vlw, d *alpha, d *beta, d *dsigma, int *idx, int *idxp, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *c, d *s, int *info) noexcept nogil +cdef void dlasd8(int *icompq, int *k, d *d, d *z, d *vf, d *vl, d *difl, d *difr, int *lddifr, d *dsigma, d *work, int *info) noexcept nogil +cdef void dlasda(int *icompq, int *smlsiz, int *n, int *sqre, d *d, d *e, d *u, int *ldu, d *vt, int *k, d *difl, d *difr, d *z, d *poles, int *givptr, int *givcol, int *ldgcol, int *perm, d *givnum, d *c, d *s, d *work, int *iwork, int *info) noexcept nogil +cdef void dlasdq(char *uplo, int *sqre, int *n, int *ncvt, int *nru, int *ncc, d *d, d *e, d *vt, int *ldvt, d *u, int *ldu, d *c, int *ldc, d *work, int *info) noexcept nogil +cdef void dlasdt(int *n, int *lvl, int *nd, int *inode, int *ndiml, int *ndimr, int *msub) noexcept nogil +cdef void dlaset(char *uplo, int *m, int *n, d *alpha, d *beta, d *a, int *lda) noexcept nogil +cdef void dlasq1(int *n, d *d, d *e, d *work, int *info) noexcept nogil +cdef void dlasq2(int *n, d *z, int *info) noexcept nogil +cdef void dlasq3(int *i0, int *n0, d *z, int *pp, d *dmin, d *sigma, d *desig, d *qmax, int *nfail, int *iter, int *ndiv, bint *ieee, int *ttype, d *dmin1, d *dmin2, d *dn, d *dn1, d *dn2, d *g, d *tau) noexcept nogil +cdef void dlasq4(int *i0, int *n0, d *z, int *pp, int *n0in, d *dmin, d *dmin1, d *dmin2, d *dn, d *dn1, d *dn2, d *tau, int *ttype, d *g) noexcept nogil +cdef void dlasq6(int *i0, int *n0, d *z, int *pp, d *dmin, d *dmin1, d *dmin2, d *dn, d *dnm1, d *dnm2) noexcept nogil +cdef void dlasr(char *side, char *pivot, char *direct, int *m, int *n, d *c, d *s, d *a, int *lda) noexcept nogil +cdef void dlasrt(char *id, int *n, d *d, int *info) noexcept nogil +cdef void dlassq(int *n, d *x, int *incx, d *scale, d *sumsq) noexcept nogil +cdef void dlasv2(d *f, d *g, d *h, d *ssmin, d *ssmax, d *snr, d *csr, d *snl, d *csl) noexcept nogil +cdef void dlaswp(int *n, d *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) noexcept nogil +cdef void dlasy2(bint *ltranl, bint *ltranr, int *isgn, int *n1, int *n2, d *tl, int *ldtl, d *tr, int *ldtr, d *b, int *ldb, d *scale, d *x, int *ldx, d *xnorm, int *info) noexcept nogil +cdef void dlasyf(char *uplo, int *n, int *nb, int *kb, d *a, int *lda, int *ipiv, d *w, int *ldw, int *info) noexcept nogil +cdef void dlat2s(char *uplo, int *n, d *a, int *lda, s *sa, int *ldsa, int *info) noexcept nogil +cdef void dlatbs(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, d *ab, int *ldab, d *x, d *scale, d *cnorm, int *info) noexcept nogil +cdef void dlatdf(int *ijob, int *n, d *z, int *ldz, d *rhs, d *rdsum, d *rdscal, int *ipiv, int *jpiv) noexcept nogil +cdef void dlatps(char *uplo, char *trans, char *diag, char *normin, int *n, d *ap, d *x, d *scale, d *cnorm, int *info) noexcept nogil +cdef void dlatrd(char *uplo, int *n, int *nb, d *a, int *lda, d *e, d *tau, d *w, int *ldw) noexcept nogil +cdef void dlatrs(char *uplo, char *trans, char *diag, char *normin, int *n, d *a, int *lda, d *x, d *scale, d *cnorm, int *info) noexcept nogil +cdef void dlatrz(int *m, int *n, int *l, d *a, int *lda, d *tau, d *work) noexcept nogil +cdef void dlauu2(char *uplo, int *n, d *a, int *lda, int *info) noexcept nogil +cdef void dlauum(char *uplo, int *n, d *a, int *lda, int *info) noexcept nogil +cdef void dopgtr(char *uplo, int *n, d *ap, d *tau, d *q, int *ldq, d *work, int *info) noexcept nogil +cdef void dopmtr(char *side, char *uplo, char *trans, int *m, int *n, d *ap, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil +cdef void dorbdb(char *trans, char *signs, int *m, int *p, int *q, d *x11, int *ldx11, d *x12, int *ldx12, d *x21, int *ldx21, d *x22, int *ldx22, d *theta, d *phi, d *taup1, d *taup2, d *tauq1, d *tauq2, d *work, int *lwork, int *info) noexcept nogil +cdef void dorcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, d *x11, int *ldx11, d *x12, int *ldx12, d *x21, int *ldx21, d *x22, int *ldx22, d *theta, d *u1, int *ldu1, d *u2, int *ldu2, d *v1t, int *ldv1t, d *v2t, int *ldv2t, d *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void dorg2l(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil +cdef void dorg2r(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil +cdef void dorgbr(char *vect, int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dorghr(int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dorgl2(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil +cdef void dorglq(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dorgql(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dorgqr(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dorgr2(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil +cdef void dorgrq(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dorgtr(char *uplo, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dorm2l(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil +cdef void dorm2r(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil +cdef void dormbr(char *vect, char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil +cdef void dormhr(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil +cdef void dorml2(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil +cdef void dormlq(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil +cdef void dormql(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil +cdef void dormqr(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil +cdef void dormr2(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil +cdef void dormr3(char *side, char *trans, int *m, int *n, int *k, int *l, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil +cdef void dormrq(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil +cdef void dormrz(char *side, char *trans, int *m, int *n, int *k, int *l, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil +cdef void dormtr(char *side, char *uplo, char *trans, int *m, int *n, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil +cdef void dpbcon(char *uplo, int *n, int *kd, d *ab, int *ldab, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dpbequ(char *uplo, int *n, int *kd, d *ab, int *ldab, d *s, d *scond, d *amax, int *info) noexcept nogil +cdef void dpbrfs(char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dpbstf(char *uplo, int *n, int *kd, d *ab, int *ldab, int *info) noexcept nogil +cdef void dpbsv(char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, int *info) noexcept nogil +cdef void dpbsvx(char *fact, char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, char *equed, d *s, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dpbtf2(char *uplo, int *n, int *kd, d *ab, int *ldab, int *info) noexcept nogil +cdef void dpbtrf(char *uplo, int *n, int *kd, d *ab, int *ldab, int *info) noexcept nogil +cdef void dpbtrs(char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, int *info) noexcept nogil +cdef void dpftrf(char *transr, char *uplo, int *n, d *a, int *info) noexcept nogil +cdef void dpftri(char *transr, char *uplo, int *n, d *a, int *info) noexcept nogil +cdef void dpftrs(char *transr, char *uplo, int *n, int *nrhs, d *a, d *b, int *ldb, int *info) noexcept nogil +cdef void dpocon(char *uplo, int *n, d *a, int *lda, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dpoequ(int *n, d *a, int *lda, d *s, d *scond, d *amax, int *info) noexcept nogil +cdef void dpoequb(int *n, d *a, int *lda, d *s, d *scond, d *amax, int *info) noexcept nogil +cdef void dporfs(char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dposv(char *uplo, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *info) noexcept nogil +cdef void dposvx(char *fact, char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, char *equed, d *s, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dpotf2(char *uplo, int *n, d *a, int *lda, int *info) noexcept nogil +cdef void dpotrf(char *uplo, int *n, d *a, int *lda, int *info) noexcept nogil +cdef void dpotri(char *uplo, int *n, d *a, int *lda, int *info) noexcept nogil +cdef void dpotrs(char *uplo, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *info) noexcept nogil +cdef void dppcon(char *uplo, int *n, d *ap, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dppequ(char *uplo, int *n, d *ap, d *s, d *scond, d *amax, int *info) noexcept nogil +cdef void dpprfs(char *uplo, int *n, int *nrhs, d *ap, d *afp, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dppsv(char *uplo, int *n, int *nrhs, d *ap, d *b, int *ldb, int *info) noexcept nogil +cdef void dppsvx(char *fact, char *uplo, int *n, int *nrhs, d *ap, d *afp, char *equed, d *s, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dpptrf(char *uplo, int *n, d *ap, int *info) noexcept nogil +cdef void dpptri(char *uplo, int *n, d *ap, int *info) noexcept nogil +cdef void dpptrs(char *uplo, int *n, int *nrhs, d *ap, d *b, int *ldb, int *info) noexcept nogil +cdef void dpstf2(char *uplo, int *n, d *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) noexcept nogil +cdef void dpstrf(char *uplo, int *n, d *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) noexcept nogil +cdef void dptcon(int *n, d *d, d *e, d *anorm, d *rcond, d *work, int *info) noexcept nogil +cdef void dpteqr(char *compz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *info) noexcept nogil +cdef void dptrfs(int *n, int *nrhs, d *d, d *e, d *df, d *ef, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *info) noexcept nogil +cdef void dptsv(int *n, int *nrhs, d *d, d *e, d *b, int *ldb, int *info) noexcept nogil +cdef void dptsvx(char *fact, int *n, int *nrhs, d *d, d *e, d *df, d *ef, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *info) noexcept nogil +cdef void dpttrf(int *n, d *d, d *e, int *info) noexcept nogil +cdef void dpttrs(int *n, int *nrhs, d *d, d *e, d *b, int *ldb, int *info) noexcept nogil +cdef void dptts2(int *n, int *nrhs, d *d, d *e, d *b, int *ldb) noexcept nogil +cdef void drscl(int *n, d *sa, d *sx, int *incx) noexcept nogil +cdef void dsbev(char *jobz, char *uplo, int *n, int *kd, d *ab, int *ldab, d *w, d *z, int *ldz, d *work, int *info) noexcept nogil +cdef void dsbevd(char *jobz, char *uplo, int *n, int *kd, d *ab, int *ldab, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dsbevx(char *jobz, char *range, char *uplo, int *n, int *kd, d *ab, int *ldab, d *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void dsbgst(char *vect, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *x, int *ldx, d *work, int *info) noexcept nogil +cdef void dsbgv(char *jobz, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *w, d *z, int *ldz, d *work, int *info) noexcept nogil +cdef void dsbgvd(char *jobz, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dsbgvx(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void dsbtrd(char *vect, char *uplo, int *n, int *kd, d *ab, int *ldab, d *d, d *e, d *q, int *ldq, d *work, int *info) noexcept nogil +cdef void dsfrk(char *transr, char *uplo, char *trans, int *n, int *k, d *alpha, d *a, int *lda, d *beta, d *c) noexcept nogil +cdef void dsgesv(int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *work, s *swork, int *iter, int *info) noexcept nogil +cdef void dspcon(char *uplo, int *n, d *ap, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dspev(char *jobz, char *uplo, int *n, d *ap, d *w, d *z, int *ldz, d *work, int *info) noexcept nogil +cdef void dspevd(char *jobz, char *uplo, int *n, d *ap, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dspevx(char *jobz, char *range, char *uplo, int *n, d *ap, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void dspgst(int *itype, char *uplo, int *n, d *ap, d *bp, int *info) noexcept nogil +cdef void dspgv(int *itype, char *jobz, char *uplo, int *n, d *ap, d *bp, d *w, d *z, int *ldz, d *work, int *info) noexcept nogil +cdef void dspgvd(int *itype, char *jobz, char *uplo, int *n, d *ap, d *bp, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dspgvx(int *itype, char *jobz, char *range, char *uplo, int *n, d *ap, d *bp, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void dsposv(char *uplo, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *x, int *ldx, d *work, s *swork, int *iter, int *info) noexcept nogil +cdef void dsprfs(char *uplo, int *n, int *nrhs, d *ap, d *afp, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dspsv(char *uplo, int *n, int *nrhs, d *ap, int *ipiv, d *b, int *ldb, int *info) noexcept nogil +cdef void dspsvx(char *fact, char *uplo, int *n, int *nrhs, d *ap, d *afp, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dsptrd(char *uplo, int *n, d *ap, d *d, d *e, d *tau, int *info) noexcept nogil +cdef void dsptrf(char *uplo, int *n, d *ap, int *ipiv, int *info) noexcept nogil +cdef void dsptri(char *uplo, int *n, d *ap, int *ipiv, d *work, int *info) noexcept nogil +cdef void dsptrs(char *uplo, int *n, int *nrhs, d *ap, int *ipiv, d *b, int *ldb, int *info) noexcept nogil +cdef void dstebz(char *range, char *order, int *n, d *vl, d *vu, int *il, int *iu, d *abstol, d *d, d *e, int *m, int *nsplit, d *w, int *iblock, int *isplit, d *work, int *iwork, int *info) noexcept nogil +cdef void dstedc(char *compz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dstegr(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dstein(int *n, d *d, d *e, int *m, d *w, int *iblock, int *isplit, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void dstemr(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, int *m, d *w, d *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dsteqr(char *compz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *info) noexcept nogil +cdef void dsterf(int *n, d *d, d *e, int *info) noexcept nogil +cdef void dstev(char *jobz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *info) noexcept nogil +cdef void dstevd(char *jobz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dstevr(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dstevx(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void dsycon(char *uplo, int *n, d *a, int *lda, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dsyconv(char *uplo, char *way, int *n, d *a, int *lda, int *ipiv, d *work, int *info) noexcept nogil +cdef void dsyequb(char *uplo, int *n, d *a, int *lda, d *s, d *scond, d *amax, d *work, int *info) noexcept nogil +cdef void dsyev(char *jobz, char *uplo, int *n, d *a, int *lda, d *w, d *work, int *lwork, int *info) noexcept nogil +cdef void dsyevd(char *jobz, char *uplo, int *n, d *a, int *lda, d *w, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dsyevr(char *jobz, char *range, char *uplo, int *n, d *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dsyevx(char *jobz, char *range, char *uplo, int *n, d *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void dsygs2(int *itype, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, int *info) noexcept nogil +cdef void dsygst(int *itype, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, int *info) noexcept nogil +cdef void dsygv(int *itype, char *jobz, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, d *w, d *work, int *lwork, int *info) noexcept nogil +cdef void dsygvd(int *itype, char *jobz, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, d *w, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dsygvx(int *itype, char *jobz, char *range, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void dsyrfs(char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dsysv(char *uplo, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, d *work, int *lwork, int *info) noexcept nogil +cdef void dsysvx(char *fact, char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void dsyswapr(char *uplo, int *n, d *a, int *lda, int *i1, int *i2) noexcept nogil +cdef void dsytd2(char *uplo, int *n, d *a, int *lda, d *d, d *e, d *tau, int *info) noexcept nogil +cdef void dsytf2(char *uplo, int *n, d *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void dsytrd(char *uplo, int *n, d *a, int *lda, d *d, d *e, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef void dsytrf(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *lwork, int *info) noexcept nogil +cdef void dsytri(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *info) noexcept nogil +cdef void dsytri2(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *lwork, int *info) noexcept nogil +cdef void dsytri2x(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *nb, int *info) noexcept nogil +cdef void dsytrs(char *uplo, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, int *info) noexcept nogil +cdef void dsytrs2(char *uplo, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, d *work, int *info) noexcept nogil +cdef void dtbcon(char *norm, char *uplo, char *diag, int *n, int *kd, d *ab, int *ldab, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dtbrfs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dtbtrs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, int *info) noexcept nogil +cdef void dtfsm(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, d *alpha, d *a, d *b, int *ldb) noexcept nogil +cdef void dtftri(char *transr, char *uplo, char *diag, int *n, d *a, int *info) noexcept nogil +cdef void dtfttp(char *transr, char *uplo, int *n, d *arf, d *ap, int *info) noexcept nogil +cdef void dtfttr(char *transr, char *uplo, int *n, d *arf, d *a, int *lda, int *info) noexcept nogil +cdef void dtgevc(char *side, char *howmny, bint *select, int *n, d *s, int *lds, d *p, int *ldp, d *vl, int *ldvl, d *vr, int *ldvr, int *mm, int *m, d *work, int *info) noexcept nogil +cdef void dtgex2(bint *wantq, bint *wantz, int *n, d *a, int *lda, d *b, int *ldb, d *q, int *ldq, d *z, int *ldz, int *j1, int *n1, int *n2, d *work, int *lwork, int *info) noexcept nogil +cdef void dtgexc(bint *wantq, bint *wantz, int *n, d *a, int *lda, d *b, int *ldb, d *q, int *ldq, d *z, int *ldz, int *ifst, int *ilst, d *work, int *lwork, int *info) noexcept nogil +cdef void dtgsen(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *q, int *ldq, d *z, int *ldz, int *m, d *pl, d *pr, d *dif, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dtgsja(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, d *a, int *lda, d *b, int *ldb, d *tola, d *tolb, d *alpha, d *beta, d *u, int *ldu, d *v, int *ldv, d *q, int *ldq, d *work, int *ncycle, int *info) noexcept nogil +cdef void dtgsna(char *job, char *howmny, bint *select, int *n, d *a, int *lda, d *b, int *ldb, d *vl, int *ldvl, d *vr, int *ldvr, d *s, d *dif, int *mm, int *m, d *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void dtgsy2(char *trans, int *ijob, int *m, int *n, d *a, int *lda, d *b, int *ldb, d *c, int *ldc, d *d, int *ldd, d *e, int *lde, d *f, int *ldf, d *scale, d *rdsum, d *rdscal, int *iwork, int *pq, int *info) noexcept nogil +cdef void dtgsyl(char *trans, int *ijob, int *m, int *n, d *a, int *lda, d *b, int *ldb, d *c, int *ldc, d *d, int *ldd, d *e, int *lde, d *f, int *ldf, d *scale, d *dif, d *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void dtpcon(char *norm, char *uplo, char *diag, int *n, d *ap, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dtpmqrt(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, d *v, int *ldv, d *t, int *ldt, d *a, int *lda, d *b, int *ldb, d *work, int *info) noexcept nogil +cdef void dtpqrt(int *m, int *n, int *l, int *nb, d *a, int *lda, d *b, int *ldb, d *t, int *ldt, d *work, int *info) noexcept nogil +cdef void dtpqrt2(int *m, int *n, int *l, d *a, int *lda, d *b, int *ldb, d *t, int *ldt, int *info) noexcept nogil +cdef void dtprfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, d *v, int *ldv, d *t, int *ldt, d *a, int *lda, d *b, int *ldb, d *work, int *ldwork) noexcept nogil +cdef void dtprfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *ap, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dtptri(char *uplo, char *diag, int *n, d *ap, int *info) noexcept nogil +cdef void dtptrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *ap, d *b, int *ldb, int *info) noexcept nogil +cdef void dtpttf(char *transr, char *uplo, int *n, d *ap, d *arf, int *info) noexcept nogil +cdef void dtpttr(char *uplo, int *n, d *ap, d *a, int *lda, int *info) noexcept nogil +cdef void dtrcon(char *norm, char *uplo, char *diag, int *n, d *a, int *lda, d *rcond, d *work, int *iwork, int *info) noexcept nogil +cdef void dtrevc(char *side, char *howmny, bint *select, int *n, d *t, int *ldt, d *vl, int *ldvl, d *vr, int *ldvr, int *mm, int *m, d *work, int *info) noexcept nogil +cdef void dtrexc(char *compq, int *n, d *t, int *ldt, d *q, int *ldq, int *ifst, int *ilst, d *work, int *info) noexcept nogil +cdef void dtrrfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil +cdef void dtrsen(char *job, char *compq, bint *select, int *n, d *t, int *ldt, d *q, int *ldq, d *wr, d *wi, int *m, d *s, d *sep, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void dtrsna(char *job, char *howmny, bint *select, int *n, d *t, int *ldt, d *vl, int *ldvl, d *vr, int *ldvr, d *s, d *sep, int *mm, int *m, d *work, int *ldwork, int *iwork, int *info) noexcept nogil +cdef void dtrsyl(char *trana, char *tranb, int *isgn, int *m, int *n, d *a, int *lda, d *b, int *ldb, d *c, int *ldc, d *scale, int *info) noexcept nogil +cdef void dtrti2(char *uplo, char *diag, int *n, d *a, int *lda, int *info) noexcept nogil +cdef void dtrtri(char *uplo, char *diag, int *n, d *a, int *lda, int *info) noexcept nogil +cdef void dtrtrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *info) noexcept nogil +cdef void dtrttf(char *transr, char *uplo, int *n, d *a, int *lda, d *arf, int *info) noexcept nogil +cdef void dtrttp(char *uplo, int *n, d *a, int *lda, d *ap, int *info) noexcept nogil +cdef void dtzrzf(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil +cdef d dzsum1(int *n, z *cx, int *incx) noexcept nogil +cdef int icmax1(int *n, c *cx, int *incx) noexcept nogil +cdef int ieeeck(int *ispec, s *zero, s *one) noexcept nogil +cdef int ilaclc(int *m, int *n, c *a, int *lda) noexcept nogil +cdef int ilaclr(int *m, int *n, c *a, int *lda) noexcept nogil +cdef int iladiag(char *diag) noexcept nogil +cdef int iladlc(int *m, int *n, d *a, int *lda) noexcept nogil +cdef int iladlr(int *m, int *n, d *a, int *lda) noexcept nogil +cdef int ilaprec(char *prec) noexcept nogil +cdef int ilaslc(int *m, int *n, s *a, int *lda) noexcept nogil +cdef int ilaslr(int *m, int *n, s *a, int *lda) noexcept nogil +cdef int ilatrans(char *trans) noexcept nogil +cdef int ilauplo(char *uplo) noexcept nogil +cdef void ilaver(int *vers_major, int *vers_minor, int *vers_patch) noexcept nogil +cdef int ilazlc(int *m, int *n, z *a, int *lda) noexcept nogil +cdef int ilazlr(int *m, int *n, z *a, int *lda) noexcept nogil +cdef int izmax1(int *n, z *cx, int *incx) noexcept nogil +cdef void sbbcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, s *theta, s *phi, s *u1, int *ldu1, s *u2, int *ldu2, s *v1t, int *ldv1t, s *v2t, int *ldv2t, s *b11d, s *b11e, s *b12d, s *b12e, s *b21d, s *b21e, s *b22d, s *b22e, s *work, int *lwork, int *info) noexcept nogil +cdef void sbdsdc(char *uplo, char *compq, int *n, s *d, s *e, s *u, int *ldu, s *vt, int *ldvt, s *q, int *iq, s *work, int *iwork, int *info) noexcept nogil +cdef void sbdsqr(char *uplo, int *n, int *ncvt, int *nru, int *ncc, s *d, s *e, s *vt, int *ldvt, s *u, int *ldu, s *c, int *ldc, s *work, int *info) noexcept nogil +cdef s scsum1(int *n, c *cx, int *incx) noexcept nogil +cdef void sdisna(char *job, int *m, int *n, s *d, s *sep, int *info) noexcept nogil +cdef void sgbbrd(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, s *ab, int *ldab, s *d, s *e, s *q, int *ldq, s *pt, int *ldpt, s *c, int *ldc, s *work, int *info) noexcept nogil +cdef void sgbcon(char *norm, int *n, int *kl, int *ku, s *ab, int *ldab, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void sgbequ(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil +cdef void sgbequb(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil +cdef void sgbrfs(char *trans, int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void sgbsv(int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, int *ipiv, s *b, int *ldb, int *info) noexcept nogil +cdef void sgbsvx(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, int *ipiv, char *equed, s *r, s *c, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void sgbtf2(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, int *ipiv, int *info) noexcept nogil +cdef void sgbtrf(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, int *ipiv, int *info) noexcept nogil +cdef void sgbtrs(char *trans, int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, int *ipiv, s *b, int *ldb, int *info) noexcept nogil +cdef void sgebak(char *job, char *side, int *n, int *ilo, int *ihi, s *scale, int *m, s *v, int *ldv, int *info) noexcept nogil +cdef void sgebal(char *job, int *n, s *a, int *lda, int *ilo, int *ihi, s *scale, int *info) noexcept nogil +cdef void sgebd2(int *m, int *n, s *a, int *lda, s *d, s *e, s *tauq, s *taup, s *work, int *info) noexcept nogil +cdef void sgebrd(int *m, int *n, s *a, int *lda, s *d, s *e, s *tauq, s *taup, s *work, int *lwork, int *info) noexcept nogil +cdef void sgecon(char *norm, int *n, s *a, int *lda, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void sgeequ(int *m, int *n, s *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil +cdef void sgeequb(int *m, int *n, s *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil +cdef void sgees(char *jobvs, char *sort, sselect2 *select, int *n, s *a, int *lda, int *sdim, s *wr, s *wi, s *vs, int *ldvs, s *work, int *lwork, bint *bwork, int *info) noexcept nogil +cdef void sgeesx(char *jobvs, char *sort, sselect2 *select, char *sense, int *n, s *a, int *lda, int *sdim, s *wr, s *wi, s *vs, int *ldvs, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil +cdef void sgeev(char *jobvl, char *jobvr, int *n, s *a, int *lda, s *wr, s *wi, s *vl, int *ldvl, s *vr, int *ldvr, s *work, int *lwork, int *info) noexcept nogil +cdef void sgeevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, s *a, int *lda, s *wr, s *wi, s *vl, int *ldvl, s *vr, int *ldvr, int *ilo, int *ihi, s *scale, s *abnrm, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void sgehd2(int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil +cdef void sgehrd(int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sgejsv(char *joba, char *jobu, char *jobv, char *jobr, char *jobt, char *jobp, int *m, int *n, s *a, int *lda, s *sva, s *u, int *ldu, s *v, int *ldv, s *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void sgelq2(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil +cdef void sgelqf(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sgels(char *trans, int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *work, int *lwork, int *info) noexcept nogil +cdef void sgelsd(int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *s, s *rcond, int *rank, s *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void sgelss(int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *s, s *rcond, int *rank, s *work, int *lwork, int *info) noexcept nogil +cdef void sgelsy(int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *jpvt, s *rcond, int *rank, s *work, int *lwork, int *info) noexcept nogil +cdef void sgemqrt(char *side, char *trans, int *m, int *n, int *k, int *nb, s *v, int *ldv, s *t, int *ldt, s *c, int *ldc, s *work, int *info) noexcept nogil +cdef void sgeql2(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil +cdef void sgeqlf(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sgeqp3(int *m, int *n, s *a, int *lda, int *jpvt, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sgeqr2(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil +cdef void sgeqr2p(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil +cdef void sgeqrf(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sgeqrfp(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sgeqrt(int *m, int *n, int *nb, s *a, int *lda, s *t, int *ldt, s *work, int *info) noexcept nogil +cdef void sgeqrt2(int *m, int *n, s *a, int *lda, s *t, int *ldt, int *info) noexcept nogil +cdef void sgeqrt3(int *m, int *n, s *a, int *lda, s *t, int *ldt, int *info) noexcept nogil +cdef void sgerfs(char *trans, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void sgerq2(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil +cdef void sgerqf(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sgesc2(int *n, s *a, int *lda, s *rhs, int *ipiv, int *jpiv, s *scale) noexcept nogil +cdef void sgesdd(char *jobz, int *m, int *n, s *a, int *lda, s *s, s *u, int *ldu, s *vt, int *ldvt, s *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void sgesv(int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, int *info) noexcept nogil +cdef void sgesvd(char *jobu, char *jobvt, int *m, int *n, s *a, int *lda, s *s, s *u, int *ldu, s *vt, int *ldvt, s *work, int *lwork, int *info) noexcept nogil +cdef void sgesvj(char *joba, char *jobu, char *jobv, int *m, int *n, s *a, int *lda, s *sva, int *mv, s *v, int *ldv, s *work, int *lwork, int *info) noexcept nogil +cdef void sgesvx(char *fact, char *trans, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, char *equed, s *r, s *c, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void sgetc2(int *n, s *a, int *lda, int *ipiv, int *jpiv, int *info) noexcept nogil +cdef void sgetf2(int *m, int *n, s *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void sgetrf(int *m, int *n, s *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void sgetri(int *n, s *a, int *lda, int *ipiv, s *work, int *lwork, int *info) noexcept nogil +cdef void sgetrs(char *trans, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, int *info) noexcept nogil +cdef void sggbak(char *job, char *side, int *n, int *ilo, int *ihi, s *lscale, s *rscale, int *m, s *v, int *ldv, int *info) noexcept nogil +cdef void sggbal(char *job, int *n, s *a, int *lda, s *b, int *ldb, int *ilo, int *ihi, s *lscale, s *rscale, s *work, int *info) noexcept nogil +cdef void sgges(char *jobvsl, char *jobvsr, char *sort, sselect3 *selctg, int *n, s *a, int *lda, s *b, int *ldb, int *sdim, s *alphar, s *alphai, s *beta, s *vsl, int *ldvsl, s *vsr, int *ldvsr, s *work, int *lwork, bint *bwork, int *info) noexcept nogil +cdef void sggesx(char *jobvsl, char *jobvsr, char *sort, sselect3 *selctg, char *sense, int *n, s *a, int *lda, s *b, int *ldb, int *sdim, s *alphar, s *alphai, s *beta, s *vsl, int *ldvsl, s *vsr, int *ldvsr, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil +cdef void sggev(char *jobvl, char *jobvr, int *n, s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *vl, int *ldvl, s *vr, int *ldvr, s *work, int *lwork, int *info) noexcept nogil +cdef void sggevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *vl, int *ldvl, s *vr, int *ldvr, int *ilo, int *ihi, s *lscale, s *rscale, s *abnrm, s *bbnrm, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, bint *bwork, int *info) noexcept nogil +cdef void sggglm(int *n, int *m, int *p, s *a, int *lda, s *b, int *ldb, s *d, s *x, s *y, s *work, int *lwork, int *info) noexcept nogil +cdef void sgghrd(char *compq, char *compz, int *n, int *ilo, int *ihi, s *a, int *lda, s *b, int *ldb, s *q, int *ldq, s *z, int *ldz, int *info) noexcept nogil +cdef void sgglse(int *m, int *n, int *p, s *a, int *lda, s *b, int *ldb, s *c, s *d, s *x, s *work, int *lwork, int *info) noexcept nogil +cdef void sggqrf(int *n, int *m, int *p, s *a, int *lda, s *taua, s *b, int *ldb, s *taub, s *work, int *lwork, int *info) noexcept nogil +cdef void sggrqf(int *m, int *p, int *n, s *a, int *lda, s *taua, s *b, int *ldb, s *taub, s *work, int *lwork, int *info) noexcept nogil +cdef void sgsvj0(char *jobv, int *m, int *n, s *a, int *lda, s *d, s *sva, int *mv, s *v, int *ldv, s *eps, s *sfmin, s *tol, int *nsweep, s *work, int *lwork, int *info) noexcept nogil +cdef void sgsvj1(char *jobv, int *m, int *n, int *n1, s *a, int *lda, s *d, s *sva, int *mv, s *v, int *ldv, s *eps, s *sfmin, s *tol, int *nsweep, s *work, int *lwork, int *info) noexcept nogil +cdef void sgtcon(char *norm, int *n, s *dl, s *d, s *du, s *du2, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void sgtrfs(char *trans, int *n, int *nrhs, s *dl, s *d, s *du, s *dlf, s *df, s *duf, s *du2, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void sgtsv(int *n, int *nrhs, s *dl, s *d, s *du, s *b, int *ldb, int *info) noexcept nogil +cdef void sgtsvx(char *fact, char *trans, int *n, int *nrhs, s *dl, s *d, s *du, s *dlf, s *df, s *duf, s *du2, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void sgttrf(int *n, s *dl, s *d, s *du, s *du2, int *ipiv, int *info) noexcept nogil +cdef void sgttrs(char *trans, int *n, int *nrhs, s *dl, s *d, s *du, s *du2, int *ipiv, s *b, int *ldb, int *info) noexcept nogil +cdef void sgtts2(int *itrans, int *n, int *nrhs, s *dl, s *d, s *du, s *du2, int *ipiv, s *b, int *ldb) noexcept nogil +cdef void shgeqz(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *t, int *ldt, s *alphar, s *alphai, s *beta, s *q, int *ldq, s *z, int *ldz, s *work, int *lwork, int *info) noexcept nogil +cdef void shsein(char *side, char *eigsrc, char *initv, bint *select, int *n, s *h, int *ldh, s *wr, s *wi, s *vl, int *ldvl, s *vr, int *ldvr, int *mm, int *m, s *work, int *ifaill, int *ifailr, int *info) noexcept nogil +cdef void shseqr(char *job, char *compz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, s *z, int *ldz, s *work, int *lwork, int *info) noexcept nogil +cdef void slabad(s *small, s *large) noexcept nogil +cdef void slabrd(int *m, int *n, int *nb, s *a, int *lda, s *d, s *e, s *tauq, s *taup, s *x, int *ldx, s *y, int *ldy) noexcept nogil +cdef void slacn2(int *n, s *v, s *x, int *isgn, s *est, int *kase, int *isave) noexcept nogil +cdef void slacon(int *n, s *v, s *x, int *isgn, s *est, int *kase) noexcept nogil +cdef void slacpy(char *uplo, int *m, int *n, s *a, int *lda, s *b, int *ldb) noexcept nogil +cdef void sladiv(s *a, s *b, s *c, s *d, s *p, s *q) noexcept nogil +cdef void slae2(s *a, s *b, s *c, s *rt1, s *rt2) noexcept nogil +cdef void slaebz(int *ijob, int *nitmax, int *n, int *mmax, int *minp, int *nbmin, s *abstol, s *reltol, s *pivmin, s *d, s *e, s *e2, int *nval, s *ab, s *c, int *mout, int *nab, s *work, int *iwork, int *info) noexcept nogil +cdef void slaed0(int *icompq, int *qsiz, int *n, s *d, s *e, s *q, int *ldq, s *qstore, int *ldqs, s *work, int *iwork, int *info) noexcept nogil +cdef void slaed1(int *n, s *d, s *q, int *ldq, int *indxq, s *rho, int *cutpnt, s *work, int *iwork, int *info) noexcept nogil +cdef void slaed2(int *k, int *n, int *n1, s *d, s *q, int *ldq, int *indxq, s *rho, s *z, s *dlamda, s *w, s *q2, int *indx, int *indxc, int *indxp, int *coltyp, int *info) noexcept nogil +cdef void slaed3(int *k, int *n, int *n1, s *d, s *q, int *ldq, s *rho, s *dlamda, s *q2, int *indx, int *ctot, s *w, s *s, int *info) noexcept nogil +cdef void slaed4(int *n, int *i, s *d, s *z, s *delta, s *rho, s *dlam, int *info) noexcept nogil +cdef void slaed5(int *i, s *d, s *z, s *delta, s *rho, s *dlam) noexcept nogil +cdef void slaed6(int *kniter, bint *orgati, s *rho, s *d, s *z, s *finit, s *tau, int *info) noexcept nogil +cdef void slaed7(int *icompq, int *n, int *qsiz, int *tlvls, int *curlvl, int *curpbm, s *d, s *q, int *ldq, int *indxq, s *rho, int *cutpnt, s *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, s *givnum, s *work, int *iwork, int *info) noexcept nogil +cdef void slaed8(int *icompq, int *k, int *n, int *qsiz, s *d, s *q, int *ldq, int *indxq, s *rho, int *cutpnt, s *z, s *dlamda, s *q2, int *ldq2, s *w, int *perm, int *givptr, int *givcol, s *givnum, int *indxp, int *indx, int *info) noexcept nogil +cdef void slaed9(int *k, int *kstart, int *kstop, int *n, s *d, s *q, int *ldq, s *rho, s *dlamda, s *w, s *s, int *lds, int *info) noexcept nogil +cdef void slaeda(int *n, int *tlvls, int *curlvl, int *curpbm, int *prmptr, int *perm, int *givptr, int *givcol, s *givnum, s *q, int *qptr, s *z, s *ztemp, int *info) noexcept nogil +cdef void slaein(bint *rightv, bint *noinit, int *n, s *h, int *ldh, s *wr, s *wi, s *vr, s *vi, s *b, int *ldb, s *work, s *eps3, s *smlnum, s *bignum, int *info) noexcept nogil +cdef void slaev2(s *a, s *b, s *c, s *rt1, s *rt2, s *cs1, s *sn1) noexcept nogil +cdef void slaexc(bint *wantq, int *n, s *t, int *ldt, s *q, int *ldq, int *j1, int *n1, int *n2, s *work, int *info) noexcept nogil +cdef void slag2(s *a, int *lda, s *b, int *ldb, s *safmin, s *scale1, s *scale2, s *wr1, s *wr2, s *wi) noexcept nogil +cdef void slag2d(int *m, int *n, s *sa, int *ldsa, d *a, int *lda, int *info) noexcept nogil +cdef void slags2(bint *upper, s *a1, s *a2, s *a3, s *b1, s *b2, s *b3, s *csu, s *snu, s *csv, s *snv, s *csq, s *snq) noexcept nogil +cdef void slagtf(int *n, s *a, s *lambda_, s *b, s *c, s *tol, s *d, int *in_, int *info) noexcept nogil +cdef void slagtm(char *trans, int *n, int *nrhs, s *alpha, s *dl, s *d, s *du, s *x, int *ldx, s *beta, s *b, int *ldb) noexcept nogil +cdef void slagts(int *job, int *n, s *a, s *b, s *c, s *d, int *in_, s *y, s *tol, int *info) noexcept nogil +cdef void slagv2(s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *csl, s *snl, s *csr, s *snr) noexcept nogil +cdef void slahqr(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, int *iloz, int *ihiz, s *z, int *ldz, int *info) noexcept nogil +cdef void slahr2(int *n, int *k, int *nb, s *a, int *lda, s *tau, s *t, int *ldt, s *y, int *ldy) noexcept nogil +cdef void slaic1(int *job, int *j, s *x, s *sest, s *w, s *gamma, s *sestpr, s *s, s *c) noexcept nogil +cdef void slaln2(bint *ltrans, int *na, int *nw, s *smin, s *ca, s *a, int *lda, s *d1, s *d2, s *b, int *ldb, s *wr, s *wi, s *x, int *ldx, s *scale, s *xnorm, int *info) noexcept nogil +cdef void slals0(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, s *b, int *ldb, s *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *poles, s *difl, s *difr, s *z, int *k, s *c, s *s, s *work, int *info) noexcept nogil +cdef void slalsa(int *icompq, int *smlsiz, int *n, int *nrhs, s *b, int *ldb, s *bx, int *ldbx, s *u, int *ldu, s *vt, int *k, s *difl, s *difr, s *z, s *poles, int *givptr, int *givcol, int *ldgcol, int *perm, s *givnum, s *c, s *s, s *work, int *iwork, int *info) noexcept nogil +cdef void slalsd(char *uplo, int *smlsiz, int *n, int *nrhs, s *d, s *e, s *b, int *ldb, s *rcond, int *rank, s *work, int *iwork, int *info) noexcept nogil +cdef s slamch(char *cmach) noexcept nogil +cdef void slamrg(int *n1, int *n2, s *a, int *strd1, int *strd2, int *index_bn) noexcept nogil +cdef s slangb(char *norm, int *n, int *kl, int *ku, s *ab, int *ldab, s *work) noexcept nogil +cdef s slange(char *norm, int *m, int *n, s *a, int *lda, s *work) noexcept nogil +cdef s slangt(char *norm, int *n, s *dl, s *d, s *du) noexcept nogil +cdef s slanhs(char *norm, int *n, s *a, int *lda, s *work) noexcept nogil +cdef s slansb(char *norm, char *uplo, int *n, int *k, s *ab, int *ldab, s *work) noexcept nogil +cdef s slansf(char *norm, char *transr, char *uplo, int *n, s *a, s *work) noexcept nogil +cdef s slansp(char *norm, char *uplo, int *n, s *ap, s *work) noexcept nogil +cdef s slanst(char *norm, int *n, s *d, s *e) noexcept nogil +cdef s slansy(char *norm, char *uplo, int *n, s *a, int *lda, s *work) noexcept nogil +cdef s slantb(char *norm, char *uplo, char *diag, int *n, int *k, s *ab, int *ldab, s *work) noexcept nogil +cdef s slantp(char *norm, char *uplo, char *diag, int *n, s *ap, s *work) noexcept nogil +cdef s slantr(char *norm, char *uplo, char *diag, int *m, int *n, s *a, int *lda, s *work) noexcept nogil +cdef void slanv2(s *a, s *b, s *c, s *d, s *rt1r, s *rt1i, s *rt2r, s *rt2i, s *cs, s *sn) noexcept nogil +cdef void slapll(int *n, s *x, int *incx, s *y, int *incy, s *ssmin) noexcept nogil +cdef void slapmr(bint *forwrd, int *m, int *n, s *x, int *ldx, int *k) noexcept nogil +cdef void slapmt(bint *forwrd, int *m, int *n, s *x, int *ldx, int *k) noexcept nogil +cdef s slapy2(s *x, s *y) noexcept nogil +cdef s slapy3(s *x, s *y, s *z) noexcept nogil +cdef void slaqgb(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) noexcept nogil +cdef void slaqge(int *m, int *n, s *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) noexcept nogil +cdef void slaqp2(int *m, int *n, int *offset, s *a, int *lda, int *jpvt, s *tau, s *vn1, s *vn2, s *work) noexcept nogil +cdef void slaqps(int *m, int *n, int *offset, int *nb, int *kb, s *a, int *lda, int *jpvt, s *tau, s *vn1, s *vn2, s *auxv, s *f, int *ldf) noexcept nogil +cdef void slaqr0(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, int *iloz, int *ihiz, s *z, int *ldz, s *work, int *lwork, int *info) noexcept nogil +cdef void slaqr1(int *n, s *h, int *ldh, s *sr1, s *si1, s *sr2, s *si2, s *v) noexcept nogil +cdef void slaqr2(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, s *h, int *ldh, int *iloz, int *ihiz, s *z, int *ldz, int *ns, int *nd, s *sr, s *si, s *v, int *ldv, int *nh, s *t, int *ldt, int *nv, s *wv, int *ldwv, s *work, int *lwork) noexcept nogil +cdef void slaqr3(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, s *h, int *ldh, int *iloz, int *ihiz, s *z, int *ldz, int *ns, int *nd, s *sr, s *si, s *v, int *ldv, int *nh, s *t, int *ldt, int *nv, s *wv, int *ldwv, s *work, int *lwork) noexcept nogil +cdef void slaqr4(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, int *iloz, int *ihiz, s *z, int *ldz, s *work, int *lwork, int *info) noexcept nogil +cdef void slaqr5(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, s *sr, s *si, s *h, int *ldh, int *iloz, int *ihiz, s *z, int *ldz, s *v, int *ldv, s *u, int *ldu, int *nv, s *wv, int *ldwv, int *nh, s *wh, int *ldwh) noexcept nogil +cdef void slaqsb(char *uplo, int *n, int *kd, s *ab, int *ldab, s *s, s *scond, s *amax, char *equed) noexcept nogil +cdef void slaqsp(char *uplo, int *n, s *ap, s *s, s *scond, s *amax, char *equed) noexcept nogil +cdef void slaqsy(char *uplo, int *n, s *a, int *lda, s *s, s *scond, s *amax, char *equed) noexcept nogil +cdef void slaqtr(bint *ltran, bint *lreal, int *n, s *t, int *ldt, s *b, s *w, s *scale, s *x, s *work, int *info) noexcept nogil +cdef void slar1v(int *n, int *b1, int *bn, s *lambda_, s *d, s *l, s *ld, s *lld, s *pivmin, s *gaptol, s *z, bint *wantnc, int *negcnt, s *ztz, s *mingma, int *r, int *isuppz, s *nrminv, s *resid, s *rqcorr, s *work) noexcept nogil +cdef void slar2v(int *n, s *x, s *y, s *z, int *incx, s *c, s *s, int *incc) noexcept nogil +cdef void slarf(char *side, int *m, int *n, s *v, int *incv, s *tau, s *c, int *ldc, s *work) noexcept nogil +cdef void slarfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, s *v, int *ldv, s *t, int *ldt, s *c, int *ldc, s *work, int *ldwork) noexcept nogil +cdef void slarfg(int *n, s *alpha, s *x, int *incx, s *tau) noexcept nogil +cdef void slarfgp(int *n, s *alpha, s *x, int *incx, s *tau) noexcept nogil +cdef void slarft(char *direct, char *storev, int *n, int *k, s *v, int *ldv, s *tau, s *t, int *ldt) noexcept nogil +cdef void slarfx(char *side, int *m, int *n, s *v, s *tau, s *c, int *ldc, s *work) noexcept nogil +cdef void slargv(int *n, s *x, int *incx, s *y, int *incy, s *c, int *incc) noexcept nogil +cdef void slarnv(int *idist, int *iseed, int *n, s *x) noexcept nogil +cdef void slarra(int *n, s *d, s *e, s *e2, s *spltol, s *tnrm, int *nsplit, int *isplit, int *info) noexcept nogil +cdef void slarrb(int *n, s *d, s *lld, int *ifirst, int *ilast, s *rtol1, s *rtol2, int *offset, s *w, s *wgap, s *werr, s *work, int *iwork, s *pivmin, s *spdiam, int *twist, int *info) noexcept nogil +cdef void slarrc(char *jobt, int *n, s *vl, s *vu, s *d, s *e, s *pivmin, int *eigcnt, int *lcnt, int *rcnt, int *info) noexcept nogil +cdef void slarrd(char *range, char *order, int *n, s *vl, s *vu, int *il, int *iu, s *gers, s *reltol, s *d, s *e, s *e2, s *pivmin, int *nsplit, int *isplit, int *m, s *w, s *werr, s *wl, s *wu, int *iblock, int *indexw, s *work, int *iwork, int *info) noexcept nogil +cdef void slarre(char *range, int *n, s *vl, s *vu, int *il, int *iu, s *d, s *e, s *e2, s *rtol1, s *rtol2, s *spltol, int *nsplit, int *isplit, int *m, s *w, s *werr, s *wgap, int *iblock, int *indexw, s *gers, s *pivmin, s *work, int *iwork, int *info) noexcept nogil +cdef void slarrf(int *n, s *d, s *l, s *ld, int *clstrt, int *clend, s *w, s *wgap, s *werr, s *spdiam, s *clgapl, s *clgapr, s *pivmin, s *sigma, s *dplus, s *lplus, s *work, int *info) noexcept nogil +cdef void slarrj(int *n, s *d, s *e2, int *ifirst, int *ilast, s *rtol, int *offset, s *w, s *werr, s *work, int *iwork, s *pivmin, s *spdiam, int *info) noexcept nogil +cdef void slarrk(int *n, int *iw, s *gl, s *gu, s *d, s *e2, s *pivmin, s *reltol, s *w, s *werr, int *info) noexcept nogil +cdef void slarrr(int *n, s *d, s *e, int *info) noexcept nogil +cdef void slarrv(int *n, s *vl, s *vu, s *d, s *l, s *pivmin, int *isplit, int *m, int *dol, int *dou, s *minrgp, s *rtol1, s *rtol2, s *w, s *werr, s *wgap, int *iblock, int *indexw, s *gers, s *z, int *ldz, int *isuppz, s *work, int *iwork, int *info) noexcept nogil +cdef void slartg(s *f, s *g, s *cs, s *sn, s *r) noexcept nogil +cdef void slartgp(s *f, s *g, s *cs, s *sn, s *r) noexcept nogil +cdef void slartgs(s *x, s *y, s *sigma, s *cs, s *sn) noexcept nogil +cdef void slartv(int *n, s *x, int *incx, s *y, int *incy, s *c, s *s, int *incc) noexcept nogil +cdef void slaruv(int *iseed, int *n, s *x) noexcept nogil +cdef void slarz(char *side, int *m, int *n, int *l, s *v, int *incv, s *tau, s *c, int *ldc, s *work) noexcept nogil +cdef void slarzb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, s *v, int *ldv, s *t, int *ldt, s *c, int *ldc, s *work, int *ldwork) noexcept nogil +cdef void slarzt(char *direct, char *storev, int *n, int *k, s *v, int *ldv, s *tau, s *t, int *ldt) noexcept nogil +cdef void slas2(s *f, s *g, s *h, s *ssmin, s *ssmax) noexcept nogil +cdef void slascl(char *type_bn, int *kl, int *ku, s *cfrom, s *cto, int *m, int *n, s *a, int *lda, int *info) noexcept nogil +cdef void slasd0(int *n, int *sqre, s *d, s *e, s *u, int *ldu, s *vt, int *ldvt, int *smlsiz, int *iwork, s *work, int *info) noexcept nogil +cdef void slasd1(int *nl, int *nr, int *sqre, s *d, s *alpha, s *beta, s *u, int *ldu, s *vt, int *ldvt, int *idxq, int *iwork, s *work, int *info) noexcept nogil +cdef void slasd2(int *nl, int *nr, int *sqre, int *k, s *d, s *z, s *alpha, s *beta, s *u, int *ldu, s *vt, int *ldvt, s *dsigma, s *u2, int *ldu2, s *vt2, int *ldvt2, int *idxp, int *idx, int *idxc, int *idxq, int *coltyp, int *info) noexcept nogil +cdef void slasd3(int *nl, int *nr, int *sqre, int *k, s *d, s *q, int *ldq, s *dsigma, s *u, int *ldu, s *u2, int *ldu2, s *vt, int *ldvt, s *vt2, int *ldvt2, int *idxc, int *ctot, s *z, int *info) noexcept nogil +cdef void slasd4(int *n, int *i, s *d, s *z, s *delta, s *rho, s *sigma, s *work, int *info) noexcept nogil +cdef void slasd5(int *i, s *d, s *z, s *delta, s *rho, s *dsigma, s *work) noexcept nogil +cdef void slasd6(int *icompq, int *nl, int *nr, int *sqre, s *d, s *vf, s *vl, s *alpha, s *beta, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *poles, s *difl, s *difr, s *z, int *k, s *c, s *s, s *work, int *iwork, int *info) noexcept nogil +cdef void slasd7(int *icompq, int *nl, int *nr, int *sqre, int *k, s *d, s *z, s *zw, s *vf, s *vfw, s *vl, s *vlw, s *alpha, s *beta, s *dsigma, int *idx, int *idxp, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *c, s *s, int *info) noexcept nogil +cdef void slasd8(int *icompq, int *k, s *d, s *z, s *vf, s *vl, s *difl, s *difr, int *lddifr, s *dsigma, s *work, int *info) noexcept nogil +cdef void slasda(int *icompq, int *smlsiz, int *n, int *sqre, s *d, s *e, s *u, int *ldu, s *vt, int *k, s *difl, s *difr, s *z, s *poles, int *givptr, int *givcol, int *ldgcol, int *perm, s *givnum, s *c, s *s, s *work, int *iwork, int *info) noexcept nogil +cdef void slasdq(char *uplo, int *sqre, int *n, int *ncvt, int *nru, int *ncc, s *d, s *e, s *vt, int *ldvt, s *u, int *ldu, s *c, int *ldc, s *work, int *info) noexcept nogil +cdef void slasdt(int *n, int *lvl, int *nd, int *inode, int *ndiml, int *ndimr, int *msub) noexcept nogil +cdef void slaset(char *uplo, int *m, int *n, s *alpha, s *beta, s *a, int *lda) noexcept nogil +cdef void slasq1(int *n, s *d, s *e, s *work, int *info) noexcept nogil +cdef void slasq2(int *n, s *z, int *info) noexcept nogil +cdef void slasq3(int *i0, int *n0, s *z, int *pp, s *dmin, s *sigma, s *desig, s *qmax, int *nfail, int *iter, int *ndiv, bint *ieee, int *ttype, s *dmin1, s *dmin2, s *dn, s *dn1, s *dn2, s *g, s *tau) noexcept nogil +cdef void slasq4(int *i0, int *n0, s *z, int *pp, int *n0in, s *dmin, s *dmin1, s *dmin2, s *dn, s *dn1, s *dn2, s *tau, int *ttype, s *g) noexcept nogil +cdef void slasq6(int *i0, int *n0, s *z, int *pp, s *dmin, s *dmin1, s *dmin2, s *dn, s *dnm1, s *dnm2) noexcept nogil +cdef void slasr(char *side, char *pivot, char *direct, int *m, int *n, s *c, s *s, s *a, int *lda) noexcept nogil +cdef void slasrt(char *id, int *n, s *d, int *info) noexcept nogil +cdef void slassq(int *n, s *x, int *incx, s *scale, s *sumsq) noexcept nogil +cdef void slasv2(s *f, s *g, s *h, s *ssmin, s *ssmax, s *snr, s *csr, s *snl, s *csl) noexcept nogil +cdef void slaswp(int *n, s *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) noexcept nogil +cdef void slasy2(bint *ltranl, bint *ltranr, int *isgn, int *n1, int *n2, s *tl, int *ldtl, s *tr, int *ldtr, s *b, int *ldb, s *scale, s *x, int *ldx, s *xnorm, int *info) noexcept nogil +cdef void slasyf(char *uplo, int *n, int *nb, int *kb, s *a, int *lda, int *ipiv, s *w, int *ldw, int *info) noexcept nogil +cdef void slatbs(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, s *ab, int *ldab, s *x, s *scale, s *cnorm, int *info) noexcept nogil +cdef void slatdf(int *ijob, int *n, s *z, int *ldz, s *rhs, s *rdsum, s *rdscal, int *ipiv, int *jpiv) noexcept nogil +cdef void slatps(char *uplo, char *trans, char *diag, char *normin, int *n, s *ap, s *x, s *scale, s *cnorm, int *info) noexcept nogil +cdef void slatrd(char *uplo, int *n, int *nb, s *a, int *lda, s *e, s *tau, s *w, int *ldw) noexcept nogil +cdef void slatrs(char *uplo, char *trans, char *diag, char *normin, int *n, s *a, int *lda, s *x, s *scale, s *cnorm, int *info) noexcept nogil +cdef void slatrz(int *m, int *n, int *l, s *a, int *lda, s *tau, s *work) noexcept nogil +cdef void slauu2(char *uplo, int *n, s *a, int *lda, int *info) noexcept nogil +cdef void slauum(char *uplo, int *n, s *a, int *lda, int *info) noexcept nogil +cdef void sopgtr(char *uplo, int *n, s *ap, s *tau, s *q, int *ldq, s *work, int *info) noexcept nogil +cdef void sopmtr(char *side, char *uplo, char *trans, int *m, int *n, s *ap, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil +cdef void sorbdb(char *trans, char *signs, int *m, int *p, int *q, s *x11, int *ldx11, s *x12, int *ldx12, s *x21, int *ldx21, s *x22, int *ldx22, s *theta, s *phi, s *taup1, s *taup2, s *tauq1, s *tauq2, s *work, int *lwork, int *info) noexcept nogil +cdef void sorcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, s *x11, int *ldx11, s *x12, int *ldx12, s *x21, int *ldx21, s *x22, int *ldx22, s *theta, s *u1, int *ldu1, s *u2, int *ldu2, s *v1t, int *ldv1t, s *v2t, int *ldv2t, s *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void sorg2l(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil +cdef void sorg2r(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil +cdef void sorgbr(char *vect, int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sorghr(int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sorgl2(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil +cdef void sorglq(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sorgql(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sorgqr(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sorgr2(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil +cdef void sorgrq(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sorgtr(char *uplo, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void sorm2l(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil +cdef void sorm2r(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil +cdef void sormbr(char *vect, char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil +cdef void sormhr(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil +cdef void sorml2(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil +cdef void sormlq(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil +cdef void sormql(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil +cdef void sormqr(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil +cdef void sormr2(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil +cdef void sormr3(char *side, char *trans, int *m, int *n, int *k, int *l, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil +cdef void sormrq(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil +cdef void sormrz(char *side, char *trans, int *m, int *n, int *k, int *l, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil +cdef void sormtr(char *side, char *uplo, char *trans, int *m, int *n, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil +cdef void spbcon(char *uplo, int *n, int *kd, s *ab, int *ldab, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void spbequ(char *uplo, int *n, int *kd, s *ab, int *ldab, s *s, s *scond, s *amax, int *info) noexcept nogil +cdef void spbrfs(char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void spbstf(char *uplo, int *n, int *kd, s *ab, int *ldab, int *info) noexcept nogil +cdef void spbsv(char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, int *info) noexcept nogil +cdef void spbsvx(char *fact, char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, char *equed, s *s, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void spbtf2(char *uplo, int *n, int *kd, s *ab, int *ldab, int *info) noexcept nogil +cdef void spbtrf(char *uplo, int *n, int *kd, s *ab, int *ldab, int *info) noexcept nogil +cdef void spbtrs(char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, int *info) noexcept nogil +cdef void spftrf(char *transr, char *uplo, int *n, s *a, int *info) noexcept nogil +cdef void spftri(char *transr, char *uplo, int *n, s *a, int *info) noexcept nogil +cdef void spftrs(char *transr, char *uplo, int *n, int *nrhs, s *a, s *b, int *ldb, int *info) noexcept nogil +cdef void spocon(char *uplo, int *n, s *a, int *lda, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void spoequ(int *n, s *a, int *lda, s *s, s *scond, s *amax, int *info) noexcept nogil +cdef void spoequb(int *n, s *a, int *lda, s *s, s *scond, s *amax, int *info) noexcept nogil +cdef void sporfs(char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void sposv(char *uplo, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *info) noexcept nogil +cdef void sposvx(char *fact, char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, char *equed, s *s, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void spotf2(char *uplo, int *n, s *a, int *lda, int *info) noexcept nogil +cdef void spotrf(char *uplo, int *n, s *a, int *lda, int *info) noexcept nogil +cdef void spotri(char *uplo, int *n, s *a, int *lda, int *info) noexcept nogil +cdef void spotrs(char *uplo, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *info) noexcept nogil +cdef void sppcon(char *uplo, int *n, s *ap, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void sppequ(char *uplo, int *n, s *ap, s *s, s *scond, s *amax, int *info) noexcept nogil +cdef void spprfs(char *uplo, int *n, int *nrhs, s *ap, s *afp, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void sppsv(char *uplo, int *n, int *nrhs, s *ap, s *b, int *ldb, int *info) noexcept nogil +cdef void sppsvx(char *fact, char *uplo, int *n, int *nrhs, s *ap, s *afp, char *equed, s *s, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void spptrf(char *uplo, int *n, s *ap, int *info) noexcept nogil +cdef void spptri(char *uplo, int *n, s *ap, int *info) noexcept nogil +cdef void spptrs(char *uplo, int *n, int *nrhs, s *ap, s *b, int *ldb, int *info) noexcept nogil +cdef void spstf2(char *uplo, int *n, s *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) noexcept nogil +cdef void spstrf(char *uplo, int *n, s *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) noexcept nogil +cdef void sptcon(int *n, s *d, s *e, s *anorm, s *rcond, s *work, int *info) noexcept nogil +cdef void spteqr(char *compz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *info) noexcept nogil +cdef void sptrfs(int *n, int *nrhs, s *d, s *e, s *df, s *ef, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *info) noexcept nogil +cdef void sptsv(int *n, int *nrhs, s *d, s *e, s *b, int *ldb, int *info) noexcept nogil +cdef void sptsvx(char *fact, int *n, int *nrhs, s *d, s *e, s *df, s *ef, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *info) noexcept nogil +cdef void spttrf(int *n, s *d, s *e, int *info) noexcept nogil +cdef void spttrs(int *n, int *nrhs, s *d, s *e, s *b, int *ldb, int *info) noexcept nogil +cdef void sptts2(int *n, int *nrhs, s *d, s *e, s *b, int *ldb) noexcept nogil +cdef void srscl(int *n, s *sa, s *sx, int *incx) noexcept nogil +cdef void ssbev(char *jobz, char *uplo, int *n, int *kd, s *ab, int *ldab, s *w, s *z, int *ldz, s *work, int *info) noexcept nogil +cdef void ssbevd(char *jobz, char *uplo, int *n, int *kd, s *ab, int *ldab, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void ssbevx(char *jobz, char *range, char *uplo, int *n, int *kd, s *ab, int *ldab, s *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void ssbgst(char *vect, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *x, int *ldx, s *work, int *info) noexcept nogil +cdef void ssbgv(char *jobz, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *w, s *z, int *ldz, s *work, int *info) noexcept nogil +cdef void ssbgvd(char *jobz, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void ssbgvx(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void ssbtrd(char *vect, char *uplo, int *n, int *kd, s *ab, int *ldab, s *d, s *e, s *q, int *ldq, s *work, int *info) noexcept nogil +cdef void ssfrk(char *transr, char *uplo, char *trans, int *n, int *k, s *alpha, s *a, int *lda, s *beta, s *c) noexcept nogil +cdef void sspcon(char *uplo, int *n, s *ap, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void sspev(char *jobz, char *uplo, int *n, s *ap, s *w, s *z, int *ldz, s *work, int *info) noexcept nogil +cdef void sspevd(char *jobz, char *uplo, int *n, s *ap, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void sspevx(char *jobz, char *range, char *uplo, int *n, s *ap, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void sspgst(int *itype, char *uplo, int *n, s *ap, s *bp, int *info) noexcept nogil +cdef void sspgv(int *itype, char *jobz, char *uplo, int *n, s *ap, s *bp, s *w, s *z, int *ldz, s *work, int *info) noexcept nogil +cdef void sspgvd(int *itype, char *jobz, char *uplo, int *n, s *ap, s *bp, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void sspgvx(int *itype, char *jobz, char *range, char *uplo, int *n, s *ap, s *bp, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void ssprfs(char *uplo, int *n, int *nrhs, s *ap, s *afp, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void sspsv(char *uplo, int *n, int *nrhs, s *ap, int *ipiv, s *b, int *ldb, int *info) noexcept nogil +cdef void sspsvx(char *fact, char *uplo, int *n, int *nrhs, s *ap, s *afp, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void ssptrd(char *uplo, int *n, s *ap, s *d, s *e, s *tau, int *info) noexcept nogil +cdef void ssptrf(char *uplo, int *n, s *ap, int *ipiv, int *info) noexcept nogil +cdef void ssptri(char *uplo, int *n, s *ap, int *ipiv, s *work, int *info) noexcept nogil +cdef void ssptrs(char *uplo, int *n, int *nrhs, s *ap, int *ipiv, s *b, int *ldb, int *info) noexcept nogil +cdef void sstebz(char *range, char *order, int *n, s *vl, s *vu, int *il, int *iu, s *abstol, s *d, s *e, int *m, int *nsplit, s *w, int *iblock, int *isplit, s *work, int *iwork, int *info) noexcept nogil +cdef void sstedc(char *compz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void sstegr(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void sstein(int *n, s *d, s *e, int *m, s *w, int *iblock, int *isplit, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void sstemr(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, int *m, s *w, s *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void ssteqr(char *compz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *info) noexcept nogil +cdef void ssterf(int *n, s *d, s *e, int *info) noexcept nogil +cdef void sstev(char *jobz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *info) noexcept nogil +cdef void sstevd(char *jobz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void sstevr(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void sstevx(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void ssycon(char *uplo, int *n, s *a, int *lda, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void ssyconv(char *uplo, char *way, int *n, s *a, int *lda, int *ipiv, s *work, int *info) noexcept nogil +cdef void ssyequb(char *uplo, int *n, s *a, int *lda, s *s, s *scond, s *amax, s *work, int *info) noexcept nogil +cdef void ssyev(char *jobz, char *uplo, int *n, s *a, int *lda, s *w, s *work, int *lwork, int *info) noexcept nogil +cdef void ssyevd(char *jobz, char *uplo, int *n, s *a, int *lda, s *w, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void ssyevr(char *jobz, char *range, char *uplo, int *n, s *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void ssyevx(char *jobz, char *range, char *uplo, int *n, s *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void ssygs2(int *itype, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, int *info) noexcept nogil +cdef void ssygst(int *itype, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, int *info) noexcept nogil +cdef void ssygv(int *itype, char *jobz, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, s *w, s *work, int *lwork, int *info) noexcept nogil +cdef void ssygvd(int *itype, char *jobz, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, s *w, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void ssygvx(int *itype, char *jobz, char *range, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void ssyrfs(char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void ssysv(char *uplo, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, s *work, int *lwork, int *info) noexcept nogil +cdef void ssysvx(char *fact, char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void ssyswapr(char *uplo, int *n, s *a, int *lda, int *i1, int *i2) noexcept nogil +cdef void ssytd2(char *uplo, int *n, s *a, int *lda, s *d, s *e, s *tau, int *info) noexcept nogil +cdef void ssytf2(char *uplo, int *n, s *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void ssytrd(char *uplo, int *n, s *a, int *lda, s *d, s *e, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void ssytrf(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *lwork, int *info) noexcept nogil +cdef void ssytri(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *info) noexcept nogil +cdef void ssytri2(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *lwork, int *info) noexcept nogil +cdef void ssytri2x(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *nb, int *info) noexcept nogil +cdef void ssytrs(char *uplo, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, int *info) noexcept nogil +cdef void ssytrs2(char *uplo, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, s *work, int *info) noexcept nogil +cdef void stbcon(char *norm, char *uplo, char *diag, int *n, int *kd, s *ab, int *ldab, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void stbrfs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void stbtrs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, int *info) noexcept nogil +cdef void stfsm(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, s *alpha, s *a, s *b, int *ldb) noexcept nogil +cdef void stftri(char *transr, char *uplo, char *diag, int *n, s *a, int *info) noexcept nogil +cdef void stfttp(char *transr, char *uplo, int *n, s *arf, s *ap, int *info) noexcept nogil +cdef void stfttr(char *transr, char *uplo, int *n, s *arf, s *a, int *lda, int *info) noexcept nogil +cdef void stgevc(char *side, char *howmny, bint *select, int *n, s *s, int *lds, s *p, int *ldp, s *vl, int *ldvl, s *vr, int *ldvr, int *mm, int *m, s *work, int *info) noexcept nogil +cdef void stgex2(bint *wantq, bint *wantz, int *n, s *a, int *lda, s *b, int *ldb, s *q, int *ldq, s *z, int *ldz, int *j1, int *n1, int *n2, s *work, int *lwork, int *info) noexcept nogil +cdef void stgexc(bint *wantq, bint *wantz, int *n, s *a, int *lda, s *b, int *ldb, s *q, int *ldq, s *z, int *ldz, int *ifst, int *ilst, s *work, int *lwork, int *info) noexcept nogil +cdef void stgsen(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *q, int *ldq, s *z, int *ldz, int *m, s *pl, s *pr, s *dif, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void stgsja(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, s *a, int *lda, s *b, int *ldb, s *tola, s *tolb, s *alpha, s *beta, s *u, int *ldu, s *v, int *ldv, s *q, int *ldq, s *work, int *ncycle, int *info) noexcept nogil +cdef void stgsna(char *job, char *howmny, bint *select, int *n, s *a, int *lda, s *b, int *ldb, s *vl, int *ldvl, s *vr, int *ldvr, s *s, s *dif, int *mm, int *m, s *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void stgsy2(char *trans, int *ijob, int *m, int *n, s *a, int *lda, s *b, int *ldb, s *c, int *ldc, s *d, int *ldd, s *e, int *lde, s *f, int *ldf, s *scale, s *rdsum, s *rdscal, int *iwork, int *pq, int *info) noexcept nogil +cdef void stgsyl(char *trans, int *ijob, int *m, int *n, s *a, int *lda, s *b, int *ldb, s *c, int *ldc, s *d, int *ldd, s *e, int *lde, s *f, int *ldf, s *scale, s *dif, s *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void stpcon(char *norm, char *uplo, char *diag, int *n, s *ap, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void stpmqrt(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, s *v, int *ldv, s *t, int *ldt, s *a, int *lda, s *b, int *ldb, s *work, int *info) noexcept nogil +cdef void stpqrt(int *m, int *n, int *l, int *nb, s *a, int *lda, s *b, int *ldb, s *t, int *ldt, s *work, int *info) noexcept nogil +cdef void stpqrt2(int *m, int *n, int *l, s *a, int *lda, s *b, int *ldb, s *t, int *ldt, int *info) noexcept nogil +cdef void stprfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, s *v, int *ldv, s *t, int *ldt, s *a, int *lda, s *b, int *ldb, s *work, int *ldwork) noexcept nogil +cdef void stprfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *ap, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void stptri(char *uplo, char *diag, int *n, s *ap, int *info) noexcept nogil +cdef void stptrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *ap, s *b, int *ldb, int *info) noexcept nogil +cdef void stpttf(char *transr, char *uplo, int *n, s *ap, s *arf, int *info) noexcept nogil +cdef void stpttr(char *uplo, int *n, s *ap, s *a, int *lda, int *info) noexcept nogil +cdef void strcon(char *norm, char *uplo, char *diag, int *n, s *a, int *lda, s *rcond, s *work, int *iwork, int *info) noexcept nogil +cdef void strevc(char *side, char *howmny, bint *select, int *n, s *t, int *ldt, s *vl, int *ldvl, s *vr, int *ldvr, int *mm, int *m, s *work, int *info) noexcept nogil +cdef void strexc(char *compq, int *n, s *t, int *ldt, s *q, int *ldq, int *ifst, int *ilst, s *work, int *info) noexcept nogil +cdef void strrfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil +cdef void strsen(char *job, char *compq, bint *select, int *n, s *t, int *ldt, s *q, int *ldq, s *wr, s *wi, int *m, s *s, s *sep, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void strsna(char *job, char *howmny, bint *select, int *n, s *t, int *ldt, s *vl, int *ldvl, s *vr, int *ldvr, s *s, s *sep, int *mm, int *m, s *work, int *ldwork, int *iwork, int *info) noexcept nogil +cdef void strsyl(char *trana, char *tranb, int *isgn, int *m, int *n, s *a, int *lda, s *b, int *ldb, s *c, int *ldc, s *scale, int *info) noexcept nogil +cdef void strti2(char *uplo, char *diag, int *n, s *a, int *lda, int *info) noexcept nogil +cdef void strtri(char *uplo, char *diag, int *n, s *a, int *lda, int *info) noexcept nogil +cdef void strtrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *info) noexcept nogil +cdef void strttf(char *transr, char *uplo, int *n, s *a, int *lda, s *arf, int *info) noexcept nogil +cdef void strttp(char *uplo, int *n, s *a, int *lda, s *ap, int *info) noexcept nogil +cdef void stzrzf(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil +cdef void xerbla_array(char *srname_array, int *srname_len, int *info) noexcept nogil +cdef void zbbcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, d *theta, d *phi, z *u1, int *ldu1, z *u2, int *ldu2, z *v1t, int *ldv1t, z *v2t, int *ldv2t, d *b11d, d *b11e, d *b12d, d *b12e, d *b21d, d *b21e, d *b22d, d *b22e, d *rwork, int *lrwork, int *info) noexcept nogil +cdef void zbdsqr(char *uplo, int *n, int *ncvt, int *nru, int *ncc, d *d, d *e, z *vt, int *ldvt, z *u, int *ldu, z *c, int *ldc, d *rwork, int *info) noexcept nogil +cdef void zcgesv(int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, z *x, int *ldx, z *work, c *swork, d *rwork, int *iter, int *info) noexcept nogil +cdef void zcposv(char *uplo, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, z *x, int *ldx, z *work, c *swork, d *rwork, int *iter, int *info) noexcept nogil +cdef void zdrscl(int *n, d *sa, z *sx, int *incx) noexcept nogil +cdef void zgbbrd(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, z *ab, int *ldab, d *d, d *e, z *q, int *ldq, z *pt, int *ldpt, z *c, int *ldc, z *work, d *rwork, int *info) noexcept nogil +cdef void zgbcon(char *norm, int *n, int *kl, int *ku, z *ab, int *ldab, int *ipiv, d *anorm, d *rcond, z *work, d *rwork, int *info) noexcept nogil +cdef void zgbequ(int *m, int *n, int *kl, int *ku, z *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil +cdef void zgbequb(int *m, int *n, int *kl, int *ku, z *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil +cdef void zgbrfs(char *trans, int *n, int *kl, int *ku, int *nrhs, z *ab, int *ldab, z *afb, int *ldafb, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zgbsv(int *n, int *kl, int *ku, int *nrhs, z *ab, int *ldab, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zgbsvx(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, z *ab, int *ldab, z *afb, int *ldafb, int *ipiv, char *equed, d *r, d *c, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zgbtf2(int *m, int *n, int *kl, int *ku, z *ab, int *ldab, int *ipiv, int *info) noexcept nogil +cdef void zgbtrf(int *m, int *n, int *kl, int *ku, z *ab, int *ldab, int *ipiv, int *info) noexcept nogil +cdef void zgbtrs(char *trans, int *n, int *kl, int *ku, int *nrhs, z *ab, int *ldab, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zgebak(char *job, char *side, int *n, int *ilo, int *ihi, d *scale, int *m, z *v, int *ldv, int *info) noexcept nogil +cdef void zgebal(char *job, int *n, z *a, int *lda, int *ilo, int *ihi, d *scale, int *info) noexcept nogil +cdef void zgebd2(int *m, int *n, z *a, int *lda, d *d, d *e, z *tauq, z *taup, z *work, int *info) noexcept nogil +cdef void zgebrd(int *m, int *n, z *a, int *lda, d *d, d *e, z *tauq, z *taup, z *work, int *lwork, int *info) noexcept nogil +cdef void zgecon(char *norm, int *n, z *a, int *lda, d *anorm, d *rcond, z *work, d *rwork, int *info) noexcept nogil +cdef void zgeequ(int *m, int *n, z *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil +cdef void zgeequb(int *m, int *n, z *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil +cdef void zgees(char *jobvs, char *sort, zselect1 *select, int *n, z *a, int *lda, int *sdim, z *w, z *vs, int *ldvs, z *work, int *lwork, d *rwork, bint *bwork, int *info) noexcept nogil +cdef void zgeesx(char *jobvs, char *sort, zselect1 *select, char *sense, int *n, z *a, int *lda, int *sdim, z *w, z *vs, int *ldvs, d *rconde, d *rcondv, z *work, int *lwork, d *rwork, bint *bwork, int *info) noexcept nogil +cdef void zgeev(char *jobvl, char *jobvr, int *n, z *a, int *lda, z *w, z *vl, int *ldvl, z *vr, int *ldvr, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zgeevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, z *a, int *lda, z *w, z *vl, int *ldvl, z *vr, int *ldvr, int *ilo, int *ihi, d *scale, d *abnrm, d *rconde, d *rcondv, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zgehd2(int *n, int *ilo, int *ihi, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil +cdef void zgehrd(int *n, int *ilo, int *ihi, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zgelq2(int *m, int *n, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil +cdef void zgelqf(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zgels(char *trans, int *m, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, z *work, int *lwork, int *info) noexcept nogil +cdef void zgelsd(int *m, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, d *s, d *rcond, int *rank, z *work, int *lwork, d *rwork, int *iwork, int *info) noexcept nogil +cdef void zgelss(int *m, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, d *s, d *rcond, int *rank, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zgelsy(int *m, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, int *jpvt, d *rcond, int *rank, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zgemqrt(char *side, char *trans, int *m, int *n, int *k, int *nb, z *v, int *ldv, z *t, int *ldt, z *c, int *ldc, z *work, int *info) noexcept nogil +cdef void zgeql2(int *m, int *n, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil +cdef void zgeqlf(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zgeqp3(int *m, int *n, z *a, int *lda, int *jpvt, z *tau, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zgeqr2(int *m, int *n, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil +cdef void zgeqr2p(int *m, int *n, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil +cdef void zgeqrf(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zgeqrfp(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zgeqrt(int *m, int *n, int *nb, z *a, int *lda, z *t, int *ldt, z *work, int *info) noexcept nogil +cdef void zgeqrt2(int *m, int *n, z *a, int *lda, z *t, int *ldt, int *info) noexcept nogil +cdef void zgeqrt3(int *m, int *n, z *a, int *lda, z *t, int *ldt, int *info) noexcept nogil +cdef void zgerfs(char *trans, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zgerq2(int *m, int *n, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil +cdef void zgerqf(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zgesc2(int *n, z *a, int *lda, z *rhs, int *ipiv, int *jpiv, d *scale) noexcept nogil +cdef void zgesdd(char *jobz, int *m, int *n, z *a, int *lda, d *s, z *u, int *ldu, z *vt, int *ldvt, z *work, int *lwork, d *rwork, int *iwork, int *info) noexcept nogil +cdef void zgesv(int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zgesvd(char *jobu, char *jobvt, int *m, int *n, z *a, int *lda, d *s, z *u, int *ldu, z *vt, int *ldvt, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zgesvx(char *fact, char *trans, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, char *equed, d *r, d *c, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zgetc2(int *n, z *a, int *lda, int *ipiv, int *jpiv, int *info) noexcept nogil +cdef void zgetf2(int *m, int *n, z *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void zgetrf(int *m, int *n, z *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void zgetri(int *n, z *a, int *lda, int *ipiv, z *work, int *lwork, int *info) noexcept nogil +cdef void zgetrs(char *trans, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zggbak(char *job, char *side, int *n, int *ilo, int *ihi, d *lscale, d *rscale, int *m, z *v, int *ldv, int *info) noexcept nogil +cdef void zggbal(char *job, int *n, z *a, int *lda, z *b, int *ldb, int *ilo, int *ihi, d *lscale, d *rscale, d *work, int *info) noexcept nogil +cdef void zgges(char *jobvsl, char *jobvsr, char *sort, zselect2 *selctg, int *n, z *a, int *lda, z *b, int *ldb, int *sdim, z *alpha, z *beta, z *vsl, int *ldvsl, z *vsr, int *ldvsr, z *work, int *lwork, d *rwork, bint *bwork, int *info) noexcept nogil +cdef void zggesx(char *jobvsl, char *jobvsr, char *sort, zselect2 *selctg, char *sense, int *n, z *a, int *lda, z *b, int *ldb, int *sdim, z *alpha, z *beta, z *vsl, int *ldvsl, z *vsr, int *ldvsr, d *rconde, d *rcondv, z *work, int *lwork, d *rwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil +cdef void zggev(char *jobvl, char *jobvr, int *n, z *a, int *lda, z *b, int *ldb, z *alpha, z *beta, z *vl, int *ldvl, z *vr, int *ldvr, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zggevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, z *a, int *lda, z *b, int *ldb, z *alpha, z *beta, z *vl, int *ldvl, z *vr, int *ldvr, int *ilo, int *ihi, d *lscale, d *rscale, d *abnrm, d *bbnrm, d *rconde, d *rcondv, z *work, int *lwork, d *rwork, int *iwork, bint *bwork, int *info) noexcept nogil +cdef void zggglm(int *n, int *m, int *p, z *a, int *lda, z *b, int *ldb, z *d, z *x, z *y, z *work, int *lwork, int *info) noexcept nogil +cdef void zgghrd(char *compq, char *compz, int *n, int *ilo, int *ihi, z *a, int *lda, z *b, int *ldb, z *q, int *ldq, z *z, int *ldz, int *info) noexcept nogil +cdef void zgglse(int *m, int *n, int *p, z *a, int *lda, z *b, int *ldb, z *c, z *d, z *x, z *work, int *lwork, int *info) noexcept nogil +cdef void zggqrf(int *n, int *m, int *p, z *a, int *lda, z *taua, z *b, int *ldb, z *taub, z *work, int *lwork, int *info) noexcept nogil +cdef void zggrqf(int *m, int *p, int *n, z *a, int *lda, z *taua, z *b, int *ldb, z *taub, z *work, int *lwork, int *info) noexcept nogil +cdef void zgtcon(char *norm, int *n, z *dl, z *d, z *du, z *du2, int *ipiv, d *anorm, d *rcond, z *work, int *info) noexcept nogil +cdef void zgtrfs(char *trans, int *n, int *nrhs, z *dl, z *d, z *du, z *dlf, z *df, z *duf, z *du2, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zgtsv(int *n, int *nrhs, z *dl, z *d, z *du, z *b, int *ldb, int *info) noexcept nogil +cdef void zgtsvx(char *fact, char *trans, int *n, int *nrhs, z *dl, z *d, z *du, z *dlf, z *df, z *duf, z *du2, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zgttrf(int *n, z *dl, z *d, z *du, z *du2, int *ipiv, int *info) noexcept nogil +cdef void zgttrs(char *trans, int *n, int *nrhs, z *dl, z *d, z *du, z *du2, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zgtts2(int *itrans, int *n, int *nrhs, z *dl, z *d, z *du, z *du2, int *ipiv, z *b, int *ldb) noexcept nogil +cdef void zhbev(char *jobz, char *uplo, int *n, int *kd, z *ab, int *ldab, d *w, z *z, int *ldz, z *work, d *rwork, int *info) noexcept nogil +cdef void zhbevd(char *jobz, char *uplo, int *n, int *kd, z *ab, int *ldab, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void zhbevx(char *jobz, char *range, char *uplo, int *n, int *kd, z *ab, int *ldab, z *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void zhbgst(char *vect, char *uplo, int *n, int *ka, int *kb, z *ab, int *ldab, z *bb, int *ldbb, z *x, int *ldx, z *work, d *rwork, int *info) noexcept nogil +cdef void zhbgv(char *jobz, char *uplo, int *n, int *ka, int *kb, z *ab, int *ldab, z *bb, int *ldbb, d *w, z *z, int *ldz, z *work, d *rwork, int *info) noexcept nogil +cdef void zhbgvd(char *jobz, char *uplo, int *n, int *ka, int *kb, z *ab, int *ldab, z *bb, int *ldbb, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void zhbgvx(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, z *ab, int *ldab, z *bb, int *ldbb, z *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void zhbtrd(char *vect, char *uplo, int *n, int *kd, z *ab, int *ldab, d *d, d *e, z *q, int *ldq, z *work, int *info) noexcept nogil +cdef void zhecon(char *uplo, int *n, z *a, int *lda, int *ipiv, d *anorm, d *rcond, z *work, int *info) noexcept nogil +cdef void zheequb(char *uplo, int *n, z *a, int *lda, d *s, d *scond, d *amax, z *work, int *info) noexcept nogil +cdef void zheev(char *jobz, char *uplo, int *n, z *a, int *lda, d *w, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zheevd(char *jobz, char *uplo, int *n, z *a, int *lda, d *w, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void zheevr(char *jobz, char *range, char *uplo, int *n, z *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, int *isuppz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void zheevx(char *jobz, char *range, char *uplo, int *n, z *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void zhegs2(int *itype, char *uplo, int *n, z *a, int *lda, z *b, int *ldb, int *info) noexcept nogil +cdef void zhegst(int *itype, char *uplo, int *n, z *a, int *lda, z *b, int *ldb, int *info) noexcept nogil +cdef void zhegv(int *itype, char *jobz, char *uplo, int *n, z *a, int *lda, z *b, int *ldb, d *w, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zhegvd(int *itype, char *jobz, char *uplo, int *n, z *a, int *lda, z *b, int *ldb, d *w, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void zhegvx(int *itype, char *jobz, char *range, char *uplo, int *n, z *a, int *lda, z *b, int *ldb, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void zherfs(char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zhesv(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, z *work, int *lwork, int *info) noexcept nogil +cdef void zhesvx(char *fact, char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zheswapr(char *uplo, int *n, z *a, int *lda, int *i1, int *i2) noexcept nogil +cdef void zhetd2(char *uplo, int *n, z *a, int *lda, d *d, d *e, z *tau, int *info) noexcept nogil +cdef void zhetf2(char *uplo, int *n, z *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void zhetrd(char *uplo, int *n, z *a, int *lda, d *d, d *e, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zhetrf(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *lwork, int *info) noexcept nogil +cdef void zhetri(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *info) noexcept nogil +cdef void zhetri2(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *lwork, int *info) noexcept nogil +cdef void zhetri2x(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *nb, int *info) noexcept nogil +cdef void zhetrs(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zhetrs2(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, z *work, int *info) noexcept nogil +cdef void zhfrk(char *transr, char *uplo, char *trans, int *n, int *k, d *alpha, z *a, int *lda, d *beta, z *c) noexcept nogil +cdef void zhgeqz(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, z *h, int *ldh, z *t, int *ldt, z *alpha, z *beta, z *q, int *ldq, z *z, int *ldz, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zhpcon(char *uplo, int *n, z *ap, int *ipiv, d *anorm, d *rcond, z *work, int *info) noexcept nogil +cdef void zhpev(char *jobz, char *uplo, int *n, z *ap, d *w, z *z, int *ldz, z *work, d *rwork, int *info) noexcept nogil +cdef void zhpevd(char *jobz, char *uplo, int *n, z *ap, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void zhpevx(char *jobz, char *range, char *uplo, int *n, z *ap, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void zhpgst(int *itype, char *uplo, int *n, z *ap, z *bp, int *info) noexcept nogil +cdef void zhpgv(int *itype, char *jobz, char *uplo, int *n, z *ap, z *bp, d *w, z *z, int *ldz, z *work, d *rwork, int *info) noexcept nogil +cdef void zhpgvd(int *itype, char *jobz, char *uplo, int *n, z *ap, z *bp, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void zhpgvx(int *itype, char *jobz, char *range, char *uplo, int *n, z *ap, z *bp, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil +cdef void zhprfs(char *uplo, int *n, int *nrhs, z *ap, z *afp, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zhpsv(char *uplo, int *n, int *nrhs, z *ap, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zhpsvx(char *fact, char *uplo, int *n, int *nrhs, z *ap, z *afp, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zhptrd(char *uplo, int *n, z *ap, d *d, d *e, z *tau, int *info) noexcept nogil +cdef void zhptrf(char *uplo, int *n, z *ap, int *ipiv, int *info) noexcept nogil +cdef void zhptri(char *uplo, int *n, z *ap, int *ipiv, z *work, int *info) noexcept nogil +cdef void zhptrs(char *uplo, int *n, int *nrhs, z *ap, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zhsein(char *side, char *eigsrc, char *initv, bint *select, int *n, z *h, int *ldh, z *w, z *vl, int *ldvl, z *vr, int *ldvr, int *mm, int *m, z *work, d *rwork, int *ifaill, int *ifailr, int *info) noexcept nogil +cdef void zhseqr(char *job, char *compz, int *n, int *ilo, int *ihi, z *h, int *ldh, z *w, z *z, int *ldz, z *work, int *lwork, int *info) noexcept nogil +cdef void zlabrd(int *m, int *n, int *nb, z *a, int *lda, d *d, d *e, z *tauq, z *taup, z *x, int *ldx, z *y, int *ldy) noexcept nogil +cdef void zlacgv(int *n, z *x, int *incx) noexcept nogil +cdef void zlacn2(int *n, z *v, z *x, d *est, int *kase, int *isave) noexcept nogil +cdef void zlacon(int *n, z *v, z *x, d *est, int *kase) noexcept nogil +cdef void zlacp2(char *uplo, int *m, int *n, d *a, int *lda, z *b, int *ldb) noexcept nogil +cdef void zlacpy(char *uplo, int *m, int *n, z *a, int *lda, z *b, int *ldb) noexcept nogil +cdef void zlacrm(int *m, int *n, z *a, int *lda, d *b, int *ldb, z *c, int *ldc, d *rwork) noexcept nogil +cdef void zlacrt(int *n, z *cx, int *incx, z *cy, int *incy, z *c, z *s) noexcept nogil +cdef z zladiv(z *x, z *y) noexcept nogil +cdef void zlaed0(int *qsiz, int *n, d *d, d *e, z *q, int *ldq, z *qstore, int *ldqs, d *rwork, int *iwork, int *info) noexcept nogil +cdef void zlaed7(int *n, int *cutpnt, int *qsiz, int *tlvls, int *curlvl, int *curpbm, d *d, z *q, int *ldq, d *rho, int *indxq, d *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, d *givnum, z *work, d *rwork, int *iwork, int *info) noexcept nogil +cdef void zlaed8(int *k, int *n, int *qsiz, z *q, int *ldq, d *d, d *rho, int *cutpnt, d *z, d *dlamda, z *q2, int *ldq2, d *w, int *indxp, int *indx, int *indxq, int *perm, int *givptr, int *givcol, d *givnum, int *info) noexcept nogil +cdef void zlaein(bint *rightv, bint *noinit, int *n, z *h, int *ldh, z *w, z *v, z *b, int *ldb, d *rwork, d *eps3, d *smlnum, int *info) noexcept nogil +cdef void zlaesy(z *a, z *b, z *c, z *rt1, z *rt2, z *evscal, z *cs1, z *sn1) noexcept nogil +cdef void zlaev2(z *a, z *b, z *c, d *rt1, d *rt2, d *cs1, z *sn1) noexcept nogil +cdef void zlag2c(int *m, int *n, z *a, int *lda, c *sa, int *ldsa, int *info) noexcept nogil +cdef void zlags2(bint *upper, d *a1, z *a2, d *a3, d *b1, z *b2, d *b3, d *csu, z *snu, d *csv, z *snv, d *csq, z *snq) noexcept nogil +cdef void zlagtm(char *trans, int *n, int *nrhs, d *alpha, z *dl, z *d, z *du, z *x, int *ldx, d *beta, z *b, int *ldb) noexcept nogil +cdef void zlahef(char *uplo, int *n, int *nb, int *kb, z *a, int *lda, int *ipiv, z *w, int *ldw, int *info) noexcept nogil +cdef void zlahqr(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, z *h, int *ldh, z *w, int *iloz, int *ihiz, z *z, int *ldz, int *info) noexcept nogil +cdef void zlahr2(int *n, int *k, int *nb, z *a, int *lda, z *tau, z *t, int *ldt, z *y, int *ldy) noexcept nogil +cdef void zlaic1(int *job, int *j, z *x, d *sest, z *w, z *gamma, d *sestpr, z *s, z *c) noexcept nogil +cdef void zlals0(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, z *b, int *ldb, z *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *poles, d *difl, d *difr, d *z, int *k, d *c, d *s, d *rwork, int *info) noexcept nogil +cdef void zlalsa(int *icompq, int *smlsiz, int *n, int *nrhs, z *b, int *ldb, z *bx, int *ldbx, d *u, int *ldu, d *vt, int *k, d *difl, d *difr, d *z, d *poles, int *givptr, int *givcol, int *ldgcol, int *perm, d *givnum, d *c, d *s, d *rwork, int *iwork, int *info) noexcept nogil +cdef void zlalsd(char *uplo, int *smlsiz, int *n, int *nrhs, d *d, d *e, z *b, int *ldb, d *rcond, int *rank, z *work, d *rwork, int *iwork, int *info) noexcept nogil +cdef d zlangb(char *norm, int *n, int *kl, int *ku, z *ab, int *ldab, d *work) noexcept nogil +cdef d zlange(char *norm, int *m, int *n, z *a, int *lda, d *work) noexcept nogil +cdef d zlangt(char *norm, int *n, z *dl, z *d_, z *du) noexcept nogil +cdef d zlanhb(char *norm, char *uplo, int *n, int *k, z *ab, int *ldab, d *work) noexcept nogil +cdef d zlanhe(char *norm, char *uplo, int *n, z *a, int *lda, d *work) noexcept nogil +cdef d zlanhf(char *norm, char *transr, char *uplo, int *n, z *a, d *work) noexcept nogil +cdef d zlanhp(char *norm, char *uplo, int *n, z *ap, d *work) noexcept nogil +cdef d zlanhs(char *norm, int *n, z *a, int *lda, d *work) noexcept nogil +cdef d zlanht(char *norm, int *n, d *d_, z *e) noexcept nogil +cdef d zlansb(char *norm, char *uplo, int *n, int *k, z *ab, int *ldab, d *work) noexcept nogil +cdef d zlansp(char *norm, char *uplo, int *n, z *ap, d *work) noexcept nogil +cdef d zlansy(char *norm, char *uplo, int *n, z *a, int *lda, d *work) noexcept nogil +cdef d zlantb(char *norm, char *uplo, char *diag, int *n, int *k, z *ab, int *ldab, d *work) noexcept nogil +cdef d zlantp(char *norm, char *uplo, char *diag, int *n, z *ap, d *work) noexcept nogil +cdef d zlantr(char *norm, char *uplo, char *diag, int *m, int *n, z *a, int *lda, d *work) noexcept nogil +cdef void zlapll(int *n, z *x, int *incx, z *y, int *incy, d *ssmin) noexcept nogil +cdef void zlapmr(bint *forwrd, int *m, int *n, z *x, int *ldx, int *k) noexcept nogil +cdef void zlapmt(bint *forwrd, int *m, int *n, z *x, int *ldx, int *k) noexcept nogil +cdef void zlaqgb(int *m, int *n, int *kl, int *ku, z *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) noexcept nogil +cdef void zlaqge(int *m, int *n, z *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) noexcept nogil +cdef void zlaqhb(char *uplo, int *n, int *kd, z *ab, int *ldab, d *s, d *scond, d *amax, char *equed) noexcept nogil +cdef void zlaqhe(char *uplo, int *n, z *a, int *lda, d *s, d *scond, d *amax, char *equed) noexcept nogil +cdef void zlaqhp(char *uplo, int *n, z *ap, d *s, d *scond, d *amax, char *equed) noexcept nogil +cdef void zlaqp2(int *m, int *n, int *offset, z *a, int *lda, int *jpvt, z *tau, d *vn1, d *vn2, z *work) noexcept nogil +cdef void zlaqps(int *m, int *n, int *offset, int *nb, int *kb, z *a, int *lda, int *jpvt, z *tau, d *vn1, d *vn2, z *auxv, z *f, int *ldf) noexcept nogil +cdef void zlaqr0(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, z *h, int *ldh, z *w, int *iloz, int *ihiz, z *z, int *ldz, z *work, int *lwork, int *info) noexcept nogil +cdef void zlaqr1(int *n, z *h, int *ldh, z *s1, z *s2, z *v) noexcept nogil +cdef void zlaqr2(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, z *h, int *ldh, int *iloz, int *ihiz, z *z, int *ldz, int *ns, int *nd, z *sh, z *v, int *ldv, int *nh, z *t, int *ldt, int *nv, z *wv, int *ldwv, z *work, int *lwork) noexcept nogil +cdef void zlaqr3(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, z *h, int *ldh, int *iloz, int *ihiz, z *z, int *ldz, int *ns, int *nd, z *sh, z *v, int *ldv, int *nh, z *t, int *ldt, int *nv, z *wv, int *ldwv, z *work, int *lwork) noexcept nogil +cdef void zlaqr4(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, z *h, int *ldh, z *w, int *iloz, int *ihiz, z *z, int *ldz, z *work, int *lwork, int *info) noexcept nogil +cdef void zlaqr5(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, z *s, z *h, int *ldh, int *iloz, int *ihiz, z *z, int *ldz, z *v, int *ldv, z *u, int *ldu, int *nv, z *wv, int *ldwv, int *nh, z *wh, int *ldwh) noexcept nogil +cdef void zlaqsb(char *uplo, int *n, int *kd, z *ab, int *ldab, d *s, d *scond, d *amax, char *equed) noexcept nogil +cdef void zlaqsp(char *uplo, int *n, z *ap, d *s, d *scond, d *amax, char *equed) noexcept nogil +cdef void zlaqsy(char *uplo, int *n, z *a, int *lda, d *s, d *scond, d *amax, char *equed) noexcept nogil +cdef void zlar1v(int *n, int *b1, int *bn, d *lambda_, d *d, d *l, d *ld, d *lld, d *pivmin, d *gaptol, z *z, bint *wantnc, int *negcnt, d *ztz, d *mingma, int *r, int *isuppz, d *nrminv, d *resid, d *rqcorr, d *work) noexcept nogil +cdef void zlar2v(int *n, z *x, z *y, z *z, int *incx, d *c, z *s, int *incc) noexcept nogil +cdef void zlarcm(int *m, int *n, d *a, int *lda, z *b, int *ldb, z *c, int *ldc, d *rwork) noexcept nogil +cdef void zlarf(char *side, int *m, int *n, z *v, int *incv, z *tau, z *c, int *ldc, z *work) noexcept nogil +cdef void zlarfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, z *v, int *ldv, z *t, int *ldt, z *c, int *ldc, z *work, int *ldwork) noexcept nogil +cdef void zlarfg(int *n, z *alpha, z *x, int *incx, z *tau) noexcept nogil +cdef void zlarfgp(int *n, z *alpha, z *x, int *incx, z *tau) noexcept nogil +cdef void zlarft(char *direct, char *storev, int *n, int *k, z *v, int *ldv, z *tau, z *t, int *ldt) noexcept nogil +cdef void zlarfx(char *side, int *m, int *n, z *v, z *tau, z *c, int *ldc, z *work) noexcept nogil +cdef void zlargv(int *n, z *x, int *incx, z *y, int *incy, d *c, int *incc) noexcept nogil +cdef void zlarnv(int *idist, int *iseed, int *n, z *x) noexcept nogil +cdef void zlarrv(int *n, d *vl, d *vu, d *d, d *l, d *pivmin, int *isplit, int *m, int *dol, int *dou, d *minrgp, d *rtol1, d *rtol2, d *w, d *werr, d *wgap, int *iblock, int *indexw, d *gers, z *z, int *ldz, int *isuppz, d *work, int *iwork, int *info) noexcept nogil +cdef void zlartg(z *f, z *g, d *cs, z *sn, z *r) noexcept nogil +cdef void zlartv(int *n, z *x, int *incx, z *y, int *incy, d *c, z *s, int *incc) noexcept nogil +cdef void zlarz(char *side, int *m, int *n, int *l, z *v, int *incv, z *tau, z *c, int *ldc, z *work) noexcept nogil +cdef void zlarzb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, z *v, int *ldv, z *t, int *ldt, z *c, int *ldc, z *work, int *ldwork) noexcept nogil +cdef void zlarzt(char *direct, char *storev, int *n, int *k, z *v, int *ldv, z *tau, z *t, int *ldt) noexcept nogil +cdef void zlascl(char *type_bn, int *kl, int *ku, d *cfrom, d *cto, int *m, int *n, z *a, int *lda, int *info) noexcept nogil +cdef void zlaset(char *uplo, int *m, int *n, z *alpha, z *beta, z *a, int *lda) noexcept nogil +cdef void zlasr(char *side, char *pivot, char *direct, int *m, int *n, d *c, d *s, z *a, int *lda) noexcept nogil +cdef void zlassq(int *n, z *x, int *incx, d *scale, d *sumsq) noexcept nogil +cdef void zlaswp(int *n, z *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) noexcept nogil +cdef void zlasyf(char *uplo, int *n, int *nb, int *kb, z *a, int *lda, int *ipiv, z *w, int *ldw, int *info) noexcept nogil +cdef void zlat2c(char *uplo, int *n, z *a, int *lda, c *sa, int *ldsa, int *info) noexcept nogil +cdef void zlatbs(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, z *ab, int *ldab, z *x, d *scale, d *cnorm, int *info) noexcept nogil +cdef void zlatdf(int *ijob, int *n, z *z, int *ldz, z *rhs, d *rdsum, d *rdscal, int *ipiv, int *jpiv) noexcept nogil +cdef void zlatps(char *uplo, char *trans, char *diag, char *normin, int *n, z *ap, z *x, d *scale, d *cnorm, int *info) noexcept nogil +cdef void zlatrd(char *uplo, int *n, int *nb, z *a, int *lda, d *e, z *tau, z *w, int *ldw) noexcept nogil +cdef void zlatrs(char *uplo, char *trans, char *diag, char *normin, int *n, z *a, int *lda, z *x, d *scale, d *cnorm, int *info) noexcept nogil +cdef void zlatrz(int *m, int *n, int *l, z *a, int *lda, z *tau, z *work) noexcept nogil +cdef void zlauu2(char *uplo, int *n, z *a, int *lda, int *info) noexcept nogil +cdef void zlauum(char *uplo, int *n, z *a, int *lda, int *info) noexcept nogil +cdef void zpbcon(char *uplo, int *n, int *kd, z *ab, int *ldab, d *anorm, d *rcond, z *work, d *rwork, int *info) noexcept nogil +cdef void zpbequ(char *uplo, int *n, int *kd, z *ab, int *ldab, d *s, d *scond, d *amax, int *info) noexcept nogil +cdef void zpbrfs(char *uplo, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *afb, int *ldafb, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zpbstf(char *uplo, int *n, int *kd, z *ab, int *ldab, int *info) noexcept nogil +cdef void zpbsv(char *uplo, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *b, int *ldb, int *info) noexcept nogil +cdef void zpbsvx(char *fact, char *uplo, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *afb, int *ldafb, char *equed, d *s, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zpbtf2(char *uplo, int *n, int *kd, z *ab, int *ldab, int *info) noexcept nogil +cdef void zpbtrf(char *uplo, int *n, int *kd, z *ab, int *ldab, int *info) noexcept nogil +cdef void zpbtrs(char *uplo, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *b, int *ldb, int *info) noexcept nogil +cdef void zpftrf(char *transr, char *uplo, int *n, z *a, int *info) noexcept nogil +cdef void zpftri(char *transr, char *uplo, int *n, z *a, int *info) noexcept nogil +cdef void zpftrs(char *transr, char *uplo, int *n, int *nrhs, z *a, z *b, int *ldb, int *info) noexcept nogil +cdef void zpocon(char *uplo, int *n, z *a, int *lda, d *anorm, d *rcond, z *work, d *rwork, int *info) noexcept nogil +cdef void zpoequ(int *n, z *a, int *lda, d *s, d *scond, d *amax, int *info) noexcept nogil +cdef void zpoequb(int *n, z *a, int *lda, d *s, d *scond, d *amax, int *info) noexcept nogil +cdef void zporfs(char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zposv(char *uplo, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, int *info) noexcept nogil +cdef void zposvx(char *fact, char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, char *equed, d *s, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zpotf2(char *uplo, int *n, z *a, int *lda, int *info) noexcept nogil +cdef void zpotrf(char *uplo, int *n, z *a, int *lda, int *info) noexcept nogil +cdef void zpotri(char *uplo, int *n, z *a, int *lda, int *info) noexcept nogil +cdef void zpotrs(char *uplo, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, int *info) noexcept nogil +cdef void zppcon(char *uplo, int *n, z *ap, d *anorm, d *rcond, z *work, d *rwork, int *info) noexcept nogil +cdef void zppequ(char *uplo, int *n, z *ap, d *s, d *scond, d *amax, int *info) noexcept nogil +cdef void zpprfs(char *uplo, int *n, int *nrhs, z *ap, z *afp, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zppsv(char *uplo, int *n, int *nrhs, z *ap, z *b, int *ldb, int *info) noexcept nogil +cdef void zppsvx(char *fact, char *uplo, int *n, int *nrhs, z *ap, z *afp, char *equed, d *s, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zpptrf(char *uplo, int *n, z *ap, int *info) noexcept nogil +cdef void zpptri(char *uplo, int *n, z *ap, int *info) noexcept nogil +cdef void zpptrs(char *uplo, int *n, int *nrhs, z *ap, z *b, int *ldb, int *info) noexcept nogil +cdef void zpstf2(char *uplo, int *n, z *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) noexcept nogil +cdef void zpstrf(char *uplo, int *n, z *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) noexcept nogil +cdef void zptcon(int *n, d *d, z *e, d *anorm, d *rcond, d *rwork, int *info) noexcept nogil +cdef void zpteqr(char *compz, int *n, d *d, d *e, z *z, int *ldz, d *work, int *info) noexcept nogil +cdef void zptrfs(char *uplo, int *n, int *nrhs, d *d, z *e, d *df, z *ef, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zptsv(int *n, int *nrhs, d *d, z *e, z *b, int *ldb, int *info) noexcept nogil +cdef void zptsvx(char *fact, int *n, int *nrhs, d *d, z *e, d *df, z *ef, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zpttrf(int *n, d *d, z *e, int *info) noexcept nogil +cdef void zpttrs(char *uplo, int *n, int *nrhs, d *d, z *e, z *b, int *ldb, int *info) noexcept nogil +cdef void zptts2(int *iuplo, int *n, int *nrhs, d *d, z *e, z *b, int *ldb) noexcept nogil +cdef void zrot(int *n, z *cx, int *incx, z *cy, int *incy, d *c, z *s) noexcept nogil +cdef void zspcon(char *uplo, int *n, z *ap, int *ipiv, d *anorm, d *rcond, z *work, int *info) noexcept nogil +cdef void zspmv(char *uplo, int *n, z *alpha, z *ap, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil +cdef void zspr(char *uplo, int *n, z *alpha, z *x, int *incx, z *ap) noexcept nogil +cdef void zsprfs(char *uplo, int *n, int *nrhs, z *ap, z *afp, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zspsv(char *uplo, int *n, int *nrhs, z *ap, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zspsvx(char *fact, char *uplo, int *n, int *nrhs, z *ap, z *afp, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zsptrf(char *uplo, int *n, z *ap, int *ipiv, int *info) noexcept nogil +cdef void zsptri(char *uplo, int *n, z *ap, int *ipiv, z *work, int *info) noexcept nogil +cdef void zsptrs(char *uplo, int *n, int *nrhs, z *ap, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zstedc(char *compz, int *n, d *d, d *e, z *z, int *ldz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void zstegr(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void zstein(int *n, d *d, d *e, int *m, d *w, int *iblock, int *isplit, z *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil +cdef void zstemr(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, int *m, d *w, z *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void zsteqr(char *compz, int *n, d *d, d *e, z *z, int *ldz, d *work, int *info) noexcept nogil +cdef void zsycon(char *uplo, int *n, z *a, int *lda, int *ipiv, d *anorm, d *rcond, z *work, int *info) noexcept nogil +cdef void zsyconv(char *uplo, char *way, int *n, z *a, int *lda, int *ipiv, z *work, int *info) noexcept nogil +cdef void zsyequb(char *uplo, int *n, z *a, int *lda, d *s, d *scond, d *amax, z *work, int *info) noexcept nogil +cdef void zsymv(char *uplo, int *n, z *alpha, z *a, int *lda, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil +cdef void zsyr(char *uplo, int *n, z *alpha, z *x, int *incx, z *a, int *lda) noexcept nogil +cdef void zsyrfs(char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void zsysv(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, z *work, int *lwork, int *info) noexcept nogil +cdef void zsysvx(char *fact, char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, int *lwork, d *rwork, int *info) noexcept nogil +cdef void zsyswapr(char *uplo, int *n, z *a, int *lda, int *i1, int *i2) noexcept nogil +cdef void zsytf2(char *uplo, int *n, z *a, int *lda, int *ipiv, int *info) noexcept nogil +cdef void zsytrf(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *lwork, int *info) noexcept nogil +cdef void zsytri(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *info) noexcept nogil +cdef void zsytri2(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *lwork, int *info) noexcept nogil +cdef void zsytri2x(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *nb, int *info) noexcept nogil +cdef void zsytrs(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, int *info) noexcept nogil +cdef void zsytrs2(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, z *work, int *info) noexcept nogil +cdef void ztbcon(char *norm, char *uplo, char *diag, int *n, int *kd, z *ab, int *ldab, d *rcond, z *work, d *rwork, int *info) noexcept nogil +cdef void ztbrfs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void ztbtrs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *b, int *ldb, int *info) noexcept nogil +cdef void ztfsm(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, z *alpha, z *a, z *b, int *ldb) noexcept nogil +cdef void ztftri(char *transr, char *uplo, char *diag, int *n, z *a, int *info) noexcept nogil +cdef void ztfttp(char *transr, char *uplo, int *n, z *arf, z *ap, int *info) noexcept nogil +cdef void ztfttr(char *transr, char *uplo, int *n, z *arf, z *a, int *lda, int *info) noexcept nogil +cdef void ztgevc(char *side, char *howmny, bint *select, int *n, z *s, int *lds, z *p, int *ldp, z *vl, int *ldvl, z *vr, int *ldvr, int *mm, int *m, z *work, d *rwork, int *info) noexcept nogil +cdef void ztgex2(bint *wantq, bint *wantz, int *n, z *a, int *lda, z *b, int *ldb, z *q, int *ldq, z *z, int *ldz, int *j1, int *info) noexcept nogil +cdef void ztgexc(bint *wantq, bint *wantz, int *n, z *a, int *lda, z *b, int *ldb, z *q, int *ldq, z *z, int *ldz, int *ifst, int *ilst, int *info) noexcept nogil +cdef void ztgsen(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, z *a, int *lda, z *b, int *ldb, z *alpha, z *beta, z *q, int *ldq, z *z, int *ldz, int *m, d *pl, d *pr, d *dif, z *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil +cdef void ztgsja(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, z *a, int *lda, z *b, int *ldb, d *tola, d *tolb, d *alpha, d *beta, z *u, int *ldu, z *v, int *ldv, z *q, int *ldq, z *work, int *ncycle, int *info) noexcept nogil +cdef void ztgsna(char *job, char *howmny, bint *select, int *n, z *a, int *lda, z *b, int *ldb, z *vl, int *ldvl, z *vr, int *ldvr, d *s, d *dif, int *mm, int *m, z *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void ztgsy2(char *trans, int *ijob, int *m, int *n, z *a, int *lda, z *b, int *ldb, z *c, int *ldc, z *d, int *ldd, z *e, int *lde, z *f, int *ldf, d *scale, d *rdsum, d *rdscal, int *info) noexcept nogil +cdef void ztgsyl(char *trans, int *ijob, int *m, int *n, z *a, int *lda, z *b, int *ldb, z *c, int *ldc, z *d, int *ldd, z *e, int *lde, z *f, int *ldf, d *scale, d *dif, z *work, int *lwork, int *iwork, int *info) noexcept nogil +cdef void ztpcon(char *norm, char *uplo, char *diag, int *n, z *ap, d *rcond, z *work, d *rwork, int *info) noexcept nogil +cdef void ztpmqrt(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, z *v, int *ldv, z *t, int *ldt, z *a, int *lda, z *b, int *ldb, z *work, int *info) noexcept nogil +cdef void ztpqrt(int *m, int *n, int *l, int *nb, z *a, int *lda, z *b, int *ldb, z *t, int *ldt, z *work, int *info) noexcept nogil +cdef void ztpqrt2(int *m, int *n, int *l, z *a, int *lda, z *b, int *ldb, z *t, int *ldt, int *info) noexcept nogil +cdef void ztprfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, z *v, int *ldv, z *t, int *ldt, z *a, int *lda, z *b, int *ldb, z *work, int *ldwork) noexcept nogil +cdef void ztprfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, z *ap, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void ztptri(char *uplo, char *diag, int *n, z *ap, int *info) noexcept nogil +cdef void ztptrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, z *ap, z *b, int *ldb, int *info) noexcept nogil +cdef void ztpttf(char *transr, char *uplo, int *n, z *ap, z *arf, int *info) noexcept nogil +cdef void ztpttr(char *uplo, int *n, z *ap, z *a, int *lda, int *info) noexcept nogil +cdef void ztrcon(char *norm, char *uplo, char *diag, int *n, z *a, int *lda, d *rcond, z *work, d *rwork, int *info) noexcept nogil +cdef void ztrevc(char *side, char *howmny, bint *select, int *n, z *t, int *ldt, z *vl, int *ldvl, z *vr, int *ldvr, int *mm, int *m, z *work, d *rwork, int *info) noexcept nogil +cdef void ztrexc(char *compq, int *n, z *t, int *ldt, z *q, int *ldq, int *ifst, int *ilst, int *info) noexcept nogil +cdef void ztrrfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil +cdef void ztrsen(char *job, char *compq, bint *select, int *n, z *t, int *ldt, z *q, int *ldq, z *w, int *m, d *s, d *sep, z *work, int *lwork, int *info) noexcept nogil +cdef void ztrsna(char *job, char *howmny, bint *select, int *n, z *t, int *ldt, z *vl, int *ldvl, z *vr, int *ldvr, d *s, d *sep, int *mm, int *m, z *work, int *ldwork, d *rwork, int *info) noexcept nogil +cdef void ztrsyl(char *trana, char *tranb, int *isgn, int *m, int *n, z *a, int *lda, z *b, int *ldb, z *c, int *ldc, d *scale, int *info) noexcept nogil +cdef void ztrti2(char *uplo, char *diag, int *n, z *a, int *lda, int *info) noexcept nogil +cdef void ztrtri(char *uplo, char *diag, int *n, z *a, int *lda, int *info) noexcept nogil +cdef void ztrtrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, int *info) noexcept nogil +cdef void ztrttf(char *transr, char *uplo, int *n, z *a, int *lda, z *arf, int *info) noexcept nogil +cdef void ztrttp(char *uplo, int *n, z *a, int *lda, z *ap, int *info) noexcept nogil +cdef void ztzrzf(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zunbdb(char *trans, char *signs, int *m, int *p, int *q, z *x11, int *ldx11, z *x12, int *ldx12, z *x21, int *ldx21, z *x22, int *ldx22, d *theta, d *phi, z *taup1, z *taup2, z *tauq1, z *tauq2, z *work, int *lwork, int *info) noexcept nogil +cdef void zuncsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, z *x11, int *ldx11, z *x12, int *ldx12, z *x21, int *ldx21, z *x22, int *ldx22, d *theta, z *u1, int *ldu1, z *u2, int *ldu2, z *v1t, int *ldv1t, z *v2t, int *ldv2t, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *info) noexcept nogil +cdef void zung2l(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil +cdef void zung2r(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil +cdef void zungbr(char *vect, int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zunghr(int *n, int *ilo, int *ihi, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zungl2(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil +cdef void zunglq(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zungql(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zungqr(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zungr2(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil +cdef void zungrq(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zungtr(char *uplo, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil +cdef void zunm2l(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil +cdef void zunm2r(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil +cdef void zunmbr(char *vect, char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil +cdef void zunmhr(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil +cdef void zunml2(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil +cdef void zunmlq(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil +cdef void zunmql(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil +cdef void zunmqr(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil +cdef void zunmr2(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil +cdef void zunmr3(char *side, char *trans, int *m, int *n, int *k, int *l, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil +cdef void zunmrq(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil +cdef void zunmrz(char *side, char *trans, int *m, int *n, int *k, int *l, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil +cdef void zunmtr(char *side, char *uplo, char *trans, int *m, int *n, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil +cdef void zupgtr(char *uplo, int *n, z *ap, z *tau, z *q, int *ldq, z *work, int *info) noexcept nogil +cdef void zupmtr(char *side, char *uplo, char *trans, int *m, int *n, z *ap, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_lapack.pyx b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_lapack.pyx new file mode 100644 index 0000000000000000000000000000000000000000..7f9cbfbb519603d4107af51ac353e0650720cf8c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/cython_lapack.pyx @@ -0,0 +1,12045 @@ +# This file was generated by _generate_pyx.py. +# Do not edit this file directly. +""" +LAPACK functions for Cython +=========================== + +Usable from Cython via:: + + cimport scipy.linalg.cython_lapack + +This module provides Cython-level wrappers for all primary routines included +in LAPACK 3.4.0 except for ``zcgesv`` since its interface is not consistent +from LAPACK 3.4.0 to 3.6.0. It also provides some of the +fixed-api auxiliary routines. + +These wrappers do not check for alignment of arrays. +Alignment should be checked before these wrappers are used. + +Raw function pointers (Fortran-style pointer arguments): + +- cbbcsd +- cbdsqr +- cgbbrd +- cgbcon +- cgbequ +- cgbequb +- cgbrfs +- cgbsv +- cgbsvx +- cgbtf2 +- cgbtrf +- cgbtrs +- cgebak +- cgebal +- cgebd2 +- cgebrd +- cgecon +- cgeequ +- cgeequb +- cgees +- cgeesx +- cgeev +- cgeevx +- cgehd2 +- cgehrd +- cgelq2 +- cgelqf +- cgels +- cgelsd +- cgelss +- cgelsy +- cgemqrt +- cgeql2 +- cgeqlf +- cgeqp3 +- cgeqr2 +- cgeqr2p +- cgeqrf +- cgeqrfp +- cgeqrt +- cgeqrt2 +- cgeqrt3 +- cgerfs +- cgerq2 +- cgerqf +- cgesc2 +- cgesdd +- cgesv +- cgesvd +- cgesvx +- cgetc2 +- cgetf2 +- cgetrf +- cgetri +- cgetrs +- cggbak +- cggbal +- cgges +- cggesx +- cggev +- cggevx +- cggglm +- cgghrd +- cgglse +- cggqrf +- cggrqf +- cgtcon +- cgtrfs +- cgtsv +- cgtsvx +- cgttrf +- cgttrs +- cgtts2 +- chbev +- chbevd +- chbevx +- chbgst +- chbgv +- chbgvd +- chbgvx +- chbtrd +- checon +- cheequb +- cheev +- cheevd +- cheevr +- cheevx +- chegs2 +- chegst +- chegv +- chegvd +- chegvx +- cherfs +- chesv +- chesvx +- cheswapr +- chetd2 +- chetf2 +- chetrd +- chetrf +- chetri +- chetri2 +- chetri2x +- chetrs +- chetrs2 +- chfrk +- chgeqz +- chla_transtype +- chpcon +- chpev +- chpevd +- chpevx +- chpgst +- chpgv +- chpgvd +- chpgvx +- chprfs +- chpsv +- chpsvx +- chptrd +- chptrf +- chptri +- chptrs +- chsein +- chseqr +- clabrd +- clacgv +- clacn2 +- clacon +- clacp2 +- clacpy +- clacrm +- clacrt +- cladiv +- claed0 +- claed7 +- claed8 +- claein +- claesy +- claev2 +- clag2z +- clags2 +- clagtm +- clahef +- clahqr +- clahr2 +- claic1 +- clals0 +- clalsa +- clalsd +- clangb +- clange +- clangt +- clanhb +- clanhe +- clanhf +- clanhp +- clanhs +- clanht +- clansb +- clansp +- clansy +- clantb +- clantp +- clantr +- clapll +- clapmr +- clapmt +- claqgb +- claqge +- claqhb +- claqhe +- claqhp +- claqp2 +- claqps +- claqr0 +- claqr1 +- claqr2 +- claqr3 +- claqr4 +- claqr5 +- claqsb +- claqsp +- claqsy +- clar1v +- clar2v +- clarcm +- clarf +- clarfb +- clarfg +- clarfgp +- clarft +- clarfx +- clargv +- clarnv +- clarrv +- clartg +- clartv +- clarz +- clarzb +- clarzt +- clascl +- claset +- clasr +- classq +- claswp +- clasyf +- clatbs +- clatdf +- clatps +- clatrd +- clatrs +- clatrz +- clauu2 +- clauum +- cpbcon +- cpbequ +- cpbrfs +- cpbstf +- cpbsv +- cpbsvx +- cpbtf2 +- cpbtrf +- cpbtrs +- cpftrf +- cpftri +- cpftrs +- cpocon +- cpoequ +- cpoequb +- cporfs +- cposv +- cposvx +- cpotf2 +- cpotrf +- cpotri +- cpotrs +- cppcon +- cppequ +- cpprfs +- cppsv +- cppsvx +- cpptrf +- cpptri +- cpptrs +- cpstf2 +- cpstrf +- cptcon +- cpteqr +- cptrfs +- cptsv +- cptsvx +- cpttrf +- cpttrs +- cptts2 +- crot +- cspcon +- cspmv +- cspr +- csprfs +- cspsv +- cspsvx +- csptrf +- csptri +- csptrs +- csrscl +- cstedc +- cstegr +- cstein +- cstemr +- csteqr +- csycon +- csyconv +- csyequb +- csymv +- csyr +- csyrfs +- csysv +- csysvx +- csyswapr +- csytf2 +- csytrf +- csytri +- csytri2 +- csytri2x +- csytrs +- csytrs2 +- ctbcon +- ctbrfs +- ctbtrs +- ctfsm +- ctftri +- ctfttp +- ctfttr +- ctgevc +- ctgex2 +- ctgexc +- ctgsen +- ctgsja +- ctgsna +- ctgsy2 +- ctgsyl +- ctpcon +- ctpmqrt +- ctpqrt +- ctpqrt2 +- ctprfb +- ctprfs +- ctptri +- ctptrs +- ctpttf +- ctpttr +- ctrcon +- ctrevc +- ctrexc +- ctrrfs +- ctrsen +- ctrsna +- ctrsyl +- ctrti2 +- ctrtri +- ctrtrs +- ctrttf +- ctrttp +- ctzrzf +- cunbdb +- cuncsd +- cung2l +- cung2r +- cungbr +- cunghr +- cungl2 +- cunglq +- cungql +- cungqr +- cungr2 +- cungrq +- cungtr +- cunm2l +- cunm2r +- cunmbr +- cunmhr +- cunml2 +- cunmlq +- cunmql +- cunmqr +- cunmr2 +- cunmr3 +- cunmrq +- cunmrz +- cunmtr +- cupgtr +- cupmtr +- dbbcsd +- dbdsdc +- dbdsqr +- ddisna +- dgbbrd +- dgbcon +- dgbequ +- dgbequb +- dgbrfs +- dgbsv +- dgbsvx +- dgbtf2 +- dgbtrf +- dgbtrs +- dgebak +- dgebal +- dgebd2 +- dgebrd +- dgecon +- dgeequ +- dgeequb +- dgees +- dgeesx +- dgeev +- dgeevx +- dgehd2 +- dgehrd +- dgejsv +- dgelq2 +- dgelqf +- dgels +- dgelsd +- dgelss +- dgelsy +- dgemqrt +- dgeql2 +- dgeqlf +- dgeqp3 +- dgeqr2 +- dgeqr2p +- dgeqrf +- dgeqrfp +- dgeqrt +- dgeqrt2 +- dgeqrt3 +- dgerfs +- dgerq2 +- dgerqf +- dgesc2 +- dgesdd +- dgesv +- dgesvd +- dgesvj +- dgesvx +- dgetc2 +- dgetf2 +- dgetrf +- dgetri +- dgetrs +- dggbak +- dggbal +- dgges +- dggesx +- dggev +- dggevx +- dggglm +- dgghrd +- dgglse +- dggqrf +- dggrqf +- dgsvj0 +- dgsvj1 +- dgtcon +- dgtrfs +- dgtsv +- dgtsvx +- dgttrf +- dgttrs +- dgtts2 +- dhgeqz +- dhsein +- dhseqr +- disnan +- dlabad +- dlabrd +- dlacn2 +- dlacon +- dlacpy +- dladiv +- dlae2 +- dlaebz +- dlaed0 +- dlaed1 +- dlaed2 +- dlaed3 +- dlaed4 +- dlaed5 +- dlaed6 +- dlaed7 +- dlaed8 +- dlaed9 +- dlaeda +- dlaein +- dlaev2 +- dlaexc +- dlag2 +- dlag2s +- dlags2 +- dlagtf +- dlagtm +- dlagts +- dlagv2 +- dlahqr +- dlahr2 +- dlaic1 +- dlaln2 +- dlals0 +- dlalsa +- dlalsd +- dlamch +- dlamrg +- dlaneg +- dlangb +- dlange +- dlangt +- dlanhs +- dlansb +- dlansf +- dlansp +- dlanst +- dlansy +- dlantb +- dlantp +- dlantr +- dlanv2 +- dlapll +- dlapmr +- dlapmt +- dlapy2 +- dlapy3 +- dlaqgb +- dlaqge +- dlaqp2 +- dlaqps +- dlaqr0 +- dlaqr1 +- dlaqr2 +- dlaqr3 +- dlaqr4 +- dlaqr5 +- dlaqsb +- dlaqsp +- dlaqsy +- dlaqtr +- dlar1v +- dlar2v +- dlarf +- dlarfb +- dlarfg +- dlarfgp +- dlarft +- dlarfx +- dlargv +- dlarnv +- dlarra +- dlarrb +- dlarrc +- dlarrd +- dlarre +- dlarrf +- dlarrj +- dlarrk +- dlarrr +- dlarrv +- dlartg +- dlartgp +- dlartgs +- dlartv +- dlaruv +- dlarz +- dlarzb +- dlarzt +- dlas2 +- dlascl +- dlasd0 +- dlasd1 +- dlasd2 +- dlasd3 +- dlasd4 +- dlasd5 +- dlasd6 +- dlasd7 +- dlasd8 +- dlasda +- dlasdq +- dlasdt +- dlaset +- dlasq1 +- dlasq2 +- dlasq3 +- dlasq4 +- dlasq6 +- dlasr +- dlasrt +- dlassq +- dlasv2 +- dlaswp +- dlasy2 +- dlasyf +- dlat2s +- dlatbs +- dlatdf +- dlatps +- dlatrd +- dlatrs +- dlatrz +- dlauu2 +- dlauum +- dopgtr +- dopmtr +- dorbdb +- dorcsd +- dorg2l +- dorg2r +- dorgbr +- dorghr +- dorgl2 +- dorglq +- dorgql +- dorgqr +- dorgr2 +- dorgrq +- dorgtr +- dorm2l +- dorm2r +- dormbr +- dormhr +- dorml2 +- dormlq +- dormql +- dormqr +- dormr2 +- dormr3 +- dormrq +- dormrz +- dormtr +- dpbcon +- dpbequ +- dpbrfs +- dpbstf +- dpbsv +- dpbsvx +- dpbtf2 +- dpbtrf +- dpbtrs +- dpftrf +- dpftri +- dpftrs +- dpocon +- dpoequ +- dpoequb +- dporfs +- dposv +- dposvx +- dpotf2 +- dpotrf +- dpotri +- dpotrs +- dppcon +- dppequ +- dpprfs +- dppsv +- dppsvx +- dpptrf +- dpptri +- dpptrs +- dpstf2 +- dpstrf +- dptcon +- dpteqr +- dptrfs +- dptsv +- dptsvx +- dpttrf +- dpttrs +- dptts2 +- drscl +- dsbev +- dsbevd +- dsbevx +- dsbgst +- dsbgv +- dsbgvd +- dsbgvx +- dsbtrd +- dsfrk +- dsgesv +- dspcon +- dspev +- dspevd +- dspevx +- dspgst +- dspgv +- dspgvd +- dspgvx +- dsposv +- dsprfs +- dspsv +- dspsvx +- dsptrd +- dsptrf +- dsptri +- dsptrs +- dstebz +- dstedc +- dstegr +- dstein +- dstemr +- dsteqr +- dsterf +- dstev +- dstevd +- dstevr +- dstevx +- dsycon +- dsyconv +- dsyequb +- dsyev +- dsyevd +- dsyevr +- dsyevx +- dsygs2 +- dsygst +- dsygv +- dsygvd +- dsygvx +- dsyrfs +- dsysv +- dsysvx +- dsyswapr +- dsytd2 +- dsytf2 +- dsytrd +- dsytrf +- dsytri +- dsytri2 +- dsytri2x +- dsytrs +- dsytrs2 +- dtbcon +- dtbrfs +- dtbtrs +- dtfsm +- dtftri +- dtfttp +- dtfttr +- dtgevc +- dtgex2 +- dtgexc +- dtgsen +- dtgsja +- dtgsna +- dtgsy2 +- dtgsyl +- dtpcon +- dtpmqrt +- dtpqrt +- dtpqrt2 +- dtprfb +- dtprfs +- dtptri +- dtptrs +- dtpttf +- dtpttr +- dtrcon +- dtrevc +- dtrexc +- dtrrfs +- dtrsen +- dtrsna +- dtrsyl +- dtrti2 +- dtrtri +- dtrtrs +- dtrttf +- dtrttp +- dtzrzf +- dzsum1 +- icmax1 +- ieeeck +- ilaclc +- ilaclr +- iladiag +- iladlc +- iladlr +- ilaprec +- ilaslc +- ilaslr +- ilatrans +- ilauplo +- ilaver +- ilazlc +- ilazlr +- izmax1 +- sbbcsd +- sbdsdc +- sbdsqr +- scsum1 +- sdisna +- sgbbrd +- sgbcon +- sgbequ +- sgbequb +- sgbrfs +- sgbsv +- sgbsvx +- sgbtf2 +- sgbtrf +- sgbtrs +- sgebak +- sgebal +- sgebd2 +- sgebrd +- sgecon +- sgeequ +- sgeequb +- sgees +- sgeesx +- sgeev +- sgeevx +- sgehd2 +- sgehrd +- sgejsv +- sgelq2 +- sgelqf +- sgels +- sgelsd +- sgelss +- sgelsy +- sgemqrt +- sgeql2 +- sgeqlf +- sgeqp3 +- sgeqr2 +- sgeqr2p +- sgeqrf +- sgeqrfp +- sgeqrt +- sgeqrt2 +- sgeqrt3 +- sgerfs +- sgerq2 +- sgerqf +- sgesc2 +- sgesdd +- sgesv +- sgesvd +- sgesvj +- sgesvx +- sgetc2 +- sgetf2 +- sgetrf +- sgetri +- sgetrs +- sggbak +- sggbal +- sgges +- sggesx +- sggev +- sggevx +- sggglm +- sgghrd +- sgglse +- sggqrf +- sggrqf +- sgsvj0 +- sgsvj1 +- sgtcon +- sgtrfs +- sgtsv +- sgtsvx +- sgttrf +- sgttrs +- sgtts2 +- shgeqz +- shsein +- shseqr +- slabad +- slabrd +- slacn2 +- slacon +- slacpy +- sladiv +- slae2 +- slaebz +- slaed0 +- slaed1 +- slaed2 +- slaed3 +- slaed4 +- slaed5 +- slaed6 +- slaed7 +- slaed8 +- slaed9 +- slaeda +- slaein +- slaev2 +- slaexc +- slag2 +- slag2d +- slags2 +- slagtf +- slagtm +- slagts +- slagv2 +- slahqr +- slahr2 +- slaic1 +- slaln2 +- slals0 +- slalsa +- slalsd +- slamch +- slamrg +- slangb +- slange +- slangt +- slanhs +- slansb +- slansf +- slansp +- slanst +- slansy +- slantb +- slantp +- slantr +- slanv2 +- slapll +- slapmr +- slapmt +- slapy2 +- slapy3 +- slaqgb +- slaqge +- slaqp2 +- slaqps +- slaqr0 +- slaqr1 +- slaqr2 +- slaqr3 +- slaqr4 +- slaqr5 +- slaqsb +- slaqsp +- slaqsy +- slaqtr +- slar1v +- slar2v +- slarf +- slarfb +- slarfg +- slarfgp +- slarft +- slarfx +- slargv +- slarnv +- slarra +- slarrb +- slarrc +- slarrd +- slarre +- slarrf +- slarrj +- slarrk +- slarrr +- slarrv +- slartg +- slartgp +- slartgs +- slartv +- slaruv +- slarz +- slarzb +- slarzt +- slas2 +- slascl +- slasd0 +- slasd1 +- slasd2 +- slasd3 +- slasd4 +- slasd5 +- slasd6 +- slasd7 +- slasd8 +- slasda +- slasdq +- slasdt +- slaset +- slasq1 +- slasq2 +- slasq3 +- slasq4 +- slasq6 +- slasr +- slasrt +- slassq +- slasv2 +- slaswp +- slasy2 +- slasyf +- slatbs +- slatdf +- slatps +- slatrd +- slatrs +- slatrz +- slauu2 +- slauum +- sopgtr +- sopmtr +- sorbdb +- sorcsd +- sorg2l +- sorg2r +- sorgbr +- sorghr +- sorgl2 +- sorglq +- sorgql +- sorgqr +- sorgr2 +- sorgrq +- sorgtr +- sorm2l +- sorm2r +- sormbr +- sormhr +- sorml2 +- sormlq +- sormql +- sormqr +- sormr2 +- sormr3 +- sormrq +- sormrz +- sormtr +- spbcon +- spbequ +- spbrfs +- spbstf +- spbsv +- spbsvx +- spbtf2 +- spbtrf +- spbtrs +- spftrf +- spftri +- spftrs +- spocon +- spoequ +- spoequb +- sporfs +- sposv +- sposvx +- spotf2 +- spotrf +- spotri +- spotrs +- sppcon +- sppequ +- spprfs +- sppsv +- sppsvx +- spptrf +- spptri +- spptrs +- spstf2 +- spstrf +- sptcon +- spteqr +- sptrfs +- sptsv +- sptsvx +- spttrf +- spttrs +- sptts2 +- srscl +- ssbev +- ssbevd +- ssbevx +- ssbgst +- ssbgv +- ssbgvd +- ssbgvx +- ssbtrd +- ssfrk +- sspcon +- sspev +- sspevd +- sspevx +- sspgst +- sspgv +- sspgvd +- sspgvx +- ssprfs +- sspsv +- sspsvx +- ssptrd +- ssptrf +- ssptri +- ssptrs +- sstebz +- sstedc +- sstegr +- sstein +- sstemr +- ssteqr +- ssterf +- sstev +- sstevd +- sstevr +- sstevx +- ssycon +- ssyconv +- ssyequb +- ssyev +- ssyevd +- ssyevr +- ssyevx +- ssygs2 +- ssygst +- ssygv +- ssygvd +- ssygvx +- ssyrfs +- ssysv +- ssysvx +- ssyswapr +- ssytd2 +- ssytf2 +- ssytrd +- ssytrf +- ssytri +- ssytri2 +- ssytri2x +- ssytrs +- ssytrs2 +- stbcon +- stbrfs +- stbtrs +- stfsm +- stftri +- stfttp +- stfttr +- stgevc +- stgex2 +- stgexc +- stgsen +- stgsja +- stgsna +- stgsy2 +- stgsyl +- stpcon +- stpmqrt +- stpqrt +- stpqrt2 +- stprfb +- stprfs +- stptri +- stptrs +- stpttf +- stpttr +- strcon +- strevc +- strexc +- strrfs +- strsen +- strsna +- strsyl +- strti2 +- strtri +- strtrs +- strttf +- strttp +- stzrzf +- xerbla_array +- zbbcsd +- zbdsqr +- zcgesv +- zcposv +- zdrscl +- zgbbrd +- zgbcon +- zgbequ +- zgbequb +- zgbrfs +- zgbsv +- zgbsvx +- zgbtf2 +- zgbtrf +- zgbtrs +- zgebak +- zgebal +- zgebd2 +- zgebrd +- zgecon +- zgeequ +- zgeequb +- zgees +- zgeesx +- zgeev +- zgeevx +- zgehd2 +- zgehrd +- zgelq2 +- zgelqf +- zgels +- zgelsd +- zgelss +- zgelsy +- zgemqrt +- zgeql2 +- zgeqlf +- zgeqp3 +- zgeqr2 +- zgeqr2p +- zgeqrf +- zgeqrfp +- zgeqrt +- zgeqrt2 +- zgeqrt3 +- zgerfs +- zgerq2 +- zgerqf +- zgesc2 +- zgesdd +- zgesv +- zgesvd +- zgesvx +- zgetc2 +- zgetf2 +- zgetrf +- zgetri +- zgetrs +- zggbak +- zggbal +- zgges +- zggesx +- zggev +- zggevx +- zggglm +- zgghrd +- zgglse +- zggqrf +- zggrqf +- zgtcon +- zgtrfs +- zgtsv +- zgtsvx +- zgttrf +- zgttrs +- zgtts2 +- zhbev +- zhbevd +- zhbevx +- zhbgst +- zhbgv +- zhbgvd +- zhbgvx +- zhbtrd +- zhecon +- zheequb +- zheev +- zheevd +- zheevr +- zheevx +- zhegs2 +- zhegst +- zhegv +- zhegvd +- zhegvx +- zherfs +- zhesv +- zhesvx +- zheswapr +- zhetd2 +- zhetf2 +- zhetrd +- zhetrf +- zhetri +- zhetri2 +- zhetri2x +- zhetrs +- zhetrs2 +- zhfrk +- zhgeqz +- zhpcon +- zhpev +- zhpevd +- zhpevx +- zhpgst +- zhpgv +- zhpgvd +- zhpgvx +- zhprfs +- zhpsv +- zhpsvx +- zhptrd +- zhptrf +- zhptri +- zhptrs +- zhsein +- zhseqr +- zlabrd +- zlacgv +- zlacn2 +- zlacon +- zlacp2 +- zlacpy +- zlacrm +- zlacrt +- zladiv +- zlaed0 +- zlaed7 +- zlaed8 +- zlaein +- zlaesy +- zlaev2 +- zlag2c +- zlags2 +- zlagtm +- zlahef +- zlahqr +- zlahr2 +- zlaic1 +- zlals0 +- zlalsa +- zlalsd +- zlangb +- zlange +- zlangt +- zlanhb +- zlanhe +- zlanhf +- zlanhp +- zlanhs +- zlanht +- zlansb +- zlansp +- zlansy +- zlantb +- zlantp +- zlantr +- zlapll +- zlapmr +- zlapmt +- zlaqgb +- zlaqge +- zlaqhb +- zlaqhe +- zlaqhp +- zlaqp2 +- zlaqps +- zlaqr0 +- zlaqr1 +- zlaqr2 +- zlaqr3 +- zlaqr4 +- zlaqr5 +- zlaqsb +- zlaqsp +- zlaqsy +- zlar1v +- zlar2v +- zlarcm +- zlarf +- zlarfb +- zlarfg +- zlarfgp +- zlarft +- zlarfx +- zlargv +- zlarnv +- zlarrv +- zlartg +- zlartv +- zlarz +- zlarzb +- zlarzt +- zlascl +- zlaset +- zlasr +- zlassq +- zlaswp +- zlasyf +- zlat2c +- zlatbs +- zlatdf +- zlatps +- zlatrd +- zlatrs +- zlatrz +- zlauu2 +- zlauum +- zpbcon +- zpbequ +- zpbrfs +- zpbstf +- zpbsv +- zpbsvx +- zpbtf2 +- zpbtrf +- zpbtrs +- zpftrf +- zpftri +- zpftrs +- zpocon +- zpoequ +- zpoequb +- zporfs +- zposv +- zposvx +- zpotf2 +- zpotrf +- zpotri +- zpotrs +- zppcon +- zppequ +- zpprfs +- zppsv +- zppsvx +- zpptrf +- zpptri +- zpptrs +- zpstf2 +- zpstrf +- zptcon +- zpteqr +- zptrfs +- zptsv +- zptsvx +- zpttrf +- zpttrs +- zptts2 +- zrot +- zspcon +- zspmv +- zspr +- zsprfs +- zspsv +- zspsvx +- zsptrf +- zsptri +- zsptrs +- zstedc +- zstegr +- zstein +- zstemr +- zsteqr +- zsycon +- zsyconv +- zsyequb +- zsymv +- zsyr +- zsyrfs +- zsysv +- zsysvx +- zsyswapr +- zsytf2 +- zsytrf +- zsytri +- zsytri2 +- zsytri2x +- zsytrs +- zsytrs2 +- ztbcon +- ztbrfs +- ztbtrs +- ztfsm +- ztftri +- ztfttp +- ztfttr +- ztgevc +- ztgex2 +- ztgexc +- ztgsen +- ztgsja +- ztgsna +- ztgsy2 +- ztgsyl +- ztpcon +- ztpmqrt +- ztpqrt +- ztpqrt2 +- ztprfb +- ztprfs +- ztptri +- ztptrs +- ztpttf +- ztpttr +- ztrcon +- ztrevc +- ztrexc +- ztrrfs +- ztrsen +- ztrsna +- ztrsyl +- ztrti2 +- ztrtri +- ztrtrs +- ztrttf +- ztrttp +- ztzrzf +- zunbdb +- zuncsd +- zung2l +- zung2r +- zungbr +- zunghr +- zungl2 +- zunglq +- zungql +- zungqr +- zungr2 +- zungrq +- zungtr +- zunm2l +- zunm2r +- zunmbr +- zunmhr +- zunml2 +- zunmlq +- zunmql +- zunmqr +- zunmr2 +- zunmr3 +- zunmrq +- zunmrz +- zunmtr +- zupgtr +- zupmtr + + +""" + +# Within SciPy, these wrappers can be used via relative or absolute cimport. +# Examples: +# from ..linalg cimport cython_lapack +# from scipy.linalg cimport cython_lapack +# cimport scipy.linalg.cython_lapack as cython_lapack +# cimport ..linalg.cython_lapack as cython_lapack + +# Within SciPy, if LAPACK functions are needed in C/C++/Fortran, +# these wrappers should not be used. +# The original libraries should be linked directly. + +cdef extern from "fortran_defs.h": + pass + +from numpy cimport npy_complex64, npy_complex128 + +cdef extern from "_lapack_subroutines.h": + # Function pointer type declarations for + # gees and gges families of functions. + ctypedef bint _cselect1(npy_complex64*) + ctypedef bint _cselect2(npy_complex64*, npy_complex64*) + ctypedef bint _dselect2(d*, d*) + ctypedef bint _dselect3(d*, d*, d*) + ctypedef bint _sselect2(s*, s*) + ctypedef bint _sselect3(s*, s*, s*) + ctypedef bint _zselect1(npy_complex128*) + ctypedef bint _zselect2(npy_complex128*, npy_complex128*) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cbbcsd "BLAS_FUNC(cbbcsd)"(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, s *theta, s *phi, npy_complex64 *u1, int *ldu1, npy_complex64 *u2, int *ldu2, npy_complex64 *v1t, int *ldv1t, npy_complex64 *v2t, int *ldv2t, s *b11d, s *b11e, s *b12d, s *b12e, s *b21d, s *b21e, s *b22d, s *b22e, s *rwork, int *lrwork, int *info) nogil +cdef void cbbcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, s *theta, s *phi, c *u1, int *ldu1, c *u2, int *ldu2, c *v1t, int *ldv1t, c *v2t, int *ldv2t, s *b11d, s *b11e, s *b12d, s *b12e, s *b21d, s *b21e, s *b22d, s *b22e, s *rwork, int *lrwork, int *info) noexcept nogil: + + _fortran_cbbcsd(jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta, phi, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, b11d, b11e, b12d, b12e, b21d, b21e, b22d, b22e, rwork, lrwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cbdsqr "BLAS_FUNC(cbdsqr)"(char *uplo, int *n, int *ncvt, int *nru, int *ncc, s *d, s *e, npy_complex64 *vt, int *ldvt, npy_complex64 *u, int *ldu, npy_complex64 *c, int *ldc, s *rwork, int *info) nogil +cdef void cbdsqr(char *uplo, int *n, int *ncvt, int *nru, int *ncc, s *d, s *e, c *vt, int *ldvt, c *u, int *ldu, c *c, int *ldc, s *rwork, int *info) noexcept nogil: + + _fortran_cbdsqr(uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgbbrd "BLAS_FUNC(cgbbrd)"(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, npy_complex64 *ab, int *ldab, s *d, s *e, npy_complex64 *q, int *ldq, npy_complex64 *pt, int *ldpt, npy_complex64 *c, int *ldc, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cgbbrd(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, c *ab, int *ldab, s *d, s *e, c *q, int *ldq, c *pt, int *ldpt, c *c, int *ldc, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cgbbrd(vect, m, n, ncc, kl, ku, ab, ldab, d, e, q, ldq, pt, ldpt, c, ldc, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgbcon "BLAS_FUNC(cgbcon)"(char *norm, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, int *ipiv, s *anorm, s *rcond, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cgbcon(char *norm, int *n, int *kl, int *ku, c *ab, int *ldab, int *ipiv, s *anorm, s *rcond, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cgbcon(norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgbequ "BLAS_FUNC(cgbequ)"(int *m, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) nogil +cdef void cgbequ(int *m, int *n, int *kl, int *ku, c *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil: + + _fortran_cgbequ(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgbequb "BLAS_FUNC(cgbequb)"(int *m, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) nogil +cdef void cgbequb(int *m, int *n, int *kl, int *ku, c *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil: + + _fortran_cgbequb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgbrfs "BLAS_FUNC(cgbrfs)"(char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *afb, int *ldafb, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cgbrfs(char *trans, int *n, int *kl, int *ku, int *nrhs, c *ab, int *ldab, c *afb, int *ldafb, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cgbrfs(trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgbsv "BLAS_FUNC(cgbsv)"(int *n, int *kl, int *ku, int *nrhs, npy_complex64 *ab, int *ldab, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cgbsv(int *n, int *kl, int *ku, int *nrhs, c *ab, int *ldab, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cgbsv(n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgbsvx "BLAS_FUNC(cgbsvx)"(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *afb, int *ldafb, int *ipiv, char *equed, s *r, s *c, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cgbsvx(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, c *ab, int *ldab, c *afb, int *ldafb, int *ipiv, char *equed, s *r, s *c, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cgbsvx(fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgbtf2 "BLAS_FUNC(cgbtf2)"(int *m, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, int *ipiv, int *info) nogil +cdef void cgbtf2(int *m, int *n, int *kl, int *ku, c *ab, int *ldab, int *ipiv, int *info) noexcept nogil: + + _fortran_cgbtf2(m, n, kl, ku, ab, ldab, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgbtrf "BLAS_FUNC(cgbtrf)"(int *m, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, int *ipiv, int *info) nogil +cdef void cgbtrf(int *m, int *n, int *kl, int *ku, c *ab, int *ldab, int *ipiv, int *info) noexcept nogil: + + _fortran_cgbtrf(m, n, kl, ku, ab, ldab, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgbtrs "BLAS_FUNC(cgbtrs)"(char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex64 *ab, int *ldab, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cgbtrs(char *trans, int *n, int *kl, int *ku, int *nrhs, c *ab, int *ldab, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cgbtrs(trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgebak "BLAS_FUNC(cgebak)"(char *job, char *side, int *n, int *ilo, int *ihi, s *scale, int *m, npy_complex64 *v, int *ldv, int *info) nogil +cdef void cgebak(char *job, char *side, int *n, int *ilo, int *ihi, s *scale, int *m, c *v, int *ldv, int *info) noexcept nogil: + + _fortran_cgebak(job, side, n, ilo, ihi, scale, m, v, ldv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgebal "BLAS_FUNC(cgebal)"(char *job, int *n, npy_complex64 *a, int *lda, int *ilo, int *ihi, s *scale, int *info) nogil +cdef void cgebal(char *job, int *n, c *a, int *lda, int *ilo, int *ihi, s *scale, int *info) noexcept nogil: + + _fortran_cgebal(job, n, a, lda, ilo, ihi, scale, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgebd2 "BLAS_FUNC(cgebd2)"(int *m, int *n, npy_complex64 *a, int *lda, s *d, s *e, npy_complex64 *tauq, npy_complex64 *taup, npy_complex64 *work, int *info) nogil +cdef void cgebd2(int *m, int *n, c *a, int *lda, s *d, s *e, c *tauq, c *taup, c *work, int *info) noexcept nogil: + + _fortran_cgebd2(m, n, a, lda, d, e, tauq, taup, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgebrd "BLAS_FUNC(cgebrd)"(int *m, int *n, npy_complex64 *a, int *lda, s *d, s *e, npy_complex64 *tauq, npy_complex64 *taup, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cgebrd(int *m, int *n, c *a, int *lda, s *d, s *e, c *tauq, c *taup, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cgebrd(m, n, a, lda, d, e, tauq, taup, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgecon "BLAS_FUNC(cgecon)"(char *norm, int *n, npy_complex64 *a, int *lda, s *anorm, s *rcond, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cgecon(char *norm, int *n, c *a, int *lda, s *anorm, s *rcond, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cgecon(norm, n, a, lda, anorm, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeequ "BLAS_FUNC(cgeequ)"(int *m, int *n, npy_complex64 *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) nogil +cdef void cgeequ(int *m, int *n, c *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil: + + _fortran_cgeequ(m, n, a, lda, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeequb "BLAS_FUNC(cgeequb)"(int *m, int *n, npy_complex64 *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) nogil +cdef void cgeequb(int *m, int *n, c *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil: + + _fortran_cgeequb(m, n, a, lda, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgees "BLAS_FUNC(cgees)"(char *jobvs, char *sort, _cselect1 *select, int *n, npy_complex64 *a, int *lda, int *sdim, npy_complex64 *w, npy_complex64 *vs, int *ldvs, npy_complex64 *work, int *lwork, s *rwork, bint *bwork, int *info) nogil +cdef void cgees(char *jobvs, char *sort, cselect1 *select, int *n, c *a, int *lda, int *sdim, c *w, c *vs, int *ldvs, c *work, int *lwork, s *rwork, bint *bwork, int *info) noexcept nogil: + + _fortran_cgees(jobvs, sort, <_cselect1*>select, n, a, lda, sdim, w, vs, ldvs, work, lwork, rwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeesx "BLAS_FUNC(cgeesx)"(char *jobvs, char *sort, _cselect1 *select, char *sense, int *n, npy_complex64 *a, int *lda, int *sdim, npy_complex64 *w, npy_complex64 *vs, int *ldvs, s *rconde, s *rcondv, npy_complex64 *work, int *lwork, s *rwork, bint *bwork, int *info) nogil +cdef void cgeesx(char *jobvs, char *sort, cselect1 *select, char *sense, int *n, c *a, int *lda, int *sdim, c *w, c *vs, int *ldvs, s *rconde, s *rcondv, c *work, int *lwork, s *rwork, bint *bwork, int *info) noexcept nogil: + + _fortran_cgeesx(jobvs, sort, <_cselect1*>select, sense, n, a, lda, sdim, w, vs, ldvs, rconde, rcondv, work, lwork, rwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeev "BLAS_FUNC(cgeev)"(char *jobvl, char *jobvr, int *n, npy_complex64 *a, int *lda, npy_complex64 *w, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void cgeev(char *jobvl, char *jobvr, int *n, c *a, int *lda, c *w, c *vl, int *ldvl, c *vr, int *ldvr, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_cgeev(jobvl, jobvr, n, a, lda, w, vl, ldvl, vr, ldvr, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeevx "BLAS_FUNC(cgeevx)"(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, npy_complex64 *a, int *lda, npy_complex64 *w, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, int *ilo, int *ihi, s *scale, s *abnrm, s *rconde, s *rcondv, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void cgeevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, c *a, int *lda, c *w, c *vl, int *ldvl, c *vr, int *ldvr, int *ilo, int *ihi, s *scale, s *abnrm, s *rconde, s *rcondv, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_cgeevx(balanc, jobvl, jobvr, sense, n, a, lda, w, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgehd2 "BLAS_FUNC(cgehd2)"(int *n, int *ilo, int *ihi, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info) nogil +cdef void cgehd2(int *n, int *ilo, int *ihi, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil: + + _fortran_cgehd2(n, ilo, ihi, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgehrd "BLAS_FUNC(cgehrd)"(int *n, int *ilo, int *ihi, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cgehrd(int *n, int *ilo, int *ihi, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cgehrd(n, ilo, ihi, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgelq2 "BLAS_FUNC(cgelq2)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info) nogil +cdef void cgelq2(int *m, int *n, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil: + + _fortran_cgelq2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgelqf "BLAS_FUNC(cgelqf)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cgelqf(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cgelqf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgels "BLAS_FUNC(cgels)"(char *trans, int *m, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cgels(char *trans, int *m, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cgels(trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgelsd "BLAS_FUNC(cgelsd)"(int *m, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, s *s, s *rcond, int *rank, npy_complex64 *work, int *lwork, s *rwork, int *iwork, int *info) nogil +cdef void cgelsd(int *m, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, s *s, s *rcond, int *rank, c *work, int *lwork, s *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_cgelsd(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgelss "BLAS_FUNC(cgelss)"(int *m, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, s *s, s *rcond, int *rank, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void cgelss(int *m, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, s *s, s *rcond, int *rank, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_cgelss(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgelsy "BLAS_FUNC(cgelsy)"(int *m, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *jpvt, s *rcond, int *rank, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void cgelsy(int *m, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, int *jpvt, s *rcond, int *rank, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_cgelsy(m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgemqrt "BLAS_FUNC(cgemqrt)"(char *side, char *trans, int *m, int *n, int *k, int *nb, npy_complex64 *v, int *ldv, npy_complex64 *t, int *ldt, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info) nogil +cdef void cgemqrt(char *side, char *trans, int *m, int *n, int *k, int *nb, c *v, int *ldv, c *t, int *ldt, c *c, int *ldc, c *work, int *info) noexcept nogil: + + _fortran_cgemqrt(side, trans, m, n, k, nb, v, ldv, t, ldt, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeql2 "BLAS_FUNC(cgeql2)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info) nogil +cdef void cgeql2(int *m, int *n, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil: + + _fortran_cgeql2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeqlf "BLAS_FUNC(cgeqlf)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cgeqlf(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cgeqlf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeqp3 "BLAS_FUNC(cgeqp3)"(int *m, int *n, npy_complex64 *a, int *lda, int *jpvt, npy_complex64 *tau, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void cgeqp3(int *m, int *n, c *a, int *lda, int *jpvt, c *tau, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_cgeqp3(m, n, a, lda, jpvt, tau, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeqr2 "BLAS_FUNC(cgeqr2)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info) nogil +cdef void cgeqr2(int *m, int *n, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil: + + _fortran_cgeqr2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeqr2p "BLAS_FUNC(cgeqr2p)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info) nogil +cdef void cgeqr2p(int *m, int *n, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil: + + _fortran_cgeqr2p(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeqrf "BLAS_FUNC(cgeqrf)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cgeqrf(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cgeqrf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeqrfp "BLAS_FUNC(cgeqrfp)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cgeqrfp(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cgeqrfp(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeqrt "BLAS_FUNC(cgeqrt)"(int *m, int *n, int *nb, npy_complex64 *a, int *lda, npy_complex64 *t, int *ldt, npy_complex64 *work, int *info) nogil +cdef void cgeqrt(int *m, int *n, int *nb, c *a, int *lda, c *t, int *ldt, c *work, int *info) noexcept nogil: + + _fortran_cgeqrt(m, n, nb, a, lda, t, ldt, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeqrt2 "BLAS_FUNC(cgeqrt2)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *t, int *ldt, int *info) nogil +cdef void cgeqrt2(int *m, int *n, c *a, int *lda, c *t, int *ldt, int *info) noexcept nogil: + + _fortran_cgeqrt2(m, n, a, lda, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgeqrt3 "BLAS_FUNC(cgeqrt3)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *t, int *ldt, int *info) nogil +cdef void cgeqrt3(int *m, int *n, c *a, int *lda, c *t, int *ldt, int *info) noexcept nogil: + + _fortran_cgeqrt3(m, n, a, lda, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgerfs "BLAS_FUNC(cgerfs)"(char *trans, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cgerfs(char *trans, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cgerfs(trans, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgerq2 "BLAS_FUNC(cgerq2)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info) nogil +cdef void cgerq2(int *m, int *n, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil: + + _fortran_cgerq2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgerqf "BLAS_FUNC(cgerqf)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cgerqf(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cgerqf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgesc2 "BLAS_FUNC(cgesc2)"(int *n, npy_complex64 *a, int *lda, npy_complex64 *rhs, int *ipiv, int *jpiv, s *scale) nogil +cdef void cgesc2(int *n, c *a, int *lda, c *rhs, int *ipiv, int *jpiv, s *scale) noexcept nogil: + + _fortran_cgesc2(n, a, lda, rhs, ipiv, jpiv, scale) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgesdd "BLAS_FUNC(cgesdd)"(char *jobz, int *m, int *n, npy_complex64 *a, int *lda, s *s, npy_complex64 *u, int *ldu, npy_complex64 *vt, int *ldvt, npy_complex64 *work, int *lwork, s *rwork, int *iwork, int *info) nogil +cdef void cgesdd(char *jobz, int *m, int *n, c *a, int *lda, s *s, c *u, int *ldu, c *vt, int *ldvt, c *work, int *lwork, s *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_cgesdd(jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgesv "BLAS_FUNC(cgesv)"(int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cgesv(int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cgesv(n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgesvd "BLAS_FUNC(cgesvd)"(char *jobu, char *jobvt, int *m, int *n, npy_complex64 *a, int *lda, s *s, npy_complex64 *u, int *ldu, npy_complex64 *vt, int *ldvt, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void cgesvd(char *jobu, char *jobvt, int *m, int *n, c *a, int *lda, s *s, c *u, int *ldu, c *vt, int *ldvt, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_cgesvd(jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgesvx "BLAS_FUNC(cgesvx)"(char *fact, char *trans, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, char *equed, s *r, s *c, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cgesvx(char *fact, char *trans, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, char *equed, s *r, s *c, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cgesvx(fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgetc2 "BLAS_FUNC(cgetc2)"(int *n, npy_complex64 *a, int *lda, int *ipiv, int *jpiv, int *info) nogil +cdef void cgetc2(int *n, c *a, int *lda, int *ipiv, int *jpiv, int *info) noexcept nogil: + + _fortran_cgetc2(n, a, lda, ipiv, jpiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgetf2 "BLAS_FUNC(cgetf2)"(int *m, int *n, npy_complex64 *a, int *lda, int *ipiv, int *info) nogil +cdef void cgetf2(int *m, int *n, c *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_cgetf2(m, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgetrf "BLAS_FUNC(cgetrf)"(int *m, int *n, npy_complex64 *a, int *lda, int *ipiv, int *info) nogil +cdef void cgetrf(int *m, int *n, c *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_cgetrf(m, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgetri "BLAS_FUNC(cgetri)"(int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cgetri(int *n, c *a, int *lda, int *ipiv, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cgetri(n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgetrs "BLAS_FUNC(cgetrs)"(char *trans, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cgetrs(char *trans, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cgetrs(trans, n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cggbak "BLAS_FUNC(cggbak)"(char *job, char *side, int *n, int *ilo, int *ihi, s *lscale, s *rscale, int *m, npy_complex64 *v, int *ldv, int *info) nogil +cdef void cggbak(char *job, char *side, int *n, int *ilo, int *ihi, s *lscale, s *rscale, int *m, c *v, int *ldv, int *info) noexcept nogil: + + _fortran_cggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cggbal "BLAS_FUNC(cggbal)"(char *job, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *ilo, int *ihi, s *lscale, s *rscale, s *work, int *info) nogil +cdef void cggbal(char *job, int *n, c *a, int *lda, c *b, int *ldb, int *ilo, int *ihi, s *lscale, s *rscale, s *work, int *info) noexcept nogil: + + _fortran_cggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgges "BLAS_FUNC(cgges)"(char *jobvsl, char *jobvsr, char *sort, _cselect2 *selctg, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *sdim, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *vsl, int *ldvsl, npy_complex64 *vsr, int *ldvsr, npy_complex64 *work, int *lwork, s *rwork, bint *bwork, int *info) nogil +cdef void cgges(char *jobvsl, char *jobvsr, char *sort, cselect2 *selctg, int *n, c *a, int *lda, c *b, int *ldb, int *sdim, c *alpha, c *beta, c *vsl, int *ldvsl, c *vsr, int *ldvsr, c *work, int *lwork, s *rwork, bint *bwork, int *info) noexcept nogil: + + _fortran_cgges(jobvsl, jobvsr, sort, <_cselect2*>selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cggesx "BLAS_FUNC(cggesx)"(char *jobvsl, char *jobvsr, char *sort, _cselect2 *selctg, char *sense, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *sdim, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *vsl, int *ldvsl, npy_complex64 *vsr, int *ldvsr, s *rconde, s *rcondv, npy_complex64 *work, int *lwork, s *rwork, int *iwork, int *liwork, bint *bwork, int *info) nogil +cdef void cggesx(char *jobvsl, char *jobvsr, char *sort, cselect2 *selctg, char *sense, int *n, c *a, int *lda, c *b, int *ldb, int *sdim, c *alpha, c *beta, c *vsl, int *ldvsl, c *vsr, int *ldvsr, s *rconde, s *rcondv, c *work, int *lwork, s *rwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil: + + _fortran_cggesx(jobvsl, jobvsr, sort, <_cselect2*>selctg, sense, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, rwork, iwork, liwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cggev "BLAS_FUNC(cggev)"(char *jobvl, char *jobvr, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void cggev(char *jobvl, char *jobvr, int *n, c *a, int *lda, c *b, int *ldb, c *alpha, c *beta, c *vl, int *ldvl, c *vr, int *ldvr, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_cggev(jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cggevx "BLAS_FUNC(cggevx)"(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, int *ilo, int *ihi, s *lscale, s *rscale, s *abnrm, s *bbnrm, s *rconde, s *rcondv, npy_complex64 *work, int *lwork, s *rwork, int *iwork, bint *bwork, int *info) nogil +cdef void cggevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, c *a, int *lda, c *b, int *ldb, c *alpha, c *beta, c *vl, int *ldvl, c *vr, int *ldvr, int *ilo, int *ihi, s *lscale, s *rscale, s *abnrm, s *bbnrm, s *rconde, s *rcondv, c *work, int *lwork, s *rwork, int *iwork, bint *bwork, int *info) noexcept nogil: + + _fortran_cggevx(balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cggglm "BLAS_FUNC(cggglm)"(int *n, int *m, int *p, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *d, npy_complex64 *x, npy_complex64 *y, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cggglm(int *n, int *m, int *p, c *a, int *lda, c *b, int *ldb, c *d, c *x, c *y, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cggglm(n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgghrd "BLAS_FUNC(cgghrd)"(char *compq, char *compz, int *n, int *ilo, int *ihi, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *q, int *ldq, npy_complex64 *z, int *ldz, int *info) nogil +cdef void cgghrd(char *compq, char *compz, int *n, int *ilo, int *ihi, c *a, int *lda, c *b, int *ldb, c *q, int *ldq, c *z, int *ldz, int *info) noexcept nogil: + + _fortran_cgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgglse "BLAS_FUNC(cgglse)"(int *m, int *n, int *p, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *c, npy_complex64 *d, npy_complex64 *x, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cgglse(int *m, int *n, int *p, c *a, int *lda, c *b, int *ldb, c *c, c *d, c *x, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cgglse(m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cggqrf "BLAS_FUNC(cggqrf)"(int *n, int *m, int *p, npy_complex64 *a, int *lda, npy_complex64 *taua, npy_complex64 *b, int *ldb, npy_complex64 *taub, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cggqrf(int *n, int *m, int *p, c *a, int *lda, c *taua, c *b, int *ldb, c *taub, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cggqrf(n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cggrqf "BLAS_FUNC(cggrqf)"(int *m, int *p, int *n, npy_complex64 *a, int *lda, npy_complex64 *taua, npy_complex64 *b, int *ldb, npy_complex64 *taub, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cggrqf(int *m, int *p, int *n, c *a, int *lda, c *taua, c *b, int *ldb, c *taub, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cggrqf(m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgtcon "BLAS_FUNC(cgtcon)"(char *norm, int *n, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *du2, int *ipiv, s *anorm, s *rcond, npy_complex64 *work, int *info) nogil +cdef void cgtcon(char *norm, int *n, c *dl, c *d, c *du, c *du2, int *ipiv, s *anorm, s *rcond, c *work, int *info) noexcept nogil: + + _fortran_cgtcon(norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgtrfs "BLAS_FUNC(cgtrfs)"(char *trans, int *n, int *nrhs, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *dlf, npy_complex64 *df, npy_complex64 *duf, npy_complex64 *du2, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cgtrfs(char *trans, int *n, int *nrhs, c *dl, c *d, c *du, c *dlf, c *df, c *duf, c *du2, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cgtrfs(trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgtsv "BLAS_FUNC(cgtsv)"(int *n, int *nrhs, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cgtsv(int *n, int *nrhs, c *dl, c *d, c *du, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cgtsv(n, nrhs, dl, d, du, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgtsvx "BLAS_FUNC(cgtsvx)"(char *fact, char *trans, int *n, int *nrhs, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *dlf, npy_complex64 *df, npy_complex64 *duf, npy_complex64 *du2, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cgtsvx(char *fact, char *trans, int *n, int *nrhs, c *dl, c *d, c *du, c *dlf, c *df, c *duf, c *du2, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cgtsvx(fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgttrf "BLAS_FUNC(cgttrf)"(int *n, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *du2, int *ipiv, int *info) nogil +cdef void cgttrf(int *n, c *dl, c *d, c *du, c *du2, int *ipiv, int *info) noexcept nogil: + + _fortran_cgttrf(n, dl, d, du, du2, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgttrs "BLAS_FUNC(cgttrs)"(char *trans, int *n, int *nrhs, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *du2, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cgttrs(char *trans, int *n, int *nrhs, c *dl, c *d, c *du, c *du2, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cgtts2 "BLAS_FUNC(cgtts2)"(int *itrans, int *n, int *nrhs, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *du2, int *ipiv, npy_complex64 *b, int *ldb) nogil +cdef void cgtts2(int *itrans, int *n, int *nrhs, c *dl, c *d, c *du, c *du2, int *ipiv, c *b, int *ldb) noexcept nogil: + + _fortran_cgtts2(itrans, n, nrhs, dl, d, du, du2, ipiv, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chbev "BLAS_FUNC(chbev)"(char *jobz, char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, s *rwork, int *info) nogil +cdef void chbev(char *jobz, char *uplo, int *n, int *kd, c *ab, int *ldab, s *w, c *z, int *ldz, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_chbev(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chbevd "BLAS_FUNC(chbevd)"(char *jobz, char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void chbevd(char *jobz, char *uplo, int *n, int *kd, c *ab, int *ldab, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_chbevd(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chbevx "BLAS_FUNC(chbevx)"(char *jobz, char *range, char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, npy_complex64 *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, s *rwork, int *iwork, int *ifail, int *info) nogil +cdef void chbevx(char *jobz, char *range, char *uplo, int *n, int *kd, c *ab, int *ldab, c *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_chbevx(jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chbgst "BLAS_FUNC(chbgst)"(char *vect, char *uplo, int *n, int *ka, int *kb, npy_complex64 *ab, int *ldab, npy_complex64 *bb, int *ldbb, npy_complex64 *x, int *ldx, npy_complex64 *work, s *rwork, int *info) nogil +cdef void chbgst(char *vect, char *uplo, int *n, int *ka, int *kb, c *ab, int *ldab, c *bb, int *ldbb, c *x, int *ldx, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_chbgst(vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chbgv "BLAS_FUNC(chbgv)"(char *jobz, char *uplo, int *n, int *ka, int *kb, npy_complex64 *ab, int *ldab, npy_complex64 *bb, int *ldbb, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, s *rwork, int *info) nogil +cdef void chbgv(char *jobz, char *uplo, int *n, int *ka, int *kb, c *ab, int *ldab, c *bb, int *ldbb, s *w, c *z, int *ldz, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_chbgv(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chbgvd "BLAS_FUNC(chbgvd)"(char *jobz, char *uplo, int *n, int *ka, int *kb, npy_complex64 *ab, int *ldab, npy_complex64 *bb, int *ldbb, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void chbgvd(char *jobz, char *uplo, int *n, int *ka, int *kb, c *ab, int *ldab, c *bb, int *ldbb, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_chbgvd(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chbgvx "BLAS_FUNC(chbgvx)"(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, npy_complex64 *ab, int *ldab, npy_complex64 *bb, int *ldbb, npy_complex64 *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, s *rwork, int *iwork, int *ifail, int *info) nogil +cdef void chbgvx(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, c *ab, int *ldab, c *bb, int *ldbb, c *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_chbgvx(jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chbtrd "BLAS_FUNC(chbtrd)"(char *vect, char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, s *d, s *e, npy_complex64 *q, int *ldq, npy_complex64 *work, int *info) nogil +cdef void chbtrd(char *vect, char *uplo, int *n, int *kd, c *ab, int *ldab, s *d, s *e, c *q, int *ldq, c *work, int *info) noexcept nogil: + + _fortran_chbtrd(vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_checon "BLAS_FUNC(checon)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, s *anorm, s *rcond, npy_complex64 *work, int *info) nogil +cdef void checon(char *uplo, int *n, c *a, int *lda, int *ipiv, s *anorm, s *rcond, c *work, int *info) noexcept nogil: + + _fortran_checon(uplo, n, a, lda, ipiv, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cheequb "BLAS_FUNC(cheequb)"(char *uplo, int *n, npy_complex64 *a, int *lda, s *s, s *scond, s *amax, npy_complex64 *work, int *info) nogil +cdef void cheequb(char *uplo, int *n, c *a, int *lda, s *s, s *scond, s *amax, c *work, int *info) noexcept nogil: + + _fortran_cheequb(uplo, n, a, lda, s, scond, amax, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cheev "BLAS_FUNC(cheev)"(char *jobz, char *uplo, int *n, npy_complex64 *a, int *lda, s *w, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void cheev(char *jobz, char *uplo, int *n, c *a, int *lda, s *w, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_cheev(jobz, uplo, n, a, lda, w, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cheevd "BLAS_FUNC(cheevd)"(char *jobz, char *uplo, int *n, npy_complex64 *a, int *lda, s *w, npy_complex64 *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void cheevd(char *jobz, char *uplo, int *n, c *a, int *lda, s *w, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_cheevd(jobz, uplo, n, a, lda, w, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cheevr "BLAS_FUNC(cheevr)"(char *jobz, char *range, char *uplo, int *n, npy_complex64 *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, npy_complex64 *z, int *ldz, int *isuppz, npy_complex64 *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void cheevr(char *jobz, char *range, char *uplo, int *n, c *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, int *isuppz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_cheevr(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cheevx "BLAS_FUNC(cheevx)"(char *jobz, char *range, char *uplo, int *n, npy_complex64 *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, s *rwork, int *iwork, int *ifail, int *info) nogil +cdef void cheevx(char *jobz, char *range, char *uplo, int *n, c *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_cheevx(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chegs2 "BLAS_FUNC(chegs2)"(int *itype, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *info) nogil +cdef void chegs2(int *itype, char *uplo, int *n, c *a, int *lda, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_chegs2(itype, uplo, n, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chegst "BLAS_FUNC(chegst)"(int *itype, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *info) nogil +cdef void chegst(int *itype, char *uplo, int *n, c *a, int *lda, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_chegst(itype, uplo, n, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chegv "BLAS_FUNC(chegv)"(int *itype, char *jobz, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, s *w, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void chegv(int *itype, char *jobz, char *uplo, int *n, c *a, int *lda, c *b, int *ldb, s *w, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_chegv(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chegvd "BLAS_FUNC(chegvd)"(int *itype, char *jobz, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, s *w, npy_complex64 *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void chegvd(int *itype, char *jobz, char *uplo, int *n, c *a, int *lda, c *b, int *ldb, s *w, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_chegvd(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chegvx "BLAS_FUNC(chegvx)"(int *itype, char *jobz, char *range, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, s *rwork, int *iwork, int *ifail, int *info) nogil +cdef void chegvx(int *itype, char *jobz, char *range, char *uplo, int *n, c *a, int *lda, c *b, int *ldb, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_chegvx(itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cherfs "BLAS_FUNC(cherfs)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cherfs(char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cherfs(uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chesv "BLAS_FUNC(chesv)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *work, int *lwork, int *info) nogil +cdef void chesv(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_chesv(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chesvx "BLAS_FUNC(chesvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void chesvx(char *fact, char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_chesvx(fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cheswapr "BLAS_FUNC(cheswapr)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *i1, int *i2) nogil +cdef void cheswapr(char *uplo, int *n, c *a, int *lda, int *i1, int *i2) noexcept nogil: + + _fortran_cheswapr(uplo, n, a, lda, i1, i2) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chetd2 "BLAS_FUNC(chetd2)"(char *uplo, int *n, npy_complex64 *a, int *lda, s *d, s *e, npy_complex64 *tau, int *info) nogil +cdef void chetd2(char *uplo, int *n, c *a, int *lda, s *d, s *e, c *tau, int *info) noexcept nogil: + + _fortran_chetd2(uplo, n, a, lda, d, e, tau, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chetf2 "BLAS_FUNC(chetf2)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, int *info) nogil +cdef void chetf2(char *uplo, int *n, c *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_chetf2(uplo, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chetrd "BLAS_FUNC(chetrd)"(char *uplo, int *n, npy_complex64 *a, int *lda, s *d, s *e, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void chetrd(char *uplo, int *n, c *a, int *lda, s *d, s *e, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_chetrd(uplo, n, a, lda, d, e, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chetrf "BLAS_FUNC(chetrf)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *lwork, int *info) nogil +cdef void chetrf(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_chetrf(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chetri "BLAS_FUNC(chetri)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *info) nogil +cdef void chetri(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *info) noexcept nogil: + + _fortran_chetri(uplo, n, a, lda, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chetri2 "BLAS_FUNC(chetri2)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *lwork, int *info) nogil +cdef void chetri2(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_chetri2(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chetri2x "BLAS_FUNC(chetri2x)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *nb, int *info) nogil +cdef void chetri2x(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *nb, int *info) noexcept nogil: + + _fortran_chetri2x(uplo, n, a, lda, ipiv, work, nb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chetrs "BLAS_FUNC(chetrs)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void chetrs(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_chetrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chetrs2 "BLAS_FUNC(chetrs2)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *work, int *info) nogil +cdef void chetrs2(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, c *work, int *info) noexcept nogil: + + _fortran_chetrs2(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chfrk "BLAS_FUNC(chfrk)"(char *transr, char *uplo, char *trans, int *n, int *k, s *alpha, npy_complex64 *a, int *lda, s *beta, npy_complex64 *c) nogil +cdef void chfrk(char *transr, char *uplo, char *trans, int *n, int *k, s *alpha, c *a, int *lda, s *beta, c *c) noexcept nogil: + + _fortran_chfrk(transr, uplo, trans, n, k, alpha, a, lda, beta, c) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chgeqz "BLAS_FUNC(chgeqz)"(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, npy_complex64 *h, int *ldh, npy_complex64 *t, int *ldt, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *q, int *ldq, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void chgeqz(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, c *h, int *ldh, c *t, int *ldt, c *alpha, c *beta, c *q, int *ldq, c *z, int *ldz, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_chgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + char _fortran_chla_transtype "BLAS_FUNC(chla_transtype)"(int *trans) nogil +cdef char chla_transtype(int *trans) noexcept nogil: + + return _fortran_chla_transtype(trans) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chpcon "BLAS_FUNC(chpcon)"(char *uplo, int *n, npy_complex64 *ap, int *ipiv, s *anorm, s *rcond, npy_complex64 *work, int *info) nogil +cdef void chpcon(char *uplo, int *n, c *ap, int *ipiv, s *anorm, s *rcond, c *work, int *info) noexcept nogil: + + _fortran_chpcon(uplo, n, ap, ipiv, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chpev "BLAS_FUNC(chpev)"(char *jobz, char *uplo, int *n, npy_complex64 *ap, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, s *rwork, int *info) nogil +cdef void chpev(char *jobz, char *uplo, int *n, c *ap, s *w, c *z, int *ldz, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_chpev(jobz, uplo, n, ap, w, z, ldz, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chpevd "BLAS_FUNC(chpevd)"(char *jobz, char *uplo, int *n, npy_complex64 *ap, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void chpevd(char *jobz, char *uplo, int *n, c *ap, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_chpevd(jobz, uplo, n, ap, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chpevx "BLAS_FUNC(chpevx)"(char *jobz, char *range, char *uplo, int *n, npy_complex64 *ap, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, s *rwork, int *iwork, int *ifail, int *info) nogil +cdef void chpevx(char *jobz, char *range, char *uplo, int *n, c *ap, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_chpevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chpgst "BLAS_FUNC(chpgst)"(int *itype, char *uplo, int *n, npy_complex64 *ap, npy_complex64 *bp, int *info) nogil +cdef void chpgst(int *itype, char *uplo, int *n, c *ap, c *bp, int *info) noexcept nogil: + + _fortran_chpgst(itype, uplo, n, ap, bp, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chpgv "BLAS_FUNC(chpgv)"(int *itype, char *jobz, char *uplo, int *n, npy_complex64 *ap, npy_complex64 *bp, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, s *rwork, int *info) nogil +cdef void chpgv(int *itype, char *jobz, char *uplo, int *n, c *ap, c *bp, s *w, c *z, int *ldz, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_chpgv(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chpgvd "BLAS_FUNC(chpgvd)"(int *itype, char *jobz, char *uplo, int *n, npy_complex64 *ap, npy_complex64 *bp, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void chpgvd(int *itype, char *jobz, char *uplo, int *n, c *ap, c *bp, s *w, c *z, int *ldz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_chpgvd(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chpgvx "BLAS_FUNC(chpgvx)"(int *itype, char *jobz, char *range, char *uplo, int *n, npy_complex64 *ap, npy_complex64 *bp, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, npy_complex64 *z, int *ldz, npy_complex64 *work, s *rwork, int *iwork, int *ifail, int *info) nogil +cdef void chpgvx(int *itype, char *jobz, char *range, char *uplo, int *n, c *ap, c *bp, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, c *work, s *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_chpgvx(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chprfs "BLAS_FUNC(chprfs)"(char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void chprfs(char *uplo, int *n, int *nrhs, c *ap, c *afp, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_chprfs(uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chpsv "BLAS_FUNC(chpsv)"(char *uplo, int *n, int *nrhs, npy_complex64 *ap, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void chpsv(char *uplo, int *n, int *nrhs, c *ap, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_chpsv(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chpsvx "BLAS_FUNC(chpsvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void chpsvx(char *fact, char *uplo, int *n, int *nrhs, c *ap, c *afp, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_chpsvx(fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chptrd "BLAS_FUNC(chptrd)"(char *uplo, int *n, npy_complex64 *ap, s *d, s *e, npy_complex64 *tau, int *info) nogil +cdef void chptrd(char *uplo, int *n, c *ap, s *d, s *e, c *tau, int *info) noexcept nogil: + + _fortran_chptrd(uplo, n, ap, d, e, tau, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chptrf "BLAS_FUNC(chptrf)"(char *uplo, int *n, npy_complex64 *ap, int *ipiv, int *info) nogil +cdef void chptrf(char *uplo, int *n, c *ap, int *ipiv, int *info) noexcept nogil: + + _fortran_chptrf(uplo, n, ap, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chptri "BLAS_FUNC(chptri)"(char *uplo, int *n, npy_complex64 *ap, int *ipiv, npy_complex64 *work, int *info) nogil +cdef void chptri(char *uplo, int *n, c *ap, int *ipiv, c *work, int *info) noexcept nogil: + + _fortran_chptri(uplo, n, ap, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chptrs "BLAS_FUNC(chptrs)"(char *uplo, int *n, int *nrhs, npy_complex64 *ap, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void chptrs(char *uplo, int *n, int *nrhs, c *ap, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_chptrs(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chsein "BLAS_FUNC(chsein)"(char *side, char *eigsrc, char *initv, bint *select, int *n, npy_complex64 *h, int *ldh, npy_complex64 *w, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, int *mm, int *m, npy_complex64 *work, s *rwork, int *ifaill, int *ifailr, int *info) nogil +cdef void chsein(char *side, char *eigsrc, char *initv, bint *select, int *n, c *h, int *ldh, c *w, c *vl, int *ldvl, c *vr, int *ldvr, int *mm, int *m, c *work, s *rwork, int *ifaill, int *ifailr, int *info) noexcept nogil: + + _fortran_chsein(side, eigsrc, initv, select, n, h, ldh, w, vl, ldvl, vr, ldvr, mm, m, work, rwork, ifaill, ifailr, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_chseqr "BLAS_FUNC(chseqr)"(char *job, char *compz, int *n, int *ilo, int *ihi, npy_complex64 *h, int *ldh, npy_complex64 *w, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, int *info) nogil +cdef void chseqr(char *job, char *compz, int *n, int *ilo, int *ihi, c *h, int *ldh, c *w, c *z, int *ldz, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_chseqr(job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clabrd "BLAS_FUNC(clabrd)"(int *m, int *n, int *nb, npy_complex64 *a, int *lda, s *d, s *e, npy_complex64 *tauq, npy_complex64 *taup, npy_complex64 *x, int *ldx, npy_complex64 *y, int *ldy) nogil +cdef void clabrd(int *m, int *n, int *nb, c *a, int *lda, s *d, s *e, c *tauq, c *taup, c *x, int *ldx, c *y, int *ldy) noexcept nogil: + + _fortran_clabrd(m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clacgv "BLAS_FUNC(clacgv)"(int *n, npy_complex64 *x, int *incx) nogil +cdef void clacgv(int *n, c *x, int *incx) noexcept nogil: + + _fortran_clacgv(n, x, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clacn2 "BLAS_FUNC(clacn2)"(int *n, npy_complex64 *v, npy_complex64 *x, s *est, int *kase, int *isave) nogil +cdef void clacn2(int *n, c *v, c *x, s *est, int *kase, int *isave) noexcept nogil: + + _fortran_clacn2(n, v, x, est, kase, isave) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clacon "BLAS_FUNC(clacon)"(int *n, npy_complex64 *v, npy_complex64 *x, s *est, int *kase) nogil +cdef void clacon(int *n, c *v, c *x, s *est, int *kase) noexcept nogil: + + _fortran_clacon(n, v, x, est, kase) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clacp2 "BLAS_FUNC(clacp2)"(char *uplo, int *m, int *n, s *a, int *lda, npy_complex64 *b, int *ldb) nogil +cdef void clacp2(char *uplo, int *m, int *n, s *a, int *lda, c *b, int *ldb) noexcept nogil: + + _fortran_clacp2(uplo, m, n, a, lda, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clacpy "BLAS_FUNC(clacpy)"(char *uplo, int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb) nogil +cdef void clacpy(char *uplo, int *m, int *n, c *a, int *lda, c *b, int *ldb) noexcept nogil: + + _fortran_clacpy(uplo, m, n, a, lda, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clacrm "BLAS_FUNC(clacrm)"(int *m, int *n, npy_complex64 *a, int *lda, s *b, int *ldb, npy_complex64 *c, int *ldc, s *rwork) nogil +cdef void clacrm(int *m, int *n, c *a, int *lda, s *b, int *ldb, c *c, int *ldc, s *rwork) noexcept nogil: + + _fortran_clacrm(m, n, a, lda, b, ldb, c, ldc, rwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clacrt "BLAS_FUNC(clacrt)"(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy, npy_complex64 *c, npy_complex64 *s) nogil +cdef void clacrt(int *n, c *cx, int *incx, c *cy, int *incy, c *c, c *s) noexcept nogil: + + _fortran_clacrt(n, cx, incx, cy, incy, c, s) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cladiv "F_FUNC(cladivwrp,CLADIVWRP)"(npy_complex64 *out, npy_complex64 *x, npy_complex64 *y) nogil +cdef c cladiv(c *x, c *y) noexcept nogil: + cdef c out + _fortran_cladiv(&out, x, y) + return out + +cdef extern from "_lapack_subroutines.h": + void _fortran_claed0 "BLAS_FUNC(claed0)"(int *qsiz, int *n, s *d, s *e, npy_complex64 *q, int *ldq, npy_complex64 *qstore, int *ldqs, s *rwork, int *iwork, int *info) nogil +cdef void claed0(int *qsiz, int *n, s *d, s *e, c *q, int *ldq, c *qstore, int *ldqs, s *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_claed0(qsiz, n, d, e, q, ldq, qstore, ldqs, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claed7 "BLAS_FUNC(claed7)"(int *n, int *cutpnt, int *qsiz, int *tlvls, int *curlvl, int *curpbm, s *d, npy_complex64 *q, int *ldq, s *rho, int *indxq, s *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, s *givnum, npy_complex64 *work, s *rwork, int *iwork, int *info) nogil +cdef void claed7(int *n, int *cutpnt, int *qsiz, int *tlvls, int *curlvl, int *curpbm, s *d, c *q, int *ldq, s *rho, int *indxq, s *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, s *givnum, c *work, s *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_claed7(n, cutpnt, qsiz, tlvls, curlvl, curpbm, d, q, ldq, rho, indxq, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claed8 "BLAS_FUNC(claed8)"(int *k, int *n, int *qsiz, npy_complex64 *q, int *ldq, s *d, s *rho, int *cutpnt, s *z, s *dlamda, npy_complex64 *q2, int *ldq2, s *w, int *indxp, int *indx, int *indxq, int *perm, int *givptr, int *givcol, s *givnum, int *info) nogil +cdef void claed8(int *k, int *n, int *qsiz, c *q, int *ldq, s *d, s *rho, int *cutpnt, s *z, s *dlamda, c *q2, int *ldq2, s *w, int *indxp, int *indx, int *indxq, int *perm, int *givptr, int *givcol, s *givnum, int *info) noexcept nogil: + + _fortran_claed8(k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlamda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claein "BLAS_FUNC(claein)"(bint *rightv, bint *noinit, int *n, npy_complex64 *h, int *ldh, npy_complex64 *w, npy_complex64 *v, npy_complex64 *b, int *ldb, s *rwork, s *eps3, s *smlnum, int *info) nogil +cdef void claein(bint *rightv, bint *noinit, int *n, c *h, int *ldh, c *w, c *v, c *b, int *ldb, s *rwork, s *eps3, s *smlnum, int *info) noexcept nogil: + + _fortran_claein(rightv, noinit, n, h, ldh, w, v, b, ldb, rwork, eps3, smlnum, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claesy "BLAS_FUNC(claesy)"(npy_complex64 *a, npy_complex64 *b, npy_complex64 *c, npy_complex64 *rt1, npy_complex64 *rt2, npy_complex64 *evscal, npy_complex64 *cs1, npy_complex64 *sn1) nogil +cdef void claesy(c *a, c *b, c *c, c *rt1, c *rt2, c *evscal, c *cs1, c *sn1) noexcept nogil: + + _fortran_claesy(a, b, c, rt1, rt2, evscal, cs1, sn1) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claev2 "BLAS_FUNC(claev2)"(npy_complex64 *a, npy_complex64 *b, npy_complex64 *c, s *rt1, s *rt2, s *cs1, npy_complex64 *sn1) nogil +cdef void claev2(c *a, c *b, c *c, s *rt1, s *rt2, s *cs1, c *sn1) noexcept nogil: + + _fortran_claev2(a, b, c, rt1, rt2, cs1, sn1) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clag2z "BLAS_FUNC(clag2z)"(int *m, int *n, npy_complex64 *sa, int *ldsa, npy_complex128 *a, int *lda, int *info) nogil +cdef void clag2z(int *m, int *n, c *sa, int *ldsa, z *a, int *lda, int *info) noexcept nogil: + + _fortran_clag2z(m, n, sa, ldsa, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clags2 "BLAS_FUNC(clags2)"(bint *upper, s *a1, npy_complex64 *a2, s *a3, s *b1, npy_complex64 *b2, s *b3, s *csu, npy_complex64 *snu, s *csv, npy_complex64 *snv, s *csq, npy_complex64 *snq) nogil +cdef void clags2(bint *upper, s *a1, c *a2, s *a3, s *b1, c *b2, s *b3, s *csu, c *snu, s *csv, c *snv, s *csq, c *snq) noexcept nogil: + + _fortran_clags2(upper, a1, a2, a3, b1, b2, b3, csu, snu, csv, snv, csq, snq) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clagtm "BLAS_FUNC(clagtm)"(char *trans, int *n, int *nrhs, s *alpha, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du, npy_complex64 *x, int *ldx, s *beta, npy_complex64 *b, int *ldb) nogil +cdef void clagtm(char *trans, int *n, int *nrhs, s *alpha, c *dl, c *d, c *du, c *x, int *ldx, s *beta, c *b, int *ldb) noexcept nogil: + + _fortran_clagtm(trans, n, nrhs, alpha, dl, d, du, x, ldx, beta, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clahef "BLAS_FUNC(clahef)"(char *uplo, int *n, int *nb, int *kb, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *w, int *ldw, int *info) nogil +cdef void clahef(char *uplo, int *n, int *nb, int *kb, c *a, int *lda, int *ipiv, c *w, int *ldw, int *info) noexcept nogil: + + _fortran_clahef(uplo, n, nb, kb, a, lda, ipiv, w, ldw, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clahqr "BLAS_FUNC(clahqr)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, npy_complex64 *h, int *ldh, npy_complex64 *w, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, int *info) nogil +cdef void clahqr(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, c *h, int *ldh, c *w, int *iloz, int *ihiz, c *z, int *ldz, int *info) noexcept nogil: + + _fortran_clahqr(wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clahr2 "BLAS_FUNC(clahr2)"(int *n, int *k, int *nb, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *t, int *ldt, npy_complex64 *y, int *ldy) nogil +cdef void clahr2(int *n, int *k, int *nb, c *a, int *lda, c *tau, c *t, int *ldt, c *y, int *ldy) noexcept nogil: + + _fortran_clahr2(n, k, nb, a, lda, tau, t, ldt, y, ldy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claic1 "BLAS_FUNC(claic1)"(int *job, int *j, npy_complex64 *x, s *sest, npy_complex64 *w, npy_complex64 *gamma, s *sestpr, npy_complex64 *s, npy_complex64 *c) nogil +cdef void claic1(int *job, int *j, c *x, s *sest, c *w, c *gamma, s *sestpr, c *s, c *c) noexcept nogil: + + _fortran_claic1(job, j, x, sest, w, gamma, sestpr, s, c) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clals0 "BLAS_FUNC(clals0)"(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, npy_complex64 *b, int *ldb, npy_complex64 *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *poles, s *difl, s *difr, s *z, int *k, s *c, s *s, s *rwork, int *info) nogil +cdef void clals0(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, c *b, int *ldb, c *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *poles, s *difl, s *difr, s *z, int *k, s *c, s *s, s *rwork, int *info) noexcept nogil: + + _fortran_clals0(icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clalsa "BLAS_FUNC(clalsa)"(int *icompq, int *smlsiz, int *n, int *nrhs, npy_complex64 *b, int *ldb, npy_complex64 *bx, int *ldbx, s *u, int *ldu, s *vt, int *k, s *difl, s *difr, s *z, s *poles, int *givptr, int *givcol, int *ldgcol, int *perm, s *givnum, s *c, s *s, s *rwork, int *iwork, int *info) nogil +cdef void clalsa(int *icompq, int *smlsiz, int *n, int *nrhs, c *b, int *ldb, c *bx, int *ldbx, s *u, int *ldu, s *vt, int *k, s *difl, s *difr, s *z, s *poles, int *givptr, int *givcol, int *ldgcol, int *perm, s *givnum, s *c, s *s, s *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_clalsa(icompq, smlsiz, n, nrhs, b, ldb, bx, ldbx, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clalsd "BLAS_FUNC(clalsd)"(char *uplo, int *smlsiz, int *n, int *nrhs, s *d, s *e, npy_complex64 *b, int *ldb, s *rcond, int *rank, npy_complex64 *work, s *rwork, int *iwork, int *info) nogil +cdef void clalsd(char *uplo, int *smlsiz, int *n, int *nrhs, s *d, s *e, c *b, int *ldb, s *rcond, int *rank, c *work, s *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_clalsd(uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clangb "BLAS_FUNC(clangb)"(char *norm, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, s *work) nogil +cdef s clangb(char *norm, int *n, int *kl, int *ku, c *ab, int *ldab, s *work) noexcept nogil: + + return _fortran_clangb(norm, n, kl, ku, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clange "BLAS_FUNC(clange)"(char *norm, int *m, int *n, npy_complex64 *a, int *lda, s *work) nogil +cdef s clange(char *norm, int *m, int *n, c *a, int *lda, s *work) noexcept nogil: + + return _fortran_clange(norm, m, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clangt "BLAS_FUNC(clangt)"(char *norm, int *n, npy_complex64 *dl, npy_complex64 *d, npy_complex64 *du) nogil +cdef s clangt(char *norm, int *n, c *dl, c *d, c *du) noexcept nogil: + + return _fortran_clangt(norm, n, dl, d, du) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clanhb "BLAS_FUNC(clanhb)"(char *norm, char *uplo, int *n, int *k, npy_complex64 *ab, int *ldab, s *work) nogil +cdef s clanhb(char *norm, char *uplo, int *n, int *k, c *ab, int *ldab, s *work) noexcept nogil: + + return _fortran_clanhb(norm, uplo, n, k, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clanhe "BLAS_FUNC(clanhe)"(char *norm, char *uplo, int *n, npy_complex64 *a, int *lda, s *work) nogil +cdef s clanhe(char *norm, char *uplo, int *n, c *a, int *lda, s *work) noexcept nogil: + + return _fortran_clanhe(norm, uplo, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clanhf "BLAS_FUNC(clanhf)"(char *norm, char *transr, char *uplo, int *n, npy_complex64 *a, s *work) nogil +cdef s clanhf(char *norm, char *transr, char *uplo, int *n, c *a, s *work) noexcept nogil: + + return _fortran_clanhf(norm, transr, uplo, n, a, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clanhp "BLAS_FUNC(clanhp)"(char *norm, char *uplo, int *n, npy_complex64 *ap, s *work) nogil +cdef s clanhp(char *norm, char *uplo, int *n, c *ap, s *work) noexcept nogil: + + return _fortran_clanhp(norm, uplo, n, ap, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clanhs "BLAS_FUNC(clanhs)"(char *norm, int *n, npy_complex64 *a, int *lda, s *work) nogil +cdef s clanhs(char *norm, int *n, c *a, int *lda, s *work) noexcept nogil: + + return _fortran_clanhs(norm, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clanht "BLAS_FUNC(clanht)"(char *norm, int *n, s *d, npy_complex64 *e) nogil +cdef s clanht(char *norm, int *n, s *d, c *e) noexcept nogil: + + return _fortran_clanht(norm, n, d, e) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clansb "BLAS_FUNC(clansb)"(char *norm, char *uplo, int *n, int *k, npy_complex64 *ab, int *ldab, s *work) nogil +cdef s clansb(char *norm, char *uplo, int *n, int *k, c *ab, int *ldab, s *work) noexcept nogil: + + return _fortran_clansb(norm, uplo, n, k, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clansp "BLAS_FUNC(clansp)"(char *norm, char *uplo, int *n, npy_complex64 *ap, s *work) nogil +cdef s clansp(char *norm, char *uplo, int *n, c *ap, s *work) noexcept nogil: + + return _fortran_clansp(norm, uplo, n, ap, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clansy "BLAS_FUNC(clansy)"(char *norm, char *uplo, int *n, npy_complex64 *a, int *lda, s *work) nogil +cdef s clansy(char *norm, char *uplo, int *n, c *a, int *lda, s *work) noexcept nogil: + + return _fortran_clansy(norm, uplo, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clantb "BLAS_FUNC(clantb)"(char *norm, char *uplo, char *diag, int *n, int *k, npy_complex64 *ab, int *ldab, s *work) nogil +cdef s clantb(char *norm, char *uplo, char *diag, int *n, int *k, c *ab, int *ldab, s *work) noexcept nogil: + + return _fortran_clantb(norm, uplo, diag, n, k, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clantp "BLAS_FUNC(clantp)"(char *norm, char *uplo, char *diag, int *n, npy_complex64 *ap, s *work) nogil +cdef s clantp(char *norm, char *uplo, char *diag, int *n, c *ap, s *work) noexcept nogil: + + return _fortran_clantp(norm, uplo, diag, n, ap, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_clantr "BLAS_FUNC(clantr)"(char *norm, char *uplo, char *diag, int *m, int *n, npy_complex64 *a, int *lda, s *work) nogil +cdef s clantr(char *norm, char *uplo, char *diag, int *m, int *n, c *a, int *lda, s *work) noexcept nogil: + + return _fortran_clantr(norm, uplo, diag, m, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clapll "BLAS_FUNC(clapll)"(int *n, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, s *ssmin) nogil +cdef void clapll(int *n, c *x, int *incx, c *y, int *incy, s *ssmin) noexcept nogil: + + _fortran_clapll(n, x, incx, y, incy, ssmin) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clapmr "BLAS_FUNC(clapmr)"(bint *forwrd, int *m, int *n, npy_complex64 *x, int *ldx, int *k) nogil +cdef void clapmr(bint *forwrd, int *m, int *n, c *x, int *ldx, int *k) noexcept nogil: + + _fortran_clapmr(forwrd, m, n, x, ldx, k) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clapmt "BLAS_FUNC(clapmt)"(bint *forwrd, int *m, int *n, npy_complex64 *x, int *ldx, int *k) nogil +cdef void clapmt(bint *forwrd, int *m, int *n, c *x, int *ldx, int *k) noexcept nogil: + + _fortran_clapmt(forwrd, m, n, x, ldx, k) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqgb "BLAS_FUNC(claqgb)"(int *m, int *n, int *kl, int *ku, npy_complex64 *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) nogil +cdef void claqgb(int *m, int *n, int *kl, int *ku, c *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) noexcept nogil: + + _fortran_claqgb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqge "BLAS_FUNC(claqge)"(int *m, int *n, npy_complex64 *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) nogil +cdef void claqge(int *m, int *n, c *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) noexcept nogil: + + _fortran_claqge(m, n, a, lda, r, c, rowcnd, colcnd, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqhb "BLAS_FUNC(claqhb)"(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, s *s, s *scond, s *amax, char *equed) nogil +cdef void claqhb(char *uplo, int *n, int *kd, c *ab, int *ldab, s *s, s *scond, s *amax, char *equed) noexcept nogil: + + _fortran_claqhb(uplo, n, kd, ab, ldab, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqhe "BLAS_FUNC(claqhe)"(char *uplo, int *n, npy_complex64 *a, int *lda, s *s, s *scond, s *amax, char *equed) nogil +cdef void claqhe(char *uplo, int *n, c *a, int *lda, s *s, s *scond, s *amax, char *equed) noexcept nogil: + + _fortran_claqhe(uplo, n, a, lda, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqhp "BLAS_FUNC(claqhp)"(char *uplo, int *n, npy_complex64 *ap, s *s, s *scond, s *amax, char *equed) nogil +cdef void claqhp(char *uplo, int *n, c *ap, s *s, s *scond, s *amax, char *equed) noexcept nogil: + + _fortran_claqhp(uplo, n, ap, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqp2 "BLAS_FUNC(claqp2)"(int *m, int *n, int *offset, npy_complex64 *a, int *lda, int *jpvt, npy_complex64 *tau, s *vn1, s *vn2, npy_complex64 *work) nogil +cdef void claqp2(int *m, int *n, int *offset, c *a, int *lda, int *jpvt, c *tau, s *vn1, s *vn2, c *work) noexcept nogil: + + _fortran_claqp2(m, n, offset, a, lda, jpvt, tau, vn1, vn2, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqps "BLAS_FUNC(claqps)"(int *m, int *n, int *offset, int *nb, int *kb, npy_complex64 *a, int *lda, int *jpvt, npy_complex64 *tau, s *vn1, s *vn2, npy_complex64 *auxv, npy_complex64 *f, int *ldf) nogil +cdef void claqps(int *m, int *n, int *offset, int *nb, int *kb, c *a, int *lda, int *jpvt, c *tau, s *vn1, s *vn2, c *auxv, c *f, int *ldf) noexcept nogil: + + _fortran_claqps(m, n, offset, nb, kb, a, lda, jpvt, tau, vn1, vn2, auxv, f, ldf) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqr0 "BLAS_FUNC(claqr0)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, npy_complex64 *h, int *ldh, npy_complex64 *w, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, int *info) nogil +cdef void claqr0(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, c *h, int *ldh, c *w, int *iloz, int *ihiz, c *z, int *ldz, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_claqr0(wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqr1 "BLAS_FUNC(claqr1)"(int *n, npy_complex64 *h, int *ldh, npy_complex64 *s1, npy_complex64 *s2, npy_complex64 *v) nogil +cdef void claqr1(int *n, c *h, int *ldh, c *s1, c *s2, c *v) noexcept nogil: + + _fortran_claqr1(n, h, ldh, s1, s2, v) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqr2 "BLAS_FUNC(claqr2)"(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, npy_complex64 *h, int *ldh, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, int *ns, int *nd, npy_complex64 *sh, npy_complex64 *v, int *ldv, int *nh, npy_complex64 *t, int *ldt, int *nv, npy_complex64 *wv, int *ldwv, npy_complex64 *work, int *lwork) nogil +cdef void claqr2(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, c *h, int *ldh, int *iloz, int *ihiz, c *z, int *ldz, int *ns, int *nd, c *sh, c *v, int *ldv, int *nh, c *t, int *ldt, int *nv, c *wv, int *ldwv, c *work, int *lwork) noexcept nogil: + + _fortran_claqr2(wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqr3 "BLAS_FUNC(claqr3)"(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, npy_complex64 *h, int *ldh, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, int *ns, int *nd, npy_complex64 *sh, npy_complex64 *v, int *ldv, int *nh, npy_complex64 *t, int *ldt, int *nv, npy_complex64 *wv, int *ldwv, npy_complex64 *work, int *lwork) nogil +cdef void claqr3(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, c *h, int *ldh, int *iloz, int *ihiz, c *z, int *ldz, int *ns, int *nd, c *sh, c *v, int *ldv, int *nh, c *t, int *ldt, int *nv, c *wv, int *ldwv, c *work, int *lwork) noexcept nogil: + + _fortran_claqr3(wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqr4 "BLAS_FUNC(claqr4)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, npy_complex64 *h, int *ldh, npy_complex64 *w, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, int *info) nogil +cdef void claqr4(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, c *h, int *ldh, c *w, int *iloz, int *ihiz, c *z, int *ldz, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_claqr4(wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqr5 "BLAS_FUNC(claqr5)"(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, npy_complex64 *s, npy_complex64 *h, int *ldh, int *iloz, int *ihiz, npy_complex64 *z, int *ldz, npy_complex64 *v, int *ldv, npy_complex64 *u, int *ldu, int *nv, npy_complex64 *wv, int *ldwv, int *nh, npy_complex64 *wh, int *ldwh) nogil +cdef void claqr5(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, c *s, c *h, int *ldh, int *iloz, int *ihiz, c *z, int *ldz, c *v, int *ldv, c *u, int *ldu, int *nv, c *wv, int *ldwv, int *nh, c *wh, int *ldwh) noexcept nogil: + + _fortran_claqr5(wantt, wantz, kacc22, n, ktop, kbot, nshfts, s, h, ldh, iloz, ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqsb "BLAS_FUNC(claqsb)"(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, s *s, s *scond, s *amax, char *equed) nogil +cdef void claqsb(char *uplo, int *n, int *kd, c *ab, int *ldab, s *s, s *scond, s *amax, char *equed) noexcept nogil: + + _fortran_claqsb(uplo, n, kd, ab, ldab, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqsp "BLAS_FUNC(claqsp)"(char *uplo, int *n, npy_complex64 *ap, s *s, s *scond, s *amax, char *equed) nogil +cdef void claqsp(char *uplo, int *n, c *ap, s *s, s *scond, s *amax, char *equed) noexcept nogil: + + _fortran_claqsp(uplo, n, ap, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claqsy "BLAS_FUNC(claqsy)"(char *uplo, int *n, npy_complex64 *a, int *lda, s *s, s *scond, s *amax, char *equed) nogil +cdef void claqsy(char *uplo, int *n, c *a, int *lda, s *s, s *scond, s *amax, char *equed) noexcept nogil: + + _fortran_claqsy(uplo, n, a, lda, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clar1v "BLAS_FUNC(clar1v)"(int *n, int *b1, int *bn, s *lambda_, s *d, s *l, s *ld, s *lld, s *pivmin, s *gaptol, npy_complex64 *z, bint *wantnc, int *negcnt, s *ztz, s *mingma, int *r, int *isuppz, s *nrminv, s *resid, s *rqcorr, s *work) nogil +cdef void clar1v(int *n, int *b1, int *bn, s *lambda_, s *d, s *l, s *ld, s *lld, s *pivmin, s *gaptol, c *z, bint *wantnc, int *negcnt, s *ztz, s *mingma, int *r, int *isuppz, s *nrminv, s *resid, s *rqcorr, s *work) noexcept nogil: + + _fortran_clar1v(n, b1, bn, lambda_, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clar2v "BLAS_FUNC(clar2v)"(int *n, npy_complex64 *x, npy_complex64 *y, npy_complex64 *z, int *incx, s *c, npy_complex64 *s, int *incc) nogil +cdef void clar2v(int *n, c *x, c *y, c *z, int *incx, s *c, c *s, int *incc) noexcept nogil: + + _fortran_clar2v(n, x, y, z, incx, c, s, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarcm "BLAS_FUNC(clarcm)"(int *m, int *n, s *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *c, int *ldc, s *rwork) nogil +cdef void clarcm(int *m, int *n, s *a, int *lda, c *b, int *ldb, c *c, int *ldc, s *rwork) noexcept nogil: + + _fortran_clarcm(m, n, a, lda, b, ldb, c, ldc, rwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarf "BLAS_FUNC(clarf)"(char *side, int *m, int *n, npy_complex64 *v, int *incv, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work) nogil +cdef void clarf(char *side, int *m, int *n, c *v, int *incv, c *tau, c *c, int *ldc, c *work) noexcept nogil: + + _fortran_clarf(side, m, n, v, incv, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarfb "BLAS_FUNC(clarfb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, npy_complex64 *v, int *ldv, npy_complex64 *t, int *ldt, npy_complex64 *c, int *ldc, npy_complex64 *work, int *ldwork) nogil +cdef void clarfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, c *v, int *ldv, c *t, int *ldt, c *c, int *ldc, c *work, int *ldwork) noexcept nogil: + + _fortran_clarfb(side, trans, direct, storev, m, n, k, v, ldv, t, ldt, c, ldc, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarfg "BLAS_FUNC(clarfg)"(int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *tau) nogil +cdef void clarfg(int *n, c *alpha, c *x, int *incx, c *tau) noexcept nogil: + + _fortran_clarfg(n, alpha, x, incx, tau) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarfgp "BLAS_FUNC(clarfgp)"(int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *tau) nogil +cdef void clarfgp(int *n, c *alpha, c *x, int *incx, c *tau) noexcept nogil: + + _fortran_clarfgp(n, alpha, x, incx, tau) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarft "BLAS_FUNC(clarft)"(char *direct, char *storev, int *n, int *k, npy_complex64 *v, int *ldv, npy_complex64 *tau, npy_complex64 *t, int *ldt) nogil +cdef void clarft(char *direct, char *storev, int *n, int *k, c *v, int *ldv, c *tau, c *t, int *ldt) noexcept nogil: + + _fortran_clarft(direct, storev, n, k, v, ldv, tau, t, ldt) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarfx "BLAS_FUNC(clarfx)"(char *side, int *m, int *n, npy_complex64 *v, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work) nogil +cdef void clarfx(char *side, int *m, int *n, c *v, c *tau, c *c, int *ldc, c *work) noexcept nogil: + + _fortran_clarfx(side, m, n, v, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clargv "BLAS_FUNC(clargv)"(int *n, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, s *c, int *incc) nogil +cdef void clargv(int *n, c *x, int *incx, c *y, int *incy, s *c, int *incc) noexcept nogil: + + _fortran_clargv(n, x, incx, y, incy, c, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarnv "BLAS_FUNC(clarnv)"(int *idist, int *iseed, int *n, npy_complex64 *x) nogil +cdef void clarnv(int *idist, int *iseed, int *n, c *x) noexcept nogil: + + _fortran_clarnv(idist, iseed, n, x) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarrv "BLAS_FUNC(clarrv)"(int *n, s *vl, s *vu, s *d, s *l, s *pivmin, int *isplit, int *m, int *dol, int *dou, s *minrgp, s *rtol1, s *rtol2, s *w, s *werr, s *wgap, int *iblock, int *indexw, s *gers, npy_complex64 *z, int *ldz, int *isuppz, s *work, int *iwork, int *info) nogil +cdef void clarrv(int *n, s *vl, s *vu, s *d, s *l, s *pivmin, int *isplit, int *m, int *dol, int *dou, s *minrgp, s *rtol1, s *rtol2, s *w, s *werr, s *wgap, int *iblock, int *indexw, s *gers, c *z, int *ldz, int *isuppz, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_clarrv(n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clartg "BLAS_FUNC(clartg)"(npy_complex64 *f, npy_complex64 *g, s *cs, npy_complex64 *sn, npy_complex64 *r) nogil +cdef void clartg(c *f, c *g, s *cs, c *sn, c *r) noexcept nogil: + + _fortran_clartg(f, g, cs, sn, r) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clartv "BLAS_FUNC(clartv)"(int *n, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, s *c, npy_complex64 *s, int *incc) nogil +cdef void clartv(int *n, c *x, int *incx, c *y, int *incy, s *c, c *s, int *incc) noexcept nogil: + + _fortran_clartv(n, x, incx, y, incy, c, s, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarz "BLAS_FUNC(clarz)"(char *side, int *m, int *n, int *l, npy_complex64 *v, int *incv, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work) nogil +cdef void clarz(char *side, int *m, int *n, int *l, c *v, int *incv, c *tau, c *c, int *ldc, c *work) noexcept nogil: + + _fortran_clarz(side, m, n, l, v, incv, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarzb "BLAS_FUNC(clarzb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, npy_complex64 *v, int *ldv, npy_complex64 *t, int *ldt, npy_complex64 *c, int *ldc, npy_complex64 *work, int *ldwork) nogil +cdef void clarzb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, c *v, int *ldv, c *t, int *ldt, c *c, int *ldc, c *work, int *ldwork) noexcept nogil: + + _fortran_clarzb(side, trans, direct, storev, m, n, k, l, v, ldv, t, ldt, c, ldc, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clarzt "BLAS_FUNC(clarzt)"(char *direct, char *storev, int *n, int *k, npy_complex64 *v, int *ldv, npy_complex64 *tau, npy_complex64 *t, int *ldt) nogil +cdef void clarzt(char *direct, char *storev, int *n, int *k, c *v, int *ldv, c *tau, c *t, int *ldt) noexcept nogil: + + _fortran_clarzt(direct, storev, n, k, v, ldv, tau, t, ldt) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clascl "BLAS_FUNC(clascl)"(char *type_bn, int *kl, int *ku, s *cfrom, s *cto, int *m, int *n, npy_complex64 *a, int *lda, int *info) nogil +cdef void clascl(char *type_bn, int *kl, int *ku, s *cfrom, s *cto, int *m, int *n, c *a, int *lda, int *info) noexcept nogil: + + _fortran_clascl(type_bn, kl, ku, cfrom, cto, m, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claset "BLAS_FUNC(claset)"(char *uplo, int *m, int *n, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *a, int *lda) nogil +cdef void claset(char *uplo, int *m, int *n, c *alpha, c *beta, c *a, int *lda) noexcept nogil: + + _fortran_claset(uplo, m, n, alpha, beta, a, lda) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clasr "BLAS_FUNC(clasr)"(char *side, char *pivot, char *direct, int *m, int *n, s *c, s *s, npy_complex64 *a, int *lda) nogil +cdef void clasr(char *side, char *pivot, char *direct, int *m, int *n, s *c, s *s, c *a, int *lda) noexcept nogil: + + _fortran_clasr(side, pivot, direct, m, n, c, s, a, lda) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_classq "BLAS_FUNC(classq)"(int *n, npy_complex64 *x, int *incx, s *scale, s *sumsq) nogil +cdef void classq(int *n, c *x, int *incx, s *scale, s *sumsq) noexcept nogil: + + _fortran_classq(n, x, incx, scale, sumsq) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_claswp "BLAS_FUNC(claswp)"(int *n, npy_complex64 *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) nogil +cdef void claswp(int *n, c *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) noexcept nogil: + + _fortran_claswp(n, a, lda, k1, k2, ipiv, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clasyf "BLAS_FUNC(clasyf)"(char *uplo, int *n, int *nb, int *kb, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *w, int *ldw, int *info) nogil +cdef void clasyf(char *uplo, int *n, int *nb, int *kb, c *a, int *lda, int *ipiv, c *w, int *ldw, int *info) noexcept nogil: + + _fortran_clasyf(uplo, n, nb, kb, a, lda, ipiv, w, ldw, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clatbs "BLAS_FUNC(clatbs)"(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, npy_complex64 *ab, int *ldab, npy_complex64 *x, s *scale, s *cnorm, int *info) nogil +cdef void clatbs(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, c *ab, int *ldab, c *x, s *scale, s *cnorm, int *info) noexcept nogil: + + _fortran_clatbs(uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clatdf "BLAS_FUNC(clatdf)"(int *ijob, int *n, npy_complex64 *z, int *ldz, npy_complex64 *rhs, s *rdsum, s *rdscal, int *ipiv, int *jpiv) nogil +cdef void clatdf(int *ijob, int *n, c *z, int *ldz, c *rhs, s *rdsum, s *rdscal, int *ipiv, int *jpiv) noexcept nogil: + + _fortran_clatdf(ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clatps "BLAS_FUNC(clatps)"(char *uplo, char *trans, char *diag, char *normin, int *n, npy_complex64 *ap, npy_complex64 *x, s *scale, s *cnorm, int *info) nogil +cdef void clatps(char *uplo, char *trans, char *diag, char *normin, int *n, c *ap, c *x, s *scale, s *cnorm, int *info) noexcept nogil: + + _fortran_clatps(uplo, trans, diag, normin, n, ap, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clatrd "BLAS_FUNC(clatrd)"(char *uplo, int *n, int *nb, npy_complex64 *a, int *lda, s *e, npy_complex64 *tau, npy_complex64 *w, int *ldw) nogil +cdef void clatrd(char *uplo, int *n, int *nb, c *a, int *lda, s *e, c *tau, c *w, int *ldw) noexcept nogil: + + _fortran_clatrd(uplo, n, nb, a, lda, e, tau, w, ldw) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clatrs "BLAS_FUNC(clatrs)"(char *uplo, char *trans, char *diag, char *normin, int *n, npy_complex64 *a, int *lda, npy_complex64 *x, s *scale, s *cnorm, int *info) nogil +cdef void clatrs(char *uplo, char *trans, char *diag, char *normin, int *n, c *a, int *lda, c *x, s *scale, s *cnorm, int *info) noexcept nogil: + + _fortran_clatrs(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clatrz "BLAS_FUNC(clatrz)"(int *m, int *n, int *l, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work) nogil +cdef void clatrz(int *m, int *n, int *l, c *a, int *lda, c *tau, c *work) noexcept nogil: + + _fortran_clatrz(m, n, l, a, lda, tau, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clauu2 "BLAS_FUNC(clauu2)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *info) nogil +cdef void clauu2(char *uplo, int *n, c *a, int *lda, int *info) noexcept nogil: + + _fortran_clauu2(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_clauum "BLAS_FUNC(clauum)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *info) nogil +cdef void clauum(char *uplo, int *n, c *a, int *lda, int *info) noexcept nogil: + + _fortran_clauum(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpbcon "BLAS_FUNC(cpbcon)"(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, s *anorm, s *rcond, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cpbcon(char *uplo, int *n, int *kd, c *ab, int *ldab, s *anorm, s *rcond, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cpbcon(uplo, n, kd, ab, ldab, anorm, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpbequ "BLAS_FUNC(cpbequ)"(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, s *s, s *scond, s *amax, int *info) nogil +cdef void cpbequ(char *uplo, int *n, int *kd, c *ab, int *ldab, s *s, s *scond, s *amax, int *info) noexcept nogil: + + _fortran_cpbequ(uplo, n, kd, ab, ldab, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpbrfs "BLAS_FUNC(cpbrfs)"(char *uplo, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *afb, int *ldafb, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cpbrfs(char *uplo, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *afb, int *ldafb, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cpbrfs(uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpbstf "BLAS_FUNC(cpbstf)"(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, int *info) nogil +cdef void cpbstf(char *uplo, int *n, int *kd, c *ab, int *ldab, int *info) noexcept nogil: + + _fortran_cpbstf(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpbsv "BLAS_FUNC(cpbsv)"(char *uplo, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cpbsv(char *uplo, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cpbsv(uplo, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpbsvx "BLAS_FUNC(cpbsvx)"(char *fact, char *uplo, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *afb, int *ldafb, char *equed, s *s, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cpbsvx(char *fact, char *uplo, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *afb, int *ldafb, char *equed, s *s, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cpbsvx(fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpbtf2 "BLAS_FUNC(cpbtf2)"(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, int *info) nogil +cdef void cpbtf2(char *uplo, int *n, int *kd, c *ab, int *ldab, int *info) noexcept nogil: + + _fortran_cpbtf2(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpbtrf "BLAS_FUNC(cpbtrf)"(char *uplo, int *n, int *kd, npy_complex64 *ab, int *ldab, int *info) nogil +cdef void cpbtrf(char *uplo, int *n, int *kd, c *ab, int *ldab, int *info) noexcept nogil: + + _fortran_cpbtrf(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpbtrs "BLAS_FUNC(cpbtrs)"(char *uplo, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cpbtrs(char *uplo, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cpbtrs(uplo, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpftrf "BLAS_FUNC(cpftrf)"(char *transr, char *uplo, int *n, npy_complex64 *a, int *info) nogil +cdef void cpftrf(char *transr, char *uplo, int *n, c *a, int *info) noexcept nogil: + + _fortran_cpftrf(transr, uplo, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpftri "BLAS_FUNC(cpftri)"(char *transr, char *uplo, int *n, npy_complex64 *a, int *info) nogil +cdef void cpftri(char *transr, char *uplo, int *n, c *a, int *info) noexcept nogil: + + _fortran_cpftri(transr, uplo, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpftrs "BLAS_FUNC(cpftrs)"(char *transr, char *uplo, int *n, int *nrhs, npy_complex64 *a, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cpftrs(char *transr, char *uplo, int *n, int *nrhs, c *a, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cpftrs(transr, uplo, n, nrhs, a, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpocon "BLAS_FUNC(cpocon)"(char *uplo, int *n, npy_complex64 *a, int *lda, s *anorm, s *rcond, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cpocon(char *uplo, int *n, c *a, int *lda, s *anorm, s *rcond, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cpocon(uplo, n, a, lda, anorm, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpoequ "BLAS_FUNC(cpoequ)"(int *n, npy_complex64 *a, int *lda, s *s, s *scond, s *amax, int *info) nogil +cdef void cpoequ(int *n, c *a, int *lda, s *s, s *scond, s *amax, int *info) noexcept nogil: + + _fortran_cpoequ(n, a, lda, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpoequb "BLAS_FUNC(cpoequb)"(int *n, npy_complex64 *a, int *lda, s *s, s *scond, s *amax, int *info) nogil +cdef void cpoequb(int *n, c *a, int *lda, s *s, s *scond, s *amax, int *info) noexcept nogil: + + _fortran_cpoequb(n, a, lda, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cporfs "BLAS_FUNC(cporfs)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cporfs(char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cporfs(uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cposv "BLAS_FUNC(cposv)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cposv(char *uplo, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cposv(uplo, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cposvx "BLAS_FUNC(cposvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, char *equed, s *s, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cposvx(char *fact, char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, char *equed, s *s, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cposvx(fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpotf2 "BLAS_FUNC(cpotf2)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *info) nogil +cdef void cpotf2(char *uplo, int *n, c *a, int *lda, int *info) noexcept nogil: + + _fortran_cpotf2(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpotrf "BLAS_FUNC(cpotrf)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *info) nogil +cdef void cpotrf(char *uplo, int *n, c *a, int *lda, int *info) noexcept nogil: + + _fortran_cpotrf(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpotri "BLAS_FUNC(cpotri)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *info) nogil +cdef void cpotri(char *uplo, int *n, c *a, int *lda, int *info) noexcept nogil: + + _fortran_cpotri(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpotrs "BLAS_FUNC(cpotrs)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cpotrs(char *uplo, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cpotrs(uplo, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cppcon "BLAS_FUNC(cppcon)"(char *uplo, int *n, npy_complex64 *ap, s *anorm, s *rcond, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cppcon(char *uplo, int *n, c *ap, s *anorm, s *rcond, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cppcon(uplo, n, ap, anorm, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cppequ "BLAS_FUNC(cppequ)"(char *uplo, int *n, npy_complex64 *ap, s *s, s *scond, s *amax, int *info) nogil +cdef void cppequ(char *uplo, int *n, c *ap, s *s, s *scond, s *amax, int *info) noexcept nogil: + + _fortran_cppequ(uplo, n, ap, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpprfs "BLAS_FUNC(cpprfs)"(char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cpprfs(char *uplo, int *n, int *nrhs, c *ap, c *afp, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cpprfs(uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cppsv "BLAS_FUNC(cppsv)"(char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cppsv(char *uplo, int *n, int *nrhs, c *ap, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cppsv(uplo, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cppsvx "BLAS_FUNC(cppsvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, char *equed, s *s, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cppsvx(char *fact, char *uplo, int *n, int *nrhs, c *ap, c *afp, char *equed, s *s, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cppsvx(fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpptrf "BLAS_FUNC(cpptrf)"(char *uplo, int *n, npy_complex64 *ap, int *info) nogil +cdef void cpptrf(char *uplo, int *n, c *ap, int *info) noexcept nogil: + + _fortran_cpptrf(uplo, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpptri "BLAS_FUNC(cpptri)"(char *uplo, int *n, npy_complex64 *ap, int *info) nogil +cdef void cpptri(char *uplo, int *n, c *ap, int *info) noexcept nogil: + + _fortran_cpptri(uplo, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpptrs "BLAS_FUNC(cpptrs)"(char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cpptrs(char *uplo, int *n, int *nrhs, c *ap, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cpptrs(uplo, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpstf2 "BLAS_FUNC(cpstf2)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) nogil +cdef void cpstf2(char *uplo, int *n, c *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) noexcept nogil: + + _fortran_cpstf2(uplo, n, a, lda, piv, rank, tol, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpstrf "BLAS_FUNC(cpstrf)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) nogil +cdef void cpstrf(char *uplo, int *n, c *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) noexcept nogil: + + _fortran_cpstrf(uplo, n, a, lda, piv, rank, tol, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cptcon "BLAS_FUNC(cptcon)"(int *n, s *d, npy_complex64 *e, s *anorm, s *rcond, s *rwork, int *info) nogil +cdef void cptcon(int *n, s *d, c *e, s *anorm, s *rcond, s *rwork, int *info) noexcept nogil: + + _fortran_cptcon(n, d, e, anorm, rcond, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpteqr "BLAS_FUNC(cpteqr)"(char *compz, int *n, s *d, s *e, npy_complex64 *z, int *ldz, s *work, int *info) nogil +cdef void cpteqr(char *compz, int *n, s *d, s *e, c *z, int *ldz, s *work, int *info) noexcept nogil: + + _fortran_cpteqr(compz, n, d, e, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cptrfs "BLAS_FUNC(cptrfs)"(char *uplo, int *n, int *nrhs, s *d, npy_complex64 *e, s *df, npy_complex64 *ef, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cptrfs(char *uplo, int *n, int *nrhs, s *d, c *e, s *df, c *ef, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cptrfs(uplo, n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cptsv "BLAS_FUNC(cptsv)"(int *n, int *nrhs, s *d, npy_complex64 *e, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cptsv(int *n, int *nrhs, s *d, c *e, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cptsv(n, nrhs, d, e, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cptsvx "BLAS_FUNC(cptsvx)"(char *fact, int *n, int *nrhs, s *d, npy_complex64 *e, s *df, npy_complex64 *ef, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cptsvx(char *fact, int *n, int *nrhs, s *d, c *e, s *df, c *ef, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cptsvx(fact, n, nrhs, d, e, df, ef, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpttrf "BLAS_FUNC(cpttrf)"(int *n, s *d, npy_complex64 *e, int *info) nogil +cdef void cpttrf(int *n, s *d, c *e, int *info) noexcept nogil: + + _fortran_cpttrf(n, d, e, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cpttrs "BLAS_FUNC(cpttrs)"(char *uplo, int *n, int *nrhs, s *d, npy_complex64 *e, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cpttrs(char *uplo, int *n, int *nrhs, s *d, c *e, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cpttrs(uplo, n, nrhs, d, e, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cptts2 "BLAS_FUNC(cptts2)"(int *iuplo, int *n, int *nrhs, s *d, npy_complex64 *e, npy_complex64 *b, int *ldb) nogil +cdef void cptts2(int *iuplo, int *n, int *nrhs, s *d, c *e, c *b, int *ldb) noexcept nogil: + + _fortran_cptts2(iuplo, n, nrhs, d, e, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_crot "BLAS_FUNC(crot)"(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy, s *c, npy_complex64 *s) nogil +cdef void crot(int *n, c *cx, int *incx, c *cy, int *incy, s *c, c *s) noexcept nogil: + + _fortran_crot(n, cx, incx, cy, incy, c, s) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cspcon "BLAS_FUNC(cspcon)"(char *uplo, int *n, npy_complex64 *ap, int *ipiv, s *anorm, s *rcond, npy_complex64 *work, int *info) nogil +cdef void cspcon(char *uplo, int *n, c *ap, int *ipiv, s *anorm, s *rcond, c *work, int *info) noexcept nogil: + + _fortran_cspcon(uplo, n, ap, ipiv, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cspmv "BLAS_FUNC(cspmv)"(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *ap, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy) nogil +cdef void cspmv(char *uplo, int *n, c *alpha, c *ap, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil: + + _fortran_cspmv(uplo, n, alpha, ap, x, incx, beta, y, incy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cspr "BLAS_FUNC(cspr)"(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *ap) nogil +cdef void cspr(char *uplo, int *n, c *alpha, c *x, int *incx, c *ap) noexcept nogil: + + _fortran_cspr(uplo, n, alpha, x, incx, ap) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csprfs "BLAS_FUNC(csprfs)"(char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void csprfs(char *uplo, int *n, int *nrhs, c *ap, c *afp, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_csprfs(uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cspsv "BLAS_FUNC(cspsv)"(char *uplo, int *n, int *nrhs, npy_complex64 *ap, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void cspsv(char *uplo, int *n, int *nrhs, c *ap, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_cspsv(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cspsvx "BLAS_FUNC(cspsvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *afp, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void cspsvx(char *fact, char *uplo, int *n, int *nrhs, c *ap, c *afp, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_cspsvx(fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csptrf "BLAS_FUNC(csptrf)"(char *uplo, int *n, npy_complex64 *ap, int *ipiv, int *info) nogil +cdef void csptrf(char *uplo, int *n, c *ap, int *ipiv, int *info) noexcept nogil: + + _fortran_csptrf(uplo, n, ap, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csptri "BLAS_FUNC(csptri)"(char *uplo, int *n, npy_complex64 *ap, int *ipiv, npy_complex64 *work, int *info) nogil +cdef void csptri(char *uplo, int *n, c *ap, int *ipiv, c *work, int *info) noexcept nogil: + + _fortran_csptri(uplo, n, ap, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csptrs "BLAS_FUNC(csptrs)"(char *uplo, int *n, int *nrhs, npy_complex64 *ap, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void csptrs(char *uplo, int *n, int *nrhs, c *ap, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_csptrs(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csrscl "BLAS_FUNC(csrscl)"(int *n, s *sa, npy_complex64 *sx, int *incx) nogil +cdef void csrscl(int *n, s *sa, c *sx, int *incx) noexcept nogil: + + _fortran_csrscl(n, sa, sx, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cstedc "BLAS_FUNC(cstedc)"(char *compz, int *n, s *d, s *e, npy_complex64 *z, int *ldz, npy_complex64 *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void cstedc(char *compz, int *n, s *d, s *e, c *z, int *ldz, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_cstedc(compz, n, d, e, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cstegr "BLAS_FUNC(cstegr)"(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, npy_complex64 *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void cstegr(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, c *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_cstegr(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cstein "BLAS_FUNC(cstein)"(int *n, s *d, s *e, int *m, s *w, int *iblock, int *isplit, npy_complex64 *z, int *ldz, s *work, int *iwork, int *ifail, int *info) nogil +cdef void cstein(int *n, s *d, s *e, int *m, s *w, int *iblock, int *isplit, c *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_cstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cstemr "BLAS_FUNC(cstemr)"(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, int *m, s *w, npy_complex64 *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void cstemr(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, int *m, s *w, c *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_cstemr(jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csteqr "BLAS_FUNC(csteqr)"(char *compz, int *n, s *d, s *e, npy_complex64 *z, int *ldz, s *work, int *info) nogil +cdef void csteqr(char *compz, int *n, s *d, s *e, c *z, int *ldz, s *work, int *info) noexcept nogil: + + _fortran_csteqr(compz, n, d, e, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csycon "BLAS_FUNC(csycon)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, s *anorm, s *rcond, npy_complex64 *work, int *info) nogil +cdef void csycon(char *uplo, int *n, c *a, int *lda, int *ipiv, s *anorm, s *rcond, c *work, int *info) noexcept nogil: + + _fortran_csycon(uplo, n, a, lda, ipiv, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csyconv "BLAS_FUNC(csyconv)"(char *uplo, char *way, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *info) nogil +cdef void csyconv(char *uplo, char *way, int *n, c *a, int *lda, int *ipiv, c *work, int *info) noexcept nogil: + + _fortran_csyconv(uplo, way, n, a, lda, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csyequb "BLAS_FUNC(csyequb)"(char *uplo, int *n, npy_complex64 *a, int *lda, s *s, s *scond, s *amax, npy_complex64 *work, int *info) nogil +cdef void csyequb(char *uplo, int *n, c *a, int *lda, s *s, s *scond, s *amax, c *work, int *info) noexcept nogil: + + _fortran_csyequb(uplo, n, a, lda, s, scond, amax, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csymv "BLAS_FUNC(csymv)"(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy) nogil +cdef void csymv(char *uplo, int *n, c *alpha, c *a, int *lda, c *x, int *incx, c *beta, c *y, int *incy) noexcept nogil: + + _fortran_csymv(uplo, n, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csyr "BLAS_FUNC(csyr)"(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *a, int *lda) nogil +cdef void csyr(char *uplo, int *n, c *alpha, c *x, int *incx, c *a, int *lda) noexcept nogil: + + _fortran_csyr(uplo, n, alpha, x, incx, a, lda) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csyrfs "BLAS_FUNC(csyrfs)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void csyrfs(char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_csyrfs(uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csysv "BLAS_FUNC(csysv)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *work, int *lwork, int *info) nogil +cdef void csysv(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_csysv(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csysvx "BLAS_FUNC(csysvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *af, int *ldaf, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *rcond, s *ferr, s *berr, npy_complex64 *work, int *lwork, s *rwork, int *info) nogil +cdef void csysvx(char *fact, char *uplo, int *n, int *nrhs, c *a, int *lda, c *af, int *ldaf, int *ipiv, c *b, int *ldb, c *x, int *ldx, s *rcond, s *ferr, s *berr, c *work, int *lwork, s *rwork, int *info) noexcept nogil: + + _fortran_csysvx(fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csyswapr "BLAS_FUNC(csyswapr)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *i1, int *i2) nogil +cdef void csyswapr(char *uplo, int *n, c *a, int *lda, int *i1, int *i2) noexcept nogil: + + _fortran_csyswapr(uplo, n, a, lda, i1, i2) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csytf2 "BLAS_FUNC(csytf2)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, int *info) nogil +cdef void csytf2(char *uplo, int *n, c *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_csytf2(uplo, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csytrf "BLAS_FUNC(csytrf)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *lwork, int *info) nogil +cdef void csytrf(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_csytrf(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csytri "BLAS_FUNC(csytri)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *info) nogil +cdef void csytri(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *info) noexcept nogil: + + _fortran_csytri(uplo, n, a, lda, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csytri2 "BLAS_FUNC(csytri2)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *lwork, int *info) nogil +cdef void csytri2(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_csytri2(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csytri2x "BLAS_FUNC(csytri2x)"(char *uplo, int *n, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *work, int *nb, int *info) nogil +cdef void csytri2x(char *uplo, int *n, c *a, int *lda, int *ipiv, c *work, int *nb, int *info) noexcept nogil: + + _fortran_csytri2x(uplo, n, a, lda, ipiv, work, nb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csytrs "BLAS_FUNC(csytrs)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, int *info) nogil +cdef void csytrs(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_csytrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_csytrs2 "BLAS_FUNC(csytrs2)"(char *uplo, int *n, int *nrhs, npy_complex64 *a, int *lda, int *ipiv, npy_complex64 *b, int *ldb, npy_complex64 *work, int *info) nogil +cdef void csytrs2(char *uplo, int *n, int *nrhs, c *a, int *lda, int *ipiv, c *b, int *ldb, c *work, int *info) noexcept nogil: + + _fortran_csytrs2(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctbcon "BLAS_FUNC(ctbcon)"(char *norm, char *uplo, char *diag, int *n, int *kd, npy_complex64 *ab, int *ldab, s *rcond, npy_complex64 *work, s *rwork, int *info) nogil +cdef void ctbcon(char *norm, char *uplo, char *diag, int *n, int *kd, c *ab, int *ldab, s *rcond, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_ctbcon(norm, uplo, diag, n, kd, ab, ldab, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctbrfs "BLAS_FUNC(ctbrfs)"(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void ctbrfs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_ctbrfs(uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctbtrs "BLAS_FUNC(ctbtrs)"(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, npy_complex64 *ab, int *ldab, npy_complex64 *b, int *ldb, int *info) nogil +cdef void ctbtrs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, c *ab, int *ldab, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_ctbtrs(uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctfsm "BLAS_FUNC(ctfsm)"(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, npy_complex64 *b, int *ldb) nogil +cdef void ctfsm(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, c *alpha, c *a, c *b, int *ldb) noexcept nogil: + + _fortran_ctfsm(transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctftri "BLAS_FUNC(ctftri)"(char *transr, char *uplo, char *diag, int *n, npy_complex64 *a, int *info) nogil +cdef void ctftri(char *transr, char *uplo, char *diag, int *n, c *a, int *info) noexcept nogil: + + _fortran_ctftri(transr, uplo, diag, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctfttp "BLAS_FUNC(ctfttp)"(char *transr, char *uplo, int *n, npy_complex64 *arf, npy_complex64 *ap, int *info) nogil +cdef void ctfttp(char *transr, char *uplo, int *n, c *arf, c *ap, int *info) noexcept nogil: + + _fortran_ctfttp(transr, uplo, n, arf, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctfttr "BLAS_FUNC(ctfttr)"(char *transr, char *uplo, int *n, npy_complex64 *arf, npy_complex64 *a, int *lda, int *info) nogil +cdef void ctfttr(char *transr, char *uplo, int *n, c *arf, c *a, int *lda, int *info) noexcept nogil: + + _fortran_ctfttr(transr, uplo, n, arf, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctgevc "BLAS_FUNC(ctgevc)"(char *side, char *howmny, bint *select, int *n, npy_complex64 *s, int *lds, npy_complex64 *p, int *ldp, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, int *mm, int *m, npy_complex64 *work, s *rwork, int *info) nogil +cdef void ctgevc(char *side, char *howmny, bint *select, int *n, c *s, int *lds, c *p, int *ldp, c *vl, int *ldvl, c *vr, int *ldvr, int *mm, int *m, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_ctgevc(side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctgex2 "BLAS_FUNC(ctgex2)"(bint *wantq, bint *wantz, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *q, int *ldq, npy_complex64 *z, int *ldz, int *j1, int *info) nogil +cdef void ctgex2(bint *wantq, bint *wantz, int *n, c *a, int *lda, c *b, int *ldb, c *q, int *ldq, c *z, int *ldz, int *j1, int *info) noexcept nogil: + + _fortran_ctgex2(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctgexc "BLAS_FUNC(ctgexc)"(bint *wantq, bint *wantz, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *q, int *ldq, npy_complex64 *z, int *ldz, int *ifst, int *ilst, int *info) nogil +cdef void ctgexc(bint *wantq, bint *wantz, int *n, c *a, int *lda, c *b, int *ldb, c *q, int *ldq, c *z, int *ldz, int *ifst, int *ilst, int *info) noexcept nogil: + + _fortran_ctgexc(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctgsen "BLAS_FUNC(ctgsen)"(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *alpha, npy_complex64 *beta, npy_complex64 *q, int *ldq, npy_complex64 *z, int *ldz, int *m, s *pl, s *pr, s *dif, npy_complex64 *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void ctgsen(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, c *a, int *lda, c *b, int *ldb, c *alpha, c *beta, c *q, int *ldq, c *z, int *ldz, int *m, s *pl, s *pr, s *dif, c *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_ctgsen(ijob, wantq, wantz, select, n, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctgsja "BLAS_FUNC(ctgsja)"(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, s *tola, s *tolb, s *alpha, s *beta, npy_complex64 *u, int *ldu, npy_complex64 *v, int *ldv, npy_complex64 *q, int *ldq, npy_complex64 *work, int *ncycle, int *info) nogil +cdef void ctgsja(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, c *a, int *lda, c *b, int *ldb, s *tola, s *tolb, s *alpha, s *beta, c *u, int *ldu, c *v, int *ldv, c *q, int *ldq, c *work, int *ncycle, int *info) noexcept nogil: + + _fortran_ctgsja(jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb, tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq, work, ncycle, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctgsna "BLAS_FUNC(ctgsna)"(char *job, char *howmny, bint *select, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, s *s, s *dif, int *mm, int *m, npy_complex64 *work, int *lwork, int *iwork, int *info) nogil +cdef void ctgsna(char *job, char *howmny, bint *select, int *n, c *a, int *lda, c *b, int *ldb, c *vl, int *ldvl, c *vr, int *ldvr, s *s, s *dif, int *mm, int *m, c *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_ctgsna(job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctgsy2 "BLAS_FUNC(ctgsy2)"(char *trans, int *ijob, int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *c, int *ldc, npy_complex64 *d, int *ldd, npy_complex64 *e, int *lde, npy_complex64 *f, int *ldf, s *scale, s *rdsum, s *rdscal, int *info) nogil +cdef void ctgsy2(char *trans, int *ijob, int *m, int *n, c *a, int *lda, c *b, int *ldb, c *c, int *ldc, c *d, int *ldd, c *e, int *lde, c *f, int *ldf, s *scale, s *rdsum, s *rdscal, int *info) noexcept nogil: + + _fortran_ctgsy2(trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctgsyl "BLAS_FUNC(ctgsyl)"(char *trans, int *ijob, int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *c, int *ldc, npy_complex64 *d, int *ldd, npy_complex64 *e, int *lde, npy_complex64 *f, int *ldf, s *scale, s *dif, npy_complex64 *work, int *lwork, int *iwork, int *info) nogil +cdef void ctgsyl(char *trans, int *ijob, int *m, int *n, c *a, int *lda, c *b, int *ldb, c *c, int *ldc, c *d, int *ldd, c *e, int *lde, c *f, int *ldf, s *scale, s *dif, c *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_ctgsyl(trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctpcon "BLAS_FUNC(ctpcon)"(char *norm, char *uplo, char *diag, int *n, npy_complex64 *ap, s *rcond, npy_complex64 *work, s *rwork, int *info) nogil +cdef void ctpcon(char *norm, char *uplo, char *diag, int *n, c *ap, s *rcond, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_ctpcon(norm, uplo, diag, n, ap, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctpmqrt "BLAS_FUNC(ctpmqrt)"(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, npy_complex64 *v, int *ldv, npy_complex64 *t, int *ldt, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *work, int *info) nogil +cdef void ctpmqrt(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, c *v, int *ldv, c *t, int *ldt, c *a, int *lda, c *b, int *ldb, c *work, int *info) noexcept nogil: + + _fortran_ctpmqrt(side, trans, m, n, k, l, nb, v, ldv, t, ldt, a, lda, b, ldb, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctpqrt "BLAS_FUNC(ctpqrt)"(int *m, int *n, int *l, int *nb, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *t, int *ldt, npy_complex64 *work, int *info) nogil +cdef void ctpqrt(int *m, int *n, int *l, int *nb, c *a, int *lda, c *b, int *ldb, c *t, int *ldt, c *work, int *info) noexcept nogil: + + _fortran_ctpqrt(m, n, l, nb, a, lda, b, ldb, t, ldt, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctpqrt2 "BLAS_FUNC(ctpqrt2)"(int *m, int *n, int *l, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *t, int *ldt, int *info) nogil +cdef void ctpqrt2(int *m, int *n, int *l, c *a, int *lda, c *b, int *ldb, c *t, int *ldt, int *info) noexcept nogil: + + _fortran_ctpqrt2(m, n, l, a, lda, b, ldb, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctprfb "BLAS_FUNC(ctprfb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, npy_complex64 *v, int *ldv, npy_complex64 *t, int *ldt, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *work, int *ldwork) nogil +cdef void ctprfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, c *v, int *ldv, c *t, int *ldt, c *a, int *lda, c *b, int *ldb, c *work, int *ldwork) noexcept nogil: + + _fortran_ctprfb(side, trans, direct, storev, m, n, k, l, v, ldv, t, ldt, a, lda, b, ldb, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctprfs "BLAS_FUNC(ctprfs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void ctprfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, c *ap, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_ctprfs(uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctptri "BLAS_FUNC(ctptri)"(char *uplo, char *diag, int *n, npy_complex64 *ap, int *info) nogil +cdef void ctptri(char *uplo, char *diag, int *n, c *ap, int *info) noexcept nogil: + + _fortran_ctptri(uplo, diag, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctptrs "BLAS_FUNC(ctptrs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex64 *ap, npy_complex64 *b, int *ldb, int *info) nogil +cdef void ctptrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, c *ap, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_ctptrs(uplo, trans, diag, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctpttf "BLAS_FUNC(ctpttf)"(char *transr, char *uplo, int *n, npy_complex64 *ap, npy_complex64 *arf, int *info) nogil +cdef void ctpttf(char *transr, char *uplo, int *n, c *ap, c *arf, int *info) noexcept nogil: + + _fortran_ctpttf(transr, uplo, n, ap, arf, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctpttr "BLAS_FUNC(ctpttr)"(char *uplo, int *n, npy_complex64 *ap, npy_complex64 *a, int *lda, int *info) nogil +cdef void ctpttr(char *uplo, int *n, c *ap, c *a, int *lda, int *info) noexcept nogil: + + _fortran_ctpttr(uplo, n, ap, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrcon "BLAS_FUNC(ctrcon)"(char *norm, char *uplo, char *diag, int *n, npy_complex64 *a, int *lda, s *rcond, npy_complex64 *work, s *rwork, int *info) nogil +cdef void ctrcon(char *norm, char *uplo, char *diag, int *n, c *a, int *lda, s *rcond, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_ctrcon(norm, uplo, diag, n, a, lda, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrevc "BLAS_FUNC(ctrevc)"(char *side, char *howmny, bint *select, int *n, npy_complex64 *t, int *ldt, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, int *mm, int *m, npy_complex64 *work, s *rwork, int *info) nogil +cdef void ctrevc(char *side, char *howmny, bint *select, int *n, c *t, int *ldt, c *vl, int *ldvl, c *vr, int *ldvr, int *mm, int *m, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_ctrevc(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrexc "BLAS_FUNC(ctrexc)"(char *compq, int *n, npy_complex64 *t, int *ldt, npy_complex64 *q, int *ldq, int *ifst, int *ilst, int *info) nogil +cdef void ctrexc(char *compq, int *n, c *t, int *ldt, c *q, int *ldq, int *ifst, int *ilst, int *info) noexcept nogil: + + _fortran_ctrexc(compq, n, t, ldt, q, ldq, ifst, ilst, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrrfs "BLAS_FUNC(ctrrfs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *x, int *ldx, s *ferr, s *berr, npy_complex64 *work, s *rwork, int *info) nogil +cdef void ctrrfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, c *x, int *ldx, s *ferr, s *berr, c *work, s *rwork, int *info) noexcept nogil: + + _fortran_ctrrfs(uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrsen "BLAS_FUNC(ctrsen)"(char *job, char *compq, bint *select, int *n, npy_complex64 *t, int *ldt, npy_complex64 *q, int *ldq, npy_complex64 *w, int *m, s *s, s *sep, npy_complex64 *work, int *lwork, int *info) nogil +cdef void ctrsen(char *job, char *compq, bint *select, int *n, c *t, int *ldt, c *q, int *ldq, c *w, int *m, s *s, s *sep, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_ctrsen(job, compq, select, n, t, ldt, q, ldq, w, m, s, sep, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrsna "BLAS_FUNC(ctrsna)"(char *job, char *howmny, bint *select, int *n, npy_complex64 *t, int *ldt, npy_complex64 *vl, int *ldvl, npy_complex64 *vr, int *ldvr, s *s, s *sep, int *mm, int *m, npy_complex64 *work, int *ldwork, s *rwork, int *info) nogil +cdef void ctrsna(char *job, char *howmny, bint *select, int *n, c *t, int *ldt, c *vl, int *ldvl, c *vr, int *ldvr, s *s, s *sep, int *mm, int *m, c *work, int *ldwork, s *rwork, int *info) noexcept nogil: + + _fortran_ctrsna(job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrsyl "BLAS_FUNC(ctrsyl)"(char *trana, char *tranb, int *isgn, int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *c, int *ldc, s *scale, int *info) nogil +cdef void ctrsyl(char *trana, char *tranb, int *isgn, int *m, int *n, c *a, int *lda, c *b, int *ldb, c *c, int *ldc, s *scale, int *info) noexcept nogil: + + _fortran_ctrsyl(trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrti2 "BLAS_FUNC(ctrti2)"(char *uplo, char *diag, int *n, npy_complex64 *a, int *lda, int *info) nogil +cdef void ctrti2(char *uplo, char *diag, int *n, c *a, int *lda, int *info) noexcept nogil: + + _fortran_ctrti2(uplo, diag, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrtri "BLAS_FUNC(ctrtri)"(char *uplo, char *diag, int *n, npy_complex64 *a, int *lda, int *info) nogil +cdef void ctrtri(char *uplo, char *diag, int *n, c *a, int *lda, int *info) noexcept nogil: + + _fortran_ctrtri(uplo, diag, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrtrs "BLAS_FUNC(ctrtrs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, int *info) nogil +cdef void ctrtrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, c *a, int *lda, c *b, int *ldb, int *info) noexcept nogil: + + _fortran_ctrtrs(uplo, trans, diag, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrttf "BLAS_FUNC(ctrttf)"(char *transr, char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *arf, int *info) nogil +cdef void ctrttf(char *transr, char *uplo, int *n, c *a, int *lda, c *arf, int *info) noexcept nogil: + + _fortran_ctrttf(transr, uplo, n, a, lda, arf, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctrttp "BLAS_FUNC(ctrttp)"(char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *ap, int *info) nogil +cdef void ctrttp(char *uplo, int *n, c *a, int *lda, c *ap, int *info) noexcept nogil: + + _fortran_ctrttp(uplo, n, a, lda, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ctzrzf "BLAS_FUNC(ctzrzf)"(int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void ctzrzf(int *m, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_ctzrzf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunbdb "BLAS_FUNC(cunbdb)"(char *trans, char *signs, int *m, int *p, int *q, npy_complex64 *x11, int *ldx11, npy_complex64 *x12, int *ldx12, npy_complex64 *x21, int *ldx21, npy_complex64 *x22, int *ldx22, s *theta, s *phi, npy_complex64 *taup1, npy_complex64 *taup2, npy_complex64 *tauq1, npy_complex64 *tauq2, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunbdb(char *trans, char *signs, int *m, int *p, int *q, c *x11, int *ldx11, c *x12, int *ldx12, c *x21, int *ldx21, c *x22, int *ldx22, s *theta, s *phi, c *taup1, c *taup2, c *tauq1, c *tauq2, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunbdb(trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cuncsd "BLAS_FUNC(cuncsd)"(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, npy_complex64 *x11, int *ldx11, npy_complex64 *x12, int *ldx12, npy_complex64 *x21, int *ldx21, npy_complex64 *x22, int *ldx22, s *theta, npy_complex64 *u1, int *ldu1, npy_complex64 *u2, int *ldu2, npy_complex64 *v1t, int *ldv1t, npy_complex64 *v2t, int *ldv2t, npy_complex64 *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *info) nogil +cdef void cuncsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, c *x11, int *ldx11, c *x12, int *ldx12, c *x21, int *ldx21, c *x22, int *ldx22, s *theta, c *u1, int *ldu1, c *u2, int *ldu2, c *v1t, int *ldv1t, c *v2t, int *ldv2t, c *work, int *lwork, s *rwork, int *lrwork, int *iwork, int *info) noexcept nogil: + + _fortran_cuncsd(jobu1, jobu2, jobv1t, jobv2t, trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, work, lwork, rwork, lrwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cung2l "BLAS_FUNC(cung2l)"(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info) nogil +cdef void cung2l(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil: + + _fortran_cung2l(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cung2r "BLAS_FUNC(cung2r)"(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info) nogil +cdef void cung2r(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil: + + _fortran_cung2r(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cungbr "BLAS_FUNC(cungbr)"(char *vect, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cungbr(char *vect, int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cungbr(vect, m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunghr "BLAS_FUNC(cunghr)"(int *n, int *ilo, int *ihi, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunghr(int *n, int *ilo, int *ihi, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunghr(n, ilo, ihi, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cungl2 "BLAS_FUNC(cungl2)"(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info) nogil +cdef void cungl2(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil: + + _fortran_cungl2(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunglq "BLAS_FUNC(cunglq)"(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunglq(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunglq(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cungql "BLAS_FUNC(cungql)"(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cungql(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cungql(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cungqr "BLAS_FUNC(cungqr)"(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cungqr(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cungqr(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cungr2 "BLAS_FUNC(cungr2)"(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *info) nogil +cdef void cungr2(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *info) noexcept nogil: + + _fortran_cungr2(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cungrq "BLAS_FUNC(cungrq)"(int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cungrq(int *m, int *n, int *k, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cungrq(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cungtr "BLAS_FUNC(cungtr)"(char *uplo, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cungtr(char *uplo, int *n, c *a, int *lda, c *tau, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cungtr(uplo, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunm2l "BLAS_FUNC(cunm2l)"(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info) nogil +cdef void cunm2l(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil: + + _fortran_cunm2l(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunm2r "BLAS_FUNC(cunm2r)"(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info) nogil +cdef void cunm2r(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil: + + _fortran_cunm2r(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunmbr "BLAS_FUNC(cunmbr)"(char *vect, char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunmbr(char *vect, char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunmbr(vect, side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunmhr "BLAS_FUNC(cunmhr)"(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunmhr(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunmhr(side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunml2 "BLAS_FUNC(cunml2)"(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info) nogil +cdef void cunml2(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil: + + _fortran_cunml2(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunmlq "BLAS_FUNC(cunmlq)"(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunmlq(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunmlq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunmql "BLAS_FUNC(cunmql)"(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunmql(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunmql(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunmqr "BLAS_FUNC(cunmqr)"(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunmqr(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunmqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunmr2 "BLAS_FUNC(cunmr2)"(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info) nogil +cdef void cunmr2(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil: + + _fortran_cunmr2(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunmr3 "BLAS_FUNC(cunmr3)"(char *side, char *trans, int *m, int *n, int *k, int *l, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info) nogil +cdef void cunmr3(char *side, char *trans, int *m, int *n, int *k, int *l, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil: + + _fortran_cunmr3(side, trans, m, n, k, l, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunmrq "BLAS_FUNC(cunmrq)"(char *side, char *trans, int *m, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunmrq(char *side, char *trans, int *m, int *n, int *k, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunmrq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunmrz "BLAS_FUNC(cunmrz)"(char *side, char *trans, int *m, int *n, int *k, int *l, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunmrz(char *side, char *trans, int *m, int *n, int *k, int *l, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunmrz(side, trans, m, n, k, l, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cunmtr "BLAS_FUNC(cunmtr)"(char *side, char *uplo, char *trans, int *m, int *n, npy_complex64 *a, int *lda, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *lwork, int *info) nogil +cdef void cunmtr(char *side, char *uplo, char *trans, int *m, int *n, c *a, int *lda, c *tau, c *c, int *ldc, c *work, int *lwork, int *info) noexcept nogil: + + _fortran_cunmtr(side, uplo, trans, m, n, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cupgtr "BLAS_FUNC(cupgtr)"(char *uplo, int *n, npy_complex64 *ap, npy_complex64 *tau, npy_complex64 *q, int *ldq, npy_complex64 *work, int *info) nogil +cdef void cupgtr(char *uplo, int *n, c *ap, c *tau, c *q, int *ldq, c *work, int *info) noexcept nogil: + + _fortran_cupgtr(uplo, n, ap, tau, q, ldq, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_cupmtr "BLAS_FUNC(cupmtr)"(char *side, char *uplo, char *trans, int *m, int *n, npy_complex64 *ap, npy_complex64 *tau, npy_complex64 *c, int *ldc, npy_complex64 *work, int *info) nogil +cdef void cupmtr(char *side, char *uplo, char *trans, int *m, int *n, c *ap, c *tau, c *c, int *ldc, c *work, int *info) noexcept nogil: + + _fortran_cupmtr(side, uplo, trans, m, n, ap, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dbbcsd "BLAS_FUNC(dbbcsd)"(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, d *theta, d *phi, d *u1, int *ldu1, d *u2, int *ldu2, d *v1t, int *ldv1t, d *v2t, int *ldv2t, d *b11d, d *b11e, d *b12d, d *b12e, d *b21d, d *b21e, d *b22d, d *b22e, d *work, int *lwork, int *info) nogil +cdef void dbbcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, d *theta, d *phi, d *u1, int *ldu1, d *u2, int *ldu2, d *v1t, int *ldv1t, d *v2t, int *ldv2t, d *b11d, d *b11e, d *b12d, d *b12e, d *b21d, d *b21e, d *b22d, d *b22e, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dbbcsd(jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta, phi, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, b11d, b11e, b12d, b12e, b21d, b21e, b22d, b22e, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dbdsdc "BLAS_FUNC(dbdsdc)"(char *uplo, char *compq, int *n, d *d, d *e, d *u, int *ldu, d *vt, int *ldvt, d *q, int *iq, d *work, int *iwork, int *info) nogil +cdef void dbdsdc(char *uplo, char *compq, int *n, d *d, d *e, d *u, int *ldu, d *vt, int *ldvt, d *q, int *iq, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dbdsdc(uplo, compq, n, d, e, u, ldu, vt, ldvt, q, iq, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dbdsqr "BLAS_FUNC(dbdsqr)"(char *uplo, int *n, int *ncvt, int *nru, int *ncc, d *d, d *e, d *vt, int *ldvt, d *u, int *ldu, d *c, int *ldc, d *work, int *info) nogil +cdef void dbdsqr(char *uplo, int *n, int *ncvt, int *nru, int *ncc, d *d, d *e, d *vt, int *ldvt, d *u, int *ldu, d *c, int *ldc, d *work, int *info) noexcept nogil: + + _fortran_dbdsqr(uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ddisna "BLAS_FUNC(ddisna)"(char *job, int *m, int *n, d *d, d *sep, int *info) nogil +cdef void ddisna(char *job, int *m, int *n, d *d, d *sep, int *info) noexcept nogil: + + _fortran_ddisna(job, m, n, d, sep, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgbbrd "BLAS_FUNC(dgbbrd)"(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, d *ab, int *ldab, d *d, d *e, d *q, int *ldq, d *pt, int *ldpt, d *c, int *ldc, d *work, int *info) nogil +cdef void dgbbrd(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, d *ab, int *ldab, d *d, d *e, d *q, int *ldq, d *pt, int *ldpt, d *c, int *ldc, d *work, int *info) noexcept nogil: + + _fortran_dgbbrd(vect, m, n, ncc, kl, ku, ab, ldab, d, e, q, ldq, pt, ldpt, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgbcon "BLAS_FUNC(dgbcon)"(char *norm, int *n, int *kl, int *ku, d *ab, int *ldab, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dgbcon(char *norm, int *n, int *kl, int *ku, d *ab, int *ldab, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dgbcon(norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgbequ "BLAS_FUNC(dgbequ)"(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) nogil +cdef void dgbequ(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil: + + _fortran_dgbequ(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgbequb "BLAS_FUNC(dgbequb)"(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) nogil +cdef void dgbequb(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil: + + _fortran_dgbequb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgbrfs "BLAS_FUNC(dgbrfs)"(char *trans, int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dgbrfs(char *trans, int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dgbrfs(trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgbsv "BLAS_FUNC(dgbsv)"(int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, int *ipiv, d *b, int *ldb, int *info) nogil +cdef void dgbsv(int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, int *ipiv, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dgbsv(n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgbsvx "BLAS_FUNC(dgbsvx)"(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, int *ipiv, char *equed, d *r, d *c, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dgbsvx(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, int *ipiv, char *equed, d *r, d *c, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dgbsvx(fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgbtf2 "BLAS_FUNC(dgbtf2)"(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, int *ipiv, int *info) nogil +cdef void dgbtf2(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, int *ipiv, int *info) noexcept nogil: + + _fortran_dgbtf2(m, n, kl, ku, ab, ldab, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgbtrf "BLAS_FUNC(dgbtrf)"(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, int *ipiv, int *info) nogil +cdef void dgbtrf(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, int *ipiv, int *info) noexcept nogil: + + _fortran_dgbtrf(m, n, kl, ku, ab, ldab, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgbtrs "BLAS_FUNC(dgbtrs)"(char *trans, int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, int *ipiv, d *b, int *ldb, int *info) nogil +cdef void dgbtrs(char *trans, int *n, int *kl, int *ku, int *nrhs, d *ab, int *ldab, int *ipiv, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dgbtrs(trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgebak "BLAS_FUNC(dgebak)"(char *job, char *side, int *n, int *ilo, int *ihi, d *scale, int *m, d *v, int *ldv, int *info) nogil +cdef void dgebak(char *job, char *side, int *n, int *ilo, int *ihi, d *scale, int *m, d *v, int *ldv, int *info) noexcept nogil: + + _fortran_dgebak(job, side, n, ilo, ihi, scale, m, v, ldv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgebal "BLAS_FUNC(dgebal)"(char *job, int *n, d *a, int *lda, int *ilo, int *ihi, d *scale, int *info) nogil +cdef void dgebal(char *job, int *n, d *a, int *lda, int *ilo, int *ihi, d *scale, int *info) noexcept nogil: + + _fortran_dgebal(job, n, a, lda, ilo, ihi, scale, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgebd2 "BLAS_FUNC(dgebd2)"(int *m, int *n, d *a, int *lda, d *d, d *e, d *tauq, d *taup, d *work, int *info) nogil +cdef void dgebd2(int *m, int *n, d *a, int *lda, d *d, d *e, d *tauq, d *taup, d *work, int *info) noexcept nogil: + + _fortran_dgebd2(m, n, a, lda, d, e, tauq, taup, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgebrd "BLAS_FUNC(dgebrd)"(int *m, int *n, d *a, int *lda, d *d, d *e, d *tauq, d *taup, d *work, int *lwork, int *info) nogil +cdef void dgebrd(int *m, int *n, d *a, int *lda, d *d, d *e, d *tauq, d *taup, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgebrd(m, n, a, lda, d, e, tauq, taup, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgecon "BLAS_FUNC(dgecon)"(char *norm, int *n, d *a, int *lda, d *anorm, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dgecon(char *norm, int *n, d *a, int *lda, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dgecon(norm, n, a, lda, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeequ "BLAS_FUNC(dgeequ)"(int *m, int *n, d *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) nogil +cdef void dgeequ(int *m, int *n, d *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil: + + _fortran_dgeequ(m, n, a, lda, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeequb "BLAS_FUNC(dgeequb)"(int *m, int *n, d *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) nogil +cdef void dgeequb(int *m, int *n, d *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil: + + _fortran_dgeequb(m, n, a, lda, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgees "BLAS_FUNC(dgees)"(char *jobvs, char *sort, _dselect2 *select, int *n, d *a, int *lda, int *sdim, d *wr, d *wi, d *vs, int *ldvs, d *work, int *lwork, bint *bwork, int *info) nogil +cdef void dgees(char *jobvs, char *sort, dselect2 *select, int *n, d *a, int *lda, int *sdim, d *wr, d *wi, d *vs, int *ldvs, d *work, int *lwork, bint *bwork, int *info) noexcept nogil: + + _fortran_dgees(jobvs, sort, <_dselect2*>select, n, a, lda, sdim, wr, wi, vs, ldvs, work, lwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeesx "BLAS_FUNC(dgeesx)"(char *jobvs, char *sort, _dselect2 *select, char *sense, int *n, d *a, int *lda, int *sdim, d *wr, d *wi, d *vs, int *ldvs, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) nogil +cdef void dgeesx(char *jobvs, char *sort, dselect2 *select, char *sense, int *n, d *a, int *lda, int *sdim, d *wr, d *wi, d *vs, int *ldvs, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil: + + _fortran_dgeesx(jobvs, sort, <_dselect2*>select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde, rcondv, work, lwork, iwork, liwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeev "BLAS_FUNC(dgeev)"(char *jobvl, char *jobvr, int *n, d *a, int *lda, d *wr, d *wi, d *vl, int *ldvl, d *vr, int *ldvr, d *work, int *lwork, int *info) nogil +cdef void dgeev(char *jobvl, char *jobvr, int *n, d *a, int *lda, d *wr, d *wi, d *vl, int *ldvl, d *vr, int *ldvr, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgeev(jobvl, jobvr, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeevx "BLAS_FUNC(dgeevx)"(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, d *a, int *lda, d *wr, d *wi, d *vl, int *ldvl, d *vr, int *ldvr, int *ilo, int *ihi, d *scale, d *abnrm, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, int *info) nogil +cdef void dgeevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, d *a, int *lda, d *wr, d *wi, d *vl, int *ldvl, d *vr, int *ldvr, int *ilo, int *ihi, d *scale, d *abnrm, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_dgeevx(balanc, jobvl, jobvr, sense, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgehd2 "BLAS_FUNC(dgehd2)"(int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *work, int *info) nogil +cdef void dgehd2(int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil: + + _fortran_dgehd2(n, ilo, ihi, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgehrd "BLAS_FUNC(dgehrd)"(int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dgehrd(int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgehrd(n, ilo, ihi, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgejsv "BLAS_FUNC(dgejsv)"(char *joba, char *jobu, char *jobv, char *jobr, char *jobt, char *jobp, int *m, int *n, d *a, int *lda, d *sva, d *u, int *ldu, d *v, int *ldv, d *work, int *lwork, int *iwork, int *info) nogil +cdef void dgejsv(char *joba, char *jobu, char *jobv, char *jobr, char *jobt, char *jobp, int *m, int *n, d *a, int *lda, d *sva, d *u, int *ldu, d *v, int *ldv, d *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_dgejsv(joba, jobu, jobv, jobr, jobt, jobp, m, n, a, lda, sva, u, ldu, v, ldv, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgelq2 "BLAS_FUNC(dgelq2)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) nogil +cdef void dgelq2(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil: + + _fortran_dgelq2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgelqf "BLAS_FUNC(dgelqf)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dgelqf(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgelqf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgels "BLAS_FUNC(dgels)"(char *trans, int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *work, int *lwork, int *info) nogil +cdef void dgels(char *trans, int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgels(trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgelsd "BLAS_FUNC(dgelsd)"(int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *s, d *rcond, int *rank, d *work, int *lwork, int *iwork, int *info) nogil +cdef void dgelsd(int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *s, d *rcond, int *rank, d *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_dgelsd(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgelss "BLAS_FUNC(dgelss)"(int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *s, d *rcond, int *rank, d *work, int *lwork, int *info) nogil +cdef void dgelss(int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *s, d *rcond, int *rank, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgelss(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgelsy "BLAS_FUNC(dgelsy)"(int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *jpvt, d *rcond, int *rank, d *work, int *lwork, int *info) nogil +cdef void dgelsy(int *m, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *jpvt, d *rcond, int *rank, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgelsy(m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgemqrt "BLAS_FUNC(dgemqrt)"(char *side, char *trans, int *m, int *n, int *k, int *nb, d *v, int *ldv, d *t, int *ldt, d *c, int *ldc, d *work, int *info) nogil +cdef void dgemqrt(char *side, char *trans, int *m, int *n, int *k, int *nb, d *v, int *ldv, d *t, int *ldt, d *c, int *ldc, d *work, int *info) noexcept nogil: + + _fortran_dgemqrt(side, trans, m, n, k, nb, v, ldv, t, ldt, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeql2 "BLAS_FUNC(dgeql2)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) nogil +cdef void dgeql2(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil: + + _fortran_dgeql2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeqlf "BLAS_FUNC(dgeqlf)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dgeqlf(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgeqlf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeqp3 "BLAS_FUNC(dgeqp3)"(int *m, int *n, d *a, int *lda, int *jpvt, d *tau, d *work, int *lwork, int *info) nogil +cdef void dgeqp3(int *m, int *n, d *a, int *lda, int *jpvt, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgeqp3(m, n, a, lda, jpvt, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeqr2 "BLAS_FUNC(dgeqr2)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) nogil +cdef void dgeqr2(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil: + + _fortran_dgeqr2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeqr2p "BLAS_FUNC(dgeqr2p)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) nogil +cdef void dgeqr2p(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil: + + _fortran_dgeqr2p(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeqrf "BLAS_FUNC(dgeqrf)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dgeqrf(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgeqrf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeqrfp "BLAS_FUNC(dgeqrfp)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dgeqrfp(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgeqrfp(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeqrt "BLAS_FUNC(dgeqrt)"(int *m, int *n, int *nb, d *a, int *lda, d *t, int *ldt, d *work, int *info) nogil +cdef void dgeqrt(int *m, int *n, int *nb, d *a, int *lda, d *t, int *ldt, d *work, int *info) noexcept nogil: + + _fortran_dgeqrt(m, n, nb, a, lda, t, ldt, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeqrt2 "BLAS_FUNC(dgeqrt2)"(int *m, int *n, d *a, int *lda, d *t, int *ldt, int *info) nogil +cdef void dgeqrt2(int *m, int *n, d *a, int *lda, d *t, int *ldt, int *info) noexcept nogil: + + _fortran_dgeqrt2(m, n, a, lda, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgeqrt3 "BLAS_FUNC(dgeqrt3)"(int *m, int *n, d *a, int *lda, d *t, int *ldt, int *info) nogil +cdef void dgeqrt3(int *m, int *n, d *a, int *lda, d *t, int *ldt, int *info) noexcept nogil: + + _fortran_dgeqrt3(m, n, a, lda, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgerfs "BLAS_FUNC(dgerfs)"(char *trans, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dgerfs(char *trans, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dgerfs(trans, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgerq2 "BLAS_FUNC(dgerq2)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) nogil +cdef void dgerq2(int *m, int *n, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil: + + _fortran_dgerq2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgerqf "BLAS_FUNC(dgerqf)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dgerqf(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgerqf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgesc2 "BLAS_FUNC(dgesc2)"(int *n, d *a, int *lda, d *rhs, int *ipiv, int *jpiv, d *scale) nogil +cdef void dgesc2(int *n, d *a, int *lda, d *rhs, int *ipiv, int *jpiv, d *scale) noexcept nogil: + + _fortran_dgesc2(n, a, lda, rhs, ipiv, jpiv, scale) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgesdd "BLAS_FUNC(dgesdd)"(char *jobz, int *m, int *n, d *a, int *lda, d *s, d *u, int *ldu, d *vt, int *ldvt, d *work, int *lwork, int *iwork, int *info) nogil +cdef void dgesdd(char *jobz, int *m, int *n, d *a, int *lda, d *s, d *u, int *ldu, d *vt, int *ldvt, d *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_dgesdd(jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgesv "BLAS_FUNC(dgesv)"(int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, int *info) nogil +cdef void dgesv(int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dgesv(n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgesvd "BLAS_FUNC(dgesvd)"(char *jobu, char *jobvt, int *m, int *n, d *a, int *lda, d *s, d *u, int *ldu, d *vt, int *ldvt, d *work, int *lwork, int *info) nogil +cdef void dgesvd(char *jobu, char *jobvt, int *m, int *n, d *a, int *lda, d *s, d *u, int *ldu, d *vt, int *ldvt, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgesvd(jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgesvj "BLAS_FUNC(dgesvj)"(char *joba, char *jobu, char *jobv, int *m, int *n, d *a, int *lda, d *sva, int *mv, d *v, int *ldv, d *work, int *lwork, int *info) nogil +cdef void dgesvj(char *joba, char *jobu, char *jobv, int *m, int *n, d *a, int *lda, d *sva, int *mv, d *v, int *ldv, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgesvj(joba, jobu, jobv, m, n, a, lda, sva, mv, v, ldv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgesvx "BLAS_FUNC(dgesvx)"(char *fact, char *trans, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, char *equed, d *r, d *c, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dgesvx(char *fact, char *trans, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, char *equed, d *r, d *c, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dgesvx(fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgetc2 "BLAS_FUNC(dgetc2)"(int *n, d *a, int *lda, int *ipiv, int *jpiv, int *info) nogil +cdef void dgetc2(int *n, d *a, int *lda, int *ipiv, int *jpiv, int *info) noexcept nogil: + + _fortran_dgetc2(n, a, lda, ipiv, jpiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgetf2 "BLAS_FUNC(dgetf2)"(int *m, int *n, d *a, int *lda, int *ipiv, int *info) nogil +cdef void dgetf2(int *m, int *n, d *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_dgetf2(m, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgetrf "BLAS_FUNC(dgetrf)"(int *m, int *n, d *a, int *lda, int *ipiv, int *info) nogil +cdef void dgetrf(int *m, int *n, d *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_dgetrf(m, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgetri "BLAS_FUNC(dgetri)"(int *n, d *a, int *lda, int *ipiv, d *work, int *lwork, int *info) nogil +cdef void dgetri(int *n, d *a, int *lda, int *ipiv, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgetri(n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgetrs "BLAS_FUNC(dgetrs)"(char *trans, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, int *info) nogil +cdef void dgetrs(char *trans, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dgetrs(trans, n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dggbak "BLAS_FUNC(dggbak)"(char *job, char *side, int *n, int *ilo, int *ihi, d *lscale, d *rscale, int *m, d *v, int *ldv, int *info) nogil +cdef void dggbak(char *job, char *side, int *n, int *ilo, int *ihi, d *lscale, d *rscale, int *m, d *v, int *ldv, int *info) noexcept nogil: + + _fortran_dggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dggbal "BLAS_FUNC(dggbal)"(char *job, int *n, d *a, int *lda, d *b, int *ldb, int *ilo, int *ihi, d *lscale, d *rscale, d *work, int *info) nogil +cdef void dggbal(char *job, int *n, d *a, int *lda, d *b, int *ldb, int *ilo, int *ihi, d *lscale, d *rscale, d *work, int *info) noexcept nogil: + + _fortran_dggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgges "BLAS_FUNC(dgges)"(char *jobvsl, char *jobvsr, char *sort, _dselect3 *selctg, int *n, d *a, int *lda, d *b, int *ldb, int *sdim, d *alphar, d *alphai, d *beta, d *vsl, int *ldvsl, d *vsr, int *ldvsr, d *work, int *lwork, bint *bwork, int *info) nogil +cdef void dgges(char *jobvsl, char *jobvsr, char *sort, dselect3 *selctg, int *n, d *a, int *lda, d *b, int *ldb, int *sdim, d *alphar, d *alphai, d *beta, d *vsl, int *ldvsl, d *vsr, int *ldvsr, d *work, int *lwork, bint *bwork, int *info) noexcept nogil: + + _fortran_dgges(jobvsl, jobvsr, sort, <_dselect3*>selctg, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dggesx "BLAS_FUNC(dggesx)"(char *jobvsl, char *jobvsr, char *sort, _dselect3 *selctg, char *sense, int *n, d *a, int *lda, d *b, int *ldb, int *sdim, d *alphar, d *alphai, d *beta, d *vsl, int *ldvsl, d *vsr, int *ldvsr, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) nogil +cdef void dggesx(char *jobvsl, char *jobvsr, char *sort, dselect3 *selctg, char *sense, int *n, d *a, int *lda, d *b, int *ldb, int *sdim, d *alphar, d *alphai, d *beta, d *vsl, int *ldvsl, d *vsr, int *ldvsr, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil: + + _fortran_dggesx(jobvsl, jobvsr, sort, <_dselect3*>selctg, sense, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, iwork, liwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dggev "BLAS_FUNC(dggev)"(char *jobvl, char *jobvr, int *n, d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *vl, int *ldvl, d *vr, int *ldvr, d *work, int *lwork, int *info) nogil +cdef void dggev(char *jobvl, char *jobvr, int *n, d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *vl, int *ldvl, d *vr, int *ldvr, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dggev(jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dggevx "BLAS_FUNC(dggevx)"(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *vl, int *ldvl, d *vr, int *ldvr, int *ilo, int *ihi, d *lscale, d *rscale, d *abnrm, d *bbnrm, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, bint *bwork, int *info) nogil +cdef void dggevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *vl, int *ldvl, d *vr, int *ldvr, int *ilo, int *ihi, d *lscale, d *rscale, d *abnrm, d *bbnrm, d *rconde, d *rcondv, d *work, int *lwork, int *iwork, bint *bwork, int *info) noexcept nogil: + + _fortran_dggevx(balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dggglm "BLAS_FUNC(dggglm)"(int *n, int *m, int *p, d *a, int *lda, d *b, int *ldb, d *d, d *x, d *y, d *work, int *lwork, int *info) nogil +cdef void dggglm(int *n, int *m, int *p, d *a, int *lda, d *b, int *ldb, d *d, d *x, d *y, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dggglm(n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgghrd "BLAS_FUNC(dgghrd)"(char *compq, char *compz, int *n, int *ilo, int *ihi, d *a, int *lda, d *b, int *ldb, d *q, int *ldq, d *z, int *ldz, int *info) nogil +cdef void dgghrd(char *compq, char *compz, int *n, int *ilo, int *ihi, d *a, int *lda, d *b, int *ldb, d *q, int *ldq, d *z, int *ldz, int *info) noexcept nogil: + + _fortran_dgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgglse "BLAS_FUNC(dgglse)"(int *m, int *n, int *p, d *a, int *lda, d *b, int *ldb, d *c, d *d, d *x, d *work, int *lwork, int *info) nogil +cdef void dgglse(int *m, int *n, int *p, d *a, int *lda, d *b, int *ldb, d *c, d *d, d *x, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgglse(m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dggqrf "BLAS_FUNC(dggqrf)"(int *n, int *m, int *p, d *a, int *lda, d *taua, d *b, int *ldb, d *taub, d *work, int *lwork, int *info) nogil +cdef void dggqrf(int *n, int *m, int *p, d *a, int *lda, d *taua, d *b, int *ldb, d *taub, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dggqrf(n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dggrqf "BLAS_FUNC(dggrqf)"(int *m, int *p, int *n, d *a, int *lda, d *taua, d *b, int *ldb, d *taub, d *work, int *lwork, int *info) nogil +cdef void dggrqf(int *m, int *p, int *n, d *a, int *lda, d *taua, d *b, int *ldb, d *taub, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dggrqf(m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgsvj0 "BLAS_FUNC(dgsvj0)"(char *jobv, int *m, int *n, d *a, int *lda, d *d, d *sva, int *mv, d *v, int *ldv, d *eps, d *sfmin, d *tol, int *nsweep, d *work, int *lwork, int *info) nogil +cdef void dgsvj0(char *jobv, int *m, int *n, d *a, int *lda, d *d, d *sva, int *mv, d *v, int *ldv, d *eps, d *sfmin, d *tol, int *nsweep, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgsvj0(jobv, m, n, a, lda, d, sva, mv, v, ldv, eps, sfmin, tol, nsweep, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgsvj1 "BLAS_FUNC(dgsvj1)"(char *jobv, int *m, int *n, int *n1, d *a, int *lda, d *d, d *sva, int *mv, d *v, int *ldv, d *eps, d *sfmin, d *tol, int *nsweep, d *work, int *lwork, int *info) nogil +cdef void dgsvj1(char *jobv, int *m, int *n, int *n1, d *a, int *lda, d *d, d *sva, int *mv, d *v, int *ldv, d *eps, d *sfmin, d *tol, int *nsweep, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dgsvj1(jobv, m, n, n1, a, lda, d, sva, mv, v, ldv, eps, sfmin, tol, nsweep, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgtcon "BLAS_FUNC(dgtcon)"(char *norm, int *n, d *dl, d *d, d *du, d *du2, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dgtcon(char *norm, int *n, d *dl, d *d, d *du, d *du2, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dgtcon(norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgtrfs "BLAS_FUNC(dgtrfs)"(char *trans, int *n, int *nrhs, d *dl, d *d, d *du, d *dlf, d *df, d *duf, d *du2, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dgtrfs(char *trans, int *n, int *nrhs, d *dl, d *d, d *du, d *dlf, d *df, d *duf, d *du2, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dgtrfs(trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgtsv "BLAS_FUNC(dgtsv)"(int *n, int *nrhs, d *dl, d *d, d *du, d *b, int *ldb, int *info) nogil +cdef void dgtsv(int *n, int *nrhs, d *dl, d *d, d *du, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dgtsv(n, nrhs, dl, d, du, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgtsvx "BLAS_FUNC(dgtsvx)"(char *fact, char *trans, int *n, int *nrhs, d *dl, d *d, d *du, d *dlf, d *df, d *duf, d *du2, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dgtsvx(char *fact, char *trans, int *n, int *nrhs, d *dl, d *d, d *du, d *dlf, d *df, d *duf, d *du2, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dgtsvx(fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgttrf "BLAS_FUNC(dgttrf)"(int *n, d *dl, d *d, d *du, d *du2, int *ipiv, int *info) nogil +cdef void dgttrf(int *n, d *dl, d *d, d *du, d *du2, int *ipiv, int *info) noexcept nogil: + + _fortran_dgttrf(n, dl, d, du, du2, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgttrs "BLAS_FUNC(dgttrs)"(char *trans, int *n, int *nrhs, d *dl, d *d, d *du, d *du2, int *ipiv, d *b, int *ldb, int *info) nogil +cdef void dgttrs(char *trans, int *n, int *nrhs, d *dl, d *d, d *du, d *du2, int *ipiv, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dgtts2 "BLAS_FUNC(dgtts2)"(int *itrans, int *n, int *nrhs, d *dl, d *d, d *du, d *du2, int *ipiv, d *b, int *ldb) nogil +cdef void dgtts2(int *itrans, int *n, int *nrhs, d *dl, d *d, d *du, d *du2, int *ipiv, d *b, int *ldb) noexcept nogil: + + _fortran_dgtts2(itrans, n, nrhs, dl, d, du, du2, ipiv, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dhgeqz "BLAS_FUNC(dhgeqz)"(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *t, int *ldt, d *alphar, d *alphai, d *beta, d *q, int *ldq, d *z, int *ldz, d *work, int *lwork, int *info) nogil +cdef void dhgeqz(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *t, int *ldt, d *alphar, d *alphai, d *beta, d *q, int *ldq, d *z, int *ldz, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dhgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dhsein "BLAS_FUNC(dhsein)"(char *side, char *eigsrc, char *initv, bint *select, int *n, d *h, int *ldh, d *wr, d *wi, d *vl, int *ldvl, d *vr, int *ldvr, int *mm, int *m, d *work, int *ifaill, int *ifailr, int *info) nogil +cdef void dhsein(char *side, char *eigsrc, char *initv, bint *select, int *n, d *h, int *ldh, d *wr, d *wi, d *vl, int *ldvl, d *vr, int *ldvr, int *mm, int *m, d *work, int *ifaill, int *ifailr, int *info) noexcept nogil: + + _fortran_dhsein(side, eigsrc, initv, select, n, h, ldh, wr, wi, vl, ldvl, vr, ldvr, mm, m, work, ifaill, ifailr, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dhseqr "BLAS_FUNC(dhseqr)"(char *job, char *compz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, d *z, int *ldz, d *work, int *lwork, int *info) nogil +cdef void dhseqr(char *job, char *compz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, d *z, int *ldz, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dhseqr(job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + bint _fortran_disnan "BLAS_FUNC(disnan)"(d *din) nogil +cdef bint disnan(d *din) noexcept nogil: + + return _fortran_disnan(din) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlabad "BLAS_FUNC(dlabad)"(d *small, d *large) nogil +cdef void dlabad(d *small, d *large) noexcept nogil: + + _fortran_dlabad(small, large) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlabrd "BLAS_FUNC(dlabrd)"(int *m, int *n, int *nb, d *a, int *lda, d *d, d *e, d *tauq, d *taup, d *x, int *ldx, d *y, int *ldy) nogil +cdef void dlabrd(int *m, int *n, int *nb, d *a, int *lda, d *d, d *e, d *tauq, d *taup, d *x, int *ldx, d *y, int *ldy) noexcept nogil: + + _fortran_dlabrd(m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlacn2 "BLAS_FUNC(dlacn2)"(int *n, d *v, d *x, int *isgn, d *est, int *kase, int *isave) nogil +cdef void dlacn2(int *n, d *v, d *x, int *isgn, d *est, int *kase, int *isave) noexcept nogil: + + _fortran_dlacn2(n, v, x, isgn, est, kase, isave) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlacon "BLAS_FUNC(dlacon)"(int *n, d *v, d *x, int *isgn, d *est, int *kase) nogil +cdef void dlacon(int *n, d *v, d *x, int *isgn, d *est, int *kase) noexcept nogil: + + _fortran_dlacon(n, v, x, isgn, est, kase) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlacpy "BLAS_FUNC(dlacpy)"(char *uplo, int *m, int *n, d *a, int *lda, d *b, int *ldb) nogil +cdef void dlacpy(char *uplo, int *m, int *n, d *a, int *lda, d *b, int *ldb) noexcept nogil: + + _fortran_dlacpy(uplo, m, n, a, lda, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dladiv "BLAS_FUNC(dladiv)"(d *a, d *b, d *c, d *d, d *p, d *q) nogil +cdef void dladiv(d *a, d *b, d *c, d *d, d *p, d *q) noexcept nogil: + + _fortran_dladiv(a, b, c, d, p, q) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlae2 "BLAS_FUNC(dlae2)"(d *a, d *b, d *c, d *rt1, d *rt2) nogil +cdef void dlae2(d *a, d *b, d *c, d *rt1, d *rt2) noexcept nogil: + + _fortran_dlae2(a, b, c, rt1, rt2) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaebz "BLAS_FUNC(dlaebz)"(int *ijob, int *nitmax, int *n, int *mmax, int *minp, int *nbmin, d *abstol, d *reltol, d *pivmin, d *d, d *e, d *e2, int *nval, d *ab, d *c, int *mout, int *nab, d *work, int *iwork, int *info) nogil +cdef void dlaebz(int *ijob, int *nitmax, int *n, int *mmax, int *minp, int *nbmin, d *abstol, d *reltol, d *pivmin, d *d, d *e, d *e2, int *nval, d *ab, d *c, int *mout, int *nab, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlaebz(ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol, pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaed0 "BLAS_FUNC(dlaed0)"(int *icompq, int *qsiz, int *n, d *d, d *e, d *q, int *ldq, d *qstore, int *ldqs, d *work, int *iwork, int *info) nogil +cdef void dlaed0(int *icompq, int *qsiz, int *n, d *d, d *e, d *q, int *ldq, d *qstore, int *ldqs, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlaed0(icompq, qsiz, n, d, e, q, ldq, qstore, ldqs, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaed1 "BLAS_FUNC(dlaed1)"(int *n, d *d, d *q, int *ldq, int *indxq, d *rho, int *cutpnt, d *work, int *iwork, int *info) nogil +cdef void dlaed1(int *n, d *d, d *q, int *ldq, int *indxq, d *rho, int *cutpnt, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlaed1(n, d, q, ldq, indxq, rho, cutpnt, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaed2 "BLAS_FUNC(dlaed2)"(int *k, int *n, int *n1, d *d, d *q, int *ldq, int *indxq, d *rho, d *z, d *dlamda, d *w, d *q2, int *indx, int *indxc, int *indxp, int *coltyp, int *info) nogil +cdef void dlaed2(int *k, int *n, int *n1, d *d, d *q, int *ldq, int *indxq, d *rho, d *z, d *dlamda, d *w, d *q2, int *indx, int *indxc, int *indxp, int *coltyp, int *info) noexcept nogil: + + _fortran_dlaed2(k, n, n1, d, q, ldq, indxq, rho, z, dlamda, w, q2, indx, indxc, indxp, coltyp, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaed3 "BLAS_FUNC(dlaed3)"(int *k, int *n, int *n1, d *d, d *q, int *ldq, d *rho, d *dlamda, d *q2, int *indx, int *ctot, d *w, d *s, int *info) nogil +cdef void dlaed3(int *k, int *n, int *n1, d *d, d *q, int *ldq, d *rho, d *dlamda, d *q2, int *indx, int *ctot, d *w, d *s, int *info) noexcept nogil: + + _fortran_dlaed3(k, n, n1, d, q, ldq, rho, dlamda, q2, indx, ctot, w, s, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaed4 "BLAS_FUNC(dlaed4)"(int *n, int *i, d *d, d *z, d *delta, d *rho, d *dlam, int *info) nogil +cdef void dlaed4(int *n, int *i, d *d, d *z, d *delta, d *rho, d *dlam, int *info) noexcept nogil: + + _fortran_dlaed4(n, i, d, z, delta, rho, dlam, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaed5 "BLAS_FUNC(dlaed5)"(int *i, d *d, d *z, d *delta, d *rho, d *dlam) nogil +cdef void dlaed5(int *i, d *d, d *z, d *delta, d *rho, d *dlam) noexcept nogil: + + _fortran_dlaed5(i, d, z, delta, rho, dlam) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaed6 "BLAS_FUNC(dlaed6)"(int *kniter, bint *orgati, d *rho, d *d, d *z, d *finit, d *tau, int *info) nogil +cdef void dlaed6(int *kniter, bint *orgati, d *rho, d *d, d *z, d *finit, d *tau, int *info) noexcept nogil: + + _fortran_dlaed6(kniter, orgati, rho, d, z, finit, tau, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaed7 "BLAS_FUNC(dlaed7)"(int *icompq, int *n, int *qsiz, int *tlvls, int *curlvl, int *curpbm, d *d, d *q, int *ldq, int *indxq, d *rho, int *cutpnt, d *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, d *givnum, d *work, int *iwork, int *info) nogil +cdef void dlaed7(int *icompq, int *n, int *qsiz, int *tlvls, int *curlvl, int *curpbm, d *d, d *q, int *ldq, int *indxq, d *rho, int *cutpnt, d *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, d *givnum, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlaed7(icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq, indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaed8 "BLAS_FUNC(dlaed8)"(int *icompq, int *k, int *n, int *qsiz, d *d, d *q, int *ldq, int *indxq, d *rho, int *cutpnt, d *z, d *dlamda, d *q2, int *ldq2, d *w, int *perm, int *givptr, int *givcol, d *givnum, int *indxp, int *indx, int *info) nogil +cdef void dlaed8(int *icompq, int *k, int *n, int *qsiz, d *d, d *q, int *ldq, int *indxq, d *rho, int *cutpnt, d *z, d *dlamda, d *q2, int *ldq2, d *w, int *perm, int *givptr, int *givcol, d *givnum, int *indxp, int *indx, int *info) noexcept nogil: + + _fortran_dlaed8(icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlamda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaed9 "BLAS_FUNC(dlaed9)"(int *k, int *kstart, int *kstop, int *n, d *d, d *q, int *ldq, d *rho, d *dlamda, d *w, d *s, int *lds, int *info) nogil +cdef void dlaed9(int *k, int *kstart, int *kstop, int *n, d *d, d *q, int *ldq, d *rho, d *dlamda, d *w, d *s, int *lds, int *info) noexcept nogil: + + _fortran_dlaed9(k, kstart, kstop, n, d, q, ldq, rho, dlamda, w, s, lds, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaeda "BLAS_FUNC(dlaeda)"(int *n, int *tlvls, int *curlvl, int *curpbm, int *prmptr, int *perm, int *givptr, int *givcol, d *givnum, d *q, int *qptr, d *z, d *ztemp, int *info) nogil +cdef void dlaeda(int *n, int *tlvls, int *curlvl, int *curpbm, int *prmptr, int *perm, int *givptr, int *givcol, d *givnum, d *q, int *qptr, d *z, d *ztemp, int *info) noexcept nogil: + + _fortran_dlaeda(n, tlvls, curlvl, curpbm, prmptr, perm, givptr, givcol, givnum, q, qptr, z, ztemp, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaein "BLAS_FUNC(dlaein)"(bint *rightv, bint *noinit, int *n, d *h, int *ldh, d *wr, d *wi, d *vr, d *vi, d *b, int *ldb, d *work, d *eps3, d *smlnum, d *bignum, int *info) nogil +cdef void dlaein(bint *rightv, bint *noinit, int *n, d *h, int *ldh, d *wr, d *wi, d *vr, d *vi, d *b, int *ldb, d *work, d *eps3, d *smlnum, d *bignum, int *info) noexcept nogil: + + _fortran_dlaein(rightv, noinit, n, h, ldh, wr, wi, vr, vi, b, ldb, work, eps3, smlnum, bignum, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaev2 "BLAS_FUNC(dlaev2)"(d *a, d *b, d *c, d *rt1, d *rt2, d *cs1, d *sn1) nogil +cdef void dlaev2(d *a, d *b, d *c, d *rt1, d *rt2, d *cs1, d *sn1) noexcept nogil: + + _fortran_dlaev2(a, b, c, rt1, rt2, cs1, sn1) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaexc "BLAS_FUNC(dlaexc)"(bint *wantq, int *n, d *t, int *ldt, d *q, int *ldq, int *j1, int *n1, int *n2, d *work, int *info) nogil +cdef void dlaexc(bint *wantq, int *n, d *t, int *ldt, d *q, int *ldq, int *j1, int *n1, int *n2, d *work, int *info) noexcept nogil: + + _fortran_dlaexc(wantq, n, t, ldt, q, ldq, j1, n1, n2, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlag2 "BLAS_FUNC(dlag2)"(d *a, int *lda, d *b, int *ldb, d *safmin, d *scale1, d *scale2, d *wr1, d *wr2, d *wi) nogil +cdef void dlag2(d *a, int *lda, d *b, int *ldb, d *safmin, d *scale1, d *scale2, d *wr1, d *wr2, d *wi) noexcept nogil: + + _fortran_dlag2(a, lda, b, ldb, safmin, scale1, scale2, wr1, wr2, wi) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlag2s "BLAS_FUNC(dlag2s)"(int *m, int *n, d *a, int *lda, s *sa, int *ldsa, int *info) nogil +cdef void dlag2s(int *m, int *n, d *a, int *lda, s *sa, int *ldsa, int *info) noexcept nogil: + + _fortran_dlag2s(m, n, a, lda, sa, ldsa, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlags2 "BLAS_FUNC(dlags2)"(bint *upper, d *a1, d *a2, d *a3, d *b1, d *b2, d *b3, d *csu, d *snu, d *csv, d *snv, d *csq, d *snq) nogil +cdef void dlags2(bint *upper, d *a1, d *a2, d *a3, d *b1, d *b2, d *b3, d *csu, d *snu, d *csv, d *snv, d *csq, d *snq) noexcept nogil: + + _fortran_dlags2(upper, a1, a2, a3, b1, b2, b3, csu, snu, csv, snv, csq, snq) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlagtf "BLAS_FUNC(dlagtf)"(int *n, d *a, d *lambda_, d *b, d *c, d *tol, d *d, int *in_, int *info) nogil +cdef void dlagtf(int *n, d *a, d *lambda_, d *b, d *c, d *tol, d *d, int *in_, int *info) noexcept nogil: + + _fortran_dlagtf(n, a, lambda_, b, c, tol, d, in_, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlagtm "BLAS_FUNC(dlagtm)"(char *trans, int *n, int *nrhs, d *alpha, d *dl, d *d, d *du, d *x, int *ldx, d *beta, d *b, int *ldb) nogil +cdef void dlagtm(char *trans, int *n, int *nrhs, d *alpha, d *dl, d *d, d *du, d *x, int *ldx, d *beta, d *b, int *ldb) noexcept nogil: + + _fortran_dlagtm(trans, n, nrhs, alpha, dl, d, du, x, ldx, beta, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlagts "BLAS_FUNC(dlagts)"(int *job, int *n, d *a, d *b, d *c, d *d, int *in_, d *y, d *tol, int *info) nogil +cdef void dlagts(int *job, int *n, d *a, d *b, d *c, d *d, int *in_, d *y, d *tol, int *info) noexcept nogil: + + _fortran_dlagts(job, n, a, b, c, d, in_, y, tol, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlagv2 "BLAS_FUNC(dlagv2)"(d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *csl, d *snl, d *csr, d *snr) nogil +cdef void dlagv2(d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *csl, d *snl, d *csr, d *snr) noexcept nogil: + + _fortran_dlagv2(a, lda, b, ldb, alphar, alphai, beta, csl, snl, csr, snr) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlahqr "BLAS_FUNC(dlahqr)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, int *iloz, int *ihiz, d *z, int *ldz, int *info) nogil +cdef void dlahqr(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, int *iloz, int *ihiz, d *z, int *ldz, int *info) noexcept nogil: + + _fortran_dlahqr(wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlahr2 "BLAS_FUNC(dlahr2)"(int *n, int *k, int *nb, d *a, int *lda, d *tau, d *t, int *ldt, d *y, int *ldy) nogil +cdef void dlahr2(int *n, int *k, int *nb, d *a, int *lda, d *tau, d *t, int *ldt, d *y, int *ldy) noexcept nogil: + + _fortran_dlahr2(n, k, nb, a, lda, tau, t, ldt, y, ldy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaic1 "BLAS_FUNC(dlaic1)"(int *job, int *j, d *x, d *sest, d *w, d *gamma, d *sestpr, d *s, d *c) nogil +cdef void dlaic1(int *job, int *j, d *x, d *sest, d *w, d *gamma, d *sestpr, d *s, d *c) noexcept nogil: + + _fortran_dlaic1(job, j, x, sest, w, gamma, sestpr, s, c) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaln2 "BLAS_FUNC(dlaln2)"(bint *ltrans, int *na, int *nw, d *smin, d *ca, d *a, int *lda, d *d1, d *d2, d *b, int *ldb, d *wr, d *wi, d *x, int *ldx, d *scale, d *xnorm, int *info) nogil +cdef void dlaln2(bint *ltrans, int *na, int *nw, d *smin, d *ca, d *a, int *lda, d *d1, d *d2, d *b, int *ldb, d *wr, d *wi, d *x, int *ldx, d *scale, d *xnorm, int *info) noexcept nogil: + + _fortran_dlaln2(ltrans, na, nw, smin, ca, a, lda, d1, d2, b, ldb, wr, wi, x, ldx, scale, xnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlals0 "BLAS_FUNC(dlals0)"(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, d *b, int *ldb, d *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *poles, d *difl, d *difr, d *z, int *k, d *c, d *s, d *work, int *info) nogil +cdef void dlals0(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, d *b, int *ldb, d *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *poles, d *difl, d *difr, d *z, int *k, d *c, d *s, d *work, int *info) noexcept nogil: + + _fortran_dlals0(icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlalsa "BLAS_FUNC(dlalsa)"(int *icompq, int *smlsiz, int *n, int *nrhs, d *b, int *ldb, d *bx, int *ldbx, d *u, int *ldu, d *vt, int *k, d *difl, d *difr, d *z, d *poles, int *givptr, int *givcol, int *ldgcol, int *perm, d *givnum, d *c, d *s, d *work, int *iwork, int *info) nogil +cdef void dlalsa(int *icompq, int *smlsiz, int *n, int *nrhs, d *b, int *ldb, d *bx, int *ldbx, d *u, int *ldu, d *vt, int *k, d *difl, d *difr, d *z, d *poles, int *givptr, int *givcol, int *ldgcol, int *perm, d *givnum, d *c, d *s, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlalsa(icompq, smlsiz, n, nrhs, b, ldb, bx, ldbx, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlalsd "BLAS_FUNC(dlalsd)"(char *uplo, int *smlsiz, int *n, int *nrhs, d *d, d *e, d *b, int *ldb, d *rcond, int *rank, d *work, int *iwork, int *info) nogil +cdef void dlalsd(char *uplo, int *smlsiz, int *n, int *nrhs, d *d, d *e, d *b, int *ldb, d *rcond, int *rank, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlalsd(uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlamch "BLAS_FUNC(dlamch)"(char *cmach) nogil +cdef d dlamch(char *cmach) noexcept nogil: + + return _fortran_dlamch(cmach) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlamrg "BLAS_FUNC(dlamrg)"(int *n1, int *n2, d *a, int *dtrd1, int *dtrd2, int *index_bn) nogil +cdef void dlamrg(int *n1, int *n2, d *a, int *dtrd1, int *dtrd2, int *index_bn) noexcept nogil: + + _fortran_dlamrg(n1, n2, a, dtrd1, dtrd2, index_bn) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_dlaneg "BLAS_FUNC(dlaneg)"(int *n, d *d, d *lld, d *sigma, d *pivmin, int *r) nogil +cdef int dlaneg(int *n, d *d, d *lld, d *sigma, d *pivmin, int *r) noexcept nogil: + + return _fortran_dlaneg(n, d, lld, sigma, pivmin, r) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlangb "BLAS_FUNC(dlangb)"(char *norm, int *n, int *kl, int *ku, d *ab, int *ldab, d *work) nogil +cdef d dlangb(char *norm, int *n, int *kl, int *ku, d *ab, int *ldab, d *work) noexcept nogil: + + return _fortran_dlangb(norm, n, kl, ku, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlange "BLAS_FUNC(dlange)"(char *norm, int *m, int *n, d *a, int *lda, d *work) nogil +cdef d dlange(char *norm, int *m, int *n, d *a, int *lda, d *work) noexcept nogil: + + return _fortran_dlange(norm, m, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlangt "BLAS_FUNC(dlangt)"(char *norm, int *n, d *dl, d *d_, d *du) nogil +cdef d dlangt(char *norm, int *n, d *dl, d *d_, d *du) noexcept nogil: + + return _fortran_dlangt(norm, n, dl, d_, du) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlanhs "BLAS_FUNC(dlanhs)"(char *norm, int *n, d *a, int *lda, d *work) nogil +cdef d dlanhs(char *norm, int *n, d *a, int *lda, d *work) noexcept nogil: + + return _fortran_dlanhs(norm, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlansb "BLAS_FUNC(dlansb)"(char *norm, char *uplo, int *n, int *k, d *ab, int *ldab, d *work) nogil +cdef d dlansb(char *norm, char *uplo, int *n, int *k, d *ab, int *ldab, d *work) noexcept nogil: + + return _fortran_dlansb(norm, uplo, n, k, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlansf "BLAS_FUNC(dlansf)"(char *norm, char *transr, char *uplo, int *n, d *a, d *work) nogil +cdef d dlansf(char *norm, char *transr, char *uplo, int *n, d *a, d *work) noexcept nogil: + + return _fortran_dlansf(norm, transr, uplo, n, a, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlansp "BLAS_FUNC(dlansp)"(char *norm, char *uplo, int *n, d *ap, d *work) nogil +cdef d dlansp(char *norm, char *uplo, int *n, d *ap, d *work) noexcept nogil: + + return _fortran_dlansp(norm, uplo, n, ap, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlanst "BLAS_FUNC(dlanst)"(char *norm, int *n, d *d_, d *e) nogil +cdef d dlanst(char *norm, int *n, d *d_, d *e) noexcept nogil: + + return _fortran_dlanst(norm, n, d_, e) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlansy "BLAS_FUNC(dlansy)"(char *norm, char *uplo, int *n, d *a, int *lda, d *work) nogil +cdef d dlansy(char *norm, char *uplo, int *n, d *a, int *lda, d *work) noexcept nogil: + + return _fortran_dlansy(norm, uplo, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlantb "BLAS_FUNC(dlantb)"(char *norm, char *uplo, char *diag, int *n, int *k, d *ab, int *ldab, d *work) nogil +cdef d dlantb(char *norm, char *uplo, char *diag, int *n, int *k, d *ab, int *ldab, d *work) noexcept nogil: + + return _fortran_dlantb(norm, uplo, diag, n, k, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlantp "BLAS_FUNC(dlantp)"(char *norm, char *uplo, char *diag, int *n, d *ap, d *work) nogil +cdef d dlantp(char *norm, char *uplo, char *diag, int *n, d *ap, d *work) noexcept nogil: + + return _fortran_dlantp(norm, uplo, diag, n, ap, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlantr "BLAS_FUNC(dlantr)"(char *norm, char *uplo, char *diag, int *m, int *n, d *a, int *lda, d *work) nogil +cdef d dlantr(char *norm, char *uplo, char *diag, int *m, int *n, d *a, int *lda, d *work) noexcept nogil: + + return _fortran_dlantr(norm, uplo, diag, m, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlanv2 "BLAS_FUNC(dlanv2)"(d *a, d *b, d *c, d *d, d *rt1r, d *rt1i, d *rt2r, d *rt2i, d *cs, d *sn) nogil +cdef void dlanv2(d *a, d *b, d *c, d *d, d *rt1r, d *rt1i, d *rt2r, d *rt2i, d *cs, d *sn) noexcept nogil: + + _fortran_dlanv2(a, b, c, d, rt1r, rt1i, rt2r, rt2i, cs, sn) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlapll "BLAS_FUNC(dlapll)"(int *n, d *x, int *incx, d *y, int *incy, d *ssmin) nogil +cdef void dlapll(int *n, d *x, int *incx, d *y, int *incy, d *ssmin) noexcept nogil: + + _fortran_dlapll(n, x, incx, y, incy, ssmin) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlapmr "BLAS_FUNC(dlapmr)"(bint *forwrd, int *m, int *n, d *x, int *ldx, int *k) nogil +cdef void dlapmr(bint *forwrd, int *m, int *n, d *x, int *ldx, int *k) noexcept nogil: + + _fortran_dlapmr(forwrd, m, n, x, ldx, k) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlapmt "BLAS_FUNC(dlapmt)"(bint *forwrd, int *m, int *n, d *x, int *ldx, int *k) nogil +cdef void dlapmt(bint *forwrd, int *m, int *n, d *x, int *ldx, int *k) noexcept nogil: + + _fortran_dlapmt(forwrd, m, n, x, ldx, k) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlapy2 "BLAS_FUNC(dlapy2)"(d *x, d *y) nogil +cdef d dlapy2(d *x, d *y) noexcept nogil: + + return _fortran_dlapy2(x, y) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dlapy3 "BLAS_FUNC(dlapy3)"(d *x, d *y, d *z) nogil +cdef d dlapy3(d *x, d *y, d *z) noexcept nogil: + + return _fortran_dlapy3(x, y, z) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqgb "BLAS_FUNC(dlaqgb)"(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) nogil +cdef void dlaqgb(int *m, int *n, int *kl, int *ku, d *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) noexcept nogil: + + _fortran_dlaqgb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqge "BLAS_FUNC(dlaqge)"(int *m, int *n, d *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) nogil +cdef void dlaqge(int *m, int *n, d *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) noexcept nogil: + + _fortran_dlaqge(m, n, a, lda, r, c, rowcnd, colcnd, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqp2 "BLAS_FUNC(dlaqp2)"(int *m, int *n, int *offset, d *a, int *lda, int *jpvt, d *tau, d *vn1, d *vn2, d *work) nogil +cdef void dlaqp2(int *m, int *n, int *offset, d *a, int *lda, int *jpvt, d *tau, d *vn1, d *vn2, d *work) noexcept nogil: + + _fortran_dlaqp2(m, n, offset, a, lda, jpvt, tau, vn1, vn2, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqps "BLAS_FUNC(dlaqps)"(int *m, int *n, int *offset, int *nb, int *kb, d *a, int *lda, int *jpvt, d *tau, d *vn1, d *vn2, d *auxv, d *f, int *ldf) nogil +cdef void dlaqps(int *m, int *n, int *offset, int *nb, int *kb, d *a, int *lda, int *jpvt, d *tau, d *vn1, d *vn2, d *auxv, d *f, int *ldf) noexcept nogil: + + _fortran_dlaqps(m, n, offset, nb, kb, a, lda, jpvt, tau, vn1, vn2, auxv, f, ldf) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqr0 "BLAS_FUNC(dlaqr0)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, int *iloz, int *ihiz, d *z, int *ldz, d *work, int *lwork, int *info) nogil +cdef void dlaqr0(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, int *iloz, int *ihiz, d *z, int *ldz, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dlaqr0(wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqr1 "BLAS_FUNC(dlaqr1)"(int *n, d *h, int *ldh, d *sr1, d *si1, d *sr2, d *si2, d *v) nogil +cdef void dlaqr1(int *n, d *h, int *ldh, d *sr1, d *si1, d *sr2, d *si2, d *v) noexcept nogil: + + _fortran_dlaqr1(n, h, ldh, sr1, si1, sr2, si2, v) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqr2 "BLAS_FUNC(dlaqr2)"(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, d *h, int *ldh, int *iloz, int *ihiz, d *z, int *ldz, int *ns, int *nd, d *sr, d *si, d *v, int *ldv, int *nh, d *t, int *ldt, int *nv, d *wv, int *ldwv, d *work, int *lwork) nogil +cdef void dlaqr2(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, d *h, int *ldh, int *iloz, int *ihiz, d *z, int *ldz, int *ns, int *nd, d *sr, d *si, d *v, int *ldv, int *nh, d *t, int *ldt, int *nv, d *wv, int *ldwv, d *work, int *lwork) noexcept nogil: + + _fortran_dlaqr2(wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqr3 "BLAS_FUNC(dlaqr3)"(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, d *h, int *ldh, int *iloz, int *ihiz, d *z, int *ldz, int *ns, int *nd, d *sr, d *si, d *v, int *ldv, int *nh, d *t, int *ldt, int *nv, d *wv, int *ldwv, d *work, int *lwork) nogil +cdef void dlaqr3(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, d *h, int *ldh, int *iloz, int *ihiz, d *z, int *ldz, int *ns, int *nd, d *sr, d *si, d *v, int *ldv, int *nh, d *t, int *ldt, int *nv, d *wv, int *ldwv, d *work, int *lwork) noexcept nogil: + + _fortran_dlaqr3(wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqr4 "BLAS_FUNC(dlaqr4)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, int *iloz, int *ihiz, d *z, int *ldz, d *work, int *lwork, int *info) nogil +cdef void dlaqr4(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, d *h, int *ldh, d *wr, d *wi, int *iloz, int *ihiz, d *z, int *ldz, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dlaqr4(wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqr5 "BLAS_FUNC(dlaqr5)"(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, d *sr, d *si, d *h, int *ldh, int *iloz, int *ihiz, d *z, int *ldz, d *v, int *ldv, d *u, int *ldu, int *nv, d *wv, int *ldwv, int *nh, d *wh, int *ldwh) nogil +cdef void dlaqr5(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, d *sr, d *si, d *h, int *ldh, int *iloz, int *ihiz, d *z, int *ldz, d *v, int *ldv, d *u, int *ldu, int *nv, d *wv, int *ldwv, int *nh, d *wh, int *ldwh) noexcept nogil: + + _fortran_dlaqr5(wantt, wantz, kacc22, n, ktop, kbot, nshfts, sr, si, h, ldh, iloz, ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqsb "BLAS_FUNC(dlaqsb)"(char *uplo, int *n, int *kd, d *ab, int *ldab, d *s, d *scond, d *amax, char *equed) nogil +cdef void dlaqsb(char *uplo, int *n, int *kd, d *ab, int *ldab, d *s, d *scond, d *amax, char *equed) noexcept nogil: + + _fortran_dlaqsb(uplo, n, kd, ab, ldab, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqsp "BLAS_FUNC(dlaqsp)"(char *uplo, int *n, d *ap, d *s, d *scond, d *amax, char *equed) nogil +cdef void dlaqsp(char *uplo, int *n, d *ap, d *s, d *scond, d *amax, char *equed) noexcept nogil: + + _fortran_dlaqsp(uplo, n, ap, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqsy "BLAS_FUNC(dlaqsy)"(char *uplo, int *n, d *a, int *lda, d *s, d *scond, d *amax, char *equed) nogil +cdef void dlaqsy(char *uplo, int *n, d *a, int *lda, d *s, d *scond, d *amax, char *equed) noexcept nogil: + + _fortran_dlaqsy(uplo, n, a, lda, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaqtr "BLAS_FUNC(dlaqtr)"(bint *ltran, bint *lreal, int *n, d *t, int *ldt, d *b, d *w, d *scale, d *x, d *work, int *info) nogil +cdef void dlaqtr(bint *ltran, bint *lreal, int *n, d *t, int *ldt, d *b, d *w, d *scale, d *x, d *work, int *info) noexcept nogil: + + _fortran_dlaqtr(ltran, lreal, n, t, ldt, b, w, scale, x, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlar1v "BLAS_FUNC(dlar1v)"(int *n, int *b1, int *bn, d *lambda_, d *d, d *l, d *ld, d *lld, d *pivmin, d *gaptol, d *z, bint *wantnc, int *negcnt, d *ztz, d *mingma, int *r, int *isuppz, d *nrminv, d *resid, d *rqcorr, d *work) nogil +cdef void dlar1v(int *n, int *b1, int *bn, d *lambda_, d *d, d *l, d *ld, d *lld, d *pivmin, d *gaptol, d *z, bint *wantnc, int *negcnt, d *ztz, d *mingma, int *r, int *isuppz, d *nrminv, d *resid, d *rqcorr, d *work) noexcept nogil: + + _fortran_dlar1v(n, b1, bn, lambda_, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlar2v "BLAS_FUNC(dlar2v)"(int *n, d *x, d *y, d *z, int *incx, d *c, d *s, int *incc) nogil +cdef void dlar2v(int *n, d *x, d *y, d *z, int *incx, d *c, d *s, int *incc) noexcept nogil: + + _fortran_dlar2v(n, x, y, z, incx, c, s, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarf "BLAS_FUNC(dlarf)"(char *side, int *m, int *n, d *v, int *incv, d *tau, d *c, int *ldc, d *work) nogil +cdef void dlarf(char *side, int *m, int *n, d *v, int *incv, d *tau, d *c, int *ldc, d *work) noexcept nogil: + + _fortran_dlarf(side, m, n, v, incv, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarfb "BLAS_FUNC(dlarfb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, d *v, int *ldv, d *t, int *ldt, d *c, int *ldc, d *work, int *ldwork) nogil +cdef void dlarfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, d *v, int *ldv, d *t, int *ldt, d *c, int *ldc, d *work, int *ldwork) noexcept nogil: + + _fortran_dlarfb(side, trans, direct, storev, m, n, k, v, ldv, t, ldt, c, ldc, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarfg "BLAS_FUNC(dlarfg)"(int *n, d *alpha, d *x, int *incx, d *tau) nogil +cdef void dlarfg(int *n, d *alpha, d *x, int *incx, d *tau) noexcept nogil: + + _fortran_dlarfg(n, alpha, x, incx, tau) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarfgp "BLAS_FUNC(dlarfgp)"(int *n, d *alpha, d *x, int *incx, d *tau) nogil +cdef void dlarfgp(int *n, d *alpha, d *x, int *incx, d *tau) noexcept nogil: + + _fortran_dlarfgp(n, alpha, x, incx, tau) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarft "BLAS_FUNC(dlarft)"(char *direct, char *storev, int *n, int *k, d *v, int *ldv, d *tau, d *t, int *ldt) nogil +cdef void dlarft(char *direct, char *storev, int *n, int *k, d *v, int *ldv, d *tau, d *t, int *ldt) noexcept nogil: + + _fortran_dlarft(direct, storev, n, k, v, ldv, tau, t, ldt) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarfx "BLAS_FUNC(dlarfx)"(char *side, int *m, int *n, d *v, d *tau, d *c, int *ldc, d *work) nogil +cdef void dlarfx(char *side, int *m, int *n, d *v, d *tau, d *c, int *ldc, d *work) noexcept nogil: + + _fortran_dlarfx(side, m, n, v, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlargv "BLAS_FUNC(dlargv)"(int *n, d *x, int *incx, d *y, int *incy, d *c, int *incc) nogil +cdef void dlargv(int *n, d *x, int *incx, d *y, int *incy, d *c, int *incc) noexcept nogil: + + _fortran_dlargv(n, x, incx, y, incy, c, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarnv "BLAS_FUNC(dlarnv)"(int *idist, int *iseed, int *n, d *x) nogil +cdef void dlarnv(int *idist, int *iseed, int *n, d *x) noexcept nogil: + + _fortran_dlarnv(idist, iseed, n, x) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarra "BLAS_FUNC(dlarra)"(int *n, d *d, d *e, d *e2, d *spltol, d *tnrm, int *nsplit, int *isplit, int *info) nogil +cdef void dlarra(int *n, d *d, d *e, d *e2, d *spltol, d *tnrm, int *nsplit, int *isplit, int *info) noexcept nogil: + + _fortran_dlarra(n, d, e, e2, spltol, tnrm, nsplit, isplit, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarrb "BLAS_FUNC(dlarrb)"(int *n, d *d, d *lld, int *ifirst, int *ilast, d *rtol1, d *rtol2, int *offset, d *w, d *wgap, d *werr, d *work, int *iwork, d *pivmin, d *spdiam, int *twist, int *info) nogil +cdef void dlarrb(int *n, d *d, d *lld, int *ifirst, int *ilast, d *rtol1, d *rtol2, int *offset, d *w, d *wgap, d *werr, d *work, int *iwork, d *pivmin, d *spdiam, int *twist, int *info) noexcept nogil: + + _fortran_dlarrb(n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, spdiam, twist, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarrc "BLAS_FUNC(dlarrc)"(char *jobt, int *n, d *vl, d *vu, d *d, d *e, d *pivmin, int *eigcnt, int *lcnt, int *rcnt, int *info) nogil +cdef void dlarrc(char *jobt, int *n, d *vl, d *vu, d *d, d *e, d *pivmin, int *eigcnt, int *lcnt, int *rcnt, int *info) noexcept nogil: + + _fortran_dlarrc(jobt, n, vl, vu, d, e, pivmin, eigcnt, lcnt, rcnt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarrd "BLAS_FUNC(dlarrd)"(char *range, char *order, int *n, d *vl, d *vu, int *il, int *iu, d *gers, d *reltol, d *d, d *e, d *e2, d *pivmin, int *nsplit, int *isplit, int *m, d *w, d *werr, d *wl, d *wu, int *iblock, int *indexw, d *work, int *iwork, int *info) nogil +cdef void dlarrd(char *range, char *order, int *n, d *vl, d *vu, int *il, int *iu, d *gers, d *reltol, d *d, d *e, d *e2, d *pivmin, int *nsplit, int *isplit, int *m, d *w, d *werr, d *wl, d *wu, int *iblock, int *indexw, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlarrd(range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarre "BLAS_FUNC(dlarre)"(char *range, int *n, d *vl, d *vu, int *il, int *iu, d *d, d *e, d *e2, d *rtol1, d *rtol2, d *spltol, int *nsplit, int *isplit, int *m, d *w, d *werr, d *wgap, int *iblock, int *indexw, d *gers, d *pivmin, d *work, int *iwork, int *info) nogil +cdef void dlarre(char *range, int *n, d *vl, d *vu, int *il, int *iu, d *d, d *e, d *e2, d *rtol1, d *rtol2, d *spltol, int *nsplit, int *isplit, int *m, d *w, d *werr, d *wgap, int *iblock, int *indexw, d *gers, d *pivmin, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlarre(range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarrf "BLAS_FUNC(dlarrf)"(int *n, d *d, d *l, d *ld, int *clstrt, int *clend, d *w, d *wgap, d *werr, d *spdiam, d *clgapl, d *clgapr, d *pivmin, d *sigma, d *dplus, d *lplus, d *work, int *info) nogil +cdef void dlarrf(int *n, d *d, d *l, d *ld, int *clstrt, int *clend, d *w, d *wgap, d *werr, d *spdiam, d *clgapl, d *clgapr, d *pivmin, d *sigma, d *dplus, d *lplus, d *work, int *info) noexcept nogil: + + _fortran_dlarrf(n, d, l, ld, clstrt, clend, w, wgap, werr, spdiam, clgapl, clgapr, pivmin, sigma, dplus, lplus, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarrj "BLAS_FUNC(dlarrj)"(int *n, d *d, d *e2, int *ifirst, int *ilast, d *rtol, int *offset, d *w, d *werr, d *work, int *iwork, d *pivmin, d *spdiam, int *info) nogil +cdef void dlarrj(int *n, d *d, d *e2, int *ifirst, int *ilast, d *rtol, int *offset, d *w, d *werr, d *work, int *iwork, d *pivmin, d *spdiam, int *info) noexcept nogil: + + _fortran_dlarrj(n, d, e2, ifirst, ilast, rtol, offset, w, werr, work, iwork, pivmin, spdiam, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarrk "BLAS_FUNC(dlarrk)"(int *n, int *iw, d *gl, d *gu, d *d, d *e2, d *pivmin, d *reltol, d *w, d *werr, int *info) nogil +cdef void dlarrk(int *n, int *iw, d *gl, d *gu, d *d, d *e2, d *pivmin, d *reltol, d *w, d *werr, int *info) noexcept nogil: + + _fortran_dlarrk(n, iw, gl, gu, d, e2, pivmin, reltol, w, werr, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarrr "BLAS_FUNC(dlarrr)"(int *n, d *d, d *e, int *info) nogil +cdef void dlarrr(int *n, d *d, d *e, int *info) noexcept nogil: + + _fortran_dlarrr(n, d, e, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarrv "BLAS_FUNC(dlarrv)"(int *n, d *vl, d *vu, d *d, d *l, d *pivmin, int *isplit, int *m, int *dol, int *dou, d *minrgp, d *rtol1, d *rtol2, d *w, d *werr, d *wgap, int *iblock, int *indexw, d *gers, d *z, int *ldz, int *isuppz, d *work, int *iwork, int *info) nogil +cdef void dlarrv(int *n, d *vl, d *vu, d *d, d *l, d *pivmin, int *isplit, int *m, int *dol, int *dou, d *minrgp, d *rtol1, d *rtol2, d *w, d *werr, d *wgap, int *iblock, int *indexw, d *gers, d *z, int *ldz, int *isuppz, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlarrv(n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlartg "BLAS_FUNC(dlartg)"(d *f, d *g, d *cs, d *sn, d *r) nogil +cdef void dlartg(d *f, d *g, d *cs, d *sn, d *r) noexcept nogil: + + _fortran_dlartg(f, g, cs, sn, r) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlartgp "BLAS_FUNC(dlartgp)"(d *f, d *g, d *cs, d *sn, d *r) nogil +cdef void dlartgp(d *f, d *g, d *cs, d *sn, d *r) noexcept nogil: + + _fortran_dlartgp(f, g, cs, sn, r) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlartgs "BLAS_FUNC(dlartgs)"(d *x, d *y, d *sigma, d *cs, d *sn) nogil +cdef void dlartgs(d *x, d *y, d *sigma, d *cs, d *sn) noexcept nogil: + + _fortran_dlartgs(x, y, sigma, cs, sn) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlartv "BLAS_FUNC(dlartv)"(int *n, d *x, int *incx, d *y, int *incy, d *c, d *s, int *incc) nogil +cdef void dlartv(int *n, d *x, int *incx, d *y, int *incy, d *c, d *s, int *incc) noexcept nogil: + + _fortran_dlartv(n, x, incx, y, incy, c, s, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaruv "BLAS_FUNC(dlaruv)"(int *iseed, int *n, d *x) nogil +cdef void dlaruv(int *iseed, int *n, d *x) noexcept nogil: + + _fortran_dlaruv(iseed, n, x) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarz "BLAS_FUNC(dlarz)"(char *side, int *m, int *n, int *l, d *v, int *incv, d *tau, d *c, int *ldc, d *work) nogil +cdef void dlarz(char *side, int *m, int *n, int *l, d *v, int *incv, d *tau, d *c, int *ldc, d *work) noexcept nogil: + + _fortran_dlarz(side, m, n, l, v, incv, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarzb "BLAS_FUNC(dlarzb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, d *v, int *ldv, d *t, int *ldt, d *c, int *ldc, d *work, int *ldwork) nogil +cdef void dlarzb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, d *v, int *ldv, d *t, int *ldt, d *c, int *ldc, d *work, int *ldwork) noexcept nogil: + + _fortran_dlarzb(side, trans, direct, storev, m, n, k, l, v, ldv, t, ldt, c, ldc, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlarzt "BLAS_FUNC(dlarzt)"(char *direct, char *storev, int *n, int *k, d *v, int *ldv, d *tau, d *t, int *ldt) nogil +cdef void dlarzt(char *direct, char *storev, int *n, int *k, d *v, int *ldv, d *tau, d *t, int *ldt) noexcept nogil: + + _fortran_dlarzt(direct, storev, n, k, v, ldv, tau, t, ldt) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlas2 "BLAS_FUNC(dlas2)"(d *f, d *g, d *h, d *ssmin, d *ssmax) nogil +cdef void dlas2(d *f, d *g, d *h, d *ssmin, d *ssmax) noexcept nogil: + + _fortran_dlas2(f, g, h, ssmin, ssmax) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlascl "BLAS_FUNC(dlascl)"(char *type_bn, int *kl, int *ku, d *cfrom, d *cto, int *m, int *n, d *a, int *lda, int *info) nogil +cdef void dlascl(char *type_bn, int *kl, int *ku, d *cfrom, d *cto, int *m, int *n, d *a, int *lda, int *info) noexcept nogil: + + _fortran_dlascl(type_bn, kl, ku, cfrom, cto, m, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasd0 "BLAS_FUNC(dlasd0)"(int *n, int *sqre, d *d, d *e, d *u, int *ldu, d *vt, int *ldvt, int *smlsiz, int *iwork, d *work, int *info) nogil +cdef void dlasd0(int *n, int *sqre, d *d, d *e, d *u, int *ldu, d *vt, int *ldvt, int *smlsiz, int *iwork, d *work, int *info) noexcept nogil: + + _fortran_dlasd0(n, sqre, d, e, u, ldu, vt, ldvt, smlsiz, iwork, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasd1 "BLAS_FUNC(dlasd1)"(int *nl, int *nr, int *sqre, d *d, d *alpha, d *beta, d *u, int *ldu, d *vt, int *ldvt, int *idxq, int *iwork, d *work, int *info) nogil +cdef void dlasd1(int *nl, int *nr, int *sqre, d *d, d *alpha, d *beta, d *u, int *ldu, d *vt, int *ldvt, int *idxq, int *iwork, d *work, int *info) noexcept nogil: + + _fortran_dlasd1(nl, nr, sqre, d, alpha, beta, u, ldu, vt, ldvt, idxq, iwork, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasd2 "BLAS_FUNC(dlasd2)"(int *nl, int *nr, int *sqre, int *k, d *d, d *z, d *alpha, d *beta, d *u, int *ldu, d *vt, int *ldvt, d *dsigma, d *u2, int *ldu2, d *vt2, int *ldvt2, int *idxp, int *idx, int *idxc, int *idxq, int *coltyp, int *info) nogil +cdef void dlasd2(int *nl, int *nr, int *sqre, int *k, d *d, d *z, d *alpha, d *beta, d *u, int *ldu, d *vt, int *ldvt, d *dsigma, d *u2, int *ldu2, d *vt2, int *ldvt2, int *idxp, int *idx, int *idxc, int *idxq, int *coltyp, int *info) noexcept nogil: + + _fortran_dlasd2(nl, nr, sqre, k, d, z, alpha, beta, u, ldu, vt, ldvt, dsigma, u2, ldu2, vt2, ldvt2, idxp, idx, idxc, idxq, coltyp, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasd3 "BLAS_FUNC(dlasd3)"(int *nl, int *nr, int *sqre, int *k, d *d, d *q, int *ldq, d *dsigma, d *u, int *ldu, d *u2, int *ldu2, d *vt, int *ldvt, d *vt2, int *ldvt2, int *idxc, int *ctot, d *z, int *info) nogil +cdef void dlasd3(int *nl, int *nr, int *sqre, int *k, d *d, d *q, int *ldq, d *dsigma, d *u, int *ldu, d *u2, int *ldu2, d *vt, int *ldvt, d *vt2, int *ldvt2, int *idxc, int *ctot, d *z, int *info) noexcept nogil: + + _fortran_dlasd3(nl, nr, sqre, k, d, q, ldq, dsigma, u, ldu, u2, ldu2, vt, ldvt, vt2, ldvt2, idxc, ctot, z, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasd4 "BLAS_FUNC(dlasd4)"(int *n, int *i, d *d, d *z, d *delta, d *rho, d *sigma, d *work, int *info) nogil +cdef void dlasd4(int *n, int *i, d *d, d *z, d *delta, d *rho, d *sigma, d *work, int *info) noexcept nogil: + + _fortran_dlasd4(n, i, d, z, delta, rho, sigma, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasd5 "BLAS_FUNC(dlasd5)"(int *i, d *d, d *z, d *delta, d *rho, d *dsigma, d *work) nogil +cdef void dlasd5(int *i, d *d, d *z, d *delta, d *rho, d *dsigma, d *work) noexcept nogil: + + _fortran_dlasd5(i, d, z, delta, rho, dsigma, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasd6 "BLAS_FUNC(dlasd6)"(int *icompq, int *nl, int *nr, int *sqre, d *d, d *vf, d *vl, d *alpha, d *beta, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *poles, d *difl, d *difr, d *z, int *k, d *c, d *s, d *work, int *iwork, int *info) nogil +cdef void dlasd6(int *icompq, int *nl, int *nr, int *sqre, d *d, d *vf, d *vl, d *alpha, d *beta, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *poles, d *difl, d *difr, d *z, int *k, d *c, d *s, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlasd6(icompq, nl, nr, sqre, d, vf, vl, alpha, beta, idxq, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasd7 "BLAS_FUNC(dlasd7)"(int *icompq, int *nl, int *nr, int *sqre, int *k, d *d, d *z, d *zw, d *vf, d *vfw, d *vl, d *vlw, d *alpha, d *beta, d *dsigma, int *idx, int *idxp, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *c, d *s, int *info) nogil +cdef void dlasd7(int *icompq, int *nl, int *nr, int *sqre, int *k, d *d, d *z, d *zw, d *vf, d *vfw, d *vl, d *vlw, d *alpha, d *beta, d *dsigma, int *idx, int *idxp, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *c, d *s, int *info) noexcept nogil: + + _fortran_dlasd7(icompq, nl, nr, sqre, k, d, z, zw, vf, vfw, vl, vlw, alpha, beta, dsigma, idx, idxp, idxq, perm, givptr, givcol, ldgcol, givnum, ldgnum, c, s, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasd8 "BLAS_FUNC(dlasd8)"(int *icompq, int *k, d *d, d *z, d *vf, d *vl, d *difl, d *difr, int *lddifr, d *dsigma, d *work, int *info) nogil +cdef void dlasd8(int *icompq, int *k, d *d, d *z, d *vf, d *vl, d *difl, d *difr, int *lddifr, d *dsigma, d *work, int *info) noexcept nogil: + + _fortran_dlasd8(icompq, k, d, z, vf, vl, difl, difr, lddifr, dsigma, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasda "BLAS_FUNC(dlasda)"(int *icompq, int *smlsiz, int *n, int *sqre, d *d, d *e, d *u, int *ldu, d *vt, int *k, d *difl, d *difr, d *z, d *poles, int *givptr, int *givcol, int *ldgcol, int *perm, d *givnum, d *c, d *s, d *work, int *iwork, int *info) nogil +cdef void dlasda(int *icompq, int *smlsiz, int *n, int *sqre, d *d, d *e, d *u, int *ldu, d *vt, int *k, d *difl, d *difr, d *z, d *poles, int *givptr, int *givcol, int *ldgcol, int *perm, d *givnum, d *c, d *s, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dlasda(icompq, smlsiz, n, sqre, d, e, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasdq "BLAS_FUNC(dlasdq)"(char *uplo, int *sqre, int *n, int *ncvt, int *nru, int *ncc, d *d, d *e, d *vt, int *ldvt, d *u, int *ldu, d *c, int *ldc, d *work, int *info) nogil +cdef void dlasdq(char *uplo, int *sqre, int *n, int *ncvt, int *nru, int *ncc, d *d, d *e, d *vt, int *ldvt, d *u, int *ldu, d *c, int *ldc, d *work, int *info) noexcept nogil: + + _fortran_dlasdq(uplo, sqre, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasdt "BLAS_FUNC(dlasdt)"(int *n, int *lvl, int *nd, int *inode, int *ndiml, int *ndimr, int *msub) nogil +cdef void dlasdt(int *n, int *lvl, int *nd, int *inode, int *ndiml, int *ndimr, int *msub) noexcept nogil: + + _fortran_dlasdt(n, lvl, nd, inode, ndiml, ndimr, msub) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaset "BLAS_FUNC(dlaset)"(char *uplo, int *m, int *n, d *alpha, d *beta, d *a, int *lda) nogil +cdef void dlaset(char *uplo, int *m, int *n, d *alpha, d *beta, d *a, int *lda) noexcept nogil: + + _fortran_dlaset(uplo, m, n, alpha, beta, a, lda) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasq1 "BLAS_FUNC(dlasq1)"(int *n, d *d, d *e, d *work, int *info) nogil +cdef void dlasq1(int *n, d *d, d *e, d *work, int *info) noexcept nogil: + + _fortran_dlasq1(n, d, e, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasq2 "BLAS_FUNC(dlasq2)"(int *n, d *z, int *info) nogil +cdef void dlasq2(int *n, d *z, int *info) noexcept nogil: + + _fortran_dlasq2(n, z, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasq3 "BLAS_FUNC(dlasq3)"(int *i0, int *n0, d *z, int *pp, d *dmin, d *sigma, d *desig, d *qmax, int *nfail, int *iter, int *ndiv, bint *ieee, int *ttype, d *dmin1, d *dmin2, d *dn, d *dn1, d *dn2, d *g, d *tau) nogil +cdef void dlasq3(int *i0, int *n0, d *z, int *pp, d *dmin, d *sigma, d *desig, d *qmax, int *nfail, int *iter, int *ndiv, bint *ieee, int *ttype, d *dmin1, d *dmin2, d *dn, d *dn1, d *dn2, d *g, d *tau) noexcept nogil: + + _fortran_dlasq3(i0, n0, z, pp, dmin, sigma, desig, qmax, nfail, iter, ndiv, ieee, ttype, dmin1, dmin2, dn, dn1, dn2, g, tau) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasq4 "BLAS_FUNC(dlasq4)"(int *i0, int *n0, d *z, int *pp, int *n0in, d *dmin, d *dmin1, d *dmin2, d *dn, d *dn1, d *dn2, d *tau, int *ttype, d *g) nogil +cdef void dlasq4(int *i0, int *n0, d *z, int *pp, int *n0in, d *dmin, d *dmin1, d *dmin2, d *dn, d *dn1, d *dn2, d *tau, int *ttype, d *g) noexcept nogil: + + _fortran_dlasq4(i0, n0, z, pp, n0in, dmin, dmin1, dmin2, dn, dn1, dn2, tau, ttype, g) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasq6 "BLAS_FUNC(dlasq6)"(int *i0, int *n0, d *z, int *pp, d *dmin, d *dmin1, d *dmin2, d *dn, d *dnm1, d *dnm2) nogil +cdef void dlasq6(int *i0, int *n0, d *z, int *pp, d *dmin, d *dmin1, d *dmin2, d *dn, d *dnm1, d *dnm2) noexcept nogil: + + _fortran_dlasq6(i0, n0, z, pp, dmin, dmin1, dmin2, dn, dnm1, dnm2) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasr "BLAS_FUNC(dlasr)"(char *side, char *pivot, char *direct, int *m, int *n, d *c, d *s, d *a, int *lda) nogil +cdef void dlasr(char *side, char *pivot, char *direct, int *m, int *n, d *c, d *s, d *a, int *lda) noexcept nogil: + + _fortran_dlasr(side, pivot, direct, m, n, c, s, a, lda) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasrt "BLAS_FUNC(dlasrt)"(char *id, int *n, d *d, int *info) nogil +cdef void dlasrt(char *id, int *n, d *d, int *info) noexcept nogil: + + _fortran_dlasrt(id, n, d, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlassq "BLAS_FUNC(dlassq)"(int *n, d *x, int *incx, d *scale, d *sumsq) nogil +cdef void dlassq(int *n, d *x, int *incx, d *scale, d *sumsq) noexcept nogil: + + _fortran_dlassq(n, x, incx, scale, sumsq) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasv2 "BLAS_FUNC(dlasv2)"(d *f, d *g, d *h, d *ssmin, d *ssmax, d *snr, d *csr, d *snl, d *csl) nogil +cdef void dlasv2(d *f, d *g, d *h, d *ssmin, d *ssmax, d *snr, d *csr, d *snl, d *csl) noexcept nogil: + + _fortran_dlasv2(f, g, h, ssmin, ssmax, snr, csr, snl, csl) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlaswp "BLAS_FUNC(dlaswp)"(int *n, d *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) nogil +cdef void dlaswp(int *n, d *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) noexcept nogil: + + _fortran_dlaswp(n, a, lda, k1, k2, ipiv, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasy2 "BLAS_FUNC(dlasy2)"(bint *ltranl, bint *ltranr, int *isgn, int *n1, int *n2, d *tl, int *ldtl, d *tr, int *ldtr, d *b, int *ldb, d *scale, d *x, int *ldx, d *xnorm, int *info) nogil +cdef void dlasy2(bint *ltranl, bint *ltranr, int *isgn, int *n1, int *n2, d *tl, int *ldtl, d *tr, int *ldtr, d *b, int *ldb, d *scale, d *x, int *ldx, d *xnorm, int *info) noexcept nogil: + + _fortran_dlasy2(ltranl, ltranr, isgn, n1, n2, tl, ldtl, tr, ldtr, b, ldb, scale, x, ldx, xnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlasyf "BLAS_FUNC(dlasyf)"(char *uplo, int *n, int *nb, int *kb, d *a, int *lda, int *ipiv, d *w, int *ldw, int *info) nogil +cdef void dlasyf(char *uplo, int *n, int *nb, int *kb, d *a, int *lda, int *ipiv, d *w, int *ldw, int *info) noexcept nogil: + + _fortran_dlasyf(uplo, n, nb, kb, a, lda, ipiv, w, ldw, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlat2s "BLAS_FUNC(dlat2s)"(char *uplo, int *n, d *a, int *lda, s *sa, int *ldsa, int *info) nogil +cdef void dlat2s(char *uplo, int *n, d *a, int *lda, s *sa, int *ldsa, int *info) noexcept nogil: + + _fortran_dlat2s(uplo, n, a, lda, sa, ldsa, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlatbs "BLAS_FUNC(dlatbs)"(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, d *ab, int *ldab, d *x, d *scale, d *cnorm, int *info) nogil +cdef void dlatbs(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, d *ab, int *ldab, d *x, d *scale, d *cnorm, int *info) noexcept nogil: + + _fortran_dlatbs(uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlatdf "BLAS_FUNC(dlatdf)"(int *ijob, int *n, d *z, int *ldz, d *rhs, d *rdsum, d *rdscal, int *ipiv, int *jpiv) nogil +cdef void dlatdf(int *ijob, int *n, d *z, int *ldz, d *rhs, d *rdsum, d *rdscal, int *ipiv, int *jpiv) noexcept nogil: + + _fortran_dlatdf(ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlatps "BLAS_FUNC(dlatps)"(char *uplo, char *trans, char *diag, char *normin, int *n, d *ap, d *x, d *scale, d *cnorm, int *info) nogil +cdef void dlatps(char *uplo, char *trans, char *diag, char *normin, int *n, d *ap, d *x, d *scale, d *cnorm, int *info) noexcept nogil: + + _fortran_dlatps(uplo, trans, diag, normin, n, ap, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlatrd "BLAS_FUNC(dlatrd)"(char *uplo, int *n, int *nb, d *a, int *lda, d *e, d *tau, d *w, int *ldw) nogil +cdef void dlatrd(char *uplo, int *n, int *nb, d *a, int *lda, d *e, d *tau, d *w, int *ldw) noexcept nogil: + + _fortran_dlatrd(uplo, n, nb, a, lda, e, tau, w, ldw) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlatrs "BLAS_FUNC(dlatrs)"(char *uplo, char *trans, char *diag, char *normin, int *n, d *a, int *lda, d *x, d *scale, d *cnorm, int *info) nogil +cdef void dlatrs(char *uplo, char *trans, char *diag, char *normin, int *n, d *a, int *lda, d *x, d *scale, d *cnorm, int *info) noexcept nogil: + + _fortran_dlatrs(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlatrz "BLAS_FUNC(dlatrz)"(int *m, int *n, int *l, d *a, int *lda, d *tau, d *work) nogil +cdef void dlatrz(int *m, int *n, int *l, d *a, int *lda, d *tau, d *work) noexcept nogil: + + _fortran_dlatrz(m, n, l, a, lda, tau, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlauu2 "BLAS_FUNC(dlauu2)"(char *uplo, int *n, d *a, int *lda, int *info) nogil +cdef void dlauu2(char *uplo, int *n, d *a, int *lda, int *info) noexcept nogil: + + _fortran_dlauu2(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dlauum "BLAS_FUNC(dlauum)"(char *uplo, int *n, d *a, int *lda, int *info) nogil +cdef void dlauum(char *uplo, int *n, d *a, int *lda, int *info) noexcept nogil: + + _fortran_dlauum(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dopgtr "BLAS_FUNC(dopgtr)"(char *uplo, int *n, d *ap, d *tau, d *q, int *ldq, d *work, int *info) nogil +cdef void dopgtr(char *uplo, int *n, d *ap, d *tau, d *q, int *ldq, d *work, int *info) noexcept nogil: + + _fortran_dopgtr(uplo, n, ap, tau, q, ldq, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dopmtr "BLAS_FUNC(dopmtr)"(char *side, char *uplo, char *trans, int *m, int *n, d *ap, d *tau, d *c, int *ldc, d *work, int *info) nogil +cdef void dopmtr(char *side, char *uplo, char *trans, int *m, int *n, d *ap, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil: + + _fortran_dopmtr(side, uplo, trans, m, n, ap, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorbdb "BLAS_FUNC(dorbdb)"(char *trans, char *signs, int *m, int *p, int *q, d *x11, int *ldx11, d *x12, int *ldx12, d *x21, int *ldx21, d *x22, int *ldx22, d *theta, d *phi, d *taup1, d *taup2, d *tauq1, d *tauq2, d *work, int *lwork, int *info) nogil +cdef void dorbdb(char *trans, char *signs, int *m, int *p, int *q, d *x11, int *ldx11, d *x12, int *ldx12, d *x21, int *ldx21, d *x22, int *ldx22, d *theta, d *phi, d *taup1, d *taup2, d *tauq1, d *tauq2, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dorbdb(trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorcsd "BLAS_FUNC(dorcsd)"(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, d *x11, int *ldx11, d *x12, int *ldx12, d *x21, int *ldx21, d *x22, int *ldx22, d *theta, d *u1, int *ldu1, d *u2, int *ldu2, d *v1t, int *ldv1t, d *v2t, int *ldv2t, d *work, int *lwork, int *iwork, int *info) nogil +cdef void dorcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, d *x11, int *ldx11, d *x12, int *ldx12, d *x21, int *ldx21, d *x22, int *ldx22, d *theta, d *u1, int *ldu1, d *u2, int *ldu2, d *v1t, int *ldv1t, d *v2t, int *ldv2t, d *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_dorcsd(jobu1, jobu2, jobv1t, jobv2t, trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorg2l "BLAS_FUNC(dorg2l)"(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) nogil +cdef void dorg2l(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil: + + _fortran_dorg2l(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorg2r "BLAS_FUNC(dorg2r)"(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) nogil +cdef void dorg2r(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil: + + _fortran_dorg2r(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorgbr "BLAS_FUNC(dorgbr)"(char *vect, int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dorgbr(char *vect, int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dorgbr(vect, m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorghr "BLAS_FUNC(dorghr)"(int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dorghr(int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dorghr(n, ilo, ihi, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorgl2 "BLAS_FUNC(dorgl2)"(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) nogil +cdef void dorgl2(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil: + + _fortran_dorgl2(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorglq "BLAS_FUNC(dorglq)"(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dorglq(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dorglq(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorgql "BLAS_FUNC(dorgql)"(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dorgql(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dorgql(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorgqr "BLAS_FUNC(dorgqr)"(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dorgqr(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dorgqr(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorgr2 "BLAS_FUNC(dorgr2)"(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) nogil +cdef void dorgr2(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *info) noexcept nogil: + + _fortran_dorgr2(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorgrq "BLAS_FUNC(dorgrq)"(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dorgrq(int *m, int *n, int *k, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dorgrq(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorgtr "BLAS_FUNC(dorgtr)"(char *uplo, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dorgtr(char *uplo, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dorgtr(uplo, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorm2l "BLAS_FUNC(dorm2l)"(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) nogil +cdef void dorm2l(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil: + + _fortran_dorm2l(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorm2r "BLAS_FUNC(dorm2r)"(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) nogil +cdef void dorm2r(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil: + + _fortran_dorm2r(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dormbr "BLAS_FUNC(dormbr)"(char *vect, char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) nogil +cdef void dormbr(char *vect, char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dormbr(vect, side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dormhr "BLAS_FUNC(dormhr)"(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) nogil +cdef void dormhr(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dormhr(side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dorml2 "BLAS_FUNC(dorml2)"(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) nogil +cdef void dorml2(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil: + + _fortran_dorml2(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dormlq "BLAS_FUNC(dormlq)"(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) nogil +cdef void dormlq(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dormlq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dormql "BLAS_FUNC(dormql)"(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) nogil +cdef void dormql(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dormql(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dormqr "BLAS_FUNC(dormqr)"(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) nogil +cdef void dormqr(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dormqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dormr2 "BLAS_FUNC(dormr2)"(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) nogil +cdef void dormr2(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil: + + _fortran_dormr2(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dormr3 "BLAS_FUNC(dormr3)"(char *side, char *trans, int *m, int *n, int *k, int *l, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) nogil +cdef void dormr3(char *side, char *trans, int *m, int *n, int *k, int *l, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *info) noexcept nogil: + + _fortran_dormr3(side, trans, m, n, k, l, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dormrq "BLAS_FUNC(dormrq)"(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) nogil +cdef void dormrq(char *side, char *trans, int *m, int *n, int *k, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dormrq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dormrz "BLAS_FUNC(dormrz)"(char *side, char *trans, int *m, int *n, int *k, int *l, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) nogil +cdef void dormrz(char *side, char *trans, int *m, int *n, int *k, int *l, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dormrz(side, trans, m, n, k, l, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dormtr "BLAS_FUNC(dormtr)"(char *side, char *uplo, char *trans, int *m, int *n, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) nogil +cdef void dormtr(char *side, char *uplo, char *trans, int *m, int *n, d *a, int *lda, d *tau, d *c, int *ldc, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dormtr(side, uplo, trans, m, n, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpbcon "BLAS_FUNC(dpbcon)"(char *uplo, int *n, int *kd, d *ab, int *ldab, d *anorm, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dpbcon(char *uplo, int *n, int *kd, d *ab, int *ldab, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dpbcon(uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpbequ "BLAS_FUNC(dpbequ)"(char *uplo, int *n, int *kd, d *ab, int *ldab, d *s, d *scond, d *amax, int *info) nogil +cdef void dpbequ(char *uplo, int *n, int *kd, d *ab, int *ldab, d *s, d *scond, d *amax, int *info) noexcept nogil: + + _fortran_dpbequ(uplo, n, kd, ab, ldab, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpbrfs "BLAS_FUNC(dpbrfs)"(char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dpbrfs(char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dpbrfs(uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpbstf "BLAS_FUNC(dpbstf)"(char *uplo, int *n, int *kd, d *ab, int *ldab, int *info) nogil +cdef void dpbstf(char *uplo, int *n, int *kd, d *ab, int *ldab, int *info) noexcept nogil: + + _fortran_dpbstf(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpbsv "BLAS_FUNC(dpbsv)"(char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, int *info) nogil +cdef void dpbsv(char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dpbsv(uplo, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpbsvx "BLAS_FUNC(dpbsvx)"(char *fact, char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, char *equed, d *s, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dpbsvx(char *fact, char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *afb, int *ldafb, char *equed, d *s, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dpbsvx(fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpbtf2 "BLAS_FUNC(dpbtf2)"(char *uplo, int *n, int *kd, d *ab, int *ldab, int *info) nogil +cdef void dpbtf2(char *uplo, int *n, int *kd, d *ab, int *ldab, int *info) noexcept nogil: + + _fortran_dpbtf2(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpbtrf "BLAS_FUNC(dpbtrf)"(char *uplo, int *n, int *kd, d *ab, int *ldab, int *info) nogil +cdef void dpbtrf(char *uplo, int *n, int *kd, d *ab, int *ldab, int *info) noexcept nogil: + + _fortran_dpbtrf(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpbtrs "BLAS_FUNC(dpbtrs)"(char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, int *info) nogil +cdef void dpbtrs(char *uplo, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dpbtrs(uplo, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpftrf "BLAS_FUNC(dpftrf)"(char *transr, char *uplo, int *n, d *a, int *info) nogil +cdef void dpftrf(char *transr, char *uplo, int *n, d *a, int *info) noexcept nogil: + + _fortran_dpftrf(transr, uplo, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpftri "BLAS_FUNC(dpftri)"(char *transr, char *uplo, int *n, d *a, int *info) nogil +cdef void dpftri(char *transr, char *uplo, int *n, d *a, int *info) noexcept nogil: + + _fortran_dpftri(transr, uplo, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpftrs "BLAS_FUNC(dpftrs)"(char *transr, char *uplo, int *n, int *nrhs, d *a, d *b, int *ldb, int *info) nogil +cdef void dpftrs(char *transr, char *uplo, int *n, int *nrhs, d *a, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dpftrs(transr, uplo, n, nrhs, a, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpocon "BLAS_FUNC(dpocon)"(char *uplo, int *n, d *a, int *lda, d *anorm, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dpocon(char *uplo, int *n, d *a, int *lda, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dpocon(uplo, n, a, lda, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpoequ "BLAS_FUNC(dpoequ)"(int *n, d *a, int *lda, d *s, d *scond, d *amax, int *info) nogil +cdef void dpoequ(int *n, d *a, int *lda, d *s, d *scond, d *amax, int *info) noexcept nogil: + + _fortran_dpoequ(n, a, lda, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpoequb "BLAS_FUNC(dpoequb)"(int *n, d *a, int *lda, d *s, d *scond, d *amax, int *info) nogil +cdef void dpoequb(int *n, d *a, int *lda, d *s, d *scond, d *amax, int *info) noexcept nogil: + + _fortran_dpoequb(n, a, lda, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dporfs "BLAS_FUNC(dporfs)"(char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dporfs(char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dporfs(uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dposv "BLAS_FUNC(dposv)"(char *uplo, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *info) nogil +cdef void dposv(char *uplo, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dposv(uplo, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dposvx "BLAS_FUNC(dposvx)"(char *fact, char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, char *equed, d *s, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dposvx(char *fact, char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, char *equed, d *s, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dposvx(fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpotf2 "BLAS_FUNC(dpotf2)"(char *uplo, int *n, d *a, int *lda, int *info) nogil +cdef void dpotf2(char *uplo, int *n, d *a, int *lda, int *info) noexcept nogil: + + _fortran_dpotf2(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpotrf "BLAS_FUNC(dpotrf)"(char *uplo, int *n, d *a, int *lda, int *info) nogil +cdef void dpotrf(char *uplo, int *n, d *a, int *lda, int *info) noexcept nogil: + + _fortran_dpotrf(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpotri "BLAS_FUNC(dpotri)"(char *uplo, int *n, d *a, int *lda, int *info) nogil +cdef void dpotri(char *uplo, int *n, d *a, int *lda, int *info) noexcept nogil: + + _fortran_dpotri(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpotrs "BLAS_FUNC(dpotrs)"(char *uplo, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *info) nogil +cdef void dpotrs(char *uplo, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dpotrs(uplo, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dppcon "BLAS_FUNC(dppcon)"(char *uplo, int *n, d *ap, d *anorm, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dppcon(char *uplo, int *n, d *ap, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dppcon(uplo, n, ap, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dppequ "BLAS_FUNC(dppequ)"(char *uplo, int *n, d *ap, d *s, d *scond, d *amax, int *info) nogil +cdef void dppequ(char *uplo, int *n, d *ap, d *s, d *scond, d *amax, int *info) noexcept nogil: + + _fortran_dppequ(uplo, n, ap, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpprfs "BLAS_FUNC(dpprfs)"(char *uplo, int *n, int *nrhs, d *ap, d *afp, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dpprfs(char *uplo, int *n, int *nrhs, d *ap, d *afp, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dpprfs(uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dppsv "BLAS_FUNC(dppsv)"(char *uplo, int *n, int *nrhs, d *ap, d *b, int *ldb, int *info) nogil +cdef void dppsv(char *uplo, int *n, int *nrhs, d *ap, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dppsv(uplo, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dppsvx "BLAS_FUNC(dppsvx)"(char *fact, char *uplo, int *n, int *nrhs, d *ap, d *afp, char *equed, d *s, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dppsvx(char *fact, char *uplo, int *n, int *nrhs, d *ap, d *afp, char *equed, d *s, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dppsvx(fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpptrf "BLAS_FUNC(dpptrf)"(char *uplo, int *n, d *ap, int *info) nogil +cdef void dpptrf(char *uplo, int *n, d *ap, int *info) noexcept nogil: + + _fortran_dpptrf(uplo, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpptri "BLAS_FUNC(dpptri)"(char *uplo, int *n, d *ap, int *info) nogil +cdef void dpptri(char *uplo, int *n, d *ap, int *info) noexcept nogil: + + _fortran_dpptri(uplo, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpptrs "BLAS_FUNC(dpptrs)"(char *uplo, int *n, int *nrhs, d *ap, d *b, int *ldb, int *info) nogil +cdef void dpptrs(char *uplo, int *n, int *nrhs, d *ap, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dpptrs(uplo, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpstf2 "BLAS_FUNC(dpstf2)"(char *uplo, int *n, d *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) nogil +cdef void dpstf2(char *uplo, int *n, d *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) noexcept nogil: + + _fortran_dpstf2(uplo, n, a, lda, piv, rank, tol, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpstrf "BLAS_FUNC(dpstrf)"(char *uplo, int *n, d *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) nogil +cdef void dpstrf(char *uplo, int *n, d *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) noexcept nogil: + + _fortran_dpstrf(uplo, n, a, lda, piv, rank, tol, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dptcon "BLAS_FUNC(dptcon)"(int *n, d *d, d *e, d *anorm, d *rcond, d *work, int *info) nogil +cdef void dptcon(int *n, d *d, d *e, d *anorm, d *rcond, d *work, int *info) noexcept nogil: + + _fortran_dptcon(n, d, e, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpteqr "BLAS_FUNC(dpteqr)"(char *compz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *info) nogil +cdef void dpteqr(char *compz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *info) noexcept nogil: + + _fortran_dpteqr(compz, n, d, e, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dptrfs "BLAS_FUNC(dptrfs)"(int *n, int *nrhs, d *d, d *e, d *df, d *ef, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *info) nogil +cdef void dptrfs(int *n, int *nrhs, d *d, d *e, d *df, d *ef, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *info) noexcept nogil: + + _fortran_dptrfs(n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dptsv "BLAS_FUNC(dptsv)"(int *n, int *nrhs, d *d, d *e, d *b, int *ldb, int *info) nogil +cdef void dptsv(int *n, int *nrhs, d *d, d *e, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dptsv(n, nrhs, d, e, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dptsvx "BLAS_FUNC(dptsvx)"(char *fact, int *n, int *nrhs, d *d, d *e, d *df, d *ef, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *info) nogil +cdef void dptsvx(char *fact, int *n, int *nrhs, d *d, d *e, d *df, d *ef, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *info) noexcept nogil: + + _fortran_dptsvx(fact, n, nrhs, d, e, df, ef, b, ldb, x, ldx, rcond, ferr, berr, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpttrf "BLAS_FUNC(dpttrf)"(int *n, d *d, d *e, int *info) nogil +cdef void dpttrf(int *n, d *d, d *e, int *info) noexcept nogil: + + _fortran_dpttrf(n, d, e, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dpttrs "BLAS_FUNC(dpttrs)"(int *n, int *nrhs, d *d, d *e, d *b, int *ldb, int *info) nogil +cdef void dpttrs(int *n, int *nrhs, d *d, d *e, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dpttrs(n, nrhs, d, e, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dptts2 "BLAS_FUNC(dptts2)"(int *n, int *nrhs, d *d, d *e, d *b, int *ldb) nogil +cdef void dptts2(int *n, int *nrhs, d *d, d *e, d *b, int *ldb) noexcept nogil: + + _fortran_dptts2(n, nrhs, d, e, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_drscl "BLAS_FUNC(drscl)"(int *n, d *sa, d *sx, int *incx) nogil +cdef void drscl(int *n, d *sa, d *sx, int *incx) noexcept nogil: + + _fortran_drscl(n, sa, sx, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsbev "BLAS_FUNC(dsbev)"(char *jobz, char *uplo, int *n, int *kd, d *ab, int *ldab, d *w, d *z, int *ldz, d *work, int *info) nogil +cdef void dsbev(char *jobz, char *uplo, int *n, int *kd, d *ab, int *ldab, d *w, d *z, int *ldz, d *work, int *info) noexcept nogil: + + _fortran_dsbev(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsbevd "BLAS_FUNC(dsbevd)"(char *jobz, char *uplo, int *n, int *kd, d *ab, int *ldab, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dsbevd(char *jobz, char *uplo, int *n, int *kd, d *ab, int *ldab, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dsbevd(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsbevx "BLAS_FUNC(dsbevx)"(char *jobz, char *range, char *uplo, int *n, int *kd, d *ab, int *ldab, d *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) nogil +cdef void dsbevx(char *jobz, char *range, char *uplo, int *n, int *kd, d *ab, int *ldab, d *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_dsbevx(jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsbgst "BLAS_FUNC(dsbgst)"(char *vect, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *x, int *ldx, d *work, int *info) nogil +cdef void dsbgst(char *vect, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *x, int *ldx, d *work, int *info) noexcept nogil: + + _fortran_dsbgst(vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsbgv "BLAS_FUNC(dsbgv)"(char *jobz, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *w, d *z, int *ldz, d *work, int *info) nogil +cdef void dsbgv(char *jobz, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *w, d *z, int *ldz, d *work, int *info) noexcept nogil: + + _fortran_dsbgv(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsbgvd "BLAS_FUNC(dsbgvd)"(char *jobz, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dsbgvd(char *jobz, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dsbgvd(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsbgvx "BLAS_FUNC(dsbgvx)"(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) nogil +cdef void dsbgvx(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, d *ab, int *ldab, d *bb, int *ldbb, d *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_dsbgvx(jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsbtrd "BLAS_FUNC(dsbtrd)"(char *vect, char *uplo, int *n, int *kd, d *ab, int *ldab, d *d, d *e, d *q, int *ldq, d *work, int *info) nogil +cdef void dsbtrd(char *vect, char *uplo, int *n, int *kd, d *ab, int *ldab, d *d, d *e, d *q, int *ldq, d *work, int *info) noexcept nogil: + + _fortran_dsbtrd(vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsfrk "BLAS_FUNC(dsfrk)"(char *transr, char *uplo, char *trans, int *n, int *k, d *alpha, d *a, int *lda, d *beta, d *c) nogil +cdef void dsfrk(char *transr, char *uplo, char *trans, int *n, int *k, d *alpha, d *a, int *lda, d *beta, d *c) noexcept nogil: + + _fortran_dsfrk(transr, uplo, trans, n, k, alpha, a, lda, beta, c) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsgesv "BLAS_FUNC(dsgesv)"(int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *work, s *swork, int *iter, int *info) nogil +cdef void dsgesv(int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *work, s *swork, int *iter, int *info) noexcept nogil: + + _fortran_dsgesv(n, nrhs, a, lda, ipiv, b, ldb, x, ldx, work, swork, iter, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dspcon "BLAS_FUNC(dspcon)"(char *uplo, int *n, d *ap, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dspcon(char *uplo, int *n, d *ap, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dspcon(uplo, n, ap, ipiv, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dspev "BLAS_FUNC(dspev)"(char *jobz, char *uplo, int *n, d *ap, d *w, d *z, int *ldz, d *work, int *info) nogil +cdef void dspev(char *jobz, char *uplo, int *n, d *ap, d *w, d *z, int *ldz, d *work, int *info) noexcept nogil: + + _fortran_dspev(jobz, uplo, n, ap, w, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dspevd "BLAS_FUNC(dspevd)"(char *jobz, char *uplo, int *n, d *ap, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dspevd(char *jobz, char *uplo, int *n, d *ap, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dspevd(jobz, uplo, n, ap, w, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dspevx "BLAS_FUNC(dspevx)"(char *jobz, char *range, char *uplo, int *n, d *ap, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) nogil +cdef void dspevx(char *jobz, char *range, char *uplo, int *n, d *ap, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_dspevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dspgst "BLAS_FUNC(dspgst)"(int *itype, char *uplo, int *n, d *ap, d *bp, int *info) nogil +cdef void dspgst(int *itype, char *uplo, int *n, d *ap, d *bp, int *info) noexcept nogil: + + _fortran_dspgst(itype, uplo, n, ap, bp, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dspgv "BLAS_FUNC(dspgv)"(int *itype, char *jobz, char *uplo, int *n, d *ap, d *bp, d *w, d *z, int *ldz, d *work, int *info) nogil +cdef void dspgv(int *itype, char *jobz, char *uplo, int *n, d *ap, d *bp, d *w, d *z, int *ldz, d *work, int *info) noexcept nogil: + + _fortran_dspgv(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dspgvd "BLAS_FUNC(dspgvd)"(int *itype, char *jobz, char *uplo, int *n, d *ap, d *bp, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dspgvd(int *itype, char *jobz, char *uplo, int *n, d *ap, d *bp, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dspgvd(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dspgvx "BLAS_FUNC(dspgvx)"(int *itype, char *jobz, char *range, char *uplo, int *n, d *ap, d *bp, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) nogil +cdef void dspgvx(int *itype, char *jobz, char *range, char *uplo, int *n, d *ap, d *bp, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_dspgvx(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsposv "BLAS_FUNC(dsposv)"(char *uplo, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *x, int *ldx, d *work, s *swork, int *iter, int *info) nogil +cdef void dsposv(char *uplo, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *x, int *ldx, d *work, s *swork, int *iter, int *info) noexcept nogil: + + _fortran_dsposv(uplo, n, nrhs, a, lda, b, ldb, x, ldx, work, swork, iter, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsprfs "BLAS_FUNC(dsprfs)"(char *uplo, int *n, int *nrhs, d *ap, d *afp, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dsprfs(char *uplo, int *n, int *nrhs, d *ap, d *afp, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dsprfs(uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dspsv "BLAS_FUNC(dspsv)"(char *uplo, int *n, int *nrhs, d *ap, int *ipiv, d *b, int *ldb, int *info) nogil +cdef void dspsv(char *uplo, int *n, int *nrhs, d *ap, int *ipiv, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dspsv(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dspsvx "BLAS_FUNC(dspsvx)"(char *fact, char *uplo, int *n, int *nrhs, d *ap, d *afp, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dspsvx(char *fact, char *uplo, int *n, int *nrhs, d *ap, d *afp, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dspsvx(fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsptrd "BLAS_FUNC(dsptrd)"(char *uplo, int *n, d *ap, d *d, d *e, d *tau, int *info) nogil +cdef void dsptrd(char *uplo, int *n, d *ap, d *d, d *e, d *tau, int *info) noexcept nogil: + + _fortran_dsptrd(uplo, n, ap, d, e, tau, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsptrf "BLAS_FUNC(dsptrf)"(char *uplo, int *n, d *ap, int *ipiv, int *info) nogil +cdef void dsptrf(char *uplo, int *n, d *ap, int *ipiv, int *info) noexcept nogil: + + _fortran_dsptrf(uplo, n, ap, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsptri "BLAS_FUNC(dsptri)"(char *uplo, int *n, d *ap, int *ipiv, d *work, int *info) nogil +cdef void dsptri(char *uplo, int *n, d *ap, int *ipiv, d *work, int *info) noexcept nogil: + + _fortran_dsptri(uplo, n, ap, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsptrs "BLAS_FUNC(dsptrs)"(char *uplo, int *n, int *nrhs, d *ap, int *ipiv, d *b, int *ldb, int *info) nogil +cdef void dsptrs(char *uplo, int *n, int *nrhs, d *ap, int *ipiv, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dsptrs(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dstebz "BLAS_FUNC(dstebz)"(char *range, char *order, int *n, d *vl, d *vu, int *il, int *iu, d *abstol, d *d, d *e, int *m, int *nsplit, d *w, int *iblock, int *isplit, d *work, int *iwork, int *info) nogil +cdef void dstebz(char *range, char *order, int *n, d *vl, d *vu, int *il, int *iu, d *abstol, d *d, d *e, int *m, int *nsplit, d *w, int *iblock, int *isplit, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dstebz(range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dstedc "BLAS_FUNC(dstedc)"(char *compz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dstedc(char *compz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dstedc(compz, n, d, e, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dstegr "BLAS_FUNC(dstegr)"(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dstegr(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dstegr(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dstein "BLAS_FUNC(dstein)"(int *n, d *d, d *e, int *m, d *w, int *iblock, int *isplit, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) nogil +cdef void dstein(int *n, d *d, d *e, int *m, d *w, int *iblock, int *isplit, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_dstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dstemr "BLAS_FUNC(dstemr)"(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, int *m, d *w, d *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dstemr(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, int *m, d *w, d *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dstemr(jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsteqr "BLAS_FUNC(dsteqr)"(char *compz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *info) nogil +cdef void dsteqr(char *compz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *info) noexcept nogil: + + _fortran_dsteqr(compz, n, d, e, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsterf "BLAS_FUNC(dsterf)"(int *n, d *d, d *e, int *info) nogil +cdef void dsterf(int *n, d *d, d *e, int *info) noexcept nogil: + + _fortran_dsterf(n, d, e, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dstev "BLAS_FUNC(dstev)"(char *jobz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *info) nogil +cdef void dstev(char *jobz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *info) noexcept nogil: + + _fortran_dstev(jobz, n, d, e, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dstevd "BLAS_FUNC(dstevd)"(char *jobz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dstevd(char *jobz, int *n, d *d, d *e, d *z, int *ldz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dstevd(jobz, n, d, e, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dstevr "BLAS_FUNC(dstevr)"(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dstevr(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dstevr(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dstevx "BLAS_FUNC(dstevx)"(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) nogil +cdef void dstevx(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_dstevx(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsycon "BLAS_FUNC(dsycon)"(char *uplo, int *n, d *a, int *lda, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dsycon(char *uplo, int *n, d *a, int *lda, int *ipiv, d *anorm, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dsycon(uplo, n, a, lda, ipiv, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsyconv "BLAS_FUNC(dsyconv)"(char *uplo, char *way, int *n, d *a, int *lda, int *ipiv, d *work, int *info) nogil +cdef void dsyconv(char *uplo, char *way, int *n, d *a, int *lda, int *ipiv, d *work, int *info) noexcept nogil: + + _fortran_dsyconv(uplo, way, n, a, lda, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsyequb "BLAS_FUNC(dsyequb)"(char *uplo, int *n, d *a, int *lda, d *s, d *scond, d *amax, d *work, int *info) nogil +cdef void dsyequb(char *uplo, int *n, d *a, int *lda, d *s, d *scond, d *amax, d *work, int *info) noexcept nogil: + + _fortran_dsyequb(uplo, n, a, lda, s, scond, amax, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsyev "BLAS_FUNC(dsyev)"(char *jobz, char *uplo, int *n, d *a, int *lda, d *w, d *work, int *lwork, int *info) nogil +cdef void dsyev(char *jobz, char *uplo, int *n, d *a, int *lda, d *w, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dsyev(jobz, uplo, n, a, lda, w, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsyevd "BLAS_FUNC(dsyevd)"(char *jobz, char *uplo, int *n, d *a, int *lda, d *w, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dsyevd(char *jobz, char *uplo, int *n, d *a, int *lda, d *w, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dsyevd(jobz, uplo, n, a, lda, w, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsyevr "BLAS_FUNC(dsyevr)"(char *jobz, char *range, char *uplo, int *n, d *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dsyevr(char *jobz, char *range, char *uplo, int *n, d *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dsyevr(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsyevx "BLAS_FUNC(dsyevx)"(char *jobz, char *range, char *uplo, int *n, d *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *ifail, int *info) nogil +cdef void dsyevx(char *jobz, char *range, char *uplo, int *n, d *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_dsyevx(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsygs2 "BLAS_FUNC(dsygs2)"(int *itype, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, int *info) nogil +cdef void dsygs2(int *itype, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dsygs2(itype, uplo, n, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsygst "BLAS_FUNC(dsygst)"(int *itype, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, int *info) nogil +cdef void dsygst(int *itype, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dsygst(itype, uplo, n, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsygv "BLAS_FUNC(dsygv)"(int *itype, char *jobz, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, d *w, d *work, int *lwork, int *info) nogil +cdef void dsygv(int *itype, char *jobz, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, d *w, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dsygv(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsygvd "BLAS_FUNC(dsygvd)"(int *itype, char *jobz, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, d *w, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dsygvd(int *itype, char *jobz, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, d *w, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dsygvd(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsygvx "BLAS_FUNC(dsygvx)"(int *itype, char *jobz, char *range, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *ifail, int *info) nogil +cdef void dsygvx(int *itype, char *jobz, char *range, char *uplo, int *n, d *a, int *lda, d *b, int *ldb, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, d *z, int *ldz, d *work, int *lwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_dsygvx(itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsyrfs "BLAS_FUNC(dsyrfs)"(char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dsyrfs(char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dsyrfs(uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsysv "BLAS_FUNC(dsysv)"(char *uplo, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, d *work, int *lwork, int *info) nogil +cdef void dsysv(char *uplo, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dsysv(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsysvx "BLAS_FUNC(dsysvx)"(char *fact, char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *lwork, int *iwork, int *info) nogil +cdef void dsysvx(char *fact, char *uplo, int *n, int *nrhs, d *a, int *lda, d *af, int *ldaf, int *ipiv, d *b, int *ldb, d *x, int *ldx, d *rcond, d *ferr, d *berr, d *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_dsysvx(fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsyswapr "BLAS_FUNC(dsyswapr)"(char *uplo, int *n, d *a, int *lda, int *i1, int *i2) nogil +cdef void dsyswapr(char *uplo, int *n, d *a, int *lda, int *i1, int *i2) noexcept nogil: + + _fortran_dsyswapr(uplo, n, a, lda, i1, i2) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsytd2 "BLAS_FUNC(dsytd2)"(char *uplo, int *n, d *a, int *lda, d *d, d *e, d *tau, int *info) nogil +cdef void dsytd2(char *uplo, int *n, d *a, int *lda, d *d, d *e, d *tau, int *info) noexcept nogil: + + _fortran_dsytd2(uplo, n, a, lda, d, e, tau, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsytf2 "BLAS_FUNC(dsytf2)"(char *uplo, int *n, d *a, int *lda, int *ipiv, int *info) nogil +cdef void dsytf2(char *uplo, int *n, d *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_dsytf2(uplo, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsytrd "BLAS_FUNC(dsytrd)"(char *uplo, int *n, d *a, int *lda, d *d, d *e, d *tau, d *work, int *lwork, int *info) nogil +cdef void dsytrd(char *uplo, int *n, d *a, int *lda, d *d, d *e, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dsytrd(uplo, n, a, lda, d, e, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsytrf "BLAS_FUNC(dsytrf)"(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *lwork, int *info) nogil +cdef void dsytrf(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dsytrf(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsytri "BLAS_FUNC(dsytri)"(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *info) nogil +cdef void dsytri(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *info) noexcept nogil: + + _fortran_dsytri(uplo, n, a, lda, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsytri2 "BLAS_FUNC(dsytri2)"(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *lwork, int *info) nogil +cdef void dsytri2(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dsytri2(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsytri2x "BLAS_FUNC(dsytri2x)"(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *nb, int *info) nogil +cdef void dsytri2x(char *uplo, int *n, d *a, int *lda, int *ipiv, d *work, int *nb, int *info) noexcept nogil: + + _fortran_dsytri2x(uplo, n, a, lda, ipiv, work, nb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsytrs "BLAS_FUNC(dsytrs)"(char *uplo, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, int *info) nogil +cdef void dsytrs(char *uplo, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dsytrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dsytrs2 "BLAS_FUNC(dsytrs2)"(char *uplo, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, d *work, int *info) nogil +cdef void dsytrs2(char *uplo, int *n, int *nrhs, d *a, int *lda, int *ipiv, d *b, int *ldb, d *work, int *info) noexcept nogil: + + _fortran_dsytrs2(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtbcon "BLAS_FUNC(dtbcon)"(char *norm, char *uplo, char *diag, int *n, int *kd, d *ab, int *ldab, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dtbcon(char *norm, char *uplo, char *diag, int *n, int *kd, d *ab, int *ldab, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dtbcon(norm, uplo, diag, n, kd, ab, ldab, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtbrfs "BLAS_FUNC(dtbrfs)"(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dtbrfs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dtbrfs(uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtbtrs "BLAS_FUNC(dtbtrs)"(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, int *info) nogil +cdef void dtbtrs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, d *ab, int *ldab, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dtbtrs(uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtfsm "BLAS_FUNC(dtfsm)"(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, d *alpha, d *a, d *b, int *ldb) nogil +cdef void dtfsm(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, d *alpha, d *a, d *b, int *ldb) noexcept nogil: + + _fortran_dtfsm(transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtftri "BLAS_FUNC(dtftri)"(char *transr, char *uplo, char *diag, int *n, d *a, int *info) nogil +cdef void dtftri(char *transr, char *uplo, char *diag, int *n, d *a, int *info) noexcept nogil: + + _fortran_dtftri(transr, uplo, diag, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtfttp "BLAS_FUNC(dtfttp)"(char *transr, char *uplo, int *n, d *arf, d *ap, int *info) nogil +cdef void dtfttp(char *transr, char *uplo, int *n, d *arf, d *ap, int *info) noexcept nogil: + + _fortran_dtfttp(transr, uplo, n, arf, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtfttr "BLAS_FUNC(dtfttr)"(char *transr, char *uplo, int *n, d *arf, d *a, int *lda, int *info) nogil +cdef void dtfttr(char *transr, char *uplo, int *n, d *arf, d *a, int *lda, int *info) noexcept nogil: + + _fortran_dtfttr(transr, uplo, n, arf, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtgevc "BLAS_FUNC(dtgevc)"(char *side, char *howmny, bint *select, int *n, d *s, int *lds, d *p, int *ldp, d *vl, int *ldvl, d *vr, int *ldvr, int *mm, int *m, d *work, int *info) nogil +cdef void dtgevc(char *side, char *howmny, bint *select, int *n, d *s, int *lds, d *p, int *ldp, d *vl, int *ldvl, d *vr, int *ldvr, int *mm, int *m, d *work, int *info) noexcept nogil: + + _fortran_dtgevc(side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtgex2 "BLAS_FUNC(dtgex2)"(bint *wantq, bint *wantz, int *n, d *a, int *lda, d *b, int *ldb, d *q, int *ldq, d *z, int *ldz, int *j1, int *n1, int *n2, d *work, int *lwork, int *info) nogil +cdef void dtgex2(bint *wantq, bint *wantz, int *n, d *a, int *lda, d *b, int *ldb, d *q, int *ldq, d *z, int *ldz, int *j1, int *n1, int *n2, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dtgex2(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, n1, n2, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtgexc "BLAS_FUNC(dtgexc)"(bint *wantq, bint *wantz, int *n, d *a, int *lda, d *b, int *ldb, d *q, int *ldq, d *z, int *ldz, int *ifst, int *ilst, d *work, int *lwork, int *info) nogil +cdef void dtgexc(bint *wantq, bint *wantz, int *n, d *a, int *lda, d *b, int *ldb, d *q, int *ldq, d *z, int *ldz, int *ifst, int *ilst, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dtgexc(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtgsen "BLAS_FUNC(dtgsen)"(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *q, int *ldq, d *z, int *ldz, int *m, d *pl, d *pr, d *dif, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dtgsen(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, d *a, int *lda, d *b, int *ldb, d *alphar, d *alphai, d *beta, d *q, int *ldq, d *z, int *ldz, int *m, d *pl, d *pr, d *dif, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dtgsen(ijob, wantq, wantz, select, n, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtgsja "BLAS_FUNC(dtgsja)"(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, d *a, int *lda, d *b, int *ldb, d *tola, d *tolb, d *alpha, d *beta, d *u, int *ldu, d *v, int *ldv, d *q, int *ldq, d *work, int *ncycle, int *info) nogil +cdef void dtgsja(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, d *a, int *lda, d *b, int *ldb, d *tola, d *tolb, d *alpha, d *beta, d *u, int *ldu, d *v, int *ldv, d *q, int *ldq, d *work, int *ncycle, int *info) noexcept nogil: + + _fortran_dtgsja(jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb, tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq, work, ncycle, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtgsna "BLAS_FUNC(dtgsna)"(char *job, char *howmny, bint *select, int *n, d *a, int *lda, d *b, int *ldb, d *vl, int *ldvl, d *vr, int *ldvr, d *s, d *dif, int *mm, int *m, d *work, int *lwork, int *iwork, int *info) nogil +cdef void dtgsna(char *job, char *howmny, bint *select, int *n, d *a, int *lda, d *b, int *ldb, d *vl, int *ldvl, d *vr, int *ldvr, d *s, d *dif, int *mm, int *m, d *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_dtgsna(job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtgsy2 "BLAS_FUNC(dtgsy2)"(char *trans, int *ijob, int *m, int *n, d *a, int *lda, d *b, int *ldb, d *c, int *ldc, d *d, int *ldd, d *e, int *lde, d *f, int *ldf, d *scale, d *rdsum, d *rdscal, int *iwork, int *pq, int *info) nogil +cdef void dtgsy2(char *trans, int *ijob, int *m, int *n, d *a, int *lda, d *b, int *ldb, d *c, int *ldc, d *d, int *ldd, d *e, int *lde, d *f, int *ldf, d *scale, d *rdsum, d *rdscal, int *iwork, int *pq, int *info) noexcept nogil: + + _fortran_dtgsy2(trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, iwork, pq, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtgsyl "BLAS_FUNC(dtgsyl)"(char *trans, int *ijob, int *m, int *n, d *a, int *lda, d *b, int *ldb, d *c, int *ldc, d *d, int *ldd, d *e, int *lde, d *f, int *ldf, d *scale, d *dif, d *work, int *lwork, int *iwork, int *info) nogil +cdef void dtgsyl(char *trans, int *ijob, int *m, int *n, d *a, int *lda, d *b, int *ldb, d *c, int *ldc, d *d, int *ldd, d *e, int *lde, d *f, int *ldf, d *scale, d *dif, d *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_dtgsyl(trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtpcon "BLAS_FUNC(dtpcon)"(char *norm, char *uplo, char *diag, int *n, d *ap, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dtpcon(char *norm, char *uplo, char *diag, int *n, d *ap, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dtpcon(norm, uplo, diag, n, ap, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtpmqrt "BLAS_FUNC(dtpmqrt)"(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, d *v, int *ldv, d *t, int *ldt, d *a, int *lda, d *b, int *ldb, d *work, int *info) nogil +cdef void dtpmqrt(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, d *v, int *ldv, d *t, int *ldt, d *a, int *lda, d *b, int *ldb, d *work, int *info) noexcept nogil: + + _fortran_dtpmqrt(side, trans, m, n, k, l, nb, v, ldv, t, ldt, a, lda, b, ldb, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtpqrt "BLAS_FUNC(dtpqrt)"(int *m, int *n, int *l, int *nb, d *a, int *lda, d *b, int *ldb, d *t, int *ldt, d *work, int *info) nogil +cdef void dtpqrt(int *m, int *n, int *l, int *nb, d *a, int *lda, d *b, int *ldb, d *t, int *ldt, d *work, int *info) noexcept nogil: + + _fortran_dtpqrt(m, n, l, nb, a, lda, b, ldb, t, ldt, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtpqrt2 "BLAS_FUNC(dtpqrt2)"(int *m, int *n, int *l, d *a, int *lda, d *b, int *ldb, d *t, int *ldt, int *info) nogil +cdef void dtpqrt2(int *m, int *n, int *l, d *a, int *lda, d *b, int *ldb, d *t, int *ldt, int *info) noexcept nogil: + + _fortran_dtpqrt2(m, n, l, a, lda, b, ldb, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtprfb "BLAS_FUNC(dtprfb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, d *v, int *ldv, d *t, int *ldt, d *a, int *lda, d *b, int *ldb, d *work, int *ldwork) nogil +cdef void dtprfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, d *v, int *ldv, d *t, int *ldt, d *a, int *lda, d *b, int *ldb, d *work, int *ldwork) noexcept nogil: + + _fortran_dtprfb(side, trans, direct, storev, m, n, k, l, v, ldv, t, ldt, a, lda, b, ldb, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtprfs "BLAS_FUNC(dtprfs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *ap, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dtprfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *ap, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dtprfs(uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtptri "BLAS_FUNC(dtptri)"(char *uplo, char *diag, int *n, d *ap, int *info) nogil +cdef void dtptri(char *uplo, char *diag, int *n, d *ap, int *info) noexcept nogil: + + _fortran_dtptri(uplo, diag, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtptrs "BLAS_FUNC(dtptrs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *ap, d *b, int *ldb, int *info) nogil +cdef void dtptrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *ap, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dtptrs(uplo, trans, diag, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtpttf "BLAS_FUNC(dtpttf)"(char *transr, char *uplo, int *n, d *ap, d *arf, int *info) nogil +cdef void dtpttf(char *transr, char *uplo, int *n, d *ap, d *arf, int *info) noexcept nogil: + + _fortran_dtpttf(transr, uplo, n, ap, arf, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtpttr "BLAS_FUNC(dtpttr)"(char *uplo, int *n, d *ap, d *a, int *lda, int *info) nogil +cdef void dtpttr(char *uplo, int *n, d *ap, d *a, int *lda, int *info) noexcept nogil: + + _fortran_dtpttr(uplo, n, ap, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrcon "BLAS_FUNC(dtrcon)"(char *norm, char *uplo, char *diag, int *n, d *a, int *lda, d *rcond, d *work, int *iwork, int *info) nogil +cdef void dtrcon(char *norm, char *uplo, char *diag, int *n, d *a, int *lda, d *rcond, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dtrcon(norm, uplo, diag, n, a, lda, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrevc "BLAS_FUNC(dtrevc)"(char *side, char *howmny, bint *select, int *n, d *t, int *ldt, d *vl, int *ldvl, d *vr, int *ldvr, int *mm, int *m, d *work, int *info) nogil +cdef void dtrevc(char *side, char *howmny, bint *select, int *n, d *t, int *ldt, d *vl, int *ldvl, d *vr, int *ldvr, int *mm, int *m, d *work, int *info) noexcept nogil: + + _fortran_dtrevc(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrexc "BLAS_FUNC(dtrexc)"(char *compq, int *n, d *t, int *ldt, d *q, int *ldq, int *ifst, int *ilst, d *work, int *info) nogil +cdef void dtrexc(char *compq, int *n, d *t, int *ldt, d *q, int *ldq, int *ifst, int *ilst, d *work, int *info) noexcept nogil: + + _fortran_dtrexc(compq, n, t, ldt, q, ldq, ifst, ilst, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrrfs "BLAS_FUNC(dtrrfs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) nogil +cdef void dtrrfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, d *x, int *ldx, d *ferr, d *berr, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_dtrrfs(uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrsen "BLAS_FUNC(dtrsen)"(char *job, char *compq, bint *select, int *n, d *t, int *ldt, d *q, int *ldq, d *wr, d *wi, int *m, d *s, d *sep, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void dtrsen(char *job, char *compq, bint *select, int *n, d *t, int *ldt, d *q, int *ldq, d *wr, d *wi, int *m, d *s, d *sep, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_dtrsen(job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s, sep, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrsna "BLAS_FUNC(dtrsna)"(char *job, char *howmny, bint *select, int *n, d *t, int *ldt, d *vl, int *ldvl, d *vr, int *ldvr, d *s, d *sep, int *mm, int *m, d *work, int *ldwork, int *iwork, int *info) nogil +cdef void dtrsna(char *job, char *howmny, bint *select, int *n, d *t, int *ldt, d *vl, int *ldvl, d *vr, int *ldvr, d *s, d *sep, int *mm, int *m, d *work, int *ldwork, int *iwork, int *info) noexcept nogil: + + _fortran_dtrsna(job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrsyl "BLAS_FUNC(dtrsyl)"(char *trana, char *tranb, int *isgn, int *m, int *n, d *a, int *lda, d *b, int *ldb, d *c, int *ldc, d *scale, int *info) nogil +cdef void dtrsyl(char *trana, char *tranb, int *isgn, int *m, int *n, d *a, int *lda, d *b, int *ldb, d *c, int *ldc, d *scale, int *info) noexcept nogil: + + _fortran_dtrsyl(trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrti2 "BLAS_FUNC(dtrti2)"(char *uplo, char *diag, int *n, d *a, int *lda, int *info) nogil +cdef void dtrti2(char *uplo, char *diag, int *n, d *a, int *lda, int *info) noexcept nogil: + + _fortran_dtrti2(uplo, diag, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrtri "BLAS_FUNC(dtrtri)"(char *uplo, char *diag, int *n, d *a, int *lda, int *info) nogil +cdef void dtrtri(char *uplo, char *diag, int *n, d *a, int *lda, int *info) noexcept nogil: + + _fortran_dtrtri(uplo, diag, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrtrs "BLAS_FUNC(dtrtrs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *info) nogil +cdef void dtrtrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, d *a, int *lda, d *b, int *ldb, int *info) noexcept nogil: + + _fortran_dtrtrs(uplo, trans, diag, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrttf "BLAS_FUNC(dtrttf)"(char *transr, char *uplo, int *n, d *a, int *lda, d *arf, int *info) nogil +cdef void dtrttf(char *transr, char *uplo, int *n, d *a, int *lda, d *arf, int *info) noexcept nogil: + + _fortran_dtrttf(transr, uplo, n, a, lda, arf, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtrttp "BLAS_FUNC(dtrttp)"(char *uplo, int *n, d *a, int *lda, d *ap, int *info) nogil +cdef void dtrttp(char *uplo, int *n, d *a, int *lda, d *ap, int *info) noexcept nogil: + + _fortran_dtrttp(uplo, n, a, lda, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_dtzrzf "BLAS_FUNC(dtzrzf)"(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) nogil +cdef void dtzrzf(int *m, int *n, d *a, int *lda, d *tau, d *work, int *lwork, int *info) noexcept nogil: + + _fortran_dtzrzf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_dzsum1 "BLAS_FUNC(dzsum1)"(int *n, npy_complex128 *cx, int *incx) nogil +cdef d dzsum1(int *n, z *cx, int *incx) noexcept nogil: + + return _fortran_dzsum1(n, cx, incx) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_icmax1 "BLAS_FUNC(icmax1)"(int *n, npy_complex64 *cx, int *incx) nogil +cdef int icmax1(int *n, c *cx, int *incx) noexcept nogil: + + return _fortran_icmax1(n, cx, incx) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_ieeeck "BLAS_FUNC(ieeeck)"(int *ispec, s *zero, s *one) nogil +cdef int ieeeck(int *ispec, s *zero, s *one) noexcept nogil: + + return _fortran_ieeeck(ispec, zero, one) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_ilaclc "BLAS_FUNC(ilaclc)"(int *m, int *n, npy_complex64 *a, int *lda) nogil +cdef int ilaclc(int *m, int *n, c *a, int *lda) noexcept nogil: + + return _fortran_ilaclc(m, n, a, lda) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_ilaclr "BLAS_FUNC(ilaclr)"(int *m, int *n, npy_complex64 *a, int *lda) nogil +cdef int ilaclr(int *m, int *n, c *a, int *lda) noexcept nogil: + + return _fortran_ilaclr(m, n, a, lda) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_iladiag "BLAS_FUNC(iladiag)"(char *diag) nogil +cdef int iladiag(char *diag) noexcept nogil: + + return _fortran_iladiag(diag) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_iladlc "BLAS_FUNC(iladlc)"(int *m, int *n, d *a, int *lda) nogil +cdef int iladlc(int *m, int *n, d *a, int *lda) noexcept nogil: + + return _fortran_iladlc(m, n, a, lda) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_iladlr "BLAS_FUNC(iladlr)"(int *m, int *n, d *a, int *lda) nogil +cdef int iladlr(int *m, int *n, d *a, int *lda) noexcept nogil: + + return _fortran_iladlr(m, n, a, lda) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_ilaprec "BLAS_FUNC(ilaprec)"(char *prec) nogil +cdef int ilaprec(char *prec) noexcept nogil: + + return _fortran_ilaprec(prec) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_ilaslc "BLAS_FUNC(ilaslc)"(int *m, int *n, s *a, int *lda) nogil +cdef int ilaslc(int *m, int *n, s *a, int *lda) noexcept nogil: + + return _fortran_ilaslc(m, n, a, lda) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_ilaslr "BLAS_FUNC(ilaslr)"(int *m, int *n, s *a, int *lda) nogil +cdef int ilaslr(int *m, int *n, s *a, int *lda) noexcept nogil: + + return _fortran_ilaslr(m, n, a, lda) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_ilatrans "BLAS_FUNC(ilatrans)"(char *trans) nogil +cdef int ilatrans(char *trans) noexcept nogil: + + return _fortran_ilatrans(trans) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_ilauplo "BLAS_FUNC(ilauplo)"(char *uplo) nogil +cdef int ilauplo(char *uplo) noexcept nogil: + + return _fortran_ilauplo(uplo) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ilaver "BLAS_FUNC(ilaver)"(int *vers_major, int *vers_minor, int *vers_patch) nogil +cdef void ilaver(int *vers_major, int *vers_minor, int *vers_patch) noexcept nogil: + + _fortran_ilaver(vers_major, vers_minor, vers_patch) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_ilazlc "BLAS_FUNC(ilazlc)"(int *m, int *n, npy_complex128 *a, int *lda) nogil +cdef int ilazlc(int *m, int *n, z *a, int *lda) noexcept nogil: + + return _fortran_ilazlc(m, n, a, lda) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_ilazlr "BLAS_FUNC(ilazlr)"(int *m, int *n, npy_complex128 *a, int *lda) nogil +cdef int ilazlr(int *m, int *n, z *a, int *lda) noexcept nogil: + + return _fortran_ilazlr(m, n, a, lda) + + +cdef extern from "_lapack_subroutines.h": + int _fortran_izmax1 "BLAS_FUNC(izmax1)"(int *n, npy_complex128 *cx, int *incx) nogil +cdef int izmax1(int *n, z *cx, int *incx) noexcept nogil: + + return _fortran_izmax1(n, cx, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sbbcsd "BLAS_FUNC(sbbcsd)"(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, s *theta, s *phi, s *u1, int *ldu1, s *u2, int *ldu2, s *v1t, int *ldv1t, s *v2t, int *ldv2t, s *b11d, s *b11e, s *b12d, s *b12e, s *b21d, s *b21e, s *b22d, s *b22e, s *work, int *lwork, int *info) nogil +cdef void sbbcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, s *theta, s *phi, s *u1, int *ldu1, s *u2, int *ldu2, s *v1t, int *ldv1t, s *v2t, int *ldv2t, s *b11d, s *b11e, s *b12d, s *b12e, s *b21d, s *b21e, s *b22d, s *b22e, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sbbcsd(jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta, phi, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, b11d, b11e, b12d, b12e, b21d, b21e, b22d, b22e, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sbdsdc "BLAS_FUNC(sbdsdc)"(char *uplo, char *compq, int *n, s *d, s *e, s *u, int *ldu, s *vt, int *ldvt, s *q, int *iq, s *work, int *iwork, int *info) nogil +cdef void sbdsdc(char *uplo, char *compq, int *n, s *d, s *e, s *u, int *ldu, s *vt, int *ldvt, s *q, int *iq, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sbdsdc(uplo, compq, n, d, e, u, ldu, vt, ldvt, q, iq, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sbdsqr "BLAS_FUNC(sbdsqr)"(char *uplo, int *n, int *ncvt, int *nru, int *ncc, s *d, s *e, s *vt, int *ldvt, s *u, int *ldu, s *c, int *ldc, s *work, int *info) nogil +cdef void sbdsqr(char *uplo, int *n, int *ncvt, int *nru, int *ncc, s *d, s *e, s *vt, int *ldvt, s *u, int *ldu, s *c, int *ldc, s *work, int *info) noexcept nogil: + + _fortran_sbdsqr(uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_scsum1 "BLAS_FUNC(scsum1)"(int *n, npy_complex64 *cx, int *incx) nogil +cdef s scsum1(int *n, c *cx, int *incx) noexcept nogil: + + return _fortran_scsum1(n, cx, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sdisna "BLAS_FUNC(sdisna)"(char *job, int *m, int *n, s *d, s *sep, int *info) nogil +cdef void sdisna(char *job, int *m, int *n, s *d, s *sep, int *info) noexcept nogil: + + _fortran_sdisna(job, m, n, d, sep, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgbbrd "BLAS_FUNC(sgbbrd)"(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, s *ab, int *ldab, s *d, s *e, s *q, int *ldq, s *pt, int *ldpt, s *c, int *ldc, s *work, int *info) nogil +cdef void sgbbrd(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, s *ab, int *ldab, s *d, s *e, s *q, int *ldq, s *pt, int *ldpt, s *c, int *ldc, s *work, int *info) noexcept nogil: + + _fortran_sgbbrd(vect, m, n, ncc, kl, ku, ab, ldab, d, e, q, ldq, pt, ldpt, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgbcon "BLAS_FUNC(sgbcon)"(char *norm, int *n, int *kl, int *ku, s *ab, int *ldab, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) nogil +cdef void sgbcon(char *norm, int *n, int *kl, int *ku, s *ab, int *ldab, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sgbcon(norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgbequ "BLAS_FUNC(sgbequ)"(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) nogil +cdef void sgbequ(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil: + + _fortran_sgbequ(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgbequb "BLAS_FUNC(sgbequb)"(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) nogil +cdef void sgbequb(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil: + + _fortran_sgbequb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgbrfs "BLAS_FUNC(sgbrfs)"(char *trans, int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void sgbrfs(char *trans, int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sgbrfs(trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgbsv "BLAS_FUNC(sgbsv)"(int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, int *ipiv, s *b, int *ldb, int *info) nogil +cdef void sgbsv(int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, int *ipiv, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_sgbsv(n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgbsvx "BLAS_FUNC(sgbsvx)"(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, int *ipiv, char *equed, s *r, s *c, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void sgbsvx(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, int *ipiv, char *equed, s *r, s *c, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sgbsvx(fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgbtf2 "BLAS_FUNC(sgbtf2)"(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, int *ipiv, int *info) nogil +cdef void sgbtf2(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, int *ipiv, int *info) noexcept nogil: + + _fortran_sgbtf2(m, n, kl, ku, ab, ldab, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgbtrf "BLAS_FUNC(sgbtrf)"(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, int *ipiv, int *info) nogil +cdef void sgbtrf(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, int *ipiv, int *info) noexcept nogil: + + _fortran_sgbtrf(m, n, kl, ku, ab, ldab, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgbtrs "BLAS_FUNC(sgbtrs)"(char *trans, int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, int *ipiv, s *b, int *ldb, int *info) nogil +cdef void sgbtrs(char *trans, int *n, int *kl, int *ku, int *nrhs, s *ab, int *ldab, int *ipiv, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_sgbtrs(trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgebak "BLAS_FUNC(sgebak)"(char *job, char *side, int *n, int *ilo, int *ihi, s *scale, int *m, s *v, int *ldv, int *info) nogil +cdef void sgebak(char *job, char *side, int *n, int *ilo, int *ihi, s *scale, int *m, s *v, int *ldv, int *info) noexcept nogil: + + _fortran_sgebak(job, side, n, ilo, ihi, scale, m, v, ldv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgebal "BLAS_FUNC(sgebal)"(char *job, int *n, s *a, int *lda, int *ilo, int *ihi, s *scale, int *info) nogil +cdef void sgebal(char *job, int *n, s *a, int *lda, int *ilo, int *ihi, s *scale, int *info) noexcept nogil: + + _fortran_sgebal(job, n, a, lda, ilo, ihi, scale, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgebd2 "BLAS_FUNC(sgebd2)"(int *m, int *n, s *a, int *lda, s *d, s *e, s *tauq, s *taup, s *work, int *info) nogil +cdef void sgebd2(int *m, int *n, s *a, int *lda, s *d, s *e, s *tauq, s *taup, s *work, int *info) noexcept nogil: + + _fortran_sgebd2(m, n, a, lda, d, e, tauq, taup, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgebrd "BLAS_FUNC(sgebrd)"(int *m, int *n, s *a, int *lda, s *d, s *e, s *tauq, s *taup, s *work, int *lwork, int *info) nogil +cdef void sgebrd(int *m, int *n, s *a, int *lda, s *d, s *e, s *tauq, s *taup, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgebrd(m, n, a, lda, d, e, tauq, taup, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgecon "BLAS_FUNC(sgecon)"(char *norm, int *n, s *a, int *lda, s *anorm, s *rcond, s *work, int *iwork, int *info) nogil +cdef void sgecon(char *norm, int *n, s *a, int *lda, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sgecon(norm, n, a, lda, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeequ "BLAS_FUNC(sgeequ)"(int *m, int *n, s *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) nogil +cdef void sgeequ(int *m, int *n, s *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil: + + _fortran_sgeequ(m, n, a, lda, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeequb "BLAS_FUNC(sgeequb)"(int *m, int *n, s *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) nogil +cdef void sgeequb(int *m, int *n, s *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, int *info) noexcept nogil: + + _fortran_sgeequb(m, n, a, lda, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgees "BLAS_FUNC(sgees)"(char *jobvs, char *sort, _sselect2 *select, int *n, s *a, int *lda, int *sdim, s *wr, s *wi, s *vs, int *ldvs, s *work, int *lwork, bint *bwork, int *info) nogil +cdef void sgees(char *jobvs, char *sort, sselect2 *select, int *n, s *a, int *lda, int *sdim, s *wr, s *wi, s *vs, int *ldvs, s *work, int *lwork, bint *bwork, int *info) noexcept nogil: + + _fortran_sgees(jobvs, sort, <_sselect2*>select, n, a, lda, sdim, wr, wi, vs, ldvs, work, lwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeesx "BLAS_FUNC(sgeesx)"(char *jobvs, char *sort, _sselect2 *select, char *sense, int *n, s *a, int *lda, int *sdim, s *wr, s *wi, s *vs, int *ldvs, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) nogil +cdef void sgeesx(char *jobvs, char *sort, sselect2 *select, char *sense, int *n, s *a, int *lda, int *sdim, s *wr, s *wi, s *vs, int *ldvs, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil: + + _fortran_sgeesx(jobvs, sort, <_sselect2*>select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde, rcondv, work, lwork, iwork, liwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeev "BLAS_FUNC(sgeev)"(char *jobvl, char *jobvr, int *n, s *a, int *lda, s *wr, s *wi, s *vl, int *ldvl, s *vr, int *ldvr, s *work, int *lwork, int *info) nogil +cdef void sgeev(char *jobvl, char *jobvr, int *n, s *a, int *lda, s *wr, s *wi, s *vl, int *ldvl, s *vr, int *ldvr, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgeev(jobvl, jobvr, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeevx "BLAS_FUNC(sgeevx)"(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, s *a, int *lda, s *wr, s *wi, s *vl, int *ldvl, s *vr, int *ldvr, int *ilo, int *ihi, s *scale, s *abnrm, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, int *info) nogil +cdef void sgeevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, s *a, int *lda, s *wr, s *wi, s *vl, int *ldvl, s *vr, int *ldvr, int *ilo, int *ihi, s *scale, s *abnrm, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_sgeevx(balanc, jobvl, jobvr, sense, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgehd2 "BLAS_FUNC(sgehd2)"(int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *work, int *info) nogil +cdef void sgehd2(int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil: + + _fortran_sgehd2(n, ilo, ihi, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgehrd "BLAS_FUNC(sgehrd)"(int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sgehrd(int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgehrd(n, ilo, ihi, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgejsv "BLAS_FUNC(sgejsv)"(char *joba, char *jobu, char *jobv, char *jobr, char *jobt, char *jobp, int *m, int *n, s *a, int *lda, s *sva, s *u, int *ldu, s *v, int *ldv, s *work, int *lwork, int *iwork, int *info) nogil +cdef void sgejsv(char *joba, char *jobu, char *jobv, char *jobr, char *jobt, char *jobp, int *m, int *n, s *a, int *lda, s *sva, s *u, int *ldu, s *v, int *ldv, s *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_sgejsv(joba, jobu, jobv, jobr, jobt, jobp, m, n, a, lda, sva, u, ldu, v, ldv, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgelq2 "BLAS_FUNC(sgelq2)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) nogil +cdef void sgelq2(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil: + + _fortran_sgelq2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgelqf "BLAS_FUNC(sgelqf)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sgelqf(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgelqf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgels "BLAS_FUNC(sgels)"(char *trans, int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *work, int *lwork, int *info) nogil +cdef void sgels(char *trans, int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgels(trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgelsd "BLAS_FUNC(sgelsd)"(int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *s, s *rcond, int *rank, s *work, int *lwork, int *iwork, int *info) nogil +cdef void sgelsd(int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *s, s *rcond, int *rank, s *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_sgelsd(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgelss "BLAS_FUNC(sgelss)"(int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *s, s *rcond, int *rank, s *work, int *lwork, int *info) nogil +cdef void sgelss(int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *s, s *rcond, int *rank, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgelss(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgelsy "BLAS_FUNC(sgelsy)"(int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *jpvt, s *rcond, int *rank, s *work, int *lwork, int *info) nogil +cdef void sgelsy(int *m, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *jpvt, s *rcond, int *rank, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgelsy(m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgemqrt "BLAS_FUNC(sgemqrt)"(char *side, char *trans, int *m, int *n, int *k, int *nb, s *v, int *ldv, s *t, int *ldt, s *c, int *ldc, s *work, int *info) nogil +cdef void sgemqrt(char *side, char *trans, int *m, int *n, int *k, int *nb, s *v, int *ldv, s *t, int *ldt, s *c, int *ldc, s *work, int *info) noexcept nogil: + + _fortran_sgemqrt(side, trans, m, n, k, nb, v, ldv, t, ldt, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeql2 "BLAS_FUNC(sgeql2)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) nogil +cdef void sgeql2(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil: + + _fortran_sgeql2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeqlf "BLAS_FUNC(sgeqlf)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sgeqlf(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgeqlf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeqp3 "BLAS_FUNC(sgeqp3)"(int *m, int *n, s *a, int *lda, int *jpvt, s *tau, s *work, int *lwork, int *info) nogil +cdef void sgeqp3(int *m, int *n, s *a, int *lda, int *jpvt, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgeqp3(m, n, a, lda, jpvt, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeqr2 "BLAS_FUNC(sgeqr2)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) nogil +cdef void sgeqr2(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil: + + _fortran_sgeqr2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeqr2p "BLAS_FUNC(sgeqr2p)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) nogil +cdef void sgeqr2p(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil: + + _fortran_sgeqr2p(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeqrf "BLAS_FUNC(sgeqrf)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sgeqrf(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgeqrf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeqrfp "BLAS_FUNC(sgeqrfp)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sgeqrfp(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgeqrfp(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeqrt "BLAS_FUNC(sgeqrt)"(int *m, int *n, int *nb, s *a, int *lda, s *t, int *ldt, s *work, int *info) nogil +cdef void sgeqrt(int *m, int *n, int *nb, s *a, int *lda, s *t, int *ldt, s *work, int *info) noexcept nogil: + + _fortran_sgeqrt(m, n, nb, a, lda, t, ldt, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeqrt2 "BLAS_FUNC(sgeqrt2)"(int *m, int *n, s *a, int *lda, s *t, int *ldt, int *info) nogil +cdef void sgeqrt2(int *m, int *n, s *a, int *lda, s *t, int *ldt, int *info) noexcept nogil: + + _fortran_sgeqrt2(m, n, a, lda, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgeqrt3 "BLAS_FUNC(sgeqrt3)"(int *m, int *n, s *a, int *lda, s *t, int *ldt, int *info) nogil +cdef void sgeqrt3(int *m, int *n, s *a, int *lda, s *t, int *ldt, int *info) noexcept nogil: + + _fortran_sgeqrt3(m, n, a, lda, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgerfs "BLAS_FUNC(sgerfs)"(char *trans, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void sgerfs(char *trans, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sgerfs(trans, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgerq2 "BLAS_FUNC(sgerq2)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) nogil +cdef void sgerq2(int *m, int *n, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil: + + _fortran_sgerq2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgerqf "BLAS_FUNC(sgerqf)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sgerqf(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgerqf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgesc2 "BLAS_FUNC(sgesc2)"(int *n, s *a, int *lda, s *rhs, int *ipiv, int *jpiv, s *scale) nogil +cdef void sgesc2(int *n, s *a, int *lda, s *rhs, int *ipiv, int *jpiv, s *scale) noexcept nogil: + + _fortran_sgesc2(n, a, lda, rhs, ipiv, jpiv, scale) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgesdd "BLAS_FUNC(sgesdd)"(char *jobz, int *m, int *n, s *a, int *lda, s *s, s *u, int *ldu, s *vt, int *ldvt, s *work, int *lwork, int *iwork, int *info) nogil +cdef void sgesdd(char *jobz, int *m, int *n, s *a, int *lda, s *s, s *u, int *ldu, s *vt, int *ldvt, s *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_sgesdd(jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgesv "BLAS_FUNC(sgesv)"(int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, int *info) nogil +cdef void sgesv(int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_sgesv(n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgesvd "BLAS_FUNC(sgesvd)"(char *jobu, char *jobvt, int *m, int *n, s *a, int *lda, s *s, s *u, int *ldu, s *vt, int *ldvt, s *work, int *lwork, int *info) nogil +cdef void sgesvd(char *jobu, char *jobvt, int *m, int *n, s *a, int *lda, s *s, s *u, int *ldu, s *vt, int *ldvt, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgesvd(jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgesvj "BLAS_FUNC(sgesvj)"(char *joba, char *jobu, char *jobv, int *m, int *n, s *a, int *lda, s *sva, int *mv, s *v, int *ldv, s *work, int *lwork, int *info) nogil +cdef void sgesvj(char *joba, char *jobu, char *jobv, int *m, int *n, s *a, int *lda, s *sva, int *mv, s *v, int *ldv, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgesvj(joba, jobu, jobv, m, n, a, lda, sva, mv, v, ldv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgesvx "BLAS_FUNC(sgesvx)"(char *fact, char *trans, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, char *equed, s *r, s *c, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void sgesvx(char *fact, char *trans, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, char *equed, s *r, s *c, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sgesvx(fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgetc2 "BLAS_FUNC(sgetc2)"(int *n, s *a, int *lda, int *ipiv, int *jpiv, int *info) nogil +cdef void sgetc2(int *n, s *a, int *lda, int *ipiv, int *jpiv, int *info) noexcept nogil: + + _fortran_sgetc2(n, a, lda, ipiv, jpiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgetf2 "BLAS_FUNC(sgetf2)"(int *m, int *n, s *a, int *lda, int *ipiv, int *info) nogil +cdef void sgetf2(int *m, int *n, s *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_sgetf2(m, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgetrf "BLAS_FUNC(sgetrf)"(int *m, int *n, s *a, int *lda, int *ipiv, int *info) nogil +cdef void sgetrf(int *m, int *n, s *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_sgetrf(m, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgetri "BLAS_FUNC(sgetri)"(int *n, s *a, int *lda, int *ipiv, s *work, int *lwork, int *info) nogil +cdef void sgetri(int *n, s *a, int *lda, int *ipiv, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgetri(n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgetrs "BLAS_FUNC(sgetrs)"(char *trans, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, int *info) nogil +cdef void sgetrs(char *trans, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_sgetrs(trans, n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sggbak "BLAS_FUNC(sggbak)"(char *job, char *side, int *n, int *ilo, int *ihi, s *lscale, s *rscale, int *m, s *v, int *ldv, int *info) nogil +cdef void sggbak(char *job, char *side, int *n, int *ilo, int *ihi, s *lscale, s *rscale, int *m, s *v, int *ldv, int *info) noexcept nogil: + + _fortran_sggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sggbal "BLAS_FUNC(sggbal)"(char *job, int *n, s *a, int *lda, s *b, int *ldb, int *ilo, int *ihi, s *lscale, s *rscale, s *work, int *info) nogil +cdef void sggbal(char *job, int *n, s *a, int *lda, s *b, int *ldb, int *ilo, int *ihi, s *lscale, s *rscale, s *work, int *info) noexcept nogil: + + _fortran_sggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgges "BLAS_FUNC(sgges)"(char *jobvsl, char *jobvsr, char *sort, _sselect3 *selctg, int *n, s *a, int *lda, s *b, int *ldb, int *sdim, s *alphar, s *alphai, s *beta, s *vsl, int *ldvsl, s *vsr, int *ldvsr, s *work, int *lwork, bint *bwork, int *info) nogil +cdef void sgges(char *jobvsl, char *jobvsr, char *sort, sselect3 *selctg, int *n, s *a, int *lda, s *b, int *ldb, int *sdim, s *alphar, s *alphai, s *beta, s *vsl, int *ldvsl, s *vsr, int *ldvsr, s *work, int *lwork, bint *bwork, int *info) noexcept nogil: + + _fortran_sgges(jobvsl, jobvsr, sort, <_sselect3*>selctg, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sggesx "BLAS_FUNC(sggesx)"(char *jobvsl, char *jobvsr, char *sort, _sselect3 *selctg, char *sense, int *n, s *a, int *lda, s *b, int *ldb, int *sdim, s *alphar, s *alphai, s *beta, s *vsl, int *ldvsl, s *vsr, int *ldvsr, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) nogil +cdef void sggesx(char *jobvsl, char *jobvsr, char *sort, sselect3 *selctg, char *sense, int *n, s *a, int *lda, s *b, int *ldb, int *sdim, s *alphar, s *alphai, s *beta, s *vsl, int *ldvsl, s *vsr, int *ldvsr, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil: + + _fortran_sggesx(jobvsl, jobvsr, sort, <_sselect3*>selctg, sense, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, iwork, liwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sggev "BLAS_FUNC(sggev)"(char *jobvl, char *jobvr, int *n, s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *vl, int *ldvl, s *vr, int *ldvr, s *work, int *lwork, int *info) nogil +cdef void sggev(char *jobvl, char *jobvr, int *n, s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *vl, int *ldvl, s *vr, int *ldvr, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sggev(jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sggevx "BLAS_FUNC(sggevx)"(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *vl, int *ldvl, s *vr, int *ldvr, int *ilo, int *ihi, s *lscale, s *rscale, s *abnrm, s *bbnrm, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, bint *bwork, int *info) nogil +cdef void sggevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *vl, int *ldvl, s *vr, int *ldvr, int *ilo, int *ihi, s *lscale, s *rscale, s *abnrm, s *bbnrm, s *rconde, s *rcondv, s *work, int *lwork, int *iwork, bint *bwork, int *info) noexcept nogil: + + _fortran_sggevx(balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sggglm "BLAS_FUNC(sggglm)"(int *n, int *m, int *p, s *a, int *lda, s *b, int *ldb, s *d, s *x, s *y, s *work, int *lwork, int *info) nogil +cdef void sggglm(int *n, int *m, int *p, s *a, int *lda, s *b, int *ldb, s *d, s *x, s *y, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sggglm(n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgghrd "BLAS_FUNC(sgghrd)"(char *compq, char *compz, int *n, int *ilo, int *ihi, s *a, int *lda, s *b, int *ldb, s *q, int *ldq, s *z, int *ldz, int *info) nogil +cdef void sgghrd(char *compq, char *compz, int *n, int *ilo, int *ihi, s *a, int *lda, s *b, int *ldb, s *q, int *ldq, s *z, int *ldz, int *info) noexcept nogil: + + _fortran_sgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgglse "BLAS_FUNC(sgglse)"(int *m, int *n, int *p, s *a, int *lda, s *b, int *ldb, s *c, s *d, s *x, s *work, int *lwork, int *info) nogil +cdef void sgglse(int *m, int *n, int *p, s *a, int *lda, s *b, int *ldb, s *c, s *d, s *x, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgglse(m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sggqrf "BLAS_FUNC(sggqrf)"(int *n, int *m, int *p, s *a, int *lda, s *taua, s *b, int *ldb, s *taub, s *work, int *lwork, int *info) nogil +cdef void sggqrf(int *n, int *m, int *p, s *a, int *lda, s *taua, s *b, int *ldb, s *taub, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sggqrf(n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sggrqf "BLAS_FUNC(sggrqf)"(int *m, int *p, int *n, s *a, int *lda, s *taua, s *b, int *ldb, s *taub, s *work, int *lwork, int *info) nogil +cdef void sggrqf(int *m, int *p, int *n, s *a, int *lda, s *taua, s *b, int *ldb, s *taub, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sggrqf(m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgsvj0 "BLAS_FUNC(sgsvj0)"(char *jobv, int *m, int *n, s *a, int *lda, s *d, s *sva, int *mv, s *v, int *ldv, s *eps, s *sfmin, s *tol, int *nsweep, s *work, int *lwork, int *info) nogil +cdef void sgsvj0(char *jobv, int *m, int *n, s *a, int *lda, s *d, s *sva, int *mv, s *v, int *ldv, s *eps, s *sfmin, s *tol, int *nsweep, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgsvj0(jobv, m, n, a, lda, d, sva, mv, v, ldv, eps, sfmin, tol, nsweep, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgsvj1 "BLAS_FUNC(sgsvj1)"(char *jobv, int *m, int *n, int *n1, s *a, int *lda, s *d, s *sva, int *mv, s *v, int *ldv, s *eps, s *sfmin, s *tol, int *nsweep, s *work, int *lwork, int *info) nogil +cdef void sgsvj1(char *jobv, int *m, int *n, int *n1, s *a, int *lda, s *d, s *sva, int *mv, s *v, int *ldv, s *eps, s *sfmin, s *tol, int *nsweep, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sgsvj1(jobv, m, n, n1, a, lda, d, sva, mv, v, ldv, eps, sfmin, tol, nsweep, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgtcon "BLAS_FUNC(sgtcon)"(char *norm, int *n, s *dl, s *d, s *du, s *du2, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) nogil +cdef void sgtcon(char *norm, int *n, s *dl, s *d, s *du, s *du2, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sgtcon(norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgtrfs "BLAS_FUNC(sgtrfs)"(char *trans, int *n, int *nrhs, s *dl, s *d, s *du, s *dlf, s *df, s *duf, s *du2, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void sgtrfs(char *trans, int *n, int *nrhs, s *dl, s *d, s *du, s *dlf, s *df, s *duf, s *du2, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sgtrfs(trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgtsv "BLAS_FUNC(sgtsv)"(int *n, int *nrhs, s *dl, s *d, s *du, s *b, int *ldb, int *info) nogil +cdef void sgtsv(int *n, int *nrhs, s *dl, s *d, s *du, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_sgtsv(n, nrhs, dl, d, du, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgtsvx "BLAS_FUNC(sgtsvx)"(char *fact, char *trans, int *n, int *nrhs, s *dl, s *d, s *du, s *dlf, s *df, s *duf, s *du2, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void sgtsvx(char *fact, char *trans, int *n, int *nrhs, s *dl, s *d, s *du, s *dlf, s *df, s *duf, s *du2, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sgtsvx(fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgttrf "BLAS_FUNC(sgttrf)"(int *n, s *dl, s *d, s *du, s *du2, int *ipiv, int *info) nogil +cdef void sgttrf(int *n, s *dl, s *d, s *du, s *du2, int *ipiv, int *info) noexcept nogil: + + _fortran_sgttrf(n, dl, d, du, du2, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgttrs "BLAS_FUNC(sgttrs)"(char *trans, int *n, int *nrhs, s *dl, s *d, s *du, s *du2, int *ipiv, s *b, int *ldb, int *info) nogil +cdef void sgttrs(char *trans, int *n, int *nrhs, s *dl, s *d, s *du, s *du2, int *ipiv, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_sgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sgtts2 "BLAS_FUNC(sgtts2)"(int *itrans, int *n, int *nrhs, s *dl, s *d, s *du, s *du2, int *ipiv, s *b, int *ldb) nogil +cdef void sgtts2(int *itrans, int *n, int *nrhs, s *dl, s *d, s *du, s *du2, int *ipiv, s *b, int *ldb) noexcept nogil: + + _fortran_sgtts2(itrans, n, nrhs, dl, d, du, du2, ipiv, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_shgeqz "BLAS_FUNC(shgeqz)"(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *t, int *ldt, s *alphar, s *alphai, s *beta, s *q, int *ldq, s *z, int *ldz, s *work, int *lwork, int *info) nogil +cdef void shgeqz(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *t, int *ldt, s *alphar, s *alphai, s *beta, s *q, int *ldq, s *z, int *ldz, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_shgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_shsein "BLAS_FUNC(shsein)"(char *side, char *eigsrc, char *initv, bint *select, int *n, s *h, int *ldh, s *wr, s *wi, s *vl, int *ldvl, s *vr, int *ldvr, int *mm, int *m, s *work, int *ifaill, int *ifailr, int *info) nogil +cdef void shsein(char *side, char *eigsrc, char *initv, bint *select, int *n, s *h, int *ldh, s *wr, s *wi, s *vl, int *ldvl, s *vr, int *ldvr, int *mm, int *m, s *work, int *ifaill, int *ifailr, int *info) noexcept nogil: + + _fortran_shsein(side, eigsrc, initv, select, n, h, ldh, wr, wi, vl, ldvl, vr, ldvr, mm, m, work, ifaill, ifailr, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_shseqr "BLAS_FUNC(shseqr)"(char *job, char *compz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, s *z, int *ldz, s *work, int *lwork, int *info) nogil +cdef void shseqr(char *job, char *compz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, s *z, int *ldz, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_shseqr(job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slabad "BLAS_FUNC(slabad)"(s *small, s *large) nogil +cdef void slabad(s *small, s *large) noexcept nogil: + + _fortran_slabad(small, large) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slabrd "BLAS_FUNC(slabrd)"(int *m, int *n, int *nb, s *a, int *lda, s *d, s *e, s *tauq, s *taup, s *x, int *ldx, s *y, int *ldy) nogil +cdef void slabrd(int *m, int *n, int *nb, s *a, int *lda, s *d, s *e, s *tauq, s *taup, s *x, int *ldx, s *y, int *ldy) noexcept nogil: + + _fortran_slabrd(m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slacn2 "BLAS_FUNC(slacn2)"(int *n, s *v, s *x, int *isgn, s *est, int *kase, int *isave) nogil +cdef void slacn2(int *n, s *v, s *x, int *isgn, s *est, int *kase, int *isave) noexcept nogil: + + _fortran_slacn2(n, v, x, isgn, est, kase, isave) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slacon "BLAS_FUNC(slacon)"(int *n, s *v, s *x, int *isgn, s *est, int *kase) nogil +cdef void slacon(int *n, s *v, s *x, int *isgn, s *est, int *kase) noexcept nogil: + + _fortran_slacon(n, v, x, isgn, est, kase) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slacpy "BLAS_FUNC(slacpy)"(char *uplo, int *m, int *n, s *a, int *lda, s *b, int *ldb) nogil +cdef void slacpy(char *uplo, int *m, int *n, s *a, int *lda, s *b, int *ldb) noexcept nogil: + + _fortran_slacpy(uplo, m, n, a, lda, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sladiv "BLAS_FUNC(sladiv)"(s *a, s *b, s *c, s *d, s *p, s *q) nogil +cdef void sladiv(s *a, s *b, s *c, s *d, s *p, s *q) noexcept nogil: + + _fortran_sladiv(a, b, c, d, p, q) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slae2 "BLAS_FUNC(slae2)"(s *a, s *b, s *c, s *rt1, s *rt2) nogil +cdef void slae2(s *a, s *b, s *c, s *rt1, s *rt2) noexcept nogil: + + _fortran_slae2(a, b, c, rt1, rt2) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaebz "BLAS_FUNC(slaebz)"(int *ijob, int *nitmax, int *n, int *mmax, int *minp, int *nbmin, s *abstol, s *reltol, s *pivmin, s *d, s *e, s *e2, int *nval, s *ab, s *c, int *mout, int *nab, s *work, int *iwork, int *info) nogil +cdef void slaebz(int *ijob, int *nitmax, int *n, int *mmax, int *minp, int *nbmin, s *abstol, s *reltol, s *pivmin, s *d, s *e, s *e2, int *nval, s *ab, s *c, int *mout, int *nab, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slaebz(ijob, nitmax, n, mmax, minp, nbmin, abstol, reltol, pivmin, d, e, e2, nval, ab, c, mout, nab, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaed0 "BLAS_FUNC(slaed0)"(int *icompq, int *qsiz, int *n, s *d, s *e, s *q, int *ldq, s *qstore, int *ldqs, s *work, int *iwork, int *info) nogil +cdef void slaed0(int *icompq, int *qsiz, int *n, s *d, s *e, s *q, int *ldq, s *qstore, int *ldqs, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slaed0(icompq, qsiz, n, d, e, q, ldq, qstore, ldqs, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaed1 "BLAS_FUNC(slaed1)"(int *n, s *d, s *q, int *ldq, int *indxq, s *rho, int *cutpnt, s *work, int *iwork, int *info) nogil +cdef void slaed1(int *n, s *d, s *q, int *ldq, int *indxq, s *rho, int *cutpnt, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slaed1(n, d, q, ldq, indxq, rho, cutpnt, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaed2 "BLAS_FUNC(slaed2)"(int *k, int *n, int *n1, s *d, s *q, int *ldq, int *indxq, s *rho, s *z, s *dlamda, s *w, s *q2, int *indx, int *indxc, int *indxp, int *coltyp, int *info) nogil +cdef void slaed2(int *k, int *n, int *n1, s *d, s *q, int *ldq, int *indxq, s *rho, s *z, s *dlamda, s *w, s *q2, int *indx, int *indxc, int *indxp, int *coltyp, int *info) noexcept nogil: + + _fortran_slaed2(k, n, n1, d, q, ldq, indxq, rho, z, dlamda, w, q2, indx, indxc, indxp, coltyp, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaed3 "BLAS_FUNC(slaed3)"(int *k, int *n, int *n1, s *d, s *q, int *ldq, s *rho, s *dlamda, s *q2, int *indx, int *ctot, s *w, s *s, int *info) nogil +cdef void slaed3(int *k, int *n, int *n1, s *d, s *q, int *ldq, s *rho, s *dlamda, s *q2, int *indx, int *ctot, s *w, s *s, int *info) noexcept nogil: + + _fortran_slaed3(k, n, n1, d, q, ldq, rho, dlamda, q2, indx, ctot, w, s, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaed4 "BLAS_FUNC(slaed4)"(int *n, int *i, s *d, s *z, s *delta, s *rho, s *dlam, int *info) nogil +cdef void slaed4(int *n, int *i, s *d, s *z, s *delta, s *rho, s *dlam, int *info) noexcept nogil: + + _fortran_slaed4(n, i, d, z, delta, rho, dlam, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaed5 "BLAS_FUNC(slaed5)"(int *i, s *d, s *z, s *delta, s *rho, s *dlam) nogil +cdef void slaed5(int *i, s *d, s *z, s *delta, s *rho, s *dlam) noexcept nogil: + + _fortran_slaed5(i, d, z, delta, rho, dlam) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaed6 "BLAS_FUNC(slaed6)"(int *kniter, bint *orgati, s *rho, s *d, s *z, s *finit, s *tau, int *info) nogil +cdef void slaed6(int *kniter, bint *orgati, s *rho, s *d, s *z, s *finit, s *tau, int *info) noexcept nogil: + + _fortran_slaed6(kniter, orgati, rho, d, z, finit, tau, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaed7 "BLAS_FUNC(slaed7)"(int *icompq, int *n, int *qsiz, int *tlvls, int *curlvl, int *curpbm, s *d, s *q, int *ldq, int *indxq, s *rho, int *cutpnt, s *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, s *givnum, s *work, int *iwork, int *info) nogil +cdef void slaed7(int *icompq, int *n, int *qsiz, int *tlvls, int *curlvl, int *curpbm, s *d, s *q, int *ldq, int *indxq, s *rho, int *cutpnt, s *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, s *givnum, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slaed7(icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq, indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaed8 "BLAS_FUNC(slaed8)"(int *icompq, int *k, int *n, int *qsiz, s *d, s *q, int *ldq, int *indxq, s *rho, int *cutpnt, s *z, s *dlamda, s *q2, int *ldq2, s *w, int *perm, int *givptr, int *givcol, s *givnum, int *indxp, int *indx, int *info) nogil +cdef void slaed8(int *icompq, int *k, int *n, int *qsiz, s *d, s *q, int *ldq, int *indxq, s *rho, int *cutpnt, s *z, s *dlamda, s *q2, int *ldq2, s *w, int *perm, int *givptr, int *givcol, s *givnum, int *indxp, int *indx, int *info) noexcept nogil: + + _fortran_slaed8(icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlamda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaed9 "BLAS_FUNC(slaed9)"(int *k, int *kstart, int *kstop, int *n, s *d, s *q, int *ldq, s *rho, s *dlamda, s *w, s *s, int *lds, int *info) nogil +cdef void slaed9(int *k, int *kstart, int *kstop, int *n, s *d, s *q, int *ldq, s *rho, s *dlamda, s *w, s *s, int *lds, int *info) noexcept nogil: + + _fortran_slaed9(k, kstart, kstop, n, d, q, ldq, rho, dlamda, w, s, lds, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaeda "BLAS_FUNC(slaeda)"(int *n, int *tlvls, int *curlvl, int *curpbm, int *prmptr, int *perm, int *givptr, int *givcol, s *givnum, s *q, int *qptr, s *z, s *ztemp, int *info) nogil +cdef void slaeda(int *n, int *tlvls, int *curlvl, int *curpbm, int *prmptr, int *perm, int *givptr, int *givcol, s *givnum, s *q, int *qptr, s *z, s *ztemp, int *info) noexcept nogil: + + _fortran_slaeda(n, tlvls, curlvl, curpbm, prmptr, perm, givptr, givcol, givnum, q, qptr, z, ztemp, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaein "BLAS_FUNC(slaein)"(bint *rightv, bint *noinit, int *n, s *h, int *ldh, s *wr, s *wi, s *vr, s *vi, s *b, int *ldb, s *work, s *eps3, s *smlnum, s *bignum, int *info) nogil +cdef void slaein(bint *rightv, bint *noinit, int *n, s *h, int *ldh, s *wr, s *wi, s *vr, s *vi, s *b, int *ldb, s *work, s *eps3, s *smlnum, s *bignum, int *info) noexcept nogil: + + _fortran_slaein(rightv, noinit, n, h, ldh, wr, wi, vr, vi, b, ldb, work, eps3, smlnum, bignum, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaev2 "BLAS_FUNC(slaev2)"(s *a, s *b, s *c, s *rt1, s *rt2, s *cs1, s *sn1) nogil +cdef void slaev2(s *a, s *b, s *c, s *rt1, s *rt2, s *cs1, s *sn1) noexcept nogil: + + _fortran_slaev2(a, b, c, rt1, rt2, cs1, sn1) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaexc "BLAS_FUNC(slaexc)"(bint *wantq, int *n, s *t, int *ldt, s *q, int *ldq, int *j1, int *n1, int *n2, s *work, int *info) nogil +cdef void slaexc(bint *wantq, int *n, s *t, int *ldt, s *q, int *ldq, int *j1, int *n1, int *n2, s *work, int *info) noexcept nogil: + + _fortran_slaexc(wantq, n, t, ldt, q, ldq, j1, n1, n2, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slag2 "BLAS_FUNC(slag2)"(s *a, int *lda, s *b, int *ldb, s *safmin, s *scale1, s *scale2, s *wr1, s *wr2, s *wi) nogil +cdef void slag2(s *a, int *lda, s *b, int *ldb, s *safmin, s *scale1, s *scale2, s *wr1, s *wr2, s *wi) noexcept nogil: + + _fortran_slag2(a, lda, b, ldb, safmin, scale1, scale2, wr1, wr2, wi) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slag2d "BLAS_FUNC(slag2d)"(int *m, int *n, s *sa, int *ldsa, d *a, int *lda, int *info) nogil +cdef void slag2d(int *m, int *n, s *sa, int *ldsa, d *a, int *lda, int *info) noexcept nogil: + + _fortran_slag2d(m, n, sa, ldsa, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slags2 "BLAS_FUNC(slags2)"(bint *upper, s *a1, s *a2, s *a3, s *b1, s *b2, s *b3, s *csu, s *snu, s *csv, s *snv, s *csq, s *snq) nogil +cdef void slags2(bint *upper, s *a1, s *a2, s *a3, s *b1, s *b2, s *b3, s *csu, s *snu, s *csv, s *snv, s *csq, s *snq) noexcept nogil: + + _fortran_slags2(upper, a1, a2, a3, b1, b2, b3, csu, snu, csv, snv, csq, snq) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slagtf "BLAS_FUNC(slagtf)"(int *n, s *a, s *lambda_, s *b, s *c, s *tol, s *d, int *in_, int *info) nogil +cdef void slagtf(int *n, s *a, s *lambda_, s *b, s *c, s *tol, s *d, int *in_, int *info) noexcept nogil: + + _fortran_slagtf(n, a, lambda_, b, c, tol, d, in_, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slagtm "BLAS_FUNC(slagtm)"(char *trans, int *n, int *nrhs, s *alpha, s *dl, s *d, s *du, s *x, int *ldx, s *beta, s *b, int *ldb) nogil +cdef void slagtm(char *trans, int *n, int *nrhs, s *alpha, s *dl, s *d, s *du, s *x, int *ldx, s *beta, s *b, int *ldb) noexcept nogil: + + _fortran_slagtm(trans, n, nrhs, alpha, dl, d, du, x, ldx, beta, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slagts "BLAS_FUNC(slagts)"(int *job, int *n, s *a, s *b, s *c, s *d, int *in_, s *y, s *tol, int *info) nogil +cdef void slagts(int *job, int *n, s *a, s *b, s *c, s *d, int *in_, s *y, s *tol, int *info) noexcept nogil: + + _fortran_slagts(job, n, a, b, c, d, in_, y, tol, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slagv2 "BLAS_FUNC(slagv2)"(s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *csl, s *snl, s *csr, s *snr) nogil +cdef void slagv2(s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *csl, s *snl, s *csr, s *snr) noexcept nogil: + + _fortran_slagv2(a, lda, b, ldb, alphar, alphai, beta, csl, snl, csr, snr) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slahqr "BLAS_FUNC(slahqr)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, int *iloz, int *ihiz, s *z, int *ldz, int *info) nogil +cdef void slahqr(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, int *iloz, int *ihiz, s *z, int *ldz, int *info) noexcept nogil: + + _fortran_slahqr(wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slahr2 "BLAS_FUNC(slahr2)"(int *n, int *k, int *nb, s *a, int *lda, s *tau, s *t, int *ldt, s *y, int *ldy) nogil +cdef void slahr2(int *n, int *k, int *nb, s *a, int *lda, s *tau, s *t, int *ldt, s *y, int *ldy) noexcept nogil: + + _fortran_slahr2(n, k, nb, a, lda, tau, t, ldt, y, ldy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaic1 "BLAS_FUNC(slaic1)"(int *job, int *j, s *x, s *sest, s *w, s *gamma, s *sestpr, s *s, s *c) nogil +cdef void slaic1(int *job, int *j, s *x, s *sest, s *w, s *gamma, s *sestpr, s *s, s *c) noexcept nogil: + + _fortran_slaic1(job, j, x, sest, w, gamma, sestpr, s, c) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaln2 "BLAS_FUNC(slaln2)"(bint *ltrans, int *na, int *nw, s *smin, s *ca, s *a, int *lda, s *d1, s *d2, s *b, int *ldb, s *wr, s *wi, s *x, int *ldx, s *scale, s *xnorm, int *info) nogil +cdef void slaln2(bint *ltrans, int *na, int *nw, s *smin, s *ca, s *a, int *lda, s *d1, s *d2, s *b, int *ldb, s *wr, s *wi, s *x, int *ldx, s *scale, s *xnorm, int *info) noexcept nogil: + + _fortran_slaln2(ltrans, na, nw, smin, ca, a, lda, d1, d2, b, ldb, wr, wi, x, ldx, scale, xnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slals0 "BLAS_FUNC(slals0)"(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, s *b, int *ldb, s *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *poles, s *difl, s *difr, s *z, int *k, s *c, s *s, s *work, int *info) nogil +cdef void slals0(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, s *b, int *ldb, s *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *poles, s *difl, s *difr, s *z, int *k, s *c, s *s, s *work, int *info) noexcept nogil: + + _fortran_slals0(icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slalsa "BLAS_FUNC(slalsa)"(int *icompq, int *smlsiz, int *n, int *nrhs, s *b, int *ldb, s *bx, int *ldbx, s *u, int *ldu, s *vt, int *k, s *difl, s *difr, s *z, s *poles, int *givptr, int *givcol, int *ldgcol, int *perm, s *givnum, s *c, s *s, s *work, int *iwork, int *info) nogil +cdef void slalsa(int *icompq, int *smlsiz, int *n, int *nrhs, s *b, int *ldb, s *bx, int *ldbx, s *u, int *ldu, s *vt, int *k, s *difl, s *difr, s *z, s *poles, int *givptr, int *givcol, int *ldgcol, int *perm, s *givnum, s *c, s *s, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slalsa(icompq, smlsiz, n, nrhs, b, ldb, bx, ldbx, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slalsd "BLAS_FUNC(slalsd)"(char *uplo, int *smlsiz, int *n, int *nrhs, s *d, s *e, s *b, int *ldb, s *rcond, int *rank, s *work, int *iwork, int *info) nogil +cdef void slalsd(char *uplo, int *smlsiz, int *n, int *nrhs, s *d, s *e, s *b, int *ldb, s *rcond, int *rank, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slalsd(uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slamch "BLAS_FUNC(slamch)"(char *cmach) nogil +cdef s slamch(char *cmach) noexcept nogil: + + return _fortran_slamch(cmach) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slamrg "BLAS_FUNC(slamrg)"(int *n1, int *n2, s *a, int *strd1, int *strd2, int *index_bn) nogil +cdef void slamrg(int *n1, int *n2, s *a, int *strd1, int *strd2, int *index_bn) noexcept nogil: + + _fortran_slamrg(n1, n2, a, strd1, strd2, index_bn) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slangb "BLAS_FUNC(slangb)"(char *norm, int *n, int *kl, int *ku, s *ab, int *ldab, s *work) nogil +cdef s slangb(char *norm, int *n, int *kl, int *ku, s *ab, int *ldab, s *work) noexcept nogil: + + return _fortran_slangb(norm, n, kl, ku, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slange "BLAS_FUNC(slange)"(char *norm, int *m, int *n, s *a, int *lda, s *work) nogil +cdef s slange(char *norm, int *m, int *n, s *a, int *lda, s *work) noexcept nogil: + + return _fortran_slange(norm, m, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slangt "BLAS_FUNC(slangt)"(char *norm, int *n, s *dl, s *d, s *du) nogil +cdef s slangt(char *norm, int *n, s *dl, s *d, s *du) noexcept nogil: + + return _fortran_slangt(norm, n, dl, d, du) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slanhs "BLAS_FUNC(slanhs)"(char *norm, int *n, s *a, int *lda, s *work) nogil +cdef s slanhs(char *norm, int *n, s *a, int *lda, s *work) noexcept nogil: + + return _fortran_slanhs(norm, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slansb "BLAS_FUNC(slansb)"(char *norm, char *uplo, int *n, int *k, s *ab, int *ldab, s *work) nogil +cdef s slansb(char *norm, char *uplo, int *n, int *k, s *ab, int *ldab, s *work) noexcept nogil: + + return _fortran_slansb(norm, uplo, n, k, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slansf "BLAS_FUNC(slansf)"(char *norm, char *transr, char *uplo, int *n, s *a, s *work) nogil +cdef s slansf(char *norm, char *transr, char *uplo, int *n, s *a, s *work) noexcept nogil: + + return _fortran_slansf(norm, transr, uplo, n, a, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slansp "BLAS_FUNC(slansp)"(char *norm, char *uplo, int *n, s *ap, s *work) nogil +cdef s slansp(char *norm, char *uplo, int *n, s *ap, s *work) noexcept nogil: + + return _fortran_slansp(norm, uplo, n, ap, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slanst "BLAS_FUNC(slanst)"(char *norm, int *n, s *d, s *e) nogil +cdef s slanst(char *norm, int *n, s *d, s *e) noexcept nogil: + + return _fortran_slanst(norm, n, d, e) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slansy "BLAS_FUNC(slansy)"(char *norm, char *uplo, int *n, s *a, int *lda, s *work) nogil +cdef s slansy(char *norm, char *uplo, int *n, s *a, int *lda, s *work) noexcept nogil: + + return _fortran_slansy(norm, uplo, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slantb "BLAS_FUNC(slantb)"(char *norm, char *uplo, char *diag, int *n, int *k, s *ab, int *ldab, s *work) nogil +cdef s slantb(char *norm, char *uplo, char *diag, int *n, int *k, s *ab, int *ldab, s *work) noexcept nogil: + + return _fortran_slantb(norm, uplo, diag, n, k, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slantp "BLAS_FUNC(slantp)"(char *norm, char *uplo, char *diag, int *n, s *ap, s *work) nogil +cdef s slantp(char *norm, char *uplo, char *diag, int *n, s *ap, s *work) noexcept nogil: + + return _fortran_slantp(norm, uplo, diag, n, ap, work) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slantr "BLAS_FUNC(slantr)"(char *norm, char *uplo, char *diag, int *m, int *n, s *a, int *lda, s *work) nogil +cdef s slantr(char *norm, char *uplo, char *diag, int *m, int *n, s *a, int *lda, s *work) noexcept nogil: + + return _fortran_slantr(norm, uplo, diag, m, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slanv2 "BLAS_FUNC(slanv2)"(s *a, s *b, s *c, s *d, s *rt1r, s *rt1i, s *rt2r, s *rt2i, s *cs, s *sn) nogil +cdef void slanv2(s *a, s *b, s *c, s *d, s *rt1r, s *rt1i, s *rt2r, s *rt2i, s *cs, s *sn) noexcept nogil: + + _fortran_slanv2(a, b, c, d, rt1r, rt1i, rt2r, rt2i, cs, sn) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slapll "BLAS_FUNC(slapll)"(int *n, s *x, int *incx, s *y, int *incy, s *ssmin) nogil +cdef void slapll(int *n, s *x, int *incx, s *y, int *incy, s *ssmin) noexcept nogil: + + _fortran_slapll(n, x, incx, y, incy, ssmin) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slapmr "BLAS_FUNC(slapmr)"(bint *forwrd, int *m, int *n, s *x, int *ldx, int *k) nogil +cdef void slapmr(bint *forwrd, int *m, int *n, s *x, int *ldx, int *k) noexcept nogil: + + _fortran_slapmr(forwrd, m, n, x, ldx, k) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slapmt "BLAS_FUNC(slapmt)"(bint *forwrd, int *m, int *n, s *x, int *ldx, int *k) nogil +cdef void slapmt(bint *forwrd, int *m, int *n, s *x, int *ldx, int *k) noexcept nogil: + + _fortran_slapmt(forwrd, m, n, x, ldx, k) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slapy2 "BLAS_FUNC(slapy2)"(s *x, s *y) nogil +cdef s slapy2(s *x, s *y) noexcept nogil: + + return _fortran_slapy2(x, y) + + +cdef extern from "_lapack_subroutines.h": + s _fortran_slapy3 "BLAS_FUNC(slapy3)"(s *x, s *y, s *z) nogil +cdef s slapy3(s *x, s *y, s *z) noexcept nogil: + + return _fortran_slapy3(x, y, z) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqgb "BLAS_FUNC(slaqgb)"(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) nogil +cdef void slaqgb(int *m, int *n, int *kl, int *ku, s *ab, int *ldab, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) noexcept nogil: + + _fortran_slaqgb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqge "BLAS_FUNC(slaqge)"(int *m, int *n, s *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) nogil +cdef void slaqge(int *m, int *n, s *a, int *lda, s *r, s *c, s *rowcnd, s *colcnd, s *amax, char *equed) noexcept nogil: + + _fortran_slaqge(m, n, a, lda, r, c, rowcnd, colcnd, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqp2 "BLAS_FUNC(slaqp2)"(int *m, int *n, int *offset, s *a, int *lda, int *jpvt, s *tau, s *vn1, s *vn2, s *work) nogil +cdef void slaqp2(int *m, int *n, int *offset, s *a, int *lda, int *jpvt, s *tau, s *vn1, s *vn2, s *work) noexcept nogil: + + _fortran_slaqp2(m, n, offset, a, lda, jpvt, tau, vn1, vn2, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqps "BLAS_FUNC(slaqps)"(int *m, int *n, int *offset, int *nb, int *kb, s *a, int *lda, int *jpvt, s *tau, s *vn1, s *vn2, s *auxv, s *f, int *ldf) nogil +cdef void slaqps(int *m, int *n, int *offset, int *nb, int *kb, s *a, int *lda, int *jpvt, s *tau, s *vn1, s *vn2, s *auxv, s *f, int *ldf) noexcept nogil: + + _fortran_slaqps(m, n, offset, nb, kb, a, lda, jpvt, tau, vn1, vn2, auxv, f, ldf) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqr0 "BLAS_FUNC(slaqr0)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, int *iloz, int *ihiz, s *z, int *ldz, s *work, int *lwork, int *info) nogil +cdef void slaqr0(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, int *iloz, int *ihiz, s *z, int *ldz, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_slaqr0(wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqr1 "BLAS_FUNC(slaqr1)"(int *n, s *h, int *ldh, s *sr1, s *si1, s *sr2, s *si2, s *v) nogil +cdef void slaqr1(int *n, s *h, int *ldh, s *sr1, s *si1, s *sr2, s *si2, s *v) noexcept nogil: + + _fortran_slaqr1(n, h, ldh, sr1, si1, sr2, si2, v) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqr2 "BLAS_FUNC(slaqr2)"(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, s *h, int *ldh, int *iloz, int *ihiz, s *z, int *ldz, int *ns, int *nd, s *sr, s *si, s *v, int *ldv, int *nh, s *t, int *ldt, int *nv, s *wv, int *ldwv, s *work, int *lwork) nogil +cdef void slaqr2(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, s *h, int *ldh, int *iloz, int *ihiz, s *z, int *ldz, int *ns, int *nd, s *sr, s *si, s *v, int *ldv, int *nh, s *t, int *ldt, int *nv, s *wv, int *ldwv, s *work, int *lwork) noexcept nogil: + + _fortran_slaqr2(wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqr3 "BLAS_FUNC(slaqr3)"(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, s *h, int *ldh, int *iloz, int *ihiz, s *z, int *ldz, int *ns, int *nd, s *sr, s *si, s *v, int *ldv, int *nh, s *t, int *ldt, int *nv, s *wv, int *ldwv, s *work, int *lwork) nogil +cdef void slaqr3(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, s *h, int *ldh, int *iloz, int *ihiz, s *z, int *ldz, int *ns, int *nd, s *sr, s *si, s *v, int *ldv, int *nh, s *t, int *ldt, int *nv, s *wv, int *ldwv, s *work, int *lwork) noexcept nogil: + + _fortran_slaqr3(wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqr4 "BLAS_FUNC(slaqr4)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, int *iloz, int *ihiz, s *z, int *ldz, s *work, int *lwork, int *info) nogil +cdef void slaqr4(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, s *h, int *ldh, s *wr, s *wi, int *iloz, int *ihiz, s *z, int *ldz, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_slaqr4(wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqr5 "BLAS_FUNC(slaqr5)"(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, s *sr, s *si, s *h, int *ldh, int *iloz, int *ihiz, s *z, int *ldz, s *v, int *ldv, s *u, int *ldu, int *nv, s *wv, int *ldwv, int *nh, s *wh, int *ldwh) nogil +cdef void slaqr5(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, s *sr, s *si, s *h, int *ldh, int *iloz, int *ihiz, s *z, int *ldz, s *v, int *ldv, s *u, int *ldu, int *nv, s *wv, int *ldwv, int *nh, s *wh, int *ldwh) noexcept nogil: + + _fortran_slaqr5(wantt, wantz, kacc22, n, ktop, kbot, nshfts, sr, si, h, ldh, iloz, ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqsb "BLAS_FUNC(slaqsb)"(char *uplo, int *n, int *kd, s *ab, int *ldab, s *s, s *scond, s *amax, char *equed) nogil +cdef void slaqsb(char *uplo, int *n, int *kd, s *ab, int *ldab, s *s, s *scond, s *amax, char *equed) noexcept nogil: + + _fortran_slaqsb(uplo, n, kd, ab, ldab, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqsp "BLAS_FUNC(slaqsp)"(char *uplo, int *n, s *ap, s *s, s *scond, s *amax, char *equed) nogil +cdef void slaqsp(char *uplo, int *n, s *ap, s *s, s *scond, s *amax, char *equed) noexcept nogil: + + _fortran_slaqsp(uplo, n, ap, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqsy "BLAS_FUNC(slaqsy)"(char *uplo, int *n, s *a, int *lda, s *s, s *scond, s *amax, char *equed) nogil +cdef void slaqsy(char *uplo, int *n, s *a, int *lda, s *s, s *scond, s *amax, char *equed) noexcept nogil: + + _fortran_slaqsy(uplo, n, a, lda, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaqtr "BLAS_FUNC(slaqtr)"(bint *ltran, bint *lreal, int *n, s *t, int *ldt, s *b, s *w, s *scale, s *x, s *work, int *info) nogil +cdef void slaqtr(bint *ltran, bint *lreal, int *n, s *t, int *ldt, s *b, s *w, s *scale, s *x, s *work, int *info) noexcept nogil: + + _fortran_slaqtr(ltran, lreal, n, t, ldt, b, w, scale, x, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slar1v "BLAS_FUNC(slar1v)"(int *n, int *b1, int *bn, s *lambda_, s *d, s *l, s *ld, s *lld, s *pivmin, s *gaptol, s *z, bint *wantnc, int *negcnt, s *ztz, s *mingma, int *r, int *isuppz, s *nrminv, s *resid, s *rqcorr, s *work) nogil +cdef void slar1v(int *n, int *b1, int *bn, s *lambda_, s *d, s *l, s *ld, s *lld, s *pivmin, s *gaptol, s *z, bint *wantnc, int *negcnt, s *ztz, s *mingma, int *r, int *isuppz, s *nrminv, s *resid, s *rqcorr, s *work) noexcept nogil: + + _fortran_slar1v(n, b1, bn, lambda_, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slar2v "BLAS_FUNC(slar2v)"(int *n, s *x, s *y, s *z, int *incx, s *c, s *s, int *incc) nogil +cdef void slar2v(int *n, s *x, s *y, s *z, int *incx, s *c, s *s, int *incc) noexcept nogil: + + _fortran_slar2v(n, x, y, z, incx, c, s, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarf "BLAS_FUNC(slarf)"(char *side, int *m, int *n, s *v, int *incv, s *tau, s *c, int *ldc, s *work) nogil +cdef void slarf(char *side, int *m, int *n, s *v, int *incv, s *tau, s *c, int *ldc, s *work) noexcept nogil: + + _fortran_slarf(side, m, n, v, incv, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarfb "BLAS_FUNC(slarfb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, s *v, int *ldv, s *t, int *ldt, s *c, int *ldc, s *work, int *ldwork) nogil +cdef void slarfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, s *v, int *ldv, s *t, int *ldt, s *c, int *ldc, s *work, int *ldwork) noexcept nogil: + + _fortran_slarfb(side, trans, direct, storev, m, n, k, v, ldv, t, ldt, c, ldc, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarfg "BLAS_FUNC(slarfg)"(int *n, s *alpha, s *x, int *incx, s *tau) nogil +cdef void slarfg(int *n, s *alpha, s *x, int *incx, s *tau) noexcept nogil: + + _fortran_slarfg(n, alpha, x, incx, tau) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarfgp "BLAS_FUNC(slarfgp)"(int *n, s *alpha, s *x, int *incx, s *tau) nogil +cdef void slarfgp(int *n, s *alpha, s *x, int *incx, s *tau) noexcept nogil: + + _fortran_slarfgp(n, alpha, x, incx, tau) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarft "BLAS_FUNC(slarft)"(char *direct, char *storev, int *n, int *k, s *v, int *ldv, s *tau, s *t, int *ldt) nogil +cdef void slarft(char *direct, char *storev, int *n, int *k, s *v, int *ldv, s *tau, s *t, int *ldt) noexcept nogil: + + _fortran_slarft(direct, storev, n, k, v, ldv, tau, t, ldt) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarfx "BLAS_FUNC(slarfx)"(char *side, int *m, int *n, s *v, s *tau, s *c, int *ldc, s *work) nogil +cdef void slarfx(char *side, int *m, int *n, s *v, s *tau, s *c, int *ldc, s *work) noexcept nogil: + + _fortran_slarfx(side, m, n, v, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slargv "BLAS_FUNC(slargv)"(int *n, s *x, int *incx, s *y, int *incy, s *c, int *incc) nogil +cdef void slargv(int *n, s *x, int *incx, s *y, int *incy, s *c, int *incc) noexcept nogil: + + _fortran_slargv(n, x, incx, y, incy, c, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarnv "BLAS_FUNC(slarnv)"(int *idist, int *iseed, int *n, s *x) nogil +cdef void slarnv(int *idist, int *iseed, int *n, s *x) noexcept nogil: + + _fortran_slarnv(idist, iseed, n, x) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarra "BLAS_FUNC(slarra)"(int *n, s *d, s *e, s *e2, s *spltol, s *tnrm, int *nsplit, int *isplit, int *info) nogil +cdef void slarra(int *n, s *d, s *e, s *e2, s *spltol, s *tnrm, int *nsplit, int *isplit, int *info) noexcept nogil: + + _fortran_slarra(n, d, e, e2, spltol, tnrm, nsplit, isplit, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarrb "BLAS_FUNC(slarrb)"(int *n, s *d, s *lld, int *ifirst, int *ilast, s *rtol1, s *rtol2, int *offset, s *w, s *wgap, s *werr, s *work, int *iwork, s *pivmin, s *spdiam, int *twist, int *info) nogil +cdef void slarrb(int *n, s *d, s *lld, int *ifirst, int *ilast, s *rtol1, s *rtol2, int *offset, s *w, s *wgap, s *werr, s *work, int *iwork, s *pivmin, s *spdiam, int *twist, int *info) noexcept nogil: + + _fortran_slarrb(n, d, lld, ifirst, ilast, rtol1, rtol2, offset, w, wgap, werr, work, iwork, pivmin, spdiam, twist, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarrc "BLAS_FUNC(slarrc)"(char *jobt, int *n, s *vl, s *vu, s *d, s *e, s *pivmin, int *eigcnt, int *lcnt, int *rcnt, int *info) nogil +cdef void slarrc(char *jobt, int *n, s *vl, s *vu, s *d, s *e, s *pivmin, int *eigcnt, int *lcnt, int *rcnt, int *info) noexcept nogil: + + _fortran_slarrc(jobt, n, vl, vu, d, e, pivmin, eigcnt, lcnt, rcnt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarrd "BLAS_FUNC(slarrd)"(char *range, char *order, int *n, s *vl, s *vu, int *il, int *iu, s *gers, s *reltol, s *d, s *e, s *e2, s *pivmin, int *nsplit, int *isplit, int *m, s *w, s *werr, s *wl, s *wu, int *iblock, int *indexw, s *work, int *iwork, int *info) nogil +cdef void slarrd(char *range, char *order, int *n, s *vl, s *vu, int *il, int *iu, s *gers, s *reltol, s *d, s *e, s *e2, s *pivmin, int *nsplit, int *isplit, int *m, s *w, s *werr, s *wl, s *wu, int *iblock, int *indexw, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slarrd(range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarre "BLAS_FUNC(slarre)"(char *range, int *n, s *vl, s *vu, int *il, int *iu, s *d, s *e, s *e2, s *rtol1, s *rtol2, s *spltol, int *nsplit, int *isplit, int *m, s *w, s *werr, s *wgap, int *iblock, int *indexw, s *gers, s *pivmin, s *work, int *iwork, int *info) nogil +cdef void slarre(char *range, int *n, s *vl, s *vu, int *il, int *iu, s *d, s *e, s *e2, s *rtol1, s *rtol2, s *spltol, int *nsplit, int *isplit, int *m, s *w, s *werr, s *wgap, int *iblock, int *indexw, s *gers, s *pivmin, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slarre(range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarrf "BLAS_FUNC(slarrf)"(int *n, s *d, s *l, s *ld, int *clstrt, int *clend, s *w, s *wgap, s *werr, s *spdiam, s *clgapl, s *clgapr, s *pivmin, s *sigma, s *dplus, s *lplus, s *work, int *info) nogil +cdef void slarrf(int *n, s *d, s *l, s *ld, int *clstrt, int *clend, s *w, s *wgap, s *werr, s *spdiam, s *clgapl, s *clgapr, s *pivmin, s *sigma, s *dplus, s *lplus, s *work, int *info) noexcept nogil: + + _fortran_slarrf(n, d, l, ld, clstrt, clend, w, wgap, werr, spdiam, clgapl, clgapr, pivmin, sigma, dplus, lplus, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarrj "BLAS_FUNC(slarrj)"(int *n, s *d, s *e2, int *ifirst, int *ilast, s *rtol, int *offset, s *w, s *werr, s *work, int *iwork, s *pivmin, s *spdiam, int *info) nogil +cdef void slarrj(int *n, s *d, s *e2, int *ifirst, int *ilast, s *rtol, int *offset, s *w, s *werr, s *work, int *iwork, s *pivmin, s *spdiam, int *info) noexcept nogil: + + _fortran_slarrj(n, d, e2, ifirst, ilast, rtol, offset, w, werr, work, iwork, pivmin, spdiam, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarrk "BLAS_FUNC(slarrk)"(int *n, int *iw, s *gl, s *gu, s *d, s *e2, s *pivmin, s *reltol, s *w, s *werr, int *info) nogil +cdef void slarrk(int *n, int *iw, s *gl, s *gu, s *d, s *e2, s *pivmin, s *reltol, s *w, s *werr, int *info) noexcept nogil: + + _fortran_slarrk(n, iw, gl, gu, d, e2, pivmin, reltol, w, werr, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarrr "BLAS_FUNC(slarrr)"(int *n, s *d, s *e, int *info) nogil +cdef void slarrr(int *n, s *d, s *e, int *info) noexcept nogil: + + _fortran_slarrr(n, d, e, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarrv "BLAS_FUNC(slarrv)"(int *n, s *vl, s *vu, s *d, s *l, s *pivmin, int *isplit, int *m, int *dol, int *dou, s *minrgp, s *rtol1, s *rtol2, s *w, s *werr, s *wgap, int *iblock, int *indexw, s *gers, s *z, int *ldz, int *isuppz, s *work, int *iwork, int *info) nogil +cdef void slarrv(int *n, s *vl, s *vu, s *d, s *l, s *pivmin, int *isplit, int *m, int *dol, int *dou, s *minrgp, s *rtol1, s *rtol2, s *w, s *werr, s *wgap, int *iblock, int *indexw, s *gers, s *z, int *ldz, int *isuppz, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slarrv(n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slartg "BLAS_FUNC(slartg)"(s *f, s *g, s *cs, s *sn, s *r) nogil +cdef void slartg(s *f, s *g, s *cs, s *sn, s *r) noexcept nogil: + + _fortran_slartg(f, g, cs, sn, r) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slartgp "BLAS_FUNC(slartgp)"(s *f, s *g, s *cs, s *sn, s *r) nogil +cdef void slartgp(s *f, s *g, s *cs, s *sn, s *r) noexcept nogil: + + _fortran_slartgp(f, g, cs, sn, r) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slartgs "BLAS_FUNC(slartgs)"(s *x, s *y, s *sigma, s *cs, s *sn) nogil +cdef void slartgs(s *x, s *y, s *sigma, s *cs, s *sn) noexcept nogil: + + _fortran_slartgs(x, y, sigma, cs, sn) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slartv "BLAS_FUNC(slartv)"(int *n, s *x, int *incx, s *y, int *incy, s *c, s *s, int *incc) nogil +cdef void slartv(int *n, s *x, int *incx, s *y, int *incy, s *c, s *s, int *incc) noexcept nogil: + + _fortran_slartv(n, x, incx, y, incy, c, s, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaruv "BLAS_FUNC(slaruv)"(int *iseed, int *n, s *x) nogil +cdef void slaruv(int *iseed, int *n, s *x) noexcept nogil: + + _fortran_slaruv(iseed, n, x) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarz "BLAS_FUNC(slarz)"(char *side, int *m, int *n, int *l, s *v, int *incv, s *tau, s *c, int *ldc, s *work) nogil +cdef void slarz(char *side, int *m, int *n, int *l, s *v, int *incv, s *tau, s *c, int *ldc, s *work) noexcept nogil: + + _fortran_slarz(side, m, n, l, v, incv, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarzb "BLAS_FUNC(slarzb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, s *v, int *ldv, s *t, int *ldt, s *c, int *ldc, s *work, int *ldwork) nogil +cdef void slarzb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, s *v, int *ldv, s *t, int *ldt, s *c, int *ldc, s *work, int *ldwork) noexcept nogil: + + _fortran_slarzb(side, trans, direct, storev, m, n, k, l, v, ldv, t, ldt, c, ldc, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slarzt "BLAS_FUNC(slarzt)"(char *direct, char *storev, int *n, int *k, s *v, int *ldv, s *tau, s *t, int *ldt) nogil +cdef void slarzt(char *direct, char *storev, int *n, int *k, s *v, int *ldv, s *tau, s *t, int *ldt) noexcept nogil: + + _fortran_slarzt(direct, storev, n, k, v, ldv, tau, t, ldt) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slas2 "BLAS_FUNC(slas2)"(s *f, s *g, s *h, s *ssmin, s *ssmax) nogil +cdef void slas2(s *f, s *g, s *h, s *ssmin, s *ssmax) noexcept nogil: + + _fortran_slas2(f, g, h, ssmin, ssmax) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slascl "BLAS_FUNC(slascl)"(char *type_bn, int *kl, int *ku, s *cfrom, s *cto, int *m, int *n, s *a, int *lda, int *info) nogil +cdef void slascl(char *type_bn, int *kl, int *ku, s *cfrom, s *cto, int *m, int *n, s *a, int *lda, int *info) noexcept nogil: + + _fortran_slascl(type_bn, kl, ku, cfrom, cto, m, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasd0 "BLAS_FUNC(slasd0)"(int *n, int *sqre, s *d, s *e, s *u, int *ldu, s *vt, int *ldvt, int *smlsiz, int *iwork, s *work, int *info) nogil +cdef void slasd0(int *n, int *sqre, s *d, s *e, s *u, int *ldu, s *vt, int *ldvt, int *smlsiz, int *iwork, s *work, int *info) noexcept nogil: + + _fortran_slasd0(n, sqre, d, e, u, ldu, vt, ldvt, smlsiz, iwork, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasd1 "BLAS_FUNC(slasd1)"(int *nl, int *nr, int *sqre, s *d, s *alpha, s *beta, s *u, int *ldu, s *vt, int *ldvt, int *idxq, int *iwork, s *work, int *info) nogil +cdef void slasd1(int *nl, int *nr, int *sqre, s *d, s *alpha, s *beta, s *u, int *ldu, s *vt, int *ldvt, int *idxq, int *iwork, s *work, int *info) noexcept nogil: + + _fortran_slasd1(nl, nr, sqre, d, alpha, beta, u, ldu, vt, ldvt, idxq, iwork, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasd2 "BLAS_FUNC(slasd2)"(int *nl, int *nr, int *sqre, int *k, s *d, s *z, s *alpha, s *beta, s *u, int *ldu, s *vt, int *ldvt, s *dsigma, s *u2, int *ldu2, s *vt2, int *ldvt2, int *idxp, int *idx, int *idxc, int *idxq, int *coltyp, int *info) nogil +cdef void slasd2(int *nl, int *nr, int *sqre, int *k, s *d, s *z, s *alpha, s *beta, s *u, int *ldu, s *vt, int *ldvt, s *dsigma, s *u2, int *ldu2, s *vt2, int *ldvt2, int *idxp, int *idx, int *idxc, int *idxq, int *coltyp, int *info) noexcept nogil: + + _fortran_slasd2(nl, nr, sqre, k, d, z, alpha, beta, u, ldu, vt, ldvt, dsigma, u2, ldu2, vt2, ldvt2, idxp, idx, idxc, idxq, coltyp, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasd3 "BLAS_FUNC(slasd3)"(int *nl, int *nr, int *sqre, int *k, s *d, s *q, int *ldq, s *dsigma, s *u, int *ldu, s *u2, int *ldu2, s *vt, int *ldvt, s *vt2, int *ldvt2, int *idxc, int *ctot, s *z, int *info) nogil +cdef void slasd3(int *nl, int *nr, int *sqre, int *k, s *d, s *q, int *ldq, s *dsigma, s *u, int *ldu, s *u2, int *ldu2, s *vt, int *ldvt, s *vt2, int *ldvt2, int *idxc, int *ctot, s *z, int *info) noexcept nogil: + + _fortran_slasd3(nl, nr, sqre, k, d, q, ldq, dsigma, u, ldu, u2, ldu2, vt, ldvt, vt2, ldvt2, idxc, ctot, z, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasd4 "BLAS_FUNC(slasd4)"(int *n, int *i, s *d, s *z, s *delta, s *rho, s *sigma, s *work, int *info) nogil +cdef void slasd4(int *n, int *i, s *d, s *z, s *delta, s *rho, s *sigma, s *work, int *info) noexcept nogil: + + _fortran_slasd4(n, i, d, z, delta, rho, sigma, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasd5 "BLAS_FUNC(slasd5)"(int *i, s *d, s *z, s *delta, s *rho, s *dsigma, s *work) nogil +cdef void slasd5(int *i, s *d, s *z, s *delta, s *rho, s *dsigma, s *work) noexcept nogil: + + _fortran_slasd5(i, d, z, delta, rho, dsigma, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasd6 "BLAS_FUNC(slasd6)"(int *icompq, int *nl, int *nr, int *sqre, s *d, s *vf, s *vl, s *alpha, s *beta, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *poles, s *difl, s *difr, s *z, int *k, s *c, s *s, s *work, int *iwork, int *info) nogil +cdef void slasd6(int *icompq, int *nl, int *nr, int *sqre, s *d, s *vf, s *vl, s *alpha, s *beta, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *poles, s *difl, s *difr, s *z, int *k, s *c, s *s, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slasd6(icompq, nl, nr, sqre, d, vf, vl, alpha, beta, idxq, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasd7 "BLAS_FUNC(slasd7)"(int *icompq, int *nl, int *nr, int *sqre, int *k, s *d, s *z, s *zw, s *vf, s *vfw, s *vl, s *vlw, s *alpha, s *beta, s *dsigma, int *idx, int *idxp, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *c, s *s, int *info) nogil +cdef void slasd7(int *icompq, int *nl, int *nr, int *sqre, int *k, s *d, s *z, s *zw, s *vf, s *vfw, s *vl, s *vlw, s *alpha, s *beta, s *dsigma, int *idx, int *idxp, int *idxq, int *perm, int *givptr, int *givcol, int *ldgcol, s *givnum, int *ldgnum, s *c, s *s, int *info) noexcept nogil: + + _fortran_slasd7(icompq, nl, nr, sqre, k, d, z, zw, vf, vfw, vl, vlw, alpha, beta, dsigma, idx, idxp, idxq, perm, givptr, givcol, ldgcol, givnum, ldgnum, c, s, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasd8 "BLAS_FUNC(slasd8)"(int *icompq, int *k, s *d, s *z, s *vf, s *vl, s *difl, s *difr, int *lddifr, s *dsigma, s *work, int *info) nogil +cdef void slasd8(int *icompq, int *k, s *d, s *z, s *vf, s *vl, s *difl, s *difr, int *lddifr, s *dsigma, s *work, int *info) noexcept nogil: + + _fortran_slasd8(icompq, k, d, z, vf, vl, difl, difr, lddifr, dsigma, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasda "BLAS_FUNC(slasda)"(int *icompq, int *smlsiz, int *n, int *sqre, s *d, s *e, s *u, int *ldu, s *vt, int *k, s *difl, s *difr, s *z, s *poles, int *givptr, int *givcol, int *ldgcol, int *perm, s *givnum, s *c, s *s, s *work, int *iwork, int *info) nogil +cdef void slasda(int *icompq, int *smlsiz, int *n, int *sqre, s *d, s *e, s *u, int *ldu, s *vt, int *k, s *difl, s *difr, s *z, s *poles, int *givptr, int *givcol, int *ldgcol, int *perm, s *givnum, s *c, s *s, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_slasda(icompq, smlsiz, n, sqre, d, e, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasdq "BLAS_FUNC(slasdq)"(char *uplo, int *sqre, int *n, int *ncvt, int *nru, int *ncc, s *d, s *e, s *vt, int *ldvt, s *u, int *ldu, s *c, int *ldc, s *work, int *info) nogil +cdef void slasdq(char *uplo, int *sqre, int *n, int *ncvt, int *nru, int *ncc, s *d, s *e, s *vt, int *ldvt, s *u, int *ldu, s *c, int *ldc, s *work, int *info) noexcept nogil: + + _fortran_slasdq(uplo, sqre, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasdt "BLAS_FUNC(slasdt)"(int *n, int *lvl, int *nd, int *inode, int *ndiml, int *ndimr, int *msub) nogil +cdef void slasdt(int *n, int *lvl, int *nd, int *inode, int *ndiml, int *ndimr, int *msub) noexcept nogil: + + _fortran_slasdt(n, lvl, nd, inode, ndiml, ndimr, msub) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaset "BLAS_FUNC(slaset)"(char *uplo, int *m, int *n, s *alpha, s *beta, s *a, int *lda) nogil +cdef void slaset(char *uplo, int *m, int *n, s *alpha, s *beta, s *a, int *lda) noexcept nogil: + + _fortran_slaset(uplo, m, n, alpha, beta, a, lda) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasq1 "BLAS_FUNC(slasq1)"(int *n, s *d, s *e, s *work, int *info) nogil +cdef void slasq1(int *n, s *d, s *e, s *work, int *info) noexcept nogil: + + _fortran_slasq1(n, d, e, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasq2 "BLAS_FUNC(slasq2)"(int *n, s *z, int *info) nogil +cdef void slasq2(int *n, s *z, int *info) noexcept nogil: + + _fortran_slasq2(n, z, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasq3 "BLAS_FUNC(slasq3)"(int *i0, int *n0, s *z, int *pp, s *dmin, s *sigma, s *desig, s *qmax, int *nfail, int *iter, int *ndiv, bint *ieee, int *ttype, s *dmin1, s *dmin2, s *dn, s *dn1, s *dn2, s *g, s *tau) nogil +cdef void slasq3(int *i0, int *n0, s *z, int *pp, s *dmin, s *sigma, s *desig, s *qmax, int *nfail, int *iter, int *ndiv, bint *ieee, int *ttype, s *dmin1, s *dmin2, s *dn, s *dn1, s *dn2, s *g, s *tau) noexcept nogil: + + _fortran_slasq3(i0, n0, z, pp, dmin, sigma, desig, qmax, nfail, iter, ndiv, ieee, ttype, dmin1, dmin2, dn, dn1, dn2, g, tau) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasq4 "BLAS_FUNC(slasq4)"(int *i0, int *n0, s *z, int *pp, int *n0in, s *dmin, s *dmin1, s *dmin2, s *dn, s *dn1, s *dn2, s *tau, int *ttype, s *g) nogil +cdef void slasq4(int *i0, int *n0, s *z, int *pp, int *n0in, s *dmin, s *dmin1, s *dmin2, s *dn, s *dn1, s *dn2, s *tau, int *ttype, s *g) noexcept nogil: + + _fortran_slasq4(i0, n0, z, pp, n0in, dmin, dmin1, dmin2, dn, dn1, dn2, tau, ttype, g) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasq6 "BLAS_FUNC(slasq6)"(int *i0, int *n0, s *z, int *pp, s *dmin, s *dmin1, s *dmin2, s *dn, s *dnm1, s *dnm2) nogil +cdef void slasq6(int *i0, int *n0, s *z, int *pp, s *dmin, s *dmin1, s *dmin2, s *dn, s *dnm1, s *dnm2) noexcept nogil: + + _fortran_slasq6(i0, n0, z, pp, dmin, dmin1, dmin2, dn, dnm1, dnm2) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasr "BLAS_FUNC(slasr)"(char *side, char *pivot, char *direct, int *m, int *n, s *c, s *s, s *a, int *lda) nogil +cdef void slasr(char *side, char *pivot, char *direct, int *m, int *n, s *c, s *s, s *a, int *lda) noexcept nogil: + + _fortran_slasr(side, pivot, direct, m, n, c, s, a, lda) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasrt "BLAS_FUNC(slasrt)"(char *id, int *n, s *d, int *info) nogil +cdef void slasrt(char *id, int *n, s *d, int *info) noexcept nogil: + + _fortran_slasrt(id, n, d, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slassq "BLAS_FUNC(slassq)"(int *n, s *x, int *incx, s *scale, s *sumsq) nogil +cdef void slassq(int *n, s *x, int *incx, s *scale, s *sumsq) noexcept nogil: + + _fortran_slassq(n, x, incx, scale, sumsq) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasv2 "BLAS_FUNC(slasv2)"(s *f, s *g, s *h, s *ssmin, s *ssmax, s *snr, s *csr, s *snl, s *csl) nogil +cdef void slasv2(s *f, s *g, s *h, s *ssmin, s *ssmax, s *snr, s *csr, s *snl, s *csl) noexcept nogil: + + _fortran_slasv2(f, g, h, ssmin, ssmax, snr, csr, snl, csl) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slaswp "BLAS_FUNC(slaswp)"(int *n, s *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) nogil +cdef void slaswp(int *n, s *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) noexcept nogil: + + _fortran_slaswp(n, a, lda, k1, k2, ipiv, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasy2 "BLAS_FUNC(slasy2)"(bint *ltranl, bint *ltranr, int *isgn, int *n1, int *n2, s *tl, int *ldtl, s *tr, int *ldtr, s *b, int *ldb, s *scale, s *x, int *ldx, s *xnorm, int *info) nogil +cdef void slasy2(bint *ltranl, bint *ltranr, int *isgn, int *n1, int *n2, s *tl, int *ldtl, s *tr, int *ldtr, s *b, int *ldb, s *scale, s *x, int *ldx, s *xnorm, int *info) noexcept nogil: + + _fortran_slasy2(ltranl, ltranr, isgn, n1, n2, tl, ldtl, tr, ldtr, b, ldb, scale, x, ldx, xnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slasyf "BLAS_FUNC(slasyf)"(char *uplo, int *n, int *nb, int *kb, s *a, int *lda, int *ipiv, s *w, int *ldw, int *info) nogil +cdef void slasyf(char *uplo, int *n, int *nb, int *kb, s *a, int *lda, int *ipiv, s *w, int *ldw, int *info) noexcept nogil: + + _fortran_slasyf(uplo, n, nb, kb, a, lda, ipiv, w, ldw, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slatbs "BLAS_FUNC(slatbs)"(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, s *ab, int *ldab, s *x, s *scale, s *cnorm, int *info) nogil +cdef void slatbs(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, s *ab, int *ldab, s *x, s *scale, s *cnorm, int *info) noexcept nogil: + + _fortran_slatbs(uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slatdf "BLAS_FUNC(slatdf)"(int *ijob, int *n, s *z, int *ldz, s *rhs, s *rdsum, s *rdscal, int *ipiv, int *jpiv) nogil +cdef void slatdf(int *ijob, int *n, s *z, int *ldz, s *rhs, s *rdsum, s *rdscal, int *ipiv, int *jpiv) noexcept nogil: + + _fortran_slatdf(ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slatps "BLAS_FUNC(slatps)"(char *uplo, char *trans, char *diag, char *normin, int *n, s *ap, s *x, s *scale, s *cnorm, int *info) nogil +cdef void slatps(char *uplo, char *trans, char *diag, char *normin, int *n, s *ap, s *x, s *scale, s *cnorm, int *info) noexcept nogil: + + _fortran_slatps(uplo, trans, diag, normin, n, ap, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slatrd "BLAS_FUNC(slatrd)"(char *uplo, int *n, int *nb, s *a, int *lda, s *e, s *tau, s *w, int *ldw) nogil +cdef void slatrd(char *uplo, int *n, int *nb, s *a, int *lda, s *e, s *tau, s *w, int *ldw) noexcept nogil: + + _fortran_slatrd(uplo, n, nb, a, lda, e, tau, w, ldw) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slatrs "BLAS_FUNC(slatrs)"(char *uplo, char *trans, char *diag, char *normin, int *n, s *a, int *lda, s *x, s *scale, s *cnorm, int *info) nogil +cdef void slatrs(char *uplo, char *trans, char *diag, char *normin, int *n, s *a, int *lda, s *x, s *scale, s *cnorm, int *info) noexcept nogil: + + _fortran_slatrs(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slatrz "BLAS_FUNC(slatrz)"(int *m, int *n, int *l, s *a, int *lda, s *tau, s *work) nogil +cdef void slatrz(int *m, int *n, int *l, s *a, int *lda, s *tau, s *work) noexcept nogil: + + _fortran_slatrz(m, n, l, a, lda, tau, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slauu2 "BLAS_FUNC(slauu2)"(char *uplo, int *n, s *a, int *lda, int *info) nogil +cdef void slauu2(char *uplo, int *n, s *a, int *lda, int *info) noexcept nogil: + + _fortran_slauu2(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_slauum "BLAS_FUNC(slauum)"(char *uplo, int *n, s *a, int *lda, int *info) nogil +cdef void slauum(char *uplo, int *n, s *a, int *lda, int *info) noexcept nogil: + + _fortran_slauum(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sopgtr "BLAS_FUNC(sopgtr)"(char *uplo, int *n, s *ap, s *tau, s *q, int *ldq, s *work, int *info) nogil +cdef void sopgtr(char *uplo, int *n, s *ap, s *tau, s *q, int *ldq, s *work, int *info) noexcept nogil: + + _fortran_sopgtr(uplo, n, ap, tau, q, ldq, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sopmtr "BLAS_FUNC(sopmtr)"(char *side, char *uplo, char *trans, int *m, int *n, s *ap, s *tau, s *c, int *ldc, s *work, int *info) nogil +cdef void sopmtr(char *side, char *uplo, char *trans, int *m, int *n, s *ap, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil: + + _fortran_sopmtr(side, uplo, trans, m, n, ap, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorbdb "BLAS_FUNC(sorbdb)"(char *trans, char *signs, int *m, int *p, int *q, s *x11, int *ldx11, s *x12, int *ldx12, s *x21, int *ldx21, s *x22, int *ldx22, s *theta, s *phi, s *taup1, s *taup2, s *tauq1, s *tauq2, s *work, int *lwork, int *info) nogil +cdef void sorbdb(char *trans, char *signs, int *m, int *p, int *q, s *x11, int *ldx11, s *x12, int *ldx12, s *x21, int *ldx21, s *x22, int *ldx22, s *theta, s *phi, s *taup1, s *taup2, s *tauq1, s *tauq2, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sorbdb(trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorcsd "BLAS_FUNC(sorcsd)"(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, s *x11, int *ldx11, s *x12, int *ldx12, s *x21, int *ldx21, s *x22, int *ldx22, s *theta, s *u1, int *ldu1, s *u2, int *ldu2, s *v1t, int *ldv1t, s *v2t, int *ldv2t, s *work, int *lwork, int *iwork, int *info) nogil +cdef void sorcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, s *x11, int *ldx11, s *x12, int *ldx12, s *x21, int *ldx21, s *x22, int *ldx22, s *theta, s *u1, int *ldu1, s *u2, int *ldu2, s *v1t, int *ldv1t, s *v2t, int *ldv2t, s *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_sorcsd(jobu1, jobu2, jobv1t, jobv2t, trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorg2l "BLAS_FUNC(sorg2l)"(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) nogil +cdef void sorg2l(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil: + + _fortran_sorg2l(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorg2r "BLAS_FUNC(sorg2r)"(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) nogil +cdef void sorg2r(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil: + + _fortran_sorg2r(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorgbr "BLAS_FUNC(sorgbr)"(char *vect, int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sorgbr(char *vect, int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sorgbr(vect, m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorghr "BLAS_FUNC(sorghr)"(int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sorghr(int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sorghr(n, ilo, ihi, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorgl2 "BLAS_FUNC(sorgl2)"(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) nogil +cdef void sorgl2(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil: + + _fortran_sorgl2(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorglq "BLAS_FUNC(sorglq)"(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sorglq(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sorglq(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorgql "BLAS_FUNC(sorgql)"(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sorgql(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sorgql(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorgqr "BLAS_FUNC(sorgqr)"(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sorgqr(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sorgqr(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorgr2 "BLAS_FUNC(sorgr2)"(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) nogil +cdef void sorgr2(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *info) noexcept nogil: + + _fortran_sorgr2(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorgrq "BLAS_FUNC(sorgrq)"(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sorgrq(int *m, int *n, int *k, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sorgrq(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorgtr "BLAS_FUNC(sorgtr)"(char *uplo, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void sorgtr(char *uplo, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sorgtr(uplo, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorm2l "BLAS_FUNC(sorm2l)"(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) nogil +cdef void sorm2l(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil: + + _fortran_sorm2l(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorm2r "BLAS_FUNC(sorm2r)"(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) nogil +cdef void sorm2r(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil: + + _fortran_sorm2r(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sormbr "BLAS_FUNC(sormbr)"(char *vect, char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) nogil +cdef void sormbr(char *vect, char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sormbr(vect, side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sormhr "BLAS_FUNC(sormhr)"(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) nogil +cdef void sormhr(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sormhr(side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sorml2 "BLAS_FUNC(sorml2)"(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) nogil +cdef void sorml2(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil: + + _fortran_sorml2(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sormlq "BLAS_FUNC(sormlq)"(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) nogil +cdef void sormlq(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sormlq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sormql "BLAS_FUNC(sormql)"(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) nogil +cdef void sormql(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sormql(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sormqr "BLAS_FUNC(sormqr)"(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) nogil +cdef void sormqr(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sormqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sormr2 "BLAS_FUNC(sormr2)"(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) nogil +cdef void sormr2(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil: + + _fortran_sormr2(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sormr3 "BLAS_FUNC(sormr3)"(char *side, char *trans, int *m, int *n, int *k, int *l, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) nogil +cdef void sormr3(char *side, char *trans, int *m, int *n, int *k, int *l, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *info) noexcept nogil: + + _fortran_sormr3(side, trans, m, n, k, l, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sormrq "BLAS_FUNC(sormrq)"(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) nogil +cdef void sormrq(char *side, char *trans, int *m, int *n, int *k, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sormrq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sormrz "BLAS_FUNC(sormrz)"(char *side, char *trans, int *m, int *n, int *k, int *l, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) nogil +cdef void sormrz(char *side, char *trans, int *m, int *n, int *k, int *l, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sormrz(side, trans, m, n, k, l, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sormtr "BLAS_FUNC(sormtr)"(char *side, char *uplo, char *trans, int *m, int *n, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) nogil +cdef void sormtr(char *side, char *uplo, char *trans, int *m, int *n, s *a, int *lda, s *tau, s *c, int *ldc, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_sormtr(side, uplo, trans, m, n, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spbcon "BLAS_FUNC(spbcon)"(char *uplo, int *n, int *kd, s *ab, int *ldab, s *anorm, s *rcond, s *work, int *iwork, int *info) nogil +cdef void spbcon(char *uplo, int *n, int *kd, s *ab, int *ldab, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_spbcon(uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spbequ "BLAS_FUNC(spbequ)"(char *uplo, int *n, int *kd, s *ab, int *ldab, s *s, s *scond, s *amax, int *info) nogil +cdef void spbequ(char *uplo, int *n, int *kd, s *ab, int *ldab, s *s, s *scond, s *amax, int *info) noexcept nogil: + + _fortran_spbequ(uplo, n, kd, ab, ldab, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spbrfs "BLAS_FUNC(spbrfs)"(char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void spbrfs(char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_spbrfs(uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spbstf "BLAS_FUNC(spbstf)"(char *uplo, int *n, int *kd, s *ab, int *ldab, int *info) nogil +cdef void spbstf(char *uplo, int *n, int *kd, s *ab, int *ldab, int *info) noexcept nogil: + + _fortran_spbstf(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spbsv "BLAS_FUNC(spbsv)"(char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, int *info) nogil +cdef void spbsv(char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_spbsv(uplo, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spbsvx "BLAS_FUNC(spbsvx)"(char *fact, char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, char *equed, s *s, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void spbsvx(char *fact, char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *afb, int *ldafb, char *equed, s *s, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_spbsvx(fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spbtf2 "BLAS_FUNC(spbtf2)"(char *uplo, int *n, int *kd, s *ab, int *ldab, int *info) nogil +cdef void spbtf2(char *uplo, int *n, int *kd, s *ab, int *ldab, int *info) noexcept nogil: + + _fortran_spbtf2(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spbtrf "BLAS_FUNC(spbtrf)"(char *uplo, int *n, int *kd, s *ab, int *ldab, int *info) nogil +cdef void spbtrf(char *uplo, int *n, int *kd, s *ab, int *ldab, int *info) noexcept nogil: + + _fortran_spbtrf(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spbtrs "BLAS_FUNC(spbtrs)"(char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, int *info) nogil +cdef void spbtrs(char *uplo, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_spbtrs(uplo, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spftrf "BLAS_FUNC(spftrf)"(char *transr, char *uplo, int *n, s *a, int *info) nogil +cdef void spftrf(char *transr, char *uplo, int *n, s *a, int *info) noexcept nogil: + + _fortran_spftrf(transr, uplo, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spftri "BLAS_FUNC(spftri)"(char *transr, char *uplo, int *n, s *a, int *info) nogil +cdef void spftri(char *transr, char *uplo, int *n, s *a, int *info) noexcept nogil: + + _fortran_spftri(transr, uplo, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spftrs "BLAS_FUNC(spftrs)"(char *transr, char *uplo, int *n, int *nrhs, s *a, s *b, int *ldb, int *info) nogil +cdef void spftrs(char *transr, char *uplo, int *n, int *nrhs, s *a, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_spftrs(transr, uplo, n, nrhs, a, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spocon "BLAS_FUNC(spocon)"(char *uplo, int *n, s *a, int *lda, s *anorm, s *rcond, s *work, int *iwork, int *info) nogil +cdef void spocon(char *uplo, int *n, s *a, int *lda, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_spocon(uplo, n, a, lda, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spoequ "BLAS_FUNC(spoequ)"(int *n, s *a, int *lda, s *s, s *scond, s *amax, int *info) nogil +cdef void spoequ(int *n, s *a, int *lda, s *s, s *scond, s *amax, int *info) noexcept nogil: + + _fortran_spoequ(n, a, lda, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spoequb "BLAS_FUNC(spoequb)"(int *n, s *a, int *lda, s *s, s *scond, s *amax, int *info) nogil +cdef void spoequb(int *n, s *a, int *lda, s *s, s *scond, s *amax, int *info) noexcept nogil: + + _fortran_spoequb(n, a, lda, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sporfs "BLAS_FUNC(sporfs)"(char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void sporfs(char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sporfs(uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sposv "BLAS_FUNC(sposv)"(char *uplo, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *info) nogil +cdef void sposv(char *uplo, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_sposv(uplo, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sposvx "BLAS_FUNC(sposvx)"(char *fact, char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, char *equed, s *s, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void sposvx(char *fact, char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, char *equed, s *s, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sposvx(fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spotf2 "BLAS_FUNC(spotf2)"(char *uplo, int *n, s *a, int *lda, int *info) nogil +cdef void spotf2(char *uplo, int *n, s *a, int *lda, int *info) noexcept nogil: + + _fortran_spotf2(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spotrf "BLAS_FUNC(spotrf)"(char *uplo, int *n, s *a, int *lda, int *info) nogil +cdef void spotrf(char *uplo, int *n, s *a, int *lda, int *info) noexcept nogil: + + _fortran_spotrf(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spotri "BLAS_FUNC(spotri)"(char *uplo, int *n, s *a, int *lda, int *info) nogil +cdef void spotri(char *uplo, int *n, s *a, int *lda, int *info) noexcept nogil: + + _fortran_spotri(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spotrs "BLAS_FUNC(spotrs)"(char *uplo, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *info) nogil +cdef void spotrs(char *uplo, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_spotrs(uplo, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sppcon "BLAS_FUNC(sppcon)"(char *uplo, int *n, s *ap, s *anorm, s *rcond, s *work, int *iwork, int *info) nogil +cdef void sppcon(char *uplo, int *n, s *ap, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sppcon(uplo, n, ap, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sppequ "BLAS_FUNC(sppequ)"(char *uplo, int *n, s *ap, s *s, s *scond, s *amax, int *info) nogil +cdef void sppequ(char *uplo, int *n, s *ap, s *s, s *scond, s *amax, int *info) noexcept nogil: + + _fortran_sppequ(uplo, n, ap, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spprfs "BLAS_FUNC(spprfs)"(char *uplo, int *n, int *nrhs, s *ap, s *afp, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void spprfs(char *uplo, int *n, int *nrhs, s *ap, s *afp, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_spprfs(uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sppsv "BLAS_FUNC(sppsv)"(char *uplo, int *n, int *nrhs, s *ap, s *b, int *ldb, int *info) nogil +cdef void sppsv(char *uplo, int *n, int *nrhs, s *ap, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_sppsv(uplo, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sppsvx "BLAS_FUNC(sppsvx)"(char *fact, char *uplo, int *n, int *nrhs, s *ap, s *afp, char *equed, s *s, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void sppsvx(char *fact, char *uplo, int *n, int *nrhs, s *ap, s *afp, char *equed, s *s, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sppsvx(fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spptrf "BLAS_FUNC(spptrf)"(char *uplo, int *n, s *ap, int *info) nogil +cdef void spptrf(char *uplo, int *n, s *ap, int *info) noexcept nogil: + + _fortran_spptrf(uplo, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spptri "BLAS_FUNC(spptri)"(char *uplo, int *n, s *ap, int *info) nogil +cdef void spptri(char *uplo, int *n, s *ap, int *info) noexcept nogil: + + _fortran_spptri(uplo, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spptrs "BLAS_FUNC(spptrs)"(char *uplo, int *n, int *nrhs, s *ap, s *b, int *ldb, int *info) nogil +cdef void spptrs(char *uplo, int *n, int *nrhs, s *ap, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_spptrs(uplo, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spstf2 "BLAS_FUNC(spstf2)"(char *uplo, int *n, s *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) nogil +cdef void spstf2(char *uplo, int *n, s *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) noexcept nogil: + + _fortran_spstf2(uplo, n, a, lda, piv, rank, tol, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spstrf "BLAS_FUNC(spstrf)"(char *uplo, int *n, s *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) nogil +cdef void spstrf(char *uplo, int *n, s *a, int *lda, int *piv, int *rank, s *tol, s *work, int *info) noexcept nogil: + + _fortran_spstrf(uplo, n, a, lda, piv, rank, tol, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sptcon "BLAS_FUNC(sptcon)"(int *n, s *d, s *e, s *anorm, s *rcond, s *work, int *info) nogil +cdef void sptcon(int *n, s *d, s *e, s *anorm, s *rcond, s *work, int *info) noexcept nogil: + + _fortran_sptcon(n, d, e, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spteqr "BLAS_FUNC(spteqr)"(char *compz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *info) nogil +cdef void spteqr(char *compz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *info) noexcept nogil: + + _fortran_spteqr(compz, n, d, e, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sptrfs "BLAS_FUNC(sptrfs)"(int *n, int *nrhs, s *d, s *e, s *df, s *ef, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *info) nogil +cdef void sptrfs(int *n, int *nrhs, s *d, s *e, s *df, s *ef, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *info) noexcept nogil: + + _fortran_sptrfs(n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sptsv "BLAS_FUNC(sptsv)"(int *n, int *nrhs, s *d, s *e, s *b, int *ldb, int *info) nogil +cdef void sptsv(int *n, int *nrhs, s *d, s *e, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_sptsv(n, nrhs, d, e, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sptsvx "BLAS_FUNC(sptsvx)"(char *fact, int *n, int *nrhs, s *d, s *e, s *df, s *ef, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *info) nogil +cdef void sptsvx(char *fact, int *n, int *nrhs, s *d, s *e, s *df, s *ef, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *info) noexcept nogil: + + _fortran_sptsvx(fact, n, nrhs, d, e, df, ef, b, ldb, x, ldx, rcond, ferr, berr, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spttrf "BLAS_FUNC(spttrf)"(int *n, s *d, s *e, int *info) nogil +cdef void spttrf(int *n, s *d, s *e, int *info) noexcept nogil: + + _fortran_spttrf(n, d, e, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_spttrs "BLAS_FUNC(spttrs)"(int *n, int *nrhs, s *d, s *e, s *b, int *ldb, int *info) nogil +cdef void spttrs(int *n, int *nrhs, s *d, s *e, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_spttrs(n, nrhs, d, e, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sptts2 "BLAS_FUNC(sptts2)"(int *n, int *nrhs, s *d, s *e, s *b, int *ldb) nogil +cdef void sptts2(int *n, int *nrhs, s *d, s *e, s *b, int *ldb) noexcept nogil: + + _fortran_sptts2(n, nrhs, d, e, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_srscl "BLAS_FUNC(srscl)"(int *n, s *sa, s *sx, int *incx) nogil +cdef void srscl(int *n, s *sa, s *sx, int *incx) noexcept nogil: + + _fortran_srscl(n, sa, sx, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssbev "BLAS_FUNC(ssbev)"(char *jobz, char *uplo, int *n, int *kd, s *ab, int *ldab, s *w, s *z, int *ldz, s *work, int *info) nogil +cdef void ssbev(char *jobz, char *uplo, int *n, int *kd, s *ab, int *ldab, s *w, s *z, int *ldz, s *work, int *info) noexcept nogil: + + _fortran_ssbev(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssbevd "BLAS_FUNC(ssbevd)"(char *jobz, char *uplo, int *n, int *kd, s *ab, int *ldab, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void ssbevd(char *jobz, char *uplo, int *n, int *kd, s *ab, int *ldab, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_ssbevd(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssbevx "BLAS_FUNC(ssbevx)"(char *jobz, char *range, char *uplo, int *n, int *kd, s *ab, int *ldab, s *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) nogil +cdef void ssbevx(char *jobz, char *range, char *uplo, int *n, int *kd, s *ab, int *ldab, s *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_ssbevx(jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssbgst "BLAS_FUNC(ssbgst)"(char *vect, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *x, int *ldx, s *work, int *info) nogil +cdef void ssbgst(char *vect, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *x, int *ldx, s *work, int *info) noexcept nogil: + + _fortran_ssbgst(vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssbgv "BLAS_FUNC(ssbgv)"(char *jobz, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *w, s *z, int *ldz, s *work, int *info) nogil +cdef void ssbgv(char *jobz, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *w, s *z, int *ldz, s *work, int *info) noexcept nogil: + + _fortran_ssbgv(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssbgvd "BLAS_FUNC(ssbgvd)"(char *jobz, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void ssbgvd(char *jobz, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_ssbgvd(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssbgvx "BLAS_FUNC(ssbgvx)"(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) nogil +cdef void ssbgvx(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, s *ab, int *ldab, s *bb, int *ldbb, s *q, int *ldq, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_ssbgvx(jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssbtrd "BLAS_FUNC(ssbtrd)"(char *vect, char *uplo, int *n, int *kd, s *ab, int *ldab, s *d, s *e, s *q, int *ldq, s *work, int *info) nogil +cdef void ssbtrd(char *vect, char *uplo, int *n, int *kd, s *ab, int *ldab, s *d, s *e, s *q, int *ldq, s *work, int *info) noexcept nogil: + + _fortran_ssbtrd(vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssfrk "BLAS_FUNC(ssfrk)"(char *transr, char *uplo, char *trans, int *n, int *k, s *alpha, s *a, int *lda, s *beta, s *c) nogil +cdef void ssfrk(char *transr, char *uplo, char *trans, int *n, int *k, s *alpha, s *a, int *lda, s *beta, s *c) noexcept nogil: + + _fortran_ssfrk(transr, uplo, trans, n, k, alpha, a, lda, beta, c) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sspcon "BLAS_FUNC(sspcon)"(char *uplo, int *n, s *ap, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) nogil +cdef void sspcon(char *uplo, int *n, s *ap, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sspcon(uplo, n, ap, ipiv, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sspev "BLAS_FUNC(sspev)"(char *jobz, char *uplo, int *n, s *ap, s *w, s *z, int *ldz, s *work, int *info) nogil +cdef void sspev(char *jobz, char *uplo, int *n, s *ap, s *w, s *z, int *ldz, s *work, int *info) noexcept nogil: + + _fortran_sspev(jobz, uplo, n, ap, w, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sspevd "BLAS_FUNC(sspevd)"(char *jobz, char *uplo, int *n, s *ap, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void sspevd(char *jobz, char *uplo, int *n, s *ap, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_sspevd(jobz, uplo, n, ap, w, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sspevx "BLAS_FUNC(sspevx)"(char *jobz, char *range, char *uplo, int *n, s *ap, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) nogil +cdef void sspevx(char *jobz, char *range, char *uplo, int *n, s *ap, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_sspevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sspgst "BLAS_FUNC(sspgst)"(int *itype, char *uplo, int *n, s *ap, s *bp, int *info) nogil +cdef void sspgst(int *itype, char *uplo, int *n, s *ap, s *bp, int *info) noexcept nogil: + + _fortran_sspgst(itype, uplo, n, ap, bp, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sspgv "BLAS_FUNC(sspgv)"(int *itype, char *jobz, char *uplo, int *n, s *ap, s *bp, s *w, s *z, int *ldz, s *work, int *info) nogil +cdef void sspgv(int *itype, char *jobz, char *uplo, int *n, s *ap, s *bp, s *w, s *z, int *ldz, s *work, int *info) noexcept nogil: + + _fortran_sspgv(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sspgvd "BLAS_FUNC(sspgvd)"(int *itype, char *jobz, char *uplo, int *n, s *ap, s *bp, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void sspgvd(int *itype, char *jobz, char *uplo, int *n, s *ap, s *bp, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_sspgvd(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sspgvx "BLAS_FUNC(sspgvx)"(int *itype, char *jobz, char *range, char *uplo, int *n, s *ap, s *bp, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) nogil +cdef void sspgvx(int *itype, char *jobz, char *range, char *uplo, int *n, s *ap, s *bp, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_sspgvx(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssprfs "BLAS_FUNC(ssprfs)"(char *uplo, int *n, int *nrhs, s *ap, s *afp, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void ssprfs(char *uplo, int *n, int *nrhs, s *ap, s *afp, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_ssprfs(uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sspsv "BLAS_FUNC(sspsv)"(char *uplo, int *n, int *nrhs, s *ap, int *ipiv, s *b, int *ldb, int *info) nogil +cdef void sspsv(char *uplo, int *n, int *nrhs, s *ap, int *ipiv, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_sspsv(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sspsvx "BLAS_FUNC(sspsvx)"(char *fact, char *uplo, int *n, int *nrhs, s *ap, s *afp, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void sspsvx(char *fact, char *uplo, int *n, int *nrhs, s *ap, s *afp, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sspsvx(fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssptrd "BLAS_FUNC(ssptrd)"(char *uplo, int *n, s *ap, s *d, s *e, s *tau, int *info) nogil +cdef void ssptrd(char *uplo, int *n, s *ap, s *d, s *e, s *tau, int *info) noexcept nogil: + + _fortran_ssptrd(uplo, n, ap, d, e, tau, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssptrf "BLAS_FUNC(ssptrf)"(char *uplo, int *n, s *ap, int *ipiv, int *info) nogil +cdef void ssptrf(char *uplo, int *n, s *ap, int *ipiv, int *info) noexcept nogil: + + _fortran_ssptrf(uplo, n, ap, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssptri "BLAS_FUNC(ssptri)"(char *uplo, int *n, s *ap, int *ipiv, s *work, int *info) nogil +cdef void ssptri(char *uplo, int *n, s *ap, int *ipiv, s *work, int *info) noexcept nogil: + + _fortran_ssptri(uplo, n, ap, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssptrs "BLAS_FUNC(ssptrs)"(char *uplo, int *n, int *nrhs, s *ap, int *ipiv, s *b, int *ldb, int *info) nogil +cdef void ssptrs(char *uplo, int *n, int *nrhs, s *ap, int *ipiv, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_ssptrs(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sstebz "BLAS_FUNC(sstebz)"(char *range, char *order, int *n, s *vl, s *vu, int *il, int *iu, s *abstol, s *d, s *e, int *m, int *nsplit, s *w, int *iblock, int *isplit, s *work, int *iwork, int *info) nogil +cdef void sstebz(char *range, char *order, int *n, s *vl, s *vu, int *il, int *iu, s *abstol, s *d, s *e, int *m, int *nsplit, s *w, int *iblock, int *isplit, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_sstebz(range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sstedc "BLAS_FUNC(sstedc)"(char *compz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void sstedc(char *compz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_sstedc(compz, n, d, e, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sstegr "BLAS_FUNC(sstegr)"(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void sstegr(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_sstegr(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sstein "BLAS_FUNC(sstein)"(int *n, s *d, s *e, int *m, s *w, int *iblock, int *isplit, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) nogil +cdef void sstein(int *n, s *d, s *e, int *m, s *w, int *iblock, int *isplit, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_sstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sstemr "BLAS_FUNC(sstemr)"(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, int *m, s *w, s *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void sstemr(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, int *m, s *w, s *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_sstemr(jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssteqr "BLAS_FUNC(ssteqr)"(char *compz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *info) nogil +cdef void ssteqr(char *compz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *info) noexcept nogil: + + _fortran_ssteqr(compz, n, d, e, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssterf "BLAS_FUNC(ssterf)"(int *n, s *d, s *e, int *info) nogil +cdef void ssterf(int *n, s *d, s *e, int *info) noexcept nogil: + + _fortran_ssterf(n, d, e, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sstev "BLAS_FUNC(sstev)"(char *jobz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *info) nogil +cdef void sstev(char *jobz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *info) noexcept nogil: + + _fortran_sstev(jobz, n, d, e, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sstevd "BLAS_FUNC(sstevd)"(char *jobz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void sstevd(char *jobz, int *n, s *d, s *e, s *z, int *ldz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_sstevd(jobz, n, d, e, z, ldz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sstevr "BLAS_FUNC(sstevr)"(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void sstevr(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_sstevr(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_sstevx "BLAS_FUNC(sstevx)"(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) nogil +cdef void sstevx(char *jobz, char *range, int *n, s *d, s *e, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_sstevx(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssycon "BLAS_FUNC(ssycon)"(char *uplo, int *n, s *a, int *lda, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) nogil +cdef void ssycon(char *uplo, int *n, s *a, int *lda, int *ipiv, s *anorm, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_ssycon(uplo, n, a, lda, ipiv, anorm, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssyconv "BLAS_FUNC(ssyconv)"(char *uplo, char *way, int *n, s *a, int *lda, int *ipiv, s *work, int *info) nogil +cdef void ssyconv(char *uplo, char *way, int *n, s *a, int *lda, int *ipiv, s *work, int *info) noexcept nogil: + + _fortran_ssyconv(uplo, way, n, a, lda, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssyequb "BLAS_FUNC(ssyequb)"(char *uplo, int *n, s *a, int *lda, s *s, s *scond, s *amax, s *work, int *info) nogil +cdef void ssyequb(char *uplo, int *n, s *a, int *lda, s *s, s *scond, s *amax, s *work, int *info) noexcept nogil: + + _fortran_ssyequb(uplo, n, a, lda, s, scond, amax, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssyev "BLAS_FUNC(ssyev)"(char *jobz, char *uplo, int *n, s *a, int *lda, s *w, s *work, int *lwork, int *info) nogil +cdef void ssyev(char *jobz, char *uplo, int *n, s *a, int *lda, s *w, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_ssyev(jobz, uplo, n, a, lda, w, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssyevd "BLAS_FUNC(ssyevd)"(char *jobz, char *uplo, int *n, s *a, int *lda, s *w, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void ssyevd(char *jobz, char *uplo, int *n, s *a, int *lda, s *w, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_ssyevd(jobz, uplo, n, a, lda, w, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssyevr "BLAS_FUNC(ssyevr)"(char *jobz, char *range, char *uplo, int *n, s *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void ssyevr(char *jobz, char *range, char *uplo, int *n, s *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, int *isuppz, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_ssyevr(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssyevx "BLAS_FUNC(ssyevx)"(char *jobz, char *range, char *uplo, int *n, s *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *ifail, int *info) nogil +cdef void ssyevx(char *jobz, char *range, char *uplo, int *n, s *a, int *lda, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_ssyevx(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssygs2 "BLAS_FUNC(ssygs2)"(int *itype, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, int *info) nogil +cdef void ssygs2(int *itype, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_ssygs2(itype, uplo, n, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssygst "BLAS_FUNC(ssygst)"(int *itype, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, int *info) nogil +cdef void ssygst(int *itype, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_ssygst(itype, uplo, n, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssygv "BLAS_FUNC(ssygv)"(int *itype, char *jobz, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, s *w, s *work, int *lwork, int *info) nogil +cdef void ssygv(int *itype, char *jobz, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, s *w, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_ssygv(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssygvd "BLAS_FUNC(ssygvd)"(int *itype, char *jobz, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, s *w, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void ssygvd(int *itype, char *jobz, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, s *w, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_ssygvd(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssygvx "BLAS_FUNC(ssygvx)"(int *itype, char *jobz, char *range, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *ifail, int *info) nogil +cdef void ssygvx(int *itype, char *jobz, char *range, char *uplo, int *n, s *a, int *lda, s *b, int *ldb, s *vl, s *vu, int *il, int *iu, s *abstol, int *m, s *w, s *z, int *ldz, s *work, int *lwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_ssygvx(itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssyrfs "BLAS_FUNC(ssyrfs)"(char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void ssyrfs(char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_ssyrfs(uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssysv "BLAS_FUNC(ssysv)"(char *uplo, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, s *work, int *lwork, int *info) nogil +cdef void ssysv(char *uplo, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_ssysv(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssysvx "BLAS_FUNC(ssysvx)"(char *fact, char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *lwork, int *iwork, int *info) nogil +cdef void ssysvx(char *fact, char *uplo, int *n, int *nrhs, s *a, int *lda, s *af, int *ldaf, int *ipiv, s *b, int *ldb, s *x, int *ldx, s *rcond, s *ferr, s *berr, s *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_ssysvx(fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssyswapr "BLAS_FUNC(ssyswapr)"(char *uplo, int *n, s *a, int *lda, int *i1, int *i2) nogil +cdef void ssyswapr(char *uplo, int *n, s *a, int *lda, int *i1, int *i2) noexcept nogil: + + _fortran_ssyswapr(uplo, n, a, lda, i1, i2) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssytd2 "BLAS_FUNC(ssytd2)"(char *uplo, int *n, s *a, int *lda, s *d, s *e, s *tau, int *info) nogil +cdef void ssytd2(char *uplo, int *n, s *a, int *lda, s *d, s *e, s *tau, int *info) noexcept nogil: + + _fortran_ssytd2(uplo, n, a, lda, d, e, tau, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssytf2 "BLAS_FUNC(ssytf2)"(char *uplo, int *n, s *a, int *lda, int *ipiv, int *info) nogil +cdef void ssytf2(char *uplo, int *n, s *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_ssytf2(uplo, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssytrd "BLAS_FUNC(ssytrd)"(char *uplo, int *n, s *a, int *lda, s *d, s *e, s *tau, s *work, int *lwork, int *info) nogil +cdef void ssytrd(char *uplo, int *n, s *a, int *lda, s *d, s *e, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_ssytrd(uplo, n, a, lda, d, e, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssytrf "BLAS_FUNC(ssytrf)"(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *lwork, int *info) nogil +cdef void ssytrf(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_ssytrf(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssytri "BLAS_FUNC(ssytri)"(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *info) nogil +cdef void ssytri(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *info) noexcept nogil: + + _fortran_ssytri(uplo, n, a, lda, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssytri2 "BLAS_FUNC(ssytri2)"(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *lwork, int *info) nogil +cdef void ssytri2(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_ssytri2(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssytri2x "BLAS_FUNC(ssytri2x)"(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *nb, int *info) nogil +cdef void ssytri2x(char *uplo, int *n, s *a, int *lda, int *ipiv, s *work, int *nb, int *info) noexcept nogil: + + _fortran_ssytri2x(uplo, n, a, lda, ipiv, work, nb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssytrs "BLAS_FUNC(ssytrs)"(char *uplo, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, int *info) nogil +cdef void ssytrs(char *uplo, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_ssytrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ssytrs2 "BLAS_FUNC(ssytrs2)"(char *uplo, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, s *work, int *info) nogil +cdef void ssytrs2(char *uplo, int *n, int *nrhs, s *a, int *lda, int *ipiv, s *b, int *ldb, s *work, int *info) noexcept nogil: + + _fortran_ssytrs2(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stbcon "BLAS_FUNC(stbcon)"(char *norm, char *uplo, char *diag, int *n, int *kd, s *ab, int *ldab, s *rcond, s *work, int *iwork, int *info) nogil +cdef void stbcon(char *norm, char *uplo, char *diag, int *n, int *kd, s *ab, int *ldab, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_stbcon(norm, uplo, diag, n, kd, ab, ldab, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stbrfs "BLAS_FUNC(stbrfs)"(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void stbrfs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_stbrfs(uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stbtrs "BLAS_FUNC(stbtrs)"(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, int *info) nogil +cdef void stbtrs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, s *ab, int *ldab, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_stbtrs(uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stfsm "BLAS_FUNC(stfsm)"(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, s *alpha, s *a, s *b, int *ldb) nogil +cdef void stfsm(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, s *alpha, s *a, s *b, int *ldb) noexcept nogil: + + _fortran_stfsm(transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stftri "BLAS_FUNC(stftri)"(char *transr, char *uplo, char *diag, int *n, s *a, int *info) nogil +cdef void stftri(char *transr, char *uplo, char *diag, int *n, s *a, int *info) noexcept nogil: + + _fortran_stftri(transr, uplo, diag, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stfttp "BLAS_FUNC(stfttp)"(char *transr, char *uplo, int *n, s *arf, s *ap, int *info) nogil +cdef void stfttp(char *transr, char *uplo, int *n, s *arf, s *ap, int *info) noexcept nogil: + + _fortran_stfttp(transr, uplo, n, arf, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stfttr "BLAS_FUNC(stfttr)"(char *transr, char *uplo, int *n, s *arf, s *a, int *lda, int *info) nogil +cdef void stfttr(char *transr, char *uplo, int *n, s *arf, s *a, int *lda, int *info) noexcept nogil: + + _fortran_stfttr(transr, uplo, n, arf, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stgevc "BLAS_FUNC(stgevc)"(char *side, char *howmny, bint *select, int *n, s *s, int *lds, s *p, int *ldp, s *vl, int *ldvl, s *vr, int *ldvr, int *mm, int *m, s *work, int *info) nogil +cdef void stgevc(char *side, char *howmny, bint *select, int *n, s *s, int *lds, s *p, int *ldp, s *vl, int *ldvl, s *vr, int *ldvr, int *mm, int *m, s *work, int *info) noexcept nogil: + + _fortran_stgevc(side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stgex2 "BLAS_FUNC(stgex2)"(bint *wantq, bint *wantz, int *n, s *a, int *lda, s *b, int *ldb, s *q, int *ldq, s *z, int *ldz, int *j1, int *n1, int *n2, s *work, int *lwork, int *info) nogil +cdef void stgex2(bint *wantq, bint *wantz, int *n, s *a, int *lda, s *b, int *ldb, s *q, int *ldq, s *z, int *ldz, int *j1, int *n1, int *n2, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_stgex2(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, n1, n2, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stgexc "BLAS_FUNC(stgexc)"(bint *wantq, bint *wantz, int *n, s *a, int *lda, s *b, int *ldb, s *q, int *ldq, s *z, int *ldz, int *ifst, int *ilst, s *work, int *lwork, int *info) nogil +cdef void stgexc(bint *wantq, bint *wantz, int *n, s *a, int *lda, s *b, int *ldb, s *q, int *ldq, s *z, int *ldz, int *ifst, int *ilst, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_stgexc(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stgsen "BLAS_FUNC(stgsen)"(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *q, int *ldq, s *z, int *ldz, int *m, s *pl, s *pr, s *dif, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void stgsen(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, s *a, int *lda, s *b, int *ldb, s *alphar, s *alphai, s *beta, s *q, int *ldq, s *z, int *ldz, int *m, s *pl, s *pr, s *dif, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_stgsen(ijob, wantq, wantz, select, n, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stgsja "BLAS_FUNC(stgsja)"(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, s *a, int *lda, s *b, int *ldb, s *tola, s *tolb, s *alpha, s *beta, s *u, int *ldu, s *v, int *ldv, s *q, int *ldq, s *work, int *ncycle, int *info) nogil +cdef void stgsja(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, s *a, int *lda, s *b, int *ldb, s *tola, s *tolb, s *alpha, s *beta, s *u, int *ldu, s *v, int *ldv, s *q, int *ldq, s *work, int *ncycle, int *info) noexcept nogil: + + _fortran_stgsja(jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb, tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq, work, ncycle, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stgsna "BLAS_FUNC(stgsna)"(char *job, char *howmny, bint *select, int *n, s *a, int *lda, s *b, int *ldb, s *vl, int *ldvl, s *vr, int *ldvr, s *s, s *dif, int *mm, int *m, s *work, int *lwork, int *iwork, int *info) nogil +cdef void stgsna(char *job, char *howmny, bint *select, int *n, s *a, int *lda, s *b, int *ldb, s *vl, int *ldvl, s *vr, int *ldvr, s *s, s *dif, int *mm, int *m, s *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_stgsna(job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stgsy2 "BLAS_FUNC(stgsy2)"(char *trans, int *ijob, int *m, int *n, s *a, int *lda, s *b, int *ldb, s *c, int *ldc, s *d, int *ldd, s *e, int *lde, s *f, int *ldf, s *scale, s *rdsum, s *rdscal, int *iwork, int *pq, int *info) nogil +cdef void stgsy2(char *trans, int *ijob, int *m, int *n, s *a, int *lda, s *b, int *ldb, s *c, int *ldc, s *d, int *ldd, s *e, int *lde, s *f, int *ldf, s *scale, s *rdsum, s *rdscal, int *iwork, int *pq, int *info) noexcept nogil: + + _fortran_stgsy2(trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, iwork, pq, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stgsyl "BLAS_FUNC(stgsyl)"(char *trans, int *ijob, int *m, int *n, s *a, int *lda, s *b, int *ldb, s *c, int *ldc, s *d, int *ldd, s *e, int *lde, s *f, int *ldf, s *scale, s *dif, s *work, int *lwork, int *iwork, int *info) nogil +cdef void stgsyl(char *trans, int *ijob, int *m, int *n, s *a, int *lda, s *b, int *ldb, s *c, int *ldc, s *d, int *ldd, s *e, int *lde, s *f, int *ldf, s *scale, s *dif, s *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_stgsyl(trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stpcon "BLAS_FUNC(stpcon)"(char *norm, char *uplo, char *diag, int *n, s *ap, s *rcond, s *work, int *iwork, int *info) nogil +cdef void stpcon(char *norm, char *uplo, char *diag, int *n, s *ap, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_stpcon(norm, uplo, diag, n, ap, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stpmqrt "BLAS_FUNC(stpmqrt)"(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, s *v, int *ldv, s *t, int *ldt, s *a, int *lda, s *b, int *ldb, s *work, int *info) nogil +cdef void stpmqrt(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, s *v, int *ldv, s *t, int *ldt, s *a, int *lda, s *b, int *ldb, s *work, int *info) noexcept nogil: + + _fortran_stpmqrt(side, trans, m, n, k, l, nb, v, ldv, t, ldt, a, lda, b, ldb, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stpqrt "BLAS_FUNC(stpqrt)"(int *m, int *n, int *l, int *nb, s *a, int *lda, s *b, int *ldb, s *t, int *ldt, s *work, int *info) nogil +cdef void stpqrt(int *m, int *n, int *l, int *nb, s *a, int *lda, s *b, int *ldb, s *t, int *ldt, s *work, int *info) noexcept nogil: + + _fortran_stpqrt(m, n, l, nb, a, lda, b, ldb, t, ldt, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stpqrt2 "BLAS_FUNC(stpqrt2)"(int *m, int *n, int *l, s *a, int *lda, s *b, int *ldb, s *t, int *ldt, int *info) nogil +cdef void stpqrt2(int *m, int *n, int *l, s *a, int *lda, s *b, int *ldb, s *t, int *ldt, int *info) noexcept nogil: + + _fortran_stpqrt2(m, n, l, a, lda, b, ldb, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stprfb "BLAS_FUNC(stprfb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, s *v, int *ldv, s *t, int *ldt, s *a, int *lda, s *b, int *ldb, s *work, int *ldwork) nogil +cdef void stprfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, s *v, int *ldv, s *t, int *ldt, s *a, int *lda, s *b, int *ldb, s *work, int *ldwork) noexcept nogil: + + _fortran_stprfb(side, trans, direct, storev, m, n, k, l, v, ldv, t, ldt, a, lda, b, ldb, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stprfs "BLAS_FUNC(stprfs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *ap, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void stprfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *ap, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_stprfs(uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stptri "BLAS_FUNC(stptri)"(char *uplo, char *diag, int *n, s *ap, int *info) nogil +cdef void stptri(char *uplo, char *diag, int *n, s *ap, int *info) noexcept nogil: + + _fortran_stptri(uplo, diag, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stptrs "BLAS_FUNC(stptrs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *ap, s *b, int *ldb, int *info) nogil +cdef void stptrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *ap, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_stptrs(uplo, trans, diag, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stpttf "BLAS_FUNC(stpttf)"(char *transr, char *uplo, int *n, s *ap, s *arf, int *info) nogil +cdef void stpttf(char *transr, char *uplo, int *n, s *ap, s *arf, int *info) noexcept nogil: + + _fortran_stpttf(transr, uplo, n, ap, arf, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stpttr "BLAS_FUNC(stpttr)"(char *uplo, int *n, s *ap, s *a, int *lda, int *info) nogil +cdef void stpttr(char *uplo, int *n, s *ap, s *a, int *lda, int *info) noexcept nogil: + + _fortran_stpttr(uplo, n, ap, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strcon "BLAS_FUNC(strcon)"(char *norm, char *uplo, char *diag, int *n, s *a, int *lda, s *rcond, s *work, int *iwork, int *info) nogil +cdef void strcon(char *norm, char *uplo, char *diag, int *n, s *a, int *lda, s *rcond, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_strcon(norm, uplo, diag, n, a, lda, rcond, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strevc "BLAS_FUNC(strevc)"(char *side, char *howmny, bint *select, int *n, s *t, int *ldt, s *vl, int *ldvl, s *vr, int *ldvr, int *mm, int *m, s *work, int *info) nogil +cdef void strevc(char *side, char *howmny, bint *select, int *n, s *t, int *ldt, s *vl, int *ldvl, s *vr, int *ldvr, int *mm, int *m, s *work, int *info) noexcept nogil: + + _fortran_strevc(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strexc "BLAS_FUNC(strexc)"(char *compq, int *n, s *t, int *ldt, s *q, int *ldq, int *ifst, int *ilst, s *work, int *info) nogil +cdef void strexc(char *compq, int *n, s *t, int *ldt, s *q, int *ldq, int *ifst, int *ilst, s *work, int *info) noexcept nogil: + + _fortran_strexc(compq, n, t, ldt, q, ldq, ifst, ilst, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strrfs "BLAS_FUNC(strrfs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) nogil +cdef void strrfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, s *x, int *ldx, s *ferr, s *berr, s *work, int *iwork, int *info) noexcept nogil: + + _fortran_strrfs(uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strsen "BLAS_FUNC(strsen)"(char *job, char *compq, bint *select, int *n, s *t, int *ldt, s *q, int *ldq, s *wr, s *wi, int *m, s *s, s *sep, s *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void strsen(char *job, char *compq, bint *select, int *n, s *t, int *ldt, s *q, int *ldq, s *wr, s *wi, int *m, s *s, s *sep, s *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_strsen(job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s, sep, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strsna "BLAS_FUNC(strsna)"(char *job, char *howmny, bint *select, int *n, s *t, int *ldt, s *vl, int *ldvl, s *vr, int *ldvr, s *s, s *sep, int *mm, int *m, s *work, int *ldwork, int *iwork, int *info) nogil +cdef void strsna(char *job, char *howmny, bint *select, int *n, s *t, int *ldt, s *vl, int *ldvl, s *vr, int *ldvr, s *s, s *sep, int *mm, int *m, s *work, int *ldwork, int *iwork, int *info) noexcept nogil: + + _fortran_strsna(job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strsyl "BLAS_FUNC(strsyl)"(char *trana, char *tranb, int *isgn, int *m, int *n, s *a, int *lda, s *b, int *ldb, s *c, int *ldc, s *scale, int *info) nogil +cdef void strsyl(char *trana, char *tranb, int *isgn, int *m, int *n, s *a, int *lda, s *b, int *ldb, s *c, int *ldc, s *scale, int *info) noexcept nogil: + + _fortran_strsyl(trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strti2 "BLAS_FUNC(strti2)"(char *uplo, char *diag, int *n, s *a, int *lda, int *info) nogil +cdef void strti2(char *uplo, char *diag, int *n, s *a, int *lda, int *info) noexcept nogil: + + _fortran_strti2(uplo, diag, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strtri "BLAS_FUNC(strtri)"(char *uplo, char *diag, int *n, s *a, int *lda, int *info) nogil +cdef void strtri(char *uplo, char *diag, int *n, s *a, int *lda, int *info) noexcept nogil: + + _fortran_strtri(uplo, diag, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strtrs "BLAS_FUNC(strtrs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *info) nogil +cdef void strtrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, s *a, int *lda, s *b, int *ldb, int *info) noexcept nogil: + + _fortran_strtrs(uplo, trans, diag, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strttf "BLAS_FUNC(strttf)"(char *transr, char *uplo, int *n, s *a, int *lda, s *arf, int *info) nogil +cdef void strttf(char *transr, char *uplo, int *n, s *a, int *lda, s *arf, int *info) noexcept nogil: + + _fortran_strttf(transr, uplo, n, a, lda, arf, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_strttp "BLAS_FUNC(strttp)"(char *uplo, int *n, s *a, int *lda, s *ap, int *info) nogil +cdef void strttp(char *uplo, int *n, s *a, int *lda, s *ap, int *info) noexcept nogil: + + _fortran_strttp(uplo, n, a, lda, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_stzrzf "BLAS_FUNC(stzrzf)"(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) nogil +cdef void stzrzf(int *m, int *n, s *a, int *lda, s *tau, s *work, int *lwork, int *info) noexcept nogil: + + _fortran_stzrzf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_xerbla_array "BLAS_FUNC(xerbla_array)"(char *srname_array, int *srname_len, int *info) nogil +cdef void xerbla_array(char *srname_array, int *srname_len, int *info) noexcept nogil: + + _fortran_xerbla_array(srname_array, srname_len, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zbbcsd "BLAS_FUNC(zbbcsd)"(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, d *theta, d *phi, npy_complex128 *u1, int *ldu1, npy_complex128 *u2, int *ldu2, npy_complex128 *v1t, int *ldv1t, npy_complex128 *v2t, int *ldv2t, d *b11d, d *b11e, d *b12d, d *b12e, d *b21d, d *b21e, d *b22d, d *b22e, d *rwork, int *lrwork, int *info) nogil +cdef void zbbcsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, int *m, int *p, int *q, d *theta, d *phi, z *u1, int *ldu1, z *u2, int *ldu2, z *v1t, int *ldv1t, z *v2t, int *ldv2t, d *b11d, d *b11e, d *b12d, d *b12e, d *b21d, d *b21e, d *b22d, d *b22e, d *rwork, int *lrwork, int *info) noexcept nogil: + + _fortran_zbbcsd(jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta, phi, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, b11d, b11e, b12d, b12e, b21d, b21e, b22d, b22e, rwork, lrwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zbdsqr "BLAS_FUNC(zbdsqr)"(char *uplo, int *n, int *ncvt, int *nru, int *ncc, d *d, d *e, npy_complex128 *vt, int *ldvt, npy_complex128 *u, int *ldu, npy_complex128 *c, int *ldc, d *rwork, int *info) nogil +cdef void zbdsqr(char *uplo, int *n, int *ncvt, int *nru, int *ncc, d *d, d *e, z *vt, int *ldvt, z *u, int *ldu, z *c, int *ldc, d *rwork, int *info) noexcept nogil: + + _fortran_zbdsqr(uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zcgesv "BLAS_FUNC(zcgesv)"(int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, npy_complex128 *work, npy_complex64 *swork, d *rwork, int *iter, int *info) nogil +cdef void zcgesv(int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, z *x, int *ldx, z *work, c *swork, d *rwork, int *iter, int *info) noexcept nogil: + + _fortran_zcgesv(n, nrhs, a, lda, ipiv, b, ldb, x, ldx, work, swork, rwork, iter, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zcposv "BLAS_FUNC(zcposv)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, npy_complex128 *work, npy_complex64 *swork, d *rwork, int *iter, int *info) nogil +cdef void zcposv(char *uplo, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, z *x, int *ldx, z *work, c *swork, d *rwork, int *iter, int *info) noexcept nogil: + + _fortran_zcposv(uplo, n, nrhs, a, lda, b, ldb, x, ldx, work, swork, rwork, iter, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zdrscl "BLAS_FUNC(zdrscl)"(int *n, d *sa, npy_complex128 *sx, int *incx) nogil +cdef void zdrscl(int *n, d *sa, z *sx, int *incx) noexcept nogil: + + _fortran_zdrscl(n, sa, sx, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgbbrd "BLAS_FUNC(zgbbrd)"(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, npy_complex128 *ab, int *ldab, d *d, d *e, npy_complex128 *q, int *ldq, npy_complex128 *pt, int *ldpt, npy_complex128 *c, int *ldc, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zgbbrd(char *vect, int *m, int *n, int *ncc, int *kl, int *ku, z *ab, int *ldab, d *d, d *e, z *q, int *ldq, z *pt, int *ldpt, z *c, int *ldc, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zgbbrd(vect, m, n, ncc, kl, ku, ab, ldab, d, e, q, ldq, pt, ldpt, c, ldc, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgbcon "BLAS_FUNC(zgbcon)"(char *norm, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, int *ipiv, d *anorm, d *rcond, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zgbcon(char *norm, int *n, int *kl, int *ku, z *ab, int *ldab, int *ipiv, d *anorm, d *rcond, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zgbcon(norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgbequ "BLAS_FUNC(zgbequ)"(int *m, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) nogil +cdef void zgbequ(int *m, int *n, int *kl, int *ku, z *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil: + + _fortran_zgbequ(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgbequb "BLAS_FUNC(zgbequb)"(int *m, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) nogil +cdef void zgbequb(int *m, int *n, int *kl, int *ku, z *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil: + + _fortran_zgbequb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgbrfs "BLAS_FUNC(zgbrfs)"(char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *afb, int *ldafb, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zgbrfs(char *trans, int *n, int *kl, int *ku, int *nrhs, z *ab, int *ldab, z *afb, int *ldafb, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zgbrfs(trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgbsv "BLAS_FUNC(zgbsv)"(int *n, int *kl, int *ku, int *nrhs, npy_complex128 *ab, int *ldab, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zgbsv(int *n, int *kl, int *ku, int *nrhs, z *ab, int *ldab, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zgbsv(n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgbsvx "BLAS_FUNC(zgbsvx)"(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *afb, int *ldafb, int *ipiv, char *equed, d *r, d *c, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zgbsvx(char *fact, char *trans, int *n, int *kl, int *ku, int *nrhs, z *ab, int *ldab, z *afb, int *ldafb, int *ipiv, char *equed, d *r, d *c, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zgbsvx(fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgbtf2 "BLAS_FUNC(zgbtf2)"(int *m, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, int *ipiv, int *info) nogil +cdef void zgbtf2(int *m, int *n, int *kl, int *ku, z *ab, int *ldab, int *ipiv, int *info) noexcept nogil: + + _fortran_zgbtf2(m, n, kl, ku, ab, ldab, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgbtrf "BLAS_FUNC(zgbtrf)"(int *m, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, int *ipiv, int *info) nogil +cdef void zgbtrf(int *m, int *n, int *kl, int *ku, z *ab, int *ldab, int *ipiv, int *info) noexcept nogil: + + _fortran_zgbtrf(m, n, kl, ku, ab, ldab, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgbtrs "BLAS_FUNC(zgbtrs)"(char *trans, int *n, int *kl, int *ku, int *nrhs, npy_complex128 *ab, int *ldab, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zgbtrs(char *trans, int *n, int *kl, int *ku, int *nrhs, z *ab, int *ldab, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zgbtrs(trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgebak "BLAS_FUNC(zgebak)"(char *job, char *side, int *n, int *ilo, int *ihi, d *scale, int *m, npy_complex128 *v, int *ldv, int *info) nogil +cdef void zgebak(char *job, char *side, int *n, int *ilo, int *ihi, d *scale, int *m, z *v, int *ldv, int *info) noexcept nogil: + + _fortran_zgebak(job, side, n, ilo, ihi, scale, m, v, ldv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgebal "BLAS_FUNC(zgebal)"(char *job, int *n, npy_complex128 *a, int *lda, int *ilo, int *ihi, d *scale, int *info) nogil +cdef void zgebal(char *job, int *n, z *a, int *lda, int *ilo, int *ihi, d *scale, int *info) noexcept nogil: + + _fortran_zgebal(job, n, a, lda, ilo, ihi, scale, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgebd2 "BLAS_FUNC(zgebd2)"(int *m, int *n, npy_complex128 *a, int *lda, d *d, d *e, npy_complex128 *tauq, npy_complex128 *taup, npy_complex128 *work, int *info) nogil +cdef void zgebd2(int *m, int *n, z *a, int *lda, d *d, d *e, z *tauq, z *taup, z *work, int *info) noexcept nogil: + + _fortran_zgebd2(m, n, a, lda, d, e, tauq, taup, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgebrd "BLAS_FUNC(zgebrd)"(int *m, int *n, npy_complex128 *a, int *lda, d *d, d *e, npy_complex128 *tauq, npy_complex128 *taup, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zgebrd(int *m, int *n, z *a, int *lda, d *d, d *e, z *tauq, z *taup, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zgebrd(m, n, a, lda, d, e, tauq, taup, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgecon "BLAS_FUNC(zgecon)"(char *norm, int *n, npy_complex128 *a, int *lda, d *anorm, d *rcond, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zgecon(char *norm, int *n, z *a, int *lda, d *anorm, d *rcond, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zgecon(norm, n, a, lda, anorm, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeequ "BLAS_FUNC(zgeequ)"(int *m, int *n, npy_complex128 *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) nogil +cdef void zgeequ(int *m, int *n, z *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil: + + _fortran_zgeequ(m, n, a, lda, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeequb "BLAS_FUNC(zgeequb)"(int *m, int *n, npy_complex128 *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) nogil +cdef void zgeequb(int *m, int *n, z *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, int *info) noexcept nogil: + + _fortran_zgeequb(m, n, a, lda, r, c, rowcnd, colcnd, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgees "BLAS_FUNC(zgees)"(char *jobvs, char *sort, _zselect1 *select, int *n, npy_complex128 *a, int *lda, int *sdim, npy_complex128 *w, npy_complex128 *vs, int *ldvs, npy_complex128 *work, int *lwork, d *rwork, bint *bwork, int *info) nogil +cdef void zgees(char *jobvs, char *sort, zselect1 *select, int *n, z *a, int *lda, int *sdim, z *w, z *vs, int *ldvs, z *work, int *lwork, d *rwork, bint *bwork, int *info) noexcept nogil: + + _fortran_zgees(jobvs, sort, <_zselect1*>select, n, a, lda, sdim, w, vs, ldvs, work, lwork, rwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeesx "BLAS_FUNC(zgeesx)"(char *jobvs, char *sort, _zselect1 *select, char *sense, int *n, npy_complex128 *a, int *lda, int *sdim, npy_complex128 *w, npy_complex128 *vs, int *ldvs, d *rconde, d *rcondv, npy_complex128 *work, int *lwork, d *rwork, bint *bwork, int *info) nogil +cdef void zgeesx(char *jobvs, char *sort, zselect1 *select, char *sense, int *n, z *a, int *lda, int *sdim, z *w, z *vs, int *ldvs, d *rconde, d *rcondv, z *work, int *lwork, d *rwork, bint *bwork, int *info) noexcept nogil: + + _fortran_zgeesx(jobvs, sort, <_zselect1*>select, sense, n, a, lda, sdim, w, vs, ldvs, rconde, rcondv, work, lwork, rwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeev "BLAS_FUNC(zgeev)"(char *jobvl, char *jobvr, int *n, npy_complex128 *a, int *lda, npy_complex128 *w, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zgeev(char *jobvl, char *jobvr, int *n, z *a, int *lda, z *w, z *vl, int *ldvl, z *vr, int *ldvr, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zgeev(jobvl, jobvr, n, a, lda, w, vl, ldvl, vr, ldvr, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeevx "BLAS_FUNC(zgeevx)"(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, npy_complex128 *a, int *lda, npy_complex128 *w, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, int *ilo, int *ihi, d *scale, d *abnrm, d *rconde, d *rcondv, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zgeevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, z *a, int *lda, z *w, z *vl, int *ldvl, z *vr, int *ldvr, int *ilo, int *ihi, d *scale, d *abnrm, d *rconde, d *rcondv, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zgeevx(balanc, jobvl, jobvr, sense, n, a, lda, w, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgehd2 "BLAS_FUNC(zgehd2)"(int *n, int *ilo, int *ihi, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info) nogil +cdef void zgehd2(int *n, int *ilo, int *ihi, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil: + + _fortran_zgehd2(n, ilo, ihi, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgehrd "BLAS_FUNC(zgehrd)"(int *n, int *ilo, int *ihi, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zgehrd(int *n, int *ilo, int *ihi, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zgehrd(n, ilo, ihi, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgelq2 "BLAS_FUNC(zgelq2)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info) nogil +cdef void zgelq2(int *m, int *n, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil: + + _fortran_zgelq2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgelqf "BLAS_FUNC(zgelqf)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zgelqf(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zgelqf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgels "BLAS_FUNC(zgels)"(char *trans, int *m, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zgels(char *trans, int *m, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zgels(trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgelsd "BLAS_FUNC(zgelsd)"(int *m, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, d *s, d *rcond, int *rank, npy_complex128 *work, int *lwork, d *rwork, int *iwork, int *info) nogil +cdef void zgelsd(int *m, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, d *s, d *rcond, int *rank, z *work, int *lwork, d *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_zgelsd(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgelss "BLAS_FUNC(zgelss)"(int *m, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, d *s, d *rcond, int *rank, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zgelss(int *m, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, d *s, d *rcond, int *rank, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zgelss(m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgelsy "BLAS_FUNC(zgelsy)"(int *m, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *jpvt, d *rcond, int *rank, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zgelsy(int *m, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, int *jpvt, d *rcond, int *rank, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zgelsy(m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgemqrt "BLAS_FUNC(zgemqrt)"(char *side, char *trans, int *m, int *n, int *k, int *nb, npy_complex128 *v, int *ldv, npy_complex128 *t, int *ldt, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info) nogil +cdef void zgemqrt(char *side, char *trans, int *m, int *n, int *k, int *nb, z *v, int *ldv, z *t, int *ldt, z *c, int *ldc, z *work, int *info) noexcept nogil: + + _fortran_zgemqrt(side, trans, m, n, k, nb, v, ldv, t, ldt, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeql2 "BLAS_FUNC(zgeql2)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info) nogil +cdef void zgeql2(int *m, int *n, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil: + + _fortran_zgeql2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeqlf "BLAS_FUNC(zgeqlf)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zgeqlf(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zgeqlf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeqp3 "BLAS_FUNC(zgeqp3)"(int *m, int *n, npy_complex128 *a, int *lda, int *jpvt, npy_complex128 *tau, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zgeqp3(int *m, int *n, z *a, int *lda, int *jpvt, z *tau, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zgeqp3(m, n, a, lda, jpvt, tau, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeqr2 "BLAS_FUNC(zgeqr2)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info) nogil +cdef void zgeqr2(int *m, int *n, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil: + + _fortran_zgeqr2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeqr2p "BLAS_FUNC(zgeqr2p)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info) nogil +cdef void zgeqr2p(int *m, int *n, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil: + + _fortran_zgeqr2p(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeqrf "BLAS_FUNC(zgeqrf)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zgeqrf(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zgeqrf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeqrfp "BLAS_FUNC(zgeqrfp)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zgeqrfp(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zgeqrfp(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeqrt "BLAS_FUNC(zgeqrt)"(int *m, int *n, int *nb, npy_complex128 *a, int *lda, npy_complex128 *t, int *ldt, npy_complex128 *work, int *info) nogil +cdef void zgeqrt(int *m, int *n, int *nb, z *a, int *lda, z *t, int *ldt, z *work, int *info) noexcept nogil: + + _fortran_zgeqrt(m, n, nb, a, lda, t, ldt, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeqrt2 "BLAS_FUNC(zgeqrt2)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *t, int *ldt, int *info) nogil +cdef void zgeqrt2(int *m, int *n, z *a, int *lda, z *t, int *ldt, int *info) noexcept nogil: + + _fortran_zgeqrt2(m, n, a, lda, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgeqrt3 "BLAS_FUNC(zgeqrt3)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *t, int *ldt, int *info) nogil +cdef void zgeqrt3(int *m, int *n, z *a, int *lda, z *t, int *ldt, int *info) noexcept nogil: + + _fortran_zgeqrt3(m, n, a, lda, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgerfs "BLAS_FUNC(zgerfs)"(char *trans, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zgerfs(char *trans, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zgerfs(trans, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgerq2 "BLAS_FUNC(zgerq2)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info) nogil +cdef void zgerq2(int *m, int *n, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil: + + _fortran_zgerq2(m, n, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgerqf "BLAS_FUNC(zgerqf)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zgerqf(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zgerqf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgesc2 "BLAS_FUNC(zgesc2)"(int *n, npy_complex128 *a, int *lda, npy_complex128 *rhs, int *ipiv, int *jpiv, d *scale) nogil +cdef void zgesc2(int *n, z *a, int *lda, z *rhs, int *ipiv, int *jpiv, d *scale) noexcept nogil: + + _fortran_zgesc2(n, a, lda, rhs, ipiv, jpiv, scale) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgesdd "BLAS_FUNC(zgesdd)"(char *jobz, int *m, int *n, npy_complex128 *a, int *lda, d *s, npy_complex128 *u, int *ldu, npy_complex128 *vt, int *ldvt, npy_complex128 *work, int *lwork, d *rwork, int *iwork, int *info) nogil +cdef void zgesdd(char *jobz, int *m, int *n, z *a, int *lda, d *s, z *u, int *ldu, z *vt, int *ldvt, z *work, int *lwork, d *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_zgesdd(jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgesv "BLAS_FUNC(zgesv)"(int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zgesv(int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zgesv(n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgesvd "BLAS_FUNC(zgesvd)"(char *jobu, char *jobvt, int *m, int *n, npy_complex128 *a, int *lda, d *s, npy_complex128 *u, int *ldu, npy_complex128 *vt, int *ldvt, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zgesvd(char *jobu, char *jobvt, int *m, int *n, z *a, int *lda, d *s, z *u, int *ldu, z *vt, int *ldvt, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zgesvd(jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgesvx "BLAS_FUNC(zgesvx)"(char *fact, char *trans, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, char *equed, d *r, d *c, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zgesvx(char *fact, char *trans, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, char *equed, d *r, d *c, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zgesvx(fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgetc2 "BLAS_FUNC(zgetc2)"(int *n, npy_complex128 *a, int *lda, int *ipiv, int *jpiv, int *info) nogil +cdef void zgetc2(int *n, z *a, int *lda, int *ipiv, int *jpiv, int *info) noexcept nogil: + + _fortran_zgetc2(n, a, lda, ipiv, jpiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgetf2 "BLAS_FUNC(zgetf2)"(int *m, int *n, npy_complex128 *a, int *lda, int *ipiv, int *info) nogil +cdef void zgetf2(int *m, int *n, z *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_zgetf2(m, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgetrf "BLAS_FUNC(zgetrf)"(int *m, int *n, npy_complex128 *a, int *lda, int *ipiv, int *info) nogil +cdef void zgetrf(int *m, int *n, z *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_zgetrf(m, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgetri "BLAS_FUNC(zgetri)"(int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zgetri(int *n, z *a, int *lda, int *ipiv, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zgetri(n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgetrs "BLAS_FUNC(zgetrs)"(char *trans, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zgetrs(char *trans, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zgetrs(trans, n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zggbak "BLAS_FUNC(zggbak)"(char *job, char *side, int *n, int *ilo, int *ihi, d *lscale, d *rscale, int *m, npy_complex128 *v, int *ldv, int *info) nogil +cdef void zggbak(char *job, char *side, int *n, int *ilo, int *ihi, d *lscale, d *rscale, int *m, z *v, int *ldv, int *info) noexcept nogil: + + _fortran_zggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zggbal "BLAS_FUNC(zggbal)"(char *job, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *ilo, int *ihi, d *lscale, d *rscale, d *work, int *info) nogil +cdef void zggbal(char *job, int *n, z *a, int *lda, z *b, int *ldb, int *ilo, int *ihi, d *lscale, d *rscale, d *work, int *info) noexcept nogil: + + _fortran_zggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgges "BLAS_FUNC(zgges)"(char *jobvsl, char *jobvsr, char *sort, _zselect2 *selctg, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *sdim, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *vsl, int *ldvsl, npy_complex128 *vsr, int *ldvsr, npy_complex128 *work, int *lwork, d *rwork, bint *bwork, int *info) nogil +cdef void zgges(char *jobvsl, char *jobvsr, char *sort, zselect2 *selctg, int *n, z *a, int *lda, z *b, int *ldb, int *sdim, z *alpha, z *beta, z *vsl, int *ldvsl, z *vsr, int *ldvsr, z *work, int *lwork, d *rwork, bint *bwork, int *info) noexcept nogil: + + _fortran_zgges(jobvsl, jobvsr, sort, <_zselect2*>selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zggesx "BLAS_FUNC(zggesx)"(char *jobvsl, char *jobvsr, char *sort, _zselect2 *selctg, char *sense, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *sdim, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *vsl, int *ldvsl, npy_complex128 *vsr, int *ldvsr, d *rconde, d *rcondv, npy_complex128 *work, int *lwork, d *rwork, int *iwork, int *liwork, bint *bwork, int *info) nogil +cdef void zggesx(char *jobvsl, char *jobvsr, char *sort, zselect2 *selctg, char *sense, int *n, z *a, int *lda, z *b, int *ldb, int *sdim, z *alpha, z *beta, z *vsl, int *ldvsl, z *vsr, int *ldvsr, d *rconde, d *rcondv, z *work, int *lwork, d *rwork, int *iwork, int *liwork, bint *bwork, int *info) noexcept nogil: + + _fortran_zggesx(jobvsl, jobvsr, sort, <_zselect2*>selctg, sense, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, rwork, iwork, liwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zggev "BLAS_FUNC(zggev)"(char *jobvl, char *jobvr, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zggev(char *jobvl, char *jobvr, int *n, z *a, int *lda, z *b, int *ldb, z *alpha, z *beta, z *vl, int *ldvl, z *vr, int *ldvr, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zggev(jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zggevx "BLAS_FUNC(zggevx)"(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, int *ilo, int *ihi, d *lscale, d *rscale, d *abnrm, d *bbnrm, d *rconde, d *rcondv, npy_complex128 *work, int *lwork, d *rwork, int *iwork, bint *bwork, int *info) nogil +cdef void zggevx(char *balanc, char *jobvl, char *jobvr, char *sense, int *n, z *a, int *lda, z *b, int *ldb, z *alpha, z *beta, z *vl, int *ldvl, z *vr, int *ldvr, int *ilo, int *ihi, d *lscale, d *rscale, d *abnrm, d *bbnrm, d *rconde, d *rcondv, z *work, int *lwork, d *rwork, int *iwork, bint *bwork, int *info) noexcept nogil: + + _fortran_zggevx(balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zggglm "BLAS_FUNC(zggglm)"(int *n, int *m, int *p, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *d, npy_complex128 *x, npy_complex128 *y, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zggglm(int *n, int *m, int *p, z *a, int *lda, z *b, int *ldb, z *d, z *x, z *y, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zggglm(n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgghrd "BLAS_FUNC(zgghrd)"(char *compq, char *compz, int *n, int *ilo, int *ihi, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *q, int *ldq, npy_complex128 *z, int *ldz, int *info) nogil +cdef void zgghrd(char *compq, char *compz, int *n, int *ilo, int *ihi, z *a, int *lda, z *b, int *ldb, z *q, int *ldq, z *z, int *ldz, int *info) noexcept nogil: + + _fortran_zgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgglse "BLAS_FUNC(zgglse)"(int *m, int *n, int *p, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *c, npy_complex128 *d, npy_complex128 *x, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zgglse(int *m, int *n, int *p, z *a, int *lda, z *b, int *ldb, z *c, z *d, z *x, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zgglse(m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zggqrf "BLAS_FUNC(zggqrf)"(int *n, int *m, int *p, npy_complex128 *a, int *lda, npy_complex128 *taua, npy_complex128 *b, int *ldb, npy_complex128 *taub, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zggqrf(int *n, int *m, int *p, z *a, int *lda, z *taua, z *b, int *ldb, z *taub, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zggqrf(n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zggrqf "BLAS_FUNC(zggrqf)"(int *m, int *p, int *n, npy_complex128 *a, int *lda, npy_complex128 *taua, npy_complex128 *b, int *ldb, npy_complex128 *taub, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zggrqf(int *m, int *p, int *n, z *a, int *lda, z *taua, z *b, int *ldb, z *taub, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zggrqf(m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgtcon "BLAS_FUNC(zgtcon)"(char *norm, int *n, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *du2, int *ipiv, d *anorm, d *rcond, npy_complex128 *work, int *info) nogil +cdef void zgtcon(char *norm, int *n, z *dl, z *d, z *du, z *du2, int *ipiv, d *anorm, d *rcond, z *work, int *info) noexcept nogil: + + _fortran_zgtcon(norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgtrfs "BLAS_FUNC(zgtrfs)"(char *trans, int *n, int *nrhs, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *dlf, npy_complex128 *df, npy_complex128 *duf, npy_complex128 *du2, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zgtrfs(char *trans, int *n, int *nrhs, z *dl, z *d, z *du, z *dlf, z *df, z *duf, z *du2, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zgtrfs(trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgtsv "BLAS_FUNC(zgtsv)"(int *n, int *nrhs, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zgtsv(int *n, int *nrhs, z *dl, z *d, z *du, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zgtsv(n, nrhs, dl, d, du, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgtsvx "BLAS_FUNC(zgtsvx)"(char *fact, char *trans, int *n, int *nrhs, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *dlf, npy_complex128 *df, npy_complex128 *duf, npy_complex128 *du2, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zgtsvx(char *fact, char *trans, int *n, int *nrhs, z *dl, z *d, z *du, z *dlf, z *df, z *duf, z *du2, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zgtsvx(fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgttrf "BLAS_FUNC(zgttrf)"(int *n, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *du2, int *ipiv, int *info) nogil +cdef void zgttrf(int *n, z *dl, z *d, z *du, z *du2, int *ipiv, int *info) noexcept nogil: + + _fortran_zgttrf(n, dl, d, du, du2, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgttrs "BLAS_FUNC(zgttrs)"(char *trans, int *n, int *nrhs, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *du2, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zgttrs(char *trans, int *n, int *nrhs, z *dl, z *d, z *du, z *du2, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zgtts2 "BLAS_FUNC(zgtts2)"(int *itrans, int *n, int *nrhs, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *du2, int *ipiv, npy_complex128 *b, int *ldb) nogil +cdef void zgtts2(int *itrans, int *n, int *nrhs, z *dl, z *d, z *du, z *du2, int *ipiv, z *b, int *ldb) noexcept nogil: + + _fortran_zgtts2(itrans, n, nrhs, dl, d, du, du2, ipiv, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhbev "BLAS_FUNC(zhbev)"(char *jobz, char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zhbev(char *jobz, char *uplo, int *n, int *kd, z *ab, int *ldab, d *w, z *z, int *ldz, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zhbev(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhbevd "BLAS_FUNC(zhbevd)"(char *jobz, char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void zhbevd(char *jobz, char *uplo, int *n, int *kd, z *ab, int *ldab, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_zhbevd(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhbevx "BLAS_FUNC(zhbevx)"(char *jobz, char *range, char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, npy_complex128 *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, d *rwork, int *iwork, int *ifail, int *info) nogil +cdef void zhbevx(char *jobz, char *range, char *uplo, int *n, int *kd, z *ab, int *ldab, z *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_zhbevx(jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhbgst "BLAS_FUNC(zhbgst)"(char *vect, char *uplo, int *n, int *ka, int *kb, npy_complex128 *ab, int *ldab, npy_complex128 *bb, int *ldbb, npy_complex128 *x, int *ldx, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zhbgst(char *vect, char *uplo, int *n, int *ka, int *kb, z *ab, int *ldab, z *bb, int *ldbb, z *x, int *ldx, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zhbgst(vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhbgv "BLAS_FUNC(zhbgv)"(char *jobz, char *uplo, int *n, int *ka, int *kb, npy_complex128 *ab, int *ldab, npy_complex128 *bb, int *ldbb, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zhbgv(char *jobz, char *uplo, int *n, int *ka, int *kb, z *ab, int *ldab, z *bb, int *ldbb, d *w, z *z, int *ldz, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zhbgv(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhbgvd "BLAS_FUNC(zhbgvd)"(char *jobz, char *uplo, int *n, int *ka, int *kb, npy_complex128 *ab, int *ldab, npy_complex128 *bb, int *ldbb, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void zhbgvd(char *jobz, char *uplo, int *n, int *ka, int *kb, z *ab, int *ldab, z *bb, int *ldbb, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_zhbgvd(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhbgvx "BLAS_FUNC(zhbgvx)"(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, npy_complex128 *ab, int *ldab, npy_complex128 *bb, int *ldbb, npy_complex128 *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, d *rwork, int *iwork, int *ifail, int *info) nogil +cdef void zhbgvx(char *jobz, char *range, char *uplo, int *n, int *ka, int *kb, z *ab, int *ldab, z *bb, int *ldbb, z *q, int *ldq, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_zhbgvx(jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhbtrd "BLAS_FUNC(zhbtrd)"(char *vect, char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, d *d, d *e, npy_complex128 *q, int *ldq, npy_complex128 *work, int *info) nogil +cdef void zhbtrd(char *vect, char *uplo, int *n, int *kd, z *ab, int *ldab, d *d, d *e, z *q, int *ldq, z *work, int *info) noexcept nogil: + + _fortran_zhbtrd(vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhecon "BLAS_FUNC(zhecon)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, d *anorm, d *rcond, npy_complex128 *work, int *info) nogil +cdef void zhecon(char *uplo, int *n, z *a, int *lda, int *ipiv, d *anorm, d *rcond, z *work, int *info) noexcept nogil: + + _fortran_zhecon(uplo, n, a, lda, ipiv, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zheequb "BLAS_FUNC(zheequb)"(char *uplo, int *n, npy_complex128 *a, int *lda, d *s, d *scond, d *amax, npy_complex128 *work, int *info) nogil +cdef void zheequb(char *uplo, int *n, z *a, int *lda, d *s, d *scond, d *amax, z *work, int *info) noexcept nogil: + + _fortran_zheequb(uplo, n, a, lda, s, scond, amax, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zheev "BLAS_FUNC(zheev)"(char *jobz, char *uplo, int *n, npy_complex128 *a, int *lda, d *w, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zheev(char *jobz, char *uplo, int *n, z *a, int *lda, d *w, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zheev(jobz, uplo, n, a, lda, w, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zheevd "BLAS_FUNC(zheevd)"(char *jobz, char *uplo, int *n, npy_complex128 *a, int *lda, d *w, npy_complex128 *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void zheevd(char *jobz, char *uplo, int *n, z *a, int *lda, d *w, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_zheevd(jobz, uplo, n, a, lda, w, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zheevr "BLAS_FUNC(zheevr)"(char *jobz, char *range, char *uplo, int *n, npy_complex128 *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, npy_complex128 *z, int *ldz, int *isuppz, npy_complex128 *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void zheevr(char *jobz, char *range, char *uplo, int *n, z *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, int *isuppz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_zheevr(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zheevx "BLAS_FUNC(zheevx)"(char *jobz, char *range, char *uplo, int *n, npy_complex128 *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, d *rwork, int *iwork, int *ifail, int *info) nogil +cdef void zheevx(char *jobz, char *range, char *uplo, int *n, z *a, int *lda, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_zheevx(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhegs2 "BLAS_FUNC(zhegs2)"(int *itype, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zhegs2(int *itype, char *uplo, int *n, z *a, int *lda, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zhegs2(itype, uplo, n, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhegst "BLAS_FUNC(zhegst)"(int *itype, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zhegst(int *itype, char *uplo, int *n, z *a, int *lda, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zhegst(itype, uplo, n, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhegv "BLAS_FUNC(zhegv)"(int *itype, char *jobz, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, d *w, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zhegv(int *itype, char *jobz, char *uplo, int *n, z *a, int *lda, z *b, int *ldb, d *w, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zhegv(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhegvd "BLAS_FUNC(zhegvd)"(int *itype, char *jobz, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, d *w, npy_complex128 *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void zhegvd(int *itype, char *jobz, char *uplo, int *n, z *a, int *lda, z *b, int *ldb, d *w, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_zhegvd(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhegvx "BLAS_FUNC(zhegvx)"(int *itype, char *jobz, char *range, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, d *rwork, int *iwork, int *ifail, int *info) nogil +cdef void zhegvx(int *itype, char *jobz, char *range, char *uplo, int *n, z *a, int *lda, z *b, int *ldb, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_zhegvx(itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zherfs "BLAS_FUNC(zherfs)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zherfs(char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zherfs(uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhesv "BLAS_FUNC(zhesv)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zhesv(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zhesv(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhesvx "BLAS_FUNC(zhesvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zhesvx(char *fact, char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zhesvx(fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zheswapr "BLAS_FUNC(zheswapr)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *i1, int *i2) nogil +cdef void zheswapr(char *uplo, int *n, z *a, int *lda, int *i1, int *i2) noexcept nogil: + + _fortran_zheswapr(uplo, n, a, lda, i1, i2) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhetd2 "BLAS_FUNC(zhetd2)"(char *uplo, int *n, npy_complex128 *a, int *lda, d *d, d *e, npy_complex128 *tau, int *info) nogil +cdef void zhetd2(char *uplo, int *n, z *a, int *lda, d *d, d *e, z *tau, int *info) noexcept nogil: + + _fortran_zhetd2(uplo, n, a, lda, d, e, tau, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhetf2 "BLAS_FUNC(zhetf2)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, int *info) nogil +cdef void zhetf2(char *uplo, int *n, z *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_zhetf2(uplo, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhetrd "BLAS_FUNC(zhetrd)"(char *uplo, int *n, npy_complex128 *a, int *lda, d *d, d *e, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zhetrd(char *uplo, int *n, z *a, int *lda, d *d, d *e, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zhetrd(uplo, n, a, lda, d, e, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhetrf "BLAS_FUNC(zhetrf)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zhetrf(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zhetrf(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhetri "BLAS_FUNC(zhetri)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *info) nogil +cdef void zhetri(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *info) noexcept nogil: + + _fortran_zhetri(uplo, n, a, lda, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhetri2 "BLAS_FUNC(zhetri2)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zhetri2(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zhetri2(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhetri2x "BLAS_FUNC(zhetri2x)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *nb, int *info) nogil +cdef void zhetri2x(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *nb, int *info) noexcept nogil: + + _fortran_zhetri2x(uplo, n, a, lda, ipiv, work, nb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhetrs "BLAS_FUNC(zhetrs)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zhetrs(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zhetrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhetrs2 "BLAS_FUNC(zhetrs2)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *work, int *info) nogil +cdef void zhetrs2(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, z *work, int *info) noexcept nogil: + + _fortran_zhetrs2(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhfrk "BLAS_FUNC(zhfrk)"(char *transr, char *uplo, char *trans, int *n, int *k, d *alpha, npy_complex128 *a, int *lda, d *beta, npy_complex128 *c) nogil +cdef void zhfrk(char *transr, char *uplo, char *trans, int *n, int *k, d *alpha, z *a, int *lda, d *beta, z *c) noexcept nogil: + + _fortran_zhfrk(transr, uplo, trans, n, k, alpha, a, lda, beta, c) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhgeqz "BLAS_FUNC(zhgeqz)"(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, npy_complex128 *h, int *ldh, npy_complex128 *t, int *ldt, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *q, int *ldq, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zhgeqz(char *job, char *compq, char *compz, int *n, int *ilo, int *ihi, z *h, int *ldh, z *t, int *ldt, z *alpha, z *beta, z *q, int *ldq, z *z, int *ldz, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zhgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhpcon "BLAS_FUNC(zhpcon)"(char *uplo, int *n, npy_complex128 *ap, int *ipiv, d *anorm, d *rcond, npy_complex128 *work, int *info) nogil +cdef void zhpcon(char *uplo, int *n, z *ap, int *ipiv, d *anorm, d *rcond, z *work, int *info) noexcept nogil: + + _fortran_zhpcon(uplo, n, ap, ipiv, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhpev "BLAS_FUNC(zhpev)"(char *jobz, char *uplo, int *n, npy_complex128 *ap, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zhpev(char *jobz, char *uplo, int *n, z *ap, d *w, z *z, int *ldz, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zhpev(jobz, uplo, n, ap, w, z, ldz, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhpevd "BLAS_FUNC(zhpevd)"(char *jobz, char *uplo, int *n, npy_complex128 *ap, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void zhpevd(char *jobz, char *uplo, int *n, z *ap, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_zhpevd(jobz, uplo, n, ap, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhpevx "BLAS_FUNC(zhpevx)"(char *jobz, char *range, char *uplo, int *n, npy_complex128 *ap, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, d *rwork, int *iwork, int *ifail, int *info) nogil +cdef void zhpevx(char *jobz, char *range, char *uplo, int *n, z *ap, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_zhpevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhpgst "BLAS_FUNC(zhpgst)"(int *itype, char *uplo, int *n, npy_complex128 *ap, npy_complex128 *bp, int *info) nogil +cdef void zhpgst(int *itype, char *uplo, int *n, z *ap, z *bp, int *info) noexcept nogil: + + _fortran_zhpgst(itype, uplo, n, ap, bp, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhpgv "BLAS_FUNC(zhpgv)"(int *itype, char *jobz, char *uplo, int *n, npy_complex128 *ap, npy_complex128 *bp, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zhpgv(int *itype, char *jobz, char *uplo, int *n, z *ap, z *bp, d *w, z *z, int *ldz, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zhpgv(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhpgvd "BLAS_FUNC(zhpgvd)"(int *itype, char *jobz, char *uplo, int *n, npy_complex128 *ap, npy_complex128 *bp, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void zhpgvd(int *itype, char *jobz, char *uplo, int *n, z *ap, z *bp, d *w, z *z, int *ldz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_zhpgvd(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhpgvx "BLAS_FUNC(zhpgvx)"(int *itype, char *jobz, char *range, char *uplo, int *n, npy_complex128 *ap, npy_complex128 *bp, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, npy_complex128 *z, int *ldz, npy_complex128 *work, d *rwork, int *iwork, int *ifail, int *info) nogil +cdef void zhpgvx(int *itype, char *jobz, char *range, char *uplo, int *n, z *ap, z *bp, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, z *work, d *rwork, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_zhpgvx(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhprfs "BLAS_FUNC(zhprfs)"(char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zhprfs(char *uplo, int *n, int *nrhs, z *ap, z *afp, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zhprfs(uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhpsv "BLAS_FUNC(zhpsv)"(char *uplo, int *n, int *nrhs, npy_complex128 *ap, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zhpsv(char *uplo, int *n, int *nrhs, z *ap, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zhpsv(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhpsvx "BLAS_FUNC(zhpsvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zhpsvx(char *fact, char *uplo, int *n, int *nrhs, z *ap, z *afp, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zhpsvx(fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhptrd "BLAS_FUNC(zhptrd)"(char *uplo, int *n, npy_complex128 *ap, d *d, d *e, npy_complex128 *tau, int *info) nogil +cdef void zhptrd(char *uplo, int *n, z *ap, d *d, d *e, z *tau, int *info) noexcept nogil: + + _fortran_zhptrd(uplo, n, ap, d, e, tau, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhptrf "BLAS_FUNC(zhptrf)"(char *uplo, int *n, npy_complex128 *ap, int *ipiv, int *info) nogil +cdef void zhptrf(char *uplo, int *n, z *ap, int *ipiv, int *info) noexcept nogil: + + _fortran_zhptrf(uplo, n, ap, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhptri "BLAS_FUNC(zhptri)"(char *uplo, int *n, npy_complex128 *ap, int *ipiv, npy_complex128 *work, int *info) nogil +cdef void zhptri(char *uplo, int *n, z *ap, int *ipiv, z *work, int *info) noexcept nogil: + + _fortran_zhptri(uplo, n, ap, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhptrs "BLAS_FUNC(zhptrs)"(char *uplo, int *n, int *nrhs, npy_complex128 *ap, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zhptrs(char *uplo, int *n, int *nrhs, z *ap, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zhptrs(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhsein "BLAS_FUNC(zhsein)"(char *side, char *eigsrc, char *initv, bint *select, int *n, npy_complex128 *h, int *ldh, npy_complex128 *w, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, int *mm, int *m, npy_complex128 *work, d *rwork, int *ifaill, int *ifailr, int *info) nogil +cdef void zhsein(char *side, char *eigsrc, char *initv, bint *select, int *n, z *h, int *ldh, z *w, z *vl, int *ldvl, z *vr, int *ldvr, int *mm, int *m, z *work, d *rwork, int *ifaill, int *ifailr, int *info) noexcept nogil: + + _fortran_zhsein(side, eigsrc, initv, select, n, h, ldh, w, vl, ldvl, vr, ldvr, mm, m, work, rwork, ifaill, ifailr, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zhseqr "BLAS_FUNC(zhseqr)"(char *job, char *compz, int *n, int *ilo, int *ihi, npy_complex128 *h, int *ldh, npy_complex128 *w, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zhseqr(char *job, char *compz, int *n, int *ilo, int *ihi, z *h, int *ldh, z *w, z *z, int *ldz, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zhseqr(job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlabrd "BLAS_FUNC(zlabrd)"(int *m, int *n, int *nb, npy_complex128 *a, int *lda, d *d, d *e, npy_complex128 *tauq, npy_complex128 *taup, npy_complex128 *x, int *ldx, npy_complex128 *y, int *ldy) nogil +cdef void zlabrd(int *m, int *n, int *nb, z *a, int *lda, d *d, d *e, z *tauq, z *taup, z *x, int *ldx, z *y, int *ldy) noexcept nogil: + + _fortran_zlabrd(m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlacgv "BLAS_FUNC(zlacgv)"(int *n, npy_complex128 *x, int *incx) nogil +cdef void zlacgv(int *n, z *x, int *incx) noexcept nogil: + + _fortran_zlacgv(n, x, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlacn2 "BLAS_FUNC(zlacn2)"(int *n, npy_complex128 *v, npy_complex128 *x, d *est, int *kase, int *isave) nogil +cdef void zlacn2(int *n, z *v, z *x, d *est, int *kase, int *isave) noexcept nogil: + + _fortran_zlacn2(n, v, x, est, kase, isave) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlacon "BLAS_FUNC(zlacon)"(int *n, npy_complex128 *v, npy_complex128 *x, d *est, int *kase) nogil +cdef void zlacon(int *n, z *v, z *x, d *est, int *kase) noexcept nogil: + + _fortran_zlacon(n, v, x, est, kase) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlacp2 "BLAS_FUNC(zlacp2)"(char *uplo, int *m, int *n, d *a, int *lda, npy_complex128 *b, int *ldb) nogil +cdef void zlacp2(char *uplo, int *m, int *n, d *a, int *lda, z *b, int *ldb) noexcept nogil: + + _fortran_zlacp2(uplo, m, n, a, lda, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlacpy "BLAS_FUNC(zlacpy)"(char *uplo, int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb) nogil +cdef void zlacpy(char *uplo, int *m, int *n, z *a, int *lda, z *b, int *ldb) noexcept nogil: + + _fortran_zlacpy(uplo, m, n, a, lda, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlacrm "BLAS_FUNC(zlacrm)"(int *m, int *n, npy_complex128 *a, int *lda, d *b, int *ldb, npy_complex128 *c, int *ldc, d *rwork) nogil +cdef void zlacrm(int *m, int *n, z *a, int *lda, d *b, int *ldb, z *c, int *ldc, d *rwork) noexcept nogil: + + _fortran_zlacrm(m, n, a, lda, b, ldb, c, ldc, rwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlacrt "BLAS_FUNC(zlacrt)"(int *n, npy_complex128 *cx, int *incx, npy_complex128 *cy, int *incy, npy_complex128 *c, npy_complex128 *s) nogil +cdef void zlacrt(int *n, z *cx, int *incx, z *cy, int *incy, z *c, z *s) noexcept nogil: + + _fortran_zlacrt(n, cx, incx, cy, incy, c, s) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zladiv "F_FUNC(zladivwrp,ZLADIVWRP)"(npy_complex128 *out, npy_complex128 *x, npy_complex128 *y) nogil +cdef z zladiv(z *x, z *y) noexcept nogil: + cdef z out + _fortran_zladiv(&out, x, y) + return out + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaed0 "BLAS_FUNC(zlaed0)"(int *qsiz, int *n, d *d, d *e, npy_complex128 *q, int *ldq, npy_complex128 *qstore, int *ldqs, d *rwork, int *iwork, int *info) nogil +cdef void zlaed0(int *qsiz, int *n, d *d, d *e, z *q, int *ldq, z *qstore, int *ldqs, d *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_zlaed0(qsiz, n, d, e, q, ldq, qstore, ldqs, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaed7 "BLAS_FUNC(zlaed7)"(int *n, int *cutpnt, int *qsiz, int *tlvls, int *curlvl, int *curpbm, d *d, npy_complex128 *q, int *ldq, d *rho, int *indxq, d *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, d *givnum, npy_complex128 *work, d *rwork, int *iwork, int *info) nogil +cdef void zlaed7(int *n, int *cutpnt, int *qsiz, int *tlvls, int *curlvl, int *curpbm, d *d, z *q, int *ldq, d *rho, int *indxq, d *qstore, int *qptr, int *prmptr, int *perm, int *givptr, int *givcol, d *givnum, z *work, d *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_zlaed7(n, cutpnt, qsiz, tlvls, curlvl, curpbm, d, q, ldq, rho, indxq, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaed8 "BLAS_FUNC(zlaed8)"(int *k, int *n, int *qsiz, npy_complex128 *q, int *ldq, d *d, d *rho, int *cutpnt, d *z, d *dlamda, npy_complex128 *q2, int *ldq2, d *w, int *indxp, int *indx, int *indxq, int *perm, int *givptr, int *givcol, d *givnum, int *info) nogil +cdef void zlaed8(int *k, int *n, int *qsiz, z *q, int *ldq, d *d, d *rho, int *cutpnt, d *z, d *dlamda, z *q2, int *ldq2, d *w, int *indxp, int *indx, int *indxq, int *perm, int *givptr, int *givcol, d *givnum, int *info) noexcept nogil: + + _fortran_zlaed8(k, n, qsiz, q, ldq, d, rho, cutpnt, z, dlamda, q2, ldq2, w, indxp, indx, indxq, perm, givptr, givcol, givnum, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaein "BLAS_FUNC(zlaein)"(bint *rightv, bint *noinit, int *n, npy_complex128 *h, int *ldh, npy_complex128 *w, npy_complex128 *v, npy_complex128 *b, int *ldb, d *rwork, d *eps3, d *smlnum, int *info) nogil +cdef void zlaein(bint *rightv, bint *noinit, int *n, z *h, int *ldh, z *w, z *v, z *b, int *ldb, d *rwork, d *eps3, d *smlnum, int *info) noexcept nogil: + + _fortran_zlaein(rightv, noinit, n, h, ldh, w, v, b, ldb, rwork, eps3, smlnum, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaesy "BLAS_FUNC(zlaesy)"(npy_complex128 *a, npy_complex128 *b, npy_complex128 *c, npy_complex128 *rt1, npy_complex128 *rt2, npy_complex128 *evscal, npy_complex128 *cs1, npy_complex128 *sn1) nogil +cdef void zlaesy(z *a, z *b, z *c, z *rt1, z *rt2, z *evscal, z *cs1, z *sn1) noexcept nogil: + + _fortran_zlaesy(a, b, c, rt1, rt2, evscal, cs1, sn1) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaev2 "BLAS_FUNC(zlaev2)"(npy_complex128 *a, npy_complex128 *b, npy_complex128 *c, d *rt1, d *rt2, d *cs1, npy_complex128 *sn1) nogil +cdef void zlaev2(z *a, z *b, z *c, d *rt1, d *rt2, d *cs1, z *sn1) noexcept nogil: + + _fortran_zlaev2(a, b, c, rt1, rt2, cs1, sn1) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlag2c "BLAS_FUNC(zlag2c)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex64 *sa, int *ldsa, int *info) nogil +cdef void zlag2c(int *m, int *n, z *a, int *lda, c *sa, int *ldsa, int *info) noexcept nogil: + + _fortran_zlag2c(m, n, a, lda, sa, ldsa, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlags2 "BLAS_FUNC(zlags2)"(bint *upper, d *a1, npy_complex128 *a2, d *a3, d *b1, npy_complex128 *b2, d *b3, d *csu, npy_complex128 *snu, d *csv, npy_complex128 *snv, d *csq, npy_complex128 *snq) nogil +cdef void zlags2(bint *upper, d *a1, z *a2, d *a3, d *b1, z *b2, d *b3, d *csu, z *snu, d *csv, z *snv, d *csq, z *snq) noexcept nogil: + + _fortran_zlags2(upper, a1, a2, a3, b1, b2, b3, csu, snu, csv, snv, csq, snq) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlagtm "BLAS_FUNC(zlagtm)"(char *trans, int *n, int *nrhs, d *alpha, npy_complex128 *dl, npy_complex128 *d, npy_complex128 *du, npy_complex128 *x, int *ldx, d *beta, npy_complex128 *b, int *ldb) nogil +cdef void zlagtm(char *trans, int *n, int *nrhs, d *alpha, z *dl, z *d, z *du, z *x, int *ldx, d *beta, z *b, int *ldb) noexcept nogil: + + _fortran_zlagtm(trans, n, nrhs, alpha, dl, d, du, x, ldx, beta, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlahef "BLAS_FUNC(zlahef)"(char *uplo, int *n, int *nb, int *kb, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *w, int *ldw, int *info) nogil +cdef void zlahef(char *uplo, int *n, int *nb, int *kb, z *a, int *lda, int *ipiv, z *w, int *ldw, int *info) noexcept nogil: + + _fortran_zlahef(uplo, n, nb, kb, a, lda, ipiv, w, ldw, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlahqr "BLAS_FUNC(zlahqr)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, npy_complex128 *h, int *ldh, npy_complex128 *w, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, int *info) nogil +cdef void zlahqr(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, z *h, int *ldh, z *w, int *iloz, int *ihiz, z *z, int *ldz, int *info) noexcept nogil: + + _fortran_zlahqr(wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlahr2 "BLAS_FUNC(zlahr2)"(int *n, int *k, int *nb, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *t, int *ldt, npy_complex128 *y, int *ldy) nogil +cdef void zlahr2(int *n, int *k, int *nb, z *a, int *lda, z *tau, z *t, int *ldt, z *y, int *ldy) noexcept nogil: + + _fortran_zlahr2(n, k, nb, a, lda, tau, t, ldt, y, ldy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaic1 "BLAS_FUNC(zlaic1)"(int *job, int *j, npy_complex128 *x, d *sest, npy_complex128 *w, npy_complex128 *gamma, d *sestpr, npy_complex128 *s, npy_complex128 *c) nogil +cdef void zlaic1(int *job, int *j, z *x, d *sest, z *w, z *gamma, d *sestpr, z *s, z *c) noexcept nogil: + + _fortran_zlaic1(job, j, x, sest, w, gamma, sestpr, s, c) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlals0 "BLAS_FUNC(zlals0)"(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, npy_complex128 *b, int *ldb, npy_complex128 *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *poles, d *difl, d *difr, d *z, int *k, d *c, d *s, d *rwork, int *info) nogil +cdef void zlals0(int *icompq, int *nl, int *nr, int *sqre, int *nrhs, z *b, int *ldb, z *bx, int *ldbx, int *perm, int *givptr, int *givcol, int *ldgcol, d *givnum, int *ldgnum, d *poles, d *difl, d *difr, d *z, int *k, d *c, d *s, d *rwork, int *info) noexcept nogil: + + _fortran_zlals0(icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlalsa "BLAS_FUNC(zlalsa)"(int *icompq, int *smlsiz, int *n, int *nrhs, npy_complex128 *b, int *ldb, npy_complex128 *bx, int *ldbx, d *u, int *ldu, d *vt, int *k, d *difl, d *difr, d *z, d *poles, int *givptr, int *givcol, int *ldgcol, int *perm, d *givnum, d *c, d *s, d *rwork, int *iwork, int *info) nogil +cdef void zlalsa(int *icompq, int *smlsiz, int *n, int *nrhs, z *b, int *ldb, z *bx, int *ldbx, d *u, int *ldu, d *vt, int *k, d *difl, d *difr, d *z, d *poles, int *givptr, int *givcol, int *ldgcol, int *perm, d *givnum, d *c, d *s, d *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_zlalsa(icompq, smlsiz, n, nrhs, b, ldb, bx, ldbx, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlalsd "BLAS_FUNC(zlalsd)"(char *uplo, int *smlsiz, int *n, int *nrhs, d *d, d *e, npy_complex128 *b, int *ldb, d *rcond, int *rank, npy_complex128 *work, d *rwork, int *iwork, int *info) nogil +cdef void zlalsd(char *uplo, int *smlsiz, int *n, int *nrhs, d *d, d *e, z *b, int *ldb, d *rcond, int *rank, z *work, d *rwork, int *iwork, int *info) noexcept nogil: + + _fortran_zlalsd(uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, rwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlangb "BLAS_FUNC(zlangb)"(char *norm, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, d *work) nogil +cdef d zlangb(char *norm, int *n, int *kl, int *ku, z *ab, int *ldab, d *work) noexcept nogil: + + return _fortran_zlangb(norm, n, kl, ku, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlange "BLAS_FUNC(zlange)"(char *norm, int *m, int *n, npy_complex128 *a, int *lda, d *work) nogil +cdef d zlange(char *norm, int *m, int *n, z *a, int *lda, d *work) noexcept nogil: + + return _fortran_zlange(norm, m, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlangt "BLAS_FUNC(zlangt)"(char *norm, int *n, npy_complex128 *dl, npy_complex128 *d_, npy_complex128 *du) nogil +cdef d zlangt(char *norm, int *n, z *dl, z *d_, z *du) noexcept nogil: + + return _fortran_zlangt(norm, n, dl, d_, du) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlanhb "BLAS_FUNC(zlanhb)"(char *norm, char *uplo, int *n, int *k, npy_complex128 *ab, int *ldab, d *work) nogil +cdef d zlanhb(char *norm, char *uplo, int *n, int *k, z *ab, int *ldab, d *work) noexcept nogil: + + return _fortran_zlanhb(norm, uplo, n, k, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlanhe "BLAS_FUNC(zlanhe)"(char *norm, char *uplo, int *n, npy_complex128 *a, int *lda, d *work) nogil +cdef d zlanhe(char *norm, char *uplo, int *n, z *a, int *lda, d *work) noexcept nogil: + + return _fortran_zlanhe(norm, uplo, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlanhf "BLAS_FUNC(zlanhf)"(char *norm, char *transr, char *uplo, int *n, npy_complex128 *a, d *work) nogil +cdef d zlanhf(char *norm, char *transr, char *uplo, int *n, z *a, d *work) noexcept nogil: + + return _fortran_zlanhf(norm, transr, uplo, n, a, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlanhp "BLAS_FUNC(zlanhp)"(char *norm, char *uplo, int *n, npy_complex128 *ap, d *work) nogil +cdef d zlanhp(char *norm, char *uplo, int *n, z *ap, d *work) noexcept nogil: + + return _fortran_zlanhp(norm, uplo, n, ap, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlanhs "BLAS_FUNC(zlanhs)"(char *norm, int *n, npy_complex128 *a, int *lda, d *work) nogil +cdef d zlanhs(char *norm, int *n, z *a, int *lda, d *work) noexcept nogil: + + return _fortran_zlanhs(norm, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlanht "BLAS_FUNC(zlanht)"(char *norm, int *n, d *d_, npy_complex128 *e) nogil +cdef d zlanht(char *norm, int *n, d *d_, z *e) noexcept nogil: + + return _fortran_zlanht(norm, n, d_, e) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlansb "BLAS_FUNC(zlansb)"(char *norm, char *uplo, int *n, int *k, npy_complex128 *ab, int *ldab, d *work) nogil +cdef d zlansb(char *norm, char *uplo, int *n, int *k, z *ab, int *ldab, d *work) noexcept nogil: + + return _fortran_zlansb(norm, uplo, n, k, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlansp "BLAS_FUNC(zlansp)"(char *norm, char *uplo, int *n, npy_complex128 *ap, d *work) nogil +cdef d zlansp(char *norm, char *uplo, int *n, z *ap, d *work) noexcept nogil: + + return _fortran_zlansp(norm, uplo, n, ap, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlansy "BLAS_FUNC(zlansy)"(char *norm, char *uplo, int *n, npy_complex128 *a, int *lda, d *work) nogil +cdef d zlansy(char *norm, char *uplo, int *n, z *a, int *lda, d *work) noexcept nogil: + + return _fortran_zlansy(norm, uplo, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlantb "BLAS_FUNC(zlantb)"(char *norm, char *uplo, char *diag, int *n, int *k, npy_complex128 *ab, int *ldab, d *work) nogil +cdef d zlantb(char *norm, char *uplo, char *diag, int *n, int *k, z *ab, int *ldab, d *work) noexcept nogil: + + return _fortran_zlantb(norm, uplo, diag, n, k, ab, ldab, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlantp "BLAS_FUNC(zlantp)"(char *norm, char *uplo, char *diag, int *n, npy_complex128 *ap, d *work) nogil +cdef d zlantp(char *norm, char *uplo, char *diag, int *n, z *ap, d *work) noexcept nogil: + + return _fortran_zlantp(norm, uplo, diag, n, ap, work) + + +cdef extern from "_lapack_subroutines.h": + d _fortran_zlantr "BLAS_FUNC(zlantr)"(char *norm, char *uplo, char *diag, int *m, int *n, npy_complex128 *a, int *lda, d *work) nogil +cdef d zlantr(char *norm, char *uplo, char *diag, int *m, int *n, z *a, int *lda, d *work) noexcept nogil: + + return _fortran_zlantr(norm, uplo, diag, m, n, a, lda, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlapll "BLAS_FUNC(zlapll)"(int *n, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, d *ssmin) nogil +cdef void zlapll(int *n, z *x, int *incx, z *y, int *incy, d *ssmin) noexcept nogil: + + _fortran_zlapll(n, x, incx, y, incy, ssmin) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlapmr "BLAS_FUNC(zlapmr)"(bint *forwrd, int *m, int *n, npy_complex128 *x, int *ldx, int *k) nogil +cdef void zlapmr(bint *forwrd, int *m, int *n, z *x, int *ldx, int *k) noexcept nogil: + + _fortran_zlapmr(forwrd, m, n, x, ldx, k) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlapmt "BLAS_FUNC(zlapmt)"(bint *forwrd, int *m, int *n, npy_complex128 *x, int *ldx, int *k) nogil +cdef void zlapmt(bint *forwrd, int *m, int *n, z *x, int *ldx, int *k) noexcept nogil: + + _fortran_zlapmt(forwrd, m, n, x, ldx, k) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqgb "BLAS_FUNC(zlaqgb)"(int *m, int *n, int *kl, int *ku, npy_complex128 *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) nogil +cdef void zlaqgb(int *m, int *n, int *kl, int *ku, z *ab, int *ldab, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) noexcept nogil: + + _fortran_zlaqgb(m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqge "BLAS_FUNC(zlaqge)"(int *m, int *n, npy_complex128 *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) nogil +cdef void zlaqge(int *m, int *n, z *a, int *lda, d *r, d *c, d *rowcnd, d *colcnd, d *amax, char *equed) noexcept nogil: + + _fortran_zlaqge(m, n, a, lda, r, c, rowcnd, colcnd, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqhb "BLAS_FUNC(zlaqhb)"(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, d *s, d *scond, d *amax, char *equed) nogil +cdef void zlaqhb(char *uplo, int *n, int *kd, z *ab, int *ldab, d *s, d *scond, d *amax, char *equed) noexcept nogil: + + _fortran_zlaqhb(uplo, n, kd, ab, ldab, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqhe "BLAS_FUNC(zlaqhe)"(char *uplo, int *n, npy_complex128 *a, int *lda, d *s, d *scond, d *amax, char *equed) nogil +cdef void zlaqhe(char *uplo, int *n, z *a, int *lda, d *s, d *scond, d *amax, char *equed) noexcept nogil: + + _fortran_zlaqhe(uplo, n, a, lda, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqhp "BLAS_FUNC(zlaqhp)"(char *uplo, int *n, npy_complex128 *ap, d *s, d *scond, d *amax, char *equed) nogil +cdef void zlaqhp(char *uplo, int *n, z *ap, d *s, d *scond, d *amax, char *equed) noexcept nogil: + + _fortran_zlaqhp(uplo, n, ap, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqp2 "BLAS_FUNC(zlaqp2)"(int *m, int *n, int *offset, npy_complex128 *a, int *lda, int *jpvt, npy_complex128 *tau, d *vn1, d *vn2, npy_complex128 *work) nogil +cdef void zlaqp2(int *m, int *n, int *offset, z *a, int *lda, int *jpvt, z *tau, d *vn1, d *vn2, z *work) noexcept nogil: + + _fortran_zlaqp2(m, n, offset, a, lda, jpvt, tau, vn1, vn2, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqps "BLAS_FUNC(zlaqps)"(int *m, int *n, int *offset, int *nb, int *kb, npy_complex128 *a, int *lda, int *jpvt, npy_complex128 *tau, d *vn1, d *vn2, npy_complex128 *auxv, npy_complex128 *f, int *ldf) nogil +cdef void zlaqps(int *m, int *n, int *offset, int *nb, int *kb, z *a, int *lda, int *jpvt, z *tau, d *vn1, d *vn2, z *auxv, z *f, int *ldf) noexcept nogil: + + _fortran_zlaqps(m, n, offset, nb, kb, a, lda, jpvt, tau, vn1, vn2, auxv, f, ldf) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqr0 "BLAS_FUNC(zlaqr0)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, npy_complex128 *h, int *ldh, npy_complex128 *w, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zlaqr0(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, z *h, int *ldh, z *w, int *iloz, int *ihiz, z *z, int *ldz, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zlaqr0(wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqr1 "BLAS_FUNC(zlaqr1)"(int *n, npy_complex128 *h, int *ldh, npy_complex128 *s1, npy_complex128 *s2, npy_complex128 *v) nogil +cdef void zlaqr1(int *n, z *h, int *ldh, z *s1, z *s2, z *v) noexcept nogil: + + _fortran_zlaqr1(n, h, ldh, s1, s2, v) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqr2 "BLAS_FUNC(zlaqr2)"(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, npy_complex128 *h, int *ldh, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, int *ns, int *nd, npy_complex128 *sh, npy_complex128 *v, int *ldv, int *nh, npy_complex128 *t, int *ldt, int *nv, npy_complex128 *wv, int *ldwv, npy_complex128 *work, int *lwork) nogil +cdef void zlaqr2(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, z *h, int *ldh, int *iloz, int *ihiz, z *z, int *ldz, int *ns, int *nd, z *sh, z *v, int *ldv, int *nh, z *t, int *ldt, int *nv, z *wv, int *ldwv, z *work, int *lwork) noexcept nogil: + + _fortran_zlaqr2(wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqr3 "BLAS_FUNC(zlaqr3)"(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, npy_complex128 *h, int *ldh, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, int *ns, int *nd, npy_complex128 *sh, npy_complex128 *v, int *ldv, int *nh, npy_complex128 *t, int *ldt, int *nv, npy_complex128 *wv, int *ldwv, npy_complex128 *work, int *lwork) nogil +cdef void zlaqr3(bint *wantt, bint *wantz, int *n, int *ktop, int *kbot, int *nw, z *h, int *ldh, int *iloz, int *ihiz, z *z, int *ldz, int *ns, int *nd, z *sh, z *v, int *ldv, int *nh, z *t, int *ldt, int *nv, z *wv, int *ldwv, z *work, int *lwork) noexcept nogil: + + _fortran_zlaqr3(wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqr4 "BLAS_FUNC(zlaqr4)"(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, npy_complex128 *h, int *ldh, npy_complex128 *w, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zlaqr4(bint *wantt, bint *wantz, int *n, int *ilo, int *ihi, z *h, int *ldh, z *w, int *iloz, int *ihiz, z *z, int *ldz, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zlaqr4(wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqr5 "BLAS_FUNC(zlaqr5)"(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, npy_complex128 *s, npy_complex128 *h, int *ldh, int *iloz, int *ihiz, npy_complex128 *z, int *ldz, npy_complex128 *v, int *ldv, npy_complex128 *u, int *ldu, int *nv, npy_complex128 *wv, int *ldwv, int *nh, npy_complex128 *wh, int *ldwh) nogil +cdef void zlaqr5(bint *wantt, bint *wantz, int *kacc22, int *n, int *ktop, int *kbot, int *nshfts, z *s, z *h, int *ldh, int *iloz, int *ihiz, z *z, int *ldz, z *v, int *ldv, z *u, int *ldu, int *nv, z *wv, int *ldwv, int *nh, z *wh, int *ldwh) noexcept nogil: + + _fortran_zlaqr5(wantt, wantz, kacc22, n, ktop, kbot, nshfts, s, h, ldh, iloz, ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqsb "BLAS_FUNC(zlaqsb)"(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, d *s, d *scond, d *amax, char *equed) nogil +cdef void zlaqsb(char *uplo, int *n, int *kd, z *ab, int *ldab, d *s, d *scond, d *amax, char *equed) noexcept nogil: + + _fortran_zlaqsb(uplo, n, kd, ab, ldab, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqsp "BLAS_FUNC(zlaqsp)"(char *uplo, int *n, npy_complex128 *ap, d *s, d *scond, d *amax, char *equed) nogil +cdef void zlaqsp(char *uplo, int *n, z *ap, d *s, d *scond, d *amax, char *equed) noexcept nogil: + + _fortran_zlaqsp(uplo, n, ap, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaqsy "BLAS_FUNC(zlaqsy)"(char *uplo, int *n, npy_complex128 *a, int *lda, d *s, d *scond, d *amax, char *equed) nogil +cdef void zlaqsy(char *uplo, int *n, z *a, int *lda, d *s, d *scond, d *amax, char *equed) noexcept nogil: + + _fortran_zlaqsy(uplo, n, a, lda, s, scond, amax, equed) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlar1v "BLAS_FUNC(zlar1v)"(int *n, int *b1, int *bn, d *lambda_, d *d, d *l, d *ld, d *lld, d *pivmin, d *gaptol, npy_complex128 *z, bint *wantnc, int *negcnt, d *ztz, d *mingma, int *r, int *isuppz, d *nrminv, d *resid, d *rqcorr, d *work) nogil +cdef void zlar1v(int *n, int *b1, int *bn, d *lambda_, d *d, d *l, d *ld, d *lld, d *pivmin, d *gaptol, z *z, bint *wantnc, int *negcnt, d *ztz, d *mingma, int *r, int *isuppz, d *nrminv, d *resid, d *rqcorr, d *work) noexcept nogil: + + _fortran_zlar1v(n, b1, bn, lambda_, d, l, ld, lld, pivmin, gaptol, z, wantnc, negcnt, ztz, mingma, r, isuppz, nrminv, resid, rqcorr, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlar2v "BLAS_FUNC(zlar2v)"(int *n, npy_complex128 *x, npy_complex128 *y, npy_complex128 *z, int *incx, d *c, npy_complex128 *s, int *incc) nogil +cdef void zlar2v(int *n, z *x, z *y, z *z, int *incx, d *c, z *s, int *incc) noexcept nogil: + + _fortran_zlar2v(n, x, y, z, incx, c, s, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarcm "BLAS_FUNC(zlarcm)"(int *m, int *n, d *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *c, int *ldc, d *rwork) nogil +cdef void zlarcm(int *m, int *n, d *a, int *lda, z *b, int *ldb, z *c, int *ldc, d *rwork) noexcept nogil: + + _fortran_zlarcm(m, n, a, lda, b, ldb, c, ldc, rwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarf "BLAS_FUNC(zlarf)"(char *side, int *m, int *n, npy_complex128 *v, int *incv, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work) nogil +cdef void zlarf(char *side, int *m, int *n, z *v, int *incv, z *tau, z *c, int *ldc, z *work) noexcept nogil: + + _fortran_zlarf(side, m, n, v, incv, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarfb "BLAS_FUNC(zlarfb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, npy_complex128 *v, int *ldv, npy_complex128 *t, int *ldt, npy_complex128 *c, int *ldc, npy_complex128 *work, int *ldwork) nogil +cdef void zlarfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, z *v, int *ldv, z *t, int *ldt, z *c, int *ldc, z *work, int *ldwork) noexcept nogil: + + _fortran_zlarfb(side, trans, direct, storev, m, n, k, v, ldv, t, ldt, c, ldc, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarfg "BLAS_FUNC(zlarfg)"(int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *tau) nogil +cdef void zlarfg(int *n, z *alpha, z *x, int *incx, z *tau) noexcept nogil: + + _fortran_zlarfg(n, alpha, x, incx, tau) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarfgp "BLAS_FUNC(zlarfgp)"(int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *tau) nogil +cdef void zlarfgp(int *n, z *alpha, z *x, int *incx, z *tau) noexcept nogil: + + _fortran_zlarfgp(n, alpha, x, incx, tau) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarft "BLAS_FUNC(zlarft)"(char *direct, char *storev, int *n, int *k, npy_complex128 *v, int *ldv, npy_complex128 *tau, npy_complex128 *t, int *ldt) nogil +cdef void zlarft(char *direct, char *storev, int *n, int *k, z *v, int *ldv, z *tau, z *t, int *ldt) noexcept nogil: + + _fortran_zlarft(direct, storev, n, k, v, ldv, tau, t, ldt) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarfx "BLAS_FUNC(zlarfx)"(char *side, int *m, int *n, npy_complex128 *v, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work) nogil +cdef void zlarfx(char *side, int *m, int *n, z *v, z *tau, z *c, int *ldc, z *work) noexcept nogil: + + _fortran_zlarfx(side, m, n, v, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlargv "BLAS_FUNC(zlargv)"(int *n, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, d *c, int *incc) nogil +cdef void zlargv(int *n, z *x, int *incx, z *y, int *incy, d *c, int *incc) noexcept nogil: + + _fortran_zlargv(n, x, incx, y, incy, c, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarnv "BLAS_FUNC(zlarnv)"(int *idist, int *iseed, int *n, npy_complex128 *x) nogil +cdef void zlarnv(int *idist, int *iseed, int *n, z *x) noexcept nogil: + + _fortran_zlarnv(idist, iseed, n, x) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarrv "BLAS_FUNC(zlarrv)"(int *n, d *vl, d *vu, d *d, d *l, d *pivmin, int *isplit, int *m, int *dol, int *dou, d *minrgp, d *rtol1, d *rtol2, d *w, d *werr, d *wgap, int *iblock, int *indexw, d *gers, npy_complex128 *z, int *ldz, int *isuppz, d *work, int *iwork, int *info) nogil +cdef void zlarrv(int *n, d *vl, d *vu, d *d, d *l, d *pivmin, int *isplit, int *m, int *dol, int *dou, d *minrgp, d *rtol1, d *rtol2, d *w, d *werr, d *wgap, int *iblock, int *indexw, d *gers, z *z, int *ldz, int *isuppz, d *work, int *iwork, int *info) noexcept nogil: + + _fortran_zlarrv(n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlartg "BLAS_FUNC(zlartg)"(npy_complex128 *f, npy_complex128 *g, d *cs, npy_complex128 *sn, npy_complex128 *r) nogil +cdef void zlartg(z *f, z *g, d *cs, z *sn, z *r) noexcept nogil: + + _fortran_zlartg(f, g, cs, sn, r) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlartv "BLAS_FUNC(zlartv)"(int *n, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, d *c, npy_complex128 *s, int *incc) nogil +cdef void zlartv(int *n, z *x, int *incx, z *y, int *incy, d *c, z *s, int *incc) noexcept nogil: + + _fortran_zlartv(n, x, incx, y, incy, c, s, incc) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarz "BLAS_FUNC(zlarz)"(char *side, int *m, int *n, int *l, npy_complex128 *v, int *incv, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work) nogil +cdef void zlarz(char *side, int *m, int *n, int *l, z *v, int *incv, z *tau, z *c, int *ldc, z *work) noexcept nogil: + + _fortran_zlarz(side, m, n, l, v, incv, tau, c, ldc, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarzb "BLAS_FUNC(zlarzb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, npy_complex128 *v, int *ldv, npy_complex128 *t, int *ldt, npy_complex128 *c, int *ldc, npy_complex128 *work, int *ldwork) nogil +cdef void zlarzb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, z *v, int *ldv, z *t, int *ldt, z *c, int *ldc, z *work, int *ldwork) noexcept nogil: + + _fortran_zlarzb(side, trans, direct, storev, m, n, k, l, v, ldv, t, ldt, c, ldc, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlarzt "BLAS_FUNC(zlarzt)"(char *direct, char *storev, int *n, int *k, npy_complex128 *v, int *ldv, npy_complex128 *tau, npy_complex128 *t, int *ldt) nogil +cdef void zlarzt(char *direct, char *storev, int *n, int *k, z *v, int *ldv, z *tau, z *t, int *ldt) noexcept nogil: + + _fortran_zlarzt(direct, storev, n, k, v, ldv, tau, t, ldt) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlascl "BLAS_FUNC(zlascl)"(char *type_bn, int *kl, int *ku, d *cfrom, d *cto, int *m, int *n, npy_complex128 *a, int *lda, int *info) nogil +cdef void zlascl(char *type_bn, int *kl, int *ku, d *cfrom, d *cto, int *m, int *n, z *a, int *lda, int *info) noexcept nogil: + + _fortran_zlascl(type_bn, kl, ku, cfrom, cto, m, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaset "BLAS_FUNC(zlaset)"(char *uplo, int *m, int *n, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *a, int *lda) nogil +cdef void zlaset(char *uplo, int *m, int *n, z *alpha, z *beta, z *a, int *lda) noexcept nogil: + + _fortran_zlaset(uplo, m, n, alpha, beta, a, lda) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlasr "BLAS_FUNC(zlasr)"(char *side, char *pivot, char *direct, int *m, int *n, d *c, d *s, npy_complex128 *a, int *lda) nogil +cdef void zlasr(char *side, char *pivot, char *direct, int *m, int *n, d *c, d *s, z *a, int *lda) noexcept nogil: + + _fortran_zlasr(side, pivot, direct, m, n, c, s, a, lda) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlassq "BLAS_FUNC(zlassq)"(int *n, npy_complex128 *x, int *incx, d *scale, d *sumsq) nogil +cdef void zlassq(int *n, z *x, int *incx, d *scale, d *sumsq) noexcept nogil: + + _fortran_zlassq(n, x, incx, scale, sumsq) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlaswp "BLAS_FUNC(zlaswp)"(int *n, npy_complex128 *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) nogil +cdef void zlaswp(int *n, z *a, int *lda, int *k1, int *k2, int *ipiv, int *incx) noexcept nogil: + + _fortran_zlaswp(n, a, lda, k1, k2, ipiv, incx) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlasyf "BLAS_FUNC(zlasyf)"(char *uplo, int *n, int *nb, int *kb, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *w, int *ldw, int *info) nogil +cdef void zlasyf(char *uplo, int *n, int *nb, int *kb, z *a, int *lda, int *ipiv, z *w, int *ldw, int *info) noexcept nogil: + + _fortran_zlasyf(uplo, n, nb, kb, a, lda, ipiv, w, ldw, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlat2c "BLAS_FUNC(zlat2c)"(char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex64 *sa, int *ldsa, int *info) nogil +cdef void zlat2c(char *uplo, int *n, z *a, int *lda, c *sa, int *ldsa, int *info) noexcept nogil: + + _fortran_zlat2c(uplo, n, a, lda, sa, ldsa, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlatbs "BLAS_FUNC(zlatbs)"(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, npy_complex128 *ab, int *ldab, npy_complex128 *x, d *scale, d *cnorm, int *info) nogil +cdef void zlatbs(char *uplo, char *trans, char *diag, char *normin, int *n, int *kd, z *ab, int *ldab, z *x, d *scale, d *cnorm, int *info) noexcept nogil: + + _fortran_zlatbs(uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlatdf "BLAS_FUNC(zlatdf)"(int *ijob, int *n, npy_complex128 *z, int *ldz, npy_complex128 *rhs, d *rdsum, d *rdscal, int *ipiv, int *jpiv) nogil +cdef void zlatdf(int *ijob, int *n, z *z, int *ldz, z *rhs, d *rdsum, d *rdscal, int *ipiv, int *jpiv) noexcept nogil: + + _fortran_zlatdf(ijob, n, z, ldz, rhs, rdsum, rdscal, ipiv, jpiv) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlatps "BLAS_FUNC(zlatps)"(char *uplo, char *trans, char *diag, char *normin, int *n, npy_complex128 *ap, npy_complex128 *x, d *scale, d *cnorm, int *info) nogil +cdef void zlatps(char *uplo, char *trans, char *diag, char *normin, int *n, z *ap, z *x, d *scale, d *cnorm, int *info) noexcept nogil: + + _fortran_zlatps(uplo, trans, diag, normin, n, ap, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlatrd "BLAS_FUNC(zlatrd)"(char *uplo, int *n, int *nb, npy_complex128 *a, int *lda, d *e, npy_complex128 *tau, npy_complex128 *w, int *ldw) nogil +cdef void zlatrd(char *uplo, int *n, int *nb, z *a, int *lda, d *e, z *tau, z *w, int *ldw) noexcept nogil: + + _fortran_zlatrd(uplo, n, nb, a, lda, e, tau, w, ldw) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlatrs "BLAS_FUNC(zlatrs)"(char *uplo, char *trans, char *diag, char *normin, int *n, npy_complex128 *a, int *lda, npy_complex128 *x, d *scale, d *cnorm, int *info) nogil +cdef void zlatrs(char *uplo, char *trans, char *diag, char *normin, int *n, z *a, int *lda, z *x, d *scale, d *cnorm, int *info) noexcept nogil: + + _fortran_zlatrs(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlatrz "BLAS_FUNC(zlatrz)"(int *m, int *n, int *l, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work) nogil +cdef void zlatrz(int *m, int *n, int *l, z *a, int *lda, z *tau, z *work) noexcept nogil: + + _fortran_zlatrz(m, n, l, a, lda, tau, work) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlauu2 "BLAS_FUNC(zlauu2)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *info) nogil +cdef void zlauu2(char *uplo, int *n, z *a, int *lda, int *info) noexcept nogil: + + _fortran_zlauu2(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zlauum "BLAS_FUNC(zlauum)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *info) nogil +cdef void zlauum(char *uplo, int *n, z *a, int *lda, int *info) noexcept nogil: + + _fortran_zlauum(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpbcon "BLAS_FUNC(zpbcon)"(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, d *anorm, d *rcond, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zpbcon(char *uplo, int *n, int *kd, z *ab, int *ldab, d *anorm, d *rcond, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zpbcon(uplo, n, kd, ab, ldab, anorm, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpbequ "BLAS_FUNC(zpbequ)"(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, d *s, d *scond, d *amax, int *info) nogil +cdef void zpbequ(char *uplo, int *n, int *kd, z *ab, int *ldab, d *s, d *scond, d *amax, int *info) noexcept nogil: + + _fortran_zpbequ(uplo, n, kd, ab, ldab, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpbrfs "BLAS_FUNC(zpbrfs)"(char *uplo, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *afb, int *ldafb, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zpbrfs(char *uplo, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *afb, int *ldafb, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zpbrfs(uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpbstf "BLAS_FUNC(zpbstf)"(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, int *info) nogil +cdef void zpbstf(char *uplo, int *n, int *kd, z *ab, int *ldab, int *info) noexcept nogil: + + _fortran_zpbstf(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpbsv "BLAS_FUNC(zpbsv)"(char *uplo, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zpbsv(char *uplo, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zpbsv(uplo, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpbsvx "BLAS_FUNC(zpbsvx)"(char *fact, char *uplo, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *afb, int *ldafb, char *equed, d *s, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zpbsvx(char *fact, char *uplo, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *afb, int *ldafb, char *equed, d *s, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zpbsvx(fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpbtf2 "BLAS_FUNC(zpbtf2)"(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, int *info) nogil +cdef void zpbtf2(char *uplo, int *n, int *kd, z *ab, int *ldab, int *info) noexcept nogil: + + _fortran_zpbtf2(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpbtrf "BLAS_FUNC(zpbtrf)"(char *uplo, int *n, int *kd, npy_complex128 *ab, int *ldab, int *info) nogil +cdef void zpbtrf(char *uplo, int *n, int *kd, z *ab, int *ldab, int *info) noexcept nogil: + + _fortran_zpbtrf(uplo, n, kd, ab, ldab, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpbtrs "BLAS_FUNC(zpbtrs)"(char *uplo, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zpbtrs(char *uplo, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zpbtrs(uplo, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpftrf "BLAS_FUNC(zpftrf)"(char *transr, char *uplo, int *n, npy_complex128 *a, int *info) nogil +cdef void zpftrf(char *transr, char *uplo, int *n, z *a, int *info) noexcept nogil: + + _fortran_zpftrf(transr, uplo, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpftri "BLAS_FUNC(zpftri)"(char *transr, char *uplo, int *n, npy_complex128 *a, int *info) nogil +cdef void zpftri(char *transr, char *uplo, int *n, z *a, int *info) noexcept nogil: + + _fortran_zpftri(transr, uplo, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpftrs "BLAS_FUNC(zpftrs)"(char *transr, char *uplo, int *n, int *nrhs, npy_complex128 *a, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zpftrs(char *transr, char *uplo, int *n, int *nrhs, z *a, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zpftrs(transr, uplo, n, nrhs, a, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpocon "BLAS_FUNC(zpocon)"(char *uplo, int *n, npy_complex128 *a, int *lda, d *anorm, d *rcond, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zpocon(char *uplo, int *n, z *a, int *lda, d *anorm, d *rcond, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zpocon(uplo, n, a, lda, anorm, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpoequ "BLAS_FUNC(zpoequ)"(int *n, npy_complex128 *a, int *lda, d *s, d *scond, d *amax, int *info) nogil +cdef void zpoequ(int *n, z *a, int *lda, d *s, d *scond, d *amax, int *info) noexcept nogil: + + _fortran_zpoequ(n, a, lda, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpoequb "BLAS_FUNC(zpoequb)"(int *n, npy_complex128 *a, int *lda, d *s, d *scond, d *amax, int *info) nogil +cdef void zpoequb(int *n, z *a, int *lda, d *s, d *scond, d *amax, int *info) noexcept nogil: + + _fortran_zpoequb(n, a, lda, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zporfs "BLAS_FUNC(zporfs)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zporfs(char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zporfs(uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zposv "BLAS_FUNC(zposv)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zposv(char *uplo, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zposv(uplo, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zposvx "BLAS_FUNC(zposvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, char *equed, d *s, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zposvx(char *fact, char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, char *equed, d *s, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zposvx(fact, uplo, n, nrhs, a, lda, af, ldaf, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpotf2 "BLAS_FUNC(zpotf2)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *info) nogil +cdef void zpotf2(char *uplo, int *n, z *a, int *lda, int *info) noexcept nogil: + + _fortran_zpotf2(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpotrf "BLAS_FUNC(zpotrf)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *info) nogil +cdef void zpotrf(char *uplo, int *n, z *a, int *lda, int *info) noexcept nogil: + + _fortran_zpotrf(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpotri "BLAS_FUNC(zpotri)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *info) nogil +cdef void zpotri(char *uplo, int *n, z *a, int *lda, int *info) noexcept nogil: + + _fortran_zpotri(uplo, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpotrs "BLAS_FUNC(zpotrs)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zpotrs(char *uplo, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zpotrs(uplo, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zppcon "BLAS_FUNC(zppcon)"(char *uplo, int *n, npy_complex128 *ap, d *anorm, d *rcond, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zppcon(char *uplo, int *n, z *ap, d *anorm, d *rcond, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zppcon(uplo, n, ap, anorm, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zppequ "BLAS_FUNC(zppequ)"(char *uplo, int *n, npy_complex128 *ap, d *s, d *scond, d *amax, int *info) nogil +cdef void zppequ(char *uplo, int *n, z *ap, d *s, d *scond, d *amax, int *info) noexcept nogil: + + _fortran_zppequ(uplo, n, ap, s, scond, amax, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpprfs "BLAS_FUNC(zpprfs)"(char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zpprfs(char *uplo, int *n, int *nrhs, z *ap, z *afp, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zpprfs(uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zppsv "BLAS_FUNC(zppsv)"(char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zppsv(char *uplo, int *n, int *nrhs, z *ap, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zppsv(uplo, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zppsvx "BLAS_FUNC(zppsvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, char *equed, d *s, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zppsvx(char *fact, char *uplo, int *n, int *nrhs, z *ap, z *afp, char *equed, d *s, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zppsvx(fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpptrf "BLAS_FUNC(zpptrf)"(char *uplo, int *n, npy_complex128 *ap, int *info) nogil +cdef void zpptrf(char *uplo, int *n, z *ap, int *info) noexcept nogil: + + _fortran_zpptrf(uplo, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpptri "BLAS_FUNC(zpptri)"(char *uplo, int *n, npy_complex128 *ap, int *info) nogil +cdef void zpptri(char *uplo, int *n, z *ap, int *info) noexcept nogil: + + _fortran_zpptri(uplo, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpptrs "BLAS_FUNC(zpptrs)"(char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zpptrs(char *uplo, int *n, int *nrhs, z *ap, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zpptrs(uplo, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpstf2 "BLAS_FUNC(zpstf2)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) nogil +cdef void zpstf2(char *uplo, int *n, z *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) noexcept nogil: + + _fortran_zpstf2(uplo, n, a, lda, piv, rank, tol, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpstrf "BLAS_FUNC(zpstrf)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) nogil +cdef void zpstrf(char *uplo, int *n, z *a, int *lda, int *piv, int *rank, d *tol, d *work, int *info) noexcept nogil: + + _fortran_zpstrf(uplo, n, a, lda, piv, rank, tol, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zptcon "BLAS_FUNC(zptcon)"(int *n, d *d, npy_complex128 *e, d *anorm, d *rcond, d *rwork, int *info) nogil +cdef void zptcon(int *n, d *d, z *e, d *anorm, d *rcond, d *rwork, int *info) noexcept nogil: + + _fortran_zptcon(n, d, e, anorm, rcond, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpteqr "BLAS_FUNC(zpteqr)"(char *compz, int *n, d *d, d *e, npy_complex128 *z, int *ldz, d *work, int *info) nogil +cdef void zpteqr(char *compz, int *n, d *d, d *e, z *z, int *ldz, d *work, int *info) noexcept nogil: + + _fortran_zpteqr(compz, n, d, e, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zptrfs "BLAS_FUNC(zptrfs)"(char *uplo, int *n, int *nrhs, d *d, npy_complex128 *e, d *df, npy_complex128 *ef, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zptrfs(char *uplo, int *n, int *nrhs, d *d, z *e, d *df, z *ef, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zptrfs(uplo, n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zptsv "BLAS_FUNC(zptsv)"(int *n, int *nrhs, d *d, npy_complex128 *e, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zptsv(int *n, int *nrhs, d *d, z *e, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zptsv(n, nrhs, d, e, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zptsvx "BLAS_FUNC(zptsvx)"(char *fact, int *n, int *nrhs, d *d, npy_complex128 *e, d *df, npy_complex128 *ef, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zptsvx(char *fact, int *n, int *nrhs, d *d, z *e, d *df, z *ef, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zptsvx(fact, n, nrhs, d, e, df, ef, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpttrf "BLAS_FUNC(zpttrf)"(int *n, d *d, npy_complex128 *e, int *info) nogil +cdef void zpttrf(int *n, d *d, z *e, int *info) noexcept nogil: + + _fortran_zpttrf(n, d, e, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zpttrs "BLAS_FUNC(zpttrs)"(char *uplo, int *n, int *nrhs, d *d, npy_complex128 *e, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zpttrs(char *uplo, int *n, int *nrhs, d *d, z *e, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zpttrs(uplo, n, nrhs, d, e, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zptts2 "BLAS_FUNC(zptts2)"(int *iuplo, int *n, int *nrhs, d *d, npy_complex128 *e, npy_complex128 *b, int *ldb) nogil +cdef void zptts2(int *iuplo, int *n, int *nrhs, d *d, z *e, z *b, int *ldb) noexcept nogil: + + _fortran_zptts2(iuplo, n, nrhs, d, e, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zrot "BLAS_FUNC(zrot)"(int *n, npy_complex128 *cx, int *incx, npy_complex128 *cy, int *incy, d *c, npy_complex128 *s) nogil +cdef void zrot(int *n, z *cx, int *incx, z *cy, int *incy, d *c, z *s) noexcept nogil: + + _fortran_zrot(n, cx, incx, cy, incy, c, s) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zspcon "BLAS_FUNC(zspcon)"(char *uplo, int *n, npy_complex128 *ap, int *ipiv, d *anorm, d *rcond, npy_complex128 *work, int *info) nogil +cdef void zspcon(char *uplo, int *n, z *ap, int *ipiv, d *anorm, d *rcond, z *work, int *info) noexcept nogil: + + _fortran_zspcon(uplo, n, ap, ipiv, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zspmv "BLAS_FUNC(zspmv)"(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *ap, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy) nogil +cdef void zspmv(char *uplo, int *n, z *alpha, z *ap, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil: + + _fortran_zspmv(uplo, n, alpha, ap, x, incx, beta, y, incy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zspr "BLAS_FUNC(zspr)"(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *ap) nogil +cdef void zspr(char *uplo, int *n, z *alpha, z *x, int *incx, z *ap) noexcept nogil: + + _fortran_zspr(uplo, n, alpha, x, incx, ap) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsprfs "BLAS_FUNC(zsprfs)"(char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zsprfs(char *uplo, int *n, int *nrhs, z *ap, z *afp, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zsprfs(uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zspsv "BLAS_FUNC(zspsv)"(char *uplo, int *n, int *nrhs, npy_complex128 *ap, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zspsv(char *uplo, int *n, int *nrhs, z *ap, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zspsv(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zspsvx "BLAS_FUNC(zspsvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *afp, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zspsvx(char *fact, char *uplo, int *n, int *nrhs, z *ap, z *afp, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zspsvx(fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsptrf "BLAS_FUNC(zsptrf)"(char *uplo, int *n, npy_complex128 *ap, int *ipiv, int *info) nogil +cdef void zsptrf(char *uplo, int *n, z *ap, int *ipiv, int *info) noexcept nogil: + + _fortran_zsptrf(uplo, n, ap, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsptri "BLAS_FUNC(zsptri)"(char *uplo, int *n, npy_complex128 *ap, int *ipiv, npy_complex128 *work, int *info) nogil +cdef void zsptri(char *uplo, int *n, z *ap, int *ipiv, z *work, int *info) noexcept nogil: + + _fortran_zsptri(uplo, n, ap, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsptrs "BLAS_FUNC(zsptrs)"(char *uplo, int *n, int *nrhs, npy_complex128 *ap, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zsptrs(char *uplo, int *n, int *nrhs, z *ap, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zsptrs(uplo, n, nrhs, ap, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zstedc "BLAS_FUNC(zstedc)"(char *compz, int *n, d *d, d *e, npy_complex128 *z, int *ldz, npy_complex128 *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) nogil +cdef void zstedc(char *compz, int *n, d *d, d *e, z *z, int *ldz, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_zstedc(compz, n, d, e, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zstegr "BLAS_FUNC(zstegr)"(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, npy_complex128 *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void zstegr(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, d *abstol, int *m, d *w, z *z, int *ldz, int *isuppz, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_zstegr(jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zstein "BLAS_FUNC(zstein)"(int *n, d *d, d *e, int *m, d *w, int *iblock, int *isplit, npy_complex128 *z, int *ldz, d *work, int *iwork, int *ifail, int *info) nogil +cdef void zstein(int *n, d *d, d *e, int *m, d *w, int *iblock, int *isplit, z *z, int *ldz, d *work, int *iwork, int *ifail, int *info) noexcept nogil: + + _fortran_zstein(n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zstemr "BLAS_FUNC(zstemr)"(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, int *m, d *w, npy_complex128 *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, d *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void zstemr(char *jobz, char *range, int *n, d *d, d *e, d *vl, d *vu, int *il, int *iu, int *m, d *w, z *z, int *ldz, int *nzc, int *isuppz, bint *tryrac, d *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_zstemr(jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsteqr "BLAS_FUNC(zsteqr)"(char *compz, int *n, d *d, d *e, npy_complex128 *z, int *ldz, d *work, int *info) nogil +cdef void zsteqr(char *compz, int *n, d *d, d *e, z *z, int *ldz, d *work, int *info) noexcept nogil: + + _fortran_zsteqr(compz, n, d, e, z, ldz, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsycon "BLAS_FUNC(zsycon)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, d *anorm, d *rcond, npy_complex128 *work, int *info) nogil +cdef void zsycon(char *uplo, int *n, z *a, int *lda, int *ipiv, d *anorm, d *rcond, z *work, int *info) noexcept nogil: + + _fortran_zsycon(uplo, n, a, lda, ipiv, anorm, rcond, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsyconv "BLAS_FUNC(zsyconv)"(char *uplo, char *way, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *info) nogil +cdef void zsyconv(char *uplo, char *way, int *n, z *a, int *lda, int *ipiv, z *work, int *info) noexcept nogil: + + _fortran_zsyconv(uplo, way, n, a, lda, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsyequb "BLAS_FUNC(zsyequb)"(char *uplo, int *n, npy_complex128 *a, int *lda, d *s, d *scond, d *amax, npy_complex128 *work, int *info) nogil +cdef void zsyequb(char *uplo, int *n, z *a, int *lda, d *s, d *scond, d *amax, z *work, int *info) noexcept nogil: + + _fortran_zsyequb(uplo, n, a, lda, s, scond, amax, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsymv "BLAS_FUNC(zsymv)"(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy) nogil +cdef void zsymv(char *uplo, int *n, z *alpha, z *a, int *lda, z *x, int *incx, z *beta, z *y, int *incy) noexcept nogil: + + _fortran_zsymv(uplo, n, alpha, a, lda, x, incx, beta, y, incy) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsyr "BLAS_FUNC(zsyr)"(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *a, int *lda) nogil +cdef void zsyr(char *uplo, int *n, z *alpha, z *x, int *incx, z *a, int *lda) noexcept nogil: + + _fortran_zsyr(uplo, n, alpha, x, incx, a, lda) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsyrfs "BLAS_FUNC(zsyrfs)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void zsyrfs(char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_zsyrfs(uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsysv "BLAS_FUNC(zsysv)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zsysv(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zsysv(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsysvx "BLAS_FUNC(zsysvx)"(char *fact, char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *af, int *ldaf, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *rcond, d *ferr, d *berr, npy_complex128 *work, int *lwork, d *rwork, int *info) nogil +cdef void zsysvx(char *fact, char *uplo, int *n, int *nrhs, z *a, int *lda, z *af, int *ldaf, int *ipiv, z *b, int *ldb, z *x, int *ldx, d *rcond, d *ferr, d *berr, z *work, int *lwork, d *rwork, int *info) noexcept nogil: + + _fortran_zsysvx(fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsyswapr "BLAS_FUNC(zsyswapr)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *i1, int *i2) nogil +cdef void zsyswapr(char *uplo, int *n, z *a, int *lda, int *i1, int *i2) noexcept nogil: + + _fortran_zsyswapr(uplo, n, a, lda, i1, i2) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsytf2 "BLAS_FUNC(zsytf2)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, int *info) nogil +cdef void zsytf2(char *uplo, int *n, z *a, int *lda, int *ipiv, int *info) noexcept nogil: + + _fortran_zsytf2(uplo, n, a, lda, ipiv, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsytrf "BLAS_FUNC(zsytrf)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zsytrf(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zsytrf(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsytri "BLAS_FUNC(zsytri)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *info) nogil +cdef void zsytri(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *info) noexcept nogil: + + _fortran_zsytri(uplo, n, a, lda, ipiv, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsytri2 "BLAS_FUNC(zsytri2)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zsytri2(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zsytri2(uplo, n, a, lda, ipiv, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsytri2x "BLAS_FUNC(zsytri2x)"(char *uplo, int *n, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *work, int *nb, int *info) nogil +cdef void zsytri2x(char *uplo, int *n, z *a, int *lda, int *ipiv, z *work, int *nb, int *info) noexcept nogil: + + _fortran_zsytri2x(uplo, n, a, lda, ipiv, work, nb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsytrs "BLAS_FUNC(zsytrs)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, int *info) nogil +cdef void zsytrs(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_zsytrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zsytrs2 "BLAS_FUNC(zsytrs2)"(char *uplo, int *n, int *nrhs, npy_complex128 *a, int *lda, int *ipiv, npy_complex128 *b, int *ldb, npy_complex128 *work, int *info) nogil +cdef void zsytrs2(char *uplo, int *n, int *nrhs, z *a, int *lda, int *ipiv, z *b, int *ldb, z *work, int *info) noexcept nogil: + + _fortran_zsytrs2(uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztbcon "BLAS_FUNC(ztbcon)"(char *norm, char *uplo, char *diag, int *n, int *kd, npy_complex128 *ab, int *ldab, d *rcond, npy_complex128 *work, d *rwork, int *info) nogil +cdef void ztbcon(char *norm, char *uplo, char *diag, int *n, int *kd, z *ab, int *ldab, d *rcond, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_ztbcon(norm, uplo, diag, n, kd, ab, ldab, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztbrfs "BLAS_FUNC(ztbrfs)"(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void ztbrfs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_ztbrfs(uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztbtrs "BLAS_FUNC(ztbtrs)"(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, npy_complex128 *ab, int *ldab, npy_complex128 *b, int *ldb, int *info) nogil +cdef void ztbtrs(char *uplo, char *trans, char *diag, int *n, int *kd, int *nrhs, z *ab, int *ldab, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_ztbtrs(uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztfsm "BLAS_FUNC(ztfsm)"(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, npy_complex128 *b, int *ldb) nogil +cdef void ztfsm(char *transr, char *side, char *uplo, char *trans, char *diag, int *m, int *n, z *alpha, z *a, z *b, int *ldb) noexcept nogil: + + _fortran_ztfsm(transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztftri "BLAS_FUNC(ztftri)"(char *transr, char *uplo, char *diag, int *n, npy_complex128 *a, int *info) nogil +cdef void ztftri(char *transr, char *uplo, char *diag, int *n, z *a, int *info) noexcept nogil: + + _fortran_ztftri(transr, uplo, diag, n, a, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztfttp "BLAS_FUNC(ztfttp)"(char *transr, char *uplo, int *n, npy_complex128 *arf, npy_complex128 *ap, int *info) nogil +cdef void ztfttp(char *transr, char *uplo, int *n, z *arf, z *ap, int *info) noexcept nogil: + + _fortran_ztfttp(transr, uplo, n, arf, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztfttr "BLAS_FUNC(ztfttr)"(char *transr, char *uplo, int *n, npy_complex128 *arf, npy_complex128 *a, int *lda, int *info) nogil +cdef void ztfttr(char *transr, char *uplo, int *n, z *arf, z *a, int *lda, int *info) noexcept nogil: + + _fortran_ztfttr(transr, uplo, n, arf, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztgevc "BLAS_FUNC(ztgevc)"(char *side, char *howmny, bint *select, int *n, npy_complex128 *s, int *lds, npy_complex128 *p, int *ldp, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, int *mm, int *m, npy_complex128 *work, d *rwork, int *info) nogil +cdef void ztgevc(char *side, char *howmny, bint *select, int *n, z *s, int *lds, z *p, int *ldp, z *vl, int *ldvl, z *vr, int *ldvr, int *mm, int *m, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_ztgevc(side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztgex2 "BLAS_FUNC(ztgex2)"(bint *wantq, bint *wantz, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *q, int *ldq, npy_complex128 *z, int *ldz, int *j1, int *info) nogil +cdef void ztgex2(bint *wantq, bint *wantz, int *n, z *a, int *lda, z *b, int *ldb, z *q, int *ldq, z *z, int *ldz, int *j1, int *info) noexcept nogil: + + _fortran_ztgex2(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztgexc "BLAS_FUNC(ztgexc)"(bint *wantq, bint *wantz, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *q, int *ldq, npy_complex128 *z, int *ldz, int *ifst, int *ilst, int *info) nogil +cdef void ztgexc(bint *wantq, bint *wantz, int *n, z *a, int *lda, z *b, int *ldb, z *q, int *ldq, z *z, int *ldz, int *ifst, int *ilst, int *info) noexcept nogil: + + _fortran_ztgexc(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztgsen "BLAS_FUNC(ztgsen)"(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *alpha, npy_complex128 *beta, npy_complex128 *q, int *ldq, npy_complex128 *z, int *ldz, int *m, d *pl, d *pr, d *dif, npy_complex128 *work, int *lwork, int *iwork, int *liwork, int *info) nogil +cdef void ztgsen(int *ijob, bint *wantq, bint *wantz, bint *select, int *n, z *a, int *lda, z *b, int *ldb, z *alpha, z *beta, z *q, int *ldq, z *z, int *ldz, int *m, d *pl, d *pr, d *dif, z *work, int *lwork, int *iwork, int *liwork, int *info) noexcept nogil: + + _fortran_ztgsen(ijob, wantq, wantz, select, n, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztgsja "BLAS_FUNC(ztgsja)"(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, d *tola, d *tolb, d *alpha, d *beta, npy_complex128 *u, int *ldu, npy_complex128 *v, int *ldv, npy_complex128 *q, int *ldq, npy_complex128 *work, int *ncycle, int *info) nogil +cdef void ztgsja(char *jobu, char *jobv, char *jobq, int *m, int *p, int *n, int *k, int *l, z *a, int *lda, z *b, int *ldb, d *tola, d *tolb, d *alpha, d *beta, z *u, int *ldu, z *v, int *ldv, z *q, int *ldq, z *work, int *ncycle, int *info) noexcept nogil: + + _fortran_ztgsja(jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb, tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq, work, ncycle, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztgsna "BLAS_FUNC(ztgsna)"(char *job, char *howmny, bint *select, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, d *s, d *dif, int *mm, int *m, npy_complex128 *work, int *lwork, int *iwork, int *info) nogil +cdef void ztgsna(char *job, char *howmny, bint *select, int *n, z *a, int *lda, z *b, int *ldb, z *vl, int *ldvl, z *vr, int *ldvr, d *s, d *dif, int *mm, int *m, z *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_ztgsna(job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztgsy2 "BLAS_FUNC(ztgsy2)"(char *trans, int *ijob, int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *c, int *ldc, npy_complex128 *d, int *ldd, npy_complex128 *e, int *lde, npy_complex128 *f, int *ldf, d *scale, d *rdsum, d *rdscal, int *info) nogil +cdef void ztgsy2(char *trans, int *ijob, int *m, int *n, z *a, int *lda, z *b, int *ldb, z *c, int *ldc, z *d, int *ldd, z *e, int *lde, z *f, int *ldf, d *scale, d *rdsum, d *rdscal, int *info) noexcept nogil: + + _fortran_ztgsy2(trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztgsyl "BLAS_FUNC(ztgsyl)"(char *trans, int *ijob, int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *c, int *ldc, npy_complex128 *d, int *ldd, npy_complex128 *e, int *lde, npy_complex128 *f, int *ldf, d *scale, d *dif, npy_complex128 *work, int *lwork, int *iwork, int *info) nogil +cdef void ztgsyl(char *trans, int *ijob, int *m, int *n, z *a, int *lda, z *b, int *ldb, z *c, int *ldc, z *d, int *ldd, z *e, int *lde, z *f, int *ldf, d *scale, d *dif, z *work, int *lwork, int *iwork, int *info) noexcept nogil: + + _fortran_ztgsyl(trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztpcon "BLAS_FUNC(ztpcon)"(char *norm, char *uplo, char *diag, int *n, npy_complex128 *ap, d *rcond, npy_complex128 *work, d *rwork, int *info) nogil +cdef void ztpcon(char *norm, char *uplo, char *diag, int *n, z *ap, d *rcond, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_ztpcon(norm, uplo, diag, n, ap, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztpmqrt "BLAS_FUNC(ztpmqrt)"(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, npy_complex128 *v, int *ldv, npy_complex128 *t, int *ldt, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *work, int *info) nogil +cdef void ztpmqrt(char *side, char *trans, int *m, int *n, int *k, int *l, int *nb, z *v, int *ldv, z *t, int *ldt, z *a, int *lda, z *b, int *ldb, z *work, int *info) noexcept nogil: + + _fortran_ztpmqrt(side, trans, m, n, k, l, nb, v, ldv, t, ldt, a, lda, b, ldb, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztpqrt "BLAS_FUNC(ztpqrt)"(int *m, int *n, int *l, int *nb, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *t, int *ldt, npy_complex128 *work, int *info) nogil +cdef void ztpqrt(int *m, int *n, int *l, int *nb, z *a, int *lda, z *b, int *ldb, z *t, int *ldt, z *work, int *info) noexcept nogil: + + _fortran_ztpqrt(m, n, l, nb, a, lda, b, ldb, t, ldt, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztpqrt2 "BLAS_FUNC(ztpqrt2)"(int *m, int *n, int *l, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *t, int *ldt, int *info) nogil +cdef void ztpqrt2(int *m, int *n, int *l, z *a, int *lda, z *b, int *ldb, z *t, int *ldt, int *info) noexcept nogil: + + _fortran_ztpqrt2(m, n, l, a, lda, b, ldb, t, ldt, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztprfb "BLAS_FUNC(ztprfb)"(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, npy_complex128 *v, int *ldv, npy_complex128 *t, int *ldt, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *work, int *ldwork) nogil +cdef void ztprfb(char *side, char *trans, char *direct, char *storev, int *m, int *n, int *k, int *l, z *v, int *ldv, z *t, int *ldt, z *a, int *lda, z *b, int *ldb, z *work, int *ldwork) noexcept nogil: + + _fortran_ztprfb(side, trans, direct, storev, m, n, k, l, v, ldv, t, ldt, a, lda, b, ldb, work, ldwork) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztprfs "BLAS_FUNC(ztprfs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void ztprfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, z *ap, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_ztprfs(uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztptri "BLAS_FUNC(ztptri)"(char *uplo, char *diag, int *n, npy_complex128 *ap, int *info) nogil +cdef void ztptri(char *uplo, char *diag, int *n, z *ap, int *info) noexcept nogil: + + _fortran_ztptri(uplo, diag, n, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztptrs "BLAS_FUNC(ztptrs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex128 *ap, npy_complex128 *b, int *ldb, int *info) nogil +cdef void ztptrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, z *ap, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_ztptrs(uplo, trans, diag, n, nrhs, ap, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztpttf "BLAS_FUNC(ztpttf)"(char *transr, char *uplo, int *n, npy_complex128 *ap, npy_complex128 *arf, int *info) nogil +cdef void ztpttf(char *transr, char *uplo, int *n, z *ap, z *arf, int *info) noexcept nogil: + + _fortran_ztpttf(transr, uplo, n, ap, arf, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztpttr "BLAS_FUNC(ztpttr)"(char *uplo, int *n, npy_complex128 *ap, npy_complex128 *a, int *lda, int *info) nogil +cdef void ztpttr(char *uplo, int *n, z *ap, z *a, int *lda, int *info) noexcept nogil: + + _fortran_ztpttr(uplo, n, ap, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrcon "BLAS_FUNC(ztrcon)"(char *norm, char *uplo, char *diag, int *n, npy_complex128 *a, int *lda, d *rcond, npy_complex128 *work, d *rwork, int *info) nogil +cdef void ztrcon(char *norm, char *uplo, char *diag, int *n, z *a, int *lda, d *rcond, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_ztrcon(norm, uplo, diag, n, a, lda, rcond, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrevc "BLAS_FUNC(ztrevc)"(char *side, char *howmny, bint *select, int *n, npy_complex128 *t, int *ldt, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, int *mm, int *m, npy_complex128 *work, d *rwork, int *info) nogil +cdef void ztrevc(char *side, char *howmny, bint *select, int *n, z *t, int *ldt, z *vl, int *ldvl, z *vr, int *ldvr, int *mm, int *m, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_ztrevc(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrexc "BLAS_FUNC(ztrexc)"(char *compq, int *n, npy_complex128 *t, int *ldt, npy_complex128 *q, int *ldq, int *ifst, int *ilst, int *info) nogil +cdef void ztrexc(char *compq, int *n, z *t, int *ldt, z *q, int *ldq, int *ifst, int *ilst, int *info) noexcept nogil: + + _fortran_ztrexc(compq, n, t, ldt, q, ldq, ifst, ilst, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrrfs "BLAS_FUNC(ztrrfs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *x, int *ldx, d *ferr, d *berr, npy_complex128 *work, d *rwork, int *info) nogil +cdef void ztrrfs(char *uplo, char *trans, char *diag, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, z *x, int *ldx, d *ferr, d *berr, z *work, d *rwork, int *info) noexcept nogil: + + _fortran_ztrrfs(uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrsen "BLAS_FUNC(ztrsen)"(char *job, char *compq, bint *select, int *n, npy_complex128 *t, int *ldt, npy_complex128 *q, int *ldq, npy_complex128 *w, int *m, d *s, d *sep, npy_complex128 *work, int *lwork, int *info) nogil +cdef void ztrsen(char *job, char *compq, bint *select, int *n, z *t, int *ldt, z *q, int *ldq, z *w, int *m, d *s, d *sep, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_ztrsen(job, compq, select, n, t, ldt, q, ldq, w, m, s, sep, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrsna "BLAS_FUNC(ztrsna)"(char *job, char *howmny, bint *select, int *n, npy_complex128 *t, int *ldt, npy_complex128 *vl, int *ldvl, npy_complex128 *vr, int *ldvr, d *s, d *sep, int *mm, int *m, npy_complex128 *work, int *ldwork, d *rwork, int *info) nogil +cdef void ztrsna(char *job, char *howmny, bint *select, int *n, z *t, int *ldt, z *vl, int *ldvl, z *vr, int *ldvr, d *s, d *sep, int *mm, int *m, z *work, int *ldwork, d *rwork, int *info) noexcept nogil: + + _fortran_ztrsna(job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrsyl "BLAS_FUNC(ztrsyl)"(char *trana, char *tranb, int *isgn, int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *c, int *ldc, d *scale, int *info) nogil +cdef void ztrsyl(char *trana, char *tranb, int *isgn, int *m, int *n, z *a, int *lda, z *b, int *ldb, z *c, int *ldc, d *scale, int *info) noexcept nogil: + + _fortran_ztrsyl(trana, tranb, isgn, m, n, a, lda, b, ldb, c, ldc, scale, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrti2 "BLAS_FUNC(ztrti2)"(char *uplo, char *diag, int *n, npy_complex128 *a, int *lda, int *info) nogil +cdef void ztrti2(char *uplo, char *diag, int *n, z *a, int *lda, int *info) noexcept nogil: + + _fortran_ztrti2(uplo, diag, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrtri "BLAS_FUNC(ztrtri)"(char *uplo, char *diag, int *n, npy_complex128 *a, int *lda, int *info) nogil +cdef void ztrtri(char *uplo, char *diag, int *n, z *a, int *lda, int *info) noexcept nogil: + + _fortran_ztrtri(uplo, diag, n, a, lda, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrtrs "BLAS_FUNC(ztrtrs)"(char *uplo, char *trans, char *diag, int *n, int *nrhs, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, int *info) nogil +cdef void ztrtrs(char *uplo, char *trans, char *diag, int *n, int *nrhs, z *a, int *lda, z *b, int *ldb, int *info) noexcept nogil: + + _fortran_ztrtrs(uplo, trans, diag, n, nrhs, a, lda, b, ldb, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrttf "BLAS_FUNC(ztrttf)"(char *transr, char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *arf, int *info) nogil +cdef void ztrttf(char *transr, char *uplo, int *n, z *a, int *lda, z *arf, int *info) noexcept nogil: + + _fortran_ztrttf(transr, uplo, n, a, lda, arf, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztrttp "BLAS_FUNC(ztrttp)"(char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *ap, int *info) nogil +cdef void ztrttp(char *uplo, int *n, z *a, int *lda, z *ap, int *info) noexcept nogil: + + _fortran_ztrttp(uplo, n, a, lda, ap, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_ztzrzf "BLAS_FUNC(ztzrzf)"(int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void ztzrzf(int *m, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_ztzrzf(m, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunbdb "BLAS_FUNC(zunbdb)"(char *trans, char *signs, int *m, int *p, int *q, npy_complex128 *x11, int *ldx11, npy_complex128 *x12, int *ldx12, npy_complex128 *x21, int *ldx21, npy_complex128 *x22, int *ldx22, d *theta, d *phi, npy_complex128 *taup1, npy_complex128 *taup2, npy_complex128 *tauq1, npy_complex128 *tauq2, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunbdb(char *trans, char *signs, int *m, int *p, int *q, z *x11, int *ldx11, z *x12, int *ldx12, z *x21, int *ldx21, z *x22, int *ldx22, d *theta, d *phi, z *taup1, z *taup2, z *tauq1, z *tauq2, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunbdb(trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zuncsd "BLAS_FUNC(zuncsd)"(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, npy_complex128 *x11, int *ldx11, npy_complex128 *x12, int *ldx12, npy_complex128 *x21, int *ldx21, npy_complex128 *x22, int *ldx22, d *theta, npy_complex128 *u1, int *ldu1, npy_complex128 *u2, int *ldu2, npy_complex128 *v1t, int *ldv1t, npy_complex128 *v2t, int *ldv2t, npy_complex128 *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *info) nogil +cdef void zuncsd(char *jobu1, char *jobu2, char *jobv1t, char *jobv2t, char *trans, char *signs, int *m, int *p, int *q, z *x11, int *ldx11, z *x12, int *ldx12, z *x21, int *ldx21, z *x22, int *ldx22, d *theta, z *u1, int *ldu1, z *u2, int *ldu2, z *v1t, int *ldv1t, z *v2t, int *ldv2t, z *work, int *lwork, d *rwork, int *lrwork, int *iwork, int *info) noexcept nogil: + + _fortran_zuncsd(jobu1, jobu2, jobv1t, jobv2t, trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, work, lwork, rwork, lrwork, iwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zung2l "BLAS_FUNC(zung2l)"(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info) nogil +cdef void zung2l(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil: + + _fortran_zung2l(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zung2r "BLAS_FUNC(zung2r)"(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info) nogil +cdef void zung2r(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil: + + _fortran_zung2r(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zungbr "BLAS_FUNC(zungbr)"(char *vect, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zungbr(char *vect, int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zungbr(vect, m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunghr "BLAS_FUNC(zunghr)"(int *n, int *ilo, int *ihi, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunghr(int *n, int *ilo, int *ihi, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunghr(n, ilo, ihi, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zungl2 "BLAS_FUNC(zungl2)"(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info) nogil +cdef void zungl2(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil: + + _fortran_zungl2(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunglq "BLAS_FUNC(zunglq)"(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunglq(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunglq(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zungql "BLAS_FUNC(zungql)"(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zungql(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zungql(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zungqr "BLAS_FUNC(zungqr)"(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zungqr(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zungqr(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zungr2 "BLAS_FUNC(zungr2)"(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *info) nogil +cdef void zungr2(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *info) noexcept nogil: + + _fortran_zungr2(m, n, k, a, lda, tau, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zungrq "BLAS_FUNC(zungrq)"(int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zungrq(int *m, int *n, int *k, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zungrq(m, n, k, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zungtr "BLAS_FUNC(zungtr)"(char *uplo, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zungtr(char *uplo, int *n, z *a, int *lda, z *tau, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zungtr(uplo, n, a, lda, tau, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunm2l "BLAS_FUNC(zunm2l)"(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info) nogil +cdef void zunm2l(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil: + + _fortran_zunm2l(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunm2r "BLAS_FUNC(zunm2r)"(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info) nogil +cdef void zunm2r(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil: + + _fortran_zunm2r(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunmbr "BLAS_FUNC(zunmbr)"(char *vect, char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunmbr(char *vect, char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunmbr(vect, side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunmhr "BLAS_FUNC(zunmhr)"(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunmhr(char *side, char *trans, int *m, int *n, int *ilo, int *ihi, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunmhr(side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunml2 "BLAS_FUNC(zunml2)"(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info) nogil +cdef void zunml2(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil: + + _fortran_zunml2(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunmlq "BLAS_FUNC(zunmlq)"(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunmlq(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunmlq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunmql "BLAS_FUNC(zunmql)"(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunmql(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunmql(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunmqr "BLAS_FUNC(zunmqr)"(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunmqr(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunmqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunmr2 "BLAS_FUNC(zunmr2)"(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info) nogil +cdef void zunmr2(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil: + + _fortran_zunmr2(side, trans, m, n, k, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunmr3 "BLAS_FUNC(zunmr3)"(char *side, char *trans, int *m, int *n, int *k, int *l, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info) nogil +cdef void zunmr3(char *side, char *trans, int *m, int *n, int *k, int *l, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil: + + _fortran_zunmr3(side, trans, m, n, k, l, a, lda, tau, c, ldc, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunmrq "BLAS_FUNC(zunmrq)"(char *side, char *trans, int *m, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunmrq(char *side, char *trans, int *m, int *n, int *k, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunmrq(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunmrz "BLAS_FUNC(zunmrz)"(char *side, char *trans, int *m, int *n, int *k, int *l, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunmrz(char *side, char *trans, int *m, int *n, int *k, int *l, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunmrz(side, trans, m, n, k, l, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zunmtr "BLAS_FUNC(zunmtr)"(char *side, char *uplo, char *trans, int *m, int *n, npy_complex128 *a, int *lda, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *lwork, int *info) nogil +cdef void zunmtr(char *side, char *uplo, char *trans, int *m, int *n, z *a, int *lda, z *tau, z *c, int *ldc, z *work, int *lwork, int *info) noexcept nogil: + + _fortran_zunmtr(side, uplo, trans, m, n, a, lda, tau, c, ldc, work, lwork, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zupgtr "BLAS_FUNC(zupgtr)"(char *uplo, int *n, npy_complex128 *ap, npy_complex128 *tau, npy_complex128 *q, int *ldq, npy_complex128 *work, int *info) nogil +cdef void zupgtr(char *uplo, int *n, z *ap, z *tau, z *q, int *ldq, z *work, int *info) noexcept nogil: + + _fortran_zupgtr(uplo, n, ap, tau, q, ldq, work, info) + + +cdef extern from "_lapack_subroutines.h": + void _fortran_zupmtr "BLAS_FUNC(zupmtr)"(char *side, char *uplo, char *trans, int *m, int *n, npy_complex128 *ap, npy_complex128 *tau, npy_complex128 *c, int *ldc, npy_complex128 *work, int *info) nogil +cdef void zupmtr(char *side, char *uplo, char *trans, int *m, int *n, z *ap, z *tau, z *c, int *ldc, z *work, int *info) noexcept nogil: + + _fortran_zupmtr(side, uplo, trans, m, n, ap, tau, c, ldc, work, info) + + + +# Python accessible wrappers for testing: + +def _test_dlamch(cmach): + # This conversion is necessary to handle Python 3 strings. + cmach_bytes = bytes(cmach) + # Now that it is a bytes representation, a non-temporary variable + # must be passed as a part of the function call. + cdef char* cmach_char = cmach_bytes + return dlamch(cmach_char) + +def _test_slamch(cmach): + # This conversion is necessary to handle Python 3 strings. + cmach_bytes = bytes(cmach) + # Now that it is a bytes representation, a non-temporary variable + # must be passed as a part of the function call. + cdef char* cmach_char = cmach_bytes + return slamch(cmach_char) + +cpdef double complex _test_zladiv(double complex zx, double complex zy) noexcept nogil: + return zladiv(&zx, &zy) + +cpdef float complex _test_cladiv(float complex cx, float complex cy) noexcept nogil: + return cladiv(&cx, &cy) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp.py new file mode 100644 index 0000000000000000000000000000000000000000..0d82ab157ce3be763f63e453c9a6fee064557c85 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp.py @@ -0,0 +1,23 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'eig', 'eigvals', 'eigh', 'eigvalsh', + 'eig_banded', 'eigvals_banded', + 'eigh_tridiagonal', 'eigvalsh_tridiagonal', 'hessenberg', 'cdf2rdf', + 'LinAlgError', 'norm', 'get_lapack_funcs' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="linalg", module="decomp", + private_modules=["_decomp"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_cholesky.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_cholesky.py new file mode 100644 index 0000000000000000000000000000000000000000..92545a5c6af5fe7a4de13f8746b96696d68b5bd2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_cholesky.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'cholesky', 'cho_factor', 'cho_solve', 'cholesky_banded', + 'cho_solve_banded', 'LinAlgError', 'get_lapack_funcs' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="linalg", module="decomp_cholesky", + private_modules=["_decomp_cholesky"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_lu.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_lu.py new file mode 100644 index 0000000000000000000000000000000000000000..9d5d9a98a04a689fb735f81299e129dc7f307590 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_lu.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'lu', 'lu_solve', 'lu_factor', + 'LinAlgWarning', 'get_lapack_funcs', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="linalg", module="decomp_lu", + private_modules=["_decomp_lu"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_qr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_qr.py new file mode 100644 index 0000000000000000000000000000000000000000..4ef58729412ce2c83310b7817a143d14b8f28c19 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_qr.py @@ -0,0 +1,20 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'qr', 'qr_multiply', 'rq', 'get_lapack_funcs' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="linalg", module="decomp_qr", + private_modules=["_decomp_qr"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_schur.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_schur.py new file mode 100644 index 0000000000000000000000000000000000000000..c3c6cc494db9b35dce8e4007c8c30d823b03881f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_schur.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'schur', 'rsf2csf', 'norm', 'LinAlgError', 'get_lapack_funcs', 'eigvals', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="linalg", module="decomp_schur", + private_modules=["_decomp_schur"], all=__all__, + attribute=name) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_svd.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_svd.py new file mode 100644 index 0000000000000000000000000000000000000000..64d0ce8562f06a3837df050f0ea6b8b15a2b359e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/decomp_svd.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'svd', 'svdvals', 'diagsvd', 'orth', 'subspace_angles', 'null_space', + 'LinAlgError', 'get_lapack_funcs' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="linalg", module="decomp_svd", + private_modules=["_decomp_svd"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/interpolative.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/interpolative.py new file mode 100644 index 0000000000000000000000000000000000000000..38070863aa515266b0b130dbbdbd3d645da4aa0c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/interpolative.py @@ -0,0 +1,989 @@ +# ****************************************************************************** +# Copyright (C) 2013 Kenneth L. Ho +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions are met: +# +# Redistributions of source code must retain the above copyright notice, this +# list of conditions and the following disclaimer. Redistributions in binary +# form must reproduce the above copyright notice, this list of conditions and +# the following disclaimer in the documentation and/or other materials +# provided with the distribution. +# +# None of the names of the copyright holders may be used to endorse or +# promote products derived from this software without specific prior written +# permission. +# +# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" +# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE +# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE +# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR +# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF +# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN +# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) +# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE +# POSSIBILITY OF SUCH DAMAGE. +# ****************************************************************************** + +r""" +====================================================================== +Interpolative matrix decomposition (:mod:`scipy.linalg.interpolative`) +====================================================================== + +.. versionadded:: 0.13 + +.. versionchanged:: 1.15.0 + The underlying algorithms have been ported to Python from the original Fortran77 + code. See references below for more details. + +.. currentmodule:: scipy.linalg.interpolative + +An interpolative decomposition (ID) of a matrix :math:`A \in +\mathbb{C}^{m \times n}` of rank :math:`k \leq \min \{ m, n \}` is a +factorization + +.. math:: + A \Pi = + \begin{bmatrix} + A \Pi_{1} & A \Pi_{2} + \end{bmatrix} = + A \Pi_{1} + \begin{bmatrix} + I & T + \end{bmatrix}, + +where :math:`\Pi = [\Pi_{1}, \Pi_{2}]` is a permutation matrix with +:math:`\Pi_{1} \in \{ 0, 1 \}^{n \times k}`, i.e., :math:`A \Pi_{2} = +A \Pi_{1} T`. This can equivalently be written as :math:`A = BP`, +where :math:`B = A \Pi_{1}` and :math:`P = [I, T] \Pi^{\mathsf{T}}` +are the *skeleton* and *interpolation matrices*, respectively. + +If :math:`A` does not have exact rank :math:`k`, then there exists an +approximation in the form of an ID such that :math:`A = BP + E`, where +:math:`\| E \| \sim \sigma_{k + 1}` is on the order of the :math:`(k + +1)`-th largest singular value of :math:`A`. Note that :math:`\sigma_{k ++ 1}` is the best possible error for a rank-:math:`k` approximation +and, in fact, is achieved by the singular value decomposition (SVD) +:math:`A \approx U S V^{*}`, where :math:`U \in \mathbb{C}^{m \times +k}` and :math:`V \in \mathbb{C}^{n \times k}` have orthonormal columns +and :math:`S = \mathop{\mathrm{diag}} (\sigma_{i}) \in \mathbb{C}^{k +\times k}` is diagonal with nonnegative entries. The principal +advantages of using an ID over an SVD are that: + +- it is cheaper to construct; +- it preserves the structure of :math:`A`; and +- it is more efficient to compute with in light of the identity submatrix of :math:`P`. + +Routines +======== + +Main functionality: + +.. autosummary:: + :toctree: generated/ + + interp_decomp + reconstruct_matrix_from_id + reconstruct_interp_matrix + reconstruct_skel_matrix + id_to_svd + svd + estimate_spectral_norm + estimate_spectral_norm_diff + estimate_rank + +Following support functions are deprecated and will be removed in SciPy 1.17.0: + +.. autosummary:: + :toctree: generated/ + + seed + rand + + +References +========== + +This module uses the algorithms found in ID software package [1]_ by Martinsson, +Rokhlin, Shkolnisky, and Tygert, which is a Fortran library for computing IDs using +various algorithms, including the rank-revealing QR approach of [2]_ and the more +recent randomized methods described in [3]_, [4]_, and [5]_. + +We advise the user to consult also the documentation for the `ID package +`_. + +.. [1] P.G. Martinsson, V. Rokhlin, Y. Shkolnisky, M. Tygert. "ID: a + software package for low-rank approximation of matrices via interpolative + decompositions, version 0.2." http://tygert.com/id_doc.4.pdf. + +.. [2] H. Cheng, Z. Gimbutas, P.G. Martinsson, V. Rokhlin. "On the + compression of low rank matrices." *SIAM J. Sci. Comput.* 26 (4): 1389--1404, + 2005. :doi:`10.1137/030602678`. + +.. [3] E. Liberty, F. Woolfe, P.G. Martinsson, V. Rokhlin, M. + Tygert. "Randomized algorithms for the low-rank approximation of matrices." + *Proc. Natl. Acad. Sci. U.S.A.* 104 (51): 20167--20172, 2007. + :doi:`10.1073/pnas.0709640104`. + +.. [4] P.G. Martinsson, V. Rokhlin, M. Tygert. "A randomized + algorithm for the decomposition of matrices." *Appl. Comput. Harmon. Anal.* 30 + (1): 47--68, 2011. :doi:`10.1016/j.acha.2010.02.003`. + +.. [5] F. Woolfe, E. Liberty, V. Rokhlin, M. Tygert. "A fast + randomized algorithm for the approximation of matrices." *Appl. Comput. + Harmon. Anal.* 25 (3): 335--366, 2008. :doi:`10.1016/j.acha.2007.12.002`. + + +Tutorial +======== + +Initializing +------------ + +The first step is to import :mod:`scipy.linalg.interpolative` by issuing the +command: + +>>> import scipy.linalg.interpolative as sli + +Now let's build a matrix. For this, we consider a Hilbert matrix, which is well +know to have low rank: + +>>> from scipy.linalg import hilbert +>>> n = 1000 +>>> A = hilbert(n) + +We can also do this explicitly via: + +>>> import numpy as np +>>> n = 1000 +>>> A = np.empty((n, n), order='F') +>>> for j in range(n): +... for i in range(n): +... A[i,j] = 1. / (i + j + 1) + +Note the use of the flag ``order='F'`` in :func:`numpy.empty`. This +instantiates the matrix in Fortran-contiguous order and is important for +avoiding data copying when passing to the backend. + +We then define multiplication routines for the matrix by regarding it as a +:class:`scipy.sparse.linalg.LinearOperator`: + +>>> from scipy.sparse.linalg import aslinearoperator +>>> L = aslinearoperator(A) + +This automatically sets up methods describing the action of the matrix and its +adjoint on a vector. + +Computing an ID +--------------- + +We have several choices of algorithm to compute an ID. These fall largely +according to two dichotomies: + +1. how the matrix is represented, i.e., via its entries or via its action on a + vector; and +2. whether to approximate it to a fixed relative precision or to a fixed rank. + +We step through each choice in turn below. + +In all cases, the ID is represented by three parameters: + +1. a rank ``k``; +2. an index array ``idx``; and +3. interpolation coefficients ``proj``. + +The ID is specified by the relation +``np.dot(A[:,idx[:k]], proj) == A[:,idx[k:]]``. + +From matrix entries +................... + +We first consider a matrix given in terms of its entries. + +To compute an ID to a fixed precision, type: + +>>> eps = 1e-3 +>>> k, idx, proj = sli.interp_decomp(A, eps) + +where ``eps < 1`` is the desired precision. + +To compute an ID to a fixed rank, use: + +>>> idx, proj = sli.interp_decomp(A, k) + +where ``k >= 1`` is the desired rank. + +Both algorithms use random sampling and are usually faster than the +corresponding older, deterministic algorithms, which can be accessed via the +commands: + +>>> k, idx, proj = sli.interp_decomp(A, eps, rand=False) + +and: + +>>> idx, proj = sli.interp_decomp(A, k, rand=False) + +respectively. + +From matrix action +.................. + +Now consider a matrix given in terms of its action on a vector as a +:class:`scipy.sparse.linalg.LinearOperator`. + +To compute an ID to a fixed precision, type: + +>>> k, idx, proj = sli.interp_decomp(L, eps) + +To compute an ID to a fixed rank, use: + +>>> idx, proj = sli.interp_decomp(L, k) + +These algorithms are randomized. + +Reconstructing an ID +-------------------- + +The ID routines above do not output the skeleton and interpolation matrices +explicitly but instead return the relevant information in a more compact (and +sometimes more useful) form. To build these matrices, write: + +>>> B = sli.reconstruct_skel_matrix(A, k, idx) + +for the skeleton matrix and: + +>>> P = sli.reconstruct_interp_matrix(idx, proj) + +for the interpolation matrix. The ID approximation can then be computed as: + +>>> C = np.dot(B, P) + +This can also be constructed directly using: + +>>> C = sli.reconstruct_matrix_from_id(B, idx, proj) + +without having to first compute ``P``. + +Alternatively, this can be done explicitly as well using: + +>>> B = A[:,idx[:k]] +>>> P = np.hstack([np.eye(k), proj])[:,np.argsort(idx)] +>>> C = np.dot(B, P) + +Computing an SVD +---------------- + +An ID can be converted to an SVD via the command: + +>>> U, S, V = sli.id_to_svd(B, idx, proj) + +The SVD approximation is then: + +>>> approx = U @ np.diag(S) @ V.conj().T + +The SVD can also be computed "fresh" by combining both the ID and conversion +steps into one command. Following the various ID algorithms above, there are +correspondingly various SVD algorithms that one can employ. + +From matrix entries +................... + +We consider first SVD algorithms for a matrix given in terms of its entries. + +To compute an SVD to a fixed precision, type: + +>>> U, S, V = sli.svd(A, eps) + +To compute an SVD to a fixed rank, use: + +>>> U, S, V = sli.svd(A, k) + +Both algorithms use random sampling; for the deterministic versions, issue the +keyword ``rand=False`` as above. + +From matrix action +.................. + +Now consider a matrix given in terms of its action on a vector. + +To compute an SVD to a fixed precision, type: + +>>> U, S, V = sli.svd(L, eps) + +To compute an SVD to a fixed rank, use: + +>>> U, S, V = sli.svd(L, k) + +Utility routines +---------------- + +Several utility routines are also available. + +To estimate the spectral norm of a matrix, use: + +>>> snorm = sli.estimate_spectral_norm(A) + +This algorithm is based on the randomized power method and thus requires only +matrix-vector products. The number of iterations to take can be set using the +keyword ``its`` (default: ``its=20``). The matrix is interpreted as a +:class:`scipy.sparse.linalg.LinearOperator`, but it is also valid to supply it +as a :class:`numpy.ndarray`, in which case it is trivially converted using +:func:`scipy.sparse.linalg.aslinearoperator`. + +The same algorithm can also estimate the spectral norm of the difference of two +matrices ``A1`` and ``A2`` as follows: + +>>> A1, A2 = A**2, A +>>> diff = sli.estimate_spectral_norm_diff(A1, A2) + +This is often useful for checking the accuracy of a matrix approximation. + +Some routines in :mod:`scipy.linalg.interpolative` require estimating the rank +of a matrix as well. This can be done with either: + +>>> k = sli.estimate_rank(A, eps) + +or: + +>>> k = sli.estimate_rank(L, eps) + +depending on the representation. The parameter ``eps`` controls the definition +of the numerical rank. + +Finally, the random number generation required for all randomized routines can +be controlled via providing NumPy pseudo-random generators with a fixed seed. See +:class:`numpy.random.Generator` and :func:`numpy.random.default_rng` for more details. + +Remarks +------- + +The above functions all automatically detect the appropriate interface and work +with both real and complex data types, passing input arguments to the proper +backend routine. + +""" + +import scipy.linalg._decomp_interpolative as _backend +import numpy as np +import warnings + +__all__ = [ + 'estimate_rank', + 'estimate_spectral_norm', + 'estimate_spectral_norm_diff', + 'id_to_svd', + 'interp_decomp', + 'rand', + 'reconstruct_interp_matrix', + 'reconstruct_matrix_from_id', + 'reconstruct_skel_matrix', + 'seed', + 'svd', +] + +_DTYPE_ERROR = ValueError("invalid input dtype (input must be float64 or complex128)") +_TYPE_ERROR = TypeError("invalid input type (must be array or LinearOperator)") + + +def _C_contiguous_copy(A): + """ + Same as np.ascontiguousarray, but ensure a copy + """ + A = np.asarray(A) + if A.flags.c_contiguous: + A = A.copy() + else: + A = np.ascontiguousarray(A) + return A + + +def _is_real(A): + try: + if A.dtype == np.complex128: + return False + elif A.dtype == np.float64: + return True + else: + raise _DTYPE_ERROR + except AttributeError as e: + raise _TYPE_ERROR from e + + +def seed(seed=None): + """ + This function, historically, used to set the seed of the randomization algorithms + used in the `scipy.linalg.interpolative` functions written in Fortran77. + + The library has been ported to Python and now the functions use the native NumPy + generators and this function has no content and returns None. Thus this function + should not be used and will be removed in SciPy version 1.17.0. + """ + warnings.warn("`scipy.linalg.interpolative.seed` is deprecated and will be " + "removed in SciPy 1.17.0.", DeprecationWarning, stacklevel=3) + + +def rand(*shape): + """ + This function, historically, used to generate uniformly distributed random number + for the randomization algorithms used in the `scipy.linalg.interpolative` functions + written in Fortran77. + + The library has been ported to Python and now the functions use the native NumPy + generators. Thus this function should not be used and will be removed in the + SciPy version 1.17.0. + + If pseudo-random numbers are needed, NumPy pseudo-random generators should be used + instead. + + Parameters + ---------- + *shape + Shape of output array + + """ + warnings.warn("`scipy.linalg.interpolative.rand` is deprecated and will be " + "removed in SciPy 1.17.0.", DeprecationWarning, stacklevel=3) + rng = np.random.default_rng() + return rng.uniform(low=0., high=1.0, size=shape) + + +def interp_decomp(A, eps_or_k, rand=True, rng=None): + """ + Compute ID of a matrix. + + An ID of a matrix `A` is a factorization defined by a rank `k`, a column + index array `idx`, and interpolation coefficients `proj` such that:: + + numpy.dot(A[:,idx[:k]], proj) = A[:,idx[k:]] + + The original matrix can then be reconstructed as:: + + numpy.hstack([A[:,idx[:k]], + numpy.dot(A[:,idx[:k]], proj)] + )[:,numpy.argsort(idx)] + + or via the routine :func:`reconstruct_matrix_from_id`. This can + equivalently be written as:: + + numpy.dot(A[:,idx[:k]], + numpy.hstack([numpy.eye(k), proj]) + )[:,np.argsort(idx)] + + in terms of the skeleton and interpolation matrices:: + + B = A[:,idx[:k]] + + and:: + + P = numpy.hstack([numpy.eye(k), proj])[:,np.argsort(idx)] + + respectively. See also :func:`reconstruct_interp_matrix` and + :func:`reconstruct_skel_matrix`. + + The ID can be computed to any relative precision or rank (depending on the + value of `eps_or_k`). If a precision is specified (`eps_or_k < 1`), then + this function has the output signature:: + + k, idx, proj = interp_decomp(A, eps_or_k) + + Otherwise, if a rank is specified (`eps_or_k >= 1`), then the output + signature is:: + + idx, proj = interp_decomp(A, eps_or_k) + + .. This function automatically detects the form of the input parameters + and passes them to the appropriate backend. For details, see + :func:`_backend.iddp_id`, :func:`_backend.iddp_aid`, + :func:`_backend.iddp_rid`, :func:`_backend.iddr_id`, + :func:`_backend.iddr_aid`, :func:`_backend.iddr_rid`, + :func:`_backend.idzp_id`, :func:`_backend.idzp_aid`, + :func:`_backend.idzp_rid`, :func:`_backend.idzr_id`, + :func:`_backend.idzr_aid`, and :func:`_backend.idzr_rid`. + + Parameters + ---------- + A : :class:`numpy.ndarray` or :class:`scipy.sparse.linalg.LinearOperator` with `rmatvec` + Matrix to be factored + eps_or_k : float or int + Relative error (if ``eps_or_k < 1``) or rank (if ``eps_or_k >= 1``) of + approximation. + rand : bool, optional + Whether to use random sampling if `A` is of type :class:`numpy.ndarray` + (randomized algorithms are always used if `A` is of type + :class:`scipy.sparse.linalg.LinearOperator`). + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + If `rand` is ``False``, the argument is ignored. + + Returns + ------- + k : int + Rank required to achieve specified relative precision if + ``eps_or_k < 1``. + idx : :class:`numpy.ndarray` + Column index array. + proj : :class:`numpy.ndarray` + Interpolation coefficients. + """ # numpy/numpydoc#87 # noqa: E501 + from scipy.sparse.linalg import LinearOperator + rng = np.random.default_rng(rng) + real = _is_real(A) + + if isinstance(A, np.ndarray): + A = _C_contiguous_copy(A) + if eps_or_k < 1: + eps = eps_or_k + if rand: + if real: + k, idx, proj = _backend.iddp_aid(A, eps, rng=rng) + else: + k, idx, proj = _backend.idzp_aid(A, eps, rng=rng) + else: + if real: + k, idx, proj = _backend.iddp_id(A, eps) + else: + k, idx, proj = _backend.idzp_id(A, eps) + return k, idx, proj + else: + k = int(eps_or_k) + if rand: + if real: + idx, proj = _backend.iddr_aid(A, k, rng=rng) + else: + idx, proj = _backend.idzr_aid(A, k, rng=rng) + else: + if real: + idx, proj = _backend.iddr_id(A, k) + else: + idx, proj = _backend.idzr_id(A, k) + return idx, proj + elif isinstance(A, LinearOperator): + + if eps_or_k < 1: + eps = eps_or_k + if real: + k, idx, proj = _backend.iddp_rid(A, eps, rng=rng) + else: + k, idx, proj = _backend.idzp_rid(A, eps, rng=rng) + return k, idx, proj + else: + k = int(eps_or_k) + if real: + idx, proj = _backend.iddr_rid(A, k, rng=rng) + else: + idx, proj = _backend.idzr_rid(A, k, rng=rng) + return idx, proj + else: + raise _TYPE_ERROR + + +def reconstruct_matrix_from_id(B, idx, proj): + """ + Reconstruct matrix from its ID. + + A matrix `A` with skeleton matrix `B` and ID indices and coefficients `idx` + and `proj`, respectively, can be reconstructed as:: + + numpy.hstack([B, numpy.dot(B, proj)])[:,numpy.argsort(idx)] + + See also :func:`reconstruct_interp_matrix` and + :func:`reconstruct_skel_matrix`. + + .. This function automatically detects the matrix data type and calls the + appropriate backend. For details, see :func:`_backend.idd_reconid` and + :func:`_backend.idz_reconid`. + + Parameters + ---------- + B : :class:`numpy.ndarray` + Skeleton matrix. + idx : :class:`numpy.ndarray` + Column index array. + proj : :class:`numpy.ndarray` + Interpolation coefficients. + + Returns + ------- + :class:`numpy.ndarray` + Reconstructed matrix. + """ + if _is_real(B): + return _backend.idd_reconid(B, idx, proj) + else: + return _backend.idz_reconid(B, idx, proj) + + +def reconstruct_interp_matrix(idx, proj): + """ + Reconstruct interpolation matrix from ID. + + The interpolation matrix can be reconstructed from the ID indices and + coefficients `idx` and `proj`, respectively, as:: + + P = numpy.hstack([numpy.eye(proj.shape[0]), proj])[:,numpy.argsort(idx)] + + The original matrix can then be reconstructed from its skeleton matrix ``B`` + via ``A = B @ P`` + + See also :func:`reconstruct_matrix_from_id` and + :func:`reconstruct_skel_matrix`. + + .. This function automatically detects the matrix data type and calls the + appropriate backend. For details, see :func:`_backend.idd_reconint` and + :func:`_backend.idz_reconint`. + + Parameters + ---------- + idx : :class:`numpy.ndarray` + 1D column index array. + proj : :class:`numpy.ndarray` + Interpolation coefficients. + + Returns + ------- + :class:`numpy.ndarray` + Interpolation matrix. + """ + n, krank = len(idx), proj.shape[0] + if _is_real(proj): + p = np.zeros([krank, n], dtype=np.float64) + else: + p = np.zeros([krank, n], dtype=np.complex128) + + for ci in range(krank): + p[ci, idx[ci]] = 1.0 + p[:, idx[krank:]] = proj[:, :] + + return p + + +def reconstruct_skel_matrix(A, k, idx): + """ + Reconstruct skeleton matrix from ID. + + The skeleton matrix can be reconstructed from the original matrix `A` and its + ID rank and indices `k` and `idx`, respectively, as:: + + B = A[:,idx[:k]] + + The original matrix can then be reconstructed via:: + + numpy.hstack([B, numpy.dot(B, proj)])[:,numpy.argsort(idx)] + + See also :func:`reconstruct_matrix_from_id` and + :func:`reconstruct_interp_matrix`. + + .. This function automatically detects the matrix data type and calls the + appropriate backend. For details, see :func:`_backend.idd_copycols` and + :func:`_backend.idz_copycols`. + + Parameters + ---------- + A : :class:`numpy.ndarray` + Original matrix. + k : int + Rank of ID. + idx : :class:`numpy.ndarray` + Column index array. + + Returns + ------- + :class:`numpy.ndarray` + Skeleton matrix. + """ + return A[:, idx[:k]] + + +def id_to_svd(B, idx, proj): + """ + Convert ID to SVD. + + The SVD reconstruction of a matrix with skeleton matrix `B` and ID indices and + coefficients `idx` and `proj`, respectively, is:: + + U, S, V = id_to_svd(B, idx, proj) + A = numpy.dot(U, numpy.dot(numpy.diag(S), V.conj().T)) + + See also :func:`svd`. + + .. This function automatically detects the matrix data type and calls the + appropriate backend. For details, see :func:`_backend.idd_id2svd` and + :func:`_backend.idz_id2svd`. + + Parameters + ---------- + B : :class:`numpy.ndarray` + Skeleton matrix. + idx : :class:`numpy.ndarray` + 1D column index array. + proj : :class:`numpy.ndarray` + Interpolation coefficients. + + Returns + ------- + U : :class:`numpy.ndarray` + Left singular vectors. + S : :class:`numpy.ndarray` + Singular values. + V : :class:`numpy.ndarray` + Right singular vectors. + """ + B = _C_contiguous_copy(B) + if _is_real(B): + U, S, V = _backend.idd_id2svd(B, idx, proj) + else: + U, S, V = _backend.idz_id2svd(B, idx, proj) + + return U, S, V + + +def estimate_spectral_norm(A, its=20, rng=None): + """ + Estimate spectral norm of a matrix by the randomized power method. + + .. This function automatically detects the matrix data type and calls the + appropriate backend. For details, see :func:`_backend.idd_snorm` and + :func:`_backend.idz_snorm`. + + Parameters + ---------- + A : :class:`scipy.sparse.linalg.LinearOperator` + Matrix given as a :class:`scipy.sparse.linalg.LinearOperator` with the + `matvec` and `rmatvec` methods (to apply the matrix and its adjoint). + its : int, optional + Number of power method iterations. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + If `rand` is ``False``, the argument is ignored. + + Returns + ------- + float + Spectral norm estimate. + """ + from scipy.sparse.linalg import aslinearoperator + rng = np.random.default_rng(rng) + A = aslinearoperator(A) + + if _is_real(A): + return _backend.idd_snorm(A, its=its, rng=rng) + else: + return _backend.idz_snorm(A, its=its, rng=rng) + + +def estimate_spectral_norm_diff(A, B, its=20, rng=None): + """ + Estimate spectral norm of the difference of two matrices by the randomized + power method. + + .. This function automatically detects the matrix data type and calls the + appropriate backend. For details, see :func:`_backend.idd_diffsnorm` and + :func:`_backend.idz_diffsnorm`. + + Parameters + ---------- + A : :class:`scipy.sparse.linalg.LinearOperator` + First matrix given as a :class:`scipy.sparse.linalg.LinearOperator` with the + `matvec` and `rmatvec` methods (to apply the matrix and its adjoint). + B : :class:`scipy.sparse.linalg.LinearOperator` + Second matrix given as a :class:`scipy.sparse.linalg.LinearOperator` with + the `matvec` and `rmatvec` methods (to apply the matrix and its adjoint). + its : int, optional + Number of power method iterations. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + If `rand` is ``False``, the argument is ignored. + + Returns + ------- + float + Spectral norm estimate of matrix difference. + """ + from scipy.sparse.linalg import aslinearoperator + rng = np.random.default_rng(rng) + A = aslinearoperator(A) + B = aslinearoperator(B) + + if _is_real(A): + return _backend.idd_diffsnorm(A, B, its=its, rng=rng) + else: + return _backend.idz_diffsnorm(A, B, its=its, rng=rng) + + +def svd(A, eps_or_k, rand=True, rng=None): + """ + Compute SVD of a matrix via an ID. + + An SVD of a matrix `A` is a factorization:: + + A = U @ np.diag(S) @ V.conj().T + + where `U` and `V` have orthonormal columns and `S` is nonnegative. + + The SVD can be computed to any relative precision or rank (depending on the + value of `eps_or_k`). + + See also :func:`interp_decomp` and :func:`id_to_svd`. + + .. This function automatically detects the form of the input parameters and + passes them to the appropriate backend. For details, see + :func:`_backend.iddp_svd`, :func:`_backend.iddp_asvd`, + :func:`_backend.iddp_rsvd`, :func:`_backend.iddr_svd`, + :func:`_backend.iddr_asvd`, :func:`_backend.iddr_rsvd`, + :func:`_backend.idzp_svd`, :func:`_backend.idzp_asvd`, + :func:`_backend.idzp_rsvd`, :func:`_backend.idzr_svd`, + :func:`_backend.idzr_asvd`, and :func:`_backend.idzr_rsvd`. + + Parameters + ---------- + A : :class:`numpy.ndarray` or :class:`scipy.sparse.linalg.LinearOperator` + Matrix to be factored, given as either a :class:`numpy.ndarray` or a + :class:`scipy.sparse.linalg.LinearOperator` with the `matvec` and + `rmatvec` methods (to apply the matrix and its adjoint). + eps_or_k : float or int + Relative error (if ``eps_or_k < 1``) or rank (if ``eps_or_k >= 1``) of + approximation. + rand : bool, optional + Whether to use random sampling if `A` is of type :class:`numpy.ndarray` + (randomized algorithms are always used if `A` is of type + :class:`scipy.sparse.linalg.LinearOperator`). + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + If `rand` is ``False``, the argument is ignored. + + Returns + ------- + U : :class:`numpy.ndarray` + 2D array of left singular vectors. + S : :class:`numpy.ndarray` + 1D array of singular values. + V : :class:`numpy.ndarray` + 2D array right singular vectors. + """ + from scipy.sparse.linalg import LinearOperator + rng = np.random.default_rng(rng) + + real = _is_real(A) + + if isinstance(A, np.ndarray): + A = _C_contiguous_copy(A) + if eps_or_k < 1: + eps = eps_or_k + if rand: + if real: + U, S, V = _backend.iddp_asvd(A, eps, rng=rng) + else: + U, S, V = _backend.idzp_asvd(A, eps, rng=rng) + else: + if real: + U, S, V = _backend.iddp_svd(A, eps) + V = V.T.conj() + else: + U, S, V = _backend.idzp_svd(A, eps) + V = V.T.conj() + else: + k = int(eps_or_k) + if k > min(A.shape): + raise ValueError(f"Approximation rank {k} exceeds min(A.shape) = " + f" {min(A.shape)} ") + if rand: + if real: + U, S, V = _backend.iddr_asvd(A, k, rng=rng) + else: + U, S, V = _backend.idzr_asvd(A, k, rng=rng) + else: + if real: + U, S, V = _backend.iddr_svd(A, k) + V = V.T.conj() + else: + U, S, V = _backend.idzr_svd(A, k) + V = V.T.conj() + elif isinstance(A, LinearOperator): + if eps_or_k < 1: + eps = eps_or_k + if real: + U, S, V = _backend.iddp_rsvd(A, eps, rng=rng) + else: + U, S, V = _backend.idzp_rsvd(A, eps, rng=rng) + else: + k = int(eps_or_k) + if real: + U, S, V = _backend.iddr_rsvd(A, k, rng=rng) + else: + U, S, V = _backend.idzr_rsvd(A, k, rng=rng) + else: + raise _TYPE_ERROR + return U, S, V + + +def estimate_rank(A, eps, rng=None): + """ + Estimate matrix rank to a specified relative precision using randomized + methods. + + The matrix `A` can be given as either a :class:`numpy.ndarray` or a + :class:`scipy.sparse.linalg.LinearOperator`, with different algorithms used + for each case. If `A` is of type :class:`numpy.ndarray`, then the output + rank is typically about 8 higher than the actual numerical rank. + + .. This function automatically detects the form of the input parameters and + passes them to the appropriate backend. For details, + see :func:`_backend.idd_estrank`, :func:`_backend.idd_findrank`, + :func:`_backend.idz_estrank`, and :func:`_backend.idz_findrank`. + + Parameters + ---------- + A : :class:`numpy.ndarray` or :class:`scipy.sparse.linalg.LinearOperator` + Matrix whose rank is to be estimated, given as either a + :class:`numpy.ndarray` or a :class:`scipy.sparse.linalg.LinearOperator` + with the `rmatvec` method (to apply the matrix adjoint). + eps : float + Relative error for numerical rank definition. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + If `rand` is ``False``, the argument is ignored. + + Returns + ------- + int + Estimated matrix rank. + """ + from scipy.sparse.linalg import LinearOperator + + rng = np.random.default_rng(rng) + real = _is_real(A) + + if isinstance(A, np.ndarray): + A = _C_contiguous_copy(A) + if real: + rank, _ = _backend.idd_estrank(A, eps, rng=rng) + else: + rank, _ = _backend.idz_estrank(A, eps, rng=rng) + if rank == 0: + # special return value for nearly full rank + rank = min(A.shape) + return rank + elif isinstance(A, LinearOperator): + if real: + return _backend.idd_findrank(A, eps, rng=rng)[0] + else: + return _backend.idz_findrank(A, eps, rng=rng)[0] + else: + raise _TYPE_ERROR diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/lapack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/lapack.py new file mode 100644 index 0000000000000000000000000000000000000000..2d15cf4d72d19f4b7949cd80bc77aa31553841c1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/lapack.py @@ -0,0 +1,1061 @@ +""" +Low-level LAPACK functions (:mod:`scipy.linalg.lapack`) +======================================================= + +This module contains low-level functions from the LAPACK library. + +.. versionadded:: 0.12.0 + +.. note:: + + The common ``overwrite_<>`` option in many routines, allows the + input arrays to be overwritten to avoid extra memory allocation. + However this requires the array to satisfy two conditions + which are memory order and the data type to match exactly the + order and the type expected by the routine. + + As an example, if you pass a double precision float array to any + ``S....`` routine which expects single precision arguments, f2py + will create an intermediate array to match the argument types and + overwriting will be performed on that intermediate array. + + Similarly, if a C-contiguous array is passed, f2py will pass a + FORTRAN-contiguous array internally. Please make sure that these + details are satisfied. More information can be found in the f2py + documentation. + +.. warning:: + + These functions do little to no error checking. + It is possible to cause crashes by mis-using them, + so prefer using the higher-level routines in `scipy.linalg`. + +Finding functions +----------------- + +.. autosummary:: + :toctree: generated/ + + get_lapack_funcs + +All functions +------------- + +.. autosummary:: + :toctree: generated/ + + sgbsv + dgbsv + cgbsv + zgbsv + + sgbtrf + dgbtrf + cgbtrf + zgbtrf + + sgbtrs + dgbtrs + cgbtrs + zgbtrs + + sgebal + dgebal + cgebal + zgebal + + sgecon + dgecon + cgecon + zgecon + + sgeequ + dgeequ + cgeequ + zgeequ + + sgeequb + dgeequb + cgeequb + zgeequb + + sgees + dgees + cgees + zgees + + sgeev + dgeev + cgeev + zgeev + + sgeev_lwork + dgeev_lwork + cgeev_lwork + zgeev_lwork + + sgehrd + dgehrd + cgehrd + zgehrd + + sgehrd_lwork + dgehrd_lwork + cgehrd_lwork + zgehrd_lwork + + sgejsv + dgejsv + + sgels + dgels + cgels + zgels + + sgels_lwork + dgels_lwork + cgels_lwork + zgels_lwork + + sgelsd + dgelsd + cgelsd + zgelsd + + sgelsd_lwork + dgelsd_lwork + cgelsd_lwork + zgelsd_lwork + + sgelss + dgelss + cgelss + zgelss + + sgelss_lwork + dgelss_lwork + cgelss_lwork + zgelss_lwork + + sgelsy + dgelsy + cgelsy + zgelsy + + sgelsy_lwork + dgelsy_lwork + cgelsy_lwork + zgelsy_lwork + + sgeqp3 + dgeqp3 + cgeqp3 + zgeqp3 + + sgeqrf + dgeqrf + cgeqrf + zgeqrf + + sgeqrf_lwork + dgeqrf_lwork + cgeqrf_lwork + zgeqrf_lwork + + sgeqrfp + dgeqrfp + cgeqrfp + zgeqrfp + + sgeqrfp_lwork + dgeqrfp_lwork + cgeqrfp_lwork + zgeqrfp_lwork + + sgerqf + dgerqf + cgerqf + zgerqf + + sgesdd + dgesdd + cgesdd + zgesdd + + sgesdd_lwork + dgesdd_lwork + cgesdd_lwork + zgesdd_lwork + + sgesv + dgesv + cgesv + zgesv + + sgesvd + dgesvd + cgesvd + zgesvd + + sgesvd_lwork + dgesvd_lwork + cgesvd_lwork + zgesvd_lwork + + sgesvx + dgesvx + cgesvx + zgesvx + + sgetrf + dgetrf + cgetrf + zgetrf + + sgetc2 + dgetc2 + cgetc2 + zgetc2 + + sgetri + dgetri + cgetri + zgetri + + sgetri_lwork + dgetri_lwork + cgetri_lwork + zgetri_lwork + + sgetrs + dgetrs + cgetrs + zgetrs + + sgesc2 + dgesc2 + cgesc2 + zgesc2 + + sgges + dgges + cgges + zgges + + sggev + dggev + cggev + zggev + + sgglse + dgglse + cgglse + zgglse + + sgglse_lwork + dgglse_lwork + cgglse_lwork + zgglse_lwork + + sgtsv + dgtsv + cgtsv + zgtsv + + sgtsvx + dgtsvx + cgtsvx + zgtsvx + + chbevd + zhbevd + + chbevx + zhbevx + + checon + zhecon + + cheequb + zheequb + + cheev + zheev + + cheev_lwork + zheev_lwork + + cheevd + zheevd + + cheevd_lwork + zheevd_lwork + + cheevr + zheevr + + cheevr_lwork + zheevr_lwork + + cheevx + zheevx + + cheevx_lwork + zheevx_lwork + + chegst + zhegst + + chegv + zhegv + + chegv_lwork + zhegv_lwork + + chegvd + zhegvd + + chegvx + zhegvx + + chegvx_lwork + zhegvx_lwork + + chesv + zhesv + + chesv_lwork + zhesv_lwork + + chesvx + zhesvx + + chesvx_lwork + zhesvx_lwork + + chetrd + zhetrd + + chetrd_lwork + zhetrd_lwork + + chetrf + zhetrf + + chetrf_lwork + zhetrf_lwork + + chetrs + zhetrs + + chfrk + zhfrk + + slamch + dlamch + + slange + dlange + clange + zlange + + slantr + dlantr + clantr + zlantr + + slarf + dlarf + clarf + zlarf + + slarfg + dlarfg + clarfg + zlarfg + + slartg + dlartg + clartg + zlartg + + slasd4 + dlasd4 + + slaswp + dlaswp + claswp + zlaswp + + slauum + dlauum + clauum + zlauum + + sorcsd + dorcsd + sorcsd_lwork + dorcsd_lwork + + sorghr + dorghr + sorghr_lwork + dorghr_lwork + + sorgqr + dorgqr + + sorgrq + dorgrq + + sormqr + dormqr + + sormrz + dormrz + + sormrz_lwork + dormrz_lwork + + spbsv + dpbsv + cpbsv + zpbsv + + spbtrf + dpbtrf + cpbtrf + zpbtrf + + spbtrs + dpbtrs + cpbtrs + zpbtrs + + spftrf + dpftrf + cpftrf + zpftrf + + spftri + dpftri + cpftri + zpftri + + spftrs + dpftrs + cpftrs + zpftrs + + spocon + dpocon + cpocon + zpocon + + spstrf + dpstrf + cpstrf + zpstrf + + spstf2 + dpstf2 + cpstf2 + zpstf2 + + sposv + dposv + cposv + zposv + + sposvx + dposvx + cposvx + zposvx + + spotrf + dpotrf + cpotrf + zpotrf + + spotri + dpotri + cpotri + zpotri + + spotrs + dpotrs + cpotrs + zpotrs + + sppcon + dppcon + cppcon + zppcon + + sppsv + dppsv + cppsv + zppsv + + spptrf + dpptrf + cpptrf + zpptrf + + spptri + dpptri + cpptri + zpptri + + spptrs + dpptrs + cpptrs + zpptrs + + sptsv + dptsv + cptsv + zptsv + + sptsvx + dptsvx + cptsvx + zptsvx + + spttrf + dpttrf + cpttrf + zpttrf + + spttrs + dpttrs + cpttrs + zpttrs + + spteqr + dpteqr + cpteqr + zpteqr + + crot + zrot + + ssbev + dsbev + + ssbevd + dsbevd + + ssbevx + dsbevx + + ssfrk + dsfrk + + sstebz + dstebz + + sstein + dstein + + sstemr + dstemr + + sstemr_lwork + dstemr_lwork + + ssterf + dsterf + + sstev + dstev + + ssycon + dsycon + csycon + zsycon + + ssyconv + dsyconv + csyconv + zsyconv + + ssyequb + dsyequb + csyequb + zsyequb + + ssyev + dsyev + + ssyev_lwork + dsyev_lwork + + ssyevd + dsyevd + + ssyevd_lwork + dsyevd_lwork + + ssyevr + dsyevr + + ssyevr_lwork + dsyevr_lwork + + ssyevx + dsyevx + + ssyevx_lwork + dsyevx_lwork + + ssygst + dsygst + + ssygv + dsygv + + ssygv_lwork + dsygv_lwork + + ssygvd + dsygvd + + ssygvx + dsygvx + + ssygvx_lwork + dsygvx_lwork + + ssysv + dsysv + csysv + zsysv + + ssysv_lwork + dsysv_lwork + csysv_lwork + zsysv_lwork + + ssysvx + dsysvx + csysvx + zsysvx + + ssysvx_lwork + dsysvx_lwork + csysvx_lwork + zsysvx_lwork + + ssytf2 + dsytf2 + csytf2 + zsytf2 + + ssytrd + dsytrd + + ssytrd_lwork + dsytrd_lwork + + ssytrf + dsytrf + csytrf + zsytrf + + ssytrf_lwork + dsytrf_lwork + csytrf_lwork + zsytrf_lwork + + ssytrs + dsytrs + csytrs + zsytrs + + stbtrs + dtbtrs + ctbtrs + ztbtrs + + stfsm + dtfsm + ctfsm + ztfsm + + stfttp + dtfttp + ctfttp + ztfttp + + stfttr + dtfttr + ctfttr + ztfttr + + stgexc + dtgexc + ctgexc + ztgexc + + stgsen + dtgsen + ctgsen + ztgsen + + stgsen_lwork + dtgsen_lwork + ctgsen_lwork + ztgsen_lwork + + stgsyl + dtgsyl + + stpttf + dtpttf + ctpttf + ztpttf + + stpttr + dtpttr + ctpttr + ztpttr + + strcon + dtrcon + ctrcon + ztrcon + + strexc + dtrexc + ctrexc + ztrexc + + strsen + dtrsen + ctrsen + ztrsen + + strsen_lwork + dtrsen_lwork + ctrsen_lwork + ztrsen_lwork + + strsyl + dtrsyl + ctrsyl + ztrsyl + + strtri + dtrtri + ctrtri + ztrtri + + strtrs + dtrtrs + ctrtrs + ztrtrs + + strttf + dtrttf + ctrttf + ztrttf + + strttp + dtrttp + ctrttp + ztrttp + + stzrzf + dtzrzf + ctzrzf + ztzrzf + + stzrzf_lwork + dtzrzf_lwork + ctzrzf_lwork + ztzrzf_lwork + + cunghr + zunghr + + cunghr_lwork + zunghr_lwork + + cungqr + zungqr + + cungrq + zungrq + + cunmqr + zunmqr + + sgeqrt + dgeqrt + cgeqrt + zgeqrt + + sgemqrt + dgemqrt + cgemqrt + zgemqrt + + sgttrf + dgttrf + cgttrf + zgttrf + + sgttrs + dgttrs + cgttrs + zgttrs + + sgtcon + dgtcon + cgtcon + zgtcon + + stpqrt + dtpqrt + ctpqrt + ztpqrt + + stpmqrt + dtpmqrt + ctpmqrt + ztpmqrt + + cuncsd + zuncsd + + cuncsd_lwork + zuncsd_lwork + + cunmrz + zunmrz + + cunmrz_lwork + zunmrz_lwork + + ilaver + +""" +# +# Author: Pearu Peterson, March 2002 +# + +import numpy as np +from .blas import _get_funcs, _memoize_get_funcs +from scipy.linalg import _flapack +from re import compile as regex_compile +try: + from scipy.linalg import _clapack +except ImportError: + _clapack = None + +try: + from scipy.linalg import _flapack_64 + HAS_ILP64 = True +except ImportError: + HAS_ILP64 = False + _flapack_64 = None + + +# Expose all functions (only flapack --- clapack is an implementation detail) +empty_module = None +from scipy.linalg._flapack import * # noqa: E402, F403 +del empty_module + +__all__ = ['get_lapack_funcs'] + +# some convenience alias for complex functions +_lapack_alias = { + 'corghr': 'cunghr', 'zorghr': 'zunghr', + 'corghr_lwork': 'cunghr_lwork', 'zorghr_lwork': 'zunghr_lwork', + 'corgqr': 'cungqr', 'zorgqr': 'zungqr', + 'cormqr': 'cunmqr', 'zormqr': 'zunmqr', + 'corgrq': 'cungrq', 'zorgrq': 'zungrq', +} + + +# Place guards against docstring rendering issues with special characters +p1 = regex_compile(r'with bounds (?P.*?)( and (?P.*?) storage){0,1}\n') +p2 = regex_compile(r'Default: (?P.*?)\n') + + +def backtickrepl(m): + if m.group('s'): + return (f"with bounds ``{m.group('b')}`` with ``{m.group('s')}`` storage\n") + else: + return f"with bounds ``{m.group('b')}``\n" + + +for routine in [ssyevr, dsyevr, cheevr, zheevr, + ssyevx, dsyevx, cheevx, zheevx, + ssygvd, dsygvd, chegvd, zhegvd]: + if routine.__doc__: + routine.__doc__ = p1.sub(backtickrepl, routine.__doc__) + routine.__doc__ = p2.sub('Default ``\\1``\n', routine.__doc__) + else: + continue + +del regex_compile, p1, p2, backtickrepl + + +@_memoize_get_funcs +def get_lapack_funcs(names, arrays=(), dtype=None, ilp64=False): + """Return available LAPACK function objects from names. + + Arrays are used to determine the optimal prefix of LAPACK routines. + + Parameters + ---------- + names : str or sequence of str + Name(s) of LAPACK functions without type prefix. + + arrays : sequence of ndarrays, optional + Arrays can be given to determine optimal prefix of LAPACK + routines. If not given, double-precision routines will be + used, otherwise the most generic type in arrays will be used. + + dtype : str or dtype, optional + Data-type specifier. Not used if `arrays` is non-empty. + + ilp64 : {True, False, 'preferred'}, optional + Whether to return ILP64 routine variant. + Choosing 'preferred' returns ILP64 routine if available, and + otherwise the 32-bit routine. Default: False + + Returns + ------- + funcs : list + List containing the found function(s). + + Notes + ----- + This routine automatically chooses between Fortran/C + interfaces. Fortran code is used whenever possible for arrays with + column major order. In all other cases, C code is preferred. + + In LAPACK, the naming convention is that all functions start with a + type prefix, which depends on the type of the principal + matrix. These can be one of {'s', 'd', 'c', 'z'} for the NumPy + types {float32, float64, complex64, complex128} respectively, and + are stored in attribute ``typecode`` of the returned functions. + + Examples + -------- + Suppose we would like to use '?lange' routine which computes the selected + norm of an array. We pass our array in order to get the correct 'lange' + flavor. + + >>> import numpy as np + >>> import scipy.linalg as LA + >>> rng = np.random.default_rng() + + >>> a = rng.random((3,2)) + >>> x_lange = LA.get_lapack_funcs('lange', (a,)) + >>> x_lange.typecode + 'd' + >>> x_lange = LA.get_lapack_funcs('lange',(a*1j,)) + >>> x_lange.typecode + 'z' + + Several LAPACK routines work best when its internal WORK array has + the optimal size (big enough for fast computation and small enough to + avoid waste of memory). This size is determined also by a dedicated query + to the function which is often wrapped as a standalone function and + commonly denoted as ``###_lwork``. Below is an example for ``?sysv`` + + >>> a = rng.random((1000, 1000)) + >>> b = rng.random((1000, 1)) * 1j + >>> # We pick up zsysv and zsysv_lwork due to b array + ... xsysv, xlwork = LA.get_lapack_funcs(('sysv', 'sysv_lwork'), (a, b)) + >>> opt_lwork, _ = xlwork(a.shape[0]) # returns a complex for 'z' prefix + >>> udut, ipiv, x, info = xsysv(a, b, lwork=int(opt_lwork.real)) + + """ + if isinstance(ilp64, str): + if ilp64 == 'preferred': + ilp64 = HAS_ILP64 + else: + raise ValueError("Invalid value for 'ilp64'") + + if not ilp64: + return _get_funcs(names, arrays, dtype, + "LAPACK", _flapack, _clapack, + "flapack", "clapack", _lapack_alias, + ilp64=False) + else: + if not HAS_ILP64: + raise RuntimeError("LAPACK ILP64 routine requested, but Scipy " + "compiled only with 32-bit BLAS") + return _get_funcs(names, arrays, dtype, + "LAPACK", _flapack_64, None, + "flapack_64", None, _lapack_alias, + ilp64=True) + + +_int32_max = np.iinfo(np.int32).max +_int64_max = np.iinfo(np.int64).max + + +def _compute_lwork(routine, *args, **kwargs): + """ + Round floating-point lwork returned by lapack to integer. + + Several LAPACK routines compute optimal values for LWORK, which + they return in a floating-point variable. However, for large + values of LWORK, single-precision floating point is not sufficient + to hold the exact value --- some LAPACK versions (<= 3.5.0 at + least) truncate the returned integer to single precision and in + some cases this can be smaller than the required value. + + Examples + -------- + >>> from scipy.linalg import lapack + >>> n = 5000 + >>> s_r, s_lw = lapack.get_lapack_funcs(('sysvx', 'sysvx_lwork')) + >>> lwork = lapack._compute_lwork(s_lw, n) + >>> lwork + 32000 + + """ + dtype = getattr(routine, 'dtype', None) + int_dtype = getattr(routine, 'int_dtype', None) + ret = routine(*args, **kwargs) + if ret[-1] != 0: + raise ValueError("Internal work array size computation failed: " + "%d" % (ret[-1],)) + + if len(ret) == 2: + return _check_work_float(ret[0].real, dtype, int_dtype) + else: + return tuple(_check_work_float(x.real, dtype, int_dtype) + for x in ret[:-1]) + + +def _check_work_float(value, dtype, int_dtype): + """ + Convert LAPACK-returned work array size float to integer, + carefully for single-precision types. + """ + + if dtype == np.float32 or dtype == np.complex64: + # Single-precision routine -- take next fp value to work + # around possible truncation in LAPACK code + value = np.nextafter(value, np.inf, dtype=np.float32) + + value = int(value) + if int_dtype.itemsize == 4: + if value < 0 or value > _int32_max: + raise ValueError("Too large work array required -- computation " + "cannot be performed with standard 32-bit" + " LAPACK.") + elif int_dtype.itemsize == 8: + if value < 0 or value > _int64_max: + raise ValueError("Too large work array required -- computation" + " cannot be performed with standard 64-bit" + " LAPACK.") + return value diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/matfuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/matfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..9ec8123b3ad8df096d5791c29bb28cce8271d4ad --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/matfuncs.py @@ -0,0 +1,23 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'expm', 'cosm', 'sinm', 'tanm', 'coshm', 'sinhm', + 'tanhm', 'logm', 'funm', 'signm', 'sqrtm', + 'expm_frechet', 'expm_cond', 'fractional_matrix_power', + 'khatri_rao', 'norm', 'solve', 'inv', 'svd', 'schur', 'rsf2csf' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="linalg", module="matfuncs", + private_modules=["_matfuncs"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/misc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/misc.py new file mode 100644 index 0000000000000000000000000000000000000000..1fad087489c6a24c8e33df54b811b6c37a3a46d4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/misc.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'LinAlgError', 'LinAlgWarning', 'norm', 'get_blas_funcs', + 'get_lapack_funcs' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="linalg", module="misc", + private_modules=["_misc"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/special_matrices.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/special_matrices.py new file mode 100644 index 0000000000000000000000000000000000000000..a881ce765dfa3a3c4c2853c405f8129aafc615b5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/special_matrices.py @@ -0,0 +1,22 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'toeplitz', 'circulant', 'hankel', + 'hadamard', 'leslie', 'kron', 'block_diag', 'companion', + 'helmert', 'hilbert', 'invhilbert', 'pascal', 'invpascal', 'dft', + 'fiedler', 'fiedler_companion', 'convolution_matrix' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="linalg", module="special_matrices", + private_modules=["_special_matrices"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/_cython_examples/extending.pyx b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/_cython_examples/extending.pyx new file mode 100644 index 0000000000000000000000000000000000000000..3954d08791cceb3a2b66669fe3c0ec4180089208 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/_cython_examples/extending.pyx @@ -0,0 +1,23 @@ +#!/usr/bin/env python3 +#cython: language_level=3 +#cython: boundscheck=False +#cython: wraparound=False + +cimport scipy.linalg +from scipy.linalg.cython_blas cimport cdotu +from scipy.linalg.cython_lapack cimport dgtsv + +cpdef tridiag(double[:] a, double[:] b, double[:] c, double[:] x): + """ Solve the system A y = x for y where A is the tridiagonal matrix with + subdiagonal 'a', diagonal 'b', and superdiagonal 'c'. """ + cdef int n=b.shape[0], nrhs=1, info + # Solution is written over the values in x. + dgtsv(&n, &nrhs, &a[0], &b[0], &c[0], &x[0], &n, &info) + +cpdef float complex complex_dot(float complex[:] cx, float complex[:] cy): + """ Take dot product of two complex vectors """ + cdef: + int n = cx.shape[0] + int incx = cx.strides[0] // sizeof(cx[0]) + int incy = cy.strides[0] // sizeof(cy[0]) + return cdotu(&n, &cx[0], &incx, &cy[0], &incy) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/_cython_examples/meson.build b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/_cython_examples/meson.build new file mode 100644 index 0000000000000000000000000000000000000000..88f23170ac0bbe382d8470bd22a42a92d9473008 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/_cython_examples/meson.build @@ -0,0 +1,27 @@ +project('random-build-examples', 'c', 'cpp', 'cython') + +fs = import('fs') + +py3 = import('python').find_installation(pure: false) + +cy = meson.get_compiler('cython') + +if not cy.version().version_compare('>=3.0.8') + error('tests requires Cython >= 3.0.8') +endif + +py3.extension_module( + 'extending', + 'extending.pyx', + install: false, + c_args: ['-DCYTHON_CCOMPLEX=0'] # see gh-18975 for why we need this +) + +extending_cpp = fs.copyfile('extending.pyx', 'extending_cpp.pyx') +py3.extension_module( + 'extending_cpp', + extending_cpp, + install: false, + override_options : ['cython_language=cpp'], + cpp_args: ['-DCYTHON_CCOMPLEX=0'] # see gh-18975 for why we need this +) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_basic.py new file mode 100644 index 0000000000000000000000000000000000000000..fe51dcc21824ae122a17376940d43e34ec9ac22c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_basic.py @@ -0,0 +1,2059 @@ +import itertools + +import numpy as np +from numpy import (arange, array, dot, zeros, identity, conjugate, transpose, + float32) +from numpy.random import random + +from numpy.testing import (assert_equal, assert_almost_equal, assert_, + assert_array_almost_equal, assert_allclose, + assert_array_equal, suppress_warnings) +import pytest +from pytest import raises as assert_raises + +from scipy.linalg import (solve, inv, det, lstsq, pinv, pinvh, norm, + solve_banded, solveh_banded, solve_triangular, + solve_circulant, circulant, LinAlgError, block_diag, + matrix_balance, qr, LinAlgWarning) + +from scipy.linalg._testutils import assert_no_overwrite +from scipy._lib._testutils import check_free_memory, IS_MUSL +from scipy.linalg.blas import HAS_ILP64 + +REAL_DTYPES = (np.float32, np.float64, np.longdouble) +COMPLEX_DTYPES = (np.complex64, np.complex128, np.clongdouble) +DTYPES = REAL_DTYPES + COMPLEX_DTYPES + + +def _eps_cast(dtyp): + """Get the epsilon for dtype, possibly downcast to BLAS types.""" + dt = dtyp + if dt == np.longdouble: + dt = np.float64 + elif dt == np.clongdouble: + dt = np.complex128 + return np.finfo(dt).eps + + +class TestSolveBanded: + + def test_real(self): + a = array([[1.0, 20, 0, 0], + [-30, 4, 6, 0], + [2, 1, 20, 2], + [0, -1, 7, 14]]) + ab = array([[0.0, 20, 6, 2], + [1, 4, 20, 14], + [-30, 1, 7, 0], + [2, -1, 0, 0]]) + l, u = 2, 1 + b4 = array([10.0, 0.0, 2.0, 14.0]) + b4by1 = b4.reshape(-1, 1) + b4by2 = array([[2, 1], + [-30, 4], + [2, 3], + [1, 3]]) + b4by4 = array([[1, 0, 0, 0], + [0, 0, 0, 1], + [0, 1, 0, 0], + [0, 1, 0, 0]]) + for b in [b4, b4by1, b4by2, b4by4]: + x = solve_banded((l, u), ab, b) + assert_array_almost_equal(dot(a, x), b) + + def test_complex(self): + a = array([[1.0, 20, 0, 0], + [-30, 4, 6, 0], + [2j, 1, 20, 2j], + [0, -1, 7, 14]]) + ab = array([[0.0, 20, 6, 2j], + [1, 4, 20, 14], + [-30, 1, 7, 0], + [2j, -1, 0, 0]]) + l, u = 2, 1 + b4 = array([10.0, 0.0, 2.0, 14.0j]) + b4by1 = b4.reshape(-1, 1) + b4by2 = array([[2, 1], + [-30, 4], + [2, 3], + [1, 3]]) + b4by4 = array([[1, 0, 0, 0], + [0, 0, 0, 1j], + [0, 1, 0, 0], + [0, 1, 0, 0]]) + for b in [b4, b4by1, b4by2, b4by4]: + x = solve_banded((l, u), ab, b) + assert_array_almost_equal(dot(a, x), b) + + def test_tridiag_real(self): + ab = array([[0.0, 20, 6, 2], + [1, 4, 20, 14], + [-30, 1, 7, 0]]) + a = np.diag(ab[0, 1:], 1) + np.diag(ab[1, :], 0) + np.diag( + ab[2, :-1], -1) + b4 = array([10.0, 0.0, 2.0, 14.0]) + b4by1 = b4.reshape(-1, 1) + b4by2 = array([[2, 1], + [-30, 4], + [2, 3], + [1, 3]]) + b4by4 = array([[1, 0, 0, 0], + [0, 0, 0, 1], + [0, 1, 0, 0], + [0, 1, 0, 0]]) + for b in [b4, b4by1, b4by2, b4by4]: + x = solve_banded((1, 1), ab, b) + assert_array_almost_equal(dot(a, x), b) + + def test_tridiag_complex(self): + ab = array([[0.0, 20, 6, 2j], + [1, 4, 20, 14], + [-30, 1, 7, 0]]) + a = np.diag(ab[0, 1:], 1) + np.diag(ab[1, :], 0) + np.diag( + ab[2, :-1], -1) + b4 = array([10.0, 0.0, 2.0, 14.0j]) + b4by1 = b4.reshape(-1, 1) + b4by2 = array([[2, 1], + [-30, 4], + [2, 3], + [1, 3]]) + b4by4 = array([[1, 0, 0, 0], + [0, 0, 0, 1], + [0, 1, 0, 0], + [0, 1, 0, 0]]) + for b in [b4, b4by1, b4by2, b4by4]: + x = solve_banded((1, 1), ab, b) + assert_array_almost_equal(dot(a, x), b) + + def test_check_finite(self): + a = array([[1.0, 20, 0, 0], + [-30, 4, 6, 0], + [2, 1, 20, 2], + [0, -1, 7, 14]]) + ab = array([[0.0, 20, 6, 2], + [1, 4, 20, 14], + [-30, 1, 7, 0], + [2, -1, 0, 0]]) + l, u = 2, 1 + b4 = array([10.0, 0.0, 2.0, 14.0]) + x = solve_banded((l, u), ab, b4, check_finite=False) + assert_array_almost_equal(dot(a, x), b4) + + def test_bad_shape(self): + ab = array([[0.0, 20, 6, 2], + [1, 4, 20, 14], + [-30, 1, 7, 0], + [2, -1, 0, 0]]) + l, u = 2, 1 + bad = array([1.0, 2.0, 3.0, 4.0]).reshape(-1, 4) + assert_raises(ValueError, solve_banded, (l, u), ab, bad) + assert_raises(ValueError, solve_banded, (l, u), ab, [1.0, 2.0]) + + # Values of (l,u) are not compatible with ab. + assert_raises(ValueError, solve_banded, (1, 1), ab, [1.0, 2.0]) + + def test_1x1(self): + # gh-8906 noted that the case of A@x = b with 1x1 A was handled + # incorrectly; check that this is resolved. Typical case: + # nupper == nlower == 0 + # A = [[2]] + b = array([[1., 2., 3.]]) + ref = array([[0.5, 1.0, 1.5]]) + x = solve_banded((0, 0), [[2]], b) + assert_allclose(x, ref, rtol=1e-15) + + # However, the user *can* represent the same system with garbage rows + # in `ab`. Test the case with `nupper == 1, nlower == 1`. + x = solve_banded((1, 1), [[0], [2], [0]], b) + assert_allclose(x, ref, rtol=1e-15) + assert_equal(x.dtype, np.dtype('f8')) + assert_array_equal(b, [[1.0, 2.0, 3.0]]) + + def test_native_list_arguments(self): + a = [[1.0, 20, 0, 0], + [-30, 4, 6, 0], + [2, 1, 20, 2], + [0, -1, 7, 14]] + ab = [[0.0, 20, 6, 2], + [1, 4, 20, 14], + [-30, 1, 7, 0], + [2, -1, 0, 0]] + l, u = 2, 1 + b = [10.0, 0.0, 2.0, 14.0] + x = solve_banded((l, u), ab, b) + assert_array_almost_equal(dot(a, x), b) + + @pytest.mark.thread_unsafe # due to Cython fused types, see cython#6506 + @pytest.mark.parametrize('dt_ab', [int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt_ab, dt_b): + # ab contains one empty row corresponding to the diagonal + ab = np.array([[]], dtype=dt_ab) + b = np.array([], dtype=dt_b) + x = solve_banded((0, 0), ab, b) + + assert x.shape == (0,) + assert x.dtype == solve(np.eye(1, dtype=dt_ab), np.ones(1, dtype=dt_b)).dtype + + b = np.empty((0, 0), dtype=dt_b) + x = solve_banded((0, 0), ab, b) + + assert x.shape == (0, 0) + assert x.dtype == solve(np.eye(1, dtype=dt_ab), np.ones(1, dtype=dt_b)).dtype + + +class TestSolveHBanded: + + def test_01_upper(self): + # Solve + # [ 4 1 2 0] [1] + # [ 1 4 1 2] X = [4] + # [ 2 1 4 1] [1] + # [ 0 2 1 4] [2] + # with the RHS as a 1D array. + ab = array([[0.0, 0.0, 2.0, 2.0], + [-99, 1.0, 1.0, 1.0], + [4.0, 4.0, 4.0, 4.0]]) + b = array([1.0, 4.0, 1.0, 2.0]) + x = solveh_banded(ab, b) + assert_array_almost_equal(x, [0.0, 1.0, 0.0, 0.0]) + + def test_02_upper(self): + # Solve + # [ 4 1 2 0] [1 6] + # [ 1 4 1 2] X = [4 2] + # [ 2 1 4 1] [1 6] + # [ 0 2 1 4] [2 1] + # + ab = array([[0.0, 0.0, 2.0, 2.0], + [-99, 1.0, 1.0, 1.0], + [4.0, 4.0, 4.0, 4.0]]) + b = array([[1.0, 6.0], + [4.0, 2.0], + [1.0, 6.0], + [2.0, 1.0]]) + x = solveh_banded(ab, b) + expected = array([[0.0, 1.0], + [1.0, 0.0], + [0.0, 1.0], + [0.0, 0.0]]) + assert_array_almost_equal(x, expected) + + def test_03_upper(self): + # Solve + # [ 4 1 2 0] [1] + # [ 1 4 1 2] X = [4] + # [ 2 1 4 1] [1] + # [ 0 2 1 4] [2] + # with the RHS as a 2D array with shape (3,1). + ab = array([[0.0, 0.0, 2.0, 2.0], + [-99, 1.0, 1.0, 1.0], + [4.0, 4.0, 4.0, 4.0]]) + b = array([1.0, 4.0, 1.0, 2.0]).reshape(-1, 1) + x = solveh_banded(ab, b) + assert_array_almost_equal(x, array([0., 1., 0., 0.]).reshape(-1, 1)) + + def test_01_lower(self): + # Solve + # [ 4 1 2 0] [1] + # [ 1 4 1 2] X = [4] + # [ 2 1 4 1] [1] + # [ 0 2 1 4] [2] + # + ab = array([[4.0, 4.0, 4.0, 4.0], + [1.0, 1.0, 1.0, -99], + [2.0, 2.0, 0.0, 0.0]]) + b = array([1.0, 4.0, 1.0, 2.0]) + x = solveh_banded(ab, b, lower=True) + assert_array_almost_equal(x, [0.0, 1.0, 0.0, 0.0]) + + def test_02_lower(self): + # Solve + # [ 4 1 2 0] [1 6] + # [ 1 4 1 2] X = [4 2] + # [ 2 1 4 1] [1 6] + # [ 0 2 1 4] [2 1] + # + ab = array([[4.0, 4.0, 4.0, 4.0], + [1.0, 1.0, 1.0, -99], + [2.0, 2.0, 0.0, 0.0]]) + b = array([[1.0, 6.0], + [4.0, 2.0], + [1.0, 6.0], + [2.0, 1.0]]) + x = solveh_banded(ab, b, lower=True) + expected = array([[0.0, 1.0], + [1.0, 0.0], + [0.0, 1.0], + [0.0, 0.0]]) + assert_array_almost_equal(x, expected) + + def test_01_float32(self): + # Solve + # [ 4 1 2 0] [1] + # [ 1 4 1 2] X = [4] + # [ 2 1 4 1] [1] + # [ 0 2 1 4] [2] + # + ab = array([[0.0, 0.0, 2.0, 2.0], + [-99, 1.0, 1.0, 1.0], + [4.0, 4.0, 4.0, 4.0]], dtype=float32) + b = array([1.0, 4.0, 1.0, 2.0], dtype=float32) + x = solveh_banded(ab, b) + assert_array_almost_equal(x, [0.0, 1.0, 0.0, 0.0]) + + def test_02_float32(self): + # Solve + # [ 4 1 2 0] [1 6] + # [ 1 4 1 2] X = [4 2] + # [ 2 1 4 1] [1 6] + # [ 0 2 1 4] [2 1] + # + ab = array([[0.0, 0.0, 2.0, 2.0], + [-99, 1.0, 1.0, 1.0], + [4.0, 4.0, 4.0, 4.0]], dtype=float32) + b = array([[1.0, 6.0], + [4.0, 2.0], + [1.0, 6.0], + [2.0, 1.0]], dtype=float32) + x = solveh_banded(ab, b) + expected = array([[0.0, 1.0], + [1.0, 0.0], + [0.0, 1.0], + [0.0, 0.0]]) + assert_array_almost_equal(x, expected) + + def test_01_complex(self): + # Solve + # [ 4 -j 2 0] [2-j] + # [ j 4 -j 2] X = [4-j] + # [ 2 j 4 -j] [4+j] + # [ 0 2 j 4] [2+j] + # + ab = array([[0.0, 0.0, 2.0, 2.0], + [-99, -1.0j, -1.0j, -1.0j], + [4.0, 4.0, 4.0, 4.0]]) + b = array([2-1.0j, 4.0-1j, 4+1j, 2+1j]) + x = solveh_banded(ab, b) + assert_array_almost_equal(x, [0.0, 1.0, 1.0, 0.0]) + + def test_02_complex(self): + # Solve + # [ 4 -j 2 0] [2-j 2+4j] + # [ j 4 -j 2] X = [4-j -1-j] + # [ 2 j 4 -j] [4+j 4+2j] + # [ 0 2 j 4] [2+j j] + # + ab = array([[0.0, 0.0, 2.0, 2.0], + [-99, -1.0j, -1.0j, -1.0j], + [4.0, 4.0, 4.0, 4.0]]) + b = array([[2-1j, 2+4j], + [4.0-1j, -1-1j], + [4.0+1j, 4+2j], + [2+1j, 1j]]) + x = solveh_banded(ab, b) + expected = array([[0.0, 1.0j], + [1.0, 0.0], + [1.0, 1.0], + [0.0, 0.0]]) + assert_array_almost_equal(x, expected) + + def test_tridiag_01_upper(self): + # Solve + # [ 4 1 0] [1] + # [ 1 4 1] X = [4] + # [ 0 1 4] [1] + # with the RHS as a 1D array. + ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]]) + b = array([1.0, 4.0, 1.0]) + x = solveh_banded(ab, b) + assert_array_almost_equal(x, [0.0, 1.0, 0.0]) + + def test_tridiag_02_upper(self): + # Solve + # [ 4 1 0] [1 4] + # [ 1 4 1] X = [4 2] + # [ 0 1 4] [1 4] + # + ab = array([[-99, 1.0, 1.0], + [4.0, 4.0, 4.0]]) + b = array([[1.0, 4.0], + [4.0, 2.0], + [1.0, 4.0]]) + x = solveh_banded(ab, b) + expected = array([[0.0, 1.0], + [1.0, 0.0], + [0.0, 1.0]]) + assert_array_almost_equal(x, expected) + + def test_tridiag_03_upper(self): + # Solve + # [ 4 1 0] [1] + # [ 1 4 1] X = [4] + # [ 0 1 4] [1] + # with the RHS as a 2D array with shape (3,1). + ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]]) + b = array([1.0, 4.0, 1.0]).reshape(-1, 1) + x = solveh_banded(ab, b) + assert_array_almost_equal(x, array([0.0, 1.0, 0.0]).reshape(-1, 1)) + + def test_tridiag_01_lower(self): + # Solve + # [ 4 1 0] [1] + # [ 1 4 1] X = [4] + # [ 0 1 4] [1] + # + ab = array([[4.0, 4.0, 4.0], + [1.0, 1.0, -99]]) + b = array([1.0, 4.0, 1.0]) + x = solveh_banded(ab, b, lower=True) + assert_array_almost_equal(x, [0.0, 1.0, 0.0]) + + def test_tridiag_02_lower(self): + # Solve + # [ 4 1 0] [1 4] + # [ 1 4 1] X = [4 2] + # [ 0 1 4] [1 4] + # + ab = array([[4.0, 4.0, 4.0], + [1.0, 1.0, -99]]) + b = array([[1.0, 4.0], + [4.0, 2.0], + [1.0, 4.0]]) + x = solveh_banded(ab, b, lower=True) + expected = array([[0.0, 1.0], + [1.0, 0.0], + [0.0, 1.0]]) + assert_array_almost_equal(x, expected) + + def test_tridiag_01_float32(self): + # Solve + # [ 4 1 0] [1] + # [ 1 4 1] X = [4] + # [ 0 1 4] [1] + # + ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]], dtype=float32) + b = array([1.0, 4.0, 1.0], dtype=float32) + x = solveh_banded(ab, b) + assert_array_almost_equal(x, [0.0, 1.0, 0.0]) + + def test_tridiag_02_float32(self): + # Solve + # [ 4 1 0] [1 4] + # [ 1 4 1] X = [4 2] + # [ 0 1 4] [1 4] + # + ab = array([[-99, 1.0, 1.0], + [4.0, 4.0, 4.0]], dtype=float32) + b = array([[1.0, 4.0], + [4.0, 2.0], + [1.0, 4.0]], dtype=float32) + x = solveh_banded(ab, b) + expected = array([[0.0, 1.0], + [1.0, 0.0], + [0.0, 1.0]]) + assert_array_almost_equal(x, expected) + + def test_tridiag_01_complex(self): + # Solve + # [ 4 -j 0] [ -j] + # [ j 4 -j] X = [4-j] + # [ 0 j 4] [4+j] + # + ab = array([[-99, -1.0j, -1.0j], [4.0, 4.0, 4.0]]) + b = array([-1.0j, 4.0-1j, 4+1j]) + x = solveh_banded(ab, b) + assert_array_almost_equal(x, [0.0, 1.0, 1.0]) + + def test_tridiag_02_complex(self): + # Solve + # [ 4 -j 0] [ -j 4j] + # [ j 4 -j] X = [4-j -1-j] + # [ 0 j 4] [4+j 4 ] + # + ab = array([[-99, -1.0j, -1.0j], + [4.0, 4.0, 4.0]]) + b = array([[-1j, 4.0j], + [4.0-1j, -1.0-1j], + [4.0+1j, 4.0]]) + x = solveh_banded(ab, b) + expected = array([[0.0, 1.0j], + [1.0, 0.0], + [1.0, 1.0]]) + assert_array_almost_equal(x, expected) + + def test_check_finite(self): + # Solve + # [ 4 1 0] [1] + # [ 1 4 1] X = [4] + # [ 0 1 4] [1] + # with the RHS as a 1D array. + ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]]) + b = array([1.0, 4.0, 1.0]) + x = solveh_banded(ab, b, check_finite=False) + assert_array_almost_equal(x, [0.0, 1.0, 0.0]) + + def test_bad_shapes(self): + ab = array([[-99, 1.0, 1.0], + [4.0, 4.0, 4.0]]) + b = array([[1.0, 4.0], + [4.0, 2.0]]) + assert_raises(ValueError, solveh_banded, ab, b) + assert_raises(ValueError, solveh_banded, ab, [1.0, 2.0]) + assert_raises(ValueError, solveh_banded, ab, [1.0]) + + def test_1x1(self): + x = solveh_banded([[1]], [[1, 2, 3]]) + assert_array_equal(x, [[1.0, 2.0, 3.0]]) + assert_equal(x.dtype, np.dtype('f8')) + + def test_native_list_arguments(self): + # Same as test_01_upper, using python's native list. + ab = [[0.0, 0.0, 2.0, 2.0], + [-99, 1.0, 1.0, 1.0], + [4.0, 4.0, 4.0, 4.0]] + b = [1.0, 4.0, 1.0, 2.0] + x = solveh_banded(ab, b) + assert_array_almost_equal(x, [0.0, 1.0, 0.0, 0.0]) + + @pytest.mark.parametrize('dt_ab', [int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt_ab, dt_b): + # ab contains one empty row corresponding to the diagonal + ab = np.array([[]], dtype=dt_ab) + b = np.array([], dtype=dt_b) + x = solveh_banded(ab, b) + + assert x.shape == (0,) + assert x.dtype == solve(np.eye(1, dtype=dt_ab), np.ones(1, dtype=dt_b)).dtype + + b = np.empty((0, 0), dtype=dt_b) + x = solveh_banded(ab, b) + + assert x.shape == (0, 0) + assert x.dtype == solve(np.eye(1, dtype=dt_ab), np.ones(1, dtype=dt_b)).dtype + + +class TestSolve: + def setup_method(self): + np.random.seed(1234) + + @pytest.mark.thread_unsafe + def test_20Feb04_bug(self): + a = [[1, 1], [1.0, 0]] # ok + x0 = solve(a, [1, 0j]) + assert_array_almost_equal(dot(a, x0), [1, 0]) + + # gives failure with clapack.zgesv(..,rowmajor=0) + a = [[1, 1], [1.2, 0]] + b = [1, 0j] + x0 = solve(a, b) + assert_array_almost_equal(dot(a, x0), [1, 0]) + + def test_simple(self): + a = [[1, 20], [-30, 4]] + for b in ([[1, 0], [0, 1]], + [1, 0], + [[2, 1], [-30, 4]] + ): + x = solve(a, b) + assert_array_almost_equal(dot(a, x), b) + + def test_simple_complex(self): + a = array([[5, 2], [2j, 4]], 'D') + for b in ([1j, 0], + [[1j, 1j], [0, 2]], + [1, 0j], + array([1, 0], 'D'), + ): + x = solve(a, b) + assert_array_almost_equal(dot(a, x), b) + + def test_simple_pos(self): + a = [[2, 3], [3, 5]] + for lower in [0, 1]: + for b in ([[1, 0], [0, 1]], + [1, 0] + ): + x = solve(a, b, assume_a='pos', lower=lower) + assert_array_almost_equal(dot(a, x), b) + + def test_simple_pos_complexb(self): + a = [[5, 2], [2, 4]] + for b in ([1j, 0], + [[1j, 1j], [0, 2]], + ): + x = solve(a, b, assume_a='pos') + assert_array_almost_equal(dot(a, x), b) + + def test_simple_sym(self): + a = [[2, 3], [3, -5]] + for lower in [0, 1]: + for b in ([[1, 0], [0, 1]], + [1, 0] + ): + x = solve(a, b, assume_a='sym', lower=lower) + assert_array_almost_equal(dot(a, x), b) + + def test_simple_sym_complexb(self): + a = [[5, 2], [2, -4]] + for b in ([1j, 0], + [[1j, 1j], [0, 2]] + ): + x = solve(a, b, assume_a='sym') + assert_array_almost_equal(dot(a, x), b) + + def test_simple_sym_complex(self): + a = [[5, 2+1j], [2+1j, -4]] + for b in ([1j, 0], + [1, 0], + [[1j, 1j], [0, 2]] + ): + x = solve(a, b, assume_a='sym') + assert_array_almost_equal(dot(a, x), b) + + def test_simple_her_actuallysym(self): + a = [[2, 3], [3, -5]] + for lower in [0, 1]: + for b in ([[1, 0], [0, 1]], + [1, 0], + [1j, 0], + ): + x = solve(a, b, assume_a='her', lower=lower) + assert_array_almost_equal(dot(a, x), b) + + def test_simple_her(self): + a = [[5, 2+1j], [2-1j, -4]] + for b in ([1j, 0], + [1, 0], + [[1j, 1j], [0, 2]] + ): + x = solve(a, b, assume_a='her') + assert_array_almost_equal(dot(a, x), b) + + def test_nils_20Feb04(self): + n = 2 + A = random([n, n])+random([n, n])*1j + X = zeros((n, n), 'D') + Ainv = inv(A) + R = identity(n)+identity(n)*0j + for i in arange(0, n): + r = R[:, i] + X[:, i] = solve(A, r) + assert_array_almost_equal(X, Ainv) + + def test_random(self): + + n = 20 + a = random([n, n]) + for i in range(n): + a[i, i] = 20*(.1+a[i, i]) + for i in range(4): + b = random([n, 3]) + x = solve(a, b) + assert_array_almost_equal(dot(a, x), b) + + def test_random_complex(self): + n = 20 + a = random([n, n]) + 1j * random([n, n]) + for i in range(n): + a[i, i] = 20*(.1+a[i, i]) + for i in range(2): + b = random([n, 3]) + x = solve(a, b) + assert_array_almost_equal(dot(a, x), b) + + def test_random_sym(self): + n = 20 + a = random([n, n]) + for i in range(n): + a[i, i] = abs(20*(.1+a[i, i])) + for j in range(i): + a[i, j] = a[j, i] + for i in range(4): + b = random([n]) + x = solve(a, b, assume_a="pos") + assert_array_almost_equal(dot(a, x), b) + + def test_random_sym_complex(self): + n = 20 + a = random([n, n]) + a = a + 1j*random([n, n]) + for i in range(n): + a[i, i] = abs(20*(.1+a[i, i])) + for j in range(i): + a[i, j] = conjugate(a[j, i]) + b = random([n])+2j*random([n]) + for i in range(2): + x = solve(a, b, assume_a="pos") + assert_array_almost_equal(dot(a, x), b) + + def test_check_finite(self): + a = [[1, 20], [-30, 4]] + for b in ([[1, 0], [0, 1]], [1, 0], + [[2, 1], [-30, 4]]): + x = solve(a, b, check_finite=False) + assert_array_almost_equal(dot(a, x), b) + + def test_scalar_a_and_1D_b(self): + a = 1 + b = [1, 2, 3] + x = solve(a, b) + assert_array_almost_equal(x.ravel(), b) + assert_(x.shape == (3,), 'Scalar_a_1D_b test returned wrong shape') + + def test_simple2(self): + a = np.array([[1.80, 2.88, 2.05, -0.89], + [525.00, -295.00, -95.00, -380.00], + [1.58, -2.69, -2.90, -1.04], + [-1.11, -0.66, -0.59, 0.80]]) + + b = np.array([[9.52, 18.47], + [2435.00, 225.00], + [0.77, -13.28], + [-6.22, -6.21]]) + + x = solve(a, b) + assert_array_almost_equal(x, np.array([[1., -1, 3, -5], + [3, 2, 4, 1]]).T) + + def test_simple_complex2(self): + a = np.array([[-1.34+2.55j, 0.28+3.17j, -6.39-2.20j, 0.72-0.92j], + [-1.70-14.10j, 33.10-1.50j, -1.50+13.40j, 12.90+13.80j], + [-3.29-2.39j, -1.91+4.42j, -0.14-1.35j, 1.72+1.35j], + [2.41+0.39j, -0.56+1.47j, -0.83-0.69j, -1.96+0.67j]]) + + b = np.array([[26.26+51.78j, 31.32-6.70j], + [64.30-86.80j, 158.60-14.20j], + [-5.75+25.31j, -2.15+30.19j], + [1.16+2.57j, -2.56+7.55j]]) + + x = solve(a, b) + assert_array_almost_equal(x, np. array([[1+1.j, -1-2.j], + [2-3.j, 5+1.j], + [-4-5.j, -3+4.j], + [6.j, 2-3.j]])) + + def test_hermitian(self): + # An upper triangular matrix will be used for hermitian matrix a + a = np.array([[-1.84, 0.11-0.11j, -1.78-1.18j, 3.91-1.50j], + [0, -4.63, -1.84+0.03j, 2.21+0.21j], + [0, 0, -8.87, 1.58-0.90j], + [0, 0, 0, -1.36]]) + b = np.array([[2.98-10.18j, 28.68-39.89j], + [-9.58+3.88j, -24.79-8.40j], + [-0.77-16.05j, 4.23-70.02j], + [7.79+5.48j, -35.39+18.01j]]) + res = np.array([[2.+1j, -8+6j], + [3.-2j, 7-2j], + [-1+2j, -1+5j], + [1.-1j, 3-4j]]) + x = solve(a, b, assume_a='her') + assert_array_almost_equal(x, res) + # Also conjugate a and test for lower triangular data + x = solve(a.conj().T, b, assume_a='her', lower=True) + assert_array_almost_equal(x, res) + + def test_pos_and_sym(self): + A = np.arange(1, 10).reshape(3, 3) + x = solve(np.tril(A)/9, np.ones(3), assume_a='pos') + assert_array_almost_equal(x, [9., 1.8, 1.]) + x = solve(np.tril(A)/9, np.ones(3), assume_a='sym') + assert_array_almost_equal(x, [9., 1.8, 1.]) + + def test_singularity(self): + a = np.array([[1, 0, 0, 0, 0, 0, 1, 0, 1], + [1, 1, 1, 0, 0, 0, 1, 0, 1], + [0, 1, 1, 0, 0, 0, 1, 0, 1], + [1, 0, 1, 1, 1, 1, 0, 0, 0], + [1, 0, 1, 1, 1, 1, 0, 0, 0], + [1, 0, 1, 1, 1, 1, 0, 0, 0], + [1, 0, 1, 1, 1, 1, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 1, 1, 1], + [1, 1, 1, 1, 1, 1, 1, 1, 1]]) + b = np.arange(9)[:, None] + assert_raises(LinAlgError, solve, a, b) + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize('structure', + ('diagonal', 'tridiagonal', 'lower triangular', + 'upper triangular', 'symmetric', 'hermitian', + 'positive definite', 'general', None)) + def test_ill_condition_warning(self, structure): + rng = np.random.default_rng(234859349452) + n = 10 + d = np.logspace(0, 50, n) + A = np.diag(d) + b = rng.random(size=n) + message = "Ill-conditioned matrix..." + with pytest.warns(LinAlgWarning, match=message): + solve(A, b, assume_a=structure) + + def test_multiple_rhs(self): + a = np.eye(2) + b = np.random.rand(2, 3, 4) + x = solve(a, b) + assert_array_almost_equal(x, b) + + def test_transposed_keyword(self): + A = np.arange(9).reshape(3, 3) + 1 + x = solve(np.tril(A)/9, np.ones(3), transposed=True) + assert_array_almost_equal(x, [1.2, 0.2, 1]) + x = solve(np.tril(A)/9, np.ones(3), transposed=False) + assert_array_almost_equal(x, [9, -5.4, -1.2]) + + def test_transposed_notimplemented(self): + a = np.eye(3).astype(complex) + with assert_raises(NotImplementedError): + solve(a, a, transposed=True) + + def test_nonsquare_a(self): + assert_raises(ValueError, solve, [1, 2], 1) + + def test_size_mismatch_with_1D_b(self): + assert_array_almost_equal(solve(np.eye(3), np.ones(3)), np.ones(3)) + assert_raises(ValueError, solve, np.eye(3), np.ones(4)) + + def test_assume_a_keyword(self): + assert_raises(ValueError, solve, 1, 1, assume_a='zxcv') + + @pytest.mark.skip(reason="Failure on OS X (gh-7500), " + "crash on Windows (gh-8064)") + def test_all_type_size_routine_combinations(self): + sizes = [10, 100] + assume_as = ['gen', 'sym', 'pos', 'her'] + dtypes = [np.float32, np.float64, np.complex64, np.complex128] + for size, assume_a, dtype in itertools.product(sizes, assume_as, + dtypes): + is_complex = dtype in (np.complex64, np.complex128) + if assume_a == 'her' and not is_complex: + continue + + err_msg = (f"Failed for size: {size}, assume_a: {assume_a}," + f"dtype: {dtype}") + + a = np.random.randn(size, size).astype(dtype) + b = np.random.randn(size).astype(dtype) + if is_complex: + a = a + (1j*np.random.randn(size, size)).astype(dtype) + + if assume_a == 'sym': # Can still be complex but only symmetric + a = a + a.T + elif assume_a == 'her': # Handle hermitian matrices here instead + a = a + a.T.conj() + elif assume_a == 'pos': + a = a.conj().T.dot(a) + 0.1*np.eye(size) + + tol = 1e-12 if dtype in (np.float64, np.complex128) else 1e-6 + + if assume_a in ['gen', 'sym', 'her']: + # We revert the tolerance from before + # 4b4a6e7c34fa4060533db38f9a819b98fa81476c + if dtype in (np.float32, np.complex64): + tol *= 10 + + x = solve(a, b, assume_a=assume_a) + assert_allclose(a.dot(x), b, + atol=tol * size, + rtol=tol * size, + err_msg=err_msg) + + if assume_a == 'sym' and dtype not in (np.complex64, + np.complex128): + x = solve(a, b, assume_a=assume_a, transposed=True) + assert_allclose(a.dot(x), b, + atol=tol * size, + rtol=tol * size, + err_msg=err_msg) + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize('dt_a', [int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt_a, dt_b): + a = np.empty((0, 0), dtype=dt_a) + b = np.empty(0, dtype=dt_b) + x = solve(a, b) + + assert x.size == 0 + dt_nonempty = solve(np.eye(2, dtype=dt_a), np.ones(2, dtype=dt_b)).dtype + assert x.dtype == dt_nonempty + + def test_empty_rhs(self): + a = np.eye(2) + b = [[], []] + x = solve(a, b) + assert_(x.size == 0, 'Returned array is not empty') + assert_(x.shape == (2, 0), 'Returned empty array shape is wrong') + + @pytest.mark.parametrize('dtype', [np.float64, np.complex128]) + # "pos" and "positive definite" need to be added + @pytest.mark.parametrize('assume_a', ['diagonal', 'tridiagonal', 'banded', + 'lower triangular', 'upper triangular', + 'symmetric', 'hermitian', + 'general', 'sym', 'her', 'gen']) + @pytest.mark.parametrize('nrhs', [(), (5,)]) + @pytest.mark.parametrize('transposed', [True, False]) + @pytest.mark.parametrize('overwrite', [True, False]) + @pytest.mark.parametrize('fortran', [True, False]) + def test_structure_detection(self, dtype, assume_a, nrhs, transposed, + overwrite, fortran): + rng = np.random.default_rng(982345982439826) + n = 5 if not assume_a == 'banded' else 20 + b = rng.random(size=(n,) + nrhs) + A = rng.random(size=(n, n)) + + if np.issubdtype(dtype, np.complexfloating): + b = b + rng.random(size=(n,) + nrhs) * 1j + A = A + rng.random(size=(n, n)) * 1j + + if assume_a == 'diagonal': + A = np.diag(np.diag(A)) + elif assume_a == 'lower triangular': + A = np.tril(A) + elif assume_a == 'upper triangular': + A = np.triu(A) + elif assume_a == 'tridiagonal': + A = (np.diag(np.diag(A)) + + np.diag(np.diag(A, -1), -1) + + np.diag(np.diag(A, 1), 1)) + elif assume_a == 'banded': + A = np.triu(np.tril(A, 2), -1) + elif assume_a in {'symmetric', 'sym'}: + A = A + A.T + elif assume_a in {'hermitian', 'her'}: + A = A + A.conj().T + elif assume_a in {'positive definite', 'pos'}: + A = A + A.T + A += np.diag(A.sum(axis=1)) + + if fortran: + A = np.asfortranarray(A) + + A_copy = A.copy(order='A') + b_copy = b.copy() + + if np.issubdtype(dtype, np.complexfloating) and transposed: + message = "scipy.linalg.solve can currently..." + with pytest.raises(NotImplementedError, match=message): + solve(A, b, overwrite_a=overwrite, overwrite_b=overwrite, + transposed=transposed) + return + + res = solve(A, b, overwrite_a=overwrite, overwrite_b=overwrite, + transposed=transposed, assume_a=assume_a) + + # Check that solution this solution is *correct* + ref = np.linalg.solve(A_copy.T if transposed else A_copy, b_copy) + assert_allclose(res, ref) + + # Check that `solve` correctly identifies the structure and returns + # *exactly* the same solution whether `assume_a` is specified or not + if assume_a != 'banded': # structure detection removed for banded + assert_equal(solve(A_copy, b_copy, transposed=transposed), res) + + # Check that overwrite was respected + if not overwrite: + assert_equal(A, A_copy) + assert_equal(b, b_copy) + + +class TestSolveTriangular: + + def test_simple(self): + """ + solve_triangular on a simple 2x2 matrix. + """ + A = array([[1, 0], [1, 2]]) + b = [1, 1] + sol = solve_triangular(A, b, lower=True) + assert_array_almost_equal(sol, [1, 0]) + + # check that it works also for non-contiguous matrices + sol = solve_triangular(A.T, b, lower=False) + assert_array_almost_equal(sol, [.5, .5]) + + # and that it gives the same result as trans=1 + sol = solve_triangular(A, b, lower=True, trans=1) + assert_array_almost_equal(sol, [.5, .5]) + + b = identity(2) + sol = solve_triangular(A, b, lower=True, trans=1) + assert_array_almost_equal(sol, [[1., -.5], [0, 0.5]]) + + def test_simple_complex(self): + """ + solve_triangular on a simple 2x2 complex matrix + """ + A = array([[1+1j, 0], [1j, 2]]) + b = identity(2) + sol = solve_triangular(A, b, lower=True, trans=1) + assert_array_almost_equal(sol, [[.5-.5j, -.25-.25j], [0, 0.5]]) + + # check other option combinations with complex rhs + b = np.diag([1+1j, 1+2j]) + sol = solve_triangular(A, b, lower=True, trans=0) + assert_array_almost_equal(sol, [[1, 0], [-0.5j, 0.5+1j]]) + + sol = solve_triangular(A, b, lower=True, trans=1) + assert_array_almost_equal(sol, [[1, 0.25-0.75j], [0, 0.5+1j]]) + + sol = solve_triangular(A, b, lower=True, trans=2) + assert_array_almost_equal(sol, [[1j, -0.75-0.25j], [0, 0.5+1j]]) + + sol = solve_triangular(A.T, b, lower=False, trans=0) + assert_array_almost_equal(sol, [[1, 0.25-0.75j], [0, 0.5+1j]]) + + sol = solve_triangular(A.T, b, lower=False, trans=1) + assert_array_almost_equal(sol, [[1, 0], [-0.5j, 0.5+1j]]) + + sol = solve_triangular(A.T, b, lower=False, trans=2) + assert_array_almost_equal(sol, [[1j, 0], [-0.5, 0.5+1j]]) + + def test_check_finite(self): + """ + solve_triangular on a simple 2x2 matrix. + """ + A = array([[1, 0], [1, 2]]) + b = [1, 1] + sol = solve_triangular(A, b, lower=True, check_finite=False) + assert_array_almost_equal(sol, [1, 0]) + + @pytest.mark.parametrize('dt_a', [int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt_a, dt_b): + a = np.empty((0, 0), dtype=dt_a) + b = np.empty(0, dtype=dt_b) + x = solve_triangular(a, b) + + assert x.size == 0 + dt_nonempty = solve_triangular( + np.eye(2, dtype=dt_a), np.ones(2, dtype=dt_b) + ).dtype + assert x.dtype == dt_nonempty + + def test_empty_rhs(self): + a = np.eye(2) + b = [[], []] + x = solve_triangular(a, b) + assert_(x.size == 0, 'Returned array is not empty') + assert_(x.shape == (2, 0), 'Returned empty array shape is wrong') + + +class TestInv: + def setup_method(self): + np.random.seed(1234) + + def test_simple(self): + a = [[1, 2], [3, 4]] + a_inv = inv(a) + assert_array_almost_equal(dot(a, a_inv), np.eye(2)) + a = [[1, 2, 3], [4, 5, 6], [7, 8, 10]] + a_inv = inv(a) + assert_array_almost_equal(dot(a, a_inv), np.eye(3)) + + def test_random(self): + n = 20 + for i in range(4): + a = random([n, n]) + for i in range(n): + a[i, i] = 20*(.1+a[i, i]) + a_inv = inv(a) + assert_array_almost_equal(dot(a, a_inv), + identity(n)) + + def test_simple_complex(self): + a = [[1, 2], [3, 4j]] + a_inv = inv(a) + assert_array_almost_equal(dot(a, a_inv), [[1, 0], [0, 1]]) + + def test_random_complex(self): + n = 20 + for i in range(4): + a = random([n, n])+2j*random([n, n]) + for i in range(n): + a[i, i] = 20*(.1+a[i, i]) + a_inv = inv(a) + assert_array_almost_equal(dot(a, a_inv), + identity(n)) + + def test_check_finite(self): + a = [[1, 2], [3, 4]] + a_inv = inv(a, check_finite=False) + assert_array_almost_equal(dot(a, a_inv), [[1, 0], [0, 1]]) + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + a_inv = inv(a) + assert a_inv.size == 0 + assert a_inv.dtype == inv(np.eye(2, dtype=dt)).dtype + + +class TestDet: + def setup_method(self): + self.rng = np.random.default_rng(1680305949878959) + + def test_1x1_all_singleton_dims(self): + a = np.array([[1]]) + deta = det(a) + assert deta.dtype.char == 'd' + assert np.isscalar(deta) + assert deta == 1. + a = np.array([[[[1]]]], dtype='f') + deta = det(a) + assert deta.dtype.char == 'd' + assert deta.shape == (1, 1) + assert_equal(deta, [[1.0]]) + a = np.array([[[1 + 3.j]]], dtype=np.complex64) + deta = det(a) + assert deta.dtype.char == 'D' + assert deta.shape == (1,) + assert_equal(deta, [1.+3.j]) + + def test_1by1_stacked_input_output(self): + a = self.rng.random([4, 5, 1, 1], dtype=np.float32) + deta = det(a) + assert deta.dtype.char == 'd' + assert deta.shape == (4, 5) + assert_allclose(deta, np.squeeze(a)) + + a = self.rng.random([4, 5, 1, 1], dtype=np.float32)*np.complex64(1.j) + deta = det(a) + assert deta.dtype.char == 'D' + assert deta.shape == (4, 5) + assert_allclose(deta, np.squeeze(a)) + + @pytest.mark.parametrize('shape', [[2, 2], [20, 20], [3, 2, 20, 20]]) + def test_simple_det_shapes_real_complex(self, shape): + a = self.rng.uniform(-1., 1., size=shape) + d1, d2 = det(a), np.linalg.det(a) + assert_allclose(d1, d2) + + b = self.rng.uniform(-1., 1., size=shape)*1j + b += self.rng.uniform(-0.5, 0.5, size=shape) + d3, d4 = det(b), np.linalg.det(b) + assert_allclose(d3, d4) + + def test_for_known_det_values(self): + # Hadamard8 + a = np.array([[1, 1, 1, 1, 1, 1, 1, 1], + [1, -1, 1, -1, 1, -1, 1, -1], + [1, 1, -1, -1, 1, 1, -1, -1], + [1, -1, -1, 1, 1, -1, -1, 1], + [1, 1, 1, 1, -1, -1, -1, -1], + [1, -1, 1, -1, -1, 1, -1, 1], + [1, 1, -1, -1, -1, -1, 1, 1], + [1, -1, -1, 1, -1, 1, 1, -1]]) + assert_allclose(det(a), 4096.) + + # consecutive number array always singular + assert_allclose(det(np.arange(25).reshape(5, 5)), 0.) + + # simple anti-diagonal block array + # Upper right has det (-2+1j) and lower right has (-2-1j) + # det(a) = - (-2+1j) (-2-1j) = 5. + a = np.array([[0.+0.j, 0.+0.j, 0.-1.j, 1.-1.j], + [0.+0.j, 0.+0.j, 1.+0.j, 0.-1.j], + [0.+1.j, 1.+1.j, 0.+0.j, 0.+0.j], + [1.+0.j, 0.+1.j, 0.+0.j, 0.+0.j]], dtype=np.complex64) + assert_allclose(det(a), 5.+0.j) + + # Fiedler companion complexified + # >>> a = scipy.linalg.fiedler_companion(np.arange(1, 10)) + a = np.array([[-2., -3., 1., 0., 0., 0., 0., 0.], + [1., 0., 0., 0., 0., 0., 0., 0.], + [0., -4., 0., -5., 1., 0., 0., 0.], + [0., 1., 0., 0., 0., 0., 0., 0.], + [0., 0., 0., -6., 0., -7., 1., 0.], + [0., 0., 0., 1., 0., 0., 0., 0.], + [0., 0., 0., 0., 0., -8., 0., -9.], + [0., 0., 0., 0., 0., 1., 0., 0.]])*1.j + assert_allclose(det(a), 9.) + + # g and G dtypes are handled differently in windows and other platforms + @pytest.mark.parametrize('typ', [x for x in np.typecodes['All'][:20] + if x not in 'gG']) + def test_sample_compatible_dtype_input(self, typ): + n = 4 + a = self.rng.random([n, n]).astype(typ) # value is not important + assert isinstance(det(a), (np.float64 | np.complex128)) + + def test_incompatible_dtype_input(self): + # Double backslashes needed for escaping pytest regex. + msg = 'cannot be cast to float\\(32, 64\\)' + + for c, t in zip('SUO', ['bytes8', 'str32', 'object']): + with assert_raises(TypeError, match=msg): + det(np.array([['a', 'b']]*2, dtype=c)) + with assert_raises(TypeError, match=msg): + det(np.array([[b'a', b'b']]*2, dtype='V')) + with assert_raises(TypeError, match=msg): + det(np.array([[100, 200]]*2, dtype='datetime64[s]')) + with assert_raises(TypeError, match=msg): + det(np.array([[100, 200]]*2, dtype='timedelta64[s]')) + + def test_empty_edge_cases(self): + assert_allclose(det(np.empty([0, 0])), 1.) + assert_allclose(det(np.empty([0, 0, 0])), np.array([])) + assert_allclose(det(np.empty([3, 0, 0])), np.array([1., 1., 1.])) + with assert_raises(ValueError, match='Last 2 dimensions'): + det(np.empty([0, 0, 3])) + with assert_raises(ValueError, match='at least two-dimensional'): + det(np.array([])) + with assert_raises(ValueError, match='Last 2 dimensions'): + det(np.array([[]])) + with assert_raises(ValueError, match='Last 2 dimensions'): + det(np.array([[[]]])) + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty_dtype(self, dt): + a = np.empty((0, 0), dtype=dt) + d = det(a) + assert d.shape == () + assert d.dtype == det(np.eye(2, dtype=dt)).dtype + + a = np.empty((3, 0, 0), dtype=dt) + d = det(a) + assert d.shape == (3,) + assert d.dtype == det(np.zeros((3, 1, 1), dtype=dt)).dtype + + def test_overwrite_a(self): + # If all conditions are met then input should be overwritten; + # - dtype is one of 'fdFD' + # - C-contiguous + # - writeable + a = np.arange(9).reshape(3, 3).astype(np.float32) + ac = a.copy() + deta = det(ac, overwrite_a=True) + assert_allclose(deta, 0.) + assert not (a == ac).all() + + def test_readonly_array(self): + a = np.array([[2., 0., 1.], [5., 3., -1.], [1., 1., 1.]]) + a.setflags(write=False) + # overwrite_a will be overridden + assert_allclose(det(a, overwrite_a=True), 10.) + + def test_simple_check_finite(self): + a = [[1, 2], [3, np.inf]] + with assert_raises(ValueError, match='array must not contain'): + det(a) + + +def direct_lstsq(a, b, cmplx=0): + at = transpose(a) + if cmplx: + at = conjugate(at) + a1 = dot(at, a) + b1 = dot(at, b) + return solve(a1, b1) + + +class TestLstsq: + lapack_drivers = ('gelsd', 'gelss', 'gelsy', None) + + def test_simple_exact(self): + for dtype in REAL_DTYPES: + a = np.array([[1, 20], [-30, 4]], dtype=dtype) + for lapack_driver in TestLstsq.lapack_drivers: + for overwrite in (True, False): + for bt in (((1, 0), (0, 1)), (1, 0), + ((2, 1), (-30, 4))): + # Store values in case they are overwritten + # later + a1 = a.copy() + b = np.array(bt, dtype=dtype) + b1 = b.copy() + out = lstsq(a1, b1, + lapack_driver=lapack_driver, + overwrite_a=overwrite, + overwrite_b=overwrite) + x = out[0] + r = out[2] + assert_(r == 2, + f'expected efficient rank 2, got {r}') + assert_allclose(dot(a, x), b, + atol=25 * _eps_cast(a1.dtype), + rtol=25 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + + def test_simple_overdet(self): + for dtype in REAL_DTYPES: + a = np.array([[1, 2], [4, 5], [3, 4]], dtype=dtype) + b = np.array([1, 2, 3], dtype=dtype) + for lapack_driver in TestLstsq.lapack_drivers: + for overwrite in (True, False): + # Store values in case they are overwritten later + a1 = a.copy() + b1 = b.copy() + out = lstsq(a1, b1, lapack_driver=lapack_driver, + overwrite_a=overwrite, + overwrite_b=overwrite) + x = out[0] + if lapack_driver == 'gelsy': + residuals = np.sum((b - a.dot(x))**2) + else: + residuals = out[1] + r = out[2] + assert_(r == 2, f'expected efficient rank 2, got {r}') + assert_allclose(abs((dot(a, x) - b)**2).sum(axis=0), + residuals, + rtol=25 * _eps_cast(a1.dtype), + atol=25 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + assert_allclose(x, (-0.428571428571429, 0.85714285714285), + rtol=25 * _eps_cast(a1.dtype), + atol=25 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + + def test_simple_overdet_complex(self): + for dtype in COMPLEX_DTYPES: + a = np.array([[1+2j, 2], [4, 5], [3, 4]], dtype=dtype) + b = np.array([1, 2+4j, 3], dtype=dtype) + for lapack_driver in TestLstsq.lapack_drivers: + for overwrite in (True, False): + # Store values in case they are overwritten later + a1 = a.copy() + b1 = b.copy() + out = lstsq(a1, b1, lapack_driver=lapack_driver, + overwrite_a=overwrite, + overwrite_b=overwrite) + + x = out[0] + if lapack_driver == 'gelsy': + res = b - a.dot(x) + residuals = np.sum(res * res.conj()) + else: + residuals = out[1] + r = out[2] + assert_(r == 2, f'expected efficient rank 2, got {r}') + assert_allclose(abs((dot(a, x) - b)**2).sum(axis=0), + residuals, + rtol=25 * _eps_cast(a1.dtype), + atol=25 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + assert_allclose( + x, (-0.4831460674157303 + 0.258426966292135j, + 0.921348314606741 + 0.292134831460674j), + rtol=25 * _eps_cast(a1.dtype), + atol=25 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + + def test_simple_underdet(self): + for dtype in REAL_DTYPES: + a = np.array([[1, 2, 3], [4, 5, 6]], dtype=dtype) + b = np.array([1, 2], dtype=dtype) + for lapack_driver in TestLstsq.lapack_drivers: + for overwrite in (True, False): + # Store values in case they are overwritten later + a1 = a.copy() + b1 = b.copy() + out = lstsq(a1, b1, lapack_driver=lapack_driver, + overwrite_a=overwrite, + overwrite_b=overwrite) + + x = out[0] + r = out[2] + assert_(r == 2, f'expected efficient rank 2, got {r}') + assert_allclose(x, (-0.055555555555555, 0.111111111111111, + 0.277777777777777), + rtol=25 * _eps_cast(a1.dtype), + atol=25 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + + @pytest.mark.parametrize("dtype", REAL_DTYPES) + @pytest.mark.parametrize("n", (20, 200)) + @pytest.mark.parametrize("lapack_driver", lapack_drivers) + @pytest.mark.parametrize("overwrite", (True, False)) + def test_random_exact(self, dtype, n, lapack_driver, overwrite): + rng = np.random.RandomState(1234) + + a = np.asarray(rng.random([n, n]), dtype=dtype) + for i in range(n): + a[i, i] = 20 * (0.1 + a[i, i]) + for i in range(4): + b = np.asarray(rng.random([n, 3]), dtype=dtype) + # Store values in case they are overwritten later + a1 = a.copy() + b1 = b.copy() + out = lstsq(a1, b1, + lapack_driver=lapack_driver, + overwrite_a=overwrite, + overwrite_b=overwrite) + x = out[0] + r = out[2] + assert_(r == n, f'expected efficient rank {n}, ' + f'got {r}') + if dtype is np.float32: + assert_allclose( + dot(a, x), b, + rtol=500 * _eps_cast(a1.dtype), + atol=500 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + else: + assert_allclose( + dot(a, x), b, + rtol=1000 * _eps_cast(a1.dtype), + atol=1000 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + + @pytest.mark.skipif(IS_MUSL, reason="may segfault on Alpine, see gh-17630") + @pytest.mark.parametrize("dtype", COMPLEX_DTYPES) + @pytest.mark.parametrize("n", (20, 200)) + @pytest.mark.parametrize("lapack_driver", lapack_drivers) + @pytest.mark.parametrize("overwrite", (True, False)) + def test_random_complex_exact(self, dtype, n, lapack_driver, overwrite): + rng = np.random.RandomState(1234) + + a = np.asarray(rng.random([n, n]) + 1j*rng.random([n, n]), + dtype=dtype) + for i in range(n): + a[i, i] = 20 * (0.1 + a[i, i]) + for i in range(2): + b = np.asarray(rng.random([n, 3]), dtype=dtype) + # Store values in case they are overwritten later + a1 = a.copy() + b1 = b.copy() + out = lstsq(a1, b1, lapack_driver=lapack_driver, + overwrite_a=overwrite, + overwrite_b=overwrite) + x = out[0] + r = out[2] + assert_(r == n, f'expected efficient rank {n}, ' + f'got {r}') + if dtype is np.complex64: + assert_allclose( + dot(a, x), b, + rtol=400 * _eps_cast(a1.dtype), + atol=400 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + else: + assert_allclose( + dot(a, x), b, + rtol=1000 * _eps_cast(a1.dtype), + atol=1000 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + + def test_random_overdet(self): + rng = np.random.RandomState(1234) + for dtype in REAL_DTYPES: + for (n, m) in ((20, 15), (200, 2)): + for lapack_driver in TestLstsq.lapack_drivers: + for overwrite in (True, False): + a = np.asarray(rng.random([n, m]), dtype=dtype) + for i in range(m): + a[i, i] = 20 * (0.1 + a[i, i]) + for i in range(4): + b = np.asarray(rng.random([n, 3]), dtype=dtype) + # Store values in case they are overwritten later + a1 = a.copy() + b1 = b.copy() + out = lstsq(a1, b1, + lapack_driver=lapack_driver, + overwrite_a=overwrite, + overwrite_b=overwrite) + x = out[0] + r = out[2] + assert_(r == m, f'expected efficient rank {m}, ' + f'got {r}') + assert_allclose( + x, direct_lstsq(a, b, cmplx=0), + rtol=25 * _eps_cast(a1.dtype), + atol=25 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + + def test_random_complex_overdet(self): + rng = np.random.RandomState(1234) + for dtype in COMPLEX_DTYPES: + for (n, m) in ((20, 15), (200, 2)): + for lapack_driver in TestLstsq.lapack_drivers: + for overwrite in (True, False): + a = np.asarray(rng.random([n, m]) + 1j*rng.random([n, m]), + dtype=dtype) + for i in range(m): + a[i, i] = 20 * (0.1 + a[i, i]) + for i in range(2): + b = np.asarray(rng.random([n, 3]), dtype=dtype) + # Store values in case they are overwritten + # later + a1 = a.copy() + b1 = b.copy() + out = lstsq(a1, b1, + lapack_driver=lapack_driver, + overwrite_a=overwrite, + overwrite_b=overwrite) + x = out[0] + r = out[2] + assert_(r == m, f'expected efficient rank {m}, ' + f'got {r}') + assert_allclose( + x, direct_lstsq(a, b, cmplx=1), + rtol=25 * _eps_cast(a1.dtype), + atol=25 * _eps_cast(a1.dtype), + err_msg=f"driver: {lapack_driver}") + + def test_check_finite(self): + with suppress_warnings() as sup: + # On (some) OSX this tests triggers a warning (gh-7538) + sup.filter(RuntimeWarning, + "internal gelsd driver lwork query error,.*" + "Falling back to 'gelss' driver.") + + at = np.array(((1, 20), (-30, 4))) + for dtype, bt, lapack_driver, overwrite, check_finite in \ + itertools.product(REAL_DTYPES, + (((1, 0), (0, 1)), (1, 0), ((2, 1), (-30, 4))), + TestLstsq.lapack_drivers, + (True, False), + (True, False)): + + a = at.astype(dtype) + b = np.array(bt, dtype=dtype) + # Store values in case they are overwritten + # later + a1 = a.copy() + b1 = b.copy() + out = lstsq(a1, b1, lapack_driver=lapack_driver, + check_finite=check_finite, overwrite_a=overwrite, + overwrite_b=overwrite) + x = out[0] + r = out[2] + assert_(r == 2, f'expected efficient rank 2, got {r}') + assert_allclose(dot(a, x), b, + rtol=25 * _eps_cast(a.dtype), + atol=25 * _eps_cast(a.dtype), + err_msg=f"driver: {lapack_driver}") + + def test_empty(self): + for a_shape, b_shape in (((0, 2), (0,)), + ((0, 4), (0, 2)), + ((4, 0), (4,)), + ((4, 0), (4, 2))): + b = np.ones(b_shape) + x, residues, rank, s = lstsq(np.zeros(a_shape), b) + assert_equal(x, np.zeros((a_shape[1],) + b_shape[1:])) + residues_should_be = (np.empty((0,)) if a_shape[1] + else np.linalg.norm(b, axis=0)**2) + assert_equal(residues, residues_should_be) + assert_(rank == 0, 'expected rank 0') + assert_equal(s, np.empty((0,))) + + @pytest.mark.parametrize('dt_a', [int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64]) + def test_empty_dtype(self, dt_a, dt_b): + a = np.empty((0, 0), dtype=dt_a) + b = np.empty(0, dtype=dt_b) + x, residues, rank, s = lstsq(a, b) + + assert x.size == 0 + dt_nonempty = lstsq(np.eye(2, dtype=dt_a), np.ones(2, dtype=dt_b))[0].dtype + assert x.dtype == dt_nonempty + + +class TestPinv: + def setup_method(self): + np.random.seed(1234) + + def test_simple_real(self): + a = array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], dtype=float) + a_pinv = pinv(a) + assert_array_almost_equal(dot(a, a_pinv), np.eye(3)) + + def test_simple_complex(self): + a = (array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], + dtype=float) + 1j * array([[10, 8, 7], [6, 5, 4], [3, 2, 1]], + dtype=float)) + a_pinv = pinv(a) + assert_array_almost_equal(dot(a, a_pinv), np.eye(3)) + + def test_simple_singular(self): + a = array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=float) + a_pinv = pinv(a) + expected = array([[-6.38888889e-01, -1.66666667e-01, 3.05555556e-01], + [-5.55555556e-02, 1.30136518e-16, 5.55555556e-02], + [5.27777778e-01, 1.66666667e-01, -1.94444444e-01]]) + assert_array_almost_equal(a_pinv, expected) + + def test_simple_cols(self): + a = array([[1, 2, 3], [4, 5, 6]], dtype=float) + a_pinv = pinv(a) + expected = array([[-0.94444444, 0.44444444], + [-0.11111111, 0.11111111], + [0.72222222, -0.22222222]]) + assert_array_almost_equal(a_pinv, expected) + + def test_simple_rows(self): + a = array([[1, 2], [3, 4], [5, 6]], dtype=float) + a_pinv = pinv(a) + expected = array([[-1.33333333, -0.33333333, 0.66666667], + [1.08333333, 0.33333333, -0.41666667]]) + assert_array_almost_equal(a_pinv, expected) + + def test_check_finite(self): + a = array([[1, 2, 3], [4, 5, 6.], [7, 8, 10]]) + a_pinv = pinv(a, check_finite=False) + assert_array_almost_equal(dot(a, a_pinv), np.eye(3)) + + def test_native_list_argument(self): + a = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] + a_pinv = pinv(a) + expected = array([[-6.38888889e-01, -1.66666667e-01, 3.05555556e-01], + [-5.55555556e-02, 1.30136518e-16, 5.55555556e-02], + [5.27777778e-01, 1.66666667e-01, -1.94444444e-01]]) + assert_array_almost_equal(a_pinv, expected) + + def test_atol_rtol(self): + n = 12 + # get a random ortho matrix for shuffling + q, _ = qr(np.random.rand(n, n)) + a_m = np.arange(35.0).reshape(7, 5) + a = a_m.copy() + a[0, 0] = 0.001 + atol = 1e-5 + rtol = 0.05 + # svds of a_m is ~ [116.906, 4.234, tiny, tiny, tiny] + # svds of a is ~ [116.906, 4.234, 4.62959e-04, tiny, tiny] + # Just abs cutoff such that we arrive at a_modified + a_p = pinv(a_m, atol=atol, rtol=0.) + adiff1 = a @ a_p @ a - a + adiff2 = a_m @ a_p @ a_m - a_m + # Now adiff1 should be around atol value while adiff2 should be + # relatively tiny + assert_allclose(np.linalg.norm(adiff1), 5e-4, atol=5.e-4) + assert_allclose(np.linalg.norm(adiff2), 5e-14, atol=5.e-14) + + # Now do the same but remove another sv ~4.234 via rtol + a_p = pinv(a_m, atol=atol, rtol=rtol) + adiff1 = a @ a_p @ a - a + adiff2 = a_m @ a_p @ a_m - a_m + assert_allclose(np.linalg.norm(adiff1), 4.233, rtol=0.01) + assert_allclose(np.linalg.norm(adiff2), 4.233, rtol=0.01) + + @pytest.mark.parametrize('dt', [float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + a_pinv = pinv(a) + assert a_pinv.size == 0 + assert a_pinv.dtype == pinv(np.eye(2, dtype=dt)).dtype + + +class TestPinvSymmetric: + + def setup_method(self): + np.random.seed(1234) + + def test_simple_real(self): + a = array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], dtype=float) + a = np.dot(a, a.T) + a_pinv = pinvh(a) + assert_array_almost_equal(np.dot(a, a_pinv), np.eye(3)) + + def test_nonpositive(self): + a = array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=float) + a = np.dot(a, a.T) + u, s, vt = np.linalg.svd(a) + s[0] *= -1 + a = np.dot(u * s, vt) # a is now symmetric non-positive and singular + a_pinv = pinv(a) + a_pinvh = pinvh(a) + assert_array_almost_equal(a_pinv, a_pinvh) + + def test_simple_complex(self): + a = (array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], + dtype=float) + 1j * array([[10, 8, 7], [6, 5, 4], [3, 2, 1]], + dtype=float)) + a = np.dot(a, a.conj().T) + a_pinv = pinvh(a) + assert_array_almost_equal(np.dot(a, a_pinv), np.eye(3)) + + def test_native_list_argument(self): + a = array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], dtype=float) + a = np.dot(a, a.T) + a_pinv = pinvh(a.tolist()) + assert_array_almost_equal(np.dot(a, a_pinv), np.eye(3)) + + def test_zero_eigenvalue(self): + # https://github.com/scipy/scipy/issues/12515 + # the SYEVR eigh driver may give the zero eigenvalue > eps + a = np.array([[1, -1, 0], [-1, 2, -1], [0, -1, 1]]) + p = pinvh(a) + assert_allclose(p @ a @ p, p, atol=1e-15) + assert_allclose(a @ p @ a, a, atol=1e-15) + + def test_atol_rtol(self): + n = 12 + # get a random ortho matrix for shuffling + q, _ = qr(np.random.rand(n, n)) + a = np.diag([4, 3, 2, 1, 0.99e-4, 0.99e-5] + [0.99e-6]*(n-6)) + a = q.T @ a @ q + a_m = np.diag([4, 3, 2, 1, 0.99e-4, 0.] + [0.]*(n-6)) + a_m = q.T @ a_m @ q + atol = 1e-5 + rtol = (4.01e-4 - 4e-5)/4 + # Just abs cutoff such that we arrive at a_modified + a_p = pinvh(a, atol=atol, rtol=0.) + adiff1 = a @ a_p @ a - a + adiff2 = a_m @ a_p @ a_m - a_m + # Now adiff1 should dance around atol value since truncation + # while adiff2 should be relatively tiny + assert_allclose(norm(adiff1), atol, rtol=0.1) + assert_allclose(norm(adiff2), 1e-12, atol=1e-11) + + # Now do the same but through rtol cancelling atol value + a_p = pinvh(a, atol=atol, rtol=rtol) + adiff1 = a @ a_p @ a - a + adiff2 = a_m @ a_p @ a_m - a_m + # adiff1 and adiff2 should be elevated to ~1e-4 due to mismatch + assert_allclose(norm(adiff1), 1e-4, rtol=0.1) + assert_allclose(norm(adiff2), 1e-4, rtol=0.1) + + @pytest.mark.parametrize('dt', [float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + a_pinv = pinvh(a) + assert a_pinv.size == 0 + assert a_pinv.dtype == pinv(np.eye(2, dtype=dt)).dtype + + +@pytest.mark.parametrize('scale', (1e-20, 1., 1e20)) +@pytest.mark.parametrize('pinv_', (pinv, pinvh)) +def test_auto_rcond(scale, pinv_): + x = np.array([[1, 0], [0, 1e-10]]) * scale + expected = np.diag(1. / np.diag(x)) + x_inv = pinv_(x) + assert_allclose(x_inv, expected) + + +class TestVectorNorms: + + def test_types(self): + for dtype in np.typecodes['AllFloat']: + x = np.array([1, 2, 3], dtype=dtype) + tol = max(1e-15, np.finfo(dtype).eps.real * 20) + assert_allclose(norm(x), np.sqrt(14), rtol=tol) + assert_allclose(norm(x, 2), np.sqrt(14), rtol=tol) + + for dtype in np.typecodes['Complex']: + x = np.array([1j, 2j, 3j], dtype=dtype) + tol = max(1e-15, np.finfo(dtype).eps.real * 20) + assert_allclose(norm(x), np.sqrt(14), rtol=tol) + assert_allclose(norm(x, 2), np.sqrt(14), rtol=tol) + + def test_overflow(self): + # unlike numpy's norm, this one is + # safer on overflow + a = array([1e20], dtype=float32) + assert_almost_equal(norm(a), a) + + def test_stable(self): + # more stable than numpy's norm + a = array([1e4] + [1]*10000, dtype=float32) + try: + # snrm in double precision; we obtain the same as for float64 + # -- large atol needed due to varying blas implementations + assert_allclose(norm(a) - 1e4, 0.5, atol=1e-2) + except AssertionError: + # snrm implemented in single precision, == np.linalg.norm result + msg = ": Result should equal either 0.0 or 0.5 (depending on " \ + "implementation of snrm2)." + assert_almost_equal(norm(a) - 1e4, 0.0, err_msg=msg) + + def test_zero_norm(self): + assert_equal(norm([1, 0, 3], 0), 2) + assert_equal(norm([1, 2, 3], 0), 3) + + def test_axis_kwd(self): + a = np.array([[[2, 1], [3, 4]]] * 2, 'd') + assert_allclose(norm(a, axis=1), [[3.60555128, 4.12310563]] * 2) + assert_allclose(norm(a, 1, axis=1), [[5.] * 2] * 2) + + def test_keepdims_kwd(self): + a = np.array([[[2, 1], [3, 4]]] * 2, 'd') + b = norm(a, axis=1, keepdims=True) + assert_allclose(b, [[[3.60555128, 4.12310563]]] * 2) + assert_(b.shape == (2, 1, 2)) + assert_allclose(norm(a, 1, axis=2, keepdims=True), [[[3.], [7.]]] * 2) + + @pytest.mark.skipif(not HAS_ILP64, reason="64-bit BLAS required") + def test_large_vector(self): + check_free_memory(free_mb=17000) + x = np.zeros([2**31], dtype=np.float64) + x[-1] = 1 + res = norm(x) + del x + assert_allclose(res, 1.0) + + +class TestMatrixNorms: + + def test_matrix_norms(self): + # Not all of these are matrix norms in the most technical sense. + np.random.seed(1234) + for n, m in (1, 1), (1, 3), (3, 1), (4, 4), (4, 5), (5, 4): + for t in np.float32, np.float64, np.complex64, np.complex128, np.int64: + A = 10 * np.random.randn(n, m).astype(t) + if np.issubdtype(A.dtype, np.complexfloating): + A = (A + 10j * np.random.randn(n, m)).astype(t) + t_high = np.complex128 + else: + t_high = np.float64 + for order in (None, 'fro', 1, -1, 2, -2, np.inf, -np.inf): + actual = norm(A, ord=order) + desired = np.linalg.norm(A, ord=order) + # SciPy may return higher precision matrix norms. + # This is a consequence of using LAPACK. + if not np.allclose(actual, desired): + desired = np.linalg.norm(A.astype(t_high), ord=order) + assert_allclose(actual, desired) + + def test_axis_kwd(self): + a = np.array([[[2, 1], [3, 4]]] * 2, 'd') + b = norm(a, ord=np.inf, axis=(1, 0)) + c = norm(np.swapaxes(a, 0, 1), ord=np.inf, axis=(0, 1)) + d = norm(a, ord=1, axis=(0, 1)) + assert_allclose(b, c) + assert_allclose(c, d) + assert_allclose(b, d) + assert_(b.shape == c.shape == d.shape) + b = norm(a, ord=1, axis=(1, 0)) + c = norm(np.swapaxes(a, 0, 1), ord=1, axis=(0, 1)) + d = norm(a, ord=np.inf, axis=(0, 1)) + assert_allclose(b, c) + assert_allclose(c, d) + assert_allclose(b, d) + assert_(b.shape == c.shape == d.shape) + + def test_keepdims_kwd(self): + a = np.arange(120, dtype='d').reshape(2, 3, 4, 5) + b = norm(a, ord=np.inf, axis=(1, 0), keepdims=True) + c = norm(a, ord=1, axis=(0, 1), keepdims=True) + assert_allclose(b, c) + assert_(b.shape == c.shape) + + def test_empty(self): + a = np.empty((0, 0)) + assert_allclose(norm(a), 0.) + assert_allclose(norm(a, axis=0), np.zeros((0,))) + assert_allclose(norm(a, keepdims=True), np.zeros((1, 1))) + + a = np.empty((0, 3)) + assert_allclose(norm(a), 0.) + assert_allclose(norm(a, axis=0), np.zeros((3,))) + assert_allclose(norm(a, keepdims=True), np.zeros((1, 1))) + + +class TestOverwrite: + def test_solve(self): + assert_no_overwrite(solve, [(3, 3), (3,)]) + + def test_solve_triangular(self): + assert_no_overwrite(solve_triangular, [(3, 3), (3,)]) + + def test_solve_banded(self): + assert_no_overwrite(lambda ab, b: solve_banded((2, 1), ab, b), + [(4, 6), (6,)]) + + def test_solveh_banded(self): + assert_no_overwrite(solveh_banded, [(2, 6), (6,)]) + + def test_inv(self): + assert_no_overwrite(inv, [(3, 3)]) + + def test_det(self): + assert_no_overwrite(det, [(3, 3)]) + + def test_lstsq(self): + assert_no_overwrite(lstsq, [(3, 2), (3,)]) + + def test_pinv(self): + assert_no_overwrite(pinv, [(3, 3)]) + + def test_pinvh(self): + assert_no_overwrite(pinvh, [(3, 3)]) + + +class TestSolveCirculant: + + def test_basic1(self): + c = np.array([1, 2, 3, 5]) + b = np.array([1, -1, 1, 0]) + x = solve_circulant(c, b) + y = solve(circulant(c), b) + assert_allclose(x, y) + + def test_basic2(self): + # b is a 2-d matrix. + c = np.array([1, 2, -3, -5]) + b = np.arange(12).reshape(4, 3) + x = solve_circulant(c, b) + y = solve(circulant(c), b) + assert_allclose(x, y) + + def test_basic3(self): + # b is a 3-d matrix. + c = np.array([1, 2, -3, -5]) + b = np.arange(24).reshape(4, 3, 2) + x = solve_circulant(c, b) + y = solve(circulant(c), b) + assert_allclose(x, y) + + def test_complex(self): + # Complex b and c + c = np.array([1+2j, -3, 4j, 5]) + b = np.arange(8).reshape(4, 2) + 0.5j + x = solve_circulant(c, b) + y = solve(circulant(c), b) + assert_allclose(x, y) + + def test_random_b_and_c(self): + # Random b and c + rng = np.random.RandomState(54321) + c = rng.randn(50) + b = rng.randn(50) + x = solve_circulant(c, b) + y = solve(circulant(c), b) + assert_allclose(x, y) + + def test_singular(self): + # c gives a singular circulant matrix. + c = np.array([1, 1, 0, 0]) + b = np.array([1, 2, 3, 4]) + x = solve_circulant(c, b, singular='lstsq') + y, res, rnk, s = lstsq(circulant(c), b) + assert_allclose(x, y) + assert_raises(LinAlgError, solve_circulant, x, y) + + def test_axis_args(self): + # Test use of caxis, baxis and outaxis. + + # c has shape (2, 1, 4) + c = np.array([[[-1, 2.5, 3, 3.5]], [[1, 6, 6, 6.5]]]) + + # b has shape (3, 4) + b = np.array([[0, 0, 1, 1], [1, 1, 0, 0], [1, -1, 0, 0]]) + + x = solve_circulant(c, b, baxis=1) + assert_equal(x.shape, (4, 2, 3)) + expected = np.empty_like(x) + expected[:, 0, :] = solve(circulant(c[0].ravel()), b.T) + expected[:, 1, :] = solve(circulant(c[1].ravel()), b.T) + assert_allclose(x, expected) + + x = solve_circulant(c, b, baxis=1, outaxis=-1) + assert_equal(x.shape, (2, 3, 4)) + assert_allclose(np.moveaxis(x, -1, 0), expected) + + # np.swapaxes(c, 1, 2) has shape (2, 4, 1); b.T has shape (4, 3). + x = solve_circulant(np.swapaxes(c, 1, 2), b.T, caxis=1) + assert_equal(x.shape, (4, 2, 3)) + assert_allclose(x, expected) + + def test_native_list_arguments(self): + # Same as test_basic1 using python's native list. + c = [1, 2, 3, 5] + b = [1, -1, 1, 0] + x = solve_circulant(c, b) + y = solve(circulant(c), b) + assert_allclose(x, y) + + @pytest.mark.parametrize('dt_c', [int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt_c, dt_b): + c = np.array([], dtype=dt_c) + b = np.array([], dtype=dt_b) + x = solve_circulant(c, b) + assert x.shape == (0,) + assert x.dtype == solve_circulant(np.arange(3, dtype=dt_c), + np.ones(3, dtype=dt_b)).dtype + + b = np.empty((0, 0), dtype=dt_b) + x1 = solve_circulant(c, b) + assert x1.shape == (0, 0) + assert x1.dtype == x.dtype + + +class TestMatrix_Balance: + + def test_string_arg(self): + assert_raises(ValueError, matrix_balance, 'Some string for fail') + + def test_infnan_arg(self): + assert_raises(ValueError, matrix_balance, + np.array([[1, 2], [3, np.inf]])) + assert_raises(ValueError, matrix_balance, + np.array([[1, 2], [3, np.nan]])) + + def test_scaling(self): + _, y = matrix_balance(np.array([[1000, 1], [1000, 0]])) + # Pre/post LAPACK 3.5.0 gives the same result up to an offset + # since in each case col norm is x1000 greater and + # 1000 / 32 ~= 1 * 32 hence balanced with 2 ** 5. + assert_allclose(np.diff(np.log2(np.diag(y))), [5]) + + def test_scaling_order(self): + A = np.array([[1, 0, 1e-4], [1, 1, 1e-2], [1e4, 1e2, 1]]) + x, y = matrix_balance(A) + assert_allclose(solve(y, A).dot(y), x) + + def test_separate(self): + _, (y, z) = matrix_balance(np.array([[1000, 1], [1000, 0]]), + separate=1) + assert_equal(np.diff(np.log2(y)), [5]) + assert_allclose(z, np.arange(2)) + + def test_permutation(self): + A = block_diag(np.ones((2, 2)), np.tril(np.ones((2, 2))), + np.ones((3, 3))) + x, (y, z) = matrix_balance(A, separate=1) + assert_allclose(y, np.ones_like(y)) + assert_allclose(z, np.array([0, 1, 6, 5, 4, 3, 2])) + + def test_perm_and_scaling(self): + # Matrix with its diagonal removed + cases = ( # Case 0 + np.array([[0., 0., 0., 0., 0.000002], + [0., 0., 0., 0., 0.], + [2., 2., 0., 0., 0.], + [2., 2., 0., 0., 0.], + [0., 0., 0.000002, 0., 0.]]), + # Case 1 user reported GH-7258 + np.array([[-0.5, 0., 0., 0.], + [0., -1., 0., 0.], + [1., 0., -0.5, 0.], + [0., 1., 0., -1.]]), + # Case 2 user reported GH-7258 + np.array([[-3., 0., 1., 0.], + [-1., -1., -0., 1.], + [-3., -0., -0., 0.], + [-1., -0., 1., -1.]]) + ) + + for A in cases: + x, y = matrix_balance(A) + x, (s, p) = matrix_balance(A, separate=1) + ip = np.empty_like(p) + ip[p] = np.arange(A.shape[0]) + assert_allclose(y, np.diag(s)[ip, :]) + assert_allclose(solve(y, A).dot(y), x) + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + b, t = matrix_balance(a) + + assert b.size == 0 + assert t.size == 0 + + b_n, t_n = matrix_balance(np.eye(2, dtype=dt)) + assert b.dtype == b_n.dtype + assert t.dtype == t_n.dtype + + b, (scale, perm) = matrix_balance(a, separate=True) + assert b.size == 0 + assert scale.size == 0 + assert perm.size == 0 + + b_n, (scale_n, perm_n) = matrix_balance(a, separate=True) + assert b.dtype == b_n.dtype + assert scale.dtype == scale_n.dtype + assert perm.dtype == perm_n.dtype diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_blas.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_blas.py new file mode 100644 index 0000000000000000000000000000000000000000..b6645d0ad5d967ca00a2a5193d51e7ee74b8ad73 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_blas.py @@ -0,0 +1,1127 @@ +# +# Created by: Pearu Peterson, April 2002 +# + +import math +import pytest +import numpy as np +import numpy.random +from numpy.testing import (assert_equal, assert_almost_equal, assert_, + assert_array_almost_equal, assert_allclose) +from pytest import raises as assert_raises + +from numpy import float32, float64, complex64, complex128, arange, triu, \ + tril, zeros, tril_indices, ones, mod, diag, append, eye, \ + nonzero + +import scipy +from scipy.linalg import _fblas as fblas, get_blas_funcs, toeplitz, solve + +try: + from scipy.linalg import _cblas as cblas +except ImportError: + cblas = None + +REAL_DTYPES = [float32, float64] +COMPLEX_DTYPES = [complex64, complex128] +DTYPES = REAL_DTYPES + COMPLEX_DTYPES + + +def test_get_blas_funcs(): + # check that it returns Fortran code for arrays that are + # fortran-ordered + f1, f2, f3 = get_blas_funcs( + ('axpy', 'axpy', 'axpy'), + (np.empty((2, 2), dtype=np.complex64, order='F'), + np.empty((2, 2), dtype=np.complex128, order='C')) + ) + + # get_blas_funcs will choose libraries depending on most generic + # array + assert_equal(f1.typecode, 'z') + assert_equal(f2.typecode, 'z') + if cblas is not None: + assert_equal(f1.module_name, 'cblas') + assert_equal(f2.module_name, 'cblas') + + # check defaults. + f1 = get_blas_funcs('rotg') + assert_equal(f1.typecode, 'd') + + # check also dtype interface + f1 = get_blas_funcs('gemm', dtype=np.complex64) + assert_equal(f1.typecode, 'c') + f1 = get_blas_funcs('gemm', dtype='F') + assert_equal(f1.typecode, 'c') + + # extended precision complex + f1 = get_blas_funcs('gemm', dtype=np.clongdouble) + assert_equal(f1.typecode, 'z') + + # check safe complex upcasting + f1 = get_blas_funcs('axpy', + (np.empty((2, 2), dtype=np.float64), + np.empty((2, 2), dtype=np.complex64)) + ) + assert_equal(f1.typecode, 'z') + + +def test_get_blas_funcs_alias(): + # check alias for get_blas_funcs + f, g = get_blas_funcs(('nrm2', 'dot'), dtype=np.complex64) + assert f.typecode == 'c' + assert g.typecode == 'c' + + f, g, h = get_blas_funcs(('dot', 'dotc', 'dotu'), dtype=np.float64) + assert f is g + assert f is h + + +class TestCBLAS1Simple: + + def test_axpy(self): + for p in 'sd': + f = getattr(cblas, p+'axpy', None) + if f is None: + continue + assert_array_almost_equal(f([1, 2, 3], [2, -1, 3], a=5), + [7, 9, 18]) + for p in 'cz': + f = getattr(cblas, p+'axpy', None) + if f is None: + continue + assert_array_almost_equal(f([1, 2j, 3], [2, -1, 3], a=5), + [7, 10j-1, 18]) + + +class TestFBLAS1Simple: + + def test_axpy(self): + for p in 'sd': + f = getattr(fblas, p+'axpy', None) + if f is None: + continue + assert_array_almost_equal(f([1, 2, 3], [2, -1, 3], a=5), + [7, 9, 18]) + for p in 'cz': + f = getattr(fblas, p+'axpy', None) + if f is None: + continue + assert_array_almost_equal(f([1, 2j, 3], [2, -1, 3], a=5), + [7, 10j-1, 18]) + + def test_copy(self): + for p in 'sd': + f = getattr(fblas, p+'copy', None) + if f is None: + continue + assert_array_almost_equal(f([3, 4, 5], [8]*3), [3, 4, 5]) + for p in 'cz': + f = getattr(fblas, p+'copy', None) + if f is None: + continue + assert_array_almost_equal(f([3, 4j, 5+3j], [8]*3), [3, 4j, 5+3j]) + + def test_asum(self): + for p in 'sd': + f = getattr(fblas, p+'asum', None) + if f is None: + continue + assert_almost_equal(f([3, -4, 5]), 12) + for p in ['sc', 'dz']: + f = getattr(fblas, p+'asum', None) + if f is None: + continue + assert_almost_equal(f([3j, -4, 3-4j]), 14) + + def test_dot(self): + for p in 'sd': + f = getattr(fblas, p+'dot', None) + if f is None: + continue + assert_almost_equal(f([3, -4, 5], [2, 5, 1]), -9) + + def test_complex_dotu(self): + for p in 'cz': + f = getattr(fblas, p+'dotu', None) + if f is None: + continue + assert_almost_equal(f([3j, -4, 3-4j], [2, 3, 1]), -9+2j) + + def test_complex_dotc(self): + for p in 'cz': + f = getattr(fblas, p+'dotc', None) + if f is None: + continue + assert_almost_equal(f([3j, -4, 3-4j], [2, 3j, 1]), 3-14j) + + def test_nrm2(self): + for p in 'sd': + f = getattr(fblas, p+'nrm2', None) + if f is None: + continue + assert_almost_equal(f([3, -4, 5]), math.sqrt(50)) + for p in ['c', 'z', 'sc', 'dz']: + f = getattr(fblas, p+'nrm2', None) + if f is None: + continue + assert_almost_equal(f([3j, -4, 3-4j]), math.sqrt(50)) + + def test_scal(self): + for p in 'sd': + f = getattr(fblas, p+'scal', None) + if f is None: + continue + assert_array_almost_equal(f(2, [3, -4, 5]), [6, -8, 10]) + for p in 'cz': + f = getattr(fblas, p+'scal', None) + if f is None: + continue + assert_array_almost_equal(f(3j, [3j, -4, 3-4j]), [-9, -12j, 12+9j]) + for p in ['cs', 'zd']: + f = getattr(fblas, p+'scal', None) + if f is None: + continue + assert_array_almost_equal(f(3, [3j, -4, 3-4j]), [9j, -12, 9-12j]) + + def test_swap(self): + for p in 'sd': + f = getattr(fblas, p+'swap', None) + if f is None: + continue + x, y = [2, 3, 1], [-2, 3, 7] + x1, y1 = f(x, y) + assert_array_almost_equal(x1, y) + assert_array_almost_equal(y1, x) + for p in 'cz': + f = getattr(fblas, p+'swap', None) + if f is None: + continue + x, y = [2, 3j, 1], [-2, 3, 7-3j] + x1, y1 = f(x, y) + assert_array_almost_equal(x1, y) + assert_array_almost_equal(y1, x) + + def test_amax(self): + for p in 'sd': + f = getattr(fblas, 'i'+p+'amax') + assert_equal(f([-2, 4, 3]), 1) + for p in 'cz': + f = getattr(fblas, 'i'+p+'amax') + assert_equal(f([-5, 4+3j, 6]), 1) + # XXX: need tests for rot,rotm,rotg,rotmg + + +class TestFBLAS2Simple: + + def test_gemv(self): + for p in 'sd': + f = getattr(fblas, p+'gemv', None) + if f is None: + continue + assert_array_almost_equal(f(3, [[3]], [-4]), [-36]) + assert_array_almost_equal(f(3, [[3]], [-4], 3, [5]), [-21]) + for p in 'cz': + f = getattr(fblas, p+'gemv', None) + if f is None: + continue + assert_array_almost_equal(f(3j, [[3-4j]], [-4]), [-48-36j]) + assert_array_almost_equal(f(3j, [[3-4j]], [-4], 3, [5j]), + [-48-21j]) + + # All of these *ger* functions are segfaulting when called from multiple + # threads under free-threaded CPython, see gh-21936. + @pytest.mark.thread_unsafe + def test_ger(self): + + for p in 'sd': + f = getattr(fblas, p+'ger', None) + if f is None: + continue + assert_array_almost_equal(f(1, [1, 2], [3, 4]), [[3, 4], [6, 8]]) + assert_array_almost_equal(f(2, [1, 2, 3], [3, 4]), + [[6, 8], [12, 16], [18, 24]]) + + assert_array_almost_equal(f(1, [1, 2], [3, 4], + a=[[1, 2], [3, 4]]), [[4, 6], [9, 12]]) + + for p in 'cz': + f = getattr(fblas, p+'geru', None) + if f is None: + continue + assert_array_almost_equal(f(1, [1j, 2], [3, 4]), + [[3j, 4j], [6, 8]]) + assert_array_almost_equal(f(-2, [1j, 2j, 3j], [3j, 4j]), + [[6, 8], [12, 16], [18, 24]]) + + for p in 'cz': + for name in ('ger', 'gerc'): + f = getattr(fblas, p+name, None) + if f is None: + continue + assert_array_almost_equal(f(1, [1j, 2], [3, 4]), + [[3j, 4j], [6, 8]]) + assert_array_almost_equal(f(2, [1j, 2j, 3j], [3j, 4j]), + [[6, 8], [12, 16], [18, 24]]) + + def test_syr_her(self): + x = np.arange(1, 5, dtype='d') + resx = np.triu(x[:, np.newaxis] * x) + resx_reverse = np.triu(x[::-1, np.newaxis] * x[::-1]) + + y = np.linspace(0, 8.5, 17, endpoint=False) + + z = np.arange(1, 9, dtype='d').view('D') + resz = np.triu(z[:, np.newaxis] * z) + resz_reverse = np.triu(z[::-1, np.newaxis] * z[::-1]) + rehz = np.triu(z[:, np.newaxis] * z.conj()) + rehz_reverse = np.triu(z[::-1, np.newaxis] * z[::-1].conj()) + + w = np.c_[np.zeros(4), z, np.zeros(4)].ravel() + + for p, rtol in zip('sd', [1e-7, 1e-14]): + f = getattr(fblas, p+'syr', None) + if f is None: + continue + assert_allclose(f(1.0, x), resx, rtol=rtol) + assert_allclose(f(1.0, x, lower=True), resx.T, rtol=rtol) + assert_allclose(f(1.0, y, incx=2, offx=2, n=4), resx, rtol=rtol) + # negative increments imply reversed vectors in blas + assert_allclose(f(1.0, y, incx=-2, offx=2, n=4), + resx_reverse, rtol=rtol) + + a = np.zeros((4, 4), 'f' if p == 's' else 'd', 'F') + b = f(1.0, x, a=a, overwrite_a=True) + assert_allclose(a, resx, rtol=rtol) + + b = f(2.0, x, a=a) + assert_(a is not b) + assert_allclose(b, 3*resx, rtol=rtol) + + assert_raises(Exception, f, 1.0, x, incx=0) + assert_raises(Exception, f, 1.0, x, offx=5) + assert_raises(Exception, f, 1.0, x, offx=-2) + assert_raises(Exception, f, 1.0, x, n=-2) + assert_raises(Exception, f, 1.0, x, n=5) + assert_raises(Exception, f, 1.0, x, lower=2) + assert_raises(Exception, f, 1.0, x, a=np.zeros((2, 2), 'd', 'F')) + + for p, rtol in zip('cz', [1e-7, 1e-14]): + f = getattr(fblas, p+'syr', None) + if f is None: + continue + assert_allclose(f(1.0, z), resz, rtol=rtol) + assert_allclose(f(1.0, z, lower=True), resz.T, rtol=rtol) + assert_allclose(f(1.0, w, incx=3, offx=1, n=4), resz, rtol=rtol) + # negative increments imply reversed vectors in blas + assert_allclose(f(1.0, w, incx=-3, offx=1, n=4), + resz_reverse, rtol=rtol) + + a = np.zeros((4, 4), 'F' if p == 'c' else 'D', 'F') + b = f(1.0, z, a=a, overwrite_a=True) + assert_allclose(a, resz, rtol=rtol) + + b = f(2.0, z, a=a) + assert_(a is not b) + assert_allclose(b, 3*resz, rtol=rtol) + + assert_raises(Exception, f, 1.0, x, incx=0) + assert_raises(Exception, f, 1.0, x, offx=5) + assert_raises(Exception, f, 1.0, x, offx=-2) + assert_raises(Exception, f, 1.0, x, n=-2) + assert_raises(Exception, f, 1.0, x, n=5) + assert_raises(Exception, f, 1.0, x, lower=2) + assert_raises(Exception, f, 1.0, x, a=np.zeros((2, 2), 'd', 'F')) + + for p, rtol in zip('cz', [1e-7, 1e-14]): + f = getattr(fblas, p+'her', None) + if f is None: + continue + assert_allclose(f(1.0, z), rehz, rtol=rtol) + assert_allclose(f(1.0, z, lower=True), rehz.T.conj(), rtol=rtol) + assert_allclose(f(1.0, w, incx=3, offx=1, n=4), rehz, rtol=rtol) + # negative increments imply reversed vectors in blas + assert_allclose(f(1.0, w, incx=-3, offx=1, n=4), + rehz_reverse, rtol=rtol) + + a = np.zeros((4, 4), 'F' if p == 'c' else 'D', 'F') + b = f(1.0, z, a=a, overwrite_a=True) + assert_allclose(a, rehz, rtol=rtol) + + b = f(2.0, z, a=a) + assert_(a is not b) + assert_allclose(b, 3*rehz, rtol=rtol) + + assert_raises(Exception, f, 1.0, x, incx=0) + assert_raises(Exception, f, 1.0, x, offx=5) + assert_raises(Exception, f, 1.0, x, offx=-2) + assert_raises(Exception, f, 1.0, x, n=-2) + assert_raises(Exception, f, 1.0, x, n=5) + assert_raises(Exception, f, 1.0, x, lower=2) + assert_raises(Exception, f, 1.0, x, a=np.zeros((2, 2), 'd', 'F')) + + def test_syr2(self): + x = np.arange(1, 5, dtype='d') + y = np.arange(5, 9, dtype='d') + resxy = np.triu(x[:, np.newaxis] * y + y[:, np.newaxis] * x) + resxy_reverse = np.triu(x[::-1, np.newaxis] * y[::-1] + + y[::-1, np.newaxis] * x[::-1]) + + q = np.linspace(0, 8.5, 17, endpoint=False) + + for p, rtol in zip('sd', [1e-7, 1e-14]): + f = getattr(fblas, p+'syr2', None) + if f is None: + continue + assert_allclose(f(1.0, x, y), resxy, rtol=rtol) + assert_allclose(f(1.0, x, y, n=3), resxy[:3, :3], rtol=rtol) + assert_allclose(f(1.0, x, y, lower=True), resxy.T, rtol=rtol) + + assert_allclose(f(1.0, q, q, incx=2, offx=2, incy=2, offy=10), + resxy, rtol=rtol) + assert_allclose(f(1.0, q, q, incx=2, offx=2, incy=2, offy=10, n=3), + resxy[:3, :3], rtol=rtol) + # negative increments imply reversed vectors in blas + assert_allclose(f(1.0, q, q, incx=-2, offx=2, incy=-2, offy=10), + resxy_reverse, rtol=rtol) + + a = np.zeros((4, 4), 'f' if p == 's' else 'd', 'F') + b = f(1.0, x, y, a=a, overwrite_a=True) + assert_allclose(a, resxy, rtol=rtol) + + b = f(2.0, x, y, a=a) + assert_(a is not b) + assert_allclose(b, 3*resxy, rtol=rtol) + + assert_raises(Exception, f, 1.0, x, y, incx=0) + assert_raises(Exception, f, 1.0, x, y, offx=5) + assert_raises(Exception, f, 1.0, x, y, offx=-2) + assert_raises(Exception, f, 1.0, x, y, incy=0) + assert_raises(Exception, f, 1.0, x, y, offy=5) + assert_raises(Exception, f, 1.0, x, y, offy=-2) + assert_raises(Exception, f, 1.0, x, y, n=-2) + assert_raises(Exception, f, 1.0, x, y, n=5) + assert_raises(Exception, f, 1.0, x, y, lower=2) + assert_raises(Exception, f, 1.0, x, y, + a=np.zeros((2, 2), 'd', 'F')) + + def test_her2(self): + x = np.arange(1, 9, dtype='d').view('D') + y = np.arange(9, 17, dtype='d').view('D') + resxy = x[:, np.newaxis] * y.conj() + y[:, np.newaxis] * x.conj() + resxy = np.triu(resxy) + + resxy_reverse = x[::-1, np.newaxis] * y[::-1].conj() + resxy_reverse += y[::-1, np.newaxis] * x[::-1].conj() + resxy_reverse = np.triu(resxy_reverse) + + u = np.c_[np.zeros(4), x, np.zeros(4)].ravel() + v = np.c_[np.zeros(4), y, np.zeros(4)].ravel() + + for p, rtol in zip('cz', [1e-7, 1e-14]): + f = getattr(fblas, p+'her2', None) + if f is None: + continue + assert_allclose(f(1.0, x, y), resxy, rtol=rtol) + assert_allclose(f(1.0, x, y, n=3), resxy[:3, :3], rtol=rtol) + assert_allclose(f(1.0, x, y, lower=True), resxy.T.conj(), + rtol=rtol) + + assert_allclose(f(1.0, u, v, incx=3, offx=1, incy=3, offy=1), + resxy, rtol=rtol) + assert_allclose(f(1.0, u, v, incx=3, offx=1, incy=3, offy=1, n=3), + resxy[:3, :3], rtol=rtol) + # negative increments imply reversed vectors in blas + assert_allclose(f(1.0, u, v, incx=-3, offx=1, incy=-3, offy=1), + resxy_reverse, rtol=rtol) + + a = np.zeros((4, 4), 'F' if p == 'c' else 'D', 'F') + b = f(1.0, x, y, a=a, overwrite_a=True) + assert_allclose(a, resxy, rtol=rtol) + + b = f(2.0, x, y, a=a) + assert_(a is not b) + assert_allclose(b, 3*resxy, rtol=rtol) + + assert_raises(Exception, f, 1.0, x, y, incx=0) + assert_raises(Exception, f, 1.0, x, y, offx=5) + assert_raises(Exception, f, 1.0, x, y, offx=-2) + assert_raises(Exception, f, 1.0, x, y, incy=0) + assert_raises(Exception, f, 1.0, x, y, offy=5) + assert_raises(Exception, f, 1.0, x, y, offy=-2) + assert_raises(Exception, f, 1.0, x, y, n=-2) + assert_raises(Exception, f, 1.0, x, y, n=5) + assert_raises(Exception, f, 1.0, x, y, lower=2) + assert_raises(Exception, f, 1.0, x, y, + a=np.zeros((2, 2), 'd', 'F')) + + def test_gbmv(self): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES): + n = 7 + m = 5 + kl = 1 + ku = 2 + # fake a banded matrix via toeplitz + A = toeplitz(append(rng.random(kl+1), zeros(m-kl-1)), + append(rng.random(ku+1), zeros(n-ku-1))) + A = A.astype(dtype) + Ab = zeros((kl+ku+1, n), dtype=dtype) + + # Form the banded storage + Ab[2, :5] = A[0, 0] # diag + Ab[1, 1:6] = A[0, 1] # sup1 + Ab[0, 2:7] = A[0, 2] # sup2 + Ab[3, :4] = A[1, 0] # sub1 + + x = rng.random(n).astype(dtype) + y = rng.random(m).astype(dtype) + alpha, beta = dtype(3), dtype(-5) + + func, = get_blas_funcs(('gbmv',), dtype=dtype) + y1 = func(m=m, n=n, ku=ku, kl=kl, alpha=alpha, a=Ab, + x=x, y=y, beta=beta) + y2 = alpha * A.dot(x) + beta * y + assert_array_almost_equal(y1, y2) + + y1 = func(m=m, n=n, ku=ku, kl=kl, alpha=alpha, a=Ab, + x=y, y=x, beta=beta, trans=1) + y2 = alpha * A.T.dot(y) + beta * x + assert_array_almost_equal(y1, y2) + + def test_sbmv_hbmv(self): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES): + n = 6 + k = 2 + A = zeros((n, n), dtype=dtype) + Ab = zeros((k+1, n), dtype=dtype) + + # Form the array and its packed banded storage + A[arange(n), arange(n)] = rng.random(n) + for ind2 in range(1, k+1): + temp = rng.random(n-ind2) + A[arange(n-ind2), arange(ind2, n)] = temp + Ab[-1-ind2, ind2:] = temp + A = A.astype(dtype) + A = A + A.T if ind < 2 else A + A.conj().T + Ab[-1, :] = diag(A) + x = rng.random(n).astype(dtype) + y = rng.random(n).astype(dtype) + alpha, beta = dtype(1.25), dtype(3) + + if ind > 1: + func, = get_blas_funcs(('hbmv',), dtype=dtype) + else: + func, = get_blas_funcs(('sbmv',), dtype=dtype) + y1 = func(k=k, alpha=alpha, a=Ab, x=x, y=y, beta=beta) + y2 = alpha * A.dot(x) + beta * y + assert_array_almost_equal(y1, y2) + + def test_spmv_hpmv(self): + rng = np.random.default_rng(12345698) + for ind, dtype in enumerate(DTYPES+COMPLEX_DTYPES): + n = 3 + A = rng.random((n, n)).astype(dtype) + if ind > 1: + A += rng.random((n, n))*1j + A = A.astype(dtype) + A = A + A.T if ind < 4 else A + A.conj().T + c, r = tril_indices(n) + Ap = A[r, c] + x = rng.random(n).astype(dtype) + y = rng.random(n).astype(dtype) + xlong = arange(2*n).astype(dtype) + ylong = ones(2*n).astype(dtype) + alpha, beta = dtype(1.25), dtype(2) + + if ind > 3: + func, = get_blas_funcs(('hpmv',), dtype=dtype) + else: + func, = get_blas_funcs(('spmv',), dtype=dtype) + y1 = func(n=n, alpha=alpha, ap=Ap, x=x, y=y, beta=beta) + y2 = alpha * A.dot(x) + beta * y + assert_array_almost_equal(y1, y2) + + # Test inc and offsets + y1 = func(n=n-1, alpha=alpha, beta=beta, x=xlong, y=ylong, ap=Ap, + incx=2, incy=2, offx=n, offy=n) + y2 = (alpha * A[:-1, :-1]).dot(xlong[3::2]) + beta * ylong[3::2] + assert_array_almost_equal(y1[3::2], y2) + assert_almost_equal(y1[4], ylong[4]) + + def test_spr_hpr(self): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES+COMPLEX_DTYPES): + n = 3 + A = rng.random((n, n)).astype(dtype) + if ind > 1: + A += rng.random((n, n))*1j + A = A.astype(dtype) + A = A + A.T if ind < 4 else A + A.conj().T + c, r = tril_indices(n) + Ap = A[r, c] + x = rng.random(n).astype(dtype) + alpha = (DTYPES+COMPLEX_DTYPES)[mod(ind, 4)](2.5) + + if ind > 3: + func, = get_blas_funcs(('hpr',), dtype=dtype) + y2 = alpha * x[:, None].dot(x[None, :].conj()) + A + else: + func, = get_blas_funcs(('spr',), dtype=dtype) + y2 = alpha * x[:, None].dot(x[None, :]) + A + + y1 = func(n=n, alpha=alpha, ap=Ap, x=x) + y1f = zeros((3, 3), dtype=dtype) + y1f[r, c] = y1 + y1f[c, r] = y1.conj() if ind > 3 else y1 + assert_array_almost_equal(y1f, y2) + + def test_spr2_hpr2(self): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES): + n = 3 + A = rng.random((n, n)).astype(dtype) + if ind > 1: + A += rng.random((n, n))*1j + A = A.astype(dtype) + A = A + A.T if ind < 2 else A + A.conj().T + c, r = tril_indices(n) + Ap = A[r, c] + x = rng.random(n).astype(dtype) + y = rng.random(n).astype(dtype) + alpha = dtype(2) + + if ind > 1: + func, = get_blas_funcs(('hpr2',), dtype=dtype) + else: + func, = get_blas_funcs(('spr2',), dtype=dtype) + + u = alpha.conj() * x[:, None].dot(y[None, :].conj()) + y2 = A + u + u.conj().T + y1 = func(n=n, alpha=alpha, x=x, y=y, ap=Ap) + y1f = zeros((3, 3), dtype=dtype) + y1f[r, c] = y1 + y1f[[1, 2, 2], [0, 0, 1]] = y1[[1, 3, 4]].conj() + assert_array_almost_equal(y1f, y2) + + def test_tbmv(self): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES): + n = 10 + k = 3 + x = rng.random(n).astype(dtype) + A = zeros((n, n), dtype=dtype) + # Banded upper triangular array + for sup in range(k+1): + A[arange(n-sup), arange(sup, n)] = rng.random(n-sup) + + # Add complex parts for c,z + if ind > 1: + A[nonzero(A)] += 1j * rng.random((k+1)*n-(k*(k+1)//2)).astype(dtype) + + # Form the banded storage + Ab = zeros((k+1, n), dtype=dtype) + for row in range(k+1): + Ab[-row-1, row:] = diag(A, k=row) + func, = get_blas_funcs(('tbmv',), dtype=dtype) + + y1 = func(k=k, a=Ab, x=x) + y2 = A.dot(x) + assert_array_almost_equal(y1, y2) + + y1 = func(k=k, a=Ab, x=x, diag=1) + A[arange(n), arange(n)] = dtype(1) + y2 = A.dot(x) + assert_array_almost_equal(y1, y2) + + y1 = func(k=k, a=Ab, x=x, diag=1, trans=1) + y2 = A.T.dot(x) + assert_array_almost_equal(y1, y2) + + y1 = func(k=k, a=Ab, x=x, diag=1, trans=2) + y2 = A.conj().T.dot(x) + assert_array_almost_equal(y1, y2) + + def test_tbsv(self): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES): + n = 6 + k = 3 + x = rng.random(n).astype(dtype) + A = zeros((n, n), dtype=dtype) + # Banded upper triangular array + for sup in range(k+1): + A[arange(n-sup), arange(sup, n)] = rng.random(n-sup) + + # Add complex parts for c,z + if ind > 1: + A[nonzero(A)] += 1j * rng.random((k+1)*n-(k*(k+1)//2)).astype(dtype) + + # Form the banded storage + Ab = zeros((k+1, n), dtype=dtype) + for row in range(k+1): + Ab[-row-1, row:] = diag(A, k=row) + func, = get_blas_funcs(('tbsv',), dtype=dtype) + + y1 = func(k=k, a=Ab, x=x) + y2 = solve(A, x) + assert_array_almost_equal(y1, y2) + + y1 = func(k=k, a=Ab, x=x, diag=1) + A[arange(n), arange(n)] = dtype(1) + y2 = solve(A, x) + assert_array_almost_equal(y1, y2) + + y1 = func(k=k, a=Ab, x=x, diag=1, trans=1) + y2 = solve(A.T, x) + assert_array_almost_equal(y1, y2) + + y1 = func(k=k, a=Ab, x=x, diag=1, trans=2) + y2 = solve(A.conj().T, x) + assert_array_almost_equal(y1, y2) + + def test_tpmv(self): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES): + n = 10 + x = rng.random(n).astype(dtype) + # Upper triangular array + if ind < 2: + A = triu(rng.random((n, n))) + else: + A = triu(rng.random((n, n)) + rng.random((n, n))*1j) + + # Form the packed storage + c, r = tril_indices(n) + Ap = A[r, c] + func, = get_blas_funcs(('tpmv',), dtype=dtype) + + y1 = func(n=n, ap=Ap, x=x) + y2 = A.dot(x) + assert_array_almost_equal(y1, y2) + + y1 = func(n=n, ap=Ap, x=x, diag=1) + A[arange(n), arange(n)] = dtype(1) + y2 = A.dot(x) + assert_array_almost_equal(y1, y2) + + y1 = func(n=n, ap=Ap, x=x, diag=1, trans=1) + y2 = A.T.dot(x) + assert_array_almost_equal(y1, y2) + + y1 = func(n=n, ap=Ap, x=x, diag=1, trans=2) + y2 = A.conj().T.dot(x) + assert_array_almost_equal(y1, y2) + + def test_tpsv(self): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES): + n = 10 + x = rng.random(n).astype(dtype) + # Upper triangular array + if ind < 2: + A = triu(rng.random((n, n))) + else: + A = triu(rng.random((n, n)) + rng.random((n, n))*1j) + A += eye(n) + # Form the packed storage + c, r = tril_indices(n) + Ap = A[r, c] + func, = get_blas_funcs(('tpsv',), dtype=dtype) + + y1 = func(n=n, ap=Ap, x=x) + y2 = solve(A, x) + assert_array_almost_equal(y1, y2) + + y1 = func(n=n, ap=Ap, x=x, diag=1) + A[arange(n), arange(n)] = dtype(1) + y2 = solve(A, x) + assert_array_almost_equal(y1, y2) + + y1 = func(n=n, ap=Ap, x=x, diag=1, trans=1) + y2 = solve(A.T, x) + assert_array_almost_equal(y1, y2) + + y1 = func(n=n, ap=Ap, x=x, diag=1, trans=2) + y2 = solve(A.conj().T, x) + assert_array_almost_equal(y1, y2) + + def test_trmv(self): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES): + n = 3 + A = (rng.random((n, n))+eye(n)).astype(dtype) + x = rng.random(3).astype(dtype) + func, = get_blas_funcs(('trmv',), dtype=dtype) + + y1 = func(a=A, x=x) + y2 = triu(A).dot(x) + assert_array_almost_equal(y1, y2) + + y1 = func(a=A, x=x, diag=1) + A[arange(n), arange(n)] = dtype(1) + y2 = triu(A).dot(x) + assert_array_almost_equal(y1, y2) + + y1 = func(a=A, x=x, diag=1, trans=1) + y2 = triu(A).T.dot(x) + assert_array_almost_equal(y1, y2) + + y1 = func(a=A, x=x, diag=1, trans=2) + y2 = triu(A).conj().T.dot(x) + assert_array_almost_equal(y1, y2) + + def test_trsv(self): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES): + n = 15 + A = (rng.random((n, n))+eye(n)).astype(dtype) + x = rng.random(n).astype(dtype) + func, = get_blas_funcs(('trsv',), dtype=dtype) + + y1 = func(a=A, x=x) + y2 = solve(triu(A), x) + assert_array_almost_equal(y1, y2) + + y1 = func(a=A, x=x, lower=1) + y2 = solve(tril(A), x) + assert_array_almost_equal(y1, y2) + + y1 = func(a=A, x=x, diag=1) + A[arange(n), arange(n)] = dtype(1) + y2 = solve(triu(A), x) + assert_array_almost_equal(y1, y2) + + y1 = func(a=A, x=x, diag=1, trans=1) + y2 = solve(triu(A).T, x) + assert_array_almost_equal(y1, y2) + + y1 = func(a=A, x=x, diag=1, trans=2) + y2 = solve(triu(A).conj().T, x) + assert_array_almost_equal(y1, y2) + + +class TestFBLAS3Simple: + + def test_gemm(self): + for p in 'sd': + f = getattr(fblas, p+'gemm', None) + if f is None: + continue + assert_array_almost_equal(f(3, [3], [-4]), [[-36]]) + assert_array_almost_equal(f(3, [3], [-4], 3, [5]), [-21]) + for p in 'cz': + f = getattr(fblas, p+'gemm', None) + if f is None: + continue + assert_array_almost_equal(f(3j, [3-4j], [-4]), [[-48-36j]]) + assert_array_almost_equal(f(3j, [3-4j], [-4], 3, [5j]), [-48-21j]) + + +def _get_func(func, ps='sdzc'): + """Just a helper: return a specified BLAS function w/typecode.""" + for p in ps: + f = getattr(fblas, p+func, None) + if f is None: + continue + yield f + + +class TestBLAS3Symm: + + def setup_method(self): + self.a = np.array([[1., 2.], + [0., 1.]]) + self.b = np.array([[1., 0., 3.], + [0., -1., 2.]]) + self.c = np.ones((2, 3)) + self.t = np.array([[2., -1., 8.], + [3., 0., 9.]]) + + def test_symm(self): + for f in _get_func('symm'): + res = f(a=self.a, b=self.b, c=self.c, alpha=1., beta=1.) + assert_array_almost_equal(res, self.t) + + res = f(a=self.a.T, b=self.b, lower=1, c=self.c, alpha=1., beta=1.) + assert_array_almost_equal(res, self.t) + + res = f(a=self.a, b=self.b.T, side=1, c=self.c.T, + alpha=1., beta=1.) + assert_array_almost_equal(res, self.t.T) + + def test_summ_wrong_side(self): + f = getattr(fblas, 'dsymm', None) + if f is not None: + assert_raises(Exception, f, **{'a': self.a, 'b': self.b, + 'alpha': 1, 'side': 1}) + # `side=1` means C <- B*A, hence shapes of A and B are to be + # compatible. Otherwise, f2py exception is raised + + def test_symm_wrong_uplo(self): + """SYMM only considers the upper/lower part of A. Hence setting + wrong value for `lower` (default is lower=0, meaning upper triangle) + gives a wrong result. + """ + f = getattr(fblas, 'dsymm', None) + if f is not None: + res = f(a=self.a, b=self.b, c=self.c, alpha=1., beta=1.) + assert np.allclose(res, self.t) + + res = f(a=self.a, b=self.b, lower=1, c=self.c, alpha=1., beta=1.) + assert not np.allclose(res, self.t) + + +class TestBLAS3Syrk: + def setup_method(self): + self.a = np.array([[1., 0.], + [0., -2.], + [2., 3.]]) + self.t = np.array([[1., 0., 2.], + [0., 4., -6.], + [2., -6., 13.]]) + self.tt = np.array([[5., 6.], + [6., 13.]]) + + def test_syrk(self): + for f in _get_func('syrk'): + c = f(a=self.a, alpha=1.) + assert_array_almost_equal(np.triu(c), np.triu(self.t)) + + c = f(a=self.a, alpha=1., lower=1) + assert_array_almost_equal(np.tril(c), np.tril(self.t)) + + c0 = np.ones(self.t.shape) + c = f(a=self.a, alpha=1., beta=1., c=c0) + assert_array_almost_equal(np.triu(c), np.triu(self.t+c0)) + + c = f(a=self.a, alpha=1., trans=1) + assert_array_almost_equal(np.triu(c), np.triu(self.tt)) + + # prints '0-th dimension must be fixed to 3 but got 5', + # FIXME: suppress? + # FIXME: how to catch the _fblas.error? + def test_syrk_wrong_c(self): + f = getattr(fblas, 'dsyrk', None) + if f is not None: + assert_raises(Exception, f, **{'a': self.a, 'alpha': 1., + 'c': np.ones((5, 8))}) + # if C is supplied, it must have compatible dimensions + + +class TestBLAS3Syr2k: + def setup_method(self): + self.a = np.array([[1., 0.], + [0., -2.], + [2., 3.]]) + self.b = np.array([[0., 1.], + [1., 0.], + [0, 1.]]) + self.t = np.array([[0., -1., 3.], + [-1., 0., 0.], + [3., 0., 6.]]) + self.tt = np.array([[0., 1.], + [1., 6]]) + + def test_syr2k(self): + for f in _get_func('syr2k'): + c = f(a=self.a, b=self.b, alpha=1.) + assert_array_almost_equal(np.triu(c), np.triu(self.t)) + + c = f(a=self.a, b=self.b, alpha=1., lower=1) + assert_array_almost_equal(np.tril(c), np.tril(self.t)) + + c0 = np.ones(self.t.shape) + c = f(a=self.a, b=self.b, alpha=1., beta=1., c=c0) + assert_array_almost_equal(np.triu(c), np.triu(self.t+c0)) + + c = f(a=self.a, b=self.b, alpha=1., trans=1) + assert_array_almost_equal(np.triu(c), np.triu(self.tt)) + + # prints '0-th dimension must be fixed to 3 but got 5', FIXME: suppress? + def test_syr2k_wrong_c(self): + f = getattr(fblas, 'dsyr2k', None) + if f is not None: + assert_raises(Exception, f, **{'a': self.a, + 'b': self.b, + 'alpha': 1., + 'c': np.zeros((15, 8))}) + # if C is supplied, it must have compatible dimensions + + +class TestSyHe: + """Quick and simple tests for (zc)-symm, syrk, syr2k.""" + + def setup_method(self): + self.sigma_y = np.array([[0., -1.j], + [1.j, 0.]]) + + def test_symm_zc(self): + for f in _get_func('symm', 'zc'): + # NB: a is symmetric w/upper diag of ONLY + res = f(a=self.sigma_y, b=self.sigma_y, alpha=1.) + assert_array_almost_equal(np.triu(res), np.diag([1, -1])) + + def test_hemm_zc(self): + for f in _get_func('hemm', 'zc'): + # NB: a is hermitian w/upper diag of ONLY + res = f(a=self.sigma_y, b=self.sigma_y, alpha=1.) + assert_array_almost_equal(np.triu(res), np.diag([1, 1])) + + def test_syrk_zr(self): + for f in _get_func('syrk', 'zc'): + res = f(a=self.sigma_y, alpha=1.) + assert_array_almost_equal(np.triu(res), np.diag([-1, -1])) + + def test_herk_zr(self): + for f in _get_func('herk', 'zc'): + res = f(a=self.sigma_y, alpha=1.) + assert_array_almost_equal(np.triu(res), np.diag([1, 1])) + + def test_syr2k_zr(self): + for f in _get_func('syr2k', 'zc'): + res = f(a=self.sigma_y, b=self.sigma_y, alpha=1.) + assert_array_almost_equal(np.triu(res), 2.*np.diag([-1, -1])) + + def test_her2k_zr(self): + for f in _get_func('her2k', 'zc'): + res = f(a=self.sigma_y, b=self.sigma_y, alpha=1.) + assert_array_almost_equal(np.triu(res), 2.*np.diag([1, 1])) + + +class TestTRMM: + """Quick and simple tests for dtrmm.""" + + def setup_method(self): + self.a = np.array([[1., 2., ], + [-2., 1.]]) + self.b = np.array([[3., 4., -1.], + [5., 6., -2.]]) + + self.a2 = np.array([[1, 1, 2, 3], + [0, 1, 4, 5], + [0, 0, 1, 6], + [0, 0, 0, 1]], order="f") + self.b2 = np.array([[1, 4], [2, 5], [3, 6], [7, 8], [9, 10]], + order="f") + + @pytest.mark.parametrize("dtype_", DTYPES) + def test_side(self, dtype_): + trmm = get_blas_funcs("trmm", dtype=dtype_) + # Provide large A array that works for side=1 but not 0 (see gh-10841) + assert_raises(Exception, trmm, 1.0, self.a2, self.b2) + res = trmm(1.0, self.a2.astype(dtype_), self.b2.astype(dtype_), + side=1) + k = self.b2.shape[1] + assert_allclose(res, self.b2 @ self.a2[:k, :k], rtol=0., + atol=100*np.finfo(dtype_).eps) + + def test_ab(self): + f = getattr(fblas, 'dtrmm', None) + if f is not None: + result = f(1., self.a, self.b) + # default a is upper triangular + expected = np.array([[13., 16., -5.], + [5., 6., -2.]]) + assert_array_almost_equal(result, expected) + + def test_ab_lower(self): + f = getattr(fblas, 'dtrmm', None) + if f is not None: + result = f(1., self.a, self.b, lower=True) + expected = np.array([[3., 4., -1.], + [-1., -2., 0.]]) # now a is lower triangular + assert_array_almost_equal(result, expected) + + def test_b_overwrites(self): + # BLAS dtrmm modifies B argument in-place. + # Here the default is to copy, but this can be overridden + f = getattr(fblas, 'dtrmm', None) + if f is not None: + for overwr in [True, False]: + bcopy = self.b.copy() + result = f(1., self.a, bcopy, overwrite_b=overwr) + # C-contiguous arrays are copied + assert_(bcopy.flags.f_contiguous is False and + np.may_share_memory(bcopy, result) is False) + assert_equal(bcopy, self.b) + + bcopy = np.asfortranarray(self.b.copy()) # or just transpose it + result = f(1., self.a, bcopy, overwrite_b=True) + assert_(bcopy.flags.f_contiguous is True and + np.may_share_memory(bcopy, result) is True) + assert_array_almost_equal(bcopy, result) + + +def test_trsm(): + rng = np.random.default_rng(1234) + for ind, dtype in enumerate(DTYPES): + tol = np.finfo(dtype).eps*1000 + func, = get_blas_funcs(('trsm',), dtype=dtype) + + # Test protection against size mismatches + A = rng.random((4, 5)).astype(dtype) + B = rng.random((4, 4)).astype(dtype) + alpha = dtype(1) + assert_raises(Exception, func, alpha, A, B) + assert_raises(Exception, func, alpha, A.T, B) + + n = 8 + m = 7 + alpha = dtype(-2.5) + if ind < 2: + A = rng.random((m, m)) + eye(m) + else: + A = (rng.random((m, m)) + rng.random((m, m))*1j) + eye(m) + A = A.astype(dtype) + Au = triu(A) + Al = tril(A) + B1 = rng.random((m, n)).astype(dtype) + B2 = rng.random((n, m)).astype(dtype) + + x1 = func(alpha=alpha, a=A, b=B1) + assert_equal(B1.shape, x1.shape) + x2 = solve(Au, alpha*B1) + assert_allclose(x1, x2, atol=tol) + + x1 = func(alpha=alpha, a=A, b=B1, trans_a=1) + x2 = solve(Au.T, alpha*B1) + assert_allclose(x1, x2, atol=tol) + + x1 = func(alpha=alpha, a=A, b=B1, trans_a=2) + x2 = solve(Au.conj().T, alpha*B1) + assert_allclose(x1, x2, atol=tol) + + x1 = func(alpha=alpha, a=A, b=B1, diag=1) + Au[arange(m), arange(m)] = dtype(1) + x2 = solve(Au, alpha*B1) + assert_allclose(x1, x2, atol=tol) + + x1 = func(alpha=alpha, a=A, b=B2, diag=1, side=1) + x2 = solve(Au.conj().T, alpha*B2.conj().T) + assert_allclose(x1, x2.conj().T, atol=tol) + + x1 = func(alpha=alpha, a=A, b=B2, diag=1, side=1, lower=1) + Al[arange(m), arange(m)] = dtype(1) + x2 = solve(Al.conj().T, alpha*B2.conj().T) + assert_allclose(x1, x2.conj().T, atol=tol) + + +@pytest.mark.xfail(run=False, + reason="gh-16930") +def test_gh_169309(): + x = np.repeat(10, 9) + actual = scipy.linalg.blas.dnrm2(x, 5, 3, -1) + expected = math.sqrt(500) + assert_allclose(actual, expected) + + +def test_dnrm2_neg_incx(): + # check that dnrm2(..., incx < 0) raises + # XXX: remove the test after the lowest supported BLAS implements + # negative incx (new in LAPACK 3.10) + x = np.repeat(10, 9) + incx = -1 + with assert_raises(fblas.__fblas_error): + scipy.linalg.blas.dnrm2(x, 5, 3, incx) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_cython_blas.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_cython_blas.py new file mode 100644 index 0000000000000000000000000000000000000000..284e214d38ed331cf0493d1e3bba6e1214939b2c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_cython_blas.py @@ -0,0 +1,118 @@ +import numpy as np +from numpy.testing import (assert_allclose, + assert_equal) +import scipy.linalg.cython_blas as blas + +class TestDGEMM: + + def test_transposes(self): + + a = np.arange(12, dtype='d').reshape((3, 4))[:2,:2] + b = np.arange(1, 13, dtype='d').reshape((4, 3))[:2,:2] + c = np.empty((2, 4))[:2,:2] + + blas._test_dgemm(1., a, b, 0., c) + assert_allclose(c, a.dot(b)) + + blas._test_dgemm(1., a.T, b, 0., c) + assert_allclose(c, a.T.dot(b)) + + blas._test_dgemm(1., a, b.T, 0., c) + assert_allclose(c, a.dot(b.T)) + + blas._test_dgemm(1., a.T, b.T, 0., c) + assert_allclose(c, a.T.dot(b.T)) + + blas._test_dgemm(1., a, b, 0., c.T) + assert_allclose(c, a.dot(b).T) + + blas._test_dgemm(1., a.T, b, 0., c.T) + assert_allclose(c, a.T.dot(b).T) + + blas._test_dgemm(1., a, b.T, 0., c.T) + assert_allclose(c, a.dot(b.T).T) + + blas._test_dgemm(1., a.T, b.T, 0., c.T) + assert_allclose(c, a.T.dot(b.T).T) + + def test_shapes(self): + a = np.arange(6, dtype='d').reshape((3, 2)) + b = np.arange(-6, 2, dtype='d').reshape((2, 4)) + c = np.empty((3, 4)) + + blas._test_dgemm(1., a, b, 0., c) + assert_allclose(c, a.dot(b)) + + blas._test_dgemm(1., b.T, a.T, 0., c.T) + assert_allclose(c, b.T.dot(a.T).T) + +class TestWfuncPointers: + """ Test the function pointers that are expected to fail on + Mac OS X without the additional entry statement in their definitions + in fblas_l1.pyf.src. """ + + def test_complex_args(self): + + cx = np.array([.5 + 1.j, .25 - .375j, 12.5 - 4.j], np.complex64) + cy = np.array([.8 + 2.j, .875 - .625j, -1. + 2.j], np.complex64) + + assert_allclose(blas._test_cdotc(cx, cy), + -17.6468753815+21.3718757629j) + assert_allclose(blas._test_cdotu(cx, cy), + -6.11562538147+30.3156242371j) + + assert_equal(blas._test_icamax(cx), 3) + + assert_allclose(blas._test_scasum(cx), 18.625) + assert_allclose(blas._test_scnrm2(cx), 13.1796483994) + + assert_allclose(blas._test_cdotc(cx[::2], cy[::2]), + -18.1000003815+21.2000007629j) + assert_allclose(blas._test_cdotu(cx[::2], cy[::2]), + -6.10000038147+30.7999992371j) + assert_allclose(blas._test_scasum(cx[::2]), 18.) + assert_allclose(blas._test_scnrm2(cx[::2]), 13.1719398499) + + def test_double_args(self): + + x = np.array([5., -3, -.5], np.float64) + y = np.array([2, 1, .5], np.float64) + + assert_allclose(blas._test_dasum(x), 8.5) + assert_allclose(blas._test_ddot(x, y), 6.75) + assert_allclose(blas._test_dnrm2(x), 5.85234975815) + + assert_allclose(blas._test_dasum(x[::2]), 5.5) + assert_allclose(blas._test_ddot(x[::2], y[::2]), 9.75) + assert_allclose(blas._test_dnrm2(x[::2]), 5.0249376297) + + assert_equal(blas._test_idamax(x), 1) + + def test_float_args(self): + + x = np.array([5., -3, -.5], np.float32) + y = np.array([2, 1, .5], np.float32) + + assert_equal(blas._test_isamax(x), 1) + + assert_allclose(blas._test_sasum(x), 8.5) + assert_allclose(blas._test_sdot(x, y), 6.75) + assert_allclose(blas._test_snrm2(x), 5.85234975815) + + assert_allclose(blas._test_sasum(x[::2]), 5.5) + assert_allclose(blas._test_sdot(x[::2], y[::2]), 9.75) + assert_allclose(blas._test_snrm2(x[::2]), 5.0249376297) + + def test_double_complex_args(self): + + cx = np.array([.5 + 1.j, .25 - .375j, 13. - 4.j], np.complex128) + cy = np.array([.875 + 2.j, .875 - .625j, -1. + 2.j], np.complex128) + + assert_equal(blas._test_izamax(cx), 3) + + assert_allclose(blas._test_zdotc(cx, cy), -18.109375+22.296875j) + assert_allclose(blas._test_zdotu(cx, cy), -6.578125+31.390625j) + + assert_allclose(blas._test_zdotc(cx[::2], cy[::2]), -18.5625+22.125j) + assert_allclose(blas._test_zdotu(cx[::2], cy[::2]), -6.5625+31.875j) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_cython_lapack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_cython_lapack.py new file mode 100644 index 0000000000000000000000000000000000000000..2a4e7b34b62042efdb0ce0f8ee61ce0189320995 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_cython_lapack.py @@ -0,0 +1,22 @@ +from numpy.testing import assert_allclose +from scipy.linalg import cython_lapack as cython_lapack +from scipy.linalg import lapack + + +class TestLamch: + + def test_slamch(self): + for c in [b'e', b's', b'b', b'p', b'n', b'r', b'm', b'u', b'l', b'o']: + assert_allclose(cython_lapack._test_slamch(c), + lapack.slamch(c)) + + def test_dlamch(self): + for c in [b'e', b's', b'b', b'p', b'n', b'r', b'm', b'u', b'l', b'o']: + assert_allclose(cython_lapack._test_dlamch(c), + lapack.dlamch(c)) + + def test_complex_ladiv(self): + cx = .5 + 1.j + cy = .875 + 2.j + assert_allclose(cython_lapack._test_zladiv(cy, cx), 1.95+0.1j) + assert_allclose(cython_lapack._test_cladiv(cy, cx), 1.95+0.1j) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_cythonized_array_utils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_cythonized_array_utils.py new file mode 100644 index 0000000000000000000000000000000000000000..d52c93950b6398c010b8bb8e5312153b3102fdf4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_cythonized_array_utils.py @@ -0,0 +1,132 @@ +import numpy as np +from scipy.linalg import bandwidth, issymmetric, ishermitian +import pytest +from pytest import raises + + +def test_bandwidth_dtypes(): + n = 5 + for t in np.typecodes['All']: + A = np.zeros([n, n], dtype=t) + if t in 'eUVOMm': + raises(TypeError, bandwidth, A) + elif t == 'G': # No-op test. On win these pass on others fail. + pass + else: + _ = bandwidth(A) + + +def test_bandwidth_non2d_input(): + A = np.array([1, 2, 3]) + raises(ValueError, bandwidth, A) + A = np.array([[[1, 2, 3], [4, 5, 6]]]) + raises(ValueError, bandwidth, A) + + +@pytest.mark.parametrize('T', [x for x in np.typecodes['All'] + if x not in 'eGUVOMm']) +def test_bandwidth_square_inputs(T): + n = 20 + k = 4 + R = np.zeros([n, n], dtype=T, order='F') + # form a banded matrix inplace + R[[x for x in range(n)], [x for x in range(n)]] = 1 + R[[x for x in range(n-k)], [x for x in range(k, n)]] = 1 + R[[x for x in range(1, n)], [x for x in range(n-1)]] = 1 + R[[x for x in range(k, n)], [x for x in range(n-k)]] = 1 + assert bandwidth(R) == (k, k) + A = np.array([ + [1, 1, 0, 0, 0, 0, 0, 0], + [1, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 1, 1], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + ]) + assert bandwidth(A) == (2, 2) + + +@pytest.mark.parametrize('T', [x for x in np.typecodes['All'] + if x not in 'eGUVOMm']) +def test_bandwidth_rect_inputs(T): + n, m = 10, 20 + k = 5 + R = np.zeros([n, m], dtype=T, order='F') + # form a banded matrix inplace + R[[x for x in range(n)], [x for x in range(n)]] = 1 + R[[x for x in range(n-k)], [x for x in range(k, n)]] = 1 + R[[x for x in range(1, n)], [x for x in range(n-1)]] = 1 + R[[x for x in range(k, n)], [x for x in range(n-k)]] = 1 + assert bandwidth(R) == (k, k) + + +def test_issymetric_ishermitian_dtypes(): + n = 5 + for t in np.typecodes['All']: + A = np.zeros([n, n], dtype=t) + if t in 'eUVOMm': + raises(TypeError, issymmetric, A) + raises(TypeError, ishermitian, A) + elif t == 'G': # No-op test. On win these pass on others fail. + pass + else: + assert issymmetric(A) + assert ishermitian(A) + + +def test_issymmetric_ishermitian_invalid_input(): + A = np.array([1, 2, 3]) + raises(ValueError, issymmetric, A) + raises(ValueError, ishermitian, A) + A = np.array([[[1, 2, 3], [4, 5, 6]]]) + raises(ValueError, issymmetric, A) + raises(ValueError, ishermitian, A) + A = np.array([[1, 2, 3], [4, 5, 6]]) + raises(ValueError, issymmetric, A) + raises(ValueError, ishermitian, A) + + +def test_issymetric_complex_decimals(): + A = np.arange(1, 10).astype(complex).reshape(3, 3) + A += np.arange(-4, 5).astype(complex).reshape(3, 3)*1j + # make entries decimal + A /= np.pi + A = A + A.T + assert issymmetric(A) + + +def test_ishermitian_complex_decimals(): + A = np.arange(1, 10).astype(complex).reshape(3, 3) + A += np.arange(-4, 5).astype(complex).reshape(3, 3)*1j + # make entries decimal + A /= np.pi + A = A + A.T.conj() + assert ishermitian(A) + + +def test_issymmetric_approximate_results(): + n = 20 + rng = np.random.RandomState(123456789) + x = rng.uniform(high=5., size=[n, n]) + y = x @ x.T # symmetric + p = rng.standard_normal([n, n]) + z = p @ y @ p.T + assert issymmetric(z, atol=1e-10) + assert issymmetric(z, atol=1e-10, rtol=0.) + assert issymmetric(z, atol=0., rtol=1e-12) + assert issymmetric(z, atol=1e-13, rtol=1e-12) + + +def test_ishermitian_approximate_results(): + n = 20 + rng = np.random.RandomState(987654321) + x = rng.uniform(high=5., size=[n, n]) + y = x @ x.T # symmetric + p = rng.standard_normal([n, n]) + rng.standard_normal([n, n])*1j + z = p @ y @ p.conj().T + assert ishermitian(z, atol=1e-10) + assert ishermitian(z, atol=1e-10, rtol=0.) + assert ishermitian(z, atol=0., rtol=1e-12) + assert ishermitian(z, atol=1e-13, rtol=1e-12) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp.py new file mode 100644 index 0000000000000000000000000000000000000000..605496721f8eec7a522fbba0a77ed33d6c1fdeaf --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp.py @@ -0,0 +1,3152 @@ +import itertools +import platform +import sys + +import numpy as np +from numpy.testing import (assert_equal, assert_almost_equal, + assert_array_almost_equal, assert_array_equal, + assert_, assert_allclose) + +import pytest +from pytest import raises as assert_raises + +from scipy.linalg import (eig, eigvals, lu, svd, svdvals, cholesky, qr, + schur, rsf2csf, lu_solve, lu_factor, solve, diagsvd, + hessenberg, rq, eig_banded, eigvals_banded, eigh, + eigvalsh, qr_multiply, qz, orth, ordqz, + subspace_angles, hadamard, eigvalsh_tridiagonal, + eigh_tridiagonal, null_space, cdf2rdf, LinAlgError) + +from scipy.linalg.lapack import (dgbtrf, dgbtrs, zgbtrf, zgbtrs, dsbev, + dsbevd, dsbevx, zhbevd, zhbevx) + +from scipy.linalg._misc import norm +from scipy.linalg._decomp_qz import _select_function +from scipy.stats import ortho_group + +from numpy import (array, diag, full, linalg, argsort, zeros, arange, + float32, complex64, ravel, sqrt, iscomplex, shape, sort, + sign, asarray, isfinite, ndarray, eye,) + +from scipy.linalg._testutils import assert_no_overwrite +from scipy.sparse._sputils import matrix + +from scipy._lib._testutils import check_free_memory +from scipy.linalg.blas import HAS_ILP64 +try: + from scipy.__config__ import CONFIG +except ImportError: + CONFIG = None + +IS_WASM = (sys.platform == "emscripten" or platform.machine() in ["wasm32", "wasm64"]) + + +def _random_hermitian_matrix(n, posdef=False, dtype=float): + "Generate random sym/hermitian array of the given size n" + if dtype in COMPLEX_DTYPES: + A = np.random.rand(n, n) + np.random.rand(n, n)*1.0j + A = (A + A.conj().T)/2 + else: + A = np.random.rand(n, n) + A = (A + A.T)/2 + + if posdef: + A += sqrt(2*n)*np.eye(n) + + return A.astype(dtype) + + +REAL_DTYPES = [np.float32, np.float64] +COMPLEX_DTYPES = [np.complex64, np.complex128] +DTYPES = REAL_DTYPES + COMPLEX_DTYPES + + +# XXX: This function should not be defined here, but somewhere in +# scipy.linalg namespace +def symrand(dim_or_eigv, rng): + """Return a random symmetric (Hermitian) matrix. + + If 'dim_or_eigv' is an integer N, return a NxN matrix, with eigenvalues + uniformly distributed on (-1,1). + + If 'dim_or_eigv' is 1-D real array 'a', return a matrix whose + eigenvalues are 'a'. + """ + if isinstance(dim_or_eigv, int): + dim = dim_or_eigv + d = rng.random(dim)*2 - 1 + elif (isinstance(dim_or_eigv, ndarray) and + len(dim_or_eigv.shape) == 1): + dim = dim_or_eigv.shape[0] + d = dim_or_eigv + else: + raise TypeError("input type not supported.") + + v = ortho_group.rvs(dim) + h = v.T.conj() @ diag(d) @ v + # to avoid roundoff errors, symmetrize the matrix (again) + h = 0.5*(h.T+h) + return h + + +class TestEigVals: + + def test_simple(self): + a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]] + w = eigvals(a) + exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2] + assert_array_almost_equal(w, exact_w) + + def test_simple_tr(self): + a = array([[1, 2, 3], [1, 2, 3], [2, 5, 6]], 'd').T + a = a.copy() + a = a.T + w = eigvals(a) + exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2] + assert_array_almost_equal(w, exact_w) + + def test_simple_complex(self): + a = [[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]] + w = eigvals(a) + exact_w = [(9+1j+sqrt(92+6j))/2, + 0, + (9+1j-sqrt(92+6j))/2] + assert_array_almost_equal(w, exact_w) + + def test_finite(self): + a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]] + w = eigvals(a, check_finite=False) + exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2] + assert_array_almost_equal(w, exact_w) + + @pytest.mark.parametrize('dt', [int, float, float32, complex, complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + w = eigvals(a) + assert w.shape == (0,) + assert w.dtype == eigvals(np.eye(2, dtype=dt)).dtype + + w = eigvals(a, homogeneous_eigvals=True) + assert w.shape == (2, 0) + assert w.dtype == eigvals(np.eye(2, dtype=dt)).dtype + + +class TestEig: + + def test_simple(self): + a = array([[1, 2, 3], [1, 2, 3], [2, 5, 6]]) + w, v = eig(a) + exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2] + v0 = array([1, 1, (1+sqrt(93)/3)/2]) + v1 = array([3., 0, -1]) + v2 = array([1, 1, (1-sqrt(93)/3)/2]) + v0 = v0 / norm(v0) + v1 = v1 / norm(v1) + v2 = v2 / norm(v2) + assert_array_almost_equal(w, exact_w) + assert_array_almost_equal(v0, v[:, 0]*sign(v[0, 0])) + assert_array_almost_equal(v1, v[:, 1]*sign(v[0, 1])) + assert_array_almost_equal(v2, v[:, 2]*sign(v[0, 2])) + for i in range(3): + assert_array_almost_equal(a @ v[:, i], w[i]*v[:, i]) + w, v = eig(a, left=1, right=0) + for i in range(3): + assert_array_almost_equal(a.T @ v[:, i], w[i]*v[:, i]) + + def test_simple_complex_eig(self): + a = array([[1, 2], [-2, 1]]) + w, vl, vr = eig(a, left=1, right=1) + assert_array_almost_equal(w, array([1+2j, 1-2j])) + for i in range(2): + assert_array_almost_equal(a @ vr[:, i], w[i]*vr[:, i]) + for i in range(2): + assert_array_almost_equal(a.conj().T @ vl[:, i], + w[i].conj()*vl[:, i]) + + def test_simple_complex(self): + a = array([[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]]) + w, vl, vr = eig(a, left=1, right=1) + for i in range(3): + assert_array_almost_equal(a @ vr[:, i], w[i]*vr[:, i]) + for i in range(3): + assert_array_almost_equal(a.conj().T @ vl[:, i], + w[i].conj()*vl[:, i]) + + def test_gh_3054(self): + a = [[1]] + b = [[0]] + w, vr = eig(a, b, homogeneous_eigvals=True) + assert_allclose(w[1, 0], 0) + assert_(w[0, 0] != 0) + assert_allclose(vr, 1) + + w, vr = eig(a, b) + assert_equal(w, np.inf) + assert_allclose(vr, 1) + + def _check_gen_eig(self, A, B, atol_homog=1e-13, rtol_homog=1e-13, + atol=1e-13, rtol=1e-13): + if B is not None: + A, B = asarray(A), asarray(B) + B0 = B + else: + A = asarray(A) + B0 = B + B = np.eye(*A.shape) + msg = f"\n{A!r}\n{B!r}" + + # Eigenvalues in homogeneous coordinates + w, vr = eig(A, B0, homogeneous_eigvals=True) + wt = eigvals(A, B0, homogeneous_eigvals=True) + val1 = A @ vr * w[1, :] + val2 = B @ vr * w[0, :] + for i in range(val1.shape[1]): + assert_allclose(val1[:, i], val2[:, i], + rtol=rtol_homog, atol=atol_homog, err_msg=msg) + + if B0 is None: + assert_allclose(w[1, :], 1) + assert_allclose(wt[1, :], 1) + + perm = np.lexsort(w) + permt = np.lexsort(wt) + assert_allclose(w[:, perm], wt[:, permt], atol=1e-7, rtol=1e-7, + err_msg=msg) + + length = np.empty(len(vr)) + + for i in range(len(vr)): + length[i] = norm(vr[:, i]) + + assert_allclose(length, np.ones(length.size), err_msg=msg, + atol=1e-7, rtol=1e-7) + + # Convert homogeneous coordinates + beta_nonzero = (w[1, :] != 0) + wh = w[0, beta_nonzero] / w[1, beta_nonzero] + + # Eigenvalues in standard coordinates + w, vr = eig(A, B0) + wt = eigvals(A, B0) + val1 = A @ vr + val2 = B @ vr * w + res = val1 - val2 + for i in range(res.shape[1]): + if np.all(isfinite(res[:, i])): + assert_allclose(res[:, i], 0, + rtol=rtol, atol=atol, err_msg=msg) + + # try to consistently order eigenvalues, including complex conjugate pairs + w_fin = w[isfinite(w)] + wt_fin = wt[isfinite(wt)] + + # prune noise in the real parts + w_fin = -1j * np.real_if_close(1j*w_fin, tol=1e-10) + wt_fin = -1j * np.real_if_close(1j*wt_fin, tol=1e-10) + + perm = argsort(abs(w_fin) + w_fin.imag) + permt = argsort(abs(wt_fin) + wt_fin.imag) + + assert_allclose(w_fin[perm], wt_fin[permt], + atol=1e-7, rtol=1e-7, err_msg=msg) + + length = np.empty(len(vr)) + for i in range(len(vr)): + length[i] = norm(vr[:, i]) + assert_allclose(length, np.ones(length.size), err_msg=msg) + + # Compare homogeneous and nonhomogeneous versions + assert_allclose(sort(wh), sort(w[np.isfinite(w)])) + + def test_singular(self): + # Example taken from + # https://web.archive.org/web/20040903121217/http://www.cs.umu.se/research/nla/singular_pairs/guptri/matlab.html + A = array([[22, 34, 31, 31, 17], + [45, 45, 42, 19, 29], + [39, 47, 49, 26, 34], + [27, 31, 26, 21, 15], + [38, 44, 44, 24, 30]]) + B = array([[13, 26, 25, 17, 24], + [31, 46, 40, 26, 37], + [26, 40, 19, 25, 25], + [16, 25, 27, 14, 23], + [24, 35, 18, 21, 22]]) + + with np.errstate(all='ignore'): + self._check_gen_eig(A, B, atol_homog=5e-13, atol=5e-13) + + def test_falker(self): + # Test matrices giving some Nan generalized eigenvalues. + M = diag(array([1, 0, 3])) + K = array(([2, -1, -1], [-1, 2, -1], [-1, -1, 2])) + D = array(([1, -1, 0], [-1, 1, 0], [0, 0, 0])) + Z = zeros((3, 3)) + I3 = eye(3) + A = np.block([[I3, Z], [Z, -K]]) + B = np.block([[Z, I3], [M, D]]) + + with np.errstate(all='ignore'): + self._check_gen_eig(A, B) + + def test_bad_geneig(self): + # Ticket #709 (strange return values from DGGEV) + + def matrices(omega): + c1 = -9 + omega**2 + c2 = 2*omega + A = [[1, 0, 0, 0], + [0, 1, 0, 0], + [0, 0, c1, 0], + [0, 0, 0, c1]] + B = [[0, 0, 1, 0], + [0, 0, 0, 1], + [1, 0, 0, -c2], + [0, 1, c2, 0]] + return A, B + + # With a buggy LAPACK, this can fail for different omega on different + # machines -- so we need to test several values + with np.errstate(all='ignore'): + for k in range(100): + A, B = matrices(omega=k*5./100) + self._check_gen_eig(A, B) + + def test_make_eigvals(self): + # Step through all paths in _make_eigvals + # Real eigenvalues + rng = np.random.RandomState(1234) + A = symrand(3, rng) + self._check_gen_eig(A, None) + B = symrand(3, rng) + self._check_gen_eig(A, B) + # Complex eigenvalues + A = rng.random((3, 3)) + 1j*rng.random((3, 3)) + self._check_gen_eig(A, None) + B = rng.random((3, 3)) + 1j*rng.random((3, 3)) + self._check_gen_eig(A, B) + + def test_check_finite(self): + a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]] + w, v = eig(a, check_finite=False) + exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2] + v0 = array([1, 1, (1+sqrt(93)/3)/2]) + v1 = array([3., 0, -1]) + v2 = array([1, 1, (1-sqrt(93)/3)/2]) + v0 = v0 / norm(v0) + v1 = v1 / norm(v1) + v2 = v2 / norm(v2) + assert_array_almost_equal(w, exact_w) + assert_array_almost_equal(v0, v[:, 0]*sign(v[0, 0])) + assert_array_almost_equal(v1, v[:, 1]*sign(v[0, 1])) + assert_array_almost_equal(v2, v[:, 2]*sign(v[0, 2])) + for i in range(3): + assert_array_almost_equal(a @ v[:, i], w[i]*v[:, i]) + + def test_not_square_error(self): + """Check that passing a non-square array raises a ValueError.""" + A = np.arange(6).reshape(3, 2) + assert_raises(ValueError, eig, A) + + def test_shape_mismatch(self): + """Check that passing arrays of with different shapes + raises a ValueError.""" + A = eye(2) + B = np.arange(9.0).reshape(3, 3) + assert_raises(ValueError, eig, A, B) + assert_raises(ValueError, eig, B, A) + + def test_gh_11577(self): + # https://github.com/scipy/scipy/issues/11577 + # `A - lambda B` should have 4 and 8 among the eigenvalues, and this + # was apparently broken on some platforms + A = np.array([[12.0, 28.0, 76.0, 220.0], + [16.0, 32.0, 80.0, 224.0], + [24.0, 40.0, 88.0, 232.0], + [40.0, 56.0, 104.0, 248.0]], dtype='float64') + B = np.array([[2.0, 4.0, 10.0, 28.0], + [3.0, 5.0, 11.0, 29.0], + [5.0, 7.0, 13.0, 31.0], + [9.0, 11.0, 17.0, 35.0]], dtype='float64') + + D, V = eig(A, B) + + # The problem is ill-conditioned, and two other eigenvalues + # depend on ATLAS/OpenBLAS version, compiler version etc + # see gh-11577 for discussion + # + # NB: it is tempting to use `assert_allclose(D[:2], [4, 8])` instead but + # the ordering of eigenvalues also comes out different on different + # systems depending on who knows what. + with np.testing.suppress_warnings() as sup: + # isclose chokes on inf/nan values + sup.filter(RuntimeWarning, "invalid value encountered in multiply") + assert np.isclose(D, 4.0, atol=1e-14).any() + assert np.isclose(D, 8.0, atol=1e-14).any() + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + w, vr = eig(a) + + w_n, vr_n = eig(np.eye(2, dtype=dt)) + + assert w.shape == (0,) + assert w.dtype == w_n.dtype #eigvals(np.eye(2, dtype=dt)).dtype + + assert_allclose(vr, np.empty((0, 0))) + assert vr.shape == (0, 0) + assert vr.dtype == vr_n.dtype + + w, vr = eig(a, homogeneous_eigvals=True) + assert w.shape == (2, 0) + assert w.dtype == w_n.dtype + + assert vr.shape == (0, 0) + assert vr.dtype == vr_n.dtype + + + +class TestEigBanded: + def setup_method(self): + self.create_bandmat() + + def create_bandmat(self): + """Create the full matrix `self.fullmat` and + the corresponding band matrix `self.bandmat`.""" + N = 10 + self.KL = 2 # number of subdiagonals (below the diagonal) + self.KU = 2 # number of superdiagonals (above the diagonal) + + # symmetric band matrix + self.sym_mat = (diag(full(N, 1.0)) + + diag(full(N-1, -1.0), -1) + diag(full(N-1, -1.0), 1) + + diag(full(N-2, -2.0), -2) + diag(full(N-2, -2.0), 2)) + + # hermitian band matrix + self.herm_mat = (diag(full(N, -1.0)) + + 1j*diag(full(N-1, 1.0), -1) + - 1j*diag(full(N-1, 1.0), 1) + + diag(full(N-2, -2.0), -2) + + diag(full(N-2, -2.0), 2)) + + # general real band matrix + self.real_mat = (diag(full(N, 1.0)) + + diag(full(N-1, -1.0), -1) + diag(full(N-1, -3.0), 1) + + diag(full(N-2, 2.0), -2) + diag(full(N-2, -2.0), 2)) + + # general complex band matrix + self.comp_mat = (1j*diag(full(N, 1.0)) + + diag(full(N-1, -1.0), -1) + + 1j*diag(full(N-1, -3.0), 1) + + diag(full(N-2, 2.0), -2) + + diag(full(N-2, -2.0), 2)) + + # Eigenvalues and -vectors from linalg.eig + ew, ev = linalg.eig(self.sym_mat) + ew = ew.real + args = argsort(ew) + self.w_sym_lin = ew[args] + self.evec_sym_lin = ev[:, args] + + ew, ev = linalg.eig(self.herm_mat) + ew = ew.real + args = argsort(ew) + self.w_herm_lin = ew[args] + self.evec_herm_lin = ev[:, args] + + # Extract upper bands from symmetric and hermitian band matrices + # (for use in dsbevd, dsbevx, zhbevd, zhbevx + # and their single precision versions) + LDAB = self.KU + 1 + self.bandmat_sym = zeros((LDAB, N), dtype=float) + self.bandmat_herm = zeros((LDAB, N), dtype=complex) + for i in range(LDAB): + self.bandmat_sym[LDAB-i-1, i:N] = diag(self.sym_mat, i) + self.bandmat_herm[LDAB-i-1, i:N] = diag(self.herm_mat, i) + + # Extract bands from general real and complex band matrix + # (for use in dgbtrf, dgbtrs and their single precision versions) + LDAB = 2*self.KL + self.KU + 1 + self.bandmat_real = zeros((LDAB, N), dtype=float) + self.bandmat_real[2*self.KL, :] = diag(self.real_mat) # diagonal + for i in range(self.KL): + # superdiagonals + self.bandmat_real[2*self.KL-1-i, i+1:N] = diag(self.real_mat, i+1) + # subdiagonals + self.bandmat_real[2*self.KL+1+i, 0:N-1-i] = diag(self.real_mat, + -i-1) + + self.bandmat_comp = zeros((LDAB, N), dtype=complex) + self.bandmat_comp[2*self.KL, :] = diag(self.comp_mat) # diagonal + for i in range(self.KL): + # superdiagonals + self.bandmat_comp[2*self.KL-1-i, i+1:N] = diag(self.comp_mat, i+1) + # subdiagonals + self.bandmat_comp[2*self.KL+1+i, 0:N-1-i] = diag(self.comp_mat, + -i-1) + + # absolute value for linear equation system A*x = b + self.b = 1.0*arange(N) + self.bc = self.b * (1 + 1j) + + ##################################################################### + + def test_dsbev(self): + """Compare dsbev eigenvalues and eigenvectors with + the result of linalg.eig.""" + w, evec, info = dsbev(self.bandmat_sym, compute_v=1) + evec_ = evec[:, argsort(w)] + assert_array_almost_equal(sort(w), self.w_sym_lin) + assert_array_almost_equal(abs(evec_), abs(self.evec_sym_lin)) + + def test_dsbevd(self): + """Compare dsbevd eigenvalues and eigenvectors with + the result of linalg.eig.""" + w, evec, info = dsbevd(self.bandmat_sym, compute_v=1) + evec_ = evec[:, argsort(w)] + assert_array_almost_equal(sort(w), self.w_sym_lin) + assert_array_almost_equal(abs(evec_), abs(self.evec_sym_lin)) + + def test_dsbevx(self): + """Compare dsbevx eigenvalues and eigenvectors + with the result of linalg.eig.""" + N, N = shape(self.sym_mat) + # Achtung: Argumente 0.0,0.0,range? + w, evec, num, ifail, info = dsbevx(self.bandmat_sym, 0.0, 0.0, 1, N, + compute_v=1, range=2) + evec_ = evec[:, argsort(w)] + assert_array_almost_equal(sort(w), self.w_sym_lin) + assert_array_almost_equal(abs(evec_), abs(self.evec_sym_lin)) + + def test_zhbevd(self): + """Compare zhbevd eigenvalues and eigenvectors + with the result of linalg.eig.""" + w, evec, info = zhbevd(self.bandmat_herm, compute_v=1) + evec_ = evec[:, argsort(w)] + assert_array_almost_equal(sort(w), self.w_herm_lin) + assert_array_almost_equal(abs(evec_), abs(self.evec_herm_lin)) + + def test_zhbevx(self): + """Compare zhbevx eigenvalues and eigenvectors + with the result of linalg.eig.""" + N, N = shape(self.herm_mat) + # Achtung: Argumente 0.0,0.0,range? + w, evec, num, ifail, info = zhbevx(self.bandmat_herm, 0.0, 0.0, 1, N, + compute_v=1, range=2) + evec_ = evec[:, argsort(w)] + assert_array_almost_equal(sort(w), self.w_herm_lin) + assert_array_almost_equal(abs(evec_), abs(self.evec_herm_lin)) + + def test_eigvals_banded(self): + """Compare eigenvalues of eigvals_banded with those of linalg.eig.""" + w_sym = eigvals_banded(self.bandmat_sym) + w_sym = w_sym.real + assert_array_almost_equal(sort(w_sym), self.w_sym_lin) + + w_herm = eigvals_banded(self.bandmat_herm) + w_herm = w_herm.real + assert_array_almost_equal(sort(w_herm), self.w_herm_lin) + + # extracting eigenvalues with respect to an index range + ind1 = 2 + ind2 = np.longlong(6) + w_sym_ind = eigvals_banded(self.bandmat_sym, + select='i', select_range=(ind1, ind2)) + assert_array_almost_equal(sort(w_sym_ind), + self.w_sym_lin[ind1:ind2+1]) + w_herm_ind = eigvals_banded(self.bandmat_herm, + select='i', select_range=(ind1, ind2)) + assert_array_almost_equal(sort(w_herm_ind), + self.w_herm_lin[ind1:ind2+1]) + + # extracting eigenvalues with respect to a value range + v_lower = self.w_sym_lin[ind1] - 1.0e-5 + v_upper = self.w_sym_lin[ind2] + 1.0e-5 + w_sym_val = eigvals_banded(self.bandmat_sym, + select='v', select_range=(v_lower, v_upper)) + assert_array_almost_equal(sort(w_sym_val), + self.w_sym_lin[ind1:ind2+1]) + + v_lower = self.w_herm_lin[ind1] - 1.0e-5 + v_upper = self.w_herm_lin[ind2] + 1.0e-5 + w_herm_val = eigvals_banded(self.bandmat_herm, + select='v', + select_range=(v_lower, v_upper)) + assert_array_almost_equal(sort(w_herm_val), + self.w_herm_lin[ind1:ind2+1]) + + w_sym = eigvals_banded(self.bandmat_sym, check_finite=False) + w_sym = w_sym.real + assert_array_almost_equal(sort(w_sym), self.w_sym_lin) + + def test_eig_banded(self): + """Compare eigenvalues and eigenvectors of eig_banded + with those of linalg.eig. """ + w_sym, evec_sym = eig_banded(self.bandmat_sym) + evec_sym_ = evec_sym[:, argsort(w_sym.real)] + assert_array_almost_equal(sort(w_sym), self.w_sym_lin) + assert_array_almost_equal(abs(evec_sym_), abs(self.evec_sym_lin)) + + w_herm, evec_herm = eig_banded(self.bandmat_herm) + evec_herm_ = evec_herm[:, argsort(w_herm.real)] + assert_array_almost_equal(sort(w_herm), self.w_herm_lin) + assert_array_almost_equal(abs(evec_herm_), abs(self.evec_herm_lin)) + + # extracting eigenvalues with respect to an index range + ind1 = 2 + ind2 = 6 + w_sym_ind, evec_sym_ind = eig_banded(self.bandmat_sym, + select='i', + select_range=(ind1, ind2)) + assert_array_almost_equal(sort(w_sym_ind), + self.w_sym_lin[ind1:ind2+1]) + assert_array_almost_equal(abs(evec_sym_ind), + abs(self.evec_sym_lin[:, ind1:ind2+1])) + + w_herm_ind, evec_herm_ind = eig_banded(self.bandmat_herm, + select='i', + select_range=(ind1, ind2)) + assert_array_almost_equal(sort(w_herm_ind), + self.w_herm_lin[ind1:ind2+1]) + assert_array_almost_equal(abs(evec_herm_ind), + abs(self.evec_herm_lin[:, ind1:ind2+1])) + + # extracting eigenvalues with respect to a value range + v_lower = self.w_sym_lin[ind1] - 1.0e-5 + v_upper = self.w_sym_lin[ind2] + 1.0e-5 + w_sym_val, evec_sym_val = eig_banded(self.bandmat_sym, + select='v', + select_range=(v_lower, v_upper)) + assert_array_almost_equal(sort(w_sym_val), + self.w_sym_lin[ind1:ind2+1]) + assert_array_almost_equal(abs(evec_sym_val), + abs(self.evec_sym_lin[:, ind1:ind2+1])) + + v_lower = self.w_herm_lin[ind1] - 1.0e-5 + v_upper = self.w_herm_lin[ind2] + 1.0e-5 + w_herm_val, evec_herm_val = eig_banded(self.bandmat_herm, + select='v', + select_range=(v_lower, v_upper)) + assert_array_almost_equal(sort(w_herm_val), + self.w_herm_lin[ind1:ind2+1]) + assert_array_almost_equal(abs(evec_herm_val), + abs(self.evec_herm_lin[:, ind1:ind2+1])) + + w_sym, evec_sym = eig_banded(self.bandmat_sym, check_finite=False) + evec_sym_ = evec_sym[:, argsort(w_sym.real)] + assert_array_almost_equal(sort(w_sym), self.w_sym_lin) + assert_array_almost_equal(abs(evec_sym_), abs(self.evec_sym_lin)) + + def test_dgbtrf(self): + """Compare dgbtrf LU factorisation with the LU factorisation result + of linalg.lu.""" + M, N = shape(self.real_mat) + lu_symm_band, ipiv, info = dgbtrf(self.bandmat_real, self.KL, self.KU) + + # extract matrix u from lu_symm_band + u = diag(lu_symm_band[2*self.KL, :]) + for i in range(self.KL + self.KU): + u += diag(lu_symm_band[2*self.KL-1-i, i+1:N], i+1) + + p_lin, l_lin, u_lin = lu(self.real_mat, permute_l=0) + assert_array_almost_equal(u, u_lin) + + def test_zgbtrf(self): + """Compare zgbtrf LU factorisation with the LU factorisation result + of linalg.lu.""" + M, N = shape(self.comp_mat) + lu_symm_band, ipiv, info = zgbtrf(self.bandmat_comp, self.KL, self.KU) + + # extract matrix u from lu_symm_band + u = diag(lu_symm_band[2*self.KL, :]) + for i in range(self.KL + self.KU): + u += diag(lu_symm_band[2*self.KL-1-i, i+1:N], i+1) + + p_lin, l_lin, u_lin = lu(self.comp_mat, permute_l=0) + assert_array_almost_equal(u, u_lin) + + def test_dgbtrs(self): + """Compare dgbtrs solutions for linear equation system A*x = b + with solutions of linalg.solve.""" + + lu_symm_band, ipiv, info = dgbtrf(self.bandmat_real, self.KL, self.KU) + y, info = dgbtrs(lu_symm_band, self.KL, self.KU, self.b, ipiv) + + y_lin = linalg.solve(self.real_mat, self.b) + assert_array_almost_equal(y, y_lin) + + def test_zgbtrs(self): + """Compare zgbtrs solutions for linear equation system A*x = b + with solutions of linalg.solve.""" + + lu_symm_band, ipiv, info = zgbtrf(self.bandmat_comp, self.KL, self.KU) + y, info = zgbtrs(lu_symm_band, self.KL, self.KU, self.bc, ipiv) + + y_lin = linalg.solve(self.comp_mat, self.bc) + assert_array_almost_equal(y, y_lin) + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a_band = np.empty((0, 0), dtype=dt) + w, v = eig_banded(a_band) + + w_n, v_n = eig_banded(np.array([[0, 0], [1, 1]], dtype=dt)) + + assert w.shape == (0,) + assert w.dtype == w_n.dtype + + assert v.shape == (0, 0) + assert v.dtype == v_n.dtype + + w = eig_banded(a_band, eigvals_only=True) + assert w.shape == (0,) + assert w.dtype == w_n.dtype + +class TestEigTridiagonal: + def setup_method(self): + self.create_trimat() + + def create_trimat(self): + """Create the full matrix `self.fullmat`, `self.d`, and `self.e`.""" + N = 10 + + # symmetric band matrix + self.d = full(N, 1.0) + self.e = full(N-1, -1.0) + self.full_mat = (diag(self.d) + diag(self.e, -1) + diag(self.e, 1)) + + ew, ev = linalg.eig(self.full_mat) + ew = ew.real + args = argsort(ew) + self.w = ew[args] + self.evec = ev[:, args] + + def test_degenerate(self): + """Test error conditions.""" + # Wrong sizes + assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e[:-1]) + # Must be real + assert_raises(TypeError, eigvalsh_tridiagonal, self.d, self.e * 1j) + # Bad driver + assert_raises(TypeError, eigvalsh_tridiagonal, self.d, self.e, + lapack_driver=1.) + assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e, + lapack_driver='foo') + # Bad bounds + assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e, + select='i', select_range=(0, -1)) + + def test_eigvalsh_tridiagonal(self): + """Compare eigenvalues of eigvalsh_tridiagonal with those of eig.""" + # can't use ?STERF with subselection + for driver in ('sterf', 'stev', 'stebz', 'stemr', 'auto'): + w = eigvalsh_tridiagonal(self.d, self.e, lapack_driver=driver) + assert_array_almost_equal(sort(w), self.w) + + for driver in ('sterf', 'stev'): + assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e, + lapack_driver=driver, select='i', + select_range=(0, 1)) + for driver in ('stebz', 'stemr', 'auto'): + # extracting eigenvalues with respect to the full index range + w_ind = eigvalsh_tridiagonal( + self.d, self.e, select='i', select_range=(0, len(self.d)-1), + lapack_driver=driver) + assert_array_almost_equal(sort(w_ind), self.w) + + # extracting eigenvalues with respect to an index range + ind1 = 2 + ind2 = 6 + w_ind = eigvalsh_tridiagonal( + self.d, self.e, select='i', select_range=(ind1, ind2), + lapack_driver=driver) + assert_array_almost_equal(sort(w_ind), self.w[ind1:ind2+1]) + + # extracting eigenvalues with respect to a value range + v_lower = self.w[ind1] - 1.0e-5 + v_upper = self.w[ind2] + 1.0e-5 + w_val = eigvalsh_tridiagonal( + self.d, self.e, select='v', select_range=(v_lower, v_upper), + lapack_driver=driver) + assert_array_almost_equal(sort(w_val), self.w[ind1:ind2+1]) + + def test_eigh_tridiagonal(self): + """Compare eigenvalues and eigenvectors of eigh_tridiagonal + with those of eig. """ + # can't use ?STERF when eigenvectors are requested + assert_raises(ValueError, eigh_tridiagonal, self.d, self.e, + lapack_driver='sterf') + for driver in ('stebz', 'stev', 'stemr', 'auto'): + w, evec = eigh_tridiagonal(self.d, self.e, lapack_driver=driver) + evec_ = evec[:, argsort(w)] + assert_array_almost_equal(sort(w), self.w) + assert_array_almost_equal(abs(evec_), abs(self.evec)) + + assert_raises(ValueError, eigh_tridiagonal, self.d, self.e, + lapack_driver='stev', select='i', select_range=(0, 1)) + for driver in ('stebz', 'stemr', 'auto'): + # extracting eigenvalues with respect to an index range + ind1 = 0 + ind2 = len(self.d)-1 + w, evec = eigh_tridiagonal( + self.d, self.e, select='i', select_range=(ind1, ind2), + lapack_driver=driver) + assert_array_almost_equal(sort(w), self.w) + assert_array_almost_equal(abs(evec), abs(self.evec)) + ind1 = 2 + ind2 = 6 + w, evec = eigh_tridiagonal( + self.d, self.e, select='i', select_range=(ind1, ind2), + lapack_driver=driver) + assert_array_almost_equal(sort(w), self.w[ind1:ind2+1]) + assert_array_almost_equal(abs(evec), + abs(self.evec[:, ind1:ind2+1])) + + # extracting eigenvalues with respect to a value range + v_lower = self.w[ind1] - 1.0e-5 + v_upper = self.w[ind2] + 1.0e-5 + w, evec = eigh_tridiagonal( + self.d, self.e, select='v', select_range=(v_lower, v_upper), + lapack_driver=driver) + assert_array_almost_equal(sort(w), self.w[ind1:ind2+1]) + assert_array_almost_equal(abs(evec), + abs(self.evec[:, ind1:ind2+1])) + + def test_eigh_tridiagonal_1x1(self): + """See gh-20075""" + a = np.array([-2.0]) + b = np.array([]) + x = eigh_tridiagonal(a, b, eigvals_only=True) + assert x.ndim == 1 + assert_allclose(x, a) + x, V = eigh_tridiagonal(a, b, select="i", select_range=(0, 0)) + assert x.ndim == 1 + assert V.ndim == 2 + assert_allclose(x, a) + assert_allclose(V, array([[1.]])) + + x, V = eigh_tridiagonal(a, b, select="v", select_range=(-2, 0)) + assert x.size == 0 + assert x.shape == (0,) + assert V.shape == (1, 0) + + +class TestEigh: + def setup_class(self): + np.random.seed(1234) + + def test_wrong_inputs(self): + # Nonsquare a + assert_raises(ValueError, eigh, np.ones([1, 2])) + # Nonsquare b + assert_raises(ValueError, eigh, np.ones([2, 2]), np.ones([2, 1])) + # Incompatible a, b sizes + assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([2, 2])) + # Wrong type parameter for generalized problem + assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]), + type=4) + # Both value and index subsets requested + assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]), + subset_by_value=[1, 2], subset_by_index=[2, 4]) + # Invalid upper index spec + assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]), + subset_by_index=[0, 4]) + # Invalid lower index + assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]), + subset_by_index=[-2, 2]) + # Invalid index spec #2 + assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]), + subset_by_index=[2, 0]) + # Invalid value spec + assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]), + subset_by_value=[2, 0]) + # Invalid driver name + assert_raises(ValueError, eigh, np.ones([2, 2]), driver='wrong') + # Generalized driver selection without b + assert_raises(ValueError, eigh, np.ones([3, 3]), None, driver='gvx') + # Standard driver with b + assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]), + driver='evr') + # Subset request from invalid driver + assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]), + driver='gvd', subset_by_index=[1, 2]) + assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]), + driver='gvd', subset_by_index=[1, 2]) + + def test_nonpositive_b(self): + assert_raises(LinAlgError, eigh, np.ones([3, 3]), np.ones([3, 3])) + + # index based subsets are done in the legacy test_eigh() + def test_value_subsets(self): + for ind, dt in enumerate(DTYPES): + + a = _random_hermitian_matrix(20, dtype=dt) + w, v = eigh(a, subset_by_value=[-2, 2]) + assert_equal(v.shape[1], len(w)) + assert all((w > -2) & (w < 2)) + + b = _random_hermitian_matrix(20, posdef=True, dtype=dt) + w, v = eigh(a, b, subset_by_value=[-2, 2]) + assert_equal(v.shape[1], len(w)) + assert all((w > -2) & (w < 2)) + + def test_eigh_integer(self): + a = array([[1, 2], [2, 7]]) + b = array([[3, 1], [1, 5]]) + w, z = eigh(a) + w, z = eigh(a, b) + + def test_eigh_of_sparse(self): + # This tests the rejection of inputs that eigh cannot currently handle. + import scipy.sparse + a = scipy.sparse.identity(2).tocsc() + b = np.atleast_2d(a) + assert_raises(ValueError, eigh, a) + assert_raises(ValueError, eigh, b) + + @pytest.mark.parametrize('dtype_', DTYPES) + @pytest.mark.parametrize('driver', ("ev", "evd", "evr", "evx")) + def test_various_drivers_standard(self, driver, dtype_): + a = _random_hermitian_matrix(n=20, dtype=dtype_) + w, v = eigh(a, driver=driver) + assert_allclose(a @ v - (v * w), 0., + atol=1000*np.finfo(dtype_).eps, + rtol=0.) + + @pytest.mark.parametrize('driver', ("ev", "evd", "evr", "evx")) + def test_1x1_lwork(self, driver): + w, v = eigh([[1]], driver=driver) + assert_allclose(w, array([1.]), atol=1e-15) + assert_allclose(v, array([[1.]]), atol=1e-15) + + # complex case now + w, v = eigh([[1j]], driver=driver) + assert_allclose(w, array([0]), atol=1e-15) + assert_allclose(v, array([[1.]]), atol=1e-15) + + @pytest.mark.parametrize('type', (1, 2, 3)) + @pytest.mark.parametrize('driver', ("gv", "gvd", "gvx")) + def test_various_drivers_generalized(self, driver, type): + atol = np.spacing(5000.) + a = _random_hermitian_matrix(20) + b = _random_hermitian_matrix(20, posdef=True) + w, v = eigh(a=a, b=b, driver=driver, type=type) + if type == 1: + assert_allclose(a @ v - w*(b @ v), 0., atol=atol, rtol=0.) + elif type == 2: + assert_allclose(a @ b @ v - v * w, 0., atol=atol, rtol=0.) + else: + assert_allclose(b @ a @ v - v * w, 0., atol=atol, rtol=0.) + + def test_eigvalsh_new_args(self): + a = _random_hermitian_matrix(5) + w = eigvalsh(a, subset_by_index=[1, 2]) + assert_equal(len(w), 2) + + w2 = eigvalsh(a, subset_by_index=[1, 2]) + assert_equal(len(w2), 2) + assert_allclose(w, w2) + + b = np.diag([1, 1.2, 1.3, 1.5, 2]) + w3 = eigvalsh(b, subset_by_value=[1, 1.4]) + assert_equal(len(w3), 2) + assert_allclose(w3, np.array([1.2, 1.3])) + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + w, v = eigh(a) + + w_n, v_n = eigh(np.eye(2, dtype=dt)) + + assert w.shape == (0,) + assert w.dtype == w_n.dtype + + assert v.shape == (0, 0) + assert v.dtype == v_n.dtype + + w = eigh(a, eigvals_only=True) + assert_allclose(w, np.empty((0,))) + + assert w.shape == (0,) + assert w.dtype == w_n.dtype + +class TestSVD_GESDD: + lapack_driver = 'gesdd' + + def test_degenerate(self): + assert_raises(TypeError, svd, [[1.]], lapack_driver=1.) + assert_raises(ValueError, svd, [[1.]], lapack_driver='foo') + + def test_simple(self): + a = [[1, 2, 3], [1, 20, 3], [2, 5, 6]] + for full_matrices in (True, False): + u, s, vh = svd(a, full_matrices=full_matrices, + lapack_driver=self.lapack_driver) + assert_array_almost_equal(u.T @ u, eye(3)) + assert_array_almost_equal(vh.T @ vh, eye(3)) + sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char) + for i in range(len(s)): + sigma[i, i] = s[i] + assert_array_almost_equal(u @ sigma @ vh, a) + + def test_simple_singular(self): + a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]] + for full_matrices in (True, False): + u, s, vh = svd(a, full_matrices=full_matrices, + lapack_driver=self.lapack_driver) + assert_array_almost_equal(u.T @ u, eye(3)) + assert_array_almost_equal(vh.T @ vh, eye(3)) + sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char) + for i in range(len(s)): + sigma[i, i] = s[i] + assert_array_almost_equal(u @ sigma @ vh, a) + + def test_simple_underdet(self): + a = [[1, 2, 3], [4, 5, 6]] + for full_matrices in (True, False): + u, s, vh = svd(a, full_matrices=full_matrices, + lapack_driver=self.lapack_driver) + assert_array_almost_equal(u.T @ u, eye(u.shape[0])) + sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char) + for i in range(len(s)): + sigma[i, i] = s[i] + assert_array_almost_equal(u @ sigma @ vh, a) + + def test_simple_overdet(self): + a = [[1, 2], [4, 5], [3, 4]] + for full_matrices in (True, False): + u, s, vh = svd(a, full_matrices=full_matrices, + lapack_driver=self.lapack_driver) + assert_array_almost_equal(u.T @ u, eye(u.shape[1])) + assert_array_almost_equal(vh.T @ vh, eye(2)) + sigma = zeros((u.shape[1], vh.shape[0]), s.dtype.char) + for i in range(len(s)): + sigma[i, i] = s[i] + assert_array_almost_equal(u @ sigma @ vh, a) + + def test_random(self): + rng = np.random.RandomState(1234) + n = 20 + m = 15 + for i in range(3): + for a in [rng.random([n, m]), rng.random([m, n])]: + for full_matrices in (True, False): + u, s, vh = svd(a, full_matrices=full_matrices, + lapack_driver=self.lapack_driver) + assert_array_almost_equal(u.T @ u, eye(u.shape[1])) + assert_array_almost_equal(vh @ vh.T, eye(vh.shape[0])) + sigma = zeros((u.shape[1], vh.shape[0]), s.dtype.char) + for i in range(len(s)): + sigma[i, i] = s[i] + assert_array_almost_equal(u @ sigma @ vh, a) + + def test_simple_complex(self): + a = [[1, 2, 3], [1, 2j, 3], [2, 5, 6]] + for full_matrices in (True, False): + u, s, vh = svd(a, full_matrices=full_matrices, + lapack_driver=self.lapack_driver) + assert_array_almost_equal(u.conj().T @ u, eye(u.shape[1])) + assert_array_almost_equal(vh.conj().T @ vh, eye(vh.shape[0])) + sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char) + for i in range(len(s)): + sigma[i, i] = s[i] + assert_array_almost_equal(u @ sigma @ vh, a) + + def test_random_complex(self): + rng = np.random.RandomState(1234) + n = 20 + m = 15 + for i in range(3): + for full_matrices in (True, False): + for a in [rng.random([n, m]), rng.random([m, n])]: + a = a + 1j*rng.random(list(a.shape)) + u, s, vh = svd(a, full_matrices=full_matrices, + lapack_driver=self.lapack_driver) + assert_array_almost_equal(u.conj().T @ u, + eye(u.shape[1])) + # This fails when [m,n] + # assert_array_almost_equal(vh.conj().T @ vh, + # eye(len(vh),dtype=vh.dtype.char)) + sigma = zeros((u.shape[1], vh.shape[0]), s.dtype.char) + for i in range(len(s)): + sigma[i, i] = s[i] + assert_array_almost_equal(u @ sigma @ vh, a) + + def test_crash_1580(self): + rng = np.random.RandomState(1234) + sizes = [(13, 23), (30, 50), (60, 100)] + for sz in sizes: + for dt in [np.float32, np.float64, np.complex64, np.complex128]: + a = rng.rand(*sz).astype(dt) + # should not crash + svd(a, lapack_driver=self.lapack_driver) + + def test_check_finite(self): + a = [[1, 2, 3], [1, 20, 3], [2, 5, 6]] + u, s, vh = svd(a, check_finite=False, lapack_driver=self.lapack_driver) + assert_array_almost_equal(u.T @ u, eye(3)) + assert_array_almost_equal(vh.T @ vh, eye(3)) + sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char) + for i in range(len(s)): + sigma[i, i] = s[i] + assert_array_almost_equal(u @ sigma @ vh, a) + + def test_gh_5039(self): + # This is a smoke test for https://github.com/scipy/scipy/issues/5039 + # + # The following is reported to raise "ValueError: On entry to DGESDD + # parameter number 12 had an illegal value". + # `interp1d([1,2,3,4], [1,2,3,4], kind='cubic')` + # This is reported to only show up on LAPACK 3.0.3. + # + # The matrix below is taken from the call to + # `B = _fitpack._bsplmat(order, xk)` in interpolate._find_smoothest + b = np.array( + [[0.16666667, 0.66666667, 0.16666667, 0., 0., 0.], + [0., 0.16666667, 0.66666667, 0.16666667, 0., 0.], + [0., 0., 0.16666667, 0.66666667, 0.16666667, 0.], + [0., 0., 0., 0.16666667, 0.66666667, 0.16666667]]) + svd(b, lapack_driver=self.lapack_driver) + + @pytest.mark.skipif(not HAS_ILP64, reason="64-bit LAPACK required") + @pytest.mark.slow + def test_large_matrix(self): + check_free_memory(free_mb=17000) + A = np.zeros([1, 2**31], dtype=np.float32) + A[0, -1] = 1 + u, s, vh = svd(A, full_matrices=False) + assert_allclose(s[0], 1.0) + assert_allclose(u[0, 0] * vh[0, -1], 1.0) + + @pytest.mark.parametrize("m", [0, 1, 2]) + @pytest.mark.parametrize("n", [0, 1, 2]) + @pytest.mark.parametrize('dtype', DTYPES) + def test_shape_dtype(self, m, n, dtype): + a = np.zeros((m, n), dtype=dtype) + k = min(m, n) + dchar = a.dtype.char + real_dchar = dchar.lower() if dchar in 'FD' else dchar + + u, s, v = svd(a) + assert_equal(u.shape, (m, m)) + assert_equal(u.dtype, dtype) + assert_equal(s.shape, (k,)) + assert_equal(s.dtype, np.dtype(real_dchar)) + assert_equal(v.shape, (n, n)) + assert_equal(v.dtype, dtype) + + u, s, v = svd(a, full_matrices=False) + assert_equal(u.shape, (m, k)) + assert_equal(u.dtype, dtype) + assert_equal(s.shape, (k,)) + assert_equal(s.dtype, np.dtype(real_dchar)) + assert_equal(v.shape, (k, n)) + assert_equal(v.dtype, dtype) + + s = svd(a, compute_uv=False) + assert_equal(s.shape, (k,)) + assert_equal(s.dtype, np.dtype(real_dchar)) + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize(("m", "n"), [(0, 0), (0, 2), (2, 0)]) + def test_empty(self, dt, m, n): + a0 = np.eye(3, dtype=dt) + u0, s0, v0 = svd(a0) + + a = np.empty((m, n), dtype=dt) + u, s, v = svd(a) + assert_allclose(u, np.identity(m)) + assert_allclose(s, np.empty((0,))) + assert_allclose(v, np.identity(n)) + + assert u.dtype == u0.dtype + assert v.dtype == v0.dtype + assert s.dtype == s0.dtype + + u, s, v = svd(a, full_matrices=False) + assert_allclose(u, np.empty((m, 0))) + assert_allclose(s, np.empty((0,))) + assert_allclose(v, np.empty((0, n))) + + assert u.dtype == u0.dtype + assert v.dtype == v0.dtype + assert s.dtype == s0.dtype + + s = svd(a, compute_uv=False) + assert_allclose(s, np.empty((0,))) + + assert s.dtype == s0.dtype + +class TestSVD_GESVD(TestSVD_GESDD): + lapack_driver = 'gesvd' + + +# Allocating an array of such a size leads to _ArrayMemoryError(s) +# since the maximum memory that can be in 32-bit (WASM) is 4GB +@pytest.mark.skipif(IS_WASM, reason="out of memory in WASM") +@pytest.mark.fail_slow(10) +def test_svd_gesdd_nofegfault(): + # svd(a) with {U,VT}.size > INT_MAX does not segfault + # cf https://github.com/scipy/scipy/issues/14001 + df=np.ones((4799, 53130), dtype=np.float64) + with assert_raises(ValueError): + svd(df) + + +class TestSVDVals: + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + for a in [[]], np.empty((2, 0)), np.ones((0, 3)): + a = np.array(a, dtype=dt) + s = svdvals(a) + assert_equal(s, np.empty(0)) + + s0 = svdvals(np.eye(2, dtype=dt)) + assert s.dtype == s0.dtype + + def test_simple(self): + a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]] + s = svdvals(a) + assert_(len(s) == 3) + assert_(s[0] >= s[1] >= s[2]) + + def test_simple_underdet(self): + a = [[1, 2, 3], [4, 5, 6]] + s = svdvals(a) + assert_(len(s) == 2) + assert_(s[0] >= s[1]) + + def test_simple_overdet(self): + a = [[1, 2], [4, 5], [3, 4]] + s = svdvals(a) + assert_(len(s) == 2) + assert_(s[0] >= s[1]) + + def test_simple_complex(self): + a = [[1, 2, 3], [1, 20, 3j], [2, 5, 6]] + s = svdvals(a) + assert_(len(s) == 3) + assert_(s[0] >= s[1] >= s[2]) + + def test_simple_underdet_complex(self): + a = [[1, 2, 3], [4, 5j, 6]] + s = svdvals(a) + assert_(len(s) == 2) + assert_(s[0] >= s[1]) + + def test_simple_overdet_complex(self): + a = [[1, 2], [4, 5], [3j, 4]] + s = svdvals(a) + assert_(len(s) == 2) + assert_(s[0] >= s[1]) + + def test_check_finite(self): + a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]] + s = svdvals(a, check_finite=False) + assert_(len(s) == 3) + assert_(s[0] >= s[1] >= s[2]) + + @pytest.mark.slow + def test_crash_2609(self): + np.random.seed(1234) + a = np.random.rand(1500, 2800) + # Shouldn't crash: + svdvals(a) + + +class TestDiagSVD: + + def test_simple(self): + assert_array_almost_equal(diagsvd([1, 0, 0], 3, 3), + [[1, 0, 0], [0, 0, 0], [0, 0, 0]]) + + +class TestQR: + def test_simple(self): + a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]] + q, r = qr(a) + assert_array_almost_equal(q.T @ q, eye(3)) + assert_array_almost_equal(q @ r, a) + + def test_simple_left(self): + a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]] + q, r = qr(a) + c = [1, 2, 3] + qc, r2 = qr_multiply(a, c, "left") + assert_array_almost_equal(q @ c, qc) + assert_array_almost_equal(r, r2) + qc, r2 = qr_multiply(a, eye(3), "left") + assert_array_almost_equal(q, qc) + + def test_simple_right(self): + a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]] + q, r = qr(a) + c = [1, 2, 3] + qc, r2 = qr_multiply(a, c) + assert_array_almost_equal(c @ q, qc) + assert_array_almost_equal(r, r2) + qc, r = qr_multiply(a, eye(3)) + assert_array_almost_equal(q, qc) + + def test_simple_pivoting(self): + a = np.asarray([[8, 2, 3], [2, 9, 3], [5, 3, 6]]) + q, r, p = qr(a, pivoting=True) + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.T @ q, eye(3)) + assert_array_almost_equal(q @ r, a[:, p]) + q2, r2 = qr(a[:, p]) + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_simple_left_pivoting(self): + a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]] + q, r, jpvt = qr(a, pivoting=True) + c = [1, 2, 3] + qc, r, jpvt = qr_multiply(a, c, "left", True) + assert_array_almost_equal(q @ c, qc) + + def test_simple_right_pivoting(self): + a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]] + q, r, jpvt = qr(a, pivoting=True) + c = [1, 2, 3] + qc, r, jpvt = qr_multiply(a, c, pivoting=True) + assert_array_almost_equal(c @ q, qc) + + def test_simple_trap(self): + a = [[8, 2, 3], [2, 9, 3]] + q, r = qr(a) + assert_array_almost_equal(q.T @ q, eye(2)) + assert_array_almost_equal(q @ r, a) + + def test_simple_trap_pivoting(self): + a = np.asarray([[8, 2, 3], [2, 9, 3]]) + q, r, p = qr(a, pivoting=True) + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.T @ q, eye(2)) + assert_array_almost_equal(q @ r, a[:, p]) + q2, r2 = qr(a[:, p]) + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_simple_tall(self): + # full version + a = [[8, 2], [2, 9], [5, 3]] + q, r = qr(a) + assert_array_almost_equal(q.T @ q, eye(3)) + assert_array_almost_equal(q @ r, a) + + def test_simple_tall_pivoting(self): + # full version pivoting + a = np.asarray([[8, 2], [2, 9], [5, 3]]) + q, r, p = qr(a, pivoting=True) + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.T @ q, eye(3)) + assert_array_almost_equal(q @ r, a[:, p]) + q2, r2 = qr(a[:, p]) + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_simple_tall_e(self): + # economy version + a = [[8, 2], [2, 9], [5, 3]] + q, r = qr(a, mode='economic') + assert_array_almost_equal(q.T @ q, eye(2)) + assert_array_almost_equal(q @ r, a) + assert_equal(q.shape, (3, 2)) + assert_equal(r.shape, (2, 2)) + + def test_simple_tall_e_pivoting(self): + # economy version pivoting + a = np.asarray([[8, 2], [2, 9], [5, 3]]) + q, r, p = qr(a, pivoting=True, mode='economic') + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.T @ q, eye(2)) + assert_array_almost_equal(q @ r, a[:, p]) + q2, r2 = qr(a[:, p], mode='economic') + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_simple_tall_left(self): + a = [[8, 2], [2, 9], [5, 3]] + q, r = qr(a, mode="economic") + c = [1, 2] + qc, r2 = qr_multiply(a, c, "left") + assert_array_almost_equal(q @ c, qc) + assert_array_almost_equal(r, r2) + c = array([1, 2, 0]) + qc, r2 = qr_multiply(a, c, "left", overwrite_c=True) + assert_array_almost_equal(q @ c[:2], qc) + qc, r = qr_multiply(a, eye(2), "left") + assert_array_almost_equal(qc, q) + + def test_simple_tall_left_pivoting(self): + a = [[8, 2], [2, 9], [5, 3]] + q, r, jpvt = qr(a, mode="economic", pivoting=True) + c = [1, 2] + qc, r, kpvt = qr_multiply(a, c, "left", True) + assert_array_equal(jpvt, kpvt) + assert_array_almost_equal(q @ c, qc) + qc, r, jpvt = qr_multiply(a, eye(2), "left", True) + assert_array_almost_equal(qc, q) + + def test_simple_tall_right(self): + a = [[8, 2], [2, 9], [5, 3]] + q, r = qr(a, mode="economic") + c = [1, 2, 3] + cq, r2 = qr_multiply(a, c) + assert_array_almost_equal(c @ q, cq) + assert_array_almost_equal(r, r2) + cq, r = qr_multiply(a, eye(3)) + assert_array_almost_equal(cq, q) + + def test_simple_tall_right_pivoting(self): + a = [[8, 2], [2, 9], [5, 3]] + q, r, jpvt = qr(a, pivoting=True, mode="economic") + c = [1, 2, 3] + cq, r, jpvt = qr_multiply(a, c, pivoting=True) + assert_array_almost_equal(c @ q, cq) + cq, r, jpvt = qr_multiply(a, eye(3), pivoting=True) + assert_array_almost_equal(cq, q) + + def test_simple_fat(self): + # full version + a = [[8, 2, 5], [2, 9, 3]] + q, r = qr(a) + assert_array_almost_equal(q.T @ q, eye(2)) + assert_array_almost_equal(q @ r, a) + assert_equal(q.shape, (2, 2)) + assert_equal(r.shape, (2, 3)) + + def test_simple_fat_pivoting(self): + # full version pivoting + a = np.asarray([[8, 2, 5], [2, 9, 3]]) + q, r, p = qr(a, pivoting=True) + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.T @ q, eye(2)) + assert_array_almost_equal(q @ r, a[:, p]) + assert_equal(q.shape, (2, 2)) + assert_equal(r.shape, (2, 3)) + q2, r2 = qr(a[:, p]) + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_simple_fat_e(self): + # economy version + a = [[8, 2, 3], [2, 9, 5]] + q, r = qr(a, mode='economic') + assert_array_almost_equal(q.T @ q, eye(2)) + assert_array_almost_equal(q @ r, a) + assert_equal(q.shape, (2, 2)) + assert_equal(r.shape, (2, 3)) + + def test_simple_fat_e_pivoting(self): + # economy version pivoting + a = np.asarray([[8, 2, 3], [2, 9, 5]]) + q, r, p = qr(a, pivoting=True, mode='economic') + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.T @ q, eye(2)) + assert_array_almost_equal(q @ r, a[:, p]) + assert_equal(q.shape, (2, 2)) + assert_equal(r.shape, (2, 3)) + q2, r2 = qr(a[:, p], mode='economic') + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_simple_fat_left(self): + a = [[8, 2, 3], [2, 9, 5]] + q, r = qr(a, mode="economic") + c = [1, 2] + qc, r2 = qr_multiply(a, c, "left") + assert_array_almost_equal(q @ c, qc) + assert_array_almost_equal(r, r2) + qc, r = qr_multiply(a, eye(2), "left") + assert_array_almost_equal(qc, q) + + def test_simple_fat_left_pivoting(self): + a = [[8, 2, 3], [2, 9, 5]] + q, r, jpvt = qr(a, mode="economic", pivoting=True) + c = [1, 2] + qc, r, jpvt = qr_multiply(a, c, "left", True) + assert_array_almost_equal(q @ c, qc) + qc, r, jpvt = qr_multiply(a, eye(2), "left", True) + assert_array_almost_equal(qc, q) + + def test_simple_fat_right(self): + a = [[8, 2, 3], [2, 9, 5]] + q, r = qr(a, mode="economic") + c = [1, 2] + cq, r2 = qr_multiply(a, c) + assert_array_almost_equal(c @ q, cq) + assert_array_almost_equal(r, r2) + cq, r = qr_multiply(a, eye(2)) + assert_array_almost_equal(cq, q) + + def test_simple_fat_right_pivoting(self): + a = [[8, 2, 3], [2, 9, 5]] + q, r, jpvt = qr(a, pivoting=True, mode="economic") + c = [1, 2] + cq, r, jpvt = qr_multiply(a, c, pivoting=True) + assert_array_almost_equal(c @ q, cq) + cq, r, jpvt = qr_multiply(a, eye(2), pivoting=True) + assert_array_almost_equal(cq, q) + + def test_simple_complex(self): + a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]] + q, r = qr(a) + assert_array_almost_equal(q.conj().T @ q, eye(3)) + assert_array_almost_equal(q @ r, a) + + def test_simple_complex_left(self): + a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]] + q, r = qr(a) + c = [1, 2, 3+4j] + qc, r = qr_multiply(a, c, "left") + assert_array_almost_equal(q @ c, qc) + qc, r = qr_multiply(a, eye(3), "left") + assert_array_almost_equal(q, qc) + + def test_simple_complex_right(self): + a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]] + q, r = qr(a) + c = [1, 2, 3+4j] + qc, r = qr_multiply(a, c) + assert_array_almost_equal(c @ q, qc) + qc, r = qr_multiply(a, eye(3)) + assert_array_almost_equal(q, qc) + + def test_simple_tall_complex_left(self): + a = [[8, 2+3j], [2, 9], [5+7j, 3]] + q, r = qr(a, mode="economic") + c = [1, 2+2j] + qc, r2 = qr_multiply(a, c, "left") + assert_array_almost_equal(q @ c, qc) + assert_array_almost_equal(r, r2) + c = array([1, 2, 0]) + qc, r2 = qr_multiply(a, c, "left", overwrite_c=True) + assert_array_almost_equal(q @ c[:2], qc) + qc, r = qr_multiply(a, eye(2), "left") + assert_array_almost_equal(qc, q) + + def test_simple_complex_left_conjugate(self): + a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]] + q, r = qr(a) + c = [1, 2, 3+4j] + qc, r = qr_multiply(a, c, "left", conjugate=True) + assert_array_almost_equal(q.conj() @ c, qc) + + def test_simple_complex_tall_left_conjugate(self): + a = [[3, 3+4j], [5, 2+2j], [3, 2]] + q, r = qr(a, mode='economic') + c = [1, 3+4j] + qc, r = qr_multiply(a, c, "left", conjugate=True) + assert_array_almost_equal(q.conj() @ c, qc) + + def test_simple_complex_right_conjugate(self): + a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]] + q, r = qr(a) + c = np.array([1, 2, 3+4j]) + qc, r = qr_multiply(a, c, conjugate=True) + assert_array_almost_equal(c @ q.conj(), qc) + + def test_simple_complex_pivoting(self): + a = array([[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]) + q, r, p = qr(a, pivoting=True) + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.conj().T @ q, eye(3)) + assert_array_almost_equal(q @ r, a[:, p]) + q2, r2 = qr(a[:, p]) + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_simple_complex_left_pivoting(self): + a = array([[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]) + q, r, jpvt = qr(a, pivoting=True) + c = [1, 2, 3+4j] + qc, r, jpvt = qr_multiply(a, c, "left", True) + assert_array_almost_equal(q @ c, qc) + + def test_simple_complex_right_pivoting(self): + a = array([[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]) + q, r, jpvt = qr(a, pivoting=True) + c = [1, 2, 3+4j] + qc, r, jpvt = qr_multiply(a, c, pivoting=True) + assert_array_almost_equal(c @ q, qc) + + def test_random(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + q, r = qr(a) + assert_array_almost_equal(q.T @ q, eye(n)) + assert_array_almost_equal(q @ r, a) + + def test_random_left(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + q, r = qr(a) + c = rng.random([n]) + qc, r = qr_multiply(a, c, "left") + assert_array_almost_equal(q @ c, qc) + qc, r = qr_multiply(a, eye(n), "left") + assert_array_almost_equal(q, qc) + + def test_random_right(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + q, r = qr(a) + c = rng.random([n]) + cq, r = qr_multiply(a, c) + assert_array_almost_equal(c @ q, cq) + cq, r = qr_multiply(a, eye(n)) + assert_array_almost_equal(q, cq) + + def test_random_pivoting(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + q, r, p = qr(a, pivoting=True) + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.T @ q, eye(n)) + assert_array_almost_equal(q @ r, a[:, p]) + q2, r2 = qr(a[:, p]) + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_random_tall(self): + rng = np.random.RandomState(1234) + # full version + m = 200 + n = 100 + for k in range(2): + a = rng.random([m, n]) + q, r = qr(a) + assert_array_almost_equal(q.T @ q, eye(m)) + assert_array_almost_equal(q @ r, a) + + def test_random_tall_left(self): + rng = np.random.RandomState(1234) + # full version + m = 200 + n = 100 + for k in range(2): + a = rng.random([m, n]) + q, r = qr(a, mode="economic") + c = rng.random([n]) + qc, r = qr_multiply(a, c, "left") + assert_array_almost_equal(q @ c, qc) + qc, r = qr_multiply(a, eye(n), "left") + assert_array_almost_equal(qc, q) + + def test_random_tall_right(self): + rng = np.random.RandomState(1234) + # full version + m = 200 + n = 100 + for k in range(2): + a = rng.random([m, n]) + q, r = qr(a, mode="economic") + c = rng.random([m]) + cq, r = qr_multiply(a, c) + assert_array_almost_equal(c @ q, cq) + cq, r = qr_multiply(a, eye(m)) + assert_array_almost_equal(cq, q) + + def test_random_tall_pivoting(self): + rng = np.random.RandomState(1234) + # full version pivoting + m = 200 + n = 100 + for k in range(2): + a = rng.random([m, n]) + q, r, p = qr(a, pivoting=True) + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.T @ q, eye(m)) + assert_array_almost_equal(q @ r, a[:, p]) + q2, r2 = qr(a[:, p]) + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_random_tall_e(self): + rng = np.random.RandomState(1234) + # economy version + m = 200 + n = 100 + for k in range(2): + a = rng.random([m, n]) + q, r = qr(a, mode='economic') + assert_array_almost_equal(q.T @ q, eye(n)) + assert_array_almost_equal(q @ r, a) + assert_equal(q.shape, (m, n)) + assert_equal(r.shape, (n, n)) + + def test_random_tall_e_pivoting(self): + rng = np.random.RandomState(1234) + # economy version pivoting + m = 200 + n = 100 + for k in range(2): + a = rng.random([m, n]) + q, r, p = qr(a, pivoting=True, mode='economic') + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.T @ q, eye(n)) + assert_array_almost_equal(q @ r, a[:, p]) + assert_equal(q.shape, (m, n)) + assert_equal(r.shape, (n, n)) + q2, r2 = qr(a[:, p], mode='economic') + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_random_trap(self): + rng = np.random.RandomState(1234) + m = 100 + n = 200 + for k in range(2): + a = rng.random([m, n]) + q, r = qr(a) + assert_array_almost_equal(q.T @ q, eye(m)) + assert_array_almost_equal(q @ r, a) + + def test_random_trap_pivoting(self): + rng = np.random.RandomState(1234) + m = 100 + n = 200 + for k in range(2): + a = rng.random([m, n]) + q, r, p = qr(a, pivoting=True) + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.T @ q, eye(m)) + assert_array_almost_equal(q @ r, a[:, p]) + q2, r2 = qr(a[:, p]) + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_random_complex(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + 1j*rng.random([n, n]) + q, r = qr(a) + assert_array_almost_equal(q.conj().T @ q, eye(n)) + assert_array_almost_equal(q @ r, a) + + def test_random_complex_left(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + 1j*rng.random([n, n]) + q, r = qr(a) + c = rng.random([n]) + 1j*rng.random([n]) + qc, r = qr_multiply(a, c, "left") + assert_array_almost_equal(q @ c, qc) + qc, r = qr_multiply(a, eye(n), "left") + assert_array_almost_equal(q, qc) + + def test_random_complex_right(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + 1j*rng.random([n, n]) + q, r = qr(a) + c = rng.random([n]) + 1j*rng.random([n]) + cq, r = qr_multiply(a, c) + assert_array_almost_equal(c @ q, cq) + cq, r = qr_multiply(a, eye(n)) + assert_array_almost_equal(q, cq) + + def test_random_complex_pivoting(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + 1j*rng.random([n, n]) + q, r, p = qr(a, pivoting=True) + d = abs(diag(r)) + assert_(np.all(d[1:] <= d[:-1])) + assert_array_almost_equal(q.conj().T @ q, eye(n)) + assert_array_almost_equal(q @ r, a[:, p]) + q2, r2 = qr(a[:, p]) + assert_array_almost_equal(q, q2) + assert_array_almost_equal(r, r2) + + def test_check_finite(self): + a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]] + q, r = qr(a, check_finite=False) + assert_array_almost_equal(q.T @ q, eye(3)) + assert_array_almost_equal(q @ r, a) + + def test_lwork(self): + a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]] + # Get comparison values + q, r = qr(a, lwork=None) + + # Test against minimum valid lwork + q2, r2 = qr(a, lwork=3) + assert_array_almost_equal(q2, q) + assert_array_almost_equal(r2, r) + + # Test against larger lwork + q3, r3 = qr(a, lwork=10) + assert_array_almost_equal(q3, q) + assert_array_almost_equal(r3, r) + + # Test against explicit lwork=-1 + q4, r4 = qr(a, lwork=-1) + assert_array_almost_equal(q4, q) + assert_array_almost_equal(r4, r) + + # Test against invalid lwork + assert_raises(Exception, qr, (a,), {'lwork': 0}) + assert_raises(Exception, qr, (a,), {'lwork': 2}) + + @pytest.mark.parametrize("m", [0, 1, 2]) + @pytest.mark.parametrize("n", [0, 1, 2]) + @pytest.mark.parametrize("pivoting", [False, True]) + @pytest.mark.parametrize('dtype', DTYPES) + def test_shape_dtype(self, m, n, pivoting, dtype): + k = min(m, n) + + a = np.zeros((m, n), dtype=dtype) + q, r, *other = qr(a, pivoting=pivoting) + assert_equal(q.shape, (m, m)) + assert_equal(q.dtype, dtype) + assert_equal(r.shape, (m, n)) + assert_equal(r.dtype, dtype) + assert len(other) == (1 if pivoting else 0) + if pivoting: + p, = other + assert_equal(p.shape, (n,)) + assert_equal(p.dtype, np.int32) + + r, *other = qr(a, mode='r', pivoting=pivoting) + assert_equal(r.shape, (m, n)) + assert_equal(r.dtype, dtype) + assert len(other) == (1 if pivoting else 0) + if pivoting: + p, = other + assert_equal(p.shape, (n,)) + assert_equal(p.dtype, np.int32) + + q, r, *other = qr(a, mode='economic', pivoting=pivoting) + assert_equal(q.shape, (m, k)) + assert_equal(q.dtype, dtype) + assert_equal(r.shape, (k, n)) + assert_equal(r.dtype, dtype) + assert len(other) == (1 if pivoting else 0) + if pivoting: + p, = other + assert_equal(p.shape, (n,)) + assert_equal(p.dtype, np.int32) + + (raw, tau), r, *other = qr(a, mode='raw', pivoting=pivoting) + assert_equal(raw.shape, (m, n)) + assert_equal(raw.dtype, dtype) + assert_equal(tau.shape, (k,)) + assert_equal(tau.dtype, dtype) + assert_equal(r.shape, (k, n)) + assert_equal(r.dtype, dtype) + assert len(other) == (1 if pivoting else 0) + if pivoting: + p, = other + assert_equal(p.shape, (n,)) + assert_equal(p.dtype, np.int32) + + @pytest.mark.parametrize(("m", "n"), [(0, 0), (0, 2), (2, 0)]) + def test_empty(self, m, n): + k = min(m, n) + + a = np.empty((m, n)) + q, r = qr(a) + assert_allclose(q, np.identity(m)) + assert_allclose(r, np.empty((m, n))) + + q, r, p = qr(a, pivoting=True) + assert_allclose(q, np.identity(m)) + assert_allclose(r, np.empty((m, n))) + assert_allclose(p, np.arange(n)) + + r, = qr(a, mode='r') + assert_allclose(r, np.empty((m, n))) + + q, r = qr(a, mode='economic') + assert_allclose(q, np.empty((m, k))) + assert_allclose(r, np.empty((k, n))) + + (raw, tau), r = qr(a, mode='raw') + assert_allclose(raw, np.empty((m, n))) + assert_allclose(tau, np.empty((k,))) + assert_allclose(r, np.empty((k, n))) + + def test_multiply_empty(self): + a = np.empty((0, 0)) + c = np.empty((0, 0)) + cq, r = qr_multiply(a, c) + assert_allclose(cq, np.empty((0, 0))) + + a = np.empty((0, 2)) + c = np.empty((2, 0)) + cq, r = qr_multiply(a, c) + assert_allclose(cq, np.empty((2, 0))) + + a = np.empty((2, 0)) + c = np.empty((0, 2)) + cq, r = qr_multiply(a, c) + assert_allclose(cq, np.empty((0, 2))) + + +class TestRQ: + def test_simple(self): + a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]] + r, q = rq(a) + assert_array_almost_equal(q @ q.T, eye(3)) + assert_array_almost_equal(r @ q, a) + + def test_r(self): + a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]] + r, q = rq(a) + r2 = rq(a, mode='r') + assert_array_almost_equal(r, r2) + + def test_random(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + r, q = rq(a) + assert_array_almost_equal(q @ q.T, eye(n)) + assert_array_almost_equal(r @ q, a) + + def test_simple_trap(self): + a = [[8, 2, 3], [2, 9, 3]] + r, q = rq(a) + assert_array_almost_equal(q.T @ q, eye(3)) + assert_array_almost_equal(r @ q, a) + + def test_simple_tall(self): + a = [[8, 2], [2, 9], [5, 3]] + r, q = rq(a) + assert_array_almost_equal(q.T @ q, eye(2)) + assert_array_almost_equal(r @ q, a) + + def test_simple_fat(self): + a = [[8, 2, 5], [2, 9, 3]] + r, q = rq(a) + assert_array_almost_equal(q @ q.T, eye(3)) + assert_array_almost_equal(r @ q, a) + + def test_simple_complex(self): + a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]] + r, q = rq(a) + assert_array_almost_equal(q @ q.conj().T, eye(3)) + assert_array_almost_equal(r @ q, a) + + def test_random_tall(self): + rng = np.random.RandomState(1234) + m = 200 + n = 100 + for k in range(2): + a = rng.random([m, n]) + r, q = rq(a) + assert_array_almost_equal(q @ q.T, eye(n)) + assert_array_almost_equal(r @ q, a) + + def test_random_trap(self): + rng = np.random.RandomState(1234) + m = 100 + n = 200 + for k in range(2): + a = rng.random([m, n]) + r, q = rq(a) + assert_array_almost_equal(q @ q.T, eye(n)) + assert_array_almost_equal(r @ q, a) + + def test_random_trap_economic(self): + rng = np.random.RandomState(1234) + m = 100 + n = 200 + for k in range(2): + a = rng.random([m, n]) + r, q = rq(a, mode='economic') + assert_array_almost_equal(q @ q.T, eye(m)) + assert_array_almost_equal(r @ q, a) + assert_equal(q.shape, (m, n)) + assert_equal(r.shape, (m, m)) + + def test_random_complex(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + 1j*rng.random([n, n]) + r, q = rq(a) + assert_array_almost_equal(q @ q.conj().T, eye(n)) + assert_array_almost_equal(r @ q, a) + + def test_random_complex_economic(self): + rng = np.random.RandomState(1234) + m = 100 + n = 200 + for k in range(2): + a = rng.random([m, n]) + 1j*rng.random([m, n]) + r, q = rq(a, mode='economic') + assert_array_almost_equal(q @ q.conj().T, eye(m)) + assert_array_almost_equal(r @ q, a) + assert_equal(q.shape, (m, n)) + assert_equal(r.shape, (m, m)) + + def test_check_finite(self): + a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]] + r, q = rq(a, check_finite=False) + assert_array_almost_equal(q @ q.T, eye(3)) + assert_array_almost_equal(r @ q, a) + + @pytest.mark.parametrize("m", [0, 1, 2]) + @pytest.mark.parametrize("n", [0, 1, 2]) + @pytest.mark.parametrize('dtype', DTYPES) + def test_shape_dtype(self, m, n, dtype): + k = min(m, n) + + a = np.zeros((m, n), dtype=dtype) + r, q = rq(a) + assert_equal(q.shape, (n, n)) + assert_equal(r.shape, (m, n)) + assert_equal(r.dtype, dtype) + assert_equal(q.dtype, dtype) + + r = rq(a, mode='r') + assert_equal(r.shape, (m, n)) + assert_equal(r.dtype, dtype) + + r, q = rq(a, mode='economic') + assert_equal(r.shape, (m, k)) + assert_equal(r.dtype, dtype) + assert_equal(q.shape, (k, n)) + assert_equal(q.dtype, dtype) + + @pytest.mark.parametrize(("m", "n"), [(0, 0), (0, 2), (2, 0)]) + def test_empty(self, m, n): + k = min(m, n) + + a = np.empty((m, n)) + r, q = rq(a) + assert_allclose(r, np.empty((m, n))) + assert_allclose(q, np.identity(n)) + + r = rq(a, mode='r') + assert_allclose(r, np.empty((m, n))) + + r, q = rq(a, mode='economic') + assert_allclose(r, np.empty((m, k))) + assert_allclose(q, np.empty((k, n))) + + +class TestSchur: + + def check_schur(self, a, t, u, rtol, atol): + # Check that the Schur decomposition is correct. + assert_allclose(u @ t @ u.conj().T, a, rtol=rtol, atol=atol, + err_msg="Schur decomposition does not match 'a'") + # The expected value of u @ u.H - I is all zeros, so test + # with absolute tolerance only. + assert_allclose(u @ u.conj().T - np.eye(len(u)), 0, rtol=0, atol=atol, + err_msg="u is not unitary") + + def test_simple(self): + a = [[8, 12, 3], [2, 9, 3], [10, 3, 6]] + t, z = schur(a) + self.check_schur(a, t, z, rtol=1e-14, atol=5e-15) + tc, zc = schur(a, 'complex') + assert_(np.any(ravel(iscomplex(zc))) and np.any(ravel(iscomplex(tc)))) + self.check_schur(a, tc, zc, rtol=1e-14, atol=5e-15) + tc2, zc2 = rsf2csf(tc, zc) + self.check_schur(a, tc2, zc2, rtol=1e-14, atol=5e-15) + + @pytest.mark.parametrize( + 'sort, expected_diag', + [('lhp', [-np.sqrt(2), -0.5, np.sqrt(2), 0.5]), + ('rhp', [np.sqrt(2), 0.5, -np.sqrt(2), -0.5]), + ('iuc', [-0.5, 0.5, np.sqrt(2), -np.sqrt(2)]), + ('ouc', [np.sqrt(2), -np.sqrt(2), -0.5, 0.5]), + (lambda x: x >= 0.0, [np.sqrt(2), 0.5, -np.sqrt(2), -0.5])] + ) + def test_sort(self, sort, expected_diag): + # The exact eigenvalues of this matrix are + # -sqrt(2), sqrt(2), -1/2, 1/2. + a = [[4., 3., 1., -1.], + [-4.5, -3.5, -1., 1.], + [9., 6., -4., 4.5], + [6., 4., -3., 3.5]] + t, u, sdim = schur(a, sort=sort) + self.check_schur(a, t, u, rtol=1e-14, atol=5e-15) + assert_allclose(np.diag(t), expected_diag, rtol=1e-12) + assert_equal(2, sdim) + + def test_sort_errors(self): + a = [[4., 3., 1., -1.], + [-4.5, -3.5, -1., 1.], + [9., 6., -4., 4.5], + [6., 4., -3., 3.5]] + assert_raises(ValueError, schur, a, sort='unsupported') + assert_raises(ValueError, schur, a, sort=1) + + def test_check_finite(self): + a = [[8, 12, 3], [2, 9, 3], [10, 3, 6]] + t, z = schur(a, check_finite=False) + assert_array_almost_equal(z @ t @ z.conj().T, a) + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + t, z = schur(a) + t0, z0 = schur(np.eye(2, dtype=dt)) + assert_allclose(t, np.empty((0, 0))) + assert_allclose(z, np.empty((0, 0))) + assert t.dtype == t0.dtype + assert z.dtype == z0.dtype + + t, z, sdim = schur(a, sort='lhp') + assert_allclose(t, np.empty((0, 0))) + assert_allclose(z, np.empty((0, 0))) + assert_equal(sdim, 0) + assert t.dtype == t0.dtype + assert z.dtype == z0.dtype + + @pytest.mark.parametrize('sort', ['iuc', 'ouc']) + @pytest.mark.parametrize('output', ['real', 'complex']) + @pytest.mark.parametrize('dtype', [np.float32, np.float64, + np.complex64, np.complex128]) + def test_gh_13137_sort_str(self, sort, output, dtype): + # gh-13137 reported that sort values 'iuc' and 'ouc' were not + # correct because the callables assumed that the eigenvalues would + # always be expressed as a single complex number. + # In fact, when `output='real'` and the dtype is real, the + # eigenvalues are passed as separate real and imaginary components + # (yet no error is raised if the callable accepts only one argument). + # + # This tests these sort values by counting the number of eigenvalues + # `schur` reports as being inside/outside the unit circle. + + # Real matrix with eigenvalues 0.1 +- 2j + A = np.asarray([[0.1, -2], [2, 0.1]]) + + # Previously, this would fail for `output='real'` with real dtypes + sdim = schur(A.astype(dtype), sort=sort, output=output)[-1] + assert sdim == 0 if sort == 'iuc' else sdim == 2 + + @pytest.mark.parametrize('output', ['real', 'complex']) + @pytest.mark.parametrize('dtype', [np.float32, np.float64, + np.complex64, np.complex128]) + def test_gh_13137_sort_custom(self, output, dtype): + # This simply tests our understanding of how eigenvalues are + # passed to a sort callable. If `output='real'` and the dtype is real, + # real and imaginary parts are passed as separate real arguments; + # otherwise, they are passed a single complex argument. + # Also, if `output='real'` and the dtype is real, when either + # eigenvalue in a complex conjugate pair satisfies the sort condition, + # `sdim` is incremented by TWO. + + # Real matrix with eigenvalues 0.1 +- 2j + A = np.asarray([[0.1, -2], [2, 0.1]]) + + all_real = output=='real' and dtype in {np.float32, np.float64} + + def sort(x, y=None): + if all_real: + assert not np.iscomplexobj(x) + assert y is not None and np.isreal(y) + z = x + y*1j + else: + assert np.iscomplexobj(x) + assert y is None + z = x + return z.imag > 1e-15 + + # Only one complex eigenvalue satisfies the condition, but when + # `all_real` applies, both eigenvalues in the complex conjugate pair + # are counted. + sdim = schur(A.astype(dtype), sort=sort, output=output)[-1] + assert sdim == 2 if all_real else sdim == 1 + + +class TestHessenberg: + + def test_simple(self): + a = [[-149, -50, -154], + [537, 180, 546], + [-27, -9, -25]] + h1 = [[-149.0000, 42.2037, -156.3165], + [-537.6783, 152.5511, -554.9272], + [0, 0.0728, 2.4489]] + h, q = hessenberg(a, calc_q=1) + assert_array_almost_equal(q.T @ a @ q, h) + assert_array_almost_equal(h, h1, decimal=4) + + def test_simple_complex(self): + a = [[-149, -50, -154], + [537, 180j, 546], + [-27j, -9, -25]] + h, q = hessenberg(a, calc_q=1) + assert_array_almost_equal(q.conj().T @ a @ q, h) + + def test_simple2(self): + a = [[1, 2, 3, 4, 5, 6, 7], + [0, 2, 3, 4, 6, 7, 2], + [0, 2, 2, 3, 0, 3, 2], + [0, 0, 2, 8, 0, 0, 2], + [0, 3, 1, 2, 0, 1, 2], + [0, 1, 2, 3, 0, 1, 0], + [0, 0, 0, 0, 0, 1, 2]] + h, q = hessenberg(a, calc_q=1) + assert_array_almost_equal(q.T @ a @ q, h) + + def test_simple3(self): + a = np.eye(3) + a[-1, 0] = 2 + h, q = hessenberg(a, calc_q=1) + assert_array_almost_equal(q.T @ a @ q, h) + + def test_random(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + h, q = hessenberg(a, calc_q=1) + assert_array_almost_equal(q.T @ a @ q, h) + + def test_random_complex(self): + rng = np.random.RandomState(1234) + n = 20 + for k in range(2): + a = rng.random([n, n]) + 1j*rng.random([n, n]) + h, q = hessenberg(a, calc_q=1) + assert_array_almost_equal(q.conj().T @ a @ q, h) + + def test_check_finite(self): + a = [[-149, -50, -154], + [537, 180, 546], + [-27, -9, -25]] + h1 = [[-149.0000, 42.2037, -156.3165], + [-537.6783, 152.5511, -554.9272], + [0, 0.0728, 2.4489]] + h, q = hessenberg(a, calc_q=1, check_finite=False) + assert_array_almost_equal(q.T @ a @ q, h) + assert_array_almost_equal(h, h1, decimal=4) + + def test_2x2(self): + a = [[2, 1], [7, 12]] + + h, q = hessenberg(a, calc_q=1) + assert_array_almost_equal(q, np.eye(2)) + assert_array_almost_equal(h, a) + + b = [[2-7j, 1+2j], [7+3j, 12-2j]] + h2, q2 = hessenberg(b, calc_q=1) + assert_array_almost_equal(q2, np.eye(2)) + assert_array_almost_equal(h2, b) + + @pytest.mark.parametrize('dt', [int, float, float32, complex, complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + h = hessenberg(a) + assert h.shape == (0, 0) + assert h.dtype == hessenberg(np.eye(3, dtype=dt)).dtype + + h, q = hessenberg(a, calc_q=True) + h3, q3 = hessenberg(a, calc_q=True) + assert h.shape == (0, 0) + assert h.dtype == h3.dtype + + assert q.shape == (0, 0) + assert q.dtype == q3.dtype + + +blas_provider = blas_version = None +if CONFIG is not None: + blas_provider = CONFIG['Build Dependencies']['blas']['name'] + blas_version = CONFIG['Build Dependencies']['blas']['version'] + + +class TestQZ: + def test_qz_single(self): + rng = np.random.RandomState(12345) + n = 5 + A = rng.random([n, n]).astype(float32) + B = rng.random([n, n]).astype(float32) + AA, BB, Q, Z = qz(A, B) + assert_array_almost_equal(Q @ AA @ Z.T, A, decimal=5) + assert_array_almost_equal(Q @ BB @ Z.T, B, decimal=5) + assert_array_almost_equal(Q @ Q.T, eye(n), decimal=5) + assert_array_almost_equal(Z @ Z.T, eye(n), decimal=5) + assert_(np.all(diag(BB) >= 0)) + + def test_qz_double(self): + rng = np.random.RandomState(12345) + n = 5 + A = rng.random([n, n]) + B = rng.random([n, n]) + AA, BB, Q, Z = qz(A, B) + assert_array_almost_equal(Q @ AA @ Z.T, A) + assert_array_almost_equal(Q @ BB @ Z.T, B) + assert_array_almost_equal(Q @ Q.T, eye(n)) + assert_array_almost_equal(Z @ Z.T, eye(n)) + assert_(np.all(diag(BB) >= 0)) + + def test_qz_complex(self): + rng = np.random.RandomState(12345) + n = 5 + A = rng.random([n, n]) + 1j*rng.random([n, n]) + B = rng.random([n, n]) + 1j*rng.random([n, n]) + AA, BB, Q, Z = qz(A, B) + assert_array_almost_equal(Q @ AA @ Z.conj().T, A) + assert_array_almost_equal(Q @ BB @ Z.conj().T, B) + assert_array_almost_equal(Q @ Q.conj().T, eye(n)) + assert_array_almost_equal(Z @ Z.conj().T, eye(n)) + assert_(np.all(diag(BB) >= 0)) + assert_(np.all(diag(BB).imag == 0)) + + def test_qz_complex64(self): + rng = np.random.RandomState(12345) + n = 5 + A = (rng.random([n, n]) + 1j*rng.random([n, n])).astype(complex64) + B = (rng.random([n, n]) + 1j*rng.random([n, n])).astype(complex64) + AA, BB, Q, Z = qz(A, B) + assert_array_almost_equal(Q @ AA @ Z.conj().T, A, decimal=5) + assert_array_almost_equal(Q @ BB @ Z.conj().T, B, decimal=5) + assert_array_almost_equal(Q @ Q.conj().T, eye(n), decimal=5) + assert_array_almost_equal(Z @ Z.conj().T, eye(n), decimal=5) + assert_(np.all(diag(BB) >= 0)) + assert_(np.all(diag(BB).imag == 0)) + + def test_qz_double_complex(self): + rng = np.random.RandomState(12345) + n = 5 + A = rng.random([n, n]) + B = rng.random([n, n]) + AA, BB, Q, Z = qz(A, B, output='complex') + aa = Q @ AA @ Z.conj().T + assert_array_almost_equal(aa.real, A) + assert_array_almost_equal(aa.imag, 0) + bb = Q @ BB @ Z.conj().T + assert_array_almost_equal(bb.real, B) + assert_array_almost_equal(bb.imag, 0) + assert_array_almost_equal(Q @ Q.conj().T, eye(n)) + assert_array_almost_equal(Z @ Z.conj().T, eye(n)) + assert_(np.all(diag(BB) >= 0)) + + def test_qz_double_sort(self): + # from https://www.nag.com/lapack-ex/node119.html + # NOTE: These matrices may be ill-conditioned and lead to a + # seg fault on certain python versions when compiled with + # sse2 or sse3 older ATLAS/LAPACK binaries for windows + # A = np.array([[3.9, 12.5, -34.5, -0.5], + # [ 4.3, 21.5, -47.5, 7.5], + # [ 4.3, 21.5, -43.5, 3.5], + # [ 4.4, 26.0, -46.0, 6.0 ]]) + + # B = np.array([[ 1.0, 2.0, -3.0, 1.0], + # [1.0, 3.0, -5.0, 4.0], + # [1.0, 3.0, -4.0, 3.0], + # [1.0, 3.0, -4.0, 4.0]]) + A = np.array([[3.9, 12.5, -34.5, 2.5], + [4.3, 21.5, -47.5, 7.5], + [4.3, 1.5, -43.5, 3.5], + [4.4, 6.0, -46.0, 6.0]]) + + B = np.array([[1.0, 1.0, -3.0, 1.0], + [1.0, 3.0, -5.0, 4.4], + [1.0, 2.0, -4.0, 1.0], + [1.2, 3.0, -4.0, 4.0]]) + + assert_raises(ValueError, qz, A, B, sort=lambda ar, ai, beta: ai == 0) + if False: + AA, BB, Q, Z, sdim = qz(A, B, sort=lambda ar, ai, beta: ai == 0) + # assert_(sdim == 2) + assert_(sdim == 4) + assert_array_almost_equal(Q @ AA @ Z.T, A) + assert_array_almost_equal(Q @ BB @ Z.T, B) + + # test absolute values bc the sign is ambiguous and + # might be platform dependent + assert_array_almost_equal(np.abs(AA), np.abs(np.array( + [[35.7864, -80.9061, -12.0629, -9.498], + [0., 2.7638, -2.3505, 7.3256], + [0., 0., 0.6258, -0.0398], + [0., 0., 0., -12.8217]])), 4) + assert_array_almost_equal(np.abs(BB), np.abs(np.array( + [[4.5324, -8.7878, 3.2357, -3.5526], + [0., 1.4314, -2.1894, 0.9709], + [0., 0., 1.3126, -0.3468], + [0., 0., 0., 0.559]])), 4) + assert_array_almost_equal(np.abs(Q), np.abs(np.array( + [[-0.4193, -0.605, -0.1894, -0.6498], + [-0.5495, 0.6987, 0.2654, -0.3734], + [-0.4973, -0.3682, 0.6194, 0.4832], + [-0.5243, 0.1008, -0.7142, 0.4526]])), 4) + assert_array_almost_equal(np.abs(Z), np.abs(np.array( + [[-0.9471, -0.2971, -0.1217, 0.0055], + [-0.0367, 0.1209, 0.0358, 0.9913], + [0.3171, -0.9041, -0.2547, 0.1312], + [0.0346, 0.2824, -0.9587, 0.0014]])), 4) + + # test absolute values bc the sign is ambiguous and might be platform + # dependent + # assert_array_almost_equal(abs(AA), abs(np.array([ + # [3.8009, -69.4505, 50.3135, -43.2884], + # [0.0000, 9.2033, -0.2001, 5.9881], + # [0.0000, 0.0000, 1.4279, 4.4453], + # [0.0000, 0.0000, 0.9019, -1.1962]])), 4) + # assert_array_almost_equal(abs(BB), abs(np.array([ + # [1.9005, -10.2285, 0.8658, -5.2134], + # [0.0000, 2.3008, 0.7915, 0.4262], + # [0.0000, 0.0000, 0.8101, 0.0000], + # [0.0000, 0.0000, 0.0000, -0.2823]])), 4) + # assert_array_almost_equal(abs(Q), abs(np.array([ + # [0.4642, 0.7886, 0.2915, -0.2786], + # [0.5002, -0.5986, 0.5638, -0.2713], + # [0.5002, 0.0154, -0.0107, 0.8657], + # [0.5331, -0.1395, -0.7727, -0.3151]])), 4) + # assert_array_almost_equal(dot(Q,Q.T), eye(4)) + # assert_array_almost_equal(abs(Z), abs(np.array([ + # [0.9961, -0.0014, 0.0887, -0.0026], + # [0.0057, -0.0404, -0.0938, -0.9948], + # [0.0626, 0.7194, -0.6908, 0.0363], + # [0.0626, -0.6934, -0.7114, 0.0956]])), 4) + # assert_array_almost_equal(dot(Z,Z.T), eye(4)) + + # def test_qz_complex_sort(self): + # cA = np.array([ + # [-21.10+22.50*1j, 53.50+-50.50*1j, -34.50+127.50*1j, 7.50+ 0.50*1j], + # [-0.46+ -7.78*1j, -3.50+-37.50*1j, -15.50+ 58.50*1j,-10.50+ -1.50*1j], + # [ 4.30+ -5.50*1j, 39.70+-17.10*1j, -68.50+ 12.50*1j, -7.50+ -3.50*1j], + # [ 5.50+ 4.40*1j, 14.40+ 43.30*1j, -32.50+-46.00*1j,-19.00+-32.50*1j]]) + + # cB = np.array([ + # [1.00+ -5.00*1j, 1.60+ 1.20*1j,-3.00+ 0.00*1j, 0.00+ -1.00*1j], + # [0.80+ -0.60*1j, 3.00+ -5.00*1j,-4.00+ 3.00*1j,-2.40+ -3.20*1j], + # [1.00+ 0.00*1j, 2.40+ 1.80*1j,-4.00+ -5.00*1j, 0.00+ -3.00*1j], + # [0.00+ 1.00*1j,-1.80+ 2.40*1j, 0.00+ -4.00*1j, 4.00+ -5.00*1j]]) + + # AAS,BBS,QS,ZS,sdim = qz(cA,cB,sort='lhp') + + # eigenvalues = diag(AAS)/diag(BBS) + # assert_(np.all(np.real(eigenvalues[:sdim] < 0))) + # assert_(np.all(np.real(eigenvalues[sdim:] > 0))) + + def test_check_finite(self): + rng = np.random.RandomState(12345) + n = 5 + A = rng.random([n, n]) + B = rng.random([n, n]) + AA, BB, Q, Z = qz(A, B, check_finite=False) + assert_array_almost_equal(Q @ AA @ Z.T, A) + assert_array_almost_equal(Q @ BB @ Z.T, B) + assert_array_almost_equal(Q @ Q.T, eye(n)) + assert_array_almost_equal(Z @ Z.T, eye(n)) + assert_(np.all(diag(BB) >= 0)) + + +class TestOrdQZ: + @classmethod + def setup_class(cls): + # https://www.nag.com/lapack-ex/node119.html + A1 = np.array([[-21.10 - 22.50j, 53.5 - 50.5j, -34.5 + 127.5j, + 7.5 + 0.5j], + [-0.46 - 7.78j, -3.5 - 37.5j, -15.5 + 58.5j, + -10.5 - 1.5j], + [4.30 - 5.50j, 39.7 - 17.1j, -68.5 + 12.5j, + -7.5 - 3.5j], + [5.50 + 4.40j, 14.4 + 43.3j, -32.5 - 46.0j, + -19.0 - 32.5j]]) + + B1 = np.array([[1.0 - 5.0j, 1.6 + 1.2j, -3 + 0j, 0.0 - 1.0j], + [0.8 - 0.6j, .0 - 5.0j, -4 + 3j, -2.4 - 3.2j], + [1.0 + 0.0j, 2.4 + 1.8j, -4 - 5j, 0.0 - 3.0j], + [0.0 + 1.0j, -1.8 + 2.4j, 0 - 4j, 4.0 - 5.0j]]) + + # https://www.nag.com/numeric/fl/nagdoc_fl23/xhtml/F08/f08yuf.xml + A2 = np.array([[3.9, 12.5, -34.5, -0.5], + [4.3, 21.5, -47.5, 7.5], + [4.3, 21.5, -43.5, 3.5], + [4.4, 26.0, -46.0, 6.0]]) + + B2 = np.array([[1, 2, -3, 1], + [1, 3, -5, 4], + [1, 3, -4, 3], + [1, 3, -4, 4]]) + + # example with the eigenvalues + # -0.33891648, 1.61217396+0.74013521j, 1.61217396-0.74013521j, + # 0.61244091 + # thus featuring: + # * one complex conjugate eigenvalue pair, + # * one eigenvalue in the lhp + # * 2 eigenvalues in the unit circle + # * 2 non-real eigenvalues + A3 = np.array([[5., 1., 3., 3.], + [4., 4., 2., 7.], + [7., 4., 1., 3.], + [0., 4., 8., 7.]]) + B3 = np.array([[8., 10., 6., 10.], + [7., 7., 2., 9.], + [9., 1., 6., 6.], + [5., 1., 4., 7.]]) + + # example with infinite eigenvalues + A4 = np.eye(2) + B4 = np.diag([0, 1]) + + # example with (alpha, beta) = (0, 0) + A5 = np.diag([1, 0]) + + cls.A = [A1, A2, A3, A4, A5] + cls.B = [B1, B2, B3, B4, A5] + + def qz_decomp(self, sort): + with np.errstate(all='raise'): + ret = [ordqz(Ai, Bi, sort=sort) for Ai, Bi in zip(self.A, self.B)] + return tuple(ret) + + def check(self, A, B, sort, AA, BB, alpha, beta, Q, Z): + Id = np.eye(*A.shape) + # make sure Q and Z are orthogonal + assert_array_almost_equal(Q @ Q.T.conj(), Id) + assert_array_almost_equal(Z @ Z.T.conj(), Id) + # check factorization + assert_array_almost_equal(Q @ AA, A @ Z) + assert_array_almost_equal(Q @ BB, B @ Z) + # check shape of AA and BB + assert_array_equal(np.tril(AA, -2), np.zeros(AA.shape)) + assert_array_equal(np.tril(BB, -1), np.zeros(BB.shape)) + # check eigenvalues + for i in range(A.shape[0]): + # does the current diagonal element belong to a 2-by-2 block + # that was already checked? + if i > 0 and A[i, i - 1] != 0: + continue + # take care of 2-by-2 blocks + if i < AA.shape[0] - 1 and AA[i + 1, i] != 0: + evals, _ = eig(AA[i:i + 2, i:i + 2], BB[i:i + 2, i:i + 2]) + # make sure the pair of complex conjugate eigenvalues + # is ordered consistently (positive imaginary part first) + if evals[0].imag < 0: + evals = evals[[1, 0]] + tmp = alpha[i:i + 2]/beta[i:i + 2] + if tmp[0].imag < 0: + tmp = tmp[[1, 0]] + assert_array_almost_equal(evals, tmp) + else: + if alpha[i] == 0 and beta[i] == 0: + assert_equal(AA[i, i], 0) + assert_equal(BB[i, i], 0) + elif beta[i] == 0: + assert_equal(BB[i, i], 0) + else: + assert_almost_equal(AA[i, i]/BB[i, i], alpha[i]/beta[i]) + sortfun = _select_function(sort) + lastsort = True + for i in range(A.shape[0]): + cursort = sortfun(np.array([alpha[i]]), np.array([beta[i]])) + # once the sorting criterion was not matched all subsequent + # eigenvalues also shouldn't match + if not lastsort: + assert not cursort + lastsort = cursort + + def check_all(self, sort): + ret = self.qz_decomp(sort) + + for reti, Ai, Bi in zip(ret, self.A, self.B): + self.check(Ai, Bi, sort, *reti) + + def test_lhp(self): + self.check_all('lhp') + + def test_rhp(self): + self.check_all('rhp') + + def test_iuc(self): + self.check_all('iuc') + + def test_ouc(self): + self.check_all('ouc') + + def test_ref(self): + # real eigenvalues first (top-left corner) + def sort(x, y): + out = np.empty_like(x, dtype=bool) + nonzero = (y != 0) + out[~nonzero] = False + out[nonzero] = (x[nonzero]/y[nonzero]).imag == 0 + return out + + self.check_all(sort) + + def test_cef(self): + # complex eigenvalues first (top-left corner) + def sort(x, y): + out = np.empty_like(x, dtype=bool) + nonzero = (y != 0) + out[~nonzero] = False + out[nonzero] = (x[nonzero]/y[nonzero]).imag != 0 + return out + + self.check_all(sort) + + def test_diff_input_types(self): + ret = ordqz(self.A[1], self.B[2], sort='lhp') + self.check(self.A[1], self.B[2], 'lhp', *ret) + + ret = ordqz(self.B[2], self.A[1], sort='lhp') + self.check(self.B[2], self.A[1], 'lhp', *ret) + + def test_sort_explicit(self): + # Test order of the eigenvalues in the 2 x 2 case where we can + # explicitly compute the solution + A1 = np.eye(2) + B1 = np.diag([-2, 0.5]) + expected1 = [('lhp', [-0.5, 2]), + ('rhp', [2, -0.5]), + ('iuc', [-0.5, 2]), + ('ouc', [2, -0.5])] + A2 = np.eye(2) + B2 = np.diag([-2 + 1j, 0.5 + 0.5j]) + expected2 = [('lhp', [1/(-2 + 1j), 1/(0.5 + 0.5j)]), + ('rhp', [1/(0.5 + 0.5j), 1/(-2 + 1j)]), + ('iuc', [1/(-2 + 1j), 1/(0.5 + 0.5j)]), + ('ouc', [1/(0.5 + 0.5j), 1/(-2 + 1j)])] + # 'lhp' is ambiguous so don't test it + A3 = np.eye(2) + B3 = np.diag([2, 0]) + expected3 = [('rhp', [0.5, np.inf]), + ('iuc', [0.5, np.inf]), + ('ouc', [np.inf, 0.5])] + # 'rhp' is ambiguous so don't test it + A4 = np.eye(2) + B4 = np.diag([-2, 0]) + expected4 = [('lhp', [-0.5, np.inf]), + ('iuc', [-0.5, np.inf]), + ('ouc', [np.inf, -0.5])] + A5 = np.diag([0, 1]) + B5 = np.diag([0, 0.5]) + # 'lhp' and 'iuc' are ambiguous so don't test them + expected5 = [('rhp', [2, np.nan]), + ('ouc', [2, np.nan])] + + A = [A1, A2, A3, A4, A5] + B = [B1, B2, B3, B4, B5] + expected = [expected1, expected2, expected3, expected4, expected5] + for Ai, Bi, expectedi in zip(A, B, expected): + for sortstr, expected_eigvals in expectedi: + _, _, alpha, beta, _, _ = ordqz(Ai, Bi, sort=sortstr) + azero = (alpha == 0) + bzero = (beta == 0) + x = np.empty_like(alpha) + x[azero & bzero] = np.nan + x[~azero & bzero] = np.inf + x[~bzero] = alpha[~bzero]/beta[~bzero] + assert_allclose(expected_eigvals, x) + + +class TestOrdQZWorkspaceSize: + @pytest.mark.fail_slow(5) + def test_decompose(self): + rng = np.random.RandomState(12345) + N = 202 + # raises error if lwork parameter to dtrsen is too small + for ddtype in [np.float32, np.float64]: + A = rng.random((N, N)).astype(ddtype) + B = rng.random((N, N)).astype(ddtype) + # sort = lambda ar, ai, b: ar**2 + ai**2 < b**2 + _ = ordqz(A, B, sort=lambda alpha, beta: alpha < beta, + output='real') + + for ddtype in [np.complex128, np.complex64]: + A = rng.random((N, N)).astype(ddtype) + B = rng.random((N, N)).astype(ddtype) + _ = ordqz(A, B, sort=lambda alpha, beta: alpha < beta, + output='complex') + + @pytest.mark.slow + def test_decompose_ouc(self): + rng = np.random.RandomState(12345) + N = 202 + # segfaults if lwork parameter to dtrsen is too small + for ddtype in [np.float32, np.float64, np.complex128, np.complex64]: + A = rng.random((N, N)).astype(ddtype) + B = rng.random((N, N)).astype(ddtype) + S, T, alpha, beta, U, V = ordqz(A, B, sort='ouc') + + +class TestDatacopied: + + def test_datacopied(self): + from scipy.linalg._decomp import _datacopied + + M = matrix([[0, 1], [2, 3]]) + A = asarray(M) + L = M.tolist() + M2 = M.copy() + + class Fake1: + def __array__(self, dtype=None, copy=None): + return A + + class Fake2: + __array_interface__ = A.__array_interface__ + + F1 = Fake1() + F2 = Fake2() + + for item, status in [(M, False), (A, False), (L, True), + (M2, False), (F1, False), (F2, False)]: + arr = asarray(item) + assert_equal(_datacopied(arr, item), status, + err_msg=repr(item)) + + +def test_aligned_mem_float(): + """Check linalg works with non-aligned memory (float32)""" + # Allocate 402 bytes of memory (allocated on boundary) + a = arange(402, dtype=np.uint8) + + # Create an array with boundary offset 4 + z = np.frombuffer(a.data, offset=2, count=100, dtype=float32) + z.shape = 10, 10 + + eig(z, overwrite_a=True) + eig(z.T, overwrite_a=True) + + +@pytest.mark.skipif(platform.machine() == 'ppc64le', + reason="crashes on ppc64le") +def test_aligned_mem(): + """Check linalg works with non-aligned memory (float64)""" + # Allocate 804 bytes of memory (allocated on boundary) + a = arange(804, dtype=np.uint8) + + # Create an array with boundary offset 4 + z = np.frombuffer(a.data, offset=4, count=100, dtype=float) + z.shape = 10, 10 + + eig(z, overwrite_a=True) + eig(z.T, overwrite_a=True) + + +def test_aligned_mem_complex(): + """Check that complex objects don't need to be completely aligned""" + # Allocate 1608 bytes of memory (allocated on boundary) + a = zeros(1608, dtype=np.uint8) + + # Create an array with boundary offset 8 + z = np.frombuffer(a.data, offset=8, count=100, dtype=complex) + z.shape = 10, 10 + + eig(z, overwrite_a=True) + # This does not need special handling + eig(z.T, overwrite_a=True) + + +def check_lapack_misaligned(func, args, kwargs): + args = list(args) + for i in range(len(args)): + a = args[:] + if isinstance(a[i], np.ndarray): + # Try misaligning a[i] + aa = np.zeros(a[i].size*a[i].dtype.itemsize+8, dtype=np.uint8) + aa = np.frombuffer(aa.data, offset=4, count=a[i].size, + dtype=a[i].dtype) + aa.shape = a[i].shape + aa[...] = a[i] + a[i] = aa + func(*a, **kwargs) + if len(a[i].shape) > 1: + a[i] = a[i].T + func(*a, **kwargs) + + +@pytest.mark.xfail(run=False, + reason="Ticket #1152, triggers a segfault in rare cases.") +def test_lapack_misaligned(): + M = np.eye(10, dtype=float) + R = np.arange(100) + R.shape = 10, 10 + S = np.arange(20000, dtype=np.uint8) + S = np.frombuffer(S.data, offset=4, count=100, dtype=float) + S.shape = 10, 10 + b = np.ones(10) + LU, piv = lu_factor(S) + for (func, args, kwargs) in [ + (eig, (S,), dict(overwrite_a=True)), # crash + (eigvals, (S,), dict(overwrite_a=True)), # no crash + (lu, (S,), dict(overwrite_a=True)), # no crash + (lu_factor, (S,), dict(overwrite_a=True)), # no crash + (lu_solve, ((LU, piv), b), dict(overwrite_b=True)), + (solve, (S, b), dict(overwrite_a=True, overwrite_b=True)), + (svd, (M,), dict(overwrite_a=True)), # no crash + (svd, (R,), dict(overwrite_a=True)), # no crash + (svd, (S,), dict(overwrite_a=True)), # crash + (svdvals, (S,), dict()), # no crash + (svdvals, (S,), dict(overwrite_a=True)), # crash + (cholesky, (M,), dict(overwrite_a=True)), # no crash + (qr, (S,), dict(overwrite_a=True)), # crash + (rq, (S,), dict(overwrite_a=True)), # crash + (hessenberg, (S,), dict(overwrite_a=True)), # crash + (schur, (S,), dict(overwrite_a=True)), # crash + ]: + check_lapack_misaligned(func, args, kwargs) +# not properly tested +# cholesky, rsf2csf, lu_solve, solve, eig_banded, eigvals_banded, eigh, diagsvd + + +class TestOverwrite: + def test_eig(self): + assert_no_overwrite(eig, [(3, 3)]) + assert_no_overwrite(eig, [(3, 3), (3, 3)]) + + def test_eigh(self): + assert_no_overwrite(eigh, [(3, 3)]) + assert_no_overwrite(eigh, [(3, 3), (3, 3)]) + + def test_eig_banded(self): + assert_no_overwrite(eig_banded, [(3, 2)]) + + def test_eigvals(self): + assert_no_overwrite(eigvals, [(3, 3)]) + + def test_eigvalsh(self): + assert_no_overwrite(eigvalsh, [(3, 3)]) + + def test_eigvals_banded(self): + assert_no_overwrite(eigvals_banded, [(3, 2)]) + + def test_hessenberg(self): + assert_no_overwrite(hessenberg, [(3, 3)]) + + def test_lu_factor(self): + assert_no_overwrite(lu_factor, [(3, 3)]) + + def test_lu_solve(self): + x = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 8]]) + xlu = lu_factor(x) + assert_no_overwrite(lambda b: lu_solve(xlu, b), [(3,)]) + + def test_lu(self): + assert_no_overwrite(lu, [(3, 3)]) + + def test_qr(self): + assert_no_overwrite(qr, [(3, 3)]) + + def test_rq(self): + assert_no_overwrite(rq, [(3, 3)]) + + def test_schur(self): + assert_no_overwrite(schur, [(3, 3)]) + + def test_schur_complex(self): + assert_no_overwrite(lambda a: schur(a, 'complex'), [(3, 3)], + dtypes=[np.float32, np.float64]) + + def test_svd(self): + assert_no_overwrite(svd, [(3, 3)]) + assert_no_overwrite(lambda a: svd(a, lapack_driver='gesvd'), [(3, 3)]) + + def test_svdvals(self): + assert_no_overwrite(svdvals, [(3, 3)]) + + +def _check_orth(n, dtype, skip_big=False): + X = np.ones((n, 2), dtype=float).astype(dtype) + + eps = np.finfo(dtype).eps + tol = 1000 * eps + + Y = orth(X) + assert_equal(Y.shape, (n, 1)) + assert_allclose(Y, Y.mean(), atol=tol) + + Y = orth(X.T) + assert_equal(Y.shape, (2, 1)) + assert_allclose(Y, Y.mean(), atol=tol) + + if n > 5 and not skip_big: + rng = np.random.RandomState(1) + X = rng.rand(n, 5) @ rng.rand(5, n) + X = X + 1e-4 * rng.rand(n, 1) @ rng.rand(1, n) + X = X.astype(dtype) + + Y = orth(X, rcond=1e-3) + assert_equal(Y.shape, (n, 5)) + + Y = orth(X, rcond=1e-6) + assert_equal(Y.shape, (n, 5 + 1)) + + +@pytest.mark.slow +@pytest.mark.skipif(np.dtype(np.intp).itemsize < 8, + reason="test only on 64-bit, else too slow") +def test_orth_memory_efficiency(): + # Pick n so that 16*n bytes is reasonable but 8*n*n bytes is unreasonable. + # Keep in mind that @pytest.mark.slow tests are likely to be running + # under configurations that support 4Gb+ memory for tests related to + # 32 bit overflow. + n = 10*1000*1000 + try: + _check_orth(n, np.float64, skip_big=True) + except MemoryError as e: + raise AssertionError( + 'memory error perhaps caused by orth regression' + ) from e + + +def test_orth(): + dtypes = [np.float32, np.float64, np.complex64, np.complex128] + sizes = [1, 2, 3, 10, 100] + for dt, n in itertools.product(dtypes, sizes): + _check_orth(n, dt) + +@pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) +def test_orth_empty(dt): + a = np.empty((0, 0), dtype=dt) + a0 = np.eye(2, dtype=dt) + + oa = orth(a) + assert oa.dtype == orth(a0).dtype + assert oa.shape == (0, 0) + + +class TestNullSpace: + def test_null_space(self): + rng = np.random.RandomState(1) + + dtypes = [np.float32, np.float64, np.complex64, np.complex128] + sizes = [1, 2, 3, 10, 100] + + for dt, n in itertools.product(dtypes, sizes): + X = np.ones((2, n), dtype=dt) + + eps = np.finfo(dt).eps + tol = 1000 * eps + + Y = null_space(X) + assert_equal(Y.shape, (n, n-1)) + assert_allclose(X @ Y, 0, atol=tol) + + Y = null_space(X.T) + assert_equal(Y.shape, (2, 1)) + assert_allclose(X.T @ Y, 0, atol=tol) + + X = rng.randn(1 + n//2, n) + Y = null_space(X) + assert_equal(Y.shape, (n, n - 1 - n//2)) + assert_allclose(X @ Y, 0, atol=tol) + + if n > 5: + rng = np.random.RandomState(1) + X = rng.rand(n, 5) @ rng.rand(5, n) + X = X + 1e-4 * rng.rand(n, 1) @ rng.rand(1, n) + X = X.astype(dt) + + Y = null_space(X, rcond=1e-3) + assert_equal(Y.shape, (n, n - 5)) + + Y = null_space(X, rcond=1e-6) + assert_equal(Y.shape, (n, n - 6)) + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_null_space_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + a0 = np.eye(2, dtype=dt) + nsa = null_space(a) + + assert nsa.shape == (0, 0) + assert nsa.dtype == null_space(a0).dtype + + @pytest.mark.parametrize("overwrite_a", [True, False]) + @pytest.mark.parametrize("check_finite", [True, False]) + @pytest.mark.parametrize("lapack_driver", ["gesdd", "gesvd"]) + def test_null_space_options(self, overwrite_a, check_finite, lapack_driver): + rng = np.random.default_rng(42887289350573064398746) + n = 10 + X = rng.standard_normal((1 + n//2, n)) + Y = null_space(X.copy(), overwrite_a=overwrite_a, check_finite=check_finite, + lapack_driver=lapack_driver) + assert_allclose(X @ Y, 0, atol=np.finfo(X.dtype).eps*100) + + +def test_subspace_angles(): + H = hadamard(8, float) + A = H[:, :3] + B = H[:, 3:] + assert_allclose(subspace_angles(A, B), [np.pi / 2.] * 3, atol=1e-14) + assert_allclose(subspace_angles(B, A), [np.pi / 2.] * 3, atol=1e-14) + for x in (A, B): + assert_allclose(subspace_angles(x, x), np.zeros(x.shape[1]), + atol=1e-14) + # From MATLAB function "subspace", which effectively only returns the + # last value that we calculate + x = np.array( + [[0.537667139546100, 0.318765239858981, 3.578396939725760, 0.725404224946106], # noqa: E501 + [1.833885014595086, -1.307688296305273, 2.769437029884877, -0.063054873189656], # noqa: E501 + [-2.258846861003648, -0.433592022305684, -1.349886940156521, 0.714742903826096], # noqa: E501 + [0.862173320368121, 0.342624466538650, 3.034923466331855, -0.204966058299775]]) # noqa: E501 + expected = 1.481454682101605 + assert_allclose(subspace_angles(x[:, :2], x[:, 2:])[0], expected, + rtol=1e-12) + assert_allclose(subspace_angles(x[:, 2:], x[:, :2])[0], expected, + rtol=1e-12) + expected = 0.746361174247302 + assert_allclose(subspace_angles(x[:, :2], x[:, [2]]), expected, rtol=1e-12) + assert_allclose(subspace_angles(x[:, [2]], x[:, :2]), expected, rtol=1e-12) + expected = 0.487163718534313 + assert_allclose(subspace_angles(x[:, :3], x[:, [3]]), expected, rtol=1e-12) + assert_allclose(subspace_angles(x[:, [3]], x[:, :3]), expected, rtol=1e-12) + expected = 0.328950515907756 + assert_allclose(subspace_angles(x[:, :2], x[:, 1:]), [expected, 0], + atol=1e-12) + # Degenerate conditions + assert_raises(ValueError, subspace_angles, x[0], x) + assert_raises(ValueError, subspace_angles, x, x[0]) + assert_raises(ValueError, subspace_angles, x[:-1], x) + + # Test branch if mask.any is True: + A = np.array([[1, 0, 0], + [0, 1, 0], + [0, 0, 1], + [0, 0, 0], + [0, 0, 0]]) + B = np.array([[1, 0, 0], + [0, 1, 0], + [0, 0, 0], + [0, 0, 0], + [0, 0, 1]]) + expected = np.array([np.pi/2, 0, 0]) + assert_allclose(subspace_angles(A, B), expected, rtol=1e-12) + + # Complex + # second column in "b" does not affect result, just there so that + # b can have more cols than a, and vice-versa (both conditional code paths) + a = [[1 + 1j], [0]] + b = [[1 - 1j, 0], [0, 1]] + assert_allclose(subspace_angles(a, b), 0., atol=1e-14) + assert_allclose(subspace_angles(b, a), 0., atol=1e-14) + + # Empty + a = np.empty((0, 0)) + b = np.empty((0, 0)) + assert_allclose(subspace_angles(a, b), np.empty((0,))) + a = np.empty((2, 0)) + b = np.empty((2, 0)) + assert_allclose(subspace_angles(a, b), np.empty((0,))) + a = np.empty((0, 2)) + b = np.empty((0, 3)) + assert_allclose(subspace_angles(a, b), np.empty((0,))) + + +class TestCDF2RDF: + + def matmul(self, a, b): + return np.einsum('...ij,...jk->...ik', a, b) + + def assert_eig_valid(self, w, v, x): + assert_array_almost_equal( + self.matmul(v, w), + self.matmul(x, v) + ) + + def test_single_array0x0real(self): + # eig doesn't support 0x0 in old versions of numpy + X = np.empty((0, 0)) + w, v = np.empty(0), np.empty((0, 0)) + wr, vr = cdf2rdf(w, v) + self.assert_eig_valid(wr, vr, X) + + def test_single_array2x2_real(self): + X = np.array([[1, 2], [3, -1]]) + w, v = np.linalg.eig(X) + wr, vr = cdf2rdf(w, v) + self.assert_eig_valid(wr, vr, X) + + def test_single_array2x2_complex(self): + X = np.array([[1, 2], [-2, 1]]) + w, v = np.linalg.eig(X) + wr, vr = cdf2rdf(w, v) + self.assert_eig_valid(wr, vr, X) + + def test_single_array3x3_real(self): + X = np.array([[1, 2, 3], [1, 2, 3], [2, 5, 6]]) + w, v = np.linalg.eig(X) + wr, vr = cdf2rdf(w, v) + self.assert_eig_valid(wr, vr, X) + + def test_single_array3x3_complex(self): + X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]]) + w, v = np.linalg.eig(X) + wr, vr = cdf2rdf(w, v) + self.assert_eig_valid(wr, vr, X) + + def test_random_1d_stacked_arrays(self): + # cannot test M == 0 due to bug in old numpy + for M in range(1, 7): + np.random.seed(999999999) + X = np.random.rand(100, M, M) + w, v = np.linalg.eig(X) + wr, vr = cdf2rdf(w, v) + self.assert_eig_valid(wr, vr, X) + + def test_random_2d_stacked_arrays(self): + # cannot test M == 0 due to bug in old numpy + for M in range(1, 7): + X = np.random.rand(10, 10, M, M) + w, v = np.linalg.eig(X) + wr, vr = cdf2rdf(w, v) + self.assert_eig_valid(wr, vr, X) + + def test_low_dimensionality_error(self): + w, v = np.empty(()), np.array((2,)) + assert_raises(ValueError, cdf2rdf, w, v) + + def test_not_square_error(self): + # Check that passing a non-square array raises a ValueError. + w, v = np.arange(3), np.arange(6).reshape(3, 2) + assert_raises(ValueError, cdf2rdf, w, v) + + def test_swapped_v_w_error(self): + # Check that exchanging places of w and v raises ValueError. + X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]]) + w, v = np.linalg.eig(X) + assert_raises(ValueError, cdf2rdf, v, w) + + def test_non_associated_error(self): + # Check that passing non-associated eigenvectors raises a ValueError. + w, v = np.arange(3), np.arange(16).reshape(4, 4) + assert_raises(ValueError, cdf2rdf, w, v) + + def test_not_conjugate_pairs(self): + # Check that passing non-conjugate pairs raises a ValueError. + X = np.array([[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]]) + w, v = np.linalg.eig(X) + assert_raises(ValueError, cdf2rdf, w, v) + + # different arrays in the stack, so not conjugate + X = np.array([ + [[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]], + [[1, 2, 3], [1, 2, 3], [2, 5, 6-1j]], + ]) + w, v = np.linalg.eig(X) + assert_raises(ValueError, cdf2rdf, w, v) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_cholesky.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_cholesky.py new file mode 100644 index 0000000000000000000000000000000000000000..61bbc7e544f10fc834034fbadd7141f6deb1d423 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_cholesky.py @@ -0,0 +1,268 @@ +import pytest +import numpy as np +from numpy.testing import assert_array_almost_equal +from pytest import raises as assert_raises + +from numpy import array, transpose, dot, conjugate, zeros_like, empty +from numpy.random import random +from scipy.linalg import (cholesky, cholesky_banded, cho_solve_banded, + cho_factor, cho_solve) + +from scipy.linalg._testutils import assert_no_overwrite + + +class TestCholesky: + + def test_simple(self): + a = [[8, 2, 3], [2, 9, 3], [3, 3, 6]] + c = cholesky(a) + assert_array_almost_equal(dot(transpose(c), c), a) + c = transpose(c) + a = dot(c, transpose(c)) + assert_array_almost_equal(cholesky(a, lower=1), c) + + def test_check_finite(self): + a = [[8, 2, 3], [2, 9, 3], [3, 3, 6]] + c = cholesky(a, check_finite=False) + assert_array_almost_equal(dot(transpose(c), c), a) + c = transpose(c) + a = dot(c, transpose(c)) + assert_array_almost_equal(cholesky(a, lower=1, check_finite=False), c) + + def test_simple_complex(self): + m = array([[3+1j, 3+4j, 5], [0, 2+2j, 2+7j], [0, 0, 7+4j]]) + a = dot(transpose(conjugate(m)), m) + c = cholesky(a) + a1 = dot(transpose(conjugate(c)), c) + assert_array_almost_equal(a, a1) + c = transpose(c) + a = dot(c, transpose(conjugate(c))) + assert_array_almost_equal(cholesky(a, lower=1), c) + + def test_random(self): + n = 20 + for k in range(2): + m = random([n, n]) + for i in range(n): + m[i, i] = 20*(.1+m[i, i]) + a = dot(transpose(m), m) + c = cholesky(a) + a1 = dot(transpose(c), c) + assert_array_almost_equal(a, a1) + c = transpose(c) + a = dot(c, transpose(c)) + assert_array_almost_equal(cholesky(a, lower=1), c) + + def test_random_complex(self): + n = 20 + for k in range(2): + m = random([n, n])+1j*random([n, n]) + for i in range(n): + m[i, i] = 20*(.1+abs(m[i, i])) + a = dot(transpose(conjugate(m)), m) + c = cholesky(a) + a1 = dot(transpose(conjugate(c)), c) + assert_array_almost_equal(a, a1) + c = transpose(c) + a = dot(c, transpose(conjugate(c))) + assert_array_almost_equal(cholesky(a, lower=1), c) + + @pytest.mark.xslow + def test_int_overflow(self): + # regression test for + # https://github.com/scipy/scipy/issues/17436 + # the problem was an int overflow in zeroing out + # the unused triangular part + n = 47_000 + x = np.eye(n, dtype=np.float64, order='F') + x[:4, :4] = np.array([[4, -2, 3, -1], + [-2, 4, -3, 1], + [3, -3, 5, 0], + [-1, 1, 0, 5]]) + + cholesky(x, check_finite=False, overwrite_a=True) # should not segfault + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt, dt_b): + a = empty((0, 0), dtype=dt) + + c = cholesky(a) + assert c.shape == (0, 0) + assert c.dtype == cholesky(np.eye(2, dtype=dt)).dtype + + c_and_lower = (c, True) + b = np.asarray([], dtype=dt_b) + x = cho_solve(c_and_lower, b) + assert x.shape == (0,) + assert x.dtype == cho_solve((np.eye(2, dtype=dt), True), + np.ones(2, dtype=dt_b)).dtype + + b = empty((0, 0), dtype=dt_b) + x = cho_solve(c_and_lower, b) + assert x.shape == (0, 0) + assert x.dtype == cho_solve((np.eye(2, dtype=dt), True), + np.ones(2, dtype=dt_b)).dtype + + a1 = array([]) + a2 = array([[]]) + a3 = [] + a4 = [[]] + for x in ([a1, a2, a3, a4]): + assert_raises(ValueError, cholesky, x) + + +class TestCholeskyBanded: + """Tests for cholesky_banded() and cho_solve_banded.""" + + def test_check_finite(self): + # Symmetric positive definite banded matrix `a` + a = array([[4.0, 1.0, 0.0, 0.0], + [1.0, 4.0, 0.5, 0.0], + [0.0, 0.5, 4.0, 0.2], + [0.0, 0.0, 0.2, 4.0]]) + # Banded storage form of `a`. + ab = array([[-1.0, 1.0, 0.5, 0.2], + [4.0, 4.0, 4.0, 4.0]]) + c = cholesky_banded(ab, lower=False, check_finite=False) + ufac = zeros_like(a) + ufac[list(range(4)), list(range(4))] = c[-1] + ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:] + assert_array_almost_equal(a, dot(ufac.T, ufac)) + + b = array([0.0, 0.5, 4.2, 4.2]) + x = cho_solve_banded((c, False), b, check_finite=False) + assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0]) + + def test_upper_real(self): + # Symmetric positive definite banded matrix `a` + a = array([[4.0, 1.0, 0.0, 0.0], + [1.0, 4.0, 0.5, 0.0], + [0.0, 0.5, 4.0, 0.2], + [0.0, 0.0, 0.2, 4.0]]) + # Banded storage form of `a`. + ab = array([[-1.0, 1.0, 0.5, 0.2], + [4.0, 4.0, 4.0, 4.0]]) + c = cholesky_banded(ab, lower=False) + ufac = zeros_like(a) + ufac[list(range(4)), list(range(4))] = c[-1] + ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:] + assert_array_almost_equal(a, dot(ufac.T, ufac)) + + b = array([0.0, 0.5, 4.2, 4.2]) + x = cho_solve_banded((c, False), b) + assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0]) + + def test_upper_complex(self): + # Hermitian positive definite banded matrix `a` + a = array([[4.0, 1.0, 0.0, 0.0], + [1.0, 4.0, 0.5, 0.0], + [0.0, 0.5, 4.0, -0.2j], + [0.0, 0.0, 0.2j, 4.0]]) + # Banded storage form of `a`. + ab = array([[-1.0, 1.0, 0.5, -0.2j], + [4.0, 4.0, 4.0, 4.0]]) + c = cholesky_banded(ab, lower=False) + ufac = zeros_like(a) + ufac[list(range(4)), list(range(4))] = c[-1] + ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:] + assert_array_almost_equal(a, dot(ufac.conj().T, ufac)) + + b = array([0.0, 0.5, 4.0-0.2j, 0.2j + 4.0]) + x = cho_solve_banded((c, False), b) + assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0]) + + def test_lower_real(self): + # Symmetric positive definite banded matrix `a` + a = array([[4.0, 1.0, 0.0, 0.0], + [1.0, 4.0, 0.5, 0.0], + [0.0, 0.5, 4.0, 0.2], + [0.0, 0.0, 0.2, 4.0]]) + # Banded storage form of `a`. + ab = array([[4.0, 4.0, 4.0, 4.0], + [1.0, 0.5, 0.2, -1.0]]) + c = cholesky_banded(ab, lower=True) + lfac = zeros_like(a) + lfac[list(range(4)), list(range(4))] = c[0] + lfac[(1, 2, 3), (0, 1, 2)] = c[1, :3] + assert_array_almost_equal(a, dot(lfac, lfac.T)) + + b = array([0.0, 0.5, 4.2, 4.2]) + x = cho_solve_banded((c, True), b) + assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0]) + + def test_lower_complex(self): + # Hermitian positive definite banded matrix `a` + a = array([[4.0, 1.0, 0.0, 0.0], + [1.0, 4.0, 0.5, 0.0], + [0.0, 0.5, 4.0, -0.2j], + [0.0, 0.0, 0.2j, 4.0]]) + # Banded storage form of `a`. + ab = array([[4.0, 4.0, 4.0, 4.0], + [1.0, 0.5, 0.2j, -1.0]]) + c = cholesky_banded(ab, lower=True) + lfac = zeros_like(a) + lfac[list(range(4)), list(range(4))] = c[0] + lfac[(1, 2, 3), (0, 1, 2)] = c[1, :3] + assert_array_almost_equal(a, dot(lfac, lfac.conj().T)) + + b = array([0.0, 0.5j, 3.8j, 3.8]) + x = cho_solve_banded((c, True), b) + assert_array_almost_equal(x, [0.0, 0.0, 1.0j, 1.0]) + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt, dt_b): + ab = empty((0, 0), dtype=dt) + + cb = cholesky_banded(ab) + assert cb.shape == (0, 0) + + m = cholesky_banded(np.array([[0, 0], [1, 1]], dtype=dt)) + assert cb.dtype == m.dtype + + cb_and_lower = (cb, True) + b = np.asarray([], dtype=dt_b) + x = cho_solve_banded(cb_and_lower, b) + assert x.shape == (0,) + + dtype_nonempty = cho_solve_banded((m, True), np.ones(2, dtype=dt_b)).dtype + assert x.dtype == dtype_nonempty + + b = empty((0, 0), dtype=dt_b) + x = cho_solve_banded(cb_and_lower, b) + assert x.shape == (0, 0) + assert x.dtype == dtype_nonempty + + +class TestOverwrite: + def test_cholesky(self): + assert_no_overwrite(cholesky, [(3, 3)]) + + def test_cho_factor(self): + assert_no_overwrite(cho_factor, [(3, 3)]) + + def test_cho_solve(self): + x = array([[2, -1, 0], [-1, 2, -1], [0, -1, 2]]) + xcho = cho_factor(x) + assert_no_overwrite(lambda b: cho_solve(xcho, b), [(3,)]) + + def test_cholesky_banded(self): + assert_no_overwrite(cholesky_banded, [(2, 3)]) + + def test_cho_solve_banded(self): + x = array([[0, -1, -1], [2, 2, 2]]) + xcho = cholesky_banded(x) + assert_no_overwrite(lambda b: cho_solve_banded((xcho, False), b), + [(3,)]) + +class TestChoFactor: + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + x, lower = cho_factor(a) + + assert x.shape == (0, 0) + + xx, lower = cho_factor(np.eye(2, dtype=dt)) + assert x.dtype == xx.dtype diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_cossin.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_cossin.py new file mode 100644 index 0000000000000000000000000000000000000000..df112f0e4cf75d03d2787c24306665b7027967ee --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_cossin.py @@ -0,0 +1,300 @@ +import pytest +import numpy as np +from numpy.random import default_rng +from numpy.testing import assert_allclose + +from scipy import linalg +from scipy.linalg.lapack import _compute_lwork +from scipy.stats import ortho_group, unitary_group +from scipy.linalg import cossin, get_lapack_funcs + +REAL_DTYPES = (np.float32, np.float64) +COMPLEX_DTYPES = (np.complex64, np.complex128) +DTYPES = REAL_DTYPES + COMPLEX_DTYPES + + +@pytest.mark.parametrize('dtype_', DTYPES) +@pytest.mark.parametrize('m, p, q', + [ + (2, 1, 1), + (3, 2, 1), + (3, 1, 2), + (4, 2, 2), + (4, 1, 2), + (40, 12, 20), + (40, 30, 1), + (40, 1, 30), + (100, 50, 1), + (100, 50, 50), + ]) +@pytest.mark.parametrize('swap_sign', [True, False]) +def test_cossin(dtype_, m, p, q, swap_sign): + rng = default_rng(1708093570726217) + if dtype_ in COMPLEX_DTYPES: + x = np.array(unitary_group.rvs(m, random_state=rng), dtype=dtype_) + else: + x = np.array(ortho_group.rvs(m, random_state=rng), dtype=dtype_) + + u, cs, vh = cossin(x, p, q, + swap_sign=swap_sign) + assert_allclose(x, u @ cs @ vh, rtol=0., atol=m*1e3*np.finfo(dtype_).eps) + assert u.dtype == dtype_ + # Test for float32 or float 64 + assert cs.dtype == np.real(u).dtype + assert vh.dtype == dtype_ + + u, cs, vh = cossin([x[:p, :q], x[:p, q:], x[p:, :q], x[p:, q:]], + swap_sign=swap_sign) + assert_allclose(x, u @ cs @ vh, rtol=0., atol=m*1e3*np.finfo(dtype_).eps) + assert u.dtype == dtype_ + assert cs.dtype == np.real(u).dtype + assert vh.dtype == dtype_ + + _, cs2, vh2 = cossin(x, p, q, + compute_u=False, + swap_sign=swap_sign) + assert_allclose(cs, cs2, rtol=0., atol=10*np.finfo(dtype_).eps) + assert_allclose(vh, vh2, rtol=0., atol=10*np.finfo(dtype_).eps) + + u2, cs2, _ = cossin(x, p, q, + compute_vh=False, + swap_sign=swap_sign) + assert_allclose(u, u2, rtol=0., atol=10*np.finfo(dtype_).eps) + assert_allclose(cs, cs2, rtol=0., atol=10*np.finfo(dtype_).eps) + + _, cs2, _ = cossin(x, p, q, + compute_u=False, + compute_vh=False, + swap_sign=swap_sign) + assert_allclose(cs, cs2, rtol=0., atol=10*np.finfo(dtype_).eps) + + +def test_cossin_mixed_types(): + rng = default_rng(1708093736390459) + x = np.array(ortho_group.rvs(4, random_state=rng), dtype=np.float64) + u, cs, vh = cossin([x[:2, :2], + np.array(x[:2, 2:], dtype=np.complex128), + x[2:, :2], + x[2:, 2:]]) + + assert u.dtype == np.complex128 + assert cs.dtype == np.float64 + assert vh.dtype == np.complex128 + assert_allclose(x, u @ cs @ vh, rtol=0., + atol=1e4 * np.finfo(np.complex128).eps) + + +def test_cossin_error_incorrect_subblocks(): + with pytest.raises(ValueError, match="be due to missing p, q arguments."): + cossin(([1, 2], [3, 4, 5], [6, 7], [8, 9, 10])) + + +def test_cossin_error_empty_subblocks(): + with pytest.raises(ValueError, match="x11.*empty"): + cossin(([], [], [], [])) + with pytest.raises(ValueError, match="x12.*empty"): + cossin(([1, 2], [], [6, 7], [8, 9, 10])) + with pytest.raises(ValueError, match="x21.*empty"): + cossin(([1, 2], [3, 4, 5], [], [8, 9, 10])) + with pytest.raises(ValueError, match="x22.*empty"): + cossin(([1, 2], [3, 4, 5], [2], [])) + + +def test_cossin_error_missing_partitioning(): + with pytest.raises(ValueError, match=".*exactly four arrays.* got 2"): + cossin(unitary_group.rvs(2)) + + with pytest.raises(ValueError, match=".*might be due to missing p, q"): + cossin(unitary_group.rvs(4)) + + +def test_cossin_error_non_iterable(): + with pytest.raises(ValueError, match="containing the subblocks of X"): + cossin(12j) + + +def test_cossin_error_non_square(): + with pytest.raises(ValueError, match="only supports square"): + cossin(np.array([[1, 2]]), 1, 1) + + +def test_cossin_error_partitioning(): + x = np.array(ortho_group.rvs(4), dtype=np.float64) + with pytest.raises(ValueError, match="invalid p=0.*0= m) or (q >= m): + pytest.skip("`0 < p < m` and `0 < q < m` must hold") + + # Generate unitary input + rng = np.random.default_rng(329548272348596421) + X = unitary_group.rvs(m, random_state=rng) + np.testing.assert_allclose(X @ X.conj().T, np.eye(m), atol=1e-15) + + # Perform the decomposition + u0, cs0, vh0 = linalg.cossin(X, p=p, q=q, separate=True, swap_sign=swap_sign) + u1, u2 = u0 + v1, v2 = vh0 + v1, v2 = v1.conj().T, v2.conj().T + + # "U1, U2, V1, V2 are square orthogonal/unitary matrices + # of dimensions (p,p), (m-p,m-p), (q,q), and (m-q,m-q) respectively" + np.testing.assert_allclose(u1 @ u1.conj().T, np.eye(p), atol=1e-13) + np.testing.assert_allclose(u2 @ u2.conj().T, np.eye(m-p), atol=1e-13) + np.testing.assert_allclose(v1 @ v1.conj().T, np.eye(q), atol=1e-13) + np.testing.assert_allclose(v2 @ v2.conj().T, np.eye(m-q), atol=1e-13) + + # "and C and S are (r, r) nonnegative diagonal matrices..." + C = np.diag(np.cos(cs0)) + S = np.diag(np.sin(cs0)) + # "...satisfying C^2 + S^2 = I where r = min(p, m-p, q, m-q)." + r = min(p, m-p, q, m-q) + np.testing.assert_allclose(C**2 + S**2, np.eye(r)) + + # "Moreover, the rank of the identity matrices are + # min(p, q) - r, min(p, m - q) - r, min(m - p, q) - r, + # and min(m - p, m - q) - r respectively." + I11 = np.eye(min(p, q) - r) + I12 = np.eye(min(p, m - q) - r) + I21 = np.eye(min(m - p, q) - r) + I22 = np.eye(min(m - p, m - q) - r) + + # From: + # ┌ ┐ + # │ I 0 0 │ 0 0 0 │ + # ┌ ┐ ┌ ┐│ 0 C 0 │ 0 -S 0 │┌ ┐* + # │ X11 │ X12 │ │ U1 │ ││ 0 0 0 │ 0 0 -I ││ V1 │ │ + # │ ────┼──── │ = │────┼────││─────────┼─────────││────┼────│ + # │ X21 │ X22 │ │ │ U2 ││ 0 0 0 │ I 0 0 ││ │ V2 │ + # └ ┘ └ ┘│ 0 S 0 │ 0 C 0 │└ ┘ + # │ 0 0 I │ 0 0 0 │ + # └ ┘ + + # We can see that U and V are block diagonal matrices like so: + U = linalg.block_diag(u1, u2) + V = linalg.block_diag(v1, v2) + + # And the center matrix, which we'll call Q here, must be: + Q11 = np.zeros((u1.shape[1], v1.shape[0])) + IC11 = linalg.block_diag(I11, C) + Q11[:IC11.shape[0], :IC11.shape[1]] = IC11 + + Q12 = np.zeros((u1.shape[1], v2.shape[0])) + SI12 = linalg.block_diag(S, I12) if swap_sign else linalg.block_diag(-S, -I12) + Q12[-SI12.shape[0]:, -SI12.shape[1]:] = SI12 + + Q21 = np.zeros((u2.shape[1], v1.shape[0])) + SI21 = linalg.block_diag(-S, -I21) if swap_sign else linalg.block_diag(S, I21) + Q21[-SI21.shape[0]:, -SI21.shape[1]:] = SI21 + + Q22 = np.zeros((u2.shape[1], v2.shape[0])) + IC22 = linalg.block_diag(I22, C) + Q22[:IC22.shape[0], :IC22.shape[1]] = IC22 + + Q = np.block([[Q11, Q12], [Q21, Q22]]) + + # Confirm that `cossin` decomposes `X` as shown + np.testing.assert_allclose(X, U @ Q @ V.conj().T) + + # And check that `separate=False` agrees + U0, CS0, Vh0 = linalg.cossin(X, p=p, q=q, swap_sign=swap_sign) + np.testing.assert_allclose(U, U0) + np.testing.assert_allclose(Q, CS0) + np.testing.assert_allclose(V, Vh0.conj().T) + + # Confirm that `compute_u`/`compute_vh` don't affect the results + kwargs = dict(p=p, q=q, swap_sign=swap_sign) + + # `compute_u=False` + u, cs, vh = linalg.cossin(X, separate=True, compute_u=False, **kwargs) + assert u[0].shape == (0, 0) # probably not ideal, but this is what it does + assert u[1].shape == (0, 0) + assert_allclose(cs, cs0, rtol=1e-15) + assert_allclose(vh[0], vh0[0], rtol=1e-15) + assert_allclose(vh[1], vh0[1], rtol=1e-15) + + U, CS, Vh = linalg.cossin(X, compute_u=False, **kwargs) + assert U.shape == (0, 0) + assert_allclose(CS, CS0, rtol=1e-15) + assert_allclose(Vh, Vh0, rtol=1e-15) + + # `compute_vh=False` + u, cs, vh = linalg.cossin(X, separate=True, compute_vh=False, **kwargs) + assert_allclose(u[0], u[0], rtol=1e-15) + assert_allclose(u[1], u[1], rtol=1e-15) + assert_allclose(cs, cs0, rtol=1e-15) + assert vh[0].shape == (0, 0) + assert vh[1].shape == (0, 0) + + U, CS, Vh = linalg.cossin(X, compute_vh=False, **kwargs) + assert_allclose(U, U0, rtol=1e-15) + assert_allclose(CS, CS0, rtol=1e-15) + assert Vh.shape == (0, 0) + + # `compute_u=False, compute_vh=False` + u, cs, vh = linalg.cossin(X, separate=True, compute_u=False, + compute_vh=False, **kwargs) + assert u[0].shape == (0, 0) + assert u[1].shape == (0, 0) + assert_allclose(cs, cs0, rtol=1e-15) + assert vh[0].shape == (0, 0) + assert vh[1].shape == (0, 0) + + U, CS, Vh = linalg.cossin(X, compute_u=False, compute_vh=False, **kwargs) + assert U.shape == (0, 0) + assert_allclose(CS, CS0, rtol=1e-15) + assert Vh.shape == (0, 0) + + +def test_indexing_bug_gh19365(): + # Regression test for gh-19365, which reported a bug with `separate=False` + rng = np.random.default_rng(32954827234421) + m = rng.integers(50, high=100) + p = rng.integers(10, 40) # always p < m + q = rng.integers(m - p + 1, m - 1) # always m-p < q < m + X = unitary_group.rvs(m, random_state=rng) # random unitary matrix + U, D, Vt = linalg.cossin(X, p=p, q=q, separate=False) + assert np.allclose(U @ D @ Vt, X) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_ldl.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_ldl.py new file mode 100644 index 0000000000000000000000000000000000000000..2d74a746b4dd7bb6367500b4c893aa9f767de51f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_ldl.py @@ -0,0 +1,137 @@ +from numpy.testing import assert_array_almost_equal, assert_allclose, assert_ +from numpy import (array, eye, zeros, empty_like, empty, tril_indices_from, + tril, triu_indices_from, spacing, float32, float64, + complex64, complex128) +from numpy.random import rand, randint, seed +from scipy.linalg import ldl +from scipy._lib._util import ComplexWarning +import pytest +from pytest import raises as assert_raises, warns + + +@pytest.mark.thread_unsafe +def test_args(): + A = eye(3) + # Nonsquare array + assert_raises(ValueError, ldl, A[:, :2]) + # Complex matrix with imaginary diagonal entries with "hermitian=True" + with warns(ComplexWarning): + ldl(A*1j) + + +def test_empty_array(): + a = empty((0, 0), dtype=complex) + l, d, p = ldl(empty((0, 0))) + assert_array_almost_equal(l, empty_like(a)) + assert_array_almost_equal(d, empty_like(a)) + assert_array_almost_equal(p, array([], dtype=int)) + + +def test_simple(): + a = array([[-0.39-0.71j, 5.14-0.64j, -7.86-2.96j, 3.80+0.92j], + [5.14-0.64j, 8.86+1.81j, -3.52+0.58j, 5.32-1.59j], + [-7.86-2.96j, -3.52+0.58j, -2.83-0.03j, -1.54-2.86j], + [3.80+0.92j, 5.32-1.59j, -1.54-2.86j, -0.56+0.12j]]) + b = array([[5., 10, 1, 18], + [10., 2, 11, 1], + [1., 11, 19, 9], + [18., 1, 9, 0]]) + c = array([[52., 97, 112, 107, 50], + [97., 114, 89, 98, 13], + [112., 89, 64, 33, 6], + [107., 98, 33, 60, 73], + [50., 13, 6, 73, 77]]) + + d = array([[2., 2, -4, 0, 4], + [2., -2, -2, 10, -8], + [-4., -2, 6, -8, -4], + [0., 10, -8, 6, -6], + [4., -8, -4, -6, 10]]) + e = array([[-1.36+0.00j, 0+0j, 0+0j, 0+0j], + [1.58-0.90j, -8.87+0j, 0+0j, 0+0j], + [2.21+0.21j, -1.84+0.03j, -4.63+0j, 0+0j], + [3.91-1.50j, -1.78-1.18j, 0.11-0.11j, -1.84+0.00j]]) + for x in (b, c, d): + l, d, p = ldl(x) + assert_allclose(l.dot(d).dot(l.T), x, atol=spacing(1000.), rtol=0) + + u, d, p = ldl(x, lower=False) + assert_allclose(u.dot(d).dot(u.T), x, atol=spacing(1000.), rtol=0) + + l, d, p = ldl(a, hermitian=False) + assert_allclose(l.dot(d).dot(l.T), a, atol=spacing(1000.), rtol=0) + + u, d, p = ldl(a, lower=False, hermitian=False) + assert_allclose(u.dot(d).dot(u.T), a, atol=spacing(1000.), rtol=0) + + # Use upper part for the computation and use the lower part for comparison + l, d, p = ldl(e.conj().T, lower=0) + assert_allclose(tril(l.dot(d).dot(l.conj().T)-e), zeros((4, 4)), + atol=spacing(1000.), rtol=0) + + +def test_permutations(): + seed(1234) + for _ in range(10): + n = randint(1, 100) + # Random real/complex array + x = rand(n, n) if randint(2) else rand(n, n) + rand(n, n)*1j + x = x + x.conj().T + x += eye(n)*randint(5, 1e6) + l_ind = tril_indices_from(x, k=-1) + u_ind = triu_indices_from(x, k=1) + + # Test whether permutations lead to a triangular array + u, d, p = ldl(x, lower=0) + # lower part should be zero + assert_(not any(u[p, :][l_ind]), f'Spin {_} failed') + + l, d, p = ldl(x, lower=1) + # upper part should be zero + assert_(not any(l[p, :][u_ind]), f'Spin {_} failed') + + +@pytest.mark.parametrize("dtype", [float32, float64]) +@pytest.mark.parametrize("n", [30, 150]) +def test_ldl_type_size_combinations_real(n, dtype): + seed(1234) + msg = (f"Failed for size: {n}, dtype: {dtype}") + + x = rand(n, n).astype(dtype) + x = x + x.T + x += eye(n, dtype=dtype)*dtype(randint(5, 1e6)) + + l, d1, p = ldl(x) + u, d2, p = ldl(x, lower=0) + rtol = 1e-4 if dtype is float32 else 1e-10 + assert_allclose(l.dot(d1).dot(l.T), x, rtol=rtol, err_msg=msg) + assert_allclose(u.dot(d2).dot(u.T), x, rtol=rtol, err_msg=msg) + + +@pytest.mark.parametrize("dtype", [complex64, complex128]) +@pytest.mark.parametrize("n", [30, 150]) +def test_ldl_type_size_combinations_complex(n, dtype): + seed(1234) + msg1 = (f"Her failed for size: {n}, dtype: {dtype}") + msg2 = (f"Sym failed for size: {n}, dtype: {dtype}") + + # Complex hermitian upper/lower + x = (rand(n, n)+1j*rand(n, n)).astype(dtype) + x = x+x.conj().T + x += eye(n, dtype=dtype)*dtype(randint(5, 1e6)) + + l, d1, p = ldl(x) + u, d2, p = ldl(x, lower=0) + rtol = 2e-4 if dtype is complex64 else 1e-10 + assert_allclose(l.dot(d1).dot(l.conj().T), x, rtol=rtol, err_msg=msg1) + assert_allclose(u.dot(d2).dot(u.conj().T), x, rtol=rtol, err_msg=msg1) + + # Complex symmetric upper/lower + x = (rand(n, n)+1j*rand(n, n)).astype(dtype) + x = x+x.T + x += eye(n, dtype=dtype)*dtype(randint(5, 1e6)) + + l, d1, p = ldl(x, hermitian=0) + u, d2, p = ldl(x, lower=0, hermitian=0) + assert_allclose(l.dot(d1).dot(l.T), x, rtol=rtol, err_msg=msg2) + assert_allclose(u.dot(d2).dot(u.T), x, rtol=rtol, err_msg=msg2) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_lu.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_lu.py new file mode 100644 index 0000000000000000000000000000000000000000..da0beccf1f0e66baf4ac4ec80d7ff7129b2df345 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_lu.py @@ -0,0 +1,308 @@ +import pytest +from pytest import raises as assert_raises + +import numpy as np +from scipy.linalg import lu, lu_factor, lu_solve, get_lapack_funcs, solve +from numpy.testing import assert_allclose, assert_array_equal, assert_equal + + +REAL_DTYPES = [np.float32, np.float64] +COMPLEX_DTYPES = [np.complex64, np.complex128] +DTYPES = REAL_DTYPES + COMPLEX_DTYPES + + +class TestLU: + def setup_method(self): + self.rng = np.random.default_rng(1682281250228846) + + def test_old_lu_smoke_tests(self): + "Tests from old fortran based lu test suite" + a = np.array([[1, 2, 3], [1, 2, 3], [2, 5, 6]]) + p, l, u = lu(a) + result_lu = np.array([[2., 5., 6.], [0.5, -0.5, 0.], [0.5, 1., 0.]]) + assert_allclose(p, np.rot90(np.eye(3))) + assert_allclose(l, np.tril(result_lu, k=-1)+np.eye(3)) + assert_allclose(u, np.triu(result_lu)) + + a = np.array([[1, 2, 3], [1, 2, 3], [2, 5j, 6]]) + p, l, u = lu(a) + result_lu = np.array([[2., 5.j, 6.], [0.5, 2-2.5j, 0.], [0.5, 1., 0.]]) + assert_allclose(p, np.rot90(np.eye(3))) + assert_allclose(l, np.tril(result_lu, k=-1)+np.eye(3)) + assert_allclose(u, np.triu(result_lu)) + + b = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) + p, l, u = lu(b) + assert_allclose(p, np.array([[0, 1, 0], [0, 0, 1], [1, 0, 0]])) + assert_allclose(l, np.array([[1, 0, 0], [1/7, 1, 0], [4/7, 0.5, 1]])) + assert_allclose(u, np.array([[7, 8, 9], [0, 6/7, 12/7], [0, 0, 0]]), + rtol=0., atol=1e-14) + + cb = np.array([[1.j, 2.j, 3.j], [4j, 5j, 6j], [7j, 8j, 9j]]) + p, l, u = lu(cb) + assert_allclose(p, np.array([[0, 1, 0], [0, 0, 1], [1, 0, 0]])) + assert_allclose(l, np.array([[1, 0, 0], [1/7, 1, 0], [4/7, 0.5, 1]])) + assert_allclose(u, np.array([[7, 8, 9], [0, 6/7, 12/7], [0, 0, 0]])*1j, + rtol=0., atol=1e-14) + + # Rectangular matrices + hrect = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 12, 12]]) + p, l, u = lu(hrect) + assert_allclose(p, np.array([[0, 1, 0], [0, 0, 1], [1, 0, 0]])) + assert_allclose(l, np.array([[1, 0, 0], [1/9, 1, 0], [5/9, 0.5, 1]])) + assert_allclose(u, np.array([[9, 10, 12, 12], [0, 8/9, 15/9, 24/9], + [0, 0, -0.5, 0]]), rtol=0., atol=1e-14) + + chrect = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 12, 12]])*1.j + p, l, u = lu(chrect) + assert_allclose(p, np.array([[0, 1, 0], [0, 0, 1], [1, 0, 0]])) + assert_allclose(l, np.array([[1, 0, 0], [1/9, 1, 0], [5/9, 0.5, 1]])) + assert_allclose(u, np.array([[9, 10, 12, 12], [0, 8/9, 15/9, 24/9], + [0, 0, -0.5, 0]])*1j, rtol=0., atol=1e-14) + + vrect = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 12, 12]]) + p, l, u = lu(vrect) + assert_allclose(p, np.eye(4)[[1, 3, 2, 0], :]) + assert_allclose(l, np.array([[1., 0, 0], [0.1, 1, 0], [0.7, -0.5, 1], + [0.4, 0.25, 0.5]])) + assert_allclose(u, np.array([[10, 12, 12], + [0, 0.8, 1.8], + [0, 0, 1.5]])) + + cvrect = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 12, 12]])*1j + p, l, u = lu(cvrect) + assert_allclose(p, np.eye(4)[[1, 3, 2, 0], :]) + assert_allclose(l, np.array([[1., 0, 0], + [0.1, 1, 0], + [0.7, -0.5, 1], + [0.4, 0.25, 0.5]])) + assert_allclose(u, np.array([[10, 12, 12], + [0, 0.8, 1.8], + [0, 0, 1.5]])*1j) + + @pytest.mark.parametrize('shape', [[2, 2], [2, 4], [4, 2], [20, 20], + [20, 4], [4, 20], [3, 2, 9, 9], + [2, 2, 17, 5], [2, 2, 11, 7]]) + def test_simple_lu_shapes_real_complex(self, shape): + a = self.rng.uniform(-10., 10., size=shape) + p, l, u = lu(a) + assert_allclose(a, p @ l @ u) + pl, u = lu(a, permute_l=True) + assert_allclose(a, pl @ u) + + b = self.rng.uniform(-10., 10., size=shape)*1j + b += self.rng.uniform(-10, 10, size=shape) + pl, u = lu(b, permute_l=True) + assert_allclose(b, pl @ u) + + @pytest.mark.parametrize('shape', [[2, 2], [2, 4], [4, 2], [20, 20], + [20, 4], [4, 20]]) + def test_simple_lu_shapes_real_complex_2d_indices(self, shape): + a = self.rng.uniform(-10., 10., size=shape) + p, l, u = lu(a, p_indices=True) + assert_allclose(a, l[p, :] @ u) + + def test_1by1_input_output(self): + a = self.rng.random([4, 5, 1, 1], dtype=np.float32) + p, l, u = lu(a, p_indices=True) + assert_allclose(p, np.zeros(shape=(4, 5, 1), dtype=int)) + assert_allclose(l, np.ones(shape=(4, 5, 1, 1), dtype=np.float32)) + assert_allclose(u, a) + + a = self.rng.random([4, 5, 1, 1], dtype=np.float32) + p, l, u = lu(a) + assert_allclose(p, np.ones(shape=(4, 5, 1, 1), dtype=np.float32)) + assert_allclose(l, np.ones(shape=(4, 5, 1, 1), dtype=np.float32)) + assert_allclose(u, a) + + pl, u = lu(a, permute_l=True) + assert_allclose(pl, np.ones(shape=(4, 5, 1, 1), dtype=np.float32)) + assert_allclose(u, a) + + a = self.rng.random([4, 5, 1, 1], dtype=np.float32)*np.complex64(1.j) + p, l, u = lu(a) + assert_allclose(p, np.ones(shape=(4, 5, 1, 1), dtype=np.complex64)) + assert_allclose(l, np.ones(shape=(4, 5, 1, 1), dtype=np.complex64)) + assert_allclose(u, a) + + def test_empty_edge_cases(self): + a = np.empty([0, 0]) + p, l, u = lu(a) + assert_allclose(p, np.empty(shape=(0, 0), dtype=np.float64)) + assert_allclose(l, np.empty(shape=(0, 0), dtype=np.float64)) + assert_allclose(u, np.empty(shape=(0, 0), dtype=np.float64)) + + a = np.empty([0, 3], dtype=np.float16) + p, l, u = lu(a) + assert_allclose(p, np.empty(shape=(0, 0), dtype=np.float32)) + assert_allclose(l, np.empty(shape=(0, 0), dtype=np.float32)) + assert_allclose(u, np.empty(shape=(0, 3), dtype=np.float32)) + + a = np.empty([3, 0], dtype=np.complex64) + p, l, u = lu(a) + assert_allclose(p, np.empty(shape=(0, 0), dtype=np.float32)) + assert_allclose(l, np.empty(shape=(3, 0), dtype=np.complex64)) + assert_allclose(u, np.empty(shape=(0, 0), dtype=np.complex64)) + p, l, u = lu(a, p_indices=True) + assert_allclose(p, np.empty(shape=(0,), dtype=int)) + assert_allclose(l, np.empty(shape=(3, 0), dtype=np.complex64)) + assert_allclose(u, np.empty(shape=(0, 0), dtype=np.complex64)) + pl, u = lu(a, permute_l=True) + assert_allclose(pl, np.empty(shape=(3, 0), dtype=np.complex64)) + assert_allclose(u, np.empty(shape=(0, 0), dtype=np.complex64)) + + a = np.empty([3, 0, 0], dtype=np.complex64) + p, l, u = lu(a) + assert_allclose(p, np.empty(shape=(3, 0, 0), dtype=np.float32)) + assert_allclose(l, np.empty(shape=(3, 0, 0), dtype=np.complex64)) + assert_allclose(u, np.empty(shape=(3, 0, 0), dtype=np.complex64)) + + a = np.empty([0, 0, 3]) + p, l, u = lu(a) + assert_allclose(p, np.empty(shape=(0, 0, 0))) + assert_allclose(l, np.empty(shape=(0, 0, 0))) + assert_allclose(u, np.empty(shape=(0, 0, 3))) + + with assert_raises(ValueError, match='at least two-dimensional'): + lu(np.array([])) + + a = np.array([[]]) + p, l, u = lu(a) + assert_allclose(p, np.empty(shape=(0, 0))) + assert_allclose(l, np.empty(shape=(1, 0))) + assert_allclose(u, np.empty(shape=(0, 0))) + + a = np.array([[[]]]) + p, l, u = lu(a) + assert_allclose(p, np.empty(shape=(1, 0, 0))) + assert_allclose(l, np.empty(shape=(1, 1, 0))) + assert_allclose(u, np.empty(shape=(1, 0, 0))) + + +class TestLUFactor: + def setup_method(self): + self.rng = np.random.default_rng(1682281250228846) + + self.a = np.array([[1, 2, 3], [1, 2, 3], [2, 5, 6]]) + self.ca = np.array([[1, 2, 3], [1, 2, 3], [2, 5j, 6]]) + # Those matrices are more robust to detect problems in permutation + # matrices than the ones above + self.b = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) + self.cb = np.array([[1j, 2j, 3j], [4j, 5j, 6j], [7j, 8j, 9j]]) + + # Rectangular matrices + self.hrect = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 12, 12]]) + self.chrect = np.array([[1, 2, 3, 4], [5, 6, 7, 8], + [9, 10, 12, 12]]) * 1.j + + self.vrect = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 12, 12]]) + self.cvrect = 1.j * np.array([[1, 2, 3], + [4, 5, 6], + [7, 8, 9], + [10, 12, 12]]) + + # Medium sizes matrices + self.med = self.rng.random((30, 40)) + self.cmed = self.rng.random((30, 40)) + 1.j*self.rng.random((30, 40)) + + def _test_common_lu_factor(self, data): + l_and_u1, piv1 = lu_factor(data) + (getrf,) = get_lapack_funcs(("getrf",), (data,)) + l_and_u2, piv2, _ = getrf(data, overwrite_a=False) + assert_allclose(l_and_u1, l_and_u2) + assert_allclose(piv1, piv2) + + # Simple tests. + # For lu_factor gives a LinAlgWarning because these matrices are singular + def test_hrectangular(self): + self._test_common_lu_factor(self.hrect) + + def test_vrectangular(self): + self._test_common_lu_factor(self.vrect) + + def test_hrectangular_complex(self): + self._test_common_lu_factor(self.chrect) + + def test_vrectangular_complex(self): + self._test_common_lu_factor(self.cvrect) + + # Bigger matrices + def test_medium1(self): + """Check lu decomposition on medium size, rectangular matrix.""" + self._test_common_lu_factor(self.med) + + def test_medium1_complex(self): + """Check lu decomposition on medium size, rectangular matrix.""" + self._test_common_lu_factor(self.cmed) + + def test_check_finite(self): + p, l, u = lu(self.a, check_finite=False) + assert_allclose(p @ l @ u, self.a) + + def test_simple_known(self): + # Ticket #1458 + for order in ['C', 'F']: + A = np.array([[2, 1], [0, 1.]], order=order) + LU, P = lu_factor(A) + assert_allclose(LU, np.array([[2, 1], [0, 1]])) + assert_array_equal(P, np.array([0, 1])) + + @pytest.mark.parametrize("m", [0, 1, 2]) + @pytest.mark.parametrize("n", [0, 1, 2]) + @pytest.mark.parametrize('dtype', DTYPES) + def test_shape_dtype(self, m, n, dtype): + k = min(m, n) + + a = np.eye(m, n, dtype=dtype) + lu, p = lu_factor(a) + assert_equal(lu.shape, (m, n)) + assert_equal(lu.dtype, dtype) + assert_equal(p.shape, (k,)) + assert_equal(p.dtype, np.int32) + + @pytest.mark.parametrize(("m", "n"), [(0, 0), (0, 2), (2, 0)]) + def test_empty(self, m, n): + a = np.zeros((m, n)) + lu, p = lu_factor(a) + assert_allclose(lu, np.empty((m, n))) + assert_allclose(p, np.arange(0)) + + +class TestLUSolve: + def setup_method(self): + self.rng = np.random.default_rng(1682281250228846) + + def test_lu(self): + a0 = self.rng.random((10, 10)) + b = self.rng.random((10,)) + + for order in ['C', 'F']: + a = np.array(a0, order=order) + x1 = solve(a, b) + lu_a = lu_factor(a) + x2 = lu_solve(lu_a, b) + assert_allclose(x1, x2) + + def test_check_finite(self): + a = self.rng.random((10, 10)) + b = self.rng.random((10,)) + x1 = solve(a, b) + lu_a = lu_factor(a, check_finite=False) + x2 = lu_solve(lu_a, b, check_finite=False) + assert_allclose(x1, x2) + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt, dt_b): + lu_and_piv = (np.empty((0, 0), dtype=dt), np.array([])) + b = np.asarray([], dtype=dt_b) + x = lu_solve(lu_and_piv, b) + assert x.shape == (0,) + + m = lu_solve((np.eye(2, dtype=dt), [0, 1]), np.ones(2, dtype=dt_b)) + assert x.dtype == m.dtype + + b = np.empty((0, 0), dtype=dt_b) + x = lu_solve(lu_and_piv, b) + assert x.shape == (0, 0) + assert x.dtype == m.dtype diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_polar.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_polar.py new file mode 100644 index 0000000000000000000000000000000000000000..607238842b3cc643d9665e40f29e41b15d8951a1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_polar.py @@ -0,0 +1,110 @@ +import pytest +import numpy as np +from numpy.linalg import norm +from numpy.testing import (assert_, assert_allclose, assert_equal) +from scipy.linalg import polar, eigh + + +diag2 = np.array([[2, 0], [0, 3]]) +a13 = np.array([[1, 2, 2]]) + +precomputed_cases = [ + [[[0]], 'right', [[1]], [[0]]], + [[[0]], 'left', [[1]], [[0]]], + [[[9]], 'right', [[1]], [[9]]], + [[[9]], 'left', [[1]], [[9]]], + [diag2, 'right', np.eye(2), diag2], + [diag2, 'left', np.eye(2), diag2], + [a13, 'right', a13/norm(a13[0]), a13.T.dot(a13)/norm(a13[0])], +] + +verify_cases = [ + [[1, 2], [3, 4]], + [[1, 2, 3]], + [[1], [2], [3]], + [[1, 2, 3], [3, 4, 0]], + [[1, 2], [3, 4], [5, 5]], + [[1, 2], [3, 4+5j]], + [[1, 2, 3j]], + [[1], [2], [3j]], + [[1, 2, 3+2j], [3, 4-1j, -4j]], + [[1, 2], [3-2j, 4+0.5j], [5, 5]], + [[10000, 10, 1], [-1, 2, 3j], [0, 1, 2]], + np.empty((0, 0)), + np.empty((0, 2)), + np.empty((2, 0)), +] + + +def check_precomputed_polar(a, side, expected_u, expected_p): + # Compare the result of the polar decomposition to a + # precomputed result. + u, p = polar(a, side=side) + assert_allclose(u, expected_u, atol=1e-15) + assert_allclose(p, expected_p, atol=1e-15) + + +def verify_polar(a): + # Compute the polar decomposition, and then verify that + # the result has all the expected properties. + product_atol = np.sqrt(np.finfo(float).eps) + + aa = np.asarray(a) + m, n = aa.shape + + u, p = polar(a, side='right') + assert_equal(u.shape, (m, n)) + assert_equal(p.shape, (n, n)) + # a = up + assert_allclose(u.dot(p), a, atol=product_atol) + if m >= n: + assert_allclose(u.conj().T.dot(u), np.eye(n), atol=1e-15) + else: + assert_allclose(u.dot(u.conj().T), np.eye(m), atol=1e-15) + # p is Hermitian positive semidefinite. + assert_allclose(p.conj().T, p) + evals = eigh(p, eigvals_only=True) + nonzero_evals = evals[abs(evals) > 1e-14] + assert_((nonzero_evals >= 0).all()) + + u, p = polar(a, side='left') + assert_equal(u.shape, (m, n)) + assert_equal(p.shape, (m, m)) + # a = pu + assert_allclose(p.dot(u), a, atol=product_atol) + if m >= n: + assert_allclose(u.conj().T.dot(u), np.eye(n), atol=1e-15) + else: + assert_allclose(u.dot(u.conj().T), np.eye(m), atol=1e-15) + # p is Hermitian positive semidefinite. + assert_allclose(p.conj().T, p) + evals = eigh(p, eigvals_only=True) + nonzero_evals = evals[abs(evals) > 1e-14] + assert_((nonzero_evals >= 0).all()) + + +def test_precomputed_cases(): + for a, side, expected_u, expected_p in precomputed_cases: + check_precomputed_polar(a, side, expected_u, expected_p) + + +def test_verify_cases(): + for a in verify_cases: + verify_polar(a) + +@pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) +@pytest.mark.parametrize('shape', [(0, 0), (0, 2), (2, 0)]) +@pytest.mark.parametrize('side', ['left', 'right']) +def test_empty(dt, shape, side): + a = np.empty(shape, dtype=dt) + m, n = shape + p_shape = (m, m) if side == 'left' else (n, n) + + u, p = polar(a, side=side) + u_n, p_n = polar(np.eye(5, dtype=dt)) + + assert_equal(u.dtype, u_n.dtype) + assert_equal(p.dtype, p_n.dtype) + assert u.shape == shape + assert p.shape == p_shape + assert np.all(p == 0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_update.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_update.py new file mode 100644 index 0000000000000000000000000000000000000000..7553e21d61ceaa774d24c48d9d4bc2e3a8e3cc00 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_decomp_update.py @@ -0,0 +1,1701 @@ +import itertools + +import numpy as np +from numpy.testing import assert_, assert_allclose, assert_equal +from pytest import raises as assert_raises +from scipy import linalg +import scipy.linalg._decomp_update as _decomp_update +from scipy.linalg._decomp_update import qr_delete, qr_update, qr_insert + +def assert_unitary(a, rtol=None, atol=None, assert_sqr=True): + if rtol is None: + rtol = 10.0 ** -(np.finfo(a.dtype).precision-2) + if atol is None: + atol = 10*np.finfo(a.dtype).eps + + if assert_sqr: + assert_(a.shape[0] == a.shape[1], 'unitary matrices must be square') + aTa = np.dot(a.T.conj(), a) + assert_allclose(aTa, np.eye(a.shape[1]), rtol=rtol, atol=atol) + +def assert_upper_tri(a, rtol=None, atol=None): + if rtol is None: + rtol = 10.0 ** -(np.finfo(a.dtype).precision-2) + if atol is None: + atol = 2*np.finfo(a.dtype).eps + mask = np.tri(a.shape[0], a.shape[1], -1, np.bool_) + assert_allclose(a[mask], 0.0, rtol=rtol, atol=atol) + +def check_qr(q, r, a, rtol, atol, assert_sqr=True): + assert_unitary(q, rtol, atol, assert_sqr) + assert_upper_tri(r, rtol, atol) + assert_allclose(q.dot(r), a, rtol=rtol, atol=atol) + +def make_strided(arrs): + strides = [(3, 7), (2, 2), (3, 4), (4, 2), (5, 4), (2, 3), (2, 1), (4, 5)] + kmax = len(strides) + k = 0 + ret = [] + for a in arrs: + if a.ndim == 1: + s = strides[k % kmax] + k += 1 + base = np.zeros(s[0]*a.shape[0]+s[1], a.dtype) + view = base[s[1]::s[0]] + view[...] = a + elif a.ndim == 2: + s = strides[k % kmax] + t = strides[(k+1) % kmax] + k += 2 + base = np.zeros((s[0]*a.shape[0]+s[1], t[0]*a.shape[1]+t[1]), + a.dtype) + view = base[s[1]::s[0], t[1]::t[0]] + view[...] = a + else: + raise ValueError('make_strided only works for ndim = 1 or' + ' 2 arrays') + ret.append(view) + return ret + +def negate_strides(arrs): + ret = [] + for a in arrs: + b = np.zeros_like(a) + if b.ndim == 2: + b = b[::-1, ::-1] + elif b.ndim == 1: + b = b[::-1] + else: + raise ValueError('negate_strides only works for ndim = 1 or' + ' 2 arrays') + b[...] = a + ret.append(b) + return ret + +def nonitemsize_strides(arrs): + out = [] + for a in arrs: + a_dtype = a.dtype + b = np.zeros(a.shape, [('a', a_dtype), ('junk', 'S1')]) + c = b.getfield(a_dtype) + c[...] = a + out.append(c) + return out + + +def make_nonnative(arrs): + return [a.astype(a.dtype.newbyteorder()) for a in arrs] + + +class BaseQRdeltas: + def setup_method(self): + self.rtol = 10.0 ** -(np.finfo(self.dtype).precision-2) + self.atol = 10 * np.finfo(self.dtype).eps + + def generate(self, type, mode='full'): + np.random.seed(29382) + shape = {'sqr': (8, 8), 'tall': (12, 7), 'fat': (7, 12), + 'Mx1': (8, 1), '1xN': (1, 8), '1x1': (1, 1)}[type] + a = np.random.random(shape) + if np.iscomplexobj(self.dtype.type(1)): + b = np.random.random(shape) + a = a + 1j * b + a = a.astype(self.dtype) + q, r = linalg.qr(a, mode=mode) + return a, q, r + +class BaseQRdelete(BaseQRdeltas): + def test_sqr_1_row(self): + a, q, r = self.generate('sqr') + for row in range(r.shape[0]): + q1, r1 = qr_delete(q, r, row, overwrite_qr=False) + a1 = np.delete(a, row, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_sqr_p_row(self): + a, q, r = self.generate('sqr') + for ndel in range(2, 6): + for row in range(a.shape[0]-ndel): + q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False) + a1 = np.delete(a, slice(row, row+ndel), 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_sqr_1_col(self): + a, q, r = self.generate('sqr') + for col in range(r.shape[1]): + q1, r1 = qr_delete(q, r, col, which='col', overwrite_qr=False) + a1 = np.delete(a, col, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_sqr_p_col(self): + a, q, r = self.generate('sqr') + for ndel in range(2, 6): + for col in range(r.shape[1]-ndel): + q1, r1 = qr_delete(q, r, col, ndel, which='col', + overwrite_qr=False) + a1 = np.delete(a, slice(col, col+ndel), 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_tall_1_row(self): + a, q, r = self.generate('tall') + for row in range(r.shape[0]): + q1, r1 = qr_delete(q, r, row, overwrite_qr=False) + a1 = np.delete(a, row, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_tall_p_row(self): + a, q, r = self.generate('tall') + for ndel in range(2, 6): + for row in range(a.shape[0]-ndel): + q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False) + a1 = np.delete(a, slice(row, row+ndel), 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_tall_1_col(self): + a, q, r = self.generate('tall') + for col in range(r.shape[1]): + q1, r1 = qr_delete(q, r, col, which='col', overwrite_qr=False) + a1 = np.delete(a, col, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_tall_p_col(self): + a, q, r = self.generate('tall') + for ndel in range(2, 6): + for col in range(r.shape[1]-ndel): + q1, r1 = qr_delete(q, r, col, ndel, which='col', + overwrite_qr=False) + a1 = np.delete(a, slice(col, col+ndel), 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_fat_1_row(self): + a, q, r = self.generate('fat') + for row in range(r.shape[0]): + q1, r1 = qr_delete(q, r, row, overwrite_qr=False) + a1 = np.delete(a, row, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_fat_p_row(self): + a, q, r = self.generate('fat') + for ndel in range(2, 6): + for row in range(a.shape[0]-ndel): + q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False) + a1 = np.delete(a, slice(row, row+ndel), 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_fat_1_col(self): + a, q, r = self.generate('fat') + for col in range(r.shape[1]): + q1, r1 = qr_delete(q, r, col, which='col', overwrite_qr=False) + a1 = np.delete(a, col, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_fat_p_col(self): + a, q, r = self.generate('fat') + for ndel in range(2, 6): + for col in range(r.shape[1]-ndel): + q1, r1 = qr_delete(q, r, col, ndel, which='col', + overwrite_qr=False) + a1 = np.delete(a, slice(col, col+ndel), 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_economic_1_row(self): + # this test always starts and ends with an economic decomp. + a, q, r = self.generate('tall', 'economic') + for row in range(r.shape[0]): + q1, r1 = qr_delete(q, r, row, overwrite_qr=False) + a1 = np.delete(a, row, 0) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + # for economic row deletes + # eco - prow = eco + # eco - prow = sqr + # eco - prow = fat + def base_economic_p_row_xxx(self, ndel): + a, q, r = self.generate('tall', 'economic') + for row in range(a.shape[0]-ndel): + q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False) + a1 = np.delete(a, slice(row, row+ndel), 0) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_economic_p_row_economic(self): + # (12, 7) - (3, 7) = (9,7) --> stays economic + self.base_economic_p_row_xxx(3) + + def test_economic_p_row_sqr(self): + # (12, 7) - (5, 7) = (7, 7) --> becomes square + self.base_economic_p_row_xxx(5) + + def test_economic_p_row_fat(self): + # (12, 7) - (7,7) = (5, 7) --> becomes fat + self.base_economic_p_row_xxx(7) + + def test_economic_1_col(self): + a, q, r = self.generate('tall', 'economic') + for col in range(r.shape[1]): + q1, r1 = qr_delete(q, r, col, which='col', overwrite_qr=False) + a1 = np.delete(a, col, 1) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_economic_p_col(self): + a, q, r = self.generate('tall', 'economic') + for ndel in range(2, 6): + for col in range(r.shape[1]-ndel): + q1, r1 = qr_delete(q, r, col, ndel, which='col', + overwrite_qr=False) + a1 = np.delete(a, slice(col, col+ndel), 1) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_Mx1_1_row(self): + a, q, r = self.generate('Mx1') + for row in range(r.shape[0]): + q1, r1 = qr_delete(q, r, row, overwrite_qr=False) + a1 = np.delete(a, row, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_Mx1_p_row(self): + a, q, r = self.generate('Mx1') + for ndel in range(2, 6): + for row in range(a.shape[0]-ndel): + q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False) + a1 = np.delete(a, slice(row, row+ndel), 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1xN_1_col(self): + a, q, r = self.generate('1xN') + for col in range(r.shape[1]): + q1, r1 = qr_delete(q, r, col, which='col', overwrite_qr=False) + a1 = np.delete(a, col, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1xN_p_col(self): + a, q, r = self.generate('1xN') + for ndel in range(2, 6): + for col in range(r.shape[1]-ndel): + q1, r1 = qr_delete(q, r, col, ndel, which='col', + overwrite_qr=False) + a1 = np.delete(a, slice(col, col+ndel), 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_Mx1_economic_1_row(self): + a, q, r = self.generate('Mx1', 'economic') + for row in range(r.shape[0]): + q1, r1 = qr_delete(q, r, row, overwrite_qr=False) + a1 = np.delete(a, row, 0) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_Mx1_economic_p_row(self): + a, q, r = self.generate('Mx1', 'economic') + for ndel in range(2, 6): + for row in range(a.shape[0]-ndel): + q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False) + a1 = np.delete(a, slice(row, row+ndel), 0) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_delete_last_1_row(self): + # full and eco are the same for 1xN + a, q, r = self.generate('1xN') + q1, r1 = qr_delete(q, r, 0, 1, 'row') + assert_equal(q1, np.ndarray(shape=(0, 0), dtype=q.dtype)) + assert_equal(r1, np.ndarray(shape=(0, r.shape[1]), dtype=r.dtype)) + + def test_delete_last_p_row(self): + a, q, r = self.generate('tall', 'full') + q1, r1 = qr_delete(q, r, 0, a.shape[0], 'row') + assert_equal(q1, np.ndarray(shape=(0, 0), dtype=q.dtype)) + assert_equal(r1, np.ndarray(shape=(0, r.shape[1]), dtype=r.dtype)) + + a, q, r = self.generate('tall', 'economic') + q1, r1 = qr_delete(q, r, 0, a.shape[0], 'row') + assert_equal(q1, np.ndarray(shape=(0, 0), dtype=q.dtype)) + assert_equal(r1, np.ndarray(shape=(0, r.shape[1]), dtype=r.dtype)) + + def test_delete_last_1_col(self): + a, q, r = self.generate('Mx1', 'economic') + q1, r1 = qr_delete(q, r, 0, 1, 'col') + assert_equal(q1, np.ndarray(shape=(q.shape[0], 0), dtype=q.dtype)) + assert_equal(r1, np.ndarray(shape=(0, 0), dtype=r.dtype)) + + a, q, r = self.generate('Mx1', 'full') + q1, r1 = qr_delete(q, r, 0, 1, 'col') + assert_unitary(q1) + assert_(q1.dtype == q.dtype) + assert_(q1.shape == q.shape) + assert_equal(r1, np.ndarray(shape=(r.shape[0], 0), dtype=r.dtype)) + + def test_delete_last_p_col(self): + a, q, r = self.generate('tall', 'full') + q1, r1 = qr_delete(q, r, 0, a.shape[1], 'col') + assert_unitary(q1) + assert_(q1.dtype == q.dtype) + assert_(q1.shape == q.shape) + assert_equal(r1, np.ndarray(shape=(r.shape[0], 0), dtype=r.dtype)) + + a, q, r = self.generate('tall', 'economic') + q1, r1 = qr_delete(q, r, 0, a.shape[1], 'col') + assert_equal(q1, np.ndarray(shape=(q.shape[0], 0), dtype=q.dtype)) + assert_equal(r1, np.ndarray(shape=(0, 0), dtype=r.dtype)) + + def test_delete_1x1_row_col(self): + a, q, r = self.generate('1x1') + q1, r1 = qr_delete(q, r, 0, 1, 'row') + assert_equal(q1, np.ndarray(shape=(0, 0), dtype=q.dtype)) + assert_equal(r1, np.ndarray(shape=(0, r.shape[1]), dtype=r.dtype)) + + a, q, r = self.generate('1x1') + q1, r1 = qr_delete(q, r, 0, 1, 'col') + assert_unitary(q1) + assert_(q1.dtype == q.dtype) + assert_(q1.shape == q.shape) + assert_equal(r1, np.ndarray(shape=(r.shape[0], 0), dtype=r.dtype)) + + # all full qr, row deletes and single column deletes should be able to + # handle any non negative strides. (only row and column vector + # operations are used.) p column delete require fortran ordered + # Q and R and will make a copy as necessary. Economic qr row deletes + # require a contiguous q. + + def base_non_simple_strides(self, adjust_strides, ks, p, which, + overwriteable): + if which == 'row': + qind = (slice(p,None), slice(p,None)) + rind = (slice(p,None), slice(None)) + else: + qind = (slice(None), slice(None)) + rind = (slice(None), slice(None,-p)) + + for type, k in itertools.product(['sqr', 'tall', 'fat'], ks): + a, q0, r0, = self.generate(type) + qs, rs = adjust_strides((q0, r0)) + if p == 1: + a1 = np.delete(a, k, 0 if which == 'row' else 1) + else: + s = slice(k,k+p) + if k < 0: + s = slice(k, k + p + + (a.shape[0] if which == 'row' else a.shape[1])) + a1 = np.delete(a, s, 0 if which == 'row' else 1) + + # for each variable, q, r we try with it strided and + # overwrite=False. Then we try with overwrite=True, and make + # sure that q and r are still overwritten. + + q = q0.copy('F') + r = r0.copy('F') + q1, r1 = qr_delete(qs, r, k, p, which, False) + check_qr(q1, r1, a1, self.rtol, self.atol) + q1o, r1o = qr_delete(qs, r, k, p, which, True) + check_qr(q1o, r1o, a1, self.rtol, self.atol) + if overwriteable: + assert_allclose(q1o, qs[qind], rtol=self.rtol, atol=self.atol) + assert_allclose(r1o, r[rind], rtol=self.rtol, atol=self.atol) + + q = q0.copy('F') + r = r0.copy('F') + q2, r2 = qr_delete(q, rs, k, p, which, False) + check_qr(q2, r2, a1, self.rtol, self.atol) + q2o, r2o = qr_delete(q, rs, k, p, which, True) + check_qr(q2o, r2o, a1, self.rtol, self.atol) + if overwriteable: + assert_allclose(q2o, q[qind], rtol=self.rtol, atol=self.atol) + assert_allclose(r2o, rs[rind], rtol=self.rtol, atol=self.atol) + + q = q0.copy('F') + r = r0.copy('F') + # since some of these were consumed above + qs, rs = adjust_strides((q, r)) + q3, r3 = qr_delete(qs, rs, k, p, which, False) + check_qr(q3, r3, a1, self.rtol, self.atol) + q3o, r3o = qr_delete(qs, rs, k, p, which, True) + check_qr(q3o, r3o, a1, self.rtol, self.atol) + if overwriteable: + assert_allclose(q2o, qs[qind], rtol=self.rtol, atol=self.atol) + assert_allclose(r3o, rs[rind], rtol=self.rtol, atol=self.atol) + + def test_non_unit_strides_1_row(self): + self.base_non_simple_strides(make_strided, [0], 1, 'row', True) + + def test_non_unit_strides_p_row(self): + self.base_non_simple_strides(make_strided, [0], 3, 'row', True) + + def test_non_unit_strides_1_col(self): + self.base_non_simple_strides(make_strided, [0], 1, 'col', True) + + def test_non_unit_strides_p_col(self): + self.base_non_simple_strides(make_strided, [0], 3, 'col', False) + + def test_neg_strides_1_row(self): + self.base_non_simple_strides(negate_strides, [0], 1, 'row', False) + + def test_neg_strides_p_row(self): + self.base_non_simple_strides(negate_strides, [0], 3, 'row', False) + + def test_neg_strides_1_col(self): + self.base_non_simple_strides(negate_strides, [0], 1, 'col', False) + + def test_neg_strides_p_col(self): + self.base_non_simple_strides(negate_strides, [0], 3, 'col', False) + + def test_non_itemize_strides_1_row(self): + self.base_non_simple_strides(nonitemsize_strides, [0], 1, 'row', False) + + def test_non_itemize_strides_p_row(self): + self.base_non_simple_strides(nonitemsize_strides, [0], 3, 'row', False) + + def test_non_itemize_strides_1_col(self): + self.base_non_simple_strides(nonitemsize_strides, [0], 1, 'col', False) + + def test_non_itemize_strides_p_col(self): + self.base_non_simple_strides(nonitemsize_strides, [0], 3, 'col', False) + + def test_non_native_byte_order_1_row(self): + self.base_non_simple_strides(make_nonnative, [0], 1, 'row', False) + + def test_non_native_byte_order_p_row(self): + self.base_non_simple_strides(make_nonnative, [0], 3, 'row', False) + + def test_non_native_byte_order_1_col(self): + self.base_non_simple_strides(make_nonnative, [0], 1, 'col', False) + + def test_non_native_byte_order_p_col(self): + self.base_non_simple_strides(make_nonnative, [0], 3, 'col', False) + + def test_neg_k(self): + a, q, r = self.generate('sqr') + for k, p, w in itertools.product([-3, -7], [1, 3], ['row', 'col']): + q1, r1 = qr_delete(q, r, k, p, w, overwrite_qr=False) + if w == 'row': + a1 = np.delete(a, slice(k+a.shape[0], k+p+a.shape[0]), 0) + else: + a1 = np.delete(a, slice(k+a.shape[0], k+p+a.shape[1]), 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def base_overwrite_qr(self, which, p, test_C, test_F, mode='full'): + assert_sqr = True if mode == 'full' else False + if which == 'row': + qind = (slice(p,None), slice(p,None)) + rind = (slice(p,None), slice(None)) + else: + qind = (slice(None), slice(None)) + rind = (slice(None), slice(None,-p)) + a, q0, r0 = self.generate('sqr', mode) + if p == 1: + a1 = np.delete(a, 3, 0 if which == 'row' else 1) + else: + a1 = np.delete(a, slice(3, 3+p), 0 if which == 'row' else 1) + + # don't overwrite + q = q0.copy('F') + r = r0.copy('F') + q1, r1 = qr_delete(q, r, 3, p, which, False) + check_qr(q1, r1, a1, self.rtol, self.atol, assert_sqr) + check_qr(q, r, a, self.rtol, self.atol, assert_sqr) + + if test_F: + q = q0.copy('F') + r = r0.copy('F') + q2, r2 = qr_delete(q, r, 3, p, which, True) + check_qr(q2, r2, a1, self.rtol, self.atol, assert_sqr) + # verify the overwriting + assert_allclose(q2, q[qind], rtol=self.rtol, atol=self.atol) + assert_allclose(r2, r[rind], rtol=self.rtol, atol=self.atol) + + if test_C: + q = q0.copy('C') + r = r0.copy('C') + q3, r3 = qr_delete(q, r, 3, p, which, True) + check_qr(q3, r3, a1, self.rtol, self.atol, assert_sqr) + assert_allclose(q3, q[qind], rtol=self.rtol, atol=self.atol) + assert_allclose(r3, r[rind], rtol=self.rtol, atol=self.atol) + + def test_overwrite_qr_1_row(self): + # any positively strided q and r. + self.base_overwrite_qr('row', 1, True, True) + + def test_overwrite_economic_qr_1_row(self): + # Any contiguous q and positively strided r. + self.base_overwrite_qr('row', 1, True, True, 'economic') + + def test_overwrite_qr_1_col(self): + # any positively strided q and r. + # full and eco share code paths + self.base_overwrite_qr('col', 1, True, True) + + def test_overwrite_qr_p_row(self): + # any positively strided q and r. + self.base_overwrite_qr('row', 3, True, True) + + def test_overwrite_economic_qr_p_row(self): + # any contiguous q and positively strided r + self.base_overwrite_qr('row', 3, True, True, 'economic') + + def test_overwrite_qr_p_col(self): + # only F ordered q and r can be overwritten for cols + # full and eco share code paths + self.base_overwrite_qr('col', 3, False, True) + + def test_bad_which(self): + a, q, r = self.generate('sqr') + assert_raises(ValueError, qr_delete, q, r, 0, which='foo') + + def test_bad_k(self): + a, q, r = self.generate('tall') + assert_raises(ValueError, qr_delete, q, r, q.shape[0], 1) + assert_raises(ValueError, qr_delete, q, r, -q.shape[0]-1, 1) + assert_raises(ValueError, qr_delete, q, r, r.shape[0], 1, 'col') + assert_raises(ValueError, qr_delete, q, r, -r.shape[0]-1, 1, 'col') + + def test_bad_p(self): + a, q, r = self.generate('tall') + # p must be positive + assert_raises(ValueError, qr_delete, q, r, 0, -1) + assert_raises(ValueError, qr_delete, q, r, 0, -1, 'col') + + # and nonzero + assert_raises(ValueError, qr_delete, q, r, 0, 0) + assert_raises(ValueError, qr_delete, q, r, 0, 0, 'col') + + # must have at least k+p rows or cols, depending. + assert_raises(ValueError, qr_delete, q, r, 3, q.shape[0]-2) + assert_raises(ValueError, qr_delete, q, r, 3, r.shape[1]-2, 'col') + + def test_empty_q(self): + a, q, r = self.generate('tall') + # same code path for 'row' and 'col' + assert_raises(ValueError, qr_delete, np.array([]), r, 0, 1) + + def test_empty_r(self): + a, q, r = self.generate('tall') + # same code path for 'row' and 'col' + assert_raises(ValueError, qr_delete, q, np.array([]), 0, 1) + + def test_mismatched_q_and_r(self): + a, q, r = self.generate('tall') + r = r[1:] + assert_raises(ValueError, qr_delete, q, r, 0, 1) + + def test_unsupported_dtypes(self): + dts = ['int8', 'int16', 'int32', 'int64', + 'uint8', 'uint16', 'uint32', 'uint64', + 'float16', 'longdouble', 'clongdouble', + 'bool'] + a, q0, r0 = self.generate('tall') + for dtype in dts: + q = q0.real.astype(dtype) + with np.errstate(invalid="ignore"): + r = r0.real.astype(dtype) + assert_raises(ValueError, qr_delete, q, r0, 0, 1, 'row') + assert_raises(ValueError, qr_delete, q, r0, 0, 2, 'row') + assert_raises(ValueError, qr_delete, q, r0, 0, 1, 'col') + assert_raises(ValueError, qr_delete, q, r0, 0, 2, 'col') + + assert_raises(ValueError, qr_delete, q0, r, 0, 1, 'row') + assert_raises(ValueError, qr_delete, q0, r, 0, 2, 'row') + assert_raises(ValueError, qr_delete, q0, r, 0, 1, 'col') + assert_raises(ValueError, qr_delete, q0, r, 0, 2, 'col') + + def test_check_finite(self): + a0, q0, r0 = self.generate('tall') + + q = q0.copy('F') + q[1,1] = np.nan + assert_raises(ValueError, qr_delete, q, r0, 0, 1, 'row') + assert_raises(ValueError, qr_delete, q, r0, 0, 3, 'row') + assert_raises(ValueError, qr_delete, q, r0, 0, 1, 'col') + assert_raises(ValueError, qr_delete, q, r0, 0, 3, 'col') + + r = r0.copy('F') + r[1,1] = np.nan + assert_raises(ValueError, qr_delete, q0, r, 0, 1, 'row') + assert_raises(ValueError, qr_delete, q0, r, 0, 3, 'row') + assert_raises(ValueError, qr_delete, q0, r, 0, 1, 'col') + assert_raises(ValueError, qr_delete, q0, r, 0, 3, 'col') + + def test_qr_scalar(self): + a, q, r = self.generate('1x1') + assert_raises(ValueError, qr_delete, q[0, 0], r, 0, 1, 'row') + assert_raises(ValueError, qr_delete, q, r[0, 0], 0, 1, 'row') + assert_raises(ValueError, qr_delete, q[0, 0], r, 0, 1, 'col') + assert_raises(ValueError, qr_delete, q, r[0, 0], 0, 1, 'col') + +class TestQRdelete_f(BaseQRdelete): + dtype = np.dtype('f') + +class TestQRdelete_F(BaseQRdelete): + dtype = np.dtype('F') + +class TestQRdelete_d(BaseQRdelete): + dtype = np.dtype('d') + +class TestQRdelete_D(BaseQRdelete): + dtype = np.dtype('D') + +class BaseQRinsert(BaseQRdeltas): + def generate(self, type, mode='full', which='row', p=1): + a, q, r = super().generate(type, mode) + + assert_(p > 0) + rng = np.random.RandomState(1234) + + # super call set the seed... + if which == 'row': + if p == 1: + u = rng.random(a.shape[1]) + else: + u = rng.random((p, a.shape[1])) + elif which == 'col': + if p == 1: + u = rng.random(a.shape[0]) + else: + u = rng.random((a.shape[0], p)) + else: + ValueError('which should be either "row" or "col"') + + if np.iscomplexobj(self.dtype.type(1)): + b = rng.random(u.shape) + u = u + 1j * b + + u = u.astype(self.dtype) + return a, q, r, u + + def test_sqr_1_row(self): + a, q, r, u = self.generate('sqr', which='row') + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, row, u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_sqr_p_row(self): + # sqr + rows --> fat always + a, q, r, u = self.generate('sqr', which='row', p=3) + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, np.full(3, row, np.intp), u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_sqr_1_col(self): + a, q, r, u = self.generate('sqr', which='col') + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, col, u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_sqr_p_col(self): + # sqr + cols --> fat always + a, q, r, u = self.generate('sqr', which='col', p=3) + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, np.full(3, col, np.intp), u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_tall_1_row(self): + a, q, r, u = self.generate('tall', which='row') + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, row, u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_tall_p_row(self): + # tall + rows --> tall always + a, q, r, u = self.generate('tall', which='row', p=3) + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, np.full(3, row, np.intp), u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_tall_1_col(self): + a, q, r, u = self.generate('tall', which='col') + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, col, u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + # for column adds to tall matrices there are three cases to test + # tall + pcol --> tall + # tall + pcol --> sqr + # tall + pcol --> fat + def base_tall_p_col_xxx(self, p): + a, q, r, u = self.generate('tall', which='col', p=p) + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, np.full(p, col, np.intp), u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_tall_p_col_tall(self): + # 12x7 + 12x3 = 12x10 --> stays tall + self.base_tall_p_col_xxx(3) + + def test_tall_p_col_sqr(self): + # 12x7 + 12x5 = 12x12 --> becomes sqr + self.base_tall_p_col_xxx(5) + + def test_tall_p_col_fat(self): + # 12x7 + 12x7 = 12x14 --> becomes fat + self.base_tall_p_col_xxx(7) + + def test_fat_1_row(self): + a, q, r, u = self.generate('fat', which='row') + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, row, u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + # for row adds to fat matrices there are three cases to test + # fat + prow --> fat + # fat + prow --> sqr + # fat + prow --> tall + def base_fat_p_row_xxx(self, p): + a, q, r, u = self.generate('fat', which='row', p=p) + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, np.full(p, row, np.intp), u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_fat_p_row_fat(self): + # 7x12 + 3x12 = 10x12 --> stays fat + self.base_fat_p_row_xxx(3) + + def test_fat_p_row_sqr(self): + # 7x12 + 5x12 = 12x12 --> becomes sqr + self.base_fat_p_row_xxx(5) + + def test_fat_p_row_tall(self): + # 7x12 + 7x12 = 14x12 --> becomes tall + self.base_fat_p_row_xxx(7) + + def test_fat_1_col(self): + a, q, r, u = self.generate('fat', which='col') + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, col, u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_fat_p_col(self): + # fat + cols --> fat always + a, q, r, u = self.generate('fat', which='col', p=3) + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, np.full(3, col, np.intp), u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_economic_1_row(self): + a, q, r, u = self.generate('tall', 'economic', 'row') + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row, overwrite_qru=False) + a1 = np.insert(a, row, u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_economic_p_row(self): + # tall + rows --> tall always + a, q, r, u = self.generate('tall', 'economic', 'row', 3) + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row, overwrite_qru=False) + a1 = np.insert(a, np.full(3, row, np.intp), u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_economic_1_col(self): + a, q, r, u = self.generate('tall', 'economic', which='col') + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u.copy(), col, 'col', overwrite_qru=False) + a1 = np.insert(a, col, u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_economic_1_col_bad_update(self): + # When the column to be added lies in the span of Q, the update is + # not meaningful. This is detected, and a LinAlgError is issued. + q = np.eye(5, 3, dtype=self.dtype) + r = np.eye(3, dtype=self.dtype) + u = np.array([1, 0, 0, 0, 0], self.dtype) + assert_raises(linalg.LinAlgError, qr_insert, q, r, u, 0, 'col') + + # for column adds to economic matrices there are three cases to test + # eco + pcol --> eco + # eco + pcol --> sqr + # eco + pcol --> fat + def base_economic_p_col_xxx(self, p): + a, q, r, u = self.generate('tall', 'economic', which='col', p=p) + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, np.full(p, col, np.intp), u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_economic_p_col_eco(self): + # 12x7 + 12x3 = 12x10 --> stays eco + self.base_economic_p_col_xxx(3) + + def test_economic_p_col_sqr(self): + # 12x7 + 12x5 = 12x12 --> becomes sqr + self.base_economic_p_col_xxx(5) + + def test_economic_p_col_fat(self): + # 12x7 + 12x7 = 12x14 --> becomes fat + self.base_economic_p_col_xxx(7) + + def test_Mx1_1_row(self): + a, q, r, u = self.generate('Mx1', which='row') + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, row, u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_Mx1_p_row(self): + a, q, r, u = self.generate('Mx1', which='row', p=3) + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, np.full(3, row, np.intp), u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_Mx1_1_col(self): + a, q, r, u = self.generate('Mx1', which='col') + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, col, u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_Mx1_p_col(self): + a, q, r, u = self.generate('Mx1', which='col', p=3) + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, np.full(3, col, np.intp), u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_Mx1_economic_1_row(self): + a, q, r, u = self.generate('Mx1', 'economic', 'row') + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, row, u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_Mx1_economic_p_row(self): + a, q, r, u = self.generate('Mx1', 'economic', 'row', 3) + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, np.full(3, row, np.intp), u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_Mx1_economic_1_col(self): + a, q, r, u = self.generate('Mx1', 'economic', 'col') + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, col, u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_Mx1_economic_p_col(self): + a, q, r, u = self.generate('Mx1', 'economic', 'col', 3) + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, np.full(3, col, np.intp), u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_1xN_1_row(self): + a, q, r, u = self.generate('1xN', which='row') + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, row, u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1xN_p_row(self): + a, q, r, u = self.generate('1xN', which='row', p=3) + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, np.full(3, row, np.intp), u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1xN_1_col(self): + a, q, r, u = self.generate('1xN', which='col') + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, col, u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1xN_p_col(self): + a, q, r, u = self.generate('1xN', which='col', p=3) + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, np.full(3, col, np.intp), u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1x1_1_row(self): + a, q, r, u = self.generate('1x1', which='row') + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, row, u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1x1_p_row(self): + a, q, r, u = self.generate('1x1', which='row', p=3) + for row in range(r.shape[0] + 1): + q1, r1 = qr_insert(q, r, u, row) + a1 = np.insert(a, np.full(3, row, np.intp), u, 0) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1x1_1_col(self): + a, q, r, u = self.generate('1x1', which='col') + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, col, u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1x1_p_col(self): + a, q, r, u = self.generate('1x1', which='col', p=3) + for col in range(r.shape[1] + 1): + q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False) + a1 = np.insert(a, np.full(3, col, np.intp), u, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1x1_1_scalar(self): + a, q, r, u = self.generate('1x1', which='row') + assert_raises(ValueError, qr_insert, q[0, 0], r, u, 0, 'row') + assert_raises(ValueError, qr_insert, q, r[0, 0], u, 0, 'row') + assert_raises(ValueError, qr_insert, q, r, u[0], 0, 'row') + + assert_raises(ValueError, qr_insert, q[0, 0], r, u, 0, 'col') + assert_raises(ValueError, qr_insert, q, r[0, 0], u, 0, 'col') + assert_raises(ValueError, qr_insert, q, r, u[0], 0, 'col') + + def base_non_simple_strides(self, adjust_strides, k, p, which): + for type in ['sqr', 'tall', 'fat']: + a, q0, r0, u0 = self.generate(type, which=which, p=p) + qs, rs, us = adjust_strides((q0, r0, u0)) + if p == 1: + ai = np.insert(a, k, u0, 0 if which == 'row' else 1) + else: + ai = np.insert(a, np.full(p, k, np.intp), + u0 if which == 'row' else u0, + 0 if which == 'row' else 1) + + # for each variable, q, r, u we try with it strided and + # overwrite=False. Then we try with overwrite=True. Nothing + # is checked to see if it can be overwritten, since only + # F ordered Q can be overwritten when adding columns. + + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + q1, r1 = qr_insert(qs, r, u, k, which, overwrite_qru=False) + check_qr(q1, r1, ai, self.rtol, self.atol) + q1o, r1o = qr_insert(qs, r, u, k, which, overwrite_qru=True) + check_qr(q1o, r1o, ai, self.rtol, self.atol) + + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + q2, r2 = qr_insert(q, rs, u, k, which, overwrite_qru=False) + check_qr(q2, r2, ai, self.rtol, self.atol) + q2o, r2o = qr_insert(q, rs, u, k, which, overwrite_qru=True) + check_qr(q2o, r2o, ai, self.rtol, self.atol) + + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + q3, r3 = qr_insert(q, r, us, k, which, overwrite_qru=False) + check_qr(q3, r3, ai, self.rtol, self.atol) + q3o, r3o = qr_insert(q, r, us, k, which, overwrite_qru=True) + check_qr(q3o, r3o, ai, self.rtol, self.atol) + + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + # since some of these were consumed above + qs, rs, us = adjust_strides((q, r, u)) + q5, r5 = qr_insert(qs, rs, us, k, which, overwrite_qru=False) + check_qr(q5, r5, ai, self.rtol, self.atol) + q5o, r5o = qr_insert(qs, rs, us, k, which, overwrite_qru=True) + check_qr(q5o, r5o, ai, self.rtol, self.atol) + + def test_non_unit_strides_1_row(self): + self.base_non_simple_strides(make_strided, 0, 1, 'row') + + def test_non_unit_strides_p_row(self): + self.base_non_simple_strides(make_strided, 0, 3, 'row') + + def test_non_unit_strides_1_col(self): + self.base_non_simple_strides(make_strided, 0, 1, 'col') + + def test_non_unit_strides_p_col(self): + self.base_non_simple_strides(make_strided, 0, 3, 'col') + + def test_neg_strides_1_row(self): + self.base_non_simple_strides(negate_strides, 0, 1, 'row') + + def test_neg_strides_p_row(self): + self.base_non_simple_strides(negate_strides, 0, 3, 'row') + + def test_neg_strides_1_col(self): + self.base_non_simple_strides(negate_strides, 0, 1, 'col') + + def test_neg_strides_p_col(self): + self.base_non_simple_strides(negate_strides, 0, 3, 'col') + + def test_non_itemsize_strides_1_row(self): + self.base_non_simple_strides(nonitemsize_strides, 0, 1, 'row') + + def test_non_itemsize_strides_p_row(self): + self.base_non_simple_strides(nonitemsize_strides, 0, 3, 'row') + + def test_non_itemsize_strides_1_col(self): + self.base_non_simple_strides(nonitemsize_strides, 0, 1, 'col') + + def test_non_itemsize_strides_p_col(self): + self.base_non_simple_strides(nonitemsize_strides, 0, 3, 'col') + + def test_non_native_byte_order_1_row(self): + self.base_non_simple_strides(make_nonnative, 0, 1, 'row') + + def test_non_native_byte_order_p_row(self): + self.base_non_simple_strides(make_nonnative, 0, 3, 'row') + + def test_non_native_byte_order_1_col(self): + self.base_non_simple_strides(make_nonnative, 0, 1, 'col') + + def test_non_native_byte_order_p_col(self): + self.base_non_simple_strides(make_nonnative, 0, 3, 'col') + + def test_overwrite_qu_rank_1(self): + # when inserting rows, the size of both Q and R change, so only + # column inserts can overwrite q. Only complex column inserts + # with C ordered Q overwrite u. Any contiguous Q is overwritten + # when inserting 1 column + a, q0, r, u, = self.generate('sqr', which='col', p=1) + q = q0.copy('C') + u0 = u.copy() + # don't overwrite + q1, r1 = qr_insert(q, r, u, 0, 'col', overwrite_qru=False) + a1 = np.insert(a, 0, u0, 1) + check_qr(q1, r1, a1, self.rtol, self.atol) + check_qr(q, r, a, self.rtol, self.atol) + + # try overwriting + q2, r2 = qr_insert(q, r, u, 0, 'col', overwrite_qru=True) + check_qr(q2, r2, a1, self.rtol, self.atol) + # verify the overwriting + assert_allclose(q2, q, rtol=self.rtol, atol=self.atol) + assert_allclose(u, u0.conj(), self.rtol, self.atol) + + # now try with a fortran ordered Q + qF = q0.copy('F') + u1 = u0.copy() + q3, r3 = qr_insert(qF, r, u1, 0, 'col', overwrite_qru=False) + check_qr(q3, r3, a1, self.rtol, self.atol) + check_qr(qF, r, a, self.rtol, self.atol) + + # try overwriting + q4, r4 = qr_insert(qF, r, u1, 0, 'col', overwrite_qru=True) + check_qr(q4, r4, a1, self.rtol, self.atol) + assert_allclose(q4, qF, rtol=self.rtol, atol=self.atol) + + def test_overwrite_qu_rank_p(self): + # when inserting rows, the size of both Q and R change, so only + # column inserts can potentially overwrite Q. In practice, only + # F ordered Q are overwritten with a rank p update. + a, q0, r, u, = self.generate('sqr', which='col', p=3) + q = q0.copy('F') + a1 = np.insert(a, np.zeros(3, np.intp), u, 1) + + # don't overwrite + q1, r1 = qr_insert(q, r, u, 0, 'col', overwrite_qru=False) + check_qr(q1, r1, a1, self.rtol, self.atol) + check_qr(q, r, a, self.rtol, self.atol) + + # try overwriting + q2, r2 = qr_insert(q, r, u, 0, 'col', overwrite_qru=True) + check_qr(q2, r2, a1, self.rtol, self.atol) + assert_allclose(q2, q, rtol=self.rtol, atol=self.atol) + + def test_empty_inputs(self): + a, q, r, u = self.generate('sqr', which='row') + assert_raises(ValueError, qr_insert, np.array([]), r, u, 0, 'row') + assert_raises(ValueError, qr_insert, q, np.array([]), u, 0, 'row') + assert_raises(ValueError, qr_insert, q, r, np.array([]), 0, 'row') + assert_raises(ValueError, qr_insert, np.array([]), r, u, 0, 'col') + assert_raises(ValueError, qr_insert, q, np.array([]), u, 0, 'col') + assert_raises(ValueError, qr_insert, q, r, np.array([]), 0, 'col') + + def test_mismatched_shapes(self): + a, q, r, u = self.generate('tall', which='row') + assert_raises(ValueError, qr_insert, q, r[1:], u, 0, 'row') + assert_raises(ValueError, qr_insert, q[:-2], r, u, 0, 'row') + assert_raises(ValueError, qr_insert, q, r, u[1:], 0, 'row') + assert_raises(ValueError, qr_insert, q, r[1:], u, 0, 'col') + assert_raises(ValueError, qr_insert, q[:-2], r, u, 0, 'col') + assert_raises(ValueError, qr_insert, q, r, u[1:], 0, 'col') + + def test_unsupported_dtypes(self): + dts = ['int8', 'int16', 'int32', 'int64', + 'uint8', 'uint16', 'uint32', 'uint64', + 'float16', 'longdouble', 'clongdouble', + 'bool'] + a, q0, r0, u0 = self.generate('sqr', which='row') + for dtype in dts: + q = q0.real.astype(dtype) + with np.errstate(invalid="ignore"): + r = r0.real.astype(dtype) + u = u0.real.astype(dtype) + assert_raises(ValueError, qr_insert, q, r0, u0, 0, 'row') + assert_raises(ValueError, qr_insert, q, r0, u0, 0, 'col') + assert_raises(ValueError, qr_insert, q0, r, u0, 0, 'row') + assert_raises(ValueError, qr_insert, q0, r, u0, 0, 'col') + assert_raises(ValueError, qr_insert, q0, r0, u, 0, 'row') + assert_raises(ValueError, qr_insert, q0, r0, u, 0, 'col') + + def test_check_finite(self): + a0, q0, r0, u0 = self.generate('sqr', which='row', p=3) + + q = q0.copy('F') + q[1,1] = np.nan + assert_raises(ValueError, qr_insert, q, r0, u0[:,0], 0, 'row') + assert_raises(ValueError, qr_insert, q, r0, u0, 0, 'row') + assert_raises(ValueError, qr_insert, q, r0, u0[:,0], 0, 'col') + assert_raises(ValueError, qr_insert, q, r0, u0, 0, 'col') + + r = r0.copy('F') + r[1,1] = np.nan + assert_raises(ValueError, qr_insert, q0, r, u0[:,0], 0, 'row') + assert_raises(ValueError, qr_insert, q0, r, u0, 0, 'row') + assert_raises(ValueError, qr_insert, q0, r, u0[:,0], 0, 'col') + assert_raises(ValueError, qr_insert, q0, r, u0, 0, 'col') + + u = u0.copy('F') + u[0,0] = np.nan + assert_raises(ValueError, qr_insert, q0, r0, u[:,0], 0, 'row') + assert_raises(ValueError, qr_insert, q0, r0, u, 0, 'row') + assert_raises(ValueError, qr_insert, q0, r0, u[:,0], 0, 'col') + assert_raises(ValueError, qr_insert, q0, r0, u, 0, 'col') + +class TestQRinsert_f(BaseQRinsert): + dtype = np.dtype('f') + +class TestQRinsert_F(BaseQRinsert): + dtype = np.dtype('F') + +class TestQRinsert_d(BaseQRinsert): + dtype = np.dtype('d') + +class TestQRinsert_D(BaseQRinsert): + dtype = np.dtype('D') + +class BaseQRupdate(BaseQRdeltas): + def generate(self, type, mode='full', p=1): + a, q, r = super().generate(type, mode) + + # super call set the seed... + if p == 1: + u = np.random.random(q.shape[0]) + v = np.random.random(r.shape[1]) + else: + u = np.random.random((q.shape[0], p)) + v = np.random.random((r.shape[1], p)) + + if np.iscomplexobj(self.dtype.type(1)): + b = np.random.random(u.shape) + u = u + 1j * b + + c = np.random.random(v.shape) + v = v + 1j * c + + u = u.astype(self.dtype) + v = v.astype(self.dtype) + return a, q, r, u, v + + def test_sqr_rank_1(self): + a, q, r, u, v = self.generate('sqr') + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.outer(u, v.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_sqr_rank_p(self): + # test ndim = 2, rank 1 updates here too + for p in [1, 2, 3, 5]: + a, q, r, u, v = self.generate('sqr', p=p) + if p == 1: + u = u.reshape(u.size, 1) + v = v.reshape(v.size, 1) + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.dot(u, v.T.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_tall_rank_1(self): + a, q, r, u, v = self.generate('tall') + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.outer(u, v.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_tall_rank_p(self): + for p in [1, 2, 3, 5]: + a, q, r, u, v = self.generate('tall', p=p) + if p == 1: + u = u.reshape(u.size, 1) + v = v.reshape(v.size, 1) + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.dot(u, v.T.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_fat_rank_1(self): + a, q, r, u, v = self.generate('fat') + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.outer(u, v.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_fat_rank_p(self): + for p in [1, 2, 3, 5]: + a, q, r, u, v = self.generate('fat', p=p) + if p == 1: + u = u.reshape(u.size, 1) + v = v.reshape(v.size, 1) + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.dot(u, v.T.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_economic_rank_1(self): + a, q, r, u, v = self.generate('tall', 'economic') + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.outer(u, v.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_economic_rank_p(self): + for p in [1, 2, 3, 5]: + a, q, r, u, v = self.generate('tall', 'economic', p) + if p == 1: + u = u.reshape(u.size, 1) + v = v.reshape(v.size, 1) + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.dot(u, v.T.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_Mx1_rank_1(self): + a, q, r, u, v = self.generate('Mx1') + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.outer(u, v.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_Mx1_rank_p(self): + # when M or N == 1, only a rank 1 update is allowed. This isn't + # fundamental limitation, but the code does not support it. + a, q, r, u, v = self.generate('Mx1', p=1) + u = u.reshape(u.size, 1) + v = v.reshape(v.size, 1) + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.dot(u, v.T.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_Mx1_economic_rank_1(self): + a, q, r, u, v = self.generate('Mx1', 'economic') + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.outer(u, v.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_Mx1_economic_rank_p(self): + # when M or N == 1, only a rank 1 update is allowed. This isn't + # fundamental limitation, but the code does not support it. + a, q, r, u, v = self.generate('Mx1', 'economic', p=1) + u = u.reshape(u.size, 1) + v = v.reshape(v.size, 1) + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.dot(u, v.T.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + + def test_1xN_rank_1(self): + a, q, r, u, v = self.generate('1xN') + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.outer(u, v.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1xN_rank_p(self): + # when M or N == 1, only a rank 1 update is allowed. This isn't + # fundamental limitation, but the code does not support it. + a, q, r, u, v = self.generate('1xN', p=1) + u = u.reshape(u.size, 1) + v = v.reshape(v.size, 1) + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.dot(u, v.T.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1x1_rank_1(self): + a, q, r, u, v = self.generate('1x1') + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.outer(u, v.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1x1_rank_p(self): + # when M or N == 1, only a rank 1 update is allowed. This isn't + # fundamental limitation, but the code does not support it. + a, q, r, u, v = self.generate('1x1', p=1) + u = u.reshape(u.size, 1) + v = v.reshape(v.size, 1) + q1, r1 = qr_update(q, r, u, v, False) + a1 = a + np.dot(u, v.T.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol) + + def test_1x1_rank_1_scalar(self): + a, q, r, u, v = self.generate('1x1') + assert_raises(ValueError, qr_update, q[0, 0], r, u, v) + assert_raises(ValueError, qr_update, q, r[0, 0], u, v) + assert_raises(ValueError, qr_update, q, r, u[0], v) + assert_raises(ValueError, qr_update, q, r, u, v[0]) + + def base_non_simple_strides(self, adjust_strides, mode, p, overwriteable): + assert_sqr = False if mode == 'economic' else True + for type in ['sqr', 'tall', 'fat']: + a, q0, r0, u0, v0 = self.generate(type, mode, p) + qs, rs, us, vs = adjust_strides((q0, r0, u0, v0)) + if p == 1: + aup = a + np.outer(u0, v0.conj()) + else: + aup = a + np.dot(u0, v0.T.conj()) + + # for each variable, q, r, u, v we try with it strided and + # overwrite=False. Then we try with overwrite=True, and make + # sure that if p == 1, r and v are still overwritten. + # a strided q and u must always be copied. + + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + v = v0.copy('C') + q1, r1 = qr_update(qs, r, u, v, False) + check_qr(q1, r1, aup, self.rtol, self.atol, assert_sqr) + q1o, r1o = qr_update(qs, r, u, v, True) + check_qr(q1o, r1o, aup, self.rtol, self.atol, assert_sqr) + if overwriteable: + assert_allclose(r1o, r, rtol=self.rtol, atol=self.atol) + assert_allclose(v, v0.conj(), rtol=self.rtol, atol=self.atol) + + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + v = v0.copy('C') + q2, r2 = qr_update(q, rs, u, v, False) + check_qr(q2, r2, aup, self.rtol, self.atol, assert_sqr) + q2o, r2o = qr_update(q, rs, u, v, True) + check_qr(q2o, r2o, aup, self.rtol, self.atol, assert_sqr) + if overwriteable: + assert_allclose(r2o, rs, rtol=self.rtol, atol=self.atol) + assert_allclose(v, v0.conj(), rtol=self.rtol, atol=self.atol) + + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + v = v0.copy('C') + q3, r3 = qr_update(q, r, us, v, False) + check_qr(q3, r3, aup, self.rtol, self.atol, assert_sqr) + q3o, r3o = qr_update(q, r, us, v, True) + check_qr(q3o, r3o, aup, self.rtol, self.atol, assert_sqr) + if overwriteable: + assert_allclose(r3o, r, rtol=self.rtol, atol=self.atol) + assert_allclose(v, v0.conj(), rtol=self.rtol, atol=self.atol) + + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + v = v0.copy('C') + q4, r4 = qr_update(q, r, u, vs, False) + check_qr(q4, r4, aup, self.rtol, self.atol, assert_sqr) + q4o, r4o = qr_update(q, r, u, vs, True) + check_qr(q4o, r4o, aup, self.rtol, self.atol, assert_sqr) + if overwriteable: + assert_allclose(r4o, r, rtol=self.rtol, atol=self.atol) + assert_allclose(vs, v0.conj(), rtol=self.rtol, atol=self.atol) + + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + v = v0.copy('C') + # since some of these were consumed above + qs, rs, us, vs = adjust_strides((q, r, u, v)) + q5, r5 = qr_update(qs, rs, us, vs, False) + check_qr(q5, r5, aup, self.rtol, self.atol, assert_sqr) + q5o, r5o = qr_update(qs, rs, us, vs, True) + check_qr(q5o, r5o, aup, self.rtol, self.atol, assert_sqr) + if overwriteable: + assert_allclose(r5o, rs, rtol=self.rtol, atol=self.atol) + assert_allclose(vs, v0.conj(), rtol=self.rtol, atol=self.atol) + + def test_non_unit_strides_rank_1(self): + self.base_non_simple_strides(make_strided, 'full', 1, True) + + def test_non_unit_strides_economic_rank_1(self): + self.base_non_simple_strides(make_strided, 'economic', 1, True) + + def test_non_unit_strides_rank_p(self): + self.base_non_simple_strides(make_strided, 'full', 3, False) + + def test_non_unit_strides_economic_rank_p(self): + self.base_non_simple_strides(make_strided, 'economic', 3, False) + + def test_neg_strides_rank_1(self): + self.base_non_simple_strides(negate_strides, 'full', 1, False) + + def test_neg_strides_economic_rank_1(self): + self.base_non_simple_strides(negate_strides, 'economic', 1, False) + + def test_neg_strides_rank_p(self): + self.base_non_simple_strides(negate_strides, 'full', 3, False) + + def test_neg_strides_economic_rank_p(self): + self.base_non_simple_strides(negate_strides, 'economic', 3, False) + + def test_non_itemsize_strides_rank_1(self): + self.base_non_simple_strides(nonitemsize_strides, 'full', 1, False) + + def test_non_itemsize_strides_economic_rank_1(self): + self.base_non_simple_strides(nonitemsize_strides, 'economic', 1, False) + + def test_non_itemsize_strides_rank_p(self): + self.base_non_simple_strides(nonitemsize_strides, 'full', 3, False) + + def test_non_itemsize_strides_economic_rank_p(self): + self.base_non_simple_strides(nonitemsize_strides, 'economic', 3, False) + + def test_non_native_byte_order_rank_1(self): + self.base_non_simple_strides(make_nonnative, 'full', 1, False) + + def test_non_native_byte_order_economic_rank_1(self): + self.base_non_simple_strides(make_nonnative, 'economic', 1, False) + + def test_non_native_byte_order_rank_p(self): + self.base_non_simple_strides(make_nonnative, 'full', 3, False) + + def test_non_native_byte_order_economic_rank_p(self): + self.base_non_simple_strides(make_nonnative, 'economic', 3, False) + + def test_overwrite_qruv_rank_1(self): + # Any positive strided q, r, u, and v can be overwritten for a rank 1 + # update, only checking C and F contiguous. + a, q0, r0, u0, v0 = self.generate('sqr') + a1 = a + np.outer(u0, v0.conj()) + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + v = v0.copy('F') + + # don't overwrite + q1, r1 = qr_update(q, r, u, v, False) + check_qr(q1, r1, a1, self.rtol, self.atol) + check_qr(q, r, a, self.rtol, self.atol) + + q2, r2 = qr_update(q, r, u, v, True) + check_qr(q2, r2, a1, self.rtol, self.atol) + # verify the overwriting, no good way to check u and v. + assert_allclose(q2, q, rtol=self.rtol, atol=self.atol) + assert_allclose(r2, r, rtol=self.rtol, atol=self.atol) + + q = q0.copy('C') + r = r0.copy('C') + u = u0.copy('C') + v = v0.copy('C') + q3, r3 = qr_update(q, r, u, v, True) + check_qr(q3, r3, a1, self.rtol, self.atol) + assert_allclose(q3, q, rtol=self.rtol, atol=self.atol) + assert_allclose(r3, r, rtol=self.rtol, atol=self.atol) + + def test_overwrite_qruv_rank_1_economic(self): + # updating economic decompositions can overwrite any contiguous r, + # and positively strided r and u. V is only ever read. + # only checking C and F contiguous. + a, q0, r0, u0, v0 = self.generate('tall', 'economic') + a1 = a + np.outer(u0, v0.conj()) + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + v = v0.copy('F') + + # don't overwrite + q1, r1 = qr_update(q, r, u, v, False) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + check_qr(q, r, a, self.rtol, self.atol, False) + + q2, r2 = qr_update(q, r, u, v, True) + check_qr(q2, r2, a1, self.rtol, self.atol, False) + # verify the overwriting, no good way to check u and v. + assert_allclose(q2, q, rtol=self.rtol, atol=self.atol) + assert_allclose(r2, r, rtol=self.rtol, atol=self.atol) + + q = q0.copy('C') + r = r0.copy('C') + u = u0.copy('C') + v = v0.copy('C') + q3, r3 = qr_update(q, r, u, v, True) + check_qr(q3, r3, a1, self.rtol, self.atol, False) + assert_allclose(q3, q, rtol=self.rtol, atol=self.atol) + assert_allclose(r3, r, rtol=self.rtol, atol=self.atol) + + def test_overwrite_qruv_rank_p(self): + # for rank p updates, q r must be F contiguous, v must be C (v.T --> F) + # and u can be C or F, but is only overwritten if Q is C and complex + a, q0, r0, u0, v0 = self.generate('sqr', p=3) + a1 = a + np.dot(u0, v0.T.conj()) + q = q0.copy('F') + r = r0.copy('F') + u = u0.copy('F') + v = v0.copy('C') + + # don't overwrite + q1, r1 = qr_update(q, r, u, v, False) + check_qr(q1, r1, a1, self.rtol, self.atol) + check_qr(q, r, a, self.rtol, self.atol) + + q2, r2 = qr_update(q, r, u, v, True) + check_qr(q2, r2, a1, self.rtol, self.atol) + # verify the overwriting, no good way to check u and v. + assert_allclose(q2, q, rtol=self.rtol, atol=self.atol) + assert_allclose(r2, r, rtol=self.rtol, atol=self.atol) + + def test_empty_inputs(self): + a, q, r, u, v = self.generate('tall') + assert_raises(ValueError, qr_update, np.array([]), r, u, v) + assert_raises(ValueError, qr_update, q, np.array([]), u, v) + assert_raises(ValueError, qr_update, q, r, np.array([]), v) + assert_raises(ValueError, qr_update, q, r, u, np.array([])) + + def test_mismatched_shapes(self): + a, q, r, u, v = self.generate('tall') + assert_raises(ValueError, qr_update, q, r[1:], u, v) + assert_raises(ValueError, qr_update, q[:-2], r, u, v) + assert_raises(ValueError, qr_update, q, r, u[1:], v) + assert_raises(ValueError, qr_update, q, r, u, v[1:]) + + def test_unsupported_dtypes(self): + dts = ['int8', 'int16', 'int32', 'int64', + 'uint8', 'uint16', 'uint32', 'uint64', + 'float16', 'longdouble', 'clongdouble', + 'bool'] + a, q0, r0, u0, v0 = self.generate('tall') + for dtype in dts: + q = q0.real.astype(dtype) + with np.errstate(invalid="ignore"): + r = r0.real.astype(dtype) + u = u0.real.astype(dtype) + v = v0.real.astype(dtype) + assert_raises(ValueError, qr_update, q, r0, u0, v0) + assert_raises(ValueError, qr_update, q0, r, u0, v0) + assert_raises(ValueError, qr_update, q0, r0, u, v0) + assert_raises(ValueError, qr_update, q0, r0, u0, v) + + def test_integer_input(self): + q = np.arange(16).reshape(4, 4) + r = q.copy() # doesn't matter + u = q[:, 0].copy() + v = r[0, :].copy() + assert_raises(ValueError, qr_update, q, r, u, v) + + def test_check_finite(self): + a0, q0, r0, u0, v0 = self.generate('tall', p=3) + + q = q0.copy('F') + q[1,1] = np.nan + assert_raises(ValueError, qr_update, q, r0, u0[:,0], v0[:,0]) + assert_raises(ValueError, qr_update, q, r0, u0, v0) + + r = r0.copy('F') + r[1,1] = np.nan + assert_raises(ValueError, qr_update, q0, r, u0[:,0], v0[:,0]) + assert_raises(ValueError, qr_update, q0, r, u0, v0) + + u = u0.copy('F') + u[0,0] = np.nan + assert_raises(ValueError, qr_update, q0, r0, u[:,0], v0[:,0]) + assert_raises(ValueError, qr_update, q0, r0, u, v0) + + v = v0.copy('F') + v[0,0] = np.nan + assert_raises(ValueError, qr_update, q0, r0, u[:,0], v[:,0]) + assert_raises(ValueError, qr_update, q0, r0, u, v) + + def test_economic_check_finite(self): + a0, q0, r0, u0, v0 = self.generate('tall', mode='economic', p=3) + + q = q0.copy('F') + q[1,1] = np.nan + assert_raises(ValueError, qr_update, q, r0, u0[:,0], v0[:,0]) + assert_raises(ValueError, qr_update, q, r0, u0, v0) + + r = r0.copy('F') + r[1,1] = np.nan + assert_raises(ValueError, qr_update, q0, r, u0[:,0], v0[:,0]) + assert_raises(ValueError, qr_update, q0, r, u0, v0) + + u = u0.copy('F') + u[0,0] = np.nan + assert_raises(ValueError, qr_update, q0, r0, u[:,0], v0[:,0]) + assert_raises(ValueError, qr_update, q0, r0, u, v0) + + v = v0.copy('F') + v[0,0] = np.nan + assert_raises(ValueError, qr_update, q0, r0, u[:,0], v[:,0]) + assert_raises(ValueError, qr_update, q0, r0, u, v) + + def test_u_exactly_in_span_q(self): + q = np.array([[0, 0], [0, 0], [1, 0], [0, 1]], self.dtype) + r = np.array([[1, 0], [0, 1]], self.dtype) + u = np.array([0, 0, 0, -1], self.dtype) + v = np.array([1, 2], self.dtype) + q1, r1 = qr_update(q, r, u, v) + a1 = np.dot(q, r) + np.outer(u, v.conj()) + check_qr(q1, r1, a1, self.rtol, self.atol, False) + +class TestQRupdate_f(BaseQRupdate): + dtype = np.dtype('f') + +class TestQRupdate_F(BaseQRupdate): + dtype = np.dtype('F') + +class TestQRupdate_d(BaseQRupdate): + dtype = np.dtype('d') + +class TestQRupdate_D(BaseQRupdate): + dtype = np.dtype('D') + +def test_form_qTu(): + # We want to ensure that all of the code paths through this function are + # tested. Most of them should be hit with the rest of test suite, but + # explicit tests make clear precisely what is being tested. + # + # This function expects that Q is either C or F contiguous and square. + # Economic mode decompositions (Q is (M, N), M != N) do not go through this + # function. U may have any positive strides. + # + # Some of these test are duplicates, since contiguous 1d arrays are both C + # and F. + + q_order = ['F', 'C'] + q_shape = [(8, 8), ] + u_order = ['F', 'C', 'A'] # here A means is not F not C + u_shape = [1, 3] + dtype = ['f', 'd', 'F', 'D'] + + for qo, qs, uo, us, d in \ + itertools.product(q_order, q_shape, u_order, u_shape, dtype): + if us == 1: + check_form_qTu(qo, qs, uo, us, 1, d) + check_form_qTu(qo, qs, uo, us, 2, d) + else: + check_form_qTu(qo, qs, uo, us, 2, d) + +def check_form_qTu(q_order, q_shape, u_order, u_shape, u_ndim, dtype): + np.random.seed(47) + if u_shape == 1 and u_ndim == 1: + u_shape = (q_shape[0],) + else: + u_shape = (q_shape[0], u_shape) + dtype = np.dtype(dtype) + + if dtype.char in 'fd': + q = np.random.random(q_shape) + u = np.random.random(u_shape) + elif dtype.char in 'FD': + q = np.random.random(q_shape) + 1j*np.random.random(q_shape) + u = np.random.random(u_shape) + 1j*np.random.random(u_shape) + else: + ValueError("form_qTu doesn't support this dtype") + + q = np.require(q, dtype, q_order) + if u_order != 'A': + u = np.require(u, dtype, u_order) + else: + u, = make_strided((u.astype(dtype),)) + + rtol = 10.0 ** -(np.finfo(dtype).precision-2) + atol = 2*np.finfo(dtype).eps + + expected = np.dot(q.T.conj(), u) + res = _decomp_update._form_qTu(q, u) + assert_allclose(res, expected, rtol=rtol, atol=atol) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_extending.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_extending.py new file mode 100644 index 0000000000000000000000000000000000000000..36e4692cd9717a221cc683a663e8ee23a81aa5b6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_extending.py @@ -0,0 +1,46 @@ +import os +import platform +import sysconfig + +import numpy as np +import pytest + +from scipy._lib._testutils import IS_EDITABLE, _test_cython_extension, cython +from scipy.linalg.blas import cdotu # type: ignore[attr-defined] +from scipy.linalg.lapack import dgtsv # type: ignore[attr-defined] + + +@pytest.mark.fail_slow(120) +# essential per https://github.com/scipy/scipy/pull/20487#discussion_r1567057247 +@pytest.mark.skipif(IS_EDITABLE, + reason='Editable install cannot find .pxd headers.') +@pytest.mark.skipif((platform.system() == 'Windows' and + sysconfig.get_config_var('Py_GIL_DISABLED')), + reason='gh-22039') +@pytest.mark.skipif(platform.machine() in ["wasm32", "wasm64"], + reason="Can't start subprocess") +@pytest.mark.skipif(cython is None, reason="requires cython") +def test_cython(tmp_path): + srcdir = os.path.dirname(os.path.dirname(__file__)) + extensions, extensions_cpp = _test_cython_extension(tmp_path, srcdir) + # actually test the cython c-extensions + a = np.ones(8) * 3 + b = np.ones(9) + c = np.ones(8) * 4 + x = np.ones(9) + _, _, _, x, _ = dgtsv(a, b, c, x) + a = np.ones(8) * 3 + b = np.ones(9) + c = np.ones(8) * 4 + x_c = np.ones(9) + extensions.tridiag(a, b, c, x_c) + a = np.ones(8) * 3 + b = np.ones(9) + c = np.ones(8) * 4 + x_cpp = np.ones(9) + extensions_cpp.tridiag(a, b, c, x_cpp) + np.testing.assert_array_equal(x, x_cpp) + cx = np.array([1-1j, 2+2j, 3-3j], dtype=np.complex64) + cy = np.array([4+4j, 5-5j, 6+6j], dtype=np.complex64) + np.testing.assert_array_equal(cdotu(cx, cy), extensions.complex_dot(cx, cy)) + np.testing.assert_array_equal(cdotu(cx, cy), extensions_cpp.complex_dot(cx, cy)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_fblas.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_fblas.py new file mode 100644 index 0000000000000000000000000000000000000000..7c5ada830043af0eecb6d04bf39aef13d29d777c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_fblas.py @@ -0,0 +1,607 @@ +# Test interfaces to fortran blas. +# +# The tests are more of interface than they are of the underlying blas. +# Only very small matrices checked -- N=3 or so. +# +# !! Complex calculations really aren't checked that carefully. +# !! Only real valued complex numbers are used in tests. + +from numpy import float32, float64, complex64, complex128, arange, array, \ + zeros, shape, transpose, newaxis, common_type, conjugate + +from scipy.linalg import _fblas as fblas + +from numpy.testing import assert_array_equal, \ + assert_allclose, assert_array_almost_equal, assert_ + +import pytest + +# decimal accuracy to require between Python and LAPACK/BLAS calculations +accuracy = 5 + +# Since numpy.dot likely uses the same blas, use this routine +# to check. + + +def matrixmultiply(a, b): + if len(b.shape) == 1: + b_is_vector = True + b = b[:, newaxis] + else: + b_is_vector = False + assert_(a.shape[1] == b.shape[0]) + c = zeros((a.shape[0], b.shape[1]), common_type(a, b)) + for i in range(a.shape[0]): + for j in range(b.shape[1]): + s = 0 + for k in range(a.shape[1]): + s += a[i, k] * b[k, j] + c[i, j] = s + if b_is_vector: + c = c.reshape((a.shape[0],)) + return c + +################################################## +# Test blas ?axpy + + +class BaseAxpy: + ''' Mixin class for axpy tests ''' + + def test_default_a(self): + x = arange(3., dtype=self.dtype) + y = arange(3., dtype=x.dtype) + real_y = x*1.+y + y = self.blas_func(x, y) + assert_array_equal(real_y, y) + + def test_simple(self): + x = arange(3., dtype=self.dtype) + y = arange(3., dtype=x.dtype) + real_y = x*3.+y + y = self.blas_func(x, y, a=3.) + assert_array_equal(real_y, y) + + def test_x_stride(self): + x = arange(6., dtype=self.dtype) + y = zeros(3, x.dtype) + y = arange(3., dtype=x.dtype) + real_y = x[::2]*3.+y + y = self.blas_func(x, y, a=3., n=3, incx=2) + assert_array_equal(real_y, y) + + def test_y_stride(self): + x = arange(3., dtype=self.dtype) + y = zeros(6, x.dtype) + real_y = x*3.+y[::2] + y = self.blas_func(x, y, a=3., n=3, incy=2) + assert_array_equal(real_y, y[::2]) + + def test_x_and_y_stride(self): + x = arange(12., dtype=self.dtype) + y = zeros(6, x.dtype) + real_y = x[::4]*3.+y[::2] + y = self.blas_func(x, y, a=3., n=3, incx=4, incy=2) + assert_array_equal(real_y, y[::2]) + + def test_x_bad_size(self): + x = arange(12., dtype=self.dtype) + y = zeros(6, x.dtype) + with pytest.raises(Exception, match='failed for 1st keyword'): + self.blas_func(x, y, n=4, incx=5) + + def test_y_bad_size(self): + x = arange(12., dtype=self.dtype) + y = zeros(6, x.dtype) + with pytest.raises(Exception, match='failed for 1st keyword'): + self.blas_func(x, y, n=3, incy=5) + + +try: + class TestSaxpy(BaseAxpy): + blas_func = fblas.saxpy + dtype = float32 +except AttributeError: + class TestSaxpy: + pass + + +class TestDaxpy(BaseAxpy): + blas_func = fblas.daxpy + dtype = float64 + + +try: + class TestCaxpy(BaseAxpy): + blas_func = fblas.caxpy + dtype = complex64 +except AttributeError: + class TestCaxpy: + pass + + +class TestZaxpy(BaseAxpy): + blas_func = fblas.zaxpy + dtype = complex128 + + +################################################## +# Test blas ?scal + +class BaseScal: + ''' Mixin class for scal testing ''' + + def test_simple(self): + x = arange(3., dtype=self.dtype) + real_x = x*3. + x = self.blas_func(3., x) + assert_array_equal(real_x, x) + + def test_x_stride(self): + x = arange(6., dtype=self.dtype) + real_x = x.copy() + real_x[::2] = x[::2]*array(3., self.dtype) + x = self.blas_func(3., x, n=3, incx=2) + assert_array_equal(real_x, x) + + def test_x_bad_size(self): + x = arange(12., dtype=self.dtype) + with pytest.raises(Exception, match='failed for 1st keyword'): + self.blas_func(2., x, n=4, incx=5) + + +try: + class TestSscal(BaseScal): + blas_func = fblas.sscal + dtype = float32 +except AttributeError: + class TestSscal: + pass + + +class TestDscal(BaseScal): + blas_func = fblas.dscal + dtype = float64 + + +try: + class TestCscal(BaseScal): + blas_func = fblas.cscal + dtype = complex64 +except AttributeError: + class TestCscal: + pass + + +class TestZscal(BaseScal): + blas_func = fblas.zscal + dtype = complex128 + + +################################################## +# Test blas ?copy + +class BaseCopy: + ''' Mixin class for copy testing ''' + + def test_simple(self): + x = arange(3., dtype=self.dtype) + y = zeros(shape(x), x.dtype) + y = self.blas_func(x, y) + assert_array_equal(x, y) + + def test_x_stride(self): + x = arange(6., dtype=self.dtype) + y = zeros(3, x.dtype) + y = self.blas_func(x, y, n=3, incx=2) + assert_array_equal(x[::2], y) + + def test_y_stride(self): + x = arange(3., dtype=self.dtype) + y = zeros(6, x.dtype) + y = self.blas_func(x, y, n=3, incy=2) + assert_array_equal(x, y[::2]) + + def test_x_and_y_stride(self): + x = arange(12., dtype=self.dtype) + y = zeros(6, x.dtype) + y = self.blas_func(x, y, n=3, incx=4, incy=2) + assert_array_equal(x[::4], y[::2]) + + def test_x_bad_size(self): + x = arange(12., dtype=self.dtype) + y = zeros(6, x.dtype) + with pytest.raises(Exception, match='failed for 1st keyword'): + self.blas_func(x, y, n=4, incx=5) + + def test_y_bad_size(self): + x = arange(12., dtype=self.dtype) + y = zeros(6, x.dtype) + with pytest.raises(Exception, match='failed for 1st keyword'): + self.blas_func(x, y, n=3, incy=5) + + # def test_y_bad_type(self): + ## Hmmm. Should this work? What should be the output. + # x = arange(3.,dtype=self.dtype) + # y = zeros(shape(x)) + # self.blas_func(x,y) + # assert_array_equal(x,y) + + +try: + class TestScopy(BaseCopy): + blas_func = fblas.scopy + dtype = float32 +except AttributeError: + class TestScopy: + pass + + +class TestDcopy(BaseCopy): + blas_func = fblas.dcopy + dtype = float64 + + +try: + class TestCcopy(BaseCopy): + blas_func = fblas.ccopy + dtype = complex64 +except AttributeError: + class TestCcopy: + pass + + +class TestZcopy(BaseCopy): + blas_func = fblas.zcopy + dtype = complex128 + + +################################################## +# Test blas ?swap + +class BaseSwap: + ''' Mixin class for swap tests ''' + + def test_simple(self): + x = arange(3., dtype=self.dtype) + y = zeros(shape(x), x.dtype) + desired_x = y.copy() + desired_y = x.copy() + x, y = self.blas_func(x, y) + assert_array_equal(desired_x, x) + assert_array_equal(desired_y, y) + + def test_x_stride(self): + x = arange(6., dtype=self.dtype) + y = zeros(3, x.dtype) + desired_x = y.copy() + desired_y = x.copy()[::2] + x, y = self.blas_func(x, y, n=3, incx=2) + assert_array_equal(desired_x, x[::2]) + assert_array_equal(desired_y, y) + + def test_y_stride(self): + x = arange(3., dtype=self.dtype) + y = zeros(6, x.dtype) + desired_x = y.copy()[::2] + desired_y = x.copy() + x, y = self.blas_func(x, y, n=3, incy=2) + assert_array_equal(desired_x, x) + assert_array_equal(desired_y, y[::2]) + + def test_x_and_y_stride(self): + x = arange(12., dtype=self.dtype) + y = zeros(6, x.dtype) + desired_x = y.copy()[::2] + desired_y = x.copy()[::4] + x, y = self.blas_func(x, y, n=3, incx=4, incy=2) + assert_array_equal(desired_x, x[::4]) + assert_array_equal(desired_y, y[::2]) + + def test_x_bad_size(self): + x = arange(12., dtype=self.dtype) + y = zeros(6, x.dtype) + with pytest.raises(Exception, match='failed for 1st keyword'): + self.blas_func(x, y, n=4, incx=5) + + def test_y_bad_size(self): + x = arange(12., dtype=self.dtype) + y = zeros(6, x.dtype) + with pytest.raises(Exception, match='failed for 1st keyword'): + self.blas_func(x, y, n=3, incy=5) + + +try: + class TestSswap(BaseSwap): + blas_func = fblas.sswap + dtype = float32 +except AttributeError: + class TestSswap: + pass + + +class TestDswap(BaseSwap): + blas_func = fblas.dswap + dtype = float64 + + +try: + class TestCswap(BaseSwap): + blas_func = fblas.cswap + dtype = complex64 +except AttributeError: + class TestCswap: + pass + + +class TestZswap(BaseSwap): + blas_func = fblas.zswap + dtype = complex128 + +################################################## +# Test blas ?gemv +# This will be a mess to test all cases. + + +class BaseGemv: + ''' Mixin class for gemv tests ''' + + def get_data(self, x_stride=1, y_stride=1): + mult = array(1, dtype=self.dtype) + if self.dtype in [complex64, complex128]: + mult = array(1+1j, dtype=self.dtype) + from numpy.random import normal, seed + seed(1234) + alpha = array(1., dtype=self.dtype) * mult + beta = array(1., dtype=self.dtype) * mult + a = normal(0., 1., (3, 3)).astype(self.dtype) * mult + x = arange(shape(a)[0]*x_stride, dtype=self.dtype) * mult + y = arange(shape(a)[1]*y_stride, dtype=self.dtype) * mult + return alpha, beta, a, x, y + + def test_simple(self): + alpha, beta, a, x, y = self.get_data() + desired_y = alpha*matrixmultiply(a, x)+beta*y + y = self.blas_func(alpha, a, x, beta, y) + assert_array_almost_equal(desired_y, y) + + def test_default_beta_y(self): + alpha, beta, a, x, y = self.get_data() + desired_y = matrixmultiply(a, x) + y = self.blas_func(1, a, x) + assert_array_almost_equal(desired_y, y) + + def test_simple_transpose(self): + alpha, beta, a, x, y = self.get_data() + desired_y = alpha*matrixmultiply(transpose(a), x)+beta*y + y = self.blas_func(alpha, a, x, beta, y, trans=1) + assert_array_almost_equal(desired_y, y) + + def test_simple_transpose_conj(self): + alpha, beta, a, x, y = self.get_data() + desired_y = alpha*matrixmultiply(transpose(conjugate(a)), x)+beta*y + y = self.blas_func(alpha, a, x, beta, y, trans=2) + assert_array_almost_equal(desired_y, y) + + def test_x_stride(self): + alpha, beta, a, x, y = self.get_data(x_stride=2) + desired_y = alpha*matrixmultiply(a, x[::2])+beta*y + y = self.blas_func(alpha, a, x, beta, y, incx=2) + assert_array_almost_equal(desired_y, y) + + def test_x_stride_transpose(self): + alpha, beta, a, x, y = self.get_data(x_stride=2) + desired_y = alpha*matrixmultiply(transpose(a), x[::2])+beta*y + y = self.blas_func(alpha, a, x, beta, y, trans=1, incx=2) + assert_array_almost_equal(desired_y, y) + + def test_x_stride_assert(self): + # What is the use of this test? + alpha, beta, a, x, y = self.get_data(x_stride=2) + with pytest.raises(Exception, match='failed for 3rd argument'): + y = self.blas_func(1, a, x, 1, y, trans=0, incx=3) + with pytest.raises(Exception, match='failed for 3rd argument'): + y = self.blas_func(1, a, x, 1, y, trans=1, incx=3) + + def test_y_stride(self): + alpha, beta, a, x, y = self.get_data(y_stride=2) + desired_y = y.copy() + desired_y[::2] = alpha*matrixmultiply(a, x)+beta*y[::2] + y = self.blas_func(alpha, a, x, beta, y, incy=2) + assert_array_almost_equal(desired_y, y) + + def test_y_stride_transpose(self): + alpha, beta, a, x, y = self.get_data(y_stride=2) + desired_y = y.copy() + desired_y[::2] = alpha*matrixmultiply(transpose(a), x)+beta*y[::2] + y = self.blas_func(alpha, a, x, beta, y, trans=1, incy=2) + assert_array_almost_equal(desired_y, y) + + def test_y_stride_assert(self): + # What is the use of this test? + alpha, beta, a, x, y = self.get_data(y_stride=2) + with pytest.raises(Exception, match='failed for 2nd keyword'): + y = self.blas_func(1, a, x, 1, y, trans=0, incy=3) + with pytest.raises(Exception, match='failed for 2nd keyword'): + y = self.blas_func(1, a, x, 1, y, trans=1, incy=3) + + +try: + class TestSgemv(BaseGemv): + blas_func = fblas.sgemv + dtype = float32 + + def test_sgemv_on_osx(self): + from itertools import product + import sys + import numpy as np + + if sys.platform != 'darwin': + return + + def aligned_array(shape, align, dtype, order='C'): + # Make array shape `shape` with aligned at `align` bytes + d = dtype() + # Make array of correct size with `align` extra bytes + N = np.prod(shape) + tmp = np.zeros(N * d.nbytes + align, dtype=np.uint8) + address = tmp.__array_interface__["data"][0] + # Find offset into array giving desired alignment + for offset in range(align): + if (address + offset) % align == 0: + break + tmp = tmp[offset:offset+N*d.nbytes].view(dtype=dtype) + return tmp.reshape(shape, order=order) + + def as_aligned(arr, align, dtype, order='C'): + # Copy `arr` into an aligned array with same shape + aligned = aligned_array(arr.shape, align, dtype, order) + aligned[:] = arr[:] + return aligned + + def assert_dot_close(A, X, desired): + assert_allclose(self.blas_func(1.0, A, X), desired, + rtol=1e-5, atol=1e-7) + + testdata = product((15, 32), (10000,), (200, 89), ('C', 'F')) + for align, m, n, a_order in testdata: + A_d = np.random.rand(m, n) + X_d = np.random.rand(n) + desired = np.dot(A_d, X_d) + # Calculation with aligned single precision + A_f = as_aligned(A_d, align, np.float32, order=a_order) + X_f = as_aligned(X_d, align, np.float32, order=a_order) + assert_dot_close(A_f, X_f, desired) + +except AttributeError: + class TestSgemv: + pass + + +class TestDgemv(BaseGemv): + blas_func = fblas.dgemv + dtype = float64 + + +try: + class TestCgemv(BaseGemv): + blas_func = fblas.cgemv + dtype = complex64 +except AttributeError: + class TestCgemv: + pass + + +class TestZgemv(BaseGemv): + blas_func = fblas.zgemv + dtype = complex128 + + +""" +################################################## +### Test blas ?ger +### This will be a mess to test all cases. + +class BaseGer: + def get_data(self,x_stride=1,y_stride=1): + from numpy.random import normal, seed + seed(1234) + alpha = array(1., dtype = self.dtype) + a = normal(0.,1.,(3,3)).astype(self.dtype) + x = arange(shape(a)[0]*x_stride,dtype=self.dtype) + y = arange(shape(a)[1]*y_stride,dtype=self.dtype) + return alpha,a,x,y + def test_simple(self): + alpha,a,x,y = self.get_data() + # transpose takes care of Fortran vs. C(and Python) memory layout + desired_a = alpha*transpose(x[:,newaxis]*y) + a + self.blas_func(x,y,a) + assert_array_almost_equal(desired_a,a) + def test_x_stride(self): + alpha,a,x,y = self.get_data(x_stride=2) + desired_a = alpha*transpose(x[::2,newaxis]*y) + a + self.blas_func(x,y,a,incx=2) + assert_array_almost_equal(desired_a,a) + def test_x_stride_assert(self): + alpha,a,x,y = self.get_data(x_stride=2) + with pytest.raises(ValueError, match='foo'): + self.blas_func(x,y,a,incx=3) + def test_y_stride(self): + alpha,a,x,y = self.get_data(y_stride=2) + desired_a = alpha*transpose(x[:,newaxis]*y[::2]) + a + self.blas_func(x,y,a,incy=2) + assert_array_almost_equal(desired_a,a) + + def test_y_stride_assert(self): + alpha,a,x,y = self.get_data(y_stride=2) + with pytest.raises(ValueError, match='foo'): + self.blas_func(a,x,y,incy=3) + +class TestSger(BaseGer): + blas_func = fblas.sger + dtype = float32 +class TestDger(BaseGer): + blas_func = fblas.dger + dtype = float64 +""" +################################################## +# Test blas ?gerc +# This will be a mess to test all cases. + +""" +class BaseGerComplex(BaseGer): + def get_data(self,x_stride=1,y_stride=1): + from numpy.random import normal, seed + seed(1234) + alpha = array(1+1j, dtype = self.dtype) + a = normal(0.,1.,(3,3)).astype(self.dtype) + a = a + normal(0.,1.,(3,3)) * array(1j, dtype = self.dtype) + x = normal(0.,1.,shape(a)[0]*x_stride).astype(self.dtype) + x = x + x * array(1j, dtype = self.dtype) + y = normal(0.,1.,shape(a)[1]*y_stride).astype(self.dtype) + y = y + y * array(1j, dtype = self.dtype) + return alpha,a,x,y + def test_simple(self): + alpha,a,x,y = self.get_data() + # transpose takes care of Fortran vs. C(and Python) memory layout + a = a * array(0.,dtype = self.dtype) + #desired_a = alpha*transpose(x[:,newaxis]*self.transform(y)) + a + desired_a = alpha*transpose(x[:,newaxis]*y) + a + #self.blas_func(x,y,a,alpha = alpha) + fblas.cgeru(x,y,a,alpha = alpha) + assert_array_almost_equal(desired_a,a) + + #def test_x_stride(self): + # alpha,a,x,y = self.get_data(x_stride=2) + # desired_a = alpha*transpose(x[::2,newaxis]*self.transform(y)) + a + # self.blas_func(x,y,a,incx=2) + # assert_array_almost_equal(desired_a,a) + #def test_y_stride(self): + # alpha,a,x,y = self.get_data(y_stride=2) + # desired_a = alpha*transpose(x[:,newaxis]*self.transform(y[::2])) + a + # self.blas_func(x,y,a,incy=2) + # assert_array_almost_equal(desired_a,a) + +class TestCgeru(BaseGerComplex): + blas_func = fblas.cgeru + dtype = complex64 + def transform(self,x): + return x +class TestZgeru(BaseGerComplex): + blas_func = fblas.zgeru + dtype = complex128 + def transform(self,x): + return x + +class TestCgerc(BaseGerComplex): + blas_func = fblas.cgerc + dtype = complex64 + def transform(self,x): + return conjugate(x) + +class TestZgerc(BaseGerComplex): + blas_func = fblas.zgerc + dtype = complex128 + def transform(self,x): + return conjugate(x) +""" diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_interpolative.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_interpolative.py new file mode 100644 index 0000000000000000000000000000000000000000..6e1cc5496eafe19ced81ea546e3bad386148ae7b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_interpolative.py @@ -0,0 +1,232 @@ +# ****************************************************************************** +# Copyright (C) 2013 Kenneth L. Ho +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions are met: +# +# Redistributions of source code must retain the above copyright notice, this +# list of conditions and the following disclaimer. Redistributions in binary +# form must reproduce the above copyright notice, this list of conditions and +# the following disclaimer in the documentation and/or other materials +# provided with the distribution. +# +# None of the names of the copyright holders may be used to endorse or +# promote products derived from this software without specific prior written +# permission. +# +# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" +# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE +# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE +# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR +# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF +# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN +# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) +# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE +# POSSIBILITY OF SUCH DAMAGE. +# ****************************************************************************** + +import scipy.linalg.interpolative as pymatrixid +import numpy as np +from scipy.linalg import hilbert, svdvals, norm +from scipy.sparse.linalg import aslinearoperator +from scipy.linalg.interpolative import interp_decomp + +from numpy.testing import (assert_, assert_allclose, assert_equal, + assert_array_equal) +import pytest +from pytest import raises as assert_raises + + +@pytest.fixture() +def eps(): + yield 1e-12 + + +@pytest.fixture() +def rng(): + rng = np.random.default_rng(1718313768084012) + yield rng + + +@pytest.fixture(params=[np.float64, np.complex128]) +def A(request): + # construct Hilbert matrix + # set parameters + n = 300 + yield hilbert(n).astype(request.param) + + +@pytest.fixture() +def L(A): + yield aslinearoperator(A) + + +@pytest.fixture() +def rank(A, eps): + S = np.linalg.svd(A, compute_uv=False) + try: + rank = np.nonzero(S < eps)[0][0] + except IndexError: + rank = A.shape[0] + return rank + + +class TestInterpolativeDecomposition: + + @pytest.mark.parametrize( + "rand,lin_op", + [(False, False), (True, False), (True, True)]) + def test_real_id_fixed_precision(self, A, L, eps, rand, lin_op, rng): + # Test ID routines on a Hilbert matrix. + A_or_L = A if not lin_op else L + + k, idx, proj = pymatrixid.interp_decomp(A_or_L, eps, rand=rand, rng=rng) + B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) + assert_allclose(A, B, rtol=eps, atol=1e-08) + + @pytest.mark.parametrize( + "rand,lin_op", + [(False, False), (True, False), (True, True)]) + def test_real_id_fixed_rank(self, A, L, eps, rank, rand, lin_op, rng): + k = rank + A_or_L = A if not lin_op else L + + idx, proj = pymatrixid.interp_decomp(A_or_L, k, rand=rand, rng=rng) + B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) + assert_allclose(A, B, rtol=eps, atol=1e-08) + + @pytest.mark.parametrize("rand,lin_op", [(False, False)]) + def test_real_id_skel_and_interp_matrices( + self, A, L, eps, rank, rand, lin_op, rng): + k = rank + A_or_L = A if not lin_op else L + + idx, proj = pymatrixid.interp_decomp(A_or_L, k, rand=rand, rng=rng) + P = pymatrixid.reconstruct_interp_matrix(idx, proj) + B = pymatrixid.reconstruct_skel_matrix(A, k, idx) + assert_allclose(B, A[:, idx[:k]], rtol=eps, atol=1e-08) + assert_allclose(B @ P, A, rtol=eps, atol=1e-08) + + @pytest.mark.parametrize( + "rand,lin_op", + [(False, False), (True, False), (True, True)]) + def test_svd_fixed_precision(self, A, L, eps, rand, lin_op, rng): + A_or_L = A if not lin_op else L + + U, S, V = pymatrixid.svd(A_or_L, eps, rand=rand, rng=rng) + B = U * S @ V.T.conj() + assert_allclose(A, B, rtol=eps, atol=1e-08) + + @pytest.mark.parametrize( + "rand,lin_op", + [(False, False), (True, False), (True, True)]) + def test_svd_fixed_rank(self, A, L, eps, rank, rand, lin_op, rng): + k = rank + A_or_L = A if not lin_op else L + + U, S, V = pymatrixid.svd(A_or_L, k, rand=rand, rng=rng) + B = U * S @ V.T.conj() + assert_allclose(A, B, rtol=eps, atol=1e-08) + + def test_id_to_svd(self, A, eps, rank): + k = rank + + idx, proj = pymatrixid.interp_decomp(A, k, rand=False) + U, S, V = pymatrixid.id_to_svd(A[:, idx[:k]], idx, proj) + B = U * S @ V.T.conj() + assert_allclose(A, B, rtol=eps, atol=1e-08) + + def test_estimate_spectral_norm(self, A, rng): + s = svdvals(A) + norm_2_est = pymatrixid.estimate_spectral_norm(A, rng=rng) + assert_allclose(norm_2_est, s[0], rtol=1e-6, atol=1e-8) + + def test_estimate_spectral_norm_diff(self, A, rng): + B = A.copy() + B[:, 0] *= 1.2 + s = svdvals(A - B) + norm_2_est = pymatrixid.estimate_spectral_norm_diff(A, B, rng=rng) + assert_allclose(norm_2_est, s[0], rtol=1e-6, atol=1e-8) + + def test_rank_estimates_array(self, A, rng): + B = np.array([[1, 1, 0], [0, 0, 1], [0, 0, 1]], dtype=A.dtype) + + for M in [A, B]: + rank_tol = 1e-9 + rank_np = np.linalg.matrix_rank(M, norm(M, 2) * rank_tol) + rank_est = pymatrixid.estimate_rank(M, rank_tol, rng=rng) + assert_(rank_est >= rank_np) + assert_(rank_est <= rank_np + 10) + + def test_rank_estimates_lin_op(self, A, rng): + B = np.array([[1, 1, 0], [0, 0, 1], [0, 0, 1]], dtype=A.dtype) + + for M in [A, B]: + ML = aslinearoperator(M) + rank_tol = 1e-9 + rank_np = np.linalg.matrix_rank(M, norm(M, 2) * rank_tol) + rank_est = pymatrixid.estimate_rank(ML, rank_tol, rng=rng) + assert_(rank_est >= rank_np - 4) + assert_(rank_est <= rank_np + 4) + + def test_badcall(self): + A = hilbert(5).astype(np.float32) + with assert_raises(ValueError): + pymatrixid.interp_decomp(A, 1e-6, rand=False) + + def test_rank_too_large(self): + # svd(array, k) should not segfault + a = np.ones((4, 3)) + with assert_raises(ValueError): + pymatrixid.svd(a, 4) + + def test_full_rank(self): + eps = 1.0e-12 + + # fixed precision + A = np.random.rand(16, 8) + k, idx, proj = pymatrixid.interp_decomp(A, eps) + assert_equal(k, A.shape[1]) + + P = pymatrixid.reconstruct_interp_matrix(idx, proj) + B = pymatrixid.reconstruct_skel_matrix(A, k, idx) + assert_allclose(A, B @ P) + + # fixed rank + idx, proj = pymatrixid.interp_decomp(A, k) + + P = pymatrixid.reconstruct_interp_matrix(idx, proj) + B = pymatrixid.reconstruct_skel_matrix(A, k, idx) + assert_allclose(A, B @ P) + + @pytest.mark.parametrize("dtype", [np.float64, np.complex128]) + @pytest.mark.parametrize("rand", [True, False]) + @pytest.mark.parametrize("eps", [1, 0.1]) + def test_bug_9793(self, dtype, rand, eps): + A = np.array([[-1, -1, -1, 0, 0, 0], + [0, 0, 0, 1, 1, 1], + [1, 0, 0, 1, 0, 0], + [0, 1, 0, 0, 1, 0], + [0, 0, 1, 0, 0, 1]], + dtype=dtype, order="C") + B = A.copy() + interp_decomp(A.T, eps, rand=rand) + assert_array_equal(A, B) + + def test_svd_aslinearoperator_shape_check(self): + # See gh-issue #22451 + rng = np.random.default_rng(1744580941832515) + x = rng.uniform(size=[7, 5]) + xl = aslinearoperator(x) + u, s, v = pymatrixid.svd(xl, 3) + assert_equal(u.shape, (7, 3)) + assert_equal(s.shape, (3,)) + assert_equal(v.shape, (5, 3)) + + x = rng.uniform(size=[4, 9]) + xl = aslinearoperator(x) + u, s, v = pymatrixid.svd(xl, 2) + assert_equal(u.shape, (4, 2)) + assert_equal(s.shape, (2,)) + assert_equal(v.shape, (9, 2)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_lapack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_lapack.py new file mode 100644 index 0000000000000000000000000000000000000000..86555d6c19916c8ae1f6a796fb009a6b803b2159 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_lapack.py @@ -0,0 +1,3508 @@ +# +# Created by: Pearu Peterson, September 2002 +# + +from functools import reduce +import random + +from numpy.testing import (assert_equal, assert_array_almost_equal, assert_, + assert_allclose, assert_almost_equal, + assert_array_equal) +import pytest +from pytest import raises as assert_raises + +import numpy as np +from numpy import (eye, ones, zeros, zeros_like, triu, tril, tril_indices, + triu_indices) + +from numpy.random import rand, randint, seed + +from scipy.linalg import (_flapack as flapack, lapack, inv, svd, cholesky, + solve, ldl, norm, block_diag, qr, eigh, qz) + +from scipy.linalg.lapack import _compute_lwork +from scipy.stats import ortho_group, unitary_group + +import scipy.sparse as sps +try: + from scipy.__config__ import CONFIG +except ImportError: + CONFIG = None + +try: + from scipy.linalg import _clapack as clapack +except ImportError: + clapack = None +from scipy.linalg.lapack import get_lapack_funcs +from scipy.linalg.blas import get_blas_funcs + +REAL_DTYPES = [np.float32, np.float64] +COMPLEX_DTYPES = [np.complex64, np.complex128] +DTYPES = REAL_DTYPES + COMPLEX_DTYPES + +blas_provider = blas_version = None +if CONFIG is not None: + blas_provider = CONFIG['Build Dependencies']['blas']['name'] + blas_version = CONFIG['Build Dependencies']['blas']['version'] + + +def generate_random_dtype_array(shape, dtype, rng): + # generates a random matrix of desired data type of shape + if dtype in COMPLEX_DTYPES: + return (rng.rand(*shape) + + rng.rand(*shape)*1.0j).astype(dtype) + return rng.rand(*shape).astype(dtype) + + +def test_lapack_documented(): + """Test that all entries are in the doc.""" + if lapack.__doc__ is None: # just in case there is a python -OO + pytest.skip('lapack.__doc__ is None') + names = set(lapack.__doc__.split()) + ignore_list = { + "absolute_import", + "clapack", + "division", + "find_best_lapack_type", + "flapack", + "print_function", + "HAS_ILP64", + "np", + } + missing = list() + for name in dir(lapack): + if (not name.startswith('_') and name not in ignore_list and + name not in names): + missing.append(name) + assert missing == [], 'Name(s) missing from lapack.__doc__ or ignore_list' + + +class TestFlapackSimple: + + def test_gebal(self): + a = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] + a1 = [[1, 0, 0, 3e-4], + [4, 0, 0, 2e-3], + [7, 1, 0, 0], + [0, 1, 0, 0]] + for p in 'sdzc': + f = getattr(flapack, p+'gebal', None) + if f is None: + continue + ba, lo, hi, pivscale, info = f(a) + assert_(not info, repr(info)) + assert_array_almost_equal(ba, a) + assert_equal((lo, hi), (0, len(a[0])-1)) + assert_array_almost_equal(pivscale, np.ones(len(a))) + + ba, lo, hi, pivscale, info = f(a1, permute=1, scale=1) + assert_(not info, repr(info)) + # print(a1) + # print(ba, lo, hi, pivscale) + + def test_gehrd(self): + a = [[-149, -50, -154], + [537, 180, 546], + [-27, -9, -25]] + for p in 'd': + f = getattr(flapack, p+'gehrd', None) + if f is None: + continue + ht, tau, info = f(a) + assert_(not info, repr(info)) + + def test_trsyl(self): + a = np.array([[1, 2], [0, 4]]) + b = np.array([[5, 6], [0, 8]]) + c = np.array([[9, 10], [11, 12]]) + trans = 'T' + + # Test single and double implementations, including most + # of the options + for dtype in 'fdFD': + a1, b1, c1 = a.astype(dtype), b.astype(dtype), c.astype(dtype) + trsyl, = get_lapack_funcs(('trsyl',), (a1,)) + if dtype.isupper(): # is complex dtype + a1[0] += 1j + trans = 'C' + + x, scale, info = trsyl(a1, b1, c1) + assert_array_almost_equal(np.dot(a1, x) + np.dot(x, b1), + scale * c1) + + x, scale, info = trsyl(a1, b1, c1, trana=trans, tranb=trans) + assert_array_almost_equal( + np.dot(a1.conjugate().T, x) + np.dot(x, b1.conjugate().T), + scale * c1, decimal=4) + + x, scale, info = trsyl(a1, b1, c1, isgn=-1) + assert_array_almost_equal(np.dot(a1, x) - np.dot(x, b1), + scale * c1, decimal=4) + + def test_lange(self): + a = np.array([ + [-149, -50, -154], + [537, 180, 546], + [-27, -9, -25]]) + + for dtype in 'fdFD': + for norm_str in 'Mm1OoIiFfEe': + a1 = a.astype(dtype) + if dtype.isupper(): + # is complex dtype + a1[0, 0] += 1j + + lange, = get_lapack_funcs(('lange',), (a1,)) + value = lange(norm_str, a1) + + if norm_str in 'FfEe': + if dtype in 'Ff': + decimal = 3 + else: + decimal = 7 + ref = np.sqrt(np.sum(np.square(np.abs(a1)))) + assert_almost_equal(value, ref, decimal) + else: + if norm_str in 'Mm': + ref = np.max(np.abs(a1)) + elif norm_str in '1Oo': + ref = np.max(np.sum(np.abs(a1), axis=0)) + elif norm_str in 'Ii': + ref = np.max(np.sum(np.abs(a1), axis=1)) + + assert_equal(value, ref) + + +class TestLapack: + + def test_flapack(self): + if hasattr(flapack, 'empty_module'): + # flapack module is empty + pass + + def test_clapack(self): + if hasattr(clapack, 'empty_module'): + # clapack module is empty + pass + + +class TestLeastSquaresSolvers: + + def test_gels(self): + seed(1234) + # Test fat/tall matrix argument handling - gh-issue #8329 + for ind, dtype in enumerate(DTYPES): + m = 10 + n = 20 + nrhs = 1 + a1 = rand(m, n).astype(dtype) + b1 = rand(n).astype(dtype) + gls, glslw = get_lapack_funcs(('gels', 'gels_lwork'), dtype=dtype) + + # Request of sizes + lwork = _compute_lwork(glslw, m, n, nrhs) + _, _, info = gls(a1, b1, lwork=lwork) + assert_(info >= 0) + _, _, info = gls(a1, b1, trans='TTCC'[ind], lwork=lwork) + assert_(info >= 0) + + for dtype in REAL_DTYPES: + a1 = np.array([[1.0, 2.0], + [4.0, 5.0], + [7.0, 8.0]], dtype=dtype) + b1 = np.array([16.0, 17.0, 20.0], dtype=dtype) + gels, gels_lwork, geqrf = get_lapack_funcs( + ('gels', 'gels_lwork', 'geqrf'), (a1, b1)) + + m, n = a1.shape + if len(b1.shape) == 2: + nrhs = b1.shape[1] + else: + nrhs = 1 + + # Request of sizes + lwork = _compute_lwork(gels_lwork, m, n, nrhs) + + lqr, x, info = gels(a1, b1, lwork=lwork) + assert_allclose(x[:-1], np.array([-14.333333333333323, + 14.999999999999991], + dtype=dtype), + rtol=25*np.finfo(dtype).eps) + lqr_truth, _, _, _ = geqrf(a1) + assert_array_equal(lqr, lqr_truth) + + for dtype in COMPLEX_DTYPES: + a1 = np.array([[1.0+4.0j, 2.0], + [4.0+0.5j, 5.0-3.0j], + [7.0-2.0j, 8.0+0.7j]], dtype=dtype) + b1 = np.array([16.0, 17.0+2.0j, 20.0-4.0j], dtype=dtype) + gels, gels_lwork, geqrf = get_lapack_funcs( + ('gels', 'gels_lwork', 'geqrf'), (a1, b1)) + + m, n = a1.shape + if len(b1.shape) == 2: + nrhs = b1.shape[1] + else: + nrhs = 1 + + # Request of sizes + lwork = _compute_lwork(gels_lwork, m, n, nrhs) + + lqr, x, info = gels(a1, b1, lwork=lwork) + assert_allclose(x[:-1], + np.array([1.161753632288328-1.901075709391912j, + 1.735882340522193+1.521240901196909j], + dtype=dtype), rtol=25*np.finfo(dtype).eps) + lqr_truth, _, _, _ = geqrf(a1) + assert_array_equal(lqr, lqr_truth) + + def test_gelsd(self): + for dtype in REAL_DTYPES: + a1 = np.array([[1.0, 2.0], + [4.0, 5.0], + [7.0, 8.0]], dtype=dtype) + b1 = np.array([16.0, 17.0, 20.0], dtype=dtype) + gelsd, gelsd_lwork = get_lapack_funcs(('gelsd', 'gelsd_lwork'), + (a1, b1)) + + m, n = a1.shape + if len(b1.shape) == 2: + nrhs = b1.shape[1] + else: + nrhs = 1 + + # Request of sizes + work, iwork, info = gelsd_lwork(m, n, nrhs, -1) + lwork = int(np.real(work)) + iwork_size = iwork + + x, s, rank, info = gelsd(a1, b1, lwork, iwork_size, + -1, False, False) + assert_allclose(x[:-1], np.array([-14.333333333333323, + 14.999999999999991], + dtype=dtype), + rtol=25*np.finfo(dtype).eps) + assert_allclose(s, np.array([12.596017180511966, + 0.583396253199685], dtype=dtype), + rtol=25*np.finfo(dtype).eps) + + for dtype in COMPLEX_DTYPES: + a1 = np.array([[1.0+4.0j, 2.0], + [4.0+0.5j, 5.0-3.0j], + [7.0-2.0j, 8.0+0.7j]], dtype=dtype) + b1 = np.array([16.0, 17.0+2.0j, 20.0-4.0j], dtype=dtype) + gelsd, gelsd_lwork = get_lapack_funcs(('gelsd', 'gelsd_lwork'), + (a1, b1)) + + m, n = a1.shape + if len(b1.shape) == 2: + nrhs = b1.shape[1] + else: + nrhs = 1 + + # Request of sizes + work, rwork, iwork, info = gelsd_lwork(m, n, nrhs, -1) + lwork = int(np.real(work)) + rwork_size = int(rwork) + iwork_size = iwork + + x, s, rank, info = gelsd(a1, b1, lwork, rwork_size, iwork_size, + -1, False, False) + assert_allclose(x[:-1], + np.array([1.161753632288328-1.901075709391912j, + 1.735882340522193+1.521240901196909j], + dtype=dtype), rtol=25*np.finfo(dtype).eps) + assert_allclose(s, + np.array([13.035514762572043, 4.337666985231382], + dtype=dtype), rtol=25*np.finfo(dtype).eps) + + def test_gelss(self): + + for dtype in REAL_DTYPES: + a1 = np.array([[1.0, 2.0], + [4.0, 5.0], + [7.0, 8.0]], dtype=dtype) + b1 = np.array([16.0, 17.0, 20.0], dtype=dtype) + gelss, gelss_lwork = get_lapack_funcs(('gelss', 'gelss_lwork'), + (a1, b1)) + + m, n = a1.shape + if len(b1.shape) == 2: + nrhs = b1.shape[1] + else: + nrhs = 1 + + # Request of sizes + work, info = gelss_lwork(m, n, nrhs, -1) + lwork = int(np.real(work)) + + v, x, s, rank, work, info = gelss(a1, b1, -1, lwork, False, False) + assert_allclose(x[:-1], np.array([-14.333333333333323, + 14.999999999999991], + dtype=dtype), + rtol=25*np.finfo(dtype).eps) + assert_allclose(s, np.array([12.596017180511966, + 0.583396253199685], dtype=dtype), + rtol=25*np.finfo(dtype).eps) + + for dtype in COMPLEX_DTYPES: + a1 = np.array([[1.0+4.0j, 2.0], + [4.0+0.5j, 5.0-3.0j], + [7.0-2.0j, 8.0+0.7j]], dtype=dtype) + b1 = np.array([16.0, 17.0+2.0j, 20.0-4.0j], dtype=dtype) + gelss, gelss_lwork = get_lapack_funcs(('gelss', 'gelss_lwork'), + (a1, b1)) + + m, n = a1.shape + if len(b1.shape) == 2: + nrhs = b1.shape[1] + else: + nrhs = 1 + + # Request of sizes + work, info = gelss_lwork(m, n, nrhs, -1) + lwork = int(np.real(work)) + + v, x, s, rank, work, info = gelss(a1, b1, -1, lwork, False, False) + assert_allclose(x[:-1], + np.array([1.161753632288328-1.901075709391912j, + 1.735882340522193+1.521240901196909j], + dtype=dtype), + rtol=25*np.finfo(dtype).eps) + assert_allclose(s, np.array([13.035514762572043, + 4.337666985231382], dtype=dtype), + rtol=25*np.finfo(dtype).eps) + + def test_gelsy(self): + + for dtype in REAL_DTYPES: + a1 = np.array([[1.0, 2.0], + [4.0, 5.0], + [7.0, 8.0]], dtype=dtype) + b1 = np.array([16.0, 17.0, 20.0], dtype=dtype) + gelsy, gelsy_lwork = get_lapack_funcs(('gelsy', 'gelss_lwork'), + (a1, b1)) + + m, n = a1.shape + if len(b1.shape) == 2: + nrhs = b1.shape[1] + else: + nrhs = 1 + + # Request of sizes + work, info = gelsy_lwork(m, n, nrhs, 10*np.finfo(dtype).eps) + lwork = int(np.real(work)) + + jptv = np.zeros((a1.shape[1], 1), dtype=np.int32) + v, x, j, rank, info = gelsy(a1, b1, jptv, np.finfo(dtype).eps, + lwork, False, False) + assert_allclose(x[:-1], np.array([-14.333333333333323, + 14.999999999999991], + dtype=dtype), + rtol=25*np.finfo(dtype).eps) + + for dtype in COMPLEX_DTYPES: + a1 = np.array([[1.0+4.0j, 2.0], + [4.0+0.5j, 5.0-3.0j], + [7.0-2.0j, 8.0+0.7j]], dtype=dtype) + b1 = np.array([16.0, 17.0+2.0j, 20.0-4.0j], dtype=dtype) + gelsy, gelsy_lwork = get_lapack_funcs(('gelsy', 'gelss_lwork'), + (a1, b1)) + + m, n = a1.shape + if len(b1.shape) == 2: + nrhs = b1.shape[1] + else: + nrhs = 1 + + # Request of sizes + work, info = gelsy_lwork(m, n, nrhs, 10*np.finfo(dtype).eps) + lwork = int(np.real(work)) + + jptv = np.zeros((a1.shape[1], 1), dtype=np.int32) + v, x, j, rank, info = gelsy(a1, b1, jptv, np.finfo(dtype).eps, + lwork, False, False) + assert_allclose(x[:-1], + np.array([1.161753632288328-1.901075709391912j, + 1.735882340522193+1.521240901196909j], + dtype=dtype), + rtol=25*np.finfo(dtype).eps) + + +@pytest.mark.parametrize('dtype', DTYPES) +@pytest.mark.parametrize('shape', [(3, 4), (5, 2), (2**18, 2**18)]) +def test_geqrf_lwork(dtype, shape): + geqrf_lwork = get_lapack_funcs(('geqrf_lwork'), dtype=dtype) + m, n = shape + lwork, info = geqrf_lwork(m=m, n=n) + assert_equal(info, 0) + + +class TestRegression: + + def test_ticket_1645(self): + # Check that RQ routines have correct lwork + for dtype in DTYPES: + a = np.zeros((300, 2), dtype=dtype) + + gerqf, = get_lapack_funcs(['gerqf'], [a]) + assert_raises(Exception, gerqf, a, lwork=2) + rq, tau, work, info = gerqf(a) + + if dtype in REAL_DTYPES: + orgrq, = get_lapack_funcs(['orgrq'], [a]) + assert_raises(Exception, orgrq, rq[-2:], tau, lwork=1) + orgrq(rq[-2:], tau, lwork=2) + elif dtype in COMPLEX_DTYPES: + ungrq, = get_lapack_funcs(['ungrq'], [a]) + assert_raises(Exception, ungrq, rq[-2:], tau, lwork=1) + ungrq(rq[-2:], tau, lwork=2) + + +class TestDpotr: + def test_gh_2691(self): + # 'lower' argument of dportf/dpotri + for lower in [True, False]: + for clean in [True, False]: + np.random.seed(42) + x = np.random.normal(size=(3, 3)) + a = x.dot(x.T) + + dpotrf, dpotri = get_lapack_funcs(("potrf", "potri"), (a, )) + + c, info = dpotrf(a, lower, clean=clean) + dpt = dpotri(c, lower)[0] + + if lower: + assert_allclose(np.tril(dpt), np.tril(inv(a))) + else: + assert_allclose(np.triu(dpt), np.triu(inv(a))) + + +class TestDlasd4: + def test_sing_val_update(self): + + sigmas = np.array([4., 3., 2., 0]) + m_vec = np.array([3.12, 5.7, -4.8, -2.2]) + + M = np.hstack((np.vstack((np.diag(sigmas[0:-1]), + np.zeros((1, len(m_vec) - 1)))), + m_vec[:, np.newaxis])) + SM = svd(M, full_matrices=False, compute_uv=False, overwrite_a=False, + check_finite=False) + + it_len = len(sigmas) + sgm = np.concatenate((sigmas[::-1], [sigmas[0] + it_len*norm(m_vec)])) + mvc = np.concatenate((m_vec[::-1], (0,))) + + lasd4 = get_lapack_funcs('lasd4', (sigmas,)) + + roots = [] + for i in range(0, it_len): + res = lasd4(i, sgm, mvc) + roots.append(res[1]) + + assert_((res[3] <= 0), "LAPACK root finding dlasd4 failed to find \ + the singular value %i" % i) + roots = np.array(roots)[::-1] + + assert_((not np.any(np.isnan(roots)), "There are NaN roots")) + assert_allclose(SM, roots, atol=100*np.finfo(np.float64).eps, + rtol=100*np.finfo(np.float64).eps) + + +class TestTbtrs: + + @pytest.mark.parametrize('dtype', DTYPES) + def test_nag_example_f07vef_f07vsf(self, dtype): + """Test real (f07vef) and complex (f07vsf) examples from NAG + + Examples available from: + * https://www.nag.com/numeric/fl/nagdoc_latest/html/f07/f07vef.html + * https://www.nag.com/numeric/fl/nagdoc_latest/html/f07/f07vsf.html + + """ + if dtype in REAL_DTYPES: + ab = np.array([[-4.16, 4.78, 6.32, 0.16], + [-2.25, 5.86, -4.82, 0]], + dtype=dtype) + b = np.array([[-16.64, -4.16], + [-13.78, -16.59], + [13.10, -4.94], + [-14.14, -9.96]], + dtype=dtype) + x_out = np.array([[4, 1], + [-1, -3], + [3, 2], + [2, -2]], + dtype=dtype) + elif dtype in COMPLEX_DTYPES: + ab = np.array([[-1.94+4.43j, 4.12-4.27j, 0.43-2.66j, 0.44+0.1j], + [-3.39+3.44j, -1.84+5.52j, 1.74 - 0.04j, 0], + [1.62+3.68j, -2.77-1.93j, 0, 0]], + dtype=dtype) + b = np.array([[-8.86 - 3.88j, -24.09 - 5.27j], + [-15.57 - 23.41j, -57.97 + 8.14j], + [-7.63 + 22.78j, 19.09 - 29.51j], + [-14.74 - 2.40j, 19.17 + 21.33j]], + dtype=dtype) + x_out = np.array([[2j, 1 + 5j], + [1 - 3j, -7 - 2j], + [-4.001887 - 4.988417j, 3.026830 + 4.003182j], + [1.996158 - 1.045105j, -6.103357 - 8.986653j]], + dtype=dtype) + else: + raise ValueError(f"Datatype {dtype} not understood.") + + tbtrs = get_lapack_funcs(('tbtrs'), dtype=dtype) + x, info = tbtrs(ab=ab, b=b, uplo='L') + assert_equal(info, 0) + assert_allclose(x, x_out, rtol=0, atol=1e-5) + + @pytest.mark.parametrize('dtype,trans', + [(dtype, trans) + for dtype in DTYPES for trans in ['N', 'T', 'C'] + if not (trans == 'C' and dtype in REAL_DTYPES)]) + @pytest.mark.parametrize('uplo', ['U', 'L']) + @pytest.mark.parametrize('diag', ['N', 'U']) + def test_random_matrices(self, dtype, trans, uplo, diag): + rng = np.random.RandomState(1724) + + # n, nrhs, kd are used to specify A and b. + # A is of shape n x n with kd super/sub-diagonals + # b is of shape n x nrhs matrix + n, nrhs, kd = 4, 3, 2 + tbtrs = get_lapack_funcs('tbtrs', dtype=dtype) + + is_upper = (uplo == 'U') + ku = kd * is_upper + kl = kd - ku + + # Construct the diagonal and kd super/sub diagonals of A with + # the corresponding offsets. + band_offsets = range(ku, -kl - 1, -1) + band_widths = [n - abs(x) for x in band_offsets] + bands = [generate_random_dtype_array((width,), dtype, rng) + for width in band_widths] + + if diag == 'U': # A must be unit triangular + bands[ku] = np.ones(n, dtype=dtype) + + # Construct the diagonal banded matrix A from the bands and offsets. + a = sps.diags(bands, band_offsets, format='dia') + + # Convert A into banded storage form + ab = np.zeros((kd + 1, n), dtype) + for row, k in enumerate(band_offsets): + ab[row, max(k, 0):min(n+k, n)] = a.diagonal(k) + + # The RHS values. + b = generate_random_dtype_array((n, nrhs), dtype, rng) + + x, info = tbtrs(ab=ab, b=b, uplo=uplo, trans=trans, diag=diag) + assert_equal(info, 0) + + if trans == 'N': + assert_allclose(a @ x, b, rtol=5e-5) + elif trans == 'T': + assert_allclose(a.T @ x, b, rtol=5e-5) + elif trans == 'C': + assert_allclose(a.T.conjugate() @ x, b, rtol=5e-5) + else: + raise ValueError('Invalid trans argument') + + @pytest.mark.parametrize('uplo,trans,diag', + [['U', 'N', 'Invalid'], + ['U', 'Invalid', 'N'], + ['Invalid', 'N', 'N']]) + def test_invalid_argument_raises_exception(self, uplo, trans, diag): + """Test if invalid values of uplo, trans and diag raise exceptions""" + # Argument checks occur independently of used datatype. + # This mean we must not parameterize all available datatypes. + tbtrs = get_lapack_funcs('tbtrs', dtype=np.float64) + ab = rand(4, 2) + b = rand(2, 4) + assert_raises(Exception, tbtrs, ab, b, uplo, trans, diag) + + def test_zero_element_in_diagonal(self): + """Test if a matrix with a zero diagonal element is singular + + If the i-th diagonal of A is zero, ?tbtrs should return `i` in `info` + indicating the provided matrix is singular. + + Note that ?tbtrs requires the matrix A to be stored in banded form. + In this form the diagonal corresponds to the last row.""" + ab = np.ones((3, 4), dtype=float) + b = np.ones(4, dtype=float) + tbtrs = get_lapack_funcs('tbtrs', dtype=float) + + ab[-1, 3] = 0 + _, info = tbtrs(ab=ab, b=b, uplo='U') + assert_equal(info, 4) + + @pytest.mark.parametrize('ldab,n,ldb,nrhs', [ + (5, 5, 0, 5), + (5, 5, 3, 5) + ]) + def test_invalid_matrix_shapes(self, ldab, n, ldb, nrhs): + """Test ?tbtrs fails correctly if shapes are invalid.""" + ab = np.ones((ldab, n), dtype=float) + b = np.ones((ldb, nrhs), dtype=float) + tbtrs = get_lapack_funcs('tbtrs', dtype=float) + assert_raises(Exception, tbtrs, ab, b) + + + +@pytest.mark.parametrize('dtype', DTYPES) +@pytest.mark.parametrize('norm', ['I', '1', 'O']) +@pytest.mark.parametrize('uplo', ['U', 'L']) +@pytest.mark.parametrize('diag', ['N', 'U']) +@pytest.mark.parametrize('n', [3, 10]) +def test_trcon(dtype, norm, uplo, diag, n): + # Simple way to get deterministic (unlike `hash`) integer seed based on arguments + random.seed(f"{dtype}{norm}{uplo}{diag}{n}") + rng = np.random.default_rng(random.randint(0, 9999999999999)) + + A = rng.random(size=(n, n)) + rng.random(size=(n, n))*1j + # make the condition numbers more interesting + offset = rng.permuted(np.logspace(0, rng.integers(0, 10), n)) + A += offset + A = A.real if np.issubdtype(dtype, np.floating) else A + A = np.triu(A) if uplo == 'U' else np.tril(A) + if diag == 'U': + A /= np.diag(A)[:, np.newaxis] + A = A.astype(dtype) + + trcon = get_lapack_funcs('trcon', (A,)) + res, _ = trcon(A, norm=norm, uplo=uplo, diag=diag) + + if norm == 'I': + norm_A = np.linalg.norm(A, ord=np.inf) + norm_inv_A = np.linalg.norm(np.linalg.inv(A), ord=np.inf) + ref = 1 / (norm_A * norm_inv_A) + else: + anorm = np.abs(A).sum(axis=0).max() + gecon, getrf = get_lapack_funcs(('gecon', 'getrf'), (A,)) + lu, ipvt, info = getrf(A) + ref, _ = gecon(lu, anorm, norm=norm) + + # This is an estimate of reciprocal condition number; we just need order of + # magnitude. In testing, we observed that much smaller rtol is OK in almost + # all cases... but sometimes it isn't. + rtol = 1 # np.finfo(dtype).eps**0.75 + assert_allclose(res, ref, rtol=rtol) + + +def test_lartg(): + for dtype in 'fdFD': + lartg = get_lapack_funcs('lartg', dtype=dtype) + + f = np.array(3, dtype) + g = np.array(4, dtype) + + if np.iscomplexobj(g): + g *= 1j + + cs, sn, r = lartg(f, g) + + assert_allclose(cs, 3.0/5.0) + assert_allclose(r, 5.0) + + if np.iscomplexobj(g): + assert_allclose(sn, -4.0j/5.0) + assert_(isinstance(r, complex)) + assert_(isinstance(cs, float)) + else: + assert_allclose(sn, 4.0/5.0) + + +def test_rot(): + # srot, drot from blas and crot and zrot from lapack. + + for dtype in 'fdFD': + c = 0.6 + s = 0.8 + + u = np.full(4, 3, dtype) + v = np.full(4, 4, dtype) + atol = 10**-(np.finfo(dtype).precision-1) + + if dtype in 'fd': + rot = get_blas_funcs('rot', dtype=dtype) + f = 4 + else: + rot = get_lapack_funcs('rot', dtype=dtype) + s *= -1j + v *= 1j + f = 4j + + assert_allclose(rot(u, v, c, s), [[5, 5, 5, 5], + [0, 0, 0, 0]], atol=atol) + assert_allclose(rot(u, v, c, s, n=2), [[5, 5, 3, 3], + [0, 0, f, f]], atol=atol) + assert_allclose(rot(u, v, c, s, offx=2, offy=2), + [[3, 3, 5, 5], [f, f, 0, 0]], atol=atol) + assert_allclose(rot(u, v, c, s, incx=2, offy=2, n=2), + [[5, 3, 5, 3], [f, f, 0, 0]], atol=atol) + assert_allclose(rot(u, v, c, s, offx=2, incy=2, n=2), + [[3, 3, 5, 5], [0, f, 0, f]], atol=atol) + assert_allclose(rot(u, v, c, s, offx=2, incx=2, offy=2, incy=2, n=1), + [[3, 3, 5, 3], [f, f, 0, f]], atol=atol) + assert_allclose(rot(u, v, c, s, incx=-2, incy=-2, n=2), + [[5, 3, 5, 3], [0, f, 0, f]], atol=atol) + + a, b = rot(u, v, c, s, overwrite_x=1, overwrite_y=1) + assert_(a is u) + assert_(b is v) + assert_allclose(a, [5, 5, 5, 5], atol=atol) + assert_allclose(b, [0, 0, 0, 0], atol=atol) + + +def test_larfg_larf(): + np.random.seed(1234) + a0 = np.random.random((4, 4)) + a0 = a0.T.dot(a0) + + a0j = np.random.random((4, 4)) + 1j*np.random.random((4, 4)) + a0j = a0j.T.conj().dot(a0j) + + # our test here will be to do one step of reducing a hermetian matrix to + # tridiagonal form using householder transforms. + + for dtype in 'fdFD': + larfg, larf = get_lapack_funcs(['larfg', 'larf'], dtype=dtype) + + if dtype in 'FD': + a = a0j.copy() + else: + a = a0.copy() + + # generate a householder transform to clear a[2:,0] + alpha, x, tau = larfg(a.shape[0]-1, a[1, 0], a[2:, 0]) + + # create expected output + expected = np.zeros_like(a[:, 0]) + expected[0] = a[0, 0] + expected[1] = alpha + + # assemble householder vector + v = np.zeros_like(a[1:, 0]) + v[0] = 1.0 + v[1:] = x + + # apply transform from the left + a[1:, :] = larf(v, tau.conjugate(), a[1:, :], np.zeros(a.shape[1])) + + # apply transform from the right + a[:, 1:] = larf(v, tau, a[:, 1:], np.zeros(a.shape[0]), side='R') + + assert_allclose(a[:, 0], expected, atol=1e-5) + assert_allclose(a[0, :], expected, atol=1e-5) + + +def test_sgesdd_lwork_bug_workaround(): + # Test that SGESDD lwork is sufficiently large for LAPACK. + # + # This checks that _compute_lwork() correctly works around a bug in + # LAPACK versions older than 3.10.1. + + sgesdd_lwork = get_lapack_funcs('gesdd_lwork', dtype=np.float32, + ilp64='preferred') + n = 9537 + lwork = _compute_lwork(sgesdd_lwork, n, n, + compute_uv=True, full_matrices=True) + # If we called the Fortran function SGESDD directly with IWORK=-1, the + # LAPACK bug would result in lwork being 272929856, which was too small. + # (The result was returned in a single precision float, which does not + # have sufficient precision to represent the exact integer value that it + # computed internally.) The work-around implemented in _compute_lwork() + # will convert that to 272929888. If we are using LAPACK 3.10.1 or later + # (such as in OpenBLAS 0.3.21 or later), the work-around will return + # 272929920, because it does not know which version of LAPACK is being + # used, so it always applies the correction to whatever it is given. We + # will accept either 272929888 or 272929920. + # Note that the acceptable values are a LAPACK implementation detail. + # If a future version of LAPACK changes how SGESDD works, and therefore + # changes the required LWORK size, the acceptable values might have to + # be updated. + assert lwork == 272929888 or lwork == 272929920 + + +class TestSytrd: + @pytest.mark.parametrize('dtype', REAL_DTYPES) + def test_sytrd_with_zero_dim_array(self, dtype): + # Assert that a 0x0 matrix raises an error + A = np.zeros((0, 0), dtype=dtype) + sytrd = get_lapack_funcs('sytrd', (A,)) + assert_raises(ValueError, sytrd, A) + + @pytest.mark.parametrize('dtype', REAL_DTYPES) + @pytest.mark.parametrize('n', (1, 3)) + def test_sytrd(self, dtype, n): + A = np.zeros((n, n), dtype=dtype) + + sytrd, sytrd_lwork = \ + get_lapack_funcs(('sytrd', 'sytrd_lwork'), (A,)) + + # some upper triangular array + A[np.triu_indices_from(A)] = \ + np.arange(1, n*(n+1)//2+1, dtype=dtype) + + # query lwork + lwork, info = sytrd_lwork(n) + assert_equal(info, 0) + + # check lower=1 behavior (shouldn't do much since the matrix is + # upper triangular) + data, d, e, tau, info = sytrd(A, lower=1, lwork=lwork) + assert_equal(info, 0) + + assert_allclose(data, A, atol=5*np.finfo(dtype).eps, rtol=1.0) + assert_allclose(d, np.diag(A)) + assert_allclose(e, 0.0) + assert_allclose(tau, 0.0) + + # and now for the proper test (lower=0 is the default) + data, d, e, tau, info = sytrd(A, lwork=lwork) + assert_equal(info, 0) + + # assert Q^T*A*Q = tridiag(e, d, e) + + # build tridiagonal matrix + T = np.zeros_like(A, dtype=dtype) + k = np.arange(A.shape[0]) + T[k, k] = d + k2 = np.arange(A.shape[0]-1) + T[k2+1, k2] = e + T[k2, k2+1] = e + + # build Q + Q = np.eye(n, n, dtype=dtype) + for i in range(n-1): + v = np.zeros(n, dtype=dtype) + v[:i] = data[:i, i+1] + v[i] = 1.0 + H = np.eye(n, n, dtype=dtype) - tau[i] * np.outer(v, v) + Q = np.dot(H, Q) + + # Make matrix fully symmetric + i_lower = np.tril_indices(n, -1) + A[i_lower] = A.T[i_lower] + + QTAQ = np.dot(Q.T, np.dot(A, Q)) + + # disable rtol here since some values in QTAQ and T are very close + # to 0. + assert_allclose(QTAQ, T, atol=5*np.finfo(dtype).eps, rtol=1.0) + + +class TestHetrd: + @pytest.mark.parametrize('complex_dtype', COMPLEX_DTYPES) + def test_hetrd_with_zero_dim_array(self, complex_dtype): + # Assert that a 0x0 matrix raises an error + A = np.zeros((0, 0), dtype=complex_dtype) + hetrd = get_lapack_funcs('hetrd', (A,)) + assert_raises(ValueError, hetrd, A) + + @pytest.mark.parametrize('real_dtype,complex_dtype', + zip(REAL_DTYPES, COMPLEX_DTYPES)) + @pytest.mark.parametrize('n', (1, 3)) + def test_hetrd(self, n, real_dtype, complex_dtype): + A = np.zeros((n, n), dtype=complex_dtype) + hetrd, hetrd_lwork = \ + get_lapack_funcs(('hetrd', 'hetrd_lwork'), (A,)) + + # some upper triangular array + A[np.triu_indices_from(A)] = ( + np.arange(1, n*(n+1)//2+1, dtype=real_dtype) + + 1j * np.arange(1, n*(n+1)//2+1, dtype=real_dtype) + ) + np.fill_diagonal(A, np.real(np.diag(A))) + + # test query lwork + for x in [0, 1]: + _, info = hetrd_lwork(n, lower=x) + assert_equal(info, 0) + # lwork returns complex which segfaults hetrd call (gh-10388) + # use the safe and recommended option + lwork = _compute_lwork(hetrd_lwork, n) + + # check lower=1 behavior (shouldn't do much since the matrix is + # upper triangular) + data, d, e, tau, info = hetrd(A, lower=1, lwork=lwork) + assert_equal(info, 0) + + assert_allclose(data, A, atol=5*np.finfo(real_dtype).eps, rtol=1.0) + + assert_allclose(d, np.real(np.diag(A))) + assert_allclose(e, 0.0) + assert_allclose(tau, 0.0) + + # and now for the proper test (lower=0 is the default) + data, d, e, tau, info = hetrd(A, lwork=lwork) + assert_equal(info, 0) + + # assert Q^T*A*Q = tridiag(e, d, e) + + # build tridiagonal matrix + T = np.zeros_like(A, dtype=real_dtype) + k = np.arange(A.shape[0], dtype=int) + T[k, k] = d + k2 = np.arange(A.shape[0]-1, dtype=int) + T[k2+1, k2] = e + T[k2, k2+1] = e + + # build Q + Q = np.eye(n, n, dtype=complex_dtype) + for i in range(n-1): + v = np.zeros(n, dtype=complex_dtype) + v[:i] = data[:i, i+1] + v[i] = 1.0 + H = np.eye(n, n, dtype=complex_dtype) \ + - tau[i] * np.outer(v, np.conj(v)) + Q = np.dot(H, Q) + + # Make matrix fully Hermitian + i_lower = np.tril_indices(n, -1) + A[i_lower] = np.conj(A.T[i_lower]) + + QHAQ = np.dot(np.conj(Q.T), np.dot(A, Q)) + + # disable rtol here since some values in QTAQ and T are very close + # to 0. + assert_allclose( + QHAQ, T, atol=10*np.finfo(real_dtype).eps, rtol=1.0 + ) + + +def test_gglse(): + # Example data taken from NAG manual + for ind, dtype in enumerate(DTYPES): + # DTYPES = gglse + func, func_lwork = get_lapack_funcs(('gglse', 'gglse_lwork'), + dtype=dtype) + lwork = _compute_lwork(func_lwork, m=6, n=4, p=2) + # For gglse + if ind < 2: + a = np.array([[-0.57, -1.28, -0.39, 0.25], + [-1.93, 1.08, -0.31, -2.14], + [2.30, 0.24, 0.40, -0.35], + [-1.93, 0.64, -0.66, 0.08], + [0.15, 0.30, 0.15, -2.13], + [-0.02, 1.03, -1.43, 0.50]], dtype=dtype) + c = np.array([-1.50, -2.14, 1.23, -0.54, -1.68, 0.82], dtype=dtype) + d = np.array([0., 0.], dtype=dtype) + # For gglse + else: + a = np.array([[0.96-0.81j, -0.03+0.96j, -0.91+2.06j, -0.05+0.41j], + [-0.98+1.98j, -1.20+0.19j, -0.66+0.42j, -0.81+0.56j], + [0.62-0.46j, 1.01+0.02j, 0.63-0.17j, -1.11+0.60j], + [0.37+0.38j, 0.19-0.54j, -0.98-0.36j, 0.22-0.20j], + [0.83+0.51j, 0.20+0.01j, -0.17-0.46j, 1.47+1.59j], + [1.08-0.28j, 0.20-0.12j, -0.07+1.23j, 0.26+0.26j]]) + c = np.array([[-2.54+0.09j], + [1.65-2.26j], + [-2.11-3.96j], + [1.82+3.30j], + [-6.41+3.77j], + [2.07+0.66j]]) + d = np.zeros(2, dtype=dtype) + + b = np.array([[1., 0., -1., 0.], [0., 1., 0., -1.]], dtype=dtype) + + _, _, _, result, _ = func(a, b, c, d, lwork=lwork) + if ind < 2: + expected = np.array([0.48904455, + 0.99754786, + 0.48904455, + 0.99754786]) + else: + expected = np.array([1.08742917-1.96205783j, + -0.74093902+3.72973919j, + 1.08742917-1.96205759j, + -0.74093896+3.72973895j]) + assert_array_almost_equal(result, expected, decimal=4) + + +def test_sycon_hecon(): + seed(1234) + for ind, dtype in enumerate(DTYPES+COMPLEX_DTYPES): + # DTYPES + COMPLEX DTYPES = sycon + hecon + n = 10 + # For sycon + if ind < 4: + func_lwork = get_lapack_funcs('sytrf_lwork', dtype=dtype) + funcon, functrf = get_lapack_funcs(('sycon', 'sytrf'), dtype=dtype) + A = (rand(n, n)).astype(dtype) + # For hecon + else: + func_lwork = get_lapack_funcs('hetrf_lwork', dtype=dtype) + funcon, functrf = get_lapack_funcs(('hecon', 'hetrf'), dtype=dtype) + A = (rand(n, n) + rand(n, n)*1j).astype(dtype) + + # Since sycon only refers to upper/lower part, conj() is safe here. + A = (A + A.conj().T)/2 + 2*np.eye(n, dtype=dtype) + + anorm = norm(A, 1) + lwork = _compute_lwork(func_lwork, n) + ldu, ipiv, _ = functrf(A, lwork=lwork, lower=1) + rcond, _ = funcon(a=ldu, ipiv=ipiv, anorm=anorm, lower=1) + # The error is at most 1-fold + assert_(abs(1/rcond - np.linalg.cond(A, p=1))*rcond < 1) + + +def test_sygst(): + seed(1234) + for ind, dtype in enumerate(REAL_DTYPES): + # DTYPES = sygst + n = 10 + + potrf, sygst, syevd, sygvd = get_lapack_funcs(('potrf', 'sygst', + 'syevd', 'sygvd'), + dtype=dtype) + + A = rand(n, n).astype(dtype) + A = (A + A.T)/2 + # B must be positive definite + B = rand(n, n).astype(dtype) + B = (B + B.T)/2 + 2 * np.eye(n, dtype=dtype) + + # Perform eig (sygvd) + eig_gvd, _, info = sygvd(A, B) + assert_(info == 0) + + # Convert to std problem potrf + b, info = potrf(B) + assert_(info == 0) + a, info = sygst(A, b) + assert_(info == 0) + + eig, _, info = syevd(a) + assert_(info == 0) + assert_allclose(eig, eig_gvd, rtol=1.2e-4) + + +def test_hegst(): + seed(1234) + for ind, dtype in enumerate(COMPLEX_DTYPES): + # DTYPES = hegst + n = 10 + + potrf, hegst, heevd, hegvd = get_lapack_funcs(('potrf', 'hegst', + 'heevd', 'hegvd'), + dtype=dtype) + + A = rand(n, n).astype(dtype) + 1j * rand(n, n).astype(dtype) + A = (A + A.conj().T)/2 + # B must be positive definite + B = rand(n, n).astype(dtype) + 1j * rand(n, n).astype(dtype) + B = (B + B.conj().T)/2 + 2 * np.eye(n, dtype=dtype) + + # Perform eig (hegvd) + eig_gvd, _, info = hegvd(A, B) + assert_(info == 0) + + # Convert to std problem potrf + b, info = potrf(B) + assert_(info == 0) + a, info = hegst(A, b) + assert_(info == 0) + + eig, _, info = heevd(a) + assert_(info == 0) + assert_allclose(eig, eig_gvd, rtol=1e-4) + + +def test_tzrzf(): + """ + This test performs an RZ decomposition in which an m x n upper trapezoidal + array M (m <= n) is factorized as M = [R 0] * Z where R is upper triangular + and Z is unitary. + """ + rng = np.random.RandomState(1234) + m, n = 10, 15 + for ind, dtype in enumerate(DTYPES): + tzrzf, tzrzf_lw = get_lapack_funcs(('tzrzf', 'tzrzf_lwork'), + dtype=dtype) + lwork = _compute_lwork(tzrzf_lw, m, n) + + if ind < 2: + A = triu(rng.rand(m, n).astype(dtype)) + else: + A = triu((rng.rand(m, n) + rng.rand(m, n)*1j).astype(dtype)) + + # assert wrong shape arg, f2py returns generic error + assert_raises(Exception, tzrzf, A.T) + rz, tau, info = tzrzf(A, lwork=lwork) + # Check success + assert_(info == 0) + + # Get Z manually for comparison + R = np.hstack((rz[:, :m], np.zeros((m, n-m), dtype=dtype))) + V = np.hstack((np.eye(m, dtype=dtype), rz[:, m:])) + Id = np.eye(n, dtype=dtype) + ref = [Id-tau[x]*V[[x], :].T.dot(V[[x], :].conj()) for x in range(m)] + Z = reduce(np.dot, ref) + assert_allclose(R.dot(Z) - A, zeros_like(A, dtype=dtype), + atol=10*np.spacing(dtype(1.0).real), rtol=0.) + + +def test_tfsm(): + """ + Test for solving a linear system with the coefficient matrix is a + triangular array stored in Full Packed (RFP) format. + """ + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + n = 20 + if ind > 1: + A = triu(rng.rand(n, n) + rng.rand(n, n)*1j + eye(n)).astype(dtype) + trans = 'C' + else: + A = triu(rng.rand(n, n) + eye(n)).astype(dtype) + trans = 'T' + + trttf, tfttr, tfsm = get_lapack_funcs(('trttf', 'tfttr', 'tfsm'), + dtype=dtype) + + Afp, _ = trttf(A) + B = rng.rand(n, 2).astype(dtype) + soln = tfsm(-1, Afp, B) + assert_array_almost_equal(soln, solve(-A, B), + decimal=4 if ind % 2 == 0 else 6) + + soln = tfsm(-1, Afp, B, trans=trans) + assert_array_almost_equal(soln, solve(-A.conj().T, B), + decimal=4 if ind % 2 == 0 else 6) + + # Make A, unit diagonal + A[np.arange(n), np.arange(n)] = dtype(1.) + soln = tfsm(-1, Afp, B, trans=trans, diag='U') + assert_array_almost_equal(soln, solve(-A.conj().T, B), + decimal=4 if ind % 2 == 0 else 6) + + # Change side + B2 = rng.rand(3, n).astype(dtype) + soln = tfsm(-1, Afp, B2, trans=trans, diag='U', side='R') + assert_array_almost_equal(soln, solve(-A, B2.T).conj().T, + decimal=4 if ind % 2 == 0 else 6) + + +def test_ormrz_unmrz(): + """ + This test performs a matrix multiplication with an arbitrary m x n matrix C + and a unitary matrix Q without explicitly forming the array. The array data + is encoded in the rectangular part of A which is obtained from ?TZRZF. Q + size is inferred by m, n, side keywords. + """ + rng = np.random.RandomState(1234) + qm, qn, cn = 10, 15, 15 + for ind, dtype in enumerate(DTYPES): + tzrzf, tzrzf_lw = get_lapack_funcs(('tzrzf', 'tzrzf_lwork'), + dtype=dtype) + lwork_rz = _compute_lwork(tzrzf_lw, qm, qn) + + if ind < 2: + A = triu(rng.rand(qm, qn).astype(dtype)) + C = rng.rand(cn, cn).astype(dtype) + orun_mrz, orun_mrz_lw = get_lapack_funcs(('ormrz', 'ormrz_lwork'), + dtype=dtype) + else: + A = triu((rng.rand(qm, qn) + rng.rand(qm, qn)*1j).astype(dtype)) + C = (rng.rand(cn, cn) + rand(cn, cn)*1j).astype(dtype) + orun_mrz, orun_mrz_lw = get_lapack_funcs(('unmrz', 'unmrz_lwork'), + dtype=dtype) + + lwork_mrz = _compute_lwork(orun_mrz_lw, cn, cn) + rz, tau, info = tzrzf(A, lwork=lwork_rz) + + # Get Q manually for comparison + V = np.hstack((np.eye(qm, dtype=dtype), rz[:, qm:])) + Id = np.eye(qn, dtype=dtype) + ref = [Id-tau[x]*V[[x], :].T.dot(V[[x], :].conj()) for x in range(qm)] + Q = reduce(np.dot, ref) + + # Now that we have Q, we can test whether lapack results agree with + # each case of CQ, CQ^H, QC, and QC^H + trans = 'T' if ind < 2 else 'C' + tol = 10*np.spacing(dtype(1.0).real) + + cq, info = orun_mrz(rz, tau, C, lwork=lwork_mrz) + assert_(info == 0) + assert_allclose(cq - Q.dot(C), zeros_like(C), atol=tol, rtol=0.) + + cq, info = orun_mrz(rz, tau, C, trans=trans, lwork=lwork_mrz) + assert_(info == 0) + assert_allclose(cq - Q.conj().T.dot(C), zeros_like(C), atol=tol, + rtol=0.) + + cq, info = orun_mrz(rz, tau, C, side='R', lwork=lwork_mrz) + assert_(info == 0) + assert_allclose(cq - C.dot(Q), zeros_like(C), atol=tol, rtol=0.) + + cq, info = orun_mrz(rz, tau, C, side='R', trans=trans, lwork=lwork_mrz) + assert_(info == 0) + assert_allclose(cq - C.dot(Q.conj().T), zeros_like(C), atol=tol, + rtol=0.) + + +def test_tfttr_trttf(): + """ + Test conversion routines between the Rectangular Full Packed (RFP) format + and Standard Triangular Array (TR) + """ + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + n = 20 + if ind > 1: + A_full = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + transr = 'C' + else: + A_full = (rng.rand(n, n)).astype(dtype) + transr = 'T' + + trttf, tfttr = get_lapack_funcs(('trttf', 'tfttr'), dtype=dtype) + A_tf_U, info = trttf(A_full) + assert_(info == 0) + A_tf_L, info = trttf(A_full, uplo='L') + assert_(info == 0) + A_tf_U_T, info = trttf(A_full, transr=transr, uplo='U') + assert_(info == 0) + A_tf_L_T, info = trttf(A_full, transr=transr, uplo='L') + assert_(info == 0) + + # Create the RFP array manually (n is even!) + A_tf_U_m = zeros((n+1, n//2), dtype=dtype) + A_tf_U_m[:-1, :] = triu(A_full)[:, n//2:] + A_tf_U_m[n//2+1:, :] += triu(A_full)[:n//2, :n//2].conj().T + + A_tf_L_m = zeros((n+1, n//2), dtype=dtype) + A_tf_L_m[1:, :] = tril(A_full)[:, :n//2] + A_tf_L_m[:n//2, :] += tril(A_full)[n//2:, n//2:].conj().T + + assert_array_almost_equal(A_tf_U, A_tf_U_m.reshape(-1, order='F')) + assert_array_almost_equal(A_tf_U_T, + A_tf_U_m.conj().T.reshape(-1, order='F')) + + assert_array_almost_equal(A_tf_L, A_tf_L_m.reshape(-1, order='F')) + assert_array_almost_equal(A_tf_L_T, + A_tf_L_m.conj().T.reshape(-1, order='F')) + + # Get the original array from RFP + A_tr_U, info = tfttr(n, A_tf_U) + assert_(info == 0) + A_tr_L, info = tfttr(n, A_tf_L, uplo='L') + assert_(info == 0) + A_tr_U_T, info = tfttr(n, A_tf_U_T, transr=transr, uplo='U') + assert_(info == 0) + A_tr_L_T, info = tfttr(n, A_tf_L_T, transr=transr, uplo='L') + assert_(info == 0) + + assert_array_almost_equal(A_tr_U, triu(A_full)) + assert_array_almost_equal(A_tr_U_T, triu(A_full)) + assert_array_almost_equal(A_tr_L, tril(A_full)) + assert_array_almost_equal(A_tr_L_T, tril(A_full)) + + +def test_tpttr_trttp(): + """ + Test conversion routines between the Rectangular Full Packed (RFP) format + and Standard Triangular Array (TR) + """ + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + n = 20 + if ind > 1: + A_full = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + else: + A_full = (rng.rand(n, n)).astype(dtype) + + trttp, tpttr = get_lapack_funcs(('trttp', 'tpttr'), dtype=dtype) + A_tp_U, info = trttp(A_full) + assert_(info == 0) + A_tp_L, info = trttp(A_full, uplo='L') + assert_(info == 0) + + # Create the TP array manually + inds = tril_indices(n) + A_tp_U_m = zeros(n*(n+1)//2, dtype=dtype) + A_tp_U_m[:] = (triu(A_full).T)[inds] + + inds = triu_indices(n) + A_tp_L_m = zeros(n*(n+1)//2, dtype=dtype) + A_tp_L_m[:] = (tril(A_full).T)[inds] + + assert_array_almost_equal(A_tp_U, A_tp_U_m) + assert_array_almost_equal(A_tp_L, A_tp_L_m) + + # Get the original array from TP + A_tr_U, info = tpttr(n, A_tp_U) + assert_(info == 0) + A_tr_L, info = tpttr(n, A_tp_L, uplo='L') + assert_(info == 0) + + assert_array_almost_equal(A_tr_U, triu(A_full)) + assert_array_almost_equal(A_tr_L, tril(A_full)) + + +def test_pftrf(): + """ + Test Cholesky factorization of a positive definite Rectangular Full + Packed (RFP) format array + """ + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + n = 20 + if ind > 1: + A = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + A = A + A.conj().T + n*eye(n) + else: + A = (rng.rand(n, n)).astype(dtype) + A = A + A.T + n*eye(n) + + pftrf, trttf, tfttr = get_lapack_funcs(('pftrf', 'trttf', 'tfttr'), + dtype=dtype) + + # Get the original array from TP + Afp, info = trttf(A) + Achol_rfp, info = pftrf(n, Afp) + assert_(info == 0) + A_chol_r, _ = tfttr(n, Achol_rfp) + Achol = cholesky(A) + assert_array_almost_equal(A_chol_r, Achol) + + +def test_pftri(): + """ + Test Cholesky factorization of a positive definite Rectangular Full + Packed (RFP) format array to find its inverse + """ + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + n = 20 + if ind > 1: + A = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + A = A + A.conj().T + n*eye(n) + else: + A = (rng.rand(n, n)).astype(dtype) + A = A + A.T + n*eye(n) + + pftri, pftrf, trttf, tfttr = get_lapack_funcs(('pftri', + 'pftrf', + 'trttf', + 'tfttr'), + dtype=dtype) + + # Get the original array from TP + Afp, info = trttf(A) + A_chol_rfp, info = pftrf(n, Afp) + A_inv_rfp, info = pftri(n, A_chol_rfp) + assert_(info == 0) + A_inv_r, _ = tfttr(n, A_inv_rfp) + Ainv = inv(A) + assert_array_almost_equal(A_inv_r, triu(Ainv), + decimal=4 if ind % 2 == 0 else 6) + + +def test_pftrs(): + """ + Test Cholesky factorization of a positive definite Rectangular Full + Packed (RFP) format array and solve a linear system + """ + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + n = 20 + if ind > 1: + A = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + A = A + A.conj().T + n*eye(n) + else: + A = (rng.rand(n, n)).astype(dtype) + A = A + A.T + n*eye(n) + + B = ones((n, 3), dtype=dtype) + Bf1 = ones((n+2, 3), dtype=dtype) + Bf2 = ones((n-2, 3), dtype=dtype) + pftrs, pftrf, trttf, tfttr = get_lapack_funcs(('pftrs', + 'pftrf', + 'trttf', + 'tfttr'), + dtype=dtype) + + # Get the original array from TP + Afp, info = trttf(A) + A_chol_rfp, info = pftrf(n, Afp) + # larger B arrays shouldn't segfault + soln, info = pftrs(n, A_chol_rfp, Bf1) + assert_(info == 0) + assert_raises(Exception, pftrs, n, A_chol_rfp, Bf2) + soln, info = pftrs(n, A_chol_rfp, B) + assert_(info == 0) + assert_array_almost_equal(solve(A, B), soln, + decimal=4 if ind % 2 == 0 else 6) + + +def test_sfrk_hfrk(): + """ + Test for performing a symmetric rank-k operation for matrix in RFP format. + """ + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + n = 20 + if ind > 1: + A = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + A = A + A.conj().T + n*eye(n) + else: + A = (rng.rand(n, n)).astype(dtype) + A = A + A.T + n*eye(n) + + prefix = 's'if ind < 2 else 'h' + trttf, tfttr, shfrk = get_lapack_funcs(('trttf', 'tfttr', f'{prefix}frk'), + dtype=dtype) + + Afp, _ = trttf(A) + C = rng.rand(n, 2).astype(dtype) + Afp_out = shfrk(n, 2, -1, C, 2, Afp) + A_out, _ = tfttr(n, Afp_out) + assert_array_almost_equal(A_out, triu(-C.dot(C.conj().T) + 2*A), + decimal=4 if ind % 2 == 0 else 6) + + +def test_syconv(): + """ + Test for going back and forth between the returned format of he/sytrf to + L and D factors/permutations. + """ + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + n = 10 + + if ind > 1: + A = (rng.randint(-30, 30, (n, n)) + + rng.randint(-30, 30, (n, n))*1j).astype(dtype) + + A = A + A.conj().T + else: + A = rng.randint(-30, 30, (n, n)).astype(dtype) + A = A + A.T + n*eye(n) + + tol = 100*np.spacing(dtype(1.0).real) + syconv, trf, trf_lwork = get_lapack_funcs(('syconv', 'sytrf', + 'sytrf_lwork'), dtype=dtype) + lw = _compute_lwork(trf_lwork, n, lower=1) + L, D, perm = ldl(A, lower=1, hermitian=False) + lw = _compute_lwork(trf_lwork, n, lower=1) + ldu, ipiv, info = trf(A, lower=1, lwork=lw) + a, e, info = syconv(ldu, ipiv, lower=1) + assert_allclose(tril(a, -1,), tril(L[perm, :], -1), atol=tol, rtol=0.) + + # Test also upper + U, D, perm = ldl(A, lower=0, hermitian=False) + ldu, ipiv, info = trf(A, lower=0) + a, e, info = syconv(ldu, ipiv, lower=0) + assert_allclose(triu(a, 1), triu(U[perm, :], 1), atol=tol, rtol=0.) + + +class TestBlockedQR: + """ + Tests for the blocked QR factorization, namely through geqrt, gemqrt, tpqrt + and tpmqr. + """ + + def test_geqrt_gemqrt(self): + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + n = 20 + + if ind > 1: + A = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + else: + A = (rng.rand(n, n)).astype(dtype) + + tol = 100*np.spacing(dtype(1.0).real) + geqrt, gemqrt = get_lapack_funcs(('geqrt', 'gemqrt'), dtype=dtype) + + a, t, info = geqrt(n, A) + assert info == 0 + + # Extract elementary reflectors from lower triangle, adding the + # main diagonal of ones. + v = np.tril(a, -1) + np.eye(n, dtype=dtype) + # Generate the block Householder transform I - VTV^H + Q = np.eye(n, dtype=dtype) - v @ t @ v.T.conj() + R = np.triu(a) + + # Test columns of Q are orthogonal + assert_allclose(Q.T.conj() @ Q, np.eye(n, dtype=dtype), atol=tol, + rtol=0.) + assert_allclose(Q @ R, A, atol=tol, rtol=0.) + + if ind > 1: + C = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + transpose = 'C' + else: + C = (rng.rand(n, n)).astype(dtype) + transpose = 'T' + + for side in ('L', 'R'): + for trans in ('N', transpose): + c, info = gemqrt(a, t, C, side=side, trans=trans) + assert info == 0 + + if trans == transpose: + q = Q.T.conj() + else: + q = Q + + if side == 'L': + qC = q @ C + else: + qC = C @ q + + assert_allclose(c, qC, atol=tol, rtol=0.) + + # Test default arguments + if (side, trans) == ('L', 'N'): + c_default, info = gemqrt(a, t, C) + assert info == 0 + assert_equal(c_default, c) + + # Test invalid side/trans + assert_raises(Exception, gemqrt, a, t, C, side='A') + assert_raises(Exception, gemqrt, a, t, C, trans='A') + + def test_tpqrt_tpmqrt(self): + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + n = 20 + + if ind > 1: + A = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + B = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + else: + A = (rng.rand(n, n)).astype(dtype) + B = (rng.rand(n, n)).astype(dtype) + + tol = 100*np.spacing(dtype(1.0).real) + tpqrt, tpmqrt = get_lapack_funcs(('tpqrt', 'tpmqrt'), dtype=dtype) + + # Test for the range of pentagonal B, from square to upper + # triangular + for l in (0, n // 2, n): + a, b, t, info = tpqrt(l, n, A, B) + assert info == 0 + + # Check that lower triangular part of A has not been modified + assert_equal(np.tril(a, -1), np.tril(A, -1)) + # Check that elements not part of the pentagonal portion of B + # have not been modified. + assert_equal(np.tril(b, l - n - 1), np.tril(B, l - n - 1)) + + # Extract pentagonal portion of B + B_pent, b_pent = np.triu(B, l - n), np.triu(b, l - n) + + # Generate elementary reflectors + v = np.concatenate((np.eye(n, dtype=dtype), b_pent)) + # Generate the block Householder transform I - VTV^H + Q = np.eye(2 * n, dtype=dtype) - v @ t @ v.T.conj() + R = np.concatenate((np.triu(a), np.zeros_like(a))) + + # Test columns of Q are orthogonal + assert_allclose(Q.T.conj() @ Q, np.eye(2 * n, dtype=dtype), + atol=tol, rtol=0.) + assert_allclose(Q @ R, np.concatenate((np.triu(A), B_pent)), + atol=tol, rtol=0.) + + if ind > 1: + C = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + D = (rng.rand(n, n) + rng.rand(n, n)*1j).astype(dtype) + transpose = 'C' + else: + C = (rng.rand(n, n)).astype(dtype) + D = (rng.rand(n, n)).astype(dtype) + transpose = 'T' + + for side in ('L', 'R'): + for trans in ('N', transpose): + c, d, info = tpmqrt(l, b, t, C, D, side=side, + trans=trans) + assert info == 0 + + if trans == transpose: + q = Q.T.conj() + else: + q = Q + + if side == 'L': + cd = np.concatenate((c, d), axis=0) + CD = np.concatenate((C, D), axis=0) + qCD = q @ CD + else: + cd = np.concatenate((c, d), axis=1) + CD = np.concatenate((C, D), axis=1) + qCD = CD @ q + + assert_allclose(cd, qCD, atol=tol, rtol=0.) + + if (side, trans) == ('L', 'N'): + c_default, d_default, info = tpmqrt(l, b, t, C, D) + assert info == 0 + assert_equal(c_default, c) + assert_equal(d_default, d) + + # Test invalid side/trans + assert_raises(Exception, tpmqrt, l, b, t, C, D, side='A') + assert_raises(Exception, tpmqrt, l, b, t, C, D, trans='A') + + +def test_pstrf(): + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + # DTYPES = pstrf + n = 10 + r = 2 + pstrf = get_lapack_funcs('pstrf', dtype=dtype) + + # Create positive semidefinite A + if ind > 1: + A = rng.rand(n, n-r).astype(dtype) + 1j * rng.rand(n, n-r).astype(dtype) + A = A @ A.conj().T + else: + A = rng.rand(n, n-r).astype(dtype) + A = A @ A.T + + c, piv, r_c, info = pstrf(A) + U = triu(c) + U[r_c - n:, r_c - n:] = 0. + + assert_equal(info, 1) + # python-dbg 3.5.2 runs cause trouble with the following assertion. + # assert_equal(r_c, n - r) + single_atol = 1000 * np.finfo(np.float32).eps + double_atol = 1000 * np.finfo(np.float64).eps + atol = single_atol if ind in [0, 2] else double_atol + assert_allclose(A[piv-1][:, piv-1], U.conj().T @ U, rtol=0., atol=atol) + + c, piv, r_c, info = pstrf(A, lower=1) + L = tril(c) + L[r_c - n:, r_c - n:] = 0. + + assert_equal(info, 1) + # assert_equal(r_c, n - r) + single_atol = 1000 * np.finfo(np.float32).eps + double_atol = 1000 * np.finfo(np.float64).eps + atol = single_atol if ind in [0, 2] else double_atol + assert_allclose(A[piv-1][:, piv-1], L @ L.conj().T, rtol=0., atol=atol) + + +def test_pstf2(): + rng = np.random.RandomState(1234) + for ind, dtype in enumerate(DTYPES): + # DTYPES = pstf2 + n = 10 + r = 2 + pstf2 = get_lapack_funcs('pstf2', dtype=dtype) + + # Create positive semidefinite A + if ind > 1: + A = rng.rand(n, n-r).astype(dtype) + 1j * rng.rand(n, n-r).astype(dtype) + A = A @ A.conj().T + else: + A = rng.rand(n, n-r).astype(dtype) + A = A @ A.T + + c, piv, r_c, info = pstf2(A) + U = triu(c) + U[r_c - n:, r_c - n:] = 0. + + assert_equal(info, 1) + # python-dbg 3.5.2 runs cause trouble with the commented assertions. + # assert_equal(r_c, n - r) + single_atol = 1000 * np.finfo(np.float32).eps + double_atol = 1000 * np.finfo(np.float64).eps + atol = single_atol if ind in [0, 2] else double_atol + assert_allclose(A[piv-1][:, piv-1], U.conj().T @ U, rtol=0., atol=atol) + + c, piv, r_c, info = pstf2(A, lower=1) + L = tril(c) + L[r_c - n:, r_c - n:] = 0. + + assert_equal(info, 1) + # assert_equal(r_c, n - r) + single_atol = 1000 * np.finfo(np.float32).eps + double_atol = 1000 * np.finfo(np.float64).eps + atol = single_atol if ind in [0, 2] else double_atol + assert_allclose(A[piv-1][:, piv-1], L @ L.conj().T, rtol=0., atol=atol) + + +def test_geequ(): + desired_real = np.array([[0.6250, 1.0000, 0.0393, -0.4269], + [1.0000, -0.5619, -1.0000, -1.0000], + [0.5874, -1.0000, -0.0596, -0.5341], + [-1.0000, -0.5946, -0.0294, 0.9957]]) + + desired_cplx = np.array([[-0.2816+0.5359*1j, + 0.0812+0.9188*1j, + -0.7439-0.2561*1j], + [-0.3562-0.2954*1j, + 0.9566-0.0434*1j, + -0.0174+0.1555*1j], + [0.8607+0.1393*1j, + -0.2759+0.7241*1j, + -0.1642-0.1365*1j]]) + + for ind, dtype in enumerate(DTYPES): + if ind < 2: + # Use examples from the NAG documentation + A = np.array([[1.80e+10, 2.88e+10, 2.05e+00, -8.90e+09], + [5.25e+00, -2.95e+00, -9.50e-09, -3.80e+00], + [1.58e+00, -2.69e+00, -2.90e-10, -1.04e+00], + [-1.11e+00, -6.60e-01, -5.90e-11, 8.00e-01]]) + A = A.astype(dtype) + else: + A = np.array([[-1.34e+00, 0.28e+10, -6.39e+00], + [-1.70e+00, 3.31e+10, -0.15e+00], + [2.41e-10, -0.56e+00, -0.83e-10]], dtype=dtype) + A += np.array([[2.55e+00, 3.17e+10, -2.20e+00], + [-1.41e+00, -0.15e+10, 1.34e+00], + [0.39e-10, 1.47e+00, -0.69e-10]])*1j + + A = A.astype(dtype) + + geequ = get_lapack_funcs('geequ', dtype=dtype) + r, c, rowcnd, colcnd, amax, info = geequ(A) + + if ind < 2: + assert_allclose(desired_real.astype(dtype), r[:, None]*A*c, + rtol=0, atol=1e-4) + else: + assert_allclose(desired_cplx.astype(dtype), r[:, None]*A*c, + rtol=0, atol=1e-4) + + +def test_syequb(): + desired_log2s = np.array([0, 0, 0, 0, 0, 0, -1, -1, -2, -3]) + + for ind, dtype in enumerate(DTYPES): + A = np.eye(10, dtype=dtype) + alpha = dtype(1. if ind < 2 else 1.j) + d = np.array([alpha * 2.**x for x in range(-5, 5)], dtype=dtype) + A += np.rot90(np.diag(d)) + + syequb = get_lapack_funcs('syequb', dtype=dtype) + s, scond, amax, info = syequb(A) + + assert_equal(np.log2(s).astype(int), desired_log2s) + + +@pytest.mark.skipif(True, + reason="Failing on some OpenBLAS version, see gh-12276") +def test_heequb(): + # zheequb has a bug for versions =< LAPACK 3.9.0 + # See Reference-LAPACK gh-61 and gh-408 + # Hence the zheequb test is customized accordingly to avoid + # work scaling. + A = np.diag([2]*5 + [1002]*5) + np.diag(np.ones(9), k=1)*1j + s, scond, amax, info = lapack.zheequb(A) + assert_equal(info, 0) + assert_allclose(np.log2(s), [0., -1.]*2 + [0.] + [-4]*5) + + A = np.diag(2**np.abs(np.arange(-5, 6)) + 0j) + A[5, 5] = 1024 + A[5, 0] = 16j + s, scond, amax, info = lapack.cheequb(A.astype(np.complex64), lower=1) + assert_equal(info, 0) + assert_allclose(np.log2(s), [-2, -1, -1, 0, 0, -5, 0, -1, -1, -2, -2]) + + +def test_getc2_gesc2(): + rng = np.random.RandomState(42) + n = 10 + desired_real = rng.rand(n) + desired_cplx = rng.rand(n) + rng.rand(n)*1j + + for ind, dtype in enumerate(DTYPES): + if ind < 2: + A = rng.rand(n, n) + A = A.astype(dtype) + b = A @ desired_real + b = b.astype(dtype) + else: + A = rng.rand(n, n) + rng.rand(n, n)*1j + A = A.astype(dtype) + b = A @ desired_cplx + b = b.astype(dtype) + + getc2 = get_lapack_funcs('getc2', dtype=dtype) + gesc2 = get_lapack_funcs('gesc2', dtype=dtype) + lu, ipiv, jpiv, info = getc2(A, overwrite_a=0) + x, scale = gesc2(lu, b, ipiv, jpiv, overwrite_rhs=0) + + if ind < 2: + assert_array_almost_equal(desired_real.astype(dtype), + x/scale, decimal=4) + else: + assert_array_almost_equal(desired_cplx.astype(dtype), + x/scale, decimal=4) + + +@pytest.mark.parametrize('size', [(6, 5), (5, 5)]) +@pytest.mark.parametrize('dtype', REAL_DTYPES) +@pytest.mark.parametrize('joba', range(6)) # 'C', 'E', 'F', 'G', 'A', 'R' +@pytest.mark.parametrize('jobu', range(4)) # 'U', 'F', 'W', 'N' +@pytest.mark.parametrize('jobv', range(4)) # 'V', 'J', 'W', 'N' +@pytest.mark.parametrize('jobr', [0, 1]) +@pytest.mark.parametrize('jobp', [0, 1]) +def test_gejsv_general(size, dtype, joba, jobu, jobv, jobr, jobp, jobt=0): + """Test the lapack routine ?gejsv. + + This function tests that a singular value decomposition can be performed + on the random M-by-N matrix A. The test performs the SVD using ?gejsv + then performs the following checks: + + * ?gejsv exist successfully (info == 0) + * The returned singular values are correct + * `A` can be reconstructed from `u`, `SIGMA`, `v` + * Ensure that u.T @ u is the identity matrix + * Ensure that v.T @ v is the identity matrix + * The reported matrix rank + * The reported number of singular values + * If denormalized floats are required + + Notes + ----- + joba specifies several choices effecting the calculation's accuracy + Although all arguments are tested, the tests only check that the correct + solution is returned - NOT that the prescribed actions are performed + internally. + + jobt is, as of v3.9.0, still experimental and removed to cut down number of + test cases. However keyword itself is tested externally. + """ + rng = np.random.RandomState(42) + + # Define some constants for later use: + m, n = size + atol = 100 * np.finfo(dtype).eps + A = generate_random_dtype_array(size, dtype, rng) + gejsv = get_lapack_funcs('gejsv', dtype=dtype) + + # Set up checks for invalid job? combinations + # if an invalid combination occurs we set the appropriate + # exit status. + lsvec = jobu < 2 # Calculate left singular vectors + rsvec = jobv < 2 # Calculate right singular vectors + l2tran = (jobt == 1) and (m == n) + is_complex = np.iscomplexobj(A) + + invalid_real_jobv = (jobv == 1) and (not lsvec) and (not is_complex) + invalid_cplx_jobu = (jobu == 2) and not (rsvec and l2tran) and is_complex + invalid_cplx_jobv = (jobv == 2) and not (lsvec and l2tran) and is_complex + + # Set the exit status to the expected value. + # Here we only check for invalid combinations, not individual + # parameters. + if invalid_cplx_jobu: + exit_status = -2 + elif invalid_real_jobv or invalid_cplx_jobv: + exit_status = -3 + else: + exit_status = 0 + + if (jobu > 1) and (jobv == 1): + assert_raises(Exception, gejsv, A, joba, jobu, jobv, jobr, jobt, jobp) + else: + sva, u, v, work, iwork, info = gejsv(A, + joba=joba, + jobu=jobu, + jobv=jobv, + jobr=jobr, + jobt=jobt, + jobp=jobp) + + # Check that ?gejsv exited successfully/as expected + assert_equal(info, exit_status) + + # If exit_status is non-zero the combination of jobs is invalid. + # We test this above but no calculations are performed. + if not exit_status: + + # Check the returned singular values + sigma = (work[0] / work[1]) * sva[:n] + assert_allclose(sigma, svd(A, compute_uv=False), atol=atol) + + if jobu == 1: + # If JOBU = 'F', then u contains the M-by-M matrix of + # the left singular vectors, including an ONB of the orthogonal + # complement of the Range(A) + # However, to recalculate A we are concerned about the + # first n singular values and so can ignore the latter. + # TODO: Add a test for ONB? + u = u[:, :n] + + if lsvec and rsvec: + assert_allclose(u @ np.diag(sigma) @ v.conj().T, A, atol=atol) + if lsvec: + assert_allclose(u.conj().T @ u, np.identity(n), atol=atol) + if rsvec: + assert_allclose(v.conj().T @ v, np.identity(n), atol=atol) + + assert_equal(iwork[0], np.linalg.matrix_rank(A)) + assert_equal(iwork[1], np.count_nonzero(sigma)) + # iwork[2] is non-zero if requested accuracy is not warranted for + # the data. This should never occur for these tests. + assert_equal(iwork[2], 0) + + +@pytest.mark.parametrize('dtype', REAL_DTYPES) +def test_gejsv_edge_arguments(dtype): + """Test edge arguments return expected status""" + gejsv = get_lapack_funcs('gejsv', dtype=dtype) + + # scalar A + sva, u, v, work, iwork, info = gejsv(1.) + assert_equal(info, 0) + assert_equal(u.shape, (1, 1)) + assert_equal(v.shape, (1, 1)) + assert_equal(sva, np.array([1.], dtype=dtype)) + + # 1d A + A = np.ones((1,), dtype=dtype) + sva, u, v, work, iwork, info = gejsv(A) + assert_equal(info, 0) + assert_equal(u.shape, (1, 1)) + assert_equal(v.shape, (1, 1)) + assert_equal(sva, np.array([1.], dtype=dtype)) + + # 2d empty A + A = np.ones((1, 0), dtype=dtype) + sva, u, v, work, iwork, info = gejsv(A) + assert_equal(info, 0) + assert_equal(u.shape, (1, 0)) + assert_equal(v.shape, (1, 0)) + assert_equal(sva, np.array([], dtype=dtype)) + + # make sure "overwrite_a" is respected - user reported in gh-13191 + A = np.sin(np.arange(100).reshape(10, 10)).astype(dtype) + A = np.asfortranarray(A + A.T) # make it symmetric and column major + Ac = A.copy('A') + _ = gejsv(A) + assert_allclose(A, Ac) + + +@pytest.mark.parametrize(('kwargs'), + ({'joba': 9}, + {'jobu': 9}, + {'jobv': 9}, + {'jobr': 9}, + {'jobt': 9}, + {'jobp': 9}) + ) +def test_gejsv_invalid_job_arguments(kwargs): + """Test invalid job arguments raise an Exception""" + A = np.ones((2, 2), dtype=float) + gejsv = get_lapack_funcs('gejsv', dtype=float) + assert_raises(Exception, gejsv, A, **kwargs) + + +@pytest.mark.parametrize("A,sva_expect,u_expect,v_expect", + [(np.array([[2.27, -1.54, 1.15, -1.94], + [0.28, -1.67, 0.94, -0.78], + [-0.48, -3.09, 0.99, -0.21], + [1.07, 1.22, 0.79, 0.63], + [-2.35, 2.93, -1.45, 2.30], + [0.62, -7.39, 1.03, -2.57]]), + np.array([9.9966, 3.6831, 1.3569, 0.5000]), + np.array([[0.2774, -0.6003, -0.1277, 0.1323], + [0.2020, -0.0301, 0.2805, 0.7034], + [0.2918, 0.3348, 0.6453, 0.1906], + [-0.0938, -0.3699, 0.6781, -0.5399], + [-0.4213, 0.5266, 0.0413, -0.0575], + [0.7816, 0.3353, -0.1645, -0.3957]]), + np.array([[0.1921, -0.8030, 0.0041, -0.5642], + [-0.8794, -0.3926, -0.0752, 0.2587], + [0.2140, -0.2980, 0.7827, 0.5027], + [-0.3795, 0.3351, 0.6178, -0.6017]]))]) +def test_gejsv_NAG(A, sva_expect, u_expect, v_expect): + """ + This test implements the example found in the NAG manual, f08khf. + An example was not found for the complex case. + """ + # NAG manual provides accuracy up to 4 decimals + atol = 1e-4 + gejsv = get_lapack_funcs('gejsv', dtype=A.dtype) + + sva, u, v, work, iwork, info = gejsv(A) + + assert_allclose(sva_expect, sva, atol=atol) + assert_allclose(u_expect, u, atol=atol) + assert_allclose(v_expect, v, atol=atol) + + +@pytest.mark.parametrize("dtype", DTYPES) +def test_gttrf_gttrs(dtype): + # The test uses ?gttrf and ?gttrs to solve a random system for each dtype, + # tests that the output of ?gttrf define LU matrices, that input + # parameters are unmodified, transposal options function correctly, that + # incompatible matrix shapes raise an error, and singular matrices return + # non zero info. + + rng = np.random.RandomState(42) + n = 10 + atol = 100 * np.finfo(dtype).eps + + # create the matrix in accordance with the data type + du = generate_random_dtype_array((n-1,), dtype=dtype, rng=rng) + d = generate_random_dtype_array((n,), dtype=dtype, rng=rng) + dl = generate_random_dtype_array((n-1,), dtype=dtype, rng=rng) + + diag_cpy = [dl.copy(), d.copy(), du.copy()] + + A = np.diag(d) + np.diag(dl, -1) + np.diag(du, 1) + x = np.random.rand(n) + b = A @ x + + gttrf, gttrs = get_lapack_funcs(('gttrf', 'gttrs'), dtype=dtype) + + _dl, _d, _du, du2, ipiv, info = gttrf(dl, d, du) + # test to assure that the inputs of ?gttrf are unmodified + assert_array_equal(dl, diag_cpy[0]) + assert_array_equal(d, diag_cpy[1]) + assert_array_equal(du, diag_cpy[2]) + + # generate L and U factors from ?gttrf return values + # L/U are lower/upper triangular by construction (initially and at end) + U = np.diag(_d, 0) + np.diag(_du, 1) + np.diag(du2, 2) + L = np.eye(n, dtype=dtype) + + for i, m in enumerate(_dl): + # L is given in a factored form. + # See + # www.hpcavf.uclan.ac.uk/softwaredoc/sgi_scsl_html/sgi_html/ch03.html + piv = ipiv[i] - 1 + # right multiply by permutation matrix + L[:, [i, piv]] = L[:, [piv, i]] + # right multiply by Li, rank-one modification of identity + L[:, i] += L[:, i+1]*m + + # one last permutation + i, piv = -1, ipiv[-1] - 1 + # right multiply by final permutation matrix + L[:, [i, piv]] = L[:, [piv, i]] + + # check that the outputs of ?gttrf define an LU decomposition of A + assert_allclose(A, L @ U, atol=atol) + + b_cpy = b.copy() + x_gttrs, info = gttrs(_dl, _d, _du, du2, ipiv, b) + # test that the inputs of ?gttrs are unmodified + assert_array_equal(b, b_cpy) + # test that the result of ?gttrs matches the expected input + assert_allclose(x, x_gttrs, atol=atol) + + # test that ?gttrf and ?gttrs work with transposal options + if dtype in REAL_DTYPES: + trans = "T" + b_trans = A.T @ x + else: + trans = "C" + b_trans = A.conj().T @ x + + x_gttrs, info = gttrs(_dl, _d, _du, du2, ipiv, b_trans, trans=trans) + assert_allclose(x, x_gttrs, atol=atol) + + # test that ValueError is raised with incompatible matrix shapes + with assert_raises(ValueError): + gttrf(dl[:-1], d, du) + with assert_raises(ValueError): + gttrf(dl, d[:-1], du) + with assert_raises(ValueError): + gttrf(dl, d, du[:-1]) + + # test that matrix of size n=2 raises exception + with assert_raises(ValueError): + gttrf(dl[0], d[:1], du[0]) + + # test that singular (row of all zeroes) matrix fails via info + du[0] = 0 + d[0] = 0 + __dl, __d, __du, _du2, _ipiv, _info = gttrf(dl, d, du) + np.testing.assert_(__d[info - 1] == 0, (f"?gttrf: _d[info-1] is {__d[info - 1]}," + " not the illegal value :0.")) + + +@pytest.mark.parametrize("du, d, dl, du_exp, d_exp, du2_exp, ipiv_exp, b, x", + [(np.array([2.1, -1.0, 1.9, 8.0]), + np.array([3.0, 2.3, -5.0, -.9, 7.1]), + np.array([3.4, 3.6, 7.0, -6.0]), + np.array([2.3, -5, -.9, 7.1]), + np.array([3.4, 3.6, 7, -6, -1.015373]), + np.array([-1, 1.9, 8]), + np.array([2, 3, 4, 5, 5]), + np.array([[2.7, 6.6], + [-0.5, 10.8], + [2.6, -3.2], + [0.6, -11.2], + [2.7, 19.1] + ]), + np.array([[-4, 5], + [7, -4], + [3, -3], + [-4, -2], + [-3, 1]])), + ( + np.array([2 - 1j, 2 + 1j, -1 + 1j, 1 - 1j]), + np.array([-1.3 + 1.3j, -1.3 + 1.3j, + -1.3 + 3.3j, - .3 + 4.3j, + -3.3 + 1.3j]), + np.array([1 - 2j, 1 + 1j, 2 - 3j, 1 + 1j]), + # du exp + np.array([-1.3 + 1.3j, -1.3 + 3.3j, + -0.3 + 4.3j, -3.3 + 1.3j]), + np.array([1 - 2j, 1 + 1j, 2 - 3j, 1 + 1j, + -1.3399 + 0.2875j]), + np.array([2 + 1j, -1 + 1j, 1 - 1j]), + np.array([2, 3, 4, 5, 5]), + np.array([[2.4 - 5j, 2.7 + 6.9j], + [3.4 + 18.2j, - 6.9 - 5.3j], + [-14.7 + 9.7j, - 6 - .6j], + [31.9 - 7.7j, -3.9 + 9.3j], + [-1 + 1.6j, -3 + 12.2j]]), + np.array([[1 + 1j, 2 - 1j], + [3 - 1j, 1 + 2j], + [4 + 5j, -1 + 1j], + [-1 - 2j, 2 + 1j], + [1 - 1j, 2 - 2j]]) + )]) +def test_gttrf_gttrs_NAG_f07cdf_f07cef_f07crf_f07csf(du, d, dl, du_exp, d_exp, + du2_exp, ipiv_exp, b, x): + # test to assure that wrapper is consistent with NAG Library Manual Mark 26 + # example problems: f07cdf and f07cef (real) + # examples: f07crf and f07csf (complex) + # (Links may expire, so search for "NAG Library Manual Mark 26" online) + + gttrf, gttrs = get_lapack_funcs(('gttrf', "gttrs"), (du[0], du[0])) + + _dl, _d, _du, du2, ipiv, info = gttrf(dl, d, du) + assert_allclose(du2, du2_exp) + assert_allclose(_du, du_exp) + assert_allclose(_d, d_exp, atol=1e-4) # NAG examples provide 4 decimals. + assert_allclose(ipiv, ipiv_exp) + + x_gttrs, info = gttrs(_dl, _d, _du, du2, ipiv, b) + + assert_allclose(x_gttrs, x) + + +@pytest.mark.parametrize('dtype', DTYPES) +@pytest.mark.parametrize('norm', ['1', 'I', 'O']) +@pytest.mark.parametrize('n', [3, 10]) +def test_gtcon(dtype, norm, n): + rng = np.random.default_rng(23498324) + + d = rng.random(n) + rng.random(n)*1j + dl = rng.random(n - 1) + rng.random(n - 1)*1j + du = rng.random(n - 1) + rng.random(n - 1)*1j + A = np.diag(d) + np.diag(dl, -1) + np.diag(du, 1) + if np.issubdtype(dtype, np.floating): + A, d, dl, du = A.real, d.real, dl.real, du.real + A, d, dl, du = A.astype(dtype), d.astype(dtype), dl.astype(dtype), du.astype(dtype) + + anorm = np.abs(A).sum(axis=0).max() + + gttrf, gtcon = get_lapack_funcs(('gttrf', 'gtcon'), (A,)) + dl, d, du, du2, ipiv, info = gttrf(dl, d, du) + res, _ = gtcon(dl, d, du, du2, ipiv, anorm, norm=norm) + + gecon, getrf = get_lapack_funcs(('gecon', 'getrf'), (A,)) + lu, ipvt, info = getrf(A) + ref, _ = gecon(lu, anorm, norm=norm) + + rtol = np.finfo(dtype).eps**0.75 + assert_allclose(res, ref, rtol=rtol) + + +@pytest.mark.parametrize('dtype', DTYPES) +@pytest.mark.parametrize('shape', [(3, 7), (7, 3), (2**18, 2**18)]) +def test_geqrfp_lwork(dtype, shape): + geqrfp_lwork = get_lapack_funcs(('geqrfp_lwork'), dtype=dtype) + m, n = shape + lwork, info = geqrfp_lwork(m=m, n=n) + assert_equal(info, 0) + + +@pytest.mark.parametrize("ddtype,dtype", + zip(REAL_DTYPES + REAL_DTYPES, DTYPES)) +def test_pttrf_pttrs(ddtype, dtype): + rng = np.random.RandomState(42) + # set test tolerance appropriate for dtype + atol = 100*np.finfo(dtype).eps + # n is the length diagonal of A + n = 10 + # create diagonals according to size and dtype + + # diagonal d should always be real. + # add 4 to d so it will be dominant for all dtypes + d = generate_random_dtype_array((n,), ddtype, rng) + 4 + # diagonal e may be real or complex. + e = generate_random_dtype_array((n-1,), dtype, rng) + + # assemble diagonals together into matrix + A = np.diag(d) + np.diag(e, -1) + np.diag(np.conj(e), 1) + # store a copy of diagonals to later verify + diag_cpy = [d.copy(), e.copy()] + + pttrf = get_lapack_funcs('pttrf', dtype=dtype) + + _d, _e, info = pttrf(d, e) + # test to assure that the inputs of ?pttrf are unmodified + assert_array_equal(d, diag_cpy[0]) + assert_array_equal(e, diag_cpy[1]) + assert_equal(info, 0, err_msg=f"pttrf: info = {info}, should be 0") + + # test that the factors from pttrf can be recombined to make A + L = np.diag(_e, -1) + np.diag(np.ones(n)) + D = np.diag(_d) + + assert_allclose(A, L@D@L.conjugate().T, atol=atol) + + # generate random solution x + x = generate_random_dtype_array((n,), dtype, rng) + # determine accompanying b to get soln x + b = A@x + + # determine _x from pttrs + pttrs = get_lapack_funcs('pttrs', dtype=dtype) + _x, info = pttrs(_d, _e.conj(), b) + assert_equal(info, 0, err_msg=f"pttrs: info = {info}, should be 0") + + # test that _x from pttrs matches the expected x + assert_allclose(x, _x, atol=atol) + + +@pytest.mark.parametrize("ddtype,dtype", + zip(REAL_DTYPES + REAL_DTYPES, DTYPES)) +def test_pttrf_pttrs_errors_incompatible_shape(ddtype, dtype): + n = 10 + rng = np.random.RandomState(1234) + pttrf = get_lapack_funcs('pttrf', dtype=dtype) + d = generate_random_dtype_array((n,), ddtype, rng) + 2 + e = generate_random_dtype_array((n-1,), dtype, rng) + # test that ValueError is raised with incompatible matrix shapes + assert_raises(ValueError, pttrf, d[:-1], e) + assert_raises(ValueError, pttrf, d, e[:-1]) + + +@pytest.mark.parametrize("ddtype,dtype", + zip(REAL_DTYPES + REAL_DTYPES, DTYPES)) +def test_pttrf_pttrs_errors_singular_nonSPD(ddtype, dtype): + n = 10 + rng = np.random.RandomState(42) + pttrf = get_lapack_funcs('pttrf', dtype=dtype) + d = generate_random_dtype_array((n,), ddtype, rng) + 2 + e = generate_random_dtype_array((n-1,), dtype, rng) + # test that singular (row of all zeroes) matrix fails via info + d[0] = 0 + e[0] = 0 + _d, _e, info = pttrf(d, e) + assert_equal(_d[info - 1], 0, + f"?pttrf: _d[info-1] is {_d[info - 1]}, not the illegal value :0.") + + # test with non-spd matrix + d = generate_random_dtype_array((n,), ddtype, rng) + _d, _e, info = pttrf(d, e) + assert_(info != 0, "?pttrf should fail with non-spd matrix, but didn't") + + +@pytest.mark.parametrize(("d, e, d_expect, e_expect, b, x_expect"), [ + (np.array([4, 10, 29, 25, 5]), + np.array([-2, -6, 15, 8]), + np.array([4, 9, 25, 16, 1]), + np.array([-.5, -.6667, .6, .5]), + np.array([[6, 10], [9, 4], [2, 9], [14, 65], + [7, 23]]), + np.array([[2.5, 2], [2, -1], [1, -3], [-1, 6], + [3, -5]]) + ), ( + np.array([16, 41, 46, 21]), + np.array([16 + 16j, 18 - 9j, 1 - 4j]), + np.array([16, 9, 1, 4]), + np.array([1+1j, 2-1j, 1-4j]), + np.array([[64+16j, -16-32j], [93+62j, 61-66j], + [78-80j, 71-74j], [14-27j, 35+15j]]), + np.array([[2+1j, -3-2j], [1+1j, 1+1j], [1-2j, 1-2j], + [1-1j, 2+1j]]) + )]) +def test_pttrf_pttrs_NAG(d, e, d_expect, e_expect, b, x_expect): + # test to assure that wrapper is consistent with NAG Manual Mark 26 + # example problems: f07jdf and f07jef (real) + # examples: f07jrf and f07csf (complex) + # NAG examples provide 4 decimals. + # (Links expire, so please search for "NAG Library Manual Mark 26" online) + + atol = 1e-4 + pttrf = get_lapack_funcs('pttrf', dtype=e[0]) + _d, _e, info = pttrf(d, e) + assert_allclose(_d, d_expect, atol=atol) + assert_allclose(_e, e_expect, atol=atol) + + pttrs = get_lapack_funcs('pttrs', dtype=e[0]) + _x, info = pttrs(_d, _e.conj(), b) + assert_allclose(_x, x_expect, atol=atol) + + # also test option `lower` + if e.dtype in COMPLEX_DTYPES: + _x, info = pttrs(_d, _e, b, lower=1) + assert_allclose(_x, x_expect, atol=atol) + + +def pteqr_get_d_e_A_z(dtype, realtype, n, compute_z): + # used by ?pteqr tests to build parameters + # returns tuple of (d, e, A, z) + rng = np.random.RandomState(42) + if compute_z == 1: + # build Hermitian A from Q**T * tri * Q = A by creating Q and tri + A_eig = generate_random_dtype_array((n, n), dtype, rng) + A_eig = A_eig + np.diag(np.zeros(n) + 4*n) + A_eig = (A_eig + A_eig.conj().T) / 2 + # obtain right eigenvectors (orthogonal) + vr = eigh(A_eig)[1] + # create tridiagonal matrix + d = generate_random_dtype_array((n,), realtype, rng) + 4 + e = generate_random_dtype_array((n-1,), realtype, rng) + tri = np.diag(d) + np.diag(e, 1) + np.diag(e, -1) + # Build A using these factors that sytrd would: (Q**T * tri * Q = A) + A = vr @ tri @ vr.conj().T + # vr is orthogonal + z = vr + + else: + # d and e are always real per lapack docs. + d = generate_random_dtype_array((n,), realtype, rng) + e = generate_random_dtype_array((n-1,), realtype, rng) + + # make SPD + d = d + 4 + A = np.diag(d) + np.diag(e, 1) + np.diag(e, -1) + z = np.diag(d) + np.diag(e, -1) + np.diag(e, 1) + return (d, e, A, z) + + +@pytest.mark.parametrize("dtype,realtype", + zip(DTYPES, REAL_DTYPES + REAL_DTYPES)) +@pytest.mark.parametrize("compute_z", range(3)) +def test_pteqr(dtype, realtype, compute_z): + ''' + Tests the ?pteqr lapack routine for all dtypes and compute_z parameters. + It generates random SPD matrix diagonals d and e, and then confirms + correct eigenvalues with scipy.linalg.eig. With applicable compute_z=2 it + tests that z can reform A. + ''' + seed(42) + atol = 1000*np.finfo(dtype).eps + pteqr = get_lapack_funcs(('pteqr'), dtype=dtype) + + n = 10 + + d, e, A, z = pteqr_get_d_e_A_z(dtype, realtype, n, compute_z) + + d_pteqr, e_pteqr, z_pteqr, info = pteqr(d=d, e=e, z=z, compute_z=compute_z) + assert_equal(info, 0, f"info = {info}, should be 0.") + + # compare the routine's eigenvalues with scipy.linalg.eig's. + assert_allclose(np.sort(eigh(A)[0]), np.sort(d_pteqr), atol=atol) + + if compute_z: + # verify z_pteqr as orthogonal + assert_allclose(z_pteqr @ np.conj(z_pteqr).T, np.identity(n), + atol=atol) + # verify that z_pteqr recombines to A + assert_allclose(z_pteqr @ np.diag(d_pteqr) @ np.conj(z_pteqr).T, + A, atol=atol) + + +@pytest.mark.parametrize("dtype,realtype", + zip(DTYPES, REAL_DTYPES + REAL_DTYPES)) +@pytest.mark.parametrize("compute_z", range(3)) +def test_pteqr_error_non_spd(dtype, realtype, compute_z): + seed(42) + pteqr = get_lapack_funcs(('pteqr'), dtype=dtype) + + n = 10 + d, e, A, z = pteqr_get_d_e_A_z(dtype, realtype, n, compute_z) + + # test with non-spd matrix + d_pteqr, e_pteqr, z_pteqr, info = pteqr(d - 4, e, z=z, compute_z=compute_z) + assert info > 0 + + +@pytest.mark.parametrize("dtype,realtype", + zip(DTYPES, REAL_DTYPES + REAL_DTYPES)) +@pytest.mark.parametrize("compute_z", range(3)) +def test_pteqr_raise_error_wrong_shape(dtype, realtype, compute_z): + seed(42) + pteqr = get_lapack_funcs(('pteqr'), dtype=dtype) + n = 10 + d, e, A, z = pteqr_get_d_e_A_z(dtype, realtype, n, compute_z) + # test with incorrect/incompatible array sizes + assert_raises(ValueError, pteqr, d[:-1], e, z=z, compute_z=compute_z) + assert_raises(ValueError, pteqr, d, e[:-1], z=z, compute_z=compute_z) + if compute_z: + assert_raises(ValueError, pteqr, d, e, z=z[:-1], compute_z=compute_z) + + +@pytest.mark.parametrize("dtype,realtype", + zip(DTYPES, REAL_DTYPES + REAL_DTYPES)) +@pytest.mark.parametrize("compute_z", range(3)) +def test_pteqr_error_singular(dtype, realtype, compute_z): + seed(42) + pteqr = get_lapack_funcs(('pteqr'), dtype=dtype) + n = 10 + d, e, A, z = pteqr_get_d_e_A_z(dtype, realtype, n, compute_z) + # test with singular matrix + d[0] = 0 + e[0] = 0 + d_pteqr, e_pteqr, z_pteqr, info = pteqr(d, e, z=z, compute_z=compute_z) + assert info > 0 + + +@pytest.mark.parametrize("compute_z,d,e,d_expect,z_expect", + [(2, # "I" + np.array([4.16, 5.25, 1.09, .62]), + np.array([3.17, -.97, .55]), + np.array([8.0023, 1.9926, 1.0014, 0.1237]), + np.array([[0.6326, 0.6245, -0.4191, 0.1847], + [0.7668, -0.4270, 0.4176, -0.2352], + [-0.1082, 0.6071, 0.4594, -0.6393], + [-0.0081, 0.2432, 0.6625, 0.7084]])), + ]) +def test_pteqr_NAG_f08jgf(compute_z, d, e, d_expect, z_expect): + ''' + Implements real (f08jgf) example from NAG Manual Mark 26. + Tests for correct outputs. + ''' + # the NAG manual has 4 decimals accuracy + atol = 1e-4 + pteqr = get_lapack_funcs(('pteqr'), dtype=d.dtype) + + z = np.diag(d) + np.diag(e, 1) + np.diag(e, -1) + _d, _e, _z, info = pteqr(d=d, e=e, z=z, compute_z=compute_z) + assert_allclose(_d, d_expect, atol=atol) + assert_allclose(np.abs(_z), np.abs(z_expect), atol=atol) + + +@pytest.mark.parametrize('dtype', DTYPES) +@pytest.mark.parametrize('matrix_size', [(3, 4), (7, 6), (6, 6)]) +def test_geqrfp(dtype, matrix_size): + # Tests for all dytpes, tall, wide, and square matrices. + # Using the routine with random matrix A, Q and R are obtained and then + # tested such that R is upper triangular and non-negative on the diagonal, + # and Q is an orthogonal matrix. Verifies that A=Q@R. It also + # tests against a matrix that for which the linalg.qr method returns + # negative diagonals, and for error messaging. + + # set test tolerance appropriate for dtype + rng = np.random.RandomState(42) + rtol = 250*np.finfo(dtype).eps + atol = 100*np.finfo(dtype).eps + # get appropriate ?geqrfp for dtype + geqrfp = get_lapack_funcs(('geqrfp'), dtype=dtype) + gqr = get_lapack_funcs(("orgqr"), dtype=dtype) + + m, n = matrix_size + + # create random matrix of dimensions m x n + A = generate_random_dtype_array((m, n), dtype=dtype, rng=rng) + # create qr matrix using geqrfp + qr_A, tau, info = geqrfp(A) + + # obtain r from the upper triangular area + r = np.triu(qr_A) + + # obtain q from the orgqr lapack routine + # based on linalg.qr's extraction strategy of q with orgqr + + if m > n: + # this adds an extra column to the end of qr_A + # let qqr be an empty m x m matrix + qqr = np.zeros((m, m), dtype=dtype) + # set first n columns of qqr to qr_A + qqr[:, :n] = qr_A + # determine q from this qqr + # note that m is a sufficient for lwork based on LAPACK documentation + q = gqr(qqr, tau=tau, lwork=m)[0] + else: + q = gqr(qr_A[:, :m], tau=tau, lwork=m)[0] + + # test that q and r still make A + assert_allclose(q@r, A, rtol=rtol) + # ensure that q is orthogonal (that q @ transposed q is the identity) + assert_allclose(np.eye(q.shape[0]), q@(q.conj().T), rtol=rtol, + atol=atol) + # ensure r is upper tri by comparing original r to r as upper triangular + assert_allclose(r, np.triu(r), rtol=rtol) + # make sure diagonals of r are positive for this random solution + assert_(np.all(np.diag(r) > np.zeros(len(np.diag(r))))) + # ensure that info is zero for this success + assert_(info == 0) + + # test that this routine gives r diagonals that are positive for a + # matrix that returns negatives in the diagonal with scipy.linalg.rq + A_negative = generate_random_dtype_array((n, m), dtype=dtype, rng=rng) * -1 + r_rq_neg, q_rq_neg = qr(A_negative) + rq_A_neg, tau_neg, info_neg = geqrfp(A_negative) + # assert that any of the entries on the diagonal from linalg.qr + # are negative and that all of geqrfp are positive. + assert_(np.any(np.diag(r_rq_neg) < 0) and + np.all(np.diag(r) > 0)) + + +def test_geqrfp_errors_with_empty_array(): + # check that empty array raises good error message + A_empty = np.array([]) + geqrfp = get_lapack_funcs('geqrfp', dtype=A_empty.dtype) + assert_raises(Exception, geqrfp, A_empty) + + +@pytest.mark.parametrize("driver", ['ev', 'evd', 'evr', 'evx']) +@pytest.mark.parametrize("pfx", ['sy', 'he']) +def test_standard_eigh_lworks(pfx, driver): + n = 1200 # Some sufficiently big arbitrary number + dtype = REAL_DTYPES if pfx == 'sy' else COMPLEX_DTYPES + sc_dlw = get_lapack_funcs(pfx+driver+'_lwork', dtype=dtype[0]) + dz_dlw = get_lapack_funcs(pfx+driver+'_lwork', dtype=dtype[1]) + try: + _compute_lwork(sc_dlw, n, lower=1) + _compute_lwork(dz_dlw, n, lower=1) + except Exception as e: + pytest.fail(f"{pfx+driver}_lwork raised unexpected exception: {e}") + + +@pytest.mark.parametrize("driver", ['gv', 'gvx']) +@pytest.mark.parametrize("pfx", ['sy', 'he']) +def test_generalized_eigh_lworks(pfx, driver): + n = 1200 # Some sufficiently big arbitrary number + dtype = REAL_DTYPES if pfx == 'sy' else COMPLEX_DTYPES + sc_dlw = get_lapack_funcs(pfx+driver+'_lwork', dtype=dtype[0]) + dz_dlw = get_lapack_funcs(pfx+driver+'_lwork', dtype=dtype[1]) + # Shouldn't raise any exceptions + try: + _compute_lwork(sc_dlw, n, uplo="L") + _compute_lwork(dz_dlw, n, uplo="L") + except Exception as e: + pytest.fail(f"{pfx+driver}_lwork raised unexpected exception: {e}") + + +@pytest.mark.parametrize("dtype_", DTYPES) +@pytest.mark.parametrize("m", [1, 10, 100, 1000]) +def test_orcsd_uncsd_lwork(dtype_, m): + seed(1234) + p = randint(0, m) + q = m - p + pfx = 'or' if dtype_ in REAL_DTYPES else 'un' + dlw = pfx + 'csd_lwork' + lw = get_lapack_funcs(dlw, dtype=dtype_) + lwval = _compute_lwork(lw, m, p, q) + lwval = lwval if pfx == 'un' else (lwval,) + assert all([x > 0 for x in lwval]) + + +@pytest.mark.parametrize("dtype_", DTYPES) +def test_orcsd_uncsd(dtype_): + m, p, q = 250, 80, 170 + + pfx = 'or' if dtype_ in REAL_DTYPES else 'un' + X = ortho_group.rvs(m) if pfx == 'or' else unitary_group.rvs(m) + + drv, dlw = get_lapack_funcs((pfx + 'csd', pfx + 'csd_lwork'), dtype=dtype_) + lwval = _compute_lwork(dlw, m, p, q) + lwvals = {'lwork': lwval} if pfx == 'or' else dict(zip(['lwork', + 'lrwork'], lwval)) + + cs11, cs12, cs21, cs22, theta, u1, u2, v1t, v2t, info =\ + drv(X[:p, :q], X[:p, q:], X[p:, :q], X[p:, q:], **lwvals) + + assert info == 0 + + U = block_diag(u1, u2) + VH = block_diag(v1t, v2t) + r = min(min(p, q), min(m-p, m-q)) + n11 = min(p, q) - r + n12 = min(p, m-q) - r + n21 = min(m-p, q) - r + n22 = min(m-p, m-q) - r + + S = np.zeros((m, m), dtype=dtype_) + one = dtype_(1.) + for i in range(n11): + S[i, i] = one + for i in range(n22): + S[p+i, q+i] = one + for i in range(n12): + S[i+n11+r, i+n11+r+n21+n22+r] = -one + for i in range(n21): + S[p+n22+r+i, n11+r+i] = one + + for i in range(r): + S[i+n11, i+n11] = np.cos(theta[i]) + S[p+n22+i, i+r+n21+n22] = np.cos(theta[i]) + + S[i+n11, i+n11+n21+n22+r] = -np.sin(theta[i]) + S[p+n22+i, i+n11] = np.sin(theta[i]) + + Xc = U @ S @ VH + assert_allclose(X, Xc, rtol=0., atol=1e4*np.finfo(dtype_).eps) + + +@pytest.mark.parametrize("dtype", DTYPES) +@pytest.mark.parametrize("trans_bool", [False, True]) +@pytest.mark.parametrize("fact", ["F", "N"]) +def test_gtsvx(dtype, trans_bool, fact): + """ + These tests uses ?gtsvx to solve a random Ax=b system for each dtype. + It tests that the outputs define an LU matrix, that inputs are unmodified, + transposal options, incompatible shapes, singular matrices, and + singular factorizations. It parametrizes DTYPES and the 'fact' value along + with the fact related inputs. + """ + rng = np.random.RandomState(42) + # set test tolerance appropriate for dtype + atol = 100 * np.finfo(dtype).eps + # obtain routine + gtsvx, gttrf = get_lapack_funcs(('gtsvx', 'gttrf'), dtype=dtype) + # Generate random tridiagonal matrix A + n = 10 + dl = generate_random_dtype_array((n-1,), dtype=dtype, rng=rng) + d = generate_random_dtype_array((n,), dtype=dtype, rng=rng) + du = generate_random_dtype_array((n-1,), dtype=dtype, rng=rng) + A = np.diag(dl, -1) + np.diag(d) + np.diag(du, 1) + # generate random solution x + x = generate_random_dtype_array((n, 2), dtype=dtype, rng=rng) + # create b from x for equation Ax=b + trans = ("T" if dtype in REAL_DTYPES else "C") if trans_bool else "N" + b = (A.conj().T if trans_bool else A) @ x + + # store a copy of the inputs to check they haven't been modified later + inputs_cpy = [dl.copy(), d.copy(), du.copy(), b.copy()] + + # set these to None if fact = 'N', or the output of gttrf is fact = 'F' + dlf_, df_, duf_, du2f_, ipiv_, info_ = \ + gttrf(dl, d, du) if fact == 'F' else [None]*6 + + gtsvx_out = gtsvx(dl, d, du, b, fact=fact, trans=trans, dlf=dlf_, df=df_, + duf=duf_, du2=du2f_, ipiv=ipiv_) + dlf, df, duf, du2f, ipiv, x_soln, rcond, ferr, berr, info = gtsvx_out + assert_(info == 0, f"?gtsvx info = {info}, should be zero") + + # assure that inputs are unmodified + assert_array_equal(dl, inputs_cpy[0]) + assert_array_equal(d, inputs_cpy[1]) + assert_array_equal(du, inputs_cpy[2]) + assert_array_equal(b, inputs_cpy[3]) + + # test that x_soln matches the expected x + assert_allclose(x, x_soln, atol=atol) + + # assert that the outputs are of correct type or shape + # rcond should be a scalar + assert_(hasattr(rcond, "__len__") is not True, + f"rcond should be scalar but is {rcond}") + # ferr should be length of # of cols in x + assert_(ferr.shape[0] == b.shape[1], (f"ferr.shape is {ferr.shape[0]} but should" + f" be {b.shape[1]}")) + # berr should be length of # of cols in x + assert_(berr.shape[0] == b.shape[1], (f"berr.shape is {berr.shape[0]} but should" + f" be {b.shape[1]}")) + + +@pytest.mark.parametrize("dtype", DTYPES) +@pytest.mark.parametrize("trans_bool", [0, 1]) +@pytest.mark.parametrize("fact", ["F", "N"]) +def test_gtsvx_error_singular(dtype, trans_bool, fact): + rng = np.random.RandomState(42) + # obtain routine + gtsvx, gttrf = get_lapack_funcs(('gtsvx', 'gttrf'), dtype=dtype) + # Generate random tridiagonal matrix A + n = 10 + dl = generate_random_dtype_array((n-1,), dtype=dtype, rng=rng) + d = generate_random_dtype_array((n,), dtype=dtype, rng=rng) + du = generate_random_dtype_array((n-1,), dtype=dtype, rng=rng) + A = np.diag(dl, -1) + np.diag(d) + np.diag(du, 1) + # generate random solution x + x = generate_random_dtype_array((n, 2), dtype=dtype, rng=rng) + # create b from x for equation Ax=b + trans = "T" if dtype in REAL_DTYPES else "C" + b = (A.conj().T if trans_bool else A) @ x + + # set these to None if fact = 'N', or the output of gttrf is fact = 'F' + dlf_, df_, duf_, du2f_, ipiv_, info_ = \ + gttrf(dl, d, du) if fact == 'F' else [None]*6 + + gtsvx_out = gtsvx(dl, d, du, b, fact=fact, trans=trans, dlf=dlf_, df=df_, + duf=duf_, du2=du2f_, ipiv=ipiv_) + dlf, df, duf, du2f, ipiv, x_soln, rcond, ferr, berr, info = gtsvx_out + # test with singular matrix + # no need to test inputs with fact "F" since ?gttrf already does. + if fact == "N": + # Construct a singular example manually + d[-1] = 0 + dl[-1] = 0 + # solve using routine + gtsvx_out = gtsvx(dl, d, du, b) + dlf, df, duf, du2f, ipiv, x_soln, rcond, ferr, berr, info = gtsvx_out + # test for the singular matrix. + assert info > 0, "info should be > 0 for singular matrix" + + elif fact == 'F': + # assuming that a singular factorization is input + df_[-1] = 0 + duf_[-1] = 0 + du2f_[-1] = 0 + + gtsvx_out = gtsvx(dl, d, du, b, fact=fact, dlf=dlf_, df=df_, duf=duf_, + du2=du2f_, ipiv=ipiv_) + dlf, df, duf, du2f, ipiv, x_soln, rcond, ferr, berr, info = gtsvx_out + # info should not be zero and should provide index of illegal value + assert info > 0, "info should be > 0 for singular matrix" + + +@pytest.mark.parametrize("dtype", DTYPES*2) +@pytest.mark.parametrize("trans_bool", [False, True]) +@pytest.mark.parametrize("fact", ["F", "N"]) +def test_gtsvx_error_incompatible_size(dtype, trans_bool, fact): + rng = np.random.RandomState(42) + # obtain routine + gtsvx, gttrf = get_lapack_funcs(('gtsvx', 'gttrf'), dtype=dtype) + # Generate random tridiagonal matrix A + n = 10 + dl = generate_random_dtype_array((n-1,), dtype=dtype, rng=rng) + d = generate_random_dtype_array((n,), dtype=dtype, rng=rng) + du = generate_random_dtype_array((n-1,), dtype=dtype, rng=rng) + A = np.diag(dl, -1) + np.diag(d) + np.diag(du, 1) + # generate random solution x + x = generate_random_dtype_array((n, 2), dtype=dtype, rng=rng) + # create b from x for equation Ax=b + trans = "T" if dtype in REAL_DTYPES else "C" + b = (A.conj().T if trans_bool else A) @ x + + # set these to None if fact = 'N', or the output of gttrf is fact = 'F' + dlf_, df_, duf_, du2f_, ipiv_, info_ = \ + gttrf(dl, d, du) if fact == 'F' else [None]*6 + + if fact == "N": + assert_raises(ValueError, gtsvx, dl[:-1], d, du, b, + fact=fact, trans=trans, dlf=dlf_, df=df_, + duf=duf_, du2=du2f_, ipiv=ipiv_) + assert_raises(ValueError, gtsvx, dl, d[:-1], du, b, + fact=fact, trans=trans, dlf=dlf_, df=df_, + duf=duf_, du2=du2f_, ipiv=ipiv_) + assert_raises(ValueError, gtsvx, dl, d, du[:-1], b, + fact=fact, trans=trans, dlf=dlf_, df=df_, + duf=duf_, du2=du2f_, ipiv=ipiv_) + assert_raises(Exception, gtsvx, dl, d, du, b[:-1], + fact=fact, trans=trans, dlf=dlf_, df=df_, + duf=duf_, du2=du2f_, ipiv=ipiv_) + else: + assert_raises(ValueError, gtsvx, dl, d, du, b, + fact=fact, trans=trans, dlf=dlf_[:-1], df=df_, + duf=duf_, du2=du2f_, ipiv=ipiv_) + assert_raises(ValueError, gtsvx, dl, d, du, b, + fact=fact, trans=trans, dlf=dlf_, df=df_[:-1], + duf=duf_, du2=du2f_, ipiv=ipiv_) + assert_raises(ValueError, gtsvx, dl, d, du, b, + fact=fact, trans=trans, dlf=dlf_, df=df_, + duf=duf_[:-1], du2=du2f_, ipiv=ipiv_) + assert_raises(ValueError, gtsvx, dl, d, du, b, + fact=fact, trans=trans, dlf=dlf_, df=df_, + duf=duf_, du2=du2f_[:-1], ipiv=ipiv_) + + +@pytest.mark.parametrize("du,d,dl,b,x", + [(np.array([2.1, -1.0, 1.9, 8.0]), + np.array([3.0, 2.3, -5.0, -0.9, 7.1]), + np.array([3.4, 3.6, 7.0, -6.0]), + np.array([[2.7, 6.6], [-.5, 10.8], [2.6, -3.2], + [.6, -11.2], [2.7, 19.1]]), + np.array([[-4, 5], [7, -4], [3, -3], [-4, -2], + [-3, 1]])), + (np.array([2 - 1j, 2 + 1j, -1 + 1j, 1 - 1j]), + np.array([-1.3 + 1.3j, -1.3 + 1.3j, -1.3 + 3.3j, + -.3 + 4.3j, -3.3 + 1.3j]), + np.array([1 - 2j, 1 + 1j, 2 - 3j, 1 + 1j]), + np.array([[2.4 - 5j, 2.7 + 6.9j], + [3.4 + 18.2j, -6.9 - 5.3j], + [-14.7 + 9.7j, -6 - .6j], + [31.9 - 7.7j, -3.9 + 9.3j], + [-1 + 1.6j, -3 + 12.2j]]), + np.array([[1 + 1j, 2 - 1j], [3 - 1j, 1 + 2j], + [4 + 5j, -1 + 1j], [-1 - 2j, 2 + 1j], + [1 - 1j, 2 - 2j]]))]) +def test_gtsvx_NAG(du, d, dl, b, x): + # Test to ensure wrapper is consistent with NAG Manual Mark 26 + # example problems: real (f07cbf) and complex (f07cpf) + gtsvx = get_lapack_funcs('gtsvx', dtype=d.dtype) + + gtsvx_out = gtsvx(dl, d, du, b) + dlf, df, duf, du2f, ipiv, x_soln, rcond, ferr, berr, info = gtsvx_out + + assert_array_almost_equal(x, x_soln) + + +@pytest.mark.parametrize("dtype,realtype", zip(DTYPES, REAL_DTYPES + + REAL_DTYPES)) +@pytest.mark.parametrize("fact,df_de_lambda", + [("F", + lambda d, e: get_lapack_funcs('pttrf', + dtype=e.dtype)(d, e)), + ("N", lambda d, e: (None, None, None))]) +def test_ptsvx(dtype, realtype, fact, df_de_lambda): + ''' + This tests the ?ptsvx lapack routine wrapper to solve a random system + Ax = b for all dtypes and input variations. Tests for: unmodified + input parameters, fact options, incompatible matrix shapes raise an error, + and singular matrices return info of illegal value. + ''' + rng = np.random.RandomState(42) + # set test tolerance appropriate for dtype + atol = 100 * np.finfo(dtype).eps + ptsvx = get_lapack_funcs('ptsvx', dtype=dtype) + n = 5 + # create diagonals according to size and dtype + d = generate_random_dtype_array((n,), realtype, rng) + 4 + e = generate_random_dtype_array((n-1,), dtype, rng) + A = np.diag(d) + np.diag(e, -1) + np.diag(np.conj(e), 1) + x_soln = generate_random_dtype_array((n, 2), dtype=dtype, rng=rng) + b = A @ x_soln + + # use lambda to determine what df, ef are + df, ef, info = df_de_lambda(d, e) + + # create copy to later test that they are unmodified + diag_cpy = [d.copy(), e.copy(), b.copy()] + + # solve using routine + df, ef, x, rcond, ferr, berr, info = ptsvx(d, e, b, fact=fact, + df=df, ef=ef) + # d, e, and b should be unmodified + assert_array_equal(d, diag_cpy[0]) + assert_array_equal(e, diag_cpy[1]) + assert_array_equal(b, diag_cpy[2]) + assert_(info == 0, f"info should be 0 but is {info}.") + assert_array_almost_equal(x_soln, x) + + # test that the factors from ptsvx can be recombined to make A + L = np.diag(ef, -1) + np.diag(np.ones(n)) + D = np.diag(df) + assert_allclose(A, L@D@(np.conj(L).T), atol=atol) + + # assert that the outputs are of correct type or shape + # rcond should be a scalar + assert not hasattr(rcond, "__len__"), \ + f"rcond should be scalar but is {rcond}" + # ferr should be length of # of cols in x + assert_(ferr.shape == (2,), (f"ferr.shape is {ferr.shape} but should be " + "({x_soln.shape[1]},)")) + # berr should be length of # of cols in x + assert_(berr.shape == (2,), (f"berr.shape is {berr.shape} but should be " + "({x_soln.shape[1]},)")) + + +@pytest.mark.parametrize("dtype,realtype", zip(DTYPES, REAL_DTYPES + + REAL_DTYPES)) +@pytest.mark.parametrize("fact,df_de_lambda", + [("F", + lambda d, e: get_lapack_funcs('pttrf', + dtype=e.dtype)(d, e)), + ("N", lambda d, e: (None, None, None))]) +def test_ptsvx_error_raise_errors(dtype, realtype, fact, df_de_lambda): + rng = np.random.RandomState(42) + ptsvx = get_lapack_funcs('ptsvx', dtype=dtype) + n = 5 + # create diagonals according to size and dtype + d = generate_random_dtype_array((n,), realtype, rng) + 4 + e = generate_random_dtype_array((n-1,), dtype, rng) + A = np.diag(d) + np.diag(e, -1) + np.diag(np.conj(e), 1) + x_soln = generate_random_dtype_array((n, 2), dtype=dtype, rng=rng) + b = A @ x_soln + + # use lambda to determine what df, ef are + df, ef, info = df_de_lambda(d, e) + + # test with malformatted array sizes + assert_raises(ValueError, ptsvx, d[:-1], e, b, fact=fact, df=df, ef=ef) + assert_raises(ValueError, ptsvx, d, e[:-1], b, fact=fact, df=df, ef=ef) + assert_raises(Exception, ptsvx, d, e, b[:-1], fact=fact, df=df, ef=ef) + + +@pytest.mark.parametrize("dtype,realtype", zip(DTYPES, REAL_DTYPES + + REAL_DTYPES)) +@pytest.mark.parametrize("fact,df_de_lambda", + [("F", + lambda d, e: get_lapack_funcs('pttrf', + dtype=e.dtype)(d, e)), + ("N", lambda d, e: (None, None, None))]) +def test_ptsvx_non_SPD_singular(dtype, realtype, fact, df_de_lambda): + rng = np.random.RandomState(42) + ptsvx = get_lapack_funcs('ptsvx', dtype=dtype) + n = 5 + # create diagonals according to size and dtype + d = generate_random_dtype_array((n,), realtype, rng) + 4 + e = generate_random_dtype_array((n-1,), dtype, rng) + A = np.diag(d) + np.diag(e, -1) + np.diag(np.conj(e), 1) + x_soln = generate_random_dtype_array((n, 2), dtype=dtype, rng=rng) + b = A @ x_soln + + # use lambda to determine what df, ef are + df, ef, info = df_de_lambda(d, e) + + if fact == "N": + d[3] = 0 + # obtain new df, ef + df, ef, info = df_de_lambda(d, e) + # solve using routine + df, ef, x, rcond, ferr, berr, info = ptsvx(d, e, b) + # test for the singular matrix. + assert info > 0 and info <= n + + # non SPD matrix + d = generate_random_dtype_array((n,), realtype, rng) + df, ef, x, rcond, ferr, berr, info = ptsvx(d, e, b) + assert info > 0 and info <= n + else: + # assuming that someone is using a singular factorization + df, ef, info = df_de_lambda(d, e) + df[0] = 0 + ef[0] = 0 + df, ef, x, rcond, ferr, berr, info = ptsvx(d, e, b, fact=fact, + df=df, ef=ef) + assert info > 0 + + +@pytest.mark.parametrize('d,e,b,x', + [(np.array([4, 10, 29, 25, 5]), + np.array([-2, -6, 15, 8]), + np.array([[6, 10], [9, 4], [2, 9], [14, 65], + [7, 23]]), + np.array([[2.5, 2], [2, -1], [1, -3], + [-1, 6], [3, -5]])), + (np.array([16, 41, 46, 21]), + np.array([16 + 16j, 18 - 9j, 1 - 4j]), + np.array([[64 + 16j, -16 - 32j], + [93 + 62j, 61 - 66j], + [78 - 80j, 71 - 74j], + [14 - 27j, 35 + 15j]]), + np.array([[2 + 1j, -3 - 2j], + [1 + 1j, 1 + 1j], + [1 - 2j, 1 - 2j], + [1 - 1j, 2 + 1j]]))]) +def test_ptsvx_NAG(d, e, b, x): + # test to assure that wrapper is consistent with NAG Manual Mark 26 + # example problems: f07jbf, f07jpf + # (Links expire, so please search for "NAG Library Manual Mark 26" online) + + # obtain routine with correct type based on e.dtype + ptsvx = get_lapack_funcs('ptsvx', dtype=e.dtype) + # solve using routine + df, ef, x_ptsvx, rcond, ferr, berr, info = ptsvx(d, e, b) + # determine ptsvx's solution and x are the same. + assert_array_almost_equal(x, x_ptsvx) + + +@pytest.mark.parametrize('lower', [False, True]) +@pytest.mark.parametrize('dtype', DTYPES) +def test_pptrs_pptri_pptrf_ppsv_ppcon(dtype, lower): + rng = np.random.RandomState(1234) + atol = np.finfo(dtype).eps*100 + # Manual conversion to/from packed format is feasible here. + n, nrhs = 10, 4 + a = generate_random_dtype_array([n, n], dtype=dtype, rng=rng) + b = generate_random_dtype_array([n, nrhs], dtype=dtype, rng=rng) + + a = a.conj().T + a + np.eye(n, dtype=dtype) * dtype(5.) + if lower: + inds = ([x for y in range(n) for x in range(y, n)], + [y for y in range(n) for x in range(y, n)]) + else: + inds = ([x for y in range(1, n+1) for x in range(y)], + [y-1 for y in range(1, n+1) for x in range(y)]) + ap = a[inds] + ppsv, pptrf, pptrs, pptri, ppcon = get_lapack_funcs( + ('ppsv', 'pptrf', 'pptrs', 'pptri', 'ppcon'), + dtype=dtype, + ilp64="preferred") + + ul, info = pptrf(n, ap, lower=lower) + assert_equal(info, 0) + aul = cholesky(a, lower=lower)[inds] + assert_allclose(ul, aul, rtol=0, atol=atol) + + uli, info = pptri(n, ul, lower=lower) + assert_equal(info, 0) + auli = inv(a)[inds] + assert_allclose(uli, auli, rtol=0, atol=atol) + + x, info = pptrs(n, ul, b, lower=lower) + assert_equal(info, 0) + bx = solve(a, b) + assert_allclose(x, bx, rtol=0, atol=atol) + + xv, info = ppsv(n, ap, b, lower=lower) + assert_equal(info, 0) + assert_allclose(xv, bx, rtol=0, atol=atol) + + anorm = np.linalg.norm(a, 1) + rcond, info = ppcon(n, ap, anorm=anorm, lower=lower) + assert_equal(info, 0) + assert_(abs(1/rcond - np.linalg.cond(a, p=1))*rcond < 1) + + +@pytest.mark.parametrize('dtype', DTYPES) +def test_gees_trexc(dtype): + rng = np.random.RandomState(1234) + atol = np.finfo(dtype).eps*100 + + n = 10 + a = generate_random_dtype_array([n, n], dtype=dtype, rng=rng) + + gees, trexc = get_lapack_funcs(('gees', 'trexc'), dtype=dtype) + + result = gees(lambda x: None, a, overwrite_a=False) + assert_equal(result[-1], 0) + + t = result[0] + z = result[-3] + + d2 = t[6, 6] + + if dtype in COMPLEX_DTYPES: + assert_allclose(t, np.triu(t), rtol=0, atol=atol) + + assert_allclose(z @ t @ z.conj().T, a, rtol=0, atol=atol) + + result = trexc(t, z, 7, 1) + assert_equal(result[-1], 0) + + t = result[0] + z = result[-2] + + if dtype in COMPLEX_DTYPES: + assert_allclose(t, np.triu(t), rtol=0, atol=atol) + + assert_allclose(z @ t @ z.conj().T, a, rtol=0, atol=atol) + + assert_allclose(t[0, 0], d2, rtol=0, atol=atol) + + +@pytest.mark.parametrize( + "t, expect, ifst, ilst", + [(np.array([[0.80, -0.11, 0.01, 0.03], + [0.00, -0.10, 0.25, 0.35], + [0.00, -0.65, -0.10, 0.20], + [0.00, 0.00, 0.00, -0.10]]), + np.array([[-0.1000, -0.6463, 0.0874, 0.2010], + [0.2514, -0.1000, 0.0927, 0.3505], + [0.0000, 0.0000, 0.8000, -0.0117], + [0.0000, 0.0000, 0.0000, -0.1000]]), + 2, 1), + (np.array([[-6.00 - 7.00j, 0.36 - 0.36j, -0.19 + 0.48j, 0.88 - 0.25j], + [0.00 + 0.00j, -5.00 + 2.00j, -0.03 - 0.72j, -0.23 + 0.13j], + [0.00 + 0.00j, 0.00 + 0.00j, 8.00 - 1.00j, 0.94 + 0.53j], + [0.00 + 0.00j, 0.00 + 0.00j, 0.00 + 0.00j, 3.00 - 4.00j]]), + np.array([[-5.0000 + 2.0000j, -0.1574 + 0.7143j, + 0.1781 - 0.1913j, 0.3950 + 0.3861j], + [0.0000 + 0.0000j, 8.0000 - 1.0000j, + 1.0742 + 0.1447j, 0.2515 - 0.3397j], + [0.0000 + 0.0000j, 0.0000 + 0.0000j, + 3.0000 - 4.0000j, 0.2264 + 0.8962j], + [0.0000 + 0.0000j, 0.0000 + 0.0000j, + 0.0000 + 0.0000j, -6.0000 - 7.0000j]]), + 1, 4)]) +def test_trexc_NAG(t, ifst, ilst, expect): + """ + This test implements the example found in the NAG manual, + f08qfc, f08qtc, f08qgc, f08quc. + """ + # NAG manual provides accuracy up to 4 decimals + atol = 1e-4 + trexc = get_lapack_funcs('trexc', dtype=t.dtype) + + result = trexc(t, t, ifst, ilst, wantq=0) + assert_equal(result[-1], 0) + + t = result[0] + assert_allclose(expect, t, atol=atol) + + +@pytest.mark.parametrize('dtype', DTYPES) +def test_gges_tgexc(dtype): + rng = np.random.RandomState(1234) + atol = np.finfo(dtype).eps*100 + + n = 10 + a = generate_random_dtype_array([n, n], dtype=dtype, rng=rng) + b = generate_random_dtype_array([n, n], dtype=dtype, rng=rng) + + gges, tgexc = get_lapack_funcs(('gges', 'tgexc'), dtype=dtype) + + result = gges(lambda x: None, a, b, overwrite_a=False, overwrite_b=False) + assert_equal(result[-1], 0) + + s = result[0] + t = result[1] + q = result[-4] + z = result[-3] + + d1 = s[0, 0] / t[0, 0] + d2 = s[6, 6] / t[6, 6] + + if dtype in COMPLEX_DTYPES: + assert_allclose(s, np.triu(s), rtol=0, atol=atol) + assert_allclose(t, np.triu(t), rtol=0, atol=atol) + + assert_allclose(q @ s @ z.conj().T, a, rtol=0, atol=atol) + assert_allclose(q @ t @ z.conj().T, b, rtol=0, atol=atol) + + result = tgexc(s, t, q, z, 7, 1) + assert_equal(result[-1], 0) + + s = result[0] + t = result[1] + q = result[2] + z = result[3] + + if dtype in COMPLEX_DTYPES: + assert_allclose(s, np.triu(s), rtol=0, atol=atol) + assert_allclose(t, np.triu(t), rtol=0, atol=atol) + + assert_allclose(q @ s @ z.conj().T, a, rtol=0, atol=atol) + assert_allclose(q @ t @ z.conj().T, b, rtol=0, atol=atol) + + assert_allclose(s[0, 0] / t[0, 0], d2, rtol=0, atol=atol) + assert_allclose(s[1, 1] / t[1, 1], d1, rtol=0, atol=atol) + + +@pytest.mark.parametrize('dtype', DTYPES) +def test_gees_trsen(dtype): + rng = np.random.RandomState(1234) + atol = np.finfo(dtype).eps*100 + + n = 10 + a = generate_random_dtype_array([n, n], dtype=dtype, rng=rng) + + gees, trsen, trsen_lwork = get_lapack_funcs( + ('gees', 'trsen', 'trsen_lwork'), dtype=dtype) + + result = gees(lambda x: None, a, overwrite_a=False) + assert_equal(result[-1], 0) + + t = result[0] + z = result[-3] + + d2 = t[6, 6] + + if dtype in COMPLEX_DTYPES: + assert_allclose(t, np.triu(t), rtol=0, atol=atol) + + assert_allclose(z @ t @ z.conj().T, a, rtol=0, atol=atol) + + select = np.zeros(n) + select[6] = 1 + + lwork = _compute_lwork(trsen_lwork, select, t) + + if dtype in COMPLEX_DTYPES: + result = trsen(select, t, z, lwork=lwork) + else: + result = trsen(select, t, z, lwork=lwork, liwork=lwork[1]) + assert_equal(result[-1], 0) + + t = result[0] + z = result[1] + + if dtype in COMPLEX_DTYPES: + assert_allclose(t, np.triu(t), rtol=0, atol=atol) + + assert_allclose(z @ t @ z.conj().T, a, rtol=0, atol=atol) + + assert_allclose(t[0, 0], d2, rtol=0, atol=atol) + + +@pytest.mark.parametrize( + "t, q, expect, select, expect_s, expect_sep", + [(np.array([[0.7995, -0.1144, 0.0060, 0.0336], + [0.0000, -0.0994, 0.2478, 0.3474], + [0.0000, -0.6483, -0.0994, 0.2026], + [0.0000, 0.0000, 0.0000, -0.1007]]), + np.array([[0.6551, 0.1037, 0.3450, 0.6641], + [0.5236, -0.5807, -0.6141, -0.1068], + [-0.5362, -0.3073, -0.2935, 0.7293], + [0.0956, 0.7467, -0.6463, 0.1249]]), + np.array([[0.3500, 0.4500, -0.1400, -0.1700], + [0.0900, 0.0700, -0.5399, 0.3500], + [-0.4400, -0.3300, -0.0300, 0.1700], + [0.2500, -0.3200, -0.1300, 0.1100]]), + np.array([1, 0, 0, 1]), + 1.75e+00, 3.22e+00), + (np.array([[-6.0004 - 6.9999j, 0.3637 - 0.3656j, + -0.1880 + 0.4787j, 0.8785 - 0.2539j], + [0.0000 + 0.0000j, -5.0000 + 2.0060j, + -0.0307 - 0.7217j, -0.2290 + 0.1313j], + [0.0000 + 0.0000j, 0.0000 + 0.0000j, + 7.9982 - 0.9964j, 0.9357 + 0.5359j], + [0.0000 + 0.0000j, 0.0000 + 0.0000j, + 0.0000 + 0.0000j, 3.0023 - 3.9998j]]), + np.array([[-0.8347 - 0.1364j, -0.0628 + 0.3806j, + 0.2765 - 0.0846j, 0.0633 - 0.2199j], + [0.0664 - 0.2968j, 0.2365 + 0.5240j, + -0.5877 - 0.4208j, 0.0835 + 0.2183j], + [-0.0362 - 0.3215j, 0.3143 - 0.5473j, + 0.0576 - 0.5736j, 0.0057 - 0.4058j], + [0.0086 + 0.2958j, -0.3416 - 0.0757j, + -0.1900 - 0.1600j, 0.8327 - 0.1868j]]), + np.array([[-3.9702 - 5.0406j, -4.1108 + 3.7002j, + -0.3403 + 1.0098j, 1.2899 - 0.8590j], + [0.3397 - 1.5006j, 1.5201 - 0.4301j, + 1.8797 - 5.3804j, 3.3606 + 0.6498j], + [3.3101 - 3.8506j, 2.4996 + 3.4504j, + 0.8802 - 1.0802j, 0.6401 - 1.4800j], + [-1.0999 + 0.8199j, 1.8103 - 1.5905j, + 3.2502 + 1.3297j, 1.5701 - 3.4397j]]), + np.array([1, 0, 0, 1]), + 1.02e+00, 1.82e-01)]) +def test_trsen_NAG(t, q, select, expect, expect_s, expect_sep): + """ + This test implements the example found in the NAG manual, + f08qgc, f08quc. + """ + # NAG manual provides accuracy up to 4 and 2 decimals + atol = 1e-4 + atol2 = 1e-2 + trsen, trsen_lwork = get_lapack_funcs( + ('trsen', 'trsen_lwork'), dtype=t.dtype) + + lwork = _compute_lwork(trsen_lwork, select, t) + + if t.dtype in COMPLEX_DTYPES: + result = trsen(select, t, q, lwork=lwork) + else: + result = trsen(select, t, q, lwork=lwork, liwork=lwork[1]) + assert_equal(result[-1], 0) + + t = result[0] + q = result[1] + if t.dtype in COMPLEX_DTYPES: + s = result[4] + sep = result[5] + else: + s = result[5] + sep = result[6] + + assert_allclose(expect, q @ t @ q.conj().T, atol=atol) + assert_allclose(expect_s, 1 / s, atol=atol2) + assert_allclose(expect_sep, 1 / sep, atol=atol2) + + +@pytest.mark.parametrize('dtype', DTYPES) +def test_gges_tgsen(dtype): + rng = np.random.RandomState(1234) + atol = np.finfo(dtype).eps*100 + + n = 10 + a = generate_random_dtype_array([n, n], dtype=dtype, rng=rng) + b = generate_random_dtype_array([n, n], dtype=dtype, rng=rng) + + gges, tgsen, tgsen_lwork = get_lapack_funcs( + ('gges', 'tgsen', 'tgsen_lwork'), dtype=dtype) + + result = gges(lambda x: None, a, b, overwrite_a=False, overwrite_b=False) + assert_equal(result[-1], 0) + + s = result[0] + t = result[1] + q = result[-4] + z = result[-3] + + d1 = s[0, 0] / t[0, 0] + d2 = s[6, 6] / t[6, 6] + + if dtype in COMPLEX_DTYPES: + assert_allclose(s, np.triu(s), rtol=0, atol=atol) + assert_allclose(t, np.triu(t), rtol=0, atol=atol) + + assert_allclose(q @ s @ z.conj().T, a, rtol=0, atol=atol) + assert_allclose(q @ t @ z.conj().T, b, rtol=0, atol=atol) + + select = np.zeros(n) + select[6] = 1 + + lwork = _compute_lwork(tgsen_lwork, select, s, t) + + # off-by-one error in LAPACK, see gh-issue #13397 + lwork = (lwork[0]+1, lwork[1]) + + result = tgsen(select, s, t, q, z, lwork=lwork) + assert_equal(result[-1], 0) + + s = result[0] + t = result[1] + q = result[-7] + z = result[-6] + + if dtype in COMPLEX_DTYPES: + assert_allclose(s, np.triu(s), rtol=0, atol=atol) + assert_allclose(t, np.triu(t), rtol=0, atol=atol) + + assert_allclose(q @ s @ z.conj().T, a, rtol=0, atol=atol) + assert_allclose(q @ t @ z.conj().T, b, rtol=0, atol=atol) + + assert_allclose(s[0, 0] / t[0, 0], d2, rtol=0, atol=atol) + assert_allclose(s[1, 1] / t[1, 1], d1, rtol=0, atol=atol) + + +@pytest.mark.parametrize( + "a, b, c, d, e, f, rans, lans", + [(np.array([[4.0, 1.0, 1.0, 2.0], + [0.0, 3.0, 4.0, 1.0], + [0.0, 1.0, 3.0, 1.0], + [0.0, 0.0, 0.0, 6.0]]), + np.array([[1.0, 1.0, 1.0, 1.0], + [0.0, 3.0, 4.0, 1.0], + [0.0, 1.0, 3.0, 1.0], + [0.0, 0.0, 0.0, 4.0]]), + np.array([[-4.0, 7.0, 1.0, 12.0], + [-9.0, 2.0, -2.0, -2.0], + [-4.0, 2.0, -2.0, 8.0], + [-7.0, 7.0, -6.0, 19.0]]), + np.array([[2.0, 1.0, 1.0, 3.0], + [0.0, 1.0, 2.0, 1.0], + [0.0, 0.0, 1.0, 1.0], + [0.0, 0.0, 0.0, 2.0]]), + np.array([[1.0, 1.0, 1.0, 2.0], + [0.0, 1.0, 4.0, 1.0], + [0.0, 0.0, 1.0, 1.0], + [0.0, 0.0, 0.0, 1.0]]), + np.array([[-7.0, 5.0, 0.0, 7.0], + [-5.0, 1.0, -8.0, 0.0], + [-1.0, 2.0, -3.0, 5.0], + [-3.0, 2.0, 0.0, 5.0]]), + np.array([[1.0, 1.0, 1.0, 1.0], + [-1.0, 2.0, -1.0, -1.0], + [-1.0, 1.0, 3.0, 1.0], + [-1.0, 1.0, -1.0, 4.0]]), + np.array([[4.0, -1.0, 1.0, -1.0], + [1.0, 3.0, -1.0, 1.0], + [-1.0, 1.0, 2.0, -1.0], + [1.0, -1.0, 1.0, 1.0]]))]) +@pytest.mark.parametrize('dtype', REAL_DTYPES) +def test_tgsyl_NAG(a, b, c, d, e, f, rans, lans, dtype): + atol = 1e-4 + + tgsyl = get_lapack_funcs(('tgsyl'), dtype=dtype) + rout, lout, scale, dif, info = tgsyl(a, b, c, d, e, f) + + assert_equal(info, 0) + assert_allclose(scale, 1.0, rtol=0, atol=np.finfo(dtype).eps*100, + err_msg="SCALE must be 1.0") + assert_allclose(dif, 0.0, rtol=0, atol=np.finfo(dtype).eps*100, + err_msg="DIF must be nearly 0") + assert_allclose(rout, rans, atol=atol, + err_msg="Solution for R is incorrect") + assert_allclose(lout, lans, atol=atol, + err_msg="Solution for L is incorrect") + + +@pytest.mark.parametrize('dtype', REAL_DTYPES) +@pytest.mark.parametrize('trans', ('N', 'T')) +@pytest.mark.parametrize('ijob', [0, 1, 2, 3, 4]) +def test_tgsyl(dtype, trans, ijob): + + atol = 1e-3 if dtype == np.float32 else 1e-10 + rng = np.random.default_rng(1685779866898198) + m, n = 10, 15 + + a, d, *_ = qz(rng.uniform(-10, 10, [m, m]).astype(dtype), + rng.uniform(-10, 10, [m, m]).astype(dtype), + output='real') + + b, e, *_ = qz(rng.uniform(-10, 10, [n, n]).astype(dtype), + rng.uniform(-10, 10, [n, n]).astype(dtype), + output='real') + + c = rng.uniform(-2, 2, [m, n]).astype(dtype) + f = rng.uniform(-2, 2, [m, n]).astype(dtype) + + tgsyl = get_lapack_funcs(('tgsyl'), dtype=dtype) + rout, lout, scale, dif, info = tgsyl(a, b, c, d, e, f, + trans=trans, ijob=ijob) + + assert info == 0, "INFO is non-zero" + assert scale >= 0.0, "SCALE must be non-negative" + if ijob == 0: + assert_allclose(dif, 0.0, rtol=0, atol=np.finfo(dtype).eps*100, + err_msg="DIF must be 0 for ijob =0") + else: + assert dif >= 0.0, "DIF must be non-negative" + + # Only DIF is calculated for ijob = 3/4 + if ijob <= 2: + if trans == 'N': + lhs1 = a @ rout - lout @ b + rhs1 = scale*c + lhs2 = d @ rout - lout @ e + rhs2 = scale*f + elif trans == 'T': + lhs1 = np.transpose(a) @ rout + np.transpose(d) @ lout + rhs1 = scale*c + lhs2 = rout @ np.transpose(b) + lout @ np.transpose(e) + rhs2 = -1.0*scale*f + + assert_allclose(lhs1, rhs1, atol=atol, rtol=0., + err_msg='lhs1 and rhs1 do not match') + assert_allclose(lhs2, rhs2, atol=atol, rtol=0., + err_msg='lhs2 and rhs2 do not match') + + +@pytest.mark.parametrize('mtype', ['sy', 'he']) # matrix type +@pytest.mark.parametrize('dtype', DTYPES) +@pytest.mark.parametrize('lower', (0, 1)) +def test_sy_hetrs(mtype, dtype, lower): + if mtype == 'he' and dtype in REAL_DTYPES: + pytest.skip("hetrs not for real dtypes.") + rng = np.random.default_rng(1723059677121834) + n, nrhs = 20, 5 + if dtype in COMPLEX_DTYPES: + A = (rng.uniform(size=(n, n)) + rng.uniform(size=(n, n))*1j).astype(dtype) + else: + A = rng.uniform(size=(n, n)).astype(dtype) + + A = A + A.T if mtype == 'sy' else A + A.conj().T + b = rng.uniform(size=(n, nrhs)).astype(dtype) + names = f'{mtype}trf', f'{mtype}trf_lwork', f'{mtype}trs' + trf, trf_lwork, trs = get_lapack_funcs(names, dtype=dtype) + lwork = trf_lwork(n, lower=lower) + ldu, ipiv, info = trf(A, lwork=lwork) + assert info == 0 + x, info = trs(a=ldu, ipiv=ipiv, b=b) + assert info == 0 + eps = np.finfo(dtype).eps + assert_allclose(A@x, b, atol=100*n*eps) + + +@pytest.mark.parametrize('norm', list('Mm1OoIiFfEe')) +@pytest.mark.parametrize('uplo, m, n', [('U', 5, 10), ('U', 10, 10), + ('L', 10, 5), ('L', 10, 10)]) +@pytest.mark.parametrize('diag', ['N', 'U']) +@pytest.mark.parametrize('dtype', DTYPES) +def test_lantr(norm, uplo, m, n, diag, dtype): + rng = np.random.default_rng(98426598246982456) + A = rng.random(size=(m, n)).astype(dtype) + lantr, lange = get_lapack_funcs(('lantr', 'lange'), (A,)) + res = lantr(norm, A, uplo=uplo, diag=diag) + + # now modify the matrix according to assumptions made by `lantr` + A = np.triu(A) if uplo == 'U' else np.tril(A) + if diag == 'U': + i = np.arange(min(m, n)) + A[i, i] = 1 + ref = lange(norm, A) + + assert_allclose(res, ref, rtol=2e-6) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_matfuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_matfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..9e87333c40b64d3f54024779dfb959be7a599178 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_matfuncs.py @@ -0,0 +1,1063 @@ +# +# Created by: Pearu Peterson, March 2002 +# +""" Test functions for linalg.matfuncs module + +""" +import functools + +import numpy as np +from numpy import array, identity, dot, sqrt +from numpy.testing import (assert_array_almost_equal, assert_allclose, assert_, + assert_array_less, assert_array_equal, assert_warns) +import pytest + +import scipy.linalg +from scipy.linalg import (funm, signm, logm, sqrtm, fractional_matrix_power, + expm, expm_frechet, expm_cond, norm, khatri_rao, + cosm, sinm, tanm, coshm, sinhm, tanhm) +from scipy.linalg import _matfuncs_inv_ssq +from scipy.linalg._matfuncs import pick_pade_structure +from scipy.linalg._matfuncs_inv_ssq import LogmExactlySingularWarning +import scipy.linalg._expm_frechet + +from scipy.optimize import minimize + + +def _get_al_mohy_higham_2012_experiment_1(): + """ + Return the test matrix from Experiment (1) of [1]_. + + References + ---------- + .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012) + "Improved Inverse Scaling and Squaring Algorithms + for the Matrix Logarithm." + SIAM Journal on Scientific Computing, 34 (4). C152-C169. + ISSN 1095-7197 + + """ + A = np.array([ + [3.2346e-1, 3e4, 3e4, 3e4], + [0, 3.0089e-1, 3e4, 3e4], + [0, 0, 3.2210e-1, 3e4], + [0, 0, 0, 3.0744e-1]], dtype=float) + return A + + +class TestSignM: + + def test_nils(self): + a = array([[29.2, -24.2, 69.5, 49.8, 7.], + [-9.2, 5.2, -18., -16.8, -2.], + [-10., 6., -20., -18., -2.], + [-9.6, 9.6, -25.5, -15.4, -2.], + [9.8, -4.8, 18., 18.2, 2.]]) + cr = array([[11.94933333,-2.24533333,15.31733333,21.65333333,-2.24533333], + [-3.84266667,0.49866667,-4.59066667,-7.18666667,0.49866667], + [-4.08,0.56,-4.92,-7.6,0.56], + [-4.03466667,1.04266667,-5.59866667,-7.02666667,1.04266667], + [4.15733333,-0.50133333,4.90933333,7.81333333,-0.50133333]]) + r = signm(a) + assert_array_almost_equal(r,cr) + + def test_defective1(self): + a = array([[0.0,1,0,0],[1,0,1,0],[0,0,0,1],[0,0,1,0]]) + signm(a, disp=False) + #XXX: what would be the correct result? + + def test_defective2(self): + a = array(( + [29.2,-24.2,69.5,49.8,7.0], + [-9.2,5.2,-18.0,-16.8,-2.0], + [-10.0,6.0,-20.0,-18.0,-2.0], + [-9.6,9.6,-25.5,-15.4,-2.0], + [9.8,-4.8,18.0,18.2,2.0])) + signm(a, disp=False) + #XXX: what would be the correct result? + + def test_defective3(self): + a = array([[-2., 25., 0., 0., 0., 0., 0.], + [0., -3., 10., 3., 3., 3., 0.], + [0., 0., 2., 15., 3., 3., 0.], + [0., 0., 0., 0., 15., 3., 0.], + [0., 0., 0., 0., 3., 10., 0.], + [0., 0., 0., 0., 0., -2., 25.], + [0., 0., 0., 0., 0., 0., -3.]]) + signm(a, disp=False) + #XXX: what would be the correct result? + + +class TestLogM: + + def test_nils(self): + a = array([[-2., 25., 0., 0., 0., 0., 0.], + [0., -3., 10., 3., 3., 3., 0.], + [0., 0., 2., 15., 3., 3., 0.], + [0., 0., 0., 0., 15., 3., 0.], + [0., 0., 0., 0., 3., 10., 0.], + [0., 0., 0., 0., 0., -2., 25.], + [0., 0., 0., 0., 0., 0., -3.]]) + m = (identity(7)*3.1+0j)-a + logm(m, disp=False) + #XXX: what would be the correct result? + + def test_al_mohy_higham_2012_experiment_1_logm(self): + # The logm completes the round trip successfully. + # Note that the expm leg of the round trip is badly conditioned. + A = _get_al_mohy_higham_2012_experiment_1() + A_logm, info = logm(A, disp=False) + A_round_trip = expm(A_logm) + assert_allclose(A_round_trip, A, rtol=5e-5, atol=1e-14) + + def test_al_mohy_higham_2012_experiment_1_funm_log(self): + # The raw funm with np.log does not complete the round trip. + # Note that the expm leg of the round trip is badly conditioned. + A = _get_al_mohy_higham_2012_experiment_1() + A_funm_log, info = funm(A, np.log, disp=False) + A_round_trip = expm(A_funm_log) + assert_(not np.allclose(A_round_trip, A, rtol=1e-5, atol=1e-14)) + + def test_round_trip_random_float(self): + np.random.seed(1234) + for n in range(1, 6): + M_unscaled = np.random.randn(n, n) + for scale in np.logspace(-4, 4, 9): + M = M_unscaled * scale + + # Eigenvalues are related to the branch cut. + W = np.linalg.eigvals(M) + err_msg = f'M:{M} eivals:{W}' + + # Check sqrtm round trip because it is used within logm. + M_sqrtm, info = sqrtm(M, disp=False) + M_sqrtm_round_trip = M_sqrtm.dot(M_sqrtm) + assert_allclose(M_sqrtm_round_trip, M) + + # Check logm round trip. + M_logm, info = logm(M, disp=False) + M_logm_round_trip = expm(M_logm) + assert_allclose(M_logm_round_trip, M, err_msg=err_msg) + + def test_round_trip_random_complex(self): + np.random.seed(1234) + for n in range(1, 6): + M_unscaled = np.random.randn(n, n) + 1j * np.random.randn(n, n) + for scale in np.logspace(-4, 4, 9): + M = M_unscaled * scale + M_logm, info = logm(M, disp=False) + M_round_trip = expm(M_logm) + assert_allclose(M_round_trip, M) + + def test_logm_type_preservation_and_conversion(self): + # The logm matrix function should preserve the type of a matrix + # whose eigenvalues are positive with zero imaginary part. + # Test this preservation for variously structured matrices. + complex_dtype_chars = ('F', 'D', 'G') + for matrix_as_list in ( + [[1, 0], [0, 1]], + [[1, 0], [1, 1]], + [[2, 1], [1, 1]], + [[2, 3], [1, 2]]): + + # check that the spectrum has the expected properties + W = scipy.linalg.eigvals(matrix_as_list) + assert_(not any(w.imag or w.real < 0 for w in W)) + + # check float type preservation + A = np.array(matrix_as_list, dtype=float) + A_logm, info = logm(A, disp=False) + assert_(A_logm.dtype.char not in complex_dtype_chars) + + # check complex type preservation + A = np.array(matrix_as_list, dtype=complex) + A_logm, info = logm(A, disp=False) + assert_(A_logm.dtype.char in complex_dtype_chars) + + # check float->complex type conversion for the matrix negation + A = -np.array(matrix_as_list, dtype=float) + A_logm, info = logm(A, disp=False) + assert_(A_logm.dtype.char in complex_dtype_chars) + + def test_complex_spectrum_real_logm(self): + # This matrix has complex eigenvalues and real logm. + # Its output dtype depends on its input dtype. + M = [[1, 1, 2], [2, 1, 1], [1, 2, 1]] + for dt in float, complex: + X = np.array(M, dtype=dt) + w = scipy.linalg.eigvals(X) + assert_(1e-2 < np.absolute(w.imag).sum()) + Y, info = logm(X, disp=False) + assert_(np.issubdtype(Y.dtype, np.inexact)) + assert_allclose(expm(Y), X) + + def test_real_mixed_sign_spectrum(self): + # These matrices have real eigenvalues with mixed signs. + # The output logm dtype is complex, regardless of input dtype. + for M in ( + [[1, 0], [0, -1]], + [[0, 1], [1, 0]]): + for dt in float, complex: + A = np.array(M, dtype=dt) + A_logm, info = logm(A, disp=False) + assert_(np.issubdtype(A_logm.dtype, np.complexfloating)) + + @pytest.mark.thread_unsafe + def test_exactly_singular(self): + A = np.array([[0, 0], [1j, 1j]]) + B = np.asarray([[1, 1], [0, 0]]) + for M in A, A.T, B, B.T: + expected_warning = _matfuncs_inv_ssq.LogmExactlySingularWarning + L, info = assert_warns(expected_warning, logm, M, disp=False) + E = expm(L) + assert_allclose(E, M, atol=1e-14) + + @pytest.mark.thread_unsafe + def test_nearly_singular(self): + M = np.array([[1e-100]]) + expected_warning = _matfuncs_inv_ssq.LogmNearlySingularWarning + L, info = assert_warns(expected_warning, logm, M, disp=False) + E = expm(L) + assert_allclose(E, M, atol=1e-14) + + def test_opposite_sign_complex_eigenvalues(self): + # See gh-6113 + E = [[0, 1], [-1, 0]] + L = [[0, np.pi*0.5], [-np.pi*0.5, 0]] + assert_allclose(expm(L), E, atol=1e-14) + assert_allclose(logm(E), L, atol=1e-14) + E = [[1j, 4], [0, -1j]] + L = [[1j*np.pi*0.5, 2*np.pi], [0, -1j*np.pi*0.5]] + assert_allclose(expm(L), E, atol=1e-14) + assert_allclose(logm(E), L, atol=1e-14) + E = [[1j, 0], [0, -1j]] + L = [[1j*np.pi*0.5, 0], [0, -1j*np.pi*0.5]] + assert_allclose(expm(L), E, atol=1e-14) + assert_allclose(logm(E), L, atol=1e-14) + + def test_readonly(self): + n = 5 + a = np.ones((n, n)) + np.identity(n) + a.flags.writeable = False + logm(a) + + @pytest.mark.xfail(reason="ValueError: attempt to get argmax of an empty sequence") + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + log_a = logm(a) + a0 = np.eye(2, dtype=dt) + log_a0 = logm(a0) + + assert log_a.shape == (0, 0) + assert log_a.dtype == log_a0.dtype + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize('dtype', [int, float, np.float32, complex, np.complex64]) + def test_no_ZeroDivisionError(self, dtype): + # gh-17136 reported inconsistent behavior in `logm` depending on input dtype: + # sometimes it raised an error, and sometimes it printed a warning message. + # check that this is resolved and that the warning is emitted properly. + with (pytest.warns(RuntimeWarning, match="logm result may be inaccurate"), + pytest.warns(LogmExactlySingularWarning)): + logm(np.zeros((2, 2), dtype=dtype)) + + +class TestSqrtM: + def test_round_trip_random_float(self): + rng = np.random.RandomState(1234) + for n in range(1, 6): + M_unscaled = rng.randn(n, n) + for scale in np.logspace(-4, 4, 9): + M = M_unscaled * scale + M_sqrtm, info = sqrtm(M, disp=False) + M_sqrtm_round_trip = M_sqrtm.dot(M_sqrtm) + assert_allclose(M_sqrtm_round_trip, M) + + def test_round_trip_random_complex(self): + rng = np.random.RandomState(1234) + for n in range(1, 6): + M_unscaled = rng.randn(n, n) + 1j * rng.randn(n, n) + for scale in np.logspace(-4, 4, 9): + M = M_unscaled * scale + M_sqrtm, info = sqrtm(M, disp=False) + M_sqrtm_round_trip = M_sqrtm.dot(M_sqrtm) + assert_allclose(M_sqrtm_round_trip, M) + + def test_bad(self): + # See https://web.archive.org/web/20051220232650/http://www.maths.man.ac.uk/~nareports/narep336.ps.gz + e = 2**-5 + se = sqrt(e) + a = array([[1.0,0,0,1], + [0,e,0,0], + [0,0,e,0], + [0,0,0,1]]) + sa = array([[1,0,0,0.5], + [0,se,0,0], + [0,0,se,0], + [0,0,0,1]]) + n = a.shape[0] + assert_array_almost_equal(dot(sa,sa),a) + # Check default sqrtm. + esa = sqrtm(a, disp=False, blocksize=n)[0] + assert_array_almost_equal(dot(esa,esa),a) + # Check sqrtm with 2x2 blocks. + esa = sqrtm(a, disp=False, blocksize=2)[0] + assert_array_almost_equal(dot(esa,esa),a) + + def test_sqrtm_type_preservation_and_conversion(self): + # The sqrtm matrix function should preserve the type of a matrix + # whose eigenvalues are nonnegative with zero imaginary part. + # Test this preservation for variously structured matrices. + complex_dtype_chars = ('F', 'D', 'G') + for matrix_as_list in ( + [[1, 0], [0, 1]], + [[1, 0], [1, 1]], + [[2, 1], [1, 1]], + [[2, 3], [1, 2]], + [[1, 1], [1, 1]]): + + # check that the spectrum has the expected properties + W = scipy.linalg.eigvals(matrix_as_list) + assert_(not any(w.imag or w.real < 0 for w in W)) + + # check float type preservation + A = np.array(matrix_as_list, dtype=float) + A_sqrtm, info = sqrtm(A, disp=False) + assert_(A_sqrtm.dtype.char not in complex_dtype_chars) + + # check complex type preservation + A = np.array(matrix_as_list, dtype=complex) + A_sqrtm, info = sqrtm(A, disp=False) + assert_(A_sqrtm.dtype.char in complex_dtype_chars) + + # check float->complex type conversion for the matrix negation + A = -np.array(matrix_as_list, dtype=float) + A_sqrtm, info = sqrtm(A, disp=False) + assert_(A_sqrtm.dtype.char in complex_dtype_chars) + + def test_sqrtm_type_conversion_mixed_sign_or_complex_spectrum(self): + complex_dtype_chars = ('F', 'D', 'G') + for matrix_as_list in ( + [[1, 0], [0, -1]], + [[0, 1], [1, 0]], + [[0, 1, 0], [0, 0, 1], [1, 0, 0]]): + + # check that the spectrum has the expected properties + W = scipy.linalg.eigvals(matrix_as_list) + assert_(any(w.imag or w.real < 0 for w in W)) + + # check complex->complex + A = np.array(matrix_as_list, dtype=complex) + A_sqrtm, info = sqrtm(A, disp=False) + assert_(A_sqrtm.dtype.char in complex_dtype_chars) + + # check float->complex + A = np.array(matrix_as_list, dtype=float) + A_sqrtm, info = sqrtm(A, disp=False) + assert_(A_sqrtm.dtype.char in complex_dtype_chars) + + def test_blocksizes(self): + # Make sure I do not goof up the blocksizes when they do not divide n. + np.random.seed(1234) + for n in range(1, 8): + A = np.random.rand(n, n) + 1j*np.random.randn(n, n) + A_sqrtm_default, info = sqrtm(A, disp=False, blocksize=n) + assert_allclose(A, np.linalg.matrix_power(A_sqrtm_default, 2)) + for blocksize in range(1, 10): + A_sqrtm_new, info = sqrtm(A, disp=False, blocksize=blocksize) + assert_allclose(A_sqrtm_default, A_sqrtm_new) + + def test_al_mohy_higham_2012_experiment_1(self): + # Matrix square root of a tricky upper triangular matrix. + A = _get_al_mohy_higham_2012_experiment_1() + A_sqrtm, info = sqrtm(A, disp=False) + A_round_trip = A_sqrtm.dot(A_sqrtm) + assert_allclose(A_round_trip, A, rtol=1e-5) + assert_allclose(np.tril(A_round_trip), np.tril(A)) + + def test_strict_upper_triangular(self): + # This matrix has no square root. + for dt in int, float: + A = np.array([ + [0, 3, 0, 0], + [0, 0, 3, 0], + [0, 0, 0, 3], + [0, 0, 0, 0]], dtype=dt) + A_sqrtm, info = sqrtm(A, disp=False) + assert_(np.isnan(A_sqrtm).all()) + + def test_weird_matrix(self): + # The square root of matrix B exists. + for dt in int, float: + A = np.array([ + [0, 0, 1], + [0, 0, 0], + [0, 1, 0]], dtype=dt) + B = np.array([ + [0, 1, 0], + [0, 0, 0], + [0, 0, 0]], dtype=dt) + assert_array_equal(B, A.dot(A)) + + # But scipy sqrtm is not clever enough to find it. + B_sqrtm, info = sqrtm(B, disp=False) + assert_(np.isnan(B_sqrtm).all()) + + def test_disp(self): + np.random.seed(1234) + + A = np.random.rand(3, 3) + B = sqrtm(A, disp=True) + assert_allclose(B.dot(B), A) + + def test_opposite_sign_complex_eigenvalues(self): + M = [[2j, 4], [0, -2j]] + R = [[1+1j, 2], [0, 1-1j]] + assert_allclose(np.dot(R, R), M, atol=1e-14) + assert_allclose(sqrtm(M), R, atol=1e-14) + + def test_gh4866(self): + M = np.array([[1, 0, 0, 1], + [0, 0, 0, 0], + [0, 0, 0, 0], + [1, 0, 0, 1]]) + R = np.array([[sqrt(0.5), 0, 0, sqrt(0.5)], + [0, 0, 0, 0], + [0, 0, 0, 0], + [sqrt(0.5), 0, 0, sqrt(0.5)]]) + assert_allclose(np.dot(R, R), M, atol=1e-14) + assert_allclose(sqrtm(M), R, atol=1e-14) + + def test_gh5336(self): + M = np.diag([2, 1, 0]) + R = np.diag([sqrt(2), 1, 0]) + assert_allclose(np.dot(R, R), M, atol=1e-14) + assert_allclose(sqrtm(M), R, atol=1e-14) + + def test_gh7839(self): + M = np.zeros((2, 2)) + R = np.zeros((2, 2)) + assert_allclose(np.dot(R, R), M, atol=1e-14) + assert_allclose(sqrtm(M), R, atol=1e-14) + + @pytest.mark.xfail(reason="failing on macOS after gh-20212") + def test_gh17918(self): + M = np.empty((19, 19)) + M.fill(0.94) + np.fill_diagonal(M, 1) + assert np.isrealobj(sqrtm(M)) + + def test_data_size_preservation_uint_in_float_out(self): + M = np.zeros((10, 10), dtype=np.uint8) + assert sqrtm(M).dtype == np.float64 + M = np.zeros((10, 10), dtype=np.uint16) + assert sqrtm(M).dtype == np.float64 + M = np.zeros((10, 10), dtype=np.uint32) + assert sqrtm(M).dtype == np.float64 + M = np.zeros((10, 10), dtype=np.uint64) + assert sqrtm(M).dtype == np.float64 + + def test_data_size_preservation_int_in_float_out(self): + M = np.zeros((10, 10), dtype=np.int8) + assert sqrtm(M).dtype == np.float64 + M = np.zeros((10, 10), dtype=np.int16) + assert sqrtm(M).dtype == np.float64 + M = np.zeros((10, 10), dtype=np.int32) + assert sqrtm(M).dtype == np.float64 + M = np.zeros((10, 10), dtype=np.int64) + assert sqrtm(M).dtype == np.float64 + + def test_data_size_preservation_int_in_comp_out(self): + M = np.array([[2, 4], [0, -2]], dtype=np.int8) + assert sqrtm(M).dtype == np.complex128 + M = np.array([[2, 4], [0, -2]], dtype=np.int16) + assert sqrtm(M).dtype == np.complex128 + M = np.array([[2, 4], [0, -2]], dtype=np.int32) + assert sqrtm(M).dtype == np.complex128 + M = np.array([[2, 4], [0, -2]], dtype=np.int64) + assert sqrtm(M).dtype == np.complex128 + + def test_data_size_preservation_float_in_float_out(self): + M = np.zeros((10, 10), dtype=np.float16) + assert sqrtm(M).dtype == np.float32 + M = np.zeros((10, 10), dtype=np.float32) + assert sqrtm(M).dtype == np.float32 + M = np.zeros((10, 10), dtype=np.float64) + assert sqrtm(M).dtype == np.float64 + if hasattr(np, 'float128'): + M = np.zeros((10, 10), dtype=np.float128) + assert sqrtm(M).dtype == np.float64 + + def test_data_size_preservation_float_in_comp_out(self): + M = np.array([[2, 4], [0, -2]], dtype=np.float16) + assert sqrtm(M).dtype == np.complex64 + M = np.array([[2, 4], [0, -2]], dtype=np.float32) + assert sqrtm(M).dtype == np.complex64 + M = np.array([[2, 4], [0, -2]], dtype=np.float64) + assert sqrtm(M).dtype == np.complex128 + if hasattr(np, 'float128') and hasattr(np, 'complex256'): + M = np.array([[2, 4], [0, -2]], dtype=np.float128) + assert sqrtm(M).dtype == np.complex128 + + def test_data_size_preservation_comp_in_comp_out(self): + M = np.array([[2j, 4], [0, -2j]], dtype=np.complex64) + assert sqrtm(M).dtype == np.complex64 + M = np.array([[2j, 4], [0, -2j]], dtype=np.complex128) + assert sqrtm(M).dtype == np.complex128 + if hasattr(np, 'complex256'): + M = np.array([[2j, 4], [0, -2j]], dtype=np.complex256) + assert sqrtm(M).dtype == np.complex128 + + @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64]) + def test_empty(self, dt): + a = np.empty((0, 0), dtype=dt) + s = sqrtm(a) + a0 = np.eye(2, dtype=dt) + s0 = sqrtm(a0) + + assert s.shape == (0, 0) + assert s.dtype == s0.dtype + + +class TestFractionalMatrixPower: + def test_round_trip_random_complex(self): + np.random.seed(1234) + for p in range(1, 5): + for n in range(1, 5): + M_unscaled = np.random.randn(n, n) + 1j * np.random.randn(n, n) + for scale in np.logspace(-4, 4, 9): + M = M_unscaled * scale + M_root = fractional_matrix_power(M, 1/p) + M_round_trip = np.linalg.matrix_power(M_root, p) + assert_allclose(M_round_trip, M) + + def test_round_trip_random_float(self): + # This test is more annoying because it can hit the branch cut; + # this happens when the matrix has an eigenvalue + # with no imaginary component and with a real negative component, + # and it means that the principal branch does not exist. + np.random.seed(1234) + for p in range(1, 5): + for n in range(1, 5): + M_unscaled = np.random.randn(n, n) + for scale in np.logspace(-4, 4, 9): + M = M_unscaled * scale + M_root = fractional_matrix_power(M, 1/p) + M_round_trip = np.linalg.matrix_power(M_root, p) + assert_allclose(M_round_trip, M) + + def test_larger_abs_fractional_matrix_powers(self): + np.random.seed(1234) + for n in (2, 3, 5): + for i in range(10): + M = np.random.randn(n, n) + 1j * np.random.randn(n, n) + M_one_fifth = fractional_matrix_power(M, 0.2) + # Test the round trip. + M_round_trip = np.linalg.matrix_power(M_one_fifth, 5) + assert_allclose(M, M_round_trip) + # Test a large abs fractional power. + X = fractional_matrix_power(M, -5.4) + Y = np.linalg.matrix_power(M_one_fifth, -27) + assert_allclose(X, Y) + # Test another large abs fractional power. + X = fractional_matrix_power(M, 3.8) + Y = np.linalg.matrix_power(M_one_fifth, 19) + assert_allclose(X, Y) + + def test_random_matrices_and_powers(self): + # Each independent iteration of this fuzz test picks random parameters. + # It tries to hit some edge cases. + rng = np.random.default_rng(1726500458620605) + nsamples = 20 + for i in range(nsamples): + # Sample a matrix size and a random real power. + n = rng.integers(1, 5) + p = rng.random() + + # Sample a random real or complex matrix. + matrix_scale = np.exp(rng.integers(-4, 5)) + A = rng.random(size=[n, n]) + if [True, False][rng.choice(2)]: + A = A + 1j * rng.random(size=[n, n]) + A = A * matrix_scale + + # Check a couple of analytically equivalent ways + # to compute the fractional matrix power. + # These can be compared because they both use the principal branch. + A_power = fractional_matrix_power(A, p) + A_logm, info = logm(A, disp=False) + A_power_expm_logm = expm(A_logm * p) + assert_allclose(A_power, A_power_expm_logm) + + def test_al_mohy_higham_2012_experiment_1(self): + # Fractional powers of a tricky upper triangular matrix. + A = _get_al_mohy_higham_2012_experiment_1() + + # Test remainder matrix power. + A_funm_sqrt, info = funm(A, np.sqrt, disp=False) + A_sqrtm, info = sqrtm(A, disp=False) + A_rem_power = _matfuncs_inv_ssq._remainder_matrix_power(A, 0.5) + A_power = fractional_matrix_power(A, 0.5) + assert_allclose(A_rem_power, A_power, rtol=1e-11) + assert_allclose(A_sqrtm, A_power) + assert_allclose(A_sqrtm, A_funm_sqrt) + + # Test more fractional powers. + for p in (1/2, 5/3): + A_power = fractional_matrix_power(A, p) + A_round_trip = fractional_matrix_power(A_power, 1/p) + assert_allclose(A_round_trip, A, rtol=1e-2) + assert_allclose(np.tril(A_round_trip, 1), np.tril(A, 1)) + + def test_briggs_helper_function(self): + np.random.seed(1234) + for a in np.random.randn(10) + 1j * np.random.randn(10): + for k in range(5): + x_observed = _matfuncs_inv_ssq._briggs_helper_function(a, k) + x_expected = a ** np.exp2(-k) - 1 + assert_allclose(x_observed, x_expected) + + def test_type_preservation_and_conversion(self): + # The fractional_matrix_power matrix function should preserve + # the type of a matrix whose eigenvalues + # are positive with zero imaginary part. + # Test this preservation for variously structured matrices. + complex_dtype_chars = ('F', 'D', 'G') + for matrix_as_list in ( + [[1, 0], [0, 1]], + [[1, 0], [1, 1]], + [[2, 1], [1, 1]], + [[2, 3], [1, 2]]): + + # check that the spectrum has the expected properties + W = scipy.linalg.eigvals(matrix_as_list) + assert_(not any(w.imag or w.real < 0 for w in W)) + + # Check various positive and negative powers + # with absolute values bigger and smaller than 1. + for p in (-2.4, -0.9, 0.2, 3.3): + + # check float type preservation + A = np.array(matrix_as_list, dtype=float) + A_power = fractional_matrix_power(A, p) + assert_(A_power.dtype.char not in complex_dtype_chars) + + # check complex type preservation + A = np.array(matrix_as_list, dtype=complex) + A_power = fractional_matrix_power(A, p) + assert_(A_power.dtype.char in complex_dtype_chars) + + # check float->complex for the matrix negation + A = -np.array(matrix_as_list, dtype=float) + A_power = fractional_matrix_power(A, p) + assert_(A_power.dtype.char in complex_dtype_chars) + + def test_type_conversion_mixed_sign_or_complex_spectrum(self): + complex_dtype_chars = ('F', 'D', 'G') + for matrix_as_list in ( + [[1, 0], [0, -1]], + [[0, 1], [1, 0]], + [[0, 1, 0], [0, 0, 1], [1, 0, 0]]): + + # check that the spectrum has the expected properties + W = scipy.linalg.eigvals(matrix_as_list) + assert_(any(w.imag or w.real < 0 for w in W)) + + # Check various positive and negative powers + # with absolute values bigger and smaller than 1. + for p in (-2.4, -0.9, 0.2, 3.3): + + # check complex->complex + A = np.array(matrix_as_list, dtype=complex) + A_power = fractional_matrix_power(A, p) + assert_(A_power.dtype.char in complex_dtype_chars) + + # check float->complex + A = np.array(matrix_as_list, dtype=float) + A_power = fractional_matrix_power(A, p) + assert_(A_power.dtype.char in complex_dtype_chars) + + @pytest.mark.xfail(reason='Too unstable across LAPACKs.') + def test_singular(self): + # Negative fractional powers do not work with singular matrices. + for matrix_as_list in ( + [[0, 0], [0, 0]], + [[1, 1], [1, 1]], + [[1, 2], [3, 6]], + [[0, 0, 0], [0, 1, 1], [0, -1, 1]]): + + # Check fractional powers both for float and for complex types. + for newtype in (float, complex): + A = np.array(matrix_as_list, dtype=newtype) + for p in (-0.7, -0.9, -2.4, -1.3): + A_power = fractional_matrix_power(A, p) + assert_(np.isnan(A_power).all()) + for p in (0.2, 1.43): + A_power = fractional_matrix_power(A, p) + A_round_trip = fractional_matrix_power(A_power, 1/p) + assert_allclose(A_round_trip, A) + + def test_opposite_sign_complex_eigenvalues(self): + M = [[2j, 4], [0, -2j]] + R = [[1+1j, 2], [0, 1-1j]] + assert_allclose(np.dot(R, R), M, atol=1e-14) + assert_allclose(fractional_matrix_power(M, 0.5), R, atol=1e-14) + + +class TestExpM: + def test_zero(self): + a = array([[0.,0],[0,0]]) + assert_array_almost_equal(expm(a),[[1,0],[0,1]]) + + def test_single_elt(self): + elt = expm(1) + assert_allclose(elt, np.array([[np.e]])) + + @pytest.mark.parametrize('func', [expm, cosm, sinm, tanm, coshm, sinhm, tanhm]) + @pytest.mark.parametrize('dt',[int, float, np.float32, complex, np.complex64]) + @pytest.mark.parametrize('shape', [(0, 0), (1, 1)]) + def test_small_empty_matrix_input(self, func, dt, shape): + # regression test for gh-11082 / gh-20372 - test behavior of expm + # and related functions for small and zero-sized arrays. + A = np.zeros(shape, dtype=dt) + A0 = np.zeros((10, 10), dtype=dt) + result = func(A) + result0 = func(A0) + assert result.shape == shape + assert result.dtype == result0.dtype + + def test_2x2_input(self): + E = np.e + a = array([[1, 4], [1, 1]]) + aa = (E**4 + 1)/(2*E) + bb = (E**4 - 1)/E + assert_allclose(expm(a), array([[aa, bb], [bb/4, aa]])) + assert expm(a.astype(np.complex64)).dtype.char == 'F' + assert expm(a.astype(np.float32)).dtype.char == 'f' + + def test_nx2x2_input(self): + E = np.e + # These are integer matrices with integer eigenvalues + a = np.array([[[1, 4], [1, 1]], + [[1, 3], [1, -1]], + [[1, 3], [4, 5]], + [[1, 3], [5, 3]], + [[4, 5], [-3, -4]]], order='F') + # Exact results are computed symbolically + a_res = np.array([ + [[(E**4+1)/(2*E), (E**4-1)/E], + [(E**4-1)/4/E, (E**4+1)/(2*E)]], + [[1/(4*E**2)+(3*E**2)/4, (3*E**2)/4-3/(4*E**2)], + [E**2/4-1/(4*E**2), 3/(4*E**2)+E**2/4]], + [[3/(4*E)+E**7/4, -3/(8*E)+(3*E**7)/8], + [-1/(2*E)+E**7/2, 1/(4*E)+(3*E**7)/4]], + [[5/(8*E**2)+(3*E**6)/8, -3/(8*E**2)+(3*E**6)/8], + [-5/(8*E**2)+(5*E**6)/8, 3/(8*E**2)+(5*E**6)/8]], + [[-3/(2*E)+(5*E)/2, -5/(2*E)+(5*E)/2], + [3/(2*E)-(3*E)/2, 5/(2*E)-(3*E)/2]] + ]) + assert_allclose(expm(a), a_res) + + def test_readonly(self): + n = 7 + a = np.ones((n, n)) + a.flags.writeable = False + expm(a) + + @pytest.mark.thread_unsafe + @pytest.mark.fail_slow(5) + def test_gh18086(self): + A = np.zeros((400, 400), dtype=float) + rng = np.random.default_rng(100) + i = rng.integers(0, 399, 500) + j = rng.integers(0, 399, 500) + A[i, j] = rng.random(500) + # Problem appears when m = 9 + Am = np.empty((5, 400, 400), dtype=float) + Am[0] = A.copy() + m, s = pick_pade_structure(Am) + assert m == 9 + # Check that result is accurate + first_res = expm(A) + np.testing.assert_array_almost_equal(logm(first_res), A) + # Check that result is consistent + for i in range(5): + next_res = expm(A) + np.testing.assert_array_almost_equal(first_res, next_res) + + +class TestExpmFrechet: + + def test_expm_frechet(self): + # a test of the basic functionality + M = np.array([ + [1, 2, 3, 4], + [5, 6, 7, 8], + [0, 0, 1, 2], + [0, 0, 5, 6], + ], dtype=float) + A = np.array([ + [1, 2], + [5, 6], + ], dtype=float) + E = np.array([ + [3, 4], + [7, 8], + ], dtype=float) + expected_expm = scipy.linalg.expm(A) + expected_frechet = scipy.linalg.expm(M)[:2, 2:] + for kwargs in ({}, {'method':'SPS'}, {'method':'blockEnlarge'}): + observed_expm, observed_frechet = expm_frechet(A, E, **kwargs) + assert_allclose(expected_expm, observed_expm) + assert_allclose(expected_frechet, observed_frechet) + + def test_small_norm_expm_frechet(self): + # methodically test matrices with a range of norms, for better coverage + M_original = np.array([ + [1, 2, 3, 4], + [5, 6, 7, 8], + [0, 0, 1, 2], + [0, 0, 5, 6], + ], dtype=float) + A_original = np.array([ + [1, 2], + [5, 6], + ], dtype=float) + E_original = np.array([ + [3, 4], + [7, 8], + ], dtype=float) + A_original_norm_1 = scipy.linalg.norm(A_original, 1) + selected_m_list = [1, 3, 5, 7, 9, 11, 13, 15] + m_neighbor_pairs = zip(selected_m_list[:-1], selected_m_list[1:]) + for ma, mb in m_neighbor_pairs: + ell_a = scipy.linalg._expm_frechet.ell_table_61[ma] + ell_b = scipy.linalg._expm_frechet.ell_table_61[mb] + target_norm_1 = 0.5 * (ell_a + ell_b) + scale = target_norm_1 / A_original_norm_1 + M = scale * M_original + A = scale * A_original + E = scale * E_original + expected_expm = scipy.linalg.expm(A) + expected_frechet = scipy.linalg.expm(M)[:2, 2:] + observed_expm, observed_frechet = expm_frechet(A, E) + assert_allclose(expected_expm, observed_expm) + assert_allclose(expected_frechet, observed_frechet) + + def test_fuzz(self): + rng = np.random.default_rng(1726500908359153) + # try a bunch of crazy inputs + rfuncs = ( + np.random.uniform, + np.random.normal, + np.random.standard_cauchy, + np.random.exponential) + ntests = 100 + for i in range(ntests): + rfunc = rfuncs[rng.choice(4)] + target_norm_1 = rng.exponential() + n = rng.integers(2, 16) + A_original = rfunc(size=(n,n)) + E_original = rfunc(size=(n,n)) + A_original_norm_1 = scipy.linalg.norm(A_original, 1) + scale = target_norm_1 / A_original_norm_1 + A = scale * A_original + E = scale * E_original + M = np.vstack([ + np.hstack([A, E]), + np.hstack([np.zeros_like(A), A])]) + expected_expm = scipy.linalg.expm(A) + expected_frechet = scipy.linalg.expm(M)[:n, n:] + observed_expm, observed_frechet = expm_frechet(A, E) + assert_allclose(expected_expm, observed_expm, atol=5e-8) + assert_allclose(expected_frechet, observed_frechet, atol=1e-7) + + def test_problematic_matrix(self): + # this test case uncovered a bug which has since been fixed + A = np.array([ + [1.50591997, 1.93537998], + [0.41203263, 0.23443516], + ], dtype=float) + E = np.array([ + [1.87864034, 2.07055038], + [1.34102727, 0.67341123], + ], dtype=float) + scipy.linalg.norm(A, 1) + sps_expm, sps_frechet = expm_frechet( + A, E, method='SPS') + blockEnlarge_expm, blockEnlarge_frechet = expm_frechet( + A, E, method='blockEnlarge') + assert_allclose(sps_expm, blockEnlarge_expm) + assert_allclose(sps_frechet, blockEnlarge_frechet) + + @pytest.mark.slow + @pytest.mark.skip(reason='this test is deliberately slow') + def test_medium_matrix(self): + # profile this to see the speed difference + n = 1000 + A = np.random.exponential(size=(n, n)) + E = np.random.exponential(size=(n, n)) + sps_expm, sps_frechet = expm_frechet( + A, E, method='SPS') + blockEnlarge_expm, blockEnlarge_frechet = expm_frechet( + A, E, method='blockEnlarge') + assert_allclose(sps_expm, blockEnlarge_expm) + assert_allclose(sps_frechet, blockEnlarge_frechet) + + +def _help_expm_cond_search(A, A_norm, X, X_norm, eps, p): + p = np.reshape(p, A.shape) + p_norm = norm(p) + perturbation = eps * p * (A_norm / p_norm) + X_prime = expm(A + perturbation) + scaled_relative_error = norm(X_prime - X) / (X_norm * eps) + return -scaled_relative_error + + +def _normalized_like(A, B): + return A * (scipy.linalg.norm(B) / scipy.linalg.norm(A)) + + +def _relative_error(f, A, perturbation): + X = f(A) + X_prime = f(A + perturbation) + return norm(X_prime - X) / norm(X) + + +class TestExpmConditionNumber: + def test_expm_cond_smoke(self): + np.random.seed(1234) + for n in range(1, 4): + A = np.random.randn(n, n) + kappa = expm_cond(A) + assert_array_less(0, kappa) + + def test_expm_bad_condition_number(self): + A = np.array([ + [-1.128679820, 9.614183771e4, -4.524855739e9, 2.924969411e14], + [0, -1.201010529, 9.634696872e4, -4.681048289e9], + [0, 0, -1.132893222, 9.532491830e4], + [0, 0, 0, -1.179475332], + ]) + kappa = expm_cond(A) + assert_array_less(1e36, kappa) + + def test_univariate(self): + np.random.seed(12345) + for x in np.linspace(-5, 5, num=11): + A = np.array([[x]]) + assert_allclose(expm_cond(A), abs(x)) + for x in np.logspace(-2, 2, num=11): + A = np.array([[x]]) + assert_allclose(expm_cond(A), abs(x)) + for i in range(10): + A = np.random.randn(1, 1) + assert_allclose(expm_cond(A), np.absolute(A)[0, 0]) + + @pytest.mark.slow + def test_expm_cond_fuzz(self): + rng = np.random.RandomState(12345) + eps = 1e-5 + nsamples = 10 + for i in range(nsamples): + n = rng.randint(2, 5) + A = rng.randn(n, n) + A_norm = scipy.linalg.norm(A) + X = expm(A) + X_norm = scipy.linalg.norm(X) + kappa = expm_cond(A) + + # Look for the small perturbation that gives the greatest + # relative error. + f = functools.partial(_help_expm_cond_search, + A, A_norm, X, X_norm, eps) + guess = np.ones(n*n) + out = minimize(f, guess, method='L-BFGS-B') + xopt = out.x + yopt = f(xopt) + p_best = eps * _normalized_like(np.reshape(xopt, A.shape), A) + p_best_relerr = _relative_error(expm, A, p_best) + assert_allclose(p_best_relerr, -yopt * eps) + + # Check that the identified perturbation indeed gives greater + # relative error than random perturbations with similar norms. + for j in range(5): + p_rand = eps * _normalized_like(rng.randn(*A.shape), A) + assert_allclose(norm(p_best), norm(p_rand)) + p_rand_relerr = _relative_error(expm, A, p_rand) + assert_array_less(p_rand_relerr, p_best_relerr) + + # The greatest relative error should not be much greater than + # eps times the condition number kappa. + # In the limit as eps approaches zero it should never be greater. + assert_array_less(p_best_relerr, (1 + 2*eps) * eps * kappa) + + +class TestKhatriRao: + + def test_basic(self): + a = khatri_rao(array([[1, 2], [3, 4]]), + array([[5, 6], [7, 8]])) + + assert_array_equal(a, array([[5, 12], + [7, 16], + [15, 24], + [21, 32]])) + + b = khatri_rao(np.empty([2, 2]), np.empty([2, 2])) + assert_array_equal(b.shape, (4, 2)) + + def test_number_of_columns_equality(self): + with pytest.raises(ValueError): + a = array([[1, 2, 3], + [4, 5, 6]]) + b = array([[1, 2], + [3, 4]]) + khatri_rao(a, b) + + def test_to_assure_2d_array(self): + with pytest.raises(ValueError): + # both arrays are 1-D + a = array([1, 2, 3]) + b = array([4, 5, 6]) + khatri_rao(a, b) + + with pytest.raises(ValueError): + # first array is 1-D + a = array([1, 2, 3]) + b = array([ + [1, 2, 3], + [4, 5, 6] + ]) + khatri_rao(a, b) + + with pytest.raises(ValueError): + # second array is 1-D + a = array([ + [1, 2, 3], + [7, 8, 9] + ]) + b = array([4, 5, 6]) + khatri_rao(a, b) + + def test_equality_of_two_equations(self): + a = array([[1, 2], [3, 4]]) + b = array([[5, 6], [7, 8]]) + + res1 = khatri_rao(a, b) + res2 = np.vstack([np.kron(a[:, k], b[:, k]) + for k in range(b.shape[1])]).T + + assert_array_equal(res1, res2) + + def test_empty(self): + a = np.empty((0, 2)) + b = np.empty((3, 2)) + res = khatri_rao(a, b) + assert_allclose(res, np.empty((0, 2))) + + a = np.empty((3, 0)) + b = np.empty((5, 0)) + res = khatri_rao(a, b) + assert_allclose(res, np.empty((15, 0))) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_matmul_toeplitz.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_matmul_toeplitz.py new file mode 100644 index 0000000000000000000000000000000000000000..22f8f94fd10a5404d4013adf995bba54f76ff803 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_matmul_toeplitz.py @@ -0,0 +1,136 @@ +"""Test functions for linalg.matmul_toeplitz function +""" + +import numpy as np +from scipy.linalg import toeplitz, matmul_toeplitz + +from pytest import raises as assert_raises +from numpy.testing import assert_allclose + + +class TestMatmulToeplitz: + + def setup_method(self): + self.rng = np.random.RandomState(42) + self.tolerance = 1.5e-13 + + def test_real(self): + cases = [] + + n = 1 + c = self.rng.normal(size=n) + r = self.rng.normal(size=n) + x = self.rng.normal(size=(n, 1)) + cases.append((x, c, r, False)) + + n = 2 + c = self.rng.normal(size=n) + r = self.rng.normal(size=n) + x = self.rng.normal(size=(n, 1)) + cases.append((x, c, r, False)) + + n = 101 + c = self.rng.normal(size=n) + r = self.rng.normal(size=n) + x = self.rng.normal(size=(n, 1)) + cases.append((x, c, r, True)) + + n = 1000 + c = self.rng.normal(size=n) + r = self.rng.normal(size=n) + x = self.rng.normal(size=(n, 1)) + cases.append((x, c, r, False)) + + n = 100 + c = self.rng.normal(size=n) + r = self.rng.normal(size=n) + x = self.rng.normal(size=(n, self.rng.randint(1, 10))) + cases.append((x, c, r, False)) + + n = 100 + c = self.rng.normal(size=(n, 1)) + r = self.rng.normal(size=(n, 1)) + x = self.rng.normal(size=(n, self.rng.randint(1, 10))) + cases.append((x, c, r, True)) + + n = 100 + c = self.rng.normal(size=(n, 1)) + r = None + x = self.rng.normal(size=(n, self.rng.randint(1, 10))) + cases.append((x, c, r, True, -1)) + + n = 100 + c = self.rng.normal(size=(n, 1)) + r = None + x = self.rng.normal(size=n) + cases.append((x, c, r, False)) + + n = 101 + c = self.rng.normal(size=n) + r = self.rng.normal(size=n-27) + x = self.rng.normal(size=(n-27, 1)) + cases.append((x, c, r, True)) + + n = 100 + c = self.rng.normal(size=n) + r = self.rng.normal(size=n//4) + x = self.rng.normal(size=(n//4, self.rng.randint(1, 10))) + cases.append((x, c, r, True)) + + [self.do(*i) for i in cases] + + def test_complex(self): + n = 127 + c = self.rng.normal(size=(n, 1)) + self.rng.normal(size=(n, 1))*1j + r = self.rng.normal(size=(n, 1)) + self.rng.normal(size=(n, 1))*1j + x = self.rng.normal(size=(n, 3)) + self.rng.normal(size=(n, 3))*1j + self.do(x, c, r, False) + + n = 100 + c = self.rng.normal(size=(n, 1)) + self.rng.normal(size=(n, 1))*1j + r = self.rng.normal(size=(n//2, 1)) +\ + self.rng.normal(size=(n//2, 1))*1j + x = self.rng.normal(size=(n//2, 3)) +\ + self.rng.normal(size=(n//2, 3))*1j + self.do(x, c, r, False) + + def test_empty(self): + c = [] + r = [] + x = [] + self.do(x, c, r, False) + + x = np.empty((0, 0)) + self.do(x, c, r, False) + + def test_exceptions(self): + + n = 100 + c = self.rng.normal(size=n) + r = self.rng.normal(size=2*n) + x = self.rng.normal(size=n) + assert_raises(ValueError, matmul_toeplitz, (c, r), x, True) + + n = 100 + c = self.rng.normal(size=n) + r = self.rng.normal(size=n) + x = self.rng.normal(size=n-1) + assert_raises(ValueError, matmul_toeplitz, (c, r), x, True) + + n = 100 + c = self.rng.normal(size=n) + r = self.rng.normal(size=n//2) + x = self.rng.normal(size=n//2-1) + assert_raises(ValueError, matmul_toeplitz, (c, r), x, True) + + # For toeplitz matrices, matmul_toeplitz() should be equivalent to @. + def do(self, x, c, r=None, check_finite=False, workers=None): + c = np.ravel(c) + if r is None: + actual = matmul_toeplitz(c, x, check_finite, workers) + else: + r = np.ravel(r) + actual = matmul_toeplitz((c, r), x, check_finite) + desired = toeplitz(c, r) @ x + assert_allclose(actual, desired, + rtol=self.tolerance, atol=self.tolerance) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_procrustes.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_procrustes.py new file mode 100644 index 0000000000000000000000000000000000000000..4efa433c2cab01a4f77ef1c4f1bde3ab4d5c421b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_procrustes.py @@ -0,0 +1,221 @@ +from itertools import product, permutations + +import numpy as np +import pytest +from numpy.testing import assert_array_less, assert_allclose +from pytest import raises as assert_raises + +from scipy.linalg import inv, eigh, norm, svd +from scipy.linalg import orthogonal_procrustes +from scipy.sparse._sputils import matrix + + +def test_orthogonal_procrustes_ndim_too_large(): + rng = np.random.RandomState(1234) + A = rng.randn(3, 4, 5) + B = rng.randn(3, 4, 5) + assert_raises(ValueError, orthogonal_procrustes, A, B) + + +def test_orthogonal_procrustes_ndim_too_small(): + rng = np.random.RandomState(1234) + A = rng.randn(3) + B = rng.randn(3) + assert_raises(ValueError, orthogonal_procrustes, A, B) + + +def test_orthogonal_procrustes_shape_mismatch(): + rng = np.random.RandomState(1234) + shapes = ((3, 3), (3, 4), (4, 3), (4, 4)) + for a, b in permutations(shapes, 2): + A = rng.randn(*a) + B = rng.randn(*b) + assert_raises(ValueError, orthogonal_procrustes, A, B) + + +def test_orthogonal_procrustes_checkfinite_exception(): + rng = np.random.RandomState(1234) + m, n = 2, 3 + A_good = rng.randn(m, n) + B_good = rng.randn(m, n) + for bad_value in np.inf, -np.inf, np.nan: + A_bad = A_good.copy() + A_bad[1, 2] = bad_value + B_bad = B_good.copy() + B_bad[1, 2] = bad_value + for A, B in ((A_good, B_bad), (A_bad, B_good), (A_bad, B_bad)): + assert_raises(ValueError, orthogonal_procrustes, A, B) + + +def test_orthogonal_procrustes_scale_invariance(): + rng = np.random.RandomState(1234) + m, n = 4, 3 + for i in range(3): + A_orig = rng.randn(m, n) + B_orig = rng.randn(m, n) + R_orig, s = orthogonal_procrustes(A_orig, B_orig) + for A_scale in np.square(rng.randn(3)): + for B_scale in np.square(rng.randn(3)): + R, s = orthogonal_procrustes(A_orig * A_scale, B_orig * B_scale) + assert_allclose(R, R_orig) + + +def test_orthogonal_procrustes_array_conversion(): + rng = np.random.RandomState(1234) + for m, n in ((6, 4), (4, 4), (4, 6)): + A_arr = rng.randn(m, n) + B_arr = rng.randn(m, n) + As = (A_arr, A_arr.tolist(), matrix(A_arr)) + Bs = (B_arr, B_arr.tolist(), matrix(B_arr)) + R_arr, s = orthogonal_procrustes(A_arr, B_arr) + AR_arr = A_arr.dot(R_arr) + for A, B in product(As, Bs): + R, s = orthogonal_procrustes(A, B) + AR = A_arr.dot(R) + assert_allclose(AR, AR_arr) + + +def test_orthogonal_procrustes(): + rng = np.random.RandomState(1234) + for m, n in ((6, 4), (4, 4), (4, 6)): + # Sample a random target matrix. + B = rng.randn(m, n) + # Sample a random orthogonal matrix + # by computing eigh of a sampled symmetric matrix. + X = rng.randn(n, n) + w, V = eigh(X.T + X) + assert_allclose(inv(V), V.T) + # Compute a matrix with a known orthogonal transformation that gives B. + A = np.dot(B, V.T) + # Check that an orthogonal transformation from A to B can be recovered. + R, s = orthogonal_procrustes(A, B) + assert_allclose(inv(R), R.T) + assert_allclose(A.dot(R), B) + # Create a perturbed input matrix. + A_perturbed = A + 1e-2 * rng.randn(m, n) + # Check that the orthogonal procrustes function can find an orthogonal + # transformation that is better than the orthogonal transformation + # computed from the original input matrix. + R_prime, s = orthogonal_procrustes(A_perturbed, B) + assert_allclose(inv(R_prime), R_prime.T) + # Compute the naive and optimal transformations of the perturbed input. + naive_approx = A_perturbed.dot(R) + optim_approx = A_perturbed.dot(R_prime) + # Compute the Frobenius norm errors of the matrix approximations. + naive_approx_error = norm(naive_approx - B, ord='fro') + optim_approx_error = norm(optim_approx - B, ord='fro') + # Check that the orthogonal Procrustes approximation is better. + assert_array_less(optim_approx_error, naive_approx_error) + + +def _centered(A): + mu = A.mean(axis=0) + return A - mu, mu + + +def test_orthogonal_procrustes_exact_example(): + # Check a small application. + # It uses translation, scaling, reflection, and rotation. + # + # | + # a b | + # | + # d c | w + # | + # --------+--- x ----- z --- + # | + # | y + # | + # + A_orig = np.array([[-3, 3], [-2, 3], [-2, 2], [-3, 2]], dtype=float) + B_orig = np.array([[3, 2], [1, 0], [3, -2], [5, 0]], dtype=float) + A, A_mu = _centered(A_orig) + B, B_mu = _centered(B_orig) + R, s = orthogonal_procrustes(A, B) + scale = s / np.square(norm(A)) + B_approx = scale * np.dot(A, R) + B_mu + assert_allclose(B_approx, B_orig, atol=1e-8) + + +def test_orthogonal_procrustes_stretched_example(): + # Try again with a target with a stretched y axis. + A_orig = np.array([[-3, 3], [-2, 3], [-2, 2], [-3, 2]], dtype=float) + B_orig = np.array([[3, 40], [1, 0], [3, -40], [5, 0]], dtype=float) + A, A_mu = _centered(A_orig) + B, B_mu = _centered(B_orig) + R, s = orthogonal_procrustes(A, B) + scale = s / np.square(norm(A)) + B_approx = scale * np.dot(A, R) + B_mu + expected = np.array([[3, 21], [-18, 0], [3, -21], [24, 0]], dtype=float) + assert_allclose(B_approx, expected, atol=1e-8) + # Check disparity symmetry. + expected_disparity = 0.4501246882793018 + AB_disparity = np.square(norm(B_approx - B_orig) / norm(B)) + assert_allclose(AB_disparity, expected_disparity) + R, s = orthogonal_procrustes(B, A) + scale = s / np.square(norm(B)) + A_approx = scale * np.dot(B, R) + A_mu + BA_disparity = np.square(norm(A_approx - A_orig) / norm(A)) + assert_allclose(BA_disparity, expected_disparity) + + +def test_orthogonal_procrustes_skbio_example(): + # This transformation is also exact. + # It uses translation, scaling, and reflection. + # + # | + # | a + # | b + # | c d + # --+--------- + # | + # | w + # | + # | x + # | + # | z y + # | + # + A_orig = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], dtype=float) + B_orig = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], dtype=float) + B_standardized = np.array([ + [-0.13363062, 0.6681531], + [-0.13363062, 0.13363062], + [-0.13363062, -0.40089186], + [0.40089186, -0.40089186]]) + A, A_mu = _centered(A_orig) + B, B_mu = _centered(B_orig) + R, s = orthogonal_procrustes(A, B) + scale = s / np.square(norm(A)) + B_approx = scale * np.dot(A, R) + B_mu + assert_allclose(B_approx, B_orig) + assert_allclose(B / norm(B), B_standardized) + + +def test_empty(): + a = np.empty((0, 0)) + r, s = orthogonal_procrustes(a, a) + assert_allclose(r, np.empty((0, 0))) + + a = np.empty((0, 3)) + r, s = orthogonal_procrustes(a, a) + assert_allclose(r, np.identity(3)) + + +@pytest.mark.parametrize('shape', [(4, 5), (5, 5), (5, 4)]) +def test_unitary(shape): + # gh-12071 added support for unitary matrices; check that it + # works as intended. + m, n = shape + rng = np.random.default_rng(589234981235) + A = rng.random(shape) + rng.random(shape) * 1j + Q = rng.random((n, n)) + rng.random((n, n)) * 1j + Q, _ = np.linalg.qr(Q) + B = A @ Q + R, scale = orthogonal_procrustes(A, B) + assert_allclose(R @ R.conj().T, np.eye(n), atol=1e-14) + assert_allclose(A @ Q, B) + if shape != (4, 5): # solution is unique + assert_allclose(R, Q) + _, s, _ = svd(A.conj().T @ B) + assert_allclose(scale, np.sum(s)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_sketches.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_sketches.py new file mode 100644 index 0000000000000000000000000000000000000000..7fc5a8540510f57a2b00334b5e190d4ddd474d09 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_sketches.py @@ -0,0 +1,118 @@ +"""Tests for _sketches.py.""" + +import numpy as np +from numpy.testing import assert_, assert_equal +from scipy.linalg import clarkson_woodruff_transform +from scipy.linalg._sketches import cwt_matrix +from scipy.sparse import issparse, rand +from scipy.sparse.linalg import norm + + +class TestClarksonWoodruffTransform: + """ + Testing the Clarkson Woodruff Transform + """ + # set seed for generating test matrices + rng = np.random.default_rng(1179103485) + + # Test matrix parameters + n_rows = 2000 + n_cols = 100 + density = 0.1 + + # Sketch matrix dimensions + n_sketch_rows = 200 + + # Seeds to test with + seeds = [1755490010, 934377150, 1391612830, 1752708722, 2008891431, + 1302443994, 1521083269, 1501189312, 1126232505, 1533465685] + + A_dense = rng.random((n_rows, n_cols)) + A_csc = rand( + n_rows, n_cols, density=density, format='csc', random_state=rng, + ) + A_csr = rand( + n_rows, n_cols, density=density, format='csr', random_state=rng, + ) + A_coo = rand( + n_rows, n_cols, density=density, format='coo', random_state=rng, + ) + + # Collect the test matrices + test_matrices = [ + A_dense, A_csc, A_csr, A_coo, + ] + + # Test vector with norm ~1 + x = rng.random((n_rows, 1)) / np.sqrt(n_rows) + + def test_sketch_dimensions(self): + for A in self.test_matrices: + for seed in self.seeds: + # seed to ensure backwards compatibility post SPEC7 + sketch = clarkson_woodruff_transform( + A, self.n_sketch_rows, seed=seed + ) + assert_(sketch.shape == (self.n_sketch_rows, self.n_cols)) + + def test_seed_returns_identical_transform_matrix(self): + for seed in self.seeds: + S1 = cwt_matrix( + self.n_sketch_rows, self.n_rows, rng=seed + ).toarray() + S2 = cwt_matrix( + self.n_sketch_rows, self.n_rows, rng=seed + ).toarray() + assert_equal(S1, S2) + + def test_seed_returns_identically(self): + for A in self.test_matrices: + for seed in self.seeds: + sketch1 = clarkson_woodruff_transform( + A, self.n_sketch_rows, rng=seed + ) + sketch2 = clarkson_woodruff_transform( + A, self.n_sketch_rows, rng=seed + ) + if issparse(sketch1): + sketch1 = sketch1.toarray() + if issparse(sketch2): + sketch2 = sketch2.toarray() + assert_equal(sketch1, sketch2) + + def test_sketch_preserves_frobenius_norm(self): + # Given the probabilistic nature of the sketches + # we run the test multiple times and check that + # we pass all/almost all the tries. + n_errors = 0 + for A in self.test_matrices: + if issparse(A): + true_norm = norm(A) + else: + true_norm = np.linalg.norm(A) + for seed in self.seeds: + sketch = clarkson_woodruff_transform( + A, self.n_sketch_rows, rng=seed, + ) + if issparse(sketch): + sketch_norm = norm(sketch) + else: + sketch_norm = np.linalg.norm(sketch) + + if np.abs(true_norm - sketch_norm) > 0.1 * true_norm: + n_errors += 1 + assert_(n_errors == 0) + + def test_sketch_preserves_vector_norm(self): + n_errors = 0 + n_sketch_rows = int(np.ceil(2. / (0.01 * 0.5**2))) + true_norm = np.linalg.norm(self.x) + for seed in self.seeds: + sketch = clarkson_woodruff_transform( + self.x, n_sketch_rows, rng=seed, + ) + sketch_norm = np.linalg.norm(sketch) + + if np.abs(true_norm - sketch_norm) > 0.5 * true_norm: + n_errors += 1 + assert_(n_errors == 0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_solve_toeplitz.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_solve_toeplitz.py new file mode 100644 index 0000000000000000000000000000000000000000..440a73abc8c83bc32887c37c75577b790f3f1be9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_solve_toeplitz.py @@ -0,0 +1,150 @@ +"""Test functions for linalg._solve_toeplitz module +""" +import numpy as np +from scipy.linalg._solve_toeplitz import levinson +from scipy.linalg import solve, toeplitz, solve_toeplitz, matmul_toeplitz +from numpy.testing import assert_equal, assert_allclose + +import pytest +from pytest import raises as assert_raises + + +def test_solve_equivalence(): + # For toeplitz matrices, solve_toeplitz() should be equivalent to solve(). + random = np.random.RandomState(1234) + for n in (1, 2, 3, 10): + c = random.randn(n) + if random.rand() < 0.5: + c = c + 1j * random.randn(n) + r = random.randn(n) + if random.rand() < 0.5: + r = r + 1j * random.randn(n) + y = random.randn(n) + if random.rand() < 0.5: + y = y + 1j * random.randn(n) + + # Check equivalence when both the column and row are provided. + actual = solve_toeplitz((c,r), y) + desired = solve(toeplitz(c, r=r), y) + assert_allclose(actual, desired) + + # Check equivalence when the column is provided but not the row. + actual = solve_toeplitz(c, b=y) + desired = solve(toeplitz(c), y) + assert_allclose(actual, desired) + + +def test_multiple_rhs(): + random = np.random.RandomState(1234) + c = random.randn(4) + r = random.randn(4) + for offset in [0, 1j]: + for yshape in ((4,), (4, 3), (4, 3, 2)): + y = random.randn(*yshape) + offset + actual = solve_toeplitz((c,r), b=y) + desired = solve(toeplitz(c, r=r), y) + assert_equal(actual.shape, yshape) + assert_equal(desired.shape, yshape) + assert_allclose(actual, desired) + + +def test_native_list_arguments(): + c = [1,2,4,7] + r = [1,3,9,12] + y = [5,1,4,2] + actual = solve_toeplitz((c,r), y) + desired = solve(toeplitz(c, r=r), y) + assert_allclose(actual, desired) + + +def test_zero_diag_error(): + # The Levinson-Durbin implementation fails when the diagonal is zero. + random = np.random.RandomState(1234) + n = 4 + c = random.randn(n) + r = random.randn(n) + y = random.randn(n) + c[0] = 0 + assert_raises(np.linalg.LinAlgError, + solve_toeplitz, (c, r), b=y) + + +def test_wikipedia_counterexample(): + # The Levinson-Durbin implementation also fails in other cases. + # This example is from the talk page of the wikipedia article. + random = np.random.RandomState(1234) + c = [2, 2, 1] + y = random.randn(3) + assert_raises(np.linalg.LinAlgError, solve_toeplitz, c, b=y) + + +def test_reflection_coeffs(): + # check that the partial solutions are given by the reflection + # coefficients + + random = np.random.RandomState(1234) + y_d = random.randn(10) + y_z = random.randn(10) + 1j + reflection_coeffs_d = [1] + reflection_coeffs_z = [1] + for i in range(2, 10): + reflection_coeffs_d.append(solve_toeplitz(y_d[:(i-1)], b=y_d[1:i])[-1]) + reflection_coeffs_z.append(solve_toeplitz(y_z[:(i-1)], b=y_z[1:i])[-1]) + + y_d_concat = np.concatenate((y_d[-2:0:-1], y_d[:-1])) + y_z_concat = np.concatenate((y_z[-2:0:-1].conj(), y_z[:-1])) + _, ref_d = levinson(y_d_concat, b=y_d[1:]) + _, ref_z = levinson(y_z_concat, b=y_z[1:]) + + assert_allclose(reflection_coeffs_d, ref_d[:-1]) + assert_allclose(reflection_coeffs_z, ref_z[:-1]) + + +@pytest.mark.xfail(reason='Instability of Levinson iteration') +def test_unstable(): + # this is a "Gaussian Toeplitz matrix", as mentioned in Example 2 of + # I. Gohbert, T. Kailath and V. Olshevsky "Fast Gaussian Elimination with + # Partial Pivoting for Matrices with Displacement Structure" + # Mathematics of Computation, 64, 212 (1995), pp 1557-1576 + # which can be unstable for levinson recursion. + + # other fast toeplitz solvers such as GKO or Burg should be better. + random = np.random.RandomState(1234) + n = 100 + c = 0.9 ** (np.arange(n)**2) + y = random.randn(n) + + solution1 = solve_toeplitz(c, b=y) + solution2 = solve(toeplitz(c), y) + + assert_allclose(solution1, solution2) + + +@pytest.mark.parametrize('dt_c', [int, float, np.float32, complex, np.complex64]) +@pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64]) +def test_empty(dt_c, dt_b): + c = np.array([], dtype=dt_c) + b = np.array([], dtype=dt_b) + x = solve_toeplitz(c, b) + assert x.shape == (0,) + assert x.dtype == solve_toeplitz(np.array([2, 1], dtype=dt_c), + np.ones(2, dtype=dt_b)).dtype + + b = np.empty((0, 0), dtype=dt_b) + x1 = solve_toeplitz(c, b) + assert x1.shape == (0, 0) + assert x1.dtype == x.dtype + + +@pytest.mark.parametrize('fun', [solve_toeplitz, matmul_toeplitz]) +def test_nd_FutureWarning(fun): + # Test future warnings with n-D `c`/`r` + rng = np.random.default_rng(283592436523456) + c = rng.random((2, 3, 4)) + r = rng.random((2, 3, 4)) + b_or_x = rng.random(24) + message = "Beginning in SciPy 1.17, multidimensional input will be..." + with pytest.warns(FutureWarning, match=message): + fun(c, b_or_x) + with pytest.warns(FutureWarning, match=message): + fun((c, r), b_or_x) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_solvers.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_solvers.py new file mode 100644 index 0000000000000000000000000000000000000000..a4a39c5e86939bddabafaaf87c643c0a4ad570fe --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_solvers.py @@ -0,0 +1,844 @@ +import os +import numpy as np + +from numpy.testing import assert_array_almost_equal, assert_allclose +import pytest +from pytest import raises as assert_raises + +from scipy.linalg import solve_sylvester +from scipy.linalg import solve_continuous_lyapunov, solve_discrete_lyapunov +from scipy.linalg import solve_continuous_are, solve_discrete_are +from scipy.linalg import block_diag, solve, LinAlgError +from scipy.sparse._sputils import matrix + + +# dtypes for testing size-0 case following precedent set in gh-20295 +dtypes = [int, float, np.float32, complex, np.complex64] + + +def _load_data(name): + """ + Load npz data file under data/ + Returns a copy of the data, rather than keeping the npz file open. + """ + filename = os.path.join(os.path.abspath(os.path.dirname(__file__)), + 'data', name) + with np.load(filename) as f: + return dict(f.items()) + + +class TestSolveLyapunov: + + cases = [ + # empty case + (np.empty((0, 0)), + np.empty((0, 0))), + (np.array([[1, 2], [3, 4]]), + np.array([[9, 10], [11, 12]])), + # a, q all complex. + (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]), + np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])), + # a real; q complex. + (np.array([[1.0, 2.0], [3.0, 5.0]]), + np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])), + # a complex; q real. + (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]), + np.array([[2.0, 2.0], [-1.0, 2.0]])), + # An example from Kitagawa, 1977 + (np.array([[3, 9, 5, 1, 4], [1, 2, 3, 8, 4], [4, 6, 6, 6, 3], + [1, 5, 2, 0, 7], [5, 3, 3, 1, 5]]), + np.array([[2, 4, 1, 0, 1], [4, 1, 0, 2, 0], [1, 0, 3, 0, 3], + [0, 2, 0, 1, 0], [1, 0, 3, 0, 4]])), + # Companion matrix example. a complex; q real; a.shape[0] = 11 + (np.array([[0.100+0.j, 0.091+0.j, 0.082+0.j, 0.073+0.j, 0.064+0.j, + 0.055+0.j, 0.046+0.j, 0.037+0.j, 0.028+0.j, 0.019+0.j, + 0.010+0.j], + [1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j], + [0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j], + [0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j], + [0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, + 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j], + [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, + 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j], + [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j], + [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j], + [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j], + [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, + 0.000+0.j], + [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, + 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, + 0.000+0.j]]), + np.eye(11)), + # https://github.com/scipy/scipy/issues/4176 + (matrix([[0, 1], [-1/2, -1]]), + (matrix([0, 3]).T @ matrix([0, 3]).T.T)), + # https://github.com/scipy/scipy/issues/4176 + (matrix([[0, 1], [-1/2, -1]]), + (np.array(matrix([0, 3]).T @ matrix([0, 3]).T.T))), + ] + + def test_continuous_squareness_and_shape(self): + nsq = np.ones((3, 2)) + sq = np.eye(3) + assert_raises(ValueError, solve_continuous_lyapunov, nsq, sq) + assert_raises(ValueError, solve_continuous_lyapunov, sq, nsq) + assert_raises(ValueError, solve_continuous_lyapunov, sq, np.eye(2)) + + def check_continuous_case(self, a, q): + x = solve_continuous_lyapunov(a, q) + assert_array_almost_equal( + np.dot(a, x) + np.dot(x, a.conj().transpose()), q) + + def check_discrete_case(self, a, q, method=None): + x = solve_discrete_lyapunov(a, q, method=method) + assert_array_almost_equal( + np.dot(np.dot(a, x), a.conj().transpose()) - x, -1.0*q) + + def test_cases(self): + for case in self.cases: + self.check_continuous_case(case[0], case[1]) + self.check_discrete_case(case[0], case[1]) + self.check_discrete_case(case[0], case[1], method='direct') + self.check_discrete_case(case[0], case[1], method='bilinear') + + @pytest.mark.parametrize("dtype_a", dtypes) + @pytest.mark.parametrize("dtype_q", dtypes) + def test_size_0(self, dtype_a, dtype_q): + rng = np.random.default_rng(234598235) + + a = np.zeros((0, 0), dtype=dtype_a) + q = np.zeros((0, 0), dtype=dtype_q) + res = solve_continuous_lyapunov(a, q) + + a = (rng.random((5, 5))*100).astype(dtype_a) + q = (rng.random((5, 5))*100).astype(dtype_q) + ref = solve_continuous_lyapunov(a, q) + + assert res.shape == (0, 0) + assert res.dtype == ref.dtype + + +class TestSolveContinuousAre: + mat6 = _load_data('carex_6_data.npz') + mat15 = _load_data('carex_15_data.npz') + mat18 = _load_data('carex_18_data.npz') + mat19 = _load_data('carex_19_data.npz') + mat20 = _load_data('carex_20_data.npz') + cases = [ + # Carex examples taken from (with default parameters): + # [1] P.BENNER, A.J. LAUB, V. MEHRMANN: 'A Collection of Benchmark + # Examples for the Numerical Solution of Algebraic Riccati + # Equations II: Continuous-Time Case', Tech. Report SPC 95_23, + # Fak. f. Mathematik, TU Chemnitz-Zwickau (Germany), 1995. + # + # The format of the data is (a, b, q, r, knownfailure), where + # knownfailure is None if the test passes or a string + # indicating the reason for failure. + # + # Test Case 0: carex #1 + (np.diag([1.], 1), + np.array([[0], [1]]), + block_diag(1., 2.), + 1, + None), + # Test Case 1: carex #2 + (np.array([[4, 3], [-4.5, -3.5]]), + np.array([[1], [-1]]), + np.array([[9, 6], [6, 4.]]), + 1, + None), + # Test Case 2: carex #3 + (np.array([[0, 1, 0, 0], + [0, -1.89, 0.39, -5.53], + [0, -0.034, -2.98, 2.43], + [0.034, -0.0011, -0.99, -0.21]]), + np.array([[0, 0], [0.36, -1.6], [-0.95, -0.032], [0.03, 0]]), + np.array([[2.313, 2.727, 0.688, 0.023], + [2.727, 4.271, 1.148, 0.323], + [0.688, 1.148, 0.313, 0.102], + [0.023, 0.323, 0.102, 0.083]]), + np.eye(2), + None), + # Test Case 3: carex #4 + (np.array([[-0.991, 0.529, 0, 0, 0, 0, 0, 0], + [0.522, -1.051, 0.596, 0, 0, 0, 0, 0], + [0, 0.522, -1.118, 0.596, 0, 0, 0, 0], + [0, 0, 0.522, -1.548, 0.718, 0, 0, 0], + [0, 0, 0, 0.922, -1.64, 0.799, 0, 0], + [0, 0, 0, 0, 0.922, -1.721, 0.901, 0], + [0, 0, 0, 0, 0, 0.922, -1.823, 1.021], + [0, 0, 0, 0, 0, 0, 0.922, -1.943]]), + np.array([[3.84, 4.00, 37.60, 3.08, 2.36, 2.88, 3.08, 3.00], + [-2.88, -3.04, -2.80, -2.32, -3.32, -3.82, -4.12, -3.96]] + ).T * 0.001, + np.array([[1.0, 0.0, 0.0, 0.0, 0.5, 0.0, 0.0, 0.1], + [0.0, 1.0, 0.0, 0.0, 0.1, 0.0, 0.0, 0.0], + [0.0, 0.0, 1.0, 0.0, 0.0, 0.5, 0.0, 0.0], + [0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0], + [0.5, 0.1, 0.0, 0.0, 0.1, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.5, 0.0, 0.0, 0.1, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.1, 0.0], + [0.1, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.1]]), + np.eye(2), + None), + # Test Case 4: carex #5 + (np.array( + [[-4.019, 5.120, 0., 0., -2.082, 0., 0., 0., 0.870], + [-0.346, 0.986, 0., 0., -2.340, 0., 0., 0., 0.970], + [-7.909, 15.407, -4.069, 0., -6.450, 0., 0., 0., 2.680], + [-21.816, 35.606, -0.339, -3.870, -17.800, 0., 0., 0., 7.390], + [-60.196, 98.188, -7.907, 0.340, -53.008, 0., 0., 0., 20.400], + [0, 0, 0, 0, 94.000, -147.200, 0., 53.200, 0.], + [0, 0, 0, 0, 0, 94.000, -147.200, 0, 0], + [0, 0, 0, 0, 0, 12.800, 0.000, -31.600, 0], + [0, 0, 0, 0, 12.800, 0.000, 0.000, 18.800, -31.600]]), + np.array([[0.010, -0.011, -0.151], + [0.003, -0.021, 0.000], + [0.009, -0.059, 0.000], + [0.024, -0.162, 0.000], + [0.068, -0.445, 0.000], + [0.000, 0.000, 0.000], + [0.000, 0.000, 0.000], + [0.000, 0.000, 0.000], + [0.000, 0.000, 0.000]]), + np.eye(9), + np.eye(3), + None), + # Test Case 5: carex #6 + (mat6['A'], mat6['B'], mat6['Q'], mat6['R'], None), + # Test Case 6: carex #7 + (np.array([[1, 0], [0, -2.]]), + np.array([[1e-6], [0]]), + np.ones((2, 2)), + 1., + 'Bad residual accuracy'), + # Test Case 7: carex #8 + (block_diag(-0.1, -0.02), + np.array([[0.100, 0.000], [0.001, 0.010]]), + np.array([[100, 1000], [1000, 10000]]), + np.ones((2, 2)) + block_diag(1e-6, 0), + None), + # Test Case 8: carex #9 + (np.array([[0, 1e6], [0, 0]]), + np.array([[0], [1.]]), + np.eye(2), + 1., + None), + # Test Case 9: carex #10 + (np.array([[1.0000001, 1], [1., 1.0000001]]), + np.eye(2), + np.eye(2), + np.eye(2), + None), + # Test Case 10: carex #11 + (np.array([[3, 1.], [4, 2]]), + np.array([[1], [1]]), + np.array([[-11, -5], [-5, -2.]]), + 1., + None), + # Test Case 11: carex #12 + (np.array([[7000000., 2000000., -0.], + [2000000., 6000000., -2000000.], + [0., -2000000., 5000000.]]) / 3, + np.eye(3), + np.array([[1., -2., -2.], [-2., 1., -2.], [-2., -2., 1.]]).dot( + np.diag([1e-6, 1, 1e6])).dot( + np.array([[1., -2., -2.], [-2., 1., -2.], [-2., -2., 1.]])) / 9, + np.eye(3) * 1e6, + 'Bad Residual Accuracy'), + # Test Case 12: carex #13 + (np.array([[0, 0.4, 0, 0], + [0, 0, 0.345, 0], + [0, -0.524e6, -0.465e6, 0.262e6], + [0, 0, 0, -1e6]]), + np.array([[0, 0, 0, 1e6]]).T, + np.diag([1, 0, 1, 0]), + 1., + None), + # Test Case 13: carex #14 + (np.array([[-1e-6, 1, 0, 0], + [-1, -1e-6, 0, 0], + [0, 0, 1e-6, 1], + [0, 0, -1, 1e-6]]), + np.ones((4, 1)), + np.ones((4, 4)), + 1., + None), + # Test Case 14: carex #15 + (mat15['A'], mat15['B'], mat15['Q'], mat15['R'], None), + # Test Case 15: carex #16 + (np.eye(64, 64, k=-1) + np.eye(64, 64)*(-2.) + np.rot90( + block_diag(1, np.zeros((62, 62)), 1)) + np.eye(64, 64, k=1), + np.eye(64), + np.eye(64), + np.eye(64), + None), + # Test Case 16: carex #17 + (np.diag(np.ones((20, )), 1), + np.flipud(np.eye(21, 1)), + np.eye(21, 1) * np.eye(21, 1).T, + 1, + 'Bad Residual Accuracy'), + # Test Case 17: carex #18 + (mat18['A'], mat18['B'], mat18['Q'], mat18['R'], None), + # Test Case 18: carex #19 + (mat19['A'], mat19['B'], mat19['Q'], mat19['R'], + 'Bad Residual Accuracy'), + # Test Case 19: carex #20 + (mat20['A'], mat20['B'], mat20['Q'], mat20['R'], + 'Bad Residual Accuracy') + ] + # Makes the minimum precision requirements customized to the test. + # Here numbers represent the number of decimals that agrees with zero + # matrix when the solution x is plugged in to the equation. + # + # res = array([[8e-3,1e-16],[1e-16,1e-20]]) --> min_decimal[k] = 2 + # + # If the test is failing use "None" for that entry. + # + min_decimal = (14, 12, 13, 14, 11, 6, None, 5, 7, 14, 14, + None, 9, 14, 13, 14, None, 12, None, None) + + @pytest.mark.parametrize("j, case", enumerate(cases)) + def test_solve_continuous_are(self, j, case): + """Checks if 0 = XA + A'X - XB(R)^{-1} B'X + Q is true""" + a, b, q, r, knownfailure = case + if knownfailure: + pytest.xfail(reason=knownfailure) + + dec = self.min_decimal[j] + x = solve_continuous_are(a, b, q, r) + res = x @ a + a.conj().T @ x + q + out_fact = x @ b + res -= out_fact @ solve(np.atleast_2d(r), out_fact.conj().T) + assert_array_almost_equal(res, np.zeros_like(res), decimal=dec) + + +class TestSolveDiscreteAre: + cases = [ + # Darex examples taken from (with default parameters): + # [1] P.BENNER, A.J. LAUB, V. MEHRMANN: 'A Collection of Benchmark + # Examples for the Numerical Solution of Algebraic Riccati + # Equations II: Discrete-Time Case', Tech. Report SPC 95_23, + # Fak. f. Mathematik, TU Chemnitz-Zwickau (Germany), 1995. + # [2] T. GUDMUNDSSON, C. KENNEY, A.J. LAUB: 'Scaling of the + # Discrete-Time Algebraic Riccati Equation to Enhance Stability + # of the Schur Solution Method', IEEE Trans.Aut.Cont., vol.37(4) + # + # The format of the data is (a, b, q, r, knownfailure), where + # knownfailure is None if the test passes or a string + # indicating the reason for failure. + # + # TEST CASE 0 : Complex a; real b, q, r + (np.array([[2, 1-2j], [0, -3j]]), + np.array([[0], [1]]), + np.array([[1, 0], [0, 2]]), + np.array([[1]]), + None), + # TEST CASE 1 :Real a, q, r; complex b + (np.array([[2, 1], [0, -1]]), + np.array([[-2j], [1j]]), + np.array([[1, 0], [0, 2]]), + np.array([[1]]), + None), + # TEST CASE 2 : Real a, b; complex q, r + (np.array([[3, 1], [0, -1]]), + np.array([[1, 2], [1, 3]]), + np.array([[1, 1+1j], [1-1j, 2]]), + np.array([[2, -2j], [2j, 3]]), + None), + # TEST CASE 3 : User-reported gh-2251 (Trac #1732) + (np.array([[0.63399379, 0.54906824, 0.76253406], + [0.5404729, 0.53745766, 0.08731853], + [0.27524045, 0.84922129, 0.4681622]]), + np.array([[0.96861695], [0.05532739], [0.78934047]]), + np.eye(3), + np.eye(1), + None), + # TEST CASE 4 : darex #1 + (np.array([[4, 3], [-4.5, -3.5]]), + np.array([[1], [-1]]), + np.array([[9, 6], [6, 4]]), + np.array([[1]]), + None), + # TEST CASE 5 : darex #2 + (np.array([[0.9512, 0], [0, 0.9048]]), + np.array([[4.877, 4.877], [-1.1895, 3.569]]), + np.array([[0.005, 0], [0, 0.02]]), + np.array([[1/3, 0], [0, 3]]), + None), + # TEST CASE 6 : darex #3 + (np.array([[2, -1], [1, 0]]), + np.array([[1], [0]]), + np.array([[0, 0], [0, 1]]), + np.array([[0]]), + None), + # TEST CASE 7 : darex #4 (skipped the gen. Ric. term S) + (np.array([[0, 1], [0, -1]]), + np.array([[1, 0], [2, 1]]), + np.array([[-4, -4], [-4, 7]]) * (1/11), + np.array([[9, 3], [3, 1]]), + None), + # TEST CASE 8 : darex #5 + (np.array([[0, 1], [0, 0]]), + np.array([[0], [1]]), + np.array([[1, 2], [2, 4]]), + np.array([[1]]), + None), + # TEST CASE 9 : darex #6 + (np.array([[0.998, 0.067, 0, 0], + [-.067, 0.998, 0, 0], + [0, 0, 0.998, 0.153], + [0, 0, -.153, 0.998]]), + np.array([[0.0033, 0.0200], + [0.1000, -.0007], + [0.0400, 0.0073], + [-.0028, 0.1000]]), + np.array([[1.87, 0, 0, -0.244], + [0, 0.744, 0.205, 0], + [0, 0.205, 0.589, 0], + [-0.244, 0, 0, 1.048]]), + np.eye(2), + None), + # TEST CASE 10 : darex #7 + (np.array([[0.984750, -.079903, 0.0009054, -.0010765], + [0.041588, 0.998990, -.0358550, 0.0126840], + [-.546620, 0.044916, -.3299100, 0.1931800], + [2.662400, -.100450, -.9245500, -.2632500]]), + np.array([[0.0037112, 0.0007361], + [-.0870510, 9.3411e-6], + [-1.198440, -4.1378e-4], + [-3.192700, 9.2535e-4]]), + np.eye(4)*1e-2, + np.eye(2), + None), + # TEST CASE 11 : darex #8 + (np.array([[-0.6000000, -2.2000000, -3.6000000, -5.4000180], + [1.0000000, 0.6000000, 0.8000000, 3.3999820], + [0.0000000, 1.0000000, 1.8000000, 3.7999820], + [0.0000000, 0.0000000, 0.0000000, -0.9999820]]), + np.array([[1.0, -1.0, -1.0, -1.0], + [0.0, 1.0, -1.0, -1.0], + [0.0, 0.0, 1.0, -1.0], + [0.0, 0.0, 0.0, 1.0]]), + np.array([[2, 1, 3, 6], + [1, 2, 2, 5], + [3, 2, 6, 11], + [6, 5, 11, 22]]), + np.eye(4), + None), + # TEST CASE 12 : darex #9 + (np.array([[95.4070, 1.9643, 0.3597, 0.0673, 0.0190], + [40.8490, 41.3170, 16.0840, 4.4679, 1.1971], + [12.2170, 26.3260, 36.1490, 15.9300, 12.3830], + [4.1118, 12.8580, 27.2090, 21.4420, 40.9760], + [0.1305, 0.5808, 1.8750, 3.6162, 94.2800]]) * 0.01, + np.array([[0.0434, -0.0122], + [2.6606, -1.0453], + [3.7530, -5.5100], + [3.6076, -6.6000], + [0.4617, -0.9148]]) * 0.01, + np.eye(5), + np.eye(2), + None), + # TEST CASE 13 : darex #10 + (np.kron(np.eye(2), np.diag([1, 1], k=1)), + np.kron(np.eye(2), np.array([[0], [0], [1]])), + np.array([[1, 1, 0, 0, 0, 0], + [1, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, -1, 0], + [0, 0, 0, -1, 1, 0], + [0, 0, 0, 0, 0, 0]]), + np.array([[3, 0], [0, 1]]), + None), + # TEST CASE 14 : darex #11 + (0.001 * np.array( + [[870.1, 135.0, 11.59, .5014, -37.22, .3484, 0, 4.242, 7.249], + [76.55, 897.4, 12.72, 0.5504, -40.16, .3743, 0, 4.53, 7.499], + [-127.2, 357.5, 817, 1.455, -102.8, .987, 0, 11.85, 18.72], + [-363.5, 633.9, 74.91, 796.6, -273.5, 2.653, 0, 31.72, 48.82], + [-960, 1645.9, -128.9, -5.597, 71.42, 7.108, 0, 84.52, 125.9], + [-664.4, 112.96, -88.89, -3.854, 84.47, 13.6, 0, 144.3, 101.6], + [-410.2, 693, -54.71, -2.371, 66.49, 12.49, .1063, 99.97, 69.67], + [-179.9, 301.7, -23.93, -1.035, 60.59, 22.16, 0, 213.9, 35.54], + [-345.1, 580.4, -45.96, -1.989, 105.6, 19.86, 0, 219.1, 215.2]]), + np.array([[4.7600, -0.5701, -83.6800], + [0.8790, -4.7730, -2.7300], + [1.4820, -13.1200, 8.8760], + [3.8920, -35.1300, 24.8000], + [10.3400, -92.7500, 66.8000], + [7.2030, -61.5900, 38.3400], + [4.4540, -36.8300, 20.2900], + [1.9710, -15.5400, 6.9370], + [3.7730, -30.2800, 14.6900]]) * 0.001, + np.diag([50, 0, 0, 0, 50, 0, 0, 0, 0]), + np.eye(3), + None), + # TEST CASE 15 : darex #12 - numerically least accurate example + (np.array([[0, 1e6], [0, 0]]), + np.array([[0], [1]]), + np.eye(2), + np.array([[1]]), + None), + # TEST CASE 16 : darex #13 + (np.array([[16, 10, -2], + [10, 13, -8], + [-2, -8, 7]]) * (1/9), + np.eye(3), + 1e6 * np.eye(3), + 1e6 * np.eye(3), + None), + # TEST CASE 17 : darex #14 + (np.array([[1 - 1/1e8, 0, 0, 0], + [1, 0, 0, 0], + [0, 1, 0, 0], + [0, 0, 1, 0]]), + np.array([[1e-08], [0], [0], [0]]), + np.diag([0, 0, 0, 1]), + np.array([[0.25]]), + None), + # TEST CASE 18 : darex #15 + (np.eye(100, k=1), + np.flipud(np.eye(100, 1)), + np.eye(100), + np.array([[1]]), + None) + ] + + # Makes the minimum precision requirements customized to the test. + # Here numbers represent the number of decimals that agrees with zero + # matrix when the solution x is plugged in to the equation. + # + # res = array([[8e-3,1e-16],[1e-16,1e-20]]) --> min_decimal[k] = 2 + # + # If the test is failing use "None" for that entry. + # + min_decimal = (12, 14, 13, 14, 13, 16, 18, 14, 14, 13, + 14, 13, 13, 14, 12, 2, 4, 6, 10) + max_tol = [1.5 * 10**-ind for ind in min_decimal] + # relaxed tolerance in gh-18012 after bump to OpenBLAS + max_tol[11] = 2.5e-13 + + # relaxed tolerance in gh-20335 for linux-aarch64 build on Cirrus + # with OpenBLAS from ubuntu jammy + max_tol[15] = 2.0e-2 + + # relaxed tolerance in gh-20335 for OpenBLAS 3.20 on ubuntu jammy + # bump not needed for OpenBLAS 3.26 + max_tol[16] = 2.0e-4 + + @pytest.mark.parametrize("j, case", enumerate(cases)) + def test_solve_discrete_are(self, j, case): + """Checks if X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q) is true""" + a, b, q, r, knownfailure = case + if knownfailure: + pytest.xfail(reason=knownfailure) + + atol = self.max_tol[j] + + x = solve_discrete_are(a, b, q, r) + bH = b.conj().T + xa, xb = x @ a, x @ b + + res = a.conj().T @ xa - x + q + res -= a.conj().T @ xb @ (solve(r + bH @ xb, bH) @ xa) + + # changed from + # assert_array_almost_equal(res, np.zeros_like(res), decimal=dec) + # in gh-18012 as it's easier to relax a tolerance and allclose is + # preferred + assert_allclose(res, np.zeros_like(res), atol=atol) + + def test_infeasible(self): + # An infeasible example taken from https://arxiv.org/abs/1505.04861v1 + A = np.triu(np.ones((3, 3))) + A[0, 1] = -1 + B = np.array([[1, 1, 0], [0, 0, 1]]).T + Q = np.full_like(A, -2) + np.diag([8, -1, -1.9]) + R = np.diag([-10, 0.1]) + assert_raises(LinAlgError, solve_continuous_are, A, B, Q, R) + + +def test_solve_generalized_continuous_are(): + cases = [ + # Two random examples differ by s term + # in the absence of any literature for demanding examples. + (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01], + [4.617139e-02, 6.948286e-01, 3.444608e-02], + [9.713178e-02, 3.170995e-01, 4.387444e-01]]), + np.array([[3.815585e-01, 1.868726e-01], + [7.655168e-01, 4.897644e-01], + [7.951999e-01, 4.455862e-01]]), + np.eye(3), + np.eye(2), + np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01], + [7.093648e-01, 6.797027e-01, 1.189977e-01], + [7.546867e-01, 6.550980e-01, 4.983641e-01]]), + np.zeros((3, 2)), + None), + (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01], + [4.617139e-02, 6.948286e-01, 3.444608e-02], + [9.713178e-02, 3.170995e-01, 4.387444e-01]]), + np.array([[3.815585e-01, 1.868726e-01], + [7.655168e-01, 4.897644e-01], + [7.951999e-01, 4.455862e-01]]), + np.eye(3), + np.eye(2), + np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01], + [7.093648e-01, 6.797027e-01, 1.189977e-01], + [7.546867e-01, 6.550980e-01, 4.983641e-01]]), + np.ones((3, 2)), + None) + ] + + min_decimal = (10, 10) + + def _test_factory(case, dec): + """Checks if X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q) is true""" + a, b, q, r, e, s, knownfailure = case + if knownfailure: + pytest.xfail(reason=knownfailure) + + x = solve_continuous_are(a, b, q, r, e, s) + res = a.conj().T.dot(x.dot(e)) + e.conj().T.dot(x.dot(a)) + q + out_fact = e.conj().T.dot(x).dot(b) + s + res -= out_fact.dot(solve(np.atleast_2d(r), out_fact.conj().T)) + assert_array_almost_equal(res, np.zeros_like(res), decimal=dec) + + for ind, case in enumerate(cases): + _test_factory(case, min_decimal[ind]) + + +def test_solve_generalized_discrete_are(): + mat20170120 = _load_data('gendare_20170120_data.npz') + + cases = [ + # Two random examples differ by s term + # in the absence of any literature for demanding examples. + (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01], + [4.617139e-02, 6.948286e-01, 3.444608e-02], + [9.713178e-02, 3.170995e-01, 4.387444e-01]]), + np.array([[3.815585e-01, 1.868726e-01], + [7.655168e-01, 4.897644e-01], + [7.951999e-01, 4.455862e-01]]), + np.eye(3), + np.eye(2), + np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01], + [7.093648e-01, 6.797027e-01, 1.189977e-01], + [7.546867e-01, 6.550980e-01, 4.983641e-01]]), + np.zeros((3, 2)), + None), + (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01], + [4.617139e-02, 6.948286e-01, 3.444608e-02], + [9.713178e-02, 3.170995e-01, 4.387444e-01]]), + np.array([[3.815585e-01, 1.868726e-01], + [7.655168e-01, 4.897644e-01], + [7.951999e-01, 4.455862e-01]]), + np.eye(3), + np.eye(2), + np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01], + [7.093648e-01, 6.797027e-01, 1.189977e-01], + [7.546867e-01, 6.550980e-01, 4.983641e-01]]), + np.ones((3, 2)), + None), + # user-reported (under PR-6616) 20-Jan-2017 + # tests against the case where E is None but S is provided + (mat20170120['A'], + mat20170120['B'], + mat20170120['Q'], + mat20170120['R'], + None, + mat20170120['S'], + None), + ] + + max_atol = (1.5e-11, 1.5e-11, 3.5e-16) + + def _test_factory(case, atol): + """Checks if X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q) is true""" + a, b, q, r, e, s, knownfailure = case + if knownfailure: + pytest.xfail(reason=knownfailure) + + x = solve_discrete_are(a, b, q, r, e, s) + if e is None: + e = np.eye(a.shape[0]) + if s is None: + s = np.zeros_like(b) + res = a.conj().T.dot(x.dot(a)) - e.conj().T.dot(x.dot(e)) + q + res -= (a.conj().T.dot(x.dot(b)) + s).dot( + solve(r+b.conj().T.dot(x.dot(b)), + (b.conj().T.dot(x.dot(a)) + s.conj().T) + ) + ) + # changed from: + # assert_array_almost_equal(res, np.zeros_like(res), decimal=dec) + # in gh-17950 because of a Linux 32 bit fail. + assert_allclose(res, np.zeros_like(res), atol=atol) + + for ind, case in enumerate(cases): + _test_factory(case, max_atol[ind]) + + +def test_are_validate_args(): + + def test_square_shape(): + nsq = np.ones((3, 2)) + sq = np.eye(3) + for x in (solve_continuous_are, solve_discrete_are): + assert_raises(ValueError, x, nsq, 1, 1, 1) + assert_raises(ValueError, x, sq, sq, nsq, 1) + assert_raises(ValueError, x, sq, sq, sq, nsq) + assert_raises(ValueError, x, sq, sq, sq, sq, nsq) + + def test_compatible_sizes(): + nsq = np.ones((3, 2)) + sq = np.eye(4) + for x in (solve_continuous_are, solve_discrete_are): + assert_raises(ValueError, x, sq, nsq, 1, 1) + assert_raises(ValueError, x, sq, sq, sq, sq, sq, nsq) + assert_raises(ValueError, x, sq, sq, np.eye(3), sq) + assert_raises(ValueError, x, sq, sq, sq, np.eye(3)) + assert_raises(ValueError, x, sq, sq, sq, sq, np.eye(3)) + + def test_symmetry(): + nsym = np.arange(9).reshape(3, 3) + sym = np.eye(3) + for x in (solve_continuous_are, solve_discrete_are): + assert_raises(ValueError, x, sym, sym, nsym, sym) + assert_raises(ValueError, x, sym, sym, sym, nsym) + + def test_singularity(): + sing = np.full((3, 3), 1e12) + sing[2, 2] -= 1 + sq = np.eye(3) + for x in (solve_continuous_are, solve_discrete_are): + assert_raises(ValueError, x, sq, sq, sq, sq, sing) + + assert_raises(ValueError, solve_continuous_are, sq, sq, sq, sing) + + def test_finiteness(): + nm = np.full((2, 2), np.nan) + sq = np.eye(2) + for x in (solve_continuous_are, solve_discrete_are): + assert_raises(ValueError, x, nm, sq, sq, sq) + assert_raises(ValueError, x, sq, nm, sq, sq) + assert_raises(ValueError, x, sq, sq, nm, sq) + assert_raises(ValueError, x, sq, sq, sq, nm) + assert_raises(ValueError, x, sq, sq, sq, sq, nm) + assert_raises(ValueError, x, sq, sq, sq, sq, sq, nm) + + +class TestSolveSylvester: + cases = [ + # empty cases + (np.empty((0, 0)), + np.empty((0, 0)), + np.empty((0, 0))), + (np.empty((0, 0)), + np.empty((2, 2)), + np.empty((0, 2))), + (np.empty((2, 2)), + np.empty((0, 0)), + np.empty((2, 0))), + # a, b, c all real. + (np.array([[1, 2], [0, 4]]), + np.array([[5, 6], [0, 8]]), + np.array([[9, 10], [11, 12]])), + # a, b, c all real, 4x4. a and b have non-trivial 2x2 blocks in their + # quasi-triangular form. + (np.array([[1.0, 0, 0, 0], + [0, 1.0, 2.0, 0.0], + [0, 0, 3.0, -4], + [0, 0, 2, 5]]), + np.array([[2.0, 0, 0, 1.0], + [0, 1.0, 0.0, 0.0], + [0, 0, 1.0, -1], + [0, 0, 1, 1]]), + np.array([[1.0, 0, 0, 0], + [0, 1.0, 0, 0], + [0, 0, 1.0, 0], + [0, 0, 0, 1.0]])), + # a, b, c all complex. + (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]), + np.array([[-1.0, 2j], [3.0, 4.0]]), + np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])), + # a and b real; c complex. + (np.array([[1.0, 2.0], [3.0, 5.0]]), + np.array([[-1.0, 0], [3.0, 4.0]]), + np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])), + # a and c complex; b real. + (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]), + np.array([[-1.0, 0], [3.0, 4.0]]), + np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])), + # a complex; b and c real. + (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]), + np.array([[-1.0, 0], [3.0, 4.0]]), + np.array([[2.0, 2.0], [-1.0, 2.0]])), + # not square matrices, real + (np.array([[8, 1, 6], [3, 5, 7], [4, 9, 2]]), + np.array([[2, 3], [4, 5]]), + np.array([[1, 2], [3, 4], [5, 6]])), + # not square matrices, complex + (np.array([[8, 1j, 6+2j], [3, 5, 7], [4, 9, 2]]), + np.array([[2, 3], [4, 5-1j]]), + np.array([[1, 2j], [3, 4j], [5j, 6+7j]])), + ] + + def check_case(self, a, b, c): + x = solve_sylvester(a, b, c) + assert_array_almost_equal(np.dot(a, x) + np.dot(x, b), c) + + def test_cases(self): + for case in self.cases: + self.check_case(case[0], case[1], case[2]) + + def test_trivial(self): + a = np.array([[1.0, 0.0], [0.0, 1.0]]) + b = np.array([[1.0]]) + c = np.array([2.0, 2.0]).reshape(-1, 1) + x = solve_sylvester(a, b, c) + assert_array_almost_equal(x, np.array([1.0, 1.0]).reshape(-1, 1)) + + # Feel free to adjust this to test fewer dtypes or random selections rather than + # the Cartesian product. It doesn't take very long to test all combinations, + # though, so we'll start there and trim it down as we see fit. + @pytest.mark.parametrize("dtype_a", dtypes) + @pytest.mark.parametrize("dtype_b", dtypes) + @pytest.mark.parametrize("dtype_q", dtypes) + @pytest.mark.parametrize("m", [0, 3]) + @pytest.mark.parametrize("n", [0, 3]) + def test_size_0(self, m, n, dtype_a, dtype_b, dtype_q): + if m == n != 0: + pytest.skip('m = n != 0 is not a case that needs to be tested here.') + + rng = np.random.default_rng(598435298262546) + + a = np.zeros((m, m), dtype=dtype_a) + b = np.zeros((n, n), dtype=dtype_b) + q = np.zeros((m, n), dtype=dtype_q) + res = solve_sylvester(a, b, q) + + a = (rng.random((5, 5))*100).astype(dtype_a) + b = (rng.random((6, 6))*100).astype(dtype_b) + q = (rng.random((5, 6))*100).astype(dtype_q) + ref = solve_sylvester(a, b, q) + + assert res.shape == (m, n) + assert res.dtype == ref.dtype diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_special_matrices.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_special_matrices.py new file mode 100644 index 0000000000000000000000000000000000000000..d32e7ed4b4016924c60b3ed6287888a0372e8791 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/linalg/tests/test_special_matrices.py @@ -0,0 +1,640 @@ +import pytest +import numpy as np +from numpy import arange, array, eye, copy, sqrt +from numpy.testing import (assert_equal, assert_array_equal, + assert_array_almost_equal, assert_allclose) +from pytest import raises as assert_raises + +from scipy.fft import fft +from scipy.special import comb +from scipy.linalg import (toeplitz, hankel, circulant, hadamard, leslie, dft, + companion, kron, block_diag, + helmert, hilbert, invhilbert, pascal, invpascal, + fiedler, fiedler_companion, eigvals, + convolution_matrix) +from numpy.linalg import cond + + +class TestToeplitz: + + def test_basic(self): + y = toeplitz([1, 2, 3]) + assert_array_equal(y, [[1, 2, 3], [2, 1, 2], [3, 2, 1]]) + y = toeplitz([1, 2, 3], [1, 4, 5]) + assert_array_equal(y, [[1, 4, 5], [2, 1, 4], [3, 2, 1]]) + + def test_complex_01(self): + data = (1.0 + arange(3.0)) * (1.0 + 1.0j) + x = copy(data) + t = toeplitz(x) + # Calling toeplitz should not change x. + assert_array_equal(x, data) + # According to the docstring, x should be the first column of t. + col0 = t[:, 0] + assert_array_equal(col0, data) + assert_array_equal(t[0, 1:], data[1:].conj()) + + def test_scalar_00(self): + """Scalar arguments still produce a 2D array.""" + t = toeplitz(10) + assert_array_equal(t, [[10]]) + t = toeplitz(10, 20) + assert_array_equal(t, [[10]]) + + def test_scalar_01(self): + c = array([1, 2, 3]) + t = toeplitz(c, 1) + assert_array_equal(t, [[1], [2], [3]]) + + def test_scalar_02(self): + c = array([1, 2, 3]) + t = toeplitz(c, array(1)) + assert_array_equal(t, [[1], [2], [3]]) + + def test_scalar_03(self): + c = array([1, 2, 3]) + t = toeplitz(c, array([1])) + assert_array_equal(t, [[1], [2], [3]]) + + def test_scalar_04(self): + r = array([10, 2, 3]) + t = toeplitz(1, r) + assert_array_equal(t, [[1, 2, 3]]) + + +class TestHankel: + def test_basic(self): + y = hankel([1, 2, 3]) + assert_array_equal(y, [[1, 2, 3], [2, 3, 0], [3, 0, 0]]) + y = hankel([1, 2, 3], [3, 4, 5]) + assert_array_equal(y, [[1, 2, 3], [2, 3, 4], [3, 4, 5]]) + + +class TestCirculant: + def test_basic(self): + y = circulant([1, 2, 3]) + assert_array_equal(y, [[1, 3, 2], [2, 1, 3], [3, 2, 1]]) + + +class TestHadamard: + + def test_basic(self): + + y = hadamard(1) + assert_array_equal(y, [[1]]) + + y = hadamard(2, dtype=float) + assert_array_equal(y, [[1.0, 1.0], [1.0, -1.0]]) + + y = hadamard(4) + assert_array_equal(y, [[1, 1, 1, 1], + [1, -1, 1, -1], + [1, 1, -1, -1], + [1, -1, -1, 1]]) + + assert_raises(ValueError, hadamard, 0) + assert_raises(ValueError, hadamard, 5) + + +class TestLeslie: + + def test_bad_shapes(self): + assert_raises(ValueError, leslie, [[1, 1], [2, 2]], [3, 4, 5]) + assert_raises(ValueError, leslie, [1, 2], [1, 2]) + assert_raises(ValueError, leslie, [1], []) + + def test_basic(self): + a = leslie([1, 2, 3], [0.25, 0.5]) + expected = array([[1.0, 2.0, 3.0], + [0.25, 0.0, 0.0], + [0.0, 0.5, 0.0]]) + assert_array_equal(a, expected) + + +class TestCompanion: + + def test_bad_shapes(self): + assert_raises(ValueError, companion, [0, 4, 5]) + assert_raises(ValueError, companion, [1]) + assert_raises(ValueError, companion, []) + + def test_basic(self): + c = companion([1, 2, 3]) + expected = array([ + [-2.0, -3.0], + [1.0, 0.0]]) + assert_array_equal(c, expected) + + c = companion([2.0, 5.0, -10.0]) + expected = array([ + [-2.5, 5.0], + [1.0, 0.0]]) + assert_array_equal(c, expected) + + c = companion([(1.0, 2.0, 3.0), + (4.0, 5.0, 6.0)]) + expected = array([ + ([-2.00, -3.00], + [+1.00, +0.00]), + ([-1.25, -1.50], + [+1.00, +0.00]) + ]) + assert_array_equal(c, expected) + + +class TestBlockDiag: + def test_basic(self): + x = block_diag(eye(2), [[1, 2], [3, 4], [5, 6]], [[1, 2, 3]]) + assert_array_equal(x, [[1, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0], + [0, 0, 1, 2, 0, 0, 0], + [0, 0, 3, 4, 0, 0, 0], + [0, 0, 5, 6, 0, 0, 0], + [0, 0, 0, 0, 1, 2, 3]]) + + def test_dtype(self): + x = block_diag([[1.5]]) + assert_equal(x.dtype, float) + + x = block_diag([[True]]) + assert_equal(x.dtype, bool) + + def test_mixed_dtypes(self): + actual = block_diag([[1]], [[1j]]) + desired = np.array([[1, 0], [0, 1j]]) + assert_array_equal(actual, desired) + + def test_scalar_and_1d_args(self): + a = block_diag(1) + assert_equal(a.shape, (1, 1)) + assert_array_equal(a, [[1]]) + + a = block_diag([2, 3], 4) + assert_array_equal(a, [[2, 3, 0], [0, 0, 4]]) + + def test_bad_arg(self): + assert_raises(ValueError, block_diag, [[[1]]]) + + def test_no_args(self): + a = block_diag() + assert_equal(a.ndim, 2) + assert_equal(a.nbytes, 0) + + def test_empty_matrix_arg(self): + # regression test for gh-4596: check the shape of the result + # for empty matrix inputs. Empty matrices are no longer ignored + # (gh-4908) it is viewed as a shape (1, 0) matrix. + a = block_diag([[1, 0], [0, 1]], + [], + [[2, 3], [4, 5], [6, 7]]) + assert_array_equal(a, [[1, 0, 0, 0], + [0, 1, 0, 0], + [0, 0, 0, 0], + [0, 0, 2, 3], + [0, 0, 4, 5], + [0, 0, 6, 7]]) + + def test_zerosized_matrix_arg(self): + # test for gh-4908: check the shape of the result for + # zero-sized matrix inputs, i.e. matrices with shape (0,n) or (n,0). + # note that [[]] takes shape (1,0) + a = block_diag([[1, 0], [0, 1]], + [[]], + [[2, 3], [4, 5], [6, 7]], + np.zeros([0, 2], dtype='int32')) + assert_array_equal(a, [[1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 2, 3, 0, 0], + [0, 0, 4, 5, 0, 0], + [0, 0, 6, 7, 0, 0]]) + + +class TestKron: + @pytest.mark.thread_unsafe + def test_dep(self): + with pytest.deprecated_call(match="`kron`"): + kron(np.array([[1, 2],[3, 4]]),np.array([[1, 1, 1]])) + + @pytest.mark.filterwarnings('ignore::DeprecationWarning') + def test_basic(self): + + a = kron(array([[1, 2], [3, 4]]), array([[1, 1, 1]])) + assert_array_equal(a, array([[1, 1, 1, 2, 2, 2], + [3, 3, 3, 4, 4, 4]])) + + m1 = array([[1, 2], [3, 4]]) + m2 = array([[10], [11]]) + a = kron(m1, m2) + expected = array([[10, 20], + [11, 22], + [30, 40], + [33, 44]]) + assert_array_equal(a, expected) + + @pytest.mark.filterwarnings('ignore::DeprecationWarning') + def test_empty(self): + m1 = np.empty((0, 2)) + m2 = np.empty((1, 3)) + a = kron(m1, m2) + assert_allclose(a, np.empty((0, 6))) + + +class TestHelmert: + + def test_orthogonality(self): + for n in range(1, 7): + H = helmert(n, full=True) + Id = np.eye(n) + assert_allclose(H.dot(H.T), Id, atol=1e-12) + assert_allclose(H.T.dot(H), Id, atol=1e-12) + + def test_subspace(self): + for n in range(2, 7): + H_full = helmert(n, full=True) + H_partial = helmert(n) + for U in H_full[1:, :].T, H_partial.T: + C = np.eye(n) - np.full((n, n), 1 / n) + assert_allclose(U.dot(U.T), C) + assert_allclose(U.T.dot(U), np.eye(n-1), atol=1e-12) + + +class TestHilbert: + + def test_basic(self): + h3 = array([[1.0, 1/2., 1/3.], + [1/2., 1/3., 1/4.], + [1/3., 1/4., 1/5.]]) + assert_array_almost_equal(hilbert(3), h3) + + assert_array_equal(hilbert(1), [[1.0]]) + + h0 = hilbert(0) + assert_equal(h0.shape, (0, 0)) + + +class TestInvHilbert: + + def test_basic(self): + invh1 = array([[1]]) + assert_array_equal(invhilbert(1, exact=True), invh1) + assert_array_equal(invhilbert(1), invh1) + + invh2 = array([[4, -6], + [-6, 12]]) + assert_array_equal(invhilbert(2, exact=True), invh2) + assert_array_almost_equal(invhilbert(2), invh2) + + invh3 = array([[9, -36, 30], + [-36, 192, -180], + [30, -180, 180]]) + assert_array_equal(invhilbert(3, exact=True), invh3) + assert_array_almost_equal(invhilbert(3), invh3) + + invh4 = array([[16, -120, 240, -140], + [-120, 1200, -2700, 1680], + [240, -2700, 6480, -4200], + [-140, 1680, -4200, 2800]]) + assert_array_equal(invhilbert(4, exact=True), invh4) + assert_array_almost_equal(invhilbert(4), invh4) + + invh5 = array([[25, -300, 1050, -1400, 630], + [-300, 4800, -18900, 26880, -12600], + [1050, -18900, 79380, -117600, 56700], + [-1400, 26880, -117600, 179200, -88200], + [630, -12600, 56700, -88200, 44100]]) + assert_array_equal(invhilbert(5, exact=True), invh5) + assert_array_almost_equal(invhilbert(5), invh5) + + invh17 = array([ + [289, -41616, 1976760, -46124400, 629598060, -5540462928, + 33374693352, -143034400080, 446982500250, -1033026222800, + 1774926873720, -2258997839280, 2099709530100, -1384423866000, + 613101997800, -163493866080, 19835652870], + [-41616, 7990272, -426980160, 10627061760, -151103534400, + 1367702848512, -8410422724704, 36616806420480, -115857864064800, + 270465047424000, -468580694662080, 600545887119360, + -561522320049600, 372133135180800, -165537539406000, + 44316454993920, -5395297580640], + [1976760, -426980160, 24337869120, -630981792000, 9228108708000, + -85267724461920, 532660105897920, -2348052711713280, + 7504429831470000, -17664748409880000, 30818191841236800, + -39732544853164800, 37341234283298400, -24857330514030000, + 11100752642520000, -2982128117299200, 364182586693200], + [-46124400, 10627061760, -630981792000, 16826181120000, + -251209625940000, 2358021022156800, -14914482965141760, + 66409571644416000, -214015221119700000, 507295338950400000, + -890303319857952000, 1153715376477081600, -1089119333262870000, + 727848632044800000, -326170262829600000, 87894302404608000, + -10763618673376800], + [629598060, -151103534400, 9228108708000, + -251209625940000, 3810012660090000, -36210360321495360, + 231343968720664800, -1038687206500944000, 3370739732635275000, + -8037460526495400000, 14178080368737885600, -18454939322943942000, + 17489975175339030000, -11728977435138600000, 5272370630081100000, + -1424711708039692800, 174908803442373000], + [-5540462928, 1367702848512, -85267724461920, 2358021022156800, + -36210360321495360, 347619459086355456, -2239409617216035264, + 10124803292907663360, -33052510749726468000, + 79217210949138662400, -140362995650505067440, + 183420385176741672960, -174433352415381259200, + 117339159519533952000, -52892422160973595200, + 14328529177999196160, -1763080738699119840], + [33374693352, -8410422724704, 532660105897920, + -14914482965141760, 231343968720664800, -2239409617216035264, + 14527452132196331328, -66072377044391477760, + 216799987176909536400, -521925895055522958000, + 928414062734059661760, -1217424500995626443520, + 1161358898976091015200, -783401860847777371200, + 354015418167362952000, -96120549902411274240, + 11851820521255194480], + [-143034400080, 36616806420480, -2348052711713280, + 66409571644416000, -1038687206500944000, 10124803292907663360, + -66072377044391477760, 302045152202932469760, + -995510145200094810000, 2405996923185123840000, + -4294704507885446054400, 5649058909023744614400, + -5403874060541811254400, 3654352703663101440000, + -1655137020003255360000, 450325202737117593600, + -55630994283442749600], + [446982500250, -115857864064800, 7504429831470000, + -214015221119700000, 3370739732635275000, -33052510749726468000, + 216799987176909536400, -995510145200094810000, + 3293967392206196062500, -7988661659013106500000, + 14303908928401362270000, -18866974090684772052000, + 18093328327706957325000, -12263364009096700500000, + 5565847995255512250000, -1517208935002984080000, + 187754605706619279900], + [-1033026222800, 270465047424000, -17664748409880000, + 507295338950400000, -8037460526495400000, 79217210949138662400, + -521925895055522958000, 2405996923185123840000, + -7988661659013106500000, 19434404971634224000000, + -34894474126569249192000, 46141453390504792320000, + -44349976506971935800000, 30121928988527376000000, + -13697025107665828500000, 3740200989399948902400, + -463591619028689580000], + [1774926873720, -468580694662080, + 30818191841236800, -890303319857952000, 14178080368737885600, + -140362995650505067440, 928414062734059661760, + -4294704507885446054400, 14303908928401362270000, + -34894474126569249192000, 62810053427824648545600, + -83243376594051600326400, 80177044485212743068000, + -54558343880470209780000, 24851882355348879230400, + -6797096028813368678400, 843736746632215035600], + [-2258997839280, 600545887119360, -39732544853164800, + 1153715376477081600, -18454939322943942000, 183420385176741672960, + -1217424500995626443520, 5649058909023744614400, + -18866974090684772052000, 46141453390504792320000, + -83243376594051600326400, 110552468520163390156800, + -106681852579497947388000, 72720410752415168870400, + -33177973900974346080000, 9087761081682520473600, + -1129631016152221783200], + [2099709530100, -561522320049600, 37341234283298400, + -1089119333262870000, 17489975175339030000, + -174433352415381259200, 1161358898976091015200, + -5403874060541811254400, 18093328327706957325000, + -44349976506971935800000, 80177044485212743068000, + -106681852579497947388000, 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-2982128117299200, + 87894302404608000, -1424711708039692800, + 14328529177999196160, -96120549902411274240, + 450325202737117593600, -1517208935002984080000, + 3740200989399948902400, -6797096028813368678400, + 9087761081682520473600, -8824053728865840192000, + 6049545098753157120000, -2774765838662800128000, + 763806510427609497600, -95382575704033754400], + [19835652870, -5395297580640, 364182586693200, -10763618673376800, + 174908803442373000, -1763080738699119840, 11851820521255194480, + -55630994283442749600, 187754605706619279900, + -463591619028689580000, 843736746632215035600, + -1129631016152221783200, 1098252376814660067000, + -753830033789944188000, 346146444087219270000, + -95382575704033754400, 11922821963004219300] + ]) + assert_array_equal(invhilbert(17, exact=True), invh17) + assert_allclose(invhilbert(17), invh17.astype(float), rtol=1e-12) + + def test_inverse(self): + for n in range(1, 10): + a = hilbert(n) + b = invhilbert(n) + # The Hilbert matrix is increasingly badly conditioned, + # so take that into account in the test + c = cond(a) + assert_allclose(a.dot(b), eye(n), atol=1e-15*c, rtol=1e-15*c) + + +class TestPascal: + + cases = [ + (1, array([[1]]), array([[1]])), + (2, array([[1, 1], + [1, 2]]), + array([[1, 0], + [1, 1]])), + (3, array([[1, 1, 1], + [1, 2, 3], + [1, 3, 6]]), + array([[1, 0, 0], + [1, 1, 0], + [1, 2, 1]])), + (4, array([[1, 1, 1, 1], + [1, 2, 3, 4], + [1, 3, 6, 10], + [1, 4, 10, 20]]), + array([[1, 0, 0, 0], + [1, 1, 0, 0], + [1, 2, 1, 0], + [1, 3, 3, 1]])), + ] + + def check_case(self, n, sym, low): + assert_array_equal(pascal(n), sym) + assert_array_equal(pascal(n, kind='lower'), low) + assert_array_equal(pascal(n, kind='upper'), low.T) + assert_array_almost_equal(pascal(n, exact=False), sym) + assert_array_almost_equal(pascal(n, exact=False, kind='lower'), low) + assert_array_almost_equal(pascal(n, exact=False, kind='upper'), low.T) + + def test_cases(self): + for n, sym, low in self.cases: + self.check_case(n, sym, low) + + def test_big(self): + p = pascal(50) + assert p[-1, -1] == comb(98, 49, exact=True) + + def test_threshold(self): + # Regression test. An early version of `pascal` returned an + # array of type np.uint64 for n=35, but that data type is too small + # to hold p[-1, -1]. The second assert_equal below would fail + # because p[-1, -1] overflowed. + p = pascal(34) + assert_equal(2*p.item(-1, -2), p.item(-1, -1), err_msg="n = 34") + p = pascal(35) + assert_equal(2.*p.item(-1, -2), 1.*p.item(-1, -1), err_msg="n = 35") + + +def test_invpascal(): + + def check_invpascal(n, kind, exact): + ip = invpascal(n, kind=kind, exact=exact) + p = pascal(n, kind=kind, exact=exact) + # Matrix-multiply ip and p, and check that we get the identity matrix. + # We can't use the simple expression e = ip.dot(p), because when + # n < 35 and exact is True, p.dtype is np.uint64 and ip.dtype is + # np.int64. The product of those dtypes is np.float64, which loses + # precision when n is greater than 18. Instead we'll cast both to + # object arrays, and then multiply. + e = ip.astype(object).dot(p.astype(object)) + assert_array_equal(e, eye(n), err_msg="n=%d kind=%r exact=%r" % + (n, kind, exact)) + + kinds = ['symmetric', 'lower', 'upper'] + + ns = [1, 2, 5, 18] + for n in ns: + for kind in kinds: + for exact in [True, False]: + check_invpascal(n, kind, exact) + + ns = [19, 34, 35, 50] + for n in ns: + for kind in kinds: + check_invpascal(n, kind, True) + + +def test_dft(): + m = dft(2) + expected = array([[1.0, 1.0], [1.0, -1.0]]) + assert_array_almost_equal(m, expected) + m = dft(2, scale='n') + assert_array_almost_equal(m, expected/2.0) + m = dft(2, scale='sqrtn') + assert_array_almost_equal(m, expected/sqrt(2.0)) + + x = array([0, 1, 2, 3, 4, 5, 0, 1]) + m = dft(8) + mx = m.dot(x) + fx = fft(x) + assert_array_almost_equal(mx, fx) + + +def test_fiedler(): + f = fiedler([]) + assert_equal(f.size, 0) + f = fiedler([123.]) + assert_array_equal(f, np.array([[0.]])) + f = fiedler(np.arange(1, 7)) + des = np.array([[0, 1, 2, 3, 4, 5], + [1, 0, 1, 2, 3, 4], + [2, 1, 0, 1, 2, 3], + [3, 2, 1, 0, 1, 2], + [4, 3, 2, 1, 0, 1], + [5, 4, 3, 2, 1, 0]]) + assert_array_equal(f, des) + + +def test_fiedler_companion(): + fc = fiedler_companion([]) + assert_equal(fc.size, 0) + fc = fiedler_companion([1.]) + assert_equal(fc.size, 0) + fc = fiedler_companion([1., 2.]) + assert_array_equal(fc, np.array([[-2.]])) + fc = fiedler_companion([1e-12, 2., 3.]) + assert_array_almost_equal(fc, companion([1e-12, 2., 3.])) + with assert_raises(ValueError): + fiedler_companion([0, 1, 2]) + fc = fiedler_companion([1., -16., 86., -176., 105.]) + assert_array_almost_equal(eigvals(fc), + np.array([7., 5., 3., 1.])) + + +class TestConvolutionMatrix: + """ + Test convolution_matrix vs. numpy.convolve for various parameters. + """ + + def create_vector(self, n, cpx): + """Make a complex or real test vector of length n.""" + x = np.linspace(-2.5, 2.2, n) + if cpx: + x = x + 1j*np.linspace(-1.5, 3.1, n) + return x + + def test_bad_n(self): + # n must be a positive integer + with pytest.raises(ValueError, match='n must be a positive integer'): + convolution_matrix([1, 2, 3], 0) + + def test_empty_first_arg(self): + # first arg must have at least one value + with pytest.raises(ValueError, match=r'len\(a\)'): + convolution_matrix([], 4) + + def test_bad_mode(self): + # mode must be in ('full', 'valid', 'same') + with pytest.raises(ValueError, match='mode.*must be one of'): + convolution_matrix((1, 1), 4, mode='invalid argument') + + @pytest.mark.parametrize('cpx', [False, True]) + @pytest.mark.parametrize('na', [1, 2, 9]) + @pytest.mark.parametrize('nv', [1, 2, 9]) + @pytest.mark.parametrize('mode', [None, 'full', 'valid', 'same']) + def test_against_numpy_convolve(self, cpx, na, nv, mode): + a = self.create_vector(na, cpx) + v = self.create_vector(nv, cpx) + if mode is None: + y1 = np.convolve(v, a) + A = convolution_matrix(a, nv) + else: + y1 = np.convolve(v, a, mode) + A = convolution_matrix(a, nv, mode) + y2 = A @ v + assert_array_almost_equal(y1, y2) + + +@pytest.mark.thread_unsafe +@pytest.mark.fail_slow(5) # `leslie` has an import in the function +@pytest.mark.parametrize('f, args', [(circulant, ()), + (companion, ()), + (convolution_matrix, (5, 'same')), + (fiedler, ()), + (fiedler_companion, ()), + (leslie, (np.arange(9),)), + (toeplitz, (np.arange(9),)), + ]) +def test_batch(f, args): + rng = np.random.default_rng(283592436523456) + batch_shape = (2, 3) + m = 10 + A = rng.random(batch_shape + (m,)) + + if f in {toeplitz}: + message = "Beginning in SciPy 1.17, multidimensional input will be..." + with pytest.warns(FutureWarning, match=message): + f(A, *args) + return + + res = f(A, *args) + ref = np.asarray([f(a, *args) for a in A.reshape(-1, m)]) + ref = ref.reshape(A.shape[:-1] + ref.shape[-2:]) + assert_allclose(res, ref) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/misc/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/misc/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..2eb2f71afd54a67b54d7012347e5d1a983fac7be --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/misc/__init__.py @@ -0,0 +1,6 @@ +import warnings +warnings.warn( + "scipy.misc is deprecated and will be removed in 2.0.0", + DeprecationWarning, + stacklevel=2 +) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/misc/common.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/misc/common.py new file mode 100644 index 0000000000000000000000000000000000000000..e85acca3ac49d1cb84792bdf369cffe69a5d8ad8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/misc/common.py @@ -0,0 +1,6 @@ +import warnings +warnings.warn( + "scipy.misc.common is deprecated and will be removed in 2.0.0", + DeprecationWarning, + stacklevel=2 +) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/misc/doccer.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/misc/doccer.py new file mode 100644 index 0000000000000000000000000000000000000000..74cabc8c2fd14fe6424b8aad828c329ecdaee4b2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/misc/doccer.py @@ -0,0 +1,6 @@ +import warnings +warnings.warn( + "scipy.misc.doccer is deprecated and will be removed in 2.0.0", + DeprecationWarning, + stacklevel=2 +) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..2e9d9f6ff99218088fd9e693aaca00ca8a070040 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/__init__.py @@ -0,0 +1,173 @@ +""" +========================================================= +Multidimensional image processing (:mod:`scipy.ndimage`) +========================================================= + +.. currentmodule:: scipy.ndimage + +This package contains various functions for multidimensional image +processing. + + +Filters +======= + +.. autosummary:: + :toctree: generated/ + + convolve - Multidimensional convolution + convolve1d - 1-D convolution along the given axis + correlate - Multidimensional correlation + correlate1d - 1-D correlation along the given axis + gaussian_filter + gaussian_filter1d + gaussian_gradient_magnitude + gaussian_laplace + generic_filter - Multidimensional filter using a given function + generic_filter1d - 1-D generic filter along the given axis + generic_gradient_magnitude + generic_laplace + laplace - N-D Laplace filter based on approximate second derivatives + maximum_filter + maximum_filter1d + median_filter - Calculates a multidimensional median filter + minimum_filter + minimum_filter1d + percentile_filter - Calculates a multidimensional percentile filter + prewitt + rank_filter - Calculates a multidimensional rank filter + sobel + uniform_filter - Multidimensional uniform filter + uniform_filter1d - 1-D uniform filter along the given axis + +Fourier filters +=============== + +.. autosummary:: + :toctree: generated/ + + fourier_ellipsoid + fourier_gaussian + fourier_shift + fourier_uniform + +Interpolation +============= + +.. autosummary:: + :toctree: generated/ + + affine_transform - Apply an affine transformation + geometric_transform - Apply an arbitrary geometric transform + map_coordinates - Map input array to new coordinates by interpolation + rotate - Rotate an array + shift - Shift an array + spline_filter + spline_filter1d + zoom - Zoom an array + +Measurements +============ + +.. autosummary:: + :toctree: generated/ + + center_of_mass - The center of mass of the values of an array at labels + extrema - Min's and max's of an array at labels, with their positions + find_objects - Find objects in a labeled array + histogram - Histogram of the values of an array, optionally at labels + label - Label features in an array + labeled_comprehension + maximum + maximum_position + mean - Mean of the values of an array at labels + median + minimum + minimum_position + standard_deviation - Standard deviation of an N-D image array + sum_labels - Sum of the values of the array + value_indices - Find indices of each distinct value in given array + variance - Variance of the values of an N-D image array + watershed_ift + +Morphology +========== + +.. autosummary:: + :toctree: generated/ + + binary_closing + binary_dilation + binary_erosion + binary_fill_holes + binary_hit_or_miss + binary_opening + binary_propagation + black_tophat + distance_transform_bf + distance_transform_cdt + distance_transform_edt + generate_binary_structure + grey_closing + grey_dilation + grey_erosion + grey_opening + iterate_structure + morphological_gradient + morphological_laplace + white_tophat + +""" + +# Copyright (C) 2003-2005 Peter J. Verveer +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions +# are met: +# +# 1. Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# +# 2. Redistributions in binary form must reproduce the above +# copyright notice, this list of conditions and the following +# disclaimer in the documentation and/or other materials provided +# with the distribution. +# +# 3. The name of the author may not be used to endorse or promote +# products derived from this software without specific prior +# written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS +# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY +# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE +# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, +# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +# bring in the public functionality from private namespaces + +# mypy: ignore-errors + +from ._support_alternative_backends import * + +# adjust __all__ and do not leak implementation details +from . import _support_alternative_backends +__all__ = _support_alternative_backends.__all__ +del _support_alternative_backends, _ndimage_api, _delegators # noqa: F821 + + +# Deprecated namespaces, to be removed in v2.0.0 +from . import filters +from . import fourier +from . import interpolation +from . import measurements +from . import morphology + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ctest.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ctest.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..0d05e123ba1f7f45c1f37795e7ba5cd0257018b4 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ctest.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_cytest.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_cytest.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..7898248eb4d089b5ffa726ff39de4d0ec4637272 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_cytest.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_delegators.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_delegators.py new file mode 100644 index 0000000000000000000000000000000000000000..9647ea6456426c9a62178ff277b0f35017a8310b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_delegators.py @@ -0,0 +1,297 @@ +"""Delegators for alternative backends in scipy.ndimage. + +The signature of `func_signature` must match the signature of ndimage.func. +The job of a `func_signature` is to know which arguments of `ndimage.func` +are arrays. + +* signatures are generated by + +-------------- +import inspect +from scipy import ndimage + +names = [x for x in dir(ndimage) if not x.startswith('_')] +objs = [getattr(ndimage, name) for name in names] +funcs = [obj for obj in objs if inspect.isroutine(obj)] + +for func in funcs: + sig = inspect.signature(func) + print(f"def {func.__name__}_signature{sig}:\n\tpass\n\n") +--------------- + +* which arguments to delegate on: manually trawled the documentation for + array-like and array arguments + +""" +import numpy as np +from scipy._lib._array_api import array_namespace +from scipy.ndimage._ni_support import _skip_if_dtype, _skip_if_int + + +def affine_transform_signature( + input, matrix, offset=0.0, output_shape=None, output=None, *args, **kwds +): + return array_namespace(input, matrix, _skip_if_dtype(output)) + + +def binary_closing_signature( + input, structure=None, iterations=1, output=None, *args, **kwds +): + return array_namespace(input, structure, _skip_if_dtype(output)) + +binary_opening_signature = binary_closing_signature + + +def binary_dilation_signature( + input, structure=None, iterations=1, mask=None, output=None, *args, **kwds +): + return array_namespace(input, structure, _skip_if_dtype(output), mask) + +binary_erosion_signature = binary_dilation_signature + + +def binary_fill_holes_signature( + input, structure=None, output=None, origin=0, *args, **kwargs +): + return array_namespace(input, structure, _skip_if_dtype(output)) + + +def label_signature(input, structure=None, output=None, origin=0): + return array_namespace(input, structure, _skip_if_dtype(output)) + + +def binary_hit_or_miss_signature( + input, structure1=None, structure2=None, output=None, *args, **kwds +): + return array_namespace(input, structure1, structure2, _skip_if_dtype(output)) + + +def binary_propagation_signature( + input, structure=None, mask=None, output=None, *args, **kwds +): + return array_namespace(input, structure, mask, _skip_if_dtype(output)) + + +def convolve_signature(input, weights, output=None, *args, **kwds): + return array_namespace(input, weights, _skip_if_dtype(output)) + +correlate_signature = convolve_signature + + +def convolve1d_signature(input, weights, axis=-1, output=None, *args, **kwds): + return array_namespace(input, weights, _skip_if_dtype(output)) + +correlate1d_signature = convolve1d_signature + + +def distance_transform_bf_signature( + input, metric='euclidean', sampling=None, return_distances=True, + return_indices=False, distances=None, indices=None +): + return array_namespace(input, distances, indices) + + +def distance_transform_cdt_signature( + input, metric='chessboard', return_distances=True, return_indices=False, + distances=None, indices=None +): + return array_namespace(input, distances, indices) + + +def distance_transform_edt_signature( + input, sampling=None, return_distances=True, return_indices=False, + distances=None, indices=None +): + return array_namespace(input, distances, indices) + + +def find_objects_signature(input, max_label=0): + return array_namespace(input) + + +def fourier_ellipsoid_signature(input, size, n=-1, axis=-1, output=None): + return array_namespace(input, _skip_if_dtype(output)) + +fourier_uniform_signature = fourier_ellipsoid_signature + + +def fourier_gaussian_signature(input, sigma, n=-1, axis=-1, output=None): + return array_namespace(input, _skip_if_dtype(output)) + +def fourier_shift_signature(input, shift, n=-1, axis=-1, output=None): + return array_namespace(input, _skip_if_dtype(output)) + + +def gaussian_filter_signature(input, sigma, order=0, output=None, *args, **kwds): + return array_namespace(input, _skip_if_dtype(output)) + + +def gaussian_filter1d_signature( + input, sigma, axis=-1, order=0, output=None, *args, **kwds +): + return array_namespace(input, _skip_if_dtype(output)) + + +def gaussian_gradient_magnitude_signature(input, sigma, output=None, *args, **kwds): + return array_namespace(input, _skip_if_dtype(output)) + +gaussian_laplace_signature = gaussian_gradient_magnitude_signature + + +def generate_binary_structure_signature(rank, connectivity): + # XXX: no input arrays; always return numpy + return np + + +def generic_filter_signature( + input, function, size=None, footprint=None, output=None, *args, **kwds +): + # XXX: function LowLevelCallable w/backends + return array_namespace(input, footprint, _skip_if_dtype(output)) + + +def generic_filter1d_signature( + input, function, filter_size, axis=-1, output=None, *args, **kwds +): + return array_namespace(input, _skip_if_dtype(output)) + + +def generic_gradient_magnitude_signature( + input, derivative, output=None, *args, **kwds +): + # XXX: function LowLevelCallable w/backends + return array_namespace(input, _skip_if_dtype(output)) + + +def generic_laplace_signature(input, derivative2, output=None, *args, **kwds): + # XXX: function LowLevelCallable w/backends + return array_namespace(input, _skip_if_dtype(output)) + + +def geometric_transform_signature( + input, mapping, output_shape=None, output=None, *args, **kwds +): + return array_namespace(input, _skip_if_dtype(output)) + + +def histogram_signature(input, min, max, bins, labels=None, index=None): + return array_namespace(input, labels) + + +def iterate_structure_signature(structure, iterations, origin=None): + return array_namespace(structure) + + +def labeled_comprehension_signature(input, labels, *args, **kwds): + return array_namespace(input, labels) + + +def laplace_signature(input, output=None, *args, **kwds): + return array_namespace(input, _skip_if_dtype(output)) + + +def map_coordinates_signature(input, coordinates, output=None, *args, **kwds): + return array_namespace(input, coordinates, _skip_if_dtype(output)) + + +def maximum_filter1d_signature(input, size, axis=-1, output=None, *args, **kwds): + return array_namespace(input, _skip_if_dtype(output)) + +minimum_filter1d_signature = maximum_filter1d_signature +uniform_filter1d_signature = maximum_filter1d_signature + + +def maximum_signature(input, labels=None, index=None): + return array_namespace(input, labels, _skip_if_int(index)) + +minimum_signature = maximum_signature +median_signature = maximum_signature +mean_signature = maximum_signature +variance_signature = maximum_signature +standard_deviation_signature = maximum_signature +sum_labels_signature = maximum_signature +sum_signature = maximum_signature # ndimage.sum is sum_labels + +maximum_position_signature = maximum_signature +minimum_position_signature = maximum_signature + +extrema_signature = maximum_signature +center_of_mass_signature = extrema_signature + + +def median_filter_signature( + input, size=None, footprint=None, output=None, *args, **kwds +): + return array_namespace(input, footprint, _skip_if_dtype(output)) + +minimum_filter_signature = median_filter_signature +maximum_filter_signature = median_filter_signature + + +def morphological_gradient_signature( + input, size=None, footprint=None, structure=None, output=None, *args, **kwds +): + return array_namespace(input, footprint, structure, _skip_if_dtype(output)) + +morphological_laplace_signature = morphological_gradient_signature +white_tophat_signature = morphological_gradient_signature +black_tophat_signature = morphological_gradient_signature +grey_closing_signature = morphological_gradient_signature +grey_dilation_signature = morphological_gradient_signature +grey_erosion_signature = morphological_gradient_signature +grey_opening_signature = morphological_gradient_signature + + +def percentile_filter_signature( + input, percentile, size=None, footprint=None, output=None, *args, **kwds +): + return array_namespace(input, footprint, _skip_if_dtype(output)) + + +def prewitt_signature(input, axis=-1, output=None, *args, **kwds): + return array_namespace(input, _skip_if_dtype(output)) + +sobel_signature = prewitt_signature + + +def rank_filter_signature( + input, rank, size=None, footprint=None, output=None, *args, **kwds +): + return array_namespace(input, footprint, _skip_if_dtype(output)) + + +def rotate_signature( + input, angle, axes=(1, 0), reshape=True, output=None , *args, **kwds +): + return array_namespace(input, _skip_if_dtype(output)) + + +def shift_signature(input, shift, output=None, *args, **kwds): + return array_namespace(input, _skip_if_dtype(output)) + + +def spline_filter_signature(input, order=3, output=np.float64, *args, **kwds): + return array_namespace(input, _skip_if_dtype(output)) + + +def spline_filter1d_signature( + input, order=3, axis=-1, output=np.float64, *args, **kwds +): + return array_namespace(input, _skip_if_dtype(output)) + + +def uniform_filter_signature(input, size=3, output=None, *args, **kwds): + return array_namespace(input, _skip_if_dtype(output)) + + +def value_indices_signature(arr, *args, **kwds): + return array_namespace(arr) + + +def watershed_ift_signature(input, markers, structure=None, output=None): + return array_namespace(input, markers, structure, _skip_if_dtype(output)) + + +def zoom_signature(input, zoom, output=None, *args, **kwds): + return array_namespace(input, _skip_if_dtype(output)) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_filters.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_filters.py new file mode 100644 index 0000000000000000000000000000000000000000..710ea60c03653cc80ac3bd1eefd425b4268a5246 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_filters.py @@ -0,0 +1,1965 @@ +# Copyright (C) 2003-2005 Peter J. Verveer +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions +# are met: +# +# 1. Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# +# 2. Redistributions in binary form must reproduce the above +# copyright notice, this list of conditions and the following +# disclaimer in the documentation and/or other materials provided +# with the distribution. +# +# 3. The name of the author may not be used to endorse or promote +# products derived from this software without specific prior +# written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS +# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY +# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE +# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, +# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +from collections.abc import Iterable +import numbers +import warnings +import numpy as np +import operator + +from scipy._lib._util import normalize_axis_index +from . import _ni_support +from . import _nd_image +from . import _ni_docstrings +from . import _rank_filter_1d + +__all__ = ['correlate1d', 'convolve1d', 'gaussian_filter1d', 'gaussian_filter', + 'prewitt', 'sobel', 'generic_laplace', 'laplace', + 'gaussian_laplace', 'generic_gradient_magnitude', + 'gaussian_gradient_magnitude', 'correlate', 'convolve', + 'uniform_filter1d', 'uniform_filter', 'minimum_filter1d', + 'maximum_filter1d', 'minimum_filter', 'maximum_filter', + 'rank_filter', 'median_filter', 'percentile_filter', + 'generic_filter1d', 'generic_filter'] + + +def _invalid_origin(origin, lenw): + return (origin < -(lenw // 2)) or (origin > (lenw - 1) // 2) + + +def _complex_via_real_components(func, input, weights, output, cval, **kwargs): + """Complex convolution via a linear combination of real convolutions.""" + complex_input = input.dtype.kind == 'c' + complex_weights = weights.dtype.kind == 'c' + if complex_input and complex_weights: + # real component of the output + func(input.real, weights.real, output=output.real, + cval=np.real(cval), **kwargs) + output.real -= func(input.imag, weights.imag, output=None, + cval=np.imag(cval), **kwargs) + # imaginary component of the output + func(input.real, weights.imag, output=output.imag, + cval=np.real(cval), **kwargs) + output.imag += func(input.imag, weights.real, output=None, + cval=np.imag(cval), **kwargs) + elif complex_input: + func(input.real, weights, output=output.real, cval=np.real(cval), + **kwargs) + func(input.imag, weights, output=output.imag, cval=np.imag(cval), + **kwargs) + else: + if np.iscomplexobj(cval): + raise ValueError("Cannot provide a complex-valued cval when the " + "input is real.") + func(input, weights.real, output=output.real, cval=cval, **kwargs) + func(input, weights.imag, output=output.imag, cval=cval, **kwargs) + return output + + +def _expand_origin(ndim_image, axes, origin): + num_axes = len(axes) + origins = _ni_support._normalize_sequence(origin, num_axes) + if num_axes < ndim_image: + # set origin = 0 for any axes not being filtered + origins_temp = [0,] * ndim_image + for o, ax in zip(origins, axes): + origins_temp[ax] = o + origins = origins_temp + return origins + + +def _expand_footprint(ndim_image, axes, footprint, + footprint_name="footprint"): + num_axes = len(axes) + if num_axes < ndim_image: + if footprint.ndim != num_axes: + raise RuntimeError(f"{footprint_name}.ndim ({footprint.ndim}) " + f"must match len(axes) ({num_axes})") + + footprint = np.expand_dims( + footprint, + tuple(ax for ax in range(ndim_image) if ax not in axes) + ) + return footprint + + +def _expand_mode(ndim_image, axes, mode): + num_axes = len(axes) + if not isinstance(mode, str) and isinstance(mode, Iterable): + # set mode = 'constant' for any axes not being filtered + modes = _ni_support._normalize_sequence(mode, num_axes) + modes_temp = ['constant'] * ndim_image + for m, ax in zip(modes, axes): + modes_temp[ax] = m + mode = modes_temp + return mode + + +@_ni_docstrings.docfiller +def correlate1d(input, weights, axis=-1, output=None, mode="reflect", + cval=0.0, origin=0): + """Calculate a 1-D correlation along the given axis. + + The lines of the array along the given axis are correlated with the + given weights. + + Parameters + ---------- + %(input)s + weights : array + 1-D sequence of numbers. + %(axis)s + %(output)s + %(mode_reflect)s + %(cval)s + %(origin)s + + Returns + ------- + result : ndarray + Correlation result. Has the same shape as `input`. + + Examples + -------- + >>> from scipy.ndimage import correlate1d + >>> correlate1d([2, 8, 0, 4, 1, 9, 9, 0], weights=[1, 3]) + array([ 8, 26, 8, 12, 7, 28, 36, 9]) + """ + input = np.asarray(input) + weights = np.asarray(weights) + complex_input = input.dtype.kind == 'c' + complex_weights = weights.dtype.kind == 'c' + if complex_input or complex_weights: + if complex_weights: + weights = weights.conj() + weights = weights.astype(np.complex128, copy=False) + kwargs = dict(axis=axis, mode=mode, origin=origin) + output = _ni_support._get_output(output, input, complex_output=True) + return _complex_via_real_components(correlate1d, input, weights, + output, cval, **kwargs) + + output = _ni_support._get_output(output, input) + weights = np.asarray(weights, dtype=np.float64) + if weights.ndim != 1 or weights.shape[0] < 1: + raise RuntimeError('no filter weights given') + if not weights.flags.contiguous: + weights = weights.copy() + axis = normalize_axis_index(axis, input.ndim) + if _invalid_origin(origin, len(weights)): + raise ValueError('Invalid origin; origin must satisfy ' + '-(len(weights) // 2) <= origin <= ' + '(len(weights)-1) // 2') + mode = _ni_support._extend_mode_to_code(mode) + _nd_image.correlate1d(input, weights, axis, output, mode, cval, + origin) + return output + + +@_ni_docstrings.docfiller +def convolve1d(input, weights, axis=-1, output=None, mode="reflect", + cval=0.0, origin=0): + """Calculate a 1-D convolution along the given axis. + + The lines of the array along the given axis are convolved with the + given weights. + + Parameters + ---------- + %(input)s + weights : ndarray + 1-D sequence of numbers. + %(axis)s + %(output)s + %(mode_reflect)s + %(cval)s + %(origin)s + + Returns + ------- + convolve1d : ndarray + Convolved array with same shape as input + + Examples + -------- + >>> from scipy.ndimage import convolve1d + >>> convolve1d([2, 8, 0, 4, 1, 9, 9, 0], weights=[1, 3]) + array([14, 24, 4, 13, 12, 36, 27, 0]) + """ + weights = np.asarray(weights) + weights = weights[::-1] + origin = -origin + if not weights.shape[0] & 1: + origin -= 1 + if weights.dtype.kind == 'c': + # pre-conjugate here to counteract the conjugation in correlate1d + weights = weights.conj() + return correlate1d(input, weights, axis, output, mode, cval, origin) + + +def _gaussian_kernel1d(sigma, order, radius): + """ + Computes a 1-D Gaussian convolution kernel. + """ + if order < 0: + raise ValueError('order must be non-negative') + exponent_range = np.arange(order + 1) + sigma2 = sigma * sigma + x = np.arange(-radius, radius+1) + phi_x = np.exp(-0.5 / sigma2 * x ** 2) + phi_x = phi_x / phi_x.sum() + + if order == 0: + return phi_x + else: + # f(x) = q(x) * phi(x) = q(x) * exp(p(x)) + # f'(x) = (q'(x) + q(x) * p'(x)) * phi(x) + # p'(x) = -1 / sigma ** 2 + # Implement q'(x) + q(x) * p'(x) as a matrix operator and apply to the + # coefficients of q(x) + q = np.zeros(order + 1) + q[0] = 1 + D = np.diag(exponent_range[1:], 1) # D @ q(x) = q'(x) + P = np.diag(np.ones(order)/-sigma2, -1) # P @ q(x) = q(x) * p'(x) + Q_deriv = D + P + for _ in range(order): + q = Q_deriv.dot(q) + q = (x[:, None] ** exponent_range).dot(q) + return q * phi_x + + +@_ni_docstrings.docfiller +def gaussian_filter1d(input, sigma, axis=-1, order=0, output=None, + mode="reflect", cval=0.0, truncate=4.0, *, radius=None): + """1-D Gaussian filter. + + Parameters + ---------- + %(input)s + sigma : scalar + standard deviation for Gaussian kernel + %(axis)s + order : int, optional + An order of 0 corresponds to convolution with a Gaussian + kernel. A positive order corresponds to convolution with + that derivative of a Gaussian. + %(output)s + %(mode_reflect)s + %(cval)s + truncate : float, optional + Truncate the filter at this many standard deviations. + Default is 4.0. + radius : None or int, optional + Radius of the Gaussian kernel. If specified, the size of + the kernel will be ``2*radius + 1``, and `truncate` is ignored. + Default is None. + + Returns + ------- + gaussian_filter1d : ndarray + + Notes + ----- + The Gaussian kernel will have size ``2*radius + 1`` along each axis. If + `radius` is None, a default ``radius = round(truncate * sigma)`` will be + used. + + Examples + -------- + >>> from scipy.ndimage import gaussian_filter1d + >>> import numpy as np + >>> gaussian_filter1d([1.0, 2.0, 3.0, 4.0, 5.0], 1) + array([ 1.42704095, 2.06782203, 3. , 3.93217797, 4.57295905]) + >>> gaussian_filter1d([1.0, 2.0, 3.0, 4.0, 5.0], 4) + array([ 2.91948343, 2.95023502, 3. , 3.04976498, 3.08051657]) + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + >>> x = rng.standard_normal(101).cumsum() + >>> y3 = gaussian_filter1d(x, 3) + >>> y6 = gaussian_filter1d(x, 6) + >>> plt.plot(x, 'k', label='original data') + >>> plt.plot(y3, '--', label='filtered, sigma=3') + >>> plt.plot(y6, ':', label='filtered, sigma=6') + >>> plt.legend() + >>> plt.grid() + >>> plt.show() + + """ + sd = float(sigma) + # make the radius of the filter equal to truncate standard deviations + lw = int(truncate * sd + 0.5) + if radius is not None: + lw = radius + if not isinstance(lw, numbers.Integral) or lw < 0: + raise ValueError('Radius must be a nonnegative integer.') + # Since we are calling correlate, not convolve, revert the kernel + weights = _gaussian_kernel1d(sigma, order, lw)[::-1] + return correlate1d(input, weights, axis, output, mode, cval, 0) + + +@_ni_docstrings.docfiller +def gaussian_filter(input, sigma, order=0, output=None, + mode="reflect", cval=0.0, truncate=4.0, *, radius=None, + axes=None): + """Multidimensional Gaussian filter. + + Parameters + ---------- + %(input)s + sigma : scalar or sequence of scalars + Standard deviation for Gaussian kernel. The standard + deviations of the Gaussian filter are given for each axis as a + sequence, or as a single number, in which case it is equal for + all axes. + order : int or sequence of ints, optional + The order of the filter along each axis is given as a sequence + of integers, or as a single number. An order of 0 corresponds + to convolution with a Gaussian kernel. A positive order + corresponds to convolution with that derivative of a Gaussian. + %(output)s + %(mode_multiple)s + %(cval)s + truncate : float, optional + Truncate the filter at this many standard deviations. + Default is 4.0. + radius : None or int or sequence of ints, optional + Radius of the Gaussian kernel. The radius are given for each axis + as a sequence, or as a single number, in which case it is equal + for all axes. If specified, the size of the kernel along each axis + will be ``2*radius + 1``, and `truncate` is ignored. + Default is None. + axes : tuple of int or None, optional + If None, `input` is filtered along all axes. Otherwise, + `input` is filtered along the specified axes. When `axes` is + specified, any tuples used for `sigma`, `order`, `mode` and/or `radius` + must match the length of `axes`. The ith entry in any of these tuples + corresponds to the ith entry in `axes`. + + Returns + ------- + gaussian_filter : ndarray + Returned array of same shape as `input`. + + Notes + ----- + The multidimensional filter is implemented as a sequence of + 1-D convolution filters. The intermediate arrays are + stored in the same data type as the output. Therefore, for output + types with a limited precision, the results may be imprecise + because intermediate results may be stored with insufficient + precision. + + The Gaussian kernel will have size ``2*radius + 1`` along each axis. If + `radius` is None, the default ``radius = round(truncate * sigma)`` will be + used. + + Examples + -------- + >>> from scipy.ndimage import gaussian_filter + >>> import numpy as np + >>> a = np.arange(50, step=2).reshape((5,5)) + >>> a + array([[ 0, 2, 4, 6, 8], + [10, 12, 14, 16, 18], + [20, 22, 24, 26, 28], + [30, 32, 34, 36, 38], + [40, 42, 44, 46, 48]]) + >>> gaussian_filter(a, sigma=1) + array([[ 4, 6, 8, 9, 11], + [10, 12, 14, 15, 17], + [20, 22, 24, 25, 27], + [29, 31, 33, 34, 36], + [35, 37, 39, 40, 42]]) + + >>> from scipy import datasets + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> plt.gray() # show the filtered result in grayscale + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + >>> ascent = datasets.ascent() + >>> result = gaussian_filter(ascent, sigma=5) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result) + >>> plt.show() + """ + input = np.asarray(input) + output = _ni_support._get_output(output, input) + + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + orders = _ni_support._normalize_sequence(order, num_axes) + sigmas = _ni_support._normalize_sequence(sigma, num_axes) + modes = _ni_support._normalize_sequence(mode, num_axes) + radiuses = _ni_support._normalize_sequence(radius, num_axes) + axes = [(axes[ii], sigmas[ii], orders[ii], modes[ii], radiuses[ii]) + for ii in range(num_axes) if sigmas[ii] > 1e-15] + if len(axes) > 0: + for axis, sigma, order, mode, radius in axes: + gaussian_filter1d(input, sigma, axis, order, output, + mode, cval, truncate, radius=radius) + input = output + else: + output[...] = input[...] + return output + + +@_ni_docstrings.docfiller +def prewitt(input, axis=-1, output=None, mode="reflect", cval=0.0): + """Calculate a Prewitt filter. + + Parameters + ---------- + %(input)s + %(axis)s + %(output)s + %(mode_multiple)s + %(cval)s + + Returns + ------- + prewitt : ndarray + Filtered array. Has the same shape as `input`. + + See Also + -------- + sobel: Sobel filter + + Notes + ----- + This function computes the one-dimensional Prewitt filter. + Horizontal edges are emphasised with the horizontal transform (axis=0), + vertical edges with the vertical transform (axis=1), and so on for higher + dimensions. These can be combined to give the magnitude. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> ascent = datasets.ascent() + >>> prewitt_h = ndimage.prewitt(ascent, axis=0) + >>> prewitt_v = ndimage.prewitt(ascent, axis=1) + >>> magnitude = np.sqrt(prewitt_h ** 2 + prewitt_v ** 2) + >>> magnitude *= 255 / np.max(magnitude) # Normalization + >>> fig, axes = plt.subplots(2, 2, figsize = (8, 8)) + >>> plt.gray() + >>> axes[0, 0].imshow(ascent) + >>> axes[0, 1].imshow(prewitt_h) + >>> axes[1, 0].imshow(prewitt_v) + >>> axes[1, 1].imshow(magnitude) + >>> titles = ["original", "horizontal", "vertical", "magnitude"] + >>> for i, ax in enumerate(axes.ravel()): + ... ax.set_title(titles[i]) + ... ax.axis("off") + >>> plt.show() + + """ + input = np.asarray(input) + axis = normalize_axis_index(axis, input.ndim) + output = _ni_support._get_output(output, input) + modes = _ni_support._normalize_sequence(mode, input.ndim) + correlate1d(input, [-1, 0, 1], axis, output, modes[axis], cval, 0) + axes = [ii for ii in range(input.ndim) if ii != axis] + for ii in axes: + correlate1d(output, [1, 1, 1], ii, output, modes[ii], cval, 0,) + return output + + +@_ni_docstrings.docfiller +def sobel(input, axis=-1, output=None, mode="reflect", cval=0.0): + """Calculate a Sobel filter. + + Parameters + ---------- + %(input)s + %(axis)s + %(output)s + %(mode_multiple)s + %(cval)s + + Returns + ------- + sobel : ndarray + Filtered array. Has the same shape as `input`. + + Notes + ----- + This function computes the axis-specific Sobel gradient. + The horizontal edges can be emphasised with the horizontal transform (axis=0), + the vertical edges with the vertical transform (axis=1) and so on for higher + dimensions. These can be combined to give the magnitude. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> ascent = datasets.ascent().astype('int32') + >>> sobel_h = ndimage.sobel(ascent, 0) # horizontal gradient + >>> sobel_v = ndimage.sobel(ascent, 1) # vertical gradient + >>> magnitude = np.sqrt(sobel_h**2 + sobel_v**2) + >>> magnitude *= 255.0 / np.max(magnitude) # normalization + >>> fig, axs = plt.subplots(2, 2, figsize=(8, 8)) + >>> plt.gray() # show the filtered result in grayscale + >>> axs[0, 0].imshow(ascent) + >>> axs[0, 1].imshow(sobel_h) + >>> axs[1, 0].imshow(sobel_v) + >>> axs[1, 1].imshow(magnitude) + >>> titles = ["original", "horizontal", "vertical", "magnitude"] + >>> for i, ax in enumerate(axs.ravel()): + ... ax.set_title(titles[i]) + ... ax.axis("off") + >>> plt.show() + + """ + input = np.asarray(input) + axis = normalize_axis_index(axis, input.ndim) + output = _ni_support._get_output(output, input) + modes = _ni_support._normalize_sequence(mode, input.ndim) + correlate1d(input, [-1, 0, 1], axis, output, modes[axis], cval, 0) + axes = [ii for ii in range(input.ndim) if ii != axis] + for ii in axes: + correlate1d(output, [1, 2, 1], ii, output, modes[ii], cval, 0) + return output + + +@_ni_docstrings.docfiller +def generic_laplace(input, derivative2, output=None, mode="reflect", + cval=0.0, + extra_arguments=(), + extra_keywords=None, + *, axes=None): + """ + N-D Laplace filter using a provided second derivative function. + + Parameters + ---------- + %(input)s + derivative2 : callable + Callable with the following signature:: + + derivative2(input, axis, output, mode, cval, + *extra_arguments, **extra_keywords) + + See `extra_arguments`, `extra_keywords` below. + %(output)s + %(mode_multiple)s + %(cval)s + %(extra_keywords)s + %(extra_arguments)s + axes : tuple of int or None + The axes over which to apply the filter. If a `mode` tuple is + provided, its length must match the number of axes. + + Returns + ------- + generic_laplace : ndarray + Filtered array. Has the same shape as `input`. + + """ + if extra_keywords is None: + extra_keywords = {} + input = np.asarray(input) + output = _ni_support._get_output(output, input) + axes = _ni_support._check_axes(axes, input.ndim) + if len(axes) > 0: + modes = _ni_support._normalize_sequence(mode, len(axes)) + derivative2(input, axes[0], output, modes[0], cval, + *extra_arguments, **extra_keywords) + for ii in range(1, len(axes)): + tmp = derivative2(input, axes[ii], output.dtype, modes[ii], cval, + *extra_arguments, **extra_keywords) + output += tmp + else: + output[...] = input[...] + return output + + +@_ni_docstrings.docfiller +def laplace(input, output=None, mode="reflect", cval=0.0, *, axes=None): + """N-D Laplace filter based on approximate second derivatives. + + Parameters + ---------- + %(input)s + %(output)s + %(mode_multiple)s + %(cval)s + axes : tuple of int or None + The axes over which to apply the filter. If a `mode` tuple is + provided, its length must match the number of axes. + + Returns + ------- + laplace : ndarray + Filtered array. Has the same shape as `input`. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> plt.gray() # show the filtered result in grayscale + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + >>> ascent = datasets.ascent() + >>> result = ndimage.laplace(ascent) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result) + >>> plt.show() + """ + def derivative2(input, axis, output, mode, cval): + return correlate1d(input, [1, -2, 1], axis, output, mode, cval, 0) + return generic_laplace(input, derivative2, output, mode, cval, axes=axes) + + +@_ni_docstrings.docfiller +def gaussian_laplace(input, sigma, output=None, mode="reflect", + cval=0.0, *, axes=None, **kwargs): + """Multidimensional Laplace filter using Gaussian second derivatives. + + Parameters + ---------- + %(input)s + sigma : scalar or sequence of scalars + The standard deviations of the Gaussian filter are given for + each axis as a sequence, or as a single number, in which case + it is equal for all axes. + %(output)s + %(mode_multiple)s + %(cval)s + axes : tuple of int or None + The axes over which to apply the filter. If `sigma` or `mode` tuples + are provided, their length must match the number of axes. + Extra keyword arguments will be passed to gaussian_filter(). + + Returns + ------- + gaussian_laplace : ndarray + Filtered array. Has the same shape as `input`. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> ascent = datasets.ascent() + + >>> fig = plt.figure() + >>> plt.gray() # show the filtered result in grayscale + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + + >>> result = ndimage.gaussian_laplace(ascent, sigma=1) + >>> ax1.imshow(result) + + >>> result = ndimage.gaussian_laplace(ascent, sigma=3) + >>> ax2.imshow(result) + >>> plt.show() + """ + input = np.asarray(input) + + def derivative2(input, axis, output, mode, cval, sigma, **kwargs): + order = [0] * input.ndim + order[axis] = 2 + return gaussian_filter(input, sigma, order, output, mode, cval, + **kwargs) + + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + sigma = _ni_support._normalize_sequence(sigma, num_axes) + if num_axes < input.ndim: + # set sigma = 0 for any axes not being filtered + sigma_temp = [0,] * input.ndim + for s, ax in zip(sigma, axes): + sigma_temp[ax] = s + sigma = sigma_temp + + return generic_laplace(input, derivative2, output, mode, cval, + extra_arguments=(sigma,), + extra_keywords=kwargs, + axes=axes) + + +@_ni_docstrings.docfiller +def generic_gradient_magnitude(input, derivative, output=None, + mode="reflect", cval=0.0, + extra_arguments=(), extra_keywords=None, + *, axes=None): + """Gradient magnitude using a provided gradient function. + + Parameters + ---------- + %(input)s + derivative : callable + Callable with the following signature:: + + derivative(input, axis, output, mode, cval, + *extra_arguments, **extra_keywords) + + See `extra_arguments`, `extra_keywords` below. + `derivative` can assume that `input` and `output` are ndarrays. + Note that the output from `derivative` is modified inplace; + be careful to copy important inputs before returning them. + %(output)s + %(mode_multiple)s + %(cval)s + %(extra_keywords)s + %(extra_arguments)s + axes : tuple of int or None + The axes over which to apply the filter. If a `mode` tuple is + provided, its length must match the number of axes. + + Returns + ------- + generic_gradient_matnitude : ndarray + Filtered array. Has the same shape as `input`. + + """ + if extra_keywords is None: + extra_keywords = {} + input = np.asarray(input) + output = _ni_support._get_output(output, input) + axes = _ni_support._check_axes(axes, input.ndim) + if len(axes) > 0: + modes = _ni_support._normalize_sequence(mode, len(axes)) + derivative(input, axes[0], output, modes[0], cval, + *extra_arguments, **extra_keywords) + np.multiply(output, output, output) + for ii in range(1, len(axes)): + tmp = derivative(input, axes[ii], output.dtype, modes[ii], cval, + *extra_arguments, **extra_keywords) + np.multiply(tmp, tmp, tmp) + output += tmp + # This allows the sqrt to work with a different default casting + np.sqrt(output, output, casting='unsafe') + else: + output[...] = input[...] + return output + + +@_ni_docstrings.docfiller +def gaussian_gradient_magnitude(input, sigma, output=None, + mode="reflect", cval=0.0, *, axes=None, + **kwargs): + """Multidimensional gradient magnitude using Gaussian derivatives. + + Parameters + ---------- + %(input)s + sigma : scalar or sequence of scalars + The standard deviations of the Gaussian filter are given for + each axis as a sequence, or as a single number, in which case + it is equal for all axes. + %(output)s + %(mode_multiple)s + %(cval)s + axes : tuple of int or None + The axes over which to apply the filter. If `sigma` or `mode` tuples + are provided, their length must match the number of axes. + Extra keyword arguments will be passed to gaussian_filter(). + + Returns + ------- + gaussian_gradient_magnitude : ndarray + Filtered array. Has the same shape as `input`. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> plt.gray() # show the filtered result in grayscale + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + >>> ascent = datasets.ascent() + >>> result = ndimage.gaussian_gradient_magnitude(ascent, sigma=5) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result) + >>> plt.show() + """ + input = np.asarray(input) + + def derivative(input, axis, output, mode, cval, sigma, **kwargs): + order = [0] * input.ndim + order[axis] = 1 + return gaussian_filter(input, sigma, order, output, mode, + cval, **kwargs) + + return generic_gradient_magnitude(input, derivative, output, mode, + cval, extra_arguments=(sigma,), + extra_keywords=kwargs, axes=axes) + + +def _correlate_or_convolve(input, weights, output, mode, cval, origin, + convolution, axes): + input = np.asarray(input) + weights = np.asarray(weights) + complex_input = input.dtype.kind == 'c' + complex_weights = weights.dtype.kind == 'c' + if complex_input or complex_weights: + if complex_weights and not convolution: + # As for np.correlate, conjugate weights rather than input. + weights = weights.conj() + kwargs = dict( + mode=mode, origin=origin, convolution=convolution, axes=axes + ) + output = _ni_support._get_output(output, input, complex_output=True) + + return _complex_via_real_components(_correlate_or_convolve, input, + weights, output, cval, **kwargs) + + axes = _ni_support._check_axes(axes, input.ndim) + weights = np.asarray(weights, dtype=np.float64) + + # expand weights and origins if num_axes < input.ndim + weights = _expand_footprint(input.ndim, axes, weights, "weights") + origins = _expand_origin(input.ndim, axes, origin) + + wshape = [ii for ii in weights.shape if ii > 0] + if len(wshape) != input.ndim: + raise RuntimeError(f"weights.ndim ({len(wshape)}) must match " + f"len(axes) ({len(axes)})") + if convolution: + weights = weights[tuple([slice(None, None, -1)] * weights.ndim)] + for ii in range(len(origins)): + origins[ii] = -origins[ii] + if not weights.shape[ii] & 1: + origins[ii] -= 1 + for origin, lenw in zip(origins, wshape): + if _invalid_origin(origin, lenw): + raise ValueError('Invalid origin; origin must satisfy ' + '-(weights.shape[k] // 2) <= origin[k] <= ' + '(weights.shape[k]-1) // 2') + + if not weights.flags.contiguous: + weights = weights.copy() + output = _ni_support._get_output(output, input) + temp_needed = np.may_share_memory(input, output) + if temp_needed: + # input and output arrays cannot share memory + temp = output + output = _ni_support._get_output(output.dtype, input) + if not isinstance(mode, str) and isinstance(mode, Iterable): + raise RuntimeError("A sequence of modes is not supported") + mode = _ni_support._extend_mode_to_code(mode) + _nd_image.correlate(input, weights, output, mode, cval, origins) + if temp_needed: + temp[...] = output + output = temp + return output + + +@_ni_docstrings.docfiller +def correlate(input, weights, output=None, mode='reflect', cval=0.0, + origin=0, *, axes=None): + """ + Multidimensional correlation. + + The array is correlated with the given kernel. + + Parameters + ---------- + %(input)s + weights : ndarray + array of weights, same number of dimensions as input + %(output)s + %(mode_reflect)s + %(cval)s + %(origin_multiple)s + axes : tuple of int or None, optional + If None, `input` is filtered along all axes. Otherwise, + `input` is filtered along the specified axes. When `axes` is + specified, any tuples used for `mode` or `origin` must match the length + of `axes`. The ith entry in any of these tuples corresponds to the ith + entry in `axes`. + + Returns + ------- + result : ndarray + The result of correlation of `input` with `weights`. + + See Also + -------- + convolve : Convolve an image with a kernel. + + Examples + -------- + Correlation is the process of moving a filter mask often referred to + as kernel over the image and computing the sum of products at each location. + + >>> from scipy.ndimage import correlate + >>> import numpy as np + >>> input_img = np.arange(25).reshape(5,5) + >>> print(input_img) + [[ 0 1 2 3 4] + [ 5 6 7 8 9] + [10 11 12 13 14] + [15 16 17 18 19] + [20 21 22 23 24]] + + Define a kernel (weights) for correlation. In this example, it is for sum of + center and up, down, left and right next elements. + + >>> weights = [[0, 1, 0], + ... [1, 1, 1], + ... [0, 1, 0]] + + We can calculate a correlation result: + For example, element ``[2,2]`` is ``7 + 11 + 12 + 13 + 17 = 60``. + + >>> correlate(input_img, weights) + array([[ 6, 10, 15, 20, 24], + [ 26, 30, 35, 40, 44], + [ 51, 55, 60, 65, 69], + [ 76, 80, 85, 90, 94], + [ 96, 100, 105, 110, 114]]) + + """ + return _correlate_or_convolve(input, weights, output, mode, cval, + origin, False, axes) + + +@_ni_docstrings.docfiller +def convolve(input, weights, output=None, mode='reflect', cval=0.0, + origin=0, *, axes=None): + """ + Multidimensional convolution. + + The array is convolved with the given kernel. + + Parameters + ---------- + %(input)s + weights : array_like + Array of weights, same number of dimensions as input + %(output)s + %(mode_reflect)s + cval : scalar, optional + Value to fill past edges of input if `mode` is 'constant'. Default + is 0.0 + origin : int or sequence, optional + Controls the placement of the filter on the input array's pixels. + A value of 0 (the default) centers the filter over the pixel, with + positive values shifting the filter to the right, and negative ones + to the left. By passing a sequence of origins with length equal to + the number of dimensions of the input array, different shifts can + be specified along each axis. + axes : tuple of int or None, optional + If None, `input` is filtered along all axes. Otherwise, + `input` is filtered along the specified axes. When `axes` is + specified, any tuples used for `mode` or `origin` must match the length + of `axes`. The ith entry in any of these tuples corresponds to the ith + entry in `axes`. + + Returns + ------- + result : ndarray + The result of convolution of `input` with `weights`. + + See Also + -------- + correlate : Correlate an image with a kernel. + + Notes + ----- + Each value in result is :math:`C_i = \\sum_j{I_{i+k-j} W_j}`, where + W is the `weights` kernel, + j is the N-D spatial index over :math:`W`, + I is the `input` and k is the coordinate of the center of + W, specified by `origin` in the input parameters. + + Examples + -------- + Perhaps the simplest case to understand is ``mode='constant', cval=0.0``, + because in this case borders (i.e., where the `weights` kernel, centered + on any one value, extends beyond an edge of `input`) are treated as zeros. + + >>> import numpy as np + >>> a = np.array([[1, 2, 0, 0], + ... [5, 3, 0, 4], + ... [0, 0, 0, 7], + ... [9, 3, 0, 0]]) + >>> k = np.array([[1,1,1],[1,1,0],[1,0,0]]) + >>> from scipy import ndimage + >>> ndimage.convolve(a, k, mode='constant', cval=0.0) + array([[11, 10, 7, 4], + [10, 3, 11, 11], + [15, 12, 14, 7], + [12, 3, 7, 0]]) + + Setting ``cval=1.0`` is equivalent to padding the outer edge of `input` + with 1.0's (and then extracting only the original region of the result). + + >>> ndimage.convolve(a, k, mode='constant', cval=1.0) + array([[13, 11, 8, 7], + [11, 3, 11, 14], + [16, 12, 14, 10], + [15, 6, 10, 5]]) + + With ``mode='reflect'`` (the default), outer values are reflected at the + edge of `input` to fill in missing values. + + >>> b = np.array([[2, 0, 0], + ... [1, 0, 0], + ... [0, 0, 0]]) + >>> k = np.array([[0,1,0], [0,1,0], [0,1,0]]) + >>> ndimage.convolve(b, k, mode='reflect') + array([[5, 0, 0], + [3, 0, 0], + [1, 0, 0]]) + + This includes diagonally at the corners. + + >>> k = np.array([[1,0,0],[0,1,0],[0,0,1]]) + >>> ndimage.convolve(b, k) + array([[4, 2, 0], + [3, 2, 0], + [1, 1, 0]]) + + With ``mode='nearest'``, the single nearest value in to an edge in + `input` is repeated as many times as needed to match the overlapping + `weights`. + + >>> c = np.array([[2, 0, 1], + ... [1, 0, 0], + ... [0, 0, 0]]) + >>> k = np.array([[0, 1, 0], + ... [0, 1, 0], + ... [0, 1, 0], + ... [0, 1, 0], + ... [0, 1, 0]]) + >>> ndimage.convolve(c, k, mode='nearest') + array([[7, 0, 3], + [5, 0, 2], + [3, 0, 1]]) + + """ + return _correlate_or_convolve(input, weights, output, mode, cval, + origin, True, axes) + + +@_ni_docstrings.docfiller +def uniform_filter1d(input, size, axis=-1, output=None, + mode="reflect", cval=0.0, origin=0): + """Calculate a 1-D uniform filter along the given axis. + + The lines of the array along the given axis are filtered with a + uniform filter of given size. + + Parameters + ---------- + %(input)s + size : int + length of uniform filter + %(axis)s + %(output)s + %(mode_reflect)s + %(cval)s + %(origin)s + + Returns + ------- + result : ndarray + Filtered array. Has same shape as `input`. + + Examples + -------- + >>> from scipy.ndimage import uniform_filter1d + >>> uniform_filter1d([2, 8, 0, 4, 1, 9, 9, 0], size=3) + array([4, 3, 4, 1, 4, 6, 6, 3]) + """ + input = np.asarray(input) + axis = normalize_axis_index(axis, input.ndim) + if size < 1: + raise RuntimeError('incorrect filter size') + complex_output = input.dtype.kind == 'c' + output = _ni_support._get_output(output, input, + complex_output=complex_output) + if (size // 2 + origin < 0) or (size // 2 + origin >= size): + raise ValueError('invalid origin') + mode = _ni_support._extend_mode_to_code(mode) + if not complex_output: + _nd_image.uniform_filter1d(input, size, axis, output, mode, cval, + origin) + else: + _nd_image.uniform_filter1d(input.real, size, axis, output.real, mode, + np.real(cval), origin) + _nd_image.uniform_filter1d(input.imag, size, axis, output.imag, mode, + np.imag(cval), origin) + return output + + +@_ni_docstrings.docfiller +def uniform_filter(input, size=3, output=None, mode="reflect", + cval=0.0, origin=0, *, axes=None): + """Multidimensional uniform filter. + + Parameters + ---------- + %(input)s + size : int or sequence of ints, optional + The sizes of the uniform filter are given for each axis as a + sequence, or as a single number, in which case the size is + equal for all axes. + %(output)s + %(mode_multiple)s + %(cval)s + %(origin_multiple)s + axes : tuple of int or None, optional + If None, `input` is filtered along all axes. Otherwise, + `input` is filtered along the specified axes. When `axes` is + specified, any tuples used for `size`, `origin`, and/or `mode` + must match the length of `axes`. The ith entry in any of these tuples + corresponds to the ith entry in `axes`. + + Returns + ------- + uniform_filter : ndarray + Filtered array. Has the same shape as `input`. + + Notes + ----- + The multidimensional filter is implemented as a sequence of + 1-D uniform filters. The intermediate arrays are stored + in the same data type as the output. Therefore, for output types + with a limited precision, the results may be imprecise because + intermediate results may be stored with insufficient precision. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> plt.gray() # show the filtered result in grayscale + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + >>> ascent = datasets.ascent() + >>> result = ndimage.uniform_filter(ascent, size=20) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result) + >>> plt.show() + """ + input = np.asarray(input) + output = _ni_support._get_output(output, input, + complex_output=input.dtype.kind == 'c') + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + sizes = _ni_support._normalize_sequence(size, num_axes) + origins = _ni_support._normalize_sequence(origin, num_axes) + modes = _ni_support._normalize_sequence(mode, num_axes) + axes = [(axes[ii], sizes[ii], origins[ii], modes[ii]) + for ii in range(num_axes) if sizes[ii] > 1] + if len(axes) > 0: + for axis, size, origin, mode in axes: + uniform_filter1d(input, int(size), axis, output, mode, + cval, origin) + input = output + else: + output[...] = input[...] + return output + + +@_ni_docstrings.docfiller +def minimum_filter1d(input, size, axis=-1, output=None, + mode="reflect", cval=0.0, origin=0): + """Calculate a 1-D minimum filter along the given axis. + + The lines of the array along the given axis are filtered with a + minimum filter of given size. + + Parameters + ---------- + %(input)s + size : int + length along which to calculate 1D minimum + %(axis)s + %(output)s + %(mode_reflect)s + %(cval)s + %(origin)s + + Returns + ------- + result : ndarray. + Filtered image. Has the same shape as `input`. + + Notes + ----- + This function implements the MINLIST algorithm [1]_, as described by + Richard Harter [2]_, and has a guaranteed O(n) performance, `n` being + the `input` length, regardless of filter size. + + References + ---------- + .. [1] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.2777 + .. [2] http://www.richardhartersworld.com/cri/2001/slidingmin.html + + + Examples + -------- + >>> from scipy.ndimage import minimum_filter1d + >>> minimum_filter1d([2, 8, 0, 4, 1, 9, 9, 0], size=3) + array([2, 0, 0, 0, 1, 1, 0, 0]) + """ + input = np.asarray(input) + if np.iscomplexobj(input): + raise TypeError('Complex type not supported') + axis = normalize_axis_index(axis, input.ndim) + if size < 1: + raise RuntimeError('incorrect filter size') + output = _ni_support._get_output(output, input) + if (size // 2 + origin < 0) or (size // 2 + origin >= size): + raise ValueError('invalid origin') + mode = _ni_support._extend_mode_to_code(mode) + _nd_image.min_or_max_filter1d(input, size, axis, output, mode, cval, + origin, 1) + return output + + +@_ni_docstrings.docfiller +def maximum_filter1d(input, size, axis=-1, output=None, + mode="reflect", cval=0.0, origin=0): + """Calculate a 1-D maximum filter along the given axis. + + The lines of the array along the given axis are filtered with a + maximum filter of given size. + + Parameters + ---------- + %(input)s + size : int + Length along which to calculate the 1-D maximum. + %(axis)s + %(output)s + %(mode_reflect)s + %(cval)s + %(origin)s + + Returns + ------- + maximum1d : ndarray, None + Maximum-filtered array with same shape as input. + None if `output` is not None + + Notes + ----- + This function implements the MAXLIST algorithm [1]_, as described by + Richard Harter [2]_, and has a guaranteed O(n) performance, `n` being + the `input` length, regardless of filter size. + + References + ---------- + .. [1] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.2777 + .. [2] http://www.richardhartersworld.com/cri/2001/slidingmin.html + + Examples + -------- + >>> from scipy.ndimage import maximum_filter1d + >>> maximum_filter1d([2, 8, 0, 4, 1, 9, 9, 0], size=3) + array([8, 8, 8, 4, 9, 9, 9, 9]) + """ + input = np.asarray(input) + if np.iscomplexobj(input): + raise TypeError('Complex type not supported') + axis = normalize_axis_index(axis, input.ndim) + if size < 1: + raise RuntimeError('incorrect filter size') + output = _ni_support._get_output(output, input) + if (size // 2 + origin < 0) or (size // 2 + origin >= size): + raise ValueError('invalid origin') + mode = _ni_support._extend_mode_to_code(mode) + _nd_image.min_or_max_filter1d(input, size, axis, output, mode, cval, + origin, 0) + return output + + +def _min_or_max_filter(input, size, footprint, structure, output, mode, + cval, origin, minimum, axes=None): + if (size is not None) and (footprint is not None): + warnings.warn("ignoring size because footprint is set", + UserWarning, stacklevel=3) + if structure is None: + if footprint is None: + if size is None: + raise RuntimeError("no footprint provided") + separable = True + else: + footprint = np.asarray(footprint, dtype=bool) + if not footprint.any(): + raise ValueError("All-zero footprint is not supported.") + if footprint.all(): + size = footprint.shape + footprint = None + separable = True + else: + separable = False + else: + structure = np.asarray(structure, dtype=np.float64) + separable = False + if footprint is None: + footprint = np.ones(structure.shape, bool) + else: + footprint = np.asarray(footprint, dtype=bool) + input = np.asarray(input) + if np.iscomplexobj(input): + raise TypeError("Complex type not supported") + output = _ni_support._get_output(output, input) + temp_needed = np.may_share_memory(input, output) + if temp_needed: + # input and output arrays cannot share memory + temp = output + output = _ni_support._get_output(output.dtype, input) + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + if separable: + origins = _ni_support._normalize_sequence(origin, num_axes) + sizes = _ni_support._normalize_sequence(size, num_axes) + modes = _ni_support._normalize_sequence(mode, num_axes) + axes = [(axes[ii], sizes[ii], origins[ii], modes[ii]) + for ii in range(len(axes)) if sizes[ii] > 1] + if minimum: + filter_ = minimum_filter1d + else: + filter_ = maximum_filter1d + if len(axes) > 0: + for axis, size, origin, mode in axes: + filter_(input, int(size), axis, output, mode, cval, origin) + input = output + else: + output[...] = input[...] + else: + # expand origins and footprint if num_axes < input.ndim + footprint = _expand_footprint(input.ndim, axes, footprint) + origins = _expand_origin(input.ndim, axes, origin) + + fshape = [ii for ii in footprint.shape if ii > 0] + if len(fshape) != input.ndim: + raise RuntimeError(f"footprint.ndim ({footprint.ndim}) must match " + f"len(axes) ({len(axes)})") + for origin, lenf in zip(origins, fshape): + if (lenf // 2 + origin < 0) or (lenf // 2 + origin >= lenf): + raise ValueError("invalid origin") + if not footprint.flags.contiguous: + footprint = footprint.copy() + if structure is not None: + if len(structure.shape) != num_axes: + raise RuntimeError("structure array has incorrect shape") + if num_axes != structure.ndim: + structure = np.expand_dims( + structure, + tuple(ax for ax in range(structure.ndim) if ax not in axes) + ) + if not structure.flags.contiguous: + structure = structure.copy() + if not isinstance(mode, str) and isinstance(mode, Iterable): + raise RuntimeError( + "A sequence of modes is not supported for non-separable " + "footprints") + mode = _ni_support._extend_mode_to_code(mode) + _nd_image.min_or_max_filter(input, footprint, structure, output, + mode, cval, origins, minimum) + if temp_needed: + temp[...] = output + output = temp + return output + + +@_ni_docstrings.docfiller +def minimum_filter(input, size=None, footprint=None, output=None, + mode="reflect", cval=0.0, origin=0, *, axes=None): + """Calculate a multidimensional minimum filter. + + Parameters + ---------- + %(input)s + %(size_foot)s + %(output)s + %(mode_multiple)s + %(cval)s + %(origin_multiple)s + axes : tuple of int or None, optional + If None, `input` is filtered along all axes. Otherwise, + `input` is filtered along the specified axes. When `axes` is + specified, any tuples used for `size`, `origin`, and/or `mode` + must match the length of `axes`. The ith entry in any of these tuples + corresponds to the ith entry in `axes`. + + Returns + ------- + minimum_filter : ndarray + Filtered array. Has the same shape as `input`. + + Notes + ----- + A sequence of modes (one per axis) is only supported when the footprint is + separable. Otherwise, a single mode string must be provided. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> plt.gray() # show the filtered result in grayscale + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + >>> ascent = datasets.ascent() + >>> result = ndimage.minimum_filter(ascent, size=20) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result) + >>> plt.show() + """ + return _min_or_max_filter(input, size, footprint, None, output, mode, + cval, origin, 1, axes) + + +@_ni_docstrings.docfiller +def maximum_filter(input, size=None, footprint=None, output=None, + mode="reflect", cval=0.0, origin=0, *, axes=None): + """Calculate a multidimensional maximum filter. + + Parameters + ---------- + %(input)s + %(size_foot)s + %(output)s + %(mode_multiple)s + %(cval)s + %(origin_multiple)s + axes : tuple of int or None, optional + If None, `input` is filtered along all axes. Otherwise, + `input` is filtered along the specified axes. When `axes` is + specified, any tuples used for `size`, `origin`, and/or `mode` + must match the length of `axes`. The ith entry in any of these tuples + corresponds to the ith entry in `axes`. + + Returns + ------- + maximum_filter : ndarray + Filtered array. Has the same shape as `input`. + + Notes + ----- + A sequence of modes (one per axis) is only supported when the footprint is + separable. Otherwise, a single mode string must be provided. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> plt.gray() # show the filtered result in grayscale + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + >>> ascent = datasets.ascent() + >>> result = ndimage.maximum_filter(ascent, size=20) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result) + >>> plt.show() + """ + return _min_or_max_filter(input, size, footprint, None, output, mode, + cval, origin, 0, axes) + + +@_ni_docstrings.docfiller +def _rank_filter(input, rank, size=None, footprint=None, output=None, + mode="reflect", cval=0.0, origin=0, operation='rank', + axes=None): + if (size is not None) and (footprint is not None): + warnings.warn("ignoring size because footprint is set", + UserWarning, stacklevel=3) + input = np.asarray(input) + if np.iscomplexobj(input): + raise TypeError('Complex type not supported') + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + if footprint is None: + if size is None: + raise RuntimeError("no footprint or filter size provided") + sizes = _ni_support._normalize_sequence(size, num_axes) + footprint = np.ones(sizes, dtype=bool) + else: + footprint = np.asarray(footprint, dtype=bool) + # expand origins, footprint and modes if num_axes < input.ndim + footprint = _expand_footprint(input.ndim, axes, footprint) + origins = _expand_origin(input.ndim, axes, origin) + mode = _expand_mode(input.ndim, axes, mode) + + fshape = [ii for ii in footprint.shape if ii > 0] + if len(fshape) != input.ndim: + raise RuntimeError(f"footprint.ndim ({footprint.ndim}) must match " + f"len(axes) ({len(axes)})") + for origin, lenf in zip(origins, fshape): + if (lenf // 2 + origin < 0) or (lenf // 2 + origin >= lenf): + raise ValueError('invalid origin') + if not footprint.flags.contiguous: + footprint = footprint.copy() + filter_size = np.where(footprint, 1, 0).sum() + if operation == 'median': + rank = filter_size // 2 + elif operation == 'percentile': + percentile = rank + if percentile < 0.0: + percentile += 100.0 + if percentile < 0 or percentile > 100: + raise RuntimeError('invalid percentile') + if percentile == 100.0: + rank = filter_size - 1 + else: + rank = int(float(filter_size) * percentile / 100.0) + if rank < 0: + rank += filter_size + if rank < 0 or rank >= filter_size: + raise RuntimeError('rank not within filter footprint size') + if rank == 0: + return minimum_filter(input, None, footprint, output, mode, cval, + origins, axes=None) + elif rank == filter_size - 1: + return maximum_filter(input, None, footprint, output, mode, cval, + origins, axes=None) + else: + output = _ni_support._get_output(output, input) + temp_needed = np.may_share_memory(input, output) + if temp_needed: + # input and output arrays cannot share memory + temp = output + output = _ni_support._get_output(output.dtype, input) + if not isinstance(mode, str) and isinstance(mode, Iterable): + raise RuntimeError( + "A sequence of modes is not supported by non-separable rank " + "filters") + mode = _ni_support._extend_mode_to_code(mode, is_filter=True) + if input.ndim == 1: + if input.dtype in (np.int64, np.float64, np.float32): + x = input + x_out = output + elif input.dtype == np.float16: + x = input.astype('float32') + x_out = np.empty(x.shape, dtype='float32') + elif np.result_type(input, np.int64) == np.int64: + x = input.astype('int64') + x_out = np.empty(x.shape, dtype='int64') + elif input.dtype.kind in 'biu': + # cast any other boolean, integer or unsigned type to int64 + x = input.astype('int64') + x_out = np.empty(x.shape, dtype='int64') + else: + raise RuntimeError('Unsupported array type') + cval = x.dtype.type(cval) + _rank_filter_1d.rank_filter(x, rank, footprint.size, x_out, mode, cval, + origin) + if input.dtype not in (np.int64, np.float64, np.float32): + np.copyto(output, x_out, casting='unsafe') + else: + _nd_image.rank_filter(input, rank, footprint, output, mode, cval, origins) + if temp_needed: + temp[...] = output + output = temp + return output + + +@_ni_docstrings.docfiller +def rank_filter(input, rank, size=None, footprint=None, output=None, + mode="reflect", cval=0.0, origin=0, *, axes=None): + """Calculate a multidimensional rank filter. + + Parameters + ---------- + %(input)s + rank : int + The rank parameter may be less than zero, i.e., rank = -1 + indicates the largest element. + %(size_foot)s + %(output)s + %(mode_reflect)s + %(cval)s + %(origin_multiple)s + axes : tuple of int or None, optional + If None, `input` is filtered along all axes. Otherwise, + `input` is filtered along the specified axes. When `axes` is + specified, any tuples used for `size`, `origin`, and/or `mode` + must match the length of `axes`. The ith entry in any of these tuples + corresponds to the ith entry in `axes`. + + Returns + ------- + rank_filter : ndarray + Filtered array. Has the same shape as `input`. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> plt.gray() # show the filtered result in grayscale + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + >>> ascent = datasets.ascent() + >>> result = ndimage.rank_filter(ascent, rank=42, size=20) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result) + >>> plt.show() + """ + rank = operator.index(rank) + return _rank_filter(input, rank, size, footprint, output, mode, cval, + origin, 'rank', axes=axes) + + +@_ni_docstrings.docfiller +def median_filter(input, size=None, footprint=None, output=None, + mode="reflect", cval=0.0, origin=0, *, axes=None): + """ + Calculate a multidimensional median filter. + + Parameters + ---------- + %(input)s + %(size_foot)s + %(output)s + %(mode_reflect)s + %(cval)s + %(origin_multiple)s + axes : tuple of int or None, optional + If None, `input` is filtered along all axes. Otherwise, + `input` is filtered along the specified axes. When `axes` is + specified, any tuples used for `size`, `origin`, and/or `mode` + must match the length of `axes`. The ith entry in any of these tuples + corresponds to the ith entry in `axes`. + + Returns + ------- + median_filter : ndarray + Filtered array. Has the same shape as `input`. + + See Also + -------- + scipy.signal.medfilt2d + + Notes + ----- + For 2-dimensional images with ``uint8``, ``float32`` or ``float64`` dtypes + the specialised function `scipy.signal.medfilt2d` may be faster. It is + however limited to constant mode with ``cval=0``. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> plt.gray() # show the filtered result in grayscale + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + >>> ascent = datasets.ascent() + >>> result = ndimage.median_filter(ascent, size=20) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result) + >>> plt.show() + """ + return _rank_filter(input, 0, size, footprint, output, mode, cval, + origin, 'median', axes=axes) + + +@_ni_docstrings.docfiller +def percentile_filter(input, percentile, size=None, footprint=None, + output=None, mode="reflect", cval=0.0, origin=0, *, + axes=None): + """Calculate a multidimensional percentile filter. + + Parameters + ---------- + %(input)s + percentile : scalar + The percentile parameter may be less than zero, i.e., + percentile = -20 equals percentile = 80 + %(size_foot)s + %(output)s + %(mode_reflect)s + %(cval)s + %(origin_multiple)s + axes : tuple of int or None, optional + If None, `input` is filtered along all axes. Otherwise, + `input` is filtered along the specified axes. When `axes` is + specified, any tuples used for `size`, `origin`, and/or `mode` + must match the length of `axes`. The ith entry in any of these tuples + corresponds to the ith entry in `axes`. + + Returns + ------- + percentile_filter : ndarray + Filtered array. Has the same shape as `input`. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> plt.gray() # show the filtered result in grayscale + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + >>> ascent = datasets.ascent() + >>> result = ndimage.percentile_filter(ascent, percentile=20, size=20) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result) + >>> plt.show() + """ + return _rank_filter(input, percentile, size, footprint, output, mode, + cval, origin, 'percentile', axes=axes) + + +@_ni_docstrings.docfiller +def generic_filter1d(input, function, filter_size, axis=-1, + output=None, mode="reflect", cval=0.0, origin=0, + extra_arguments=(), extra_keywords=None): + """Calculate a 1-D filter along the given axis. + + `generic_filter1d` iterates over the lines of the array, calling the + given function at each line. The arguments of the line are the + input line, and the output line. The input and output lines are 1-D + double arrays. The input line is extended appropriately according + to the filter size and origin. The output line must be modified + in-place with the result. + + Parameters + ---------- + %(input)s + function : {callable, scipy.LowLevelCallable} + Function to apply along given axis. + filter_size : scalar + Length of the filter. + %(axis)s + %(output)s + %(mode_reflect)s + %(cval)s + %(origin)s + %(extra_arguments)s + %(extra_keywords)s + + Returns + ------- + generic_filter1d : ndarray + Filtered array. Has the same shape as `input`. + + Notes + ----- + This function also accepts low-level callback functions with one of + the following signatures and wrapped in `scipy.LowLevelCallable`: + + .. code:: c + + int function(double *input_line, npy_intp input_length, + double *output_line, npy_intp output_length, + void *user_data) + int function(double *input_line, intptr_t input_length, + double *output_line, intptr_t output_length, + void *user_data) + + The calling function iterates over the lines of the input and output + arrays, calling the callback function at each line. The current line + is extended according to the border conditions set by the calling + function, and the result is copied into the array that is passed + through ``input_line``. The length of the input line (after extension) + is passed through ``input_length``. The callback function should apply + the filter and store the result in the array passed through + ``output_line``. The length of the output line is passed through + ``output_length``. ``user_data`` is the data pointer provided + to `scipy.LowLevelCallable` as-is. + + The callback function must return an integer error status that is zero + if something went wrong and one otherwise. If an error occurs, you should + normally set the python error status with an informative message + before returning, otherwise a default error message is set by the + calling function. + + In addition, some other low-level function pointer specifications + are accepted, but these are for backward compatibility only and should + not be used in new code. + + """ + if extra_keywords is None: + extra_keywords = {} + input = np.asarray(input) + if np.iscomplexobj(input): + raise TypeError('Complex type not supported') + output = _ni_support._get_output(output, input) + if filter_size < 1: + raise RuntimeError('invalid filter size') + axis = normalize_axis_index(axis, input.ndim) + if (filter_size // 2 + origin < 0) or (filter_size // 2 + origin >= + filter_size): + raise ValueError('invalid origin') + mode = _ni_support._extend_mode_to_code(mode) + _nd_image.generic_filter1d(input, function, filter_size, axis, output, + mode, cval, origin, extra_arguments, + extra_keywords) + return output + + +@_ni_docstrings.docfiller +def generic_filter(input, function, size=None, footprint=None, + output=None, mode="reflect", cval=0.0, origin=0, + extra_arguments=(), extra_keywords=None, *, axes=None): + """Calculate a multidimensional filter using the given function. + + At each element the provided function is called. The input values + within the filter footprint at that element are passed to the function + as a 1-D array of double values. + + Parameters + ---------- + %(input)s + function : {callable, scipy.LowLevelCallable} + Function to apply at each element. + %(size_foot)s + %(output)s + %(mode_reflect)s + %(cval)s + %(origin_multiple)s + %(extra_arguments)s + %(extra_keywords)s + axes : tuple of int or None, optional + If None, `input` is filtered along all axes. Otherwise, + `input` is filtered along the specified axes. When `axes` is + specified, any tuples used for `size` or `origin` must match the length + of `axes`. The ith entry in any of these tuples corresponds to the ith + entry in `axes`. + + Returns + ------- + generic_filter : ndarray + Filtered array. Has the same shape as `input`. + + Notes + ----- + This function also accepts low-level callback functions with one of + the following signatures and wrapped in `scipy.LowLevelCallable`: + + .. code:: c + + int callback(double *buffer, npy_intp filter_size, + double *return_value, void *user_data) + int callback(double *buffer, intptr_t filter_size, + double *return_value, void *user_data) + + The calling function iterates over the elements of the input and + output arrays, calling the callback function at each element. The + elements within the footprint of the filter at the current element are + passed through the ``buffer`` parameter, and the number of elements + within the footprint through ``filter_size``. The calculated value is + returned in ``return_value``. ``user_data`` is the data pointer provided + to `scipy.LowLevelCallable` as-is. + + The callback function must return an integer error status that is zero + if something went wrong and one otherwise. If an error occurs, you should + normally set the python error status with an informative message + before returning, otherwise a default error message is set by the + calling function. + + In addition, some other low-level function pointer specifications + are accepted, but these are for backward compatibility only and should + not be used in new code. + + Examples + -------- + Import the necessary modules and load the example image used for + filtering. + + >>> import numpy as np + >>> from scipy import datasets + >>> from scipy.ndimage import zoom, generic_filter + >>> import matplotlib.pyplot as plt + >>> ascent = zoom(datasets.ascent(), 0.5) + + Compute a maximum filter with kernel size 5 by passing a simple NumPy + aggregation function as argument to `function`. + + >>> maximum_filter_result = generic_filter(ascent, np.amax, [5, 5]) + + While a maximum filter could also directly be obtained using + `maximum_filter`, `generic_filter` allows generic Python function or + `scipy.LowLevelCallable` to be used as a filter. Here, we compute the + range between maximum and minimum value as an example for a kernel size + of 5. + + >>> def custom_filter(image): + ... return np.amax(image) - np.amin(image) + >>> custom_filter_result = generic_filter(ascent, custom_filter, [5, 5]) + + Plot the original and filtered images. + + >>> fig, axes = plt.subplots(3, 1, figsize=(3, 9)) + >>> plt.gray() # show the filtered result in grayscale + >>> top, middle, bottom = axes + >>> for ax in axes: + ... ax.set_axis_off() # remove coordinate system + >>> top.imshow(ascent) + >>> top.set_title("Original image") + >>> middle.imshow(maximum_filter_result) + >>> middle.set_title("Maximum filter, Kernel: 5x5") + >>> bottom.imshow(custom_filter_result) + >>> bottom.set_title("Custom filter, Kernel: 5x5") + >>> fig.tight_layout() + + """ + if (size is not None) and (footprint is not None): + warnings.warn("ignoring size because footprint is set", + UserWarning, stacklevel=2) + if extra_keywords is None: + extra_keywords = {} + input = np.asarray(input) + if np.iscomplexobj(input): + raise TypeError('Complex type not supported') + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + if footprint is None: + if size is None: + raise RuntimeError("no footprint or filter size provided") + sizes = _ni_support._normalize_sequence(size, num_axes) + footprint = np.ones(sizes, dtype=bool) + else: + footprint = np.asarray(footprint, dtype=bool) + + # expand origins, footprint if num_axes < input.ndim + footprint = _expand_footprint(input.ndim, axes, footprint) + origins = _expand_origin(input.ndim, axes, origin) + + fshape = [ii for ii in footprint.shape if ii > 0] + if len(fshape) != input.ndim: + raise RuntimeError(f"footprint.ndim ({footprint.ndim}) " + f"must match len(axes) ({num_axes})") + for origin, lenf in zip(origins, fshape): + if (lenf // 2 + origin < 0) or (lenf // 2 + origin >= lenf): + raise ValueError('invalid origin') + if not footprint.flags.contiguous: + footprint = footprint.copy() + output = _ni_support._get_output(output, input) + + mode = _ni_support._extend_mode_to_code(mode) + _nd_image.generic_filter(input, function, footprint, output, mode, + cval, origins, extra_arguments, extra_keywords) + return output diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_fourier.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_fourier.py new file mode 100644 index 0000000000000000000000000000000000000000..bb5ffa6b9287cb740611aefba5f1f322011518cf --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_fourier.py @@ -0,0 +1,306 @@ +# Copyright (C) 2003-2005 Peter J. Verveer +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions +# are met: +# +# 1. Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# +# 2. Redistributions in binary form must reproduce the above +# copyright notice, this list of conditions and the following +# disclaimer in the documentation and/or other materials provided +# with the distribution. +# +# 3. The name of the author may not be used to endorse or promote +# products derived from this software without specific prior +# written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS +# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY +# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE +# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, +# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +import numpy as np +from scipy._lib._util import normalize_axis_index +from . import _ni_support +from . import _nd_image + +__all__ = ['fourier_gaussian', 'fourier_uniform', 'fourier_ellipsoid', + 'fourier_shift'] + + +def _get_output_fourier(output, input): + if output is None: + if input.dtype.type in [np.complex64, np.complex128, np.float32]: + output = np.zeros(input.shape, dtype=input.dtype) + else: + output = np.zeros(input.shape, dtype=np.float64) + elif type(output) is type: + if output not in [np.complex64, np.complex128, + np.float32, np.float64]: + raise RuntimeError("output type not supported") + output = np.zeros(input.shape, dtype=output) + elif output.shape != input.shape: + raise RuntimeError("output shape not correct") + return output + + +def _get_output_fourier_complex(output, input): + if output is None: + if input.dtype.type in [np.complex64, np.complex128]: + output = np.zeros(input.shape, dtype=input.dtype) + else: + output = np.zeros(input.shape, dtype=np.complex128) + elif type(output) is type: + if output not in [np.complex64, np.complex128]: + raise RuntimeError("output type not supported") + output = np.zeros(input.shape, dtype=output) + elif output.shape != input.shape: + raise RuntimeError("output shape not correct") + return output + + +def fourier_gaussian(input, sigma, n=-1, axis=-1, output=None): + """ + Multidimensional Gaussian fourier filter. + + The array is multiplied with the fourier transform of a Gaussian + kernel. + + Parameters + ---------- + input : array_like + The input array. + sigma : float or sequence + The sigma of the Gaussian kernel. If a float, `sigma` is the same for + all axes. If a sequence, `sigma` has to contain one value for each + axis. + n : int, optional + If `n` is negative (default), then the input is assumed to be the + result of a complex fft. + If `n` is larger than or equal to zero, the input is assumed to be the + result of a real fft, and `n` gives the length of the array before + transformation along the real transform direction. + axis : int, optional + The axis of the real transform. + output : ndarray, optional + If given, the result of filtering the input is placed in this array. + + Returns + ------- + fourier_gaussian : ndarray + The filtered input. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import numpy.fft + >>> import matplotlib.pyplot as plt + >>> fig, (ax1, ax2) = plt.subplots(1, 2) + >>> plt.gray() # show the filtered result in grayscale + >>> ascent = datasets.ascent() + >>> input_ = numpy.fft.fft2(ascent) + >>> result = ndimage.fourier_gaussian(input_, sigma=4) + >>> result = numpy.fft.ifft2(result) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result.real) # the imaginary part is an artifact + >>> plt.show() + """ + input = np.asarray(input) + output = _get_output_fourier(output, input) + axis = normalize_axis_index(axis, input.ndim) + sigmas = _ni_support._normalize_sequence(sigma, input.ndim) + sigmas = np.asarray(sigmas, dtype=np.float64) + if not sigmas.flags.contiguous: + sigmas = sigmas.copy() + + _nd_image.fourier_filter(input, sigmas, n, axis, output, 0) + return output + + +def fourier_uniform(input, size, n=-1, axis=-1, output=None): + """ + Multidimensional uniform fourier filter. + + The array is multiplied with the Fourier transform of a box of given + size. + + Parameters + ---------- + input : array_like + The input array. + size : float or sequence + The size of the box used for filtering. + If a float, `size` is the same for all axes. If a sequence, `size` has + to contain one value for each axis. + n : int, optional + If `n` is negative (default), then the input is assumed to be the + result of a complex fft. + If `n` is larger than or equal to zero, the input is assumed to be the + result of a real fft, and `n` gives the length of the array before + transformation along the real transform direction. + axis : int, optional + The axis of the real transform. + output : ndarray, optional + If given, the result of filtering the input is placed in this array. + + Returns + ------- + fourier_uniform : ndarray + The filtered input. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import numpy.fft + >>> import matplotlib.pyplot as plt + >>> fig, (ax1, ax2) = plt.subplots(1, 2) + >>> plt.gray() # show the filtered result in grayscale + >>> ascent = datasets.ascent() + >>> input_ = numpy.fft.fft2(ascent) + >>> result = ndimage.fourier_uniform(input_, size=20) + >>> result = numpy.fft.ifft2(result) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result.real) # the imaginary part is an artifact + >>> plt.show() + """ + input = np.asarray(input) + output = _get_output_fourier(output, input) + axis = normalize_axis_index(axis, input.ndim) + sizes = _ni_support._normalize_sequence(size, input.ndim) + sizes = np.asarray(sizes, dtype=np.float64) + if not sizes.flags.contiguous: + sizes = sizes.copy() + _nd_image.fourier_filter(input, sizes, n, axis, output, 1) + return output + + +def fourier_ellipsoid(input, size, n=-1, axis=-1, output=None): + """ + Multidimensional ellipsoid Fourier filter. + + The array is multiplied with the fourier transform of an ellipsoid of + given sizes. + + Parameters + ---------- + input : array_like + The input array. + size : float or sequence + The size of the box used for filtering. + If a float, `size` is the same for all axes. If a sequence, `size` has + to contain one value for each axis. + n : int, optional + If `n` is negative (default), then the input is assumed to be the + result of a complex fft. + If `n` is larger than or equal to zero, the input is assumed to be the + result of a real fft, and `n` gives the length of the array before + transformation along the real transform direction. + axis : int, optional + The axis of the real transform. + output : ndarray, optional + If given, the result of filtering the input is placed in this array. + + Returns + ------- + fourier_ellipsoid : ndarray + The filtered input. + + Notes + ----- + This function is implemented for arrays of rank 1, 2, or 3. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import numpy.fft + >>> import matplotlib.pyplot as plt + >>> fig, (ax1, ax2) = plt.subplots(1, 2) + >>> plt.gray() # show the filtered result in grayscale + >>> ascent = datasets.ascent() + >>> input_ = numpy.fft.fft2(ascent) + >>> result = ndimage.fourier_ellipsoid(input_, size=20) + >>> result = numpy.fft.ifft2(result) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result.real) # the imaginary part is an artifact + >>> plt.show() + """ + input = np.asarray(input) + if input.ndim > 3: + raise NotImplementedError("Only 1d, 2d and 3d inputs are supported") + output = _get_output_fourier(output, input) + if output.size == 0: + # The C code has a bug that can result in a segfault with arrays + # that have size 0 (gh-17270), so check here. + return output + axis = normalize_axis_index(axis, input.ndim) + sizes = _ni_support._normalize_sequence(size, input.ndim) + sizes = np.asarray(sizes, dtype=np.float64) + if not sizes.flags.contiguous: + sizes = sizes.copy() + _nd_image.fourier_filter(input, sizes, n, axis, output, 2) + return output + + +def fourier_shift(input, shift, n=-1, axis=-1, output=None): + """ + Multidimensional Fourier shift filter. + + The array is multiplied with the Fourier transform of a shift operation. + + Parameters + ---------- + input : array_like + The input array. + shift : float or sequence + The size of the box used for filtering. + If a float, `shift` is the same for all axes. If a sequence, `shift` + has to contain one value for each axis. + n : int, optional + If `n` is negative (default), then the input is assumed to be the + result of a complex fft. + If `n` is larger than or equal to zero, the input is assumed to be the + result of a real fft, and `n` gives the length of the array before + transformation along the real transform direction. + axis : int, optional + The axis of the real transform. + output : ndarray, optional + If given, the result of shifting the input is placed in this array. + + Returns + ------- + fourier_shift : ndarray + The shifted input. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> import numpy.fft + >>> fig, (ax1, ax2) = plt.subplots(1, 2) + >>> plt.gray() # show the filtered result in grayscale + >>> ascent = datasets.ascent() + >>> input_ = numpy.fft.fft2(ascent) + >>> result = ndimage.fourier_shift(input_, shift=200) + >>> result = numpy.fft.ifft2(result) + >>> ax1.imshow(ascent) + >>> ax2.imshow(result.real) # the imaginary part is an artifact + >>> plt.show() + """ + input = np.asarray(input) + output = _get_output_fourier_complex(output, input) + axis = normalize_axis_index(axis, input.ndim) + shifts = _ni_support._normalize_sequence(shift, input.ndim) + shifts = np.asarray(shifts, dtype=np.float64) + if not shifts.flags.contiguous: + shifts = shifts.copy() + _nd_image.fourier_shift(input, shifts, n, axis, output) + return output diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_interpolation.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_interpolation.py new file mode 100644 index 0000000000000000000000000000000000000000..4e4ea94184871fe87f848532b21e2def29bd406b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_interpolation.py @@ -0,0 +1,1003 @@ +# Copyright (C) 2003-2005 Peter J. Verveer +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions +# are met: +# +# 1. Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# +# 2. Redistributions in binary form must reproduce the above +# copyright notice, this list of conditions and the following +# disclaimer in the documentation and/or other materials provided +# with the distribution. +# +# 3. The name of the author may not be used to endorse or promote +# products derived from this software without specific prior +# written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS +# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY +# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE +# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, +# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +import itertools +import warnings + +import numpy as np +from scipy._lib._util import normalize_axis_index + +from scipy import special +from . import _ni_support +from . import _nd_image +from ._ni_docstrings import docfiller + + +__all__ = ['spline_filter1d', 'spline_filter', 'geometric_transform', + 'map_coordinates', 'affine_transform', 'shift', 'zoom', 'rotate'] + + +@docfiller +def spline_filter1d(input, order=3, axis=-1, output=np.float64, + mode='mirror'): + """ + Calculate a 1-D spline filter along the given axis. + + The lines of the array along the given axis are filtered by a + spline filter. The order of the spline must be >= 2 and <= 5. + + Parameters + ---------- + %(input)s + order : int, optional + The order of the spline, default is 3. + axis : int, optional + The axis along which the spline filter is applied. Default is the last + axis. + output : ndarray or dtype, optional + The array in which to place the output, or the dtype of the returned + array. Default is ``numpy.float64``. + %(mode_interp_mirror)s + + Returns + ------- + spline_filter1d : ndarray + The filtered input. + + See Also + -------- + spline_filter : Multidimensional spline filter. + + Notes + ----- + All of the interpolation functions in `ndimage` do spline interpolation of + the input image. If using B-splines of `order > 1`, the input image + values have to be converted to B-spline coefficients first, which is + done by applying this 1-D filter sequentially along all + axes of the input. All functions that require B-spline coefficients + will automatically filter their inputs, a behavior controllable with + the `prefilter` keyword argument. For functions that accept a `mode` + parameter, the result will only be correct if it matches the `mode` + used when filtering. + + For complex-valued `input`, this function processes the real and imaginary + components independently. + + .. versionadded:: 1.6.0 + Complex-valued support added. + + Examples + -------- + We can filter an image using 1-D spline along the given axis: + + >>> from scipy.ndimage import spline_filter1d + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> orig_img = np.eye(20) # create an image + >>> orig_img[10, :] = 1.0 + >>> sp_filter_axis_0 = spline_filter1d(orig_img, axis=0) + >>> sp_filter_axis_1 = spline_filter1d(orig_img, axis=1) + >>> f, ax = plt.subplots(1, 3, sharex=True) + >>> for ind, data in enumerate([[orig_img, "original image"], + ... [sp_filter_axis_0, "spline filter (axis=0)"], + ... [sp_filter_axis_1, "spline filter (axis=1)"]]): + ... ax[ind].imshow(data[0], cmap='gray_r') + ... ax[ind].set_title(data[1]) + >>> plt.tight_layout() + >>> plt.show() + + """ + if order < 0 or order > 5: + raise RuntimeError('spline order not supported') + input = np.asarray(input) + complex_output = np.iscomplexobj(input) + output = _ni_support._get_output(output, input, + complex_output=complex_output) + if complex_output: + spline_filter1d(input.real, order, axis, output.real, mode) + spline_filter1d(input.imag, order, axis, output.imag, mode) + return output + if order in [0, 1]: + output[...] = np.array(input) + else: + mode = _ni_support._extend_mode_to_code(mode) + axis = normalize_axis_index(axis, input.ndim) + _nd_image.spline_filter1d(input, order, axis, output, mode) + return output + +@docfiller +def spline_filter(input, order=3, output=np.float64, mode='mirror'): + """ + Multidimensional spline filter. + + Parameters + ---------- + %(input)s + order : int, optional + The order of the spline, default is 3. + output : ndarray or dtype, optional + The array in which to place the output, or the dtype of the returned + array. Default is ``numpy.float64``. + %(mode_interp_mirror)s + + Returns + ------- + spline_filter : ndarray + Filtered array. Has the same shape as `input`. + + See Also + -------- + spline_filter1d : Calculate a 1-D spline filter along the given axis. + + Notes + ----- + The multidimensional filter is implemented as a sequence of + 1-D spline filters. The intermediate arrays are stored + in the same data type as the output. Therefore, for output types + with a limited precision, the results may be imprecise because + intermediate results may be stored with insufficient precision. + + For complex-valued `input`, this function processes the real and imaginary + components independently. + + .. versionadded:: 1.6.0 + Complex-valued support added. + + Examples + -------- + We can filter an image using multidimensional splines: + + >>> from scipy.ndimage import spline_filter + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> orig_img = np.eye(20) # create an image + >>> orig_img[10, :] = 1.0 + >>> sp_filter = spline_filter(orig_img, order=3) + >>> f, ax = plt.subplots(1, 2, sharex=True) + >>> for ind, data in enumerate([[orig_img, "original image"], + ... [sp_filter, "spline filter"]]): + ... ax[ind].imshow(data[0], cmap='gray_r') + ... ax[ind].set_title(data[1]) + >>> plt.tight_layout() + >>> plt.show() + + """ + if order < 2 or order > 5: + raise RuntimeError('spline order not supported') + input = np.asarray(input) + complex_output = np.iscomplexobj(input) + output = _ni_support._get_output(output, input, + complex_output=complex_output) + if complex_output: + spline_filter(input.real, order, output.real, mode) + spline_filter(input.imag, order, output.imag, mode) + return output + if order not in [0, 1] and input.ndim > 0: + for axis in range(input.ndim): + spline_filter1d(input, order, axis, output=output, mode=mode) + input = output + else: + output[...] = input[...] + return output + + +def _prepad_for_spline_filter(input, mode, cval): + if mode in ['nearest', 'grid-constant']: + npad = 12 + if mode == 'grid-constant': + padded = np.pad(input, npad, mode='constant', + constant_values=cval) + elif mode == 'nearest': + padded = np.pad(input, npad, mode='edge') + else: + # other modes have exact boundary conditions implemented so + # no prepadding is needed + npad = 0 + padded = input + return padded, npad + + +@docfiller +def geometric_transform(input, mapping, output_shape=None, + output=None, order=3, + mode='constant', cval=0.0, prefilter=True, + extra_arguments=(), extra_keywords=None): + """ + Apply an arbitrary geometric transform. + + The given mapping function is used to find, for each point in the + output, the corresponding coordinates in the input. The value of the + input at those coordinates is determined by spline interpolation of + the requested order. + + Parameters + ---------- + %(input)s + mapping : {callable, scipy.LowLevelCallable} + A callable object that accepts a tuple of length equal to the output + array rank, and returns the corresponding input coordinates as a tuple + of length equal to the input array rank. + output_shape : tuple of ints, optional + Shape tuple. + %(output)s + order : int, optional + The order of the spline interpolation, default is 3. + The order has to be in the range 0-5. + %(mode_interp_constant)s + %(cval)s + %(prefilter)s + extra_arguments : tuple, optional + Extra arguments passed to `mapping`. + extra_keywords : dict, optional + Extra keywords passed to `mapping`. + + Returns + ------- + output : ndarray + The filtered input. + + See Also + -------- + map_coordinates, affine_transform, spline_filter1d + + + Notes + ----- + This function also accepts low-level callback functions with one + the following signatures and wrapped in `scipy.LowLevelCallable`: + + .. code:: c + + int mapping(npy_intp *output_coordinates, double *input_coordinates, + int output_rank, int input_rank, void *user_data) + int mapping(intptr_t *output_coordinates, double *input_coordinates, + int output_rank, int input_rank, void *user_data) + + The calling function iterates over the elements of the output array, + calling the callback function at each element. The coordinates of the + current output element are passed through ``output_coordinates``. The + callback function must return the coordinates at which the input must + be interpolated in ``input_coordinates``. The rank of the input and + output arrays are given by ``input_rank`` and ``output_rank`` + respectively. ``user_data`` is the data pointer provided + to `scipy.LowLevelCallable` as-is. + + The callback function must return an integer error status that is zero + if something went wrong and one otherwise. If an error occurs, you should + normally set the Python error status with an informative message + before returning, otherwise a default error message is set by the + calling function. + + In addition, some other low-level function pointer specifications + are accepted, but these are for backward compatibility only and should + not be used in new code. + + For complex-valued `input`, this function transforms the real and imaginary + components independently. + + .. versionadded:: 1.6.0 + Complex-valued support added. + + Examples + -------- + >>> import numpy as np + >>> from scipy.ndimage import geometric_transform + >>> a = np.arange(12.).reshape((4, 3)) + >>> def shift_func(output_coords): + ... return (output_coords[0] - 0.5, output_coords[1] - 0.5) + ... + >>> geometric_transform(a, shift_func) + array([[ 0. , 0. , 0. ], + [ 0. , 1.362, 2.738], + [ 0. , 4.812, 6.187], + [ 0. , 8.263, 9.637]]) + + >>> b = [1, 2, 3, 4, 5] + >>> def shift_func(output_coords): + ... return (output_coords[0] - 3,) + ... + >>> geometric_transform(b, shift_func, mode='constant') + array([0, 0, 0, 1, 2]) + >>> geometric_transform(b, shift_func, mode='nearest') + array([1, 1, 1, 1, 2]) + >>> geometric_transform(b, shift_func, mode='reflect') + array([3, 2, 1, 1, 2]) + >>> geometric_transform(b, shift_func, mode='wrap') + array([2, 3, 4, 1, 2]) + + """ + if extra_keywords is None: + extra_keywords = {} + if order < 0 or order > 5: + raise RuntimeError('spline order not supported') + input = np.asarray(input) + if output_shape is None: + output_shape = input.shape + if input.ndim < 1 or len(output_shape) < 1: + raise RuntimeError('input and output rank must be > 0') + complex_output = np.iscomplexobj(input) + output = _ni_support._get_output(output, input, shape=output_shape, + complex_output=complex_output) + if complex_output: + kwargs = dict(order=order, mode=mode, prefilter=prefilter, + output_shape=output_shape, + extra_arguments=extra_arguments, + extra_keywords=extra_keywords) + geometric_transform(input.real, mapping, output=output.real, + cval=np.real(cval), **kwargs) + geometric_transform(input.imag, mapping, output=output.imag, + cval=np.imag(cval), **kwargs) + return output + + if prefilter and order > 1: + padded, npad = _prepad_for_spline_filter(input, mode, cval) + filtered = spline_filter(padded, order, output=np.float64, + mode=mode) + else: + npad = 0 + filtered = input + mode = _ni_support._extend_mode_to_code(mode) + _nd_image.geometric_transform(filtered, mapping, None, None, None, output, + order, mode, cval, npad, extra_arguments, + extra_keywords) + return output + + +@docfiller +def map_coordinates(input, coordinates, output=None, order=3, + mode='constant', cval=0.0, prefilter=True): + """ + Map the input array to new coordinates by interpolation. + + The array of coordinates is used to find, for each point in the output, + the corresponding coordinates in the input. The value of the input at + those coordinates is determined by spline interpolation of the + requested order. + + The shape of the output is derived from that of the coordinate + array by dropping the first axis. The values of the array along + the first axis are the coordinates in the input array at which the + output value is found. + + Parameters + ---------- + %(input)s + coordinates : array_like + The coordinates at which `input` is evaluated. + %(output)s + order : int, optional + The order of the spline interpolation, default is 3. + The order has to be in the range 0-5. + %(mode_interp_constant)s + %(cval)s + %(prefilter)s + + Returns + ------- + map_coordinates : ndarray + The result of transforming the input. The shape of the output is + derived from that of `coordinates` by dropping the first axis. + + See Also + -------- + spline_filter, geometric_transform, scipy.interpolate + + Notes + ----- + For complex-valued `input`, this function maps the real and imaginary + components independently. + + .. versionadded:: 1.6.0 + Complex-valued support added. + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.arange(12.).reshape((4, 3)) + >>> a + array([[ 0., 1., 2.], + [ 3., 4., 5.], + [ 6., 7., 8.], + [ 9., 10., 11.]]) + >>> ndimage.map_coordinates(a, [[0.5, 2], [0.5, 1]], order=1) + array([ 2., 7.]) + + Above, the interpolated value of a[0.5, 0.5] gives output[0], while + a[2, 1] is output[1]. + + >>> inds = np.array([[0.5, 2], [0.5, 4]]) + >>> ndimage.map_coordinates(a, inds, order=1, cval=-33.3) + array([ 2. , -33.3]) + >>> ndimage.map_coordinates(a, inds, order=1, mode='nearest') + array([ 2., 8.]) + >>> ndimage.map_coordinates(a, inds, order=1, cval=0, output=bool) + array([ True, False], dtype=bool) + + """ + if order < 0 or order > 5: + raise RuntimeError('spline order not supported') + input = np.asarray(input) + coordinates = np.asarray(coordinates) + if np.iscomplexobj(coordinates): + raise TypeError('Complex type not supported') + output_shape = coordinates.shape[1:] + if input.ndim < 1 or len(output_shape) < 1: + raise RuntimeError('input and output rank must be > 0') + if coordinates.shape[0] != input.ndim: + raise RuntimeError('invalid shape for coordinate array') + complex_output = np.iscomplexobj(input) + output = _ni_support._get_output(output, input, shape=output_shape, + complex_output=complex_output) + if complex_output: + kwargs = dict(order=order, mode=mode, prefilter=prefilter) + map_coordinates(input.real, coordinates, output=output.real, + cval=np.real(cval), **kwargs) + map_coordinates(input.imag, coordinates, output=output.imag, + cval=np.imag(cval), **kwargs) + return output + if prefilter and order > 1: + padded, npad = _prepad_for_spline_filter(input, mode, cval) + filtered = spline_filter(padded, order, output=np.float64, mode=mode) + else: + npad = 0 + filtered = input + mode = _ni_support._extend_mode_to_code(mode) + _nd_image.geometric_transform(filtered, None, coordinates, None, None, + output, order, mode, cval, npad, None, None) + return output + + +@docfiller +def affine_transform(input, matrix, offset=0.0, output_shape=None, + output=None, order=3, + mode='constant', cval=0.0, prefilter=True): + """ + Apply an affine transformation. + + Given an output image pixel index vector ``o``, the pixel value + is determined from the input image at position + ``np.dot(matrix, o) + offset``. + + This does 'pull' (or 'backward') resampling, transforming the output space + to the input to locate data. Affine transformations are often described in + the 'push' (or 'forward') direction, transforming input to output. If you + have a matrix for the 'push' transformation, use its inverse + (:func:`numpy.linalg.inv`) in this function. + + Parameters + ---------- + %(input)s + matrix : ndarray + The inverse coordinate transformation matrix, mapping output + coordinates to input coordinates. If ``ndim`` is the number of + dimensions of ``input``, the given matrix must have one of the + following shapes: + + - ``(ndim, ndim)``: the linear transformation matrix for each + output coordinate. + - ``(ndim,)``: assume that the 2-D transformation matrix is + diagonal, with the diagonal specified by the given value. A more + efficient algorithm is then used that exploits the separability + of the problem. + - ``(ndim + 1, ndim + 1)``: assume that the transformation is + specified using homogeneous coordinates [1]_. In this case, any + value passed to ``offset`` is ignored. + - ``(ndim, ndim + 1)``: as above, but the bottom row of a + homogeneous transformation matrix is always ``[0, 0, ..., 1]``, + and may be omitted. + + offset : float or sequence, optional + The offset into the array where the transform is applied. If a float, + `offset` is the same for each axis. If a sequence, `offset` should + contain one value for each axis. + output_shape : tuple of ints, optional + Shape tuple. + %(output)s + order : int, optional + The order of the spline interpolation, default is 3. + The order has to be in the range 0-5. + %(mode_interp_constant)s + %(cval)s + %(prefilter)s + + Returns + ------- + affine_transform : ndarray + The transformed input. + + Notes + ----- + The given matrix and offset are used to find for each point in the + output the corresponding coordinates in the input by an affine + transformation. The value of the input at those coordinates is + determined by spline interpolation of the requested order. Points + outside the boundaries of the input are filled according to the given + mode. + + .. versionchanged:: 0.18.0 + Previously, the exact interpretation of the affine transformation + depended on whether the matrix was supplied as a 1-D or a + 2-D array. If a 1-D array was supplied + to the matrix parameter, the output pixel value at index ``o`` + was determined from the input image at position + ``matrix * (o + offset)``. + + For complex-valued `input`, this function transforms the real and imaginary + components independently. + + .. versionadded:: 1.6.0 + Complex-valued support added. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Homogeneous_coordinates + """ + if order < 0 or order > 5: + raise RuntimeError('spline order not supported') + input = np.asarray(input) + if output_shape is None: + if isinstance(output, np.ndarray): + output_shape = output.shape + else: + output_shape = input.shape + if input.ndim < 1 or len(output_shape) < 1: + raise RuntimeError('input and output rank must be > 0') + complex_output = np.iscomplexobj(input) + output = _ni_support._get_output(output, input, shape=output_shape, + complex_output=complex_output) + if complex_output: + kwargs = dict(offset=offset, output_shape=output_shape, order=order, + mode=mode, prefilter=prefilter) + affine_transform(input.real, matrix, output=output.real, + cval=np.real(cval), **kwargs) + affine_transform(input.imag, matrix, output=output.imag, + cval=np.imag(cval), **kwargs) + return output + if prefilter and order > 1: + padded, npad = _prepad_for_spline_filter(input, mode, cval) + filtered = spline_filter(padded, order, output=np.float64, mode=mode) + else: + npad = 0 + filtered = input + mode = _ni_support._extend_mode_to_code(mode) + matrix = np.asarray(matrix, dtype=np.float64) + if matrix.ndim not in [1, 2] or matrix.shape[0] < 1: + raise RuntimeError('no proper affine matrix provided') + if (matrix.ndim == 2 and matrix.shape[1] == input.ndim + 1 and + (matrix.shape[0] in [input.ndim, input.ndim + 1])): + if matrix.shape[0] == input.ndim + 1: + exptd = [0] * input.ndim + [1] + if not np.all(matrix[input.ndim] == exptd): + msg = (f'Expected homogeneous transformation matrix with ' + f'shape {matrix.shape} for image shape {input.shape}, ' + f'but bottom row was not equal to {exptd}') + raise ValueError(msg) + # assume input is homogeneous coordinate transformation matrix + offset = matrix[:input.ndim, input.ndim] + matrix = matrix[:input.ndim, :input.ndim] + if matrix.shape[0] != input.ndim: + raise RuntimeError('affine matrix has wrong number of rows') + if matrix.ndim == 2 and matrix.shape[1] != output.ndim: + raise RuntimeError('affine matrix has wrong number of columns') + if not matrix.flags.contiguous: + matrix = matrix.copy() + offset = _ni_support._normalize_sequence(offset, input.ndim) + offset = np.asarray(offset, dtype=np.float64) + if offset.ndim != 1 or offset.shape[0] < 1: + raise RuntimeError('no proper offset provided') + if not offset.flags.contiguous: + offset = offset.copy() + if matrix.ndim == 1: + warnings.warn( + "The behavior of affine_transform with a 1-D " + "array supplied for the matrix parameter has changed in " + "SciPy 0.18.0.", + stacklevel=2 + ) + _nd_image.zoom_shift(filtered, matrix, offset/matrix, output, order, + mode, cval, npad, False) + else: + _nd_image.geometric_transform(filtered, None, None, matrix, offset, + output, order, mode, cval, npad, None, + None) + return output + + +@docfiller +def shift(input, shift, output=None, order=3, mode='constant', cval=0.0, + prefilter=True): + """ + Shift an array. + + The array is shifted using spline interpolation of the requested order. + Points outside the boundaries of the input are filled according to the + given mode. + + Parameters + ---------- + %(input)s + shift : float or sequence + The shift along the axes. If a float, `shift` is the same for each + axis. If a sequence, `shift` should contain one value for each axis. + %(output)s + order : int, optional + The order of the spline interpolation, default is 3. + The order has to be in the range 0-5. + %(mode_interp_constant)s + %(cval)s + %(prefilter)s + + Returns + ------- + shift : ndarray + The shifted input. + + See Also + -------- + affine_transform : Affine transformations + + Notes + ----- + For complex-valued `input`, this function shifts the real and imaginary + components independently. + + .. versionadded:: 1.6.0 + Complex-valued support added. + + Examples + -------- + Import the necessary modules and an exemplary image. + + >>> from scipy.ndimage import shift + >>> import matplotlib.pyplot as plt + >>> from scipy import datasets + >>> image = datasets.ascent() + + Shift the image vertically by 20 pixels. + + >>> image_shifted_vertically = shift(image, (20, 0)) + + Shift the image vertically by -200 pixels and horizontally by 100 pixels. + + >>> image_shifted_both_directions = shift(image, (-200, 100)) + + Plot the original and the shifted images. + + >>> fig, axes = plt.subplots(3, 1, figsize=(4, 12)) + >>> plt.gray() # show the filtered result in grayscale + >>> top, middle, bottom = axes + >>> for ax in axes: + ... ax.set_axis_off() # remove coordinate system + >>> top.imshow(image) + >>> top.set_title("Original image") + >>> middle.imshow(image_shifted_vertically) + >>> middle.set_title("Vertically shifted image") + >>> bottom.imshow(image_shifted_both_directions) + >>> bottom.set_title("Image shifted in both directions") + >>> fig.tight_layout() + """ + if order < 0 or order > 5: + raise RuntimeError('spline order not supported') + input = np.asarray(input) + if input.ndim < 1: + raise RuntimeError('input and output rank must be > 0') + complex_output = np.iscomplexobj(input) + output = _ni_support._get_output(output, input, complex_output=complex_output) + if complex_output: + # import under different name to avoid confusion with shift parameter + from scipy.ndimage._interpolation import shift as _shift + + kwargs = dict(order=order, mode=mode, prefilter=prefilter) + _shift(input.real, shift, output=output.real, cval=np.real(cval), **kwargs) + _shift(input.imag, shift, output=output.imag, cval=np.imag(cval), **kwargs) + return output + if prefilter and order > 1: + padded, npad = _prepad_for_spline_filter(input, mode, cval) + filtered = spline_filter(padded, order, output=np.float64, mode=mode) + else: + npad = 0 + filtered = input + mode = _ni_support._extend_mode_to_code(mode) + shift = _ni_support._normalize_sequence(shift, input.ndim) + shift = [-ii for ii in shift] + shift = np.asarray(shift, dtype=np.float64) + if not shift.flags.contiguous: + shift = shift.copy() + _nd_image.zoom_shift(filtered, None, shift, output, order, mode, cval, + npad, False) + return output + + +@docfiller +def zoom(input, zoom, output=None, order=3, mode='constant', cval=0.0, + prefilter=True, *, grid_mode=False): + """ + Zoom an array. + + The array is zoomed using spline interpolation of the requested order. + + Parameters + ---------- + %(input)s + zoom : float or sequence + The zoom factor along the axes. If a float, `zoom` is the same for each + axis. If a sequence, `zoom` should contain one value for each axis. + %(output)s + order : int, optional + The order of the spline interpolation, default is 3. + The order has to be in the range 0-5. + %(mode_interp_constant)s + %(cval)s + %(prefilter)s + grid_mode : bool, optional + If False, the distance from the pixel centers is zoomed. Otherwise, the + distance including the full pixel extent is used. For example, a 1d + signal of length 5 is considered to have length 4 when `grid_mode` is + False, but length 5 when `grid_mode` is True. See the following + visual illustration: + + .. code-block:: text + + | pixel 1 | pixel 2 | pixel 3 | pixel 4 | pixel 5 | + |<-------------------------------------->| + vs. + |<----------------------------------------------->| + + The starting point of the arrow in the diagram above corresponds to + coordinate location 0 in each mode. + + Returns + ------- + zoom : ndarray + The zoomed input. + + Notes + ----- + For complex-valued `input`, this function zooms the real and imaginary + components independently. + + .. versionadded:: 1.6.0 + Complex-valued support added. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + + >>> fig = plt.figure() + >>> ax1 = fig.add_subplot(121) # left side + >>> ax2 = fig.add_subplot(122) # right side + >>> ascent = datasets.ascent() + >>> result = ndimage.zoom(ascent, 3.0) + >>> ax1.imshow(ascent, vmin=0, vmax=255) + >>> ax2.imshow(result, vmin=0, vmax=255) + >>> plt.show() + + >>> print(ascent.shape) + (512, 512) + + >>> print(result.shape) + (1536, 1536) + """ + if order < 0 or order > 5: + raise RuntimeError('spline order not supported') + input = np.asarray(input) + if input.ndim < 1: + raise RuntimeError('input and output rank must be > 0') + zoom = _ni_support._normalize_sequence(zoom, input.ndim) + output_shape = tuple( + [int(round(ii * jj)) for ii, jj in zip(input.shape, zoom)]) + complex_output = np.iscomplexobj(input) + output = _ni_support._get_output(output, input, shape=output_shape, + complex_output=complex_output) + if complex_output: + # import under different name to avoid confusion with zoom parameter + from scipy.ndimage._interpolation import zoom as _zoom + + kwargs = dict(order=order, mode=mode, prefilter=prefilter) + _zoom(input.real, zoom, output=output.real, cval=np.real(cval), **kwargs) + _zoom(input.imag, zoom, output=output.imag, cval=np.imag(cval), **kwargs) + return output + if prefilter and order > 1: + padded, npad = _prepad_for_spline_filter(input, mode, cval) + filtered = spline_filter(padded, order, output=np.float64, mode=mode) + else: + npad = 0 + filtered = input + if grid_mode: + # warn about modes that may have surprising behavior + suggest_mode = None + if mode == 'constant': + suggest_mode = 'grid-constant' + elif mode == 'wrap': + suggest_mode = 'grid-wrap' + if suggest_mode is not None: + warnings.warn( + (f"It is recommended to use mode = {suggest_mode} instead of {mode} " + f"when grid_mode is True."), + stacklevel=2 + ) + mode = _ni_support._extend_mode_to_code(mode) + + zoom_div = np.array(output_shape) + zoom_nominator = np.array(input.shape) + if not grid_mode: + zoom_div -= 1 + zoom_nominator -= 1 + + # Zooming to infinite values is unpredictable, so just choose + # zoom factor 1 instead + zoom = np.divide(zoom_nominator, zoom_div, + out=np.ones_like(input.shape, dtype=np.float64), + where=zoom_div != 0) + zoom = np.ascontiguousarray(zoom) + _nd_image.zoom_shift(filtered, zoom, None, output, order, mode, cval, npad, + grid_mode) + return output + + +@docfiller +def rotate(input, angle, axes=(1, 0), reshape=True, output=None, order=3, + mode='constant', cval=0.0, prefilter=True): + """ + Rotate an array. + + The array is rotated in the plane defined by the two axes given by the + `axes` parameter using spline interpolation of the requested order. + + Parameters + ---------- + %(input)s + angle : float + The rotation angle in degrees. + axes : tuple of 2 ints, optional + The two axes that define the plane of rotation. Default is the first + two axes. + reshape : bool, optional + If `reshape` is true, the output shape is adapted so that the input + array is contained completely in the output. Default is True. + %(output)s + order : int, optional + The order of the spline interpolation, default is 3. + The order has to be in the range 0-5. + %(mode_interp_constant)s + %(cval)s + %(prefilter)s + + Returns + ------- + rotate : ndarray + The rotated input. + + Notes + ----- + For complex-valued `input`, this function rotates the real and imaginary + components independently. + + .. versionadded:: 1.6.0 + Complex-valued support added. + + Examples + -------- + >>> from scipy import ndimage, datasets + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure(figsize=(10, 3)) + >>> ax1, ax2, ax3 = fig.subplots(1, 3) + >>> img = datasets.ascent() + >>> img_45 = ndimage.rotate(img, 45, reshape=False) + >>> full_img_45 = ndimage.rotate(img, 45, reshape=True) + >>> ax1.imshow(img, cmap='gray') + >>> ax1.set_axis_off() + >>> ax2.imshow(img_45, cmap='gray') + >>> ax2.set_axis_off() + >>> ax3.imshow(full_img_45, cmap='gray') + >>> ax3.set_axis_off() + >>> fig.set_layout_engine('tight') + >>> plt.show() + >>> print(img.shape) + (512, 512) + >>> print(img_45.shape) + (512, 512) + >>> print(full_img_45.shape) + (724, 724) + + """ + input_arr = np.asarray(input) + ndim = input_arr.ndim + + if ndim < 2: + raise ValueError('input array should be at least 2D') + + axes = list(axes) + + if len(axes) != 2: + raise ValueError('axes should contain exactly two values') + + if not all([float(ax).is_integer() for ax in axes]): + raise ValueError('axes should contain only integer values') + + if axes[0] < 0: + axes[0] += ndim + if axes[1] < 0: + axes[1] += ndim + if axes[0] < 0 or axes[1] < 0 or axes[0] >= ndim or axes[1] >= ndim: + raise ValueError('invalid rotation plane specified') + + axes.sort() + + c, s = special.cosdg(angle), special.sindg(angle) + + rot_matrix = np.array([[c, s], + [-s, c]]) + + img_shape = np.asarray(input_arr.shape) + in_plane_shape = img_shape[axes] + if reshape: + # Compute transformed input bounds + iy, ix = in_plane_shape + out_bounds = rot_matrix @ [[0, 0, iy, iy], + [0, ix, 0, ix]] + # Compute the shape of the transformed input plane + out_plane_shape = (np.ptp(out_bounds, axis=1) + 0.5).astype(int) + else: + out_plane_shape = img_shape[axes] + + out_center = rot_matrix @ ((out_plane_shape - 1) / 2) + in_center = (in_plane_shape - 1) / 2 + offset = in_center - out_center + + output_shape = img_shape + output_shape[axes] = out_plane_shape + output_shape = tuple(output_shape) + + complex_output = np.iscomplexobj(input_arr) + output = _ni_support._get_output(output, input_arr, shape=output_shape, + complex_output=complex_output) + + if ndim <= 2: + affine_transform(input_arr, rot_matrix, offset, output_shape, output, + order, mode, cval, prefilter) + else: + # If ndim > 2, the rotation is applied over all the planes + # parallel to axes + planes_coord = itertools.product( + *[[slice(None)] if ax in axes else range(img_shape[ax]) + for ax in range(ndim)]) + + out_plane_shape = tuple(out_plane_shape) + + for coordinates in planes_coord: + ia = input_arr[coordinates] + oa = output[coordinates] + affine_transform(ia, rot_matrix, offset, out_plane_shape, + oa, order, mode, cval, prefilter) + + return output diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_measurements.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_measurements.py new file mode 100644 index 0000000000000000000000000000000000000000..67ec12870ccf1dfe52624da05787b197542a0253 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_measurements.py @@ -0,0 +1,1687 @@ +# Copyright (C) 2003-2005 Peter J. Verveer +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions +# are met: +# +# 1. Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# +# 2. Redistributions in binary form must reproduce the above +# copyright notice, this list of conditions and the following +# disclaimer in the documentation and/or other materials provided +# with the distribution. +# +# 3. The name of the author may not be used to endorse or promote +# products derived from this software without specific prior +# written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS +# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY +# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE +# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, +# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +import numpy as np +from . import _ni_support +from . import _ni_label +from . import _nd_image +from . import _morphology + +__all__ = ['label', 'find_objects', 'labeled_comprehension', 'sum', 'mean', + 'variance', 'standard_deviation', 'minimum', 'maximum', 'median', + 'minimum_position', 'maximum_position', 'extrema', 'center_of_mass', + 'histogram', 'watershed_ift', 'sum_labels', 'value_indices'] + + +def label(input, structure=None, output=None): + """ + Label features in an array. + + Parameters + ---------- + input : array_like + An array-like object to be labeled. Any non-zero values in `input` are + counted as features and zero values are considered the background. + structure : array_like, optional + A structuring element that defines feature connections. + `structure` must be centrosymmetric + (see Notes). + If no structuring element is provided, + one is automatically generated with a squared connectivity equal to + one. That is, for a 2-D `input` array, the default structuring element + is:: + + [[0,1,0], + [1,1,1], + [0,1,0]] + + output : (None, data-type, array_like), optional + If `output` is a data type, it specifies the type of the resulting + labeled feature array. + If `output` is an array-like object, then `output` will be updated + with the labeled features from this function. This function can + operate in-place, by passing output=input. + Note that the output must be able to store the largest label, or this + function will raise an Exception. + + Returns + ------- + label : ndarray or int + An integer ndarray where each unique feature in `input` has a unique + label in the returned array. + num_features : int + How many objects were found. + + If `output` is None, this function returns a tuple of + (`labeled_array`, `num_features`). + + If `output` is a ndarray, then it will be updated with values in + `labeled_array` and only `num_features` will be returned by this + function. + + See Also + -------- + find_objects : generate a list of slices for the labeled features (or + objects); useful for finding features' position or + dimensions + + Notes + ----- + A centrosymmetric matrix is a matrix that is symmetric about the center. + See [1]_ for more information. + + The `structure` matrix must be centrosymmetric to ensure + two-way connections. + For instance, if the `structure` matrix is not centrosymmetric + and is defined as:: + + [[0,1,0], + [1,1,0], + [0,0,0]] + + and the `input` is:: + + [[1,2], + [0,3]] + + then the structure matrix would indicate the + entry 2 in the input is connected to 1, + but 1 is not connected to 2. + + References + ---------- + .. [1] James R. Weaver, "Centrosymmetric (cross-symmetric) + matrices, their basic properties, eigenvalues, and + eigenvectors." The American Mathematical Monthly 92.10 + (1985): 711-717. + + Examples + -------- + Create an image with some features, then label it using the default + (cross-shaped) structuring element: + + >>> from scipy.ndimage import label, generate_binary_structure + >>> import numpy as np + >>> a = np.array([[0,0,1,1,0,0], + ... [0,0,0,1,0,0], + ... [1,1,0,0,1,0], + ... [0,0,0,1,0,0]]) + >>> labeled_array, num_features = label(a) + + Each of the 4 features are labeled with a different integer: + + >>> num_features + 4 + >>> labeled_array + array([[0, 0, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0], + [2, 2, 0, 0, 3, 0], + [0, 0, 0, 4, 0, 0]], dtype=int32) + + Generate a structuring element that will consider features connected even + if they touch diagonally: + + >>> s = generate_binary_structure(2,2) + + or, + + >>> s = [[1,1,1], + ... [1,1,1], + ... [1,1,1]] + + Label the image using the new structuring element: + + >>> labeled_array, num_features = label(a, structure=s) + + Show the 2 labeled features (note that features 1, 3, and 4 from above are + now considered a single feature): + + >>> num_features + 2 + >>> labeled_array + array([[0, 0, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0], + [2, 2, 0, 0, 1, 0], + [0, 0, 0, 1, 0, 0]], dtype=int32) + + """ + input = np.asarray(input) + if np.iscomplexobj(input): + raise TypeError('Complex type not supported') + if structure is None: + structure = _morphology.generate_binary_structure(input.ndim, 1) + structure = np.asarray(structure, dtype=bool) + if structure.ndim != input.ndim: + raise RuntimeError('structure and input must have equal rank') + for ii in structure.shape: + if ii != 3: + raise ValueError('structure dimensions must be equal to 3') + + # Use 32 bits if it's large enough for this image. + # _ni_label.label() needs two entries for background and + # foreground tracking + need_64bits = input.size >= (2**31 - 2) + + if isinstance(output, np.ndarray): + if output.shape != input.shape: + raise ValueError("output shape not correct") + caller_provided_output = True + else: + caller_provided_output = False + if output is None: + output = np.empty(input.shape, np.intp if need_64bits else np.int32) + else: + output = np.empty(input.shape, output) + + # handle scalars, 0-D arrays + if input.ndim == 0 or input.size == 0: + if input.ndim == 0: + # scalar + maxlabel = 1 if (input != 0) else 0 + output[...] = maxlabel + else: + # 0-D + maxlabel = 0 + if caller_provided_output: + return maxlabel + else: + return output, maxlabel + + try: + max_label = _ni_label._label(input, structure, output) + except _ni_label.NeedMoreBits as e: + # Make another attempt with enough bits, then try to cast to the + # new type. + tmp_output = np.empty(input.shape, np.intp if need_64bits else np.int32) + max_label = _ni_label._label(input, structure, tmp_output) + output[...] = tmp_output[...] + if not np.all(output == tmp_output): + # refuse to return bad results + raise RuntimeError( + "insufficient bit-depth in requested output type" + ) from e + + if caller_provided_output: + # result was written in-place + return max_label + else: + return output, max_label + + +def find_objects(input, max_label=0): + """ + Find objects in a labeled array. + + Parameters + ---------- + input : ndarray of ints + Array containing objects defined by different labels. Labels with + value 0 are ignored. + max_label : int, optional + Maximum label to be searched for in `input`. If max_label is not + given, the positions of all objects are returned. + + Returns + ------- + object_slices : list of tuples + A list of tuples, with each tuple containing N slices (with N the + dimension of the input array). Slices correspond to the minimal + parallelepiped that contains the object. If a number is missing, + None is returned instead of a slice. The label ``l`` corresponds to + the index ``l-1`` in the returned list. + + See Also + -------- + label, center_of_mass + + Notes + ----- + This function is very useful for isolating a volume of interest inside + a 3-D array, that cannot be "seen through". + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.zeros((6,6), dtype=int) + >>> a[2:4, 2:4] = 1 + >>> a[4, 4] = 1 + >>> a[:2, :3] = 2 + >>> a[0, 5] = 3 + >>> a + array([[2, 2, 2, 0, 0, 3], + [2, 2, 2, 0, 0, 0], + [0, 0, 1, 1, 0, 0], + [0, 0, 1, 1, 0, 0], + [0, 0, 0, 0, 1, 0], + [0, 0, 0, 0, 0, 0]]) + >>> ndimage.find_objects(a) + [(slice(2, 5, None), slice(2, 5, None)), + (slice(0, 2, None), slice(0, 3, None)), + (slice(0, 1, None), slice(5, 6, None))] + >>> ndimage.find_objects(a, max_label=2) + [(slice(2, 5, None), slice(2, 5, None)), (slice(0, 2, None), slice(0, 3, None))] + >>> ndimage.find_objects(a == 1, max_label=2) + [(slice(2, 5, None), slice(2, 5, None)), None] + + >>> loc = ndimage.find_objects(a)[0] + >>> a[loc] + array([[1, 1, 0], + [1, 1, 0], + [0, 0, 1]]) + + """ + input = np.asarray(input) + if np.iscomplexobj(input): + raise TypeError('Complex type not supported') + + if max_label < 1: + max_label = input.max() + + return _nd_image.find_objects(input, max_label) + + +def value_indices(arr, *, ignore_value=None): + """ + Find indices of each distinct value in given array. + + Parameters + ---------- + arr : ndarray of ints + Array containing integer values. + ignore_value : int, optional + This value will be ignored in searching the `arr` array. If not + given, all values found will be included in output. Default + is None. + + Returns + ------- + indices : dictionary + A Python dictionary of array indices for each distinct value. The + dictionary is keyed by the distinct values, the entries are array + index tuples covering all occurrences of the value within the + array. + + This dictionary can occupy significant memory, usually several times + the size of the input array. + + See Also + -------- + label, maximum, median, minimum_position, extrema, sum, mean, variance, + standard_deviation, numpy.where, numpy.unique + + Notes + ----- + For a small array with few distinct values, one might use + `numpy.unique()` to find all possible values, and ``(arr == val)`` to + locate each value within that array. However, for large arrays, + with many distinct values, this can become extremely inefficient, + as locating each value would require a new search through the entire + array. Using this function, there is essentially one search, with + the indices saved for all distinct values. + + This is useful when matching a categorical image (e.g. a segmentation + or classification) to an associated image of other data, allowing + any per-class statistic(s) to then be calculated. Provides a + more flexible alternative to functions like ``scipy.ndimage.mean()`` + and ``scipy.ndimage.variance()``. + + Some other closely related functionality, with different strengths and + weaknesses, can also be found in ``scipy.stats.binned_statistic()`` and + the `scikit-image `_ function + ``skimage.measure.regionprops()``. + + Note for IDL users: this provides functionality equivalent to IDL's + REVERSE_INDICES option (as per the IDL documentation for the + `HISTOGRAM `_ + function). + + .. versionadded:: 1.10.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy import ndimage + >>> a = np.zeros((6, 6), dtype=int) + >>> a[2:4, 2:4] = 1 + >>> a[4, 4] = 1 + >>> a[:2, :3] = 2 + >>> a[0, 5] = 3 + >>> a + array([[2, 2, 2, 0, 0, 3], + [2, 2, 2, 0, 0, 0], + [0, 0, 1, 1, 0, 0], + [0, 0, 1, 1, 0, 0], + [0, 0, 0, 0, 1, 0], + [0, 0, 0, 0, 0, 0]]) + >>> val_indices = ndimage.value_indices(a) + + The dictionary `val_indices` will have an entry for each distinct + value in the input array. + + >>> val_indices.keys() + dict_keys([np.int64(0), np.int64(1), np.int64(2), np.int64(3)]) + + The entry for each value is an index tuple, locating the elements + with that value. + + >>> ndx1 = val_indices[1] + >>> ndx1 + (array([2, 2, 3, 3, 4]), array([2, 3, 2, 3, 4])) + + This can be used to index into the original array, or any other + array with the same shape. + + >>> a[ndx1] + array([1, 1, 1, 1, 1]) + + If the zeros were to be ignored, then the resulting dictionary + would no longer have an entry for zero. + + >>> val_indices = ndimage.value_indices(a, ignore_value=0) + >>> val_indices.keys() + dict_keys([np.int64(1), np.int64(2), np.int64(3)]) + + """ + # Cope with ignore_value being None, without too much extra complexity + # in the C code. If not None, the value is passed in as a numpy array + # with the same dtype as arr. + arr = np.asarray(arr) + ignore_value_arr = np.zeros((1,), dtype=arr.dtype) + ignoreIsNone = (ignore_value is None) + if not ignoreIsNone: + ignore_value_arr[0] = ignore_value_arr.dtype.type(ignore_value) + + val_indices = _nd_image.value_indices(arr, ignoreIsNone, ignore_value_arr) + return val_indices + + +def labeled_comprehension(input, labels, index, func, out_dtype, default, + pass_positions=False): + """ + Roughly equivalent to [func(input[labels == i]) for i in index]. + + Sequentially applies an arbitrary function (that works on array_like input) + to subsets of an N-D image array specified by `labels` and `index`. + The option exists to provide the function with positional parameters as the + second argument. + + Parameters + ---------- + input : array_like + Data from which to select `labels` to process. + labels : array_like or None + Labels to objects in `input`. + If not None, array must be same shape as `input`. + If None, `func` is applied to raveled `input`. + index : int, sequence of ints or None + Subset of `labels` to which to apply `func`. + If a scalar, a single value is returned. + If None, `func` is applied to all non-zero values of `labels`. + func : callable + Python function to apply to `labels` from `input`. + out_dtype : dtype + Dtype to use for `result`. + default : int, float or None + Default return value when a element of `index` does not exist + in `labels`. + pass_positions : bool, optional + If True, pass linear indices to `func` as a second argument. + Default is False. + + Returns + ------- + result : ndarray + Result of applying `func` to each of `labels` to `input` in `index`. + + Examples + -------- + >>> import numpy as np + >>> a = np.array([[1, 2, 0, 0], + ... [5, 3, 0, 4], + ... [0, 0, 0, 7], + ... [9, 3, 0, 0]]) + >>> from scipy import ndimage + >>> lbl, nlbl = ndimage.label(a) + >>> lbls = np.arange(1, nlbl+1) + >>> ndimage.labeled_comprehension(a, lbl, lbls, np.mean, float, 0) + array([ 2.75, 5.5 , 6. ]) + + Falling back to `default`: + + >>> lbls = np.arange(1, nlbl+2) + >>> ndimage.labeled_comprehension(a, lbl, lbls, np.mean, float, -1) + array([ 2.75, 5.5 , 6. , -1. ]) + + Passing positions: + + >>> def fn(val, pos): + ... print("fn says: %s : %s" % (val, pos)) + ... return (val.sum()) if (pos.sum() % 2 == 0) else (-val.sum()) + ... + >>> ndimage.labeled_comprehension(a, lbl, lbls, fn, float, 0, True) + fn says: [1 2 5 3] : [0 1 4 5] + fn says: [4 7] : [ 7 11] + fn says: [9 3] : [12 13] + array([ 11., 11., -12., 0.]) + + """ + + as_scalar = np.isscalar(index) + input = np.asarray(input) + + if pass_positions: + positions = np.arange(input.size).reshape(input.shape) + + if labels is None: + if index is not None: + raise ValueError("index without defined labels") + if not pass_positions: + return func(input.ravel()) + else: + return func(input.ravel(), positions.ravel()) + + labels = np.asarray(labels) + + try: + input, labels = np.broadcast_arrays(input, labels) + except ValueError as e: + raise ValueError("input and labels must have the same shape " + "(excepting dimensions with width 1)") from e + + if index is None: + if not pass_positions: + return func(input[labels > 0]) + else: + return func(input[labels > 0], positions[labels > 0]) + + index = np.atleast_1d(index) + if np.any(index.astype(labels.dtype).astype(index.dtype) != index): + raise ValueError(f"Cannot convert index values from <{index.dtype}> to " + f"<{labels.dtype}> (labels' type) without loss of precision") + + index = index.astype(labels.dtype) + + # optimization: find min/max in index, + # and select those parts of labels, input, and positions + lo = index.min() + hi = index.max() + mask = (labels >= lo) & (labels <= hi) + + # this also ravels the arrays + labels = labels[mask] + input = input[mask] + if pass_positions: + positions = positions[mask] + + # sort everything by labels + label_order = labels.argsort() + labels = labels[label_order] + input = input[label_order] + if pass_positions: + positions = positions[label_order] + + index_order = index.argsort() + sorted_index = index[index_order] + + def do_map(inputs, output): + """labels must be sorted""" + nidx = sorted_index.size + + # Find boundaries for each stretch of constant labels + # This could be faster, but we already paid N log N to sort labels. + lo = np.searchsorted(labels, sorted_index, side='left') + hi = np.searchsorted(labels, sorted_index, side='right') + + for i, l, h in zip(range(nidx), lo, hi): + if l == h: + continue + output[i] = func(*[inp[l:h] for inp in inputs]) + + temp = np.empty(index.shape, out_dtype) + temp[:] = default + if not pass_positions: + do_map([input], temp) + else: + do_map([input, positions], temp) + + output = np.zeros(index.shape, out_dtype) + output[index_order] = temp + if as_scalar: + output = output[0] + + return output + + +def _safely_castable_to_int(dt): + """Test whether the NumPy data type `dt` can be safely cast to an int.""" + int_size = np.dtype(int).itemsize + safe = ((np.issubdtype(dt, np.signedinteger) and dt.itemsize <= int_size) or + (np.issubdtype(dt, np.unsignedinteger) and dt.itemsize < int_size)) + return safe + + +def _stats(input, labels=None, index=None, centered=False): + """Count, sum, and optionally compute (sum - centre)^2 of input by label + + Parameters + ---------- + input : array_like, N-D + The input data to be analyzed. + labels : array_like (N-D), optional + The labels of the data in `input`. This array must be broadcast + compatible with `input`; typically, it is the same shape as `input`. + If `labels` is None, all nonzero values in `input` are treated as + the single labeled group. + index : label or sequence of labels, optional + These are the labels of the groups for which the stats are computed. + If `index` is None, the stats are computed for the single group where + `labels` is greater than 0. + centered : bool, optional + If True, the centered sum of squares for each labeled group is + also returned. Default is False. + + Returns + ------- + counts : int or ndarray of ints + The number of elements in each labeled group. + sums : scalar or ndarray of scalars + The sums of the values in each labeled group. + sums_c : scalar or ndarray of scalars, optional + The sums of mean-centered squares of the values in each labeled group. + This is only returned if `centered` is True. + + """ + def single_group(vals): + if centered: + vals_c = vals - vals.mean() + return vals.size, vals.sum(), (vals_c * vals_c.conjugate()).sum() + else: + return vals.size, vals.sum() + + input = np.asarray(input) + if labels is None: + return single_group(input) + + # ensure input and labels match sizes + input, labels = np.broadcast_arrays(input, labels) + + if index is None: + return single_group(input[labels > 0]) + + if np.isscalar(index): + return single_group(input[labels == index]) + + def _sum_centered(labels): + # `labels` is expected to be an ndarray with the same shape as `input`. + # It must contain the label indices (which are not necessarily the labels + # themselves). + means = sums / counts + centered_input = input - means[labels] + # bincount expects 1-D inputs, so we ravel the arguments. + bc = np.bincount(labels.ravel(), + weights=(centered_input * + centered_input.conjugate()).ravel()) + return bc + + # Remap labels to unique integers if necessary, or if the largest + # label is larger than the number of values. + + if (not _safely_castable_to_int(labels.dtype) or + labels.min() < 0 or labels.max() > labels.size): + # Use np.unique to generate the label indices. `new_labels` will + # be 1-D, but it should be interpreted as the flattened N-D array of + # label indices. + unique_labels, new_labels = np.unique(labels, return_inverse=True) + new_labels = np.reshape(new_labels, (-1,)) # flatten, since it may be >1-D + counts = np.bincount(new_labels) + sums = np.bincount(new_labels, weights=input.ravel()) + if centered: + # Compute the sum of the mean-centered squares. + # We must reshape new_labels to the N-D shape of `input` before + # passing it _sum_centered. + sums_c = _sum_centered(new_labels.reshape(labels.shape)) + idxs = np.searchsorted(unique_labels, index) + # make all of idxs valid + idxs[idxs >= unique_labels.size] = 0 + found = (unique_labels[idxs] == index) + else: + # labels are an integer type allowed by bincount, and there aren't too + # many, so call bincount directly. + counts = np.bincount(labels.ravel()) + sums = np.bincount(labels.ravel(), weights=input.ravel()) + if centered: + sums_c = _sum_centered(labels) + # make sure all index values are valid + idxs = np.asanyarray(index, np.int_).copy() + found = (idxs >= 0) & (idxs < counts.size) + idxs[~found] = 0 + + counts = counts[idxs] + counts[~found] = 0 + sums = sums[idxs] + sums[~found] = 0 + + if not centered: + return (counts, sums) + else: + sums_c = sums_c[idxs] + sums_c[~found] = 0 + return (counts, sums, sums_c) + + +def sum(input, labels=None, index=None): + """ + Calculate the sum of the values of the array. + + Notes + ----- + This is an alias for `ndimage.sum_labels` kept for backwards compatibility + reasons, for new code please prefer `sum_labels`. See the `sum_labels` + docstring for more details. + + """ + return sum_labels(input, labels, index) + + +def sum_labels(input, labels=None, index=None): + """ + Calculate the sum of the values of the array. + + Parameters + ---------- + input : array_like + Values of `input` inside the regions defined by `labels` + are summed together. + labels : array_like of ints, optional + Assign labels to the values of the array. Has to have the same shape as + `input`. + index : array_like, optional + A single label number or a sequence of label numbers of + the objects to be measured. + + Returns + ------- + sum : ndarray or scalar + An array of the sums of values of `input` inside the regions defined + by `labels` with the same shape as `index`. If 'index' is None or scalar, + a scalar is returned. + + See Also + -------- + mean, median + + Examples + -------- + >>> from scipy import ndimage + >>> input = [0,1,2,3] + >>> labels = [1,1,2,2] + >>> ndimage.sum_labels(input, labels, index=[1,2]) + [1.0, 5.0] + >>> ndimage.sum_labels(input, labels, index=1) + 1 + >>> ndimage.sum_labels(input, labels) + 6 + + + """ + count, sum = _stats(input, labels, index) + return sum + + +def mean(input, labels=None, index=None): + """ + Calculate the mean of the values of an array at labels. + + Parameters + ---------- + input : array_like + Array on which to compute the mean of elements over distinct + regions. + labels : array_like, optional + Array of labels of same shape, or broadcastable to the same shape as + `input`. All elements sharing the same label form one region over + which the mean of the elements is computed. + index : int or sequence of ints, optional + Labels of the objects over which the mean is to be computed. + Default is None, in which case the mean for all values where label is + greater than 0 is calculated. + + Returns + ------- + out : list + Sequence of same length as `index`, with the mean of the different + regions labeled by the labels in `index`. + + See Also + -------- + variance, standard_deviation, minimum, maximum, sum, label + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.arange(25).reshape((5,5)) + >>> labels = np.zeros_like(a) + >>> labels[3:5,3:5] = 1 + >>> index = np.unique(labels) + >>> labels + array([[0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 1, 1], + [0, 0, 0, 1, 1]]) + >>> index + array([0, 1]) + >>> ndimage.mean(a, labels=labels, index=index) + [10.285714285714286, 21.0] + + """ + + count, sum = _stats(input, labels, index) + return sum / np.asanyarray(count).astype(np.float64) + + +def variance(input, labels=None, index=None): + """ + Calculate the variance of the values of an N-D image array, optionally at + specified sub-regions. + + Parameters + ---------- + input : array_like + Nd-image data to process. + labels : array_like, optional + Labels defining sub-regions in `input`. + If not None, must be same shape as `input`. + index : int or sequence of ints, optional + `labels` to include in output. If None (default), all values where + `labels` is non-zero are used. + + Returns + ------- + variance : float or ndarray + Values of variance, for each sub-region if `labels` and `index` are + specified. + + See Also + -------- + label, standard_deviation, maximum, minimum, extrema + + Examples + -------- + >>> import numpy as np + >>> a = np.array([[1, 2, 0, 0], + ... [5, 3, 0, 4], + ... [0, 0, 0, 7], + ... [9, 3, 0, 0]]) + >>> from scipy import ndimage + >>> ndimage.variance(a) + 7.609375 + + Features to process can be specified using `labels` and `index`: + + >>> lbl, nlbl = ndimage.label(a) + >>> ndimage.variance(a, lbl, index=np.arange(1, nlbl+1)) + array([ 2.1875, 2.25 , 9. ]) + + If no index is given, all non-zero `labels` are processed: + + >>> ndimage.variance(a, lbl) + 6.1875 + + """ + count, sum, sum_c_sq = _stats(input, labels, index, centered=True) + return sum_c_sq / np.asanyarray(count).astype(float) + + +def standard_deviation(input, labels=None, index=None): + """ + Calculate the standard deviation of the values of an N-D image array, + optionally at specified sub-regions. + + Parameters + ---------- + input : array_like + N-D image data to process. + labels : array_like, optional + Labels to identify sub-regions in `input`. + If not None, must be same shape as `input`. + index : int or sequence of ints, optional + `labels` to include in output. If None (default), all values where + `labels` is non-zero are used. + + Returns + ------- + standard_deviation : float or ndarray + Values of standard deviation, for each sub-region if `labels` and + `index` are specified. + + See Also + -------- + label, variance, maximum, minimum, extrema + + Examples + -------- + >>> import numpy as np + >>> a = np.array([[1, 2, 0, 0], + ... [5, 3, 0, 4], + ... [0, 0, 0, 7], + ... [9, 3, 0, 0]]) + >>> from scipy import ndimage + >>> ndimage.standard_deviation(a) + 2.7585095613392387 + + Features to process can be specified using `labels` and `index`: + + >>> lbl, nlbl = ndimage.label(a) + >>> ndimage.standard_deviation(a, lbl, index=np.arange(1, nlbl+1)) + array([ 1.479, 1.5 , 3. ]) + + If no index is given, non-zero `labels` are processed: + + >>> ndimage.standard_deviation(a, lbl) + 2.4874685927665499 + + """ + return np.sqrt(variance(input, labels, index)) + + +def _select(input, labels=None, index=None, find_min=False, find_max=False, + find_min_positions=False, find_max_positions=False, + find_median=False): + """Returns min, max, or both, plus their positions (if requested), and + median.""" + + input = np.asanyarray(input) + + find_positions = find_min_positions or find_max_positions + positions = None + if find_positions: + positions = np.arange(input.size).reshape(input.shape) + + def single_group(vals, positions): + result = [] + if find_min: + result += [vals.min()] + if find_min_positions: + result += [positions[vals == vals.min()][0]] + if find_max: + result += [vals.max()] + if find_max_positions: + result += [positions[vals == vals.max()][0]] + if find_median: + result += [np.median(vals)] + return result + + if labels is None: + return single_group(input, positions) + + # ensure input and labels match sizes + input, labels = np.broadcast_arrays(input, labels) + + if index is None: + mask = (labels > 0) + masked_positions = None + if find_positions: + masked_positions = positions[mask] + return single_group(input[mask], masked_positions) + + if np.isscalar(index): + mask = (labels == index) + masked_positions = None + if find_positions: + masked_positions = positions[mask] + return single_group(input[mask], masked_positions) + + index = np.asarray(index) + + # remap labels to unique integers if necessary, or if the largest + # label is larger than the number of values. + if (not _safely_castable_to_int(labels.dtype) or + labels.min() < 0 or labels.max() > labels.size): + # remap labels, and indexes + unique_labels, labels = np.unique(labels, return_inverse=True) + idxs = np.searchsorted(unique_labels, index) + + # make all of idxs valid + idxs[idxs >= unique_labels.size] = 0 + found = (unique_labels[idxs] == index) + else: + # labels are an integer type, and there aren't too many + idxs = np.asanyarray(index, np.int_).copy() + found = (idxs >= 0) & (idxs <= labels.max()) + + idxs[~ found] = labels.max() + 1 + + if find_median: + order = np.lexsort((input.ravel(), labels.ravel())) + else: + order = input.ravel().argsort() + input = input.ravel()[order] + labels = labels.ravel()[order] + if find_positions: + positions = positions.ravel()[order] + + result = [] + if find_min: + mins = np.zeros(labels.max() + 2, input.dtype) + mins[labels[::-1]] = input[::-1] + result += [mins[idxs]] + if find_min_positions: + minpos = np.zeros(labels.max() + 2, int) + minpos[labels[::-1]] = positions[::-1] + result += [minpos[idxs]] + if find_max: + maxs = np.zeros(labels.max() + 2, input.dtype) + maxs[labels] = input + result += [maxs[idxs]] + if find_max_positions: + maxpos = np.zeros(labels.max() + 2, int) + maxpos[labels] = positions + result += [maxpos[idxs]] + if find_median: + locs = np.arange(len(labels)) + lo = np.zeros(labels.max() + 2, np.int_) + lo[labels[::-1]] = locs[::-1] + hi = np.zeros(labels.max() + 2, np.int_) + hi[labels] = locs + lo = lo[idxs] + hi = hi[idxs] + # lo is an index to the lowest value in input for each label, + # hi is an index to the largest value. + # move them to be either the same ((hi - lo) % 2 == 0) or next + # to each other ((hi - lo) % 2 == 1), then average. + step = (hi - lo) // 2 + lo += step + hi -= step + if (np.issubdtype(input.dtype, np.integer) + or np.issubdtype(input.dtype, np.bool_)): + # avoid integer overflow or boolean addition (gh-12836) + result += [(input[lo].astype('d') + input[hi].astype('d')) / 2.0] + else: + result += [(input[lo] + input[hi]) / 2.0] + + return result + + +def minimum(input, labels=None, index=None): + """ + Calculate the minimum of the values of an array over labeled regions. + + Parameters + ---------- + input : array_like + Array_like of values. For each region specified by `labels`, the + minimal values of `input` over the region is computed. + labels : array_like, optional + An array_like of integers marking different regions over which the + minimum value of `input` is to be computed. `labels` must have the + same shape as `input`. If `labels` is not specified, the minimum + over the whole array is returned. + index : array_like, optional + A list of region labels that are taken into account for computing the + minima. If index is None, the minimum over all elements where `labels` + is non-zero is returned. + + Returns + ------- + minimum : float or list of floats + List of minima of `input` over the regions determined by `labels` and + whose index is in `index`. If `index` or `labels` are not specified, a + float is returned: the minimal value of `input` if `labels` is None, + and the minimal value of elements where `labels` is greater than zero + if `index` is None. + + See Also + -------- + label, maximum, median, minimum_position, extrema, sum, mean, variance, + standard_deviation + + Notes + ----- + The function returns a Python list and not a NumPy array, use + `np.array` to convert the list to an array. + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.array([[1, 2, 0, 0], + ... [5, 3, 0, 4], + ... [0, 0, 0, 7], + ... [9, 3, 0, 0]]) + >>> labels, labels_nb = ndimage.label(a) + >>> labels + array([[1, 1, 0, 0], + [1, 1, 0, 2], + [0, 0, 0, 2], + [3, 3, 0, 0]], dtype=int32) + >>> ndimage.minimum(a, labels=labels, index=np.arange(1, labels_nb + 1)) + [1, 4, 3] + >>> ndimage.minimum(a) + 0 + >>> ndimage.minimum(a, labels=labels) + 1 + + """ + return _select(input, labels, index, find_min=True)[0] + + +def maximum(input, labels=None, index=None): + """ + Calculate the maximum of the values of an array over labeled regions. + + Parameters + ---------- + input : array_like + Array_like of values. For each region specified by `labels`, the + maximal values of `input` over the region is computed. + labels : array_like, optional + An array of integers marking different regions over which the + maximum value of `input` is to be computed. `labels` must have the + same shape as `input`. If `labels` is not specified, the maximum + over the whole array is returned. + index : array_like, optional + A list of region labels that are taken into account for computing the + maxima. If index is None, the maximum over all elements where `labels` + is non-zero is returned. + + Returns + ------- + output : float or list of floats + List of maxima of `input` over the regions determined by `labels` and + whose index is in `index`. If `index` or `labels` are not specified, a + float is returned: the maximal value of `input` if `labels` is None, + and the maximal value of elements where `labels` is greater than zero + if `index` is None. + + See Also + -------- + label, minimum, median, maximum_position, extrema, sum, mean, variance, + standard_deviation + + Notes + ----- + The function returns a Python list and not a NumPy array, use + `np.array` to convert the list to an array. + + Examples + -------- + >>> import numpy as np + >>> a = np.arange(16).reshape((4,4)) + >>> a + array([[ 0, 1, 2, 3], + [ 4, 5, 6, 7], + [ 8, 9, 10, 11], + [12, 13, 14, 15]]) + >>> labels = np.zeros_like(a) + >>> labels[:2,:2] = 1 + >>> labels[2:, 1:3] = 2 + >>> labels + array([[1, 1, 0, 0], + [1, 1, 0, 0], + [0, 2, 2, 0], + [0, 2, 2, 0]]) + >>> from scipy import ndimage + >>> ndimage.maximum(a) + 15 + >>> ndimage.maximum(a, labels=labels, index=[1,2]) + [5, 14] + >>> ndimage.maximum(a, labels=labels) + 14 + + >>> b = np.array([[1, 2, 0, 0], + ... [5, 3, 0, 4], + ... [0, 0, 0, 7], + ... [9, 3, 0, 0]]) + >>> labels, labels_nb = ndimage.label(b) + >>> labels + array([[1, 1, 0, 0], + [1, 1, 0, 2], + [0, 0, 0, 2], + [3, 3, 0, 0]], dtype=int32) + >>> ndimage.maximum(b, labels=labels, index=np.arange(1, labels_nb + 1)) + [5, 7, 9] + + """ + return _select(input, labels, index, find_max=True)[0] + + +def median(input, labels=None, index=None): + """ + Calculate the median of the values of an array over labeled regions. + + Parameters + ---------- + input : array_like + Array_like of values. For each region specified by `labels`, the + median value of `input` over the region is computed. + labels : array_like, optional + An array_like of integers marking different regions over which the + median value of `input` is to be computed. `labels` must have the + same shape as `input`. If `labels` is not specified, the median + over the whole array is returned. + index : array_like, optional + A list of region labels that are taken into account for computing the + medians. If index is None, the median over all elements where `labels` + is non-zero is returned. + + Returns + ------- + median : float or list of floats + List of medians of `input` over the regions determined by `labels` and + whose index is in `index`. If `index` or `labels` are not specified, a + float is returned: the median value of `input` if `labels` is None, + and the median value of elements where `labels` is greater than zero + if `index` is None. + + See Also + -------- + label, minimum, maximum, extrema, sum, mean, variance, standard_deviation + + Notes + ----- + The function returns a Python list and not a NumPy array, use + `np.array` to convert the list to an array. + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.array([[1, 2, 0, 1], + ... [5, 3, 0, 4], + ... [0, 0, 0, 7], + ... [9, 3, 0, 0]]) + >>> labels, labels_nb = ndimage.label(a) + >>> labels + array([[1, 1, 0, 2], + [1, 1, 0, 2], + [0, 0, 0, 2], + [3, 3, 0, 0]], dtype=int32) + >>> ndimage.median(a, labels=labels, index=np.arange(1, labels_nb + 1)) + [2.5, 4.0, 6.0] + >>> ndimage.median(a) + 1.0 + >>> ndimage.median(a, labels=labels) + 3.0 + + """ + return _select(input, labels, index, find_median=True)[0] + + +def minimum_position(input, labels=None, index=None): + """ + Find the positions of the minimums of the values of an array at labels. + + Parameters + ---------- + input : array_like + Array_like of values. + labels : array_like, optional + An array of integers marking different regions over which the + position of the minimum value of `input` is to be computed. + `labels` must have the same shape as `input`. If `labels` is not + specified, the location of the first minimum over the whole + array is returned. + + The `labels` argument only works when `index` is specified. + index : array_like, optional + A list of region labels that are taken into account for finding the + location of the minima. If `index` is None, the ``first`` minimum + over all elements where `labels` is non-zero is returned. + + The `index` argument only works when `labels` is specified. + + Returns + ------- + output : list of tuples of ints + Tuple of ints or list of tuples of ints that specify the location + of minima of `input` over the regions determined by `labels` and + whose index is in `index`. + + If `index` or `labels` are not specified, a tuple of ints is + returned specifying the location of the first minimal value of `input`. + + See Also + -------- + label, minimum, median, maximum_position, extrema, sum, mean, variance, + standard_deviation + + Examples + -------- + >>> import numpy as np + >>> a = np.array([[10, 20, 30], + ... [40, 80, 100], + ... [1, 100, 200]]) + >>> b = np.array([[1, 2, 0, 1], + ... [5, 3, 0, 4], + ... [0, 0, 0, 7], + ... [9, 3, 0, 0]]) + + >>> from scipy import ndimage + + >>> ndimage.minimum_position(a) + (2, 0) + >>> ndimage.minimum_position(b) + (0, 2) + + Features to process can be specified using `labels` and `index`: + + >>> label, pos = ndimage.label(a) + >>> ndimage.minimum_position(a, label, index=np.arange(1, pos+1)) + [(2, 0)] + + >>> label, pos = ndimage.label(b) + >>> ndimage.minimum_position(b, label, index=np.arange(1, pos+1)) + [(0, 0), (0, 3), (3, 1)] + + """ + dims = np.array(np.asarray(input).shape) + # see np.unravel_index to understand this line. + dim_prod = np.cumprod([1] + list(dims[:0:-1]))[::-1] + + result = _select(input, labels, index, find_min_positions=True)[0] + + if np.isscalar(result): + return tuple((result // dim_prod) % dims) + + return [tuple(v) for v in (result.reshape(-1, 1) // dim_prod) % dims] + + +def maximum_position(input, labels=None, index=None): + """ + Find the positions of the maximums of the values of an array at labels. + + For each region specified by `labels`, the position of the maximum + value of `input` within the region is returned. + + Parameters + ---------- + input : array_like + Array_like of values. + labels : array_like, optional + An array of integers marking different regions over which the + position of the maximum value of `input` is to be computed. + `labels` must have the same shape as `input`. If `labels` is not + specified, the location of the first maximum over the whole + array is returned. + + The `labels` argument only works when `index` is specified. + index : array_like, optional + A list of region labels that are taken into account for finding the + location of the maxima. If `index` is None, the first maximum + over all elements where `labels` is non-zero is returned. + + The `index` argument only works when `labels` is specified. + + Returns + ------- + output : list of tuples of ints + List of tuples of ints that specify the location of maxima of + `input` over the regions determined by `labels` and whose index + is in `index`. + + If `index` or `labels` are not specified, a tuple of ints is + returned specifying the location of the ``first`` maximal value + of `input`. + + See Also + -------- + label, minimum, median, maximum_position, extrema, sum, mean, variance, + standard_deviation + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.array([[1, 2, 0, 0], + ... [5, 3, 0, 4], + ... [0, 0, 0, 7], + ... [9, 3, 0, 0]]) + >>> ndimage.maximum_position(a) + (3, 0) + + Features to process can be specified using `labels` and `index`: + + >>> lbl = np.array([[0, 1, 2, 3], + ... [0, 1, 2, 3], + ... [0, 1, 2, 3], + ... [0, 1, 2, 3]]) + >>> ndimage.maximum_position(a, lbl, 1) + (1, 1) + + If no index is given, non-zero `labels` are processed: + + >>> ndimage.maximum_position(a, lbl) + (2, 3) + + If there are no maxima, the position of the first element is returned: + + >>> ndimage.maximum_position(a, lbl, 2) + (0, 2) + + """ + dims = np.array(np.asarray(input).shape) + # see np.unravel_index to understand this line. + dim_prod = np.cumprod([1] + list(dims[:0:-1]))[::-1] + + result = _select(input, labels, index, find_max_positions=True)[0] + + if np.isscalar(result): + return tuple((result // dim_prod) % dims) + + return [tuple(v) for v in (result.reshape(-1, 1) // dim_prod) % dims] + + +def extrema(input, labels=None, index=None): + """ + Calculate the minimums and maximums of the values of an array + at labels, along with their positions. + + Parameters + ---------- + input : ndarray + N-D image data to process. + labels : ndarray, optional + Labels of features in input. + If not None, must be same shape as `input`. + index : int or sequence of ints, optional + Labels to include in output. If None (default), all values where + non-zero `labels` are used. + + Returns + ------- + minimums, maximums : int or ndarray + Values of minimums and maximums in each feature. + min_positions, max_positions : tuple or list of tuples + Each tuple gives the N-D coordinates of the corresponding minimum + or maximum. + + See Also + -------- + maximum, minimum, maximum_position, minimum_position, center_of_mass + + Examples + -------- + >>> import numpy as np + >>> a = np.array([[1, 2, 0, 0], + ... [5, 3, 0, 4], + ... [0, 0, 0, 7], + ... [9, 3, 0, 0]]) + >>> from scipy import ndimage + >>> ndimage.extrema(a) + (0, 9, (0, 2), (3, 0)) + + Features to process can be specified using `labels` and `index`: + + >>> lbl, nlbl = ndimage.label(a) + >>> ndimage.extrema(a, lbl, index=np.arange(1, nlbl+1)) + (array([1, 4, 3]), + array([5, 7, 9]), + [(0, 0), (1, 3), (3, 1)], + [(1, 0), (2, 3), (3, 0)]) + + If no index is given, non-zero `labels` are processed: + + >>> ndimage.extrema(a, lbl) + (1, 9, (0, 0), (3, 0)) + + """ + dims = np.array(np.asarray(input).shape) + # see np.unravel_index to understand this line. + dim_prod = np.cumprod([1] + list(dims[:0:-1]))[::-1] + + minimums, min_positions, maximums, max_positions = _select(input, labels, + index, + find_min=True, + find_max=True, + find_min_positions=True, + find_max_positions=True) + + if np.isscalar(minimums): + return (minimums, maximums, tuple((min_positions // dim_prod) % dims), + tuple((max_positions // dim_prod) % dims)) + + min_positions = [ + tuple(v) for v in (min_positions.reshape(-1, 1) // dim_prod) % dims + ] + max_positions = [ + tuple(v) for v in (max_positions.reshape(-1, 1) // dim_prod) % dims + ] + + return minimums, maximums, min_positions, max_positions + + +def center_of_mass(input, labels=None, index=None): + """ + Calculate the center of mass of the values of an array at labels. + + Parameters + ---------- + input : ndarray + Data from which to calculate center-of-mass. The masses can either + be positive or negative. + labels : ndarray, optional + Labels for objects in `input`, as generated by `ndimage.label`. + Only used with `index`. Dimensions must be the same as `input`. + index : int or sequence of ints, optional + Labels for which to calculate centers-of-mass. If not specified, + the combined center of mass of all labels greater than zero + will be calculated. Only used with `labels`. + + Returns + ------- + center_of_mass : tuple, or list of tuples + Coordinates of centers-of-mass. + + Examples + -------- + >>> import numpy as np + >>> a = np.array(([0,0,0,0], + ... [0,1,1,0], + ... [0,1,1,0], + ... [0,1,1,0])) + >>> from scipy import ndimage + >>> ndimage.center_of_mass(a) + (2.0, 1.5) + + Calculation of multiple objects in an image + + >>> b = np.array(([0,1,1,0], + ... [0,1,0,0], + ... [0,0,0,0], + ... [0,0,1,1], + ... [0,0,1,1])) + >>> lbl = ndimage.label(b)[0] + >>> ndimage.center_of_mass(b, lbl, [1,2]) + [(0.33333333333333331, 1.3333333333333333), (3.5, 2.5)] + + Negative masses are also accepted, which can occur for example when + bias is removed from measured data due to random noise. + + >>> c = np.array(([-1,0,0,0], + ... [0,-1,-1,0], + ... [0,1,-1,0], + ... [0,1,1,0])) + >>> ndimage.center_of_mass(c) + (-4.0, 1.0) + + If there are division by zero issues, the function does not raise an + error but rather issues a RuntimeWarning before returning inf and/or NaN. + + >>> d = np.array([-1, 1]) + >>> ndimage.center_of_mass(d) + (inf,) + """ + input = np.asarray(input) + normalizer = sum_labels(input, labels, index) + grids = np.ogrid[[slice(0, i) for i in input.shape]] + + results = [sum_labels(input * grids[dir].astype(float), labels, index) / normalizer + for dir in range(input.ndim)] + + if np.isscalar(results[0]): + return tuple(results) + + return [tuple(v) for v in np.array(results).T] + + +def histogram(input, min, max, bins, labels=None, index=None): + """ + Calculate the histogram of the values of an array, optionally at labels. + + Histogram calculates the frequency of values in an array within bins + determined by `min`, `max`, and `bins`. The `labels` and `index` + keywords can limit the scope of the histogram to specified sub-regions + within the array. + + Parameters + ---------- + input : array_like + Data for which to calculate histogram. + min, max : int + Minimum and maximum values of range of histogram bins. + bins : int + Number of bins. + labels : array_like, optional + Labels for objects in `input`. + If not None, must be same shape as `input`. + index : int or sequence of ints, optional + Label or labels for which to calculate histogram. If None, all values + where label is greater than zero are used + + Returns + ------- + hist : ndarray + Histogram counts. + + Examples + -------- + >>> import numpy as np + >>> a = np.array([[ 0. , 0.2146, 0.5962, 0. ], + ... [ 0. , 0.7778, 0. , 0. ], + ... [ 0. , 0. , 0. , 0. ], + ... [ 0. , 0. , 0.7181, 0.2787], + ... [ 0. , 0. , 0.6573, 0.3094]]) + >>> from scipy import ndimage + >>> ndimage.histogram(a, 0, 1, 10) + array([13, 0, 2, 1, 0, 1, 1, 2, 0, 0]) + + With labels and no indices, non-zero elements are counted: + + >>> lbl, nlbl = ndimage.label(a) + >>> ndimage.histogram(a, 0, 1, 10, lbl) + array([0, 0, 2, 1, 0, 1, 1, 2, 0, 0]) + + Indices can be used to count only certain objects: + + >>> ndimage.histogram(a, 0, 1, 10, lbl, 2) + array([0, 0, 1, 1, 0, 0, 1, 1, 0, 0]) + + """ + _bins = np.linspace(min, max, bins + 1) + + def _hist(vals): + return np.histogram(vals, _bins)[0] + + return labeled_comprehension(input, labels, index, _hist, object, None, + pass_positions=False) + + +def watershed_ift(input, markers, structure=None, output=None): + """ + Apply watershed from markers using image foresting transform algorithm. + + Parameters + ---------- + input : array_like + Input. + markers : array_like + Markers are points within each watershed that form the beginning + of the process. Negative markers are considered background markers + which are processed after the other markers. + structure : structure element, optional + A structuring element defining the connectivity of the object can be + provided. If None, an element is generated with a squared + connectivity equal to one. + output : ndarray, optional + An output array can optionally be provided. The same shape as input. + + Returns + ------- + watershed_ift : ndarray + Output. Same shape as `input`. + + References + ---------- + .. [1] A.X. Falcao, J. Stolfi and R. de Alencar Lotufo, "The image + foresting transform: theory, algorithms, and applications", + Pattern Analysis and Machine Intelligence, vol. 26, pp. 19-29, 2004. + + """ + input = np.asarray(input) + if input.dtype.type not in [np.uint8, np.uint16]: + raise TypeError('only 8 and 16 unsigned inputs are supported') + + if structure is None: + structure = _morphology.generate_binary_structure(input.ndim, 1) + structure = np.asarray(structure, dtype=bool) + if structure.ndim != input.ndim: + raise RuntimeError('structure and input must have equal rank') + for ii in structure.shape: + if ii != 3: + raise RuntimeError('structure dimensions must be equal to 3') + + if not structure.flags.contiguous: + structure = structure.copy() + markers = np.asarray(markers) + if input.shape != markers.shape: + raise RuntimeError('input and markers must have equal shape') + + integral_types = [np.int8, + np.int16, + np.int32, + np.int64, + np.intc, + np.intp] + + if markers.dtype.type not in integral_types: + raise RuntimeError('marker should be of integer type') + + if isinstance(output, np.ndarray): + if output.dtype.type not in integral_types: + raise RuntimeError('output should be of integer type') + else: + output = markers.dtype + + output = _ni_support._get_output(output, input) + _nd_image.watershed_ift(input, markers, structure, output) + return output diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_morphology.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_morphology.py new file mode 100644 index 0000000000000000000000000000000000000000..12972c09a7cd5de0ca059814281fb9d210fbd395 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_morphology.py @@ -0,0 +1,2629 @@ +# Copyright (C) 2003-2005 Peter J. Verveer +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions +# are met: +# +# 1. Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# +# 2. Redistributions in binary form must reproduce the above +# copyright notice, this list of conditions and the following +# disclaimer in the documentation and/or other materials provided +# with the distribution. +# +# 3. The name of the author may not be used to endorse or promote +# products derived from this software without specific prior +# written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS +# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY +# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE +# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, +# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +import warnings +import operator + +import numpy as np +from . import _ni_support +from . import _nd_image +from . import _filters + +__all__ = ['iterate_structure', 'generate_binary_structure', 'binary_erosion', + 'binary_dilation', 'binary_opening', 'binary_closing', + 'binary_hit_or_miss', 'binary_propagation', 'binary_fill_holes', + 'grey_erosion', 'grey_dilation', 'grey_opening', 'grey_closing', + 'morphological_gradient', 'morphological_laplace', 'white_tophat', + 'black_tophat', 'distance_transform_bf', 'distance_transform_cdt', + 'distance_transform_edt'] + + +def _center_is_true(structure, origin): + structure = np.asarray(structure) + coor = tuple([oo + ss // 2 for ss, oo in zip(structure.shape, + origin)]) + return bool(structure[coor]) + + +def iterate_structure(structure, iterations, origin=None): + """ + Iterate a structure by dilating it with itself. + + Parameters + ---------- + structure : array_like + Structuring element (an array of bools, for example), to be dilated with + itself. + iterations : int + number of dilations performed on the structure with itself + origin : optional + If origin is None, only the iterated structure is returned. If + not, a tuple of the iterated structure and the modified origin is + returned. + + Returns + ------- + iterate_structure : ndarray of bools + A new structuring element obtained by dilating `structure` + (`iterations` - 1) times with itself. + + See Also + -------- + generate_binary_structure + + Examples + -------- + >>> from scipy import ndimage + >>> struct = ndimage.generate_binary_structure(2, 1) + >>> struct.astype(int) + array([[0, 1, 0], + [1, 1, 1], + [0, 1, 0]]) + >>> ndimage.iterate_structure(struct, 2).astype(int) + array([[0, 0, 1, 0, 0], + [0, 1, 1, 1, 0], + [1, 1, 1, 1, 1], + [0, 1, 1, 1, 0], + [0, 0, 1, 0, 0]]) + >>> ndimage.iterate_structure(struct, 3).astype(int) + array([[0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [1, 1, 1, 1, 1, 1, 1], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0]]) + + """ + structure = np.asarray(structure) + if iterations < 2: + return structure.copy() + ni = iterations - 1 + shape = [ii + ni * (ii - 1) for ii in structure.shape] + pos = [ni * (structure.shape[ii] // 2) for ii in range(len(shape))] + slc = tuple(slice(pos[ii], pos[ii] + structure.shape[ii], None) + for ii in range(len(shape))) + out = np.zeros(shape, bool) + out[slc] = structure != 0 + out = binary_dilation(out, structure, iterations=ni) + if origin is None: + return out + else: + origin = _ni_support._normalize_sequence(origin, structure.ndim) + origin = [iterations * o for o in origin] + return out, origin + + +def generate_binary_structure(rank, connectivity): + """ + Generate a binary structure for binary morphological operations. + + Parameters + ---------- + rank : int + Number of dimensions of the array to which the structuring element + will be applied, as returned by `np.ndim`. + connectivity : int + `connectivity` determines which elements of the output array belong + to the structure, i.e., are considered as neighbors of the central + element. Elements up to a squared distance of `connectivity` from + the center are considered neighbors. `connectivity` may range from 1 + (no diagonal elements are neighbors) to `rank` (all elements are + neighbors). + + Returns + ------- + output : ndarray of bools + Structuring element which may be used for binary morphological + operations, with `rank` dimensions and all dimensions equal to 3. + + See Also + -------- + iterate_structure, binary_dilation, binary_erosion + + Notes + ----- + `generate_binary_structure` can only create structuring elements with + dimensions equal to 3, i.e., minimal dimensions. For larger structuring + elements, that are useful e.g., for eroding large objects, one may either + use `iterate_structure`, or create directly custom arrays with + numpy functions such as `numpy.ones`. + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> struct = ndimage.generate_binary_structure(2, 1) + >>> struct + array([[False, True, False], + [ True, True, True], + [False, True, False]], dtype=bool) + >>> a = np.zeros((5,5)) + >>> a[2, 2] = 1 + >>> a + array([[ 0., 0., 0., 0., 0.], + [ 0., 0., 0., 0., 0.], + [ 0., 0., 1., 0., 0.], + [ 0., 0., 0., 0., 0.], + [ 0., 0., 0., 0., 0.]]) + >>> b = ndimage.binary_dilation(a, structure=struct).astype(a.dtype) + >>> b + array([[ 0., 0., 0., 0., 0.], + [ 0., 0., 1., 0., 0.], + [ 0., 1., 1., 1., 0.], + [ 0., 0., 1., 0., 0.], + [ 0., 0., 0., 0., 0.]]) + >>> ndimage.binary_dilation(b, structure=struct).astype(a.dtype) + array([[ 0., 0., 1., 0., 0.], + [ 0., 1., 1., 1., 0.], + [ 1., 1., 1., 1., 1.], + [ 0., 1., 1., 1., 0.], + [ 0., 0., 1., 0., 0.]]) + >>> struct = ndimage.generate_binary_structure(2, 2) + >>> struct + array([[ True, True, True], + [ True, True, True], + [ True, True, True]], dtype=bool) + >>> struct = ndimage.generate_binary_structure(3, 1) + >>> struct # no diagonal elements + array([[[False, False, False], + [False, True, False], + [False, False, False]], + [[False, True, False], + [ True, True, True], + [False, True, False]], + [[False, False, False], + [False, True, False], + [False, False, False]]], dtype=bool) + + """ + if connectivity < 1: + connectivity = 1 + if rank < 1: + return np.array(True, dtype=bool) + output = np.fabs(np.indices([3] * rank) - 1) + output = np.add.reduce(output, 0) + return output <= connectivity + + +def _binary_erosion(input, structure, iterations, mask, output, + border_value, origin, invert, brute_force, axes): + try: + iterations = operator.index(iterations) + except TypeError as e: + raise TypeError('iterations parameter should be an integer') from e + + input = np.asarray(input) + ndim = input.ndim + if np.iscomplexobj(input): + raise TypeError('Complex type not supported') + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + if structure is None: + structure = generate_binary_structure(num_axes, 1) + else: + structure = np.asarray(structure, dtype=bool) + if ndim > num_axes: + structure = _filters._expand_footprint(ndim, axes, structure, + footprint_name="structure") + + if structure.ndim != input.ndim: + raise RuntimeError('structure and input must have same dimensionality') + if not structure.flags.contiguous: + structure = structure.copy() + if structure.size < 1: + raise RuntimeError('structure must not be empty') + if mask is not None: + mask = np.asarray(mask) + if mask.shape != input.shape: + raise RuntimeError('mask and input must have equal sizes') + origin = _ni_support._normalize_sequence(origin, num_axes) + origin = _filters._expand_origin(ndim, axes, origin) + cit = _center_is_true(structure, origin) + if isinstance(output, np.ndarray): + if np.iscomplexobj(output): + raise TypeError('Complex output type not supported') + else: + output = bool + output = _ni_support._get_output(output, input) + temp_needed = np.may_share_memory(input, output) + if temp_needed: + # input and output arrays cannot share memory + temp = output + output = _ni_support._get_output(output.dtype, input) + if iterations == 1: + _nd_image.binary_erosion(input, structure, mask, output, + border_value, origin, invert, cit, 0) + elif cit and not brute_force: + changed, coordinate_list = _nd_image.binary_erosion( + input, structure, mask, output, + border_value, origin, invert, cit, 1) + structure = structure[tuple([slice(None, None, -1)] * + structure.ndim)] + for ii in range(len(origin)): + origin[ii] = -origin[ii] + if not structure.shape[ii] & 1: + origin[ii] -= 1 + if mask is not None: + mask = np.asarray(mask, dtype=np.int8) + if not structure.flags.contiguous: + structure = structure.copy() + _nd_image.binary_erosion2(output, structure, mask, iterations - 1, + origin, invert, coordinate_list) + else: + tmp_in = np.empty_like(input, dtype=bool) + tmp_out = output + if iterations >= 1 and not iterations & 1: + tmp_in, tmp_out = tmp_out, tmp_in + changed = _nd_image.binary_erosion( + input, structure, mask, tmp_out, + border_value, origin, invert, cit, 0) + ii = 1 + while ii < iterations or (iterations < 1 and changed): + tmp_in, tmp_out = tmp_out, tmp_in + changed = _nd_image.binary_erosion( + tmp_in, structure, mask, tmp_out, + border_value, origin, invert, cit, 0) + ii += 1 + if temp_needed: + temp[...] = output + output = temp + return output + + +def binary_erosion(input, structure=None, iterations=1, mask=None, output=None, + border_value=0, origin=0, brute_force=False, *, axes=None): + """ + Multidimensional binary erosion with a given structuring element. + + Binary erosion is a mathematical morphology operation used for image + processing. + + Parameters + ---------- + input : array_like + Binary image to be eroded. Non-zero (True) elements form + the subset to be eroded. + structure : array_like, optional + Structuring element used for the erosion. Non-zero elements are + considered True. If no structuring element is provided, an element + is generated with a square connectivity equal to one. + iterations : int, optional + The erosion is repeated `iterations` times (one, by default). + If iterations is less than 1, the erosion is repeated until the + result does not change anymore. + mask : array_like, optional + If a mask is given, only those elements with a True value at + the corresponding mask element are modified at each iteration. + output : ndarray, optional + Array of the same shape as input, into which the output is placed. + By default, a new array is created. + border_value : int (cast to 0 or 1), optional + Value at the border in the output array. + origin : int or tuple of ints, optional + Placement of the filter, by default 0. + brute_force : boolean, optional + Memory condition: if False, only the pixels whose value was changed in + the last iteration are tracked as candidates to be updated (eroded) in + the current iteration; if True all pixels are considered as candidates + for erosion, regardless of what happened in the previous iteration. + False by default. + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + binary_erosion : ndarray of bools + Erosion of the input by the structuring element. + + See Also + -------- + grey_erosion, binary_dilation, binary_closing, binary_opening, + generate_binary_structure + + Notes + ----- + Erosion [1]_ is a mathematical morphology operation [2]_ that uses a + structuring element for shrinking the shapes in an image. The binary + erosion of an image by a structuring element is the locus of the points + where a superimposition of the structuring element centered on the point + is entirely contained in the set of non-zero elements of the image. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Erosion_%28morphology%29 + .. [2] https://en.wikipedia.org/wiki/Mathematical_morphology + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.zeros((7,7), dtype=int) + >>> a[1:6, 2:5] = 1 + >>> a + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> ndimage.binary_erosion(a).astype(a.dtype) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> #Erosion removes objects smaller than the structure + >>> ndimage.binary_erosion(a, structure=np.ones((5,5))).astype(a.dtype) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + + """ + return _binary_erosion(input, structure, iterations, mask, + output, border_value, origin, 0, brute_force, axes) + + +def binary_dilation(input, structure=None, iterations=1, mask=None, + output=None, border_value=0, origin=0, + brute_force=False, *, axes=None): + """ + Multidimensional binary dilation with the given structuring element. + + Parameters + ---------- + input : array_like + Binary array_like to be dilated. Non-zero (True) elements form + the subset to be dilated. + structure : array_like, optional + Structuring element used for the dilation. Non-zero elements are + considered True. If no structuring element is provided an element + is generated with a square connectivity equal to one. + iterations : int, optional + The dilation is repeated `iterations` times (one, by default). + If iterations is less than 1, the dilation is repeated until the + result does not change anymore. Only an integer of iterations is + accepted. + mask : array_like, optional + If a mask is given, only those elements with a True value at + the corresponding mask element are modified at each iteration. + output : ndarray, optional + Array of the same shape as input, into which the output is placed. + By default, a new array is created. + border_value : int (cast to 0 or 1), optional + Value at the border in the output array. + origin : int or tuple of ints, optional + Placement of the filter, by default 0. + brute_force : boolean, optional + Memory condition: if False, only the pixels whose value was changed in + the last iteration are tracked as candidates to be updated (dilated) + in the current iteration; if True all pixels are considered as + candidates for dilation, regardless of what happened in the previous + iteration. False by default. + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + binary_dilation : ndarray of bools + Dilation of the input by the structuring element. + + See Also + -------- + grey_dilation, binary_erosion, binary_closing, binary_opening, + generate_binary_structure + + Notes + ----- + Dilation [1]_ is a mathematical morphology operation [2]_ that uses a + structuring element for expanding the shapes in an image. The binary + dilation of an image by a structuring element is the locus of the points + covered by the structuring element, when its center lies within the + non-zero points of the image. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Dilation_%28morphology%29 + .. [2] https://en.wikipedia.org/wiki/Mathematical_morphology + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.zeros((5, 5)) + >>> a[2, 2] = 1 + >>> a + array([[ 0., 0., 0., 0., 0.], + [ 0., 0., 0., 0., 0.], + [ 0., 0., 1., 0., 0.], + [ 0., 0., 0., 0., 0.], + [ 0., 0., 0., 0., 0.]]) + >>> ndimage.binary_dilation(a) + array([[False, False, False, False, False], + [False, False, True, False, False], + [False, True, True, True, False], + [False, False, True, False, False], + [False, False, False, False, False]], dtype=bool) + >>> ndimage.binary_dilation(a).astype(a.dtype) + array([[ 0., 0., 0., 0., 0.], + [ 0., 0., 1., 0., 0.], + [ 0., 1., 1., 1., 0.], + [ 0., 0., 1., 0., 0.], + [ 0., 0., 0., 0., 0.]]) + >>> # 3x3 structuring element with connectivity 1, used by default + >>> struct1 = ndimage.generate_binary_structure(2, 1) + >>> struct1 + array([[False, True, False], + [ True, True, True], + [False, True, False]], dtype=bool) + >>> # 3x3 structuring element with connectivity 2 + >>> struct2 = ndimage.generate_binary_structure(2, 2) + >>> struct2 + array([[ True, True, True], + [ True, True, True], + [ True, True, True]], dtype=bool) + >>> ndimage.binary_dilation(a, structure=struct1).astype(a.dtype) + array([[ 0., 0., 0., 0., 0.], + [ 0., 0., 1., 0., 0.], + [ 0., 1., 1., 1., 0.], + [ 0., 0., 1., 0., 0.], + [ 0., 0., 0., 0., 0.]]) + >>> ndimage.binary_dilation(a, structure=struct2).astype(a.dtype) + array([[ 0., 0., 0., 0., 0.], + [ 0., 1., 1., 1., 0.], + [ 0., 1., 1., 1., 0.], + [ 0., 1., 1., 1., 0.], + [ 0., 0., 0., 0., 0.]]) + >>> ndimage.binary_dilation(a, structure=struct1,\\ + ... iterations=2).astype(a.dtype) + array([[ 0., 0., 1., 0., 0.], + [ 0., 1., 1., 1., 0.], + [ 1., 1., 1., 1., 1.], + [ 0., 1., 1., 1., 0.], + [ 0., 0., 1., 0., 0.]]) + + """ + input = np.asarray(input) + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + if structure is None: + structure = generate_binary_structure(num_axes, 1) + origin = _ni_support._normalize_sequence(origin, num_axes) + structure = np.asarray(structure) + structure = structure[tuple([slice(None, None, -1)] * + structure.ndim)] + for ii in range(len(origin)): + origin[ii] = -origin[ii] + if not structure.shape[ii] & 1: + origin[ii] -= 1 + + return _binary_erosion(input, structure, iterations, mask, + output, border_value, origin, 1, brute_force, axes) + + +def binary_opening(input, structure=None, iterations=1, output=None, + origin=0, mask=None, border_value=0, brute_force=False, *, + axes=None): + """ + Multidimensional binary opening with the given structuring element. + + The *opening* of an input image by a structuring element is the + *dilation* of the *erosion* of the image by the structuring element. + + Parameters + ---------- + input : array_like + Binary array_like to be opened. Non-zero (True) elements form + the subset to be opened. + structure : array_like, optional + Structuring element used for the opening. Non-zero elements are + considered True. If no structuring element is provided an element + is generated with a square connectivity equal to one (i.e., only + nearest neighbors are connected to the center, diagonally-connected + elements are not considered neighbors). + iterations : int, optional + The erosion step of the opening, then the dilation step are each + repeated `iterations` times (one, by default). If `iterations` is + less than 1, each operation is repeated until the result does + not change anymore. Only an integer of iterations is accepted. + output : ndarray, optional + Array of the same shape as input, into which the output is placed. + By default, a new array is created. + origin : int or tuple of ints, optional + Placement of the filter, by default 0. + mask : array_like, optional + If a mask is given, only those elements with a True value at + the corresponding mask element are modified at each iteration. + + .. versionadded:: 1.1.0 + border_value : int (cast to 0 or 1), optional + Value at the border in the output array. + + .. versionadded:: 1.1.0 + brute_force : boolean, optional + Memory condition: if False, only the pixels whose value was changed in + the last iteration are tracked as candidates to be updated in the + current iteration; if true all pixels are considered as candidates for + update, regardless of what happened in the previous iteration. + False by default. + + .. versionadded:: 1.1.0 + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + binary_opening : ndarray of bools + Opening of the input by the structuring element. + + See Also + -------- + grey_opening, binary_closing, binary_erosion, binary_dilation, + generate_binary_structure + + Notes + ----- + *Opening* [1]_ is a mathematical morphology operation [2]_ that + consists in the succession of an erosion and a dilation of the + input with the same structuring element. Opening, therefore, removes + objects smaller than the structuring element. + + Together with *closing* (`binary_closing`), opening can be used for + noise removal. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Opening_%28morphology%29 + .. [2] https://en.wikipedia.org/wiki/Mathematical_morphology + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.zeros((5,5), dtype=int) + >>> a[1:4, 1:4] = 1; a[4, 4] = 1 + >>> a + array([[0, 0, 0, 0, 0], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 0, 0, 0, 1]]) + >>> # Opening removes small objects + >>> ndimage.binary_opening(a, structure=np.ones((3,3))).astype(int) + array([[0, 0, 0, 0, 0], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 0, 0, 0, 0]]) + >>> # Opening can also smooth corners + >>> ndimage.binary_opening(a).astype(int) + array([[0, 0, 0, 0, 0], + [0, 0, 1, 0, 0], + [0, 1, 1, 1, 0], + [0, 0, 1, 0, 0], + [0, 0, 0, 0, 0]]) + >>> # Opening is the dilation of the erosion of the input + >>> ndimage.binary_erosion(a).astype(int) + array([[0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 1, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0]]) + >>> ndimage.binary_dilation(ndimage.binary_erosion(a)).astype(int) + array([[0, 0, 0, 0, 0], + [0, 0, 1, 0, 0], + [0, 1, 1, 1, 0], + [0, 0, 1, 0, 0], + [0, 0, 0, 0, 0]]) + + """ + input = np.asarray(input) + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + if structure is None: + structure = generate_binary_structure(num_axes, 1) + + tmp = binary_erosion(input, structure, iterations, mask, None, + border_value, origin, brute_force, axes=axes) + return binary_dilation(tmp, structure, iterations, mask, output, + border_value, origin, brute_force, axes=axes) + + +def binary_closing(input, structure=None, iterations=1, output=None, + origin=0, mask=None, border_value=0, brute_force=False, *, + axes=None): + """ + Multidimensional binary closing with the given structuring element. + + The *closing* of an input image by a structuring element is the + *erosion* of the *dilation* of the image by the structuring element. + + Parameters + ---------- + input : array_like + Binary array_like to be closed. Non-zero (True) elements form + the subset to be closed. + structure : array_like, optional + Structuring element used for the closing. Non-zero elements are + considered True. If no structuring element is provided an element + is generated with a square connectivity equal to one (i.e., only + nearest neighbors are connected to the center, diagonally-connected + elements are not considered neighbors). + iterations : int, optional + The dilation step of the closing, then the erosion step are each + repeated `iterations` times (one, by default). If iterations is + less than 1, each operations is repeated until the result does + not change anymore. Only an integer of iterations is accepted. + output : ndarray, optional + Array of the same shape as input, into which the output is placed. + By default, a new array is created. + origin : int or tuple of ints, optional + Placement of the filter, by default 0. + mask : array_like, optional + If a mask is given, only those elements with a True value at + the corresponding mask element are modified at each iteration. + + .. versionadded:: 1.1.0 + border_value : int (cast to 0 or 1), optional + Value at the border in the output array. + + .. versionadded:: 1.1.0 + brute_force : boolean, optional + Memory condition: if False, only the pixels whose value was changed in + the last iteration are tracked as candidates to be updated in the + current iteration; if true al pixels are considered as candidates for + update, regardless of what happened in the previous iteration. + False by default. + + .. versionadded:: 1.1.0 + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + binary_closing : ndarray of bools + Closing of the input by the structuring element. + + See Also + -------- + grey_closing, binary_opening, binary_dilation, binary_erosion, + generate_binary_structure + + Notes + ----- + *Closing* [1]_ is a mathematical morphology operation [2]_ that + consists in the succession of a dilation and an erosion of the + input with the same structuring element. Closing therefore fills + holes smaller than the structuring element. + + Together with *opening* (`binary_opening`), closing can be used for + noise removal. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Closing_%28morphology%29 + .. [2] https://en.wikipedia.org/wiki/Mathematical_morphology + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.zeros((5,5), dtype=int) + >>> a[1:-1, 1:-1] = 1; a[2,2] = 0 + >>> a + array([[0, 0, 0, 0, 0], + [0, 1, 1, 1, 0], + [0, 1, 0, 1, 0], + [0, 1, 1, 1, 0], + [0, 0, 0, 0, 0]]) + >>> # Closing removes small holes + >>> ndimage.binary_closing(a).astype(int) + array([[0, 0, 0, 0, 0], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 0, 0, 0, 0]]) + >>> # Closing is the erosion of the dilation of the input + >>> ndimage.binary_dilation(a).astype(int) + array([[0, 1, 1, 1, 0], + [1, 1, 1, 1, 1], + [1, 1, 1, 1, 1], + [1, 1, 1, 1, 1], + [0, 1, 1, 1, 0]]) + >>> ndimage.binary_erosion(ndimage.binary_dilation(a)).astype(int) + array([[0, 0, 0, 0, 0], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 0, 0, 0, 0]]) + + + >>> a = np.zeros((7,7), dtype=int) + >>> a[1:6, 2:5] = 1; a[1:3,3] = 0 + >>> a + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 1, 0, 0], + [0, 0, 1, 0, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> # In addition to removing holes, closing can also + >>> # coarsen boundaries with fine hollows. + >>> ndimage.binary_closing(a).astype(int) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> ndimage.binary_closing(a, structure=np.ones((2,2))).astype(int) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + + """ + input = np.asarray(input) + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + if structure is None: + structure = generate_binary_structure(num_axes, 1) + + tmp = binary_dilation(input, structure, iterations, mask, None, + border_value, origin, brute_force, axes=axes) + return binary_erosion(tmp, structure, iterations, mask, output, + border_value, origin, brute_force, axes=axes) + + +def binary_hit_or_miss(input, structure1=None, structure2=None, + output=None, origin1=0, origin2=None, *, axes=None): + """ + Multidimensional binary hit-or-miss transform. + + The hit-or-miss transform finds the locations of a given pattern + inside the input image. + + Parameters + ---------- + input : array_like (cast to booleans) + Binary image where a pattern is to be detected. + structure1 : array_like (cast to booleans), optional + Part of the structuring element to be fitted to the foreground + (non-zero elements) of `input`. If no value is provided, a + structure of square connectivity 1 is chosen. + structure2 : array_like (cast to booleans), optional + Second part of the structuring element that has to miss completely + the foreground. If no value is provided, the complementary of + `structure1` is taken. + output : ndarray, optional + Array of the same shape as input, into which the output is placed. + By default, a new array is created. + origin1 : int or tuple of ints, optional + Placement of the first part of the structuring element `structure1`, + by default 0 for a centered structure. + origin2 : int or tuple of ints, optional + Placement of the second part of the structuring element `structure2`, + by default 0 for a centered structure. If a value is provided for + `origin1` and not for `origin2`, then `origin2` is set to `origin1`. + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If `origin1` or `origin2` tuples are provided, their + length must match the number of axes. + + Returns + ------- + binary_hit_or_miss : ndarray + Hit-or-miss transform of `input` with the given structuring + element (`structure1`, `structure2`). + + See Also + -------- + binary_erosion + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Hit-or-miss_transform + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.zeros((7,7), dtype=int) + >>> a[1, 1] = 1; a[2:4, 2:4] = 1; a[4:6, 4:6] = 1 + >>> a + array([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 1, 1, 0], + [0, 0, 0, 0, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> structure1 = np.array([[1, 0, 0], [0, 1, 1], [0, 1, 1]]) + >>> structure1 + array([[1, 0, 0], + [0, 1, 1], + [0, 1, 1]]) + >>> # Find the matches of structure1 in the array a + >>> ndimage.binary_hit_or_miss(a, structure1=structure1).astype(int) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> # Change the origin of the filter + >>> # origin1=1 is equivalent to origin1=(1,1) here + >>> ndimage.binary_hit_or_miss(a, structure1=structure1,\\ + ... origin1=1).astype(int) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0], + [0, 0, 0, 0, 0, 0, 0]]) + + """ + input = np.asarray(input) + axes = _ni_support._check_axes(axes, input.ndim) + num_axes = len(axes) + if structure1 is None: + structure1 = generate_binary_structure(num_axes, 1) + else: + structure1 = np.asarray(structure1) + if structure2 is None: + structure2 = np.logical_not(structure1) + origin1 = _ni_support._normalize_sequence(origin1, num_axes) + if origin2 is None: + origin2 = origin1 + else: + origin2 = _ni_support._normalize_sequence(origin2, num_axes) + + tmp1 = _binary_erosion(input, structure1, 1, None, None, 0, origin1, + 0, False, axes) + inplace = isinstance(output, np.ndarray) + result = _binary_erosion(input, structure2, 1, None, output, 0, + origin2, 1, False, axes) + if inplace: + np.logical_not(output, output) + np.logical_and(tmp1, output, output) + else: + np.logical_not(result, result) + return np.logical_and(tmp1, result) + + +def binary_propagation(input, structure=None, mask=None, + output=None, border_value=0, origin=0, *, axes=None): + """ + Multidimensional binary propagation with the given structuring element. + + Parameters + ---------- + input : array_like + Binary image to be propagated inside `mask`. + structure : array_like, optional + Structuring element used in the successive dilations. The output + may depend on the structuring element, especially if `mask` has + several connex components. If no structuring element is + provided, an element is generated with a squared connectivity equal + to one. + mask : array_like, optional + Binary mask defining the region into which `input` is allowed to + propagate. + output : ndarray, optional + Array of the same shape as input, into which the output is placed. + By default, a new array is created. + border_value : int (cast to 0 or 1), optional + Value at the border in the output array. + origin : int or tuple of ints, optional + Placement of the filter, by default 0. + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + binary_propagation : ndarray + Binary propagation of `input` inside `mask`. + + Notes + ----- + This function is functionally equivalent to calling binary_dilation + with the number of iterations less than one: iterative dilation until + the result does not change anymore. + + The succession of an erosion and propagation inside the original image + can be used instead of an *opening* for deleting small objects while + keeping the contours of larger objects untouched. + + References + ---------- + .. [1] http://cmm.ensmp.fr/~serra/cours/pdf/en/ch6en.pdf, slide 15. + .. [2] I.T. Young, J.J. Gerbrands, and L.J. van Vliet, "Fundamentals of + image processing", 1998 + ftp://qiftp.tudelft.nl/DIPimage/docs/FIP2.3.pdf + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> input = np.zeros((8, 8), dtype=int) + >>> input[2, 2] = 1 + >>> mask = np.zeros((8, 8), dtype=int) + >>> mask[1:4, 1:4] = mask[4, 4] = mask[6:8, 6:8] = 1 + >>> input + array([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]]) + >>> mask + array([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 1, 1], + [0, 0, 0, 0, 0, 0, 1, 1]]) + >>> ndimage.binary_propagation(input, mask=mask).astype(int) + array([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]]) + >>> ndimage.binary_propagation(input, mask=mask,\\ + ... structure=np.ones((3,3))).astype(int) + array([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]]) + + >>> # Comparison between opening and erosion+propagation + >>> a = np.zeros((6,6), dtype=int) + >>> a[2:5, 2:5] = 1; a[0, 0] = 1; a[5, 5] = 1 + >>> a + array([[1, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 1]]) + >>> ndimage.binary_opening(a).astype(int) + array([[0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0], + [0, 0, 1, 1, 1, 0], + [0, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0]]) + >>> b = ndimage.binary_erosion(a) + >>> b.astype(int) + array([[0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0]]) + >>> ndimage.binary_propagation(b, mask=a).astype(int) + array([[0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0]]) + + """ + return binary_dilation(input, structure, -1, mask, output, + border_value, origin, axes=axes) + + +def binary_fill_holes(input, structure=None, output=None, origin=0, *, + axes=None): + """ + Fill the holes in binary objects. + + + Parameters + ---------- + input : array_like + N-D binary array with holes to be filled + structure : array_like, optional + Structuring element used in the computation; large-size elements + make computations faster but may miss holes separated from the + background by thin regions. The default element (with a square + connectivity equal to one) yields the intuitive result where all + holes in the input have been filled. + output : ndarray, optional + Array of the same shape as input, into which the output is placed. + By default, a new array is created. + origin : int, tuple of ints, optional + Position of the structuring element. + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + out : ndarray + Transformation of the initial image `input` where holes have been + filled. + + See Also + -------- + binary_dilation, binary_propagation, label + + Notes + ----- + The algorithm used in this function consists in invading the complementary + of the shapes in `input` from the outer boundary of the image, + using binary dilations. Holes are not connected to the boundary and are + therefore not invaded. The result is the complementary subset of the + invaded region. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Mathematical_morphology + + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.zeros((5, 5), dtype=int) + >>> a[1:4, 1:4] = 1 + >>> a[2,2] = 0 + >>> a + array([[0, 0, 0, 0, 0], + [0, 1, 1, 1, 0], + [0, 1, 0, 1, 0], + [0, 1, 1, 1, 0], + [0, 0, 0, 0, 0]]) + >>> ndimage.binary_fill_holes(a).astype(int) + array([[0, 0, 0, 0, 0], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 0, 0, 0, 0]]) + >>> # Too big structuring element + >>> ndimage.binary_fill_holes(a, structure=np.ones((5,5))).astype(int) + array([[0, 0, 0, 0, 0], + [0, 1, 1, 1, 0], + [0, 1, 0, 1, 0], + [0, 1, 1, 1, 0], + [0, 0, 0, 0, 0]]) + + """ + input = np.asarray(input) + mask = np.logical_not(input) + tmp = np.zeros(mask.shape, bool) + inplace = isinstance(output, np.ndarray) + if inplace: + binary_dilation(tmp, structure, -1, mask, output, 1, origin, axes=axes) + np.logical_not(output, output) + else: + output = binary_dilation(tmp, structure, -1, mask, None, 1, + origin, axes=axes) + np.logical_not(output, output) + return output + + +def grey_erosion(input, size=None, footprint=None, structure=None, + output=None, mode="reflect", cval=0.0, origin=0, *, + axes=None): + """ + Calculate a greyscale erosion, using either a structuring element, + or a footprint corresponding to a flat structuring element. + + Grayscale erosion is a mathematical morphology operation. For the + simple case of a full and flat structuring element, it can be viewed + as a minimum filter over a sliding window. + + Parameters + ---------- + input : array_like + Array over which the grayscale erosion is to be computed. + size : tuple of ints + Shape of a flat and full structuring element used for the grayscale + erosion. Optional if `footprint` or `structure` is provided. + footprint : array of ints, optional + Positions of non-infinite elements of a flat structuring element + used for the grayscale erosion. Non-zero values give the set of + neighbors of the center over which the minimum is chosen. + structure : array of ints, optional + Structuring element used for the grayscale erosion. `structure` + may be a non-flat structuring element. The `structure` array applies a + subtractive offset for each pixel in the neighborhood. + output : array, optional + An array used for storing the output of the erosion may be provided. + mode : {'reflect','constant','nearest','mirror', 'wrap'}, optional + The `mode` parameter determines how the array borders are + handled, where `cval` is the value when mode is equal to + 'constant'. Default is 'reflect' + cval : scalar, optional + Value to fill past edges of input if `mode` is 'constant'. Default + is 0.0. + origin : scalar, optional + The `origin` parameter controls the placement of the filter. + Default 0 + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + output : ndarray + Grayscale erosion of `input`. + + See Also + -------- + binary_erosion, grey_dilation, grey_opening, grey_closing + generate_binary_structure, minimum_filter + + Notes + ----- + The grayscale erosion of an image input by a structuring element s defined + over a domain E is given by: + + (input+s)(x) = min {input(y) - s(x-y), for y in E} + + In particular, for structuring elements defined as + s(y) = 0 for y in E, the grayscale erosion computes the minimum of the + input image inside a sliding window defined by E. + + Grayscale erosion [1]_ is a *mathematical morphology* operation [2]_. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Erosion_%28morphology%29 + .. [2] https://en.wikipedia.org/wiki/Mathematical_morphology + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.zeros((7,7), dtype=int) + >>> a[1:6, 1:6] = 3 + >>> a[4,4] = 2; a[2,3] = 1 + >>> a + array([[0, 0, 0, 0, 0, 0, 0], + [0, 3, 3, 3, 3, 3, 0], + [0, 3, 3, 1, 3, 3, 0], + [0, 3, 3, 3, 3, 3, 0], + [0, 3, 3, 3, 2, 3, 0], + [0, 3, 3, 3, 3, 3, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> ndimage.grey_erosion(a, size=(3,3)) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 3, 2, 2, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> footprint = ndimage.generate_binary_structure(2, 1) + >>> footprint + array([[False, True, False], + [ True, True, True], + [False, True, False]], dtype=bool) + >>> # Diagonally-connected elements are not considered neighbors + >>> ndimage.grey_erosion(a, footprint=footprint) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 3, 1, 2, 0, 0], + [0, 0, 3, 2, 2, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + + """ + if size is None and footprint is None and structure is None: + raise ValueError("size, footprint, or structure must be specified") + + return _filters._min_or_max_filter(input, size, footprint, structure, + output, mode, cval, origin, 1, + axes=axes) + + +def grey_dilation(input, size=None, footprint=None, structure=None, + output=None, mode="reflect", cval=0.0, origin=0, *, + axes=None): + """ + Calculate a greyscale dilation, using either a structuring element, + or a footprint corresponding to a flat structuring element. + + Grayscale dilation is a mathematical morphology operation. For the + simple case of a full and flat structuring element, it can be viewed + as a maximum filter over a sliding window. + + Parameters + ---------- + input : array_like + Array over which the grayscale dilation is to be computed. + size : tuple of ints + Shape of a flat and full structuring element used for the grayscale + dilation. Optional if `footprint` or `structure` is provided. + footprint : array of ints, optional + Positions of non-infinite elements of a flat structuring element + used for the grayscale dilation. Non-zero values give the set of + neighbors of the center over which the maximum is chosen. + structure : array of ints, optional + Structuring element used for the grayscale dilation. `structure` + may be a non-flat structuring element. The `structure` array applies an + additive offset for each pixel in the neighborhood. + output : array, optional + An array used for storing the output of the dilation may be provided. + mode : {'reflect','constant','nearest','mirror', 'wrap'}, optional + The `mode` parameter determines how the array borders are + handled, where `cval` is the value when mode is equal to + 'constant'. Default is 'reflect' + cval : scalar, optional + Value to fill past edges of input if `mode` is 'constant'. Default + is 0.0. + origin : scalar, optional + The `origin` parameter controls the placement of the filter. + Default 0 + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + grey_dilation : ndarray + Grayscale dilation of `input`. + + See Also + -------- + binary_dilation, grey_erosion, grey_closing, grey_opening + generate_binary_structure, maximum_filter + + Notes + ----- + The grayscale dilation of an image input by a structuring element s defined + over a domain E is given by: + + (input+s)(x) = max {input(y) + s(x-y), for y in E} + + In particular, for structuring elements defined as + s(y) = 0 for y in E, the grayscale dilation computes the maximum of the + input image inside a sliding window defined by E. + + Grayscale dilation [1]_ is a *mathematical morphology* operation [2]_. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Dilation_%28morphology%29 + .. [2] https://en.wikipedia.org/wiki/Mathematical_morphology + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.zeros((7,7), dtype=int) + >>> a[2:5, 2:5] = 1 + >>> a[4,4] = 2; a[2,3] = 3 + >>> a + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 3, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 2, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> ndimage.grey_dilation(a, size=(3,3)) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 3, 3, 3, 1, 0], + [0, 1, 3, 3, 3, 1, 0], + [0, 1, 3, 3, 3, 2, 0], + [0, 1, 1, 2, 2, 2, 0], + [0, 1, 1, 2, 2, 2, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> ndimage.grey_dilation(a, footprint=np.ones((3,3))) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 3, 3, 3, 1, 0], + [0, 1, 3, 3, 3, 1, 0], + [0, 1, 3, 3, 3, 2, 0], + [0, 1, 1, 2, 2, 2, 0], + [0, 1, 1, 2, 2, 2, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> s = ndimage.generate_binary_structure(2,1) + >>> s + array([[False, True, False], + [ True, True, True], + [False, True, False]], dtype=bool) + >>> ndimage.grey_dilation(a, footprint=s) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 3, 1, 0, 0], + [0, 1, 3, 3, 3, 1, 0], + [0, 1, 1, 3, 2, 1, 0], + [0, 1, 1, 2, 2, 2, 0], + [0, 0, 1, 1, 2, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> ndimage.grey_dilation(a, size=(3,3), structure=np.ones((3,3))) + array([[1, 1, 1, 1, 1, 1, 1], + [1, 2, 4, 4, 4, 2, 1], + [1, 2, 4, 4, 4, 2, 1], + [1, 2, 4, 4, 4, 3, 1], + [1, 2, 2, 3, 3, 3, 1], + [1, 2, 2, 3, 3, 3, 1], + [1, 1, 1, 1, 1, 1, 1]]) + + """ + if size is None and footprint is None and structure is None: + raise ValueError("size, footprint, or structure must be specified") + if structure is not None: + structure = np.asarray(structure) + structure = structure[tuple([slice(None, None, -1)] * + structure.ndim)] + if footprint is not None: + footprint = np.asarray(footprint) + footprint = footprint[tuple([slice(None, None, -1)] * + footprint.ndim)] + + input = np.asarray(input) + axes = _ni_support._check_axes(axes, input.ndim) + origin = _ni_support._normalize_sequence(origin, len(axes)) + for ii in range(len(origin)): + origin[ii] = -origin[ii] + if footprint is not None: + sz = footprint.shape[ii] + elif structure is not None: + sz = structure.shape[ii] + elif np.isscalar(size): + sz = size + else: + sz = size[ii] + if not sz & 1: + origin[ii] -= 1 + + return _filters._min_or_max_filter(input, size, footprint, structure, + output, mode, cval, origin, 0, + axes=axes) + + +def grey_opening(input, size=None, footprint=None, structure=None, + output=None, mode="reflect", cval=0.0, origin=0, *, + axes=None): + """ + Multidimensional grayscale opening. + + A grayscale opening consists in the succession of a grayscale erosion, + and a grayscale dilation. + + Parameters + ---------- + input : array_like + Array over which the grayscale opening is to be computed. + size : tuple of ints + Shape of a flat and full structuring element used for the grayscale + opening. Optional if `footprint` or `structure` is provided. + footprint : array of ints, optional + Positions of non-infinite elements of a flat structuring element + used for the grayscale opening. + structure : array of ints, optional + Structuring element used for the grayscale opening. `structure` + may be a non-flat structuring element. The `structure` array applies + offsets to the pixels in a neighborhood (the offset is additive during + dilation and subtractive during erosion). + output : array, optional + An array used for storing the output of the opening may be provided. + mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap'}, optional + The `mode` parameter determines how the array borders are + handled, where `cval` is the value when mode is equal to + 'constant'. Default is 'reflect' + cval : scalar, optional + Value to fill past edges of input if `mode` is 'constant'. Default + is 0.0. + origin : scalar, optional + The `origin` parameter controls the placement of the filter. + Default 0 + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + grey_opening : ndarray + Result of the grayscale opening of `input` with `structure`. + + See Also + -------- + binary_opening, grey_dilation, grey_erosion, grey_closing + generate_binary_structure + + Notes + ----- + The action of a grayscale opening with a flat structuring element amounts + to smoothen high local maxima, whereas binary opening erases small objects. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Mathematical_morphology + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.arange(36).reshape((6,6)) + >>> a[3, 3] = 50 + >>> a + array([[ 0, 1, 2, 3, 4, 5], + [ 6, 7, 8, 9, 10, 11], + [12, 13, 14, 15, 16, 17], + [18, 19, 20, 50, 22, 23], + [24, 25, 26, 27, 28, 29], + [30, 31, 32, 33, 34, 35]]) + >>> ndimage.grey_opening(a, size=(3,3)) + array([[ 0, 1, 2, 3, 4, 4], + [ 6, 7, 8, 9, 10, 10], + [12, 13, 14, 15, 16, 16], + [18, 19, 20, 22, 22, 22], + [24, 25, 26, 27, 28, 28], + [24, 25, 26, 27, 28, 28]]) + >>> # Note that the local maximum a[3,3] has disappeared + + """ + if (size is not None) and (footprint is not None): + warnings.warn("ignoring size because footprint is set", + UserWarning, stacklevel=2) + tmp = grey_erosion(input, size, footprint, structure, None, mode, + cval, origin, axes=axes) + return grey_dilation(tmp, size, footprint, structure, output, mode, + cval, origin, axes=axes) + + +def grey_closing(input, size=None, footprint=None, structure=None, + output=None, mode="reflect", cval=0.0, origin=0, *, + axes=None): + """ + Multidimensional grayscale closing. + + A grayscale closing consists in the succession of a grayscale dilation, + and a grayscale erosion. + + Parameters + ---------- + input : array_like + Array over which the grayscale closing is to be computed. + size : tuple of ints + Shape of a flat and full structuring element used for the grayscale + closing. Optional if `footprint` or `structure` is provided. + footprint : array of ints, optional + Positions of non-infinite elements of a flat structuring element + used for the grayscale closing. + structure : array of ints, optional + Structuring element used for the grayscale closing. `structure` + may be a non-flat structuring element. The `structure` array applies + offsets to the pixels in a neighborhood (the offset is additive during + dilation and subtractive during erosion) + output : array, optional + An array used for storing the output of the closing may be provided. + mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap'}, optional + The `mode` parameter determines how the array borders are + handled, where `cval` is the value when mode is equal to + 'constant'. Default is 'reflect' + cval : scalar, optional + Value to fill past edges of input if `mode` is 'constant'. Default + is 0.0. + origin : scalar, optional + The `origin` parameter controls the placement of the filter. + Default 0 + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + grey_closing : ndarray + Result of the grayscale closing of `input` with `structure`. + + See Also + -------- + binary_closing, grey_dilation, grey_erosion, grey_opening, + generate_binary_structure + + Notes + ----- + The action of a grayscale closing with a flat structuring element amounts + to smoothen deep local minima, whereas binary closing fills small holes. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Mathematical_morphology + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.arange(36).reshape((6,6)) + >>> a[3,3] = 0 + >>> a + array([[ 0, 1, 2, 3, 4, 5], + [ 6, 7, 8, 9, 10, 11], + [12, 13, 14, 15, 16, 17], + [18, 19, 20, 0, 22, 23], + [24, 25, 26, 27, 28, 29], + [30, 31, 32, 33, 34, 35]]) + >>> ndimage.grey_closing(a, size=(3,3)) + array([[ 7, 7, 8, 9, 10, 11], + [ 7, 7, 8, 9, 10, 11], + [13, 13, 14, 15, 16, 17], + [19, 19, 20, 20, 22, 23], + [25, 25, 26, 27, 28, 29], + [31, 31, 32, 33, 34, 35]]) + >>> # Note that the local minimum a[3,3] has disappeared + + """ + if (size is not None) and (footprint is not None): + warnings.warn("ignoring size because footprint is set", + UserWarning, stacklevel=2) + tmp = grey_dilation(input, size, footprint, structure, None, mode, + cval, origin, axes=axes) + return grey_erosion(tmp, size, footprint, structure, output, mode, + cval, origin, axes=axes) + + +def morphological_gradient(input, size=None, footprint=None, structure=None, + output=None, mode="reflect", cval=0.0, origin=0, *, + axes=None): + """ + Multidimensional morphological gradient. + + The morphological gradient is calculated as the difference between a + dilation and an erosion of the input with a given structuring element. + + Parameters + ---------- + input : array_like + Array over which to compute the morphlogical gradient. + size : tuple of ints + Shape of a flat and full structuring element used for the mathematical + morphology operations. Optional if `footprint` or `structure` is + provided. A larger `size` yields a more blurred gradient. + footprint : array of ints, optional + Positions of non-infinite elements of a flat structuring element + used for the morphology operations. Larger footprints + give a more blurred morphological gradient. + structure : array of ints, optional + Structuring element used for the morphology operations. `structure` may + be a non-flat structuring element. The `structure` array applies + offsets to the pixels in a neighborhood (the offset is additive during + dilation and subtractive during erosion) + output : array, optional + An array used for storing the output of the morphological gradient + may be provided. + mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap'}, optional + The `mode` parameter determines how the array borders are + handled, where `cval` is the value when mode is equal to + 'constant'. Default is 'reflect' + cval : scalar, optional + Value to fill past edges of input if `mode` is 'constant'. Default + is 0.0. + origin : scalar, optional + The `origin` parameter controls the placement of the filter. + Default 0 + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + morphological_gradient : ndarray + Morphological gradient of `input`. + + See Also + -------- + grey_dilation, grey_erosion, gaussian_gradient_magnitude + + Notes + ----- + For a flat structuring element, the morphological gradient + computed at a given point corresponds to the maximal difference + between elements of the input among the elements covered by the + structuring element centered on the point. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Mathematical_morphology + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.zeros((7,7), dtype=int) + >>> a[2:5, 2:5] = 1 + >>> ndimage.morphological_gradient(a, size=(3,3)) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 0, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> # The morphological gradient is computed as the difference + >>> # between a dilation and an erosion + >>> ndimage.grey_dilation(a, size=(3,3)) -\\ + ... ndimage.grey_erosion(a, size=(3,3)) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 0, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> a = np.zeros((7,7), dtype=int) + >>> a[2:5, 2:5] = 1 + >>> a[4,4] = 2; a[2,3] = 3 + >>> a + array([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 3, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 2, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]]) + >>> ndimage.morphological_gradient(a, size=(3,3)) + array([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 3, 3, 3, 1, 0], + [0, 1, 3, 3, 3, 1, 0], + [0, 1, 3, 2, 3, 2, 0], + [0, 1, 1, 2, 2, 2, 0], + [0, 1, 1, 2, 2, 2, 0], + [0, 0, 0, 0, 0, 0, 0]]) + + """ + tmp = grey_dilation(input, size, footprint, structure, None, mode, + cval, origin, axes=axes) + if isinstance(output, np.ndarray): + grey_erosion(input, size, footprint, structure, output, mode, + cval, origin, axes=axes) + return np.subtract(tmp, output, output) + else: + return (tmp - grey_erosion(input, size, footprint, structure, + None, mode, cval, origin, axes=axes)) + + +def morphological_laplace(input, size=None, footprint=None, structure=None, + output=None, mode="reflect", cval=0.0, origin=0, *, + axes=None): + """ + Multidimensional morphological laplace. + + Parameters + ---------- + input : array_like + Input. + size : tuple of ints + Shape of a flat and full structuring element used for the mathematical + morphology operations. Optional if `footprint` or `structure` is + provided. + footprint : array of ints, optional + Positions of non-infinite elements of a flat structuring element + used for the morphology operations. + structure : array of ints, optional + Structuring element used for the morphology operations. `structure` may + be a non-flat structuring element. The `structure` array applies + offsets to the pixels in a neighborhood (the offset is additive during + dilation and subtractive during erosion) + output : ndarray, optional + An output array can optionally be provided. + mode : {'reflect','constant','nearest','mirror', 'wrap'}, optional + The mode parameter determines how the array borders are handled. + For 'constant' mode, values beyond borders are set to be `cval`. + Default is 'reflect'. + cval : scalar, optional + Value to fill past edges of input if mode is 'constant'. + Default is 0.0 + origin : origin, optional + The origin parameter controls the placement of the filter. + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + morphological_laplace : ndarray + Output + + """ + tmp1 = grey_dilation(input, size, footprint, structure, None, mode, + cval, origin, axes=axes) + if isinstance(output, np.ndarray): + grey_erosion(input, size, footprint, structure, output, mode, + cval, origin, axes=axes) + np.add(tmp1, output, output) + np.subtract(output, input, output) + return np.subtract(output, input, output) + else: + tmp2 = grey_erosion(input, size, footprint, structure, None, mode, + cval, origin, axes=axes) + np.add(tmp1, tmp2, tmp2) + np.subtract(tmp2, input, tmp2) + np.subtract(tmp2, input, tmp2) + return tmp2 + + +def white_tophat(input, size=None, footprint=None, structure=None, + output=None, mode="reflect", cval=0.0, origin=0, *, + axes=None): + """ + Multidimensional white tophat filter. + + Parameters + ---------- + input : array_like + Input. + size : tuple of ints + Shape of a flat and full structuring element used for the filter. + Optional if `footprint` or `structure` is provided. + footprint : array of ints, optional + Positions of elements of a flat structuring element + used for the white tophat filter. + structure : array of ints, optional + Structuring element used for the filter. `structure` may be a non-flat + structuring element. The `structure` array applies offsets to the + pixels in a neighborhood (the offset is additive during dilation and + subtractive during erosion) + output : array, optional + An array used for storing the output of the filter may be provided. + mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap'}, optional + The `mode` parameter determines how the array borders are + handled, where `cval` is the value when mode is equal to + 'constant'. Default is 'reflect' + cval : scalar, optional + Value to fill past edges of input if `mode` is 'constant'. + Default is 0.0. + origin : scalar, optional + The `origin` parameter controls the placement of the filter. + Default is 0. + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + output : ndarray + Result of the filter of `input` with `structure`. + + See Also + -------- + black_tophat + + Examples + -------- + Subtract gray background from a bright peak. + + >>> from scipy.ndimage import generate_binary_structure, white_tophat + >>> import numpy as np + >>> square = generate_binary_structure(rank=2, connectivity=3) + >>> bright_on_gray = np.array([[2, 3, 3, 3, 2], + ... [3, 4, 5, 4, 3], + ... [3, 5, 9, 5, 3], + ... [3, 4, 5, 4, 3], + ... [2, 3, 3, 3, 2]]) + >>> white_tophat(input=bright_on_gray, structure=square) + array([[0, 0, 0, 0, 0], + [0, 0, 1, 0, 0], + [0, 1, 5, 1, 0], + [0, 0, 1, 0, 0], + [0, 0, 0, 0, 0]]) + + """ + input = np.asarray(input) + + if (size is not None) and (footprint is not None): + warnings.warn("ignoring size because footprint is set", + UserWarning, stacklevel=2) + tmp = grey_erosion(input, size, footprint, structure, None, mode, + cval, origin, axes=axes) + tmp = grey_dilation(tmp, size, footprint, structure, output, mode, + cval, origin, axes=axes) + if tmp is None: + tmp = output + + if input.dtype == np.bool_ and tmp.dtype == np.bool_: + np.bitwise_xor(input, tmp, out=tmp) + else: + np.subtract(input, tmp, out=tmp) + return tmp + + +def black_tophat(input, size=None, footprint=None, structure=None, output=None, + mode="reflect", cval=0.0, origin=0, *, axes=None): + """ + Multidimensional black tophat filter. + + Parameters + ---------- + input : array_like + Input. + size : tuple of ints, optional + Shape of a flat and full structuring element used for the filter. + Optional if `footprint` or `structure` is provided. + footprint : array of ints, optional + Positions of non-infinite elements of a flat structuring element + used for the black tophat filter. + structure : array of ints, optional + Structuring element used for the filter. `structure` may be a non-flat + structuring element. The `structure` array applies offsets to the + pixels in a neighborhood (the offset is additive during dilation and + subtractive during erosion) + output : array, optional + An array used for storing the output of the filter may be provided. + mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap'}, optional + The `mode` parameter determines how the array borders are + handled, where `cval` is the value when mode is equal to + 'constant'. Default is 'reflect' + cval : scalar, optional + Value to fill past edges of input if `mode` is 'constant'. Default + is 0.0. + origin : scalar, optional + The `origin` parameter controls the placement of the filter. + Default 0 + axes : tuple of int or None + The axes over which to apply the filter. If None, `input` is filtered + along all axes. If an `origin` tuple is provided, its length must match + the number of axes. + + Returns + ------- + black_tophat : ndarray + Result of the filter of `input` with `structure`. + + See Also + -------- + white_tophat, grey_opening, grey_closing + + Examples + -------- + Change dark peak to bright peak and subtract background. + + >>> from scipy.ndimage import generate_binary_structure, black_tophat + >>> import numpy as np + >>> square = generate_binary_structure(rank=2, connectivity=3) + >>> dark_on_gray = np.array([[7, 6, 6, 6, 7], + ... [6, 5, 4, 5, 6], + ... [6, 4, 0, 4, 6], + ... [6, 5, 4, 5, 6], + ... [7, 6, 6, 6, 7]]) + >>> black_tophat(input=dark_on_gray, structure=square) + array([[0, 0, 0, 0, 0], + [0, 0, 1, 0, 0], + [0, 1, 5, 1, 0], + [0, 0, 1, 0, 0], + [0, 0, 0, 0, 0]]) + + """ + input = np.asarray(input) + + if (size is not None) and (footprint is not None): + warnings.warn("ignoring size because footprint is set", + UserWarning, stacklevel=2) + tmp = grey_dilation(input, size, footprint, structure, None, mode, + cval, origin, axes=axes) + tmp = grey_erosion(tmp, size, footprint, structure, output, mode, + cval, origin, axes=axes) + if tmp is None: + tmp = output + + if input.dtype == np.bool_ and tmp.dtype == np.bool_: + np.bitwise_xor(tmp, input, out=tmp) + else: + np.subtract(tmp, input, out=tmp) + return tmp + + +def distance_transform_bf(input, metric="euclidean", sampling=None, + return_distances=True, return_indices=False, + distances=None, indices=None): + """ + Distance transform function by a brute force algorithm. + + This function calculates the distance transform of the `input`, by + replacing each foreground (non-zero) element, with its + shortest distance to the background (any zero-valued element). + + In addition to the distance transform, the feature transform can + be calculated. In this case the index of the closest background + element to each foreground element is returned in a separate array. + + Parameters + ---------- + input : array_like + Input + metric : {'euclidean', 'taxicab', 'chessboard'}, optional + 'cityblock' and 'manhattan' are also valid, and map to 'taxicab'. + The default is 'euclidean'. + sampling : float, or sequence of float, optional + This parameter is only used when `metric` is 'euclidean'. + Spacing of elements along each dimension. If a sequence, must be of + length equal to the input rank; if a single number, this is used for + all axes. If not specified, a grid spacing of unity is implied. + return_distances : bool, optional + Whether to calculate the distance transform. + Default is True. + return_indices : bool, optional + Whether to calculate the feature transform. + Default is False. + distances : ndarray, optional + An output array to store the calculated distance transform, instead of + returning it. + `return_distances` must be True. + It must be the same shape as `input`, and of type float64 if `metric` + is 'euclidean', uint32 otherwise. + indices : int32 ndarray, optional + An output array to store the calculated feature transform, instead of + returning it. + `return_indicies` must be True. + Its shape must be ``(input.ndim,) + input.shape``. + + Returns + ------- + distances : ndarray, optional + The calculated distance transform. Returned only when + `return_distances` is True and `distances` is not supplied. + It will have the same shape as the input array. + indices : int32 ndarray, optional + The calculated feature transform. It has an input-shaped array for each + dimension of the input. See distance_transform_edt documentation for an + example. + Returned only when `return_indices` is True and `indices` is not + supplied. + + See Also + -------- + distance_transform_cdt : Faster distance transform for taxicab and + chessboard metrics + distance_transform_edt : Faster distance transform for euclidean metric + + Notes + ----- + This function employs a slow brute force algorithm. See also the + function `distance_transform_cdt` for more efficient taxicab [1]_ and + chessboard algorithms [2]_. + + References + ---------- + .. [1] Taxicab distance. Wikipedia, 2023. + https://en.wikipedia.org/wiki/Taxicab_geometry + .. [2] Chessboard distance. Wikipedia, 2023. + https://en.wikipedia.org/wiki/Chebyshev_distance + + Examples + -------- + Import the necessary modules. + + >>> import numpy as np + >>> from scipy.ndimage import distance_transform_bf + >>> import matplotlib.pyplot as plt + >>> from mpl_toolkits.axes_grid1 import ImageGrid + + First, we create a toy binary image. + + >>> def add_circle(center_x, center_y, radius, image, fillvalue=1): + ... # fill circular area with 1 + ... xx, yy = np.mgrid[:image.shape[0], :image.shape[1]] + ... circle = (xx - center_x) ** 2 + (yy - center_y) ** 2 + ... circle_shape = np.sqrt(circle) < radius + ... image[circle_shape] = fillvalue + ... return image + >>> image = np.zeros((100, 100), dtype=np.uint8) + >>> image[35:65, 20:80] = 1 + >>> image = add_circle(28, 65, 10, image) + >>> image = add_circle(37, 30, 10, image) + >>> image = add_circle(70, 45, 20, image) + >>> image = add_circle(45, 80, 10, image) + + Next, we set up the figure. + + >>> fig = plt.figure(figsize=(8, 8)) # set up the figure structure + >>> grid = ImageGrid(fig, 111, nrows_ncols=(2, 2), axes_pad=(0.4, 0.3), + ... label_mode="1", share_all=True, + ... cbar_location="right", cbar_mode="each", + ... cbar_size="7%", cbar_pad="2%") + >>> for ax in grid: + ... ax.axis('off') # remove axes from images + + The top left image is the original binary image. + + >>> binary_image = grid[0].imshow(image, cmap='gray') + >>> cbar_binary_image = grid.cbar_axes[0].colorbar(binary_image) + >>> cbar_binary_image.set_ticks([0, 1]) + >>> grid[0].set_title("Binary image: foreground in white") + + The distance transform calculates the distance between foreground pixels + and the image background according to a distance metric. Available metrics + in `distance_transform_bf` are: ``euclidean`` (default), ``taxicab`` + and ``chessboard``. The top right image contains the distance transform + based on the ``euclidean`` metric. + + >>> distance_transform_euclidean = distance_transform_bf(image) + >>> euclidean_transform = grid[1].imshow(distance_transform_euclidean, + ... cmap='gray') + >>> cbar_euclidean = grid.cbar_axes[1].colorbar(euclidean_transform) + >>> colorbar_ticks = [0, 10, 20] + >>> cbar_euclidean.set_ticks(colorbar_ticks) + >>> grid[1].set_title("Euclidean distance") + + The lower left image contains the distance transform using the ``taxicab`` + metric. + + >>> distance_transform_taxicab = distance_transform_bf(image, + ... metric='taxicab') + >>> taxicab_transformation = grid[2].imshow(distance_transform_taxicab, + ... cmap='gray') + >>> cbar_taxicab = grid.cbar_axes[2].colorbar(taxicab_transformation) + >>> cbar_taxicab.set_ticks(colorbar_ticks) + >>> grid[2].set_title("Taxicab distance") + + Finally, the lower right image contains the distance transform using the + ``chessboard`` metric. + + >>> distance_transform_cb = distance_transform_bf(image, + ... metric='chessboard') + >>> chessboard_transformation = grid[3].imshow(distance_transform_cb, + ... cmap='gray') + >>> cbar_taxicab = grid.cbar_axes[3].colorbar(chessboard_transformation) + >>> cbar_taxicab.set_ticks(colorbar_ticks) + >>> grid[3].set_title("Chessboard distance") + >>> plt.show() + + """ + ft_inplace = isinstance(indices, np.ndarray) + dt_inplace = isinstance(distances, np.ndarray) + _distance_tranform_arg_check( + dt_inplace, ft_inplace, return_distances, return_indices + ) + + tmp1 = np.asarray(input) != 0 + struct = generate_binary_structure(tmp1.ndim, tmp1.ndim) + tmp2 = binary_dilation(tmp1, struct) + tmp2 = np.logical_xor(tmp1, tmp2) + tmp1 = tmp1.astype(np.int8) - tmp2.astype(np.int8) + metric = metric.lower() + if metric == 'euclidean': + metric = 1 + elif metric in ['taxicab', 'cityblock', 'manhattan']: + metric = 2 + elif metric == 'chessboard': + metric = 3 + else: + raise RuntimeError('distance metric not supported') + if sampling is not None: + sampling = _ni_support._normalize_sequence(sampling, tmp1.ndim) + sampling = np.asarray(sampling, dtype=np.float64) + if not sampling.flags.contiguous: + sampling = sampling.copy() + if return_indices: + ft = np.zeros(tmp1.shape, dtype=np.int32) + else: + ft = None + if return_distances: + if distances is None: + if metric == 1: + dt = np.zeros(tmp1.shape, dtype=np.float64) + else: + dt = np.zeros(tmp1.shape, dtype=np.uint32) + else: + if distances.shape != tmp1.shape: + raise RuntimeError('distances array has wrong shape') + if metric == 1: + if distances.dtype.type != np.float64: + raise RuntimeError('distances array must be float64') + else: + if distances.dtype.type != np.uint32: + raise RuntimeError('distances array must be uint32') + dt = distances + else: + dt = None + + _nd_image.distance_transform_bf(tmp1, metric, sampling, dt, ft) + if return_indices: + if isinstance(indices, np.ndarray): + if indices.dtype.type != np.int32: + raise RuntimeError('indices array must be int32') + if indices.shape != (tmp1.ndim,) + tmp1.shape: + raise RuntimeError('indices array has wrong shape') + tmp2 = indices + else: + tmp2 = np.indices(tmp1.shape, dtype=np.int32) + ft = np.ravel(ft) + for ii in range(tmp2.shape[0]): + rtmp = np.ravel(tmp2[ii, ...])[ft] + rtmp.shape = tmp1.shape + tmp2[ii, ...] = rtmp + ft = tmp2 + + # construct and return the result + result = [] + if return_distances and not dt_inplace: + result.append(dt) + if return_indices and not ft_inplace: + result.append(ft) + + if len(result) == 2: + return tuple(result) + elif len(result) == 1: + return result[0] + else: + return None + + +def distance_transform_cdt(input, metric='chessboard', return_distances=True, + return_indices=False, distances=None, indices=None): + """ + Distance transform for chamfer type of transforms. + + This function calculates the distance transform of the `input`, by + replacing each foreground (non-zero) element, with its + shortest distance to the background (any zero-valued element). + + In addition to the distance transform, the feature transform can + be calculated. In this case the index of the closest background + element to each foreground element is returned in a separate array. + + Parameters + ---------- + input : array_like + Input. Values of 0 are treated as background. + metric : {'chessboard', 'taxicab'} or array_like, optional + The `metric` determines the type of chamfering that is done. If the + `metric` is equal to 'taxicab' a structure is generated using + `generate_binary_structure` with a squared distance equal to 1. If + the `metric` is equal to 'chessboard', a `metric` is generated + using `generate_binary_structure` with a squared distance equal to + the dimensionality of the array. These choices correspond to the + common interpretations of the 'taxicab' and the 'chessboard' + distance metrics in two dimensions. + A custom metric may be provided, in the form of a matrix where + each dimension has a length of three. + 'cityblock' and 'manhattan' are also valid, and map to 'taxicab'. + The default is 'chessboard'. + return_distances : bool, optional + Whether to calculate the distance transform. + Default is True. + return_indices : bool, optional + Whether to calculate the feature transform. + Default is False. + distances : int32 ndarray, optional + An output array to store the calculated distance transform, instead of + returning it. + `return_distances` must be True. + It must be the same shape as `input`. + indices : int32 ndarray, optional + An output array to store the calculated feature transform, instead of + returning it. + `return_indicies` must be True. + Its shape must be ``(input.ndim,) + input.shape``. + + Returns + ------- + distances : int32 ndarray, optional + The calculated distance transform. Returned only when + `return_distances` is True, and `distances` is not supplied. + It will have the same shape as the input array. + indices : int32 ndarray, optional + The calculated feature transform. It has an input-shaped array for each + dimension of the input. See distance_transform_edt documentation for an + example. + Returned only when `return_indices` is True, and `indices` is not + supplied. + + See Also + -------- + distance_transform_edt : Fast distance transform for euclidean metric + distance_transform_bf : Distance transform for different metrics using + a slower brute force algorithm + + Examples + -------- + Import the necessary modules. + + >>> import numpy as np + >>> from scipy.ndimage import distance_transform_cdt + >>> import matplotlib.pyplot as plt + >>> from mpl_toolkits.axes_grid1 import ImageGrid + + First, we create a toy binary image. + + >>> def add_circle(center_x, center_y, radius, image, fillvalue=1): + ... # fill circular area with 1 + ... xx, yy = np.mgrid[:image.shape[0], :image.shape[1]] + ... circle = (xx - center_x) ** 2 + (yy - center_y) ** 2 + ... circle_shape = np.sqrt(circle) < radius + ... image[circle_shape] = fillvalue + ... return image + >>> image = np.zeros((100, 100), dtype=np.uint8) + >>> image[35:65, 20:80] = 1 + >>> image = add_circle(28, 65, 10, image) + >>> image = add_circle(37, 30, 10, image) + >>> image = add_circle(70, 45, 20, image) + >>> image = add_circle(45, 80, 10, image) + + Next, we set up the figure. + + >>> fig = plt.figure(figsize=(5, 15)) + >>> grid = ImageGrid(fig, 111, nrows_ncols=(3, 1), axes_pad=(0.5, 0.3), + ... label_mode="1", share_all=True, + ... cbar_location="right", cbar_mode="each", + ... cbar_size="7%", cbar_pad="2%") + >>> for ax in grid: + ... ax.axis('off') + >>> top, middle, bottom = grid + >>> colorbar_ticks = [0, 10, 20] + + The top image contains the original binary image. + + >>> binary_image = top.imshow(image, cmap='gray') + >>> cbar_binary_image = top.cax.colorbar(binary_image) + >>> cbar_binary_image.set_ticks([0, 1]) + >>> top.set_title("Binary image: foreground in white") + + The middle image contains the distance transform using the ``taxicab`` + metric. + + >>> distance_taxicab = distance_transform_cdt(image, metric="taxicab") + >>> taxicab_transform = middle.imshow(distance_taxicab, cmap='gray') + >>> cbar_taxicab = middle.cax.colorbar(taxicab_transform) + >>> cbar_taxicab.set_ticks(colorbar_ticks) + >>> middle.set_title("Taxicab metric") + + The bottom image contains the distance transform using the ``chessboard`` + metric. + + >>> distance_chessboard = distance_transform_cdt(image, + ... metric="chessboard") + >>> chessboard_transform = bottom.imshow(distance_chessboard, cmap='gray') + >>> cbar_chessboard = bottom.cax.colorbar(chessboard_transform) + >>> cbar_chessboard.set_ticks(colorbar_ticks) + >>> bottom.set_title("Chessboard metric") + >>> plt.tight_layout() + >>> plt.show() + + """ + ft_inplace = isinstance(indices, np.ndarray) + dt_inplace = isinstance(distances, np.ndarray) + _distance_tranform_arg_check( + dt_inplace, ft_inplace, return_distances, return_indices + ) + input = np.asarray(input) + if isinstance(metric, str): + if metric in ['taxicab', 'cityblock', 'manhattan']: + rank = input.ndim + metric = generate_binary_structure(rank, 1) + elif metric == 'chessboard': + rank = input.ndim + metric = generate_binary_structure(rank, rank) + else: + raise ValueError('invalid metric provided') + else: + try: + metric = np.asarray(metric) + except Exception as e: + raise ValueError('invalid metric provided') from e + for s in metric.shape: + if s != 3: + raise ValueError('metric sizes must be equal to 3') + + if not metric.flags.contiguous: + metric = metric.copy() + if dt_inplace: + if distances.dtype.type != np.int32: + raise ValueError('distances must be of int32 type') + if distances.shape != input.shape: + raise ValueError('distances has wrong shape') + dt = distances + dt[...] = np.where(input, -1, 0).astype(np.int32) + else: + dt = np.where(input, -1, 0).astype(np.int32) + + rank = dt.ndim + if return_indices: + ft = np.arange(dt.size, dtype=np.int32) + ft.shape = dt.shape + else: + ft = None + + _nd_image.distance_transform_op(metric, dt, ft) + dt = dt[tuple([slice(None, None, -1)] * rank)] + if return_indices: + ft = ft[tuple([slice(None, None, -1)] * rank)] + _nd_image.distance_transform_op(metric, dt, ft) + dt = dt[tuple([slice(None, None, -1)] * rank)] + if return_indices: + ft = ft[tuple([slice(None, None, -1)] * rank)] + ft = np.ravel(ft) + if ft_inplace: + if indices.dtype.type != np.int32: + raise ValueError('indices array must be int32') + if indices.shape != (dt.ndim,) + dt.shape: + raise ValueError('indices array has wrong shape') + tmp = indices + else: + tmp = np.indices(dt.shape, dtype=np.int32) + for ii in range(tmp.shape[0]): + rtmp = np.ravel(tmp[ii, ...])[ft] + rtmp.shape = dt.shape + tmp[ii, ...] = rtmp + ft = tmp + + # construct and return the result + result = [] + if return_distances and not dt_inplace: + result.append(dt) + if return_indices and not ft_inplace: + result.append(ft) + + if len(result) == 2: + return tuple(result) + elif len(result) == 1: + return result[0] + else: + return None + + +def distance_transform_edt(input, sampling=None, return_distances=True, + return_indices=False, distances=None, indices=None): + """ + Exact Euclidean distance transform. + + This function calculates the distance transform of the `input`, by + replacing each foreground (non-zero) element, with its + shortest distance to the background (any zero-valued element). + + In addition to the distance transform, the feature transform can + be calculated. In this case the index of the closest background + element to each foreground element is returned in a separate array. + + Parameters + ---------- + input : array_like + Input data to transform. Can be any type but will be converted + into binary: 1 wherever input equates to True, 0 elsewhere. + sampling : float, or sequence of float, optional + Spacing of elements along each dimension. If a sequence, must be of + length equal to the input rank; if a single number, this is used for + all axes. If not specified, a grid spacing of unity is implied. + return_distances : bool, optional + Whether to calculate the distance transform. + Default is True. + return_indices : bool, optional + Whether to calculate the feature transform. + Default is False. + distances : float64 ndarray, optional + An output array to store the calculated distance transform, instead of + returning it. + `return_distances` must be True. + It must be the same shape as `input`. + indices : int32 ndarray, optional + An output array to store the calculated feature transform, instead of + returning it. + `return_indicies` must be True. + Its shape must be ``(input.ndim,) + input.shape``. + + Returns + ------- + distances : float64 ndarray, optional + The calculated distance transform. Returned only when + `return_distances` is True and `distances` is not supplied. + It will have the same shape as the input array. + indices : int32 ndarray, optional + The calculated feature transform. It has an input-shaped array for each + dimension of the input. See example below. + Returned only when `return_indices` is True and `indices` is not + supplied. + + Notes + ----- + The Euclidean distance transform gives values of the Euclidean + distance:: + + n + y_i = sqrt(sum (x[i]-b[i])**2) + i + + where b[i] is the background point (value 0) with the smallest + Euclidean distance to input points x[i], and n is the + number of dimensions. + + Examples + -------- + >>> from scipy import ndimage + >>> import numpy as np + >>> a = np.array(([0,1,1,1,1], + ... [0,0,1,1,1], + ... [0,1,1,1,1], + ... [0,1,1,1,0], + ... [0,1,1,0,0])) + >>> ndimage.distance_transform_edt(a) + array([[ 0. , 1. , 1.4142, 2.2361, 3. ], + [ 0. , 0. , 1. , 2. , 2. ], + [ 0. , 1. , 1.4142, 1.4142, 1. ], + [ 0. , 1. , 1.4142, 1. , 0. ], + [ 0. , 1. , 1. , 0. , 0. ]]) + + With a sampling of 2 units along x, 1 along y: + + >>> ndimage.distance_transform_edt(a, sampling=[2,1]) + array([[ 0. , 1. , 2. , 2.8284, 3.6056], + [ 0. , 0. , 1. , 2. , 3. ], + [ 0. , 1. , 2. , 2.2361, 2. ], + [ 0. , 1. , 2. , 1. , 0. ], + [ 0. , 1. , 1. , 0. , 0. ]]) + + Asking for indices as well: + + >>> edt, inds = ndimage.distance_transform_edt(a, return_indices=True) + >>> inds + array([[[0, 0, 1, 1, 3], + [1, 1, 1, 1, 3], + [2, 2, 1, 3, 3], + [3, 3, 4, 4, 3], + [4, 4, 4, 4, 4]], + [[0, 0, 1, 1, 4], + [0, 1, 1, 1, 4], + [0, 0, 1, 4, 4], + [0, 0, 3, 3, 4], + [0, 0, 3, 3, 4]]], dtype=int32) + + With arrays provided for inplace outputs: + + >>> indices = np.zeros(((np.ndim(a),) + a.shape), dtype=np.int32) + >>> ndimage.distance_transform_edt(a, return_indices=True, indices=indices) + array([[ 0. , 1. , 1.4142, 2.2361, 3. ], + [ 0. , 0. , 1. , 2. , 2. ], + [ 0. , 1. , 1.4142, 1.4142, 1. ], + [ 0. , 1. , 1.4142, 1. , 0. ], + [ 0. , 1. , 1. , 0. , 0. ]]) + >>> indices + array([[[0, 0, 1, 1, 3], + [1, 1, 1, 1, 3], + [2, 2, 1, 3, 3], + [3, 3, 4, 4, 3], + [4, 4, 4, 4, 4]], + [[0, 0, 1, 1, 4], + [0, 1, 1, 1, 4], + [0, 0, 1, 4, 4], + [0, 0, 3, 3, 4], + [0, 0, 3, 3, 4]]], dtype=int32) + + """ + ft_inplace = isinstance(indices, np.ndarray) + dt_inplace = isinstance(distances, np.ndarray) + _distance_tranform_arg_check( + dt_inplace, ft_inplace, return_distances, return_indices + ) + + # calculate the feature transform + input = np.atleast_1d(np.where(input, 1, 0).astype(np.int8)) + if sampling is not None: + sampling = _ni_support._normalize_sequence(sampling, input.ndim) + sampling = np.asarray(sampling, dtype=np.float64) + if not sampling.flags.contiguous: + sampling = sampling.copy() + + if ft_inplace: + ft = indices + if ft.shape != (input.ndim,) + input.shape: + raise RuntimeError('indices array has wrong shape') + if ft.dtype.type != np.int32: + raise RuntimeError('indices array must be int32') + else: + ft = np.zeros((input.ndim,) + input.shape, dtype=np.int32) + + _nd_image.euclidean_feature_transform(input, sampling, ft) + # if requested, calculate the distance transform + if return_distances: + dt = ft - np.indices(input.shape, dtype=ft.dtype) + dt = dt.astype(np.float64) + if sampling is not None: + for ii in range(len(sampling)): + dt[ii, ...] *= sampling[ii] + np.multiply(dt, dt, dt) + if dt_inplace: + dt = np.add.reduce(dt, axis=0) + if distances.shape != dt.shape: + raise RuntimeError('distances array has wrong shape') + if distances.dtype.type != np.float64: + raise RuntimeError('distances array must be float64') + np.sqrt(dt, distances) + else: + dt = np.add.reduce(dt, axis=0) + dt = np.sqrt(dt) + + # construct and return the result + result = [] + if return_distances and not dt_inplace: + result.append(dt) + if return_indices and not ft_inplace: + result.append(ft) + + if len(result) == 2: + return tuple(result) + elif len(result) == 1: + return result[0] + else: + return None + + +def _distance_tranform_arg_check(distances_out, indices_out, + return_distances, return_indices): + """Raise a RuntimeError if the arguments are invalid""" + error_msgs = [] + if (not return_distances) and (not return_indices): + error_msgs.append( + 'at least one of return_distances/return_indices must be True') + if distances_out and not return_distances: + error_msgs.append( + 'return_distances must be True if distances is supplied' + ) + if indices_out and not return_indices: + error_msgs.append('return_indices must be True if indices is supplied') + if error_msgs: + raise RuntimeError(', '.join(error_msgs)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ndimage_api.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ndimage_api.py new file mode 100644 index 0000000000000000000000000000000000000000..1673391726a4070af4814d04be16e29e01d9f29b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ndimage_api.py @@ -0,0 +1,16 @@ +"""This is the 'bare' ndimage API. + +This --- private! --- module only collects implementations of public ndimage API +for _support_alternative_backends. +The latter --- also private! --- module adds delegation to CuPy etc and +re-exports decorated names to __init__.py +""" + +from ._filters import * # noqa: F403 +from ._fourier import * # noqa: F403 +from ._interpolation import * # noqa: F403 +from ._measurements import * # noqa: F403 +from ._morphology import * # noqa: F403 + +# '@' due to pytest bug, scipy/scipy#22236 +__all__ = [s for s in dir() if not s.startswith(('_', '@'))] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ni_docstrings.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ni_docstrings.py new file mode 100644 index 0000000000000000000000000000000000000000..2c0d977500574b73f168296645fb36d10c1320de --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ni_docstrings.py @@ -0,0 +1,210 @@ +"""Docstring components common to several ndimage functions.""" +from typing import Final + +from scipy._lib import doccer + +__all__ = ['docfiller'] + + +_input_doc = ( +"""input : array_like + The input array.""") +_axis_doc = ( +"""axis : int, optional + The axis of `input` along which to calculate. Default is -1.""") +_output_doc = ( +"""output : array or dtype, optional + The array in which to place the output, or the dtype of the + returned array. By default an array of the same dtype as input + will be created.""") +_size_foot_doc = ( +"""size : scalar or tuple, optional + See footprint, below. Ignored if footprint is given. +footprint : array, optional + Either `size` or `footprint` must be defined. `size` gives + the shape that is taken from the input array, at every element + position, to define the input to the filter function. + `footprint` is a boolean array that specifies (implicitly) a + shape, but also which of the elements within this shape will get + passed to the filter function. Thus ``size=(n,m)`` is equivalent + to ``footprint=np.ones((n,m))``. We adjust `size` to the number + of dimensions of the input array, so that, if the input array is + shape (10,10,10), and `size` is 2, then the actual size used is + (2,2,2). When `footprint` is given, `size` is ignored.""") +_mode_reflect_doc = ( +"""mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap'}, optional + The `mode` parameter determines how the input array is extended + beyond its boundaries. Default is 'reflect'. Behavior for each valid + value is as follows: + + 'reflect' (`d c b a | a b c d | d c b a`) + The input is extended by reflecting about the edge of the last + pixel. This mode is also sometimes referred to as half-sample + symmetric. + + 'constant' (`k k k k | a b c d | k k k k`) + The input is extended by filling all values beyond the edge with + the same constant value, defined by the `cval` parameter. + + 'nearest' (`a a a a | a b c d | d d d d`) + The input is extended by replicating the last pixel. + + 'mirror' (`d c b | a b c d | c b a`) + The input is extended by reflecting about the center of the last + pixel. This mode is also sometimes referred to as whole-sample + symmetric. + + 'wrap' (`a b c d | a b c d | a b c d`) + The input is extended by wrapping around to the opposite edge. + + For consistency with the interpolation functions, the following mode + names can also be used: + + 'grid-mirror' + This is a synonym for 'reflect'. + + 'grid-constant' + This is a synonym for 'constant'. + + 'grid-wrap' + This is a synonym for 'wrap'.""") + +_mode_interp_constant_doc = ( +"""mode : {'reflect', 'grid-mirror', 'constant', 'grid-constant', 'nearest', \ +'mirror', 'grid-wrap', 'wrap'}, optional + The `mode` parameter determines how the input array is extended + beyond its boundaries. Default is 'constant'. Behavior for each valid + value is as follows (see additional plots and details on + :ref:`boundary modes `): + + 'reflect' (`d c b a | a b c d | d c b a`) + The input is extended by reflecting about the edge of the last + pixel. This mode is also sometimes referred to as half-sample + symmetric. + + 'grid-mirror' + This is a synonym for 'reflect'. + + 'constant' (`k k k k | a b c d | k k k k`) + The input is extended by filling all values beyond the edge with + the same constant value, defined by the `cval` parameter. No + interpolation is performed beyond the edges of the input. + + 'grid-constant' (`k k k k | a b c d | k k k k`) + The input is extended by filling all values beyond the edge with + the same constant value, defined by the `cval` parameter. Interpolation + occurs for samples outside the input's extent as well. + + 'nearest' (`a a a a | a b c d | d d d d`) + The input is extended by replicating the last pixel. + + 'mirror' (`d c b | a b c d | c b a`) + The input is extended by reflecting about the center of the last + pixel. This mode is also sometimes referred to as whole-sample + symmetric. + + 'grid-wrap' (`a b c d | a b c d | a b c d`) + The input is extended by wrapping around to the opposite edge. + + 'wrap' (`d b c d | a b c d | b c a b`) + The input is extended by wrapping around to the opposite edge, but in a + way such that the last point and initial point exactly overlap. In this + case it is not well defined which sample will be chosen at the point of + overlap.""") +_mode_interp_mirror_doc = ( + _mode_interp_constant_doc.replace("Default is 'constant'", + "Default is 'mirror'") +) +assert _mode_interp_mirror_doc != _mode_interp_constant_doc, \ + 'Default not replaced' + +_mode_multiple_doc = ( +"""mode : str or sequence, optional + The `mode` parameter determines how the input array is extended + when the filter overlaps a border. By passing a sequence of modes + with length equal to the number of dimensions of the input array, + different modes can be specified along each axis. Default value is + 'reflect'. The valid values and their behavior is as follows: + + 'reflect' (`d c b a | a b c d | d c b a`) + The input is extended by reflecting about the edge of the last + pixel. This mode is also sometimes referred to as half-sample + symmetric. + + 'constant' (`k k k k | a b c d | k k k k`) + The input is extended by filling all values beyond the edge with + the same constant value, defined by the `cval` parameter. + + 'nearest' (`a a a a | a b c d | d d d d`) + The input is extended by replicating the last pixel. + + 'mirror' (`d c b | a b c d | c b a`) + The input is extended by reflecting about the center of the last + pixel. This mode is also sometimes referred to as whole-sample + symmetric. + + 'wrap' (`a b c d | a b c d | a b c d`) + The input is extended by wrapping around to the opposite edge. + + For consistency with the interpolation functions, the following mode + names can also be used: + + 'grid-constant' + This is a synonym for 'constant'. + + 'grid-mirror' + This is a synonym for 'reflect'. + + 'grid-wrap' + This is a synonym for 'wrap'.""") +_cval_doc = ( +"""cval : scalar, optional + Value to fill past edges of input if `mode` is 'constant'. Default + is 0.0.""") +_origin_doc = ( +"""origin : int, optional + Controls the placement of the filter on the input array's pixels. + A value of 0 (the default) centers the filter over the pixel, with + positive values shifting the filter to the left, and negative ones + to the right.""") +_origin_multiple_doc = ( +"""origin : int or sequence, optional + Controls the placement of the filter on the input array's pixels. + A value of 0 (the default) centers the filter over the pixel, with + positive values shifting the filter to the left, and negative ones + to the right. By passing a sequence of origins with length equal to + the number of dimensions of the input array, different shifts can + be specified along each axis.""") +_extra_arguments_doc = ( +"""extra_arguments : sequence, optional + Sequence of extra positional arguments to pass to passed function.""") +_extra_keywords_doc = ( +"""extra_keywords : dict, optional + dict of extra keyword arguments to pass to passed function.""") +_prefilter_doc = ( +"""prefilter : bool, optional + Determines if the input array is prefiltered with `spline_filter` + before interpolation. The default is True, which will create a + temporary `float64` array of filtered values if ``order > 1``. If + setting this to False, the output will be slightly blurred if + ``order > 1``, unless the input is prefiltered, i.e. it is the result + of calling `spline_filter` on the original input.""") + +docdict = { + 'input': _input_doc, + 'axis': _axis_doc, + 'output': _output_doc, + 'size_foot': _size_foot_doc, + 'mode_interp_constant': _mode_interp_constant_doc, + 'mode_interp_mirror': _mode_interp_mirror_doc, + 'mode_reflect': _mode_reflect_doc, + 'mode_multiple': _mode_multiple_doc, + 'cval': _cval_doc, + 'origin': _origin_doc, + 'origin_multiple': _origin_multiple_doc, + 'extra_arguments': _extra_arguments_doc, + 'extra_keywords': _extra_keywords_doc, + 'prefilter': _prefilter_doc + } + +docfiller: Final = doccer.filldoc(docdict) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ni_support.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ni_support.py new file mode 100644 index 0000000000000000000000000000000000000000..f8d41d00d9edf8d347c4ffc95598210489fed5e9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_ni_support.py @@ -0,0 +1,143 @@ +# Copyright (C) 2003-2005 Peter J. Verveer +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions +# are met: +# +# 1. Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# +# 2. Redistributions in binary form must reproduce the above +# copyright notice, this list of conditions and the following +# disclaimer in the documentation and/or other materials provided +# with the distribution. +# +# 3. The name of the author may not be used to endorse or promote +# products derived from this software without specific prior +# written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS +# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY +# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE +# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, +# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +from collections.abc import Iterable +import operator +import warnings +import numpy as np + + +def _extend_mode_to_code(mode, is_filter=False): + """Convert an extension mode to the corresponding integer code. + """ + if mode == 'nearest': + return 0 + elif mode == 'wrap': + return 1 + elif mode in ['reflect', 'grid-mirror']: + return 2 + elif mode == 'mirror': + return 3 + elif mode == 'constant': + return 4 + elif mode == 'grid-wrap' and is_filter: + return 1 + elif mode == 'grid-wrap': + return 5 + elif mode == 'grid-constant' and is_filter: + return 4 + elif mode == 'grid-constant': + return 6 + else: + raise RuntimeError('boundary mode not supported') + + +def _normalize_sequence(input, rank): + """If input is a scalar, create a sequence of length equal to the + rank by duplicating the input. If input is a sequence, + check if its length is equal to the length of array. + """ + is_str = isinstance(input, str) + if not is_str and np.iterable(input): + normalized = list(input) + if len(normalized) != rank: + err = "sequence argument must have length equal to input rank" + raise RuntimeError(err) + else: + normalized = [input] * rank + return normalized + + +def _get_output(output, input, shape=None, complex_output=False): + if shape is None: + shape = input.shape + if output is None: + if not complex_output: + output = np.zeros(shape, dtype=input.dtype.name) + else: + complex_type = np.promote_types(input.dtype, np.complex64) + output = np.zeros(shape, dtype=complex_type) + elif isinstance(output, (type, np.dtype)): + # Classes (like `np.float32`) and dtypes are interpreted as dtype + if complex_output and np.dtype(output).kind != 'c': + warnings.warn("promoting specified output dtype to complex", stacklevel=3) + output = np.promote_types(output, np.complex64) + output = np.zeros(shape, dtype=output) + elif isinstance(output, str): + output = np.dtype(output) + if complex_output and output.kind != 'c': + raise RuntimeError("output must have complex dtype") + elif not issubclass(output.type, np.number): + raise RuntimeError("output must have numeric dtype") + output = np.zeros(shape, dtype=output) + else: + # output was supplied as an array + output = np.asarray(output) + if output.shape != shape: + raise RuntimeError("output shape not correct") + elif complex_output and output.dtype.kind != 'c': + raise RuntimeError("output must have complex dtype") + return output + + +def _check_axes(axes, ndim): + if axes is None: + return tuple(range(ndim)) + elif np.isscalar(axes): + axes = (operator.index(axes),) + elif isinstance(axes, Iterable): + for ax in axes: + axes = tuple(operator.index(ax) for ax in axes) + if ax < -ndim or ax > ndim - 1: + raise ValueError(f"specified axis: {ax} is out of range") + axes = tuple(ax % ndim if ax < 0 else ax for ax in axes) + else: + message = "axes must be an integer, iterable of integers, or None" + raise ValueError(message) + if len(tuple(set(axes))) != len(axes): + raise ValueError("axes must be unique") + return axes + +def _skip_if_dtype(arg): + """'array or dtype' polymorphism. + + Return None for np.int8, dtype('float32') or 'f' etc + arg for np.empty(3) etc + """ + if isinstance(arg, str): + return None + if type(arg) is type: + return None if issubclass(arg, np.generic) else arg + else: + return None if isinstance(arg, np.dtype) else arg + + +def _skip_if_int(arg): + return None if (arg is None or isinstance(arg, int)) else arg diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_rank_filter_1d.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_rank_filter_1d.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..6d998aa648b493336d9871117e687e8f8d5279aa Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_rank_filter_1d.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_support_alternative_backends.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_support_alternative_backends.py new file mode 100644 index 0000000000000000000000000000000000000000..fbb913b14c76873202fce8eaabe2d08f778abe94 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/_support_alternative_backends.py @@ -0,0 +1,72 @@ +import functools +from scipy._lib._array_api import ( + is_cupy, is_jax, scipy_namespace_for, SCIPY_ARRAY_API +) + +import numpy as np +from ._ndimage_api import * # noqa: F403 +from . import _ndimage_api +from . import _delegators +__all__ = _ndimage_api.__all__ + + +MODULE_NAME = 'ndimage' + + +def delegate_xp(delegator, module_name): + def inner(func): + @functools.wraps(func) + def wrapper(*args, **kwds): + xp = delegator(*args, **kwds) + + # try delegating to a cupyx/jax namesake + if is_cupy(xp): + # https://github.com/cupy/cupy/issues/8336 + import importlib + cupyx_module = importlib.import_module(f"cupyx.scipy.{module_name}") + cupyx_func = getattr(cupyx_module, func.__name__) + return cupyx_func(*args, **kwds) + elif is_jax(xp) and func.__name__ == "map_coordinates": + spx = scipy_namespace_for(xp) + jax_module = getattr(spx, module_name) + jax_func = getattr(jax_module, func.__name__) + return jax_func(*args, **kwds) + else: + # the original function (does all np.asarray internally) + # XXX: output arrays + result = func(*args, **kwds) + + if isinstance(result, (np.ndarray, np.generic)): + # XXX: np.int32->np.array_0D + return xp.asarray(result) + elif isinstance(result, int): + return result + elif isinstance(result, dict): + # value_indices: result is {np.int64(1): (array(0), array(1))} etc + return { + k.item(): tuple(xp.asarray(vv) for vv in v) + for k,v in result.items() + } + elif result is None: + # inplace operations + return result + else: + # lists/tuples + return type(result)( + xp.asarray(x) if isinstance(x, np.ndarray) else x + for x in result + ) + return wrapper + return inner + +# ### decorate ### +for func_name in _ndimage_api.__all__: + bare_func = getattr(_ndimage_api, func_name) + delegator = getattr(_delegators, func_name + "_signature") + + f = (delegate_xp(delegator, MODULE_NAME)(bare_func) + if SCIPY_ARRAY_API + else bare_func) + + # add the decorated function to the namespace, to be imported in __init__.py + vars()[func_name] = f diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/filters.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/filters.py new file mode 100644 index 0000000000000000000000000000000000000000..e16d9d279a9585b2454c46ee09cf22143de833a6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/filters.py @@ -0,0 +1,27 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.ndimage` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'correlate1d', 'convolve1d', 'gaussian_filter1d', + 'gaussian_filter', 'prewitt', 'sobel', 'generic_laplace', + 'laplace', 'gaussian_laplace', 'generic_gradient_magnitude', + 'gaussian_gradient_magnitude', 'correlate', 'convolve', + 'uniform_filter1d', 'uniform_filter', 'minimum_filter1d', + 'maximum_filter1d', 'minimum_filter', 'maximum_filter', + 'rank_filter', 'median_filter', 'percentile_filter', + 'generic_filter1d', 'generic_filter' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package='ndimage', module='filters', + private_modules=['_filters'], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/fourier.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/fourier.py new file mode 100644 index 0000000000000000000000000000000000000000..73c49bd52d9a446ce0fe25d9e15b8de68fbd46fb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/fourier.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.ndimage` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'fourier_gaussian', 'fourier_uniform', + 'fourier_ellipsoid', 'fourier_shift' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package='ndimage', module='fourier', + private_modules=['_fourier'], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/interpolation.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/interpolation.py new file mode 100644 index 0000000000000000000000000000000000000000..a2739c60c51037487ae8892c407e2f3d7870d5da --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/interpolation.py @@ -0,0 +1,22 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.ndimage` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'spline_filter1d', 'spline_filter', + 'geometric_transform', 'map_coordinates', + 'affine_transform', 'shift', 'zoom', 'rotate', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package='ndimage', module='interpolation', + private_modules=['_interpolation'], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/measurements.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/measurements.py new file mode 100644 index 0000000000000000000000000000000000000000..22f76b01840ffb829205bd1d28a7ad1f9ac5db61 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/measurements.py @@ -0,0 +1,24 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.ndimage` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'label', 'find_objects', 'labeled_comprehension', + 'sum', 'mean', 'variance', 'standard_deviation', + 'minimum', 'maximum', 'median', 'minimum_position', + 'maximum_position', 'extrema', 'center_of_mass', + 'histogram', 'watershed_ift', 'sum_labels' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package='ndimage', module='measurements', + private_modules=['_measurements'], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/morphology.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/morphology.py new file mode 100644 index 0000000000000000000000000000000000000000..e522e7df3a4b06b7e04ed8c2d0ecaff2a98b951d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/morphology.py @@ -0,0 +1,27 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.ndimage` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'iterate_structure', 'generate_binary_structure', + 'binary_erosion', 'binary_dilation', 'binary_opening', + 'binary_closing', 'binary_hit_or_miss', 'binary_propagation', + 'binary_fill_holes', 'grey_erosion', 'grey_dilation', + 'grey_opening', 'grey_closing', 'morphological_gradient', + 'morphological_laplace', 'white_tophat', 'black_tophat', + 'distance_transform_bf', 'distance_transform_cdt', + 'distance_transform_edt' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package='ndimage', module='morphology', + private_modules=['_morphology'], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..8d8fd292b537a84fe48d0c8ae8bee75bab2b3353 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/__init__.py @@ -0,0 +1,12 @@ +import numpy as np + +# list of numarray data types +integer_types: list[str] = [ + "int8", "uint8", "int16", "uint16", + "int32", "uint32", "int64", "uint64"] + +float_types: list[str] = ["float32", "float64"] + +complex_types: list[str] = ["complex64", "complex128"] + +types: list[str] = integer_types + float_types diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/data/label_inputs.txt b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/data/label_inputs.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c3cff3b12cec4ad050b31cc5d5c327f32784447 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/data/label_inputs.txt @@ -0,0 +1,21 @@ +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 0 1 1 1 +1 1 0 0 0 1 1 +1 0 1 0 1 0 1 +0 0 0 1 0 0 0 +1 0 1 0 1 0 1 +1 1 0 0 0 1 1 +1 1 1 0 1 1 1 +1 0 1 1 1 0 1 +0 0 0 1 0 0 0 +1 0 0 1 0 0 1 +1 1 1 1 1 1 1 +1 0 0 1 0 0 1 +0 0 0 1 0 0 0 +1 0 1 1 1 0 1 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/data/label_results.txt b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/data/label_results.txt new file mode 100644 index 0000000000000000000000000000000000000000..c239b0369c9df3e06df9a2fbf048faec2f84941f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/data/label_results.txt @@ -0,0 +1,294 @@ +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +1 1 1 1 1 1 1 +2 2 2 2 2 2 2 +3 3 3 3 3 3 3 +4 4 4 4 4 4 4 +5 5 5 5 5 5 5 +6 6 6 6 6 6 6 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0 0 6 0 0 7 +8 5 6 9 6 7 10 +5 0 0 6 0 0 7 +0 0 0 11 0 0 0 +12 0 11 13 11 0 14 +1 0 2 3 4 0 5 +0 0 0 4 0 0 0 +6 0 0 7 0 0 8 +9 10 7 11 12 8 13 +10 0 0 12 0 0 14 +0 0 0 15 0 0 0 +16 0 15 17 18 0 19 +1 0 2 2 2 0 3 +0 0 0 2 0 0 0 +2 0 0 2 0 0 2 +2 2 2 2 2 2 2 +2 0 0 2 0 0 2 +0 0 0 2 0 0 0 +4 0 2 2 2 0 5 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/data/label_strels.txt b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/data/label_strels.txt new file mode 100644 index 0000000000000000000000000000000000000000..35ae8121364d4fb3292c11f2a72333f456fa9c0a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/data/label_strels.txt @@ -0,0 +1,42 @@ +0 0 1 +1 1 1 +1 0 0 +1 0 0 +1 1 1 +0 0 1 +0 0 0 +1 1 1 +0 0 0 +0 1 1 +0 1 0 +1 1 0 +0 0 0 +0 0 0 +0 0 0 +0 1 1 +1 1 1 +1 1 0 +0 1 0 +1 1 1 +0 1 0 +1 0 0 +0 1 0 +0 0 1 +0 1 0 +0 1 0 +0 1 0 +1 1 1 +1 1 1 +1 1 1 +1 1 0 +0 1 0 +0 1 1 +1 0 1 +0 1 0 +1 0 1 +0 0 1 +0 1 0 +1 0 0 +1 1 0 +1 1 1 +0 1 1 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_c_api.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_c_api.py new file mode 100644 index 0000000000000000000000000000000000000000..61a5a0f70262ef9f21fe8593a64f155bd583cab1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_c_api.py @@ -0,0 +1,102 @@ +import numpy as np +from scipy._lib._array_api import xp_assert_close + +from scipy import ndimage +from scipy.ndimage import _ctest +from scipy.ndimage import _cytest +from scipy._lib._ccallback import LowLevelCallable + +FILTER1D_FUNCTIONS = [ + lambda filter_size: _ctest.filter1d(filter_size), + lambda filter_size: _cytest.filter1d(filter_size, with_signature=False), + lambda filter_size: LowLevelCallable( + _cytest.filter1d(filter_size, with_signature=True) + ), + lambda filter_size: LowLevelCallable.from_cython( + _cytest, "_filter1d", + _cytest.filter1d_capsule(filter_size), + ), +] + +FILTER2D_FUNCTIONS = [ + lambda weights: _ctest.filter2d(weights), + lambda weights: _cytest.filter2d(weights, with_signature=False), + lambda weights: LowLevelCallable(_cytest.filter2d(weights, with_signature=True)), + lambda weights: LowLevelCallable.from_cython(_cytest, + "_filter2d", + _cytest.filter2d_capsule(weights),), +] + +TRANSFORM_FUNCTIONS = [ + lambda shift: _ctest.transform(shift), + lambda shift: _cytest.transform(shift, with_signature=False), + lambda shift: LowLevelCallable(_cytest.transform(shift, with_signature=True)), + lambda shift: LowLevelCallable.from_cython(_cytest, + "_transform", + _cytest.transform_capsule(shift),), +] + + +def test_generic_filter(): + def filter2d(footprint_elements, weights): + return (weights*footprint_elements).sum() + + def check(j): + func = FILTER2D_FUNCTIONS[j] + + im = np.ones((20, 20)) + im[:10,:10] = 0 + footprint = np.array([[0, 1, 0], [1, 1, 1], [0, 1, 0]]) + footprint_size = np.count_nonzero(footprint) + weights = np.ones(footprint_size)/footprint_size + + res = ndimage.generic_filter(im, func(weights), + footprint=footprint) + std = ndimage.generic_filter(im, filter2d, footprint=footprint, + extra_arguments=(weights,)) + xp_assert_close(res, std, err_msg=f"#{j} failed") + + for j, func in enumerate(FILTER2D_FUNCTIONS): + check(j) + + +def test_generic_filter1d(): + def filter1d(input_line, output_line, filter_size): + for i in range(output_line.size): + output_line[i] = 0 + for j in range(filter_size): + output_line[i] += input_line[i+j] + output_line /= filter_size + + def check(j): + func = FILTER1D_FUNCTIONS[j] + + im = np.tile(np.hstack((np.zeros(10), np.ones(10))), (10, 1)) + filter_size = 3 + + res = ndimage.generic_filter1d(im, func(filter_size), + filter_size) + std = ndimage.generic_filter1d(im, filter1d, filter_size, + extra_arguments=(filter_size,)) + xp_assert_close(res, std, err_msg=f"#{j} failed") + + for j, func in enumerate(FILTER1D_FUNCTIONS): + check(j) + + +def test_geometric_transform(): + def transform(output_coordinates, shift): + return output_coordinates[0] - shift, output_coordinates[1] - shift + + def check(j): + func = TRANSFORM_FUNCTIONS[j] + + im = np.arange(12).reshape(4, 3).astype(np.float64) + shift = 0.5 + + res = ndimage.geometric_transform(im, func(shift)) + std = ndimage.geometric_transform(im, transform, extra_arguments=(shift,)) + xp_assert_close(res, std, err_msg=f"#{j} failed") + + for j, func in enumerate(TRANSFORM_FUNCTIONS): + check(j) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_datatypes.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_datatypes.py new file mode 100644 index 0000000000000000000000000000000000000000..a82de456bb92c96d5b9a599d4b33c987e134fdc8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_datatypes.py @@ -0,0 +1,67 @@ +""" Testing data types for ndimage calls +""" +import numpy as np + +from scipy._lib._array_api import assert_array_almost_equal +import pytest + +from scipy import ndimage + + +def test_map_coordinates_dts(): + # check that ndimage accepts different data types for interpolation + data = np.array([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + shifted_data = np.array([[0, 0, 0, 0], + [0, 4, 1, 3], + [0, 7, 6, 8]]) + idx = np.indices(data.shape) + dts = (np.uint8, np.uint16, np.uint32, np.uint64, + np.int8, np.int16, np.int32, np.int64, + np.intp, np.uintp, np.float32, np.float64) + for order in range(0, 6): + for data_dt in dts: + these_data = data.astype(data_dt) + for coord_dt in dts: + # affine mapping + mat = np.eye(2, dtype=coord_dt) + off = np.zeros((2,), dtype=coord_dt) + out = ndimage.affine_transform(these_data, mat, off) + assert_array_almost_equal(these_data, out) + # map coordinates + coords_m1 = idx.astype(coord_dt) - 1 + coords_p10 = idx.astype(coord_dt) + 10 + out = ndimage.map_coordinates(these_data, coords_m1, order=order) + assert_array_almost_equal(out, shifted_data) + # check constant fill works + out = ndimage.map_coordinates(these_data, coords_p10, order=order) + assert_array_almost_equal(out, np.zeros((3,4))) + # check shift and zoom + out = ndimage.shift(these_data, 1) + assert_array_almost_equal(out, shifted_data) + out = ndimage.zoom(these_data, 1) + assert_array_almost_equal(these_data, out) + + +@pytest.mark.xfail(True, reason="Broken on many platforms") +def test_uint64_max(): + # Test interpolation respects uint64 max. Reported to fail at least on + # win32 (due to the 32 bit visual C compiler using signed int64 when + # converting between uint64 to double) and Debian on s390x. + # Interpolation is always done in double precision floating point, so + # we use the largest uint64 value for which int(float(big)) still fits + # in a uint64. + # This test was last enabled on macOS only, and there it started failing + # on arm64 as well (see gh-19117). + big = 2**64 - 1025 + arr = np.array([big, big, big], dtype=np.uint64) + # Tests geometric transform (map_coordinates, affine_transform) + inds = np.indices(arr.shape) - 0.1 + x = ndimage.map_coordinates(arr, inds) + assert x[1] == int(float(big)) + assert x[2] == int(float(big)) + # Tests zoom / shift + x = ndimage.shift(arr, 0.1) + assert x[1] == int(float(big)) + assert x[2] == int(float(big)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_filters.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_filters.py new file mode 100644 index 0000000000000000000000000000000000000000..1d5cb39f566827f41037e34a5a63ce874996c36f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_filters.py @@ -0,0 +1,2920 @@ +''' Some tests for filters ''' +import functools +import itertools +import re + +import numpy as np +import pytest +from numpy.testing import suppress_warnings, assert_allclose, assert_array_equal +from hypothesis import strategies as st +from hypothesis import given +import hypothesis.extra.numpy as npst +from pytest import raises as assert_raises +from scipy import ndimage +from scipy._lib._array_api import ( + assert_almost_equal, + assert_array_almost_equal, + xp_assert_close, + xp_assert_equal, +) +from scipy._lib._array_api import is_cupy, is_numpy, is_torch, array_namespace +from scipy.conftest import array_api_compatible +from scipy.ndimage._filters import _gaussian_kernel1d + +from . import types, float_types, complex_types + + +skip_xp_backends = pytest.mark.skip_xp_backends +xfail_xp_backends = pytest.mark.xfail_xp_backends +pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends"), + pytest.mark.usefixtures("xfail_xp_backends"), + skip_xp_backends(cpu_only=True, exceptions=['cupy', 'jax.numpy']),] + + +def sumsq(a, b, xp=None): + xp = array_namespace(a, b) if xp is None else xp + return xp.sqrt(xp.sum((a - b)**2)) + + +def _complex_correlate(xp, array, kernel, real_dtype, convolve=False, + mode="reflect", cval=0, ): + """Utility to perform a reference complex-valued convolutions. + + When convolve==False, correlation is performed instead + """ + array = xp.asarray(array) + kernel = xp.asarray(kernel) + isdtype = array_namespace(array, kernel).isdtype + complex_array = isdtype(array.dtype, 'complex floating') + complex_kernel = isdtype(kernel.dtype, 'complex floating') + if array.ndim == 1: + func = ndimage.convolve1d if convolve else ndimage.correlate1d + else: + func = ndimage.convolve if convolve else ndimage.correlate + if not convolve: + kernel = xp.conj(kernel) + if complex_array and complex_kernel: + # use: real(cval) for array.real component + # imag(cval) for array.imag component + re_cval = cval.real if isinstance(cval, complex) else xp.real(cval) + im_cval = cval.imag if isinstance(cval, complex) else xp.imag(cval) + + output = ( + func(xp.real(array), xp.real(kernel), output=real_dtype, + mode=mode, cval=re_cval) - + func(xp.imag(array), xp.imag(kernel), output=real_dtype, + mode=mode, cval=im_cval) + + 1j * func(xp.imag(array), xp.real(kernel), output=real_dtype, + mode=mode, cval=im_cval) + + 1j * func(xp.real(array), xp.imag(kernel), output=real_dtype, + mode=mode, cval=re_cval) + ) + elif complex_array: + re_cval = xp.real(cval) + re_cval = re_cval.item() if isinstance(re_cval, xp.ndarray) else re_cval + im_cval = xp.imag(cval) + im_cval = im_cval.item() if isinstance(im_cval, xp.ndarray) else im_cval + + output = ( + func(xp.real(array), kernel, output=real_dtype, mode=mode, + cval=re_cval) + + 1j * func(xp.imag(array), kernel, output=real_dtype, mode=mode, + cval=im_cval) + ) + elif complex_kernel: + # real array so cval is real too + output = ( + func(array, xp.real(kernel), output=real_dtype, mode=mode, cval=cval) + + 1j * func(array, xp.imag(kernel), output=real_dtype, mode=mode, + cval=cval) + ) + return output + + +def _cases_axes_tuple_length_mismatch(): + # Generate combinations of filter function, valid kwargs, and + # keyword-value pairs for which the value will become with mismatched + # (invalid) size + filter_func = ndimage.gaussian_filter + kwargs = dict(radius=3, mode='constant', sigma=1.0, order=0) + for key, val in kwargs.items(): + yield filter_func, kwargs, key, val + + filter_funcs = [ndimage.uniform_filter, ndimage.minimum_filter, + ndimage.maximum_filter] + kwargs = dict(size=3, mode='constant', origin=0) + for filter_func in filter_funcs: + for key, val in kwargs.items(): + yield filter_func, kwargs, key, val + + filter_funcs = [ndimage.correlate, ndimage.convolve] + # sequence of mode not supported for correlate or convolve + kwargs = dict(origin=0) + for filter_func in filter_funcs: + for key, val in kwargs.items(): + yield filter_func, kwargs, key, val + + +class TestNdimageFilters: + + def _validate_complex(self, xp, array, kernel, type2, mode='reflect', + cval=0, check_warnings=True): + # utility for validating complex-valued correlations + real_dtype = xp.real(xp.asarray([], dtype=type2)).dtype + expected = _complex_correlate( + xp, array, kernel, real_dtype, convolve=False, mode=mode, cval=cval + ) + + if array.ndim == 1: + correlate = functools.partial(ndimage.correlate1d, axis=-1, + mode=mode, cval=cval) + convolve = functools.partial(ndimage.convolve1d, axis=-1, + mode=mode, cval=cval) + else: + correlate = functools.partial(ndimage.correlate, mode=mode, + cval=cval) + convolve = functools.partial(ndimage.convolve, mode=mode, + cval=cval) + + # test correlate output dtype + output = correlate(array, kernel, output=type2) + assert_array_almost_equal(expected, output) + assert output.dtype.type == type2 + + # test correlate with pre-allocated output + output = xp.zeros_like(array, dtype=type2) + correlate(array, kernel, output=output) + assert_array_almost_equal(expected, output) + + # test convolve output dtype + output = convolve(array, kernel, output=type2) + expected = _complex_correlate( + xp, array, kernel, real_dtype, convolve=True, mode=mode, cval=cval, + ) + assert_array_almost_equal(expected, output) + assert output.dtype.type == type2 + + # convolve with pre-allocated output + convolve(array, kernel, output=output) + assert_array_almost_equal(expected, output) + assert output.dtype.type == type2 + + if check_warnings: + # warns if the output is not a complex dtype + with pytest.warns(UserWarning, + match="promoting specified output dtype to " + "complex"): + correlate(array, kernel, output=real_dtype) + + with pytest.warns(UserWarning, + match="promoting specified output dtype to " + "complex"): + convolve(array, kernel, output=real_dtype) + + # raises if output array is provided, but is not complex-valued + output_real = xp.zeros_like(array, dtype=real_dtype) + with assert_raises(RuntimeError): + correlate(array, kernel, output=output_real) + + with assert_raises(RuntimeError): + convolve(array, kernel, output=output_real) + + def test_correlate01(self, xp): + array = xp.asarray([1, 2]) + weights = xp.asarray([2]) + expected = xp.asarray([2, 4]) + + output = ndimage.correlate(array, weights) + assert_array_almost_equal(output, expected) + + output = ndimage.convolve(array, weights) + assert_array_almost_equal(output, expected) + + output = ndimage.correlate1d(array, weights) + assert_array_almost_equal(output, expected) + + output = ndimage.convolve1d(array, weights) + assert_array_almost_equal(output, expected) + + @xfail_xp_backends('cupy', reason="Differs by a factor of two?") + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + def test_correlate01_overlap(self, xp): + array = xp.reshape(xp.arange(256), (16, 16)) + weights = xp.asarray([2]) + expected = 2 * array + + ndimage.correlate1d(array, weights, output=array) + assert_array_almost_equal(array, expected) + + def test_correlate02(self, xp): + array = xp.asarray([1, 2, 3]) + kernel = xp.asarray([1]) + + output = ndimage.correlate(array, kernel) + assert_array_almost_equal(array, output) + + output = ndimage.convolve(array, kernel) + assert_array_almost_equal(array, output) + + output = ndimage.correlate1d(array, kernel) + assert_array_almost_equal(array, output) + + output = ndimage.convolve1d(array, kernel) + assert_array_almost_equal(array, output) + + def test_correlate03(self, xp): + array = xp.asarray([1]) + weights = xp.asarray([1, 1]) + expected = xp.asarray([2]) + + output = ndimage.correlate(array, weights) + assert_array_almost_equal(output, expected) + + output = ndimage.convolve(array, weights) + assert_array_almost_equal(output, expected) + + output = ndimage.correlate1d(array, weights) + assert_array_almost_equal(output, expected) + + output = ndimage.convolve1d(array, weights) + assert_array_almost_equal(output, expected) + + def test_correlate04(self, xp): + array = xp.asarray([1, 2]) + tcor = xp.asarray([2, 3]) + tcov = xp.asarray([3, 4]) + weights = xp.asarray([1, 1]) + output = ndimage.correlate(array, weights) + assert_array_almost_equal(output, tcor) + output = ndimage.convolve(array, weights) + assert_array_almost_equal(output, tcov) + output = ndimage.correlate1d(array, weights) + assert_array_almost_equal(output, tcor) + output = ndimage.convolve1d(array, weights) + assert_array_almost_equal(output, tcov) + + def test_correlate05(self, xp): + array = xp.asarray([1, 2, 3]) + tcor = xp.asarray([2, 3, 5]) + tcov = xp.asarray([3, 5, 6]) + kernel = xp.asarray([1, 1]) + output = ndimage.correlate(array, kernel) + assert_array_almost_equal(tcor, output) + output = ndimage.convolve(array, kernel) + assert_array_almost_equal(tcov, output) + output = ndimage.correlate1d(array, kernel) + assert_array_almost_equal(tcor, output) + output = ndimage.convolve1d(array, kernel) + assert_array_almost_equal(tcov, output) + + def test_correlate06(self, xp): + array = xp.asarray([1, 2, 3]) + tcor = xp.asarray([9, 14, 17]) + tcov = xp.asarray([7, 10, 15]) + weights = xp.asarray([1, 2, 3]) + output = ndimage.correlate(array, weights) + assert_array_almost_equal(output, tcor) + output = ndimage.convolve(array, weights) + assert_array_almost_equal(output, tcov) + output = ndimage.correlate1d(array, weights) + assert_array_almost_equal(output, tcor) + output = ndimage.convolve1d(array, weights) + assert_array_almost_equal(output, tcov) + + def test_correlate07(self, xp): + array = xp.asarray([1, 2, 3]) + expected = xp.asarray([5, 8, 11]) + weights = xp.asarray([1, 2, 1]) + output = ndimage.correlate(array, weights) + assert_array_almost_equal(output, expected) + output = ndimage.convolve(array, weights) + assert_array_almost_equal(output, expected) + output = ndimage.correlate1d(array, weights) + assert_array_almost_equal(output, expected) + output = ndimage.convolve1d(array, weights) + assert_array_almost_equal(output, expected) + + def test_correlate08(self, xp): + array = xp.asarray([1, 2, 3]) + tcor = xp.asarray([1, 2, 5]) + tcov = xp.asarray([3, 6, 7]) + weights = xp.asarray([1, 2, -1]) + output = ndimage.correlate(array, weights) + assert_array_almost_equal(output, tcor) + output = ndimage.convolve(array, weights) + assert_array_almost_equal(output, tcov) + output = ndimage.correlate1d(array, weights) + assert_array_almost_equal(output, tcor) + output = ndimage.convolve1d(array, weights) + assert_array_almost_equal(output, tcov) + + def test_correlate09(self, xp): + array = xp.asarray([]) + kernel = xp.asarray([1, 1]) + output = ndimage.correlate(array, kernel) + assert_array_almost_equal(array, output) + output = ndimage.convolve(array, kernel) + assert_array_almost_equal(array, output) + output = ndimage.correlate1d(array, kernel) + assert_array_almost_equal(array, output) + output = ndimage.convolve1d(array, kernel) + assert_array_almost_equal(array, output) + + def test_correlate10(self, xp): + array = xp.asarray([[]]) + kernel = xp.asarray([[1, 1]]) + output = ndimage.correlate(array, kernel) + assert_array_almost_equal(array, output) + output = ndimage.convolve(array, kernel) + assert_array_almost_equal(array, output) + + def test_correlate11(self, xp): + array = xp.asarray([[1, 2, 3], + [4, 5, 6]]) + kernel = xp.asarray([[1, 1], + [1, 1]]) + output = ndimage.correlate(array, kernel) + assert_array_almost_equal(xp.asarray([[4, 6, 10], [10, 12, 16]]), output) + output = ndimage.convolve(array, kernel) + assert_array_almost_equal(xp.asarray([[12, 16, 18], [18, 22, 24]]), output) + + def test_correlate12(self, xp): + array = xp.asarray([[1, 2, 3], + [4, 5, 6]]) + kernel = xp.asarray([[1, 0], + [0, 1]]) + output = ndimage.correlate(array, kernel) + assert_array_almost_equal(xp.asarray([[2, 3, 5], [5, 6, 8]]), output) + output = ndimage.convolve(array, kernel) + assert_array_almost_equal(xp.asarray([[6, 8, 9], [9, 11, 12]]), output) + + @xfail_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype_array', types) + @pytest.mark.parametrize('dtype_kernel', types) + def test_correlate13(self, dtype_array, dtype_kernel, xp): + dtype_array = getattr(xp, dtype_array) + dtype_kernel = getattr(xp, dtype_kernel) + + kernel = xp.asarray([[1, 0], + [0, 1]]) + array = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=dtype_array) + output = ndimage.correlate(array, kernel, output=dtype_kernel) + assert_array_almost_equal(xp.asarray([[2, 3, 5], [5, 6, 8]]), output) + assert output.dtype.type == dtype_kernel + + output = ndimage.convolve(array, kernel, + output=dtype_kernel) + assert_array_almost_equal(xp.asarray([[6, 8, 9], [9, 11, 12]]), output) + assert output.dtype.type == dtype_kernel + + @xfail_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype_array', types) + @pytest.mark.parametrize('dtype_output', types) + def test_correlate14(self, dtype_array, dtype_output, xp): + dtype_array = getattr(xp, dtype_array) + dtype_output = getattr(xp, dtype_output) + + kernel = xp.asarray([[1, 0], + [0, 1]]) + array = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=dtype_array) + output = xp.zeros(array.shape, dtype=dtype_output) + ndimage.correlate(array, kernel, output=output) + assert_array_almost_equal(xp.asarray([[2, 3, 5], [5, 6, 8]]), output) + assert output.dtype.type == dtype_output + + ndimage.convolve(array, kernel, output=output) + assert_array_almost_equal(xp.asarray([[6, 8, 9], [9, 11, 12]]), output) + assert output.dtype.type == dtype_output + + @xfail_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype_array', types) + def test_correlate15(self, dtype_array, xp): + dtype_array = getattr(xp, dtype_array) + + kernel = xp.asarray([[1, 0], + [0, 1]]) + array = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=dtype_array) + output = ndimage.correlate(array, kernel, output=xp.float32) + assert_array_almost_equal(xp.asarray([[2, 3, 5], [5, 6, 8]]), output) + assert output.dtype.type == xp.float32 + + output = ndimage.convolve(array, kernel, output=xp.float32) + assert_array_almost_equal(xp.asarray([[6, 8, 9], [9, 11, 12]]), output) + assert output.dtype.type == xp.float32 + + @xfail_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype_array', types) + def test_correlate16(self, dtype_array, xp): + dtype_array = getattr(xp, dtype_array) + + kernel = xp.asarray([[0.5, 0], + [0, 0.5]]) + array = xp.asarray([[1, 2, 3], [4, 5, 6]], dtype=dtype_array) + output = ndimage.correlate(array, kernel, output=xp.float32) + assert_array_almost_equal(xp.asarray([[1, 1.5, 2.5], [2.5, 3, 4]]), output) + assert output.dtype.type == xp.float32 + + output = ndimage.convolve(array, kernel, output=xp.float32) + assert_array_almost_equal(xp.asarray([[3, 4, 4.5], [4.5, 5.5, 6]]), output) + assert output.dtype.type == xp.float32 + + def test_correlate17(self, xp): + array = xp.asarray([1, 2, 3]) + tcor = xp.asarray([3, 5, 6]) + tcov = xp.asarray([2, 3, 5]) + kernel = xp.asarray([1, 1]) + output = ndimage.correlate(array, kernel, origin=-1) + assert_array_almost_equal(tcor, output) + output = ndimage.convolve(array, kernel, origin=-1) + assert_array_almost_equal(tcov, output) + output = ndimage.correlate1d(array, kernel, origin=-1) + assert_array_almost_equal(tcor, output) + output = ndimage.convolve1d(array, kernel, origin=-1) + assert_array_almost_equal(tcov, output) + + @xfail_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype_array', types) + def test_correlate18(self, dtype_array, xp): + dtype_array = getattr(xp, dtype_array) + + kernel = xp.asarray([[1, 0], + [0, 1]]) + array = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=dtype_array) + output = ndimage.correlate(array, kernel, + output=xp.float32, + mode='nearest', origin=-1) + assert_array_almost_equal(xp.asarray([[6, 8, 9], [9, 11, 12]]), output) + assert output.dtype.type == xp.float32 + + output = ndimage.convolve(array, kernel, + output=xp.float32, + mode='nearest', origin=-1) + assert_array_almost_equal(xp.asarray([[2, 3, 5], [5, 6, 8]]), output) + assert output.dtype.type == xp.float32 + + def test_correlate_mode_sequence(self, xp): + kernel = xp.ones((2, 2)) + array = xp.ones((3, 3), dtype=xp.float64) + with assert_raises(RuntimeError): + ndimage.correlate(array, kernel, mode=['nearest', 'reflect']) + with assert_raises(RuntimeError): + ndimage.convolve(array, kernel, mode=['nearest', 'reflect']) + + @xfail_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype_array', types) + def test_correlate19(self, dtype_array, xp): + dtype_array = getattr(xp, dtype_array) + + kernel = xp.asarray([[1, 0], + [0, 1]]) + array = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=dtype_array) + output = ndimage.correlate(array, kernel, + output=xp.float32, + mode='nearest', origin=[-1, 0]) + assert_array_almost_equal(xp.asarray([[5, 6, 8], [8, 9, 11]]), output) + assert output.dtype.type == xp.float32 + + output = ndimage.convolve(array, kernel, + output=xp.float32, + mode='nearest', origin=[-1, 0]) + assert_array_almost_equal(xp.asarray([[3, 5, 6], [6, 8, 9]]), output) + assert output.dtype.type == xp.float32 + + @xfail_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype_array', types) + @pytest.mark.parametrize('dtype_output', types) + def test_correlate20(self, dtype_array, dtype_output, xp): + dtype_array = getattr(xp, dtype_array) + dtype_output = getattr(xp, dtype_output) + + weights = xp.asarray([1, 2, 1]) + expected = xp.asarray([[5, 10, 15], [7, 14, 21]]) + array = xp.asarray([[1, 2, 3], + [2, 4, 6]], dtype=dtype_array) + output = xp.zeros((2, 3), dtype=dtype_output) + ndimage.correlate1d(array, weights, axis=0, output=output) + assert_array_almost_equal(output, expected) + ndimage.convolve1d(array, weights, axis=0, output=output) + assert_array_almost_equal(output, expected) + + def test_correlate21(self, xp): + array = xp.asarray([[1, 2, 3], + [2, 4, 6]]) + expected = xp.asarray([[5, 10, 15], [7, 14, 21]]) + weights = xp.asarray([1, 2, 1]) + output = ndimage.correlate1d(array, weights, axis=0) + assert_array_almost_equal(output, expected) + output = ndimage.convolve1d(array, weights, axis=0) + assert_array_almost_equal(output, expected) + + @xfail_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype_array', types) + @pytest.mark.parametrize('dtype_output', types) + def test_correlate22(self, dtype_array, dtype_output, xp): + dtype_array = getattr(xp, dtype_array) + dtype_output = getattr(xp, dtype_output) + + weights = xp.asarray([1, 2, 1]) + expected = xp.asarray([[6, 12, 18], [6, 12, 18]]) + array = xp.asarray([[1, 2, 3], + [2, 4, 6]], dtype=dtype_array) + output = xp.zeros((2, 3), dtype=dtype_output) + ndimage.correlate1d(array, weights, axis=0, + mode='wrap', output=output) + assert_array_almost_equal(output, expected) + ndimage.convolve1d(array, weights, axis=0, + mode='wrap', output=output) + assert_array_almost_equal(output, expected) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + @pytest.mark.parametrize('dtype_array', types) + @pytest.mark.parametrize('dtype_output', types) + def test_correlate23(self, dtype_array, dtype_output, xp): + dtype_array = getattr(xp, dtype_array) + dtype_output = getattr(xp, dtype_output) + + weights = xp.asarray([1, 2, 1]) + expected = xp.asarray([[5, 10, 15], [7, 14, 21]]) + array = xp.asarray([[1, 2, 3], + [2, 4, 6]], dtype=dtype_array) + output = xp.zeros((2, 3), dtype=dtype_output) + ndimage.correlate1d(array, weights, axis=0, + mode='nearest', output=output) + assert_array_almost_equal(output, expected) + ndimage.convolve1d(array, weights, axis=0, + mode='nearest', output=output) + assert_array_almost_equal(output, expected) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + @pytest.mark.parametrize('dtype_array', types) + @pytest.mark.parametrize('dtype_output', types) + def test_correlate24(self, dtype_array, dtype_output, xp): + dtype_array = getattr(xp, dtype_array) + dtype_output = getattr(xp, dtype_output) + + weights = xp.asarray([1, 2, 1]) + tcor = xp.asarray([[7, 14, 21], [8, 16, 24]]) + tcov = xp.asarray([[4, 8, 12], [5, 10, 15]]) + array = xp.asarray([[1, 2, 3], + [2, 4, 6]], dtype=dtype_array) + output = xp.zeros((2, 3), dtype=dtype_output) + ndimage.correlate1d(array, weights, axis=0, + mode='nearest', output=output, origin=-1) + assert_array_almost_equal(output, tcor) + ndimage.convolve1d(array, weights, axis=0, + mode='nearest', output=output, origin=-1) + assert_array_almost_equal(output, tcov) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + @pytest.mark.parametrize('dtype_array', types) + @pytest.mark.parametrize('dtype_output', types) + def test_correlate25(self, dtype_array, dtype_output, xp): + dtype_array = getattr(xp, dtype_array) + dtype_output = getattr(xp, dtype_output) + + weights = xp.asarray([1, 2, 1]) + tcor = xp.asarray([[4, 8, 12], [5, 10, 15]]) + tcov = xp.asarray([[7, 14, 21], [8, 16, 24]]) + array = xp.asarray([[1, 2, 3], + [2, 4, 6]], dtype=dtype_array) + output = xp.zeros((2, 3), dtype=dtype_output) + ndimage.correlate1d(array, weights, axis=0, + mode='nearest', output=output, origin=1) + assert_array_almost_equal(output, tcor) + ndimage.convolve1d(array, weights, axis=0, + mode='nearest', output=output, origin=1) + assert_array_almost_equal(output, tcov) + + def test_correlate26(self, xp): + # test fix for gh-11661 (mirror extension of a length 1 signal) + y = ndimage.convolve1d(xp.ones(1), xp.ones(5), mode='mirror') + xp_assert_equal(y, xp.asarray([5.])) + + y = ndimage.correlate1d(xp.ones(1), xp.ones(5), mode='mirror') + xp_assert_equal(y, xp.asarray([5.])) + + @xfail_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype_kernel', complex_types) + @pytest.mark.parametrize('dtype_input', types) + @pytest.mark.parametrize('dtype_output', complex_types) + def test_correlate_complex_kernel(self, dtype_input, dtype_kernel, + dtype_output, xp, num_parallel_threads): + dtype_input = getattr(xp, dtype_input) + dtype_kernel = getattr(xp, dtype_kernel) + dtype_output = getattr(xp, dtype_output) + + kernel = xp.asarray([[1, 0], + [0, 1 + 1j]], dtype=dtype_kernel) + array = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=dtype_input) + self._validate_complex(xp, array, kernel, dtype_output, + check_warnings=num_parallel_threads == 1) + + @xfail_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype_kernel', complex_types) + @pytest.mark.parametrize('dtype_input', types) + @pytest.mark.parametrize('dtype_output', complex_types) + @pytest.mark.parametrize('mode', ['grid-constant', 'constant']) + def test_correlate_complex_kernel_cval(self, dtype_input, dtype_kernel, + dtype_output, mode, xp, + num_parallel_threads): + dtype_input = getattr(xp, dtype_input) + dtype_kernel = getattr(xp, dtype_kernel) + dtype_output = getattr(xp, dtype_output) + + if is_cupy(xp) and mode == 'grid-constant': + pytest.xfail('https://github.com/cupy/cupy/issues/8404') + + # test use of non-zero cval with complex inputs + # also verifies that mode 'grid-constant' does not segfault + kernel = xp.asarray([[1, 0], + [0, 1 + 1j]], dtype=dtype_kernel) + array = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=dtype_input) + self._validate_complex(xp, array, kernel, dtype_output, mode=mode, + cval=5.0, + check_warnings=num_parallel_threads == 1) + + @xfail_xp_backends('cupy', reason="cupy/cupy#8405") + @pytest.mark.parametrize('dtype_kernel', complex_types) + @pytest.mark.parametrize('dtype_input', types) + @pytest.mark.thread_unsafe + def test_correlate_complex_kernel_invalid_cval(self, dtype_input, + dtype_kernel, xp): + dtype_input = getattr(xp, dtype_input) + dtype_kernel = getattr(xp, dtype_kernel) + + # cannot give complex cval with a real image + kernel = xp.asarray([[1, 0], + [0, 1 + 1j]], dtype=dtype_kernel) + array = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=dtype_input) + for func in [ndimage.convolve, ndimage.correlate, ndimage.convolve1d, + ndimage.correlate1d]: + with pytest.raises((ValueError, TypeError)): + func(array, kernel, mode='constant', cval=5.0 + 1.0j, + output=xp.complex64) + + @skip_xp_backends(np_only=True, reason='output=dtype is numpy-specific') + @pytest.mark.parametrize('dtype_kernel', complex_types) + @pytest.mark.parametrize('dtype_input', types) + @pytest.mark.parametrize('dtype_output', complex_types) + def test_correlate1d_complex_kernel(self, dtype_input, dtype_kernel, + dtype_output, xp, num_parallel_threads): + dtype_input = getattr(xp, dtype_input) + dtype_kernel = getattr(xp, dtype_kernel) + dtype_output = getattr(xp, dtype_output) + + kernel = xp.asarray([1, 1 + 1j], dtype=dtype_kernel) + array = xp.asarray([1, 2, 3, 4, 5, 6], dtype=dtype_input) + self._validate_complex(xp, array, kernel, dtype_output, + check_warnings=num_parallel_threads == 1) + + @skip_xp_backends(np_only=True, reason='output=dtype is numpy-specific') + @pytest.mark.parametrize('dtype_kernel', complex_types) + @pytest.mark.parametrize('dtype_input', types) + @pytest.mark.parametrize('dtype_output', complex_types) + def test_correlate1d_complex_kernel_cval(self, dtype_input, dtype_kernel, + dtype_output, xp, + num_parallel_threads): + dtype_input = getattr(xp, dtype_input) + dtype_kernel = getattr(xp, dtype_kernel) + dtype_output = getattr(xp, dtype_output) + + kernel = xp.asarray([1, 1 + 1j], dtype=dtype_kernel) + array = xp.asarray([1, 2, 3, 4, 5, 6], dtype=dtype_input) + self._validate_complex(xp, array, kernel, dtype_output, mode='constant', + cval=5.0, + check_warnings=num_parallel_threads == 1) + + @skip_xp_backends(np_only=True, reason='output=dtype is numpy-specific') + @pytest.mark.parametrize('dtype_kernel', types) + @pytest.mark.parametrize('dtype_input', complex_types) + @pytest.mark.parametrize('dtype_output', complex_types) + def test_correlate_complex_input(self, dtype_input, dtype_kernel, + dtype_output, xp, num_parallel_threads): + dtype_input = getattr(xp, dtype_input) + dtype_kernel = getattr(xp, dtype_kernel) + dtype_output = getattr(xp, dtype_output) + + kernel = xp.asarray([[1, 0], + [0, 1]], dtype=dtype_kernel) + array = xp.asarray([[1, 2j, 3], + [1 + 4j, 5, 6j]], dtype=dtype_input) + self._validate_complex(xp, array, kernel, dtype_output, + check_warnings=num_parallel_threads == 1) + + @skip_xp_backends(np_only=True, reason='output=dtype is numpy-specific') + @pytest.mark.parametrize('dtype_kernel', types) + @pytest.mark.parametrize('dtype_input', complex_types) + @pytest.mark.parametrize('dtype_output', complex_types) + def test_correlate1d_complex_input(self, dtype_input, dtype_kernel, + dtype_output, xp, num_parallel_threads): + dtype_input = getattr(xp, dtype_input) + dtype_kernel = getattr(xp, dtype_kernel) + dtype_output = getattr(xp, dtype_output) + + kernel = xp.asarray([1, 0, 1], dtype=dtype_kernel) + array = xp.asarray([1, 2j, 3, 1 + 4j, 5, 6j], dtype=dtype_input) + self._validate_complex(xp, array, kernel, dtype_output, + check_warnings=num_parallel_threads == 1) + + @xfail_xp_backends('cupy', reason="cupy/cupy#8405") + @skip_xp_backends(np_only=True, + reason='output=dtype is numpy-specific', + exceptions=['cupy']) + @pytest.mark.parametrize('dtype_kernel', types) + @pytest.mark.parametrize('dtype_input', complex_types) + @pytest.mark.parametrize('dtype_output', complex_types) + def test_correlate1d_complex_input_cval(self, dtype_input, dtype_kernel, + dtype_output, xp, + num_parallel_threads): + dtype_input = getattr(xp, dtype_input) + dtype_kernel = getattr(xp, dtype_kernel) + dtype_output = getattr(xp, dtype_output) + + kernel = xp.asarray([1, 0, 1], dtype=dtype_kernel) + array = xp.asarray([1, 2j, 3, 1 + 4j, 5, 6j], dtype=dtype_input) + self._validate_complex(xp, array, kernel, dtype_output, mode='constant', + cval=5 - 3j, + check_warnings=num_parallel_threads == 1) + + @skip_xp_backends(np_only=True, reason='output=dtype is numpy-specific') + @pytest.mark.parametrize('dtype', complex_types) + @pytest.mark.parametrize('dtype_output', complex_types) + def test_correlate_complex_input_and_kernel(self, dtype, dtype_output, xp, + num_parallel_threads): + dtype = getattr(xp, dtype) + dtype_output = getattr(xp, dtype_output) + + kernel = xp.asarray([[1, 0], + [0, 1 + 1j]], dtype=dtype) + array = xp.asarray([[1, 2j, 3], + [1 + 4j, 5, 6j]], dtype=dtype) + self._validate_complex(xp, array, kernel, dtype_output, + check_warnings=num_parallel_threads == 1) + + @xfail_xp_backends('cupy', reason="cupy/cupy#8405") + @skip_xp_backends(np_only=True, + reason="output=dtype is numpy-specific", + exceptions=['cupy'],) + @pytest.mark.parametrize('dtype', complex_types) + @pytest.mark.parametrize('dtype_output', complex_types) + def test_correlate_complex_input_and_kernel_cval(self, dtype, + dtype_output, xp, + num_parallel_threads): + dtype = getattr(xp, dtype) + dtype_output = getattr(xp, dtype_output) + + kernel = xp.asarray([[1, 0], + [0, 1 + 1j]], dtype=dtype) + array = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=dtype) + self._validate_complex(xp, array, kernel, dtype_output, mode='constant', + cval=5.0 + 2.0j, + check_warnings=num_parallel_threads == 1) + + @skip_xp_backends(np_only=True, reason="output=dtype is numpy-specific") + @pytest.mark.parametrize('dtype', complex_types) + @pytest.mark.parametrize('dtype_output', complex_types) + @pytest.mark.thread_unsafe + def test_correlate1d_complex_input_and_kernel(self, dtype, dtype_output, xp, + num_parallel_threads): + dtype = getattr(xp, dtype) + dtype_output = getattr(xp, dtype_output) + + kernel = xp.asarray([1, 1 + 1j], dtype=dtype) + array = xp.asarray([1, 2j, 3, 1 + 4j, 5, 6j], dtype=dtype) + self._validate_complex(xp, array, kernel, dtype_output, + check_warnings=num_parallel_threads == 1) + + @pytest.mark.parametrize('dtype', complex_types) + @pytest.mark.parametrize('dtype_output', complex_types) + def test_correlate1d_complex_input_and_kernel_cval(self, dtype, + dtype_output, xp, + num_parallel_threads): + if not (is_numpy(xp) or is_cupy(xp)): + pytest.xfail("output=dtype is numpy-specific") + + dtype = getattr(xp, dtype) + dtype_output = getattr(xp, dtype_output) + + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/issues/8405") + + kernel = xp.asarray([1, 1 + 1j], dtype=dtype) + array = xp.asarray([1, 2j, 3, 1 + 4j, 5, 6j], dtype=dtype) + self._validate_complex(xp, array, kernel, dtype_output, mode='constant', + cval=5.0 + 2.0j, + check_warnings=num_parallel_threads == 1) + + def test_gauss01(self, xp): + input = xp.asarray([[1, 2, 3], + [2, 4, 6]], dtype=xp.float32) + output = ndimage.gaussian_filter(input, 0) + assert_array_almost_equal(output, input) + + def test_gauss02(self, xp): + input = xp.asarray([[1, 2, 3], + [2, 4, 6]], dtype=xp.float32) + output = ndimage.gaussian_filter(input, 1.0) + assert input.dtype == output.dtype + assert input.shape == output.shape + + def test_gauss03(self, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/issues/8403") + + # single precision data + input = xp.arange(100 * 100, dtype=xp.float32) + input = xp.reshape(input, (100, 100)) + output = ndimage.gaussian_filter(input, [1.0, 1.0]) + + assert input.dtype == output.dtype + assert input.shape == output.shape + + # input.sum() is 49995000.0. With single precision floats, we can't + # expect more than 8 digits of accuracy, so use decimal=0 in this test. + o_sum = xp.sum(output, dtype=xp.float64) + i_sum = xp.sum(input, dtype=xp.float64) + assert_almost_equal(o_sum, i_sum, decimal=0) + assert sumsq(input, output) > 1.0 + + def test_gauss04(self, xp): + if not (is_numpy(xp) or is_cupy(xp)): + pytest.xfail("output=dtype is numpy-specific") + + input = xp.arange(100 * 100, dtype=xp.float32) + input = xp.reshape(input, (100, 100)) + otype = xp.float64 + output = ndimage.gaussian_filter(input, [1.0, 1.0], output=otype) + assert output.dtype.type == xp.float64 + assert input.shape == output.shape + assert sumsq(input, output) > 1.0 + + def test_gauss05(self, xp): + if not (is_numpy(xp) or is_cupy(xp)): + pytest.xfail("output=dtype is numpy-specific") + + input = xp.arange(100 * 100, dtype=xp.float32) + input = xp.reshape(input, (100, 100)) + otype = xp.float64 + output = ndimage.gaussian_filter(input, [1.0, 1.0], + order=1, output=otype) + assert output.dtype.type == xp.float64 + assert input.shape == output.shape + assert sumsq(input, output) > 1.0 + + def test_gauss06(self, xp): + if not (is_numpy(xp) or is_cupy(xp)): + pytest.xfail("output=dtype is numpy-specific") + + input = xp.arange(100 * 100, dtype=xp.float32) + input = xp.reshape(input, (100, 100)) + otype = xp.float64 + output1 = ndimage.gaussian_filter(input, [1.0, 1.0], output=otype) + output2 = ndimage.gaussian_filter(input, 1.0, output=otype) + assert_array_almost_equal(output1, output2) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + def test_gauss_memory_overlap(self, xp): + input = xp.arange(100 * 100, dtype=xp.float32) + input = xp.reshape(input, (100, 100)) + output1 = ndimage.gaussian_filter(input, 1.0) + ndimage.gaussian_filter(input, 1.0, output=input) + assert_array_almost_equal(output1, input) + + @pytest.mark.parametrize(('filter_func', 'extra_args', 'size0', 'size'), + [(ndimage.gaussian_filter, (), 0, 1.0), + (ndimage.uniform_filter, (), 1, 3), + (ndimage.minimum_filter, (), 1, 3), + (ndimage.maximum_filter, (), 1, 3), + (ndimage.median_filter, (), 1, 3), + (ndimage.rank_filter, (1,), 1, 3), + (ndimage.percentile_filter, (40,), 1, 3)]) + @pytest.mark.parametrize( + 'axes', + tuple(itertools.combinations(range(-3, 3), 1)) + + tuple(itertools.combinations(range(-3, 3), 2)) + + ((0, 1, 2),)) + def test_filter_axes(self, filter_func, extra_args, size0, size, axes, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/pull/8339") + + # Note: `size` is called `sigma` in `gaussian_filter` + array = xp.arange(6 * 8 * 12, dtype=xp.float64) + array = xp.reshape(array, (6, 8, 12)) + + if len(set(ax % array.ndim for ax in axes)) != len(axes): + # parametrized cases with duplicate axes raise an error + with pytest.raises(ValueError, match="axes must be unique"): + filter_func(array, *extra_args, size, axes=axes) + return + output = filter_func(array, *extra_args, size, axes=axes) + + # result should be equivalent to sigma=0.0/size=1 on unfiltered axes + axes = xp.asarray(axes) + all_sizes = tuple(size if ax in (axes % array.ndim) else size0 + for ax in range(array.ndim)) + expected = filter_func(array, *extra_args, all_sizes) + xp_assert_close(output, expected) + + @skip_xp_backends("cupy", + reason="these filters do not yet have axes support", + ) + @pytest.mark.parametrize(('filter_func', 'kwargs'), + [(ndimage.laplace, {}), + (ndimage.gaussian_gradient_magnitude, + {"sigma": 1.0}), + (ndimage.gaussian_laplace, {"sigma": 0.5})]) + def test_derivative_filter_axes(self, xp, filter_func, kwargs): + array = xp.arange(6 * 8 * 12, dtype=xp.float64) + array = xp.reshape(array, (6, 8, 12)) + + # duplicate axes raises an error + with pytest.raises(ValueError, match="axes must be unique"): + filter_func(array, axes=(1, 1), **kwargs) + + # compare results to manually looping over the non-filtered axes + output = filter_func(array, axes=(1, 2), **kwargs) + expected = xp.empty_like(output) + expected = [] + for i in range(array.shape[0]): + expected.append(filter_func(array[i, ...], **kwargs)) + expected = xp.stack(expected, axis=0) + xp_assert_close(output, expected) + + output = filter_func(array, axes=(0, -1), **kwargs) + expected = [] + for i in range(array.shape[1]): + expected.append(filter_func(array[:, i, :], **kwargs)) + expected = xp.stack(expected, axis=1) + xp_assert_close(output, expected) + + output = filter_func(array, axes=(1), **kwargs) + expected = [] + for i in range(array.shape[0]): + exp_inner = [] + for j in range(array.shape[2]): + exp_inner.append(filter_func(array[i, :, j], **kwargs)) + expected.append(xp.stack(exp_inner, axis=-1)) + expected = xp.stack(expected, axis=0) + xp_assert_close(output, expected) + + @skip_xp_backends("cupy", + reason="generic_filter does not yet have axes support", + ) + @pytest.mark.parametrize( + 'axes', + tuple(itertools.combinations(range(-3, 3), 1)) + + tuple(itertools.combinations(range(-3, 3), 2)) + + ((0, 1, 2),)) + def test_generic_filter_axes(self, xp, axes): + array = xp.arange(6 * 8 * 12, dtype=xp.float64) + array = xp.reshape(array, (6, 8, 12)) + size = 3 + if len(set(ax % array.ndim for ax in axes)) != len(axes): + # parametrized cases with duplicate axes raise an error + with pytest.raises(ValueError, match="axes must be unique"): + ndimage.generic_filter(array, np.amax, size=size, axes=axes) + return + + # choose np.amax as the function so we can compare to maximum_filter + output = ndimage.generic_filter(array, np.amax, size=size, axes=axes) + expected = ndimage.maximum_filter(array, size=size, axes=axes) + xp_assert_close(output, expected) + + @skip_xp_backends("cupy", + reason="https://github.com/cupy/cupy/pull/8339", + ) + @pytest.mark.parametrize('func', [ndimage.correlate, ndimage.convolve]) + @pytest.mark.parametrize( + 'dtype', [np.float32, np.float64, np.complex64, np.complex128] + ) + @pytest.mark.parametrize( + 'axes', tuple(itertools.combinations(range(-3, 3), 2)) + ) + @pytest.mark.parametrize('origin', [(0, 0), (-1, 1)]) + def test_correlate_convolve_axes(self, xp, func, dtype, axes, origin): + array = xp.asarray(np.arange(6 * 8 * 12, dtype=dtype).reshape(6, 8, 12)) + weights = xp.arange(3 * 5) + weights = xp.reshape(weights, (3, 5)) + axes = tuple(ax % array.ndim for ax in axes) + if len(tuple(set(axes))) != len(axes): + # parametrized cases with duplicate axes raise an error + with pytest.raises(ValueError): + func(array, weights=weights, axes=axes, origin=origin) + return + output = func(array, weights=weights, axes=axes, origin=origin) + + missing_axis = tuple(set(range(3)) - set(axes))[0] + # module 'torch' has no attribute 'expand_dims' so use reshape instead + # weights_3d = xp.expand_dims(weights, axis=missing_axis) + shape_3d = ( + weights.shape[:missing_axis] + (1,) + weights.shape[missing_axis:] + ) + weights_3d = xp.reshape(weights, shape_3d) + origin_3d = [0, 0, 0] + for i, ax in enumerate(axes): + origin_3d[ax] = origin[i] + expected = func(array, weights=weights_3d, origin=origin_3d) + xp_assert_close(output, expected) + + kwargs_gauss = dict(radius=[4, 2, 3], order=[0, 1, 2], + mode=['reflect', 'nearest', 'constant']) + kwargs_other = dict(origin=(-1, 0, 1), + mode=['reflect', 'nearest', 'constant']) + kwargs_rank = dict(origin=(-1, 0, 1)) + + @skip_xp_backends("array_api_strict", + reason="fancy indexing is only available in 2024 version", + ) + @pytest.mark.parametrize("filter_func, size0, size, kwargs", + [(ndimage.gaussian_filter, 0, 1.0, kwargs_gauss), + (ndimage.uniform_filter, 1, 3, kwargs_other), + (ndimage.maximum_filter, 1, 3, kwargs_other), + (ndimage.minimum_filter, 1, 3, kwargs_other), + (ndimage.median_filter, 1, 3, kwargs_rank), + (ndimage.rank_filter, 1, 3, kwargs_rank), + (ndimage.percentile_filter, 1, 3, kwargs_rank)]) + @pytest.mark.parametrize('axes', itertools.combinations(range(-3, 3), 2)) + def test_filter_axes_kwargs(self, filter_func, size0, size, kwargs, axes, xp): + + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/pull/8339") + + array = xp.arange(6 * 8 * 12, dtype=xp.float64) + array = xp.reshape(array, (6, 8, 12)) + + kwargs = {key: np.array(val) for key, val in kwargs.items()} + axes = np.array(axes) + n_axes = axes.size + + if filter_func == ndimage.rank_filter: + args = (2,) # (rank,) + elif filter_func == ndimage.percentile_filter: + args = (30,) # (percentile,) + else: + args = () + + # form kwargs that specify only the axes in `axes` + reduced_kwargs = {key: val[axes] for key, val in kwargs.items()} + if len(set(axes % array.ndim)) != len(axes): + # parametrized cases with duplicate axes raise an error + with pytest.raises(ValueError, match="axes must be unique"): + filter_func(array, *args, [size]*n_axes, axes=axes, + **reduced_kwargs) + return + + output = filter_func(array, *args, [size]*n_axes, axes=axes, + **reduced_kwargs) + + # result should be equivalent to sigma=0.0/size=1 on unfiltered axes + size_3d = np.full(array.ndim, fill_value=size0) + size_3d[axes] = size + size_3d = [size_3d[i] for i in range(size_3d.shape[0])] + if 'origin' in kwargs: + # origin should be zero on the axis that has size 0 + origin = np.asarray([0, 0, 0]) + origin[axes] = reduced_kwargs['origin'] + origin = xp.asarray(origin) + kwargs['origin'] = origin + expected = filter_func(array, *args, size_3d, **kwargs) + xp_assert_close(output, expected) + + + @pytest.mark.parametrize("filter_func, kwargs", + [(ndimage.convolve, {}), + (ndimage.correlate, {}), + (ndimage.minimum_filter, {}), + (ndimage.maximum_filter, {}), + (ndimage.median_filter, {}), + (ndimage.rank_filter, {"rank": 1}), + (ndimage.percentile_filter, {"percentile": 30})]) + def test_filter_weights_subset_axes_origins(self, filter_func, kwargs, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/pull/8339") + + axes = (-2, -1) + origins = (0, 1) + array = xp.arange(6 * 8 * 12, dtype=xp.float64) + array = xp.reshape(array, (6, 8, 12)) + + # weights with ndim matching len(axes) + footprint = np.ones((3, 5), dtype=bool) + footprint[0, 1] = 0 # make non-separable + footprint = xp.asarray(footprint) + + if filter_func in (ndimage.convolve, ndimage.correlate): + kwargs["weights"] = footprint + else: + kwargs["footprint"] = footprint + kwargs["axes"] = axes + + output = filter_func(array, origin=origins, **kwargs) + + output0 = filter_func(array, origin=0, **kwargs) + + # output has origin shift on last axis relative to output0, so + # expect shifted arrays to be equal. + if filter_func == ndimage.convolve: + # shift is in the opposite direction for convolve because it + # flips the weights array and negates the origin values. + xp_assert_equal( + output[:, :, :-origins[1]], output0[:, :, origins[1]:]) + else: + xp_assert_equal( + output[:, :, origins[1]:], output0[:, :, :-origins[1]]) + + + @pytest.mark.parametrize( + 'filter_func, args', + [(ndimage.convolve, (np.ones((3, 3, 3)),)), # args = (weights,) + (ndimage.correlate,(np.ones((3, 3, 3)),)), # args = (weights,) + (ndimage.gaussian_filter, (1.0,)), # args = (sigma,) + (ndimage.uniform_filter, (3,)), # args = (size,) + (ndimage.minimum_filter, (3,)), # args = (size,) + (ndimage.maximum_filter, (3,)), # args = (size,) + (ndimage.median_filter, (3,)), # args = (size,) + (ndimage.rank_filter, (2, 3)), # args = (rank, size) + (ndimage.percentile_filter, (30, 3))]) # args = (percentile, size) + @pytest.mark.parametrize( + 'axes', [(1.5,), (0, 1, 2, 3), (3,), (-4,)] + ) + def test_filter_invalid_axes(self, filter_func, args, axes, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/pull/8339") + + array = xp.arange(6 * 8 * 12, dtype=xp.float64) + array = xp.reshape(array, (6, 8, 12)) + args = [ + xp.asarray(arg) if isinstance(arg, np.ndarray) else arg + for arg in args + ] + if any(isinstance(ax, float) for ax in axes): + error_class = TypeError + match = "cannot be interpreted as an integer" + else: + error_class = ValueError + match = "out of range" + with pytest.raises(error_class, match=match): + filter_func(array, *args, axes=axes) + + @pytest.mark.parametrize( + 'filter_func, kwargs', + [(ndimage.convolve, {}), + (ndimage.correlate, {}), + (ndimage.minimum_filter, {}), + (ndimage.maximum_filter, {}), + (ndimage.median_filter, {}), + (ndimage.rank_filter, dict(rank=3)), + (ndimage.percentile_filter, dict(percentile=30))]) + @pytest.mark.parametrize( + 'axes', [(0, ), (1, 2), (0, 1, 2)] + ) + @pytest.mark.parametrize('separable_footprint', [False, True]) + def test_filter_invalid_footprint_ndim(self, filter_func, kwargs, axes, + separable_footprint, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/pull/8339") + + array = xp.arange(6 * 8 * 12, dtype=xp.float64) + array = xp.reshape(array, (6, 8, 12)) + # create a footprint with one too many dimensions + footprint = np.ones((3,) * (len(axes) + 1)) + if not separable_footprint: + footprint[(0,) * footprint.ndim] = 0 + footprint = xp.asarray(footprint) + if (filter_func in [ndimage.minimum_filter, ndimage.maximum_filter] + and separable_footprint): + match = "sequence argument must have length equal to input rank" + elif filter_func in [ndimage.convolve, ndimage.correlate]: + match = re.escape(f"weights.ndim ({footprint.ndim}) must match " + f"len(axes) ({len(axes)})") + else: + match = re.escape(f"footprint.ndim ({footprint.ndim}) must match " + f"len(axes) ({len(axes)})") + if filter_func in [ndimage.convolve, ndimage.correlate]: + kwargs["weights"] = footprint + else: + kwargs["footprint"] = footprint + with pytest.raises(RuntimeError, match=match): + filter_func(array, axes=axes, **kwargs) + + @pytest.mark.parametrize('n_mismatch', [1, 3]) + @pytest.mark.parametrize('filter_func, kwargs, key, val', + _cases_axes_tuple_length_mismatch()) + def test_filter_tuple_length_mismatch(self, n_mismatch, filter_func, + kwargs, key, val, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/pull/8339") + + # Test for the intended RuntimeError when a kwargs has an invalid size + array = xp.arange(6 * 8 * 12, dtype=xp.float64) + array = xp.reshape(array, (6, 8, 12)) + axes = (0, 1) + kwargs = dict(**kwargs, axes=axes) + kwargs[key] = (val,) * n_mismatch + if filter_func in [ndimage.convolve, ndimage.correlate]: + kwargs["weights"] = xp.ones((5,) * len(axes)) + err_msg = "sequence argument must have length equal to input rank" + with pytest.raises(RuntimeError, match=err_msg): + filter_func(array, **kwargs) + + @pytest.mark.parametrize('dtype', types + complex_types) + def test_prewitt01(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + t = ndimage.correlate1d(array, xp.asarray([-1.0, 0.0, 1.0]), 0) + t = ndimage.correlate1d(t, xp.asarray([1.0, 1.0, 1.0]), 1) + output = ndimage.prewitt(array, 0) + assert_array_almost_equal(t, output) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + @pytest.mark.parametrize('dtype', types + complex_types) + def test_prewitt02(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + t = ndimage.correlate1d(array, xp.asarray([-1.0, 0.0, 1.0]), 0) + t = ndimage.correlate1d(t, xp.asarray([1.0, 1.0, 1.0]), 1) + output = xp.zeros(array.shape, dtype=dtype) + ndimage.prewitt(array, 0, output) + assert_array_almost_equal(t, output) + + @pytest.mark.parametrize('dtype', types + complex_types) + def test_prewitt03(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + if is_cupy(xp) and dtype in [xp.uint32, xp.uint64]: + pytest.xfail("uint UB? XXX") + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + t = ndimage.correlate1d(array, xp.asarray([-1.0, 0.0, 1.0]), 1) + t = ndimage.correlate1d(t, xp.asarray([1.0, 1.0, 1.0]), 0) + output = ndimage.prewitt(array, 1) + assert_array_almost_equal(t, output) + + @pytest.mark.parametrize('dtype', types + complex_types) + def test_prewitt04(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + t = ndimage.prewitt(array, -1) + output = ndimage.prewitt(array, 1) + assert_array_almost_equal(t, output) + + @pytest.mark.parametrize('dtype', types + complex_types) + def test_sobel01(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + t = ndimage.correlate1d(array, xp.asarray([-1.0, 0.0, 1.0]), 0) + t = ndimage.correlate1d(t, xp.asarray([1.0, 2.0, 1.0]), 1) + output = ndimage.sobel(array, 0) + assert_array_almost_equal(t, output) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.",) + @pytest.mark.parametrize('dtype', types + complex_types) + def test_sobel02(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + t = ndimage.correlate1d(array, xp.asarray([-1.0, 0.0, 1.0]), 0) + t = ndimage.correlate1d(t, xp.asarray([1.0, 2.0, 1.0]), 1) + output = xp.zeros(array.shape, dtype=dtype) + ndimage.sobel(array, 0, output) + assert_array_almost_equal(t, output) + + @pytest.mark.parametrize('dtype', types + complex_types) + def test_sobel03(self, dtype, xp): + if is_cupy(xp) and dtype in ["uint32", "uint64"]: + pytest.xfail("uint UB? XXX") + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + t = ndimage.correlate1d(array, xp.asarray([-1.0, 0.0, 1.0]), 1) + t = ndimage.correlate1d(t, xp.asarray([1.0, 2.0, 1.0]), 0) + output = xp.zeros(array.shape, dtype=dtype) + output = ndimage.sobel(array, 1) + assert_array_almost_equal(t, output) + + @pytest.mark.parametrize('dtype', types + complex_types) + def test_sobel04(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + t = ndimage.sobel(array, -1) + output = ndimage.sobel(array, 1) + assert_array_almost_equal(t, output) + + @pytest.mark.parametrize('dtype', + ["int32", "float32", "float64", + "complex64", "complex128"]) + def test_laplace01(self, dtype, xp): + dtype = getattr(xp, dtype) + + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) * 100 + tmp1 = ndimage.correlate1d(array, xp.asarray([1, -2, 1]), 0) + tmp2 = ndimage.correlate1d(array, xp.asarray([1, -2, 1]), 1) + output = ndimage.laplace(array) + assert_array_almost_equal(tmp1 + tmp2, output) + + @skip_xp_backends("jax.numpy", reason="output array is read-only",) + @pytest.mark.parametrize('dtype', + ["int32", "float32", "float64", + "complex64", "complex128"]) + def test_laplace02(self, dtype, xp): + dtype = getattr(xp, dtype) + + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) * 100 + tmp1 = ndimage.correlate1d(array, xp.asarray([1, -2, 1]), 0) + tmp2 = ndimage.correlate1d(array, xp.asarray([1, -2, 1]), 1) + output = xp.zeros(array.shape, dtype=dtype) + ndimage.laplace(array, output=output) + assert_array_almost_equal(tmp1 + tmp2, output) + + @pytest.mark.parametrize('dtype', + ["int32", "float32", "float64", + "complex64", "complex128"]) + def test_gaussian_laplace01(self, dtype, xp): + dtype = getattr(xp, dtype) + + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) * 100 + tmp1 = ndimage.gaussian_filter(array, 1.0, [2, 0]) + tmp2 = ndimage.gaussian_filter(array, 1.0, [0, 2]) + output = ndimage.gaussian_laplace(array, 1.0) + assert_array_almost_equal(tmp1 + tmp2, output) + + @skip_xp_backends("jax.numpy", reason="output array is read-only") + @pytest.mark.parametrize('dtype', + ["int32", "float32", "float64", + "complex64", "complex128"]) + def test_gaussian_laplace02(self, dtype, xp): + dtype = getattr(xp, dtype) + + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) * 100 + tmp1 = ndimage.gaussian_filter(array, 1.0, [2, 0]) + tmp2 = ndimage.gaussian_filter(array, 1.0, [0, 2]) + output = xp.zeros(array.shape, dtype=dtype) + ndimage.gaussian_laplace(array, 1.0, output) + assert_array_almost_equal(tmp1 + tmp2, output) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + @pytest.mark.parametrize('dtype', types + complex_types) + def test_generic_laplace01(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + def derivative2(input, axis, output, mode, cval, a, b): + sigma = np.asarray([a, b / 2.0]) + order = [0] * input.ndim + order[axis] = 2 + return ndimage.gaussian_filter(input, sigma, order, + output, mode, cval) + + dtype = getattr(xp, dtype) + + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + output = xp.zeros(array.shape, dtype=dtype) + tmp = ndimage.generic_laplace(array, derivative2, + extra_arguments=(1.0,), + extra_keywords={'b': 2.0}) + ndimage.gaussian_laplace(array, 1.0, output) + assert_array_almost_equal(tmp, output) + + @skip_xp_backends("jax.numpy", reason="output array is read-only") + @pytest.mark.parametrize('dtype', + ["int32", "float32", "float64", + "complex64", "complex128"]) + def test_gaussian_gradient_magnitude01(self, dtype, xp): + is_int_dtype = dtype == "int32" + dtype = getattr(xp, dtype) + + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) * 100 + tmp1 = ndimage.gaussian_filter(array, 1.0, [1, 0]) + tmp2 = ndimage.gaussian_filter(array, 1.0, [0, 1]) + output = ndimage.gaussian_gradient_magnitude(array, 1.0) + expected = tmp1 * tmp1 + tmp2 * tmp2 + + astype = array_namespace(expected).astype + expected_float = astype(expected, xp.float64) if is_int_dtype else expected + expected = astype(xp.sqrt(expected_float), dtype) + xp_assert_close(output, expected, rtol=1e-6, atol=1e-6) + + @skip_xp_backends("jax.numpy", reason="output array is read-only") + @pytest.mark.parametrize('dtype', + ["int32", "float32", "float64", + "complex64", "complex128"]) + def test_gaussian_gradient_magnitude02(self, dtype, xp): + is_int_dtype = dtype == 'int32' + dtype = getattr(xp, dtype) + + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) * 100 + tmp1 = ndimage.gaussian_filter(array, 1.0, [1, 0]) + tmp2 = ndimage.gaussian_filter(array, 1.0, [0, 1]) + output = xp.zeros(array.shape, dtype=dtype) + ndimage.gaussian_gradient_magnitude(array, 1.0, output) + expected = tmp1 * tmp1 + tmp2 * tmp2 + + astype = array_namespace(expected).astype + fl_expected = astype(expected, xp.float64) if is_int_dtype else expected + + expected = astype(xp.sqrt(fl_expected), dtype) + xp_assert_close(output, expected, rtol=1e-6, atol=1e-6) + + def test_generic_gradient_magnitude01(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=xp.float64) + + def derivative(input, axis, output, mode, cval, a, b): + sigma = [a, b / 2.0] + order = [0] * input.ndim + order[axis] = 1 + return ndimage.gaussian_filter(input, sigma, order, output, mode, cval) + + tmp1 = ndimage.gaussian_gradient_magnitude(array, 1.0) + tmp2 = ndimage.generic_gradient_magnitude( + array, derivative, extra_arguments=(1.0,), + extra_keywords={'b': 2.0}) + assert_array_almost_equal(tmp1, tmp2) + + @skip_xp_backends("cupy", + reason="https://github.com/cupy/cupy/pull/8430", + ) + def test_uniform01(self, xp): + array = xp.asarray([2, 4, 6]) + size = 2 + output = ndimage.uniform_filter1d(array, size, origin=-1) + assert_array_almost_equal(xp.asarray([3, 5, 6]), output) + + @skip_xp_backends("cupy", + reason="https://github.com/cupy/cupy/pull/8430", + ) + def test_uniform01_complex(self, xp): + array = xp.asarray([2 + 1j, 4 + 2j, 6 + 3j], dtype=xp.complex128) + size = 2 + output = ndimage.uniform_filter1d(array, size, origin=-1) + assert_array_almost_equal(xp.real(output), xp.asarray([3., 5, 6])) + assert_array_almost_equal(xp.imag(output), xp.asarray([1.5, 2.5, 3])) + + def test_uniform02(self, xp): + array = xp.asarray([1, 2, 3]) + filter_shape = [0] + output = ndimage.uniform_filter(array, filter_shape) + assert_array_almost_equal(array, output) + + def test_uniform03(self, xp): + array = xp.asarray([1, 2, 3]) + filter_shape = [1] + output = ndimage.uniform_filter(array, filter_shape) + assert_array_almost_equal(array, output) + + @skip_xp_backends("cupy", + reason="https://github.com/cupy/cupy/pull/8430", + ) + def test_uniform04(self, xp): + array = xp.asarray([2, 4, 6]) + filter_shape = [2] + output = ndimage.uniform_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([2, 3, 5]), output) + + def test_uniform05(self, xp): + array = xp.asarray([]) + filter_shape = [1] + output = ndimage.uniform_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([]), output) + + @skip_xp_backends("cupy", + reason="https://github.com/cupy/cupy/pull/8430", + ) + @pytest.mark.parametrize('dtype_array', types) + @pytest.mark.parametrize('dtype_output', types) + def test_uniform06(self, dtype_array, dtype_output, xp): + if not (is_numpy(xp) or is_cupy(xp)): + pytest.xfail("output=dtype is numpy-specific") + + dtype_array = getattr(xp, dtype_array) + dtype_output = getattr(xp, dtype_output) + + filter_shape = [2, 2] + array = xp.asarray([[4, 8, 12], + [16, 20, 24]], dtype=dtype_array) + output = ndimage.uniform_filter( + array, filter_shape, output=dtype_output) + assert_array_almost_equal(xp.asarray([[4, 6, 10], [10, 12, 16]]), output) + assert output.dtype.type == dtype_output + + @skip_xp_backends("cupy", + reason="https://github.com/cupy/cupy/pull/8430", + ) + @pytest.mark.parametrize('dtype_array', complex_types) + @pytest.mark.parametrize('dtype_output', complex_types) + def test_uniform06_complex(self, dtype_array, dtype_output, xp): + if not (is_numpy(xp) or is_cupy(xp)): + pytest.xfail("output=dtype is numpy-specific") + + dtype_array = getattr(xp, dtype_array) + dtype_output = getattr(xp, dtype_output) + + filter_shape = [2, 2] + array = xp.asarray([[4, 8 + 5j, 12], + [16, 20, 24]], dtype=dtype_array) + output = ndimage.uniform_filter( + array, filter_shape, output=dtype_output) + assert_array_almost_equal(xp.asarray([[4, 6, 10], [10, 12, 16]]), output.real) + assert output.dtype.type == dtype_output + + def test_minimum_filter01(self, xp): + array = xp.asarray([1, 2, 3, 4, 5]) + filter_shape = xp.asarray([2]) + output = ndimage.minimum_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([1, 1, 2, 3, 4]), output) + + def test_minimum_filter02(self, xp): + array = xp.asarray([1, 2, 3, 4, 5]) + filter_shape = xp.asarray([3]) + output = ndimage.minimum_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([1, 1, 2, 3, 4]), output) + + def test_minimum_filter03(self, xp): + array = xp.asarray([3, 2, 5, 1, 4]) + filter_shape = xp.asarray([2]) + output = ndimage.minimum_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([3, 2, 2, 1, 1]), output) + + def test_minimum_filter04(self, xp): + array = xp.asarray([3, 2, 5, 1, 4]) + filter_shape = xp.asarray([3]) + output = ndimage.minimum_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([2, 2, 1, 1, 1]), output) + + def test_minimum_filter05(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + filter_shape = xp.asarray([2, 3]) + output = ndimage.minimum_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([[2, 2, 1, 1, 1], + [2, 2, 1, 1, 1], + [5, 3, 3, 1, 1]]), output) + + @skip_xp_backends("jax.numpy", reason="assignment destination is read-only") + def test_minimum_filter05_overlap(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + filter_shape = xp.asarray([2, 3]) + ndimage.minimum_filter(array, filter_shape, output=array) + assert_array_almost_equal(xp.asarray([[2, 2, 1, 1, 1], + [2, 2, 1, 1, 1], + [5, 3, 3, 1, 1]]), array) + + def test_minimum_filter06(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 1, 1], [1, 1, 1]]) + output = ndimage.minimum_filter(array, footprint=footprint) + assert_array_almost_equal(xp.asarray([[2, 2, 1, 1, 1], + [2, 2, 1, 1, 1], + [5, 3, 3, 1, 1]]), output) + # separable footprint should allow mode sequence + output2 = ndimage.minimum_filter(array, footprint=footprint, + mode=['reflect', 'reflect']) + assert_array_almost_equal(output2, output) + + def test_minimum_filter07(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + output = ndimage.minimum_filter(array, footprint=footprint) + assert_array_almost_equal(xp.asarray([[2, 2, 1, 1, 1], + [2, 3, 1, 3, 1], + [5, 5, 3, 3, 1]]), output) + with assert_raises(RuntimeError): + ndimage.minimum_filter(array, footprint=footprint, + mode=['reflect', 'constant']) + + def test_minimum_filter08(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + output = ndimage.minimum_filter(array, footprint=footprint, origin=-1) + assert_array_almost_equal(xp.asarray([[3, 1, 3, 1, 1], + [5, 3, 3, 1, 1], + [3, 3, 1, 1, 1]]), output) + + def test_minimum_filter09(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + output = ndimage.minimum_filter(array, footprint=footprint, + origin=[-1, 0]) + assert_array_almost_equal(xp.asarray([[2, 3, 1, 3, 1], + [5, 5, 3, 3, 1], + [5, 3, 3, 1, 1]]), output) + + def test_maximum_filter01(self, xp): + array = xp.asarray([1, 2, 3, 4, 5]) + filter_shape = xp.asarray([2]) + output = ndimage.maximum_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([1, 2, 3, 4, 5]), output) + + def test_maximum_filter02(self, xp): + array = xp.asarray([1, 2, 3, 4, 5]) + filter_shape = xp.asarray([3]) + output = ndimage.maximum_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([2, 3, 4, 5, 5]), output) + + def test_maximum_filter03(self, xp): + array = xp.asarray([3, 2, 5, 1, 4]) + filter_shape = xp.asarray([2]) + output = ndimage.maximum_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([3, 3, 5, 5, 4]), output) + + def test_maximum_filter04(self, xp): + array = xp.asarray([3, 2, 5, 1, 4]) + filter_shape = xp.asarray([3]) + output = ndimage.maximum_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([3, 5, 5, 5, 4]), output) + + def test_maximum_filter05(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + filter_shape = xp.asarray([2, 3]) + output = ndimage.maximum_filter(array, filter_shape) + assert_array_almost_equal(xp.asarray([[3, 5, 5, 5, 4], + [7, 9, 9, 9, 5], + [8, 9, 9, 9, 7]]), output) + + def test_maximum_filter06(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 1, 1], [1, 1, 1]]) + output = ndimage.maximum_filter(array, footprint=footprint) + assert_array_almost_equal(xp.asarray([[3, 5, 5, 5, 4], + [7, 9, 9, 9, 5], + [8, 9, 9, 9, 7]]), output) + # separable footprint should allow mode sequence + output2 = ndimage.maximum_filter(array, footprint=footprint, + mode=['reflect', 'reflect']) + assert_array_almost_equal(output2, output) + + def test_maximum_filter07(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + output = ndimage.maximum_filter(array, footprint=footprint) + assert_array_almost_equal(xp.asarray([[3, 5, 5, 5, 4], + [7, 7, 9, 9, 5], + [7, 9, 8, 9, 7]]), output) + # non-separable footprint should not allow mode sequence + with assert_raises(RuntimeError): + ndimage.maximum_filter(array, footprint=footprint, + mode=['reflect', 'reflect']) + + def test_maximum_filter08(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + output = ndimage.maximum_filter(array, footprint=footprint, origin=-1) + assert_array_almost_equal(xp.asarray([[7, 9, 9, 5, 5], + [9, 8, 9, 7, 5], + [8, 8, 7, 7, 7]]), output) + + def test_maximum_filter09(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + output = ndimage.maximum_filter(array, footprint=footprint, + origin=[-1, 0]) + assert_array_almost_equal(xp.asarray([[7, 7, 9, 9, 5], + [7, 9, 8, 9, 7], + [8, 8, 8, 7, 7]]), output) + + @pytest.mark.parametrize( + 'axes', tuple(itertools.combinations(range(-3, 3), 2)) + ) + @pytest.mark.parametrize( + 'filter_func, kwargs', + [(ndimage.minimum_filter, {}), + (ndimage.maximum_filter, {}), + (ndimage.median_filter, {}), + (ndimage.rank_filter, dict(rank=3)), + (ndimage.percentile_filter, dict(percentile=60))] + ) + def test_minmax_nonseparable_axes(self, filter_func, axes, kwargs, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/pull/8339") + + array = xp.arange(6 * 8 * 12, dtype=xp.float32) + array = xp.reshape(array, (6, 8, 12)) + # use 2D triangular footprint because it is non-separable + footprint = xp.asarray(np.tri(5)) + axes = np.asarray(axes) + + if len(set(axes % array.ndim)) != len(axes): + # parametrized cases with duplicate axes raise an error + with pytest.raises(ValueError): + filter_func(array, footprint=footprint, axes=axes, **kwargs) + return + output = filter_func(array, footprint=footprint, axes=axes, **kwargs) + + missing_axis = tuple(set(range(3)) - set(axes % array.ndim))[0] + + expand_dims = array_namespace(footprint).expand_dims + footprint_3d = expand_dims(footprint, axis=missing_axis) + expected = filter_func(array, footprint=footprint_3d, **kwargs) + xp_assert_close(output, expected) + + def test_rank01(self, xp): + array = xp.asarray([1, 2, 3, 4, 5]) + output = ndimage.rank_filter(array, 1, size=2) + xp_assert_equal(array, output) + output = ndimage.percentile_filter(array, 100, size=2) + xp_assert_equal(array, output) + output = ndimage.median_filter(array, 2) + xp_assert_equal(array, output) + + def test_rank02(self, xp): + array = xp.asarray([1, 2, 3, 4, 5]) + output = ndimage.rank_filter(array, 1, size=[3]) + xp_assert_equal(array, output) + output = ndimage.percentile_filter(array, 50, size=3) + xp_assert_equal(array, output) + output = ndimage.median_filter(array, (3,)) + xp_assert_equal(array, output) + + def test_rank03(self, xp): + array = xp.asarray([3, 2, 5, 1, 4]) + output = ndimage.rank_filter(array, 1, size=[2]) + xp_assert_equal(xp.asarray([3, 3, 5, 5, 4]), output) + output = ndimage.percentile_filter(array, 100, size=2) + xp_assert_equal(xp.asarray([3, 3, 5, 5, 4]), output) + + def test_rank04(self, xp): + array = xp.asarray([3, 2, 5, 1, 4]) + expected = xp.asarray([3, 3, 2, 4, 4]) + output = ndimage.rank_filter(array, 1, size=3) + xp_assert_equal(expected, output) + output = ndimage.percentile_filter(array, 50, size=3) + xp_assert_equal(expected, output) + output = ndimage.median_filter(array, size=3) + xp_assert_equal(expected, output) + + def test_rank05(self, xp): + array = xp.asarray([3, 2, 5, 1, 4]) + expected = xp.asarray([3, 3, 2, 4, 4]) + output = ndimage.rank_filter(array, -2, size=3) + xp_assert_equal(expected, output) + + def test_rank06(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]]) + expected = [[2, 2, 1, 1, 1], + [3, 3, 2, 1, 1], + [5, 5, 3, 3, 1]] + expected = xp.asarray(expected) + output = ndimage.rank_filter(array, 1, size=[2, 3]) + xp_assert_equal(expected, output) + output = ndimage.percentile_filter(array, 17, size=(2, 3)) + xp_assert_equal(expected, output) + + @skip_xp_backends("jax.numpy", + reason="assignment destination is read-only", + ) + def test_rank06_overlap(self, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/issues/8406") + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]]) + + asarray = array_namespace(array).asarray + array_copy = asarray(array, copy=True) + expected = [[2, 2, 1, 1, 1], + [3, 3, 2, 1, 1], + [5, 5, 3, 3, 1]] + expected = xp.asarray(expected) + ndimage.rank_filter(array, 1, size=[2, 3], output=array) + xp_assert_equal(expected, array) + + ndimage.percentile_filter(array_copy, 17, size=(2, 3), + output=array_copy) + xp_assert_equal(expected, array_copy) + + def test_rank07(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]]) + expected = [[3, 5, 5, 5, 4], + [5, 5, 7, 5, 4], + [6, 8, 8, 7, 5]] + expected = xp.asarray(expected) + output = ndimage.rank_filter(array, -2, size=[2, 3]) + xp_assert_equal(expected, output) + + def test_rank08(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]]) + expected = [[3, 3, 2, 4, 4], + [5, 5, 5, 4, 4], + [5, 6, 7, 5, 5]] + expected = xp.asarray(expected) + output = ndimage.percentile_filter(array, 50.0, size=(2, 3)) + xp_assert_equal(expected, output) + output = ndimage.rank_filter(array, 3, size=(2, 3)) + xp_assert_equal(expected, output) + output = ndimage.median_filter(array, size=(2, 3)) + xp_assert_equal(expected, output) + + # non-separable: does not allow mode sequence + with assert_raises(RuntimeError): + ndimage.percentile_filter(array, 50.0, size=(2, 3), + mode=['reflect', 'constant']) + with assert_raises(RuntimeError): + ndimage.rank_filter(array, 3, size=(2, 3), mode=['reflect']*2) + with assert_raises(RuntimeError): + ndimage.median_filter(array, size=(2, 3), mode=['reflect']*2) + + @pytest.mark.parametrize('dtype', types) + def test_rank09(self, dtype, xp): + dtype = getattr(xp, dtype) + expected = [[3, 3, 2, 4, 4], + [3, 5, 2, 5, 1], + [5, 5, 8, 3, 5]] + expected = xp.asarray(expected) + footprint = xp.asarray([[1, 0, 1], [0, 1, 0]]) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + output = ndimage.rank_filter(array, 1, footprint=footprint) + assert_array_almost_equal(expected, output) + output = ndimage.percentile_filter(array, 35, footprint=footprint) + assert_array_almost_equal(expected, output) + + def test_rank10(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + expected = [[2, 2, 1, 1, 1], + [2, 3, 1, 3, 1], + [5, 5, 3, 3, 1]] + expected = xp.asarray(expected) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + output = ndimage.rank_filter(array, 0, footprint=footprint) + xp_assert_equal(expected, output) + output = ndimage.percentile_filter(array, 0.0, footprint=footprint) + xp_assert_equal(expected, output) + + def test_rank11(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + expected = [[3, 5, 5, 5, 4], + [7, 7, 9, 9, 5], + [7, 9, 8, 9, 7]] + expected = xp.asarray(expected) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + output = ndimage.rank_filter(array, -1, footprint=footprint) + xp_assert_equal(expected, output) + output = ndimage.percentile_filter(array, 100.0, footprint=footprint) + xp_assert_equal(expected, output) + + @pytest.mark.parametrize('dtype', types) + def test_rank12(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + expected = [[3, 3, 2, 4, 4], + [3, 5, 2, 5, 1], + [5, 5, 8, 3, 5]] + expected = xp.asarray(expected, dtype=dtype) + footprint = xp.asarray([[1, 0, 1], [0, 1, 0]]) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + output = ndimage.rank_filter(array, 1, footprint=footprint) + assert_array_almost_equal(expected, output) + output = ndimage.percentile_filter(array, 50.0, + footprint=footprint) + xp_assert_equal(expected, output) + output = ndimage.median_filter(array, footprint=footprint) + xp_assert_equal(expected, output) + + @pytest.mark.parametrize('dtype', types) + def test_rank13(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + expected = [[5, 2, 5, 1, 1], + [5, 8, 3, 5, 5], + [6, 6, 5, 5, 5]] + expected = xp.asarray(expected, dtype=dtype) + footprint = xp.asarray([[1, 0, 1], [0, 1, 0]]) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + output = ndimage.rank_filter(array, 1, footprint=footprint, + origin=-1) + xp_assert_equal(expected, output) + + @pytest.mark.parametrize('dtype', types) + def test_rank14(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + expected = [[3, 5, 2, 5, 1], + [5, 5, 8, 3, 5], + [5, 6, 6, 5, 5]] + expected = xp.asarray(expected, dtype=dtype) + footprint = xp.asarray([[1, 0, 1], [0, 1, 0]]) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + output = ndimage.rank_filter(array, 1, footprint=footprint, + origin=[-1, 0]) + xp_assert_equal(expected, output) + + @pytest.mark.parametrize('dtype', types) + def test_rank15(self, dtype, xp): + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, dtype) + expected = [[2, 3, 1, 4, 1], + [5, 3, 7, 1, 1], + [5, 5, 3, 3, 3]] + expected = xp.asarray(expected, dtype=dtype) + footprint = xp.asarray([[1, 0, 1], [0, 1, 0]]) + array = xp.asarray([[3, 2, 5, 1, 4], + [5, 8, 3, 7, 1], + [5, 6, 9, 3, 5]], dtype=dtype) + output = ndimage.rank_filter(array, 0, footprint=footprint, + origin=[-1, 0]) + xp_assert_equal(expected, output) + + def test_rank16(self, xp): + # test that lists are accepted and interpreted as numpy arrays + array = [3, 2, 5, 1, 4] + # expected values are: median(3, 2, 5) = 3, median(2, 5, 1) = 2, etc + expected = np.asarray([3, 3, 2, 4, 4]) + output = ndimage.rank_filter(array, -2, size=3) + xp_assert_equal(expected, output) + + def test_rank17(self, xp): + array = xp.asarray([3, 2, 5, 1, 4]) + if not hasattr(array, 'flags'): + return + array.flags.writeable = False + expected = xp.asarray([3, 3, 2, 4, 4]) + output = ndimage.rank_filter(array, -2, size=3) + xp_assert_equal(expected, output) + + def test_rank18(self, xp): + # module 'array_api_strict' has no attribute 'float16' + tested_dtypes = ['int8', 'int16', 'int32', 'int64', 'float32', 'float64', + 'uint8', 'uint16', 'uint32', 'uint64'] + for dtype_str in tested_dtypes: + dtype = getattr(xp, dtype_str) + x = xp.asarray([3, 2, 5, 1, 4], dtype=dtype) + y = ndimage.rank_filter(x, -2, size=3) + assert y.dtype == x.dtype + + def test_rank19(self, xp): + # module 'array_api_strict' has no attribute 'float16' + tested_dtypes = ['int8', 'int16', 'int32', 'int64', 'float32', 'float64', + 'uint8', 'uint16', 'uint32', 'uint64'] + for dtype_str in tested_dtypes: + dtype = getattr(xp, dtype_str) + x = xp.asarray([[3, 2, 5, 1, 4], [3, 2, 5, 1, 4]], dtype=dtype) + y = ndimage.rank_filter(x, -2, size=3) + assert y.dtype == x.dtype + + @skip_xp_backends(np_only=True, reason="off-by-ones on alt backends") + @pytest.mark.parametrize('dtype', types) + def test_generic_filter1d01(self, dtype, xp): + weights = xp.asarray([1.1, 2.2, 3.3]) + + if is_cupy(xp): + pytest.xfail("CuPy does not support extra_arguments") + + def _filter_func(input, output, fltr, total): + fltr = fltr / total + for ii in range(input.shape[0] - 2): + output[ii] = input[ii] * fltr[0] + output[ii] += input[ii + 1] * fltr[1] + output[ii] += input[ii + 2] * fltr[2] + a = np.arange(12, dtype=dtype).reshape(3, 4) + a = xp.asarray(a) + dtype = getattr(xp, dtype) + + r1 = ndimage.correlate1d(a, weights / xp.sum(weights), 0, origin=-1) + r2 = ndimage.generic_filter1d( + a, _filter_func, 3, axis=0, origin=-1, + extra_arguments=(weights,), + extra_keywords={'total': xp.sum(weights)}) + assert_array_almost_equal(r1, r2) + + @pytest.mark.parametrize('dtype', types) + def test_generic_filter01(self, dtype, xp): + if is_cupy(xp): + pytest.xfail("CuPy does not support extra_arguments") + if is_torch(xp) and dtype in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype_str = dtype + dtype = getattr(xp, dtype_str) + + filter_ = xp.asarray([[1.0, 2.0], [3.0, 4.0]]) + footprint = xp.asarray([[1.0, 0.0], [0.0, 1.0]]) + cf = xp.asarray([1., 4.]) + + def _filter_func(buffer, weights, total=1.0): + weights = np.asarray(cf) / np.asarray(total) + return np.sum(buffer * weights) + + a = np.arange(12, dtype=dtype_str).reshape(3, 4) + a = xp.asarray(a) + r1 = ndimage.correlate(a, filter_ * footprint) + if dtype_str in float_types: + r1 /= 5 + else: + r1 //= 5 + r2 = ndimage.generic_filter( + a, _filter_func, footprint=footprint, extra_arguments=(cf,), + extra_keywords={'total': xp.sum(cf)}) + assert_array_almost_equal(r1, r2) + + # generic_filter doesn't allow mode sequence + with assert_raises(RuntimeError): + r2 = ndimage.generic_filter( + a, _filter_func, mode=['reflect', 'reflect'], + footprint=footprint, extra_arguments=(cf,), + extra_keywords={'total': xp.sum(cf)}) + + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [1, 1, 2]), + ('wrap', [3, 1, 2]), + ('reflect', [1, 1, 2]), + ('mirror', [2, 1, 2]), + ('constant', [0, 1, 2])] + ) + def test_extend01(self, mode, expected_value, xp): + array = xp.asarray([1, 2, 3]) + weights = xp.asarray([1, 0]) + output = ndimage.correlate1d(array, weights, 0, mode=mode, cval=0) + expected_value = xp.asarray(expected_value) + xp_assert_equal(output, expected_value) + + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [1, 1, 1]), + ('wrap', [3, 1, 2]), + ('reflect', [3, 3, 2]), + ('mirror', [1, 2, 3]), + ('constant', [0, 0, 0])] + ) + def test_extend02(self, mode, expected_value, xp): + array = xp.asarray([1, 2, 3]) + weights = xp.asarray([1, 0, 0, 0, 0, 0, 0, 0]) + output = ndimage.correlate1d(array, weights, 0, mode=mode, cval=0) + expected_value = xp.asarray(expected_value) + xp_assert_equal(output, expected_value) + + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [2, 3, 3]), + ('wrap', [2, 3, 1]), + ('reflect', [2, 3, 3]), + ('mirror', [2, 3, 2]), + ('constant', [2, 3, 0])] + ) + def test_extend03(self, mode, expected_value, xp): + array = xp.asarray([1, 2, 3]) + weights = xp.asarray([0, 0, 1]) + output = ndimage.correlate1d(array, weights, 0, mode=mode, cval=0) + expected_value = xp.asarray(expected_value) + xp_assert_equal(output, expected_value) + + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [3, 3, 3]), + ('wrap', [2, 3, 1]), + ('reflect', [2, 1, 1]), + ('mirror', [1, 2, 3]), + ('constant', [0, 0, 0])] + ) + def test_extend04(self, mode, expected_value, xp): + array = xp.asarray([1, 2, 3]) + weights = xp.asarray([0, 0, 0, 0, 0, 0, 0, 0, 1]) + output = ndimage.correlate1d(array, weights, 0, mode=mode, cval=0) + expected_value = xp.asarray(expected_value) + xp_assert_equal(output, expected_value) + + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [[1, 1, 2], [1, 1, 2], [4, 4, 5]]), + ('wrap', [[9, 7, 8], [3, 1, 2], [6, 4, 5]]), + ('reflect', [[1, 1, 2], [1, 1, 2], [4, 4, 5]]), + ('mirror', [[5, 4, 5], [2, 1, 2], [5, 4, 5]]), + ('constant', [[0, 0, 0], [0, 1, 2], [0, 4, 5]])] + ) + def test_extend05(self, mode, expected_value, xp): + array = xp.asarray([[1, 2, 3], + [4, 5, 6], + [7, 8, 9]]) + weights = xp.asarray([[1, 0], [0, 0]]) + output = ndimage.correlate(array, weights, mode=mode, cval=0) + expected_value = xp.asarray(expected_value) + xp_assert_equal(output, expected_value) + + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [[5, 6, 6], [8, 9, 9], [8, 9, 9]]), + ('wrap', [[5, 6, 4], [8, 9, 7], [2, 3, 1]]), + ('reflect', [[5, 6, 6], [8, 9, 9], [8, 9, 9]]), + ('mirror', [[5, 6, 5], [8, 9, 8], [5, 6, 5]]), + ('constant', [[5, 6, 0], [8, 9, 0], [0, 0, 0]])] + ) + def test_extend06(self, mode, expected_value, xp): + array = xp.asarray([[1, 2, 3], + [4, 5, 6], + [7, 8, 9]]) + weights = xp.asarray([[0, 0, 0], [0, 0, 0], [0, 0, 1]]) + output = ndimage.correlate(array, weights, mode=mode, cval=0) + expected_value = xp.asarray(expected_value) + xp_assert_equal(output, expected_value) + + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [3, 3, 3]), + ('wrap', [2, 3, 1]), + ('reflect', [2, 1, 1]), + ('mirror', [1, 2, 3]), + ('constant', [0, 0, 0])] + ) + def test_extend07(self, mode, expected_value, xp): + array = xp.asarray([1, 2, 3]) + weights = xp.asarray([0, 0, 0, 0, 0, 0, 0, 0, 1]) + output = ndimage.correlate(array, weights, mode=mode, cval=0) + expected_value = xp.asarray(expected_value) + xp_assert_equal(output, expected_value) + + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [[3], [3], [3]]), + ('wrap', [[2], [3], [1]]), + ('reflect', [[2], [1], [1]]), + ('mirror', [[1], [2], [3]]), + ('constant', [[0], [0], [0]])] + ) + def test_extend08(self, mode, expected_value, xp): + array = xp.asarray([[1], [2], [3]]) + weights = xp.asarray([[0], [0], [0], [0], [0], [0], [0], [0], [1]]) + output = ndimage.correlate(array, weights, mode=mode, cval=0) + expected_value = xp.asarray(expected_value) + xp_assert_equal(output, expected_value) + + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [3, 3, 3]), + ('wrap', [2, 3, 1]), + ('reflect', [2, 1, 1]), + ('mirror', [1, 2, 3]), + ('constant', [0, 0, 0])] + ) + def test_extend09(self, mode, expected_value, xp): + array = xp.asarray([1, 2, 3]) + weights = xp.asarray([0, 0, 0, 0, 0, 0, 0, 0, 1]) + output = ndimage.correlate(array, weights, mode=mode, cval=0) + expected_value = xp.asarray(expected_value) + xp_assert_equal(output, expected_value) + + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [[3], [3], [3]]), + ('wrap', [[2], [3], [1]]), + ('reflect', [[2], [1], [1]]), + ('mirror', [[1], [2], [3]]), + ('constant', [[0], [0], [0]])] + ) + def test_extend10(self, mode, expected_value, xp): + array = xp.asarray([[1], [2], [3]]) + weights = xp.asarray([[0], [0], [0], [0], [0], [0], [0], [0], [1]]) + output = ndimage.correlate(array, weights, mode=mode, cval=0) + expected_value = xp.asarray(expected_value) + xp_assert_equal(output, expected_value) + + +def test_ticket_701(xp): + if is_cupy(xp): + pytest.xfail("CuPy raises a TypeError.") + + # Test generic filter sizes + arr = xp.asarray(np.arange(4).reshape(2, 2)) + def func(x): + return np.min(x) # NB: np.min not xp.min for callables + res = ndimage.generic_filter(arr, func, size=(1, 1)) + # The following raises an error unless ticket 701 is fixed + res2 = ndimage.generic_filter(arr, func, size=1) + xp_assert_equal(res, res2) + + +def test_gh_5430(): + # At least one of these raises an error unless gh-5430 is + # fixed. In py2k an int is implemented using a C long, so + # which one fails depends on your system. In py3k there is only + # one arbitrary precision integer type, so both should fail. + sigma = np.int32(1) + out = ndimage._ni_support._normalize_sequence(sigma, 1) + assert out == [sigma] + sigma = np.int64(1) + out = ndimage._ni_support._normalize_sequence(sigma, 1) + assert out == [sigma] + # This worked before; make sure it still works + sigma = 1 + out = ndimage._ni_support._normalize_sequence(sigma, 1) + assert out == [sigma] + # This worked before; make sure it still works + sigma = [1, 1] + out = ndimage._ni_support._normalize_sequence(sigma, 2) + assert out == sigma + # Also include the OPs original example to make sure we fixed the issue + x = np.random.normal(size=(256, 256)) + perlin = np.zeros_like(x) + for i in 2**np.arange(6): + perlin += ndimage.gaussian_filter(x, i, mode="wrap") * i**2 + # This also fixes gh-4106, show that the OPs example now runs. + x = np.int64(21) + ndimage._ni_support._normalize_sequence(x, 0) + + +def test_gaussian_kernel1d(xp): + if is_cupy(xp): + pytest.skip("This test tests a private scipy utility.") + radius = 10 + sigma = 2 + sigma2 = sigma * sigma + x = np.arange(-radius, radius + 1, dtype=np.float64) + x = xp.asarray(x) + phi_x = xp.exp(-0.5 * x * x / sigma2) + phi_x /= xp.sum(phi_x) + xp_assert_close(phi_x, + xp.asarray(_gaussian_kernel1d(sigma, 0, radius))) + xp_assert_close(-phi_x * x / sigma2, + xp.asarray(_gaussian_kernel1d(sigma, 1, radius))) + xp_assert_close(phi_x * (x * x / sigma2 - 1) / sigma2, + xp.asarray(_gaussian_kernel1d(sigma, 2, radius))) + xp_assert_close(phi_x * (3 - x * x / sigma2) * x / (sigma2 * sigma2), + xp.asarray(_gaussian_kernel1d(sigma, 3, radius))) + + +def test_orders_gauss(xp): + # Check order inputs to Gaussians + arr = xp.zeros((1,)) + xp_assert_equal(ndimage.gaussian_filter(arr, 1, order=0), xp.asarray([0.])) + xp_assert_equal(ndimage.gaussian_filter(arr, 1, order=3), xp.asarray([0.])) + assert_raises(ValueError, ndimage.gaussian_filter, arr, 1, -1) + xp_assert_equal(ndimage.gaussian_filter1d(arr, 1, axis=-1, order=0), + xp.asarray([0.])) + xp_assert_equal(ndimage.gaussian_filter1d(arr, 1, axis=-1, order=3), + xp.asarray([0.])) + assert_raises(ValueError, ndimage.gaussian_filter1d, arr, 1, -1, -1) + + +def test_valid_origins(xp): + """Regression test for #1311.""" + if is_cupy(xp): + pytest.xfail("CuPy raises a TypeError.") + + def func(x): + return xp.mean(x) + data = xp.asarray([1, 2, 3, 4, 5], dtype=xp.float64) + assert_raises(ValueError, ndimage.generic_filter, data, func, size=3, + origin=2) + assert_raises(ValueError, ndimage.generic_filter1d, data, func, + filter_size=3, origin=2) + assert_raises(ValueError, ndimage.percentile_filter, data, 0.2, size=3, + origin=2) + + for filter in [ndimage.uniform_filter, ndimage.minimum_filter, + ndimage.maximum_filter, ndimage.maximum_filter1d, + ndimage.median_filter, ndimage.minimum_filter1d]: + # This should work, since for size == 3, the valid range for origin is + # -1 to 1. + list(filter(data, 3, origin=-1)) + list(filter(data, 3, origin=1)) + # Just check this raises an error instead of silently accepting or + # segfaulting. + assert_raises(ValueError, filter, data, 3, origin=2) + + +def test_bad_convolve_and_correlate_origins(xp): + """Regression test for gh-822.""" + # Before gh-822 was fixed, these would generate seg. faults or + # other crashes on many system. + assert_raises(ValueError, ndimage.correlate1d, + [0, 1, 2, 3, 4, 5], [1, 1, 2, 0], origin=2) + assert_raises(ValueError, ndimage.correlate, + [0, 1, 2, 3, 4, 5], [0, 1, 2], origin=[2]) + assert_raises(ValueError, ndimage.correlate, + xp.ones((3, 5)), xp.ones((2, 2)), origin=[0, 1]) + + assert_raises(ValueError, ndimage.convolve1d, + xp.arange(10), xp.ones(3), origin=-2) + assert_raises(ValueError, ndimage.convolve, + xp.arange(10), xp.ones(3), origin=[-2]) + assert_raises(ValueError, ndimage.convolve, + xp.ones((3, 5)), xp.ones((2, 2)), origin=[0, -2]) + +@skip_xp_backends("cupy", + reason="https://github.com/cupy/cupy/pull/8430", +) +def test_multiple_modes(xp): + # Test that the filters with multiple mode capabilities for different + # dimensions give the same result as applying a single mode. + arr = xp.asarray([[1., 0., 0.], + [1., 1., 0.], + [0., 0., 0.]]) + + mode1 = 'reflect' + mode2 = ['reflect', 'reflect'] + + xp_assert_equal(ndimage.gaussian_filter(arr, 1, mode=mode1), + ndimage.gaussian_filter(arr, 1, mode=mode2)) + xp_assert_equal(ndimage.prewitt(arr, mode=mode1), + ndimage.prewitt(arr, mode=mode2)) + xp_assert_equal(ndimage.sobel(arr, mode=mode1), + ndimage.sobel(arr, mode=mode2)) + xp_assert_equal(ndimage.laplace(arr, mode=mode1), + ndimage.laplace(arr, mode=mode2)) + xp_assert_equal(ndimage.gaussian_laplace(arr, 1, mode=mode1), + ndimage.gaussian_laplace(arr, 1, mode=mode2)) + xp_assert_equal(ndimage.maximum_filter(arr, size=5, mode=mode1), + ndimage.maximum_filter(arr, size=5, mode=mode2)) + xp_assert_equal(ndimage.minimum_filter(arr, size=5, mode=mode1), + ndimage.minimum_filter(arr, size=5, mode=mode2)) + xp_assert_equal(ndimage.gaussian_gradient_magnitude(arr, 1, mode=mode1), + ndimage.gaussian_gradient_magnitude(arr, 1, mode=mode2)) + xp_assert_equal(ndimage.uniform_filter(arr, 5, mode=mode1), + ndimage.uniform_filter(arr, 5, mode=mode2)) + + +@skip_xp_backends("cupy", reason="https://github.com/cupy/cupy/pull/8430") +@skip_xp_backends("jax.numpy", reason="output array is read-only.") +def test_multiple_modes_sequentially(xp): + # Test that the filters with multiple mode capabilities for different + # dimensions give the same result as applying the filters with + # different modes sequentially + arr = xp.asarray([[1., 0., 0.], + [1., 1., 0.], + [0., 0., 0.]]) + + modes = ['reflect', 'wrap'] + + expected = ndimage.gaussian_filter1d(arr, 1, axis=0, mode=modes[0]) + expected = ndimage.gaussian_filter1d(expected, 1, axis=1, mode=modes[1]) + xp_assert_equal(expected, + ndimage.gaussian_filter(arr, 1, mode=modes)) + + expected = ndimage.uniform_filter1d(arr, 5, axis=0, mode=modes[0]) + expected = ndimage.uniform_filter1d(expected, 5, axis=1, mode=modes[1]) + xp_assert_equal(expected, + ndimage.uniform_filter(arr, 5, mode=modes)) + + expected = ndimage.maximum_filter1d(arr, size=5, axis=0, mode=modes[0]) + expected = ndimage.maximum_filter1d(expected, size=5, axis=1, + mode=modes[1]) + xp_assert_equal(expected, + ndimage.maximum_filter(arr, size=5, mode=modes)) + + expected = ndimage.minimum_filter1d(arr, size=5, axis=0, mode=modes[0]) + expected = ndimage.minimum_filter1d(expected, size=5, axis=1, + mode=modes[1]) + xp_assert_equal(expected, + ndimage.minimum_filter(arr, size=5, mode=modes)) + + +def test_multiple_modes_prewitt(xp): + # Test prewitt filter for multiple extrapolation modes + arr = xp.asarray([[1., 0., 0.], + [1., 1., 0.], + [0., 0., 0.]]) + + expected = xp.asarray([[1., -3., 2.], + [1., -2., 1.], + [1., -1., 0.]]) + + modes = ['reflect', 'wrap'] + + xp_assert_equal(expected, + ndimage.prewitt(arr, mode=modes)) + + +def test_multiple_modes_sobel(xp): + # Test sobel filter for multiple extrapolation modes + arr = xp.asarray([[1., 0., 0.], + [1., 1., 0.], + [0., 0., 0.]]) + + expected = xp.asarray([[1., -4., 3.], + [2., -3., 1.], + [1., -1., 0.]]) + + modes = ['reflect', 'wrap'] + + xp_assert_equal(expected, + ndimage.sobel(arr, mode=modes)) + + +def test_multiple_modes_laplace(xp): + # Test laplace filter for multiple extrapolation modes + arr = xp.asarray([[1., 0., 0.], + [1., 1., 0.], + [0., 0., 0.]]) + + expected = xp.asarray([[-2., 2., 1.], + [-2., -3., 2.], + [1., 1., 0.]]) + + modes = ['reflect', 'wrap'] + + xp_assert_equal(expected, + ndimage.laplace(arr, mode=modes)) + + +def test_multiple_modes_gaussian_laplace(xp): + # Test gaussian_laplace filter for multiple extrapolation modes + arr = xp.asarray([[1., 0., 0.], + [1., 1., 0.], + [0., 0., 0.]]) + + expected = xp.asarray([[-0.28438687, 0.01559809, 0.19773499], + [-0.36630503, -0.20069774, 0.07483620], + [0.15849176, 0.18495566, 0.21934094]]) + + modes = ['reflect', 'wrap'] + + assert_almost_equal(expected, + ndimage.gaussian_laplace(arr, 1, mode=modes)) + + +def test_multiple_modes_gaussian_gradient_magnitude(xp): + # Test gaussian_gradient_magnitude filter for multiple + # extrapolation modes + arr = xp.asarray([[1., 0., 0.], + [1., 1., 0.], + [0., 0., 0.]]) + + expected = xp.asarray([[0.04928965, 0.09745625, 0.06405368], + [0.23056905, 0.14025305, 0.04550846], + [0.19894369, 0.14950060, 0.06796850]]) + + modes = ['reflect', 'wrap'] + + calculated = ndimage.gaussian_gradient_magnitude(arr, 1, mode=modes) + + assert_almost_equal(expected, calculated) + +@skip_xp_backends("cupy", + reason="https://github.com/cupy/cupy/pull/8430", +) +def test_multiple_modes_uniform(xp): + # Test uniform filter for multiple extrapolation modes + arr = xp.asarray([[1., 0., 0.], + [1., 1., 0.], + [0., 0., 0.]]) + + expected = xp.asarray([[0.32, 0.40, 0.48], + [0.20, 0.28, 0.32], + [0.28, 0.32, 0.40]]) + + modes = ['reflect', 'wrap'] + + assert_almost_equal(expected, + ndimage.uniform_filter(arr, 5, mode=modes)) + + +def _count_nonzero(arr): + # XXX: a simplified count_nonzero replacement; replace once + # https://github.com/data-apis/array-api/pull/803/ is in + + # this assumes arr.dtype == xp.bool + xp = array_namespace(arr) + return xp.sum(xp.astype(arr, xp.int8)) + + +def test_gaussian_truncate(xp): + # Test that Gaussian filters can be truncated at different widths. + # These tests only check that the result has the expected number + # of nonzero elements. + arr = np.zeros((100, 100), dtype=np.float64) + arr[50, 50] = 1 + arr = xp.asarray(arr) + num_nonzeros_2 = _count_nonzero(ndimage.gaussian_filter(arr, 5, truncate=2) > 0) + assert num_nonzeros_2 == 21**2 + + num_nonzeros_5 = _count_nonzero( + ndimage.gaussian_filter(arr, 5, truncate=5) > 0 + ) + assert num_nonzeros_5 == 51**2 + + nnz_kw = {'as_tuple': True} if is_torch(xp) else {} + + # Test truncate when sigma is a sequence. + f = ndimage.gaussian_filter(arr, [0.5, 2.5], truncate=3.5) + fpos = f > 0 + n0 = _count_nonzero(xp.any(fpos, axis=0)) + assert n0 == 19 + n1 = _count_nonzero(xp.any(fpos, axis=1)) + assert n1 == 5 + + # Test gaussian_filter1d. + x = np.zeros(51) + x[25] = 1 + x = xp.asarray(x) + f = ndimage.gaussian_filter1d(x, sigma=2, truncate=3.5) + n = _count_nonzero(f > 0) + assert n == 15 + + # Test gaussian_laplace + y = ndimage.gaussian_laplace(x, sigma=2, truncate=3.5) + nonzero_indices = xp.nonzero(y != 0, **nnz_kw)[0] + + n = xp.max(nonzero_indices) - xp.min(nonzero_indices) + 1 + assert n == 15 + + # Test gaussian_gradient_magnitude + y = ndimage.gaussian_gradient_magnitude(x, sigma=2, truncate=3.5) + nonzero_indices = xp.nonzero(y != 0, **nnz_kw)[0] + n = xp.max(nonzero_indices) - xp.min(nonzero_indices) + 1 + assert n == 15 + + +def test_gaussian_radius(xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/issues/8402") + + # Test that Gaussian filters with radius argument produce the same + # results as the filters with corresponding truncate argument. + # radius = int(truncate * sigma + 0.5) + # Test gaussian_filter1d + x = np.zeros(7) + x[3] = 1 + x = xp.asarray(x) + f1 = ndimage.gaussian_filter1d(x, sigma=2, truncate=1.5) + f2 = ndimage.gaussian_filter1d(x, sigma=2, radius=3) + xp_assert_equal(f1, f2) + + # Test gaussian_filter when sigma is a number. + a = np.zeros((9, 9)) + a[4, 4] = 1 + a = xp.asarray(a) + f1 = ndimage.gaussian_filter(a, sigma=0.5, truncate=3.5) + f2 = ndimage.gaussian_filter(a, sigma=0.5, radius=2) + xp_assert_equal(f1, f2) + + # Test gaussian_filter when sigma is a sequence. + a = np.zeros((50, 50)) + a[25, 25] = 1 + a = xp.asarray(a) + f1 = ndimage.gaussian_filter(a, sigma=[0.5, 2.5], truncate=3.5) + f2 = ndimage.gaussian_filter(a, sigma=[0.5, 2.5], radius=[2, 9]) + xp_assert_equal(f1, f2) + + +def test_gaussian_radius_invalid(xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/issues/8402") + + # radius must be a nonnegative integer + with assert_raises(ValueError): + ndimage.gaussian_filter1d(xp.zeros(8), sigma=1, radius=-1) + with assert_raises(ValueError): + ndimage.gaussian_filter1d(xp.zeros(8), sigma=1, radius=1.1) + + +@skip_xp_backends("jax.numpy", reason="output array is read-only") +class TestThreading: + def check_func_thread(self, n, fun, args, out): + from threading import Thread + thrds = [Thread(target=fun, args=args, kwargs={'output': out[x, ...]}) + for x in range(n)] + [t.start() for t in thrds] + [t.join() for t in thrds] + + def check_func_serial(self, n, fun, args, out): + for i in range(n): + fun(*args, output=out[i, ...]) + + def test_correlate1d(self, xp): + if is_cupy(xp): + pytest.xfail("XXX thread exception; cannot repro outside of pytest") + + d = np.random.randn(5000) + os = np.empty((4, d.size)) + ot = np.empty_like(os) + d = xp.asarray(d) + os = xp.asarray(os) + ot = xp.asarray(ot) + k = xp.arange(5) + self.check_func_serial(4, ndimage.correlate1d, (d, k), os) + self.check_func_thread(4, ndimage.correlate1d, (d, k), ot) + xp_assert_equal(os, ot) + + def test_correlate(self, xp): + if is_cupy(xp): + pytest.xfail("XXX thread exception; cannot repro outside of pytest") + + d = xp.asarray(np.random.randn(500, 500)) + k = xp.asarray(np.random.randn(10, 10)) + os = xp.empty([4] + list(d.shape)) + ot = xp.empty_like(os) + self.check_func_serial(4, ndimage.correlate, (d, k), os) + self.check_func_thread(4, ndimage.correlate, (d, k), ot) + xp_assert_equal(os, ot) + + def test_median_filter(self, xp): + if is_cupy(xp): + pytest.xfail("XXX thread exception; cannot repro outside of pytest") + + d = xp.asarray(np.random.randn(500, 500)) + os = xp.empty([4] + list(d.shape)) + ot = xp.empty_like(os) + self.check_func_serial(4, ndimage.median_filter, (d, 3), os) + self.check_func_thread(4, ndimage.median_filter, (d, 3), ot) + xp_assert_equal(os, ot) + + def test_uniform_filter1d(self, xp): + if is_cupy(xp): + pytest.xfail("XXX thread exception; cannot repro outside of pytest") + + d = np.random.randn(5000) + os = np.empty((4, d.size)) + ot = np.empty_like(os) + d = xp.asarray(d) + os = xp.asarray(os) + ot = xp.asarray(ot) + self.check_func_serial(4, ndimage.uniform_filter1d, (d, 5), os) + self.check_func_thread(4, ndimage.uniform_filter1d, (d, 5), ot) + xp_assert_equal(os, ot) + + def test_minmax_filter(self, xp): + if is_cupy(xp): + pytest.xfail("XXX thread exception; cannot repro outside of pytest") + + d = xp.asarray(np.random.randn(500, 500)) + os = xp.empty([4] + list(d.shape)) + ot = xp.empty_like(os) + self.check_func_serial(4, ndimage.maximum_filter, (d, 3), os) + self.check_func_thread(4, ndimage.maximum_filter, (d, 3), ot) + xp_assert_equal(os, ot) + self.check_func_serial(4, ndimage.minimum_filter, (d, 3), os) + self.check_func_thread(4, ndimage.minimum_filter, (d, 3), ot) + xp_assert_equal(os, ot) + + +def test_minmaximum_filter1d(xp): + # Regression gh-3898 + in_ = xp.arange(10) + out = ndimage.minimum_filter1d(in_, 1) + xp_assert_equal(in_, out) + out = ndimage.maximum_filter1d(in_, 1) + xp_assert_equal(in_, out) + # Test reflect + out = ndimage.minimum_filter1d(in_, 5, mode='reflect') + xp_assert_equal(xp.asarray([0, 0, 0, 1, 2, 3, 4, 5, 6, 7]), out) + out = ndimage.maximum_filter1d(in_, 5, mode='reflect') + xp_assert_equal(xp.asarray([2, 3, 4, 5, 6, 7, 8, 9, 9, 9]), out) + # Test constant + out = ndimage.minimum_filter1d(in_, 5, mode='constant', cval=-1) + xp_assert_equal(xp.asarray([-1, -1, 0, 1, 2, 3, 4, 5, -1, -1]), out) + out = ndimage.maximum_filter1d(in_, 5, mode='constant', cval=10) + xp_assert_equal(xp.asarray([10, 10, 4, 5, 6, 7, 8, 9, 10, 10]), out) + # Test nearest + out = ndimage.minimum_filter1d(in_, 5, mode='nearest') + xp_assert_equal(xp.asarray([0, 0, 0, 1, 2, 3, 4, 5, 6, 7]), out) + out = ndimage.maximum_filter1d(in_, 5, mode='nearest') + xp_assert_equal(xp.asarray([2, 3, 4, 5, 6, 7, 8, 9, 9, 9]), out) + # Test wrap + out = ndimage.minimum_filter1d(in_, 5, mode='wrap') + xp_assert_equal(xp.asarray([0, 0, 0, 1, 2, 3, 4, 5, 0, 0]), out) + out = ndimage.maximum_filter1d(in_, 5, mode='wrap') + xp_assert_equal(xp.asarray([9, 9, 4, 5, 6, 7, 8, 9, 9, 9]), out) + + +def test_uniform_filter1d_roundoff_errors(xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/issues/8401") + # gh-6930 + in_ = np.repeat([0, 1, 0], [9, 9, 9]) + in_ = xp.asarray(in_) + + for filter_size in range(3, 10): + out = ndimage.uniform_filter1d(in_, filter_size) + xp_assert_equal(xp.sum(out), xp.asarray(10 - filter_size), check_0d=False) + + +def test_footprint_all_zeros(xp): + # regression test for gh-6876: footprint of all zeros segfaults + arr = xp.asarray(np.random.randint(0, 100, (100, 100))) + kernel = xp.asarray(np.zeros((3, 3), dtype=bool)) + with assert_raises(ValueError): + ndimage.maximum_filter(arr, footprint=kernel) + + +def test_gaussian_filter(xp): + if is_cupy(xp): + pytest.xfail("CuPy does not raise") + + if not hasattr(xp, "float16"): + pytest.xfail(f"{xp} does not have float16") + + # Test gaussian filter with xp.float16 + # gh-8207 + data = xp.asarray([1], dtype=xp.float16) + sigma = 1.0 + with assert_raises(RuntimeError): + ndimage.gaussian_filter(data, sigma) + + +def test_rank_filter_noninteger_rank(xp): + if is_cupy(xp): + pytest.xfail("CuPy does not raise") + + # regression test for issue 9388: ValueError for + # non integer rank when performing rank_filter + arr = xp.asarray(np.random.random((10, 20, 30))) + footprint = xp.asarray(np.ones((1, 1, 10), dtype=bool)) + assert_raises(TypeError, ndimage.rank_filter, arr, 0.5, + footprint=footprint) + + +def test_size_footprint_both_set(xp): + # test for input validation, expect user warning when + # size and footprint is set + with suppress_warnings() as sup: + sup.filter(UserWarning, + "ignoring size because footprint is set") + arr = xp.asarray(np.random.random((10, 20, 30))) + footprint = xp.asarray(np.ones((1, 1, 10), dtype=bool)) + ndimage.rank_filter( + arr, 5, size=2, footprint=footprint + ) + + +@skip_xp_backends(np_only=True, reason='byteorder is numpy-specific') +def test_byte_order_median(xp): + """Regression test for #413: median_filter does not handle bytes orders.""" + a = xp.arange(9, dtype='1 makes sense too + (3, + np.array([0.25266576, 0.30958242, 0.27894721, 0.27894721, 0.27894721, 0.30445588, + 0.31442572, 0.30445588, 0.18015438, 0.14831921, 0.18015438, 0.25754605, + 0.32910465, 0.25754605, 0.17736568, 0.17736568, 0.09089549, 0.22183391, + 0.25266576, 0.30958242]), + ), + (15, + np.array([0.27894721, 0.25266576, 0.25266576, 0.25266576, 0.27894721, 0.27894721, + 0.27894721, 0.27894721, 0.25754605, 0.25754605, 0.22183391, 0.22183391, + 0.25266576, 0.25266576, 0.22183391, 0.22183391, 0.25266576, 0.25266576, + 0.25754605, 0.25754605]), + ), +]) +def test_gh_22250(filter_size, exp): + rng = np.random.default_rng(42) + image = np.zeros((20,)) + noisy_image = image + 0.4 * rng.random(image.shape) + result = ndimage.median_filter(noisy_image, size=filter_size, mode='wrap') + assert_allclose(result, exp) + + +def test_gh_22333(): + x = np.array([272, 58, 67, 163, 463, 608, 87, 108, 1378]) + expected = [58, 67, 87, 108, 163, 108, 108, 108, 87] + actual = ndimage.median_filter(x, size=9, mode='constant') + assert_array_equal(actual, expected) + + +@given(x=npst.arrays(dtype=np.float64, + shape=st.integers(min_value=1, max_value=1000)), + size=st.integers(min_value=1, max_value=50), + mode=st.sampled_from(["constant", "mirror", "wrap", "reflect", + "nearest"]), + ) +def test_gh_22586_crash_property(x, size, mode): + # property-based test for median_filter resilience to hard crashing + ndimage.median_filter(x, size=size, mode=mode) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_fourier.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_fourier.py new file mode 100644 index 0000000000000000000000000000000000000000..be544eaab9ce00b2e9802cf8f9a4819c4f1d2731 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_fourier.py @@ -0,0 +1,189 @@ +import math +import numpy as np + +from scipy._lib._array_api import ( + xp_assert_equal, + assert_array_almost_equal, + assert_almost_equal, + is_cupy, +) + +import pytest + +from scipy import ndimage + +from scipy.conftest import array_api_compatible +skip_xp_backends = pytest.mark.skip_xp_backends +pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends"), + skip_xp_backends(cpu_only=True, exceptions=['cupy', 'jax.numpy'],)] + + +@skip_xp_backends('jax.numpy', reason="jax-ml/jax#23827") +class TestNdimageFourier: + + @pytest.mark.parametrize('shape', [(32, 16), (31, 15), (1, 10)]) + @pytest.mark.parametrize('dtype, dec', [("float32", 6), ("float64", 14)]) + def test_fourier_gaussian_real01(self, shape, dtype, dec, xp): + fft = getattr(xp, 'fft') + + a = np.zeros(shape, dtype=dtype) + a[0, 0] = 1.0 + a = xp.asarray(a) + + a = fft.rfft(a, n=shape[0], axis=0) + a = fft.fft(a, n=shape[1], axis=1) + a = ndimage.fourier_gaussian(a, [5.0, 2.5], shape[0], 0) + a = fft.ifft(a, n=shape[1], axis=1) + a = fft.irfft(a, n=shape[0], axis=0) + assert_almost_equal(ndimage.sum(a), xp.asarray(1), decimal=dec, + check_0d=False) + + @pytest.mark.parametrize('shape', [(32, 16), (31, 15)]) + @pytest.mark.parametrize('dtype, dec', [("complex64", 6), ("complex128", 14)]) + def test_fourier_gaussian_complex01(self, shape, dtype, dec, xp): + fft = getattr(xp, 'fft') + + a = np.zeros(shape, dtype=dtype) + a[0, 0] = 1.0 + a = xp.asarray(a) + + a = fft.fft(a, n=shape[0], axis=0) + a = fft.fft(a, n=shape[1], axis=1) + a = ndimage.fourier_gaussian(a, [5.0, 2.5], -1, 0) + a = fft.ifft(a, n=shape[1], axis=1) + a = fft.ifft(a, n=shape[0], axis=0) + assert_almost_equal(ndimage.sum(xp.real(a)), xp.asarray(1.0), decimal=dec, + check_0d=False) + + @pytest.mark.parametrize('shape', [(32, 16), (31, 15), (1, 10)]) + @pytest.mark.parametrize('dtype, dec', [("float32", 6), ("float64", 14)]) + def test_fourier_uniform_real01(self, shape, dtype, dec, xp): + fft = getattr(xp, 'fft') + + a = np.zeros(shape, dtype=dtype) + a[0, 0] = 1.0 + a = xp.asarray(a) + + a = fft.rfft(a, n=shape[0], axis=0) + a = fft.fft(a, n=shape[1], axis=1) + a = ndimage.fourier_uniform(a, [5.0, 2.5], shape[0], 0) + a = fft.ifft(a, n=shape[1], axis=1) + a = fft.irfft(a, n=shape[0], axis=0) + assert_almost_equal(ndimage.sum(a), xp.asarray(1.0), decimal=dec, + check_0d=False) + + @pytest.mark.parametrize('shape', [(32, 16), (31, 15)]) + @pytest.mark.parametrize('dtype, dec', [("complex64", 6), ("complex128", 14)]) + def test_fourier_uniform_complex01(self, shape, dtype, dec, xp): + fft = getattr(xp, 'fft') + + a = np.zeros(shape, dtype=dtype) + a[0, 0] = 1.0 + a = xp.asarray(a) + + a = fft.fft(a, n=shape[0], axis=0) + a = fft.fft(a, n=shape[1], axis=1) + a = ndimage.fourier_uniform(a, [5.0, 2.5], -1, 0) + a = fft.ifft(a, n=shape[1], axis=1) + a = fft.ifft(a, n=shape[0], axis=0) + assert_almost_equal(ndimage.sum(xp.real(a)), xp.asarray(1.0), decimal=dec, + check_0d=False) + + @pytest.mark.parametrize('shape', [(32, 16), (31, 15)]) + @pytest.mark.parametrize('dtype, dec', [("float32", 4), ("float64", 11)]) + def test_fourier_shift_real01(self, shape, dtype, dec, xp): + fft = getattr(xp, 'fft') + + expected = np.arange(shape[0] * shape[1], dtype=dtype).reshape(shape) + expected = xp.asarray(expected) + + a = fft.rfft(expected, n=shape[0], axis=0) + a = fft.fft(a, n=shape[1], axis=1) + a = ndimage.fourier_shift(a, [1, 1], shape[0], 0) + a = fft.ifft(a, n=shape[1], axis=1) + a = fft.irfft(a, n=shape[0], axis=0) + assert_array_almost_equal(a[1:, 1:], expected[:-1, :-1], decimal=dec) + + @pytest.mark.parametrize('shape', [(32, 16), (31, 15)]) + @pytest.mark.parametrize('dtype, dec', [("complex64", 4), ("complex128", 11)]) + def test_fourier_shift_complex01(self, shape, dtype, dec, xp): + fft = getattr(xp, 'fft') + + expected = np.arange(shape[0] * shape[1], dtype=dtype).reshape(shape) + expected = xp.asarray(expected) + + a = fft.fft(expected, n=shape[0], axis=0) + a = fft.fft(a, n=shape[1], axis=1) + a = ndimage.fourier_shift(a, [1, 1], -1, 0) + a = fft.ifft(a, n=shape[1], axis=1) + a = fft.ifft(a, n=shape[0], axis=0) + assert_array_almost_equal(xp.real(a)[1:, 1:], expected[:-1, :-1], decimal=dec) + assert_array_almost_equal(xp.imag(a), xp.zeros(shape), decimal=dec) + + @pytest.mark.parametrize('shape', [(32, 16), (31, 15), (1, 10)]) + @pytest.mark.parametrize('dtype, dec', [("float32", 5), ("float64", 14)]) + def test_fourier_ellipsoid_real01(self, shape, dtype, dec, xp): + fft = getattr(xp, 'fft') + + a = np.zeros(shape, dtype=dtype) + a[0, 0] = 1.0 + a = xp.asarray(a) + + a = fft.rfft(a, n=shape[0], axis=0) + a = fft.fft(a, n=shape[1], axis=1) + a = ndimage.fourier_ellipsoid(a, [5.0, 2.5], shape[0], 0) + a = fft.ifft(a, n=shape[1], axis=1) + a = fft.irfft(a, n=shape[0], axis=0) + assert_almost_equal(ndimage.sum(a), xp.asarray(1.0), decimal=dec, + check_0d=False) + + @pytest.mark.parametrize('shape', [(32, 16), (31, 15)]) + @pytest.mark.parametrize('dtype, dec', [("complex64", 5), ("complex128", 14)]) + def test_fourier_ellipsoid_complex01(self, shape, dtype, dec, xp): + fft = getattr(xp, 'fft') + + a = np.zeros(shape, dtype=dtype) + a[0, 0] = 1.0 + a = xp.asarray(a) + + a = fft.fft(a, n=shape[0], axis=0) + a = fft.fft(a, n=shape[1], axis=1) + a = ndimage.fourier_ellipsoid(a, [5.0, 2.5], -1, 0) + a = fft.ifft(a, n=shape[1], axis=1) + a = fft.ifft(a, n=shape[0], axis=0) + assert_almost_equal(ndimage.sum(xp.real(a)), xp.asarray(1.0), decimal=dec, + check_0d=False) + + def test_fourier_ellipsoid_unimplemented_ndim(self, xp): + # arrays with ndim > 3 raise NotImplementedError + x = xp.ones((4, 6, 8, 10), dtype=xp.complex128) + with pytest.raises(NotImplementedError): + ndimage.fourier_ellipsoid(x, 3) + + def test_fourier_ellipsoid_1d_complex(self, xp): + # expected result of 1d ellipsoid is the same as for fourier_uniform + for shape in [(32, ), (31, )]: + for type_, dec in zip([xp.complex64, xp.complex128], [5, 14]): + x = xp.ones(shape, dtype=type_) + a = ndimage.fourier_ellipsoid(x, 5, -1, 0) + b = ndimage.fourier_uniform(x, 5, -1, 0) + assert_array_almost_equal(a, b, decimal=dec) + + @pytest.mark.parametrize('shape', [(0, ), (0, 10), (10, 0)]) + @pytest.mark.parametrize('dtype', ["float32", "float64", + "complex64", "complex128"]) + @pytest.mark.parametrize('test_func', + [ndimage.fourier_ellipsoid, + ndimage.fourier_gaussian, + ndimage.fourier_uniform]) + def test_fourier_zero_length_dims(self, shape, dtype, test_func, xp): + if is_cupy(xp): + if (test_func.__name__ == "fourier_ellipsoid" and + math.prod(shape) == 0): + pytest.xfail( + "CuPy's fourier_ellipsoid does not accept size==0 arrays" + ) + dtype = getattr(xp, dtype) + a = xp.ones(shape, dtype=dtype) + b = test_func(a, 3) + xp_assert_equal(a, b) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_interpolation.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_interpolation.py new file mode 100644 index 0000000000000000000000000000000000000000..51e8441e244f46642a07102e297b4d72513514d0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_interpolation.py @@ -0,0 +1,1484 @@ +import sys + +import numpy as np +from numpy.testing import suppress_warnings +from scipy._lib._array_api import ( + xp_assert_equal, xp_assert_close, + assert_array_almost_equal, +) +from scipy._lib._array_api import is_cupy, is_jax, _asarray, array_namespace + +import pytest +from pytest import raises as assert_raises +import scipy.ndimage as ndimage + +from . import types + +from scipy.conftest import array_api_compatible +skip_xp_backends = pytest.mark.skip_xp_backends +pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends"), + skip_xp_backends(cpu_only=True, exceptions=['cupy', 'jax.numpy'],)] + + +eps = 1e-12 + +ndimage_to_numpy_mode = { + 'mirror': 'reflect', + 'reflect': 'symmetric', + 'grid-mirror': 'symmetric', + 'grid-wrap': 'wrap', + 'nearest': 'edge', + 'grid-constant': 'constant', +} + + +class TestBoundaries: + + @skip_xp_backends("cupy", reason="CuPy does not have geometric_transform") + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [1.5, 2.5, 3.5, 4, 4, 4, 4]), + ('wrap', [1.5, 2.5, 3.5, 1.5, 2.5, 3.5, 1.5]), + ('grid-wrap', [1.5, 2.5, 3.5, 2.5, 1.5, 2.5, 3.5]), + ('mirror', [1.5, 2.5, 3.5, 3.5, 2.5, 1.5, 1.5]), + ('reflect', [1.5, 2.5, 3.5, 4, 3.5, 2.5, 1.5]), + ('constant', [1.5, 2.5, 3.5, -1, -1, -1, -1]), + ('grid-constant', [1.5, 2.5, 3.5, 1.5, -1, -1, -1])] + ) + def test_boundaries(self, mode, expected_value, xp): + def shift(x): + return (x[0] + 0.5,) + + data = xp.asarray([1, 2, 3, 4.]) + xp_assert_equal( + ndimage.geometric_transform(data, shift, cval=-1, mode=mode, + output_shape=(7,), order=1), + xp.asarray(expected_value)) + + @skip_xp_backends("cupy", reason="CuPy does not have geometric_transform") + @pytest.mark.parametrize( + 'mode, expected_value', + [('nearest', [1, 1, 2, 3]), + ('wrap', [3, 1, 2, 3]), + ('grid-wrap', [4, 1, 2, 3]), + ('mirror', [2, 1, 2, 3]), + ('reflect', [1, 1, 2, 3]), + ('constant', [-1, 1, 2, 3]), + ('grid-constant', [-1, 1, 2, 3])] + ) + def test_boundaries2(self, mode, expected_value, xp): + def shift(x): + return (x[0] - 0.9,) + + data = xp.asarray([1, 2, 3, 4]) + xp_assert_equal( + ndimage.geometric_transform(data, shift, cval=-1, mode=mode, + output_shape=(4,)), + xp.asarray(expected_value)) + + @pytest.mark.parametrize('mode', ['mirror', 'reflect', 'grid-mirror', + 'grid-wrap', 'grid-constant', + 'nearest']) + @pytest.mark.parametrize('order', range(6)) + def test_boundary_spline_accuracy(self, mode, order, xp): + """Tests based on examples from gh-2640""" + if (is_jax(xp) and + (mode not in ['mirror', 'reflect', 'constant', 'wrap', 'nearest'] + or order > 1) + ): + pytest.xfail("Jax does not support grid- modes or order > 1") + + np_data = np.arange(-6, 7, dtype=np.float64) + data = xp.asarray(np_data) + x = xp.asarray(np.linspace(-8, 15, num=1000)) + newaxis = array_namespace(x).newaxis + y = ndimage.map_coordinates(data, x[newaxis, ...], order=order, mode=mode) + + # compute expected value using explicit padding via np.pad + npad = 32 + pad_mode = ndimage_to_numpy_mode.get(mode) + padded = xp.asarray(np.pad(np_data, npad, mode=pad_mode)) + coords = xp.asarray(npad + x)[newaxis, ...] + expected = ndimage.map_coordinates(padded, coords, order=order, mode=mode) + + atol = 1e-5 if mode == 'grid-constant' else 1e-12 + xp_assert_close(y, expected, rtol=1e-7, atol=atol) + + +@pytest.mark.parametrize('order', range(2, 6)) +@pytest.mark.parametrize('dtype', types) +class TestSpline: + + def test_spline01(self, dtype, order, xp): + dtype = getattr(xp, dtype) + data = xp.ones([], dtype=dtype) + out = ndimage.spline_filter(data, order=order) + assert out == xp.asarray(1, dtype=out.dtype) + + def test_spline02(self, dtype, order, xp): + dtype = getattr(xp, dtype) + data = xp.asarray([1], dtype=dtype) + out = ndimage.spline_filter(data, order=order) + assert_array_almost_equal(out, xp.asarray([1])) + + @skip_xp_backends(np_only=True, reason='output=dtype is numpy-specific') + def test_spline03(self, dtype, order, xp): + dtype = getattr(xp, dtype) + data = xp.ones([], dtype=dtype) + out = ndimage.spline_filter(data, order, output=dtype) + assert out == xp.asarray(1, dtype=out.dtype) + + def test_spline04(self, dtype, order, xp): + dtype = getattr(xp, dtype) + data = xp.ones([4], dtype=dtype) + out = ndimage.spline_filter(data, order) + assert_array_almost_equal(out, xp.asarray([1, 1, 1, 1])) + + def test_spline05(self, dtype, order, xp): + dtype = getattr(xp, dtype) + data = xp.ones([4, 4], dtype=dtype) + out = ndimage.spline_filter(data, order=order) + expected = xp.asarray([[1, 1, 1, 1], + [1, 1, 1, 1], + [1, 1, 1, 1], + [1, 1, 1, 1]]) + assert_array_almost_equal(out, expected) + + +@skip_xp_backends("cupy", reason="CuPy does not have geometric_transform") +@pytest.mark.parametrize('order', range(0, 6)) +class TestGeometricTransform: + + def test_geometric_transform01(self, order, xp): + data = xp.asarray([1]) + + def mapping(x): + return x + + out = ndimage.geometric_transform(data, mapping, data.shape, + order=order) + assert_array_almost_equal(out, xp.asarray([1], dtype=out.dtype)) + + def test_geometric_transform02(self, order, xp): + data = xp.ones([4]) + + def mapping(x): + return x + + out = ndimage.geometric_transform(data, mapping, data.shape, + order=order) + assert_array_almost_equal(out, xp.asarray([1, 1, 1, 1], dtype=out.dtype)) + + def test_geometric_transform03(self, order, xp): + data = xp.ones([4]) + + def mapping(x): + return (x[0] - 1,) + + out = ndimage.geometric_transform(data, mapping, data.shape, + order=order) + assert_array_almost_equal(out, xp.asarray([0, 1, 1, 1], dtype=out.dtype)) + + def test_geometric_transform04(self, order, xp): + data = xp.asarray([4, 1, 3, 2]) + + def mapping(x): + return (x[0] - 1,) + + out = ndimage.geometric_transform(data, mapping, data.shape, + order=order) + assert_array_almost_equal(out, xp.asarray([0, 4, 1, 3], dtype=out.dtype)) + + @pytest.mark.parametrize('dtype', ["float64", "complex128"]) + def test_geometric_transform05(self, order, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.asarray([[1, 1, 1, 1], + [1, 1, 1, 1], + [1, 1, 1, 1]], dtype=dtype) + expected = xp.asarray([[0, 1, 1, 1], + [0, 1, 1, 1], + [0, 1, 1, 1]], dtype=dtype) + + isdtype = array_namespace(data).isdtype + if isdtype(data.dtype, 'complex floating'): + data -= 1j * data + expected -= 1j * expected + + def mapping(x): + return (x[0], x[1] - 1) + + out = ndimage.geometric_transform(data, mapping, data.shape, + order=order) + assert_array_almost_equal(out, expected) + + def test_geometric_transform06(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + + def mapping(x): + return (x[0], x[1] - 1) + + out = ndimage.geometric_transform(data, mapping, data.shape, + order=order) + expected = xp.asarray([[0, 4, 1, 3], + [0, 7, 6, 8], + [0, 3, 5, 3]], dtype=out.dtype) + assert_array_almost_equal(out, expected) + + def test_geometric_transform07(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + + def mapping(x): + return (x[0] - 1, x[1]) + + out = ndimage.geometric_transform(data, mapping, data.shape, + order=order) + expected = xp.asarray([[0, 0, 0, 0], + [4, 1, 3, 2], + [7, 6, 8, 5]], dtype=out.dtype) + assert_array_almost_equal(out, expected) + + def test_geometric_transform08(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + + def mapping(x): + return (x[0] - 1, x[1] - 1) + + out = ndimage.geometric_transform(data, mapping, data.shape, + order=order) + expected = xp.asarray([[0, 0, 0, 0], + [0, 4, 1, 3], + [0, 7, 6, 8]], dtype=out.dtype) + assert_array_almost_equal(out, expected) + + def test_geometric_transform10(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + + def mapping(x): + return (x[0] - 1, x[1] - 1) + + if (order > 1): + filtered = ndimage.spline_filter(data, order=order) + else: + filtered = data + out = ndimage.geometric_transform(filtered, mapping, data.shape, + order=order, prefilter=False) + expected = xp.asarray([[0, 0, 0, 0], + [0, 4, 1, 3], + [0, 7, 6, 8]], dtype=out.dtype) + assert_array_almost_equal(out, expected) + + def test_geometric_transform13(self, order, xp): + data = xp.ones([2], dtype=xp.float64) + + def mapping(x): + return (x[0] // 2,) + + out = ndimage.geometric_transform(data, mapping, [4], order=order) + assert_array_almost_equal(out, xp.asarray([1, 1, 1, 1], dtype=out.dtype)) + + def test_geometric_transform14(self, order, xp): + data = xp.asarray([1, 5, 2, 6, 3, 7, 4, 4]) + + def mapping(x): + return (2 * x[0],) + + out = ndimage.geometric_transform(data, mapping, [4], order=order) + assert_array_almost_equal(out, xp.asarray([1, 2, 3, 4], dtype=out.dtype)) + + def test_geometric_transform15(self, order, xp): + data = [1, 2, 3, 4] + + def mapping(x): + return (x[0] / 2,) + + out = ndimage.geometric_transform(data, mapping, [8], order=order) + assert_array_almost_equal(out[::2], [1, 2, 3, 4]) + + def test_geometric_transform16(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9.0, 10, 11, 12]] + + def mapping(x): + return (x[0], x[1] * 2) + + out = ndimage.geometric_transform(data, mapping, (3, 2), + order=order) + assert_array_almost_equal(out, [[1, 3], [5, 7], [9, 11]]) + + def test_geometric_transform17(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + + def mapping(x): + return (x[0] * 2, x[1]) + + out = ndimage.geometric_transform(data, mapping, (1, 4), + order=order) + assert_array_almost_equal(out, [[1, 2, 3, 4]]) + + def test_geometric_transform18(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + + def mapping(x): + return (x[0] * 2, x[1] * 2) + + out = ndimage.geometric_transform(data, mapping, (1, 2), + order=order) + assert_array_almost_equal(out, [[1, 3]]) + + def test_geometric_transform19(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + + def mapping(x): + return (x[0], x[1] / 2) + + out = ndimage.geometric_transform(data, mapping, (3, 8), + order=order) + assert_array_almost_equal(out[..., ::2], data) + + def test_geometric_transform20(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + + def mapping(x): + return (x[0] / 2, x[1]) + + out = ndimage.geometric_transform(data, mapping, (6, 4), + order=order) + assert_array_almost_equal(out[::2, ...], data) + + def test_geometric_transform21(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + + def mapping(x): + return (x[0] / 2, x[1] / 2) + + out = ndimage.geometric_transform(data, mapping, (6, 8), + order=order) + assert_array_almost_equal(out[::2, ::2], data) + + def test_geometric_transform22(self, order, xp): + data = xp.asarray([[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]], dtype=xp.float64) + + def mapping1(x): + return (x[0] / 2, x[1] / 2) + + def mapping2(x): + return (x[0] * 2, x[1] * 2) + + out = ndimage.geometric_transform(data, mapping1, + (6, 8), order=order) + out = ndimage.geometric_transform(out, mapping2, + (3, 4), order=order) + assert_array_almost_equal(out, data) + + def test_geometric_transform23(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + + def mapping(x): + return (1, x[0] * 2) + + out = ndimage.geometric_transform(data, mapping, (2,), order=order) + out = out.astype(np.int32) + assert_array_almost_equal(out, [5, 7]) + + def test_geometric_transform24(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + + def mapping(x, a, b): + return (a, x[0] * b) + + out = ndimage.geometric_transform( + data, mapping, (2,), order=order, extra_arguments=(1,), + extra_keywords={'b': 2}) + assert_array_almost_equal(out, [5, 7]) + + +@skip_xp_backends("cupy", reason="CuPy does not have geometric_transform") +class TestGeometricTransformExtra: + + def test_geometric_transform_grid_constant_order1(self, xp): + + # verify interpolation outside the original bounds + x = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=xp.float64) + + def mapping(x): + return (x[0] - 0.5), (x[1] - 0.5) + + expected_result = xp.asarray([[0.25, 0.75, 1.25], + [1.25, 3.00, 4.00]]) + assert_array_almost_equal( + ndimage.geometric_transform(x, mapping, mode='grid-constant', + order=1), + expected_result, + ) + + @pytest.mark.parametrize('mode', ['grid-constant', 'grid-wrap', 'nearest', + 'mirror', 'reflect']) + @pytest.mark.parametrize('order', range(6)) + def test_geometric_transform_vs_padded(self, order, mode, xp): + + def mapping(x): + return (x[0] - 0.4), (x[1] + 2.3) + + # Manually pad and then extract center after the transform to get the + # expected result. + x = np.arange(144, dtype=float).reshape(12, 12) + npad = 24 + pad_mode = ndimage_to_numpy_mode.get(mode) + x_padded = np.pad(x, npad, mode=pad_mode) + + x = xp.asarray(x) + x_padded = xp.asarray(x_padded) + + center_slice = tuple([slice(npad, -npad)] * x.ndim) + expected_result = ndimage.geometric_transform( + x_padded, mapping, mode=mode, order=order)[center_slice] + + xp_assert_close( + ndimage.geometric_transform(x, mapping, mode=mode, + order=order), + expected_result, + rtol=1e-7, + ) + + @skip_xp_backends(np_only=True, reason='endianness is numpy-specific') + def test_geometric_transform_endianness_with_output_parameter(self, xp): + # geometric transform given output ndarray or dtype with + # non-native endianness. see issue #4127 + data = np.asarray([1]) + + def mapping(x): + return x + + for out in [data.dtype, data.dtype.newbyteorder(), + np.empty_like(data), + np.empty_like(data).astype(data.dtype.newbyteorder())]: + returned = ndimage.geometric_transform(data, mapping, data.shape, + output=out) + result = out if returned is None else returned + assert_array_almost_equal(result, [1]) + + @skip_xp_backends(np_only=True, reason='string `output` is numpy-specific') + def test_geometric_transform_with_string_output(self, xp): + data = xp.asarray([1]) + + def mapping(x): + return x + + out = ndimage.geometric_transform(data, mapping, output='f') + assert out.dtype is np.dtype('f') + assert_array_almost_equal(out, [1]) + + +class TestMapCoordinates: + + @pytest.mark.parametrize('order', range(0, 6)) + @pytest.mark.parametrize('dtype', [np.float64, np.complex128]) + def test_map_coordinates01(self, order, dtype, xp): + if is_jax(xp) and order > 1: + pytest.xfail("jax map_coordinates requires order <= 1") + + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + expected = xp.asarray([[0, 0, 0, 0], + [0, 4, 1, 3], + [0, 7, 6, 8]]) + isdtype = array_namespace(data).isdtype + if isdtype(data.dtype, 'complex floating'): + data = data - 1j * data + expected = expected - 1j * expected + + idx = np.indices(data.shape) + idx -= 1 + idx = xp.asarray(idx) + + out = ndimage.map_coordinates(data, idx, order=order) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_map_coordinates02(self, order, xp): + if is_jax(xp): + if order > 1: + pytest.xfail("jax map_coordinates requires order <= 1") + if order == 1: + pytest.xfail("output differs. jax bug?") + + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + idx = np.indices(data.shape, np.float64) + idx -= 0.5 + idx = xp.asarray(idx) + + out1 = ndimage.shift(data, 0.5, order=order) + out2 = ndimage.map_coordinates(data, idx, order=order) + assert_array_almost_equal(out1, out2) + + @skip_xp_backends("jax.numpy", reason="`order` is required in jax") + def test_map_coordinates03(self, xp): + data = _asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]], order='F', xp=xp) + idx = np.indices(data.shape) - 1 + idx = xp.asarray(idx) + out = ndimage.map_coordinates(data, idx) + expected = xp.asarray([[0, 0, 0, 0], + [0, 4, 1, 3], + [0, 7, 6, 8]]) + assert_array_almost_equal(out, expected) + assert_array_almost_equal(out, ndimage.shift(data, (1, 1))) + + idx = np.indices(data[::2, ...].shape) - 1 + idx = xp.asarray(idx) + out = ndimage.map_coordinates(data[::2, ...], idx) + assert_array_almost_equal(out, xp.asarray([[0, 0, 0, 0], + [0, 4, 1, 3]])) + assert_array_almost_equal(out, ndimage.shift(data[::2, ...], (1, 1))) + + idx = np.indices(data[:, ::2].shape) - 1 + idx = xp.asarray(idx) + out = ndimage.map_coordinates(data[:, ::2], idx) + assert_array_almost_equal(out, xp.asarray([[0, 0], [0, 4], [0, 7]])) + assert_array_almost_equal(out, ndimage.shift(data[:, ::2], (1, 1))) + + @skip_xp_backends(np_only=True) + def test_map_coordinates_endianness_with_output_parameter(self, xp): + # output parameter given as array or dtype with either endianness + # see issue #4127 + # NB: NumPy-only + + data = np.asarray([[1, 2], [7, 6]]) + expected = np.asarray([[0, 0], [0, 1]]) + idx = np.indices(data.shape) + idx -= 1 + for out in [ + data.dtype, + data.dtype.newbyteorder(), + np.empty_like(expected), + np.empty_like(expected).astype(expected.dtype.newbyteorder()) + ]: + returned = ndimage.map_coordinates(data, idx, output=out) + result = out if returned is None else returned + assert_array_almost_equal(result, expected) + + @skip_xp_backends(np_only=True, reason='string `output` is numpy-specific') + def test_map_coordinates_with_string_output(self, xp): + data = xp.asarray([[1]]) + idx = np.indices(data.shape) + idx = xp.asarray(idx) + out = ndimage.map_coordinates(data, idx, output='f') + assert out.dtype is np.dtype('f') + assert_array_almost_equal(out, xp.asarray([[1]])) + + @pytest.mark.skipif('win32' in sys.platform or np.intp(0).itemsize < 8, + reason='do not run on 32 bit or windows ' + '(no sparse memory)') + def test_map_coordinates_large_data(self, xp): + # check crash on large data + try: + n = 30000 + # a = xp.reshape(xp.empty(n**2, dtype=xp.float32), (n, n)) + a = np.empty(n**2, dtype=np.float32).reshape(n, n) + # fill the part we might read + a[n - 3:, n - 3:] = 0 + ndimage.map_coordinates( + xp.asarray(a), xp.asarray([[n - 1.5], [n - 1.5]]), order=1 + ) + except MemoryError as e: + raise pytest.skip('Not enough memory available') from e + + +class TestAffineTransform: + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform01(self, order, xp): + data = xp.asarray([1]) + out = ndimage.affine_transform(data, xp.asarray([[1]]), order=order) + assert_array_almost_equal(out, xp.asarray([1])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform02(self, order, xp): + data = xp.ones([4]) + out = ndimage.affine_transform(data, xp.asarray([[1]]), order=order) + assert_array_almost_equal(out, xp.asarray([1, 1, 1, 1])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform03(self, order, xp): + data = xp.ones([4]) + out = ndimage.affine_transform(data, xp.asarray([[1]]), -1, order=order) + assert_array_almost_equal(out, xp.asarray([0, 1, 1, 1])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform04(self, order, xp): + data = xp.asarray([4, 1, 3, 2]) + out = ndimage.affine_transform(data, xp.asarray([[1]]), -1, order=order) + assert_array_almost_equal(out, xp.asarray([0, 4, 1, 3])) + + @pytest.mark.parametrize('order', range(0, 6)) + @pytest.mark.parametrize('dtype', ["float64", "complex128"]) + def test_affine_transform05(self, order, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.asarray([[1, 1, 1, 1], + [1, 1, 1, 1], + [1, 1, 1, 1]], dtype=dtype) + expected = xp.asarray([[0, 1, 1, 1], + [0, 1, 1, 1], + [0, 1, 1, 1]], dtype=dtype) + isdtype = array_namespace(data).isdtype + if isdtype(data.dtype, 'complex floating'): + data -= 1j * data + expected -= 1j * expected + out = ndimage.affine_transform(data, xp.asarray([[1, 0], [0, 1]]), + [0, -1], order=order) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform06(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + out = ndimage.affine_transform(data, xp.asarray([[1, 0], [0, 1]]), + [0, -1], order=order) + assert_array_almost_equal(out, xp.asarray([[0, 4, 1, 3], + [0, 7, 6, 8], + [0, 3, 5, 3]])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform07(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + out = ndimage.affine_transform(data, xp.asarray([[1, 0], [0, 1]]), + [-1, 0], order=order) + assert_array_almost_equal(out, xp.asarray([[0, 0, 0, 0], + [4, 1, 3, 2], + [7, 6, 8, 5]])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform08(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + out = ndimage.affine_transform(data, xp.asarray([[1, 0], [0, 1]]), + [-1, -1], order=order) + assert_array_almost_equal(out, xp.asarray([[0, 0, 0, 0], + [0, 4, 1, 3], + [0, 7, 6, 8]])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform09(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + if (order > 1): + filtered = ndimage.spline_filter(data, order=order) + else: + filtered = data + out = ndimage.affine_transform(filtered, xp.asarray([[1, 0], [0, 1]]), + [-1, -1], order=order, + prefilter=False) + assert_array_almost_equal(out, xp.asarray([[0, 0, 0, 0], + [0, 4, 1, 3], + [0, 7, 6, 8]])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform10(self, order, xp): + data = xp.ones([2], dtype=xp.float64) + out = ndimage.affine_transform(data, xp.asarray([[0.5]]), output_shape=(4,), + order=order) + assert_array_almost_equal(out, xp.asarray([1, 1, 1, 0])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform11(self, order, xp): + data = xp.asarray([1, 5, 2, 6, 3, 7, 4, 4]) + out = ndimage.affine_transform(data, xp.asarray([[2]]), 0, (4,), order=order) + assert_array_almost_equal(out, xp.asarray([1, 2, 3, 4])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform12(self, order, xp): + data = xp.asarray([1, 2, 3, 4]) + out = ndimage.affine_transform(data, xp.asarray([[0.5]]), 0, (8,), order=order) + assert_array_almost_equal(out[::2], xp.asarray([1, 2, 3, 4])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform13(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9.0, 10, 11, 12]] + data = xp.asarray(data) + out = ndimage.affine_transform(data, xp.asarray([[1, 0], [0, 2]]), 0, (3, 2), + order=order) + assert_array_almost_equal(out, xp.asarray([[1, 3], [5, 7], [9, 11]])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform14(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + data = xp.asarray(data) + out = ndimage.affine_transform(data, xp.asarray([[2, 0], [0, 1]]), 0, (1, 4), + order=order) + assert_array_almost_equal(out, xp.asarray([[1, 2, 3, 4]])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform15(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + data = xp.asarray(data) + out = ndimage.affine_transform(data, xp.asarray([[2, 0], [0, 2]]), 0, (1, 2), + order=order) + assert_array_almost_equal(out, xp.asarray([[1, 3]])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform16(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + data = xp.asarray(data) + out = ndimage.affine_transform(data, xp.asarray([[1, 0.0], [0, 0.5]]), 0, + (3, 8), order=order) + assert_array_almost_equal(out[..., ::2], data) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform17(self, order, xp): + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + data = xp.asarray(data) + out = ndimage.affine_transform(data, xp.asarray([[0.5, 0], [0, 1]]), 0, + (6, 4), order=order) + assert_array_almost_equal(out[::2, ...], data) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform18(self, order, xp): + data = xp.asarray([[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]]) + out = ndimage.affine_transform(data, xp.asarray([[0.5, 0], [0, 0.5]]), 0, + (6, 8), order=order) + assert_array_almost_equal(out[::2, ::2], data) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform19(self, order, xp): + data = xp.asarray([[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]], dtype=xp.float64) + out = ndimage.affine_transform(data, xp.asarray([[0.5, 0], [0, 0.5]]), 0, + (6, 8), order=order) + out = ndimage.affine_transform(out, xp.asarray([[2.0, 0], [0, 2.0]]), 0, + (3, 4), order=order) + assert_array_almost_equal(out, data) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform20(self, order, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/issues/8394") + + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + data = xp.asarray(data) + out = ndimage.affine_transform(data, xp.asarray([[0], [2]]), 0, (2,), + order=order) + assert_array_almost_equal(out, xp.asarray([1, 3])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform21(self, order, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/issues/8394") + + data = [[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]] + data = xp.asarray(data) + out = ndimage.affine_transform(data, xp.asarray([[2], [0]]), 0, (2,), + order=order) + assert_array_almost_equal(out, xp.asarray([1, 9])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform22(self, order, xp): + # shift and offset interaction; see issue #1547 + data = xp.asarray([4, 1, 3, 2]) + out = ndimage.affine_transform(data, xp.asarray([[2]]), [-1], (3,), + order=order) + assert_array_almost_equal(out, xp.asarray([0, 1, 2])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform23(self, order, xp): + # shift and offset interaction; see issue #1547 + data = xp.asarray([4, 1, 3, 2]) + out = ndimage.affine_transform(data, xp.asarray([[0.5]]), [-1], (8,), + order=order) + assert_array_almost_equal(out[::2], xp.asarray([0, 4, 1, 3])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform24(self, order, xp): + # consistency between diagonal and non-diagonal case; see issue #1547 + data = xp.asarray([4, 1, 3, 2]) + with suppress_warnings() as sup: + sup.filter(UserWarning, + 'The behavior of affine_transform with a 1-D array .* ' + 'has changed') + out1 = ndimage.affine_transform(data, xp.asarray([2]), -1, order=order) + out2 = ndimage.affine_transform(data, xp.asarray([[2]]), -1, order=order) + assert_array_almost_equal(out1, out2) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform25(self, order, xp): + # consistency between diagonal and non-diagonal case; see issue #1547 + data = xp.asarray([4, 1, 3, 2]) + with suppress_warnings() as sup: + sup.filter(UserWarning, + 'The behavior of affine_transform with a 1-D array .* ' + 'has changed') + out1 = ndimage.affine_transform(data, xp.asarray([0.5]), -1, order=order) + out2 = ndimage.affine_transform(data, xp.asarray([[0.5]]), -1, order=order) + assert_array_almost_equal(out1, out2) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform26(self, order, xp): + # test homogeneous coordinates + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + if (order > 1): + filtered = ndimage.spline_filter(data, order=order) + else: + filtered = data + tform_original = xp.eye(2) + offset_original = -xp.ones((2, 1)) + + concat = array_namespace(tform_original, offset_original).concat + tform_h1 = concat((tform_original, offset_original), axis=1) # hstack + tform_h2 = concat( (tform_h1, xp.asarray([[0.0, 0, 1]])), axis=0) # vstack + + offs = [float(x) for x in xp.reshape(offset_original, (-1,))] + + out1 = ndimage.affine_transform(filtered, tform_original, + offs, + order=order, prefilter=False) + out2 = ndimage.affine_transform(filtered, tform_h1, order=order, + prefilter=False) + out3 = ndimage.affine_transform(filtered, tform_h2, order=order, + prefilter=False) + for out in [out1, out2, out3]: + assert_array_almost_equal(out, xp.asarray([[0, 0, 0, 0], + [0, 4, 1, 3], + [0, 7, 6, 8]])) + + def test_affine_transform27(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy does not raise") + + # test valid homogeneous transformation matrix + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + concat = array_namespace(data).concat + tform_h1 = concat( (xp.eye(2), -xp.ones((2, 1))) , axis=1) # vstack + tform_h2 = concat((tform_h1, xp.asarray([[5.0, 2, 1]])), axis=0) # hstack + + assert_raises(ValueError, ndimage.affine_transform, data, tform_h2) + + @skip_xp_backends(np_only=True, reason='byteorder is numpy-specific') + def test_affine_transform_1d_endianness_with_output_parameter(self, xp): + # 1d affine transform given output ndarray or dtype with + # either endianness. see issue #7388 + data = xp.ones((2, 2)) + for out in [xp.empty_like(data), + xp.empty_like(data).astype(data.dtype.newbyteorder()), + data.dtype, data.dtype.newbyteorder()]: + with suppress_warnings() as sup: + sup.filter(UserWarning, + 'The behavior of affine_transform with a 1-D array ' + '.* has changed') + matrix = xp.asarray([1, 1]) + returned = ndimage.affine_transform(data, matrix, output=out) + result = out if returned is None else returned + assert_array_almost_equal(result, xp.asarray([[1, 1], [1, 1]])) + + @skip_xp_backends(np_only=True, reason='byteorder is numpy-specific') + def test_affine_transform_multi_d_endianness_with_output_parameter(self, xp): + # affine transform given output ndarray or dtype with either endianness + # see issue #4127 + # NB: byteorder is numpy-specific + data = np.asarray([1]) + for out in [data.dtype, data.dtype.newbyteorder(), + np.empty_like(data), + np.empty_like(data).astype(data.dtype.newbyteorder())]: + returned = ndimage.affine_transform(data, np.asarray([[1]]), output=out) + result = out if returned is None else returned + assert_array_almost_equal(result, np.asarray([1])) + + @skip_xp_backends(np_only=True, + reason='`out` of a different size is numpy-specific' + ) + def test_affine_transform_output_shape(self, xp): + # don't require output_shape when out of a different size is given + data = xp.arange(8, dtype=xp.float64) + out = xp.ones((16,)) + + ndimage.affine_transform(data, xp.asarray([[1]]), output=out) + assert_array_almost_equal(out[:8], data) + + # mismatched output shape raises an error + with pytest.raises(RuntimeError): + ndimage.affine_transform( + data, [[1]], output=out, output_shape=(12,)) + + @skip_xp_backends(np_only=True, reason='string `output` is numpy-specific') + def test_affine_transform_with_string_output(self, xp): + data = xp.asarray([1]) + out = ndimage.affine_transform(data, xp.asarray([[1]]), output='f') + assert out.dtype is np.dtype('f') + assert_array_almost_equal(out, xp.asarray([1])) + + @pytest.mark.parametrize('shift', + [(1, 0), (0, 1), (-1, 1), (3, -5), (2, 7)]) + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform_shift_via_grid_wrap(self, shift, order, xp): + # For mode 'grid-wrap', integer shifts should match np.roll + x = np.asarray([[0, 1], + [2, 3]]) + affine = np.zeros((2, 3)) + affine[:2, :2] = np.eye(2) + affine[:, 2] = np.asarray(shift) + + expected = np.roll(x, shift, axis=(0, 1)) + + x = xp.asarray(x) + affine = xp.asarray(affine) + expected = xp.asarray(expected) + + assert_array_almost_equal( + ndimage.affine_transform(x, affine, mode='grid-wrap', order=order), + expected + ) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_affine_transform_shift_reflect(self, order, xp): + # shift by x.shape results in reflection + x = np.asarray([[0, 1, 2], + [3, 4, 5]]) + expected = x[::-1, ::-1].copy() # strides >0 for torch + x = xp.asarray(x) + expected = xp.asarray(expected) + + affine = np.zeros([2, 3]) + affine[:2, :2] = np.eye(2) + affine[:, 2] = np.asarray(x.shape) + affine = xp.asarray(affine) + + assert_array_almost_equal( + ndimage.affine_transform(x, affine, mode='reflect', order=order), + expected, + ) + + +class TestShift: + + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift01(self, order, xp): + data = xp.asarray([1]) + out = ndimage.shift(data, [1], order=order) + assert_array_almost_equal(out, xp.asarray([0])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift02(self, order, xp): + data = xp.ones([4]) + out = ndimage.shift(data, [1], order=order) + assert_array_almost_equal(out, xp.asarray([0, 1, 1, 1])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift03(self, order, xp): + data = xp.ones([4]) + out = ndimage.shift(data, -1, order=order) + assert_array_almost_equal(out, xp.asarray([1, 1, 1, 0])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift04(self, order, xp): + data = xp.asarray([4, 1, 3, 2]) + out = ndimage.shift(data, 1, order=order) + assert_array_almost_equal(out, xp.asarray([0, 4, 1, 3])) + + @pytest.mark.parametrize('order', range(0, 6)) + @pytest.mark.parametrize('dtype', ["float64", "complex128"]) + def test_shift05(self, order, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.asarray([[1, 1, 1, 1], + [1, 1, 1, 1], + [1, 1, 1, 1]], dtype=dtype) + expected = xp.asarray([[0, 1, 1, 1], + [0, 1, 1, 1], + [0, 1, 1, 1]], dtype=dtype) + isdtype = array_namespace(data).isdtype + if isdtype(data.dtype, 'complex floating'): + data -= 1j * data + expected -= 1j * expected + out = ndimage.shift(data, [0, 1], order=order) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('order', range(0, 6)) + @pytest.mark.parametrize('mode', ['constant', 'grid-constant']) + @pytest.mark.parametrize('dtype', ['float64', 'complex128']) + def test_shift_with_nonzero_cval(self, order, mode, dtype, xp): + data = np.asarray([[1, 1, 1, 1], + [1, 1, 1, 1], + [1, 1, 1, 1]], dtype=dtype) + + expected = np.asarray([[0, 1, 1, 1], + [0, 1, 1, 1], + [0, 1, 1, 1]], dtype=dtype) + + isdtype = array_namespace(data).isdtype + if isdtype(data.dtype, 'complex floating'): + data -= 1j * data + expected -= 1j * expected + cval = 5.0 + expected[:, 0] = cval # specific to shift of [0, 1] used below + + data = xp.asarray(data) + expected = xp.asarray(expected) + out = ndimage.shift(data, [0, 1], order=order, mode=mode, cval=cval) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift06(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + out = ndimage.shift(data, [0, 1], order=order) + assert_array_almost_equal(out, xp.asarray([[0, 4, 1, 3], + [0, 7, 6, 8], + [0, 3, 5, 3]])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift07(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + out = ndimage.shift(data, [1, 0], order=order) + assert_array_almost_equal(out, xp.asarray([[0, 0, 0, 0], + [4, 1, 3, 2], + [7, 6, 8, 5]])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift08(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + out = ndimage.shift(data, [1, 1], order=order) + assert_array_almost_equal(out, xp.asarray([[0, 0, 0, 0], + [0, 4, 1, 3], + [0, 7, 6, 8]])) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift09(self, order, xp): + data = xp.asarray([[4, 1, 3, 2], + [7, 6, 8, 5], + [3, 5, 3, 6]]) + if (order > 1): + filtered = ndimage.spline_filter(data, order=order) + else: + filtered = data + out = ndimage.shift(filtered, [1, 1], order=order, prefilter=False) + assert_array_almost_equal(out, xp.asarray([[0, 0, 0, 0], + [0, 4, 1, 3], + [0, 7, 6, 8]])) + + @pytest.mark.parametrize('shift', + [(1, 0), (0, 1), (-1, 1), (3, -5), (2, 7)]) + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift_grid_wrap(self, shift, order, xp): + # For mode 'grid-wrap', integer shifts should match np.roll + x = np.asarray([[0, 1], + [2, 3]]) + expected = np.roll(x, shift, axis=(0,1)) + + x = xp.asarray(x) + expected = xp.asarray(expected) + + assert_array_almost_equal( + ndimage.shift(x, shift, mode='grid-wrap', order=order), + expected + ) + + @pytest.mark.parametrize('shift', + [(1, 0), (0, 1), (-1, 1), (3, -5), (2, 7)]) + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift_grid_constant1(self, shift, order, xp): + # For integer shifts, 'constant' and 'grid-constant' should be equal + x = xp.reshape(xp.arange(20), (5, 4)) + assert_array_almost_equal( + ndimage.shift(x, shift, mode='grid-constant', order=order), + ndimage.shift(x, shift, mode='constant', order=order), + ) + + def test_shift_grid_constant_order1(self, xp): + x = xp.asarray([[1, 2, 3], + [4, 5, 6]], dtype=xp.float64) + expected_result = xp.asarray([[0.25, 0.75, 1.25], + [1.25, 3.00, 4.00]]) + assert_array_almost_equal( + ndimage.shift(x, (0.5, 0.5), mode='grid-constant', order=1), + expected_result, + ) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_shift_reflect(self, order, xp): + # shift by x.shape results in reflection + x = np.asarray([[0, 1, 2], + [3, 4, 5]]) + expected = x[::-1, ::-1].copy() # strides > 0 for torch + + x = xp.asarray(x) + expected = xp.asarray(expected) + assert_array_almost_equal( + ndimage.shift(x, x.shape, mode='reflect', order=order), + expected, + ) + + @pytest.mark.parametrize('order', range(0, 6)) + @pytest.mark.parametrize('prefilter', [False, True]) + def test_shift_nearest_boundary(self, order, prefilter, xp): + # verify that shifting at least order // 2 beyond the end of the array + # gives a value equal to the edge value. + x = xp.arange(16) + kwargs = dict(mode='nearest', order=order, prefilter=prefilter) + assert_array_almost_equal( + ndimage.shift(x, order // 2 + 1, **kwargs)[0], x[0], + ) + assert_array_almost_equal( + ndimage.shift(x, -order // 2 - 1, **kwargs)[-1], x[-1], + ) + + @pytest.mark.parametrize('mode', ['grid-constant', 'grid-wrap', 'nearest', + 'mirror', 'reflect']) + @pytest.mark.parametrize('order', range(6)) + def test_shift_vs_padded(self, order, mode, xp): + x_np = np.arange(144, dtype=float).reshape(12, 12) + shift = (0.4, -2.3) + + # manually pad and then extract center to get expected result + npad = 32 + pad_mode = ndimage_to_numpy_mode.get(mode) + x_padded = xp.asarray(np.pad(x_np, npad, mode=pad_mode)) + x = xp.asarray(x_np) + + center_slice = tuple([slice(npad, -npad)] * x.ndim) + expected_result = ndimage.shift( + x_padded, shift, mode=mode, order=order)[center_slice] + + xp_assert_close( + ndimage.shift(x, shift, mode=mode, order=order), + expected_result, + rtol=1e-7, + ) + + +class TestZoom: + + @pytest.mark.parametrize('order', range(0, 6)) + def test_zoom1(self, order, xp): + for z in [2, [2, 2]]: + arr = xp.reshape(xp.arange(25, dtype=xp.float64), (5, 5)) + arr = ndimage.zoom(arr, z, order=order) + assert arr.shape == (10, 10) + assert xp.all(arr[-1, :] != 0) + assert xp.all(arr[-1, :] >= (20 - eps)) + assert xp.all(arr[0, :] <= (5 + eps)) + assert xp.all(arr >= (0 - eps)) + assert xp.all(arr <= (24 + eps)) + + def test_zoom2(self, xp): + arr = xp.reshape(xp.arange(12), (3, 4)) + out = ndimage.zoom(ndimage.zoom(arr, 2), 0.5) + xp_assert_equal(out, arr) + + def test_zoom3(self, xp): + arr = xp.asarray([[1, 2]]) + out1 = ndimage.zoom(arr, (2, 1)) + out2 = ndimage.zoom(arr, (1, 2)) + + assert_array_almost_equal(out1, xp.asarray([[1, 2], [1, 2]])) + assert_array_almost_equal(out2, xp.asarray([[1, 1, 2, 2]])) + + @pytest.mark.parametrize('order', range(0, 6)) + @pytest.mark.parametrize('dtype', ["float64", "complex128"]) + def test_zoom_affine01(self, order, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.asarray([[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]], dtype=dtype) + isdtype = array_namespace(data).isdtype + if isdtype(data.dtype, 'complex floating'): + data -= 1j * data + with suppress_warnings() as sup: + sup.filter(UserWarning, + 'The behavior of affine_transform with a 1-D array .* ' + 'has changed') + out = ndimage.affine_transform(data, xp.asarray([0.5, 0.5]), 0, + (6, 8), order=order) + assert_array_almost_equal(out[::2, ::2], data) + + def test_zoom_infinity(self, xp): + # Ticket #1419 regression test + dim = 8 + ndimage.zoom(xp.zeros((dim, dim)), 1. / dim, mode='nearest') + + def test_zoom_zoomfactor_one(self, xp): + # Ticket #1122 regression test + arr = xp.zeros((1, 5, 5)) + zoom = (1.0, 2.0, 2.0) + + out = ndimage.zoom(arr, zoom, cval=7) + ref = xp.zeros((1, 10, 10)) + assert_array_almost_equal(out, ref) + + def test_zoom_output_shape_roundoff(self, xp): + arr = xp.zeros((3, 11, 25)) + zoom = (4.0 / 3, 15.0 / 11, 29.0 / 25) + out = ndimage.zoom(arr, zoom) + assert out.shape == (4, 15, 29) + + @pytest.mark.parametrize('zoom', [(1, 1), (3, 5), (8, 2), (8, 8)]) + @pytest.mark.parametrize('mode', ['nearest', 'constant', 'wrap', 'reflect', + 'mirror', 'grid-wrap', 'grid-mirror', + 'grid-constant']) + def test_zoom_by_int_order0(self, zoom, mode, xp): + # order 0 zoom should be the same as replication via np.kron + # Note: This is not True for general x shapes when grid_mode is False, + # but works here for all modes because the size ratio happens to + # always be an integer when x.shape = (2, 2). + x_np = np.asarray([[0, 1], + [2, 3]], dtype=np.float64) + expected = np.kron(x_np, np.ones(zoom)) + + x = xp.asarray(x_np) + expected = xp.asarray(expected) + + assert_array_almost_equal( + ndimage.zoom(x, zoom, order=0, mode=mode), + expected + ) + + @pytest.mark.parametrize('shape', [(2, 3), (4, 4)]) + @pytest.mark.parametrize('zoom', [(1, 1), (3, 5), (8, 2), (8, 8)]) + @pytest.mark.parametrize('mode', ['nearest', 'reflect', 'mirror', + 'grid-wrap', 'grid-constant']) + def test_zoom_grid_by_int_order0(self, shape, zoom, mode, xp): + # When grid_mode is True, order 0 zoom should be the same as + # replication via np.kron. The only exceptions to this are the + # non-grid modes 'constant' and 'wrap'. + x_np = np.arange(np.prod(shape), dtype=float).reshape(shape) + + x = xp.asarray(x_np) + assert_array_almost_equal( + ndimage.zoom(x, zoom, order=0, mode=mode, grid_mode=True), + xp.asarray(np.kron(x_np, np.ones(zoom))) + ) + + @pytest.mark.parametrize('mode', ['constant', 'wrap']) + @pytest.mark.thread_unsafe + def test_zoom_grid_mode_warnings(self, mode, xp): + # Warn on use of non-grid modes when grid_mode is True + x = xp.reshape(xp.arange(9, dtype=xp.float64), (3, 3)) + with pytest.warns(UserWarning, + match="It is recommended to use mode"): + ndimage.zoom(x, 2, mode=mode, grid_mode=True), + + @skip_xp_backends(np_only=True, reason='inplace output= is numpy-specific') + def test_zoom_output_shape(self, xp): + """Ticket #643""" + x = xp.reshape(xp.arange(12), (3, 4)) + ndimage.zoom(x, 2, output=xp.zeros((6, 8))) + + def test_zoom_0d_array(self, xp): + # Ticket #21670 regression test + a = xp.arange(10.) + factor = 2 + actual = ndimage.zoom(a, np.array(factor)) + expected = ndimage.zoom(a, factor) + xp_assert_close(actual, expected) + + +class TestRotate: + + @pytest.mark.parametrize('order', range(0, 6)) + def test_rotate01(self, order, xp): + data = xp.asarray([[0, 0, 0, 0], + [0, 1, 1, 0], + [0, 0, 0, 0]], dtype=xp.float64) + out = ndimage.rotate(data, 0, order=order) + assert_array_almost_equal(out, data) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_rotate02(self, order, xp): + data = xp.asarray([[0, 0, 0, 0], + [0, 1, 0, 0], + [0, 0, 0, 0]], dtype=xp.float64) + expected = xp.asarray([[0, 0, 0], + [0, 0, 0], + [0, 1, 0], + [0, 0, 0]], dtype=xp.float64) + out = ndimage.rotate(data, 90, order=order) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('order', range(0, 6)) + @pytest.mark.parametrize('dtype', ["float64", "complex128"]) + def test_rotate03(self, order, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.asarray([[0, 0, 0, 0, 0], + [0, 1, 1, 0, 0], + [0, 0, 0, 0, 0]], dtype=dtype) + expected = xp.asarray([[0, 0, 0], + [0, 0, 0], + [0, 1, 0], + [0, 1, 0], + [0, 0, 0]], dtype=dtype) + isdtype = array_namespace(data).isdtype + if isdtype(data.dtype, 'complex floating'): + data -= 1j * data + expected -= 1j * expected + out = ndimage.rotate(data, 90, order=order) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_rotate04(self, order, xp): + data = xp.asarray([[0, 0, 0, 0, 0], + [0, 1, 1, 0, 0], + [0, 0, 0, 0, 0]], dtype=xp.float64) + expected = xp.asarray([[0, 0, 0, 0, 0], + [0, 0, 1, 0, 0], + [0, 0, 1, 0, 0]], dtype=xp.float64) + out = ndimage.rotate(data, 90, reshape=False, order=order) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_rotate05(self, order, xp): + data = np.empty((4, 3, 3)) + for i in range(3): + data[:, :, i] = np.asarray([[0, 0, 0], + [0, 1, 0], + [0, 1, 0], + [0, 0, 0]], dtype=np.float64) + data = xp.asarray(data) + expected = xp.asarray([[0, 0, 0, 0], + [0, 1, 1, 0], + [0, 0, 0, 0]], dtype=xp.float64) + out = ndimage.rotate(data, 90, order=order) + for i in range(3): + assert_array_almost_equal(out[:, :, i], expected) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_rotate06(self, order, xp): + data = np.empty((3, 4, 3)) + for i in range(3): + data[:, :, i] = np.asarray([[0, 0, 0, 0], + [0, 1, 1, 0], + [0, 0, 0, 0]], dtype=np.float64) + data = xp.asarray(data) + expected = xp.asarray([[0, 0, 0], + [0, 1, 0], + [0, 1, 0], + [0, 0, 0]], dtype=xp.float64) + out = ndimage.rotate(data, 90, order=order) + for i in range(3): + assert_array_almost_equal(out[:, :, i], expected) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_rotate07(self, order, xp): + data = xp.asarray([[[0, 0, 0, 0, 0], + [0, 1, 1, 0, 0], + [0, 0, 0, 0, 0]]] * 2, dtype=xp.float64) + permute_dims = array_namespace(data).permute_dims + data = permute_dims(data, (2, 1, 0)) + expected = xp.asarray([[[0, 0, 0], + [0, 1, 0], + [0, 1, 0], + [0, 0, 0], + [0, 0, 0]]] * 2, dtype=xp.float64) + expected = permute_dims(expected, (2, 1, 0)) + out = ndimage.rotate(data, 90, axes=(0, 1), order=order) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('order', range(0, 6)) + def test_rotate08(self, order, xp): + data = xp.asarray([[[0, 0, 0, 0, 0], + [0, 1, 1, 0, 0], + [0, 0, 0, 0, 0]]] * 2, dtype=xp.float64) + permute_dims = array_namespace(data).permute_dims + data = permute_dims(data, (2, 1, 0)) # == np.transpose + expected = xp.asarray([[[0, 0, 1, 0, 0], + [0, 0, 1, 0, 0], + [0, 0, 0, 0, 0]]] * 2, dtype=xp.float64) + permute_dims = array_namespace(data).permute_dims + expected = permute_dims(expected, (2, 1, 0)) + out = ndimage.rotate(data, 90, axes=(0, 1), reshape=False, order=order) + assert_array_almost_equal(out, expected) + + def test_rotate09(self, xp): + data = xp.asarray([[0, 0, 0, 0, 0], + [0, 1, 1, 0, 0], + [0, 0, 0, 0, 0]] * 2, dtype=xp.float64) + with assert_raises(ValueError): + ndimage.rotate(data, 90, axes=(0, data.ndim)) + + def test_rotate10(self, xp): + data = xp.reshape(xp.arange(45, dtype=xp.float64), (3, 5, 3)) + + # The output of ndimage.rotate before refactoring + expected = xp.asarray([[[0.0, 0.0, 0.0], + [0.0, 0.0, 0.0], + [6.54914793, 7.54914793, 8.54914793], + [10.84520162, 11.84520162, 12.84520162], + [0.0, 0.0, 0.0]], + [[6.19286575, 7.19286575, 8.19286575], + [13.4730712, 14.4730712, 15.4730712], + [21.0, 22.0, 23.0], + [28.5269288, 29.5269288, 30.5269288], + [35.80713425, 36.80713425, 37.80713425]], + [[0.0, 0.0, 0.0], + [31.15479838, 32.15479838, 33.15479838], + [35.45085207, 36.45085207, 37.45085207], + [0.0, 0.0, 0.0], + [0.0, 0.0, 0.0]]], dtype=xp.float64) + + out = ndimage.rotate(data, angle=12, reshape=False) + #assert_array_almost_equal(out, expected) + xp_assert_close(out, expected, rtol=1e-6, atol=2e-6) + + def test_rotate_exact_180(self, xp): + if is_cupy(xp): + pytest.xfail("https://github.com/cupy/cupy/issues/8400") + + a = np.tile(xp.arange(5), (5, 1)) + b = ndimage.rotate(ndimage.rotate(a, 180), -180) + xp_assert_equal(a, b) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_measurements.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_measurements.py new file mode 100644 index 0000000000000000000000000000000000000000..c8175ba309dd223a6d0fd46df017ea1fbc797a4e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_measurements.py @@ -0,0 +1,1609 @@ +import os +import os.path + +import numpy as np +from numpy.testing import suppress_warnings + +from scipy._lib._array_api import ( + is_jax, + is_torch, + array_namespace, + xp_assert_equal, + xp_assert_close, + assert_array_almost_equal, + assert_almost_equal, +) + +import pytest +from pytest import raises as assert_raises + +import scipy.ndimage as ndimage + +from . import types + +from scipy.conftest import array_api_compatible +skip_xp_backends = pytest.mark.skip_xp_backends +pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends"), + skip_xp_backends(cpu_only=True, exceptions=['cupy', 'jax.numpy'],)] + +IS_WINDOWS_AND_NP1 = os.name == 'nt' and np.__version__ < '2' + + +@skip_xp_backends(np_only=True, reason='test internal numpy-only helpers') +class Test_measurements_stats: + """ndimage._measurements._stats() is a utility used by other functions. + + Since internal ndimage/_measurements.py code is NumPy-only, + so is this this test class. + """ + def test_a(self, xp): + x = [0, 1, 2, 6] + labels = [0, 0, 1, 1] + index = [0, 1] + for shp in [(4,), (2, 2)]: + x = np.array(x).reshape(shp) + labels = np.array(labels).reshape(shp) + counts, sums = ndimage._measurements._stats( + x, labels=labels, index=index) + + dtype_arg = {'dtype': np.int64} if IS_WINDOWS_AND_NP1 else {} + xp_assert_equal(counts, np.asarray([2, 2], **dtype_arg)) + xp_assert_equal(sums, np.asarray([1.0, 8.0])) + + def test_b(self, xp): + # Same data as test_a, but different labels. The label 9 exceeds the + # length of 'labels', so this test will follow a different code path. + x = [0, 1, 2, 6] + labels = [0, 0, 9, 9] + index = [0, 9] + for shp in [(4,), (2, 2)]: + x = np.array(x).reshape(shp) + labels = np.array(labels).reshape(shp) + counts, sums = ndimage._measurements._stats( + x, labels=labels, index=index) + + dtype_arg = {'dtype': np.int64} if IS_WINDOWS_AND_NP1 else {} + xp_assert_equal(counts, np.asarray([2, 2], **dtype_arg)) + xp_assert_equal(sums, np.asarray([1.0, 8.0])) + + def test_a_centered(self, xp): + x = [0, 1, 2, 6] + labels = [0, 0, 1, 1] + index = [0, 1] + for shp in [(4,), (2, 2)]: + x = np.array(x).reshape(shp) + labels = np.array(labels).reshape(shp) + counts, sums, centers = ndimage._measurements._stats( + x, labels=labels, index=index, centered=True) + + dtype_arg = {'dtype': np.int64} if IS_WINDOWS_AND_NP1 else {} + xp_assert_equal(counts, np.asarray([2, 2], **dtype_arg)) + xp_assert_equal(sums, np.asarray([1.0, 8.0])) + xp_assert_equal(centers, np.asarray([0.5, 8.0])) + + def test_b_centered(self, xp): + x = [0, 1, 2, 6] + labels = [0, 0, 9, 9] + index = [0, 9] + for shp in [(4,), (2, 2)]: + x = np.array(x).reshape(shp) + labels = np.array(labels).reshape(shp) + counts, sums, centers = ndimage._measurements._stats( + x, labels=labels, index=index, centered=True) + + dtype_arg = {'dtype': np.int64} if IS_WINDOWS_AND_NP1 else {} + xp_assert_equal(counts, np.asarray([2, 2], **dtype_arg)) + xp_assert_equal(sums, np.asarray([1.0, 8.0])) + xp_assert_equal(centers, np.asarray([0.5, 8.0])) + + def test_nonint_labels(self, xp): + x = [0, 1, 2, 6] + labels = [0.0, 0.0, 9.0, 9.0] + index = [0.0, 9.0] + for shp in [(4,), (2, 2)]: + x = np.array(x).reshape(shp) + labels = np.array(labels).reshape(shp) + counts, sums, centers = ndimage._measurements._stats( + x, labels=labels, index=index, centered=True) + + dtype_arg = {'dtype': np.int64} if IS_WINDOWS_AND_NP1 else {} + xp_assert_equal(counts, np.asarray([2, 2], **dtype_arg)) + xp_assert_equal(sums, np.asarray([1.0, 8.0])) + xp_assert_equal(centers, np.asarray([0.5, 8.0])) + + +class Test_measurements_select: + """ndimage._measurements._select() is a utility used by other functions.""" + + def test_basic(self, xp): + x = [0, 1, 6, 2] + cases = [ + ([0, 0, 1, 1], [0, 1]), # "Small" integer labels + ([0, 0, 9, 9], [0, 9]), # A label larger than len(labels) + ([0.0, 0.0, 7.0, 7.0], [0.0, 7.0]), # Non-integer labels + ] + for labels, index in cases: + result = ndimage._measurements._select( + x, labels=labels, index=index) + assert len(result) == 0 + result = ndimage._measurements._select( + x, labels=labels, index=index, find_max=True) + assert len(result) == 1 + xp_assert_equal(result[0], [1, 6]) + result = ndimage._measurements._select( + x, labels=labels, index=index, find_min=True) + assert len(result) == 1 + xp_assert_equal(result[0], [0, 2]) + result = ndimage._measurements._select( + x, labels=labels, index=index, find_min=True, + find_min_positions=True) + assert len(result) == 2 + xp_assert_equal(result[0], [0, 2]) + xp_assert_equal(result[1], [0, 3]) + assert result[1].dtype.kind == 'i' + result = ndimage._measurements._select( + x, labels=labels, index=index, find_max=True, + find_max_positions=True) + assert len(result) == 2 + xp_assert_equal(result[0], [1, 6]) + xp_assert_equal(result[1], [1, 2]) + assert result[1].dtype.kind == 'i' + + +def test_label01(xp): + data = xp.ones([]) + out, n = ndimage.label(data) + assert out == 1 + assert n == 1 + + +def test_label02(xp): + data = xp.zeros([]) + out, n = ndimage.label(data) + assert out == 0 + assert n == 0 + + +@pytest.mark.thread_unsafe # due to Cython fused types, see cython#6506 +def test_label03(xp): + data = xp.ones([1]) + out, n = ndimage.label(data) + assert_array_almost_equal(out, xp.asarray([1])) + assert n == 1 + + +def test_label04(xp): + data = xp.zeros([1]) + out, n = ndimage.label(data) + assert_array_almost_equal(out, xp.asarray([0])) + assert n == 0 + + +def test_label05(xp): + data = xp.ones([5]) + out, n = ndimage.label(data) + assert_array_almost_equal(out, xp.asarray([1, 1, 1, 1, 1])) + assert n == 1 + + +def test_label06(xp): + data = xp.asarray([1, 0, 1, 1, 0, 1]) + out, n = ndimage.label(data) + assert_array_almost_equal(out, xp.asarray([1, 0, 2, 2, 0, 3])) + assert n == 3 + + +def test_label07(xp): + data = xp.asarray([[0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0]]) + out, n = ndimage.label(data) + assert_array_almost_equal(out, xp.asarray( + [[0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0]])) + assert n == 0 + + +def test_label08(xp): + data = xp.asarray([[1, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0], + [1, 1, 0, 0, 0, 0], + [1, 1, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 0]]) + out, n = ndimage.label(data) + assert_array_almost_equal(out, xp.asarray([[1, 0, 0, 0, 0, 0], + [0, 0, 2, 2, 0, 0], + [0, 0, 2, 2, 2, 0], + [3, 3, 0, 0, 0, 0], + [3, 3, 0, 0, 0, 0], + [0, 0, 0, 4, 4, 0]])) + assert n == 4 + + +def test_label09(xp): + data = xp.asarray([[1, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0], + [1, 1, 0, 0, 0, 0], + [1, 1, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 0]]) + struct = ndimage.generate_binary_structure(2, 2) + struct = xp.asarray(struct) + out, n = ndimage.label(data, struct) + assert_array_almost_equal(out, xp.asarray([[1, 0, 0, 0, 0, 0], + [0, 0, 2, 2, 0, 0], + [0, 0, 2, 2, 2, 0], + [2, 2, 0, 0, 0, 0], + [2, 2, 0, 0, 0, 0], + [0, 0, 0, 3, 3, 0]])) + assert n == 3 + + +def test_label10(xp): + data = xp.asarray([[0, 0, 0, 0, 0, 0], + [0, 1, 1, 0, 1, 0], + [0, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0]]) + struct = ndimage.generate_binary_structure(2, 2) + struct = xp.asarray(struct) + out, n = ndimage.label(data, struct) + assert_array_almost_equal(out, xp.asarray([[0, 0, 0, 0, 0, 0], + [0, 1, 1, 0, 1, 0], + [0, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0]])) + assert n == 1 + + +def test_label11(xp): + for type in types: + dtype = getattr(xp, type) + data = xp.asarray([[1, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0], + [1, 1, 0, 0, 0, 0], + [1, 1, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 0]], dtype=dtype) + out, n = ndimage.label(data) + expected = [[1, 0, 0, 0, 0, 0], + [0, 0, 2, 2, 0, 0], + [0, 0, 2, 2, 2, 0], + [3, 3, 0, 0, 0, 0], + [3, 3, 0, 0, 0, 0], + [0, 0, 0, 4, 4, 0]] + expected = xp.asarray(expected) + assert_array_almost_equal(out, expected) + assert n == 4 + + +@skip_xp_backends(np_only=True, reason='inplace output is numpy-specific') +def test_label11_inplace(xp): + for type in types: + dtype = getattr(xp, type) + data = xp.asarray([[1, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0], + [1, 1, 0, 0, 0, 0], + [1, 1, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 0]], dtype=dtype) + n = ndimage.label(data, output=data) + expected = [[1, 0, 0, 0, 0, 0], + [0, 0, 2, 2, 0, 0], + [0, 0, 2, 2, 2, 0], + [3, 3, 0, 0, 0, 0], + [3, 3, 0, 0, 0, 0], + [0, 0, 0, 4, 4, 0]] + expected = xp.asarray(expected) + assert_array_almost_equal(data, expected) + assert n == 4 + + +def test_label12(xp): + for type in types: + dtype = getattr(xp, type) + data = xp.asarray([[0, 0, 0, 0, 1, 1], + [0, 0, 0, 0, 0, 1], + [0, 0, 1, 0, 1, 1], + [0, 0, 1, 1, 1, 1], + [0, 0, 0, 1, 1, 0]], dtype=dtype) + out, n = ndimage.label(data) + expected = [[0, 0, 0, 0, 1, 1], + [0, 0, 0, 0, 0, 1], + [0, 0, 1, 0, 1, 1], + [0, 0, 1, 1, 1, 1], + [0, 0, 0, 1, 1, 0]] + expected = xp.asarray(expected) + assert_array_almost_equal(out, expected) + assert n == 1 + + +def test_label13(xp): + for type in types: + dtype = getattr(xp, type) + data = xp.asarray([[1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1], + [1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1], + [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], + [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]], + dtype=dtype) + out, n = ndimage.label(data) + expected = [[1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1], + [1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1], + [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], + [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] + expected = xp.asarray(expected) + assert_array_almost_equal(out, expected) + assert n == 1 + + +@skip_xp_backends(np_only=True, reason='output=dtype is numpy-specific') +def test_label_output_typed(xp): + data = xp.ones([5]) + for t in types: + dtype = getattr(xp, t) + output = xp.zeros([5], dtype=dtype) + n = ndimage.label(data, output=output) + assert_array_almost_equal(output, + xp.ones(output.shape, dtype=output.dtype)) + assert n == 1 + + +@skip_xp_backends(np_only=True, reason='output=dtype is numpy-specific') +def test_label_output_dtype(xp): + data = xp.ones([5]) + for t in types: + dtype = getattr(xp, t) + output, n = ndimage.label(data, output=dtype) + assert_array_almost_equal(output, + xp.ones(output.shape, dtype=output.dtype)) + assert output.dtype == t + + +def test_label_output_wrong_size(xp): + if is_jax(xp): + pytest.xfail("JAX does not raise") + + data = xp.ones([5]) + for t in types: + dtype = getattr(xp, t) + output = xp.zeros([10], dtype=dtype) + # TypeError is from non-numpy arrays as output + assert_raises((ValueError, TypeError), + ndimage.label, data, output=output) + + +def test_label_structuring_elements(xp): + data = np.loadtxt(os.path.join(os.path.dirname( + __file__), "data", "label_inputs.txt")) + strels = np.loadtxt(os.path.join( + os.path.dirname(__file__), "data", "label_strels.txt")) + results = np.loadtxt(os.path.join( + os.path.dirname(__file__), "data", "label_results.txt")) + data = data.reshape((-1, 7, 7)) + strels = strels.reshape((-1, 3, 3)) + results = results.reshape((-1, 7, 7)) + + data = xp.asarray(data) + strels = xp.asarray(strels) + results = xp.asarray(results) + r = 0 + for i in range(data.shape[0]): + d = data[i, :, :] + for j in range(strels.shape[0]): + s = strels[j, :, :] + xp_assert_equal(ndimage.label(d, s)[0], results[r, :, :], check_dtype=False) + r += 1 + +@skip_xp_backends("cupy", + reason="`cupyx.scipy.ndimage` does not have `find_objects`" +) +def test_ticket_742(xp): + def SE(img, thresh=.7, size=4): + mask = img > thresh + rank = len(mask.shape) + struct = ndimage.generate_binary_structure(rank, rank) + struct = xp.asarray(struct) + la, co = ndimage.label(mask, + struct) + _ = ndimage.find_objects(la) + + if np.dtype(np.intp) != np.dtype('i'): + shape = (3, 1240, 1240) + a = np.random.rand(np.prod(shape)).reshape(shape) + a = xp.asarray(a) + # shouldn't crash + SE(a) + + +def test_gh_issue_3025(xp): + """Github issue #3025 - improper merging of labels""" + d = np.zeros((60, 320)) + d[:, :257] = 1 + d[:, 260:] = 1 + d[36, 257] = 1 + d[35, 258] = 1 + d[35, 259] = 1 + d = xp.asarray(d) + assert ndimage.label(d, xp.ones((3, 3)))[1] == 1 + + +@skip_xp_backends("cupy", reason="cupyx.scipy.ndimage does not have find_object") +class TestFindObjects: + def test_label_default_dtype(self, xp): + test_array = np.random.rand(10, 10) + test_array = xp.asarray(test_array) + label, no_features = ndimage.label(test_array > 0.5) + assert label.dtype in (xp.int32, xp.int64) + # Shouldn't raise an exception + ndimage.find_objects(label) + + + def test_find_objects01(self, xp): + data = xp.ones([], dtype=xp.int64) + out = ndimage.find_objects(data) + assert out == [()] + + + def test_find_objects02(self, xp): + data = xp.zeros([], dtype=xp.int64) + out = ndimage.find_objects(data) + assert out == [] + + + def test_find_objects03(self, xp): + data = xp.ones([1], dtype=xp.int64) + out = ndimage.find_objects(data) + assert out == [(slice(0, 1, None),)] + + + def test_find_objects04(self, xp): + data = xp.zeros([1], dtype=xp.int64) + out = ndimage.find_objects(data) + assert out == [] + + + def test_find_objects05(self, xp): + data = xp.ones([5], dtype=xp.int64) + out = ndimage.find_objects(data) + assert out == [(slice(0, 5, None),)] + + + def test_find_objects06(self, xp): + data = xp.asarray([1, 0, 2, 2, 0, 3]) + out = ndimage.find_objects(data) + assert out == [(slice(0, 1, None),), + (slice(2, 4, None),), + (slice(5, 6, None),)] + + + def test_find_objects07(self, xp): + data = xp.asarray([[0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0]]) + out = ndimage.find_objects(data) + assert out == [] + + + def test_find_objects08(self, xp): + data = xp.asarray([[1, 0, 0, 0, 0, 0], + [0, 0, 2, 2, 0, 0], + [0, 0, 2, 2, 2, 0], + [3, 3, 0, 0, 0, 0], + [3, 3, 0, 0, 0, 0], + [0, 0, 0, 4, 4, 0]]) + out = ndimage.find_objects(data) + assert out == [(slice(0, 1, None), slice(0, 1, None)), + (slice(1, 3, None), slice(2, 5, None)), + (slice(3, 5, None), slice(0, 2, None)), + (slice(5, 6, None), slice(3, 5, None))] + + + def test_find_objects09(self, xp): + data = xp.asarray([[1, 0, 0, 0, 0, 0], + [0, 0, 2, 2, 0, 0], + [0, 0, 2, 2, 2, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 4, 4, 0]]) + out = ndimage.find_objects(data) + assert out == [(slice(0, 1, None), slice(0, 1, None)), + (slice(1, 3, None), slice(2, 5, None)), + None, + (slice(5, 6, None), slice(3, 5, None))] + + +def test_value_indices01(xp): + "Test dictionary keys and entries" + data = xp.asarray([[1, 0, 0, 0, 0, 0], + [0, 0, 2, 2, 0, 0], + [0, 0, 2, 2, 2, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 4, 4, 0]]) + vi = ndimage.value_indices(data, ignore_value=0) + true_keys = [1, 2, 4] + assert list(vi.keys()) == true_keys + + nnz_kwd = {'as_tuple': True} if is_torch(xp) else {} + + truevi = {} + for k in true_keys: + truevi[k] = xp.nonzero(data == k, **nnz_kwd) + + vi = ndimage.value_indices(data, ignore_value=0) + assert vi.keys() == truevi.keys() + for key in vi.keys(): + assert len(vi[key]) == len(truevi[key]) + for v, true_v in zip(vi[key], truevi[key]): + xp_assert_equal(v, true_v) + + +def test_value_indices02(xp): + "Test input checking" + data = xp.zeros((5, 4), dtype=xp.float32) + msg = "Parameter 'arr' must be an integer array" + with assert_raises(ValueError, match=msg): + ndimage.value_indices(data) + + +def test_value_indices03(xp): + "Test different input array shapes, from 1-D to 4-D" + for shape in [(36,), (18, 2), (3, 3, 4), (3, 3, 2, 2)]: + a = xp.asarray((12*[1]+12*[2]+12*[3]), dtype=xp.int32) + a = xp.reshape(a, shape) + + nnz_kwd = {'as_tuple': True} if is_torch(xp) else {} + + unique_values = array_namespace(a).unique_values + trueKeys = unique_values(a) + vi = ndimage.value_indices(a) + assert list(vi.keys()) == list(trueKeys) + for k in [int(x) for x in trueKeys]: + trueNdx = xp.nonzero(a == k, **nnz_kwd) + assert len(vi[k]) == len(trueNdx) + for vik, true_vik in zip(vi[k], trueNdx): + xp_assert_equal(vik, true_vik) + + +def test_sum01(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([], dtype=dtype) + output = ndimage.sum(input) + assert output == 0 + + +def test_sum02(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.zeros([0, 4], dtype=dtype) + output = ndimage.sum(input) + assert output == 0 + + +def test_sum03(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.ones([], dtype=dtype) + output = ndimage.sum(input) + assert_almost_equal(output, xp.asarray(1.0), check_0d=False) + + +def test_sum04(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([1, 2], dtype=dtype) + output = ndimage.sum(input) + assert_almost_equal(output, xp.asarray(3.0), check_0d=False) + + +def test_sum05(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.sum(input) + assert_almost_equal(output, xp.asarray(10.0), check_0d=False) + + +def test_sum06(xp): + labels = np.asarray([], dtype=bool) + labels = xp.asarray(labels) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([], dtype=dtype) + output = ndimage.sum(input, labels=labels) + assert output == 0 + + +def test_sum07(xp): + labels = np.ones([0, 4], dtype=bool) + labels = xp.asarray(labels) + for type in types: + dtype = getattr(xp, type) + input = xp.zeros([0, 4], dtype=dtype) + output = ndimage.sum(input, labels=labels) + assert output == 0 + + +def test_sum08(xp): + labels = np.asarray([1, 0], dtype=bool) + labels = xp.asarray(labels) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([1, 2], dtype=dtype) + output = ndimage.sum(input, labels=labels) + assert output == 1 + + +def test_sum09(xp): + labels = np.asarray([1, 0], dtype=bool) + labels = xp.asarray(labels) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.sum(input, labels=labels) + assert_almost_equal(output, xp.asarray(4.0), check_0d=False) + + +def test_sum10(xp): + labels = np.asarray([1, 0], dtype=bool) + input = np.asarray([[1, 2], [3, 4]], dtype=bool) + + labels = xp.asarray(labels) + input = xp.asarray(input) + output = ndimage.sum(input, labels=labels) + assert_almost_equal(output, xp.asarray(2.0), check_0d=False) + + +def test_sum11(xp): + labels = xp.asarray([1, 2], dtype=xp.int8) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.sum(input, labels=labels, + index=2) + assert_almost_equal(output, xp.asarray(6.0), check_0d=False) + + +def test_sum12(xp): + labels = xp.asarray([[1, 2], [2, 4]], dtype=xp.int8) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.sum(input, labels=labels, index=xp.asarray([4, 8, 2])) + assert_array_almost_equal(output, xp.asarray([4.0, 0.0, 5.0])) + + +def test_sum_labels(xp): + labels = xp.asarray([[1, 2], [2, 4]], dtype=xp.int8) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output_sum = ndimage.sum(input, labels=labels, index=xp.asarray([4, 8, 2])) + output_labels = ndimage.sum_labels( + input, labels=labels, index=xp.asarray([4, 8, 2])) + + assert xp.all(output_sum == output_labels) + assert_array_almost_equal(output_labels, xp.asarray([4.0, 0.0, 5.0])) + + +def test_mean01(xp): + labels = np.asarray([1, 0], dtype=bool) + labels = xp.asarray(labels) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.mean(input, labels=labels) + assert_almost_equal(output, xp.asarray(2.0), check_0d=False) + + +def test_mean02(xp): + labels = np.asarray([1, 0], dtype=bool) + input = np.asarray([[1, 2], [3, 4]], dtype=bool) + + labels = xp.asarray(labels) + input = xp.asarray(input) + output = ndimage.mean(input, labels=labels) + assert_almost_equal(output, xp.asarray(1.0), check_0d=False) + + +def test_mean03(xp): + labels = xp.asarray([1, 2]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.mean(input, labels=labels, + index=2) + assert_almost_equal(output, xp.asarray(3.0), check_0d=False) + + +def test_mean04(xp): + labels = xp.asarray([[1, 2], [2, 4]], dtype=xp.int8) + with np.errstate(all='ignore'): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.mean(input, labels=labels, + index=xp.asarray([4, 8, 2])) + # XXX: output[[0, 2]] does not work in array-api-strict; annoying + # assert_array_almost_equal(output[[0, 2]], xp.asarray([4.0, 2.5])) + assert output[0] == 4.0 + assert output[2] == 2.5 + assert xp.isnan(output[1]) + + +def test_minimum01(xp): + labels = np.asarray([1, 0], dtype=bool) + labels = xp.asarray(labels) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.minimum(input, labels=labels) + assert_almost_equal(output, xp.asarray(1.0), check_0d=False) + + +def test_minimum02(xp): + labels = np.asarray([1, 0], dtype=bool) + input = np.asarray([[2, 2], [2, 4]], dtype=bool) + + labels = xp.asarray(labels) + input = xp.asarray(input) + output = ndimage.minimum(input, labels=labels) + assert_almost_equal(output, xp.asarray(1.0), check_0d=False) + + +def test_minimum03(xp): + labels = xp.asarray([1, 2]) + for type in types: + dtype = getattr(xp, type) + + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.minimum(input, labels=labels, + index=2) + assert_almost_equal(output, xp.asarray(2.0), check_0d=False) + + +def test_minimum04(xp): + labels = xp.asarray([[1, 2], [2, 3]]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.minimum(input, labels=labels, + index=xp.asarray([2, 3, 8])) + assert_array_almost_equal(output, xp.asarray([2.0, 4.0, 0.0])) + + +def test_maximum01(xp): + labels = np.asarray([1, 0], dtype=bool) + labels = xp.asarray(labels) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.maximum(input, labels=labels) + assert_almost_equal(output, xp.asarray(3.0), check_0d=False) + + +def test_maximum02(xp): + labels = np.asarray([1, 0], dtype=bool) + input = np.asarray([[2, 2], [2, 4]], dtype=bool) + labels = xp.asarray(labels) + input = xp.asarray(input) + output = ndimage.maximum(input, labels=labels) + assert_almost_equal(output, xp.asarray(1.0), check_0d=False) + + +def test_maximum03(xp): + labels = xp.asarray([1, 2]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.maximum(input, labels=labels, + index=2) + assert_almost_equal(output, xp.asarray(4.0), check_0d=False) + + +def test_maximum04(xp): + labels = xp.asarray([[1, 2], [2, 3]]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.maximum(input, labels=labels, + index=xp.asarray([2, 3, 8])) + assert_array_almost_equal(output, xp.asarray([3.0, 4.0, 0.0])) + + +def test_maximum05(xp): + # Regression test for ticket #501 (Trac) + x = xp.asarray([-3, -2, -1]) + assert ndimage.maximum(x) == -1 + + +def test_median01(xp): + a = xp.asarray([[1, 2, 0, 1], + [5, 3, 0, 4], + [0, 0, 0, 7], + [9, 3, 0, 0]]) + labels = xp.asarray([[1, 1, 0, 2], + [1, 1, 0, 2], + [0, 0, 0, 2], + [3, 3, 0, 0]]) + output = ndimage.median(a, labels=labels, index=xp.asarray([1, 2, 3])) + assert_array_almost_equal(output, xp.asarray([2.5, 4.0, 6.0])) + + +def test_median02(xp): + a = xp.asarray([[1, 2, 0, 1], + [5, 3, 0, 4], + [0, 0, 0, 7], + [9, 3, 0, 0]]) + output = ndimage.median(a) + assert_almost_equal(output, xp.asarray(1.0), check_0d=False) + + +def test_median03(xp): + a = xp.asarray([[1, 2, 0, 1], + [5, 3, 0, 4], + [0, 0, 0, 7], + [9, 3, 0, 0]]) + labels = xp.asarray([[1, 1, 0, 2], + [1, 1, 0, 2], + [0, 0, 0, 2], + [3, 3, 0, 0]]) + output = ndimage.median(a, labels=labels) + assert_almost_equal(output, xp.asarray(3.0), check_0d=False) + + +def test_median_gh12836_bool(xp): + # test boolean addition fix on example from gh-12836 + a = np.asarray([1, 1], dtype=bool) + a = xp.asarray(a) + output = ndimage.median(a, labels=xp.ones((2,)), index=xp.asarray([1])) + assert_array_almost_equal(output, xp.asarray([1.0])) + + +def test_median_no_int_overflow(xp): + # test integer overflow fix on example from gh-12836 + a = xp.asarray([65, 70], dtype=xp.int8) + output = ndimage.median(a, labels=xp.ones((2,)), index=xp.asarray([1])) + assert_array_almost_equal(output, xp.asarray([67.5])) + + +def test_variance01(xp): + with np.errstate(all='ignore'): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([], dtype=dtype) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "Mean of empty slice") + output = ndimage.variance(input) + assert xp.isnan(output) + + +def test_variance02(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([1], dtype=dtype) + output = ndimage.variance(input) + assert_almost_equal(output, xp.asarray(0.0), check_0d=False) + + +def test_variance03(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([1, 3], dtype=dtype) + output = ndimage.variance(input) + assert_almost_equal(output, xp.asarray(1.0), check_0d=False) + + +def test_variance04(xp): + input = np.asarray([1, 0], dtype=bool) + input = xp.asarray(input) + output = ndimage.variance(input) + assert_almost_equal(output, xp.asarray(0.25), check_0d=False) + + +def test_variance05(xp): + labels = xp.asarray([2, 2, 3]) + for type in types: + dtype = getattr(xp, type) + + input = xp.asarray([1, 3, 8], dtype=dtype) + output = ndimage.variance(input, labels, 2) + assert_almost_equal(output, xp.asarray(1.0), check_0d=False) + + +def test_variance06(xp): + labels = xp.asarray([2, 2, 3, 3, 4]) + with np.errstate(all='ignore'): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([1, 3, 8, 10, 8], dtype=dtype) + output = ndimage.variance(input, labels, xp.asarray([2, 3, 4])) + assert_array_almost_equal(output, xp.asarray([1.0, 1.0, 0.0])) + + +def test_standard_deviation01(xp): + with np.errstate(all='ignore'): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([], dtype=dtype) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "Mean of empty slice") + output = ndimage.standard_deviation(input) + assert xp.isnan(output) + + +def test_standard_deviation02(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([1], dtype=dtype) + output = ndimage.standard_deviation(input) + assert_almost_equal(output, xp.asarray(0.0), check_0d=False) + + +def test_standard_deviation03(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([1, 3], dtype=dtype) + output = ndimage.standard_deviation(input) + assert_almost_equal(output, xp.asarray(1.0), check_0d=False) + + +def test_standard_deviation04(xp): + input = np.asarray([1, 0], dtype=bool) + input = xp.asarray(input) + output = ndimage.standard_deviation(input) + assert_almost_equal(output, xp.asarray(0.5), check_0d=False) + + +def test_standard_deviation05(xp): + labels = xp.asarray([2, 2, 3]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([1, 3, 8], dtype=dtype) + output = ndimage.standard_deviation(input, labels, 2) + assert_almost_equal(output, xp.asarray(1.0), check_0d=False) + + +def test_standard_deviation06(xp): + labels = xp.asarray([2, 2, 3, 3, 4]) + with np.errstate(all='ignore'): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([1, 3, 8, 10, 8], dtype=dtype) + output = ndimage.standard_deviation( + input, labels, xp.asarray([2, 3, 4]) + ) + assert_array_almost_equal(output, xp.asarray([1.0, 1.0, 0.0])) + + +def test_standard_deviation07(xp): + labels = xp.asarray([1]) + with np.errstate(all='ignore'): + for type in types: + if is_torch(xp) and type == 'uint8': + pytest.xfail("value cannot be converted to type uint8 " + "without overflow") + dtype = getattr(xp, type) + input = xp.asarray([-0.00619519], dtype=dtype) + output = ndimage.standard_deviation(input, labels, xp.asarray([1])) + assert_array_almost_equal(output, xp.asarray([0])) + + +def test_minimum_position01(xp): + labels = np.asarray([1, 0], dtype=bool) + labels = xp.asarray(labels) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.minimum_position(input, labels=labels) + assert output == (0, 0) + + +def test_minimum_position02(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[5, 4, 2, 5], + [3, 7, 0, 2], + [1, 5, 1, 1]], dtype=dtype) + output = ndimage.minimum_position(input) + assert output == (1, 2) + + +def test_minimum_position03(xp): + input = np.asarray([[5, 4, 2, 5], + [3, 7, 0, 2], + [1, 5, 1, 1]], dtype=bool) + input = xp.asarray(input) + output = ndimage.minimum_position(input) + assert output == (1, 2) + + +def test_minimum_position04(xp): + input = np.asarray([[5, 4, 2, 5], + [3, 7, 1, 2], + [1, 5, 1, 1]], dtype=bool) + input = xp.asarray(input) + output = ndimage.minimum_position(input) + assert output == (0, 0) + + +def test_minimum_position05(xp): + labels = xp.asarray([1, 2, 0, 4]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[5, 4, 2, 5], + [3, 7, 0, 2], + [1, 5, 2, 3]], dtype=dtype) + output = ndimage.minimum_position(input, labels) + assert output == (2, 0) + + +def test_minimum_position06(xp): + labels = xp.asarray([1, 2, 3, 4]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[5, 4, 2, 5], + [3, 7, 0, 2], + [1, 5, 1, 1]], dtype=dtype) + output = ndimage.minimum_position(input, labels, 2) + assert output == (0, 1) + + +def test_minimum_position07(xp): + labels = xp.asarray([1, 2, 3, 4]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[5, 4, 2, 5], + [3, 7, 0, 2], + [1, 5, 1, 1]], dtype=dtype) + output = ndimage.minimum_position(input, labels, + xp.asarray([2, 3])) + assert output[0] == (0, 1) + assert output[1] == (1, 2) + + +def test_maximum_position01(xp): + labels = np.asarray([1, 0], dtype=bool) + labels = xp.asarray(labels) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output = ndimage.maximum_position(input, + labels=labels) + assert output == (1, 0) + + +def test_maximum_position02(xp): + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[5, 4, 2, 5], + [3, 7, 8, 2], + [1, 5, 1, 1]], dtype=dtype) + output = ndimage.maximum_position(input) + assert output == (1, 2) + + +def test_maximum_position03(xp): + input = np.asarray([[5, 4, 2, 5], + [3, 7, 8, 2], + [1, 5, 1, 1]], dtype=bool) + input = xp.asarray(input) + output = ndimage.maximum_position(input) + assert output == (0, 0) + + +def test_maximum_position04(xp): + labels = xp.asarray([1, 2, 0, 4]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[5, 4, 2, 5], + [3, 7, 8, 2], + [1, 5, 1, 1]], dtype=dtype) + output = ndimage.maximum_position(input, labels) + assert output == (1, 1) + + +def test_maximum_position05(xp): + labels = xp.asarray([1, 2, 0, 4]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[5, 4, 2, 5], + [3, 7, 8, 2], + [1, 5, 1, 1]], dtype=dtype) + output = ndimage.maximum_position(input, labels, 1) + assert output == (0, 0) + + +def test_maximum_position06(xp): + labels = xp.asarray([1, 2, 0, 4]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[5, 4, 2, 5], + [3, 7, 8, 2], + [1, 5, 1, 1]], dtype=dtype) + output = ndimage.maximum_position(input, labels, + xp.asarray([1, 2])) + assert output[0] == (0, 0) + assert output[1] == (1, 1) + + +def test_maximum_position07(xp): + # Test float labels + if is_torch(xp): + pytest.xfail("output[1] is wrong on pytorch") + + labels = xp.asarray([1.0, 2.5, 0.0, 4.5]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[5, 4, 2, 5], + [3, 7, 8, 2], + [1, 5, 1, 1]], dtype=dtype) + output = ndimage.maximum_position(input, labels, + xp.asarray([1.0, 4.5])) + assert output[0] == (0, 0) + assert output[1] == (0, 3) + + +def test_extrema01(xp): + labels = np.asarray([1, 0], dtype=bool) + labels = xp.asarray(labels) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output1 = ndimage.extrema(input, labels=labels) + output2 = ndimage.minimum(input, labels=labels) + output3 = ndimage.maximum(input, labels=labels) + output4 = ndimage.minimum_position(input, + labels=labels) + output5 = ndimage.maximum_position(input, + labels=labels) + assert output1 == (output2, output3, output4, output5) + + +def test_extrema02(xp): + labels = xp.asarray([1, 2]) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output1 = ndimage.extrema(input, labels=labels, + index=2) + output2 = ndimage.minimum(input, labels=labels, + index=2) + output3 = ndimage.maximum(input, labels=labels, + index=2) + output4 = ndimage.minimum_position(input, + labels=labels, index=2) + output5 = ndimage.maximum_position(input, + labels=labels, index=2) + assert output1 == (output2, output3, output4, output5) + + +def test_extrema03(xp): + labels = xp.asarray([[1, 2], [2, 3]]) + for type in types: + if is_torch(xp) and type in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, type) + input = xp.asarray([[1, 2], [3, 4]], dtype=dtype) + output1 = ndimage.extrema(input, + labels=labels, + index=xp.asarray([2, 3, 8])) + output2 = ndimage.minimum(input, + labels=labels, + index=xp.asarray([2, 3, 8])) + output3 = ndimage.maximum(input, labels=labels, + index=xp.asarray([2, 3, 8])) + output4 = ndimage.minimum_position(input, + labels=labels, + index=xp.asarray([2, 3, 8])) + output5 = ndimage.maximum_position(input, + labels=labels, + index=xp.asarray([2, 3, 8])) + assert_array_almost_equal(output1[0], output2) + assert_array_almost_equal(output1[1], output3) + assert output1[2] == output4 + assert output1[3] == output5 + + +def test_extrema04(xp): + labels = xp.asarray([1, 2, 0, 4]) + for type in types: + if is_torch(xp) and type in ("uint16", "uint32", "uint64"): + pytest.xfail("https://github.com/pytorch/pytorch/issues/58734") + + dtype = getattr(xp, type) + input = xp.asarray([[5, 4, 2, 5], + [3, 7, 8, 2], + [1, 5, 1, 1]], dtype=dtype) + output1 = ndimage.extrema(input, labels, xp.asarray([1, 2])) + output2 = ndimage.minimum(input, labels, xp.asarray([1, 2])) + output3 = ndimage.maximum(input, labels, xp.asarray([1, 2])) + output4 = ndimage.minimum_position(input, labels, + xp.asarray([1, 2])) + output5 = ndimage.maximum_position(input, labels, + xp.asarray([1, 2])) + assert_array_almost_equal(output1[0], output2) + assert_array_almost_equal(output1[1], output3) + assert output1[2] == output4 + assert output1[3] == output5 + + +def test_center_of_mass01(xp): + expected = (0.0, 0.0) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 0], [0, 0]], dtype=dtype) + output = ndimage.center_of_mass(input) + assert output == expected + + +def test_center_of_mass02(xp): + expected = (1, 0) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[0, 0], [1, 0]], dtype=dtype) + output = ndimage.center_of_mass(input) + assert output == expected + + +def test_center_of_mass03(xp): + expected = (0, 1) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[0, 1], [0, 0]], dtype=dtype) + output = ndimage.center_of_mass(input) + assert output == expected + + +def test_center_of_mass04(xp): + expected = (1, 1) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[0, 0], [0, 1]], dtype=dtype) + output = ndimage.center_of_mass(input) + assert output == expected + + +def test_center_of_mass05(xp): + expected = (0.5, 0.5) + for type in types: + dtype = getattr(xp, type) + input = xp.asarray([[1, 1], [1, 1]], dtype=dtype) + output = ndimage.center_of_mass(input) + assert output == expected + + +def test_center_of_mass06(xp): + expected = (0.5, 0.5) + input = np.asarray([[1, 2], [3, 1]], dtype=bool) + input = xp.asarray(input) + output = ndimage.center_of_mass(input) + assert output == expected + + +def test_center_of_mass07(xp): + labels = xp.asarray([1, 0]) + expected = (0.5, 0.0) + input = np.asarray([[1, 2], [3, 1]], dtype=bool) + input = xp.asarray(input) + output = ndimage.center_of_mass(input, labels) + assert output == expected + + +def test_center_of_mass08(xp): + labels = xp.asarray([1, 2]) + expected = (0.5, 1.0) + input = np.asarray([[5, 2], [3, 1]], dtype=bool) + input = xp.asarray(input) + output = ndimage.center_of_mass(input, labels, 2) + assert output == expected + + +def test_center_of_mass09(xp): + labels = xp.asarray((1, 2)) + expected = xp.asarray([(0.5, 0.0), (0.5, 1.0)], dtype=xp.float64) + input = np.asarray([[1, 2], [1, 1]], dtype=bool) + input = xp.asarray(input) + output = ndimage.center_of_mass(input, labels, xp.asarray([1, 2])) + xp_assert_equal(xp.asarray(output), xp.asarray(expected)) + + +def test_histogram01(xp): + expected = xp.ones(10) + input = xp.arange(10) + output = ndimage.histogram(input, 0, 10, 10) + assert_array_almost_equal(output, expected) + + +def test_histogram02(xp): + labels = xp.asarray([1, 1, 1, 1, 2, 2, 2, 2]) + expected = xp.asarray([0, 2, 0, 1, 1]) + input = xp.asarray([1, 1, 3, 4, 3, 3, 3, 3]) + output = ndimage.histogram(input, 0, 4, 5, labels, 1) + assert_array_almost_equal(output, expected) + + +@skip_xp_backends(np_only=True, reason='object arrays') +def test_histogram03(xp): + labels = xp.asarray([1, 0, 1, 1, 2, 2, 2, 2]) + expected1 = xp.asarray([0, 1, 0, 1, 1]) + expected2 = xp.asarray([0, 0, 0, 3, 0]) + input = xp.asarray([1, 1, 3, 4, 3, 5, 3, 3]) + + output = ndimage.histogram(input, 0, 4, 5, labels, (1, 2)) + + assert_array_almost_equal(output[0], expected1) + assert_array_almost_equal(output[1], expected2) + + +def test_stat_funcs_2d(xp): + a = xp.asarray([[5, 6, 0, 0, 0], [8, 9, 0, 0, 0], [0, 0, 0, 3, 5]]) + lbl = xp.asarray([[1, 1, 0, 0, 0], [1, 1, 0, 0, 0], [0, 0, 0, 2, 2]]) + + mean = ndimage.mean(a, labels=lbl, index=xp.asarray([1, 2])) + xp_assert_equal(mean, xp.asarray([7.0, 4.0], dtype=xp.float64)) + + var = ndimage.variance(a, labels=lbl, index=xp.asarray([1, 2])) + xp_assert_equal(var, xp.asarray([2.5, 1.0], dtype=xp.float64)) + + std = ndimage.standard_deviation(a, labels=lbl, index=xp.asarray([1, 2])) + assert_array_almost_equal(std, xp.sqrt(xp.asarray([2.5, 1.0], dtype=xp.float64))) + + med = ndimage.median(a, labels=lbl, index=xp.asarray([1, 2])) + xp_assert_equal(med, xp.asarray([7.0, 4.0], dtype=xp.float64)) + + min = ndimage.minimum(a, labels=lbl, index=xp.asarray([1, 2])) + xp_assert_equal(min, xp.asarray([5, 3]), check_dtype=False) + + max = ndimage.maximum(a, labels=lbl, index=xp.asarray([1, 2])) + xp_assert_equal(max, xp.asarray([9, 5]), check_dtype=False) + + +@skip_xp_backends("cupy", reason="no watershed_ift on CuPy") +class TestWatershedIft: + + def test_watershed_ift01(self, xp): + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 0, 0, 0, 1, 0], + [0, 1, 0, 0, 0, 1, 0], + [0, 1, 0, 0, 0, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=xp.uint8) + markers = xp.asarray([[-1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=xp.int8) + structure=xp.asarray([[1, 1, 1], + [1, 1, 1], + [1, 1, 1]]) + out = ndimage.watershed_ift(data, markers, structure=structure) + expected = [[-1, -1, -1, -1, -1, -1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, -1, -1, -1, -1, -1, -1], + [-1, -1, -1, -1, -1, -1, -1]] + assert_array_almost_equal(out, xp.asarray(expected)) + + def test_watershed_ift02(self, xp): + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 0, 0, 0, 1, 0], + [0, 1, 0, 0, 0, 1, 0], + [0, 1, 0, 0, 0, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=xp.uint8) + markers = xp.asarray([[-1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=xp.int8) + out = ndimage.watershed_ift(data, markers) + expected = [[-1, -1, -1, -1, -1, -1, -1], + [-1, -1, 1, 1, 1, -1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, -1, 1, 1, 1, -1, -1], + [-1, -1, -1, -1, -1, -1, -1], + [-1, -1, -1, -1, -1, -1, -1]] + assert_array_almost_equal(out, xp.asarray(expected)) + + def test_watershed_ift03(self, xp): + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 0, 1, 0, 1, 0], + [0, 1, 0, 1, 0, 1, 0], + [0, 1, 0, 1, 0, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=xp.uint8) + markers = xp.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 2, 0, 3, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, -1]], dtype=xp.int8) + out = ndimage.watershed_ift(data, markers) + expected = [[-1, -1, -1, -1, -1, -1, -1], + [-1, -1, 2, -1, 3, -1, -1], + [-1, 2, 2, 3, 3, 3, -1], + [-1, 2, 2, 3, 3, 3, -1], + [-1, 2, 2, 3, 3, 3, -1], + [-1, -1, 2, -1, 3, -1, -1], + [-1, -1, -1, -1, -1, -1, -1]] + assert_array_almost_equal(out, xp.asarray(expected)) + + def test_watershed_ift04(self, xp): + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 0, 1, 0, 1, 0], + [0, 1, 0, 1, 0, 1, 0], + [0, 1, 0, 1, 0, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=xp.uint8) + markers = xp.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 2, 0, 3, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, -1]], + dtype=xp.int8) + + structure=xp.asarray([[1, 1, 1], + [1, 1, 1], + [1, 1, 1]]) + out = ndimage.watershed_ift(data, markers, structure=structure) + expected = [[-1, -1, -1, -1, -1, -1, -1], + [-1, 2, 2, 3, 3, 3, -1], + [-1, 2, 2, 3, 3, 3, -1], + [-1, 2, 2, 3, 3, 3, -1], + [-1, 2, 2, 3, 3, 3, -1], + [-1, 2, 2, 3, 3, 3, -1], + [-1, -1, -1, -1, -1, -1, -1]] + assert_array_almost_equal(out, xp.asarray(expected)) + + def test_watershed_ift05(self, xp): + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 0, 1, 0, 1, 0], + [0, 1, 0, 1, 0, 1, 0], + [0, 1, 0, 1, 0, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=xp.uint8) + markers = xp.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 3, 0, 2, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, -1]], + dtype=xp.int8) + structure = xp.asarray([[1, 1, 1], + [1, 1, 1], + [1, 1, 1]]) + out = ndimage.watershed_ift(data, markers, structure=structure) + expected = [[-1, -1, -1, -1, -1, -1, -1], + [-1, 3, 3, 2, 2, 2, -1], + [-1, 3, 3, 2, 2, 2, -1], + [-1, 3, 3, 2, 2, 2, -1], + [-1, 3, 3, 2, 2, 2, -1], + [-1, 3, 3, 2, 2, 2, -1], + [-1, -1, -1, -1, -1, -1, -1]] + assert_array_almost_equal(out, xp.asarray(expected)) + + def test_watershed_ift06(self, xp): + data = xp.asarray([[0, 1, 0, 0, 0, 1, 0], + [0, 1, 0, 0, 0, 1, 0], + [0, 1, 0, 0, 0, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=xp.uint8) + markers = xp.asarray([[-1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=xp.int8) + structure=xp.asarray([[1, 1, 1], + [1, 1, 1], + [1, 1, 1]]) + out = ndimage.watershed_ift(data, markers, structure=structure) + expected = [[-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, -1, -1, -1, -1, -1, -1], + [-1, -1, -1, -1, -1, -1, -1]] + assert_array_almost_equal(out, xp.asarray(expected)) + + @skip_xp_backends(np_only=True, reason="inplace ops are numpy-specific") + def test_watershed_ift07(self, xp): + shape = (7, 6) + data = np.zeros(shape, dtype=np.uint8) + data = data.transpose() + data[...] = np.asarray([[0, 1, 0, 0, 0, 1, 0], + [0, 1, 0, 0, 0, 1, 0], + [0, 1, 0, 0, 0, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=np.uint8) + data = xp.asarray(data) + markers = xp.asarray([[-1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=xp.int8) + out = xp.zeros(shape, dtype=xp.int16) + out = out.T + structure=xp.asarray([[1, 1, 1], + [1, 1, 1], + [1, 1, 1]]) + ndimage.watershed_ift(data, markers, structure=structure, + output=out) + expected = [[-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, 1, 1, 1, 1, 1, -1], + [-1, -1, -1, -1, -1, -1, -1], + [-1, -1, -1, -1, -1, -1, -1]] + assert_array_almost_equal(out, xp.asarray(expected)) + + @skip_xp_backends("cupy", reason="no watershed_ift on CuPy") + def test_watershed_ift08(self, xp): + # Test cost larger than uint8. See gh-10069. + data = xp.asarray([[256, 0], + [0, 0]], dtype=xp.uint16) + markers = xp.asarray([[1, 0], + [0, 0]], dtype=xp.int8) + out = ndimage.watershed_ift(data, markers) + expected = [[1, 1], + [1, 1]] + assert_array_almost_equal(out, xp.asarray(expected)) + + @skip_xp_backends("cupy", reason="no watershed_ift on CuPy" ) + def test_watershed_ift09(self, xp): + # Test large cost. See gh-19575 + data = xp.asarray([[xp.iinfo(xp.uint16).max, 0], + [0, 0]], dtype=xp.uint16) + markers = xp.asarray([[1, 0], + [0, 0]], dtype=xp.int8) + out = ndimage.watershed_ift(data, markers) + expected = [[1, 1], + [1, 1]] + xp_assert_close(out, xp.asarray(expected), check_dtype=False) + + +@skip_xp_backends(np_only=True) +@pytest.mark.parametrize("dt", [np.intc, np.uintc]) +def test_gh_19423(dt, xp): + rng = np.random.default_rng(123) + max_val = 8 + image = rng.integers(low=0, high=max_val, size=(10, 12)).astype(dtype=dt) + val_idx = ndimage.value_indices(image) + assert len(val_idx.keys()) == max_val diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_morphology.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_morphology.py new file mode 100644 index 0000000000000000000000000000000000000000..9eff9a2c0f4a05295b7565761292b4ecaac007ac --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_morphology.py @@ -0,0 +1,2938 @@ +import numpy as np +from scipy._lib._array_api import ( + is_cupy, is_numpy, is_torch, array_namespace, + xp_assert_close, xp_assert_equal, assert_array_almost_equal +) +import pytest +from pytest import raises as assert_raises + +from scipy import ndimage + +from . import types + +from scipy.conftest import array_api_compatible +skip_xp_backends = pytest.mark.skip_xp_backends +xfail_xp_backends = pytest.mark.xfail_xp_backends +pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends"), + pytest.mark.usefixtures("xfail_xp_backends"), + skip_xp_backends(cpu_only=True, exceptions=['cupy', 'jax.numpy'],)] + + +class TestNdimageMorphology: + + @xfail_xp_backends('cupy', reason='CuPy does not have distance_transform_bf.') + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_bf01(self, dtype, xp): + dtype = getattr(xp, dtype) + + # brute force (bf) distance transform + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out, ft = ndimage.distance_transform_bf(data, 'euclidean', + return_indices=True) + expected = [[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 2, 4, 2, 1, 0, 0], + [0, 0, 1, 4, 8, 4, 1, 0, 0], + [0, 0, 1, 2, 4, 2, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + assert_array_almost_equal(out * out, expected) + + expected = [[[0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 1, 1, 1], + [2, 2, 2, 2, 1, 2, 2, 2, 2], + [3, 3, 3, 2, 1, 2, 3, 3, 3], + [4, 4, 4, 4, 6, 4, 4, 4, 4], + [5, 5, 6, 6, 7, 6, 6, 5, 5], + [6, 6, 6, 7, 7, 7, 6, 6, 6], + [7, 7, 7, 7, 7, 7, 7, 7, 7], + [8, 8, 8, 8, 8, 8, 8, 8, 8]], + [[0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 2, 4, 6, 6, 7, 8], + [0, 1, 1, 2, 4, 6, 7, 7, 8], + [0, 1, 1, 1, 6, 7, 7, 7, 8], + [0, 1, 2, 2, 4, 6, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8]]] + expected = xp.asarray(expected) + assert_array_almost_equal(ft, expected) + + @xfail_xp_backends('cupy', reason='CuPy does not have distance_transform_bf.') + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_bf02(self, dtype, xp): + dtype = getattr(xp, dtype) + + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out, ft = ndimage.distance_transform_bf(data, 'cityblock', + return_indices=True) + + expected = [[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 2, 2, 2, 1, 0, 0], + [0, 0, 1, 2, 3, 2, 1, 0, 0], + [0, 0, 1, 2, 2, 2, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + assert_array_almost_equal(out, expected) + + expected = [[[0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 1, 1, 1], + [2, 2, 2, 2, 1, 2, 2, 2, 2], + [3, 3, 3, 3, 1, 3, 3, 3, 3], + [4, 4, 4, 4, 7, 4, 4, 4, 4], + [5, 5, 6, 7, 7, 7, 6, 5, 5], + [6, 6, 6, 7, 7, 7, 6, 6, 6], + [7, 7, 7, 7, 7, 7, 7, 7, 7], + [8, 8, 8, 8, 8, 8, 8, 8, 8]], + [[0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 2, 4, 6, 6, 7, 8], + [0, 1, 1, 1, 4, 7, 7, 7, 8], + [0, 1, 1, 1, 4, 7, 7, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8]]] + expected = xp.asarray(expected) + assert_array_almost_equal(expected, ft) + + @xfail_xp_backends('cupy', reason='CuPy does not have distance_transform_bf.') + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_bf03(self, dtype, xp): + dtype = getattr(xp, dtype) + + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out, ft = ndimage.distance_transform_bf(data, 'chessboard', + return_indices=True) + + expected = [[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 2, 1, 1, 0, 0], + [0, 0, 1, 2, 2, 2, 1, 0, 0], + [0, 0, 1, 1, 2, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + assert_array_almost_equal(out, expected) + + expected = [[[0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 1, 1, 1], + [2, 2, 2, 2, 1, 2, 2, 2, 2], + [3, 3, 4, 2, 2, 2, 4, 3, 3], + [4, 4, 5, 6, 6, 6, 5, 4, 4], + [5, 5, 6, 6, 7, 6, 6, 5, 5], + [6, 6, 6, 7, 7, 7, 6, 6, 6], + [7, 7, 7, 7, 7, 7, 7, 7, 7], + [8, 8, 8, 8, 8, 8, 8, 8, 8]], + [[0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 2, 5, 6, 6, 7, 8], + [0, 1, 1, 2, 6, 6, 7, 7, 8], + [0, 1, 1, 2, 6, 7, 7, 7, 8], + [0, 1, 2, 2, 6, 6, 7, 7, 8], + [0, 1, 2, 4, 5, 6, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8]]] + expected = xp.asarray(expected) + assert_array_almost_equal(ft, expected) + + @skip_xp_backends( + np_only=True, reason='inplace distances= arrays are numpy-specific' + ) + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_bf04(self, dtype, xp): + dtype = getattr(xp, dtype) + + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + tdt, tft = ndimage.distance_transform_bf(data, return_indices=1) + dts = [] + fts = [] + dt = xp.zeros(data.shape, dtype=xp.float64) + ndimage.distance_transform_bf(data, distances=dt) + dts.append(dt) + ft = ndimage.distance_transform_bf( + data, return_distances=False, return_indices=1) + fts.append(ft) + ft = np.indices(data.shape, dtype=xp.int32) + ndimage.distance_transform_bf( + data, return_distances=False, return_indices=True, indices=ft) + fts.append(ft) + dt, ft = ndimage.distance_transform_bf( + data, return_indices=1) + dts.append(dt) + fts.append(ft) + dt = xp.zeros(data.shape, dtype=xp.float64) + ft = ndimage.distance_transform_bf( + data, distances=dt, return_indices=True) + dts.append(dt) + fts.append(ft) + ft = np.indices(data.shape, dtype=xp.int32) + dt = ndimage.distance_transform_bf( + data, return_indices=True, indices=ft) + dts.append(dt) + fts.append(ft) + dt = xp.zeros(data.shape, dtype=xp.float64) + ft = np.indices(data.shape, dtype=xp.int32) + ndimage.distance_transform_bf( + data, distances=dt, return_indices=True, indices=ft) + dts.append(dt) + fts.append(ft) + for dt in dts: + assert_array_almost_equal(tdt, dt) + for ft in fts: + assert_array_almost_equal(tft, ft) + + @xfail_xp_backends('cupy', reason='CuPy does not have distance_transform_bf.') + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_bf05(self, dtype, xp): + dtype = getattr(xp, dtype) + + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out, ft = ndimage.distance_transform_bf( + data, 'euclidean', return_indices=True, sampling=[2, 2]) + expected = [[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 4, 4, 4, 0, 0, 0], + [0, 0, 4, 8, 16, 8, 4, 0, 0], + [0, 0, 4, 16, 32, 16, 4, 0, 0], + [0, 0, 4, 8, 16, 8, 4, 0, 0], + [0, 0, 0, 4, 4, 4, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + assert_array_almost_equal(out * out, expected) + + expected = [[[0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 1, 1, 1], + [2, 2, 2, 2, 1, 2, 2, 2, 2], + [3, 3, 3, 2, 1, 2, 3, 3, 3], + [4, 4, 4, 4, 6, 4, 4, 4, 4], + [5, 5, 6, 6, 7, 6, 6, 5, 5], + [6, 6, 6, 7, 7, 7, 6, 6, 6], + [7, 7, 7, 7, 7, 7, 7, 7, 7], + [8, 8, 8, 8, 8, 8, 8, 8, 8]], + [[0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 2, 4, 6, 6, 7, 8], + [0, 1, 1, 2, 4, 6, 7, 7, 8], + [0, 1, 1, 1, 6, 7, 7, 7, 8], + [0, 1, 2, 2, 4, 6, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8]]] + expected = xp.asarray(expected) + assert_array_almost_equal(ft, expected) + + @xfail_xp_backends('cupy', reason='CuPy does not have distance_transform_bf.') + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_bf06(self, dtype, xp): + dtype = getattr(xp, dtype) + + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out, ft = ndimage.distance_transform_bf( + data, 'euclidean', return_indices=True, sampling=[2, 1]) + expected = [[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 4, 1, 0, 0, 0], + [0, 0, 1, 4, 8, 4, 1, 0, 0], + [0, 0, 1, 4, 9, 4, 1, 0, 0], + [0, 0, 1, 4, 8, 4, 1, 0, 0], + [0, 0, 0, 1, 4, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + assert_array_almost_equal(out * out, expected) + + expected = [[[0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 1, 1, 1], + [2, 2, 2, 2, 2, 2, 2, 2, 2], + [3, 3, 3, 3, 2, 3, 3, 3, 3], + [4, 4, 4, 4, 4, 4, 4, 4, 4], + [5, 5, 5, 5, 6, 5, 5, 5, 5], + [6, 6, 6, 6, 7, 6, 6, 6, 6], + [7, 7, 7, 7, 7, 7, 7, 7, 7], + [8, 8, 8, 8, 8, 8, 8, 8, 8]], + [[0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 2, 6, 6, 6, 7, 8], + [0, 1, 1, 1, 6, 7, 7, 7, 8], + [0, 1, 1, 1, 7, 7, 7, 7, 8], + [0, 1, 1, 1, 6, 7, 7, 7, 8], + [0, 1, 2, 2, 4, 6, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8]]] + expected = xp.asarray(expected) + assert_array_almost_equal(ft, expected) + + def test_distance_transform_bf07(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy does not have distance_transform_bf.") + + # test input validation per discussion on PR #13302 + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]]) + with assert_raises(RuntimeError): + ndimage.distance_transform_bf( + data, return_distances=False, return_indices=False + ) + + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_cdt01(self, dtype, xp): + dtype = getattr(xp, dtype) + if is_cupy(xp): + pytest.xfail("CuPy does not have distance_transform_cdt.") + + # chamfer type distance (cdt) transform + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out, ft = ndimage.distance_transform_cdt( + data, 'cityblock', return_indices=True) + bf = ndimage.distance_transform_bf(data, 'cityblock') + assert_array_almost_equal(bf, out) + + expected = [[[0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 1, 1, 1], + [2, 2, 2, 1, 1, 1, 2, 2, 2], + [3, 3, 2, 1, 1, 1, 2, 3, 3], + [4, 4, 4, 4, 1, 4, 4, 4, 4], + [5, 5, 5, 5, 7, 7, 6, 5, 5], + [6, 6, 6, 6, 7, 7, 6, 6, 6], + [7, 7, 7, 7, 7, 7, 7, 7, 7], + [8, 8, 8, 8, 8, 8, 8, 8, 8]], + [[0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 1, 1, 4, 7, 7, 7, 8], + [0, 1, 1, 1, 4, 5, 6, 7, 8], + [0, 1, 2, 2, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8]]] + expected = xp.asarray(expected) + assert_array_almost_equal(ft, expected) + + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_cdt02(self, dtype, xp): + dtype = getattr(xp, dtype) + if is_cupy(xp): + pytest.xfail("CuPy does not have distance_transform_cdt.") + + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out, ft = ndimage.distance_transform_cdt(data, 'chessboard', + return_indices=True) + bf = ndimage.distance_transform_bf(data, 'chessboard') + assert_array_almost_equal(bf, out) + + expected = [[[0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 1, 1, 1], + [2, 2, 2, 1, 1, 1, 2, 2, 2], + [3, 3, 2, 2, 1, 2, 2, 3, 3], + [4, 4, 3, 2, 2, 2, 3, 4, 4], + [5, 5, 4, 6, 7, 6, 4, 5, 5], + [6, 6, 6, 6, 7, 7, 6, 6, 6], + [7, 7, 7, 7, 7, 7, 7, 7, 7], + [8, 8, 8, 8, 8, 8, 8, 8, 8]], + [[0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 2, 3, 4, 6, 7, 8], + [0, 1, 1, 2, 2, 6, 6, 7, 8], + [0, 1, 1, 1, 2, 6, 7, 7, 8], + [0, 1, 1, 2, 6, 6, 7, 7, 8], + [0, 1, 2, 2, 5, 6, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8], + [0, 1, 2, 3, 4, 5, 6, 7, 8]]] + expected = xp.asarray(expected) + assert_array_almost_equal(ft, expected) + + @skip_xp_backends( + np_only=True, reason='inplace indices= arrays are numpy-specific' + ) + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_cdt03(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + tdt, tft = ndimage.distance_transform_cdt(data, return_indices=True) + dts = [] + fts = [] + dt = xp.zeros(data.shape, dtype=xp.int32) + ndimage.distance_transform_cdt(data, distances=dt) + dts.append(dt) + ft = ndimage.distance_transform_cdt( + data, return_distances=False, return_indices=True) + fts.append(ft) + ft = xp.asarray(np.indices(data.shape, dtype=np.int32)) + ndimage.distance_transform_cdt( + data, return_distances=False, return_indices=True, indices=ft) + fts.append(ft) + dt, ft = ndimage.distance_transform_cdt( + data, return_indices=True) + dts.append(dt) + fts.append(ft) + dt = xp.zeros(data.shape, dtype=xp.int32) + ft = ndimage.distance_transform_cdt( + data, distances=dt, return_indices=True) + dts.append(dt) + fts.append(ft) + ft = xp.asarray(np.indices(data.shape, dtype=np.int32)) + dt = ndimage.distance_transform_cdt( + data, return_indices=True, indices=ft) + dts.append(dt) + fts.append(ft) + dt = xp.zeros(data.shape, dtype=xp.int32) + ft = xp.asarray(np.indices(data.shape, dtype=np.int32)) + ndimage.distance_transform_cdt(data, distances=dt, + return_indices=True, indices=ft) + dts.append(dt) + fts.append(ft) + for dt in dts: + assert_array_almost_equal(tdt, dt) + for ft in fts: + assert_array_almost_equal(tft, ft) + + @skip_xp_backends( + np_only=True, reason='XXX: does not raise unless indices is a numpy array' + ) + def test_distance_transform_cdt04(self, xp): + # test input validation per discussion on PR #13302 + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]]) + indices_out = xp.zeros((data.ndim,) + data.shape, dtype=xp.int32) + with assert_raises(RuntimeError): + ndimage.distance_transform_bf( + data, + return_distances=True, + return_indices=False, + indices=indices_out + ) + + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_cdt05(self, dtype, xp): + dtype = getattr(xp, dtype) + if is_cupy(xp): + pytest.xfail("CuPy does not have distance_transform_cdt.") + elif is_torch(xp): + pytest.xfail("int overflow") + + # test custom metric type per discussion on issue #17381 + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + metric_arg = xp.ones((3, 3)) + actual = ndimage.distance_transform_cdt(data, metric=metric_arg) + assert xp.sum(actual) == -21 + + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_edt01(self, dtype, xp): + dtype = getattr(xp, dtype) + if is_cupy(xp): + pytest.xfail("CuPy does not have distance_transform_bf") + + # euclidean distance transform (edt) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out, ft = ndimage.distance_transform_edt(data, return_indices=True) + bf = ndimage.distance_transform_bf(data, 'euclidean') + assert_array_almost_equal(bf, out) + + # np-specific check + np_ft = np.asarray(ft) + dt = np_ft - np.indices(np_ft.shape[1:], dtype=np_ft.dtype) + dt = dt.astype(np.float64) + np.multiply(dt, dt, dt) + dt = np.add.reduce(dt, axis=0) + np.sqrt(dt, dt) + + dt = xp.asarray(dt) + assert_array_almost_equal(bf, dt) + + @skip_xp_backends( + np_only=True, reason='inplace distances= are numpy-specific' + ) + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_edt02(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + tdt, tft = ndimage.distance_transform_edt(data, return_indices=True) + dts = [] + fts = [] + + dt = xp.zeros(data.shape, dtype=xp.float64) + ndimage.distance_transform_edt(data, distances=dt) + dts.append(dt) + + ft = ndimage.distance_transform_edt( + data, return_distances=0, return_indices=True) + fts.append(ft) + + ft = np.indices(data.shape, dtype=xp.int32) + ft = xp.asarray(ft) + ndimage.distance_transform_edt( + data, return_distances=False, return_indices=True, indices=ft) + fts.append(ft) + + dt, ft = ndimage.distance_transform_edt( + data, return_indices=True) + dts.append(dt) + fts.append(ft) + + dt = xp.zeros(data.shape, dtype=xp.float64) + ft = ndimage.distance_transform_edt( + data, distances=dt, return_indices=True) + dts.append(dt) + fts.append(ft) + + ft = np.indices(data.shape, dtype=xp.int32) + ft = xp.asarray(ft) + dt = ndimage.distance_transform_edt( + data, return_indices=True, indices=ft) + dts.append(dt) + fts.append(ft) + + dt = xp.zeros(data.shape, dtype=xp.float64) + ft = np.indices(data.shape, dtype=xp.int32) + ft = xp.asarray(ft) + ndimage.distance_transform_edt( + data, distances=dt, return_indices=True, indices=ft) + dts.append(dt) + fts.append(ft) + + for dt in dts: + assert_array_almost_equal(tdt, dt) + for ft in fts: + assert_array_almost_equal(tft, ft) + + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_edt03(self, dtype, xp): + dtype = getattr(xp, dtype) + if is_cupy(xp): + pytest.xfail("CuPy does not have distance_transform_bf") + + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + ref = ndimage.distance_transform_bf(data, 'euclidean', sampling=[2, 2]) + out = ndimage.distance_transform_edt(data, sampling=[2, 2]) + assert_array_almost_equal(ref, out) + + @pytest.mark.parametrize('dtype', types) + def test_distance_transform_edt4(self, dtype, xp): + dtype = getattr(xp, dtype) + if is_cupy(xp): + pytest.xfail("CuPy does not have distance_transform_bf") + + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + ref = ndimage.distance_transform_bf(data, 'euclidean', sampling=[2, 1]) + out = ndimage.distance_transform_edt(data, sampling=[2, 1]) + assert_array_almost_equal(ref, out) + + def test_distance_transform_edt5(self, xp): + # Ticket #954 regression test + out = ndimage.distance_transform_edt(False) + assert_array_almost_equal(out, [0.]) + + @skip_xp_backends( + np_only=True, reason='XXX: does not raise unless indices is a numpy array' + ) + def test_distance_transform_edt6(self, xp): + # test input validation per discussion on PR #13302 + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0]]) + distances_out = xp.zeros(data.shape, dtype=xp.float64) + with assert_raises(RuntimeError): + ndimage.distance_transform_bf( + data, + return_indices=True, + return_distances=False, + distances=distances_out + ) + + def test_generate_structure01(self, xp): + struct = ndimage.generate_binary_structure(0, 1) + assert struct == 1 + + def test_generate_structure02(self, xp): + struct = ndimage.generate_binary_structure(1, 1) + assert_array_almost_equal(struct, [1, 1, 1]) + + def test_generate_structure03(self, xp): + struct = ndimage.generate_binary_structure(2, 1) + assert_array_almost_equal(struct, [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]]) + + def test_generate_structure04(self, xp): + struct = ndimage.generate_binary_structure(2, 2) + assert_array_almost_equal(struct, [[1, 1, 1], + [1, 1, 1], + [1, 1, 1]]) + + def test_iterate_structure01(self, xp): + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + out = ndimage.iterate_structure(struct, 2) + expected = np.asarray([[0, 0, 1, 0, 0], + [0, 1, 1, 1, 0], + [1, 1, 1, 1, 1], + [0, 1, 1, 1, 0], + [0, 0, 1, 0, 0]], dtype=bool) + expected = xp.asarray(expected) + assert_array_almost_equal(out, expected) + + def test_iterate_structure02(self, xp): + struct = [[0, 1], + [1, 1], + [0, 1]] + struct = xp.asarray(struct) + out = ndimage.iterate_structure(struct, 2) + expected = np.asarray([[0, 0, 1], + [0, 1, 1], + [1, 1, 1], + [0, 1, 1], + [0, 0, 1]], dtype=bool) + expected = xp.asarray(expected) + + assert_array_almost_equal(out, expected) + + def test_iterate_structure03(self, xp): + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + out = ndimage.iterate_structure(struct, 2, 1) + expected = [[0, 0, 1, 0, 0], + [0, 1, 1, 1, 0], + [1, 1, 1, 1, 1], + [0, 1, 1, 1, 0], + [0, 0, 1, 0, 0]] + expected = np.asarray(expected, dtype=bool) + expected = xp.asarray(expected) + assert_array_almost_equal(out[0], expected) + assert out[1] == [2, 2] + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion01(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([], dtype=dtype) + out = ndimage.binary_erosion(data) + assert out == xp.asarray(1, dtype=out.dtype) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion02(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([], dtype=dtype) + out = ndimage.binary_erosion(data, border_value=1) + assert out == xp.asarray(1, dtype=out.dtype) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion03(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([1], dtype=dtype) + out = ndimage.binary_erosion(data) + assert_array_almost_equal(out, xp.asarray([0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion04(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([1], dtype=dtype) + out = ndimage.binary_erosion(data, border_value=1) + assert_array_almost_equal(out, xp.asarray([1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion05(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([3], dtype=dtype) + out = ndimage.binary_erosion(data) + assert_array_almost_equal(out, xp.asarray([0, 1, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion06(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([3], dtype=dtype) + out = ndimage.binary_erosion(data, border_value=1) + assert_array_almost_equal(out, xp.asarray([1, 1, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion07(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([5], dtype=dtype) + out = ndimage.binary_erosion(data) + assert_array_almost_equal(out, xp.asarray([0, 1, 1, 1, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion08(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([5], dtype=dtype) + out = ndimage.binary_erosion(data, border_value=1) + assert_array_almost_equal(out, xp.asarray([1, 1, 1, 1, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion09(self, dtype, xp): + data = np.ones([5], dtype=dtype) + data[2] = 0 + data = xp.asarray(data) + out = ndimage.binary_erosion(data) + assert_array_almost_equal(out, xp.asarray([0, 0, 0, 0, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion10(self, dtype, xp): + data = np.ones([5], dtype=dtype) + data[2] = 0 + data = xp.asarray(data) + out = ndimage.binary_erosion(data, border_value=1) + assert_array_almost_equal(out, xp.asarray([1, 0, 0, 0, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion11(self, dtype, xp): + data = np.ones([5], dtype=dtype) + data[2] = 0 + data = xp.asarray(data) + struct = xp.asarray([1, 0, 1]) + out = ndimage.binary_erosion(data, struct, border_value=1) + assert_array_almost_equal(out, xp.asarray([1, 0, 1, 0, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion12(self, dtype, xp): + data = np.ones([5], dtype=dtype) + data[2] = 0 + data = xp.asarray(data) + struct = xp.asarray([1, 0, 1]) + out = ndimage.binary_erosion(data, struct, border_value=1, origin=-1) + assert_array_almost_equal(out, xp.asarray([0, 1, 0, 1, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion13(self, dtype, xp): + data = np.ones([5], dtype=dtype) + data[2] = 0 + data = xp.asarray(data) + struct = xp.asarray([1, 0, 1]) + out = ndimage.binary_erosion(data, struct, border_value=1, origin=1) + assert_array_almost_equal(out, xp.asarray([1, 1, 0, 1, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion14(self, dtype, xp): + data = np.ones([5], dtype=dtype) + data[2] = 0 + data = xp.asarray(data) + struct = xp.asarray([1, 1]) + out = ndimage.binary_erosion(data, struct, border_value=1) + assert_array_almost_equal(out, xp.asarray([1, 1, 0, 0, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion15(self, dtype, xp): + data = np.ones([5], dtype=dtype) + data[2] = 0 + data = xp.asarray(data) + struct = xp.asarray([1, 1]) + out = ndimage.binary_erosion(data, struct, border_value=1, origin=-1) + assert_array_almost_equal(out, xp.asarray([1, 0, 0, 1, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion16(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([1, 1], dtype=dtype) + out = ndimage.binary_erosion(data, border_value=1) + assert_array_almost_equal(out, xp.asarray([[1]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion17(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([1, 1], dtype=dtype) + out = ndimage.binary_erosion(data) + assert_array_almost_equal(out, xp.asarray([[0]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion18(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([1, 3], dtype=dtype) + out = ndimage.binary_erosion(data) + assert_array_almost_equal(out, xp.asarray([[0, 0, 0]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion19(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([1, 3], dtype=dtype) + out = ndimage.binary_erosion(data, border_value=1) + assert_array_almost_equal(out, xp.asarray([[1, 1, 1]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion20(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([3, 3], dtype=dtype) + out = ndimage.binary_erosion(data) + assert_array_almost_equal(out, xp.asarray([[0, 0, 0], + [0, 1, 0], + [0, 0, 0]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion21(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([3, 3], dtype=dtype) + out = ndimage.binary_erosion(data, border_value=1) + assert_array_almost_equal(out, xp.asarray([[1, 1, 1], + [1, 1, 1], + [1, 1, 1]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion22(self, dtype, xp): + dtype = getattr(xp, dtype) + expected = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 1, 1], + [0, 0, 1, 1, 1, 1, 1, 1], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 0, 0, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_erosion(data, border_value=1) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion23(self, dtype, xp): + dtype = getattr(xp, dtype) + struct = ndimage.generate_binary_structure(2, 2) + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 1, 1], + [0, 0, 1, 1, 1, 1, 1, 1], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 0, 0, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_erosion(data, struct, border_value=1) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion24(self, dtype, xp): + dtype = getattr(xp, dtype) + struct = xp.asarray([[0, 1], + [1, 1]]) + expected = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 1, 1], + [0, 0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 0, 0, 0, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 1, 1], + [0, 0, 1, 1, 1, 1, 1, 1], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 0, 0, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_erosion(data, struct, border_value=1) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion25(self, dtype, xp): + dtype = getattr(xp, dtype) + struct = [[0, 1, 0], + [1, 0, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 1, 1], + [0, 0, 1, 1, 1, 0, 1, 1], + [0, 0, 1, 0, 1, 1, 0, 0], + [0, 1, 0, 1, 1, 1, 1, 0], + [0, 1, 1, 0, 0, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_erosion(data, struct, border_value=1) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_erosion26(self, dtype, xp): + dtype = getattr(xp, dtype) + struct = [[0, 1, 0], + [1, 0, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 1], + [0, 0, 0, 0, 1, 0, 0, 1], + [0, 0, 1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 1]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 1, 1], + [0, 0, 1, 1, 1, 0, 1, 1], + [0, 0, 1, 0, 1, 1, 0, 0], + [0, 1, 0, 1, 1, 1, 1, 0], + [0, 1, 1, 0, 0, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_erosion(data, struct, border_value=1, + origin=(-1, -1)) + assert_array_almost_equal(out, expected) + + def test_binary_erosion27(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_erosion(data, struct, border_value=1, + iterations=2) + assert_array_almost_equal(out, expected) + + @skip_xp_backends( + np_only=True, reason='inplace out= arguments are numpy-specific' + ) + def test_binary_erosion28(self, xp): + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + expected = np.asarray(expected, dtype=bool) + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = np.zeros(data.shape, dtype=bool) + out = xp.asarray(out) + ndimage.binary_erosion(data, struct, border_value=1, + iterations=2, output=out) + assert_array_almost_equal(out, expected) + + def test_binary_erosion29(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [1, 1, 1, 1, 1, 1, 1], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_erosion(data, struct, + border_value=1, iterations=3) + assert_array_almost_equal(out, expected) + + @skip_xp_backends( + np_only=True, reason='inplace out= arguments are numpy-specific' + ) + def test_binary_erosion30(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + expected = np.asarray(expected, dtype=bool) + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [1, 1, 1, 1, 1, 1, 1], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = np.zeros(data.shape, dtype=bool) + out = xp.asarray(out) + ndimage.binary_erosion(data, struct, border_value=1, + iterations=3, output=out) + assert_array_almost_equal(out, expected) + + # test with output memory overlap + ndimage.binary_erosion(data, struct, border_value=1, + iterations=3, output=data) + assert_array_almost_equal(data, expected) + + @skip_xp_backends( + np_only=True, reason='inplace out= arguments are numpy-specific' + ) + def test_binary_erosion31(self, xp): + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 1, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0], + [1, 1, 1, 1, 1, 0, 1], + [0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 1]] + expected = np.asarray(expected, dtype=bool) + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [1, 1, 1, 1, 1, 1, 1], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = np.zeros(data.shape, dtype=bool) + out = xp.asarray(out) + ndimage.binary_erosion(data, struct, border_value=1, + iterations=1, output=out, origin=(-1, -1)) + assert_array_almost_equal(out, expected) + + def test_binary_erosion32(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_erosion(data, struct, + border_value=1, iterations=2) + assert_array_almost_equal(out, expected) + + def test_binary_erosion33(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 1, 1], + [0, 0, 0, 0, 0, 0, 1], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + mask = [[1, 1, 1, 1, 1, 0, 0], + [1, 1, 1, 1, 1, 1, 0], + [1, 1, 1, 1, 1, 1, 1], + [1, 1, 1, 1, 1, 1, 1], + [1, 1, 1, 1, 1, 1, 1], + [1, 1, 1, 1, 1, 1, 1], + [1, 1, 1, 1, 1, 1, 1]] + mask = xp.asarray(mask) + data = np.asarray([[0, 0, 0, 0, 0, 1, 1], + [0, 0, 0, 1, 0, 0, 1], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_erosion(data, struct, + border_value=1, mask=mask, iterations=-1) + assert_array_almost_equal(out, expected) + + def test_binary_erosion34(self, xp): + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + mask = [[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 0, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + mask = xp.asarray(mask) + data = np.asarray([[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_erosion(data, struct, + border_value=1, mask=mask) + assert_array_almost_equal(out, expected) + + @skip_xp_backends( + np_only=True, reason='inplace out= arguments are numpy-specific' + ) + def test_binary_erosion35(self, xp): + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + mask = [[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 0, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + mask = np.asarray(mask, dtype=bool) + mask = xp.asarray(mask) + data = np.asarray([[0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [1, 1, 1, 1, 1, 1, 1], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + tmp = [[0, 0, 1, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0], + [1, 1, 1, 1, 1, 0, 1], + [0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 1]] + tmp = np.asarray(tmp, dtype=bool) + tmp = xp.asarray(tmp) + expected = xp.logical_and(tmp, mask) + tmp = xp.logical_and(data, xp.logical_not(mask)) + expected = xp.logical_or(expected, tmp) + out = np.zeros(data.shape, dtype=bool) + out = xp.asarray(out) + ndimage.binary_erosion(data, struct, border_value=1, + iterations=1, output=out, + origin=(-1, -1), mask=mask) + assert_array_almost_equal(out, expected) + + def test_binary_erosion36(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + struct = [[0, 1, 0], + [1, 0, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + mask = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + mask = np.asarray(mask, dtype=bool) + mask = xp.asarray(mask) + tmp = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 1], + [0, 0, 0, 0, 1, 0, 0, 1], + [0, 0, 1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 1]] + tmp = np.asarray(tmp, dtype=bool) + tmp = xp.asarray(tmp) + data = np.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 1, 1], + [0, 0, 1, 1, 1, 0, 1, 1], + [0, 0, 1, 0, 1, 1, 0, 0], + [0, 1, 0, 1, 1, 1, 1, 0], + [0, 1, 1, 0, 0, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + expected = xp.logical_and(tmp, mask) + tmp = xp.logical_and(data, xp.logical_not(mask)) + expected = xp.logical_or(expected, tmp) + out = ndimage.binary_erosion(data, struct, mask=mask, + border_value=1, origin=(-1, -1)) + assert_array_almost_equal(out, expected) + + @skip_xp_backends( + np_only=True, reason='inplace out= arguments are numpy-specific' + ) + def test_binary_erosion37(self, xp): + a = np.asarray([[1, 0, 1], + [0, 1, 0], + [1, 0, 1]], dtype=bool) + a = xp.asarray(a) + b = xp.zeros_like(a) + out = ndimage.binary_erosion(a, structure=a, output=b, iterations=0, + border_value=True, brute_force=True) + assert out is b + xp_assert_equal( + ndimage.binary_erosion(a, structure=a, iterations=0, + border_value=True), + b) + + def test_binary_erosion38(self, xp): + data = np.asarray([[1, 0, 1], + [0, 1, 0], + [1, 0, 1]], dtype=bool) + data = xp.asarray(data) + iterations = 2.0 + with assert_raises(TypeError): + _ = ndimage.binary_erosion(data, iterations=iterations) + + @skip_xp_backends( + np_only=True, reason='inplace out= arguments are numpy-specific' + ) + def test_binary_erosion39(self, xp): + iterations = np.int32(3) + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected, dtype=bool) + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [1, 1, 1, 1, 1, 1, 1], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = np.zeros(data.shape, dtype=bool) + out = xp.asarray(out) + ndimage.binary_erosion(data, struct, border_value=1, + iterations=iterations, output=out) + assert_array_almost_equal(out, expected) + + @skip_xp_backends( + np_only=True, reason='inplace out= arguments are numpy-specific' + ) + def test_binary_erosion40(self, xp): + iterations = np.int64(3) + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0]] + expected = np.asarray(expected, dtype=bool) + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [1, 1, 1, 1, 1, 1, 1], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = np.zeros(data.shape, dtype=bool) + out = xp.asarray(out) + ndimage.binary_erosion(data, struct, border_value=1, + iterations=iterations, output=out) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation01(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([], dtype=dtype) + out = ndimage.binary_dilation(data) + assert out == xp.asarray(1, dtype=out.dtype) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation02(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.zeros([], dtype=dtype) + out = ndimage.binary_dilation(data) + assert out == xp.asarray(False) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation03(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([1], dtype=dtype) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([1], dtype=out.dtype)) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation04(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.zeros([1], dtype=dtype) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation05(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([3], dtype=dtype) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([1, 1, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation06(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.zeros([3], dtype=dtype) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([0, 0, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation07(self, dtype, xp): + data = np.zeros([3], dtype=dtype) + data[1] = 1 + data = xp.asarray(data) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([1, 1, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation08(self, dtype, xp): + data = np.zeros([5], dtype=dtype) + data[1] = 1 + data[3] = 1 + data = xp.asarray(data) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([1, 1, 1, 1, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation09(self, dtype, xp): + data = np.zeros([5], dtype=dtype) + data[1] = 1 + data = xp.asarray(data) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([1, 1, 1, 0, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation10(self, dtype, xp): + data = np.zeros([5], dtype=dtype) + data[1] = 1 + data = xp.asarray(data) + out = ndimage.binary_dilation(data, origin=-1) + assert_array_almost_equal(out, xp.asarray([0, 1, 1, 1, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation11(self, dtype, xp): + data = np.zeros([5], dtype=dtype) + data[1] = 1 + data = xp.asarray(data) + out = ndimage.binary_dilation(data, origin=1) + assert_array_almost_equal(out, xp.asarray([1, 1, 0, 0, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation12(self, dtype, xp): + data = np.zeros([5], dtype=dtype) + data[1] = 1 + data = xp.asarray(data) + struct = xp.asarray([1, 0, 1]) + out = ndimage.binary_dilation(data, struct) + assert_array_almost_equal(out, xp.asarray([1, 0, 1, 0, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation13(self, dtype, xp): + data = np.zeros([5], dtype=dtype) + data[1] = 1 + data = xp.asarray(data) + struct = xp.asarray([1, 0, 1]) + out = ndimage.binary_dilation(data, struct, border_value=1) + assert_array_almost_equal(out, xp.asarray([1, 0, 1, 0, 1])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation14(self, dtype, xp): + data = np.zeros([5], dtype=dtype) + data[1] = 1 + data = xp.asarray(data) + struct = xp.asarray([1, 0, 1]) + out = ndimage.binary_dilation(data, struct, origin=-1) + assert_array_almost_equal(out, xp.asarray([0, 1, 0, 1, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation15(self, dtype, xp): + data = np.zeros([5], dtype=dtype) + data[1] = 1 + data = xp.asarray(data) + struct = xp.asarray([1, 0, 1]) + out = ndimage.binary_dilation(data, struct, + origin=-1, border_value=1) + assert_array_almost_equal(out, xp.asarray([1, 1, 0, 1, 0])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation16(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([1, 1], dtype=dtype) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([[1]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation17(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.zeros([1, 1], dtype=dtype) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([[0]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation18(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([1, 3], dtype=dtype) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([[1, 1, 1]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation19(self, dtype, xp): + dtype = getattr(xp, dtype) + data = xp.ones([3, 3], dtype=dtype) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([[1, 1, 1], + [1, 1, 1], + [1, 1, 1]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation20(self, dtype, xp): + data = np.zeros([3, 3], dtype=dtype) + data[1, 1] = 1 + data = xp.asarray(data) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, xp.asarray([[0, 1, 0], + [1, 1, 1], + [0, 1, 0]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation21(self, dtype, xp): + struct = ndimage.generate_binary_structure(2, 2) + struct = xp.asarray(struct) + data = np.zeros([3, 3], dtype=dtype) + data[1, 1] = 1 + data = xp.asarray(data) + out = ndimage.binary_dilation(data, struct) + assert_array_almost_equal(out, xp.asarray([[1, 1, 1], + [1, 1, 1], + [1, 1, 1]])) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation22(self, dtype, xp): + dtype = getattr(xp, dtype) + expected = [[0, 1, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_dilation(data) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation23(self, dtype, xp): + dtype = getattr(xp, dtype) + expected = [[1, 1, 1, 1, 1, 1, 1, 1], + [1, 1, 1, 0, 0, 0, 0, 1], + [1, 1, 0, 0, 0, 1, 0, 1], + [1, 0, 0, 1, 1, 1, 1, 1], + [1, 0, 1, 1, 1, 1, 0, 1], + [1, 1, 1, 1, 1, 1, 1, 1], + [1, 0, 1, 0, 0, 1, 0, 1], + [1, 1, 1, 1, 1, 1, 1, 1]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_dilation(data, border_value=1) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation24(self, dtype, xp): + dtype = getattr(xp, dtype) + expected = [[1, 1, 0, 0, 0, 0, 0, 0], + [1, 0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 0, 0], + [0, 1, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_dilation(data, origin=(1, 1)) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation25(self, dtype, xp): + dtype = getattr(xp, dtype) + expected = [[1, 1, 0, 0, 0, 0, 1, 1], + [1, 0, 0, 0, 1, 0, 1, 1], + [0, 0, 1, 1, 1, 1, 1, 1], + [0, 1, 1, 1, 1, 0, 1, 1], + [1, 1, 1, 1, 1, 1, 1, 1], + [0, 1, 0, 0, 1, 0, 1, 1], + [1, 1, 1, 1, 1, 1, 1, 1], + [1, 1, 1, 1, 1, 1, 1, 1]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_dilation(data, origin=(1, 1), border_value=1) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation26(self, dtype, xp): + dtype = getattr(xp, dtype) + struct = ndimage.generate_binary_structure(2, 2) + expected = [[1, 1, 1, 0, 0, 0, 0, 0], + [1, 1, 1, 0, 0, 0, 0, 0], + [1, 1, 1, 0, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + struct = xp.asarray(struct) + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_dilation(data, struct) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation27(self, dtype, xp): + dtype = getattr(xp, dtype) + struct = [[0, 1], + [1, 1]] + expected = [[0, 1, 0, 0, 0, 0, 0, 0], + [1, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 1, 1, 0, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + struct = xp.asarray(struct) + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_dilation(data, struct) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation28(self, dtype, xp): + dtype = getattr(xp, dtype) + expected = [[1, 1, 1, 1], + [1, 0, 0, 1], + [1, 0, 0, 1], + [1, 1, 1, 1]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_dilation(data, border_value=1) + assert_array_almost_equal(out, expected) + + def test_binary_dilation29(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + struct = [[0, 1], + [1, 1]] + expected = [[0, 0, 0, 0, 0], + [0, 0, 0, 1, 0], + [0, 0, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 0, 0, 0, 0]] + struct = xp.asarray(struct) + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 1, 0], + [0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_dilation(data, struct, iterations=2) + assert_array_almost_equal(out, expected) + + @skip_xp_backends(np_only=True, reason='output= arrays are numpy-specific') + def test_binary_dilation30(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + struct = [[0, 1], + [1, 1]] + expected = [[0, 0, 0, 0, 0], + [0, 0, 0, 1, 0], + [0, 0, 1, 1, 0], + [0, 1, 1, 1, 0], + [0, 0, 0, 0, 0]] + struct = xp.asarray(struct) + expected = xp.asarray(expected) + data = xp.asarray([[0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 1, 0], + [0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = np.zeros(data.shape, dtype=bool) + out = xp.asarray(out) + ndimage.binary_dilation(data, struct, iterations=2, output=out) + assert_array_almost_equal(out, expected) + + def test_binary_dilation31(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + struct = [[0, 1], + [1, 1]] + expected = [[0, 0, 0, 1, 0], + [0, 0, 1, 1, 0], + [0, 1, 1, 1, 0], + [1, 1, 1, 1, 0], + [0, 0, 0, 0, 0]] + struct = xp.asarray(struct) + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 1, 0], + [0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_dilation(data, struct, iterations=3) + assert_array_almost_equal(out, expected) + + @skip_xp_backends(np_only=True, reason='output= arrays are numpy-specific') + def test_binary_dilation32(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + struct = [[0, 1], + [1, 1]] + expected = [[0, 0, 0, 1, 0], + [0, 0, 1, 1, 0], + [0, 1, 1, 1, 0], + [1, 1, 1, 1, 0], + [0, 0, 0, 0, 0]] + struct = xp.asarray(struct) + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 1, 0], + [0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = np.zeros(data.shape, dtype=bool) + out = xp.asarray(out) + ndimage.binary_dilation(data, struct, iterations=3, output=out) + assert_array_almost_equal(out, expected) + + def test_binary_dilation33(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = np.asarray([[0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0, 0], + [0, 1, 1, 0, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + expected = xp.asarray(expected) + mask = np.asarray([[0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 1, 0], + [0, 0, 0, 0, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0, 0], + [0, 1, 1, 0, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + mask = xp.asarray(mask) + data = np.asarray([[0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + + out = ndimage.binary_dilation(data, struct, iterations=-1, + mask=mask, border_value=0) + assert_array_almost_equal(out, expected) + + @skip_xp_backends( + np_only=True, reason='inplace output= arrays are numpy-specific', + ) + def test_binary_dilation34(self, xp): + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 1, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + mask = np.asarray([[0, 1, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + mask = xp.asarray(mask) + data = np.zeros(mask.shape, dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_dilation(data, struct, iterations=-1, + mask=mask, border_value=1) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_dilation35(self, dtype, xp): + dtype = getattr(xp, dtype) + tmp = [[1, 1, 0, 0, 0, 0, 1, 1], + [1, 0, 0, 0, 1, 0, 1, 1], + [0, 0, 1, 1, 1, 1, 1, 1], + [0, 1, 1, 1, 1, 0, 1, 1], + [1, 1, 1, 1, 1, 1, 1, 1], + [0, 1, 0, 0, 1, 0, 1, 1], + [1, 1, 1, 1, 1, 1, 1, 1], + [1, 1, 1, 1, 1, 1, 1, 1]] + + data = np.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]]) + mask = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + mask = np.asarray(mask, dtype=bool) + + expected = np.logical_and(tmp, mask) + tmp = np.logical_and(data, np.logical_not(mask)) + expected = np.logical_or(expected, tmp) + + mask = xp.asarray(mask) + expected = xp.asarray(expected) + + data = xp.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_dilation(data, mask=mask, + origin=(1, 1), border_value=1) + assert_array_almost_equal(out, expected) + + def test_binary_dilation36(self, xp): + # gh-21009 + data = np.zeros([], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_dilation(data, iterations=-1) + assert out == xp.asarray(False) + + def test_binary_propagation01(self, xp): + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = np.asarray([[0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0, 0], + [0, 1, 1, 0, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + expected = xp.asarray(expected) + mask = np.asarray([[0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 1, 0], + [0, 0, 0, 0, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0, 0, 0], + [0, 1, 1, 0, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + mask = xp.asarray(mask) + data = np.asarray([[0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_propagation(data, struct, + mask=mask, border_value=0) + assert_array_almost_equal(out, expected) + + def test_binary_propagation02(self, xp): + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + expected = [[0, 1, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + struct = xp.asarray(struct) + mask = np.asarray([[0, 1, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + mask = xp.asarray(mask) + data = np.zeros(mask.shape, dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_propagation(data, struct, + mask=mask, border_value=1) + assert_array_almost_equal(out, expected) + + def test_binary_propagation03(self, xp): + # gh-21009 + data = xp.asarray(np.zeros([], dtype=bool)) + expected = xp.asarray(np.zeros([], dtype=bool)) + out = ndimage.binary_propagation(data) + assert out == expected + + @pytest.mark.parametrize('dtype', types) + def test_binary_opening01(self, dtype, xp): + dtype = getattr(xp, dtype) + expected = [[0, 1, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 1, 1, 1, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 1, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 0, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_opening(data) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_opening02(self, dtype, xp): + dtype = getattr(xp, dtype) + struct = ndimage.generate_binary_structure(2, 2) + expected = [[1, 1, 1, 0, 0, 0, 0, 0], + [1, 1, 1, 0, 0, 0, 0, 0], + [1, 1, 1, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + struct = xp.asarray(struct) + data = xp.asarray([[1, 1, 1, 0, 0, 0, 0, 0], + [1, 1, 1, 0, 0, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 0, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_opening(data, struct) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_closing01(self, dtype, xp): + dtype = getattr(xp, dtype) + expected = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 1, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 0, 1, 0, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_closing(data) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_binary_closing02(self, dtype, xp): + dtype = getattr(xp, dtype) + struct = ndimage.generate_binary_structure(2, 2) + expected = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + struct = xp.asarray(struct) + data = xp.asarray([[1, 1, 1, 0, 0, 0, 0, 0], + [1, 1, 1, 0, 0, 0, 0, 0], + [1, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 0, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_closing(data, struct) + assert_array_almost_equal(out, expected) + + def test_binary_fill_holes01(self, xp): + expected = np.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + expected = xp.asarray(expected) + + data = np.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + + out = ndimage.binary_fill_holes(data) + assert_array_almost_equal(out, expected) + + def test_binary_fill_holes02(self, xp): + expected = np.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 1, 0, 0, 1, 0, 0], + [0, 0, 0, 1, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_fill_holes(data) + assert_array_almost_equal(out, expected) + + def test_binary_fill_holes03(self, xp): + expected = np.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 0, 1, 1, 1], + [0, 1, 1, 1, 0, 1, 1, 1], + [0, 1, 1, 1, 0, 1, 1, 1], + [0, 0, 1, 0, 0, 1, 1, 1], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + expected = xp.asarray(expected) + data = np.asarray([[0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0, 0], + [0, 1, 0, 1, 0, 1, 1, 1], + [0, 1, 0, 1, 0, 1, 0, 1], + [0, 1, 0, 1, 0, 1, 0, 1], + [0, 0, 1, 0, 0, 1, 1, 1], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=bool) + data = xp.asarray(data) + out = ndimage.binary_fill_holes(data) + assert_array_almost_equal(out, expected) + + @skip_xp_backends(cpu_only=True) + @skip_xp_backends( + "cupy", reason="these filters do not yet have axes support in CuPy") + @skip_xp_backends( + "jax.numpy", reason="these filters are not implemented in JAX.numpy") + @pytest.mark.parametrize('border_value',[0, 1]) + @pytest.mark.parametrize('origin', [(0, 0), (-1, 0)]) + @pytest.mark.parametrize('expand_axis', [0, 1, 2]) + @pytest.mark.parametrize('func_name', ["binary_erosion", + "binary_dilation", + "binary_opening", + "binary_closing", + "binary_hit_or_miss", + "binary_propagation", + "binary_fill_holes"]) + def test_binary_axes(self, xp, func_name, expand_axis, origin, border_value): + struct = np.asarray([[0, 1, 0], + [1, 1, 1], + [0, 1, 0]], bool) + struct = xp.asarray(struct) + + data = np.asarray([[0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0], + [0, 0, 1, 1, 0, 1, 0], + [0, 1, 0, 1, 1, 0, 1], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0]], bool) + data = xp.asarray(data) + if func_name == "binary_hit_or_miss": + kwargs = dict(origin1=origin, origin2=origin) + else: + kwargs = dict(origin=origin) + border_supported = func_name not in ["binary_hit_or_miss", + "binary_fill_holes"] + if border_supported: + kwargs['border_value'] = border_value + elif border_value != 0: + pytest.skip('border_value !=0 unsupported by this function') + func = getattr(ndimage, func_name) + expected = func(data, struct, **kwargs) + + # replicate data and expected result along a new axis + n_reps = 5 + expected = xp.stack([expected] * n_reps, axis=expand_axis) + data = xp.stack([data] * n_reps, axis=expand_axis) + + # filter all axes except expand_axis + axes = [0, 1, 2] + axes.remove(expand_axis) + if is_numpy(xp) or is_cupy(xp): + out = xp.asarray(np.zeros(data.shape, bool)) + func(data, struct, output=out, axes=axes, **kwargs) + else: + # inplace output= is unsupported by JAX + out = func(data, struct, axes=axes, **kwargs) + xp_assert_close(out, expected) + + def test_grey_erosion01(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + output = ndimage.grey_erosion(array, footprint=footprint) + assert_array_almost_equal(output, + xp.asarray([[2, 2, 1, 1, 1], + [2, 3, 1, 3, 1], + [5, 5, 3, 3, 1]])) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + @xfail_xp_backends("cupy", reason="https://github.com/cupy/cupy/issues/8398") + def test_grey_erosion01_overlap(self, xp): + + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + ndimage.grey_erosion(array, footprint=footprint, output=array) + assert_array_almost_equal(array, + xp.asarray([[2, 2, 1, 1, 1], + [2, 3, 1, 3, 1], + [5, 5, 3, 3, 1]]) + ) + + def test_grey_erosion02(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + output = ndimage.grey_erosion(array, footprint=footprint, + structure=structure) + assert_array_almost_equal(output, + xp.asarray([[2, 2, 1, 1, 1], + [2, 3, 1, 3, 1], + [5, 5, 3, 3, 1]]) + ) + + def test_grey_erosion03(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[1, 1, 1], [1, 1, 1]]) + output = ndimage.grey_erosion(array, footprint=footprint, + structure=structure) + assert_array_almost_equal(output, + xp.asarray([[1, 1, 0, 0, 0], + [1, 2, 0, 2, 0], + [4, 4, 2, 2, 0]]) + ) + + def test_grey_dilation01(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[0, 1, 1], [1, 0, 1]]) + output = ndimage.grey_dilation(array, footprint=footprint) + assert_array_almost_equal(output, + xp.asarray([[7, 7, 9, 9, 5], + [7, 9, 8, 9, 7], + [8, 8, 8, 7, 7]]), + ) + + def test_grey_dilation02(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[0, 1, 1], [1, 0, 1]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + output = ndimage.grey_dilation(array, footprint=footprint, + structure=structure) + assert_array_almost_equal(output, + xp.asarray([[7, 7, 9, 9, 5], + [7, 9, 8, 9, 7], + [8, 8, 8, 7, 7]]), + ) + + def test_grey_dilation03(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[0, 1, 1], [1, 0, 1]]) + structure = xp.asarray([[1, 1, 1], [1, 1, 1]]) + output = ndimage.grey_dilation(array, footprint=footprint, + structure=structure) + assert_array_almost_equal(output, + xp.asarray([[8, 8, 10, 10, 6], + [8, 10, 9, 10, 8], + [9, 9, 9, 8, 8]]), + ) + + def test_grey_opening01(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + tmp = ndimage.grey_erosion(array, footprint=footprint) + expected = ndimage.grey_dilation(tmp, footprint=footprint) + output = ndimage.grey_opening(array, footprint=footprint) + assert_array_almost_equal(output, expected) + + def test_grey_opening02(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + tmp = ndimage.grey_erosion(array, footprint=footprint, + structure=structure) + expected = ndimage.grey_dilation(tmp, footprint=footprint, + structure=structure) + output = ndimage.grey_opening(array, footprint=footprint, + structure=structure) + assert_array_almost_equal(output, expected) + + def test_grey_closing01(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + tmp = ndimage.grey_dilation(array, footprint=footprint) + expected = ndimage.grey_erosion(tmp, footprint=footprint) + output = ndimage.grey_closing(array, footprint=footprint) + assert_array_almost_equal(expected, output) + + def test_grey_closing02(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + tmp = ndimage.grey_dilation(array, footprint=footprint, + structure=structure) + expected = ndimage.grey_erosion(tmp, footprint=footprint, + structure=structure) + output = ndimage.grey_closing(array, footprint=footprint, + structure=structure) + assert_array_almost_equal(expected, output) + + @skip_xp_backends(np_only=True, reason='output= arrays are numpy-specific') + def test_morphological_gradient01(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + tmp1 = ndimage.grey_dilation(array, footprint=footprint, + structure=structure) + tmp2 = ndimage.grey_erosion(array, footprint=footprint, + structure=structure) + expected = tmp1 - tmp2 + output = xp.zeros(array.shape, dtype=array.dtype) + ndimage.morphological_gradient(array, footprint=footprint, + structure=structure, output=output) + assert_array_almost_equal(expected, output) + + def test_morphological_gradient02(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + tmp1 = ndimage.grey_dilation(array, footprint=footprint, + structure=structure) + tmp2 = ndimage.grey_erosion(array, footprint=footprint, + structure=structure) + expected = tmp1 - tmp2 + output = ndimage.morphological_gradient(array, footprint=footprint, + structure=structure) + assert_array_almost_equal(expected, output) + + @skip_xp_backends(np_only=True, reason='output= arrays are numpy-specific') + def test_morphological_laplace01(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + tmp1 = ndimage.grey_dilation(array, footprint=footprint, + structure=structure) + tmp2 = ndimage.grey_erosion(array, footprint=footprint, + structure=structure) + expected = tmp1 + tmp2 - 2 * array + output = xp.zeros(array.shape, dtype=array.dtype) + ndimage.morphological_laplace(array, footprint=footprint, + structure=structure, output=output) + assert_array_almost_equal(expected, output) + + def test_morphological_laplace02(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + tmp1 = ndimage.grey_dilation(array, footprint=footprint, + structure=structure) + tmp2 = ndimage.grey_erosion(array, footprint=footprint, + structure=structure) + expected = tmp1 + tmp2 - 2 * array + output = ndimage.morphological_laplace(array, footprint=footprint, + structure=structure) + assert_array_almost_equal(expected, output) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + def test_white_tophat01(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + tmp = ndimage.grey_opening(array, footprint=footprint, + structure=structure) + expected = array - tmp + output = xp.zeros(array.shape, dtype=array.dtype) + ndimage.white_tophat(array, footprint=footprint, + structure=structure, output=output) + assert_array_almost_equal(expected, output) + + def test_white_tophat02(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + tmp = ndimage.grey_opening(array, footprint=footprint, + structure=structure) + expected = array - tmp + output = ndimage.white_tophat(array, footprint=footprint, + structure=structure) + assert_array_almost_equal(expected, output) + + @xfail_xp_backends('cupy', reason="cupy#8399") + def test_white_tophat03(self, xp): + + array = np.asarray([[1, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 0, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 1]], dtype=bool) + array = xp.asarray(array) + structure = np.ones((3, 3), dtype=bool) + structure = xp.asarray(structure) + expected = np.asarray([[0, 1, 1, 0, 0, 0, 0], + [1, 0, 0, 1, 1, 1, 0], + [1, 0, 0, 1, 1, 1, 0], + [0, 1, 1, 0, 0, 0, 1], + [0, 1, 1, 0, 1, 0, 1], + [0, 1, 1, 0, 0, 0, 1], + [0, 0, 0, 1, 1, 1, 1]], dtype=bool) + expected = xp.asarray(expected) + + output = ndimage.white_tophat(array, structure=structure) + xp_assert_equal(expected, output) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + def test_white_tophat04(self, xp): + array = np.eye(5, dtype=bool) + structure = np.ones((3, 3), dtype=bool) + + array = xp.asarray(array) + structure = xp.asarray(structure) + + # Check that type mismatch is properly handled + output = xp.empty_like(array, dtype=xp.float64) + ndimage.white_tophat(array, structure=structure, output=output) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + def test_black_tophat01(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + tmp = ndimage.grey_closing(array, footprint=footprint, + structure=structure) + expected = tmp - array + output = xp.zeros(array.shape, dtype=array.dtype) + ndimage.black_tophat(array, footprint=footprint, + structure=structure, output=output) + assert_array_almost_equal(expected, output) + + def test_black_tophat02(self, xp): + array = xp.asarray([[3, 2, 5, 1, 4], + [7, 6, 9, 3, 5], + [5, 8, 3, 7, 1]]) + footprint = xp.asarray([[1, 0, 1], [1, 1, 0]]) + structure = xp.asarray([[0, 0, 0], [0, 0, 0]]) + tmp = ndimage.grey_closing(array, footprint=footprint, + structure=structure) + expected = tmp - array + output = ndimage.black_tophat(array, footprint=footprint, + structure=structure) + assert_array_almost_equal(expected, output) + + @xfail_xp_backends('cupy', reason="cupy/cupy#8399") + def test_black_tophat03(self, xp): + + array = np.asarray([[1, 0, 0, 0, 0, 0, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 0, 1, 0], + [0, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 1]], dtype=bool) + array = xp.asarray(array) + structure = np.ones((3, 3), dtype=bool) + structure = xp.asarray(structure) + expected = np.asarray([[0, 1, 1, 1, 1, 1, 1], + [1, 0, 0, 0, 0, 0, 1], + [1, 0, 0, 0, 0, 0, 1], + [1, 0, 0, 0, 0, 0, 1], + [1, 0, 0, 0, 1, 0, 1], + [1, 0, 0, 0, 0, 0, 1], + [1, 1, 1, 1, 1, 1, 0]], dtype=bool) + expected = xp.asarray(expected) + + output = ndimage.black_tophat(array, structure=structure) + xp_assert_equal(expected, output) + + @skip_xp_backends("jax.numpy", reason="output array is read-only.") + def test_black_tophat04(self, xp): + array = xp.asarray(np.eye(5, dtype=bool)) + structure = xp.asarray(np.ones((3, 3), dtype=bool)) + + # Check that type mismatch is properly handled + output = xp.empty_like(array, dtype=xp.float64) + ndimage.black_tophat(array, structure=structure, output=output) + + @skip_xp_backends(cpu_only=True) + @skip_xp_backends( + "cupy", reason="these filters do not yet have axes support in CuPy") + @skip_xp_backends( + "jax.numpy", reason="these filters are not implemented in JAX.numpy") + @pytest.mark.parametrize('origin', [(0, 0), (-1, 0)]) + @pytest.mark.parametrize('expand_axis', [0, 1, 2]) + @pytest.mark.parametrize('mode', ['reflect', 'constant', 'nearest', + 'mirror', 'wrap']) + @pytest.mark.parametrize('footprint_mode', ['size', 'footprint', + 'structure']) + @pytest.mark.parametrize('func_name', ["grey_erosion", + "grey_dilation", + "grey_opening", + "grey_closing", + "morphological_laplace", + "morphological_gradient", + "white_tophat", + "black_tophat"]) + def test_grey_axes(self, xp, func_name, expand_axis, origin, footprint_mode, + mode): + + data = xp.asarray([[0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 4, 0, 0, 0], + [0, 0, 2, 1, 0, 2, 0], + [0, 3, 0, 6, 5, 0, 1], + [0, 4, 5, 3, 3, 4, 0], + [0, 0, 9, 3, 0, 0, 0], + [0, 0, 0, 2, 0, 0, 0]]) + kwargs = dict(origin=origin, mode=mode) + if footprint_mode == 'size': + kwargs['size'] = (2, 3) + else: + kwargs['footprint'] = xp.asarray([[1, 0, 1], [1, 1, 0]]) + if footprint_mode == 'structure': + kwargs['structure'] = xp.ones_like(kwargs['footprint']) + func = getattr(ndimage, func_name) + expected = func(data, **kwargs) + + # replicate data and expected result along a new axis + n_reps = 5 + expected = xp.stack([expected] * n_reps, axis=expand_axis) + data = xp.stack([data] * n_reps, axis=expand_axis) + + # filter all axes except expand_axis + axes = [0, 1, 2] + axes.remove(expand_axis) + + if is_numpy(xp) or is_cupy(xp): + out = xp.zeros(expected.shape, dtype=expected.dtype) + func(data, output=out, axes=axes, **kwargs) + else: + # inplace output= is unsupported by JAX + out = func(data, axes=axes, **kwargs) + xp_assert_close(out, expected) + + @pytest.mark.parametrize('dtype', types) + def test_hit_or_miss01(self, dtype, xp): + if not (is_numpy(xp) or is_cupy(xp)): + pytest.xfail("inplace output= is numpy-specific") + + dtype = getattr(xp, dtype) + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + struct = xp.asarray(struct) + expected = [[0, 0, 0, 0, 0], + [0, 1, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0]] + expected = xp.asarray(expected) + data = xp.asarray([[0, 1, 0, 0, 0], + [1, 1, 1, 0, 0], + [0, 1, 0, 1, 1], + [0, 0, 1, 1, 1], + [0, 1, 1, 1, 0], + [0, 1, 1, 1, 1], + [0, 1, 1, 1, 1], + [0, 0, 0, 0, 0]], dtype=dtype) + out = xp.asarray(np.zeros(data.shape, dtype=bool)) + ndimage.binary_hit_or_miss(data, struct, output=out) + assert_array_almost_equal(expected, out) + + @pytest.mark.parametrize('dtype', types) + def test_hit_or_miss02(self, dtype, xp): + dtype = getattr(xp, dtype) + struct = [[0, 1, 0], + [1, 1, 1], + [0, 1, 0]] + expected = [[0, 0, 0, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + struct = xp.asarray(struct) + expected = xp.asarray(expected) + data = xp.asarray([[0, 1, 0, 0, 1, 1, 1, 0], + [1, 1, 1, 0, 0, 1, 0, 0], + [0, 1, 0, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_hit_or_miss(data, struct) + assert_array_almost_equal(expected, out) + + @pytest.mark.parametrize('dtype', types) + def test_hit_or_miss03(self, dtype, xp): + dtype = getattr(xp, dtype) + struct1 = [[0, 0, 0], + [1, 1, 1], + [0, 0, 0]] + struct2 = [[1, 1, 1], + [0, 0, 0], + [1, 1, 1]] + expected = [[0, 0, 0, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0]] + struct1 = xp.asarray(struct1) + struct2 = xp.asarray(struct2) + expected = xp.asarray(expected) + data = xp.asarray([[0, 1, 0, 0, 1, 1, 1, 0], + [1, 1, 1, 0, 0, 0, 0, 0], + [0, 1, 0, 1, 1, 1, 1, 0], + [0, 0, 1, 1, 1, 1, 1, 0], + [0, 1, 1, 1, 0, 1, 1, 0], + [0, 0, 0, 0, 1, 1, 1, 0], + [0, 1, 1, 1, 1, 1, 1, 0], + [0, 0, 0, 0, 0, 0, 0, 0]], dtype=dtype) + out = ndimage.binary_hit_or_miss(data, struct1, struct2) + assert_array_almost_equal(expected, out) + + +class TestDilateFix: + + # pytest's setup_method seems to clash with the autouse `xp` fixture + # so call _setup manually from all methods + def _setup(self, xp): + # dilation related setup + self.array = xp.asarray([[0, 0, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 1, 0], + [0, 0, 1, 1, 0], + [0, 0, 0, 0, 0]], dtype=xp.uint8) + + self.sq3x3 = xp.ones((3, 3)) + dilated3x3 = ndimage.binary_dilation(self.array, structure=self.sq3x3) + + if is_numpy(xp): + self.dilated3x3 = dilated3x3.view(xp.uint8) + else: + astype = array_namespace(dilated3x3).astype + self.dilated3x3 = astype(dilated3x3, xp.uint8) + + + def test_dilation_square_structure(self, xp): + self._setup(xp) + result = ndimage.grey_dilation(self.array, structure=self.sq3x3) + # +1 accounts for difference between grey and binary dilation + assert_array_almost_equal(result, self.dilated3x3 + 1) + + def test_dilation_scalar_size(self, xp): + self._setup(xp) + result = ndimage.grey_dilation(self.array, size=3) + assert_array_almost_equal(result, self.dilated3x3) + + +class TestBinaryOpeningClosing: + + def _setup(self, xp): + a = np.zeros((5, 5), dtype=bool) + a[1:4, 1:4] = True + a[4, 4] = True + self.array = xp.asarray(a) + self.sq3x3 = xp.ones((3, 3)) + self.opened_old = ndimage.binary_opening(self.array, self.sq3x3, + 1, None, 0) + self.closed_old = ndimage.binary_closing(self.array, self.sq3x3, + 1, None, 0) + + def test_opening_new_arguments(self, xp): + self._setup(xp) + opened_new = ndimage.binary_opening(self.array, self.sq3x3, 1, None, + 0, None, 0, False) + xp_assert_equal(opened_new, self.opened_old) + + def test_closing_new_arguments(self, xp): + self._setup(xp) + closed_new = ndimage.binary_closing(self.array, self.sq3x3, 1, None, + 0, None, 0, False) + xp_assert_equal(closed_new, self.closed_old) + + +def test_binary_erosion_noninteger_iterations(xp): + # regression test for gh-9905, gh-9909: ValueError for + # non integer iterations + data = xp.ones([1]) + assert_raises(TypeError, ndimage.binary_erosion, data, iterations=0.5) + assert_raises(TypeError, ndimage.binary_erosion, data, iterations=1.5) + + +def test_binary_dilation_noninteger_iterations(xp): + # regression test for gh-9905, gh-9909: ValueError for + # non integer iterations + data = xp.ones([1]) + assert_raises(TypeError, ndimage.binary_dilation, data, iterations=0.5) + assert_raises(TypeError, ndimage.binary_dilation, data, iterations=1.5) + + +def test_binary_opening_noninteger_iterations(xp): + # regression test for gh-9905, gh-9909: ValueError for + # non integer iterations + data = xp.ones([1]) + assert_raises(TypeError, ndimage.binary_opening, data, iterations=0.5) + assert_raises(TypeError, ndimage.binary_opening, data, iterations=1.5) + + +def test_binary_closing_noninteger_iterations(xp): + # regression test for gh-9905, gh-9909: ValueError for + # non integer iterations + data = xp.ones([1]) + assert_raises(TypeError, ndimage.binary_closing, data, iterations=0.5) + assert_raises(TypeError, ndimage.binary_closing, data, iterations=1.5) + + +def test_binary_closing_noninteger_brute_force_passes_when_true(xp): + # regression test for gh-9905, gh-9909: ValueError for + # non integer iterations + if is_cupy(xp): + pytest.xfail("CuPy: NotImplementedError: only brute_force iteration") + + data = xp.ones([1]) + + xp_assert_equal(ndimage.binary_erosion(data, iterations=2, brute_force=1.5), + ndimage.binary_erosion(data, iterations=2, brute_force=bool(1.5)) + ) + xp_assert_equal(ndimage.binary_erosion(data, iterations=2, brute_force=0.0), + ndimage.binary_erosion(data, iterations=2, brute_force=bool(0.0)) + ) + + +@pytest.mark.parametrize( + 'function', + ['binary_erosion', 'binary_dilation', 'binary_opening', 'binary_closing'], +) +@pytest.mark.parametrize('iterations', [1, 5]) +@pytest.mark.parametrize('brute_force', [False, True]) +def test_binary_input_as_output(function, iterations, brute_force, xp): + rstate = np.random.RandomState(123) + data = rstate.randint(low=0, high=2, size=100).astype(bool) + ndi_func = getattr(ndimage, function) + + # input data is not modified + data_orig = data.copy() + expected = ndi_func(data, brute_force=brute_force, iterations=iterations) + xp_assert_equal(data, data_orig) + + # data should now contain the expected result + ndi_func(data, brute_force=brute_force, iterations=iterations, output=data) + xp_assert_equal(expected, data) + + +def test_binary_hit_or_miss_input_as_output(xp): + if not (is_numpy(xp) or is_cupy(xp)): + pytest.xfail("inplace output= is numpy-specific") + + rstate = np.random.RandomState(123) + data = rstate.randint(low=0, high=2, size=100).astype(bool) + + # input data is not modified + data_orig = data.copy() + expected = ndimage.binary_hit_or_miss(data) + xp_assert_equal(data, data_orig) + + # data should now contain the expected result + ndimage.binary_hit_or_miss(data, output=data) + xp_assert_equal(expected, data) + + +def test_distance_transform_cdt_invalid_metric(xp): + if is_cupy(xp): + pytest.xfail("CuPy does not have distance_transform_cdt") + + msg = 'invalid metric provided' + with pytest.raises(ValueError, match=msg): + ndimage.distance_transform_cdt(xp.ones((5, 5)), + metric="garbage") diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_ni_support.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_ni_support.py new file mode 100644 index 0000000000000000000000000000000000000000..426d1cf0eccd1ac0f10a90412f008f4b3463c333 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_ni_support.py @@ -0,0 +1,78 @@ +import pytest + +import numpy as np +from .._ni_support import _get_output + + +@pytest.mark.parametrize( + 'dtype', + [ + # String specifiers + 'f4', 'float32', 'complex64', 'complex128', + # Type and dtype specifiers + np.float32, float, np.dtype('f4'), + # Derive from input + None, + ], +) +def test_get_output_basic(dtype): + shape = (2, 3) + + input_ = np.zeros(shape, dtype='float32') + + # For None, derive dtype from input + expected_dtype = 'float32' if dtype is None else dtype + + # Output is dtype-specifier, retrieve shape from input + result = _get_output(dtype, input_) + assert result.shape == shape + assert result.dtype == np.dtype(expected_dtype) + + # Output is dtype specifier, with explicit shape, overriding input + result = _get_output(dtype, input_, shape=(3, 2)) + assert result.shape == (3, 2) + assert result.dtype == np.dtype(expected_dtype) + + # Output is pre-allocated array, return directly + output = np.zeros(shape, dtype=dtype) + result = _get_output(output, input_) + assert result is output + + +@pytest.mark.thread_unsafe +def test_get_output_complex(): + shape = (2, 3) + + input_ = np.zeros(shape) + + # None, promote input type to complex + result = _get_output(None, input_, complex_output=True) + assert result.shape == shape + assert result.dtype == np.dtype('complex128') + + # Explicit type, promote type to complex + with pytest.warns(UserWarning, match='promoting specified output dtype to complex'): + result = _get_output(float, input_, complex_output=True) + assert result.shape == shape + assert result.dtype == np.dtype('complex128') + + # String specifier, simply verify complex output + result = _get_output('complex64', input_, complex_output=True) + assert result.shape == shape + assert result.dtype == np.dtype('complex64') + + +def test_get_output_error_cases(): + input_ = np.zeros((2, 3), 'float32') + + # Two separate paths can raise the same error + with pytest.raises(RuntimeError, match='output must have complex dtype'): + _get_output('float32', input_, complex_output=True) + with pytest.raises(RuntimeError, match='output must have complex dtype'): + _get_output(np.zeros((2, 3)), input_, complex_output=True) + + with pytest.raises(RuntimeError, match='output must have numeric dtype'): + _get_output('void', input_) + + with pytest.raises(RuntimeError, match='shape not correct'): + _get_output(np.zeros((3, 2)), input_) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_splines.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_splines.py new file mode 100644 index 0000000000000000000000000000000000000000..2561ba5acef20ac340e06164bae96f187486c06a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/ndimage/tests/test_splines.py @@ -0,0 +1,72 @@ +"""Tests for spline filtering.""" +import pytest + +import numpy as np +from scipy._lib._array_api import assert_almost_equal + +from scipy import ndimage + +from scipy.conftest import array_api_compatible +skip_xp_backends = pytest.mark.skip_xp_backends +pytestmark = [array_api_compatible, pytest.mark.usefixtures("skip_xp_backends"), + skip_xp_backends(cpu_only=True, exceptions=['cupy', 'jax.numpy'],)] + + +def get_spline_knot_values(order): + """Knot values to the right of a B-spline's center.""" + knot_values = {0: [1], + 1: [1], + 2: [6, 1], + 3: [4, 1], + 4: [230, 76, 1], + 5: [66, 26, 1]} + + return knot_values[order] + + +def make_spline_knot_matrix(xp, n, order, mode='mirror'): + """Matrix to invert to find the spline coefficients.""" + knot_values = get_spline_knot_values(order) + + # NB: do computations with numpy, convert to xp as the last step only + + matrix = np.zeros((n, n)) + for diag, knot_value in enumerate(knot_values): + indices = np.arange(diag, n) + if diag == 0: + matrix[indices, indices] = knot_value + else: + matrix[indices, indices - diag] = knot_value + matrix[indices - diag, indices] = knot_value + + knot_values_sum = knot_values[0] + 2 * sum(knot_values[1:]) + + if mode == 'mirror': + start, step = 1, 1 + elif mode == 'reflect': + start, step = 0, 1 + elif mode == 'grid-wrap': + start, step = -1, -1 + else: + raise ValueError(f'unsupported mode {mode}') + + for row in range(len(knot_values) - 1): + for idx, knot_value in enumerate(knot_values[row + 1:]): + matrix[row, start + step*idx] += knot_value + matrix[-row - 1, -start - 1 - step*idx] += knot_value + + return xp.asarray(matrix / knot_values_sum) + + +@pytest.mark.parametrize('order', [0, 1, 2, 3, 4, 5]) +@pytest.mark.parametrize('mode', ['mirror', 'grid-wrap', 'reflect']) +def test_spline_filter_vs_matrix_solution(order, mode, xp): + n = 100 + eye = xp.eye(n, dtype=xp.float64) + spline_filter_axis_0 = ndimage.spline_filter1d(eye, axis=0, order=order, + mode=mode) + spline_filter_axis_1 = ndimage.spline_filter1d(eye, axis=1, order=order, + mode=mode) + matrix = make_spline_knot_matrix(xp, n, order, mode=mode) + assert_almost_equal(eye, spline_filter_axis_0 @ matrix) + assert_almost_equal(eye, spline_filter_axis_1 @ matrix.T) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..a44a8c133b674aea416efeb4da469241b50a547f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/__init__.py @@ -0,0 +1,131 @@ +""" +================================================= +Orthogonal distance regression (:mod:`scipy.odr`) +================================================= + +.. currentmodule:: scipy.odr + +Package Content +=============== + +.. autosummary:: + :toctree: generated/ + + Data -- The data to fit. + RealData -- Data with weights as actual std. dev.s and/or covariances. + Model -- Stores information about the function to be fit. + ODR -- Gathers all info & manages the main fitting routine. + Output -- Result from the fit. + odr -- Low-level function for ODR. + + OdrWarning -- Warning about potential problems when running ODR. + OdrError -- Error exception. + OdrStop -- Stop exception. + + polynomial -- Factory function for a general polynomial model. + exponential -- Exponential model + multilinear -- Arbitrary-dimensional linear model + unilinear -- Univariate linear model + quadratic -- Quadratic model + +Usage information +================= + +Introduction +------------ + +Why Orthogonal Distance Regression (ODR)? Sometimes one has +measurement errors in the explanatory (a.k.a., "independent") +variable(s), not just the response (a.k.a., "dependent") variable(s). +Ordinary Least Squares (OLS) fitting procedures treat the data for +explanatory variables as fixed, i.e., not subject to error of any kind. +Furthermore, OLS procedures require that the response variables be an +explicit function of the explanatory variables; sometimes making the +equation explicit is impractical and/or introduces errors. ODR can +handle both of these cases with ease, and can even reduce to the OLS +case if that is sufficient for the problem. + +ODRPACK is a FORTRAN-77 library for performing ODR with possibly +non-linear fitting functions. It uses a modified trust-region +Levenberg-Marquardt-type algorithm [1]_ to estimate the function +parameters. The fitting functions are provided by Python functions +operating on NumPy arrays. The required derivatives may be provided +by Python functions as well, or may be estimated numerically. ODRPACK +can do explicit or implicit ODR fits, or it can do OLS. Input and +output variables may be multidimensional. Weights can be provided to +account for different variances of the observations, and even +covariances between dimensions of the variables. + +The `scipy.odr` package offers an object-oriented interface to +ODRPACK, in addition to the low-level `odr` function. + +Additional background information about ODRPACK can be found in the +`ODRPACK User's Guide +`_, reading +which is recommended. + +Basic usage +----------- + +1. Define the function you want to fit against.:: + + def f(B, x): + '''Linear function y = m*x + b''' + # B is a vector of the parameters. + # x is an array of the current x values. + # x is in the same format as the x passed to Data or RealData. + # + # Return an array in the same format as y passed to Data or RealData. + return B[0]*x + B[1] + +2. Create a Model.:: + + linear = Model(f) + +3. Create a Data or RealData instance.:: + + mydata = Data(x, y, wd=1./power(sx,2), we=1./power(sy,2)) + + or, when the actual covariances are known:: + + mydata = RealData(x, y, sx=sx, sy=sy) + +4. Instantiate ODR with your data, model and initial parameter estimate.:: + + myodr = ODR(mydata, linear, beta0=[1., 2.]) + +5. Run the fit.:: + + myoutput = myodr.run() + +6. Examine output.:: + + myoutput.pprint() + + +References +---------- +.. [1] P. T. Boggs and J. E. Rogers, "Orthogonal Distance Regression," + in "Statistical analysis of measurement error models and + applications: proceedings of the AMS-IMS-SIAM joint summer research + conference held June 10-16, 1989," Contemporary Mathematics, + vol. 112, pg. 186, 1990. + +""" +# version: 0.7 +# author: Robert Kern +# date: 2006-09-21 + +from ._odrpack import * +from ._models import * +from . import _add_newdocs + +# Deprecated namespaces, to be removed in v2.0.0 +from . import models, odrpack + +__all__ = [s for s in dir() + if not (s.startswith('_') or s in ('odr_stop', 'odr_error'))] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/_add_newdocs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/_add_newdocs.py new file mode 100644 index 0000000000000000000000000000000000000000..e09fb6cc8c5f1523dfbeaef466a5b76bd22c01bb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/_add_newdocs.py @@ -0,0 +1,34 @@ +from numpy.lib import add_newdoc + +add_newdoc('scipy.odr', 'odr', + """ + odr(fcn, beta0, y, x, we=None, wd=None, fjacb=None, fjacd=None, extra_args=None, + ifixx=None, ifixb=None, job=0, iprint=0, errfile=None, rptfile=None, ndigit=0, + taufac=0.0, sstol=-1.0, partol=-1.0, maxit=-1, stpb=None, stpd=None, sclb=None, + scld=None, work=None, iwork=None, full_output=0) + + Low-level function for ODR. + + See Also + -------- + ODR : The ODR class gathers all information and coordinates the running of the + main fitting routine. + Model : The Model class stores information about the function you wish to fit. + Data : The data to fit. + RealData : Data with weights as actual std. dev.s and/or covariances. + + Notes + ----- + This is a function performing the same operation as the `ODR`, + `Model`, and `Data` classes together. The parameters of this + function are explained in the class documentation. + + """) + +add_newdoc('scipy.odr.__odrpack', '_set_exceptions', + """ + _set_exceptions(odr_error, odr_stop) + + Internal function: set exception classes. + + """) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/_models.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/_models.py new file mode 100644 index 0000000000000000000000000000000000000000..e0a8d2275dcc4698a9ea61be5871d62069be2599 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/_models.py @@ -0,0 +1,315 @@ +""" Collection of Model instances for use with the odrpack fitting package. +""" +import numpy as np +from scipy.odr._odrpack import Model + +__all__ = ['Model', 'exponential', 'multilinear', 'unilinear', 'quadratic', + 'polynomial'] + + +def _lin_fcn(B, x): + a, b = B[0], B[1:] + b.shape = (b.shape[0], 1) + + return a + (x*b).sum(axis=0) + + +def _lin_fjb(B, x): + a = np.ones(x.shape[-1], float) + res = np.concatenate((a, x.ravel())) + res.shape = (B.shape[-1], x.shape[-1]) + return res + + +def _lin_fjd(B, x): + b = B[1:] + b = np.repeat(b, (x.shape[-1],)*b.shape[-1], axis=0) + b.shape = x.shape + return b + + +def _lin_est(data): + # Eh. The answer is analytical, so just return all ones. + # Don't return zeros since that will interfere with + # ODRPACK's auto-scaling procedures. + + if len(data.x.shape) == 2: + m = data.x.shape[0] + else: + m = 1 + + return np.ones((m + 1,), float) + + +def _poly_fcn(B, x, powers): + a, b = B[0], B[1:] + b.shape = (b.shape[0], 1) + + return a + np.sum(b * np.power(x, powers), axis=0) + + +def _poly_fjacb(B, x, powers): + res = np.concatenate((np.ones(x.shape[-1], float), + np.power(x, powers).flat)) + res.shape = (B.shape[-1], x.shape[-1]) + return res + + +def _poly_fjacd(B, x, powers): + b = B[1:] + b.shape = (b.shape[0], 1) + + b = b * powers + + return np.sum(b * np.power(x, powers-1), axis=0) + + +def _exp_fcn(B, x): + return B[0] + np.exp(B[1] * x) + + +def _exp_fjd(B, x): + return B[1] * np.exp(B[1] * x) + + +def _exp_fjb(B, x): + res = np.concatenate((np.ones(x.shape[-1], float), x * np.exp(B[1] * x))) + res.shape = (2, x.shape[-1]) + return res + + +def _exp_est(data): + # Eh. + return np.array([1., 1.]) + + +class _MultilinearModel(Model): + r""" + Arbitrary-dimensional linear model + + This model is defined by :math:`y=\beta_0 + \sum_{i=1}^m \beta_i x_i` + + Examples + -------- + We can calculate orthogonal distance regression with an arbitrary + dimensional linear model: + + >>> from scipy import odr + >>> import numpy as np + >>> x = np.linspace(0.0, 5.0) + >>> y = 10.0 + 5.0 * x + >>> data = odr.Data(x, y) + >>> odr_obj = odr.ODR(data, odr.multilinear) + >>> output = odr_obj.run() + >>> print(output.beta) + [10. 5.] + + """ + + def __init__(self): + super().__init__( + _lin_fcn, fjacb=_lin_fjb, fjacd=_lin_fjd, estimate=_lin_est, + meta={'name': 'Arbitrary-dimensional Linear', + 'equ': 'y = B_0 + Sum[i=1..m, B_i * x_i]', + 'TeXequ': r'$y=\beta_0 + \sum_{i=1}^m \beta_i x_i$'}) + + +multilinear = _MultilinearModel() + + +def polynomial(order): + """ + Factory function for a general polynomial model. + + Parameters + ---------- + order : int or sequence + If an integer, it becomes the order of the polynomial to fit. If + a sequence of numbers, then these are the explicit powers in the + polynomial. + A constant term (power 0) is always included, so don't include 0. + Thus, polynomial(n) is equivalent to polynomial(range(1, n+1)). + + Returns + ------- + polynomial : Model instance + Model instance. + + Examples + -------- + We can fit an input data using orthogonal distance regression (ODR) with + a polynomial model: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy import odr + >>> x = np.linspace(0.0, 5.0) + >>> y = np.sin(x) + >>> poly_model = odr.polynomial(3) # using third order polynomial model + >>> data = odr.Data(x, y) + >>> odr_obj = odr.ODR(data, poly_model) + >>> output = odr_obj.run() # running ODR fitting + >>> poly = np.poly1d(output.beta[::-1]) + >>> poly_y = poly(x) + >>> plt.plot(x, y, label="input data") + >>> plt.plot(x, poly_y, label="polynomial ODR") + >>> plt.legend() + >>> plt.show() + + """ + + powers = np.asarray(order) + if powers.shape == (): + # Scalar. + powers = np.arange(1, powers + 1) + + powers.shape = (len(powers), 1) + len_beta = len(powers) + 1 + + def _poly_est(data, len_beta=len_beta): + # Eh. Ignore data and return all ones. + return np.ones((len_beta,), float) + + return Model(_poly_fcn, fjacd=_poly_fjacd, fjacb=_poly_fjacb, + estimate=_poly_est, extra_args=(powers,), + meta={'name': 'Sorta-general Polynomial', + 'equ': 'y = B_0 + Sum[i=1..%s, B_i * (x**i)]' % (len_beta-1), + 'TeXequ': r'$y=\beta_0 + \sum_{i=1}^{%s} \beta_i x^i$' % + (len_beta-1)}) + + +class _ExponentialModel(Model): + r""" + Exponential model + + This model is defined by :math:`y=\beta_0 + e^{\beta_1 x}` + + Examples + -------- + We can calculate orthogonal distance regression with an exponential model: + + >>> from scipy import odr + >>> import numpy as np + >>> x = np.linspace(0.0, 5.0) + >>> y = -10.0 + np.exp(0.5*x) + >>> data = odr.Data(x, y) + >>> odr_obj = odr.ODR(data, odr.exponential) + >>> output = odr_obj.run() + >>> print(output.beta) + [-10. 0.5] + + """ + + def __init__(self): + super().__init__(_exp_fcn, fjacd=_exp_fjd, fjacb=_exp_fjb, + estimate=_exp_est, + meta={'name': 'Exponential', + 'equ': 'y= B_0 + exp(B_1 * x)', + 'TeXequ': r'$y=\beta_0 + e^{\beta_1 x}$'}) + + +exponential = _ExponentialModel() + + +def _unilin(B, x): + return x*B[0] + B[1] + + +def _unilin_fjd(B, x): + return np.ones(x.shape, float) * B[0] + + +def _unilin_fjb(B, x): + _ret = np.concatenate((x, np.ones(x.shape, float))) + _ret.shape = (2,) + x.shape + + return _ret + + +def _unilin_est(data): + return (1., 1.) + + +def _quadratic(B, x): + return x*(x*B[0] + B[1]) + B[2] + + +def _quad_fjd(B, x): + return 2*x*B[0] + B[1] + + +def _quad_fjb(B, x): + _ret = np.concatenate((x*x, x, np.ones(x.shape, float))) + _ret.shape = (3,) + x.shape + + return _ret + + +def _quad_est(data): + return (1.,1.,1.) + + +class _UnilinearModel(Model): + r""" + Univariate linear model + + This model is defined by :math:`y = \beta_0 x + \beta_1` + + Examples + -------- + We can calculate orthogonal distance regression with an unilinear model: + + >>> from scipy import odr + >>> import numpy as np + >>> x = np.linspace(0.0, 5.0) + >>> y = 1.0 * x + 2.0 + >>> data = odr.Data(x, y) + >>> odr_obj = odr.ODR(data, odr.unilinear) + >>> output = odr_obj.run() + >>> print(output.beta) + [1. 2.] + + """ + + def __init__(self): + super().__init__(_unilin, fjacd=_unilin_fjd, fjacb=_unilin_fjb, + estimate=_unilin_est, + meta={'name': 'Univariate Linear', + 'equ': 'y = B_0 * x + B_1', + 'TeXequ': '$y = \\beta_0 x + \\beta_1$'}) + + +unilinear = _UnilinearModel() + + +class _QuadraticModel(Model): + r""" + Quadratic model + + This model is defined by :math:`y = \beta_0 x^2 + \beta_1 x + \beta_2` + + Examples + -------- + We can calculate orthogonal distance regression with a quadratic model: + + >>> from scipy import odr + >>> import numpy as np + >>> x = np.linspace(0.0, 5.0) + >>> y = 1.0 * x ** 2 + 2.0 * x + 3.0 + >>> data = odr.Data(x, y) + >>> odr_obj = odr.ODR(data, odr.quadratic) + >>> output = odr_obj.run() + >>> print(output.beta) + [1. 2. 3.] + + """ + + def __init__(self): + super().__init__( + _quadratic, fjacd=_quad_fjd, fjacb=_quad_fjb, estimate=_quad_est, + meta={'name': 'Quadratic', + 'equ': 'y = B_0*x**2 + B_1*x + B_2', + 'TeXequ': '$y = \\beta_0 x^2 + \\beta_1 x + \\beta_2'}) + + +quadratic = _QuadraticModel() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/_odrpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/_odrpack.py new file mode 100644 index 0000000000000000000000000000000000000000..30d46aa3f4465f31b32e5f13f0b01b940981d489 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/_odrpack.py @@ -0,0 +1,1154 @@ +""" +Python wrappers for Orthogonal Distance Regression (ODRPACK). + +Notes +===== + +* Array formats -- FORTRAN stores its arrays in memory column first, i.e., an + array element A(i, j, k) will be next to A(i+1, j, k). In C and, consequently, + NumPy, arrays are stored row first: A[i, j, k] is next to A[i, j, k+1]. For + efficiency and convenience, the input and output arrays of the fitting + function (and its Jacobians) are passed to FORTRAN without transposition. + Therefore, where the ODRPACK documentation says that the X array is of shape + (N, M), it will be passed to the Python function as an array of shape (M, N). + If M==1, the 1-D case, then nothing matters; if M>1, then your + Python functions will be dealing with arrays that are indexed in reverse of + the ODRPACK documentation. No real issue, but watch out for your indexing of + the Jacobians: the i,jth elements (@f_i/@x_j) evaluated at the nth + observation will be returned as jacd[j, i, n]. Except for the Jacobians, it + really is easier to deal with x[0] and x[1] than x[:,0] and x[:,1]. Of course, + you can always use the transpose() function from SciPy explicitly. + +* Examples -- See the accompanying file test/test.py for examples of how to set + up fits of your own. Some are taken from the User's Guide; some are from + other sources. + +* Models -- Some common models are instantiated in the accompanying module + models.py . Contributions are welcome. + +Credits +======= + +* Thanks to Arnold Moene and Gerard Vermeulen for fixing some killer bugs. + +Robert Kern +robert.kern@gmail.com + +""" +import os +from threading import Lock + +import numpy as np +from warnings import warn +from scipy.odr import __odrpack + +__all__ = ['odr', 'OdrWarning', 'OdrError', 'OdrStop', + 'Data', 'RealData', 'Model', 'Output', 'ODR', + 'odr_error', 'odr_stop'] + +odr = __odrpack.odr +ODR_LOCK = Lock() + + +class OdrWarning(UserWarning): + """ + Warning indicating that the data passed into + ODR will cause problems when passed into 'odr' + that the user should be aware of. + """ + pass + + +class OdrError(Exception): + """ + Exception indicating an error in fitting. + + This is raised by `~scipy.odr.odr` if an error occurs during fitting. + """ + pass + + +class OdrStop(Exception): + """ + Exception stopping fitting. + + You can raise this exception in your objective function to tell + `~scipy.odr.odr` to stop fitting. + """ + pass + + +# Backwards compatibility +odr_error = OdrError +odr_stop = OdrStop + +__odrpack._set_exceptions(OdrError, OdrStop) + + +def _conv(obj, dtype=None): + """ Convert an object to the preferred form for input to the odr routine. + """ + + if obj is None: + return obj + else: + if dtype is None: + obj = np.asarray(obj) + else: + obj = np.asarray(obj, dtype) + if obj.shape == (): + # Scalar. + return obj.dtype.type(obj) + else: + return obj + + +def _report_error(info): + """ Interprets the return code of the odr routine. + + Parameters + ---------- + info : int + The return code of the odr routine. + + Returns + ------- + problems : list(str) + A list of messages about why the odr() routine stopped. + """ + + stopreason = ('Blank', + 'Sum of squares convergence', + 'Parameter convergence', + 'Both sum of squares and parameter convergence', + 'Iteration limit reached')[info % 5] + + if info >= 5: + # questionable results or fatal error + + I = (info//10000 % 10, + info//1000 % 10, + info//100 % 10, + info//10 % 10, + info % 10) + problems = [] + + if I[0] == 0: + if I[1] != 0: + problems.append('Derivatives possibly not correct') + if I[2] != 0: + problems.append('Error occurred in callback') + if I[3] != 0: + problems.append('Problem is not full rank at solution') + problems.append(stopreason) + elif I[0] == 1: + if I[1] != 0: + problems.append('N < 1') + if I[2] != 0: + problems.append('M < 1') + if I[3] != 0: + problems.append('NP < 1 or NP > N') + if I[4] != 0: + problems.append('NQ < 1') + elif I[0] == 2: + if I[1] != 0: + problems.append('LDY and/or LDX incorrect') + if I[2] != 0: + problems.append('LDWE, LD2WE, LDWD, and/or LD2WD incorrect') + if I[3] != 0: + problems.append('LDIFX, LDSTPD, and/or LDSCLD incorrect') + if I[4] != 0: + problems.append('LWORK and/or LIWORK too small') + elif I[0] == 3: + if I[1] != 0: + problems.append('STPB and/or STPD incorrect') + if I[2] != 0: + problems.append('SCLB and/or SCLD incorrect') + if I[3] != 0: + problems.append('WE incorrect') + if I[4] != 0: + problems.append('WD incorrect') + elif I[0] == 4: + problems.append('Error in derivatives') + elif I[0] == 5: + problems.append('Error occurred in callback') + elif I[0] == 6: + problems.append('Numerical error detected') + + return problems + + else: + return [stopreason] + + +class Data: + """ + The data to fit. + + Parameters + ---------- + x : array_like + Observed data for the independent variable of the regression + y : array_like, optional + If array-like, observed data for the dependent variable of the + regression. A scalar input implies that the model to be used on + the data is implicit. + we : array_like, optional + If `we` is a scalar, then that value is used for all data points (and + all dimensions of the response variable). + If `we` is a rank-1 array of length q (the dimensionality of the + response variable), then this vector is the diagonal of the covariant + weighting matrix for all data points. + If `we` is a rank-1 array of length n (the number of data points), then + the i'th element is the weight for the i'th response variable + observation (single-dimensional only). + If `we` is a rank-2 array of shape (q, q), then this is the full + covariant weighting matrix broadcast to each observation. + If `we` is a rank-2 array of shape (q, n), then `we[:,i]` is the + diagonal of the covariant weighting matrix for the i'th observation. + If `we` is a rank-3 array of shape (q, q, n), then `we[:,:,i]` is the + full specification of the covariant weighting matrix for each + observation. + If the fit is implicit, then only a positive scalar value is used. + wd : array_like, optional + If `wd` is a scalar, then that value is used for all data points + (and all dimensions of the input variable). If `wd` = 0, then the + covariant weighting matrix for each observation is set to the identity + matrix (so each dimension of each observation has the same weight). + If `wd` is a rank-1 array of length m (the dimensionality of the input + variable), then this vector is the diagonal of the covariant weighting + matrix for all data points. + If `wd` is a rank-1 array of length n (the number of data points), then + the i'th element is the weight for the ith input variable observation + (single-dimensional only). + If `wd` is a rank-2 array of shape (m, m), then this is the full + covariant weighting matrix broadcast to each observation. + If `wd` is a rank-2 array of shape (m, n), then `wd[:,i]` is the + diagonal of the covariant weighting matrix for the ith observation. + If `wd` is a rank-3 array of shape (m, m, n), then `wd[:,:,i]` is the + full specification of the covariant weighting matrix for each + observation. + fix : array_like of ints, optional + The `fix` argument is the same as ifixx in the class ODR. It is an + array of integers with the same shape as data.x that determines which + input observations are treated as fixed. One can use a sequence of + length m (the dimensionality of the input observations) to fix some + dimensions for all observations. A value of 0 fixes the observation, + a value > 0 makes it free. + meta : dict, optional + Free-form dictionary for metadata. + + Notes + ----- + Each argument is attached to the member of the instance of the same name. + The structures of `x` and `y` are described in the Model class docstring. + If `y` is an integer, then the Data instance can only be used to fit with + implicit models where the dimensionality of the response is equal to the + specified value of `y`. + + The `we` argument weights the effect a deviation in the response variable + has on the fit. The `wd` argument weights the effect a deviation in the + input variable has on the fit. To handle multidimensional inputs and + responses easily, the structure of these arguments has the n'th + dimensional axis first. These arguments heavily use the structured + arguments feature of ODRPACK to conveniently and flexibly support all + options. See the ODRPACK User's Guide for a full explanation of how these + weights are used in the algorithm. Basically, a higher value of the weight + for a particular data point makes a deviation at that point more + detrimental to the fit. + + """ + + def __init__(self, x, y=None, we=None, wd=None, fix=None, meta=None): + self.x = _conv(x) + + if not isinstance(self.x, np.ndarray): + raise ValueError("Expected an 'ndarray' of data for 'x', " + f"but instead got data of type '{type(self.x).__name__}'") + + self.y = _conv(y) + self.we = _conv(we) + self.wd = _conv(wd) + self.fix = _conv(fix) + self.meta = {} if meta is None else meta + + def set_meta(self, **kwds): + """ Update the metadata dictionary with the keywords and data provided + by keywords. + + Examples + -------- + :: + + data.set_meta(lab="Ph 7; Lab 26", title="Ag110 + Ag108 Decay") + """ + + self.meta.update(kwds) + + def __getattr__(self, attr): + """ Dispatch attribute access to the metadata dictionary. + """ + if attr != "meta" and attr in self.meta: + return self.meta[attr] + else: + raise AttributeError(f"'{attr}' not in metadata") + + +class RealData(Data): + """ + The data, with weightings as actual standard deviations and/or + covariances. + + Parameters + ---------- + x : array_like + Observed data for the independent variable of the regression + y : array_like, optional + If array-like, observed data for the dependent variable of the + regression. A scalar input implies that the model to be used on + the data is implicit. + sx : array_like, optional + Standard deviations of `x`. + `sx` are standard deviations of `x` and are converted to weights by + dividing 1.0 by their squares. + sy : array_like, optional + Standard deviations of `y`. + `sy` are standard deviations of `y` and are converted to weights by + dividing 1.0 by their squares. + covx : array_like, optional + Covariance of `x` + `covx` is an array of covariance matrices of `x` and are converted to + weights by performing a matrix inversion on each observation's + covariance matrix. + covy : array_like, optional + Covariance of `y` + `covy` is an array of covariance matrices and are converted to + weights by performing a matrix inversion on each observation's + covariance matrix. + fix : array_like, optional + The argument and member fix is the same as Data.fix and ODR.ifixx: + It is an array of integers with the same shape as `x` that + determines which input observations are treated as fixed. One can + use a sequence of length m (the dimensionality of the input + observations) to fix some dimensions for all observations. A value + of 0 fixes the observation, a value > 0 makes it free. + meta : dict, optional + Free-form dictionary for metadata. + + Notes + ----- + The weights `wd` and `we` are computed from provided values as follows: + + `sx` and `sy` are converted to weights by dividing 1.0 by their squares. + For example, ``wd = 1./np.power(`sx`, 2)``. + + `covx` and `covy` are arrays of covariance matrices and are converted to + weights by performing a matrix inversion on each observation's covariance + matrix. For example, ``we[i] = np.linalg.inv(covy[i])``. + + These arguments follow the same structured argument conventions as wd and + we only restricted by their natures: `sx` and `sy` can't be rank-3, but + `covx` and `covy` can be. + + Only set *either* `sx` or `covx` (not both). Setting both will raise an + exception. Same with `sy` and `covy`. + + """ + + def __init__(self, x, y=None, sx=None, sy=None, covx=None, covy=None, + fix=None, meta=None): + if (sx is not None) and (covx is not None): + raise ValueError("cannot set both sx and covx") + if (sy is not None) and (covy is not None): + raise ValueError("cannot set both sy and covy") + + # Set flags for __getattr__ + self._ga_flags = {} + if sx is not None: + self._ga_flags['wd'] = 'sx' + else: + self._ga_flags['wd'] = 'covx' + if sy is not None: + self._ga_flags['we'] = 'sy' + else: + self._ga_flags['we'] = 'covy' + + self.x = _conv(x) + + if not isinstance(self.x, np.ndarray): + raise ValueError("Expected an 'ndarray' of data for 'x', " + f"but instead got data of type '{type(self.x).__name__}'") + + self.y = _conv(y) + self.sx = _conv(sx) + self.sy = _conv(sy) + self.covx = _conv(covx) + self.covy = _conv(covy) + self.fix = _conv(fix) + self.meta = {} if meta is None else meta + + def _sd2wt(self, sd): + """ Convert standard deviation to weights. + """ + + return 1./np.power(sd, 2) + + def _cov2wt(self, cov): + """ Convert covariance matrix(-ices) to weights. + """ + + from scipy.linalg import inv + + if len(cov.shape) == 2: + return inv(cov) + else: + weights = np.zeros(cov.shape, float) + + for i in range(cov.shape[-1]): # n + weights[:,:,i] = inv(cov[:,:,i]) + + return weights + + def __getattr__(self, attr): + + if attr not in ('wd', 'we'): + if attr != "meta" and attr in self.meta: + return self.meta[attr] + else: + raise AttributeError(f"'{attr}' not in metadata") + else: + lookup_tbl = {('wd', 'sx'): (self._sd2wt, self.sx), + ('wd', 'covx'): (self._cov2wt, self.covx), + ('we', 'sy'): (self._sd2wt, self.sy), + ('we', 'covy'): (self._cov2wt, self.covy)} + + func, arg = lookup_tbl[(attr, self._ga_flags[attr])] + + if arg is not None: + return func(*(arg,)) + else: + return None + + +class Model: + """ + The Model class stores information about the function you wish to fit. + + It stores the function itself, at the least, and optionally stores + functions which compute the Jacobians used during fitting. Also, one + can provide a function that will provide reasonable starting values + for the fit parameters possibly given the set of data. + + Parameters + ---------- + fcn : function + fcn(beta, x) --> y + fjacb : function + Jacobian of fcn wrt the fit parameters beta. + + fjacb(beta, x) --> @f_i(x,B)/@B_j + fjacd : function + Jacobian of fcn wrt the (possibly multidimensional) input + variable. + + fjacd(beta, x) --> @f_i(x,B)/@x_j + extra_args : tuple, optional + If specified, `extra_args` should be a tuple of extra + arguments to pass to `fcn`, `fjacb`, and `fjacd`. Each will be called + by `apply(fcn, (beta, x) + extra_args)` + estimate : array_like of rank-1 + Provides estimates of the fit parameters from the data + + estimate(data) --> estbeta + implicit : boolean + If TRUE, specifies that the model + is implicit; i.e `fcn(beta, x)` ~= 0 and there is no y data to fit + against + meta : dict, optional + freeform dictionary of metadata for the model + + Notes + ----- + Note that the `fcn`, `fjacb`, and `fjacd` operate on NumPy arrays and + return a NumPy array. The `estimate` object takes an instance of the + Data class. + + Here are the rules for the shapes of the argument and return + arrays of the callback functions: + + `x` + if the input data is single-dimensional, then `x` is rank-1 + array; i.e., ``x = array([1, 2, 3, ...]); x.shape = (n,)`` + If the input data is multi-dimensional, then `x` is a rank-2 array; + i.e., ``x = array([[1, 2, ...], [2, 4, ...]]); x.shape = (m, n)``. + In all cases, it has the same shape as the input data array passed to + `~scipy.odr.odr`. `m` is the dimensionality of the input data, + `n` is the number of observations. + `y` + if the response variable is single-dimensional, then `y` is a + rank-1 array, i.e., ``y = array([2, 4, ...]); y.shape = (n,)``. + If the response variable is multi-dimensional, then `y` is a rank-2 + array, i.e., ``y = array([[2, 4, ...], [3, 6, ...]]); y.shape = + (q, n)`` where `q` is the dimensionality of the response variable. + `beta` + rank-1 array of length `p` where `p` is the number of parameters; + i.e. ``beta = array([B_1, B_2, ..., B_p])`` + `fjacb` + if the response variable is multi-dimensional, then the + return array's shape is ``(q, p, n)`` such that ``fjacb(x,beta)[l,k,i] = + d f_l(X,B)/d B_k`` evaluated at the ith data point. If ``q == 1``, then + the return array is only rank-2 and with shape ``(p, n)``. + `fjacd` + as with fjacb, only the return array's shape is ``(q, m, n)`` + such that ``fjacd(x,beta)[l,j,i] = d f_l(X,B)/d X_j`` at the ith data + point. If ``q == 1``, then the return array's shape is ``(m, n)``. If + ``m == 1``, the shape is (q, n). If `m == q == 1`, the shape is ``(n,)``. + + """ + + def __init__(self, fcn, fjacb=None, fjacd=None, + extra_args=None, estimate=None, implicit=0, meta=None): + + self.fcn = fcn + self.fjacb = fjacb + self.fjacd = fjacd + + if extra_args is not None: + extra_args = tuple(extra_args) + + self.extra_args = extra_args + self.estimate = estimate + self.implicit = implicit + self.meta = meta if meta is not None else {} + + def set_meta(self, **kwds): + """ Update the metadata dictionary with the keywords and data provided + here. + + Examples + -------- + set_meta(name="Exponential", equation="y = a exp(b x) + c") + """ + + self.meta.update(kwds) + + def __getattr__(self, attr): + """ Dispatch attribute access to the metadata. + """ + + if attr != "meta" and attr in self.meta: + return self.meta[attr] + else: + raise AttributeError(f"'{attr}' not in metadata") + + +class Output: + """ + The Output class stores the output of an ODR run. + + Attributes + ---------- + beta : ndarray + Estimated parameter values, of shape (q,). + sd_beta : ndarray + Standard deviations of the estimated parameters, of shape (p,). + cov_beta : ndarray + Covariance matrix of the estimated parameters, of shape (p,p). + Note that this `cov_beta` is not scaled by the residual variance + `res_var`, whereas `sd_beta` is. This means + ``np.sqrt(np.diag(output.cov_beta * output.res_var))`` is the same + result as `output.sd_beta`. + delta : ndarray, optional + Array of estimated errors in input variables, of same shape as `x`. + eps : ndarray, optional + Array of estimated errors in response variables, of same shape as `y`. + xplus : ndarray, optional + Array of ``x + delta``. + y : ndarray, optional + Array ``y = fcn(x + delta)``. + res_var : float, optional + Residual variance. + sum_square : float, optional + Sum of squares error. + sum_square_delta : float, optional + Sum of squares of delta error. + sum_square_eps : float, optional + Sum of squares of eps error. + inv_condnum : float, optional + Inverse condition number (cf. ODRPACK UG p. 77). + rel_error : float, optional + Relative error in function values computed within fcn. + work : ndarray, optional + Final work array. + work_ind : dict, optional + Indices into work for drawing out values (cf. ODRPACK UG p. 83). + info : int, optional + Reason for returning, as output by ODRPACK (cf. ODRPACK UG p. 38). + stopreason : list of str, optional + `info` interpreted into English. + + Notes + ----- + Takes one argument for initialization, the return value from the + function `~scipy.odr.odr`. The attributes listed as "optional" above are + only present if `~scipy.odr.odr` was run with ``full_output=1``. + + """ + + def __init__(self, output): + self.beta = output[0] + self.sd_beta = output[1] + self.cov_beta = output[2] + + if len(output) == 4: + # full output + self.__dict__.update(output[3]) + self.stopreason = _report_error(self.info) + + def pprint(self): + """ Pretty-print important results. + """ + + print('Beta:', self.beta) + print('Beta Std Error:', self.sd_beta) + print('Beta Covariance:', self.cov_beta) + if hasattr(self, 'info'): + print('Residual Variance:',self.res_var) + print('Inverse Condition #:', self.inv_condnum) + print('Reason(s) for Halting:') + for r in self.stopreason: + print(f' {r}') + + +class ODR: + """ + The ODR class gathers all information and coordinates the running of the + main fitting routine. + + Members of instances of the ODR class have the same names as the arguments + to the initialization routine. + + Parameters + ---------- + data : Data class instance + instance of the Data class + model : Model class instance + instance of the Model class + + Other Parameters + ---------------- + beta0 : array_like of rank-1 + a rank-1 sequence of initial parameter values. Optional if + model provides an "estimate" function to estimate these values. + delta0 : array_like of floats of rank-1, optional + a (double-precision) float array to hold the initial values of + the errors in the input variables. Must be same shape as data.x + ifixb : array_like of ints of rank-1, optional + sequence of integers with the same length as beta0 that determines + which parameters are held fixed. A value of 0 fixes the parameter, + a value > 0 makes the parameter free. + ifixx : array_like of ints with same shape as data.x, optional + an array of integers with the same shape as data.x that determines + which input observations are treated as fixed. One can use a sequence + of length m (the dimensionality of the input observations) to fix some + dimensions for all observations. A value of 0 fixes the observation, + a value > 0 makes it free. + job : int, optional + an integer telling ODRPACK what tasks to perform. See p. 31 of the + ODRPACK User's Guide if you absolutely must set the value here. Use the + method set_job post-initialization for a more readable interface. + iprint : int, optional + an integer telling ODRPACK what to print. See pp. 33-34 of the + ODRPACK User's Guide if you absolutely must set the value here. Use the + method set_iprint post-initialization for a more readable interface. + errfile : str, optional + string with the filename to print ODRPACK errors to. If the file already + exists, an error will be thrown. The `overwrite` argument can be used to + prevent this. *Do Not Open This File Yourself!* + rptfile : str, optional + string with the filename to print ODRPACK summaries to. If the file + already exists, an error will be thrown. The `overwrite` argument can be + used to prevent this. *Do Not Open This File Yourself!* + ndigit : int, optional + integer specifying the number of reliable digits in the computation + of the function. + taufac : float, optional + float specifying the initial trust region. The default value is 1. + The initial trust region is equal to taufac times the length of the + first computed Gauss-Newton step. taufac must be less than 1. + sstol : float, optional + float specifying the tolerance for convergence based on the relative + change in the sum-of-squares. The default value is eps**(1/2) where eps + is the smallest value such that 1 + eps > 1 for double precision + computation on the machine. sstol must be less than 1. + partol : float, optional + float specifying the tolerance for convergence based on the relative + change in the estimated parameters. The default value is eps**(2/3) for + explicit models and ``eps**(1/3)`` for implicit models. partol must be less + than 1. + maxit : int, optional + integer specifying the maximum number of iterations to perform. For + first runs, maxit is the total number of iterations performed and + defaults to 50. For restarts, maxit is the number of additional + iterations to perform and defaults to 10. + stpb : array_like, optional + sequence (``len(stpb) == len(beta0)``) of relative step sizes to compute + finite difference derivatives wrt the parameters. + stpd : optional + array (``stpd.shape == data.x.shape`` or ``stpd.shape == (m,)``) of relative + step sizes to compute finite difference derivatives wrt the input + variable errors. If stpd is a rank-1 array with length m (the + dimensionality of the input variable), then the values are broadcast to + all observations. + sclb : array_like, optional + sequence (``len(stpb) == len(beta0)``) of scaling factors for the + parameters. The purpose of these scaling factors are to scale all of + the parameters to around unity. Normally appropriate scaling factors + are computed if this argument is not specified. Specify them yourself + if the automatic procedure goes awry. + scld : array_like, optional + array (scld.shape == data.x.shape or scld.shape == (m,)) of scaling + factors for the *errors* in the input variables. Again, these factors + are automatically computed if you do not provide them. If scld.shape == + (m,), then the scaling factors are broadcast to all observations. + work : ndarray, optional + array to hold the double-valued working data for ODRPACK. When + restarting, takes the value of self.output.work. + iwork : ndarray, optional + array to hold the integer-valued working data for ODRPACK. When + restarting, takes the value of self.output.iwork. + overwrite : bool, optional + If it is True, output files defined by `errfile` and `rptfile` are + overwritten. The default is False. + + Attributes + ---------- + data : Data + The data for this fit + model : Model + The model used in fit + output : Output + An instance if the Output class containing all of the returned + data from an invocation of ODR.run() or ODR.restart() + + """ + + def __init__(self, data, model, beta0=None, delta0=None, ifixb=None, + ifixx=None, job=None, iprint=None, errfile=None, rptfile=None, + ndigit=None, taufac=None, sstol=None, partol=None, maxit=None, + stpb=None, stpd=None, sclb=None, scld=None, work=None, iwork=None, + overwrite=False): + + self.data = data + self.model = model + + if beta0 is None: + if self.model.estimate is not None: + self.beta0 = _conv(self.model.estimate(self.data)) + else: + raise ValueError( + "must specify beta0 or provide an estimator with the model" + ) + else: + self.beta0 = _conv(beta0) + + if ifixx is None and data.fix is not None: + ifixx = data.fix + + if overwrite: + # remove output files for overwriting. + if rptfile is not None and os.path.exists(rptfile): + os.remove(rptfile) + if errfile is not None and os.path.exists(errfile): + os.remove(errfile) + + self.delta0 = _conv(delta0) + # These really are 32-bit integers in FORTRAN (gfortran), even on 64-bit + # platforms. + # XXX: some other FORTRAN compilers may not agree. + self.ifixx = _conv(ifixx, dtype=np.int32) + self.ifixb = _conv(ifixb, dtype=np.int32) + self.job = job + self.iprint = iprint + self.errfile = errfile + self.rptfile = rptfile + self.ndigit = ndigit + self.taufac = taufac + self.sstol = sstol + self.partol = partol + self.maxit = maxit + self.stpb = _conv(stpb) + self.stpd = _conv(stpd) + self.sclb = _conv(sclb) + self.scld = _conv(scld) + self.work = _conv(work) + self.iwork = _conv(iwork) + + self.output = None + + self._check() + + def _check(self): + """ Check the inputs for consistency, but don't bother checking things + that the builtin function odr will check. + """ + + x_s = list(self.data.x.shape) + + if isinstance(self.data.y, np.ndarray): + y_s = list(self.data.y.shape) + if self.model.implicit: + raise OdrError("an implicit model cannot use response data") + else: + # implicit model with q == self.data.y + y_s = [self.data.y, x_s[-1]] + if not self.model.implicit: + raise OdrError("an explicit model needs response data") + self.set_job(fit_type=1) + + if x_s[-1] != y_s[-1]: + raise OdrError("number of observations do not match") + + n = x_s[-1] + + if len(x_s) == 2: + m = x_s[0] + else: + m = 1 + if len(y_s) == 2: + q = y_s[0] + else: + q = 1 + + p = len(self.beta0) + + # permissible output array shapes + + fcn_perms = [(q, n)] + fjacd_perms = [(q, m, n)] + fjacb_perms = [(q, p, n)] + + if q == 1: + fcn_perms.append((n,)) + fjacd_perms.append((m, n)) + fjacb_perms.append((p, n)) + if m == 1: + fjacd_perms.append((q, n)) + if p == 1: + fjacb_perms.append((q, n)) + if m == q == 1: + fjacd_perms.append((n,)) + if p == q == 1: + fjacb_perms.append((n,)) + + # try evaluating the supplied functions to make sure they provide + # sensible outputs + + arglist = (self.beta0, self.data.x) + if self.model.extra_args is not None: + arglist = arglist + self.model.extra_args + res = self.model.fcn(*arglist) + + if res.shape not in fcn_perms: + print(res.shape) + print(fcn_perms) + raise OdrError(f"fcn does not output {y_s}-shaped array") + + if self.model.fjacd is not None: + res = self.model.fjacd(*arglist) + if res.shape not in fjacd_perms: + raise OdrError( + f"fjacd does not output {repr((q, m, n))}-shaped array") + if self.model.fjacb is not None: + res = self.model.fjacb(*arglist) + if res.shape not in fjacb_perms: + raise OdrError( + f"fjacb does not output {repr((q, p, n))}-shaped array") + + # check shape of delta0 + + if self.delta0 is not None and self.delta0.shape != self.data.x.shape: + raise OdrError( + f"delta0 is not a {repr(self.data.x.shape)}-shaped array") + + if self.data.x.size == 0: + warn("Empty data detected for ODR instance. " + "Do not expect any fitting to occur", + OdrWarning, stacklevel=3) + + def _gen_work(self): + """ Generate a suitable work array if one does not already exist. + """ + + n = self.data.x.shape[-1] + p = self.beta0.shape[0] + + if len(self.data.x.shape) == 2: + m = self.data.x.shape[0] + else: + m = 1 + + if self.model.implicit: + q = self.data.y + elif len(self.data.y.shape) == 2: + q = self.data.y.shape[0] + else: + q = 1 + + if self.data.we is None: + ldwe = ld2we = 1 + elif len(self.data.we.shape) == 3: + ld2we, ldwe = self.data.we.shape[1:] + else: + we = self.data.we + ldwe = 1 + ld2we = 1 + if we.ndim == 1 and q == 1: + ldwe = n + elif we.ndim == 2: + if we.shape == (q, q): + ld2we = q + elif we.shape == (q, n): + ldwe = n + + if self.job % 10 < 2: + # ODR not OLS + lwork = (18 + 11*p + p*p + m + m*m + 4*n*q + 6*n*m + 2*n*q*p + + 2*n*q*m + q*q + 5*q + q*(p+m) + ldwe*ld2we*q) + else: + # OLS not ODR + lwork = (18 + 11*p + p*p + m + m*m + 4*n*q + 2*n*m + 2*n*q*p + + 5*q + q*(p+m) + ldwe*ld2we*q) + + if isinstance(self.work, np.ndarray) and self.work.shape == (lwork,)\ + and self.work.dtype.str.endswith('f8'): + # the existing array is fine + return + else: + self.work = np.zeros((lwork,), float) + + def set_job(self, fit_type=None, deriv=None, var_calc=None, + del_init=None, restart=None): + """ + Sets the "job" parameter is a hopefully comprehensible way. + + If an argument is not specified, then the value is left as is. The + default value from class initialization is for all of these options set + to 0. + + Parameters + ---------- + fit_type : {0, 1, 2} int + 0 -> explicit ODR + + 1 -> implicit ODR + + 2 -> ordinary least-squares + deriv : {0, 1, 2, 3} int + 0 -> forward finite differences + + 1 -> central finite differences + + 2 -> user-supplied derivatives (Jacobians) with results + checked by ODRPACK + + 3 -> user-supplied derivatives, no checking + var_calc : {0, 1, 2} int + 0 -> calculate asymptotic covariance matrix and fit + parameter uncertainties (V_B, s_B) using derivatives + recomputed at the final solution + + 1 -> calculate V_B and s_B using derivatives from last iteration + + 2 -> do not calculate V_B and s_B + del_init : {0, 1} int + 0 -> initial input variable offsets set to 0 + + 1 -> initial offsets provided by user in variable "work" + restart : {0, 1} int + 0 -> fit is not a restart + + 1 -> fit is a restart + + Notes + ----- + The permissible values are different from those given on pg. 31 of the + ODRPACK User's Guide only in that one cannot specify numbers greater than + the last value for each variable. + + If one does not supply functions to compute the Jacobians, the fitting + procedure will change deriv to 0, finite differences, as a default. To + initialize the input variable offsets by yourself, set del_init to 1 and + put the offsets into the "work" variable correctly. + + """ + + if self.job is None: + job_l = [0, 0, 0, 0, 0] + else: + job_l = [self.job // 10000 % 10, + self.job // 1000 % 10, + self.job // 100 % 10, + self.job // 10 % 10, + self.job % 10] + + if fit_type in (0, 1, 2): + job_l[4] = fit_type + if deriv in (0, 1, 2, 3): + job_l[3] = deriv + if var_calc in (0, 1, 2): + job_l[2] = var_calc + if del_init in (0, 1): + job_l[1] = del_init + if restart in (0, 1): + job_l[0] = restart + + self.job = (job_l[0]*10000 + job_l[1]*1000 + + job_l[2]*100 + job_l[3]*10 + job_l[4]) + + def set_iprint(self, init=None, so_init=None, + iter=None, so_iter=None, iter_step=None, final=None, so_final=None): + """ Set the iprint parameter for the printing of computation reports. + + If any of the arguments are specified here, then they are set in the + iprint member. If iprint is not set manually or with this method, then + ODRPACK defaults to no printing. If no filename is specified with the + member rptfile, then ODRPACK prints to stdout. One can tell ODRPACK to + print to stdout in addition to the specified filename by setting the + so_* arguments to this function, but one cannot specify to print to + stdout but not a file since one can do that by not specifying a rptfile + filename. + + There are three reports: initialization, iteration, and final reports. + They are represented by the arguments init, iter, and final + respectively. The permissible values are 0, 1, and 2 representing "no + report", "short report", and "long report" respectively. + + The argument iter_step (0 <= iter_step <= 9) specifies how often to make + the iteration report; the report will be made for every iter_step'th + iteration starting with iteration one. If iter_step == 0, then no + iteration report is made, regardless of the other arguments. + + If the rptfile is None, then any so_* arguments supplied will raise an + exception. + """ + if self.iprint is None: + self.iprint = 0 + + ip = [self.iprint // 1000 % 10, + self.iprint // 100 % 10, + self.iprint // 10 % 10, + self.iprint % 10] + + # make a list to convert iprint digits to/from argument inputs + # rptfile, stdout + ip2arg = [[0, 0], # none, none + [1, 0], # short, none + [2, 0], # long, none + [1, 1], # short, short + [2, 1], # long, short + [1, 2], # short, long + [2, 2]] # long, long + + if (self.rptfile is None and + (so_init is not None or + so_iter is not None or + so_final is not None)): + raise OdrError( + "no rptfile specified, cannot output to stdout twice") + + iprint_l = ip2arg[ip[0]] + ip2arg[ip[1]] + ip2arg[ip[3]] + + if init is not None: + iprint_l[0] = init + if so_init is not None: + iprint_l[1] = so_init + if iter is not None: + iprint_l[2] = iter + if so_iter is not None: + iprint_l[3] = so_iter + if final is not None: + iprint_l[4] = final + if so_final is not None: + iprint_l[5] = so_final + + if iter_step in range(10): + # 0..9 + ip[2] = iter_step + + ip[0] = ip2arg.index(iprint_l[0:2]) + ip[1] = ip2arg.index(iprint_l[2:4]) + ip[3] = ip2arg.index(iprint_l[4:6]) + + self.iprint = ip[0]*1000 + ip[1]*100 + ip[2]*10 + ip[3] + + def run(self): + """ Run the fitting routine with all of the information given and with ``full_output=1``. + + Returns + ------- + output : Output instance + This object is also assigned to the attribute .output . + """ # noqa: E501 + + args = (self.model.fcn, self.beta0, self.data.y, self.data.x) + kwds = {'full_output': 1} + kwd_l = ['ifixx', 'ifixb', 'job', 'iprint', 'errfile', 'rptfile', + 'ndigit', 'taufac', 'sstol', 'partol', 'maxit', 'stpb', + 'stpd', 'sclb', 'scld', 'work', 'iwork'] + + if self.delta0 is not None and (self.job // 10000) % 10 == 0: + # delta0 provided and fit is not a restart + self._gen_work() + + d0 = np.ravel(self.delta0) + + self.work[:len(d0)] = d0 + + # set the kwds from other objects explicitly + if self.model.fjacb is not None: + kwds['fjacb'] = self.model.fjacb + if self.model.fjacd is not None: + kwds['fjacd'] = self.model.fjacd + if self.data.we is not None: + kwds['we'] = self.data.we + if self.data.wd is not None: + kwds['wd'] = self.data.wd + if self.model.extra_args is not None: + kwds['extra_args'] = self.model.extra_args + + # implicitly set kwds from self's members + for attr in kwd_l: + obj = getattr(self, attr) + if obj is not None: + kwds[attr] = obj + + with ODR_LOCK: + self.output = Output(odr(*args, **kwds)) + + return self.output + + def restart(self, iter=None): + """ Restarts the run with iter more iterations. + + Parameters + ---------- + iter : int, optional + ODRPACK's default for the number of new iterations is 10. + + Returns + ------- + output : Output instance + This object is also assigned to the attribute .output . + """ + + if self.output is None: + raise OdrError("cannot restart: run() has not been called before") + + self.set_job(restart=1) + self.work = self.output.work + self.iwork = self.output.iwork + + self.maxit = iter + + return self.run() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/models.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/models.py new file mode 100644 index 0000000000000000000000000000000000000000..0289b59747bb68a4954e58732ac69d7df144f5f6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/models.py @@ -0,0 +1,20 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.odr` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'Model', 'exponential', 'multilinear', 'unilinear', + 'quadratic', 'polynomial' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="odr", module="models", + private_modules=["_models"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/odrpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/odrpack.py new file mode 100644 index 0000000000000000000000000000000000000000..192fb3342b7957703996957c882d44656706e41b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/odrpack.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.odr` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'odr', 'OdrWarning', 'OdrError', 'OdrStop', + 'Data', 'RealData', 'Model', 'Output', 'ODR', + 'odr_error', 'odr_stop' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="odr", module="odrpack", + private_modules=["_odrpack"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/tests/test_odr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/tests/test_odr.py new file mode 100644 index 0000000000000000000000000000000000000000..971cce6c55a84e08a182e3b25bf9a7e362937e01 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/odr/tests/test_odr.py @@ -0,0 +1,607 @@ +import pickle +import tempfile +import shutil +import os + +import numpy as np +from numpy import pi +from numpy.testing import (assert_array_almost_equal, + assert_equal, assert_warns, + assert_allclose) +import pytest +from pytest import raises as assert_raises + +from scipy.odr import (Data, Model, ODR, RealData, OdrStop, OdrWarning, + multilinear, exponential, unilinear, quadratic, + polynomial) + + +class TestODR: + + # Bad Data for 'x' + + def test_bad_data(self): + assert_raises(ValueError, Data, 2, 1) + assert_raises(ValueError, RealData, 2, 1) + + # Empty Data for 'x' + def empty_data_func(self, B, x): + return B[0]*x + B[1] + + @pytest.mark.thread_unsafe + def test_empty_data(self): + beta0 = [0.02, 0.0] + linear = Model(self.empty_data_func) + + empty_dat = Data([], []) + assert_warns(OdrWarning, ODR, + empty_dat, linear, beta0=beta0) + + empty_dat = RealData([], []) + assert_warns(OdrWarning, ODR, + empty_dat, linear, beta0=beta0) + + # Explicit Example + + def explicit_fcn(self, B, x): + ret = B[0] + B[1] * np.power(np.exp(B[2]*x) - 1.0, 2) + return ret + + def explicit_fjd(self, B, x): + eBx = np.exp(B[2]*x) + ret = B[1] * 2.0 * (eBx-1.0) * B[2] * eBx + return ret + + def explicit_fjb(self, B, x): + eBx = np.exp(B[2]*x) + res = np.vstack([np.ones(x.shape[-1]), + np.power(eBx-1.0, 2), + B[1]*2.0*(eBx-1.0)*eBx*x]) + return res + + def test_explicit(self): + explicit_mod = Model( + self.explicit_fcn, + fjacb=self.explicit_fjb, + fjacd=self.explicit_fjd, + meta=dict(name='Sample Explicit Model', + ref='ODRPACK UG, pg. 39'), + ) + explicit_dat = Data([0.,0.,5.,7.,7.5,10.,16.,26.,30.,34.,34.5,100.], + [1265.,1263.6,1258.,1254.,1253.,1249.8,1237.,1218.,1220.6, + 1213.8,1215.5,1212.]) + explicit_odr = ODR(explicit_dat, explicit_mod, beta0=[1500.0, -50.0, -0.1], + ifixx=[0,0,1,1,1,1,1,1,1,1,1,0]) + explicit_odr.set_job(deriv=2) + explicit_odr.set_iprint(init=0, iter=0, final=0) + + out = explicit_odr.run() + assert_array_almost_equal( + out.beta, + np.array([1.2646548050648876e+03, -5.4018409956678255e+01, + -8.7849712165253724e-02]), + ) + assert_array_almost_equal( + out.sd_beta, + np.array([1.0349270280543437, 1.583997785262061, 0.0063321988657267]), + ) + assert_array_almost_equal( + out.cov_beta, + np.array([[4.4949592379003039e-01, -3.7421976890364739e-01, + -8.0978217468468912e-04], + [-3.7421976890364739e-01, 1.0529686462751804e+00, + -1.9453521827942002e-03], + [-8.0978217468468912e-04, -1.9453521827942002e-03, + 1.6827336938454476e-05]]), + ) + + # Implicit Example + + def implicit_fcn(self, B, x): + return (B[2]*np.power(x[0]-B[0], 2) + + 2.0*B[3]*(x[0]-B[0])*(x[1]-B[1]) + + B[4]*np.power(x[1]-B[1], 2) - 1.0) + + def test_implicit(self): + implicit_mod = Model( + self.implicit_fcn, + implicit=1, + meta=dict(name='Sample Implicit Model', + ref='ODRPACK UG, pg. 49'), + ) + implicit_dat = Data([ + [0.5,1.2,1.6,1.86,2.12,2.36,2.44,2.36,2.06,1.74,1.34,0.9,-0.28, + -0.78,-1.36,-1.9,-2.5,-2.88,-3.18,-3.44], + [-0.12,-0.6,-1.,-1.4,-2.54,-3.36,-4.,-4.75,-5.25,-5.64,-5.97,-6.32, + -6.44,-6.44,-6.41,-6.25,-5.88,-5.5,-5.24,-4.86]], + 1, + ) + implicit_odr = ODR(implicit_dat, implicit_mod, + beta0=[-1.0, -3.0, 0.09, 0.02, 0.08]) + + out = implicit_odr.run() + assert_array_almost_equal( + out.beta, + np.array([-0.9993809167281279, -2.9310484652026476, 0.0875730502693354, + 0.0162299708984738, 0.0797537982976416]), + ) + assert_array_almost_equal( + out.sd_beta, + np.array([0.1113840353364371, 0.1097673310686467, 0.0041060738314314, + 0.0027500347539902, 0.0034962501532468]), + ) + assert_allclose( + out.cov_beta, + np.array([[2.1089274602333052e+00, -1.9437686411979040e+00, + 7.0263550868344446e-02, -4.7175267373474862e-02, + 5.2515575927380355e-02], + [-1.9437686411979040e+00, 2.0481509222414456e+00, + -6.1600515853057307e-02, 4.6268827806232933e-02, + -5.8822307501391467e-02], + [7.0263550868344446e-02, -6.1600515853057307e-02, + 2.8659542561579308e-03, -1.4628662260014491e-03, + 1.4528860663055824e-03], + [-4.7175267373474862e-02, 4.6268827806232933e-02, + -1.4628662260014491e-03, 1.2855592885514335e-03, + -1.2692942951415293e-03], + [5.2515575927380355e-02, -5.8822307501391467e-02, + 1.4528860663055824e-03, -1.2692942951415293e-03, + 2.0778813389755596e-03]]), + rtol=1e-6, atol=2e-6, + ) + + # Multi-variable Example + + def multi_fcn(self, B, x): + if (x < 0.0).any(): + raise OdrStop + theta = pi*B[3]/2. + ctheta = np.cos(theta) + stheta = np.sin(theta) + omega = np.power(2.*pi*x*np.exp(-B[2]), B[3]) + phi = np.arctan2((omega*stheta), (1.0 + omega*ctheta)) + r = (B[0] - B[1]) * np.power(np.sqrt(np.power(1.0 + omega*ctheta, 2) + + np.power(omega*stheta, 2)), -B[4]) + ret = np.vstack([B[1] + r*np.cos(B[4]*phi), + r*np.sin(B[4]*phi)]) + return ret + + def test_multi(self): + multi_mod = Model( + self.multi_fcn, + meta=dict(name='Sample Multi-Response Model', + ref='ODRPACK UG, pg. 56'), + ) + + multi_x = np.array([30.0, 50.0, 70.0, 100.0, 150.0, 200.0, 300.0, 500.0, + 700.0, 1000.0, 1500.0, 2000.0, 3000.0, 5000.0, 7000.0, 10000.0, + 15000.0, 20000.0, 30000.0, 50000.0, 70000.0, 100000.0, 150000.0]) + multi_y = np.array([ + [4.22, 4.167, 4.132, 4.038, 4.019, 3.956, 3.884, 3.784, 3.713, + 3.633, 3.54, 3.433, 3.358, 3.258, 3.193, 3.128, 3.059, 2.984, + 2.934, 2.876, 2.838, 2.798, 2.759], + [0.136, 0.167, 0.188, 0.212, 0.236, 0.257, 0.276, 0.297, 0.309, + 0.311, 0.314, 0.311, 0.305, 0.289, 0.277, 0.255, 0.24, 0.218, + 0.202, 0.182, 0.168, 0.153, 0.139], + ]) + n = len(multi_x) + multi_we = np.zeros((2, 2, n), dtype=float) + multi_ifixx = np.ones(n, dtype=int) + multi_delta = np.zeros(n, dtype=float) + + multi_we[0,0,:] = 559.6 + multi_we[1,0,:] = multi_we[0,1,:] = -1634.0 + multi_we[1,1,:] = 8397.0 + + for i in range(n): + if multi_x[i] < 100.0: + multi_ifixx[i] = 0 + elif multi_x[i] <= 150.0: + pass # defaults are fine + elif multi_x[i] <= 1000.0: + multi_delta[i] = 25.0 + elif multi_x[i] <= 10000.0: + multi_delta[i] = 560.0 + elif multi_x[i] <= 100000.0: + multi_delta[i] = 9500.0 + else: + multi_delta[i] = 144000.0 + if multi_x[i] == 100.0 or multi_x[i] == 150.0: + multi_we[:,:,i] = 0.0 + + multi_dat = Data(multi_x, multi_y, wd=1e-4/np.power(multi_x, 2), + we=multi_we) + multi_odr = ODR(multi_dat, multi_mod, beta0=[4.,2.,7.,.4,.5], + delta0=multi_delta, ifixx=multi_ifixx) + multi_odr.set_job(deriv=1, del_init=1) + + out = multi_odr.run() + assert_array_almost_equal( + out.beta, + np.array([4.3799880305938963, 2.4333057577497703, 8.0028845899503978, + 0.5101147161764654, 0.5173902330489161]), + ) + assert_array_almost_equal( + out.sd_beta, + np.array([0.0130625231081944, 0.0130499785273277, 0.1167085962217757, + 0.0132642749596149, 0.0288529201353984]), + ) + assert_array_almost_equal( + out.cov_beta, + np.array([[0.0064918418231375, 0.0036159705923791, 0.0438637051470406, + -0.0058700836512467, 0.011281212888768], + [0.0036159705923791, 0.0064793789429006, 0.0517610978353126, + -0.0051181304940204, 0.0130726943624117], + [0.0438637051470406, 0.0517610978353126, 0.5182263323095322, + -0.0563083340093696, 0.1269490939468611], + [-0.0058700836512467, -0.0051181304940204, -0.0563083340093696, + 0.0066939246261263, -0.0140184391377962], + [0.011281212888768, 0.0130726943624117, 0.1269490939468611, + -0.0140184391377962, 0.0316733013820852]]), + ) + + # Pearson's Data + # K. Pearson, Philosophical Magazine, 2, 559 (1901) + + def pearson_fcn(self, B, x): + return B[0] + B[1]*x + + def test_pearson(self): + p_x = np.array([0.,.9,1.8,2.6,3.3,4.4,5.2,6.1,6.5,7.4]) + p_y = np.array([5.9,5.4,4.4,4.6,3.5,3.7,2.8,2.8,2.4,1.5]) + p_sx = np.array([.03,.03,.04,.035,.07,.11,.13,.22,.74,1.]) + p_sy = np.array([1.,.74,.5,.35,.22,.22,.12,.12,.1,.04]) + + p_dat = RealData(p_x, p_y, sx=p_sx, sy=p_sy) + + # Reverse the data to test invariance of results + pr_dat = RealData(p_y, p_x, sx=p_sy, sy=p_sx) + + p_mod = Model(self.pearson_fcn, meta=dict(name='Uni-linear Fit')) + + p_odr = ODR(p_dat, p_mod, beta0=[1.,1.]) + pr_odr = ODR(pr_dat, p_mod, beta0=[1.,1.]) + + out = p_odr.run() + assert_array_almost_equal( + out.beta, + np.array([5.4767400299231674, -0.4796082367610305]), + ) + assert_array_almost_equal( + out.sd_beta, + np.array([0.3590121690702467, 0.0706291186037444]), + ) + assert_array_almost_equal( + out.cov_beta, + np.array([[0.0854275622946333, -0.0161807025443155], + [-0.0161807025443155, 0.003306337993922]]), + ) + + rout = pr_odr.run() + assert_array_almost_equal( + rout.beta, + np.array([11.4192022410781231, -2.0850374506165474]), + ) + assert_array_almost_equal( + rout.sd_beta, + np.array([0.9820231665657161, 0.3070515616198911]), + ) + assert_array_almost_equal( + rout.cov_beta, + np.array([[0.6391799462548782, -0.1955657291119177], + [-0.1955657291119177, 0.0624888159223392]]), + ) + + # Lorentz Peak + # The data is taken from one of the undergraduate physics labs I performed. + + def lorentz(self, beta, x): + return (beta[0]*beta[1]*beta[2] / np.sqrt(np.power(x*x - + beta[2]*beta[2], 2.0) + np.power(beta[1]*x, 2.0))) + + def test_lorentz(self): + l_sy = np.array([.29]*18) + l_sx = np.array([.000972971,.000948268,.000707632,.000706679, + .000706074, .000703918,.000698955,.000456856, + .000455207,.000662717,.000654619,.000652694, + .000000859202,.00106589,.00106378,.00125483, .00140818,.00241839]) + + l_dat = RealData( + [3.9094, 3.85945, 3.84976, 3.84716, 3.84551, 3.83964, 3.82608, + 3.78847, 3.78163, 3.72558, 3.70274, 3.6973, 3.67373, 3.65982, + 3.6562, 3.62498, 3.55525, 3.41886], + [652, 910.5, 984, 1000, 1007.5, 1053, 1160.5, 1409.5, 1430, 1122, + 957.5, 920, 777.5, 709.5, 698, 578.5, 418.5, 275.5], + sx=l_sx, + sy=l_sy, + ) + l_mod = Model(self.lorentz, meta=dict(name='Lorentz Peak')) + l_odr = ODR(l_dat, l_mod, beta0=(1000., .1, 3.8)) + + out = l_odr.run() + assert_array_almost_equal( + out.beta, + np.array([1.4306780846149925e+03, 1.3390509034538309e-01, + 3.7798193600109009e+00]), + ) + assert_array_almost_equal( + out.sd_beta, + np.array([7.3621186811330963e-01, 3.5068899941471650e-04, + 2.4451209281408992e-04]), + ) + assert_array_almost_equal( + out.cov_beta, + np.array([[2.4714409064597873e-01, -6.9067261911110836e-05, + -3.1236953270424990e-05], + [-6.9067261911110836e-05, 5.6077531517333009e-08, + 3.6133261832722601e-08], + [-3.1236953270424990e-05, 3.6133261832722601e-08, + 2.7261220025171730e-08]]), + ) + + def test_ticket_1253(self): + def linear(c, x): + return c[0]*x+c[1] + + c = [2.0, 3.0] + x = np.linspace(0, 10) + y = linear(c, x) + + model = Model(linear) + data = Data(x, y, wd=1.0, we=1.0) + job = ODR(data, model, beta0=[1.0, 1.0]) + result = job.run() + assert_equal(result.info, 2) + + # Verify fix for gh-9140 + + def test_ifixx(self): + x1 = [-2.01, -0.99, -0.001, 1.02, 1.98] + x2 = [3.98, 1.01, 0.001, 0.998, 4.01] + fix = np.vstack((np.zeros_like(x1, dtype=int), np.ones_like(x2, dtype=int))) + data = Data(np.vstack((x1, x2)), y=1, fix=fix) + model = Model(lambda beta, x: x[1, :] - beta[0] * x[0, :]**2., implicit=True) + + odr1 = ODR(data, model, beta0=np.array([1.])) + sol1 = odr1.run() + odr2 = ODR(data, model, beta0=np.array([1.]), ifixx=fix) + sol2 = odr2.run() + assert_equal(sol1.beta, sol2.beta) + + # verify bugfix for #11800 in #11802 + def test_ticket_11800(self): + # parameters + beta_true = np.array([1.0, 2.3, 1.1, -1.0, 1.3, 0.5]) + nr_measurements = 10 + + std_dev_x = 0.01 + x_error = np.array([[0.00063445, 0.00515731, 0.00162719, 0.01022866, + -0.01624845, 0.00482652, 0.00275988, -0.00714734, -0.00929201, -0.00687301], + [-0.00831623, -0.00821211, -0.00203459, 0.00938266, -0.00701829, + 0.0032169, 0.00259194, -0.00581017, -0.0030283, 0.01014164]]) + + std_dev_y = 0.05 + y_error = np.array([[0.05275304, 0.04519563, -0.07524086, 0.03575642, + 0.04745194, 0.03806645, 0.07061601, -0.00753604, -0.02592543, -0.02394929], + [0.03632366, 0.06642266, 0.08373122, 0.03988822, -0.0092536, + -0.03750469, -0.03198903, 0.01642066, 0.01293648, -0.05627085]]) + + beta_solution = np.array([ + 2.62920235756665876536e+00, -1.26608484996299608838e+02, + 1.29703572775403074502e+02, -1.88560985401185465804e+00, + 7.83834160771274923718e+01, -7.64124076838087091801e+01]) + + # model's function and Jacobians + def func(beta, x): + y0 = beta[0] + beta[1] * x[0, :] + beta[2] * x[1, :] + y1 = beta[3] + beta[4] * x[0, :] + beta[5] * x[1, :] + + return np.vstack((y0, y1)) + + def df_dbeta_odr(beta, x): + nr_meas = np.shape(x)[1] + zeros = np.zeros(nr_meas) + ones = np.ones(nr_meas) + + dy0 = np.array([ones, x[0, :], x[1, :], zeros, zeros, zeros]) + dy1 = np.array([zeros, zeros, zeros, ones, x[0, :], x[1, :]]) + + return np.stack((dy0, dy1)) + + def df_dx_odr(beta, x): + nr_meas = np.shape(x)[1] + ones = np.ones(nr_meas) + + dy0 = np.array([beta[1] * ones, beta[2] * ones]) + dy1 = np.array([beta[4] * ones, beta[5] * ones]) + return np.stack((dy0, dy1)) + + # do measurements with errors in independent and dependent variables + x0_true = np.linspace(1, 10, nr_measurements) + x1_true = np.linspace(1, 10, nr_measurements) + x_true = np.array([x0_true, x1_true]) + + y_true = func(beta_true, x_true) + + x_meas = x_true + x_error + y_meas = y_true + y_error + + # estimate model's parameters + model_f = Model(func, fjacb=df_dbeta_odr, fjacd=df_dx_odr) + + data = RealData(x_meas, y_meas, sx=std_dev_x, sy=std_dev_y) + + odr_obj = ODR(data, model_f, beta0=0.9 * beta_true, maxit=100) + #odr_obj.set_iprint(init=2, iter=0, iter_step=1, final=1) + odr_obj.set_job(deriv=3) + + odr_out = odr_obj.run() + + # check results + assert_equal(odr_out.info, 1) + assert_array_almost_equal(odr_out.beta, beta_solution) + + def test_multilinear_model(self): + x = np.linspace(0.0, 5.0) + y = 10.0 + 5.0 * x + data = Data(x, y) + odr_obj = ODR(data, multilinear) + output = odr_obj.run() + assert_array_almost_equal(output.beta, [10.0, 5.0]) + + def test_exponential_model(self): + x = np.linspace(0.0, 5.0) + y = -10.0 + np.exp(0.5*x) + data = Data(x, y) + odr_obj = ODR(data, exponential) + output = odr_obj.run() + assert_array_almost_equal(output.beta, [-10.0, 0.5]) + + def test_polynomial_model(self): + x = np.linspace(0.0, 5.0) + y = 1.0 + 2.0 * x + 3.0 * x ** 2 + 4.0 * x ** 3 + poly_model = polynomial(3) + data = Data(x, y) + odr_obj = ODR(data, poly_model) + output = odr_obj.run() + assert_array_almost_equal(output.beta, [1.0, 2.0, 3.0, 4.0]) + + def test_unilinear_model(self): + x = np.linspace(0.0, 5.0) + y = 1.0 * x + 2.0 + data = Data(x, y) + odr_obj = ODR(data, unilinear) + output = odr_obj.run() + assert_array_almost_equal(output.beta, [1.0, 2.0]) + + def test_quadratic_model(self): + x = np.linspace(0.0, 5.0) + y = 1.0 * x ** 2 + 2.0 * x + 3.0 + data = Data(x, y) + odr_obj = ODR(data, quadratic) + output = odr_obj.run() + assert_array_almost_equal(output.beta, [1.0, 2.0, 3.0]) + + def test_work_ind(self): + + def func(par, x): + b0, b1 = par + return b0 + b1 * x + + # generate some data + n_data = 4 + x = np.arange(n_data) + y = np.where(x % 2, x + 0.1, x - 0.1) + x_err = np.full(n_data, 0.1) + y_err = np.full(n_data, 0.1) + + # do the fitting + linear_model = Model(func) + real_data = RealData(x, y, sx=x_err, sy=y_err) + odr_obj = ODR(real_data, linear_model, beta0=[0.4, 0.4]) + odr_obj.set_job(fit_type=0) + out = odr_obj.run() + + sd_ind = out.work_ind['sd'] + assert_array_almost_equal(out.sd_beta, + out.work[sd_ind:sd_ind + len(out.sd_beta)]) + + @pytest.mark.skipif(True, reason="Fortran I/O prone to crashing so better " + "not to run this test, see gh-13127") + def test_output_file_overwrite(self): + """ + Verify fix for gh-1892 + """ + def func(b, x): + return b[0] + b[1] * x + + p = Model(func) + data = Data(np.arange(10), 12 * np.arange(10)) + tmp_dir = tempfile.mkdtemp() + error_file_path = os.path.join(tmp_dir, "error.dat") + report_file_path = os.path.join(tmp_dir, "report.dat") + try: + ODR(data, p, beta0=[0.1, 13], errfile=error_file_path, + rptfile=report_file_path).run() + ODR(data, p, beta0=[0.1, 13], errfile=error_file_path, + rptfile=report_file_path, overwrite=True).run() + finally: + # remove output files for clean up + shutil.rmtree(tmp_dir) + + def test_odr_model_default_meta(self): + def func(b, x): + return b[0] + b[1] * x + + p = Model(func) + p.set_meta(name='Sample Model Meta', ref='ODRPACK') + assert_equal(p.meta, {'name': 'Sample Model Meta', 'ref': 'ODRPACK'}) + + def test_work_array_del_init(self): + """ + Verify fix for gh-18739 where del_init=1 fails. + """ + def func(b, x): + return b[0] + b[1] * x + + # generate some data + n_data = 4 + x = np.arange(n_data) + y = np.where(x % 2, x + 0.1, x - 0.1) + x_err = np.full(n_data, 0.1) + y_err = np.full(n_data, 0.1) + + linear_model = Model(func) + # Try various shapes of the `we` array from various `sy` and `covy` + rd0 = RealData(x, y, sx=x_err, sy=y_err) + rd1 = RealData(x, y, sx=x_err, sy=0.1) + rd2 = RealData(x, y, sx=x_err, sy=[0.1]) + rd3 = RealData(x, y, sx=x_err, sy=np.full((1, n_data), 0.1)) + rd4 = RealData(x, y, sx=x_err, covy=[[0.01]]) + rd5 = RealData(x, y, sx=x_err, covy=np.full((1, 1, n_data), 0.01)) + for rd in [rd0, rd1, rd2, rd3, rd4, rd5]: + odr_obj = ODR(rd, linear_model, beta0=[0.4, 0.4], + delta0=np.full(n_data, -0.1)) + odr_obj.set_job(fit_type=0, del_init=1) + # Just make sure that it runs without raising an exception. + odr_obj.run() + + def test_pickling_data(self): + x = np.linspace(0.0, 5.0) + y = 1.0 * x + 2.0 + data = Data(x, y) + + obj_pickle = pickle.dumps(data) + del data + pickle.loads(obj_pickle) + + def test_pickling_real_data(self): + x = np.linspace(0.0, 5.0) + y = 1.0 * x + 2.0 + data = RealData(x, y) + + obj_pickle = pickle.dumps(data) + del data + pickle.loads(obj_pickle) + + def test_pickling_model(self): + obj_pickle = pickle.dumps(unilinear) + pickle.loads(obj_pickle) + + def test_pickling_odr(self): + x = np.linspace(0.0, 5.0) + y = 1.0 * x + 2.0 + odr_obj = ODR(Data(x, y), unilinear) + + obj_pickle = pickle.dumps(odr_obj) + del odr_obj + pickle.loads(obj_pickle) + + def test_pickling_output(self): + x = np.linspace(0.0, 5.0) + y = 1.0 * x + 2.0 + output = ODR(Data(x, y), unilinear).run + + obj_pickle = pickle.dumps(output) + del output + pickle.loads(obj_pickle) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/__init__.pxd b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/__init__.pxd new file mode 100644 index 0000000000000000000000000000000000000000..2402eeb020d34ad8b82e287e32545423911ff66c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/__init__.pxd @@ -0,0 +1 @@ +from .optimize cimport cython_optimize diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..fce4cecd22b165f9975160fe7c1ed718ed358853 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/__init__.py @@ -0,0 +1,460 @@ +""" +===================================================== +Optimization and root finding (:mod:`scipy.optimize`) +===================================================== + +.. currentmodule:: scipy.optimize + +.. toctree:: + :hidden: + + optimize.cython_optimize + +SciPy ``optimize`` provides functions for minimizing (or maximizing) +objective functions, possibly subject to constraints. It includes +solvers for nonlinear problems (with support for both local and global +optimization algorithms), linear programming, constrained +and nonlinear least-squares, root finding, and curve fitting. + +Common functions and objects, shared across different solvers, are: + +.. autosummary:: + :toctree: generated/ + + show_options - Show specific options optimization solvers. + OptimizeResult - The optimization result returned by some optimizers. + OptimizeWarning - The optimization encountered problems. + + +Optimization +============ + +Scalar functions optimization +----------------------------- + +.. autosummary:: + :toctree: generated/ + + minimize_scalar - Interface for minimizers of univariate functions + +The `minimize_scalar` function supports the following methods: + +.. toctree:: + + optimize.minimize_scalar-brent + optimize.minimize_scalar-bounded + optimize.minimize_scalar-golden + +Local (multivariate) optimization +--------------------------------- + +.. autosummary:: + :toctree: generated/ + + minimize - Interface for minimizers of multivariate functions. + +The `minimize` function supports the following methods: + +.. toctree:: + + optimize.minimize-neldermead + optimize.minimize-powell + optimize.minimize-cg + optimize.minimize-bfgs + optimize.minimize-newtoncg + optimize.minimize-lbfgsb + optimize.minimize-tnc + optimize.minimize-cobyla + optimize.minimize-cobyqa + optimize.minimize-slsqp + optimize.minimize-trustconstr + optimize.minimize-dogleg + optimize.minimize-trustncg + optimize.minimize-trustkrylov + optimize.minimize-trustexact + +Constraints are passed to `minimize` function as a single object or +as a list of objects from the following classes: + +.. autosummary:: + :toctree: generated/ + + NonlinearConstraint - Class defining general nonlinear constraints. + LinearConstraint - Class defining general linear constraints. + +Simple bound constraints are handled separately and there is a special class +for them: + +.. autosummary:: + :toctree: generated/ + + Bounds - Bound constraints. + +Quasi-Newton strategies implementing `HessianUpdateStrategy` +interface can be used to approximate the Hessian in `minimize` +function (available only for the 'trust-constr' method). Available +quasi-Newton methods implementing this interface are: + +.. autosummary:: + :toctree: generated/ + + BFGS - Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian update strategy. + SR1 - Symmetric-rank-1 Hessian update strategy. + +.. _global_optimization: + +Global optimization +------------------- + +.. autosummary:: + :toctree: generated/ + + basinhopping - Basinhopping stochastic optimizer. + brute - Brute force searching optimizer. + differential_evolution - Stochastic optimizer using differential evolution. + + shgo - Simplicial homology global optimizer. + dual_annealing - Dual annealing stochastic optimizer. + direct - DIRECT (Dividing Rectangles) optimizer. + +Least-squares and curve fitting +=============================== + +Nonlinear least-squares +----------------------- + +.. autosummary:: + :toctree: generated/ + + least_squares - Solve a nonlinear least-squares problem with bounds on the variables. + +Linear least-squares +-------------------- + +.. autosummary:: + :toctree: generated/ + + nnls - Linear least-squares problem with non-negativity constraint. + lsq_linear - Linear least-squares problem with bound constraints. + isotonic_regression - Least squares problem of isotonic regression via PAVA. + +Curve fitting +------------- + +.. autosummary:: + :toctree: generated/ + + curve_fit -- Fit curve to a set of points. + +Root finding +============ + +Scalar functions +---------------- +.. autosummary:: + :toctree: generated/ + + root_scalar - Unified interface for nonlinear solvers of scalar functions. + brentq - quadratic interpolation Brent method. + brenth - Brent method, modified by Harris with hyperbolic extrapolation. + ridder - Ridder's method. + bisect - Bisection method. + newton - Newton's method (also Secant and Halley's methods). + toms748 - Alefeld, Potra & Shi Algorithm 748. + RootResults - The root finding result returned by some root finders. + +The `root_scalar` function supports the following methods: + +.. toctree:: + + optimize.root_scalar-brentq + optimize.root_scalar-brenth + optimize.root_scalar-bisect + optimize.root_scalar-ridder + optimize.root_scalar-newton + optimize.root_scalar-toms748 + optimize.root_scalar-secant + optimize.root_scalar-halley + + + +The table below lists situations and appropriate methods, along with +*asymptotic* convergence rates per iteration (and per function evaluation) +for successful convergence to a simple root(*). +Bisection is the slowest of them all, adding one bit of accuracy for each +function evaluation, but is guaranteed to converge. +The other bracketing methods all (eventually) increase the number of accurate +bits by about 50% for every function evaluation. +The derivative-based methods, all built on `newton`, can converge quite quickly +if the initial value is close to the root. They can also be applied to +functions defined on (a subset of) the complex plane. + ++-------------+----------+----------+-----------+-------------+-------------+----------------+ +| Domain of f | Bracket? | Derivatives? | Solvers | Convergence | ++ + +----------+-----------+ +-------------+----------------+ +| | | `fprime` | `fprime2` | | Guaranteed? | Rate(s)(*) | ++=============+==========+==========+===========+=============+=============+================+ +| `R` | Yes | N/A | N/A | - bisection | - Yes | - 1 "Linear" | +| | | | | - brentq | - Yes | - >=1, <= 1.62 | +| | | | | - brenth | - Yes | - >=1, <= 1.62 | +| | | | | - ridder | - Yes | - 2.0 (1.41) | +| | | | | - toms748 | - Yes | - 2.7 (1.65) | ++-------------+----------+----------+-----------+-------------+-------------+----------------+ +| `R` or `C` | No | No | No | secant | No | 1.62 (1.62) | ++-------------+----------+----------+-----------+-------------+-------------+----------------+ +| `R` or `C` | No | Yes | No | newton | No | 2.00 (1.41) | ++-------------+----------+----------+-----------+-------------+-------------+----------------+ +| `R` or `C` | No | Yes | Yes | halley | No | 3.00 (1.44) | ++-------------+----------+----------+-----------+-------------+-------------+----------------+ + +.. seealso:: + + `scipy.optimize.cython_optimize` -- Typed Cython versions of root finding functions + +Fixed point finding: + +.. autosummary:: + :toctree: generated/ + + fixed_point - Single-variable fixed-point solver. + +Multidimensional +---------------- + +.. autosummary:: + :toctree: generated/ + + root - Unified interface for nonlinear solvers of multivariate functions. + +The `root` function supports the following methods: + +.. toctree:: + + optimize.root-hybr + optimize.root-lm + optimize.root-broyden1 + optimize.root-broyden2 + optimize.root-anderson + optimize.root-linearmixing + optimize.root-diagbroyden + optimize.root-excitingmixing + optimize.root-krylov + optimize.root-dfsane + +Elementwise Minimization and Root Finding +========================================= + +.. toctree:: + :maxdepth: 3 + + optimize.elementwise + +Linear programming / MILP +========================= + +.. autosummary:: + :toctree: generated/ + + milp -- Mixed integer linear programming. + linprog -- Unified interface for minimizers of linear programming problems. + +The `linprog` function supports the following methods: + +.. toctree:: + + optimize.linprog-simplex + optimize.linprog-interior-point + optimize.linprog-revised_simplex + optimize.linprog-highs-ipm + optimize.linprog-highs-ds + optimize.linprog-highs + +The simplex, interior-point, and revised simplex methods support callback +functions, such as: + +.. autosummary:: + :toctree: generated/ + + linprog_verbose_callback -- Sample callback function for linprog (simplex). + +Assignment problems +=================== + +.. autosummary:: + :toctree: generated/ + + linear_sum_assignment -- Solves the linear-sum assignment problem. + quadratic_assignment -- Solves the quadratic assignment problem. + +The `quadratic_assignment` function supports the following methods: + +.. toctree:: + + optimize.qap-faq + optimize.qap-2opt + +Utilities +========= + +Finite-difference approximation +------------------------------- + +.. autosummary:: + :toctree: generated/ + + approx_fprime - Approximate the gradient of a scalar function. + check_grad - Check the supplied derivative using finite differences. + + +Line search +----------- + +.. autosummary:: + :toctree: generated/ + + bracket - Bracket a minimum, given two starting points. + line_search - Return a step that satisfies the strong Wolfe conditions. + +Hessian approximation +--------------------- + +.. autosummary:: + :toctree: generated/ + + LbfgsInvHessProduct - Linear operator for L-BFGS approximate inverse Hessian. + HessianUpdateStrategy - Interface for implementing Hessian update strategies + +Benchmark problems +------------------ + +.. autosummary:: + :toctree: generated/ + + rosen - The Rosenbrock function. + rosen_der - The derivative of the Rosenbrock function. + rosen_hess - The Hessian matrix of the Rosenbrock function. + rosen_hess_prod - Product of the Rosenbrock Hessian with a vector. + +Legacy functions +================ + +The functions below are not recommended for use in new scripts; +all of these methods are accessible via a newer, more consistent +interfaces, provided by the interfaces above. + +Optimization +------------ + +General-purpose multivariate methods: + +.. autosummary:: + :toctree: generated/ + + fmin - Nelder-Mead Simplex algorithm. + fmin_powell - Powell's (modified) conjugate direction method. + fmin_cg - Non-linear (Polak-Ribiere) conjugate gradient algorithm. + fmin_bfgs - Quasi-Newton method (Broydon-Fletcher-Goldfarb-Shanno). + fmin_ncg - Line-search Newton Conjugate Gradient. + +Constrained multivariate methods: + +.. autosummary:: + :toctree: generated/ + + fmin_l_bfgs_b - Zhu, Byrd, and Nocedal's constrained optimizer. + fmin_tnc - Truncated Newton code. + fmin_cobyla - Constrained optimization by linear approximation. + fmin_slsqp - Minimization using sequential least-squares programming. + +Univariate (scalar) minimization methods: + +.. autosummary:: + :toctree: generated/ + + fminbound - Bounded minimization of a scalar function. + brent - 1-D function minimization using Brent method. + golden - 1-D function minimization using Golden Section method. + +Least-squares +------------- + +.. autosummary:: + :toctree: generated/ + + leastsq - Minimize the sum of squares of M equations in N unknowns. + +Root finding +------------ + +General nonlinear solvers: + +.. autosummary:: + :toctree: generated/ + + fsolve - Non-linear multivariable equation solver. + broyden1 - Broyden's first method. + broyden2 - Broyden's second method. + NoConvergence - Exception raised when nonlinear solver does not converge. + +Large-scale nonlinear solvers: + +.. autosummary:: + :toctree: generated/ + + newton_krylov + anderson + + BroydenFirst + InverseJacobian + KrylovJacobian + +Simple iteration solvers: + +.. autosummary:: + :toctree: generated/ + + excitingmixing + linearmixing + diagbroyden + +""" # noqa: E501 + +from ._optimize import * +from ._minimize import * +from ._root import * +from ._root_scalar import * +from ._minpack_py import * +from ._zeros_py import * +from ._lbfgsb_py import fmin_l_bfgs_b, LbfgsInvHessProduct +from ._tnc import fmin_tnc +from ._cobyla_py import fmin_cobyla +from ._nonlin import * +from ._slsqp_py import fmin_slsqp +from ._nnls import nnls +from ._basinhopping import basinhopping +from ._linprog import linprog, linprog_verbose_callback +from ._lsap import linear_sum_assignment +from ._differentialevolution import differential_evolution +from ._lsq import least_squares, lsq_linear +from ._isotonic import isotonic_regression +from ._constraints import (NonlinearConstraint, + LinearConstraint, + Bounds) +from ._hessian_update_strategy import HessianUpdateStrategy, BFGS, SR1 +from ._shgo import shgo +from ._dual_annealing import dual_annealing +from ._qap import quadratic_assignment +from ._direct_py import direct +from ._milp import milp + +# Deprecated namespaces, to be removed in v2.0.0 +from . import ( + cobyla, lbfgsb, linesearch, minpack, minpack2, moduleTNC, nonlin, optimize, + slsqp, tnc, zeros +) + +__all__ = [s for s in dir() if not s.startswith('_')] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_basinhopping.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_basinhopping.py new file mode 100644 index 0000000000000000000000000000000000000000..90498155887fc45ba0748c0d798bda17caef39f0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_basinhopping.py @@ -0,0 +1,735 @@ +""" +basinhopping: The basinhopping global optimization algorithm +""" +import numpy as np +import math +import inspect +import scipy.optimize +from scipy._lib._util import check_random_state, _transition_to_rng + +__all__ = ['basinhopping'] + + +_params = (inspect.Parameter('res_new', kind=inspect.Parameter.KEYWORD_ONLY), + inspect.Parameter('res_old', kind=inspect.Parameter.KEYWORD_ONLY)) +_new_accept_test_signature = inspect.Signature(parameters=_params) + + +class Storage: + """ + Class used to store the lowest energy structure + """ + def __init__(self, minres): + self._add(minres) + + def _add(self, minres): + self.minres = minres + self.minres.x = np.copy(minres.x) + + def update(self, minres): + if minres.success and (minres.fun < self.minres.fun + or not self.minres.success): + self._add(minres) + return True + else: + return False + + def get_lowest(self): + return self.minres + + +class BasinHoppingRunner: + """This class implements the core of the basinhopping algorithm. + + x0 : ndarray + The starting coordinates. + minimizer : callable + The local minimizer, with signature ``result = minimizer(x)``. + The return value is an `optimize.OptimizeResult` object. + step_taking : callable + This function displaces the coordinates randomly. Signature should + be ``x_new = step_taking(x)``. Note that `x` may be modified in-place. + accept_tests : list of callables + Each test is passed the kwargs `f_new`, `x_new`, `f_old` and + `x_old`. These tests will be used to judge whether or not to accept + the step. The acceptable return values are True, False, or ``"force + accept"``. If any of the tests return False then the step is rejected. + If ``"force accept"``, then this will override any other tests in + order to accept the step. This can be used, for example, to forcefully + escape from a local minimum that ``basinhopping`` is trapped in. + disp : bool, optional + Display status messages. + + """ + def __init__(self, x0, minimizer, step_taking, accept_tests, disp=False): + self.x = np.copy(x0) + self.minimizer = minimizer + self.step_taking = step_taking + self.accept_tests = accept_tests + self.disp = disp + + self.nstep = 0 + + # initialize return object + self.res = scipy.optimize.OptimizeResult() + self.res.minimization_failures = 0 + + # do initial minimization + minres = minimizer(self.x) + if not minres.success: + self.res.minimization_failures += 1 + if self.disp: + print("warning: basinhopping: local minimization failure") + self.x = np.copy(minres.x) + self.energy = minres.fun + self.incumbent_minres = minres # best minimize result found so far + if self.disp: + print("basinhopping step %d: f %g" % (self.nstep, self.energy)) + + # initialize storage class + self.storage = Storage(minres) + + if hasattr(minres, "nfev"): + self.res.nfev = minres.nfev + if hasattr(minres, "njev"): + self.res.njev = minres.njev + if hasattr(minres, "nhev"): + self.res.nhev = minres.nhev + + def _monte_carlo_step(self): + """Do one Monte Carlo iteration + + Randomly displace the coordinates, minimize, and decide whether + or not to accept the new coordinates. + """ + # Take a random step. Make a copy of x because the step_taking + # algorithm might change x in place + x_after_step = np.copy(self.x) + x_after_step = self.step_taking(x_after_step) + + # do a local minimization + minres = self.minimizer(x_after_step) + x_after_quench = minres.x + energy_after_quench = minres.fun + if not minres.success: + self.res.minimization_failures += 1 + if self.disp: + print("warning: basinhopping: local minimization failure") + if hasattr(minres, "nfev"): + self.res.nfev += minres.nfev + if hasattr(minres, "njev"): + self.res.njev += minres.njev + if hasattr(minres, "nhev"): + self.res.nhev += minres.nhev + + # accept the move based on self.accept_tests. If any test is False, + # then reject the step. If any test returns the special string + # 'force accept', then accept the step regardless. This can be used + # to forcefully escape from a local minimum if normal basin hopping + # steps are not sufficient. + accept = True + for test in self.accept_tests: + if inspect.signature(test) == _new_accept_test_signature: + testres = test(res_new=minres, res_old=self.incumbent_minres) + else: + testres = test(f_new=energy_after_quench, x_new=x_after_quench, + f_old=self.energy, x_old=self.x) + + if testres == 'force accept': + accept = True + break + elif testres is None: + raise ValueError("accept_tests must return True, False, or " + "'force accept'") + elif not testres: + accept = False + + # Report the result of the acceptance test to the take step class. + # This is for adaptive step taking + if hasattr(self.step_taking, "report"): + self.step_taking.report(accept, f_new=energy_after_quench, + x_new=x_after_quench, f_old=self.energy, + x_old=self.x) + + return accept, minres + + def one_cycle(self): + """Do one cycle of the basinhopping algorithm + """ + self.nstep += 1 + new_global_min = False + + accept, minres = self._monte_carlo_step() + + if accept: + self.energy = minres.fun + self.x = np.copy(minres.x) + self.incumbent_minres = minres # best minimize result found so far + new_global_min = self.storage.update(minres) + + # print some information + if self.disp: + self.print_report(minres.fun, accept) + if new_global_min: + print("found new global minimum on step %d with function" + " value %g" % (self.nstep, self.energy)) + + # save some variables as BasinHoppingRunner attributes + self.xtrial = minres.x + self.energy_trial = minres.fun + self.accept = accept + + return new_global_min + + def print_report(self, energy_trial, accept): + """print a status update""" + minres = self.storage.get_lowest() + print("basinhopping step %d: f %g trial_f %g accepted %d " + " lowest_f %g" % (self.nstep, self.energy, energy_trial, + accept, minres.fun)) + + +class AdaptiveStepsize: + """ + Class to implement adaptive stepsize. + + This class wraps the step taking class and modifies the stepsize to + ensure the true acceptance rate is as close as possible to the target. + + Parameters + ---------- + takestep : callable + The step taking routine. Must contain modifiable attribute + takestep.stepsize + accept_rate : float, optional + The target step acceptance rate + interval : int, optional + Interval for how often to update the stepsize + factor : float, optional + The step size is multiplied or divided by this factor upon each + update. + verbose : bool, optional + Print information about each update + + """ + def __init__(self, takestep, accept_rate=0.5, interval=50, factor=0.9, + verbose=True): + self.takestep = takestep + self.target_accept_rate = accept_rate + self.interval = interval + self.factor = factor + self.verbose = verbose + + self.nstep = 0 + self.nstep_tot = 0 + self.naccept = 0 + + def __call__(self, x): + return self.take_step(x) + + def _adjust_step_size(self): + old_stepsize = self.takestep.stepsize + accept_rate = float(self.naccept) / self.nstep + if accept_rate > self.target_accept_rate: + # We're accepting too many steps. This generally means we're + # trapped in a basin. Take bigger steps. + self.takestep.stepsize /= self.factor + else: + # We're not accepting enough steps. Take smaller steps. + self.takestep.stepsize *= self.factor + if self.verbose: + print(f"adaptive stepsize: acceptance rate {accept_rate:f} target " + f"{self.target_accept_rate:f} new stepsize " + f"{self.takestep.stepsize:g} old stepsize {old_stepsize:g}") + + def take_step(self, x): + self.nstep += 1 + self.nstep_tot += 1 + if self.nstep % self.interval == 0: + self._adjust_step_size() + return self.takestep(x) + + def report(self, accept, **kwargs): + "called by basinhopping to report the result of the step" + if accept: + self.naccept += 1 + + +class RandomDisplacement: + """Add a random displacement of maximum size `stepsize` to each coordinate. + + Calling this updates `x` in-place. + + Parameters + ---------- + stepsize : float, optional + Maximum stepsize in any dimension + rng : {None, int, `numpy.random.Generator`}, optional + Random number generator + """ + + def __init__(self, stepsize=0.5, rng=None): + self.stepsize = stepsize + self.rng = check_random_state(rng) + + def __call__(self, x): + x += self.rng.uniform(-self.stepsize, self.stepsize, + np.shape(x)) + return x + + +class MinimizerWrapper: + """ + wrap a minimizer function as a minimizer class + """ + def __init__(self, minimizer, func=None, **kwargs): + self.minimizer = minimizer + self.func = func + self.kwargs = kwargs + + def __call__(self, x0): + if self.func is None: + return self.minimizer(x0, **self.kwargs) + else: + return self.minimizer(self.func, x0, **self.kwargs) + + +class Metropolis: + """Metropolis acceptance criterion. + + Parameters + ---------- + T : float + The "temperature" parameter for the accept or reject criterion. + rng : {None, int, `numpy.random.Generator`}, optional + Random number generator used for acceptance test. + + """ + + def __init__(self, T, rng=None): + # Avoid ZeroDivisionError since "MBH can be regarded as a special case + # of the BH framework with the Metropolis criterion, where temperature + # T = 0." (Reject all steps that increase energy.) + self.beta = 1.0 / T if T != 0 else float('inf') + self.rng = check_random_state(rng) + + def accept_reject(self, res_new, res_old): + """ + Assuming the local search underlying res_new was successful: + If new energy is lower than old, it will always be accepted. + If new is higher than old, there is a chance it will be accepted, + less likely for larger differences. + """ + with np.errstate(invalid='ignore'): + # The energy values being fed to Metropolis are 1-length arrays, and if + # they are equal, their difference is 0, which gets multiplied by beta, + # which is inf, and array([0]) * float('inf') causes + # + # RuntimeWarning: invalid value encountered in multiply + # + # Ignore this warning so when the algorithm is on a flat plane, it always + # accepts the step, to try to move off the plane. + prod = -(res_new.fun - res_old.fun) * self.beta + w = math.exp(min(0, prod)) + + rand = self.rng.uniform() + return w >= rand and (res_new.success or not res_old.success) + + def __call__(self, *, res_new, res_old): + """ + f_new and f_old are mandatory in kwargs + """ + return bool(self.accept_reject(res_new, res_old)) + + +@_transition_to_rng("seed", position_num=12, replace_doc=True) +def basinhopping(func, x0, niter=100, T=1.0, stepsize=0.5, + minimizer_kwargs=None, take_step=None, accept_test=None, + callback=None, interval=50, disp=False, niter_success=None, + rng=None, *, target_accept_rate=0.5, stepwise_factor=0.9): + """Find the global minimum of a function using the basin-hopping algorithm. + + Basin-hopping is a two-phase method that combines a global stepping + algorithm with local minimization at each step. Designed to mimic + the natural process of energy minimization of clusters of atoms, it works + well for similar problems with "funnel-like, but rugged" energy landscapes + [5]_. + + As the step-taking, step acceptance, and minimization methods are all + customizable, this function can also be used to implement other two-phase + methods. + + Parameters + ---------- + func : callable ``f(x, *args)`` + Function to be optimized. ``args`` can be passed as an optional item + in the dict `minimizer_kwargs` + x0 : array_like + Initial guess. + niter : integer, optional + The number of basin-hopping iterations. There will be a total of + ``niter + 1`` runs of the local minimizer. + T : float, optional + The "temperature" parameter for the acceptance or rejection criterion. + Higher "temperatures" mean that larger jumps in function value will be + accepted. For best results `T` should be comparable to the + separation (in function value) between local minima. + stepsize : float, optional + Maximum step size for use in the random displacement. + minimizer_kwargs : dict, optional + Extra keyword arguments to be passed to the local minimizer + `scipy.optimize.minimize` Some important options could be: + + method : str + The minimization method (e.g. ``"L-BFGS-B"``) + args : tuple + Extra arguments passed to the objective function (`func`) and + its derivatives (Jacobian, Hessian). + + take_step : callable ``take_step(x)``, optional + Replace the default step-taking routine with this routine. The default + step-taking routine is a random displacement of the coordinates, but + other step-taking algorithms may be better for some systems. + `take_step` can optionally have the attribute ``take_step.stepsize``. + If this attribute exists, then `basinhopping` will adjust + ``take_step.stepsize`` in order to try to optimize the global minimum + search. + accept_test : callable, ``accept_test(f_new=f_new, x_new=x_new, f_old=fold, x_old=x_old)``, optional + Define a test which will be used to judge whether to accept the + step. This will be used in addition to the Metropolis test based on + "temperature" `T`. The acceptable return values are True, + False, or ``"force accept"``. If any of the tests return False + then the step is rejected. If the latter, then this will override any + other tests in order to accept the step. This can be used, for example, + to forcefully escape from a local minimum that `basinhopping` is + trapped in. + callback : callable, ``callback(x, f, accept)``, optional + A callback function which will be called for all minima found. ``x`` + and ``f`` are the coordinates and function value of the trial minimum, + and ``accept`` is whether that minimum was accepted. This can + be used, for example, to save the lowest N minima found. Also, + `callback` can be used to specify a user defined stop criterion by + optionally returning True to stop the `basinhopping` routine. + interval : integer, optional + interval for how often to update the `stepsize` + disp : bool, optional + Set to True to print status messages + niter_success : integer, optional + Stop the run if the global minimum candidate remains the same for this + number of iterations. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + + The random numbers generated only affect the default Metropolis + `accept_test` and the default `take_step`. If you supply your own + `take_step` and `accept_test`, and these functions use random + number generation, then those functions are responsible for the state + of their random number generator. + target_accept_rate : float, optional + The target acceptance rate that is used to adjust the `stepsize`. + If the current acceptance rate is greater than the target, + then the `stepsize` is increased. Otherwise, it is decreased. + Range is (0, 1). Default is 0.5. + + .. versionadded:: 1.8.0 + + stepwise_factor : float, optional + The `stepsize` is multiplied or divided by this stepwise factor upon + each update. Range is (0, 1). Default is 0.9. + + .. versionadded:: 1.8.0 + + Returns + ------- + res : OptimizeResult + The optimization result represented as a `OptimizeResult` object. + Important attributes are: ``x`` the solution array, ``fun`` the value + of the function at the solution, and ``message`` which describes the + cause of the termination. The ``OptimizeResult`` object returned by the + selected minimizer at the lowest minimum is also contained within this + object and can be accessed through the ``lowest_optimization_result`` + attribute. See `OptimizeResult` for a description of other attributes. + + See Also + -------- + minimize : + The local minimization function called once for each basinhopping step. + `minimizer_kwargs` is passed to this routine. + + Notes + ----- + Basin-hopping is a stochastic algorithm which attempts to find the global + minimum of a smooth scalar function of one or more variables [1]_ [2]_ [3]_ + [4]_. The algorithm in its current form was described by David Wales and + Jonathan Doye [2]_ http://www-wales.ch.cam.ac.uk/. + + The algorithm is iterative with each cycle composed of the following + features + + 1) random perturbation of the coordinates + + 2) local minimization + + 3) accept or reject the new coordinates based on the minimized function + value + + The acceptance test used here is the Metropolis criterion of standard Monte + Carlo algorithms, although there are many other possibilities [3]_. + + This global minimization method has been shown to be extremely efficient + for a wide variety of problems in physics and chemistry. It is + particularly useful when the function has many minima separated by large + barriers. See the `Cambridge Cluster Database + `_ for databases of molecular + systems that have been optimized primarily using basin-hopping. This + database includes minimization problems exceeding 300 degrees of freedom. + + See the free software program `GMIN `_ + for a Fortran implementation of basin-hopping. This implementation has many + variations of the procedure described above, including more + advanced step taking algorithms and alternate acceptance criterion. + + For stochastic global optimization there is no way to determine if the true + global minimum has actually been found. Instead, as a consistency check, + the algorithm can be run from a number of different random starting points + to ensure the lowest minimum found in each example has converged to the + global minimum. For this reason, `basinhopping` will by default simply + run for the number of iterations `niter` and return the lowest minimum + found. It is left to the user to ensure that this is in fact the global + minimum. + + Choosing `stepsize`: This is a crucial parameter in `basinhopping` and + depends on the problem being solved. The step is chosen uniformly in the + region from x0-stepsize to x0+stepsize, in each dimension. Ideally, it + should be comparable to the typical separation (in argument values) between + local minima of the function being optimized. `basinhopping` will, by + default, adjust `stepsize` to find an optimal value, but this may take + many iterations. You will get quicker results if you set a sensible + initial value for ``stepsize``. + + Choosing `T`: The parameter `T` is the "temperature" used in the + Metropolis criterion. Basinhopping steps are always accepted if + ``func(xnew) < func(xold)``. Otherwise, they are accepted with + probability:: + + exp( -(func(xnew) - func(xold)) / T ) + + So, for best results, `T` should to be comparable to the typical + difference (in function values) between local minima. (The height of + "walls" between local minima is irrelevant.) + + If `T` is 0, the algorithm becomes Monotonic Basin-Hopping, in which all + steps that increase energy are rejected. + + .. versionadded:: 0.12.0 + + References + ---------- + .. [1] Wales, David J. 2003, Energy Landscapes, Cambridge University Press, + Cambridge, UK. + .. [2] Wales, D J, and Doye J P K, Global Optimization by Basin-Hopping and + the Lowest Energy Structures of Lennard-Jones Clusters Containing up to + 110 Atoms. Journal of Physical Chemistry A, 1997, 101, 5111. + .. [3] Li, Z. and Scheraga, H. A., Monte Carlo-minimization approach to the + multiple-minima problem in protein folding, Proc. Natl. Acad. Sci. USA, + 1987, 84, 6611. + .. [4] Wales, D. J. and Scheraga, H. A., Global optimization of clusters, + crystals, and biomolecules, Science, 1999, 285, 1368. + .. [5] Olson, B., Hashmi, I., Molloy, K., and Shehu1, A., Basin Hopping as + a General and Versatile Optimization Framework for the Characterization + of Biological Macromolecules, Advances in Artificial Intelligence, + Volume 2012 (2012), Article ID 674832, :doi:`10.1155/2012/674832` + + Examples + -------- + The following example is a 1-D minimization problem, with many + local minima superimposed on a parabola. + + >>> import numpy as np + >>> from scipy.optimize import basinhopping + >>> func = lambda x: np.cos(14.5 * x - 0.3) + (x + 0.2) * x + >>> x0 = [1.] + + Basinhopping, internally, uses a local minimization algorithm. We will use + the parameter `minimizer_kwargs` to tell basinhopping which algorithm to + use and how to set up that minimizer. This parameter will be passed to + `scipy.optimize.minimize`. + + >>> minimizer_kwargs = {"method": "BFGS"} + >>> ret = basinhopping(func, x0, minimizer_kwargs=minimizer_kwargs, + ... niter=200) + >>> # the global minimum is: + >>> ret.x, ret.fun + -0.1951, -1.0009 + + Next consider a 2-D minimization problem. Also, this time, we + will use gradient information to significantly speed up the search. + + >>> def func2d(x): + ... f = np.cos(14.5 * x[0] - 0.3) + (x[1] + 0.2) * x[1] + (x[0] + + ... 0.2) * x[0] + ... df = np.zeros(2) + ... df[0] = -14.5 * np.sin(14.5 * x[0] - 0.3) + 2. * x[0] + 0.2 + ... df[1] = 2. * x[1] + 0.2 + ... return f, df + + We'll also use a different local minimization algorithm. Also, we must tell + the minimizer that our function returns both energy and gradient (Jacobian). + + >>> minimizer_kwargs = {"method":"L-BFGS-B", "jac":True} + >>> x0 = [1.0, 1.0] + >>> ret = basinhopping(func2d, x0, minimizer_kwargs=minimizer_kwargs, + ... niter=200) + >>> print("global minimum: x = [%.4f, %.4f], f(x) = %.4f" % (ret.x[0], + ... ret.x[1], + ... ret.fun)) + global minimum: x = [-0.1951, -0.1000], f(x) = -1.0109 + + Here is an example using a custom step-taking routine. Imagine you want + the first coordinate to take larger steps than the rest of the coordinates. + This can be implemented like so: + + >>> class MyTakeStep: + ... def __init__(self, stepsize=0.5): + ... self.stepsize = stepsize + ... self.rng = np.random.default_rng() + ... def __call__(self, x): + ... s = self.stepsize + ... x[0] += self.rng.uniform(-2.*s, 2.*s) + ... x[1:] += self.rng.uniform(-s, s, x[1:].shape) + ... return x + + Since ``MyTakeStep.stepsize`` exists basinhopping will adjust the magnitude + of `stepsize` to optimize the search. We'll use the same 2-D function as + before + + >>> mytakestep = MyTakeStep() + >>> ret = basinhopping(func2d, x0, minimizer_kwargs=minimizer_kwargs, + ... niter=200, take_step=mytakestep) + >>> print("global minimum: x = [%.4f, %.4f], f(x) = %.4f" % (ret.x[0], + ... ret.x[1], + ... ret.fun)) + global minimum: x = [-0.1951, -0.1000], f(x) = -1.0109 + + Now, let's do an example using a custom callback function which prints the + value of every minimum found + + >>> def print_fun(x, f, accepted): + ... print("at minimum %.4f accepted %d" % (f, int(accepted))) + + We'll run it for only 10 basinhopping steps this time. + + >>> rng = np.random.default_rng() + >>> ret = basinhopping(func2d, x0, minimizer_kwargs=minimizer_kwargs, + ... niter=10, callback=print_fun, rng=rng) + at minimum 0.4159 accepted 1 + at minimum -0.4317 accepted 1 + at minimum -1.0109 accepted 1 + at minimum -0.9073 accepted 1 + at minimum -0.4317 accepted 0 + at minimum -0.1021 accepted 1 + at minimum -0.7425 accepted 1 + at minimum -0.9073 accepted 1 + at minimum -0.4317 accepted 0 + at minimum -0.7425 accepted 1 + at minimum -0.9073 accepted 1 + + The minimum at -1.0109 is actually the global minimum, found already on the + 8th iteration. + + """ # numpy/numpydoc#87 # noqa: E501 + if target_accept_rate <= 0. or target_accept_rate >= 1.: + raise ValueError('target_accept_rate has to be in range (0, 1)') + if stepwise_factor <= 0. or stepwise_factor >= 1.: + raise ValueError('stepwise_factor has to be in range (0, 1)') + + x0 = np.array(x0) + + # set up the np.random generator + rng = check_random_state(rng) + + # set up minimizer + if minimizer_kwargs is None: + minimizer_kwargs = dict() + wrapped_minimizer = MinimizerWrapper(scipy.optimize.minimize, func, + **minimizer_kwargs) + + # set up step-taking algorithm + if take_step is not None: + if not callable(take_step): + raise TypeError("take_step must be callable") + # if take_step.stepsize exists then use AdaptiveStepsize to control + # take_step.stepsize + if hasattr(take_step, "stepsize"): + take_step_wrapped = AdaptiveStepsize( + take_step, interval=interval, + accept_rate=target_accept_rate, + factor=stepwise_factor, + verbose=disp) + else: + take_step_wrapped = take_step + else: + # use default + displace = RandomDisplacement(stepsize=stepsize, rng=rng) + take_step_wrapped = AdaptiveStepsize(displace, interval=interval, + accept_rate=target_accept_rate, + factor=stepwise_factor, + verbose=disp) + + # set up accept tests + accept_tests = [] + if accept_test is not None: + if not callable(accept_test): + raise TypeError("accept_test must be callable") + accept_tests = [accept_test] + + # use default + metropolis = Metropolis(T, rng=rng) + accept_tests.append(metropolis) + + if niter_success is None: + niter_success = niter + 2 + + bh = BasinHoppingRunner(x0, wrapped_minimizer, take_step_wrapped, + accept_tests, disp=disp) + + # The wrapped minimizer is called once during construction of + # BasinHoppingRunner, so run the callback + if callable(callback): + callback(bh.storage.minres.x, bh.storage.minres.fun, True) + + # start main iteration loop + count, i = 0, 0 + message = ["requested number of basinhopping iterations completed" + " successfully"] + for i in range(niter): + new_global_min = bh.one_cycle() + + if callable(callback): + # should we pass a copy of x? + val = callback(bh.xtrial, bh.energy_trial, bh.accept) + if val is not None: + if val: + message = ["callback function requested stop early by" + "returning True"] + break + + count += 1 + if new_global_min: + count = 0 + elif count > niter_success: + message = ["success condition satisfied"] + break + + # prepare return object + res = bh.res + res.lowest_optimization_result = bh.storage.get_lowest() + res.x = np.copy(res.lowest_optimization_result.x) + res.fun = res.lowest_optimization_result.fun + res.message = message + res.nit = i + 1 + res.success = res.lowest_optimization_result.success + return res diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_bracket.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_bracket.py new file mode 100644 index 0000000000000000000000000000000000000000..263243c612d08ebdc9939cc892771b49ac766d0c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_bracket.py @@ -0,0 +1,713 @@ +import numpy as np +import scipy._lib._elementwise_iterative_method as eim +from scipy._lib._util import _RichResult +from scipy._lib._array_api import array_namespace, xp_ravel, xp_default_dtype + +_ELIMITS = -1 # used in _bracket_root +_ESTOPONESIDE = 2 # used in _bracket_root + +def _bracket_root_iv(func, xl0, xr0, xmin, xmax, factor, args, maxiter): + + if not callable(func): + raise ValueError('`func` must be callable.') + + if not np.iterable(args): + args = (args,) + + xp = array_namespace(xl0) + xl0 = xp.asarray(xl0)[()] + if (not xp.isdtype(xl0.dtype, "numeric") + or xp.isdtype(xl0.dtype, "complex floating")): + raise ValueError('`xl0` must be numeric and real.') + if not xp.isdtype(xl0.dtype, "real floating"): + xl0 = xp.asarray(xl0, dtype=xp_default_dtype(xp)) + + # If xr0 is not supplied, fill with a dummy value for the sake of + # broadcasting. We need to wait until xmax has been validated to + # compute the default value. + xr0_not_supplied = False + if xr0 is None: + xr0 = xp.nan + xr0_not_supplied = True + + xmin = -xp.inf if xmin is None else xmin + xmax = xp.inf if xmax is None else xmax + factor = 2. if factor is None else factor + xl0, xr0, xmin, xmax, factor = xp.broadcast_arrays( + xl0, xp.asarray(xr0), xp.asarray(xmin), xp.asarray(xmax), xp.asarray(factor)) + + if (not xp.isdtype(xr0.dtype, "numeric") + or xp.isdtype(xr0.dtype, "complex floating")): + raise ValueError('`xr0` must be numeric and real.') + + if (not xp.isdtype(xmin.dtype, "numeric") + or xp.isdtype(xmin.dtype, "complex floating")): + raise ValueError('`xmin` must be numeric and real.') + + if (not xp.isdtype(xmax.dtype, "numeric") + or xp.isdtype(xmax.dtype, "complex floating")): + raise ValueError('`xmax` must be numeric and real.') + + if (not xp.isdtype(factor.dtype, "numeric") + or xp.isdtype(factor.dtype, "complex floating")): + raise ValueError('`factor` must be numeric and real.') + if not xp.all(factor > 1): + raise ValueError('All elements of `factor` must be greater than 1.') + + # Calculate the default value of xr0 if a value has not been supplied. + # Be careful to ensure xr0 is not larger than xmax. + if xr0_not_supplied: + xr0 = xl0 + xp.minimum((xmax - xl0)/ 8, xp.asarray(1.0)) + xr0 = xp.astype(xr0, xl0.dtype, copy=False) + + maxiter = xp.asarray(maxiter) + message = '`maxiter` must be a non-negative integer.' + if (not xp.isdtype(maxiter.dtype, "numeric") or maxiter.shape != tuple() + or xp.isdtype(maxiter.dtype, "complex floating")): + raise ValueError(message) + maxiter_int = int(maxiter[()]) + if not maxiter == maxiter_int or maxiter < 0: + raise ValueError(message) + + return func, xl0, xr0, xmin, xmax, factor, args, maxiter, xp + + +def _bracket_root(func, xl0, xr0=None, *, xmin=None, xmax=None, factor=None, + args=(), maxiter=1000): + """Bracket the root of a monotonic scalar function of one variable + + This function works elementwise when `xl0`, `xr0`, `xmin`, `xmax`, `factor`, and + the elements of `args` are broadcastable arrays. + + Parameters + ---------- + func : callable + The function for which the root is to be bracketed. + The signature must be:: + + func(x: ndarray, *args) -> ndarray + + where each element of ``x`` is a finite real and ``args`` is a tuple, + which may contain an arbitrary number of arrays that are broadcastable + with `x`. ``func`` must be an elementwise function: each element + ``func(x)[i]`` must equal ``func(x[i])`` for all indices ``i``. + xl0, xr0: float array_like + Starting guess of bracket, which need not contain a root. If `xr0` is + not provided, ``xr0 = xl0 + 1``. Must be broadcastable with one another. + xmin, xmax : float array_like, optional + Minimum and maximum allowable endpoints of the bracket, inclusive. Must + be broadcastable with `xl0` and `xr0`. + factor : float array_like, default: 2 + The factor used to grow the bracket. See notes for details. + args : tuple, optional + Additional positional arguments to be passed to `func`. Must be arrays + broadcastable with `xl0`, `xr0`, `xmin`, and `xmax`. If the callable to be + bracketed requires arguments that are not broadcastable with these + arrays, wrap that callable with `func` such that `func` accepts + only `x` and broadcastable arrays. + maxiter : int, optional + The maximum number of iterations of the algorithm to perform. + + Returns + ------- + res : _RichResult + An instance of `scipy._lib._util._RichResult` with the following + attributes. The descriptions are written as though the values will be + scalars; however, if `func` returns an array, the outputs will be + arrays of the same shape. + + xl, xr : float + The lower and upper ends of the bracket, if the algorithm + terminated successfully. + fl, fr : float + The function value at the lower and upper ends of the bracket. + nfev : int + The number of function evaluations required to find the bracket. + This is distinct from the number of times `func` is *called* + because the function may evaluated at multiple points in a single + call. + nit : int + The number of iterations of the algorithm that were performed. + status : int + An integer representing the exit status of the algorithm. + + - ``0`` : The algorithm produced a valid bracket. + - ``-1`` : The bracket expanded to the allowable limits without finding a bracket. + - ``-2`` : The maximum number of iterations was reached. + - ``-3`` : A non-finite value was encountered. + - ``-4`` : Iteration was terminated by `callback`. + - ``-5``: The initial bracket does not satisfy `xmin <= xl0 < xr0 < xmax`. + - ``1`` : The algorithm is proceeding normally (in `callback` only). + - ``2`` : A bracket was found in the opposite search direction (in `callback` only). + + success : bool + ``True`` when the algorithm terminated successfully (status ``0``). + + Notes + ----- + This function generalizes an algorithm found in pieces throughout + `scipy.stats`. The strategy is to iteratively grow the bracket ``(l, r)`` + until ``func(l) < 0 < func(r)``. The bracket grows to the left as follows. + + - If `xmin` is not provided, the distance between `xl0` and `l` is iteratively + increased by `factor`. + - If `xmin` is provided, the distance between `xmin` and `l` is iteratively + decreased by `factor`. Note that this also *increases* the bracket size. + + Growth of the bracket to the right is analogous. + + Growth of the bracket in one direction stops when the endpoint is no longer + finite, the function value at the endpoint is no longer finite, or the + endpoint reaches its limiting value (`xmin` or `xmax`). Iteration terminates + when the bracket stops growing in both directions, the bracket surrounds + the root, or a root is found (accidentally). + + If two brackets are found - that is, a bracket is found on both sides in + the same iteration, the smaller of the two is returned. + If roots of the function are found, both `l` and `r` are set to the + leftmost root. + + """ # noqa: E501 + # Todo: + # - find bracket with sign change in specified direction + # - Add tolerance + # - allow factor < 1? + + callback = None # works; I just don't want to test it + temp = _bracket_root_iv(func, xl0, xr0, xmin, xmax, factor, args, maxiter) + func, xl0, xr0, xmin, xmax, factor, args, maxiter, xp = temp + + xs = (xl0, xr0) + temp = eim._initialize(func, xs, args) + func, xs, fs, args, shape, dtype, xp = temp # line split for PEP8 + xl0, xr0 = xs + xmin = xp_ravel(xp.astype(xp.broadcast_to(xmin, shape), dtype, copy=False), xp=xp) + xmax = xp_ravel(xp.astype(xp.broadcast_to(xmax, shape), dtype, copy=False), xp=xp) + invalid_bracket = ~((xmin <= xl0) & (xl0 < xr0) & (xr0 <= xmax)) + + # The approach is to treat the left and right searches as though they were + # (almost) totally independent one-sided bracket searches. (The interaction + # is considered when checking for termination and preparing the result + # object.) + # `x` is the "moving" end of the bracket + x = xp.concat(xs) + f = xp.concat(fs) + invalid_bracket = xp.concat((invalid_bracket, invalid_bracket)) + n = x.shape[0] // 2 + + # `x_last` is the previous location of the moving end of the bracket. If + # the signs of `f` and `f_last` are different, `x` and `x_last` form a + # bracket. + x_last = xp.concat((x[n:], x[:n])) + f_last = xp.concat((f[n:], f[:n])) + # `x0` is the "fixed" end of the bracket. + x0 = x_last + # We don't need to retain the corresponding function value, since the + # fixed end of the bracket is only needed to compute the new value of the + # moving end; it is never returned. + limit = xp.concat((xmin, xmax)) + + factor = xp_ravel(xp.broadcast_to(factor, shape), xp=xp) + factor = xp.astype(factor, dtype, copy=False) + factor = xp.concat((factor, factor)) + + active = xp.arange(2*n) + args = [xp.concat((arg, arg)) for arg in args] + + # This is needed due to inner workings of `eim._loop`. + # We're abusing it a tiny bit. + shape = shape + (2,) + + # `d` is for "distance". + # For searches without a limit, the distance between the fixed end of the + # bracket `x0` and the moving end `x` will grow by `factor` each iteration. + # For searches with a limit, the distance between the `limit` and moving + # end of the bracket `x` will shrink by `factor` each iteration. + i = xp.isinf(limit) + ni = ~i + d = xp.zeros_like(x) + d[i] = x[i] - x0[i] + d[ni] = limit[ni] - x[ni] + + status = xp.full_like(x, eim._EINPROGRESS, dtype=xp.int32) # in progress + status[invalid_bracket] = eim._EINPUTERR + nit, nfev = 0, 1 # one function evaluation per side performed above + + work = _RichResult(x=x, x0=x0, f=f, limit=limit, factor=factor, + active=active, d=d, x_last=x_last, f_last=f_last, + nit=nit, nfev=nfev, status=status, args=args, + xl=xp.nan, xr=xp.nan, fl=xp.nan, fr=xp.nan, n=n) + res_work_pairs = [('status', 'status'), ('xl', 'xl'), ('xr', 'xr'), + ('nit', 'nit'), ('nfev', 'nfev'), ('fl', 'fl'), + ('fr', 'fr'), ('x', 'x'), ('f', 'f'), + ('x_last', 'x_last'), ('f_last', 'f_last')] + + def pre_func_eval(work): + # Initialize moving end of bracket + x = xp.zeros_like(work.x) + + # Unlimited brackets grow by `factor` by increasing distance from fixed + # end to moving end. + i = xp.isinf(work.limit) # indices of unlimited brackets + work.d[i] *= work.factor[i] + x[i] = work.x0[i] + work.d[i] + + # Limited brackets grow by decreasing the distance from the limit to + # the moving end. + ni = ~i # indices of limited brackets + work.d[ni] /= work.factor[ni] + x[ni] = work.limit[ni] - work.d[ni] + + return x + + def post_func_eval(x, f, work): + # Keep track of the previous location of the moving end so that we can + # return a narrower bracket. (The alternative is to remember the + # original fixed end, but then the bracket would be wider than needed.) + work.x_last = work.x + work.f_last = work.f + work.x = x + work.f = f + + def check_termination(work): + # Condition 0: initial bracket is invalid + stop = (work.status == eim._EINPUTERR) + + # Condition 1: a valid bracket (or the root itself) has been found + sf = xp.sign(work.f) + sf_last = xp.sign(work.f_last) + i = ((sf_last == -sf) | (sf_last == 0) | (sf == 0)) & ~stop + work.status[i] = eim._ECONVERGED + stop[i] = True + + # Condition 2: the other side's search found a valid bracket. + # (If we just found a bracket with the rightward search, we can stop + # the leftward search, and vice-versa.) + # To do this, we need to set the status of the other side's search; + # this is tricky because `work.status` contains only the *active* + # elements, so we don't immediately know the index of the element we + # need to set - or even if it's still there. (That search may have + # terminated already, e.g. by reaching its `limit`.) + # To facilitate this, `work.active` contains a unit integer index of + # each search. Index `k` (`k < n)` and `k + n` correspond with a + # leftward and rightward search, respectively. Elements are removed + # from `work.active` just as they are removed from `work.status`, so + # we use `work.active` to help find the right location in + # `work.status`. + # Get the integer indices of the elements that can also stop + also_stop = (work.active[i] + work.n) % (2*work.n) + # Check whether they are still active. We want to find the indices + # in work.active where the associated values in work.active are + # contained in also_stop. xp.searchsorted let's us take advantage + # of work.active being sorted, but requires some hackery because + # searchsorted solves the separate but related problem of finding + # the indices where the values in also_stop should be added to + # maintain sorted order. + j = xp.searchsorted(work.active, also_stop) + # If the location exceeds the length of the `work.active`, they are + # not there. This happens when a value in also_stop is larger than + # the greatest value in work.active. This case needs special handling + # because we cannot simply check that also_stop == work.active[j]. + mask = j < work.active.shape[0] + # Note that we also have to use the mask to filter also_stop to ensure + # that also_stop and j will still have the same shape. + j, also_stop = j[mask], also_stop[mask] + j = j[also_stop == work.active[j]] + # Now convert these to boolean indices to use with `work.status`. + i = xp.zeros_like(stop) + i[j] = True # boolean indices of elements that can also stop + i = i & ~stop + work.status[i] = _ESTOPONESIDE + stop[i] = True + + # Condition 3: moving end of bracket reaches limit + i = (work.x == work.limit) & ~stop + work.status[i] = _ELIMITS + stop[i] = True + + # Condition 4: non-finite value encountered + i = ~(xp.isfinite(work.x) & xp.isfinite(work.f)) & ~stop + work.status[i] = eim._EVALUEERR + stop[i] = True + + return stop + + def post_termination_check(work): + pass + + def customize_result(res, shape): + n = res['x'].shape[0] // 2 + + # To avoid ambiguity, below we refer to `xl0`, the initial left endpoint + # as `a` and `xr0`, the initial right endpoint, as `b`. + # Because we treat the two one-sided searches as though they were + # independent, what we keep track of in `work` and what we want to + # return in `res` look quite different. Combine the results from the + # two one-sided searches before reporting the results to the user. + # - "a" refers to the leftward search (the moving end started at `a`) + # - "b" refers to the rightward search (the moving end started at `b`) + # - "l" refers to the left end of the bracket (closer to -oo) + # - "r" refers to the right end of the bracket (closer to +oo) + xal = res['x'][:n] + xar = res['x_last'][:n] + xbl = res['x_last'][n:] + xbr = res['x'][n:] + + fal = res['f'][:n] + far = res['f_last'][:n] + fbl = res['f_last'][n:] + fbr = res['f'][n:] + + # Initialize the brackets and corresponding function values to return + # to the user. Brackets may not be valid (e.g. there is no root, + # there weren't enough iterations, NaN encountered), but we still need + # to return something. One option would be all NaNs, but what I've + # chosen here is the left- and right-most points at which the function + # has been evaluated. This gives the user some information about what + # interval of the real line has been searched and shows that there is + # no sign change between the two ends. + xl = xp.asarray(xal, copy=True) + fl = xp.asarray(fal, copy=True) + xr = xp.asarray(xbr, copy=True) + fr = xp.asarray(fbr, copy=True) + + # `status` indicates whether the bracket is valid or not. If so, + # we want to adjust the bracket we return to be the narrowest possible + # given the points at which we evaluated the function. + # For example if bracket "a" is valid and smaller than bracket "b" OR + # if bracket "a" is valid and bracket "b" is not valid, we want to + # return bracket "a" (and vice versa). + sa = res['status'][:n] + sb = res['status'][n:] + + da = xar - xal + db = xbr - xbl + + i1 = ((da <= db) & (sa == 0)) | ((sa == 0) & (sb != 0)) + i2 = ((db <= da) & (sb == 0)) | ((sb == 0) & (sa != 0)) + + xr[i1] = xar[i1] + fr[i1] = far[i1] + xl[i2] = xbl[i2] + fl[i2] = fbl[i2] + + # Finish assembling the result object + res['xl'] = xl + res['xr'] = xr + res['fl'] = fl + res['fr'] = fr + + res['nit'] = xp.maximum(res['nit'][:n], res['nit'][n:]) + res['nfev'] = res['nfev'][:n] + res['nfev'][n:] + # If the status on one side is zero, the status is zero. In any case, + # report the status from one side only. + res['status'] = xp.where(sa == 0, sa, sb) + res['success'] = (res['status'] == 0) + + del res['x'] + del res['f'] + del res['x_last'] + del res['f_last'] + + return shape[:-1] + + return eim._loop(work, callback, shape, maxiter, func, args, dtype, + pre_func_eval, post_func_eval, check_termination, + post_termination_check, customize_result, res_work_pairs, + xp) + + +def _bracket_minimum_iv(func, xm0, xl0, xr0, xmin, xmax, factor, args, maxiter): + + if not callable(func): + raise ValueError('`func` must be callable.') + + if not np.iterable(args): + args = (args,) + + xp = array_namespace(xm0) + xm0 = xp.asarray(xm0)[()] + if (not xp.isdtype(xm0.dtype, "numeric") + or xp.isdtype(xm0.dtype, "complex floating")): + raise ValueError('`xm0` must be numeric and real.') + if not xp.isdtype(xm0.dtype, "real floating"): + xm0 = xp.asarray(xm0, dtype=xp_default_dtype(xp)) + + xmin = -xp.inf if xmin is None else xmin + xmax = xp.inf if xmax is None else xmax + + # If xl0 (xr0) is not supplied, fill with a dummy value for the sake + # of broadcasting. We need to wait until xmin (xmax) has been validated + # to compute the default values. + xl0_not_supplied = False + if xl0 is None: + xl0 = xp.nan + xl0_not_supplied = True + + xr0_not_supplied = False + if xr0 is None: + xr0 = xp.nan + xr0_not_supplied = True + + factor = 2.0 if factor is None else factor + xl0, xm0, xr0, xmin, xmax, factor = xp.broadcast_arrays( + xp.asarray(xl0), xm0, xp.asarray(xr0), xp.asarray(xmin), + xp.asarray(xmax), xp.asarray(factor) + ) + + if (not xp.isdtype(xl0.dtype, "numeric") + or xp.isdtype(xl0.dtype, "complex floating")): + raise ValueError('`xl0` must be numeric and real.') + + if (not xp.isdtype(xr0.dtype, "numeric") + or xp.isdtype(xr0.dtype, "complex floating")): + raise ValueError('`xr0` must be numeric and real.') + + if (not xp.isdtype(xmin.dtype, "numeric") + or xp.isdtype(xmin.dtype, "complex floating")): + raise ValueError('`xmin` must be numeric and real.') + + if (not xp.isdtype(xmax.dtype, "numeric") + or xp.isdtype(xmax.dtype, "complex floating")): + raise ValueError('`xmax` must be numeric and real.') + + if (not xp.isdtype(factor.dtype, "numeric") + or xp.isdtype(factor.dtype, "complex floating")): + raise ValueError('`factor` must be numeric and real.') + if not xp.all(factor > 1): + raise ValueError('All elements of `factor` must be greater than 1.') + + # Calculate default values of xl0 and/or xr0 if they have not been supplied + # by the user. We need to be careful to ensure xl0 and xr0 are not outside + # of (xmin, xmax). + if xl0_not_supplied: + xl0 = xm0 - xp.minimum((xm0 - xmin)/16, xp.asarray(0.5)) + xl0 = xp.astype(xl0, xm0.dtype, copy=False) + if xr0_not_supplied: + xr0 = xm0 + xp.minimum((xmax - xm0)/16, xp.asarray(0.5)) + xr0 = xp.astype(xr0, xm0.dtype, copy=False) + + maxiter = xp.asarray(maxiter) + message = '`maxiter` must be a non-negative integer.' + if (not xp.isdtype(maxiter.dtype, "numeric") or maxiter.shape != tuple() + or xp.isdtype(maxiter.dtype, "complex floating")): + raise ValueError(message) + maxiter_int = int(maxiter[()]) + if not maxiter == maxiter_int or maxiter < 0: + raise ValueError(message) + + return func, xm0, xl0, xr0, xmin, xmax, factor, args, maxiter, xp + + +def _bracket_minimum(func, xm0, *, xl0=None, xr0=None, xmin=None, xmax=None, + factor=None, args=(), maxiter=1000): + """Bracket the minimum of a unimodal scalar function of one variable + + This function works elementwise when `xm0`, `xl0`, `xr0`, `xmin`, `xmax`, + and the elements of `args` are broadcastable arrays. + + Parameters + ---------- + func : callable + The function for which the minimum is to be bracketed. + The signature must be:: + + func(x: ndarray, *args) -> ndarray + + where each element of ``x`` is a finite real and ``args`` is a tuple, + which may contain an arbitrary number of arrays that are broadcastable + with ``x``. `func` must be an elementwise function: each element + ``func(x)[i]`` must equal ``func(x[i])`` for all indices `i`. + xm0: float array_like + Starting guess for middle point of bracket. + xl0, xr0: float array_like, optional + Starting guesses for left and right endpoints of the bracket. Must be + broadcastable with one another and with `xm0`. + xmin, xmax : float array_like, optional + Minimum and maximum allowable endpoints of the bracket, inclusive. Must + be broadcastable with `xl0`, `xm0`, and `xr0`. + factor : float array_like, optional + Controls expansion of bracket endpoint in downhill direction. Works + differently in the cases where a limit is set in the downhill direction + with `xmax` or `xmin`. See Notes. + args : tuple, optional + Additional positional arguments to be passed to `func`. Must be arrays + broadcastable with `xl0`, `xm0`, `xr0`, `xmin`, and `xmax`. If the + callable to be bracketed requires arguments that are not broadcastable + with these arrays, wrap that callable with `func` such that `func` + accepts only ``x`` and broadcastable arrays. + maxiter : int, optional + The maximum number of iterations of the algorithm to perform. The number + of function evaluations is three greater than the number of iterations. + + Returns + ------- + res : _RichResult + An instance of `scipy._lib._util._RichResult` with the following + attributes. The descriptions are written as though the values will be + scalars; however, if `func` returns an array, the outputs will be + arrays of the same shape. + + xl, xm, xr : float + The left, middle, and right points of the bracket, if the algorithm + terminated successfully. + fl, fm, fr : float + The function value at the left, middle, and right points of the bracket. + nfev : int + The number of function evaluations required to find the bracket. + nit : int + The number of iterations of the algorithm that were performed. + status : int + An integer representing the exit status of the algorithm. + + - ``0`` : The algorithm produced a valid bracket. + - ``-1`` : The bracket expanded to the allowable limits. Assuming + unimodality, this implies the endpoint at the limit is a + minimizer. + - ``-2`` : The maximum number of iterations was reached. + - ``-3`` : A non-finite value was encountered. + - ``-4`` : ``None`` shall pass. + - ``-5`` : The initial bracket does not satisfy + `xmin <= xl0 < xm0 < xr0 <= xmax`. + + success : bool + ``True`` when the algorithm terminated successfully (status ``0``). + + Notes + ----- + Similar to `scipy.optimize.bracket`, this function seeks to find real + points ``xl < xm < xr`` such that ``f(xl) >= f(xm)`` and ``f(xr) >= f(xm)``, + where at least one of the inequalities is strict. Unlike `scipy.optimize.bracket`, + this function can operate in a vectorized manner on array input, so long as + the input arrays are broadcastable with each other. Also unlike + `scipy.optimize.bracket`, users may specify minimum and maximum endpoints + for the desired bracket. + + Given an initial trio of points ``xl = xl0``, ``xm = xm0``, ``xr = xr0``, + the algorithm checks if these points already give a valid bracket. If not, + a new endpoint, ``w`` is chosen in the "downhill" direction, ``xm`` becomes the new + opposite endpoint, and either `xl` or `xr` becomes the new middle point, + depending on which direction is downhill. The algorithm repeats from here. + + The new endpoint `w` is chosen differently depending on whether or not a + boundary `xmin` or `xmax` has been set in the downhill direction. Without + loss of generality, suppose the downhill direction is to the right, so that + ``f(xl) > f(xm) > f(xr)``. If there is no boundary to the right, then `w` + is chosen to be ``xr + factor * (xr - xm)`` where `factor` is controlled by + the user (defaults to 2.0) so that step sizes increase in geometric proportion. + If there is a boundary, `xmax` in this case, then `w` is chosen to be + ``xmax - (xmax - xr)/factor``, with steps slowing to a stop at + `xmax`. This cautious approach ensures that a minimum near but distinct from + the boundary isn't missed while also detecting whether or not the `xmax` is + a minimizer when `xmax` is reached after a finite number of steps. + """ # noqa: E501 + callback = None # works; I just don't want to test it + + temp = _bracket_minimum_iv(func, xm0, xl0, xr0, xmin, xmax, factor, args, maxiter) + func, xm0, xl0, xr0, xmin, xmax, factor, args, maxiter, xp = temp + + xs = (xl0, xm0, xr0) + temp = eim._initialize(func, xs, args) + func, xs, fs, args, shape, dtype, xp = temp + + xl0, xm0, xr0 = xs + fl0, fm0, fr0 = fs + xmin = xp.astype(xp.broadcast_to(xmin, shape), dtype, copy=False) + xmin = xp_ravel(xmin, xp=xp) + xmax = xp.astype(xp.broadcast_to(xmax, shape), dtype, copy=False) + xmax = xp_ravel(xmax, xp=xp) + invalid_bracket = ~((xmin <= xl0) & (xl0 < xm0) & (xm0 < xr0) & (xr0 <= xmax)) + # We will modify factor later on so make a copy. np.broadcast_to returns + # a read-only view. + factor = xp.astype(xp.broadcast_to(factor, shape), dtype, copy=True) + factor = xp_ravel(factor) + + # To simplify the logic, swap xl and xr if f(xl) < f(xr). We should always be + # marching downhill in the direction from xl to xr. + comp = fl0 < fr0 + xl0[comp], xr0[comp] = xr0[comp], xl0[comp] + fl0[comp], fr0[comp] = fr0[comp], fl0[comp] + # We only need the boundary in the direction we're traveling. + limit = xp.where(comp, xmin, xmax) + + unlimited = xp.isinf(limit) + limited = ~unlimited + step = xp.empty_like(xl0) + + step[unlimited] = (xr0[unlimited] - xm0[unlimited]) + step[limited] = (limit[limited] - xr0[limited]) + + # Step size is divided by factor for case where there is a limit. + factor[limited] = 1 / factor[limited] + + status = xp.full_like(xl0, eim._EINPROGRESS, dtype=xp.int32) + status[invalid_bracket] = eim._EINPUTERR + nit, nfev = 0, 3 + + work = _RichResult(xl=xl0, xm=xm0, xr=xr0, xr0=xr0, fl=fl0, fm=fm0, fr=fr0, + step=step, limit=limit, limited=limited, factor=factor, nit=nit, + nfev=nfev, status=status, args=args) + + res_work_pairs = [('status', 'status'), ('xl', 'xl'), ('xm', 'xm'), ('xr', 'xr'), + ('nit', 'nit'), ('nfev', 'nfev'), ('fl', 'fl'), ('fm', 'fm'), + ('fr', 'fr')] + + def pre_func_eval(work): + work.step *= work.factor + x = xp.empty_like(work.xr) + x[~work.limited] = work.xr0[~work.limited] + work.step[~work.limited] + x[work.limited] = work.limit[work.limited] - work.step[work.limited] + # Since the new bracket endpoint is calculated from an offset with the + # limit, it may be the case that the new endpoint equals the old endpoint, + # when the old endpoint is sufficiently close to the limit. We use the + # limit itself as the new endpoint in these cases. + x[work.limited] = xp.where( + x[work.limited] == work.xr[work.limited], + work.limit[work.limited], + x[work.limited], + ) + return x + + def post_func_eval(x, f, work): + work.xl, work.xm, work.xr = work.xm, work.xr, x + work.fl, work.fm, work.fr = work.fm, work.fr, f + + def check_termination(work): + # Condition 0: Initial bracket is invalid. + stop = (work.status == eim._EINPUTERR) + + # Condition 1: A valid bracket has been found. + i = ( + (work.fl >= work.fm) & (work.fr > work.fm) + | (work.fl > work.fm) & (work.fr >= work.fm) + ) & ~stop + work.status[i] = eim._ECONVERGED + stop[i] = True + + # Condition 2: Moving end of bracket reaches limit. + i = (work.xr == work.limit) & ~stop + work.status[i] = _ELIMITS + stop[i] = True + + # Condition 3: non-finite value encountered + i = ~(xp.isfinite(work.xr) & xp.isfinite(work.fr)) & ~stop + work.status[i] = eim._EVALUEERR + stop[i] = True + + return stop + + def post_termination_check(work): + pass + + def customize_result(res, shape): + # Reorder entries of xl and xr if they were swapped due to f(xl0) < f(xr0). + comp = res['xl'] > res['xr'] + res['xl'][comp], res['xr'][comp] = res['xr'][comp], res['xl'][comp] + res['fl'][comp], res['fr'][comp] = res['fr'][comp], res['fl'][comp] + return shape + + return eim._loop(work, callback, shape, + maxiter, func, args, dtype, + pre_func_eval, post_func_eval, + check_termination, post_termination_check, + customize_result, res_work_pairs, xp) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_chandrupatla.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_chandrupatla.py new file mode 100644 index 0000000000000000000000000000000000000000..5a4b70098919b9fba626bfecd5c1bcc559ab7702 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_chandrupatla.py @@ -0,0 +1,552 @@ +import math +import numpy as np +import scipy._lib._elementwise_iterative_method as eim +from scipy._lib._util import _RichResult +from scipy._lib._array_api import xp_sign, xp_copy, xp_take_along_axis + +# TODO: +# - (maybe?) don't use fancy indexing assignment +# - figure out how to replace the new `try`/`except`s + + +def _chandrupatla(func, a, b, *, args=(), xatol=None, xrtol=None, + fatol=None, frtol=0, maxiter=None, callback=None): + """Find the root of an elementwise function using Chandrupatla's algorithm. + + For each element of the output of `func`, `chandrupatla` seeks the scalar + root that makes the element 0. This function allows for `a`, `b`, and the + output of `func` to be of any broadcastable shapes. + + Parameters + ---------- + func : callable + The function whose root is desired. The signature must be:: + + func(x: ndarray, *args) -> ndarray + + where each element of ``x`` is a finite real and ``args`` is a tuple, + which may contain an arbitrary number of components of any type(s). + ``func`` must be an elementwise function: each element ``func(x)[i]`` + must equal ``func(x[i])`` for all indices ``i``. `_chandrupatla` + seeks an array ``x`` such that ``func(x)`` is an array of zeros. + a, b : array_like + The lower and upper bounds of the root of the function. Must be + broadcastable with one another. + args : tuple, optional + Additional positional arguments to be passed to `func`. + xatol, xrtol, fatol, frtol : float, optional + Absolute and relative tolerances on the root and function value. + See Notes for details. + maxiter : int, optional + The maximum number of iterations of the algorithm to perform. + The default is the maximum possible number of bisections within + the (normal) floating point numbers of the relevant dtype. + callback : callable, optional + An optional user-supplied function to be called before the first + iteration and after each iteration. + Called as ``callback(res)``, where ``res`` is a ``_RichResult`` + similar to that returned by `_chandrupatla` (but containing the current + iterate's values of all variables). If `callback` raises a + ``StopIteration``, the algorithm will terminate immediately and + `_chandrupatla` will return a result. + + Returns + ------- + res : _RichResult + An instance of `scipy._lib._util._RichResult` with the following + attributes. The descriptions are written as though the values will be + scalars; however, if `func` returns an array, the outputs will be + arrays of the same shape. + + x : float + The root of the function, if the algorithm terminated successfully. + nfev : int + The number of times the function was called to find the root. + nit : int + The number of iterations of Chandrupatla's algorithm performed. + status : int + An integer representing the exit status of the algorithm. + ``0`` : The algorithm converged to the specified tolerances. + ``-1`` : The algorithm encountered an invalid bracket. + ``-2`` : The maximum number of iterations was reached. + ``-3`` : A non-finite value was encountered. + ``-4`` : Iteration was terminated by `callback`. + ``1`` : The algorithm is proceeding normally (in `callback` only). + success : bool + ``True`` when the algorithm terminated successfully (status ``0``). + fun : float + The value of `func` evaluated at `x`. + xl, xr : float + The lower and upper ends of the bracket. + fl, fr : float + The function value at the lower and upper ends of the bracket. + + Notes + ----- + Implemented based on Chandrupatla's original paper [1]_. + + If ``xl`` and ``xr`` are the left and right ends of the bracket, + ``xmin = xl if abs(func(xl)) <= abs(func(xr)) else xr``, + and ``fmin0 = min(func(a), func(b))``, then the algorithm is considered to + have converged when ``abs(xr - xl) < xatol + abs(xmin) * xrtol`` or + ``fun(xmin) <= fatol + abs(fmin0) * frtol``. This is equivalent to the + termination condition described in [1]_ with ``xrtol = 4e-10``, + ``xatol = 1e-5``, and ``fatol = frtol = 0``. The default values are + ``xatol = 4*tiny``, ``xrtol = 4*eps``, ``frtol = 0``, and ``fatol = tiny``, + where ``eps`` and ``tiny`` are the precision and smallest normal number + of the result ``dtype`` of function inputs and outputs. + + References + ---------- + + .. [1] Chandrupatla, Tirupathi R. + "A new hybrid quadratic/bisection algorithm for finding the zero of a + nonlinear function without using derivatives". + Advances in Engineering Software, 28(3), 145-149. + https://doi.org/10.1016/s0965-9978(96)00051-8 + + See Also + -------- + brentq, brenth, ridder, bisect, newton + + Examples + -------- + >>> from scipy import optimize + >>> def f(x, c): + ... return x**3 - 2*x - c + >>> c = 5 + >>> res = optimize._chandrupatla._chandrupatla(f, 0, 3, args=(c,)) + >>> res.x + 2.0945514818937463 + + >>> c = [3, 4, 5] + >>> res = optimize._chandrupatla._chandrupatla(f, 0, 3, args=(c,)) + >>> res.x + array([1.8932892 , 2. , 2.09455148]) + + """ + res = _chandrupatla_iv(func, args, xatol, xrtol, + fatol, frtol, maxiter, callback) + func, args, xatol, xrtol, fatol, frtol, maxiter, callback = res + + # Initialization + temp = eim._initialize(func, (a, b), args) + func, xs, fs, args, shape, dtype, xp = temp + x1, x2 = xs + f1, f2 = fs + status = xp.full_like(x1, xp.asarray(eim._EINPROGRESS), + dtype=xp.int32) # in progress + nit, nfev = 0, 2 # two function evaluations performed above + finfo = xp.finfo(dtype) + xatol = 4*finfo.smallest_normal if xatol is None else xatol + xrtol = 4*finfo.eps if xrtol is None else xrtol + fatol = finfo.smallest_normal if fatol is None else fatol + frtol = frtol * xp.minimum(xp.abs(f1), xp.abs(f2)) + maxiter = (math.log2(finfo.max) - math.log2(finfo.smallest_normal) + if maxiter is None else maxiter) + work = _RichResult(x1=x1, f1=f1, x2=x2, f2=f2, x3=None, f3=None, t=0.5, + xatol=xatol, xrtol=xrtol, fatol=fatol, frtol=frtol, + nit=nit, nfev=nfev, status=status) + res_work_pairs = [('status', 'status'), ('x', 'xmin'), ('fun', 'fmin'), + ('nit', 'nit'), ('nfev', 'nfev'), ('xl', 'x1'), + ('fl', 'f1'), ('xr', 'x2'), ('fr', 'f2')] + + def pre_func_eval(work): + # [1] Figure 1 (first box) + x = work.x1 + work.t * (work.x2 - work.x1) + return x + + def post_func_eval(x, f, work): + # [1] Figure 1 (first diamond and boxes) + # Note: y/n are reversed in figure; compare to BASIC in appendix + work.x3, work.f3 = (xp.asarray(work.x2, copy=True), + xp.asarray(work.f2, copy=True)) + j = xp.sign(f) == xp.sign(work.f1) + nj = ~j + work.x3[j], work.f3[j] = work.x1[j], work.f1[j] + work.x2[nj], work.f2[nj] = work.x1[nj], work.f1[nj] + work.x1, work.f1 = x, f + + def check_termination(work): + # [1] Figure 1 (second diamond) + # Check for all terminal conditions and record statuses. + + # See [1] Section 4 (first two sentences) + i = xp.abs(work.f1) < xp.abs(work.f2) + work.xmin = xp.where(i, work.x1, work.x2) + work.fmin = xp.where(i, work.f1, work.f2) + stop = xp.zeros_like(work.x1, dtype=xp.bool) # termination condition met + + # If function value tolerance is met, report successful convergence, + # regardless of other conditions. Note that `frtol` has been redefined + # as `frtol = frtol * minimum(f1, f2)`, where `f1` and `f2` are the + # function evaluated at the original ends of the bracket. + i = xp.abs(work.fmin) <= work.fatol + work.frtol + work.status[i] = eim._ECONVERGED + stop[i] = True + + # If the bracket is no longer valid, report failure (unless a function + # tolerance is met, as detected above). + i = (xp_sign(work.f1) == xp_sign(work.f2)) & ~stop + NaN = xp.asarray(xp.nan, dtype=work.xmin.dtype) + work.xmin[i], work.fmin[i], work.status[i] = NaN, NaN, eim._ESIGNERR + stop[i] = True + + # If the abscissae are non-finite or either function value is NaN, + # report failure. + x_nonfinite = ~(xp.isfinite(work.x1) & xp.isfinite(work.x2)) + f_nan = xp.isnan(work.f1) & xp.isnan(work.f2) + i = (x_nonfinite | f_nan) & ~stop + work.xmin[i], work.fmin[i], work.status[i] = NaN, NaN, eim._EVALUEERR + stop[i] = True + + # This is the convergence criterion used in bisect. Chandrupatla's + # criterion is equivalent to this except with a factor of 4 on `xrtol`. + work.dx = xp.abs(work.x2 - work.x1) + work.tol = xp.abs(work.xmin) * work.xrtol + work.xatol + i = work.dx < work.tol + work.status[i] = eim._ECONVERGED + stop[i] = True + + return stop + + def post_termination_check(work): + # [1] Figure 1 (third diamond and boxes / Equation 1) + xi1 = (work.x1 - work.x2) / (work.x3 - work.x2) + with np.errstate(divide='ignore', invalid='ignore'): + phi1 = (work.f1 - work.f2) / (work.f3 - work.f2) + alpha = (work.x3 - work.x1) / (work.x2 - work.x1) + j = ((1 - xp.sqrt(1 - xi1)) < phi1) & (phi1 < xp.sqrt(xi1)) + + f1j, f2j, f3j, alphaj = work.f1[j], work.f2[j], work.f3[j], alpha[j] + t = xp.full_like(alpha, xp.asarray(0.5)) + t[j] = (f1j / (f1j - f2j) * f3j / (f3j - f2j) + - alphaj * f1j / (f3j - f1j) * f2j / (f2j - f3j)) + + # [1] Figure 1 (last box; see also BASIC in appendix with comment + # "Adjust T Away from the Interval Boundary") + tl = 0.5 * work.tol / work.dx + work.t = xp.clip(t, tl, 1 - tl) + + def customize_result(res, shape): + xl, xr, fl, fr = res['xl'], res['xr'], res['fl'], res['fr'] + i = res['xl'] < res['xr'] + res['xl'] = xp.where(i, xl, xr) + res['xr'] = xp.where(i, xr, xl) + res['fl'] = xp.where(i, fl, fr) + res['fr'] = xp.where(i, fr, fl) + return shape + + return eim._loop(work, callback, shape, maxiter, func, args, dtype, + pre_func_eval, post_func_eval, check_termination, + post_termination_check, customize_result, res_work_pairs, + xp=xp) + + +def _chandrupatla_iv(func, args, xatol, xrtol, + fatol, frtol, maxiter, callback): + # Input validation for `_chandrupatla` + + if not callable(func): + raise ValueError('`func` must be callable.') + + if not np.iterable(args): + args = (args,) + + # tolerances are floats, not arrays; OK to use NumPy + tols = np.asarray([xatol if xatol is not None else 1, + xrtol if xrtol is not None else 1, + fatol if fatol is not None else 1, + frtol if frtol is not None else 1]) + if (not np.issubdtype(tols.dtype, np.number) or np.any(tols < 0) + or np.any(np.isnan(tols)) or tols.shape != (4,)): + raise ValueError('Tolerances must be non-negative scalars.') + + if maxiter is not None: + maxiter_int = int(maxiter) + if maxiter != maxiter_int or maxiter < 0: + raise ValueError('`maxiter` must be a non-negative integer.') + + if callback is not None and not callable(callback): + raise ValueError('`callback` must be callable.') + + return func, args, xatol, xrtol, fatol, frtol, maxiter, callback + + +def _chandrupatla_minimize(func, x1, x2, x3, *, args=(), xatol=None, + xrtol=None, fatol=None, frtol=None, maxiter=100, + callback=None): + """Find the minimizer of an elementwise function. + + For each element of the output of `func`, `_chandrupatla_minimize` seeks + the scalar minimizer that minimizes the element. This function allows for + `x1`, `x2`, `x3`, and the elements of `args` to be arrays of any + broadcastable shapes. + + Parameters + ---------- + func : callable + The function whose minimizer is desired. The signature must be:: + + func(x: ndarray, *args) -> ndarray + + where each element of ``x`` is a finite real and ``args`` is a tuple, + which may contain an arbitrary number of arrays that are broadcastable + with `x`. ``func`` must be an elementwise function: each element + ``func(x)[i]`` must equal ``func(x[i])`` for all indices ``i``. + `_chandrupatla` seeks an array ``x`` such that ``func(x)`` is an array + of minima. + x1, x2, x3 : array_like + The abscissae of a standard scalar minimization bracket. A bracket is + valid if ``x1 < x2 < x3`` and ``func(x1) > func(x2) <= func(x3)``. + Must be broadcastable with one another and `args`. + args : tuple, optional + Additional positional arguments to be passed to `func`. Must be arrays + broadcastable with `x1`, `x2`, and `x3`. If the callable to be + differentiated requires arguments that are not broadcastable with `x`, + wrap that callable with `func` such that `func` accepts only `x` and + broadcastable arrays. + xatol, xrtol, fatol, frtol : float, optional + Absolute and relative tolerances on the minimizer and function value. + See Notes for details. + maxiter : int, optional + The maximum number of iterations of the algorithm to perform. + callback : callable, optional + An optional user-supplied function to be called before the first + iteration and after each iteration. + Called as ``callback(res)``, where ``res`` is a ``_RichResult`` + similar to that returned by `_chandrupatla_minimize` (but containing + the current iterate's values of all variables). If `callback` raises a + ``StopIteration``, the algorithm will terminate immediately and + `_chandrupatla_minimize` will return a result. + + Returns + ------- + res : _RichResult + An instance of `scipy._lib._util._RichResult` with the following + attributes. (The descriptions are written as though the values will be + scalars; however, if `func` returns an array, the outputs will be + arrays of the same shape.) + + success : bool + ``True`` when the algorithm terminated successfully (status ``0``). + status : int + An integer representing the exit status of the algorithm. + ``0`` : The algorithm converged to the specified tolerances. + ``-1`` : The algorithm encountered an invalid bracket. + ``-2`` : The maximum number of iterations was reached. + ``-3`` : A non-finite value was encountered. + ``-4`` : Iteration was terminated by `callback`. + ``1`` : The algorithm is proceeding normally (in `callback` only). + x : float + The minimizer of the function, if the algorithm terminated + successfully. + fun : float + The value of `func` evaluated at `x`. + nfev : int + The number of points at which `func` was evaluated. + nit : int + The number of iterations of the algorithm that were performed. + xl, xm, xr : float + The final three-point bracket. + fl, fm, fr : float + The function value at the bracket points. + + Notes + ----- + Implemented based on Chandrupatla's original paper [1]_. + + If ``x1 < x2 < x3`` are the points of the bracket and ``f1 > f2 <= f3`` + are the values of ``func`` at those points, then the algorithm is + considered to have converged when ``x3 - x1 <= abs(x2)*xrtol + xatol`` + or ``(f1 - 2*f2 + f3)/2 <= abs(f2)*frtol + fatol``. Note that first of + these differs from the termination conditions described in [1]_. The + default values of `xrtol` is the square root of the precision of the + appropriate dtype, and ``xatol = fatol = frtol`` is the smallest normal + number of the appropriate dtype. + + References + ---------- + .. [1] Chandrupatla, Tirupathi R. (1998). + "An efficient quadratic fit-sectioning algorithm for minimization + without derivatives". + Computer Methods in Applied Mechanics and Engineering, 152 (1-2), + 211-217. https://doi.org/10.1016/S0045-7825(97)00190-4 + + See Also + -------- + golden, brent, bounded + + Examples + -------- + >>> from scipy.optimize._chandrupatla import _chandrupatla_minimize + >>> def f(x, args=1): + ... return (x - args)**2 + >>> res = _chandrupatla_minimize(f, -5, 0, 5) + >>> res.x + 1.0 + >>> c = [1, 1.5, 2] + >>> res = _chandrupatla_minimize(f, -5, 0, 5, args=(c,)) + >>> res.x + array([1. , 1.5, 2. ]) + """ + res = _chandrupatla_iv(func, args, xatol, xrtol, + fatol, frtol, maxiter, callback) + func, args, xatol, xrtol, fatol, frtol, maxiter, callback = res + + # Initialization + xs = (x1, x2, x3) + temp = eim._initialize(func, xs, args) + func, xs, fs, args, shape, dtype, xp = temp # line split for PEP8 + x1, x2, x3 = xs + f1, f2, f3 = fs + phi = xp.asarray(0.5 + 0.5*5**0.5, dtype=dtype)[()] # golden ratio + status = xp.full_like(x1, xp.asarray(eim._EINPROGRESS), + dtype=xp.int32) # in progress + nit, nfev = 0, 3 # three function evaluations performed above + fatol = xp.finfo(dtype).smallest_normal if fatol is None else fatol + frtol = xp.finfo(dtype).smallest_normal if frtol is None else frtol + xatol = xp.finfo(dtype).smallest_normal if xatol is None else xatol + xrtol = math.sqrt(xp.finfo(dtype).eps) if xrtol is None else xrtol + + # Ensure that x1 < x2 < x3 initially. + xs, fs = xp.stack((x1, x2, x3)), xp.stack((f1, f2, f3)) + i = xp.argsort(xs, axis=0) + x1, x2, x3 = xp_take_along_axis(xs, i, axis=0) # data-apis/array-api#808 + f1, f2, f3 = xp_take_along_axis(fs, i, axis=0) # data-apis/array-api#808 + q0 = xp_copy(x3) # "At the start, q0 is set at x3..." ([1] after (7)) + + work = _RichResult(x1=x1, f1=f1, x2=x2, f2=f2, x3=x3, f3=f3, phi=phi, + xatol=xatol, xrtol=xrtol, fatol=fatol, frtol=frtol, + nit=nit, nfev=nfev, status=status, q0=q0, args=args) + res_work_pairs = [('status', 'status'), + ('x', 'x2'), ('fun', 'f2'), + ('nit', 'nit'), ('nfev', 'nfev'), + ('xl', 'x1'), ('xm', 'x2'), ('xr', 'x3'), + ('fl', 'f1'), ('fm', 'f2'), ('fr', 'f3')] + + def pre_func_eval(work): + # `_check_termination` is called first -> `x3 - x2 > x2 - x1` + # But let's calculate a few terms that we'll reuse + x21 = work.x2 - work.x1 + x32 = work.x3 - work.x2 + + # [1] Section 3. "The quadratic minimum point Q1 is calculated using + # the relations developed in the previous section." [1] Section 2 (5/6) + A = x21 * (work.f3 - work.f2) + B = x32 * (work.f1 - work.f2) + C = A / (A + B) + # q1 = C * (work.x1 + work.x2) / 2 + (1 - C) * (work.x2 + work.x3) / 2 + q1 = 0.5 * (C*(work.x1 - work.x3) + work.x2 + work.x3) # much faster + # this is an array, so multiplying by 0.5 does not change dtype + + # "If Q1 and Q0 are sufficiently close... Q1 is accepted if it is + # sufficiently away from the inside point x2" + i = xp.abs(q1 - work.q0) < 0.5 * xp.abs(x21) # [1] (7) + xi = q1[i] + # Later, after (9), "If the point Q1 is in a +/- xtol neighborhood of + # x2, the new point is chosen in the larger interval at a distance + # tol away from x2." + # See also QBASIC code after "Accept Ql adjust if close to X2". + j = xp.abs(q1[i] - work.x2[i]) <= work.xtol[i] + xi[j] = work.x2[i][j] + xp_sign(x32[i][j]) * work.xtol[i][j] + + # "If condition (7) is not satisfied, golden sectioning of the larger + # interval is carried out to introduce the new point." + # (For simplicity, we go ahead and calculate it for all points, but we + # change the elements for which the condition was satisfied.) + x = work.x2 + (2 - work.phi) * x32 + x[i] = xi + + # "We define Q0 as the value of Q1 at the previous iteration." + work.q0 = q1 + return x + + def post_func_eval(x, f, work): + # Standard logic for updating a three-point bracket based on a new + # point. In QBASIC code, see "IF SGN(X-X2) = SGN(X3-X2) THEN...". + # There is an awful lot of data copying going on here; this would + # probably benefit from code optimization or implementation in Pythran. + i = xp_sign(x - work.x2) == xp_sign(work.x3 - work.x2) + xi, x1i, x2i, x3i = x[i], work.x1[i], work.x2[i], work.x3[i], + fi, f1i, f2i, f3i = f[i], work.f1[i], work.f2[i], work.f3[i] + j = fi > f2i + x3i[j], f3i[j] = xi[j], fi[j] + j = ~j + x1i[j], f1i[j], x2i[j], f2i[j] = x2i[j], f2i[j], xi[j], fi[j] + + ni = ~i + xni, x1ni, x2ni, x3ni = x[ni], work.x1[ni], work.x2[ni], work.x3[ni], + fni, f1ni, f2ni, f3ni = f[ni], work.f1[ni], work.f2[ni], work.f3[ni] + j = fni > f2ni + x1ni[j], f1ni[j] = xni[j], fni[j] + j = ~j + x3ni[j], f3ni[j], x2ni[j], f2ni[j] = x2ni[j], f2ni[j], xni[j], fni[j] + + work.x1[i], work.x2[i], work.x3[i] = x1i, x2i, x3i + work.f1[i], work.f2[i], work.f3[i] = f1i, f2i, f3i + work.x1[ni], work.x2[ni], work.x3[ni] = x1ni, x2ni, x3ni, + work.f1[ni], work.f2[ni], work.f3[ni] = f1ni, f2ni, f3ni + + def check_termination(work): + # Check for all terminal conditions and record statuses. + stop = xp.zeros_like(work.x1, dtype=bool) # termination condition met + + # Bracket is invalid; stop and don't return minimizer/minimum + i = ((work.f2 > work.f1) | (work.f2 > work.f3)) + work.x2[i], work.f2[i] = xp.nan, xp.nan + stop[i], work.status[i] = True, eim._ESIGNERR + + # Non-finite values; stop and don't return minimizer/minimum + finite = xp.isfinite(work.x1+work.x2+work.x3+work.f1+work.f2+work.f3) + i = ~(finite | stop) + work.x2[i], work.f2[i] = xp.nan, xp.nan + stop[i], work.status[i] = True, eim._EVALUEERR + + # [1] Section 3 "Points 1 and 3 are interchanged if necessary to make + # the (x2, x3) the larger interval." + # Note: I had used np.choose; this is much faster. This would be a good + # place to save e.g. `work.x3 - work.x2` for reuse, but I tried and + # didn't notice a speed boost, so let's keep it simple. + i = xp.abs(work.x3 - work.x2) < xp.abs(work.x2 - work.x1) + temp = work.x1[i] + work.x1[i] = work.x3[i] + work.x3[i] = temp + temp = work.f1[i] + work.f1[i] = work.f3[i] + work.f3[i] = temp + + # [1] Section 3 (bottom of page 212) + # "We set a tolerance value xtol..." + work.xtol = xp.abs(work.x2) * work.xrtol + work.xatol # [1] (8) + # "The convergence based on interval is achieved when..." + # Note: Equality allowed in case of `xtol=0` + i = xp.abs(work.x3 - work.x2) <= 2 * work.xtol # [1] (9) + + # "We define ftol using..." + ftol = xp.abs(work.f2) * work.frtol + work.fatol # [1] (10) + # "The convergence based on function values is achieved when..." + # Note 1: modify in place to incorporate tolerance on function value. + # Note 2: factor of 2 is not in the text; see QBASIC start of DO loop + i |= (work.f1 - 2 * work.f2 + work.f3) <= 2*ftol # [1] (11) + i &= ~stop + stop[i], work.status[i] = True, eim._ECONVERGED + + return stop + + def post_termination_check(work): + pass + + def customize_result(res, shape): + xl, xr, fl, fr = res['xl'], res['xr'], res['fl'], res['fr'] + i = res['xl'] >= res['xr'] + res['xl'] = xp.where(i, xr, xl) + res['xr'] = xp.where(i, xl, xr) + res['fl'] = xp.where(i, fr, fl) + res['fr'] = xp.where(i, fl, fr) + return shape + + return eim._loop(work, callback, shape, maxiter, func, args, dtype, + pre_func_eval, post_func_eval, check_termination, + post_termination_check, customize_result, res_work_pairs, + xp=xp) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_cobyla_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_cobyla_py.py new file mode 100644 index 0000000000000000000000000000000000000000..7e99acf373df59524f66e19f625f50b8d5d3cc76 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_cobyla_py.py @@ -0,0 +1,316 @@ +""" +Interface to Constrained Optimization By Linear Approximation + +Functions +--------- +.. autosummary:: + :toctree: generated/ + + fmin_cobyla + +""" + +import functools +from threading import RLock + +import numpy as np +from scipy.optimize import _cobyla as cobyla +from ._optimize import (OptimizeResult, _check_unknown_options, + _prepare_scalar_function) +try: + from itertools import izip +except ImportError: + izip = zip + +__all__ = ['fmin_cobyla'] + +# Workaround as _cobyla.minimize is not threadsafe +# due to an unknown f2py bug and can segfault, +# see gh-9658. +_module_lock = RLock() +def synchronized(func): + @functools.wraps(func) + def wrapper(*args, **kwargs): + with _module_lock: + return func(*args, **kwargs) + return wrapper + +@synchronized +def fmin_cobyla(func, x0, cons, args=(), consargs=None, rhobeg=1.0, + rhoend=1e-4, maxfun=1000, disp=None, catol=2e-4, + *, callback=None): + """ + Minimize a function using the Constrained Optimization By Linear + Approximation (COBYLA) method. This method wraps a FORTRAN + implementation of the algorithm. + + Parameters + ---------- + func : callable + Function to minimize. In the form func(x, \\*args). + x0 : ndarray + Initial guess. + cons : sequence + Constraint functions; must all be ``>=0`` (a single function + if only 1 constraint). Each function takes the parameters `x` + as its first argument, and it can return either a single number or + an array or list of numbers. + args : tuple, optional + Extra arguments to pass to function. + consargs : tuple, optional + Extra arguments to pass to constraint functions (default of None means + use same extra arguments as those passed to func). + Use ``()`` for no extra arguments. + rhobeg : float, optional + Reasonable initial changes to the variables. + rhoend : float, optional + Final accuracy in the optimization (not precisely guaranteed). This + is a lower bound on the size of the trust region. + disp : {0, 1, 2, 3}, optional + Controls the frequency of output; 0 implies no output. + maxfun : int, optional + Maximum number of function evaluations. + catol : float, optional + Absolute tolerance for constraint violations. + callback : callable, optional + Called after each iteration, as ``callback(x)``, where ``x`` is the + current parameter vector. + + Returns + ------- + x : ndarray + The argument that minimises `f`. + + See also + -------- + minimize: Interface to minimization algorithms for multivariate + functions. See the 'COBYLA' `method` in particular. + + Notes + ----- + This algorithm is based on linear approximations to the objective + function and each constraint. We briefly describe the algorithm. + + Suppose the function is being minimized over k variables. At the + jth iteration the algorithm has k+1 points v_1, ..., v_(k+1), + an approximate solution x_j, and a radius RHO_j. + (i.e., linear plus a constant) approximations to the objective + function and constraint functions such that their function values + agree with the linear approximation on the k+1 points v_1,.., v_(k+1). + This gives a linear program to solve (where the linear approximations + of the constraint functions are constrained to be non-negative). + + However, the linear approximations are likely only good + approximations near the current simplex, so the linear program is + given the further requirement that the solution, which + will become x_(j+1), must be within RHO_j from x_j. RHO_j only + decreases, never increases. The initial RHO_j is rhobeg and the + final RHO_j is rhoend. In this way COBYLA's iterations behave + like a trust region algorithm. + + Additionally, the linear program may be inconsistent, or the + approximation may give poor improvement. For details about + how these issues are resolved, as well as how the points v_i are + updated, refer to the source code or the references below. + + + References + ---------- + Powell M.J.D. (1994), "A direct search optimization method that models + the objective and constraint functions by linear interpolation.", in + Advances in Optimization and Numerical Analysis, eds. S. Gomez and + J-P Hennart, Kluwer Academic (Dordrecht), pp. 51-67 + + Powell M.J.D. (1998), "Direct search algorithms for optimization + calculations", Acta Numerica 7, 287-336 + + Powell M.J.D. (2007), "A view of algorithms for optimization without + derivatives", Cambridge University Technical Report DAMTP 2007/NA03 + + + Examples + -------- + Minimize the objective function f(x,y) = x*y subject + to the constraints x**2 + y**2 < 1 and y > 0:: + + >>> def objective(x): + ... return x[0]*x[1] + ... + >>> def constr1(x): + ... return 1 - (x[0]**2 + x[1]**2) + ... + >>> def constr2(x): + ... return x[1] + ... + >>> from scipy.optimize import fmin_cobyla + >>> fmin_cobyla(objective, [0.0, 0.1], [constr1, constr2], rhoend=1e-7) + array([-0.70710685, 0.70710671]) + + The exact solution is (-sqrt(2)/2, sqrt(2)/2). + + + + """ + err = "cons must be a sequence of callable functions or a single"\ + " callable function." + try: + len(cons) + except TypeError as e: + if callable(cons): + cons = [cons] + else: + raise TypeError(err) from e + else: + for thisfunc in cons: + if not callable(thisfunc): + raise TypeError(err) + + if consargs is None: + consargs = args + + # build constraints + con = tuple({'type': 'ineq', 'fun': c, 'args': consargs} for c in cons) + + # options + opts = {'rhobeg': rhobeg, + 'tol': rhoend, + 'disp': disp, + 'maxiter': maxfun, + 'catol': catol, + 'callback': callback} + + sol = _minimize_cobyla(func, x0, args, constraints=con, + **opts) + if disp and not sol['success']: + print(f"COBYLA failed to find a solution: {sol.message}") + return sol['x'] + + +@synchronized +def _minimize_cobyla(fun, x0, args=(), constraints=(), + rhobeg=1.0, tol=1e-4, maxiter=1000, + disp=False, catol=2e-4, callback=None, bounds=None, + **unknown_options): + """ + Minimize a scalar function of one or more variables using the + Constrained Optimization BY Linear Approximation (COBYLA) algorithm. + + Options + ------- + rhobeg : float + Reasonable initial changes to the variables. + tol : float + Final accuracy in the optimization (not precisely guaranteed). + This is a lower bound on the size of the trust region. + disp : bool + Set to True to print convergence messages. If False, + `verbosity` is ignored as set to 0. + maxiter : int + Maximum number of function evaluations. + catol : float + Tolerance (absolute) for constraint violations + + """ + _check_unknown_options(unknown_options) + maxfun = maxiter + rhoend = tol + iprint = int(bool(disp)) + + # check constraints + if isinstance(constraints, dict): + constraints = (constraints, ) + + if bounds: + i_lb = np.isfinite(bounds.lb) + if np.any(i_lb): + def lb_constraint(x, *args, **kwargs): + return x[i_lb] - bounds.lb[i_lb] + + constraints.append({'type': 'ineq', 'fun': lb_constraint}) + + i_ub = np.isfinite(bounds.ub) + if np.any(i_ub): + def ub_constraint(x): + return bounds.ub[i_ub] - x[i_ub] + + constraints.append({'type': 'ineq', 'fun': ub_constraint}) + + for ic, con in enumerate(constraints): + # check type + try: + ctype = con['type'].lower() + except KeyError as e: + raise KeyError('Constraint %d has no type defined.' % ic) from e + except TypeError as e: + raise TypeError('Constraints must be defined using a ' + 'dictionary.') from e + except AttributeError as e: + raise TypeError("Constraint's type must be a string.") from e + else: + if ctype != 'ineq': + raise ValueError(f"Constraints of type '{con['type']}' not handled by " + "COBYLA.") + + # check function + if 'fun' not in con: + raise KeyError('Constraint %d has no function defined.' % ic) + + # check extra arguments + if 'args' not in con: + con['args'] = () + + # m is the total number of constraint values + # it takes into account that some constraints may be vector-valued + cons_lengths = [] + for c in constraints: + f = c['fun'](x0, *c['args']) + try: + cons_length = len(f) + except TypeError: + cons_length = 1 + cons_lengths.append(cons_length) + m = sum(cons_lengths) + + # create the ScalarFunction, cobyla doesn't require derivative function + def _jac(x, *args): + return None + + sf = _prepare_scalar_function(fun, x0, args=args, jac=_jac) + + def calcfc(x, con): + f = sf.fun(x) + i = 0 + for size, c in izip(cons_lengths, constraints): + con[i: i + size] = c['fun'](x, *c['args']) + i += size + return f + + def wrapped_callback(x): + if callback is not None: + callback(np.copy(x)) + + info = np.zeros(4, np.float64) + xopt, info = cobyla.minimize(calcfc, m=m, x=np.copy(x0), rhobeg=rhobeg, + rhoend=rhoend, iprint=iprint, maxfun=maxfun, + dinfo=info, callback=wrapped_callback) + + if info[3] > catol: + # Check constraint violation + info[0] = 4 + + return OptimizeResult(x=xopt, + status=int(info[0]), + success=info[0] == 1, + message={1: 'Optimization terminated successfully.', + 2: 'Maximum number of function evaluations ' + 'has been exceeded.', + 3: 'Rounding errors are becoming damaging ' + 'in COBYLA subroutine.', + 4: 'Did not converge to a solution ' + 'satisfying the constraints. See ' + '`maxcv` for magnitude of violation.', + 5: 'NaN result encountered.' + }.get(info[0], 'Unknown exit status.'), + nfev=int(info[1]), + fun=info[2], + maxcv=info[3]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_cobyqa_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_cobyqa_py.py new file mode 100644 index 0000000000000000000000000000000000000000..38ae0477ca38e28dda80e0bf2dd1f0905eacff6e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_cobyqa_py.py @@ -0,0 +1,72 @@ +import numpy as np +from threading import Lock + +from ._optimize import _check_unknown_options + + +COBYQA_LOCK = Lock() + + +def _minimize_cobyqa(fun, x0, args=(), bounds=None, constraints=(), + callback=None, disp=False, maxfev=None, maxiter=None, + f_target=-np.inf, feasibility_tol=1e-8, + initial_tr_radius=1.0, final_tr_radius=1e-6, scale=False, + **unknown_options): + """ + Minimize a scalar function of one or more variables using the + Constrained Optimization BY Quadratic Approximations (COBYQA) algorithm [1]_. + + .. versionadded:: 1.14.0 + + Options + ------- + disp : bool + Set to True to print information about the optimization procedure. + Default is ``False``. + maxfev : int + Maximum number of function evaluations. Default is ``500 * n``, where + ``n`` is the number of variables. + maxiter : int + Maximum number of iterations. Default is ``1000 * n``, where ``n`` is + the number of variables. + f_target : float + Target value for the objective function. The optimization procedure is + terminated when the objective function value of a feasible point (see + `feasibility_tol` below) is less than or equal to this target. Default + is ``-numpy.inf``. + feasibility_tol : float + Absolute tolerance for the constraint violation. Default is ``1e-8``. + initial_tr_radius : float + Initial trust-region radius. Typically, this value should be in the + order of one tenth of the greatest expected change to the variables. + Default is ``1.0``. + final_tr_radius : float + Final trust-region radius. It should indicate the accuracy required in + the final values of the variables. If provided, this option overrides + the value of `tol` in the `minimize` function. Default is ``1e-6``. + scale : bool + Set to True to scale the variables according to the bounds. If True and + if all the lower and upper bounds are finite, the variables are scaled + to be within the range :math:`[-1, 1]`. If any of the lower or upper + bounds is infinite, the variables are not scaled. Default is ``False``. + + References + ---------- + .. [1] COBYQA + https://www.cobyqa.com/stable/ + """ + from .._lib.cobyqa import minimize # import here to avoid circular imports + + _check_unknown_options(unknown_options) + options = { + 'disp': bool(disp), + 'maxfev': int(maxfev) if maxfev is not None else 500 * len(x0), + 'maxiter': int(maxiter) if maxiter is not None else 1000 * len(x0), + 'target': float(f_target), + 'feasibility_tol': float(feasibility_tol), + 'radius_init': float(initial_tr_radius), + 'radius_final': float(final_tr_radius), + 'scale': bool(scale), + } + with COBYQA_LOCK: + return minimize(fun, x0, args, bounds, constraints, callback, options) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_constraints.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_constraints.py new file mode 100644 index 0000000000000000000000000000000000000000..1bae893e231eb9bd89308e441b8abf841f4605bb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_constraints.py @@ -0,0 +1,594 @@ +"""Constraints definition for minimize.""" +import numpy as np +from ._hessian_update_strategy import BFGS +from ._differentiable_functions import ( + VectorFunction, LinearVectorFunction, IdentityVectorFunction) +from ._optimize import OptimizeWarning +from warnings import warn, catch_warnings, simplefilter, filterwarnings +from scipy.sparse import issparse + + +def _arr_to_scalar(x): + # If x is a numpy array, return x.item(). This will + # fail if the array has more than one element. + return x.item() if isinstance(x, np.ndarray) else x + + +class NonlinearConstraint: + """Nonlinear constraint on the variables. + + The constraint has the general inequality form:: + + lb <= fun(x) <= ub + + Here the vector of independent variables x is passed as ndarray of shape + (n,) and ``fun`` returns a vector with m components. + + It is possible to use equal bounds to represent an equality constraint or + infinite bounds to represent a one-sided constraint. + + Parameters + ---------- + fun : callable + The function defining the constraint. + The signature is ``fun(x) -> array_like, shape (m,)``. + lb, ub : array_like + Lower and upper bounds on the constraint. Each array must have the + shape (m,) or be a scalar, in the latter case a bound will be the same + for all components of the constraint. Use ``np.inf`` with an + appropriate sign to specify a one-sided constraint. + Set components of `lb` and `ub` equal to represent an equality + constraint. Note that you can mix constraints of different types: + interval, one-sided or equality, by setting different components of + `lb` and `ub` as necessary. + jac : {callable, '2-point', '3-point', 'cs'}, optional + Method of computing the Jacobian matrix (an m-by-n matrix, + where element (i, j) is the partial derivative of f[i] with + respect to x[j]). The keywords {'2-point', '3-point', + 'cs'} select a finite difference scheme for the numerical estimation. + A callable must have the following signature:: + + jac(x) -> {ndarray, sparse matrix}, shape (m, n) + + Default is '2-point'. + hess : {callable, '2-point', '3-point', 'cs', HessianUpdateStrategy, None}, optional + Method for computing the Hessian matrix. The keywords + {'2-point', '3-point', 'cs'} select a finite difference scheme for + numerical estimation. Alternatively, objects implementing + `HessianUpdateStrategy` interface can be used to approximate the + Hessian. Currently available implementations are: + + - `BFGS` (default option) + - `SR1` + + A callable must return the Hessian matrix of ``dot(fun, v)`` and + must have the following signature: + ``hess(x, v) -> {LinearOperator, sparse matrix, array_like}, shape (n, n)``. + Here ``v`` is ndarray with shape (m,) containing Lagrange multipliers. + keep_feasible : array_like of bool, optional + Whether to keep the constraint components feasible throughout + iterations. A single value set this property for all components. + Default is False. Has no effect for equality constraints. + finite_diff_rel_step: None or array_like, optional + Relative step size for the finite difference approximation. Default is + None, which will select a reasonable value automatically depending + on a finite difference scheme. + finite_diff_jac_sparsity: {None, array_like, sparse matrix}, optional + Defines the sparsity structure of the Jacobian matrix for finite + difference estimation, its shape must be (m, n). If the Jacobian has + only few non-zero elements in *each* row, providing the sparsity + structure will greatly speed up the computations. A zero entry means + that a corresponding element in the Jacobian is identically zero. + If provided, forces the use of 'lsmr' trust-region solver. + If None (default) then dense differencing will be used. + + Notes + ----- + Finite difference schemes {'2-point', '3-point', 'cs'} may be used for + approximating either the Jacobian or the Hessian. We, however, do not allow + its use for approximating both simultaneously. Hence whenever the Jacobian + is estimated via finite-differences, we require the Hessian to be estimated + using one of the quasi-Newton strategies. + + The scheme 'cs' is potentially the most accurate, but requires the function + to correctly handles complex inputs and be analytically continuable to the + complex plane. The scheme '3-point' is more accurate than '2-point' but + requires twice as many operations. + + Examples + -------- + Constrain ``x[0] < sin(x[1]) + 1.9`` + + >>> from scipy.optimize import NonlinearConstraint + >>> import numpy as np + >>> con = lambda x: x[0] - np.sin(x[1]) + >>> nlc = NonlinearConstraint(con, -np.inf, 1.9) + + """ + def __init__(self, fun, lb, ub, jac='2-point', hess=None, + keep_feasible=False, finite_diff_rel_step=None, + finite_diff_jac_sparsity=None): + if hess is None: + hess = BFGS() + self.fun = fun + self.lb = lb + self.ub = ub + self.finite_diff_rel_step = finite_diff_rel_step + self.finite_diff_jac_sparsity = finite_diff_jac_sparsity + self.jac = jac + self.hess = hess + self.keep_feasible = keep_feasible + + +class LinearConstraint: + """Linear constraint on the variables. + + The constraint has the general inequality form:: + + lb <= A.dot(x) <= ub + + Here the vector of independent variables x is passed as ndarray of shape + (n,) and the matrix A has shape (m, n). + + It is possible to use equal bounds to represent an equality constraint or + infinite bounds to represent a one-sided constraint. + + Parameters + ---------- + A : {array_like, sparse matrix}, shape (m, n) + Matrix defining the constraint. + lb, ub : dense array_like, optional + Lower and upper limits on the constraint. Each array must have the + shape (m,) or be a scalar, in the latter case a bound will be the same + for all components of the constraint. Use ``np.inf`` with an + appropriate sign to specify a one-sided constraint. + Set components of `lb` and `ub` equal to represent an equality + constraint. Note that you can mix constraints of different types: + interval, one-sided or equality, by setting different components of + `lb` and `ub` as necessary. Defaults to ``lb = -np.inf`` + and ``ub = np.inf`` (no limits). + keep_feasible : dense array_like of bool, optional + Whether to keep the constraint components feasible throughout + iterations. A single value set this property for all components. + Default is False. Has no effect for equality constraints. + """ + def _input_validation(self): + if self.A.ndim != 2: + message = "`A` must have exactly two dimensions." + raise ValueError(message) + + try: + shape = self.A.shape[0:1] + self.lb = np.broadcast_to(self.lb, shape) + self.ub = np.broadcast_to(self.ub, shape) + self.keep_feasible = np.broadcast_to(self.keep_feasible, shape) + except ValueError: + message = ("`lb`, `ub`, and `keep_feasible` must be broadcastable " + "to shape `A.shape[0:1]`") + raise ValueError(message) + + def __init__(self, A, lb=-np.inf, ub=np.inf, keep_feasible=False): + if not issparse(A): + # In some cases, if the constraint is not valid, this emits a + # VisibleDeprecationWarning about ragged nested sequences + # before eventually causing an error. `scipy.optimize.milp` would + # prefer that this just error out immediately so it can handle it + # rather than concerning the user. + with catch_warnings(): + simplefilter("error") + self.A = np.atleast_2d(A).astype(np.float64) + else: + self.A = A + if issparse(lb) or issparse(ub): + raise ValueError("Constraint limits must be dense arrays.") + self.lb = np.atleast_1d(lb).astype(np.float64) + self.ub = np.atleast_1d(ub).astype(np.float64) + + if issparse(keep_feasible): + raise ValueError("`keep_feasible` must be a dense array.") + self.keep_feasible = np.atleast_1d(keep_feasible).astype(bool) + self._input_validation() + + def residual(self, x): + """ + Calculate the residual between the constraint function and the limits + + For a linear constraint of the form:: + + lb <= A@x <= ub + + the lower and upper residuals between ``A@x`` and the limits are values + ``sl`` and ``sb`` such that:: + + lb + sl == A@x == ub - sb + + When all elements of ``sl`` and ``sb`` are positive, all elements of + the constraint are satisfied; a negative element in ``sl`` or ``sb`` + indicates that the corresponding element of the constraint is not + satisfied. + + Parameters + ---------- + x: array_like + Vector of independent variables + + Returns + ------- + sl, sb : array-like + The lower and upper residuals + """ + return self.A@x - self.lb, self.ub - self.A@x + + +class Bounds: + """Bounds constraint on the variables. + + The constraint has the general inequality form:: + + lb <= x <= ub + + It is possible to use equal bounds to represent an equality constraint or + infinite bounds to represent a one-sided constraint. + + Parameters + ---------- + lb, ub : dense array_like, optional + Lower and upper bounds on independent variables. `lb`, `ub`, and + `keep_feasible` must be the same shape or broadcastable. + Set components of `lb` and `ub` equal + to fix a variable. Use ``np.inf`` with an appropriate sign to disable + bounds on all or some variables. Note that you can mix constraints of + different types: interval, one-sided or equality, by setting different + components of `lb` and `ub` as necessary. Defaults to ``lb = -np.inf`` + and ``ub = np.inf`` (no bounds). + keep_feasible : dense array_like of bool, optional + Whether to keep the constraint components feasible throughout + iterations. Must be broadcastable with `lb` and `ub`. + Default is False. Has no effect for equality constraints. + """ + def _input_validation(self): + try: + res = np.broadcast_arrays(self.lb, self.ub, self.keep_feasible) + self.lb, self.ub, self.keep_feasible = res + except ValueError: + message = "`lb`, `ub`, and `keep_feasible` must be broadcastable." + raise ValueError(message) + + def __init__(self, lb=-np.inf, ub=np.inf, keep_feasible=False): + if issparse(lb) or issparse(ub): + raise ValueError("Lower and upper bounds must be dense arrays.") + self.lb = np.atleast_1d(lb) + self.ub = np.atleast_1d(ub) + + if issparse(keep_feasible): + raise ValueError("`keep_feasible` must be a dense array.") + self.keep_feasible = np.atleast_1d(keep_feasible).astype(bool) + self._input_validation() + + def __repr__(self): + start = f"{type(self).__name__}({self.lb!r}, {self.ub!r}" + if np.any(self.keep_feasible): + end = f", keep_feasible={self.keep_feasible!r})" + else: + end = ")" + return start + end + + def residual(self, x): + """Calculate the residual (slack) between the input and the bounds + + For a bound constraint of the form:: + + lb <= x <= ub + + the lower and upper residuals between `x` and the bounds are values + ``sl`` and ``sb`` such that:: + + lb + sl == x == ub - sb + + When all elements of ``sl`` and ``sb`` are positive, all elements of + ``x`` lie within the bounds; a negative element in ``sl`` or ``sb`` + indicates that the corresponding element of ``x`` is out of bounds. + + Parameters + ---------- + x: array_like + Vector of independent variables + + Returns + ------- + sl, sb : array-like + The lower and upper residuals + """ + return x - self.lb, self.ub - x + + +class PreparedConstraint: + """Constraint prepared from a user defined constraint. + + On creation it will check whether a constraint definition is valid and + the initial point is feasible. If created successfully, it will contain + the attributes listed below. + + Parameters + ---------- + constraint : {NonlinearConstraint, LinearConstraint`, Bounds} + Constraint to check and prepare. + x0 : array_like + Initial vector of independent variables. + sparse_jacobian : bool or None, optional + If bool, then the Jacobian of the constraint will be converted + to the corresponded format if necessary. If None (default), such + conversion is not made. + finite_diff_bounds : 2-tuple, optional + Lower and upper bounds on the independent variables for the finite + difference approximation, if applicable. Defaults to no bounds. + + Attributes + ---------- + fun : {VectorFunction, LinearVectorFunction, IdentityVectorFunction} + Function defining the constraint wrapped by one of the convenience + classes. + bounds : 2-tuple + Contains lower and upper bounds for the constraints --- lb and ub. + These are converted to ndarray and have a size equal to the number of + the constraints. + keep_feasible : ndarray + Array indicating which components must be kept feasible with a size + equal to the number of the constraints. + """ + def __init__(self, constraint, x0, sparse_jacobian=None, + finite_diff_bounds=(-np.inf, np.inf)): + if isinstance(constraint, NonlinearConstraint): + fun = VectorFunction(constraint.fun, x0, + constraint.jac, constraint.hess, + constraint.finite_diff_rel_step, + constraint.finite_diff_jac_sparsity, + finite_diff_bounds, sparse_jacobian) + elif isinstance(constraint, LinearConstraint): + fun = LinearVectorFunction(constraint.A, x0, sparse_jacobian) + elif isinstance(constraint, Bounds): + fun = IdentityVectorFunction(x0, sparse_jacobian) + else: + raise ValueError("`constraint` of an unknown type is passed.") + + m = fun.m + + lb = np.asarray(constraint.lb, dtype=float) + ub = np.asarray(constraint.ub, dtype=float) + keep_feasible = np.asarray(constraint.keep_feasible, dtype=bool) + + lb = np.broadcast_to(lb, m) + ub = np.broadcast_to(ub, m) + keep_feasible = np.broadcast_to(keep_feasible, m) + + if keep_feasible.shape != (m,): + raise ValueError("`keep_feasible` has a wrong shape.") + + mask = keep_feasible & (lb != ub) + f0 = fun.f + if np.any(f0[mask] < lb[mask]) or np.any(f0[mask] > ub[mask]): + raise ValueError("`x0` is infeasible with respect to some " + "inequality constraint with `keep_feasible` " + "set to True.") + + self.fun = fun + self.bounds = (lb, ub) + self.keep_feasible = keep_feasible + + def violation(self, x): + """How much the constraint is exceeded by. + + Parameters + ---------- + x : array-like + Vector of independent variables + + Returns + ------- + excess : array-like + How much the constraint is exceeded by, for each of the + constraints specified by `PreparedConstraint.fun`. + """ + with catch_warnings(): + # Ignore the following warning, it's not important when + # figuring out total violation + # UserWarning: delta_grad == 0.0. Check if the approximated + # function is linear + filterwarnings("ignore", "delta_grad", UserWarning) + ev = self.fun.fun(np.asarray(x)) + + excess_lb = np.maximum(self.bounds[0] - ev, 0) + excess_ub = np.maximum(ev - self.bounds[1], 0) + + return excess_lb + excess_ub + + +def new_bounds_to_old(lb, ub, n): + """Convert the new bounds representation to the old one. + + The new representation is a tuple (lb, ub) and the old one is a list + containing n tuples, ith containing lower and upper bound on a ith + variable. + If any of the entries in lb/ub are -np.inf/np.inf they are replaced by + None. + """ + lb = np.broadcast_to(lb, n) + ub = np.broadcast_to(ub, n) + + lb = [float(x) if x > -np.inf else None for x in lb] + ub = [float(x) if x < np.inf else None for x in ub] + + return list(zip(lb, ub)) + + +def old_bound_to_new(bounds): + """Convert the old bounds representation to the new one. + + The new representation is a tuple (lb, ub) and the old one is a list + containing n tuples, ith containing lower and upper bound on a ith + variable. + If any of the entries in lb/ub are None they are replaced by + -np.inf/np.inf. + """ + lb, ub = zip(*bounds) + + # Convert occurrences of None to -inf or inf, and replace occurrences of + # any numpy array x with x.item(). Then wrap the results in numpy arrays. + lb = np.array([float(_arr_to_scalar(x)) if x is not None else -np.inf + for x in lb]) + ub = np.array([float(_arr_to_scalar(x)) if x is not None else np.inf + for x in ub]) + + return lb, ub + + +def strict_bounds(lb, ub, keep_feasible, n_vars): + """Remove bounds which are not asked to be kept feasible.""" + strict_lb = np.resize(lb, n_vars).astype(float) + strict_ub = np.resize(ub, n_vars).astype(float) + keep_feasible = np.resize(keep_feasible, n_vars) + strict_lb[~keep_feasible] = -np.inf + strict_ub[~keep_feasible] = np.inf + return strict_lb, strict_ub + + +def new_constraint_to_old(con, x0): + """ + Converts new-style constraint objects to old-style constraint dictionaries. + """ + if isinstance(con, NonlinearConstraint): + if (con.finite_diff_jac_sparsity is not None or + con.finite_diff_rel_step is not None or + not isinstance(con.hess, BFGS) or # misses user specified BFGS + con.keep_feasible): + warn("Constraint options `finite_diff_jac_sparsity`, " + "`finite_diff_rel_step`, `keep_feasible`, and `hess`" + "are ignored by this method.", + OptimizeWarning, stacklevel=3) + + fun = con.fun + if callable(con.jac): + jac = con.jac + else: + jac = None + + else: # LinearConstraint + if np.any(con.keep_feasible): + warn("Constraint option `keep_feasible` is ignored by this method.", + OptimizeWarning, stacklevel=3) + + A = con.A + if issparse(A): + A = A.toarray() + def fun(x): + return np.dot(A, x) + def jac(x): + return A + + # FIXME: when bugs in VectorFunction/LinearVectorFunction are worked out, + # use pcon.fun.fun and pcon.fun.jac. Until then, get fun/jac above. + pcon = PreparedConstraint(con, x0) + lb, ub = pcon.bounds + + i_eq = lb == ub + i_bound_below = np.logical_xor(lb != -np.inf, i_eq) + i_bound_above = np.logical_xor(ub != np.inf, i_eq) + i_unbounded = np.logical_and(lb == -np.inf, ub == np.inf) + + if np.any(i_unbounded): + warn("At least one constraint is unbounded above and below. Such " + "constraints are ignored.", + OptimizeWarning, stacklevel=3) + + ceq = [] + if np.any(i_eq): + def f_eq(x): + y = np.array(fun(x)).flatten() + return y[i_eq] - lb[i_eq] + ceq = [{"type": "eq", "fun": f_eq}] + + if jac is not None: + def j_eq(x): + dy = jac(x) + if issparse(dy): + dy = dy.toarray() + dy = np.atleast_2d(dy) + return dy[i_eq, :] + ceq[0]["jac"] = j_eq + + cineq = [] + n_bound_below = np.sum(i_bound_below) + n_bound_above = np.sum(i_bound_above) + if n_bound_below + n_bound_above: + def f_ineq(x): + y = np.zeros(n_bound_below + n_bound_above) + y_all = np.array(fun(x)).flatten() + y[:n_bound_below] = y_all[i_bound_below] - lb[i_bound_below] + y[n_bound_below:] = -(y_all[i_bound_above] - ub[i_bound_above]) + return y + cineq = [{"type": "ineq", "fun": f_ineq}] + + if jac is not None: + def j_ineq(x): + dy = np.zeros((n_bound_below + n_bound_above, len(x0))) + dy_all = jac(x) + if issparse(dy_all): + dy_all = dy_all.toarray() + dy_all = np.atleast_2d(dy_all) + dy[:n_bound_below, :] = dy_all[i_bound_below] + dy[n_bound_below:, :] = -dy_all[i_bound_above] + return dy + cineq[0]["jac"] = j_ineq + + old_constraints = ceq + cineq + + if len(old_constraints) > 1: + warn("Equality and inequality constraints are specified in the same " + "element of the constraint list. For efficient use with this " + "method, equality and inequality constraints should be specified " + "in separate elements of the constraint list. ", + OptimizeWarning, stacklevel=3) + return old_constraints + + +def old_constraint_to_new(ic, con): + """ + Converts old-style constraint dictionaries to new-style constraint objects. + """ + # check type + try: + ctype = con['type'].lower() + except KeyError as e: + raise KeyError('Constraint %d has no type defined.' % ic) from e + except TypeError as e: + raise TypeError( + 'Constraints must be a sequence of dictionaries.' + ) from e + except AttributeError as e: + raise TypeError("Constraint's type must be a string.") from e + else: + if ctype not in ['eq', 'ineq']: + raise ValueError(f"Unknown constraint type '{con['type']}'.") + if 'fun' not in con: + raise ValueError('Constraint %d has no function defined.' % ic) + + lb = 0 + if ctype == 'eq': + ub = 0 + else: + ub = np.inf + + jac = '2-point' + if 'args' in con: + args = con['args'] + def fun(x): + return con["fun"](x, *args) + if 'jac' in con: + def jac(x): + return con["jac"](x, *args) + else: + fun = con['fun'] + if 'jac' in con: + jac = con['jac'] + + return NonlinearConstraint(fun, lb, ub, jac) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_dcsrch.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_dcsrch.py new file mode 100644 index 0000000000000000000000000000000000000000..f8b4df4763ba4f699869431a0b6528383c2f0328 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_dcsrch.py @@ -0,0 +1,728 @@ +import numpy as np + +""" +# 2023 - ported from minpack2.dcsrch, dcstep (Fortran) to Python +c MINPACK-1 Project. June 1983. +c Argonne National Laboratory. +c Jorge J. More' and David J. Thuente. +c +c MINPACK-2 Project. November 1993. +c Argonne National Laboratory and University of Minnesota. +c Brett M. Averick, Richard G. Carter, and Jorge J. More'. +""" + +# NOTE this file was linted by black on first commit, and can be kept that way. + + +class DCSRCH: + """ + Parameters + ---------- + phi : callable phi(alpha) + Function at point `alpha` + derphi : callable phi'(alpha) + Objective function derivative. Returns a scalar. + ftol : float + A nonnegative tolerance for the sufficient decrease condition. + gtol : float + A nonnegative tolerance for the curvature condition. + xtol : float + A nonnegative relative tolerance for an acceptable step. The + subroutine exits with a warning if the relative difference between + sty and stx is less than xtol. + stpmin : float + A nonnegative lower bound for the step. + stpmax : + A nonnegative upper bound for the step. + + Notes + ----- + + This subroutine finds a step that satisfies a sufficient + decrease condition and a curvature condition. + + Each call of the subroutine updates an interval with + endpoints stx and sty. The interval is initially chosen + so that it contains a minimizer of the modified function + + psi(stp) = f(stp) - f(0) - ftol*stp*f'(0). + + If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the + interval is chosen so that it contains a minimizer of f. + + The algorithm is designed to find a step that satisfies + the sufficient decrease condition + + f(stp) <= f(0) + ftol*stp*f'(0), + + and the curvature condition + + abs(f'(stp)) <= gtol*abs(f'(0)). + + If ftol is less than gtol and if, for example, the function + is bounded below, then there is always a step which satisfies + both conditions. + + If no step can be found that satisfies both conditions, then + the algorithm stops with a warning. In this case stp only + satisfies the sufficient decrease condition. + + A typical invocation of dcsrch has the following outline: + + Evaluate the function at stp = 0.0d0; store in f. + Evaluate the gradient at stp = 0.0d0; store in g. + Choose a starting step stp. + + task = 'START' + 10 continue + call dcsrch(stp,f,g,ftol,gtol,xtol,task,stpmin,stpmax, + isave,dsave) + if (task .eq. 'FG') then + Evaluate the function and the gradient at stp + go to 10 + end if + + NOTE: The user must not alter work arrays between calls. + + The subroutine statement is + + subroutine dcsrch(f,g,stp,ftol,gtol,xtol,stpmin,stpmax, + task,isave,dsave) + where + + stp is a double precision variable. + On entry stp is the current estimate of a satisfactory + step. On initial entry, a positive initial estimate + must be provided. + On exit stp is the current estimate of a satisfactory step + if task = 'FG'. If task = 'CONV' then stp satisfies + the sufficient decrease and curvature condition. + + f is a double precision variable. + On initial entry f is the value of the function at 0. + On subsequent entries f is the value of the + function at stp. + On exit f is the value of the function at stp. + + g is a double precision variable. + On initial entry g is the derivative of the function at 0. + On subsequent entries g is the derivative of the + function at stp. + On exit g is the derivative of the function at stp. + + ftol is a double precision variable. + On entry ftol specifies a nonnegative tolerance for the + sufficient decrease condition. + On exit ftol is unchanged. + + gtol is a double precision variable. + On entry gtol specifies a nonnegative tolerance for the + curvature condition. + On exit gtol is unchanged. + + xtol is a double precision variable. + On entry xtol specifies a nonnegative relative tolerance + for an acceptable step. The subroutine exits with a + warning if the relative difference between sty and stx + is less than xtol. + + On exit xtol is unchanged. + + task is a character variable of length at least 60. + On initial entry task must be set to 'START'. + On exit task indicates the required action: + + If task(1:2) = 'FG' then evaluate the function and + derivative at stp and call dcsrch again. + + If task(1:4) = 'CONV' then the search is successful. + + If task(1:4) = 'WARN' then the subroutine is not able + to satisfy the convergence conditions. The exit value of + stp contains the best point found during the search. + + If task(1:5) = 'ERROR' then there is an error in the + input arguments. + + On exit with convergence, a warning or an error, the + variable task contains additional information. + + stpmin is a double precision variable. + On entry stpmin is a nonnegative lower bound for the step. + On exit stpmin is unchanged. + + stpmax is a double precision variable. + On entry stpmax is a nonnegative upper bound for the step. + On exit stpmax is unchanged. + + isave is an integer work array of dimension 2. + + dsave is a double precision work array of dimension 13. + + Subprograms called + + MINPACK-2 ... dcstep + MINPACK-1 Project. June 1983. + Argonne National Laboratory. + Jorge J. More' and David J. Thuente. + + MINPACK-2 Project. November 1993. + Argonne National Laboratory and University of Minnesota. + Brett M. Averick, Richard G. Carter, and Jorge J. More'. + """ + + def __init__(self, phi, derphi, ftol, gtol, xtol, stpmin, stpmax): + self.stage = None + self.ginit = None + self.gtest = None + self.gx = None + self.gy = None + self.finit = None + self.fx = None + self.fy = None + self.stx = None + self.sty = None + self.stmin = None + self.stmax = None + self.width = None + self.width1 = None + + # leave all assessment of tolerances/limits to the first call of + # this object + self.ftol = ftol + self.gtol = gtol + self.xtol = xtol + self.stpmin = stpmin + self.stpmax = stpmax + + self.phi = phi + self.derphi = derphi + + def __call__(self, alpha1, phi0=None, derphi0=None, maxiter=100): + """ + Parameters + ---------- + alpha1 : float + alpha1 is the current estimate of a satisfactory + step. A positive initial estimate must be provided. + phi0 : float + the value of `phi` at 0 (if known). + derphi0 : float + the derivative of `derphi` at 0 (if known). + maxiter : int + + Returns + ------- + alpha : float + Step size, or None if no suitable step was found. + phi : float + Value of `phi` at the new point `alpha`. + phi0 : float + Value of `phi` at `alpha=0`. + task : bytes + On exit task indicates status information. + + If task[:4] == b'CONV' then the search is successful. + + If task[:4] == b'WARN' then the subroutine is not able + to satisfy the convergence conditions. The exit value of + stp contains the best point found during the search. + + If task[:5] == b'ERROR' then there is an error in the + input arguments. + """ + if phi0 is None: + phi0 = self.phi(0.0) + if derphi0 is None: + derphi0 = self.derphi(0.0) + + phi1 = phi0 + derphi1 = derphi0 + + task = b"START" + for i in range(maxiter): + stp, phi1, derphi1, task = self._iterate( + alpha1, phi1, derphi1, task + ) + + if not np.isfinite(stp): + task = b"WARN" + stp = None + break + + if task[:2] == b"FG": + alpha1 = stp + phi1 = self.phi(stp) + derphi1 = self.derphi(stp) + else: + break + else: + # maxiter reached, the line search did not converge + stp = None + task = b"WARNING: dcsrch did not converge within max iterations" + + if task[:5] == b"ERROR" or task[:4] == b"WARN": + stp = None # failed + + return stp, phi1, phi0, task + + def _iterate(self, stp, f, g, task): + """ + Parameters + ---------- + stp : float + The current estimate of a satisfactory step. On initial entry, a + positive initial estimate must be provided. + f : float + On first call f is the value of the function at 0. On subsequent + entries f should be the value of the function at stp. + g : float + On initial entry g is the derivative of the function at 0. On + subsequent entries g is the derivative of the function at stp. + task : bytes + On initial entry task must be set to 'START'. + + On exit with convergence, a warning or an error, the + variable task contains additional information. + + + Returns + ------- + stp, f, g, task: tuple + + stp : float + the current estimate of a satisfactory step if task = 'FG'. If + task = 'CONV' then stp satisfies the sufficient decrease and + curvature condition. + f : float + the value of the function at stp. + g : float + the derivative of the function at stp. + task : bytes + On exit task indicates the required action: + + If task(1:2) == b'FG' then evaluate the function and + derivative at stp and call dcsrch again. + + If task(1:4) == b'CONV' then the search is successful. + + If task(1:4) == b'WARN' then the subroutine is not able + to satisfy the convergence conditions. The exit value of + stp contains the best point found during the search. + + If task(1:5) == b'ERROR' then there is an error in the + input arguments. + """ + p5 = 0.5 + p66 = 0.66 + xtrapl = 1.1 + xtrapu = 4.0 + + if task[:5] == b"START": + if stp < self.stpmin: + task = b"ERROR: STP .LT. STPMIN" + if stp > self.stpmax: + task = b"ERROR: STP .GT. STPMAX" + if g >= 0: + task = b"ERROR: INITIAL G .GE. ZERO" + if self.ftol < 0: + task = b"ERROR: FTOL .LT. ZERO" + if self.gtol < 0: + task = b"ERROR: GTOL .LT. ZERO" + if self.xtol < 0: + task = b"ERROR: XTOL .LT. ZERO" + if self.stpmin < 0: + task = b"ERROR: STPMIN .LT. ZERO" + if self.stpmax < self.stpmin: + task = b"ERROR: STPMAX .LT. STPMIN" + + if task[:5] == b"ERROR": + return stp, f, g, task + + # Initialize local variables. + + self.brackt = False + self.stage = 1 + self.finit = f + self.ginit = g + self.gtest = self.ftol * self.ginit + self.width = self.stpmax - self.stpmin + self.width1 = self.width / p5 + + # The variables stx, fx, gx contain the values of the step, + # function, and derivative at the best step. + # The variables sty, fy, gy contain the value of the step, + # function, and derivative at sty. + # The variables stp, f, g contain the values of the step, + # function, and derivative at stp. + + self.stx = 0.0 + self.fx = self.finit + self.gx = self.ginit + self.sty = 0.0 + self.fy = self.finit + self.gy = self.ginit + self.stmin = 0 + self.stmax = stp + xtrapu * stp + task = b"FG" + return stp, f, g, task + + # in the original Fortran this was a location to restore variables + # we don't need to do that because they're attributes. + + # If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the + # algorithm enters the second stage. + ftest = self.finit + stp * self.gtest + + if self.stage == 1 and f <= ftest and g >= 0: + self.stage = 2 + + # test for warnings + if self.brackt and (stp <= self.stmin or stp >= self.stmax): + task = b"WARNING: ROUNDING ERRORS PREVENT PROGRESS" + if self.brackt and self.stmax - self.stmin <= self.xtol * self.stmax: + task = b"WARNING: XTOL TEST SATISFIED" + if stp == self.stpmax and f <= ftest and g <= self.gtest: + task = b"WARNING: STP = STPMAX" + if stp == self.stpmin and (f > ftest or g >= self.gtest): + task = b"WARNING: STP = STPMIN" + + # test for convergence + if f <= ftest and abs(g) <= self.gtol * -self.ginit: + task = b"CONVERGENCE" + + # test for termination + if task[:4] == b"WARN" or task[:4] == b"CONV": + return stp, f, g, task + + # A modified function is used to predict the step during the + # first stage if a lower function value has been obtained but + # the decrease is not sufficient. + if self.stage == 1 and f <= self.fx and f > ftest: + # Define the modified function and derivative values. + fm = f - stp * self.gtest + fxm = self.fx - self.stx * self.gtest + fym = self.fy - self.sty * self.gtest + gm = g - self.gtest + gxm = self.gx - self.gtest + gym = self.gy - self.gtest + + # Call dcstep to update stx, sty, and to compute the new step. + # dcstep can have several operations which can produce NaN + # e.g. inf/inf. Filter these out. + with np.errstate(invalid="ignore", over="ignore"): + tup = dcstep( + self.stx, + fxm, + gxm, + self.sty, + fym, + gym, + stp, + fm, + gm, + self.brackt, + self.stmin, + self.stmax, + ) + self.stx, fxm, gxm, self.sty, fym, gym, stp, self.brackt = tup + + # Reset the function and derivative values for f + self.fx = fxm + self.stx * self.gtest + self.fy = fym + self.sty * self.gtest + self.gx = gxm + self.gtest + self.gy = gym + self.gtest + + else: + # Call dcstep to update stx, sty, and to compute the new step. + # dcstep can have several operations which can produce NaN + # e.g. inf/inf. Filter these out. + + with np.errstate(invalid="ignore", over="ignore"): + tup = dcstep( + self.stx, + self.fx, + self.gx, + self.sty, + self.fy, + self.gy, + stp, + f, + g, + self.brackt, + self.stmin, + self.stmax, + ) + ( + self.stx, + self.fx, + self.gx, + self.sty, + self.fy, + self.gy, + stp, + self.brackt, + ) = tup + + # Decide if a bisection step is needed + if self.brackt: + if abs(self.sty - self.stx) >= p66 * self.width1: + stp = self.stx + p5 * (self.sty - self.stx) + self.width1 = self.width + self.width = abs(self.sty - self.stx) + + # Set the minimum and maximum steps allowed for stp. + if self.brackt: + self.stmin = min(self.stx, self.sty) + self.stmax = max(self.stx, self.sty) + else: + self.stmin = stp + xtrapl * (stp - self.stx) + self.stmax = stp + xtrapu * (stp - self.stx) + + # Force the step to be within the bounds stpmax and stpmin. + stp = np.clip(stp, self.stpmin, self.stpmax) + + # If further progress is not possible, let stp be the best + # point obtained during the search. + if ( + self.brackt + and (stp <= self.stmin or stp >= self.stmax) + or ( + self.brackt + and self.stmax - self.stmin <= self.xtol * self.stmax + ) + ): + stp = self.stx + + # Obtain another function and derivative + task = b"FG" + return stp, f, g, task + + +def dcstep(stx, fx, dx, sty, fy, dy, stp, fp, dp, brackt, stpmin, stpmax): + """ + Subroutine dcstep + + This subroutine computes a safeguarded step for a search + procedure and updates an interval that contains a step that + satisfies a sufficient decrease and a curvature condition. + + The parameter stx contains the step with the least function + value. If brackt is set to .true. then a minimizer has + been bracketed in an interval with endpoints stx and sty. + The parameter stp contains the current step. + The subroutine assumes that if brackt is set to .true. then + + min(stx,sty) < stp < max(stx,sty), + + and that the derivative at stx is negative in the direction + of the step. + + The subroutine statement is + + subroutine dcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,brackt, + stpmin,stpmax) + + where + + stx is a double precision variable. + On entry stx is the best step obtained so far and is an + endpoint of the interval that contains the minimizer. + On exit stx is the updated best step. + + fx is a double precision variable. + On entry fx is the function at stx. + On exit fx is the function at stx. + + dx is a double precision variable. + On entry dx is the derivative of the function at + stx. The derivative must be negative in the direction of + the step, that is, dx and stp - stx must have opposite + signs. + On exit dx is the derivative of the function at stx. + + sty is a double precision variable. + On entry sty is the second endpoint of the interval that + contains the minimizer. + On exit sty is the updated endpoint of the interval that + contains the minimizer. + + fy is a double precision variable. + On entry fy is the function at sty. + On exit fy is the function at sty. + + dy is a double precision variable. + On entry dy is the derivative of the function at sty. + On exit dy is the derivative of the function at the exit sty. + + stp is a double precision variable. + On entry stp is the current step. If brackt is set to .true. + then on input stp must be between stx and sty. + On exit stp is a new trial step. + + fp is a double precision variable. + On entry fp is the function at stp + On exit fp is unchanged. + + dp is a double precision variable. + On entry dp is the derivative of the function at stp. + On exit dp is unchanged. + + brackt is an logical variable. + On entry brackt specifies if a minimizer has been bracketed. + Initially brackt must be set to .false. + On exit brackt specifies if a minimizer has been bracketed. + When a minimizer is bracketed brackt is set to .true. + + stpmin is a double precision variable. + On entry stpmin is a lower bound for the step. + On exit stpmin is unchanged. + + stpmax is a double precision variable. + On entry stpmax is an upper bound for the step. + On exit stpmax is unchanged. + + MINPACK-1 Project. June 1983 + Argonne National Laboratory. + Jorge J. More' and David J. Thuente. + + MINPACK-2 Project. November 1993. + Argonne National Laboratory and University of Minnesota. + Brett M. Averick and Jorge J. More'. + + """ + sgn_dp = np.sign(dp) + sgn_dx = np.sign(dx) + + # sgnd = dp * (dx / abs(dx)) + sgnd = sgn_dp * sgn_dx + + # First case: A higher function value. The minimum is bracketed. + # If the cubic step is closer to stx than the quadratic step, the + # cubic step is taken, otherwise the average of the cubic and + # quadratic steps is taken. + if fp > fx: + theta = 3.0 * (fx - fp) / (stp - stx) + dx + dp + s = max(abs(theta), abs(dx), abs(dp)) + gamma = s * np.sqrt((theta / s) ** 2 - (dx / s) * (dp / s)) + if stp < stx: + gamma *= -1 + p = (gamma - dx) + theta + q = ((gamma - dx) + gamma) + dp + r = p / q + stpc = stx + r * (stp - stx) + stpq = stx + ((dx / ((fx - fp) / (stp - stx) + dx)) / 2.0) * (stp - stx) + if abs(stpc - stx) <= abs(stpq - stx): + stpf = stpc + else: + stpf = stpc + (stpq - stpc) / 2.0 + brackt = True + elif sgnd < 0.0: + # Second case: A lower function value and derivatives of opposite + # sign. The minimum is bracketed. If the cubic step is farther from + # stp than the secant step, the cubic step is taken, otherwise the + # secant step is taken. + theta = 3 * (fx - fp) / (stp - stx) + dx + dp + s = max(abs(theta), abs(dx), abs(dp)) + gamma = s * np.sqrt((theta / s) ** 2 - (dx / s) * (dp / s)) + if stp > stx: + gamma *= -1 + p = (gamma - dp) + theta + q = ((gamma - dp) + gamma) + dx + r = p / q + stpc = stp + r * (stx - stp) + stpq = stp + (dp / (dp - dx)) * (stx - stp) + if abs(stpc - stp) > abs(stpq - stp): + stpf = stpc + else: + stpf = stpq + brackt = True + elif abs(dp) < abs(dx): + # Third case: A lower function value, derivatives of the same sign, + # and the magnitude of the derivative decreases. + + # The cubic step is computed only if the cubic tends to infinity + # in the direction of the step or if the minimum of the cubic + # is beyond stp. Otherwise the cubic step is defined to be the + # secant step. + theta = 3 * (fx - fp) / (stp - stx) + dx + dp + s = max(abs(theta), abs(dx), abs(dp)) + + # The case gamma = 0 only arises if the cubic does not tend + # to infinity in the direction of the step. + gamma = s * np.sqrt(max(0, (theta / s) ** 2 - (dx / s) * (dp / s))) + if stp > stx: + gamma = -gamma + p = (gamma - dp) + theta + q = (gamma + (dx - dp)) + gamma + r = p / q + if r < 0 and gamma != 0: + stpc = stp + r * (stx - stp) + elif stp > stx: + stpc = stpmax + else: + stpc = stpmin + stpq = stp + (dp / (dp - dx)) * (stx - stp) + + if brackt: + # A minimizer has been bracketed. If the cubic step is + # closer to stp than the secant step, the cubic step is + # taken, otherwise the secant step is taken. + if abs(stpc - stp) < abs(stpq - stp): + stpf = stpc + else: + stpf = stpq + + if stp > stx: + stpf = min(stp + 0.66 * (sty - stp), stpf) + else: + stpf = max(stp + 0.66 * (sty - stp), stpf) + else: + # A minimizer has not been bracketed. If the cubic step is + # farther from stp than the secant step, the cubic step is + # taken, otherwise the secant step is taken. + if abs(stpc - stp) > abs(stpq - stp): + stpf = stpc + else: + stpf = stpq + stpf = np.clip(stpf, stpmin, stpmax) + + else: + # Fourth case: A lower function value, derivatives of the same sign, + # and the magnitude of the derivative does not decrease. If the + # minimum is not bracketed, the step is either stpmin or stpmax, + # otherwise the cubic step is taken. + if brackt: + theta = 3.0 * (fp - fy) / (sty - stp) + dy + dp + s = max(abs(theta), abs(dy), abs(dp)) + gamma = s * np.sqrt((theta / s) ** 2 - (dy / s) * (dp / s)) + if stp > sty: + gamma = -gamma + p = (gamma - dp) + theta + q = ((gamma - dp) + gamma) + dy + r = p / q + stpc = stp + r * (sty - stp) + stpf = stpc + elif stp > stx: + stpf = stpmax + else: + stpf = stpmin + + # Update the interval which contains a minimizer. + if fp > fx: + sty = stp + fy = fp + dy = dp + else: + if sgnd < 0: + sty = stx + fy = fx + dy = dx + stx = stp + fx = fp + dx = dp + + # Compute the new step. + stp = stpf + + return stx, fx, dx, sty, fy, dy, stp, brackt diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_differentiable_functions.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_differentiable_functions.py new file mode 100644 index 0000000000000000000000000000000000000000..afbb2152c21a7837e77a6a77b3d8f1f6b0114270 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_differentiable_functions.py @@ -0,0 +1,694 @@ +import numpy as np +import scipy.sparse as sps +from ._numdiff import approx_derivative, group_columns +from ._hessian_update_strategy import HessianUpdateStrategy +from scipy.sparse.linalg import LinearOperator +from scipy._lib._array_api import array_namespace +from scipy._lib import array_api_extra as xpx + + +FD_METHODS = ('2-point', '3-point', 'cs') + + +def _wrapper_fun(fun, args=()): + ncalls = [0] + + def wrapped(x): + ncalls[0] += 1 + # Send a copy because the user may overwrite it. + # Overwriting results in undefined behaviour because + # fun(self.x) will change self.x, with the two no longer linked. + fx = fun(np.copy(x), *args) + # Make sure the function returns a true scalar + if not np.isscalar(fx): + try: + fx = np.asarray(fx).item() + except (TypeError, ValueError) as e: + raise ValueError( + "The user-provided objective function " + "must return a scalar value." + ) from e + return fx + return wrapped, ncalls + + +def _wrapper_grad(grad, fun=None, args=(), finite_diff_options=None): + ncalls = [0] + + if callable(grad): + def wrapped(x, **kwds): + # kwds present to give function same signature as numdiff variant + ncalls[0] += 1 + return np.atleast_1d(grad(np.copy(x), *args)) + return wrapped, ncalls + + elif grad in FD_METHODS: + def wrapped1(x, f0=None): + ncalls[0] += 1 + return approx_derivative( + fun, x, f0=f0, **finite_diff_options + ) + + return wrapped1, ncalls + + +def _wrapper_hess(hess, grad=None, x0=None, args=(), finite_diff_options=None): + if callable(hess): + H = hess(np.copy(x0), *args) + ncalls = [1] + + if sps.issparse(H): + def wrapped(x, **kwds): + ncalls[0] += 1 + return sps.csr_matrix(hess(np.copy(x), *args)) + + H = sps.csr_matrix(H) + + elif isinstance(H, LinearOperator): + def wrapped(x, **kwds): + ncalls[0] += 1 + return hess(np.copy(x), *args) + + else: # dense + def wrapped(x, **kwds): + ncalls[0] += 1 + return np.atleast_2d(np.asarray(hess(np.copy(x), *args))) + + H = np.atleast_2d(np.asarray(H)) + + return wrapped, ncalls, H + elif hess in FD_METHODS: + ncalls = [0] + + def wrapped1(x, f0=None): + return approx_derivative( + grad, x, f0=f0, **finite_diff_options + ) + + return wrapped1, ncalls, None + + +class ScalarFunction: + """Scalar function and its derivatives. + + This class defines a scalar function F: R^n->R and methods for + computing or approximating its first and second derivatives. + + Parameters + ---------- + fun : callable + evaluates the scalar function. Must be of the form ``fun(x, *args)``, + where ``x`` is the argument in the form of a 1-D array and ``args`` is + a tuple of any additional fixed parameters needed to completely specify + the function. Should return a scalar. + x0 : array-like + Provides an initial set of variables for evaluating fun. Array of real + elements of size (n,), where 'n' is the number of independent + variables. + args : tuple, optional + Any additional fixed parameters needed to completely specify the scalar + function. + grad : {callable, '2-point', '3-point', 'cs'} + Method for computing the gradient vector. + If it is a callable, it should be a function that returns the gradient + vector: + + ``grad(x, *args) -> array_like, shape (n,)`` + + where ``x`` is an array with shape (n,) and ``args`` is a tuple with + the fixed parameters. + Alternatively, the keywords {'2-point', '3-point', 'cs'} can be used + to select a finite difference scheme for numerical estimation of the + gradient with a relative step size. These finite difference schemes + obey any specified `bounds`. + hess : {callable, '2-point', '3-point', 'cs', HessianUpdateStrategy} + Method for computing the Hessian matrix. If it is callable, it should + return the Hessian matrix: + + ``hess(x, *args) -> {LinearOperator, spmatrix, array}, (n, n)`` + + where x is a (n,) ndarray and `args` is a tuple with the fixed + parameters. Alternatively, the keywords {'2-point', '3-point', 'cs'} + select a finite difference scheme for numerical estimation. Or, objects + implementing `HessianUpdateStrategy` interface can be used to + approximate the Hessian. + Whenever the gradient is estimated via finite-differences, the Hessian + cannot be estimated with options {'2-point', '3-point', 'cs'} and needs + to be estimated using one of the quasi-Newton strategies. + finite_diff_rel_step : None or array_like + Relative step size to use. The absolute step size is computed as + ``h = finite_diff_rel_step * sign(x0) * max(1, abs(x0))``, possibly + adjusted to fit into the bounds. For ``method='3-point'`` the sign + of `h` is ignored. If None then finite_diff_rel_step is selected + automatically, + finite_diff_bounds : tuple of array_like + Lower and upper bounds on independent variables. Defaults to no bounds, + (-np.inf, np.inf). Each bound must match the size of `x0` or be a + scalar, in the latter case the bound will be the same for all + variables. Use it to limit the range of function evaluation. + epsilon : None or array_like, optional + Absolute step size to use, possibly adjusted to fit into the bounds. + For ``method='3-point'`` the sign of `epsilon` is ignored. By default + relative steps are used, only if ``epsilon is not None`` are absolute + steps used. + + Notes + ----- + This class implements a memoization logic. There are methods `fun`, + `grad`, hess` and corresponding attributes `f`, `g` and `H`. The following + things should be considered: + + 1. Use only public methods `fun`, `grad` and `hess`. + 2. After one of the methods is called, the corresponding attribute + will be set. However, a subsequent call with a different argument + of *any* of the methods may overwrite the attribute. + """ + def __init__(self, fun, x0, args, grad, hess, finite_diff_rel_step, + finite_diff_bounds, epsilon=None): + if not callable(grad) and grad not in FD_METHODS: + raise ValueError( + f"`grad` must be either callable or one of {FD_METHODS}." + ) + + if not (callable(hess) or hess in FD_METHODS + or isinstance(hess, HessianUpdateStrategy)): + raise ValueError( + f"`hess` must be either callable, HessianUpdateStrategy" + f" or one of {FD_METHODS}." + ) + + if grad in FD_METHODS and hess in FD_METHODS: + raise ValueError("Whenever the gradient is estimated via " + "finite-differences, we require the Hessian " + "to be estimated using one of the " + "quasi-Newton strategies.") + + self.xp = xp = array_namespace(x0) + _x = xpx.atleast_nd(xp.asarray(x0), ndim=1, xp=xp) + _dtype = xp.float64 + if xp.isdtype(_x.dtype, "real floating"): + _dtype = _x.dtype + + # original arguments + self._wrapped_fun, self._nfev = _wrapper_fun(fun, args=args) + self._orig_fun = fun + self._orig_grad = grad + self._orig_hess = hess + self._args = args + + # promotes to floating + self.x = xp.astype(_x, _dtype) + self.x_dtype = _dtype + self.n = self.x.size + self.f_updated = False + self.g_updated = False + self.H_updated = False + + self._lowest_x = None + self._lowest_f = np.inf + + finite_diff_options = {} + if grad in FD_METHODS: + finite_diff_options["method"] = grad + finite_diff_options["rel_step"] = finite_diff_rel_step + finite_diff_options["abs_step"] = epsilon + finite_diff_options["bounds"] = finite_diff_bounds + if hess in FD_METHODS: + finite_diff_options["method"] = hess + finite_diff_options["rel_step"] = finite_diff_rel_step + finite_diff_options["abs_step"] = epsilon + finite_diff_options["as_linear_operator"] = True + + # Initial function evaluation + self._update_fun() + + # Initial gradient evaluation + self._wrapped_grad, self._ngev = _wrapper_grad( + grad, + fun=self._wrapped_fun, + args=args, + finite_diff_options=finite_diff_options + ) + self._update_grad() + + # Hessian evaluation + if callable(hess): + self._wrapped_hess, self._nhev, self.H = _wrapper_hess( + hess, x0=x0, args=args + ) + self.H_updated = True + elif hess in FD_METHODS: + self._wrapped_hess, self._nhev, self.H = _wrapper_hess( + hess, + grad=self._wrapped_grad, + x0=x0, + finite_diff_options=finite_diff_options + ) + self._update_grad() + self.H = self._wrapped_hess(self.x, f0=self.g) + self.H_updated = True + elif isinstance(hess, HessianUpdateStrategy): + self.H = hess + self.H.initialize(self.n, 'hess') + self.H_updated = True + self.x_prev = None + self.g_prev = None + self._nhev = [0] + + @property + def nfev(self): + return self._nfev[0] + + @property + def ngev(self): + return self._ngev[0] + + @property + def nhev(self): + return self._nhev[0] + + def _update_x(self, x): + if isinstance(self._orig_hess, HessianUpdateStrategy): + self._update_grad() + self.x_prev = self.x + self.g_prev = self.g + # ensure that self.x is a copy of x. Don't store a reference + # otherwise the memoization doesn't work properly. + + _x = xpx.atleast_nd(self.xp.asarray(x), ndim=1, xp=self.xp) + self.x = self.xp.astype(_x, self.x_dtype) + self.f_updated = False + self.g_updated = False + self.H_updated = False + self._update_hess() + else: + # ensure that self.x is a copy of x. Don't store a reference + # otherwise the memoization doesn't work properly. + _x = xpx.atleast_nd(self.xp.asarray(x), ndim=1, xp=self.xp) + self.x = self.xp.astype(_x, self.x_dtype) + self.f_updated = False + self.g_updated = False + self.H_updated = False + + def _update_fun(self): + if not self.f_updated: + fx = self._wrapped_fun(self.x) + if fx < self._lowest_f: + self._lowest_x = self.x + self._lowest_f = fx + + self.f = fx + self.f_updated = True + + def _update_grad(self): + if not self.g_updated: + if self._orig_grad in FD_METHODS: + self._update_fun() + self.g = self._wrapped_grad(self.x, f0=self.f) + self.g_updated = True + + def _update_hess(self): + if not self.H_updated: + if self._orig_hess in FD_METHODS: + self._update_grad() + self.H = self._wrapped_hess(self.x, f0=self.g) + elif isinstance(self._orig_hess, HessianUpdateStrategy): + self._update_grad() + self.H.update(self.x - self.x_prev, self.g - self.g_prev) + else: # should be callable(hess) + self.H = self._wrapped_hess(self.x) + + self.H_updated = True + + def fun(self, x): + if not np.array_equal(x, self.x): + self._update_x(x) + self._update_fun() + return self.f + + def grad(self, x): + if not np.array_equal(x, self.x): + self._update_x(x) + self._update_grad() + return self.g + + def hess(self, x): + if not np.array_equal(x, self.x): + self._update_x(x) + self._update_hess() + return self.H + + def fun_and_grad(self, x): + if not np.array_equal(x, self.x): + self._update_x(x) + self._update_fun() + self._update_grad() + return self.f, self.g + + +class VectorFunction: + """Vector function and its derivatives. + + This class defines a vector function F: R^n->R^m and methods for + computing or approximating its first and second derivatives. + + Notes + ----- + This class implements a memoization logic. There are methods `fun`, + `jac`, hess` and corresponding attributes `f`, `J` and `H`. The following + things should be considered: + + 1. Use only public methods `fun`, `jac` and `hess`. + 2. After one of the methods is called, the corresponding attribute + will be set. However, a subsequent call with a different argument + of *any* of the methods may overwrite the attribute. + """ + def __init__(self, fun, x0, jac, hess, + finite_diff_rel_step, finite_diff_jac_sparsity, + finite_diff_bounds, sparse_jacobian): + if not callable(jac) and jac not in FD_METHODS: + raise ValueError(f"`jac` must be either callable or one of {FD_METHODS}.") + + if not (callable(hess) or hess in FD_METHODS + or isinstance(hess, HessianUpdateStrategy)): + raise ValueError("`hess` must be either callable," + f"HessianUpdateStrategy or one of {FD_METHODS}.") + + if jac in FD_METHODS and hess in FD_METHODS: + raise ValueError("Whenever the Jacobian is estimated via " + "finite-differences, we require the Hessian to " + "be estimated using one of the quasi-Newton " + "strategies.") + + self.xp = xp = array_namespace(x0) + _x = xpx.atleast_nd(xp.asarray(x0), ndim=1, xp=xp) + _dtype = xp.float64 + if xp.isdtype(_x.dtype, "real floating"): + _dtype = _x.dtype + + # promotes to floating + self.x = xp.astype(_x, _dtype) + self.x_dtype = _dtype + + self.n = self.x.size + self.nfev = 0 + self.njev = 0 + self.nhev = 0 + self.f_updated = False + self.J_updated = False + self.H_updated = False + + finite_diff_options = {} + if jac in FD_METHODS: + finite_diff_options["method"] = jac + finite_diff_options["rel_step"] = finite_diff_rel_step + if finite_diff_jac_sparsity is not None: + sparsity_groups = group_columns(finite_diff_jac_sparsity) + finite_diff_options["sparsity"] = (finite_diff_jac_sparsity, + sparsity_groups) + finite_diff_options["bounds"] = finite_diff_bounds + self.x_diff = np.copy(self.x) + if hess in FD_METHODS: + finite_diff_options["method"] = hess + finite_diff_options["rel_step"] = finite_diff_rel_step + finite_diff_options["as_linear_operator"] = True + self.x_diff = np.copy(self.x) + if jac in FD_METHODS and hess in FD_METHODS: + raise ValueError("Whenever the Jacobian is estimated via " + "finite-differences, we require the Hessian to " + "be estimated using one of the quasi-Newton " + "strategies.") + + # Function evaluation + def fun_wrapped(x): + self.nfev += 1 + return np.atleast_1d(fun(x)) + + def update_fun(): + self.f = fun_wrapped(self.x) + + self._update_fun_impl = update_fun + update_fun() + + self.v = np.zeros_like(self.f) + self.m = self.v.size + + # Jacobian Evaluation + if callable(jac): + self.J = jac(self.x) + self.J_updated = True + self.njev += 1 + + if (sparse_jacobian or + sparse_jacobian is None and sps.issparse(self.J)): + def jac_wrapped(x): + self.njev += 1 + return sps.csr_matrix(jac(x)) + self.J = sps.csr_matrix(self.J) + self.sparse_jacobian = True + + elif sps.issparse(self.J): + def jac_wrapped(x): + self.njev += 1 + return jac(x).toarray() + self.J = self.J.toarray() + self.sparse_jacobian = False + + else: + def jac_wrapped(x): + self.njev += 1 + return np.atleast_2d(jac(x)) + self.J = np.atleast_2d(self.J) + self.sparse_jacobian = False + + def update_jac(): + self.J = jac_wrapped(self.x) + + elif jac in FD_METHODS: + self.J = approx_derivative(fun_wrapped, self.x, f0=self.f, + **finite_diff_options) + self.J_updated = True + + if (sparse_jacobian or + sparse_jacobian is None and sps.issparse(self.J)): + def update_jac(): + self._update_fun() + self.J = sps.csr_matrix( + approx_derivative(fun_wrapped, self.x, f0=self.f, + **finite_diff_options)) + self.J = sps.csr_matrix(self.J) + self.sparse_jacobian = True + + elif sps.issparse(self.J): + def update_jac(): + self._update_fun() + self.J = approx_derivative(fun_wrapped, self.x, f0=self.f, + **finite_diff_options).toarray() + self.J = self.J.toarray() + self.sparse_jacobian = False + + else: + def update_jac(): + self._update_fun() + self.J = np.atleast_2d( + approx_derivative(fun_wrapped, self.x, f0=self.f, + **finite_diff_options)) + self.J = np.atleast_2d(self.J) + self.sparse_jacobian = False + + self._update_jac_impl = update_jac + + # Define Hessian + if callable(hess): + self.H = hess(self.x, self.v) + self.H_updated = True + self.nhev += 1 + + if sps.issparse(self.H): + def hess_wrapped(x, v): + self.nhev += 1 + return sps.csr_matrix(hess(x, v)) + self.H = sps.csr_matrix(self.H) + + elif isinstance(self.H, LinearOperator): + def hess_wrapped(x, v): + self.nhev += 1 + return hess(x, v) + + else: + def hess_wrapped(x, v): + self.nhev += 1 + return np.atleast_2d(np.asarray(hess(x, v))) + self.H = np.atleast_2d(np.asarray(self.H)) + + def update_hess(): + self.H = hess_wrapped(self.x, self.v) + elif hess in FD_METHODS: + def jac_dot_v(x, v): + return jac_wrapped(x).T.dot(v) + + def update_hess(): + self._update_jac() + self.H = approx_derivative(jac_dot_v, self.x, + f0=self.J.T.dot(self.v), + args=(self.v,), + **finite_diff_options) + update_hess() + self.H_updated = True + elif isinstance(hess, HessianUpdateStrategy): + self.H = hess + self.H.initialize(self.n, 'hess') + self.H_updated = True + self.x_prev = None + self.J_prev = None + + def update_hess(): + self._update_jac() + # When v is updated before x was updated, then x_prev and + # J_prev are None and we need this check. + if self.x_prev is not None and self.J_prev is not None: + delta_x = self.x - self.x_prev + delta_g = self.J.T.dot(self.v) - self.J_prev.T.dot(self.v) + self.H.update(delta_x, delta_g) + + self._update_hess_impl = update_hess + + if isinstance(hess, HessianUpdateStrategy): + def update_x(x): + self._update_jac() + self.x_prev = self.x + self.J_prev = self.J + _x = xpx.atleast_nd(self.xp.asarray(x), ndim=1, xp=self.xp) + self.x = self.xp.astype(_x, self.x_dtype) + self.f_updated = False + self.J_updated = False + self.H_updated = False + self._update_hess() + else: + def update_x(x): + _x = xpx.atleast_nd(self.xp.asarray(x), ndim=1, xp=self.xp) + self.x = self.xp.astype(_x, self.x_dtype) + self.f_updated = False + self.J_updated = False + self.H_updated = False + + self._update_x_impl = update_x + + def _update_v(self, v): + if not np.array_equal(v, self.v): + self.v = v + self.H_updated = False + + def _update_x(self, x): + if not np.array_equal(x, self.x): + self._update_x_impl(x) + + def _update_fun(self): + if not self.f_updated: + self._update_fun_impl() + self.f_updated = True + + def _update_jac(self): + if not self.J_updated: + self._update_jac_impl() + self.J_updated = True + + def _update_hess(self): + if not self.H_updated: + self._update_hess_impl() + self.H_updated = True + + def fun(self, x): + self._update_x(x) + self._update_fun() + return self.f + + def jac(self, x): + self._update_x(x) + self._update_jac() + return self.J + + def hess(self, x, v): + # v should be updated before x. + self._update_v(v) + self._update_x(x) + self._update_hess() + return self.H + + +class LinearVectorFunction: + """Linear vector function and its derivatives. + + Defines a linear function F = A x, where x is N-D vector and + A is m-by-n matrix. The Jacobian is constant and equals to A. The Hessian + is identically zero and it is returned as a csr matrix. + """ + def __init__(self, A, x0, sparse_jacobian): + if sparse_jacobian or sparse_jacobian is None and sps.issparse(A): + self.J = sps.csr_matrix(A) + self.sparse_jacobian = True + elif sps.issparse(A): + self.J = A.toarray() + self.sparse_jacobian = False + else: + # np.asarray makes sure A is ndarray and not matrix + self.J = np.atleast_2d(np.asarray(A)) + self.sparse_jacobian = False + + self.m, self.n = self.J.shape + + self.xp = xp = array_namespace(x0) + _x = xpx.atleast_nd(xp.asarray(x0), ndim=1, xp=xp) + _dtype = xp.float64 + if xp.isdtype(_x.dtype, "real floating"): + _dtype = _x.dtype + + # promotes to floating + self.x = xp.astype(_x, _dtype) + self.x_dtype = _dtype + + self.f = self.J.dot(self.x) + self.f_updated = True + + self.v = np.zeros(self.m, dtype=float) + self.H = sps.csr_matrix((self.n, self.n)) + + def _update_x(self, x): + if not np.array_equal(x, self.x): + _x = xpx.atleast_nd(self.xp.asarray(x), ndim=1, xp=self.xp) + self.x = self.xp.astype(_x, self.x_dtype) + self.f_updated = False + + def fun(self, x): + self._update_x(x) + if not self.f_updated: + self.f = self.J.dot(x) + self.f_updated = True + return self.f + + def jac(self, x): + self._update_x(x) + return self.J + + def hess(self, x, v): + self._update_x(x) + self.v = v + return self.H + + +class IdentityVectorFunction(LinearVectorFunction): + """Identity vector function and its derivatives. + + The Jacobian is the identity matrix, returned as a dense array when + `sparse_jacobian=False` and as a csr matrix otherwise. The Hessian is + identically zero and it is returned as a csr matrix. + """ + def __init__(self, x0, sparse_jacobian): + n = len(x0) + if sparse_jacobian or sparse_jacobian is None: + A = sps.eye(n, format='csr') + sparse_jacobian = True + else: + A = np.eye(n) + sparse_jacobian = False + super().__init__(A, x0, sparse_jacobian) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_differentialevolution.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_differentialevolution.py new file mode 100644 index 0000000000000000000000000000000000000000..70097b7aea61d3eec9f9dabadd4820e7576b44ce --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_differentialevolution.py @@ -0,0 +1,1969 @@ +""" +differential_evolution: The differential evolution global optimization algorithm +Added by Andrew Nelson 2014 +""" +import warnings + +import numpy as np +from scipy.optimize import OptimizeResult, minimize +from scipy.optimize._optimize import _status_message, _wrap_callback +from scipy._lib._util import (check_random_state, MapWrapper, _FunctionWrapper, + rng_integers, _transition_to_rng) + +from scipy.optimize._constraints import (Bounds, new_bounds_to_old, + NonlinearConstraint, LinearConstraint) +from scipy.sparse import issparse + +__all__ = ['differential_evolution'] + + +_MACHEPS = np.finfo(np.float64).eps + + +@_transition_to_rng("seed", position_num=9) +def differential_evolution(func, bounds, args=(), strategy='best1bin', + maxiter=1000, popsize=15, tol=0.01, + mutation=(0.5, 1), recombination=0.7, rng=None, + callback=None, disp=False, polish=True, + init='latinhypercube', atol=0, updating='immediate', + workers=1, constraints=(), x0=None, *, + integrality=None, vectorized=False): + r"""Finds the global minimum of a multivariate function. + + The differential evolution method [1]_ is stochastic in nature. It does + not use gradient methods to find the minimum, and can search large areas + of candidate space, but often requires larger numbers of function + evaluations than conventional gradient-based techniques. + + The algorithm is due to Storn and Price [2]_. + + Parameters + ---------- + func : callable + The objective function to be minimized. Must be in the form + ``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array + and ``args`` is a tuple of any additional fixed parameters needed to + completely specify the function. The number of parameters, N, is equal + to ``len(x)``. + bounds : sequence or `Bounds` + Bounds for variables. There are two ways to specify the bounds: + + 1. Instance of `Bounds` class. + 2. ``(min, max)`` pairs for each element in ``x``, defining the + finite lower and upper bounds for the optimizing argument of + `func`. + + The total number of bounds is used to determine the number of + parameters, N. If there are parameters whose bounds are equal the total + number of free parameters is ``N - N_equal``. + + args : tuple, optional + Any additional fixed parameters needed to + completely specify the objective function. + strategy : {str, callable}, optional + The differential evolution strategy to use. Should be one of: + + - 'best1bin' + - 'best1exp' + - 'rand1bin' + - 'rand1exp' + - 'rand2bin' + - 'rand2exp' + - 'randtobest1bin' + - 'randtobest1exp' + - 'currenttobest1bin' + - 'currenttobest1exp' + - 'best2exp' + - 'best2bin' + + The default is 'best1bin'. Strategies that may be implemented are + outlined in 'Notes'. + Alternatively the differential evolution strategy can be customized by + providing a callable that constructs a trial vector. The callable must + have the form ``strategy(candidate: int, population: np.ndarray, rng=None)``, + where ``candidate`` is an integer specifying which entry of the + population is being evolved, ``population`` is an array of shape + ``(S, N)`` containing all the population members (where S is the + total population size), and ``rng`` is the random number generator + being used within the solver. + ``candidate`` will be in the range ``[0, S)``. + ``strategy`` must return a trial vector with shape ``(N,)``. The + fitness of this trial vector is compared against the fitness of + ``population[candidate]``. + + .. versionchanged:: 1.12.0 + Customization of evolution strategy via a callable. + + maxiter : int, optional + The maximum number of generations over which the entire population is + evolved. The maximum number of function evaluations (with no polishing) + is: ``(maxiter + 1) * popsize * (N - N_equal)`` + popsize : int, optional + A multiplier for setting the total population size. The population has + ``popsize * (N - N_equal)`` individuals. This keyword is overridden if + an initial population is supplied via the `init` keyword. When using + ``init='sobol'`` the population size is calculated as the next power + of 2 after ``popsize * (N - N_equal)``. + tol : float, optional + Relative tolerance for convergence, the solving stops when + ``np.std(population_energies) <= atol + tol * np.abs(np.mean(population_energies))``, + where and `atol` and `tol` are the absolute and relative tolerance + respectively. + mutation : float or tuple(float, float), optional + The mutation constant. In the literature this is also known as + differential weight, being denoted by :math:`F`. + If specified as a float it should be in the range [0, 2). + If specified as a tuple ``(min, max)`` dithering is employed. Dithering + randomly changes the mutation constant on a generation by generation + basis. The mutation constant for that generation is taken from + ``U[min, max)``. Dithering can help speed convergence significantly. + Increasing the mutation constant increases the search radius, but will + slow down convergence. + recombination : float, optional + The recombination constant, should be in the range [0, 1]. In the + literature this is also known as the crossover probability, being + denoted by CR. Increasing this value allows a larger number of mutants + to progress into the next generation, but at the risk of population + stability. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + disp : bool, optional + Prints the evaluated `func` at every iteration. + callback : callable, optional + A callable called after each iteration. Has the signature:: + + callback(intermediate_result: OptimizeResult) + + where ``intermediate_result`` is a keyword parameter containing an + `OptimizeResult` with attributes ``x`` and ``fun``, the best solution + found so far and the objective function. Note that the name + of the parameter must be ``intermediate_result`` for the callback + to be passed an `OptimizeResult`. + + The callback also supports a signature like:: + + callback(x, convergence: float=val) + + ``val`` represents the fractional value of the population convergence. + When ``val`` is greater than ``1.0``, the function halts. + + Introspection is used to determine which of the signatures is invoked. + + Global minimization will halt if the callback raises ``StopIteration`` + or returns ``True``; any polishing is still carried out. + + .. versionchanged:: 1.12.0 + callback accepts the ``intermediate_result`` keyword. + + polish : bool, optional + If True (default), then `scipy.optimize.minimize` with the `L-BFGS-B` + method is used to polish the best population member at the end, which + can improve the minimization slightly. If a constrained problem is + being studied then the `trust-constr` method is used instead. For large + problems with many constraints, polishing can take a long time due to + the Jacobian computations. + + .. versionchanged:: 1.15.0 + If `workers` is specified then the map-like callable that wraps + `func` is supplied to `minimize` instead of it using `func` + directly. This allows the caller to control how and where the + invocations actually run. + + init : str or array-like, optional + Specify which type of population initialization is performed. Should be + one of: + + - 'latinhypercube' + - 'sobol' + - 'halton' + - 'random' + - array specifying the initial population. The array should have + shape ``(S, N)``, where S is the total population size and N is + the number of parameters. + + `init` is clipped to `bounds` before use. + + The default is 'latinhypercube'. Latin Hypercube sampling tries to + maximize coverage of the available parameter space. + + 'sobol' and 'halton' are superior alternatives and maximize even more + the parameter space. 'sobol' will enforce an initial population + size which is calculated as the next power of 2 after + ``popsize * (N - N_equal)``. 'halton' has no requirements but is a bit + less efficient. See `scipy.stats.qmc` for more details. + + 'random' initializes the population randomly - this has the drawback + that clustering can occur, preventing the whole of parameter space + being covered. Use of an array to specify a population could be used, + for example, to create a tight bunch of initial guesses in an location + where the solution is known to exist, thereby reducing time for + convergence. + atol : float, optional + Absolute tolerance for convergence, the solving stops when + ``np.std(pop) <= atol + tol * np.abs(np.mean(population_energies))``, + where and `atol` and `tol` are the absolute and relative tolerance + respectively. + updating : {'immediate', 'deferred'}, optional + If ``'immediate'``, the best solution vector is continuously updated + within a single generation [4]_. This can lead to faster convergence as + trial vectors can take advantage of continuous improvements in the best + solution. + With ``'deferred'``, the best solution vector is updated once per + generation. Only ``'deferred'`` is compatible with parallelization or + vectorization, and the `workers` and `vectorized` keywords can + over-ride this option. + + .. versionadded:: 1.2.0 + + workers : int or map-like callable, optional + If `workers` is an int the population is subdivided into `workers` + sections and evaluated in parallel + (uses `multiprocessing.Pool `). + Supply -1 to use all available CPU cores. + Alternatively supply a map-like callable, such as + `multiprocessing.Pool.map` for evaluating the population in parallel. + This evaluation is carried out as ``workers(func, iterable)``. + This option will override the `updating` keyword to + ``updating='deferred'`` if ``workers != 1``. + This option overrides the `vectorized` keyword if ``workers != 1``. + Requires that `func` be pickleable. + + .. versionadded:: 1.2.0 + + constraints : {NonLinearConstraint, LinearConstraint, Bounds} + Constraints on the solver, over and above those applied by the `bounds` + kwd. Uses the approach by Lampinen [5]_. + + .. versionadded:: 1.4.0 + + x0 : None or array-like, optional + Provides an initial guess to the minimization. Once the population has + been initialized this vector replaces the first (best) member. This + replacement is done even if `init` is given an initial population. + ``x0.shape == (N,)``. + + .. versionadded:: 1.7.0 + + integrality : 1-D array, optional + For each decision variable, a boolean value indicating whether the + decision variable is constrained to integer values. The array is + broadcast to ``(N,)``. + If any decision variables are constrained to be integral, they will not + be changed during polishing. + Only integer values lying between the lower and upper bounds are used. + If there are no integer values lying between the bounds then a + `ValueError` is raised. + + .. versionadded:: 1.9.0 + + vectorized : bool, optional + If ``vectorized is True``, `func` is sent an `x` array with + ``x.shape == (N, S)``, and is expected to return an array of shape + ``(S,)``, where `S` is the number of solution vectors to be calculated. + If constraints are applied, each of the functions used to construct + a `Constraint` object should accept an `x` array with + ``x.shape == (N, S)``, and return an array of shape ``(M, S)``, where + `M` is the number of constraint components. + This option is an alternative to the parallelization offered by + `workers`, and may help in optimization speed by reducing interpreter + overhead from multiple function calls. This keyword is ignored if + ``workers != 1``. + This option will override the `updating` keyword to + ``updating='deferred'``. + See the notes section for further discussion on when to use + ``'vectorized'``, and when to use ``'workers'``. + + .. versionadded:: 1.9.0 + + Returns + ------- + res : OptimizeResult + The optimization result represented as a `OptimizeResult` object. + Important attributes are: ``x`` the solution array, ``success`` a + Boolean flag indicating if the optimizer exited successfully, + ``message`` which describes the cause of the termination, + ``population`` the solution vectors present in the population, and + ``population_energies`` the value of the objective function for each + entry in ``population``. + See `OptimizeResult` for a description of other attributes. If `polish` + was employed, and a lower minimum was obtained by the polishing, then + OptimizeResult also contains the ``jac`` attribute. + If the eventual solution does not satisfy the applied constraints + ``success`` will be `False`. + + Notes + ----- + Differential evolution is a stochastic population based method that is + useful for global optimization problems. At each pass through the + population the algorithm mutates each candidate solution by mixing with + other candidate solutions to create a trial candidate. There are several + strategies [3]_ for creating trial candidates, which suit some problems + more than others. The 'best1bin' strategy is a good starting point for + many systems. In this strategy two members of the population are randomly + chosen. Their difference is used to mutate the best member (the 'best' in + 'best1bin'), :math:`x_0`, so far: + + .. math:: + + b' = x_0 + F \cdot (x_{r_0} - x_{r_1}) + + where :math:`F` is the `mutation` parameter. + A trial vector is then constructed. Starting with a randomly chosen ith + parameter the trial is sequentially filled (in modulo) with parameters + from ``b'`` or the original candidate. The choice of whether to use ``b'`` + or the original candidate is made with a binomial distribution (the 'bin' + in 'best1bin') - a random number in [0, 1) is generated. If this number is + less than the `recombination` constant then the parameter is loaded from + ``b'``, otherwise it is loaded from the original candidate. The final + parameter is always loaded from ``b'``. Once the trial candidate is built + its fitness is assessed. If the trial is better than the original candidate + then it takes its place. If it is also better than the best overall + candidate it also replaces that. + + The other strategies available are outlined in Qiang and + Mitchell (2014) [3]_. + + + - ``rand1`` : :math:`b' = x_{r_0} + F \cdot (x_{r_1} - x_{r_2})` + - ``rand2`` : :math:`b' = x_{r_0} + F \cdot (x_{r_1} + x_{r_2} - x_{r_3} - x_{r_4})` + - ``best1`` : :math:`b' = x_0 + F \cdot (x_{r_0} - x_{r_1})` + - ``best2`` : :math:`b' = x_0 + F \cdot (x_{r_0} + x_{r_1} - x_{r_2} - x_{r_3})` + - ``currenttobest1`` : :math:`b' = x_i + F \cdot (x_0 - x_i + x_{r_0} - x_{r_1})` + - ``randtobest1`` : :math:`b' = x_{r_0} + F \cdot (x_0 - x_{r_0} + x_{r_1} - x_{r_2})` + + where the integers :math:`r_0, r_1, r_2, r_3, r_4` are chosen randomly + from the interval [0, NP) with `NP` being the total population size and + the original candidate having index `i`. The user can fully customize the + generation of the trial candidates by supplying a callable to ``strategy``. + + To improve your chances of finding a global minimum use higher `popsize` + values, with higher `mutation` and (dithering), but lower `recombination` + values. This has the effect of widening the search radius, but slowing + convergence. + + By default the best solution vector is updated continuously within a single + iteration (``updating='immediate'``). This is a modification [4]_ of the + original differential evolution algorithm which can lead to faster + convergence as trial vectors can immediately benefit from improved + solutions. To use the original Storn and Price behaviour, updating the best + solution once per iteration, set ``updating='deferred'``. + The ``'deferred'`` approach is compatible with both parallelization and + vectorization (``'workers'`` and ``'vectorized'`` keywords). These may + improve minimization speed by using computer resources more efficiently. + The ``'workers'`` distribute calculations over multiple processors. By + default the Python `multiprocessing` module is used, but other approaches + are also possible, such as the Message Passing Interface (MPI) used on + clusters [6]_ [7]_. The overhead from these approaches (creating new + Processes, etc) may be significant, meaning that computational speed + doesn't necessarily scale with the number of processors used. + Parallelization is best suited to computationally expensive objective + functions. If the objective function is less expensive, then + ``'vectorized'`` may aid by only calling the objective function once per + iteration, rather than multiple times for all the population members; the + interpreter overhead is reduced. + + .. versionadded:: 0.15.0 + + References + ---------- + .. [1] Differential evolution, Wikipedia, + http://en.wikipedia.org/wiki/Differential_evolution + .. [2] Storn, R and Price, K, Differential Evolution - a Simple and + Efficient Heuristic for Global Optimization over Continuous Spaces, + Journal of Global Optimization, 1997, 11, 341 - 359. + .. [3] Qiang, J., Mitchell, C., A Unified Differential Evolution Algorithm + for Global Optimization, 2014, https://www.osti.gov/servlets/purl/1163659 + .. [4] Wormington, M., Panaccione, C., Matney, K. M., Bowen, D. K., - + Characterization of structures from X-ray scattering data using + genetic algorithms, Phil. Trans. R. Soc. Lond. A, 1999, 357, + 2827-2848 + .. [5] Lampinen, J., A constraint handling approach for the differential + evolution algorithm. Proceedings of the 2002 Congress on + Evolutionary Computation. CEC'02 (Cat. No. 02TH8600). Vol. 2. IEEE, + 2002. + .. [6] https://mpi4py.readthedocs.io/en/stable/ + .. [7] https://schwimmbad.readthedocs.io/en/latest/ + + + Examples + -------- + Let us consider the problem of minimizing the Rosenbrock function. This + function is implemented in `rosen` in `scipy.optimize`. + + >>> import numpy as np + >>> from scipy.optimize import rosen, differential_evolution + >>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)] + >>> result = differential_evolution(rosen, bounds) + >>> result.x, result.fun + (array([1., 1., 1., 1., 1.]), 1.9216496320061384e-19) + + Now repeat, but with parallelization. + + >>> result = differential_evolution(rosen, bounds, updating='deferred', + ... workers=2) + >>> result.x, result.fun + (array([1., 1., 1., 1., 1.]), 1.9216496320061384e-19) + + Let's do a constrained minimization. + + >>> from scipy.optimize import LinearConstraint, Bounds + + We add the constraint that the sum of ``x[0]`` and ``x[1]`` must be less + than or equal to 1.9. This is a linear constraint, which may be written + ``A @ x <= 1.9``, where ``A = array([[1, 1]])``. This can be encoded as + a `LinearConstraint` instance: + + >>> lc = LinearConstraint([[1, 1]], -np.inf, 1.9) + + Specify limits using a `Bounds` object. + + >>> bounds = Bounds([0., 0.], [2., 2.]) + >>> result = differential_evolution(rosen, bounds, constraints=lc, + ... rng=1) + >>> result.x, result.fun + (array([0.96632622, 0.93367155]), 0.0011352416852625719) + + Next find the minimum of the Ackley function + (https://en.wikipedia.org/wiki/Test_functions_for_optimization). + + >>> def ackley(x): + ... arg1 = -0.2 * np.sqrt(0.5 * (x[0] ** 2 + x[1] ** 2)) + ... arg2 = 0.5 * (np.cos(2. * np.pi * x[0]) + np.cos(2. * np.pi * x[1])) + ... return -20. * np.exp(arg1) - np.exp(arg2) + 20. + np.e + >>> bounds = [(-5, 5), (-5, 5)] + >>> result = differential_evolution(ackley, bounds, rng=1) + >>> result.x, result.fun + (array([0., 0.]), 4.440892098500626e-16) + + The Ackley function is written in a vectorized manner, so the + ``'vectorized'`` keyword can be employed. Note the reduced number of + function evaluations. + + >>> result = differential_evolution( + ... ackley, bounds, vectorized=True, updating='deferred', rng=1 + ... ) + >>> result.x, result.fun + (array([0., 0.]), 4.440892098500626e-16) + + The following custom strategy function mimics 'best1bin': + + >>> def custom_strategy_fn(candidate, population, rng=None): + ... parameter_count = population.shape(-1) + ... mutation, recombination = 0.7, 0.9 + ... trial = np.copy(population[candidate]) + ... fill_point = rng.choice(parameter_count) + ... + ... pool = np.arange(len(population)) + ... rng.shuffle(pool) + ... + ... # two unique random numbers that aren't the same, and + ... # aren't equal to candidate. + ... idxs = [] + ... while len(idxs) < 2 and len(pool) > 0: + ... idx = pool[0] + ... pool = pool[1:] + ... if idx != candidate: + ... idxs.append(idx) + ... + ... r0, r1 = idxs[:2] + ... + ... bprime = (population[0] + mutation * + ... (population[r0] - population[r1])) + ... + ... crossovers = rng.uniform(size=parameter_count) + ... crossovers = crossovers < recombination + ... crossovers[fill_point] = True + ... trial = np.where(crossovers, bprime, trial) + ... return trial + + """# noqa: E501 + + # using a context manager means that any created Pool objects are + # cleared up. + with DifferentialEvolutionSolver(func, bounds, args=args, + strategy=strategy, + maxiter=maxiter, + popsize=popsize, tol=tol, + mutation=mutation, + recombination=recombination, + rng=rng, polish=polish, + callback=callback, + disp=disp, init=init, atol=atol, + updating=updating, + workers=workers, + constraints=constraints, + x0=x0, + integrality=integrality, + vectorized=vectorized) as solver: + ret = solver.solve() + + return ret + + +class DifferentialEvolutionSolver: + + """This class implements the differential evolution solver + + Parameters + ---------- + func : callable + The objective function to be minimized. Must be in the form + ``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array + and ``args`` is a tuple of any additional fixed parameters needed to + completely specify the function. The number of parameters, N, is equal + to ``len(x)``. + bounds : sequence or `Bounds` + Bounds for variables. There are two ways to specify the bounds: + + 1. Instance of `Bounds` class. + 2. ``(min, max)`` pairs for each element in ``x``, defining the + finite lower and upper bounds for the optimizing argument of + `func`. + + The total number of bounds is used to determine the number of + parameters, N. If there are parameters whose bounds are equal the total + number of free parameters is ``N - N_equal``. + args : tuple, optional + Any additional fixed parameters needed to + completely specify the objective function. + strategy : {str, callable}, optional + The differential evolution strategy to use. Should be one of: + + - 'best1bin' + - 'best1exp' + - 'rand1bin' + - 'rand1exp' + - 'rand2bin' + - 'rand2exp' + - 'randtobest1bin' + - 'randtobest1exp' + - 'currenttobest1bin' + - 'currenttobest1exp' + - 'best2exp' + - 'best2bin' + + The default is 'best1bin'. Strategies that may be + implemented are outlined in 'Notes'. + + Alternatively the differential evolution strategy can be customized + by providing a callable that constructs a trial vector. The callable + must have the form + ``strategy(candidate: int, population: np.ndarray, rng=None)``, + where ``candidate`` is an integer specifying which entry of the + population is being evolved, ``population`` is an array of shape + ``(S, N)`` containing all the population members (where S is the + total population size), and ``rng`` is the random number generator + being used within the solver. + ``candidate`` will be in the range ``[0, S)``. + ``strategy`` must return a trial vector with shape ``(N,)``. The + fitness of this trial vector is compared against the fitness of + ``population[candidate]``. + maxiter : int, optional + The maximum number of generations over which the entire population is + evolved. The maximum number of function evaluations (with no polishing) + is: ``(maxiter + 1) * popsize * (N - N_equal)`` + popsize : int, optional + A multiplier for setting the total population size. The population has + ``popsize * (N - N_equal)`` individuals. This keyword is overridden if + an initial population is supplied via the `init` keyword. When using + ``init='sobol'`` the population size is calculated as the next power + of 2 after ``popsize * (N - N_equal)``. + tol : float, optional + Relative tolerance for convergence, the solving stops when + ``np.std(population_energies) <= atol + tol * np.abs(np.mean(population_energies))``, + where and `atol` and `tol` are the absolute and relative tolerance + respectively. + mutation : float or tuple(float, float), optional + The mutation constant. In the literature this is also known as + differential weight, being denoted by F. + If specified as a float it should be in the range [0, 2]. + If specified as a tuple ``(min, max)`` dithering is employed. Dithering + randomly changes the mutation constant on a generation by generation + basis. The mutation constant for that generation is taken from + U[min, max). Dithering can help speed convergence significantly. + Increasing the mutation constant increases the search radius, but will + slow down convergence. + recombination : float, optional + The recombination constant, should be in the range [0, 1]. In the + literature this is also known as the crossover probability, being + denoted by CR. Increasing this value allows a larger number of mutants + to progress into the next generation, but at the risk of population + stability. + + rng : {None, int, `numpy.random.Generator`}, optional + + ..versionchanged:: 1.15.0 + As part of the `SPEC-007 `_ + transition from use of `numpy.random.RandomState` to + `numpy.random.Generator` this keyword was changed from `seed` to `rng`. + For an interim period both keywords will continue to work (only specify + one of them). After the interim period using the `seed` keyword will emit + warnings. The behavior of the `seed` and `rng` keywords is outlined below. + + If `rng` is passed by keyword, types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a `Generator`. + If `rng` is already a `Generator` instance, then the provided instance is + used. + + If this argument is passed by position or `seed` is passed by keyword, the + behavior is: + + - If `seed` is None (or `np.random`), the `numpy.random.RandomState` + singleton is used. + - If `seed` is an int, a new `RandomState` instance is used, + seeded with `seed`. + - If `seed` is already a `Generator` or `RandomState` instance then + that instance is used. + + Specify `seed`/`rng` for repeatable minimizations. + disp : bool, optional + Prints the evaluated `func` at every iteration. + callback : callable, optional + A callable called after each iteration. Has the signature: + + ``callback(intermediate_result: OptimizeResult)`` + + where ``intermediate_result`` is a keyword parameter containing an + `OptimizeResult` with attributes ``x`` and ``fun``, the best solution + found so far and the objective function. Note that the name + of the parameter must be ``intermediate_result`` for the callback + to be passed an `OptimizeResult`. + + The callback also supports a signature like: + + ``callback(x, convergence: float=val)`` + + ``val`` represents the fractional value of the population convergence. + When ``val`` is greater than ``1.0``, the function halts. + + Introspection is used to determine which of the signatures is invoked. + + Global minimization will halt if the callback raises ``StopIteration`` + or returns ``True``; any polishing is still carried out. + + .. versionchanged:: 1.12.0 + callback accepts the ``intermediate_result`` keyword. + + polish : bool, optional + If True (default), then `scipy.optimize.minimize` with the `L-BFGS-B` + method is used to polish the best population member at the end, which + can improve the minimization slightly. If a constrained problem is + being studied then the `trust-constr` method is used instead. For large + problems with many constraints, polishing can take a long time due to + the Jacobian computations. + maxfun : int, optional + Set the maximum number of function evaluations. However, it probably + makes more sense to set `maxiter` instead. + init : str or array-like, optional + Specify which type of population initialization is performed. Should be + one of: + + - 'latinhypercube' + - 'sobol' + - 'halton' + - 'random' + - array specifying the initial population. The array should have + shape ``(S, N)``, where S is the total population size and + N is the number of parameters. + `init` is clipped to `bounds` before use. + + The default is 'latinhypercube'. Latin Hypercube sampling tries to + maximize coverage of the available parameter space. + + 'sobol' and 'halton' are superior alternatives and maximize even more + the parameter space. 'sobol' will enforce an initial population + size which is calculated as the next power of 2 after + ``popsize * (N - N_equal)``. 'halton' has no requirements but is a bit + less efficient. See `scipy.stats.qmc` for more details. + + 'random' initializes the population randomly - this has the drawback + that clustering can occur, preventing the whole of parameter space + being covered. Use of an array to specify a population could be used, + for example, to create a tight bunch of initial guesses in an location + where the solution is known to exist, thereby reducing time for + convergence. + atol : float, optional + Absolute tolerance for convergence, the solving stops when + ``np.std(pop) <= atol + tol * np.abs(np.mean(population_energies))``, + where and `atol` and `tol` are the absolute and relative tolerance + respectively. + updating : {'immediate', 'deferred'}, optional + If ``'immediate'``, the best solution vector is continuously updated + within a single generation [4]_. This can lead to faster convergence as + trial vectors can take advantage of continuous improvements in the best + solution. + With ``'deferred'``, the best solution vector is updated once per + generation. Only ``'deferred'`` is compatible with parallelization or + vectorization, and the `workers` and `vectorized` keywords can + over-ride this option. + workers : int or map-like callable, optional + If `workers` is an int the population is subdivided into `workers` + sections and evaluated in parallel + (uses `multiprocessing.Pool `). + Supply `-1` to use all cores available to the Process. + Alternatively supply a map-like callable, such as + `multiprocessing.Pool.map` for evaluating the population in parallel. + This evaluation is carried out as ``workers(func, iterable)``. + This option will override the `updating` keyword to + `updating='deferred'` if `workers != 1`. + Requires that `func` be pickleable. + constraints : {NonLinearConstraint, LinearConstraint, Bounds} + Constraints on the solver, over and above those applied by the `bounds` + kwd. Uses the approach by Lampinen. + x0 : None or array-like, optional + Provides an initial guess to the minimization. Once the population has + been initialized this vector replaces the first (best) member. This + replacement is done even if `init` is given an initial population. + ``x0.shape == (N,)``. + integrality : 1-D array, optional + For each decision variable, a boolean value indicating whether the + decision variable is constrained to integer values. The array is + broadcast to ``(N,)``. + If any decision variables are constrained to be integral, they will not + be changed during polishing. + Only integer values lying between the lower and upper bounds are used. + If there are no integer values lying between the bounds then a + `ValueError` is raised. + vectorized : bool, optional + If ``vectorized is True``, `func` is sent an `x` array with + ``x.shape == (N, S)``, and is expected to return an array of shape + ``(S,)``, where `S` is the number of solution vectors to be calculated. + If constraints are applied, each of the functions used to construct + a `Constraint` object should accept an `x` array with + ``x.shape == (N, S)``, and return an array of shape ``(M, S)``, where + `M` is the number of constraint components. + This option is an alternative to the parallelization offered by + `workers`, and may help in optimization speed. This keyword is + ignored if ``workers != 1``. + This option will override the `updating` keyword to + ``updating='deferred'``. + """ # noqa: E501 + + # Dispatch of mutation strategy method (binomial or exponential). + _binomial = {'best1bin': '_best1', + 'randtobest1bin': '_randtobest1', + 'currenttobest1bin': '_currenttobest1', + 'best2bin': '_best2', + 'rand2bin': '_rand2', + 'rand1bin': '_rand1'} + _exponential = {'best1exp': '_best1', + 'rand1exp': '_rand1', + 'randtobest1exp': '_randtobest1', + 'currenttobest1exp': '_currenttobest1', + 'best2exp': '_best2', + 'rand2exp': '_rand2'} + + __init_error_msg = ("The population initialization method must be one of " + "'latinhypercube' or 'random', or an array of shape " + "(S, N) where N is the number of parameters and S>5") + + def __init__(self, func, bounds, args=(), + strategy='best1bin', maxiter=1000, popsize=15, + tol=0.01, mutation=(0.5, 1), recombination=0.7, rng=None, + maxfun=np.inf, callback=None, disp=False, polish=True, + init='latinhypercube', atol=0, updating='immediate', + workers=1, constraints=(), x0=None, *, integrality=None, + vectorized=False): + + if callable(strategy): + # a callable strategy is going to be stored in self.strategy anyway + pass + elif strategy in self._binomial: + self.mutation_func = getattr(self, self._binomial[strategy]) + elif strategy in self._exponential: + self.mutation_func = getattr(self, self._exponential[strategy]) + else: + raise ValueError("Please select a valid mutation strategy") + self.strategy = strategy + + self.callback = _wrap_callback(callback, "differential_evolution") + self.polish = polish + + # set the updating / parallelisation options + if updating in ['immediate', 'deferred']: + self._updating = updating + + self.vectorized = vectorized + + # want to use parallelisation, but updating is immediate + if workers != 1 and updating == 'immediate': + warnings.warn("differential_evolution: the 'workers' keyword has" + " overridden updating='immediate' to" + " updating='deferred'", UserWarning, stacklevel=2) + self._updating = 'deferred' + + if vectorized and workers != 1: + warnings.warn("differential_evolution: the 'workers' keyword" + " overrides the 'vectorized' keyword", stacklevel=2) + self.vectorized = vectorized = False + + if vectorized and updating == 'immediate': + warnings.warn("differential_evolution: the 'vectorized' keyword" + " has overridden updating='immediate' to updating" + "='deferred'", UserWarning, stacklevel=2) + self._updating = 'deferred' + + # an object with a map method. + if vectorized: + def maplike_for_vectorized_func(func, x): + # send an array (N, S) to the user func, + # expect to receive (S,). Transposition is required because + # internally the population is held as (S, N) + return np.atleast_1d(func(x.T)) + workers = maplike_for_vectorized_func + + self._mapwrapper = MapWrapper(workers) + + # relative and absolute tolerances for convergence + self.tol, self.atol = tol, atol + + # Mutation constant should be in [0, 2). If specified as a sequence + # then dithering is performed. + self.scale = mutation + if (not np.all(np.isfinite(mutation)) or + np.any(np.array(mutation) >= 2) or + np.any(np.array(mutation) < 0)): + raise ValueError('The mutation constant must be a float in ' + 'U[0, 2), or specified as a tuple(min, max)' + ' where min < max and min, max are in U[0, 2).') + + self.dither = None + if hasattr(mutation, '__iter__') and len(mutation) > 1: + self.dither = [mutation[0], mutation[1]] + self.dither.sort() + + self.cross_over_probability = recombination + + # we create a wrapped function to allow the use of map (and Pool.map + # in the future) + self.func = _FunctionWrapper(func, args) + self.args = args + + # convert tuple of lower and upper bounds to limits + # [(low_0, high_0), ..., (low_n, high_n] + # -> [[low_0, ..., low_n], [high_0, ..., high_n]] + if isinstance(bounds, Bounds): + self.limits = np.array(new_bounds_to_old(bounds.lb, + bounds.ub, + len(bounds.lb)), + dtype=float).T + else: + self.limits = np.array(bounds, dtype='float').T + + if (np.size(self.limits, 0) != 2 or not + np.all(np.isfinite(self.limits))): + raise ValueError('bounds should be a sequence containing finite ' + 'real valued (min, max) pairs for each value' + ' in x') + + if maxiter is None: # the default used to be None + maxiter = 1000 + self.maxiter = maxiter + if maxfun is None: # the default used to be None + maxfun = np.inf + self.maxfun = maxfun + + # population is scaled to between [0, 1]. + # We have to scale between parameter <-> population + # save these arguments for _scale_parameter and + # _unscale_parameter. This is an optimization + self.__scale_arg1 = 0.5 * (self.limits[0] + self.limits[1]) + self.__scale_arg2 = np.fabs(self.limits[0] - self.limits[1]) + with np.errstate(divide='ignore'): + # if lb == ub then the following line will be 1/0, which is why + # we ignore the divide by zero warning. The result from 1/0 is + # inf, so replace those values by 0. + self.__recip_scale_arg2 = 1 / self.__scale_arg2 + self.__recip_scale_arg2[~np.isfinite(self.__recip_scale_arg2)] = 0 + + self.parameter_count = np.size(self.limits, 1) + + self.random_number_generator = check_random_state(rng) + + # Which parameters are going to be integers? + if np.any(integrality): + # # user has provided a truth value for integer constraints + integrality = np.broadcast_to( + integrality, + self.parameter_count + ) + integrality = np.asarray(integrality, bool) + # For integrality parameters change the limits to only allow + # integer values lying between the limits. + lb, ub = np.copy(self.limits) + + lb = np.ceil(lb) + ub = np.floor(ub) + if not (lb[integrality] <= ub[integrality]).all(): + # there's a parameter that doesn't have an integer value + # lying between the limits + raise ValueError("One of the integrality constraints does not" + " have any possible integer values between" + " the lower/upper bounds.") + nlb = np.nextafter(lb[integrality] - 0.5, np.inf) + nub = np.nextafter(ub[integrality] + 0.5, -np.inf) + + self.integrality = integrality + self.limits[0, self.integrality] = nlb + self.limits[1, self.integrality] = nub + else: + self.integrality = False + + # check for equal bounds + eb = self.limits[0] == self.limits[1] + eb_count = np.count_nonzero(eb) + + # default population initialization is a latin hypercube design, but + # there are other population initializations possible. + # the minimum is 5 because 'best2bin' requires a population that's at + # least 5 long + # 202301 - reduced population size to account for parameters with + # equal bounds. If there are no varying parameters set N to at least 1 + self.num_population_members = max( + 5, + popsize * max(1, self.parameter_count - eb_count) + ) + self.population_shape = (self.num_population_members, + self.parameter_count) + + self._nfev = 0 + # check first str otherwise will fail to compare str with array + if isinstance(init, str): + if init == 'latinhypercube': + self.init_population_lhs() + elif init == 'sobol': + # must be Ns = 2**m for Sobol' + n_s = int(2 ** np.ceil(np.log2(self.num_population_members))) + self.num_population_members = n_s + self.population_shape = (self.num_population_members, + self.parameter_count) + self.init_population_qmc(qmc_engine='sobol') + elif init == 'halton': + self.init_population_qmc(qmc_engine='halton') + elif init == 'random': + self.init_population_random() + else: + raise ValueError(self.__init_error_msg) + else: + self.init_population_array(init) + + if x0 is not None: + # scale to within unit interval and + # ensure parameters are within bounds. + x0_scaled = self._unscale_parameters(np.asarray(x0)) + if ((x0_scaled > 1.0) | (x0_scaled < 0.0)).any(): + raise ValueError( + "Some entries in x0 lay outside the specified bounds" + ) + self.population[0] = x0_scaled + + # infrastructure for constraints + self.constraints = constraints + self._wrapped_constraints = [] + + if hasattr(constraints, '__len__'): + # sequence of constraints, this will also deal with default + # keyword parameter + for c in constraints: + self._wrapped_constraints.append( + _ConstraintWrapper(c, self.x) + ) + else: + self._wrapped_constraints = [ + _ConstraintWrapper(constraints, self.x) + ] + self.total_constraints = np.sum( + [c.num_constr for c in self._wrapped_constraints] + ) + self.constraint_violation = np.zeros((self.num_population_members, 1)) + self.feasible = np.ones(self.num_population_members, bool) + + # an array to shuffle when selecting candidates. Create it here + # rather than repeatedly creating it in _select_samples. + self._random_population_index = np.arange(self.num_population_members) + self.disp = disp + + def init_population_lhs(self): + """ + Initializes the population with Latin Hypercube Sampling. + Latin Hypercube Sampling ensures that each parameter is uniformly + sampled over its range. + """ + rng = self.random_number_generator + + # Each parameter range needs to be sampled uniformly. The scaled + # parameter range ([0, 1)) needs to be split into + # `self.num_population_members` segments, each of which has the following + # size: + segsize = 1.0 / self.num_population_members + + # Within each segment we sample from a uniform random distribution. + # We need to do this sampling for each parameter. + samples = (segsize * rng.uniform(size=self.population_shape) + + # Offset each segment to cover the entire parameter range [0, 1) + + np.linspace(0., 1., self.num_population_members, + endpoint=False)[:, np.newaxis]) + + # Create an array for population of candidate solutions. + self.population = np.zeros_like(samples) + + # Initialize population of candidate solutions by permutation of the + # random samples. + for j in range(self.parameter_count): + order = rng.permutation(range(self.num_population_members)) + self.population[:, j] = samples[order, j] + + # reset population energies + self.population_energies = np.full(self.num_population_members, + np.inf) + + # reset number of function evaluations counter + self._nfev = 0 + + def init_population_qmc(self, qmc_engine): + """Initializes the population with a QMC method. + + QMC methods ensures that each parameter is uniformly + sampled over its range. + + Parameters + ---------- + qmc_engine : str + The QMC method to use for initialization. Can be one of + ``latinhypercube``, ``sobol`` or ``halton``. + + """ + from scipy.stats import qmc + + rng = self.random_number_generator + + # Create an array for population of candidate solutions. + if qmc_engine == 'latinhypercube': + sampler = qmc.LatinHypercube(d=self.parameter_count, seed=rng) + elif qmc_engine == 'sobol': + sampler = qmc.Sobol(d=self.parameter_count, seed=rng) + elif qmc_engine == 'halton': + sampler = qmc.Halton(d=self.parameter_count, seed=rng) + else: + raise ValueError(self.__init_error_msg) + + self.population = sampler.random(n=self.num_population_members) + + # reset population energies + self.population_energies = np.full(self.num_population_members, + np.inf) + + # reset number of function evaluations counter + self._nfev = 0 + + def init_population_random(self): + """ + Initializes the population at random. This type of initialization + can possess clustering, Latin Hypercube sampling is generally better. + """ + rng = self.random_number_generator + self.population = rng.uniform(size=self.population_shape) + + # reset population energies + self.population_energies = np.full(self.num_population_members, + np.inf) + + # reset number of function evaluations counter + self._nfev = 0 + + def init_population_array(self, init): + """ + Initializes the population with a user specified population. + + Parameters + ---------- + init : np.ndarray + Array specifying subset of the initial population. The array should + have shape (S, N), where N is the number of parameters. + The population is clipped to the lower and upper bounds. + """ + # make sure you're using a float array + popn = np.asarray(init, dtype=np.float64) + + if (np.size(popn, 0) < 5 or + popn.shape[1] != self.parameter_count or + len(popn.shape) != 2): + raise ValueError("The population supplied needs to have shape" + " (S, len(x)), where S > 4.") + + # scale values and clip to bounds, assigning to population + self.population = np.clip(self._unscale_parameters(popn), 0, 1) + + self.num_population_members = np.size(self.population, 0) + + self.population_shape = (self.num_population_members, + self.parameter_count) + + # reset population energies + self.population_energies = np.full(self.num_population_members, + np.inf) + + # reset number of function evaluations counter + self._nfev = 0 + + @property + def x(self): + """ + The best solution from the solver + """ + return self._scale_parameters(self.population[0]) + + @property + def convergence(self): + """ + The standard deviation of the population energies divided by their + mean. + """ + if np.any(np.isinf(self.population_energies)): + return np.inf + return (np.std(self.population_energies) / + (np.abs(np.mean(self.population_energies)) + _MACHEPS)) + + def converged(self): + """ + Return True if the solver has converged. + """ + if np.any(np.isinf(self.population_energies)): + return False + + return (np.std(self.population_energies) <= + self.atol + + self.tol * np.abs(np.mean(self.population_energies))) + + def solve(self): + """ + Runs the DifferentialEvolutionSolver. + + Returns + ------- + res : OptimizeResult + The optimization result represented as a `OptimizeResult` object. + Important attributes are: ``x`` the solution array, ``success`` a + Boolean flag indicating if the optimizer exited successfully, + ``message`` which describes the cause of the termination, + ``population`` the solution vectors present in the population, and + ``population_energies`` the value of the objective function for + each entry in ``population``. + See `OptimizeResult` for a description of other attributes. If + `polish` was employed, and a lower minimum was obtained by the + polishing, then OptimizeResult also contains the ``jac`` attribute. + If the eventual solution does not satisfy the applied constraints + ``success`` will be `False`. + """ + nit, warning_flag = 0, False + status_message = _status_message['success'] + + # The population may have just been initialized (all entries are + # np.inf). If it has you have to calculate the initial energies. + # Although this is also done in the evolve generator it's possible + # that someone can set maxiter=0, at which point we still want the + # initial energies to be calculated (the following loop isn't run). + if np.all(np.isinf(self.population_energies)): + self.feasible, self.constraint_violation = ( + self._calculate_population_feasibilities(self.population)) + + # only work out population energies for feasible solutions + self.population_energies[self.feasible] = ( + self._calculate_population_energies( + self.population[self.feasible])) + + self._promote_lowest_energy() + + # do the optimization. + for nit in range(1, self.maxiter + 1): + # evolve the population by a generation + try: + next(self) + except StopIteration: + warning_flag = True + if self._nfev > self.maxfun: + status_message = _status_message['maxfev'] + elif self._nfev == self.maxfun: + status_message = ('Maximum number of function evaluations' + ' has been reached.') + break + + if self.disp: + print(f"differential_evolution step {nit}: f(x)=" + f" {self.population_energies[0]}" + ) + + if self.callback: + c = self.tol / (self.convergence + _MACHEPS) + res = self._result(nit=nit, message="in progress") + res.convergence = c + try: + warning_flag = bool(self.callback(res)) + except StopIteration: + warning_flag = True + + if warning_flag: + status_message = 'callback function requested stop early' + + # should the solver terminate? + if warning_flag or self.converged(): + break + + else: + status_message = _status_message['maxiter'] + warning_flag = True + + DE_result = self._result( + nit=nit, message=status_message, warning_flag=warning_flag + ) + + if self.polish and not np.all(self.integrality): + # can't polish if all the parameters are integers + if np.any(self.integrality): + # set the lower/upper bounds equal so that any integrality + # constraints work. + limits, integrality = self.limits, self.integrality + limits[0, integrality] = DE_result.x[integrality] + limits[1, integrality] = DE_result.x[integrality] + + polish_method = 'L-BFGS-B' + + if self._wrapped_constraints: + polish_method = 'trust-constr' + + constr_violation = self._constraint_violation_fn(DE_result.x) + if np.any(constr_violation > 0.): + warnings.warn("differential evolution didn't find a " + "solution satisfying the constraints, " + "attempting to polish from the least " + "infeasible solution", + UserWarning, stacklevel=2) + if self.disp: + print(f"Polishing solution with '{polish_method}'") + result = minimize(lambda x: + list(self._mapwrapper(self.func, np.atleast_2d(x)))[0], + np.copy(DE_result.x), + method=polish_method, + bounds=self.limits.T, + constraints=self.constraints) + + self._nfev += result.nfev + DE_result.nfev = self._nfev + + # Polishing solution is only accepted if there is an improvement in + # cost function, the polishing was successful and the solution lies + # within the bounds. + if (result.fun < DE_result.fun and + result.success and + np.all(result.x <= self.limits[1]) and + np.all(self.limits[0] <= result.x)): + DE_result.fun = result.fun + DE_result.x = result.x + DE_result.jac = result.jac + # to keep internal state consistent + self.population_energies[0] = result.fun + self.population[0] = self._unscale_parameters(result.x) + + if self._wrapped_constraints: + DE_result.constr = [c.violation(DE_result.x) for + c in self._wrapped_constraints] + DE_result.constr_violation = np.max( + np.concatenate(DE_result.constr)) + DE_result.maxcv = DE_result.constr_violation + if DE_result.maxcv > 0: + # if the result is infeasible then success must be False + DE_result.success = False + DE_result.message = ("The solution does not satisfy the " + f"constraints, MAXCV = {DE_result.maxcv}") + + return DE_result + + def _result(self, **kwds): + # form an intermediate OptimizeResult + nit = kwds.get('nit', None) + message = kwds.get('message', None) + warning_flag = kwds.get('warning_flag', False) + result = OptimizeResult( + x=self.x, + fun=self.population_energies[0], + nfev=self._nfev, + nit=nit, + message=message, + success=(warning_flag is not True), + population=self._scale_parameters(self.population), + population_energies=self.population_energies + ) + if self._wrapped_constraints: + result.constr = [c.violation(result.x) + for c in self._wrapped_constraints] + result.constr_violation = np.max(np.concatenate(result.constr)) + result.maxcv = result.constr_violation + if result.maxcv > 0: + result.success = False + + return result + + def _calculate_population_energies(self, population): + """ + Calculate the energies of a population. + + Parameters + ---------- + population : ndarray + An array of parameter vectors normalised to [0, 1] using lower + and upper limits. Has shape ``(np.size(population, 0), N)``. + + Returns + ------- + energies : ndarray + An array of energies corresponding to each population member. If + maxfun will be exceeded during this call, then the number of + function evaluations will be reduced and energies will be + right-padded with np.inf. Has shape ``(np.size(population, 0),)`` + """ + num_members = np.size(population, 0) + # S is the number of function evals left to stay under the + # maxfun budget + S = min(num_members, self.maxfun - self._nfev) + + energies = np.full(num_members, np.inf) + + parameters_pop = self._scale_parameters(population) + try: + calc_energies = list( + self._mapwrapper(self.func, parameters_pop[0:S]) + ) + calc_energies = np.squeeze(calc_energies) + except (TypeError, ValueError) as e: + # wrong number of arguments for _mapwrapper + # or wrong length returned from the mapper + raise RuntimeError( + "The map-like callable must be of the form f(func, iterable), " + "returning a sequence of numbers the same length as 'iterable'" + ) from e + + if calc_energies.size != S: + if self.vectorized: + raise RuntimeError("The vectorized function must return an" + " array of shape (S,) when given an array" + " of shape (len(x), S)") + raise RuntimeError("func(x, *args) must return a scalar value") + + energies[0:S] = calc_energies + + if self.vectorized: + self._nfev += 1 + else: + self._nfev += S + + return energies + + def _promote_lowest_energy(self): + # swaps 'best solution' into first population entry + + idx = np.arange(self.num_population_members) + feasible_solutions = idx[self.feasible] + if feasible_solutions.size: + # find the best feasible solution + idx_t = np.argmin(self.population_energies[feasible_solutions]) + l = feasible_solutions[idx_t] + else: + # no solution was feasible, use 'best' infeasible solution, which + # will violate constraints the least + l = np.argmin(np.sum(self.constraint_violation, axis=1)) + + self.population_energies[[0, l]] = self.population_energies[[l, 0]] + self.population[[0, l], :] = self.population[[l, 0], :] + self.feasible[[0, l]] = self.feasible[[l, 0]] + self.constraint_violation[[0, l], :] = ( + self.constraint_violation[[l, 0], :]) + + def _constraint_violation_fn(self, x): + """ + Calculates total constraint violation for all the constraints, for a + set of solutions. + + Parameters + ---------- + x : ndarray + Solution vector(s). Has shape (S, N), or (N,), where S is the + number of solutions to investigate and N is the number of + parameters. + + Returns + ------- + cv : ndarray + Total violation of constraints. Has shape ``(S, M)``, where M is + the total number of constraint components (which is not necessarily + equal to len(self._wrapped_constraints)). + """ + # how many solution vectors you're calculating constraint violations + # for + S = np.size(x) // self.parameter_count + _out = np.zeros((S, self.total_constraints)) + offset = 0 + for con in self._wrapped_constraints: + # the input/output of the (vectorized) constraint function is + # {(N, S), (N,)} --> (M, S) + # The input to _constraint_violation_fn is (S, N) or (N,), so + # transpose to pass it to the constraint. The output is transposed + # from (M, S) to (S, M) for further use. + c = con.violation(x.T).T + + # The shape of c should be (M,), (1, M), or (S, M). Check for + # those shapes, as an incorrect shape indicates that the + # user constraint function didn't return the right thing, and + # the reshape operation will fail. Intercept the wrong shape + # to give a reasonable error message. I'm not sure what failure + # modes an inventive user will come up with. + if c.shape[-1] != con.num_constr or (S > 1 and c.shape[0] != S): + raise RuntimeError("An array returned from a Constraint has" + " the wrong shape. If `vectorized is False`" + " the Constraint should return an array of" + " shape (M,). If `vectorized is True` then" + " the Constraint must return an array of" + " shape (M, S), where S is the number of" + " solution vectors and M is the number of" + " constraint components in a given" + " Constraint object.") + + # the violation function may return a 1D array, but is it a + # sequence of constraints for one solution (S=1, M>=1), or the + # value of a single constraint for a sequence of solutions + # (S>=1, M=1) + c = np.reshape(c, (S, con.num_constr)) + _out[:, offset:offset + con.num_constr] = c + offset += con.num_constr + + return _out + + def _calculate_population_feasibilities(self, population): + """ + Calculate the feasibilities of a population. + + Parameters + ---------- + population : ndarray + An array of parameter vectors normalised to [0, 1] using lower + and upper limits. Has shape ``(np.size(population, 0), N)``. + + Returns + ------- + feasible, constraint_violation : ndarray, ndarray + Boolean array of feasibility for each population member, and an + array of the constraint violation for each population member. + constraint_violation has shape ``(np.size(population, 0), M)``, + where M is the number of constraints. + """ + num_members = np.size(population, 0) + if not self._wrapped_constraints: + # shortcut for no constraints + return np.ones(num_members, bool), np.zeros((num_members, 1)) + + # (S, N) + parameters_pop = self._scale_parameters(population) + + if self.vectorized: + # (S, M) + constraint_violation = np.array( + self._constraint_violation_fn(parameters_pop) + ) + else: + # (S, 1, M) + constraint_violation = np.array([self._constraint_violation_fn(x) + for x in parameters_pop]) + # if you use the list comprehension in the line above it will + # create an array of shape (S, 1, M), because each iteration + # generates an array of (1, M). In comparison the vectorized + # version returns (S, M). It's therefore necessary to remove axis 1 + constraint_violation = constraint_violation[:, 0] + + feasible = ~(np.sum(constraint_violation, axis=1) > 0) + + return feasible, constraint_violation + + def __iter__(self): + return self + + def __enter__(self): + return self + + def __exit__(self, *args): + return self._mapwrapper.__exit__(*args) + + def _accept_trial(self, energy_trial, feasible_trial, cv_trial, + energy_orig, feasible_orig, cv_orig): + """ + Trial is accepted if: + * it satisfies all constraints and provides a lower or equal objective + function value, while both the compared solutions are feasible + - or - + * it is feasible while the original solution is infeasible, + - or - + * it is infeasible, but provides a lower or equal constraint violation + for all constraint functions. + + This test corresponds to section III of Lampinen [1]_. + + Parameters + ---------- + energy_trial : float + Energy of the trial solution + feasible_trial : float + Feasibility of trial solution + cv_trial : array-like + Excess constraint violation for the trial solution + energy_orig : float + Energy of the original solution + feasible_orig : float + Feasibility of original solution + cv_orig : array-like + Excess constraint violation for the original solution + + Returns + ------- + accepted : bool + + """ + if feasible_orig and feasible_trial: + return energy_trial <= energy_orig + elif feasible_trial and not feasible_orig: + return True + elif not feasible_trial and (cv_trial <= cv_orig).all(): + # cv_trial < cv_orig would imply that both trial and orig are not + # feasible + return True + + return False + + def __next__(self): + """ + Evolve the population by a single generation + + Returns + ------- + x : ndarray + The best solution from the solver. + fun : float + Value of objective function obtained from the best solution. + """ + # the population may have just been initialized (all entries are + # np.inf). If it has you have to calculate the initial energies + if np.all(np.isinf(self.population_energies)): + self.feasible, self.constraint_violation = ( + self._calculate_population_feasibilities(self.population)) + + # only need to work out population energies for those that are + # feasible + self.population_energies[self.feasible] = ( + self._calculate_population_energies( + self.population[self.feasible])) + + self._promote_lowest_energy() + + if self.dither is not None: + self.scale = self.random_number_generator.uniform(self.dither[0], + self.dither[1]) + + if self._updating == 'immediate': + # update best solution immediately + for candidate in range(self.num_population_members): + if self._nfev > self.maxfun: + raise StopIteration + + # create a trial solution + trial = self._mutate(candidate) + + # ensuring that it's in the range [0, 1) + self._ensure_constraint(trial) + + # scale from [0, 1) to the actual parameter value + parameters = self._scale_parameters(trial) + + # determine the energy of the objective function + if self._wrapped_constraints: + cv = self._constraint_violation_fn(parameters) + feasible = False + energy = np.inf + if not np.sum(cv) > 0: + # solution is feasible + feasible = True + energy = self.func(parameters) + self._nfev += 1 + else: + feasible = True + cv = np.atleast_2d([0.]) + energy = self.func(parameters) + self._nfev += 1 + + # compare trial and population member + if self._accept_trial(energy, feasible, cv, + self.population_energies[candidate], + self.feasible[candidate], + self.constraint_violation[candidate]): + self.population[candidate] = trial + self.population_energies[candidate] = np.squeeze(energy) + self.feasible[candidate] = feasible + self.constraint_violation[candidate] = cv + + # if the trial candidate is also better than the best + # solution then promote it. + if self._accept_trial(energy, feasible, cv, + self.population_energies[0], + self.feasible[0], + self.constraint_violation[0]): + self._promote_lowest_energy() + + elif self._updating == 'deferred': + # update best solution once per generation + if self._nfev >= self.maxfun: + raise StopIteration + + # 'deferred' approach, vectorised form. + # create trial solutions + trial_pop = self._mutate_many( + np.arange(self.num_population_members) + ) + + # enforce bounds + self._ensure_constraint(trial_pop) + + # determine the energies of the objective function, but only for + # feasible trials + feasible, cv = self._calculate_population_feasibilities(trial_pop) + trial_energies = np.full(self.num_population_members, np.inf) + + # only calculate for feasible entries + trial_energies[feasible] = self._calculate_population_energies( + trial_pop[feasible]) + + # which solutions are 'improved'? + loc = [self._accept_trial(*val) for val in + zip(trial_energies, feasible, cv, self.population_energies, + self.feasible, self.constraint_violation)] + loc = np.array(loc) + self.population = np.where(loc[:, np.newaxis], + trial_pop, + self.population) + self.population_energies = np.where(loc, + trial_energies, + self.population_energies) + self.feasible = np.where(loc, + feasible, + self.feasible) + self.constraint_violation = np.where(loc[:, np.newaxis], + cv, + self.constraint_violation) + + # make sure the best solution is updated if updating='deferred'. + # put the lowest energy into the best solution position. + self._promote_lowest_energy() + + return self.x, self.population_energies[0] + + def _scale_parameters(self, trial): + """Scale from a number between 0 and 1 to parameters.""" + # trial either has shape (N, ) or (L, N), where L is the number of + # solutions being scaled + scaled = self.__scale_arg1 + (trial - 0.5) * self.__scale_arg2 + if np.count_nonzero(self.integrality): + i = np.broadcast_to(self.integrality, scaled.shape) + scaled[i] = np.round(scaled[i]) + return scaled + + def _unscale_parameters(self, parameters): + """Scale from parameters to a number between 0 and 1.""" + return (parameters - self.__scale_arg1) * self.__recip_scale_arg2 + 0.5 + + def _ensure_constraint(self, trial): + """Make sure the parameters lie between the limits.""" + mask = np.bitwise_or(trial > 1, trial < 0) + if oob := np.count_nonzero(mask): + trial[mask] = self.random_number_generator.uniform(size=oob) + + def _mutate_custom(self, candidate): + rng = self.random_number_generator + msg = ( + "strategy must have signature" + " f(candidate: int, population: np.ndarray, rng=None) returning an" + " array of shape (N,)" + ) + _population = self._scale_parameters(self.population) + if not len(np.shape(candidate)): + # single entry in population + trial = self.strategy(candidate, _population, rng=rng) + if trial.shape != (self.parameter_count,): + raise RuntimeError(msg) + else: + S = candidate.shape[0] + trial = np.array( + [self.strategy(c, _population, rng=rng) for c in candidate], + dtype=float + ) + if trial.shape != (S, self.parameter_count): + raise RuntimeError(msg) + return self._unscale_parameters(trial) + + def _mutate_many(self, candidates): + """Create trial vectors based on a mutation strategy.""" + rng = self.random_number_generator + + S = len(candidates) + if callable(self.strategy): + return self._mutate_custom(candidates) + + trial = np.copy(self.population[candidates]) + samples = np.array([self._select_samples(c, 5) for c in candidates]) + + if self.strategy in ['currenttobest1exp', 'currenttobest1bin']: + bprime = self.mutation_func(candidates, samples) + else: + bprime = self.mutation_func(samples) + + fill_point = rng_integers(rng, self.parameter_count, size=S) + crossovers = rng.uniform(size=(S, self.parameter_count)) + crossovers = crossovers < self.cross_over_probability + if self.strategy in self._binomial: + # the last one is always from the bprime vector for binomial + # If you fill in modulo with a loop you have to set the last one to + # true. If you don't use a loop then you can have any random entry + # be True. + i = np.arange(S) + crossovers[i, fill_point[i]] = True + trial = np.where(crossovers, bprime, trial) + return trial + + elif self.strategy in self._exponential: + crossovers[..., 0] = True + for j in range(S): + i = 0 + init_fill = fill_point[j] + while (i < self.parameter_count and crossovers[j, i]): + trial[j, init_fill] = bprime[j, init_fill] + init_fill = (init_fill + 1) % self.parameter_count + i += 1 + + return trial + + def _mutate(self, candidate): + """Create a trial vector based on a mutation strategy.""" + rng = self.random_number_generator + + if callable(self.strategy): + return self._mutate_custom(candidate) + + fill_point = rng_integers(rng, self.parameter_count) + samples = self._select_samples(candidate, 5) + + trial = np.copy(self.population[candidate]) + + if self.strategy in ['currenttobest1exp', 'currenttobest1bin']: + bprime = self.mutation_func(candidate, samples) + else: + bprime = self.mutation_func(samples) + + crossovers = rng.uniform(size=self.parameter_count) + crossovers = crossovers < self.cross_over_probability + if self.strategy in self._binomial: + # the last one is always from the bprime vector for binomial + # If you fill in modulo with a loop you have to set the last one to + # true. If you don't use a loop then you can have any random entry + # be True. + crossovers[fill_point] = True + trial = np.where(crossovers, bprime, trial) + return trial + + elif self.strategy in self._exponential: + i = 0 + crossovers[0] = True + while i < self.parameter_count and crossovers[i]: + trial[fill_point] = bprime[fill_point] + fill_point = (fill_point + 1) % self.parameter_count + i += 1 + + return trial + + def _best1(self, samples): + """best1bin, best1exp""" + # samples.shape == (S, 5) + # or + # samples.shape(5,) + r0, r1 = samples[..., :2].T + return (self.population[0] + self.scale * + (self.population[r0] - self.population[r1])) + + def _rand1(self, samples): + """rand1bin, rand1exp""" + r0, r1, r2 = samples[..., :3].T + return (self.population[r0] + self.scale * + (self.population[r1] - self.population[r2])) + + def _randtobest1(self, samples): + """randtobest1bin, randtobest1exp""" + r0, r1, r2 = samples[..., :3].T + bprime = np.copy(self.population[r0]) + bprime += self.scale * (self.population[0] - bprime) + bprime += self.scale * (self.population[r1] - + self.population[r2]) + return bprime + + def _currenttobest1(self, candidate, samples): + """currenttobest1bin, currenttobest1exp""" + r0, r1 = samples[..., :2].T + bprime = (self.population[candidate] + self.scale * + (self.population[0] - self.population[candidate] + + self.population[r0] - self.population[r1])) + return bprime + + def _best2(self, samples): + """best2bin, best2exp""" + r0, r1, r2, r3 = samples[..., :4].T + bprime = (self.population[0] + self.scale * + (self.population[r0] + self.population[r1] - + self.population[r2] - self.population[r3])) + + return bprime + + def _rand2(self, samples): + """rand2bin, rand2exp""" + r0, r1, r2, r3, r4 = samples[..., :5].T + bprime = (self.population[r0] + self.scale * + (self.population[r1] + self.population[r2] - + self.population[r3] - self.population[r4])) + + return bprime + + def _select_samples(self, candidate, number_samples): + """ + obtain random integers from range(self.num_population_members), + without replacement. You can't have the original candidate either. + """ + self.random_number_generator.shuffle(self._random_population_index) + idxs = self._random_population_index[:number_samples + 1] + return idxs[idxs != candidate][:number_samples] + + +class _ConstraintWrapper: + """Object to wrap/evaluate user defined constraints. + + Very similar in practice to `PreparedConstraint`, except that no evaluation + of jac/hess is performed (explicit or implicit). + + If created successfully, it will contain the attributes listed below. + + Parameters + ---------- + constraint : {`NonlinearConstraint`, `LinearConstraint`, `Bounds`} + Constraint to check and prepare. + x0 : array_like + Initial vector of independent variables, shape (N,) + + Attributes + ---------- + fun : callable + Function defining the constraint wrapped by one of the convenience + classes. + bounds : 2-tuple + Contains lower and upper bounds for the constraints --- lb and ub. + These are converted to ndarray and have a size equal to the number of + the constraints. + + Notes + ----- + _ConstraintWrapper.fun and _ConstraintWrapper.violation can get sent + arrays of shape (N, S) or (N,), where S is the number of vectors of shape + (N,) to consider constraints for. + """ + def __init__(self, constraint, x0): + self.constraint = constraint + + if isinstance(constraint, NonlinearConstraint): + def fun(x): + x = np.asarray(x) + return np.atleast_1d(constraint.fun(x)) + elif isinstance(constraint, LinearConstraint): + def fun(x): + if issparse(constraint.A): + A = constraint.A + else: + A = np.atleast_2d(constraint.A) + + res = A.dot(x) + # x either has shape (N, S) or (N) + # (M, N) x (N, S) --> (M, S) + # (M, N) x (N,) --> (M,) + # However, if (M, N) is a matrix then: + # (M, N) * (N,) --> (M, 1), we need this to be (M,) + if x.ndim == 1 and res.ndim == 2: + # deal with case that constraint.A is an np.matrix + # see gh20041 + res = np.asarray(res)[:, 0] + + return res + elif isinstance(constraint, Bounds): + def fun(x): + return np.asarray(x) + else: + raise ValueError("`constraint` of an unknown type is passed.") + + self.fun = fun + + lb = np.asarray(constraint.lb, dtype=float) + ub = np.asarray(constraint.ub, dtype=float) + + x0 = np.asarray(x0) + + # find out the number of constraints + f0 = fun(x0) + self.num_constr = m = f0.size + self.parameter_count = x0.size + + if lb.ndim == 0: + lb = np.resize(lb, m) + if ub.ndim == 0: + ub = np.resize(ub, m) + + self.bounds = (lb, ub) + + def __call__(self, x): + return np.atleast_1d(self.fun(x)) + + def violation(self, x): + """How much the constraint is exceeded by. + + Parameters + ---------- + x : array-like + Vector of independent variables, (N, S), where N is number of + parameters and S is the number of solutions to be investigated. + + Returns + ------- + excess : array-like + How much the constraint is exceeded by, for each of the + constraints specified by `_ConstraintWrapper.fun`. + Has shape (M, S) where M is the number of constraint components. + """ + # expect ev to have shape (num_constr, S) or (num_constr,) + ev = self.fun(np.asarray(x)) + + try: + excess_lb = np.maximum(self.bounds[0] - ev.T, 0) + excess_ub = np.maximum(ev.T - self.bounds[1], 0) + except ValueError as e: + raise RuntimeError("An array returned from a Constraint has" + " the wrong shape. If `vectorized is False`" + " the Constraint should return an array of" + " shape (M,). If `vectorized is True` then" + " the Constraint must return an array of" + " shape (M, S), where S is the number of" + " solution vectors and M is the number of" + " constraint components in a given" + " Constraint object.") from e + + v = (excess_lb + excess_ub).T + return v diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_direct.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_direct.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..fe89bac4a6f6b828def53da54b5eac17d45a4019 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_direct.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_direct_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_direct_py.py new file mode 100644 index 0000000000000000000000000000000000000000..4c01c38747dbef4b9c71e9b593316f466f484bca --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_direct_py.py @@ -0,0 +1,280 @@ +from typing import ( # noqa: UP035 + Any, Callable, Iterable +) + +import numpy as np +from scipy.optimize import OptimizeResult +from ._constraints import old_bound_to_new, Bounds +from ._direct import direct as _direct # type: ignore + +__all__ = ['direct'] + +ERROR_MESSAGES = ( + "Number of function evaluations done is larger than maxfun={}", + "Number of iterations is larger than maxiter={}", + "u[i] < l[i] for some i", + "maxfun is too large", + "Initialization failed", + "There was an error in the creation of the sample points", + "An error occurred while the function was sampled", + "Maximum number of levels has been reached.", + "Forced stop", + "Invalid arguments", + "Out of memory", +) + +SUCCESS_MESSAGES = ( + ("The best function value found is within a relative error={} " + "of the (known) global optimum f_min"), + ("The volume of the hyperrectangle containing the lowest function value " + "found is below vol_tol={}"), + ("The side length measure of the hyperrectangle containing the lowest " + "function value found is below len_tol={}"), +) + + +def direct( + func: Callable[ + [np.ndarray[tuple[int], np.dtype[np.float64]]], + float | np.floating[Any] | np.integer[Any] | np.bool_, + ], + bounds: Iterable | Bounds, + *, + args: tuple = (), + eps: float = 1e-4, + maxfun: int | None = None, + maxiter: int = 1000, + locally_biased: bool = True, + f_min: float = -np.inf, + f_min_rtol: float = 1e-4, + vol_tol: float = 1e-16, + len_tol: float = 1e-6, + callback: Callable[ + [np.ndarray[tuple[int], np.dtype[np.float64]]], + object, + ] | None = None, +) -> OptimizeResult: + """ + Finds the global minimum of a function using the + DIRECT algorithm. + + Parameters + ---------- + func : callable + The objective function to be minimized. + ``func(x, *args) -> float`` + where ``x`` is an 1-D array with shape (n,) and ``args`` is a tuple of + the fixed parameters needed to completely specify the function. + bounds : sequence or `Bounds` + Bounds for variables. There are two ways to specify the bounds: + + 1. Instance of `Bounds` class. + 2. ``(min, max)`` pairs for each element in ``x``. + + args : tuple, optional + Any additional fixed parameters needed to + completely specify the objective function. + eps : float, optional + Minimal required difference of the objective function values + between the current best hyperrectangle and the next potentially + optimal hyperrectangle to be divided. In consequence, `eps` serves as a + tradeoff between local and global search: the smaller, the more local + the search becomes. Default is 1e-4. + maxfun : int or None, optional + Approximate upper bound on objective function evaluations. + If `None`, will be automatically set to ``1000 * N`` where ``N`` + represents the number of dimensions. Will be capped if necessary to + limit DIRECT's RAM usage to app. 1GiB. This will only occur for very + high dimensional problems and excessive `max_fun`. Default is `None`. + maxiter : int, optional + Maximum number of iterations. Default is 1000. + locally_biased : bool, optional + If `True` (default), use the locally biased variant of the + algorithm known as DIRECT_L. If `False`, use the original unbiased + DIRECT algorithm. For hard problems with many local minima, + `False` is recommended. + f_min : float, optional + Function value of the global optimum. Set this value only if the + global optimum is known. Default is ``-np.inf``, so that this + termination criterion is deactivated. + f_min_rtol : float, optional + Terminate the optimization once the relative error between the + current best minimum `f` and the supplied global minimum `f_min` + is smaller than `f_min_rtol`. This parameter is only used if + `f_min` is also set. Must lie between 0 and 1. Default is 1e-4. + vol_tol : float, optional + Terminate the optimization once the volume of the hyperrectangle + containing the lowest function value is smaller than `vol_tol` + of the complete search space. Must lie between 0 and 1. + Default is 1e-16. + len_tol : float, optional + If ``locally_biased=True``, terminate the optimization once half of + the normalized maximal side length of the hyperrectangle containing + the lowest function value is smaller than `len_tol`. + If ``locally_biased=False``, terminate the optimization once half of + the normalized diagonal of the hyperrectangle containing the lowest + function value is smaller than `len_tol`. Must lie between 0 and 1. + Default is 1e-6. + callback : callable, optional + A callback function with signature ``callback(xk)`` where ``xk`` + represents the best function value found so far. + + Returns + ------- + res : OptimizeResult + The optimization result represented as a ``OptimizeResult`` object. + Important attributes are: ``x`` the solution array, ``success`` a + Boolean flag indicating if the optimizer exited successfully and + ``message`` which describes the cause of the termination. See + `OptimizeResult` for a description of other attributes. + + Notes + ----- + DIviding RECTangles (DIRECT) is a deterministic global + optimization algorithm capable of minimizing a black box function with + its variables subject to lower and upper bound constraints by sampling + potential solutions in the search space [1]_. The algorithm starts by + normalising the search space to an n-dimensional unit hypercube. + It samples the function at the center of this hypercube and at 2n + (n is the number of variables) more points, 2 in each coordinate + direction. Using these function values, DIRECT then divides the + domain into hyperrectangles, each having exactly one of the sampling + points as its center. In each iteration, DIRECT chooses, using the `eps` + parameter which defaults to 1e-4, some of the existing hyperrectangles + to be further divided. This division process continues until either the + maximum number of iterations or maximum function evaluations allowed + are exceeded, or the hyperrectangle containing the minimal value found + so far becomes small enough. If `f_min` is specified, the optimization + will stop once this function value is reached within a relative tolerance. + The locally biased variant of DIRECT (originally called DIRECT_L) [2]_ is + used by default. It makes the search more locally biased and more + efficient for cases with only a few local minima. + + A note about termination criteria: `vol_tol` refers to the volume of the + hyperrectangle containing the lowest function value found so far. This + volume decreases exponentially with increasing dimensionality of the + problem. Therefore `vol_tol` should be decreased to avoid premature + termination of the algorithm for higher dimensions. This does not hold + for `len_tol`: it refers either to half of the maximal side length + (for ``locally_biased=True``) or half of the diagonal of the + hyperrectangle (for ``locally_biased=False``). + + This code is based on the DIRECT 2.0.4 Fortran code by Gablonsky et al. at + https://ctk.math.ncsu.edu/SOFTWARE/DIRECTv204.tar.gz . + This original version was initially converted via f2c and then cleaned up + and reorganized by Steven G. Johnson, August 2007, for the NLopt project. + The `direct` function wraps the C implementation. + + .. versionadded:: 1.9.0 + + References + ---------- + .. [1] Jones, D.R., Perttunen, C.D. & Stuckman, B.E. Lipschitzian + optimization without the Lipschitz constant. J Optim Theory Appl + 79, 157-181 (1993). + .. [2] Gablonsky, J., Kelley, C. A Locally-Biased form of the DIRECT + Algorithm. Journal of Global Optimization 21, 27-37 (2001). + + Examples + -------- + The following example is a 2-D problem with four local minima: minimizing + the Styblinski-Tang function + (https://en.wikipedia.org/wiki/Test_functions_for_optimization). + + >>> from scipy.optimize import direct, Bounds + >>> def styblinski_tang(pos): + ... x, y = pos + ... return 0.5 * (x**4 - 16*x**2 + 5*x + y**4 - 16*y**2 + 5*y) + >>> bounds = Bounds([-4., -4.], [4., 4.]) + >>> result = direct(styblinski_tang, bounds) + >>> result.x, result.fun, result.nfev + array([-2.90321597, -2.90321597]), -78.3323279095383, 2011 + + The correct global minimum was found but with a huge number of function + evaluations (2011). Loosening the termination tolerances `vol_tol` and + `len_tol` can be used to stop DIRECT earlier. + + >>> result = direct(styblinski_tang, bounds, len_tol=1e-3) + >>> result.x, result.fun, result.nfev + array([-2.9044353, -2.9044353]), -78.33230330754142, 207 + + """ + # convert bounds to new Bounds class if necessary + if not isinstance(bounds, Bounds): + if isinstance(bounds, list) or isinstance(bounds, tuple): + lb, ub = old_bound_to_new(bounds) + bounds = Bounds(lb, ub) + else: + message = ("bounds must be a sequence or " + "instance of Bounds class") + raise ValueError(message) + + lb = np.ascontiguousarray(bounds.lb, dtype=np.float64) + ub = np.ascontiguousarray(bounds.ub, dtype=np.float64) + + # validate bounds + # check that lower bounds are smaller than upper bounds + if not np.all(lb < ub): + raise ValueError('Bounds are not consistent min < max') + # check for infs + if (np.any(np.isinf(lb)) or np.any(np.isinf(ub))): + raise ValueError("Bounds must not be inf.") + + # validate tolerances + if (vol_tol < 0 or vol_tol > 1): + raise ValueError("vol_tol must be between 0 and 1.") + if (len_tol < 0 or len_tol > 1): + raise ValueError("len_tol must be between 0 and 1.") + if (f_min_rtol < 0 or f_min_rtol > 1): + raise ValueError("f_min_rtol must be between 0 and 1.") + + # validate maxfun and maxiter + if maxfun is None: + maxfun = 1000 * lb.shape[0] + if not isinstance(maxfun, int): + raise ValueError("maxfun must be of type int.") + if maxfun < 0: + raise ValueError("maxfun must be > 0.") + if not isinstance(maxiter, int): + raise ValueError("maxiter must be of type int.") + if maxiter < 0: + raise ValueError("maxiter must be > 0.") + + # validate boolean parameters + if not isinstance(locally_biased, bool): + raise ValueError("locally_biased must be True or False.") + + def _func_wrap(x, args=None): + x = np.asarray(x) + if args is None: + f = func(x) + else: + f = func(x, *args) + # always return a float + return np.asarray(f).item() + + # TODO: fix disp argument + x, fun, ret_code, nfev, nit = _direct( + _func_wrap, + np.asarray(lb), np.asarray(ub), + args, + False, eps, maxfun, maxiter, + locally_biased, + f_min, f_min_rtol, + vol_tol, len_tol, callback + ) + + format_val = (maxfun, maxiter, f_min_rtol, vol_tol, len_tol) + if ret_code > 2: + message = SUCCESS_MESSAGES[ret_code - 3].format( + format_val[ret_code - 1]) + elif 0 < ret_code <= 2: + message = ERROR_MESSAGES[ret_code - 1].format(format_val[ret_code - 1]) + elif 0 > ret_code > -100: + message = ERROR_MESSAGES[abs(ret_code) + 1] + else: + message = ERROR_MESSAGES[ret_code + 99] + + return OptimizeResult(x=np.asarray(x), fun=fun, status=ret_code, + success=ret_code > 2, message=message, + nfev=nfev, nit=nit) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_dual_annealing.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_dual_annealing.py new file mode 100644 index 0000000000000000000000000000000000000000..eb480a902c593ffee1d242d79018c9175bcc6d3a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_dual_annealing.py @@ -0,0 +1,732 @@ +# Dual Annealing implementation. +# Copyright (c) 2018 Sylvain Gubian , +# Yang Xiang +# Author: Sylvain Gubian, Yang Xiang, PMP S.A. + +""" +A Dual Annealing global optimization algorithm +""" + +import numpy as np +from scipy.optimize import OptimizeResult +from scipy.optimize import minimize, Bounds +from scipy.special import gammaln +from scipy._lib._util import check_random_state, _transition_to_rng +from scipy.optimize._constraints import new_bounds_to_old + +__all__ = ['dual_annealing'] + + +class VisitingDistribution: + """ + Class used to generate new coordinates based on the distorted + Cauchy-Lorentz distribution. Depending on the steps within the strategy + chain, the class implements the strategy for generating new location + changes. + + Parameters + ---------- + lb : array_like + A 1-D NumPy ndarray containing lower bounds of the generated + components. Neither NaN or inf are allowed. + ub : array_like + A 1-D NumPy ndarray containing upper bounds for the generated + components. Neither NaN or inf are allowed. + visiting_param : float + Parameter for visiting distribution. Default value is 2.62. + Higher values give the visiting distribution a heavier tail, this + makes the algorithm jump to a more distant region. + The value range is (1, 3]. Its value is fixed for the life of the + object. + rng_gen : {`~numpy.random.Generator`} + A `~numpy.random.Generator` object for generating new locations. + (can be a `~numpy.random.RandomState` object until SPEC007 transition + is fully complete). + + """ + TAIL_LIMIT = 1.e8 + MIN_VISIT_BOUND = 1.e-10 + + def __init__(self, lb, ub, visiting_param, rng_gen): + # if you wish to make _visiting_param adjustable during the life of + # the object then _factor2, _factor3, _factor5, _d1, _factor6 will + # have to be dynamically calculated in `visit_fn`. They're factored + # out here so they don't need to be recalculated all the time. + self._visiting_param = visiting_param + self.rng_gen = rng_gen + self.lower = lb + self.upper = ub + self.bound_range = ub - lb + + # these are invariant numbers unless visiting_param changes + self._factor2 = np.exp((4.0 - self._visiting_param) * np.log( + self._visiting_param - 1.0)) + self._factor3 = np.exp((2.0 - self._visiting_param) * np.log(2.0) + / (self._visiting_param - 1.0)) + self._factor4_p = np.sqrt(np.pi) * self._factor2 / (self._factor3 * ( + 3.0 - self._visiting_param)) + + self._factor5 = 1.0 / (self._visiting_param - 1.0) - 0.5 + self._d1 = 2.0 - self._factor5 + self._factor6 = np.pi * (1.0 - self._factor5) / np.sin( + np.pi * (1.0 - self._factor5)) / np.exp(gammaln(self._d1)) + + def visiting(self, x, step, temperature): + """ Based on the step in the strategy chain, new coordinates are + generated by changing all components is the same time or only + one of them, the new values are computed with visit_fn method + """ + dim = x.size + if step < dim: + # Changing all coordinates with a new visiting value + visits = self.visit_fn(temperature, dim) + upper_sample, lower_sample = self.rng_gen.uniform(size=2) + visits[visits > self.TAIL_LIMIT] = self.TAIL_LIMIT * upper_sample + visits[visits < -self.TAIL_LIMIT] = -self.TAIL_LIMIT * lower_sample + x_visit = visits + x + a = x_visit - self.lower + b = np.fmod(a, self.bound_range) + self.bound_range + x_visit = np.fmod(b, self.bound_range) + self.lower + x_visit[np.fabs( + x_visit - self.lower) < self.MIN_VISIT_BOUND] += 1.e-10 + else: + # Changing only one coordinate at a time based on strategy + # chain step + x_visit = np.copy(x) + visit = self.visit_fn(temperature, 1)[0] + if visit > self.TAIL_LIMIT: + visit = self.TAIL_LIMIT * self.rng_gen.uniform() + elif visit < -self.TAIL_LIMIT: + visit = -self.TAIL_LIMIT * self.rng_gen.uniform() + index = step - dim + x_visit[index] = visit + x[index] + a = x_visit[index] - self.lower[index] + b = np.fmod(a, self.bound_range[index]) + self.bound_range[index] + x_visit[index] = np.fmod(b, self.bound_range[ + index]) + self.lower[index] + if np.fabs(x_visit[index] - self.lower[ + index]) < self.MIN_VISIT_BOUND: + x_visit[index] += self.MIN_VISIT_BOUND + return x_visit + + def visit_fn(self, temperature, dim): + """ Formula Visita from p. 405 of reference [2] """ + x, y = self.rng_gen.normal(size=(dim, 2)).T + + factor1 = np.exp(np.log(temperature) / (self._visiting_param - 1.0)) + factor4 = self._factor4_p * factor1 + + # sigmax + x *= np.exp(-(self._visiting_param - 1.0) * np.log( + self._factor6 / factor4) / (3.0 - self._visiting_param)) + + den = np.exp((self._visiting_param - 1.0) * np.log(np.fabs(y)) / + (3.0 - self._visiting_param)) + + return x / den + + +class EnergyState: + """ + Class used to record the energy state. At any time, it knows what is the + currently used coordinates and the most recent best location. + + Parameters + ---------- + lower : array_like + A 1-D NumPy ndarray containing lower bounds for generating an initial + random components in the `reset` method. + upper : array_like + A 1-D NumPy ndarray containing upper bounds for generating an initial + random components in the `reset` method + components. Neither NaN or inf are allowed. + callback : callable, ``callback(x, f, context)``, optional + A callback function which will be called for all minima found. + ``x`` and ``f`` are the coordinates and function value of the + latest minimum found, and `context` has value in [0, 1, 2] + """ + # Maximum number of trials for generating a valid starting point + MAX_REINIT_COUNT = 1000 + + def __init__(self, lower, upper, callback=None): + self.ebest = None + self.current_energy = None + self.current_location = None + self.xbest = None + self.lower = lower + self.upper = upper + self.callback = callback + + def reset(self, func_wrapper, rng_gen, x0=None): + """ + Initialize current location is the search domain. If `x0` is not + provided, a random location within the bounds is generated. + """ + if x0 is None: + self.current_location = rng_gen.uniform(self.lower, self.upper, + size=len(self.lower)) + else: + self.current_location = np.copy(x0) + init_error = True + reinit_counter = 0 + while init_error: + self.current_energy = func_wrapper.fun(self.current_location) + if self.current_energy is None: + raise ValueError('Objective function is returning None') + if not np.isfinite(self.current_energy): + if reinit_counter >= EnergyState.MAX_REINIT_COUNT: + init_error = False + message = ( + 'Stopping algorithm because function ' + 'create NaN or (+/-) infinity values even with ' + 'trying new random parameters' + ) + raise ValueError(message) + self.current_location = rng_gen.uniform(self.lower, + self.upper, + size=self.lower.size) + reinit_counter += 1 + else: + init_error = False + # If first time reset, initialize ebest and xbest + if self.ebest is None and self.xbest is None: + self.ebest = self.current_energy + self.xbest = np.copy(self.current_location) + # Otherwise, we keep them in case of reannealing reset + + def update_best(self, e, x, context): + self.ebest = e + self.xbest = np.copy(x) + if self.callback is not None: + val = self.callback(x, e, context) + if val is not None: + if val: + return ('Callback function requested to stop early by ' + 'returning True') + + def update_current(self, e, x): + self.current_energy = e + self.current_location = np.copy(x) + + +class StrategyChain: + """ + Class that implements within a Markov chain the strategy for location + acceptance and local search decision making. + + Parameters + ---------- + acceptance_param : float + Parameter for acceptance distribution. It is used to control the + probability of acceptance. The lower the acceptance parameter, the + smaller the probability of acceptance. Default value is -5.0 with + a range (-1e4, -5]. + visit_dist : VisitingDistribution + Instance of `VisitingDistribution` class. + func_wrapper : ObjectiveFunWrapper + Instance of `ObjectiveFunWrapper` class. + minimizer_wrapper: LocalSearchWrapper + Instance of `LocalSearchWrapper` class. + rand_gen : {None, int, `numpy.random.Generator`, + `numpy.random.RandomState`}, optional + + If `seed` is None (or `np.random`), the `numpy.random.RandomState` + singleton is used. + If `seed` is an int, a new ``RandomState`` instance is used, + seeded with `seed`. + If `seed` is already a ``Generator`` or ``RandomState`` instance then + that instance is used. + energy_state: EnergyState + Instance of `EnergyState` class. + + """ + + def __init__(self, acceptance_param, visit_dist, func_wrapper, + minimizer_wrapper, rand_gen, energy_state): + # Local strategy chain minimum energy and location + self.emin = energy_state.current_energy + self.xmin = np.array(energy_state.current_location) + # Global optimizer state + self.energy_state = energy_state + # Acceptance parameter + self.acceptance_param = acceptance_param + # Visiting distribution instance + self.visit_dist = visit_dist + # Wrapper to objective function + self.func_wrapper = func_wrapper + # Wrapper to the local minimizer + self.minimizer_wrapper = minimizer_wrapper + self.not_improved_idx = 0 + self.not_improved_max_idx = 1000 + self._rand_gen = rand_gen + self.temperature_step = 0 + self.K = 100 * len(energy_state.current_location) + + def accept_reject(self, j, e, x_visit): + r = self._rand_gen.uniform() + pqv_temp = 1.0 - ((1.0 - self.acceptance_param) * + (e - self.energy_state.current_energy) / self.temperature_step) + if pqv_temp <= 0.: + pqv = 0. + else: + pqv = np.exp(np.log(pqv_temp) / ( + 1. - self.acceptance_param)) + + if r <= pqv: + # We accept the new location and update state + self.energy_state.update_current(e, x_visit) + self.xmin = np.copy(self.energy_state.current_location) + + # No improvement for a long time + if self.not_improved_idx >= self.not_improved_max_idx: + if j == 0 or self.energy_state.current_energy < self.emin: + self.emin = self.energy_state.current_energy + self.xmin = np.copy(self.energy_state.current_location) + + def run(self, step, temperature): + self.temperature_step = temperature / float(step + 1) + self.not_improved_idx += 1 + for j in range(self.energy_state.current_location.size * 2): + if j == 0: + if step == 0: + self.energy_state_improved = True + else: + self.energy_state_improved = False + x_visit = self.visit_dist.visiting( + self.energy_state.current_location, j, temperature) + # Calling the objective function + e = self.func_wrapper.fun(x_visit) + if e < self.energy_state.current_energy: + # We have got a better energy value + self.energy_state.update_current(e, x_visit) + if e < self.energy_state.ebest: + val = self.energy_state.update_best(e, x_visit, 0) + if val is not None: + if val: + return val + self.energy_state_improved = True + self.not_improved_idx = 0 + else: + # We have not improved but do we accept the new location? + self.accept_reject(j, e, x_visit) + if self.func_wrapper.nfev >= self.func_wrapper.maxfun: + return ('Maximum number of function call reached ' + 'during annealing') + # End of StrategyChain loop + + def local_search(self): + # Decision making for performing a local search + # based on strategy chain results + # If energy has been improved or no improvement since too long, + # performing a local search with the best strategy chain location + if self.energy_state_improved: + # Global energy has improved, let's see if LS improves further + e, x = self.minimizer_wrapper.local_search(self.energy_state.xbest, + self.energy_state.ebest) + if e < self.energy_state.ebest: + self.not_improved_idx = 0 + val = self.energy_state.update_best(e, x, 1) + if val is not None: + if val: + return val + self.energy_state.update_current(e, x) + if self.func_wrapper.nfev >= self.func_wrapper.maxfun: + return ('Maximum number of function call reached ' + 'during local search') + # Check probability of a need to perform a LS even if no improvement + do_ls = False + if self.K < 90 * len(self.energy_state.current_location): + pls = np.exp(self.K * ( + self.energy_state.ebest - self.energy_state.current_energy) / + self.temperature_step) + if pls >= self._rand_gen.uniform(): + do_ls = True + # Global energy not improved, let's see what LS gives + # on the best strategy chain location + if self.not_improved_idx >= self.not_improved_max_idx: + do_ls = True + if do_ls: + e, x = self.minimizer_wrapper.local_search(self.xmin, self.emin) + self.xmin = np.copy(x) + self.emin = e + self.not_improved_idx = 0 + self.not_improved_max_idx = self.energy_state.current_location.size + if e < self.energy_state.ebest: + val = self.energy_state.update_best( + self.emin, self.xmin, 2) + if val is not None: + if val: + return val + self.energy_state.update_current(e, x) + if self.func_wrapper.nfev >= self.func_wrapper.maxfun: + return ('Maximum number of function call reached ' + 'during dual annealing') + + +class ObjectiveFunWrapper: + + def __init__(self, func, maxfun=1e7, *args): + self.func = func + self.args = args + # Number of objective function evaluations + self.nfev = 0 + # Number of gradient function evaluation if used + self.ngev = 0 + # Number of hessian of the objective function if used + self.nhev = 0 + self.maxfun = maxfun + + def fun(self, x): + self.nfev += 1 + return self.func(x, *self.args) + + +class LocalSearchWrapper: + """ + Class used to wrap around the minimizer used for local search + Default local minimizer is SciPy minimizer L-BFGS-B + """ + + LS_MAXITER_RATIO = 6 + LS_MAXITER_MIN = 100 + LS_MAXITER_MAX = 1000 + + def __init__(self, search_bounds, func_wrapper, *args, **kwargs): + self.func_wrapper = func_wrapper + self.kwargs = kwargs + self.jac = self.kwargs.get('jac', None) + self.hess = self.kwargs.get('hess', None) + self.hessp = self.kwargs.get('hessp', None) + self.kwargs.pop("args", None) + self.minimizer = minimize + bounds_list = list(zip(*search_bounds)) + self.lower = np.array(bounds_list[0]) + self.upper = np.array(bounds_list[1]) + + # If no minimizer specified, use SciPy minimize with 'L-BFGS-B' method + if not self.kwargs: + n = len(self.lower) + ls_max_iter = min(max(n * self.LS_MAXITER_RATIO, + self.LS_MAXITER_MIN), + self.LS_MAXITER_MAX) + self.kwargs['method'] = 'L-BFGS-B' + self.kwargs['options'] = { + 'maxiter': ls_max_iter, + } + self.kwargs['bounds'] = list(zip(self.lower, self.upper)) + else: + if callable(self.jac): + def wrapped_jac(x): + return self.jac(x, *args) + self.kwargs['jac'] = wrapped_jac + if callable(self.hess): + def wrapped_hess(x): + return self.hess(x, *args) + self.kwargs['hess'] = wrapped_hess + if callable(self.hessp): + def wrapped_hessp(x, p): + return self.hessp(x, p, *args) + self.kwargs['hessp'] = wrapped_hessp + + def local_search(self, x, e): + # Run local search from the given x location where energy value is e + x_tmp = np.copy(x) + mres = self.minimizer(self.func_wrapper.fun, x, **self.kwargs) + if 'njev' in mres: + self.func_wrapper.ngev += mres.njev + if 'nhev' in mres: + self.func_wrapper.nhev += mres.nhev + # Check if is valid value + is_finite = np.all(np.isfinite(mres.x)) and np.isfinite(mres.fun) + in_bounds = np.all(mres.x >= self.lower) and np.all( + mres.x <= self.upper) + is_valid = is_finite and in_bounds + + # Use the new point only if it is valid and return a better results + if is_valid and mres.fun < e: + return mres.fun, mres.x + else: + return e, x_tmp + + +@_transition_to_rng("seed", position_num=10) +def dual_annealing(func, bounds, args=(), maxiter=1000, + minimizer_kwargs=None, initial_temp=5230., + restart_temp_ratio=2.e-5, visit=2.62, accept=-5.0, + maxfun=1e7, rng=None, no_local_search=False, + callback=None, x0=None): + """ + Find the global minimum of a function using Dual Annealing. + + Parameters + ---------- + func : callable + The objective function to be minimized. Must be in the form + ``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array + and ``args`` is a tuple of any additional fixed parameters needed to + completely specify the function. + bounds : sequence or `Bounds` + Bounds for variables. There are two ways to specify the bounds: + + 1. Instance of `Bounds` class. + 2. Sequence of ``(min, max)`` pairs for each element in `x`. + + args : tuple, optional + Any additional fixed parameters needed to completely specify the + objective function. + maxiter : int, optional + The maximum number of global search iterations. Default value is 1000. + minimizer_kwargs : dict, optional + Keyword arguments to be passed to the local minimizer + (`minimize`). An important option could be ``method`` for the minimizer + method to use. + If no keyword arguments are provided, the local minimizer defaults to + 'L-BFGS-B' and uses the already supplied bounds. If `minimizer_kwargs` + is specified, then the dict must contain all parameters required to + control the local minimization. `args` is ignored in this dict, as it is + passed automatically. `bounds` is not automatically passed on to the + local minimizer as the method may not support them. + initial_temp : float, optional + The initial temperature, use higher values to facilitates a wider + search of the energy landscape, allowing dual_annealing to escape + local minima that it is trapped in. Default value is 5230. Range is + (0.01, 5.e4]. + restart_temp_ratio : float, optional + During the annealing process, temperature is decreasing, when it + reaches ``initial_temp * restart_temp_ratio``, the reannealing process + is triggered. Default value of the ratio is 2e-5. Range is (0, 1). + visit : float, optional + Parameter for visiting distribution. Default value is 2.62. Higher + values give the visiting distribution a heavier tail, this makes + the algorithm jump to a more distant region. The value range is (1, 3]. + accept : float, optional + Parameter for acceptance distribution. It is used to control the + probability of acceptance. The lower the acceptance parameter, the + smaller the probability of acceptance. Default value is -5.0 with + a range (-1e4, -5]. + maxfun : int, optional + Soft limit for the number of objective function calls. If the + algorithm is in the middle of a local search, this number will be + exceeded, the algorithm will stop just after the local search is + done. Default value is 1e7. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a `Generator`. + + Specify `rng` for repeatable minimizations. The random numbers + generated only affect the visiting distribution function + and new coordinates generation. + no_local_search : bool, optional + If `no_local_search` is set to True, a traditional Generalized + Simulated Annealing will be performed with no local search + strategy applied. + callback : callable, optional + A callback function with signature ``callback(x, f, context)``, + which will be called for all minima found. + ``x`` and ``f`` are the coordinates and function value of the + latest minimum found, and ``context`` has one of the following + values: + + - ``0``: minimum detected in the annealing process. + - ``1``: detection occurred in the local search process. + - ``2``: detection done in the dual annealing process. + + If the callback implementation returns True, the algorithm will stop. + x0 : ndarray, shape(n,), optional + Coordinates of a single N-D starting point. + + Returns + ------- + res : OptimizeResult + The optimization result represented as a `OptimizeResult` object. + Important attributes are: ``x`` the solution array, ``fun`` the value + of the function at the solution, and ``message`` which describes the + cause of the termination. + See `OptimizeResult` for a description of other attributes. + + Notes + ----- + This function implements the Dual Annealing optimization. This stochastic + approach derived from [3]_ combines the generalization of CSA (Classical + Simulated Annealing) and FSA (Fast Simulated Annealing) [1]_ [2]_ coupled + to a strategy for applying a local search on accepted locations [4]_. + An alternative implementation of this same algorithm is described in [5]_ + and benchmarks are presented in [6]_. This approach introduces an advanced + method to refine the solution found by the generalized annealing + process. This algorithm uses a distorted Cauchy-Lorentz visiting + distribution, with its shape controlled by the parameter :math:`q_{v}` + + .. math:: + + g_{q_{v}}(\\Delta x(t)) \\propto \\frac{ \\ + \\left[T_{q_{v}}(t) \\right]^{-\\frac{D}{3-q_{v}}}}{ \\ + \\left[{1+(q_{v}-1)\\frac{(\\Delta x(t))^{2}} { \\ + \\left[T_{q_{v}}(t)\\right]^{\\frac{2}{3-q_{v}}}}}\\right]^{ \\ + \\frac{1}{q_{v}-1}+\\frac{D-1}{2}}} + + Where :math:`t` is the artificial time. This visiting distribution is used + to generate a trial jump distance :math:`\\Delta x(t)` of variable + :math:`x(t)` under artificial temperature :math:`T_{q_{v}}(t)`. + + From the starting point, after calling the visiting distribution + function, the acceptance probability is computed as follows: + + .. math:: + + p_{q_{a}} = \\min{\\{1,\\left[1-(1-q_{a}) \\beta \\Delta E \\right]^{ \\ + \\frac{1}{1-q_{a}}}\\}} + + Where :math:`q_{a}` is a acceptance parameter. For :math:`q_{a}<1`, zero + acceptance probability is assigned to the cases where + + .. math:: + + [1-(1-q_{a}) \\beta \\Delta E] < 0 + + The artificial temperature :math:`T_{q_{v}}(t)` is decreased according to + + .. math:: + + T_{q_{v}}(t) = T_{q_{v}}(1) \\frac{2^{q_{v}-1}-1}{\\left( \\ + 1 + t\\right)^{q_{v}-1}-1} + + Where :math:`q_{v}` is the visiting parameter. + + .. versionadded:: 1.2.0 + + References + ---------- + .. [1] Tsallis C. Possible generalization of Boltzmann-Gibbs + statistics. Journal of Statistical Physics, 52, 479-487 (1998). + .. [2] Tsallis C, Stariolo DA. Generalized Simulated Annealing. + Physica A, 233, 395-406 (1996). + .. [3] Xiang Y, Sun DY, Fan W, Gong XG. Generalized Simulated + Annealing Algorithm and Its Application to the Thomson Model. + Physics Letters A, 233, 216-220 (1997). + .. [4] Xiang Y, Gong XG. Efficiency of Generalized Simulated + Annealing. Physical Review E, 62, 4473 (2000). + .. [5] Xiang Y, Gubian S, Suomela B, Hoeng J. Generalized + Simulated Annealing for Efficient Global Optimization: the GenSA + Package for R. The R Journal, Volume 5/1 (2013). + .. [6] Mullen, K. Continuous Global Optimization in R. Journal of + Statistical Software, 60(6), 1 - 45, (2014). + :doi:`10.18637/jss.v060.i06` + + Examples + -------- + The following example is a 10-D problem, with many local minima. + The function involved is called Rastrigin + (https://en.wikipedia.org/wiki/Rastrigin_function) + + >>> import numpy as np + >>> from scipy.optimize import dual_annealing + >>> func = lambda x: np.sum(x*x - 10*np.cos(2*np.pi*x)) + 10*np.size(x) + >>> lw = [-5.12] * 10 + >>> up = [5.12] * 10 + >>> ret = dual_annealing(func, bounds=list(zip(lw, up))) + >>> ret.x + array([-4.26437714e-09, -3.91699361e-09, -1.86149218e-09, -3.97165720e-09, + -6.29151648e-09, -6.53145322e-09, -3.93616815e-09, -6.55623025e-09, + -6.05775280e-09, -5.00668935e-09]) # random + >>> ret.fun + 0.000000 + + """ + + if isinstance(bounds, Bounds): + bounds = new_bounds_to_old(bounds.lb, bounds.ub, len(bounds.lb)) + + if x0 is not None and not len(x0) == len(bounds): + raise ValueError('Bounds size does not match x0') + + lu = list(zip(*bounds)) + lower = np.array(lu[0]) + upper = np.array(lu[1]) + # Check that restart temperature ratio is correct + if restart_temp_ratio <= 0. or restart_temp_ratio >= 1.: + raise ValueError('Restart temperature ratio has to be in range (0, 1)') + # Checking bounds are valid + if (np.any(np.isinf(lower)) or np.any(np.isinf(upper)) or np.any( + np.isnan(lower)) or np.any(np.isnan(upper))): + raise ValueError('Some bounds values are inf values or nan values') + # Checking that bounds are consistent + if not np.all(lower < upper): + raise ValueError('Bounds are not consistent min < max') + # Checking that bounds are the same length + if not len(lower) == len(upper): + raise ValueError('Bounds do not have the same dimensions') + + # Wrapper for the objective function + func_wrapper = ObjectiveFunWrapper(func, maxfun, *args) + + # minimizer_kwargs has to be a dict, not None + minimizer_kwargs = minimizer_kwargs or {} + + minimizer_wrapper = LocalSearchWrapper( + bounds, func_wrapper, *args, **minimizer_kwargs) + + # Initialization of random Generator for reproducible runs if rng provided + rng_gen = check_random_state(rng) + # Initialization of the energy state + energy_state = EnergyState(lower, upper, callback) + energy_state.reset(func_wrapper, rng_gen, x0) + # Minimum value of annealing temperature reached to perform + # re-annealing + temperature_restart = initial_temp * restart_temp_ratio + # VisitingDistribution instance + visit_dist = VisitingDistribution(lower, upper, visit, rng_gen) + # Strategy chain instance + strategy_chain = StrategyChain(accept, visit_dist, func_wrapper, + minimizer_wrapper, rng_gen, energy_state) + need_to_stop = False + iteration = 0 + message = [] + # OptimizeResult object to be returned + optimize_res = OptimizeResult() + optimize_res.success = True + optimize_res.status = 0 + + t1 = np.exp((visit - 1) * np.log(2.0)) - 1.0 + # Run the search loop + while not need_to_stop: + for i in range(maxiter): + # Compute temperature for this step + s = float(i) + 2.0 + t2 = np.exp((visit - 1) * np.log(s)) - 1.0 + temperature = initial_temp * t1 / t2 + if iteration >= maxiter: + message.append("Maximum number of iteration reached") + need_to_stop = True + break + # Need a re-annealing process? + if temperature < temperature_restart: + energy_state.reset(func_wrapper, rng_gen) + break + # starting strategy chain + val = strategy_chain.run(i, temperature) + if val is not None: + message.append(val) + need_to_stop = True + optimize_res.success = False + break + # Possible local search at the end of the strategy chain + if not no_local_search: + val = strategy_chain.local_search() + if val is not None: + message.append(val) + need_to_stop = True + optimize_res.success = False + break + iteration += 1 + + # Setting the OptimizeResult values + optimize_res.x = energy_state.xbest + optimize_res.fun = energy_state.ebest + optimize_res.nit = iteration + optimize_res.nfev = func_wrapper.nfev + optimize_res.njev = func_wrapper.ngev + optimize_res.nhev = func_wrapper.nhev + optimize_res.message = message + return optimize_res diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_elementwise.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_elementwise.py new file mode 100644 index 0000000000000000000000000000000000000000..883c644dbcbbb954ea3e3c184bd610b28f05cca8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_elementwise.py @@ -0,0 +1,801 @@ +from scipy.optimize._bracket import _bracket_root, _bracket_minimum +from scipy.optimize._chandrupatla import _chandrupatla, _chandrupatla_minimize +from scipy._lib._util import _RichResult + + +def find_root(f, init, /, *, args=(), tolerances=None, maxiter=None, callback=None): + """Find the root of a monotonic, real-valued function of a real variable. + + For each element of the output of `f`, `find_root` seeks the scalar + root that makes the element 0. This function currently uses Chandrupatla's + bracketing algorithm [1]_ and therefore requires argument `init` to + provide a bracket around the root: the function values at the two endpoints + must have opposite signs. + + Provided a valid bracket, `find_root` is guaranteed to converge to a solution + that satisfies the provided `tolerances` if the function is continuous within + the bracket. + + This function works elementwise when `init` and `args` contain (broadcastable) + arrays. + + Parameters + ---------- + f : callable + The function whose root is desired. The signature must be:: + + f(x: array, *args) -> array + + where each element of ``x`` is a finite real and ``args`` is a tuple, + which may contain an arbitrary number of arrays that are broadcastable + with ``x``. + + `f` must be an elementwise function: each element ``f(x)[i]`` + must equal ``f(x[i])`` for all indices ``i``. It must not mutate the + array ``x`` or the arrays in ``args``. + + `find_root` seeks an array ``x`` such that ``f(x)`` is an array of zeros. + init : 2-tuple of float array_like + The lower and upper endpoints of a bracket surrounding the desired root. + A bracket is valid if arrays ``xl, xr = init`` satisfy ``xl < xr`` and + ``sign(f(xl)) == -sign(f(xr))`` elementwise. Arrays be broadcastable with + one another and `args`. + args : tuple of array_like, optional + Additional positional array arguments to be passed to `f`. Arrays + must be broadcastable with one another and the arrays of `init`. + If the callable for which the root is desired requires arguments that are + not broadcastable with `x`, wrap that callable with `f` such that `f` + accepts only `x` and broadcastable ``*args``. + tolerances : dictionary of floats, optional + Absolute and relative tolerances on the root and function value. + Valid keys of the dictionary are: + + - ``xatol`` - absolute tolerance on the root + - ``xrtol`` - relative tolerance on the root + - ``fatol`` - absolute tolerance on the function value + - ``frtol`` - relative tolerance on the function value + + See Notes for default values and explicit termination conditions. + maxiter : int, optional + The maximum number of iterations of the algorithm to perform. + The default is the maximum possible number of bisections within + the (normal) floating point numbers of the relevant dtype. + callback : callable, optional + An optional user-supplied function to be called before the first + iteration and after each iteration. + Called as ``callback(res)``, where ``res`` is a ``_RichResult`` + similar to that returned by `find_root` (but containing the current + iterate's values of all variables). If `callback` raises a + ``StopIteration``, the algorithm will terminate immediately and + `find_root` will return a result. `callback` must not mutate + `res` or its attributes. + + Returns + ------- + res : _RichResult + An object similar to an instance of `scipy.optimize.OptimizeResult` with the + following attributes. The descriptions are written as though the values will + be scalars; however, if `f` returns an array, the outputs will be + arrays of the same shape. + + success : bool array + ``True`` where the algorithm terminated successfully (status ``0``); + ``False`` otherwise. + status : int array + An integer representing the exit status of the algorithm. + + - ``0`` : The algorithm converged to the specified tolerances. + - ``-1`` : The initial bracket was invalid. + - ``-2`` : The maximum number of iterations was reached. + - ``-3`` : A non-finite value was encountered. + - ``-4`` : Iteration was terminated by `callback`. + - ``1`` : The algorithm is proceeding normally (in `callback` only). + + x : float array + The root of the function, if the algorithm terminated successfully. + f_x : float array + The value of `f` evaluated at `x`. + nfev : int array + The number of abscissae at which `f` was evaluated to find the root. + This is distinct from the number of times `f` is *called* because the + the function may evaluated at multiple points in a single call. + nit : int array + The number of iterations of the algorithm that were performed. + bracket : tuple of float arrays + The lower and upper endpoints of the final bracket. + f_bracket : tuple of float arrays + The value of `f` evaluated at the lower and upper endpoints of the + bracket. + + Notes + ----- + Implemented based on Chandrupatla's original paper [1]_. + + Let: + + - ``a, b = init`` be the left and right endpoints of the initial bracket, + - ``xl`` and ``xr`` be the left and right endpoints of the final bracket, + - ``xmin = xl if abs(f(xl)) <= abs(f(xr)) else xr`` be the final bracket + endpoint with the smaller function value, and + - ``fmin0 = min(f(a), f(b))`` be the minimum of the two values of the + function evaluated at the initial bracket endpoints. + + Then the algorithm is considered to have converged when + + - ``abs(xr - xl) < xatol + abs(xmin) * xrtol`` or + - ``fun(xmin) <= fatol + abs(fmin0) * frtol``. + + This is equivalent to the termination condition described in [1]_ with + ``xrtol = 4e-10``, ``xatol = 1e-5``, and ``fatol = frtol = 0``. + However, the default values of the `tolerances` dictionary are + ``xatol = 4*tiny``, ``xrtol = 4*eps``, ``frtol = 0``, and ``fatol = tiny``, + where ``eps`` and ``tiny`` are the precision and smallest normal number + of the result ``dtype`` of function inputs and outputs. + + References + ---------- + + .. [1] Chandrupatla, Tirupathi R. + "A new hybrid quadratic/bisection algorithm for finding the zero of a + nonlinear function without using derivatives". + Advances in Engineering Software, 28(3), 145-149. + https://doi.org/10.1016/s0965-9978(96)00051-8 + + See Also + -------- + bracket_root + + Examples + -------- + Suppose we wish to find the root of the following function. + + >>> def f(x, c=5): + ... return x**3 - 2*x - c + + First, we must find a valid bracket. The function is not monotonic, + but `bracket_root` may be able to provide a bracket. + + >>> from scipy.optimize import elementwise + >>> res_bracket = elementwise.bracket_root(f, 0) + >>> res_bracket.success + True + >>> res_bracket.bracket + (2.0, 4.0) + + Indeed, the values of the function at the bracket endpoints have + opposite signs. + + >>> res_bracket.f_bracket + (-1.0, 51.0) + + Once we have a valid bracket, `find_root` can be used to provide + a precise root. + + >>> res_root = elementwise.find_root(f, res_bracket.bracket) + >>> res_root.x + 2.0945514815423265 + + The final bracket is only a few ULPs wide, so the error between + this value and the true root cannot be much smaller within values + that are representable in double precision arithmetic. + + >>> import numpy as np + >>> xl, xr = res_root.bracket + >>> (xr - xl) / np.spacing(xl) + 2.0 + >>> res_root.f_bracket + (-8.881784197001252e-16, 9.769962616701378e-15) + + `bracket_root` and `find_root` accept arrays for most arguments. + For instance, to find the root for a few values of the parameter ``c`` + at once: + + >>> c = np.asarray([3, 4, 5]) + >>> res_bracket = elementwise.bracket_root(f, 0, args=(c,)) + >>> res_bracket.bracket + (array([1., 1., 2.]), array([2., 2., 4.])) + >>> res_root = elementwise.find_root(f, res_bracket.bracket, args=(c,)) + >>> res_root.x + array([1.8932892 , 2. , 2.09455148]) + + """ + + def reformat_result(res_in): + res_out = _RichResult() + res_out.status = res_in.status + res_out.success = res_in.success + res_out.x = res_in.x + res_out.f_x = res_in.fun + res_out.nfev = res_in.nfev + res_out.nit = res_in.nit + res_out.bracket = (res_in.xl, res_in.xr) + res_out.f_bracket = (res_in.fl, res_in.fr) + res_out._order_keys = ['success', 'status', 'x', 'f_x', + 'nfev', 'nit', 'bracket', 'f_bracket'] + return res_out + + xl, xr = init + default_tolerances = dict(xatol=None, xrtol=None, fatol=None, frtol=0) + tolerances = {} if tolerances is None else tolerances + default_tolerances.update(tolerances) + tolerances = default_tolerances + + if callable(callback): + def _callback(res): + return callback(reformat_result(res)) + else: + _callback = callback + + res = _chandrupatla(f, xl, xr, args=args, **tolerances, + maxiter=maxiter, callback=_callback) + return reformat_result(res) + + +def find_minimum(f, init, /, *, args=(), tolerances=None, maxiter=100, callback=None): + """Find the minimum of an unimodal, real-valued function of a real variable. + + For each element of the output of `f`, `find_minimum` seeks the scalar minimizer + that minimizes the element. This function currently uses Chandrupatla's + bracketing minimization algorithm [1]_ and therefore requires argument `init` + to provide a three-point minimization bracket: ``x1 < x2 < x3`` such that + ``func(x1) >= func(x2) <= func(x3)``, where one of the inequalities is strict. + + Provided a valid bracket, `find_minimum` is guaranteed to converge to a local + minimum that satisfies the provided `tolerances` if the function is continuous + within the bracket. + + This function works elementwise when `init` and `args` contain (broadcastable) + arrays. + + Parameters + ---------- + f : callable + The function whose minimizer is desired. The signature must be:: + + f(x: array, *args) -> array + + where each element of ``x`` is a finite real and ``args`` is a tuple, + which may contain an arbitrary number of arrays that are broadcastable + with ``x``. + + `f` must be an elementwise function: each element ``f(x)[i]`` + must equal ``f(x[i])`` for all indices ``i``. It must not mutate the + array ``x`` or the arrays in ``args``. + + `find_minimum` seeks an array ``x`` such that ``f(x)`` is an array of + local minima. + init : 3-tuple of float array_like + The abscissae of a standard scalar minimization bracket. A bracket is + valid if arrays ``x1, x2, x3 = init`` satisfy ``x1 < x2 < x3`` and + ``func(x1) >= func(x2) <= func(x3)``, where one of the inequalities + is strict. Arrays must be broadcastable with one another and the arrays + of `args`. + args : tuple of array_like, optional + Additional positional array arguments to be passed to `f`. Arrays + must be broadcastable with one another and the arrays of `init`. + If the callable for which the root is desired requires arguments that are + not broadcastable with `x`, wrap that callable with `f` such that `f` + accepts only `x` and broadcastable ``*args``. + tolerances : dictionary of floats, optional + Absolute and relative tolerances on the root and function value. + Valid keys of the dictionary are: + + - ``xatol`` - absolute tolerance on the root + - ``xrtol`` - relative tolerance on the root + - ``fatol`` - absolute tolerance on the function value + - ``frtol`` - relative tolerance on the function value + + See Notes for default values and explicit termination conditions. + maxiter : int, default: 100 + The maximum number of iterations of the algorithm to perform. + callback : callable, optional + An optional user-supplied function to be called before the first + iteration and after each iteration. + Called as ``callback(res)``, where ``res`` is a ``_RichResult`` + similar to that returned by `find_minimum` (but containing the current + iterate's values of all variables). If `callback` raises a + ``StopIteration``, the algorithm will terminate immediately and + `find_root` will return a result. `callback` must not mutate + `res` or its attributes. + + Returns + ------- + res : _RichResult + An object similar to an instance of `scipy.optimize.OptimizeResult` with the + following attributes. The descriptions are written as though the values will + be scalars; however, if `f` returns an array, the outputs will be + arrays of the same shape. + + success : bool array + ``True`` where the algorithm terminated successfully (status ``0``); + ``False`` otherwise. + status : int array + An integer representing the exit status of the algorithm. + + - ``0`` : The algorithm converged to the specified tolerances. + - ``-1`` : The algorithm encountered an invalid bracket. + - ``-2`` : The maximum number of iterations was reached. + - ``-3`` : A non-finite value was encountered. + - ``-4`` : Iteration was terminated by `callback`. + - ``1`` : The algorithm is proceeding normally (in `callback` only). + + x : float array + The minimizer of the function, if the algorithm terminated successfully. + f_x : float array + The value of `f` evaluated at `x`. + nfev : int array + The number of abscissae at which `f` was evaluated to find the root. + This is distinct from the number of times `f` is *called* because the + the function may evaluated at multiple points in a single call. + nit : int array + The number of iterations of the algorithm that were performed. + bracket : tuple of float arrays + The final three-point bracket. + f_bracket : tuple of float arrays + The value of `f` evaluated at the bracket points. + + Notes + ----- + Implemented based on Chandrupatla's original paper [1]_. + + If ``xl < xm < xr`` are the points of the bracket and ``fl >= fm <= fr`` + (where one of the inequalities is strict) are the values of `f` evaluated + at those points, then the algorithm is considered to have converged when: + + - ``xr - xl <= abs(xm)*xrtol + xatol`` or + - ``(fl - 2*fm + fr)/2 <= abs(fm)*frtol + fatol``. + + Note that first of these differs from the termination conditions described + in [1]_. + + The default value of `xrtol` is the square root of the precision of the + appropriate dtype, and ``xatol = fatol = frtol`` is the smallest normal + number of the appropriate dtype. + + References + ---------- + + .. [1] Chandrupatla, Tirupathi R. (1998). + "An efficient quadratic fit-sectioning algorithm for minimization + without derivatives". + Computer Methods in Applied Mechanics and Engineering, 152 (1-2), + 211-217. https://doi.org/10.1016/S0045-7825(97)00190-4 + + See Also + -------- + bracket_minimum + + Examples + -------- + Suppose we wish to minimize the following function. + + >>> def f(x, c=1): + ... return (x - c)**2 + 2 + + First, we must find a valid bracket. The function is unimodal, + so `bracket_minium` will easily find a bracket. + + >>> from scipy.optimize import elementwise + >>> res_bracket = elementwise.bracket_minimum(f, 0) + >>> res_bracket.success + True + >>> res_bracket.bracket + (0.0, 0.5, 1.5) + + Indeed, the bracket points are ordered and the function value + at the middle bracket point is less than at the surrounding + points. + + >>> xl, xm, xr = res_bracket.bracket + >>> fl, fm, fr = res_bracket.f_bracket + >>> (xl < xm < xr) and (fl > fm <= fr) + True + + Once we have a valid bracket, `find_minimum` can be used to provide + an estimate of the minimizer. + + >>> res_minimum = elementwise.find_minimum(f, res_bracket.bracket) + >>> res_minimum.x + 1.0000000149011612 + + The function value changes by only a few ULPs within the bracket, so + the minimizer cannot be determined much more precisely by evaluating + the function alone (i.e. we would need its derivative to do better). + + >>> import numpy as np + >>> fl, fm, fr = res_minimum.f_bracket + >>> (fl - fm) / np.spacing(fm), (fr - fm) / np.spacing(fm) + (0.0, 2.0) + + Therefore, a precise minimum of the function is given by: + + >>> res_minimum.f_x + 2.0 + + `bracket_minimum` and `find_minimum` accept arrays for most arguments. + For instance, to find the minimizers and minima for a few values of the + parameter ``c`` at once: + + >>> c = np.asarray([1, 1.5, 2]) + >>> res_bracket = elementwise.bracket_minimum(f, 0, args=(c,)) + >>> res_bracket.bracket + (array([0. , 0.5, 0.5]), array([0.5, 1.5, 1.5]), array([1.5, 2.5, 2.5])) + >>> res_minimum = elementwise.find_minimum(f, res_bracket.bracket, args=(c,)) + >>> res_minimum.x + array([1.00000001, 1.5 , 2. ]) + >>> res_minimum.f_x + array([2., 2., 2.]) + + """ + + def reformat_result(res_in): + res_out = _RichResult() + res_out.status = res_in.status + res_out.success = res_in.success + res_out.x = res_in.x + res_out.f_x = res_in.fun + res_out.nfev = res_in.nfev + res_out.nit = res_in.nit + res_out.bracket = (res_in.xl, res_in.xm, res_in.xr) + res_out.f_bracket = (res_in.fl, res_in.fm, res_in.fr) + res_out._order_keys = ['success', 'status', 'x', 'f_x', + 'nfev', 'nit', 'bracket', 'f_bracket'] + return res_out + + xl, xm, xr = init + default_tolerances = dict(xatol=None, xrtol=None, fatol=None, frtol=None) + tolerances = {} if tolerances is None else tolerances + default_tolerances.update(tolerances) + tolerances = default_tolerances + + if callable(callback): + def _callback(res): + return callback(reformat_result(res)) + else: + _callback = callback + + res = _chandrupatla_minimize(f, xl, xm, xr, args=args, **tolerances, + maxiter=maxiter, callback=_callback) + return reformat_result(res) + + +def bracket_root(f, xl0, xr0=None, *, xmin=None, xmax=None, factor=None, args=(), + maxiter=1000): + """Bracket the root of a monotonic, real-valued function of a real variable. + + For each element of the output of `f`, `bracket_root` seeks the scalar + bracket endpoints ``xl`` and ``xr`` such that ``sign(f(xl)) == -sign(f(xr))`` + elementwise. + + The function is guaranteed to find a valid bracket if the function is monotonic, + but it may find a bracket under other conditions. + + This function works elementwise when `xl0`, `xr0`, `xmin`, `xmax`, `factor`, and + the elements of `args` are (mutually broadcastable) arrays. + + Parameters + ---------- + f : callable + The function for which the root is to be bracketed. The signature must be:: + + f(x: array, *args) -> array + + where each element of ``x`` is a finite real and ``args`` is a tuple, + which may contain an arbitrary number of arrays that are broadcastable + with ``x``. + + `f` must be an elementwise function: each element ``f(x)[i]`` + must equal ``f(x[i])`` for all indices ``i``. It must not mutate the + array ``x`` or the arrays in ``args``. + xl0, xr0: float array_like + Starting guess of bracket, which need not contain a root. If `xr0` is + not provided, ``xr0 = xl0 + 1``. Must be broadcastable with all other + array inputs. + xmin, xmax : float array_like, optional + Minimum and maximum allowable endpoints of the bracket, inclusive. Must + be broadcastable with all other array inputs. + factor : float array_like, default: 2 + The factor used to grow the bracket. See Notes. + args : tuple of array_like, optional + Additional positional array arguments to be passed to `f`. + If the callable for which the root is desired requires arguments that are + not broadcastable with `x`, wrap that callable with `f` such that `f` + accepts only `x` and broadcastable ``*args``. + maxiter : int, default: 1000 + The maximum number of iterations of the algorithm to perform. + + Returns + ------- + res : _RichResult + An object similar to an instance of `scipy.optimize.OptimizeResult` with the + following attributes. The descriptions are written as though the values will + be scalars; however, if `f` returns an array, the outputs will be + arrays of the same shape. + + success : bool array + ``True`` where the algorithm terminated successfully (status ``0``); + ``False`` otherwise. + status : int array + An integer representing the exit status of the algorithm. + + - ``0`` : The algorithm produced a valid bracket. + - ``-1`` : The bracket expanded to the allowable limits without success. + - ``-2`` : The maximum number of iterations was reached. + - ``-3`` : A non-finite value was encountered. + - ``-4`` : Iteration was terminated by `callback`. + - ``-5``: The initial bracket does not satisfy`xmin <= xl0 < xr0 < xmax`. + + bracket : 2-tuple of float arrays + The lower and upper endpoints of the bracket, if the algorithm + terminated successfully. + f_bracket : 2-tuple of float arrays + The values of `f` evaluated at the endpoints of ``res.bracket``, + respectively. + nfev : int array + The number of abscissae at which `f` was evaluated to find the root. + This is distinct from the number of times `f` is *called* because the + the function may evaluated at multiple points in a single call. + nit : int array + The number of iterations of the algorithm that were performed. + + Notes + ----- + This function generalizes an algorithm found in pieces throughout the + `scipy.stats` codebase. The strategy is to iteratively grow the bracket `(l, r)` + until ``f(l) < 0 < f(r)`` or ``f(r) < 0 < f(l)``. The bracket grows to the left + as follows. + + - If `xmin` is not provided, the distance between `xl0` and `l` is iteratively + increased by `factor`. + - If `xmin` is provided, the distance between `xmin` and `l` is iteratively + decreased by `factor`. Note that this also *increases* the bracket size. + + Growth of the bracket to the right is analogous. + + Growth of the bracket in one direction stops when the endpoint is no longer + finite, the function value at the endpoint is no longer finite, or the + endpoint reaches its limiting value (`xmin` or `xmax`). Iteration terminates + when the bracket stops growing in both directions, the bracket surrounds + the root, or a root is found (by chance). + + If two brackets are found - that is, a bracket is found on both sides in + the same iteration, the smaller of the two is returned. + + If roots of the function are found, both `xl` and `xr` are set to the + leftmost root. + + See Also + -------- + find_root + + Examples + -------- + Suppose we wish to find the root of the following function. + + >>> def f(x, c=5): + ... return x**3 - 2*x - c + + First, we must find a valid bracket. The function is not monotonic, + but `bracket_root` may be able to provide a bracket. + + >>> from scipy.optimize import elementwise + >>> res_bracket = elementwise.bracket_root(f, 0) + >>> res_bracket.success + True + >>> res_bracket.bracket + (2.0, 4.0) + + Indeed, the values of the function at the bracket endpoints have + opposite signs. + + >>> res_bracket.f_bracket + (-1.0, 51.0) + + Once we have a valid bracket, `find_root` can be used to provide + a precise root. + + >>> res_root = elementwise.find_root(f, res_bracket.bracket) + >>> res_root.x + 2.0945514815423265 + + `bracket_root` and `find_root` accept arrays for most arguments. + For instance, to find the root for a few values of the parameter ``c`` + at once: + + >>> import numpy as np + >>> c = np.asarray([3, 4, 5]) + >>> res_bracket = elementwise.bracket_root(f, 0, args=(c,)) + >>> res_bracket.bracket + (array([1., 1., 2.]), array([2., 2., 4.])) + >>> res_root = elementwise.find_root(f, res_bracket.bracket, args=(c,)) + >>> res_root.x + array([1.8932892 , 2. , 2.09455148]) + + """ # noqa: E501 + + res = _bracket_root(f, xl0, xr0=xr0, xmin=xmin, xmax=xmax, factor=factor, + args=args, maxiter=maxiter) + res.bracket = res.xl, res.xr + res.f_bracket = res.fl, res.fr + del res.xl + del res.xr + del res.fl + del res.fr + return res + + +def bracket_minimum(f, xm0, *, xl0=None, xr0=None, xmin=None, xmax=None, + factor=None, args=(), maxiter=1000): + """Bracket the minimum of a unimodal, real-valued function of a real variable. + + For each element of the output of `f`, `bracket_minimum` seeks the scalar + bracket points ``xl < xm < xr`` such that ``fl >= fm <= fr`` where one of the + inequalities is strict. + + The function is guaranteed to find a valid bracket if the function is + strongly unimodal, but it may find a bracket under other conditions. + + This function works elementwise when `xm0`, `xl0`, `xr0`, `xmin`, `xmax`, `factor`, + and the elements of `args` are (mutually broadcastable) arrays. + + Parameters + ---------- + f : callable + The function for which the root is to be bracketed. The signature must be:: + + f(x: array, *args) -> array + + where each element of ``x`` is a finite real and ``args`` is a tuple, + which may contain an arbitrary number of arrays that are broadcastable + with ``x``. + + `f` must be an elementwise function: each element ``f(x)[i]`` + must equal ``f(x[i])`` for all indices ``i``. It must not mutate the + array ``x`` or the arrays in ``args``. + xm0: float array_like + Starting guess for middle point of bracket. + xl0, xr0: float array_like, optional + Starting guesses for left and right endpoints of the bracket. Must + be broadcastable with all other array inputs. + xmin, xmax : float array_like, optional + Minimum and maximum allowable endpoints of the bracket, inclusive. Must + be broadcastable with all other array inputs. + factor : float array_like, default: 2 + The factor used to grow the bracket. See Notes. + args : tuple of array_like, optional + Additional positional array arguments to be passed to `f`. + If the callable for which the root is desired requires arguments that are + not broadcastable with `x`, wrap that callable with `f` such that `f` + accepts only `x` and broadcastable ``*args``. + maxiter : int, default: 1000 + The maximum number of iterations of the algorithm to perform. + + Returns + ------- + res : _RichResult + An object similar to an instance of `scipy.optimize.OptimizeResult` with the + following attributes. The descriptions are written as though the values will + be scalars; however, if `f` returns an array, the outputs will be + arrays of the same shape. + + success : bool array + ``True`` where the algorithm terminated successfully (status ``0``); + ``False`` otherwise. + status : int array + An integer representing the exit status of the algorithm. + + - ``0`` : The algorithm produced a valid bracket. + - ``-1`` : The bracket expanded to the allowable limits. Assuming + unimodality, this implies the endpoint at the limit is a minimizer. + - ``-2`` : The maximum number of iterations was reached. + - ``-3`` : A non-finite value was encountered. + - ``-4`` : ``None`` shall pass. + - ``-5`` : The initial bracket does not satisfy + `xmin <= xl0 < xm0 < xr0 <= xmax`. + + bracket : 3-tuple of float arrays + The left, middle, and right points of the bracket, if the algorithm + terminated successfully. + f_bracket : 3-tuple of float arrays + The function value at the left, middle, and right points of the bracket. + nfev : int array + The number of abscissae at which `f` was evaluated to find the root. + This is distinct from the number of times `f` is *called* because the + the function may evaluated at multiple points in a single call. + nit : int array + The number of iterations of the algorithm that were performed. + + Notes + ----- + Similar to `scipy.optimize.bracket`, this function seeks to find real + points ``xl < xm < xr`` such that ``f(xl) >= f(xm)`` and ``f(xr) >= f(xm)``, + where at least one of the inequalities is strict. Unlike `scipy.optimize.bracket`, + this function can operate in a vectorized manner on array input, so long as + the input arrays are broadcastable with each other. Also unlike + `scipy.optimize.bracket`, users may specify minimum and maximum endpoints + for the desired bracket. + + Given an initial trio of points ``xl = xl0``, ``xm = xm0``, ``xr = xr0``, + the algorithm checks if these points already give a valid bracket. If not, + a new endpoint, ``w`` is chosen in the "downhill" direction, ``xm`` becomes the new + opposite endpoint, and either `xl` or `xr` becomes the new middle point, + depending on which direction is downhill. The algorithm repeats from here. + + The new endpoint `w` is chosen differently depending on whether or not a + boundary `xmin` or `xmax` has been set in the downhill direction. Without + loss of generality, suppose the downhill direction is to the right, so that + ``f(xl) > f(xm) > f(xr)``. If there is no boundary to the right, then `w` + is chosen to be ``xr + factor * (xr - xm)`` where `factor` is controlled by + the user (defaults to 2.0) so that step sizes increase in geometric proportion. + If there is a boundary, `xmax` in this case, then `w` is chosen to be + ``xmax - (xmax - xr)/factor``, with steps slowing to a stop at + `xmax`. This cautious approach ensures that a minimum near but distinct from + the boundary isn't missed while also detecting whether or not the `xmax` is + a minimizer when `xmax` is reached after a finite number of steps. + + See Also + -------- + scipy.optimize.bracket + scipy.optimize.elementwise.find_minimum + + Examples + -------- + Suppose we wish to minimize the following function. + + >>> def f(x, c=1): + ... return (x - c)**2 + 2 + + First, we must find a valid bracket. The function is unimodal, + so `bracket_minium` will easily find a bracket. + + >>> from scipy.optimize import elementwise + >>> res_bracket = elementwise.bracket_minimum(f, 0) + >>> res_bracket.success + True + >>> res_bracket.bracket + (0.0, 0.5, 1.5) + + Indeed, the bracket points are ordered and the function value + at the middle bracket point is less than at the surrounding + points. + + >>> xl, xm, xr = res_bracket.bracket + >>> fl, fm, fr = res_bracket.f_bracket + >>> (xl < xm < xr) and (fl > fm <= fr) + True + + Once we have a valid bracket, `find_minimum` can be used to provide + an estimate of the minimizer. + + >>> res_minimum = elementwise.find_minimum(f, res_bracket.bracket) + >>> res_minimum.x + 1.0000000149011612 + + `bracket_minimum` and `find_minimum` accept arrays for most arguments. + For instance, to find the minimizers and minima for a few values of the + parameter ``c`` at once: + + >>> import numpy as np + >>> c = np.asarray([1, 1.5, 2]) + >>> res_bracket = elementwise.bracket_minimum(f, 0, args=(c,)) + >>> res_bracket.bracket + (array([0. , 0.5, 0.5]), array([0.5, 1.5, 1.5]), array([1.5, 2.5, 2.5])) + >>> res_minimum = elementwise.find_minimum(f, res_bracket.bracket, args=(c,)) + >>> res_minimum.x + array([1.00000001, 1.5 , 2. ]) + >>> res_minimum.f_x + array([2., 2., 2.]) + + """ # noqa: E501 + + res = _bracket_minimum(f, xm0, xl0=xl0, xr0=xr0, xmin=xmin, xmax=xmax, + factor=factor, args=args, maxiter=maxiter) + res.bracket = res.xl, res.xm, res.xr + res.f_bracket = res.fl, res.fm, res.fr + del res.xl + del res.xm + del res.xr + del res.fl + del res.fm + del res.fr + return res diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_group_columns.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_group_columns.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..426fa2565038a67cd669485576802658a2f48d8f Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_group_columns.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_hessian_update_strategy.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_hessian_update_strategy.py new file mode 100644 index 0000000000000000000000000000000000000000..15989969349025fd42fba9836b4f5d882c3f6791 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_hessian_update_strategy.py @@ -0,0 +1,479 @@ +"""Hessian update strategies for quasi-Newton optimization methods.""" +import numpy as np +from numpy.linalg import norm +from scipy.linalg import get_blas_funcs, issymmetric +from warnings import warn + + +__all__ = ['HessianUpdateStrategy', 'BFGS', 'SR1'] + + +class HessianUpdateStrategy: + """Interface for implementing Hessian update strategies. + + Many optimization methods make use of Hessian (or inverse Hessian) + approximations, such as the quasi-Newton methods BFGS, SR1, L-BFGS. + Some of these approximations, however, do not actually need to store + the entire matrix or can compute the internal matrix product with a + given vector in a very efficiently manner. This class serves as an + abstract interface between the optimization algorithm and the + quasi-Newton update strategies, giving freedom of implementation + to store and update the internal matrix as efficiently as possible. + Different choices of initialization and update procedure will result + in different quasi-Newton strategies. + + Four methods should be implemented in derived classes: ``initialize``, + ``update``, ``dot`` and ``get_matrix``. The matrix multiplication + operator ``@`` is also defined to call the ``dot`` method. + + Notes + ----- + Any instance of a class that implements this interface, + can be accepted by the method ``minimize`` and used by + the compatible solvers to approximate the Hessian (or + inverse Hessian) used by the optimization algorithms. + """ + + def initialize(self, n, approx_type): + """Initialize internal matrix. + + Allocate internal memory for storing and updating + the Hessian or its inverse. + + Parameters + ---------- + n : int + Problem dimension. + approx_type : {'hess', 'inv_hess'} + Selects either the Hessian or the inverse Hessian. + When set to 'hess' the Hessian will be stored and updated. + When set to 'inv_hess' its inverse will be used instead. + """ + raise NotImplementedError("The method ``initialize(n, approx_type)``" + " is not implemented.") + + def update(self, delta_x, delta_grad): + """Update internal matrix. + + Update Hessian matrix or its inverse (depending on how 'approx_type' + is defined) using information about the last evaluated points. + + Parameters + ---------- + delta_x : ndarray + The difference between two points the gradient + function have been evaluated at: ``delta_x = x2 - x1``. + delta_grad : ndarray + The difference between the gradients: + ``delta_grad = grad(x2) - grad(x1)``. + """ + raise NotImplementedError("The method ``update(delta_x, delta_grad)``" + " is not implemented.") + + def dot(self, p): + """Compute the product of the internal matrix with the given vector. + + Parameters + ---------- + p : array_like + 1-D array representing a vector. + + Returns + ------- + Hp : array + 1-D represents the result of multiplying the approximation matrix + by vector p. + """ + raise NotImplementedError("The method ``dot(p)``" + " is not implemented.") + + def get_matrix(self): + """Return current internal matrix. + + Returns + ------- + H : ndarray, shape (n, n) + Dense matrix containing either the Hessian + or its inverse (depending on how 'approx_type' + is defined). + """ + raise NotImplementedError("The method ``get_matrix(p)``" + " is not implemented.") + + def __matmul__(self, p): + return self.dot(p) + + +class FullHessianUpdateStrategy(HessianUpdateStrategy): + """Hessian update strategy with full dimensional internal representation. + """ + _syr = get_blas_funcs('syr', dtype='d') # Symmetric rank 1 update + _syr2 = get_blas_funcs('syr2', dtype='d') # Symmetric rank 2 update + # Symmetric matrix-vector product + _symv = get_blas_funcs('symv', dtype='d') + + def __init__(self, init_scale='auto'): + self.init_scale = init_scale + # Until initialize is called we can't really use the class, + # so it makes sense to set everything to None. + self.first_iteration = None + self.approx_type = None + self.B = None + self.H = None + + def initialize(self, n, approx_type): + """Initialize internal matrix. + + Allocate internal memory for storing and updating + the Hessian or its inverse. + + Parameters + ---------- + n : int + Problem dimension. + approx_type : {'hess', 'inv_hess'} + Selects either the Hessian or the inverse Hessian. + When set to 'hess' the Hessian will be stored and updated. + When set to 'inv_hess' its inverse will be used instead. + """ + self.first_iteration = True + self.n = n + self.approx_type = approx_type + if approx_type not in ('hess', 'inv_hess'): + raise ValueError("`approx_type` must be 'hess' or 'inv_hess'.") + # Create matrix + if self.approx_type == 'hess': + self.B = np.eye(n, dtype=float) + else: + self.H = np.eye(n, dtype=float) + + def _auto_scale(self, delta_x, delta_grad): + # Heuristic to scale matrix at first iteration. + # Described in Nocedal and Wright "Numerical Optimization" + # p.143 formula (6.20). + s_norm2 = np.dot(delta_x, delta_x) + y_norm2 = np.dot(delta_grad, delta_grad) + ys = np.abs(np.dot(delta_grad, delta_x)) + if ys == 0.0 or y_norm2 == 0 or s_norm2 == 0: + return 1 + if self.approx_type == 'hess': + return y_norm2 / ys + else: + return ys / y_norm2 + + def _update_implementation(self, delta_x, delta_grad): + raise NotImplementedError("The method ``_update_implementation``" + " is not implemented.") + + def update(self, delta_x, delta_grad): + """Update internal matrix. + + Update Hessian matrix or its inverse (depending on how 'approx_type' + is defined) using information about the last evaluated points. + + Parameters + ---------- + delta_x : ndarray + The difference between two points the gradient + function have been evaluated at: ``delta_x = x2 - x1``. + delta_grad : ndarray + The difference between the gradients: + ``delta_grad = grad(x2) - grad(x1)``. + """ + if np.all(delta_x == 0.0): + return + if np.all(delta_grad == 0.0): + warn('delta_grad == 0.0. Check if the approximated ' + 'function is linear. If the function is linear ' + 'better results can be obtained by defining the ' + 'Hessian as zero instead of using quasi-Newton ' + 'approximations.', + UserWarning, stacklevel=2) + return + if self.first_iteration: + # Get user specific scale + if isinstance(self.init_scale, str) and self.init_scale == "auto": + scale = self._auto_scale(delta_x, delta_grad) + else: + scale = self.init_scale + + # Check for complex: numpy will silently cast a complex array to + # a real one but not so for scalar as it raises a TypeError. + # Checking here brings a consistent behavior. + replace = False + if np.size(scale) == 1: + # to account for the legacy behavior having the exact same cast + scale = float(scale) + elif np.iscomplexobj(scale): + raise TypeError("init_scale contains complex elements, " + "must be real.") + else: # test explicitly for allowed shapes and values + replace = True + if self.approx_type == 'hess': + shape = np.shape(self.B) + dtype = self.B.dtype + else: + shape = np.shape(self.H) + dtype = self.H.dtype + # copy, will replace the original + scale = np.array(scale, dtype=dtype, copy=True) + + # it has to match the shape of the matrix for the multiplication, + # no implicit broadcasting is allowed + if shape != (init_shape := np.shape(scale)): + raise ValueError("If init_scale is an array, it must have the " + f"dimensions of the hess/inv_hess: {shape}." + f" Got {init_shape}.") + if not issymmetric(scale): + raise ValueError("If init_scale is an array, it must be" + " symmetric (passing scipy.linalg.issymmetric)" + " to be an approximation of a hess/inv_hess.") + + # Scale initial matrix with ``scale * np.eye(n)`` or replace + # This is not ideal, we could assign the scale directly in + # initialize, but we would need to + if self.approx_type == 'hess': + if replace: + self.B = scale + else: + self.B *= scale + else: + if replace: + self.H = scale + else: + self.H *= scale + self.first_iteration = False + self._update_implementation(delta_x, delta_grad) + + def dot(self, p): + """Compute the product of the internal matrix with the given vector. + + Parameters + ---------- + p : array_like + 1-D array representing a vector. + + Returns + ------- + Hp : array + 1-D represents the result of multiplying the approximation matrix + by vector p. + """ + if self.approx_type == 'hess': + return self._symv(1, self.B, p) + else: + return self._symv(1, self.H, p) + + def get_matrix(self): + """Return the current internal matrix. + + Returns + ------- + M : ndarray, shape (n, n) + Dense matrix containing either the Hessian or its inverse + (depending on how `approx_type` was defined). + """ + if self.approx_type == 'hess': + M = np.copy(self.B) + else: + M = np.copy(self.H) + li = np.tril_indices_from(M, k=-1) + M[li] = M.T[li] + return M + + +class BFGS(FullHessianUpdateStrategy): + """Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian update strategy. + + Parameters + ---------- + exception_strategy : {'skip_update', 'damp_update'}, optional + Define how to proceed when the curvature condition is violated. + Set it to 'skip_update' to just skip the update. Or, alternatively, + set it to 'damp_update' to interpolate between the actual BFGS + result and the unmodified matrix. Both exceptions strategies + are explained in [1]_, p.536-537. + min_curvature : float + This number, scaled by a normalization factor, defines the + minimum curvature ``dot(delta_grad, delta_x)`` allowed to go + unaffected by the exception strategy. By default is equal to + 1e-8 when ``exception_strategy = 'skip_update'`` and equal + to 0.2 when ``exception_strategy = 'damp_update'``. + init_scale : {float, np.array, 'auto'} + This parameter can be used to initialize the Hessian or its + inverse. When a float is given, the relevant array is initialized + to ``np.eye(n) * init_scale``, where ``n`` is the problem dimension. + Alternatively, if a precisely ``(n, n)`` shaped, symmetric array is given, + this array will be used. Otherwise an error is generated. + Set it to 'auto' in order to use an automatic heuristic for choosing + the initial scale. The heuristic is described in [1]_, p.143. + The default is 'auto'. + + Notes + ----- + The update is based on the description in [1]_, p.140. + + References + ---------- + .. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization" + Second Edition (2006). + """ + + def __init__(self, exception_strategy='skip_update', min_curvature=None, + init_scale='auto'): + if exception_strategy == 'skip_update': + if min_curvature is not None: + self.min_curvature = min_curvature + else: + self.min_curvature = 1e-8 + elif exception_strategy == 'damp_update': + if min_curvature is not None: + self.min_curvature = min_curvature + else: + self.min_curvature = 0.2 + else: + raise ValueError("`exception_strategy` must be 'skip_update' " + "or 'damp_update'.") + + super().__init__(init_scale) + self.exception_strategy = exception_strategy + + def _update_inverse_hessian(self, ys, Hy, yHy, s): + """Update the inverse Hessian matrix. + + BFGS update using the formula: + + ``H <- H + ((H*y).T*y + s.T*y)/(s.T*y)^2 * (s*s.T) + - 1/(s.T*y) * ((H*y)*s.T + s*(H*y).T)`` + + where ``s = delta_x`` and ``y = delta_grad``. This formula is + equivalent to (6.17) in [1]_ written in a more efficient way + for implementation. + + References + ---------- + .. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization" + Second Edition (2006). + """ + self.H = self._syr2(-1.0 / ys, s, Hy, a=self.H) + self.H = self._syr((ys + yHy) / ys ** 2, s, a=self.H) + + def _update_hessian(self, ys, Bs, sBs, y): + """Update the Hessian matrix. + + BFGS update using the formula: + + ``B <- B - (B*s)*(B*s).T/s.T*(B*s) + y*y^T/s.T*y`` + + where ``s`` is short for ``delta_x`` and ``y`` is short + for ``delta_grad``. Formula (6.19) in [1]_. + + References + ---------- + .. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization" + Second Edition (2006). + """ + self.B = self._syr(1.0 / ys, y, a=self.B) + self.B = self._syr(-1.0 / sBs, Bs, a=self.B) + + def _update_implementation(self, delta_x, delta_grad): + # Auxiliary variables w and z + if self.approx_type == 'hess': + w = delta_x + z = delta_grad + else: + w = delta_grad + z = delta_x + # Do some common operations + wz = np.dot(w, z) + Mw = self @ w + wMw = Mw.dot(w) + # Guarantee that wMw > 0 by reinitializing matrix. + # While this is always true in exact arithmetic, + # indefinite matrix may appear due to roundoff errors. + if wMw <= 0.0: + scale = self._auto_scale(delta_x, delta_grad) + # Reinitialize matrix + if self.approx_type == 'hess': + self.B = scale * np.eye(self.n, dtype=float) + else: + self.H = scale * np.eye(self.n, dtype=float) + # Do common operations for new matrix + Mw = self @ w + wMw = Mw.dot(w) + # Check if curvature condition is violated + if wz <= self.min_curvature * wMw: + # If the option 'skip_update' is set + # we just skip the update when the condition + # is violated. + if self.exception_strategy == 'skip_update': + return + # If the option 'damp_update' is set we + # interpolate between the actual BFGS + # result and the unmodified matrix. + elif self.exception_strategy == 'damp_update': + update_factor = (1-self.min_curvature) / (1 - wz/wMw) + z = update_factor*z + (1-update_factor)*Mw + wz = np.dot(w, z) + # Update matrix + if self.approx_type == 'hess': + self._update_hessian(wz, Mw, wMw, z) + else: + self._update_inverse_hessian(wz, Mw, wMw, z) + + +class SR1(FullHessianUpdateStrategy): + """Symmetric-rank-1 Hessian update strategy. + + Parameters + ---------- + min_denominator : float + This number, scaled by a normalization factor, + defines the minimum denominator magnitude allowed + in the update. When the condition is violated we skip + the update. By default uses ``1e-8``. + init_scale : {float, np.array, 'auto'}, optional + This parameter can be used to initialize the Hessian or its + inverse. When a float is given, the relevant array is initialized + to ``np.eye(n) * init_scale``, where ``n`` is the problem dimension. + Alternatively, if a precisely ``(n, n)`` shaped, symmetric array is given, + this array will be used. Otherwise an error is generated. + Set it to 'auto' in order to use an automatic heuristic for choosing + the initial scale. The heuristic is described in [1]_, p.143. + The default is 'auto'. + + Notes + ----- + The update is based on the description in [1]_, p.144-146. + + References + ---------- + .. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization" + Second Edition (2006). + """ + + def __init__(self, min_denominator=1e-8, init_scale='auto'): + self.min_denominator = min_denominator + super().__init__(init_scale) + + def _update_implementation(self, delta_x, delta_grad): + # Auxiliary variables w and z + if self.approx_type == 'hess': + w = delta_x + z = delta_grad + else: + w = delta_grad + z = delta_x + # Do some common operations + Mw = self @ w + z_minus_Mw = z - Mw + denominator = np.dot(w, z_minus_Mw) + # If the denominator is too small + # we just skip the update. + if np.abs(denominator) <= self.min_denominator*norm(w)*norm(z_minus_Mw): + return + # Update matrix + if self.approx_type == 'hess': + self.B = self._syr(1/denominator, z_minus_Mw, a=self.B) + else: + self.H = self._syr(1/denominator, z_minus_Mw, a=self.H) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_highspy/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_highspy/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_highspy/_highs_wrapper.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_highspy/_highs_wrapper.py new file mode 100644 index 0000000000000000000000000000000000000000..c88f0fb14c627b10584995fbde17f3a0e445b0cf --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_highspy/_highs_wrapper.py @@ -0,0 +1,338 @@ +from warnings import warn + +import numpy as np +import scipy.optimize._highspy._core as _h # type: ignore[import-not-found] +from scipy.optimize._highspy import _highs_options as hopt # type: ignore[attr-defined] +from scipy.optimize import OptimizeWarning + + +def _highs_wrapper(c, indptr, indices, data, lhs, rhs, lb, ub, integrality, options): + '''Solve linear programs using HiGHS [1]_. + + Assume problems of the form: + + MIN c.T @ x + s.t. lhs <= A @ x <= rhs + lb <= x <= ub + + Parameters + ---------- + c : 1-D array, (n,) + Array of objective value coefficients. + astart : 1-D array + CSC format index array. + aindex : 1-D array + CSC format index array. + avalue : 1-D array + Data array of the matrix. + lhs : 1-D array (or None), (m,) + Array of left hand side values of the inequality constraints. + If ``lhs=None``, then an array of ``-inf`` is assumed. + rhs : 1-D array, (m,) + Array of right hand side values of the inequality constraints. + lb : 1-D array (or None), (n,) + Lower bounds on solution variables x. If ``lb=None``, then an + array of all `0` is assumed. + ub : 1-D array (or None), (n,) + Upper bounds on solution variables x. If ``ub=None``, then an + array of ``inf`` is assumed. + options : dict + A dictionary of solver options + + Returns + ------- + res : dict + + If model_status is one of kOptimal, + kObjectiveBound, kTimeLimit, + kIterationLimit: + + - ``status`` : HighsModelStatus + Model status code. + + - ``message`` : str + Message corresponding to model status code. + + - ``x`` : list + Solution variables. + + - ``slack`` : list + Slack variables. + + - ``lambda`` : list + Lagrange multipliers associated with the constraints + Ax = b. + + - ``s`` : list + Lagrange multipliers associated with the constraints + x >= 0. + + - ``fun`` + Final objective value. + + - ``simplex_nit`` : int + Number of iterations accomplished by the simplex + solver. + + - ``ipm_nit`` : int + Number of iterations accomplished by the interior- + point solver. + + If model_status is not one of the above: + + - ``status`` : HighsModelStatus + Model status code. + + - ``message`` : str + Message corresponding to model status code. + + Notes + ----- + If ``options['write_solution_to_file']`` is ``True`` but + ``options['solution_file']`` is unset or ``''``, then the solution + will be printed to ``stdout``. + + If any iteration limit is reached, no solution will be + available. + + ``OptimizeWarning`` will be raised if any option value set by + the user is found to be incorrect. + + References + ---------- + .. [1] https://highs.dev/ + .. [2] https://www.maths.ed.ac.uk/hall/HiGHS/HighsOptions.html + ''' + numcol = c.size + numrow = rhs.size + isMip = integrality is not None and np.sum(integrality) > 0 + + # default "null" return values + res = { + "x": None, + "fun": None, + } + + # Fill up a HighsLp object + lp = _h.HighsLp() + lp.num_col_ = numcol + lp.num_row_ = numrow + lp.a_matrix_.num_col_ = numcol + lp.a_matrix_.num_row_ = numrow + lp.a_matrix_.format_ = _h.MatrixFormat.kColwise + lp.col_cost_ = c + lp.col_lower_ = lb + lp.col_upper_ = ub + lp.row_lower_ = lhs + lp.row_upper_ = rhs + lp.a_matrix_.start_ = indptr + lp.a_matrix_.index_ = indices + lp.a_matrix_.value_ = data + if integrality.size > 0: + lp.integrality_ = [_h.HighsVarType(i) for i in integrality] + + # Make a Highs object and pass it everything + highs = _h._Highs() + highs_options = _h.HighsOptions() + hoptmanager = hopt.HighsOptionsManager() + for key, val in options.items(): + # handle filtering of unsupported and default options + if val is None or key in ("sense",): + continue + + # ask for the option type + opt_type = hoptmanager.get_option_type(key) + if -1 == opt_type: + warn( + f"Unrecognized options detected: {dict({key: val})}", + OptimizeWarning, + stacklevel=2, + ) + continue + else: + if key in ("presolve", "parallel"): + # handle fake bools (require bool -> str conversions) + if isinstance(val, bool): + val = "on" if val else "off" + else: + warn( + f'Option f"{key}" is "{val}", but only True or False is ' + f"allowed. Using default.", + OptimizeWarning, + stacklevel=2, + ) + continue + opt_type = _h.HighsOptionType(opt_type) + status, msg = check_option(highs, key, val) + if opt_type == _h.HighsOptionType.kBool: + if not isinstance(val, bool): + warn( + f'Option f"{key}" is "{val}", but only True or False is ' + f"allowed. Using default.", + OptimizeWarning, + stacklevel=2, + ) + continue + + # warn or set option + if status != 0: + warn(msg, OptimizeWarning, stacklevel=2) + else: + setattr(highs_options, key, val) + + opt_status = highs.passOptions(highs_options) + if opt_status == _h.HighsStatus.kError: + res.update( + { + "status": highs.getModelStatus(), + "message": highs.modelStatusToString(highs.getModelStatus()), + } + ) + return res + + init_status = highs.passModel(lp) + if init_status == _h.HighsStatus.kError: + # if model fails to load, highs.getModelStatus() will be NOT_SET + err_model_status = _h.HighsModelStatus.kModelError + res.update( + { + "status": err_model_status, + "message": highs.modelStatusToString(err_model_status), + } + ) + return res + + # Solve the LP + run_status = highs.run() + if run_status == _h.HighsStatus.kError: + res.update( + { + "status": highs.getModelStatus(), + "message": highs.modelStatusToString(highs.getModelStatus()), + } + ) + return res + + # Extract what we need from the solution + model_status = highs.getModelStatus() + + # it should always be safe to get the info object + info = highs.getInfo() + + # Failure modes: + # LP: if we have anything other than an Optimal status, it + # is unsafe (and unhelpful) to read any results + # MIP: has a non-Optimal status or has timed out/reached max iterations + # 1) If not Optimal/TimedOut/MaxIter status, there is no solution + # 2) If TimedOut/MaxIter status, there may be a feasible solution. + # if the objective function value is not Infinity, then the + # current solution is feasible and can be returned. Else, there + # is no solution. + mipFailCondition = model_status not in ( + _h.HighsModelStatus.kOptimal, + _h.HighsModelStatus.kTimeLimit, + _h.HighsModelStatus.kIterationLimit, + _h.HighsModelStatus.kSolutionLimit, + ) or ( + model_status + in { + _h.HighsModelStatus.kTimeLimit, + _h.HighsModelStatus.kIterationLimit, + _h.HighsModelStatus.kSolutionLimit, + } + and (info.objective_function_value == _h.kHighsInf) + ) + lpFailCondition = model_status != _h.HighsModelStatus.kOptimal + if (isMip and mipFailCondition) or (not isMip and lpFailCondition): + res.update( + { + "status": model_status, + "message": "model_status is " + f"{highs.modelStatusToString(model_status)}; " + "primal_status is " + f"{highs.solutionStatusToString(info.primal_solution_status)}", + "simplex_nit": info.simplex_iteration_count, + "ipm_nit": info.ipm_iteration_count, + "crossover_nit": info.crossover_iteration_count, + } + ) + return res + + # Should be safe to read the solution: + solution = highs.getSolution() + basis = highs.getBasis() + + # Lagrangians for bounds based on column statuses + marg_bnds = np.zeros((2, numcol)) + basis_col_status = basis.col_status + solution_col_dual = solution.col_dual + for ii in range(numcol): + if basis_col_status[ii] == _h.HighsBasisStatus.kLower: + marg_bnds[0, ii] = solution_col_dual[ii] + elif basis_col_status[ii] == _h.HighsBasisStatus.kUpper: + marg_bnds[1, ii] = solution_col_dual[ii] + + res.update( + { + "status": model_status, + "message": highs.modelStatusToString(model_status), + # Primal solution + "x": np.array(solution.col_value), + # Ax + s = b => Ax = b - s + # Note: this is for all constraints (A_ub and A_eq) + "slack": rhs - solution.row_value, + # lambda are the lagrange multipliers associated with Ax=b + "lambda": np.array(solution.row_dual), + "marg_bnds": marg_bnds, + "fun": info.objective_function_value, + "simplex_nit": info.simplex_iteration_count, + "ipm_nit": info.ipm_iteration_count, + "crossover_nit": info.crossover_iteration_count, + } + ) + + if isMip: + res.update( + { + "mip_node_count": info.mip_node_count, + "mip_dual_bound": info.mip_dual_bound, + "mip_gap": info.mip_gap, + } + ) + + return res + + +def check_option(highs_inst, option, value): + status, option_type = highs_inst.getOptionType(option) + hoptmanager = hopt.HighsOptionsManager() + + if status != _h.HighsStatus.kOk: + return -1, "Invalid option name." + + valid_types = { + _h.HighsOptionType.kBool: bool, + _h.HighsOptionType.kInt: int, + _h.HighsOptionType.kDouble: float, + _h.HighsOptionType.kString: str, + } + + expected_type = valid_types.get(option_type, None) + + if expected_type is str: + if not hoptmanager.check_string_option(option, value): + return -1, "Invalid option value." + if expected_type is float: + if not hoptmanager.check_double_option(option, value): + return -1, "Invalid option value." + if expected_type is int: + if not hoptmanager.check_int_option(option, value): + return -1, "Invalid option value." + + if expected_type is None: + return 3, "Unknown option type." + + status, current_value = highs_inst.getOptionValue(option) + if status != _h.HighsStatus.kOk: + return 4, "Failed to validate option value." + return 0, "Check option succeeded." diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_isotonic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_isotonic.py new file mode 100644 index 0000000000000000000000000000000000000000..825576535402a9acf8bbff009a5f76282cb4f500 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_isotonic.py @@ -0,0 +1,157 @@ +from typing import TYPE_CHECKING + +import numpy as np + +from ._optimize import OptimizeResult +from ._pava_pybind import pava + +if TYPE_CHECKING: + import numpy.typing as npt + + +__all__ = ["isotonic_regression"] + + +def isotonic_regression( + y: "npt.ArrayLike", + *, + weights: "npt.ArrayLike | None" = None, + increasing: bool = True, +) -> OptimizeResult: + r"""Nonparametric isotonic regression. + + A (not strictly) monotonically increasing array `x` with the same length + as `y` is calculated by the pool adjacent violators algorithm (PAVA), see + [1]_. See the Notes section for more details. + + Parameters + ---------- + y : (N,) array_like + Response variable. + weights : (N,) array_like or None + Case weights. + increasing : bool + If True, fit monotonic increasing, i.e. isotonic, regression. + If False, fit a monotonic decreasing, i.e. antitonic, regression. + Default is True. + + Returns + ------- + res : OptimizeResult + The optimization result represented as a ``OptimizeResult`` object. + Important attributes are: + + - ``x``: The isotonic regression solution, i.e. an increasing (or + decreasing) array of the same length than y, with elements in the + range from min(y) to max(y). + - ``weights`` : Array with the sum of case weights for each block + (or pool) B. + - ``blocks``: Array of length B+1 with the indices of the start + positions of each block (or pool) B. The j-th block is given by + ``x[blocks[j]:blocks[j+1]]`` for which all values are the same. + + Notes + ----- + Given data :math:`y` and case weights :math:`w`, the isotonic regression + solves the following optimization problem: + + .. math:: + + \operatorname{argmin}_{x_i} \sum_i w_i (y_i - x_i)^2 \quad + \text{subject to } x_i \leq x_j \text{ whenever } i \leq j \,. + + For every input value :math:`y_i`, it generates a value :math:`x_i` such + that :math:`x` is increasing (but not strictly), i.e. + :math:`x_i \leq x_{i+1}`. This is accomplished by the PAVA. + The solution consists of pools or blocks, i.e. neighboring elements of + :math:`x`, e.g. :math:`x_i` and :math:`x_{i+1}`, that all have the same + value. + + Most interestingly, the solution stays the same if the squared loss is + replaced by the wide class of Bregman functions which are the unique + class of strictly consistent scoring functions for the mean, see [2]_ + and references therein. + + The implemented version of PAVA according to [1]_ has a computational + complexity of O(N) with input size N. + + References + ---------- + .. [1] Busing, F. M. T. A. (2022). + Monotone Regression: A Simple and Fast O(n) PAVA Implementation. + Journal of Statistical Software, Code Snippets, 102(1), 1-25. + :doi:`10.18637/jss.v102.c01` + .. [2] Jordan, A.I., Mühlemann, A. & Ziegel, J.F. + Characterizing the optimal solutions to the isotonic regression + problem for identifiable functionals. + Ann Inst Stat Math 74, 489-514 (2022). + :doi:`10.1007/s10463-021-00808-0` + + Examples + -------- + This example demonstrates that ``isotonic_regression`` really solves a + constrained optimization problem. + + >>> import numpy as np + >>> from scipy.optimize import isotonic_regression, minimize + >>> y = [1.5, 1.0, 4.0, 6.0, 5.7, 5.0, 7.8, 9.0, 7.5, 9.5, 9.0] + >>> def objective(yhat, y): + ... return np.sum((yhat - y)**2) + >>> def constraint(yhat, y): + ... # This is for a monotonically increasing regression. + ... return np.diff(yhat) + >>> result = minimize(objective, x0=y, args=(y,), + ... constraints=[{'type': 'ineq', + ... 'fun': lambda x: constraint(x, y)}]) + >>> result.x + array([1.25 , 1.25 , 4. , 5.56666667, 5.56666667, + 5.56666667, 7.8 , 8.25 , 8.25 , 9.25 , + 9.25 ]) + >>> result = isotonic_regression(y) + >>> result.x + array([1.25 , 1.25 , 4. , 5.56666667, 5.56666667, + 5.56666667, 7.8 , 8.25 , 8.25 , 9.25 , + 9.25 ]) + + The big advantage of ``isotonic_regression`` compared to calling + ``minimize`` is that it is more user friendly, i.e. one does not need to + define objective and constraint functions, and that it is orders of + magnitudes faster. On commodity hardware (in 2023), for normal distributed + input y of length 1000, the minimizer takes about 4 seconds, while + ``isotonic_regression`` takes about 200 microseconds. + """ + yarr = np.atleast_1d(y) # Check yarr.ndim == 1 is implicit (pybind11) in pava. + order = slice(None) if increasing else slice(None, None, -1) + x = np.array(yarr[order], order="C", dtype=np.float64, copy=True) + if weights is None: + wx = np.ones_like(yarr, dtype=np.float64) + else: + warr = np.atleast_1d(weights) + + if not (yarr.ndim == warr.ndim == 1 and yarr.shape[0] == warr.shape[0]): + raise ValueError( + "Input arrays y and w must have one dimension of equal length." + ) + if np.any(warr <= 0): + raise ValueError("Weights w must be strictly positive.") + + wx = np.array(warr[order], order="C", dtype=np.float64, copy=True) + n = x.shape[0] + r = np.full(shape=n + 1, fill_value=-1, dtype=np.intp) + x, wx, r, b = pava(x, wx, r) + # Now that we know the number of blocks b, we only keep the relevant part + # of r and wx. + # As information: Due to the pava implementation, after the last block + # index, there might be smaller numbers appended to r, e.g. + # r = [0, 10, 8, 7] which in the end should be r = [0, 10]. + r = r[:b + 1] # type: ignore[assignment] + wx = wx[:b] + if not increasing: + x = x[::-1] + wx = wx[::-1] + r = r[-1] - r[::-1] + return OptimizeResult( + x=x, + weights=wx, + blocks=r, + ) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lbfgsb_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lbfgsb_py.py new file mode 100644 index 0000000000000000000000000000000000000000..d0e206feaa9333229c3842850eee1f673c8ef02b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lbfgsb_py.py @@ -0,0 +1,578 @@ +""" +Functions +--------- +.. autosummary:: + :toctree: generated/ + + fmin_l_bfgs_b + +""" + +## License for the Python wrapper +## ============================== + +## Copyright (c) 2004 David M. Cooke + +## Permission is hereby granted, free of charge, to any person obtaining a +## copy of this software and associated documentation files (the "Software"), +## to deal in the Software without restriction, including without limitation +## the rights to use, copy, modify, merge, publish, distribute, sublicense, +## and/or sell copies of the Software, and to permit persons to whom the +## Software is furnished to do so, subject to the following conditions: + +## The above copyright notice and this permission notice shall be included in +## all copies or substantial portions of the Software. + +## THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +## IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +## FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +## AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +## LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING +## FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER +## DEALINGS IN THE SOFTWARE. + +## Modifications by Travis Oliphant and Enthought, Inc. for inclusion in SciPy + +import numpy as np +from numpy import array, asarray, float64, zeros +from . import _lbfgsb +from ._optimize import (MemoizeJac, OptimizeResult, _call_callback_maybe_halt, + _wrap_callback, _check_unknown_options, + _prepare_scalar_function) +from ._constraints import old_bound_to_new + +from scipy.sparse.linalg import LinearOperator + +__all__ = ['fmin_l_bfgs_b', 'LbfgsInvHessProduct'] + + +status_messages = { + 0 : "START", + 1 : "NEW_X", + 2 : "RESTART", + 3 : "FG", + 4 : "CONVERGENCE", + 5 : "STOP", + 6 : "WARNING", + 7 : "ERROR", + 8 : "ABNORMAL" +} + + +task_messages = { + 0 : "", + 301 : "", + 302 : "", + 401 : "NORM OF PROJECTED GRADIENT <= PGTOL", + 402 : "RELATIVE REDUCTION OF F <= FACTR*EPSMCH", + 501 : "CPU EXCEEDING THE TIME LIMIT", + 502 : "TOTAL NO. OF F,G EVALUATIONS EXCEEDS LIMIT", + 503 : "PROJECTED GRADIENT IS SUFFICIENTLY SMALL", + 504 : "TOTAL NO. OF ITERATIONS REACHED LIMIT", + 505 : "CALLBACK REQUESTED HALT", + 601 : "ROUNDING ERRORS PREVENT PROGRESS", + 602 : "STP = STPMAX", + 603 : "STP = STPMIN", + 604 : "XTOL TEST SATISFIED", + 701 : "NO FEASIBLE SOLUTION", + 702 : "FACTR < 0", + 703 : "FTOL < 0", + 704 : "GTOL < 0", + 705 : "XTOL < 0", + 706 : "STP < STPMIN", + 707 : "STP > STPMAX", + 708 : "STPMIN < 0", + 709 : "STPMAX < STPMIN", + 710 : "INITIAL G >= 0", + 711 : "M <= 0", + 712 : "N <= 0", + 713 : "INVALID NBD", +} + +def fmin_l_bfgs_b(func, x0, fprime=None, args=(), + approx_grad=0, + bounds=None, m=10, factr=1e7, pgtol=1e-5, + epsilon=1e-8, + iprint=-1, maxfun=15000, maxiter=15000, disp=None, + callback=None, maxls=20): + """ + Minimize a function func using the L-BFGS-B algorithm. + + Parameters + ---------- + func : callable f(x,*args) + Function to minimize. + x0 : ndarray + Initial guess. + fprime : callable fprime(x,*args), optional + The gradient of `func`. If None, then `func` returns the function + value and the gradient (``f, g = func(x, *args)``), unless + `approx_grad` is True in which case `func` returns only ``f``. + args : sequence, optional + Arguments to pass to `func` and `fprime`. + approx_grad : bool, optional + Whether to approximate the gradient numerically (in which case + `func` returns only the function value). + bounds : list, optional + ``(min, max)`` pairs for each element in ``x``, defining + the bounds on that parameter. Use None or +-inf for one of ``min`` or + ``max`` when there is no bound in that direction. + m : int, optional + The maximum number of variable metric corrections + used to define the limited memory matrix. (The limited memory BFGS + method does not store the full hessian but uses this many terms in an + approximation to it.) + factr : float, optional + The iteration stops when + ``(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps``, + where ``eps`` is the machine precision, which is automatically + generated by the code. Typical values for `factr` are: 1e12 for + low accuracy; 1e7 for moderate accuracy; 10.0 for extremely + high accuracy. See Notes for relationship to `ftol`, which is exposed + (instead of `factr`) by the `scipy.optimize.minimize` interface to + L-BFGS-B. + pgtol : float, optional + The iteration will stop when + ``max{|proj g_i | i = 1, ..., n} <= pgtol`` + where ``proj g_i`` is the i-th component of the projected gradient. + epsilon : float, optional + Step size used when `approx_grad` is True, for numerically + calculating the gradient + iprint : int, optional + Deprecated option that previously controlled the text printed on the + screen during the problem solution. Now the code does not emit any + output and this keyword has no function. + + .. deprecated:: 1.15.0 + This keyword is deprecated and will be removed from SciPy 1.17.0. + + disp : int, optional + Deprecated option that previously controlled the text printed on the + screen during the problem solution. Now the code does not emit any + output and this keyword has no function. + + .. deprecated:: 1.15.0 + This keyword is deprecated and will be removed from SciPy 1.17.0. + + maxfun : int, optional + Maximum number of function evaluations. Note that this function + may violate the limit because of evaluating gradients by numerical + differentiation. + maxiter : int, optional + Maximum number of iterations. + callback : callable, optional + Called after each iteration, as ``callback(xk)``, where ``xk`` is the + current parameter vector. + maxls : int, optional + Maximum number of line search steps (per iteration). Default is 20. + + Returns + ------- + x : array_like + Estimated position of the minimum. + f : float + Value of `func` at the minimum. + d : dict + Information dictionary. + + * d['warnflag'] is + + - 0 if converged, + - 1 if too many function evaluations or too many iterations, + - 2 if stopped for another reason, given in d['task'] + + * d['grad'] is the gradient at the minimum (should be 0 ish) + * d['funcalls'] is the number of function calls made. + * d['nit'] is the number of iterations. + + See also + -------- + minimize: Interface to minimization algorithms for multivariate + functions. See the 'L-BFGS-B' `method` in particular. Note that the + `ftol` option is made available via that interface, while `factr` is + provided via this interface, where `factr` is the factor multiplying + the default machine floating-point precision to arrive at `ftol`: + ``ftol = factr * numpy.finfo(float).eps``. + + Notes + ----- + SciPy uses a C-translated and modified version of the Fortran code, + L-BFGS-B v3.0 (released April 25, 2011, BSD-3 licensed). Original Fortran + version was written by Ciyou Zhu, Richard Byrd, Jorge Nocedal and, + Jose Luis Morales. + + References + ---------- + * R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound + Constrained Optimization, (1995), SIAM Journal on Scientific and + Statistical Computing, 16, 5, pp. 1190-1208. + * C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B, + FORTRAN routines for large scale bound constrained optimization (1997), + ACM Transactions on Mathematical Software, 23, 4, pp. 550 - 560. + * J.L. Morales and J. Nocedal. L-BFGS-B: Remark on Algorithm 778: L-BFGS-B, + FORTRAN routines for large scale bound constrained optimization (2011), + ACM Transactions on Mathematical Software, 38, 1. + + Examples + -------- + Solve a linear regression problem via `fmin_l_bfgs_b`. To do this, first we + define an objective function ``f(m, b) = (y - y_model)**2``, where `y` + describes the observations and `y_model` the prediction of the linear model + as ``y_model = m*x + b``. The bounds for the parameters, ``m`` and ``b``, + are arbitrarily chosen as ``(0,5)`` and ``(5,10)`` for this example. + + >>> import numpy as np + >>> from scipy.optimize import fmin_l_bfgs_b + >>> X = np.arange(0, 10, 1) + >>> M = 2 + >>> B = 3 + >>> Y = M * X + B + >>> def func(parameters, *args): + ... x = args[0] + ... y = args[1] + ... m, b = parameters + ... y_model = m*x + b + ... error = sum(np.power((y - y_model), 2)) + ... return error + + >>> initial_values = np.array([0.0, 1.0]) + + >>> x_opt, f_opt, info = fmin_l_bfgs_b(func, x0=initial_values, args=(X, Y), + ... approx_grad=True) + >>> x_opt, f_opt + array([1.99999999, 3.00000006]), 1.7746231151323805e-14 # may vary + + The optimized parameters in ``x_opt`` agree with the ground truth parameters + ``m`` and ``b``. Next, let us perform a bound constrained optimization using + the `bounds` parameter. + + >>> bounds = [(0, 5), (5, 10)] + >>> x_opt, f_op, info = fmin_l_bfgs_b(func, x0=initial_values, args=(X, Y), + ... approx_grad=True, bounds=bounds) + >>> x_opt, f_opt + array([1.65990508, 5.31649385]), 15.721334516453945 # may vary + """ + # handle fprime/approx_grad + if approx_grad: + fun = func + jac = None + elif fprime is None: + fun = MemoizeJac(func) + jac = fun.derivative + else: + fun = func + jac = fprime + + # build options + callback = _wrap_callback(callback) + opts = {'maxcor': m, + 'ftol': factr * np.finfo(float).eps, + 'gtol': pgtol, + 'eps': epsilon, + 'maxfun': maxfun, + 'maxiter': maxiter, + 'callback': callback, + 'maxls': maxls} + + res = _minimize_lbfgsb(fun, x0, args=args, jac=jac, bounds=bounds, + **opts) + d = {'grad': res['jac'], + 'task': res['message'], + 'funcalls': res['nfev'], + 'nit': res['nit'], + 'warnflag': res['status']} + f = res['fun'] + x = res['x'] + + return x, f, d + + +def _minimize_lbfgsb(fun, x0, args=(), jac=None, bounds=None, + disp=None, maxcor=10, ftol=2.2204460492503131e-09, + gtol=1e-5, eps=1e-8, maxfun=15000, maxiter=15000, + iprint=-1, callback=None, maxls=20, + finite_diff_rel_step=None, **unknown_options): + """ + Minimize a scalar function of one or more variables using the L-BFGS-B + algorithm. + + Options + ------- + disp : None or int + Deprecated option that previously controlled the text printed on the + screen during the problem solution. Now the code does not emit any + output and this keyword has no function. + + .. deprecated:: 1.15.0 + This keyword is deprecated and will be removed from SciPy 1.17.0. + + maxcor : int + The maximum number of variable metric corrections used to + define the limited memory matrix. (The limited memory BFGS + method does not store the full hessian but uses this many terms + in an approximation to it.) + ftol : float + The iteration stops when ``(f^k - + f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= ftol``. + gtol : float + The iteration will stop when ``max{|proj g_i | i = 1, ..., n} + <= gtol`` where ``proj g_i`` is the i-th component of the + projected gradient. + eps : float or ndarray + If `jac is None` the absolute step size used for numerical + approximation of the jacobian via forward differences. + maxfun : int + Maximum number of function evaluations. Note that this function + may violate the limit because of evaluating gradients by numerical + differentiation. + maxiter : int + Maximum number of iterations. + iprint : int, optional + Deprecated option that previously controlled the text printed on the + screen during the problem solution. Now the code does not emit any + output and this keyword has no function. + + .. deprecated:: 1.15.0 + This keyword is deprecated and will be removed from SciPy 1.17.0. + + maxls : int, optional + Maximum number of line search steps (per iteration). Default is 20. + finite_diff_rel_step : None or array_like, optional + If ``jac in ['2-point', '3-point', 'cs']`` the relative step size to + use for numerical approximation of the jacobian. The absolute step + size is computed as ``h = rel_step * sign(x) * max(1, abs(x))``, + possibly adjusted to fit into the bounds. For ``method='3-point'`` + the sign of `h` is ignored. If None (default) then step is selected + automatically. + + Notes + ----- + The option `ftol` is exposed via the `scipy.optimize.minimize` interface, + but calling `scipy.optimize.fmin_l_bfgs_b` directly exposes `factr`. The + relationship between the two is ``ftol = factr * numpy.finfo(float).eps``. + I.e., `factr` multiplies the default machine floating-point precision to + arrive at `ftol`. + + """ + _check_unknown_options(unknown_options) + m = maxcor + pgtol = gtol + factr = ftol / np.finfo(float).eps + + x0 = asarray(x0).ravel() + n, = x0.shape + + # historically old-style bounds were/are expected by lbfgsb. + # That's still the case but we'll deal with new-style from here on, + # it's easier + if bounds is None: + pass + elif len(bounds) != n: + raise ValueError('length of x0 != length of bounds') + else: + bounds = np.array(old_bound_to_new(bounds)) + + # check bounds + if (bounds[0] > bounds[1]).any(): + raise ValueError( + "LBFGSB - one of the lower bounds is greater than an upper bound." + ) + + # initial vector must lie within the bounds. Otherwise ScalarFunction and + # approx_derivative will cause problems + x0 = np.clip(x0, bounds[0], bounds[1]) + + # _prepare_scalar_function can use bounds=None to represent no bounds + sf = _prepare_scalar_function(fun, x0, jac=jac, args=args, epsilon=eps, + bounds=bounds, + finite_diff_rel_step=finite_diff_rel_step) + + func_and_grad = sf.fun_and_grad + + nbd = zeros(n, np.int32) + low_bnd = zeros(n, float64) + upper_bnd = zeros(n, float64) + bounds_map = {(-np.inf, np.inf): 0, + (1, np.inf): 1, + (1, 1): 2, + (-np.inf, 1): 3} + + if bounds is not None: + for i in range(0, n): + L, U = bounds[0, i], bounds[1, i] + if not np.isinf(L): + low_bnd[i] = L + L = 1 + if not np.isinf(U): + upper_bnd[i] = U + U = 1 + nbd[i] = bounds_map[L, U] + + if not maxls > 0: + raise ValueError('maxls must be positive.') + + x = array(x0, dtype=np.float64) + f = array(0.0, dtype=np.int32) + g = zeros((n,), dtype=np.int32) + wa = zeros(2*m*n + 5*n + 11*m*m + 8*m, float64) + iwa = zeros(3*n, dtype=np.int32) + task = zeros(2, dtype=np.int32) + ln_task = zeros(2, dtype=np.int32) + lsave = zeros(4, dtype=np.int32) + isave = zeros(44, dtype=np.int32) + dsave = zeros(29, dtype=float64) + + n_iterations = 0 + + while True: + # g may become float32 if a user provides a function that calculates + # the Jacobian in float32 (see gh-18730). The underlying code expects + # float64, so upcast it + g = g.astype(np.float64) + # x, f, g, wa, iwa, task, csave, lsave, isave, dsave = \ + _lbfgsb.setulb(m, x, low_bnd, upper_bnd, nbd, f, g, factr, pgtol, wa, + iwa, task, lsave, isave, dsave, maxls, ln_task) + + if task[0] == 3: + # The minimization routine wants f and g at the current x. + # Note that interruptions due to maxfun are postponed + # until the completion of the current minimization iteration. + # Overwrite f and g: + f, g = func_and_grad(x) + elif task[0] == 1: + # new iteration + n_iterations += 1 + + intermediate_result = OptimizeResult(x=x, fun=f) + if _call_callback_maybe_halt(callback, intermediate_result): + task[0] = 5 + task[1] = 505 + if n_iterations >= maxiter: + task[0] = 5 + task[1] = 504 + elif sf.nfev > maxfun: + task[0] = 5 + task[1] = 502 + else: + break + + if task[0] == 4: + warnflag = 0 + elif sf.nfev > maxfun or n_iterations >= maxiter: + warnflag = 1 + else: + warnflag = 2 + + # These two portions of the workspace are described in the mainlb + # function docstring in "__lbfgsb.c", ws and wy arguments. + s = wa[0: m*n].reshape(m, n) + y = wa[m*n: 2*m*n].reshape(m, n) + + # isave(31) = the total number of BFGS updates prior the current iteration. + n_bfgs_updates = isave[30] + + n_corrs = min(n_bfgs_updates, maxcor) + hess_inv = LbfgsInvHessProduct(s[:n_corrs], y[:n_corrs]) + + msg = status_messages[task[0]] + ": " + task_messages[task[1]] + + return OptimizeResult(fun=f, jac=g, nfev=sf.nfev, + njev=sf.ngev, + nit=n_iterations, status=warnflag, message=msg, + x=x, success=(warnflag == 0), hess_inv=hess_inv) + + +class LbfgsInvHessProduct(LinearOperator): + """Linear operator for the L-BFGS approximate inverse Hessian. + + This operator computes the product of a vector with the approximate inverse + of the Hessian of the objective function, using the L-BFGS limited + memory approximation to the inverse Hessian, accumulated during the + optimization. + + Objects of this class implement the ``scipy.sparse.linalg.LinearOperator`` + interface. + + Parameters + ---------- + sk : array_like, shape=(n_corr, n) + Array of `n_corr` most recent updates to the solution vector. + (See [1]). + yk : array_like, shape=(n_corr, n) + Array of `n_corr` most recent updates to the gradient. (See [1]). + + References + ---------- + .. [1] Nocedal, Jorge. "Updating quasi-Newton matrices with limited + storage." Mathematics of computation 35.151 (1980): 773-782. + + """ + + def __init__(self, sk, yk): + """Construct the operator.""" + if sk.shape != yk.shape or sk.ndim != 2: + raise ValueError('sk and yk must have matching shape, (n_corrs, n)') + n_corrs, n = sk.shape + + super().__init__(dtype=np.float64, shape=(n, n)) + + self.sk = sk + self.yk = yk + self.n_corrs = n_corrs + self.rho = 1 / np.einsum('ij,ij->i', sk, yk) + + def _matvec(self, x): + """Efficient matrix-vector multiply with the BFGS matrices. + + This calculation is described in Section (4) of [1]. + + Parameters + ---------- + x : ndarray + An array with shape (n,) or (n,1). + + Returns + ------- + y : ndarray + The matrix-vector product + + """ + s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho + q = np.array(x, dtype=self.dtype, copy=True) + if q.ndim == 2 and q.shape[1] == 1: + q = q.reshape(-1) + + alpha = np.empty(n_corrs) + + for i in range(n_corrs-1, -1, -1): + alpha[i] = rho[i] * np.dot(s[i], q) + q = q - alpha[i]*y[i] + + r = q + for i in range(n_corrs): + beta = rho[i] * np.dot(y[i], r) + r = r + s[i] * (alpha[i] - beta) + + return r + + def todense(self): + """Return a dense array representation of this operator. + + Returns + ------- + arr : ndarray, shape=(n, n) + An array with the same shape and containing + the same data represented by this `LinearOperator`. + + """ + s, y, n_corrs, rho = self.sk, self.yk, self.n_corrs, self.rho + I_arr = np.eye(*self.shape, dtype=self.dtype) + Hk = I_arr + + for i in range(n_corrs): + A1 = I_arr - s[i][:, np.newaxis] * y[i][np.newaxis, :] * rho[i] + A2 = I_arr - y[i][:, np.newaxis] * s[i][np.newaxis, :] * rho[i] + + Hk = np.dot(A1, np.dot(Hk, A2)) + (rho[i] * s[i][:, np.newaxis] * + s[i][np.newaxis, :]) + return Hk diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linesearch.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linesearch.py new file mode 100644 index 0000000000000000000000000000000000000000..31442e02d323e0f6d163505bf77dd30855ce1218 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linesearch.py @@ -0,0 +1,896 @@ +""" +Functions +--------- +.. autosummary:: + :toctree: generated/ + + line_search_armijo + line_search_wolfe1 + line_search_wolfe2 + scalar_search_wolfe1 + scalar_search_wolfe2 + +""" +from warnings import warn + +from ._dcsrch import DCSRCH +import numpy as np + +__all__ = ['LineSearchWarning', 'line_search_wolfe1', 'line_search_wolfe2', + 'scalar_search_wolfe1', 'scalar_search_wolfe2', + 'line_search_armijo'] + +class LineSearchWarning(RuntimeWarning): + pass + + +def _check_c1_c2(c1, c2): + if not (0 < c1 < c2 < 1): + raise ValueError("'c1' and 'c2' do not satisfy" + "'0 < c1 < c2 < 1'.") + + +#------------------------------------------------------------------------------ +# Minpack's Wolfe line and scalar searches +#------------------------------------------------------------------------------ + +def line_search_wolfe1(f, fprime, xk, pk, gfk=None, + old_fval=None, old_old_fval=None, + args=(), c1=1e-4, c2=0.9, amax=50, amin=1e-8, + xtol=1e-14): + """ + As `scalar_search_wolfe1` but do a line search to direction `pk` + + Parameters + ---------- + f : callable + Function `f(x)` + fprime : callable + Gradient of `f` + xk : array_like + Current point + pk : array_like + Search direction + gfk : array_like, optional + Gradient of `f` at point `xk` + old_fval : float, optional + Value of `f` at point `xk` + old_old_fval : float, optional + Value of `f` at point preceding `xk` + + The rest of the parameters are the same as for `scalar_search_wolfe1`. + + Returns + ------- + stp, f_count, g_count, fval, old_fval + As in `line_search_wolfe1` + gval : array + Gradient of `f` at the final point + + Notes + ----- + Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1``. + + """ + if gfk is None: + gfk = fprime(xk, *args) + + gval = [gfk] + gc = [0] + fc = [0] + + def phi(s): + fc[0] += 1 + return f(xk + s*pk, *args) + + def derphi(s): + gval[0] = fprime(xk + s*pk, *args) + gc[0] += 1 + return np.dot(gval[0], pk) + + derphi0 = np.dot(gfk, pk) + + stp, fval, old_fval = scalar_search_wolfe1( + phi, derphi, old_fval, old_old_fval, derphi0, + c1=c1, c2=c2, amax=amax, amin=amin, xtol=xtol) + + return stp, fc[0], gc[0], fval, old_fval, gval[0] + + +def scalar_search_wolfe1(phi, derphi, phi0=None, old_phi0=None, derphi0=None, + c1=1e-4, c2=0.9, + amax=50, amin=1e-8, xtol=1e-14): + """ + Scalar function search for alpha that satisfies strong Wolfe conditions + + alpha > 0 is assumed to be a descent direction. + + Parameters + ---------- + phi : callable phi(alpha) + Function at point `alpha` + derphi : callable phi'(alpha) + Objective function derivative. Returns a scalar. + phi0 : float, optional + Value of phi at 0 + old_phi0 : float, optional + Value of phi at previous point + derphi0 : float, optional + Value derphi at 0 + c1 : float, optional + Parameter for Armijo condition rule. + c2 : float, optional + Parameter for curvature condition rule. + amax, amin : float, optional + Maximum and minimum step size + xtol : float, optional + Relative tolerance for an acceptable step. + + Returns + ------- + alpha : float + Step size, or None if no suitable step was found + phi : float + Value of `phi` at the new point `alpha` + phi0 : float + Value of `phi` at `alpha=0` + + Notes + ----- + Uses routine DCSRCH from MINPACK. + + Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1`` as described in [1]_. + + References + ---------- + + .. [1] Nocedal, J., & Wright, S. J. (2006). Numerical optimization. + In Springer Series in Operations Research and Financial Engineering. + (Springer Series in Operations Research and Financial Engineering). + Springer Nature. + + """ + _check_c1_c2(c1, c2) + + if phi0 is None: + phi0 = phi(0.) + if derphi0 is None: + derphi0 = derphi(0.) + + if old_phi0 is not None and derphi0 != 0: + alpha1 = min(1.0, 1.01*2*(phi0 - old_phi0)/derphi0) + if alpha1 < 0: + alpha1 = 1.0 + else: + alpha1 = 1.0 + + maxiter = 100 + + dcsrch = DCSRCH(phi, derphi, c1, c2, xtol, amin, amax) + stp, phi1, phi0, task = dcsrch( + alpha1, phi0=phi0, derphi0=derphi0, maxiter=maxiter + ) + + return stp, phi1, phi0 + + +line_search = line_search_wolfe1 + + +#------------------------------------------------------------------------------ +# Pure-Python Wolfe line and scalar searches +#------------------------------------------------------------------------------ + +# Note: `line_search_wolfe2` is the public `scipy.optimize.line_search` + +def line_search_wolfe2(f, myfprime, xk, pk, gfk=None, old_fval=None, + old_old_fval=None, args=(), c1=1e-4, c2=0.9, amax=None, + extra_condition=None, maxiter=10): + """Find alpha that satisfies strong Wolfe conditions. + + Parameters + ---------- + f : callable f(x,*args) + Objective function. + myfprime : callable f'(x,*args) + Objective function gradient. + xk : ndarray + Starting point. + pk : ndarray + Search direction. The search direction must be a descent direction + for the algorithm to converge. + gfk : ndarray, optional + Gradient value for x=xk (xk being the current parameter + estimate). Will be recomputed if omitted. + old_fval : float, optional + Function value for x=xk. Will be recomputed if omitted. + old_old_fval : float, optional + Function value for the point preceding x=xk. + args : tuple, optional + Additional arguments passed to objective function. + c1 : float, optional + Parameter for Armijo condition rule. + c2 : float, optional + Parameter for curvature condition rule. + amax : float, optional + Maximum step size + extra_condition : callable, optional + A callable of the form ``extra_condition(alpha, x, f, g)`` + returning a boolean. Arguments are the proposed step ``alpha`` + and the corresponding ``x``, ``f`` and ``g`` values. The line search + accepts the value of ``alpha`` only if this + callable returns ``True``. If the callable returns ``False`` + for the step length, the algorithm will continue with + new iterates. The callable is only called for iterates + satisfying the strong Wolfe conditions. + maxiter : int, optional + Maximum number of iterations to perform. + + Returns + ------- + alpha : float or None + Alpha for which ``x_new = x0 + alpha * pk``, + or None if the line search algorithm did not converge. + fc : int + Number of function evaluations made. + gc : int + Number of gradient evaluations made. + new_fval : float or None + New function value ``f(x_new)=f(x0+alpha*pk)``, + or None if the line search algorithm did not converge. + old_fval : float + Old function value ``f(x0)``. + new_slope : float or None + The local slope along the search direction at the + new value ````, + or None if the line search algorithm did not converge. + + + Notes + ----- + Uses the line search algorithm to enforce strong Wolfe + conditions. See Wright and Nocedal, 'Numerical Optimization', + 1999, pp. 59-61. + + The search direction `pk` must be a descent direction (e.g. + ``-myfprime(xk)``) to find a step length that satisfies the strong Wolfe + conditions. If the search direction is not a descent direction (e.g. + ``myfprime(xk)``), then `alpha`, `new_fval`, and `new_slope` will be None. + + Examples + -------- + >>> import numpy as np + >>> from scipy.optimize import line_search + + A objective function and its gradient are defined. + + >>> def obj_func(x): + ... return (x[0])**2+(x[1])**2 + >>> def obj_grad(x): + ... return [2*x[0], 2*x[1]] + + We can find alpha that satisfies strong Wolfe conditions. + + >>> start_point = np.array([1.8, 1.7]) + >>> search_gradient = np.array([-1.0, -1.0]) + >>> line_search(obj_func, obj_grad, start_point, search_gradient) + (1.0, 2, 1, 1.1300000000000001, 6.13, [1.6, 1.4]) + + """ + fc = [0] + gc = [0] + gval = [None] + gval_alpha = [None] + + def phi(alpha): + fc[0] += 1 + return f(xk + alpha * pk, *args) + + fprime = myfprime + + def derphi(alpha): + gc[0] += 1 + gval[0] = fprime(xk + alpha * pk, *args) # store for later use + gval_alpha[0] = alpha + return np.dot(gval[0], pk) + + if gfk is None: + gfk = fprime(xk, *args) + derphi0 = np.dot(gfk, pk) + + if extra_condition is not None: + # Add the current gradient as argument, to avoid needless + # re-evaluation + def extra_condition2(alpha, phi): + if gval_alpha[0] != alpha: + derphi(alpha) + x = xk + alpha * pk + return extra_condition(alpha, x, phi, gval[0]) + else: + extra_condition2 = None + + alpha_star, phi_star, old_fval, derphi_star = scalar_search_wolfe2( + phi, derphi, old_fval, old_old_fval, derphi0, c1, c2, amax, + extra_condition2, maxiter=maxiter) + + if derphi_star is None: + warn('The line search algorithm did not converge', + LineSearchWarning, stacklevel=2) + else: + # derphi_star is a number (derphi) -- so use the most recently + # calculated gradient used in computing it derphi = gfk*pk + # this is the gradient at the next step no need to compute it + # again in the outer loop. + derphi_star = gval[0] + + return alpha_star, fc[0], gc[0], phi_star, old_fval, derphi_star + + +def scalar_search_wolfe2(phi, derphi, phi0=None, + old_phi0=None, derphi0=None, + c1=1e-4, c2=0.9, amax=None, + extra_condition=None, maxiter=10): + """Find alpha that satisfies strong Wolfe conditions. + + alpha > 0 is assumed to be a descent direction. + + Parameters + ---------- + phi : callable phi(alpha) + Objective scalar function. + derphi : callable phi'(alpha) + Objective function derivative. Returns a scalar. + phi0 : float, optional + Value of phi at 0. + old_phi0 : float, optional + Value of phi at previous point. + derphi0 : float, optional + Value of derphi at 0 + c1 : float, optional + Parameter for Armijo condition rule. + c2 : float, optional + Parameter for curvature condition rule. + amax : float, optional + Maximum step size. + extra_condition : callable, optional + A callable of the form ``extra_condition(alpha, phi_value)`` + returning a boolean. The line search accepts the value + of ``alpha`` only if this callable returns ``True``. + If the callable returns ``False`` for the step length, + the algorithm will continue with new iterates. + The callable is only called for iterates satisfying + the strong Wolfe conditions. + maxiter : int, optional + Maximum number of iterations to perform. + + Returns + ------- + alpha_star : float or None + Best alpha, or None if the line search algorithm did not converge. + phi_star : float + phi at alpha_star. + phi0 : float + phi at 0. + derphi_star : float or None + derphi at alpha_star, or None if the line search algorithm + did not converge. + + Notes + ----- + Uses the line search algorithm to enforce strong Wolfe + conditions. See Wright and Nocedal, 'Numerical Optimization', + 1999, pp. 59-61. + + """ + _check_c1_c2(c1, c2) + + if phi0 is None: + phi0 = phi(0.) + + if derphi0 is None: + derphi0 = derphi(0.) + + alpha0 = 0 + if old_phi0 is not None and derphi0 != 0: + alpha1 = min(1.0, 1.01*2*(phi0 - old_phi0)/derphi0) + else: + alpha1 = 1.0 + + if alpha1 < 0: + alpha1 = 1.0 + + if amax is not None: + alpha1 = min(alpha1, amax) + + phi_a1 = phi(alpha1) + #derphi_a1 = derphi(alpha1) evaluated below + + phi_a0 = phi0 + derphi_a0 = derphi0 + + if extra_condition is None: + def extra_condition(alpha, phi): + return True + + for i in range(maxiter): + if alpha1 == 0 or (amax is not None and alpha0 > amax): + # alpha1 == 0: This shouldn't happen. Perhaps the increment has + # slipped below machine precision? + alpha_star = None + phi_star = phi0 + phi0 = old_phi0 + derphi_star = None + + if alpha1 == 0: + msg = 'Rounding errors prevent the line search from converging' + else: + msg = "The line search algorithm could not find a solution " + \ + f"less than or equal to amax: {amax}" + + warn(msg, LineSearchWarning, stacklevel=2) + break + + not_first_iteration = i > 0 + if (phi_a1 > phi0 + c1 * alpha1 * derphi0) or \ + ((phi_a1 >= phi_a0) and not_first_iteration): + alpha_star, phi_star, derphi_star = \ + _zoom(alpha0, alpha1, phi_a0, + phi_a1, derphi_a0, phi, derphi, + phi0, derphi0, c1, c2, extra_condition) + break + + derphi_a1 = derphi(alpha1) + if (abs(derphi_a1) <= -c2*derphi0): + if extra_condition(alpha1, phi_a1): + alpha_star = alpha1 + phi_star = phi_a1 + derphi_star = derphi_a1 + break + + if (derphi_a1 >= 0): + alpha_star, phi_star, derphi_star = \ + _zoom(alpha1, alpha0, phi_a1, + phi_a0, derphi_a1, phi, derphi, + phi0, derphi0, c1, c2, extra_condition) + break + + alpha2 = 2 * alpha1 # increase by factor of two on each iteration + if amax is not None: + alpha2 = min(alpha2, amax) + alpha0 = alpha1 + alpha1 = alpha2 + phi_a0 = phi_a1 + phi_a1 = phi(alpha1) + derphi_a0 = derphi_a1 + + else: + # stopping test maxiter reached + alpha_star = alpha1 + phi_star = phi_a1 + derphi_star = None + warn('The line search algorithm did not converge', + LineSearchWarning, stacklevel=2) + + return alpha_star, phi_star, phi0, derphi_star + + +def _cubicmin(a, fa, fpa, b, fb, c, fc): + """ + Finds the minimizer for a cubic polynomial that goes through the + points (a,fa), (b,fb), and (c,fc) with derivative at a of fpa. + + If no minimizer can be found, return None. + + """ + # f(x) = A *(x-a)^3 + B*(x-a)^2 + C*(x-a) + D + + with np.errstate(divide='raise', over='raise', invalid='raise'): + try: + C = fpa + db = b - a + dc = c - a + denom = (db * dc) ** 2 * (db - dc) + d1 = np.empty((2, 2)) + d1[0, 0] = dc ** 2 + d1[0, 1] = -db ** 2 + d1[1, 0] = -dc ** 3 + d1[1, 1] = db ** 3 + [A, B] = np.dot(d1, np.asarray([fb - fa - C * db, + fc - fa - C * dc]).flatten()) + A /= denom + B /= denom + radical = B * B - 3 * A * C + xmin = a + (-B + np.sqrt(radical)) / (3 * A) + except ArithmeticError: + return None + if not np.isfinite(xmin): + return None + return xmin + + +def _quadmin(a, fa, fpa, b, fb): + """ + Finds the minimizer for a quadratic polynomial that goes through + the points (a,fa), (b,fb) with derivative at a of fpa. + + """ + # f(x) = B*(x-a)^2 + C*(x-a) + D + with np.errstate(divide='raise', over='raise', invalid='raise'): + try: + D = fa + C = fpa + db = b - a * 1.0 + B = (fb - D - C * db) / (db * db) + xmin = a - C / (2.0 * B) + except ArithmeticError: + return None + if not np.isfinite(xmin): + return None + return xmin + + +def _zoom(a_lo, a_hi, phi_lo, phi_hi, derphi_lo, + phi, derphi, phi0, derphi0, c1, c2, extra_condition): + """Zoom stage of approximate linesearch satisfying strong Wolfe conditions. + + Part of the optimization algorithm in `scalar_search_wolfe2`. + + Notes + ----- + Implements Algorithm 3.6 (zoom) in Wright and Nocedal, + 'Numerical Optimization', 1999, pp. 61. + + """ + + maxiter = 10 + i = 0 + delta1 = 0.2 # cubic interpolant check + delta2 = 0.1 # quadratic interpolant check + phi_rec = phi0 + a_rec = 0 + while True: + # interpolate to find a trial step length between a_lo and + # a_hi Need to choose interpolation here. Use cubic + # interpolation and then if the result is within delta * + # dalpha or outside of the interval bounded by a_lo or a_hi + # then use quadratic interpolation, if the result is still too + # close, then use bisection + + dalpha = a_hi - a_lo + if dalpha < 0: + a, b = a_hi, a_lo + else: + a, b = a_lo, a_hi + + # minimizer of cubic interpolant + # (uses phi_lo, derphi_lo, phi_hi, and the most recent value of phi) + # + # if the result is too close to the end points (or out of the + # interval), then use quadratic interpolation with phi_lo, + # derphi_lo and phi_hi if the result is still too close to the + # end points (or out of the interval) then use bisection + + if (i > 0): + cchk = delta1 * dalpha + a_j = _cubicmin(a_lo, phi_lo, derphi_lo, a_hi, phi_hi, + a_rec, phi_rec) + if (i == 0) or (a_j is None) or (a_j > b - cchk) or (a_j < a + cchk): + qchk = delta2 * dalpha + a_j = _quadmin(a_lo, phi_lo, derphi_lo, a_hi, phi_hi) + if (a_j is None) or (a_j > b-qchk) or (a_j < a+qchk): + a_j = a_lo + 0.5*dalpha + + # Check new value of a_j + + phi_aj = phi(a_j) + if (phi_aj > phi0 + c1*a_j*derphi0) or (phi_aj >= phi_lo): + phi_rec = phi_hi + a_rec = a_hi + a_hi = a_j + phi_hi = phi_aj + else: + derphi_aj = derphi(a_j) + if abs(derphi_aj) <= -c2*derphi0 and extra_condition(a_j, phi_aj): + a_star = a_j + val_star = phi_aj + valprime_star = derphi_aj + break + if derphi_aj*(a_hi - a_lo) >= 0: + phi_rec = phi_hi + a_rec = a_hi + a_hi = a_lo + phi_hi = phi_lo + else: + phi_rec = phi_lo + a_rec = a_lo + a_lo = a_j + phi_lo = phi_aj + derphi_lo = derphi_aj + i += 1 + if (i > maxiter): + # Failed to find a conforming step size + a_star = None + val_star = None + valprime_star = None + break + return a_star, val_star, valprime_star + + +#------------------------------------------------------------------------------ +# Armijo line and scalar searches +#------------------------------------------------------------------------------ + +def line_search_armijo(f, xk, pk, gfk, old_fval, args=(), c1=1e-4, alpha0=1): + """Minimize over alpha, the function ``f(xk+alpha pk)``. + + Parameters + ---------- + f : callable + Function to be minimized. + xk : array_like + Current point. + pk : array_like + Search direction. + gfk : array_like + Gradient of `f` at point `xk`. + old_fval : float + Value of `f` at point `xk`. + args : tuple, optional + Optional arguments. + c1 : float, optional + Value to control stopping criterion. + alpha0 : scalar, optional + Value of `alpha` at start of the optimization. + + Returns + ------- + alpha + f_count + f_val_at_alpha + + Notes + ----- + Uses the interpolation algorithm (Armijo backtracking) as suggested by + Wright and Nocedal in 'Numerical Optimization', 1999, pp. 56-57 + + """ + xk = np.atleast_1d(xk) + fc = [0] + + def phi(alpha1): + fc[0] += 1 + return f(xk + alpha1*pk, *args) + + if old_fval is None: + phi0 = phi(0.) + else: + phi0 = old_fval # compute f(xk) -- done in past loop + + derphi0 = np.dot(gfk, pk) + alpha, phi1 = scalar_search_armijo(phi, phi0, derphi0, c1=c1, + alpha0=alpha0) + return alpha, fc[0], phi1 + + +def line_search_BFGS(f, xk, pk, gfk, old_fval, args=(), c1=1e-4, alpha0=1): + """ + Compatibility wrapper for `line_search_armijo` + """ + r = line_search_armijo(f, xk, pk, gfk, old_fval, args=args, c1=c1, + alpha0=alpha0) + return r[0], r[1], 0, r[2] + + +def scalar_search_armijo(phi, phi0, derphi0, c1=1e-4, alpha0=1, amin=0): + """Minimize over alpha, the function ``phi(alpha)``. + + Uses the interpolation algorithm (Armijo backtracking) as suggested by + Wright and Nocedal in 'Numerical Optimization', 1999, pp. 56-57 + + alpha > 0 is assumed to be a descent direction. + + Returns + ------- + alpha + phi1 + + """ + phi_a0 = phi(alpha0) + if phi_a0 <= phi0 + c1*alpha0*derphi0: + return alpha0, phi_a0 + + # Otherwise, compute the minimizer of a quadratic interpolant: + + alpha1 = -(derphi0) * alpha0**2 / 2.0 / (phi_a0 - phi0 - derphi0 * alpha0) + phi_a1 = phi(alpha1) + + if (phi_a1 <= phi0 + c1*alpha1*derphi0): + return alpha1, phi_a1 + + # Otherwise, loop with cubic interpolation until we find an alpha which + # satisfies the first Wolfe condition (since we are backtracking, we will + # assume that the value of alpha is not too small and satisfies the second + # condition. + + while alpha1 > amin: # we are assuming alpha>0 is a descent direction + factor = alpha0**2 * alpha1**2 * (alpha1-alpha0) + a = alpha0**2 * (phi_a1 - phi0 - derphi0*alpha1) - \ + alpha1**2 * (phi_a0 - phi0 - derphi0*alpha0) + a = a / factor + b = -alpha0**3 * (phi_a1 - phi0 - derphi0*alpha1) + \ + alpha1**3 * (phi_a0 - phi0 - derphi0*alpha0) + b = b / factor + + alpha2 = (-b + np.sqrt(abs(b**2 - 3 * a * derphi0))) / (3.0*a) + phi_a2 = phi(alpha2) + + if (phi_a2 <= phi0 + c1*alpha2*derphi0): + return alpha2, phi_a2 + + if (alpha1 - alpha2) > alpha1 / 2.0 or (1 - alpha2/alpha1) < 0.96: + alpha2 = alpha1 / 2.0 + + alpha0 = alpha1 + alpha1 = alpha2 + phi_a0 = phi_a1 + phi_a1 = phi_a2 + + # Failed to find a suitable step length + return None, phi_a1 + + +#------------------------------------------------------------------------------ +# Non-monotone line search for DF-SANE +#------------------------------------------------------------------------------ + +def _nonmonotone_line_search_cruz(f, x_k, d, prev_fs, eta, + gamma=1e-4, tau_min=0.1, tau_max=0.5): + """ + Nonmonotone backtracking line search as described in [1]_ + + Parameters + ---------- + f : callable + Function returning a tuple ``(f, F)`` where ``f`` is the value + of a merit function and ``F`` the residual. + x_k : ndarray + Initial position. + d : ndarray + Search direction. + prev_fs : float + List of previous merit function values. Should have ``len(prev_fs) <= M`` + where ``M`` is the nonmonotonicity window parameter. + eta : float + Allowed merit function increase, see [1]_ + gamma, tau_min, tau_max : float, optional + Search parameters, see [1]_ + + Returns + ------- + alpha : float + Step length + xp : ndarray + Next position + fp : float + Merit function value at next position + Fp : ndarray + Residual at next position + + References + ---------- + [1] "Spectral residual method without gradient information for solving + large-scale nonlinear systems of equations." W. La Cruz, + J.M. Martinez, M. Raydan. Math. Comp. **75**, 1429 (2006). + + """ + f_k = prev_fs[-1] + f_bar = max(prev_fs) + + alpha_p = 1 + alpha_m = 1 + alpha = 1 + + while True: + xp = x_k + alpha_p * d + fp, Fp = f(xp) + + if fp <= f_bar + eta - gamma * alpha_p**2 * f_k: + alpha = alpha_p + break + + alpha_tp = alpha_p**2 * f_k / (fp + (2*alpha_p - 1)*f_k) + + xp = x_k - alpha_m * d + fp, Fp = f(xp) + + if fp <= f_bar + eta - gamma * alpha_m**2 * f_k: + alpha = -alpha_m + break + + alpha_tm = alpha_m**2 * f_k / (fp + (2*alpha_m - 1)*f_k) + + alpha_p = np.clip(alpha_tp, tau_min * alpha_p, tau_max * alpha_p) + alpha_m = np.clip(alpha_tm, tau_min * alpha_m, tau_max * alpha_m) + + return alpha, xp, fp, Fp + + +def _nonmonotone_line_search_cheng(f, x_k, d, f_k, C, Q, eta, + gamma=1e-4, tau_min=0.1, tau_max=0.5, + nu=0.85): + """ + Nonmonotone line search from [1] + + Parameters + ---------- + f : callable + Function returning a tuple ``(f, F)`` where ``f`` is the value + of a merit function and ``F`` the residual. + x_k : ndarray + Initial position. + d : ndarray + Search direction. + f_k : float + Initial merit function value. + C, Q : float + Control parameters. On the first iteration, give values + Q=1.0, C=f_k + eta : float + Allowed merit function increase, see [1]_ + nu, gamma, tau_min, tau_max : float, optional + Search parameters, see [1]_ + + Returns + ------- + alpha : float + Step length + xp : ndarray + Next position + fp : float + Merit function value at next position + Fp : ndarray + Residual at next position + C : float + New value for the control parameter C + Q : float + New value for the control parameter Q + + References + ---------- + .. [1] W. Cheng & D.-H. Li, ''A derivative-free nonmonotone line + search and its application to the spectral residual + method'', IMA J. Numer. Anal. 29, 814 (2009). + + """ + alpha_p = 1 + alpha_m = 1 + alpha = 1 + + while True: + xp = x_k + alpha_p * d + fp, Fp = f(xp) + + if fp <= C + eta - gamma * alpha_p**2 * f_k: + alpha = alpha_p + break + + alpha_tp = alpha_p**2 * f_k / (fp + (2*alpha_p - 1)*f_k) + + xp = x_k - alpha_m * d + fp, Fp = f(xp) + + if fp <= C + eta - gamma * alpha_m**2 * f_k: + alpha = -alpha_m + break + + alpha_tm = alpha_m**2 * f_k / (fp + (2*alpha_m - 1)*f_k) + + alpha_p = np.clip(alpha_tp, tau_min * alpha_p, tau_max * alpha_p) + alpha_m = np.clip(alpha_tm, tau_min * alpha_m, tau_max * alpha_m) + + # Update C and Q + Q_next = nu * Q + 1 + C = (nu * Q * (C + eta) + fp) / Q_next + Q = Q_next + + return alpha, xp, fp, Fp, C, Q diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog.py new file mode 100644 index 0000000000000000000000000000000000000000..054ba471dcbd4622ab9c2fb9dda313bb124c0451 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog.py @@ -0,0 +1,733 @@ +""" +A top-level linear programming interface. + +.. versionadded:: 0.15.0 + +Functions +--------- +.. autosummary:: + :toctree: generated/ + + linprog + linprog_verbose_callback + linprog_terse_callback + +""" + +import numpy as np + +from ._optimize import OptimizeResult, OptimizeWarning +from warnings import warn +from ._linprog_highs import _linprog_highs +from ._linprog_ip import _linprog_ip +from ._linprog_simplex import _linprog_simplex +from ._linprog_rs import _linprog_rs +from ._linprog_doc import (_linprog_highs_doc, _linprog_ip_doc, # noqa: F401 + _linprog_rs_doc, _linprog_simplex_doc, + _linprog_highs_ipm_doc, _linprog_highs_ds_doc) +from ._linprog_util import ( + _parse_linprog, _presolve, _get_Abc, _LPProblem, _autoscale, + _postsolve, _check_result, _display_summary) +from copy import deepcopy + +__all__ = ['linprog', 'linprog_verbose_callback', 'linprog_terse_callback'] + +__docformat__ = "restructuredtext en" + +LINPROG_METHODS = [ + 'simplex', 'revised simplex', 'interior-point', 'highs', 'highs-ds', 'highs-ipm' +] + + +def linprog_verbose_callback(res): + """ + A sample callback function demonstrating the linprog callback interface. + This callback produces detailed output to sys.stdout before each iteration + and after the final iteration of the simplex algorithm. + + Parameters + ---------- + res : A `scipy.optimize.OptimizeResult` consisting of the following fields: + + x : 1-D array + The independent variable vector which optimizes the linear + programming problem. + fun : float + Value of the objective function. + success : bool + True if the algorithm succeeded in finding an optimal solution. + slack : 1-D array + The values of the slack variables. Each slack variable corresponds + to an inequality constraint. If the slack is zero, then the + corresponding constraint is active. + con : 1-D array + The (nominally zero) residuals of the equality constraints, that is, + ``b - A_eq @ x`` + phase : int + The phase of the optimization being executed. In phase 1 a basic + feasible solution is sought and the T has an additional row + representing an alternate objective function. + status : int + An integer representing the exit status of the optimization: + + ``0`` : Optimization terminated successfully + + ``1`` : Iteration limit reached + + ``2`` : Problem appears to be infeasible + + ``3`` : Problem appears to be unbounded + + ``4`` : Serious numerical difficulties encountered + + nit : int + The number of iterations performed. + message : str + A string descriptor of the exit status of the optimization. + """ + x = res['x'] + fun = res['fun'] + phase = res['phase'] + status = res['status'] + nit = res['nit'] + message = res['message'] + complete = res['complete'] + + saved_printoptions = np.get_printoptions() + np.set_printoptions(linewidth=500, + formatter={'float': lambda x: f"{x: 12.4f}"}) + if status: + print('--------- Simplex Early Exit -------\n') + print(f'The simplex method exited early with status {status:d}') + print(message) + elif complete: + print('--------- Simplex Complete --------\n') + print(f'Iterations required: {nit}') + else: + print(f'--------- Iteration {nit:d} ---------\n') + + if nit > 0: + if phase == 1: + print('Current Pseudo-Objective Value:') + else: + print('Current Objective Value:') + print('f = ', fun) + print() + print('Current Solution Vector:') + print('x = ', x) + print() + + np.set_printoptions(**saved_printoptions) + + +def linprog_terse_callback(res): + """ + A sample callback function demonstrating the linprog callback interface. + This callback produces brief output to sys.stdout before each iteration + and after the final iteration of the simplex algorithm. + + Parameters + ---------- + res : A `scipy.optimize.OptimizeResult` consisting of the following fields: + + x : 1-D array + The independent variable vector which optimizes the linear + programming problem. + fun : float + Value of the objective function. + success : bool + True if the algorithm succeeded in finding an optimal solution. + slack : 1-D array + The values of the slack variables. Each slack variable corresponds + to an inequality constraint. If the slack is zero, then the + corresponding constraint is active. + con : 1-D array + The (nominally zero) residuals of the equality constraints, that is, + ``b - A_eq @ x``. + phase : int + The phase of the optimization being executed. In phase 1 a basic + feasible solution is sought and the T has an additional row + representing an alternate objective function. + status : int + An integer representing the exit status of the optimization: + + ``0`` : Optimization terminated successfully + + ``1`` : Iteration limit reached + + ``2`` : Problem appears to be infeasible + + ``3`` : Problem appears to be unbounded + + ``4`` : Serious numerical difficulties encountered + + nit : int + The number of iterations performed. + message : str + A string descriptor of the exit status of the optimization. + """ + nit = res['nit'] + x = res['x'] + + if nit == 0: + print("Iter: X:") + print(f"{nit: <5d} ", end="") + print(x) + + +def linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, + bounds=(0, None), method='highs', callback=None, + options=None, x0=None, integrality=None): + r""" + Linear programming: minimize a linear objective function subject to linear + equality and inequality constraints. + + Linear programming solves problems of the following form: + + .. math:: + + \min_x \ & c^T x \\ + \mbox{such that} \ & A_{ub} x \leq b_{ub},\\ + & A_{eq} x = b_{eq},\\ + & l \leq x \leq u , + + where :math:`x` is a vector of decision variables; :math:`c`, + :math:`b_{ub}`, :math:`b_{eq}`, :math:`l`, and :math:`u` are vectors; and + :math:`A_{ub}` and :math:`A_{eq}` are matrices. + + Alternatively, that's: + + - minimize :: + + c @ x + + - such that :: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + Note that by default ``lb = 0`` and ``ub = None``. Other bounds can be + specified with ``bounds``. + + Parameters + ---------- + c : 1-D array + The coefficients of the linear objective function to be minimized. + A_ub : 2-D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1-D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2-D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1-D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : sequence, optional + A sequence of ``(min, max)`` pairs for each element in ``x``, defining + the minimum and maximum values of that decision variable. + If a single tuple ``(min, max)`` is provided, then ``min`` and ``max`` + will serve as bounds for all decision variables. + Use ``None`` to indicate that there is no bound. For instance, the + default bound ``(0, None)`` means that all decision variables are + non-negative, and the pair ``(None, None)`` means no bounds at all, + i.e. all variables are allowed to be any real. + method : str, optional + The algorithm used to solve the standard form problem. + The following are supported. + + - :ref:`'highs' ` (default) + - :ref:`'highs-ds' ` + - :ref:`'highs-ipm' ` + - :ref:`'interior-point' ` (legacy) + - :ref:`'revised simplex' ` (legacy) + - :ref:`'simplex' ` (legacy) + + The legacy methods are deprecated and will be removed in SciPy 1.11.0. + callback : callable, optional + If a callback function is provided, it will be called at least once per + iteration of the algorithm. The callback function must accept a single + `scipy.optimize.OptimizeResult` consisting of the following fields: + + x : 1-D array + The current solution vector. + fun : float + The current value of the objective function ``c @ x``. + success : bool + ``True`` when the algorithm has completed successfully. + slack : 1-D array + The (nominally positive) values of the slack, + ``b_ub - A_ub @ x``. + con : 1-D array + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + phase : int + The phase of the algorithm being executed. + status : int + An integer representing the status of the algorithm. + + ``0`` : Optimization proceeding nominally. + + ``1`` : Iteration limit reached. + + ``2`` : Problem appears to be infeasible. + + ``3`` : Problem appears to be unbounded. + + ``4`` : Numerical difficulties encountered. + + nit : int + The current iteration number. + message : str + A string descriptor of the algorithm status. + + Callback functions are not currently supported by the HiGHS methods. + + options : dict, optional + A dictionary of solver options. All methods accept the following + options: + + maxiter : int + Maximum number of iterations to perform. + Default: see method-specific documentation. + disp : bool + Set to ``True`` to print convergence messages. + Default: ``False``. + presolve : bool + Set to ``False`` to disable automatic presolve. + Default: ``True``. + + All methods except the HiGHS solvers also accept: + + tol : float + A tolerance which determines when a residual is "close enough" to + zero to be considered exactly zero. + autoscale : bool + Set to ``True`` to automatically perform equilibration. + Consider using this option if the numerical values in the + constraints are separated by several orders of magnitude. + Default: ``False``. + rr : bool + Set to ``False`` to disable automatic redundancy removal. + Default: ``True``. + rr_method : string + Method used to identify and remove redundant rows from the + equality constraint matrix after presolve. For problems with + dense input, the available methods for redundancy removal are: + + ``SVD``: + Repeatedly performs singular value decomposition on + the matrix, detecting redundant rows based on nonzeros + in the left singular vectors that correspond with + zero singular values. May be fast when the matrix is + nearly full rank. + ``pivot``: + Uses the algorithm presented in [5]_ to identify + redundant rows. + ``ID``: + Uses a randomized interpolative decomposition. + Identifies columns of the matrix transpose not used in + a full-rank interpolative decomposition of the matrix. + ``None``: + Uses ``svd`` if the matrix is nearly full rank, that is, + the difference between the matrix rank and the number + of rows is less than five. If not, uses ``pivot``. The + behavior of this default is subject to change without + prior notice. + + Default: None. + For problems with sparse input, this option is ignored, and the + pivot-based algorithm presented in [5]_ is used. + + For method-specific options, see + :func:`show_options('linprog') `. + + x0 : 1-D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + :ref:`'revised simplex' ` method, + and can only be used if `x0` represents a basic feasible solution. + + integrality : 1-D array or int, optional + Indicates the type of integrality constraint on each decision variable. + + ``0`` : Continuous variable; no integrality constraint. + + ``1`` : Integer variable; decision variable must be an integer + within `bounds`. + + ``2`` : Semi-continuous variable; decision variable must be within + `bounds` or take value ``0``. + + ``3`` : Semi-integer variable; decision variable must be an integer + within `bounds` or take value ``0``. + + By default, all variables are continuous. + + For mixed integrality constraints, supply an array of shape ``c.shape``. + To infer a constraint on each decision variable from shorter inputs, + the argument will be broadcast to ``c.shape`` using `numpy.broadcast_to`. + + This argument is currently used only by the + :ref:`'highs' ` method and is ignored otherwise. + + Returns + ------- + res : OptimizeResult + A :class:`scipy.optimize.OptimizeResult` consisting of the fields + below. Note that the return types of the fields may depend on whether + the optimization was successful, therefore it is recommended to check + `OptimizeResult.status` before relying on the other fields: + + x : 1-D array + The values of the decision variables that minimizes the + objective function while satisfying the constraints. + fun : float + The optimal value of the objective function ``c @ x``. + slack : 1-D array + The (nominally positive) values of the slack variables, + ``b_ub - A_ub @ x``. + con : 1-D array + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + success : bool + ``True`` when the algorithm succeeds in finding an optimal + solution. + status : int + An integer representing the exit status of the algorithm. + + ``0`` : Optimization terminated successfully. + + ``1`` : Iteration limit reached. + + ``2`` : Problem appears to be infeasible. + + ``3`` : Problem appears to be unbounded. + + ``4`` : Numerical difficulties encountered. + + nit : int + The total number of iterations performed in all phases. + message : str + A string descriptor of the exit status of the algorithm. + + See Also + -------- + show_options : Additional options accepted by the solvers. + + Notes + ----- + This section describes the available solvers that can be selected by the + 'method' parameter. + + :ref:`'highs-ds' `, and + :ref:`'highs-ipm' ` are interfaces to the + HiGHS simplex and interior-point method solvers [13]_, respectively. + :ref:`'highs' ` (default) chooses between + the two automatically. These are the fastest linear + programming solvers in SciPy, especially for large, sparse problems; + which of these two is faster is problem-dependent. + The other solvers are legacy methods and will be removed when `callback` is + supported by the HiGHS methods. + + Method :ref:`'highs-ds' `, is a wrapper of the C++ high + performance dual revised simplex implementation (HSOL) [13]_, [14]_. + Method :ref:`'highs-ipm' ` is a wrapper of a C++ + implementation of an **i**\ nterior-\ **p**\ oint **m**\ ethod [13]_; it + features a crossover routine, so it is as accurate as a simplex solver. + Method :ref:`'highs' ` chooses between the two + automatically. + For new code involving `linprog`, we recommend explicitly choosing one of + these three method values. + + .. versionadded:: 1.6.0 + + Method :ref:`'interior-point' ` + uses the primal-dual path following algorithm + as outlined in [4]_. This algorithm supports sparse constraint matrices and + is typically faster than the simplex methods, especially for large, sparse + problems. Note, however, that the solution returned may be slightly less + accurate than those of the simplex methods and will not, in general, + correspond with a vertex of the polytope defined by the constraints. + + .. versionadded:: 1.0.0 + + Method :ref:`'revised simplex' ` + uses the revised simplex method as described in + [9]_, except that a factorization [11]_ of the basis matrix, rather than + its inverse, is efficiently maintained and used to solve the linear systems + at each iteration of the algorithm. + + .. versionadded:: 1.3.0 + + Method :ref:`'simplex' ` uses a traditional, + full-tableau implementation of + Dantzig's simplex algorithm [1]_, [2]_ (*not* the + Nelder-Mead simplex). This algorithm is included for backwards + compatibility and educational purposes. + + .. versionadded:: 0.15.0 + + Before applying :ref:`'interior-point' `, + :ref:`'revised simplex' `, or + :ref:`'simplex' `, + a presolve procedure based on [8]_ attempts + to identify trivial infeasibilities, trivial unboundedness, and potential + problem simplifications. Specifically, it checks for: + + - rows of zeros in ``A_eq`` or ``A_ub``, representing trivial constraints; + - columns of zeros in ``A_eq`` `and` ``A_ub``, representing unconstrained + variables; + - column singletons in ``A_eq``, representing fixed variables; and + - column singletons in ``A_ub``, representing simple bounds. + + If presolve reveals that the problem is unbounded (e.g. an unconstrained + and unbounded variable has negative cost) or infeasible (e.g., a row of + zeros in ``A_eq`` corresponds with a nonzero in ``b_eq``), the solver + terminates with the appropriate status code. Note that presolve terminates + as soon as any sign of unboundedness is detected; consequently, a problem + may be reported as unbounded when in reality the problem is infeasible + (but infeasibility has not been detected yet). Therefore, if it is + important to know whether the problem is actually infeasible, solve the + problem again with option ``presolve=False``. + + If neither infeasibility nor unboundedness are detected in a single pass + of the presolve, bounds are tightened where possible and fixed + variables are removed from the problem. Then, linearly dependent rows + of the ``A_eq`` matrix are removed, (unless they represent an + infeasibility) to avoid numerical difficulties in the primary solve + routine. Note that rows that are nearly linearly dependent (within a + prescribed tolerance) may also be removed, which can change the optimal + solution in rare cases. If this is a concern, eliminate redundancy from + your problem formulation and run with option ``rr=False`` or + ``presolve=False``. + + Several potential improvements can be made here: additional presolve + checks outlined in [8]_ should be implemented, the presolve routine should + be run multiple times (until no further simplifications can be made), and + more of the efficiency improvements from [5]_ should be implemented in the + redundancy removal routines. + + After presolve, the problem is transformed to standard form by converting + the (tightened) simple bounds to upper bound constraints, introducing + non-negative slack variables for inequality constraints, and expressing + unbounded variables as the difference between two non-negative variables. + Optionally, the problem is automatically scaled via equilibration [12]_. + The selected algorithm solves the standard form problem, and a + postprocessing routine converts the result to a solution to the original + problem. + + References + ---------- + .. [1] Dantzig, George B., Linear programming and extensions. Rand + Corporation Research Study Princeton Univ. Press, Princeton, NJ, + 1963 + .. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to + Mathematical Programming", McGraw-Hill, Chapter 4. + .. [3] Bland, Robert G. New finite pivoting rules for the simplex method. + Mathematics of Operations Research (2), 1977: pp. 103-107. + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + .. [5] Andersen, Erling D. "Finding all linearly dependent rows in + large-scale linear programming." Optimization Methods and Software + 6.3 (1995): 219-227. + .. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear + Programming based on Newton's Method." Unpublished Course Notes, + March 2004. Available 2/25/2017 at + https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf + .. [7] Fourer, Robert. "Solving Linear Programs by Interior-Point Methods." + Unpublished Course Notes, August 26, 2005. Available 2/25/2017 at + http://www.4er.org/CourseNotes/Book%20B/B-III.pdf + .. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear + programming." Mathematical Programming 71.2 (1995): 221-245. + .. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear + programming." Athena Scientific 1 (1997): 997. + .. [10] Andersen, Erling D., et al. Implementation of interior point + methods for large scale linear programming. HEC/Universite de + Geneve, 1996. + .. [11] Bartels, Richard H. "A stabilization of the simplex method." + Journal in Numerische Mathematik 16.5 (1971): 414-434. + .. [12] Tomlin, J. A. "On scaling linear programming problems." + Mathematical Programming Study 4 (1975): 146-166. + .. [13] Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J. + "HiGHS - high performance software for linear optimization." + https://highs.dev/ + .. [14] Huangfu, Q. and Hall, J. A. J. "Parallelizing the dual revised + simplex method." Mathematical Programming Computation, 10 (1), + 119-142, 2018. DOI: 10.1007/s12532-017-0130-5 + + Examples + -------- + Consider the following problem: + + .. math:: + + \min_{x_0, x_1} \ -x_0 + 4x_1 & \\ + \mbox{such that} \ -3x_0 + x_1 & \leq 6,\\ + -x_0 - 2x_1 & \geq -4,\\ + x_1 & \geq -3. + + The problem is not presented in the form accepted by `linprog`. This is + easily remedied by converting the "greater than" inequality + constraint to a "less than" inequality constraint by + multiplying both sides by a factor of :math:`-1`. Note also that the last + constraint is really the simple bound :math:`-3 \leq x_1 \leq \infty`. + Finally, since there are no bounds on :math:`x_0`, we must explicitly + specify the bounds :math:`-\infty \leq x_0 \leq \infty`, as the + default is for variables to be non-negative. After collecting coeffecients + into arrays and tuples, the input for this problem is: + + >>> from scipy.optimize import linprog + >>> c = [-1, 4] + >>> A = [[-3, 1], [1, 2]] + >>> b = [6, 4] + >>> x0_bounds = (None, None) + >>> x1_bounds = (-3, None) + >>> res = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds]) + >>> res.fun + -22.0 + >>> res.x + array([10., -3.]) + >>> res.message + 'Optimization terminated successfully. (HiGHS Status 7: Optimal)' + + The marginals (AKA dual values / shadow prices / Lagrange multipliers) + and residuals (slacks) are also available. + + >>> res.ineqlin + residual: [ 3.900e+01 0.000e+00] + marginals: [-0.000e+00 -1.000e+00] + + For example, because the marginal associated with the second inequality + constraint is -1, we expect the optimal value of the objective function + to decrease by ``eps`` if we add a small amount ``eps`` to the right hand + side of the second inequality constraint: + + >>> eps = 0.05 + >>> b[1] += eps + >>> linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds]).fun + -22.05 + + Also, because the residual on the first inequality constraint is 39, we + can decrease the right hand side of the first constraint by 39 without + affecting the optimal solution. + + >>> b = [6, 4] # reset to original values + >>> b[0] -= 39 + >>> linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds]).fun + -22.0 + + """ + + meth = method.lower() + methods = {"highs", "highs-ds", "highs-ipm", + "simplex", "revised simplex", "interior-point"} + + if meth not in methods: + raise ValueError(f"Unknown solver '{method}'") + + if x0 is not None and meth != "revised simplex": + warning_message = "x0 is used only when method is 'revised simplex'. " + warn(warning_message, OptimizeWarning, stacklevel=2) + + if np.any(integrality) and not meth == "highs": + integrality = None + warning_message = ("Only `method='highs'` supports integer " + "constraints. Ignoring `integrality`.") + warn(warning_message, OptimizeWarning, stacklevel=2) + elif np.any(integrality): + integrality = np.broadcast_to(integrality, np.shape(c)) + else: + integrality = None + + lp = _LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0, integrality) + lp, solver_options = _parse_linprog(lp, options, meth) + tol = solver_options.get('tol', 1e-9) + + # Give unmodified problem to HiGHS + if meth.startswith('highs'): + if callback is not None: + raise NotImplementedError("HiGHS solvers do not support the " + "callback interface.") + highs_solvers = {'highs-ipm': 'ipm', 'highs-ds': 'simplex', + 'highs': None} + + sol = _linprog_highs(lp, solver=highs_solvers[meth], + **solver_options) + sol['status'], sol['message'] = ( + _check_result(sol['x'], sol['fun'], sol['status'], sol['slack'], + sol['con'], lp.bounds, tol, sol['message'], + integrality)) + sol['success'] = sol['status'] == 0 + return OptimizeResult(sol) + + warn(f"`method='{meth}'` is deprecated and will be removed in SciPy " + "1.11.0. Please use one of the HiGHS solvers (e.g. " + "`method='highs'`) in new code.", DeprecationWarning, stacklevel=2) + + iteration = 0 + complete = False # will become True if solved in presolve + undo = [] + + # Keep the original arrays to calculate slack/residuals for original + # problem. + lp_o = deepcopy(lp) + + # Solve trivial problem, eliminate variables, tighten bounds, etc. + rr_method = solver_options.pop('rr_method', None) # need to pop these; + rr = solver_options.pop('rr', True) # they're not passed to methods + c0 = 0 # we might get a constant term in the objective + if solver_options.pop('presolve', True): + (lp, c0, x, undo, complete, status, message) = _presolve(lp, rr, + rr_method, + tol) + + C, b_scale = 1, 1 # for trivial unscaling if autoscale is not used + postsolve_args = (lp_o._replace(bounds=lp.bounds), undo, C, b_scale) + + if not complete: + A, b, c, c0, x0 = _get_Abc(lp, c0) + if solver_options.pop('autoscale', False): + A, b, c, x0, C, b_scale = _autoscale(A, b, c, x0) + postsolve_args = postsolve_args[:-2] + (C, b_scale) + + if meth == 'simplex': + x, status, message, iteration = _linprog_simplex( + c, c0=c0, A=A, b=b, callback=callback, + postsolve_args=postsolve_args, **solver_options) + elif meth == 'interior-point': + x, status, message, iteration = _linprog_ip( + c, c0=c0, A=A, b=b, callback=callback, + postsolve_args=postsolve_args, **solver_options) + elif meth == 'revised simplex': + x, status, message, iteration = _linprog_rs( + c, c0=c0, A=A, b=b, x0=x0, callback=callback, + postsolve_args=postsolve_args, **solver_options) + + # Eliminate artificial variables, re-introduce presolved variables, etc. + disp = solver_options.get('disp', False) + + x, fun, slack, con = _postsolve(x, postsolve_args, complete) + + status, message = _check_result(x, fun, status, slack, con, lp_o.bounds, + tol, message, integrality) + + if disp: + _display_summary(message, status, fun, iteration) + + sol = { + 'x': x, + 'fun': fun, + 'slack': slack, + 'con': con, + 'status': status, + 'message': message, + 'nit': iteration, + 'success': status == 0} + + return OptimizeResult(sol) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_doc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_doc.py new file mode 100644 index 0000000000000000000000000000000000000000..ba016aec6dafe74e48076875202704a3b85b822a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_doc.py @@ -0,0 +1,1434 @@ +""" +Created on Sat Aug 22 19:49:17 2020 + +@author: matth +""" + + +def _linprog_highs_doc(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, + bounds=None, method='highs', callback=None, + maxiter=None, disp=False, presolve=True, + time_limit=None, + dual_feasibility_tolerance=None, + primal_feasibility_tolerance=None, + ipm_optimality_tolerance=None, + simplex_dual_edge_weight_strategy=None, + mip_rel_gap=None, + **unknown_options): + r""" + Linear programming: minimize a linear objective function subject to linear + equality and inequality constraints using one of the HiGHS solvers. + + Linear programming solves problems of the following form: + + .. math:: + + \min_x \ & c^T x \\ + \mbox{such that} \ & A_{ub} x \leq b_{ub},\\ + & A_{eq} x = b_{eq},\\ + & l \leq x \leq u , + + where :math:`x` is a vector of decision variables; :math:`c`, + :math:`b_{ub}`, :math:`b_{eq}`, :math:`l`, and :math:`u` are vectors; and + :math:`A_{ub}` and :math:`A_{eq}` are matrices. + + Alternatively, that's: + + minimize:: + + c @ x + + such that:: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + Note that by default ``lb = 0`` and ``ub = None`` unless specified with + ``bounds``. + + Parameters + ---------- + c : 1-D array + The coefficients of the linear objective function to be minimized. + A_ub : 2-D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1-D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2-D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1-D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : sequence, optional + A sequence of ``(min, max)`` pairs for each element in ``x``, defining + the minimum and maximum values of that decision variable. Use ``None`` + to indicate that there is no bound. By default, bounds are + ``(0, None)`` (all decision variables are non-negative). + If a single tuple ``(min, max)`` is provided, then ``min`` and + ``max`` will serve as bounds for all decision variables. + method : str + + This is the method-specific documentation for 'highs', which chooses + automatically between + :ref:`'highs-ds' ` and + :ref:`'highs-ipm' `. + :ref:`'interior-point' ` (default), + :ref:`'revised simplex' `, and + :ref:`'simplex' ` (legacy) + are also available. + integrality : 1-D array or int, optional + Indicates the type of integrality constraint on each decision variable. + + ``0`` : Continuous variable; no integrality constraint. + + ``1`` : Integer variable; decision variable must be an integer + within `bounds`. + + ``2`` : Semi-continuous variable; decision variable must be within + `bounds` or take value ``0``. + + ``3`` : Semi-integer variable; decision variable must be an integer + within `bounds` or take value ``0``. + + By default, all variables are continuous. + + For mixed integrality constraints, supply an array of shape `c.shape`. + To infer a constraint on each decision variable from shorter inputs, + the argument will be broadcast to `c.shape` using `np.broadcast_to`. + + This argument is currently used only by the ``'highs'`` method and + ignored otherwise. + + Options + ------- + maxiter : int + The maximum number of iterations to perform in either phase. + For :ref:`'highs-ipm' `, this does not + include the number of crossover iterations. Default is the largest + possible value for an ``int`` on the platform. + disp : bool (default: ``False``) + Set to ``True`` if indicators of optimization status are to be + printed to the console during optimization. + presolve : bool (default: ``True``) + Presolve attempts to identify trivial infeasibilities, + identify trivial unboundedness, and simplify the problem before + sending it to the main solver. It is generally recommended + to keep the default setting ``True``; set to ``False`` if + presolve is to be disabled. + time_limit : float + The maximum time in seconds allotted to solve the problem; + default is the largest possible value for a ``double`` on the + platform. + dual_feasibility_tolerance : double (default: 1e-07) + Dual feasibility tolerance for + :ref:`'highs-ds' `. + The minimum of this and ``primal_feasibility_tolerance`` + is used for the feasibility tolerance of + :ref:`'highs-ipm' `. + primal_feasibility_tolerance : double (default: 1e-07) + Primal feasibility tolerance for + :ref:`'highs-ds' `. + The minimum of this and ``dual_feasibility_tolerance`` + is used for the feasibility tolerance of + :ref:`'highs-ipm' `. + ipm_optimality_tolerance : double (default: ``1e-08``) + Optimality tolerance for + :ref:`'highs-ipm' `. + Minimum allowable value is 1e-12. + simplex_dual_edge_weight_strategy : str (default: None) + Strategy for simplex dual edge weights. The default, ``None``, + automatically selects one of the following. + + ``'dantzig'`` uses Dantzig's original strategy of choosing the most + negative reduced cost. + + ``'devex'`` uses the strategy described in [15]_. + + ``steepest`` uses the exact steepest edge strategy as described in + [16]_. + + ``'steepest-devex'`` begins with the exact steepest edge strategy + until the computation is too costly or inexact and then switches to + the devex method. + + Currently, ``None`` always selects ``'steepest-devex'``, but this + may change as new options become available. + mip_rel_gap : double (default: None) + Termination criterion for MIP solver: solver will terminate when the + gap between the primal objective value and the dual objective bound, + scaled by the primal objective value, is <= mip_rel_gap. + unknown_options : dict + Optional arguments not used by this particular solver. If + ``unknown_options`` is non-empty, a warning is issued listing + all unused options. + + Returns + ------- + res : OptimizeResult + A :class:`scipy.optimize.OptimizeResult` consisting of the fields: + + x : 1D array + The values of the decision variables that minimizes the + objective function while satisfying the constraints. + fun : float + The optimal value of the objective function ``c @ x``. + slack : 1D array + The (nominally positive) values of the slack, + ``b_ub - A_ub @ x``. + con : 1D array + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + success : bool + ``True`` when the algorithm succeeds in finding an optimal + solution. + status : int + An integer representing the exit status of the algorithm. + + ``0`` : Optimization terminated successfully. + + ``1`` : Iteration or time limit reached. + + ``2`` : Problem appears to be infeasible. + + ``3`` : Problem appears to be unbounded. + + ``4`` : The HiGHS solver ran into a problem. + + message : str + A string descriptor of the exit status of the algorithm. + nit : int + The total number of iterations performed. + For the HiGHS simplex method, this includes iterations in all + phases. For the HiGHS interior-point method, this does not include + crossover iterations. + crossover_nit : int + The number of primal/dual pushes performed during the + crossover routine for the HiGHS interior-point method. + This is ``0`` for the HiGHS simplex method. + ineqlin : OptimizeResult + Solution and sensitivity information corresponding to the + inequality constraints, `b_ub`. A dictionary consisting of the + fields: + + residual : np.ndnarray + The (nominally positive) values of the slack variables, + ``b_ub - A_ub @ x``. This quantity is also commonly + referred to as "slack". + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the right-hand side of the + inequality constraints, `b_ub`. + + eqlin : OptimizeResult + Solution and sensitivity information corresponding to the + equality constraints, `b_eq`. A dictionary consisting of the + fields: + + residual : np.ndarray + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the right-hand side of the + equality constraints, `b_eq`. + + lower, upper : OptimizeResult + Solution and sensitivity information corresponding to the + lower and upper bounds on decision variables, `bounds`. + + residual : np.ndarray + The (nominally positive) values of the quantity + ``x - lb`` (lower) or ``ub - x`` (upper). + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the lower and upper + `bounds`. + + Notes + ----- + + Method :ref:`'highs-ds' ` is a wrapper + of the C++ high performance dual revised simplex implementation (HSOL) + [13]_, [14]_. Method :ref:`'highs-ipm' ` + is a wrapper of a C++ implementation of an **i**\ nterior-\ **p**\ oint + **m**\ ethod [13]_; it features a crossover routine, so it is as accurate + as a simplex solver. Method :ref:`'highs' ` chooses + between the two automatically. For new code involving `linprog`, we + recommend explicitly choosing one of these three method values instead of + :ref:`'interior-point' ` (default), + :ref:`'revised simplex' `, and + :ref:`'simplex' ` (legacy). + + The result fields `ineqlin`, `eqlin`, `lower`, and `upper` all contain + `marginals`, or partial derivatives of the objective function with respect + to the right-hand side of each constraint. These partial derivatives are + also referred to as "Lagrange multipliers", "dual values", and + "shadow prices". The sign convention of `marginals` is opposite that + of Lagrange multipliers produced by many nonlinear solvers. + + References + ---------- + .. [13] Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J. + "HiGHS - high performance software for linear optimization." + https://highs.dev/ + .. [14] Huangfu, Q. and Hall, J. A. J. "Parallelizing the dual revised + simplex method." Mathematical Programming Computation, 10 (1), + 119-142, 2018. DOI: 10.1007/s12532-017-0130-5 + .. [15] Harris, Paula MJ. "Pivot selection methods of the Devex LP code." + Mathematical programming 5.1 (1973): 1-28. + .. [16] Goldfarb, Donald, and John Ker Reid. "A practicable steepest-edge + simplex algorithm." Mathematical Programming 12.1 (1977): 361-371. + """ + pass + + +def _linprog_highs_ds_doc(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, + bounds=None, method='highs-ds', callback=None, + maxiter=None, disp=False, presolve=True, + time_limit=None, + dual_feasibility_tolerance=None, + primal_feasibility_tolerance=None, + simplex_dual_edge_weight_strategy=None, + **unknown_options): + r""" + Linear programming: minimize a linear objective function subject to linear + equality and inequality constraints using the HiGHS dual simplex solver. + + Linear programming solves problems of the following form: + + .. math:: + + \min_x \ & c^T x \\ + \mbox{such that} \ & A_{ub} x \leq b_{ub},\\ + & A_{eq} x = b_{eq},\\ + & l \leq x \leq u , + + where :math:`x` is a vector of decision variables; :math:`c`, + :math:`b_{ub}`, :math:`b_{eq}`, :math:`l`, and :math:`u` are vectors; and + :math:`A_{ub}` and :math:`A_{eq}` are matrices. + + Alternatively, that's: + + minimize:: + + c @ x + + such that:: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + Note that by default ``lb = 0`` and ``ub = None`` unless specified with + ``bounds``. + + Parameters + ---------- + c : 1-D array + The coefficients of the linear objective function to be minimized. + A_ub : 2-D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1-D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2-D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1-D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : sequence, optional + A sequence of ``(min, max)`` pairs for each element in ``x``, defining + the minimum and maximum values of that decision variable. Use ``None`` + to indicate that there is no bound. By default, bounds are + ``(0, None)`` (all decision variables are non-negative). + If a single tuple ``(min, max)`` is provided, then ``min`` and + ``max`` will serve as bounds for all decision variables. + method : str + + This is the method-specific documentation for 'highs-ds'. + :ref:`'highs' `, + :ref:`'highs-ipm' `, + :ref:`'interior-point' ` (default), + :ref:`'revised simplex' `, and + :ref:`'simplex' ` (legacy) + are also available. + + Options + ------- + maxiter : int + The maximum number of iterations to perform in either phase. + Default is the largest possible value for an ``int`` on the platform. + disp : bool (default: ``False``) + Set to ``True`` if indicators of optimization status are to be + printed to the console during optimization. + presolve : bool (default: ``True``) + Presolve attempts to identify trivial infeasibilities, + identify trivial unboundedness, and simplify the problem before + sending it to the main solver. It is generally recommended + to keep the default setting ``True``; set to ``False`` if + presolve is to be disabled. + time_limit : float + The maximum time in seconds allotted to solve the problem; + default is the largest possible value for a ``double`` on the + platform. + dual_feasibility_tolerance : double (default: 1e-07) + Dual feasibility tolerance for + :ref:`'highs-ds' `. + primal_feasibility_tolerance : double (default: 1e-07) + Primal feasibility tolerance for + :ref:`'highs-ds' `. + simplex_dual_edge_weight_strategy : str (default: None) + Strategy for simplex dual edge weights. The default, ``None``, + automatically selects one of the following. + + ``'dantzig'`` uses Dantzig's original strategy of choosing the most + negative reduced cost. + + ``'devex'`` uses the strategy described in [15]_. + + ``steepest`` uses the exact steepest edge strategy as described in + [16]_. + + ``'steepest-devex'`` begins with the exact steepest edge strategy + until the computation is too costly or inexact and then switches to + the devex method. + + Currently, ``None`` always selects ``'steepest-devex'``, but this + may change as new options become available. + unknown_options : dict + Optional arguments not used by this particular solver. If + ``unknown_options`` is non-empty, a warning is issued listing + all unused options. + + Returns + ------- + res : OptimizeResult + A :class:`scipy.optimize.OptimizeResult` consisting of the fields: + + x : 1D array + The values of the decision variables that minimizes the + objective function while satisfying the constraints. + fun : float + The optimal value of the objective function ``c @ x``. + slack : 1D array + The (nominally positive) values of the slack, + ``b_ub - A_ub @ x``. + con : 1D array + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + success : bool + ``True`` when the algorithm succeeds in finding an optimal + solution. + status : int + An integer representing the exit status of the algorithm. + + ``0`` : Optimization terminated successfully. + + ``1`` : Iteration or time limit reached. + + ``2`` : Problem appears to be infeasible. + + ``3`` : Problem appears to be unbounded. + + ``4`` : The HiGHS solver ran into a problem. + + message : str + A string descriptor of the exit status of the algorithm. + nit : int + The total number of iterations performed. This includes iterations + in all phases. + crossover_nit : int + This is always ``0`` for the HiGHS simplex method. + For the HiGHS interior-point method, this is the number of + primal/dual pushes performed during the crossover routine. + ineqlin : OptimizeResult + Solution and sensitivity information corresponding to the + inequality constraints, `b_ub`. A dictionary consisting of the + fields: + + residual : np.ndnarray + The (nominally positive) values of the slack variables, + ``b_ub - A_ub @ x``. This quantity is also commonly + referred to as "slack". + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the right-hand side of the + inequality constraints, `b_ub`. + + eqlin : OptimizeResult + Solution and sensitivity information corresponding to the + equality constraints, `b_eq`. A dictionary consisting of the + fields: + + residual : np.ndarray + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the right-hand side of the + equality constraints, `b_eq`. + + lower, upper : OptimizeResult + Solution and sensitivity information corresponding to the + lower and upper bounds on decision variables, `bounds`. + + residual : np.ndarray + The (nominally positive) values of the quantity + ``x - lb`` (lower) or ``ub - x`` (upper). + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the lower and upper + `bounds`. + + Notes + ----- + + Method :ref:`'highs-ds' ` is a wrapper + of the C++ high performance dual revised simplex implementation (HSOL) + [13]_, [14]_. Method :ref:`'highs-ipm' ` + is a wrapper of a C++ implementation of an **i**\ nterior-\ **p**\ oint + **m**\ ethod [13]_; it features a crossover routine, so it is as accurate + as a simplex solver. Method :ref:`'highs' ` chooses + between the two automatically. For new code involving `linprog`, we + recommend explicitly choosing one of these three method values instead of + :ref:`'interior-point' ` (default), + :ref:`'revised simplex' `, and + :ref:`'simplex' ` (legacy). + + The result fields `ineqlin`, `eqlin`, `lower`, and `upper` all contain + `marginals`, or partial derivatives of the objective function with respect + to the right-hand side of each constraint. These partial derivatives are + also referred to as "Lagrange multipliers", "dual values", and + "shadow prices". The sign convention of `marginals` is opposite that + of Lagrange multipliers produced by many nonlinear solvers. + + References + ---------- + .. [13] Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J. + "HiGHS - high performance software for linear optimization." + https://highs.dev/ + .. [14] Huangfu, Q. and Hall, J. A. J. "Parallelizing the dual revised + simplex method." Mathematical Programming Computation, 10 (1), + 119-142, 2018. DOI: 10.1007/s12532-017-0130-5 + .. [15] Harris, Paula MJ. "Pivot selection methods of the Devex LP code." + Mathematical programming 5.1 (1973): 1-28. + .. [16] Goldfarb, Donald, and John Ker Reid. "A practicable steepest-edge + simplex algorithm." Mathematical Programming 12.1 (1977): 361-371. + """ + pass + + +def _linprog_highs_ipm_doc(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, + bounds=None, method='highs-ipm', callback=None, + maxiter=None, disp=False, presolve=True, + time_limit=None, + dual_feasibility_tolerance=None, + primal_feasibility_tolerance=None, + ipm_optimality_tolerance=None, + **unknown_options): + r""" + Linear programming: minimize a linear objective function subject to linear + equality and inequality constraints using the HiGHS interior point solver. + + Linear programming solves problems of the following form: + + .. math:: + + \min_x \ & c^T x \\ + \mbox{such that} \ & A_{ub} x \leq b_{ub},\\ + & A_{eq} x = b_{eq},\\ + & l \leq x \leq u , + + where :math:`x` is a vector of decision variables; :math:`c`, + :math:`b_{ub}`, :math:`b_{eq}`, :math:`l`, and :math:`u` are vectors; and + :math:`A_{ub}` and :math:`A_{eq}` are matrices. + + Alternatively, that's: + + minimize:: + + c @ x + + such that:: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + Note that by default ``lb = 0`` and ``ub = None`` unless specified with + ``bounds``. + + Parameters + ---------- + c : 1-D array + The coefficients of the linear objective function to be minimized. + A_ub : 2-D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1-D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2-D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1-D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : sequence, optional + A sequence of ``(min, max)`` pairs for each element in ``x``, defining + the minimum and maximum values of that decision variable. Use ``None`` + to indicate that there is no bound. By default, bounds are + ``(0, None)`` (all decision variables are non-negative). + If a single tuple ``(min, max)`` is provided, then ``min`` and + ``max`` will serve as bounds for all decision variables. + method : str + + This is the method-specific documentation for 'highs-ipm'. + :ref:`'highs-ipm' `, + :ref:`'highs-ds' `, + :ref:`'interior-point' ` (default), + :ref:`'revised simplex' `, and + :ref:`'simplex' ` (legacy) + are also available. + + Options + ------- + maxiter : int + The maximum number of iterations to perform in either phase. + For :ref:`'highs-ipm' `, this does not + include the number of crossover iterations. Default is the largest + possible value for an ``int`` on the platform. + disp : bool (default: ``False``) + Set to ``True`` if indicators of optimization status are to be + printed to the console during optimization. + presolve : bool (default: ``True``) + Presolve attempts to identify trivial infeasibilities, + identify trivial unboundedness, and simplify the problem before + sending it to the main solver. It is generally recommended + to keep the default setting ``True``; set to ``False`` if + presolve is to be disabled. + time_limit : float + The maximum time in seconds allotted to solve the problem; + default is the largest possible value for a ``double`` on the + platform. + dual_feasibility_tolerance : double (default: 1e-07) + The minimum of this and ``primal_feasibility_tolerance`` + is used for the feasibility tolerance of + :ref:`'highs-ipm' `. + primal_feasibility_tolerance : double (default: 1e-07) + The minimum of this and ``dual_feasibility_tolerance`` + is used for the feasibility tolerance of + :ref:`'highs-ipm' `. + ipm_optimality_tolerance : double (default: ``1e-08``) + Optimality tolerance for + :ref:`'highs-ipm' `. + Minimum allowable value is 1e-12. + unknown_options : dict + Optional arguments not used by this particular solver. If + ``unknown_options`` is non-empty, a warning is issued listing + all unused options. + + Returns + ------- + res : OptimizeResult + A :class:`scipy.optimize.OptimizeResult` consisting of the fields: + + x : 1D array + The values of the decision variables that minimizes the + objective function while satisfying the constraints. + fun : float + The optimal value of the objective function ``c @ x``. + slack : 1D array + The (nominally positive) values of the slack, + ``b_ub - A_ub @ x``. + con : 1D array + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + success : bool + ``True`` when the algorithm succeeds in finding an optimal + solution. + status : int + An integer representing the exit status of the algorithm. + + ``0`` : Optimization terminated successfully. + + ``1`` : Iteration or time limit reached. + + ``2`` : Problem appears to be infeasible. + + ``3`` : Problem appears to be unbounded. + + ``4`` : The HiGHS solver ran into a problem. + + message : str + A string descriptor of the exit status of the algorithm. + nit : int + The total number of iterations performed. + For the HiGHS interior-point method, this does not include + crossover iterations. + crossover_nit : int + The number of primal/dual pushes performed during the + crossover routine for the HiGHS interior-point method. + ineqlin : OptimizeResult + Solution and sensitivity information corresponding to the + inequality constraints, `b_ub`. A dictionary consisting of the + fields: + + residual : np.ndnarray + The (nominally positive) values of the slack variables, + ``b_ub - A_ub @ x``. This quantity is also commonly + referred to as "slack". + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the right-hand side of the + inequality constraints, `b_ub`. + + eqlin : OptimizeResult + Solution and sensitivity information corresponding to the + equality constraints, `b_eq`. A dictionary consisting of the + fields: + + residual : np.ndarray + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the right-hand side of the + equality constraints, `b_eq`. + + lower, upper : OptimizeResult + Solution and sensitivity information corresponding to the + lower and upper bounds on decision variables, `bounds`. + + residual : np.ndarray + The (nominally positive) values of the quantity + ``x - lb`` (lower) or ``ub - x`` (upper). + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the lower and upper + `bounds`. + + Notes + ----- + + Method :ref:`'highs-ipm' ` + is a wrapper of a C++ implementation of an **i**\ nterior-\ **p**\ oint + **m**\ ethod [13]_; it features a crossover routine, so it is as accurate + as a simplex solver. + Method :ref:`'highs-ds' ` is a wrapper + of the C++ high performance dual revised simplex implementation (HSOL) + [13]_, [14]_. Method :ref:`'highs' ` chooses + between the two automatically. For new code involving `linprog`, we + recommend explicitly choosing one of these three method values instead of + :ref:`'interior-point' ` (default), + :ref:`'revised simplex' `, and + :ref:`'simplex' ` (legacy). + + The result fields `ineqlin`, `eqlin`, `lower`, and `upper` all contain + `marginals`, or partial derivatives of the objective function with respect + to the right-hand side of each constraint. These partial derivatives are + also referred to as "Lagrange multipliers", "dual values", and + "shadow prices". The sign convention of `marginals` is opposite that + of Lagrange multipliers produced by many nonlinear solvers. + + References + ---------- + .. [13] Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J. + "HiGHS - high performance software for linear optimization." + https://highs.dev/ + .. [14] Huangfu, Q. and Hall, J. A. J. "Parallelizing the dual revised + simplex method." Mathematical Programming Computation, 10 (1), + 119-142, 2018. DOI: 10.1007/s12532-017-0130-5 + """ + pass + + +def _linprog_ip_doc(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, + bounds=None, method='interior-point', callback=None, + maxiter=1000, disp=False, presolve=True, + tol=1e-8, autoscale=False, rr=True, + alpha0=.99995, beta=0.1, sparse=False, + lstsq=False, sym_pos=True, cholesky=True, pc=True, + ip=False, permc_spec='MMD_AT_PLUS_A', **unknown_options): + r""" + Linear programming: minimize a linear objective function subject to linear + equality and inequality constraints using the interior-point method of + [4]_. + + .. deprecated:: 1.9.0 + `method='interior-point'` will be removed in SciPy 1.11.0. + It is replaced by `method='highs'` because the latter is + faster and more robust. + + Linear programming solves problems of the following form: + + .. math:: + + \min_x \ & c^T x \\ + \mbox{such that} \ & A_{ub} x \leq b_{ub},\\ + & A_{eq} x = b_{eq},\\ + & l \leq x \leq u , + + where :math:`x` is a vector of decision variables; :math:`c`, + :math:`b_{ub}`, :math:`b_{eq}`, :math:`l`, and :math:`u` are vectors; and + :math:`A_{ub}` and :math:`A_{eq}` are matrices. + + Alternatively, that's: + + minimize:: + + c @ x + + such that:: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + Note that by default ``lb = 0`` and ``ub = None`` unless specified with + ``bounds``. + + Parameters + ---------- + c : 1-D array + The coefficients of the linear objective function to be minimized. + A_ub : 2-D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1-D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2-D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1-D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : sequence, optional + A sequence of ``(min, max)`` pairs for each element in ``x``, defining + the minimum and maximum values of that decision variable. Use ``None`` + to indicate that there is no bound. By default, bounds are + ``(0, None)`` (all decision variables are non-negative). + If a single tuple ``(min, max)`` is provided, then ``min`` and + ``max`` will serve as bounds for all decision variables. + method : str + This is the method-specific documentation for 'interior-point'. + :ref:`'highs' `, + :ref:`'highs-ds' `, + :ref:`'highs-ipm' `, + :ref:`'revised simplex' `, and + :ref:`'simplex' ` (legacy) + are also available. + callback : callable, optional + Callback function to be executed once per iteration. + + Options + ------- + maxiter : int (default: 1000) + The maximum number of iterations of the algorithm. + disp : bool (default: False) + Set to ``True`` if indicators of optimization status are to be printed + to the console each iteration. + presolve : bool (default: True) + Presolve attempts to identify trivial infeasibilities, + identify trivial unboundedness, and simplify the problem before + sending it to the main solver. It is generally recommended + to keep the default setting ``True``; set to ``False`` if + presolve is to be disabled. + tol : float (default: 1e-8) + Termination tolerance to be used for all termination criteria; + see [4]_ Section 4.5. + autoscale : bool (default: False) + Set to ``True`` to automatically perform equilibration. + Consider using this option if the numerical values in the + constraints are separated by several orders of magnitude. + rr : bool (default: True) + Set to ``False`` to disable automatic redundancy removal. + alpha0 : float (default: 0.99995) + The maximal step size for Mehrota's predictor-corrector search + direction; see :math:`\beta_{3}` of [4]_ Table 8.1. + beta : float (default: 0.1) + The desired reduction of the path parameter :math:`\mu` (see [6]_) + when Mehrota's predictor-corrector is not in use (uncommon). + sparse : bool (default: False) + Set to ``True`` if the problem is to be treated as sparse after + presolve. If either ``A_eq`` or ``A_ub`` is a sparse matrix, + this option will automatically be set ``True``, and the problem + will be treated as sparse even during presolve. If your constraint + matrices contain mostly zeros and the problem is not very small (less + than about 100 constraints or variables), consider setting ``True`` + or providing ``A_eq`` and ``A_ub`` as sparse matrices. + lstsq : bool (default: ``False``) + Set to ``True`` if the problem is expected to be very poorly + conditioned. This should always be left ``False`` unless severe + numerical difficulties are encountered. Leave this at the default + unless you receive a warning message suggesting otherwise. + sym_pos : bool (default: True) + Leave ``True`` if the problem is expected to yield a well conditioned + symmetric positive definite normal equation matrix + (almost always). Leave this at the default unless you receive + a warning message suggesting otherwise. + cholesky : bool (default: True) + Set to ``True`` if the normal equations are to be solved by explicit + Cholesky decomposition followed by explicit forward/backward + substitution. This is typically faster for problems + that are numerically well-behaved. + pc : bool (default: True) + Leave ``True`` if the predictor-corrector method of Mehrota is to be + used. This is almost always (if not always) beneficial. + ip : bool (default: False) + Set to ``True`` if the improved initial point suggestion due to [4]_ + Section 4.3 is desired. Whether this is beneficial or not + depends on the problem. + permc_spec : str (default: 'MMD_AT_PLUS_A') + (Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = + True``, and no SuiteSparse.) + A matrix is factorized in each iteration of the algorithm. + This option specifies how to permute the columns of the matrix for + sparsity preservation. Acceptable values are: + + - ``NATURAL``: natural ordering. + - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. + - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. + - ``COLAMD``: approximate minimum degree column ordering. + + This option can impact the convergence of the + interior point algorithm; test different values to determine which + performs best for your problem. For more information, refer to + ``scipy.sparse.linalg.splu``. + unknown_options : dict + Optional arguments not used by this particular solver. If + `unknown_options` is non-empty a warning is issued listing all + unused options. + + Returns + ------- + res : OptimizeResult + A :class:`scipy.optimize.OptimizeResult` consisting of the fields: + + x : 1-D array + The values of the decision variables that minimizes the + objective function while satisfying the constraints. + fun : float + The optimal value of the objective function ``c @ x``. + slack : 1-D array + The (nominally positive) values of the slack variables, + ``b_ub - A_ub @ x``. + con : 1-D array + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + success : bool + ``True`` when the algorithm succeeds in finding an optimal + solution. + status : int + An integer representing the exit status of the algorithm. + + ``0`` : Optimization terminated successfully. + + ``1`` : Iteration limit reached. + + ``2`` : Problem appears to be infeasible. + + ``3`` : Problem appears to be unbounded. + + ``4`` : Numerical difficulties encountered. + + message : str + A string descriptor of the exit status of the algorithm. + nit : int + The total number of iterations performed in all phases. + + + Notes + ----- + This method implements the algorithm outlined in [4]_ with ideas from [8]_ + and a structure inspired by the simpler methods of [6]_. + + The primal-dual path following method begins with initial 'guesses' of + the primal and dual variables of the standard form problem and iteratively + attempts to solve the (nonlinear) Karush-Kuhn-Tucker conditions for the + problem with a gradually reduced logarithmic barrier term added to the + objective. This particular implementation uses a homogeneous self-dual + formulation, which provides certificates of infeasibility or unboundedness + where applicable. + + The default initial point for the primal and dual variables is that + defined in [4]_ Section 4.4 Equation 8.22. Optionally (by setting initial + point option ``ip=True``), an alternate (potentially improved) starting + point can be calculated according to the additional recommendations of + [4]_ Section 4.4. + + A search direction is calculated using the predictor-corrector method + (single correction) proposed by Mehrota and detailed in [4]_ Section 4.1. + (A potential improvement would be to implement the method of multiple + corrections described in [4]_ Section 4.2.) In practice, this is + accomplished by solving the normal equations, [4]_ Section 5.1 Equations + 8.31 and 8.32, derived from the Newton equations [4]_ Section 5 Equations + 8.25 (compare to [4]_ Section 4 Equations 8.6-8.8). The advantage of + solving the normal equations rather than 8.25 directly is that the + matrices involved are symmetric positive definite, so Cholesky + decomposition can be used rather than the more expensive LU factorization. + + With default options, the solver used to perform the factorization depends + on third-party software availability and the conditioning of the problem. + + For dense problems, solvers are tried in the following order: + + 1. ``scipy.linalg.cho_factor`` + + 2. ``scipy.linalg.solve`` with option ``sym_pos=True`` + + 3. ``scipy.linalg.solve`` with option ``sym_pos=False`` + + 4. ``scipy.linalg.lstsq`` + + For sparse problems: + + 1. ``sksparse.cholmod.cholesky`` (if scikit-sparse and SuiteSparse are + installed) + + 2. ``scipy.sparse.linalg.factorized`` (if scikit-umfpack and SuiteSparse + are installed) + + 3. ``scipy.sparse.linalg.splu`` (which uses SuperLU distributed with SciPy) + + 4. ``scipy.sparse.linalg.lsqr`` + + If the solver fails for any reason, successively more robust (but slower) + solvers are attempted in the order indicated. Attempting, failing, and + re-starting factorization can be time consuming, so if the problem is + numerically challenging, options can be set to bypass solvers that are + failing. Setting ``cholesky=False`` skips to solver 2, + ``sym_pos=False`` skips to solver 3, and ``lstsq=True`` skips + to solver 4 for both sparse and dense problems. + + Potential improvements for combating issues associated with dense + columns in otherwise sparse problems are outlined in [4]_ Section 5.3 and + [10]_ Section 4.1-4.2; the latter also discusses the alleviation of + accuracy issues associated with the substitution approach to free + variables. + + After calculating the search direction, the maximum possible step size + that does not activate the non-negativity constraints is calculated, and + the smaller of this step size and unity is applied (as in [4]_ Section + 4.1.) [4]_ Section 4.3 suggests improvements for choosing the step size. + + The new point is tested according to the termination conditions of [4]_ + Section 4.5. The same tolerance, which can be set using the ``tol`` option, + is used for all checks. (A potential improvement would be to expose + the different tolerances to be set independently.) If optimality, + unboundedness, or infeasibility is detected, the solve procedure + terminates; otherwise it repeats. + + Whereas the top level ``linprog`` module expects a problem of form: + + Minimize:: + + c @ x + + Subject to:: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + where ``lb = 0`` and ``ub = None`` unless set in ``bounds``. The problem + is automatically converted to the form: + + Minimize:: + + c @ x + + Subject to:: + + A @ x == b + x >= 0 + + for solution. That is, the original problem contains equality, upper-bound + and variable constraints whereas the method specific solver requires + equality constraints and variable non-negativity. ``linprog`` converts the + original problem to standard form by converting the simple bounds to upper + bound constraints, introducing non-negative slack variables for inequality + constraints, and expressing unbounded variables as the difference between + two non-negative variables. The problem is converted back to the original + form before results are reported. + + References + ---------- + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + .. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear + Programming based on Newton's Method." Unpublished Course Notes, + March 2004. Available 2/25/2017 at + https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf + .. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear + programming." Mathematical Programming 71.2 (1995): 221-245. + .. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear + programming." Athena Scientific 1 (1997): 997. + .. [10] Andersen, Erling D., et al. Implementation of interior point + methods for large scale linear programming. HEC/Universite de + Geneve, 1996. + """ + pass + + +def _linprog_rs_doc(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, + bounds=None, method='interior-point', callback=None, + x0=None, maxiter=5000, disp=False, presolve=True, + tol=1e-12, autoscale=False, rr=True, maxupdate=10, + mast=False, pivot="mrc", **unknown_options): + r""" + Linear programming: minimize a linear objective function subject to linear + equality and inequality constraints using the revised simplex method. + + .. deprecated:: 1.9.0 + `method='revised simplex'` will be removed in SciPy 1.11.0. + It is replaced by `method='highs'` because the latter is + faster and more robust. + + Linear programming solves problems of the following form: + + .. math:: + + \min_x \ & c^T x \\ + \mbox{such that} \ & A_{ub} x \leq b_{ub},\\ + & A_{eq} x = b_{eq},\\ + & l \leq x \leq u , + + where :math:`x` is a vector of decision variables; :math:`c`, + :math:`b_{ub}`, :math:`b_{eq}`, :math:`l`, and :math:`u` are vectors; and + :math:`A_{ub}` and :math:`A_{eq}` are matrices. + + Alternatively, that's: + + minimize:: + + c @ x + + such that:: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + Note that by default ``lb = 0`` and ``ub = None`` unless specified with + ``bounds``. + + Parameters + ---------- + c : 1-D array + The coefficients of the linear objective function to be minimized. + A_ub : 2-D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1-D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2-D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1-D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : sequence, optional + A sequence of ``(min, max)`` pairs for each element in ``x``, defining + the minimum and maximum values of that decision variable. Use ``None`` + to indicate that there is no bound. By default, bounds are + ``(0, None)`` (all decision variables are non-negative). + If a single tuple ``(min, max)`` is provided, then ``min`` and + ``max`` will serve as bounds for all decision variables. + method : str + This is the method-specific documentation for 'revised simplex'. + :ref:`'highs' `, + :ref:`'highs-ds' `, + :ref:`'highs-ipm' `, + :ref:`'interior-point' ` (default), + and :ref:`'simplex' ` (legacy) + are also available. + callback : callable, optional + Callback function to be executed once per iteration. + x0 : 1-D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + 'revised simplex' method, and can only be used if `x0` represents a + basic feasible solution. + + Options + ------- + maxiter : int (default: 5000) + The maximum number of iterations to perform in either phase. + disp : bool (default: False) + Set to ``True`` if indicators of optimization status are to be printed + to the console each iteration. + presolve : bool (default: True) + Presolve attempts to identify trivial infeasibilities, + identify trivial unboundedness, and simplify the problem before + sending it to the main solver. It is generally recommended + to keep the default setting ``True``; set to ``False`` if + presolve is to be disabled. + tol : float (default: 1e-12) + The tolerance which determines when a solution is "close enough" to + zero in Phase 1 to be considered a basic feasible solution or close + enough to positive to serve as an optimal solution. + autoscale : bool (default: False) + Set to ``True`` to automatically perform equilibration. + Consider using this option if the numerical values in the + constraints are separated by several orders of magnitude. + rr : bool (default: True) + Set to ``False`` to disable automatic redundancy removal. + maxupdate : int (default: 10) + The maximum number of updates performed on the LU factorization. + After this many updates is reached, the basis matrix is factorized + from scratch. + mast : bool (default: False) + Minimize Amortized Solve Time. If enabled, the average time to solve + a linear system using the basis factorization is measured. Typically, + the average solve time will decrease with each successive solve after + initial factorization, as factorization takes much more time than the + solve operation (and updates). Eventually, however, the updated + factorization becomes sufficiently complex that the average solve time + begins to increase. When this is detected, the basis is refactorized + from scratch. Enable this option to maximize speed at the risk of + nondeterministic behavior. Ignored if ``maxupdate`` is 0. + pivot : "mrc" or "bland" (default: "mrc") + Pivot rule: Minimum Reduced Cost ("mrc") or Bland's rule ("bland"). + Choose Bland's rule if iteration limit is reached and cycling is + suspected. + unknown_options : dict + Optional arguments not used by this particular solver. If + `unknown_options` is non-empty a warning is issued listing all + unused options. + + Returns + ------- + res : OptimizeResult + A :class:`scipy.optimize.OptimizeResult` consisting of the fields: + + x : 1-D array + The values of the decision variables that minimizes the + objective function while satisfying the constraints. + fun : float + The optimal value of the objective function ``c @ x``. + slack : 1-D array + The (nominally positive) values of the slack variables, + ``b_ub - A_ub @ x``. + con : 1-D array + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + success : bool + ``True`` when the algorithm succeeds in finding an optimal + solution. + status : int + An integer representing the exit status of the algorithm. + + ``0`` : Optimization terminated successfully. + + ``1`` : Iteration limit reached. + + ``2`` : Problem appears to be infeasible. + + ``3`` : Problem appears to be unbounded. + + ``4`` : Numerical difficulties encountered. + + ``5`` : Problem has no constraints; turn presolve on. + + ``6`` : Invalid guess provided. + + message : str + A string descriptor of the exit status of the algorithm. + nit : int + The total number of iterations performed in all phases. + + + Notes + ----- + Method *revised simplex* uses the revised simplex method as described in + [9]_, except that a factorization [11]_ of the basis matrix, rather than + its inverse, is efficiently maintained and used to solve the linear systems + at each iteration of the algorithm. + + References + ---------- + .. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear + programming." Athena Scientific 1 (1997): 997. + .. [11] Bartels, Richard H. "A stabilization of the simplex method." + Journal in Numerische Mathematik 16.5 (1971): 414-434. + """ + pass + + +def _linprog_simplex_doc(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, + bounds=None, method='interior-point', callback=None, + maxiter=5000, disp=False, presolve=True, + tol=1e-12, autoscale=False, rr=True, bland=False, + **unknown_options): + r""" + Linear programming: minimize a linear objective function subject to linear + equality and inequality constraints using the tableau-based simplex method. + + .. deprecated:: 1.9.0 + `method='simplex'` will be removed in SciPy 1.11.0. + It is replaced by `method='highs'` because the latter is + faster and more robust. + + Linear programming solves problems of the following form: + + .. math:: + + \min_x \ & c^T x \\ + \mbox{such that} \ & A_{ub} x \leq b_{ub},\\ + & A_{eq} x = b_{eq},\\ + & l \leq x \leq u , + + where :math:`x` is a vector of decision variables; :math:`c`, + :math:`b_{ub}`, :math:`b_{eq}`, :math:`l`, and :math:`u` are vectors; and + :math:`A_{ub}` and :math:`A_{eq}` are matrices. + + Alternatively, that's: + + minimize:: + + c @ x + + such that:: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + Note that by default ``lb = 0`` and ``ub = None`` unless specified with + ``bounds``. + + Parameters + ---------- + c : 1-D array + The coefficients of the linear objective function to be minimized. + A_ub : 2-D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1-D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2-D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1-D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : sequence, optional + A sequence of ``(min, max)`` pairs for each element in ``x``, defining + the minimum and maximum values of that decision variable. Use ``None`` + to indicate that there is no bound. By default, bounds are + ``(0, None)`` (all decision variables are non-negative). + If a single tuple ``(min, max)`` is provided, then ``min`` and + ``max`` will serve as bounds for all decision variables. + method : str + This is the method-specific documentation for 'simplex'. + :ref:`'highs' `, + :ref:`'highs-ds' `, + :ref:`'highs-ipm' `, + :ref:`'interior-point' ` (default), + and :ref:`'revised simplex' ` + are also available. + callback : callable, optional + Callback function to be executed once per iteration. + + Options + ------- + maxiter : int (default: 5000) + The maximum number of iterations to perform in either phase. + disp : bool (default: False) + Set to ``True`` if indicators of optimization status are to be printed + to the console each iteration. + presolve : bool (default: True) + Presolve attempts to identify trivial infeasibilities, + identify trivial unboundedness, and simplify the problem before + sending it to the main solver. It is generally recommended + to keep the default setting ``True``; set to ``False`` if + presolve is to be disabled. + tol : float (default: 1e-12) + The tolerance which determines when a solution is "close enough" to + zero in Phase 1 to be considered a basic feasible solution or close + enough to positive to serve as an optimal solution. + autoscale : bool (default: False) + Set to ``True`` to automatically perform equilibration. + Consider using this option if the numerical values in the + constraints are separated by several orders of magnitude. + rr : bool (default: True) + Set to ``False`` to disable automatic redundancy removal. + bland : bool + If True, use Bland's anti-cycling rule [3]_ to choose pivots to + prevent cycling. If False, choose pivots which should lead to a + converged solution more quickly. The latter method is subject to + cycling (non-convergence) in rare instances. + unknown_options : dict + Optional arguments not used by this particular solver. If + `unknown_options` is non-empty a warning is issued listing all + unused options. + + Returns + ------- + res : OptimizeResult + A :class:`scipy.optimize.OptimizeResult` consisting of the fields: + + x : 1-D array + The values of the decision variables that minimizes the + objective function while satisfying the constraints. + fun : float + The optimal value of the objective function ``c @ x``. + slack : 1-D array + The (nominally positive) values of the slack variables, + ``b_ub - A_ub @ x``. + con : 1-D array + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + success : bool + ``True`` when the algorithm succeeds in finding an optimal + solution. + status : int + An integer representing the exit status of the algorithm. + + ``0`` : Optimization terminated successfully. + + ``1`` : Iteration limit reached. + + ``2`` : Problem appears to be infeasible. + + ``3`` : Problem appears to be unbounded. + + ``4`` : Numerical difficulties encountered. + + message : str + A string descriptor of the exit status of the algorithm. + nit : int + The total number of iterations performed in all phases. + + References + ---------- + .. [1] Dantzig, George B., Linear programming and extensions. Rand + Corporation Research Study Princeton Univ. Press, Princeton, NJ, + 1963 + .. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to + Mathematical Programming", McGraw-Hill, Chapter 4. + .. [3] Bland, Robert G. New finite pivoting rules for the simplex method. + Mathematics of Operations Research (2), 1977: pp. 103-107. + """ + pass diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_highs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_highs.py new file mode 100644 index 0000000000000000000000000000000000000000..9455cf460f96b7768c28acfbba76ad9ad08eed3f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_highs.py @@ -0,0 +1,422 @@ +"""HiGHS Linear Optimization Methods + +Interface to HiGHS linear optimization software. +https://highs.dev/ + +.. versionadded:: 1.5.0 + +References +---------- +.. [1] Q. Huangfu and J.A.J. Hall. "Parallelizing the dual revised simplex + method." Mathematical Programming Computation, 10 (1), 119-142, + 2018. DOI: 10.1007/s12532-017-0130-5 + +""" + +import inspect +import numpy as np +from ._optimize import OptimizeWarning, OptimizeResult +from warnings import warn +from ._highspy._highs_wrapper import _highs_wrapper +from ._highspy._core import( + kHighsInf, + HighsDebugLevel, + ObjSense, + HighsModelStatus, + simplex_constants as s_c, # [1] +) +from scipy.sparse import csc_matrix, vstack, issparse + +# [1]: Directly importing from "._highspy._core.simplex_constants" +# causes problems when reloading. +# See https://github.com/scipy/scipy/pull/22869 for details. + +def _highs_to_scipy_status_message(highs_status, highs_message): + """Converts HiGHS status number/message to SciPy status number/message""" + + scipy_statuses_messages = { + None: (4, "HiGHS did not provide a status code. "), + HighsModelStatus.kNotset: (4, ""), + HighsModelStatus.kLoadError: (4, ""), + HighsModelStatus.kModelError: (2, ""), + HighsModelStatus.kPresolveError: (4, ""), + HighsModelStatus.kSolveError: (4, ""), + HighsModelStatus.kPostsolveError: (4, ""), + HighsModelStatus.kModelEmpty: (4, ""), + HighsModelStatus.kObjectiveBound: (4, ""), + HighsModelStatus.kObjectiveTarget: (4, ""), + HighsModelStatus.kOptimal: (0, "Optimization terminated successfully. "), + HighsModelStatus.kTimeLimit: (1, "Time limit reached. "), + HighsModelStatus.kIterationLimit: (1, "Iteration limit reached. "), + HighsModelStatus.kInfeasible: (2, "The problem is infeasible. "), + HighsModelStatus.kUnbounded: (3, "The problem is unbounded. "), + HighsModelStatus.kUnboundedOrInfeasible: (4, "The problem is unbounded " + "or infeasible. ")} + unrecognized = (4, "The HiGHS status code was not recognized. ") + scipy_status, scipy_message = ( + scipy_statuses_messages.get(highs_status, unrecognized)) + hstat = int(highs_status) if highs_status is not None else None + scipy_message = (f"{scipy_message}" + f"(HiGHS Status {hstat}: {highs_message})") + return scipy_status, scipy_message + + +def _replace_inf(x): + # Replace `np.inf` with kHighsInf + infs = np.isinf(x) + with np.errstate(invalid="ignore"): + x[infs] = np.sign(x[infs])*kHighsInf + return x + + +def _convert_to_highs_enum(option, option_str, choices): + # If option is in the choices we can look it up, if not use + # the default value taken from function signature and warn: + try: + return choices[option.lower()] + except AttributeError: + return choices[option] + except KeyError: + sig = inspect.signature(_linprog_highs) + default_str = sig.parameters[option_str].default + warn(f"Option {option_str} is {option}, but only values in " + f"{set(choices.keys())} are allowed. Using default: " + f"{default_str}.", + OptimizeWarning, stacklevel=3) + return choices[default_str] + + +def _linprog_highs(lp, solver, time_limit=None, presolve=True, + disp=False, maxiter=None, + dual_feasibility_tolerance=None, + primal_feasibility_tolerance=None, + ipm_optimality_tolerance=None, + simplex_dual_edge_weight_strategy=None, + mip_rel_gap=None, + mip_max_nodes=None, + **unknown_options): + r""" + Solve the following linear programming problem using one of the HiGHS + solvers: + + User-facing documentation is in _linprog_doc.py. + + Parameters + ---------- + lp : _LPProblem + A ``scipy.optimize._linprog_util._LPProblem`` ``namedtuple``. + solver : "ipm" or "simplex" or None + Which HiGHS solver to use. If ``None``, "simplex" will be used. + + Options + ------- + maxiter : int + The maximum number of iterations to perform in either phase. For + ``solver='ipm'``, this does not include the number of crossover + iterations. Default is the largest possible value for an ``int`` + on the platform. + disp : bool + Set to ``True`` if indicators of optimization status are to be printed + to the console each iteration; default ``False``. + time_limit : float + The maximum time in seconds allotted to solve the problem; default is + the largest possible value for a ``double`` on the platform. + presolve : bool + Presolve attempts to identify trivial infeasibilities, + identify trivial unboundedness, and simplify the problem before + sending it to the main solver. It is generally recommended + to keep the default setting ``True``; set to ``False`` if presolve is + to be disabled. + dual_feasibility_tolerance : double + Dual feasibility tolerance. Default is 1e-07. + The minimum of this and ``primal_feasibility_tolerance`` + is used for the feasibility tolerance when ``solver='ipm'``. + primal_feasibility_tolerance : double + Primal feasibility tolerance. Default is 1e-07. + The minimum of this and ``dual_feasibility_tolerance`` + is used for the feasibility tolerance when ``solver='ipm'``. + ipm_optimality_tolerance : double + Optimality tolerance for ``solver='ipm'``. Default is 1e-08. + Minimum possible value is 1e-12 and must be smaller than the largest + possible value for a ``double`` on the platform. + simplex_dual_edge_weight_strategy : str (default: None) + Strategy for simplex dual edge weights. The default, ``None``, + automatically selects one of the following. + + ``'dantzig'`` uses Dantzig's original strategy of choosing the most + negative reduced cost. + + ``'devex'`` uses the strategy described in [15]_. + + ``steepest`` uses the exact steepest edge strategy as described in + [16]_. + + ``'steepest-devex'`` begins with the exact steepest edge strategy + until the computation is too costly or inexact and then switches to + the devex method. + + Currently, using ``None`` always selects ``'steepest-devex'``, but this + may change as new options become available. + + mip_max_nodes : int + The maximum number of nodes allotted to solve the problem; default is + the largest possible value for a ``HighsInt`` on the platform. + Ignored if not using the MIP solver. + unknown_options : dict + Optional arguments not used by this particular solver. If + ``unknown_options`` is non-empty, a warning is issued listing all + unused options. + + Returns + ------- + sol : dict + A dictionary consisting of the fields: + + x : 1D array + The values of the decision variables that minimizes the + objective function while satisfying the constraints. + fun : float + The optimal value of the objective function ``c @ x``. + slack : 1D array + The (nominally positive) values of the slack, + ``b_ub - A_ub @ x``. + con : 1D array + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + success : bool + ``True`` when the algorithm succeeds in finding an optimal + solution. + status : int + An integer representing the exit status of the algorithm. + + ``0`` : Optimization terminated successfully. + + ``1`` : Iteration or time limit reached. + + ``2`` : Problem appears to be infeasible. + + ``3`` : Problem appears to be unbounded. + + ``4`` : The HiGHS solver ran into a problem. + + message : str + A string descriptor of the exit status of the algorithm. + nit : int + The total number of iterations performed. + For ``solver='simplex'``, this includes iterations in all + phases. For ``solver='ipm'``, this does not include + crossover iterations. + crossover_nit : int + The number of primal/dual pushes performed during the + crossover routine for ``solver='ipm'``. This is ``0`` + for ``solver='simplex'``. + ineqlin : OptimizeResult + Solution and sensitivity information corresponding to the + inequality constraints, `b_ub`. A dictionary consisting of the + fields: + + residual : np.ndnarray + The (nominally positive) values of the slack variables, + ``b_ub - A_ub @ x``. This quantity is also commonly + referred to as "slack". + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the right-hand side of the + inequality constraints, `b_ub`. + + eqlin : OptimizeResult + Solution and sensitivity information corresponding to the + equality constraints, `b_eq`. A dictionary consisting of the + fields: + + residual : np.ndarray + The (nominally zero) residuals of the equality constraints, + ``b_eq - A_eq @ x``. + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the right-hand side of the + equality constraints, `b_eq`. + + lower, upper : OptimizeResult + Solution and sensitivity information corresponding to the + lower and upper bounds on decision variables, `bounds`. + + residual : np.ndarray + The (nominally positive) values of the quantity + ``x - lb`` (lower) or ``ub - x`` (upper). + + marginals : np.ndarray + The sensitivity (partial derivative) of the objective + function with respect to the lower and upper + `bounds`. + + mip_node_count : int + The number of subproblems or "nodes" solved by the MILP + solver. Only present when `integrality` is not `None`. + + mip_dual_bound : float + The MILP solver's final estimate of the lower bound on the + optimal solution. Only present when `integrality` is not + `None`. + + mip_gap : float + The difference between the final objective function value + and the final dual bound, scaled by the final objective + function value. Only present when `integrality` is not + `None`. + + Notes + ----- + The result fields `ineqlin`, `eqlin`, `lower`, and `upper` all contain + `marginals`, or partial derivatives of the objective function with respect + to the right-hand side of each constraint. These partial derivatives are + also referred to as "Lagrange multipliers", "dual values", and + "shadow prices". The sign convention of `marginals` is opposite that + of Lagrange multipliers produced by many nonlinear solvers. + + References + ---------- + .. [15] Harris, Paula MJ. "Pivot selection methods of the Devex LP code." + Mathematical programming 5.1 (1973): 1-28. + .. [16] Goldfarb, Donald, and John Ker Reid. "A practicable steepest-edge + simplex algorithm." Mathematical Programming 12.1 (1977): 361-371. + """ + if unknown_options: + message = (f"Unrecognized options detected: {unknown_options}. " + "These will be passed to HiGHS verbatim.") + warn(message, OptimizeWarning, stacklevel=3) + + # Map options to HiGHS enum values + simplex_dual_edge_weight_strategy_enum = _convert_to_highs_enum( + simplex_dual_edge_weight_strategy, + 'simplex_dual_edge_weight_strategy', + choices={'dantzig': \ + s_c.SimplexEdgeWeightStrategy.kSimplexEdgeWeightStrategyDantzig, + 'devex': \ + s_c.SimplexEdgeWeightStrategy.kSimplexEdgeWeightStrategyDevex, + 'steepest-devex': \ + s_c.SimplexEdgeWeightStrategy.kSimplexEdgeWeightStrategyChoose, + 'steepest': \ + s_c.SimplexEdgeWeightStrategy.kSimplexEdgeWeightStrategySteepestEdge, + None: None}) + + c, A_ub, b_ub, A_eq, b_eq, bounds, x0, integrality = lp + + lb, ub = bounds.T.copy() # separate bounds, copy->C-cntgs + # highs_wrapper solves LHS <= A*x <= RHS, not equality constraints + with np.errstate(invalid="ignore"): + lhs_ub = -np.ones_like(b_ub)*np.inf # LHS of UB constraints is -inf + rhs_ub = b_ub # RHS of UB constraints is b_ub + lhs_eq = b_eq # Equality constraint is inequality + rhs_eq = b_eq # constraint with LHS=RHS + lhs = np.concatenate((lhs_ub, lhs_eq)) + rhs = np.concatenate((rhs_ub, rhs_eq)) + + if issparse(A_ub) or issparse(A_eq): + A = vstack((A_ub, A_eq)) + else: + A = np.vstack((A_ub, A_eq)) + A = csc_matrix(A) + + options = { + 'presolve': presolve, + 'sense': ObjSense.kMinimize, + 'solver': solver, + 'time_limit': time_limit, + 'highs_debug_level': HighsDebugLevel.kHighsDebugLevelNone, + 'dual_feasibility_tolerance': dual_feasibility_tolerance, + 'ipm_optimality_tolerance': ipm_optimality_tolerance, + 'log_to_console': disp, + 'mip_max_nodes': mip_max_nodes, + 'output_flag': disp, + 'primal_feasibility_tolerance': primal_feasibility_tolerance, + 'simplex_dual_edge_weight_strategy': + simplex_dual_edge_weight_strategy_enum, + 'simplex_strategy': s_c.SimplexStrategy.kSimplexStrategyDual, + 'ipm_iteration_limit': maxiter, + 'simplex_iteration_limit': maxiter, + 'mip_rel_gap': mip_rel_gap, + } + options.update(unknown_options) + + # np.inf doesn't work; use very large constant + rhs = _replace_inf(rhs) + lhs = _replace_inf(lhs) + lb = _replace_inf(lb) + ub = _replace_inf(ub) + + if integrality is None or np.sum(integrality) == 0: + integrality = np.empty(0) + else: + integrality = np.array(integrality) + + res = _highs_wrapper(c, A.indptr, A.indices, A.data, lhs, rhs, + lb, ub, integrality.astype(np.uint8), options) + + # HiGHS represents constraints as lhs/rhs, so + # Ax + s = b => Ax = b - s + # and we need to split up s by A_ub and A_eq + if 'slack' in res: + slack = res['slack'] + con = np.array(slack[len(b_ub):]) + slack = np.array(slack[:len(b_ub)]) + else: + slack, con = None, None + + # lagrange multipliers for equalities/inequalities and upper/lower bounds + if 'lambda' in res: + lamda = res['lambda'] + marg_ineqlin = np.array(lamda[:len(b_ub)]) + marg_eqlin = np.array(lamda[len(b_ub):]) + marg_upper = np.array(res['marg_bnds'][1, :]) + marg_lower = np.array(res['marg_bnds'][0, :]) + else: + marg_ineqlin, marg_eqlin = None, None + marg_upper, marg_lower = None, None + + # this needs to be updated if we start choosing the solver intelligently + + # Convert to scipy-style status and message + highs_status = res.get('status', None) + highs_message = res.get('message', None) + status, message = _highs_to_scipy_status_message(highs_status, + highs_message) + + x = res['x'] # is None if not set + sol = {'x': x, + 'slack': slack, + 'con': con, + 'ineqlin': OptimizeResult({ + 'residual': slack, + 'marginals': marg_ineqlin, + }), + 'eqlin': OptimizeResult({ + 'residual': con, + 'marginals': marg_eqlin, + }), + 'lower': OptimizeResult({ + 'residual': None if x is None else x - lb, + 'marginals': marg_lower, + }), + 'upper': OptimizeResult({ + 'residual': None if x is None else ub - x, + 'marginals': marg_upper + }), + 'fun': res.get('fun'), + 'status': status, + 'success': res['status'] == HighsModelStatus.kOptimal, + 'message': message, + 'nit': res.get('simplex_nit', 0) or res.get('ipm_nit', 0), + 'crossover_nit': res.get('crossover_nit'), + } + + if np.any(x) and integrality is not None: + sol.update({ + 'mip_node_count': res.get('mip_node_count', 0), + 'mip_dual_bound': res.get('mip_dual_bound', 0.0), + 'mip_gap': res.get('mip_gap', 0.0), + }) + + return sol diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_ip.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_ip.py new file mode 100644 index 0000000000000000000000000000000000000000..4e6bf717b4d7becd46d0046cedf5f807004898e4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_ip.py @@ -0,0 +1,1126 @@ +"""Interior-point method for linear programming + +The *interior-point* method uses the primal-dual path following algorithm +outlined in [1]_. This algorithm supports sparse constraint matrices and +is typically faster than the simplex methods, especially for large, sparse +problems. Note, however, that the solution returned may be slightly less +accurate than those of the simplex methods and will not, in general, +correspond with a vertex of the polytope defined by the constraints. + + .. versionadded:: 1.0.0 + +References +---------- +.. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. +""" +# Author: Matt Haberland + +import numpy as np +import scipy as sp +import scipy.sparse as sps +from warnings import warn +from scipy.linalg import LinAlgError +from ._optimize import OptimizeWarning, OptimizeResult, _check_unknown_options +from ._linprog_util import _postsolve +has_umfpack = True +has_cholmod = True +try: + import sksparse # noqa: F401 + from sksparse.cholmod import cholesky as cholmod # noqa: F401 + from sksparse.cholmod import analyze as cholmod_analyze +except ImportError: + has_cholmod = False +try: + import scikits.umfpack # test whether to use factorized # noqa: F401 +except ImportError: + has_umfpack = False + + +def _get_solver(M, sparse=False, lstsq=False, sym_pos=True, + cholesky=True, permc_spec='MMD_AT_PLUS_A'): + """ + Given solver options, return a handle to the appropriate linear system + solver. + + Parameters + ---------- + M : 2-D array + As defined in [4] Equation 8.31 + sparse : bool (default = False) + True if the system to be solved is sparse. This is typically set + True when the original ``A_ub`` and ``A_eq`` arrays are sparse. + lstsq : bool (default = False) + True if the system is ill-conditioned and/or (nearly) singular and + thus a more robust least-squares solver is desired. This is sometimes + needed as the solution is approached. + sym_pos : bool (default = True) + True if the system matrix is symmetric positive definite + Sometimes this needs to be set false as the solution is approached, + even when the system should be symmetric positive definite, due to + numerical difficulties. + cholesky : bool (default = True) + True if the system is to be solved by Cholesky, rather than LU, + decomposition. This is typically faster unless the problem is very + small or prone to numerical difficulties. + permc_spec : str (default = 'MMD_AT_PLUS_A') + Sparsity preservation strategy used by SuperLU. Acceptable values are: + + - ``NATURAL``: natural ordering. + - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. + - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. + - ``COLAMD``: approximate minimum degree column ordering. + + See SuperLU documentation. + + Returns + ------- + solve : function + Handle to the appropriate solver function + + """ + try: + if sparse: + if lstsq: + def solve(r, sym_pos=False): + return sps.linalg.lsqr(M, r)[0] + elif cholesky: + try: + # Will raise an exception in the first call, + # or when the matrix changes due to a new problem + _get_solver.cholmod_factor.cholesky_inplace(M) + except Exception: + _get_solver.cholmod_factor = cholmod_analyze(M) + _get_solver.cholmod_factor.cholesky_inplace(M) + solve = _get_solver.cholmod_factor + else: + if has_umfpack and sym_pos: + solve = sps.linalg.factorized(M) + else: # factorized doesn't pass permc_spec + solve = sps.linalg.splu(M, permc_spec=permc_spec).solve + + else: + if lstsq: # sometimes necessary as solution is approached + def solve(r): + return sp.linalg.lstsq(M, r)[0] + elif cholesky: + L = sp.linalg.cho_factor(M) + + def solve(r): + return sp.linalg.cho_solve(L, r) + else: + # this seems to cache the matrix factorization, so solving + # with multiple right hand sides is much faster + def solve(r, sym_pos=sym_pos): + if sym_pos: + return sp.linalg.solve(M, r, assume_a="pos") + else: + return sp.linalg.solve(M, r) + # There are many things that can go wrong here, and it's hard to say + # what all of them are. It doesn't really matter: if the matrix can't be + # factorized, return None. get_solver will be called again with different + # inputs, and a new routine will try to factorize the matrix. + except KeyboardInterrupt: + raise + except Exception: + return None + return solve + + +def _get_delta(A, b, c, x, y, z, tau, kappa, gamma, eta, sparse=False, + lstsq=False, sym_pos=True, cholesky=True, pc=True, ip=False, + permc_spec='MMD_AT_PLUS_A'): + """ + Given standard form problem defined by ``A``, ``b``, and ``c``; + current variable estimates ``x``, ``y``, ``z``, ``tau``, and ``kappa``; + algorithmic parameters ``gamma and ``eta; + and options ``sparse``, ``lstsq``, ``sym_pos``, ``cholesky``, ``pc`` + (predictor-corrector), and ``ip`` (initial point improvement), + get the search direction for increments to the variable estimates. + + Parameters + ---------- + As defined in [4], except: + sparse : bool + True if the system to be solved is sparse. This is typically set + True when the original ``A_ub`` and ``A_eq`` arrays are sparse. + lstsq : bool + True if the system is ill-conditioned and/or (nearly) singular and + thus a more robust least-squares solver is desired. This is sometimes + needed as the solution is approached. + sym_pos : bool + True if the system matrix is symmetric positive definite + Sometimes this needs to be set false as the solution is approached, + even when the system should be symmetric positive definite, due to + numerical difficulties. + cholesky : bool + True if the system is to be solved by Cholesky, rather than LU, + decomposition. This is typically faster unless the problem is very + small or prone to numerical difficulties. + pc : bool + True if the predictor-corrector method of Mehrota is to be used. This + is almost always (if not always) beneficial. Even though it requires + the solution of an additional linear system, the factorization + is typically (implicitly) reused so solution is efficient, and the + number of algorithm iterations is typically reduced. + ip : bool + True if the improved initial point suggestion due to [4] section 4.3 + is desired. It's unclear whether this is beneficial. + permc_spec : str (default = 'MMD_AT_PLUS_A') + (Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = + True``.) A matrix is factorized in each iteration of the algorithm. + This option specifies how to permute the columns of the matrix for + sparsity preservation. Acceptable values are: + + - ``NATURAL``: natural ordering. + - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. + - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. + - ``COLAMD``: approximate minimum degree column ordering. + + This option can impact the convergence of the + interior point algorithm; test different values to determine which + performs best for your problem. For more information, refer to + ``scipy.sparse.linalg.splu``. + + Returns + ------- + Search directions as defined in [4] + + References + ---------- + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + + """ + if A.shape[0] == 0: + # If there are no constraints, some solvers fail (understandably) + # rather than returning empty solution. This gets the job done. + sparse, lstsq, sym_pos, cholesky = False, False, True, False + n_x = len(x) + + # [4] Equation 8.8 + r_P = b * tau - A.dot(x) + r_D = c * tau - A.T.dot(y) - z + r_G = c.dot(x) - b.transpose().dot(y) + kappa + mu = (x.dot(z) + tau * kappa) / (n_x + 1) + + # Assemble M from [4] Equation 8.31 + Dinv = x / z + + if sparse: + M = A.dot(sps.diags(Dinv, 0, format="csc").dot(A.T)) + else: + M = A.dot(Dinv.reshape(-1, 1) * A.T) + solve = _get_solver(M, sparse, lstsq, sym_pos, cholesky, permc_spec) + + # pc: "predictor-corrector" [4] Section 4.1 + # In development this option could be turned off + # but it always seems to improve performance substantially + n_corrections = 1 if pc else 0 + + i = 0 + alpha, d_x, d_z, d_tau, d_kappa = 0, 0, 0, 0, 0 + while i <= n_corrections: + # Reference [4] Eq. 8.6 + rhatp = eta(gamma) * r_P + rhatd = eta(gamma) * r_D + rhatg = eta(gamma) * r_G + + # Reference [4] Eq. 8.7 + rhatxs = gamma * mu - x * z + rhattk = gamma * mu - tau * kappa + + if i == 1: + if ip: # if the correction is to get "initial point" + # Reference [4] Eq. 8.23 + rhatxs = ((1 - alpha) * gamma * mu - + x * z - alpha**2 * d_x * d_z) + rhattk = ((1 - alpha) * gamma * mu - + tau * kappa - + alpha**2 * d_tau * d_kappa) + else: # if the correction is for "predictor-corrector" + # Reference [4] Eq. 8.13 + rhatxs -= d_x * d_z + rhattk -= d_tau * d_kappa + + # sometimes numerical difficulties arise as the solution is approached + # this loop tries to solve the equations using a sequence of functions + # for solve. For dense systems, the order is: + # 1. scipy.linalg.cho_factor/scipy.linalg.cho_solve, + # 2. scipy.linalg.solve w/ sym_pos = True, + # 3. scipy.linalg.solve w/ sym_pos = False, and if all else fails + # 4. scipy.linalg.lstsq + # For sparse systems, the order is: + # 1. sksparse.cholmod.cholesky (if available) + # 2. scipy.sparse.linalg.factorized (if umfpack available) + # 3. scipy.sparse.linalg.splu + # 4. scipy.sparse.linalg.lsqr + solved = False + while not solved: + try: + # [4] Equation 8.28 + p, q = _sym_solve(Dinv, A, c, b, solve) + # [4] Equation 8.29 + u, v = _sym_solve(Dinv, A, rhatd - + (1 / x) * rhatxs, rhatp, solve) + if np.any(np.isnan(p)) or np.any(np.isnan(q)): + raise LinAlgError + solved = True + except (LinAlgError, ValueError, TypeError) as e: + # Usually this doesn't happen. If it does, it happens when + # there are redundant constraints or when approaching the + # solution. If so, change solver. + if cholesky: + cholesky = False + warn( + "Solving system with option 'cholesky':True " + "failed. It is normal for this to happen " + "occasionally, especially as the solution is " + "approached. However, if you see this frequently, " + "consider setting option 'cholesky' to False.", + OptimizeWarning, stacklevel=5) + elif sym_pos: + sym_pos = False + warn( + "Solving system with option 'sym_pos':True " + "failed. It is normal for this to happen " + "occasionally, especially as the solution is " + "approached. However, if you see this frequently, " + "consider setting option 'sym_pos' to False.", + OptimizeWarning, stacklevel=5) + elif not lstsq: + lstsq = True + warn( + "Solving system with option 'sym_pos':False " + "failed. This may happen occasionally, " + "especially as the solution is " + "approached. However, if you see this frequently, " + "your problem may be numerically challenging. " + "If you cannot improve the formulation, consider " + "setting 'lstsq' to True. Consider also setting " + "`presolve` to True, if it is not already.", + OptimizeWarning, stacklevel=5) + else: + raise e + solve = _get_solver(M, sparse, lstsq, sym_pos, + cholesky, permc_spec) + # [4] Results after 8.29 + d_tau = ((rhatg + 1 / tau * rhattk - (-c.dot(u) + b.dot(v))) / + (1 / tau * kappa + (-c.dot(p) + b.dot(q)))) + d_x = u + p * d_tau + d_y = v + q * d_tau + + # [4] Relations between after 8.25 and 8.26 + d_z = (1 / x) * (rhatxs - z * d_x) + d_kappa = 1 / tau * (rhattk - kappa * d_tau) + + # [4] 8.12 and "Let alpha be the maximal possible step..." before 8.23 + alpha = _get_step(x, d_x, z, d_z, tau, d_tau, kappa, d_kappa, 1) + if ip: # initial point - see [4] 4.4 + gamma = 10 + else: # predictor-corrector, [4] definition after 8.12 + beta1 = 0.1 # [4] pg. 220 (Table 8.1) + gamma = (1 - alpha)**2 * min(beta1, (1 - alpha)) + i += 1 + + return d_x, d_y, d_z, d_tau, d_kappa + + +def _sym_solve(Dinv, A, r1, r2, solve): + """ + An implementation of [4] equation 8.31 and 8.32 + + References + ---------- + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + + """ + # [4] 8.31 + r = r2 + A.dot(Dinv * r1) + v = solve(r) + # [4] 8.32 + u = Dinv * (A.T.dot(v) - r1) + return u, v + + +def _get_step(x, d_x, z, d_z, tau, d_tau, kappa, d_kappa, alpha0): + """ + An implementation of [4] equation 8.21 + + References + ---------- + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + + """ + # [4] 4.3 Equation 8.21, ignoring 8.20 requirement + # same step is taken in primal and dual spaces + # alpha0 is basically beta3 from [4] Table 8.1, but instead of beta3 + # the value 1 is used in Mehrota corrector and initial point correction + i_x = d_x < 0 + i_z = d_z < 0 + alpha_x = alpha0 * np.min(x[i_x] / -d_x[i_x]) if np.any(i_x) else 1 + alpha_tau = alpha0 * tau / -d_tau if d_tau < 0 else 1 + alpha_z = alpha0 * np.min(z[i_z] / -d_z[i_z]) if np.any(i_z) else 1 + alpha_kappa = alpha0 * kappa / -d_kappa if d_kappa < 0 else 1 + alpha = np.min([1, alpha_x, alpha_tau, alpha_z, alpha_kappa]) + return alpha + + +def _get_message(status): + """ + Given problem status code, return a more detailed message. + + Parameters + ---------- + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + + Returns + ------- + message : str + A string descriptor of the exit status of the optimization. + + """ + messages = ( + ["Optimization terminated successfully.", + "The iteration limit was reached before the algorithm converged.", + "The algorithm terminated successfully and determined that the " + "problem is infeasible.", + "The algorithm terminated successfully and determined that the " + "problem is unbounded.", + "Numerical difficulties were encountered before the problem " + "converged. Please check your problem formulation for errors, " + "independence of linear equality constraints, and reasonable " + "scaling and matrix condition numbers. If you continue to " + "encounter this error, please submit a bug report." + ]) + return messages[status] + + +def _do_step(x, y, z, tau, kappa, d_x, d_y, d_z, d_tau, d_kappa, alpha): + """ + An implementation of [4] Equation 8.9 + + References + ---------- + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + + """ + x = x + alpha * d_x + tau = tau + alpha * d_tau + z = z + alpha * d_z + kappa = kappa + alpha * d_kappa + y = y + alpha * d_y + return x, y, z, tau, kappa + + +def _get_blind_start(shape): + """ + Return the starting point from [4] 4.4 + + References + ---------- + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + + """ + m, n = shape + x0 = np.ones(n) + y0 = np.zeros(m) + z0 = np.ones(n) + tau0 = 1 + kappa0 = 1 + return x0, y0, z0, tau0, kappa0 + + +def _indicators(A, b, c, c0, x, y, z, tau, kappa): + """ + Implementation of several equations from [4] used as indicators of + the status of optimization. + + References + ---------- + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + + """ + + # residuals for termination are relative to initial values + x0, y0, z0, tau0, kappa0 = _get_blind_start(A.shape) + + # See [4], Section 4 - The Homogeneous Algorithm, Equation 8.8 + def r_p(x, tau): + return b * tau - A.dot(x) + + def r_d(y, z, tau): + return c * tau - A.T.dot(y) - z + + def r_g(x, y, kappa): + return kappa + c.dot(x) - b.dot(y) + + # np.dot unpacks if they are arrays of size one + def mu(x, tau, z, kappa): + return (x.dot(z) + np.dot(tau, kappa)) / (len(x) + 1) + + obj = c.dot(x / tau) + c0 + + def norm(a): + return np.linalg.norm(a) + + # See [4], Section 4.5 - The Stopping Criteria + r_p0 = r_p(x0, tau0) + r_d0 = r_d(y0, z0, tau0) + r_g0 = r_g(x0, y0, kappa0) + mu_0 = mu(x0, tau0, z0, kappa0) + rho_A = norm(c.T.dot(x) - b.T.dot(y)) / (tau + norm(b.T.dot(y))) + rho_p = norm(r_p(x, tau)) / max(1, norm(r_p0)) + rho_d = norm(r_d(y, z, tau)) / max(1, norm(r_d0)) + rho_g = norm(r_g(x, y, kappa)) / max(1, norm(r_g0)) + rho_mu = mu(x, tau, z, kappa) / mu_0 + return rho_p, rho_d, rho_A, rho_g, rho_mu, obj + + +def _display_iter(rho_p, rho_d, rho_g, alpha, rho_mu, obj, header=False): + """ + Print indicators of optimization status to the console. + + Parameters + ---------- + rho_p : float + The (normalized) primal feasibility, see [4] 4.5 + rho_d : float + The (normalized) dual feasibility, see [4] 4.5 + rho_g : float + The (normalized) duality gap, see [4] 4.5 + alpha : float + The step size, see [4] 4.3 + rho_mu : float + The (normalized) path parameter, see [4] 4.5 + obj : float + The objective function value of the current iterate + header : bool + True if a header is to be printed + + References + ---------- + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + + """ + if header: + print("Primal Feasibility ", + "Dual Feasibility ", + "Duality Gap ", + "Step ", + "Path Parameter ", + "Objective ") + + # no clue why this works + fmt = '{0:<20.13}{1:<20.13}{2:<20.13}{3:<17.13}{4:<20.13}{5:<20.13}' + print(fmt.format( + float(rho_p), + float(rho_d), + float(rho_g), + alpha if isinstance(alpha, str) else float(alpha), + float(rho_mu), + float(obj))) + + +def _ip_hsd(A, b, c, c0, alpha0, beta, maxiter, disp, tol, sparse, lstsq, + sym_pos, cholesky, pc, ip, permc_spec, callback, postsolve_args): + r""" + Solve a linear programming problem in standard form: + + Minimize:: + + c @ x + + Subject to:: + + A @ x == b + x >= 0 + + using the interior point method of [4]. + + Parameters + ---------- + A : 2-D array + 2-D array such that ``A @ x``, gives the values of the equality + constraints at ``x``. + b : 1-D array + 1-D array of values representing the RHS of each equality constraint + (row) in ``A`` (for standard form problem). + c : 1-D array + Coefficients of the linear objective function to be minimized (for + standard form problem). + c0 : float + Constant term in objective function due to fixed (and eliminated) + variables. (Purely for display.) + alpha0 : float + The maximal step size for Mehrota's predictor-corrector search + direction; see :math:`\beta_3`of [4] Table 8.1 + beta : float + The desired reduction of the path parameter :math:`\mu` (see [6]_) + maxiter : int + The maximum number of iterations of the algorithm. + disp : bool + Set to ``True`` if indicators of optimization status are to be printed + to the console each iteration. + tol : float + Termination tolerance; see [4]_ Section 4.5. + sparse : bool + Set to ``True`` if the problem is to be treated as sparse. However, + the inputs ``A_eq`` and ``A_ub`` should nonetheless be provided as + (dense) arrays rather than sparse matrices. + lstsq : bool + Set to ``True`` if the problem is expected to be very poorly + conditioned. This should always be left as ``False`` unless severe + numerical difficulties are frequently encountered, and a better option + would be to improve the formulation of the problem. + sym_pos : bool + Leave ``True`` if the problem is expected to yield a well conditioned + symmetric positive definite normal equation matrix (almost always). + cholesky : bool + Set to ``True`` if the normal equations are to be solved by explicit + Cholesky decomposition followed by explicit forward/backward + substitution. This is typically faster for moderate, dense problems + that are numerically well-behaved. + pc : bool + Leave ``True`` if the predictor-corrector method of Mehrota is to be + used. This is almost always (if not always) beneficial. + ip : bool + Set to ``True`` if the improved initial point suggestion due to [4]_ + Section 4.3 is desired. It's unclear whether this is beneficial. + permc_spec : str (default = 'MMD_AT_PLUS_A') + (Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = + True``.) A matrix is factorized in each iteration of the algorithm. + This option specifies how to permute the columns of the matrix for + sparsity preservation. Acceptable values are: + + - ``NATURAL``: natural ordering. + - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. + - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. + - ``COLAMD``: approximate minimum degree column ordering. + + This option can impact the convergence of the + interior point algorithm; test different values to determine which + performs best for your problem. For more information, refer to + ``scipy.sparse.linalg.splu``. + callback : callable, optional + If a callback function is provided, it will be called within each + iteration of the algorithm. The callback function must accept a single + `scipy.optimize.OptimizeResult` consisting of the following fields: + + x : 1-D array + Current solution vector + fun : float + Current value of the objective function + success : bool + True only when an algorithm has completed successfully, + so this is always False as the callback function is called + only while the algorithm is still iterating. + slack : 1-D array + The values of the slack variables. Each slack variable + corresponds to an inequality constraint. If the slack is zero, + the corresponding constraint is active. + con : 1-D array + The (nominally zero) residuals of the equality constraints, + that is, ``b - A_eq @ x`` + phase : int + The phase of the algorithm being executed. This is always + 1 for the interior-point method because it has only one phase. + status : int + For revised simplex, this is always 0 because if a different + status is detected, the algorithm terminates. + nit : int + The number of iterations performed. + message : str + A string descriptor of the exit status of the optimization. + postsolve_args : tuple + Data needed by _postsolve to convert the solution to the standard-form + problem into the solution to the original problem. + + Returns + ------- + x_hat : float + Solution vector (for standard form problem). + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + + message : str + A string descriptor of the exit status of the optimization. + iteration : int + The number of iterations taken to solve the problem + + References + ---------- + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + .. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear + Programming based on Newton's Method." Unpublished Course Notes, + March 2004. Available 2/25/2017 at: + https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf + + """ + + iteration = 0 + + # default initial point + x, y, z, tau, kappa = _get_blind_start(A.shape) + + # first iteration is special improvement of initial point + ip = ip if pc else False + + # [4] 4.5 + rho_p, rho_d, rho_A, rho_g, rho_mu, obj = _indicators( + A, b, c, c0, x, y, z, tau, kappa) + go = rho_p > tol or rho_d > tol or rho_A > tol # we might get lucky : ) + + if disp: + _display_iter(rho_p, rho_d, rho_g, "-", rho_mu, obj, header=True) + if callback is not None: + x_o, fun, slack, con = _postsolve(x/tau, postsolve_args) + res = OptimizeResult({'x': x_o, 'fun': fun, 'slack': slack, + 'con': con, 'nit': iteration, 'phase': 1, + 'complete': False, 'status': 0, + 'message': "", 'success': False}) + callback(res) + + status = 0 + message = "Optimization terminated successfully." + + if sparse: + A = sps.csc_matrix(A) + + while go: + + iteration += 1 + + if ip: # initial point + # [4] Section 4.4 + gamma = 1 + + def eta(g): + return 1 + else: + # gamma = 0 in predictor step according to [4] 4.1 + # if predictor/corrector is off, use mean of complementarity [6] + # 5.1 / [4] Below Figure 10-4 + gamma = 0 if pc else beta * np.mean(z * x) + # [4] Section 4.1 + + def eta(g=gamma): + return 1 - g + + try: + # Solve [4] 8.6 and 8.7/8.13/8.23 + d_x, d_y, d_z, d_tau, d_kappa = _get_delta( + A, b, c, x, y, z, tau, kappa, gamma, eta, + sparse, lstsq, sym_pos, cholesky, pc, ip, permc_spec) + + if ip: # initial point + # [4] 4.4 + # Formula after 8.23 takes a full step regardless if this will + # take it negative + alpha = 1.0 + x, y, z, tau, kappa = _do_step( + x, y, z, tau, kappa, d_x, d_y, + d_z, d_tau, d_kappa, alpha) + x[x < 1] = 1 + z[z < 1] = 1 + tau = max(1, tau) + kappa = max(1, kappa) + ip = False # done with initial point + else: + # [4] Section 4.3 + alpha = _get_step(x, d_x, z, d_z, tau, + d_tau, kappa, d_kappa, alpha0) + # [4] Equation 8.9 + x, y, z, tau, kappa = _do_step( + x, y, z, tau, kappa, d_x, d_y, d_z, d_tau, d_kappa, alpha) + + except (LinAlgError, FloatingPointError, + ValueError, ZeroDivisionError): + # this can happen when sparse solver is used and presolve + # is turned off. Also observed ValueError in AppVeyor Python 3.6 + # Win32 build (PR #8676). I've never seen it otherwise. + status = 4 + message = _get_message(status) + break + + # [4] 4.5 + rho_p, rho_d, rho_A, rho_g, rho_mu, obj = _indicators( + A, b, c, c0, x, y, z, tau, kappa) + go = rho_p > tol or rho_d > tol or rho_A > tol + + if disp: + _display_iter(rho_p, rho_d, rho_g, alpha, rho_mu, obj) + if callback is not None: + x_o, fun, slack, con = _postsolve(x/tau, postsolve_args) + res = OptimizeResult({'x': x_o, 'fun': fun, 'slack': slack, + 'con': con, 'nit': iteration, 'phase': 1, + 'complete': False, 'status': 0, + 'message': "", 'success': False}) + callback(res) + + # [4] 4.5 + inf1 = (rho_p < tol and rho_d < tol and rho_g < tol and tau < tol * + max(1, kappa)) + inf2 = rho_mu < tol and tau < tol * min(1, kappa) + if inf1 or inf2: + # [4] Lemma 8.4 / Theorem 8.3 + if b.transpose().dot(y) > tol: + status = 2 + else: # elif c.T.dot(x) < tol: ? Probably not necessary. + status = 3 + message = _get_message(status) + break + elif iteration >= maxiter: + status = 1 + message = _get_message(status) + break + + x_hat = x / tau + # [4] Statement after Theorem 8.2 + return x_hat, status, message, iteration + + +def _linprog_ip(c, c0, A, b, callback, postsolve_args, maxiter=1000, tol=1e-8, + disp=False, alpha0=.99995, beta=0.1, sparse=False, lstsq=False, + sym_pos=True, cholesky=None, pc=True, ip=False, + permc_spec='MMD_AT_PLUS_A', **unknown_options): + r""" + Minimize a linear objective function subject to linear + equality and non-negativity constraints using the interior point method + of [4]_. Linear programming is intended to solve problems + of the following form: + + Minimize:: + + c @ x + + Subject to:: + + A @ x == b + x >= 0 + + User-facing documentation is in _linprog_doc.py. + + Parameters + ---------- + c : 1-D array + Coefficients of the linear objective function to be minimized. + c0 : float + Constant term in objective function due to fixed (and eliminated) + variables. (Purely for display.) + A : 2-D array + 2-D array such that ``A @ x``, gives the values of the equality + constraints at ``x``. + b : 1-D array + 1-D array of values representing the right hand side of each equality + constraint (row) in ``A``. + callback : callable, optional + Callback function to be executed once per iteration. + postsolve_args : tuple + Data needed by _postsolve to convert the solution to the standard-form + problem into the solution to the original problem. + + Options + ------- + maxiter : int (default = 1000) + The maximum number of iterations of the algorithm. + tol : float (default = 1e-8) + Termination tolerance to be used for all termination criteria; + see [4]_ Section 4.5. + disp : bool (default = False) + Set to ``True`` if indicators of optimization status are to be printed + to the console each iteration. + alpha0 : float (default = 0.99995) + The maximal step size for Mehrota's predictor-corrector search + direction; see :math:`\beta_{3}` of [4]_ Table 8.1. + beta : float (default = 0.1) + The desired reduction of the path parameter :math:`\mu` (see [6]_) + when Mehrota's predictor-corrector is not in use (uncommon). + sparse : bool (default = False) + Set to ``True`` if the problem is to be treated as sparse after + presolve. If either ``A_eq`` or ``A_ub`` is a sparse matrix, + this option will automatically be set ``True``, and the problem + will be treated as sparse even during presolve. If your constraint + matrices contain mostly zeros and the problem is not very small (less + than about 100 constraints or variables), consider setting ``True`` + or providing ``A_eq`` and ``A_ub`` as sparse matrices. + lstsq : bool (default = False) + Set to ``True`` if the problem is expected to be very poorly + conditioned. This should always be left ``False`` unless severe + numerical difficulties are encountered. Leave this at the default + unless you receive a warning message suggesting otherwise. + sym_pos : bool (default = True) + Leave ``True`` if the problem is expected to yield a well conditioned + symmetric positive definite normal equation matrix + (almost always). Leave this at the default unless you receive + a warning message suggesting otherwise. + cholesky : bool (default = True) + Set to ``True`` if the normal equations are to be solved by explicit + Cholesky decomposition followed by explicit forward/backward + substitution. This is typically faster for problems + that are numerically well-behaved. + pc : bool (default = True) + Leave ``True`` if the predictor-corrector method of Mehrota is to be + used. This is almost always (if not always) beneficial. + ip : bool (default = False) + Set to ``True`` if the improved initial point suggestion due to [4]_ + Section 4.3 is desired. Whether this is beneficial or not + depends on the problem. + permc_spec : str (default = 'MMD_AT_PLUS_A') + (Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = + True``, and no SuiteSparse.) + A matrix is factorized in each iteration of the algorithm. + This option specifies how to permute the columns of the matrix for + sparsity preservation. Acceptable values are: + + - ``NATURAL``: natural ordering. + - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. + - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. + - ``COLAMD``: approximate minimum degree column ordering. + + This option can impact the convergence of the + interior point algorithm; test different values to determine which + performs best for your problem. For more information, refer to + ``scipy.sparse.linalg.splu``. + unknown_options : dict + Optional arguments not used by this particular solver. If + `unknown_options` is non-empty a warning is issued listing all + unused options. + + Returns + ------- + x : 1-D array + Solution vector. + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + + message : str + A string descriptor of the exit status of the optimization. + iteration : int + The number of iterations taken to solve the problem. + + Notes + ----- + This method implements the algorithm outlined in [4]_ with ideas from [8]_ + and a structure inspired by the simpler methods of [6]_. + + The primal-dual path following method begins with initial 'guesses' of + the primal and dual variables of the standard form problem and iteratively + attempts to solve the (nonlinear) Karush-Kuhn-Tucker conditions for the + problem with a gradually reduced logarithmic barrier term added to the + objective. This particular implementation uses a homogeneous self-dual + formulation, which provides certificates of infeasibility or unboundedness + where applicable. + + The default initial point for the primal and dual variables is that + defined in [4]_ Section 4.4 Equation 8.22. Optionally (by setting initial + point option ``ip=True``), an alternate (potentially improved) starting + point can be calculated according to the additional recommendations of + [4]_ Section 4.4. + + A search direction is calculated using the predictor-corrector method + (single correction) proposed by Mehrota and detailed in [4]_ Section 4.1. + (A potential improvement would be to implement the method of multiple + corrections described in [4]_ Section 4.2.) In practice, this is + accomplished by solving the normal equations, [4]_ Section 5.1 Equations + 8.31 and 8.32, derived from the Newton equations [4]_ Section 5 Equations + 8.25 (compare to [4]_ Section 4 Equations 8.6-8.8). The advantage of + solving the normal equations rather than 8.25 directly is that the + matrices involved are symmetric positive definite, so Cholesky + decomposition can be used rather than the more expensive LU factorization. + + With default options, the solver used to perform the factorization depends + on third-party software availability and the conditioning of the problem. + + For dense problems, solvers are tried in the following order: + + 1. ``scipy.linalg.cho_factor`` + + 2. ``scipy.linalg.solve`` with option ``sym_pos=True`` + + 3. ``scipy.linalg.solve`` with option ``sym_pos=False`` + + 4. ``scipy.linalg.lstsq`` + + For sparse problems: + + 1. ``sksparse.cholmod.cholesky`` (if scikit-sparse and SuiteSparse are installed) + + 2. ``scipy.sparse.linalg.factorized`` + (if scikit-umfpack and SuiteSparse are installed) + + 3. ``scipy.sparse.linalg.splu`` (which uses SuperLU distributed with SciPy) + + 4. ``scipy.sparse.linalg.lsqr`` + + If the solver fails for any reason, successively more robust (but slower) + solvers are attempted in the order indicated. Attempting, failing, and + re-starting factorization can be time consuming, so if the problem is + numerically challenging, options can be set to bypass solvers that are + failing. Setting ``cholesky=False`` skips to solver 2, + ``sym_pos=False`` skips to solver 3, and ``lstsq=True`` skips + to solver 4 for both sparse and dense problems. + + Potential improvements for combating issues associated with dense + columns in otherwise sparse problems are outlined in [4]_ Section 5.3 and + [10]_ Section 4.1-4.2; the latter also discusses the alleviation of + accuracy issues associated with the substitution approach to free + variables. + + After calculating the search direction, the maximum possible step size + that does not activate the non-negativity constraints is calculated, and + the smaller of this step size and unity is applied (as in [4]_ Section + 4.1.) [4]_ Section 4.3 suggests improvements for choosing the step size. + + The new point is tested according to the termination conditions of [4]_ + Section 4.5. The same tolerance, which can be set using the ``tol`` option, + is used for all checks. (A potential improvement would be to expose + the different tolerances to be set independently.) If optimality, + unboundedness, or infeasibility is detected, the solve procedure + terminates; otherwise it repeats. + + The expected problem formulation differs between the top level ``linprog`` + module and the method specific solvers. The method specific solvers expect a + problem in standard form: + + Minimize:: + + c @ x + + Subject to:: + + A @ x == b + x >= 0 + + Whereas the top level ``linprog`` module expects a problem of form: + + Minimize:: + + c @ x + + Subject to:: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + where ``lb = 0`` and ``ub = None`` unless set in ``bounds``. + + The original problem contains equality, upper-bound and variable constraints + whereas the method specific solver requires equality constraints and + variable non-negativity. + + ``linprog`` module converts the original problem to standard form by + converting the simple bounds to upper bound constraints, introducing + non-negative slack variables for inequality constraints, and expressing + unbounded variables as the difference between two non-negative variables. + + + References + ---------- + .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point + optimizer for linear programming: an implementation of the + homogeneous algorithm." High performance optimization. Springer US, + 2000. 197-232. + .. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear + Programming based on Newton's Method." Unpublished Course Notes, + March 2004. Available 2/25/2017 at + https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf + .. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear + programming." Mathematical Programming 71.2 (1995): 221-245. + .. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear + programming." Athena Scientific 1 (1997): 997. + .. [10] Andersen, Erling D., et al. Implementation of interior point methods + for large scale linear programming. HEC/Universite de Geneve, 1996. + + """ + + _check_unknown_options(unknown_options) + + # These should be warnings, not errors + if (cholesky or cholesky is None) and sparse and not has_cholmod: + if cholesky: + warn("Sparse cholesky is only available with scikit-sparse. " + "Setting `cholesky = False`", + OptimizeWarning, stacklevel=3) + cholesky = False + + if sparse and lstsq: + warn("Option combination 'sparse':True and 'lstsq':True " + "is not recommended.", + OptimizeWarning, stacklevel=3) + + if lstsq and cholesky: + warn("Invalid option combination 'lstsq':True " + "and 'cholesky':True; option 'cholesky' has no effect when " + "'lstsq' is set True.", + OptimizeWarning, stacklevel=3) + + valid_permc_spec = ('NATURAL', 'MMD_ATA', 'MMD_AT_PLUS_A', 'COLAMD') + if permc_spec.upper() not in valid_permc_spec: + warn("Invalid permc_spec option: '" + str(permc_spec) + "'. " + "Acceptable values are 'NATURAL', 'MMD_ATA', 'MMD_AT_PLUS_A', " + "and 'COLAMD'. Reverting to default.", + OptimizeWarning, stacklevel=3) + permc_spec = 'MMD_AT_PLUS_A' + + # This can be an error + if not sym_pos and cholesky: + raise ValueError( + "Invalid option combination 'sym_pos':False " + "and 'cholesky':True: Cholesky decomposition is only possible " + "for symmetric positive definite matrices.") + + cholesky = cholesky or (cholesky is None and sym_pos and not lstsq) + + x, status, message, iteration = _ip_hsd(A, b, c, c0, alpha0, beta, + maxiter, disp, tol, sparse, + lstsq, sym_pos, cholesky, + pc, ip, permc_spec, callback, + postsolve_args) + + return x, status, message, iteration diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_rs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_rs.py new file mode 100644 index 0000000000000000000000000000000000000000..43fed5805c4e40f0c38de91f053e3926cf1478e4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_rs.py @@ -0,0 +1,572 @@ +"""Revised simplex method for linear programming + +The *revised simplex* method uses the method described in [1]_, except +that a factorization [2]_ of the basis matrix, rather than its inverse, +is efficiently maintained and used to solve the linear systems at each +iteration of the algorithm. + +.. versionadded:: 1.3.0 + +References +---------- +.. [1] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear + programming." Athena Scientific 1 (1997): 997. +.. [2] Bartels, Richard H. "A stabilization of the simplex method." + Journal in Numerische Mathematik 16.5 (1971): 414-434. + +""" +# Author: Matt Haberland + +import numpy as np +from numpy.linalg import LinAlgError + +from scipy.linalg import solve +from ._optimize import _check_unknown_options +from ._bglu_dense import LU +from ._bglu_dense import BGLU as BGLU +from ._linprog_util import _postsolve +from ._optimize import OptimizeResult + + +def _phase_one(A, b, x0, callback, postsolve_args, maxiter, tol, disp, + maxupdate, mast, pivot): + """ + The purpose of phase one is to find an initial basic feasible solution + (BFS) to the original problem. + + Generates an auxiliary problem with a trivial BFS and an objective that + minimizes infeasibility of the original problem. Solves the auxiliary + problem using the main simplex routine (phase two). This either yields + a BFS to the original problem or determines that the original problem is + infeasible. If feasible, phase one detects redundant rows in the original + constraint matrix and removes them, then chooses additional indices as + necessary to complete a basis/BFS for the original problem. + """ + + m, n = A.shape + status = 0 + + # generate auxiliary problem to get initial BFS + A, b, c, basis, x, status = _generate_auxiliary_problem(A, b, x0, tol) + + if status == 6: + residual = c.dot(x) + iter_k = 0 + return x, basis, A, b, residual, status, iter_k + + # solve auxiliary problem + phase_one_n = n + iter_k = 0 + x, basis, status, iter_k = _phase_two(c, A, x, basis, callback, + postsolve_args, + maxiter, tol, disp, + maxupdate, mast, pivot, + iter_k, phase_one_n) + + # check for infeasibility + residual = c.dot(x) + if status == 0 and residual > tol: + status = 2 + + # drive artificial variables out of basis + # TODO: test redundant row removal better + # TODO: make solve more efficient with BGLU? This could take a while. + keep_rows = np.ones(m, dtype=bool) + for basis_column in basis[basis >= n]: + B = A[:, basis] + try: + basis_finder = np.abs(solve(B, A)) # inefficient + pertinent_row = np.argmax(basis_finder[:, basis_column]) + eligible_columns = np.ones(n, dtype=bool) + eligible_columns[basis[basis < n]] = 0 + eligible_column_indices = np.where(eligible_columns)[0] + index = np.argmax(basis_finder[:, :n] + [pertinent_row, eligible_columns]) + new_basis_column = eligible_column_indices[index] + if basis_finder[pertinent_row, new_basis_column] < tol: + keep_rows[pertinent_row] = False + else: + basis[basis == basis_column] = new_basis_column + except LinAlgError: + status = 4 + + # form solution to original problem + A = A[keep_rows, :n] + basis = basis[keep_rows] + x = x[:n] + m = A.shape[0] + return x, basis, A, b, residual, status, iter_k + + +def _get_more_basis_columns(A, basis): + """ + Called when the auxiliary problem terminates with artificial columns in + the basis, which must be removed and replaced with non-artificial + columns. Finds additional columns that do not make the matrix singular. + """ + m, n = A.shape + + # options for inclusion are those that aren't already in the basis + a = np.arange(m+n) + bl = np.zeros(len(a), dtype=bool) + bl[basis] = 1 + options = a[~bl] + options = options[options < n] # and they have to be non-artificial + + # form basis matrix + B = np.zeros((m, m)) + B[:, 0:len(basis)] = A[:, basis] + + if (basis.size > 0 and + np.linalg.matrix_rank(B[:, :len(basis)]) < len(basis)): + raise Exception("Basis has dependent columns") + + rank = 0 # just enter the loop + for i in range(n): # somewhat arbitrary, but we need another way out + # permute the options, and take as many as needed + new_basis = np.random.permutation(options)[:m-len(basis)] + B[:, len(basis):] = A[:, new_basis] # update the basis matrix + rank = np.linalg.matrix_rank(B) # check the rank + if rank == m: + break + + return np.concatenate((basis, new_basis)) + + +def _generate_auxiliary_problem(A, b, x0, tol): + """ + Modifies original problem to create an auxiliary problem with a trivial + initial basic feasible solution and an objective that minimizes + infeasibility in the original problem. + + Conceptually, this is done by stacking an identity matrix on the right of + the original constraint matrix, adding artificial variables to correspond + with each of these new columns, and generating a cost vector that is all + zeros except for ones corresponding with each of the new variables. + + A initial basic feasible solution is trivial: all variables are zero + except for the artificial variables, which are set equal to the + corresponding element of the right hand side `b`. + + Running the simplex method on this auxiliary problem drives all of the + artificial variables - and thus the cost - to zero if the original problem + is feasible. The original problem is declared infeasible otherwise. + + Much of the complexity below is to improve efficiency by using singleton + columns in the original problem where possible, thus generating artificial + variables only as necessary, and using an initial 'guess' basic feasible + solution. + """ + status = 0 + m, n = A.shape + + if x0 is not None: + x = x0 + else: + x = np.zeros(n) + + r = b - A@x # residual; this must be all zeros for feasibility + + A[r < 0] = -A[r < 0] # express problem with RHS positive for trivial BFS + b[r < 0] = -b[r < 0] # to the auxiliary problem + r[r < 0] *= -1 + + # Rows which we will need to find a trivial way to zero. + # This should just be the rows where there is a nonzero residual. + # But then we would not necessarily have a column singleton in every row. + # This makes it difficult to find an initial basis. + if x0 is None: + nonzero_constraints = np.arange(m) + else: + nonzero_constraints = np.where(r > tol)[0] + + # these are (at least some of) the initial basis columns + basis = np.where(np.abs(x) > tol)[0] + + if len(nonzero_constraints) == 0 and len(basis) <= m: # already a BFS + c = np.zeros(n) + basis = _get_more_basis_columns(A, basis) + return A, b, c, basis, x, status + elif (len(nonzero_constraints) > m - len(basis) or + np.any(x < 0)): # can't get trivial BFS + c = np.zeros(n) + status = 6 + return A, b, c, basis, x, status + + # chooses existing columns appropriate for inclusion in initial basis + cols, rows = _select_singleton_columns(A, r) + + # find the rows we need to zero that we _can_ zero with column singletons + i_tofix = np.isin(rows, nonzero_constraints) + # these columns can't already be in the basis, though + # we are going to add them to the basis and change the corresponding x val + i_notinbasis = np.logical_not(np.isin(cols, basis)) + i_fix_without_aux = np.logical_and(i_tofix, i_notinbasis) + rows = rows[i_fix_without_aux] + cols = cols[i_fix_without_aux] + + # indices of the rows we can only zero with auxiliary variable + # these rows will get a one in each auxiliary column + arows = nonzero_constraints[np.logical_not( + np.isin(nonzero_constraints, rows))] + n_aux = len(arows) + acols = n + np.arange(n_aux) # indices of auxiliary columns + + basis_ng = np.concatenate((cols, acols)) # basis columns not from guess + basis_ng_rows = np.concatenate((rows, arows)) # rows we need to zero + + # add auxiliary singleton columns + A = np.hstack((A, np.zeros((m, n_aux)))) + A[arows, acols] = 1 + + # generate initial BFS + x = np.concatenate((x, np.zeros(n_aux))) + x[basis_ng] = r[basis_ng_rows]/A[basis_ng_rows, basis_ng] + + # generate costs to minimize infeasibility + c = np.zeros(n_aux + n) + c[acols] = 1 + + # basis columns correspond with nonzeros in guess, those with column + # singletons we used to zero remaining constraints, and any additional + # columns to get a full set (m columns) + basis = np.concatenate((basis, basis_ng)) + basis = _get_more_basis_columns(A, basis) # add columns as needed + + return A, b, c, basis, x, status + + +def _select_singleton_columns(A, b): + """ + Finds singleton columns for which the singleton entry is of the same sign + as the right-hand side; these columns are eligible for inclusion in an + initial basis. Determines the rows in which the singleton entries are + located. For each of these rows, returns the indices of the one singleton + column and its corresponding row. + """ + # find indices of all singleton columns and corresponding row indices + column_indices = np.nonzero(np.sum(np.abs(A) != 0, axis=0) == 1)[0] + columns = A[:, column_indices] # array of singleton columns + row_indices = np.zeros(len(column_indices), dtype=int) + nonzero_rows, nonzero_columns = np.nonzero(columns) + row_indices[nonzero_columns] = nonzero_rows # corresponding row indices + + # keep only singletons with entries that have same sign as RHS + # this is necessary because all elements of BFS must be non-negative + same_sign = A[row_indices, column_indices]*b[row_indices] >= 0 + column_indices = column_indices[same_sign][::-1] + row_indices = row_indices[same_sign][::-1] + # Reversing the order so that steps below select rightmost columns + # for initial basis, which will tend to be slack variables. (If the + # guess corresponds with a basic feasible solution but a constraint + # is not satisfied with the corresponding slack variable zero, the slack + # variable must be basic.) + + # for each row, keep rightmost singleton column with an entry in that row + unique_row_indices, first_columns = np.unique(row_indices, + return_index=True) + return column_indices[first_columns], unique_row_indices + + +def _find_nonzero_rows(A, tol): + """ + Returns logical array indicating the locations of rows with at least + one nonzero element. + """ + return np.any(np.abs(A) > tol, axis=1) + + +def _select_enter_pivot(c_hat, bl, a, rule="bland", tol=1e-12): + """ + Selects a pivot to enter the basis. Currently Bland's rule - the smallest + index that has a negative reduced cost - is the default. + """ + if rule.lower() == "mrc": # index with minimum reduced cost + return a[~bl][np.argmin(c_hat)] + else: # smallest index w/ negative reduced cost + return a[~bl][c_hat < -tol][0] + + +def _display_iter(phase, iteration, slack, con, fun): + """ + Print indicators of optimization status to the console. + """ + header = True if not iteration % 20 else False + + if header: + print("Phase", + "Iteration", + "Minimum Slack ", + "Constraint Residual", + "Objective ") + + # := -tol): # all reduced costs positive -> terminate + break + + j = _select_enter_pivot(c_hat, bl, a, rule=pivot, tol=tol) + u = B.solve(A[:, j]) # similar to u = solve(B, A[:, j]) + + i = u > tol # if none of the u are positive, unbounded + if not np.any(i): + status = 3 + break + + th = xb[i]/u[i] + l = np.argmin(th) # implicitly selects smallest subscript + th_star = th[l] # step size + + x[b] = x[b] - th_star*u # take step + x[j] = th_star + B.update(ab[i][l], j) # modify basis + b = B.b # similar to b[ab[i][l]] = + + else: + # If the end of the for loop is reached (without a break statement), + # then another step has been taken, so the iteration counter should + # increment, info should be displayed, and callback should be called. + iteration += 1 + status = 1 + if disp or callback is not None: + _display_and_callback(phase_one_n, x, postsolve_args, status, + iteration, disp, callback) + + return x, b, status, iteration + + +def _linprog_rs(c, c0, A, b, x0, callback, postsolve_args, + maxiter=5000, tol=1e-12, disp=False, + maxupdate=10, mast=False, pivot="mrc", + **unknown_options): + """ + Solve the following linear programming problem via a two-phase + revised simplex algorithm.:: + + minimize: c @ x + + subject to: A @ x == b + 0 <= x < oo + + User-facing documentation is in _linprog_doc.py. + + Parameters + ---------- + c : 1-D array + Coefficients of the linear objective function to be minimized. + c0 : float + Constant term in objective function due to fixed (and eliminated) + variables. (Currently unused.) + A : 2-D array + 2-D array which, when matrix-multiplied by ``x``, gives the values of + the equality constraints at ``x``. + b : 1-D array + 1-D array of values representing the RHS of each equality constraint + (row) in ``A_eq``. + x0 : 1-D array, optional + Starting values of the independent variables, which will be refined by + the optimization algorithm. For the revised simplex method, these must + correspond with a basic feasible solution. + callback : callable, optional + If a callback function is provided, it will be called within each + iteration of the algorithm. The callback function must accept a single + `scipy.optimize.OptimizeResult` consisting of the following fields: + + x : 1-D array + Current solution vector. + fun : float + Current value of the objective function ``c @ x``. + success : bool + True only when an algorithm has completed successfully, + so this is always False as the callback function is called + only while the algorithm is still iterating. + slack : 1-D array + The values of the slack variables. Each slack variable + corresponds to an inequality constraint. If the slack is zero, + the corresponding constraint is active. + con : 1-D array + The (nominally zero) residuals of the equality constraints, + that is, ``b - A_eq @ x``. + phase : int + The phase of the algorithm being executed. + status : int + For revised simplex, this is always 0 because if a different + status is detected, the algorithm terminates. + nit : int + The number of iterations performed. + message : str + A string descriptor of the exit status of the optimization. + postsolve_args : tuple + Data needed by _postsolve to convert the solution to the standard-form + problem into the solution to the original problem. + + Options + ------- + maxiter : int + The maximum number of iterations to perform in either phase. + tol : float + The tolerance which determines when a solution is "close enough" to + zero in Phase 1 to be considered a basic feasible solution or close + enough to positive to serve as an optimal solution. + disp : bool + Set to ``True`` if indicators of optimization status are to be printed + to the console each iteration. + maxupdate : int + The maximum number of updates performed on the LU factorization. + After this many updates is reached, the basis matrix is factorized + from scratch. + mast : bool + Minimize Amortized Solve Time. If enabled, the average time to solve + a linear system using the basis factorization is measured. Typically, + the average solve time will decrease with each successive solve after + initial factorization, as factorization takes much more time than the + solve operation (and updates). Eventually, however, the updated + factorization becomes sufficiently complex that the average solve time + begins to increase. When this is detected, the basis is refactorized + from scratch. Enable this option to maximize speed at the risk of + nondeterministic behavior. Ignored if ``maxupdate`` is 0. + pivot : "mrc" or "bland" + Pivot rule: Minimum Reduced Cost (default) or Bland's rule. Choose + Bland's rule if iteration limit is reached and cycling is suspected. + unknown_options : dict + Optional arguments not used by this particular solver. If + `unknown_options` is non-empty a warning is issued listing all + unused options. + + Returns + ------- + x : 1-D array + Solution vector. + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Numerical difficulties encountered + 5 : No constraints; turn presolve on + 6 : Guess x0 cannot be converted to a basic feasible solution + + message : str + A string descriptor of the exit status of the optimization. + iteration : int + The number of iterations taken to solve the problem. + """ + + _check_unknown_options(unknown_options) + + messages = ["Optimization terminated successfully.", + "Iteration limit reached.", + "The problem appears infeasible, as the phase one auxiliary " + "problem terminated successfully with a residual of {0:.1e}, " + "greater than the tolerance {1} required for the solution to " + "be considered feasible. Consider increasing the tolerance to " + "be greater than {0:.1e}. If this tolerance is unacceptably " + "large, the problem is likely infeasible.", + "The problem is unbounded, as the simplex algorithm found " + "a basic feasible solution from which there is a direction " + "with negative reduced cost in which all decision variables " + "increase.", + "Numerical difficulties encountered; consider trying " + "method='interior-point'.", + "Problems with no constraints are trivially solved; please " + "turn presolve on.", + "The guess x0 cannot be converted to a basic feasible " + "solution. " + ] + + if A.size == 0: # address test_unbounded_below_no_presolve_corrected + return np.zeros(c.shape), 5, messages[5], 0 + + x, basis, A, b, residual, status, iteration = ( + _phase_one(A, b, x0, callback, postsolve_args, + maxiter, tol, disp, maxupdate, mast, pivot)) + + if status == 0: + x, basis, status, iteration = _phase_two(c, A, x, basis, callback, + postsolve_args, + maxiter, tol, disp, + maxupdate, mast, pivot, + iteration) + + return x, status, messages[status].format(residual, tol), iteration diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_simplex.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_simplex.py new file mode 100644 index 0000000000000000000000000000000000000000..c47806c9a595f756b9f86d268c5c146ea86c77c6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_simplex.py @@ -0,0 +1,663 @@ +"""Simplex method for linear programming + +The *simplex* method uses a traditional, full-tableau implementation of +Dantzig's simplex algorithm [1]_, [2]_ (*not* the Nelder-Mead simplex). +This algorithm is included for backwards compatibility and educational +purposes. + + .. versionadded:: 0.15.0 + +Warnings +-------- + +The simplex method may encounter numerical difficulties when pivot +values are close to the specified tolerance. If encountered try +remove any redundant constraints, change the pivot strategy to Bland's +rule or increase the tolerance value. + +Alternatively, more robust methods maybe be used. See +:ref:`'interior-point' ` and +:ref:`'revised simplex' `. + +References +---------- +.. [1] Dantzig, George B., Linear programming and extensions. Rand + Corporation Research Study Princeton Univ. Press, Princeton, NJ, + 1963 +.. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to + Mathematical Programming", McGraw-Hill, Chapter 4. +""" + +import numpy as np +from warnings import warn +from ._optimize import OptimizeResult, OptimizeWarning, _check_unknown_options +from ._linprog_util import _postsolve + + +def _pivot_col(T, tol=1e-9, bland=False): + """ + Given a linear programming simplex tableau, determine the column + of the variable to enter the basis. + + Parameters + ---------- + T : 2-D array + A 2-D array representing the simplex tableau, T, corresponding to the + linear programming problem. It should have the form: + + [[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]], + [A[1, 0], A[1, 1], ..., A[1, n_total], b[1]], + . + . + . + [A[m, 0], A[m, 1], ..., A[m, n_total], b[m]], + [c[0], c[1], ..., c[n_total], 0]] + + for a Phase 2 problem, or the form: + + [[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]], + [A[1, 0], A[1, 1], ..., A[1, n_total], b[1]], + . + . + . + [A[m, 0], A[m, 1], ..., A[m, n_total], b[m]], + [c[0], c[1], ..., c[n_total], 0], + [c'[0], c'[1], ..., c'[n_total], 0]] + + for a Phase 1 problem (a problem in which a basic feasible solution is + sought prior to maximizing the actual objective. ``T`` is modified in + place by ``_solve_simplex``. + tol : float + Elements in the objective row larger than -tol will not be considered + for pivoting. Nominally this value is zero, but numerical issues + cause a tolerance about zero to be necessary. + bland : bool + If True, use Bland's rule for selection of the column (select the + first column with a negative coefficient in the objective row, + regardless of magnitude). + + Returns + ------- + status: bool + True if a suitable pivot column was found, otherwise False. + A return of False indicates that the linear programming simplex + algorithm is complete. + col: int + The index of the column of the pivot element. + If status is False, col will be returned as nan. + """ + ma = np.ma.masked_where(T[-1, :-1] >= -tol, T[-1, :-1], copy=False) + if ma.count() == 0: + return False, np.nan + if bland: + # ma.mask is sometimes 0d + return True, np.nonzero(np.logical_not(np.atleast_1d(ma.mask)))[0][0] + return True, np.ma.nonzero(ma == ma.min())[0][0] + + +def _pivot_row(T, basis, pivcol, phase, tol=1e-9, bland=False): + """ + Given a linear programming simplex tableau, determine the row for the + pivot operation. + + Parameters + ---------- + T : 2-D array + A 2-D array representing the simplex tableau, T, corresponding to the + linear programming problem. It should have the form: + + [[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]], + [A[1, 0], A[1, 1], ..., A[1, n_total], b[1]], + . + . + . + [A[m, 0], A[m, 1], ..., A[m, n_total], b[m]], + [c[0], c[1], ..., c[n_total], 0]] + + for a Phase 2 problem, or the form: + + [[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]], + [A[1, 0], A[1, 1], ..., A[1, n_total], b[1]], + . + . + . + [A[m, 0], A[m, 1], ..., A[m, n_total], b[m]], + [c[0], c[1], ..., c[n_total], 0], + [c'[0], c'[1], ..., c'[n_total], 0]] + + for a Phase 1 problem (a Problem in which a basic feasible solution is + sought prior to maximizing the actual objective. ``T`` is modified in + place by ``_solve_simplex``. + basis : array + A list of the current basic variables. + pivcol : int + The index of the pivot column. + phase : int + The phase of the simplex algorithm (1 or 2). + tol : float + Elements in the pivot column smaller than tol will not be considered + for pivoting. Nominally this value is zero, but numerical issues + cause a tolerance about zero to be necessary. + bland : bool + If True, use Bland's rule for selection of the row (if more than one + row can be used, choose the one with the lowest variable index). + + Returns + ------- + status: bool + True if a suitable pivot row was found, otherwise False. A return + of False indicates that the linear programming problem is unbounded. + row: int + The index of the row of the pivot element. If status is False, row + will be returned as nan. + """ + if phase == 1: + k = 2 + else: + k = 1 + ma = np.ma.masked_where(T[:-k, pivcol] <= tol, T[:-k, pivcol], copy=False) + if ma.count() == 0: + return False, np.nan + mb = np.ma.masked_where(T[:-k, pivcol] <= tol, T[:-k, -1], copy=False) + q = mb / ma + min_rows = np.ma.nonzero(q == q.min())[0] + if bland: + return True, min_rows[np.argmin(np.take(basis, min_rows))] + return True, min_rows[0] + + +def _apply_pivot(T, basis, pivrow, pivcol, tol=1e-9): + """ + Pivot the simplex tableau inplace on the element given by (pivrow, pivol). + The entering variable corresponds to the column given by pivcol forcing + the variable basis[pivrow] to leave the basis. + + Parameters + ---------- + T : 2-D array + A 2-D array representing the simplex tableau, T, corresponding to the + linear programming problem. It should have the form: + + [[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]], + [A[1, 0], A[1, 1], ..., A[1, n_total], b[1]], + . + . + . + [A[m, 0], A[m, 1], ..., A[m, n_total], b[m]], + [c[0], c[1], ..., c[n_total], 0]] + + for a Phase 2 problem, or the form: + + [[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]], + [A[1, 0], A[1, 1], ..., A[1, n_total], b[1]], + . + . + . + [A[m, 0], A[m, 1], ..., A[m, n_total], b[m]], + [c[0], c[1], ..., c[n_total], 0], + [c'[0], c'[1], ..., c'[n_total], 0]] + + for a Phase 1 problem (a problem in which a basic feasible solution is + sought prior to maximizing the actual objective. ``T`` is modified in + place by ``_solve_simplex``. + basis : 1-D array + An array of the indices of the basic variables, such that basis[i] + contains the column corresponding to the basic variable for row i. + Basis is modified in place by _apply_pivot. + pivrow : int + Row index of the pivot. + pivcol : int + Column index of the pivot. + """ + basis[pivrow] = pivcol + pivval = T[pivrow, pivcol] + T[pivrow] = T[pivrow] / pivval + for irow in range(T.shape[0]): + if irow != pivrow: + T[irow] = T[irow] - T[pivrow] * T[irow, pivcol] + + # The selected pivot should never lead to a pivot value less than the tol. + if np.isclose(pivval, tol, atol=0, rtol=1e4): + message = ( + f"The pivot operation produces a pivot value of:{pivval: .1e}, " + "which is only slightly greater than the specified " + f"tolerance{tol: .1e}. This may lead to issues regarding the " + "numerical stability of the simplex method. " + "Removing redundant constraints, changing the pivot strategy " + "via Bland's rule or increasing the tolerance may " + "help reduce the issue.") + warn(message, OptimizeWarning, stacklevel=5) + + +def _solve_simplex(T, n, basis, callback, postsolve_args, + maxiter=1000, tol=1e-9, phase=2, bland=False, nit0=0, + ): + """ + Solve a linear programming problem in "standard form" using the Simplex + Method. Linear Programming is intended to solve the following problem form: + + Minimize:: + + c @ x + + Subject to:: + + A @ x == b + x >= 0 + + Parameters + ---------- + T : 2-D array + A 2-D array representing the simplex tableau, T, corresponding to the + linear programming problem. It should have the form: + + [[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]], + [A[1, 0], A[1, 1], ..., A[1, n_total], b[1]], + . + . + . + [A[m, 0], A[m, 1], ..., A[m, n_total], b[m]], + [c[0], c[1], ..., c[n_total], 0]] + + for a Phase 2 problem, or the form: + + [[A[0, 0], A[0, 1], ..., A[0, n_total], b[0]], + [A[1, 0], A[1, 1], ..., A[1, n_total], b[1]], + . + . + . + [A[m, 0], A[m, 1], ..., A[m, n_total], b[m]], + [c[0], c[1], ..., c[n_total], 0], + [c'[0], c'[1], ..., c'[n_total], 0]] + + for a Phase 1 problem (a problem in which a basic feasible solution is + sought prior to maximizing the actual objective. ``T`` is modified in + place by ``_solve_simplex``. + n : int + The number of true variables in the problem. + basis : 1-D array + An array of the indices of the basic variables, such that basis[i] + contains the column corresponding to the basic variable for row i. + Basis is modified in place by _solve_simplex + callback : callable, optional + If a callback function is provided, it will be called within each + iteration of the algorithm. The callback must accept a + `scipy.optimize.OptimizeResult` consisting of the following fields: + + x : 1-D array + Current solution vector + fun : float + Current value of the objective function + success : bool + True only when a phase has completed successfully. This + will be False for most iterations. + slack : 1-D array + The values of the slack variables. Each slack variable + corresponds to an inequality constraint. If the slack is zero, + the corresponding constraint is active. + con : 1-D array + The (nominally zero) residuals of the equality constraints, + that is, ``b - A_eq @ x`` + phase : int + The phase of the optimization being executed. In phase 1 a basic + feasible solution is sought and the T has an additional row + representing an alternate objective function. + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + + nit : int + The number of iterations performed. + message : str + A string descriptor of the exit status of the optimization. + postsolve_args : tuple + Data needed by _postsolve to convert the solution to the standard-form + problem into the solution to the original problem. + maxiter : int + The maximum number of iterations to perform before aborting the + optimization. + tol : float + The tolerance which determines when a solution is "close enough" to + zero in Phase 1 to be considered a basic feasible solution or close + enough to positive to serve as an optimal solution. + phase : int + The phase of the optimization being executed. In phase 1 a basic + feasible solution is sought and the T has an additional row + representing an alternate objective function. + bland : bool + If True, choose pivots using Bland's rule [3]_. In problems which + fail to converge due to cycling, using Bland's rule can provide + convergence at the expense of a less optimal path about the simplex. + nit0 : int + The initial iteration number used to keep an accurate iteration total + in a two-phase problem. + + Returns + ------- + nit : int + The number of iterations. Used to keep an accurate iteration total + in the two-phase problem. + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + + """ + nit = nit0 + status = 0 + message = '' + complete = False + + if phase == 1: + m = T.shape[1]-2 + elif phase == 2: + m = T.shape[1]-1 + else: + raise ValueError("Argument 'phase' to _solve_simplex must be 1 or 2") + + if phase == 2: + # Check if any artificial variables are still in the basis. + # If yes, check if any coefficients from this row and a column + # corresponding to one of the non-artificial variable is non-zero. + # If found, pivot at this term. If not, start phase 2. + # Do this for all artificial variables in the basis. + # Ref: "An Introduction to Linear Programming and Game Theory" + # by Paul R. Thie, Gerard E. Keough, 3rd Ed, + # Chapter 3.7 Redundant Systems (pag 102) + for pivrow in [row for row in range(basis.size) + if basis[row] > T.shape[1] - 2]: + non_zero_row = [col for col in range(T.shape[1] - 1) + if abs(T[pivrow, col]) > tol] + if len(non_zero_row) > 0: + pivcol = non_zero_row[0] + _apply_pivot(T, basis, pivrow, pivcol, tol) + nit += 1 + + if len(basis[:m]) == 0: + solution = np.empty(T.shape[1] - 1, dtype=np.float64) + else: + solution = np.empty(max(T.shape[1] - 1, max(basis[:m]) + 1), + dtype=np.float64) + + while not complete: + # Find the pivot column + pivcol_found, pivcol = _pivot_col(T, tol, bland) + if not pivcol_found: + pivcol = np.nan + pivrow = np.nan + status = 0 + complete = True + else: + # Find the pivot row + pivrow_found, pivrow = _pivot_row(T, basis, pivcol, phase, tol, bland) + if not pivrow_found: + status = 3 + complete = True + + if callback is not None: + solution[:] = 0 + solution[basis[:n]] = T[:n, -1] + x = solution[:m] + x, fun, slack, con = _postsolve( + x, postsolve_args + ) + res = OptimizeResult({ + 'x': x, + 'fun': fun, + 'slack': slack, + 'con': con, + 'status': status, + 'message': message, + 'nit': nit, + 'success': status == 0 and complete, + 'phase': phase, + 'complete': complete, + }) + callback(res) + + if not complete: + if nit >= maxiter: + # Iteration limit exceeded + status = 1 + complete = True + else: + _apply_pivot(T, basis, pivrow, pivcol, tol) + nit += 1 + return nit, status + + +def _linprog_simplex(c, c0, A, b, callback, postsolve_args, + maxiter=1000, tol=1e-9, disp=False, bland=False, + **unknown_options): + """ + Minimize a linear objective function subject to linear equality and + non-negativity constraints using the two phase simplex method. + Linear programming is intended to solve problems of the following form: + + Minimize:: + + c @ x + + Subject to:: + + A @ x == b + x >= 0 + + User-facing documentation is in _linprog_doc.py. + + Parameters + ---------- + c : 1-D array + Coefficients of the linear objective function to be minimized. + c0 : float + Constant term in objective function due to fixed (and eliminated) + variables. (Purely for display.) + A : 2-D array + 2-D array such that ``A @ x``, gives the values of the equality + constraints at ``x``. + b : 1-D array + 1-D array of values representing the right hand side of each equality + constraint (row) in ``A``. + callback : callable, optional + If a callback function is provided, it will be called within each + iteration of the algorithm. The callback function must accept a single + `scipy.optimize.OptimizeResult` consisting of the following fields: + + x : 1-D array + Current solution vector + fun : float + Current value of the objective function + success : bool + True when an algorithm has completed successfully. + slack : 1-D array + The values of the slack variables. Each slack variable + corresponds to an inequality constraint. If the slack is zero, + the corresponding constraint is active. + con : 1-D array + The (nominally zero) residuals of the equality constraints, + that is, ``b - A_eq @ x`` + phase : int + The phase of the algorithm being executed. + status : int + An integer representing the status of the optimization:: + + 0 : Algorithm proceeding nominally + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + nit : int + The number of iterations performed. + message : str + A string descriptor of the exit status of the optimization. + postsolve_args : tuple + Data needed by _postsolve to convert the solution to the standard-form + problem into the solution to the original problem. + + Options + ------- + maxiter : int + The maximum number of iterations to perform. + disp : bool + If True, print exit status message to sys.stdout + tol : float + The tolerance which determines when a solution is "close enough" to + zero in Phase 1 to be considered a basic feasible solution or close + enough to positive to serve as an optimal solution. + bland : bool + If True, use Bland's anti-cycling rule [3]_ to choose pivots to + prevent cycling. If False, choose pivots which should lead to a + converged solution more quickly. The latter method is subject to + cycling (non-convergence) in rare instances. + unknown_options : dict + Optional arguments not used by this particular solver. If + `unknown_options` is non-empty a warning is issued listing all + unused options. + + Returns + ------- + x : 1-D array + Solution vector. + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + + message : str + A string descriptor of the exit status of the optimization. + iteration : int + The number of iterations taken to solve the problem. + + References + ---------- + .. [1] Dantzig, George B., Linear programming and extensions. Rand + Corporation Research Study Princeton Univ. Press, Princeton, NJ, + 1963 + .. [2] Hillier, S.H. and Lieberman, G.J. (1995), "Introduction to + Mathematical Programming", McGraw-Hill, Chapter 4. + .. [3] Bland, Robert G. New finite pivoting rules for the simplex method. + Mathematics of Operations Research (2), 1977: pp. 103-107. + + + Notes + ----- + The expected problem formulation differs between the top level ``linprog`` + module and the method specific solvers. The method specific solvers expect a + problem in standard form: + + Minimize:: + + c @ x + + Subject to:: + + A @ x == b + x >= 0 + + Whereas the top level ``linprog`` module expects a problem of form: + + Minimize:: + + c @ x + + Subject to:: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + where ``lb = 0`` and ``ub = None`` unless set in ``bounds``. + + The original problem contains equality, upper-bound and variable constraints + whereas the method specific solver requires equality constraints and + variable non-negativity. + + ``linprog`` module converts the original problem to standard form by + converting the simple bounds to upper bound constraints, introducing + non-negative slack variables for inequality constraints, and expressing + unbounded variables as the difference between two non-negative variables. + """ + _check_unknown_options(unknown_options) + + status = 0 + messages = {0: "Optimization terminated successfully.", + 1: "Iteration limit reached.", + 2: "Optimization failed. Unable to find a feasible" + " starting point.", + 3: "Optimization failed. The problem appears to be unbounded.", + 4: "Optimization failed. Singular matrix encountered."} + + n, m = A.shape + + # All constraints must have b >= 0. + is_negative_constraint = np.less(b, 0) + A[is_negative_constraint] *= -1 + b[is_negative_constraint] *= -1 + + # As all constraints are equality constraints the artificial variables + # will also be basic variables. + av = np.arange(n) + m + basis = av.copy() + + # Format the phase one tableau by adding artificial variables and stacking + # the constraints, the objective row and pseudo-objective row. + row_constraints = np.hstack((A, np.eye(n), b[:, np.newaxis])) + row_objective = np.hstack((c, np.zeros(n), c0)) + row_pseudo_objective = -row_constraints.sum(axis=0) + row_pseudo_objective[av] = 0 + T = np.vstack((row_constraints, row_objective, row_pseudo_objective)) + + nit1, status = _solve_simplex(T, n, basis, callback=callback, + postsolve_args=postsolve_args, + maxiter=maxiter, tol=tol, phase=1, + bland=bland + ) + # if pseudo objective is zero, remove the last row from the tableau and + # proceed to phase 2 + nit2 = nit1 + if abs(T[-1, -1]) < tol: + # Remove the pseudo-objective row from the tableau + T = T[:-1, :] + # Remove the artificial variable columns from the tableau + T = np.delete(T, av, 1) + else: + # Failure to find a feasible starting point + status = 2 + messages[status] = ( + "Phase 1 of the simplex method failed to find a feasible " + "solution. The pseudo-objective function evaluates to " + f"{abs(T[-1, -1]):.1e} " + f"which exceeds the required tolerance of {tol} for a solution to be " + "considered 'close enough' to zero to be a basic solution. " + "Consider increasing the tolerance to be greater than " + f"{abs(T[-1, -1]):.1e}. " + "If this tolerance is unacceptably large the problem may be " + "infeasible." + ) + + if status == 0: + # Phase 2 + nit2, status = _solve_simplex(T, n, basis, callback=callback, + postsolve_args=postsolve_args, + maxiter=maxiter, tol=tol, phase=2, + bland=bland, nit0=nit1 + ) + + solution = np.zeros(n + m) + solution[basis[:n]] = T[:n, -1] + x = solution[:m] + + return x, status, messages[status], int(nit2) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_util.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_util.py new file mode 100644 index 0000000000000000000000000000000000000000..405ff0feee7116712a6b0897e2f956a6e8a1760f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_linprog_util.py @@ -0,0 +1,1523 @@ +""" +Method agnostic utility functions for linear programming +""" + +import numpy as np +import scipy.sparse as sps +from warnings import warn +from ._optimize import OptimizeWarning +from scipy.optimize._remove_redundancy import ( + _remove_redundancy_svd, _remove_redundancy_pivot_sparse, + _remove_redundancy_pivot_dense, _remove_redundancy_id + ) +from collections import namedtuple + +_LPProblem = namedtuple('_LPProblem', + 'c A_ub b_ub A_eq b_eq bounds x0 integrality') +_LPProblem.__new__.__defaults__ = (None,) * 7 # make c the only required arg +_LPProblem.__doc__ = \ + """ Represents a linear-programming problem. + + Attributes + ---------- + c : 1D array + The coefficients of the linear objective function to be minimized. + A_ub : 2D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : various valid formats, optional + The bounds of ``x``, as ``min`` and ``max`` pairs. + If bounds are specified for all N variables separately, valid formats + are: + * a 2D array (N x 2); + * a sequence of N sequences, each with 2 values. + If all variables have the same bounds, the bounds can be specified as + a 1-D or 2-D array or sequence with 2 scalar values. + If all variables have a lower bound of 0 and no upper bound, the bounds + parameter can be omitted (or given as None). + Absent lower and/or upper bounds can be specified as -numpy.inf (no + lower bound), numpy.inf (no upper bound) or None (both). + x0 : 1D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + 'revised simplex' method, and can only be used if `x0` represents a + basic feasible solution. + integrality : 1-D array or int, optional + Indicates the type of integrality constraint on each decision variable. + + ``0`` : Continuous variable; no integrality constraint. + + ``1`` : Integer variable; decision variable must be an integer + within `bounds`. + + ``2`` : Semi-continuous variable; decision variable must be within + `bounds` or take value ``0``. + + ``3`` : Semi-integer variable; decision variable must be an integer + within `bounds` or take value ``0``. + + By default, all variables are continuous. + + For mixed integrality constraints, supply an array of shape `c.shape`. + To infer a constraint on each decision variable from shorter inputs, + the argument will be broadcast to `c.shape` using `np.broadcast_to`. + + This argument is currently used only by the ``'highs'`` method and + ignored otherwise. + + Notes + ----- + This namedtuple supports 2 ways of initialization: + >>> lp1 = _LPProblem(c=[-1, 4], A_ub=[[-3, 1], [1, 2]], b_ub=[6, 4]) + >>> lp2 = _LPProblem([-1, 4], [[-3, 1], [1, 2]], [6, 4]) + + Note that only ``c`` is a required argument here, whereas all other arguments + ``A_ub``, ``b_ub``, ``A_eq``, ``b_eq``, ``bounds``, ``x0`` are optional with + default values of None. + For example, ``A_eq`` and ``b_eq`` can be set without ``A_ub`` or ``b_ub``: + >>> lp3 = _LPProblem(c=[-1, 4], A_eq=[[2, 1]], b_eq=[10]) + """ + + +def _check_sparse_inputs(options, meth, A_ub, A_eq): + """ + Check the provided ``A_ub`` and ``A_eq`` matrices conform to the specified + optional sparsity variables. + + Parameters + ---------- + A_ub : 2-D array, optional + 2-D array such that ``A_ub @ x`` gives the values of the upper-bound + inequality constraints at ``x``. + A_eq : 2-D array, optional + 2-D array such that ``A_eq @ x`` gives the values of the equality + constraints at ``x``. + options : dict + A dictionary of solver options. All methods accept the following + generic options: + + maxiter : int + Maximum number of iterations to perform. + disp : bool + Set to True to print convergence messages. + + For method-specific options, see :func:`show_options('linprog')`. + method : str, optional + The algorithm used to solve the standard form problem. + + Returns + ------- + A_ub : 2-D array, optional + 2-D array such that ``A_ub @ x`` gives the values of the upper-bound + inequality constraints at ``x``. + A_eq : 2-D array, optional + 2-D array such that ``A_eq @ x`` gives the values of the equality + constraints at ``x``. + options : dict + A dictionary of solver options. All methods accept the following + generic options: + + maxiter : int + Maximum number of iterations to perform. + disp : bool + Set to True to print convergence messages. + + For method-specific options, see :func:`show_options('linprog')`. + """ + # This is an undocumented option for unit testing sparse presolve + _sparse_presolve = options.pop('_sparse_presolve', False) + if _sparse_presolve and A_eq is not None: + A_eq = sps.coo_matrix(A_eq) + if _sparse_presolve and A_ub is not None: + A_ub = sps.coo_matrix(A_ub) + + sparse_constraint = sps.issparse(A_eq) or sps.issparse(A_ub) + + preferred_methods = {"highs", "highs-ds", "highs-ipm"} + dense_methods = {"simplex", "revised simplex"} + if meth in dense_methods and sparse_constraint: + raise ValueError(f"Method '{meth}' does not support sparse " + "constraint matrices. Please consider using one of " + f"{preferred_methods}.") + + sparse = options.get('sparse', False) + if not sparse and sparse_constraint and meth == 'interior-point': + options['sparse'] = True + warn("Sparse constraint matrix detected; setting 'sparse':True.", + OptimizeWarning, stacklevel=4) + return options, A_ub, A_eq + + +def _format_A_constraints(A, n_x, sparse_lhs=False): + """Format the left hand side of the constraints to a 2-D array + + Parameters + ---------- + A : 2-D array + 2-D array such that ``A @ x`` gives the values of the upper-bound + (in)equality constraints at ``x``. + n_x : int + The number of variables in the linear programming problem. + sparse_lhs : bool + Whether either of `A_ub` or `A_eq` are sparse. If true return a + coo_matrix instead of a numpy array. + + Returns + ------- + np.ndarray or sparse.coo_matrix + 2-D array such that ``A @ x`` gives the values of the upper-bound + (in)equality constraints at ``x``. + + """ + if sparse_lhs: + return sps.coo_matrix( + (0, n_x) if A is None else A, dtype=float, copy=True + ) + elif A is None: + return np.zeros((0, n_x), dtype=float) + else: + return np.array(A, dtype=float, copy=True) + + +def _format_b_constraints(b): + """Format the upper bounds of the constraints to a 1-D array + + Parameters + ---------- + b : 1-D array + 1-D array of values representing the upper-bound of each (in)equality + constraint (row) in ``A``. + + Returns + ------- + 1-D np.array + 1-D array of values representing the upper-bound of each (in)equality + constraint (row) in ``A``. + + """ + if b is None: + return np.array([], dtype=float) + b = np.array(b, dtype=float, copy=True).squeeze() + return b if b.size != 1 else b.reshape(-1) + + +def _clean_inputs(lp): + """ + Given user inputs for a linear programming problem, return the + objective vector, upper bound constraints, equality constraints, + and simple bounds in a preferred format. + + Parameters + ---------- + lp : A `scipy.optimize._linprog_util._LPProblem` consisting of the following fields: + + c : 1D array + The coefficients of the linear objective function to be minimized. + A_ub : 2D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : various valid formats, optional + The bounds of ``x``, as ``min`` and ``max`` pairs. + If bounds are specified for all N variables separately, valid formats are: + * a 2D array (2 x N or N x 2); + * a sequence of N sequences, each with 2 values. + If all variables have the same bounds, a single pair of values can + be specified. Valid formats are: + * a sequence with 2 scalar values; + * a sequence with a single element containing 2 scalar values. + If all variables have a lower bound of 0 and no upper bound, the bounds + parameter can be omitted (or given as None). + x0 : 1D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + 'revised simplex' method, and can only be used if `x0` represents a + basic feasible solution. + + Returns + ------- + lp : A `scipy.optimize._linprog_util._LPProblem` consisting of the following fields: + + c : 1D array + The coefficients of the linear objective function to be minimized. + A_ub : 2D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : 2D array + The bounds of ``x``, as ``min`` and ``max`` pairs, one for each of the N + elements of ``x``. The N x 2 array contains lower bounds in the first + column and upper bounds in the 2nd. Unbounded variables have lower + bound -np.inf and/or upper bound np.inf. + x0 : 1D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + 'revised simplex' method, and can only be used if `x0` represents a + basic feasible solution. + + """ + c, A_ub, b_ub, A_eq, b_eq, bounds, x0, integrality = lp + + if c is None: + raise TypeError + + try: + c = np.array(c, dtype=np.float64, copy=True).squeeze() + except ValueError as e: + raise TypeError( + "Invalid input for linprog: c must be a 1-D array of numerical " + "coefficients") from e + else: + # If c is a single value, convert it to a 1-D array. + if c.size == 1: + c = c.reshape(-1) + + n_x = len(c) + if n_x == 0 or len(c.shape) != 1: + raise ValueError( + "Invalid input for linprog: c must be a 1-D array and must " + "not have more than one non-singleton dimension") + if not np.isfinite(c).all(): + raise ValueError( + "Invalid input for linprog: c must not contain values " + "inf, nan, or None") + + sparse_lhs = sps.issparse(A_eq) or sps.issparse(A_ub) + try: + A_ub = _format_A_constraints(A_ub, n_x, sparse_lhs=sparse_lhs) + except ValueError as e: + raise TypeError( + "Invalid input for linprog: A_ub must be a 2-D array " + "of numerical values") from e + else: + n_ub = A_ub.shape[0] + if len(A_ub.shape) != 2 or A_ub.shape[1] != n_x: + raise ValueError( + "Invalid input for linprog: A_ub must have exactly two " + "dimensions, and the number of columns in A_ub must be " + "equal to the size of c") + if (sps.issparse(A_ub) and not np.isfinite(A_ub.data).all() + or not sps.issparse(A_ub) and not np.isfinite(A_ub).all()): + raise ValueError( + "Invalid input for linprog: A_ub must not contain values " + "inf, nan, or None") + + try: + b_ub = _format_b_constraints(b_ub) + except ValueError as e: + raise TypeError( + "Invalid input for linprog: b_ub must be a 1-D array of " + "numerical values, each representing the upper bound of an " + "inequality constraint (row) in A_ub") from e + else: + if b_ub.shape != (n_ub,): + raise ValueError( + "Invalid input for linprog: b_ub must be a 1-D array; b_ub " + "must not have more than one non-singleton dimension and " + "the number of rows in A_ub must equal the number of values " + "in b_ub") + if not np.isfinite(b_ub).all(): + raise ValueError( + "Invalid input for linprog: b_ub must not contain values " + "inf, nan, or None") + + try: + A_eq = _format_A_constraints(A_eq, n_x, sparse_lhs=sparse_lhs) + except ValueError as e: + raise TypeError( + "Invalid input for linprog: A_eq must be a 2-D array " + "of numerical values") from e + else: + n_eq = A_eq.shape[0] + if len(A_eq.shape) != 2 or A_eq.shape[1] != n_x: + raise ValueError( + "Invalid input for linprog: A_eq must have exactly two " + "dimensions, and the number of columns in A_eq must be " + "equal to the size of c") + + if (sps.issparse(A_eq) and not np.isfinite(A_eq.data).all() + or not sps.issparse(A_eq) and not np.isfinite(A_eq).all()): + raise ValueError( + "Invalid input for linprog: A_eq must not contain values " + "inf, nan, or None") + + try: + b_eq = _format_b_constraints(b_eq) + except ValueError as e: + raise TypeError( + "Invalid input for linprog: b_eq must be a dense, 1-D array of " + "numerical values, each representing the right hand side of an " + "equality constraint (row) in A_eq") from e + else: + if b_eq.shape != (n_eq,): + raise ValueError( + "Invalid input for linprog: b_eq must be a 1-D array; b_eq " + "must not have more than one non-singleton dimension and " + "the number of rows in A_eq must equal the number of values " + "in b_eq") + if not np.isfinite(b_eq).all(): + raise ValueError( + "Invalid input for linprog: b_eq must not contain values " + "inf, nan, or None") + + # x0 gives a (optional) starting solution to the solver. If x0 is None, + # skip the checks. Initial solution will be generated automatically. + if x0 is not None: + try: + x0 = np.array(x0, dtype=float, copy=True).squeeze() + except ValueError as e: + raise TypeError( + "Invalid input for linprog: x0 must be a 1-D array of " + "numerical coefficients") from e + if x0.ndim == 0: + x0 = x0.reshape(-1) + if len(x0) == 0 or x0.ndim != 1: + raise ValueError( + "Invalid input for linprog: x0 should be a 1-D array; it " + "must not have more than one non-singleton dimension") + if not x0.size == c.size: + raise ValueError( + "Invalid input for linprog: x0 and c should contain the " + "same number of elements") + if not np.isfinite(x0).all(): + raise ValueError( + "Invalid input for linprog: x0 must not contain values " + "inf, nan, or None") + + # Bounds can be one of these formats: + # (1) a 2-D array or sequence, with shape N x 2 + # (2) a 1-D or 2-D sequence or array with 2 scalars + # (3) None (or an empty sequence or array) + # Unspecified bounds can be represented by None or (-)np.inf. + # All formats are converted into a N x 2 np.array with (-)np.inf where + # bounds are unspecified. + + # Prepare clean bounds array + bounds_clean = np.zeros((n_x, 2), dtype=float) + + # Convert to a numpy array. + # np.array(..,dtype=float) raises an error if dimensions are inconsistent + # or if there are invalid data types in bounds. Just add a linprog prefix + # to the error and re-raise. + # Creating at least a 2-D array simplifies the cases to distinguish below. + if bounds is None or np.array_equal(bounds, []) or np.array_equal(bounds, [[]]): + bounds = (0, np.inf) + try: + bounds_conv = np.atleast_2d(np.array(bounds, dtype=float)) + except ValueError as e: + raise ValueError( + "Invalid input for linprog: unable to interpret bounds, " + "check values and dimensions: " + e.args[0]) from e + except TypeError as e: + raise TypeError( + "Invalid input for linprog: unable to interpret bounds, " + "check values and dimensions: " + e.args[0]) from e + + # Check bounds options + bsh = bounds_conv.shape + if len(bsh) > 2: + # Do not try to handle multidimensional bounds input + raise ValueError( + "Invalid input for linprog: provide a 2-D array for bounds, " + f"not a {len(bsh):d}-D array.") + elif np.all(bsh == (n_x, 2)): + # Regular N x 2 array + bounds_clean = bounds_conv + elif (np.all(bsh == (2, 1)) or np.all(bsh == (1, 2))): + # 2 values: interpret as overall lower and upper bound + bounds_flat = bounds_conv.flatten() + bounds_clean[:, 0] = bounds_flat[0] + bounds_clean[:, 1] = bounds_flat[1] + elif np.all(bsh == (2, n_x)): + # Reject a 2 x N array + raise ValueError( + f"Invalid input for linprog: provide a {n_x:d} x 2 array for bounds, " + f"not a 2 x {n_x:d} array.") + else: + raise ValueError( + "Invalid input for linprog: unable to interpret bounds with this " + f"dimension tuple: {bsh}.") + + # The process above creates nan-s where the input specified None + # Convert the nan-s in the 1st column to -np.inf and in the 2nd column + # to np.inf + i_none = np.isnan(bounds_clean[:, 0]) + bounds_clean[i_none, 0] = -np.inf + i_none = np.isnan(bounds_clean[:, 1]) + bounds_clean[i_none, 1] = np.inf + + return _LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds_clean, x0, integrality) + + +def _presolve(lp, rr, rr_method, tol=1e-9): + """ + Given inputs for a linear programming problem in preferred format, + presolve the problem: identify trivial infeasibilities, redundancies, + and unboundedness, tighten bounds where possible, and eliminate fixed + variables. + + Parameters + ---------- + lp : A `scipy.optimize._linprog_util._LPProblem` consisting of the following fields: + + c : 1D array + The coefficients of the linear objective function to be minimized. + A_ub : 2D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : 2D array + The bounds of ``x``, as ``min`` and ``max`` pairs, one for each of the N + elements of ``x``. The N x 2 array contains lower bounds in the first + column and upper bounds in the 2nd. Unbounded variables have lower + bound -np.inf and/or upper bound np.inf. + x0 : 1D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + 'revised simplex' method, and can only be used if `x0` represents a + basic feasible solution. + + rr : bool + If ``True`` attempts to eliminate any redundant rows in ``A_eq``. + Set False if ``A_eq`` is known to be of full row rank, or if you are + looking for a potential speedup (at the expense of reliability). + rr_method : string + Method used to identify and remove redundant rows from the + equality constraint matrix after presolve. + tol : float + The tolerance which determines when a solution is "close enough" to + zero in Phase 1 to be considered a basic feasible solution or close + enough to positive to serve as an optimal solution. + + Returns + ------- + lp : A `scipy.optimize._linprog_util._LPProblem` consisting of the following fields: + + c : 1D array + The coefficients of the linear objective function to be minimized. + A_ub : 2D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : 2D array + The bounds of ``x``, as ``min`` and ``max`` pairs, possibly tightened. + x0 : 1D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + 'revised simplex' method, and can only be used if `x0` represents a + basic feasible solution. + + c0 : 1D array + Constant term in objective function due to fixed (and eliminated) + variables. + x : 1D array + Solution vector (when the solution is trivial and can be determined + in presolve) + revstack: list of functions + the functions in the list reverse the operations of _presolve() + the function signature is x_org = f(x_mod), where x_mod is the result + of a presolve step and x_org the value at the start of the step + (currently, the revstack contains only one function) + complete: bool + Whether the solution is complete (solved or determined to be infeasible + or unbounded in presolve) + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + + message : str + A string descriptor of the exit status of the optimization. + + References + ---------- + .. [5] Andersen, Erling D. "Finding all linearly dependent rows in + large-scale linear programming." Optimization Methods and Software + 6.3 (1995): 219-227. + .. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear + programming." Mathematical Programming 71.2 (1995): 221-245. + + """ + # ideas from Reference [5] by Andersen and Andersen + # however, unlike the reference, this is performed before converting + # problem to standard form + # There are a few advantages: + # * artificial variables have not been added, so matrices are smaller + # * bounds have not been converted to constraints yet. (It is better to + # do that after presolve because presolve may adjust the simple bounds.) + # There are many improvements that can be made, namely: + # * implement remaining checks from [5] + # * loop presolve until no additional changes are made + # * implement additional efficiency improvements in redundancy removal [2] + + c, A_ub, b_ub, A_eq, b_eq, bounds, x0, _ = lp + + revstack = [] # record of variables eliminated from problem + # constant term in cost function may be added if variables are eliminated + c0 = 0 + complete = False # complete is True if detected infeasible/unbounded + x = np.zeros(c.shape) # this is solution vector if completed in presolve + + status = 0 # all OK unless determined otherwise + message = "" + + # Lower and upper bounds. Copy to prevent feedback. + lb = bounds[:, 0].copy() + ub = bounds[:, 1].copy() + + m_eq, n = A_eq.shape + m_ub, n = A_ub.shape + + if (rr_method is not None + and rr_method.lower() not in {"svd", "pivot", "id"}): + message = ("'" + str(rr_method) + "' is not a valid option " + "for redundancy removal. Valid options are 'SVD', " + "'pivot', and 'ID'.") + raise ValueError(message) + + if sps.issparse(A_eq): + A_eq = A_eq.tocsr() + A_ub = A_ub.tocsr() + + def where(A): + return A.nonzero() + + vstack = sps.vstack + else: + where = np.where + vstack = np.vstack + + # upper bounds > lower bounds + if np.any(ub < lb) or np.any(lb == np.inf) or np.any(ub == -np.inf): + status = 2 + message = ("The problem is (trivially) infeasible since one " + "or more upper bounds are smaller than the corresponding " + "lower bounds, a lower bound is np.inf or an upper bound " + "is -np.inf.") + complete = True + return (_LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0), + c0, x, revstack, complete, status, message) + + # zero row in equality constraints + zero_row = np.array(np.sum(A_eq != 0, axis=1) == 0).flatten() + if np.any(zero_row): + if np.any( + np.logical_and( + zero_row, + np.abs(b_eq) > tol)): # test_zero_row_1 + # infeasible if RHS is not zero + status = 2 + message = ("The problem is (trivially) infeasible due to a row " + "of zeros in the equality constraint matrix with a " + "nonzero corresponding constraint value.") + complete = True + return (_LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0), + c0, x, revstack, complete, status, message) + else: # test_zero_row_2 + # if RHS is zero, we can eliminate this equation entirely + A_eq = A_eq[np.logical_not(zero_row), :] + b_eq = b_eq[np.logical_not(zero_row)] + + # zero row in inequality constraints + zero_row = np.array(np.sum(A_ub != 0, axis=1) == 0).flatten() + if np.any(zero_row): + if np.any(np.logical_and(zero_row, b_ub < -tol)): # test_zero_row_1 + # infeasible if RHS is less than zero (because LHS is zero) + status = 2 + message = ("The problem is (trivially) infeasible due to a row " + "of zeros in the equality constraint matrix with a " + "nonzero corresponding constraint value.") + complete = True + return (_LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0), + c0, x, revstack, complete, status, message) + else: # test_zero_row_2 + # if LHS is >= 0, we can eliminate this constraint entirely + A_ub = A_ub[np.logical_not(zero_row), :] + b_ub = b_ub[np.logical_not(zero_row)] + + # zero column in (both) constraints + # this indicates that a variable isn't constrained and can be removed + A = vstack((A_eq, A_ub)) + if A.shape[0] > 0: + zero_col = np.array(np.sum(A != 0, axis=0) == 0).flatten() + # variable will be at upper or lower bound, depending on objective + x[np.logical_and(zero_col, c < 0)] = ub[ + np.logical_and(zero_col, c < 0)] + x[np.logical_and(zero_col, c > 0)] = lb[ + np.logical_and(zero_col, c > 0)] + if np.any(np.isinf(x)): # if an unconstrained variable has no bound + status = 3 + message = ("If feasible, the problem is (trivially) unbounded " + "due to a zero column in the constraint matrices. If " + "you wish to check whether the problem is infeasible, " + "turn presolve off.") + complete = True + return (_LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0), + c0, x, revstack, complete, status, message) + # variables will equal upper/lower bounds will be removed later + lb[np.logical_and(zero_col, c < 0)] = ub[ + np.logical_and(zero_col, c < 0)] + ub[np.logical_and(zero_col, c > 0)] = lb[ + np.logical_and(zero_col, c > 0)] + + # row singleton in equality constraints + # this fixes a variable and removes the constraint + singleton_row = np.array(np.sum(A_eq != 0, axis=1) == 1).flatten() + rows = where(singleton_row)[0] + cols = where(A_eq[rows, :])[1] + if len(rows) > 0: + for row, col in zip(rows, cols): + val = b_eq[row] / A_eq[row, col] + if not lb[col] - tol <= val <= ub[col] + tol: + # infeasible if fixed value is not within bounds + status = 2 + message = ("The problem is (trivially) infeasible because a " + "singleton row in the equality constraints is " + "inconsistent with the bounds.") + complete = True + return (_LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0), + c0, x, revstack, complete, status, message) + else: + # sets upper and lower bounds at that fixed value - variable + # will be removed later + lb[col] = val + ub[col] = val + A_eq = A_eq[np.logical_not(singleton_row), :] + b_eq = b_eq[np.logical_not(singleton_row)] + + # row singleton in inequality constraints + # this indicates a simple bound and the constraint can be removed + # simple bounds may be adjusted here + # After all of the simple bound information is combined here, get_Abc will + # turn the simple bounds into constraints + singleton_row = np.array(np.sum(A_ub != 0, axis=1) == 1).flatten() + cols = where(A_ub[singleton_row, :])[1] + rows = where(singleton_row)[0] + if len(rows) > 0: + for row, col in zip(rows, cols): + val = b_ub[row] / A_ub[row, col] + if A_ub[row, col] > 0: # upper bound + if val < lb[col] - tol: # infeasible + complete = True + elif val < ub[col]: # new upper bound + ub[col] = val + else: # lower bound + if val > ub[col] + tol: # infeasible + complete = True + elif val > lb[col]: # new lower bound + lb[col] = val + if complete: + status = 2 + message = ("The problem is (trivially) infeasible because a " + "singleton row in the upper bound constraints is " + "inconsistent with the bounds.") + return (_LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0), + c0, x, revstack, complete, status, message) + A_ub = A_ub[np.logical_not(singleton_row), :] + b_ub = b_ub[np.logical_not(singleton_row)] + + # identical bounds indicate that variable can be removed + i_f = np.abs(lb - ub) < tol # indices of "fixed" variables + i_nf = np.logical_not(i_f) # indices of "not fixed" variables + + # test_bounds_equal_but_infeasible + if np.all(i_f): # if bounds define solution, check for consistency + residual = b_eq - A_eq.dot(lb) + slack = b_ub - A_ub.dot(lb) + if ((A_ub.size > 0 and np.any(slack < 0)) or + (A_eq.size > 0 and not np.allclose(residual, 0))): + status = 2 + message = ("The problem is (trivially) infeasible because the " + "bounds fix all variables to values inconsistent with " + "the constraints") + complete = True + return (_LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0), + c0, x, revstack, complete, status, message) + + ub_mod = ub + lb_mod = lb + if np.any(i_f): + c0 += c[i_f].dot(lb[i_f]) + b_eq = b_eq - A_eq[:, i_f].dot(lb[i_f]) + b_ub = b_ub - A_ub[:, i_f].dot(lb[i_f]) + c = c[i_nf] + x_undo = lb[i_f] # not x[i_f], x is just zeroes + x = x[i_nf] + # user guess x0 stays separate from presolve solution x + if x0 is not None: + x0 = x0[i_nf] + A_eq = A_eq[:, i_nf] + A_ub = A_ub[:, i_nf] + # modify bounds + lb_mod = lb[i_nf] + ub_mod = ub[i_nf] + + def rev(x_mod): + # Function to restore x: insert x_undo into x_mod. + # When elements have been removed at positions k1, k2, k3, ... + # then these must be replaced at (after) positions k1-1, k2-2, + # k3-3, ... in the modified array to recreate the original + i = np.flatnonzero(i_f) + # Number of variables to restore + N = len(i) + index_offset = np.arange(N) + # Create insert indices + insert_indices = i - index_offset + x_rev = np.insert(x_mod.astype(float), insert_indices, x_undo) + return x_rev + + # Use revstack as a list of functions, currently just this one. + revstack.append(rev) + + # no constraints indicates that problem is trivial + if A_eq.size == 0 and A_ub.size == 0: + b_eq = np.array([]) + b_ub = np.array([]) + # test_empty_constraint_1 + if c.size == 0: + status = 0 + message = ("The solution was determined in presolve as there are " + "no non-trivial constraints.") + elif (np.any(np.logical_and(c < 0, ub_mod == np.inf)) or + np.any(np.logical_and(c > 0, lb_mod == -np.inf))): + # test_no_constraints() + # test_unbounded_no_nontrivial_constraints_1 + # test_unbounded_no_nontrivial_constraints_2 + status = 3 + message = ("The problem is (trivially) unbounded " + "because there are no non-trivial constraints and " + "a) at least one decision variable is unbounded " + "above and its corresponding cost is negative, or " + "b) at least one decision variable is unbounded below " + "and its corresponding cost is positive. ") + else: # test_empty_constraint_2 + status = 0 + message = ("The solution was determined in presolve as there are " + "no non-trivial constraints.") + complete = True + x[c < 0] = ub_mod[c < 0] + x[c > 0] = lb_mod[c > 0] + # where c is zero, set x to a finite bound or zero + x_zero_c = ub_mod[c == 0] + x_zero_c[np.isinf(x_zero_c)] = ub_mod[c == 0][np.isinf(x_zero_c)] + x_zero_c[np.isinf(x_zero_c)] = 0 + x[c == 0] = x_zero_c + # if this is not the last step of presolve, should convert bounds back + # to array and return here + + # Convert modified lb and ub back into N x 2 bounds + bounds = np.hstack((lb_mod[:, np.newaxis], ub_mod[:, np.newaxis])) + + # remove redundant (linearly dependent) rows from equality constraints + n_rows_A = A_eq.shape[0] + redundancy_warning = ("A_eq does not appear to be of full row rank. To " + "improve performance, check the problem formulation " + "for redundant equality constraints.") + if (sps.issparse(A_eq)): + if rr and A_eq.size > 0: # TODO: Fast sparse rank check? + rr_res = _remove_redundancy_pivot_sparse(A_eq, b_eq) + A_eq, b_eq, status, message = rr_res + if A_eq.shape[0] < n_rows_A: + warn(redundancy_warning, OptimizeWarning, stacklevel=1) + if status != 0: + complete = True + return (_LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0), + c0, x, revstack, complete, status, message) + + # This is a wild guess for which redundancy removal algorithm will be + # faster. More testing would be good. + small_nullspace = 5 + if rr and A_eq.size > 0: + try: # TODO: use results of first SVD in _remove_redundancy_svd + rank = np.linalg.matrix_rank(A_eq) + # oh well, we'll have to go with _remove_redundancy_pivot_dense + except Exception: + rank = 0 + if rr and A_eq.size > 0 and rank < A_eq.shape[0]: + warn(redundancy_warning, OptimizeWarning, stacklevel=3) + dim_row_nullspace = A_eq.shape[0]-rank + if rr_method is None: + if dim_row_nullspace <= small_nullspace: + rr_res = _remove_redundancy_svd(A_eq, b_eq) + A_eq, b_eq, status, message = rr_res + if dim_row_nullspace > small_nullspace or status == 4: + rr_res = _remove_redundancy_pivot_dense(A_eq, b_eq) + A_eq, b_eq, status, message = rr_res + + else: + rr_method = rr_method.lower() + if rr_method == "svd": + rr_res = _remove_redundancy_svd(A_eq, b_eq) + A_eq, b_eq, status, message = rr_res + elif rr_method == "pivot": + rr_res = _remove_redundancy_pivot_dense(A_eq, b_eq) + A_eq, b_eq, status, message = rr_res + elif rr_method == "id": + rr_res = _remove_redundancy_id(A_eq, b_eq, rank) + A_eq, b_eq, status, message = rr_res + else: # shouldn't get here; option validity checked above + pass + if A_eq.shape[0] < rank: + message = ("Due to numerical issues, redundant equality " + "constraints could not be removed automatically. " + "Try providing your constraint matrices as sparse " + "matrices to activate sparse presolve, try turning " + "off redundancy removal, or try turning off presolve " + "altogether.") + status = 4 + if status != 0: + complete = True + return (_LPProblem(c, A_ub, b_ub, A_eq, b_eq, bounds, x0), + c0, x, revstack, complete, status, message) + + +def _parse_linprog(lp, options, meth): + """ + Parse the provided linear programming problem + + ``_parse_linprog`` employs two main steps ``_check_sparse_inputs`` and + ``_clean_inputs``. ``_check_sparse_inputs`` checks for sparsity in the + provided constraints (``A_ub`` and ``A_eq) and if these match the provided + sparsity optional values. + + ``_clean inputs`` checks of the provided inputs. If no violations are + identified the objective vector, upper bound constraints, equality + constraints, and simple bounds are returned in the expected format. + + Parameters + ---------- + lp : A `scipy.optimize._linprog_util._LPProblem` consisting of the following fields: + + c : 1D array + The coefficients of the linear objective function to be minimized. + A_ub : 2D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : various valid formats, optional + The bounds of ``x``, as ``min`` and ``max`` pairs. + If bounds are specified for all N variables separately, valid formats are: + * a 2D array (2 x N or N x 2); + * a sequence of N sequences, each with 2 values. + If all variables have the same bounds, a single pair of values can + be specified. Valid formats are: + * a sequence with 2 scalar values; + * a sequence with a single element containing 2 scalar values. + If all variables have a lower bound of 0 and no upper bound, the bounds + parameter can be omitted (or given as None). + x0 : 1D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + 'revised simplex' method, and can only be used if `x0` represents a + basic feasible solution. + + options : dict + A dictionary of solver options. All methods accept the following + generic options: + + maxiter : int + Maximum number of iterations to perform. + disp : bool + Set to True to print convergence messages. + + For method-specific options, see :func:`show_options('linprog')`. + + Returns + ------- + lp : A `scipy.optimize._linprog_util._LPProblem` consisting of the following fields: + + c : 1D array + The coefficients of the linear objective function to be minimized. + A_ub : 2D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : 2D array + The bounds of ``x``, as ``min`` and ``max`` pairs, one for each of the N + elements of ``x``. The N x 2 array contains lower bounds in the first + column and upper bounds in the 2nd. Unbounded variables have lower + bound -np.inf and/or upper bound np.inf. + x0 : 1D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + 'revised simplex' method, and can only be used if `x0` represents a + basic feasible solution. + + options : dict, optional + A dictionary of solver options. All methods accept the following + generic options: + + maxiter : int + Maximum number of iterations to perform. + disp : bool + Set to True to print convergence messages. + + For method-specific options, see :func:`show_options('linprog')`. + + """ + if options is None: + options = {} + + solver_options = {k: v for k, v in options.items()} + solver_options, A_ub, A_eq = _check_sparse_inputs(solver_options, meth, + lp.A_ub, lp.A_eq) + # Convert lists to numpy arrays, etc... + lp = _clean_inputs(lp._replace(A_ub=A_ub, A_eq=A_eq)) + return lp, solver_options + + +def _get_Abc(lp, c0): + """ + Given a linear programming problem of the form: + + Minimize:: + + c @ x + + Subject to:: + + A_ub @ x <= b_ub + A_eq @ x == b_eq + lb <= x <= ub + + where ``lb = 0`` and ``ub = None`` unless set in ``bounds``. + + Return the problem in standard form: + + Minimize:: + + c @ x + + Subject to:: + + A @ x == b + x >= 0 + + by adding slack variables and making variable substitutions as necessary. + + Parameters + ---------- + lp : A `scipy.optimize._linprog_util._LPProblem` consisting of the following fields: + + c : 1D array + The coefficients of the linear objective function to be minimized. + A_ub : 2D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : 2D array + The bounds of ``x``, lower bounds in the 1st column, upper + bounds in the 2nd column. The bounds are possibly tightened + by the presolve procedure. + x0 : 1D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + 'revised simplex' method, and can only be used if `x0` represents a + basic feasible solution. + + c0 : float + Constant term in objective function due to fixed (and eliminated) + variables. + + Returns + ------- + A : 2-D array + 2-D array such that ``A`` @ ``x``, gives the values of the equality + constraints at ``x``. + b : 1-D array + 1-D array of values representing the RHS of each equality constraint + (row) in A (for standard form problem). + c : 1-D array + Coefficients of the linear objective function to be minimized (for + standard form problem). + c0 : float + Constant term in objective function due to fixed (and eliminated) + variables. + x0 : 1-D array + Starting values of the independent variables, which will be refined by + the optimization algorithm + + References + ---------- + .. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear + programming." Athena Scientific 1 (1997): 997. + + """ + c, A_ub, b_ub, A_eq, b_eq, bounds, x0, integrality = lp + + if sps.issparse(A_eq): + sparse = True + A_eq = sps.csr_matrix(A_eq) + A_ub = sps.csr_matrix(A_ub) + + def hstack(blocks): + return sps.hstack(blocks, format="csr") + + def vstack(blocks): + return sps.vstack(blocks, format="csr") + + zeros = sps.csr_matrix + eye = sps.eye + else: + sparse = False + hstack = np.hstack + vstack = np.vstack + zeros = np.zeros + eye = np.eye + + # Variables lbs and ubs (see below) may be changed, which feeds back into + # bounds, so copy. + bounds = np.array(bounds, copy=True) + + # modify problem such that all variables have only non-negativity bounds + lbs = bounds[:, 0] + ubs = bounds[:, 1] + m_ub, n_ub = A_ub.shape + + lb_none = np.equal(lbs, -np.inf) + ub_none = np.equal(ubs, np.inf) + lb_some = np.logical_not(lb_none) + ub_some = np.logical_not(ub_none) + + # unbounded below: substitute xi = -xi' (unbounded above) + # if -inf <= xi <= ub, then -ub <= -xi <= inf, so swap and invert bounds + l_nolb_someub = np.logical_and(lb_none, ub_some) + i_nolb = np.nonzero(l_nolb_someub)[0] + lbs[l_nolb_someub], ubs[l_nolb_someub] = ( + -ubs[l_nolb_someub], -lbs[l_nolb_someub]) + lb_none = np.equal(lbs, -np.inf) + ub_none = np.equal(ubs, np.inf) + lb_some = np.logical_not(lb_none) + ub_some = np.logical_not(ub_none) + c[i_nolb] *= -1 + if x0 is not None: + x0[i_nolb] *= -1 + if len(i_nolb) > 0: + if A_ub.shape[0] > 0: # sometimes needed for sparse arrays... weird + A_ub[:, i_nolb] *= -1 + if A_eq.shape[0] > 0: + A_eq[:, i_nolb] *= -1 + + # upper bound: add inequality constraint + i_newub, = ub_some.nonzero() + ub_newub = ubs[ub_some] + n_bounds = len(i_newub) + if n_bounds > 0: + shape = (n_bounds, A_ub.shape[1]) + if sparse: + idxs = (np.arange(n_bounds), i_newub) + A_ub = vstack((A_ub, sps.csr_matrix((np.ones(n_bounds), idxs), + shape=shape))) + else: + A_ub = vstack((A_ub, np.zeros(shape))) + A_ub[np.arange(m_ub, A_ub.shape[0]), i_newub] = 1 + b_ub = np.concatenate((b_ub, np.zeros(n_bounds))) + b_ub[m_ub:] = ub_newub + + A1 = vstack((A_ub, A_eq)) + b = np.concatenate((b_ub, b_eq)) + c = np.concatenate((c, np.zeros((A_ub.shape[0],)))) + if x0 is not None: + x0 = np.concatenate((x0, np.zeros((A_ub.shape[0],)))) + # unbounded: substitute xi = xi+ + xi- + l_free = np.logical_and(lb_none, ub_none) + i_free = np.nonzero(l_free)[0] + n_free = len(i_free) + c = np.concatenate((c, np.zeros(n_free))) + if x0 is not None: + x0 = np.concatenate((x0, np.zeros(n_free))) + A1 = hstack((A1[:, :n_ub], -A1[:, i_free])) + c[n_ub:n_ub+n_free] = -c[i_free] + if x0 is not None: + i_free_neg = x0[i_free] < 0 + x0[np.arange(n_ub, A1.shape[1])[i_free_neg]] = -x0[i_free[i_free_neg]] + x0[i_free[i_free_neg]] = 0 + + # add slack variables + A2 = vstack([eye(A_ub.shape[0]), zeros((A_eq.shape[0], A_ub.shape[0]))]) + + A = hstack([A1, A2]) + + # lower bound: substitute xi = xi' + lb + # now there is a constant term in objective + i_shift = np.nonzero(lb_some)[0] + lb_shift = lbs[lb_some].astype(float) + c0 += np.sum(lb_shift * c[i_shift]) + if sparse: + b = b.reshape(-1, 1) + A = A.tocsc() + b -= (A[:, i_shift] @ sps.diags(lb_shift)).sum(axis=1) + b = b.ravel() + else: + b -= (A[:, i_shift] * lb_shift).sum(axis=1) + if x0 is not None: + x0[i_shift] -= lb_shift + + return A, b, c, c0, x0 + + +def _round_to_power_of_two(x): + """ + Round elements of the array to the nearest power of two. + """ + return 2**np.around(np.log2(x)) + + +def _autoscale(A, b, c, x0): + """ + Scales the problem according to equilibration from [12]. + Also normalizes the right hand side vector by its maximum element. + """ + m, n = A.shape + + C = 1 + R = 1 + + if A.size > 0: + + R = np.max(np.abs(A), axis=1) + if sps.issparse(A): + R = R.toarray().flatten() + R[R == 0] = 1 + R = 1/_round_to_power_of_two(R) + A = sps.diags(R)@A if sps.issparse(A) else A*R.reshape(m, 1) + b = b*R + + C = np.max(np.abs(A), axis=0) + if sps.issparse(A): + C = C.toarray().flatten() + C[C == 0] = 1 + C = 1/_round_to_power_of_two(C) + A = A@sps.diags(C) if sps.issparse(A) else A*C + c = c*C + + b_scale = np.max(np.abs(b)) if b.size > 0 else 1 + if b_scale == 0: + b_scale = 1. + b = b/b_scale + + if x0 is not None: + x0 = x0/b_scale*(1/C) + return A, b, c, x0, C, b_scale + + +def _unscale(x, C, b_scale): + """ + Converts solution to _autoscale problem -> solution to original problem. + """ + + try: + n = len(C) + # fails if sparse or scalar; that's OK. + # this is only needed for original simplex (never sparse) + except TypeError: + n = len(x) + + return x[:n]*b_scale*C + + +def _display_summary(message, status, fun, iteration): + """ + Print the termination summary of the linear program + + Parameters + ---------- + message : str + A string descriptor of the exit status of the optimization. + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + + fun : float + Value of the objective function. + iteration : iteration + The number of iterations performed. + """ + print(message) + if status in (0, 1): + print(f" Current function value: {fun: <12.6f}") + print(f" Iterations: {iteration:d}") + + +def _postsolve(x, postsolve_args, complete=False): + """ + Given solution x to presolved, standard form linear program x, add + fixed variables back into the problem and undo the variable substitutions + to get solution to original linear program. Also, calculate the objective + function value, slack in original upper bound constraints, and residuals + in original equality constraints. + + Parameters + ---------- + x : 1-D array + Solution vector to the standard-form problem. + postsolve_args : tuple + Data needed by _postsolve to convert the solution to the standard-form + problem into the solution to the original problem, including: + + lp : A `scipy.optimize._linprog_util._LPProblem` consisting of the following fields: + + c : 1D array + The coefficients of the linear objective function to be minimized. + A_ub : 2D array, optional + The inequality constraint matrix. Each row of ``A_ub`` specifies the + coefficients of a linear inequality constraint on ``x``. + b_ub : 1D array, optional + The inequality constraint vector. Each element represents an + upper bound on the corresponding value of ``A_ub @ x``. + A_eq : 2D array, optional + The equality constraint matrix. Each row of ``A_eq`` specifies the + coefficients of a linear equality constraint on ``x``. + b_eq : 1D array, optional + The equality constraint vector. Each element of ``A_eq @ x`` must equal + the corresponding element of ``b_eq``. + bounds : 2D array + The bounds of ``x``, lower bounds in the 1st column, upper + bounds in the 2nd column. The bounds are possibly tightened + by the presolve procedure. + x0 : 1D array, optional + Guess values of the decision variables, which will be refined by + the optimization algorithm. This argument is currently used only by the + 'revised simplex' method, and can only be used if `x0` represents a + basic feasible solution. + + revstack: list of functions + the functions in the list reverse the operations of _presolve() + the function signature is x_org = f(x_mod), where x_mod is the result + of a presolve step and x_org the value at the start of the step + complete : bool + Whether the solution is was determined in presolve (``True`` if so) + + Returns + ------- + x : 1-D array + Solution vector to original linear programming problem + fun: float + optimal objective value for original problem + slack : 1-D array + The (non-negative) slack in the upper bound constraints, that is, + ``b_ub - A_ub @ x`` + con : 1-D array + The (nominally zero) residuals of the equality constraints, that is, + ``b - A_eq @ x`` + """ + # note that all the inputs are the ORIGINAL, unmodified versions + # no rows, columns have been removed + + c, A_ub, b_ub, A_eq, b_eq, bounds, x0, integrality = postsolve_args[0] + revstack, C, b_scale = postsolve_args[1:] + + x = _unscale(x, C, b_scale) + + # Undo variable substitutions of _get_Abc() + # if "complete", problem was solved in presolve; don't do anything here + n_x = bounds.shape[0] + if not complete and bounds is not None: # bounds are never none, probably + n_unbounded = 0 + for i, bi in enumerate(bounds): + lbi = bi[0] + ubi = bi[1] + if lbi == -np.inf and ubi == np.inf: + n_unbounded += 1 + x[i] = x[i] - x[n_x + n_unbounded - 1] + else: + if lbi == -np.inf: + x[i] = ubi - x[i] + else: + x[i] += lbi + # all the rest of the variables were artificial + x = x[:n_x] + + # If there were variables removed from the problem, add them back into the + # solution vector + # Apply the functions in revstack (reverse direction) + for rev in reversed(revstack): + x = rev(x) + + fun = x.dot(c) + with np.errstate(invalid="ignore"): + slack = b_ub - A_ub.dot(x) # report slack for ORIGINAL UB constraints + # report residuals of ORIGINAL EQ constraints + con = b_eq - A_eq.dot(x) + + return x, fun, slack, con + + +def _check_result(x, fun, status, slack, con, bounds, tol, message, + integrality): + """ + Check the validity of the provided solution. + + A valid (optimal) solution satisfies all bounds, all slack variables are + negative and all equality constraint residuals are strictly non-zero. + Further, the lower-bounds, upper-bounds, slack and residuals contain + no nan values. + + Parameters + ---------- + x : 1-D array + Solution vector to original linear programming problem + fun: float + optimal objective value for original problem + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + + slack : 1-D array + The (non-negative) slack in the upper bound constraints, that is, + ``b_ub - A_ub @ x`` + con : 1-D array + The (nominally zero) residuals of the equality constraints, that is, + ``b - A_eq @ x`` + bounds : 2D array + The bounds on the original variables ``x`` + message : str + A string descriptor of the exit status of the optimization. + tol : float + Termination tolerance; see [1]_ Section 4.5. + + Returns + ------- + status : int + An integer representing the exit status of the optimization:: + + 0 : Optimization terminated successfully + 1 : Iteration limit reached + 2 : Problem appears to be infeasible + 3 : Problem appears to be unbounded + 4 : Serious numerical difficulties encountered + + message : str + A string descriptor of the exit status of the optimization. + """ + # Somewhat arbitrary + tol = np.sqrt(tol) * 10 + + if x is None: + # HiGHS does not provide x if infeasible/unbounded + if status == 0: # Observed with HiGHS Simplex Primal + status = 4 + message = ("The solver did not provide a solution nor did it " + "report a failure. Please submit a bug report.") + return status, message + + contains_nans = ( + np.isnan(x).any() + or np.isnan(fun) + or np.isnan(slack).any() + or np.isnan(con).any() + ) + + if contains_nans: + is_feasible = False + else: + if integrality is None: + integrality = 0 + valid_bounds = (x >= bounds[:, 0] - tol) & (x <= bounds[:, 1] + tol) + # When integrality is 2 or 3, x must be within bounds OR take value 0 + valid_bounds |= (integrality > 1) & np.isclose(x, 0, atol=tol) + invalid_bounds = not np.all(valid_bounds) + + invalid_slack = status != 3 and (slack < -tol).any() + invalid_con = status != 3 and (np.abs(con) > tol).any() + is_feasible = not (invalid_bounds or invalid_slack or invalid_con) + + if status == 0 and not is_feasible: + status = 4 + message = ("The solution does not satisfy the constraints within the " + "required tolerance of " + f"{tol:.2E}" + ", yet " + "no errors were raised and there is no certificate of " + "infeasibility or unboundedness. Check whether " + "the slack and constraint residuals are acceptable; " + "if not, consider enabling presolve, adjusting the " + "tolerance option(s), and/or using a different method. " + "Please consider submitting a bug report.") + elif status == 2 and is_feasible: + # Occurs if the simplex method exits after phase one with a very + # nearly basic feasible solution. Postsolving can make the solution + # basic, however, this solution is NOT optimal + status = 4 + message = ("The solution is feasible, but the solver did not report " + "that the solution was optimal. Please try a different " + "method.") + + return status, message diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsap.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsap.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..b2ea39a10549d1346bada989f573981804d22006 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsap.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..f60adcc891304e34ac9d85d108b6a232b4bf0c93 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/__init__.py @@ -0,0 +1,5 @@ +"""This module contains least-squares algorithms.""" +from .least_squares import least_squares +from .lsq_linear import lsq_linear + +__all__ = ['least_squares', 'lsq_linear'] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/bvls.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/bvls.py new file mode 100644 index 0000000000000000000000000000000000000000..8f34ead4a1fc4edbb3c2ab50a204aa9a3cc21cff --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/bvls.py @@ -0,0 +1,183 @@ +"""Bounded-variable least-squares algorithm.""" +import numpy as np +from numpy.linalg import norm, lstsq +from scipy.optimize import OptimizeResult + +from .common import print_header_linear, print_iteration_linear + + +def compute_kkt_optimality(g, on_bound): + """Compute the maximum violation of KKT conditions.""" + g_kkt = g * on_bound + free_set = on_bound == 0 + g_kkt[free_set] = np.abs(g[free_set]) + return np.max(g_kkt) + + +def bvls(A, b, x_lsq, lb, ub, tol, max_iter, verbose, rcond=None): + m, n = A.shape + + x = x_lsq.copy() + on_bound = np.zeros(n) + + mask = x <= lb + x[mask] = lb[mask] + on_bound[mask] = -1 + + mask = x >= ub + x[mask] = ub[mask] + on_bound[mask] = 1 + + free_set = on_bound == 0 + active_set = ~free_set + free_set, = np.nonzero(free_set) + + r = A.dot(x) - b + cost = 0.5 * np.dot(r, r) + initial_cost = cost + g = A.T.dot(r) + + cost_change = None + step_norm = None + iteration = 0 + + if verbose == 2: + print_header_linear() + + # This is the initialization loop. The requirement is that the + # least-squares solution on free variables is feasible before BVLS starts. + # One possible initialization is to set all variables to lower or upper + # bounds, but many iterations may be required from this state later on. + # The implemented ad-hoc procedure which intuitively should give a better + # initial state: find the least-squares solution on current free variables, + # if its feasible then stop, otherwise, set violating variables to + # corresponding bounds and continue on the reduced set of free variables. + + while free_set.size > 0: + if verbose == 2: + optimality = compute_kkt_optimality(g, on_bound) + print_iteration_linear(iteration, cost, cost_change, step_norm, + optimality) + + iteration += 1 + x_free_old = x[free_set].copy() + + A_free = A[:, free_set] + b_free = b - A.dot(x * active_set) + z = lstsq(A_free, b_free, rcond=rcond)[0] + + lbv = z < lb[free_set] + ubv = z > ub[free_set] + v = lbv | ubv + + if np.any(lbv): + ind = free_set[lbv] + x[ind] = lb[ind] + active_set[ind] = True + on_bound[ind] = -1 + + if np.any(ubv): + ind = free_set[ubv] + x[ind] = ub[ind] + active_set[ind] = True + on_bound[ind] = 1 + + ind = free_set[~v] + x[ind] = z[~v] + + r = A.dot(x) - b + cost_new = 0.5 * np.dot(r, r) + cost_change = cost - cost_new + cost = cost_new + g = A.T.dot(r) + step_norm = norm(x[free_set] - x_free_old) + + if np.any(v): + free_set = free_set[~v] + else: + break + + if max_iter is None: + max_iter = n + max_iter += iteration + + termination_status = None + + # Main BVLS loop. + + optimality = compute_kkt_optimality(g, on_bound) + for iteration in range(iteration, max_iter): # BVLS Loop A + if verbose == 2: + print_iteration_linear(iteration, cost, cost_change, + step_norm, optimality) + + if optimality < tol: + termination_status = 1 + + if termination_status is not None: + break + + move_to_free = np.argmax(g * on_bound) + on_bound[move_to_free] = 0 + + while True: # BVLS Loop B + + free_set = on_bound == 0 + active_set = ~free_set + free_set, = np.nonzero(free_set) + + x_free = x[free_set] + x_free_old = x_free.copy() + lb_free = lb[free_set] + ub_free = ub[free_set] + + A_free = A[:, free_set] + b_free = b - A.dot(x * active_set) + z = lstsq(A_free, b_free, rcond=rcond)[0] + + lbv, = np.nonzero(z < lb_free) + ubv, = np.nonzero(z > ub_free) + v = np.hstack((lbv, ubv)) + + if v.size > 0: + alphas = np.hstack(( + lb_free[lbv] - x_free[lbv], + ub_free[ubv] - x_free[ubv])) / (z[v] - x_free[v]) + + i = np.argmin(alphas) + i_free = v[i] + alpha = alphas[i] + + x_free *= 1 - alpha + x_free += alpha * z + x[free_set] = x_free + + if i < lbv.size: + on_bound[free_set[i_free]] = -1 + else: + on_bound[free_set[i_free]] = 1 + else: + x_free = z + x[free_set] = x_free + break + + step_norm = norm(x_free - x_free_old) + + r = A.dot(x) - b + cost_new = 0.5 * np.dot(r, r) + cost_change = cost - cost_new + + if cost_change < tol * cost: + termination_status = 2 + cost = cost_new + + g = A.T.dot(r) + optimality = compute_kkt_optimality(g, on_bound) + + if termination_status is None: + termination_status = 0 + + return OptimizeResult( + x=x, fun=r, cost=cost, optimality=optimality, active_mask=on_bound, + nit=iteration + 1, status=termination_status, + initial_cost=initial_cost) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/common.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/common.py new file mode 100644 index 0000000000000000000000000000000000000000..0f8117f23ec1111d5205537c59931b165e2bfdaf --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/common.py @@ -0,0 +1,731 @@ +"""Functions used by least-squares algorithms.""" +from math import copysign + +import numpy as np +from numpy.linalg import norm + +from scipy.linalg import cho_factor, cho_solve, LinAlgError +from scipy.sparse import issparse +from scipy.sparse.linalg import LinearOperator, aslinearoperator + + +EPS = np.finfo(float).eps + + +# Functions related to a trust-region problem. + + +def intersect_trust_region(x, s, Delta): + """Find the intersection of a line with the boundary of a trust region. + + This function solves the quadratic equation with respect to t + ||(x + s*t)||**2 = Delta**2. + + Returns + ------- + t_neg, t_pos : tuple of float + Negative and positive roots. + + Raises + ------ + ValueError + If `s` is zero or `x` is not within the trust region. + """ + a = np.dot(s, s) + if a == 0: + raise ValueError("`s` is zero.") + + b = np.dot(x, s) + + c = np.dot(x, x) - Delta**2 + if c > 0: + raise ValueError("`x` is not within the trust region.") + + d = np.sqrt(b*b - a*c) # Root from one fourth of the discriminant. + + # Computations below avoid loss of significance, see "Numerical Recipes". + q = -(b + copysign(d, b)) + t1 = q / a + t2 = c / q + + if t1 < t2: + return t1, t2 + else: + return t2, t1 + + +def solve_lsq_trust_region(n, m, uf, s, V, Delta, initial_alpha=None, + rtol=0.01, max_iter=10): + """Solve a trust-region problem arising in least-squares minimization. + + This function implements a method described by J. J. More [1]_ and used + in MINPACK, but it relies on a single SVD of Jacobian instead of series + of Cholesky decompositions. Before running this function, compute: + ``U, s, VT = svd(J, full_matrices=False)``. + + Parameters + ---------- + n : int + Number of variables. + m : int + Number of residuals. + uf : ndarray + Computed as U.T.dot(f). + s : ndarray + Singular values of J. + V : ndarray + Transpose of VT. + Delta : float + Radius of a trust region. + initial_alpha : float, optional + Initial guess for alpha, which might be available from a previous + iteration. If None, determined automatically. + rtol : float, optional + Stopping tolerance for the root-finding procedure. Namely, the + solution ``p`` will satisfy ``abs(norm(p) - Delta) < rtol * Delta``. + max_iter : int, optional + Maximum allowed number of iterations for the root-finding procedure. + + Returns + ------- + p : ndarray, shape (n,) + Found solution of a trust-region problem. + alpha : float + Positive value such that (J.T*J + alpha*I)*p = -J.T*f. + Sometimes called Levenberg-Marquardt parameter. + n_iter : int + Number of iterations made by root-finding procedure. Zero means + that Gauss-Newton step was selected as the solution. + + References + ---------- + .. [1] More, J. J., "The Levenberg-Marquardt Algorithm: Implementation + and Theory," Numerical Analysis, ed. G. A. Watson, Lecture Notes + in Mathematics 630, Springer Verlag, pp. 105-116, 1977. + """ + def phi_and_derivative(alpha, suf, s, Delta): + """Function of which to find zero. + + It is defined as "norm of regularized (by alpha) least-squares + solution minus `Delta`". Refer to [1]_. + """ + denom = s**2 + alpha + p_norm = norm(suf / denom) + phi = p_norm - Delta + phi_prime = -np.sum(suf ** 2 / denom**3) / p_norm + return phi, phi_prime + + suf = s * uf + + # Check if J has full rank and try Gauss-Newton step. + if m >= n: + threshold = EPS * m * s[0] + full_rank = s[-1] > threshold + else: + full_rank = False + + if full_rank: + p = -V.dot(uf / s) + if norm(p) <= Delta: + return p, 0.0, 0 + + alpha_upper = norm(suf) / Delta + + if full_rank: + phi, phi_prime = phi_and_derivative(0.0, suf, s, Delta) + alpha_lower = -phi / phi_prime + else: + alpha_lower = 0.0 + + if initial_alpha is None or not full_rank and initial_alpha == 0: + alpha = max(0.001 * alpha_upper, (alpha_lower * alpha_upper)**0.5) + else: + alpha = initial_alpha + + for it in range(max_iter): + if alpha < alpha_lower or alpha > alpha_upper: + alpha = max(0.001 * alpha_upper, (alpha_lower * alpha_upper)**0.5) + + phi, phi_prime = phi_and_derivative(alpha, suf, s, Delta) + + if phi < 0: + alpha_upper = alpha + + ratio = phi / phi_prime + alpha_lower = max(alpha_lower, alpha - ratio) + alpha -= (phi + Delta) * ratio / Delta + + if np.abs(phi) < rtol * Delta: + break + + p = -V.dot(suf / (s**2 + alpha)) + + # Make the norm of p equal to Delta, p is changed only slightly during + # this. It is done to prevent p lie outside the trust region (which can + # cause problems later). + p *= Delta / norm(p) + + return p, alpha, it + 1 + + +def solve_trust_region_2d(B, g, Delta): + """Solve a general trust-region problem in 2 dimensions. + + The problem is reformulated as a 4th order algebraic equation, + the solution of which is found by numpy.roots. + + Parameters + ---------- + B : ndarray, shape (2, 2) + Symmetric matrix, defines a quadratic term of the function. + g : ndarray, shape (2,) + Defines a linear term of the function. + Delta : float + Radius of a trust region. + + Returns + ------- + p : ndarray, shape (2,) + Found solution. + newton_step : bool + Whether the returned solution is the Newton step which lies within + the trust region. + """ + try: + R, lower = cho_factor(B) + p = -cho_solve((R, lower), g) + if np.dot(p, p) <= Delta**2: + return p, True + except LinAlgError: + pass + + a = B[0, 0] * Delta**2 + b = B[0, 1] * Delta**2 + c = B[1, 1] * Delta**2 + + d = g[0] * Delta + f = g[1] * Delta + + coeffs = np.array( + [-b + d, 2 * (a - c + f), 6 * b, 2 * (-a + c + f), -b - d]) + t = np.roots(coeffs) # Can handle leading zeros. + t = np.real(t[np.isreal(t)]) + + p = Delta * np.vstack((2 * t / (1 + t**2), (1 - t**2) / (1 + t**2))) + value = 0.5 * np.sum(p * B.dot(p), axis=0) + np.dot(g, p) + i = np.argmin(value) + p = p[:, i] + + return p, False + + +def update_tr_radius(Delta, actual_reduction, predicted_reduction, + step_norm, bound_hit): + """Update the radius of a trust region based on the cost reduction. + + Returns + ------- + Delta : float + New radius. + ratio : float + Ratio between actual and predicted reductions. + """ + if predicted_reduction > 0: + ratio = actual_reduction / predicted_reduction + elif predicted_reduction == actual_reduction == 0: + ratio = 1 + else: + ratio = 0 + + if ratio < 0.25: + Delta = 0.25 * step_norm + elif ratio > 0.75 and bound_hit: + Delta *= 2.0 + + return Delta, ratio + + +# Construction and minimization of quadratic functions. + + +def build_quadratic_1d(J, g, s, diag=None, s0=None): + """Parameterize a multivariate quadratic function along a line. + + The resulting univariate quadratic function is given as follows:: + + f(t) = 0.5 * (s0 + s*t).T * (J.T*J + diag) * (s0 + s*t) + + g.T * (s0 + s*t) + + Parameters + ---------- + J : ndarray, sparse matrix or LinearOperator shape (m, n) + Jacobian matrix, affects the quadratic term. + g : ndarray, shape (n,) + Gradient, defines the linear term. + s : ndarray, shape (n,) + Direction vector of a line. + diag : None or ndarray with shape (n,), optional + Addition diagonal part, affects the quadratic term. + If None, assumed to be 0. + s0 : None or ndarray with shape (n,), optional + Initial point. If None, assumed to be 0. + + Returns + ------- + a : float + Coefficient for t**2. + b : float + Coefficient for t. + c : float + Free term. Returned only if `s0` is provided. + """ + v = J.dot(s) + a = np.dot(v, v) + if diag is not None: + a += np.dot(s * diag, s) + a *= 0.5 + + b = np.dot(g, s) + + if s0 is not None: + u = J.dot(s0) + b += np.dot(u, v) + c = 0.5 * np.dot(u, u) + np.dot(g, s0) + if diag is not None: + b += np.dot(s0 * diag, s) + c += 0.5 * np.dot(s0 * diag, s0) + return a, b, c + else: + return a, b + + +def minimize_quadratic_1d(a, b, lb, ub, c=0): + """Minimize a 1-D quadratic function subject to bounds. + + The free term `c` is 0 by default. Bounds must be finite. + + Returns + ------- + t : float + Minimum point. + y : float + Minimum value. + """ + t = [lb, ub] + if a != 0: + extremum = -0.5 * b / a + if lb < extremum < ub: + t.append(extremum) + t = np.asarray(t) + y = t * (a * t + b) + c + min_index = np.argmin(y) + return t[min_index], y[min_index] + + +def evaluate_quadratic(J, g, s, diag=None): + """Compute values of a quadratic function arising in least squares. + + The function is 0.5 * s.T * (J.T * J + diag) * s + g.T * s. + + Parameters + ---------- + J : ndarray, sparse matrix or LinearOperator, shape (m, n) + Jacobian matrix, affects the quadratic term. + g : ndarray, shape (n,) + Gradient, defines the linear term. + s : ndarray, shape (k, n) or (n,) + Array containing steps as rows. + diag : ndarray, shape (n,), optional + Addition diagonal part, affects the quadratic term. + If None, assumed to be 0. + + Returns + ------- + values : ndarray with shape (k,) or float + Values of the function. If `s` was 2-D, then ndarray is + returned, otherwise, float is returned. + """ + if s.ndim == 1: + Js = J.dot(s) + q = np.dot(Js, Js) + if diag is not None: + q += np.dot(s * diag, s) + else: + Js = J.dot(s.T) + q = np.sum(Js**2, axis=0) + if diag is not None: + q += np.sum(diag * s**2, axis=1) + + l = np.dot(s, g) + + return 0.5 * q + l + + +# Utility functions to work with bound constraints. + + +def in_bounds(x, lb, ub): + """Check if a point lies within bounds.""" + return np.all((x >= lb) & (x <= ub)) + + +def step_size_to_bound(x, s, lb, ub): + """Compute a min_step size required to reach a bound. + + The function computes a positive scalar t, such that x + s * t is on + the bound. + + Returns + ------- + step : float + Computed step. Non-negative value. + hits : ndarray of int with shape of x + Each element indicates whether a corresponding variable reaches the + bound: + + * 0 - the bound was not hit. + * -1 - the lower bound was hit. + * 1 - the upper bound was hit. + """ + non_zero = np.nonzero(s) + s_non_zero = s[non_zero] + steps = np.empty_like(x) + steps.fill(np.inf) + with np.errstate(over='ignore'): + steps[non_zero] = np.maximum((lb - x)[non_zero] / s_non_zero, + (ub - x)[non_zero] / s_non_zero) + min_step = np.min(steps) + return min_step, np.equal(steps, min_step) * np.sign(s).astype(int) + + +def find_active_constraints(x, lb, ub, rtol=1e-10): + """Determine which constraints are active in a given point. + + The threshold is computed using `rtol` and the absolute value of the + closest bound. + + Returns + ------- + active : ndarray of int with shape of x + Each component shows whether the corresponding constraint is active: + + * 0 - a constraint is not active. + * -1 - a lower bound is active. + * 1 - a upper bound is active. + """ + active = np.zeros_like(x, dtype=int) + + if rtol == 0: + active[x <= lb] = -1 + active[x >= ub] = 1 + return active + + lower_dist = x - lb + upper_dist = ub - x + + lower_threshold = rtol * np.maximum(1, np.abs(lb)) + upper_threshold = rtol * np.maximum(1, np.abs(ub)) + + lower_active = (np.isfinite(lb) & + (lower_dist <= np.minimum(upper_dist, lower_threshold))) + active[lower_active] = -1 + + upper_active = (np.isfinite(ub) & + (upper_dist <= np.minimum(lower_dist, upper_threshold))) + active[upper_active] = 1 + + return active + + +def make_strictly_feasible(x, lb, ub, rstep=1e-10): + """Shift a point to the interior of a feasible region. + + Each element of the returned vector is at least at a relative distance + `rstep` from the closest bound. If ``rstep=0`` then `np.nextafter` is used. + """ + x_new = x.copy() + + active = find_active_constraints(x, lb, ub, rstep) + lower_mask = np.equal(active, -1) + upper_mask = np.equal(active, 1) + + if rstep == 0: + x_new[lower_mask] = np.nextafter(lb[lower_mask], ub[lower_mask]) + x_new[upper_mask] = np.nextafter(ub[upper_mask], lb[upper_mask]) + else: + x_new[lower_mask] = (lb[lower_mask] + + rstep * np.maximum(1, np.abs(lb[lower_mask]))) + x_new[upper_mask] = (ub[upper_mask] - + rstep * np.maximum(1, np.abs(ub[upper_mask]))) + + tight_bounds = (x_new < lb) | (x_new > ub) + x_new[tight_bounds] = 0.5 * (lb[tight_bounds] + ub[tight_bounds]) + + return x_new + + +def CL_scaling_vector(x, g, lb, ub): + """Compute Coleman-Li scaling vector and its derivatives. + + Components of a vector v are defined as follows:: + + | ub[i] - x[i], if g[i] < 0 and ub[i] < np.inf + v[i] = | x[i] - lb[i], if g[i] > 0 and lb[i] > -np.inf + | 1, otherwise + + According to this definition v[i] >= 0 for all i. It differs from the + definition in paper [1]_ (eq. (2.2)), where the absolute value of v is + used. Both definitions are equivalent down the line. + Derivatives of v with respect to x take value 1, -1 or 0 depending on a + case. + + Returns + ------- + v : ndarray with shape of x + Scaling vector. + dv : ndarray with shape of x + Derivatives of v[i] with respect to x[i], diagonal elements of v's + Jacobian. + + References + ---------- + .. [1] M.A. Branch, T.F. Coleman, and Y. Li, "A Subspace, Interior, + and Conjugate Gradient Method for Large-Scale Bound-Constrained + Minimization Problems," SIAM Journal on Scientific Computing, + Vol. 21, Number 1, pp 1-23, 1999. + """ + v = np.ones_like(x) + dv = np.zeros_like(x) + + mask = (g < 0) & np.isfinite(ub) + v[mask] = ub[mask] - x[mask] + dv[mask] = -1 + + mask = (g > 0) & np.isfinite(lb) + v[mask] = x[mask] - lb[mask] + dv[mask] = 1 + + return v, dv + + +def reflective_transformation(y, lb, ub): + """Compute reflective transformation and its gradient.""" + if in_bounds(y, lb, ub): + return y, np.ones_like(y) + + lb_finite = np.isfinite(lb) + ub_finite = np.isfinite(ub) + + x = y.copy() + g_negative = np.zeros_like(y, dtype=bool) + + mask = lb_finite & ~ub_finite + x[mask] = np.maximum(y[mask], 2 * lb[mask] - y[mask]) + g_negative[mask] = y[mask] < lb[mask] + + mask = ~lb_finite & ub_finite + x[mask] = np.minimum(y[mask], 2 * ub[mask] - y[mask]) + g_negative[mask] = y[mask] > ub[mask] + + mask = lb_finite & ub_finite + d = ub - lb + t = np.remainder(y[mask] - lb[mask], 2 * d[mask]) + x[mask] = lb[mask] + np.minimum(t, 2 * d[mask] - t) + g_negative[mask] = t > d[mask] + + g = np.ones_like(y) + g[g_negative] = -1 + + return x, g + + +# Functions to display algorithm's progress. + + +def print_header_nonlinear(): + print("{:^15}{:^15}{:^15}{:^15}{:^15}{:^15}" + .format("Iteration", "Total nfev", "Cost", "Cost reduction", + "Step norm", "Optimality")) + + +def print_iteration_nonlinear(iteration, nfev, cost, cost_reduction, + step_norm, optimality): + if cost_reduction is None: + cost_reduction = " " * 15 + else: + cost_reduction = f"{cost_reduction:^15.2e}" + + if step_norm is None: + step_norm = " " * 15 + else: + step_norm = f"{step_norm:^15.2e}" + + print(f"{iteration:^15}{nfev:^15}{cost:^15.4e}{cost_reduction}{step_norm}{optimality:^15.2e}") + + +def print_header_linear(): + print("{:^15}{:^15}{:^15}{:^15}{:^15}" + .format("Iteration", "Cost", "Cost reduction", "Step norm", + "Optimality")) + + +def print_iteration_linear(iteration, cost, cost_reduction, step_norm, + optimality): + if cost_reduction is None: + cost_reduction = " " * 15 + else: + cost_reduction = f"{cost_reduction:^15.2e}" + + if step_norm is None: + step_norm = " " * 15 + else: + step_norm = f"{step_norm:^15.2e}" + + print(f"{iteration:^15}{cost:^15.4e}{cost_reduction}{step_norm}{optimality:^15.2e}") + + +# Simple helper functions. + + +def compute_grad(J, f): + """Compute gradient of the least-squares cost function.""" + if isinstance(J, LinearOperator): + return J.rmatvec(f) + else: + return J.T.dot(f) + + +def compute_jac_scale(J, scale_inv_old=None): + """Compute variables scale based on the Jacobian matrix.""" + if issparse(J): + scale_inv = np.asarray(J.power(2).sum(axis=0)).ravel()**0.5 + else: + scale_inv = np.sum(J**2, axis=0)**0.5 + + if scale_inv_old is None: + scale_inv[scale_inv == 0] = 1 + else: + scale_inv = np.maximum(scale_inv, scale_inv_old) + + return 1 / scale_inv, scale_inv + + +def left_multiplied_operator(J, d): + """Return diag(d) J as LinearOperator.""" + J = aslinearoperator(J) + + def matvec(x): + return d * J.matvec(x) + + def matmat(X): + return d[:, np.newaxis] * J.matmat(X) + + def rmatvec(x): + return J.rmatvec(x.ravel() * d) + + return LinearOperator(J.shape, matvec=matvec, matmat=matmat, + rmatvec=rmatvec) + + +def right_multiplied_operator(J, d): + """Return J diag(d) as LinearOperator.""" + J = aslinearoperator(J) + + def matvec(x): + return J.matvec(np.ravel(x) * d) + + def matmat(X): + return J.matmat(X * d[:, np.newaxis]) + + def rmatvec(x): + return d * J.rmatvec(x) + + return LinearOperator(J.shape, matvec=matvec, matmat=matmat, + rmatvec=rmatvec) + + +def regularized_lsq_operator(J, diag): + """Return a matrix arising in regularized least squares as LinearOperator. + + The matrix is + [ J ] + [ D ] + where D is diagonal matrix with elements from `diag`. + """ + J = aslinearoperator(J) + m, n = J.shape + + def matvec(x): + return np.hstack((J.matvec(x), diag * x)) + + def rmatvec(x): + x1 = x[:m] + x2 = x[m:] + return J.rmatvec(x1) + diag * x2 + + return LinearOperator((m + n, n), matvec=matvec, rmatvec=rmatvec) + + +def right_multiply(J, d, copy=True): + """Compute J diag(d). + + If `copy` is False, `J` is modified in place (unless being LinearOperator). + """ + if copy and not isinstance(J, LinearOperator): + J = J.copy() + + if issparse(J): + J.data *= d.take(J.indices, mode='clip') # scikit-learn recipe. + elif isinstance(J, LinearOperator): + J = right_multiplied_operator(J, d) + else: + J *= d + + return J + + +def left_multiply(J, d, copy=True): + """Compute diag(d) J. + + If `copy` is False, `J` is modified in place (unless being LinearOperator). + """ + if copy and not isinstance(J, LinearOperator): + J = J.copy() + + if issparse(J): + J.data *= np.repeat(d, np.diff(J.indptr)) # scikit-learn recipe. + elif isinstance(J, LinearOperator): + J = left_multiplied_operator(J, d) + else: + J *= d[:, np.newaxis] + + return J + + +def check_termination(dF, F, dx_norm, x_norm, ratio, ftol, xtol): + """Check termination condition for nonlinear least squares.""" + ftol_satisfied = dF < ftol * F and ratio > 0.25 + xtol_satisfied = dx_norm < xtol * (xtol + x_norm) + + if ftol_satisfied and xtol_satisfied: + return 4 + elif ftol_satisfied: + return 2 + elif xtol_satisfied: + return 3 + else: + return None + + +def scale_for_robust_loss_function(J, f, rho): + """Scale Jacobian and residuals for a robust loss function. + + Arrays are modified in place. + """ + J_scale = rho[1] + 2 * rho[2] * f**2 + J_scale[J_scale < EPS] = EPS + J_scale **= 0.5 + + f *= rho[1] / J_scale + + return left_multiply(J, J_scale, copy=False), f diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/dogbox.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/dogbox.py new file mode 100644 index 0000000000000000000000000000000000000000..6bb5abbe79028afed7b110603a0d5dfd6affae7f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/dogbox.py @@ -0,0 +1,331 @@ +""" +Dogleg algorithm with rectangular trust regions for least-squares minimization. + +The description of the algorithm can be found in [Voglis]_. The algorithm does +trust-region iterations, but the shape of trust regions is rectangular as +opposed to conventional elliptical. The intersection of a trust region and +an initial feasible region is again some rectangle. Thus, on each iteration a +bound-constrained quadratic optimization problem is solved. + +A quadratic problem is solved by well-known dogleg approach, where the +function is minimized along piecewise-linear "dogleg" path [NumOpt]_, +Chapter 4. If Jacobian is not rank-deficient then the function is decreasing +along this path, and optimization amounts to simply following along this +path as long as a point stays within the bounds. A constrained Cauchy step +(along the anti-gradient) is considered for safety in rank deficient cases, +in this situations the convergence might be slow. + +If during iterations some variable hit the initial bound and the component +of anti-gradient points outside the feasible region, then a next dogleg step +won't make any progress. At this state such variables satisfy first-order +optimality conditions and they are excluded before computing a next dogleg +step. + +Gauss-Newton step can be computed exactly by `numpy.linalg.lstsq` (for dense +Jacobian matrices) or by iterative procedure `scipy.sparse.linalg.lsmr` (for +dense and sparse matrices, or Jacobian being LinearOperator). The second +option allows to solve very large problems (up to couple of millions of +residuals on a regular PC), provided the Jacobian matrix is sufficiently +sparse. But note that dogbox is not very good for solving problems with +large number of constraints, because of variables exclusion-inclusion on each +iteration (a required number of function evaluations might be high or accuracy +of a solution will be poor), thus its large-scale usage is probably limited +to unconstrained problems. + +References +---------- +.. [Voglis] C. Voglis and I. E. Lagaris, "A Rectangular Trust Region Dogleg + Approach for Unconstrained and Bound Constrained Nonlinear + Optimization", WSEAS International Conference on Applied + Mathematics, Corfu, Greece, 2004. +.. [NumOpt] J. Nocedal and S. J. Wright, "Numerical optimization, 2nd edition". +""" +import numpy as np +from numpy.linalg import lstsq, norm + +from scipy.sparse.linalg import LinearOperator, aslinearoperator, lsmr +from scipy.optimize import OptimizeResult + +from .common import ( + step_size_to_bound, in_bounds, update_tr_radius, evaluate_quadratic, + build_quadratic_1d, minimize_quadratic_1d, compute_grad, + compute_jac_scale, check_termination, scale_for_robust_loss_function, + print_header_nonlinear, print_iteration_nonlinear) + + +def lsmr_operator(Jop, d, active_set): + """Compute LinearOperator to use in LSMR by dogbox algorithm. + + `active_set` mask is used to excluded active variables from computations + of matrix-vector products. + """ + m, n = Jop.shape + + def matvec(x): + x_free = x.ravel().copy() + x_free[active_set] = 0 + return Jop.matvec(x * d) + + def rmatvec(x): + r = d * Jop.rmatvec(x) + r[active_set] = 0 + return r + + return LinearOperator((m, n), matvec=matvec, rmatvec=rmatvec, dtype=float) + + +def find_intersection(x, tr_bounds, lb, ub): + """Find intersection of trust-region bounds and initial bounds. + + Returns + ------- + lb_total, ub_total : ndarray with shape of x + Lower and upper bounds of the intersection region. + orig_l, orig_u : ndarray of bool with shape of x + True means that an original bound is taken as a corresponding bound + in the intersection region. + tr_l, tr_u : ndarray of bool with shape of x + True means that a trust-region bound is taken as a corresponding bound + in the intersection region. + """ + lb_centered = lb - x + ub_centered = ub - x + + lb_total = np.maximum(lb_centered, -tr_bounds) + ub_total = np.minimum(ub_centered, tr_bounds) + + orig_l = np.equal(lb_total, lb_centered) + orig_u = np.equal(ub_total, ub_centered) + + tr_l = np.equal(lb_total, -tr_bounds) + tr_u = np.equal(ub_total, tr_bounds) + + return lb_total, ub_total, orig_l, orig_u, tr_l, tr_u + + +def dogleg_step(x, newton_step, g, a, b, tr_bounds, lb, ub): + """Find dogleg step in a rectangular region. + + Returns + ------- + step : ndarray, shape (n,) + Computed dogleg step. + bound_hits : ndarray of int, shape (n,) + Each component shows whether a corresponding variable hits the + initial bound after the step is taken: + * 0 - a variable doesn't hit the bound. + * -1 - lower bound is hit. + * 1 - upper bound is hit. + tr_hit : bool + Whether the step hit the boundary of the trust-region. + """ + lb_total, ub_total, orig_l, orig_u, tr_l, tr_u = find_intersection( + x, tr_bounds, lb, ub + ) + bound_hits = np.zeros_like(x, dtype=int) + + if in_bounds(newton_step, lb_total, ub_total): + return newton_step, bound_hits, False + + to_bounds, _ = step_size_to_bound(np.zeros_like(x), -g, lb_total, ub_total) + + # The classical dogleg algorithm would check if Cauchy step fits into + # the bounds, and just return it constrained version if not. But in a + # rectangular trust region it makes sense to try to improve constrained + # Cauchy step too. Thus, we don't distinguish these two cases. + + cauchy_step = -minimize_quadratic_1d(a, b, 0, to_bounds)[0] * g + + step_diff = newton_step - cauchy_step + step_size, hits = step_size_to_bound(cauchy_step, step_diff, + lb_total, ub_total) + bound_hits[(hits < 0) & orig_l] = -1 + bound_hits[(hits > 0) & orig_u] = 1 + tr_hit = np.any((hits < 0) & tr_l | (hits > 0) & tr_u) + + return cauchy_step + step_size * step_diff, bound_hits, tr_hit + + +def dogbox(fun, jac, x0, f0, J0, lb, ub, ftol, xtol, gtol, max_nfev, x_scale, + loss_function, tr_solver, tr_options, verbose): + f = f0 + f_true = f.copy() + nfev = 1 + + J = J0 + njev = 1 + + if loss_function is not None: + rho = loss_function(f) + cost = 0.5 * np.sum(rho[0]) + J, f = scale_for_robust_loss_function(J, f, rho) + else: + cost = 0.5 * np.dot(f, f) + + g = compute_grad(J, f) + + jac_scale = isinstance(x_scale, str) and x_scale == 'jac' + if jac_scale: + scale, scale_inv = compute_jac_scale(J) + else: + scale, scale_inv = x_scale, 1 / x_scale + + Delta = norm(x0 * scale_inv, ord=np.inf) + if Delta == 0: + Delta = 1.0 + + on_bound = np.zeros_like(x0, dtype=int) + on_bound[np.equal(x0, lb)] = -1 + on_bound[np.equal(x0, ub)] = 1 + + x = x0 + step = np.empty_like(x0) + + if max_nfev is None: + max_nfev = x0.size * 100 + + termination_status = None + iteration = 0 + step_norm = None + actual_reduction = None + + if verbose == 2: + print_header_nonlinear() + + while True: + active_set = on_bound * g < 0 + free_set = ~active_set + + g_free = g[free_set] + g_full = g.copy() + g[active_set] = 0 + + g_norm = norm(g, ord=np.inf) + if g_norm < gtol: + termination_status = 1 + + if verbose == 2: + print_iteration_nonlinear(iteration, nfev, cost, actual_reduction, + step_norm, g_norm) + + if termination_status is not None or nfev == max_nfev: + break + + x_free = x[free_set] + lb_free = lb[free_set] + ub_free = ub[free_set] + scale_free = scale[free_set] + + # Compute (Gauss-)Newton and build quadratic model for Cauchy step. + if tr_solver == 'exact': + J_free = J[:, free_set] + newton_step = lstsq(J_free, -f, rcond=-1)[0] + + # Coefficients for the quadratic model along the anti-gradient. + a, b = build_quadratic_1d(J_free, g_free, -g_free) + elif tr_solver == 'lsmr': + Jop = aslinearoperator(J) + + # We compute lsmr step in scaled variables and then + # transform back to normal variables, if lsmr would give exact lsq + # solution, this would be equivalent to not doing any + # transformations, but from experience it's better this way. + + # We pass active_set to make computations as if we selected + # the free subset of J columns, but without actually doing any + # slicing, which is expensive for sparse matrices and impossible + # for LinearOperator. + + lsmr_op = lsmr_operator(Jop, scale, active_set) + newton_step = -lsmr(lsmr_op, f, **tr_options)[0][free_set] + newton_step *= scale_free + + # Components of g for active variables were zeroed, so this call + # is correct and equivalent to using J_free and g_free. + a, b = build_quadratic_1d(Jop, g, -g) + + actual_reduction = -1.0 + while actual_reduction <= 0 and nfev < max_nfev: + tr_bounds = Delta * scale_free + + step_free, on_bound_free, tr_hit = dogleg_step( + x_free, newton_step, g_free, a, b, tr_bounds, lb_free, ub_free) + + step.fill(0.0) + step[free_set] = step_free + + if tr_solver == 'exact': + predicted_reduction = -evaluate_quadratic(J_free, g_free, + step_free) + elif tr_solver == 'lsmr': + predicted_reduction = -evaluate_quadratic(Jop, g, step) + + # gh11403 ensure that solution is fully within bounds. + x_new = np.clip(x + step, lb, ub) + + f_new = fun(x_new) + nfev += 1 + + step_h_norm = norm(step * scale_inv, ord=np.inf) + + if not np.all(np.isfinite(f_new)): + Delta = 0.25 * step_h_norm + continue + + # Usual trust-region step quality estimation. + if loss_function is not None: + cost_new = loss_function(f_new, cost_only=True) + else: + cost_new = 0.5 * np.dot(f_new, f_new) + actual_reduction = cost - cost_new + + Delta, ratio = update_tr_radius( + Delta, actual_reduction, predicted_reduction, + step_h_norm, tr_hit + ) + + step_norm = norm(step) + termination_status = check_termination( + actual_reduction, cost, step_norm, norm(x), ratio, ftol, xtol) + + if termination_status is not None: + break + + if actual_reduction > 0: + on_bound[free_set] = on_bound_free + + x = x_new + # Set variables exactly at the boundary. + mask = on_bound == -1 + x[mask] = lb[mask] + mask = on_bound == 1 + x[mask] = ub[mask] + + f = f_new + f_true = f.copy() + + cost = cost_new + + J = jac(x, f) + njev += 1 + + if loss_function is not None: + rho = loss_function(f) + J, f = scale_for_robust_loss_function(J, f, rho) + + g = compute_grad(J, f) + + if jac_scale: + scale, scale_inv = compute_jac_scale(J, scale_inv) + else: + step_norm = 0 + actual_reduction = 0 + + iteration += 1 + + if termination_status is None: + termination_status = 0 + + return OptimizeResult( + x=x, cost=cost, fun=f_true, jac=J, grad=g_full, optimality=g_norm, + active_mask=on_bound, nfev=nfev, njev=njev, status=termination_status) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/least_squares.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/least_squares.py new file mode 100644 index 0000000000000000000000000000000000000000..1595e40d16a01b8355510c4721ca0fb6b5b23b4a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/least_squares.py @@ -0,0 +1,972 @@ +"""Generic interface for least-squares minimization.""" +from warnings import warn + +import numpy as np +from numpy.linalg import norm + +from scipy.sparse import issparse +from scipy.sparse.linalg import LinearOperator +from scipy.optimize import _minpack, OptimizeResult +from scipy.optimize._numdiff import approx_derivative, group_columns +from scipy.optimize._minimize import Bounds + +from .trf import trf +from .dogbox import dogbox +from .common import EPS, in_bounds, make_strictly_feasible + + +TERMINATION_MESSAGES = { + -1: "Improper input parameters status returned from `leastsq`", + 0: "The maximum number of function evaluations is exceeded.", + 1: "`gtol` termination condition is satisfied.", + 2: "`ftol` termination condition is satisfied.", + 3: "`xtol` termination condition is satisfied.", + 4: "Both `ftol` and `xtol` termination conditions are satisfied." +} + + +FROM_MINPACK_TO_COMMON = { + 0: -1, # Improper input parameters from MINPACK. + 1: 2, + 2: 3, + 3: 4, + 4: 1, + 5: 0 + # There are 6, 7, 8 for too small tolerance parameters, + # but we guard against it by checking ftol, xtol, gtol beforehand. +} + + +def call_minpack(fun, x0, jac, ftol, xtol, gtol, max_nfev, x_scale, diff_step): + n = x0.size + + if diff_step is None: + epsfcn = EPS + else: + epsfcn = diff_step**2 + + # Compute MINPACK's `diag`, which is inverse of our `x_scale` and + # ``x_scale='jac'`` corresponds to ``diag=None``. + if isinstance(x_scale, str) and x_scale == 'jac': + diag = None + else: + diag = 1 / x_scale + + full_output = True + col_deriv = False + factor = 100.0 + + if jac is None: + if max_nfev is None: + # n squared to account for Jacobian evaluations. + max_nfev = 100 * n * (n + 1) + x, info, status = _minpack._lmdif( + fun, x0, (), full_output, ftol, xtol, gtol, + max_nfev, epsfcn, factor, diag) + else: + if max_nfev is None: + max_nfev = 100 * n + x, info, status = _minpack._lmder( + fun, jac, x0, (), full_output, col_deriv, + ftol, xtol, gtol, max_nfev, factor, diag) + + f = info['fvec'] + + if callable(jac): + J = jac(x) + else: + J = np.atleast_2d(approx_derivative(fun, x)) + + cost = 0.5 * np.dot(f, f) + g = J.T.dot(f) + g_norm = norm(g, ord=np.inf) + + nfev = info['nfev'] + njev = info.get('njev', None) + + status = FROM_MINPACK_TO_COMMON[status] + active_mask = np.zeros_like(x0, dtype=int) + + return OptimizeResult( + x=x, cost=cost, fun=f, jac=J, grad=g, optimality=g_norm, + active_mask=active_mask, nfev=nfev, njev=njev, status=status) + + +def prepare_bounds(bounds, n): + lb, ub = (np.asarray(b, dtype=float) for b in bounds) + if lb.ndim == 0: + lb = np.resize(lb, n) + + if ub.ndim == 0: + ub = np.resize(ub, n) + + return lb, ub + + +def check_tolerance(ftol, xtol, gtol, method): + def check(tol, name): + if tol is None: + tol = 0 + elif tol < EPS: + warn(f"Setting `{name}` below the machine epsilon ({EPS:.2e}) effectively " + f"disables the corresponding termination condition.", + stacklevel=3) + return tol + + ftol = check(ftol, "ftol") + xtol = check(xtol, "xtol") + gtol = check(gtol, "gtol") + + if method == "lm" and (ftol < EPS or xtol < EPS or gtol < EPS): + raise ValueError("All tolerances must be higher than machine epsilon " + f"({EPS:.2e}) for method 'lm'.") + elif ftol < EPS and xtol < EPS and gtol < EPS: + raise ValueError("At least one of the tolerances must be higher than " + f"machine epsilon ({EPS:.2e}).") + + return ftol, xtol, gtol + + +def check_x_scale(x_scale, x0): + if isinstance(x_scale, str) and x_scale == 'jac': + return x_scale + + try: + x_scale = np.asarray(x_scale, dtype=float) + valid = np.all(np.isfinite(x_scale)) and np.all(x_scale > 0) + except (ValueError, TypeError): + valid = False + + if not valid: + raise ValueError("`x_scale` must be 'jac' or array_like with " + "positive numbers.") + + if x_scale.ndim == 0: + x_scale = np.resize(x_scale, x0.shape) + + if x_scale.shape != x0.shape: + raise ValueError("Inconsistent shapes between `x_scale` and `x0`.") + + return x_scale + + +def check_jac_sparsity(jac_sparsity, m, n): + if jac_sparsity is None: + return None + + if not issparse(jac_sparsity): + jac_sparsity = np.atleast_2d(jac_sparsity) + + if jac_sparsity.shape != (m, n): + raise ValueError("`jac_sparsity` has wrong shape.") + + return jac_sparsity, group_columns(jac_sparsity) + + +# Loss functions. + + +def huber(z, rho, cost_only): + mask = z <= 1 + rho[0, mask] = z[mask] + rho[0, ~mask] = 2 * z[~mask]**0.5 - 1 + if cost_only: + return + rho[1, mask] = 1 + rho[1, ~mask] = z[~mask]**-0.5 + rho[2, mask] = 0 + rho[2, ~mask] = -0.5 * z[~mask]**-1.5 + + +def soft_l1(z, rho, cost_only): + t = 1 + z + rho[0] = 2 * (t**0.5 - 1) + if cost_only: + return + rho[1] = t**-0.5 + rho[2] = -0.5 * t**-1.5 + + +def cauchy(z, rho, cost_only): + rho[0] = np.log1p(z) + if cost_only: + return + t = 1 + z + rho[1] = 1 / t + rho[2] = -1 / t**2 + + +def arctan(z, rho, cost_only): + rho[0] = np.arctan(z) + if cost_only: + return + t = 1 + z**2 + rho[1] = 1 / t + rho[2] = -2 * z / t**2 + + +IMPLEMENTED_LOSSES = dict(linear=None, huber=huber, soft_l1=soft_l1, + cauchy=cauchy, arctan=arctan) + + +def construct_loss_function(m, loss, f_scale): + if loss == 'linear': + return None + + if not callable(loss): + loss = IMPLEMENTED_LOSSES[loss] + rho = np.empty((3, m)) + + def loss_function(f, cost_only=False): + z = (f / f_scale) ** 2 + loss(z, rho, cost_only=cost_only) + if cost_only: + return 0.5 * f_scale ** 2 * np.sum(rho[0]) + rho[0] *= f_scale ** 2 + rho[2] /= f_scale ** 2 + return rho + else: + def loss_function(f, cost_only=False): + z = (f / f_scale) ** 2 + rho = loss(z) + if cost_only: + return 0.5 * f_scale ** 2 * np.sum(rho[0]) + rho[0] *= f_scale ** 2 + rho[2] /= f_scale ** 2 + return rho + + return loss_function + + +def least_squares( + fun, x0, jac='2-point', bounds=(-np.inf, np.inf), method='trf', + ftol=1e-8, xtol=1e-8, gtol=1e-8, x_scale=1.0, loss='linear', + f_scale=1.0, diff_step=None, tr_solver=None, tr_options=None, + jac_sparsity=None, max_nfev=None, verbose=0, args=(), kwargs=None): + """Solve a nonlinear least-squares problem with bounds on the variables. + + Given the residuals f(x) (an m-D real function of n real + variables) and the loss function rho(s) (a scalar function), `least_squares` + finds a local minimum of the cost function F(x):: + + minimize F(x) = 0.5 * sum(rho(f_i(x)**2), i = 0, ..., m - 1) + subject to lb <= x <= ub + + The purpose of the loss function rho(s) is to reduce the influence of + outliers on the solution. + + Parameters + ---------- + fun : callable + Function which computes the vector of residuals, with the signature + ``fun(x, *args, **kwargs)``, i.e., the minimization proceeds with + respect to its first argument. The argument ``x`` passed to this + function is an ndarray of shape (n,) (never a scalar, even for n=1). + It must allocate and return a 1-D array_like of shape (m,) or a scalar. + If the argument ``x`` is complex or the function ``fun`` returns + complex residuals, it must be wrapped in a real function of real + arguments, as shown at the end of the Examples section. + x0 : array_like with shape (n,) or float + Initial guess on independent variables. If float, it will be treated + as a 1-D array with one element. When `method` is 'trf', the initial + guess might be slightly adjusted to lie sufficiently within the given + `bounds`. + jac : {'2-point', '3-point', 'cs', callable}, optional + Method of computing the Jacobian matrix (an m-by-n matrix, where + element (i, j) is the partial derivative of f[i] with respect to + x[j]). The keywords select a finite difference scheme for numerical + estimation. The scheme '3-point' is more accurate, but requires + twice as many operations as '2-point' (default). The scheme 'cs' + uses complex steps, and while potentially the most accurate, it is + applicable only when `fun` correctly handles complex inputs and + can be analytically continued to the complex plane. Method 'lm' + always uses the '2-point' scheme. If callable, it is used as + ``jac(x, *args, **kwargs)`` and should return a good approximation + (or the exact value) for the Jacobian as an array_like (np.atleast_2d + is applied), a sparse matrix (csr_matrix preferred for performance) or + a `scipy.sparse.linalg.LinearOperator`. + bounds : 2-tuple of array_like or `Bounds`, optional + There are two ways to specify bounds: + + 1. Instance of `Bounds` class + 2. Lower and upper bounds on independent variables. Defaults to no + bounds. Each array must match the size of `x0` or be a scalar, + in the latter case a bound will be the same for all variables. + Use ``np.inf`` with an appropriate sign to disable bounds on all + or some variables. + + method : {'trf', 'dogbox', 'lm'}, optional + Algorithm to perform minimization. + + * 'trf' : Trust Region Reflective algorithm, particularly suitable + for large sparse problems with bounds. Generally robust method. + * 'dogbox' : dogleg algorithm with rectangular trust regions, + typical use case is small problems with bounds. Not recommended + for problems with rank-deficient Jacobian. + * 'lm' : Levenberg-Marquardt algorithm as implemented in MINPACK. + Doesn't handle bounds and sparse Jacobians. Usually the most + efficient method for small unconstrained problems. + + Default is 'trf'. See Notes for more information. + ftol : float or None, optional + Tolerance for termination by the change of the cost function. Default + is 1e-8. The optimization process is stopped when ``dF < ftol * F``, + and there was an adequate agreement between a local quadratic model and + the true model in the last step. + + If None and 'method' is not 'lm', the termination by this condition is + disabled. If 'method' is 'lm', this tolerance must be higher than + machine epsilon. + xtol : float or None, optional + Tolerance for termination by the change of the independent variables. + Default is 1e-8. The exact condition depends on the `method` used: + + * For 'trf' and 'dogbox' : ``norm(dx) < xtol * (xtol + norm(x))``. + * For 'lm' : ``Delta < xtol * norm(xs)``, where ``Delta`` is + a trust-region radius and ``xs`` is the value of ``x`` + scaled according to `x_scale` parameter (see below). + + If None and 'method' is not 'lm', the termination by this condition is + disabled. If 'method' is 'lm', this tolerance must be higher than + machine epsilon. + gtol : float or None, optional + Tolerance for termination by the norm of the gradient. Default is 1e-8. + The exact condition depends on a `method` used: + + * For 'trf' : ``norm(g_scaled, ord=np.inf) < gtol``, where + ``g_scaled`` is the value of the gradient scaled to account for + the presence of the bounds [STIR]_. + * For 'dogbox' : ``norm(g_free, ord=np.inf) < gtol``, where + ``g_free`` is the gradient with respect to the variables which + are not in the optimal state on the boundary. + * For 'lm' : the maximum absolute value of the cosine of angles + between columns of the Jacobian and the residual vector is less + than `gtol`, or the residual vector is zero. + + If None and 'method' is not 'lm', the termination by this condition is + disabled. If 'method' is 'lm', this tolerance must be higher than + machine epsilon. + x_scale : array_like or 'jac', optional + Characteristic scale of each variable. Setting `x_scale` is equivalent + to reformulating the problem in scaled variables ``xs = x / x_scale``. + An alternative view is that the size of a trust region along jth + dimension is proportional to ``x_scale[j]``. Improved convergence may + be achieved by setting `x_scale` such that a step of a given size + along any of the scaled variables has a similar effect on the cost + function. If set to 'jac', the scale is iteratively updated using the + inverse norms of the columns of the Jacobian matrix (as described in + [JJMore]_). + loss : str or callable, optional + Determines the loss function. The following keyword values are allowed: + + * 'linear' (default) : ``rho(z) = z``. Gives a standard + least-squares problem. + * 'soft_l1' : ``rho(z) = 2 * ((1 + z)**0.5 - 1)``. The smooth + approximation of l1 (absolute value) loss. Usually a good + choice for robust least squares. + * 'huber' : ``rho(z) = z if z <= 1 else 2*z**0.5 - 1``. Works + similarly to 'soft_l1'. + * 'cauchy' : ``rho(z) = ln(1 + z)``. Severely weakens outliers + influence, but may cause difficulties in optimization process. + * 'arctan' : ``rho(z) = arctan(z)``. Limits a maximum loss on + a single residual, has properties similar to 'cauchy'. + + If callable, it must take a 1-D ndarray ``z=f**2`` and return an + array_like with shape (3, m) where row 0 contains function values, + row 1 contains first derivatives and row 2 contains second + derivatives. Method 'lm' supports only 'linear' loss. + f_scale : float, optional + Value of soft margin between inlier and outlier residuals, default + is 1.0. The loss function is evaluated as follows + ``rho_(f**2) = C**2 * rho(f**2 / C**2)``, where ``C`` is `f_scale`, + and ``rho`` is determined by `loss` parameter. This parameter has + no effect with ``loss='linear'``, but for other `loss` values it is + of crucial importance. + max_nfev : None or int, optional + Maximum number of function evaluations before the termination. + If None (default), the value is chosen automatically: + + * For 'trf' and 'dogbox' : 100 * n. + * For 'lm' : 100 * n if `jac` is callable and 100 * n * (n + 1) + otherwise (because 'lm' counts function calls in Jacobian + estimation). + + diff_step : None or array_like, optional + Determines the relative step size for the finite difference + approximation of the Jacobian. The actual step is computed as + ``x * diff_step``. If None (default), then `diff_step` is taken to be + a conventional "optimal" power of machine epsilon for the finite + difference scheme used [NR]_. + tr_solver : {None, 'exact', 'lsmr'}, optional + Method for solving trust-region subproblems, relevant only for 'trf' + and 'dogbox' methods. + + * 'exact' is suitable for not very large problems with dense + Jacobian matrices. The computational complexity per iteration is + comparable to a singular value decomposition of the Jacobian + matrix. + * 'lsmr' is suitable for problems with sparse and large Jacobian + matrices. It uses the iterative procedure + `scipy.sparse.linalg.lsmr` for finding a solution of a linear + least-squares problem and only requires matrix-vector product + evaluations. + + If None (default), the solver is chosen based on the type of Jacobian + returned on the first iteration. + tr_options : dict, optional + Keyword options passed to trust-region solver. + + * ``tr_solver='exact'``: `tr_options` are ignored. + * ``tr_solver='lsmr'``: options for `scipy.sparse.linalg.lsmr`. + Additionally, ``method='trf'`` supports 'regularize' option + (bool, default is True), which adds a regularization term to the + normal equation, which improves convergence if the Jacobian is + rank-deficient [Byrd]_ (eq. 3.4). + + jac_sparsity : {None, array_like, sparse matrix}, optional + Defines the sparsity structure of the Jacobian matrix for finite + difference estimation, its shape must be (m, n). If the Jacobian has + only few non-zero elements in *each* row, providing the sparsity + structure will greatly speed up the computations [Curtis]_. A zero + entry means that a corresponding element in the Jacobian is identically + zero. If provided, forces the use of 'lsmr' trust-region solver. + If None (default), then dense differencing will be used. Has no effect + for 'lm' method. + verbose : {0, 1, 2}, optional + Level of algorithm's verbosity: + + * 0 (default) : work silently. + * 1 : display a termination report. + * 2 : display progress during iterations (not supported by 'lm' + method). + + args, kwargs : tuple and dict, optional + Additional arguments passed to `fun` and `jac`. Both empty by default. + The calling signature is ``fun(x, *args, **kwargs)`` and the same for + `jac`. + + Returns + ------- + result : OptimizeResult + `OptimizeResult` with the following fields defined: + + x : ndarray, shape (n,) + Solution found. + cost : float + Value of the cost function at the solution. + fun : ndarray, shape (m,) + Vector of residuals at the solution. + jac : ndarray, sparse matrix or LinearOperator, shape (m, n) + Modified Jacobian matrix at the solution, in the sense that J^T J + is a Gauss-Newton approximation of the Hessian of the cost function. + The type is the same as the one used by the algorithm. + grad : ndarray, shape (m,) + Gradient of the cost function at the solution. + optimality : float + First-order optimality measure. In unconstrained problems, it is + always the uniform norm of the gradient. In constrained problems, + it is the quantity which was compared with `gtol` during iterations. + active_mask : ndarray of int, shape (n,) + Each component shows whether a corresponding constraint is active + (that is, whether a variable is at the bound): + + * 0 : a constraint is not active. + * -1 : a lower bound is active. + * 1 : an upper bound is active. + + Might be somewhat arbitrary for 'trf' method as it generates a + sequence of strictly feasible iterates and `active_mask` is + determined within a tolerance threshold. + nfev : int + Number of function evaluations done. Methods 'trf' and 'dogbox' do + not count function calls for numerical Jacobian approximation, as + opposed to 'lm' method. + njev : int or None + Number of Jacobian evaluations done. If numerical Jacobian + approximation is used in 'lm' method, it is set to None. + status : int + The reason for algorithm termination: + + * -1 : improper input parameters status returned from MINPACK. + * 0 : the maximum number of function evaluations is exceeded. + * 1 : `gtol` termination condition is satisfied. + * 2 : `ftol` termination condition is satisfied. + * 3 : `xtol` termination condition is satisfied. + * 4 : Both `ftol` and `xtol` termination conditions are satisfied. + + message : str + Verbal description of the termination reason. + success : bool + True if one of the convergence criteria is satisfied (`status` > 0). + + See Also + -------- + leastsq : A legacy wrapper for the MINPACK implementation of the + Levenberg-Marquadt algorithm. + curve_fit : Least-squares minimization applied to a curve-fitting problem. + + Notes + ----- + Method 'lm' (Levenberg-Marquardt) calls a wrapper over least-squares + algorithms implemented in MINPACK (lmder, lmdif). It runs the + Levenberg-Marquardt algorithm formulated as a trust-region type algorithm. + The implementation is based on paper [JJMore]_, it is very robust and + efficient with a lot of smart tricks. It should be your first choice + for unconstrained problems. Note that it doesn't support bounds. Also, + it doesn't work when m < n. + + Method 'trf' (Trust Region Reflective) is motivated by the process of + solving a system of equations, which constitute the first-order optimality + condition for a bound-constrained minimization problem as formulated in + [STIR]_. The algorithm iteratively solves trust-region subproblems + augmented by a special diagonal quadratic term and with trust-region shape + determined by the distance from the bounds and the direction of the + gradient. This enhancements help to avoid making steps directly into bounds + and efficiently explore the whole space of variables. To further improve + convergence, the algorithm considers search directions reflected from the + bounds. To obey theoretical requirements, the algorithm keeps iterates + strictly feasible. With dense Jacobians trust-region subproblems are + solved by an exact method very similar to the one described in [JJMore]_ + (and implemented in MINPACK). The difference from the MINPACK + implementation is that a singular value decomposition of a Jacobian + matrix is done once per iteration, instead of a QR decomposition and series + of Givens rotation eliminations. For large sparse Jacobians a 2-D subspace + approach of solving trust-region subproblems is used [STIR]_, [Byrd]_. + The subspace is spanned by a scaled gradient and an approximate + Gauss-Newton solution delivered by `scipy.sparse.linalg.lsmr`. When no + constraints are imposed the algorithm is very similar to MINPACK and has + generally comparable performance. The algorithm works quite robust in + unbounded and bounded problems, thus it is chosen as a default algorithm. + + Method 'dogbox' operates in a trust-region framework, but considers + rectangular trust regions as opposed to conventional ellipsoids [Voglis]_. + The intersection of a current trust region and initial bounds is again + rectangular, so on each iteration a quadratic minimization problem subject + to bound constraints is solved approximately by Powell's dogleg method + [NumOpt]_. The required Gauss-Newton step can be computed exactly for + dense Jacobians or approximately by `scipy.sparse.linalg.lsmr` for large + sparse Jacobians. The algorithm is likely to exhibit slow convergence when + the rank of Jacobian is less than the number of variables. The algorithm + often outperforms 'trf' in bounded problems with a small number of + variables. + + Robust loss functions are implemented as described in [BA]_. The idea + is to modify a residual vector and a Jacobian matrix on each iteration + such that computed gradient and Gauss-Newton Hessian approximation match + the true gradient and Hessian approximation of the cost function. Then + the algorithm proceeds in a normal way, i.e., robust loss functions are + implemented as a simple wrapper over standard least-squares algorithms. + + .. versionadded:: 0.17.0 + + References + ---------- + .. [STIR] M. A. Branch, T. F. Coleman, and Y. Li, "A Subspace, Interior, + and Conjugate Gradient Method for Large-Scale Bound-Constrained + Minimization Problems," SIAM Journal on Scientific Computing, + Vol. 21, Number 1, pp 1-23, 1999. + .. [NR] William H. Press et. al., "Numerical Recipes. The Art of Scientific + Computing. 3rd edition", Sec. 5.7. + .. [Byrd] R. H. Byrd, R. B. Schnabel and G. A. Shultz, "Approximate + solution of the trust region problem by minimization over + two-dimensional subspaces", Math. Programming, 40, pp. 247-263, + 1988. + .. [Curtis] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of + sparse Jacobian matrices", Journal of the Institute of + Mathematics and its Applications, 13, pp. 117-120, 1974. + .. [JJMore] J. J. More, "The Levenberg-Marquardt Algorithm: Implementation + and Theory," Numerical Analysis, ed. G. A. Watson, Lecture + Notes in Mathematics 630, Springer Verlag, pp. 105-116, 1977. + .. [Voglis] C. Voglis and I. E. Lagaris, "A Rectangular Trust Region + Dogleg Approach for Unconstrained and Bound Constrained + Nonlinear Optimization", WSEAS International Conference on + Applied Mathematics, Corfu, Greece, 2004. + .. [NumOpt] J. Nocedal and S. J. Wright, "Numerical optimization, + 2nd edition", Chapter 4. + .. [BA] B. Triggs et. al., "Bundle Adjustment - A Modern Synthesis", + Proceedings of the International Workshop on Vision Algorithms: + Theory and Practice, pp. 298-372, 1999. + + Examples + -------- + In this example we find a minimum of the Rosenbrock function without bounds + on independent variables. + + >>> import numpy as np + >>> def fun_rosenbrock(x): + ... return np.array([10 * (x[1] - x[0]**2), (1 - x[0])]) + + Notice that we only provide the vector of the residuals. The algorithm + constructs the cost function as a sum of squares of the residuals, which + gives the Rosenbrock function. The exact minimum is at ``x = [1.0, 1.0]``. + + >>> from scipy.optimize import least_squares + >>> x0_rosenbrock = np.array([2, 2]) + >>> res_1 = least_squares(fun_rosenbrock, x0_rosenbrock) + >>> res_1.x + array([ 1., 1.]) + >>> res_1.cost + 9.8669242910846867e-30 + >>> res_1.optimality + 8.8928864934219529e-14 + + We now constrain the variables, in such a way that the previous solution + becomes infeasible. Specifically, we require that ``x[1] >= 1.5``, and + ``x[0]`` left unconstrained. To this end, we specify the `bounds` parameter + to `least_squares` in the form ``bounds=([-np.inf, 1.5], np.inf)``. + + We also provide the analytic Jacobian: + + >>> def jac_rosenbrock(x): + ... return np.array([ + ... [-20 * x[0], 10], + ... [-1, 0]]) + + Putting this all together, we see that the new solution lies on the bound: + + >>> res_2 = least_squares(fun_rosenbrock, x0_rosenbrock, jac_rosenbrock, + ... bounds=([-np.inf, 1.5], np.inf)) + >>> res_2.x + array([ 1.22437075, 1.5 ]) + >>> res_2.cost + 0.025213093946805685 + >>> res_2.optimality + 1.5885401433157753e-07 + + Now we solve a system of equations (i.e., the cost function should be zero + at a minimum) for a Broyden tridiagonal vector-valued function of 100000 + variables: + + >>> def fun_broyden(x): + ... f = (3 - x) * x + 1 + ... f[1:] -= x[:-1] + ... f[:-1] -= 2 * x[1:] + ... return f + + The corresponding Jacobian matrix is sparse. We tell the algorithm to + estimate it by finite differences and provide the sparsity structure of + Jacobian to significantly speed up this process. + + >>> from scipy.sparse import lil_matrix + >>> def sparsity_broyden(n): + ... sparsity = lil_matrix((n, n), dtype=int) + ... i = np.arange(n) + ... sparsity[i, i] = 1 + ... i = np.arange(1, n) + ... sparsity[i, i - 1] = 1 + ... i = np.arange(n - 1) + ... sparsity[i, i + 1] = 1 + ... return sparsity + ... + >>> n = 100000 + >>> x0_broyden = -np.ones(n) + ... + >>> res_3 = least_squares(fun_broyden, x0_broyden, + ... jac_sparsity=sparsity_broyden(n)) + >>> res_3.cost + 4.5687069299604613e-23 + >>> res_3.optimality + 1.1650454296851518e-11 + + Let's also solve a curve fitting problem using robust loss function to + take care of outliers in the data. Define the model function as + ``y = a + b * exp(c * t)``, where t is a predictor variable, y is an + observation and a, b, c are parameters to estimate. + + First, define the function which generates the data with noise and + outliers, define the model parameters, and generate data: + + >>> from numpy.random import default_rng + >>> rng = default_rng() + >>> def gen_data(t, a, b, c, noise=0., n_outliers=0, seed=None): + ... rng = default_rng(seed) + ... + ... y = a + b * np.exp(t * c) + ... + ... error = noise * rng.standard_normal(t.size) + ... outliers = rng.integers(0, t.size, n_outliers) + ... error[outliers] *= 10 + ... + ... return y + error + ... + >>> a = 0.5 + >>> b = 2.0 + >>> c = -1 + >>> t_min = 0 + >>> t_max = 10 + >>> n_points = 15 + ... + >>> t_train = np.linspace(t_min, t_max, n_points) + >>> y_train = gen_data(t_train, a, b, c, noise=0.1, n_outliers=3) + + Define function for computing residuals and initial estimate of + parameters. + + >>> def fun(x, t, y): + ... return x[0] + x[1] * np.exp(x[2] * t) - y + ... + >>> x0 = np.array([1.0, 1.0, 0.0]) + + Compute a standard least-squares solution: + + >>> res_lsq = least_squares(fun, x0, args=(t_train, y_train)) + + Now compute two solutions with two different robust loss functions. The + parameter `f_scale` is set to 0.1, meaning that inlier residuals should + not significantly exceed 0.1 (the noise level used). + + >>> res_soft_l1 = least_squares(fun, x0, loss='soft_l1', f_scale=0.1, + ... args=(t_train, y_train)) + >>> res_log = least_squares(fun, x0, loss='cauchy', f_scale=0.1, + ... args=(t_train, y_train)) + + And, finally, plot all the curves. We see that by selecting an appropriate + `loss` we can get estimates close to optimal even in the presence of + strong outliers. But keep in mind that generally it is recommended to try + 'soft_l1' or 'huber' losses first (if at all necessary) as the other two + options may cause difficulties in optimization process. + + >>> t_test = np.linspace(t_min, t_max, n_points * 10) + >>> y_true = gen_data(t_test, a, b, c) + >>> y_lsq = gen_data(t_test, *res_lsq.x) + >>> y_soft_l1 = gen_data(t_test, *res_soft_l1.x) + >>> y_log = gen_data(t_test, *res_log.x) + ... + >>> import matplotlib.pyplot as plt + >>> plt.plot(t_train, y_train, 'o') + >>> plt.plot(t_test, y_true, 'k', linewidth=2, label='true') + >>> plt.plot(t_test, y_lsq, label='linear loss') + >>> plt.plot(t_test, y_soft_l1, label='soft_l1 loss') + >>> plt.plot(t_test, y_log, label='cauchy loss') + >>> plt.xlabel("t") + >>> plt.ylabel("y") + >>> plt.legend() + >>> plt.show() + + In the next example, we show how complex-valued residual functions of + complex variables can be optimized with ``least_squares()``. Consider the + following function: + + >>> def f(z): + ... return z - (0.5 + 0.5j) + + We wrap it into a function of real variables that returns real residuals + by simply handling the real and imaginary parts as independent variables: + + >>> def f_wrap(x): + ... fx = f(x[0] + 1j*x[1]) + ... return np.array([fx.real, fx.imag]) + + Thus, instead of the original m-D complex function of n complex + variables we optimize a 2m-D real function of 2n real variables: + + >>> from scipy.optimize import least_squares + >>> res_wrapped = least_squares(f_wrap, (0.1, 0.1), bounds=([0, 0], [1, 1])) + >>> z = res_wrapped.x[0] + res_wrapped.x[1]*1j + >>> z + (0.49999999999925893+0.49999999999925893j) + + """ + if method not in ['trf', 'dogbox', 'lm']: + raise ValueError("`method` must be 'trf', 'dogbox' or 'lm'.") + + if jac not in ['2-point', '3-point', 'cs'] and not callable(jac): + raise ValueError("`jac` must be '2-point', '3-point', 'cs' or " + "callable.") + + if tr_solver not in [None, 'exact', 'lsmr']: + raise ValueError("`tr_solver` must be None, 'exact' or 'lsmr'.") + + if loss not in IMPLEMENTED_LOSSES and not callable(loss): + raise ValueError(f"`loss` must be one of {IMPLEMENTED_LOSSES.keys()}" + " or a callable.") + + if method == 'lm' and loss != 'linear': + raise ValueError("method='lm' supports only 'linear' loss function.") + + if verbose not in [0, 1, 2]: + raise ValueError("`verbose` must be in [0, 1, 2].") + + if max_nfev is not None and max_nfev <= 0: + raise ValueError("`max_nfev` must be None or positive integer.") + + if np.iscomplexobj(x0): + raise ValueError("`x0` must be real.") + + x0 = np.atleast_1d(x0).astype(float) + + if x0.ndim > 1: + raise ValueError("`x0` must have at most 1 dimension.") + + if isinstance(bounds, Bounds): + lb, ub = bounds.lb, bounds.ub + bounds = (lb, ub) + else: + if len(bounds) == 2: + lb, ub = prepare_bounds(bounds, x0.shape[0]) + else: + raise ValueError("`bounds` must contain 2 elements.") + + if method == 'lm' and not np.all((lb == -np.inf) & (ub == np.inf)): + raise ValueError("Method 'lm' doesn't support bounds.") + + if lb.shape != x0.shape or ub.shape != x0.shape: + raise ValueError("Inconsistent shapes between bounds and `x0`.") + + if np.any(lb >= ub): + raise ValueError("Each lower bound must be strictly less than each " + "upper bound.") + + if not in_bounds(x0, lb, ub): + raise ValueError("Initial guess is outside of provided bounds") + + x_scale = check_x_scale(x_scale, x0) + + ftol, xtol, gtol = check_tolerance(ftol, xtol, gtol, method) + + if method == 'trf': + x0 = make_strictly_feasible(x0, lb, ub) + + if kwargs is None: + kwargs = {} + if tr_options is None: + tr_options = {} + + def fun_wrapped(x): + return np.atleast_1d(fun(x, *args, **kwargs)) + + f0 = fun_wrapped(x0) + + if f0.ndim != 1: + raise ValueError("`fun` must return at most 1-d array_like. " + f"f0.shape: {f0.shape}") + + if not np.all(np.isfinite(f0)): + raise ValueError("Residuals are not finite in the initial point.") + + n = x0.size + m = f0.size + + if method == 'lm' and m < n: + raise ValueError("Method 'lm' doesn't work when the number of " + "residuals is less than the number of variables.") + + loss_function = construct_loss_function(m, loss, f_scale) + if callable(loss): + rho = loss_function(f0) + if rho.shape != (3, m): + raise ValueError("The return value of `loss` callable has wrong " + "shape.") + initial_cost = 0.5 * np.sum(rho[0]) + elif loss_function is not None: + initial_cost = loss_function(f0, cost_only=True) + else: + initial_cost = 0.5 * np.dot(f0, f0) + + if callable(jac): + J0 = jac(x0, *args, **kwargs) + + if issparse(J0): + J0 = J0.tocsr() + + def jac_wrapped(x, _=None): + return jac(x, *args, **kwargs).tocsr() + + elif isinstance(J0, LinearOperator): + def jac_wrapped(x, _=None): + return jac(x, *args, **kwargs) + + else: + J0 = np.atleast_2d(J0) + + def jac_wrapped(x, _=None): + return np.atleast_2d(jac(x, *args, **kwargs)) + + else: # Estimate Jacobian by finite differences. + if method == 'lm': + if jac_sparsity is not None: + raise ValueError("method='lm' does not support " + "`jac_sparsity`.") + + if jac != '2-point': + warn(f"jac='{jac}' works equivalently to '2-point' for method='lm'.", + stacklevel=2) + + J0 = jac_wrapped = None + else: + if jac_sparsity is not None and tr_solver == 'exact': + raise ValueError("tr_solver='exact' is incompatible " + "with `jac_sparsity`.") + + jac_sparsity = check_jac_sparsity(jac_sparsity, m, n) + + def jac_wrapped(x, f): + J = approx_derivative(fun, x, rel_step=diff_step, method=jac, + f0=f, bounds=bounds, args=args, + kwargs=kwargs, sparsity=jac_sparsity) + if J.ndim != 2: # J is guaranteed not sparse. + J = np.atleast_2d(J) + + return J + + J0 = jac_wrapped(x0, f0) + + if J0 is not None: + if J0.shape != (m, n): + raise ValueError( + f"The return value of `jac` has wrong shape: expected {(m, n)}, " + f"actual {J0.shape}." + ) + + if not isinstance(J0, np.ndarray): + if method == 'lm': + raise ValueError("method='lm' works only with dense " + "Jacobian matrices.") + + if tr_solver == 'exact': + raise ValueError( + "tr_solver='exact' works only with dense " + "Jacobian matrices.") + + jac_scale = isinstance(x_scale, str) and x_scale == 'jac' + if isinstance(J0, LinearOperator) and jac_scale: + raise ValueError("x_scale='jac' can't be used when `jac` " + "returns LinearOperator.") + + if tr_solver is None: + if isinstance(J0, np.ndarray): + tr_solver = 'exact' + else: + tr_solver = 'lsmr' + + if method == 'lm': + result = call_minpack(fun_wrapped, x0, jac_wrapped, ftol, xtol, gtol, + max_nfev, x_scale, diff_step) + + elif method == 'trf': + result = trf(fun_wrapped, jac_wrapped, x0, f0, J0, lb, ub, ftol, xtol, + gtol, max_nfev, x_scale, loss_function, tr_solver, + tr_options.copy(), verbose) + + elif method == 'dogbox': + if tr_solver == 'lsmr' and 'regularize' in tr_options: + warn("The keyword 'regularize' in `tr_options` is not relevant " + "for 'dogbox' method.", + stacklevel=2) + tr_options = tr_options.copy() + del tr_options['regularize'] + + result = dogbox(fun_wrapped, jac_wrapped, x0, f0, J0, lb, ub, ftol, + xtol, gtol, max_nfev, x_scale, loss_function, + tr_solver, tr_options, verbose) + + result.message = TERMINATION_MESSAGES[result.status] + result.success = result.status > 0 + + if verbose >= 1: + print(result.message) + print(f"Function evaluations {result.nfev}, initial cost {initial_cost:.4e}, " + f"final cost {result.cost:.4e}, " + f"first-order optimality {result.optimality:.2e}.") + + return result diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/lsq_linear.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/lsq_linear.py new file mode 100644 index 0000000000000000000000000000000000000000..b077c45e40874fc63490748f75f8463bc2adb08d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/lsq_linear.py @@ -0,0 +1,361 @@ +"""Linear least squares with bound constraints on independent variables.""" +import numpy as np +from numpy.linalg import norm +from scipy.sparse import issparse, csr_matrix +from scipy.sparse.linalg import LinearOperator, lsmr +from scipy.optimize import OptimizeResult +from scipy.optimize._minimize import Bounds + +from .common import in_bounds, compute_grad +from .trf_linear import trf_linear +from .bvls import bvls + + +def prepare_bounds(bounds, n): + if len(bounds) != 2: + raise ValueError("`bounds` must contain 2 elements.") + lb, ub = (np.asarray(b, dtype=float) for b in bounds) + + if lb.ndim == 0: + lb = np.resize(lb, n) + + if ub.ndim == 0: + ub = np.resize(ub, n) + + return lb, ub + + +TERMINATION_MESSAGES = { + -1: "The algorithm was not able to make progress on the last iteration.", + 0: "The maximum number of iterations is exceeded.", + 1: "The first-order optimality measure is less than `tol`.", + 2: "The relative change of the cost function is less than `tol`.", + 3: "The unconstrained solution is optimal." +} + + +def lsq_linear(A, b, bounds=(-np.inf, np.inf), method='trf', tol=1e-10, + lsq_solver=None, lsmr_tol=None, max_iter=None, + verbose=0, *, lsmr_maxiter=None,): + r"""Solve a linear least-squares problem with bounds on the variables. + + Given a m-by-n design matrix A and a target vector b with m elements, + `lsq_linear` solves the following optimization problem:: + + minimize 0.5 * ||A x - b||**2 + subject to lb <= x <= ub + + This optimization problem is convex, hence a found minimum (if iterations + have converged) is guaranteed to be global. + + Parameters + ---------- + A : array_like, sparse matrix of LinearOperator, shape (m, n) + Design matrix. Can be `scipy.sparse.linalg.LinearOperator`. + b : array_like, shape (m,) + Target vector. + bounds : 2-tuple of array_like or `Bounds`, optional + Lower and upper bounds on parameters. Defaults to no bounds. + There are two ways to specify the bounds: + + - Instance of `Bounds` class. + - 2-tuple of array_like: Each element of the tuple must be either + an array with the length equal to the number of parameters, or a + scalar (in which case the bound is taken to be the same for all + parameters). Use ``np.inf`` with an appropriate sign to disable + bounds on all or some parameters. + + method : 'trf' or 'bvls', optional + Method to perform minimization. + + * 'trf' : Trust Region Reflective algorithm adapted for a linear + least-squares problem. This is an interior-point-like method + and the required number of iterations is weakly correlated with + the number of variables. + * 'bvls' : Bounded-variable least-squares algorithm. This is + an active set method, which requires the number of iterations + comparable to the number of variables. Can't be used when `A` is + sparse or LinearOperator. + + Default is 'trf'. + tol : float, optional + Tolerance parameter. The algorithm terminates if a relative change + of the cost function is less than `tol` on the last iteration. + Additionally, the first-order optimality measure is considered: + + * ``method='trf'`` terminates if the uniform norm of the gradient, + scaled to account for the presence of the bounds, is less than + `tol`. + * ``method='bvls'`` terminates if Karush-Kuhn-Tucker conditions + are satisfied within `tol` tolerance. + + lsq_solver : {None, 'exact', 'lsmr'}, optional + Method of solving unbounded least-squares problems throughout + iterations: + + * 'exact' : Use dense QR or SVD decomposition approach. Can't be + used when `A` is sparse or LinearOperator. + * 'lsmr' : Use `scipy.sparse.linalg.lsmr` iterative procedure + which requires only matrix-vector product evaluations. Can't + be used with ``method='bvls'``. + + If None (default), the solver is chosen based on type of `A`. + lsmr_tol : None, float or 'auto', optional + Tolerance parameters 'atol' and 'btol' for `scipy.sparse.linalg.lsmr` + If None (default), it is set to ``1e-2 * tol``. If 'auto', the + tolerance will be adjusted based on the optimality of the current + iterate, which can speed up the optimization process, but is not always + reliable. + max_iter : None or int, optional + Maximum number of iterations before termination. If None (default), it + is set to 100 for ``method='trf'`` or to the number of variables for + ``method='bvls'`` (not counting iterations for 'bvls' initialization). + verbose : {0, 1, 2}, optional + Level of algorithm's verbosity: + + * 0 : work silently (default). + * 1 : display a termination report. + * 2 : display progress during iterations. + + lsmr_maxiter : None or int, optional + Maximum number of iterations for the lsmr least squares solver, + if it is used (by setting ``lsq_solver='lsmr'``). If None (default), it + uses lsmr's default of ``min(m, n)`` where ``m`` and ``n`` are the + number of rows and columns of `A`, respectively. Has no effect if + ``lsq_solver='exact'``. + + Returns + ------- + OptimizeResult with the following fields defined: + x : ndarray, shape (n,) + Solution found. + cost : float + Value of the cost function at the solution. + fun : ndarray, shape (m,) + Vector of residuals at the solution. + optimality : float + First-order optimality measure. The exact meaning depends on `method`, + refer to the description of `tol` parameter. + active_mask : ndarray of int, shape (n,) + Each component shows whether a corresponding constraint is active + (that is, whether a variable is at the bound): + + * 0 : a constraint is not active. + * -1 : a lower bound is active. + * 1 : an upper bound is active. + + Might be somewhat arbitrary for the `trf` method as it generates a + sequence of strictly feasible iterates and active_mask is determined + within a tolerance threshold. + unbounded_sol : tuple + Unbounded least squares solution tuple returned by the least squares + solver (set with `lsq_solver` option). If `lsq_solver` is not set or is + set to ``'exact'``, the tuple contains an ndarray of shape (n,) with + the unbounded solution, an ndarray with the sum of squared residuals, + an int with the rank of `A`, and an ndarray with the singular values + of `A` (see NumPy's ``linalg.lstsq`` for more information). If + `lsq_solver` is set to ``'lsmr'``, the tuple contains an ndarray of + shape (n,) with the unbounded solution, an int with the exit code, + an int with the number of iterations, and five floats with + various norms and the condition number of `A` (see SciPy's + ``sparse.linalg.lsmr`` for more information). This output can be + useful for determining the convergence of the least squares solver, + particularly the iterative ``'lsmr'`` solver. The unbounded least + squares problem is to minimize ``0.5 * ||A x - b||**2``. + nit : int + Number of iterations. Zero if the unconstrained solution is optimal. + status : int + Reason for algorithm termination: + + * -1 : the algorithm was not able to make progress on the last + iteration. + * 0 : the maximum number of iterations is exceeded. + * 1 : the first-order optimality measure is less than `tol`. + * 2 : the relative change of the cost function is less than `tol`. + * 3 : the unconstrained solution is optimal. + + message : str + Verbal description of the termination reason. + success : bool + True if one of the convergence criteria is satisfied (`status` > 0). + + See Also + -------- + nnls : Linear least squares with non-negativity constraint. + least_squares : Nonlinear least squares with bounds on the variables. + + Notes + ----- + The algorithm first computes the unconstrained least-squares solution by + `numpy.linalg.lstsq` or `scipy.sparse.linalg.lsmr` depending on + `lsq_solver`. This solution is returned as optimal if it lies within the + bounds. + + Method 'trf' runs the adaptation of the algorithm described in [STIR]_ for + a linear least-squares problem. The iterations are essentially the same as + in the nonlinear least-squares algorithm, but as the quadratic function + model is always accurate, we don't need to track or modify the radius of + a trust region. The line search (backtracking) is used as a safety net + when a selected step does not decrease the cost function. Read more + detailed description of the algorithm in `scipy.optimize.least_squares`. + + Method 'bvls' runs a Python implementation of the algorithm described in + [BVLS]_. The algorithm maintains active and free sets of variables, on + each iteration chooses a new variable to move from the active set to the + free set and then solves the unconstrained least-squares problem on free + variables. This algorithm is guaranteed to give an accurate solution + eventually, but may require up to n iterations for a problem with n + variables. Additionally, an ad-hoc initialization procedure is + implemented, that determines which variables to set free or active + initially. It takes some number of iterations before actual BVLS starts, + but can significantly reduce the number of further iterations. + + References + ---------- + .. [STIR] M. A. Branch, T. F. Coleman, and Y. Li, "A Subspace, Interior, + and Conjugate Gradient Method for Large-Scale Bound-Constrained + Minimization Problems," SIAM Journal on Scientific Computing, + Vol. 21, Number 1, pp 1-23, 1999. + .. [BVLS] P. B. Start and R. L. Parker, "Bounded-Variable Least-Squares: + an Algorithm and Applications", Computational Statistics, 10, + 129-141, 1995. + + Examples + -------- + In this example, a problem with a large sparse matrix and bounds on the + variables is solved. + + >>> import numpy as np + >>> from scipy.sparse import rand + >>> from scipy.optimize import lsq_linear + >>> rng = np.random.default_rng() + ... + >>> m = 2000 + >>> n = 1000 + ... + >>> A = rand(m, n, density=1e-4, random_state=rng) + >>> b = rng.standard_normal(m) + ... + >>> lb = rng.standard_normal(n) + >>> ub = lb + 1 + ... + >>> res = lsq_linear(A, b, bounds=(lb, ub), lsmr_tol='auto', verbose=1) + The relative change of the cost function is less than `tol`. + Number of iterations 10, initial cost 1.0070e+03, final cost 9.6602e+02, + first-order optimality 2.21e-09. # may vary + """ + if method not in ['trf', 'bvls']: + raise ValueError("`method` must be 'trf' or 'bvls'") + + if lsq_solver not in [None, 'exact', 'lsmr']: + raise ValueError("`solver` must be None, 'exact' or 'lsmr'.") + + if verbose not in [0, 1, 2]: + raise ValueError("`verbose` must be in [0, 1, 2].") + + if issparse(A): + A = csr_matrix(A) + elif not isinstance(A, LinearOperator): + A = np.atleast_2d(np.asarray(A)) + + if method == 'bvls': + if lsq_solver == 'lsmr': + raise ValueError("method='bvls' can't be used with " + "lsq_solver='lsmr'") + + if not isinstance(A, np.ndarray): + raise ValueError("method='bvls' can't be used with `A` being " + "sparse or LinearOperator.") + + if lsq_solver is None: + if isinstance(A, np.ndarray): + lsq_solver = 'exact' + else: + lsq_solver = 'lsmr' + elif lsq_solver == 'exact' and not isinstance(A, np.ndarray): + raise ValueError("`exact` solver can't be used when `A` is " + "sparse or LinearOperator.") + + if len(A.shape) != 2: # No ndim for LinearOperator. + raise ValueError("`A` must have at most 2 dimensions.") + + if max_iter is not None and max_iter <= 0: + raise ValueError("`max_iter` must be None or positive integer.") + + m, n = A.shape + + b = np.atleast_1d(b) + if b.ndim != 1: + raise ValueError("`b` must have at most 1 dimension.") + + if b.size != m: + raise ValueError("Inconsistent shapes between `A` and `b`.") + + if isinstance(bounds, Bounds): + lb = bounds.lb + ub = bounds.ub + else: + lb, ub = prepare_bounds(bounds, n) + + if lb.shape != (n,) and ub.shape != (n,): + raise ValueError("Bounds have wrong shape.") + + if np.any(lb >= ub): + raise ValueError("Each lower bound must be strictly less than each " + "upper bound.") + + if lsmr_maxiter is not None and lsmr_maxiter < 1: + raise ValueError("`lsmr_maxiter` must be None or positive integer.") + + if not ((isinstance(lsmr_tol, float) and lsmr_tol > 0) or + lsmr_tol in ('auto', None)): + raise ValueError("`lsmr_tol` must be None, 'auto', or positive float.") + + if lsq_solver == 'exact': + unbd_lsq = np.linalg.lstsq(A, b, rcond=-1) + elif lsq_solver == 'lsmr': + first_lsmr_tol = lsmr_tol # tol of first call to lsmr + if lsmr_tol is None or lsmr_tol == 'auto': + first_lsmr_tol = 1e-2 * tol # default if lsmr_tol not defined + unbd_lsq = lsmr(A, b, maxiter=lsmr_maxiter, + atol=first_lsmr_tol, btol=first_lsmr_tol) + x_lsq = unbd_lsq[0] # extract the solution from the least squares solver + + if in_bounds(x_lsq, lb, ub): + r = A @ x_lsq - b + cost = 0.5 * np.dot(r, r) + termination_status = 3 + termination_message = TERMINATION_MESSAGES[termination_status] + g = compute_grad(A, r) + g_norm = norm(g, ord=np.inf) + + if verbose > 0: + print(termination_message) + print(f"Final cost {cost:.4e}, first-order optimality {g_norm:.2e}") + + return OptimizeResult( + x=x_lsq, fun=r, cost=cost, optimality=g_norm, + active_mask=np.zeros(n), unbounded_sol=unbd_lsq, + nit=0, status=termination_status, + message=termination_message, success=True) + + if method == 'trf': + res = trf_linear(A, b, x_lsq, lb, ub, tol, lsq_solver, lsmr_tol, + max_iter, verbose, lsmr_maxiter=lsmr_maxiter) + elif method == 'bvls': + res = bvls(A, b, x_lsq, lb, ub, tol, max_iter, verbose) + + res.unbounded_sol = unbd_lsq + res.message = TERMINATION_MESSAGES[res.status] + res.success = res.status > 0 + + if verbose > 0: + print(res.message) + print( + f"Number of iterations {res.nit}, initial cost {res.initial_cost:.4e}, " + f"final cost {res.cost:.4e}, first-order optimality {res.optimality:.2e}." + ) + + del res.initial_cost + + return res diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/trf.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/trf.py new file mode 100644 index 0000000000000000000000000000000000000000..9154bdba5b2cc41883811ba1820dfc251e515d6c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/trf.py @@ -0,0 +1,560 @@ +"""Trust Region Reflective algorithm for least-squares optimization. + +The algorithm is based on ideas from paper [STIR]_. The main idea is to +account for the presence of the bounds by appropriate scaling of the variables (or, +equivalently, changing a trust-region shape). Let's introduce a vector v: + + | ub[i] - x[i], if g[i] < 0 and ub[i] < np.inf + v[i] = | x[i] - lb[i], if g[i] > 0 and lb[i] > -np.inf + | 1, otherwise + +where g is the gradient of a cost function and lb, ub are the bounds. Its +components are distances to the bounds at which the anti-gradient points (if +this distance is finite). Define a scaling matrix D = diag(v**0.5). +First-order optimality conditions can be stated as + + D^2 g(x) = 0. + +Meaning that components of the gradient should be zero for strictly interior +variables, and components must point inside the feasible region for variables +on the bound. + +Now consider this system of equations as a new optimization problem. If the +point x is strictly interior (not on the bound), then the left-hand side is +differentiable and the Newton step for it satisfies + + (D^2 H + diag(g) Jv) p = -D^2 g + +where H is the Hessian matrix (or its J^T J approximation in least squares), +Jv is the Jacobian matrix of v with components -1, 1 or 0, such that all +elements of matrix C = diag(g) Jv are non-negative. Introduce the change +of the variables x = D x_h (_h would be "hat" in LaTeX). In the new variables, +we have a Newton step satisfying + + B_h p_h = -g_h, + +where B_h = D H D + C, g_h = D g. In least squares B_h = J_h^T J_h, where +J_h = J D. Note that J_h and g_h are proper Jacobian and gradient with respect +to "hat" variables. To guarantee global convergence we formulate a +trust-region problem based on the Newton step in the new variables: + + 0.5 * p_h^T B_h p + g_h^T p_h -> min, ||p_h|| <= Delta + +In the original space B = H + D^{-1} C D^{-1}, and the equivalent trust-region +problem is + + 0.5 * p^T B p + g^T p -> min, ||D^{-1} p|| <= Delta + +Here, the meaning of the matrix D becomes more clear: it alters the shape +of a trust-region, such that large steps towards the bounds are not allowed. +In the implementation, the trust-region problem is solved in "hat" space, +but handling of the bounds is done in the original space (see below and read +the code). + +The introduction of the matrix D doesn't allow to ignore bounds, the algorithm +must keep iterates strictly feasible (to satisfy aforementioned +differentiability), the parameter theta controls step back from the boundary +(see the code for details). + +The algorithm does another important trick. If the trust-region solution +doesn't fit into the bounds, then a reflected (from a firstly encountered +bound) search direction is considered. For motivation and analysis refer to +[STIR]_ paper (and other papers of the authors). In practice, it doesn't need +a lot of justifications, the algorithm simply chooses the best step among +three: a constrained trust-region step, a reflected step and a constrained +Cauchy step (a minimizer along -g_h in "hat" space, or -D^2 g in the original +space). + +Another feature is that a trust-region radius control strategy is modified to +account for appearance of the diagonal C matrix (called diag_h in the code). + +Note that all described peculiarities are completely gone as we consider +problems without bounds (the algorithm becomes a standard trust-region type +algorithm very similar to ones implemented in MINPACK). + +The implementation supports two methods of solving the trust-region problem. +The first, called 'exact', applies SVD on Jacobian and then solves the problem +very accurately using the algorithm described in [JJMore]_. It is not +applicable to large problem. The second, called 'lsmr', uses the 2-D subspace +approach (sometimes called "indefinite dogleg"), where the problem is solved +in a subspace spanned by the gradient and the approximate Gauss-Newton step +found by ``scipy.sparse.linalg.lsmr``. A 2-D trust-region problem is +reformulated as a 4th order algebraic equation and solved very accurately by +``numpy.roots``. The subspace approach allows to solve very large problems +(up to couple of millions of residuals on a regular PC), provided the Jacobian +matrix is sufficiently sparse. + +References +---------- +.. [STIR] Branch, M.A., T.F. Coleman, and Y. Li, "A Subspace, Interior, + and Conjugate Gradient Method for Large-Scale Bound-Constrained + Minimization Problems," SIAM Journal on Scientific Computing, + Vol. 21, Number 1, pp 1-23, 1999. +.. [JJMore] More, J. J., "The Levenberg-Marquardt Algorithm: Implementation + and Theory," Numerical Analysis, ed. G. A. Watson, Lecture +""" +import numpy as np +from numpy.linalg import norm +from scipy.linalg import svd, qr +from scipy.sparse.linalg import lsmr +from scipy.optimize import OptimizeResult + +from .common import ( + step_size_to_bound, find_active_constraints, in_bounds, + make_strictly_feasible, intersect_trust_region, solve_lsq_trust_region, + solve_trust_region_2d, minimize_quadratic_1d, build_quadratic_1d, + evaluate_quadratic, right_multiplied_operator, regularized_lsq_operator, + CL_scaling_vector, compute_grad, compute_jac_scale, check_termination, + update_tr_radius, scale_for_robust_loss_function, print_header_nonlinear, + print_iteration_nonlinear) + + +def trf(fun, jac, x0, f0, J0, lb, ub, ftol, xtol, gtol, max_nfev, x_scale, + loss_function, tr_solver, tr_options, verbose): + # For efficiency, it makes sense to run the simplified version of the + # algorithm when no bounds are imposed. We decided to write the two + # separate functions. It violates the DRY principle, but the individual + # functions are kept the most readable. + if np.all(lb == -np.inf) and np.all(ub == np.inf): + return trf_no_bounds( + fun, jac, x0, f0, J0, ftol, xtol, gtol, max_nfev, x_scale, + loss_function, tr_solver, tr_options, verbose) + else: + return trf_bounds( + fun, jac, x0, f0, J0, lb, ub, ftol, xtol, gtol, max_nfev, x_scale, + loss_function, tr_solver, tr_options, verbose) + + +def select_step(x, J_h, diag_h, g_h, p, p_h, d, Delta, lb, ub, theta): + """Select the best step according to Trust Region Reflective algorithm.""" + if in_bounds(x + p, lb, ub): + p_value = evaluate_quadratic(J_h, g_h, p_h, diag=diag_h) + return p, p_h, -p_value + + p_stride, hits = step_size_to_bound(x, p, lb, ub) + + # Compute the reflected direction. + r_h = np.copy(p_h) + r_h[hits.astype(bool)] *= -1 + r = d * r_h + + # Restrict trust-region step, such that it hits the bound. + p *= p_stride + p_h *= p_stride + x_on_bound = x + p + + # Reflected direction will cross first either feasible region or trust + # region boundary. + _, to_tr = intersect_trust_region(p_h, r_h, Delta) + to_bound, _ = step_size_to_bound(x_on_bound, r, lb, ub) + + # Find lower and upper bounds on a step size along the reflected + # direction, considering the strict feasibility requirement. There is no + # single correct way to do that, the chosen approach seems to work best + # on test problems. + r_stride = min(to_bound, to_tr) + if r_stride > 0: + r_stride_l = (1 - theta) * p_stride / r_stride + if r_stride == to_bound: + r_stride_u = theta * to_bound + else: + r_stride_u = to_tr + else: + r_stride_l = 0 + r_stride_u = -1 + + # Check if reflection step is available. + if r_stride_l <= r_stride_u: + a, b, c = build_quadratic_1d(J_h, g_h, r_h, s0=p_h, diag=diag_h) + r_stride, r_value = minimize_quadratic_1d( + a, b, r_stride_l, r_stride_u, c=c) + r_h *= r_stride + r_h += p_h + r = r_h * d + else: + r_value = np.inf + + # Now correct p_h to make it strictly interior. + p *= theta + p_h *= theta + p_value = evaluate_quadratic(J_h, g_h, p_h, diag=diag_h) + + ag_h = -g_h + ag = d * ag_h + + to_tr = Delta / norm(ag_h) + to_bound, _ = step_size_to_bound(x, ag, lb, ub) + if to_bound < to_tr: + ag_stride = theta * to_bound + else: + ag_stride = to_tr + + a, b = build_quadratic_1d(J_h, g_h, ag_h, diag=diag_h) + ag_stride, ag_value = minimize_quadratic_1d(a, b, 0, ag_stride) + ag_h *= ag_stride + ag *= ag_stride + + if p_value < r_value and p_value < ag_value: + return p, p_h, -p_value + elif r_value < p_value and r_value < ag_value: + return r, r_h, -r_value + else: + return ag, ag_h, -ag_value + + +def trf_bounds(fun, jac, x0, f0, J0, lb, ub, ftol, xtol, gtol, max_nfev, + x_scale, loss_function, tr_solver, tr_options, verbose): + x = x0.copy() + + f = f0 + f_true = f.copy() + nfev = 1 + + J = J0 + njev = 1 + m, n = J.shape + + if loss_function is not None: + rho = loss_function(f) + cost = 0.5 * np.sum(rho[0]) + J, f = scale_for_robust_loss_function(J, f, rho) + else: + cost = 0.5 * np.dot(f, f) + + g = compute_grad(J, f) + + jac_scale = isinstance(x_scale, str) and x_scale == 'jac' + if jac_scale: + scale, scale_inv = compute_jac_scale(J) + else: + scale, scale_inv = x_scale, 1 / x_scale + + v, dv = CL_scaling_vector(x, g, lb, ub) + v[dv != 0] *= scale_inv[dv != 0] + Delta = norm(x0 * scale_inv / v**0.5) + if Delta == 0: + Delta = 1.0 + + g_norm = norm(g * v, ord=np.inf) + + f_augmented = np.zeros(m + n) + if tr_solver == 'exact': + J_augmented = np.empty((m + n, n)) + elif tr_solver == 'lsmr': + reg_term = 0.0 + regularize = tr_options.pop('regularize', True) + + if max_nfev is None: + max_nfev = x0.size * 100 + + alpha = 0.0 # "Levenberg-Marquardt" parameter + + termination_status = None + iteration = 0 + step_norm = None + actual_reduction = None + + if verbose == 2: + print_header_nonlinear() + + while True: + v, dv = CL_scaling_vector(x, g, lb, ub) + + g_norm = norm(g * v, ord=np.inf) + if g_norm < gtol: + termination_status = 1 + + if verbose == 2: + print_iteration_nonlinear(iteration, nfev, cost, actual_reduction, + step_norm, g_norm) + + if termination_status is not None or nfev == max_nfev: + break + + # Now compute variables in "hat" space. Here, we also account for + # scaling introduced by `x_scale` parameter. This part is a bit tricky, + # you have to write down the formulas and see how the trust-region + # problem is formulated when the two types of scaling are applied. + # The idea is that first we apply `x_scale` and then apply Coleman-Li + # approach in the new variables. + + # v is recomputed in the variables after applying `x_scale`, note that + # components which were identically 1 not affected. + v[dv != 0] *= scale_inv[dv != 0] + + # Here, we apply two types of scaling. + d = v**0.5 * scale + + # C = diag(g * scale) Jv + diag_h = g * dv * scale + + # After all this has been done, we continue normally. + + # "hat" gradient. + g_h = d * g + + f_augmented[:m] = f + if tr_solver == 'exact': + J_augmented[:m] = J * d + J_h = J_augmented[:m] # Memory view. + J_augmented[m:] = np.diag(diag_h**0.5) + U, s, V = svd(J_augmented, full_matrices=False) + V = V.T + uf = U.T.dot(f_augmented) + elif tr_solver == 'lsmr': + J_h = right_multiplied_operator(J, d) + + if regularize: + a, b = build_quadratic_1d(J_h, g_h, -g_h, diag=diag_h) + to_tr = Delta / norm(g_h) + ag_value = minimize_quadratic_1d(a, b, 0, to_tr)[1] + reg_term = -ag_value / Delta**2 + + lsmr_op = regularized_lsq_operator(J_h, (diag_h + reg_term)**0.5) + gn_h = lsmr(lsmr_op, f_augmented, **tr_options)[0] + S = np.vstack((g_h, gn_h)).T + S, _ = qr(S, mode='economic') + JS = J_h.dot(S) # LinearOperator does dot too. + B_S = np.dot(JS.T, JS) + np.dot(S.T * diag_h, S) + g_S = S.T.dot(g_h) + + # theta controls step back step ratio from the bounds. + theta = max(0.995, 1 - g_norm) + + actual_reduction = -1 + while actual_reduction <= 0 and nfev < max_nfev: + if tr_solver == 'exact': + p_h, alpha, n_iter = solve_lsq_trust_region( + n, m, uf, s, V, Delta, initial_alpha=alpha) + elif tr_solver == 'lsmr': + p_S, _ = solve_trust_region_2d(B_S, g_S, Delta) + p_h = S.dot(p_S) + + p = d * p_h # Trust-region solution in the original space. + step, step_h, predicted_reduction = select_step( + x, J_h, diag_h, g_h, p, p_h, d, Delta, lb, ub, theta) + + x_new = make_strictly_feasible(x + step, lb, ub, rstep=0) + f_new = fun(x_new) + nfev += 1 + + step_h_norm = norm(step_h) + + if not np.all(np.isfinite(f_new)): + Delta = 0.25 * step_h_norm + continue + + # Usual trust-region step quality estimation. + if loss_function is not None: + cost_new = loss_function(f_new, cost_only=True) + else: + cost_new = 0.5 * np.dot(f_new, f_new) + actual_reduction = cost - cost_new + Delta_new, ratio = update_tr_radius( + Delta, actual_reduction, predicted_reduction, + step_h_norm, step_h_norm > 0.95 * Delta) + + step_norm = norm(step) + termination_status = check_termination( + actual_reduction, cost, step_norm, norm(x), ratio, ftol, xtol) + if termination_status is not None: + break + + alpha *= Delta / Delta_new + Delta = Delta_new + + if actual_reduction > 0: + x = x_new + + f = f_new + f_true = f.copy() + + cost = cost_new + + J = jac(x, f) + njev += 1 + + if loss_function is not None: + rho = loss_function(f) + J, f = scale_for_robust_loss_function(J, f, rho) + + g = compute_grad(J, f) + + if jac_scale: + scale, scale_inv = compute_jac_scale(J, scale_inv) + else: + step_norm = 0 + actual_reduction = 0 + + iteration += 1 + + if termination_status is None: + termination_status = 0 + + active_mask = find_active_constraints(x, lb, ub, rtol=xtol) + return OptimizeResult( + x=x, cost=cost, fun=f_true, jac=J, grad=g, optimality=g_norm, + active_mask=active_mask, nfev=nfev, njev=njev, + status=termination_status) + + +def trf_no_bounds(fun, jac, x0, f0, J0, ftol, xtol, gtol, max_nfev, + x_scale, loss_function, tr_solver, tr_options, verbose): + x = x0.copy() + + f = f0 + f_true = f.copy() + nfev = 1 + + J = J0 + njev = 1 + m, n = J.shape + + if loss_function is not None: + rho = loss_function(f) + cost = 0.5 * np.sum(rho[0]) + J, f = scale_for_robust_loss_function(J, f, rho) + else: + cost = 0.5 * np.dot(f, f) + + g = compute_grad(J, f) + + jac_scale = isinstance(x_scale, str) and x_scale == 'jac' + if jac_scale: + scale, scale_inv = compute_jac_scale(J) + else: + scale, scale_inv = x_scale, 1 / x_scale + + Delta = norm(x0 * scale_inv) + if Delta == 0: + Delta = 1.0 + + if tr_solver == 'lsmr': + reg_term = 0 + damp = tr_options.pop('damp', 0.0) + regularize = tr_options.pop('regularize', True) + + if max_nfev is None: + max_nfev = x0.size * 100 + + alpha = 0.0 # "Levenberg-Marquardt" parameter + + termination_status = None + iteration = 0 + step_norm = None + actual_reduction = None + + if verbose == 2: + print_header_nonlinear() + + while True: + g_norm = norm(g, ord=np.inf) + if g_norm < gtol: + termination_status = 1 + + if verbose == 2: + print_iteration_nonlinear(iteration, nfev, cost, actual_reduction, + step_norm, g_norm) + + if termination_status is not None or nfev == max_nfev: + break + + d = scale + g_h = d * g + + if tr_solver == 'exact': + J_h = J * d + U, s, V = svd(J_h, full_matrices=False) + V = V.T + uf = U.T.dot(f) + elif tr_solver == 'lsmr': + J_h = right_multiplied_operator(J, d) + + if regularize: + a, b = build_quadratic_1d(J_h, g_h, -g_h) + to_tr = Delta / norm(g_h) + ag_value = minimize_quadratic_1d(a, b, 0, to_tr)[1] + reg_term = -ag_value / Delta**2 + + damp_full = (damp**2 + reg_term)**0.5 + gn_h = lsmr(J_h, f, damp=damp_full, **tr_options)[0] + S = np.vstack((g_h, gn_h)).T + S, _ = qr(S, mode='economic') + JS = J_h.dot(S) + B_S = np.dot(JS.T, JS) + g_S = S.T.dot(g_h) + + actual_reduction = -1 + while actual_reduction <= 0 and nfev < max_nfev: + if tr_solver == 'exact': + step_h, alpha, n_iter = solve_lsq_trust_region( + n, m, uf, s, V, Delta, initial_alpha=alpha) + elif tr_solver == 'lsmr': + p_S, _ = solve_trust_region_2d(B_S, g_S, Delta) + step_h = S.dot(p_S) + + predicted_reduction = -evaluate_quadratic(J_h, g_h, step_h) + step = d * step_h + x_new = x + step + f_new = fun(x_new) + nfev += 1 + + step_h_norm = norm(step_h) + + if not np.all(np.isfinite(f_new)): + Delta = 0.25 * step_h_norm + continue + + # Usual trust-region step quality estimation. + if loss_function is not None: + cost_new = loss_function(f_new, cost_only=True) + else: + cost_new = 0.5 * np.dot(f_new, f_new) + actual_reduction = cost - cost_new + + Delta_new, ratio = update_tr_radius( + Delta, actual_reduction, predicted_reduction, + step_h_norm, step_h_norm > 0.95 * Delta) + + step_norm = norm(step) + termination_status = check_termination( + actual_reduction, cost, step_norm, norm(x), ratio, ftol, xtol) + if termination_status is not None: + break + + alpha *= Delta / Delta_new + Delta = Delta_new + + if actual_reduction > 0: + x = x_new + + f = f_new + f_true = f.copy() + + cost = cost_new + + J = jac(x, f) + njev += 1 + + if loss_function is not None: + rho = loss_function(f) + J, f = scale_for_robust_loss_function(J, f, rho) + + g = compute_grad(J, f) + + if jac_scale: + scale, scale_inv = compute_jac_scale(J, scale_inv) + else: + step_norm = 0 + actual_reduction = 0 + + iteration += 1 + + if termination_status is None: + termination_status = 0 + + active_mask = np.zeros_like(x) + return OptimizeResult( + x=x, cost=cost, fun=f_true, jac=J, grad=g, optimality=g_norm, + active_mask=active_mask, nfev=nfev, njev=njev, + status=termination_status) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/trf_linear.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/trf_linear.py new file mode 100644 index 0000000000000000000000000000000000000000..dd752763179bcf97945c7f34ce6a9e49e85c819e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_lsq/trf_linear.py @@ -0,0 +1,249 @@ +"""The adaptation of Trust Region Reflective algorithm for a linear +least-squares problem.""" +import numpy as np +from numpy.linalg import norm +from scipy.linalg import qr, solve_triangular +from scipy.sparse.linalg import lsmr +from scipy.optimize import OptimizeResult + +from .givens_elimination import givens_elimination +from .common import ( + EPS, step_size_to_bound, find_active_constraints, in_bounds, + make_strictly_feasible, build_quadratic_1d, evaluate_quadratic, + minimize_quadratic_1d, CL_scaling_vector, reflective_transformation, + print_header_linear, print_iteration_linear, compute_grad, + regularized_lsq_operator, right_multiplied_operator) + + +def regularized_lsq_with_qr(m, n, R, QTb, perm, diag, copy_R=True): + """Solve regularized least squares using information from QR-decomposition. + + The initial problem is to solve the following system in a least-squares + sense:: + + A x = b + D x = 0 + + where D is diagonal matrix. The method is based on QR decomposition + of the form A P = Q R, where P is a column permutation matrix, Q is an + orthogonal matrix and R is an upper triangular matrix. + + Parameters + ---------- + m, n : int + Initial shape of A. + R : ndarray, shape (n, n) + Upper triangular matrix from QR decomposition of A. + QTb : ndarray, shape (n,) + First n components of Q^T b. + perm : ndarray, shape (n,) + Array defining column permutation of A, such that ith column of + P is perm[i]-th column of identity matrix. + diag : ndarray, shape (n,) + Array containing diagonal elements of D. + + Returns + ------- + x : ndarray, shape (n,) + Found least-squares solution. + """ + if copy_R: + R = R.copy() + v = QTb.copy() + + givens_elimination(R, v, diag[perm]) + + abs_diag_R = np.abs(np.diag(R)) + threshold = EPS * max(m, n) * np.max(abs_diag_R) + nns, = np.nonzero(abs_diag_R > threshold) + + R = R[np.ix_(nns, nns)] + v = v[nns] + + x = np.zeros(n) + x[perm[nns]] = solve_triangular(R, v) + + return x + + +def backtracking(A, g, x, p, theta, p_dot_g, lb, ub): + """Find an appropriate step size using backtracking line search.""" + alpha = 1 + while True: + x_new, _ = reflective_transformation(x + alpha * p, lb, ub) + step = x_new - x + cost_change = -evaluate_quadratic(A, g, step) + if cost_change > -0.1 * alpha * p_dot_g: + break + alpha *= 0.5 + + active = find_active_constraints(x_new, lb, ub) + if np.any(active != 0): + x_new, _ = reflective_transformation(x + theta * alpha * p, lb, ub) + x_new = make_strictly_feasible(x_new, lb, ub, rstep=0) + step = x_new - x + cost_change = -evaluate_quadratic(A, g, step) + + return x, step, cost_change + + +def select_step(x, A_h, g_h, c_h, p, p_h, d, lb, ub, theta): + """Select the best step according to Trust Region Reflective algorithm.""" + if in_bounds(x + p, lb, ub): + return p + + p_stride, hits = step_size_to_bound(x, p, lb, ub) + r_h = np.copy(p_h) + r_h[hits.astype(bool)] *= -1 + r = d * r_h + + # Restrict step, such that it hits the bound. + p *= p_stride + p_h *= p_stride + x_on_bound = x + p + + # Find the step size along reflected direction. + r_stride_u, _ = step_size_to_bound(x_on_bound, r, lb, ub) + + # Stay interior. + r_stride_l = (1 - theta) * r_stride_u + r_stride_u *= theta + + if r_stride_u > 0: + a, b, c = build_quadratic_1d(A_h, g_h, r_h, s0=p_h, diag=c_h) + r_stride, r_value = minimize_quadratic_1d( + a, b, r_stride_l, r_stride_u, c=c) + r_h = p_h + r_h * r_stride + r = d * r_h + else: + r_value = np.inf + + # Now correct p_h to make it strictly interior. + p_h *= theta + p *= theta + p_value = evaluate_quadratic(A_h, g_h, p_h, diag=c_h) + + ag_h = -g_h + ag = d * ag_h + ag_stride_u, _ = step_size_to_bound(x, ag, lb, ub) + ag_stride_u *= theta + a, b = build_quadratic_1d(A_h, g_h, ag_h, diag=c_h) + ag_stride, ag_value = minimize_quadratic_1d(a, b, 0, ag_stride_u) + ag *= ag_stride + + if p_value < r_value and p_value < ag_value: + return p + elif r_value < p_value and r_value < ag_value: + return r + else: + return ag + + +def trf_linear(A, b, x_lsq, lb, ub, tol, lsq_solver, lsmr_tol, + max_iter, verbose, *, lsmr_maxiter=None): + m, n = A.shape + x, _ = reflective_transformation(x_lsq, lb, ub) + x = make_strictly_feasible(x, lb, ub, rstep=0.1) + + if lsq_solver == 'exact': + QT, R, perm = qr(A, mode='economic', pivoting=True) + QT = QT.T + + if m < n: + R = np.vstack((R, np.zeros((n - m, n)))) + + QTr = np.zeros(n) + k = min(m, n) + elif lsq_solver == 'lsmr': + r_aug = np.zeros(m + n) + auto_lsmr_tol = False + if lsmr_tol is None: + lsmr_tol = 1e-2 * tol + elif lsmr_tol == 'auto': + auto_lsmr_tol = True + + r = A.dot(x) - b + g = compute_grad(A, r) + cost = 0.5 * np.dot(r, r) + initial_cost = cost + + termination_status = None + step_norm = None + cost_change = None + + if max_iter is None: + max_iter = 100 + + if verbose == 2: + print_header_linear() + + for iteration in range(max_iter): + v, dv = CL_scaling_vector(x, g, lb, ub) + g_scaled = g * v + g_norm = norm(g_scaled, ord=np.inf) + if g_norm < tol: + termination_status = 1 + + if verbose == 2: + print_iteration_linear(iteration, cost, cost_change, + step_norm, g_norm) + + if termination_status is not None: + break + + diag_h = g * dv + diag_root_h = diag_h ** 0.5 + d = v ** 0.5 + g_h = d * g + + A_h = right_multiplied_operator(A, d) + if lsq_solver == 'exact': + QTr[:k] = QT.dot(r) + p_h = -regularized_lsq_with_qr(m, n, R * d[perm], QTr, perm, + diag_root_h, copy_R=False) + elif lsq_solver == 'lsmr': + lsmr_op = regularized_lsq_operator(A_h, diag_root_h) + r_aug[:m] = r + if auto_lsmr_tol: + eta = 1e-2 * min(0.5, g_norm) + lsmr_tol = max(EPS, min(0.1, eta * g_norm)) + p_h = -lsmr(lsmr_op, r_aug, maxiter=lsmr_maxiter, + atol=lsmr_tol, btol=lsmr_tol)[0] + + p = d * p_h + + p_dot_g = np.dot(p, g) + if p_dot_g > 0: + termination_status = -1 + + theta = 1 - min(0.005, g_norm) + step = select_step(x, A_h, g_h, diag_h, p, p_h, d, lb, ub, theta) + cost_change = -evaluate_quadratic(A, g, step) + + # Perhaps almost never executed, the idea is that `p` is descent + # direction thus we must find acceptable cost decrease using simple + # "backtracking", otherwise the algorithm's logic would break. + if cost_change < 0: + x, step, cost_change = backtracking( + A, g, x, p, theta, p_dot_g, lb, ub) + else: + x = make_strictly_feasible(x + step, lb, ub, rstep=0) + + step_norm = norm(step) + r = A.dot(x) - b + g = compute_grad(A, r) + + if cost_change < tol * cost: + termination_status = 2 + + cost = 0.5 * np.dot(r, r) + + if termination_status is None: + termination_status = 0 + + active_mask = find_active_constraints(x, lb, ub, rtol=tol) + + return OptimizeResult( + x=x, fun=r, cost=cost, optimality=g_norm, active_mask=active_mask, + nit=iteration + 1, status=termination_status, + initial_cost=initial_cost) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_milp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_milp.py new file mode 100644 index 0000000000000000000000000000000000000000..b97a00d15406700cfedbe50e1b3714d36a60f8fb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_milp.py @@ -0,0 +1,392 @@ +import warnings +import numpy as np +from scipy.sparse import csc_array, vstack, issparse +from scipy._lib._util import VisibleDeprecationWarning +from ._highspy._highs_wrapper import _highs_wrapper # type: ignore[import-not-found,import-untyped] +from ._constraints import LinearConstraint, Bounds +from ._optimize import OptimizeResult +from ._linprog_highs import _highs_to_scipy_status_message + + +def _constraints_to_components(constraints): + """ + Convert sequence of constraints to a single set of components A, b_l, b_u. + + `constraints` could be + + 1. A LinearConstraint + 2. A tuple representing a LinearConstraint + 3. An invalid object + 4. A sequence of composed entirely of objects of type 1/2 + 5. A sequence containing at least one object of type 3 + + We want to accept 1, 2, and 4 and reject 3 and 5. + """ + message = ("`constraints` (or each element within `constraints`) must be " + "convertible into an instance of " + "`scipy.optimize.LinearConstraint`.") + As = [] + b_ls = [] + b_us = [] + + # Accept case 1 by standardizing as case 4 + if isinstance(constraints, LinearConstraint): + constraints = [constraints] + else: + # Reject case 3 + try: + iter(constraints) + except TypeError as exc: + raise ValueError(message) from exc + + # Accept case 2 by standardizing as case 4 + if len(constraints) == 3: + # argument could be a single tuple representing a LinearConstraint + try: + constraints = [LinearConstraint(*constraints)] + except (TypeError, ValueError, VisibleDeprecationWarning): + # argument was not a tuple representing a LinearConstraint + pass + + # Address cases 4/5 + for constraint in constraints: + # if it's not a LinearConstraint or something that represents a + # LinearConstraint at this point, it's invalid + if not isinstance(constraint, LinearConstraint): + try: + constraint = LinearConstraint(*constraint) + except TypeError as exc: + raise ValueError(message) from exc + As.append(csc_array(constraint.A)) + b_ls.append(np.atleast_1d(constraint.lb).astype(np.float64)) + b_us.append(np.atleast_1d(constraint.ub).astype(np.float64)) + + if len(As) > 1: + A = vstack(As, format="csc") + b_l = np.concatenate(b_ls) + b_u = np.concatenate(b_us) + else: # avoid unnecessary copying + A = As[0] + b_l = b_ls[0] + b_u = b_us[0] + + return A, b_l, b_u + + +def _milp_iv(c, integrality, bounds, constraints, options): + # objective IV + if issparse(c): + raise ValueError("`c` must be a dense array.") + c = np.atleast_1d(c).astype(np.float64) + if c.ndim != 1 or c.size == 0 or not np.all(np.isfinite(c)): + message = ("`c` must be a one-dimensional array of finite numbers " + "with at least one element.") + raise ValueError(message) + + # integrality IV + if issparse(integrality): + raise ValueError("`integrality` must be a dense array.") + message = ("`integrality` must contain integers 0-3 and be broadcastable " + "to `c.shape`.") + if integrality is None: + integrality = 0 + try: + integrality = np.broadcast_to(integrality, c.shape).astype(np.uint8) + except ValueError: + raise ValueError(message) + if integrality.min() < 0 or integrality.max() > 3: + raise ValueError(message) + + # bounds IV + if bounds is None: + bounds = Bounds(0, np.inf) + elif not isinstance(bounds, Bounds): + message = ("`bounds` must be convertible into an instance of " + "`scipy.optimize.Bounds`.") + try: + bounds = Bounds(*bounds) + except TypeError as exc: + raise ValueError(message) from exc + + try: + lb = np.broadcast_to(bounds.lb, c.shape).astype(np.float64) + ub = np.broadcast_to(bounds.ub, c.shape).astype(np.float64) + except (ValueError, TypeError) as exc: + message = ("`bounds.lb` and `bounds.ub` must contain reals and " + "be broadcastable to `c.shape`.") + raise ValueError(message) from exc + + # constraints IV + if not constraints: + constraints = [LinearConstraint(np.empty((0, c.size)), + np.empty((0,)), np.empty((0,)))] + try: + A, b_l, b_u = _constraints_to_components(constraints) + except ValueError as exc: + message = ("`constraints` (or each element within `constraints`) must " + "be convertible into an instance of " + "`scipy.optimize.LinearConstraint`.") + raise ValueError(message) from exc + + if A.shape != (b_l.size, c.size): + message = "The shape of `A` must be (len(b_l), len(c))." + raise ValueError(message) + indptr, indices, data = A.indptr, A.indices, A.data.astype(np.float64) + + # options IV + options = options or {} + supported_options = {'disp', 'presolve', 'time_limit', 'node_limit', + 'mip_rel_gap'} + unsupported_options = set(options).difference(supported_options) + if unsupported_options: + message = (f"Unrecognized options detected: {unsupported_options}. " + "These will be passed to HiGHS verbatim.") + warnings.warn(message, RuntimeWarning, stacklevel=3) + options_iv = {'log_to_console': options.pop("disp", False), + 'mip_max_nodes': options.pop("node_limit", None)} + options_iv.update(options) + + return c, integrality, lb, ub, indptr, indices, data, b_l, b_u, options_iv + + +def milp(c, *, integrality=None, bounds=None, constraints=None, options=None): + r""" + Mixed-integer linear programming + + Solves problems of the following form: + + .. math:: + + \min_x \ & c^T x \\ + \mbox{such that} \ & b_l \leq A x \leq b_u,\\ + & l \leq x \leq u, \\ + & x_i \in \mathbb{Z}, i \in X_i + + where :math:`x` is a vector of decision variables; + :math:`c`, :math:`b_l`, :math:`b_u`, :math:`l`, and :math:`u` are vectors; + :math:`A` is a matrix, and :math:`X_i` is the set of indices of + decision variables that must be integral. (In this context, a + variable that can assume only integer values is said to be "integral"; + it has an "integrality" constraint.) + + Alternatively, that's: + + minimize:: + + c @ x + + such that:: + + b_l <= A @ x <= b_u + l <= x <= u + Specified elements of x must be integers + + By default, ``l = 0`` and ``u = np.inf`` unless specified with + ``bounds``. + + Parameters + ---------- + c : 1D dense array_like + The coefficients of the linear objective function to be minimized. + `c` is converted to a double precision array before the problem is + solved. + integrality : 1D dense array_like, optional + Indicates the type of integrality constraint on each decision variable. + + ``0`` : Continuous variable; no integrality constraint. + + ``1`` : Integer variable; decision variable must be an integer + within `bounds`. + + ``2`` : Semi-continuous variable; decision variable must be within + `bounds` or take value ``0``. + + ``3`` : Semi-integer variable; decision variable must be an integer + within `bounds` or take value ``0``. + + By default, all variables are continuous. `integrality` is converted + to an array of integers before the problem is solved. + + bounds : scipy.optimize.Bounds, optional + Bounds on the decision variables. Lower and upper bounds are converted + to double precision arrays before the problem is solved. The + ``keep_feasible`` parameter of the `Bounds` object is ignored. If + not specified, all decision variables are constrained to be + non-negative. + constraints : sequence of scipy.optimize.LinearConstraint, optional + Linear constraints of the optimization problem. Arguments may be + one of the following: + + 1. A single `LinearConstraint` object + 2. A single tuple that can be converted to a `LinearConstraint` object + as ``LinearConstraint(*constraints)`` + 3. A sequence composed entirely of objects of type 1. and 2. + + Before the problem is solved, all values are converted to double + precision, and the matrices of constraint coefficients are converted to + instances of `scipy.sparse.csc_array`. The ``keep_feasible`` parameter + of `LinearConstraint` objects is ignored. + options : dict, optional + A dictionary of solver options. The following keys are recognized. + + disp : bool (default: ``False``) + Set to ``True`` if indicators of optimization status are to be + printed to the console during optimization. + node_limit : int, optional + The maximum number of nodes (linear program relaxations) to solve + before stopping. Default is no maximum number of nodes. + presolve : bool (default: ``True``) + Presolve attempts to identify trivial infeasibilities, + identify trivial unboundedness, and simplify the problem before + sending it to the main solver. + time_limit : float, optional + The maximum number of seconds allotted to solve the problem. + Default is no time limit. + mip_rel_gap : float, optional + Termination criterion for MIP solver: solver will terminate when + the gap between the primal objective value and the dual objective + bound, scaled by the primal objective value, is <= mip_rel_gap. + + Returns + ------- + res : OptimizeResult + An instance of :class:`scipy.optimize.OptimizeResult`. The object + is guaranteed to have the following attributes. + + status : int + An integer representing the exit status of the algorithm. + + ``0`` : Optimal solution found. + + ``1`` : Iteration or time limit reached. + + ``2`` : Problem is infeasible. + + ``3`` : Problem is unbounded. + + ``4`` : Other; see message for details. + + success : bool + ``True`` when an optimal solution is found and ``False`` otherwise. + + message : str + A string descriptor of the exit status of the algorithm. + + The following attributes will also be present, but the values may be + ``None``, depending on the solution status. + + x : ndarray + The values of the decision variables that minimize the + objective function while satisfying the constraints. + fun : float + The optimal value of the objective function ``c @ x``. + mip_node_count : int + The number of subproblems or "nodes" solved by the MILP solver. + mip_dual_bound : float + The MILP solver's final estimate of the lower bound on the optimal + solution. + mip_gap : float + The difference between the primal objective value and the dual + objective bound, scaled by the primal objective value. + + Notes + ----- + `milp` is a wrapper of the HiGHS linear optimization software [1]_. The + algorithm is deterministic, and it typically finds the global optimum of + moderately challenging mixed-integer linear programs (when it exists). + + References + ---------- + .. [1] Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J. + "HiGHS - high performance software for linear optimization." + https://highs.dev/ + .. [2] Huangfu, Q. and Hall, J. A. J. "Parallelizing the dual revised + simplex method." Mathematical Programming Computation, 10 (1), + 119-142, 2018. DOI: 10.1007/s12532-017-0130-5 + + Examples + -------- + Consider the problem at + https://en.wikipedia.org/wiki/Integer_programming#Example, which is + expressed as a maximization problem of two variables. Since `milp` requires + that the problem be expressed as a minimization problem, the objective + function coefficients on the decision variables are: + + >>> import numpy as np + >>> c = -np.array([0, 1]) + + Note the negative sign: we maximize the original objective function + by minimizing the negative of the objective function. + + We collect the coefficients of the constraints into arrays like: + + >>> A = np.array([[-1, 1], [3, 2], [2, 3]]) + >>> b_u = np.array([1, 12, 12]) + >>> b_l = np.full_like(b_u, -np.inf, dtype=float) + + Because there is no lower limit on these constraints, we have defined a + variable ``b_l`` full of values representing negative infinity. This may + be unfamiliar to users of `scipy.optimize.linprog`, which only accepts + "less than" (or "upper bound") inequality constraints of the form + ``A_ub @ x <= b_u``. By accepting both ``b_l`` and ``b_u`` of constraints + ``b_l <= A_ub @ x <= b_u``, `milp` makes it easy to specify "greater than" + inequality constraints, "less than" inequality constraints, and equality + constraints concisely. + + These arrays are collected into a single `LinearConstraint` object like: + + >>> from scipy.optimize import LinearConstraint + >>> constraints = LinearConstraint(A, b_l, b_u) + + The non-negativity bounds on the decision variables are enforced by + default, so we do not need to provide an argument for `bounds`. + + Finally, the problem states that both decision variables must be integers: + + >>> integrality = np.ones_like(c) + + We solve the problem like: + + >>> from scipy.optimize import milp + >>> res = milp(c=c, constraints=constraints, integrality=integrality) + >>> res.x + [2.0, 2.0] + + Note that had we solved the relaxed problem (without integrality + constraints): + + >>> res = milp(c=c, constraints=constraints) # OR: + >>> # from scipy.optimize import linprog; res = linprog(c, A, b_u) + >>> res.x + [1.8, 2.8] + + we would not have obtained the correct solution by rounding to the nearest + integers. + + Other examples are given :ref:`in the tutorial `. + + """ + args_iv = _milp_iv(c, integrality, bounds, constraints, options) + c, integrality, lb, ub, indptr, indices, data, b_l, b_u, options = args_iv + + highs_res = _highs_wrapper(c, indptr, indices, data, b_l, b_u, + lb, ub, integrality, options) + + res = {} + + # Convert to scipy-style status and message + highs_status = highs_res.get('status', None) + highs_message = highs_res.get('message', None) + status, message = _highs_to_scipy_status_message(highs_status, + highs_message) + res['status'] = status + res['message'] = message + res['success'] = (status == 0) + x = highs_res.get('x', None) + res['x'] = np.array(x) if x is not None else None + res['fun'] = highs_res.get('fun', None) + res['mip_node_count'] = highs_res.get('mip_node_count', None) + res['mip_dual_bound'] = highs_res.get('mip_dual_bound', None) + res['mip_gap'] = highs_res.get('mip_gap', None) + + return OptimizeResult(res) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_minimize.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_minimize.py new file mode 100644 index 0000000000000000000000000000000000000000..0b47c57cb3a12c6f4a7adb5506089c569336b4c4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_minimize.py @@ -0,0 +1,1131 @@ +""" +Unified interfaces to minimization algorithms. + +Functions +--------- +- minimize : minimization of a function of several variables. +- minimize_scalar : minimization of a function of one variable. +""" + +__all__ = ['minimize', 'minimize_scalar'] + + +from warnings import warn + +import numpy as np + +# unconstrained minimization +from ._optimize import (_minimize_neldermead, _minimize_powell, _minimize_cg, + _minimize_bfgs, _minimize_newtoncg, + _minimize_scalar_brent, _minimize_scalar_bounded, + _minimize_scalar_golden, MemoizeJac, OptimizeResult, + _wrap_callback, _recover_from_bracket_error) +from ._trustregion_dogleg import _minimize_dogleg +from ._trustregion_ncg import _minimize_trust_ncg +from ._trustregion_krylov import _minimize_trust_krylov +from ._trustregion_exact import _minimize_trustregion_exact +from ._trustregion_constr import _minimize_trustregion_constr + +# constrained minimization +from ._lbfgsb_py import _minimize_lbfgsb +from ._tnc import _minimize_tnc +from ._cobyla_py import _minimize_cobyla +from ._cobyqa_py import _minimize_cobyqa +from ._slsqp_py import _minimize_slsqp +from ._constraints import (old_bound_to_new, new_bounds_to_old, + old_constraint_to_new, new_constraint_to_old, + NonlinearConstraint, LinearConstraint, Bounds, + PreparedConstraint) +from ._differentiable_functions import FD_METHODS + +MINIMIZE_METHODS = ['nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg', + 'l-bfgs-b', 'tnc', 'cobyla', 'cobyqa', 'slsqp', + 'trust-constr', 'dogleg', 'trust-ncg', 'trust-exact', + 'trust-krylov'] + +# These methods support the new callback interface (passed an OptimizeResult) +MINIMIZE_METHODS_NEW_CB = ['nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg', + 'l-bfgs-b', 'trust-constr', 'dogleg', 'trust-ncg', + 'trust-exact', 'trust-krylov', 'cobyqa'] + +MINIMIZE_SCALAR_METHODS = ['brent', 'bounded', 'golden'] + +def minimize(fun, x0, args=(), method=None, jac=None, hess=None, + hessp=None, bounds=None, constraints=(), tol=None, + callback=None, options=None): + """Minimization of scalar function of one or more variables. + + Parameters + ---------- + fun : callable + The objective function to be minimized:: + + fun(x, *args) -> float + + where ``x`` is a 1-D array with shape (n,) and ``args`` + is a tuple of the fixed parameters needed to completely + specify the function. + + Suppose the callable has signature ``f0(x, *my_args, **my_kwargs)``, where + ``my_args`` and ``my_kwargs`` are required positional and keyword arguments. + Rather than passing ``f0`` as the callable, wrap it to accept + only ``x``; e.g., pass ``fun=lambda x: f0(x, *my_args, **my_kwargs)`` as the + callable, where ``my_args`` (tuple) and ``my_kwargs`` (dict) have been + gathered before invoking this function. + x0 : ndarray, shape (n,) + Initial guess. Array of real elements of size (n,), + where ``n`` is the number of independent variables. + args : tuple, optional + Extra arguments passed to the objective function and its + derivatives (`fun`, `jac` and `hess` functions). + method : str or callable, optional + Type of solver. Should be one of + + - 'Nelder-Mead' :ref:`(see here) ` + - 'Powell' :ref:`(see here) ` + - 'CG' :ref:`(see here) ` + - 'BFGS' :ref:`(see here) ` + - 'Newton-CG' :ref:`(see here) ` + - 'L-BFGS-B' :ref:`(see here) ` + - 'TNC' :ref:`(see here) ` + - 'COBYLA' :ref:`(see here) ` + - 'COBYQA' :ref:`(see here) ` + - 'SLSQP' :ref:`(see here) ` + - 'trust-constr':ref:`(see here) ` + - 'dogleg' :ref:`(see here) ` + - 'trust-ncg' :ref:`(see here) ` + - 'trust-exact' :ref:`(see here) ` + - 'trust-krylov' :ref:`(see here) ` + - custom - a callable object, see below for description. + + If not given, chosen to be one of ``BFGS``, ``L-BFGS-B``, ``SLSQP``, + depending on whether or not the problem has constraints or bounds. + jac : {callable, '2-point', '3-point', 'cs', bool}, optional + Method for computing the gradient vector. Only for CG, BFGS, + Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg, trust-krylov, + trust-exact and trust-constr. + If it is a callable, it should be a function that returns the gradient + vector:: + + jac(x, *args) -> array_like, shape (n,) + + where ``x`` is an array with shape (n,) and ``args`` is a tuple with + the fixed parameters. If `jac` is a Boolean and is True, `fun` is + assumed to return a tuple ``(f, g)`` containing the objective + function and the gradient. + Methods 'Newton-CG', 'trust-ncg', 'dogleg', 'trust-exact', and + 'trust-krylov' require that either a callable be supplied, or that + `fun` return the objective and gradient. + If None or False, the gradient will be estimated using 2-point finite + difference estimation with an absolute step size. + Alternatively, the keywords {'2-point', '3-point', 'cs'} can be used + to select a finite difference scheme for numerical estimation of the + gradient with a relative step size. These finite difference schemes + obey any specified `bounds`. + hess : {callable, '2-point', '3-point', 'cs', HessianUpdateStrategy}, optional + Method for computing the Hessian matrix. Only for Newton-CG, dogleg, + trust-ncg, trust-krylov, trust-exact and trust-constr. + If it is callable, it should return the Hessian matrix:: + + hess(x, *args) -> {LinearOperator, spmatrix, array}, (n, n) + + where ``x`` is a (n,) ndarray and ``args`` is a tuple with the fixed + parameters. + The keywords {'2-point', '3-point', 'cs'} can also be used to select + a finite difference scheme for numerical estimation of the hessian. + Alternatively, objects implementing the `HessianUpdateStrategy` + interface can be used to approximate the Hessian. Available + quasi-Newton methods implementing this interface are: + + - `BFGS` + - `SR1` + + Not all of the options are available for each of the methods; for + availability refer to the notes. + hessp : callable, optional + Hessian of objective function times an arbitrary vector p. Only for + Newton-CG, trust-ncg, trust-krylov, trust-constr. + Only one of `hessp` or `hess` needs to be given. If `hess` is + provided, then `hessp` will be ignored. `hessp` must compute the + Hessian times an arbitrary vector:: + + hessp(x, p, *args) -> ndarray shape (n,) + + where ``x`` is a (n,) ndarray, ``p`` is an arbitrary vector with + dimension (n,) and ``args`` is a tuple with the fixed + parameters. + bounds : sequence or `Bounds`, optional + Bounds on variables for Nelder-Mead, L-BFGS-B, TNC, SLSQP, Powell, + trust-constr, COBYLA, and COBYQA methods. There are two ways to specify + the bounds: + + 1. Instance of `Bounds` class. + 2. Sequence of ``(min, max)`` pairs for each element in `x`. None + is used to specify no bound. + + constraints : {Constraint, dict} or List of {Constraint, dict}, optional + Constraints definition. Only for COBYLA, COBYQA, SLSQP and trust-constr. + + Constraints for 'trust-constr' and 'cobyqa' are defined as a single object + or a list of objects specifying constraints to the optimization problem. + Available constraints are: + + - `LinearConstraint` + - `NonlinearConstraint` + + Constraints for COBYLA, SLSQP are defined as a list of dictionaries. + Each dictionary with fields: + + type : str + Constraint type: 'eq' for equality, 'ineq' for inequality. + fun : callable + The function defining the constraint. + jac : callable, optional + The Jacobian of `fun` (only for SLSQP). + args : sequence, optional + Extra arguments to be passed to the function and Jacobian. + + Equality constraint means that the constraint function result is to + be zero whereas inequality means that it is to be non-negative. + Note that COBYLA only supports inequality constraints. + + tol : float, optional + Tolerance for termination. When `tol` is specified, the selected + minimization algorithm sets some relevant solver-specific tolerance(s) + equal to `tol`. For detailed control, use solver-specific + options. + options : dict, optional + A dictionary of solver options. All methods except `TNC` accept the + following generic options: + + maxiter : int + Maximum number of iterations to perform. Depending on the + method each iteration may use several function evaluations. + + For `TNC` use `maxfun` instead of `maxiter`. + disp : bool + Set to True to print convergence messages. + + For method-specific options, see :func:`show_options()`. + callback : callable, optional + A callable called after each iteration. + + All methods except TNC, SLSQP, and COBYLA support a callable with + the signature:: + + callback(intermediate_result: OptimizeResult) + + where ``intermediate_result`` is a keyword parameter containing an + `OptimizeResult` with attributes ``x`` and ``fun``, the present values + of the parameter vector and objective function. Note that the name + of the parameter must be ``intermediate_result`` for the callback + to be passed an `OptimizeResult`. These methods will also terminate if + the callback raises ``StopIteration``. + + All methods except trust-constr (also) support a signature like:: + + callback(xk) + + where ``xk`` is the current parameter vector. + + Introspection is used to determine which of the signatures above to + invoke. + + Returns + ------- + res : OptimizeResult + The optimization result represented as a ``OptimizeResult`` object. + Important attributes are: ``x`` the solution array, ``success`` a + Boolean flag indicating if the optimizer exited successfully and + ``message`` which describes the cause of the termination. See + `OptimizeResult` for a description of other attributes. + + See also + -------- + minimize_scalar : Interface to minimization algorithms for scalar + univariate functions + show_options : Additional options accepted by the solvers + + Notes + ----- + This section describes the available solvers that can be selected by the + 'method' parameter. The default method is *BFGS*. + + **Unconstrained minimization** + + Method :ref:`CG ` uses a nonlinear conjugate + gradient algorithm by Polak and Ribiere, a variant of the + Fletcher-Reeves method described in [5]_ pp.120-122. Only the + first derivatives are used. + + Method :ref:`BFGS ` uses the quasi-Newton + method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) [5]_ + pp. 136. It uses the first derivatives only. BFGS has proven good + performance even for non-smooth optimizations. This method also + returns an approximation of the Hessian inverse, stored as + `hess_inv` in the OptimizeResult object. + + Method :ref:`Newton-CG ` uses a + Newton-CG algorithm [5]_ pp. 168 (also known as the truncated + Newton method). It uses a CG method to the compute the search + direction. See also *TNC* method for a box-constrained + minimization with a similar algorithm. Suitable for large-scale + problems. + + Method :ref:`dogleg ` uses the dog-leg + trust-region algorithm [5]_ for unconstrained minimization. This + algorithm requires the gradient and Hessian; furthermore the + Hessian is required to be positive definite. + + Method :ref:`trust-ncg ` uses the + Newton conjugate gradient trust-region algorithm [5]_ for + unconstrained minimization. This algorithm requires the gradient + and either the Hessian or a function that computes the product of + the Hessian with a given vector. Suitable for large-scale problems. + + Method :ref:`trust-krylov ` uses + the Newton GLTR trust-region algorithm [14]_, [15]_ for unconstrained + minimization. This algorithm requires the gradient + and either the Hessian or a function that computes the product of + the Hessian with a given vector. Suitable for large-scale problems. + On indefinite problems it requires usually less iterations than the + `trust-ncg` method and is recommended for medium and large-scale problems. + + Method :ref:`trust-exact ` + is a trust-region method for unconstrained minimization in which + quadratic subproblems are solved almost exactly [13]_. This + algorithm requires the gradient and the Hessian (which is + *not* required to be positive definite). It is, in many + situations, the Newton method to converge in fewer iterations + and the most recommended for small and medium-size problems. + + **Bound-Constrained minimization** + + Method :ref:`Nelder-Mead ` uses the + Simplex algorithm [1]_, [2]_. This algorithm is robust in many + applications. However, if numerical computation of derivative can be + trusted, other algorithms using the first and/or second derivatives + information might be preferred for their better performance in + general. + + Method :ref:`L-BFGS-B ` uses the L-BFGS-B + algorithm [6]_, [7]_ for bound constrained minimization. + + Method :ref:`Powell ` is a modification + of Powell's method [3]_, [4]_ which is a conjugate direction + method. It performs sequential one-dimensional minimizations along + each vector of the directions set (`direc` field in `options` and + `info`), which is updated at each iteration of the main + minimization loop. The function need not be differentiable, and no + derivatives are taken. If bounds are not provided, then an + unbounded line search will be used. If bounds are provided and + the initial guess is within the bounds, then every function + evaluation throughout the minimization procedure will be within + the bounds. If bounds are provided, the initial guess is outside + the bounds, and `direc` is full rank (default has full rank), then + some function evaluations during the first iteration may be + outside the bounds, but every function evaluation after the first + iteration will be within the bounds. If `direc` is not full rank, + then some parameters may not be optimized and the solution is not + guaranteed to be within the bounds. + + Method :ref:`TNC ` uses a truncated Newton + algorithm [5]_, [8]_ to minimize a function with variables subject + to bounds. This algorithm uses gradient information; it is also + called Newton Conjugate-Gradient. It differs from the *Newton-CG* + method described above as it wraps a C implementation and allows + each variable to be given upper and lower bounds. + + **Constrained Minimization** + + Method :ref:`COBYLA ` uses the + Constrained Optimization BY Linear Approximation (COBYLA) method + [9]_, [10]_, [11]_. The algorithm is based on linear + approximations to the objective function and each constraint. The + method wraps a FORTRAN implementation of the algorithm. The + constraints functions 'fun' may return either a single number + or an array or list of numbers. + + Method :ref:`COBYQA ` uses the Constrained + Optimization BY Quadratic Approximations (COBYQA) method [18]_. The + algorithm is a derivative-free trust-region SQP method based on quadratic + approximations to the objective function and each nonlinear constraint. The + bounds are treated as unrelaxable constraints, in the sense that the + algorithm always respects them throughout the optimization process. + + Method :ref:`SLSQP ` uses Sequential + Least SQuares Programming to minimize a function of several + variables with any combination of bounds, equality and inequality + constraints. The method wraps the SLSQP Optimization subroutine + originally implemented by Dieter Kraft [12]_. Note that the + wrapper handles infinite values in bounds by converting them into + large floating values. + + Method :ref:`trust-constr ` is a + trust-region algorithm for constrained optimization. It switches + between two implementations depending on the problem definition. + It is the most versatile constrained minimization algorithm + implemented in SciPy and the most appropriate for large-scale problems. + For equality constrained problems it is an implementation of Byrd-Omojokun + Trust-Region SQP method described in [17]_ and in [5]_, p. 549. When + inequality constraints are imposed as well, it switches to the trust-region + interior point method described in [16]_. This interior point algorithm, + in turn, solves inequality constraints by introducing slack variables + and solving a sequence of equality-constrained barrier problems + for progressively smaller values of the barrier parameter. + The previously described equality constrained SQP method is + used to solve the subproblems with increasing levels of accuracy + as the iterate gets closer to a solution. + + **Finite-Difference Options** + + For Method :ref:`trust-constr ` + the gradient and the Hessian may be approximated using + three finite-difference schemes: {'2-point', '3-point', 'cs'}. + The scheme 'cs' is, potentially, the most accurate but it + requires the function to correctly handle complex inputs and to + be differentiable in the complex plane. The scheme '3-point' is more + accurate than '2-point' but requires twice as many operations. If the + gradient is estimated via finite-differences the Hessian must be + estimated using one of the quasi-Newton strategies. + + **Method specific options for the** `hess` **keyword** + + +--------------+------+----------+-------------------------+-----+ + | method/Hess | None | callable | '2-point/'3-point'/'cs' | HUS | + +==============+======+==========+=========================+=====+ + | Newton-CG | x | (n, n) | x | x | + | | | LO | | | + +--------------+------+----------+-------------------------+-----+ + | dogleg | | (n, n) | | | + +--------------+------+----------+-------------------------+-----+ + | trust-ncg | | (n, n) | x | x | + +--------------+------+----------+-------------------------+-----+ + | trust-krylov | | (n, n) | x | x | + +--------------+------+----------+-------------------------+-----+ + | trust-exact | | (n, n) | | | + +--------------+------+----------+-------------------------+-----+ + | trust-constr | x | (n, n) | x | x | + | | | LO | | | + | | | sp | | | + +--------------+------+----------+-------------------------+-----+ + + where LO=LinearOperator, sp=Sparse matrix, HUS=HessianUpdateStrategy + + **Custom minimizers** + + It may be useful to pass a custom minimization method, for example + when using a frontend to this method such as `scipy.optimize.basinhopping` + or a different library. You can simply pass a callable as the ``method`` + parameter. + + The callable is called as ``method(fun, x0, args, **kwargs, **options)`` + where ``kwargs`` corresponds to any other parameters passed to `minimize` + (such as `callback`, `hess`, etc.), except the `options` dict, which has + its contents also passed as `method` parameters pair by pair. Also, if + `jac` has been passed as a bool type, `jac` and `fun` are mangled so that + `fun` returns just the function values and `jac` is converted to a function + returning the Jacobian. The method shall return an `OptimizeResult` + object. + + The provided `method` callable must be able to accept (and possibly ignore) + arbitrary parameters; the set of parameters accepted by `minimize` may + expand in future versions and then these parameters will be passed to + the method. You can find an example in the scipy.optimize tutorial. + + References + ---------- + .. [1] Nelder, J A, and R Mead. 1965. A Simplex Method for Function + Minimization. The Computer Journal 7: 308-13. + .. [2] Wright M H. 1996. Direct search methods: Once scorned, now + respectable, in Numerical Analysis 1995: Proceedings of the 1995 + Dundee Biennial Conference in Numerical Analysis (Eds. D F + Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK. + 191-208. + .. [3] Powell, M J D. 1964. An efficient method for finding the minimum of + a function of several variables without calculating derivatives. The + Computer Journal 7: 155-162. + .. [4] Press W, S A Teukolsky, W T Vetterling and B P Flannery. + Numerical Recipes (any edition), Cambridge University Press. + .. [5] Nocedal, J, and S J Wright. 2006. Numerical Optimization. + Springer New York. + .. [6] Byrd, R H and P Lu and J. Nocedal. 1995. A Limited Memory + Algorithm for Bound Constrained Optimization. SIAM Journal on + Scientific and Statistical Computing 16 (5): 1190-1208. + .. [7] Zhu, C and R H Byrd and J Nocedal. 1997. L-BFGS-B: Algorithm + 778: L-BFGS-B, FORTRAN routines for large scale bound constrained + optimization. ACM Transactions on Mathematical Software 23 (4): + 550-560. + .. [8] Nash, S G. Newton-Type Minimization Via the Lanczos Method. + 1984. SIAM Journal of Numerical Analysis 21: 770-778. + .. [9] Powell, M J D. A direct search optimization method that models + the objective and constraint functions by linear interpolation. + 1994. Advances in Optimization and Numerical Analysis, eds. S. Gomez + and J-P Hennart, Kluwer Academic (Dordrecht), 51-67. + .. [10] Powell M J D. Direct search algorithms for optimization + calculations. 1998. Acta Numerica 7: 287-336. + .. [11] Powell M J D. A view of algorithms for optimization without + derivatives. 2007.Cambridge University Technical Report DAMTP + 2007/NA03 + .. [12] Kraft, D. A software package for sequential quadratic + programming. 1988. Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace + Center -- Institute for Flight Mechanics, Koln, Germany. + .. [13] Conn, A. R., Gould, N. I., and Toint, P. L. + Trust region methods. 2000. Siam. pp. 169-200. + .. [14] F. Lenders, C. Kirches, A. Potschka: "trlib: A vector-free + implementation of the GLTR method for iterative solution of + the trust region problem", :arxiv:`1611.04718` + .. [15] N. Gould, S. Lucidi, M. Roma, P. Toint: "Solving the + Trust-Region Subproblem using the Lanczos Method", + SIAM J. Optim., 9(2), 504--525, (1999). + .. [16] Byrd, Richard H., Mary E. Hribar, and Jorge Nocedal. 1999. + An interior point algorithm for large-scale nonlinear programming. + SIAM Journal on Optimization 9.4: 877-900. + .. [17] Lalee, Marucha, Jorge Nocedal, and Todd Plantenga. 1998. On the + implementation of an algorithm for large-scale equality constrained + optimization. SIAM Journal on Optimization 8.3: 682-706. + .. [18] Ragonneau, T. M. *Model-Based Derivative-Free Optimization Methods + and Software*. PhD thesis, Department of Applied Mathematics, The Hong + Kong Polytechnic University, Hong Kong, China, 2022. URL: + https://theses.lib.polyu.edu.hk/handle/200/12294. + + Examples + -------- + Let us consider the problem of minimizing the Rosenbrock function. This + function (and its respective derivatives) is implemented in `rosen` + (resp. `rosen_der`, `rosen_hess`) in the `scipy.optimize`. + + >>> from scipy.optimize import minimize, rosen, rosen_der + + A simple application of the *Nelder-Mead* method is: + + >>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2] + >>> res = minimize(rosen, x0, method='Nelder-Mead', tol=1e-6) + >>> res.x + array([ 1., 1., 1., 1., 1.]) + + Now using the *BFGS* algorithm, using the first derivative and a few + options: + + >>> res = minimize(rosen, x0, method='BFGS', jac=rosen_der, + ... options={'gtol': 1e-6, 'disp': True}) + Optimization terminated successfully. + Current function value: 0.000000 + Iterations: 26 + Function evaluations: 31 + Gradient evaluations: 31 + >>> res.x + array([ 1., 1., 1., 1., 1.]) + >>> print(res.message) + Optimization terminated successfully. + >>> res.hess_inv + array([ + [ 0.00749589, 0.01255155, 0.02396251, 0.04750988, 0.09495377], # may vary + [ 0.01255155, 0.02510441, 0.04794055, 0.09502834, 0.18996269], + [ 0.02396251, 0.04794055, 0.09631614, 0.19092151, 0.38165151], + [ 0.04750988, 0.09502834, 0.19092151, 0.38341252, 0.7664427 ], + [ 0.09495377, 0.18996269, 0.38165151, 0.7664427, 1.53713523] + ]) + + + Next, consider a minimization problem with several constraints (namely + Example 16.4 from [5]_). The objective function is: + + >>> fun = lambda x: (x[0] - 1)**2 + (x[1] - 2.5)**2 + + There are three constraints defined as: + + >>> cons = ({'type': 'ineq', 'fun': lambda x: x[0] - 2 * x[1] + 2}, + ... {'type': 'ineq', 'fun': lambda x: -x[0] - 2 * x[1] + 6}, + ... {'type': 'ineq', 'fun': lambda x: -x[0] + 2 * x[1] + 2}) + + And variables must be positive, hence the following bounds: + + >>> bnds = ((0, None), (0, None)) + + The optimization problem is solved using the SLSQP method as: + + >>> res = minimize(fun, (2, 0), method='SLSQP', bounds=bnds, + ... constraints=cons) + + It should converge to the theoretical solution (1.4 ,1.7). + + """ + x0 = np.atleast_1d(np.asarray(x0)) + + if x0.ndim != 1: + raise ValueError("'x0' must only have one dimension.") + + if x0.dtype.kind in np.typecodes["AllInteger"]: + x0 = np.asarray(x0, dtype=float) + + if not isinstance(args, tuple): + args = (args,) + + if method is None: + # Select automatically + if constraints: + method = 'SLSQP' + elif bounds is not None: + method = 'L-BFGS-B' + else: + method = 'BFGS' + + if callable(method): + meth = "_custom" + else: + meth = method.lower() + + if options is None: + options = {} + # check if optional parameters are supported by the selected method + # - jac + if meth in ('nelder-mead', 'powell', 'cobyla', 'cobyqa') and bool(jac): + warn(f'Method {method} does not use gradient information (jac).', + RuntimeWarning, stacklevel=2) + # - hess + if meth not in ('newton-cg', 'dogleg', 'trust-ncg', 'trust-constr', + 'trust-krylov', 'trust-exact', '_custom') and hess is not None: + warn(f'Method {method} does not use Hessian information (hess).', + RuntimeWarning, stacklevel=2) + # - hessp + if meth not in ('newton-cg', 'trust-ncg', 'trust-constr', + 'trust-krylov', '_custom') \ + and hessp is not None: + warn(f'Method {method} does not use Hessian-vector product' + ' information (hessp).', + RuntimeWarning, stacklevel=2) + # - constraints or bounds + if (meth not in ('cobyla', 'cobyqa', 'slsqp', 'trust-constr', '_custom') and + np.any(constraints)): + warn(f'Method {method} cannot handle constraints.', + RuntimeWarning, stacklevel=2) + if meth not in ( + 'nelder-mead', 'powell', 'l-bfgs-b', 'cobyla', 'cobyqa', 'slsqp', + 'tnc', 'trust-constr', '_custom') and bounds is not None: + warn(f'Method {method} cannot handle bounds.', + RuntimeWarning, stacklevel=2) + # - return_all + if (meth in ('l-bfgs-b', 'tnc', 'cobyla', 'cobyqa', 'slsqp') and + options.get('return_all', False)): + warn(f'Method {method} does not support the return_all option.', + RuntimeWarning, stacklevel=2) + + # check gradient vector + if callable(jac): + pass + elif jac is True: + # fun returns func and grad + fun = MemoizeJac(fun) + jac = fun.derivative + elif (jac in FD_METHODS and + meth in ['trust-constr', 'bfgs', 'cg', 'l-bfgs-b', 'tnc', 'slsqp']): + # finite differences with relative step + pass + elif meth in ['trust-constr']: + # default jac calculation for this method + jac = '2-point' + elif jac is None or bool(jac) is False: + # this will cause e.g. LBFGS to use forward difference, absolute step + jac = None + else: + # default if jac option is not understood + jac = None + + # set default tolerances + if tol is not None: + options = dict(options) + if meth == 'nelder-mead': + options.setdefault('xatol', tol) + options.setdefault('fatol', tol) + if meth in ('newton-cg', 'powell', 'tnc'): + options.setdefault('xtol', tol) + if meth in ('powell', 'l-bfgs-b', 'tnc', 'slsqp'): + options.setdefault('ftol', tol) + if meth in ('bfgs', 'cg', 'l-bfgs-b', 'tnc', 'dogleg', + 'trust-ncg', 'trust-exact', 'trust-krylov'): + options.setdefault('gtol', tol) + if meth in ('cobyla', '_custom'): + options.setdefault('tol', tol) + if meth == 'cobyqa': + options.setdefault('final_tr_radius', tol) + if meth == 'trust-constr': + options.setdefault('xtol', tol) + options.setdefault('gtol', tol) + options.setdefault('barrier_tol', tol) + + if meth == '_custom': + # custom method called before bounds and constraints are 'standardised' + # custom method should be able to accept whatever bounds/constraints + # are provided to it. + return method(fun, x0, args=args, jac=jac, hess=hess, hessp=hessp, + bounds=bounds, constraints=constraints, + callback=callback, **options) + + constraints = standardize_constraints(constraints, x0, meth) + + remove_vars = False + if bounds is not None: + # convert to new-style bounds so we only have to consider one case + bounds = standardize_bounds(bounds, x0, 'new') + bounds = _validate_bounds(bounds, x0, meth) + + if meth in {"tnc", "slsqp", "l-bfgs-b"}: + # These methods can't take the finite-difference derivatives they + # need when a variable is fixed by the bounds. To avoid this issue, + # remove fixed variables from the problem. + # NOTE: if this list is expanded, then be sure to update the + # accompanying tests and test_optimize.eb_data. Consider also if + # default OptimizeResult will need updating. + + # determine whether any variables are fixed + i_fixed = (bounds.lb == bounds.ub) + + if np.all(i_fixed): + # all the parameters are fixed, a minimizer is not able to do + # anything + return _optimize_result_for_equal_bounds( + fun, bounds, meth, args=args, constraints=constraints + ) + + # determine whether finite differences are needed for any grad/jac + fd_needed = (not callable(jac)) + for con in constraints: + if not callable(con.get('jac', None)): + fd_needed = True + + # If finite differences are ever used, remove all fixed variables + # Always remove fixed variables for TNC; see gh-14565 + remove_vars = i_fixed.any() and (fd_needed or meth == "tnc") + if remove_vars: + x_fixed = (bounds.lb)[i_fixed] + x0 = x0[~i_fixed] + bounds = _remove_from_bounds(bounds, i_fixed) + fun = _remove_from_func(fun, i_fixed, x_fixed) + if callable(callback): + callback = _remove_from_func(callback, i_fixed, x_fixed) + if callable(jac): + jac = _remove_from_func(jac, i_fixed, x_fixed, remove=1) + + # make a copy of the constraints so the user's version doesn't + # get changed. (Shallow copy is ok) + constraints = [con.copy() for con in constraints] + for con in constraints: # yes, guaranteed to be a list + con['fun'] = _remove_from_func(con['fun'], i_fixed, + x_fixed, min_dim=1, + remove=0) + if callable(con.get('jac', None)): + con['jac'] = _remove_from_func(con['jac'], i_fixed, + x_fixed, min_dim=2, + remove=1) + bounds = standardize_bounds(bounds, x0, meth) + + callback = _wrap_callback(callback, meth) + + if meth == 'nelder-mead': + res = _minimize_neldermead(fun, x0, args, callback, bounds=bounds, + **options) + elif meth == 'powell': + res = _minimize_powell(fun, x0, args, callback, bounds, **options) + elif meth == 'cg': + res = _minimize_cg(fun, x0, args, jac, callback, **options) + elif meth == 'bfgs': + res = _minimize_bfgs(fun, x0, args, jac, callback, **options) + elif meth == 'newton-cg': + res = _minimize_newtoncg(fun, x0, args, jac, hess, hessp, callback, + **options) + elif meth == 'l-bfgs-b': + res = _minimize_lbfgsb(fun, x0, args, jac, bounds, + callback=callback, **options) + elif meth == 'tnc': + res = _minimize_tnc(fun, x0, args, jac, bounds, callback=callback, + **options) + elif meth == 'cobyla': + res = _minimize_cobyla(fun, x0, args, constraints, callback=callback, + bounds=bounds, **options) + elif meth == 'cobyqa': + res = _minimize_cobyqa(fun, x0, args, bounds, constraints, callback, + **options) + elif meth == 'slsqp': + res = _minimize_slsqp(fun, x0, args, jac, bounds, + constraints, callback=callback, **options) + elif meth == 'trust-constr': + res = _minimize_trustregion_constr(fun, x0, args, jac, hess, hessp, + bounds, constraints, + callback=callback, **options) + elif meth == 'dogleg': + res = _minimize_dogleg(fun, x0, args, jac, hess, + callback=callback, **options) + elif meth == 'trust-ncg': + res = _minimize_trust_ncg(fun, x0, args, jac, hess, hessp, + callback=callback, **options) + elif meth == 'trust-krylov': + res = _minimize_trust_krylov(fun, x0, args, jac, hess, hessp, + callback=callback, **options) + elif meth == 'trust-exact': + res = _minimize_trustregion_exact(fun, x0, args, jac, hess, + callback=callback, **options) + else: + raise ValueError(f'Unknown solver {method}') + + if remove_vars: + res.x = _add_to_array(res.x, i_fixed, x_fixed) + res.jac = _add_to_array(res.jac, i_fixed, np.nan) + if "hess_inv" in res: + res.hess_inv = None # unknown + + if getattr(callback, 'stop_iteration', False): + res.success = False + res.status = 99 + res.message = "`callback` raised `StopIteration`." + + return res + + +def minimize_scalar(fun, bracket=None, bounds=None, args=(), + method=None, tol=None, options=None): + """Local minimization of scalar function of one variable. + + Parameters + ---------- + fun : callable + Objective function. + Scalar function, must return a scalar. + + Suppose the callable has signature ``f0(x, *my_args, **my_kwargs)``, where + ``my_args`` and ``my_kwargs`` are required positional and keyword arguments. + Rather than passing ``f0`` as the callable, wrap it to accept + only ``x``; e.g., pass ``fun=lambda x: f0(x, *my_args, **my_kwargs)`` as the + callable, where ``my_args`` (tuple) and ``my_kwargs`` (dict) have been + gathered before invoking this function. + + bracket : sequence, optional + For methods 'brent' and 'golden', `bracket` defines the bracketing + interval and is required. + Either a triple ``(xa, xb, xc)`` satisfying ``xa < xb < xc`` and + ``func(xb) < func(xa) and func(xb) < func(xc)``, or a pair + ``(xa, xb)`` to be used as initial points for a downhill bracket search + (see `scipy.optimize.bracket`). + The minimizer ``res.x`` will not necessarily satisfy + ``xa <= res.x <= xb``. + bounds : sequence, optional + For method 'bounded', `bounds` is mandatory and must have two finite + items corresponding to the optimization bounds. + args : tuple, optional + Extra arguments passed to the objective function. + method : str or callable, optional + Type of solver. Should be one of: + + - :ref:`Brent ` + - :ref:`Bounded ` + - :ref:`Golden ` + - custom - a callable object (added in version 0.14.0), see below + + Default is "Bounded" if bounds are provided and "Brent" otherwise. + See the 'Notes' section for details of each solver. + + tol : float, optional + Tolerance for termination. For detailed control, use solver-specific + options. + options : dict, optional + A dictionary of solver options. + + maxiter : int + Maximum number of iterations to perform. + disp : bool + Set to True to print convergence messages. + + See :func:`show_options()` for solver-specific options. + + Returns + ------- + res : OptimizeResult + The optimization result represented as a ``OptimizeResult`` object. + Important attributes are: ``x`` the solution array, ``success`` a + Boolean flag indicating if the optimizer exited successfully and + ``message`` which describes the cause of the termination. See + `OptimizeResult` for a description of other attributes. + + See also + -------- + minimize : Interface to minimization algorithms for scalar multivariate + functions + show_options : Additional options accepted by the solvers + + Notes + ----- + This section describes the available solvers that can be selected by the + 'method' parameter. The default method is the ``"Bounded"`` Brent method if + `bounds` are passed and unbounded ``"Brent"`` otherwise. + + Method :ref:`Brent ` uses Brent's + algorithm [1]_ to find a local minimum. The algorithm uses inverse + parabolic interpolation when possible to speed up convergence of + the golden section method. + + Method :ref:`Golden ` uses the + golden section search technique [1]_. It uses analog of the bisection + method to decrease the bracketed interval. It is usually + preferable to use the *Brent* method. + + Method :ref:`Bounded ` can + perform bounded minimization [2]_ [3]_. It uses the Brent method to find a + local minimum in the interval x1 < xopt < x2. + + Note that the Brent and Golden methods do not guarantee success unless a + valid ``bracket`` triple is provided. If a three-point bracket cannot be + found, consider `scipy.optimize.minimize`. Also, all methods are intended + only for local minimization. When the function of interest has more than + one local minimum, consider :ref:`global_optimization`. + + **Custom minimizers** + + It may be useful to pass a custom minimization method, for example + when using some library frontend to minimize_scalar. You can simply + pass a callable as the ``method`` parameter. + + The callable is called as ``method(fun, args, **kwargs, **options)`` + where ``kwargs`` corresponds to any other parameters passed to `minimize` + (such as `bracket`, `tol`, etc.), except the `options` dict, which has + its contents also passed as `method` parameters pair by pair. The method + shall return an `OptimizeResult` object. + + The provided `method` callable must be able to accept (and possibly ignore) + arbitrary parameters; the set of parameters accepted by `minimize` may + expand in future versions and then these parameters will be passed to + the method. You can find an example in the scipy.optimize tutorial. + + .. versionadded:: 0.11.0 + + References + ---------- + .. [1] Press, W., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. + Numerical Recipes in C. Cambridge University Press. + .. [2] Forsythe, G.E., M. A. Malcolm, and C. B. Moler. "Computer Methods + for Mathematical Computations." Prentice-Hall Series in Automatic + Computation 259 (1977). + .. [3] Brent, Richard P. Algorithms for Minimization Without Derivatives. + Courier Corporation, 2013. + + Examples + -------- + Consider the problem of minimizing the following function. + + >>> def f(x): + ... return (x - 2) * x * (x + 2)**2 + + Using the *Brent* method, we find the local minimum as: + + >>> from scipy.optimize import minimize_scalar + >>> res = minimize_scalar(f) + >>> res.fun + -9.9149495908 + + The minimizer is: + + >>> res.x + 1.28077640403 + + Using the *Bounded* method, we find a local minimum with specified + bounds as: + + >>> res = minimize_scalar(f, bounds=(-3, -1), method='bounded') + >>> res.fun # minimum + 3.28365179850e-13 + >>> res.x # minimizer + -2.0000002026 + + """ + if not isinstance(args, tuple): + args = (args,) + + if callable(method): + meth = "_custom" + elif method is None: + meth = 'brent' if bounds is None else 'bounded' + else: + meth = method.lower() + if options is None: + options = {} + + if bounds is not None and meth in {'brent', 'golden'}: + message = f"Use of `bounds` is incompatible with 'method={method}'." + raise ValueError(message) + + if tol is not None: + options = dict(options) + if meth == 'bounded' and 'xatol' not in options: + warn("Method 'bounded' does not support relative tolerance in x; " + "defaulting to absolute tolerance.", + RuntimeWarning, stacklevel=2) + options['xatol'] = tol + elif meth == '_custom': + options.setdefault('tol', tol) + else: + options.setdefault('xtol', tol) + + # replace boolean "disp" option, if specified, by an integer value. + disp = options.get('disp') + if isinstance(disp, bool): + options['disp'] = 2 * int(disp) + + if meth == '_custom': + res = method(fun, args=args, bracket=bracket, bounds=bounds, **options) + elif meth == 'brent': + res = _recover_from_bracket_error(_minimize_scalar_brent, + fun, bracket, args, **options) + elif meth == 'bounded': + if bounds is None: + raise ValueError('The `bounds` parameter is mandatory for ' + 'method `bounded`.') + res = _minimize_scalar_bounded(fun, bounds, args, **options) + elif meth == 'golden': + res = _recover_from_bracket_error(_minimize_scalar_golden, + fun, bracket, args, **options) + else: + raise ValueError(f'Unknown solver {method}') + + # gh-16196 reported inconsistencies in the output shape of `res.x`. While + # fixing this, future-proof it for when the function is vectorized: + # the shape of `res.x` should match that of `res.fun`. + res.fun = np.asarray(res.fun)[()] + res.x = np.reshape(res.x, res.fun.shape)[()] + return res + + +def _remove_from_bounds(bounds, i_fixed): + """Removes fixed variables from a `Bounds` instance""" + lb = bounds.lb[~i_fixed] + ub = bounds.ub[~i_fixed] + return Bounds(lb, ub) # don't mutate original Bounds object + + +def _remove_from_func(fun_in, i_fixed, x_fixed, min_dim=None, remove=0): + """Wraps a function such that fixed variables need not be passed in""" + def fun_out(x_in, *args, **kwargs): + x_out = np.zeros_like(i_fixed, dtype=x_in.dtype) + x_out[i_fixed] = x_fixed + x_out[~i_fixed] = x_in + y_out = fun_in(x_out, *args, **kwargs) + y_out = np.array(y_out) + + if min_dim == 1: + y_out = np.atleast_1d(y_out) + elif min_dim == 2: + y_out = np.atleast_2d(y_out) + + if remove == 1: + y_out = y_out[..., ~i_fixed] + elif remove == 2: + y_out = y_out[~i_fixed, ~i_fixed] + + return y_out + return fun_out + + +def _add_to_array(x_in, i_fixed, x_fixed): + """Adds fixed variables back to an array""" + i_free = ~i_fixed + if x_in.ndim == 2: + i_free = i_free[:, None] @ i_free[None, :] + x_out = np.zeros_like(i_free, dtype=x_in.dtype) + x_out[~i_free] = x_fixed + x_out[i_free] = x_in.ravel() + return x_out + + +def _validate_bounds(bounds, x0, meth): + """Check that bounds are valid.""" + + msg = "An upper bound is less than the corresponding lower bound." + if np.any(bounds.ub < bounds.lb): + raise ValueError(msg) + + msg = "The number of bounds is not compatible with the length of `x0`." + try: + bounds.lb = np.broadcast_to(bounds.lb, x0.shape) + bounds.ub = np.broadcast_to(bounds.ub, x0.shape) + except Exception as e: + raise ValueError(msg) from e + + return bounds + +def standardize_bounds(bounds, x0, meth): + """Converts bounds to the form required by the solver.""" + if meth in {'trust-constr', 'powell', 'nelder-mead', 'cobyla', 'cobyqa', + 'new'}: + if not isinstance(bounds, Bounds): + lb, ub = old_bound_to_new(bounds) + bounds = Bounds(lb, ub) + elif meth in ('l-bfgs-b', 'tnc', 'slsqp', 'old'): + if isinstance(bounds, Bounds): + bounds = new_bounds_to_old(bounds.lb, bounds.ub, x0.shape[0]) + return bounds + + +def standardize_constraints(constraints, x0, meth): + """Converts constraints to the form required by the solver.""" + all_constraint_types = (NonlinearConstraint, LinearConstraint, dict) + new_constraint_types = all_constraint_types[:-1] + if constraints is None: + constraints = [] + elif isinstance(constraints, all_constraint_types): + constraints = [constraints] + else: + constraints = list(constraints) # ensure it's a mutable sequence + + if meth in ['trust-constr', 'cobyqa', 'new']: + for i, con in enumerate(constraints): + if not isinstance(con, new_constraint_types): + constraints[i] = old_constraint_to_new(i, con) + else: + # iterate over copy, changing original + for i, con in enumerate(list(constraints)): + if isinstance(con, new_constraint_types): + old_constraints = new_constraint_to_old(con, x0) + constraints[i] = old_constraints[0] + constraints.extend(old_constraints[1:]) # appends 1 if present + + return constraints + + +def _optimize_result_for_equal_bounds( + fun, bounds, method, args=(), constraints=() +): + """ + Provides a default OptimizeResult for when a bounded minimization method + has (lb == ub).all(). + + Parameters + ---------- + fun: callable + bounds: Bounds + method: str + constraints: Constraint + """ + success = True + message = 'All independent variables were fixed by bounds.' + + # bounds is new-style + x0 = bounds.lb + + if constraints: + message = ("All independent variables were fixed by bounds at values" + " that satisfy the constraints.") + constraints = standardize_constraints(constraints, x0, 'new') + + maxcv = 0 + for c in constraints: + pc = PreparedConstraint(c, x0) + violation = pc.violation(x0) + if np.sum(violation): + maxcv = max(maxcv, np.max(violation)) + success = False + message = (f"All independent variables were fixed by bounds, but " + f"the independent variables do not satisfy the " + f"constraints exactly. (Maximum violation: {maxcv}).") + + return OptimizeResult( + x=x0, fun=fun(x0, *args), success=success, message=message, nfev=1, + njev=0, nhev=0, + ) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_minpack.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_minpack.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..710d8c41decf596e7481bfb8d88d30f008d6f967 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_minpack.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_minpack_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_minpack_py.py new file mode 100644 index 0000000000000000000000000000000000000000..8cfce8aae21b7533fb1cb7385d0d5f48740e743b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_minpack_py.py @@ -0,0 +1,1171 @@ +import warnings +from . import _minpack + +import numpy as np +from numpy import (atleast_1d, triu, shape, transpose, zeros, prod, greater, + asarray, inf, + finfo, inexact, issubdtype, dtype) +from scipy import linalg +from scipy.linalg import svd, cholesky, solve_triangular, LinAlgError +from scipy._lib._util import _asarray_validated, _lazywhere, _contains_nan +from scipy._lib._util import getfullargspec_no_self as _getfullargspec +from ._optimize import OptimizeResult, _check_unknown_options, OptimizeWarning +from ._lsq import least_squares +# from ._lsq.common import make_strictly_feasible +from ._lsq.least_squares import prepare_bounds +from scipy.optimize._minimize import Bounds + +__all__ = ['fsolve', 'leastsq', 'fixed_point', 'curve_fit'] + + +def _check_func(checker, argname, thefunc, x0, args, numinputs, + output_shape=None): + res = atleast_1d(thefunc(*((x0[:numinputs],) + args))) + if (output_shape is not None) and (shape(res) != output_shape): + if (output_shape[0] != 1): + if len(output_shape) > 1: + if output_shape[1] == 1: + return shape(res) + msg = f"{checker}: there is a mismatch between the input and output " \ + f"shape of the '{argname}' argument" + func_name = getattr(thefunc, '__name__', None) + if func_name: + msg += f" '{func_name}'." + else: + msg += "." + msg += f'Shape should be {output_shape} but it is {shape(res)}.' + raise TypeError(msg) + if issubdtype(res.dtype, inexact): + dt = res.dtype + else: + dt = dtype(float) + return shape(res), dt + + +def fsolve(func, x0, args=(), fprime=None, full_output=0, + col_deriv=0, xtol=1.49012e-8, maxfev=0, band=None, + epsfcn=None, factor=100, diag=None): + """ + Find the roots of a function. + + Return the roots of the (non-linear) equations defined by + ``func(x) = 0`` given a starting estimate. + + Parameters + ---------- + func : callable ``f(x, *args)`` + A function that takes at least one (possibly vector) argument, + and returns a value of the same length. + x0 : ndarray + The starting estimate for the roots of ``func(x) = 0``. + args : tuple, optional + Any extra arguments to `func`. + fprime : callable ``f(x, *args)``, optional + A function to compute the Jacobian of `func` with derivatives + across the rows. By default, the Jacobian will be estimated. + full_output : bool, optional + If True, return optional outputs. + col_deriv : bool, optional + Specify whether the Jacobian function computes derivatives down + the columns (faster, because there is no transpose operation). + xtol : float, optional + The calculation will terminate if the relative error between two + consecutive iterates is at most `xtol`. + maxfev : int, optional + The maximum number of calls to the function. If zero, then + ``100*(N+1)`` is the maximum where N is the number of elements + in `x0`. + band : tuple, optional + If set to a two-sequence containing the number of sub- and + super-diagonals within the band of the Jacobi matrix, the + Jacobi matrix is considered banded (only for ``fprime=None``). + epsfcn : float, optional + A suitable step length for the forward-difference + approximation of the Jacobian (for ``fprime=None``). If + `epsfcn` is less than the machine precision, it is assumed + that the relative errors in the functions are of the order of + the machine precision. + factor : float, optional + A parameter determining the initial step bound + (``factor * || diag * x||``). Should be in the interval + ``(0.1, 100)``. + diag : sequence, optional + N positive entries that serve as a scale factors for the + variables. + + Returns + ------- + x : ndarray + The solution (or the result of the last iteration for + an unsuccessful call). + infodict : dict + A dictionary of optional outputs with the keys: + + ``nfev`` + number of function calls + ``njev`` + number of Jacobian calls + ``fvec`` + function evaluated at the output + ``fjac`` + the orthogonal matrix, q, produced by the QR + factorization of the final approximate Jacobian + matrix, stored column wise + ``r`` + upper triangular matrix produced by QR factorization + of the same matrix + ``qtf`` + the vector ``(transpose(q) * fvec)`` + + ier : int + An integer flag. Set to 1 if a solution was found, otherwise refer + to `mesg` for more information. + mesg : str + If no solution is found, `mesg` details the cause of failure. + + See Also + -------- + root : Interface to root finding algorithms for multivariate + functions. See the ``method='hybr'`` in particular. + + Notes + ----- + ``fsolve`` is a wrapper around MINPACK's hybrd and hybrj algorithms. + + Examples + -------- + Find a solution to the system of equations: + ``x0*cos(x1) = 4, x1*x0 - x1 = 5``. + + >>> import numpy as np + >>> from scipy.optimize import fsolve + >>> def func(x): + ... return [x[0] * np.cos(x[1]) - 4, + ... x[1] * x[0] - x[1] - 5] + >>> root = fsolve(func, [1, 1]) + >>> root + array([6.50409711, 0.90841421]) + >>> np.isclose(func(root), [0.0, 0.0]) # func(root) should be almost 0.0. + array([ True, True]) + + """ + def _wrapped_func(*fargs): + """ + Wrapped `func` to track the number of times + the function has been called. + """ + _wrapped_func.nfev += 1 + return func(*fargs) + + _wrapped_func.nfev = 0 + + options = {'col_deriv': col_deriv, + 'xtol': xtol, + 'maxfev': maxfev, + 'band': band, + 'eps': epsfcn, + 'factor': factor, + 'diag': diag} + + res = _root_hybr(_wrapped_func, x0, args, jac=fprime, **options) + res.nfev = _wrapped_func.nfev + + if full_output: + x = res['x'] + info = {k: res.get(k) + for k in ('nfev', 'njev', 'fjac', 'r', 'qtf') if k in res} + info['fvec'] = res['fun'] + return x, info, res['status'], res['message'] + else: + status = res['status'] + msg = res['message'] + if status == 0: + raise TypeError(msg) + elif status == 1: + pass + elif status in [2, 3, 4, 5]: + warnings.warn(msg, RuntimeWarning, stacklevel=2) + else: + raise TypeError(msg) + return res['x'] + + +def _root_hybr(func, x0, args=(), jac=None, + col_deriv=0, xtol=1.49012e-08, maxfev=0, band=None, eps=None, + factor=100, diag=None, **unknown_options): + """ + Find the roots of a multivariate function using MINPACK's hybrd and + hybrj routines (modified Powell method). + + Options + ------- + col_deriv : bool + Specify whether the Jacobian function computes derivatives down + the columns (faster, because there is no transpose operation). + xtol : float + The calculation will terminate if the relative error between two + consecutive iterates is at most `xtol`. + maxfev : int + The maximum number of calls to the function. If zero, then + ``100*(N+1)`` is the maximum where N is the number of elements + in `x0`. + band : tuple + If set to a two-sequence containing the number of sub- and + super-diagonals within the band of the Jacobi matrix, the + Jacobi matrix is considered banded (only for ``jac=None``). + eps : float + A suitable step length for the forward-difference + approximation of the Jacobian (for ``jac=None``). If + `eps` is less than the machine precision, it is assumed + that the relative errors in the functions are of the order of + the machine precision. + factor : float + A parameter determining the initial step bound + (``factor * || diag * x||``). Should be in the interval + ``(0.1, 100)``. + diag : sequence + N positive entries that serve as a scale factors for the + variables. + + """ + _check_unknown_options(unknown_options) + epsfcn = eps + + x0 = asarray(x0).flatten() + n = len(x0) + if not isinstance(args, tuple): + args = (args,) + shape, dtype = _check_func('fsolve', 'func', func, x0, args, n, (n,)) + if epsfcn is None: + epsfcn = finfo(dtype).eps + Dfun = jac + if Dfun is None: + if band is None: + ml, mu = -10, -10 + else: + ml, mu = band[:2] + if maxfev == 0: + maxfev = 200 * (n + 1) + retval = _minpack._hybrd(func, x0, args, 1, xtol, maxfev, + ml, mu, epsfcn, factor, diag) + else: + _check_func('fsolve', 'fprime', Dfun, x0, args, n, (n, n)) + if (maxfev == 0): + maxfev = 100 * (n + 1) + retval = _minpack._hybrj(func, Dfun, x0, args, 1, + col_deriv, xtol, maxfev, factor, diag) + + x, status = retval[0], retval[-1] + + errors = {0: "Improper input parameters were entered.", + 1: "The solution converged.", + 2: "The number of calls to function has " + "reached maxfev = %d." % maxfev, + 3: f"xtol={xtol:f} is too small, no further improvement " + "in the approximate\n solution is possible.", + 4: "The iteration is not making good progress, as measured " + "by the \n improvement from the last five " + "Jacobian evaluations.", + 5: "The iteration is not making good progress, " + "as measured by the \n improvement from the last " + "ten iterations.", + 'unknown': "An error occurred."} + + info = retval[1] + info['fun'] = info.pop('fvec') + sol = OptimizeResult(x=x, success=(status == 1), status=status, + method="hybr") + sol.update(info) + try: + sol['message'] = errors[status] + except KeyError: + sol['message'] = errors['unknown'] + + return sol + + +LEASTSQ_SUCCESS = [1, 2, 3, 4] +LEASTSQ_FAILURE = [5, 6, 7, 8] + + +def leastsq(func, x0, args=(), Dfun=None, full_output=False, + col_deriv=False, ftol=1.49012e-8, xtol=1.49012e-8, + gtol=0.0, maxfev=0, epsfcn=None, factor=100, diag=None): + """ + Minimize the sum of squares of a set of equations. + + :: + + x = arg min(sum(func(y)**2,axis=0)) + y + + Parameters + ---------- + func : callable + Should take at least one (possibly length ``N`` vector) argument and + returns ``M`` floating point numbers. It must not return NaNs or + fitting might fail. ``M`` must be greater than or equal to ``N``. + x0 : ndarray + The starting estimate for the minimization. + args : tuple, optional + Any extra arguments to func are placed in this tuple. + Dfun : callable, optional + A function or method to compute the Jacobian of func with derivatives + across the rows. If this is None, the Jacobian will be estimated. + full_output : bool, optional + If ``True``, return all optional outputs (not just `x` and `ier`). + col_deriv : bool, optional + If ``True``, specify that the Jacobian function computes derivatives + down the columns (faster, because there is no transpose operation). + ftol : float, optional + Relative error desired in the sum of squares. + xtol : float, optional + Relative error desired in the approximate solution. + gtol : float, optional + Orthogonality desired between the function vector and the columns of + the Jacobian. + maxfev : int, optional + The maximum number of calls to the function. If `Dfun` is provided, + then the default `maxfev` is 100*(N+1) where N is the number of elements + in x0, otherwise the default `maxfev` is 200*(N+1). + epsfcn : float, optional + A variable used in determining a suitable step length for the forward- + difference approximation of the Jacobian (for Dfun=None). + Normally the actual step length will be sqrt(epsfcn)*x + If epsfcn is less than the machine precision, it is assumed that the + relative errors are of the order of the machine precision. + factor : float, optional + A parameter determining the initial step bound + (``factor * || diag * x||``). Should be in interval ``(0.1, 100)``. + diag : sequence, optional + N positive entries that serve as a scale factors for the variables. + + Returns + ------- + x : ndarray + The solution (or the result of the last iteration for an unsuccessful + call). + cov_x : ndarray + The inverse of the Hessian. `fjac` and `ipvt` are used to construct an + estimate of the Hessian. A value of None indicates a singular matrix, + which means the curvature in parameters `x` is numerically flat. To + obtain the covariance matrix of the parameters `x`, `cov_x` must be + multiplied by the variance of the residuals -- see curve_fit. Only + returned if `full_output` is ``True``. + infodict : dict + a dictionary of optional outputs with the keys: + + ``nfev`` + The number of function calls + ``fvec`` + The function evaluated at the output + ``fjac`` + A permutation of the R matrix of a QR + factorization of the final approximate + Jacobian matrix, stored column wise. + Together with ipvt, the covariance of the + estimate can be approximated. + ``ipvt`` + An integer array of length N which defines + a permutation matrix, p, such that + fjac*p = q*r, where r is upper triangular + with diagonal elements of nonincreasing + magnitude. Column j of p is column ipvt(j) + of the identity matrix. + ``qtf`` + The vector (transpose(q) * fvec). + + Only returned if `full_output` is ``True``. + mesg : str + A string message giving information about the cause of failure. + Only returned if `full_output` is ``True``. + ier : int + An integer flag. If it is equal to 1, 2, 3 or 4, the solution was + found. Otherwise, the solution was not found. In either case, the + optional output variable 'mesg' gives more information. + + See Also + -------- + least_squares : Newer interface to solve nonlinear least-squares problems + with bounds on the variables. See ``method='lm'`` in particular. + + Notes + ----- + "leastsq" is a wrapper around MINPACK's lmdif and lmder algorithms. + + cov_x is a Jacobian approximation to the Hessian of the least squares + objective function. + This approximation assumes that the objective function is based on the + difference between some observed target data (ydata) and a (non-linear) + function of the parameters `f(xdata, params)` :: + + func(params) = ydata - f(xdata, params) + + so that the objective function is :: + + min sum((ydata - f(xdata, params))**2, axis=0) + params + + The solution, `x`, is always a 1-D array, regardless of the shape of `x0`, + or whether `x0` is a scalar. + + Examples + -------- + >>> from scipy.optimize import leastsq + >>> def func(x): + ... return 2*(x-3)**2+1 + >>> leastsq(func, 0) + (array([2.99999999]), 1) + + """ + x0 = asarray(x0).flatten() + n = len(x0) + if not isinstance(args, tuple): + args = (args,) + shape, dtype = _check_func('leastsq', 'func', func, x0, args, n) + m = shape[0] + + if n > m: + raise TypeError(f"Improper input: func input vector length N={n} must" + f" not exceed func output vector length M={m}") + + if epsfcn is None: + epsfcn = finfo(dtype).eps + + if Dfun is None: + if maxfev == 0: + maxfev = 200*(n + 1) + retval = _minpack._lmdif(func, x0, args, full_output, ftol, xtol, + gtol, maxfev, epsfcn, factor, diag) + else: + if col_deriv: + _check_func('leastsq', 'Dfun', Dfun, x0, args, n, (n, m)) + else: + _check_func('leastsq', 'Dfun', Dfun, x0, args, n, (m, n)) + if maxfev == 0: + maxfev = 100 * (n + 1) + retval = _minpack._lmder(func, Dfun, x0, args, full_output, + col_deriv, ftol, xtol, gtol, maxfev, + factor, diag) + + errors = {0: ["Improper input parameters.", TypeError], + 1: ["Both actual and predicted relative reductions " + f"in the sum of squares\n are at most {ftol:f}", None], + 2: ["The relative error between two consecutive " + f"iterates is at most {xtol:f}", None], + 3: ["Both actual and predicted relative reductions in " + f"the sum of squares\n are at most {ftol:f} and the " + "relative error between two consecutive " + f"iterates is at \n most {xtol:f}", None], + 4: ["The cosine of the angle between func(x) and any " + f"column of the\n Jacobian is at most {gtol:f} in " + "absolute value", None], + 5: ["Number of calls to function has reached " + "maxfev = %d." % maxfev, ValueError], + 6: [f"ftol={ftol:f} is too small, no further reduction " + "in the sum of squares\n is possible.", + ValueError], + 7: [f"xtol={xtol:f} is too small, no further improvement in " + "the approximate\n solution is possible.", + ValueError], + 8: [f"gtol={gtol:f} is too small, func(x) is orthogonal to the " + "columns of\n the Jacobian to machine precision.", ValueError]} + + # The FORTRAN return value (possible return values are >= 0 and <= 8) + info = retval[-1] + + if full_output: + cov_x = None + if info in LEASTSQ_SUCCESS: + # This was + # perm = take(eye(n), retval[1]['ipvt'] - 1, 0) + # r = triu(transpose(retval[1]['fjac'])[:n, :]) + # R = dot(r, perm) + # cov_x = inv(dot(transpose(R), R)) + # but the explicit dot product was not necessary and sometimes + # the result was not symmetric positive definite. See gh-4555. + perm = retval[1]['ipvt'] + n = len(perm) + r = triu(transpose(retval[1]['fjac'])[:n, :]) + inv_triu = linalg.get_lapack_funcs('trtri', (r,)) + try: + # inverse of permuted matrix is a permutation of matrix inverse + invR, trtri_info = inv_triu(r) # default: upper, non-unit diag + if trtri_info != 0: # explicit comparison for readability + raise LinAlgError(f'trtri returned info {trtri_info}') + invR[perm] = invR.copy() + cov_x = invR @ invR.T + except (LinAlgError, ValueError): + pass + return (retval[0], cov_x) + retval[1:-1] + (errors[info][0], info) + else: + if info in LEASTSQ_FAILURE: + warnings.warn(errors[info][0], RuntimeWarning, stacklevel=2) + elif info == 0: + raise errors[info][1](errors[info][0]) + return retval[0], info + + +def _lightweight_memoizer(f): + # very shallow memoization to address gh-13670: only remember the first set + # of parameters and corresponding function value, and only attempt to use + # them twice (the number of times the function is evaluated at x0). + def _memoized_func(params): + if _memoized_func.skip_lookup: + return f(params) + + if np.all(_memoized_func.last_params == params): + return _memoized_func.last_val + elif _memoized_func.last_params is not None: + _memoized_func.skip_lookup = True + + val = f(params) + + if _memoized_func.last_params is None: + _memoized_func.last_params = np.copy(params) + _memoized_func.last_val = val + + return val + + _memoized_func.last_params = None + _memoized_func.last_val = None + _memoized_func.skip_lookup = False + return _memoized_func + + +def _wrap_func(func, xdata, ydata, transform): + if transform is None: + def func_wrapped(params): + return func(xdata, *params) - ydata + elif transform.size == 1 or transform.ndim == 1: + def func_wrapped(params): + return transform * (func(xdata, *params) - ydata) + else: + # Chisq = (y - yd)^T C^{-1} (y-yd) + # transform = L such that C = L L^T + # C^{-1} = L^{-T} L^{-1} + # Chisq = (y - yd)^T L^{-T} L^{-1} (y-yd) + # Define (y-yd)' = L^{-1} (y-yd) + # by solving + # L (y-yd)' = (y-yd) + # and minimize (y-yd)'^T (y-yd)' + def func_wrapped(params): + return solve_triangular(transform, func(xdata, *params) - ydata, lower=True) + return func_wrapped + + +def _wrap_jac(jac, xdata, transform): + if transform is None: + def jac_wrapped(params): + return jac(xdata, *params) + elif transform.ndim == 1: + def jac_wrapped(params): + return transform[:, np.newaxis] * np.asarray(jac(xdata, *params)) + else: + def jac_wrapped(params): + return solve_triangular(transform, + np.asarray(jac(xdata, *params)), + lower=True) + return jac_wrapped + + +def _initialize_feasible(lb, ub): + p0 = np.ones_like(lb) + lb_finite = np.isfinite(lb) + ub_finite = np.isfinite(ub) + + mask = lb_finite & ub_finite + p0[mask] = 0.5 * (lb[mask] + ub[mask]) + + mask = lb_finite & ~ub_finite + p0[mask] = lb[mask] + 1 + + mask = ~lb_finite & ub_finite + p0[mask] = ub[mask] - 1 + + return p0 + + +def curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, + check_finite=None, bounds=(-np.inf, np.inf), method=None, + jac=None, *, full_output=False, nan_policy=None, + **kwargs): + """ + Use non-linear least squares to fit a function, f, to data. + + Assumes ``ydata = f(xdata, *params) + eps``. + + Parameters + ---------- + f : callable + The model function, f(x, ...). It must take the independent + variable as the first argument and the parameters to fit as + separate remaining arguments. + xdata : array_like + The independent variable where the data is measured. + Should usually be an M-length sequence or an (k,M)-shaped array for + functions with k predictors, and each element should be float + convertible if it is an array like object. + ydata : array_like + The dependent data, a length M array - nominally ``f(xdata, ...)``. + p0 : array_like, optional + Initial guess for the parameters (length N). If None, then the + initial values will all be 1 (if the number of parameters for the + function can be determined using introspection, otherwise a + ValueError is raised). + sigma : None or scalar or M-length sequence or MxM array, optional + Determines the uncertainty in `ydata`. If we define residuals as + ``r = ydata - f(xdata, *popt)``, then the interpretation of `sigma` + depends on its number of dimensions: + + - A scalar or 1-D `sigma` should contain values of standard deviations of + errors in `ydata`. In this case, the optimized function is + ``chisq = sum((r / sigma) ** 2)``. + + - A 2-D `sigma` should contain the covariance matrix of + errors in `ydata`. In this case, the optimized function is + ``chisq = r.T @ inv(sigma) @ r``. + + .. versionadded:: 0.19 + + None (default) is equivalent of 1-D `sigma` filled with ones. + absolute_sigma : bool, optional + If True, `sigma` is used in an absolute sense and the estimated parameter + covariance `pcov` reflects these absolute values. + + If False (default), only the relative magnitudes of the `sigma` values matter. + The returned parameter covariance matrix `pcov` is based on scaling + `sigma` by a constant factor. This constant is set by demanding that the + reduced `chisq` for the optimal parameters `popt` when using the + *scaled* `sigma` equals unity. In other words, `sigma` is scaled to + match the sample variance of the residuals after the fit. Default is False. + Mathematically, + ``pcov(absolute_sigma=False) = pcov(absolute_sigma=True) * chisq(popt)/(M-N)`` + check_finite : bool, optional + If True, check that the input arrays do not contain nans of infs, + and raise a ValueError if they do. Setting this parameter to + False may silently produce nonsensical results if the input arrays + do contain nans. Default is True if `nan_policy` is not specified + explicitly and False otherwise. + bounds : 2-tuple of array_like or `Bounds`, optional + Lower and upper bounds on parameters. Defaults to no bounds. + There are two ways to specify the bounds: + + - Instance of `Bounds` class. + + - 2-tuple of array_like: Each element of the tuple must be either + an array with the length equal to the number of parameters, or a + scalar (in which case the bound is taken to be the same for all + parameters). Use ``np.inf`` with an appropriate sign to disable + bounds on all or some parameters. + + method : {'lm', 'trf', 'dogbox'}, optional + Method to use for optimization. See `least_squares` for more details. + Default is 'lm' for unconstrained problems and 'trf' if `bounds` are + provided. The method 'lm' won't work when the number of observations + is less than the number of variables, use 'trf' or 'dogbox' in this + case. + + .. versionadded:: 0.17 + jac : callable, string or None, optional + Function with signature ``jac(x, ...)`` which computes the Jacobian + matrix of the model function with respect to parameters as a dense + array_like structure. It will be scaled according to provided `sigma`. + If None (default), the Jacobian will be estimated numerically. + String keywords for 'trf' and 'dogbox' methods can be used to select + a finite difference scheme, see `least_squares`. + + .. versionadded:: 0.18 + full_output : boolean, optional + If True, this function returns additional information: `infodict`, + `mesg`, and `ier`. + + .. versionadded:: 1.9 + nan_policy : {'raise', 'omit', None}, optional + Defines how to handle when input contains nan. + The following options are available (default is None): + + * 'raise': throws an error + * 'omit': performs the calculations ignoring nan values + * None: no special handling of NaNs is performed + (except what is done by check_finite); the behavior when NaNs + are present is implementation-dependent and may change. + + Note that if this value is specified explicitly (not None), + `check_finite` will be set as False. + + .. versionadded:: 1.11 + **kwargs + Keyword arguments passed to `leastsq` for ``method='lm'`` or + `least_squares` otherwise. + + Returns + ------- + popt : array + Optimal values for the parameters so that the sum of the squared + residuals of ``f(xdata, *popt) - ydata`` is minimized. + pcov : 2-D array + The estimated approximate covariance of popt. The diagonals provide + the variance of the parameter estimate. To compute one standard + deviation errors on the parameters, use + ``perr = np.sqrt(np.diag(pcov))``. Note that the relationship between + `cov` and parameter error estimates is derived based on a linear + approximation to the model function around the optimum [1]_. + When this approximation becomes inaccurate, `cov` may not provide an + accurate measure of uncertainty. + + How the `sigma` parameter affects the estimated covariance + depends on `absolute_sigma` argument, as described above. + + If the Jacobian matrix at the solution doesn't have a full rank, then + 'lm' method returns a matrix filled with ``np.inf``, on the other hand + 'trf' and 'dogbox' methods use Moore-Penrose pseudoinverse to compute + the covariance matrix. Covariance matrices with large condition numbers + (e.g. computed with `numpy.linalg.cond`) may indicate that results are + unreliable. + infodict : dict (returned only if `full_output` is True) + a dictionary of optional outputs with the keys: + + ``nfev`` + The number of function calls. Methods 'trf' and 'dogbox' do not + count function calls for numerical Jacobian approximation, + as opposed to 'lm' method. + ``fvec`` + The residual values evaluated at the solution, for a 1-D `sigma` + this is ``(f(x, *popt) - ydata)/sigma``. + ``fjac`` + A permutation of the R matrix of a QR + factorization of the final approximate + Jacobian matrix, stored column wise. + Together with ipvt, the covariance of the + estimate can be approximated. + Method 'lm' only provides this information. + ``ipvt`` + An integer array of length N which defines + a permutation matrix, p, such that + fjac*p = q*r, where r is upper triangular + with diagonal elements of nonincreasing + magnitude. Column j of p is column ipvt(j) + of the identity matrix. + Method 'lm' only provides this information. + ``qtf`` + The vector (transpose(q) * fvec). + Method 'lm' only provides this information. + + .. versionadded:: 1.9 + mesg : str (returned only if `full_output` is True) + A string message giving information about the solution. + + .. versionadded:: 1.9 + ier : int (returned only if `full_output` is True) + An integer flag. If it is equal to 1, 2, 3 or 4, the solution was + found. Otherwise, the solution was not found. In either case, the + optional output variable `mesg` gives more information. + + .. versionadded:: 1.9 + + Raises + ------ + ValueError + if either `ydata` or `xdata` contain NaNs, or if incompatible options + are used. + + RuntimeError + if the least-squares minimization fails. + + OptimizeWarning + if covariance of the parameters can not be estimated. + + See Also + -------- + least_squares : Minimize the sum of squares of nonlinear functions. + scipy.stats.linregress : Calculate a linear least squares regression for + two sets of measurements. + + Notes + ----- + Users should ensure that inputs `xdata`, `ydata`, and the output of `f` + are ``float64``, or else the optimization may return incorrect results. + + With ``method='lm'``, the algorithm uses the Levenberg-Marquardt algorithm + through `leastsq`. Note that this algorithm can only deal with + unconstrained problems. + + Box constraints can be handled by methods 'trf' and 'dogbox'. Refer to + the docstring of `least_squares` for more information. + + Parameters to be fitted must have similar scale. Differences of multiple + orders of magnitude can lead to incorrect results. For the 'trf' and + 'dogbox' methods, the `x_scale` keyword argument can be used to scale + the parameters. + + References + ---------- + .. [1] K. Vugrin et al. Confidence region estimation techniques for nonlinear + regression in groundwater flow: Three case studies. Water Resources + Research, Vol. 43, W03423, :doi:`10.1029/2005WR004804` + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.optimize import curve_fit + + >>> def func(x, a, b, c): + ... return a * np.exp(-b * x) + c + + Define the data to be fit with some noise: + + >>> xdata = np.linspace(0, 4, 50) + >>> y = func(xdata, 2.5, 1.3, 0.5) + >>> rng = np.random.default_rng() + >>> y_noise = 0.2 * rng.normal(size=xdata.size) + >>> ydata = y + y_noise + >>> plt.plot(xdata, ydata, 'b-', label='data') + + Fit for the parameters a, b, c of the function `func`: + + >>> popt, pcov = curve_fit(func, xdata, ydata) + >>> popt + array([2.56274217, 1.37268521, 0.47427475]) + >>> plt.plot(xdata, func(xdata, *popt), 'r-', + ... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt)) + + Constrain the optimization to the region of ``0 <= a <= 3``, + ``0 <= b <= 1`` and ``0 <= c <= 0.5``: + + >>> popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 1., 0.5])) + >>> popt + array([2.43736712, 1. , 0.34463856]) + >>> plt.plot(xdata, func(xdata, *popt), 'g--', + ... label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt)) + + >>> plt.xlabel('x') + >>> plt.ylabel('y') + >>> plt.legend() + >>> plt.show() + + For reliable results, the model `func` should not be overparametrized; + redundant parameters can cause unreliable covariance matrices and, in some + cases, poorer quality fits. As a quick check of whether the model may be + overparameterized, calculate the condition number of the covariance matrix: + + >>> np.linalg.cond(pcov) + 34.571092161547405 # may vary + + The value is small, so it does not raise much concern. If, however, we were + to add a fourth parameter ``d`` to `func` with the same effect as ``a``: + + >>> def func2(x, a, b, c, d): + ... return a * d * np.exp(-b * x) + c # a and d are redundant + >>> popt, pcov = curve_fit(func2, xdata, ydata) + >>> np.linalg.cond(pcov) + 1.13250718925596e+32 # may vary + + Such a large value is cause for concern. The diagonal elements of the + covariance matrix, which is related to uncertainty of the fit, gives more + information: + + >>> np.diag(pcov) + array([1.48814742e+29, 3.78596560e-02, 5.39253738e-03, 2.76417220e+28]) # may vary + + Note that the first and last terms are much larger than the other elements, + suggesting that the optimal values of these parameters are ambiguous and + that only one of these parameters is needed in the model. + + If the optimal parameters of `f` differ by multiple orders of magnitude, the + resulting fit can be inaccurate. Sometimes, `curve_fit` can fail to find any + results: + + >>> ydata = func(xdata, 500000, 0.01, 15) + >>> try: + ... popt, pcov = curve_fit(func, xdata, ydata, method = 'trf') + ... except RuntimeError as e: + ... print(e) + Optimal parameters not found: The maximum number of function evaluations is + exceeded. + + If parameter scale is roughly known beforehand, it can be defined in + `x_scale` argument: + + >>> popt, pcov = curve_fit(func, xdata, ydata, method = 'trf', + ... x_scale = [1000, 1, 1]) + >>> popt + array([5.00000000e+05, 1.00000000e-02, 1.49999999e+01]) + """ + if p0 is None: + # determine number of parameters by inspecting the function + sig = _getfullargspec(f) + args = sig.args + if len(args) < 2: + raise ValueError("Unable to determine number of fit parameters.") + n = len(args) - 1 + else: + p0 = np.atleast_1d(p0) + n = p0.size + + if isinstance(bounds, Bounds): + lb, ub = bounds.lb, bounds.ub + else: + lb, ub = prepare_bounds(bounds, n) + if p0 is None: + p0 = _initialize_feasible(lb, ub) + + bounded_problem = np.any((lb > -np.inf) | (ub < np.inf)) + if method is None: + if bounded_problem: + method = 'trf' + else: + method = 'lm' + + if method == 'lm' and bounded_problem: + raise ValueError("Method 'lm' only works for unconstrained problems. " + "Use 'trf' or 'dogbox' instead.") + + if check_finite is None: + check_finite = True if nan_policy is None else False + + # optimization may produce garbage for float32 inputs, cast them to float64 + if check_finite: + ydata = np.asarray_chkfinite(ydata, float) + else: + ydata = np.asarray(ydata, float) + + if isinstance(xdata, (list, tuple, np.ndarray)): + # `xdata` is passed straight to the user-defined `f`, so allow + # non-array_like `xdata`. + if check_finite: + xdata = np.asarray_chkfinite(xdata, float) + else: + xdata = np.asarray(xdata, float) + + if ydata.size == 0: + raise ValueError("`ydata` must not be empty!") + + # nan handling is needed only if check_finite is False because if True, + # the x-y data are already checked, and they don't contain nans. + if not check_finite and nan_policy is not None: + if nan_policy == "propagate": + raise ValueError("`nan_policy='propagate'` is not supported " + "by this function.") + + policies = [None, 'raise', 'omit'] + x_contains_nan, nan_policy = _contains_nan(xdata, nan_policy, + policies=policies) + y_contains_nan, nan_policy = _contains_nan(ydata, nan_policy, + policies=policies) + + if (x_contains_nan or y_contains_nan) and nan_policy == 'omit': + # ignore NaNs for N dimensional arrays + has_nan = np.isnan(xdata) + has_nan = has_nan.any(axis=tuple(range(has_nan.ndim-1))) + has_nan |= np.isnan(ydata) + + xdata = xdata[..., ~has_nan] + ydata = ydata[~has_nan] + + # Also omit the corresponding entries from sigma + if sigma is not None: + sigma = np.asarray(sigma) + if sigma.ndim == 1: + sigma = sigma[~has_nan] + elif sigma.ndim == 2: + sigma = sigma[~has_nan, :] + sigma = sigma[:, ~has_nan] + + # Determine type of sigma + if sigma is not None: + sigma = np.asarray(sigma) + + # if 1-D or a scalar, sigma are errors, define transform = 1/sigma + if sigma.size == 1 or sigma.shape == (ydata.size,): + transform = 1.0 / sigma + # if 2-D, sigma is the covariance matrix, + # define transform = L such that L L^T = C + elif sigma.shape == (ydata.size, ydata.size): + try: + # scipy.linalg.cholesky requires lower=True to return L L^T = A + transform = cholesky(sigma, lower=True) + except LinAlgError as e: + raise ValueError("`sigma` must be positive definite.") from e + else: + raise ValueError("`sigma` has incorrect shape.") + else: + transform = None + + func = _lightweight_memoizer(_wrap_func(f, xdata, ydata, transform)) + + if callable(jac): + jac = _lightweight_memoizer(_wrap_jac(jac, xdata, transform)) + elif jac is None and method != 'lm': + jac = '2-point' + + if 'args' in kwargs: + # The specification for the model function `f` does not support + # additional arguments. Refer to the `curve_fit` docstring for + # acceptable call signatures of `f`. + raise ValueError("'args' is not a supported keyword argument.") + + if method == 'lm': + # if ydata.size == 1, this might be used for broadcast. + if ydata.size != 1 and n > ydata.size: + raise TypeError(f"The number of func parameters={n} must not" + f" exceed the number of data points={ydata.size}") + res = leastsq(func, p0, Dfun=jac, full_output=1, **kwargs) + popt, pcov, infodict, errmsg, ier = res + ysize = len(infodict['fvec']) + cost = np.sum(infodict['fvec'] ** 2) + if ier not in [1, 2, 3, 4]: + raise RuntimeError("Optimal parameters not found: " + errmsg) + else: + # Rename maxfev (leastsq) to max_nfev (least_squares), if specified. + if 'max_nfev' not in kwargs: + kwargs['max_nfev'] = kwargs.pop('maxfev', None) + + res = least_squares(func, p0, jac=jac, bounds=bounds, method=method, + **kwargs) + + if not res.success: + raise RuntimeError("Optimal parameters not found: " + res.message) + + infodict = dict(nfev=res.nfev, fvec=res.fun) + ier = res.status + errmsg = res.message + + ysize = len(res.fun) + cost = 2 * res.cost # res.cost is half sum of squares! + popt = res.x + + # Do Moore-Penrose inverse discarding zero singular values. + _, s, VT = svd(res.jac, full_matrices=False) + threshold = np.finfo(float).eps * max(res.jac.shape) * s[0] + s = s[s > threshold] + VT = VT[:s.size] + pcov = np.dot(VT.T / s**2, VT) + + warn_cov = False + if pcov is None or np.isnan(pcov).any(): + # indeterminate covariance + pcov = zeros((len(popt), len(popt)), dtype=float) + pcov.fill(inf) + warn_cov = True + elif not absolute_sigma: + if ysize > p0.size: + s_sq = cost / (ysize - p0.size) + pcov = pcov * s_sq + else: + pcov.fill(inf) + warn_cov = True + + if warn_cov: + warnings.warn('Covariance of the parameters could not be estimated', + category=OptimizeWarning, stacklevel=2) + + if full_output: + return popt, pcov, infodict, errmsg, ier + else: + return popt, pcov + + +def check_gradient(fcn, Dfcn, x0, args=(), col_deriv=0): + """Perform a simple check on the gradient for correctness. + + """ + + x = atleast_1d(x0) + n = len(x) + x = x.reshape((n,)) + fvec = atleast_1d(fcn(x, *args)) + m = len(fvec) + fvec = fvec.reshape((m,)) + ldfjac = m + fjac = atleast_1d(Dfcn(x, *args)) + fjac = fjac.reshape((m, n)) + if col_deriv == 0: + fjac = transpose(fjac) + + xp = zeros((n,), float) + err = zeros((m,), float) + fvecp = None + _minpack._chkder(m, n, x, fvec, fjac, ldfjac, xp, fvecp, 1, err) + + fvecp = atleast_1d(fcn(xp, *args)) + fvecp = fvecp.reshape((m,)) + _minpack._chkder(m, n, x, fvec, fjac, ldfjac, xp, fvecp, 2, err) + + good = (prod(greater(err, 0.5), axis=0)) + + return (good, err) + + +def _del2(p0, p1, d): + return p0 - np.square(p1 - p0) / d + + +def _relerr(actual, desired): + return (actual - desired) / desired + + +def _fixed_point_helper(func, x0, args, xtol, maxiter, use_accel): + p0 = x0 + for i in range(maxiter): + p1 = func(p0, *args) + if use_accel: + p2 = func(p1, *args) + d = p2 - 2.0 * p1 + p0 + p = _lazywhere(d != 0, (p0, p1, d), f=_del2, fillvalue=p2) + else: + p = p1 + relerr = _lazywhere(p0 != 0, (p, p0), f=_relerr, fillvalue=p) + if np.all(np.abs(relerr) < xtol): + return p + p0 = p + msg = "Failed to converge after %d iterations, value is %s" % (maxiter, p) + raise RuntimeError(msg) + + +def fixed_point(func, x0, args=(), xtol=1e-8, maxiter=500, method='del2'): + """ + Find a fixed point of the function. + + Given a function of one or more variables and a starting point, find a + fixed point of the function: i.e., where ``func(x0) == x0``. + + Parameters + ---------- + func : function + Function to evaluate. + x0 : array_like + Fixed point of function. + args : tuple, optional + Extra arguments to `func`. + xtol : float, optional + Convergence tolerance, defaults to 1e-08. + maxiter : int, optional + Maximum number of iterations, defaults to 500. + method : {"del2", "iteration"}, optional + Method of finding the fixed-point, defaults to "del2", + which uses Steffensen's Method with Aitken's ``Del^2`` + convergence acceleration [1]_. The "iteration" method simply iterates + the function until convergence is detected, without attempting to + accelerate the convergence. + + References + ---------- + .. [1] Burden, Faires, "Numerical Analysis", 5th edition, pg. 80 + + Examples + -------- + >>> import numpy as np + >>> from scipy import optimize + >>> def func(x, c1, c2): + ... return np.sqrt(c1/(x+c2)) + >>> c1 = np.array([10,12.]) + >>> c2 = np.array([3, 5.]) + >>> optimize.fixed_point(func, [1.2, 1.3], args=(c1,c2)) + array([ 1.4920333 , 1.37228132]) + + """ + use_accel = {'del2': True, 'iteration': False}[method] + x0 = _asarray_validated(x0, as_inexact=True) + return _fixed_point_helper(func, x0, args, xtol, maxiter, use_accel) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_nnls.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_nnls.py new file mode 100644 index 0000000000000000000000000000000000000000..be904c90d715583faaf7751ce62b9c992e07e208 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_nnls.py @@ -0,0 +1,97 @@ +import numpy as np +from ._cython_nnls import _nnls + + +__all__ = ['nnls'] + + +def nnls(A, b, maxiter=None, *, atol=None): + """ + Solve ``argmin_x || Ax - b ||_2`` for ``x>=0``. + + This problem, often called as NonNegative Least Squares, is a convex + optimization problem with convex constraints. It typically arises when + the ``x`` models quantities for which only nonnegative values are + attainable; weight of ingredients, component costs and so on. + + Parameters + ---------- + A : (m, n) ndarray + Coefficient array + b : (m,) ndarray, float + Right-hand side vector. + maxiter: int, optional + Maximum number of iterations, optional. Default value is ``3 * n``. + atol: float + Tolerance value used in the algorithm to assess closeness to zero in + the projected residual ``(A.T @ (A x - b)`` entries. Increasing this + value relaxes the solution constraints. A typical relaxation value can + be selected as ``max(m, n) * np.linalg.norm(a, 1) * np.spacing(1.)``. + This value is not set as default since the norm operation becomes + expensive for large problems hence can be used only when necessary. + + Returns + ------- + x : ndarray + Solution vector. + rnorm : float + The 2-norm of the residual, ``|| Ax-b ||_2``. + + See Also + -------- + lsq_linear : Linear least squares with bounds on the variables + + Notes + ----- + The code is based on [2]_ which is an improved version of the classical + algorithm of [1]_. It utilizes an active set method and solves the KKT + (Karush-Kuhn-Tucker) conditions for the non-negative least squares problem. + + References + ---------- + .. [1] : Lawson C., Hanson R.J., "Solving Least Squares Problems", SIAM, + 1995, :doi:`10.1137/1.9781611971217` + .. [2] : Bro, Rasmus and de Jong, Sijmen, "A Fast Non-Negativity- + Constrained Least Squares Algorithm", Journal Of Chemometrics, 1997, + :doi:`10.1002/(SICI)1099-128X(199709/10)11:5<393::AID-CEM483>3.0.CO;2-L` + + Examples + -------- + >>> import numpy as np + >>> from scipy.optimize import nnls + ... + >>> A = np.array([[1, 0], [1, 0], [0, 1]]) + >>> b = np.array([2, 1, 1]) + >>> nnls(A, b) + (array([1.5, 1. ]), 0.7071067811865475) + + >>> b = np.array([-1, -1, -1]) + >>> nnls(A, b) + (array([0., 0.]), 1.7320508075688772) + + """ + + A = np.asarray_chkfinite(A, dtype=np.float64, order='C') + b = np.asarray_chkfinite(b, dtype=np.float64) + + if len(A.shape) != 2: + raise ValueError("Expected a two-dimensional array (matrix)" + + f", but the shape of A is {A.shape}") + if len(b.shape) != 1: + raise ValueError("Expected a one-dimensional array (vector)" + + f", but the shape of b is {b.shape}") + + m, n = A.shape + + if m != b.shape[0]: + raise ValueError( + "Incompatible dimensions. The first dimension of " + + f"A is {m}, while the shape of b is {(b.shape[0], )}") + + if not maxiter: + maxiter = 3*n + x, rnorm, info = _nnls(A, b, maxiter) + if info == -1: + raise RuntimeError("Maximum number of iterations reached.") + + return x, rnorm diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_nonlin.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_nonlin.py new file mode 100644 index 0000000000000000000000000000000000000000..b6e07683500bb01c195b1cdfa8a13157353b5370 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_nonlin.py @@ -0,0 +1,1603 @@ +# Copyright (C) 2009, Pauli Virtanen +# Distributed under the same license as SciPy. + +import inspect +import sys +import warnings + +import numpy as np +from numpy import asarray, dot, vdot + +from scipy.linalg import norm, solve, inv, qr, svd, LinAlgError +import scipy.sparse.linalg +import scipy.sparse +from scipy.linalg import get_blas_funcs +from scipy._lib._util import copy_if_needed +from scipy._lib._util import getfullargspec_no_self as _getfullargspec +from ._linesearch import scalar_search_wolfe1, scalar_search_armijo + + +__all__ = [ + 'broyden1', 'broyden2', 'anderson', 'linearmixing', + 'diagbroyden', 'excitingmixing', 'newton_krylov', + 'BroydenFirst', 'KrylovJacobian', 'InverseJacobian', 'NoConvergence'] + +#------------------------------------------------------------------------------ +# Utility functions +#------------------------------------------------------------------------------ + + +class NoConvergence(Exception): + """Exception raised when nonlinear solver fails to converge within the specified + `maxiter`.""" + pass + + +def maxnorm(x): + return np.absolute(x).max() + + +def _as_inexact(x): + """Return `x` as an array, of either floats or complex floats""" + x = asarray(x) + if not np.issubdtype(x.dtype, np.inexact): + return asarray(x, dtype=np.float64) + return x + + +def _array_like(x, x0): + """Return ndarray `x` as same array subclass and shape as `x0`""" + x = np.reshape(x, np.shape(x0)) + wrap = getattr(x0, '__array_wrap__', x.__array_wrap__) + return wrap(x) + + +def _safe_norm(v): + if not np.isfinite(v).all(): + return np.array(np.inf) + return norm(v) + +#------------------------------------------------------------------------------ +# Generic nonlinear solver machinery +#------------------------------------------------------------------------------ + + +_doc_parts = dict( + params_basic=""" + F : function(x) -> f + Function whose root to find; should take and return an array-like + object. + xin : array_like + Initial guess for the solution + """.strip(), + params_extra=""" + iter : int, optional + Number of iterations to make. If omitted (default), make as many + as required to meet tolerances. + verbose : bool, optional + Print status to stdout on every iteration. + maxiter : int, optional + Maximum number of iterations to make. If more are needed to + meet convergence, `NoConvergence` is raised. + f_tol : float, optional + Absolute tolerance (in max-norm) for the residual. + If omitted, default is 6e-6. + f_rtol : float, optional + Relative tolerance for the residual. If omitted, not used. + x_tol : float, optional + Absolute minimum step size, as determined from the Jacobian + approximation. If the step size is smaller than this, optimization + is terminated as successful. If omitted, not used. + x_rtol : float, optional + Relative minimum step size. If omitted, not used. + tol_norm : function(vector) -> scalar, optional + Norm to use in convergence check. Default is the maximum norm. + line_search : {None, 'armijo' (default), 'wolfe'}, optional + Which type of a line search to use to determine the step size in the + direction given by the Jacobian approximation. Defaults to 'armijo'. + callback : function, optional + Optional callback function. It is called on every iteration as + ``callback(x, f)`` where `x` is the current solution and `f` + the corresponding residual. + + Returns + ------- + sol : ndarray + An array (of similar array type as `x0`) containing the final solution. + + Raises + ------ + NoConvergence + When a solution was not found. + + """.strip() +) + + +def _set_doc(obj): + if obj.__doc__: + obj.__doc__ = obj.__doc__ % _doc_parts + + +def nonlin_solve(F, x0, jacobian='krylov', iter=None, verbose=False, + maxiter=None, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None, + tol_norm=None, line_search='armijo', callback=None, + full_output=False, raise_exception=True): + """ + Find a root of a function, in a way suitable for large-scale problems. + + Parameters + ---------- + %(params_basic)s + jacobian : Jacobian + A Jacobian approximation: `Jacobian` object or something that + `asjacobian` can transform to one. Alternatively, a string specifying + which of the builtin Jacobian approximations to use: + + krylov, broyden1, broyden2, anderson + diagbroyden, linearmixing, excitingmixing + + %(params_extra)s + full_output : bool + If true, returns a dictionary `info` containing convergence + information. + raise_exception : bool + If True, a `NoConvergence` exception is raise if no solution is found. + + See Also + -------- + asjacobian, Jacobian + + Notes + ----- + This algorithm implements the inexact Newton method, with + backtracking or full line searches. Several Jacobian + approximations are available, including Krylov and Quasi-Newton + methods. + + References + ---------- + .. [KIM] C. T. Kelley, \"Iterative Methods for Linear and Nonlinear + Equations\". Society for Industrial and Applied Mathematics. (1995) + https://archive.siam.org/books/kelley/fr16/ + + """ + # Can't use default parameters because it's being explicitly passed as None + # from the calling function, so we need to set it here. + tol_norm = maxnorm if tol_norm is None else tol_norm + condition = TerminationCondition(f_tol=f_tol, f_rtol=f_rtol, + x_tol=x_tol, x_rtol=x_rtol, + iter=iter, norm=tol_norm) + + x0 = _as_inexact(x0) + def func(z): + return _as_inexact(F(_array_like(z, x0))).flatten() + x = x0.flatten() + + dx = np.full_like(x, np.inf) + Fx = func(x) + Fx_norm = norm(Fx) + + jacobian = asjacobian(jacobian) + jacobian.setup(x.copy(), Fx, func) + + if maxiter is None: + if iter is not None: + maxiter = iter + 1 + else: + maxiter = 100*(x.size+1) + + if line_search is True: + line_search = 'armijo' + elif line_search is False: + line_search = None + + if line_search not in (None, 'armijo', 'wolfe'): + raise ValueError("Invalid line search") + + # Solver tolerance selection + gamma = 0.9 + eta_max = 0.9999 + eta_treshold = 0.1 + eta = 1e-3 + + for n in range(maxiter): + status = condition.check(Fx, x, dx) + if status: + break + + # The tolerance, as computed for scipy.sparse.linalg.* routines + tol = min(eta, eta*Fx_norm) + dx = -jacobian.solve(Fx, tol=tol) + + if norm(dx) == 0: + raise ValueError("Jacobian inversion yielded zero vector. " + "This indicates a bug in the Jacobian " + "approximation.") + + # Line search, or Newton step + if line_search: + s, x, Fx, Fx_norm_new = _nonlin_line_search(func, x, Fx, dx, + line_search) + else: + s = 1.0 + x = x + dx + Fx = func(x) + Fx_norm_new = norm(Fx) + + jacobian.update(x.copy(), Fx) + + if callback: + callback(x, Fx) + + # Adjust forcing parameters for inexact methods + eta_A = gamma * Fx_norm_new**2 / Fx_norm**2 + if gamma * eta**2 < eta_treshold: + eta = min(eta_max, eta_A) + else: + eta = min(eta_max, max(eta_A, gamma*eta**2)) + + Fx_norm = Fx_norm_new + + # Print status + if verbose: + sys.stdout.write("%d: |F(x)| = %g; step %g\n" % ( + n, tol_norm(Fx), s)) + sys.stdout.flush() + else: + if raise_exception: + raise NoConvergence(_array_like(x, x0)) + else: + status = 2 + + if full_output: + info = {'nit': condition.iteration, + 'fun': Fx, + 'status': status, + 'success': status == 1, + 'message': {1: 'A solution was found at the specified ' + 'tolerance.', + 2: 'The maximum number of iterations allowed ' + 'has been reached.' + }[status] + } + return _array_like(x, x0), info + else: + return _array_like(x, x0) + + +_set_doc(nonlin_solve) + + +def _nonlin_line_search(func, x, Fx, dx, search_type='armijo', rdiff=1e-8, + smin=1e-2): + tmp_s = [0] + tmp_Fx = [Fx] + tmp_phi = [norm(Fx)**2] + s_norm = norm(x) / norm(dx) + + def phi(s, store=True): + if s == tmp_s[0]: + return tmp_phi[0] + xt = x + s*dx + v = func(xt) + p = _safe_norm(v)**2 + if store: + tmp_s[0] = s + tmp_phi[0] = p + tmp_Fx[0] = v + return p + + def derphi(s): + ds = (abs(s) + s_norm + 1) * rdiff + return (phi(s+ds, store=False) - phi(s)) / ds + + if search_type == 'wolfe': + s, phi1, phi0 = scalar_search_wolfe1(phi, derphi, tmp_phi[0], + xtol=1e-2, amin=smin) + elif search_type == 'armijo': + s, phi1 = scalar_search_armijo(phi, tmp_phi[0], -tmp_phi[0], + amin=smin) + + if s is None: + # XXX: No suitable step length found. Take the full Newton step, + # and hope for the best. + s = 1.0 + + x = x + s*dx + if s == tmp_s[0]: + Fx = tmp_Fx[0] + else: + Fx = func(x) + Fx_norm = norm(Fx) + + return s, x, Fx, Fx_norm + + +class TerminationCondition: + """ + Termination condition for an iteration. It is terminated if + + - |F| < f_rtol*|F_0|, AND + - |F| < f_tol + + AND + + - |dx| < x_rtol*|x|, AND + - |dx| < x_tol + + """ + def __init__(self, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None, + iter=None, norm=maxnorm): + + if f_tol is None: + f_tol = np.finfo(np.float64).eps ** (1./3) + if f_rtol is None: + f_rtol = np.inf + if x_tol is None: + x_tol = np.inf + if x_rtol is None: + x_rtol = np.inf + + self.x_tol = x_tol + self.x_rtol = x_rtol + self.f_tol = f_tol + self.f_rtol = f_rtol + + self.norm = norm + + self.iter = iter + + self.f0_norm = None + self.iteration = 0 + + def check(self, f, x, dx): + self.iteration += 1 + f_norm = self.norm(f) + x_norm = self.norm(x) + dx_norm = self.norm(dx) + + if self.f0_norm is None: + self.f0_norm = f_norm + + if f_norm == 0: + return 1 + + if self.iter is not None: + # backwards compatibility with SciPy 0.6.0 + return 2 * (self.iteration > self.iter) + + # NB: condition must succeed for rtol=inf even if norm == 0 + return int((f_norm <= self.f_tol + and f_norm/self.f_rtol <= self.f0_norm) + and (dx_norm <= self.x_tol + and dx_norm/self.x_rtol <= x_norm)) + + +#------------------------------------------------------------------------------ +# Generic Jacobian approximation +#------------------------------------------------------------------------------ + +class Jacobian: + """ + Common interface for Jacobians or Jacobian approximations. + + The optional methods come useful when implementing trust region + etc., algorithms that often require evaluating transposes of the + Jacobian. + + Methods + ------- + solve + Returns J^-1 * v + update + Updates Jacobian to point `x` (where the function has residual `Fx`) + + matvec : optional + Returns J * v + rmatvec : optional + Returns A^H * v + rsolve : optional + Returns A^-H * v + matmat : optional + Returns A * V, where V is a dense matrix with dimensions (N,K). + todense : optional + Form the dense Jacobian matrix. Necessary for dense trust region + algorithms, and useful for testing. + + Attributes + ---------- + shape + Matrix dimensions (M, N) + dtype + Data type of the matrix. + func : callable, optional + Function the Jacobian corresponds to + + """ + + def __init__(self, **kw): + names = ["solve", "update", "matvec", "rmatvec", "rsolve", + "matmat", "todense", "shape", "dtype"] + for name, value in kw.items(): + if name not in names: + raise ValueError(f"Unknown keyword argument {name}") + if value is not None: + setattr(self, name, kw[name]) + + + if hasattr(self, "todense"): + def __array__(self, dtype=None, copy=None): + if dtype is not None: + raise ValueError(f"`dtype` must be None, was {dtype}") + return self.todense() + + def aspreconditioner(self): + return InverseJacobian(self) + + def solve(self, v, tol=0): + raise NotImplementedError + + def update(self, x, F): + pass + + def setup(self, x, F, func): + self.func = func + self.shape = (F.size, x.size) + self.dtype = F.dtype + if self.__class__.setup is Jacobian.setup: + # Call on the first point unless overridden + self.update(x, F) + + +class InverseJacobian: + """ + A simple wrapper that inverts the Jacobian using the `solve` method. + + .. legacy:: class + + See the newer, more consistent interfaces in :mod:`scipy.optimize`. + + Parameters + ---------- + jacobian : Jacobian + The Jacobian to invert. + + Attributes + ---------- + shape + Matrix dimensions (M, N) + dtype + Data type of the matrix. + + """ + def __init__(self, jacobian): + self.jacobian = jacobian + self.matvec = jacobian.solve + self.update = jacobian.update + if hasattr(jacobian, 'setup'): + self.setup = jacobian.setup + if hasattr(jacobian, 'rsolve'): + self.rmatvec = jacobian.rsolve + + @property + def shape(self): + return self.jacobian.shape + + @property + def dtype(self): + return self.jacobian.dtype + + +def asjacobian(J): + """ + Convert given object to one suitable for use as a Jacobian. + """ + spsolve = scipy.sparse.linalg.spsolve + if isinstance(J, Jacobian): + return J + elif inspect.isclass(J) and issubclass(J, Jacobian): + return J() + elif isinstance(J, np.ndarray): + if J.ndim > 2: + raise ValueError('array must have rank <= 2') + J = np.atleast_2d(np.asarray(J)) + if J.shape[0] != J.shape[1]: + raise ValueError('array must be square') + + return Jacobian(matvec=lambda v: dot(J, v), + rmatvec=lambda v: dot(J.conj().T, v), + solve=lambda v, tol=0: solve(J, v), + rsolve=lambda v, tol=0: solve(J.conj().T, v), + dtype=J.dtype, shape=J.shape) + elif scipy.sparse.issparse(J): + if J.shape[0] != J.shape[1]: + raise ValueError('matrix must be square') + return Jacobian(matvec=lambda v: J @ v, + rmatvec=lambda v: J.conj().T @ v, + solve=lambda v, tol=0: spsolve(J, v), + rsolve=lambda v, tol=0: spsolve(J.conj().T, v), + dtype=J.dtype, shape=J.shape) + elif hasattr(J, 'shape') and hasattr(J, 'dtype') and hasattr(J, 'solve'): + return Jacobian(matvec=getattr(J, 'matvec'), + rmatvec=getattr(J, 'rmatvec'), + solve=J.solve, + rsolve=getattr(J, 'rsolve'), + update=getattr(J, 'update'), + setup=getattr(J, 'setup'), + dtype=J.dtype, + shape=J.shape) + elif callable(J): + # Assume it's a function J(x) that returns the Jacobian + class Jac(Jacobian): + def update(self, x, F): + self.x = x + + def solve(self, v, tol=0): + m = J(self.x) + if isinstance(m, np.ndarray): + return solve(m, v) + elif scipy.sparse.issparse(m): + return spsolve(m, v) + else: + raise ValueError("Unknown matrix type") + + def matvec(self, v): + m = J(self.x) + if isinstance(m, np.ndarray): + return dot(m, v) + elif scipy.sparse.issparse(m): + return m @ v + else: + raise ValueError("Unknown matrix type") + + def rsolve(self, v, tol=0): + m = J(self.x) + if isinstance(m, np.ndarray): + return solve(m.conj().T, v) + elif scipy.sparse.issparse(m): + return spsolve(m.conj().T, v) + else: + raise ValueError("Unknown matrix type") + + def rmatvec(self, v): + m = J(self.x) + if isinstance(m, np.ndarray): + return dot(m.conj().T, v) + elif scipy.sparse.issparse(m): + return m.conj().T @ v + else: + raise ValueError("Unknown matrix type") + return Jac() + elif isinstance(J, str): + return dict(broyden1=BroydenFirst, + broyden2=BroydenSecond, + anderson=Anderson, + diagbroyden=DiagBroyden, + linearmixing=LinearMixing, + excitingmixing=ExcitingMixing, + krylov=KrylovJacobian)[J]() + else: + raise TypeError('Cannot convert object to a Jacobian') + + +#------------------------------------------------------------------------------ +# Broyden +#------------------------------------------------------------------------------ + +class GenericBroyden(Jacobian): + def setup(self, x0, f0, func): + Jacobian.setup(self, x0, f0, func) + self.last_f = f0 + self.last_x = x0 + + if hasattr(self, 'alpha') and self.alpha is None: + # Autoscale the initial Jacobian parameter + # unless we have already guessed the solution. + normf0 = norm(f0) + if normf0: + self.alpha = 0.5*max(norm(x0), 1) / normf0 + else: + self.alpha = 1.0 + + def _update(self, x, f, dx, df, dx_norm, df_norm): + raise NotImplementedError + + def update(self, x, f): + df = f - self.last_f + dx = x - self.last_x + self._update(x, f, dx, df, norm(dx), norm(df)) + self.last_f = f + self.last_x = x + + +class LowRankMatrix: + r""" + A matrix represented as + + .. math:: \alpha I + \sum_{n=0}^{n=M} c_n d_n^\dagger + + However, if the rank of the matrix reaches the dimension of the vectors, + full matrix representation will be used thereon. + + """ + + def __init__(self, alpha, n, dtype): + self.alpha = alpha + self.cs = [] + self.ds = [] + self.n = n + self.dtype = dtype + self.collapsed = None + + @staticmethod + def _matvec(v, alpha, cs, ds): + axpy, scal, dotc = get_blas_funcs(['axpy', 'scal', 'dotc'], + cs[:1] + [v]) + w = alpha * v + for c, d in zip(cs, ds): + a = dotc(d, v) + w = axpy(c, w, w.size, a) + return w + + @staticmethod + def _solve(v, alpha, cs, ds): + """Evaluate w = M^-1 v""" + if len(cs) == 0: + return v/alpha + + # (B + C D^H)^-1 = B^-1 - B^-1 C (I + D^H B^-1 C)^-1 D^H B^-1 + + axpy, dotc = get_blas_funcs(['axpy', 'dotc'], cs[:1] + [v]) + + c0 = cs[0] + A = alpha * np.identity(len(cs), dtype=c0.dtype) + for i, d in enumerate(ds): + for j, c in enumerate(cs): + A[i,j] += dotc(d, c) + + q = np.zeros(len(cs), dtype=c0.dtype) + for j, d in enumerate(ds): + q[j] = dotc(d, v) + q /= alpha + q = solve(A, q) + + w = v/alpha + for c, qc in zip(cs, q): + w = axpy(c, w, w.size, -qc) + + return w + + def matvec(self, v): + """Evaluate w = M v""" + if self.collapsed is not None: + return np.dot(self.collapsed, v) + return LowRankMatrix._matvec(v, self.alpha, self.cs, self.ds) + + def rmatvec(self, v): + """Evaluate w = M^H v""" + if self.collapsed is not None: + return np.dot(self.collapsed.T.conj(), v) + return LowRankMatrix._matvec(v, np.conj(self.alpha), self.ds, self.cs) + + def solve(self, v, tol=0): + """Evaluate w = M^-1 v""" + if self.collapsed is not None: + return solve(self.collapsed, v) + return LowRankMatrix._solve(v, self.alpha, self.cs, self.ds) + + def rsolve(self, v, tol=0): + """Evaluate w = M^-H v""" + if self.collapsed is not None: + return solve(self.collapsed.T.conj(), v) + return LowRankMatrix._solve(v, np.conj(self.alpha), self.ds, self.cs) + + def append(self, c, d): + if self.collapsed is not None: + self.collapsed += c[:,None] * d[None,:].conj() + return + + self.cs.append(c) + self.ds.append(d) + + if len(self.cs) > c.size: + self.collapse() + + def __array__(self, dtype=None, copy=None): + if dtype is not None: + warnings.warn("LowRankMatrix is scipy-internal code, `dtype` " + f"should only be None but was {dtype} (not handled)", + stacklevel=3) + if copy is not None: + warnings.warn("LowRankMatrix is scipy-internal code, `copy` " + f"should only be None but was {copy} (not handled)", + stacklevel=3) + if self.collapsed is not None: + return self.collapsed + + Gm = self.alpha*np.identity(self.n, dtype=self.dtype) + for c, d in zip(self.cs, self.ds): + Gm += c[:,None]*d[None,:].conj() + return Gm + + def collapse(self): + """Collapse the low-rank matrix to a full-rank one.""" + self.collapsed = np.array(self, copy=copy_if_needed) + self.cs = None + self.ds = None + self.alpha = None + + def restart_reduce(self, rank): + """ + Reduce the rank of the matrix by dropping all vectors. + """ + if self.collapsed is not None: + return + assert rank > 0 + if len(self.cs) > rank: + del self.cs[:] + del self.ds[:] + + def simple_reduce(self, rank): + """ + Reduce the rank of the matrix by dropping oldest vectors. + """ + if self.collapsed is not None: + return + assert rank > 0 + while len(self.cs) > rank: + del self.cs[0] + del self.ds[0] + + def svd_reduce(self, max_rank, to_retain=None): + """ + Reduce the rank of the matrix by retaining some SVD components. + + This corresponds to the \"Broyden Rank Reduction Inverse\" + algorithm described in [1]_. + + Note that the SVD decomposition can be done by solving only a + problem whose size is the effective rank of this matrix, which + is viable even for large problems. + + Parameters + ---------- + max_rank : int + Maximum rank of this matrix after reduction. + to_retain : int, optional + Number of SVD components to retain when reduction is done + (ie. rank > max_rank). Default is ``max_rank - 2``. + + References + ---------- + .. [1] B.A. van der Rotten, PhD thesis, + \"A limited memory Broyden method to solve high-dimensional + systems of nonlinear equations\". Mathematisch Instituut, + Universiteit Leiden, The Netherlands (2003). + + https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf + + """ + if self.collapsed is not None: + return + + p = max_rank + if to_retain is not None: + q = to_retain + else: + q = p - 2 + + if self.cs: + p = min(p, len(self.cs[0])) + q = max(0, min(q, p-1)) + + m = len(self.cs) + if m < p: + # nothing to do + return + + C = np.array(self.cs).T + D = np.array(self.ds).T + + D, R = qr(D, mode='economic') + C = dot(C, R.T.conj()) + + U, S, WH = svd(C, full_matrices=False) + + C = dot(C, inv(WH)) + D = dot(D, WH.T.conj()) + + for k in range(q): + self.cs[k] = C[:,k].copy() + self.ds[k] = D[:,k].copy() + + del self.cs[q:] + del self.ds[q:] + + +_doc_parts['broyden_params'] = """ + alpha : float, optional + Initial guess for the Jacobian is ``(-1/alpha)``. + reduction_method : str or tuple, optional + Method used in ensuring that the rank of the Broyden matrix + stays low. Can either be a string giving the name of the method, + or a tuple of the form ``(method, param1, param2, ...)`` + that gives the name of the method and values for additional parameters. + + Methods available: + + - ``restart``: drop all matrix columns. Has no extra parameters. + - ``simple``: drop oldest matrix column. Has no extra parameters. + - ``svd``: keep only the most significant SVD components. + Takes an extra parameter, ``to_retain``, which determines the + number of SVD components to retain when rank reduction is done. + Default is ``max_rank - 2``. + + max_rank : int, optional + Maximum rank for the Broyden matrix. + Default is infinity (i.e., no rank reduction). + """.strip() + + +class BroydenFirst(GenericBroyden): + """ + Find a root of a function, using Broyden's first Jacobian approximation. + + This method is also known as "Broyden's good method". + + Parameters + ---------- + %(params_basic)s + %(broyden_params)s + %(params_extra)s + + See Also + -------- + root : Interface to root finding algorithms for multivariate + functions. See ``method='broyden1'`` in particular. + + Notes + ----- + This algorithm implements the inverse Jacobian Quasi-Newton update + + .. math:: H_+ = H + (dx - H df) dx^\\dagger H / ( dx^\\dagger H df) + + which corresponds to Broyden's first Jacobian update + + .. math:: J_+ = J + (df - J dx) dx^\\dagger / dx^\\dagger dx + + + References + ---------- + .. [1] B.A. van der Rotten, PhD thesis, + "A limited memory Broyden method to solve high-dimensional + systems of nonlinear equations". Mathematisch Instituut, + Universiteit Leiden, The Netherlands (2003). + https://math.leidenuniv.nl/scripties/Rotten.pdf + + Examples + -------- + The following functions define a system of nonlinear equations + + >>> def fun(x): + ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, + ... 0.5 * (x[1] - x[0])**3 + x[1]] + + A solution can be obtained as follows. + + >>> from scipy import optimize + >>> sol = optimize.broyden1(fun, [0, 0]) + >>> sol + array([0.84116396, 0.15883641]) + + """ + + def __init__(self, alpha=None, reduction_method='restart', max_rank=None): + GenericBroyden.__init__(self) + self.alpha = alpha + self.Gm = None + + if max_rank is None: + max_rank = np.inf + self.max_rank = max_rank + + if isinstance(reduction_method, str): + reduce_params = () + else: + reduce_params = reduction_method[1:] + reduction_method = reduction_method[0] + reduce_params = (max_rank - 1,) + reduce_params + + if reduction_method == 'svd': + self._reduce = lambda: self.Gm.svd_reduce(*reduce_params) + elif reduction_method == 'simple': + self._reduce = lambda: self.Gm.simple_reduce(*reduce_params) + elif reduction_method == 'restart': + self._reduce = lambda: self.Gm.restart_reduce(*reduce_params) + else: + raise ValueError(f"Unknown rank reduction method '{reduction_method}'") + + def setup(self, x, F, func): + GenericBroyden.setup(self, x, F, func) + self.Gm = LowRankMatrix(-self.alpha, self.shape[0], self.dtype) + + def todense(self): + return inv(self.Gm) + + def solve(self, f, tol=0): + r = self.Gm.matvec(f) + if not np.isfinite(r).all(): + # singular; reset the Jacobian approximation + self.setup(self.last_x, self.last_f, self.func) + return self.Gm.matvec(f) + return r + + def matvec(self, f): + return self.Gm.solve(f) + + def rsolve(self, f, tol=0): + return self.Gm.rmatvec(f) + + def rmatvec(self, f): + return self.Gm.rsolve(f) + + def _update(self, x, f, dx, df, dx_norm, df_norm): + self._reduce() # reduce first to preserve secant condition + + v = self.Gm.rmatvec(dx) + c = dx - self.Gm.matvec(df) + d = v / vdot(df, v) + + self.Gm.append(c, d) + + +class BroydenSecond(BroydenFirst): + """ + Find a root of a function, using Broyden\'s second Jacobian approximation. + + This method is also known as \"Broyden's bad method\". + + Parameters + ---------- + %(params_basic)s + %(broyden_params)s + %(params_extra)s + + See Also + -------- + root : Interface to root finding algorithms for multivariate + functions. See ``method='broyden2'`` in particular. + + Notes + ----- + This algorithm implements the inverse Jacobian Quasi-Newton update + + .. math:: H_+ = H + (dx - H df) df^\\dagger / ( df^\\dagger df) + + corresponding to Broyden's second method. + + References + ---------- + .. [1] B.A. van der Rotten, PhD thesis, + \"A limited memory Broyden method to solve high-dimensional + systems of nonlinear equations\". Mathematisch Instituut, + Universiteit Leiden, The Netherlands (2003). + + https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf + + Examples + -------- + The following functions define a system of nonlinear equations + + >>> def fun(x): + ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, + ... 0.5 * (x[1] - x[0])**3 + x[1]] + + A solution can be obtained as follows. + + >>> from scipy import optimize + >>> sol = optimize.broyden2(fun, [0, 0]) + >>> sol + array([0.84116365, 0.15883529]) + + """ + + def _update(self, x, f, dx, df, dx_norm, df_norm): + self._reduce() # reduce first to preserve secant condition + + v = df + c = dx - self.Gm.matvec(df) + d = v / df_norm**2 + self.Gm.append(c, d) + + +#------------------------------------------------------------------------------ +# Broyden-like (restricted memory) +#------------------------------------------------------------------------------ + +class Anderson(GenericBroyden): + """ + Find a root of a function, using (extended) Anderson mixing. + + The Jacobian is formed by for a 'best' solution in the space + spanned by last `M` vectors. As a result, only a MxM matrix + inversions and MxN multiplications are required. [Ey]_ + + Parameters + ---------- + %(params_basic)s + alpha : float, optional + Initial guess for the Jacobian is (-1/alpha). + M : float, optional + Number of previous vectors to retain. Defaults to 5. + w0 : float, optional + Regularization parameter for numerical stability. + Compared to unity, good values of the order of 0.01. + %(params_extra)s + + See Also + -------- + root : Interface to root finding algorithms for multivariate + functions. See ``method='anderson'`` in particular. + + References + ---------- + .. [Ey] V. Eyert, J. Comp. Phys., 124, 271 (1996). + + Examples + -------- + The following functions define a system of nonlinear equations + + >>> def fun(x): + ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, + ... 0.5 * (x[1] - x[0])**3 + x[1]] + + A solution can be obtained as follows. + + >>> from scipy import optimize + >>> sol = optimize.anderson(fun, [0, 0]) + >>> sol + array([0.84116588, 0.15883789]) + + """ + + # Note: + # + # Anderson method maintains a rank M approximation of the inverse Jacobian, + # + # J^-1 v ~ -v*alpha + (dX + alpha dF) A^-1 dF^H v + # A = W + dF^H dF + # W = w0^2 diag(dF^H dF) + # + # so that for w0 = 0 the secant condition applies for last M iterates, i.e., + # + # J^-1 df_j = dx_j + # + # for all j = 0 ... M-1. + # + # Moreover, (from Sherman-Morrison-Woodbury formula) + # + # J v ~ [ b I - b^2 C (I + b dF^H A^-1 C)^-1 dF^H ] v + # C = (dX + alpha dF) A^-1 + # b = -1/alpha + # + # and after simplification + # + # J v ~ -v/alpha + (dX/alpha + dF) (dF^H dX - alpha W)^-1 dF^H v + # + + def __init__(self, alpha=None, w0=0.01, M=5): + GenericBroyden.__init__(self) + self.alpha = alpha + self.M = M + self.dx = [] + self.df = [] + self.gamma = None + self.w0 = w0 + + def solve(self, f, tol=0): + dx = -self.alpha*f + + n = len(self.dx) + if n == 0: + return dx + + df_f = np.empty(n, dtype=f.dtype) + for k in range(n): + df_f[k] = vdot(self.df[k], f) + + try: + gamma = solve(self.a, df_f) + except LinAlgError: + # singular; reset the Jacobian approximation + del self.dx[:] + del self.df[:] + return dx + + for m in range(n): + dx += gamma[m]*(self.dx[m] + self.alpha*self.df[m]) + return dx + + def matvec(self, f): + dx = -f/self.alpha + + n = len(self.dx) + if n == 0: + return dx + + df_f = np.empty(n, dtype=f.dtype) + for k in range(n): + df_f[k] = vdot(self.df[k], f) + + b = np.empty((n, n), dtype=f.dtype) + for i in range(n): + for j in range(n): + b[i,j] = vdot(self.df[i], self.dx[j]) + if i == j and self.w0 != 0: + b[i,j] -= vdot(self.df[i], self.df[i])*self.w0**2*self.alpha + gamma = solve(b, df_f) + + for m in range(n): + dx += gamma[m]*(self.df[m] + self.dx[m]/self.alpha) + return dx + + def _update(self, x, f, dx, df, dx_norm, df_norm): + if self.M == 0: + return + + self.dx.append(dx) + self.df.append(df) + + while len(self.dx) > self.M: + self.dx.pop(0) + self.df.pop(0) + + n = len(self.dx) + a = np.zeros((n, n), dtype=f.dtype) + + for i in range(n): + for j in range(i, n): + if i == j: + wd = self.w0**2 + else: + wd = 0 + a[i,j] = (1+wd)*vdot(self.df[i], self.df[j]) + + a += np.triu(a, 1).T.conj() + self.a = a + +#------------------------------------------------------------------------------ +# Simple iterations +#------------------------------------------------------------------------------ + + +class DiagBroyden(GenericBroyden): + """ + Find a root of a function, using diagonal Broyden Jacobian approximation. + + The Jacobian approximation is derived from previous iterations, by + retaining only the diagonal of Broyden matrices. + + .. warning:: + + This algorithm may be useful for specific problems, but whether + it will work may depend strongly on the problem. + + Parameters + ---------- + %(params_basic)s + alpha : float, optional + Initial guess for the Jacobian is (-1/alpha). + %(params_extra)s + + See Also + -------- + root : Interface to root finding algorithms for multivariate + functions. See ``method='diagbroyden'`` in particular. + + Examples + -------- + The following functions define a system of nonlinear equations + + >>> def fun(x): + ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, + ... 0.5 * (x[1] - x[0])**3 + x[1]] + + A solution can be obtained as follows. + + >>> from scipy import optimize + >>> sol = optimize.diagbroyden(fun, [0, 0]) + >>> sol + array([0.84116403, 0.15883384]) + + """ + + def __init__(self, alpha=None): + GenericBroyden.__init__(self) + self.alpha = alpha + + def setup(self, x, F, func): + GenericBroyden.setup(self, x, F, func) + self.d = np.full((self.shape[0],), 1 / self.alpha, dtype=self.dtype) + + def solve(self, f, tol=0): + return -f / self.d + + def matvec(self, f): + return -f * self.d + + def rsolve(self, f, tol=0): + return -f / self.d.conj() + + def rmatvec(self, f): + return -f * self.d.conj() + + def todense(self): + return np.diag(-self.d) + + def _update(self, x, f, dx, df, dx_norm, df_norm): + self.d -= (df + self.d*dx)*dx/dx_norm**2 + + +class LinearMixing(GenericBroyden): + """ + Find a root of a function, using a scalar Jacobian approximation. + + .. warning:: + + This algorithm may be useful for specific problems, but whether + it will work may depend strongly on the problem. + + Parameters + ---------- + %(params_basic)s + alpha : float, optional + The Jacobian approximation is (-1/alpha). + %(params_extra)s + + See Also + -------- + root : Interface to root finding algorithms for multivariate + functions. See ``method='linearmixing'`` in particular. + + """ + + def __init__(self, alpha=None): + GenericBroyden.__init__(self) + self.alpha = alpha + + def solve(self, f, tol=0): + return -f*self.alpha + + def matvec(self, f): + return -f/self.alpha + + def rsolve(self, f, tol=0): + return -f*np.conj(self.alpha) + + def rmatvec(self, f): + return -f/np.conj(self.alpha) + + def todense(self): + return np.diag(np.full(self.shape[0], -1/self.alpha)) + + def _update(self, x, f, dx, df, dx_norm, df_norm): + pass + + +class ExcitingMixing(GenericBroyden): + """ + Find a root of a function, using a tuned diagonal Jacobian approximation. + + The Jacobian matrix is diagonal and is tuned on each iteration. + + .. warning:: + + This algorithm may be useful for specific problems, but whether + it will work may depend strongly on the problem. + + See Also + -------- + root : Interface to root finding algorithms for multivariate + functions. See ``method='excitingmixing'`` in particular. + + Parameters + ---------- + %(params_basic)s + alpha : float, optional + Initial Jacobian approximation is (-1/alpha). + alphamax : float, optional + The entries of the diagonal Jacobian are kept in the range + ``[alpha, alphamax]``. + %(params_extra)s + """ + + def __init__(self, alpha=None, alphamax=1.0): + GenericBroyden.__init__(self) + self.alpha = alpha + self.alphamax = alphamax + self.beta = None + + def setup(self, x, F, func): + GenericBroyden.setup(self, x, F, func) + self.beta = np.full((self.shape[0],), self.alpha, dtype=self.dtype) + + def solve(self, f, tol=0): + return -f*self.beta + + def matvec(self, f): + return -f/self.beta + + def rsolve(self, f, tol=0): + return -f*self.beta.conj() + + def rmatvec(self, f): + return -f/self.beta.conj() + + def todense(self): + return np.diag(-1/self.beta) + + def _update(self, x, f, dx, df, dx_norm, df_norm): + incr = f*self.last_f > 0 + self.beta[incr] += self.alpha + self.beta[~incr] = self.alpha + np.clip(self.beta, 0, self.alphamax, out=self.beta) + + +#------------------------------------------------------------------------------ +# Iterative/Krylov approximated Jacobians +#------------------------------------------------------------------------------ + +class KrylovJacobian(Jacobian): + """ + Find a root of a function, using Krylov approximation for inverse Jacobian. + + This method is suitable for solving large-scale problems. + + Parameters + ---------- + %(params_basic)s + rdiff : float, optional + Relative step size to use in numerical differentiation. + method : str or callable, optional + Krylov method to use to approximate the Jacobian. Can be a string, + or a function implementing the same interface as the iterative + solvers in `scipy.sparse.linalg`. If a string, needs to be one of: + ``'lgmres'``, ``'gmres'``, ``'bicgstab'``, ``'cgs'``, ``'minres'``, + ``'tfqmr'``. + + The default is `scipy.sparse.linalg.lgmres`. + inner_maxiter : int, optional + Parameter to pass to the "inner" Krylov solver: maximum number of + iterations. Iteration will stop after maxiter steps even if the + specified tolerance has not been achieved. + inner_M : LinearOperator or InverseJacobian + Preconditioner for the inner Krylov iteration. + Note that you can use also inverse Jacobians as (adaptive) + preconditioners. For example, + + >>> from scipy.optimize import BroydenFirst, KrylovJacobian + >>> from scipy.optimize import InverseJacobian + >>> jac = BroydenFirst() + >>> kjac = KrylovJacobian(inner_M=InverseJacobian(jac)) + + If the preconditioner has a method named 'update', it will be called + as ``update(x, f)`` after each nonlinear step, with ``x`` giving + the current point, and ``f`` the current function value. + outer_k : int, optional + Size of the subspace kept across LGMRES nonlinear iterations. + See `scipy.sparse.linalg.lgmres` for details. + inner_kwargs : kwargs + Keyword parameters for the "inner" Krylov solver + (defined with `method`). Parameter names must start with + the `inner_` prefix which will be stripped before passing on + the inner method. See, e.g., `scipy.sparse.linalg.gmres` for details. + %(params_extra)s + + See Also + -------- + root : Interface to root finding algorithms for multivariate + functions. See ``method='krylov'`` in particular. + scipy.sparse.linalg.gmres + scipy.sparse.linalg.lgmres + + Notes + ----- + This function implements a Newton-Krylov solver. The basic idea is + to compute the inverse of the Jacobian with an iterative Krylov + method. These methods require only evaluating the Jacobian-vector + products, which are conveniently approximated by a finite difference: + + .. math:: J v \\approx (f(x + \\omega*v/|v|) - f(x)) / \\omega + + Due to the use of iterative matrix inverses, these methods can + deal with large nonlinear problems. + + SciPy's `scipy.sparse.linalg` module offers a selection of Krylov + solvers to choose from. The default here is `lgmres`, which is a + variant of restarted GMRES iteration that reuses some of the + information obtained in the previous Newton steps to invert + Jacobians in subsequent steps. + + For a review on Newton-Krylov methods, see for example [1]_, + and for the LGMRES sparse inverse method, see [2]_. + + References + ---------- + .. [1] C. T. Kelley, Solving Nonlinear Equations with Newton's Method, + SIAM, pp.57-83, 2003. + :doi:`10.1137/1.9780898718898.ch3` + .. [2] D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2004). + :doi:`10.1016/j.jcp.2003.08.010` + .. [3] A.H. Baker and E.R. Jessup and T. Manteuffel, + SIAM J. Matrix Anal. Appl. 26, 962 (2005). + :doi:`10.1137/S0895479803422014` + + Examples + -------- + The following functions define a system of nonlinear equations + + >>> def fun(x): + ... return [x[0] + 0.5 * x[1] - 1.0, + ... 0.5 * (x[1] - x[0]) ** 2] + + A solution can be obtained as follows. + + >>> from scipy import optimize + >>> sol = optimize.newton_krylov(fun, [0, 0]) + >>> sol + array([0.66731771, 0.66536458]) + + """ + + def __init__(self, rdiff=None, method='lgmres', inner_maxiter=20, + inner_M=None, outer_k=10, **kw): + self.preconditioner = inner_M + self.rdiff = rdiff + # Note that this retrieves one of the named functions, or otherwise + # uses `method` as is (i.e., for a user-provided callable). + self.method = dict( + bicgstab=scipy.sparse.linalg.bicgstab, + gmres=scipy.sparse.linalg.gmres, + lgmres=scipy.sparse.linalg.lgmres, + cgs=scipy.sparse.linalg.cgs, + minres=scipy.sparse.linalg.minres, + tfqmr=scipy.sparse.linalg.tfqmr, + ).get(method, method) + + self.method_kw = dict(maxiter=inner_maxiter, M=self.preconditioner) + + if self.method is scipy.sparse.linalg.gmres: + # Replace GMRES's outer iteration with Newton steps + self.method_kw['restart'] = inner_maxiter + self.method_kw['maxiter'] = 1 + self.method_kw.setdefault('atol', 0) + elif self.method in (scipy.sparse.linalg.gcrotmk, + scipy.sparse.linalg.bicgstab, + scipy.sparse.linalg.cgs): + self.method_kw.setdefault('atol', 0) + elif self.method is scipy.sparse.linalg.lgmres: + self.method_kw['outer_k'] = outer_k + # Replace LGMRES's outer iteration with Newton steps + self.method_kw['maxiter'] = 1 + # Carry LGMRES's `outer_v` vectors across nonlinear iterations + self.method_kw.setdefault('outer_v', []) + self.method_kw.setdefault('prepend_outer_v', True) + # But don't carry the corresponding Jacobian*v products, in case + # the Jacobian changes a lot in the nonlinear step + # + # XXX: some trust-region inspired ideas might be more efficient... + # See e.g., Brown & Saad. But needs to be implemented separately + # since it's not an inexact Newton method. + self.method_kw.setdefault('store_outer_Av', False) + self.method_kw.setdefault('atol', 0) + + for key, value in kw.items(): + if not key.startswith('inner_'): + raise ValueError(f"Unknown parameter {key}") + self.method_kw[key[6:]] = value + + def _update_diff_step(self): + mx = abs(self.x0).max() + mf = abs(self.f0).max() + self.omega = self.rdiff * max(1, mx) / max(1, mf) + + def matvec(self, v): + nv = norm(v) + if nv == 0: + return 0*v + sc = self.omega / nv + r = (self.func(self.x0 + sc*v) - self.f0) / sc + if not np.all(np.isfinite(r)) and np.all(np.isfinite(v)): + raise ValueError('Function returned non-finite results') + return r + + def solve(self, rhs, tol=0): + if 'rtol' in self.method_kw: + sol, info = self.method(self.op, rhs, **self.method_kw) + else: + sol, info = self.method(self.op, rhs, rtol=tol, **self.method_kw) + return sol + + def update(self, x, f): + self.x0 = x + self.f0 = f + self._update_diff_step() + + # Update also the preconditioner, if possible + if self.preconditioner is not None: + if hasattr(self.preconditioner, 'update'): + self.preconditioner.update(x, f) + + def setup(self, x, f, func): + Jacobian.setup(self, x, f, func) + self.x0 = x + self.f0 = f + self.op = scipy.sparse.linalg.aslinearoperator(self) + + if self.rdiff is None: + self.rdiff = np.finfo(x.dtype).eps ** (1./2) + + self._update_diff_step() + + # Setup also the preconditioner, if possible + if self.preconditioner is not None: + if hasattr(self.preconditioner, 'setup'): + self.preconditioner.setup(x, f, func) + + +#------------------------------------------------------------------------------ +# Wrapper functions +#------------------------------------------------------------------------------ + +def _nonlin_wrapper(name, jac): + """ + Construct a solver wrapper with given name and Jacobian approx. + + It inspects the keyword arguments of ``jac.__init__``, and allows to + use the same arguments in the wrapper function, in addition to the + keyword arguments of `nonlin_solve` + + """ + signature = _getfullargspec(jac.__init__) + args, varargs, varkw, defaults, kwonlyargs, kwdefaults, _ = signature + kwargs = list(zip(args[-len(defaults):], defaults)) + kw_str = ", ".join([f"{k}={v!r}" for k, v in kwargs]) + if kw_str: + kw_str = ", " + kw_str + kwkw_str = ", ".join([f"{k}={k}" for k, v in kwargs]) + if kwkw_str: + kwkw_str = kwkw_str + ", " + if kwonlyargs: + raise ValueError(f'Unexpected signature {signature}') + + # Construct the wrapper function so that its keyword arguments + # are visible in pydoc.help etc. + wrapper = """ +def %(name)s(F, xin, iter=None %(kw)s, verbose=False, maxiter=None, + f_tol=None, f_rtol=None, x_tol=None, x_rtol=None, + tol_norm=None, line_search='armijo', callback=None, **kw): + jac = %(jac)s(%(kwkw)s **kw) + return nonlin_solve(F, xin, jac, iter, verbose, maxiter, + f_tol, f_rtol, x_tol, x_rtol, tol_norm, line_search, + callback) +""" + + wrapper = wrapper % dict(name=name, kw=kw_str, jac=jac.__name__, + kwkw=kwkw_str) + ns = {} + ns.update(globals()) + exec(wrapper, ns) + func = ns[name] + func.__doc__ = jac.__doc__ + _set_doc(func) + return func + + +broyden1 = _nonlin_wrapper('broyden1', BroydenFirst) +broyden2 = _nonlin_wrapper('broyden2', BroydenSecond) +anderson = _nonlin_wrapper('anderson', Anderson) +linearmixing = _nonlin_wrapper('linearmixing', LinearMixing) +diagbroyden = _nonlin_wrapper('diagbroyden', DiagBroyden) +excitingmixing = _nonlin_wrapper('excitingmixing', ExcitingMixing) +newton_krylov = _nonlin_wrapper('newton_krylov', KrylovJacobian) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_numdiff.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_numdiff.py new file mode 100644 index 0000000000000000000000000000000000000000..6f847a8ebdaec7df7428ca1267a024fb9212d824 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_numdiff.py @@ -0,0 +1,785 @@ +"""Routines for numerical differentiation.""" +import functools +import numpy as np +from numpy.linalg import norm + +from scipy.sparse.linalg import LinearOperator +from ..sparse import issparse, csc_matrix, csr_matrix, coo_matrix, find +from ._group_columns import group_dense, group_sparse +from scipy._lib._array_api import array_namespace +from scipy._lib import array_api_extra as xpx + + +def _adjust_scheme_to_bounds(x0, h, num_steps, scheme, lb, ub): + """Adjust final difference scheme to the presence of bounds. + + Parameters + ---------- + x0 : ndarray, shape (n,) + Point at which we wish to estimate derivative. + h : ndarray, shape (n,) + Desired absolute finite difference steps. + num_steps : int + Number of `h` steps in one direction required to implement finite + difference scheme. For example, 2 means that we need to evaluate + f(x0 + 2 * h) or f(x0 - 2 * h) + scheme : {'1-sided', '2-sided'} + Whether steps in one or both directions are required. In other + words '1-sided' applies to forward and backward schemes, '2-sided' + applies to center schemes. + lb : ndarray, shape (n,) + Lower bounds on independent variables. + ub : ndarray, shape (n,) + Upper bounds on independent variables. + + Returns + ------- + h_adjusted : ndarray, shape (n,) + Adjusted absolute step sizes. Step size decreases only if a sign flip + or switching to one-sided scheme doesn't allow to take a full step. + use_one_sided : ndarray of bool, shape (n,) + Whether to switch to one-sided scheme. Informative only for + ``scheme='2-sided'``. + """ + if scheme == '1-sided': + use_one_sided = np.ones_like(h, dtype=bool) + elif scheme == '2-sided': + h = np.abs(h) + use_one_sided = np.zeros_like(h, dtype=bool) + else: + raise ValueError("`scheme` must be '1-sided' or '2-sided'.") + + if np.all((lb == -np.inf) & (ub == np.inf)): + return h, use_one_sided + + h_total = h * num_steps + h_adjusted = h.copy() + + lower_dist = x0 - lb + upper_dist = ub - x0 + + if scheme == '1-sided': + x = x0 + h_total + violated = (x < lb) | (x > ub) + fitting = np.abs(h_total) <= np.maximum(lower_dist, upper_dist) + h_adjusted[violated & fitting] *= -1 + + forward = (upper_dist >= lower_dist) & ~fitting + h_adjusted[forward] = upper_dist[forward] / num_steps + backward = (upper_dist < lower_dist) & ~fitting + h_adjusted[backward] = -lower_dist[backward] / num_steps + elif scheme == '2-sided': + central = (lower_dist >= h_total) & (upper_dist >= h_total) + + forward = (upper_dist >= lower_dist) & ~central + h_adjusted[forward] = np.minimum( + h[forward], 0.5 * upper_dist[forward] / num_steps) + use_one_sided[forward] = True + + backward = (upper_dist < lower_dist) & ~central + h_adjusted[backward] = -np.minimum( + h[backward], 0.5 * lower_dist[backward] / num_steps) + use_one_sided[backward] = True + + min_dist = np.minimum(upper_dist, lower_dist) / num_steps + adjusted_central = (~central & (np.abs(h_adjusted) <= min_dist)) + h_adjusted[adjusted_central] = min_dist[adjusted_central] + use_one_sided[adjusted_central] = False + + return h_adjusted, use_one_sided + + +@functools.lru_cache +def _eps_for_method(x0_dtype, f0_dtype, method): + """ + Calculates relative EPS step to use for a given data type + and numdiff step method. + + Progressively smaller steps are used for larger floating point types. + + Parameters + ---------- + f0_dtype: np.dtype + dtype of function evaluation + + x0_dtype: np.dtype + dtype of parameter vector + + method: {'2-point', '3-point', 'cs'} + + Returns + ------- + EPS: float + relative step size. May be np.float16, np.float32, np.float64 + + Notes + ----- + The default relative step will be np.float64. However, if x0 or f0 are + smaller floating point types (np.float16, np.float32), then the smallest + floating point type is chosen. + """ + # the default EPS value + EPS = np.finfo(np.float64).eps + + x0_is_fp = False + if np.issubdtype(x0_dtype, np.inexact): + # if you're a floating point type then over-ride the default EPS + EPS = np.finfo(x0_dtype).eps + x0_itemsize = np.dtype(x0_dtype).itemsize + x0_is_fp = True + + if np.issubdtype(f0_dtype, np.inexact): + f0_itemsize = np.dtype(f0_dtype).itemsize + # choose the smallest itemsize between x0 and f0 + if x0_is_fp and f0_itemsize < x0_itemsize: + EPS = np.finfo(f0_dtype).eps + + if method in ["2-point", "cs"]: + return EPS**0.5 + elif method in ["3-point"]: + return EPS**(1/3) + else: + raise RuntimeError("Unknown step method, should be one of " + "{'2-point', '3-point', 'cs'}") + + +def _compute_absolute_step(rel_step, x0, f0, method): + """ + Computes an absolute step from a relative step for finite difference + calculation. + + Parameters + ---------- + rel_step: None or array-like + Relative step for the finite difference calculation + x0 : np.ndarray + Parameter vector + f0 : np.ndarray or scalar + method : {'2-point', '3-point', 'cs'} + + Returns + ------- + h : float + The absolute step size + + Notes + ----- + `h` will always be np.float64. However, if `x0` or `f0` are + smaller floating point dtypes (e.g. np.float32), then the absolute + step size will be calculated from the smallest floating point size. + """ + # this is used instead of np.sign(x0) because we need + # sign_x0 to be 1 when x0 == 0. + sign_x0 = (x0 >= 0).astype(float) * 2 - 1 + + rstep = _eps_for_method(x0.dtype, f0.dtype, method) + + if rel_step is None: + abs_step = rstep * sign_x0 * np.maximum(1.0, np.abs(x0)) + else: + # User has requested specific relative steps. + # Don't multiply by max(1, abs(x0) because if x0 < 1 then their + # requested step is not used. + abs_step = rel_step * sign_x0 * np.abs(x0) + + # however we don't want an abs_step of 0, which can happen if + # rel_step is 0, or x0 is 0. Instead, substitute a realistic step + dx = ((x0 + abs_step) - x0) + abs_step = np.where(dx == 0, + rstep * sign_x0 * np.maximum(1.0, np.abs(x0)), + abs_step) + + return abs_step + + +def _prepare_bounds(bounds, x0): + """ + Prepares new-style bounds from a two-tuple specifying the lower and upper + limits for values in x0. If a value is not bound then the lower/upper bound + will be expected to be -np.inf/np.inf. + + Examples + -------- + >>> _prepare_bounds([(0, 1, 2), (1, 2, np.inf)], [0.5, 1.5, 2.5]) + (array([0., 1., 2.]), array([ 1., 2., inf])) + """ + lb, ub = (np.asarray(b, dtype=float) for b in bounds) + if lb.ndim == 0: + lb = np.resize(lb, x0.shape) + + if ub.ndim == 0: + ub = np.resize(ub, x0.shape) + + return lb, ub + + +def group_columns(A, order=0): + """Group columns of a 2-D matrix for sparse finite differencing [1]_. + + Two columns are in the same group if in each row at least one of them + has zero. A greedy sequential algorithm is used to construct groups. + + Parameters + ---------- + A : array_like or sparse matrix, shape (m, n) + Matrix of which to group columns. + order : int, iterable of int with shape (n,) or None + Permutation array which defines the order of columns enumeration. + If int or None, a random permutation is used with `order` used as + a random seed. Default is 0, that is use a random permutation but + guarantee repeatability. + + Returns + ------- + groups : ndarray of int, shape (n,) + Contains values from 0 to n_groups-1, where n_groups is the number + of found groups. Each value ``groups[i]`` is an index of a group to + which ith column assigned. The procedure was helpful only if + n_groups is significantly less than n. + + References + ---------- + .. [1] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of + sparse Jacobian matrices", Journal of the Institute of Mathematics + and its Applications, 13 (1974), pp. 117-120. + """ + if issparse(A): + A = csc_matrix(A) + else: + A = np.atleast_2d(A) + A = (A != 0).astype(np.int32) + + if A.ndim != 2: + raise ValueError("`A` must be 2-dimensional.") + + m, n = A.shape + + if order is None or np.isscalar(order): + rng = np.random.RandomState(order) + order = rng.permutation(n) + else: + order = np.asarray(order) + if order.shape != (n,): + raise ValueError("`order` has incorrect shape.") + + A = A[:, order] + + if issparse(A): + groups = group_sparse(m, n, A.indices, A.indptr) + else: + groups = group_dense(m, n, A) + + groups[order] = groups.copy() + + return groups + + +def approx_derivative(fun, x0, method='3-point', rel_step=None, abs_step=None, + f0=None, bounds=(-np.inf, np.inf), sparsity=None, + as_linear_operator=False, args=(), kwargs=None): + """Compute finite difference approximation of the derivatives of a + vector-valued function. + + If a function maps from R^n to R^m, its derivatives form m-by-n matrix + called the Jacobian, where an element (i, j) is a partial derivative of + f[i] with respect to x[j]. + + Parameters + ---------- + fun : callable + Function of which to estimate the derivatives. The argument x + passed to this function is ndarray of shape (n,) (never a scalar + even if n=1). It must return 1-D array_like of shape (m,) or a scalar. + x0 : array_like of shape (n,) or float + Point at which to estimate the derivatives. Float will be converted + to a 1-D array. + method : {'3-point', '2-point', 'cs'}, optional + Finite difference method to use: + - '2-point' - use the first order accuracy forward or backward + difference. + - '3-point' - use central difference in interior points and the + second order accuracy forward or backward difference + near the boundary. + - 'cs' - use a complex-step finite difference scheme. This assumes + that the user function is real-valued and can be + analytically continued to the complex plane. Otherwise, + produces bogus results. + rel_step : None or array_like, optional + Relative step size to use. If None (default) the absolute step size is + computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``, with + `rel_step` being selected automatically, see Notes. Otherwise + ``h = rel_step * sign(x0) * abs(x0)``. For ``method='3-point'`` the + sign of `h` is ignored. The calculated step size is possibly adjusted + to fit into the bounds. + abs_step : array_like, optional + Absolute step size to use, possibly adjusted to fit into the bounds. + For ``method='3-point'`` the sign of `abs_step` is ignored. By default + relative steps are used, only if ``abs_step is not None`` are absolute + steps used. + f0 : None or array_like, optional + If not None it is assumed to be equal to ``fun(x0)``, in this case + the ``fun(x0)`` is not called. Default is None. + bounds : tuple of array_like, optional + Lower and upper bounds on independent variables. Defaults to no bounds. + Each bound must match the size of `x0` or be a scalar, in the latter + case the bound will be the same for all variables. Use it to limit the + range of function evaluation. Bounds checking is not implemented + when `as_linear_operator` is True. + sparsity : {None, array_like, sparse matrix, 2-tuple}, optional + Defines a sparsity structure of the Jacobian matrix. If the Jacobian + matrix is known to have only few non-zero elements in each row, then + it's possible to estimate its several columns by a single function + evaluation [3]_. To perform such economic computations two ingredients + are required: + + * structure : array_like or sparse matrix of shape (m, n). A zero + element means that a corresponding element of the Jacobian + identically equals to zero. + * groups : array_like of shape (n,). A column grouping for a given + sparsity structure, use `group_columns` to obtain it. + + A single array or a sparse matrix is interpreted as a sparsity + structure, and groups are computed inside the function. A tuple is + interpreted as (structure, groups). If None (default), a standard + dense differencing will be used. + + Note, that sparse differencing makes sense only for large Jacobian + matrices where each row contains few non-zero elements. + as_linear_operator : bool, optional + When True the function returns an `scipy.sparse.linalg.LinearOperator`. + Otherwise it returns a dense array or a sparse matrix depending on + `sparsity`. The linear operator provides an efficient way of computing + ``J.dot(p)`` for any vector ``p`` of shape (n,), but does not allow + direct access to individual elements of the matrix. By default + `as_linear_operator` is False. + args, kwargs : tuple and dict, optional + Additional arguments passed to `fun`. Both empty by default. + The calling signature is ``fun(x, *args, **kwargs)``. + + Returns + ------- + J : {ndarray, sparse matrix, LinearOperator} + Finite difference approximation of the Jacobian matrix. + If `as_linear_operator` is True returns a LinearOperator + with shape (m, n). Otherwise it returns a dense array or sparse + matrix depending on how `sparsity` is defined. If `sparsity` + is None then a ndarray with shape (m, n) is returned. If + `sparsity` is not None returns a csr_matrix with shape (m, n). + For sparse matrices and linear operators it is always returned as + a 2-D structure, for ndarrays, if m=1 it is returned + as a 1-D gradient array with shape (n,). + + See Also + -------- + check_derivative : Check correctness of a function computing derivatives. + + Notes + ----- + If `rel_step` is not provided, it assigned as ``EPS**(1/s)``, where EPS is + determined from the smallest floating point dtype of `x0` or `fun(x0)`, + ``np.finfo(x0.dtype).eps``, s=2 for '2-point' method and + s=3 for '3-point' method. Such relative step approximately minimizes a sum + of truncation and round-off errors, see [1]_. Relative steps are used by + default. However, absolute steps are used when ``abs_step is not None``. + If any of the absolute or relative steps produces an indistinguishable + difference from the original `x0`, ``(x0 + dx) - x0 == 0``, then a + automatic step size is substituted for that particular entry. + + A finite difference scheme for '3-point' method is selected automatically. + The well-known central difference scheme is used for points sufficiently + far from the boundary, and 3-point forward or backward scheme is used for + points near the boundary. Both schemes have the second-order accuracy in + terms of Taylor expansion. Refer to [2]_ for the formulas of 3-point + forward and backward difference schemes. + + For dense differencing when m=1 Jacobian is returned with a shape (n,), + on the other hand when n=1 Jacobian is returned with a shape (m, 1). + Our motivation is the following: a) It handles a case of gradient + computation (m=1) in a conventional way. b) It clearly separates these two + different cases. b) In all cases np.atleast_2d can be called to get 2-D + Jacobian with correct dimensions. + + References + ---------- + .. [1] W. H. Press et. al. "Numerical Recipes. The Art of Scientific + Computing. 3rd edition", sec. 5.7. + + .. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of + sparse Jacobian matrices", Journal of the Institute of Mathematics + and its Applications, 13 (1974), pp. 117-120. + + .. [3] B. Fornberg, "Generation of Finite Difference Formulas on + Arbitrarily Spaced Grids", Mathematics of Computation 51, 1988. + + Examples + -------- + >>> import numpy as np + >>> from scipy.optimize._numdiff import approx_derivative + >>> + >>> def f(x, c1, c2): + ... return np.array([x[0] * np.sin(c1 * x[1]), + ... x[0] * np.cos(c2 * x[1])]) + ... + >>> x0 = np.array([1.0, 0.5 * np.pi]) + >>> approx_derivative(f, x0, args=(1, 2)) + array([[ 1., 0.], + [-1., 0.]]) + + Bounds can be used to limit the region of function evaluation. + In the example below we compute left and right derivative at point 1.0. + + >>> def g(x): + ... return x**2 if x >= 1 else x + ... + >>> x0 = 1.0 + >>> approx_derivative(g, x0, bounds=(-np.inf, 1.0)) + array([ 1.]) + >>> approx_derivative(g, x0, bounds=(1.0, np.inf)) + array([ 2.]) + """ + if method not in ['2-point', '3-point', 'cs']: + raise ValueError(f"Unknown method '{method}'. ") + + xp = array_namespace(x0) + _x = xpx.atleast_nd(xp.asarray(x0), ndim=1, xp=xp) + _dtype = xp.float64 + if xp.isdtype(_x.dtype, "real floating"): + _dtype = _x.dtype + + # promotes to floating + x0 = xp.astype(_x, _dtype) + + if x0.ndim > 1: + raise ValueError("`x0` must have at most 1 dimension.") + + lb, ub = _prepare_bounds(bounds, x0) + + if lb.shape != x0.shape or ub.shape != x0.shape: + raise ValueError("Inconsistent shapes between bounds and `x0`.") + + if as_linear_operator and not (np.all(np.isinf(lb)) + and np.all(np.isinf(ub))): + raise ValueError("Bounds not supported when " + "`as_linear_operator` is True.") + + if kwargs is None: + kwargs = {} + + def fun_wrapped(x): + # send user function same fp type as x0. (but only if cs is not being + # used + if xp.isdtype(x.dtype, "real floating"): + x = xp.astype(x, x0.dtype) + + f = np.atleast_1d(fun(x, *args, **kwargs)) + if f.ndim > 1: + raise RuntimeError("`fun` return value has " + "more than 1 dimension.") + return f + + if f0 is None: + f0 = fun_wrapped(x0) + else: + f0 = np.atleast_1d(f0) + if f0.ndim > 1: + raise ValueError("`f0` passed has more than 1 dimension.") + + if np.any((x0 < lb) | (x0 > ub)): + raise ValueError("`x0` violates bound constraints.") + + if as_linear_operator: + if rel_step is None: + rel_step = _eps_for_method(x0.dtype, f0.dtype, method) + + return _linear_operator_difference(fun_wrapped, x0, + f0, rel_step, method) + else: + # by default we use rel_step + if abs_step is None: + h = _compute_absolute_step(rel_step, x0, f0, method) + else: + # user specifies an absolute step + sign_x0 = (x0 >= 0).astype(float) * 2 - 1 + h = abs_step + + # cannot have a zero step. This might happen if x0 is very large + # or small. In which case fall back to relative step. + dx = ((x0 + h) - x0) + h = np.where(dx == 0, + _eps_for_method(x0.dtype, f0.dtype, method) * + sign_x0 * np.maximum(1.0, np.abs(x0)), + h) + + if method == '2-point': + h, use_one_sided = _adjust_scheme_to_bounds( + x0, h, 1, '1-sided', lb, ub) + elif method == '3-point': + h, use_one_sided = _adjust_scheme_to_bounds( + x0, h, 1, '2-sided', lb, ub) + elif method == 'cs': + use_one_sided = False + + if sparsity is None: + return _dense_difference(fun_wrapped, x0, f0, h, + use_one_sided, method) + else: + if not issparse(sparsity) and len(sparsity) == 2: + structure, groups = sparsity + else: + structure = sparsity + groups = group_columns(sparsity) + + if issparse(structure): + structure = csc_matrix(structure) + else: + structure = np.atleast_2d(structure) + + groups = np.atleast_1d(groups) + return _sparse_difference(fun_wrapped, x0, f0, h, + use_one_sided, structure, + groups, method) + + +def _linear_operator_difference(fun, x0, f0, h, method): + m = f0.size + n = x0.size + + if method == '2-point': + def matvec(p): + if np.array_equal(p, np.zeros_like(p)): + return np.zeros(m) + dx = h / norm(p) + x = x0 + dx*p + df = fun(x) - f0 + return df / dx + + elif method == '3-point': + def matvec(p): + if np.array_equal(p, np.zeros_like(p)): + return np.zeros(m) + dx = 2*h / norm(p) + x1 = x0 - (dx/2)*p + x2 = x0 + (dx/2)*p + f1 = fun(x1) + f2 = fun(x2) + df = f2 - f1 + return df / dx + + elif method == 'cs': + def matvec(p): + if np.array_equal(p, np.zeros_like(p)): + return np.zeros(m) + dx = h / norm(p) + x = x0 + dx*p*1.j + f1 = fun(x) + df = f1.imag + return df / dx + + else: + raise RuntimeError("Never be here.") + + return LinearOperator((m, n), matvec) + + +def _dense_difference(fun, x0, f0, h, use_one_sided, method): + m = f0.size + n = x0.size + J_transposed = np.empty((n, m)) + x1 = x0.copy() + x2 = x0.copy() + xc = x0.astype(complex, copy=True) + + for i in range(h.size): + if method == '2-point': + x1[i] += h[i] + dx = x1[i] - x0[i] # Recompute dx as exactly representable number. + df = fun(x1) - f0 + elif method == '3-point' and use_one_sided[i]: + x1[i] += h[i] + x2[i] += 2 * h[i] + dx = x2[i] - x0[i] + f1 = fun(x1) + f2 = fun(x2) + df = -3.0 * f0 + 4 * f1 - f2 + elif method == '3-point' and not use_one_sided[i]: + x1[i] -= h[i] + x2[i] += h[i] + dx = x2[i] - x1[i] + f1 = fun(x1) + f2 = fun(x2) + df = f2 - f1 + elif method == 'cs': + xc[i] += h[i] * 1.j + f1 = fun(xc) + df = f1.imag + dx = h[i] + else: + raise RuntimeError("Never be here.") + + J_transposed[i] = df / dx + x1[i] = x2[i] = xc[i] = x0[i] + + if m == 1: + J_transposed = np.ravel(J_transposed) + + return J_transposed.T + + +def _sparse_difference(fun, x0, f0, h, use_one_sided, + structure, groups, method): + m = f0.size + n = x0.size + row_indices = [] + col_indices = [] + fractions = [] + + n_groups = np.max(groups) + 1 + for group in range(n_groups): + # Perturb variables which are in the same group simultaneously. + e = np.equal(group, groups) + h_vec = h * e + if method == '2-point': + x = x0 + h_vec + dx = x - x0 + df = fun(x) - f0 + # The result is written to columns which correspond to perturbed + # variables. + cols, = np.nonzero(e) + # Find all non-zero elements in selected columns of Jacobian. + i, j, _ = find(structure[:, cols]) + # Restore column indices in the full array. + j = cols[j] + elif method == '3-point': + # Here we do conceptually the same but separate one-sided + # and two-sided schemes. + x1 = x0.copy() + x2 = x0.copy() + + mask_1 = use_one_sided & e + x1[mask_1] += h_vec[mask_1] + x2[mask_1] += 2 * h_vec[mask_1] + + mask_2 = ~use_one_sided & e + x1[mask_2] -= h_vec[mask_2] + x2[mask_2] += h_vec[mask_2] + + dx = np.zeros(n) + dx[mask_1] = x2[mask_1] - x0[mask_1] + dx[mask_2] = x2[mask_2] - x1[mask_2] + + f1 = fun(x1) + f2 = fun(x2) + + cols, = np.nonzero(e) + i, j, _ = find(structure[:, cols]) + j = cols[j] + + mask = use_one_sided[j] + df = np.empty(m) + + rows = i[mask] + df[rows] = -3 * f0[rows] + 4 * f1[rows] - f2[rows] + + rows = i[~mask] + df[rows] = f2[rows] - f1[rows] + elif method == 'cs': + f1 = fun(x0 + h_vec*1.j) + df = f1.imag + dx = h_vec + cols, = np.nonzero(e) + i, j, _ = find(structure[:, cols]) + j = cols[j] + else: + raise ValueError("Never be here.") + + # All that's left is to compute the fraction. We store i, j and + # fractions as separate arrays and later construct coo_matrix. + row_indices.append(i) + col_indices.append(j) + fractions.append(df[i] / dx[j]) + + row_indices = np.hstack(row_indices) + col_indices = np.hstack(col_indices) + fractions = np.hstack(fractions) + J = coo_matrix((fractions, (row_indices, col_indices)), shape=(m, n)) + return csr_matrix(J) + + +def check_derivative(fun, jac, x0, bounds=(-np.inf, np.inf), args=(), + kwargs=None): + """Check correctness of a function computing derivatives (Jacobian or + gradient) by comparison with a finite difference approximation. + + Parameters + ---------- + fun : callable + Function of which to estimate the derivatives. The argument x + passed to this function is ndarray of shape (n,) (never a scalar + even if n=1). It must return 1-D array_like of shape (m,) or a scalar. + jac : callable + Function which computes Jacobian matrix of `fun`. It must work with + argument x the same way as `fun`. The return value must be array_like + or sparse matrix with an appropriate shape. + x0 : array_like of shape (n,) or float + Point at which to estimate the derivatives. Float will be converted + to 1-D array. + bounds : 2-tuple of array_like, optional + Lower and upper bounds on independent variables. Defaults to no bounds. + Each bound must match the size of `x0` or be a scalar, in the latter + case the bound will be the same for all variables. Use it to limit the + range of function evaluation. + args, kwargs : tuple and dict, optional + Additional arguments passed to `fun` and `jac`. Both empty by default. + The calling signature is ``fun(x, *args, **kwargs)`` and the same + for `jac`. + + Returns + ------- + accuracy : float + The maximum among all relative errors for elements with absolute values + higher than 1 and absolute errors for elements with absolute values + less or equal than 1. If `accuracy` is on the order of 1e-6 or lower, + then it is likely that your `jac` implementation is correct. + + See Also + -------- + approx_derivative : Compute finite difference approximation of derivative. + + Examples + -------- + >>> import numpy as np + >>> from scipy.optimize._numdiff import check_derivative + >>> + >>> + >>> def f(x, c1, c2): + ... return np.array([x[0] * np.sin(c1 * x[1]), + ... x[0] * np.cos(c2 * x[1])]) + ... + >>> def jac(x, c1, c2): + ... return np.array([ + ... [np.sin(c1 * x[1]), c1 * x[0] * np.cos(c1 * x[1])], + ... [np.cos(c2 * x[1]), -c2 * x[0] * np.sin(c2 * x[1])] + ... ]) + ... + >>> + >>> x0 = np.array([1.0, 0.5 * np.pi]) + >>> check_derivative(f, jac, x0, args=(1, 2)) + 2.4492935982947064e-16 + """ + if kwargs is None: + kwargs = {} + J_to_test = jac(x0, *args, **kwargs) + if issparse(J_to_test): + J_diff = approx_derivative(fun, x0, bounds=bounds, sparsity=J_to_test, + args=args, kwargs=kwargs) + J_to_test = csr_matrix(J_to_test) + abs_err = J_to_test - J_diff + i, j, abs_err_data = find(abs_err) + J_diff_data = np.asarray(J_diff[i, j]).ravel() + return np.max(np.abs(abs_err_data) / + np.maximum(1, np.abs(J_diff_data))) + else: + J_diff = approx_derivative(fun, x0, bounds=bounds, + args=args, kwargs=kwargs) + abs_err = np.abs(J_to_test - J_diff) + return np.max(abs_err / np.maximum(1, np.abs(J_diff))) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_optimize.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_optimize.py new file mode 100644 index 0000000000000000000000000000000000000000..4c0214daad371c51c5b860e4773e2bb01faf123d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_optimize.py @@ -0,0 +1,4131 @@ +#__docformat__ = "restructuredtext en" +# ******NOTICE*************** +# optimize.py module by Travis E. Oliphant +# +# You may copy and use this module as you see fit with no +# guarantee implied provided you keep this notice in all copies. +# *****END NOTICE************ + +# A collection of optimization algorithms. Version 0.5 +# CHANGES +# Added fminbound (July 2001) +# Added brute (Aug. 2002) +# Finished line search satisfying strong Wolfe conditions (Mar. 2004) +# Updated strong Wolfe conditions line search to use +# cubic-interpolation (Mar. 2004) + + +# Minimization routines + +__all__ = ['fmin', 'fmin_powell', 'fmin_bfgs', 'fmin_ncg', 'fmin_cg', + 'fminbound', 'brent', 'golden', 'bracket', 'rosen', 'rosen_der', + 'rosen_hess', 'rosen_hess_prod', 'brute', 'approx_fprime', + 'line_search', 'check_grad', 'OptimizeResult', 'show_options', + 'OptimizeWarning'] + +__docformat__ = "restructuredtext en" + +import math +import warnings +import sys +import inspect +from numpy import eye, argmin, zeros, shape, asarray, sqrt +import numpy as np +from scipy.linalg import cholesky, issymmetric, LinAlgError +from scipy.sparse.linalg import LinearOperator +from ._linesearch import (line_search_wolfe1, line_search_wolfe2, + line_search_wolfe2 as line_search, + LineSearchWarning) +from ._numdiff import approx_derivative +from scipy._lib._util import getfullargspec_no_self as _getfullargspec +from scipy._lib._util import (MapWrapper, check_random_state, _RichResult, + _call_callback_maybe_halt, _transition_to_rng) +from scipy.optimize._differentiable_functions import ScalarFunction, FD_METHODS +from scipy._lib._array_api import array_namespace +from scipy._lib import array_api_extra as xpx + + +# standard status messages of optimizers +_status_message = {'success': 'Optimization terminated successfully.', + 'maxfev': 'Maximum number of function evaluations has ' + 'been exceeded.', + 'maxiter': 'Maximum number of iterations has been ' + 'exceeded.', + 'pr_loss': 'Desired error not necessarily achieved due ' + 'to precision loss.', + 'nan': 'NaN result encountered.', + 'out_of_bounds': 'The result is outside of the provided ' + 'bounds.'} + + +class MemoizeJac: + """Decorator that caches the return values of a function returning ``(fun, grad)`` + each time it is called.""" + + def __init__(self, fun): + self.fun = fun + self.jac = None + self._value = None + self.x = None + + def _compute_if_needed(self, x, *args): + if not np.all(x == self.x) or self._value is None or self.jac is None: + self.x = np.asarray(x).copy() + fg = self.fun(x, *args) + self.jac = fg[1] + self._value = fg[0] + + def __call__(self, x, *args): + """ returns the function value """ + self._compute_if_needed(x, *args) + return self._value + + def derivative(self, x, *args): + self._compute_if_needed(x, *args) + return self.jac + + +def _wrap_callback(callback, method=None): + """Wrap a user-provided callback so that attributes can be attached.""" + if callback is None or method in {'tnc', 'slsqp', 'cobyla', 'cobyqa'}: + return callback # don't wrap + + sig = inspect.signature(callback) + + if set(sig.parameters) == {'intermediate_result'}: + def wrapped_callback(res): + return callback(intermediate_result=res) + elif method == 'trust-constr': + def wrapped_callback(res): + return callback(np.copy(res.x), res) + elif method == 'differential_evolution': + def wrapped_callback(res): + return callback(np.copy(res.x), res.convergence) + else: + def wrapped_callback(res): + return callback(np.copy(res.x)) + + wrapped_callback.stop_iteration = False + return wrapped_callback + + +class OptimizeResult(_RichResult): + """ + Represents the optimization result. + + Attributes + ---------- + x : ndarray + The solution of the optimization. + success : bool + Whether or not the optimizer exited successfully. + status : int + Termination status of the optimizer. Its value depends on the + underlying solver. Refer to `message` for details. + message : str + Description of the cause of the termination. + fun : float + Value of objective function at `x`. + jac, hess : ndarray + Values of objective function's Jacobian and its Hessian at `x` (if + available). The Hessian may be an approximation, see the documentation + of the function in question. + hess_inv : object + Inverse of the objective function's Hessian; may be an approximation. + Not available for all solvers. The type of this attribute may be + either np.ndarray or scipy.sparse.linalg.LinearOperator. + nfev, njev, nhev : int + Number of evaluations of the objective functions and of its + Jacobian and Hessian. + nit : int + Number of iterations performed by the optimizer. + maxcv : float + The maximum constraint violation. + + Notes + ----- + Depending on the specific solver being used, `OptimizeResult` may + not have all attributes listed here, and they may have additional + attributes not listed here. Since this class is essentially a + subclass of dict with attribute accessors, one can see which + attributes are available using the `OptimizeResult.keys` method. + + """ + pass + + +class OptimizeWarning(UserWarning): + pass + +def _check_positive_definite(Hk): + def is_pos_def(A): + if issymmetric(A): + try: + cholesky(A) + return True + except LinAlgError: + return False + else: + return False + if Hk is not None: + if not is_pos_def(Hk): + raise ValueError("'hess_inv0' matrix isn't positive definite.") + + +def _check_unknown_options(unknown_options): + if unknown_options: + msg = ", ".join(map(str, unknown_options.keys())) + # Stack level 4: this is called from _minimize_*, which is + # called from another function in SciPy. Level 4 is the first + # level in user code. + warnings.warn(f"Unknown solver options: {msg}", OptimizeWarning, stacklevel=4) + + +def is_finite_scalar(x): + """Test whether `x` is either a finite scalar or a finite array scalar. + + """ + return np.size(x) == 1 and np.isfinite(x) + + +_epsilon = sqrt(np.finfo(float).eps) + + +def vecnorm(x, ord=2): + if ord == np.inf: + return np.amax(np.abs(x)) + elif ord == -np.inf: + return np.amin(np.abs(x)) + else: + return np.sum(np.abs(x)**ord, axis=0)**(1.0 / ord) + + +def _prepare_scalar_function(fun, x0, jac=None, args=(), bounds=None, + epsilon=None, finite_diff_rel_step=None, + hess=None): + """ + Creates a ScalarFunction object for use with scalar minimizers + (BFGS/LBFGSB/SLSQP/TNC/CG/etc). + + Parameters + ---------- + fun : callable + The objective function to be minimized. + + ``fun(x, *args) -> float`` + + where ``x`` is an 1-D array with shape (n,) and ``args`` + is a tuple of the fixed parameters needed to completely + specify the function. + x0 : ndarray, shape (n,) + Initial guess. Array of real elements of size (n,), + where 'n' is the number of independent variables. + jac : {callable, '2-point', '3-point', 'cs', None}, optional + Method for computing the gradient vector. If it is a callable, it + should be a function that returns the gradient vector: + + ``jac(x, *args) -> array_like, shape (n,)`` + + If one of `{'2-point', '3-point', 'cs'}` is selected then the gradient + is calculated with a relative step for finite differences. If `None`, + then two-point finite differences with an absolute step is used. + args : tuple, optional + Extra arguments passed to the objective function and its + derivatives (`fun`, `jac` functions). + bounds : sequence, optional + Bounds on variables. 'new-style' bounds are required. + eps : float or ndarray + If ``jac is None`` the absolute step size used for numerical + approximation of the jacobian via forward differences. + finite_diff_rel_step : None or array_like, optional + If ``jac in ['2-point', '3-point', 'cs']`` the relative step size to + use for numerical approximation of the jacobian. The absolute step + size is computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``, + possibly adjusted to fit into the bounds. For ``jac='3-point'`` + the sign of `h` is ignored. If None (default) then step is selected + automatically. + hess : {callable, '2-point', '3-point', 'cs', None} + Computes the Hessian matrix. If it is callable, it should return the + Hessian matrix: + + ``hess(x, *args) -> {LinearOperator, spmatrix, array}, (n, n)`` + + Alternatively, the keywords {'2-point', '3-point', 'cs'} select a + finite difference scheme for numerical estimation. + Whenever the gradient is estimated via finite-differences, the Hessian + cannot be estimated with options {'2-point', '3-point', 'cs'} and needs + to be estimated using one of the quasi-Newton strategies. + + Returns + ------- + sf : ScalarFunction + """ + if callable(jac): + grad = jac + elif jac in FD_METHODS: + # epsilon is set to None so that ScalarFunction is made to use + # rel_step + epsilon = None + grad = jac + else: + # default (jac is None) is to do 2-point finite differences with + # absolute step size. ScalarFunction has to be provided an + # epsilon value that is not None to use absolute steps. This is + # normally the case from most _minimize* methods. + grad = '2-point' + epsilon = epsilon + + if hess is None: + # ScalarFunction requires something for hess, so we give a dummy + # implementation here if nothing is provided, return a value of None + # so that downstream minimisers halt. The results of `fun.hess` + # should not be used. + def hess(x, *args): + return None + + if bounds is None: + bounds = (-np.inf, np.inf) + + # ScalarFunction caches. Reuse of fun(x) during grad + # calculation reduces overall function evaluations. + sf = ScalarFunction(fun, x0, args, grad, hess, + finite_diff_rel_step, bounds, epsilon=epsilon) + + return sf + + +def _clip_x_for_func(func, bounds): + # ensures that x values sent to func are clipped to bounds + + # this is used as a mitigation for gh11403, slsqp/tnc sometimes + # suggest a move that is outside the limits by 1 or 2 ULP. This + # unclean fix makes sure x is strictly within bounds. + def eval(x): + x = _check_clip_x(x, bounds) + return func(x) + + return eval + + +def _check_clip_x(x, bounds): + if (x < bounds[0]).any() or (x > bounds[1]).any(): + warnings.warn("Values in x were outside bounds during a " + "minimize step, clipping to bounds", + RuntimeWarning, stacklevel=3) + x = np.clip(x, bounds[0], bounds[1]) + return x + + return x + + +def rosen(x): + """ + The Rosenbrock function. + + The function computed is:: + + sum(100.0*(x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0) + + Parameters + ---------- + x : array_like + 1-D array of points at which the Rosenbrock function is to be computed. + + Returns + ------- + f : float + The value of the Rosenbrock function. + + See Also + -------- + rosen_der, rosen_hess, rosen_hess_prod + + Examples + -------- + >>> import numpy as np + >>> from scipy.optimize import rosen + >>> X = 0.1 * np.arange(10) + >>> rosen(X) + 76.56 + + For higher-dimensional input ``rosen`` broadcasts. + In the following example, we use this to plot a 2D landscape. + Note that ``rosen_hess`` does not broadcast in this manner. + + >>> import matplotlib.pyplot as plt + >>> from mpl_toolkits.mplot3d import Axes3D + >>> x = np.linspace(-1, 1, 50) + >>> X, Y = np.meshgrid(x, x) + >>> ax = plt.subplot(111, projection='3d') + >>> ax.plot_surface(X, Y, rosen([X, Y])) + >>> plt.show() + """ + xp = array_namespace(x) + x = xp.asarray(x) + if xp.isdtype(x.dtype, 'integral'): + x = xp.astype(x, xp.asarray(1.).dtype) + r = xp.sum(100.0 * (x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0, + axis=0, dtype=x.dtype) + return r + + +def rosen_der(x): + """ + The derivative (i.e. gradient) of the Rosenbrock function. + + Parameters + ---------- + x : array_like + 1-D array of points at which the derivative is to be computed. + + Returns + ------- + rosen_der : (N,) ndarray + The gradient of the Rosenbrock function at `x`. + + See Also + -------- + rosen, rosen_hess, rosen_hess_prod + + Examples + -------- + >>> import numpy as np + >>> from scipy.optimize import rosen_der + >>> X = 0.1 * np.arange(9) + >>> rosen_der(X) + array([ -2. , 10.6, 15.6, 13.4, 6.4, -3. , -12.4, -19.4, 62. ]) + + """ + xp = array_namespace(x) + x = xp.asarray(x) + if xp.isdtype(x.dtype, 'integral'): + x = xp.astype(x, xp.asarray(1.).dtype) + xm = x[1:-1] + xm_m1 = x[:-2] + xm_p1 = x[2:] + der = xp.zeros_like(x) + der[1:-1] = (200 * (xm - xm_m1**2) - + 400 * (xm_p1 - xm**2) * xm - 2 * (1 - xm)) + der[0] = -400 * x[0] * (x[1] - x[0]**2) - 2 * (1 - x[0]) + der[-1] = 200 * (x[-1] - x[-2]**2) + return der + + +def rosen_hess(x): + """ + The Hessian matrix of the Rosenbrock function. + + Parameters + ---------- + x : array_like + 1-D array of points at which the Hessian matrix is to be computed. + + Returns + ------- + rosen_hess : ndarray + The Hessian matrix of the Rosenbrock function at `x`. + + See Also + -------- + rosen, rosen_der, rosen_hess_prod + + Examples + -------- + >>> import numpy as np + >>> from scipy.optimize import rosen_hess + >>> X = 0.1 * np.arange(4) + >>> rosen_hess(X) + array([[-38., 0., 0., 0.], + [ 0., 134., -40., 0.], + [ 0., -40., 130., -80.], + [ 0., 0., -80., 200.]]) + + """ + xp = array_namespace(x) + x = xpx.atleast_nd(x, ndim=1, xp=xp) + if xp.isdtype(x.dtype, 'integral'): + x = xp.astype(x, xp.asarray(1.).dtype) + H = (xpx.create_diagonal(-400 * x[:-1], offset=1, xp=xp) + - xpx.create_diagonal(400 * x[:-1], offset=-1, xp=xp)) + diagonal = xp.zeros(x.shape[0], dtype=x.dtype) + diagonal[0] = 1200 * x[0]**2 - 400 * x[1] + 2 + diagonal[-1] = 200 + diagonal[1:-1] = 202 + 1200 * x[1:-1]**2 - 400 * x[2:] + return H + xpx.create_diagonal(diagonal, xp=xp) + + +def rosen_hess_prod(x, p): + """ + Product of the Hessian matrix of the Rosenbrock function with a vector. + + Parameters + ---------- + x : array_like + 1-D array of points at which the Hessian matrix is to be computed. + p : array_like + 1-D array, the vector to be multiplied by the Hessian matrix. + + Returns + ------- + rosen_hess_prod : ndarray + The Hessian matrix of the Rosenbrock function at `x` multiplied + by the vector `p`. + + See Also + -------- + rosen, rosen_der, rosen_hess + + Examples + -------- + >>> import numpy as np + >>> from scipy.optimize import rosen_hess_prod + >>> X = 0.1 * np.arange(9) + >>> p = 0.5 * np.arange(9) + >>> rosen_hess_prod(X, p) + array([ -0., 27., -10., -95., -192., -265., -278., -195., -180.]) + + """ + xp = array_namespace(x, p) + x = xpx.atleast_nd(x, ndim=1, xp=xp) + if xp.isdtype(x.dtype, 'integral'): + x = xp.astype(x, xp.asarray(1.).dtype) + p = xp.asarray(p, dtype=x.dtype) + Hp = xp.zeros(x.shape[0], dtype=x.dtype) + Hp[0] = (1200 * x[0]**2 - 400 * x[1] + 2) * p[0] - 400 * x[0] * p[1] + Hp[1:-1] = (-400 * x[:-2] * p[:-2] + + (202 + 1200 * x[1:-1]**2 - 400 * x[2:]) * p[1:-1] - + 400 * x[1:-1] * p[2:]) + Hp[-1] = -400 * x[-2] * p[-2] + 200*p[-1] + return Hp + + +def _wrap_scalar_function(function, args): + # wraps a minimizer function to count number of evaluations + # and to easily provide an args kwd. + ncalls = [0] + if function is None: + return ncalls, None + + def function_wrapper(x, *wrapper_args): + ncalls[0] += 1 + # A copy of x is sent to the user function (gh13740) + fx = function(np.copy(x), *(wrapper_args + args)) + # Ideally, we'd like to a have a true scalar returned from f(x). For + # backwards-compatibility, also allow np.array([1.3]), np.array([[1.3]]) etc. + if not np.isscalar(fx): + try: + fx = np.asarray(fx).item() + except (TypeError, ValueError) as e: + raise ValueError("The user-provided objective function " + "must return a scalar value.") from e + return fx + + return ncalls, function_wrapper + + +class _MaxFuncCallError(RuntimeError): + pass + + +def _wrap_scalar_function_maxfun_validation(function, args, maxfun): + # wraps a minimizer function to count number of evaluations + # and to easily provide an args kwd. + ncalls = [0] + if function is None: + return ncalls, None + + def function_wrapper(x, *wrapper_args): + if ncalls[0] >= maxfun: + raise _MaxFuncCallError("Too many function calls") + ncalls[0] += 1 + # A copy of x is sent to the user function (gh13740) + fx = function(np.copy(x), *(wrapper_args + args)) + # Ideally, we'd like to a have a true scalar returned from f(x). For + # backwards-compatibility, also allow np.array([1.3]), + # np.array([[1.3]]) etc. + if not np.isscalar(fx): + try: + fx = np.asarray(fx).item() + except (TypeError, ValueError) as e: + raise ValueError("The user-provided objective function " + "must return a scalar value.") from e + return fx + + return ncalls, function_wrapper + + +def fmin(func, x0, args=(), xtol=1e-4, ftol=1e-4, maxiter=None, maxfun=None, + full_output=0, disp=1, retall=0, callback=None, initial_simplex=None): + """ + Minimize a function using the downhill simplex algorithm. + + This algorithm only uses function values, not derivatives or second + derivatives. + + Parameters + ---------- + func : callable func(x,*args) + The objective function to be minimized. + x0 : ndarray + Initial guess. + args : tuple, optional + Extra arguments passed to func, i.e., ``f(x,*args)``. + xtol : float, optional + Absolute error in xopt between iterations that is acceptable for + convergence. + ftol : number, optional + Absolute error in func(xopt) between iterations that is acceptable for + convergence. + maxiter : int, optional + Maximum number of iterations to perform. + maxfun : number, optional + Maximum number of function evaluations to make. + full_output : bool, optional + Set to True if fopt and warnflag outputs are desired. + disp : bool, optional + Set to True to print convergence messages. + retall : bool, optional + Set to True to return list of solutions at each iteration. + callback : callable, optional + Called after each iteration, as callback(xk), where xk is the + current parameter vector. + initial_simplex : array_like of shape (N + 1, N), optional + Initial simplex. If given, overrides `x0`. + ``initial_simplex[j,:]`` should contain the coordinates of + the jth vertex of the ``N+1`` vertices in the simplex, where + ``N`` is the dimension. + + Returns + ------- + xopt : ndarray + Parameter that minimizes function. + fopt : float + Value of function at minimum: ``fopt = func(xopt)``. + iter : int + Number of iterations performed. + funcalls : int + Number of function calls made. + warnflag : int + 1 : Maximum number of function evaluations made. + 2 : Maximum number of iterations reached. + allvecs : list + Solution at each iteration. + + See also + -------- + minimize: Interface to minimization algorithms for multivariate + functions. See the 'Nelder-Mead' `method` in particular. + + Notes + ----- + Uses a Nelder-Mead simplex algorithm to find the minimum of function of + one or more variables. + + This algorithm has a long history of successful use in applications. + But it will usually be slower than an algorithm that uses first or + second derivative information. In practice, it can have poor + performance in high-dimensional problems and is not robust to + minimizing complicated functions. Additionally, there currently is no + complete theory describing when the algorithm will successfully + converge to the minimum, or how fast it will if it does. Both the ftol and + xtol criteria must be met for convergence. + + Examples + -------- + >>> def f(x): + ... return x**2 + + >>> from scipy import optimize + + >>> minimum = optimize.fmin(f, 1) + Optimization terminated successfully. + Current function value: 0.000000 + Iterations: 17 + Function evaluations: 34 + >>> minimum[0] + -8.8817841970012523e-16 + + References + ---------- + .. [1] Nelder, J.A. and Mead, R. (1965), "A simplex method for function + minimization", The Computer Journal, 7, pp. 308-313 + + .. [2] Wright, M.H. (1996), "Direct Search Methods: Once Scorned, Now + Respectable", in Numerical Analysis 1995, Proceedings of the + 1995 Dundee Biennial Conference in Numerical Analysis, D.F. + Griffiths and G.A. Watson (Eds.), Addison Wesley Longman, + Harlow, UK, pp. 191-208. + + """ + opts = {'xatol': xtol, + 'fatol': ftol, + 'maxiter': maxiter, + 'maxfev': maxfun, + 'disp': disp, + 'return_all': retall, + 'initial_simplex': initial_simplex} + + callback = _wrap_callback(callback) + res = _minimize_neldermead(func, x0, args, callback=callback, **opts) + if full_output: + retlist = res['x'], res['fun'], res['nit'], res['nfev'], res['status'] + if retall: + retlist += (res['allvecs'], ) + return retlist + else: + if retall: + return res['x'], res['allvecs'] + else: + return res['x'] + + +def _minimize_neldermead(func, x0, args=(), callback=None, + maxiter=None, maxfev=None, disp=False, + return_all=False, initial_simplex=None, + xatol=1e-4, fatol=1e-4, adaptive=False, bounds=None, + **unknown_options): + """ + Minimization of scalar function of one or more variables using the + Nelder-Mead algorithm. + + Options + ------- + disp : bool + Set to True to print convergence messages. + maxiter, maxfev : int + Maximum allowed number of iterations and function evaluations. + Will default to ``N*200``, where ``N`` is the number of + variables, if neither `maxiter` or `maxfev` is set. If both + `maxiter` and `maxfev` are set, minimization will stop at the + first reached. + return_all : bool, optional + Set to True to return a list of the best solution at each of the + iterations. + initial_simplex : array_like of shape (N + 1, N) + Initial simplex. If given, overrides `x0`. + ``initial_simplex[j,:]`` should contain the coordinates of + the jth vertex of the ``N+1`` vertices in the simplex, where + ``N`` is the dimension. + xatol : float, optional + Absolute error in xopt between iterations that is acceptable for + convergence. + fatol : number, optional + Absolute error in func(xopt) between iterations that is acceptable for + convergence. + adaptive : bool, optional + Adapt algorithm parameters to dimensionality of problem. Useful for + high-dimensional minimization [1]_. + bounds : sequence or `Bounds`, optional + Bounds on variables. There are two ways to specify the bounds: + + 1. Instance of `Bounds` class. + 2. Sequence of ``(min, max)`` pairs for each element in `x`. None + is used to specify no bound. + + Note that this just clips all vertices in simplex based on + the bounds. + + References + ---------- + .. [1] Gao, F. and Han, L. + Implementing the Nelder-Mead simplex algorithm with adaptive + parameters. 2012. Computational Optimization and Applications. + 51:1, pp. 259-277 + + """ + _check_unknown_options(unknown_options) + maxfun = maxfev + retall = return_all + + x0 = np.atleast_1d(x0).flatten() + dtype = x0.dtype if np.issubdtype(x0.dtype, np.inexact) else np.float64 + x0 = np.asarray(x0, dtype=dtype) + + if adaptive: + dim = float(len(x0)) + rho = 1 + chi = 1 + 2/dim + psi = 0.75 - 1/(2*dim) + sigma = 1 - 1/dim + else: + rho = 1 + chi = 2 + psi = 0.5 + sigma = 0.5 + + nonzdelt = 0.05 + zdelt = 0.00025 + + if bounds is not None: + lower_bound, upper_bound = bounds.lb, bounds.ub + # check bounds + if (lower_bound > upper_bound).any(): + raise ValueError("Nelder Mead - one of the lower bounds " + "is greater than an upper bound.") + if np.any(lower_bound > x0) or np.any(x0 > upper_bound): + warnings.warn("Initial guess is not within the specified bounds", + OptimizeWarning, stacklevel=3) + + if bounds is not None: + x0 = np.clip(x0, lower_bound, upper_bound) + + if initial_simplex is None: + N = len(x0) + + sim = np.empty((N + 1, N), dtype=x0.dtype) + sim[0] = x0 + for k in range(N): + y = np.array(x0, copy=True) + if y[k] != 0: + y[k] = (1 + nonzdelt)*y[k] + else: + y[k] = zdelt + sim[k + 1] = y + else: + sim = np.atleast_2d(initial_simplex).copy() + dtype = sim.dtype if np.issubdtype(sim.dtype, np.inexact) else np.float64 + sim = np.asarray(sim, dtype=dtype) + if sim.ndim != 2 or sim.shape[0] != sim.shape[1] + 1: + raise ValueError("`initial_simplex` should be an array of shape (N+1,N)") + if len(x0) != sim.shape[1]: + raise ValueError("Size of `initial_simplex` is not consistent with `x0`") + N = sim.shape[1] + + if retall: + allvecs = [sim[0]] + + # If neither are set, then set both to default + if maxiter is None and maxfun is None: + maxiter = N * 200 + maxfun = N * 200 + elif maxiter is None: + # Convert remaining Nones, to np.inf, unless the other is np.inf, in + # which case use the default to avoid unbounded iteration + if maxfun == np.inf: + maxiter = N * 200 + else: + maxiter = np.inf + elif maxfun is None: + if maxiter == np.inf: + maxfun = N * 200 + else: + maxfun = np.inf + + if bounds is not None: + # The default simplex construction may make all entries (for a given + # parameter) greater than an upper bound if x0 is very close to the + # upper bound. If one simply clips the simplex to the bounds this could + # make the simplex entries degenerate. If that occurs reflect into the + # interior. + msk = sim > upper_bound + # reflect into the interior + sim = np.where(msk, 2*upper_bound - sim, sim) + # but make sure the reflection is no less than the lower_bound + sim = np.clip(sim, lower_bound, upper_bound) + + one2np1 = list(range(1, N + 1)) + fsim = np.full((N + 1,), np.inf, dtype=float) + + fcalls, func = _wrap_scalar_function_maxfun_validation(func, args, maxfun) + + try: + for k in range(N + 1): + fsim[k] = func(sim[k]) + except _MaxFuncCallError: + pass + finally: + ind = np.argsort(fsim) + sim = np.take(sim, ind, 0) + fsim = np.take(fsim, ind, 0) + + ind = np.argsort(fsim) + fsim = np.take(fsim, ind, 0) + # sort so sim[0,:] has the lowest function value + sim = np.take(sim, ind, 0) + + iterations = 1 + + while (fcalls[0] < maxfun and iterations < maxiter): + try: + if (np.max(np.ravel(np.abs(sim[1:] - sim[0]))) <= xatol and + np.max(np.abs(fsim[0] - fsim[1:])) <= fatol): + break + + xbar = np.add.reduce(sim[:-1], 0) / N + xr = (1 + rho) * xbar - rho * sim[-1] + if bounds is not None: + xr = np.clip(xr, lower_bound, upper_bound) + fxr = func(xr) + doshrink = 0 + + if fxr < fsim[0]: + xe = (1 + rho * chi) * xbar - rho * chi * sim[-1] + if bounds is not None: + xe = np.clip(xe, lower_bound, upper_bound) + fxe = func(xe) + + if fxe < fxr: + sim[-1] = xe + fsim[-1] = fxe + else: + sim[-1] = xr + fsim[-1] = fxr + else: # fsim[0] <= fxr + if fxr < fsim[-2]: + sim[-1] = xr + fsim[-1] = fxr + else: # fxr >= fsim[-2] + # Perform contraction + if fxr < fsim[-1]: + xc = (1 + psi * rho) * xbar - psi * rho * sim[-1] + if bounds is not None: + xc = np.clip(xc, lower_bound, upper_bound) + fxc = func(xc) + + if fxc <= fxr: + sim[-1] = xc + fsim[-1] = fxc + else: + doshrink = 1 + else: + # Perform an inside contraction + xcc = (1 - psi) * xbar + psi * sim[-1] + if bounds is not None: + xcc = np.clip(xcc, lower_bound, upper_bound) + fxcc = func(xcc) + + if fxcc < fsim[-1]: + sim[-1] = xcc + fsim[-1] = fxcc + else: + doshrink = 1 + + if doshrink: + for j in one2np1: + sim[j] = sim[0] + sigma * (sim[j] - sim[0]) + if bounds is not None: + sim[j] = np.clip( + sim[j], lower_bound, upper_bound) + fsim[j] = func(sim[j]) + iterations += 1 + except _MaxFuncCallError: + pass + finally: + ind = np.argsort(fsim) + sim = np.take(sim, ind, 0) + fsim = np.take(fsim, ind, 0) + if retall: + allvecs.append(sim[0]) + intermediate_result = OptimizeResult(x=sim[0], fun=fsim[0]) + if _call_callback_maybe_halt(callback, intermediate_result): + break + + x = sim[0] + fval = np.min(fsim) + warnflag = 0 + + if fcalls[0] >= maxfun: + warnflag = 1 + msg = _status_message['maxfev'] + if disp: + warnings.warn(msg, RuntimeWarning, stacklevel=3) + elif iterations >= maxiter: + warnflag = 2 + msg = _status_message['maxiter'] + if disp: + warnings.warn(msg, RuntimeWarning, stacklevel=3) + else: + msg = _status_message['success'] + if disp: + print(msg) + print(f" Current function value: {fval:f}") + print(" Iterations: %d" % iterations) + print(" Function evaluations: %d" % fcalls[0]) + + result = OptimizeResult(fun=fval, nit=iterations, nfev=fcalls[0], + status=warnflag, success=(warnflag == 0), + message=msg, x=x, final_simplex=(sim, fsim)) + if retall: + result['allvecs'] = allvecs + return result + + +def approx_fprime(xk, f, epsilon=_epsilon, *args): + """Finite difference approximation of the derivatives of a + scalar or vector-valued function. + + If a function maps from :math:`R^n` to :math:`R^m`, its derivatives form + an m-by-n matrix + called the Jacobian, where an element :math:`(i, j)` is a partial + derivative of f[i] with respect to ``xk[j]``. + + Parameters + ---------- + xk : array_like + The coordinate vector at which to determine the gradient of `f`. + f : callable + Function of which to estimate the derivatives of. Has the signature + ``f(xk, *args)`` where `xk` is the argument in the form of a 1-D array + and `args` is a tuple of any additional fixed parameters needed to + completely specify the function. The argument `xk` passed to this + function is an ndarray of shape (n,) (never a scalar even if n=1). + It must return a 1-D array_like of shape (m,) or a scalar. + + Suppose the callable has signature ``f0(x, *my_args, **my_kwargs)``, where + ``my_args`` and ``my_kwargs`` are required positional and keyword arguments. + Rather than passing ``f0`` as the callable, wrap it to accept + only ``x``; e.g., pass ``fun=lambda x: f0(x, *my_args, **my_kwargs)`` as the + callable, where ``my_args`` (tuple) and ``my_kwargs`` (dict) have been + gathered before invoking this function. + + .. versionchanged:: 1.9.0 + `f` is now able to return a 1-D array-like, with the :math:`(m, n)` + Jacobian being estimated. + + epsilon : {float, array_like}, optional + Increment to `xk` to use for determining the function gradient. + If a scalar, uses the same finite difference delta for all partial + derivatives. If an array, should contain one value per element of + `xk`. Defaults to ``sqrt(np.finfo(float).eps)``, which is approximately + 1.49e-08. + \\*args : args, optional + Any other arguments that are to be passed to `f`. + + Returns + ------- + jac : ndarray + The partial derivatives of `f` to `xk`. + + See Also + -------- + check_grad : Check correctness of gradient function against approx_fprime. + + Notes + ----- + The function gradient is determined by the forward finite difference + formula:: + + f(xk[i] + epsilon[i]) - f(xk[i]) + f'[i] = --------------------------------- + epsilon[i] + + Examples + -------- + >>> import numpy as np + >>> from scipy import optimize + >>> def func(x, c0, c1): + ... "Coordinate vector `x` should be an array of size two." + ... return c0 * x[0]**2 + c1*x[1]**2 + + >>> x = np.ones(2) + >>> c0, c1 = (1, 200) + >>> eps = np.sqrt(np.finfo(float).eps) + >>> optimize.approx_fprime(x, func, [eps, np.sqrt(200) * eps], c0, c1) + array([ 2. , 400.00004208]) + + """ + xk = np.asarray(xk, float) + f0 = f(xk, *args) + + return approx_derivative(f, xk, method='2-point', abs_step=epsilon, + args=args, f0=f0) + + +@_transition_to_rng("seed", position_num=6) +def check_grad(func, grad, x0, *args, epsilon=_epsilon, + direction='all', rng=None): + r"""Check the correctness of a gradient function by comparing it against a + (forward) finite-difference approximation of the gradient. + + Parameters + ---------- + func : callable ``func(x0, *args)`` + Function whose derivative is to be checked. + grad : callable ``grad(x0, *args)`` + Jacobian of `func`. + x0 : ndarray + Points to check `grad` against forward difference approximation of grad + using `func`. + args : \\*args, optional + Extra arguments passed to `func` and `grad`. + epsilon : float, optional + Step size used for the finite difference approximation. It defaults to + ``sqrt(np.finfo(float).eps)``, which is approximately 1.49e-08. + direction : str, optional + If set to ``'random'``, then gradients along a random vector + are used to check `grad` against forward difference approximation + using `func`. By default it is ``'all'``, in which case, all + the one hot direction vectors are considered to check `grad`. + If `func` is a vector valued function then only ``'all'`` can be used. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + + The random numbers generated affect the random vector along which gradients + are computed to check ``grad``. Note that `rng` is only used when `direction` + argument is set to `'random'`. + + Returns + ------- + err : float + The square root of the sum of squares (i.e., the 2-norm) of the + difference between ``grad(x0, *args)`` and the finite difference + approximation of `grad` using func at the points `x0`. + + See Also + -------- + approx_fprime + + Examples + -------- + >>> import numpy as np + >>> def func(x): + ... return x[0]**2 - 0.5 * x[1]**3 + >>> def grad(x): + ... return [2 * x[0], -1.5 * x[1]**2] + >>> from scipy.optimize import check_grad + >>> check_grad(func, grad, [1.5, -1.5]) + 2.9802322387695312e-08 # may vary + >>> rng = np.random.default_rng() + >>> check_grad(func, grad, [1.5, -1.5], + ... direction='random', seed=rng) + 2.9802322387695312e-08 + + """ + step = epsilon + x0 = np.asarray(x0) + + def g(w, func, x0, v, *args): + return func(x0 + w*v, *args) + + if direction == 'random': + _grad = np.asanyarray(grad(x0, *args)) + if _grad.ndim > 1: + raise ValueError("'random' can only be used with scalar valued" + " func") + rng_gen = check_random_state(rng) + v = rng_gen.standard_normal(size=(x0.shape)) + _args = (func, x0, v) + args + _func = g + vars = np.zeros((1,)) + analytical_grad = np.dot(_grad, v) + elif direction == 'all': + _args = args + _func = func + vars = x0 + analytical_grad = grad(x0, *args) + else: + raise ValueError(f"{direction} is not a valid string for " + "``direction`` argument") + + return np.sqrt(np.sum(np.abs( + (analytical_grad - approx_fprime(vars, _func, step, *_args))**2 + ))) + + +def approx_fhess_p(x0, p, fprime, epsilon, *args): + # calculate fprime(x0) first, as this may be cached by ScalarFunction + f1 = fprime(*((x0,) + args)) + f2 = fprime(*((x0 + epsilon*p,) + args)) + return (f2 - f1) / epsilon + + +class _LineSearchError(RuntimeError): + pass + + +def _line_search_wolfe12(f, fprime, xk, pk, gfk, old_fval, old_old_fval, + **kwargs): + """ + Same as line_search_wolfe1, but fall back to line_search_wolfe2 if + suitable step length is not found, and raise an exception if a + suitable step length is not found. + + Raises + ------ + _LineSearchError + If no suitable step size is found + + """ + + extra_condition = kwargs.pop('extra_condition', None) + + ret = line_search_wolfe1(f, fprime, xk, pk, gfk, + old_fval, old_old_fval, + **kwargs) + + if ret[0] is not None and extra_condition is not None: + xp1 = xk + ret[0] * pk + if not extra_condition(ret[0], xp1, ret[3], ret[5]): + # Reject step if extra_condition fails + ret = (None,) + + if ret[0] is None: + # line search failed: try different one. + with warnings.catch_warnings(): + warnings.simplefilter('ignore', LineSearchWarning) + kwargs2 = {} + for key in ('c1', 'c2', 'amax'): + if key in kwargs: + kwargs2[key] = kwargs[key] + ret = line_search_wolfe2(f, fprime, xk, pk, gfk, + old_fval, old_old_fval, + extra_condition=extra_condition, + **kwargs2) + + if ret[0] is None: + raise _LineSearchError() + + return ret + + +def fmin_bfgs(f, x0, fprime=None, args=(), gtol=1e-5, norm=np.inf, + epsilon=_epsilon, maxiter=None, full_output=0, disp=1, + retall=0, callback=None, xrtol=0, c1=1e-4, c2=0.9, + hess_inv0=None): + """ + Minimize a function using the BFGS algorithm. + + Parameters + ---------- + f : callable ``f(x,*args)`` + Objective function to be minimized. + x0 : ndarray + Initial guess, shape (n,) + fprime : callable ``f'(x,*args)``, optional + Gradient of f. + args : tuple, optional + Extra arguments passed to f and fprime. + gtol : float, optional + Terminate successfully if gradient norm is less than `gtol` + norm : float, optional + Order of norm (Inf is max, -Inf is min) + epsilon : int or ndarray, optional + If `fprime` is approximated, use this value for the step size. + callback : callable, optional + An optional user-supplied function to call after each + iteration. Called as ``callback(xk)``, where ``xk`` is the + current parameter vector. + maxiter : int, optional + Maximum number of iterations to perform. + full_output : bool, optional + If True, return ``fopt``, ``func_calls``, ``grad_calls``, and + ``warnflag`` in addition to ``xopt``. + disp : bool, optional + Print convergence message if True. + retall : bool, optional + Return a list of results at each iteration if True. + xrtol : float, default: 0 + Relative tolerance for `x`. Terminate successfully if step + size is less than ``xk * xrtol`` where ``xk`` is the current + parameter vector. + c1 : float, default: 1e-4 + Parameter for Armijo condition rule. + c2 : float, default: 0.9 + Parameter for curvature condition rule. + hess_inv0 : None or ndarray, optional`` + Initial inverse hessian estimate, shape (n, n). If None (default) then + the identity matrix is used. + + Returns + ------- + xopt : ndarray + Parameters which minimize f, i.e., ``f(xopt) == fopt``. + fopt : float + Minimum value. + gopt : ndarray + Value of gradient at minimum, f'(xopt), which should be near 0. + Bopt : ndarray + Value of 1/f''(xopt), i.e., the inverse Hessian matrix. + func_calls : int + Number of function_calls made. + grad_calls : int + Number of gradient calls made. + warnflag : integer + 1 : Maximum number of iterations exceeded. + 2 : Gradient and/or function calls not changing. + 3 : NaN result encountered. + allvecs : list + The value of `xopt` at each iteration. Only returned if `retall` is + True. + + Notes + ----- + Optimize the function, `f`, whose gradient is given by `fprime` + using the quasi-Newton method of Broyden, Fletcher, Goldfarb, + and Shanno (BFGS). + + Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1``. + + See Also + -------- + minimize: Interface to minimization algorithms for multivariate + functions. See ``method='BFGS'`` in particular. + + References + ---------- + Wright, and Nocedal 'Numerical Optimization', 1999, p. 198. + + Examples + -------- + >>> import numpy as np + >>> from scipy.optimize import fmin_bfgs + >>> def quadratic_cost(x, Q): + ... return x @ Q @ x + ... + >>> x0 = np.array([-3, -4]) + >>> cost_weight = np.diag([1., 10.]) + >>> # Note that a trailing comma is necessary for a tuple with single element + >>> fmin_bfgs(quadratic_cost, x0, args=(cost_weight,)) + Optimization terminated successfully. + Current function value: 0.000000 + Iterations: 7 # may vary + Function evaluations: 24 # may vary + Gradient evaluations: 8 # may vary + array([ 2.85169950e-06, -4.61820139e-07]) + + >>> def quadratic_cost_grad(x, Q): + ... return 2 * Q @ x + ... + >>> fmin_bfgs(quadratic_cost, x0, quadratic_cost_grad, args=(cost_weight,)) + Optimization terminated successfully. + Current function value: 0.000000 + Iterations: 7 + Function evaluations: 8 + Gradient evaluations: 8 + array([ 2.85916637e-06, -4.54371951e-07]) + + """ + opts = {'gtol': gtol, + 'norm': norm, + 'eps': epsilon, + 'disp': disp, + 'maxiter': maxiter, + 'return_all': retall, + 'xrtol': xrtol, + 'c1': c1, + 'c2': c2, + 'hess_inv0': hess_inv0} + + callback = _wrap_callback(callback) + res = _minimize_bfgs(f, x0, args, fprime, callback=callback, **opts) + + if full_output: + retlist = (res['x'], res['fun'], res['jac'], res['hess_inv'], + res['nfev'], res['njev'], res['status']) + if retall: + retlist += (res['allvecs'], ) + return retlist + else: + if retall: + return res['x'], res['allvecs'] + else: + return res['x'] + + +def _minimize_bfgs(fun, x0, args=(), jac=None, callback=None, + gtol=1e-5, norm=np.inf, eps=_epsilon, maxiter=None, + disp=False, return_all=False, finite_diff_rel_step=None, + xrtol=0, c1=1e-4, c2=0.9, + hess_inv0=None, **unknown_options): + """ + Minimization of scalar function of one or more variables using the + BFGS algorithm. + + Options + ------- + disp : bool + Set to True to print convergence messages. + maxiter : int + Maximum number of iterations to perform. + gtol : float + Terminate successfully if gradient norm is less than `gtol`. + norm : float + Order of norm (Inf is max, -Inf is min). + eps : float or ndarray + If `jac is None` the absolute step size used for numerical + approximation of the jacobian via forward differences. + return_all : bool, optional + Set to True to return a list of the best solution at each of the + iterations. + finite_diff_rel_step : None or array_like, optional + If ``jac in ['2-point', '3-point', 'cs']`` the relative step size to + use for numerical approximation of the jacobian. The absolute step + size is computed as ``h = rel_step * sign(x) * max(1, abs(x))``, + possibly adjusted to fit into the bounds. For ``jac='3-point'`` + the sign of `h` is ignored. If None (default) then step is selected + automatically. + xrtol : float, default: 0 + Relative tolerance for `x`. Terminate successfully if step size is + less than ``xk * xrtol`` where ``xk`` is the current parameter vector. + c1 : float, default: 1e-4 + Parameter for Armijo condition rule. + c2 : float, default: 0.9 + Parameter for curvature condition rule. + hess_inv0 : None or ndarray, optional + Initial inverse hessian estimate, shape (n, n). If None (default) then + the identity matrix is used. + + Notes + ----- + Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1``. + + If minimization doesn't complete successfully, with an error message of + ``Desired error not necessarily achieved due to precision loss``, then + consider setting `gtol` to a higher value. This precision loss typically + occurs when the (finite difference) numerical differentiation cannot provide + sufficient precision to satisfy the `gtol` termination criterion. + This can happen when working in single precision and a callable jac is not + provided. For single precision problems a `gtol` of 1e-3 seems to work. + """ + _check_unknown_options(unknown_options) + _check_positive_definite(hess_inv0) + retall = return_all + + x0 = asarray(x0).flatten() + if x0.ndim == 0: + x0.shape = (1,) + if maxiter is None: + maxiter = len(x0) * 200 + + sf = _prepare_scalar_function(fun, x0, jac, args=args, epsilon=eps, + finite_diff_rel_step=finite_diff_rel_step) + + f = sf.fun + myfprime = sf.grad + + old_fval = f(x0) + gfk = myfprime(x0) + + k = 0 + N = len(x0) + I = np.eye(N, dtype=int) + Hk = I if hess_inv0 is None else hess_inv0 + + # Sets the initial step guess to dx ~ 1 + old_old_fval = old_fval + np.linalg.norm(gfk) / 2 + + xk = x0 + if retall: + allvecs = [x0] + warnflag = 0 + gnorm = vecnorm(gfk, ord=norm) + while (gnorm > gtol) and (k < maxiter): + pk = -np.dot(Hk, gfk) + try: + alpha_k, fc, gc, old_fval, old_old_fval, gfkp1 = \ + _line_search_wolfe12(f, myfprime, xk, pk, gfk, + old_fval, old_old_fval, amin=1e-100, + amax=1e100, c1=c1, c2=c2) + except _LineSearchError: + # Line search failed to find a better solution. + warnflag = 2 + break + + sk = alpha_k * pk + xkp1 = xk + sk + + if retall: + allvecs.append(xkp1) + xk = xkp1 + if gfkp1 is None: + gfkp1 = myfprime(xkp1) + + yk = gfkp1 - gfk + gfk = gfkp1 + k += 1 + intermediate_result = OptimizeResult(x=xk, fun=old_fval) + if _call_callback_maybe_halt(callback, intermediate_result): + break + gnorm = vecnorm(gfk, ord=norm) + if (gnorm <= gtol): + break + + # See Chapter 5 in P.E. Frandsen, K. Jonasson, H.B. Nielsen, + # O. Tingleff: "Unconstrained Optimization", IMM, DTU. 1999. + # These notes are available here: + # http://www2.imm.dtu.dk/documents/ftp/publlec.html + if (alpha_k*vecnorm(pk) <= xrtol*(xrtol + vecnorm(xk))): + break + + if not np.isfinite(old_fval): + # We correctly found +-Inf as optimal value, or something went + # wrong. + warnflag = 2 + break + + rhok_inv = np.dot(yk, sk) + # this was handled in numeric, let it remains for more safety + # Cryptic comment above is preserved for posterity. Future reader: + # consider change to condition below proposed in gh-1261/gh-17345. + if rhok_inv == 0.: + rhok = 1000.0 + if disp: + msg = "Divide-by-zero encountered: rhok assumed large" + _print_success_message_or_warn(True, msg) + else: + rhok = 1. / rhok_inv + + A1 = I - sk[:, np.newaxis] * yk[np.newaxis, :] * rhok + A2 = I - yk[:, np.newaxis] * sk[np.newaxis, :] * rhok + Hk = np.dot(A1, np.dot(Hk, A2)) + (rhok * sk[:, np.newaxis] * + sk[np.newaxis, :]) + + fval = old_fval + + if warnflag == 2: + msg = _status_message['pr_loss'] + elif k >= maxiter: + warnflag = 1 + msg = _status_message['maxiter'] + elif np.isnan(gnorm) or np.isnan(fval) or np.isnan(xk).any(): + warnflag = 3 + msg = _status_message['nan'] + else: + msg = _status_message['success'] + + if disp: + _print_success_message_or_warn(warnflag, msg) + print(f" Current function value: {fval:f}") + print(" Iterations: %d" % k) + print(" Function evaluations: %d" % sf.nfev) + print(" Gradient evaluations: %d" % sf.ngev) + + result = OptimizeResult(fun=fval, jac=gfk, hess_inv=Hk, nfev=sf.nfev, + njev=sf.ngev, status=warnflag, + success=(warnflag == 0), message=msg, x=xk, + nit=k) + if retall: + result['allvecs'] = allvecs + return result + + +def _print_success_message_or_warn(warnflag, message, warntype=None): + if not warnflag: + print(message) + else: + warnings.warn(message, warntype or OptimizeWarning, stacklevel=3) + + +def fmin_cg(f, x0, fprime=None, args=(), gtol=1e-5, norm=np.inf, + epsilon=_epsilon, maxiter=None, full_output=0, disp=1, retall=0, + callback=None, c1=1e-4, c2=0.4): + """ + Minimize a function using a nonlinear conjugate gradient algorithm. + + Parameters + ---------- + f : callable, ``f(x, *args)`` + Objective function to be minimized. Here `x` must be a 1-D array of + the variables that are to be changed in the search for a minimum, and + `args` are the other (fixed) parameters of `f`. + x0 : ndarray + A user-supplied initial estimate of `xopt`, the optimal value of `x`. + It must be a 1-D array of values. + fprime : callable, ``fprime(x, *args)``, optional + A function that returns the gradient of `f` at `x`. Here `x` and `args` + are as described above for `f`. The returned value must be a 1-D array. + Defaults to None, in which case the gradient is approximated + numerically (see `epsilon`, below). + args : tuple, optional + Parameter values passed to `f` and `fprime`. Must be supplied whenever + additional fixed parameters are needed to completely specify the + functions `f` and `fprime`. + gtol : float, optional + Stop when the norm of the gradient is less than `gtol`. + norm : float, optional + Order to use for the norm of the gradient + (``-np.inf`` is min, ``np.inf`` is max). + epsilon : float or ndarray, optional + Step size(s) to use when `fprime` is approximated numerically. Can be a + scalar or a 1-D array. Defaults to ``sqrt(eps)``, with eps the + floating point machine precision. Usually ``sqrt(eps)`` is about + 1.5e-8. + maxiter : int, optional + Maximum number of iterations to perform. Default is ``200 * len(x0)``. + full_output : bool, optional + If True, return `fopt`, `func_calls`, `grad_calls`, and `warnflag` in + addition to `xopt`. See the Returns section below for additional + information on optional return values. + disp : bool, optional + If True, return a convergence message, followed by `xopt`. + retall : bool, optional + If True, add to the returned values the results of each iteration. + callback : callable, optional + An optional user-supplied function, called after each iteration. + Called as ``callback(xk)``, where ``xk`` is the current value of `x0`. + c1 : float, default: 1e-4 + Parameter for Armijo condition rule. + c2 : float, default: 0.4 + Parameter for curvature condition rule. + + Returns + ------- + xopt : ndarray + Parameters which minimize f, i.e., ``f(xopt) == fopt``. + fopt : float, optional + Minimum value found, f(xopt). Only returned if `full_output` is True. + func_calls : int, optional + The number of function_calls made. Only returned if `full_output` + is True. + grad_calls : int, optional + The number of gradient calls made. Only returned if `full_output` is + True. + warnflag : int, optional + Integer value with warning status, only returned if `full_output` is + True. + + 0 : Success. + + 1 : The maximum number of iterations was exceeded. + + 2 : Gradient and/or function calls were not changing. May indicate + that precision was lost, i.e., the routine did not converge. + + 3 : NaN result encountered. + + allvecs : list of ndarray, optional + List of arrays, containing the results at each iteration. + Only returned if `retall` is True. + + See Also + -------- + minimize : common interface to all `scipy.optimize` algorithms for + unconstrained and constrained minimization of multivariate + functions. It provides an alternative way to call + ``fmin_cg``, by specifying ``method='CG'``. + + Notes + ----- + This conjugate gradient algorithm is based on that of Polak and Ribiere + [1]_. + + Conjugate gradient methods tend to work better when: + + 1. `f` has a unique global minimizing point, and no local minima or + other stationary points, + 2. `f` is, at least locally, reasonably well approximated by a + quadratic function of the variables, + 3. `f` is continuous and has a continuous gradient, + 4. `fprime` is not too large, e.g., has a norm less than 1000, + 5. The initial guess, `x0`, is reasonably close to `f` 's global + minimizing point, `xopt`. + + Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1``. + + References + ---------- + .. [1] Wright & Nocedal, "Numerical Optimization", 1999, pp. 120-122. + + Examples + -------- + Example 1: seek the minimum value of the expression + ``a*u**2 + b*u*v + c*v**2 + d*u + e*v + f`` for given values + of the parameters and an initial guess ``(u, v) = (0, 0)``. + + >>> import numpy as np + >>> args = (2, 3, 7, 8, 9, 10) # parameter values + >>> def f(x, *args): + ... u, v = x + ... a, b, c, d, e, f = args + ... return a*u**2 + b*u*v + c*v**2 + d*u + e*v + f + >>> def gradf(x, *args): + ... u, v = x + ... a, b, c, d, e, f = args + ... gu = 2*a*u + b*v + d # u-component of the gradient + ... gv = b*u + 2*c*v + e # v-component of the gradient + ... return np.asarray((gu, gv)) + >>> x0 = np.asarray((0, 0)) # Initial guess. + >>> from scipy import optimize + >>> res1 = optimize.fmin_cg(f, x0, fprime=gradf, args=args) + Optimization terminated successfully. + Current function value: 1.617021 + Iterations: 4 + Function evaluations: 8 + Gradient evaluations: 8 + >>> res1 + array([-1.80851064, -0.25531915]) + + Example 2: solve the same problem using the `minimize` function. + (This `myopts` dictionary shows all of the available options, + although in practice only non-default values would be needed. + The returned value will be a dictionary.) + + >>> opts = {'maxiter' : None, # default value. + ... 'disp' : True, # non-default value. + ... 'gtol' : 1e-5, # default value. + ... 'norm' : np.inf, # default value. + ... 'eps' : 1.4901161193847656e-08} # default value. + >>> res2 = optimize.minimize(f, x0, jac=gradf, args=args, + ... method='CG', options=opts) + Optimization terminated successfully. + Current function value: 1.617021 + Iterations: 4 + Function evaluations: 8 + Gradient evaluations: 8 + >>> res2.x # minimum found + array([-1.80851064, -0.25531915]) + + """ + opts = {'gtol': gtol, + 'norm': norm, + 'eps': epsilon, + 'disp': disp, + 'maxiter': maxiter, + 'return_all': retall} + + callback = _wrap_callback(callback) + res = _minimize_cg(f, x0, args, fprime, callback=callback, c1=c1, c2=c2, + **opts) + + if full_output: + retlist = res['x'], res['fun'], res['nfev'], res['njev'], res['status'] + if retall: + retlist += (res['allvecs'], ) + return retlist + else: + if retall: + return res['x'], res['allvecs'] + else: + return res['x'] + + +def _minimize_cg(fun, x0, args=(), jac=None, callback=None, + gtol=1e-5, norm=np.inf, eps=_epsilon, maxiter=None, + disp=False, return_all=False, finite_diff_rel_step=None, + c1=1e-4, c2=0.4, **unknown_options): + """ + Minimization of scalar function of one or more variables using the + conjugate gradient algorithm. + + Options + ------- + disp : bool + Set to True to print convergence messages. + maxiter : int + Maximum number of iterations to perform. + gtol : float + Gradient norm must be less than `gtol` before successful + termination. + norm : float + Order of norm (Inf is max, -Inf is min). + eps : float or ndarray + If `jac is None` the absolute step size used for numerical + approximation of the jacobian via forward differences. + return_all : bool, optional + Set to True to return a list of the best solution at each of the + iterations. + finite_diff_rel_step : None or array_like, optional + If ``jac in ['2-point', '3-point', 'cs']`` the relative step size to + use for numerical approximation of the jacobian. The absolute step + size is computed as ``h = rel_step * sign(x) * max(1, abs(x))``, + possibly adjusted to fit into the bounds. For ``jac='3-point'`` + the sign of `h` is ignored. If None (default) then step is selected + automatically. + c1 : float, default: 1e-4 + Parameter for Armijo condition rule. + c2 : float, default: 0.4 + Parameter for curvature condition rule. + + Notes + ----- + Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1``. + """ + _check_unknown_options(unknown_options) + + retall = return_all + + x0 = asarray(x0).flatten() + if maxiter is None: + maxiter = len(x0) * 200 + + sf = _prepare_scalar_function(fun, x0, jac=jac, args=args, epsilon=eps, + finite_diff_rel_step=finite_diff_rel_step) + + f = sf.fun + myfprime = sf.grad + + old_fval = f(x0) + gfk = myfprime(x0) + + k = 0 + xk = x0 + # Sets the initial step guess to dx ~ 1 + old_old_fval = old_fval + np.linalg.norm(gfk) / 2 + + if retall: + allvecs = [xk] + warnflag = 0 + pk = -gfk + gnorm = vecnorm(gfk, ord=norm) + + sigma_3 = 0.01 + + while (gnorm > gtol) and (k < maxiter): + deltak = np.dot(gfk, gfk) + + cached_step = [None] + + def polak_ribiere_powell_step(alpha, gfkp1=None): + xkp1 = xk + alpha * pk + if gfkp1 is None: + gfkp1 = myfprime(xkp1) + yk = gfkp1 - gfk + beta_k = max(0, np.dot(yk, gfkp1) / deltak) + pkp1 = -gfkp1 + beta_k * pk + gnorm = vecnorm(gfkp1, ord=norm) + return (alpha, xkp1, pkp1, gfkp1, gnorm) + + def descent_condition(alpha, xkp1, fp1, gfkp1): + # Polak-Ribiere+ needs an explicit check of a sufficient + # descent condition, which is not guaranteed by strong Wolfe. + # + # See Gilbert & Nocedal, "Global convergence properties of + # conjugate gradient methods for optimization", + # SIAM J. Optimization 2, 21 (1992). + cached_step[:] = polak_ribiere_powell_step(alpha, gfkp1) + alpha, xk, pk, gfk, gnorm = cached_step + + # Accept step if it leads to convergence. + if gnorm <= gtol: + return True + + # Accept step if sufficient descent condition applies. + return np.dot(pk, gfk) <= -sigma_3 * np.dot(gfk, gfk) + + try: + alpha_k, fc, gc, old_fval, old_old_fval, gfkp1 = \ + _line_search_wolfe12(f, myfprime, xk, pk, gfk, old_fval, + old_old_fval, c1=c1, c2=c2, amin=1e-100, + amax=1e100, extra_condition=descent_condition) + except _LineSearchError: + # Line search failed to find a better solution. + warnflag = 2 + break + + # Reuse already computed results if possible + if alpha_k == cached_step[0]: + alpha_k, xk, pk, gfk, gnorm = cached_step + else: + alpha_k, xk, pk, gfk, gnorm = polak_ribiere_powell_step(alpha_k, gfkp1) + + if retall: + allvecs.append(xk) + k += 1 + intermediate_result = OptimizeResult(x=xk, fun=old_fval) + if _call_callback_maybe_halt(callback, intermediate_result): + break + + fval = old_fval + if warnflag == 2: + msg = _status_message['pr_loss'] + elif k >= maxiter: + warnflag = 1 + msg = _status_message['maxiter'] + elif np.isnan(gnorm) or np.isnan(fval) or np.isnan(xk).any(): + warnflag = 3 + msg = _status_message['nan'] + else: + msg = _status_message['success'] + + if disp: + _print_success_message_or_warn(warnflag, msg) + print(f" Current function value: {fval:f}") + print(" Iterations: %d" % k) + print(" Function evaluations: %d" % sf.nfev) + print(" Gradient evaluations: %d" % sf.ngev) + + result = OptimizeResult(fun=fval, jac=gfk, nfev=sf.nfev, + njev=sf.ngev, status=warnflag, + success=(warnflag == 0), message=msg, x=xk, + nit=k) + if retall: + result['allvecs'] = allvecs + return result + + +def fmin_ncg(f, x0, fprime, fhess_p=None, fhess=None, args=(), avextol=1e-5, + epsilon=_epsilon, maxiter=None, full_output=0, disp=1, retall=0, + callback=None, c1=1e-4, c2=0.9): + """ + Unconstrained minimization of a function using the Newton-CG method. + + Parameters + ---------- + f : callable ``f(x, *args)`` + Objective function to be minimized. + x0 : ndarray + Initial guess. + fprime : callable ``f'(x, *args)`` + Gradient of f. + fhess_p : callable ``fhess_p(x, p, *args)``, optional + Function which computes the Hessian of f times an + arbitrary vector, p. + fhess : callable ``fhess(x, *args)``, optional + Function to compute the Hessian matrix of f. + args : tuple, optional + Extra arguments passed to f, fprime, fhess_p, and fhess + (the same set of extra arguments is supplied to all of + these functions). + epsilon : float or ndarray, optional + If fhess is approximated, use this value for the step size. + callback : callable, optional + An optional user-supplied function which is called after + each iteration. Called as callback(xk), where xk is the + current parameter vector. + avextol : float, optional + Convergence is assumed when the average relative error in + the minimizer falls below this amount. + maxiter : int, optional + Maximum number of iterations to perform. + full_output : bool, optional + If True, return the optional outputs. + disp : bool, optional + If True, print convergence message. + retall : bool, optional + If True, return a list of results at each iteration. + c1 : float, default: 1e-4 + Parameter for Armijo condition rule. + c2 : float, default: 0.9 + Parameter for curvature condition rule + + Returns + ------- + xopt : ndarray + Parameters which minimize f, i.e., ``f(xopt) == fopt``. + fopt : float + Value of the function at xopt, i.e., ``fopt = f(xopt)``. + fcalls : int + Number of function calls made. + gcalls : int + Number of gradient calls made. + hcalls : int + Number of Hessian calls made. + warnflag : int + Warnings generated by the algorithm. + 1 : Maximum number of iterations exceeded. + 2 : Line search failure (precision loss). + 3 : NaN result encountered. + allvecs : list + The result at each iteration, if retall is True (see below). + + See also + -------- + minimize: Interface to minimization algorithms for multivariate + functions. See the 'Newton-CG' `method` in particular. + + Notes + ----- + Only one of `fhess_p` or `fhess` need to be given. If `fhess` + is provided, then `fhess_p` will be ignored. If neither `fhess` + nor `fhess_p` is provided, then the hessian product will be + approximated using finite differences on `fprime`. `fhess_p` + must compute the hessian times an arbitrary vector. If it is not + given, finite-differences on `fprime` are used to compute + it. + + Newton-CG methods are also called truncated Newton methods. This + function differs from scipy.optimize.fmin_tnc because + + 1. scipy.optimize.fmin_ncg is written purely in Python using NumPy + and scipy while scipy.optimize.fmin_tnc calls a C function. + 2. scipy.optimize.fmin_ncg is only for unconstrained minimization + while scipy.optimize.fmin_tnc is for unconstrained minimization + or box constrained minimization. (Box constraints give + lower and upper bounds for each variable separately.) + + Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1``. + + References + ---------- + Wright & Nocedal, 'Numerical Optimization', 1999, p. 140. + + """ + opts = {'xtol': avextol, + 'eps': epsilon, + 'maxiter': maxiter, + 'disp': disp, + 'return_all': retall} + + callback = _wrap_callback(callback) + res = _minimize_newtoncg(f, x0, args, fprime, fhess, fhess_p, + callback=callback, c1=c1, c2=c2, **opts) + + if full_output: + retlist = (res['x'], res['fun'], res['nfev'], res['njev'], + res['nhev'], res['status']) + if retall: + retlist += (res['allvecs'], ) + return retlist + else: + if retall: + return res['x'], res['allvecs'] + else: + return res['x'] + + +def _minimize_newtoncg(fun, x0, args=(), jac=None, hess=None, hessp=None, + callback=None, xtol=1e-5, eps=_epsilon, maxiter=None, + disp=False, return_all=False, c1=1e-4, c2=0.9, + **unknown_options): + """ + Minimization of scalar function of one or more variables using the + Newton-CG algorithm. + + Note that the `jac` parameter (Jacobian) is required. + + Options + ------- + disp : bool + Set to True to print convergence messages. + xtol : float + Average relative error in solution `xopt` acceptable for + convergence. + maxiter : int + Maximum number of iterations to perform. + eps : float or ndarray + If `hessp` is approximated, use this value for the step size. + return_all : bool, optional + Set to True to return a list of the best solution at each of the + iterations. + c1 : float, default: 1e-4 + Parameter for Armijo condition rule. + c2 : float, default: 0.9 + Parameter for curvature condition rule. + + Notes + ----- + Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1``. + """ + _check_unknown_options(unknown_options) + if jac is None: + raise ValueError('Jacobian is required for Newton-CG method') + fhess_p = hessp + fhess = hess + avextol = xtol + epsilon = eps + retall = return_all + + x0 = asarray(x0).flatten() + # TODO: add hessp (callable or FD) to ScalarFunction? + sf = _prepare_scalar_function( + fun, x0, jac, args=args, epsilon=eps, hess=hess + ) + f = sf.fun + fprime = sf.grad + _h = sf.hess(x0) + + # Logic for hess/hessp + # - If a callable(hess) is provided, then use that + # - If hess is a FD_METHOD, or the output from hess(x) is a LinearOperator + # then create a hessp function using those. + # - If hess is None but you have callable(hessp) then use the hessp. + # - If hess and hessp are None then approximate hessp using the grad/jac. + + if (hess in FD_METHODS or isinstance(_h, LinearOperator)): + fhess = None + + def _hessp(x, p, *args): + return sf.hess(x).dot(p) + + fhess_p = _hessp + + def terminate(warnflag, msg): + if disp: + _print_success_message_or_warn(warnflag, msg) + print(f" Current function value: {old_fval:f}") + print(" Iterations: %d" % k) + print(" Function evaluations: %d" % sf.nfev) + print(" Gradient evaluations: %d" % sf.ngev) + print(" Hessian evaluations: %d" % hcalls) + fval = old_fval + result = OptimizeResult(fun=fval, jac=gfk, nfev=sf.nfev, + njev=sf.ngev, nhev=hcalls, status=warnflag, + success=(warnflag == 0), message=msg, x=xk, + nit=k) + if retall: + result['allvecs'] = allvecs + return result + + hcalls = 0 + if maxiter is None: + maxiter = len(x0)*200 + cg_maxiter = 20*len(x0) + + xtol = len(x0) * avextol + # Make sure we enter the while loop. + update_l1norm = np.finfo(float).max + xk = np.copy(x0) + if retall: + allvecs = [xk] + k = 0 + gfk = None + old_fval = f(x0) + old_old_fval = None + float64eps = np.finfo(np.float64).eps + while update_l1norm > xtol: + if k >= maxiter: + msg = "Warning: " + _status_message['maxiter'] + return terminate(1, msg) + # Compute a search direction pk by applying the CG method to + # del2 f(xk) p = - grad f(xk) starting from 0. + b = -fprime(xk) + maggrad = np.linalg.norm(b, ord=1) + eta = min(0.5, math.sqrt(maggrad)) + termcond = eta * maggrad + xsupi = zeros(len(x0), dtype=x0.dtype) + ri = -b + psupi = -ri + i = 0 + dri0 = np.dot(ri, ri) + + if fhess is not None: # you want to compute hessian once. + A = sf.hess(xk) + hcalls += 1 + + for k2 in range(cg_maxiter): + if np.add.reduce(np.abs(ri)) <= termcond: + break + if fhess is None: + if fhess_p is None: + Ap = approx_fhess_p(xk, psupi, fprime, epsilon) + else: + Ap = fhess_p(xk, psupi, *args) + hcalls += 1 + else: + # hess was supplied as a callable or hessian update strategy, so + # A is a dense numpy array or sparse matrix + Ap = A.dot(psupi) + # check curvature + Ap = asarray(Ap).squeeze() # get rid of matrices... + curv = np.dot(psupi, Ap) + if 0 <= curv <= 3 * float64eps: + break + elif curv < 0: + if (i > 0): + break + else: + # fall back to steepest descent direction + xsupi = dri0 / (-curv) * b + break + alphai = dri0 / curv + xsupi += alphai * psupi + ri += alphai * Ap + dri1 = np.dot(ri, ri) + betai = dri1 / dri0 + psupi = -ri + betai * psupi + i += 1 + dri0 = dri1 # update np.dot(ri,ri) for next time. + else: + # curvature keeps increasing, bail out + msg = ("Warning: CG iterations didn't converge. The Hessian is not " + "positive definite.") + return terminate(3, msg) + + pk = xsupi # search direction is solution to system. + gfk = -b # gradient at xk + + try: + alphak, fc, gc, old_fval, old_old_fval, gfkp1 = \ + _line_search_wolfe12(f, fprime, xk, pk, gfk, + old_fval, old_old_fval, c1=c1, c2=c2) + except _LineSearchError: + # Line search failed to find a better solution. + msg = "Warning: " + _status_message['pr_loss'] + return terminate(2, msg) + + update = alphak * pk + xk += update # upcast if necessary + if retall: + allvecs.append(xk) + k += 1 + intermediate_result = OptimizeResult(x=xk, fun=old_fval) + if _call_callback_maybe_halt(callback, intermediate_result): + return terminate(5, "") + update_l1norm = np.linalg.norm(update, ord=1) + + else: + if np.isnan(old_fval) or np.isnan(update_l1norm): + return terminate(3, _status_message['nan']) + + msg = _status_message['success'] + return terminate(0, msg) + + +def fminbound(func, x1, x2, args=(), xtol=1e-5, maxfun=500, + full_output=0, disp=1): + """Bounded minimization for scalar functions. + + Parameters + ---------- + func : callable f(x,*args) + Objective function to be minimized (must accept and return scalars). + x1, x2 : float or array scalar + Finite optimization bounds. + args : tuple, optional + Extra arguments passed to function. + xtol : float, optional + The convergence tolerance. + maxfun : int, optional + Maximum number of function evaluations allowed. + full_output : bool, optional + If True, return optional outputs. + disp: int, optional + If non-zero, print messages. + + ``0`` : no message printing. + + ``1`` : non-convergence notification messages only. + + ``2`` : print a message on convergence too. + + ``3`` : print iteration results. + + Returns + ------- + xopt : ndarray + Parameters (over given interval) which minimize the + objective function. + fval : number + (Optional output) The function value evaluated at the minimizer. + ierr : int + (Optional output) An error flag (0 if converged, 1 if maximum number of + function calls reached). + numfunc : int + (Optional output) The number of function calls made. + + See also + -------- + minimize_scalar: Interface to minimization algorithms for scalar + univariate functions. See the 'Bounded' `method` in particular. + + Notes + ----- + Finds a local minimizer of the scalar function `func` in the + interval x1 < xopt < x2 using Brent's method. (See `brent` + for auto-bracketing.) + + References + ---------- + .. [1] Forsythe, G.E., M. A. Malcolm, and C. B. Moler. "Computer Methods + for Mathematical Computations." Prentice-Hall Series in Automatic + Computation 259 (1977). + .. [2] Brent, Richard P. Algorithms for Minimization Without Derivatives. + Courier Corporation, 2013. + + Examples + -------- + `fminbound` finds the minimizer of the function in the given range. + The following examples illustrate this. + + >>> from scipy import optimize + >>> def f(x): + ... return (x-1)**2 + >>> minimizer = optimize.fminbound(f, -4, 4) + >>> minimizer + 1.0 + >>> minimum = f(minimizer) + >>> minimum + 0.0 + >>> res = optimize.fminbound(f, 3, 4, full_output=True) + >>> minimizer, fval, ierr, numfunc = res + >>> minimizer + 3.000005960860986 + >>> minimum = f(minimizer) + >>> minimum, fval + (4.000023843479476, 4.000023843479476) + """ + options = {'xatol': xtol, + 'maxiter': maxfun, + 'disp': disp} + + res = _minimize_scalar_bounded(func, (x1, x2), args, **options) + if full_output: + return res['x'], res['fun'], res['status'], res['nfev'] + else: + return res['x'] + + +def _minimize_scalar_bounded(func, bounds, args=(), + xatol=1e-5, maxiter=500, disp=0, + **unknown_options): + """ + Options + ------- + maxiter : int + Maximum number of iterations to perform. + disp: int, optional + If non-zero, print messages. + + ``0`` : no message printing. + + ``1`` : non-convergence notification messages only. + + ``2`` : print a message on convergence too. + + ``3`` : print iteration results. + + xatol : float + Absolute error in solution `xopt` acceptable for convergence. + + """ + _check_unknown_options(unknown_options) + maxfun = maxiter + # Test bounds are of correct form + if len(bounds) != 2: + raise ValueError('bounds must have two elements.') + x1, x2 = bounds + + if not (is_finite_scalar(x1) and is_finite_scalar(x2)): + raise ValueError("Optimization bounds must be finite scalars.") + + if x1 > x2: + raise ValueError("The lower bound exceeds the upper bound.") + + flag = 0 + header = ' Func-count x f(x) Procedure' + step = ' initial' + + sqrt_eps = sqrt(2.2e-16) + golden_mean = 0.5 * (3.0 - sqrt(5.0)) + a, b = x1, x2 + fulc = a + golden_mean * (b - a) + nfc, xf = fulc, fulc + rat = e = 0.0 + x = xf + fx = func(x, *args) + num = 1 + fmin_data = (1, xf, fx) + fu = np.inf + + ffulc = fnfc = fx + xm = 0.5 * (a + b) + tol1 = sqrt_eps * np.abs(xf) + xatol / 3.0 + tol2 = 2.0 * tol1 + + if disp > 2: + print(" ") + print(header) + print("%5.0f %12.6g %12.6g %s" % (fmin_data + (step,))) + + while (np.abs(xf - xm) > (tol2 - 0.5 * (b - a))): + golden = 1 + # Check for parabolic fit + if np.abs(e) > tol1: + golden = 0 + r = (xf - nfc) * (fx - ffulc) + q = (xf - fulc) * (fx - fnfc) + p = (xf - fulc) * q - (xf - nfc) * r + q = 2.0 * (q - r) + if q > 0.0: + p = -p + q = np.abs(q) + r = e + e = rat + + # Check for acceptability of parabola + if ((np.abs(p) < np.abs(0.5*q*r)) and (p > q*(a - xf)) and + (p < q * (b - xf))): + rat = (p + 0.0) / q + x = xf + rat + step = ' parabolic' + + if ((x - a) < tol2) or ((b - x) < tol2): + si = np.sign(xm - xf) + ((xm - xf) == 0) + rat = tol1 * si + else: # do a golden-section step + golden = 1 + + if golden: # do a golden-section step + if xf >= xm: + e = a - xf + else: + e = b - xf + rat = golden_mean*e + step = ' golden' + + si = np.sign(rat) + (rat == 0) + x = xf + si * np.maximum(np.abs(rat), tol1) + fu = func(x, *args) + num += 1 + fmin_data = (num, x, fu) + if disp > 2: + print("%5.0f %12.6g %12.6g %s" % (fmin_data + (step,))) + + if fu <= fx: + if x >= xf: + a = xf + else: + b = xf + fulc, ffulc = nfc, fnfc + nfc, fnfc = xf, fx + xf, fx = x, fu + else: + if x < xf: + a = x + else: + b = x + if (fu <= fnfc) or (nfc == xf): + fulc, ffulc = nfc, fnfc + nfc, fnfc = x, fu + elif (fu <= ffulc) or (fulc == xf) or (fulc == nfc): + fulc, ffulc = x, fu + + xm = 0.5 * (a + b) + tol1 = sqrt_eps * np.abs(xf) + xatol / 3.0 + tol2 = 2.0 * tol1 + + if num >= maxfun: + flag = 1 + break + + if np.isnan(xf) or np.isnan(fx) or np.isnan(fu): + flag = 2 + + fval = fx + if disp > 0: + _endprint(x, flag, fval, maxfun, xatol, disp) + + result = OptimizeResult(fun=fval, status=flag, success=(flag == 0), + message={0: 'Solution found.', + 1: 'Maximum number of function calls ' + 'reached.', + 2: _status_message['nan']}.get(flag, ''), + x=xf, nfev=num, nit=num) + + return result + + +class Brent: + #need to rethink design of __init__ + def __init__(self, func, args=(), tol=1.48e-8, maxiter=500, + full_output=0, disp=0): + self.func = func + self.args = args + self.tol = tol + self.maxiter = maxiter + self._mintol = 1.0e-11 + self._cg = 0.3819660 + self.xmin = None + self.fval = None + self.iter = 0 + self.funcalls = 0 + self.disp = disp + + # need to rethink design of set_bracket (new options, etc.) + def set_bracket(self, brack=None): + self.brack = brack + + def get_bracket_info(self): + #set up + func = self.func + args = self.args + brack = self.brack + ### BEGIN core bracket_info code ### + ### carefully DOCUMENT any CHANGES in core ## + if brack is None: + xa, xb, xc, fa, fb, fc, funcalls = bracket(func, args=args) + elif len(brack) == 2: + xa, xb, xc, fa, fb, fc, funcalls = bracket(func, xa=brack[0], + xb=brack[1], args=args) + elif len(brack) == 3: + xa, xb, xc = brack + if (xa > xc): # swap so xa < xc can be assumed + xc, xa = xa, xc + if not ((xa < xb) and (xb < xc)): + raise ValueError( + "Bracketing values (xa, xb, xc) do not" + " fulfill this requirement: (xa < xb) and (xb < xc)" + ) + fa = func(*((xa,) + args)) + fb = func(*((xb,) + args)) + fc = func(*((xc,) + args)) + if not ((fb < fa) and (fb < fc)): + raise ValueError( + "Bracketing values (xa, xb, xc) do not fulfill" + " this requirement: (f(xb) < f(xa)) and (f(xb) < f(xc))" + ) + + funcalls = 3 + else: + raise ValueError("Bracketing interval must be " + "length 2 or 3 sequence.") + ### END core bracket_info code ### + + return xa, xb, xc, fa, fb, fc, funcalls + + def optimize(self): + # set up for optimization + func = self.func + xa, xb, xc, fa, fb, fc, funcalls = self.get_bracket_info() + _mintol = self._mintol + _cg = self._cg + ################################# + #BEGIN CORE ALGORITHM + ################################# + x = w = v = xb + fw = fv = fx = fb + if (xa < xc): + a = xa + b = xc + else: + a = xc + b = xa + deltax = 0.0 + iter = 0 + + if self.disp > 2: + print(" ") + print(f"{'Func-count':^12} {'x':^12} {'f(x)': ^12}") + print(f"{funcalls:^12g} {x:^12.6g} {fx:^12.6g}") + + while (iter < self.maxiter): + tol1 = self.tol * np.abs(x) + _mintol + tol2 = 2.0 * tol1 + xmid = 0.5 * (a + b) + # check for convergence + if np.abs(x - xmid) < (tol2 - 0.5 * (b - a)): + break + # XXX In the first iteration, rat is only bound in the true case + # of this conditional. This used to cause an UnboundLocalError + # (gh-4140). It should be set before the if (but to what?). + if (np.abs(deltax) <= tol1): + if (x >= xmid): + deltax = a - x # do a golden section step + else: + deltax = b - x + rat = _cg * deltax + else: # do a parabolic step + tmp1 = (x - w) * (fx - fv) + tmp2 = (x - v) * (fx - fw) + p = (x - v) * tmp2 - (x - w) * tmp1 + tmp2 = 2.0 * (tmp2 - tmp1) + if (tmp2 > 0.0): + p = -p + tmp2 = np.abs(tmp2) + dx_temp = deltax + deltax = rat + # check parabolic fit + if ((p > tmp2 * (a - x)) and (p < tmp2 * (b - x)) and + (np.abs(p) < np.abs(0.5 * tmp2 * dx_temp))): + rat = p * 1.0 / tmp2 # if parabolic step is useful. + u = x + rat + if ((u - a) < tol2 or (b - u) < tol2): + if xmid - x >= 0: + rat = tol1 + else: + rat = -tol1 + else: + if (x >= xmid): + deltax = a - x # if it's not do a golden section step + else: + deltax = b - x + rat = _cg * deltax + + if (np.abs(rat) < tol1): # update by at least tol1 + if rat >= 0: + u = x + tol1 + else: + u = x - tol1 + else: + u = x + rat + fu = func(*((u,) + self.args)) # calculate new output value + funcalls += 1 + + if (fu > fx): # if it's bigger than current + if (u < x): + a = u + else: + b = u + if (fu <= fw) or (w == x): + v = w + w = u + fv = fw + fw = fu + elif (fu <= fv) or (v == x) or (v == w): + v = u + fv = fu + else: + if (u >= x): + a = x + else: + b = x + v = w + w = x + x = u + fv = fw + fw = fx + fx = fu + + if self.disp > 2: + print(f"{funcalls:^12g} {x:^12.6g} {fx:^12.6g}") + + iter += 1 + ################################# + #END CORE ALGORITHM + ################################# + + self.xmin = x + self.fval = fx + self.iter = iter + self.funcalls = funcalls + + def get_result(self, full_output=False): + if full_output: + return self.xmin, self.fval, self.iter, self.funcalls + else: + return self.xmin + + +def brent(func, args=(), brack=None, tol=1.48e-8, full_output=0, maxiter=500): + """ + Given a function of one variable and a possible bracket, return + a local minimizer of the function isolated to a fractional precision + of tol. + + Parameters + ---------- + func : callable f(x,*args) + Objective function. + args : tuple, optional + Additional arguments (if present). + brack : tuple, optional + Either a triple ``(xa, xb, xc)`` satisfying ``xa < xb < xc`` and + ``func(xb) < func(xa) and func(xb) < func(xc)``, or a pair + ``(xa, xb)`` to be used as initial points for a downhill bracket search + (see `scipy.optimize.bracket`). + The minimizer ``x`` will not necessarily satisfy ``xa <= x <= xb``. + tol : float, optional + Relative error in solution `xopt` acceptable for convergence. + full_output : bool, optional + If True, return all output args (xmin, fval, iter, + funcalls). + maxiter : int, optional + Maximum number of iterations in solution. + + Returns + ------- + xmin : ndarray + Optimum point. + fval : float + (Optional output) Optimum function value. + iter : int + (Optional output) Number of iterations. + funcalls : int + (Optional output) Number of objective function evaluations made. + + See also + -------- + minimize_scalar: Interface to minimization algorithms for scalar + univariate functions. See the 'Brent' `method` in particular. + + Notes + ----- + Uses inverse parabolic interpolation when possible to speed up + convergence of golden section method. + + Does not ensure that the minimum lies in the range specified by + `brack`. See `scipy.optimize.fminbound`. + + Examples + -------- + We illustrate the behaviour of the function when `brack` is of + size 2 and 3 respectively. In the case where `brack` is of the + form ``(xa, xb)``, we can see for the given values, the output does + not necessarily lie in the range ``(xa, xb)``. + + >>> def f(x): + ... return (x-1)**2 + + >>> from scipy import optimize + + >>> minimizer = optimize.brent(f, brack=(1, 2)) + >>> minimizer + 1 + >>> res = optimize.brent(f, brack=(-1, 0.5, 2), full_output=True) + >>> xmin, fval, iter, funcalls = res + >>> f(xmin), fval + (0.0, 0.0) + + """ + options = {'xtol': tol, + 'maxiter': maxiter} + res = _minimize_scalar_brent(func, brack, args, **options) + if full_output: + return res['x'], res['fun'], res['nit'], res['nfev'] + else: + return res['x'] + + +def _minimize_scalar_brent(func, brack=None, args=(), xtol=1.48e-8, + maxiter=500, disp=0, + **unknown_options): + """ + Options + ------- + maxiter : int + Maximum number of iterations to perform. + xtol : float + Relative error in solution `xopt` acceptable for convergence. + disp : int, optional + If non-zero, print messages. + + ``0`` : no message printing. + + ``1`` : non-convergence notification messages only. + + ``2`` : print a message on convergence too. + + ``3`` : print iteration results. + + Notes + ----- + Uses inverse parabolic interpolation when possible to speed up + convergence of golden section method. + + """ + _check_unknown_options(unknown_options) + tol = xtol + if tol < 0: + raise ValueError(f'tolerance should be >= 0, got {tol!r}') + + brent = Brent(func=func, args=args, tol=tol, + full_output=True, maxiter=maxiter, disp=disp) + brent.set_bracket(brack) + brent.optimize() + x, fval, nit, nfev = brent.get_result(full_output=True) + + success = nit < maxiter and not (np.isnan(x) or np.isnan(fval)) + + if success: + message = ("\nOptimization terminated successfully;\n" + "The returned value satisfies the termination criteria\n" + f"(using xtol = {xtol} )") + else: + if nit >= maxiter: + message = "\nMaximum number of iterations exceeded" + if np.isnan(x) or np.isnan(fval): + message = f"{_status_message['nan']}" + + if disp: + _print_success_message_or_warn(not success, message) + + return OptimizeResult(fun=fval, x=x, nit=nit, nfev=nfev, + success=success, message=message) + + +def golden(func, args=(), brack=None, tol=_epsilon, + full_output=0, maxiter=5000): + """ + Return the minimizer of a function of one variable using the golden section + method. + + Given a function of one variable and a possible bracketing interval, + return a minimizer of the function isolated to a fractional precision of + tol. + + Parameters + ---------- + func : callable func(x,*args) + Objective function to minimize. + args : tuple, optional + Additional arguments (if present), passed to func. + brack : tuple, optional + Either a triple ``(xa, xb, xc)`` where ``xa < xb < xc`` and + ``func(xb) < func(xa) and func(xb) < func(xc)``, or a pair (xa, xb) + to be used as initial points for a downhill bracket search (see + `scipy.optimize.bracket`). + The minimizer ``x`` will not necessarily satisfy ``xa <= x <= xb``. + tol : float, optional + x tolerance stop criterion + full_output : bool, optional + If True, return optional outputs. + maxiter : int + Maximum number of iterations to perform. + + Returns + ------- + xmin : ndarray + Optimum point. + fval : float + (Optional output) Optimum function value. + funcalls : int + (Optional output) Number of objective function evaluations made. + + See also + -------- + minimize_scalar: Interface to minimization algorithms for scalar + univariate functions. See the 'Golden' `method` in particular. + + Notes + ----- + Uses analog of bisection method to decrease the bracketed + interval. + + Examples + -------- + We illustrate the behaviour of the function when `brack` is of + size 2 and 3, respectively. In the case where `brack` is of the + form (xa,xb), we can see for the given values, the output need + not necessarily lie in the range ``(xa, xb)``. + + >>> def f(x): + ... return (x-1)**2 + + >>> from scipy import optimize + + >>> minimizer = optimize.golden(f, brack=(1, 2)) + >>> minimizer + 1 + >>> res = optimize.golden(f, brack=(-1, 0.5, 2), full_output=True) + >>> xmin, fval, funcalls = res + >>> f(xmin), fval + (9.925165290385052e-18, 9.925165290385052e-18) + + """ + options = {'xtol': tol, 'maxiter': maxiter} + res = _minimize_scalar_golden(func, brack, args, **options) + if full_output: + return res['x'], res['fun'], res['nfev'] + else: + return res['x'] + + +def _minimize_scalar_golden(func, brack=None, args=(), + xtol=_epsilon, maxiter=5000, disp=0, + **unknown_options): + """ + Options + ------- + xtol : float + Relative error in solution `xopt` acceptable for convergence. + maxiter : int + Maximum number of iterations to perform. + disp: int, optional + If non-zero, print messages. + + ``0`` : no message printing. + + ``1`` : non-convergence notification messages only. + + ``2`` : print a message on convergence too. + + ``3`` : print iteration results. + """ + _check_unknown_options(unknown_options) + tol = xtol + if brack is None: + xa, xb, xc, fa, fb, fc, funcalls = bracket(func, args=args) + elif len(brack) == 2: + xa, xb, xc, fa, fb, fc, funcalls = bracket(func, xa=brack[0], + xb=brack[1], args=args) + elif len(brack) == 3: + xa, xb, xc = brack + if (xa > xc): # swap so xa < xc can be assumed + xc, xa = xa, xc + if not ((xa < xb) and (xb < xc)): + raise ValueError( + "Bracketing values (xa, xb, xc) do not" + " fulfill this requirement: (xa < xb) and (xb < xc)" + ) + fa = func(*((xa,) + args)) + fb = func(*((xb,) + args)) + fc = func(*((xc,) + args)) + if not ((fb < fa) and (fb < fc)): + raise ValueError( + "Bracketing values (xa, xb, xc) do not fulfill" + " this requirement: (f(xb) < f(xa)) and (f(xb) < f(xc))" + ) + funcalls = 3 + else: + raise ValueError("Bracketing interval must be length 2 or 3 sequence.") + + _gR = 0.61803399 # golden ratio conjugate: 2.0/(1.0+sqrt(5.0)) + _gC = 1.0 - _gR + x3 = xc + x0 = xa + if (np.abs(xc - xb) > np.abs(xb - xa)): + x1 = xb + x2 = xb + _gC * (xc - xb) + else: + x2 = xb + x1 = xb - _gC * (xb - xa) + f1 = func(*((x1,) + args)) + f2 = func(*((x2,) + args)) + funcalls += 2 + nit = 0 + + if disp > 2: + print(" ") + print(f"{'Func-count':^12} {'x':^12} {'f(x)': ^12}") + + for i in range(maxiter): + if np.abs(x3 - x0) <= tol * (np.abs(x1) + np.abs(x2)): + break + if (f2 < f1): + x0 = x1 + x1 = x2 + x2 = _gR * x1 + _gC * x3 + f1 = f2 + f2 = func(*((x2,) + args)) + else: + x3 = x2 + x2 = x1 + x1 = _gR * x2 + _gC * x0 + f2 = f1 + f1 = func(*((x1,) + args)) + funcalls += 1 + if disp > 2: + if (f1 < f2): + xmin, fval = x1, f1 + else: + xmin, fval = x2, f2 + print(f"{funcalls:^12g} {xmin:^12.6g} {fval:^12.6g}") + + nit += 1 + # end of iteration loop + + if (f1 < f2): + xmin = x1 + fval = f1 + else: + xmin = x2 + fval = f2 + + success = nit < maxiter and not (np.isnan(fval) or np.isnan(xmin)) + + if success: + message = ("\nOptimization terminated successfully;\n" + "The returned value satisfies the termination criteria\n" + f"(using xtol = {xtol} )") + else: + if nit >= maxiter: + message = "\nMaximum number of iterations exceeded" + if np.isnan(xmin) or np.isnan(fval): + message = f"{_status_message['nan']}" + + if disp: + _print_success_message_or_warn(not success, message) + + return OptimizeResult(fun=fval, nfev=funcalls, x=xmin, nit=nit, + success=success, message=message) + + +def bracket(func, xa=0.0, xb=1.0, args=(), grow_limit=110.0, maxiter=1000): + """ + Bracket the minimum of a function. + + Given a function and distinct initial points, search in the + downhill direction (as defined by the initial points) and return + three points that bracket the minimum of the function. + + Parameters + ---------- + func : callable f(x,*args) + Objective function to minimize. + xa, xb : float, optional + Initial points. Defaults `xa` to 0.0, and `xb` to 1.0. + A local minimum need not be contained within this interval. + args : tuple, optional + Additional arguments (if present), passed to `func`. + grow_limit : float, optional + Maximum grow limit. Defaults to 110.0 + maxiter : int, optional + Maximum number of iterations to perform. Defaults to 1000. + + Returns + ------- + xa, xb, xc : float + Final points of the bracket. + fa, fb, fc : float + Objective function values at the bracket points. + funcalls : int + Number of function evaluations made. + + Raises + ------ + BracketError + If no valid bracket is found before the algorithm terminates. + See notes for conditions of a valid bracket. + + Notes + ----- + The algorithm attempts to find three strictly ordered points (i.e. + :math:`x_a < x_b < x_c` or :math:`x_c < x_b < x_a`) satisfying + :math:`f(x_b) ≤ f(x_a)` and :math:`f(x_b) ≤ f(x_c)`, where one of the + inequalities must be satisfied strictly and all :math:`x_i` must be + finite. + + Examples + -------- + This function can find a downward convex region of a function: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.optimize import bracket + >>> def f(x): + ... return 10*x**2 + 3*x + 5 + >>> x = np.linspace(-2, 2) + >>> y = f(x) + >>> init_xa, init_xb = 0.1, 1 + >>> xa, xb, xc, fa, fb, fc, funcalls = bracket(f, xa=init_xa, xb=init_xb) + >>> plt.axvline(x=init_xa, color="k", linestyle="--") + >>> plt.axvline(x=init_xb, color="k", linestyle="--") + >>> plt.plot(x, y, "-k") + >>> plt.plot(xa, fa, "bx") + >>> plt.plot(xb, fb, "rx") + >>> plt.plot(xc, fc, "bx") + >>> plt.show() + + Note that both initial points were to the right of the minimum, and the + third point was found in the "downhill" direction: the direction + in which the function appeared to be decreasing (to the left). + The final points are strictly ordered, and the function value + at the middle point is less than the function values at the endpoints; + it follows that a minimum must lie within the bracket. + + """ + _gold = 1.618034 # golden ratio: (1.0+sqrt(5.0))/2.0 + _verysmall_num = 1e-21 + # convert to numpy floats if not already + xa, xb = np.asarray([xa, xb]) + fa = func(*(xa,) + args) + fb = func(*(xb,) + args) + if (fa < fb): # Switch so fa > fb + xa, xb = xb, xa + fa, fb = fb, fa + xc = xb + _gold * (xb - xa) + fc = func(*((xc,) + args)) + funcalls = 3 + iter = 0 + while (fc < fb): + tmp1 = (xb - xa) * (fb - fc) + tmp2 = (xb - xc) * (fb - fa) + val = tmp2 - tmp1 + if np.abs(val) < _verysmall_num: + denom = 2.0 * _verysmall_num + else: + denom = 2.0 * val + w = xb - ((xb - xc) * tmp2 - (xb - xa) * tmp1) / denom + wlim = xb + grow_limit * (xc - xb) + msg = ("No valid bracket was found before the iteration limit was " + "reached. Consider trying different initial points or " + "increasing `maxiter`.") + if iter > maxiter: + raise RuntimeError(msg) + iter += 1 + if (w - xc) * (xb - w) > 0.0: + fw = func(*((w,) + args)) + funcalls += 1 + if (fw < fc): + xa = xb + xb = w + fa = fb + fb = fw + break + elif (fw > fb): + xc = w + fc = fw + break + w = xc + _gold * (xc - xb) + fw = func(*((w,) + args)) + funcalls += 1 + elif (w - wlim)*(wlim - xc) >= 0.0: + w = wlim + fw = func(*((w,) + args)) + funcalls += 1 + elif (w - wlim)*(xc - w) > 0.0: + fw = func(*((w,) + args)) + funcalls += 1 + if (fw < fc): + xb = xc + xc = w + w = xc + _gold * (xc - xb) + fb = fc + fc = fw + fw = func(*((w,) + args)) + funcalls += 1 + else: + w = xc + _gold * (xc - xb) + fw = func(*((w,) + args)) + funcalls += 1 + xa = xb + xb = xc + xc = w + fa = fb + fb = fc + fc = fw + + # three conditions for a valid bracket + cond1 = (fb < fc and fb <= fa) or (fb < fa and fb <= fc) + cond2 = (xa < xb < xc or xc < xb < xa) + cond3 = np.isfinite(xa) and np.isfinite(xb) and np.isfinite(xc) + msg = ("The algorithm terminated without finding a valid bracket. " + "Consider trying different initial points.") + if not (cond1 and cond2 and cond3): + e = BracketError(msg) + e.data = (xa, xb, xc, fa, fb, fc, funcalls) + raise e + + return xa, xb, xc, fa, fb, fc, funcalls + + +class BracketError(RuntimeError): + pass + + +def _recover_from_bracket_error(solver, fun, bracket, args, **options): + # `bracket` was originally written without checking whether the resulting + # bracket is valid. `brent` and `golden` built on top of it without + # checking the returned bracket for validity, and their output can be + # incorrect without warning/error if the original bracket is invalid. + # gh-14858 noticed the problem, and the following is the desired + # behavior: + # - `scipy.optimize.bracket`, `scipy.optimize.brent`, and + # `scipy.optimize.golden` should raise an error if the bracket is + # invalid, as opposed to silently returning garbage + # - `scipy.optimize.minimize_scalar` should return with `success=False` + # and other information + # The changes that would be required to achieve this the traditional + # way (`return`ing all the required information from bracket all the way + # up to `minimizer_scalar`) are extensive and invasive. (See a6aa40d.) + # We can achieve the same thing by raising the error in `bracket`, but + # storing the information needed by `minimize_scalar` in the error object, + # and intercepting it here. + try: + res = solver(fun, bracket, args, **options) + except BracketError as e: + msg = str(e) + xa, xb, xc, fa, fb, fc, funcalls = e.data + xs, fs = [xa, xb, xc], [fa, fb, fc] + if np.any(np.isnan([xs, fs])): + x, fun = np.nan, np.nan + else: + imin = np.argmin(fs) + x, fun = xs[imin], fs[imin] + return OptimizeResult(fun=fun, nfev=funcalls, x=x, + nit=0, success=False, message=msg) + return res + + +def _line_for_search(x0, alpha, lower_bound, upper_bound): + """ + Given a parameter vector ``x0`` with length ``n`` and a direction + vector ``alpha`` with length ``n``, and lower and upper bounds on + each of the ``n`` parameters, what are the bounds on a scalar + ``l`` such that ``lower_bound <= x0 + alpha * l <= upper_bound``. + + + Parameters + ---------- + x0 : np.array. + The vector representing the current location. + Note ``np.shape(x0) == (n,)``. + alpha : np.array. + The vector representing the direction. + Note ``np.shape(alpha) == (n,)``. + lower_bound : np.array. + The lower bounds for each parameter in ``x0``. If the ``i``th + parameter in ``x0`` is unbounded below, then ``lower_bound[i]`` + should be ``-np.inf``. + Note ``np.shape(lower_bound) == (n,)``. + upper_bound : np.array. + The upper bounds for each parameter in ``x0``. If the ``i``th + parameter in ``x0`` is unbounded above, then ``upper_bound[i]`` + should be ``np.inf``. + Note ``np.shape(upper_bound) == (n,)``. + + Returns + ------- + res : tuple ``(lmin, lmax)`` + The bounds for ``l`` such that + ``lower_bound[i] <= x0[i] + alpha[i] * l <= upper_bound[i]`` + for all ``i``. + + """ + # get nonzero indices of alpha so we don't get any zero division errors. + # alpha will not be all zero, since it is called from _linesearch_powell + # where we have a check for this. + nonzero, = alpha.nonzero() + lower_bound, upper_bound = lower_bound[nonzero], upper_bound[nonzero] + x0, alpha = x0[nonzero], alpha[nonzero] + low = (lower_bound - x0) / alpha + high = (upper_bound - x0) / alpha + + # positive and negative indices + pos = alpha > 0 + + lmin_pos = np.where(pos, low, 0) + lmin_neg = np.where(pos, 0, high) + lmax_pos = np.where(pos, high, 0) + lmax_neg = np.where(pos, 0, low) + + lmin = np.max(lmin_pos + lmin_neg) + lmax = np.min(lmax_pos + lmax_neg) + + # if x0 is outside the bounds, then it is possible that there is + # no way to get back in the bounds for the parameters being updated + # with the current direction alpha. + # when this happens, lmax < lmin. + # If this is the case, then we can just return (0, 0) + return (lmin, lmax) if lmax >= lmin else (0, 0) + + +def _linesearch_powell(func, p, xi, tol=1e-3, + lower_bound=None, upper_bound=None, fval=None): + """Line-search algorithm using fminbound. + + Find the minimum of the function ``func(x0 + alpha*direc)``. + + lower_bound : np.array. + The lower bounds for each parameter in ``x0``. If the ``i``th + parameter in ``x0`` is unbounded below, then ``lower_bound[i]`` + should be ``-np.inf``. + Note ``np.shape(lower_bound) == (n,)``. + upper_bound : np.array. + The upper bounds for each parameter in ``x0``. If the ``i``th + parameter in ``x0`` is unbounded above, then ``upper_bound[i]`` + should be ``np.inf``. + Note ``np.shape(upper_bound) == (n,)``. + fval : number. + ``fval`` is equal to ``func(p)``, the idea is just to avoid + recomputing it so we can limit the ``fevals``. + + """ + def myfunc(alpha): + return func(p + alpha*xi) + + # if xi is zero, then don't optimize + if not np.any(xi): + return ((fval, p, xi) if fval is not None else (func(p), p, xi)) + elif lower_bound is None and upper_bound is None: + # non-bounded minimization + res = _recover_from_bracket_error(_minimize_scalar_brent, + myfunc, None, tuple(), xtol=tol) + alpha_min, fret = res.x, res.fun + xi = alpha_min * xi + return fret, p + xi, xi + else: + bound = _line_for_search(p, xi, lower_bound, upper_bound) + if np.isneginf(bound[0]) and np.isposinf(bound[1]): + # equivalent to unbounded + return _linesearch_powell(func, p, xi, fval=fval, tol=tol) + elif not np.isneginf(bound[0]) and not np.isposinf(bound[1]): + # we can use a bounded scalar minimization + res = _minimize_scalar_bounded(myfunc, bound, xatol=tol / 100) + xi = res.x * xi + return res.fun, p + xi, xi + else: + # only bounded on one side. use the tangent function to convert + # the infinity bound to a finite bound. The new bounded region + # is a subregion of the region bounded by -np.pi/2 and np.pi/2. + bound = np.arctan(bound[0]), np.arctan(bound[1]) + res = _minimize_scalar_bounded( + lambda x: myfunc(np.tan(x)), + bound, + xatol=tol / 100) + xi = np.tan(res.x) * xi + return res.fun, p + xi, xi + + +def fmin_powell(func, x0, args=(), xtol=1e-4, ftol=1e-4, maxiter=None, + maxfun=None, full_output=0, disp=1, retall=0, callback=None, + direc=None): + """ + Minimize a function using modified Powell's method. + + This method only uses function values, not derivatives. + + Parameters + ---------- + func : callable f(x,*args) + Objective function to be minimized. + x0 : ndarray + Initial guess. + args : tuple, optional + Extra arguments passed to func. + xtol : float, optional + Line-search error tolerance. + ftol : float, optional + Relative error in ``func(xopt)`` acceptable for convergence. + maxiter : int, optional + Maximum number of iterations to perform. + maxfun : int, optional + Maximum number of function evaluations to make. + full_output : bool, optional + If True, ``fopt``, ``xi``, ``direc``, ``iter``, ``funcalls``, and + ``warnflag`` are returned. + disp : bool, optional + If True, print convergence messages. + retall : bool, optional + If True, return a list of the solution at each iteration. + callback : callable, optional + An optional user-supplied function, called after each + iteration. Called as ``callback(xk)``, where ``xk`` is the + current parameter vector. + direc : ndarray, optional + Initial fitting step and parameter order set as an (N, N) array, where N + is the number of fitting parameters in `x0`. Defaults to step size 1.0 + fitting all parameters simultaneously (``np.eye((N, N))``). To + prevent initial consideration of values in a step or to change initial + step size, set to 0 or desired step size in the Jth position in the Mth + block, where J is the position in `x0` and M is the desired evaluation + step, with steps being evaluated in index order. Step size and ordering + will change freely as minimization proceeds. + + Returns + ------- + xopt : ndarray + Parameter which minimizes `func`. + fopt : number + Value of function at minimum: ``fopt = func(xopt)``. + direc : ndarray + Current direction set. + iter : int + Number of iterations. + funcalls : int + Number of function calls made. + warnflag : int + Integer warning flag: + 1 : Maximum number of function evaluations. + 2 : Maximum number of iterations. + 3 : NaN result encountered. + 4 : The result is out of the provided bounds. + allvecs : list + List of solutions at each iteration. + + See also + -------- + minimize: Interface to unconstrained minimization algorithms for + multivariate functions. See the 'Powell' method in particular. + + Notes + ----- + Uses a modification of Powell's method to find the minimum of + a function of N variables. Powell's method is a conjugate + direction method. + + The algorithm has two loops. The outer loop merely iterates over the inner + loop. The inner loop minimizes over each current direction in the direction + set. At the end of the inner loop, if certain conditions are met, the + direction that gave the largest decrease is dropped and replaced with the + difference between the current estimated x and the estimated x from the + beginning of the inner-loop. + + The technical conditions for replacing the direction of greatest + increase amount to checking that + + 1. No further gain can be made along the direction of greatest increase + from that iteration. + 2. The direction of greatest increase accounted for a large sufficient + fraction of the decrease in the function value from that iteration of + the inner loop. + + References + ---------- + Powell M.J.D. (1964) An efficient method for finding the minimum of a + function of several variables without calculating derivatives, + Computer Journal, 7 (2):155-162. + + Press W., Teukolsky S.A., Vetterling W.T., and Flannery B.P.: + Numerical Recipes (any edition), Cambridge University Press + + Examples + -------- + >>> def f(x): + ... return x**2 + + >>> from scipy import optimize + + >>> minimum = optimize.fmin_powell(f, -1) + Optimization terminated successfully. + Current function value: 0.000000 + Iterations: 2 + Function evaluations: 16 + >>> minimum + array(0.0) + + """ + opts = {'xtol': xtol, + 'ftol': ftol, + 'maxiter': maxiter, + 'maxfev': maxfun, + 'disp': disp, + 'direc': direc, + 'return_all': retall} + + callback = _wrap_callback(callback) + res = _minimize_powell(func, x0, args, callback=callback, **opts) + + if full_output: + retlist = (res['x'], res['fun'], res['direc'], res['nit'], + res['nfev'], res['status']) + if retall: + retlist += (res['allvecs'], ) + return retlist + else: + if retall: + return res['x'], res['allvecs'] + else: + return res['x'] + + +def _minimize_powell(func, x0, args=(), callback=None, bounds=None, + xtol=1e-4, ftol=1e-4, maxiter=None, maxfev=None, + disp=False, direc=None, return_all=False, + **unknown_options): + """ + Minimization of scalar function of one or more variables using the + modified Powell algorithm. + + Parameters + ---------- + fun : callable + The objective function to be minimized:: + + fun(x, *args) -> float + + where ``x`` is a 1-D array with shape (n,) and ``args`` + is a tuple of the fixed parameters needed to completely + specify the function. + x0 : ndarray, shape (n,) + Initial guess. Array of real elements of size (n,), + where ``n`` is the number of independent variables. + args : tuple, optional + Extra arguments passed to the objective function and its + derivatives (`fun`, `jac` and `hess` functions). + method : str or callable, optional + The present documentation is specific to ``method='powell'``, but other + options are available. See documentation for `scipy.optimize.minimize`. + bounds : sequence or `Bounds`, optional + Bounds on decision variables. There are two ways to specify the bounds: + + 1. Instance of `Bounds` class. + 2. Sequence of ``(min, max)`` pairs for each element in `x`. None + is used to specify no bound. + + If bounds are not provided, then an unbounded line search will be used. + If bounds are provided and the initial guess is within the bounds, then + every function evaluation throughout the minimization procedure will be + within the bounds. If bounds are provided, the initial guess is outside + the bounds, and `direc` is full rank (or left to default), then some + function evaluations during the first iteration may be outside the + bounds, but every function evaluation after the first iteration will be + within the bounds. If `direc` is not full rank, then some parameters + may not be optimized and the solution is not guaranteed to be within + the bounds. + + options : dict, optional + A dictionary of solver options. All methods accept the following + generic options: + + maxiter : int + Maximum number of iterations to perform. Depending on the + method each iteration may use several function evaluations. + disp : bool + Set to True to print convergence messages. + + See method-specific options for ``method='powell'`` below. + callback : callable, optional + Called after each iteration. The signature is:: + + callback(xk) + + where ``xk`` is the current parameter vector. + + Returns + ------- + res : OptimizeResult + The optimization result represented as a ``OptimizeResult`` object. + Important attributes are: ``x`` the solution array, ``success`` a + Boolean flag indicating if the optimizer exited successfully and + ``message`` which describes the cause of the termination. See + `OptimizeResult` for a description of other attributes. + + Options + ------- + disp : bool + Set to True to print convergence messages. + xtol : float + Relative error in solution `xopt` acceptable for convergence. + ftol : float + Relative error in ``fun(xopt)`` acceptable for convergence. + maxiter, maxfev : int + Maximum allowed number of iterations and function evaluations. + Will default to ``N*1000``, where ``N`` is the number of + variables, if neither `maxiter` or `maxfev` is set. If both + `maxiter` and `maxfev` are set, minimization will stop at the + first reached. + direc : ndarray + Initial set of direction vectors for the Powell method. + return_all : bool, optional + Set to True to return a list of the best solution at each of the + iterations. + """ + _check_unknown_options(unknown_options) + maxfun = maxfev + retall = return_all + + x = asarray(x0).flatten() + if retall: + allvecs = [x] + N = len(x) + # If neither are set, then set both to default + if maxiter is None and maxfun is None: + maxiter = N * 1000 + maxfun = N * 1000 + elif maxiter is None: + # Convert remaining Nones, to np.inf, unless the other is np.inf, in + # which case use the default to avoid unbounded iteration + if maxfun == np.inf: + maxiter = N * 1000 + else: + maxiter = np.inf + elif maxfun is None: + if maxiter == np.inf: + maxfun = N * 1000 + else: + maxfun = np.inf + + # we need to use a mutable object here that we can update in the + # wrapper function + fcalls, func = _wrap_scalar_function_maxfun_validation(func, args, maxfun) + + if direc is None: + direc = eye(N, dtype=float) + else: + direc = asarray(direc, dtype=float) + if np.linalg.matrix_rank(direc) != direc.shape[0]: + warnings.warn("direc input is not full rank, some parameters may " + "not be optimized", + OptimizeWarning, stacklevel=3) + + if bounds is None: + # don't make these arrays of all +/- inf. because + # _linesearch_powell will do an unnecessary check of all the elements. + # just keep them None, _linesearch_powell will not have to check + # all the elements. + lower_bound, upper_bound = None, None + else: + # bounds is standardized in _minimize.py. + lower_bound, upper_bound = bounds.lb, bounds.ub + if np.any(lower_bound > x0) or np.any(x0 > upper_bound): + warnings.warn("Initial guess is not within the specified bounds", + OptimizeWarning, stacklevel=3) + + fval = func(x) + x1 = x.copy() + iter = 0 + while True: + try: + fx = fval + bigind = 0 + delta = 0.0 + for i in range(N): + direc1 = direc[i] + fx2 = fval + fval, x, direc1 = _linesearch_powell(func, x, direc1, + tol=xtol * 100, + lower_bound=lower_bound, + upper_bound=upper_bound, + fval=fval) + if (fx2 - fval) > delta: + delta = fx2 - fval + bigind = i + iter += 1 + if retall: + allvecs.append(x) + intermediate_result = OptimizeResult(x=x, fun=fval) + if _call_callback_maybe_halt(callback, intermediate_result): + break + bnd = ftol * (np.abs(fx) + np.abs(fval)) + 1e-20 + if 2.0 * (fx - fval) <= bnd: + break + if fcalls[0] >= maxfun: + break + if iter >= maxiter: + break + if np.isnan(fx) and np.isnan(fval): + # Ended up in a nan-region: bail out + break + + # Construct the extrapolated point + direc1 = x - x1 + x1 = x.copy() + # make sure that we don't go outside the bounds when extrapolating + if lower_bound is None and upper_bound is None: + lmax = 1 + else: + _, lmax = _line_for_search(x, direc1, lower_bound, upper_bound) + x2 = x + min(lmax, 1) * direc1 + fx2 = func(x2) + + if (fx > fx2): + t = 2.0*(fx + fx2 - 2.0*fval) + temp = (fx - fval - delta) + t *= temp*temp + temp = fx - fx2 + t -= delta*temp*temp + if t < 0.0: + fval, x, direc1 = _linesearch_powell( + func, x, direc1, + tol=xtol * 100, + lower_bound=lower_bound, + upper_bound=upper_bound, + fval=fval + ) + if np.any(direc1): + direc[bigind] = direc[-1] + direc[-1] = direc1 + except _MaxFuncCallError: + break + + warnflag = 0 + msg = _status_message['success'] + # out of bounds is more urgent than exceeding function evals or iters, + # but I don't want to cause inconsistencies by changing the + # established warning flags for maxfev and maxiter, so the out of bounds + # warning flag becomes 3, but is checked for first. + if bounds and (np.any(lower_bound > x) or np.any(x > upper_bound)): + warnflag = 4 + msg = _status_message['out_of_bounds'] + elif fcalls[0] >= maxfun: + warnflag = 1 + msg = _status_message['maxfev'] + elif iter >= maxiter: + warnflag = 2 + msg = _status_message['maxiter'] + elif np.isnan(fval) or np.isnan(x).any(): + warnflag = 3 + msg = _status_message['nan'] + + if disp: + _print_success_message_or_warn(warnflag, msg, RuntimeWarning) + print(f" Current function value: {fval:f}") + print(" Iterations: %d" % iter) + print(" Function evaluations: %d" % fcalls[0]) + result = OptimizeResult(fun=fval, direc=direc, nit=iter, nfev=fcalls[0], + status=warnflag, success=(warnflag == 0), + message=msg, x=x) + if retall: + result['allvecs'] = allvecs + return result + + +def _endprint(x, flag, fval, maxfun, xtol, disp): + if flag == 0: + if disp > 1: + print("\nOptimization terminated successfully;\n" + "The returned value satisfies the termination criteria\n" + "(using xtol = ", xtol, ")") + return + + if flag == 1: + msg = ("\nMaximum number of function evaluations exceeded --- " + "increase maxfun argument.\n") + elif flag == 2: + msg = f"\n{_status_message['nan']}" + + _print_success_message_or_warn(flag, msg) + return + + +def brute(func, ranges, args=(), Ns=20, full_output=0, finish=fmin, + disp=False, workers=1): + """Minimize a function over a given range by brute force. + + Uses the "brute force" method, i.e., computes the function's value + at each point of a multidimensional grid of points, to find the global + minimum of the function. + + The function is evaluated everywhere in the range with the datatype of the + first call to the function, as enforced by the ``vectorize`` NumPy + function. The value and type of the function evaluation returned when + ``full_output=True`` are affected in addition by the ``finish`` argument + (see Notes). + + The brute force approach is inefficient because the number of grid points + increases exponentially - the number of grid points to evaluate is + ``Ns ** len(x)``. Consequently, even with coarse grid spacing, even + moderately sized problems can take a long time to run, and/or run into + memory limitations. + + Parameters + ---------- + func : callable + The objective function to be minimized. Must be in the + form ``f(x, *args)``, where ``x`` is the argument in + the form of a 1-D array and ``args`` is a tuple of any + additional fixed parameters needed to completely specify + the function. + ranges : tuple + Each component of the `ranges` tuple must be either a + "slice object" or a range tuple of the form ``(low, high)``. + The program uses these to create the grid of points on which + the objective function will be computed. See `Note 2` for + more detail. + args : tuple, optional + Any additional fixed parameters needed to completely specify + the function. + Ns : int, optional + Number of grid points along the axes, if not otherwise + specified. See `Note2`. + full_output : bool, optional + If True, return the evaluation grid and the objective function's + values on it. + finish : callable, optional + An optimization function that is called with the result of brute force + minimization as initial guess. `finish` should take `func` and + the initial guess as positional arguments, and take `args` as + keyword arguments. It may additionally take `full_output` + and/or `disp` as keyword arguments. Use None if no "polishing" + function is to be used. See Notes for more details. + disp : bool, optional + Set to True to print convergence messages from the `finish` callable. + workers : int or map-like callable, optional + If `workers` is an int the grid is subdivided into `workers` + sections and evaluated in parallel (uses + `multiprocessing.Pool `). + Supply `-1` to use all cores available to the Process. + Alternatively supply a map-like callable, such as + `multiprocessing.Pool.map` for evaluating the grid in parallel. + This evaluation is carried out as ``workers(func, iterable)``. + Requires that `func` be pickleable. + + .. versionadded:: 1.3.0 + + Returns + ------- + x0 : ndarray + A 1-D array containing the coordinates of a point at which the + objective function had its minimum value. (See `Note 1` for + which point is returned.) + fval : float + Function value at the point `x0`. (Returned when `full_output` is + True.) + grid : tuple + Representation of the evaluation grid. It has the same + length as `x0`. (Returned when `full_output` is True.) + Jout : ndarray + Function values at each point of the evaluation + grid, i.e., ``Jout = func(*grid)``. (Returned + when `full_output` is True.) + + See Also + -------- + basinhopping, differential_evolution + + Notes + ----- + *Note 1*: The program finds the gridpoint at which the lowest value + of the objective function occurs. If `finish` is None, that is the + point returned. When the global minimum occurs within (or not very far + outside) the grid's boundaries, and the grid is fine enough, that + point will be in the neighborhood of the global minimum. + + However, users often employ some other optimization program to + "polish" the gridpoint values, i.e., to seek a more precise + (local) minimum near `brute's` best gridpoint. + The `brute` function's `finish` option provides a convenient way to do + that. Any polishing program used must take `brute's` output as its + initial guess as a positional argument, and take `brute's` input values + for `args` as keyword arguments, otherwise an error will be raised. + It may additionally take `full_output` and/or `disp` as keyword arguments. + + `brute` assumes that the `finish` function returns either an + `OptimizeResult` object or a tuple in the form: + ``(xmin, Jmin, ... , statuscode)``, where ``xmin`` is the minimizing + value of the argument, ``Jmin`` is the minimum value of the objective + function, "..." may be some other returned values (which are not used + by `brute`), and ``statuscode`` is the status code of the `finish` program. + + Note that when `finish` is not None, the values returned are those + of the `finish` program, *not* the gridpoint ones. Consequently, + while `brute` confines its search to the input grid points, + the `finish` program's results usually will not coincide with any + gridpoint, and may fall outside the grid's boundary. Thus, if a + minimum only needs to be found over the provided grid points, make + sure to pass in ``finish=None``. + + *Note 2*: The grid of points is a `numpy.mgrid` object. + For `brute` the `ranges` and `Ns` inputs have the following effect. + Each component of the `ranges` tuple can be either a slice object or a + two-tuple giving a range of values, such as (0, 5). If the component is a + slice object, `brute` uses it directly. If the component is a two-tuple + range, `brute` internally converts it to a slice object that interpolates + `Ns` points from its low-value to its high-value, inclusive. + + Examples + -------- + We illustrate the use of `brute` to seek the global minimum of a function + of two variables that is given as the sum of a positive-definite + quadratic and two deep "Gaussian-shaped" craters. Specifically, define + the objective function `f` as the sum of three other functions, + ``f = f1 + f2 + f3``. We suppose each of these has a signature + ``(z, *params)``, where ``z = (x, y)``, and ``params`` and the functions + are as defined below. + + >>> import numpy as np + >>> params = (2, 3, 7, 8, 9, 10, 44, -1, 2, 26, 1, -2, 0.5) + >>> def f1(z, *params): + ... x, y = z + ... a, b, c, d, e, f, g, h, i, j, k, l, scale = params + ... return (a * x**2 + b * x * y + c * y**2 + d*x + e*y + f) + + >>> def f2(z, *params): + ... x, y = z + ... a, b, c, d, e, f, g, h, i, j, k, l, scale = params + ... return (-g*np.exp(-((x-h)**2 + (y-i)**2) / scale)) + + >>> def f3(z, *params): + ... x, y = z + ... a, b, c, d, e, f, g, h, i, j, k, l, scale = params + ... return (-j*np.exp(-((x-k)**2 + (y-l)**2) / scale)) + + >>> def f(z, *params): + ... return f1(z, *params) + f2(z, *params) + f3(z, *params) + + Thus, the objective function may have local minima near the minimum + of each of the three functions of which it is composed. To + use `fmin` to polish its gridpoint result, we may then continue as + follows: + + >>> rranges = (slice(-4, 4, 0.25), slice(-4, 4, 0.25)) + >>> from scipy import optimize + >>> resbrute = optimize.brute(f, rranges, args=params, full_output=True, + ... finish=optimize.fmin) + >>> resbrute[0] # global minimum + array([-1.05665192, 1.80834843]) + >>> resbrute[1] # function value at global minimum + -3.4085818767 + + Note that if `finish` had been set to None, we would have gotten the + gridpoint [-1.0 1.75] where the rounded function value is -2.892. + + """ + N = len(ranges) + if N > 40: + raise ValueError("Brute Force not possible with more " + "than 40 variables.") + lrange = list(ranges) + for k in range(N): + if not isinstance(lrange[k], slice): + if len(lrange[k]) < 3: + lrange[k] = tuple(lrange[k]) + (complex(Ns),) + lrange[k] = slice(*lrange[k]) + if (N == 1): + lrange = lrange[0] + + grid = np.mgrid[lrange] + + # obtain an array of parameters that is iterable by a map-like callable + inpt_shape = grid.shape + if (N > 1): + grid = np.reshape(grid, (inpt_shape[0], np.prod(inpt_shape[1:]))).T + + if not np.iterable(args): + args = (args,) + + wrapped_func = _Brute_Wrapper(func, args) + + # iterate over input arrays, possibly in parallel + with MapWrapper(pool=workers) as mapper: + Jout = np.array(list(mapper(wrapped_func, grid))) + if (N == 1): + grid = (grid,) + Jout = np.squeeze(Jout) + elif (N > 1): + Jout = np.reshape(Jout, inpt_shape[1:]) + grid = np.reshape(grid.T, inpt_shape) + + Nshape = shape(Jout) + + indx = argmin(Jout.ravel(), axis=-1) + Nindx = np.empty(N, int) + xmin = np.empty(N, float) + for k in range(N - 1, -1, -1): + thisN = Nshape[k] + Nindx[k] = indx % Nshape[k] + indx = indx // thisN + for k in range(N): + xmin[k] = grid[k][tuple(Nindx)] + + Jmin = Jout[tuple(Nindx)] + if (N == 1): + grid = grid[0] + xmin = xmin[0] + + if callable(finish): + # set up kwargs for `finish` function + finish_args = _getfullargspec(finish).args + finish_kwargs = dict() + if 'full_output' in finish_args: + finish_kwargs['full_output'] = 1 + if 'disp' in finish_args: + finish_kwargs['disp'] = disp + elif 'options' in finish_args: + # pass 'disp' as `options` + # (e.g., if `finish` is `minimize`) + finish_kwargs['options'] = {'disp': disp} + + # run minimizer + res = finish(func, xmin, args=args, **finish_kwargs) + + if isinstance(res, OptimizeResult): + xmin = res.x + Jmin = res.fun + success = res.success + else: + xmin = res[0] + Jmin = res[1] + success = res[-1] == 0 + if not success: + if disp: + warnings.warn("Either final optimization did not succeed or `finish` " + "does not return `statuscode` as its last argument.", + RuntimeWarning, stacklevel=2) + + if full_output: + return xmin, Jmin, grid, Jout + else: + return xmin + + +class _Brute_Wrapper: + """ + Object to wrap user cost function for optimize.brute, allowing picklability + """ + + def __init__(self, f, args): + self.f = f + self.args = [] if args is None else args + + def __call__(self, x): + # flatten needed for one dimensional case. + return self.f(np.asarray(x).flatten(), *self.args) + + +def show_options(solver=None, method=None, disp=True): + """ + Show documentation for additional options of optimization solvers. + + These are method-specific options that can be supplied through the + ``options`` dict. + + Parameters + ---------- + solver : str + Type of optimization solver. One of 'minimize', 'minimize_scalar', + 'root', 'root_scalar', 'linprog', or 'quadratic_assignment'. + method : str, optional + If not given, shows all methods of the specified solver. Otherwise, + show only the options for the specified method. Valid values + corresponds to methods' names of respective solver (e.g., 'BFGS' for + 'minimize'). + disp : bool, optional + Whether to print the result rather than returning it. + + Returns + ------- + text + Either None (for disp=True) or the text string (disp=False) + + Notes + ----- + The solver-specific methods are: + + `scipy.optimize.minimize` + + - :ref:`Nelder-Mead ` + - :ref:`Powell ` + - :ref:`CG ` + - :ref:`BFGS ` + - :ref:`Newton-CG ` + - :ref:`L-BFGS-B ` + - :ref:`TNC ` + - :ref:`COBYLA ` + - :ref:`COBYQA ` + - :ref:`SLSQP ` + - :ref:`dogleg ` + - :ref:`trust-ncg ` + + `scipy.optimize.root` + + - :ref:`hybr ` + - :ref:`lm ` + - :ref:`broyden1 ` + - :ref:`broyden2 ` + - :ref:`anderson ` + - :ref:`linearmixing ` + - :ref:`diagbroyden ` + - :ref:`excitingmixing ` + - :ref:`krylov ` + - :ref:`df-sane ` + + `scipy.optimize.minimize_scalar` + + - :ref:`brent ` + - :ref:`golden ` + - :ref:`bounded ` + + `scipy.optimize.root_scalar` + + - :ref:`bisect ` + - :ref:`brentq ` + - :ref:`brenth ` + - :ref:`ridder ` + - :ref:`toms748 ` + - :ref:`newton ` + - :ref:`secant ` + - :ref:`halley ` + + `scipy.optimize.linprog` + + - :ref:`simplex ` + - :ref:`interior-point ` + - :ref:`revised simplex ` + - :ref:`highs ` + - :ref:`highs-ds ` + - :ref:`highs-ipm ` + + `scipy.optimize.quadratic_assignment` + + - :ref:`faq ` + - :ref:`2opt ` + + Examples + -------- + We can print documentations of a solver in stdout: + + >>> from scipy.optimize import show_options + >>> show_options(solver="minimize") + ... + + Specifying a method is possible: + + >>> show_options(solver="minimize", method="Nelder-Mead") + ... + + We can also get the documentations as a string: + + >>> show_options(solver="minimize", method="Nelder-Mead", disp=False) + Minimization of scalar function of one or more variables using the ... + + """ + import textwrap + + doc_routines = { + 'minimize': ( + ('bfgs', 'scipy.optimize._optimize._minimize_bfgs'), + ('cg', 'scipy.optimize._optimize._minimize_cg'), + ('cobyla', 'scipy.optimize._cobyla_py._minimize_cobyla'), + ('cobyqa', 'scipy.optimize._cobyqa_py._minimize_cobyqa'), + ('dogleg', 'scipy.optimize._trustregion_dogleg._minimize_dogleg'), + ('l-bfgs-b', 'scipy.optimize._lbfgsb_py._minimize_lbfgsb'), + ('nelder-mead', 'scipy.optimize._optimize._minimize_neldermead'), + ('newton-cg', 'scipy.optimize._optimize._minimize_newtoncg'), + ('powell', 'scipy.optimize._optimize._minimize_powell'), + ('slsqp', 'scipy.optimize._slsqp_py._minimize_slsqp'), + ('tnc', 'scipy.optimize._tnc._minimize_tnc'), + ('trust-ncg', + 'scipy.optimize._trustregion_ncg._minimize_trust_ncg'), + ('trust-constr', + 'scipy.optimize._trustregion_constr.' + '_minimize_trustregion_constr'), + ('trust-exact', + 'scipy.optimize._trustregion_exact._minimize_trustregion_exact'), + ('trust-krylov', + 'scipy.optimize._trustregion_krylov._minimize_trust_krylov'), + ), + 'root': ( + ('hybr', 'scipy.optimize._minpack_py._root_hybr'), + ('lm', 'scipy.optimize._root._root_leastsq'), + ('broyden1', 'scipy.optimize._root._root_broyden1_doc'), + ('broyden2', 'scipy.optimize._root._root_broyden2_doc'), + ('anderson', 'scipy.optimize._root._root_anderson_doc'), + ('diagbroyden', 'scipy.optimize._root._root_diagbroyden_doc'), + ('excitingmixing', 'scipy.optimize._root._root_excitingmixing_doc'), + ('linearmixing', 'scipy.optimize._root._root_linearmixing_doc'), + ('krylov', 'scipy.optimize._root._root_krylov_doc'), + ('df-sane', 'scipy.optimize._spectral._root_df_sane'), + ), + 'root_scalar': ( + ('bisect', 'scipy.optimize._root_scalar._root_scalar_bisect_doc'), + ('brentq', 'scipy.optimize._root_scalar._root_scalar_brentq_doc'), + ('brenth', 'scipy.optimize._root_scalar._root_scalar_brenth_doc'), + ('ridder', 'scipy.optimize._root_scalar._root_scalar_ridder_doc'), + ('toms748', 'scipy.optimize._root_scalar._root_scalar_toms748_doc'), + ('secant', 'scipy.optimize._root_scalar._root_scalar_secant_doc'), + ('newton', 'scipy.optimize._root_scalar._root_scalar_newton_doc'), + ('halley', 'scipy.optimize._root_scalar._root_scalar_halley_doc'), + ), + 'linprog': ( + ('simplex', 'scipy.optimize._linprog._linprog_simplex_doc'), + ('interior-point', 'scipy.optimize._linprog._linprog_ip_doc'), + ('revised simplex', 'scipy.optimize._linprog._linprog_rs_doc'), + ('highs-ipm', 'scipy.optimize._linprog._linprog_highs_ipm_doc'), + ('highs-ds', 'scipy.optimize._linprog._linprog_highs_ds_doc'), + ('highs', 'scipy.optimize._linprog._linprog_highs_doc'), + ), + 'quadratic_assignment': ( + ('faq', 'scipy.optimize._qap._quadratic_assignment_faq'), + ('2opt', 'scipy.optimize._qap._quadratic_assignment_2opt'), + ), + 'minimize_scalar': ( + ('brent', 'scipy.optimize._optimize._minimize_scalar_brent'), + ('bounded', 'scipy.optimize._optimize._minimize_scalar_bounded'), + ('golden', 'scipy.optimize._optimize._minimize_scalar_golden'), + ), + } + + if solver is None: + text = ["\n\n\n========\n", "minimize\n", "========\n"] + text.append(show_options('minimize', disp=False)) + text.extend(["\n\n===============\n", "minimize_scalar\n", + "===============\n"]) + text.append(show_options('minimize_scalar', disp=False)) + text.extend(["\n\n\n====\n", "root\n", + "====\n"]) + text.append(show_options('root', disp=False)) + text.extend(['\n\n\n=======\n', 'linprog\n', + '=======\n']) + text.append(show_options('linprog', disp=False)) + text = "".join(text) + else: + solver = solver.lower() + if solver not in doc_routines: + raise ValueError(f'Unknown solver {solver!r}') + + if method is None: + text = [] + for name, _ in doc_routines[solver]: + text.extend(["\n\n" + name, "\n" + "="*len(name) + "\n\n"]) + text.append(show_options(solver, name, disp=False)) + text = "".join(text) + else: + method = method.lower() + methods = dict(doc_routines[solver]) + if method not in methods: + raise ValueError(f"Unknown method {method!r}") + name = methods[method] + + # Import function object + parts = name.split('.') + mod_name = ".".join(parts[:-1]) + __import__(mod_name) + obj = getattr(sys.modules[mod_name], parts[-1]) + + # Get doc + doc = obj.__doc__ + if doc is not None: + text = textwrap.dedent(doc).strip() + else: + text = "" + + if disp: + print(text) + return + else: + return text diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_qap.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_qap.py new file mode 100644 index 0000000000000000000000000000000000000000..03fe3b128c066adb21406021ec2e34effbf65703 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_qap.py @@ -0,0 +1,760 @@ +import numpy as np +import operator +import warnings +import numbers +from . import (linear_sum_assignment, OptimizeResult) +from ._optimize import _check_unknown_options + +from scipy._lib._util import check_random_state +import itertools + +QUADRATIC_ASSIGNMENT_METHODS = ['faq', '2opt'] + + +def quadratic_assignment(A, B, method="faq", options=None): + r""" + Approximates solution to the quadratic assignment problem and + the graph matching problem. + + Quadratic assignment solves problems of the following form: + + .. math:: + + \min_P & \ {\ \text{trace}(A^T P B P^T)}\\ + \mbox{s.t. } & {P \ \epsilon \ \mathcal{P}}\\ + + where :math:`\mathcal{P}` is the set of all permutation matrices, + and :math:`A` and :math:`B` are square matrices. + + Graph matching tries to *maximize* the same objective function. + This algorithm can be thought of as finding the alignment of the + nodes of two graphs that minimizes the number of induced edge + disagreements, or, in the case of weighted graphs, the sum of squared + edge weight differences. + + Note that the quadratic assignment problem is NP-hard. The results given + here are approximations and are not guaranteed to be optimal. + + + Parameters + ---------- + A : 2-D array, square + The square matrix :math:`A` in the objective function above. + + B : 2-D array, square + The square matrix :math:`B` in the objective function above. + + method : str in {'faq', '2opt'} (default: 'faq') + The algorithm used to solve the problem. + :ref:`'faq' ` (default) and + :ref:`'2opt' ` are available. + + options : dict, optional + A dictionary of solver options. All solvers support the following: + + maximize : bool (default: False) + Maximizes the objective function if ``True``. + + partial_match : 2-D array of integers, optional (default: None) + Fixes part of the matching. Also known as a "seed" [2]_. + + Each row of `partial_match` specifies a pair of matched nodes: + node ``partial_match[i, 0]`` of `A` is matched to node + ``partial_match[i, 1]`` of `B`. The array has shape ``(m, 2)``, + where ``m`` is not greater than the number of nodes, :math:`n`. + + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + + .. versionchanged:: 1.15.0 + As part of the `SPEC-007 `_ + transition from use of `numpy.random.RandomState` to + `numpy.random.Generator` is occurring. Supplying + `np.random.RandomState` to this function will now emit a + `DeprecationWarning`. In SciPy 1.17 its use will raise an exception. + In addition relying on global state using `np.random.seed` + will emit a `FutureWarning`. In SciPy 1.17 the global random number + generator will no longer be used. + Use of an int-like seed will raise a `FutureWarning`, in SciPy 1.17 it + will be normalized via `np.random.default_rng` rather than + `np.random.RandomState`. + + For method-specific options, see + :func:`show_options('quadratic_assignment') `. + + Returns + ------- + res : OptimizeResult + `OptimizeResult` containing the following fields. + + col_ind : 1-D array + Column indices corresponding to the best permutation found of the + nodes of `B`. + fun : float + The objective value of the solution. + nit : int + The number of iterations performed during optimization. + + Notes + ----- + The default method :ref:`'faq' ` uses the Fast + Approximate QAP algorithm [1]_; it typically offers the best combination of + speed and accuracy. + Method :ref:`'2opt' ` can be computationally expensive, + but may be a useful alternative, or it can be used to refine the solution + returned by another method. + + References + ---------- + .. [1] J.T. Vogelstein, J.M. Conroy, V. Lyzinski, L.J. Podrazik, + S.G. Kratzer, E.T. Harley, D.E. Fishkind, R.J. Vogelstein, and + C.E. Priebe, "Fast approximate quadratic programming for graph + matching," PLOS one, vol. 10, no. 4, p. e0121002, 2015, + :doi:`10.1371/journal.pone.0121002` + + .. [2] D. Fishkind, S. Adali, H. Patsolic, L. Meng, D. Singh, V. Lyzinski, + C. Priebe, "Seeded graph matching", Pattern Recognit. 87 (2019): + 203-215, :doi:`10.1016/j.patcog.2018.09.014` + + .. [3] "2-opt," Wikipedia. + https://en.wikipedia.org/wiki/2-opt + + Examples + -------- + >>> import numpy as np + >>> from scipy.optimize import quadratic_assignment + >>> rng = np.random.default_rng() + >>> A = np.array([[0, 80, 150, 170], [80, 0, 130, 100], + ... [150, 130, 0, 120], [170, 100, 120, 0]]) + >>> B = np.array([[0, 5, 2, 7], [0, 0, 3, 8], + ... [0, 0, 0, 3], [0, 0, 0, 0]]) + >>> res = quadratic_assignment(A, B, options={'rng': rng}) + >>> print(res) + fun: 3260 + col_ind: [0 3 2 1] + nit: 9 + + The see the relationship between the returned ``col_ind`` and ``fun``, + use ``col_ind`` to form the best permutation matrix found, then evaluate + the objective function :math:`f(P) = trace(A^T P B P^T )`. + + >>> perm = res['col_ind'] + >>> P = np.eye(len(A), dtype=int)[perm] + >>> fun = np.trace(A.T @ P @ B @ P.T) + >>> print(fun) + 3260 + + Alternatively, to avoid constructing the permutation matrix explicitly, + directly permute the rows and columns of the distance matrix. + + >>> fun = np.trace(A.T @ B[perm][:, perm]) + >>> print(fun) + 3260 + + Although not guaranteed in general, ``quadratic_assignment`` happens to + have found the globally optimal solution. + + >>> from itertools import permutations + >>> perm_opt, fun_opt = None, np.inf + >>> for perm in permutations([0, 1, 2, 3]): + ... perm = np.array(perm) + ... fun = np.trace(A.T @ B[perm][:, perm]) + ... if fun < fun_opt: + ... fun_opt, perm_opt = fun, perm + >>> print(np.array_equal(perm_opt, res['col_ind'])) + True + + Here is an example for which the default method, + :ref:`'faq' `, does not find the global optimum. + + >>> A = np.array([[0, 5, 8, 6], [5, 0, 5, 1], + ... [8, 5, 0, 2], [6, 1, 2, 0]]) + >>> B = np.array([[0, 1, 8, 4], [1, 0, 5, 2], + ... [8, 5, 0, 5], [4, 2, 5, 0]]) + >>> res = quadratic_assignment(A, B, options={'rng': rng}) + >>> print(res) + fun: 178 + col_ind: [1 0 3 2] + nit: 13 + + If accuracy is important, consider using :ref:`'2opt' ` + to refine the solution. + + >>> guess = np.array([np.arange(len(A)), res.col_ind]).T + >>> res = quadratic_assignment(A, B, method="2opt", + ... options = {'rng': rng, 'partial_guess': guess}) + >>> print(res) + fun: 176 + col_ind: [1 2 3 0] + nit: 17 + + """ + + if options is None: + options = {} + + method = method.lower() + methods = {"faq": _quadratic_assignment_faq, + "2opt": _quadratic_assignment_2opt} + if method not in methods: + raise ValueError(f"method {method} must be in {methods}.") + + _spec007_transition(options.get("rng", None)) + res = methods[method](A, B, **options) + return res + + +def _spec007_transition(rng): + if isinstance(rng, np.random.RandomState): + warnings.warn( + "Use of `RandomState` with `quadratic_assignment` is deprecated" + " and will result in an exception in SciPy 1.17", + DeprecationWarning, + stacklevel=2 + ) + if ((rng is None or rng is np.random) and + np.random.mtrand._rand._bit_generator._seed_seq is None): + warnings.warn( + "The NumPy global RNG was seeded by calling `np.random.seed`." + " From SciPy 1.17, this function will no longer use the global RNG.", + FutureWarning, + stacklevel=2 + ) + if isinstance(rng, numbers.Integral | np.integer): + warnings.warn( + "The behavior when the rng option is an integer is changing: the value" + " will be normalized using np.random.default_rng beginning in SciPy 1.17," + " and the resulting Generator will be used to generate random numbers.", + FutureWarning, + stacklevel=2 + ) + + +def _calc_score(A, B, perm): + # equivalent to objective function but avoids matmul + return np.sum(A * B[perm][:, perm]) + + +def _common_input_validation(A, B, partial_match): + A = np.atleast_2d(A) + B = np.atleast_2d(B) + + if partial_match is None: + partial_match = np.array([[], []]).T + partial_match = np.atleast_2d(partial_match).astype(int) + + msg = None + if A.shape[0] != A.shape[1]: + msg = "`A` must be square" + elif B.shape[0] != B.shape[1]: + msg = "`B` must be square" + elif A.ndim != 2 or B.ndim != 2: + msg = "`A` and `B` must have exactly two dimensions" + elif A.shape != B.shape: + msg = "`A` and `B` matrices must be of equal size" + elif partial_match.shape[0] > A.shape[0]: + msg = "`partial_match` can have only as many seeds as there are nodes" + elif partial_match.shape[1] != 2: + msg = "`partial_match` must have two columns" + elif partial_match.ndim != 2: + msg = "`partial_match` must have exactly two dimensions" + elif (partial_match < 0).any(): + msg = "`partial_match` must contain only positive indices" + elif (partial_match >= len(A)).any(): + msg = "`partial_match` entries must be less than number of nodes" + elif (not len(set(partial_match[:, 0])) == len(partial_match[:, 0]) or + not len(set(partial_match[:, 1])) == len(partial_match[:, 1])): + msg = "`partial_match` column entries must be unique" + + if msg is not None: + raise ValueError(msg) + + return A, B, partial_match + + +def _quadratic_assignment_faq(A, B, + maximize=False, partial_match=None, rng=None, + P0="barycenter", shuffle_input=False, maxiter=30, + tol=0.03, **unknown_options): + r"""Solve the quadratic assignment problem (approximately). + + This function solves the Quadratic Assignment Problem (QAP) and the + Graph Matching Problem (GMP) using the Fast Approximate QAP Algorithm + (FAQ) [1]_. + + Quadratic assignment solves problems of the following form: + + .. math:: + + \min_P & \ {\ \text{trace}(A^T P B P^T)}\\ + \mbox{s.t. } & {P \ \epsilon \ \mathcal{P}}\\ + + where :math:`\mathcal{P}` is the set of all permutation matrices, + and :math:`A` and :math:`B` are square matrices. + + Graph matching tries to *maximize* the same objective function. + This algorithm can be thought of as finding the alignment of the + nodes of two graphs that minimizes the number of induced edge + disagreements, or, in the case of weighted graphs, the sum of squared + edge weight differences. + + Note that the quadratic assignment problem is NP-hard. The results given + here are approximations and are not guaranteed to be optimal. + + Parameters + ---------- + A : 2-D array, square + The square matrix :math:`A` in the objective function above. + B : 2-D array, square + The square matrix :math:`B` in the objective function above. + method : str in {'faq', '2opt'} (default: 'faq') + The algorithm used to solve the problem. This is the method-specific + documentation for 'faq'. + :ref:`'2opt' ` is also available. + + Options + ------- + maximize : bool (default: False) + Maximizes the objective function if ``True``. + partial_match : 2-D array of integers, optional (default: None) + Fixes part of the matching. Also known as a "seed" [2]_. + + Each row of `partial_match` specifies a pair of matched nodes: + node ``partial_match[i, 0]`` of `A` is matched to node + ``partial_match[i, 1]`` of `B`. The array has shape ``(m, 2)``, where + ``m`` is not greater than the number of nodes, :math:`n`. + + rng : {None, int, `numpy.random.Generator`}, optional + Pseudorandom number generator state. See `quadratic_assignment` for details. + P0 : 2-D array, "barycenter", or "randomized" (default: "barycenter") + Initial position. Must be a doubly-stochastic matrix [3]_. + + If the initial position is an array, it must be a doubly stochastic + matrix of size :math:`m' \times m'` where :math:`m' = n - m`. + + If ``"barycenter"`` (default), the initial position is the barycenter + of the Birkhoff polytope (the space of doubly stochastic matrices). + This is a :math:`m' \times m'` matrix with all entries equal to + :math:`1 / m'`. + + If ``"randomized"`` the initial search position is + :math:`P_0 = (J + K) / 2`, where :math:`J` is the barycenter and + :math:`K` is a random doubly stochastic matrix. + shuffle_input : bool (default: False) + Set to `True` to resolve degenerate gradients randomly. For + non-degenerate gradients this option has no effect. + maxiter : int, positive (default: 30) + Integer specifying the max number of Frank-Wolfe iterations performed. + tol : float (default: 0.03) + Tolerance for termination. Frank-Wolfe iteration terminates when + :math:`\frac{||P_{i}-P_{i+1}||_F}{\sqrt{m'}} \leq tol`, + where :math:`i` is the iteration number. + + Returns + ------- + res : OptimizeResult + `OptimizeResult` containing the following fields. + + col_ind : 1-D array + Column indices corresponding to the best permutation found of the + nodes of `B`. + fun : float + The objective value of the solution. + nit : int + The number of Frank-Wolfe iterations performed. + + Notes + ----- + The algorithm may be sensitive to the initial permutation matrix (or + search "position") due to the possibility of several local minima + within the feasible region. A barycenter initialization is more likely to + result in a better solution than a single random initialization. However, + calling ``quadratic_assignment`` several times with different random + initializations may result in a better optimum at the cost of longer + total execution time. + + Examples + -------- + As mentioned above, a barycenter initialization often results in a better + solution than a single random initialization. + + >>> from scipy.optimize import quadratic_assignment + >>> import numpy as np + >>> rng = np.random.default_rng() + >>> n = 15 + >>> A = rng.random((n, n)) + >>> B = rng.random((n, n)) + >>> options = {"rng": rng} + >>> res = quadratic_assignment(A, B, options=options) # FAQ is default method + >>> print(res.fun) + 47.797048706380636 # may vary + + >>> options = {"rng": rng, "P0": "randomized"} # use randomized initialization + >>> res = quadratic_assignment(A, B, options=options) + >>> print(res.fun) + 47.37287069769966 # may vary + + However, consider running from several randomized initializations and + keeping the best result. + + >>> res = min([quadratic_assignment(A, B, options=options) + ... for i in range(30)], key=lambda x: x.fun) + >>> print(res.fun) + 46.55974835248574 # may vary + + The '2-opt' method can be used to attempt to refine the results. + + >>> options = {"partial_guess": np.array([np.arange(n), res.col_ind]).T, "rng": rng} + >>> res = quadratic_assignment(A, B, method="2opt", options=options) + >>> print(res.fun) + 46.55974835248574 # may vary + + References + ---------- + .. [1] J.T. Vogelstein, J.M. Conroy, V. Lyzinski, L.J. Podrazik, + S.G. Kratzer, E.T. Harley, D.E. Fishkind, R.J. Vogelstein, and + C.E. Priebe, "Fast approximate quadratic programming for graph + matching," PLOS one, vol. 10, no. 4, p. e0121002, 2015, + :doi:`10.1371/journal.pone.0121002` + + .. [2] D. Fishkind, S. Adali, H. Patsolic, L. Meng, D. Singh, V. Lyzinski, + C. Priebe, "Seeded graph matching", Pattern Recognit. 87 (2019): + 203-215, :doi:`10.1016/j.patcog.2018.09.014` + + .. [3] "Doubly stochastic Matrix," Wikipedia. + https://en.wikipedia.org/wiki/Doubly_stochastic_matrix + + """ + + _check_unknown_options(unknown_options) + + maxiter = operator.index(maxiter) + + # ValueError check + A, B, partial_match = _common_input_validation(A, B, partial_match) + + msg = None + if isinstance(P0, str) and P0 not in {'barycenter', 'randomized'}: + msg = "Invalid 'P0' parameter string" + elif maxiter <= 0: + msg = "'maxiter' must be a positive integer" + elif tol <= 0: + msg = "'tol' must be a positive float" + if msg is not None: + raise ValueError(msg) + + rng = check_random_state(rng) + n = len(A) # number of vertices in graphs + n_seeds = len(partial_match) # number of seeds + n_unseed = n - n_seeds + + # [1] Algorithm 1 Line 1 - choose initialization + if not isinstance(P0, str): + P0 = np.atleast_2d(P0) + if P0.shape != (n_unseed, n_unseed): + msg = "`P0` matrix must have shape m' x m', where m'=n-m" + elif ((P0 < 0).any() or not np.allclose(np.sum(P0, axis=0), 1) + or not np.allclose(np.sum(P0, axis=1), 1)): + msg = "`P0` matrix must be doubly stochastic" + if msg is not None: + raise ValueError(msg) + elif P0 == 'barycenter': + P0 = np.ones((n_unseed, n_unseed)) / n_unseed + elif P0 == 'randomized': + J = np.ones((n_unseed, n_unseed)) / n_unseed + # generate a nxn matrix where each entry is a random number [0, 1] + # would use rand, but Generators don't have it + # would use random, but old mtrand.RandomStates don't have it + K = _doubly_stochastic(rng.uniform(size=(n_unseed, n_unseed))) + P0 = (J + K) / 2 + + # check trivial cases + if n == 0 or n_seeds == n: + score = _calc_score(A, B, partial_match[:, 1]) + res = {"col_ind": partial_match[:, 1], "fun": score, "nit": 0} + return OptimizeResult(res) + + obj_func_scalar = 1 + if maximize: + obj_func_scalar = -1 + + nonseed_B = np.setdiff1d(range(n), partial_match[:, 1]) + if shuffle_input: + nonseed_B = rng.permutation(nonseed_B) + + nonseed_A = np.setdiff1d(range(n), partial_match[:, 0]) + perm_A = np.concatenate([partial_match[:, 0], nonseed_A]) + perm_B = np.concatenate([partial_match[:, 1], nonseed_B]) + + # definitions according to Seeded Graph Matching [2]. + A11, A12, A21, A22 = _split_matrix(A[perm_A][:, perm_A], n_seeds) + B11, B12, B21, B22 = _split_matrix(B[perm_B][:, perm_B], n_seeds) + const_sum = A21 @ B21.T + A12.T @ B12 + + P = P0 + # [1] Algorithm 1 Line 2 - loop while stopping criteria not met + for n_iter in range(1, maxiter+1): + # [1] Algorithm 1 Line 3 - compute the gradient of f(P) = -tr(APB^tP^t) + grad_fp = (const_sum + A22 @ P @ B22.T + A22.T @ P @ B22) + # [1] Algorithm 1 Line 4 - get direction Q by solving Eq. 8 + _, cols = linear_sum_assignment(grad_fp, maximize=maximize) + Q = np.eye(n_unseed)[cols] + + # [1] Algorithm 1 Line 5 - compute the step size + # Noting that e.g. trace(Ax) = trace(A)*x, expand and re-collect + # terms as ax**2 + bx + c. c does not affect location of minimum + # and can be ignored. Also, note that trace(A@B) = (A.T*B).sum(); + # apply where possible for efficiency. + R = P - Q + b21 = ((R.T @ A21) * B21).sum() + b12 = ((R.T @ A12.T) * B12.T).sum() + AR22 = A22.T @ R + BR22 = B22 @ R.T + b22a = (AR22 * B22.T[cols]).sum() + b22b = (A22 * BR22[cols]).sum() + a = (AR22.T * BR22).sum() + b = b21 + b12 + b22a + b22b + # critical point of ax^2 + bx + c is at x = -d/(2*e) + # if a * obj_func_scalar > 0, it is a minimum + # if minimum is not in [0, 1], only endpoints need to be considered + if a*obj_func_scalar > 0 and 0 <= -b/(2*a) <= 1: + alpha = -b/(2*a) + else: + alpha = np.argmin([0, (b + a)*obj_func_scalar]) + + # [1] Algorithm 1 Line 6 - Update P + P_i1 = alpha * P + (1 - alpha) * Q + if np.linalg.norm(P - P_i1) / np.sqrt(n_unseed) < tol: + P = P_i1 + break + P = P_i1 + # [1] Algorithm 1 Line 7 - end main loop + + # [1] Algorithm 1 Line 8 - project onto the set of permutation matrices + _, col = linear_sum_assignment(P, maximize=True) + perm = np.concatenate((np.arange(n_seeds), col + n_seeds)) + + unshuffled_perm = np.zeros(n, dtype=int) + unshuffled_perm[perm_A] = perm_B[perm] + + score = _calc_score(A, B, unshuffled_perm) + res = {"col_ind": unshuffled_perm, "fun": score, "nit": n_iter} + return OptimizeResult(res) + + +def _split_matrix(X, n): + # definitions according to Seeded Graph Matching [2]. + upper, lower = X[:n], X[n:] + return upper[:, :n], upper[:, n:], lower[:, :n], lower[:, n:] + + +def _doubly_stochastic(P, tol=1e-3): + # Adapted from @btaba implementation + # https://github.com/btaba/sinkhorn_knopp + # of Sinkhorn-Knopp algorithm + # https://projecteuclid.org/euclid.pjm/1102992505 + + max_iter = 1000 + c = 1 / P.sum(axis=0) + r = 1 / (P @ c) + P_eps = P + + for it in range(max_iter): + if ((np.abs(P_eps.sum(axis=1) - 1) < tol).all() and + (np.abs(P_eps.sum(axis=0) - 1) < tol).all()): + # All column/row sums ~= 1 within threshold + break + + c = 1 / (r @ P) + r = 1 / (P @ c) + P_eps = r[:, None] * P * c + + return P_eps + + +def _quadratic_assignment_2opt(A, B, maximize=False, rng=None, + partial_match=None, + partial_guess=None, + **unknown_options): + r"""Solve the quadratic assignment problem (approximately). + + This function solves the Quadratic Assignment Problem (QAP) and the + Graph Matching Problem (GMP) using the 2-opt algorithm [1]_. + + Quadratic assignment solves problems of the following form: + + .. math:: + + \min_P & \ {\ \text{trace}(A^T P B P^T)}\\ + \mbox{s.t. } & {P \ \epsilon \ \mathcal{P}}\\ + + where :math:`\mathcal{P}` is the set of all permutation matrices, + and :math:`A` and :math:`B` are square matrices. + + Graph matching tries to *maximize* the same objective function. + This algorithm can be thought of as finding the alignment of the + nodes of two graphs that minimizes the number of induced edge + disagreements, or, in the case of weighted graphs, the sum of squared + edge weight differences. + + Note that the quadratic assignment problem is NP-hard. The results given + here are approximations and are not guaranteed to be optimal. + + Parameters + ---------- + A : 2-D array, square + The square matrix :math:`A` in the objective function above. + B : 2-D array, square + The square matrix :math:`B` in the objective function above. + method : str in {'faq', '2opt'} (default: 'faq') + The algorithm used to solve the problem. This is the method-specific + documentation for '2opt'. + :ref:`'faq' ` is also available. + + Options + ------- + maximize : bool (default: False) + Maximizes the objective function if ``True``. + rng : {None, int, `numpy.random.Generator`}, optional + Pseudorandom number generator state. See `quadratic_assignment` for details. + partial_match : 2-D array of integers, optional (default: None) + Fixes part of the matching. Also known as a "seed" [2]_. + + Each row of `partial_match` specifies a pair of matched nodes: node + ``partial_match[i, 0]`` of `A` is matched to node + ``partial_match[i, 1]`` of `B`. The array has shape ``(m, 2)``, + where ``m`` is not greater than the number of nodes, :math:`n`. + + .. note:: + `partial_match` must be sorted by the first column. + + partial_guess : 2-D array of integers, optional (default: None) + A guess for the matching between the two matrices. Unlike + `partial_match`, `partial_guess` does not fix the indices; they are + still free to be optimized. + + Each row of `partial_guess` specifies a pair of matched nodes: node + ``partial_guess[i, 0]`` of `A` is matched to node + ``partial_guess[i, 1]`` of `B`. The array has shape ``(m, 2)``, + where ``m`` is not greater than the number of nodes, :math:`n`. + + .. note:: + `partial_guess` must be sorted by the first column. + + Returns + ------- + res : OptimizeResult + `OptimizeResult` containing the following fields. + + col_ind : 1-D array + Column indices corresponding to the best permutation found of the + nodes of `B`. + fun : float + The objective value of the solution. + nit : int + The number of iterations performed during optimization. + + Notes + ----- + This is a greedy algorithm that works similarly to bubble sort: beginning + with an initial permutation, it iteratively swaps pairs of indices to + improve the objective function until no such improvements are possible. + + References + ---------- + .. [1] "2-opt," Wikipedia. + https://en.wikipedia.org/wiki/2-opt + + .. [2] D. Fishkind, S. Adali, H. Patsolic, L. Meng, D. Singh, V. Lyzinski, + C. Priebe, "Seeded graph matching", Pattern Recognit. 87 (2019): + 203-215, https://doi.org/10.1016/j.patcog.2018.09.014 + + """ + _check_unknown_options(unknown_options) + rng = check_random_state(rng) + A, B, partial_match = _common_input_validation(A, B, partial_match) + + N = len(A) + # check trivial cases + if N == 0 or partial_match.shape[0] == N: + score = _calc_score(A, B, partial_match[:, 1]) + res = {"col_ind": partial_match[:, 1], "fun": score, "nit": 0} + return OptimizeResult(res) + + if partial_guess is None: + partial_guess = np.array([[], []]).T + partial_guess = np.atleast_2d(partial_guess).astype(int) + + msg = None + if partial_guess.shape[0] > A.shape[0]: + msg = ("`partial_guess` can have only as " + "many entries as there are nodes") + elif partial_guess.shape[1] != 2: + msg = "`partial_guess` must have two columns" + elif partial_guess.ndim != 2: + msg = "`partial_guess` must have exactly two dimensions" + elif (partial_guess < 0).any(): + msg = "`partial_guess` must contain only positive indices" + elif (partial_guess >= len(A)).any(): + msg = "`partial_guess` entries must be less than number of nodes" + elif (not len(set(partial_guess[:, 0])) == len(partial_guess[:, 0]) or + not len(set(partial_guess[:, 1])) == len(partial_guess[:, 1])): + msg = "`partial_guess` column entries must be unique" + if msg is not None: + raise ValueError(msg) + + fixed_rows = None + if partial_match.size or partial_guess.size: + # use partial_match and partial_guess for initial permutation, + # but randomly permute the rest. + guess_rows = np.zeros(N, dtype=bool) + guess_cols = np.zeros(N, dtype=bool) + fixed_rows = np.zeros(N, dtype=bool) + fixed_cols = np.zeros(N, dtype=bool) + perm = np.zeros(N, dtype=int) + + rg, cg = partial_guess.T + guess_rows[rg] = True + guess_cols[cg] = True + perm[guess_rows] = cg + + # match overrides guess + rf, cf = partial_match.T + fixed_rows[rf] = True + fixed_cols[cf] = True + perm[fixed_rows] = cf + + random_rows = ~fixed_rows & ~guess_rows + random_cols = ~fixed_cols & ~guess_cols + perm[random_rows] = rng.permutation(np.arange(N)[random_cols]) + else: + perm = rng.permutation(np.arange(N)) + + best_score = _calc_score(A, B, perm) + + i_free = np.arange(N) + if fixed_rows is not None: + i_free = i_free[~fixed_rows] + + better = operator.gt if maximize else operator.lt + n_iter = 0 + done = False + while not done: + # equivalent to nested for loops i in range(N), j in range(i, N) + for i, j in itertools.combinations_with_replacement(i_free, 2): + n_iter += 1 + perm[i], perm[j] = perm[j], perm[i] + score = _calc_score(A, B, perm) + if better(score, best_score): + best_score = score + break + # faster to swap back than to create a new list every time + perm[i], perm[j] = perm[j], perm[i] + else: # no swaps made + done = True + + res = {"col_ind": perm, "fun": best_score, "nit": n_iter} + return OptimizeResult(res) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_remove_redundancy.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_remove_redundancy.py new file mode 100644 index 0000000000000000000000000000000000000000..cb81ad1696b768d2304b2fc42a80cc9678cbde00 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_remove_redundancy.py @@ -0,0 +1,522 @@ +""" +Routines for removing redundant (linearly dependent) equations from linear +programming equality constraints. +""" +# Author: Matt Haberland + +import numpy as np +from scipy.linalg import svd +from scipy.linalg.interpolative import interp_decomp +import scipy +from scipy.linalg.blas import dtrsm + + +def _row_count(A): + """ + Counts the number of nonzeros in each row of input array A. + Nonzeros are defined as any element with absolute value greater than + tol = 1e-13. This value should probably be an input to the function. + + Parameters + ---------- + A : 2-D array + An array representing a matrix + + Returns + ------- + rowcount : 1-D array + Number of nonzeros in each row of A + + """ + tol = 1e-13 + return np.array((abs(A) > tol).sum(axis=1)).flatten() + + +def _get_densest(A, eligibleRows): + """ + Returns the index of the densest row of A. Ignores rows that are not + eligible for consideration. + + Parameters + ---------- + A : 2-D array + An array representing a matrix + eligibleRows : 1-D logical array + Values indicate whether the corresponding row of A is eligible + to be considered + + Returns + ------- + i_densest : int + Index of the densest row in A eligible for consideration + + """ + rowCounts = _row_count(A) + return np.argmax(rowCounts * eligibleRows) + + +def _remove_zero_rows(A, b): + """ + Eliminates trivial equations from system of equations defined by Ax = b + and identifies trivial infeasibilities + + Parameters + ---------- + A : 2-D array + An array representing the left-hand side of a system of equations + b : 1-D array + An array representing the right-hand side of a system of equations + + Returns + ------- + A : 2-D array + An array representing the left-hand side of a system of equations + b : 1-D array + An array representing the right-hand side of a system of equations + status: int + An integer indicating the status of the removal operation + 0: No infeasibility identified + 2: Trivially infeasible + message : str + A string descriptor of the exit status of the optimization. + + """ + status = 0 + message = "" + i_zero = _row_count(A) == 0 + A = A[np.logical_not(i_zero), :] + if not np.allclose(b[i_zero], 0): + status = 2 + message = "There is a zero row in A_eq with a nonzero corresponding " \ + "entry in b_eq. The problem is infeasible." + b = b[np.logical_not(i_zero)] + return A, b, status, message + + +def bg_update_dense(plu, perm_r, v, j): + LU, p = plu + + vperm = v[perm_r] + u = dtrsm(1, LU, vperm, lower=1, diag=1) + LU[:j+1, j] = u[:j+1] + l = u[j+1:] + piv = LU[j, j] + LU[j+1:, j] += (l/piv) + return LU, p + + +def _remove_redundancy_pivot_dense(A, rhs, true_rank=None): + """ + Eliminates redundant equations from system of equations defined by Ax = b + and identifies infeasibilities. + + Parameters + ---------- + A : 2-D sparse matrix + An matrix representing the left-hand side of a system of equations + rhs : 1-D array + An array representing the right-hand side of a system of equations + + Returns + ------- + A : 2-D sparse matrix + A matrix representing the left-hand side of a system of equations + rhs : 1-D array + An array representing the right-hand side of a system of equations + status: int + An integer indicating the status of the system + 0: No infeasibility identified + 2: Trivially infeasible + message : str + A string descriptor of the exit status of the optimization. + + References + ---------- + .. [2] Andersen, Erling D. "Finding all linearly dependent rows in + large-scale linear programming." Optimization Methods and Software + 6.3 (1995): 219-227. + + """ + tolapiv = 1e-8 + tolprimal = 1e-8 + status = 0 + message = "" + inconsistent = ("There is a linear combination of rows of A_eq that " + "results in zero, suggesting a redundant constraint. " + "However the same linear combination of b_eq is " + "nonzero, suggesting that the constraints conflict " + "and the problem is infeasible.") + A, rhs, status, message = _remove_zero_rows(A, rhs) + + if status != 0: + return A, rhs, status, message + + m, n = A.shape + + v = list(range(m)) # Artificial column indices. + b = list(v) # Basis column indices. + # This is better as a list than a set because column order of basis matrix + # needs to be consistent. + d = [] # Indices of dependent rows + perm_r = None + + A_orig = A + A = np.zeros((m, m + n), order='F') + np.fill_diagonal(A, 1) + A[:, m:] = A_orig + e = np.zeros(m) + + js_candidates = np.arange(m, m+n, dtype=int) # candidate columns for basis + # manual masking was faster than masked array + js_mask = np.ones(js_candidates.shape, dtype=bool) + + # Implements basic algorithm from [2] + # Uses some of the suggested improvements (removing zero rows and + # Bartels-Golub update idea). + # Removing column singletons would be easy, but it is not as important + # because the procedure is performed only on the equality constraint + # matrix from the original problem - not on the canonical form matrix, + # which would have many more column singletons due to slack variables + # from the inequality constraints. + # The thoughts on "crashing" the initial basis are only really useful if + # the matrix is sparse. + + lu = np.eye(m, order='F'), np.arange(m) # initial LU is trivial + perm_r = lu[1] + for i in v: + + e[i] = 1 + if i > 0: + e[i-1] = 0 + + try: # fails for i==0 and any time it gets ill-conditioned + j = b[i-1] + lu = bg_update_dense(lu, perm_r, A[:, j], i-1) + except Exception: + lu = scipy.linalg.lu_factor(A[:, b]) + LU, p = lu + perm_r = list(range(m)) + for i1, i2 in enumerate(p): + perm_r[i1], perm_r[i2] = perm_r[i2], perm_r[i1] + + pi = scipy.linalg.lu_solve(lu, e, trans=1) + + js = js_candidates[js_mask] + batch = 50 + + # This is a tiny bit faster than looping over columns individually, + # like for j in js: if abs(A[:,j].transpose().dot(pi)) > tolapiv: + for j_index in range(0, len(js), batch): + j_indices = js[j_index: min(j_index+batch, len(js))] + + c = abs(A[:, j_indices].transpose().dot(pi)) + if (c > tolapiv).any(): + j = js[j_index + np.argmax(c)] # very independent column + b[i] = j + js_mask[j-m] = False + break + else: + bibar = pi.T.dot(rhs.reshape(-1, 1)) + bnorm = np.linalg.norm(rhs) + if abs(bibar)/(1+bnorm) > tolprimal: # inconsistent + status = 2 + message = inconsistent + return A_orig, rhs, status, message + else: # dependent + d.append(i) + if true_rank is not None and len(d) == m - true_rank: + break # found all redundancies + + keep = set(range(m)) + keep = list(keep - set(d)) + return A_orig[keep, :], rhs[keep], status, message + + +def _remove_redundancy_pivot_sparse(A, rhs): + """ + Eliminates redundant equations from system of equations defined by Ax = b + and identifies infeasibilities. + + Parameters + ---------- + A : 2-D sparse matrix + An matrix representing the left-hand side of a system of equations + rhs : 1-D array + An array representing the right-hand side of a system of equations + + Returns + ------- + A : 2-D sparse matrix + A matrix representing the left-hand side of a system of equations + rhs : 1-D array + An array representing the right-hand side of a system of equations + status: int + An integer indicating the status of the system + 0: No infeasibility identified + 2: Trivially infeasible + message : str + A string descriptor of the exit status of the optimization. + + References + ---------- + .. [2] Andersen, Erling D. "Finding all linearly dependent rows in + large-scale linear programming." Optimization Methods and Software + 6.3 (1995): 219-227. + + """ + + tolapiv = 1e-8 + tolprimal = 1e-8 + status = 0 + message = "" + inconsistent = ("There is a linear combination of rows of A_eq that " + "results in zero, suggesting a redundant constraint. " + "However the same linear combination of b_eq is " + "nonzero, suggesting that the constraints conflict " + "and the problem is infeasible.") + A, rhs, status, message = _remove_zero_rows(A, rhs) + + if status != 0: + return A, rhs, status, message + + m, n = A.shape + + v = list(range(m)) # Artificial column indices. + b = list(v) # Basis column indices. + # This is better as a list than a set because column order of basis matrix + # needs to be consistent. + k = set(range(m, m+n)) # Structural column indices. + d = [] # Indices of dependent rows + + A_orig = A + A = scipy.sparse.hstack((scipy.sparse.eye(m), A)).tocsc() + e = np.zeros(m) + + # Implements basic algorithm from [2] + # Uses only one of the suggested improvements (removing zero rows). + # Removing column singletons would be easy, but it is not as important + # because the procedure is performed only on the equality constraint + # matrix from the original problem - not on the canonical form matrix, + # which would have many more column singletons due to slack variables + # from the inequality constraints. + # The thoughts on "crashing" the initial basis sound useful, but the + # description of the procedure seems to assume a lot of familiarity with + # the subject; it is not very explicit. I already went through enough + # trouble getting the basic algorithm working, so I was not interested in + # trying to decipher this, too. (Overall, the paper is fraught with + # mistakes and ambiguities - which is strange, because the rest of + # Andersen's papers are quite good.) + # I tried and tried and tried to improve performance using the + # Bartels-Golub update. It works, but it's only practical if the LU + # factorization can be specialized as described, and that is not possible + # until the SciPy SuperLU interface permits control over column + # permutation - see issue #7700. + + for i in v: + B = A[:, b] + + e[i] = 1 + if i > 0: + e[i-1] = 0 + + pi = scipy.sparse.linalg.spsolve(B.transpose(), e).reshape(-1, 1) + + js = list(k-set(b)) # not efficient, but this is not the time sink... + + # Due to overhead, it tends to be faster (for problems tested) to + # compute the full matrix-vector product rather than individual + # vector-vector products (with the chance of terminating as soon + # as any are nonzero). For very large matrices, it might be worth + # it to compute, say, 100 or 1000 at a time and stop when a nonzero + # is found. + + c = (np.abs(A[:, js].transpose().dot(pi)) > tolapiv).nonzero()[0] + if len(c) > 0: # independent + j = js[c[0]] + # in a previous commit, the previous line was changed to choose + # index j corresponding with the maximum dot product. + # While this avoided issues with almost + # singular matrices, it slowed the routine in most NETLIB tests. + # I think this is because these columns were denser than the + # first column with nonzero dot product (c[0]). + # It would be nice to have a heuristic that balances sparsity with + # high dot product, but I don't think it's worth the time to + # develop one right now. Bartels-Golub update is a much higher + # priority. + b[i] = j # replace artificial column + else: + bibar = pi.T.dot(rhs.reshape(-1, 1)) + bnorm = np.linalg.norm(rhs) + if abs(bibar)/(1 + bnorm) > tolprimal: + status = 2 + message = inconsistent + return A_orig, rhs, status, message + else: # dependent + d.append(i) + + keep = set(range(m)) + keep = list(keep - set(d)) + return A_orig[keep, :], rhs[keep], status, message + + +def _remove_redundancy_svd(A, b): + """ + Eliminates redundant equations from system of equations defined by Ax = b + and identifies infeasibilities. + + Parameters + ---------- + A : 2-D array + An array representing the left-hand side of a system of equations + b : 1-D array + An array representing the right-hand side of a system of equations + + Returns + ------- + A : 2-D array + An array representing the left-hand side of a system of equations + b : 1-D array + An array representing the right-hand side of a system of equations + status: int + An integer indicating the status of the system + 0: No infeasibility identified + 2: Trivially infeasible + message : str + A string descriptor of the exit status of the optimization. + + References + ---------- + .. [2] Andersen, Erling D. "Finding all linearly dependent rows in + large-scale linear programming." Optimization Methods and Software + 6.3 (1995): 219-227. + + """ + + A, b, status, message = _remove_zero_rows(A, b) + + if status != 0: + return A, b, status, message + + U, s, Vh = svd(A) + eps = np.finfo(float).eps + tol = s.max() * max(A.shape) * eps + + m, n = A.shape + s_min = s[-1] if m <= n else 0 + + # this algorithm is faster than that of [2] when the nullspace is small + # but it could probably be improvement by randomized algorithms and with + # a sparse implementation. + # it relies on repeated singular value decomposition to find linearly + # dependent rows (as identified by columns of U that correspond with zero + # singular values). Unfortunately, only one row can be removed per + # decomposition (I tried otherwise; doing so can cause problems.) + # It would be nice if we could do truncated SVD like sp.sparse.linalg.svds + # but that function is unreliable at finding singular values near zero. + # Finding max eigenvalue L of A A^T, then largest eigenvalue (and + # associated eigenvector) of -A A^T + L I (I is identity) via power + # iteration would also work in theory, but is only efficient if the + # smallest nonzero eigenvalue of A A^T is close to the largest nonzero + # eigenvalue. + + while abs(s_min) < tol: + v = U[:, -1] # TODO: return these so user can eliminate from problem? + # rows need to be represented in significant amount + eligibleRows = np.abs(v) > tol * 10e6 + if not np.any(eligibleRows) or np.any(np.abs(v.dot(A)) > tol): + status = 4 + message = ("Due to numerical issues, redundant equality " + "constraints could not be removed automatically. " + "Try providing your constraint matrices as sparse " + "matrices to activate sparse presolve, try turning " + "off redundancy removal, or try turning off presolve " + "altogether.") + break + if np.any(np.abs(v.dot(b)) > tol * 100): # factor of 100 to fix 10038 and 10349 + status = 2 + message = ("There is a linear combination of rows of A_eq that " + "results in zero, suggesting a redundant constraint. " + "However the same linear combination of b_eq is " + "nonzero, suggesting that the constraints conflict " + "and the problem is infeasible.") + break + + i_remove = _get_densest(A, eligibleRows) + A = np.delete(A, i_remove, axis=0) + b = np.delete(b, i_remove) + U, s, Vh = svd(A) + m, n = A.shape + s_min = s[-1] if m <= n else 0 + + return A, b, status, message + + +def _remove_redundancy_id(A, rhs, rank=None, randomized=True): + """Eliminates redundant equations from a system of equations. + + Eliminates redundant equations from system of equations defined by Ax = b + and identifies infeasibilities. + + Parameters + ---------- + A : 2-D array + An array representing the left-hand side of a system of equations + rhs : 1-D array + An array representing the right-hand side of a system of equations + rank : int, optional + The rank of A + randomized: bool, optional + True for randomized interpolative decomposition + + Returns + ------- + A : 2-D array + An array representing the left-hand side of a system of equations + rhs : 1-D array + An array representing the right-hand side of a system of equations + status: int + An integer indicating the status of the system + 0: No infeasibility identified + 2: Trivially infeasible + message : str + A string descriptor of the exit status of the optimization. + + """ + + status = 0 + message = "" + inconsistent = ("There is a linear combination of rows of A_eq that " + "results in zero, suggesting a redundant constraint. " + "However the same linear combination of b_eq is " + "nonzero, suggesting that the constraints conflict " + "and the problem is infeasible.") + + A, rhs, status, message = _remove_zero_rows(A, rhs) + + if status != 0: + return A, rhs, status, message + + m, n = A.shape + + k = rank + if rank is None: + k = np.linalg.matrix_rank(A) + + idx, proj = interp_decomp(A.T, k, rand=randomized) + + # first k entries in idx are indices of the independent rows + # remaining entries are the indices of the m-k dependent rows + # proj provides a linear combinations of rows of A2 that form the + # remaining m-k (dependent) rows. The same linear combination of entries + # in rhs2 must give the remaining m-k entries. If not, the system is + # inconsistent, and the problem is infeasible. + if not np.allclose(rhs[idx[:k]] @ proj, rhs[idx[k:]]): + status = 2 + message = inconsistent + + # sort indices because the other redundancy removal routines leave rows + # in original order and tests were written with that in mind + idx = sorted(idx[:k]) + A2 = A[idx, :] + rhs2 = rhs[idx] + return A2, rhs2, status, message diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_root.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_root.py new file mode 100644 index 0000000000000000000000000000000000000000..fdf5a0cf409392d4c3ece21f23ad2e061860c699 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_root.py @@ -0,0 +1,732 @@ +""" +Unified interfaces to root finding algorithms. + +Functions +--------- +- root : find a root of a vector function. +""" +__all__ = ['root'] + +import numpy as np + +from warnings import warn + +from ._optimize import MemoizeJac, OptimizeResult, _check_unknown_options +from ._minpack_py import _root_hybr, leastsq +from ._spectral import _root_df_sane +from . import _nonlin as nonlin + + +ROOT_METHODS = ['hybr', 'lm', 'broyden1', 'broyden2', 'anderson', + 'linearmixing', 'diagbroyden', 'excitingmixing', 'krylov', + 'df-sane'] + + +def root(fun, x0, args=(), method='hybr', jac=None, tol=None, callback=None, + options=None): + r""" + Find a root of a vector function. + + Parameters + ---------- + fun : callable + A vector function to find a root of. + + Suppose the callable has signature ``f0(x, *my_args, **my_kwargs)``, where + ``my_args`` and ``my_kwargs`` are required positional and keyword arguments. + Rather than passing ``f0`` as the callable, wrap it to accept + only ``x``; e.g., pass ``fun=lambda x: f0(x, *my_args, **my_kwargs)`` as the + callable, where ``my_args`` (tuple) and ``my_kwargs`` (dict) have been + gathered before invoking this function. + x0 : ndarray + Initial guess. + args : tuple, optional + Extra arguments passed to the objective function and its Jacobian. + method : str, optional + Type of solver. Should be one of + + - 'hybr' :ref:`(see here) ` + - 'lm' :ref:`(see here) ` + - 'broyden1' :ref:`(see here) ` + - 'broyden2' :ref:`(see here) ` + - 'anderson' :ref:`(see here) ` + - 'linearmixing' :ref:`(see here) ` + - 'diagbroyden' :ref:`(see here) ` + - 'excitingmixing' :ref:`(see here) ` + - 'krylov' :ref:`(see here) ` + - 'df-sane' :ref:`(see here) ` + + jac : bool or callable, optional + If `jac` is a Boolean and is True, `fun` is assumed to return the + value of Jacobian along with the objective function. If False, the + Jacobian will be estimated numerically. + `jac` can also be a callable returning the Jacobian of `fun`. In + this case, it must accept the same arguments as `fun`. + tol : float, optional + Tolerance for termination. For detailed control, use solver-specific + options. + callback : function, optional + Optional callback function. It is called on every iteration as + ``callback(x, f)`` where `x` is the current solution and `f` + the corresponding residual. For all methods but 'hybr' and 'lm'. + options : dict, optional + A dictionary of solver options. E.g., `xtol` or `maxiter`, see + :obj:`show_options()` for details. + + Returns + ------- + sol : OptimizeResult + The solution represented as a ``OptimizeResult`` object. + Important attributes are: ``x`` the solution array, ``success`` a + Boolean flag indicating if the algorithm exited successfully and + ``message`` which describes the cause of the termination. See + `OptimizeResult` for a description of other attributes. + + See also + -------- + show_options : Additional options accepted by the solvers + + Notes + ----- + This section describes the available solvers that can be selected by the + 'method' parameter. The default method is *hybr*. + + Method *hybr* uses a modification of the Powell hybrid method as + implemented in MINPACK [1]_. + + Method *lm* solves the system of nonlinear equations in a least squares + sense using a modification of the Levenberg-Marquardt algorithm as + implemented in MINPACK [1]_. + + Method *df-sane* is a derivative-free spectral method. [3]_ + + Methods *broyden1*, *broyden2*, *anderson*, *linearmixing*, + *diagbroyden*, *excitingmixing*, *krylov* are inexact Newton methods, + with backtracking or full line searches [2]_. Each method corresponds + to a particular Jacobian approximations. + + - Method *broyden1* uses Broyden's first Jacobian approximation, it is + known as Broyden's good method. + - Method *broyden2* uses Broyden's second Jacobian approximation, it + is known as Broyden's bad method. + - Method *anderson* uses (extended) Anderson mixing. + - Method *Krylov* uses Krylov approximation for inverse Jacobian. It + is suitable for large-scale problem. + - Method *diagbroyden* uses diagonal Broyden Jacobian approximation. + - Method *linearmixing* uses a scalar Jacobian approximation. + - Method *excitingmixing* uses a tuned diagonal Jacobian + approximation. + + .. warning:: + + The algorithms implemented for methods *diagbroyden*, + *linearmixing* and *excitingmixing* may be useful for specific + problems, but whether they will work may depend strongly on the + problem. + + .. versionadded:: 0.11.0 + + References + ---------- + .. [1] More, Jorge J., Burton S. Garbow, and Kenneth E. Hillstrom. + 1980. User Guide for MINPACK-1. + .. [2] C. T. Kelley. 1995. Iterative Methods for Linear and Nonlinear + Equations. Society for Industrial and Applied Mathematics. + + .. [3] W. La Cruz, J.M. Martinez, M. Raydan. Math. Comp. 75, 1429 (2006). + + Examples + -------- + The following functions define a system of nonlinear equations and its + jacobian. + + >>> import numpy as np + >>> def fun(x): + ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, + ... 0.5 * (x[1] - x[0])**3 + x[1]] + + >>> def jac(x): + ... return np.array([[1 + 1.5 * (x[0] - x[1])**2, + ... -1.5 * (x[0] - x[1])**2], + ... [-1.5 * (x[1] - x[0])**2, + ... 1 + 1.5 * (x[1] - x[0])**2]]) + + A solution can be obtained as follows. + + >>> from scipy import optimize + >>> sol = optimize.root(fun, [0, 0], jac=jac, method='hybr') + >>> sol.x + array([ 0.8411639, 0.1588361]) + + **Large problem** + + Suppose that we needed to solve the following integrodifferential + equation on the square :math:`[0,1]\times[0,1]`: + + .. math:: + + \nabla^2 P = 10 \left(\int_0^1\int_0^1\cosh(P)\,dx\,dy\right)^2 + + with :math:`P(x,1) = 1` and :math:`P=0` elsewhere on the boundary of + the square. + + The solution can be found using the ``method='krylov'`` solver: + + >>> from scipy import optimize + >>> # parameters + >>> nx, ny = 75, 75 + >>> hx, hy = 1./(nx-1), 1./(ny-1) + + >>> P_left, P_right = 0, 0 + >>> P_top, P_bottom = 1, 0 + + >>> def residual(P): + ... d2x = np.zeros_like(P) + ... d2y = np.zeros_like(P) + ... + ... d2x[1:-1] = (P[2:] - 2*P[1:-1] + P[:-2]) / hx/hx + ... d2x[0] = (P[1] - 2*P[0] + P_left)/hx/hx + ... d2x[-1] = (P_right - 2*P[-1] + P[-2])/hx/hx + ... + ... d2y[:,1:-1] = (P[:,2:] - 2*P[:,1:-1] + P[:,:-2])/hy/hy + ... d2y[:,0] = (P[:,1] - 2*P[:,0] + P_bottom)/hy/hy + ... d2y[:,-1] = (P_top - 2*P[:,-1] + P[:,-2])/hy/hy + ... + ... return d2x + d2y - 10*np.cosh(P).mean()**2 + + >>> guess = np.zeros((nx, ny), float) + >>> sol = optimize.root(residual, guess, method='krylov') + >>> print('Residual: %g' % abs(residual(sol.x)).max()) + Residual: 5.7972e-06 # may vary + + >>> import matplotlib.pyplot as plt + >>> x, y = np.mgrid[0:1:(nx*1j), 0:1:(ny*1j)] + >>> plt.pcolormesh(x, y, sol.x, shading='gouraud') + >>> plt.colorbar() + >>> plt.show() + + """ + def _wrapped_fun(*fargs): + """ + Wrapped `func` to track the number of times + the function has been called. + """ + _wrapped_fun.nfev += 1 + return fun(*fargs) + + _wrapped_fun.nfev = 0 + + if not isinstance(args, tuple): + args = (args,) + + meth = method.lower() + if options is None: + options = {} + + if callback is not None and meth in ('hybr', 'lm'): + warn(f'Method {method} does not accept callback.', + RuntimeWarning, stacklevel=2) + + # fun also returns the Jacobian + if not callable(jac) and meth in ('hybr', 'lm'): + if bool(jac): + fun = MemoizeJac(fun) + jac = fun.derivative + else: + jac = None + + # set default tolerances + if tol is not None: + options = dict(options) + if meth in ('hybr', 'lm'): + options.setdefault('xtol', tol) + elif meth in ('df-sane',): + options.setdefault('ftol', tol) + elif meth in ('broyden1', 'broyden2', 'anderson', 'linearmixing', + 'diagbroyden', 'excitingmixing', 'krylov'): + options.setdefault('xtol', tol) + options.setdefault('xatol', np.inf) + options.setdefault('ftol', np.inf) + options.setdefault('fatol', np.inf) + + if meth == 'hybr': + sol = _root_hybr(_wrapped_fun, x0, args=args, jac=jac, **options) + elif meth == 'lm': + sol = _root_leastsq(_wrapped_fun, x0, args=args, jac=jac, **options) + elif meth == 'df-sane': + _warn_jac_unused(jac, method) + sol = _root_df_sane(_wrapped_fun, x0, args=args, callback=callback, + **options) + elif meth in ('broyden1', 'broyden2', 'anderson', 'linearmixing', + 'diagbroyden', 'excitingmixing', 'krylov'): + _warn_jac_unused(jac, method) + sol = _root_nonlin_solve(_wrapped_fun, x0, args=args, jac=jac, + _method=meth, _callback=callback, + **options) + else: + raise ValueError(f'Unknown solver {method}') + + sol.nfev = _wrapped_fun.nfev + return sol + + +def _warn_jac_unused(jac, method): + if jac is not None: + warn(f'Method {method} does not use the jacobian (jac).', + RuntimeWarning, stacklevel=2) + + +def _root_leastsq(fun, x0, args=(), jac=None, + col_deriv=0, xtol=1.49012e-08, ftol=1.49012e-08, + gtol=0.0, maxiter=0, eps=0.0, factor=100, diag=None, + **unknown_options): + """ + Solve for least squares with Levenberg-Marquardt + + Options + ------- + col_deriv : bool + non-zero to specify that the Jacobian function computes derivatives + down the columns (faster, because there is no transpose operation). + ftol : float + Relative error desired in the sum of squares. + xtol : float + Relative error desired in the approximate solution. + gtol : float + Orthogonality desired between the function vector and the columns + of the Jacobian. + maxiter : int + The maximum number of calls to the function. If zero, then + 100*(N+1) is the maximum where N is the number of elements in x0. + eps : float + A suitable step length for the forward-difference approximation of + the Jacobian (for Dfun=None). If `eps` is less than the machine + precision, it is assumed that the relative errors in the functions + are of the order of the machine precision. + factor : float + A parameter determining the initial step bound + (``factor * || diag * x||``). Should be in interval ``(0.1, 100)``. + diag : sequence + N positive entries that serve as a scale factors for the variables. + """ + nfev = 0 + def _wrapped_fun(*fargs): + """ + Wrapped `func` to track the number of times + the function has been called. + """ + nonlocal nfev + nfev += 1 + return fun(*fargs) + + _check_unknown_options(unknown_options) + x, cov_x, info, msg, ier = leastsq(_wrapped_fun, x0, args=args, + Dfun=jac, full_output=True, + col_deriv=col_deriv, xtol=xtol, + ftol=ftol, gtol=gtol, + maxfev=maxiter, epsfcn=eps, + factor=factor, diag=diag) + sol = OptimizeResult(x=x, message=msg, status=ier, + success=ier in (1, 2, 3, 4), cov_x=cov_x, + fun=info.pop('fvec'), method="lm") + sol.update(info) + sol.nfev = nfev + return sol + + +def _root_nonlin_solve(fun, x0, args=(), jac=None, + _callback=None, _method=None, + nit=None, disp=False, maxiter=None, + ftol=None, fatol=None, xtol=None, xatol=None, + tol_norm=None, line_search='armijo', jac_options=None, + **unknown_options): + _check_unknown_options(unknown_options) + + f_tol = fatol + f_rtol = ftol + x_tol = xatol + x_rtol = xtol + verbose = disp + if jac_options is None: + jac_options = dict() + + jacobian = {'broyden1': nonlin.BroydenFirst, + 'broyden2': nonlin.BroydenSecond, + 'anderson': nonlin.Anderson, + 'linearmixing': nonlin.LinearMixing, + 'diagbroyden': nonlin.DiagBroyden, + 'excitingmixing': nonlin.ExcitingMixing, + 'krylov': nonlin.KrylovJacobian + }[_method] + + if args: + if jac is True: + def f(x): + return fun(x, *args)[0] + else: + def f(x): + return fun(x, *args) + else: + f = fun + + x, info = nonlin.nonlin_solve(f, x0, jacobian=jacobian(**jac_options), + iter=nit, verbose=verbose, + maxiter=maxiter, f_tol=f_tol, + f_rtol=f_rtol, x_tol=x_tol, + x_rtol=x_rtol, tol_norm=tol_norm, + line_search=line_search, + callback=_callback, full_output=True, + raise_exception=False) + sol = OptimizeResult(x=x, method=_method) + sol.update(info) + return sol + +def _root_broyden1_doc(): + """ + Options + ------- + nit : int, optional + Number of iterations to make. If omitted (default), make as many + as required to meet tolerances. + disp : bool, optional + Print status to stdout on every iteration. + maxiter : int, optional + Maximum number of iterations to make. + ftol : float, optional + Relative tolerance for the residual. If omitted, not used. + fatol : float, optional + Absolute tolerance (in max-norm) for the residual. + If omitted, default is 6e-6. + xtol : float, optional + Relative minimum step size. If omitted, not used. + xatol : float, optional + Absolute minimum step size, as determined from the Jacobian + approximation. If the step size is smaller than this, optimization + is terminated as successful. If omitted, not used. + tol_norm : function(vector) -> scalar, optional + Norm to use in convergence check. Default is the maximum norm. + line_search : {None, 'armijo' (default), 'wolfe'}, optional + Which type of a line search to use to determine the step size in + the direction given by the Jacobian approximation. Defaults to + 'armijo'. + jac_options : dict, optional + Options for the respective Jacobian approximation. + + alpha : float, optional + Initial guess for the Jacobian is (-1/alpha). + reduction_method : str or tuple, optional + Method used in ensuring that the rank of the Broyden + matrix stays low. Can either be a string giving the + name of the method, or a tuple of the form ``(method, + param1, param2, ...)`` that gives the name of the + method and values for additional parameters. + + Methods available: + + - ``restart``: drop all matrix columns. Has no extra parameters. + - ``simple``: drop oldest matrix column. Has no extra parameters. + - ``svd``: keep only the most significant SVD components. + Takes an extra parameter, ``to_retain``, which determines the + number of SVD components to retain when rank reduction is done. + Default is ``max_rank - 2``. + + max_rank : int, optional + Maximum rank for the Broyden matrix. + Default is infinity (i.e., no rank reduction). + + Examples + -------- + >>> def func(x): + ... return np.cos(x) + x[::-1] - [1, 2, 3, 4] + ... + >>> from scipy import optimize + >>> res = optimize.root(func, [1, 1, 1, 1], method='broyden1', tol=1e-14) + >>> x = res.x + >>> x + array([4.04674914, 3.91158389, 2.71791677, 1.61756251]) + >>> np.cos(x) + x[::-1] + array([1., 2., 3., 4.]) + + """ + pass + + +def _root_broyden2_doc(): + """ + Options + ------- + nit : int, optional + Number of iterations to make. If omitted (default), make as many + as required to meet tolerances. + disp : bool, optional + Print status to stdout on every iteration. + maxiter : int, optional + Maximum number of iterations to make. + ftol : float, optional + Relative tolerance for the residual. If omitted, not used. + fatol : float, optional + Absolute tolerance (in max-norm) for the residual. + If omitted, default is 6e-6. + xtol : float, optional + Relative minimum step size. If omitted, not used. + xatol : float, optional + Absolute minimum step size, as determined from the Jacobian + approximation. If the step size is smaller than this, optimization + is terminated as successful. If omitted, not used. + tol_norm : function(vector) -> scalar, optional + Norm to use in convergence check. Default is the maximum norm. + line_search : {None, 'armijo' (default), 'wolfe'}, optional + Which type of a line search to use to determine the step size in + the direction given by the Jacobian approximation. Defaults to + 'armijo'. + jac_options : dict, optional + Options for the respective Jacobian approximation. + + alpha : float, optional + Initial guess for the Jacobian is (-1/alpha). + reduction_method : str or tuple, optional + Method used in ensuring that the rank of the Broyden + matrix stays low. Can either be a string giving the + name of the method, or a tuple of the form ``(method, + param1, param2, ...)`` that gives the name of the + method and values for additional parameters. + + Methods available: + + - ``restart``: drop all matrix columns. Has no extra parameters. + - ``simple``: drop oldest matrix column. Has no extra parameters. + - ``svd``: keep only the most significant SVD components. + Takes an extra parameter, ``to_retain``, which determines the + number of SVD components to retain when rank reduction is done. + Default is ``max_rank - 2``. + + max_rank : int, optional + Maximum rank for the Broyden matrix. + Default is infinity (i.e., no rank reduction). + """ + pass + + +def _root_anderson_doc(): + """ + Options + ------- + nit : int, optional + Number of iterations to make. If omitted (default), make as many + as required to meet tolerances. + disp : bool, optional + Print status to stdout on every iteration. + maxiter : int, optional + Maximum number of iterations to make. + ftol : float, optional + Relative tolerance for the residual. If omitted, not used. + fatol : float, optional + Absolute tolerance (in max-norm) for the residual. + If omitted, default is 6e-6. + xtol : float, optional + Relative minimum step size. If omitted, not used. + xatol : float, optional + Absolute minimum step size, as determined from the Jacobian + approximation. If the step size is smaller than this, optimization + is terminated as successful. If omitted, not used. + tol_norm : function(vector) -> scalar, optional + Norm to use in convergence check. Default is the maximum norm. + line_search : {None, 'armijo' (default), 'wolfe'}, optional + Which type of a line search to use to determine the step size in + the direction given by the Jacobian approximation. Defaults to + 'armijo'. + jac_options : dict, optional + Options for the respective Jacobian approximation. + + alpha : float, optional + Initial guess for the Jacobian is (-1/alpha). + M : float, optional + Number of previous vectors to retain. Defaults to 5. + w0 : float, optional + Regularization parameter for numerical stability. + Compared to unity, good values of the order of 0.01. + """ + pass + +def _root_linearmixing_doc(): + """ + Options + ------- + nit : int, optional + Number of iterations to make. If omitted (default), make as many + as required to meet tolerances. + disp : bool, optional + Print status to stdout on every iteration. + maxiter : int, optional + Maximum number of iterations to make. + ftol : float, optional + Relative tolerance for the residual. If omitted, not used. + fatol : float, optional + Absolute tolerance (in max-norm) for the residual. + If omitted, default is 6e-6. + xtol : float, optional + Relative minimum step size. If omitted, not used. + xatol : float, optional + Absolute minimum step size, as determined from the Jacobian + approximation. If the step size is smaller than this, optimization + is terminated as successful. If omitted, not used. + tol_norm : function(vector) -> scalar, optional + Norm to use in convergence check. Default is the maximum norm. + line_search : {None, 'armijo' (default), 'wolfe'}, optional + Which type of a line search to use to determine the step size in + the direction given by the Jacobian approximation. Defaults to + 'armijo'. + jac_options : dict, optional + Options for the respective Jacobian approximation. + + alpha : float, optional + initial guess for the jacobian is (-1/alpha). + """ + pass + +def _root_diagbroyden_doc(): + """ + Options + ------- + nit : int, optional + Number of iterations to make. If omitted (default), make as many + as required to meet tolerances. + disp : bool, optional + Print status to stdout on every iteration. + maxiter : int, optional + Maximum number of iterations to make. + ftol : float, optional + Relative tolerance for the residual. If omitted, not used. + fatol : float, optional + Absolute tolerance (in max-norm) for the residual. + If omitted, default is 6e-6. + xtol : float, optional + Relative minimum step size. If omitted, not used. + xatol : float, optional + Absolute minimum step size, as determined from the Jacobian + approximation. If the step size is smaller than this, optimization + is terminated as successful. If omitted, not used. + tol_norm : function(vector) -> scalar, optional + Norm to use in convergence check. Default is the maximum norm. + line_search : {None, 'armijo' (default), 'wolfe'}, optional + Which type of a line search to use to determine the step size in + the direction given by the Jacobian approximation. Defaults to + 'armijo'. + jac_options : dict, optional + Options for the respective Jacobian approximation. + + alpha : float, optional + initial guess for the jacobian is (-1/alpha). + """ + pass + +def _root_excitingmixing_doc(): + """ + Options + ------- + nit : int, optional + Number of iterations to make. If omitted (default), make as many + as required to meet tolerances. + disp : bool, optional + Print status to stdout on every iteration. + maxiter : int, optional + Maximum number of iterations to make. + ftol : float, optional + Relative tolerance for the residual. If omitted, not used. + fatol : float, optional + Absolute tolerance (in max-norm) for the residual. + If omitted, default is 6e-6. + xtol : float, optional + Relative minimum step size. If omitted, not used. + xatol : float, optional + Absolute minimum step size, as determined from the Jacobian + approximation. If the step size is smaller than this, optimization + is terminated as successful. If omitted, not used. + tol_norm : function(vector) -> scalar, optional + Norm to use in convergence check. Default is the maximum norm. + line_search : {None, 'armijo' (default), 'wolfe'}, optional + Which type of a line search to use to determine the step size in + the direction given by the Jacobian approximation. Defaults to + 'armijo'. + jac_options : dict, optional + Options for the respective Jacobian approximation. + + alpha : float, optional + Initial Jacobian approximation is (-1/alpha). + alphamax : float, optional + The entries of the diagonal Jacobian are kept in the range + ``[alpha, alphamax]``. + """ + pass + +def _root_krylov_doc(): + """ + Options + ------- + nit : int, optional + Number of iterations to make. If omitted (default), make as many + as required to meet tolerances. + disp : bool, optional + Print status to stdout on every iteration. + maxiter : int, optional + Maximum number of iterations to make. + ftol : float, optional + Relative tolerance for the residual. If omitted, not used. + fatol : float, optional + Absolute tolerance (in max-norm) for the residual. + If omitted, default is 6e-6. + xtol : float, optional + Relative minimum step size. If omitted, not used. + xatol : float, optional + Absolute minimum step size, as determined from the Jacobian + approximation. If the step size is smaller than this, optimization + is terminated as successful. If omitted, not used. + tol_norm : function(vector) -> scalar, optional + Norm to use in convergence check. Default is the maximum norm. + line_search : {None, 'armijo' (default), 'wolfe'}, optional + Which type of a line search to use to determine the step size in + the direction given by the Jacobian approximation. Defaults to + 'armijo'. + jac_options : dict, optional + Options for the respective Jacobian approximation. + + rdiff : float, optional + Relative step size to use in numerical differentiation. + method : str or callable, optional + Krylov method to use to approximate the Jacobian. Can be a string, + or a function implementing the same interface as the iterative + solvers in `scipy.sparse.linalg`. If a string, needs to be one of: + ``'lgmres'``, ``'gmres'``, ``'bicgstab'``, ``'cgs'``, ``'minres'``, + ``'tfqmr'``. + + The default is `scipy.sparse.linalg.lgmres`. + inner_M : LinearOperator or InverseJacobian + Preconditioner for the inner Krylov iteration. + Note that you can use also inverse Jacobians as (adaptive) + preconditioners. For example, + + >>> jac = BroydenFirst() + >>> kjac = KrylovJacobian(inner_M=jac.inverse). + + If the preconditioner has a method named 'update', it will + be called as ``update(x, f)`` after each nonlinear step, + with ``x`` giving the current point, and ``f`` the current + function value. + inner_rtol, inner_atol, inner_callback, ... + Parameters to pass on to the "inner" Krylov solver. + + For a full list of options, see the documentation for the + solver you are using. By default this is `scipy.sparse.linalg.lgmres`. + If the solver has been overridden through `method`, see the documentation + for that solver instead. + To use an option for that solver, prepend ``inner_`` to it. + For example, to control the ``rtol`` argument to the solver, + set the `inner_rtol` option here. + + outer_k : int, optional + Size of the subspace kept across LGMRES nonlinear + iterations. + + See `scipy.sparse.linalg.lgmres` for details. + """ + pass diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_root_scalar.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_root_scalar.py new file mode 100644 index 0000000000000000000000000000000000000000..668565de62ea36e0cc9be378875604855e489d17 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_root_scalar.py @@ -0,0 +1,538 @@ +""" +Unified interfaces to root finding algorithms for real or complex +scalar functions. + +Functions +--------- +- root : find a root of a scalar function. +""" +import numpy as np + +from . import _zeros_py as optzeros +from ._numdiff import approx_derivative + +__all__ = ['root_scalar'] + +ROOT_SCALAR_METHODS = ['bisect', 'brentq', 'brenth', 'ridder', 'toms748', + 'newton', 'secant', 'halley'] + + +class MemoizeDer: + """Decorator that caches the value and derivative(s) of function each + time it is called. + + This is a simplistic memoizer that calls and caches a single value + of ``f(x, *args)``. + It assumes that `args` does not change between invocations. + It supports the use case of a root-finder where `args` is fixed, + `x` changes, and only rarely, if at all, does x assume the same value + more than once.""" + def __init__(self, fun): + self.fun = fun + self.vals = None + self.x = None + self.n_calls = 0 + + def __call__(self, x, *args): + r"""Calculate f or use cached value if available""" + # Derivative may be requested before the function itself, always check + if self.vals is None or x != self.x: + fg = self.fun(x, *args) + self.x = x + self.n_calls += 1 + self.vals = fg[:] + return self.vals[0] + + def fprime(self, x, *args): + r"""Calculate f' or use a cached value if available""" + if self.vals is None or x != self.x: + self(x, *args) + return self.vals[1] + + def fprime2(self, x, *args): + r"""Calculate f'' or use a cached value if available""" + if self.vals is None or x != self.x: + self(x, *args) + return self.vals[2] + + def ncalls(self): + return self.n_calls + + +def root_scalar(f, args=(), method=None, bracket=None, + fprime=None, fprime2=None, + x0=None, x1=None, + xtol=None, rtol=None, maxiter=None, + options=None): + """ + Find a root of a scalar function. + + Parameters + ---------- + f : callable + A function to find a root of. + + Suppose the callable has signature ``f0(x, *my_args, **my_kwargs)``, where + ``my_args`` and ``my_kwargs`` are required positional and keyword arguments. + Rather than passing ``f0`` as the callable, wrap it to accept + only ``x``; e.g., pass ``fun=lambda x: f0(x, *my_args, **my_kwargs)`` as the + callable, where ``my_args`` (tuple) and ``my_kwargs`` (dict) have been + gathered before invoking this function. + args : tuple, optional + Extra arguments passed to the objective function and its derivative(s). + method : str, optional + Type of solver. Should be one of + + - 'bisect' :ref:`(see here) ` + - 'brentq' :ref:`(see here) ` + - 'brenth' :ref:`(see here) ` + - 'ridder' :ref:`(see here) ` + - 'toms748' :ref:`(see here) ` + - 'newton' :ref:`(see here) ` + - 'secant' :ref:`(see here) ` + - 'halley' :ref:`(see here) ` + + bracket: A sequence of 2 floats, optional + An interval bracketing a root. ``f(x, *args)`` must have different + signs at the two endpoints. + x0 : float, optional + Initial guess. + x1 : float, optional + A second guess. + fprime : bool or callable, optional + If `fprime` is a boolean and is True, `f` is assumed to return the + value of the objective function and of the derivative. + `fprime` can also be a callable returning the derivative of `f`. In + this case, it must accept the same arguments as `f`. + fprime2 : bool or callable, optional + If `fprime2` is a boolean and is True, `f` is assumed to return the + value of the objective function and of the + first and second derivatives. + `fprime2` can also be a callable returning the second derivative of `f`. + In this case, it must accept the same arguments as `f`. + xtol : float, optional + Tolerance (absolute) for termination. + rtol : float, optional + Tolerance (relative) for termination. + maxiter : int, optional + Maximum number of iterations. + options : dict, optional + A dictionary of solver options. E.g., ``k``, see + :obj:`show_options()` for details. + + Returns + ------- + sol : RootResults + The solution represented as a ``RootResults`` object. + Important attributes are: ``root`` the solution , ``converged`` a + boolean flag indicating if the algorithm exited successfully and + ``flag`` which describes the cause of the termination. See + `RootResults` for a description of other attributes. + + See also + -------- + show_options : Additional options accepted by the solvers + root : Find a root of a vector function. + + Notes + ----- + This section describes the available solvers that can be selected by the + 'method' parameter. + + The default is to use the best method available for the situation + presented. + If a bracket is provided, it may use one of the bracketing methods. + If a derivative and an initial value are specified, it may + select one of the derivative-based methods. + If no method is judged applicable, it will raise an Exception. + + Arguments for each method are as follows (x=required, o=optional). + + +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ + | method | f | args | bracket | x0 | x1 | fprime | fprime2 | xtol | rtol | maxiter | options | + +===============================================+===+======+=========+====+====+========+=========+======+======+=========+=========+ + | :ref:`bisect ` | x | o | x | | | | | o | o | o | o | + +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ + | :ref:`brentq ` | x | o | x | | | | | o | o | o | o | + +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ + | :ref:`brenth ` | x | o | x | | | | | o | o | o | o | + +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ + | :ref:`ridder ` | x | o | x | | | | | o | o | o | o | + +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ + | :ref:`toms748 ` | x | o | x | | | | | o | o | o | o | + +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ + | :ref:`secant ` | x | o | | x | o | | | o | o | o | o | + +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ + | :ref:`newton ` | x | o | | x | | o | | o | o | o | o | + +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ + | :ref:`halley ` | x | o | | x | | x | x | o | o | o | o | + +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ + + Examples + -------- + + Find the root of a simple cubic + + >>> from scipy import optimize + >>> def f(x): + ... return (x**3 - 1) # only one real root at x = 1 + + >>> def fprime(x): + ... return 3*x**2 + + The `brentq` method takes as input a bracket + + >>> sol = optimize.root_scalar(f, bracket=[0, 3], method='brentq') + >>> sol.root, sol.iterations, sol.function_calls + (1.0, 10, 11) + + The `newton` method takes as input a single point and uses the + derivative(s). + + >>> sol = optimize.root_scalar(f, x0=0.2, fprime=fprime, method='newton') + >>> sol.root, sol.iterations, sol.function_calls + (1.0, 11, 22) + + The function can provide the value and derivative(s) in a single call. + + >>> def f_p_pp(x): + ... return (x**3 - 1), 3*x**2, 6*x + + >>> sol = optimize.root_scalar( + ... f_p_pp, x0=0.2, fprime=True, method='newton' + ... ) + >>> sol.root, sol.iterations, sol.function_calls + (1.0, 11, 11) + + >>> sol = optimize.root_scalar( + ... f_p_pp, x0=0.2, fprime=True, fprime2=True, method='halley' + ... ) + >>> sol.root, sol.iterations, sol.function_calls + (1.0, 7, 8) + + + """ # noqa: E501 + if not isinstance(args, tuple): + args = (args,) + + if options is None: + options = {} + + # fun also returns the derivative(s) + is_memoized = False + if fprime2 is not None and not callable(fprime2): + if bool(fprime2): + f = MemoizeDer(f) + is_memoized = True + fprime2 = f.fprime2 + fprime = f.fprime + else: + fprime2 = None + if fprime is not None and not callable(fprime): + if bool(fprime): + f = MemoizeDer(f) + is_memoized = True + fprime = f.fprime + else: + fprime = None + + # respect solver-specific default tolerances - only pass in if actually set + kwargs = {} + for k in ['xtol', 'rtol', 'maxiter']: + v = locals().get(k) + if v is not None: + kwargs[k] = v + + # Set any solver-specific options + if options: + kwargs.update(options) + # Always request full_output from the underlying method as _root_scalar + # always returns a RootResults object + kwargs.update(full_output=True, disp=False) + + # Pick a method if not specified. + # Use the "best" method available for the situation. + if not method: + if bracket is not None: + method = 'brentq' + elif x0 is not None: + if fprime: + if fprime2: + method = 'halley' + else: + method = 'newton' + elif x1 is not None: + method = 'secant' + else: + method = 'newton' + if not method: + raise ValueError('Unable to select a solver as neither bracket ' + 'nor starting point provided.') + + meth = method.lower() + map2underlying = {'halley': 'newton', 'secant': 'newton'} + + try: + methodc = getattr(optzeros, map2underlying.get(meth, meth)) + except AttributeError as e: + raise ValueError(f'Unknown solver {meth}') from e + + if meth in ['bisect', 'ridder', 'brentq', 'brenth', 'toms748']: + if not isinstance(bracket, (list, tuple, np.ndarray)): + raise ValueError(f'Bracket needed for {method}') + + a, b = bracket[:2] + try: + r, sol = methodc(f, a, b, args=args, **kwargs) + except ValueError as e: + # gh-17622 fixed some bugs in low-level solvers by raising an error + # (rather than returning incorrect results) when the callable + # returns a NaN. It did so by wrapping the callable rather than + # modifying compiled code, so the iteration count is not available. + if hasattr(e, "_x"): + sol = optzeros.RootResults(root=e._x, + iterations=np.nan, + function_calls=e._function_calls, + flag=str(e), method=method) + else: + raise + + elif meth in ['secant']: + if x0 is None: + raise ValueError(f'x0 must not be None for {method}') + if 'xtol' in kwargs: + kwargs['tol'] = kwargs.pop('xtol') + r, sol = methodc(f, x0, args=args, fprime=None, fprime2=None, + x1=x1, **kwargs) + elif meth in ['newton']: + if x0 is None: + raise ValueError(f'x0 must not be None for {method}') + if not fprime: + # approximate fprime with finite differences + + def fprime(x, *args): + # `root_scalar` doesn't actually seem to support vectorized + # use of `newton`. In that case, `approx_derivative` will + # always get scalar input. Nonetheless, it always returns an + # array, so we extract the element to produce scalar output. + # Similarly, `approx_derivative` always passes array input, so + # we extract the element to ensure the user's function gets + # scalar input. + def f_wrapped(x, *args): + return f(x[0], *args) + return approx_derivative(f_wrapped, x, method='2-point', args=args)[0] + + if 'xtol' in kwargs: + kwargs['tol'] = kwargs.pop('xtol') + r, sol = methodc(f, x0, args=args, fprime=fprime, fprime2=None, + **kwargs) + elif meth in ['halley']: + if x0 is None: + raise ValueError(f'x0 must not be None for {method}') + if not fprime: + raise ValueError(f'fprime must be specified for {method}') + if not fprime2: + raise ValueError(f'fprime2 must be specified for {method}') + if 'xtol' in kwargs: + kwargs['tol'] = kwargs.pop('xtol') + r, sol = methodc(f, x0, args=args, fprime=fprime, fprime2=fprime2, **kwargs) + else: + raise ValueError(f'Unknown solver {method}') + + if is_memoized: + # Replace the function_calls count with the memoized count. + # Avoids double and triple-counting. + n_calls = f.n_calls + sol.function_calls = n_calls + + return sol + + +def _root_scalar_brentq_doc(): + r""" + Options + ------- + args : tuple, optional + Extra arguments passed to the objective function. + bracket: A sequence of 2 floats, optional + An interval bracketing a root. ``f(x, *args)`` must have different + signs at the two endpoints. + xtol : float, optional + Tolerance (absolute) for termination. + rtol : float, optional + Tolerance (relative) for termination. + maxiter : int, optional + Maximum number of iterations. + options: dict, optional + Specifies any method-specific options not covered above + + """ + pass + + +def _root_scalar_brenth_doc(): + r""" + Options + ------- + args : tuple, optional + Extra arguments passed to the objective function. + bracket: A sequence of 2 floats, optional + An interval bracketing a root. ``f(x, *args)`` must have different + signs at the two endpoints. + xtol : float, optional + Tolerance (absolute) for termination. + rtol : float, optional + Tolerance (relative) for termination. + maxiter : int, optional + Maximum number of iterations. + options: dict, optional + Specifies any method-specific options not covered above. + + """ + pass + +def _root_scalar_toms748_doc(): + r""" + Options + ------- + args : tuple, optional + Extra arguments passed to the objective function. + bracket: A sequence of 2 floats, optional + An interval bracketing a root. ``f(x, *args)`` must have different + signs at the two endpoints. + xtol : float, optional + Tolerance (absolute) for termination. + rtol : float, optional + Tolerance (relative) for termination. + maxiter : int, optional + Maximum number of iterations. + options: dict, optional + Specifies any method-specific options not covered above. + + """ + pass + + +def _root_scalar_secant_doc(): + r""" + Options + ------- + args : tuple, optional + Extra arguments passed to the objective function. + xtol : float, optional + Tolerance (absolute) for termination. + rtol : float, optional + Tolerance (relative) for termination. + maxiter : int, optional + Maximum number of iterations. + x0 : float, required + Initial guess. + x1 : float, optional + A second guess. Must be different from `x0`. If not specified, + a value near `x0` will be chosen. + options: dict, optional + Specifies any method-specific options not covered above. + + """ + pass + + +def _root_scalar_newton_doc(): + r""" + Options + ------- + args : tuple, optional + Extra arguments passed to the objective function and its derivative. + xtol : float, optional + Tolerance (absolute) for termination. + rtol : float, optional + Tolerance (relative) for termination. + maxiter : int, optional + Maximum number of iterations. + x0 : float, required + Initial guess. + fprime : bool or callable, optional + If `fprime` is a boolean and is True, `f` is assumed to return the + value of derivative along with the objective function. + `fprime` can also be a callable returning the derivative of `f`. In + this case, it must accept the same arguments as `f`. + options: dict, optional + Specifies any method-specific options not covered above. + + """ + pass + + +def _root_scalar_halley_doc(): + r""" + Options + ------- + args : tuple, optional + Extra arguments passed to the objective function and its derivatives. + xtol : float, optional + Tolerance (absolute) for termination. + rtol : float, optional + Tolerance (relative) for termination. + maxiter : int, optional + Maximum number of iterations. + x0 : float, required + Initial guess. + fprime : bool or callable, required + If `fprime` is a boolean and is True, `f` is assumed to return the + value of derivative along with the objective function. + `fprime` can also be a callable returning the derivative of `f`. In + this case, it must accept the same arguments as `f`. + fprime2 : bool or callable, required + If `fprime2` is a boolean and is True, `f` is assumed to return the + value of 1st and 2nd derivatives along with the objective function. + `fprime2` can also be a callable returning the 2nd derivative of `f`. + In this case, it must accept the same arguments as `f`. + options: dict, optional + Specifies any method-specific options not covered above. + + """ + pass + + +def _root_scalar_ridder_doc(): + r""" + Options + ------- + args : tuple, optional + Extra arguments passed to the objective function. + bracket: A sequence of 2 floats, optional + An interval bracketing a root. ``f(x, *args)`` must have different + signs at the two endpoints. + xtol : float, optional + Tolerance (absolute) for termination. + rtol : float, optional + Tolerance (relative) for termination. + maxiter : int, optional + Maximum number of iterations. + options: dict, optional + Specifies any method-specific options not covered above. + + """ + pass + + +def _root_scalar_bisect_doc(): + r""" + Options + ------- + args : tuple, optional + Extra arguments passed to the objective function. + bracket: A sequence of 2 floats, optional + An interval bracketing a root. ``f(x, *args)`` must have different + signs at the two endpoints. + xtol : float, optional + Tolerance (absolute) for termination. + rtol : float, optional + Tolerance (relative) for termination. + maxiter : int, optional + Maximum number of iterations. + options: dict, optional + Specifies any method-specific options not covered above. + + """ + pass diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo.py new file mode 100644 index 0000000000000000000000000000000000000000..9e9f37ce003ba42df054e25f03fbcdfe1478a423 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo.py @@ -0,0 +1,1600 @@ +"""shgo: The simplicial homology global optimisation algorithm.""" +from collections import namedtuple +import time +import logging +import warnings +import sys + +import numpy as np + +from scipy import spatial +from scipy.optimize import OptimizeResult, minimize, Bounds +from scipy.optimize._optimize import MemoizeJac +from scipy.optimize._constraints import new_bounds_to_old +from scipy.optimize._minimize import standardize_constraints +from scipy._lib._util import _FunctionWrapper + +from scipy.optimize._shgo_lib._complex import Complex + +__all__ = ['shgo'] + + +def shgo( + func, bounds, args=(), constraints=None, n=100, iters=1, callback=None, + minimizer_kwargs=None, options=None, sampling_method='simplicial', *, + workers=1 +): + """ + Finds the global minimum of a function using SHG optimization. + + SHGO stands for "simplicial homology global optimization". + + Parameters + ---------- + func : callable + The objective function to be minimized. Must be in the form + ``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array + and ``args`` is a tuple of any additional fixed parameters needed to + completely specify the function. + bounds : sequence or `Bounds` + Bounds for variables. There are two ways to specify the bounds: + + 1. Instance of `Bounds` class. + 2. Sequence of ``(min, max)`` pairs for each element in `x`. + + args : tuple, optional + Any additional fixed parameters needed to completely specify the + objective function. + constraints : {Constraint, dict} or List of {Constraint, dict}, optional + Constraints definition. Only for COBYLA, COBYQA, SLSQP and trust-constr. + See the tutorial [5]_ for further details on specifying constraints. + + .. note:: + + Only COBYLA, COBYQA, SLSQP, and trust-constr local minimize methods + currently support constraint arguments. If the ``constraints`` + sequence used in the local optimization problem is not defined in + ``minimizer_kwargs`` and a constrained method is used then the + global ``constraints`` will be used. + (Defining a ``constraints`` sequence in ``minimizer_kwargs`` + means that ``constraints`` will not be added so if equality + constraints and so forth need to be added then the inequality + functions in ``constraints`` need to be added to + ``minimizer_kwargs`` too). + COBYLA only supports inequality constraints. + + .. versionchanged:: 1.11.0 + + ``constraints`` accepts `NonlinearConstraint`, `LinearConstraint`. + + n : int, optional + Number of sampling points used in the construction of the simplicial + complex. For the default ``simplicial`` sampling method 2**dim + 1 + sampling points are generated instead of the default ``n=100``. For all + other specified values `n` sampling points are generated. For + ``sobol``, ``halton`` and other arbitrary `sampling_methods` ``n=100`` or + another specified number of sampling points are generated. + iters : int, optional + Number of iterations used in the construction of the simplicial + complex. Default is 1. + callback : callable, optional + Called after each iteration, as ``callback(xk)``, where ``xk`` is the + current parameter vector. + minimizer_kwargs : dict, optional + Extra keyword arguments to be passed to the minimizer + ``scipy.optimize.minimize``. Some important options could be: + + method : str + The minimization method. If not given, chosen to be one of + BFGS, L-BFGS-B, SLSQP, depending on whether or not the + problem has constraints or bounds. + args : tuple + Extra arguments passed to the objective function (``func``) and + its derivatives (Jacobian, Hessian). + options : dict, optional + Note that by default the tolerance is specified as + ``{ftol: 1e-12}`` + + options : dict, optional + A dictionary of solver options. Many of the options specified for the + global routine are also passed to the ``scipy.optimize.minimize`` + routine. The options that are also passed to the local routine are + marked with "(L)". + + Stopping criteria, the algorithm will terminate if any of the specified + criteria are met. However, the default algorithm does not require any + to be specified: + + maxfev : int (L) + Maximum number of function evaluations in the feasible domain. + (Note only methods that support this option will terminate + the routine at precisely exact specified value. Otherwise the + criterion will only terminate during a global iteration) + f_min : float + Specify the minimum objective function value, if it is known. + f_tol : float + Precision goal for the value of f in the stopping + criterion. Note that the global routine will also + terminate if a sampling point in the global routine is + within this tolerance. + maxiter : int + Maximum number of iterations to perform. + maxev : int + Maximum number of sampling evaluations to perform (includes + searching in infeasible points). + maxtime : float + Maximum processing runtime allowed + minhgrd : int + Minimum homology group rank differential. The homology group of the + objective function is calculated (approximately) during every + iteration. The rank of this group has a one-to-one correspondence + with the number of locally convex subdomains in the objective + function (after adequate sampling points each of these subdomains + contain a unique global minimum). If the difference in the hgr is 0 + between iterations for ``maxhgrd`` specified iterations the + algorithm will terminate. + + Objective function knowledge: + + symmetry : list or bool + Specify if the objective function contains symmetric variables. + The search space (and therefore performance) is decreased by up to + O(n!) times in the fully symmetric case. If `True` is specified + then all variables will be set symmetric to the first variable. + Default + is set to False. + + E.g. f(x) = (x_1 + x_2 + x_3) + (x_4)**2 + (x_5)**2 + (x_6)**2 + + In this equation x_2 and x_3 are symmetric to x_1, while x_5 and + x_6 are symmetric to x_4, this can be specified to the solver as:: + + symmetry = [0, # Variable 1 + 0, # symmetric to variable 1 + 0, # symmetric to variable 1 + 3, # Variable 4 + 3, # symmetric to variable 4 + 3, # symmetric to variable 4 + ] + + jac : bool or callable, optional + Jacobian (gradient) of objective function. Only for CG, BFGS, + Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg. If ``jac`` is a + boolean and is True, ``fun`` is assumed to return the gradient + along with the objective function. If False, the gradient will be + estimated numerically. ``jac`` can also be a callable returning the + gradient of the objective. In this case, it must accept the same + arguments as ``fun``. (Passed to `scipy.optimize.minimize` + automatically) + + hess, hessp : callable, optional + Hessian (matrix of second-order derivatives) of objective function + or Hessian of objective function times an arbitrary vector p. + Only for Newton-CG, dogleg, trust-ncg. Only one of ``hessp`` or + ``hess`` needs to be given. If ``hess`` is provided, then + ``hessp`` will be ignored. If neither ``hess`` nor ``hessp`` is + provided, then the Hessian product will be approximated using + finite differences on ``jac``. ``hessp`` must compute the Hessian + times an arbitrary vector. (Passed to `scipy.optimize.minimize` + automatically) + + Algorithm settings: + + minimize_every_iter : bool + If True then promising global sampling points will be passed to a + local minimization routine every iteration. If True then only the + final minimizer pool will be run. Defaults to True. + + local_iter : int + Only evaluate a few of the best minimizer pool candidates every + iteration. If False all potential points are passed to the local + minimization routine. + + infty_constraints : bool + If True then any sampling points generated which are outside will + the feasible domain will be saved and given an objective function + value of ``inf``. If False then these points will be discarded. + Using this functionality could lead to higher performance with + respect to function evaluations before the global minimum is found, + specifying False will use less memory at the cost of a slight + decrease in performance. Defaults to True. + + Feedback: + + disp : bool (L) + Set to True to print convergence messages. + + sampling_method : str or function, optional + Current built in sampling method options are ``halton``, ``sobol`` and + ``simplicial``. The default ``simplicial`` provides + the theoretical guarantee of convergence to the global minimum in + finite time. ``halton`` and ``sobol`` method are faster in terms of + sampling point generation at the cost of the loss of + guaranteed convergence. It is more appropriate for most "easier" + problems where the convergence is relatively fast. + User defined sampling functions must accept two arguments of ``n`` + sampling points of dimension ``dim`` per call and output an array of + sampling points with shape `n x dim`. + + workers : int or map-like callable, optional + Sample and run the local serial minimizations in parallel. + Supply -1 to use all available CPU cores, or an int to use + that many Processes (uses `multiprocessing.Pool `). + + Alternatively supply a map-like callable, such as + `multiprocessing.Pool.map` for parallel evaluation. + This evaluation is carried out as ``workers(func, iterable)``. + Requires that `func` be pickleable. + + .. versionadded:: 1.11.0 + + Returns + ------- + res : OptimizeResult + The optimization result represented as a `OptimizeResult` object. + Important attributes are: + ``x`` the solution array corresponding to the global minimum, + ``fun`` the function output at the global solution, + ``xl`` an ordered list of local minima solutions, + ``funl`` the function output at the corresponding local solutions, + ``success`` a Boolean flag indicating if the optimizer exited + successfully, + ``message`` which describes the cause of the termination, + ``nfev`` the total number of objective function evaluations including + the sampling calls, + ``nlfev`` the total number of objective function evaluations + culminating from all local search optimizations, + ``nit`` number of iterations performed by the global routine. + + Notes + ----- + Global optimization using simplicial homology global optimization [1]_. + Appropriate for solving general purpose NLP and blackbox optimization + problems to global optimality (low-dimensional problems). + + In general, the optimization problems are of the form:: + + minimize f(x) subject to + + g_i(x) >= 0, i = 1,...,m + h_j(x) = 0, j = 1,...,p + + where x is a vector of one or more variables. ``f(x)`` is the objective + function ``R^n -> R``, ``g_i(x)`` are the inequality constraints, and + ``h_j(x)`` are the equality constraints. + + Optionally, the lower and upper bounds for each element in x can also be + specified using the `bounds` argument. + + While most of the theoretical advantages of SHGO are only proven for when + ``f(x)`` is a Lipschitz smooth function, the algorithm is also proven to + converge to the global optimum for the more general case where ``f(x)`` is + non-continuous, non-convex and non-smooth, if the default sampling method + is used [1]_. + + The local search method may be specified using the ``minimizer_kwargs`` + parameter which is passed on to ``scipy.optimize.minimize``. By default, + the ``SLSQP`` method is used. In general, it is recommended to use the + ``SLSQP``, ``COBYLA``, or ``COBYQA`` local minimization if inequality + constraints are defined for the problem since the other methods do not use + constraints. + + The ``halton`` and ``sobol`` method points are generated using + `scipy.stats.qmc`. Any other QMC method could be used. + + References + ---------- + .. [1] Endres, SC, Sandrock, C, Focke, WW (2018) "A simplicial homology + algorithm for lipschitz optimisation", Journal of Global + Optimization. + .. [2] Joe, SW and Kuo, FY (2008) "Constructing Sobol' sequences with + better two-dimensional projections", SIAM J. Sci. Comput. 30, + 2635-2654. + .. [3] Hock, W and Schittkowski, K (1981) "Test examples for nonlinear + programming codes", Lecture Notes in Economics and Mathematical + Systems, 187. Springer-Verlag, New York. + http://www.ai7.uni-bayreuth.de/test_problem_coll.pdf + .. [4] Wales, DJ (2015) "Perspective: Insight into reaction coordinates and + dynamics from the potential energy landscape", + Journal of Chemical Physics, 142(13), 2015. + .. [5] https://docs.scipy.org/doc/scipy/tutorial/optimize.html#constrained-minimization-of-multivariate-scalar-functions-minimize + + Examples + -------- + First consider the problem of minimizing the Rosenbrock function, `rosen`: + + >>> from scipy.optimize import rosen, shgo + >>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)] + >>> result = shgo(rosen, bounds) + >>> result.x, result.fun + (array([1., 1., 1., 1., 1.]), 2.920392374190081e-18) + + Note that bounds determine the dimensionality of the objective + function and is therefore a required input, however you can specify + empty bounds using ``None`` or objects like ``np.inf`` which will be + converted to large float numbers. + + >>> bounds = [(None, None), ]*4 + >>> result = shgo(rosen, bounds) + >>> result.x + array([0.99999851, 0.99999704, 0.99999411, 0.9999882 ]) + + Next, we consider the Eggholder function, a problem with several local + minima and one global minimum. We will demonstrate the use of arguments and + the capabilities of `shgo`. + (https://en.wikipedia.org/wiki/Test_functions_for_optimization) + + >>> import numpy as np + >>> def eggholder(x): + ... return (-(x[1] + 47.0) + ... * np.sin(np.sqrt(abs(x[0]/2.0 + (x[1] + 47.0)))) + ... - x[0] * np.sin(np.sqrt(abs(x[0] - (x[1] + 47.0)))) + ... ) + ... + >>> bounds = [(-512, 512), (-512, 512)] + + `shgo` has built-in low discrepancy sampling sequences. First, we will + input 64 initial sampling points of the *Sobol'* sequence: + + >>> result = shgo(eggholder, bounds, n=64, sampling_method='sobol') + >>> result.x, result.fun + (array([512. , 404.23180824]), -959.6406627208397) + + `shgo` also has a return for any other local minima that was found, these + can be called using: + + >>> result.xl + array([[ 512. , 404.23180824], + [ 283.0759062 , -487.12565635], + [-294.66820039, -462.01964031], + [-105.87688911, 423.15323845], + [-242.97926 , 274.38030925], + [-506.25823477, 6.3131022 ], + [-408.71980731, -156.10116949], + [ 150.23207937, 301.31376595], + [ 91.00920901, -391.283763 ], + [ 202.89662724, -269.38043241], + [ 361.66623976, -106.96493868], + [-219.40612786, -244.06020508]]) + + >>> result.funl + array([-959.64066272, -718.16745962, -704.80659592, -565.99778097, + -559.78685655, -557.36868733, -507.87385942, -493.9605115 , + -426.48799655, -421.15571437, -419.31194957, -410.98477763]) + + These results are useful in applications where there are many global minima + and the values of other global minima are desired or where the local minima + can provide insight into the system (for example morphologies + in physical chemistry [4]_). + + If we want to find a larger number of local minima, we can increase the + number of sampling points or the number of iterations. We'll increase the + number of sampling points to 64 and the number of iterations from the + default of 1 to 3. Using ``simplicial`` this would have given us + 64 x 3 = 192 initial sampling points. + + >>> result_2 = shgo(eggholder, + ... bounds, n=64, iters=3, sampling_method='sobol') + >>> len(result.xl), len(result_2.xl) + (12, 23) + + Note the difference between, e.g., ``n=192, iters=1`` and ``n=64, + iters=3``. + In the first case the promising points contained in the minimiser pool + are processed only once. In the latter case it is processed every 64 + sampling points for a total of 3 times. + + To demonstrate solving problems with non-linear constraints consider the + following example from Hock and Schittkowski problem 73 (cattle-feed) + [3]_:: + + minimize: f = 24.55 * x_1 + 26.75 * x_2 + 39 * x_3 + 40.50 * x_4 + + subject to: 2.3 * x_1 + 5.6 * x_2 + 11.1 * x_3 + 1.3 * x_4 - 5 >= 0, + + 12 * x_1 + 11.9 * x_2 + 41.8 * x_3 + 52.1 * x_4 - 21 + -1.645 * sqrt(0.28 * x_1**2 + 0.19 * x_2**2 + + 20.5 * x_3**2 + 0.62 * x_4**2) >= 0, + + x_1 + x_2 + x_3 + x_4 - 1 == 0, + + 1 >= x_i >= 0 for all i + + The approximate answer given in [3]_ is:: + + f([0.6355216, -0.12e-11, 0.3127019, 0.05177655]) = 29.894378 + + >>> def f(x): # (cattle-feed) + ... return 24.55*x[0] + 26.75*x[1] + 39*x[2] + 40.50*x[3] + ... + >>> def g1(x): + ... return 2.3*x[0] + 5.6*x[1] + 11.1*x[2] + 1.3*x[3] - 5 # >=0 + ... + >>> def g2(x): + ... return (12*x[0] + 11.9*x[1] +41.8*x[2] + 52.1*x[3] - 21 + ... - 1.645 * np.sqrt(0.28*x[0]**2 + 0.19*x[1]**2 + ... + 20.5*x[2]**2 + 0.62*x[3]**2) + ... ) # >=0 + ... + >>> def h1(x): + ... return x[0] + x[1] + x[2] + x[3] - 1 # == 0 + ... + >>> cons = ({'type': 'ineq', 'fun': g1}, + ... {'type': 'ineq', 'fun': g2}, + ... {'type': 'eq', 'fun': h1}) + >>> bounds = [(0, 1.0),]*4 + >>> res = shgo(f, bounds, n=150, constraints=cons) + >>> res + message: Optimization terminated successfully. + success: True + fun: 29.894378159142136 + funl: [ 2.989e+01] + x: [ 6.355e-01 1.137e-13 3.127e-01 5.178e-02] # may vary + xl: [[ 6.355e-01 1.137e-13 3.127e-01 5.178e-02]] # may vary + nit: 1 + nfev: 142 # may vary + nlfev: 35 # may vary + nljev: 5 + nlhev: 0 + + >>> g1(res.x), g2(res.x), h1(res.x) + (-5.062616992290714e-14, -2.9594104944408173e-12, 0.0) + + """ + # if necessary, convert bounds class to old bounds + if isinstance(bounds, Bounds): + bounds = new_bounds_to_old(bounds.lb, bounds.ub, len(bounds.lb)) + + # Initiate SHGO class + # use in context manager to make sure that any parallelization + # resources are freed. + with SHGO(func, bounds, args=args, constraints=constraints, n=n, + iters=iters, callback=callback, + minimizer_kwargs=minimizer_kwargs, + options=options, sampling_method=sampling_method, + workers=workers) as shc: + # Run the algorithm, process results and test success + shc.iterate_all() + + if not shc.break_routine: + if shc.disp: + logging.info("Successfully completed construction of complex.") + + # Test post iterations success + if len(shc.LMC.xl_maps) == 0: + # If sampling failed to find pool, return lowest sampled point + # with a warning + shc.find_lowest_vertex() + shc.break_routine = True + shc.fail_routine(mes="Failed to find a feasible minimizer point. " + f"Lowest sampling point = {shc.f_lowest}") + shc.res.fun = shc.f_lowest + shc.res.x = shc.x_lowest + shc.res.nfev = shc.fn + shc.res.tnev = shc.n_sampled + else: + # Test that the optimal solutions do not violate any constraints + pass # TODO + + # Confirm the routine ran successfully + if not shc.break_routine: + shc.res.message = 'Optimization terminated successfully.' + shc.res.success = True + + # Return the final results + return shc.res + + +class SHGO: + def __init__(self, func, bounds, args=(), constraints=None, n=None, + iters=None, callback=None, minimizer_kwargs=None, + options=None, sampling_method='simplicial', workers=1): + from scipy.stats import qmc + # Input checks + methods = ['halton', 'sobol', 'simplicial'] + if isinstance(sampling_method, str) and sampling_method not in methods: + raise ValueError(("Unknown sampling_method specified." + " Valid methods: {}").format(', '.join(methods))) + + # Split obj func if given with Jac + try: + if ((minimizer_kwargs['jac'] is True) and + (not callable(minimizer_kwargs['jac']))): + self.func = MemoizeJac(func) + jac = self.func.derivative + minimizer_kwargs['jac'] = jac + func = self.func # .fun + else: + self.func = func # Normal definition of objective function + except (TypeError, KeyError): + self.func = func # Normal definition of objective function + + # Initiate class + self.func = _FunctionWrapper(func, args) + self.bounds = bounds + self.args = args + self.callback = callback + + # Bounds + abound = np.array(bounds, float) + self.dim = np.shape(abound)[0] # Dimensionality of problem + + # Set none finite values to large floats + infind = ~np.isfinite(abound) + abound[infind[:, 0], 0] = -1e50 + abound[infind[:, 1], 1] = 1e50 + + # Check if bounds are correctly specified + bnderr = abound[:, 0] > abound[:, 1] + if bnderr.any(): + raise ValueError("Error: lb > ub in bounds " + f"{', '.join(str(b) for b in bnderr)}.") + + self.bounds = abound + + # Constraints + # Process constraint dict sequence: + self.constraints = constraints + if constraints is not None: + self.min_cons = constraints + self.g_cons = [] + self.g_args = [] + + # shgo internals deals with old-style constraints + # self.constraints is used to create Complex, so need + # to be stored internally in old-style. + # `minimize` takes care of normalising these constraints + # for slsqp/cobyla/cobyqa/trust-constr. + self.constraints = standardize_constraints( + constraints, + np.empty(self.dim, float), + 'old' + ) + for cons in self.constraints: + if cons['type'] in ('ineq'): + self.g_cons.append(cons['fun']) + try: + self.g_args.append(cons['args']) + except KeyError: + self.g_args.append(()) + self.g_cons = tuple(self.g_cons) + self.g_args = tuple(self.g_args) + else: + self.g_cons = None + self.g_args = None + + # Define local minimization keyword arguments + # Start with defaults + self.minimizer_kwargs = {'method': 'SLSQP', + 'bounds': self.bounds, + 'options': {}, + 'callback': self.callback + } + if minimizer_kwargs is not None: + # Overwrite with supplied values + self.minimizer_kwargs.update(minimizer_kwargs) + + else: + self.minimizer_kwargs['options'] = {'ftol': 1e-12} + + if ( + self.minimizer_kwargs['method'].lower() in ('slsqp', 'cobyla', + 'cobyqa', + 'trust-constr') + and ( + minimizer_kwargs is not None and + 'constraints' not in minimizer_kwargs and + constraints is not None + ) or + (self.g_cons is not None) + ): + self.minimizer_kwargs['constraints'] = self.min_cons + + # Process options dict + if options is not None: + self.init_options(options) + else: # Default settings: + self.f_min_true = None + self.minimize_every_iter = True + + # Algorithm limits + self.maxiter = None + self.maxfev = None + self.maxev = None + self.maxtime = None + self.f_min_true = None + self.minhgrd = None + + # Objective function knowledge + self.symmetry = None + + # Algorithm functionality + self.infty_cons_sampl = True + self.local_iter = False + + # Feedback + self.disp = False + + # Remove unknown arguments in self.minimizer_kwargs + # Start with arguments all the solvers have in common + self.min_solver_args = ['fun', 'x0', 'args', + 'callback', 'options', 'method'] + # then add the ones unique to specific solvers + solver_args = { + '_custom': ['jac', 'hess', 'hessp', 'bounds', 'constraints'], + 'nelder-mead': [], + 'powell': [], + 'cg': ['jac'], + 'bfgs': ['jac'], + 'newton-cg': ['jac', 'hess', 'hessp'], + 'l-bfgs-b': ['jac', 'bounds'], + 'tnc': ['jac', 'bounds'], + 'cobyla': ['constraints', 'catol'], + 'cobyqa': ['bounds', 'constraints', 'feasibility_tol'], + 'slsqp': ['jac', 'bounds', 'constraints'], + 'dogleg': ['jac', 'hess'], + 'trust-ncg': ['jac', 'hess', 'hessp'], + 'trust-krylov': ['jac', 'hess', 'hessp'], + 'trust-exact': ['jac', 'hess'], + 'trust-constr': ['jac', 'hess', 'hessp', 'constraints'], + } + method = self.minimizer_kwargs['method'] + self.min_solver_args += solver_args[method.lower()] + + # Only retain the known arguments + def _restrict_to_keys(dictionary, goodkeys): + """Remove keys from dictionary if not in goodkeys - inplace""" + existingkeys = set(dictionary) + for key in existingkeys - set(goodkeys): + dictionary.pop(key, None) + + _restrict_to_keys(self.minimizer_kwargs, self.min_solver_args) + _restrict_to_keys(self.minimizer_kwargs['options'], + self.min_solver_args + ['ftol']) + + # Algorithm controls + # Global controls + self.stop_global = False # Used in the stopping_criteria method + self.break_routine = False # Break the algorithm globally + self.iters = iters # Iterations to be ran + self.iters_done = 0 # Iterations completed + self.n = n # Sampling points per iteration + self.nc = 0 # n # Sampling points to sample in current iteration + self.n_prc = 0 # Processed points (used to track Delaunay iters) + self.n_sampled = 0 # To track no. of sampling points already generated + self.fn = 0 # Number of feasible sampling points evaluations performed + self.hgr = 0 # Homology group rank + # Initially attempt to build the triangulation incrementally: + self.qhull_incremental = True + + # Default settings if no sampling criteria. + if (self.n is None) and (self.iters is None) \ + and (sampling_method == 'simplicial'): + self.n = 2 ** self.dim + 1 + self.nc = 0 # self.n + if self.iters is None: + self.iters = 1 + if (self.n is None) and not (sampling_method == 'simplicial'): + self.n = self.n = 100 + self.nc = 0 # self.n + if (self.n == 100) and (sampling_method == 'simplicial'): + self.n = 2 ** self.dim + 1 + + if not ((self.maxiter is None) and (self.maxfev is None) and ( + self.maxev is None) + and (self.minhgrd is None) and (self.f_min_true is None)): + self.iters = None + + # Set complex construction mode based on a provided stopping criteria: + # Initialise sampling Complex and function cache + # Note that sfield_args=() since args are already wrapped in self.func + # using the_FunctionWrapper class. + self.HC = Complex(dim=self.dim, domain=self.bounds, + sfield=self.func, sfield_args=(), + symmetry=self.symmetry, + constraints=self.constraints, + workers=workers) + + # Choose complex constructor + if sampling_method == 'simplicial': + self.iterate_complex = self.iterate_hypercube + self.sampling_method = sampling_method + + elif sampling_method in ['halton', 'sobol'] or \ + not isinstance(sampling_method, str): + self.iterate_complex = self.iterate_delaunay + # Sampling method used + if sampling_method in ['halton', 'sobol']: + if sampling_method == 'sobol': + self.n = int(2 ** np.ceil(np.log2(self.n))) + # self.n #TODO: Should always be self.n, this is + # unacceptable for shgo, check that nfev behaves as + # expected. + self.nc = 0 + self.sampling_method = 'sobol' + self.qmc_engine = qmc.Sobol(d=self.dim, scramble=False, + seed=0) + else: + self.sampling_method = 'halton' + self.qmc_engine = qmc.Halton(d=self.dim, scramble=True, + seed=0) + + def sampling_method(n, d): + return self.qmc_engine.random(n) + + else: + # A user defined sampling method: + self.sampling_method = 'custom' + + self.sampling = self.sampling_custom + self.sampling_function = sampling_method # F(n, d) + + # Local controls + self.stop_l_iter = False # Local minimisation iterations + self.stop_complex_iter = False # Sampling iterations + + # Initiate storage objects used in algorithm classes + self.minimizer_pool = [] + + # Cache of local minimizers mapped + self.LMC = LMapCache() + + # Initialize return object + self.res = OptimizeResult() # scipy.optimize.OptimizeResult object + self.res.nfev = 0 # Includes each sampling point as func evaluation + self.res.nlfev = 0 # Local function evals for all minimisers + self.res.nljev = 0 # Local Jacobian evals for all minimisers + self.res.nlhev = 0 # Local Hessian evals for all minimisers + + # Initiation aids + def init_options(self, options): + """ + Initiates the options. + + Can also be useful to change parameters after class initiation. + + Parameters + ---------- + options : dict + + Returns + ------- + None + + """ + # Update 'options' dict passed to optimize.minimize + # Do this first so we don't mutate `options` below. + self.minimizer_kwargs['options'].update(options) + + # Ensure that 'jac', 'hess', and 'hessp' are passed directly to + # `minimize` as keywords, not as part of its 'options' dictionary. + for opt in ['jac', 'hess', 'hessp']: + if opt in self.minimizer_kwargs['options']: + self.minimizer_kwargs[opt] = ( + self.minimizer_kwargs['options'].pop(opt)) + + # Default settings: + self.minimize_every_iter = options.get('minimize_every_iter', True) + + # Algorithm limits + # Maximum number of iterations to perform. + self.maxiter = options.get('maxiter', None) + # Maximum number of function evaluations in the feasible domain + self.maxfev = options.get('maxfev', None) + # Maximum number of sampling evaluations (includes searching in + # infeasible points + self.maxev = options.get('maxev', None) + # Maximum processing runtime allowed + self.init = time.time() + self.maxtime = options.get('maxtime', None) + if 'f_min' in options: + # Specify the minimum objective function value, if it is known. + self.f_min_true = options['f_min'] + self.f_tol = options.get('f_tol', 1e-4) + else: + self.f_min_true = None + + self.minhgrd = options.get('minhgrd', None) + + # Objective function knowledge + self.symmetry = options.get('symmetry', False) + if self.symmetry: + self.symmetry = [0, ]*len(self.bounds) + else: + self.symmetry = None + # Algorithm functionality + # Only evaluate a few of the best candidates + self.local_iter = options.get('local_iter', False) + self.infty_cons_sampl = options.get('infty_constraints', True) + + # Feedback + self.disp = options.get('disp', False) + + def __enter__(self): + return self + + def __exit__(self, *args): + return self.HC.V._mapwrapper.__exit__(*args) + + # Iteration properties + # Main construction loop: + def iterate_all(self): + """ + Construct for `iters` iterations. + + If uniform sampling is used, every iteration adds 'n' sampling points. + + Iterations if a stopping criteria (e.g., sampling points or + processing time) has been met. + + """ + if self.disp: + logging.info('Splitting first generation') + + while not self.stop_global: + if self.break_routine: + break + # Iterate complex, process minimisers + self.iterate() + self.stopping_criteria() + + # Build minimiser pool + # Final iteration only needed if pools weren't minimised every + # iteration + if not self.minimize_every_iter: + if not self.break_routine: + self.find_minima() + + self.res.nit = self.iters_done # + 1 + self.fn = self.HC.V.nfev + + def find_minima(self): + """ + Construct the minimizer pool, map the minimizers to local minima + and sort the results into a global return object. + """ + if self.disp: + logging.info('Searching for minimizer pool...') + + self.minimizers() + + if len(self.X_min) != 0: + # Minimize the pool of minimizers with local minimization methods + # Note that if Options['local_iter'] is an `int` instead of default + # value False then only that number of candidates will be minimized + self.minimise_pool(self.local_iter) + # Sort results and build the global return object + self.sort_result() + + # Lowest values used to report in case of failures + self.f_lowest = self.res.fun + self.x_lowest = self.res.x + else: + self.find_lowest_vertex() + + if self.disp: + logging.info(f"Minimiser pool = SHGO.X_min = {self.X_min}") + + def find_lowest_vertex(self): + # Find the lowest objective function value on one of + # the vertices of the simplicial complex + self.f_lowest = np.inf + for x in self.HC.V.cache: + if self.HC.V[x].f < self.f_lowest: + if self.disp: + logging.info(f'self.HC.V[x].f = {self.HC.V[x].f}') + self.f_lowest = self.HC.V[x].f + self.x_lowest = self.HC.V[x].x_a + for lmc in self.LMC.cache: + if self.LMC[lmc].f_min < self.f_lowest: + self.f_lowest = self.LMC[lmc].f_min + self.x_lowest = self.LMC[lmc].x_l + + if self.f_lowest == np.inf: # no feasible point + self.f_lowest = None + self.x_lowest = None + + # Stopping criteria functions: + def finite_iterations(self): + mi = min(x for x in [self.iters, self.maxiter] if x is not None) + if self.disp: + logging.info(f'Iterations done = {self.iters_done} / {mi}') + if self.iters is not None: + if self.iters_done >= (self.iters): + self.stop_global = True + + if self.maxiter is not None: # Stop for infeasible sampling + if self.iters_done >= (self.maxiter): + self.stop_global = True + return self.stop_global + + def finite_fev(self): + # Finite function evals in the feasible domain + if self.disp: + logging.info(f'Function evaluations done = {self.fn} / {self.maxfev}') + if self.fn >= self.maxfev: + self.stop_global = True + return self.stop_global + + def finite_ev(self): + # Finite evaluations including infeasible sampling points + if self.disp: + logging.info(f'Sampling evaluations done = {self.n_sampled} ' + f'/ {self.maxev}') + if self.n_sampled >= self.maxev: + self.stop_global = True + + def finite_time(self): + if self.disp: + logging.info(f'Time elapsed = {time.time() - self.init} ' + f'/ {self.maxtime}') + if (time.time() - self.init) >= self.maxtime: + self.stop_global = True + + def finite_precision(self): + """ + Stop the algorithm if the final function value is known + + Specify in options (with ``self.f_min_true = options['f_min']``) + and the tolerance with ``f_tol = options['f_tol']`` + """ + # If no minimizer has been found use the lowest sampling value + self.find_lowest_vertex() + if self.disp: + logging.info(f'Lowest function evaluation = {self.f_lowest}') + logging.info(f'Specified minimum = {self.f_min_true}') + # If no feasible point was return from test + if self.f_lowest is None: + return self.stop_global + + # Function to stop algorithm at specified percentage error: + if self.f_min_true == 0.0: + if self.f_lowest <= self.f_tol: + self.stop_global = True + else: + pe = (self.f_lowest - self.f_min_true) / abs(self.f_min_true) + if self.f_lowest <= self.f_min_true: + self.stop_global = True + # 2if (pe - self.f_tol) <= abs(1.0 / abs(self.f_min_true)): + if abs(pe) >= 2 * self.f_tol: + warnings.warn( + f"A much lower value than expected f* = {self.f_min_true} " + f"was found f_lowest = {self.f_lowest}", + stacklevel=3 + ) + if pe <= self.f_tol: + self.stop_global = True + + return self.stop_global + + def finite_homology_growth(self): + """ + Stop the algorithm if homology group rank did not grow in iteration. + """ + if self.LMC.size == 0: + return # pass on no reason to stop yet. + self.hgrd = self.LMC.size - self.hgr + + self.hgr = self.LMC.size + if self.hgrd <= self.minhgrd: + self.stop_global = True + if self.disp: + logging.info(f'Current homology growth = {self.hgrd} ' + f' (minimum growth = {self.minhgrd})') + return self.stop_global + + def stopping_criteria(self): + """ + Various stopping criteria ran every iteration + + Returns + ------- + stop : bool + """ + if self.maxiter is not None: + self.finite_iterations() + if self.iters is not None: + self.finite_iterations() + if self.maxfev is not None: + self.finite_fev() + if self.maxev is not None: + self.finite_ev() + if self.maxtime is not None: + self.finite_time() + if self.f_min_true is not None: + self.finite_precision() + if self.minhgrd is not None: + self.finite_homology_growth() + return self.stop_global + + def iterate(self): + self.iterate_complex() + + # Build minimizer pool + if self.minimize_every_iter: + if not self.break_routine: + self.find_minima() # Process minimizer pool + + # Algorithm updates + self.iters_done += 1 + + def iterate_hypercube(self): + """ + Iterate a subdivision of the complex + + Note: called with ``self.iterate_complex()`` after class initiation + """ + # Iterate the complex + if self.disp: + logging.info('Constructing and refining simplicial complex graph ' + 'structure') + if self.n is None: + self.HC.refine_all() + self.n_sampled = self.HC.V.size() # nevs counted + else: + self.HC.refine(self.n) + self.n_sampled += self.n + + if self.disp: + logging.info('Triangulation completed, evaluating all constraints ' + 'and objective function values.') + + # Re-add minimisers to complex + if len(self.LMC.xl_maps) > 0: + for xl in self.LMC.cache: + v = self.HC.V[xl] + v_near = v.star() + for v in v.nn: + v_near = v_near.union(v.nn) + # Reconnect vertices to complex + # if self.HC.connect_vertex_non_symm(tuple(self.LMC[xl].x_l), + # near=v_near): + # continue + # else: + # If failure to find in v_near, then search all vertices + # (very expensive operation: + # self.HC.connect_vertex_non_symm(tuple(self.LMC[xl].x_l) + # ) + + # Evaluate all constraints and functions + self.HC.V.process_pools() + if self.disp: + logging.info('Evaluations completed.') + + # feasible sampling points counted by the triangulation.py routines + self.fn = self.HC.V.nfev + return + + def iterate_delaunay(self): + """ + Build a complex of Delaunay triangulated points + + Note: called with ``self.iterate_complex()`` after class initiation + """ + self.nc += self.n + self.sampled_surface(infty_cons_sampl=self.infty_cons_sampl) + + # Add sampled points to a triangulation, construct self.Tri + if self.disp: + logging.info(f'self.n = {self.n}') + logging.info(f'self.nc = {self.nc}') + logging.info('Constructing and refining simplicial complex graph ' + 'structure from sampling points.') + + if self.dim < 2: + self.Ind_sorted = np.argsort(self.C, axis=0) + self.Ind_sorted = self.Ind_sorted.flatten() + tris = [] + for ind, ind_s in enumerate(self.Ind_sorted): + if ind > 0: + tris.append(self.Ind_sorted[ind - 1:ind + 1]) + + tris = np.array(tris) + # Store 1D triangulation: + self.Tri = namedtuple('Tri', ['points', 'simplices'])(self.C, tris) + self.points = {} + else: + if self.C.shape[0] > self.dim + 1: # Ensure a simplex can be built + self.delaunay_triangulation(n_prc=self.n_prc) + self.n_prc = self.C.shape[0] + + if self.disp: + logging.info('Triangulation completed, evaluating all ' + 'constraints and objective function values.') + + if hasattr(self, 'Tri'): + self.HC.vf_to_vv(self.Tri.points, self.Tri.simplices) + + # Process all pools + # Evaluate all constraints and functions + if self.disp: + logging.info('Triangulation completed, evaluating all constraints ' + 'and objective function values.') + + # Evaluate all constraints and functions + self.HC.V.process_pools() + if self.disp: + logging.info('Evaluations completed.') + + # feasible sampling points counted by the triangulation.py routines + self.fn = self.HC.V.nfev + self.n_sampled = self.nc # nevs counted in triangulation + return + + # Hypercube minimizers + def minimizers(self): + """ + Returns the indexes of all minimizers + """ + self.minimizer_pool = [] + # Note: Can implement parallelization here + for x in self.HC.V.cache: + in_LMC = False + if len(self.LMC.xl_maps) > 0: + for xlmi in self.LMC.xl_maps: + if np.all(np.array(x) == np.array(xlmi)): + in_LMC = True + if in_LMC: + continue + + if self.HC.V[x].minimiser(): + if self.disp: + logging.info('=' * 60) + logging.info(f'v.x = {self.HC.V[x].x_a} is minimizer') + logging.info(f'v.f = {self.HC.V[x].f} is minimizer') + logging.info('=' * 30) + + if self.HC.V[x] not in self.minimizer_pool: + self.minimizer_pool.append(self.HC.V[x]) + + if self.disp: + logging.info('Neighbors:') + logging.info('=' * 30) + for vn in self.HC.V[x].nn: + logging.info(f'x = {vn.x} || f = {vn.f}') + + logging.info('=' * 60) + self.minimizer_pool_F = [] + self.X_min = [] + # normalized tuple in the Vertex cache + self.X_min_cache = {} # Cache used in hypercube sampling + + for v in self.minimizer_pool: + self.X_min.append(v.x_a) + self.minimizer_pool_F.append(v.f) + self.X_min_cache[tuple(v.x_a)] = v.x + + self.minimizer_pool_F = np.array(self.minimizer_pool_F) + self.X_min = np.array(self.X_min) + + # TODO: Only do this if global mode + self.sort_min_pool() + + return self.X_min + + # Local minimisation + # Minimiser pool processing + def minimise_pool(self, force_iter=False): + """ + This processing method can optionally minimise only the best candidate + solutions in the minimiser pool + + Parameters + ---------- + force_iter : int + Number of starting minimizers to process (can be specified + globally or locally) + + """ + # Find first local minimum + # NOTE: Since we always minimize this value regardless it is a waste to + # build the topograph first before minimizing + lres_f_min = self.minimize(self.X_min[0], ind=self.minimizer_pool[0]) + + # Trim minimized point from current minimizer set + self.trim_min_pool(0) + + while not self.stop_l_iter: + # Global stopping criteria: + self.stopping_criteria() + + # Note first iteration is outside loop: + if force_iter: + force_iter -= 1 + if force_iter == 0: + self.stop_l_iter = True + break + + if np.shape(self.X_min)[0] == 0: + self.stop_l_iter = True + break + + # Construct topograph from current minimizer set + # (NOTE: This is a very small topograph using only the minizer pool + # , it might be worth using some graph theory tools instead. + self.g_topograph(lres_f_min.x, self.X_min) + + # Find local minimum at the miniser with the greatest Euclidean + # distance from the current solution + ind_xmin_l = self.Z[:, -1] + lres_f_min = self.minimize(self.Ss[-1, :], self.minimizer_pool[-1]) + + # Trim minimised point from current minimizer set + self.trim_min_pool(ind_xmin_l) + + # Reset controls + self.stop_l_iter = False + return + + def sort_min_pool(self): + # Sort to find minimum func value in min_pool + self.ind_f_min = np.argsort(self.minimizer_pool_F) + self.minimizer_pool = np.array(self.minimizer_pool)[self.ind_f_min] + self.minimizer_pool_F = np.array(self.minimizer_pool_F)[ + self.ind_f_min] + return + + def trim_min_pool(self, trim_ind): + self.X_min = np.delete(self.X_min, trim_ind, axis=0) + self.minimizer_pool_F = np.delete(self.minimizer_pool_F, trim_ind) + self.minimizer_pool = np.delete(self.minimizer_pool, trim_ind) + return + + def g_topograph(self, x_min, X_min): + """ + Returns the topographical vector stemming from the specified value + ``x_min`` for the current feasible set ``X_min`` with True boolean + values indicating positive entries and False values indicating + negative entries. + + """ + x_min = np.array([x_min]) + self.Y = spatial.distance.cdist(x_min, X_min, 'euclidean') + # Find sorted indexes of spatial distances: + self.Z = np.argsort(self.Y, axis=-1) + + self.Ss = X_min[self.Z][0] + self.minimizer_pool = self.minimizer_pool[self.Z] + self.minimizer_pool = self.minimizer_pool[0] + return self.Ss + + # Local bound functions + def construct_lcb_simplicial(self, v_min): + """ + Construct locally (approximately) convex bounds + + Parameters + ---------- + v_min : Vertex object + The minimizer vertex + + Returns + ------- + cbounds : list of lists + List of size dimension with length-2 list of bounds for each + dimension. + + """ + cbounds = [[x_b_i[0], x_b_i[1]] for x_b_i in self.bounds] + # Loop over all bounds + for vn in v_min.nn: + for i, x_i in enumerate(vn.x_a): + # Lower bound + if (x_i < v_min.x_a[i]) and (x_i > cbounds[i][0]): + cbounds[i][0] = x_i + + # Upper bound + if (x_i > v_min.x_a[i]) and (x_i < cbounds[i][1]): + cbounds[i][1] = x_i + + if self.disp: + logging.info(f'cbounds found for v_min.x_a = {v_min.x_a}') + logging.info(f'cbounds = {cbounds}') + + return cbounds + + def construct_lcb_delaunay(self, v_min, ind=None): + """ + Construct locally (approximately) convex bounds + + Parameters + ---------- + v_min : Vertex object + The minimizer vertex + + Returns + ------- + cbounds : list of lists + List of size dimension with length-2 list of bounds for each + dimension. + """ + cbounds = [[x_b_i[0], x_b_i[1]] for x_b_i in self.bounds] + + return cbounds + + # Minimize a starting point locally + def minimize(self, x_min, ind=None): + """ + This function is used to calculate the local minima using the specified + sampling point as a starting value. + + Parameters + ---------- + x_min : vector of floats + Current starting point to minimize. + + Returns + ------- + lres : OptimizeResult + The local optimization result represented as a `OptimizeResult` + object. + """ + # Use minima maps if vertex was already run + if self.disp: + logging.info(f'Vertex minimiser maps = {self.LMC.v_maps}') + + if self.LMC[x_min].lres is not None: + logging.info(f'Found self.LMC[x_min].lres = ' + f'{self.LMC[x_min].lres}') + return self.LMC[x_min].lres + + if self.callback is not None: + logging.info(f'Callback for minimizer starting at {x_min}:') + + if self.disp: + logging.info(f'Starting minimization at {x_min}...') + + if self.sampling_method == 'simplicial': + x_min_t = tuple(x_min) + # Find the normalized tuple in the Vertex cache: + x_min_t_norm = self.X_min_cache[tuple(x_min_t)] + x_min_t_norm = tuple(x_min_t_norm) + g_bounds = self.construct_lcb_simplicial(self.HC.V[x_min_t_norm]) + if 'bounds' in self.min_solver_args: + self.minimizer_kwargs['bounds'] = g_bounds + logging.info(self.minimizer_kwargs['bounds']) + + else: + g_bounds = self.construct_lcb_delaunay(x_min, ind=ind) + if 'bounds' in self.min_solver_args: + self.minimizer_kwargs['bounds'] = g_bounds + logging.info(self.minimizer_kwargs['bounds']) + + if self.disp and 'bounds' in self.minimizer_kwargs: + logging.info('bounds in kwarg:') + logging.info(self.minimizer_kwargs['bounds']) + + # Local minimization using scipy.optimize.minimize: + lres = minimize(self.func, x_min, **self.minimizer_kwargs) + + if self.disp: + logging.info(f'lres = {lres}') + + # Local function evals for all minimizers + self.res.nlfev += lres.nfev + if 'njev' in lres: + self.res.nljev += lres.njev + if 'nhev' in lres: + self.res.nlhev += lres.nhev + + try: # Needed because of the brain dead 1x1 NumPy arrays + lres.fun = lres.fun[0] + except (IndexError, TypeError): + lres.fun + + # Append minima maps + self.LMC[x_min] + self.LMC.add_res(x_min, lres, bounds=g_bounds) + + return lres + + # Post local minimization processing + def sort_result(self): + """ + Sort results and build the global return object + """ + # Sort results in local minima cache + results = self.LMC.sort_cache_result() + self.res.xl = results['xl'] + self.res.funl = results['funl'] + self.res.x = results['x'] + self.res.fun = results['fun'] + + # Add local func evals to sampling func evals + # Count the number of feasible vertices and add to local func evals: + self.res.nfev = self.fn + self.res.nlfev + return self.res + + # Algorithm controls + def fail_routine(self, mes=("Failed to converge")): + self.break_routine = True + self.res.success = False + self.X_min = [None] + self.res.message = mes + + def sampled_surface(self, infty_cons_sampl=False): + """ + Sample the function surface. + + There are 2 modes, if ``infty_cons_sampl`` is True then the sampled + points that are generated outside the feasible domain will be + assigned an ``inf`` value in accordance with SHGO rules. + This guarantees convergence and usually requires less objective + function evaluations at the computational costs of more Delaunay + triangulation points. + + If ``infty_cons_sampl`` is False, then the infeasible points are + discarded and only a subspace of the sampled points are used. This + comes at the cost of the loss of guaranteed convergence and usually + requires more objective function evaluations. + """ + # Generate sampling points + if self.disp: + logging.info('Generating sampling points') + self.sampling(self.nc, self.dim) + if len(self.LMC.xl_maps) > 0: + self.C = np.vstack((self.C, np.array(self.LMC.xl_maps))) + if not infty_cons_sampl: + # Find subspace of feasible points + if self.g_cons is not None: + self.sampling_subspace() + + # Sort remaining samples + self.sorted_samples() + + # Find objective function references + self.n_sampled = self.nc + + def sampling_custom(self, n, dim): + """ + Generates uniform sampling points in a hypercube and scales the points + to the bound limits. + """ + # Generate sampling points. + # Generate uniform sample points in [0, 1]^m \subset R^m + if self.n_sampled == 0: + self.C = self.sampling_function(n, dim) + else: + self.C = self.sampling_function(n, dim) + # Distribute over bounds + for i in range(len(self.bounds)): + self.C[:, i] = (self.C[:, i] * + (self.bounds[i][1] - self.bounds[i][0]) + + self.bounds[i][0]) + return self.C + + def sampling_subspace(self): + """Find subspace of feasible points from g_func definition""" + # Subspace of feasible points. + for ind, g in enumerate(self.g_cons): + # C.shape = (Z, dim) where Z is the number of sampling points to + # evaluate and dim is the dimensionality of the problem. + # the constraint function may not be vectorised so have to step + # through each sampling point sequentially. + feasible = np.array( + [np.all(g(x_C, *self.g_args[ind]) >= 0.0) for x_C in self.C], + dtype=bool + ) + self.C = self.C[feasible] + + if self.C.size == 0: + self.res.message = ('No sampling point found within the ' + + 'feasible set. Increasing sampling ' + + 'size.') + # sampling correctly for both 1-D and >1-D cases + if self.disp: + logging.info(self.res.message) + + def sorted_samples(self): # Validated + """Find indexes of the sorted sampling points""" + self.Ind_sorted = np.argsort(self.C, axis=0) + self.Xs = self.C[self.Ind_sorted] + return self.Ind_sorted, self.Xs + + def delaunay_triangulation(self, n_prc=0): + if hasattr(self, 'Tri') and self.qhull_incremental: + # TODO: Uncertain if n_prc needs to add len(self.LMC.xl_maps) + # in self.sampled_surface + self.Tri.add_points(self.C[n_prc:, :]) + else: + try: + self.Tri = spatial.Delaunay(self.C, + incremental=self.qhull_incremental, + ) + except spatial.QhullError: + if str(sys.exc_info()[1])[:6] == 'QH6239': + logging.warning('QH6239 Qhull precision error detected, ' + 'this usually occurs when no bounds are ' + 'specified, Qhull can only run with ' + 'handling cocircular/cospherical points' + ' and in this case incremental mode is ' + 'switched off. The performance of shgo ' + 'will be reduced in this mode.') + self.qhull_incremental = False + self.Tri = spatial.Delaunay(self.C, + incremental= + self.qhull_incremental) + else: + raise + + return self.Tri + + +class LMap: + def __init__(self, v): + self.v = v + self.x_l = None + self.lres = None + self.f_min = None + self.lbounds = [] + + +class LMapCache: + def __init__(self): + self.cache = {} + + # Lists for search queries + self.v_maps = [] + self.xl_maps = [] + self.xl_maps_set = set() + self.f_maps = [] + self.lbound_maps = [] + self.size = 0 + + def __getitem__(self, v): + try: + v = np.ndarray.tolist(v) + except TypeError: + pass + v = tuple(v) + try: + return self.cache[v] + except KeyError: + xval = LMap(v) + self.cache[v] = xval + + return self.cache[v] + + def add_res(self, v, lres, bounds=None): + v = np.ndarray.tolist(v) + v = tuple(v) + self.cache[v].x_l = lres.x + self.cache[v].lres = lres + self.cache[v].f_min = lres.fun + self.cache[v].lbounds = bounds + + # Update cache size + self.size += 1 + + # Cache lists for search queries + self.v_maps.append(v) + self.xl_maps.append(lres.x) + self.xl_maps_set.add(tuple(lres.x)) + self.f_maps.append(lres.fun) + self.lbound_maps.append(bounds) + + def sort_cache_result(self): + """ + Sort results and build the global return object + """ + results = {} + # Sort results and save + self.xl_maps = np.array(self.xl_maps) + self.f_maps = np.array(self.f_maps) + + # Sorted indexes in Func_min + ind_sorted = np.argsort(self.f_maps) + + # Save ordered list of minima + results['xl'] = self.xl_maps[ind_sorted] # Ordered x vals + self.f_maps = np.array(self.f_maps) + results['funl'] = self.f_maps[ind_sorted] + results['funl'] = results['funl'].T + + # Find global of all minimizers + results['x'] = self.xl_maps[ind_sorted[0]] # Save global minima + results['fun'] = self.f_maps[ind_sorted[0]] # Save global fun value + + self.xl_maps = np.ndarray.tolist(self.xl_maps) + self.f_maps = np.ndarray.tolist(self.f_maps) + return results diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo_lib/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo_lib/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo_lib/_complex.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo_lib/_complex.py new file mode 100644 index 0000000000000000000000000000000000000000..178e9daea2d03e335a6901483f574d7453fb9340 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo_lib/_complex.py @@ -0,0 +1,1225 @@ +"""Base classes for low memory simplicial complex structures.""" +import copy +import logging +import itertools +import decimal +from functools import cache + +import numpy as np + +from ._vertex import (VertexCacheField, VertexCacheIndex) + + +class Complex: + """ + Base class for a simplicial complex described as a cache of vertices + together with their connections. + + Important methods: + Domain triangulation: + Complex.triangulate, Complex.split_generation + Triangulating arbitrary points (must be traingulable, + may exist outside domain): + Complex.triangulate(sample_set) + Converting another simplicial complex structure data type to the + structure used in Complex (ex. OBJ wavefront) + Complex.convert(datatype, data) + + Important objects: + HC.V: The cache of vertices and their connection + HC.H: Storage structure of all vertex groups + + Parameters + ---------- + dim : int + Spatial dimensionality of the complex R^dim + domain : list of tuples, optional + The bounds [x_l, x_u]^dim of the hyperrectangle space + ex. The default domain is the hyperrectangle [0, 1]^dim + Note: The domain must be convex, non-convex spaces can be cut + away from this domain using the non-linear + g_cons functions to define any arbitrary domain + (these domains may also be disconnected from each other) + sfield : + A scalar function defined in the associated domain f: R^dim --> R + sfield_args : tuple + Additional arguments to be passed to `sfield` + vfield : + A scalar function defined in the associated domain + f: R^dim --> R^m + (for example a gradient function of the scalar field) + vfield_args : tuple + Additional arguments to be passed to vfield + symmetry : None or list + Specify if the objective function contains symmetric variables. + The search space (and therefore performance) is decreased by up to + O(n!) times in the fully symmetric case. + + E.g. f(x) = (x_1 + x_2 + x_3) + (x_4)**2 + (x_5)**2 + (x_6)**2 + + In this equation x_2 and x_3 are symmetric to x_1, while x_5 and + x_6 are symmetric to x_4, this can be specified to the solver as: + + symmetry = [0, # Variable 1 + 0, # symmetric to variable 1 + 0, # symmetric to variable 1 + 3, # Variable 4 + 3, # symmetric to variable 4 + 3, # symmetric to variable 4 + ] + + constraints : dict or sequence of dict, optional + Constraints definition. + Function(s) ``R**n`` in the form:: + + g(x) <= 0 applied as g : R^n -> R^m + h(x) == 0 applied as h : R^n -> R^p + + Each constraint is defined in a dictionary with fields: + + type : str + Constraint type: 'eq' for equality, 'ineq' for inequality. + fun : callable + The function defining the constraint. + jac : callable, optional + The Jacobian of `fun` (only for SLSQP). + args : sequence, optional + Extra arguments to be passed to the function and Jacobian. + + Equality constraint means that the constraint function result is to + be zero whereas inequality means that it is to be + non-negative.constraints : dict or sequence of dict, optional + Constraints definition. + Function(s) ``R**n`` in the form:: + + g(x) <= 0 applied as g : R^n -> R^m + h(x) == 0 applied as h : R^n -> R^p + + Each constraint is defined in a dictionary with fields: + + type : str + Constraint type: 'eq' for equality, 'ineq' for inequality. + fun : callable + The function defining the constraint. + jac : callable, optional + The Jacobian of `fun` (unused). + args : sequence, optional + Extra arguments to be passed to the function and Jacobian. + + Equality constraint means that the constraint function result is to + be zero whereas inequality means that it is to be non-negative. + + workers : int optional + Uses `multiprocessing.Pool `) to compute the field + functions in parallel. + """ + def __init__(self, dim, domain=None, sfield=None, sfield_args=(), + symmetry=None, constraints=None, workers=1): + self.dim = dim + + # Domains + self.domain = domain + if domain is None: + self.bounds = [(0.0, 1.0), ] * dim + else: + self.bounds = domain + self.symmetry = symmetry + # here in init to avoid if checks + + # Field functions + self.sfield = sfield + self.sfield_args = sfield_args + + # Process constraints + # Constraints + # Process constraint dict sequence: + if constraints is not None: + self.min_cons = constraints + self.g_cons = [] + self.g_args = [] + if not isinstance(constraints, (tuple, list)): + constraints = (constraints,) + + for cons in constraints: + if cons['type'] in ('ineq'): + self.g_cons.append(cons['fun']) + try: + self.g_args.append(cons['args']) + except KeyError: + self.g_args.append(()) + self.g_cons = tuple(self.g_cons) + self.g_args = tuple(self.g_args) + else: + self.g_cons = None + self.g_args = None + + # Homology properties + self.gen = 0 + self.perm_cycle = 0 + + # Every cell is stored in a list of its generation, + # ex. the initial cell is stored in self.H[0] + # 1st get new cells are stored in self.H[1] etc. + # When a cell is sub-generated it is removed from this list + + self.H = [] # Storage structure of vertex groups + + # Cache of all vertices + if (sfield is not None) or (self.g_cons is not None): + # Initiate a vertex cache and an associated field cache, note that + # the field case is always initiated inside the vertex cache if an + # associated field scalar field is defined: + if sfield is not None: + self.V = VertexCacheField(field=sfield, field_args=sfield_args, + g_cons=self.g_cons, + g_cons_args=self.g_args, + workers=workers) + elif self.g_cons is not None: + self.V = VertexCacheField(field=sfield, field_args=sfield_args, + g_cons=self.g_cons, + g_cons_args=self.g_args, + workers=workers) + else: + self.V = VertexCacheIndex() + + self.V_non_symm = [] # List of non-symmetric vertices + + def __call__(self): + return self.H + + # %% Triangulation methods + def cyclic_product(self, bounds, origin, supremum, centroid=True): + """Generate initial triangulation using cyclic product""" + # Define current hyperrectangle + vot = tuple(origin) + vut = tuple(supremum) # Hyperrectangle supremum + self.V[vot] + vo = self.V[vot] + yield vo.x + self.V[vut].connect(self.V[vot]) + yield vut + # Cyclic group approach with second x_l --- x_u operation. + + # These containers store the "lower" and "upper" vertices + # corresponding to the origin or supremum of every C2 group. + # It has the structure of `dim` times embedded lists each containing + # these vertices as the entire complex grows. Bounds[0] has to be done + # outside the loops before we have symmetric containers. + # NOTE: This means that bounds[0][1] must always exist + C0x = [[self.V[vot]]] + a_vo = copy.copy(list(origin)) + a_vo[0] = vut[0] # Update aN Origin + a_vo = self.V[tuple(a_vo)] + # self.V[vot].connect(self.V[tuple(a_vo)]) + self.V[vot].connect(a_vo) + yield a_vo.x + C1x = [[a_vo]] + # C1x = [[self.V[tuple(a_vo)]]] + ab_C = [] # Container for a + b operations + + # Loop over remaining bounds + for i, x in enumerate(bounds[1:]): + # Update lower and upper containers + C0x.append([]) + C1x.append([]) + # try to access a second bound (if not, C1 is symmetric) + try: + # Early try so that we don't have to copy the cache before + # moving on to next C1/C2: Try to add the operation of a new + # C2 product by accessing the upper bound + x[1] + # Copy lists for iteration + cC0x = [x[:] for x in C0x[:i + 1]] + cC1x = [x[:] for x in C1x[:i + 1]] + for j, (VL, VU) in enumerate(zip(cC0x, cC1x)): + for k, (vl, vu) in enumerate(zip(VL, VU)): + # Build aN vertices for each lower-upper pair in N: + a_vl = list(vl.x) + a_vu = list(vu.x) + a_vl[i + 1] = vut[i + 1] + a_vu[i + 1] = vut[i + 1] + a_vl = self.V[tuple(a_vl)] + + # Connect vertices in N to corresponding vertices + # in aN: + vl.connect(a_vl) + + yield a_vl.x + + a_vu = self.V[tuple(a_vu)] + # Connect vertices in N to corresponding vertices + # in aN: + vu.connect(a_vu) + + # Connect new vertex pair in aN: + a_vl.connect(a_vu) + + # Connect lower pair to upper (triangulation + # operation of a + b (two arbitrary operations): + vl.connect(a_vu) + ab_C.append((vl, a_vu)) + + # Update the containers + C0x[i + 1].append(vl) + C0x[i + 1].append(vu) + C1x[i + 1].append(a_vl) + C1x[i + 1].append(a_vu) + + # Update old containers + C0x[j].append(a_vl) + C1x[j].append(a_vu) + + # Yield new points + yield a_vu.x + + # Try to connect aN lower source of previous a + b + # operation with a aN vertex + ab_Cc = copy.copy(ab_C) + + for vp in ab_Cc: + b_v = list(vp[0].x) + ab_v = list(vp[1].x) + b_v[i + 1] = vut[i + 1] + ab_v[i + 1] = vut[i + 1] + b_v = self.V[tuple(b_v)] # b + vl + ab_v = self.V[tuple(ab_v)] # b + a_vl + # Note o---o is already connected + vp[0].connect(ab_v) # o-s + b_v.connect(ab_v) # s-s + + # Add new list of cross pairs + ab_C.append((vp[0], ab_v)) + ab_C.append((b_v, ab_v)) + + except IndexError: + cC0x = C0x[i] + cC1x = C1x[i] + VL, VU = cC0x, cC1x + for k, (vl, vu) in enumerate(zip(VL, VU)): + # Build aN vertices for each lower-upper pair in N: + a_vu = list(vu.x) + a_vu[i + 1] = vut[i + 1] + # Connect vertices in N to corresponding vertices + # in aN: + a_vu = self.V[tuple(a_vu)] + # Connect vertices in N to corresponding vertices + # in aN: + vu.connect(a_vu) + # Connect new vertex pair in aN: + # a_vl.connect(a_vu) + # Connect lower pair to upper (triangulation + # operation of a + b (two arbitrary operations): + vl.connect(a_vu) + ab_C.append((vl, a_vu)) + C0x[i + 1].append(vu) + C1x[i + 1].append(a_vu) + # Yield new points + a_vu.connect(self.V[vut]) + yield a_vu.x + ab_Cc = copy.copy(ab_C) + for vp in ab_Cc: + if vp[1].x[i] == vut[i]: + ab_v = list(vp[1].x) + ab_v[i + 1] = vut[i + 1] + ab_v = self.V[tuple(ab_v)] # b + a_vl + # Note o---o is already connected + vp[0].connect(ab_v) # o-s + + # Add new list of cross pairs + ab_C.append((vp[0], ab_v)) + + # Clean class trash + try: + del C0x + del cC0x + del C1x + del cC1x + del ab_C + del ab_Cc + except UnboundLocalError: + pass + + # Extra yield to ensure that the triangulation is completed + if centroid: + vo = self.V[vot] + vs = self.V[vut] + # Disconnect the origin and supremum + vo.disconnect(vs) + # Build centroid + vc = self.split_edge(vot, vut) + for v in vo.nn: + v.connect(vc) + yield vc.x + return vc.x + else: + yield vut + return vut + + def triangulate(self, n=None, symmetry=None, centroid=True, + printout=False): + """ + Triangulate the initial domain, if n is not None then a limited number + of points will be generated + + Parameters + ---------- + n : int, Number of points to be sampled. + symmetry : + + Ex. Dictionary/hashtable + f(x) = (x_1 + x_2 + x_3) + (x_4)**2 + (x_5)**2 + (x_6)**2 + + symmetry = symmetry[0]: 0, # Variable 1 + symmetry[1]: 0, # symmetric to variable 1 + symmetry[2]: 0, # symmetric to variable 1 + symmetry[3]: 3, # Variable 4 + symmetry[4]: 3, # symmetric to variable 4 + symmetry[5]: 3, # symmetric to variable 4 + } + centroid : bool, if True add a central point to the hypercube + printout : bool, if True print out results + + NOTES: + ------ + Rather than using the combinatorial algorithm to connect vertices we + make the following observation: + + The bound pairs are similar a C2 cyclic group and the structure is + formed using the cartesian product: + + H = C2 x C2 x C2 ... x C2 (dim times) + + So construct any normal subgroup N and consider H/N first, we connect + all vertices within N (ex. N is C2 (the first dimension), then we move + to a left coset aN (an operation moving around the defined H/N group by + for example moving from the lower bound in C2 (dimension 2) to the + higher bound in C2. During this operation connection all the vertices. + Now repeat the N connections. Note that these elements can be connected + in parallel. + """ + # Inherit class arguments + if symmetry is None: + symmetry = self.symmetry + # Build origin and supremum vectors + origin = [i[0] for i in self.bounds] + self.origin = origin + supremum = [i[1] for i in self.bounds] + + self.supremum = supremum + + if symmetry is None: + cbounds = self.bounds + else: + cbounds = copy.copy(self.bounds) + for i, j in enumerate(symmetry): + if i is not j: + # pop second entry on second symmetry vars + cbounds[i] = [self.bounds[symmetry[i]][0]] + # Sole (first) entry is the sup value and there is no + # origin: + cbounds[i] = [self.bounds[symmetry[i]][1]] + if (self.bounds[symmetry[i]] is not + self.bounds[symmetry[j]]): + logging.warning(f"Variable {i} was specified as " + f"symmetric to variable {j}, however" + f", the bounds {i} =" + f" {self.bounds[symmetry[i]]} and {j}" + f" =" + f" {self.bounds[symmetry[j]]} do not " + f"match, the mismatch was ignored in " + f"the initial triangulation.") + cbounds[i] = self.bounds[symmetry[j]] + + if n is None: + # Build generator + self.cp = self.cyclic_product(cbounds, origin, supremum, centroid) + for i in self.cp: + i + + try: + self.triangulated_vectors.append((tuple(self.origin), + tuple(self.supremum))) + except (AttributeError, KeyError): + self.triangulated_vectors = [(tuple(self.origin), + tuple(self.supremum))] + + else: + # Check if generator already exists + try: + self.cp + except (AttributeError, KeyError): + self.cp = self.cyclic_product(cbounds, origin, supremum, + centroid) + + try: + while len(self.V.cache) < n: + next(self.cp) + except StopIteration: + try: + self.triangulated_vectors.append((tuple(self.origin), + tuple(self.supremum))) + except (AttributeError, KeyError): + self.triangulated_vectors = [(tuple(self.origin), + tuple(self.supremum))] + + if printout: + # for v in self.C0(): + # v.print_out() + for v in self.V.cache: + self.V[v].print_out() + + return + + def refine(self, n=1): + if n is None: + try: + self.triangulated_vectors + self.refine_all() + return + except AttributeError as ae: + if str(ae) == "'Complex' object has no attribute " \ + "'triangulated_vectors'": + self.triangulate(symmetry=self.symmetry) + return + else: + raise + + nt = len(self.V.cache) + n # Target number of total vertices + # In the outer while loop we iterate until we have added an extra `n` + # vertices to the complex: + while len(self.V.cache) < nt: # while loop 1 + try: # try 1 + # Try to access triangulated_vectors, this should only be + # defined if an initial triangulation has already been + # performed: + self.triangulated_vectors + # Try a usual iteration of the current generator, if it + # does not exist or is exhausted then produce a new generator + try: # try 2 + next(self.rls) + except (AttributeError, StopIteration, KeyError): + vp = self.triangulated_vectors[0] + self.rls = self.refine_local_space(*vp, bounds=self.bounds) + next(self.rls) + + except (AttributeError, KeyError): + # If an initial triangulation has not been completed, then + # we start/continue the initial triangulation targeting `nt` + # vertices, if nt is greater than the initial number of + # vertices then the `refine` routine will move back to try 1. + self.triangulate(nt, self.symmetry) + return + + def refine_all(self, centroids=True): + """Refine the entire domain of the current complex.""" + try: + self.triangulated_vectors + tvs = copy.copy(self.triangulated_vectors) + for i, vp in enumerate(tvs): + self.rls = self.refine_local_space(*vp, bounds=self.bounds) + for i in self.rls: + i + except AttributeError as ae: + if str(ae) == "'Complex' object has no attribute " \ + "'triangulated_vectors'": + self.triangulate(symmetry=self.symmetry, centroid=centroids) + else: + raise + + # This adds a centroid to every new sub-domain generated and defined + # by self.triangulated_vectors, in addition the vertices ! to complete + # the triangulation + return + + def refine_local_space(self, origin, supremum, bounds, centroid=1): + # Copy for later removal + origin_c = copy.copy(origin) + supremum_c = copy.copy(supremum) + + # Initiate local variables redefined in later inner `for` loop: + vl, vu, a_vu = None, None, None + + # Change the vector orientation so that it is only increasing + s_ov = list(origin) + s_origin = list(origin) + s_sv = list(supremum) + s_supremum = list(supremum) + for i, vi in enumerate(s_origin): + if s_ov[i] > s_sv[i]: + s_origin[i] = s_sv[i] + s_supremum[i] = s_ov[i] + + vot = tuple(s_origin) + vut = tuple(s_supremum) # Hyperrectangle supremum + + vo = self.V[vot] # initiate if doesn't exist yet + vs = self.V[vut] + # Start by finding the old centroid of the new space: + vco = self.split_edge(vo.x, vs.x) # Split in case not centroid arg + + # Find set of extreme vertices in current local space + sup_set = copy.copy(vco.nn) + # Cyclic group approach with second x_l --- x_u operation. + + # These containers store the "lower" and "upper" vertices + # corresponding to the origin or supremum of every C2 group. + # It has the structure of `dim` times embedded lists each containing + # these vertices as the entire complex grows. Bounds[0] has to be done + # outside the loops before we have symmetric containers. + # NOTE: This means that bounds[0][1] must always exist + + a_vl = copy.copy(list(vot)) + a_vl[0] = vut[0] # Update aN Origin + if tuple(a_vl) not in self.V.cache: + vo = self.V[vot] # initiate if doesn't exist yet + vs = self.V[vut] + # Start by finding the old centroid of the new space: + vco = self.split_edge(vo.x, vs.x) # Split in case not centroid arg + + # Find set of extreme vertices in current local space + sup_set = copy.copy(vco.nn) + a_vl = copy.copy(list(vot)) + a_vl[0] = vut[0] # Update aN Origin + a_vl = self.V[tuple(a_vl)] + else: + a_vl = self.V[tuple(a_vl)] + + c_v = self.split_edge(vo.x, a_vl.x) + c_v.connect(vco) + yield c_v.x + Cox = [[vo]] + Ccx = [[c_v]] + Cux = [[a_vl]] + ab_C = [] # Container for a + b operations + s_ab_C = [] # Container for symmetric a + b operations + + # Loop over remaining bounds + for i, x in enumerate(bounds[1:]): + # Update lower and upper containers + Cox.append([]) + Ccx.append([]) + Cux.append([]) + # try to access a second bound (if not, C1 is symmetric) + try: + t_a_vl = list(vot) + t_a_vl[i + 1] = vut[i + 1] + + # New: lists are used anyway, so copy all + # %% + # Copy lists for iteration + cCox = [x[:] for x in Cox[:i + 1]] + cCcx = [x[:] for x in Ccx[:i + 1]] + cCux = [x[:] for x in Cux[:i + 1]] + # Try to connect aN lower source of previous a + b + # operation with a aN vertex + ab_Cc = copy.copy(ab_C) # NOTE: We append ab_C in the + # (VL, VC, VU) for-loop, but we use the copy of the list in the + # ab_Cc for-loop. + s_ab_Cc = copy.copy(s_ab_C) + + # Early try so that we don't have to copy the cache before + # moving on to next C1/C2: Try to add the operation of a new + # C2 product by accessing the upper bound + if tuple(t_a_vl) not in self.V.cache: + # Raise error to continue symmetric refine + raise IndexError + t_a_vu = list(vut) + t_a_vu[i + 1] = vut[i + 1] + if tuple(t_a_vu) not in self.V.cache: + # Raise error to continue symmetric refine: + raise IndexError + + for vectors in s_ab_Cc: + # s_ab_C.append([c_vc, vl, vu, a_vu]) + bc_vc = list(vectors[0].x) + b_vl = list(vectors[1].x) + b_vu = list(vectors[2].x) + ba_vu = list(vectors[3].x) + + bc_vc[i + 1] = vut[i + 1] + b_vl[i + 1] = vut[i + 1] + b_vu[i + 1] = vut[i + 1] + ba_vu[i + 1] = vut[i + 1] + + bc_vc = self.V[tuple(bc_vc)] + bc_vc.connect(vco) # NOTE: Unneeded? + yield bc_vc + + # Split to centre, call this centre group "d = 0.5*a" + d_bc_vc = self.split_edge(vectors[0].x, bc_vc.x) + d_bc_vc.connect(bc_vc) + d_bc_vc.connect(vectors[1]) # Connect all to centroid + d_bc_vc.connect(vectors[2]) # Connect all to centroid + d_bc_vc.connect(vectors[3]) # Connect all to centroid + yield d_bc_vc.x + b_vl = self.V[tuple(b_vl)] + bc_vc.connect(b_vl) # Connect aN cross pairs + d_bc_vc.connect(b_vl) # Connect all to centroid + + yield b_vl + b_vu = self.V[tuple(b_vu)] + bc_vc.connect(b_vu) # Connect aN cross pairs + d_bc_vc.connect(b_vu) # Connect all to centroid + + b_vl_c = self.split_edge(b_vu.x, b_vl.x) + bc_vc.connect(b_vl_c) + + yield b_vu + ba_vu = self.V[tuple(ba_vu)] + bc_vc.connect(ba_vu) # Connect aN cross pairs + d_bc_vc.connect(ba_vu) # Connect all to centroid + + # Split the a + b edge of the initial triangulation: + os_v = self.split_edge(vectors[1].x, ba_vu.x) # o-s + ss_v = self.split_edge(b_vl.x, ba_vu.x) # s-s + b_vu_c = self.split_edge(b_vu.x, ba_vu.x) + bc_vc.connect(b_vu_c) + yield os_v.x # often equal to vco, but not always + yield ss_v.x # often equal to bc_vu, but not always + yield ba_vu + # Split remaining to centre, call this centre group + # "d = 0.5*a" + d_bc_vc = self.split_edge(vectors[0].x, bc_vc.x) + d_bc_vc.connect(vco) # NOTE: Unneeded? + yield d_bc_vc.x + d_b_vl = self.split_edge(vectors[1].x, b_vl.x) + d_bc_vc.connect(vco) # NOTE: Unneeded? + d_bc_vc.connect(d_b_vl) # Connect dN cross pairs + yield d_b_vl.x + d_b_vu = self.split_edge(vectors[2].x, b_vu.x) + d_bc_vc.connect(vco) # NOTE: Unneeded? + d_bc_vc.connect(d_b_vu) # Connect dN cross pairs + yield d_b_vu.x + d_ba_vu = self.split_edge(vectors[3].x, ba_vu.x) + d_bc_vc.connect(vco) # NOTE: Unneeded? + d_bc_vc.connect(d_ba_vu) # Connect dN cross pairs + yield d_ba_vu + + # comb = [c_vc, vl, vu, a_vl, a_vu, + # bc_vc, b_vl, b_vu, ba_vl, ba_vu] + comb = [vl, vu, a_vu, + b_vl, b_vu, ba_vu] + comb_iter = itertools.combinations(comb, 2) + for vecs in comb_iter: + self.split_edge(vecs[0].x, vecs[1].x) + # Add new list of cross pairs + ab_C.append((d_bc_vc, vectors[1], b_vl, a_vu, ba_vu)) + ab_C.append((d_bc_vc, vl, b_vl, a_vu, ba_vu)) # = prev + + for vectors in ab_Cc: + bc_vc = list(vectors[0].x) + b_vl = list(vectors[1].x) + b_vu = list(vectors[2].x) + ba_vl = list(vectors[3].x) + ba_vu = list(vectors[4].x) + bc_vc[i + 1] = vut[i + 1] + b_vl[i + 1] = vut[i + 1] + b_vu[i + 1] = vut[i + 1] + ba_vl[i + 1] = vut[i + 1] + ba_vu[i + 1] = vut[i + 1] + bc_vc = self.V[tuple(bc_vc)] + bc_vc.connect(vco) # NOTE: Unneeded? + yield bc_vc + + # Split to centre, call this centre group "d = 0.5*a" + d_bc_vc = self.split_edge(vectors[0].x, bc_vc.x) + d_bc_vc.connect(bc_vc) + d_bc_vc.connect(vectors[1]) # Connect all to centroid + d_bc_vc.connect(vectors[2]) # Connect all to centroid + d_bc_vc.connect(vectors[3]) # Connect all to centroid + d_bc_vc.connect(vectors[4]) # Connect all to centroid + yield d_bc_vc.x + b_vl = self.V[tuple(b_vl)] + bc_vc.connect(b_vl) # Connect aN cross pairs + d_bc_vc.connect(b_vl) # Connect all to centroid + yield b_vl + b_vu = self.V[tuple(b_vu)] + bc_vc.connect(b_vu) # Connect aN cross pairs + d_bc_vc.connect(b_vu) # Connect all to centroid + yield b_vu + ba_vl = self.V[tuple(ba_vl)] + bc_vc.connect(ba_vl) # Connect aN cross pairs + d_bc_vc.connect(ba_vl) # Connect all to centroid + self.split_edge(b_vu.x, ba_vl.x) + yield ba_vl + ba_vu = self.V[tuple(ba_vu)] + bc_vc.connect(ba_vu) # Connect aN cross pairs + d_bc_vc.connect(ba_vu) # Connect all to centroid + # Split the a + b edge of the initial triangulation: + os_v = self.split_edge(vectors[1].x, ba_vu.x) # o-s + ss_v = self.split_edge(b_vl.x, ba_vu.x) # s-s + yield os_v.x # often equal to vco, but not always + yield ss_v.x # often equal to bc_vu, but not always + yield ba_vu + # Split remaining to centre, call this centre group + # "d = 0.5*a" + d_bc_vc = self.split_edge(vectors[0].x, bc_vc.x) + d_bc_vc.connect(vco) # NOTE: Unneeded? + yield d_bc_vc.x + d_b_vl = self.split_edge(vectors[1].x, b_vl.x) + d_bc_vc.connect(vco) # NOTE: Unneeded? + d_bc_vc.connect(d_b_vl) # Connect dN cross pairs + yield d_b_vl.x + d_b_vu = self.split_edge(vectors[2].x, b_vu.x) + d_bc_vc.connect(vco) # NOTE: Unneeded? + d_bc_vc.connect(d_b_vu) # Connect dN cross pairs + yield d_b_vu.x + d_ba_vl = self.split_edge(vectors[3].x, ba_vl.x) + d_bc_vc.connect(vco) # NOTE: Unneeded? + d_bc_vc.connect(d_ba_vl) # Connect dN cross pairs + yield d_ba_vl + d_ba_vu = self.split_edge(vectors[4].x, ba_vu.x) + d_bc_vc.connect(vco) # NOTE: Unneeded? + d_bc_vc.connect(d_ba_vu) # Connect dN cross pairs + yield d_ba_vu + c_vc, vl, vu, a_vl, a_vu = vectors + + comb = [vl, vu, a_vl, a_vu, + b_vl, b_vu, ba_vl, ba_vu] + comb_iter = itertools.combinations(comb, 2) + for vecs in comb_iter: + self.split_edge(vecs[0].x, vecs[1].x) + + # Add new list of cross pairs + ab_C.append((bc_vc, b_vl, b_vu, ba_vl, ba_vu)) + ab_C.append((d_bc_vc, d_b_vl, d_b_vu, d_ba_vl, d_ba_vu)) + ab_C.append((d_bc_vc, vectors[1], b_vl, a_vu, ba_vu)) + ab_C.append((d_bc_vc, vu, b_vu, a_vl, ba_vl)) + + for j, (VL, VC, VU) in enumerate(zip(cCox, cCcx, cCux)): + for k, (vl, vc, vu) in enumerate(zip(VL, VC, VU)): + # Build aN vertices for each lower-upper C3 group in N: + a_vl = list(vl.x) + a_vu = list(vu.x) + a_vl[i + 1] = vut[i + 1] + a_vu[i + 1] = vut[i + 1] + a_vl = self.V[tuple(a_vl)] + a_vu = self.V[tuple(a_vu)] + # Note, build (a + vc) later for consistent yields + # Split the a + b edge of the initial triangulation: + c_vc = self.split_edge(vl.x, a_vu.x) + self.split_edge(vl.x, vu.x) # Equal to vc + # Build cN vertices for each lower-upper C3 group in N: + c_vc.connect(vco) + c_vc.connect(vc) + c_vc.connect(vl) # Connect c + ac operations + c_vc.connect(vu) # Connect c + ac operations + c_vc.connect(a_vl) # Connect c + ac operations + c_vc.connect(a_vu) # Connect c + ac operations + yield c_vc.x + c_vl = self.split_edge(vl.x, a_vl.x) + c_vl.connect(vco) + c_vc.connect(c_vl) # Connect cN group vertices + yield c_vl.x + # yield at end of loop: + c_vu = self.split_edge(vu.x, a_vu.x) + c_vu.connect(vco) + # Connect remaining cN group vertices + c_vc.connect(c_vu) # Connect cN group vertices + yield c_vu.x + + a_vc = self.split_edge(a_vl.x, a_vu.x) # is (a + vc) ? + a_vc.connect(vco) + a_vc.connect(c_vc) + + # Storage for connecting c + ac operations: + ab_C.append((c_vc, vl, vu, a_vl, a_vu)) + + # Update the containers + Cox[i + 1].append(vl) + Cox[i + 1].append(vc) + Cox[i + 1].append(vu) + Ccx[i + 1].append(c_vl) + Ccx[i + 1].append(c_vc) + Ccx[i + 1].append(c_vu) + Cux[i + 1].append(a_vl) + Cux[i + 1].append(a_vc) + Cux[i + 1].append(a_vu) + + # Update old containers + Cox[j].append(c_vl) # ! + Cox[j].append(a_vl) + Ccx[j].append(c_vc) # ! + Ccx[j].append(a_vc) # ! + Cux[j].append(c_vu) # ! + Cux[j].append(a_vu) + + # Yield new points + yield a_vc.x + + except IndexError: + for vectors in ab_Cc: + ba_vl = list(vectors[3].x) + ba_vu = list(vectors[4].x) + ba_vl[i + 1] = vut[i + 1] + ba_vu[i + 1] = vut[i + 1] + ba_vu = self.V[tuple(ba_vu)] + yield ba_vu + d_bc_vc = self.split_edge(vectors[1].x, ba_vu.x) # o-s + yield ba_vu + d_bc_vc.connect(vectors[1]) # Connect all to centroid + d_bc_vc.connect(vectors[2]) # Connect all to centroid + d_bc_vc.connect(vectors[3]) # Connect all to centroid + d_bc_vc.connect(vectors[4]) # Connect all to centroid + yield d_bc_vc.x + ba_vl = self.V[tuple(ba_vl)] + yield ba_vl + d_ba_vl = self.split_edge(vectors[3].x, ba_vl.x) + d_ba_vu = self.split_edge(vectors[4].x, ba_vu.x) + d_ba_vc = self.split_edge(d_ba_vl.x, d_ba_vu.x) + yield d_ba_vl + yield d_ba_vu + yield d_ba_vc + c_vc, vl, vu, a_vl, a_vu = vectors + comb = [vl, vu, a_vl, a_vu, + ba_vl, + ba_vu] + comb_iter = itertools.combinations(comb, 2) + for vecs in comb_iter: + self.split_edge(vecs[0].x, vecs[1].x) + + # Copy lists for iteration + cCox = Cox[i] + cCcx = Ccx[i] + cCux = Cux[i] + VL, VC, VU = cCox, cCcx, cCux + for k, (vl, vc, vu) in enumerate(zip(VL, VC, VU)): + # Build aN vertices for each lower-upper pair in N: + a_vu = list(vu.x) + a_vu[i + 1] = vut[i + 1] + + # Connect vertices in N to corresponding vertices + # in aN: + a_vu = self.V[tuple(a_vu)] + yield a_vl.x + # Split the a + b edge of the initial triangulation: + c_vc = self.split_edge(vl.x, a_vu.x) + self.split_edge(vl.x, vu.x) # Equal to vc + c_vc.connect(vco) + c_vc.connect(vc) + c_vc.connect(vl) # Connect c + ac operations + c_vc.connect(vu) # Connect c + ac operations + c_vc.connect(a_vu) # Connect c + ac operations + yield (c_vc.x) + c_vu = self.split_edge(vu.x, + a_vu.x) # yield at end of loop + c_vu.connect(vco) + # Connect remaining cN group vertices + c_vc.connect(c_vu) # Connect cN group vertices + yield (c_vu.x) + + # Update the containers + Cox[i + 1].append(vu) + Ccx[i + 1].append(c_vu) + Cux[i + 1].append(a_vu) + + # Update old containers + s_ab_C.append([c_vc, vl, vu, a_vu]) + + yield a_vu.x + + # Clean class trash + try: + del Cox + del Ccx + del Cux + del ab_C + del ab_Cc + except UnboundLocalError: + pass + + try: + self.triangulated_vectors.remove((tuple(origin_c), + tuple(supremum_c))) + except ValueError: + # Turn this into a logging warning? + pass + # Add newly triangulated vectors: + for vs in sup_set: + self.triangulated_vectors.append((tuple(vco.x), tuple(vs.x))) + + # Extra yield to ensure that the triangulation is completed + if centroid: + vcn_set = set() + c_nn_lists = [] + for vs in sup_set: + # Build centroid + c_nn = self.vpool(vco.x, vs.x) + try: + c_nn.remove(vcn_set) + except KeyError: + pass + c_nn_lists.append(c_nn) + + for c_nn in c_nn_lists: + try: + c_nn.remove(vcn_set) + except KeyError: + pass + + for vs, c_nn in zip(sup_set, c_nn_lists): + # Build centroid + vcn = self.split_edge(vco.x, vs.x) + vcn_set.add(vcn) + try: # Shouldn't be needed? + c_nn.remove(vcn_set) + except KeyError: + pass + for vnn in c_nn: + vcn.connect(vnn) + yield vcn.x + else: + pass + + yield vut + return + + def refine_star(self, v): + """Refine the star domain of a vertex `v`.""" + # Copy lists before iteration + vnn = copy.copy(v.nn) + v1nn = [] + d_v0v1_set = set() + for v1 in vnn: + v1nn.append(copy.copy(v1.nn)) + + for v1, v1nn in zip(vnn, v1nn): + vnnu = v1nn.intersection(vnn) + + d_v0v1 = self.split_edge(v.x, v1.x) + for o_d_v0v1 in d_v0v1_set: + d_v0v1.connect(o_d_v0v1) + d_v0v1_set.add(d_v0v1) + for v2 in vnnu: + d_v1v2 = self.split_edge(v1.x, v2.x) + d_v0v1.connect(d_v1v2) + return + + @cache + def split_edge(self, v1, v2): + v1 = self.V[v1] + v2 = self.V[v2] + # Destroy original edge, if it exists: + v1.disconnect(v2) + # Compute vertex on centre of edge: + try: + vct = (v2.x_a - v1.x_a) / 2.0 + v1.x_a + except TypeError: # Allow for decimal operations + vct = (v2.x_a - v1.x_a) / decimal.Decimal(2.0) + v1.x_a + + vc = self.V[tuple(vct)] + # Connect to original 2 vertices to the new centre vertex + vc.connect(v1) + vc.connect(v2) + return vc + + def vpool(self, origin, supremum): + vot = tuple(origin) + vst = tuple(supremum) + # Initiate vertices in case they don't exist + vo = self.V[vot] + vs = self.V[vst] + + # Remove origin - supremum disconnect + + # Find the lower/upper bounds of the refinement hyperrectangle + bl = list(vot) + bu = list(vst) + for i, (voi, vsi) in enumerate(zip(vot, vst)): + if bl[i] > vsi: + bl[i] = vsi + if bu[i] < voi: + bu[i] = voi + + # NOTE: This is mostly done with sets/lists because we aren't sure + # how well the numpy arrays will scale to thousands of + # dimensions. + vn_pool = set() + vn_pool.update(vo.nn) + vn_pool.update(vs.nn) + cvn_pool = copy.copy(vn_pool) + for vn in cvn_pool: + for i, xi in enumerate(vn.x): + if bl[i] <= xi <= bu[i]: + pass + else: + try: + vn_pool.remove(vn) + except KeyError: + pass # NOTE: Not all neighbours are in initial pool + return vn_pool + + def vf_to_vv(self, vertices, simplices): + """ + Convert a vertex-face mesh to a vertex-vertex mesh used by this class + + Parameters + ---------- + vertices : list + Vertices + simplices : list + Simplices + """ + if self.dim > 1: + for s in simplices: + edges = itertools.combinations(s, self.dim) + for e in edges: + self.V[tuple(vertices[e[0]])].connect( + self.V[tuple(vertices[e[1]])]) + else: + for e in simplices: + self.V[tuple(vertices[e[0]])].connect( + self.V[tuple(vertices[e[1]])]) + return + + def connect_vertex_non_symm(self, v_x, near=None): + """ + Adds a vertex at coords v_x to the complex that is not symmetric to the + initial triangulation and sub-triangulation. + + If near is specified (for example; a star domain or collections of + cells known to contain v) then only those simplices containd in near + will be searched, this greatly speeds up the process. + + If near is not specified this method will search the entire simplicial + complex structure. + + Parameters + ---------- + v_x : tuple + Coordinates of non-symmetric vertex + near : set or list + List of vertices, these are points near v to check for + """ + if near is None: + star = self.V + else: + star = near + # Create the vertex origin + if tuple(v_x) in self.V.cache: + if self.V[v_x] in self.V_non_symm: + pass + else: + return + + self.V[v_x] + found_nn = False + S_rows = [] + for v in star: + S_rows.append(v.x) + + S_rows = np.array(S_rows) + A = np.array(S_rows) - np.array(v_x) + # Iterate through all the possible simplices of S_rows + for s_i in itertools.combinations(range(S_rows.shape[0]), + r=self.dim + 1): + # Check if connected, else s_i is not a simplex + valid_simplex = True + for i in itertools.combinations(s_i, r=2): + # Every combination of vertices must be connected, we check of + # the current iteration of all combinations of s_i are + # connected we break the loop if it is not. + if ((self.V[tuple(S_rows[i[1]])] not in + self.V[tuple(S_rows[i[0]])].nn) + and (self.V[tuple(S_rows[i[0]])] not in + self.V[tuple(S_rows[i[1]])].nn)): + valid_simplex = False + break + + S = S_rows[tuple([s_i])] + if valid_simplex: + if self.deg_simplex(S, proj=None): + valid_simplex = False + + # If s_i is a valid simplex we can test if v_x is inside si + if valid_simplex: + # Find the A_j0 value from the precalculated values + A_j0 = A[tuple([s_i])] + if self.in_simplex(S, v_x, A_j0): + found_nn = True + # breaks the main for loop, s_i is the target simplex: + break + + # Connect the simplex to point + if found_nn: + for i in s_i: + self.V[v_x].connect(self.V[tuple(S_rows[i])]) + # Attached the simplex to storage for all non-symmetric vertices + self.V_non_symm.append(self.V[v_x]) + # this bool value indicates a successful connection if True: + return found_nn + + def in_simplex(self, S, v_x, A_j0=None): + """Check if a vector v_x is in simplex `S`. + + Parameters + ---------- + S : array_like + Array containing simplex entries of vertices as rows + v_x : + A candidate vertex + A_j0 : array, optional, + Allows for A_j0 to be pre-calculated + + Returns + ------- + res : boolean + True if `v_x` is in `S` + """ + A_11 = np.delete(S, 0, 0) - S[0] + + sign_det_A_11 = np.sign(np.linalg.det(A_11)) + if sign_det_A_11 == 0: + # NOTE: We keep the variable A_11, but we loop through A_jj + # ind= + # while sign_det_A_11 == 0: + # A_11 = np.delete(S, ind, 0) - S[ind] + # sign_det_A_11 = np.sign(np.linalg.det(A_11)) + + sign_det_A_11 = -1 # TODO: Choose another det of j instead? + # TODO: Unlikely to work in many cases + + if A_j0 is None: + A_j0 = S - v_x + + for d in range(self.dim + 1): + det_A_jj = (-1)**d * sign_det_A_11 + # TODO: Note that scipy might be faster to add as an optional + # dependency + sign_det_A_j0 = np.sign(np.linalg.det(np.delete(A_j0, d, + 0))) + # TODO: Note if sign_det_A_j0 == then the point is coplanar to the + # current simplex facet, so perhaps return True and attach? + if det_A_jj == sign_det_A_j0: + continue + else: + return False + + return True + + def deg_simplex(self, S, proj=None): + """Test a simplex S for degeneracy (linear dependence in R^dim). + + Parameters + ---------- + S : np.array + Simplex with rows as vertex vectors + proj : array, optional, + If the projection S[1:] - S[0] is already + computed it can be added as an optional argument. + """ + # Strategy: we test all combination of faces, if any of the + # determinants are zero then the vectors lie on the same face and is + # therefore linearly dependent in the space of R^dim + if proj is None: + proj = S[1:] - S[0] + + # TODO: Is checking the projection of one vertex against faces of other + # vertices sufficient? Or do we need to check more vertices in + # dimensions higher than 2? + # TODO: Literature seems to suggest using proj.T, but why is this + # needed? + if np.linalg.det(proj) == 0.0: # TODO: Replace with tolerance? + return True # Simplex is degenerate + else: + return False # Simplex is not degenerate diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo_lib/_vertex.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo_lib/_vertex.py new file mode 100644 index 0000000000000000000000000000000000000000..e47558ee7b9a181638841c34bb63603b5d37e221 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_shgo_lib/_vertex.py @@ -0,0 +1,460 @@ +import collections +from abc import ABC, abstractmethod + +import numpy as np + +from scipy._lib._util import MapWrapper + + +class VertexBase(ABC): + """ + Base class for a vertex. + """ + def __init__(self, x, nn=None, index=None): + """ + Initiation of a vertex object. + + Parameters + ---------- + x : tuple or vector + The geometric location (domain). + nn : list, optional + Nearest neighbour list. + index : int, optional + Index of vertex. + """ + self.x = x + self.hash = hash(self.x) # Save precomputed hash + + if nn is not None: + self.nn = set(nn) # can use .indexupdate to add a new list + else: + self.nn = set() + + self.index = index + + def __hash__(self): + return self.hash + + def __getattr__(self, item): + if item not in ['x_a']: + raise AttributeError(f"{type(self)} object has no attribute " + f"'{item}'") + if item == 'x_a': + self.x_a = np.array(self.x) + return self.x_a + + @abstractmethod + def connect(self, v): + raise NotImplementedError("This method is only implemented with an " + "associated child of the base class.") + + @abstractmethod + def disconnect(self, v): + raise NotImplementedError("This method is only implemented with an " + "associated child of the base class.") + + def star(self): + """Returns the star domain ``st(v)`` of the vertex. + + Parameters + ---------- + v : + The vertex ``v`` in ``st(v)`` + + Returns + ------- + st : set + A set containing all the vertices in ``st(v)`` + """ + self.st = self.nn + self.st.add(self) + return self.st + + +class VertexScalarField(VertexBase): + """ + Add homology properties of a scalar field f: R^n --> R associated with + the geometry built from the VertexBase class + """ + + def __init__(self, x, field=None, nn=None, index=None, field_args=(), + g_cons=None, g_cons_args=()): + """ + Parameters + ---------- + x : tuple, + vector of vertex coordinates + field : callable, optional + a scalar field f: R^n --> R associated with the geometry + nn : list, optional + list of nearest neighbours + index : int, optional + index of the vertex + field_args : tuple, optional + additional arguments to be passed to field + g_cons : callable, optional + constraints on the vertex + g_cons_args : tuple, optional + additional arguments to be passed to g_cons + + """ + super().__init__(x, nn=nn, index=index) + + # Note Vertex is only initiated once for all x so only + # evaluated once + # self.feasible = None + + # self.f is externally defined by the cache to allow parallel + # processing + # None type that will break arithmetic operations unless defined + # self.f = None + + self.check_min = True + self.check_max = True + + def connect(self, v): + """Connects self to another vertex object v. + + Parameters + ---------- + v : VertexBase or VertexScalarField object + """ + if v is not self and v not in self.nn: + self.nn.add(v) + v.nn.add(self) + + # Flags for checking homology properties: + self.check_min = True + self.check_max = True + v.check_min = True + v.check_max = True + + def disconnect(self, v): + if v in self.nn: + self.nn.remove(v) + v.nn.remove(self) + + # Flags for checking homology properties: + self.check_min = True + self.check_max = True + v.check_min = True + v.check_max = True + + def minimiser(self): + """Check whether this vertex is strictly less than all its + neighbours""" + if self.check_min: + self._min = all(self.f < v.f for v in self.nn) + self.check_min = False + + return self._min + + def maximiser(self): + """ + Check whether this vertex is strictly greater than all its + neighbours. + """ + if self.check_max: + self._max = all(self.f > v.f for v in self.nn) + self.check_max = False + + return self._max + + +class VertexVectorField(VertexBase): + """ + Add homology properties of a scalar field f: R^n --> R^m associated with + the geometry built from the VertexBase class. + """ + + def __init__(self, x, sfield=None, vfield=None, field_args=(), + vfield_args=(), g_cons=None, + g_cons_args=(), nn=None, index=None): + super().__init__(x, nn=nn, index=index) + + raise NotImplementedError("This class is still a work in progress") + + +class VertexCacheBase: + """Base class for a vertex cache for a simplicial complex.""" + def __init__(self): + + self.cache = collections.OrderedDict() + self.nfev = 0 # Feasible points + self.index = -1 + + def __iter__(self): + for v in self.cache: + yield self.cache[v] + return + + def size(self): + """Returns the size of the vertex cache.""" + return self.index + 1 + + def print_out(self): + headlen = len(f"Vertex cache of size: {len(self.cache)}:") + print('=' * headlen) + print(f"Vertex cache of size: {len(self.cache)}:") + print('=' * headlen) + for v in self.cache: + self.cache[v].print_out() + + +class VertexCube(VertexBase): + """Vertex class to be used for a pure simplicial complex with no associated + differential geometry (single level domain that exists in R^n)""" + def __init__(self, x, nn=None, index=None): + super().__init__(x, nn=nn, index=index) + + def connect(self, v): + if v is not self and v not in self.nn: + self.nn.add(v) + v.nn.add(self) + + def disconnect(self, v): + if v in self.nn: + self.nn.remove(v) + v.nn.remove(self) + + +class VertexCacheIndex(VertexCacheBase): + def __init__(self): + """ + Class for a vertex cache for a simplicial complex without an associated + field. Useful only for building and visualising a domain complex. + + Parameters + ---------- + """ + super().__init__() + self.Vertex = VertexCube + + def __getitem__(self, x, nn=None): + try: + return self.cache[x] + except KeyError: + self.index += 1 + xval = self.Vertex(x, index=self.index) + # logging.info("New generated vertex at x = {}".format(x)) + # NOTE: Surprisingly high performance increase if logging + # is commented out + self.cache[x] = xval + return self.cache[x] + + +class VertexCacheField(VertexCacheBase): + def __init__(self, field=None, field_args=(), g_cons=None, g_cons_args=(), + workers=1): + """ + Class for a vertex cache for a simplicial complex with an associated + field. + + Parameters + ---------- + field : callable + Scalar or vector field callable. + field_args : tuple, optional + Any additional fixed parameters needed to completely specify the + field function + g_cons : dict or sequence of dict, optional + Constraints definition. + Function(s) ``R**n`` in the form:: + g_cons_args : tuple, optional + Any additional fixed parameters needed to completely specify the + constraint functions + workers : int optional + Uses `multiprocessing.Pool `) to compute the field + functions in parallel. + + """ + super().__init__() + self.index = -1 + self.Vertex = VertexScalarField + self.field = field + self.field_args = field_args + self.wfield = FieldWrapper(field, field_args) # if workers is not 1 + + self.g_cons = g_cons + self.g_cons_args = g_cons_args + self.wgcons = ConstraintWrapper(g_cons, g_cons_args) + self.gpool = set() # A set of tuples to process for feasibility + + # Field processing objects + self.fpool = set() # A set of tuples to process for scalar function + self.sfc_lock = False # True if self.fpool is non-Empty + + self.workers = workers + self._mapwrapper = MapWrapper(workers) + + if workers == 1: + self.process_gpool = self.proc_gpool + if g_cons is None: + self.process_fpool = self.proc_fpool_nog + else: + self.process_fpool = self.proc_fpool_g + else: + self.process_gpool = self.pproc_gpool + if g_cons is None: + self.process_fpool = self.pproc_fpool_nog + else: + self.process_fpool = self.pproc_fpool_g + + def __getitem__(self, x, nn=None): + try: + return self.cache[x] + except KeyError: + self.index += 1 + xval = self.Vertex(x, field=self.field, nn=nn, index=self.index, + field_args=self.field_args, + g_cons=self.g_cons, + g_cons_args=self.g_cons_args) + + self.cache[x] = xval # Define in cache + self.gpool.add(xval) # Add to pool for processing feasibility + self.fpool.add(xval) # Add to pool for processing field values + return self.cache[x] + + def __getstate__(self): + self_dict = self.__dict__.copy() + del self_dict['pool'] + return self_dict + + def process_pools(self): + if self.g_cons is not None: + self.process_gpool() + self.process_fpool() + self.proc_minimisers() + + def feasibility_check(self, v): + v.feasible = True + for g, args in zip(self.g_cons, self.g_cons_args): + # constraint may return more than 1 value. + if np.any(g(v.x_a, *args) < 0.0): + v.f = np.inf + v.feasible = False + break + + def compute_sfield(self, v): + """Compute the scalar field values of a vertex object `v`. + + Parameters + ---------- + v : VertexBase or VertexScalarField object + """ + try: + v.f = self.field(v.x_a, *self.field_args) + self.nfev += 1 + except AttributeError: + v.f = np.inf + # logging.warning(f"Field function not found at x = {self.x_a}") + if np.isnan(v.f): + v.f = np.inf + + def proc_gpool(self): + """Process all constraints.""" + if self.g_cons is not None: + for v in self.gpool: + self.feasibility_check(v) + # Clean the pool + self.gpool = set() + + def pproc_gpool(self): + """Process all constraints in parallel.""" + gpool_l = [] + for v in self.gpool: + gpool_l.append(v.x_a) + + G = self._mapwrapper(self.wgcons.gcons, gpool_l) + for v, g in zip(self.gpool, G): + v.feasible = g # set vertex object attribute v.feasible = g (bool) + + def proc_fpool_g(self): + """Process all field functions with constraints supplied.""" + for v in self.fpool: + if v.feasible: + self.compute_sfield(v) + # Clean the pool + self.fpool = set() + + def proc_fpool_nog(self): + """Process all field functions with no constraints supplied.""" + for v in self.fpool: + self.compute_sfield(v) + # Clean the pool + self.fpool = set() + + def pproc_fpool_g(self): + """ + Process all field functions with constraints supplied in parallel. + """ + self.wfield.func + fpool_l = [] + for v in self.fpool: + if v.feasible: + fpool_l.append(v.x_a) + else: + v.f = np.inf + F = self._mapwrapper(self.wfield.func, fpool_l) + for va, f in zip(fpool_l, F): + vt = tuple(va) + self[vt].f = f # set vertex object attribute v.f = f + self.nfev += 1 + # Clean the pool + self.fpool = set() + + def pproc_fpool_nog(self): + """ + Process all field functions with no constraints supplied in parallel. + """ + self.wfield.func + fpool_l = [] + for v in self.fpool: + fpool_l.append(v.x_a) + F = self._mapwrapper(self.wfield.func, fpool_l) + for va, f in zip(fpool_l, F): + vt = tuple(va) + self[vt].f = f # set vertex object attribute v.f = f + self.nfev += 1 + # Clean the pool + self.fpool = set() + + def proc_minimisers(self): + """Check for minimisers.""" + for v in self: + v.minimiser() + v.maximiser() + + +class ConstraintWrapper: + """Object to wrap constraints to pass to `multiprocessing.Pool`.""" + def __init__(self, g_cons, g_cons_args): + self.g_cons = g_cons + self.g_cons_args = g_cons_args + + def gcons(self, v_x_a): + vfeasible = True + for g, args in zip(self.g_cons, self.g_cons_args): + # constraint may return more than 1 value. + if np.any(g(v_x_a, *args) < 0.0): + vfeasible = False + break + return vfeasible + + +class FieldWrapper: + """Object to wrap field to pass to `multiprocessing.Pool`.""" + def __init__(self, field, field_args): + self.field = field + self.field_args = field_args + + def func(self, v_x_a): + try: + v_f = self.field(v_x_a, *self.field_args) + except Exception: + v_f = np.inf + if np.isnan(v_f): + v_f = np.inf + + return v_f diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_slsqp.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_slsqp.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..1f64a0469d430cfacd71e0d9a2e72eb73fb28c66 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_slsqp.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_slsqp_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_slsqp_py.py new file mode 100644 index 0000000000000000000000000000000000000000..0ab66837497a6626d042d56530ead46ea134ad9c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_slsqp_py.py @@ -0,0 +1,511 @@ +""" +This module implements the Sequential Least Squares Programming optimization +algorithm (SLSQP), originally developed by Dieter Kraft. +See http://www.netlib.org/toms/733 + +Functions +--------- +.. autosummary:: + :toctree: generated/ + + approx_jacobian + fmin_slsqp + +""" + +__all__ = ['approx_jacobian', 'fmin_slsqp'] + +import numpy as np +from scipy.optimize._slsqp import slsqp +from numpy import (zeros, array, linalg, append, concatenate, finfo, + sqrt, vstack, isfinite, atleast_1d) +from ._optimize import (OptimizeResult, _check_unknown_options, + _prepare_scalar_function, _clip_x_for_func, + _check_clip_x) +from ._numdiff import approx_derivative +from ._constraints import old_bound_to_new, _arr_to_scalar +from scipy._lib._array_api import array_namespace +from scipy._lib import array_api_extra as xpx + + +__docformat__ = "restructuredtext en" + +_epsilon = sqrt(finfo(float).eps) + + +def approx_jacobian(x, func, epsilon, *args): + """ + Approximate the Jacobian matrix of a callable function. + + Parameters + ---------- + x : array_like + The state vector at which to compute the Jacobian matrix. + func : callable f(x,*args) + The vector-valued function. + epsilon : float + The perturbation used to determine the partial derivatives. + args : sequence + Additional arguments passed to func. + + Returns + ------- + An array of dimensions ``(lenf, lenx)`` where ``lenf`` is the length + of the outputs of `func`, and ``lenx`` is the number of elements in + `x`. + + Notes + ----- + The approximation is done using forward differences. + + """ + # approx_derivative returns (m, n) == (lenf, lenx) + jac = approx_derivative(func, x, method='2-point', abs_step=epsilon, + args=args) + # if func returns a scalar jac.shape will be (lenx,). Make sure + # it's at least a 2D array. + return np.atleast_2d(jac) + + +def fmin_slsqp(func, x0, eqcons=(), f_eqcons=None, ieqcons=(), f_ieqcons=None, + bounds=(), fprime=None, fprime_eqcons=None, + fprime_ieqcons=None, args=(), iter=100, acc=1.0E-6, + iprint=1, disp=None, full_output=0, epsilon=_epsilon, + callback=None): + """ + Minimize a function using Sequential Least Squares Programming + + Python interface function for the SLSQP Optimization subroutine + originally implemented by Dieter Kraft. + + Parameters + ---------- + func : callable f(x,*args) + Objective function. Must return a scalar. + x0 : 1-D ndarray of float + Initial guess for the independent variable(s). + eqcons : list, optional + A list of functions of length n such that + eqcons[j](x,*args) == 0.0 in a successfully optimized + problem. + f_eqcons : callable f(x,*args), optional + Returns a 1-D array in which each element must equal 0.0 in a + successfully optimized problem. If f_eqcons is specified, + eqcons is ignored. + ieqcons : list, optional + A list of functions of length n such that + ieqcons[j](x,*args) >= 0.0 in a successfully optimized + problem. + f_ieqcons : callable f(x,*args), optional + Returns a 1-D ndarray in which each element must be greater or + equal to 0.0 in a successfully optimized problem. If + f_ieqcons is specified, ieqcons is ignored. + bounds : list, optional + A list of tuples specifying the lower and upper bound + for each independent variable [(xl0, xu0),(xl1, xu1),...] + Infinite values will be interpreted as large floating values. + fprime : callable ``f(x,*args)``, optional + A function that evaluates the partial derivatives of func. + fprime_eqcons : callable ``f(x,*args)``, optional + A function of the form ``f(x, *args)`` that returns the m by n + array of equality constraint normals. If not provided, + the normals will be approximated. The array returned by + fprime_eqcons should be sized as ( len(eqcons), len(x0) ). + fprime_ieqcons : callable ``f(x,*args)``, optional + A function of the form ``f(x, *args)`` that returns the m by n + array of inequality constraint normals. If not provided, + the normals will be approximated. The array returned by + fprime_ieqcons should be sized as ( len(ieqcons), len(x0) ). + args : sequence, optional + Additional arguments passed to func and fprime. + iter : int, optional + The maximum number of iterations. + acc : float, optional + Requested accuracy. + iprint : int, optional + The verbosity of fmin_slsqp : + + * iprint <= 0 : Silent operation + * iprint == 1 : Print summary upon completion (default) + * iprint >= 2 : Print status of each iterate and summary + disp : int, optional + Overrides the iprint interface (preferred). + full_output : bool, optional + If False, return only the minimizer of func (default). + Otherwise, output final objective function and summary + information. + epsilon : float, optional + The step size for finite-difference derivative estimates. + callback : callable, optional + Called after each iteration, as ``callback(x)``, where ``x`` is the + current parameter vector. + + Returns + ------- + out : ndarray of float + The final minimizer of func. + fx : ndarray of float, if full_output is true + The final value of the objective function. + its : int, if full_output is true + The number of iterations. + imode : int, if full_output is true + The exit mode from the optimizer (see below). + smode : string, if full_output is true + Message describing the exit mode from the optimizer. + + See also + -------- + minimize: Interface to minimization algorithms for multivariate + functions. See the 'SLSQP' `method` in particular. + + Notes + ----- + Exit modes are defined as follows: + + - ``-1`` : Gradient evaluation required (g & a) + - ``0`` : Optimization terminated successfully + - ``1`` : Function evaluation required (f & c) + - ``2`` : More equality constraints than independent variables + - ``3`` : More than 3*n iterations in LSQ subproblem + - ``4`` : Inequality constraints incompatible + - ``5`` : Singular matrix E in LSQ subproblem + - ``6`` : Singular matrix C in LSQ subproblem + - ``7`` : Rank-deficient equality constraint subproblem HFTI + - ``8`` : Positive directional derivative for linesearch + - ``9`` : Iteration limit reached + + Examples + -------- + Examples are given :ref:`in the tutorial `. + + """ + if disp is not None: + iprint = disp + + opts = {'maxiter': iter, + 'ftol': acc, + 'iprint': iprint, + 'disp': iprint != 0, + 'eps': epsilon, + 'callback': callback} + + # Build the constraints as a tuple of dictionaries + cons = () + # 1. constraints of the 1st kind (eqcons, ieqcons); no Jacobian; take + # the same extra arguments as the objective function. + cons += tuple({'type': 'eq', 'fun': c, 'args': args} for c in eqcons) + cons += tuple({'type': 'ineq', 'fun': c, 'args': args} for c in ieqcons) + # 2. constraints of the 2nd kind (f_eqcons, f_ieqcons) and their Jacobian + # (fprime_eqcons, fprime_ieqcons); also take the same extra arguments + # as the objective function. + if f_eqcons: + cons += ({'type': 'eq', 'fun': f_eqcons, 'jac': fprime_eqcons, + 'args': args}, ) + if f_ieqcons: + cons += ({'type': 'ineq', 'fun': f_ieqcons, 'jac': fprime_ieqcons, + 'args': args}, ) + + res = _minimize_slsqp(func, x0, args, jac=fprime, bounds=bounds, + constraints=cons, **opts) + if full_output: + return res['x'], res['fun'], res['nit'], res['status'], res['message'] + else: + return res['x'] + + +def _minimize_slsqp(func, x0, args=(), jac=None, bounds=None, + constraints=(), + maxiter=100, ftol=1.0E-6, iprint=1, disp=False, + eps=_epsilon, callback=None, finite_diff_rel_step=None, + **unknown_options): + """ + Minimize a scalar function of one or more variables using Sequential + Least Squares Programming (SLSQP). + + Options + ------- + ftol : float + Precision goal for the value of f in the stopping criterion. + eps : float + Step size used for numerical approximation of the Jacobian. + disp : bool + Set to True to print convergence messages. If False, + `verbosity` is ignored and set to 0. + maxiter : int + Maximum number of iterations. + finite_diff_rel_step : None or array_like, optional + If ``jac in ['2-point', '3-point', 'cs']`` the relative step size to + use for numerical approximation of `jac`. The absolute step + size is computed as ``h = rel_step * sign(x) * max(1, abs(x))``, + possibly adjusted to fit into the bounds. For ``method='3-point'`` + the sign of `h` is ignored. If None (default) then step is selected + automatically. + """ + _check_unknown_options(unknown_options) + iter = maxiter - 1 + acc = ftol + epsilon = eps + + if not disp: + iprint = 0 + + # Transform x0 into an array. + xp = array_namespace(x0) + x0 = xpx.atleast_nd(xp.asarray(x0), ndim=1, xp=xp) + dtype = xp.float64 + if xp.isdtype(x0.dtype, "real floating"): + dtype = x0.dtype + x = xp.reshape(xp.astype(x0, dtype), -1) + + # SLSQP is sent 'old-style' bounds, 'new-style' bounds are required by + # ScalarFunction + if bounds is None or len(bounds) == 0: + new_bounds = (-np.inf, np.inf) + else: + new_bounds = old_bound_to_new(bounds) + + # clip the initial guess to bounds, otherwise ScalarFunction doesn't work + x = np.clip(x, new_bounds[0], new_bounds[1]) + + # Constraints are triaged per type into a dictionary of tuples + if isinstance(constraints, dict): + constraints = (constraints, ) + + cons = {'eq': (), 'ineq': ()} + for ic, con in enumerate(constraints): + # check type + try: + ctype = con['type'].lower() + except KeyError as e: + raise KeyError('Constraint %d has no type defined.' % ic) from e + except TypeError as e: + raise TypeError('Constraints must be defined using a ' + 'dictionary.') from e + except AttributeError as e: + raise TypeError("Constraint's type must be a string.") from e + else: + if ctype not in ['eq', 'ineq']: + raise ValueError(f"Unknown constraint type '{con['type']}'.") + + # check function + if 'fun' not in con: + raise ValueError('Constraint %d has no function defined.' % ic) + + # check Jacobian + cjac = con.get('jac') + if cjac is None: + # approximate Jacobian function. The factory function is needed + # to keep a reference to `fun`, see gh-4240. + def cjac_factory(fun): + def cjac(x, *args): + x = _check_clip_x(x, new_bounds) + + if jac in ['2-point', '3-point', 'cs']: + return approx_derivative(fun, x, method=jac, args=args, + rel_step=finite_diff_rel_step, + bounds=new_bounds) + else: + return approx_derivative(fun, x, method='2-point', + abs_step=epsilon, args=args, + bounds=new_bounds) + + return cjac + cjac = cjac_factory(con['fun']) + + # update constraints' dictionary + cons[ctype] += ({'fun': con['fun'], + 'jac': cjac, + 'args': con.get('args', ())}, ) + + exit_modes = {-1: "Gradient evaluation required (g & a)", + 0: "Optimization terminated successfully", + 1: "Function evaluation required (f & c)", + 2: "More equality constraints than independent variables", + 3: "More than 3*n iterations in LSQ subproblem", + 4: "Inequality constraints incompatible", + 5: "Singular matrix E in LSQ subproblem", + 6: "Singular matrix C in LSQ subproblem", + 7: "Rank-deficient equality constraint subproblem HFTI", + 8: "Positive directional derivative for linesearch", + 9: "Iteration limit reached"} + + # Set the parameters that SLSQP will need + # meq, mieq: number of equality and inequality constraints + meq = sum(map(len, [atleast_1d(c['fun'](x, *c['args'])) + for c in cons['eq']])) + mieq = sum(map(len, [atleast_1d(c['fun'](x, *c['args'])) + for c in cons['ineq']])) + # m = The total number of constraints + m = meq + mieq + # la = The number of constraints, or 1 if there are no constraints + la = array([1, m]).max() + # n = The number of independent variables + n = len(x) + + # Define the workspaces for SLSQP + n1 = n + 1 + mineq = m - meq + n1 + n1 + len_w = (3*n1+m)*(n1+1)+(n1-meq+1)*(mineq+2) + 2*mineq+(n1+mineq)*(n1-meq) \ + + 2*meq + n1 + ((n+1)*n)//2 + 2*m + 3*n + 3*n1 + 1 + len_jw = mineq + w = zeros(len_w) + jw = zeros(len_jw) + + # Decompose bounds into xl and xu + if bounds is None or len(bounds) == 0: + xl = np.empty(n, dtype=float) + xu = np.empty(n, dtype=float) + xl.fill(np.nan) + xu.fill(np.nan) + else: + bnds = array([(_arr_to_scalar(l), _arr_to_scalar(u)) + for (l, u) in bounds], float) + if bnds.shape[0] != n: + raise IndexError('SLSQP Error: the length of bounds is not ' + 'compatible with that of x0.') + + with np.errstate(invalid='ignore'): + bnderr = bnds[:, 0] > bnds[:, 1] + + if bnderr.any(): + raise ValueError("SLSQP Error: lb > ub in bounds " + f"{', '.join(str(b) for b in bnderr)}.") + xl, xu = bnds[:, 0], bnds[:, 1] + + # Mark infinite bounds with nans; the Fortran code understands this + infbnd = ~isfinite(bnds) + xl[infbnd[:, 0]] = np.nan + xu[infbnd[:, 1]] = np.nan + + # ScalarFunction provides function and gradient evaluation + sf = _prepare_scalar_function(func, x, jac=jac, args=args, epsilon=eps, + finite_diff_rel_step=finite_diff_rel_step, + bounds=new_bounds) + # gh11403 SLSQP sometimes exceeds bounds by 1 or 2 ULP, make sure this + # doesn't get sent to the func/grad evaluator. + wrapped_fun = _clip_x_for_func(sf.fun, new_bounds) + wrapped_grad = _clip_x_for_func(sf.grad, new_bounds) + + # Initialize the iteration counter and the mode value + mode = array(0, int) + acc = array(acc, float) + majiter = array(iter, int) + majiter_prev = 0 + + # Initialize internal SLSQP state variables + alpha = array(0, float) + f0 = array(0, float) + gs = array(0, float) + h1 = array(0, float) + h2 = array(0, float) + h3 = array(0, float) + h4 = array(0, float) + t = array(0, float) + t0 = array(0, float) + tol = array(0, float) + iexact = array(0, int) + incons = array(0, int) + ireset = array(0, int) + itermx = array(0, int) + line = array(0, int) + n1 = array(0, int) + n2 = array(0, int) + n3 = array(0, int) + + # Print the header if iprint >= 2 + if iprint >= 2: + print("%5s %5s %16s %16s" % ("NIT", "FC", "OBJFUN", "GNORM")) + + # mode is zero on entry, so call objective, constraints and gradients + # there should be no func evaluations here because it's cached from + # ScalarFunction + fx = wrapped_fun(x) + g = append(wrapped_grad(x), 0.0) + c = _eval_constraint(x, cons) + a = _eval_con_normals(x, cons, la, n, m, meq, mieq) + + while 1: + # Call SLSQP + slsqp(m, meq, x, xl, xu, fx, c, g, a, acc, majiter, mode, w, jw, + alpha, f0, gs, h1, h2, h3, h4, t, t0, tol, + iexact, incons, ireset, itermx, line, + n1, n2, n3) + + if mode == 1: # objective and constraint evaluation required + fx = wrapped_fun(x) + c = _eval_constraint(x, cons) + + if mode == -1: # gradient evaluation required + g = append(wrapped_grad(x), 0.0) + a = _eval_con_normals(x, cons, la, n, m, meq, mieq) + + if majiter > majiter_prev: + # call callback if major iteration has incremented + if callback is not None: + callback(np.copy(x)) + + # Print the status of the current iterate if iprint > 2 + if iprint >= 2: + print("%5i %5i % 16.6E % 16.6E" % (majiter, sf.nfev, + fx, linalg.norm(g))) + + # If exit mode is not -1 or 1, slsqp has completed + if abs(mode) != 1: + break + + majiter_prev = int(majiter) + + # Optimization loop complete. Print status if requested + if iprint >= 1: + print(exit_modes[int(mode)] + " (Exit mode " + str(mode) + ')') + print(" Current function value:", fx) + print(" Iterations:", majiter) + print(" Function evaluations:", sf.nfev) + print(" Gradient evaluations:", sf.ngev) + + return OptimizeResult(x=x, fun=fx, jac=g[:-1], nit=int(majiter), + nfev=sf.nfev, njev=sf.ngev, status=int(mode), + message=exit_modes[int(mode)], success=(mode == 0)) + + +def _eval_constraint(x, cons): + # Compute constraints + if cons['eq']: + c_eq = concatenate([atleast_1d(con['fun'](x, *con['args'])) + for con in cons['eq']]) + else: + c_eq = zeros(0) + + if cons['ineq']: + c_ieq = concatenate([atleast_1d(con['fun'](x, *con['args'])) + for con in cons['ineq']]) + else: + c_ieq = zeros(0) + + # Now combine c_eq and c_ieq into a single matrix + c = concatenate((c_eq, c_ieq)) + return c + + +def _eval_con_normals(x, cons, la, n, m, meq, mieq): + # Compute the normals of the constraints + if cons['eq']: + a_eq = vstack([con['jac'](x, *con['args']) + for con in cons['eq']]) + else: # no equality constraint + a_eq = zeros((meq, n)) + + if cons['ineq']: + a_ieq = vstack([con['jac'](x, *con['args']) + for con in cons['ineq']]) + else: # no inequality constraint + a_ieq = zeros((mieq, n)) + + # Now combine a_eq and a_ieq into a single a matrix + if m == 0: # no constraints + a = zeros((la, n)) + else: + a = vstack((a_eq, a_ieq)) + a = concatenate((a, zeros([la, 1])), 1) + + return a diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_spectral.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_spectral.py new file mode 100644 index 0000000000000000000000000000000000000000..5ff5bef0283b2d6b6c018c1c8b98cd46a335d7cb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_spectral.py @@ -0,0 +1,260 @@ +""" +Spectral Algorithm for Nonlinear Equations +""" +import collections + +import numpy as np +from scipy.optimize import OptimizeResult +from scipy.optimize._optimize import _check_unknown_options +from ._linesearch import _nonmonotone_line_search_cruz, _nonmonotone_line_search_cheng + +class _NoConvergence(Exception): + pass + + +def _root_df_sane(func, x0, args=(), ftol=1e-8, fatol=1e-300, maxfev=1000, + fnorm=None, callback=None, disp=False, M=10, eta_strategy=None, + sigma_eps=1e-10, sigma_0=1.0, line_search='cruz', **unknown_options): + r""" + Solve nonlinear equation with the DF-SANE method + + Options + ------- + ftol : float, optional + Relative norm tolerance. + fatol : float, optional + Absolute norm tolerance. + Algorithm terminates when ``||func(x)|| < fatol + ftol ||func(x_0)||``. + fnorm : callable, optional + Norm to use in the convergence check. If None, 2-norm is used. + maxfev : int, optional + Maximum number of function evaluations. + disp : bool, optional + Whether to print convergence process to stdout. + eta_strategy : callable, optional + Choice of the ``eta_k`` parameter, which gives slack for growth + of ``||F||**2``. Called as ``eta_k = eta_strategy(k, x, F)`` with + `k` the iteration number, `x` the current iterate and `F` the current + residual. Should satisfy ``eta_k > 0`` and ``sum(eta, k=0..inf) < inf``. + Default: ``||F||**2 / (1 + k)**2``. + sigma_eps : float, optional + The spectral coefficient is constrained to ``sigma_eps < sigma < 1/sigma_eps``. + Default: 1e-10 + sigma_0 : float, optional + Initial spectral coefficient. + Default: 1.0 + M : int, optional + Number of iterates to include in the nonmonotonic line search. + Default: 10 + line_search : {'cruz', 'cheng'} + Type of line search to employ. 'cruz' is the original one defined in + [Martinez & Raydan. Math. Comp. 75, 1429 (2006)], 'cheng' is + a modified search defined in [Cheng & Li. IMA J. Numer. Anal. 29, 814 (2009)]. + Default: 'cruz' + + References + ---------- + .. [1] "Spectral residual method without gradient information for solving + large-scale nonlinear systems of equations." W. La Cruz, + J.M. Martinez, M. Raydan. Math. Comp. **75**, 1429 (2006). + .. [2] W. La Cruz, Opt. Meth. Software, 29, 24 (2014). + .. [3] W. Cheng, D.-H. Li. IMA J. Numer. Anal. **29**, 814 (2009). + + """ + _check_unknown_options(unknown_options) + + if line_search not in ('cheng', 'cruz'): + raise ValueError(f"Invalid value {line_search!r} for 'line_search'") + + nexp = 2 + + if eta_strategy is None: + # Different choice from [1], as their eta is not invariant + # vs. scaling of F. + def eta_strategy(k, x, F): + # Obtain squared 2-norm of the initial residual from the outer scope + return f_0 / (1 + k)**2 + + if fnorm is None: + def fnorm(F): + # Obtain squared 2-norm of the current residual from the outer scope + return f_k**(1.0/nexp) + + def fmerit(F): + return np.linalg.norm(F)**nexp + + nfev = [0] + f, x_k, x_shape, f_k, F_k, is_complex = _wrap_func(func, x0, fmerit, + nfev, maxfev, args) + + k = 0 + f_0 = f_k + sigma_k = sigma_0 + + F_0_norm = fnorm(F_k) + + # For the 'cruz' line search + prev_fs = collections.deque([f_k], M) + + # For the 'cheng' line search + Q = 1.0 + C = f_0 + + converged = False + message = "too many function evaluations required" + + while True: + F_k_norm = fnorm(F_k) + + if disp: + print("iter %d: ||F|| = %g, sigma = %g" % (k, F_k_norm, sigma_k)) + + if callback is not None: + callback(x_k, F_k) + + if F_k_norm < ftol * F_0_norm + fatol: + # Converged! + message = "successful convergence" + converged = True + break + + # Control spectral parameter, from [2] + if abs(sigma_k) > 1/sigma_eps: + sigma_k = 1/sigma_eps * np.sign(sigma_k) + elif abs(sigma_k) < sigma_eps: + sigma_k = sigma_eps + + # Line search direction + d = -sigma_k * F_k + + # Nonmonotone line search + eta = eta_strategy(k, x_k, F_k) + try: + if line_search == 'cruz': + alpha, xp, fp, Fp = _nonmonotone_line_search_cruz(f, x_k, d, prev_fs, + eta=eta) + elif line_search == 'cheng': + alpha, xp, fp, Fp, C, Q = _nonmonotone_line_search_cheng(f, x_k, d, f_k, + C, Q, eta=eta) + except _NoConvergence: + break + + # Update spectral parameter + s_k = xp - x_k + y_k = Fp - F_k + sigma_k = np.vdot(s_k, s_k) / np.vdot(s_k, y_k) + + # Take step + x_k = xp + F_k = Fp + f_k = fp + + # Store function value + if line_search == 'cruz': + prev_fs.append(fp) + + k += 1 + + x = _wrap_result(x_k, is_complex, shape=x_shape) + F = _wrap_result(F_k, is_complex) + + result = OptimizeResult(x=x, success=converged, + message=message, + fun=F, nfev=nfev[0], nit=k, method="df-sane") + + return result + + +def _wrap_func(func, x0, fmerit, nfev_list, maxfev, args=()): + """ + Wrap a function and an initial value so that (i) complex values + are wrapped to reals, and (ii) value for a merit function + fmerit(x, f) is computed at the same time, (iii) iteration count + is maintained and an exception is raised if it is exceeded. + + Parameters + ---------- + func : callable + Function to wrap + x0 : ndarray + Initial value + fmerit : callable + Merit function fmerit(f) for computing merit value from residual. + nfev_list : list + List to store number of evaluations in. Should be [0] in the beginning. + maxfev : int + Maximum number of evaluations before _NoConvergence is raised. + args : tuple + Extra arguments to func + + Returns + ------- + wrap_func : callable + Wrapped function, to be called as + ``F, fp = wrap_func(x0)`` + x0_wrap : ndarray of float + Wrapped initial value; raveled to 1-D and complex + values mapped to reals. + x0_shape : tuple + Shape of the initial value array + f : float + Merit function at F + F : ndarray of float + Residual at x0_wrap + is_complex : bool + Whether complex values were mapped to reals + + """ + x0 = np.asarray(x0) + x0_shape = x0.shape + F = np.asarray(func(x0, *args)).ravel() + is_complex = np.iscomplexobj(x0) or np.iscomplexobj(F) + x0 = x0.ravel() + + nfev_list[0] = 1 + + if is_complex: + def wrap_func(x): + if nfev_list[0] >= maxfev: + raise _NoConvergence() + nfev_list[0] += 1 + z = _real2complex(x).reshape(x0_shape) + v = np.asarray(func(z, *args)).ravel() + F = _complex2real(v) + f = fmerit(F) + return f, F + + x0 = _complex2real(x0) + F = _complex2real(F) + else: + def wrap_func(x): + if nfev_list[0] >= maxfev: + raise _NoConvergence() + nfev_list[0] += 1 + x = x.reshape(x0_shape) + F = np.asarray(func(x, *args)).ravel() + f = fmerit(F) + return f, F + + return wrap_func, x0, x0_shape, fmerit(F), F, is_complex + + +def _wrap_result(result, is_complex, shape=None): + """ + Convert from real to complex and reshape result arrays. + """ + if is_complex: + z = _real2complex(result) + else: + z = result + if shape is not None: + z = z.reshape(shape) + return z + + +def _real2complex(x): + return np.ascontiguousarray(x, dtype=float).view(np.complex128) + + +def _complex2real(z): + return np.ascontiguousarray(z, dtype=complex).view(np.float64) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_tnc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_tnc.py new file mode 100644 index 0000000000000000000000000000000000000000..327fe4262e25f2f7e2c95d7e5261b6f188e72881 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_tnc.py @@ -0,0 +1,431 @@ +# TNC Python interface +# @(#) $Jeannot: tnc.py,v 1.11 2005/01/28 18:27:31 js Exp $ + +# Copyright (c) 2004-2005, Jean-Sebastien Roy (js@jeannot.org) + +# Permission is hereby granted, free of charge, to any person obtaining a +# copy of this software and associated documentation files (the +# "Software"), to deal in the Software without restriction, including +# without limitation the rights to use, copy, modify, merge, publish, +# distribute, sublicense, and/or sell copies of the Software, and to +# permit persons to whom the Software is furnished to do so, subject to +# the following conditions: + +# The above copyright notice and this permission notice shall be included +# in all copies or substantial portions of the Software. + +# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS +# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF +# MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. +# IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY +# CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, +# TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE +# SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. + +""" +TNC: A Python interface to the TNC non-linear optimizer + +TNC is a non-linear optimizer. To use it, you must provide a function to +minimize. The function must take one argument: the list of coordinates where to +evaluate the function; and it must return either a tuple, whose first element is the +value of the function, and whose second argument is the gradient of the function +(as a list of values); or None, to abort the minimization. +""" + +from scipy.optimize import _moduleTNC as moduleTNC +from ._optimize import (MemoizeJac, OptimizeResult, _check_unknown_options, + _prepare_scalar_function) +from ._constraints import old_bound_to_new +from scipy._lib._array_api import array_namespace +from scipy._lib import array_api_extra as xpx + +from numpy import inf, array, zeros + +__all__ = ['fmin_tnc'] + + +MSG_NONE = 0 # No messages +MSG_ITER = 1 # One line per iteration +MSG_INFO = 2 # Informational messages +MSG_VERS = 4 # Version info +MSG_EXIT = 8 # Exit reasons +MSG_ALL = MSG_ITER + MSG_INFO + MSG_VERS + MSG_EXIT + +MSGS = { + MSG_NONE: "No messages", + MSG_ITER: "One line per iteration", + MSG_INFO: "Informational messages", + MSG_VERS: "Version info", + MSG_EXIT: "Exit reasons", + MSG_ALL: "All messages" +} + +INFEASIBLE = -1 # Infeasible (lower bound > upper bound) +LOCALMINIMUM = 0 # Local minimum reached (|pg| ~= 0) +FCONVERGED = 1 # Converged (|f_n-f_(n-1)| ~= 0) +XCONVERGED = 2 # Converged (|x_n-x_(n-1)| ~= 0) +MAXFUN = 3 # Max. number of function evaluations reached +LSFAIL = 4 # Linear search failed +CONSTANT = 5 # All lower bounds are equal to the upper bounds +NOPROGRESS = 6 # Unable to progress +USERABORT = 7 # User requested end of minimization + +RCSTRINGS = { + INFEASIBLE: "Infeasible (lower bound > upper bound)", + LOCALMINIMUM: "Local minimum reached (|pg| ~= 0)", + FCONVERGED: "Converged (|f_n-f_(n-1)| ~= 0)", + XCONVERGED: "Converged (|x_n-x_(n-1)| ~= 0)", + MAXFUN: "Max. number of function evaluations reached", + LSFAIL: "Linear search failed", + CONSTANT: "All lower bounds are equal to the upper bounds", + NOPROGRESS: "Unable to progress", + USERABORT: "User requested end of minimization" +} + +# Changes to interface made by Travis Oliphant, Apr. 2004 for inclusion in +# SciPy + + +def fmin_tnc(func, x0, fprime=None, args=(), approx_grad=0, + bounds=None, epsilon=1e-8, scale=None, offset=None, + messages=MSG_ALL, maxCGit=-1, maxfun=None, eta=-1, + stepmx=0, accuracy=0, fmin=0, ftol=-1, xtol=-1, pgtol=-1, + rescale=-1, disp=None, callback=None): + r""" + Minimize a function with variables subject to bounds, using + gradient information in a truncated Newton algorithm. This + method wraps a C implementation of the algorithm. + + Parameters + ---------- + func : callable ``func(x, *args)`` + Function to minimize. Must do one of: + + 1. Return f and g, where f is the value of the function and g its + gradient (a list of floats). + + 2. Return the function value but supply gradient function + separately as `fprime`. + + 3. Return the function value and set ``approx_grad=True``. + + If the function returns None, the minimization + is aborted. + x0 : array_like + Initial estimate of minimum. + fprime : callable ``fprime(x, *args)``, optional + Gradient of `func`. If None, then either `func` must return the + function value and the gradient (``f,g = func(x, *args)``) + or `approx_grad` must be True. + args : tuple, optional + Arguments to pass to function. + approx_grad : bool, optional + If true, approximate the gradient numerically. + bounds : list, optional + (min, max) pairs for each element in x0, defining the + bounds on that parameter. Use None or +/-inf for one of + min or max when there is no bound in that direction. + epsilon : float, optional + Used if approx_grad is True. The stepsize in a finite + difference approximation for fprime. + scale : array_like, optional + Scaling factors to apply to each variable. If None, the + factors are up-low for interval bounded variables and + 1+|x| for the others. Defaults to None. + offset : array_like, optional + Value to subtract from each variable. If None, the + offsets are (up+low)/2 for interval bounded variables + and x for the others. + messages : int, optional + Bit mask used to select messages display during + minimization values defined in the MSGS dict. Defaults to + MGS_ALL. + disp : int, optional + Integer interface to messages. 0 = no message, 5 = all messages + maxCGit : int, optional + Maximum number of hessian*vector evaluations per main + iteration. If maxCGit == 0, the direction chosen is + -gradient if maxCGit < 0, maxCGit is set to + max(1,min(50,n/2)). Defaults to -1. + maxfun : int, optional + Maximum number of function evaluation. If None, maxfun is + set to max(100, 10*len(x0)). Defaults to None. Note that this function + may violate the limit because of evaluating gradients by numerical + differentiation. + eta : float, optional + Severity of the line search. If < 0 or > 1, set to 0.25. + Defaults to -1. + stepmx : float, optional + Maximum step for the line search. May be increased during + call. If too small, it will be set to 10.0. Defaults to 0. + accuracy : float, optional + Relative precision for finite difference calculations. If + <= machine_precision, set to sqrt(machine_precision). + Defaults to 0. + fmin : float, optional + Minimum function value estimate. Defaults to 0. + ftol : float, optional + Precision goal for the value of f in the stopping criterion. + If ftol < 0.0, ftol is set to 0.0 defaults to -1. + xtol : float, optional + Precision goal for the value of x in the stopping + criterion (after applying x scaling factors). If xtol < + 0.0, xtol is set to sqrt(machine_precision). Defaults to + -1. + pgtol : float, optional + Precision goal for the value of the projected gradient in + the stopping criterion (after applying x scaling factors). + If pgtol < 0.0, pgtol is set to 1e-2 * sqrt(accuracy). + Setting it to 0.0 is not recommended. Defaults to -1. + rescale : float, optional + Scaling factor (in log10) used to trigger f value + rescaling. If 0, rescale at each iteration. If a large + value, never rescale. If < 0, rescale is set to 1.3. + callback : callable, optional + Called after each iteration, as callback(xk), where xk is the + current parameter vector. + + Returns + ------- + x : ndarray + The solution. + nfeval : int + The number of function evaluations. + rc : int + Return code, see below + + See also + -------- + minimize: Interface to minimization algorithms for multivariate + functions. See the 'TNC' `method` in particular. + + Notes + ----- + The underlying algorithm is truncated Newton, also called + Newton Conjugate-Gradient. This method differs from + scipy.optimize.fmin_ncg in that + + 1. it wraps a C implementation of the algorithm + 2. it allows each variable to be given an upper and lower bound. + + The algorithm incorporates the bound constraints by determining + the descent direction as in an unconstrained truncated Newton, + but never taking a step-size large enough to leave the space + of feasible x's. The algorithm keeps track of a set of + currently active constraints, and ignores them when computing + the minimum allowable step size. (The x's associated with the + active constraint are kept fixed.) If the maximum allowable + step size is zero then a new constraint is added. At the end + of each iteration one of the constraints may be deemed no + longer active and removed. A constraint is considered + no longer active is if it is currently active + but the gradient for that variable points inward from the + constraint. The specific constraint removed is the one + associated with the variable of largest index whose + constraint is no longer active. + + Return codes are defined as follows: + + - ``-1`` : Infeasible (lower bound > upper bound) + - ``0`` : Local minimum reached (:math:`|pg| \approx 0`) + - ``1`` : Converged (:math:`|f_n-f_(n-1)| \approx 0`) + - ``2`` : Converged (:math:`|x_n-x_(n-1)| \approx 0`) + - ``3`` : Max. number of function evaluations reached + - ``4`` : Linear search failed + - ``5`` : All lower bounds are equal to the upper bounds + - ``6`` : Unable to progress + - ``7`` : User requested end of minimization + + References + ---------- + Wright S., Nocedal J. (2006), 'Numerical Optimization' + + Nash S.G. (1984), "Newton-Type Minimization Via the Lanczos Method", + SIAM Journal of Numerical Analysis 21, pp. 770-778 + + """ + # handle fprime/approx_grad + if approx_grad: + fun = func + jac = None + elif fprime is None: + fun = MemoizeJac(func) + jac = fun.derivative + else: + fun = func + jac = fprime + + if disp is not None: # disp takes precedence over messages + mesg_num = disp + else: + mesg_num = {0:MSG_NONE, 1:MSG_ITER, 2:MSG_INFO, 3:MSG_VERS, + 4:MSG_EXIT, 5:MSG_ALL}.get(messages, MSG_ALL) + # build options + opts = {'eps': epsilon, + 'scale': scale, + 'offset': offset, + 'mesg_num': mesg_num, + 'maxCGit': maxCGit, + 'maxfun': maxfun, + 'eta': eta, + 'stepmx': stepmx, + 'accuracy': accuracy, + 'minfev': fmin, + 'ftol': ftol, + 'xtol': xtol, + 'gtol': pgtol, + 'rescale': rescale, + 'disp': False} + + res = _minimize_tnc(fun, x0, args, jac, bounds, callback=callback, **opts) + + return res['x'], res['nfev'], res['status'] + + +def _minimize_tnc(fun, x0, args=(), jac=None, bounds=None, + eps=1e-8, scale=None, offset=None, mesg_num=None, + maxCGit=-1, eta=-1, stepmx=0, accuracy=0, + minfev=0, ftol=-1, xtol=-1, gtol=-1, rescale=-1, disp=False, + callback=None, finite_diff_rel_step=None, maxfun=None, + **unknown_options): + """ + Minimize a scalar function of one or more variables using a truncated + Newton (TNC) algorithm. + + Options + ------- + eps : float or ndarray + If `jac is None` the absolute step size used for numerical + approximation of the jacobian via forward differences. + scale : list of floats + Scaling factors to apply to each variable. If None, the + factors are up-low for interval bounded variables and + 1+|x] for the others. Defaults to None. + offset : float + Value to subtract from each variable. If None, the + offsets are (up+low)/2 for interval bounded variables + and x for the others. + disp : bool + Set to True to print convergence messages. + maxCGit : int + Maximum number of hessian*vector evaluations per main + iteration. If maxCGit == 0, the direction chosen is + -gradient if maxCGit < 0, maxCGit is set to + max(1,min(50,n/2)). Defaults to -1. + eta : float + Severity of the line search. If < 0 or > 1, set to 0.25. + Defaults to -1. + stepmx : float + Maximum step for the line search. May be increased during + call. If too small, it will be set to 10.0. Defaults to 0. + accuracy : float + Relative precision for finite difference calculations. If + <= machine_precision, set to sqrt(machine_precision). + Defaults to 0. + minfev : float + Minimum function value estimate. Defaults to 0. + ftol : float + Precision goal for the value of f in the stopping criterion. + If ftol < 0.0, ftol is set to 0.0 defaults to -1. + xtol : float + Precision goal for the value of x in the stopping + criterion (after applying x scaling factors). If xtol < + 0.0, xtol is set to sqrt(machine_precision). Defaults to + -1. + gtol : float + Precision goal for the value of the projected gradient in + the stopping criterion (after applying x scaling factors). + If gtol < 0.0, gtol is set to 1e-2 * sqrt(accuracy). + Setting it to 0.0 is not recommended. Defaults to -1. + rescale : float + Scaling factor (in log10) used to trigger f value + rescaling. If 0, rescale at each iteration. If a large + value, never rescale. If < 0, rescale is set to 1.3. + finite_diff_rel_step : None or array_like, optional + If ``jac in ['2-point', '3-point', 'cs']`` the relative step size to + use for numerical approximation of the jacobian. The absolute step + size is computed as ``h = rel_step * sign(x) * max(1, abs(x))``, + possibly adjusted to fit into the bounds. For ``method='3-point'`` + the sign of `h` is ignored. If None (default) then step is selected + automatically. + maxfun : int + Maximum number of function evaluations. If None, `maxfun` is + set to max(100, 10*len(x0)). Defaults to None. + """ + _check_unknown_options(unknown_options) + fmin = minfev + pgtol = gtol + + xp = array_namespace(x0) + x0 = xpx.atleast_nd(xp.asarray(x0), ndim=1, xp=xp) + dtype = xp.float64 + if xp.isdtype(x0.dtype, "real floating"): + dtype = x0.dtype + x0 = xp.reshape(xp.astype(x0, dtype), -1) + + n = len(x0) + + if bounds is None: + bounds = [(None,None)] * n + if len(bounds) != n: + raise ValueError('length of x0 != length of bounds') + new_bounds = old_bound_to_new(bounds) + + if mesg_num is not None: + messages = {0:MSG_NONE, 1:MSG_ITER, 2:MSG_INFO, 3:MSG_VERS, + 4:MSG_EXIT, 5:MSG_ALL}.get(mesg_num, MSG_ALL) + elif disp: + messages = MSG_ALL + else: + messages = MSG_NONE + + sf = _prepare_scalar_function(fun, x0, jac=jac, args=args, epsilon=eps, + finite_diff_rel_step=finite_diff_rel_step, + bounds=new_bounds) + func_and_grad = sf.fun_and_grad + + """ + low, up : the bounds (lists of floats) + if low is None, the lower bounds are removed. + if up is None, the upper bounds are removed. + low and up defaults to None + """ + low = zeros(n) + up = zeros(n) + for i in range(n): + if bounds[i] is None: + l, u = -inf, inf + else: + l,u = bounds[i] + if l is None: + low[i] = -inf + else: + low[i] = l + if u is None: + up[i] = inf + else: + up[i] = u + + if scale is None: + scale = array([]) + + if offset is None: + offset = array([]) + + if maxfun is None: + maxfun = max(100, 10*len(x0)) + + rc, nf, nit, x, funv, jacv = moduleTNC.tnc_minimize( + func_and_grad, x0, low, up, scale, + offset, messages, maxCGit, maxfun, + eta, stepmx, accuracy, fmin, ftol, + xtol, pgtol, rescale, callback + ) + # the TNC documentation states: "On output, x, f and g may be very + # slightly out of sync because of scaling". Therefore re-evaluate + # func_and_grad so they are synced. + funv, jacv = func_and_grad(x) + + return OptimizeResult(x=x, fun=funv, jac=jacv, nfev=sf.nfev, + nit=nit, status=rc, message=RCSTRINGS[rc], + success=(-1 < rc < 3)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trlib/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trlib/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..537b73b3aeb36df09863a0cd24957e5612deb030 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trlib/__init__.py @@ -0,0 +1,12 @@ +from ._trlib import TRLIBQuadraticSubproblem + +__all__ = ['TRLIBQuadraticSubproblem', 'get_trlib_quadratic_subproblem'] + + +def get_trlib_quadratic_subproblem(tol_rel_i=-2.0, tol_rel_b=-3.0, disp=False): + def subproblem_factory(x, fun, jac, hess, hessp): + return TRLIBQuadraticSubproblem(x, fun, jac, hess, hessp, + tol_rel_i=tol_rel_i, + tol_rel_b=tol_rel_b, + disp=disp) + return subproblem_factory diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion.py new file mode 100644 index 0000000000000000000000000000000000000000..0dadc727e74e40b5200810191b21cdeda941c5f6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion.py @@ -0,0 +1,304 @@ +"""Trust-region optimization.""" +import math +import warnings + +import numpy as np +import scipy.linalg +from ._optimize import (_check_unknown_options, _status_message, + OptimizeResult, _prepare_scalar_function, + _call_callback_maybe_halt) +from scipy.optimize._hessian_update_strategy import HessianUpdateStrategy +from scipy.optimize._differentiable_functions import FD_METHODS +__all__ = [] + + +def _wrap_function(function, args): + # wraps a minimizer function to count number of evaluations + # and to easily provide an args kwd. + ncalls = [0] + if function is None: + return ncalls, None + + def function_wrapper(x, *wrapper_args): + ncalls[0] += 1 + # A copy of x is sent to the user function (gh13740) + return function(np.copy(x), *(wrapper_args + args)) + + return ncalls, function_wrapper + + +class BaseQuadraticSubproblem: + """ + Base/abstract class defining the quadratic model for trust-region + minimization. Child classes must implement the ``solve`` method. + + Values of the objective function, Jacobian and Hessian (if provided) at + the current iterate ``x`` are evaluated on demand and then stored as + attributes ``fun``, ``jac``, ``hess``. + """ + + def __init__(self, x, fun, jac, hess=None, hessp=None): + self._x = x + self._f = None + self._g = None + self._h = None + self._g_mag = None + self._cauchy_point = None + self._newton_point = None + self._fun = fun + self._jac = jac + self._hess = hess + self._hessp = hessp + + def __call__(self, p): + return self.fun + np.dot(self.jac, p) + 0.5 * np.dot(p, self.hessp(p)) + + @property + def fun(self): + """Value of objective function at current iteration.""" + if self._f is None: + self._f = self._fun(self._x) + return self._f + + @property + def jac(self): + """Value of Jacobian of objective function at current iteration.""" + if self._g is None: + self._g = self._jac(self._x) + return self._g + + @property + def hess(self): + """Value of Hessian of objective function at current iteration.""" + if self._h is None: + self._h = self._hess(self._x) + return self._h + + def hessp(self, p): + if self._hessp is not None: + return self._hessp(self._x, p) + else: + return np.dot(self.hess, p) + + @property + def jac_mag(self): + """Magnitude of jacobian of objective function at current iteration.""" + if self._g_mag is None: + self._g_mag = scipy.linalg.norm(self.jac) + return self._g_mag + + def get_boundaries_intersections(self, z, d, trust_radius): + """ + Solve the scalar quadratic equation ``||z + t d|| == trust_radius``. + This is like a line-sphere intersection. + Return the two values of t, sorted from low to high. + """ + a = np.dot(d, d) + b = 2 * np.dot(z, d) + c = np.dot(z, z) - trust_radius**2 + sqrt_discriminant = math.sqrt(b*b - 4*a*c) + + # The following calculation is mathematically + # equivalent to: + # ta = (-b - sqrt_discriminant) / (2*a) + # tb = (-b + sqrt_discriminant) / (2*a) + # but produce smaller round off errors. + # Look at Matrix Computation p.97 + # for a better justification. + aux = b + math.copysign(sqrt_discriminant, b) + ta = -aux / (2*a) + tb = -2*c / aux + return sorted([ta, tb]) + + def solve(self, trust_radius): + raise NotImplementedError('The solve method should be implemented by ' + 'the child class') + + +def _minimize_trust_region(fun, x0, args=(), jac=None, hess=None, hessp=None, + subproblem=None, initial_trust_radius=1.0, + max_trust_radius=1000.0, eta=0.15, gtol=1e-4, + maxiter=None, disp=False, return_all=False, + callback=None, inexact=True, **unknown_options): + """ + Minimization of scalar function of one or more variables using a + trust-region algorithm. + + Options for the trust-region algorithm are: + initial_trust_radius : float + Initial trust radius. + max_trust_radius : float + Never propose steps that are longer than this value. + eta : float + Trust region related acceptance stringency for proposed steps. + gtol : float + Gradient norm must be less than `gtol` + before successful termination. + maxiter : int + Maximum number of iterations to perform. + disp : bool + If True, print convergence message. + inexact : bool + Accuracy to solve subproblems. If True requires less nonlinear + iterations, but more vector products. Only effective for method + trust-krylov. + + This function is called by the `minimize` function. + It is not supposed to be called directly. + """ + _check_unknown_options(unknown_options) + + if jac is None: + raise ValueError('Jacobian is currently required for trust-region ' + 'methods') + if hess is None and hessp is None: + raise ValueError('Either the Hessian or the Hessian-vector product ' + 'is currently required for trust-region methods') + if subproblem is None: + raise ValueError('A subproblem solving strategy is required for ' + 'trust-region methods') + if not (0 <= eta < 0.25): + raise Exception('invalid acceptance stringency') + if max_trust_radius <= 0: + raise Exception('the max trust radius must be positive') + if initial_trust_radius <= 0: + raise ValueError('the initial trust radius must be positive') + if initial_trust_radius >= max_trust_radius: + raise ValueError('the initial trust radius must be less than the ' + 'max trust radius') + + # force the initial guess into a nice format + x0 = np.asarray(x0).flatten() + + # A ScalarFunction representing the problem. This caches calls to fun, jac, + # hess. + sf = _prepare_scalar_function(fun, x0, jac=jac, hess=hess, args=args) + fun = sf.fun + jac = sf.grad + if callable(hess): + hess = sf.hess + elif callable(hessp): + # this elif statement must come before examining whether hess + # is estimated by FD methods or a HessianUpdateStrategy + pass + elif (hess in FD_METHODS or isinstance(hess, HessianUpdateStrategy)): + # If the Hessian is being estimated by finite differences or a + # Hessian update strategy then ScalarFunction.hess returns a + # LinearOperator or a HessianUpdateStrategy. This enables the + # calculation/creation of a hessp. BUT you only want to do this + # if the user *hasn't* provided a callable(hessp) function. + hess = None + + def hessp(x, p, *args): + return sf.hess(x).dot(p) + else: + raise ValueError('Either the Hessian or the Hessian-vector product ' + 'is currently required for trust-region methods') + + # ScalarFunction doesn't represent hessp + nhessp, hessp = _wrap_function(hessp, args) + + # limit the number of iterations + if maxiter is None: + maxiter = len(x0)*200 + + # init the search status + warnflag = 0 + + # initialize the search + trust_radius = initial_trust_radius + x = x0 + if return_all: + allvecs = [x] + m = subproblem(x, fun, jac, hess, hessp) + k = 0 + + # search for the function min + # do not even start if the gradient is small enough + while m.jac_mag >= gtol: + + # Solve the sub-problem. + # This gives us the proposed step relative to the current position + # and it tells us whether the proposed step + # has reached the trust region boundary or not. + try: + p, hits_boundary = m.solve(trust_radius) + except np.linalg.LinAlgError: + warnflag = 3 + break + + # calculate the predicted value at the proposed point + predicted_value = m(p) + + # define the local approximation at the proposed point + x_proposed = x + p + m_proposed = subproblem(x_proposed, fun, jac, hess, hessp) + + # evaluate the ratio defined in equation (4.4) + actual_reduction = m.fun - m_proposed.fun + predicted_reduction = m.fun - predicted_value + if predicted_reduction <= 0: + warnflag = 2 + break + rho = actual_reduction / predicted_reduction + + # update the trust radius according to the actual/predicted ratio + if rho < 0.25: + trust_radius *= 0.25 + elif rho > 0.75 and hits_boundary: + trust_radius = min(2*trust_radius, max_trust_radius) + + # if the ratio is high enough then accept the proposed step + if rho > eta: + x = x_proposed + m = m_proposed + + # append the best guess, call back, increment the iteration count + if return_all: + allvecs.append(np.copy(x)) + k += 1 + + intermediate_result = OptimizeResult(x=x, fun=m.fun) + if _call_callback_maybe_halt(callback, intermediate_result): + break + + # check if the gradient is small enough to stop + if m.jac_mag < gtol: + warnflag = 0 + break + + # check if we have looked at enough iterations + if k >= maxiter: + warnflag = 1 + break + + # print some stuff if requested + status_messages = ( + _status_message['success'], + _status_message['maxiter'], + 'A bad approximation caused failure to predict improvement.', + 'A linalg error occurred, such as a non-psd Hessian.', + ) + if disp: + if warnflag == 0: + print(status_messages[warnflag]) + else: + warnings.warn(status_messages[warnflag], RuntimeWarning, stacklevel=3) + print(f" Current function value: {m.fun:f}") + print(" Iterations: %d" % k) + print(" Function evaluations: %d" % sf.nfev) + print(" Gradient evaluations: %d" % sf.ngev) + print(" Hessian evaluations: %d" % (sf.nhev + nhessp[0])) + + result = OptimizeResult(x=x, success=(warnflag == 0), status=warnflag, + fun=m.fun, jac=m.jac, nfev=sf.nfev, njev=sf.ngev, + nhev=sf.nhev + nhessp[0], nit=k, + message=status_messages[warnflag]) + + if hess is not None: + result['hess'] = m.hess + + if return_all: + result['allvecs'] = allvecs + + return result diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..549cfb9760dda474cb858b7b36d236af48111067 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/__init__.py @@ -0,0 +1,6 @@ +"""This module contains the equality constrained SQP solver.""" + + +from .minimize_trustregion_constr import _minimize_trustregion_constr + +__all__ = ['_minimize_trustregion_constr'] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/canonical_constraint.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/canonical_constraint.py new file mode 100644 index 0000000000000000000000000000000000000000..7e9e75f04c032e5be077f9b7db210b96072092c9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/canonical_constraint.py @@ -0,0 +1,390 @@ +import numpy as np +import scipy.sparse as sps + + +class CanonicalConstraint: + """Canonical constraint to use with trust-constr algorithm. + + It represents the set of constraints of the form:: + + f_eq(x) = 0 + f_ineq(x) <= 0 + + where ``f_eq`` and ``f_ineq`` are evaluated by a single function, see + below. + + The class is supposed to be instantiated by factory methods, which + should prepare the parameters listed below. + + Parameters + ---------- + n_eq, n_ineq : int + Number of equality and inequality constraints respectively. + fun : callable + Function defining the constraints. The signature is + ``fun(x) -> c_eq, c_ineq``, where ``c_eq`` is ndarray with `n_eq` + components and ``c_ineq`` is ndarray with `n_ineq` components. + jac : callable + Function to evaluate the Jacobian of the constraint. The signature + is ``jac(x) -> J_eq, J_ineq``, where ``J_eq`` and ``J_ineq`` are + either ndarray of csr_matrix of shapes (n_eq, n) and (n_ineq, n), + respectively. + hess : callable + Function to evaluate the Hessian of the constraints multiplied + by Lagrange multipliers, that is + ``dot(f_eq, v_eq) + dot(f_ineq, v_ineq)``. The signature is + ``hess(x, v_eq, v_ineq) -> H``, where ``H`` has an implied + shape (n, n) and provide a matrix-vector product operation + ``H.dot(p)``. + keep_feasible : ndarray, shape (n_ineq,) + Mask indicating which inequality constraints should be kept feasible. + """ + def __init__(self, n_eq, n_ineq, fun, jac, hess, keep_feasible): + self.n_eq = n_eq + self.n_ineq = n_ineq + self.fun = fun + self.jac = jac + self.hess = hess + self.keep_feasible = keep_feasible + + @classmethod + def from_PreparedConstraint(cls, constraint): + """Create an instance from `PreparedConstrained` object.""" + lb, ub = constraint.bounds + cfun = constraint.fun + keep_feasible = constraint.keep_feasible + + if np.all(lb == -np.inf) and np.all(ub == np.inf): + return cls.empty(cfun.n) + + if np.all(lb == -np.inf) and np.all(ub == np.inf): + return cls.empty(cfun.n) + elif np.all(lb == ub): + return cls._equal_to_canonical(cfun, lb) + elif np.all(lb == -np.inf): + return cls._less_to_canonical(cfun, ub, keep_feasible) + elif np.all(ub == np.inf): + return cls._greater_to_canonical(cfun, lb, keep_feasible) + else: + return cls._interval_to_canonical(cfun, lb, ub, keep_feasible) + + @classmethod + def empty(cls, n): + """Create an "empty" instance. + + This "empty" instance is required to allow working with unconstrained + problems as if they have some constraints. + """ + empty_fun = np.empty(0) + empty_jac = np.empty((0, n)) + empty_hess = sps.csr_matrix((n, n)) + + def fun(x): + return empty_fun, empty_fun + + def jac(x): + return empty_jac, empty_jac + + def hess(x, v_eq, v_ineq): + return empty_hess + + return cls(0, 0, fun, jac, hess, np.empty(0, dtype=np.bool_)) + + @classmethod + def concatenate(cls, canonical_constraints, sparse_jacobian): + """Concatenate multiple `CanonicalConstraint` into one. + + `sparse_jacobian` (bool) determines the Jacobian format of the + concatenated constraint. Note that items in `canonical_constraints` + must have their Jacobians in the same format. + """ + def fun(x): + if canonical_constraints: + eq_all, ineq_all = zip( + *[c.fun(x) for c in canonical_constraints]) + else: + eq_all, ineq_all = [], [] + + return np.hstack(eq_all), np.hstack(ineq_all) + + if sparse_jacobian: + vstack = sps.vstack + else: + vstack = np.vstack + + def jac(x): + if canonical_constraints: + eq_all, ineq_all = zip( + *[c.jac(x) for c in canonical_constraints]) + else: + eq_all, ineq_all = [], [] + + return vstack(eq_all), vstack(ineq_all) + + def hess(x, v_eq, v_ineq): + hess_all = [] + index_eq = 0 + index_ineq = 0 + for c in canonical_constraints: + vc_eq = v_eq[index_eq:index_eq + c.n_eq] + vc_ineq = v_ineq[index_ineq:index_ineq + c.n_ineq] + hess_all.append(c.hess(x, vc_eq, vc_ineq)) + index_eq += c.n_eq + index_ineq += c.n_ineq + + def matvec(p): + result = np.zeros_like(p, dtype=float) + for h in hess_all: + result += h.dot(p) + return result + + n = x.shape[0] + return sps.linalg.LinearOperator((n, n), matvec, dtype=float) + + n_eq = sum(c.n_eq for c in canonical_constraints) + n_ineq = sum(c.n_ineq for c in canonical_constraints) + keep_feasible = np.hstack([c.keep_feasible for c in + canonical_constraints]) + + return cls(n_eq, n_ineq, fun, jac, hess, keep_feasible) + + @classmethod + def _equal_to_canonical(cls, cfun, value): + empty_fun = np.empty(0) + n = cfun.n + + n_eq = value.shape[0] + n_ineq = 0 + keep_feasible = np.empty(0, dtype=bool) + + if cfun.sparse_jacobian: + empty_jac = sps.csr_matrix((0, n)) + else: + empty_jac = np.empty((0, n)) + + def fun(x): + return cfun.fun(x) - value, empty_fun + + def jac(x): + return cfun.jac(x), empty_jac + + def hess(x, v_eq, v_ineq): + return cfun.hess(x, v_eq) + + empty_fun = np.empty(0) + n = cfun.n + if cfun.sparse_jacobian: + empty_jac = sps.csr_matrix((0, n)) + else: + empty_jac = np.empty((0, n)) + + return cls(n_eq, n_ineq, fun, jac, hess, keep_feasible) + + @classmethod + def _less_to_canonical(cls, cfun, ub, keep_feasible): + empty_fun = np.empty(0) + n = cfun.n + if cfun.sparse_jacobian: + empty_jac = sps.csr_matrix((0, n)) + else: + empty_jac = np.empty((0, n)) + + finite_ub = ub < np.inf + n_eq = 0 + n_ineq = np.sum(finite_ub) + + if np.all(finite_ub): + def fun(x): + return empty_fun, cfun.fun(x) - ub + + def jac(x): + return empty_jac, cfun.jac(x) + + def hess(x, v_eq, v_ineq): + return cfun.hess(x, v_ineq) + else: + finite_ub = np.nonzero(finite_ub)[0] + keep_feasible = keep_feasible[finite_ub] + ub = ub[finite_ub] + + def fun(x): + return empty_fun, cfun.fun(x)[finite_ub] - ub + + def jac(x): + return empty_jac, cfun.jac(x)[finite_ub] + + def hess(x, v_eq, v_ineq): + v = np.zeros(cfun.m) + v[finite_ub] = v_ineq + return cfun.hess(x, v) + + return cls(n_eq, n_ineq, fun, jac, hess, keep_feasible) + + @classmethod + def _greater_to_canonical(cls, cfun, lb, keep_feasible): + empty_fun = np.empty(0) + n = cfun.n + if cfun.sparse_jacobian: + empty_jac = sps.csr_matrix((0, n)) + else: + empty_jac = np.empty((0, n)) + + finite_lb = lb > -np.inf + n_eq = 0 + n_ineq = np.sum(finite_lb) + + if np.all(finite_lb): + def fun(x): + return empty_fun, lb - cfun.fun(x) + + def jac(x): + return empty_jac, -cfun.jac(x) + + def hess(x, v_eq, v_ineq): + return cfun.hess(x, -v_ineq) + else: + finite_lb = np.nonzero(finite_lb)[0] + keep_feasible = keep_feasible[finite_lb] + lb = lb[finite_lb] + + def fun(x): + return empty_fun, lb - cfun.fun(x)[finite_lb] + + def jac(x): + return empty_jac, -cfun.jac(x)[finite_lb] + + def hess(x, v_eq, v_ineq): + v = np.zeros(cfun.m) + v[finite_lb] = -v_ineq + return cfun.hess(x, v) + + return cls(n_eq, n_ineq, fun, jac, hess, keep_feasible) + + @classmethod + def _interval_to_canonical(cls, cfun, lb, ub, keep_feasible): + lb_inf = lb == -np.inf + ub_inf = ub == np.inf + equal = lb == ub + less = lb_inf & ~ub_inf + greater = ub_inf & ~lb_inf + interval = ~equal & ~lb_inf & ~ub_inf + + equal = np.nonzero(equal)[0] + less = np.nonzero(less)[0] + greater = np.nonzero(greater)[0] + interval = np.nonzero(interval)[0] + n_less = less.shape[0] + n_greater = greater.shape[0] + n_interval = interval.shape[0] + n_ineq = n_less + n_greater + 2 * n_interval + n_eq = equal.shape[0] + + keep_feasible = np.hstack((keep_feasible[less], + keep_feasible[greater], + keep_feasible[interval], + keep_feasible[interval])) + + def fun(x): + f = cfun.fun(x) + eq = f[equal] - lb[equal] + le = f[less] - ub[less] + ge = lb[greater] - f[greater] + il = f[interval] - ub[interval] + ig = lb[interval] - f[interval] + return eq, np.hstack((le, ge, il, ig)) + + def jac(x): + J = cfun.jac(x) + eq = J[equal] + le = J[less] + ge = -J[greater] + il = J[interval] + ig = -il + if sps.issparse(J): + ineq = sps.vstack((le, ge, il, ig)) + else: + ineq = np.vstack((le, ge, il, ig)) + return eq, ineq + + def hess(x, v_eq, v_ineq): + n_start = 0 + v_l = v_ineq[n_start:n_start + n_less] + n_start += n_less + v_g = v_ineq[n_start:n_start + n_greater] + n_start += n_greater + v_il = v_ineq[n_start:n_start + n_interval] + n_start += n_interval + v_ig = v_ineq[n_start:n_start + n_interval] + + v = np.zeros_like(lb) + v[equal] = v_eq + v[less] = v_l + v[greater] = -v_g + v[interval] = v_il - v_ig + + return cfun.hess(x, v) + + return cls(n_eq, n_ineq, fun, jac, hess, keep_feasible) + + +def initial_constraints_as_canonical(n, prepared_constraints, sparse_jacobian): + """Convert initial values of the constraints to the canonical format. + + The purpose to avoid one additional call to the constraints at the initial + point. It takes saved values in `PreparedConstraint`, modifies and + concatenates them to the canonical constraint format. + """ + c_eq = [] + c_ineq = [] + J_eq = [] + J_ineq = [] + + for c in prepared_constraints: + f = c.fun.f + J = c.fun.J + lb, ub = c.bounds + if np.all(lb == ub): + c_eq.append(f - lb) + J_eq.append(J) + elif np.all(lb == -np.inf): + finite_ub = ub < np.inf + c_ineq.append(f[finite_ub] - ub[finite_ub]) + J_ineq.append(J[finite_ub]) + elif np.all(ub == np.inf): + finite_lb = lb > -np.inf + c_ineq.append(lb[finite_lb] - f[finite_lb]) + J_ineq.append(-J[finite_lb]) + else: + lb_inf = lb == -np.inf + ub_inf = ub == np.inf + equal = lb == ub + less = lb_inf & ~ub_inf + greater = ub_inf & ~lb_inf + interval = ~equal & ~lb_inf & ~ub_inf + + c_eq.append(f[equal] - lb[equal]) + c_ineq.append(f[less] - ub[less]) + c_ineq.append(lb[greater] - f[greater]) + c_ineq.append(f[interval] - ub[interval]) + c_ineq.append(lb[interval] - f[interval]) + + J_eq.append(J[equal]) + J_ineq.append(J[less]) + J_ineq.append(-J[greater]) + J_ineq.append(J[interval]) + J_ineq.append(-J[interval]) + + c_eq = np.hstack(c_eq) if c_eq else np.empty(0) + c_ineq = np.hstack(c_ineq) if c_ineq else np.empty(0) + + if sparse_jacobian: + vstack = sps.vstack + empty = sps.csr_matrix((0, n)) + else: + vstack = np.vstack + empty = np.empty((0, n)) + + J_eq = vstack(J_eq) if J_eq else empty + J_ineq = vstack(J_ineq) if J_ineq else empty + + return c_eq, c_ineq, J_eq, J_ineq diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/equality_constrained_sqp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/equality_constrained_sqp.py new file mode 100644 index 0000000000000000000000000000000000000000..88a9f8deb1abfaa82533eeb6308bbfbd5516ed4d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/equality_constrained_sqp.py @@ -0,0 +1,231 @@ +"""Byrd-Omojokun Trust-Region SQP method.""" + +from scipy.sparse import eye as speye +from .projections import projections +from .qp_subproblem import modified_dogleg, projected_cg, box_intersections +import numpy as np +from numpy.linalg import norm + +__all__ = ['equality_constrained_sqp'] + + +def default_scaling(x): + n, = np.shape(x) + return speye(n) + + +def equality_constrained_sqp(fun_and_constr, grad_and_jac, lagr_hess, + x0, fun0, grad0, constr0, + jac0, stop_criteria, + state, + initial_penalty, + initial_trust_radius, + factorization_method, + trust_lb=None, + trust_ub=None, + scaling=default_scaling): + """Solve nonlinear equality-constrained problem using trust-region SQP. + + Solve optimization problem: + + minimize fun(x) + subject to: constr(x) = 0 + + using Byrd-Omojokun Trust-Region SQP method described in [1]_. Several + implementation details are based on [2]_ and [3]_, p. 549. + + References + ---------- + .. [1] Lalee, Marucha, Jorge Nocedal, and Todd Plantenga. "On the + implementation of an algorithm for large-scale equality + constrained optimization." SIAM Journal on + Optimization 8.3 (1998): 682-706. + .. [2] Byrd, Richard H., Mary E. Hribar, and Jorge Nocedal. + "An interior point algorithm for large-scale nonlinear + programming." SIAM Journal on Optimization 9.4 (1999): 877-900. + .. [3] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization" + Second Edition (2006). + """ + PENALTY_FACTOR = 0.3 # Rho from formula (3.51), reference [2]_, p.891. + LARGE_REDUCTION_RATIO = 0.9 + INTERMEDIARY_REDUCTION_RATIO = 0.3 + SUFFICIENT_REDUCTION_RATIO = 1e-8 # Eta from reference [2]_, p.892. + TRUST_ENLARGEMENT_FACTOR_L = 7.0 + TRUST_ENLARGEMENT_FACTOR_S = 2.0 + MAX_TRUST_REDUCTION = 0.5 + MIN_TRUST_REDUCTION = 0.1 + SOC_THRESHOLD = 0.1 + TR_FACTOR = 0.8 # Zeta from formula (3.21), reference [2]_, p.885. + BOX_FACTOR = 0.5 + + n, = np.shape(x0) # Number of parameters + + # Set default lower and upper bounds. + if trust_lb is None: + trust_lb = np.full(n, -np.inf) + if trust_ub is None: + trust_ub = np.full(n, np.inf) + + # Initial values + x = np.copy(x0) + trust_radius = initial_trust_radius + penalty = initial_penalty + # Compute Values + f = fun0 + c = grad0 + b = constr0 + A = jac0 + S = scaling(x) + # Get projections + try: + Z, LS, Y = projections(A, factorization_method) + except ValueError as e: + if str(e) == "expected square matrix": + # can be the case if there are more equality + # constraints than independent variables + raise ValueError( + "The 'expected square matrix' error can occur if there are" + " more equality constraints than independent variables." + " Consider how your constraints are set up, or use" + " factorization_method='SVDFactorization'." + ) from e + else: + raise e + + # Compute least-square lagrange multipliers + v = -LS.dot(c) + # Compute Hessian + H = lagr_hess(x, v) + + # Update state parameters + optimality = norm(c + A.T.dot(v), np.inf) + constr_violation = norm(b, np.inf) if len(b) > 0 else 0 + cg_info = {'niter': 0, 'stop_cond': 0, + 'hits_boundary': False} + + last_iteration_failed = False + while not stop_criteria(state, x, last_iteration_failed, + optimality, constr_violation, + trust_radius, penalty, cg_info): + # Normal Step - `dn` + # minimize 1/2*||A dn + b||^2 + # subject to: + # ||dn|| <= TR_FACTOR * trust_radius + # BOX_FACTOR * lb <= dn <= BOX_FACTOR * ub. + dn = modified_dogleg(A, Y, b, + TR_FACTOR*trust_radius, + BOX_FACTOR*trust_lb, + BOX_FACTOR*trust_ub) + + # Tangential Step - `dt` + # Solve the QP problem: + # minimize 1/2 dt.T H dt + dt.T (H dn + c) + # subject to: + # A dt = 0 + # ||dt|| <= sqrt(trust_radius**2 - ||dn||**2) + # lb - dn <= dt <= ub - dn + c_t = H.dot(dn) + c + b_t = np.zeros_like(b) + trust_radius_t = np.sqrt(trust_radius**2 - np.linalg.norm(dn)**2) + lb_t = trust_lb - dn + ub_t = trust_ub - dn + dt, cg_info = projected_cg(H, c_t, Z, Y, b_t, + trust_radius_t, + lb_t, ub_t) + + # Compute update (normal + tangential steps). + d = dn + dt + + # Compute second order model: 1/2 d H d + c.T d + f. + quadratic_model = 1/2*(H.dot(d)).dot(d) + c.T.dot(d) + # Compute linearized constraint: l = A d + b. + linearized_constr = A.dot(d)+b + # Compute new penalty parameter according to formula (3.52), + # reference [2]_, p.891. + vpred = norm(b) - norm(linearized_constr) + # Guarantee `vpred` always positive, + # regardless of roundoff errors. + vpred = max(1e-16, vpred) + previous_penalty = penalty + if quadratic_model > 0: + new_penalty = quadratic_model / ((1-PENALTY_FACTOR)*vpred) + penalty = max(penalty, new_penalty) + # Compute predicted reduction according to formula (3.52), + # reference [2]_, p.891. + predicted_reduction = -quadratic_model + penalty*vpred + + # Compute merit function at current point + merit_function = f + penalty*norm(b) + # Evaluate function and constraints at trial point + x_next = x + S.dot(d) + f_next, b_next = fun_and_constr(x_next) + # Compute merit function at trial point + merit_function_next = f_next + penalty*norm(b_next) + # Compute actual reduction according to formula (3.54), + # reference [2]_, p.892. + actual_reduction = merit_function - merit_function_next + # Compute reduction ratio + reduction_ratio = actual_reduction / predicted_reduction + + # Second order correction (SOC), reference [2]_, p.892. + if reduction_ratio < SUFFICIENT_REDUCTION_RATIO and \ + norm(dn) <= SOC_THRESHOLD * norm(dt): + # Compute second order correction + y = -Y.dot(b_next) + # Make sure increment is inside box constraints + _, t, intersect = box_intersections(d, y, trust_lb, trust_ub) + # Compute tentative point + x_soc = x + S.dot(d + t*y) + f_soc, b_soc = fun_and_constr(x_soc) + # Recompute actual reduction + merit_function_soc = f_soc + penalty*norm(b_soc) + actual_reduction_soc = merit_function - merit_function_soc + # Recompute reduction ratio + reduction_ratio_soc = actual_reduction_soc / predicted_reduction + if intersect and reduction_ratio_soc >= SUFFICIENT_REDUCTION_RATIO: + x_next = x_soc + f_next = f_soc + b_next = b_soc + reduction_ratio = reduction_ratio_soc + + # Readjust trust region step, formula (3.55), reference [2]_, p.892. + if reduction_ratio >= LARGE_REDUCTION_RATIO: + trust_radius = max(TRUST_ENLARGEMENT_FACTOR_L * norm(d), + trust_radius) + elif reduction_ratio >= INTERMEDIARY_REDUCTION_RATIO: + trust_radius = max(TRUST_ENLARGEMENT_FACTOR_S * norm(d), + trust_radius) + # Reduce trust region step, according to reference [3]_, p.696. + elif reduction_ratio < SUFFICIENT_REDUCTION_RATIO: + trust_reduction = ((1-SUFFICIENT_REDUCTION_RATIO) / + (1-reduction_ratio)) + new_trust_radius = trust_reduction * norm(d) + if new_trust_radius >= MAX_TRUST_REDUCTION * trust_radius: + trust_radius *= MAX_TRUST_REDUCTION + elif new_trust_radius >= MIN_TRUST_REDUCTION * trust_radius: + trust_radius = new_trust_radius + else: + trust_radius *= MIN_TRUST_REDUCTION + + # Update iteration + if reduction_ratio >= SUFFICIENT_REDUCTION_RATIO: + x = x_next + f, b = f_next, b_next + c, A = grad_and_jac(x) + S = scaling(x) + # Get projections + Z, LS, Y = projections(A, factorization_method) + # Compute least-square lagrange multipliers + v = -LS.dot(c) + # Compute Hessian + H = lagr_hess(x, v) + # Set Flag + last_iteration_failed = False + # Optimality values + optimality = norm(c + A.T.dot(v), np.inf) + constr_violation = norm(b, np.inf) if len(b) > 0 else 0 + else: + penalty = previous_penalty + last_iteration_failed = True + + return x, state diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/minimize_trustregion_constr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/minimize_trustregion_constr.py new file mode 100644 index 0000000000000000000000000000000000000000..580f9ccf8eab9f22d73ae41f130b2e5e541a71da --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/minimize_trustregion_constr.py @@ -0,0 +1,576 @@ +import time +import numpy as np +from scipy.sparse.linalg import LinearOperator +from .._differentiable_functions import VectorFunction +from .._constraints import ( + NonlinearConstraint, LinearConstraint, PreparedConstraint, Bounds, strict_bounds) +from .._hessian_update_strategy import BFGS +from .._optimize import OptimizeResult +from .._differentiable_functions import ScalarFunction +from .equality_constrained_sqp import equality_constrained_sqp +from .canonical_constraint import (CanonicalConstraint, + initial_constraints_as_canonical) +from .tr_interior_point import tr_interior_point +from .report import BasicReport, SQPReport, IPReport + + +TERMINATION_MESSAGES = { + 0: "The maximum number of function evaluations is exceeded.", + 1: "`gtol` termination condition is satisfied.", + 2: "`xtol` termination condition is satisfied.", + 3: "`callback` function requested termination.", + 4: "Constraint violation exceeds 'gtol'" +} + + +class HessianLinearOperator: + """Build LinearOperator from hessp""" + def __init__(self, hessp, n): + self.hessp = hessp + self.n = n + + def __call__(self, x, *args): + def matvec(p): + return self.hessp(x, p, *args) + + return LinearOperator((self.n, self.n), matvec=matvec) + + +class LagrangianHessian: + """The Hessian of the Lagrangian as LinearOperator. + + The Lagrangian is computed as the objective function plus all the + constraints multiplied with some numbers (Lagrange multipliers). + """ + def __init__(self, n, objective_hess, constraints_hess): + self.n = n + self.objective_hess = objective_hess + self.constraints_hess = constraints_hess + + def __call__(self, x, v_eq, v_ineq=None): + if v_ineq is None: + v_ineq = np.empty(0) + H_objective = self.objective_hess(x) + H_constraints = self.constraints_hess(x, v_eq, v_ineq) + + def matvec(p): + return H_objective.dot(p) + H_constraints.dot(p) + + return LinearOperator((self.n, self.n), matvec) + + +def update_state_sqp(state, x, last_iteration_failed, objective, prepared_constraints, + start_time, tr_radius, constr_penalty, cg_info): + state.nit += 1 + state.nfev = objective.nfev + state.njev = objective.ngev + state.nhev = objective.nhev + state.constr_nfev = [c.fun.nfev if isinstance(c.fun, VectorFunction) else 0 + for c in prepared_constraints] + state.constr_njev = [c.fun.njev if isinstance(c.fun, VectorFunction) else 0 + for c in prepared_constraints] + state.constr_nhev = [c.fun.nhev if isinstance(c.fun, VectorFunction) else 0 + for c in prepared_constraints] + + if not last_iteration_failed: + state.x = x + state.fun = objective.f + state.grad = objective.g + state.v = [c.fun.v for c in prepared_constraints] + state.constr = [c.fun.f for c in prepared_constraints] + state.jac = [c.fun.J for c in prepared_constraints] + # Compute Lagrangian Gradient + state.lagrangian_grad = np.copy(state.grad) + for c in prepared_constraints: + state.lagrangian_grad += c.fun.J.T.dot(c.fun.v) + state.optimality = np.linalg.norm(state.lagrangian_grad, np.inf) + # Compute maximum constraint violation + state.constr_violation = 0 + for i in range(len(prepared_constraints)): + lb, ub = prepared_constraints[i].bounds + c = state.constr[i] + state.constr_violation = np.max([state.constr_violation, + np.max(lb - c), + np.max(c - ub)]) + + state.execution_time = time.time() - start_time + state.tr_radius = tr_radius + state.constr_penalty = constr_penalty + state.cg_niter += cg_info["niter"] + state.cg_stop_cond = cg_info["stop_cond"] + + return state + + +def update_state_ip(state, x, last_iteration_failed, objective, + prepared_constraints, start_time, + tr_radius, constr_penalty, cg_info, + barrier_parameter, barrier_tolerance): + state = update_state_sqp(state, x, last_iteration_failed, objective, + prepared_constraints, start_time, tr_radius, + constr_penalty, cg_info) + state.barrier_parameter = barrier_parameter + state.barrier_tolerance = barrier_tolerance + return state + + +def _minimize_trustregion_constr(fun, x0, args, grad, + hess, hessp, bounds, constraints, + xtol=1e-8, gtol=1e-8, + barrier_tol=1e-8, + sparse_jacobian=None, + callback=None, maxiter=1000, + verbose=0, finite_diff_rel_step=None, + initial_constr_penalty=1.0, initial_tr_radius=1.0, + initial_barrier_parameter=0.1, + initial_barrier_tolerance=0.1, + factorization_method=None, + disp=False): + """Minimize a scalar function subject to constraints. + + Parameters + ---------- + gtol : float, optional + Tolerance for termination by the norm of the Lagrangian gradient. + The algorithm will terminate when both the infinity norm (i.e., max + abs value) of the Lagrangian gradient and the constraint violation + are smaller than ``gtol``. Default is 1e-8. + xtol : float, optional + Tolerance for termination by the change of the independent variable. + The algorithm will terminate when ``tr_radius < xtol``, where + ``tr_radius`` is the radius of the trust region used in the algorithm. + Default is 1e-8. + barrier_tol : float, optional + Threshold on the barrier parameter for the algorithm termination. + When inequality constraints are present, the algorithm will terminate + only when the barrier parameter is less than `barrier_tol`. + Default is 1e-8. + sparse_jacobian : {bool, None}, optional + Determines how to represent Jacobians of the constraints. If bool, + then Jacobians of all the constraints will be converted to the + corresponding format. If None (default), then Jacobians won't be + converted, but the algorithm can proceed only if they all have the + same format. + initial_tr_radius: float, optional + Initial trust radius. The trust radius gives the maximum distance + between solution points in consecutive iterations. It reflects the + trust the algorithm puts in the local approximation of the optimization + problem. For an accurate local approximation the trust-region should be + large and for an approximation valid only close to the current point it + should be a small one. The trust radius is automatically updated throughout + the optimization process, with ``initial_tr_radius`` being its initial value. + Default is 1 (recommended in [1]_, p. 19). + initial_constr_penalty : float, optional + Initial constraints penalty parameter. The penalty parameter is used for + balancing the requirements of decreasing the objective function + and satisfying the constraints. It is used for defining the merit function: + ``merit_function(x) = fun(x) + constr_penalty * constr_norm_l2(x)``, + where ``constr_norm_l2(x)`` is the l2 norm of a vector containing all + the constraints. The merit function is used for accepting or rejecting + trial points and ``constr_penalty`` weights the two conflicting goals + of reducing objective function and constraints. The penalty is automatically + updated throughout the optimization process, with + ``initial_constr_penalty`` being its initial value. Default is 1 + (recommended in [1]_, p 19). + initial_barrier_parameter, initial_barrier_tolerance: float, optional + Initial barrier parameter and initial tolerance for the barrier subproblem. + Both are used only when inequality constraints are present. For dealing with + optimization problems ``min_x f(x)`` subject to inequality constraints + ``c(x) <= 0`` the algorithm introduces slack variables, solving the problem + ``min_(x,s) f(x) + barrier_parameter*sum(ln(s))`` subject to the equality + constraints ``c(x) + s = 0`` instead of the original problem. This subproblem + is solved for decreasing values of ``barrier_parameter`` and with decreasing + tolerances for the termination, starting with ``initial_barrier_parameter`` + for the barrier parameter and ``initial_barrier_tolerance`` for the + barrier tolerance. Default is 0.1 for both values (recommended in [1]_ p. 19). + Also note that ``barrier_parameter`` and ``barrier_tolerance`` are updated + with the same prefactor. + factorization_method : string or None, optional + Method to factorize the Jacobian of the constraints. Use None (default) + for the auto selection or one of: + + - 'NormalEquation' (requires scikit-sparse) + - 'AugmentedSystem' + - 'QRFactorization' + - 'SVDFactorization' + + The methods 'NormalEquation' and 'AugmentedSystem' can be used only + with sparse constraints. The projections required by the algorithm + will be computed using, respectively, the normal equation and the + augmented system approaches explained in [1]_. 'NormalEquation' + computes the Cholesky factorization of ``A A.T`` and 'AugmentedSystem' + performs the LU factorization of an augmented system. They usually + provide similar results. 'AugmentedSystem' is used by default for + sparse matrices. + + The methods 'QRFactorization' and 'SVDFactorization' can be used + only with dense constraints. They compute the required projections + using, respectively, QR and SVD factorizations. The 'SVDFactorization' + method can cope with Jacobian matrices with deficient row rank and will + be used whenever other factorization methods fail (which may imply the + conversion of sparse matrices to a dense format when required). + By default, 'QRFactorization' is used for dense matrices. + finite_diff_rel_step : None or array_like, optional + Relative step size for the finite difference approximation. + maxiter : int, optional + Maximum number of algorithm iterations. Default is 1000. + verbose : {0, 1, 2, 3}, optional + Level of algorithm's verbosity: + + * 0 (default) : work silently. + * 1 : display a termination report. + * 2 : display progress during iterations. + * 3 : display progress during iterations (more complete report). + + disp : bool, optional + If True (default), then `verbose` will be set to 1 if it was 0. + + Returns + ------- + `OptimizeResult` with the fields documented below. Note the following: + + 1. All values corresponding to the constraints are ordered as they + were passed to the solver. And values corresponding to `bounds` + constraints are put *after* other constraints. + 2. All numbers of function, Jacobian or Hessian evaluations correspond + to numbers of actual Python function calls. It means, for example, + that if a Jacobian is estimated by finite differences, then the + number of Jacobian evaluations will be zero and the number of + function evaluations will be incremented by all calls during the + finite difference estimation. + + x : ndarray, shape (n,) + Solution found. + optimality : float + Infinity norm of the Lagrangian gradient at the solution. + constr_violation : float + Maximum constraint violation at the solution. + fun : float + Objective function at the solution. + grad : ndarray, shape (n,) + Gradient of the objective function at the solution. + lagrangian_grad : ndarray, shape (n,) + Gradient of the Lagrangian function at the solution. + nit : int + Total number of iterations. + nfev : integer + Number of the objective function evaluations. + njev : integer + Number of the objective function gradient evaluations. + nhev : integer + Number of the objective function Hessian evaluations. + cg_niter : int + Total number of the conjugate gradient method iterations. + method : {'equality_constrained_sqp', 'tr_interior_point'} + Optimization method used. + constr : list of ndarray + List of constraint values at the solution. + jac : list of {ndarray, sparse matrix} + List of the Jacobian matrices of the constraints at the solution. + v : list of ndarray + List of the Lagrange multipliers for the constraints at the solution. + For an inequality constraint a positive multiplier means that the upper + bound is active, a negative multiplier means that the lower bound is + active and if a multiplier is zero it means the constraint is not + active. + constr_nfev : list of int + Number of constraint evaluations for each of the constraints. + constr_njev : list of int + Number of Jacobian matrix evaluations for each of the constraints. + constr_nhev : list of int + Number of Hessian evaluations for each of the constraints. + tr_radius : float + Radius of the trust region at the last iteration. + constr_penalty : float + Penalty parameter at the last iteration, see `initial_constr_penalty`. + barrier_tolerance : float + Tolerance for the barrier subproblem at the last iteration. + Only for problems with inequality constraints. + barrier_parameter : float + Barrier parameter at the last iteration. Only for problems + with inequality constraints. + execution_time : float + Total execution time. + message : str + Termination message. + status : {0, 1, 2, 3, 4} + Termination status: + + * 0 : The maximum number of function evaluations is exceeded. + * 1 : `gtol` termination condition is satisfied. + * 2 : `xtol` termination condition is satisfied. + * 3 : `callback` function requested termination. + * 4 : Constraint violation exceeds 'gtol'. + + .. versionchanged:: 1.15.0 + If the constraint violation exceeds `gtol`, then ``result.success`` + will now be False. + + cg_stop_cond : int + Reason for CG subproblem termination at the last iteration: + + * 0 : CG subproblem not evaluated. + * 1 : Iteration limit was reached. + * 2 : Reached the trust-region boundary. + * 3 : Negative curvature detected. + * 4 : Tolerance was satisfied. + + References + ---------- + .. [1] Conn, A. R., Gould, N. I., & Toint, P. L. + Trust region methods. 2000. Siam. pp. 19. + """ + x0 = np.atleast_1d(x0).astype(float) + n_vars = np.size(x0) + if hess is None: + if callable(hessp): + hess = HessianLinearOperator(hessp, n_vars) + else: + hess = BFGS() + if disp and verbose == 0: + verbose = 1 + + if bounds is not None: + modified_lb = np.nextafter(bounds.lb, -np.inf, where=bounds.lb > -np.inf) + modified_ub = np.nextafter(bounds.ub, np.inf, where=bounds.ub < np.inf) + modified_lb = np.where(np.isfinite(bounds.lb), modified_lb, bounds.lb) + modified_ub = np.where(np.isfinite(bounds.ub), modified_ub, bounds.ub) + bounds = Bounds(modified_lb, modified_ub, keep_feasible=bounds.keep_feasible) + finite_diff_bounds = strict_bounds(bounds.lb, bounds.ub, + bounds.keep_feasible, n_vars) + else: + finite_diff_bounds = (-np.inf, np.inf) + + # Define Objective Function + objective = ScalarFunction(fun, x0, args, grad, hess, + finite_diff_rel_step, finite_diff_bounds) + + # Put constraints in list format when needed. + if isinstance(constraints, (NonlinearConstraint | LinearConstraint)): + constraints = [constraints] + + # Prepare constraints. + prepared_constraints = [ + PreparedConstraint(c, x0, sparse_jacobian, finite_diff_bounds) + for c in constraints] + + # Check that all constraints are either sparse or dense. + n_sparse = sum(c.fun.sparse_jacobian for c in prepared_constraints) + if 0 < n_sparse < len(prepared_constraints): + raise ValueError("All constraints must have the same kind of the " + "Jacobian --- either all sparse or all dense. " + "You can set the sparsity globally by setting " + "`sparse_jacobian` to either True of False.") + if prepared_constraints: + sparse_jacobian = n_sparse > 0 + + if bounds is not None: + if sparse_jacobian is None: + sparse_jacobian = True + prepared_constraints.append(PreparedConstraint(bounds, x0, + sparse_jacobian)) + + # Concatenate initial constraints to the canonical form. + c_eq0, c_ineq0, J_eq0, J_ineq0 = initial_constraints_as_canonical( + n_vars, prepared_constraints, sparse_jacobian) + + # Prepare all canonical constraints and concatenate it into one. + canonical_all = [CanonicalConstraint.from_PreparedConstraint(c) + for c in prepared_constraints] + + if len(canonical_all) == 0: + canonical = CanonicalConstraint.empty(n_vars) + elif len(canonical_all) == 1: + canonical = canonical_all[0] + else: + canonical = CanonicalConstraint.concatenate(canonical_all, + sparse_jacobian) + + # Generate the Hessian of the Lagrangian. + lagrangian_hess = LagrangianHessian(n_vars, objective.hess, canonical.hess) + + # Choose appropriate method + if canonical.n_ineq == 0: + method = 'equality_constrained_sqp' + else: + method = 'tr_interior_point' + + # Construct OptimizeResult + state = OptimizeResult( + nit=0, nfev=0, njev=0, nhev=0, + cg_niter=0, cg_stop_cond=0, + fun=objective.f, grad=objective.g, + lagrangian_grad=np.copy(objective.g), + constr=[c.fun.f for c in prepared_constraints], + jac=[c.fun.J for c in prepared_constraints], + constr_nfev=[0 for c in prepared_constraints], + constr_njev=[0 for c in prepared_constraints], + constr_nhev=[0 for c in prepared_constraints], + v=[c.fun.v for c in prepared_constraints], + method=method) + + # Start counting + start_time = time.time() + + # Define stop criteria + if method == 'equality_constrained_sqp': + def stop_criteria(state, x, last_iteration_failed, + optimality, constr_violation, + tr_radius, constr_penalty, cg_info): + state = update_state_sqp(state, x, last_iteration_failed, + objective, prepared_constraints, + start_time, tr_radius, constr_penalty, + cg_info) + if verbose == 2: + BasicReport.print_iteration(state.nit, + state.nfev, + state.cg_niter, + state.fun, + state.tr_radius, + state.optimality, + state.constr_violation) + elif verbose > 2: + SQPReport.print_iteration(state.nit, + state.nfev, + state.cg_niter, + state.fun, + state.tr_radius, + state.optimality, + state.constr_violation, + state.constr_penalty, + state.cg_stop_cond) + state.status = None + state.niter = state.nit # Alias for callback (backward-compatibility) + if callback is not None: + callback_stop = False + try: + callback_stop = callback(state) + except StopIteration: + callback_stop = True + if callback_stop: + state.status = 3 + return True + if state.optimality < gtol and state.constr_violation < gtol: + state.status = 1 + elif state.tr_radius < xtol: + state.status = 2 + elif state.nit >= maxiter: + state.status = 0 + return state.status in (0, 1, 2, 3) + elif method == 'tr_interior_point': + def stop_criteria(state, x, last_iteration_failed, tr_radius, + constr_penalty, cg_info, barrier_parameter, + barrier_tolerance): + state = update_state_ip(state, x, last_iteration_failed, + objective, prepared_constraints, + start_time, tr_radius, constr_penalty, + cg_info, barrier_parameter, barrier_tolerance) + if verbose == 2: + BasicReport.print_iteration(state.nit, + state.nfev, + state.cg_niter, + state.fun, + state.tr_radius, + state.optimality, + state.constr_violation) + elif verbose > 2: + IPReport.print_iteration(state.nit, + state.nfev, + state.cg_niter, + state.fun, + state.tr_radius, + state.optimality, + state.constr_violation, + state.constr_penalty, + state.barrier_parameter, + state.cg_stop_cond) + state.status = None + state.niter = state.nit # Alias for callback (backward compatibility) + if callback is not None: + callback_stop = False + try: + callback_stop = callback(state) + except StopIteration: + callback_stop = True + if callback_stop: + state.status = 3 + return True + if state.optimality < gtol and state.constr_violation < gtol: + state.status = 1 + elif (state.tr_radius < xtol + and state.barrier_parameter < barrier_tol): + state.status = 2 + elif state.nit >= maxiter: + state.status = 0 + return state.status in (0, 1, 2, 3) + + if verbose == 2: + BasicReport.print_header() + elif verbose > 2: + if method == 'equality_constrained_sqp': + SQPReport.print_header() + elif method == 'tr_interior_point': + IPReport.print_header() + + # Call inferior function to do the optimization + if method == 'equality_constrained_sqp': + def fun_and_constr(x): + f = objective.fun(x) + c_eq, _ = canonical.fun(x) + return f, c_eq + + def grad_and_jac(x): + g = objective.grad(x) + J_eq, _ = canonical.jac(x) + return g, J_eq + + _, result = equality_constrained_sqp( + fun_and_constr, grad_and_jac, lagrangian_hess, + x0, objective.f, objective.g, + c_eq0, J_eq0, + stop_criteria, state, + initial_constr_penalty, initial_tr_radius, + factorization_method) + + elif method == 'tr_interior_point': + _, result = tr_interior_point( + objective.fun, objective.grad, lagrangian_hess, + n_vars, canonical.n_ineq, canonical.n_eq, + canonical.fun, canonical.jac, + x0, objective.f, objective.g, + c_ineq0, J_ineq0, c_eq0, J_eq0, + stop_criteria, + canonical.keep_feasible, + xtol, state, initial_barrier_parameter, + initial_barrier_tolerance, + initial_constr_penalty, initial_tr_radius, + factorization_method, finite_diff_bounds) + + # Status 4 occurs when minimize is successful but constraints are not satisfied. + if result.status in (1, 2) and state.constr_violation > gtol: + result.status = 4 + + # Status 3 occurs when the callback function requests termination, + # this is assumed to not be a success. + result.success = True if result.status in (1, 2) else False + result.message = TERMINATION_MESSAGES[result.status] + + # Alias (for backward compatibility with 1.1.0) + result.niter = result.nit + + if verbose == 2: + BasicReport.print_footer() + elif verbose > 2: + if method == 'equality_constrained_sqp': + SQPReport.print_footer() + elif method == 'tr_interior_point': + IPReport.print_footer() + if verbose >= 1: + print(result.message) + print(f"Number of iterations: {result.nit}, " + f"function evaluations: {result.nfev}, " + f"CG iterations: {result.cg_niter}, " + f"optimality: {result.optimality:.2e}, " + f"constraint violation: {result.constr_violation:.2e}, " + f"execution time: {result.execution_time:4.2} s.") + return result diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/projections.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/projections.py new file mode 100644 index 0000000000000000000000000000000000000000..a07b836bdbad688a265ae34ce91a361fd5050eb1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/projections.py @@ -0,0 +1,407 @@ +"""Basic linear factorizations needed by the solver.""" + +from scipy.sparse import (bmat, csc_matrix, eye, issparse) +from scipy.sparse.linalg import LinearOperator +import scipy.linalg +import scipy.sparse.linalg +try: + from sksparse.cholmod import cholesky_AAt + sksparse_available = True +except ImportError: + import warnings + sksparse_available = False +import numpy as np +from warnings import warn + +__all__ = [ + 'orthogonality', + 'projections', +] + + +def orthogonality(A, g): + """Measure orthogonality between a vector and the null space of a matrix. + + Compute a measure of orthogonality between the null space + of the (possibly sparse) matrix ``A`` and a given vector ``g``. + + The formula is a simplified (and cheaper) version of formula (3.13) + from [1]_. + ``orth = norm(A g, ord=2)/(norm(A, ord='fro')*norm(g, ord=2))``. + + References + ---------- + .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal. + "On the solution of equality constrained quadratic + programming problems arising in optimization." + SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395. + """ + # Compute vector norms + norm_g = np.linalg.norm(g) + # Compute Froebnius norm of the matrix A + if issparse(A): + norm_A = scipy.sparse.linalg.norm(A, ord='fro') + else: + norm_A = np.linalg.norm(A, ord='fro') + + # Check if norms are zero + if norm_g == 0 or norm_A == 0: + return 0 + + norm_A_g = np.linalg.norm(A.dot(g)) + # Orthogonality measure + orth = norm_A_g / (norm_A*norm_g) + return orth + + +def normal_equation_projections(A, m, n, orth_tol, max_refin, tol): + """Return linear operators for matrix A using ``NormalEquation`` approach. + """ + # Cholesky factorization + factor = cholesky_AAt(A) + + # z = x - A.T inv(A A.T) A x + def null_space(x): + v = factor(A.dot(x)) + z = x - A.T.dot(v) + + # Iterative refinement to improve roundoff + # errors described in [2]_, algorithm 5.1. + k = 0 + while orthogonality(A, z) > orth_tol: + if k >= max_refin: + break + # z_next = z - A.T inv(A A.T) A z + v = factor(A.dot(z)) + z = z - A.T.dot(v) + k += 1 + + return z + + # z = inv(A A.T) A x + def least_squares(x): + return factor(A.dot(x)) + + # z = A.T inv(A A.T) x + def row_space(x): + return A.T.dot(factor(x)) + + return null_space, least_squares, row_space + + +def augmented_system_projections(A, m, n, orth_tol, max_refin, tol): + """Return linear operators for matrix A - ``AugmentedSystem``.""" + # Form augmented system + K = csc_matrix(bmat([[eye(n), A.T], [A, None]])) + # LU factorization + # TODO: Use a symmetric indefinite factorization + # to solve the system twice as fast (because + # of the symmetry). + try: + solve = scipy.sparse.linalg.factorized(K) + except RuntimeError: + warn("Singular Jacobian matrix. Using dense SVD decomposition to " + "perform the factorizations.", + stacklevel=3) + return svd_factorization_projections(A.toarray(), + m, n, orth_tol, + max_refin, tol) + + # z = x - A.T inv(A A.T) A x + # is computed solving the extended system: + # [I A.T] * [ z ] = [x] + # [A O ] [aux] [0] + def null_space(x): + # v = [x] + # [0] + v = np.hstack([x, np.zeros(m)]) + # lu_sol = [ z ] + # [aux] + lu_sol = solve(v) + z = lu_sol[:n] + + # Iterative refinement to improve roundoff + # errors described in [2]_, algorithm 5.2. + k = 0 + while orthogonality(A, z) > orth_tol: + if k >= max_refin: + break + # new_v = [x] - [I A.T] * [ z ] + # [0] [A O ] [aux] + new_v = v - K.dot(lu_sol) + # [I A.T] * [delta z ] = new_v + # [A O ] [delta aux] + lu_update = solve(new_v) + # [ z ] += [delta z ] + # [aux] [delta aux] + lu_sol += lu_update + z = lu_sol[:n] + k += 1 + + # return z = x - A.T inv(A A.T) A x + return z + + # z = inv(A A.T) A x + # is computed solving the extended system: + # [I A.T] * [aux] = [x] + # [A O ] [ z ] [0] + def least_squares(x): + # v = [x] + # [0] + v = np.hstack([x, np.zeros(m)]) + # lu_sol = [aux] + # [ z ] + lu_sol = solve(v) + # return z = inv(A A.T) A x + return lu_sol[n:m+n] + + # z = A.T inv(A A.T) x + # is computed solving the extended system: + # [I A.T] * [ z ] = [0] + # [A O ] [aux] [x] + def row_space(x): + # v = [0] + # [x] + v = np.hstack([np.zeros(n), x]) + # lu_sol = [ z ] + # [aux] + lu_sol = solve(v) + # return z = A.T inv(A A.T) x + return lu_sol[:n] + + return null_space, least_squares, row_space + + +def qr_factorization_projections(A, m, n, orth_tol, max_refin, tol): + """Return linear operators for matrix A using ``QRFactorization`` approach. + """ + # QRFactorization + Q, R, P = scipy.linalg.qr(A.T, pivoting=True, mode='economic') + + if np.linalg.norm(R[-1, :], np.inf) < tol: + warn('Singular Jacobian matrix. Using SVD decomposition to ' + + 'perform the factorizations.', + stacklevel=3) + return svd_factorization_projections(A, m, n, + orth_tol, + max_refin, + tol) + + # z = x - A.T inv(A A.T) A x + def null_space(x): + # v = P inv(R) Q.T x + aux1 = Q.T.dot(x) + aux2 = scipy.linalg.solve_triangular(R, aux1, lower=False) + v = np.zeros(m) + v[P] = aux2 + z = x - A.T.dot(v) + + # Iterative refinement to improve roundoff + # errors described in [2]_, algorithm 5.1. + k = 0 + while orthogonality(A, z) > orth_tol: + if k >= max_refin: + break + # v = P inv(R) Q.T x + aux1 = Q.T.dot(z) + aux2 = scipy.linalg.solve_triangular(R, aux1, lower=False) + v[P] = aux2 + # z_next = z - A.T v + z = z - A.T.dot(v) + k += 1 + + return z + + # z = inv(A A.T) A x + def least_squares(x): + # z = P inv(R) Q.T x + aux1 = Q.T.dot(x) + aux2 = scipy.linalg.solve_triangular(R, aux1, lower=False) + z = np.zeros(m) + z[P] = aux2 + return z + + # z = A.T inv(A A.T) x + def row_space(x): + # z = Q inv(R.T) P.T x + aux1 = x[P] + aux2 = scipy.linalg.solve_triangular(R, aux1, + lower=False, + trans='T') + z = Q.dot(aux2) + return z + + return null_space, least_squares, row_space + + +def svd_factorization_projections(A, m, n, orth_tol, max_refin, tol): + """Return linear operators for matrix A using ``SVDFactorization`` approach. + """ + # SVD Factorization + U, s, Vt = scipy.linalg.svd(A, full_matrices=False) + + # Remove dimensions related with very small singular values + U = U[:, s > tol] + Vt = Vt[s > tol, :] + s = s[s > tol] + + # z = x - A.T inv(A A.T) A x + def null_space(x): + # v = U 1/s V.T x = inv(A A.T) A x + aux1 = Vt.dot(x) + aux2 = 1/s*aux1 + v = U.dot(aux2) + z = x - A.T.dot(v) + + # Iterative refinement to improve roundoff + # errors described in [2]_, algorithm 5.1. + k = 0 + while orthogonality(A, z) > orth_tol: + if k >= max_refin: + break + # v = U 1/s V.T x = inv(A A.T) A x + aux1 = Vt.dot(z) + aux2 = 1/s*aux1 + v = U.dot(aux2) + # z_next = z - A.T v + z = z - A.T.dot(v) + k += 1 + + return z + + # z = inv(A A.T) A x + def least_squares(x): + # z = U 1/s V.T x = inv(A A.T) A x + aux1 = Vt.dot(x) + aux2 = 1/s*aux1 + z = U.dot(aux2) + return z + + # z = A.T inv(A A.T) x + def row_space(x): + # z = V 1/s U.T x + aux1 = U.T.dot(x) + aux2 = 1/s*aux1 + z = Vt.T.dot(aux2) + return z + + return null_space, least_squares, row_space + + +def projections(A, method=None, orth_tol=1e-12, max_refin=3, tol=1e-15): + """Return three linear operators related with a given matrix A. + + Parameters + ---------- + A : sparse matrix (or ndarray), shape (m, n) + Matrix ``A`` used in the projection. + method : string, optional + Method used for compute the given linear + operators. Should be one of: + + - 'NormalEquation': The operators + will be computed using the + so-called normal equation approach + explained in [1]_. In order to do + so the Cholesky factorization of + ``(A A.T)`` is computed. Exclusive + for sparse matrices. + - 'AugmentedSystem': The operators + will be computed using the + so-called augmented system approach + explained in [1]_. Exclusive + for sparse matrices. + - 'QRFactorization': Compute projections + using QR factorization. Exclusive for + dense matrices. + - 'SVDFactorization': Compute projections + using SVD factorization. Exclusive for + dense matrices. + + orth_tol : float, optional + Tolerance for iterative refinements. + max_refin : int, optional + Maximum number of iterative refinements. + tol : float, optional + Tolerance for singular values. + + Returns + ------- + Z : LinearOperator, shape (n, n) + Null-space operator. For a given vector ``x``, + the null space operator is equivalent to apply + a projection matrix ``P = I - A.T inv(A A.T) A`` + to the vector. It can be shown that this is + equivalent to project ``x`` into the null space + of A. + LS : LinearOperator, shape (m, n) + Least-squares operator. For a given vector ``x``, + the least-squares operator is equivalent to apply a + pseudoinverse matrix ``pinv(A.T) = inv(A A.T) A`` + to the vector. It can be shown that this vector + ``pinv(A.T) x`` is the least_square solution to + ``A.T y = x``. + Y : LinearOperator, shape (n, m) + Row-space operator. For a given vector ``x``, + the row-space operator is equivalent to apply a + projection matrix ``Q = A.T inv(A A.T)`` + to the vector. It can be shown that this + vector ``y = Q x`` the minimum norm solution + of ``A y = x``. + + Notes + ----- + Uses iterative refinements described in [1] + during the computation of ``Z`` in order to + cope with the possibility of large roundoff errors. + + References + ---------- + .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal. + "On the solution of equality constrained quadratic + programming problems arising in optimization." + SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395. + """ + m, n = np.shape(A) + + # The factorization of an empty matrix + # only works for the sparse representation. + if m*n == 0: + A = csc_matrix(A) + + # Check Argument + if issparse(A): + if method is None: + method = "AugmentedSystem" + if method not in ("NormalEquation", "AugmentedSystem"): + raise ValueError("Method not allowed for sparse matrix.") + if method == "NormalEquation" and not sksparse_available: + warnings.warn("Only accepts 'NormalEquation' option when " + "scikit-sparse is available. Using " + "'AugmentedSystem' option instead.", + ImportWarning, stacklevel=3) + method = 'AugmentedSystem' + else: + if method is None: + method = "QRFactorization" + if method not in ("QRFactorization", "SVDFactorization"): + raise ValueError("Method not allowed for dense array.") + + if method == 'NormalEquation': + null_space, least_squares, row_space \ + = normal_equation_projections(A, m, n, orth_tol, max_refin, tol) + elif method == 'AugmentedSystem': + null_space, least_squares, row_space \ + = augmented_system_projections(A, m, n, orth_tol, max_refin, tol) + elif method == "QRFactorization": + null_space, least_squares, row_space \ + = qr_factorization_projections(A, m, n, orth_tol, max_refin, tol) + elif method == "SVDFactorization": + null_space, least_squares, row_space \ + = svd_factorization_projections(A, m, n, orth_tol, max_refin, tol) + + Z = LinearOperator((n, n), null_space) + LS = LinearOperator((m, n), least_squares) + Y = LinearOperator((n, m), row_space) + + return Z, LS, Y diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/qp_subproblem.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/qp_subproblem.py new file mode 100644 index 0000000000000000000000000000000000000000..a039a7738c283f90f30fd7c4583bf9e1a8f559d5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/qp_subproblem.py @@ -0,0 +1,637 @@ +"""Equality-constrained quadratic programming solvers.""" + +from scipy.sparse import (linalg, bmat, csc_matrix) +from math import copysign +import numpy as np +from numpy.linalg import norm + +__all__ = [ + 'eqp_kktfact', + 'sphere_intersections', + 'box_intersections', + 'box_sphere_intersections', + 'inside_box_boundaries', + 'modified_dogleg', + 'projected_cg' +] + + +# For comparison with the projected CG +def eqp_kktfact(H, c, A, b): + """Solve equality-constrained quadratic programming (EQP) problem. + + Solve ``min 1/2 x.T H x + x.t c`` subject to ``A x + b = 0`` + using direct factorization of the KKT system. + + Parameters + ---------- + H : sparse matrix, shape (n, n) + Hessian matrix of the EQP problem. + c : array_like, shape (n,) + Gradient of the quadratic objective function. + A : sparse matrix + Jacobian matrix of the EQP problem. + b : array_like, shape (m,) + Right-hand side of the constraint equation. + + Returns + ------- + x : array_like, shape (n,) + Solution of the KKT problem. + lagrange_multipliers : ndarray, shape (m,) + Lagrange multipliers of the KKT problem. + """ + n, = np.shape(c) # Number of parameters + m, = np.shape(b) # Number of constraints + + # Karush-Kuhn-Tucker matrix of coefficients. + # Defined as in Nocedal/Wright "Numerical + # Optimization" p.452 in Eq. (16.4). + kkt_matrix = csc_matrix(bmat([[H, A.T], [A, None]])) + # Vector of coefficients. + kkt_vec = np.hstack([-c, -b]) + + # TODO: Use a symmetric indefinite factorization + # to solve the system twice as fast (because + # of the symmetry). + lu = linalg.splu(kkt_matrix) + kkt_sol = lu.solve(kkt_vec) + x = kkt_sol[:n] + lagrange_multipliers = -kkt_sol[n:n+m] + + return x, lagrange_multipliers + + +def sphere_intersections(z, d, trust_radius, + entire_line=False): + """Find the intersection between segment (or line) and spherical constraints. + + Find the intersection between the segment (or line) defined by the + parametric equation ``x(t) = z + t*d`` and the ball + ``||x|| <= trust_radius``. + + Parameters + ---------- + z : array_like, shape (n,) + Initial point. + d : array_like, shape (n,) + Direction. + trust_radius : float + Ball radius. + entire_line : bool, optional + When ``True``, the function returns the intersection between the line + ``x(t) = z + t*d`` (``t`` can assume any value) and the ball + ``||x|| <= trust_radius``. When ``False``, the function returns the intersection + between the segment ``x(t) = z + t*d``, ``0 <= t <= 1``, and the ball. + + Returns + ------- + ta, tb : float + The line/segment ``x(t) = z + t*d`` is inside the ball for + for ``ta <= t <= tb``. + intersect : bool + When ``True``, there is a intersection between the line/segment + and the sphere. On the other hand, when ``False``, there is no + intersection. + """ + # Special case when d=0 + if norm(d) == 0: + return 0, 0, False + # Check for inf trust_radius + if np.isinf(trust_radius): + if entire_line: + ta = -np.inf + tb = np.inf + else: + ta = 0 + tb = 1 + intersect = True + return ta, tb, intersect + + a = np.dot(d, d) + b = 2 * np.dot(z, d) + c = np.dot(z, z) - trust_radius**2 + discriminant = b*b - 4*a*c + if discriminant < 0: + intersect = False + return 0, 0, intersect + sqrt_discriminant = np.sqrt(discriminant) + + # The following calculation is mathematically + # equivalent to: + # ta = (-b - sqrt_discriminant) / (2*a) + # tb = (-b + sqrt_discriminant) / (2*a) + # but produce smaller round off errors. + # Look at Matrix Computation p.97 + # for a better justification. + aux = b + copysign(sqrt_discriminant, b) + ta = -aux / (2*a) + tb = -2*c / aux + ta, tb = sorted([ta, tb]) + + if entire_line: + intersect = True + else: + # Checks to see if intersection happens + # within vectors length. + if tb < 0 or ta > 1: + intersect = False + ta = 0 + tb = 0 + else: + intersect = True + # Restrict intersection interval + # between 0 and 1. + ta = max(0, ta) + tb = min(1, tb) + + return ta, tb, intersect + + +def box_intersections(z, d, lb, ub, + entire_line=False): + """Find the intersection between segment (or line) and box constraints. + + Find the intersection between the segment (or line) defined by the + parametric equation ``x(t) = z + t*d`` and the rectangular box + ``lb <= x <= ub``. + + Parameters + ---------- + z : array_like, shape (n,) + Initial point. + d : array_like, shape (n,) + Direction. + lb : array_like, shape (n,) + Lower bounds to each one of the components of ``x``. Used + to delimit the rectangular box. + ub : array_like, shape (n, ) + Upper bounds to each one of the components of ``x``. Used + to delimit the rectangular box. + entire_line : bool, optional + When ``True``, the function returns the intersection between the line + ``x(t) = z + t*d`` (``t`` can assume any value) and the rectangular + box. When ``False``, the function returns the intersection between the segment + ``x(t) = z + t*d``, ``0 <= t <= 1``, and the rectangular box. + + Returns + ------- + ta, tb : float + The line/segment ``x(t) = z + t*d`` is inside the box for + for ``ta <= t <= tb``. + intersect : bool + When ``True``, there is a intersection between the line (or segment) + and the rectangular box. On the other hand, when ``False``, there is no + intersection. + """ + # Make sure it is a numpy array + z = np.asarray(z) + d = np.asarray(d) + lb = np.asarray(lb) + ub = np.asarray(ub) + # Special case when d=0 + if norm(d) == 0: + return 0, 0, False + + # Get values for which d==0 + zero_d = (d == 0) + # If the boundaries are not satisfied for some coordinate + # for which "d" is zero, there is no box-line intersection. + if (z[zero_d] < lb[zero_d]).any() or (z[zero_d] > ub[zero_d]).any(): + intersect = False + return 0, 0, intersect + # Remove values for which d is zero + not_zero_d = np.logical_not(zero_d) + z = z[not_zero_d] + d = d[not_zero_d] + lb = lb[not_zero_d] + ub = ub[not_zero_d] + + # Find a series of intervals (t_lb[i], t_ub[i]). + t_lb = (lb-z) / d + t_ub = (ub-z) / d + # Get the intersection of all those intervals. + ta = max(np.minimum(t_lb, t_ub)) + tb = min(np.maximum(t_lb, t_ub)) + + # Check if intersection is feasible + if ta <= tb: + intersect = True + else: + intersect = False + # Checks to see if intersection happens within vectors length. + if not entire_line: + if tb < 0 or ta > 1: + intersect = False + ta = 0 + tb = 0 + else: + # Restrict intersection interval between 0 and 1. + ta = max(0, ta) + tb = min(1, tb) + + return ta, tb, intersect + + +def box_sphere_intersections(z, d, lb, ub, trust_radius, + entire_line=False, + extra_info=False): + """Find the intersection between segment (or line) and box/sphere constraints. + + Find the intersection between the segment (or line) defined by the + parametric equation ``x(t) = z + t*d``, the rectangular box + ``lb <= x <= ub`` and the ball ``||x|| <= trust_radius``. + + Parameters + ---------- + z : array_like, shape (n,) + Initial point. + d : array_like, shape (n,) + Direction. + lb : array_like, shape (n,) + Lower bounds to each one of the components of ``x``. Used + to delimit the rectangular box. + ub : array_like, shape (n, ) + Upper bounds to each one of the components of ``x``. Used + to delimit the rectangular box. + trust_radius : float + Ball radius. + entire_line : bool, optional + When ``True``, the function returns the intersection between the line + ``x(t) = z + t*d`` (``t`` can assume any value) and the constraints. + When ``False``, the function returns the intersection between the segment + ``x(t) = z + t*d``, ``0 <= t <= 1`` and the constraints. + extra_info : bool, optional + When ``True``, the function returns ``intersect_sphere`` and ``intersect_box``. + + Returns + ------- + ta, tb : float + The line/segment ``x(t) = z + t*d`` is inside the rectangular box and + inside the ball for ``ta <= t <= tb``. + intersect : bool + When ``True``, there is a intersection between the line (or segment) + and both constraints. On the other hand, when ``False``, there is no + intersection. + sphere_info : dict, optional + Dictionary ``{ta, tb, intersect}`` containing the interval ``[ta, tb]`` + for which the line intercepts the ball. And a boolean value indicating + whether the sphere is intersected by the line. + box_info : dict, optional + Dictionary ``{ta, tb, intersect}`` containing the interval ``[ta, tb]`` + for which the line intercepts the box. And a boolean value indicating + whether the box is intersected by the line. + """ + ta_b, tb_b, intersect_b = box_intersections(z, d, lb, ub, + entire_line) + ta_s, tb_s, intersect_s = sphere_intersections(z, d, + trust_radius, + entire_line) + ta = np.maximum(ta_b, ta_s) + tb = np.minimum(tb_b, tb_s) + if intersect_b and intersect_s and ta <= tb: + intersect = True + else: + intersect = False + + if extra_info: + sphere_info = {'ta': ta_s, 'tb': tb_s, 'intersect': intersect_s} + box_info = {'ta': ta_b, 'tb': tb_b, 'intersect': intersect_b} + return ta, tb, intersect, sphere_info, box_info + else: + return ta, tb, intersect + + +def inside_box_boundaries(x, lb, ub): + """Check if lb <= x <= ub.""" + return (lb <= x).all() and (x <= ub).all() + + +def reinforce_box_boundaries(x, lb, ub): + """Return clipped value of x""" + return np.minimum(np.maximum(x, lb), ub) + + +def modified_dogleg(A, Y, b, trust_radius, lb, ub): + """Approximately minimize ``1/2*|| A x + b ||^2`` inside trust-region. + + Approximately solve the problem of minimizing ``1/2*|| A x + b ||^2`` + subject to ``||x|| < Delta`` and ``lb <= x <= ub`` using a modification + of the classical dogleg approach. + + Parameters + ---------- + A : LinearOperator (or sparse matrix or ndarray), shape (m, n) + Matrix ``A`` in the minimization problem. It should have + dimension ``(m, n)`` such that ``m < n``. + Y : LinearOperator (or sparse matrix or ndarray), shape (n, m) + LinearOperator that apply the projection matrix + ``Q = A.T inv(A A.T)`` to the vector. The obtained vector + ``y = Q x`` being the minimum norm solution of ``A y = x``. + b : array_like, shape (m,) + Vector ``b``in the minimization problem. + trust_radius: float + Trust radius to be considered. Delimits a sphere boundary + to the problem. + lb : array_like, shape (n,) + Lower bounds to each one of the components of ``x``. + It is expected that ``lb <= 0``, otherwise the algorithm + may fail. If ``lb[i] = -Inf``, the lower + bound for the ith component is just ignored. + ub : array_like, shape (n, ) + Upper bounds to each one of the components of ``x``. + It is expected that ``ub >= 0``, otherwise the algorithm + may fail. If ``ub[i] = Inf``, the upper bound for the ith + component is just ignored. + + Returns + ------- + x : array_like, shape (n,) + Solution to the problem. + + Notes + ----- + Based on implementations described in pp. 885-886 from [1]_. + + References + ---------- + .. [1] Byrd, Richard H., Mary E. Hribar, and Jorge Nocedal. + "An interior point algorithm for large-scale nonlinear + programming." SIAM Journal on Optimization 9.4 (1999): 877-900. + """ + # Compute minimum norm minimizer of 1/2*|| A x + b ||^2. + newton_point = -Y.dot(b) + # Check for interior point + if inside_box_boundaries(newton_point, lb, ub) \ + and norm(newton_point) <= trust_radius: + x = newton_point + return x + + # Compute gradient vector ``g = A.T b`` + g = A.T.dot(b) + # Compute Cauchy point + # `cauchy_point = g.T g / (g.T A.T A g)``. + A_g = A.dot(g) + cauchy_point = -np.dot(g, g) / np.dot(A_g, A_g) * g + # Origin + origin_point = np.zeros_like(cauchy_point) + + # Check the segment between cauchy_point and newton_point + # for a possible solution. + z = cauchy_point + p = newton_point - cauchy_point + _, alpha, intersect = box_sphere_intersections(z, p, lb, ub, + trust_radius) + if intersect: + x1 = z + alpha*p + else: + # Check the segment between the origin and cauchy_point + # for a possible solution. + z = origin_point + p = cauchy_point + _, alpha, _ = box_sphere_intersections(z, p, lb, ub, + trust_radius) + x1 = z + alpha*p + + # Check the segment between origin and newton_point + # for a possible solution. + z = origin_point + p = newton_point + _, alpha, _ = box_sphere_intersections(z, p, lb, ub, + trust_radius) + x2 = z + alpha*p + + # Return the best solution among x1 and x2. + if norm(A.dot(x1) + b) < norm(A.dot(x2) + b): + return x1 + else: + return x2 + + +def projected_cg(H, c, Z, Y, b, trust_radius=np.inf, + lb=None, ub=None, tol=None, + max_iter=None, max_infeasible_iter=None, + return_all=False): + """Solve EQP problem with projected CG method. + + Solve equality-constrained quadratic programming problem + ``min 1/2 x.T H x + x.t c`` subject to ``A x + b = 0`` and, + possibly, to trust region constraints ``||x|| < trust_radius`` + and box constraints ``lb <= x <= ub``. + + Parameters + ---------- + H : LinearOperator (or sparse matrix or ndarray), shape (n, n) + Operator for computing ``H v``. + c : array_like, shape (n,) + Gradient of the quadratic objective function. + Z : LinearOperator (or sparse matrix or ndarray), shape (n, n) + Operator for projecting ``x`` into the null space of A. + Y : LinearOperator, sparse matrix, ndarray, shape (n, m) + Operator that, for a given a vector ``b``, compute smallest + norm solution of ``A x + b = 0``. + b : array_like, shape (m,) + Right-hand side of the constraint equation. + trust_radius : float, optional + Trust radius to be considered. By default, uses ``trust_radius=inf``, + which means no trust radius at all. + lb : array_like, shape (n,), optional + Lower bounds to each one of the components of ``x``. + If ``lb[i] = -Inf`` the lower bound for the i-th + component is just ignored (default). + ub : array_like, shape (n, ), optional + Upper bounds to each one of the components of ``x``. + If ``ub[i] = Inf`` the upper bound for the i-th + component is just ignored (default). + tol : float, optional + Tolerance used to interrupt the algorithm. + max_iter : int, optional + Maximum algorithm iterations. Where ``max_inter <= n-m``. + By default, uses ``max_iter = n-m``. + max_infeasible_iter : int, optional + Maximum infeasible (regarding box constraints) iterations the + algorithm is allowed to take. + By default, uses ``max_infeasible_iter = n-m``. + return_all : bool, optional + When ``true``, return the list of all vectors through the iterations. + + Returns + ------- + x : array_like, shape (n,) + Solution of the EQP problem. + info : Dict + Dictionary containing the following: + + - niter : Number of iterations. + - stop_cond : Reason for algorithm termination: + 1. Iteration limit was reached; + 2. Reached the trust-region boundary; + 3. Negative curvature detected; + 4. Tolerance was satisfied. + - allvecs : List containing all intermediary vectors (optional). + - hits_boundary : True if the proposed step is on the boundary + of the trust region. + + Notes + ----- + Implementation of Algorithm 6.2 on [1]_. + + In the absence of spherical and box constraints, for sufficient + iterations, the method returns a truly optimal result. + In the presence of those constraints, the value returned is only + a inexpensive approximation of the optimal value. + + References + ---------- + .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal. + "On the solution of equality constrained quadratic + programming problems arising in optimization." + SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395. + """ + CLOSE_TO_ZERO = 1e-25 + + n, = np.shape(c) # Number of parameters + m, = np.shape(b) # Number of constraints + + # Initial Values + x = Y.dot(-b) + r = Z.dot(H.dot(x) + c) + g = Z.dot(r) + p = -g + + # Store ``x`` value + if return_all: + allvecs = [x] + # Values for the first iteration + H_p = H.dot(p) + rt_g = norm(g)**2 # g.T g = r.T Z g = r.T g (ref [1]_ p.1389) + + # If x > trust-region the problem does not have a solution. + tr_distance = trust_radius - norm(x) + if tr_distance < 0: + raise ValueError("Trust region problem does not have a solution.") + # If x == trust_radius, then x is the solution + # to the optimization problem, since x is the + # minimum norm solution to Ax=b. + elif tr_distance < CLOSE_TO_ZERO: + info = {'niter': 0, 'stop_cond': 2, 'hits_boundary': True} + if return_all: + allvecs.append(x) + info['allvecs'] = allvecs + return x, info + + # Set default tolerance + if tol is None: + tol = max(min(0.01 * np.sqrt(rt_g), 0.1 * rt_g), CLOSE_TO_ZERO) + # Set default lower and upper bounds + if lb is None: + lb = np.full(n, -np.inf) + if ub is None: + ub = np.full(n, np.inf) + # Set maximum iterations + if max_iter is None: + max_iter = n-m + max_iter = min(max_iter, n-m) + # Set maximum infeasible iterations + if max_infeasible_iter is None: + max_infeasible_iter = n-m + + hits_boundary = False + stop_cond = 1 + counter = 0 + last_feasible_x = np.zeros_like(x) + k = 0 + for i in range(max_iter): + # Stop criteria - Tolerance : r.T g < tol + if rt_g < tol: + stop_cond = 4 + break + k += 1 + # Compute curvature + pt_H_p = H_p.dot(p) + # Stop criteria - Negative curvature + if pt_H_p <= 0: + if np.isinf(trust_radius): + raise ValueError("Negative curvature not allowed " + "for unrestricted problems.") + else: + # Find intersection with constraints + _, alpha, intersect = box_sphere_intersections( + x, p, lb, ub, trust_radius, entire_line=True) + # Update solution + if intersect: + x = x + alpha*p + # Reinforce variables are inside box constraints. + # This is only necessary because of roundoff errors. + x = reinforce_box_boundaries(x, lb, ub) + # Attribute information + stop_cond = 3 + hits_boundary = True + break + + # Get next step + alpha = rt_g / pt_H_p + x_next = x + alpha*p + + # Stop criteria - Hits boundary + if np.linalg.norm(x_next) >= trust_radius: + # Find intersection with box constraints + _, theta, intersect = box_sphere_intersections(x, alpha*p, lb, ub, + trust_radius) + # Update solution + if intersect: + x = x + theta*alpha*p + # Reinforce variables are inside box constraints. + # This is only necessary because of roundoff errors. + x = reinforce_box_boundaries(x, lb, ub) + # Attribute information + stop_cond = 2 + hits_boundary = True + break + + # Check if ``x`` is inside the box and start counter if it is not. + if inside_box_boundaries(x_next, lb, ub): + counter = 0 + else: + counter += 1 + # Whenever outside box constraints keep looking for intersections. + if counter > 0: + _, theta, intersect = box_sphere_intersections(x, alpha*p, lb, ub, + trust_radius) + if intersect: + last_feasible_x = x + theta*alpha*p + # Reinforce variables are inside box constraints. + # This is only necessary because of roundoff errors. + last_feasible_x = reinforce_box_boundaries(last_feasible_x, + lb, ub) + counter = 0 + # Stop after too many infeasible (regarding box constraints) iteration. + if counter > max_infeasible_iter: + break + # Store ``x_next`` value + if return_all: + allvecs.append(x_next) + + # Update residual + r_next = r + alpha*H_p + # Project residual g+ = Z r+ + g_next = Z.dot(r_next) + # Compute conjugate direction step d + rt_g_next = norm(g_next)**2 # g.T g = r.T g (ref [1]_ p.1389) + beta = rt_g_next / rt_g + p = - g_next + beta*p + # Prepare for next iteration + x = x_next + g = g_next + r = g_next + rt_g = norm(g)**2 # g.T g = r.T Z g = r.T g (ref [1]_ p.1389) + H_p = H.dot(p) + + if not inside_box_boundaries(x, lb, ub): + x = last_feasible_x + hits_boundary = True + info = {'niter': k, 'stop_cond': stop_cond, + 'hits_boundary': hits_boundary} + if return_all: + info['allvecs'] = allvecs + return x, info diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/report.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/report.py new file mode 100644 index 0000000000000000000000000000000000000000..f7f997d663cd5ce6265e77f940622b6105362bf7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/report.py @@ -0,0 +1,49 @@ +"""Progress report printers.""" + +class ReportBase: + COLUMN_NAMES: list[str] = NotImplemented + COLUMN_WIDTHS: list[int] = NotImplemented + ITERATION_FORMATS: list[str] = NotImplemented + + @classmethod + def print_header(cls): + fmt = ("|" + + "|".join([f"{{:^{x}}}" for x in cls.COLUMN_WIDTHS]) + + "|") + separators = ['-' * x for x in cls.COLUMN_WIDTHS] + print(fmt.format(*cls.COLUMN_NAMES)) + print(fmt.format(*separators)) + + @classmethod + def print_iteration(cls, *args): + iteration_format = [f"{{:{x}}}" for x in cls.ITERATION_FORMATS] + fmt = "|" + "|".join(iteration_format) + "|" + print(fmt.format(*args)) + + @classmethod + def print_footer(cls): + print() + + +class BasicReport(ReportBase): + COLUMN_NAMES = ["niter", "f evals", "CG iter", "obj func", "tr radius", + "opt", "c viol"] + COLUMN_WIDTHS = [7, 7, 7, 13, 10, 10, 10] + ITERATION_FORMATS = ["^7", "^7", "^7", "^+13.4e", + "^10.2e", "^10.2e", "^10.2e"] + + +class SQPReport(ReportBase): + COLUMN_NAMES = ["niter", "f evals", "CG iter", "obj func", "tr radius", + "opt", "c viol", "penalty", "CG stop"] + COLUMN_WIDTHS = [7, 7, 7, 13, 10, 10, 10, 10, 7] + ITERATION_FORMATS = ["^7", "^7", "^7", "^+13.4e", "^10.2e", "^10.2e", + "^10.2e", "^10.2e", "^7"] + + +class IPReport(ReportBase): + COLUMN_NAMES = ["niter", "f evals", "CG iter", "obj func", "tr radius", + "opt", "c viol", "penalty", "barrier param", "CG stop"] + COLUMN_WIDTHS = [7, 7, 7, 13, 10, 10, 10, 10, 13, 7] + ITERATION_FORMATS = ["^7", "^7", "^7", "^+13.4e", "^10.2e", "^10.2e", + "^10.2e", "^10.2e", "^13.2e", "^7"] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_canonical_constraint.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_canonical_constraint.py new file mode 100644 index 0000000000000000000000000000000000000000..452b327d02da3b3bd3fab9592bdef4d56d6aff57 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_canonical_constraint.py @@ -0,0 +1,296 @@ +import numpy as np +from numpy.testing import assert_array_equal, assert_equal +from scipy.optimize._constraints import (NonlinearConstraint, Bounds, + PreparedConstraint) +from scipy.optimize._trustregion_constr.canonical_constraint \ + import CanonicalConstraint, initial_constraints_as_canonical + + +def create_quadratic_function(n, m, rng): + a = rng.rand(m) + A = rng.rand(m, n) + H = rng.rand(m, n, n) + HT = np.transpose(H, (1, 2, 0)) + + def fun(x): + return a + A.dot(x) + 0.5 * H.dot(x).dot(x) + + def jac(x): + return A + H.dot(x) + + def hess(x, v): + return HT.dot(v) + + return fun, jac, hess + + +def test_bounds_cases(): + # Test 1: no constraints. + user_constraint = Bounds(-np.inf, np.inf) + x0 = np.array([-1, 2]) + prepared_constraint = PreparedConstraint(user_constraint, x0, False) + c = CanonicalConstraint.from_PreparedConstraint(prepared_constraint) + + assert_equal(c.n_eq, 0) + assert_equal(c.n_ineq, 0) + + c_eq, c_ineq = c.fun(x0) + assert_array_equal(c_eq, []) + assert_array_equal(c_ineq, []) + + J_eq, J_ineq = c.jac(x0) + assert_array_equal(J_eq, np.empty((0, 2))) + assert_array_equal(J_ineq, np.empty((0, 2))) + + assert_array_equal(c.keep_feasible, []) + + # Test 2: infinite lower bound. + user_constraint = Bounds(-np.inf, [0, np.inf, 1], [False, True, True]) + x0 = np.array([-1, -2, -3], dtype=float) + prepared_constraint = PreparedConstraint(user_constraint, x0, False) + c = CanonicalConstraint.from_PreparedConstraint(prepared_constraint) + + assert_equal(c.n_eq, 0) + assert_equal(c.n_ineq, 2) + + c_eq, c_ineq = c.fun(x0) + assert_array_equal(c_eq, []) + assert_array_equal(c_ineq, [-1, -4]) + + J_eq, J_ineq = c.jac(x0) + assert_array_equal(J_eq, np.empty((0, 3))) + assert_array_equal(J_ineq, np.array([[1, 0, 0], [0, 0, 1]])) + + assert_array_equal(c.keep_feasible, [False, True]) + + # Test 3: infinite upper bound. + user_constraint = Bounds([0, 1, -np.inf], np.inf, [True, False, True]) + x0 = np.array([1, 2, 3], dtype=float) + prepared_constraint = PreparedConstraint(user_constraint, x0, False) + c = CanonicalConstraint.from_PreparedConstraint(prepared_constraint) + + assert_equal(c.n_eq, 0) + assert_equal(c.n_ineq, 2) + + c_eq, c_ineq = c.fun(x0) + assert_array_equal(c_eq, []) + assert_array_equal(c_ineq, [-1, -1]) + + J_eq, J_ineq = c.jac(x0) + assert_array_equal(J_eq, np.empty((0, 3))) + assert_array_equal(J_ineq, np.array([[-1, 0, 0], [0, -1, 0]])) + + assert_array_equal(c.keep_feasible, [True, False]) + + # Test 4: interval constraint. + user_constraint = Bounds([-1, -np.inf, 2, 3], [1, np.inf, 10, 3], + [False, True, True, True]) + x0 = np.array([0, 10, 8, 5]) + prepared_constraint = PreparedConstraint(user_constraint, x0, False) + c = CanonicalConstraint.from_PreparedConstraint(prepared_constraint) + + assert_equal(c.n_eq, 1) + assert_equal(c.n_ineq, 4) + + c_eq, c_ineq = c.fun(x0) + assert_array_equal(c_eq, [2]) + assert_array_equal(c_ineq, [-1, -2, -1, -6]) + + J_eq, J_ineq = c.jac(x0) + assert_array_equal(J_eq, [[0, 0, 0, 1]]) + assert_array_equal(J_ineq, [[1, 0, 0, 0], + [0, 0, 1, 0], + [-1, 0, 0, 0], + [0, 0, -1, 0]]) + + assert_array_equal(c.keep_feasible, [False, True, False, True]) + + +def test_nonlinear_constraint(): + n = 3 + m = 5 + rng = np.random.RandomState(0) + x0 = rng.rand(n) + + fun, jac, hess = create_quadratic_function(n, m, rng) + f = fun(x0) + J = jac(x0) + + lb = [-10, 3, -np.inf, -np.inf, -5] + ub = [10, 3, np.inf, 3, np.inf] + user_constraint = NonlinearConstraint( + fun, lb, ub, jac, hess, [True, False, False, True, False]) + + for sparse_jacobian in [False, True]: + prepared_constraint = PreparedConstraint(user_constraint, x0, + sparse_jacobian) + c = CanonicalConstraint.from_PreparedConstraint(prepared_constraint) + + assert_array_equal(c.n_eq, 1) + assert_array_equal(c.n_ineq, 4) + + c_eq, c_ineq = c.fun(x0) + assert_array_equal(c_eq, [f[1] - lb[1]]) + assert_array_equal(c_ineq, [f[3] - ub[3], lb[4] - f[4], + f[0] - ub[0], lb[0] - f[0]]) + + J_eq, J_ineq = c.jac(x0) + if sparse_jacobian: + J_eq = J_eq.toarray() + J_ineq = J_ineq.toarray() + + assert_array_equal(J_eq, J[1, None]) + assert_array_equal(J_ineq, np.vstack((J[3], -J[4], J[0], -J[0]))) + + v_eq = rng.rand(c.n_eq) + v_ineq = rng.rand(c.n_ineq) + v = np.zeros(m) + v[1] = v_eq[0] + v[3] = v_ineq[0] + v[4] = -v_ineq[1] + v[0] = v_ineq[2] - v_ineq[3] + assert_array_equal(c.hess(x0, v_eq, v_ineq), hess(x0, v)) + + assert_array_equal(c.keep_feasible, [True, False, True, True]) + + +def test_concatenation(): + rng = np.random.RandomState(0) + n = 4 + x0 = rng.rand(n) + + f1 = x0 + J1 = np.eye(n) + lb1 = [-1, -np.inf, -2, 3] + ub1 = [1, np.inf, np.inf, 3] + bounds = Bounds(lb1, ub1, [False, False, True, False]) + + fun, jac, hess = create_quadratic_function(n, 5, rng) + f2 = fun(x0) + J2 = jac(x0) + lb2 = [-10, 3, -np.inf, -np.inf, -5] + ub2 = [10, 3, np.inf, 5, np.inf] + nonlinear = NonlinearConstraint( + fun, lb2, ub2, jac, hess, [True, False, False, True, False]) + + for sparse_jacobian in [False, True]: + bounds_prepared = PreparedConstraint(bounds, x0, sparse_jacobian) + nonlinear_prepared = PreparedConstraint(nonlinear, x0, sparse_jacobian) + + c1 = CanonicalConstraint.from_PreparedConstraint(bounds_prepared) + c2 = CanonicalConstraint.from_PreparedConstraint(nonlinear_prepared) + c = CanonicalConstraint.concatenate([c1, c2], sparse_jacobian) + + assert_equal(c.n_eq, 2) + assert_equal(c.n_ineq, 7) + + c_eq, c_ineq = c.fun(x0) + assert_array_equal(c_eq, [f1[3] - lb1[3], f2[1] - lb2[1]]) + assert_array_equal(c_ineq, [lb1[2] - f1[2], f1[0] - ub1[0], + lb1[0] - f1[0], f2[3] - ub2[3], + lb2[4] - f2[4], f2[0] - ub2[0], + lb2[0] - f2[0]]) + + J_eq, J_ineq = c.jac(x0) + if sparse_jacobian: + J_eq = J_eq.toarray() + J_ineq = J_ineq.toarray() + + assert_array_equal(J_eq, np.vstack((J1[3], J2[1]))) + assert_array_equal(J_ineq, np.vstack((-J1[2], J1[0], -J1[0], J2[3], + -J2[4], J2[0], -J2[0]))) + + v_eq = rng.rand(c.n_eq) + v_ineq = rng.rand(c.n_ineq) + v = np.zeros(5) + v[1] = v_eq[1] + v[3] = v_ineq[3] + v[4] = -v_ineq[4] + v[0] = v_ineq[5] - v_ineq[6] + H = c.hess(x0, v_eq, v_ineq).dot(np.eye(n)) + assert_array_equal(H, hess(x0, v)) + + assert_array_equal(c.keep_feasible, + [True, False, False, True, False, True, True]) + + +def test_empty(): + x = np.array([1, 2, 3]) + c = CanonicalConstraint.empty(3) + assert_equal(c.n_eq, 0) + assert_equal(c.n_ineq, 0) + + c_eq, c_ineq = c.fun(x) + assert_array_equal(c_eq, []) + assert_array_equal(c_ineq, []) + + J_eq, J_ineq = c.jac(x) + assert_array_equal(J_eq, np.empty((0, 3))) + assert_array_equal(J_ineq, np.empty((0, 3))) + + H = c.hess(x, None, None).toarray() + assert_array_equal(H, np.zeros((3, 3))) + + +def test_initial_constraints_as_canonical(): + # rng is only used to generate the coefficients of the quadratic + # function that is used by the nonlinear constraint. + rng = np.random.RandomState(0) + + x0 = np.array([0.5, 0.4, 0.3, 0.2]) + n = len(x0) + + lb1 = [-1, -np.inf, -2, 3] + ub1 = [1, np.inf, np.inf, 3] + bounds = Bounds(lb1, ub1, [False, False, True, False]) + + fun, jac, hess = create_quadratic_function(n, 5, rng) + lb2 = [-10, 3, -np.inf, -np.inf, -5] + ub2 = [10, 3, np.inf, 5, np.inf] + nonlinear = NonlinearConstraint( + fun, lb2, ub2, jac, hess, [True, False, False, True, False]) + + for sparse_jacobian in [False, True]: + bounds_prepared = PreparedConstraint(bounds, x0, sparse_jacobian) + nonlinear_prepared = PreparedConstraint(nonlinear, x0, sparse_jacobian) + + f1 = bounds_prepared.fun.f + J1 = bounds_prepared.fun.J + f2 = nonlinear_prepared.fun.f + J2 = nonlinear_prepared.fun.J + + c_eq, c_ineq, J_eq, J_ineq = initial_constraints_as_canonical( + n, [bounds_prepared, nonlinear_prepared], sparse_jacobian) + + assert_array_equal(c_eq, [f1[3] - lb1[3], f2[1] - lb2[1]]) + assert_array_equal(c_ineq, [lb1[2] - f1[2], f1[0] - ub1[0], + lb1[0] - f1[0], f2[3] - ub2[3], + lb2[4] - f2[4], f2[0] - ub2[0], + lb2[0] - f2[0]]) + + if sparse_jacobian: + J1 = J1.toarray() + J2 = J2.toarray() + J_eq = J_eq.toarray() + J_ineq = J_ineq.toarray() + + assert_array_equal(J_eq, np.vstack((J1[3], J2[1]))) + assert_array_equal(J_ineq, np.vstack((-J1[2], J1[0], -J1[0], J2[3], + -J2[4], J2[0], -J2[0]))) + + +def test_initial_constraints_as_canonical_empty(): + n = 3 + for sparse_jacobian in [False, True]: + c_eq, c_ineq, J_eq, J_ineq = initial_constraints_as_canonical( + n, [], sparse_jacobian) + + assert_array_equal(c_eq, []) + assert_array_equal(c_ineq, []) + + if sparse_jacobian: + J_eq = J_eq.toarray() + J_ineq = J_ineq.toarray() + + assert_array_equal(J_eq, np.empty((0, n))) + assert_array_equal(J_ineq, np.empty((0, n))) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_nested_minimize.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_nested_minimize.py new file mode 100644 index 0000000000000000000000000000000000000000..f9aa57058c6523d4106b26840ef447431d615cd6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_nested_minimize.py @@ -0,0 +1,39 @@ +import pytest +import numpy as np +from scipy.optimize import minimize, NonlinearConstraint, rosen, rosen_der + + +# Ignore this warning about inefficient use of Hessians +# The bug only shows up with the default HUS +@pytest.mark.filterwarnings( + "ignore:delta_grad == 0.0. Check if the approximated function is linear." +) +def test_gh21193(): + # Test that nested minimization does not share Hessian objects + def identity(x): + return x[0] + def identity_jac(x): + a = np.zeros(len(x)) + a[0] = 1 + return a + constraint1 = NonlinearConstraint(identity, 0, 0, identity_jac) + constraint2 = NonlinearConstraint(identity, 0, 0, identity_jac) + + # The default HUS for each should be distinct + assert constraint1.hess is not constraint2.hess + + _ = minimize( + lambda x: minimize( + rosen, + x[1:], + jac=rosen_der, + constraints=constraint1, + method="trust-constr", + options={'maxiter': 2}, + ).fun, + [1, 0, 0], + constraints=constraint2, + method="trust-constr", + options={'maxiter': 2}, + ) + # This test doesn't check that the output is correct, just that it doesn't crash diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_projections.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_projections.py new file mode 100644 index 0000000000000000000000000000000000000000..6ff3c39d649d0ac663d9b71bb906f1daac021118 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_projections.py @@ -0,0 +1,214 @@ +import numpy as np +import scipy.linalg +from scipy.sparse import csc_matrix +from scipy.optimize._trustregion_constr.projections \ + import projections, orthogonality +from numpy.testing import (TestCase, assert_array_almost_equal, + assert_equal, assert_allclose) + +try: + from sksparse.cholmod import cholesky_AAt # noqa: F401 + sksparse_available = True + available_sparse_methods = ("NormalEquation", "AugmentedSystem") +except ImportError: + sksparse_available = False + available_sparse_methods = ("AugmentedSystem",) +available_dense_methods = ('QRFactorization', 'SVDFactorization') + + +class TestProjections(TestCase): + + def test_nullspace_and_least_squares_sparse(self): + A_dense = np.array([[1, 2, 3, 4, 0, 5, 0, 7], + [0, 8, 7, 0, 1, 5, 9, 0], + [1, 0, 0, 0, 0, 1, 2, 3]]) + At_dense = A_dense.T + A = csc_matrix(A_dense) + test_points = ([1, 2, 3, 4, 5, 6, 7, 8], + [1, 10, 3, 0, 1, 6, 7, 8], + [1.12, 10, 0, 0, 100000, 6, 0.7, 8]) + + for method in available_sparse_methods: + Z, LS, _ = projections(A, method) + for z in test_points: + # Test if x is in the null_space + x = Z.matvec(z) + assert_array_almost_equal(A.dot(x), 0) + # Test orthogonality + assert_array_almost_equal(orthogonality(A, x), 0) + # Test if x is the least square solution + x = LS.matvec(z) + x2 = scipy.linalg.lstsq(At_dense, z)[0] + assert_array_almost_equal(x, x2) + + def test_iterative_refinements_sparse(self): + A_dense = np.array([[1, 2, 3, 4, 0, 5, 0, 7], + [0, 8, 7, 0, 1, 5, 9, 0], + [1, 0, 0, 0, 0, 1, 2, 3]]) + A = csc_matrix(A_dense) + test_points = ([1, 2, 3, 4, 5, 6, 7, 8], + [1, 10, 3, 0, 1, 6, 7, 8], + [1.12, 10, 0, 0, 100000, 6, 0.7, 8], + [1, 0, 0, 0, 0, 1, 2, 3+1e-10]) + + for method in available_sparse_methods: + Z, LS, _ = projections(A, method, orth_tol=1e-18, max_refin=100) + for z in test_points: + # Test if x is in the null_space + x = Z.matvec(z) + atol = 1e-13 * abs(x).max() + assert_allclose(A.dot(x), 0, atol=atol) + # Test orthogonality + assert_allclose(orthogonality(A, x), 0, atol=1e-13) + + def test_rowspace_sparse(self): + A_dense = np.array([[1, 2, 3, 4, 0, 5, 0, 7], + [0, 8, 7, 0, 1, 5, 9, 0], + [1, 0, 0, 0, 0, 1, 2, 3]]) + A = csc_matrix(A_dense) + test_points = ([1, 2, 3], + [1, 10, 3], + [1.12, 10, 0]) + + for method in available_sparse_methods: + _, _, Y = projections(A, method) + for z in test_points: + # Test if x is solution of A x = z + x = Y.matvec(z) + assert_array_almost_equal(A.dot(x), z) + # Test if x is in the return row space of A + A_ext = np.vstack((A_dense, x)) + assert_equal(np.linalg.matrix_rank(A_dense), + np.linalg.matrix_rank(A_ext)) + + def test_nullspace_and_least_squares_dense(self): + A = np.array([[1, 2, 3, 4, 0, 5, 0, 7], + [0, 8, 7, 0, 1, 5, 9, 0], + [1, 0, 0, 0, 0, 1, 2, 3]]) + At = A.T + test_points = ([1, 2, 3, 4, 5, 6, 7, 8], + [1, 10, 3, 0, 1, 6, 7, 8], + [1.12, 10, 0, 0, 100000, 6, 0.7, 8]) + + for method in available_dense_methods: + Z, LS, _ = projections(A, method) + for z in test_points: + # Test if x is in the null_space + x = Z.matvec(z) + assert_array_almost_equal(A.dot(x), 0) + # Test orthogonality + assert_array_almost_equal(orthogonality(A, x), 0) + # Test if x is the least square solution + x = LS.matvec(z) + x2 = scipy.linalg.lstsq(At, z)[0] + assert_array_almost_equal(x, x2) + + def test_compare_dense_and_sparse(self): + D = np.diag(range(1, 101)) + A = np.hstack([D, D, D, D]) + A_sparse = csc_matrix(A) + np.random.seed(0) + + Z, LS, Y = projections(A) + Z_sparse, LS_sparse, Y_sparse = projections(A_sparse) + for k in range(20): + z = np.random.normal(size=(400,)) + assert_array_almost_equal(Z.dot(z), Z_sparse.dot(z)) + assert_array_almost_equal(LS.dot(z), LS_sparse.dot(z)) + x = np.random.normal(size=(100,)) + assert_array_almost_equal(Y.dot(x), Y_sparse.dot(x)) + + def test_compare_dense_and_sparse2(self): + D1 = np.diag([-1.7, 1, 0.5]) + D2 = np.diag([1, -0.6, -0.3]) + D3 = np.diag([-0.3, -1.5, 2]) + A = np.hstack([D1, D2, D3]) + A_sparse = csc_matrix(A) + np.random.seed(0) + + Z, LS, Y = projections(A) + Z_sparse, LS_sparse, Y_sparse = projections(A_sparse) + for k in range(1): + z = np.random.normal(size=(9,)) + assert_array_almost_equal(Z.dot(z), Z_sparse.dot(z)) + assert_array_almost_equal(LS.dot(z), LS_sparse.dot(z)) + x = np.random.normal(size=(3,)) + assert_array_almost_equal(Y.dot(x), Y_sparse.dot(x)) + + def test_iterative_refinements_dense(self): + A = np.array([[1, 2, 3, 4, 0, 5, 0, 7], + [0, 8, 7, 0, 1, 5, 9, 0], + [1, 0, 0, 0, 0, 1, 2, 3]]) + test_points = ([1, 2, 3, 4, 5, 6, 7, 8], + [1, 10, 3, 0, 1, 6, 7, 8], + [1, 0, 0, 0, 0, 1, 2, 3+1e-10]) + + for method in available_dense_methods: + Z, LS, _ = projections(A, method, orth_tol=1e-18, max_refin=10) + for z in test_points: + # Test if x is in the null_space + x = Z.matvec(z) + assert_allclose(A.dot(x), 0, rtol=0, atol=2.5e-14) + # Test orthogonality + assert_allclose(orthogonality(A, x), 0, rtol=0, atol=5e-16) + + def test_rowspace_dense(self): + A = np.array([[1, 2, 3, 4, 0, 5, 0, 7], + [0, 8, 7, 0, 1, 5, 9, 0], + [1, 0, 0, 0, 0, 1, 2, 3]]) + test_points = ([1, 2, 3], + [1, 10, 3], + [1.12, 10, 0]) + + for method in available_dense_methods: + _, _, Y = projections(A, method) + for z in test_points: + # Test if x is solution of A x = z + x = Y.matvec(z) + assert_array_almost_equal(A.dot(x), z) + # Test if x is in the return row space of A + A_ext = np.vstack((A, x)) + assert_equal(np.linalg.matrix_rank(A), + np.linalg.matrix_rank(A_ext)) + + +class TestOrthogonality(TestCase): + + def test_dense_matrix(self): + A = np.array([[1, 2, 3, 4, 0, 5, 0, 7], + [0, 8, 7, 0, 1, 5, 9, 0], + [1, 0, 0, 0, 0, 1, 2, 3]]) + test_vectors = ([-1.98931144, -1.56363389, + -0.84115584, 2.2864762, + 5.599141, 0.09286976, + 1.37040802, -0.28145812], + [697.92794044, -4091.65114008, + -3327.42316335, 836.86906951, + 99434.98929065, -1285.37653682, + -4109.21503806, 2935.29289083]) + test_expected_orth = (0, 0) + + for i in range(len(test_vectors)): + x = test_vectors[i] + orth = test_expected_orth[i] + assert_array_almost_equal(orthogonality(A, x), orth) + + def test_sparse_matrix(self): + A = np.array([[1, 2, 3, 4, 0, 5, 0, 7], + [0, 8, 7, 0, 1, 5, 9, 0], + [1, 0, 0, 0, 0, 1, 2, 3]]) + A = csc_matrix(A) + test_vectors = ([-1.98931144, -1.56363389, + -0.84115584, 2.2864762, + 5.599141, 0.09286976, + 1.37040802, -0.28145812], + [697.92794044, -4091.65114008, + -3327.42316335, 836.86906951, + 99434.98929065, -1285.37653682, + -4109.21503806, 2935.29289083]) + test_expected_orth = (0, 0) + + for i in range(len(test_vectors)): + x = test_vectors[i] + orth = test_expected_orth[i] + assert_array_almost_equal(orthogonality(A, x), orth) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_qp_subproblem.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_qp_subproblem.py new file mode 100644 index 0000000000000000000000000000000000000000..70e65e53b9d2389541c0aab45b94b1d30dcdd146 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_qp_subproblem.py @@ -0,0 +1,645 @@ +import numpy as np +from scipy.sparse import csc_matrix +from scipy.optimize._trustregion_constr.qp_subproblem \ + import (eqp_kktfact, + projected_cg, + box_intersections, + sphere_intersections, + box_sphere_intersections, + modified_dogleg) +from scipy.optimize._trustregion_constr.projections \ + import projections +from numpy.testing import TestCase, assert_array_almost_equal, assert_equal +import pytest + + +class TestEQPDirectFactorization(TestCase): + + # From Example 16.2 Nocedal/Wright "Numerical + # Optimization" p.452. + def test_nocedal_example(self): + H = csc_matrix([[6, 2, 1], + [2, 5, 2], + [1, 2, 4]]) + A = csc_matrix([[1, 0, 1], + [0, 1, 1]]) + c = np.array([-8, -3, -3]) + b = -np.array([3, 0]) + x, lagrange_multipliers = eqp_kktfact(H, c, A, b) + assert_array_almost_equal(x, [2, -1, 1]) + assert_array_almost_equal(lagrange_multipliers, [3, -2]) + + +class TestSphericalBoundariesIntersections(TestCase): + + def test_2d_sphere_constraints(self): + # Interior initial point + ta, tb, intersect = sphere_intersections([0, 0], + [1, 0], 0.5) + assert_array_almost_equal([ta, tb], [0, 0.5]) + assert_equal(intersect, True) + + # No intersection between line and circle + ta, tb, intersect = sphere_intersections([2, 0], + [0, 1], 1) + assert_equal(intersect, False) + + # Outside initial point pointing toward outside the circle + ta, tb, intersect = sphere_intersections([2, 0], + [1, 0], 1) + assert_equal(intersect, False) + + # Outside initial point pointing toward inside the circle + ta, tb, intersect = sphere_intersections([2, 0], + [-1, 0], 1.5) + assert_array_almost_equal([ta, tb], [0.5, 1]) + assert_equal(intersect, True) + + # Initial point on the boundary + ta, tb, intersect = sphere_intersections([2, 0], + [1, 0], 2) + assert_array_almost_equal([ta, tb], [0, 0]) + assert_equal(intersect, True) + + def test_2d_sphere_constraints_line_intersections(self): + # Interior initial point + ta, tb, intersect = sphere_intersections([0, 0], + [1, 0], 0.5, + entire_line=True) + assert_array_almost_equal([ta, tb], [-0.5, 0.5]) + assert_equal(intersect, True) + + # No intersection between line and circle + ta, tb, intersect = sphere_intersections([2, 0], + [0, 1], 1, + entire_line=True) + assert_equal(intersect, False) + + # Outside initial point pointing toward outside the circle + ta, tb, intersect = sphere_intersections([2, 0], + [1, 0], 1, + entire_line=True) + assert_array_almost_equal([ta, tb], [-3, -1]) + assert_equal(intersect, True) + + # Outside initial point pointing toward inside the circle + ta, tb, intersect = sphere_intersections([2, 0], + [-1, 0], 1.5, + entire_line=True) + assert_array_almost_equal([ta, tb], [0.5, 3.5]) + assert_equal(intersect, True) + + # Initial point on the boundary + ta, tb, intersect = sphere_intersections([2, 0], + [1, 0], 2, + entire_line=True) + assert_array_almost_equal([ta, tb], [-4, 0]) + assert_equal(intersect, True) + + +class TestBoxBoundariesIntersections(TestCase): + + def test_2d_box_constraints(self): + # Box constraint in the direction of vector d + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [1, 1], [3, 3]) + assert_array_almost_equal([ta, tb], [0.5, 1]) + assert_equal(intersect, True) + + # Negative direction + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [1, -3], [3, -1]) + assert_equal(intersect, False) + + # Some constraints are absent (set to +/- inf) + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [-np.inf, 1], + [np.inf, np.inf]) + assert_array_almost_equal([ta, tb], [0.5, 1]) + assert_equal(intersect, True) + + # Intersect on the face of the box + ta, tb, intersect = box_intersections([1, 0], [0, 1], + [1, 1], [3, 3]) + assert_array_almost_equal([ta, tb], [1, 1]) + assert_equal(intersect, True) + + # Interior initial point + ta, tb, intersect = box_intersections([0, 0], [4, 4], + [-2, -3], [3, 2]) + assert_array_almost_equal([ta, tb], [0, 0.5]) + assert_equal(intersect, True) + + # No intersection between line and box constraints + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [-3, -3], [-1, -1]) + assert_equal(intersect, False) + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [-3, 3], [-1, 1]) + assert_equal(intersect, False) + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [-3, -np.inf], + [-1, np.inf]) + assert_equal(intersect, False) + ta, tb, intersect = box_intersections([0, 0], [1, 100], + [1, 1], [3, 3]) + assert_equal(intersect, False) + ta, tb, intersect = box_intersections([0.99, 0], [0, 2], + [1, 1], [3, 3]) + assert_equal(intersect, False) + + # Initial point on the boundary + ta, tb, intersect = box_intersections([2, 2], [0, 1], + [-2, -2], [2, 2]) + assert_array_almost_equal([ta, tb], [0, 0]) + assert_equal(intersect, True) + + def test_2d_box_constraints_entire_line(self): + # Box constraint in the direction of vector d + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [1, 1], [3, 3], + entire_line=True) + assert_array_almost_equal([ta, tb], [0.5, 1.5]) + assert_equal(intersect, True) + + # Negative direction + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [1, -3], [3, -1], + entire_line=True) + assert_array_almost_equal([ta, tb], [-1.5, -0.5]) + assert_equal(intersect, True) + + # Some constraints are absent (set to +/- inf) + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [-np.inf, 1], + [np.inf, np.inf], + entire_line=True) + assert_array_almost_equal([ta, tb], [0.5, np.inf]) + assert_equal(intersect, True) + + # Intersect on the face of the box + ta, tb, intersect = box_intersections([1, 0], [0, 1], + [1, 1], [3, 3], + entire_line=True) + assert_array_almost_equal([ta, tb], [1, 3]) + assert_equal(intersect, True) + + # Interior initial point + ta, tb, intersect = box_intersections([0, 0], [4, 4], + [-2, -3], [3, 2], + entire_line=True) + assert_array_almost_equal([ta, tb], [-0.5, 0.5]) + assert_equal(intersect, True) + + # No intersection between line and box constraints + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [-3, -3], [-1, -1], + entire_line=True) + assert_equal(intersect, False) + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [-3, 3], [-1, 1], + entire_line=True) + assert_equal(intersect, False) + ta, tb, intersect = box_intersections([2, 0], [0, 2], + [-3, -np.inf], + [-1, np.inf], + entire_line=True) + assert_equal(intersect, False) + ta, tb, intersect = box_intersections([0, 0], [1, 100], + [1, 1], [3, 3], + entire_line=True) + assert_equal(intersect, False) + ta, tb, intersect = box_intersections([0.99, 0], [0, 2], + [1, 1], [3, 3], + entire_line=True) + assert_equal(intersect, False) + + # Initial point on the boundary + ta, tb, intersect = box_intersections([2, 2], [0, 1], + [-2, -2], [2, 2], + entire_line=True) + assert_array_almost_equal([ta, tb], [-4, 0]) + assert_equal(intersect, True) + + def test_3d_box_constraints(self): + # Simple case + ta, tb, intersect = box_intersections([1, 1, 0], [0, 0, 1], + [1, 1, 1], [3, 3, 3]) + assert_array_almost_equal([ta, tb], [1, 1]) + assert_equal(intersect, True) + + # Negative direction + ta, tb, intersect = box_intersections([1, 1, 0], [0, 0, -1], + [1, 1, 1], [3, 3, 3]) + assert_equal(intersect, False) + + # Interior point + ta, tb, intersect = box_intersections([2, 2, 2], [0, -1, 1], + [1, 1, 1], [3, 3, 3]) + assert_array_almost_equal([ta, tb], [0, 1]) + assert_equal(intersect, True) + + def test_3d_box_constraints_entire_line(self): + # Simple case + ta, tb, intersect = box_intersections([1, 1, 0], [0, 0, 1], + [1, 1, 1], [3, 3, 3], + entire_line=True) + assert_array_almost_equal([ta, tb], [1, 3]) + assert_equal(intersect, True) + + # Negative direction + ta, tb, intersect = box_intersections([1, 1, 0], [0, 0, -1], + [1, 1, 1], [3, 3, 3], + entire_line=True) + assert_array_almost_equal([ta, tb], [-3, -1]) + assert_equal(intersect, True) + + # Interior point + ta, tb, intersect = box_intersections([2, 2, 2], [0, -1, 1], + [1, 1, 1], [3, 3, 3], + entire_line=True) + assert_array_almost_equal([ta, tb], [-1, 1]) + assert_equal(intersect, True) + + +class TestBoxSphereBoundariesIntersections(TestCase): + + def test_2d_box_constraints(self): + # Both constraints are active + ta, tb, intersect = box_sphere_intersections([1, 1], [-2, 2], + [-1, -2], [1, 2], 2, + entire_line=False) + assert_array_almost_equal([ta, tb], [0, 0.5]) + assert_equal(intersect, True) + + # None of the constraints are active + ta, tb, intersect = box_sphere_intersections([1, 1], [-1, 1], + [-1, -3], [1, 3], 10, + entire_line=False) + assert_array_almost_equal([ta, tb], [0, 1]) + assert_equal(intersect, True) + + # Box constraints are active + ta, tb, intersect = box_sphere_intersections([1, 1], [-4, 4], + [-1, -3], [1, 3], 10, + entire_line=False) + assert_array_almost_equal([ta, tb], [0, 0.5]) + assert_equal(intersect, True) + + # Spherical constraints are active + ta, tb, intersect = box_sphere_intersections([1, 1], [-4, 4], + [-1, -3], [1, 3], 2, + entire_line=False) + assert_array_almost_equal([ta, tb], [0, 0.25]) + assert_equal(intersect, True) + + # Infeasible problems + ta, tb, intersect = box_sphere_intersections([2, 2], [-4, 4], + [-1, -3], [1, 3], 2, + entire_line=False) + assert_equal(intersect, False) + ta, tb, intersect = box_sphere_intersections([1, 1], [-4, 4], + [2, 4], [2, 4], 2, + entire_line=False) + assert_equal(intersect, False) + + def test_2d_box_constraints_entire_line(self): + # Both constraints are active + ta, tb, intersect = box_sphere_intersections([1, 1], [-2, 2], + [-1, -2], [1, 2], 2, + entire_line=True) + assert_array_almost_equal([ta, tb], [0, 0.5]) + assert_equal(intersect, True) + + # None of the constraints are active + ta, tb, intersect = box_sphere_intersections([1, 1], [-1, 1], + [-1, -3], [1, 3], 10, + entire_line=True) + assert_array_almost_equal([ta, tb], [0, 2]) + assert_equal(intersect, True) + + # Box constraints are active + ta, tb, intersect = box_sphere_intersections([1, 1], [-4, 4], + [-1, -3], [1, 3], 10, + entire_line=True) + assert_array_almost_equal([ta, tb], [0, 0.5]) + assert_equal(intersect, True) + + # Spherical constraints are active + ta, tb, intersect = box_sphere_intersections([1, 1], [-4, 4], + [-1, -3], [1, 3], 2, + entire_line=True) + assert_array_almost_equal([ta, tb], [0, 0.25]) + assert_equal(intersect, True) + + # Infeasible problems + ta, tb, intersect = box_sphere_intersections([2, 2], [-4, 4], + [-1, -3], [1, 3], 2, + entire_line=True) + assert_equal(intersect, False) + ta, tb, intersect = box_sphere_intersections([1, 1], [-4, 4], + [2, 4], [2, 4], 2, + entire_line=True) + assert_equal(intersect, False) + + +class TestModifiedDogleg(TestCase): + + def test_cauchypoint_equalsto_newtonpoint(self): + A = np.array([[1, 8]]) + b = np.array([-16]) + _, _, Y = projections(A) + newton_point = np.array([0.24615385, 1.96923077]) + + # Newton point inside boundaries + x = modified_dogleg(A, Y, b, 2, [-np.inf, -np.inf], [np.inf, np.inf]) + assert_array_almost_equal(x, newton_point) + + # Spherical constraint active + x = modified_dogleg(A, Y, b, 1, [-np.inf, -np.inf], [np.inf, np.inf]) + assert_array_almost_equal(x, newton_point/np.linalg.norm(newton_point)) + + # Box constraints active + x = modified_dogleg(A, Y, b, 2, [-np.inf, -np.inf], [0.1, np.inf]) + assert_array_almost_equal(x, (newton_point/newton_point[0]) * 0.1) + + def test_3d_example(self): + A = np.array([[1, 8, 1], + [4, 2, 2]]) + b = np.array([-16, 2]) + Z, LS, Y = projections(A) + + newton_point = np.array([-1.37090909, 2.23272727, -0.49090909]) + cauchy_point = np.array([0.11165723, 1.73068711, 0.16748585]) + origin = np.zeros_like(newton_point) + + # newton_point inside boundaries + x = modified_dogleg(A, Y, b, 3, [-np.inf, -np.inf, -np.inf], + [np.inf, np.inf, np.inf]) + assert_array_almost_equal(x, newton_point) + + # line between cauchy_point and newton_point contains best point + # (spherical constraint is active). + x = modified_dogleg(A, Y, b, 2, [-np.inf, -np.inf, -np.inf], + [np.inf, np.inf, np.inf]) + z = cauchy_point + d = newton_point-cauchy_point + t = ((x-z)/(d)) + assert_array_almost_equal(t, np.full(3, 0.40807330)) + assert_array_almost_equal(np.linalg.norm(x), 2) + + # line between cauchy_point and newton_point contains best point + # (box constraint is active). + x = modified_dogleg(A, Y, b, 5, [-1, -np.inf, -np.inf], + [np.inf, np.inf, np.inf]) + z = cauchy_point + d = newton_point-cauchy_point + t = ((x-z)/(d)) + assert_array_almost_equal(t, np.full(3, 0.7498195)) + assert_array_almost_equal(x[0], -1) + + # line between origin and cauchy_point contains best point + # (spherical constraint is active). + x = modified_dogleg(A, Y, b, 1, [-np.inf, -np.inf, -np.inf], + [np.inf, np.inf, np.inf]) + z = origin + d = cauchy_point + t = ((x-z)/(d)) + assert_array_almost_equal(t, np.full(3, 0.573936265)) + assert_array_almost_equal(np.linalg.norm(x), 1) + + # line between origin and newton_point contains best point + # (box constraint is active). + x = modified_dogleg(A, Y, b, 2, [-np.inf, -np.inf, -np.inf], + [np.inf, 1, np.inf]) + z = origin + d = newton_point + t = ((x-z)/(d)) + assert_array_almost_equal(t, np.full(3, 0.4478827364)) + assert_array_almost_equal(x[1], 1) + + +class TestProjectCG(TestCase): + + # From Example 16.2 Nocedal/Wright "Numerical + # Optimization" p.452. + def test_nocedal_example(self): + H = csc_matrix([[6, 2, 1], + [2, 5, 2], + [1, 2, 4]]) + A = csc_matrix([[1, 0, 1], + [0, 1, 1]]) + c = np.array([-8, -3, -3]) + b = -np.array([3, 0]) + Z, _, Y = projections(A) + x, info = projected_cg(H, c, Z, Y, b) + assert_equal(info["stop_cond"], 4) + assert_equal(info["hits_boundary"], False) + assert_array_almost_equal(x, [2, -1, 1]) + + def test_compare_with_direct_fact(self): + H = csc_matrix([[6, 2, 1, 3], + [2, 5, 2, 4], + [1, 2, 4, 5], + [3, 4, 5, 7]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 1, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + Z, _, Y = projections(A) + x, info = projected_cg(H, c, Z, Y, b, tol=0) + x_kkt, _ = eqp_kktfact(H, c, A, b) + assert_equal(info["stop_cond"], 1) + assert_equal(info["hits_boundary"], False) + assert_array_almost_equal(x, x_kkt) + + def test_trust_region_infeasible(self): + H = csc_matrix([[6, 2, 1, 3], + [2, 5, 2, 4], + [1, 2, 4, 5], + [3, 4, 5, 7]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 1, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + trust_radius = 1 + Z, _, Y = projections(A) + with pytest.raises(ValueError): + projected_cg(H, c, Z, Y, b, trust_radius=trust_radius) + + def test_trust_region_barely_feasible(self): + H = csc_matrix([[6, 2, 1, 3], + [2, 5, 2, 4], + [1, 2, 4, 5], + [3, 4, 5, 7]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 1, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + trust_radius = 2.32379000772445021283 + Z, _, Y = projections(A) + x, info = projected_cg(H, c, Z, Y, b, + tol=0, + trust_radius=trust_radius) + assert_equal(info["stop_cond"], 2) + assert_equal(info["hits_boundary"], True) + assert_array_almost_equal(np.linalg.norm(x), trust_radius) + assert_array_almost_equal(x, -Y.dot(b)) + + def test_hits_boundary(self): + H = csc_matrix([[6, 2, 1, 3], + [2, 5, 2, 4], + [1, 2, 4, 5], + [3, 4, 5, 7]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 1, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + trust_radius = 3 + Z, _, Y = projections(A) + x, info = projected_cg(H, c, Z, Y, b, + tol=0, + trust_radius=trust_radius) + assert_equal(info["stop_cond"], 2) + assert_equal(info["hits_boundary"], True) + assert_array_almost_equal(np.linalg.norm(x), trust_radius) + + def test_negative_curvature_unconstrained(self): + H = csc_matrix([[1, 2, 1, 3], + [2, 0, 2, 4], + [1, 2, 0, 2], + [3, 4, 2, 0]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 0, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + Z, _, Y = projections(A) + with pytest.raises(ValueError): + projected_cg(H, c, Z, Y, b, tol=0) + + def test_negative_curvature(self): + H = csc_matrix([[1, 2, 1, 3], + [2, 0, 2, 4], + [1, 2, 0, 2], + [3, 4, 2, 0]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 0, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + Z, _, Y = projections(A) + trust_radius = 1000 + x, info = projected_cg(H, c, Z, Y, b, + tol=0, + trust_radius=trust_radius) + assert_equal(info["stop_cond"], 3) + assert_equal(info["hits_boundary"], True) + assert_array_almost_equal(np.linalg.norm(x), trust_radius) + + # The box constraints are inactive at the solution but + # are active during the iterations. + def test_inactive_box_constraints(self): + H = csc_matrix([[6, 2, 1, 3], + [2, 5, 2, 4], + [1, 2, 4, 5], + [3, 4, 5, 7]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 1, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + Z, _, Y = projections(A) + x, info = projected_cg(H, c, Z, Y, b, + tol=0, + lb=[0.5, -np.inf, + -np.inf, -np.inf], + return_all=True) + x_kkt, _ = eqp_kktfact(H, c, A, b) + assert_equal(info["stop_cond"], 1) + assert_equal(info["hits_boundary"], False) + assert_array_almost_equal(x, x_kkt) + + # The box constraints active and the termination is + # by maximum iterations (infeasible interaction). + def test_active_box_constraints_maximum_iterations_reached(self): + H = csc_matrix([[6, 2, 1, 3], + [2, 5, 2, 4], + [1, 2, 4, 5], + [3, 4, 5, 7]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 1, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + Z, _, Y = projections(A) + x, info = projected_cg(H, c, Z, Y, b, + tol=0, + lb=[0.8, -np.inf, + -np.inf, -np.inf], + return_all=True) + assert_equal(info["stop_cond"], 1) + assert_equal(info["hits_boundary"], True) + assert_array_almost_equal(A.dot(x), -b) + assert_array_almost_equal(x[0], 0.8) + + # The box constraints are active and the termination is + # because it hits boundary (without infeasible interaction). + def test_active_box_constraints_hits_boundaries(self): + H = csc_matrix([[6, 2, 1, 3], + [2, 5, 2, 4], + [1, 2, 4, 5], + [3, 4, 5, 7]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 1, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + trust_radius = 3 + Z, _, Y = projections(A) + x, info = projected_cg(H, c, Z, Y, b, + tol=0, + ub=[np.inf, np.inf, 1.6, np.inf], + trust_radius=trust_radius, + return_all=True) + assert_equal(info["stop_cond"], 2) + assert_equal(info["hits_boundary"], True) + assert_array_almost_equal(x[2], 1.6) + + # The box constraints are active and the termination is + # because it hits boundary (infeasible interaction). + def test_active_box_constraints_hits_boundaries_infeasible_iter(self): + H = csc_matrix([[6, 2, 1, 3], + [2, 5, 2, 4], + [1, 2, 4, 5], + [3, 4, 5, 7]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 1, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + trust_radius = 4 + Z, _, Y = projections(A) + x, info = projected_cg(H, c, Z, Y, b, + tol=0, + ub=[np.inf, 0.1, np.inf, np.inf], + trust_radius=trust_radius, + return_all=True) + assert_equal(info["stop_cond"], 2) + assert_equal(info["hits_boundary"], True) + assert_array_almost_equal(x[1], 0.1) + + # The box constraints are active and the termination is + # because it hits boundary (no infeasible interaction). + def test_active_box_constraints_negative_curvature(self): + H = csc_matrix([[1, 2, 1, 3], + [2, 0, 2, 4], + [1, 2, 0, 2], + [3, 4, 2, 0]]) + A = csc_matrix([[1, 0, 1, 0], + [0, 1, 0, 1]]) + c = np.array([-2, -3, -3, 1]) + b = -np.array([3, 0]) + Z, _, Y = projections(A) + trust_radius = 1000 + x, info = projected_cg(H, c, Z, Y, b, + tol=0, + ub=[np.inf, np.inf, 100, np.inf], + trust_radius=trust_radius) + assert_equal(info["stop_cond"], 3) + assert_equal(info["hits_boundary"], True) + assert_array_almost_equal(x[2], 100) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_report.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_report.py new file mode 100644 index 0000000000000000000000000000000000000000..66fa5bd17f80a907db425c927c08e5dc0797028e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tests/test_report.py @@ -0,0 +1,34 @@ +import pytest +import numpy as np +from scipy.optimize import minimize, Bounds + +def test_gh10880(): + # checks that verbose reporting works with trust-constr for + # bound-constrained problems + bnds = Bounds(1, 2) + opts = {'maxiter': 1000, 'verbose': 2} + minimize(lambda x: x**2, x0=2., method='trust-constr', + bounds=bnds, options=opts) + + opts = {'maxiter': 1000, 'verbose': 3} + minimize(lambda x: x**2, x0=2., method='trust-constr', + bounds=bnds, options=opts) + +@pytest.mark.xslow +def test_gh12922(): + # checks that verbose reporting works with trust-constr for + # general constraints + def objective(x): + return np.array([(np.sum((x+1)**4))]) + + cons = {'type': 'ineq', 'fun': lambda x: -x[0]**2} + n = 25 + x0 = np.linspace(-5, 5, n) + + opts = {'maxiter': 1000, 'verbose': 2} + minimize(objective, x0=x0, method='trust-constr', + constraints=cons, options=opts) + + opts = {'maxiter': 1000, 'verbose': 3} + minimize(objective, x0=x0, method='trust-constr', + constraints=cons, options=opts) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tr_interior_point.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tr_interior_point.py new file mode 100644 index 0000000000000000000000000000000000000000..e14b3f366fba818d9174af97fa91e065bf26e826 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/tr_interior_point.py @@ -0,0 +1,361 @@ +"""Trust-region interior point method. + +References +---------- +.. [1] Byrd, Richard H., Mary E. Hribar, and Jorge Nocedal. + "An interior point algorithm for large-scale nonlinear + programming." SIAM Journal on Optimization 9.4 (1999): 877-900. +.. [2] Byrd, Richard H., Guanghui Liu, and Jorge Nocedal. + "On the local behavior of an interior point method for + nonlinear programming." Numerical analysis 1997 (1997): 37-56. +.. [3] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization" + Second Edition (2006). +""" + +import scipy.sparse as sps +import numpy as np +from .equality_constrained_sqp import equality_constrained_sqp +from scipy.sparse.linalg import LinearOperator + +__all__ = ['tr_interior_point'] + + +class BarrierSubproblem: + """ + Barrier optimization problem: + minimize fun(x) - barrier_parameter*sum(log(s)) + subject to: constr_eq(x) = 0 + constr_ineq(x) + s = 0 + """ + + def __init__(self, x0, s0, fun, grad, lagr_hess, n_vars, n_ineq, n_eq, + constr, jac, barrier_parameter, tolerance, + enforce_feasibility, global_stop_criteria, + xtol, fun0, grad0, constr_ineq0, jac_ineq0, constr_eq0, + jac_eq0, finite_diff_bounds): + # Store parameters + self.n_vars = n_vars + self.x0 = x0 + self.s0 = s0 + self.fun = fun + self.grad = grad + self.lagr_hess = lagr_hess + self.constr = constr + self.jac = jac + self.barrier_parameter = barrier_parameter + self.tolerance = tolerance + self.n_eq = n_eq + self.n_ineq = n_ineq + self.enforce_feasibility = enforce_feasibility + self.global_stop_criteria = global_stop_criteria + self.xtol = xtol + self.fun0 = self._compute_function(fun0, constr_ineq0, s0) + self.grad0 = self._compute_gradient(grad0) + self.constr0 = self._compute_constr(constr_ineq0, constr_eq0, s0) + self.jac0 = self._compute_jacobian(jac_eq0, jac_ineq0, s0) + self.terminate = False + self.lb = finite_diff_bounds[0] + self.ub = finite_diff_bounds[1] + + def update(self, barrier_parameter, tolerance): + self.barrier_parameter = barrier_parameter + self.tolerance = tolerance + + def get_slack(self, z): + return z[self.n_vars:self.n_vars+self.n_ineq] + + def get_variables(self, z): + return z[:self.n_vars] + + def function_and_constraints(self, z): + """Returns barrier function and constraints at given point. + + For z = [x, s], returns barrier function: + function(z) = fun(x) - barrier_parameter*sum(log(s)) + and barrier constraints: + constraints(z) = [ constr_eq(x) ] + [ constr_ineq(x) + s ] + + """ + # Get variables and slack variables + x = self.get_variables(z) + s = self.get_slack(z) + + # Compute function and constraints, + # making sure x is within any strict bounds + if np.any((x < self.lb) | (x > self.ub)): + # If x is out of the strict bounds, set f = inf, + # and just set both equality and inequality + # constraints to 0 since we can't evaluate + # them separately. + f = np.inf + c_eq = np.full(self.n_eq, 0.) + c_ineq = np.full(self.n_ineq, 0.) + else: + f = self.fun(x) + c_eq, c_ineq = self.constr(x) + + # Return objective function and constraints + return (self._compute_function(f, c_ineq, s), + self._compute_constr(c_ineq, c_eq, s)) + + def _compute_function(self, f, c_ineq, s): + # Use technique from Nocedal and Wright book, ref [3]_, p.576, + # to guarantee constraints from `enforce_feasibility` + # stay feasible along iterations. + s[self.enforce_feasibility] = -c_ineq[self.enforce_feasibility] + log_s = [np.log(s_i) if s_i > 0 else -np.inf for s_i in s] + # Compute barrier objective function + return f - self.barrier_parameter*np.sum(log_s) + + def _compute_constr(self, c_ineq, c_eq, s): + # Compute barrier constraint + return np.hstack((c_eq, + c_ineq + s)) + + def scaling(self, z): + """Returns scaling vector. + Given by: + scaling = [ones(n_vars), s] + """ + s = self.get_slack(z) + diag_elements = np.hstack((np.ones(self.n_vars), s)) + + # Diagonal matrix + def matvec(vec): + return diag_elements*vec + return LinearOperator((self.n_vars+self.n_ineq, + self.n_vars+self.n_ineq), + matvec) + + def gradient_and_jacobian(self, z): + """Returns scaled gradient. + + Return scaled gradient: + gradient = [ grad(x) ] + [ -barrier_parameter*ones(n_ineq) ] + and scaled Jacobian matrix: + jacobian = [ jac_eq(x) 0 ] + [ jac_ineq(x) S ] + Both of them scaled by the previously defined scaling factor. + """ + # Get variables and slack variables + x = self.get_variables(z) + s = self.get_slack(z) + # Compute first derivatives + g = self.grad(x) + J_eq, J_ineq = self.jac(x) + # Return gradient and Jacobian + return (self._compute_gradient(g), + self._compute_jacobian(J_eq, J_ineq, s)) + + def _compute_gradient(self, g): + return np.hstack((g, -self.barrier_parameter*np.ones(self.n_ineq))) + + def _compute_jacobian(self, J_eq, J_ineq, s): + if self.n_ineq == 0: + return J_eq + else: + if sps.issparse(J_eq) or sps.issparse(J_ineq): + # It is expected that J_eq and J_ineq + # are already `csr_matrix` because of + # the way ``BoxConstraint``, ``NonlinearConstraint`` + # and ``LinearConstraint`` are defined. + J_eq = sps.csr_matrix(J_eq) + J_ineq = sps.csr_matrix(J_ineq) + return self._assemble_sparse_jacobian(J_eq, J_ineq, s) + else: + S = np.diag(s) + zeros = np.zeros((self.n_eq, self.n_ineq)) + # Convert to matrix + if sps.issparse(J_ineq): + J_ineq = J_ineq.toarray() + if sps.issparse(J_eq): + J_eq = J_eq.toarray() + # Concatenate matrices + return np.block([[J_eq, zeros], + [J_ineq, S]]) + + def _assemble_sparse_jacobian(self, J_eq, J_ineq, s): + """Assemble sparse Jacobian given its components. + + Given ``J_eq``, ``J_ineq`` and ``s`` returns: + jacobian = [ J_eq, 0 ] + [ J_ineq, diag(s) ] + + It is equivalent to: + sps.bmat([[ J_eq, None ], + [ J_ineq, diag(s) ]], "csr") + but significantly more efficient for this + given structure. + """ + n_vars, n_ineq, n_eq = self.n_vars, self.n_ineq, self.n_eq + J_aux = sps.vstack([J_eq, J_ineq], "csr") + indptr, indices, data = J_aux.indptr, J_aux.indices, J_aux.data + new_indptr = indptr + np.hstack((np.zeros(n_eq, dtype=int), + np.arange(n_ineq+1, dtype=int))) + size = indices.size+n_ineq + new_indices = np.empty(size) + new_data = np.empty(size) + mask = np.full(size, False, bool) + mask[new_indptr[-n_ineq:]-1] = True + new_indices[mask] = n_vars+np.arange(n_ineq) + new_indices[~mask] = indices + new_data[mask] = s + new_data[~mask] = data + J = sps.csr_matrix((new_data, new_indices, new_indptr), + (n_eq + n_ineq, n_vars + n_ineq)) + return J + + def lagrangian_hessian_x(self, z, v): + """Returns Lagrangian Hessian (in relation to `x`) -> Hx""" + x = self.get_variables(z) + # Get lagrange multipliers related to nonlinear equality constraints + v_eq = v[:self.n_eq] + # Get lagrange multipliers related to nonlinear ineq. constraints + v_ineq = v[self.n_eq:self.n_eq+self.n_ineq] + lagr_hess = self.lagr_hess + return lagr_hess(x, v_eq, v_ineq) + + def lagrangian_hessian_s(self, z, v): + """Returns scaled Lagrangian Hessian (in relation to`s`) -> S Hs S""" + s = self.get_slack(z) + # Using the primal formulation: + # S Hs S = diag(s)*diag(barrier_parameter/s**2)*diag(s). + # Reference [1]_ p. 882, formula (3.1) + primal = self.barrier_parameter + # Using the primal-dual formulation + # S Hs S = diag(s)*diag(v/s)*diag(s) + # Reference [1]_ p. 883, formula (3.11) + primal_dual = v[-self.n_ineq:]*s + # Uses the primal-dual formulation for + # positives values of v_ineq, and primal + # formulation for the remaining ones. + return np.where(v[-self.n_ineq:] > 0, primal_dual, primal) + + def lagrangian_hessian(self, z, v): + """Returns scaled Lagrangian Hessian""" + # Compute Hessian in relation to x and s + Hx = self.lagrangian_hessian_x(z, v) + if self.n_ineq > 0: + S_Hs_S = self.lagrangian_hessian_s(z, v) + + # The scaled Lagragian Hessian is: + # [ Hx 0 ] + # [ 0 S Hs S ] + def matvec(vec): + vec_x = self.get_variables(vec) + vec_s = self.get_slack(vec) + if self.n_ineq > 0: + return np.hstack((Hx.dot(vec_x), S_Hs_S*vec_s)) + else: + return Hx.dot(vec_x) + return LinearOperator((self.n_vars+self.n_ineq, + self.n_vars+self.n_ineq), + matvec) + + def stop_criteria(self, state, z, last_iteration_failed, + optimality, constr_violation, + trust_radius, penalty, cg_info): + """Stop criteria to the barrier problem. + The criteria here proposed is similar to formula (2.3) + from [1]_, p.879. + """ + x = self.get_variables(z) + if self.global_stop_criteria(state, x, + last_iteration_failed, + trust_radius, penalty, + cg_info, + self.barrier_parameter, + self.tolerance): + self.terminate = True + return True + else: + g_cond = (optimality < self.tolerance and + constr_violation < self.tolerance) + x_cond = trust_radius < self.xtol + return g_cond or x_cond + + +def tr_interior_point(fun, grad, lagr_hess, n_vars, n_ineq, n_eq, + constr, jac, x0, fun0, grad0, + constr_ineq0, jac_ineq0, constr_eq0, + jac_eq0, stop_criteria, + enforce_feasibility, xtol, state, + initial_barrier_parameter, + initial_tolerance, + initial_penalty, + initial_trust_radius, + factorization_method, + finite_diff_bounds): + """Trust-region interior points method. + + Solve problem: + minimize fun(x) + subject to: constr_ineq(x) <= 0 + constr_eq(x) = 0 + using trust-region interior point method described in [1]_. + """ + # BOUNDARY_PARAMETER controls the decrease on the slack + # variables. Represents ``tau`` from [1]_ p.885, formula (3.18). + BOUNDARY_PARAMETER = 0.995 + # BARRIER_DECAY_RATIO controls the decay of the barrier parameter + # and of the subproblem tolerance. Represents ``theta`` from [1]_ p.879. + BARRIER_DECAY_RATIO = 0.2 + # TRUST_ENLARGEMENT controls the enlargement on trust radius + # after each iteration + TRUST_ENLARGEMENT = 5 + + # Default enforce_feasibility + if enforce_feasibility is None: + enforce_feasibility = np.zeros(n_ineq, bool) + # Initial Values + barrier_parameter = initial_barrier_parameter + tolerance = initial_tolerance + trust_radius = initial_trust_radius + # Define initial value for the slack variables + s0 = np.maximum(-1.5*constr_ineq0, np.ones(n_ineq)) + # Define barrier subproblem + subprob = BarrierSubproblem( + x0, s0, fun, grad, lagr_hess, n_vars, n_ineq, n_eq, constr, jac, + barrier_parameter, tolerance, enforce_feasibility, + stop_criteria, xtol, fun0, grad0, constr_ineq0, jac_ineq0, + constr_eq0, jac_eq0, finite_diff_bounds) + # Define initial parameter for the first iteration. + z = np.hstack((x0, s0)) + fun0_subprob, constr0_subprob = subprob.fun0, subprob.constr0 + grad0_subprob, jac0_subprob = subprob.grad0, subprob.jac0 + # Define trust region bounds + trust_lb = np.hstack((np.full(subprob.n_vars, -np.inf), + np.full(subprob.n_ineq, -BOUNDARY_PARAMETER))) + trust_ub = np.full(subprob.n_vars+subprob.n_ineq, np.inf) + + # Solves a sequence of barrier problems + while True: + # Solve SQP subproblem + z, state = equality_constrained_sqp( + subprob.function_and_constraints, + subprob.gradient_and_jacobian, + subprob.lagrangian_hessian, + z, fun0_subprob, grad0_subprob, + constr0_subprob, jac0_subprob, subprob.stop_criteria, + state, initial_penalty, trust_radius, + factorization_method, trust_lb, trust_ub, subprob.scaling) + if subprob.terminate: + break + # Update parameters + trust_radius = max(initial_trust_radius, + TRUST_ENLARGEMENT*state.tr_radius) + # TODO: Use more advanced strategies from [2]_ + # to update this parameters. + barrier_parameter *= BARRIER_DECAY_RATIO + tolerance *= BARRIER_DECAY_RATIO + # Update Barrier Problem + subprob.update(barrier_parameter, tolerance) + # Compute initial values for next iteration + fun0_subprob, constr0_subprob = subprob.function_and_constraints(z) + grad0_subprob, jac0_subprob = subprob.gradient_and_jacobian(z) + + # Get x and s + x = subprob.get_variables(z) + return x, state diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_dogleg.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_dogleg.py new file mode 100644 index 0000000000000000000000000000000000000000..a54abd60c703408d6c87cb5020d6781fdf0213c7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_dogleg.py @@ -0,0 +1,122 @@ +"""Dog-leg trust-region optimization.""" +import numpy as np +import scipy.linalg +from ._trustregion import (_minimize_trust_region, BaseQuadraticSubproblem) + +__all__ = [] + + +def _minimize_dogleg(fun, x0, args=(), jac=None, hess=None, + **trust_region_options): + """ + Minimization of scalar function of one or more variables using + the dog-leg trust-region algorithm. + + Options + ------- + initial_trust_radius : float + Initial trust-region radius. + max_trust_radius : float + Maximum value of the trust-region radius. No steps that are longer + than this value will be proposed. + eta : float + Trust region related acceptance stringency for proposed steps. + gtol : float + Gradient norm must be less than `gtol` before successful + termination. + + """ + if jac is None: + raise ValueError('Jacobian is required for dogleg minimization') + if not callable(hess): + raise ValueError('Hessian is required for dogleg minimization') + return _minimize_trust_region(fun, x0, args=args, jac=jac, hess=hess, + subproblem=DoglegSubproblem, + **trust_region_options) + + +class DoglegSubproblem(BaseQuadraticSubproblem): + """Quadratic subproblem solved by the dogleg method""" + + def cauchy_point(self): + """ + The Cauchy point is minimal along the direction of steepest descent. + """ + if self._cauchy_point is None: + g = self.jac + Bg = self.hessp(g) + self._cauchy_point = -(np.dot(g, g) / np.dot(g, Bg)) * g + return self._cauchy_point + + def newton_point(self): + """ + The Newton point is a global minimum of the approximate function. + """ + if self._newton_point is None: + g = self.jac + B = self.hess + cho_info = scipy.linalg.cho_factor(B) + self._newton_point = -scipy.linalg.cho_solve(cho_info, g) + return self._newton_point + + def solve(self, trust_radius): + """ + Minimize a function using the dog-leg trust-region algorithm. + + This algorithm requires function values and first and second derivatives. + It also performs a costly Hessian decomposition for most iterations, + and the Hessian is required to be positive definite. + + Parameters + ---------- + trust_radius : float + We are allowed to wander only this far away from the origin. + + Returns + ------- + p : ndarray + The proposed step. + hits_boundary : bool + True if the proposed step is on the boundary of the trust region. + + Notes + ----- + The Hessian is required to be positive definite. + + References + ---------- + .. [1] Jorge Nocedal and Stephen Wright, + Numerical Optimization, second edition, + Springer-Verlag, 2006, page 73. + """ + + # Compute the Newton point. + # This is the optimum for the quadratic model function. + # If it is inside the trust radius then return this point. + p_best = self.newton_point() + if scipy.linalg.norm(p_best) < trust_radius: + hits_boundary = False + return p_best, hits_boundary + + # Compute the Cauchy point. + # This is the predicted optimum along the direction of steepest descent. + p_u = self.cauchy_point() + + # If the Cauchy point is outside the trust region, + # then return the point where the path intersects the boundary. + p_u_norm = scipy.linalg.norm(p_u) + if p_u_norm >= trust_radius: + p_boundary = p_u * (trust_radius / p_u_norm) + hits_boundary = True + return p_boundary, hits_boundary + + # Compute the intersection of the trust region boundary + # and the line segment connecting the Cauchy and Newton points. + # This requires solving a quadratic equation. + # ||p_u + t*(p_best - p_u)||**2 == trust_radius**2 + # Solve this for positive time t using the quadratic formula. + _, tb = self.get_boundaries_intersections(p_u, p_best - p_u, + trust_radius) + p_boundary = p_u + tb * (p_best - p_u) + hits_boundary = True + return p_boundary, hits_boundary diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_exact.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_exact.py new file mode 100644 index 0000000000000000000000000000000000000000..956e4f261907f001cf5bbb3616f331c18a676af0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_exact.py @@ -0,0 +1,438 @@ +"""Nearly exact trust-region optimization subproblem.""" +import numpy as np +from scipy.linalg import (norm, get_lapack_funcs, solve_triangular, + cho_solve) +from ._trustregion import (_minimize_trust_region, BaseQuadraticSubproblem) + +__all__ = ['_minimize_trustregion_exact', + 'estimate_smallest_singular_value', + 'singular_leading_submatrix', + 'IterativeSubproblem'] + + +def _minimize_trustregion_exact(fun, x0, args=(), jac=None, hess=None, + **trust_region_options): + """ + Minimization of scalar function of one or more variables using + a nearly exact trust-region algorithm. + + Options + ------- + initial_trust_radius : float + Initial trust-region radius. + max_trust_radius : float + Maximum value of the trust-region radius. No steps that are longer + than this value will be proposed. + eta : float + Trust region related acceptance stringency for proposed steps. + gtol : float + Gradient norm must be less than ``gtol`` before successful + termination. + """ + + if jac is None: + raise ValueError('Jacobian is required for trust region ' + 'exact minimization.') + if not callable(hess): + raise ValueError('Hessian matrix is required for trust region ' + 'exact minimization.') + return _minimize_trust_region(fun, x0, args=args, jac=jac, hess=hess, + subproblem=IterativeSubproblem, + **trust_region_options) + + +def estimate_smallest_singular_value(U): + """Given upper triangular matrix ``U`` estimate the smallest singular + value and the correspondent right singular vector in O(n**2) operations. + + Parameters + ---------- + U : ndarray + Square upper triangular matrix. + + Returns + ------- + s_min : float + Estimated smallest singular value of the provided matrix. + z_min : ndarray + Estimated right singular vector. + + Notes + ----- + The procedure is based on [1]_ and is done in two steps. First, it finds + a vector ``e`` with components selected from {+1, -1} such that the + solution ``w`` from the system ``U.T w = e`` is as large as possible. + Next it estimate ``U v = w``. The smallest singular value is close + to ``norm(w)/norm(v)`` and the right singular vector is close + to ``v/norm(v)``. + + The estimation will be better more ill-conditioned is the matrix. + + References + ---------- + .. [1] Cline, A. K., Moler, C. B., Stewart, G. W., Wilkinson, J. H. + An estimate for the condition number of a matrix. 1979. + SIAM Journal on Numerical Analysis, 16(2), 368-375. + """ + + U = np.atleast_2d(U) + m, n = U.shape + + if m != n: + raise ValueError("A square triangular matrix should be provided.") + + # A vector `e` with components selected from {+1, -1} + # is selected so that the solution `w` to the system + # `U.T w = e` is as large as possible. Implementation + # based on algorithm 3.5.1, p. 142, from reference [2] + # adapted for lower triangular matrix. + + p = np.zeros(n) + w = np.empty(n) + + # Implemented according to: Golub, G. H., Van Loan, C. F. (2013). + # "Matrix computations". Forth Edition. JHU press. pp. 140-142. + for k in range(n): + wp = (1-p[k]) / U.T[k, k] + wm = (-1-p[k]) / U.T[k, k] + pp = p[k+1:] + U.T[k+1:, k]*wp + pm = p[k+1:] + U.T[k+1:, k]*wm + + if abs(wp) + norm(pp, 1) >= abs(wm) + norm(pm, 1): + w[k] = wp + p[k+1:] = pp + else: + w[k] = wm + p[k+1:] = pm + + # The system `U v = w` is solved using backward substitution. + v = solve_triangular(U, w) + + v_norm = norm(v) + w_norm = norm(w) + + # Smallest singular value + s_min = w_norm / v_norm + + # Associated vector + z_min = v / v_norm + + return s_min, z_min + + +def gershgorin_bounds(H): + """ + Given a square matrix ``H`` compute upper + and lower bounds for its eigenvalues (Gregoshgorin Bounds). + Defined ref. [1]. + + References + ---------- + .. [1] Conn, A. R., Gould, N. I., & Toint, P. L. + Trust region methods. 2000. Siam. pp. 19. + """ + + H_diag = np.diag(H) + H_diag_abs = np.abs(H_diag) + H_row_sums = np.sum(np.abs(H), axis=1) + lb = np.min(H_diag + H_diag_abs - H_row_sums) + ub = np.max(H_diag - H_diag_abs + H_row_sums) + + return lb, ub + + +def singular_leading_submatrix(A, U, k): + """ + Compute term that makes the leading ``k`` by ``k`` + submatrix from ``A`` singular. + + Parameters + ---------- + A : ndarray + Symmetric matrix that is not positive definite. + U : ndarray + Upper triangular matrix resulting of an incomplete + Cholesky decomposition of matrix ``A``. + k : int + Positive integer such that the leading k by k submatrix from + `A` is the first non-positive definite leading submatrix. + + Returns + ------- + delta : float + Amount that should be added to the element (k, k) of the + leading k by k submatrix of ``A`` to make it singular. + v : ndarray + A vector such that ``v.T B v = 0``. Where B is the matrix A after + ``delta`` is added to its element (k, k). + """ + + # Compute delta + delta = np.sum(U[:k-1, k-1]**2) - A[k-1, k-1] + + n = len(A) + + # Initialize v + v = np.zeros(n) + v[k-1] = 1 + + # Compute the remaining values of v by solving a triangular system. + if k != 1: + v[:k-1] = solve_triangular(U[:k-1, :k-1], -U[:k-1, k-1]) + + return delta, v + + +class IterativeSubproblem(BaseQuadraticSubproblem): + """Quadratic subproblem solved by nearly exact iterative method. + + Notes + ----- + This subproblem solver was based on [1]_, [2]_ and [3]_, + which implement similar algorithms. The algorithm is basically + that of [1]_ but ideas from [2]_ and [3]_ were also used. + + References + ---------- + .. [1] A.R. Conn, N.I. Gould, and P.L. Toint, "Trust region methods", + Siam, pp. 169-200, 2000. + .. [2] J. Nocedal and S. Wright, "Numerical optimization", + Springer Science & Business Media. pp. 83-91, 2006. + .. [3] J.J. More and D.C. Sorensen, "Computing a trust region step", + SIAM Journal on Scientific and Statistical Computing, vol. 4(3), + pp. 553-572, 1983. + """ + + # UPDATE_COEFF appears in reference [1]_ + # in formula 7.3.14 (p. 190) named as "theta". + # As recommended there it value is fixed in 0.01. + UPDATE_COEFF = 0.01 + + EPS = np.finfo(float).eps + + def __init__(self, x, fun, jac, hess, hessp=None, + k_easy=0.1, k_hard=0.2): + + super().__init__(x, fun, jac, hess) + + # When the trust-region shrinks in two consecutive + # calculations (``tr_radius < previous_tr_radius``) + # the lower bound ``lambda_lb`` may be reused, + # facilitating the convergence. To indicate no + # previous value is known at first ``previous_tr_radius`` + # is set to -1 and ``lambda_lb`` to None. + self.previous_tr_radius = -1 + self.lambda_lb = None + + self.niter = 0 + + # ``k_easy`` and ``k_hard`` are parameters used + # to determine the stop criteria to the iterative + # subproblem solver. Take a look at pp. 194-197 + # from reference _[1] for a more detailed description. + self.k_easy = k_easy + self.k_hard = k_hard + + # Get Lapack function for cholesky decomposition. + # The implemented SciPy wrapper does not return + # the incomplete factorization needed by the method. + self.cholesky, = get_lapack_funcs(('potrf',), (self.hess,)) + + # Get info about Hessian + self.dimension = len(self.hess) + self.hess_gershgorin_lb,\ + self.hess_gershgorin_ub = gershgorin_bounds(self.hess) + self.hess_inf = norm(self.hess, np.inf) + self.hess_fro = norm(self.hess, 'fro') + + # A constant such that for vectors smaller than that + # backward substitution is not reliable. It was established + # based on Golub, G. H., Van Loan, C. F. (2013). + # "Matrix computations". Forth Edition. JHU press., p.165. + self.CLOSE_TO_ZERO = self.dimension * self.EPS * self.hess_inf + + def _initial_values(self, tr_radius): + """Given a trust radius, return a good initial guess for + the damping factor, the lower bound and the upper bound. + The values were chosen accordingly to the guidelines on + section 7.3.8 (p. 192) from [1]_. + """ + + # Upper bound for the damping factor + lambda_ub = max(0, self.jac_mag/tr_radius + min(-self.hess_gershgorin_lb, + self.hess_fro, + self.hess_inf)) + + # Lower bound for the damping factor + lambda_lb = max(0, -min(self.hess.diagonal()), + self.jac_mag/tr_radius - min(self.hess_gershgorin_ub, + self.hess_fro, + self.hess_inf)) + + # Improve bounds with previous info + if tr_radius < self.previous_tr_radius: + lambda_lb = max(self.lambda_lb, lambda_lb) + + # Initial guess for the damping factor + if lambda_lb == 0: + lambda_initial = 0 + else: + lambda_initial = max(np.sqrt(lambda_lb * lambda_ub), + lambda_lb + self.UPDATE_COEFF*(lambda_ub-lambda_lb)) + + return lambda_initial, lambda_lb, lambda_ub + + def solve(self, tr_radius): + """Solve quadratic subproblem""" + + lambda_current, lambda_lb, lambda_ub = self._initial_values(tr_radius) + n = self.dimension + hits_boundary = True + already_factorized = False + self.niter = 0 + + while True: + + # Compute Cholesky factorization + if already_factorized: + already_factorized = False + else: + H = self.hess+lambda_current*np.eye(n) + U, info = self.cholesky(H, lower=False, + overwrite_a=False, + clean=True) + + self.niter += 1 + + # Check if factorization succeeded + if info == 0 and self.jac_mag > self.CLOSE_TO_ZERO: + # Successful factorization + + # Solve `U.T U p = s` + p = cho_solve((U, False), -self.jac) + + p_norm = norm(p) + + # Check for interior convergence + if p_norm <= tr_radius and lambda_current == 0: + hits_boundary = False + break + + # Solve `U.T w = p` + w = solve_triangular(U, p, trans='T') + + w_norm = norm(w) + + # Compute Newton step accordingly to + # formula (4.44) p.87 from ref [2]_. + delta_lambda = (p_norm/w_norm)**2 * (p_norm-tr_radius)/tr_radius + lambda_new = lambda_current + delta_lambda + + if p_norm < tr_radius: # Inside boundary + s_min, z_min = estimate_smallest_singular_value(U) + + ta, tb = self.get_boundaries_intersections(p, z_min, + tr_radius) + + # Choose `step_len` with the smallest magnitude. + # The reason for this choice is explained at + # ref [3]_, p. 6 (Immediately before the formula + # for `tau`). + step_len = min([ta, tb], key=abs) + + # Compute the quadratic term (p.T*H*p) + quadratic_term = np.dot(p, np.dot(H, p)) + + # Check stop criteria + relative_error = ((step_len**2 * s_min**2) + / (quadratic_term + lambda_current*tr_radius**2)) + if relative_error <= self.k_hard: + p += step_len * z_min + break + + # Update uncertainty bounds + lambda_ub = lambda_current + lambda_lb = max(lambda_lb, lambda_current - s_min**2) + + # Compute Cholesky factorization + H = self.hess + lambda_new*np.eye(n) + c, info = self.cholesky(H, lower=False, + overwrite_a=False, + clean=True) + + # Check if the factorization have succeeded + # + if info == 0: # Successful factorization + # Update damping factor + lambda_current = lambda_new + already_factorized = True + else: # Unsuccessful factorization + # Update uncertainty bounds + lambda_lb = max(lambda_lb, lambda_new) + + # Update damping factor + lambda_current = max( + np.sqrt(lambda_lb * lambda_ub), + lambda_lb + self.UPDATE_COEFF*(lambda_ub-lambda_lb) + ) + + else: # Outside boundary + # Check stop criteria + relative_error = abs(p_norm - tr_radius) / tr_radius + if relative_error <= self.k_easy: + break + + # Update uncertainty bounds + lambda_lb = lambda_current + + # Update damping factor + lambda_current = lambda_new + + elif info == 0 and self.jac_mag <= self.CLOSE_TO_ZERO: + # jac_mag very close to zero + + # Check for interior convergence + if lambda_current == 0: + p = np.zeros(n) + hits_boundary = False + break + + s_min, z_min = estimate_smallest_singular_value(U) + step_len = tr_radius + + # Check stop criteria + if (step_len**2 * s_min**2 + <= self.k_hard * lambda_current * tr_radius**2): + p = step_len * z_min + break + + # Update uncertainty bounds + lambda_ub = lambda_current + lambda_lb = max(lambda_lb, lambda_current - s_min**2) + + # Update damping factor + lambda_current = max( + np.sqrt(lambda_lb * lambda_ub), + lambda_lb + self.UPDATE_COEFF*(lambda_ub-lambda_lb) + ) + + else: # Unsuccessful factorization + + # Compute auxiliary terms + delta, v = singular_leading_submatrix(H, U, info) + v_norm = norm(v) + + # Update uncertainty interval + lambda_lb = max(lambda_lb, lambda_current + delta/v_norm**2) + + # Update damping factor + lambda_current = max( + np.sqrt(lambda_lb * lambda_ub), + lambda_lb + self.UPDATE_COEFF*(lambda_ub-lambda_lb) + ) + + self.lambda_lb = lambda_lb + self.lambda_current = lambda_current + self.previous_tr_radius = tr_radius + + return p, hits_boundary diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_krylov.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_krylov.py new file mode 100644 index 0000000000000000000000000000000000000000..54e861ae2de02164966a33c437e5fdb08ba3006c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_krylov.py @@ -0,0 +1,65 @@ +from ._trustregion import (_minimize_trust_region) +from ._trlib import (get_trlib_quadratic_subproblem) + +__all__ = ['_minimize_trust_krylov'] + +def _minimize_trust_krylov(fun, x0, args=(), jac=None, hess=None, hessp=None, + inexact=True, **trust_region_options): + """ + Minimization of a scalar function of one or more variables using + a nearly exact trust-region algorithm that only requires matrix + vector products with the hessian matrix. + + .. versionadded:: 1.0.0 + + Options + ------- + inexact : bool, optional + Accuracy to solve subproblems. If True requires less nonlinear + iterations, but more vector products. + """ + + if jac is None: + raise ValueError('Jacobian is required for trust region ', + 'exact minimization.') + if hess is None and hessp is None: + raise ValueError('Either the Hessian or the Hessian-vector product ' + 'is required for Krylov trust-region minimization') + + # tol_rel specifies the termination tolerance relative to the initial + # gradient norm in the Krylov subspace iteration. + + # - tol_rel_i specifies the tolerance for interior convergence. + # - tol_rel_b specifies the tolerance for boundary convergence. + # in nonlinear programming applications it is not necessary to solve + # the boundary case as exact as the interior case. + + # - setting tol_rel_i=-2 leads to a forcing sequence in the Krylov + # subspace iteration leading to quadratic convergence if eventually + # the trust region stays inactive. + # - setting tol_rel_b=-3 leads to a forcing sequence in the Krylov + # subspace iteration leading to superlinear convergence as long + # as the iterates hit the trust region boundary. + + # For details consult the documentation of trlib_krylov_min + # in _trlib/trlib_krylov.h + # + # Optimality of this choice of parameters among a range of possibilities + # has been tested on the unconstrained subset of the CUTEst library. + + if inexact: + return _minimize_trust_region(fun, x0, args=args, jac=jac, + hess=hess, hessp=hessp, + subproblem=get_trlib_quadratic_subproblem( + tol_rel_i=-2.0, tol_rel_b=-3.0, + disp=trust_region_options.get('disp', False) + ), + **trust_region_options) + else: + return _minimize_trust_region(fun, x0, args=args, jac=jac, + hess=hess, hessp=hessp, + subproblem=get_trlib_quadratic_subproblem( + tol_rel_i=1e-8, tol_rel_b=1e-6, + disp=trust_region_options.get('disp', False) + ), + **trust_region_options) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_ncg.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_ncg.py new file mode 100644 index 0000000000000000000000000000000000000000..fed17ff8b84eaf019c0ad69a03f260ca674477ad --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_trustregion_ncg.py @@ -0,0 +1,126 @@ +"""Newton-CG trust-region optimization.""" +import math + +import numpy as np +import scipy.linalg +from ._trustregion import (_minimize_trust_region, BaseQuadraticSubproblem) + +__all__ = [] + + +def _minimize_trust_ncg(fun, x0, args=(), jac=None, hess=None, hessp=None, + **trust_region_options): + """ + Minimization of scalar function of one or more variables using + the Newton conjugate gradient trust-region algorithm. + + Options + ------- + initial_trust_radius : float + Initial trust-region radius. + max_trust_radius : float + Maximum value of the trust-region radius. No steps that are longer + than this value will be proposed. + eta : float + Trust region related acceptance stringency for proposed steps. + gtol : float + Gradient norm must be less than `gtol` before successful + termination. + + """ + if jac is None: + raise ValueError('Jacobian is required for Newton-CG trust-region ' + 'minimization') + if hess is None and hessp is None: + raise ValueError('Either the Hessian or the Hessian-vector product ' + 'is required for Newton-CG trust-region minimization') + return _minimize_trust_region(fun, x0, args=args, jac=jac, hess=hess, + hessp=hessp, subproblem=CGSteihaugSubproblem, + **trust_region_options) + + +class CGSteihaugSubproblem(BaseQuadraticSubproblem): + """Quadratic subproblem solved by a conjugate gradient method""" + def solve(self, trust_radius): + """ + Solve the subproblem using a conjugate gradient method. + + Parameters + ---------- + trust_radius : float + We are allowed to wander only this far away from the origin. + + Returns + ------- + p : ndarray + The proposed step. + hits_boundary : bool + True if the proposed step is on the boundary of the trust region. + + Notes + ----- + This is algorithm (7.2) of Nocedal and Wright 2nd edition. + Only the function that computes the Hessian-vector product is required. + The Hessian itself is not required, and the Hessian does + not need to be positive semidefinite. + """ + + # get the norm of jacobian and define the origin + p_origin = np.zeros_like(self.jac) + + # define a default tolerance + tolerance = min(0.5, math.sqrt(self.jac_mag)) * self.jac_mag + + # Stop the method if the search direction + # is a direction of nonpositive curvature. + if self.jac_mag < tolerance: + hits_boundary = False + return p_origin, hits_boundary + + # init the state for the first iteration + z = p_origin + r = self.jac + d = -r + + # Search for the min of the approximation of the objective function. + while True: + + # do an iteration + Bd = self.hessp(d) + dBd = np.dot(d, Bd) + if dBd <= 0: + # Look at the two boundary points. + # Find both values of t to get the boundary points such that + # ||z + t d|| == trust_radius + # and then choose the one with the predicted min value. + ta, tb = self.get_boundaries_intersections(z, d, trust_radius) + pa = z + ta * d + pb = z + tb * d + if self(pa) < self(pb): + p_boundary = pa + else: + p_boundary = pb + hits_boundary = True + return p_boundary, hits_boundary + r_squared = np.dot(r, r) + alpha = r_squared / dBd + z_next = z + alpha * d + if scipy.linalg.norm(z_next) >= trust_radius: + # Find t >= 0 to get the boundary point such that + # ||z + t d|| == trust_radius + ta, tb = self.get_boundaries_intersections(z, d, trust_radius) + p_boundary = z + tb * d + hits_boundary = True + return p_boundary, hits_boundary + r_next = r + alpha * Bd + r_next_squared = np.dot(r_next, r_next) + if math.sqrt(r_next_squared) < tolerance: + hits_boundary = False + return z_next, hits_boundary + beta_next = r_next_squared / r_squared + d_next = -r_next + beta_next * d + + # update the state for the next iteration + z = z_next + r = r_next + d = d_next diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_tstutils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_tstutils.py new file mode 100644 index 0000000000000000000000000000000000000000..f56e835e345d66023efae81114a45ed29269f18d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_tstutils.py @@ -0,0 +1,972 @@ +r""" +Parameters used in test and benchmark methods. + +Collections of test cases suitable for testing 1-D root-finders + 'original': The original benchmarking functions. + Real-valued functions of real-valued inputs on an interval + with a zero. + f1, .., f3 are continuous and infinitely differentiable + f4 has a left- and right- discontinuity at the root + f5 has a root at 1 replacing a 1st order pole + f6 is randomly positive on one side of the root, + randomly negative on the other. + f4 - f6 are not continuous at the root. + + 'aps': The test problems in the 1995 paper + TOMS "Algorithm 748: Enclosing Zeros of Continuous Functions" + by Alefeld, Potra and Shi. Real-valued functions of + real-valued inputs on an interval with a zero. + Suitable for methods which start with an enclosing interval, and + derivatives up to 2nd order. + + 'complex': Some complex-valued functions of complex-valued inputs. + No enclosing bracket is provided. + Suitable for methods which use one or more starting values, and + derivatives up to 2nd order. + + The test cases are provided as a list of dictionaries. The dictionary + keys will be a subset of: + ["f", "fprime", "fprime2", "args", "bracket", "smoothness", + "a", "b", "x0", "x1", "root", "ID"] +""" + +# Sources: +# [1] Alefeld, G. E. and Potra, F. A. and Shi, Yixun, +# "Algorithm 748: Enclosing Zeros of Continuous Functions", +# ACM Trans. Math. Softw. Volume 221(1995) +# doi = {10.1145/210089.210111}, +# [2] Chandrupatla, Tirupathi R. "A new hybrid quadratic/bisection algorithm +# for finding the zero of a nonlinear function without using derivatives." +# Advances in Engineering Software 28.3 (1997): 145-149. + +from random import random + +import numpy as np + +from scipy.optimize import _zeros_py as cc +from scipy._lib._array_api import array_namespace + +# "description" refers to the original functions +description = """ +f2 is a symmetric parabola, x**2 - 1 +f3 is a quartic polynomial with large hump in interval +f4 is step function with a discontinuity at 1 +f5 is a hyperbola with vertical asymptote at 1 +f6 has random values positive to left of 1, negative to right + +Of course, these are not real problems. They just test how the +'good' solvers behave in bad circumstances where bisection is +really the best. A good solver should not be much worse than +bisection in such circumstance, while being faster for smooth +monotone sorts of functions. +""" + + +def f1(x): + r"""f1 is a quadratic with roots at 0 and 1""" + return x * (x - 1.) + + +def f1_fp(x): + return 2 * x - 1 + + +def f1_fpp(x): + return 2 + + +def f2(x): + r"""f2 is a symmetric parabola, x**2 - 1""" + return x**2 - 1 + + +def f2_fp(x): + return 2 * x + + +def f2_fpp(x): + return 2 + + +def f3(x): + r"""A quartic with roots at 0, 1, 2 and 3""" + return x * (x - 1.) * (x - 2.) * (x - 3.) # x**4 - 6x**3 + 11x**2 - 6x + + +def f3_fp(x): + return 4 * x**3 - 18 * x**2 + 22 * x - 6 + + +def f3_fpp(x): + return 12 * x**2 - 36 * x + 22 + + +def f4(x): + r"""Piecewise linear, left- and right- discontinuous at x=1, the root.""" + if x > 1: + return 1.0 + .1 * x + if x < 1: + return -1.0 + .1 * x + return 0 + + +def f5(x): + r""" + Hyperbola with a pole at x=1, but pole replaced with 0. Not continuous at root. + """ + if x != 1: + return 1.0 / (1. - x) + return 0 + + +# f6(x) returns random value. Without memoization, calling twice with the +# same x returns different values, hence a "random value", not a +# "function with random values" +_f6_cache = {} +def f6(x): + v = _f6_cache.get(x, None) + if v is None: + if x > 1: + v = random() + elif x < 1: + v = -random() + else: + v = 0 + _f6_cache[x] = v + return v + + +# Each Original test case has +# - a function and its two derivatives, +# - additional arguments, +# - a bracket enclosing a root, +# - the order of differentiability (smoothness) on this interval +# - a starting value for methods which don't require a bracket +# - the root (inside the bracket) +# - an Identifier of the test case + +_ORIGINAL_TESTS_KEYS = [ + "f", "fprime", "fprime2", "args", "bracket", "smoothness", "x0", "root", "ID" +] +_ORIGINAL_TESTS = [ + [f1, f1_fp, f1_fpp, (), [0.5, np.sqrt(3)], np.inf, 0.6, 1.0, "original.01.00"], + [f2, f2_fp, f2_fpp, (), [0.5, np.sqrt(3)], np.inf, 0.6, 1.0, "original.02.00"], + [f3, f3_fp, f3_fpp, (), [0.5, np.sqrt(3)], np.inf, 0.6, 1.0, "original.03.00"], + [f4, None, None, (), [0.5, np.sqrt(3)], -1, 0.6, 1.0, "original.04.00"], + [f5, None, None, (), [0.5, np.sqrt(3)], -1, 0.6, 1.0, "original.05.00"], + [f6, None, None, (), [0.5, np.sqrt(3)], -np.inf, 0.6, 1.0, "original.05.00"] +] + +_ORIGINAL_TESTS_DICTS = [ + dict(zip(_ORIGINAL_TESTS_KEYS, testcase)) for testcase in _ORIGINAL_TESTS +] + +# ################## +# "APS" test cases +# Functions and test cases that appear in [1] + + +def aps01_f(x): + r"""Straightforward sum of trigonometric function and polynomial""" + return np.sin(x) - x / 2 + + +def aps01_fp(x): + return np.cos(x) - 1.0 / 2 + + +def aps01_fpp(x): + return -np.sin(x) + + +def aps02_f(x): + r"""poles at x=n**2, 1st and 2nd derivatives at root are also close to 0""" + ii = np.arange(1, 21) + return -2 * np.sum((2 * ii - 5)**2 / (x - ii**2)**3) + + +def aps02_fp(x): + ii = np.arange(1, 21) + return 6 * np.sum((2 * ii - 5)**2 / (x - ii**2)**4) + + +def aps02_fpp(x): + ii = np.arange(1, 21) + return 24 * np.sum((2 * ii - 5)**2 / (x - ii**2)**5) + + +def aps03_f(x, a, b): + r"""Rapidly changing at the root""" + return a * x * np.exp(b * x) + + +def aps03_fp(x, a, b): + return a * (b * x + 1) * np.exp(b * x) + + +def aps03_fpp(x, a, b): + return a * (b * (b * x + 1) + b) * np.exp(b * x) + + +def aps04_f(x, n, a): + r"""Medium-degree polynomial""" + return x**n - a + + +def aps04_fp(x, n, a): + return n * x**(n - 1) + + +def aps04_fpp(x, n, a): + return n * (n - 1) * x**(n - 2) + + +def aps05_f(x): + r"""Simple Trigonometric function""" + return np.sin(x) - 1.0 / 2 + + +def aps05_fp(x): + return np.cos(x) + + +def aps05_fpp(x): + return -np.sin(x) + + +def aps06_f(x, n): + r"""Exponential rapidly changing from -1 to 1 at x=0""" + return 2 * x * np.exp(-n) - 2 * np.exp(-n * x) + 1 + + +def aps06_fp(x, n): + return 2 * np.exp(-n) + 2 * n * np.exp(-n * x) + + +def aps06_fpp(x, n): + return -2 * n * n * np.exp(-n * x) + + +def aps07_f(x, n): + r"""Upside down parabola with parametrizable height""" + return (1 + (1 - n)**2) * x - (1 - n * x)**2 + + +def aps07_fp(x, n): + return (1 + (1 - n)**2) + 2 * n * (1 - n * x) + + +def aps07_fpp(x, n): + return -2 * n * n + + +def aps08_f(x, n): + r"""Degree n polynomial""" + return x * x - (1 - x)**n + + +def aps08_fp(x, n): + return 2 * x + n * (1 - x)**(n - 1) + + +def aps08_fpp(x, n): + return 2 - n * (n - 1) * (1 - x)**(n - 2) + + +def aps09_f(x, n): + r"""Upside down quartic with parametrizable height""" + return (1 + (1 - n)**4) * x - (1 - n * x)**4 + + +def aps09_fp(x, n): + return (1 + (1 - n)**4) + 4 * n * (1 - n * x)**3 + + +def aps09_fpp(x, n): + return -12 * n * (1 - n * x)**2 + + +def aps10_f(x, n): + r"""Exponential plus a polynomial""" + return np.exp(-n * x) * (x - 1) + x**n + + +def aps10_fp(x, n): + return np.exp(-n * x) * (-n * (x - 1) + 1) + n * x**(n - 1) + + +def aps10_fpp(x, n): + return (np.exp(-n * x) * (-n * (-n * (x - 1) + 1) + -n * x) + + n * (n - 1) * x**(n - 2)) + + +def aps11_f(x, n): + r"""Rational function with a zero at x=1/n and a pole at x=0""" + return (n * x - 1) / ((n - 1) * x) + + +def aps11_fp(x, n): + return 1 / (n - 1) / x**2 + + +def aps11_fpp(x, n): + return -2 / (n - 1) / x**3 + + +def aps12_f(x, n): + r"""nth root of x, with a zero at x=n""" + return np.power(x, 1.0 / n) - np.power(n, 1.0 / n) + + +def aps12_fp(x, n): + return np.power(x, (1.0 - n) / n) / n + + +def aps12_fpp(x, n): + return np.power(x, (1.0 - 2 * n) / n) * (1.0 / n) * (1.0 - n) / n + + +_MAX_EXPABLE = np.log(np.finfo(float).max) + + +def aps13_f(x): + r"""Function with *all* derivatives 0 at the root""" + if x == 0: + return 0 + # x2 = 1.0/x**2 + # if x2 > 708: + # return 0 + y = 1 / x**2 + if y > _MAX_EXPABLE: + return 0 + return x / np.exp(y) + + +def aps13_fp(x): + if x == 0: + return 0 + y = 1 / x**2 + if y > _MAX_EXPABLE: + return 0 + return (1 + 2 / x**2) / np.exp(y) + + +def aps13_fpp(x): + if x == 0: + return 0 + y = 1 / x**2 + if y > _MAX_EXPABLE: + return 0 + return 2 * (2 - x**2) / x**5 / np.exp(y) + + +def aps14_f(x, n): + r"""0 for negative x-values, trigonometric+linear for x positive""" + if x <= 0: + return -n / 20.0 + return n / 20.0 * (x / 1.5 + np.sin(x) - 1) + + +def aps14_fp(x, n): + if x <= 0: + return 0 + return n / 20.0 * (1.0 / 1.5 + np.cos(x)) + + +def aps14_fpp(x, n): + if x <= 0: + return 0 + return -n / 20.0 * (np.sin(x)) + + +def aps15_f(x, n): + r"""piecewise linear, constant outside of [0, 0.002/(1+n)]""" + if x < 0: + return -0.859 + if x > 2 * 1e-3 / (1 + n): + return np.e - 1.859 + return np.exp((n + 1) * x / 2 * 1000) - 1.859 + + +def aps15_fp(x, n): + if not 0 <= x <= 2 * 1e-3 / (1 + n): + return np.e - 1.859 + return np.exp((n + 1) * x / 2 * 1000) * (n + 1) / 2 * 1000 + + +def aps15_fpp(x, n): + if not 0 <= x <= 2 * 1e-3 / (1 + n): + return np.e - 1.859 + return np.exp((n + 1) * x / 2 * 1000) * (n + 1) / 2 * 1000 * (n + 1) / 2 * 1000 + + +# Each APS test case has +# - a function and its two derivatives, +# - additional arguments, +# - a bracket enclosing a root, +# - the order of differentiability of the function on this interval +# - a starting value for methods which don't require a bracket +# - the root (inside the bracket) +# - an Identifier of the test case +# +# Algorithm 748 is a bracketing algorithm so a bracketing interval was provided +# in [1] for each test case. Newton and Halley methods need a single +# starting point x0, which was chosen to be near the middle of the interval, +# unless that would have made the problem too easy. + +_APS_TESTS_KEYS = [ + "f", "fprime", "fprime2", "args", "bracket", "smoothness", "x0", "root", "ID" +] +_APS_TESTS = [ + [aps01_f, aps01_fp, aps01_fpp, (), [np.pi / 2, np.pi], np.inf, + 3, 1.89549426703398094e+00, "aps.01.00"], + [aps02_f, aps02_fp, aps02_fpp, (), [1 + 1e-9, 4 - 1e-9], np.inf, + 2, 3.02291534727305677e+00, "aps.02.00"], + [aps02_f, aps02_fp, aps02_fpp, (), [4 + 1e-9, 9 - 1e-9], np.inf, + 5, 6.68375356080807848e+00, "aps.02.01"], + [aps02_f, aps02_fp, aps02_fpp, (), [9 + 1e-9, 16 - 1e-9], np.inf, + 10, 1.12387016550022114e+01, "aps.02.02"], + [aps02_f, aps02_fp, aps02_fpp, (), [16 + 1e-9, 25 - 1e-9], np.inf, + 17, 1.96760000806234103e+01, "aps.02.03"], + [aps02_f, aps02_fp, aps02_fpp, (), [25 + 1e-9, 36 - 1e-9], np.inf, + 26, 2.98282273265047557e+01, "aps.02.04"], + [aps02_f, aps02_fp, aps02_fpp, (), [36 + 1e-9, 49 - 1e-9], np.inf, + 37, 4.19061161952894139e+01, "aps.02.05"], + [aps02_f, aps02_fp, aps02_fpp, (), [49 + 1e-9, 64 - 1e-9], np.inf, + 50, 5.59535958001430913e+01, "aps.02.06"], + [aps02_f, aps02_fp, aps02_fpp, (), [64 + 1e-9, 81 - 1e-9], np.inf, + 65, 7.19856655865877997e+01, "aps.02.07"], + [aps02_f, aps02_fp, aps02_fpp, (), [81 + 1e-9, 100 - 1e-9], np.inf, + 82, 9.00088685391666701e+01, "aps.02.08"], + [aps02_f, aps02_fp, aps02_fpp, (), [100 + 1e-9, 121 - 1e-9], np.inf, + 101, 1.10026532748330197e+02, "aps.02.09"], + [aps03_f, aps03_fp, aps03_fpp, (-40, -1), [-9, 31], np.inf, + -2, 0, "aps.03.00"], + [aps03_f, aps03_fp, aps03_fpp, (-100, -2), [-9, 31], np.inf, + -2, 0, "aps.03.01"], + [aps03_f, aps03_fp, aps03_fpp, (-200, -3), [-9, 31], np.inf, + -2, 0, "aps.03.02"], + [aps04_f, aps04_fp, aps04_fpp, (4, 0.2), [0, 5], np.inf, + 2.5, 6.68740304976422006e-01, "aps.04.00"], + [aps04_f, aps04_fp, aps04_fpp, (6, 0.2), [0, 5], np.inf, + 2.5, 7.64724491331730039e-01, "aps.04.01"], + [aps04_f, aps04_fp, aps04_fpp, (8, 0.2), [0, 5], np.inf, + 2.5, 8.17765433957942545e-01, "aps.04.02"], + [aps04_f, aps04_fp, aps04_fpp, (10, 0.2), [0, 5], np.inf, + 2.5, 8.51339922520784609e-01, "aps.04.03"], + [aps04_f, aps04_fp, aps04_fpp, (12, 0.2), [0, 5], np.inf, + 2.5, 8.74485272221167897e-01, "aps.04.04"], + [aps04_f, aps04_fp, aps04_fpp, (4, 1), [0, 5], np.inf, + 2.5, 1, "aps.04.05"], + [aps04_f, aps04_fp, aps04_fpp, (6, 1), [0, 5], np.inf, + 2.5, 1, "aps.04.06"], + [aps04_f, aps04_fp, aps04_fpp, (8, 1), [0, 5], np.inf, + 2.5, 1, "aps.04.07"], + [aps04_f, aps04_fp, aps04_fpp, (10, 1), [0, 5], np.inf, + 2.5, 1, "aps.04.08"], + [aps04_f, aps04_fp, aps04_fpp, (12, 1), [0, 5], np.inf, + 2.5, 1, "aps.04.09"], + [aps04_f, aps04_fp, aps04_fpp, (8, 1), [-0.95, 4.05], np.inf, + 1.5, 1, "aps.04.10"], + [aps04_f, aps04_fp, aps04_fpp, (10, 1), [-0.95, 4.05], np.inf, + 1.5, 1, "aps.04.11"], + [aps04_f, aps04_fp, aps04_fpp, (12, 1), [-0.95, 4.05], np.inf, + 1.5, 1, 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[-1000, 1e-4], 0, + -2, 4.96030966991445609e-05, "aps.15.04"], + [aps15_f, aps15_fp, aps15_fpp, (25,), [-1000, 1e-4], 0, + -2, 4.76952852876389951e-05, "aps.15.05"], + [aps15_f, aps15_fp, aps15_fpp, (26,), [-1000, 1e-4], 0, + -2, 4.59287932399486662e-05, "aps.15.06"], + [aps15_f, aps15_fp, aps15_fpp, (27,), [-1000, 1e-4], 0, + -2, 4.42884791956647841e-05, "aps.15.07"], + [aps15_f, aps15_fp, aps15_fpp, (28,), [-1000, 1e-4], 0, + -2, 4.27612902578832391e-05, "aps.15.08"], + [aps15_f, aps15_fp, aps15_fpp, (29,), [-1000, 1e-4], 0, + -2, 4.13359139159538030e-05, "aps.15.09"], + [aps15_f, aps15_fp, aps15_fpp, (30,), [-1000, 1e-4], 0, + -2, 4.00024973380198076e-05, "aps.15.10"], + [aps15_f, aps15_fp, aps15_fpp, (31,), [-1000, 1e-4], 0, + -2, 3.87524192962066869e-05, "aps.15.11"], + [aps15_f, aps15_fp, aps15_fpp, (32,), [-1000, 1e-4], 0, + -2, 3.75781035599579910e-05, "aps.15.12"], + [aps15_f, aps15_fp, aps15_fpp, (33,), [-1000, 1e-4], 0, + -2, 3.64728652199592355e-05, "aps.15.13"], + [aps15_f, aps15_fp, aps15_fpp, (34,), [-1000, 1e-4], 0, + -2, 3.54307833565318273e-05, "aps.15.14"], + [aps15_f, aps15_fp, aps15_fpp, (35,), [-1000, 1e-4], 0, + -2, 3.44465949299614980e-05, "aps.15.15"], + [aps15_f, aps15_fp, aps15_fpp, (36,), [-1000, 1e-4], 0, + -2, 3.35156058778003705e-05, "aps.15.16"], + [aps15_f, aps15_fp, aps15_fpp, (37,), [-1000, 1e-4], 0, + -2, 3.26336162494372125e-05, "aps.15.17"], + [aps15_f, aps15_fp, aps15_fpp, (38,), [-1000, 1e-4], 0, + -2, 3.17968568584260013e-05, "aps.15.18"], + [aps15_f, aps15_fp, aps15_fpp, (39,), [-1000, 1e-4], 0, + -2, 3.10019354369653455e-05, "aps.15.19"], + [aps15_f, aps15_fp, aps15_fpp, (40,), [-1000, 1e-4], 0, + -2, 3.02457906702100968e-05, "aps.15.20"], + [aps15_f, aps15_fp, aps15_fpp, (100,), [-1000, 1e-4], 0, + -2, 1.22779942324615231e-05, "aps.15.21"], + [aps15_f, aps15_fp, aps15_fpp, (200,), [-1000, 1e-4], 0, + -2, 6.16953939044086617e-06, "aps.15.22"], + [aps15_f, aps15_fp, aps15_fpp, (300,), [-1000, 1e-4], 0, + -2, 4.11985852982928163e-06, "aps.15.23"], + [aps15_f, aps15_fp, aps15_fpp, (400,), [-1000, 1e-4], 0, + -2, 3.09246238772721682e-06, "aps.15.24"], + [aps15_f, aps15_fp, aps15_fpp, (500,), [-1000, 1e-4], 0, + -2, 2.47520442610501789e-06, "aps.15.25"], + [aps15_f, aps15_fp, aps15_fpp, (600,), [-1000, 1e-4], 0, + -2, 2.06335676785127107e-06, "aps.15.26"], + [aps15_f, aps15_fp, aps15_fpp, (700,), [-1000, 1e-4], 0, + -2, 1.76901200781542651e-06, "aps.15.27"], + [aps15_f, aps15_fp, aps15_fpp, (800,), [-1000, 1e-4], 0, + -2, 1.54816156988591016e-06, "aps.15.28"], + [aps15_f, aps15_fp, aps15_fpp, (900,), [-1000, 1e-4], 0, + -2, 1.37633453660223511e-06, "aps.15.29"], + [aps15_f, aps15_fp, aps15_fpp, (1000,), [-1000, 1e-4], 0, + -2, 1.23883857889971403e-06, "aps.15.30"] +] + +_APS_TESTS_DICTS = [dict(zip(_APS_TESTS_KEYS, testcase)) for testcase in _APS_TESTS] + + +# ################## +# "complex" test cases +# A few simple, complex-valued, functions, defined on the complex plane. + + +def cplx01_f(z, n, a): + r"""z**n-a: Use to find the nth root of a""" + return z**n - a + + +def cplx01_fp(z, n, a): + return n * z**(n - 1) + + +def cplx01_fpp(z, n, a): + return n * (n - 1) * z**(n - 2) + + +def cplx02_f(z, a): + r"""e**z - a: Use to find the log of a""" + return np.exp(z) - a + + +def cplx02_fp(z, a): + return np.exp(z) + + +def cplx02_fpp(z, a): + return np.exp(z) + + +# Each "complex" test case has +# - a function and its two derivatives, +# - additional arguments, +# - the order of differentiability of the function on this interval +# - two starting values x0 and x1 +# - the root +# - an Identifier of the test case +# +# Algorithm 748 is a bracketing algorithm so a bracketing interval was provided +# in [1] for each test case. Newton and Halley need a single starting point +# x0, which was chosen to be near the middle of the interval, unless that +# would make the problem too easy. + + +_COMPLEX_TESTS_KEYS = [ + "f", "fprime", "fprime2", "args", "smoothness", "x0", "x1", "root", "ID" +] +_COMPLEX_TESTS = [ + [cplx01_f, cplx01_fp, cplx01_fpp, (2, -1), np.inf, + (1 + 1j), (0.5 + 0.5j), 1j, "complex.01.00"], + [cplx01_f, cplx01_fp, cplx01_fpp, (3, 1), np.inf, + (-1 + 1j), (-0.5 + 2.0j), (-0.5 + np.sqrt(3) / 2 * 1.0j), + "complex.01.01"], + [cplx01_f, cplx01_fp, cplx01_fpp, (3, -1), np.inf, + 1j, (0.5 + 0.5j), (0.5 + np.sqrt(3) / 2 * 1.0j), + "complex.01.02"], + [cplx01_f, cplx01_fp, cplx01_fpp, (3, 8), np.inf, + 5, 4, 2, "complex.01.03"], + [cplx02_f, cplx02_fp, cplx02_fpp, (-1,), np.inf, + (1 + 2j), (0.5 + 0.5j), np.pi * 1.0j, "complex.02.00"], + [cplx02_f, cplx02_fp, cplx02_fpp, (1j,), np.inf, + (1 + 2j), (0.5 + 0.5j), np.pi * 0.5j, "complex.02.01"], +] + +_COMPLEX_TESTS_DICTS = [ + dict(zip(_COMPLEX_TESTS_KEYS, testcase)) for testcase in _COMPLEX_TESTS +] + + +def _add_a_b(tests): + r"""Add "a" and "b" keys to each test from the "bracket" value""" + for d in tests: + for k, v in zip(['a', 'b'], d.get('bracket', [])): + d[k] = v + + +_add_a_b(_ORIGINAL_TESTS_DICTS) +_add_a_b(_APS_TESTS_DICTS) +_add_a_b(_COMPLEX_TESTS_DICTS) + + +def get_tests(collection='original', smoothness=None): + r"""Return the requested collection of test cases, as an array of dicts with subset-specific keys + + Allowed values of collection: + 'original': The original benchmarking functions. + Real-valued functions of real-valued inputs on an interval with a zero. + f1, .., f3 are continuous and infinitely differentiable + f4 has a single discontinuity at the root + f5 has a root at 1 replacing a 1st order pole + f6 is randomly positive on one side of the root, randomly negative on the other + 'aps': The test problems in the TOMS "Algorithm 748: Enclosing Zeros of Continuous Functions" + paper by Alefeld, Potra and Shi. Real-valued functions of + real-valued inputs on an interval with a zero. + Suitable for methods which start with an enclosing interval, and + derivatives up to 2nd order. + 'complex': Some complex-valued functions of complex-valued inputs. + No enclosing bracket is provided. + Suitable for methods which use one or more starting values, and + derivatives up to 2nd order. + + The dictionary keys will be a subset of + ["f", "fprime", "fprime2", "args", "bracket", "a", b", "smoothness", "x0", "x1", "root", "ID"] + """ # noqa: E501 + collection = collection or "original" + subsets = {"aps": _APS_TESTS_DICTS, + "complex": _COMPLEX_TESTS_DICTS, + "original": _ORIGINAL_TESTS_DICTS, + "chandrupatla": _CHANDRUPATLA_TESTS_DICTS} + tests = subsets.get(collection, []) + if smoothness is not None: + tests = [tc for tc in tests if tc['smoothness'] >= smoothness] + return tests + + +# Backwards compatibility +methods = [cc.bisect, cc.ridder, cc.brenth, cc.brentq] +mstrings = ['cc.bisect', 'cc.ridder', 'cc.brenth', 'cc.brentq'] +functions = [f2, f3, f4, f5, f6] +fstrings = ['f2', 'f3', 'f4', 'f5', 'f6'] + +# ################## +# "Chandrupatla" test cases +# Functions and test cases that appear in [2] + +def fun1(x): + return x**3 - 2*x - 5 +fun1.root = 2.0945514815423265 # additional precision using mpmath.findroot + + +def fun2(x): + return 1 - 1/x**2 +fun2.root = 1 + + +def fun3(x): + return (x-3)**3 +fun3.root = 3 + + +def fun4(x): + return 6*(x-2)**5 +fun4.root = 2 + + +def fun5(x): + return x**9 +fun5.root = 0 + + +def fun6(x): + return x**19 +fun6.root = 0 + + +def fun7(x): + xp = array_namespace(x) + return 0 if xp.abs(x) < 3.8e-4 else x*xp.exp(-x**(-2)) +fun7.root = 0 + + +def fun8(x): + xp = array_namespace(x) + xi = 0.61489 + return -(3062*(1-xi)*xp.exp(-x))/(xi + (1-xi)*xp.exp(-x)) - 1013 + 1628/x +fun8.root = 1.0375360332870405 + + +def fun9(x): + xp = array_namespace(x) + return xp.exp(x) - 2 - 0.01/x**2 + .000002/x**3 +fun9.root = 0.7032048403631358 + +# Each "chandropatla" test case has +# - a function, +# - two starting values x0 and x1 +# - the root +# - the number of function evaluations required by Chandrupatla's algorithm +# - an Identifier of the test case +# +# Chandrupatla's is a bracketing algorithm, so a bracketing interval was +# provided in [2] for each test case. No special support for testing with +# secant/Newton/Halley is provided. + +_CHANDRUPATLA_TESTS_KEYS = ["f", "bracket", "root", "nfeval", "ID"] +_CHANDRUPATLA_TESTS = [ + [fun1, [2, 3], fun1.root, 7], + [fun1, [1, 10], fun1.root, 11], + [fun1, [1, 100], fun1.root, 14], + [fun1, [-1e4, 1e4], fun1.root, 23], + [fun1, [-1e10, 1e10], fun1.root, 43], + [fun2, [0.5, 1.51], fun2.root, 8], + [fun2, [1e-4, 1e4], fun2.root, 22], + [fun2, [1e-6, 1e6], fun2.root, 28], + [fun2, [1e-10, 1e10], fun2.root, 41], + [fun2, [1e-12, 1e12], fun2.root, 48], + [fun3, [0, 5], fun3.root, 21], + [fun3, [-10, 10], fun3.root, 23], + [fun3, [-1e4, 1e4], fun3.root, 36], + [fun3, [-1e6, 1e6], fun3.root, 45], + [fun3, [-1e10, 1e10], fun3.root, 55], + [fun4, [0, 5], fun4.root, 21], + [fun4, [-10, 10], fun4.root, 23], + [fun4, [-1e4, 1e4], fun4.root, 33], + [fun4, [-1e6, 1e6], fun4.root, 43], + [fun4, [-1e10, 1e10], fun4.root, 54], + [fun5, [-1, 4], fun5.root, 21], + [fun5, [-2, 5], fun5.root, 22], + [fun5, [-1, 10], fun5.root, 23], + [fun5, [-5, 50], fun5.root, 25], + [fun5, [-10, 100], fun5.root, 26], + [fun6, [-1., 4.], fun6.root, 21], + [fun6, [-2., 5.], fun6.root, 22], + [fun6, [-1., 10.], fun6.root, 23], + [fun6, [-5., 50.], fun6.root, 25], + [fun6, [-10., 100.], fun6.root, 26], + [fun7, [-1, 4], fun7.root, 8], + [fun7, [-2, 5], fun7.root, 8], + [fun7, [-1, 10], fun7.root, 11], + [fun7, [-5, 50], fun7.root, 18], + [fun7, [-10, 100], fun7.root, 19], + [fun8, [2e-4, 2], fun8.root, 9], + [fun8, [2e-4, 3], fun8.root, 10], + [fun8, [2e-4, 9], fun8.root, 11], + [fun8, [2e-4, 27], fun8.root, 12], + [fun8, [2e-4, 81], fun8.root, 14], + [fun9, [2e-4, 1], fun9.root, 7], + [fun9, [2e-4, 3], fun9.root, 8], + [fun9, [2e-4, 9], fun9.root, 10], + [fun9, [2e-4, 27], fun9.root, 11], + [fun9, [2e-4, 81], fun9.root, 13], +] +_CHANDRUPATLA_TESTS = [test + [f'{test[0].__name__}.{i%5+1}'] + for i, test in enumerate(_CHANDRUPATLA_TESTS)] + +_CHANDRUPATLA_TESTS_DICTS = [dict(zip(_CHANDRUPATLA_TESTS_KEYS, testcase)) + for testcase in _CHANDRUPATLA_TESTS] +_add_a_b(_CHANDRUPATLA_TESTS_DICTS) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_zeros.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_zeros.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..7af3971d3d8f44903d9da2be40e4e414cbf6463d Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_zeros.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_zeros_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_zeros_py.py new file mode 100644 index 0000000000000000000000000000000000000000..2b96902ccbd58ed8fb6de3bbafe551f53fb8bf86 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/_zeros_py.py @@ -0,0 +1,1395 @@ +import warnings +from collections import namedtuple +import operator +from . import _zeros +from ._optimize import OptimizeResult +import numpy as np + + +_iter = 100 +_xtol = 2e-12 +_rtol = 4 * np.finfo(float).eps + +__all__ = ['newton', 'bisect', 'ridder', 'brentq', 'brenth', 'toms748', + 'RootResults'] + +# Must agree with CONVERGED, SIGNERR, CONVERR, ... in zeros.h +_ECONVERGED = 0 +_ESIGNERR = -1 # used in _chandrupatla +_ECONVERR = -2 +_EVALUEERR = -3 +_ECALLBACK = -4 +_EINPROGRESS = 1 + +CONVERGED = 'converged' +SIGNERR = 'sign error' +CONVERR = 'convergence error' +VALUEERR = 'value error' +INPROGRESS = 'No error' + + +flag_map = {_ECONVERGED: CONVERGED, _ESIGNERR: SIGNERR, _ECONVERR: CONVERR, + _EVALUEERR: VALUEERR, _EINPROGRESS: INPROGRESS} + + +class RootResults(OptimizeResult): + """Represents the root finding result. + + Attributes + ---------- + root : float + Estimated root location. + iterations : int + Number of iterations needed to find the root. + function_calls : int + Number of times the function was called. + converged : bool + True if the routine converged. + flag : str + Description of the cause of termination. + method : str + Root finding method used. + + """ + + def __init__(self, root, iterations, function_calls, flag, method): + self.root = root + self.iterations = iterations + self.function_calls = function_calls + self.converged = flag == _ECONVERGED + if flag in flag_map: + self.flag = flag_map[flag] + else: + self.flag = flag + self.method = method + + +def results_c(full_output, r, method): + if full_output: + x, funcalls, iterations, flag = r + results = RootResults(root=x, + iterations=iterations, + function_calls=funcalls, + flag=flag, method=method) + return x, results + else: + return r + + +def _results_select(full_output, r, method): + """Select from a tuple of (root, funccalls, iterations, flag)""" + x, funcalls, iterations, flag = r + if full_output: + results = RootResults(root=x, + iterations=iterations, + function_calls=funcalls, + flag=flag, method=method) + return x, results + return x + + +def _wrap_nan_raise(f): + + def f_raise(x, *args): + fx = f(x, *args) + f_raise._function_calls += 1 + if np.isnan(fx): + msg = (f'The function value at x={x} is NaN; ' + 'solver cannot continue.') + err = ValueError(msg) + err._x = x + err._function_calls = f_raise._function_calls + raise err + return fx + + f_raise._function_calls = 0 + return f_raise + + +def newton(func, x0, fprime=None, args=(), tol=1.48e-8, maxiter=50, + fprime2=None, x1=None, rtol=0.0, + full_output=False, disp=True): + """ + Find a root of a real or complex function using the Newton-Raphson + (or secant or Halley's) method. + + Find a root of the scalar-valued function `func` given a nearby scalar + starting point `x0`. + The Newton-Raphson method is used if the derivative `fprime` of `func` + is provided, otherwise the secant method is used. If the second order + derivative `fprime2` of `func` is also provided, then Halley's method is + used. + + If `x0` is a sequence with more than one item, `newton` returns an array: + the roots of the function from each (scalar) starting point in `x0`. + In this case, `func` must be vectorized to return a sequence or array of + the same shape as its first argument. If `fprime` (`fprime2`) is given, + then its return must also have the same shape: each element is the first + (second) derivative of `func` with respect to its only variable evaluated + at each element of its first argument. + + `newton` is for finding roots of a scalar-valued functions of a single + variable. For problems involving several variables, see `root`. + + Parameters + ---------- + func : callable + The function whose root is wanted. It must be a function of a + single variable of the form ``f(x,a,b,c...)``, where ``a,b,c...`` + are extra arguments that can be passed in the `args` parameter. + x0 : float, sequence, or ndarray + An initial estimate of the root that should be somewhere near the + actual root. If not scalar, then `func` must be vectorized and return + a sequence or array of the same shape as its first argument. + fprime : callable, optional + The derivative of the function when available and convenient. If it + is None (default), then the secant method is used. + args : tuple, optional + Extra arguments to be used in the function call. + tol : float, optional + The allowable error of the root's value. If `func` is complex-valued, + a larger `tol` is recommended as both the real and imaginary parts + of `x` contribute to ``|x - x0|``. + maxiter : int, optional + Maximum number of iterations. + fprime2 : callable, optional + The second order derivative of the function when available and + convenient. If it is None (default), then the normal Newton-Raphson + or the secant method is used. If it is not None, then Halley's method + is used. + x1 : float, optional + Another estimate of the root that should be somewhere near the + actual root. Used if `fprime` is not provided. + rtol : float, optional + Tolerance (relative) for termination. + full_output : bool, optional + If `full_output` is False (default), the root is returned. + If True and `x0` is scalar, the return value is ``(x, r)``, where ``x`` + is the root and ``r`` is a `RootResults` object. + If True and `x0` is non-scalar, the return value is ``(x, converged, + zero_der)`` (see Returns section for details). + disp : bool, optional + If True, raise a RuntimeError if the algorithm didn't converge, with + the error message containing the number of iterations and current + function value. Otherwise, the convergence status is recorded in a + `RootResults` return object. + Ignored if `x0` is not scalar. + *Note: this has little to do with displaying, however, + the `disp` keyword cannot be renamed for backwards compatibility.* + + Returns + ------- + root : float, sequence, or ndarray + Estimated location where function is zero. + r : `RootResults`, optional + Present if ``full_output=True`` and `x0` is scalar. + Object containing information about the convergence. In particular, + ``r.converged`` is True if the routine converged. + converged : ndarray of bool, optional + Present if ``full_output=True`` and `x0` is non-scalar. + For vector functions, indicates which elements converged successfully. + zero_der : ndarray of bool, optional + Present if ``full_output=True`` and `x0` is non-scalar. + For vector functions, indicates which elements had a zero derivative. + + See Also + -------- + root_scalar : interface to root solvers for scalar functions + root : interface to root solvers for multi-input, multi-output functions + + Notes + ----- + The convergence rate of the Newton-Raphson method is quadratic, + the Halley method is cubic, and the secant method is + sub-quadratic. This means that if the function is well-behaved + the actual error in the estimated root after the nth iteration + is approximately the square (cube for Halley) of the error + after the (n-1)th step. However, the stopping criterion used + here is the step size and there is no guarantee that a root + has been found. Consequently, the result should be verified. + Safer algorithms are brentq, brenth, ridder, and bisect, + but they all require that the root first be bracketed in an + interval where the function changes sign. The brentq algorithm + is recommended for general use in one dimensional problems + when such an interval has been found. + + When `newton` is used with arrays, it is best suited for the following + types of problems: + + * The initial guesses, `x0`, are all relatively the same distance from + the roots. + * Some or all of the extra arguments, `args`, are also arrays so that a + class of similar problems can be solved together. + * The size of the initial guesses, `x0`, is larger than O(100) elements. + Otherwise, a naive loop may perform as well or better than a vector. + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy import optimize + + >>> def f(x): + ... return (x**3 - 1) # only one real root at x = 1 + + ``fprime`` is not provided, use the secant method: + + >>> root = optimize.newton(f, 1.5) + >>> root + 1.0000000000000016 + >>> root = optimize.newton(f, 1.5, fprime2=lambda x: 6 * x) + >>> root + 1.0000000000000016 + + Only ``fprime`` is provided, use the Newton-Raphson method: + + >>> root = optimize.newton(f, 1.5, fprime=lambda x: 3 * x**2) + >>> root + 1.0 + + Both ``fprime2`` and ``fprime`` are provided, use Halley's method: + + >>> root = optimize.newton(f, 1.5, fprime=lambda x: 3 * x**2, + ... fprime2=lambda x: 6 * x) + >>> root + 1.0 + + When we want to find roots for a set of related starting values and/or + function parameters, we can provide both of those as an array of inputs: + + >>> f = lambda x, a: x**3 - a + >>> fder = lambda x, a: 3 * x**2 + >>> rng = np.random.default_rng() + >>> x = rng.standard_normal(100) + >>> a = np.arange(-50, 50) + >>> vec_res = optimize.newton(f, x, fprime=fder, args=(a, ), maxiter=200) + + The above is the equivalent of solving for each value in ``(x, a)`` + separately in a for-loop, just faster: + + >>> loop_res = [optimize.newton(f, x0, fprime=fder, args=(a0,), + ... maxiter=200) + ... for x0, a0 in zip(x, a)] + >>> np.allclose(vec_res, loop_res) + True + + Plot the results found for all values of ``a``: + + >>> analytical_result = np.sign(a) * np.abs(a)**(1/3) + >>> fig, ax = plt.subplots() + >>> ax.plot(a, analytical_result, 'o') + >>> ax.plot(a, vec_res, '.') + >>> ax.set_xlabel('$a$') + >>> ax.set_ylabel('$x$ where $f(x, a)=0$') + >>> plt.show() + + """ + if tol <= 0: + raise ValueError(f"tol too small ({tol:g} <= 0)") + maxiter = operator.index(maxiter) + if maxiter < 1: + raise ValueError("maxiter must be greater than 0") + if np.size(x0) > 1: + return _array_newton(func, x0, fprime, args, tol, maxiter, fprime2, + full_output) + + # Convert to float (don't use float(x0); this works also for complex x0) + # Use np.asarray because we want x0 to be a numpy object, not a Python + # object. e.g. np.complex(1+1j) > 0 is possible, but (1 + 1j) > 0 raises + # a TypeError + x0 = np.asarray(x0)[()] * 1.0 + p0 = x0 + funcalls = 0 + if fprime is not None: + # Newton-Raphson method + method = "newton" + for itr in range(maxiter): + # first evaluate fval + fval = func(p0, *args) + funcalls += 1 + # If fval is 0, a root has been found, then terminate + if fval == 0: + return _results_select( + full_output, (p0, funcalls, itr, _ECONVERGED), method) + fder = fprime(p0, *args) + funcalls += 1 + if fder == 0: + msg = "Derivative was zero." + if disp: + msg += ( + " Failed to converge after %d iterations, value is %s." + % (itr + 1, p0)) + raise RuntimeError(msg) + warnings.warn(msg, RuntimeWarning, stacklevel=2) + return _results_select( + full_output, (p0, funcalls, itr + 1, _ECONVERR), method) + newton_step = fval / fder + if fprime2: + fder2 = fprime2(p0, *args) + funcalls += 1 + method = "halley" + # Halley's method: + # newton_step /= (1.0 - 0.5 * newton_step * fder2 / fder) + # Only do it if denominator stays close enough to 1 + # Rationale: If 1-adj < 0, then Halley sends x in the + # opposite direction to Newton. Doesn't happen if x is close + # enough to root. + adj = newton_step * fder2 / fder / 2 + if np.abs(adj) < 1: + newton_step /= 1.0 - adj + p = p0 - newton_step + if np.isclose(p, p0, rtol=rtol, atol=tol): + return _results_select( + full_output, (p, funcalls, itr + 1, _ECONVERGED), method) + p0 = p + else: + # Secant method + method = "secant" + if x1 is not None: + if x1 == x0: + raise ValueError("x1 and x0 must be different") + p1 = x1 + else: + eps = 1e-4 + p1 = x0 * (1 + eps) + p1 += (eps if p1 >= 0 else -eps) + q0 = func(p0, *args) + funcalls += 1 + q1 = func(p1, *args) + funcalls += 1 + if abs(q1) < abs(q0): + p0, p1, q0, q1 = p1, p0, q1, q0 + for itr in range(maxiter): + if q1 == q0: + if p1 != p0: + msg = f"Tolerance of {p1 - p0} reached." + if disp: + msg += ( + " Failed to converge after %d iterations, value is %s." + % (itr + 1, p1)) + raise RuntimeError(msg) + warnings.warn(msg, RuntimeWarning, stacklevel=2) + p = (p1 + p0) / 2.0 + return _results_select( + full_output, (p, funcalls, itr + 1, _ECONVERR), method) + else: + if abs(q1) > abs(q0): + p = (-q0 / q1 * p1 + p0) / (1 - q0 / q1) + else: + p = (-q1 / q0 * p0 + p1) / (1 - q1 / q0) + if np.isclose(p, p1, rtol=rtol, atol=tol): + return _results_select( + full_output, (p, funcalls, itr + 1, _ECONVERGED), method) + p0, q0 = p1, q1 + p1 = p + q1 = func(p1, *args) + funcalls += 1 + + if disp: + msg = ("Failed to converge after %d iterations, value is %s." + % (itr + 1, p)) + raise RuntimeError(msg) + + return _results_select(full_output, (p, funcalls, itr + 1, _ECONVERR), method) + + +def _array_newton(func, x0, fprime, args, tol, maxiter, fprime2, full_output): + """ + A vectorized version of Newton, Halley, and secant methods for arrays. + + Do not use this method directly. This method is called from `newton` + when ``np.size(x0) > 1`` is ``True``. For docstring, see `newton`. + """ + # Explicitly copy `x0` as `p` will be modified inplace, but the + # user's array should not be altered. + p = np.array(x0, copy=True) + + failures = np.ones_like(p, dtype=bool) + nz_der = np.ones_like(failures) + if fprime is not None: + # Newton-Raphson method + for iteration in range(maxiter): + # first evaluate fval + fval = np.asarray(func(p, *args)) + # If all fval are 0, all roots have been found, then terminate + if not fval.any(): + failures = fval.astype(bool) + break + fder = np.asarray(fprime(p, *args)) + nz_der = (fder != 0) + # stop iterating if all derivatives are zero + if not nz_der.any(): + break + # Newton step + dp = fval[nz_der] / fder[nz_der] + if fprime2 is not None: + fder2 = np.asarray(fprime2(p, *args)) + dp = dp / (1.0 - 0.5 * dp * fder2[nz_der] / fder[nz_der]) + # only update nonzero derivatives + p = np.asarray(p, dtype=np.result_type(p, dp, np.float64)) + p[nz_der] -= dp + failures[nz_der] = np.abs(dp) >= tol # items not yet converged + # stop iterating if there aren't any failures, not incl zero der + if not failures[nz_der].any(): + break + else: + # Secant method + dx = np.finfo(float).eps**0.33 + p1 = p * (1 + dx) + np.where(p >= 0, dx, -dx) + q0 = np.asarray(func(p, *args)) + q1 = np.asarray(func(p1, *args)) + active = np.ones_like(p, dtype=bool) + for iteration in range(maxiter): + nz_der = (q1 != q0) + # stop iterating if all derivatives are zero + if not nz_der.any(): + p = (p1 + p) / 2.0 + break + # Secant Step + dp = (q1 * (p1 - p))[nz_der] / (q1 - q0)[nz_der] + # only update nonzero derivatives + p = np.asarray(p, dtype=np.result_type(p, p1, dp, np.float64)) + p[nz_der] = p1[nz_der] - dp + active_zero_der = ~nz_der & active + p[active_zero_der] = (p1 + p)[active_zero_der] / 2.0 + active &= nz_der # don't assign zero derivatives again + failures[nz_der] = np.abs(dp) >= tol # not yet converged + # stop iterating if there aren't any failures, not incl zero der + if not failures[nz_der].any(): + break + p1, p = p, p1 + q0 = q1 + q1 = np.asarray(func(p1, *args)) + + zero_der = ~nz_der & failures # don't include converged with zero-ders + if zero_der.any(): + # Secant warnings + if fprime is None: + nonzero_dp = (p1 != p) + # non-zero dp, but infinite newton step + zero_der_nz_dp = (zero_der & nonzero_dp) + if zero_der_nz_dp.any(): + rms = np.sqrt( + sum((p1[zero_der_nz_dp] - p[zero_der_nz_dp]) ** 2) + ) + warnings.warn(f'RMS of {rms:g} reached', RuntimeWarning, stacklevel=3) + # Newton or Halley warnings + else: + all_or_some = 'all' if zero_der.all() else 'some' + msg = f'{all_or_some:s} derivatives were zero' + warnings.warn(msg, RuntimeWarning, stacklevel=3) + elif failures.any(): + all_or_some = 'all' if failures.all() else 'some' + msg = f'{all_or_some:s} failed to converge after {maxiter:d} iterations' + if failures.all(): + raise RuntimeError(msg) + warnings.warn(msg, RuntimeWarning, stacklevel=3) + + if full_output: + result = namedtuple('result', ('root', 'converged', 'zero_der')) + p = result(p, ~failures, zero_der) + + return p + + +def bisect(f, a, b, args=(), + xtol=_xtol, rtol=_rtol, maxiter=_iter, + full_output=False, disp=True): + """ + Find root of a function within an interval using bisection. + + Basic bisection routine to find a root of the function `f` between the + arguments `a` and `b`. `f(a)` and `f(b)` cannot have the same signs. + Slow but sure. + + Parameters + ---------- + f : function + Python function returning a number. `f` must be continuous, and + f(a) and f(b) must have opposite signs. + a : scalar + One end of the bracketing interval [a,b]. + b : scalar + The other end of the bracketing interval [a,b]. + xtol : number, optional + The computed root ``x0`` will satisfy ``np.allclose(x, x0, + atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The + parameter must be positive. + rtol : number, optional + The computed root ``x0`` will satisfy ``np.allclose(x, x0, + atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The + parameter cannot be smaller than its default value of + ``4*np.finfo(float).eps``. + maxiter : int, optional + If convergence is not achieved in `maxiter` iterations, an error is + raised. Must be >= 0. + args : tuple, optional + Containing extra arguments for the function `f`. + `f` is called by ``apply(f, (x)+args)``. + full_output : bool, optional + If `full_output` is False, the root is returned. If `full_output` is + True, the return value is ``(x, r)``, where x is the root, and r is + a `RootResults` object. + disp : bool, optional + If True, raise RuntimeError if the algorithm didn't converge. + Otherwise, the convergence status is recorded in a `RootResults` + return object. + + Returns + ------- + root : float + Root of `f` between `a` and `b`. + r : `RootResults` (present if ``full_output = True``) + Object containing information about the convergence. In particular, + ``r.converged`` is True if the routine converged. + + Examples + -------- + + >>> def f(x): + ... return (x**2 - 1) + + >>> from scipy import optimize + + >>> root = optimize.bisect(f, 0, 2) + >>> root + 1.0 + + >>> root = optimize.bisect(f, -2, 0) + >>> root + -1.0 + + See Also + -------- + brentq, brenth, bisect, newton + fixed_point : scalar fixed-point finder + fsolve : n-dimensional root-finding + + """ + if not isinstance(args, tuple): + args = (args,) + maxiter = operator.index(maxiter) + if xtol <= 0: + raise ValueError(f"xtol too small ({xtol:g} <= 0)") + if rtol < _rtol: + raise ValueError(f"rtol too small ({rtol:g} < {_rtol:g})") + f = _wrap_nan_raise(f) + r = _zeros._bisect(f, a, b, xtol, rtol, maxiter, args, full_output, disp) + return results_c(full_output, r, "bisect") + + +def ridder(f, a, b, args=(), + xtol=_xtol, rtol=_rtol, maxiter=_iter, + full_output=False, disp=True): + """ + Find a root of a function in an interval using Ridder's method. + + Parameters + ---------- + f : function + Python function returning a number. f must be continuous, and f(a) and + f(b) must have opposite signs. + a : scalar + One end of the bracketing interval [a,b]. + b : scalar + The other end of the bracketing interval [a,b]. + xtol : number, optional + The computed root ``x0`` will satisfy ``np.allclose(x, x0, + atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The + parameter must be positive. + rtol : number, optional + The computed root ``x0`` will satisfy ``np.allclose(x, x0, + atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The + parameter cannot be smaller than its default value of + ``4*np.finfo(float).eps``. + maxiter : int, optional + If convergence is not achieved in `maxiter` iterations, an error is + raised. Must be >= 0. + args : tuple, optional + Containing extra arguments for the function `f`. + `f` is called by ``apply(f, (x)+args)``. + full_output : bool, optional + If `full_output` is False, the root is returned. If `full_output` is + True, the return value is ``(x, r)``, where `x` is the root, and `r` is + a `RootResults` object. + disp : bool, optional + If True, raise RuntimeError if the algorithm didn't converge. + Otherwise, the convergence status is recorded in any `RootResults` + return object. + + Returns + ------- + root : float + Root of `f` between `a` and `b`. + r : `RootResults` (present if ``full_output = True``) + Object containing information about the convergence. + In particular, ``r.converged`` is True if the routine converged. + + See Also + -------- + brentq, brenth, bisect, newton : 1-D root-finding + fixed_point : scalar fixed-point finder + + Notes + ----- + Uses [Ridders1979]_ method to find a root of the function `f` between the + arguments `a` and `b`. Ridders' method is faster than bisection, but not + generally as fast as the Brent routines. [Ridders1979]_ provides the + classic description and source of the algorithm. A description can also be + found in any recent edition of Numerical Recipes. + + The routine used here diverges slightly from standard presentations in + order to be a bit more careful of tolerance. + + References + ---------- + .. [Ridders1979] + Ridders, C. F. J. "A New Algorithm for Computing a + Single Root of a Real Continuous Function." + IEEE Trans. Circuits Systems 26, 979-980, 1979. + + Examples + -------- + + >>> def f(x): + ... return (x**2 - 1) + + >>> from scipy import optimize + + >>> root = optimize.ridder(f, 0, 2) + >>> root + 1.0 + + >>> root = optimize.ridder(f, -2, 0) + >>> root + -1.0 + """ + if not isinstance(args, tuple): + args = (args,) + maxiter = operator.index(maxiter) + if xtol <= 0: + raise ValueError(f"xtol too small ({xtol:g} <= 0)") + if rtol < _rtol: + raise ValueError(f"rtol too small ({rtol:g} < {_rtol:g})") + f = _wrap_nan_raise(f) + r = _zeros._ridder(f, a, b, xtol, rtol, maxiter, args, full_output, disp) + return results_c(full_output, r, "ridder") + + +def brentq(f, a, b, args=(), + xtol=_xtol, rtol=_rtol, maxiter=_iter, + full_output=False, disp=True): + """ + Find a root of a function in a bracketing interval using Brent's method. + + Uses the classic Brent's method to find a root of the function `f` on + the sign changing interval [a , b]. Generally considered the best of the + rootfinding routines here. It is a safe version of the secant method that + uses inverse quadratic extrapolation. Brent's method combines root + bracketing, interval bisection, and inverse quadratic interpolation. It is + sometimes known as the van Wijngaarden-Dekker-Brent method. Brent (1973) + claims convergence is guaranteed for functions computable within [a,b]. + + [Brent1973]_ provides the classic description of the algorithm. Another + description can be found in a recent edition of Numerical Recipes, including + [PressEtal1992]_. A third description is at + http://mathworld.wolfram.com/BrentsMethod.html. It should be easy to + understand the algorithm just by reading our code. Our code diverges a bit + from standard presentations: we choose a different formula for the + extrapolation step. + + Parameters + ---------- + f : function + Python function returning a number. The function :math:`f` + must be continuous, and :math:`f(a)` and :math:`f(b)` must + have opposite signs. + a : scalar + One end of the bracketing interval :math:`[a, b]`. + b : scalar + The other end of the bracketing interval :math:`[a, b]`. + xtol : number, optional + The computed root ``x0`` will satisfy ``np.allclose(x, x0, + atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The + parameter must be positive. For nice functions, Brent's + method will often satisfy the above condition with ``xtol/2`` + and ``rtol/2``. [Brent1973]_ + rtol : number, optional + The computed root ``x0`` will satisfy ``np.allclose(x, x0, + atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The + parameter cannot be smaller than its default value of + ``4*np.finfo(float).eps``. For nice functions, Brent's + method will often satisfy the above condition with ``xtol/2`` + and ``rtol/2``. [Brent1973]_ + maxiter : int, optional + If convergence is not achieved in `maxiter` iterations, an error is + raised. Must be >= 0. + args : tuple, optional + Containing extra arguments for the function `f`. + `f` is called by ``apply(f, (x)+args)``. + full_output : bool, optional + If `full_output` is False, the root is returned. If `full_output` is + True, the return value is ``(x, r)``, where `x` is the root, and `r` is + a `RootResults` object. + disp : bool, optional + If True, raise RuntimeError if the algorithm didn't converge. + Otherwise, the convergence status is recorded in any `RootResults` + return object. + + Returns + ------- + root : float + Root of `f` between `a` and `b`. + r : `RootResults` (present if ``full_output = True``) + Object containing information about the convergence. In particular, + ``r.converged`` is True if the routine converged. + + See Also + -------- + fmin, fmin_powell, fmin_cg, fmin_bfgs, fmin_ncg : multivariate local optimizers + leastsq : nonlinear least squares minimizer + fmin_l_bfgs_b, fmin_tnc, fmin_cobyla : constrained multivariate optimizers + basinhopping, differential_evolution, brute : global optimizers + fminbound, brent, golden, bracket : local scalar minimizers + fsolve : N-D root-finding + brenth, ridder, bisect, newton : 1-D root-finding + fixed_point : scalar fixed-point finder + + Notes + ----- + `f` must be continuous. f(a) and f(b) must have opposite signs. + + References + ---------- + .. [Brent1973] + Brent, R. P., + *Algorithms for Minimization Without Derivatives*. + Englewood Cliffs, NJ: Prentice-Hall, 1973. Ch. 3-4. + + .. [PressEtal1992] + Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. + *Numerical Recipes in FORTRAN: The Art of Scientific Computing*, 2nd ed. + Cambridge, England: Cambridge University Press, pp. 352-355, 1992. + Section 9.3: "Van Wijngaarden-Dekker-Brent Method." + + Examples + -------- + >>> def f(x): + ... return (x**2 - 1) + + >>> from scipy import optimize + + >>> root = optimize.brentq(f, -2, 0) + >>> root + -1.0 + + >>> root = optimize.brentq(f, 0, 2) + >>> root + 1.0 + """ + if not isinstance(args, tuple): + args = (args,) + maxiter = operator.index(maxiter) + if xtol <= 0: + raise ValueError(f"xtol too small ({xtol:g} <= 0)") + if rtol < _rtol: + raise ValueError(f"rtol too small ({rtol:g} < {_rtol:g})") + f = _wrap_nan_raise(f) + r = _zeros._brentq(f, a, b, xtol, rtol, maxiter, args, full_output, disp) + return results_c(full_output, r, "brentq") + + +def brenth(f, a, b, args=(), + xtol=_xtol, rtol=_rtol, maxiter=_iter, + full_output=False, disp=True): + """Find a root of a function in a bracketing interval using Brent's + method with hyperbolic extrapolation. + + A variation on the classic Brent routine to find a root of the function f + between the arguments a and b that uses hyperbolic extrapolation instead of + inverse quadratic extrapolation. Bus & Dekker (1975) guarantee convergence + for this method, claiming that the upper bound of function evaluations here + is 4 or 5 times that of bisection. + f(a) and f(b) cannot have the same signs. Generally, on a par with the + brent routine, but not as heavily tested. It is a safe version of the + secant method that uses hyperbolic extrapolation. + The version here is by Chuck Harris, and implements Algorithm M of + [BusAndDekker1975]_, where further details (convergence properties, + additional remarks and such) can be found + + Parameters + ---------- + f : function + Python function returning a number. f must be continuous, and f(a) and + f(b) must have opposite signs. + a : scalar + One end of the bracketing interval [a,b]. + b : scalar + The other end of the bracketing interval [a,b]. + xtol : number, optional + The computed root ``x0`` will satisfy ``np.allclose(x, x0, + atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The + parameter must be positive. As with `brentq`, for nice + functions the method will often satisfy the above condition + with ``xtol/2`` and ``rtol/2``. + rtol : number, optional + The computed root ``x0`` will satisfy ``np.allclose(x, x0, + atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The + parameter cannot be smaller than its default value of + ``4*np.finfo(float).eps``. As with `brentq`, for nice functions + the method will often satisfy the above condition with + ``xtol/2`` and ``rtol/2``. + maxiter : int, optional + If convergence is not achieved in `maxiter` iterations, an error is + raised. Must be >= 0. + args : tuple, optional + Containing extra arguments for the function `f`. + `f` is called by ``apply(f, (x)+args)``. + full_output : bool, optional + If `full_output` is False, the root is returned. If `full_output` is + True, the return value is ``(x, r)``, where `x` is the root, and `r` is + a `RootResults` object. + disp : bool, optional + If True, raise RuntimeError if the algorithm didn't converge. + Otherwise, the convergence status is recorded in any `RootResults` + return object. + + Returns + ------- + root : float + Root of `f` between `a` and `b`. + r : `RootResults` (present if ``full_output = True``) + Object containing information about the convergence. In particular, + ``r.converged`` is True if the routine converged. + + See Also + -------- + fmin, fmin_powell, fmin_cg, fmin_bfgs, fmin_ncg : multivariate local optimizers + leastsq : nonlinear least squares minimizer + fmin_l_bfgs_b, fmin_tnc, fmin_cobyla : constrained multivariate optimizers + basinhopping, differential_evolution, brute : global optimizers + fminbound, brent, golden, bracket : local scalar minimizers + fsolve : N-D root-finding + brentq, ridder, bisect, newton : 1-D root-finding + fixed_point : scalar fixed-point finder + + References + ---------- + .. [BusAndDekker1975] + Bus, J. C. P., Dekker, T. J., + "Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero + of a Function", ACM Transactions on Mathematical Software, Vol. 1, Issue + 4, Dec. 1975, pp. 330-345. Section 3: "Algorithm M". + :doi:`10.1145/355656.355659` + + Examples + -------- + >>> def f(x): + ... return (x**2 - 1) + + >>> from scipy import optimize + + >>> root = optimize.brenth(f, -2, 0) + >>> root + -1.0 + + >>> root = optimize.brenth(f, 0, 2) + >>> root + 1.0 + + """ + if not isinstance(args, tuple): + args = (args,) + maxiter = operator.index(maxiter) + if xtol <= 0: + raise ValueError(f"xtol too small ({xtol:g} <= 0)") + if rtol < _rtol: + raise ValueError(f"rtol too small ({rtol:g} < {_rtol:g})") + f = _wrap_nan_raise(f) + r = _zeros._brenth(f, a, b, xtol, rtol, maxiter, args, full_output, disp) + return results_c(full_output, r, "brenth") + + +################################ +# TOMS "Algorithm 748: Enclosing Zeros of Continuous Functions", by +# Alefeld, G. E. and Potra, F. A. and Shi, Yixun, +# See [1] + + +def _notclose(fs, rtol=_rtol, atol=_xtol): + # Ensure not None, not 0, all finite, and not very close to each other + notclosefvals = ( + all(fs) and all(np.isfinite(fs)) and + not any(any(np.isclose(_f, fs[i + 1:], rtol=rtol, atol=atol)) + for i, _f in enumerate(fs[:-1]))) + return notclosefvals + + +def _secant(xvals, fvals): + """Perform a secant step, taking a little care""" + # Secant has many "mathematically" equivalent formulations + # x2 = x0 - (x1 - x0)/(f1 - f0) * f0 + # = x1 - (x1 - x0)/(f1 - f0) * f1 + # = (-x1 * f0 + x0 * f1) / (f1 - f0) + # = (-f0 / f1 * x1 + x0) / (1 - f0 / f1) + # = (-f1 / f0 * x0 + x1) / (1 - f1 / f0) + x0, x1 = xvals[:2] + f0, f1 = fvals[:2] + if f0 == f1: + return np.nan + if np.abs(f1) > np.abs(f0): + x2 = (-f0 / f1 * x1 + x0) / (1 - f0 / f1) + else: + x2 = (-f1 / f0 * x0 + x1) / (1 - f1 / f0) + return x2 + + +def _update_bracket(ab, fab, c, fc): + """Update a bracket given (c, fc), return the discarded endpoints.""" + fa, fb = fab + idx = (0 if np.sign(fa) * np.sign(fc) > 0 else 1) + rx, rfx = ab[idx], fab[idx] + fab[idx] = fc + ab[idx] = c + return rx, rfx + + +def _compute_divided_differences(xvals, fvals, N=None, full=True, + forward=True): + """Return a matrix of divided differences for the xvals, fvals pairs + + DD[i, j] = f[x_{i-j}, ..., x_i] for 0 <= j <= i + + If full is False, just return the main diagonal(or last row): + f[a], f[a, b] and f[a, b, c]. + If forward is False, return f[c], f[b, c], f[a, b, c].""" + if full: + if forward: + xvals = np.asarray(xvals) + else: + xvals = np.array(xvals)[::-1] + M = len(xvals) + N = M if N is None else min(N, M) + DD = np.zeros([M, N]) + DD[:, 0] = fvals[:] + for i in range(1, N): + DD[i:, i] = (np.diff(DD[i - 1:, i - 1]) / + (xvals[i:] - xvals[:M - i])) + return DD + + xvals = np.asarray(xvals) + dd = np.array(fvals) + row = np.array(fvals) + idx2Use = (0 if forward else -1) + dd[0] = fvals[idx2Use] + for i in range(1, len(xvals)): + denom = xvals[i:i + len(row) - 1] - xvals[:len(row) - 1] + row = np.diff(row)[:] / denom + dd[i] = row[idx2Use] + return dd + + +def _interpolated_poly(xvals, fvals, x): + """Compute p(x) for the polynomial passing through the specified locations. + + Use Neville's algorithm to compute p(x) where p is the minimal degree + polynomial passing through the points xvals, fvals""" + xvals = np.asarray(xvals) + N = len(xvals) + Q = np.zeros([N, N]) + D = np.zeros([N, N]) + Q[:, 0] = fvals[:] + D[:, 0] = fvals[:] + for k in range(1, N): + alpha = D[k:, k - 1] - Q[k - 1:N - 1, k - 1] + diffik = xvals[0:N - k] - xvals[k:N] + Q[k:, k] = (xvals[k:] - x) / diffik * alpha + D[k:, k] = (xvals[:N - k] - x) / diffik * alpha + # Expect Q[-1, 1:] to be small relative to Q[-1, 0] as x approaches a root + return np.sum(Q[-1, 1:]) + Q[-1, 0] + + +def _inverse_poly_zero(a, b, c, d, fa, fb, fc, fd): + """Inverse cubic interpolation f-values -> x-values + + Given four points (fa, a), (fb, b), (fc, c), (fd, d) with + fa, fb, fc, fd all distinct, find poly IP(y) through the 4 points + and compute x=IP(0). + """ + return _interpolated_poly([fa, fb, fc, fd], [a, b, c, d], 0) + + +def _newton_quadratic(ab, fab, d, fd, k): + """Apply Newton-Raphson like steps, using divided differences to approximate f' + + ab is a real interval [a, b] containing a root, + fab holds the real values of f(a), f(b) + d is a real number outside [ab, b] + k is the number of steps to apply + """ + a, b = ab + fa, fb = fab + _, B, A = _compute_divided_differences([a, b, d], [fa, fb, fd], + forward=True, full=False) + + # _P is the quadratic polynomial through the 3 points + def _P(x): + # Horner evaluation of fa + B * (x - a) + A * (x - a) * (x - b) + return (A * (x - b) + B) * (x - a) + fa + + if A == 0: + r = a - fa / B + else: + r = (a if np.sign(A) * np.sign(fa) > 0 else b) + # Apply k Newton-Raphson steps to _P(x), starting from x=r + for i in range(k): + r1 = r - _P(r) / (B + A * (2 * r - a - b)) + if not (ab[0] < r1 < ab[1]): + if (ab[0] < r < ab[1]): + return r + r = sum(ab) / 2.0 + break + r = r1 + + return r + + +class TOMS748Solver: + """Solve f(x, *args) == 0 using Algorithm748 of Alefeld, Potro & Shi. + """ + _MU = 0.5 + _K_MIN = 1 + _K_MAX = 100 # A very high value for real usage. Expect 1, 2, maybe 3. + + def __init__(self): + self.f = None + self.args = None + self.function_calls = 0 + self.iterations = 0 + self.k = 2 + # ab=[a,b] is a global interval containing a root + self.ab = [np.nan, np.nan] + # fab is function values at a, b + self.fab = [np.nan, np.nan] + self.d = None + self.fd = None + self.e = None + self.fe = None + self.disp = False + self.xtol = _xtol + self.rtol = _rtol + self.maxiter = _iter + + def configure(self, xtol, rtol, maxiter, disp, k): + self.disp = disp + self.xtol = xtol + self.rtol = rtol + self.maxiter = maxiter + # Silently replace a low value of k with 1 + self.k = max(k, self._K_MIN) + # Noisily replace a high value of k with self._K_MAX + if self.k > self._K_MAX: + msg = "toms748: Overriding k: ->%d" % self._K_MAX + warnings.warn(msg, RuntimeWarning, stacklevel=3) + self.k = self._K_MAX + + def _callf(self, x, error=True): + """Call the user-supplied function, update book-keeping""" + fx = self.f(x, *self.args) + self.function_calls += 1 + if not np.isfinite(fx) and error: + raise ValueError(f"Invalid function value: f({x:f}) -> {fx} ") + return fx + + def get_result(self, x, flag=_ECONVERGED): + r"""Package the result and statistics into a tuple.""" + return (x, self.function_calls, self.iterations, flag) + + def _update_bracket(self, c, fc): + return _update_bracket(self.ab, self.fab, c, fc) + + def start(self, f, a, b, args=()): + r"""Prepare for the iterations.""" + self.function_calls = 0 + self.iterations = 0 + + self.f = f + self.args = args + self.ab[:] = [a, b] + if not np.isfinite(a) or np.imag(a) != 0: + raise ValueError(f"Invalid x value: {a} ") + if not np.isfinite(b) or np.imag(b) != 0: + raise ValueError(f"Invalid x value: {b} ") + + fa = self._callf(a) + if not np.isfinite(fa) or np.imag(fa) != 0: + raise ValueError(f"Invalid function value: f({a:f}) -> {fa} ") + if fa == 0: + return _ECONVERGED, a + fb = self._callf(b) + if not np.isfinite(fb) or np.imag(fb) != 0: + raise ValueError(f"Invalid function value: f({b:f}) -> {fb} ") + if fb == 0: + return _ECONVERGED, b + + if np.sign(fb) * np.sign(fa) > 0: + raise ValueError("f(a) and f(b) must have different signs, but " + f"f({a:e})={fa:e}, f({b:e})={fb:e} ") + self.fab[:] = [fa, fb] + + return _EINPROGRESS, sum(self.ab) / 2.0 + + def get_status(self): + """Determine the current status.""" + a, b = self.ab[:2] + if np.isclose(a, b, rtol=self.rtol, atol=self.xtol): + return _ECONVERGED, sum(self.ab) / 2.0 + if self.iterations >= self.maxiter: + return _ECONVERR, sum(self.ab) / 2.0 + return _EINPROGRESS, sum(self.ab) / 2.0 + + def iterate(self): + """Perform one step in the algorithm. + + Implements Algorithm 4.1(k=1) or 4.2(k=2) in [APS1995] + """ + self.iterations += 1 + eps = np.finfo(float).eps + d, fd, e, fe = self.d, self.fd, self.e, self.fe + ab_width = self.ab[1] - self.ab[0] # Need the start width below + c = None + + for nsteps in range(2, self.k+2): + # If the f-values are sufficiently separated, perform an inverse + # polynomial interpolation step. Otherwise, nsteps repeats of + # an approximate Newton-Raphson step. + if _notclose(self.fab + [fd, fe], rtol=0, atol=32*eps): + c0 = _inverse_poly_zero(self.ab[0], self.ab[1], d, e, + self.fab[0], self.fab[1], fd, fe) + if self.ab[0] < c0 < self.ab[1]: + c = c0 + if c is None: + c = _newton_quadratic(self.ab, self.fab, d, fd, nsteps) + + fc = self._callf(c) + if fc == 0: + return _ECONVERGED, c + + # re-bracket + e, fe = d, fd + d, fd = self._update_bracket(c, fc) + + # u is the endpoint with the smallest f-value + uix = (0 if np.abs(self.fab[0]) < np.abs(self.fab[1]) else 1) + u, fu = self.ab[uix], self.fab[uix] + + _, A = _compute_divided_differences(self.ab, self.fab, + forward=(uix == 0), full=False) + c = u - 2 * fu / A + if np.abs(c - u) > 0.5 * (self.ab[1] - self.ab[0]): + c = sum(self.ab) / 2.0 + else: + if np.isclose(c, u, rtol=eps, atol=0): + # c didn't change (much). + # Either because the f-values at the endpoints have vastly + # differing magnitudes, or because the root is very close to + # that endpoint + frs = np.frexp(self.fab)[1] + if frs[uix] < frs[1 - uix] - 50: # Differ by more than 2**50 + c = (31 * self.ab[uix] + self.ab[1 - uix]) / 32 + else: + # Make a bigger adjustment, about the + # size of the requested tolerance. + mm = (1 if uix == 0 else -1) + adj = mm * np.abs(c) * self.rtol + mm * self.xtol + c = u + adj + if not self.ab[0] < c < self.ab[1]: + c = sum(self.ab) / 2.0 + + fc = self._callf(c) + if fc == 0: + return _ECONVERGED, c + + e, fe = d, fd + d, fd = self._update_bracket(c, fc) + + # If the width of the new interval did not decrease enough, bisect + if self.ab[1] - self.ab[0] > self._MU * ab_width: + e, fe = d, fd + z = sum(self.ab) / 2.0 + fz = self._callf(z) + if fz == 0: + return _ECONVERGED, z + d, fd = self._update_bracket(z, fz) + + # Record d and e for next iteration + self.d, self.fd = d, fd + self.e, self.fe = e, fe + + status, xn = self.get_status() + return status, xn + + def solve(self, f, a, b, args=(), + xtol=_xtol, rtol=_rtol, k=2, maxiter=_iter, disp=True): + r"""Solve f(x) = 0 given an interval containing a root.""" + self.configure(xtol=xtol, rtol=rtol, maxiter=maxiter, disp=disp, k=k) + status, xn = self.start(f, a, b, args) + if status == _ECONVERGED: + return self.get_result(xn) + + # The first step only has two x-values. + c = _secant(self.ab, self.fab) + if not self.ab[0] < c < self.ab[1]: + c = sum(self.ab) / 2.0 + fc = self._callf(c) + if fc == 0: + return self.get_result(c) + + self.d, self.fd = self._update_bracket(c, fc) + self.e, self.fe = None, None + self.iterations += 1 + + while True: + status, xn = self.iterate() + if status == _ECONVERGED: + return self.get_result(xn) + if status == _ECONVERR: + fmt = "Failed to converge after %d iterations, bracket is %s" + if disp: + msg = fmt % (self.iterations + 1, self.ab) + raise RuntimeError(msg) + return self.get_result(xn, _ECONVERR) + + +def toms748(f, a, b, args=(), k=1, + xtol=_xtol, rtol=_rtol, maxiter=_iter, + full_output=False, disp=True): + """ + Find a root using TOMS Algorithm 748 method. + + Implements the Algorithm 748 method of Alefeld, Potro and Shi to find a + root of the function `f` on the interval ``[a , b]``, where ``f(a)`` and + `f(b)` must have opposite signs. + + It uses a mixture of inverse cubic interpolation and + "Newton-quadratic" steps. [APS1995]. + + Parameters + ---------- + f : function + Python function returning a scalar. The function :math:`f` + must be continuous, and :math:`f(a)` and :math:`f(b)` + have opposite signs. + a : scalar, + lower boundary of the search interval + b : scalar, + upper boundary of the search interval + args : tuple, optional + containing extra arguments for the function `f`. + `f` is called by ``f(x, *args)``. + k : int, optional + The number of Newton quadratic steps to perform each + iteration. ``k>=1``. + xtol : scalar, optional + The computed root ``x0`` will satisfy ``np.allclose(x, x0, + atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The + parameter must be positive. + rtol : scalar, optional + The computed root ``x0`` will satisfy ``np.allclose(x, x0, + atol=xtol, rtol=rtol)``, where ``x`` is the exact root. + maxiter : int, optional + If convergence is not achieved in `maxiter` iterations, an error is + raised. Must be >= 0. + full_output : bool, optional + If `full_output` is False, the root is returned. If `full_output` is + True, the return value is ``(x, r)``, where `x` is the root, and `r` is + a `RootResults` object. + disp : bool, optional + If True, raise RuntimeError if the algorithm didn't converge. + Otherwise, the convergence status is recorded in the `RootResults` + return object. + + Returns + ------- + root : float + Approximate root of `f` + r : `RootResults` (present if ``full_output = True``) + Object containing information about the convergence. In particular, + ``r.converged`` is True if the routine converged. + + See Also + -------- + brentq, brenth, ridder, bisect, newton + fsolve : find roots in N dimensions. + + Notes + ----- + `f` must be continuous. + Algorithm 748 with ``k=2`` is asymptotically the most efficient + algorithm known for finding roots of a four times continuously + differentiable function. + In contrast with Brent's algorithm, which may only decrease the length of + the enclosing bracket on the last step, Algorithm 748 decreases it each + iteration with the same asymptotic efficiency as it finds the root. + + For easy statement of efficiency indices, assume that `f` has 4 + continuous deriviatives. + For ``k=1``, the convergence order is at least 2.7, and with about + asymptotically 2 function evaluations per iteration, the efficiency + index is approximately 1.65. + For ``k=2``, the order is about 4.6 with asymptotically 3 function + evaluations per iteration, and the efficiency index 1.66. + For higher values of `k`, the efficiency index approaches + the kth root of ``(3k-2)``, hence ``k=1`` or ``k=2`` are + usually appropriate. + + References + ---------- + .. [APS1995] + Alefeld, G. E. and Potra, F. A. and Shi, Yixun, + *Algorithm 748: Enclosing Zeros of Continuous Functions*, + ACM Trans. Math. Softw. Volume 221(1995) + doi = {10.1145/210089.210111} + + Examples + -------- + >>> def f(x): + ... return (x**3 - 1) # only one real root at x = 1 + + >>> from scipy import optimize + >>> root, results = optimize.toms748(f, 0, 2, full_output=True) + >>> root + 1.0 + >>> results + converged: True + flag: converged + function_calls: 11 + iterations: 5 + root: 1.0 + method: toms748 + """ + if xtol <= 0: + raise ValueError(f"xtol too small ({xtol:g} <= 0)") + if rtol < _rtol / 4: + raise ValueError(f"rtol too small ({rtol:g} < {_rtol/4:g})") + maxiter = operator.index(maxiter) + if maxiter < 1: + raise ValueError("maxiter must be greater than 0") + if not np.isfinite(a): + raise ValueError(f"a is not finite {a}") + if not np.isfinite(b): + raise ValueError(f"b is not finite {b}") + if a >= b: + raise ValueError(f"a and b are not an interval [{a}, {b}]") + if not k >= 1: + raise ValueError(f"k too small ({k} < 1)") + + if not isinstance(args, tuple): + args = (args,) + f = _wrap_nan_raise(f) + solver = TOMS748Solver() + result = solver.solve(f, a, b, args=args, k=k, xtol=xtol, rtol=rtol, + maxiter=maxiter, disp=disp) + x, function_calls, iterations, flag = result + return _results_select(full_output, (x, function_calls, iterations, flag), + "toms748") diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cobyla.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cobyla.py new file mode 100644 index 0000000000000000000000000000000000000000..87d111d8fc1634e54d3766a3f1c58abd37ac58cb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cobyla.py @@ -0,0 +1,19 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'OptimizeResult', + 'fmin_cobyla', +] + +def __dir__(): + return __all__ + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="cobyla", + private_modules=["_cobyla_py"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize.pxd b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize.pxd new file mode 100644 index 0000000000000000000000000000000000000000..d35f8da68b34d3a587f3a99326770d8550a2135c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize.pxd @@ -0,0 +1,11 @@ +# Public Cython API declarations +# +# See doc/source/dev/contributor/public_cython_api.rst for guidelines + + +# The following cimport statement provides legacy ABI +# support. Changing it causes an ABI forward-compatibility break +# (gh-11793), so we currently leave it as is (no further cimport +# statements should be used in this file). +from scipy.optimize.cython_optimize._zeros cimport ( + brentq, brenth, ridder, bisect, zeros_full_output) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..a07250bbeb06542721480c42005307992558fced --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize/__init__.py @@ -0,0 +1,133 @@ +""" +Cython optimize root finding API +================================ +The underlying C functions for the following root finders can be accessed +directly using Cython: + +- `~scipy.optimize.bisect` +- `~scipy.optimize.ridder` +- `~scipy.optimize.brenth` +- `~scipy.optimize.brentq` + +The Cython API for the root finding functions is similar except there is no +``disp`` argument. Import the root finding functions using ``cimport`` from +`scipy.optimize.cython_optimize`. :: + + from scipy.optimize.cython_optimize cimport bisect, ridder, brentq, brenth + + +Callback signature +------------------ +The zeros functions in `~scipy.optimize.cython_optimize` expect a callback that +takes a double for the scalar independent variable as the 1st argument and a +user defined ``struct`` with any extra parameters as the 2nd argument. :: + + double (*callback_type)(double, void*) noexcept + + +Examples +-------- +Usage of `~scipy.optimize.cython_optimize` requires Cython to write callbacks +that are compiled into C. For more information on compiling Cython, see the +`Cython Documentation `_. + +These are the basic steps: + +1. Create a Cython ``.pyx`` file, for example: ``myexample.pyx``. +2. Import the desired root finder from `~scipy.optimize.cython_optimize`. +3. Write the callback function, and call the selected root finding function + passing the callback, any extra arguments, and the other solver + parameters. :: + + from scipy.optimize.cython_optimize cimport brentq + + # import math from Cython + from libc cimport math + + myargs = {'C0': 1.0, 'C1': 0.7} # a dictionary of extra arguments + XLO, XHI = 0.5, 1.0 # lower and upper search boundaries + XTOL, RTOL, MITR = 1e-3, 1e-3, 10 # other solver parameters + + # user-defined struct for extra parameters + ctypedef struct test_params: + double C0 + double C1 + + + # user-defined callback + cdef double f(double x, void *args) noexcept: + cdef test_params *myargs = args + return myargs.C0 - math.exp(-(x - myargs.C1)) + + + # Cython wrapper function + cdef double brentq_wrapper_example(dict args, double xa, double xb, + double xtol, double rtol, int mitr): + # Cython automatically casts dictionary to struct + cdef test_params myargs = args + return brentq( + f, xa, xb, &myargs, xtol, rtol, mitr, NULL) + + + # Python function + def brentq_example(args=myargs, xa=XLO, xb=XHI, xtol=XTOL, rtol=RTOL, + mitr=MITR): + '''Calls Cython wrapper from Python.''' + return brentq_wrapper_example(args, xa, xb, xtol, rtol, mitr) + +4. If you want to call your function from Python, create a Cython wrapper, and + a Python function that calls the wrapper, or use ``cpdef``. Then, in Python, + you can import and run the example. :: + + from myexample import brentq_example + + x = brentq_example() + # 0.6999942848231314 + +5. Create a Cython ``.pxd`` file if you need to export any Cython functions. + + +Full output +----------- +The functions in `~scipy.optimize.cython_optimize` can also copy the full +output from the solver to a C ``struct`` that is passed as its last argument. +If you don't want the full output, just pass ``NULL``. The full output +``struct`` must be type ``zeros_full_output``, which is defined in +`scipy.optimize.cython_optimize` with the following fields: + +- ``int funcalls``: number of function calls +- ``int iterations``: number of iterations +- ``int error_num``: error number +- ``double root``: root of function + +The root is copied by `~scipy.optimize.cython_optimize` to the full output +``struct``. An error number of -1 means a sign error, -2 means a convergence +error, and 0 means the solver converged. Continuing from the previous example:: + + from scipy.optimize.cython_optimize cimport zeros_full_output + + + # cython brentq solver with full output + cdef zeros_full_output brentq_full_output_wrapper_example( + dict args, double xa, double xb, double xtol, double rtol, + int mitr): + cdef test_params myargs = args + cdef zeros_full_output my_full_output + # use my_full_output instead of NULL + brentq(f, xa, xb, &myargs, xtol, rtol, mitr, &my_full_output) + return my_full_output + + + # Python function + def brent_full_output_example(args=myargs, xa=XLO, xb=XHI, xtol=XTOL, + rtol=RTOL, mitr=MITR): + '''Returns full output''' + return brentq_full_output_wrapper_example(args, xa, xb, xtol, rtol, + mitr) + + result = brent_full_output_example() + # {'error_num': 0, + # 'funcalls': 6, + # 'iterations': 5, + # 'root': 0.6999942848231314} +""" diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize/_zeros.pxd b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize/_zeros.pxd new file mode 100644 index 0000000000000000000000000000000000000000..d3c9e98f0a24d80d15d1f7052f690d608f66dd80 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize/_zeros.pxd @@ -0,0 +1,33 @@ +# Legacy public Cython API declarations +# +# NOTE: due to the way Cython ABI compatibility works, **no changes +# should be made to this file** --- any API additions/changes should be +# done in `cython_optimize.pxd` (see gh-11793). + +ctypedef double (*callback_type)(double, void*) noexcept + +ctypedef struct zeros_parameters: + callback_type function + void* args + +ctypedef struct zeros_full_output: + int funcalls + int iterations + int error_num + double root + +cdef double bisect(callback_type f, double xa, double xb, void* args, + double xtol, double rtol, int iter, + zeros_full_output *full_output) noexcept nogil + +cdef double ridder(callback_type f, double xa, double xb, void* args, + double xtol, double rtol, int iter, + zeros_full_output *full_output) noexcept nogil + +cdef double brenth(callback_type f, double xa, double xb, void* args, + double xtol, double rtol, int iter, + zeros_full_output *full_output) noexcept nogil + +cdef double brentq(callback_type f, double xa, double xb, void* args, + double xtol, double rtol, int iter, + zeros_full_output *full_output) noexcept nogil diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize/c_zeros.pxd b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize/c_zeros.pxd new file mode 100644 index 0000000000000000000000000000000000000000..0d83c80eb886846ddbbd6927e37e05812911f856 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/cython_optimize/c_zeros.pxd @@ -0,0 +1,26 @@ +cdef extern from "../Zeros/zeros.h": + ctypedef double (*callback_type)(double, void*) noexcept + ctypedef struct scipy_zeros_info: + int funcalls + int iterations + int error_num + +cdef extern from "../Zeros/bisect.c" nogil: + double bisect(callback_type f, double xa, double xb, double xtol, + double rtol, int iter, void *func_data_param, + scipy_zeros_info *solver_stats) + +cdef extern from "../Zeros/ridder.c" nogil: + double ridder(callback_type f, double xa, double xb, double xtol, + double rtol, int iter, void *func_data_param, + scipy_zeros_info *solver_stats) + +cdef extern from "../Zeros/brenth.c" nogil: + double brenth(callback_type f, double xa, double xb, double xtol, + double rtol, int iter, void *func_data_param, + scipy_zeros_info *solver_stats) + +cdef extern from "../Zeros/brentq.c" nogil: + double brentq(callback_type f, double xa, double xb, double xtol, + double rtol, int iter, void *func_data_param, + scipy_zeros_info *solver_stats) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/elementwise.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/elementwise.py new file mode 100644 index 0000000000000000000000000000000000000000..f7be4484626880182a17acf883d72388937578d1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/elementwise.py @@ -0,0 +1,38 @@ +""" +=================================================================== +Elementwise Scalar Optimization (:mod:`scipy.optimize.elementwise`) +=================================================================== + +.. currentmodule:: scipy.optimize.elementwise + +This module provides a collection of functions for root finding and +minimization of scalar, real-valued functions of one variable. Unlike their +counterparts in the base :mod:`scipy.optimize` namespace, these functions work +elementwise, enabling the solution of many related problems in an efficient, +vectorized call. Furthermore, when environment variable ``SCIPY_ARRAY_API=1``, +these functions can accept non-NumPy, array API standard compatible arrays and +perform all calculations using the corresponding array library (e.g. PyTorch, +JAX, CuPy). + +Root finding +============ + +.. autosummary:: + :toctree: generated/ + + find_root + bracket_root + +Minimization +============ + +.. autosummary:: + :toctree: generated/ + + find_minimum + bracket_minimum + +""" +from ._elementwise import find_root, find_minimum, bracket_root, bracket_minimum # noqa: F401, E501 + +__all__ = ["find_root", "find_minimum", "bracket_root", "bracket_minimum"] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/lbfgsb.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/lbfgsb.py new file mode 100644 index 0000000000000000000000000000000000000000..866407cabb3decf0ff72239e6fd372f69f7550c0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/lbfgsb.py @@ -0,0 +1,23 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'LbfgsInvHessProduct', + 'OptimizeResult', + 'fmin_l_bfgs_b', + 'zeros', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="lbfgsb", + private_modules=["_lbfgsb_py"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/linesearch.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/linesearch.py new file mode 100644 index 0000000000000000000000000000000000000000..cb34b25092da34991c868683da3d6a894d1a7f80 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/linesearch.py @@ -0,0 +1,18 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = ["line_search"] # noqa: F822 + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="linesearch", + private_modules=["_linesearch"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/minpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/minpack.py new file mode 100644 index 0000000000000000000000000000000000000000..29fddef537361d8508e6343d23b2c3c7d6d12ec6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/minpack.py @@ -0,0 +1,27 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'OptimizeResult', + 'OptimizeWarning', + 'curve_fit', + 'fixed_point', + 'fsolve', + 'least_squares', + 'leastsq', + 'zeros', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="minpack", + private_modules=["_minpack_py"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/minpack2.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/minpack2.py new file mode 100644 index 0000000000000000000000000000000000000000..cdb3503e0e1e4c886c89bfb62e6a2efc3ba54549 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/minpack2.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="minpack2", + private_modules=["_minpack2"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/moduleTNC.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/moduleTNC.py new file mode 100644 index 0000000000000000000000000000000000000000..3fc5884ed5c39437b7681395419d641443a1fdb8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/moduleTNC.py @@ -0,0 +1,19 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="moduleTNC", + private_modules=["_moduleTNC"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/nonlin.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/nonlin.py new file mode 100644 index 0000000000000000000000000000000000000000..20b490b40ef790a2943d539790b45fc378df2c76 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/nonlin.py @@ -0,0 +1,29 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'BroydenFirst', + 'InverseJacobian', + 'KrylovJacobian', + 'anderson', + 'broyden1', + 'broyden2', + 'diagbroyden', + 'excitingmixing', + 'linearmixing', + 'newton_krylov', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="nonlin", + private_modules=["_nonlin"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/optimize.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/optimize.py new file mode 100644 index 0000000000000000000000000000000000000000..4db770e5f6e921906c916f2650003d92f5507791 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/optimize.py @@ -0,0 +1,40 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'OptimizeResult', + 'OptimizeWarning', + 'approx_fprime', + 'bracket', + 'brent', + 'brute', + 'check_grad', + 'fmin', + 'fmin_bfgs', + 'fmin_cg', + 'fmin_ncg', + 'fmin_powell', + 'fminbound', + 'golden', + 'line_search', + 'rosen', + 'rosen_der', + 'rosen_hess', + 'rosen_hess_prod', + 'show_options', + 'zeros', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="optimize", + private_modules=["_optimize"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/slsqp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/slsqp.py new file mode 100644 index 0000000000000000000000000000000000000000..c2b77d2eb447527cd91e92907e06ad53dd1ad3d8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/slsqp.py @@ -0,0 +1,23 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'OptimizeResult', + 'fmin_slsqp', + 'slsqp', + 'zeros', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="slsqp", + private_modules=["_slsqp_py"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/_cython_examples/extending.pyx b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/_cython_examples/extending.pyx new file mode 100644 index 0000000000000000000000000000000000000000..d831b3c7f5dcaee71371027c7ee95aa9ee51d157 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/_cython_examples/extending.pyx @@ -0,0 +1,43 @@ +#!/usr/bin/env python3 +#cython: language_level=3 +#cython: boundscheck=False +#cython: wraparound=False +""" +Taken from docstring for scipy.optimize.cython_optimize module. +""" + +from scipy.optimize.cython_optimize cimport brentq + +# import math from Cython +from libc cimport math + +myargs = {'C0': 1.0, 'C1': 0.7} # a dictionary of extra arguments +XLO, XHI = 0.5, 1.0 # lower and upper search boundaries +XTOL, RTOL, MITR = 1e-3, 1e-3, 10 # other solver parameters + +# user-defined struct for extra parameters +ctypedef struct test_params: + double C0 + double C1 + + +# user-defined callback +cdef double f(double x, void *args) noexcept: + cdef test_params *myargs = args + return myargs.C0 - math.exp(-(x - myargs.C1)) + + +# Cython wrapper function +cdef double brentq_wrapper_example(dict args, double xa, double xb, + double xtol, double rtol, int mitr): + # Cython automatically casts dictionary to struct + cdef test_params myargs = args + return brentq( + f, xa, xb, &myargs, xtol, rtol, mitr, NULL) + + +# Python function +def brentq_example(args=myargs, xa=XLO, xb=XHI, xtol=XTOL, rtol=RTOL, + mitr=MITR): + '''Calls Cython wrapper from Python.''' + return brentq_wrapper_example(args, xa, xb, xtol, rtol, mitr) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/_cython_examples/meson.build b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/_cython_examples/meson.build new file mode 100644 index 0000000000000000000000000000000000000000..1fb210fbecb84a21518ad8828a789376410f02aa --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/_cython_examples/meson.build @@ -0,0 +1,32 @@ +project('random-build-examples', 'c', 'cpp', 'cython') + +fs = import('fs') + +py3 = import('python').find_installation(pure: false) + +cy = meson.get_compiler('cython') + +if not cy.version().version_compare('>=3.0.8') + error('tests requires Cython >= 3.0.8') +endif + +cython_args = [] +if cy.version().version_compare('>=3.1.0') + cython_args += ['-Xfreethreading_compatible=True'] +endif + +py3.extension_module( + 'extending', + 'extending.pyx', + cython_args: cython_args, + install: false, +) + +extending_cpp = fs.copyfile('extending.pyx', 'extending_cpp.pyx') +py3.extension_module( + 'extending_cpp', + extending_cpp, + cython_args: cython_args, + install: false, + override_options : ['cython_language=cpp'] +) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__basinhopping.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__basinhopping.py new file mode 100644 index 0000000000000000000000000000000000000000..80729460ee6aa596d7dc2c398ce43ff4aacc7bff --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__basinhopping.py @@ -0,0 +1,535 @@ +""" +Unit tests for the basin hopping global minimization algorithm. +""" +import copy + +from numpy.testing import (assert_almost_equal, assert_equal, assert_, + assert_allclose) +import pytest +from pytest import raises as assert_raises +import numpy as np +from numpy import cos, sin + +from scipy.optimize import basinhopping, OptimizeResult +from scipy.optimize._basinhopping import ( + Storage, RandomDisplacement, Metropolis, AdaptiveStepsize) + + +def func1d(x): + f = cos(14.5 * x - 0.3) + (x + 0.2) * x + df = np.array(-14.5 * sin(14.5 * x - 0.3) + 2. * x + 0.2) + return f, df + + +def func2d_nograd(x): + f = cos(14.5 * x[0] - 0.3) + (x[1] + 0.2) * x[1] + (x[0] + 0.2) * x[0] + return f + + +def func2d(x): + f = cos(14.5 * x[0] - 0.3) + (x[1] + 0.2) * x[1] + (x[0] + 0.2) * x[0] + df = np.zeros(2) + df[0] = -14.5 * sin(14.5 * x[0] - 0.3) + 2. * x[0] + 0.2 + df[1] = 2. * x[1] + 0.2 + return f, df + + +def func2d_easyderiv(x): + f = 2.0*x[0]**2 + 2.0*x[0]*x[1] + 2.0*x[1]**2 - 6.0*x[0] + df = np.zeros(2) + df[0] = 4.0*x[0] + 2.0*x[1] - 6.0 + df[1] = 2.0*x[0] + 4.0*x[1] + + return f, df + + +class MyTakeStep1(RandomDisplacement): + """use a copy of displace, but have it set a special parameter to + make sure it's actually being used.""" + def __init__(self): + self.been_called = False + super().__init__() + + def __call__(self, x): + self.been_called = True + return super().__call__(x) + + +def myTakeStep2(x): + """redo RandomDisplacement in function form without the attribute stepsize + to make sure everything still works ok + """ + s = 0.5 + x += np.random.uniform(-s, s, np.shape(x)) + return x + + +class MyAcceptTest: + """pass a custom accept test + + This does nothing but make sure it's being used and ensure all the + possible return values are accepted + """ + def __init__(self): + self.been_called = False + self.ncalls = 0 + self.testres = [False, 'force accept', True, np.bool_(True), + np.bool_(False), [], {}, 0, 1] + + def __call__(self, **kwargs): + self.been_called = True + self.ncalls += 1 + if self.ncalls - 1 < len(self.testres): + return self.testres[self.ncalls - 1] + else: + return True + + +class MyCallBack: + """pass a custom callback function + + This makes sure it's being used. It also returns True after 10 + steps to ensure that it's stopping early. + + """ + def __init__(self): + self.been_called = False + self.ncalls = 0 + + def __call__(self, x, f, accepted): + self.been_called = True + self.ncalls += 1 + if self.ncalls == 10: + return True + + +class TestBasinHopping: + + def setup_method(self): + """ Tests setup. + + Run tests based on the 1-D and 2-D functions described above. + """ + self.x0 = (1.0, [1.0, 1.0]) + self.sol = (-0.195, np.array([-0.195, -0.1])) + + self.tol = 3 # number of decimal places + + self.niter = 100 + self.disp = False + + self.kwargs = {"method": "L-BFGS-B", "jac": True} + self.kwargs_nograd = {"method": "L-BFGS-B"} + + def test_TypeError(self): + # test the TypeErrors are raised on bad input + i = 1 + # if take_step is passed, it must be callable + assert_raises(TypeError, basinhopping, func2d, self.x0[i], + take_step=1) + # if accept_test is passed, it must be callable + assert_raises(TypeError, basinhopping, func2d, self.x0[i], + accept_test=1) + + def test_input_validation(self): + msg = 'target_accept_rate has to be in range \\(0, 1\\)' + with assert_raises(ValueError, match=msg): + basinhopping(func1d, self.x0[0], target_accept_rate=0.) + with assert_raises(ValueError, match=msg): + basinhopping(func1d, self.x0[0], target_accept_rate=1.) + + msg = 'stepwise_factor has to be in range \\(0, 1\\)' + with assert_raises(ValueError, match=msg): + basinhopping(func1d, self.x0[0], stepwise_factor=0.) + with assert_raises(ValueError, match=msg): + basinhopping(func1d, self.x0[0], stepwise_factor=1.) + + def test_1d_grad(self): + # test 1-D minimizations with gradient + i = 0 + res = basinhopping(func1d, self.x0[i], minimizer_kwargs=self.kwargs, + niter=self.niter, disp=self.disp) + assert_almost_equal(res.x, self.sol[i], self.tol) + + def test_2d(self): + # test 2d minimizations with gradient + i = 1 + res = basinhopping(func2d, self.x0[i], minimizer_kwargs=self.kwargs, + niter=self.niter, disp=self.disp) + assert_almost_equal(res.x, self.sol[i], self.tol) + assert_(res.nfev > 0) + + def test_njev(self): + # test njev is returned correctly + i = 1 + minimizer_kwargs = self.kwargs.copy() + # L-BFGS-B doesn't use njev, but BFGS does + minimizer_kwargs["method"] = "BFGS" + res = basinhopping(func2d, self.x0[i], + minimizer_kwargs=minimizer_kwargs, niter=self.niter, + disp=self.disp) + assert_(res.nfev > 0) + assert_equal(res.nfev, res.njev) + + def test_jac(self): + # test Jacobian returned + minimizer_kwargs = self.kwargs.copy() + # BFGS returns a Jacobian + minimizer_kwargs["method"] = "BFGS" + + res = basinhopping(func2d_easyderiv, [0.0, 0.0], + minimizer_kwargs=minimizer_kwargs, niter=self.niter, + disp=self.disp) + + assert_(hasattr(res.lowest_optimization_result, "jac")) + + # in this case, the Jacobian is just [df/dx, df/dy] + _, jacobian = func2d_easyderiv(res.x) + assert_almost_equal(res.lowest_optimization_result.jac, jacobian, + self.tol) + + def test_2d_nograd(self): + # test 2-D minimizations without gradient + i = 1 + res = basinhopping(func2d_nograd, self.x0[i], + minimizer_kwargs=self.kwargs_nograd, + niter=self.niter, disp=self.disp) + assert_almost_equal(res.x, self.sol[i], self.tol) + + @pytest.mark.fail_slow(10) + def test_all_minimizers(self): + # Test 2-D minimizations with gradient. Nelder-Mead, Powell, COBYLA, and + # COBYQA don't accept jac=True, so aren't included here. + i = 1 + methods = ['CG', 'BFGS', 'Newton-CG', 'L-BFGS-B', 'TNC', 'SLSQP'] + minimizer_kwargs = copy.copy(self.kwargs) + for method in methods: + minimizer_kwargs["method"] = method + res = basinhopping(func2d, self.x0[i], + minimizer_kwargs=minimizer_kwargs, + niter=self.niter, disp=self.disp) + assert_almost_equal(res.x, self.sol[i], self.tol) + + @pytest.mark.fail_slow(20) + def test_all_nograd_minimizers(self): + # Test 2-D minimizations without gradient. Newton-CG requires jac=True, + # so not included here. + i = 1 + methods = ['CG', 'BFGS', 'L-BFGS-B', 'TNC', 'SLSQP', + 'Nelder-Mead', 'Powell', 'COBYLA', 'COBYQA'] + minimizer_kwargs = copy.copy(self.kwargs_nograd) + for method in methods: + # COBYQA takes extensive amount of time on this problem + niter = 10 if method == 'COBYQA' else self.niter + minimizer_kwargs["method"] = method + res = basinhopping(func2d_nograd, self.x0[i], + minimizer_kwargs=minimizer_kwargs, + niter=niter, disp=self.disp, seed=1234) + tol = self.tol + if method == 'COBYLA': + tol = 2 + assert_almost_equal(res.x, self.sol[i], decimal=tol) + + def test_pass_takestep(self): + # test that passing a custom takestep works + # also test that the stepsize is being adjusted + takestep = MyTakeStep1() + initial_step_size = takestep.stepsize + i = 1 + res = basinhopping(func2d, self.x0[i], minimizer_kwargs=self.kwargs, + niter=self.niter, disp=self.disp, + take_step=takestep) + assert_almost_equal(res.x, self.sol[i], self.tol) + assert_(takestep.been_called) + # make sure that the build in adaptive step size has been used + assert_(initial_step_size != takestep.stepsize) + + def test_pass_simple_takestep(self): + # test that passing a custom takestep without attribute stepsize + takestep = myTakeStep2 + i = 1 + res = basinhopping(func2d_nograd, self.x0[i], + minimizer_kwargs=self.kwargs_nograd, + niter=self.niter, disp=self.disp, + take_step=takestep) + assert_almost_equal(res.x, self.sol[i], self.tol) + + def test_pass_accept_test(self): + # test passing a custom accept test + # makes sure it's being used and ensures all the possible return values + # are accepted. + accept_test = MyAcceptTest() + i = 1 + # there's no point in running it more than a few steps. + basinhopping(func2d, self.x0[i], minimizer_kwargs=self.kwargs, + niter=10, disp=self.disp, accept_test=accept_test) + assert_(accept_test.been_called) + + def test_pass_callback(self): + # test passing a custom callback function + # This makes sure it's being used. It also returns True after 10 steps + # to ensure that it's stopping early. + callback = MyCallBack() + i = 1 + # there's no point in running it more than a few steps. + res = basinhopping(func2d, self.x0[i], minimizer_kwargs=self.kwargs, + niter=30, disp=self.disp, callback=callback) + assert_(callback.been_called) + assert_("callback" in res.message[0]) + # One of the calls of MyCallBack is during BasinHoppingRunner + # construction, so there are only 9 remaining before MyCallBack stops + # the minimization. + assert_equal(res.nit, 9) + + def test_minimizer_fail(self): + # test if a minimizer fails + i = 1 + self.kwargs["options"] = dict(maxiter=0) + self.niter = 10 + res = basinhopping(func2d, self.x0[i], minimizer_kwargs=self.kwargs, + niter=self.niter, disp=self.disp) + # the number of failed minimizations should be the number of + # iterations + 1 + assert_equal(res.nit + 1, res.minimization_failures) + + def test_niter_zero(self): + # gh5915, what happens if you call basinhopping with niter=0 + i = 0 + basinhopping(func1d, self.x0[i], minimizer_kwargs=self.kwargs, + niter=0, disp=self.disp) + + def test_rng_reproducibility(self): + # rng should ensure reproducibility between runs + minimizer_kwargs = {"method": "L-BFGS-B", "jac": True} + + f_1 = [] + + def callback(x, f, accepted): + f_1.append(f) + + basinhopping(func2d, [1.0, 1.0], minimizer_kwargs=minimizer_kwargs, + niter=10, callback=callback, rng=10) + + f_2 = [] + + def callback2(x, f, accepted): + f_2.append(f) + + basinhopping(func2d, [1.0, 1.0], minimizer_kwargs=minimizer_kwargs, + niter=10, callback=callback2, rng=10) + assert_equal(np.array(f_1), np.array(f_2)) + + def test_random_gen(self): + # check that np.random.Generator can be used (numpy >= 1.17) + rng = np.random.default_rng(1) + + minimizer_kwargs = {"method": "L-BFGS-B", "jac": True} + + res1 = basinhopping(func2d, [1.0, 1.0], + minimizer_kwargs=minimizer_kwargs, + niter=10, rng=rng) + + rng = np.random.default_rng(1) + res2 = basinhopping(func2d, [1.0, 1.0], + minimizer_kwargs=minimizer_kwargs, + niter=10, rng=rng) + assert_equal(res1.x, res2.x) + + def test_monotonic_basin_hopping(self): + # test 1-D minimizations with gradient and T=0 + i = 0 + + res = basinhopping(func1d, self.x0[i], minimizer_kwargs=self.kwargs, + niter=self.niter, disp=self.disp, T=0) + assert_almost_equal(res.x, self.sol[i], self.tol) + + +@pytest.mark.thread_unsafe +class Test_Storage: + def setup_method(self): + self.x0 = np.array(1) + self.f0 = 0 + + minres = OptimizeResult(success=True) + minres.x = self.x0 + minres.fun = self.f0 + + self.storage = Storage(minres) + + def test_higher_f_rejected(self): + new_minres = OptimizeResult(success=True) + new_minres.x = self.x0 + 1 + new_minres.fun = self.f0 + 1 + + ret = self.storage.update(new_minres) + minres = self.storage.get_lowest() + assert_equal(self.x0, minres.x) + assert_equal(self.f0, minres.fun) + assert_(not ret) + + @pytest.mark.parametrize('success', [True, False]) + def test_lower_f_accepted(self, success): + new_minres = OptimizeResult(success=success) + new_minres.x = self.x0 + 1 + new_minres.fun = self.f0 - 1 + + ret = self.storage.update(new_minres) + minres = self.storage.get_lowest() + assert (self.x0 != minres.x) == success # can't use `is` + assert (self.f0 != minres.fun) == success # left side is NumPy bool + assert ret is success + + +class Test_RandomDisplacement: + def setup_method(self): + self.stepsize = 1.0 + self.N = 300000 + + def test_random(self): + # the mean should be 0 + # the variance should be (2*stepsize)**2 / 12 + # note these tests are random, they will fail from time to time + rng = np.random.RandomState(0) + x0 = np.zeros([self.N]) + displace = RandomDisplacement(stepsize=self.stepsize, rng=rng) + x = displace(x0) + v = (2. * self.stepsize) ** 2 / 12 + assert_almost_equal(np.mean(x), 0., 1) + assert_almost_equal(np.var(x), v, 1) + + +class Test_Metropolis: + def setup_method(self): + self.T = 2. + self.met = Metropolis(self.T) + self.res_new = OptimizeResult(success=True, fun=0.) + self.res_old = OptimizeResult(success=True, fun=1.) + + def test_boolean_return(self): + # the return must be a bool, else an error will be raised in + # basinhopping + ret = self.met(res_new=self.res_new, res_old=self.res_old) + assert isinstance(ret, bool) + + def test_lower_f_accepted(self): + assert_(self.met(res_new=self.res_new, res_old=self.res_old)) + + def test_accept(self): + # test that steps are randomly accepted for f_new > f_old + one_accept = False + one_reject = False + for i in range(1000): + if one_accept and one_reject: + break + res_new = OptimizeResult(success=True, fun=1.) + res_old = OptimizeResult(success=True, fun=0.5) + ret = self.met(res_new=res_new, res_old=res_old) + if ret: + one_accept = True + else: + one_reject = True + assert_(one_accept) + assert_(one_reject) + + def test_GH7495(self): + # an overflow in exp was producing a RuntimeWarning + # create own object here in case someone changes self.T + met = Metropolis(2) + res_new = OptimizeResult(success=True, fun=0.) + res_old = OptimizeResult(success=True, fun=2000) + with np.errstate(over='raise'): + met.accept_reject(res_new=res_new, res_old=res_old) + + def test_gh7799(self): + # gh-7799 reported a problem in which local search was successful but + # basinhopping returned an invalid solution. Show that this is fixed. + def func(x): + return (x**2-8)**2+(x+2)**2 + + x0 = -4 + limit = 50 # Constrain to func value >= 50 + con = {'type': 'ineq', 'fun': lambda x: func(x) - limit}, + res = basinhopping( + func, + x0, + 30, + seed=np.random.RandomState(1234), + minimizer_kwargs={'constraints': con} + ) + assert res.success + assert_allclose(res.fun, limit, rtol=1e-6) + + def test_accept_gh7799(self): + # Metropolis should not accept the result of an unsuccessful new local + # search if the old local search was successful + + met = Metropolis(0) # monotonic basin hopping + res_new = OptimizeResult(success=True, fun=0.) + res_old = OptimizeResult(success=True, fun=1.) + + # if new local search was successful and energy is lower, accept + assert met(res_new=res_new, res_old=res_old) + # if new res is unsuccessful, don't accept - even if energy is lower + res_new.success = False + assert not met(res_new=res_new, res_old=res_old) + # ...unless the old res was unsuccessful, too. In that case, why not? + res_old.success = False + assert met(res_new=res_new, res_old=res_old) + + def test_reject_all_gh7799(self): + # Test the behavior when there is no feasible solution + def fun(x): + return x@x + + def constraint(x): + return x + 1 + + kwargs = {'constraints': {'type': 'eq', 'fun': constraint}, + 'bounds': [(0, 1), (0, 1)], 'method': 'slsqp'} + res = basinhopping(fun, x0=[2, 3], niter=10, minimizer_kwargs=kwargs) + assert not res.success + + +class Test_AdaptiveStepsize: + def setup_method(self): + self.stepsize = 1. + self.ts = RandomDisplacement(stepsize=self.stepsize) + self.target_accept_rate = 0.5 + self.takestep = AdaptiveStepsize(takestep=self.ts, verbose=False, + accept_rate=self.target_accept_rate) + + def test_adaptive_increase(self): + # if few steps are rejected, the stepsize should increase + x = 0. + self.takestep(x) + self.takestep.report(False) + for i in range(self.takestep.interval): + self.takestep(x) + self.takestep.report(True) + assert_(self.ts.stepsize > self.stepsize) + + def test_adaptive_decrease(self): + # if few steps are rejected, the stepsize should increase + x = 0. + self.takestep(x) + self.takestep.report(True) + for i in range(self.takestep.interval): + self.takestep(x) + self.takestep.report(False) + assert_(self.ts.stepsize < self.stepsize) + + def test_all_accepted(self): + # test that everything works OK if all steps were accepted + x = 0. + for i in range(self.takestep.interval + 1): + self.takestep(x) + self.takestep.report(True) + assert_(self.ts.stepsize > self.stepsize) + + def test_all_rejected(self): + # test that everything works OK if all steps were rejected + x = 0. + for i in range(self.takestep.interval + 1): + self.takestep(x) + self.takestep.report(False) + assert_(self.ts.stepsize < self.stepsize) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__differential_evolution.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__differential_evolution.py new file mode 100644 index 0000000000000000000000000000000000000000..3f81877f08abe436211210d5bd15d876c9a7177c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__differential_evolution.py @@ -0,0 +1,1703 @@ +""" +Unit tests for the differential global minimization algorithm. +""" +import multiprocessing +from multiprocessing.dummy import Pool as ThreadPool +import platform + +from scipy.optimize._differentialevolution import (DifferentialEvolutionSolver, + _ConstraintWrapper) +from scipy.optimize import differential_evolution, OptimizeResult +from scipy.optimize._constraints import (Bounds, NonlinearConstraint, + LinearConstraint) +from scipy.optimize import rosen, minimize +from scipy.sparse import csr_matrix +from scipy import stats + +import numpy as np +from numpy.testing import (assert_equal, assert_allclose, assert_almost_equal, + assert_string_equal, assert_, suppress_warnings) +from pytest import raises as assert_raises, warns +import pytest + + +class TestDifferentialEvolutionSolver: + + def setup_method(self): + self.old_seterr = np.seterr(invalid='raise') + self.limits = np.array([[0., 0.], + [2., 2.]]) + self.bounds = [(0., 2.), (0., 2.)] + + self.dummy_solver = DifferentialEvolutionSolver(self.quadratic, + [(0, 100)]) + + # dummy_solver2 will be used to test mutation strategies + self.dummy_solver2 = DifferentialEvolutionSolver(self.quadratic, + [(0, 1)], + popsize=7, + mutation=0.5) + # create a population that's only 7 members long + # [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7] + population = np.atleast_2d(np.arange(0.1, 0.8, 0.1)).T + self.dummy_solver2.population = population + + def teardown_method(self): + np.seterr(**self.old_seterr) + + def quadratic(self, x): + return x[0]**2 + + def test__strategy_resolves(self): + # test that the correct mutation function is resolved by + # different requested strategy arguments + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='best1exp') + assert_equal(solver.strategy, 'best1exp') + assert_equal(solver.mutation_func.__name__, '_best1') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='best1bin') + assert_equal(solver.strategy, 'best1bin') + assert_equal(solver.mutation_func.__name__, '_best1') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='rand1bin') + assert_equal(solver.strategy, 'rand1bin') + assert_equal(solver.mutation_func.__name__, '_rand1') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='rand1exp') + assert_equal(solver.strategy, 'rand1exp') + assert_equal(solver.mutation_func.__name__, '_rand1') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='rand2exp') + assert_equal(solver.strategy, 'rand2exp') + assert_equal(solver.mutation_func.__name__, '_rand2') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='best2bin') + assert_equal(solver.strategy, 'best2bin') + assert_equal(solver.mutation_func.__name__, '_best2') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='rand2bin') + assert_equal(solver.strategy, 'rand2bin') + assert_equal(solver.mutation_func.__name__, '_rand2') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='rand2exp') + assert_equal(solver.strategy, 'rand2exp') + assert_equal(solver.mutation_func.__name__, '_rand2') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='randtobest1bin') + assert_equal(solver.strategy, 'randtobest1bin') + assert_equal(solver.mutation_func.__name__, '_randtobest1') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='randtobest1exp') + assert_equal(solver.strategy, 'randtobest1exp') + assert_equal(solver.mutation_func.__name__, '_randtobest1') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='currenttobest1bin') + assert_equal(solver.strategy, 'currenttobest1bin') + assert_equal(solver.mutation_func.__name__, '_currenttobest1') + + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='currenttobest1exp') + assert_equal(solver.strategy, 'currenttobest1exp') + assert_equal(solver.mutation_func.__name__, '_currenttobest1') + + def test__mutate1(self): + # strategies */1/*, i.e. rand/1/bin, best/1/exp, etc. + result = np.array([0.05]) + trial = self.dummy_solver2._best1(np.array([2, 3, 4, 5, 6])) + assert_allclose(trial, result) + + result = np.array([0.25]) + trial = self.dummy_solver2._rand1(np.array([2, 3, 4, 5, 6])) + assert_allclose(trial, result) + + def test__mutate2(self): + # strategies */2/*, i.e. rand/2/bin, best/2/exp, etc. + # [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7] + + result = np.array([-0.1]) + trial = self.dummy_solver2._best2(np.array([2, 3, 4, 5, 6])) + assert_allclose(trial, result) + + result = np.array([0.1]) + trial = self.dummy_solver2._rand2(np.array([2, 3, 4, 5, 6])) + assert_allclose(trial, result) + + def test__randtobest1(self): + # strategies randtobest/1/* + result = np.array([0.15]) + trial = self.dummy_solver2._randtobest1(np.array([2, 3, 4, 5, 6])) + assert_allclose(trial, result) + + def test__currenttobest1(self): + # strategies currenttobest/1/* + result = np.array([0.1]) + trial = self.dummy_solver2._currenttobest1( + 1, + np.array([2, 3, 4, 5, 6]) + ) + assert_allclose(trial, result) + + def test_can_init_with_dithering(self): + mutation = (0.5, 1) + solver = DifferentialEvolutionSolver(self.quadratic, + self.bounds, + mutation=mutation) + + assert_equal(solver.dither, list(mutation)) + + def test_invalid_mutation_values_arent_accepted(self): + func = rosen + mutation = (0.5, 3) + assert_raises(ValueError, + DifferentialEvolutionSolver, + func, + self.bounds, + mutation=mutation) + + mutation = (-1, 1) + assert_raises(ValueError, + DifferentialEvolutionSolver, + func, + self.bounds, + mutation=mutation) + + mutation = (0.1, np.nan) + assert_raises(ValueError, + DifferentialEvolutionSolver, + func, + self.bounds, + mutation=mutation) + + mutation = 0.5 + solver = DifferentialEvolutionSolver(func, + self.bounds, + mutation=mutation) + assert_equal(0.5, solver.scale) + assert_equal(None, solver.dither) + + def test_invalid_functional(self): + def func(x): + return np.array([np.sum(x ** 2), np.sum(x)]) + + with assert_raises( + RuntimeError, + match=r"func\(x, \*args\) must return a scalar value"): + differential_evolution(func, [(-2, 2), (-2, 2)]) + + def test__scale_parameters(self): + trial = np.array([0.3]) + assert_equal(30, self.dummy_solver._scale_parameters(trial)) + + # it should also work with the limits reversed + self.dummy_solver.limits = np.array([[100], [0.]]) + assert_equal(30, self.dummy_solver._scale_parameters(trial)) + + def test__unscale_parameters(self): + trial = np.array([30]) + assert_equal(0.3, self.dummy_solver._unscale_parameters(trial)) + + # it should also work with the limits reversed + self.dummy_solver.limits = np.array([[100], [0.]]) + assert_equal(0.3, self.dummy_solver._unscale_parameters(trial)) + + def test_equal_bounds(self): + with np.errstate(invalid='raise'): + solver = DifferentialEvolutionSolver( + self.quadratic, + bounds=[(2.0, 2.0), (1.0, 3.0)] + ) + v = solver._unscale_parameters([2.0, 2.0]) + assert_allclose(v, 0.5) + + res = differential_evolution(self.quadratic, [(2.0, 2.0), (3.0, 3.0)]) + assert_equal(res.x, [2.0, 3.0]) + + def test__ensure_constraint(self): + trial = np.array([1.1, -100, 0.9, 2., 300., -0.00001]) + self.dummy_solver._ensure_constraint(trial) + + assert_equal(trial[2], 0.9) + assert_(np.logical_and(trial >= 0, trial <= 1).all()) + + def test_differential_evolution(self): + # test that the Jmin of DifferentialEvolutionSolver + # is the same as the function evaluation + solver = DifferentialEvolutionSolver( + self.quadratic, [(-2, 2)], maxiter=1, polish=False + ) + result = solver.solve() + assert_equal(result.fun, self.quadratic(result.x)) + + solver = DifferentialEvolutionSolver( + self.quadratic, [(-2, 2)], maxiter=1, polish=True + ) + result = solver.solve() + assert_equal(result.fun, self.quadratic(result.x)) + + def test_best_solution_retrieval(self): + # test that the getter property method for the best solution works. + solver = DifferentialEvolutionSolver(self.quadratic, [(-2, 2)]) + result = solver.solve() + assert_equal(result.x, solver.x) + + def test_intermediate_result(self): + # Check that intermediate result object passed into the callback + # function contains the expected information and that raising + # `StopIteration` causes the expected behavior. + maxiter = 10 + + def func(x): + val = rosen(x) + if val < func.val: + func.x = x + func.val = val + return val + func.x = None + func.val = np.inf + + def callback(intermediate_result): + callback.nit += 1 + callback.intermediate_result = intermediate_result + assert intermediate_result.population.ndim == 2 + assert intermediate_result.population.shape[1] == 2 + assert intermediate_result.nit == callback.nit + + # Check that `x` and `fun` attributes are the best found so far + assert_equal(intermediate_result.x, callback.func.x) + assert_equal(intermediate_result.fun, callback.func.val) + + # Check for consistency between `fun`, `population_energies`, + # `x`, and `population` + assert_equal(intermediate_result.fun, rosen(intermediate_result.x)) + for i in range(len(intermediate_result.population_energies)): + res = intermediate_result.population_energies[i] + ref = rosen(intermediate_result.population[i]) + assert_equal(res, ref) + assert_equal(intermediate_result.x, + intermediate_result.population[0]) + assert_equal(intermediate_result.fun, + intermediate_result.population_energies[0]) + + assert intermediate_result.message == 'in progress' + assert intermediate_result.success is True + assert isinstance(intermediate_result, OptimizeResult) + if callback.nit == maxiter: + raise StopIteration + callback.nit = 0 + callback.intermediate_result = None + callback.func = func + + bounds = [(0, 2), (0, 2)] + kwargs = dict(func=func, bounds=bounds, rng=838245, polish=False) + res = differential_evolution(**kwargs, callback=callback) + ref = differential_evolution(**kwargs, maxiter=maxiter) + + # Check that final `intermediate_result` is equivalent to returned + # result object and that terminating with callback `StopIteration` + # after `maxiter` iterations is equivalent to terminating with + # `maxiter` parameter. + assert res.success is ref.success is False + assert callback.nit == res.nit == maxiter + assert res.message == 'callback function requested stop early' + assert ref.message == 'Maximum number of iterations has been exceeded.' + for field, val in ref.items(): + if field in {'message', 'success'}: # checked separately + continue + assert_equal(callback.intermediate_result[field], val) + assert_equal(res[field], val) + + # Check that polish occurs after `StopIteration` as advertised + callback.nit = 0 + func.val = np.inf + kwargs['polish'] = True + res = differential_evolution(**kwargs, callback=callback) + assert res.fun < ref.fun + + def test_callback_terminates(self): + # test that if the callback returns true, then the minimization halts + bounds = [(0, 2), (0, 2)] + expected_msg = 'callback function requested stop early' + def callback_python_true(param, convergence=0.): + return True + + result = differential_evolution( + rosen, bounds, callback=callback_python_true + ) + assert_string_equal(result.message, expected_msg) + + # if callback raises StopIteration then solve should be interrupted + def callback_stop(intermediate_result): + raise StopIteration + + result = differential_evolution(rosen, bounds, callback=callback_stop) + assert not result.success + + def callback_evaluates_true(param, convergence=0.): + # DE should stop if bool(self.callback) is True + return [10] + + result = differential_evolution(rosen, bounds, callback=callback_evaluates_true) + assert_string_equal(result.message, expected_msg) + assert not result.success + + def callback_evaluates_false(param, convergence=0.): + return [] + + result = differential_evolution(rosen, bounds, + callback=callback_evaluates_false) + assert result.success + + def test_args_tuple_is_passed(self): + # test that the args tuple is passed to the cost function properly. + bounds = [(-10, 10)] + args = (1., 2., 3.) + + def quadratic(x, *args): + if not isinstance(args, tuple): + raise ValueError('args should be a tuple') + return args[0] + args[1] * x + args[2] * x**2. + + result = differential_evolution(quadratic, + bounds, + args=args, + polish=True) + assert_almost_equal(result.fun, 2 / 3.) + + def test_init_with_invalid_strategy(self): + # test that passing an invalid strategy raises ValueError + func = rosen + bounds = [(-3, 3)] + assert_raises(ValueError, + differential_evolution, + func, + bounds, + strategy='abc') + + def test_bounds_checking(self): + # test that the bounds checking works + func = rosen + bounds = [(-3)] + assert_raises(ValueError, + differential_evolution, + func, + bounds) + bounds = [(-3, 3), (3, 4, 5)] + assert_raises(ValueError, + differential_evolution, + func, + bounds) + + # test that we can use a new-type Bounds object + result = differential_evolution(rosen, Bounds([0, 0], [2, 2])) + assert_almost_equal(result.x, (1., 1.)) + + def test_select_samples(self): + # select_samples should return 5 separate random numbers. + limits = np.arange(12., dtype='float64').reshape(2, 6) + bounds = list(zip(limits[0, :], limits[1, :])) + solver = DifferentialEvolutionSolver(None, bounds, popsize=1) + candidate = 0 + r1, r2, r3, r4, r5 = solver._select_samples(candidate, 5) + assert_equal( + len(np.unique(np.array([candidate, r1, r2, r3, r4, r5]))), 6) + + def test_maxiter_stops_solve(self): + # test that if the maximum number of iterations is exceeded + # the solver stops. + solver = DifferentialEvolutionSolver(rosen, self.bounds, maxiter=1) + result = solver.solve() + assert_equal(result.success, False) + assert_equal(result.message, + 'Maximum number of iterations has been exceeded.') + + def test_maxfun_stops_solve(self): + # test that if the maximum number of function evaluations is exceeded + # during initialisation the solver stops + solver = DifferentialEvolutionSolver(rosen, self.bounds, maxfun=1, + polish=False) + result = solver.solve() + + assert_equal(result.nfev, 2) + assert_equal(result.success, False) + assert_equal(result.message, + 'Maximum number of function evaluations has ' + 'been exceeded.') + + # test that if the maximum number of function evaluations is exceeded + # during the actual minimisation, then the solver stops. + # Have to turn polishing off, as this will still occur even if maxfun + # is reached. For popsize=5 and len(bounds)=2, then there are only 10 + # function evaluations during initialisation. + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + popsize=5, + polish=False, + maxfun=40) + result = solver.solve() + + assert_equal(result.nfev, 41) + assert_equal(result.success, False) + assert_equal(result.message, + 'Maximum number of function evaluations has ' + 'been exceeded.') + + # now repeat for updating='deferred version + # 47 function evaluations is not a multiple of the population size, + # so maxfun is reached partway through a population evaluation. + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + popsize=5, + polish=False, + maxfun=47, + updating='deferred') + result = solver.solve() + + assert_equal(result.nfev, 47) + assert_equal(result.success, False) + assert_equal(result.message, + 'Maximum number of function evaluations has ' + 'been reached.') + + def test_quadratic(self): + # test the quadratic function from object + solver = DifferentialEvolutionSolver(self.quadratic, + [(-100, 100)], + tol=0.02) + solver.solve() + assert_equal(np.argmin(solver.population_energies), 0) + + def test_quadratic_from_diff_ev(self): + # test the quadratic function from differential_evolution function + differential_evolution(self.quadratic, + [(-100, 100)], + tol=0.02, + seed=1) + + def test_rng_gives_repeatability(self): + result = differential_evolution(self.quadratic, + [(-100, 100)], + polish=False, + rng=1, + tol=0.5) + result2 = differential_evolution(self.quadratic, + [(-100, 100)], + polish=False, + rng=1, + tol=0.5) + assert_equal(result.x, result2.x) + assert_equal(result.nfev, result2.nfev) + + def test_random_generator(self): + # check that np.random.Generator can be used (numpy >= 1.17) + # obtain a np.random.Generator object + rng = np.random.default_rng() + + inits = ['random', 'latinhypercube', 'sobol', 'halton'] + for init in inits: + differential_evolution(self.quadratic, + [(-100, 100)], + polish=False, + rng=rng, + tol=0.5, + init=init) + + def test_exp_runs(self): + # test whether exponential mutation loop runs + solver = DifferentialEvolutionSolver(rosen, + self.bounds, + strategy='best1exp', + maxiter=1) + + solver.solve() + + def test_gh_4511_regression(self): + # This modification of the differential evolution docstring example + # uses a custom popsize that had triggered an off-by-one error. + # Because we do not care about solving the optimization problem in + # this test, we use maxiter=1 to reduce the testing time. + bounds = [(-5, 5), (-5, 5)] + # result = differential_evolution(rosen, bounds, popsize=1815, + # maxiter=1) + + # the original issue arose because of rounding error in arange, with + # linspace being a much better solution. 1815 is quite a large popsize + # to use and results in a long test time (~13s). I used the original + # issue to figure out the lowest number of samples that would cause + # this rounding error to occur, 49. + differential_evolution(rosen, bounds, popsize=49, maxiter=1) + + def test_calculate_population_energies(self): + # if popsize is 3, then the overall generation has size (6,) + solver = DifferentialEvolutionSolver(rosen, self.bounds, popsize=3) + solver._calculate_population_energies(solver.population) + solver._promote_lowest_energy() + assert_equal(np.argmin(solver.population_energies), 0) + + # initial calculation of the energies should require 6 nfev. + assert_equal(solver._nfev, 6) + + def test_iteration(self): + # test that DifferentialEvolutionSolver is iterable + # if popsize is 3, then the overall generation has size (6,) + solver = DifferentialEvolutionSolver(rosen, self.bounds, popsize=3, + maxfun=12) + x, fun = next(solver) + assert_equal(np.size(x, 0), 2) + + # 6 nfev are required for initial calculation of energies, 6 nfev are + # required for the evolution of the 6 population members. + assert_equal(solver._nfev, 12) + + # the next generation should halt because it exceeds maxfun + assert_raises(StopIteration, next, solver) + + # check a proper minimisation can be done by an iterable solver + solver = DifferentialEvolutionSolver(rosen, self.bounds) + _, fun_prev = next(solver) + for i, soln in enumerate(solver): + x_current, fun_current = soln + assert fun_prev >= fun_current + _, fun_prev = x_current, fun_current + # need to have this otherwise the solver would never stop. + if i == 50: + break + + def test_convergence(self): + solver = DifferentialEvolutionSolver(rosen, self.bounds, tol=0.2, + polish=False) + solver.solve() + assert_(solver.convergence < 0.2) + + def test_maxiter_none_GH5731(self): + # Pre 0.17 the previous default for maxiter and maxfun was None. + # the numerical defaults are now 1000 and np.inf. However, some scripts + # will still supply None for both of those, this will raise a TypeError + # in the solve method. + solver = DifferentialEvolutionSolver(rosen, self.bounds, maxiter=None, + maxfun=None) + solver.solve() + + def test_population_initiation(self): + # test the different modes of population initiation + + # init must be either 'latinhypercube' or 'random' + # raising ValueError is something else is passed in + assert_raises(ValueError, + DifferentialEvolutionSolver, + *(rosen, self.bounds), + **{'init': 'rubbish'}) + + solver = DifferentialEvolutionSolver(rosen, self.bounds) + + # check that population initiation: + # 1) resets _nfev to 0 + # 2) all population energies are np.inf + solver.init_population_random() + assert_equal(solver._nfev, 0) + assert_(np.all(np.isinf(solver.population_energies))) + + solver.init_population_lhs() + assert_equal(solver._nfev, 0) + assert_(np.all(np.isinf(solver.population_energies))) + + solver.init_population_qmc(qmc_engine='halton') + assert_equal(solver._nfev, 0) + assert_(np.all(np.isinf(solver.population_energies))) + + solver = DifferentialEvolutionSolver(rosen, self.bounds, init='sobol') + solver.init_population_qmc(qmc_engine='sobol') + assert_equal(solver._nfev, 0) + assert_(np.all(np.isinf(solver.population_energies))) + + # we should be able to initialize with our own array + population = np.linspace(-1, 3, 10).reshape(5, 2) + solver = DifferentialEvolutionSolver(rosen, self.bounds, + init=population, + strategy='best2bin', + atol=0.01, rng=1, popsize=5) + + assert_equal(solver._nfev, 0) + assert_(np.all(np.isinf(solver.population_energies))) + assert_(solver.num_population_members == 5) + assert_(solver.population_shape == (5, 2)) + + # check that the population was initialized correctly + unscaled_population = np.clip(solver._unscale_parameters(population), + 0, 1) + assert_almost_equal(solver.population[:5], unscaled_population) + + # population values need to be clipped to bounds + assert_almost_equal(np.min(solver.population[:5]), 0) + assert_almost_equal(np.max(solver.population[:5]), 1) + + # shouldn't be able to initialize with an array if it's the wrong shape + # this would have too many parameters + population = np.linspace(-1, 3, 15).reshape(5, 3) + assert_raises(ValueError, + DifferentialEvolutionSolver, + *(rosen, self.bounds), + **{'init': population}) + + # provide an initial solution + # bounds are [(0, 2), (0, 2)] + x0 = np.random.uniform(low=0.0, high=2.0, size=2) + solver = DifferentialEvolutionSolver( + rosen, self.bounds, x0=x0 + ) + # parameters are scaled to unit interval + assert_allclose(solver.population[0], x0 / 2.0) + + def test_x0(self): + # smoke test that checks that x0 is usable. + res = differential_evolution(rosen, self.bounds, x0=[0.2, 0.8]) + assert res.success + + # check what happens if some of the x0 lay outside the bounds + with assert_raises(ValueError): + differential_evolution(rosen, self.bounds, x0=[0.2, 2.1]) + + def test_infinite_objective_function(self): + # Test that there are no problems if the objective function + # returns inf on some runs + def sometimes_inf(x): + if x[0] < .5: + return np.inf + return x[1] + bounds = [(0, 1), (0, 1)] + differential_evolution(sometimes_inf, bounds=bounds, disp=False) + + def test_deferred_updating(self): + # check setting of deferred updating, with default workers + bounds = [(0., 2.), (0., 2.)] + solver = DifferentialEvolutionSolver(rosen, bounds, updating='deferred') + assert_(solver._updating == 'deferred') + assert_(solver._mapwrapper._mapfunc is map) + res = solver.solve() + assert res.success + + # check that deferred updating works with an exponential crossover + res = differential_evolution( + rosen, bounds, updating='deferred', strategy='best1exp' + ) + assert res.success + + @pytest.mark.thread_unsafe + def test_immediate_updating(self): + # check setting of immediate updating, with default workers + bounds = [(0., 2.), (0., 2.)] + solver = DifferentialEvolutionSolver(rosen, bounds) + assert_(solver._updating == 'immediate') + + # Safely forking from a multithreaded process is + # problematic, and deprecated in Python 3.12, so + # we use a slower but portable alternative + # see gh-19848 + ctx = multiprocessing.get_context("spawn") + with ctx.Pool(2) as p: + # should raise a UserWarning because the updating='immediate' + # is being overridden by the workers keyword + with warns(UserWarning): + with DifferentialEvolutionSolver(rosen, bounds, workers=p.map) as s: + solver.solve() + assert s._updating == 'deferred' + + @pytest.mark.fail_slow(10) + def test_parallel(self): + # smoke test for parallelization with deferred updating + bounds = [(0., 2.), (0., 2.)] + # use threads instead of Process to speed things up for this simple example + with ThreadPool(2) as p, DifferentialEvolutionSolver( + rosen, bounds, updating='deferred', workers=p.map, tol=0.1, popsize=3 + ) as solver: + assert solver._mapwrapper.pool is not None + assert solver._updating == 'deferred' + solver.solve() + + with DifferentialEvolutionSolver( + rosen, bounds, updating='deferred', workers=2, popsize=3, tol=0.1 + ) as solver: + assert solver._mapwrapper.pool is not None + assert solver._updating == 'deferred' + solver.solve() + + def test_converged(self): + solver = DifferentialEvolutionSolver(rosen, [(0, 2), (0, 2)]) + solver.solve() + assert_(solver.converged()) + + def test_constraint_violation_fn(self): + def constr_f(x): + return [x[0] + x[1]] + + def constr_f2(x): + return np.array([x[0]**2 + x[1], x[0] - x[1]]) + + nlc = NonlinearConstraint(constr_f, -np.inf, 1.9) + + solver = DifferentialEvolutionSolver(rosen, [(0, 2), (0, 2)], + constraints=(nlc,)) + + cv = solver._constraint_violation_fn(np.array([1.0, 1.0])) + assert_almost_equal(cv, 0.1) + + nlc2 = NonlinearConstraint(constr_f2, -np.inf, 1.8) + solver = DifferentialEvolutionSolver(rosen, [(0, 2), (0, 2)], + constraints=(nlc, nlc2)) + + # for multiple constraints the constraint violations should + # be concatenated. + xs = [(1.2, 1), (2.0, 2.0), (0.5, 0.5)] + vs = [(0.3, 0.64, 0.0), (2.1, 4.2, 0.0), (0, 0, 0)] + + for x, v in zip(xs, vs): + cv = solver._constraint_violation_fn(np.array(x)) + assert_allclose(cv, np.atleast_2d(v)) + + # vectorized calculation of a series of solutions + assert_allclose( + solver._constraint_violation_fn(np.array(xs)), np.array(vs) + ) + + # the following line is used in _calculate_population_feasibilities. + # _constraint_violation_fn returns an (1, M) array when + # x.shape == (N,), i.e. a single solution. Therefore this list + # comprehension should generate (S, 1, M) array. + constraint_violation = np.array([solver._constraint_violation_fn(x) + for x in np.array(xs)]) + assert constraint_violation.shape == (3, 1, 3) + + # we need reasonable error messages if the constraint function doesn't + # return the right thing + def constr_f3(x): + # returns (S, M), rather than (M, S) + return constr_f2(x).T + + nlc2 = NonlinearConstraint(constr_f3, -np.inf, 1.8) + solver = DifferentialEvolutionSolver(rosen, [(0, 2), (0, 2)], + constraints=(nlc, nlc2), + vectorized=False) + solver.vectorized = True + with pytest.raises( + RuntimeError, match="An array returned from a Constraint" + ): + solver._constraint_violation_fn(np.array(xs)) + + def test_constraint_population_feasibilities(self): + def constr_f(x): + return [x[0] + x[1]] + + def constr_f2(x): + return [x[0]**2 + x[1], x[0] - x[1]] + + nlc = NonlinearConstraint(constr_f, -np.inf, 1.9) + + solver = DifferentialEvolutionSolver(rosen, [(0, 2), (0, 2)], + constraints=(nlc,)) + + # are population feasibilities correct + # [0.5, 0.5] corresponds to scaled values of [1., 1.] + feas, cv = solver._calculate_population_feasibilities( + np.array([[0.5, 0.5], [1., 1.]])) + assert_equal(feas, [False, False]) + assert_almost_equal(cv, np.array([[0.1], [2.1]])) + assert cv.shape == (2, 1) + + nlc2 = NonlinearConstraint(constr_f2, -np.inf, 1.8) + + for vectorize in [False, True]: + solver = DifferentialEvolutionSolver(rosen, [(0, 2), (0, 2)], + constraints=(nlc, nlc2), + vectorized=vectorize, + updating='deferred') + + feas, cv = solver._calculate_population_feasibilities( + np.array([[0.5, 0.5], [0.6, 0.5]])) + assert_equal(feas, [False, False]) + assert_almost_equal(cv, np.array([[0.1, 0.2, 0], [0.3, 0.64, 0]])) + + feas, cv = solver._calculate_population_feasibilities( + np.array([[0.5, 0.5], [1., 1.]])) + assert_equal(feas, [False, False]) + assert_almost_equal(cv, np.array([[0.1, 0.2, 0], [2.1, 4.2, 0]])) + assert cv.shape == (2, 3) + + feas, cv = solver._calculate_population_feasibilities( + np.array([[0.25, 0.25], [1., 1.]])) + assert_equal(feas, [True, False]) + assert_almost_equal(cv, np.array([[0.0, 0.0, 0.], [2.1, 4.2, 0]])) + assert cv.shape == (2, 3) + + @pytest.mark.thread_unsafe + def test_constraint_solve(self): + def constr_f(x): + return np.array([x[0] + x[1]]) + + nlc = NonlinearConstraint(constr_f, -np.inf, 1.9) + + solver = DifferentialEvolutionSolver(rosen, [(0, 2), (0, 2)], + constraints=(nlc,)) + + # trust-constr warns if the constraint function is linear + with warns(UserWarning): + res = solver.solve() + + assert constr_f(res.x) <= 1.9 + assert res.success + + @pytest.mark.fail_slow(10) + @pytest.mark.thread_unsafe + def test_impossible_constraint(self): + def constr_f(x): + return np.array([x[0] + x[1]]) + + nlc = NonlinearConstraint(constr_f, -np.inf, -1) + + solver = DifferentialEvolutionSolver( + rosen, [(0, 2), (0, 2)], constraints=(nlc,), popsize=1, rng=1, maxiter=100 + ) + + # a UserWarning is issued because the 'trust-constr' polishing is + # attempted on the least infeasible solution found. + with warns(UserWarning): + res = solver.solve() + + assert res.maxcv > 0 + assert not res.success + + # test _promote_lowest_energy works when none of the population is + # feasible. In this case, the solution with the lowest constraint + # violation should be promoted. + solver = DifferentialEvolutionSolver( + rosen, [(0, 2), (0, 2)], constraints=(nlc,), polish=False) + next(solver) + assert not solver.feasible.all() + assert not np.isfinite(solver.population_energies).all() + + # now swap two of the entries in the population + l = 20 + cv = solver.constraint_violation[0] + + solver.population_energies[[0, l]] = solver.population_energies[[l, 0]] + solver.population[[0, l], :] = solver.population[[l, 0], :] + solver.constraint_violation[[0, l], :] = ( + solver.constraint_violation[[l, 0], :]) + + solver._promote_lowest_energy() + assert_equal(solver.constraint_violation[0], cv) + + def test_accept_trial(self): + # _accept_trial(self, energy_trial, feasible_trial, cv_trial, + # energy_orig, feasible_orig, cv_orig) + def constr_f(x): + return [x[0] + x[1]] + nlc = NonlinearConstraint(constr_f, -np.inf, 1.9) + solver = DifferentialEvolutionSolver(rosen, [(0, 2), (0, 2)], + constraints=(nlc,)) + fn = solver._accept_trial + # both solutions are feasible, select lower energy + assert fn(0.1, True, np.array([0.]), 1.0, True, np.array([0.])) + assert (fn(1.0, True, np.array([0.0]), 0.1, True, np.array([0.0])) is False) + assert fn(0.1, True, np.array([0.]), 0.1, True, np.array([0.])) + + # trial is feasible, original is not + assert fn(9.9, True, np.array([0.]), 1.0, False, np.array([1.])) + + # trial and original are infeasible + # cv_trial have to be <= cv_original to be better + assert (fn(0.1, False, np.array([0.5, 0.5]), + 1.0, False, np.array([1., 1.0]))) + assert (fn(0.1, False, np.array([0.5, 0.5]), + 1.0, False, np.array([1., 0.50]))) + assert not (fn(1.0, False, np.array([0.5, 0.5]), + 1.0, False, np.array([1.0, 0.4]))) + + def test_constraint_wrapper(self): + lb = np.array([0, 20, 30]) + ub = np.array([0.5, np.inf, 70]) + x0 = np.array([1, 2, 3]) + pc = _ConstraintWrapper(Bounds(lb, ub), x0) + assert (pc.violation(x0) > 0).any() + assert (pc.violation([0.25, 21, 31]) == 0).all() + + # check vectorized Bounds constraint + xs = np.arange(1, 16).reshape(5, 3) + violations = [] + for x in xs: + violations.append(pc.violation(x)) + np.testing.assert_allclose(pc.violation(xs.T), np.array(violations).T) + + x0 = np.array([1, 2, 3, 4]) + A = np.array([[1, 2, 3, 4], [5, 0, 0, 6], [7, 0, 8, 0]]) + pc = _ConstraintWrapper(LinearConstraint(A, -np.inf, 0), x0) + assert (pc.violation(x0) > 0).any() + assert (pc.violation([-10, 2, -10, 4]) == 0).all() + + # check vectorized LinearConstraint, for 7 lots of parameter vectors + # with each parameter vector being 4 long, with 3 constraints + # xs is the same shape as stored in the differential evolution + # population, but it's sent to the violation function as (len(x), M) + xs = np.arange(1, 29).reshape(7, 4) + violations = [] + for x in xs: + violations.append(pc.violation(x)) + np.testing.assert_allclose(pc.violation(xs.T), np.array(violations).T) + + pc = _ConstraintWrapper(LinearConstraint(csr_matrix(A), -np.inf, 0), + x0) + assert (pc.violation(x0) > 0).any() + assert (pc.violation([-10, 2, -10, 4]) == 0).all() + + def fun(x): + return A.dot(x) + + nonlinear = NonlinearConstraint(fun, -np.inf, 0) + pc = _ConstraintWrapper(nonlinear, [-10, 2, -10, 4]) + assert (pc.violation(x0) > 0).any() + assert (pc.violation([-10, 2, -10, 4]) == 0).all() + + def test_constraint_wrapper_violation(self): + def cons_f(x): + # written in vectorised form to accept an array of (N, S) + # returning (M, S) + # where N is the number of parameters, + # S is the number of solution vectors to be examined, + # and M is the number of constraint components + return np.array([x[0] ** 2 + x[1], + x[0] ** 2 - x[1]]) + + nlc = NonlinearConstraint(cons_f, [-1, -0.8500], [2, 2]) + pc = _ConstraintWrapper(nlc, [0.5, 1]) + assert np.size(pc.bounds[0]) == 2 + + xs = [(0.5, 1), (0.5, 1.2), (1.2, 1.2), (0.1, -1.2), (0.1, 2.0)] + vs = [(0, 0), (0, 0.1), (0.64, 0), (0.19, 0), (0.01, 1.14)] + + for x, v in zip(xs, vs): + assert_allclose(pc.violation(x), v) + + # now check that we can vectorize the constraint wrapper + assert_allclose(pc.violation(np.array(xs).T), + np.array(vs).T) + assert pc.fun(np.array(xs).T).shape == (2, len(xs)) + assert pc.violation(np.array(xs).T).shape == (2, len(xs)) + assert pc.num_constr == 2 + assert pc.parameter_count == 2 + + def test_matrix_linear_constraint(self): + # gh20041 supplying an np.matrix to construct a LinearConstraint caused + # _ConstraintWrapper to start returning constraint violations of the + # wrong shape. + with suppress_warnings() as sup: + sup.filter(PendingDeprecationWarning) + matrix = np.matrix([[1, 1, 1, 1.], + [2, 2, 2, 2.]]) + lc = LinearConstraint(matrix, 0, 1) + x0 = np.ones(4) + cw = _ConstraintWrapper(lc, x0) + # the shape of the constraint violation should be the same as the number + # of constraints applied. + assert cw.violation(x0).shape == (2,) + + # let's try a vectorised violation call. + xtrial = np.arange(4 * 5).reshape(4, 5) + assert cw.violation(xtrial).shape == (2, 5) + + @pytest.mark.fail_slow(20) + def test_L1(self): + # Lampinen ([5]) test problem 1 + + def f(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + fun = np.sum(5*x[1:5]) - 5*x[1:5]@x[1:5] - np.sum(x[5:]) + return fun + + A = np.zeros((10, 14)) # 1-indexed to match reference + A[1, [1, 2, 10, 11]] = 2, 2, 1, 1 + A[2, [1, 10]] = -8, 1 + A[3, [4, 5, 10]] = -2, -1, 1 + A[4, [1, 3, 10, 11]] = 2, 2, 1, 1 + A[5, [2, 11]] = -8, 1 + A[6, [6, 7, 11]] = -2, -1, 1 + A[7, [2, 3, 11, 12]] = 2, 2, 1, 1 + A[8, [3, 12]] = -8, 1 + A[9, [8, 9, 12]] = -2, -1, 1 + A = A[1:, 1:] + + b = np.array([10, 0, 0, 10, 0, 0, 10, 0, 0]) + + L = LinearConstraint(A, -np.inf, b) + + bounds = [(0, 1)]*9 + [(0, 100)]*3 + [(0, 1)] + + # using a lower popsize to speed the test up + res = differential_evolution( + f, bounds, strategy='best1bin', rng=12345, constraints=(L,), + popsize=5, tol=0.01 + ) + + x_opt = (1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1) + f_opt = -15 + + assert_allclose(f(x_opt), f_opt, atol=6e-4) + assert res.success + assert_allclose(res.x, x_opt, atol=6e-4) + assert_allclose(res.fun, f_opt, atol=5e-3) + assert_(np.all(A@res.x <= b)) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + # now repeat the same solve, using the same overall constraints, + # but using a sparse matrix for the LinearConstraint instead of an + # array + + L = LinearConstraint(csr_matrix(A), -np.inf, b) + + # using a lower popsize to speed the test up + res = differential_evolution( + f, bounds, strategy='best1bin', rng=1211134, constraints=(L,), + popsize=2, tol=0.05 + ) + + assert_allclose(f(x_opt), f_opt) + assert res.success + assert_allclose(res.x, x_opt, atol=5e-4) + assert_allclose(res.fun, f_opt, atol=5e-3) + assert_(np.all(A@res.x <= b)) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + # now repeat the same solve, using the same overall constraints, + # but specify half the constraints in terms of LinearConstraint, + # and the other half by NonlinearConstraint + def c1(x): + x = np.hstack(([0], x)) + return [2*x[2] + 2*x[3] + x[11] + x[12], + -8*x[3] + x[12]] + + def c2(x): + x = np.hstack(([0], x)) + return -2*x[8] - x[9] + x[12] + + L = LinearConstraint(A[:5, :], -np.inf, b[:5]) + L2 = LinearConstraint(A[5:6, :], -np.inf, b[5:6]) + N = NonlinearConstraint(c1, -np.inf, b[6:8]) + N2 = NonlinearConstraint(c2, -np.inf, b[8:9]) + constraints = (L, N, L2, N2) + + with suppress_warnings() as sup: + sup.filter(UserWarning) + res = differential_evolution( + f, bounds, strategy='best1bin', rng=1211134, + constraints=constraints, popsize=2, tol=0.05 + ) + + assert_allclose(res.x, x_opt, atol=6e-4) + assert_allclose(res.fun, f_opt, atol=5e-3) + assert_(np.all(A@res.x <= b)) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + @pytest.mark.fail_slow(10) + def test_L2(self): + # Lampinen ([5]) test problem 2 + + def f(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + fun = ((x[1]-10)**2 + 5*(x[2]-12)**2 + x[3]**4 + 3*(x[4]-11)**2 + + 10*x[5]**6 + 7*x[6]**2 + x[7]**4 - 4*x[6]*x[7] - 10*x[6] - + 8*x[7]) + return fun + + def c1(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + return [127 - 2*x[1]**2 - 3*x[2]**4 - x[3] - 4*x[4]**2 - 5*x[5], + 196 - 23*x[1] - x[2]**2 - 6*x[6]**2 + 8*x[7], + 282 - 7*x[1] - 3*x[2] - 10*x[3]**2 - x[4] + x[5], + -4*x[1]**2 - x[2]**2 + 3*x[1]*x[2] - 2*x[3]**2 - + 5*x[6] + 11*x[7]] + + N = NonlinearConstraint(c1, 0, np.inf) + bounds = [(-10, 10)]*7 + constraints = (N) + + with suppress_warnings() as sup: + sup.filter(UserWarning) + res = differential_evolution(f, bounds, strategy='best1bin', + rng=1234, constraints=constraints) + + f_opt = 680.6300599487869 + x_opt = (2.330499, 1.951372, -0.4775414, 4.365726, + -0.6244870, 1.038131, 1.594227) + + assert_allclose(f(x_opt), f_opt) + assert_allclose(res.fun, f_opt) + assert_allclose(res.x, x_opt, atol=1e-5) + assert res.success + assert_(np.all(np.array(c1(res.x)) >= 0)) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + @pytest.mark.fail_slow(10) + def test_L3(self): + # Lampinen ([5]) test problem 3 + + def f(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + fun = (x[1]**2 + x[2]**2 + x[1]*x[2] - 14*x[1] - 16*x[2] + + (x[3]-10)**2 + 4*(x[4]-5)**2 + (x[5]-3)**2 + 2*(x[6]-1)**2 + + 5*x[7]**2 + 7*(x[8]-11)**2 + 2*(x[9]-10)**2 + + (x[10] - 7)**2 + 45 + ) + return fun # maximize + + A = np.zeros((4, 11)) + A[1, [1, 2, 7, 8]] = -4, -5, 3, -9 + A[2, [1, 2, 7, 8]] = -10, 8, 17, -2 + A[3, [1, 2, 9, 10]] = 8, -2, -5, 2 + A = A[1:, 1:] + b = np.array([-105, 0, -12]) + + def c1(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + return [3*x[1] - 6*x[2] - 12*(x[9]-8)**2 + 7*x[10], + -3*(x[1]-2)**2 - 4*(x[2]-3)**2 - 2*x[3]**2 + 7*x[4] + 120, + -x[1]**2 - 2*(x[2]-2)**2 + 2*x[1]*x[2] - 14*x[5] + 6*x[6], + -5*x[1]**2 - 8*x[2] - (x[3]-6)**2 + 2*x[4] + 40, + -0.5*(x[1]-8)**2 - 2*(x[2]-4)**2 - 3*x[5]**2 + x[6] + 30] + + L = LinearConstraint(A, b, np.inf) + N = NonlinearConstraint(c1, 0, np.inf) + bounds = [(-10, 10)]*10 + constraints = (L, N) + + with suppress_warnings() as sup: + sup.filter(UserWarning) + res = differential_evolution(f, bounds, rng=1234, + constraints=constraints, popsize=3) + + x_opt = (2.171996, 2.363683, 8.773926, 5.095984, 0.9906548, + 1.430574, 1.321644, 9.828726, 8.280092, 8.375927) + f_opt = 24.3062091 + + assert_allclose(f(x_opt), f_opt, atol=1e-5) + assert_allclose(res.x, x_opt, atol=1e-6) + assert_allclose(res.fun, f_opt, atol=1e-5) + assert res.success + assert_(np.all(A @ res.x >= b)) + assert_(np.all(np.array(c1(res.x)) >= 0)) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + @pytest.mark.fail_slow(10) + def test_L4(self): + # Lampinen ([5]) test problem 4 + def f(x): + return np.sum(x[:3]) + + A = np.zeros((4, 9)) + A[1, [4, 6]] = 0.0025, 0.0025 + A[2, [5, 7, 4]] = 0.0025, 0.0025, -0.0025 + A[3, [8, 5]] = 0.01, -0.01 + A = A[1:, 1:] + b = np.array([1, 1, 1]) + + def c1(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + return [x[1]*x[6] - 833.33252*x[4] - 100*x[1] + 83333.333, + x[2]*x[7] - 1250*x[5] - x[2]*x[4] + 1250*x[4], + x[3]*x[8] - 1250000 - x[3]*x[5] + 2500*x[5]] + + L = LinearConstraint(A, -np.inf, 1) + N = NonlinearConstraint(c1, 0, np.inf) + + bounds = [(100, 10000)] + [(1000, 10000)]*2 + [(10, 1000)]*5 + constraints = (L, N) + + with suppress_warnings() as sup: + sup.filter(UserWarning) + res = differential_evolution( + f, bounds, strategy='best1bin', rng=1234, + constraints=constraints, popsize=3, tol=0.05 + ) + + f_opt = 7049.248 + + x_opt = [579.306692, 1359.97063, 5109.9707, 182.0177, 295.601172, + 217.9823, 286.416528, 395.601172] + + assert_allclose(f(x_opt), f_opt, atol=0.001) + assert_allclose(res.fun, f_opt, atol=0.001) + + # use higher tol here for 32-bit Windows, see gh-11693 + if (platform.system() == 'Windows' and np.dtype(np.intp).itemsize < 8): + assert_allclose(res.x, x_opt, rtol=2.4e-6, atol=0.0035) + else: + # tolerance determined from macOS + MKL failure, see gh-12701 + assert_allclose(res.x, x_opt, rtol=5e-6, atol=0.0024) + + assert res.success + assert_(np.all(A @ res.x <= b)) + assert_(np.all(np.array(c1(res.x)) >= 0)) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + @pytest.mark.fail_slow(10) + def test_L5(self): + # Lampinen ([5]) test problem 5 + + def f(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + fun = (np.sin(2*np.pi*x[1])**3*np.sin(2*np.pi*x[2]) / + (x[1]**3*(x[1]+x[2]))) + return -fun # maximize + + def c1(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + return [x[1]**2 - x[2] + 1, + 1 - x[1] + (x[2]-4)**2] + + N = NonlinearConstraint(c1, -np.inf, 0) + bounds = [(0, 10)]*2 + constraints = (N) + + res = differential_evolution(f, bounds, strategy='rand1bin', rng=1234, + constraints=constraints) + + x_opt = (1.22797135, 4.24537337) + f_opt = -0.095825 + assert_allclose(f(x_opt), f_opt, atol=2e-5) + assert_allclose(res.fun, f_opt, atol=1e-4) + assert res.success + assert_(np.all(np.array(c1(res.x)) <= 0)) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + @pytest.mark.fail_slow(10) + def test_L6(self): + # Lampinen ([5]) test problem 6 + def f(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + fun = (x[1]-10)**3 + (x[2] - 20)**3 + return fun + + def c1(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + return [(x[1]-5)**2 + (x[2] - 5)**2 - 100, + -(x[1]-6)**2 - (x[2] - 5)**2 + 82.81] + + N = NonlinearConstraint(c1, 0, np.inf) + bounds = [(13, 100), (0, 100)] + constraints = (N) + res = differential_evolution(f, bounds, strategy='rand1bin', rng=1234, + constraints=constraints, tol=1e-7) + x_opt = (14.095, 0.84296) + f_opt = -6961.814744 + + assert_allclose(f(x_opt), f_opt, atol=1e-6) + assert_allclose(res.fun, f_opt, atol=0.001) + assert_allclose(res.x, x_opt, atol=1e-4) + assert res.success + assert_(np.all(np.array(c1(res.x)) >= 0)) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + def test_L7(self): + # Lampinen ([5]) test problem 7 + def f(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + fun = (5.3578547*x[3]**2 + 0.8356891*x[1]*x[5] + + 37.293239*x[1] - 40792.141) + return fun + + def c1(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + return [ + 85.334407 + 0.0056858*x[2]*x[5] + 0.0006262*x[1]*x[4] - + 0.0022053*x[3]*x[5], + + 80.51249 + 0.0071317*x[2]*x[5] + 0.0029955*x[1]*x[2] + + 0.0021813*x[3]**2, + + 9.300961 + 0.0047026*x[3]*x[5] + 0.0012547*x[1]*x[3] + + 0.0019085*x[3]*x[4] + ] + + N = NonlinearConstraint(c1, [0, 90, 20], [92, 110, 25]) + + bounds = [(78, 102), (33, 45)] + [(27, 45)]*3 + constraints = (N) + + res = differential_evolution(f, bounds, strategy='rand1bin', rng=1234, + constraints=constraints) + + # using our best solution, rather than Lampinen/Koziel. Koziel solution + # doesn't satisfy constraints, Lampinen f_opt just plain wrong. + x_opt = [78.00000686, 33.00000362, 29.99526064, 44.99999971, + 36.77579979] + + f_opt = -30665.537578 + + assert_allclose(f(x_opt), f_opt) + assert_allclose(res.x, x_opt, atol=1e-3) + assert_allclose(res.fun, f_opt, atol=1e-3) + + assert res.success + assert_(np.all(np.array(c1(res.x)) >= np.array([0, 90, 20]))) + assert_(np.all(np.array(c1(res.x)) <= np.array([92, 110, 25]))) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + @pytest.mark.xslow + @pytest.mark.xfail(platform.machine() == 'ppc64le', + reason="fails on ppc64le") + def test_L8(self): + def f(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + fun = 3*x[1] + 0.000001*x[1]**3 + 2*x[2] + 0.000002/3*x[2]**3 + return fun + + A = np.zeros((3, 5)) + A[1, [4, 3]] = 1, -1 + A[2, [3, 4]] = 1, -1 + A = A[1:, 1:] + b = np.array([-.55, -.55]) + + def c1(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + return [ + 1000*np.sin(-x[3]-0.25) + 1000*np.sin(-x[4]-0.25) + + 894.8 - x[1], + 1000*np.sin(x[3]-0.25) + 1000*np.sin(x[3]-x[4]-0.25) + + 894.8 - x[2], + 1000*np.sin(x[4]-0.25) + 1000*np.sin(x[4]-x[3]-0.25) + + 1294.8 + ] + L = LinearConstraint(A, b, np.inf) + N = NonlinearConstraint(c1, np.full(3, -0.001), np.full(3, 0.001)) + + bounds = [(0, 1200)]*2+[(-.55, .55)]*2 + constraints = (L, N) + + with suppress_warnings() as sup: + sup.filter(UserWarning) + # original Lampinen test was with rand1bin, but that takes a + # huge amount of CPU time. Changing strategy to best1bin speeds + # things up a lot + res = differential_evolution(f, bounds, strategy='best1bin', + rng=1234, constraints=constraints, + maxiter=5000) + + x_opt = (679.9453, 1026.067, 0.1188764, -0.3962336) + f_opt = 5126.4981 + + assert_allclose(f(x_opt), f_opt, atol=1e-3) + assert_allclose(res.x[:2], x_opt[:2], atol=2e-3) + assert_allclose(res.x[2:], x_opt[2:], atol=2e-3) + assert_allclose(res.fun, f_opt, atol=2e-2) + assert res.success + assert_(np.all(A@res.x >= b)) + assert_(np.all(np.array(c1(res.x)) >= -0.001)) + assert_(np.all(np.array(c1(res.x)) <= 0.001)) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + @pytest.mark.fail_slow(5) + def test_L9(self): + # Lampinen ([5]) test problem 9 + + def f(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + return x[1]**2 + (x[2]-1)**2 + + def c1(x): + x = np.hstack(([0], x)) # 1-indexed to match reference + return [x[2] - x[1]**2] + + N = NonlinearConstraint(c1, [-.001], [0.001]) + + bounds = [(-1, 1)]*2 + constraints = (N) + res = differential_evolution(f, bounds, strategy='rand1bin', rng=1234, + constraints=constraints) + + x_opt = [np.sqrt(2)/2, 0.5] + f_opt = 0.75 + + assert_allclose(f(x_opt), f_opt) + assert_allclose(np.abs(res.x), x_opt, atol=1e-3) + assert_allclose(res.fun, f_opt, atol=1e-3) + assert res.success + assert_(np.all(np.array(c1(res.x)) >= -0.001)) + assert_(np.all(np.array(c1(res.x)) <= 0.001)) + assert_(np.all(res.x >= np.array(bounds)[:, 0])) + assert_(np.all(res.x <= np.array(bounds)[:, 1])) + + @pytest.mark.fail_slow(10) + def test_integrality(self): + # test fitting discrete distribution to data + rng = np.random.default_rng(6519843218105) + dist = stats.nbinom + shapes = (5, 0.5) + x = dist.rvs(*shapes, size=10000, random_state=rng) + + def func(p, *args): + dist, x = args + # negative log-likelihood function + ll = -np.log(dist.pmf(x, *p)).sum(axis=-1) + if np.isnan(ll): # occurs when x is outside of support + ll = np.inf # we don't want that + return ll + + integrality = [True, False] + bounds = [(1, 18), (0, 0.95)] + + res = differential_evolution(func, bounds, args=(dist, x), + integrality=integrality, polish=False, + rng=rng) + # tolerance has to be fairly relaxed for the second parameter + # because we're fitting a distribution to random variates. + assert res.x[0] == 5 + assert_allclose(res.x, shapes, rtol=0.025) + + # check that we can still use integrality constraints with polishing + res2 = differential_evolution(func, bounds, args=(dist, x), + integrality=integrality, polish=True, + rng=rng) + + def func2(p, *args): + n, dist, x = args + return func(np.array([n, p[0]]), dist, x) + + # compare the DE derived solution to an LBFGSB solution (that doesn't + # have to find the integral values). Note we're setting x0 to be the + # output from the first DE result, thereby making the polishing step + # and this minimisation pretty much equivalent. + LBFGSB = minimize(func2, res2.x[1], args=(5, dist, x), + bounds=[(0, 0.95)]) + assert_allclose(res2.x[1], LBFGSB.x) + assert res2.fun <= res.fun + + def test_integrality_limits(self): + def f(x): + return x + + integrality = [True, False, True] + bounds = [(0.2, 1.1), (0.9, 2.2), (3.3, 4.9)] + + # no integrality constraints + solver = DifferentialEvolutionSolver(f, bounds=bounds, polish=False, + integrality=False) + assert_allclose(solver.limits[0], [0.2, 0.9, 3.3]) + assert_allclose(solver.limits[1], [1.1, 2.2, 4.9]) + + # with integrality constraints + solver = DifferentialEvolutionSolver(f, bounds=bounds, polish=False, + integrality=integrality) + assert_allclose(solver.limits[0], [0.5, 0.9, 3.5]) + assert_allclose(solver.limits[1], [1.5, 2.2, 4.5]) + assert_equal(solver.integrality, [True, False, True]) + assert solver.polish is False + + bounds = [(-1.2, -0.9), (0.9, 2.2), (-10.3, 4.1)] + solver = DifferentialEvolutionSolver(f, bounds=bounds, polish=False, + integrality=integrality) + assert_allclose(solver.limits[0], [-1.5, 0.9, -10.5]) + assert_allclose(solver.limits[1], [-0.5, 2.2, 4.5]) + + # A lower bound of -1.2 is converted to + # np.nextafter(np.ceil(-1.2) - 0.5, np.inf) + # with a similar process to the upper bound. Check that the + # conversions work + assert_allclose(np.round(solver.limits[0]), [-1.0, 1.0, -10.0]) + assert_allclose(np.round(solver.limits[1]), [-1.0, 2.0, 4.0]) + + bounds = [(-10.2, -8.1), (0.9, 2.2), (-10.9, -9.9999)] + solver = DifferentialEvolutionSolver(f, bounds=bounds, polish=False, + integrality=integrality) + assert_allclose(solver.limits[0], [-10.5, 0.9, -10.5]) + assert_allclose(solver.limits[1], [-8.5, 2.2, -9.5]) + + bounds = [(-10.2, -10.1), (0.9, 2.2), (-10.9, -9.9999)] + with pytest.raises(ValueError, match='One of the integrality'): + DifferentialEvolutionSolver(f, bounds=bounds, polish=False, + integrality=integrality) + + @pytest.mark.thread_unsafe + @pytest.mark.fail_slow(10) + def test_vectorized(self): + def quadratic(x): + return np.sum(x**2) + + def quadratic_vec(x): + return np.sum(x**2, axis=0) + + # A vectorized function needs to accept (len(x), S) and return (S,) + with pytest.raises(RuntimeError, match='The vectorized function'): + differential_evolution(quadratic, self.bounds, + vectorized=True, updating='deferred') + + # vectorized overrides the updating keyword, check for warning + with warns(UserWarning, match="differential_evolution: the 'vector"): + differential_evolution(quadratic_vec, self.bounds, + vectorized=True) + + # vectorized defers to the workers keyword, check for warning + with warns(UserWarning, match="differential_evolution: the 'workers"): + differential_evolution(quadratic_vec, self.bounds, + vectorized=True, workers=map, + updating='deferred') + + ncalls = [0] + + def rosen_vec(x): + ncalls[0] += 1 + return rosen(x) + + bounds = [(0, 10), (0, 10)] + res1 = differential_evolution(rosen, bounds, updating='deferred', + rng=1) + res2 = differential_evolution(rosen_vec, bounds, vectorized=True, + updating='deferred', rng=1) + + # the two minimisation runs should be functionally equivalent + assert_allclose(res1.x, res2.x) + assert ncalls[0] == res2.nfev + assert res1.nit == res2.nit + + def test_vectorized_constraints(self): + def constr_f(x): + return np.array([x[0] + x[1]]) + + def constr_f2(x): + return np.array([x[0]**2 + x[1], x[0] - x[1]]) + + nlc1 = NonlinearConstraint(constr_f, -np.inf, 1.9) + nlc2 = NonlinearConstraint(constr_f2, (0.9, 0.5), (2.0, 2.0)) + + def rosen_vec(x): + # accept an (len(x0), S) array, returning a (S,) array + v = 100 * (x[1:] - x[:-1]**2.0)**2.0 + v += (1 - x[:-1])**2.0 + return np.squeeze(v) + + bounds = [(0, 10), (0, 10)] + + res1 = differential_evolution(rosen, bounds, updating='deferred', + rng=1, constraints=[nlc1, nlc2], + polish=False) + res2 = differential_evolution(rosen_vec, bounds, vectorized=True, + updating='deferred', rng=1, + constraints=[nlc1, nlc2], + polish=False) + # the two minimisation runs should be functionally equivalent + assert_allclose(res1.x, res2.x) + + def test_constraint_violation_error_message(self): + + def func(x): + return np.cos(x[0]) + np.sin(x[1]) + + # Intentionally infeasible constraints. + c0 = NonlinearConstraint(lambda x: x[1] - (x[0]-1)**2, 0, np.inf) + c1 = NonlinearConstraint(lambda x: x[1] + x[0]**2, -np.inf, 0) + + result = differential_evolution(func, + bounds=[(-1, 2), (-1, 1)], + constraints=[c0, c1], + maxiter=10, + polish=False, + rng=864197532) + assert result.success is False + # The numerical value in the error message might be sensitive to + # changes in the implementation. It can be updated if the code is + # changed. The essential part of the test is that there is a number + # after the '=', so if necessary, the text could be reduced to, say, + # "MAXCV = 0.". + assert "MAXCV = 0." in result.message + + @pytest.mark.fail_slow(20) # fail-slow exception by request - see gh-20806 + def test_strategy_fn(self): + # examines ability to customize strategy by mimicking one of the + # in-built strategies + parameter_count = 4 + popsize = 10 + bounds = [(0, 10.)] * parameter_count + total_popsize = parameter_count * popsize + mutation = 0.8 + recombination = 0.7 + + calls = [0] + def custom_strategy_fn(candidate, population, rng=None): + calls[0] += 1 + trial = np.copy(population[candidate]) + fill_point = rng.choice(parameter_count) + + pool = np.arange(total_popsize) + rng.shuffle(pool) + idxs = pool[:2 + 1] + idxs = idxs[idxs != candidate][:2] + + r0, r1 = idxs[:2] + + bprime = (population[0] + mutation * + (population[r0] - population[r1])) + + crossovers = rng.uniform(size=parameter_count) + crossovers = crossovers < recombination + crossovers[fill_point] = True + trial = np.where(crossovers, bprime, trial) + return trial + + solver = DifferentialEvolutionSolver( + rosen, + bounds, + popsize=popsize, + recombination=recombination, + mutation=mutation, + maxiter=2, + strategy=custom_strategy_fn, + rng=10, + polish=False + ) + assert solver.strategy is custom_strategy_fn + solver.solve() + assert calls[0] > 0 + + # check custom strategy works with updating='deferred' + res = differential_evolution( + rosen, bounds, strategy=custom_strategy_fn, updating='deferred' + ) + assert res.success + + def custom_strategy_fn(candidate, population, rng=None): + return np.array([1.0, 2.0]) + + with pytest.raises(RuntimeError, match="strategy*"): + differential_evolution( + rosen, + bounds, + strategy=custom_strategy_fn + ) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__dual_annealing.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__dual_annealing.py new file mode 100644 index 0000000000000000000000000000000000000000..3465be508491aae411e113167b8fdd84f6d2d70b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__dual_annealing.py @@ -0,0 +1,416 @@ +# Dual annealing unit tests implementation. +# Copyright (c) 2018 Sylvain Gubian , +# Yang Xiang +# Author: Sylvain Gubian, PMP S.A. +""" +Unit tests for the dual annealing global optimizer +""" +from scipy.optimize import dual_annealing, Bounds + +from scipy.optimize._dual_annealing import EnergyState +from scipy.optimize._dual_annealing import LocalSearchWrapper +from scipy.optimize._dual_annealing import ObjectiveFunWrapper +from scipy.optimize._dual_annealing import StrategyChain +from scipy.optimize._dual_annealing import VisitingDistribution +from scipy.optimize import rosen, rosen_der +import pytest +import numpy as np +from numpy.testing import assert_equal, assert_allclose, assert_array_less +from pytest import raises as assert_raises +from scipy._lib._util import check_random_state + +import threading + + +class TestDualAnnealing: + + def setup_method(self): + # A function that returns always infinity for initialization tests + self.weirdfunc = lambda x: np.inf + # 2-D bounds for testing function + self.ld_bounds = [(-5.12, 5.12)] * 2 + # 4-D bounds for testing function + self.hd_bounds = self.ld_bounds * 4 + # Number of values to be generated for testing visit function + self.nbtestvalues = 5000 + self.high_temperature = 5230 + self.low_temperature = 0.1 + self.qv = 2.62 + self.seed = 1234 + self.rng = check_random_state(self.seed) + self.nb_fun_call = threading.local() + self.ngev = threading.local() + + def callback(self, x, f, context): + # For testing callback mechanism. Should stop for e <= 1 as + # the callback function returns True + if f <= 1.0: + return True + + def func(self, x, args=()): + # Using Rastrigin function for performing tests + if args: + shift = args + else: + shift = 0 + y = np.sum((x - shift) ** 2 - 10 * np.cos(2 * np.pi * ( + x - shift))) + 10 * np.size(x) + shift + if not hasattr(self.nb_fun_call, 'c'): + self.nb_fun_call.c = 0 + self.nb_fun_call.c += 1 + return y + + def rosen_der_wrapper(self, x, args=()): + if not hasattr(self.ngev, 'c'): + self.ngev.c = 0 + self.ngev.c += 1 + return rosen_der(x, *args) + + # FIXME: there are some discontinuities in behaviour as a function of `qv`, + # this needs investigating - see gh-12384 + @pytest.mark.parametrize('qv', [1.1, 1.41, 2, 2.62, 2.9]) + def test_visiting_stepping(self, qv): + lu = list(zip(*self.ld_bounds)) + lower = np.array(lu[0]) + upper = np.array(lu[1]) + dim = lower.size + vd = VisitingDistribution(lower, upper, qv, self.rng) + values = np.zeros(dim) + x_step_low = vd.visiting(values, 0, self.high_temperature) + # Make sure that only the first component is changed + assert_equal(np.not_equal(x_step_low, 0), True) + values = np.zeros(dim) + x_step_high = vd.visiting(values, dim, self.high_temperature) + # Make sure that component other than at dim has changed + assert_equal(np.not_equal(x_step_high[0], 0), True) + + @pytest.mark.parametrize('qv', [2.25, 2.62, 2.9]) + def test_visiting_dist_high_temperature(self, qv): + lu = list(zip(*self.ld_bounds)) + lower = np.array(lu[0]) + upper = np.array(lu[1]) + vd = VisitingDistribution(lower, upper, qv, self.rng) + # values = np.zeros(self.nbtestvalues) + # for i in np.arange(self.nbtestvalues): + # values[i] = vd.visit_fn(self.high_temperature) + values = vd.visit_fn(self.high_temperature, self.nbtestvalues) + + # Visiting distribution is a distorted version of Cauchy-Lorentz + # distribution, and as no 1st and higher moments (no mean defined, + # no variance defined). + # Check that big tails values are generated + assert_array_less(np.min(values), 1e-10) + assert_array_less(1e+10, np.max(values)) + + def test_reset(self): + owf = ObjectiveFunWrapper(self.weirdfunc) + lu = list(zip(*self.ld_bounds)) + lower = np.array(lu[0]) + upper = np.array(lu[1]) + es = EnergyState(lower, upper) + assert_raises(ValueError, es.reset, owf, check_random_state(None)) + + def test_low_dim(self): + ret = dual_annealing( + self.func, self.ld_bounds, rng=self.seed) + assert_allclose(ret.fun, 0., atol=1e-12) + assert ret.success + + @pytest.mark.fail_slow(10) + def test_high_dim(self): + ret = dual_annealing(self.func, self.hd_bounds, rng=self.seed) + assert_allclose(ret.fun, 0., atol=1e-12) + assert ret.success + + def test_low_dim_no_ls(self): + ret = dual_annealing(self.func, self.ld_bounds, + no_local_search=True, seed=self.seed) + assert_allclose(ret.fun, 0., atol=1e-4) + + @pytest.mark.fail_slow(10) + def test_high_dim_no_ls(self): + ret = dual_annealing(self.func, self.hd_bounds, + no_local_search=True, rng=self.seed) + assert_allclose(ret.fun, 0., atol=1.2e-4) + + def test_nb_fun_call(self): + self.nb_fun_call.c = 0 + ret = dual_annealing(self.func, self.ld_bounds, rng=self.seed) + assert_equal(self.nb_fun_call.c, ret.nfev) + + def test_nb_fun_call_no_ls(self): + self.nb_fun_call.c = 0 + ret = dual_annealing(self.func, self.ld_bounds, + no_local_search=True, rng=self.seed) + assert_equal(self.nb_fun_call.c, ret.nfev) + + def test_max_reinit(self): + assert_raises(ValueError, dual_annealing, self.weirdfunc, + self.ld_bounds) + + @pytest.mark.fail_slow(10) + def test_reproduce(self): + res1 = dual_annealing(self.func, self.ld_bounds, rng=self.seed) + res2 = dual_annealing(self.func, self.ld_bounds, rng=self.seed) + res3 = dual_annealing(self.func, self.ld_bounds, rng=self.seed) + # If we have reproducible results, x components found has to + # be exactly the same, which is not the case with no seeding + assert_equal(res1.x, res2.x) + assert_equal(res1.x, res3.x) + + def test_rand_gen(self): + # check that np.random.Generator can be used (numpy >= 1.17) + # obtain a np.random.Generator object + rng = np.random.default_rng(1) + + res1 = dual_annealing(self.func, self.ld_bounds, rng=rng) + # seed again + rng = np.random.default_rng(1) + res2 = dual_annealing(self.func, self.ld_bounds, rng=rng) + # If we have reproducible results, x components found has to + # be exactly the same, which is not the case with no seeding + assert_equal(res1.x, res2.x) + + def test_bounds_integrity(self): + wrong_bounds = [(-5.12, 5.12), (1, 0), (5.12, 5.12)] + assert_raises(ValueError, dual_annealing, self.func, + wrong_bounds) + + def test_bound_validity(self): + invalid_bounds = [(-5, 5), (-np.inf, 0), (-5, 5)] + assert_raises(ValueError, dual_annealing, self.func, + invalid_bounds) + invalid_bounds = [(-5, 5), (0, np.inf), (-5, 5)] + assert_raises(ValueError, dual_annealing, self.func, + invalid_bounds) + invalid_bounds = [(-5, 5), (0, np.nan), (-5, 5)] + assert_raises(ValueError, dual_annealing, self.func, + invalid_bounds) + + @pytest.mark.thread_unsafe + def test_deprecated_local_search_options_bounds(self): + def func(x): + return np.sum((x - 5) * (x - 1)) + bounds = list(zip([-6, -5], [6, 5])) + # Test bounds can be passed (see gh-10831) + + with pytest.warns(RuntimeWarning, match=r"Method CG cannot handle "): + dual_annealing( + func, + bounds=bounds, + minimizer_kwargs={"method": "CG", "bounds": bounds}) + + @pytest.mark.thread_unsafe + def test_minimizer_kwargs_bounds(self): + def func(x): + return np.sum((x - 5) * (x - 1)) + bounds = list(zip([-6, -5], [6, 5])) + # Test bounds can be passed (see gh-10831) + dual_annealing( + func, + bounds=bounds, + minimizer_kwargs={"method": "SLSQP", "bounds": bounds}) + + with pytest.warns(RuntimeWarning, match=r"Method CG cannot handle "): + dual_annealing( + func, + bounds=bounds, + minimizer_kwargs={"method": "CG", "bounds": bounds}) + + def test_max_fun_ls(self): + ret = dual_annealing(self.func, self.ld_bounds, maxfun=100, + rng=self.seed) + + ls_max_iter = min(max( + len(self.ld_bounds) * LocalSearchWrapper.LS_MAXITER_RATIO, + LocalSearchWrapper.LS_MAXITER_MIN), + LocalSearchWrapper.LS_MAXITER_MAX) + assert ret.nfev <= 100 + ls_max_iter + assert not ret.success + + def test_max_fun_no_ls(self): + ret = dual_annealing(self.func, self.ld_bounds, + no_local_search=True, maxfun=500, rng=self.seed) + assert ret.nfev <= 500 + assert not ret.success + + def test_maxiter(self): + ret = dual_annealing(self.func, self.ld_bounds, maxiter=700, + rng=self.seed) + assert ret.nit <= 700 + + # Testing that args are passed correctly for dual_annealing + def test_fun_args_ls(self): + ret = dual_annealing(self.func, self.ld_bounds, + args=((3.14159,)), rng=self.seed) + assert_allclose(ret.fun, 3.14159, atol=1e-6) + + # Testing that args are passed correctly for pure simulated annealing + def test_fun_args_no_ls(self): + ret = dual_annealing(self.func, self.ld_bounds, + args=((3.14159, )), no_local_search=True, + rng=self.seed) + assert_allclose(ret.fun, 3.14159, atol=1e-4) + + def test_callback_stop(self): + # Testing that callback make the algorithm stop for + # fun value <= 1.0 (see callback method) + ret = dual_annealing(self.func, self.ld_bounds, + callback=self.callback, rng=self.seed) + assert ret.fun <= 1.0 + assert 'stop early' in ret.message[0] + assert not ret.success + + @pytest.mark.parametrize('method, atol', [ + ('Nelder-Mead', 2e-5), + ('COBYLA', 1e-5), + ('COBYQA', 1e-8), + ('Powell', 1e-8), + ('CG', 1e-8), + ('BFGS', 1e-8), + ('TNC', 1e-8), + ('SLSQP', 2e-7), + ]) + def test_multi_ls_minimizer(self, method, atol): + ret = dual_annealing(self.func, self.ld_bounds, + minimizer_kwargs=dict(method=method), + rng=self.seed) + assert_allclose(ret.fun, 0., atol=atol) + + def test_wrong_restart_temp(self): + assert_raises(ValueError, dual_annealing, self.func, + self.ld_bounds, restart_temp_ratio=1) + assert_raises(ValueError, dual_annealing, self.func, + self.ld_bounds, restart_temp_ratio=0) + + def test_gradient_gnev(self): + minimizer_opts = { + 'jac': self.rosen_der_wrapper, + } + ret = dual_annealing(rosen, self.ld_bounds, + minimizer_kwargs=minimizer_opts, + rng=self.seed) + assert ret.njev == self.ngev.c + + @pytest.mark.fail_slow(10) + def test_from_docstring(self): + def func(x): + return np.sum(x * x - 10 * np.cos(2 * np.pi * x)) + 10 * np.size(x) + lw = [-5.12] * 10 + up = [5.12] * 10 + ret = dual_annealing(func, bounds=list(zip(lw, up)), rng=1234) + assert_allclose(ret.x, + [-4.26437714e-09, -3.91699361e-09, -1.86149218e-09, + -3.97165720e-09, -6.29151648e-09, -6.53145322e-09, + -3.93616815e-09, -6.55623025e-09, -6.05775280e-09, + -5.00668935e-09], atol=4e-8) + assert_allclose(ret.fun, 0.000000, atol=5e-13) + + @pytest.mark.parametrize('new_e, temp_step, accepted, accept_rate', [ + (0, 100, 1000, 1.0097587941791923), + (0, 2, 1000, 1.2599210498948732), + (10, 100, 878, 0.8786035869128718), + (10, 60, 695, 0.6812920690579612), + (2, 100, 990, 0.9897404249173424), + ]) + def test_accept_reject_probabilistic( + self, new_e, temp_step, accepted, accept_rate): + # Test accepts unconditionally with e < current_energy and + # probabilistically with e > current_energy + + rs = check_random_state(123) + + count_accepted = 0 + iterations = 1000 + + accept_param = -5 + current_energy = 1 + for _ in range(iterations): + energy_state = EnergyState(lower=None, upper=None) + # Set energy state with current_energy, any location. + energy_state.update_current(current_energy, [0]) + + chain = StrategyChain( + accept_param, None, None, None, rs, energy_state) + # Normally this is set in run() + chain.temperature_step = temp_step + + # Check if update is accepted. + chain.accept_reject(j=1, e=new_e, x_visit=[2]) + if energy_state.current_energy == new_e: + count_accepted += 1 + + assert count_accepted == accepted + + # Check accept rate + pqv = 1 - (1 - accept_param) * (new_e - current_energy) / temp_step + rate = 0 if pqv <= 0 else np.exp(np.log(pqv) / (1 - accept_param)) + + assert_allclose(rate, accept_rate) + + @pytest.mark.fail_slow(10) + def test_bounds_class(self): + # test that result does not depend on the bounds type + def func(x): + f = np.sum(x * x - 10 * np.cos(2 * np.pi * x)) + 10 * np.size(x) + return f + lw = [-5.12] * 5 + up = [5.12] * 5 + + # Unbounded global minimum is all zeros. Most bounds below will force + # a DV away from unbounded minimum and be active at solution. + up[0] = -2.0 + up[1] = -1.0 + lw[3] = 1.0 + lw[4] = 2.0 + + # run optimizations + bounds = Bounds(lw, up) + ret_bounds_class = dual_annealing(func, bounds=bounds, rng=1234) + + bounds_old = list(zip(lw, up)) + ret_bounds_list = dual_annealing(func, bounds=bounds_old, rng=1234) + + # test that found minima, function evaluations and iterations match + assert_allclose(ret_bounds_class.x, ret_bounds_list.x, atol=1e-8) + assert_allclose(ret_bounds_class.x, np.arange(-2, 3), atol=1e-7) + assert_allclose(ret_bounds_list.fun, ret_bounds_class.fun, atol=1e-9) + assert ret_bounds_list.nfev == ret_bounds_class.nfev + + @pytest.mark.fail_slow(10) + def test_callable_jac_hess_with_args_gh11052(self): + # dual_annealing used to fail when `jac` was callable and `args` were + # used; check that this is resolved. Example is from gh-11052. + + # extended to hess as part of closing gh20614 + rng = np.random.default_rng(94253637693657847462) + def f(x, power): + return np.sum(np.exp(x ** power)) + + def jac(x, power): + return np.exp(x ** power) * power * x ** (power - 1) + + def hess(x, power): + # calculated using WolframAlpha as d^2/dx^2 e^(x^p) + return np.diag( + power * np.exp(x ** power) * x ** (power - 2) * + (power * x ** power + power - 1) + ) + + def hessp(x, p, power): + return hess(x, power) @ p + + res1 = dual_annealing(f, args=(2, ), bounds=[[0, 1], [0, 1]], rng=rng, + minimizer_kwargs=dict(method='L-BFGS-B')) + res2 = dual_annealing(f, args=(2, ), bounds=[[0, 1], [0, 1]], rng=rng, + minimizer_kwargs=dict(method='L-BFGS-B', + jac=jac)) + res3 = dual_annealing(f, args=(2, ), bounds=[[0, 1], [0, 1]], rng=rng, + minimizer_kwargs=dict(method='newton-cg', + jac=jac, hess=hess)) + res4 = dual_annealing(f, args=(2, ), bounds=[[0, 1], [0, 1]], rng=rng, + minimizer_kwargs=dict(method='newton-cg', + jac=jac, hessp=hessp)) + assert_allclose(res1.fun, res2.fun, rtol=1e-6) + assert_allclose(res3.fun, res2.fun, rtol=1e-6) + assert_allclose(res4.fun, res2.fun, rtol=1e-6) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__linprog_clean_inputs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__linprog_clean_inputs.py new file mode 100644 index 0000000000000000000000000000000000000000..3b0e4097bc9aadbfd3335aa3a86d063216f2c69a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__linprog_clean_inputs.py @@ -0,0 +1,310 @@ +""" +Unit test for Linear Programming via Simplex Algorithm. +""" +import numpy as np +from numpy.testing import assert_, assert_allclose, assert_equal +from pytest import raises as assert_raises +from scipy.optimize._linprog_util import _clean_inputs, _LPProblem +from scipy._lib._util import VisibleDeprecationWarning +from copy import deepcopy +from datetime import date + + +def test_aliasing(): + """ + Test for ensuring that no objects referred to by `lp` attributes, + `c`, `A_ub`, `b_ub`, `A_eq`, `b_eq`, `bounds`, have been modified + by `_clean_inputs` as a side effect. + """ + lp = _LPProblem( + c=1, + A_ub=[[1]], + b_ub=[1], + A_eq=[[1]], + b_eq=[1], + bounds=(-np.inf, np.inf) + ) + lp_copy = deepcopy(lp) + + _clean_inputs(lp) + + assert_(lp.c == lp_copy.c, "c modified by _clean_inputs") + assert_(lp.A_ub == lp_copy.A_ub, "A_ub modified by _clean_inputs") + assert_(lp.b_ub == lp_copy.b_ub, "b_ub modified by _clean_inputs") + assert_(lp.A_eq == lp_copy.A_eq, "A_eq modified by _clean_inputs") + assert_(lp.b_eq == lp_copy.b_eq, "b_eq modified by _clean_inputs") + assert_(lp.bounds == lp_copy.bounds, "bounds modified by _clean_inputs") + + +def test_aliasing2(): + """ + Similar purpose as `test_aliasing` above. + """ + lp = _LPProblem( + c=np.array([1, 1]), + A_ub=np.array([[1, 1], [2, 2]]), + b_ub=np.array([[1], [1]]), + A_eq=np.array([[1, 1]]), + b_eq=np.array([1]), + bounds=[(-np.inf, np.inf), (None, 1)] + ) + lp_copy = deepcopy(lp) + + _clean_inputs(lp) + + assert_allclose(lp.c, lp_copy.c, err_msg="c modified by _clean_inputs") + assert_allclose(lp.A_ub, lp_copy.A_ub, err_msg="A_ub modified by _clean_inputs") + assert_allclose(lp.b_ub, lp_copy.b_ub, err_msg="b_ub modified by _clean_inputs") + assert_allclose(lp.A_eq, lp_copy.A_eq, err_msg="A_eq modified by _clean_inputs") + assert_allclose(lp.b_eq, lp_copy.b_eq, err_msg="b_eq modified by _clean_inputs") + assert_(lp.bounds == lp_copy.bounds, "bounds modified by _clean_inputs") + + +def test_missing_inputs(): + c = [1, 2] + A_ub = np.array([[1, 1], [2, 2]]) + b_ub = np.array([1, 1]) + A_eq = np.array([[1, 1], [2, 2]]) + b_eq = np.array([1, 1]) + + assert_raises(TypeError, _clean_inputs) + assert_raises(TypeError, _clean_inputs, _LPProblem(c=None)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, A_ub=A_ub)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, A_ub=A_ub, b_ub=None)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, b_ub=b_ub)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, A_ub=None, b_ub=b_ub)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, A_eq=A_eq)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, A_eq=A_eq, b_eq=None)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, b_eq=b_eq)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, A_eq=None, b_eq=b_eq)) + + +def test_too_many_dimensions(): + cb = [1, 2, 3, 4] + A = np.random.rand(4, 4) + bad2D = [[1, 2], [3, 4]] + bad3D = np.random.rand(4, 4, 4) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=bad2D, A_ub=A, b_ub=cb)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=cb, A_ub=bad3D, b_ub=cb)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=cb, A_ub=A, b_ub=bad2D)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=cb, A_eq=bad3D, b_eq=cb)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=cb, A_eq=A, b_eq=bad2D)) + + +def test_too_few_dimensions(): + bad = np.random.rand(4, 4).ravel() + cb = np.random.rand(4) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=cb, A_ub=bad, b_ub=cb)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=cb, A_eq=bad, b_eq=cb)) + + +def test_inconsistent_dimensions(): + m = 2 + n = 4 + c = [1, 2, 3, 4] + + Agood = np.random.rand(m, n) + Abad = np.random.rand(m, n + 1) + bgood = np.random.rand(m) + bbad = np.random.rand(m + 1) + boundsbad = [(0, 1)] * (n + 1) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, A_ub=Abad, b_ub=bgood)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, A_ub=Agood, b_ub=bbad)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, A_eq=Abad, b_eq=bgood)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, A_eq=Agood, b_eq=bbad)) + assert_raises(ValueError, _clean_inputs, _LPProblem(c=c, bounds=boundsbad)) + with np.testing.suppress_warnings() as sup: + sup.filter(VisibleDeprecationWarning, "Creating an ndarray from ragged") + assert_raises(ValueError, _clean_inputs, + _LPProblem(c=c, bounds=[[1, 2], [2, 3], [3, 4], [4, 5, 6]])) + + +def test_type_errors(): + lp = _LPProblem( + c=[1, 2], + A_ub=np.array([[1, 1], [2, 2]]), + b_ub=np.array([1, 1]), + A_eq=np.array([[1, 1], [2, 2]]), + b_eq=np.array([1, 1]), + bounds=[(0, 1)] + ) + bad = "hello" + + assert_raises(TypeError, _clean_inputs, lp._replace(c=bad)) + assert_raises(TypeError, _clean_inputs, lp._replace(A_ub=bad)) + assert_raises(TypeError, _clean_inputs, lp._replace(b_ub=bad)) + assert_raises(TypeError, _clean_inputs, lp._replace(A_eq=bad)) + assert_raises(TypeError, _clean_inputs, lp._replace(b_eq=bad)) + + assert_raises(ValueError, _clean_inputs, lp._replace(bounds=bad)) + assert_raises(ValueError, _clean_inputs, lp._replace(bounds="hi")) + assert_raises(ValueError, _clean_inputs, lp._replace(bounds=["hi"])) + assert_raises(ValueError, _clean_inputs, lp._replace(bounds=[("hi")])) + assert_raises(ValueError, _clean_inputs, lp._replace(bounds=[(1, "")])) + assert_raises(ValueError, _clean_inputs, lp._replace(bounds=[(1, 2), (1, "")])) + assert_raises(TypeError, _clean_inputs, + lp._replace(bounds=[(1, date(2020, 2, 29))])) + assert_raises(ValueError, _clean_inputs, lp._replace(bounds=[[[1, 2]]])) + + +def test_non_finite_errors(): + lp = _LPProblem( + c=[1, 2], + A_ub=np.array([[1, 1], [2, 2]]), + b_ub=np.array([1, 1]), + A_eq=np.array([[1, 1], [2, 2]]), + b_eq=np.array([1, 1]), + bounds=[(0, 1)] + ) + assert_raises(ValueError, _clean_inputs, lp._replace(c=[0, None])) + assert_raises(ValueError, _clean_inputs, lp._replace(c=[np.inf, 0])) + assert_raises(ValueError, _clean_inputs, lp._replace(c=[0, -np.inf])) + assert_raises(ValueError, _clean_inputs, lp._replace(c=[np.nan, 0])) + + assert_raises(ValueError, _clean_inputs, lp._replace(A_ub=[[1, 2], [None, 1]])) + assert_raises(ValueError, _clean_inputs, lp._replace(b_ub=[np.inf, 1])) + assert_raises(ValueError, _clean_inputs, lp._replace(A_eq=[[1, 2], [1, -np.inf]])) + assert_raises(ValueError, _clean_inputs, lp._replace(b_eq=[1, np.nan])) + + +def test__clean_inputs1(): + lp = _LPProblem( + c=[1, 2], + A_ub=[[1, 1], [2, 2]], + b_ub=[1, 1], + A_eq=[[1, 1], [2, 2]], + b_eq=[1, 1], + bounds=None + ) + + lp_cleaned = _clean_inputs(lp) + + assert_allclose(lp_cleaned.c, np.array(lp.c)) + assert_allclose(lp_cleaned.A_ub, np.array(lp.A_ub)) + assert_allclose(lp_cleaned.b_ub, np.array(lp.b_ub)) + assert_allclose(lp_cleaned.A_eq, np.array(lp.A_eq)) + assert_allclose(lp_cleaned.b_eq, np.array(lp.b_eq)) + assert_equal(lp_cleaned.bounds, [(0, np.inf)] * 2) + + assert_(lp_cleaned.c.shape == (2,), "") + assert_(lp_cleaned.A_ub.shape == (2, 2), "") + assert_(lp_cleaned.b_ub.shape == (2,), "") + assert_(lp_cleaned.A_eq.shape == (2, 2), "") + assert_(lp_cleaned.b_eq.shape == (2,), "") + + +def test__clean_inputs2(): + lp = _LPProblem( + c=1, + A_ub=[[1]], + b_ub=1, + A_eq=[[1]], + b_eq=1, + bounds=(0, 1) + ) + + lp_cleaned = _clean_inputs(lp) + + assert_allclose(lp_cleaned.c, np.array(lp.c)) + assert_allclose(lp_cleaned.A_ub, np.array(lp.A_ub)) + assert_allclose(lp_cleaned.b_ub, np.array(lp.b_ub)) + assert_allclose(lp_cleaned.A_eq, np.array(lp.A_eq)) + assert_allclose(lp_cleaned.b_eq, np.array(lp.b_eq)) + assert_equal(lp_cleaned.bounds, [(0, 1)]) + + assert_(lp_cleaned.c.shape == (1,), "") + assert_(lp_cleaned.A_ub.shape == (1, 1), "") + assert_(lp_cleaned.b_ub.shape == (1,), "") + assert_(lp_cleaned.A_eq.shape == (1, 1), "") + assert_(lp_cleaned.b_eq.shape == (1,), "") + + +def test__clean_inputs3(): + lp = _LPProblem( + c=[[1, 2]], + A_ub=np.random.rand(2, 2), + b_ub=[[1], [2]], + A_eq=np.random.rand(2, 2), + b_eq=[[1], [2]], + bounds=[(0, 1)] + ) + + lp_cleaned = _clean_inputs(lp) + + assert_allclose(lp_cleaned.c, np.array([1, 2])) + assert_allclose(lp_cleaned.b_ub, np.array([1, 2])) + assert_allclose(lp_cleaned.b_eq, np.array([1, 2])) + assert_equal(lp_cleaned.bounds, [(0, 1)] * 2) + + assert_(lp_cleaned.c.shape == (2,), "") + assert_(lp_cleaned.b_ub.shape == (2,), "") + assert_(lp_cleaned.b_eq.shape == (2,), "") + + +def test_bad_bounds(): + lp = _LPProblem(c=[1, 2]) + + assert_raises(ValueError, _clean_inputs, lp._replace(bounds=(1, 2, 2))) + assert_raises(ValueError, _clean_inputs, lp._replace(bounds=[(1, 2, 2)])) + with np.testing.suppress_warnings() as sup: + sup.filter(VisibleDeprecationWarning, "Creating an ndarray from ragged") + assert_raises(ValueError, _clean_inputs, + lp._replace(bounds=[(1, 2), (1, 2, 2)])) + assert_raises(ValueError, _clean_inputs, + lp._replace(bounds=[(1, 2), (1, 2), (1, 2)])) + + lp = _LPProblem(c=[1, 2, 3, 4]) + + assert_raises(ValueError, _clean_inputs, + lp._replace(bounds=[(1, 2, 3, 4), (1, 2, 3, 4)])) + + +def test_good_bounds(): + lp = _LPProblem(c=[1, 2]) + + lp_cleaned = _clean_inputs(lp) # lp.bounds is None by default + assert_equal(lp_cleaned.bounds, [(0, np.inf)] * 2) + + lp_cleaned = _clean_inputs(lp._replace(bounds=[])) + assert_equal(lp_cleaned.bounds, [(0, np.inf)] * 2) + + lp_cleaned = _clean_inputs(lp._replace(bounds=[[]])) + assert_equal(lp_cleaned.bounds, [(0, np.inf)] * 2) + + lp_cleaned = _clean_inputs(lp._replace(bounds=(1, 2))) + assert_equal(lp_cleaned.bounds, [(1, 2)] * 2) + + lp_cleaned = _clean_inputs(lp._replace(bounds=[(1, 2)])) + assert_equal(lp_cleaned.bounds, [(1, 2)] * 2) + + lp_cleaned = _clean_inputs(lp._replace(bounds=[(1, None)])) + assert_equal(lp_cleaned.bounds, [(1, np.inf)] * 2) + + lp_cleaned = _clean_inputs(lp._replace(bounds=[(None, 1)])) + assert_equal(lp_cleaned.bounds, [(-np.inf, 1)] * 2) + + lp_cleaned = _clean_inputs(lp._replace(bounds=[(None, None), (-np.inf, None)])) + assert_equal(lp_cleaned.bounds, [(-np.inf, np.inf)] * 2) + + lp = _LPProblem(c=[1, 2, 3, 4]) + + lp_cleaned = _clean_inputs(lp) # lp.bounds is None by default + assert_equal(lp_cleaned.bounds, [(0, np.inf)] * 4) + + lp_cleaned = _clean_inputs(lp._replace(bounds=(1, 2))) + assert_equal(lp_cleaned.bounds, [(1, 2)] * 4) + + lp_cleaned = _clean_inputs(lp._replace(bounds=[(1, 2)])) + assert_equal(lp_cleaned.bounds, [(1, 2)] * 4) + + lp_cleaned = _clean_inputs(lp._replace(bounds=[(1, None)])) + assert_equal(lp_cleaned.bounds, [(1, np.inf)] * 4) + + lp_cleaned = _clean_inputs(lp._replace(bounds=[(None, 1)])) + assert_equal(lp_cleaned.bounds, [(-np.inf, 1)] * 4) + + lp_cleaned = _clean_inputs(lp._replace(bounds=[(None, None), + (-np.inf, None), + (None, np.inf), + (-np.inf, np.inf)])) + assert_equal(lp_cleaned.bounds, [(-np.inf, np.inf)] * 4) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__numdiff.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__numdiff.py new file mode 100644 index 0000000000000000000000000000000000000000..21fcf36b01f480ea23e15b67e4bebb1270d63c3b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__numdiff.py @@ -0,0 +1,841 @@ +import math +from itertools import product + +import numpy as np +from numpy.testing import assert_allclose, assert_equal, assert_ +from pytest import raises as assert_raises + +from scipy.sparse import csr_matrix, csc_matrix, lil_matrix + +from scipy.optimize._numdiff import ( + _adjust_scheme_to_bounds, approx_derivative, check_derivative, + group_columns, _eps_for_method, _compute_absolute_step) + + +def test_group_columns(): + structure = [ + [1, 1, 0, 0, 0, 0], + [1, 1, 1, 0, 0, 0], + [0, 1, 1, 1, 0, 0], + [0, 0, 1, 1, 1, 0], + [0, 0, 0, 1, 1, 1], + [0, 0, 0, 0, 1, 1], + [0, 0, 0, 0, 0, 0] + ] + for transform in [np.asarray, csr_matrix, csc_matrix, lil_matrix]: + A = transform(structure) + order = np.arange(6) + groups_true = np.array([0, 1, 2, 0, 1, 2]) + groups = group_columns(A, order) + assert_equal(groups, groups_true) + + order = [1, 2, 4, 3, 5, 0] + groups_true = np.array([2, 0, 1, 2, 0, 1]) + groups = group_columns(A, order) + assert_equal(groups, groups_true) + + # Test repeatability. + groups_1 = group_columns(A) + groups_2 = group_columns(A) + assert_equal(groups_1, groups_2) + + +def test_correct_fp_eps(): + # check that relative step size is correct for FP size + EPS = np.finfo(np.float64).eps + relative_step = {"2-point": EPS**0.5, + "3-point": EPS**(1/3), + "cs": EPS**0.5} + for method in ['2-point', '3-point', 'cs']: + assert_allclose( + _eps_for_method(np.float64, np.float64, method), + relative_step[method]) + assert_allclose( + _eps_for_method(np.complex128, np.complex128, method), + relative_step[method] + ) + + # check another FP size + EPS = np.finfo(np.float32).eps + relative_step = {"2-point": EPS**0.5, + "3-point": EPS**(1/3), + "cs": EPS**0.5} + + for method in ['2-point', '3-point', 'cs']: + assert_allclose( + _eps_for_method(np.float64, np.float32, method), + relative_step[method] + ) + assert_allclose( + _eps_for_method(np.float32, np.float64, method), + relative_step[method] + ) + assert_allclose( + _eps_for_method(np.float32, np.float32, method), + relative_step[method] + ) + + +class TestAdjustSchemeToBounds: + def test_no_bounds(self): + x0 = np.zeros(3) + h = np.full(3, 1e-2) + inf_lower = np.empty_like(x0) + inf_upper = np.empty_like(x0) + inf_lower.fill(-np.inf) + inf_upper.fill(np.inf) + + h_adjusted, one_sided = _adjust_scheme_to_bounds( + x0, h, 1, '1-sided', inf_lower, inf_upper) + assert_allclose(h_adjusted, h) + assert_(np.all(one_sided)) + + h_adjusted, one_sided = _adjust_scheme_to_bounds( + x0, h, 2, '1-sided', inf_lower, inf_upper) + assert_allclose(h_adjusted, h) + assert_(np.all(one_sided)) + + h_adjusted, one_sided = _adjust_scheme_to_bounds( + x0, h, 1, '2-sided', inf_lower, inf_upper) + assert_allclose(h_adjusted, h) + assert_(np.all(~one_sided)) + + h_adjusted, one_sided = _adjust_scheme_to_bounds( + x0, h, 2, '2-sided', inf_lower, inf_upper) + assert_allclose(h_adjusted, h) + assert_(np.all(~one_sided)) + + def test_with_bound(self): + x0 = np.array([0.0, 0.85, -0.85]) + lb = -np.ones(3) + ub = np.ones(3) + h = np.array([1, 1, -1]) * 1e-1 + + h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 1, '1-sided', lb, ub) + assert_allclose(h_adjusted, h) + + h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 2, '1-sided', lb, ub) + assert_allclose(h_adjusted, np.array([1, -1, 1]) * 1e-1) + + h_adjusted, one_sided = _adjust_scheme_to_bounds( + x0, h, 1, '2-sided', lb, ub) + assert_allclose(h_adjusted, np.abs(h)) + assert_(np.all(~one_sided)) + + h_adjusted, one_sided = _adjust_scheme_to_bounds( + x0, h, 2, '2-sided', lb, ub) + assert_allclose(h_adjusted, np.array([1, -1, 1]) * 1e-1) + assert_equal(one_sided, np.array([False, True, True])) + + def test_tight_bounds(self): + lb = np.array([-0.03, -0.03]) + ub = np.array([0.05, 0.05]) + x0 = np.array([0.0, 0.03]) + h = np.array([-0.1, -0.1]) + + h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 1, '1-sided', lb, ub) + assert_allclose(h_adjusted, np.array([0.05, -0.06])) + + h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 2, '1-sided', lb, ub) + assert_allclose(h_adjusted, np.array([0.025, -0.03])) + + h_adjusted, one_sided = _adjust_scheme_to_bounds( + x0, h, 1, '2-sided', lb, ub) + assert_allclose(h_adjusted, np.array([0.03, -0.03])) + assert_equal(one_sided, np.array([False, True])) + + h_adjusted, one_sided = _adjust_scheme_to_bounds( + x0, h, 2, '2-sided', lb, ub) + assert_allclose(h_adjusted, np.array([0.015, -0.015])) + assert_equal(one_sided, np.array([False, True])) + + +class TestApproxDerivativesDense: + def fun_scalar_scalar(self, x): + return np.sinh(x) + + def jac_scalar_scalar(self, x): + return np.cosh(x) + + def fun_scalar_vector(self, x): + return np.array([x[0]**2, np.tan(x[0]), np.exp(x[0])]) + + def jac_scalar_vector(self, x): + return np.array( + [2 * x[0], np.cos(x[0]) ** -2, np.exp(x[0])]).reshape(-1, 1) + + def fun_vector_scalar(self, x): + return np.sin(x[0] * x[1]) * np.log(x[0]) + + def wrong_dimensions_fun(self, x): + return np.array([x**2, np.tan(x), np.exp(x)]) + + def jac_vector_scalar(self, x): + return np.array([ + x[1] * np.cos(x[0] * x[1]) * np.log(x[0]) + + np.sin(x[0] * x[1]) / x[0], + x[0] * np.cos(x[0] * x[1]) * np.log(x[0]) + ]) + + def fun_vector_vector(self, x): + return np.array([ + x[0] * np.sin(x[1]), + x[1] * np.cos(x[0]), + x[0] ** 3 * x[1] ** -0.5 + ]) + + def fun_vector_vector_with_arg(self, x, arg): + """Used to test passing custom arguments with check_derivative()""" + assert arg == 42 + return np.array([ + x[0] * np.sin(x[1]), + x[1] * np.cos(x[0]), + x[0] ** 3 * x[1] ** -0.5 + ]) + + def jac_vector_vector(self, x): + return np.array([ + [np.sin(x[1]), x[0] * np.cos(x[1])], + [-x[1] * np.sin(x[0]), np.cos(x[0])], + [3 * x[0] ** 2 * x[1] ** -0.5, -0.5 * x[0] ** 3 * x[1] ** -1.5] + ]) + + def jac_vector_vector_with_arg(self, x, arg): + """Used to test passing custom arguments with check_derivative()""" + assert arg == 42 + return np.array([ + [np.sin(x[1]), x[0] * np.cos(x[1])], + [-x[1] * np.sin(x[0]), np.cos(x[0])], + [3 * x[0] ** 2 * x[1] ** -0.5, -0.5 * x[0] ** 3 * x[1] ** -1.5] + ]) + + def fun_parametrized(self, x, c0, c1=1.0): + return np.array([np.exp(c0 * x[0]), np.exp(c1 * x[1])]) + + def jac_parametrized(self, x, c0, c1=0.1): + return np.array([ + [c0 * np.exp(c0 * x[0]), 0], + [0, c1 * np.exp(c1 * x[1])] + ]) + + def fun_with_nan(self, x): + return x if np.abs(x) <= 1e-8 else np.nan + + def jac_with_nan(self, x): + return 1.0 if np.abs(x) <= 1e-8 else np.nan + + def fun_zero_jacobian(self, x): + return np.array([x[0] * x[1], np.cos(x[0] * x[1])]) + + def jac_zero_jacobian(self, x): + return np.array([ + [x[1], x[0]], + [-x[1] * np.sin(x[0] * x[1]), -x[0] * np.sin(x[0] * x[1])] + ]) + + def jac_non_numpy(self, x): + # x can be a scalar or an array [val]. + # Cast to true scalar before handing over to math.exp + xp = np.asarray(x).item() + return math.exp(xp) + + def test_scalar_scalar(self): + x0 = 1.0 + jac_diff_2 = approx_derivative(self.fun_scalar_scalar, x0, + method='2-point') + jac_diff_3 = approx_derivative(self.fun_scalar_scalar, x0) + jac_diff_4 = approx_derivative(self.fun_scalar_scalar, x0, + method='cs') + jac_true = self.jac_scalar_scalar(x0) + assert_allclose(jac_diff_2, jac_true, rtol=1e-6) + assert_allclose(jac_diff_3, jac_true, rtol=1e-9) + assert_allclose(jac_diff_4, jac_true, rtol=1e-12) + + def test_scalar_scalar_abs_step(self): + # can approx_derivative use abs_step? + x0 = 1.0 + jac_diff_2 = approx_derivative(self.fun_scalar_scalar, x0, + method='2-point', abs_step=1.49e-8) + jac_diff_3 = approx_derivative(self.fun_scalar_scalar, x0, + abs_step=1.49e-8) + jac_diff_4 = approx_derivative(self.fun_scalar_scalar, x0, + method='cs', abs_step=1.49e-8) + jac_true = self.jac_scalar_scalar(x0) + assert_allclose(jac_diff_2, jac_true, rtol=1e-6) + assert_allclose(jac_diff_3, jac_true, rtol=1e-9) + assert_allclose(jac_diff_4, jac_true, rtol=1e-12) + + def test_scalar_vector(self): + x0 = 0.5 + jac_diff_2 = approx_derivative(self.fun_scalar_vector, x0, + method='2-point') + jac_diff_3 = approx_derivative(self.fun_scalar_vector, x0) + jac_diff_4 = approx_derivative(self.fun_scalar_vector, x0, + method='cs') + jac_true = self.jac_scalar_vector(np.atleast_1d(x0)) + assert_allclose(jac_diff_2, jac_true, rtol=1e-6) + assert_allclose(jac_diff_3, jac_true, rtol=1e-9) + assert_allclose(jac_diff_4, jac_true, rtol=1e-12) + + def test_vector_scalar(self): + x0 = np.array([100.0, -0.5]) + jac_diff_2 = approx_derivative(self.fun_vector_scalar, x0, + method='2-point') + jac_diff_3 = approx_derivative(self.fun_vector_scalar, x0) + jac_diff_4 = approx_derivative(self.fun_vector_scalar, x0, + method='cs') + jac_true = self.jac_vector_scalar(x0) + assert_allclose(jac_diff_2, jac_true, rtol=1e-6) + assert_allclose(jac_diff_3, jac_true, rtol=1e-7) + assert_allclose(jac_diff_4, jac_true, rtol=1e-12) + + def test_vector_scalar_abs_step(self): + # can approx_derivative use abs_step? + x0 = np.array([100.0, -0.5]) + jac_diff_2 = approx_derivative(self.fun_vector_scalar, x0, + method='2-point', abs_step=1.49e-8) + jac_diff_3 = approx_derivative(self.fun_vector_scalar, x0, + abs_step=1.49e-8, rel_step=np.inf) + jac_diff_4 = approx_derivative(self.fun_vector_scalar, x0, + method='cs', abs_step=1.49e-8) + jac_true = self.jac_vector_scalar(x0) + assert_allclose(jac_diff_2, jac_true, rtol=1e-6) + assert_allclose(jac_diff_3, jac_true, rtol=3e-9) + assert_allclose(jac_diff_4, jac_true, rtol=1e-12) + + def test_vector_vector(self): + x0 = np.array([-100.0, 0.2]) + jac_diff_2 = approx_derivative(self.fun_vector_vector, x0, + method='2-point') + jac_diff_3 = approx_derivative(self.fun_vector_vector, x0) + jac_diff_4 = approx_derivative(self.fun_vector_vector, x0, + method='cs') + jac_true = self.jac_vector_vector(x0) + assert_allclose(jac_diff_2, jac_true, rtol=1e-5) + assert_allclose(jac_diff_3, jac_true, rtol=1e-6) + assert_allclose(jac_diff_4, jac_true, rtol=1e-12) + + def test_wrong_dimensions(self): + x0 = 1.0 + assert_raises(RuntimeError, approx_derivative, + self.wrong_dimensions_fun, x0) + f0 = self.wrong_dimensions_fun(np.atleast_1d(x0)) + assert_raises(ValueError, approx_derivative, + self.wrong_dimensions_fun, x0, f0=f0) + + def test_custom_rel_step(self): + x0 = np.array([-0.1, 0.1]) + jac_diff_2 = approx_derivative(self.fun_vector_vector, x0, + method='2-point', rel_step=1e-4) + jac_diff_3 = approx_derivative(self.fun_vector_vector, x0, + rel_step=1e-4) + jac_true = self.jac_vector_vector(x0) + assert_allclose(jac_diff_2, jac_true, rtol=1e-2) + assert_allclose(jac_diff_3, jac_true, rtol=1e-4) + + def test_options(self): + x0 = np.array([1.0, 1.0]) + c0 = -1.0 + c1 = 1.0 + lb = 0.0 + ub = 2.0 + f0 = self.fun_parametrized(x0, c0, c1=c1) + rel_step = np.array([-1e-6, 1e-7]) + jac_true = self.jac_parametrized(x0, c0, c1) + jac_diff_2 = approx_derivative( + self.fun_parametrized, x0, method='2-point', rel_step=rel_step, + f0=f0, args=(c0,), kwargs=dict(c1=c1), bounds=(lb, ub)) + jac_diff_3 = approx_derivative( + self.fun_parametrized, x0, rel_step=rel_step, + f0=f0, args=(c0,), kwargs=dict(c1=c1), bounds=(lb, ub)) + assert_allclose(jac_diff_2, jac_true, rtol=1e-6) + assert_allclose(jac_diff_3, jac_true, rtol=1e-9) + + def test_with_bounds_2_point(self): + lb = -np.ones(2) + ub = np.ones(2) + + x0 = np.array([-2.0, 0.2]) + assert_raises(ValueError, approx_derivative, + self.fun_vector_vector, x0, bounds=(lb, ub)) + + x0 = np.array([-1.0, 1.0]) + jac_diff = approx_derivative(self.fun_vector_vector, x0, + method='2-point', bounds=(lb, ub)) + jac_true = self.jac_vector_vector(x0) + assert_allclose(jac_diff, jac_true, rtol=1e-6) + + def test_with_bounds_3_point(self): + lb = np.array([1.0, 1.0]) + ub = np.array([2.0, 2.0]) + + x0 = np.array([1.0, 2.0]) + jac_true = self.jac_vector_vector(x0) + + jac_diff = approx_derivative(self.fun_vector_vector, x0) + assert_allclose(jac_diff, jac_true, rtol=1e-9) + + jac_diff = approx_derivative(self.fun_vector_vector, x0, + bounds=(lb, np.inf)) + assert_allclose(jac_diff, jac_true, rtol=1e-9) + + jac_diff = approx_derivative(self.fun_vector_vector, x0, + bounds=(-np.inf, ub)) + assert_allclose(jac_diff, jac_true, rtol=1e-9) + + jac_diff = approx_derivative(self.fun_vector_vector, x0, + bounds=(lb, ub)) + assert_allclose(jac_diff, jac_true, rtol=1e-9) + + def test_tight_bounds(self): + x0 = np.array([10.0, 10.0]) + lb = x0 - 3e-9 + ub = x0 + 2e-9 + jac_true = self.jac_vector_vector(x0) + jac_diff = approx_derivative( + self.fun_vector_vector, x0, method='2-point', bounds=(lb, ub)) + assert_allclose(jac_diff, jac_true, rtol=1e-6) + jac_diff = approx_derivative( + self.fun_vector_vector, x0, method='2-point', + rel_step=1e-6, bounds=(lb, ub)) + assert_allclose(jac_diff, jac_true, rtol=1e-6) + + jac_diff = approx_derivative( + self.fun_vector_vector, x0, bounds=(lb, ub)) + assert_allclose(jac_diff, jac_true, rtol=1e-6) + jac_diff = approx_derivative( + self.fun_vector_vector, x0, rel_step=1e-6, bounds=(lb, ub)) + assert_allclose(jac_true, jac_diff, rtol=1e-6) + + def test_bound_switches(self): + lb = -1e-8 + ub = 1e-8 + x0 = 0.0 + jac_true = self.jac_with_nan(x0) + jac_diff_2 = approx_derivative( + self.fun_with_nan, x0, method='2-point', rel_step=1e-6, + bounds=(lb, ub)) + jac_diff_3 = approx_derivative( + self.fun_with_nan, x0, rel_step=1e-6, bounds=(lb, ub)) + assert_allclose(jac_diff_2, jac_true, rtol=1e-6) + assert_allclose(jac_diff_3, jac_true, rtol=1e-9) + + x0 = 1e-8 + jac_true = self.jac_with_nan(x0) + jac_diff_2 = approx_derivative( + self.fun_with_nan, x0, method='2-point', rel_step=1e-6, + bounds=(lb, ub)) + jac_diff_3 = approx_derivative( + self.fun_with_nan, x0, rel_step=1e-6, bounds=(lb, ub)) + assert_allclose(jac_diff_2, jac_true, rtol=1e-6) + assert_allclose(jac_diff_3, jac_true, rtol=1e-9) + + def test_non_numpy(self): + x0 = 1.0 + jac_true = self.jac_non_numpy(x0) + jac_diff_2 = approx_derivative(self.jac_non_numpy, x0, + method='2-point') + jac_diff_3 = approx_derivative(self.jac_non_numpy, x0) + assert_allclose(jac_diff_2, jac_true, rtol=1e-6) + assert_allclose(jac_diff_3, jac_true, rtol=1e-8) + + # math.exp cannot handle complex arguments, hence this raises + assert_raises(TypeError, approx_derivative, self.jac_non_numpy, x0, + **dict(method='cs')) + + def test_fp(self): + # checks that approx_derivative works for FP size other than 64. + # Example is derived from the minimal working example in gh12991. + np.random.seed(1) + + def func(p, x): + return p[0] + p[1] * x + + def err(p, x, y): + return func(p, x) - y + + x = np.linspace(0, 1, 100, dtype=np.float64) + y = np.random.random(100).astype(np.float64) + p0 = np.array([-1.0, -1.0]) + + jac_fp64 = approx_derivative(err, p0, method='2-point', args=(x, y)) + + # parameter vector is float32, func output is float64 + jac_fp = approx_derivative(err, p0.astype(np.float32), + method='2-point', args=(x, y)) + assert err(p0, x, y).dtype == np.float64 + assert_allclose(jac_fp, jac_fp64, atol=1e-3) + + # parameter vector is float64, func output is float32 + def err_fp32(p): + assert p.dtype == np.float32 + return err(p, x, y).astype(np.float32) + + jac_fp = approx_derivative(err_fp32, p0.astype(np.float32), + method='2-point') + assert_allclose(jac_fp, jac_fp64, atol=1e-3) + + # check upper bound of error on the derivative for 2-point + def f(x): + return np.sin(x) + def g(x): + return np.cos(x) + def hess(x): + return -np.sin(x) + + def calc_atol(h, x0, f, hess, EPS): + # truncation error + t0 = h / 2 * max(np.abs(hess(x0)), np.abs(hess(x0 + h))) + # roundoff error. There may be a divisor (>1) missing from + # the following line, so this contribution is possibly + # overestimated + t1 = EPS / h * max(np.abs(f(x0)), np.abs(f(x0 + h))) + return t0 + t1 + + for dtype in [np.float16, np.float32, np.float64]: + EPS = np.finfo(dtype).eps + x0 = np.array(1.0).astype(dtype) + h = _compute_absolute_step(None, x0, f(x0), '2-point') + atol = calc_atol(h, x0, f, hess, EPS) + err = approx_derivative(f, x0, method='2-point', + abs_step=h) - g(x0) + assert abs(err) < atol + + def test_check_derivative(self): + x0 = np.array([-10.0, 10]) + accuracy = check_derivative(self.fun_vector_vector, + self.jac_vector_vector, x0) + assert_(accuracy < 1e-9) + accuracy = check_derivative(self.fun_vector_vector, + self.jac_vector_vector, x0) + assert_(accuracy < 1e-6) + + x0 = np.array([0.0, 0.0]) + accuracy = check_derivative(self.fun_zero_jacobian, + self.jac_zero_jacobian, x0) + assert_(accuracy == 0) + accuracy = check_derivative(self.fun_zero_jacobian, + self.jac_zero_jacobian, x0) + assert_(accuracy == 0) + + def test_check_derivative_with_kwargs(self): + x0 = np.array([-10.0, 10]) + accuracy = check_derivative(self.fun_vector_vector_with_arg, + self.jac_vector_vector_with_arg, + x0, + kwargs={'arg': 42}) + assert_(accuracy < 1e-9) + + +class TestApproxDerivativeSparse: + # Example from Numerical Optimization 2nd edition, p. 198. + def setup_method(self): + np.random.seed(0) + self.n = 50 + self.lb = -0.1 * (1 + np.arange(self.n)) + self.ub = 0.1 * (1 + np.arange(self.n)) + self.x0 = np.empty(self.n) + self.x0[::2] = (1 - 1e-7) * self.lb[::2] + self.x0[1::2] = (1 - 1e-7) * self.ub[1::2] + + self.J_true = self.jac(self.x0) + + def fun(self, x): + e = x[1:]**3 - x[:-1]**2 + return np.hstack((0, 3 * e)) + np.hstack((2 * e, 0)) + + def jac(self, x): + n = x.size + J = np.zeros((n, n)) + J[0, 0] = -4 * x[0] + J[0, 1] = 6 * x[1]**2 + for i in range(1, n - 1): + J[i, i - 1] = -6 * x[i-1] + J[i, i] = 9 * x[i]**2 - 4 * x[i] + J[i, i + 1] = 6 * x[i+1]**2 + J[-1, -1] = 9 * x[-1]**2 + J[-1, -2] = -6 * x[-2] + + return J + + def structure(self, n): + A = np.zeros((n, n), dtype=int) + A[0, 0] = 1 + A[0, 1] = 1 + for i in range(1, n - 1): + A[i, i - 1: i + 2] = 1 + A[-1, -1] = 1 + A[-1, -2] = 1 + + return A + + def test_all(self): + A = self.structure(self.n) + order = np.arange(self.n) + groups_1 = group_columns(A, order) + np.random.shuffle(order) + groups_2 = group_columns(A, order) + + for method, groups, l, u in product( + ['2-point', '3-point', 'cs'], [groups_1, groups_2], + [-np.inf, self.lb], [np.inf, self.ub]): + J = approx_derivative(self.fun, self.x0, method=method, + bounds=(l, u), sparsity=(A, groups)) + assert_(isinstance(J, csr_matrix)) + assert_allclose(J.toarray(), self.J_true, rtol=1e-6) + + rel_step = np.full_like(self.x0, 1e-8) + rel_step[::2] *= -1 + J = approx_derivative(self.fun, self.x0, method=method, + rel_step=rel_step, sparsity=(A, groups)) + assert_allclose(J.toarray(), self.J_true, rtol=1e-5) + + def test_no_precomputed_groups(self): + A = self.structure(self.n) + J = approx_derivative(self.fun, self.x0, sparsity=A) + assert_allclose(J.toarray(), self.J_true, rtol=1e-6) + + def test_equivalence(self): + structure = np.ones((self.n, self.n), dtype=int) + groups = np.arange(self.n) + for method in ['2-point', '3-point', 'cs']: + J_dense = approx_derivative(self.fun, self.x0, method=method) + J_sparse = approx_derivative( + self.fun, self.x0, sparsity=(structure, groups), method=method) + assert_allclose(J_dense, J_sparse.toarray(), + rtol=5e-16, atol=7e-15) + + def test_check_derivative(self): + def jac(x): + return csr_matrix(self.jac(x)) + + accuracy = check_derivative(self.fun, jac, self.x0, + bounds=(self.lb, self.ub)) + assert_(accuracy < 1e-9) + + accuracy = check_derivative(self.fun, jac, self.x0, + bounds=(self.lb, self.ub)) + assert_(accuracy < 1e-9) + + +class TestApproxDerivativeLinearOperator: + + def fun_scalar_scalar(self, x): + return np.sinh(x) + + def jac_scalar_scalar(self, x): + return np.cosh(x) + + def fun_scalar_vector(self, x): + return np.array([x[0]**2, np.tan(x[0]), np.exp(x[0])]) + + def jac_scalar_vector(self, x): + return np.array( + [2 * x[0], np.cos(x[0]) ** -2, np.exp(x[0])]).reshape(-1, 1) + + def fun_vector_scalar(self, x): + return np.sin(x[0] * x[1]) * np.log(x[0]) + + def jac_vector_scalar(self, x): + return np.array([ + x[1] * np.cos(x[0] * x[1]) * np.log(x[0]) + + np.sin(x[0] * x[1]) / x[0], + x[0] * np.cos(x[0] * x[1]) * np.log(x[0]) + ]) + + def fun_vector_vector(self, x): + return np.array([ + x[0] * np.sin(x[1]), + x[1] * np.cos(x[0]), + x[0] ** 3 * x[1] ** -0.5 + ]) + + def jac_vector_vector(self, x): + return np.array([ + [np.sin(x[1]), x[0] * np.cos(x[1])], + [-x[1] * np.sin(x[0]), np.cos(x[0])], + [3 * x[0] ** 2 * x[1] ** -0.5, -0.5 * x[0] ** 3 * x[1] ** -1.5] + ]) + + def test_scalar_scalar(self): + x0 = 1.0 + jac_diff_2 = approx_derivative(self.fun_scalar_scalar, x0, + method='2-point', + as_linear_operator=True) + jac_diff_3 = approx_derivative(self.fun_scalar_scalar, x0, + as_linear_operator=True) + jac_diff_4 = approx_derivative(self.fun_scalar_scalar, x0, + method='cs', + as_linear_operator=True) + jac_true = self.jac_scalar_scalar(x0) + np.random.seed(1) + for i in range(10): + p = np.random.uniform(-10, 10, size=(1,)) + assert_allclose(jac_diff_2.dot(p), jac_true*p, + rtol=1e-5) + assert_allclose(jac_diff_3.dot(p), jac_true*p, + rtol=5e-6) + assert_allclose(jac_diff_4.dot(p), jac_true*p, + rtol=5e-6) + + def test_scalar_vector(self): + x0 = 0.5 + jac_diff_2 = approx_derivative(self.fun_scalar_vector, x0, + method='2-point', + as_linear_operator=True) + jac_diff_3 = approx_derivative(self.fun_scalar_vector, x0, + as_linear_operator=True) + jac_diff_4 = approx_derivative(self.fun_scalar_vector, x0, + method='cs', + as_linear_operator=True) + jac_true = self.jac_scalar_vector(np.atleast_1d(x0)) + np.random.seed(1) + for i in range(10): + p = np.random.uniform(-10, 10, size=(1,)) + assert_allclose(jac_diff_2.dot(p), jac_true.dot(p), + rtol=1e-5) + assert_allclose(jac_diff_3.dot(p), jac_true.dot(p), + rtol=5e-6) + assert_allclose(jac_diff_4.dot(p), jac_true.dot(p), + rtol=5e-6) + + def test_vector_scalar(self): + x0 = np.array([100.0, -0.5]) + jac_diff_2 = approx_derivative(self.fun_vector_scalar, x0, + method='2-point', + as_linear_operator=True) + jac_diff_3 = approx_derivative(self.fun_vector_scalar, x0, + as_linear_operator=True) + jac_diff_4 = approx_derivative(self.fun_vector_scalar, x0, + method='cs', + as_linear_operator=True) + jac_true = self.jac_vector_scalar(x0) + np.random.seed(1) + for i in range(10): + p = np.random.uniform(-10, 10, size=x0.shape) + assert_allclose(jac_diff_2.dot(p), np.atleast_1d(jac_true.dot(p)), + rtol=1e-5) + assert_allclose(jac_diff_3.dot(p), np.atleast_1d(jac_true.dot(p)), + rtol=5e-6) + assert_allclose(jac_diff_4.dot(p), np.atleast_1d(jac_true.dot(p)), + rtol=1e-7) + + def test_vector_vector(self): + x0 = np.array([-100.0, 0.2]) + jac_diff_2 = approx_derivative(self.fun_vector_vector, x0, + method='2-point', + as_linear_operator=True) + jac_diff_3 = approx_derivative(self.fun_vector_vector, x0, + as_linear_operator=True) + jac_diff_4 = approx_derivative(self.fun_vector_vector, x0, + method='cs', + as_linear_operator=True) + jac_true = self.jac_vector_vector(x0) + np.random.seed(1) + for i in range(10): + p = np.random.uniform(-10, 10, size=x0.shape) + assert_allclose(jac_diff_2.dot(p), jac_true.dot(p), rtol=1e-5) + assert_allclose(jac_diff_3.dot(p), jac_true.dot(p), rtol=1e-6) + assert_allclose(jac_diff_4.dot(p), jac_true.dot(p), rtol=1e-7) + + def test_exception(self): + x0 = np.array([-100.0, 0.2]) + assert_raises(ValueError, approx_derivative, + self.fun_vector_vector, x0, + method='2-point', bounds=(1, np.inf)) + + +def test_absolute_step_sign(): + # test for gh12487 + # if an absolute step is specified for 2-point differences make sure that + # the side corresponds to the step. i.e. if step is positive then forward + # differences should be used, if step is negative then backwards + # differences should be used. + + # function has double discontinuity at x = [-1, -1] + # first component is \/, second component is /\ + def f(x): + return -np.abs(x[0] + 1) + np.abs(x[1] + 1) + + # check that the forward difference is used + grad = approx_derivative(f, [-1, -1], method='2-point', abs_step=1e-8) + assert_allclose(grad, [-1.0, 1.0]) + + # check that the backwards difference is used + grad = approx_derivative(f, [-1, -1], method='2-point', abs_step=-1e-8) + assert_allclose(grad, [1.0, -1.0]) + + # check that the forwards difference is used with a step for both + # parameters + grad = approx_derivative( + f, [-1, -1], method='2-point', abs_step=[1e-8, 1e-8] + ) + assert_allclose(grad, [-1.0, 1.0]) + + # check that we can mix forward/backwards steps. + grad = approx_derivative( + f, [-1, -1], method='2-point', abs_step=[1e-8, -1e-8] + ) + assert_allclose(grad, [-1.0, -1.0]) + grad = approx_derivative( + f, [-1, -1], method='2-point', abs_step=[-1e-8, 1e-8] + ) + assert_allclose(grad, [1.0, 1.0]) + + # the forward step should reverse to a backwards step if it runs into a + # bound + # This is kind of tested in TestAdjustSchemeToBounds, but only for a lower level + # function. + grad = approx_derivative( + f, [-1, -1], method='2-point', abs_step=1e-8, + bounds=(-np.inf, -1) + ) + assert_allclose(grad, [1.0, -1.0]) + + grad = approx_derivative( + f, [-1, -1], method='2-point', abs_step=-1e-8, bounds=(-1, np.inf) + ) + assert_allclose(grad, [-1.0, 1.0]) + + +def test__compute_absolute_step(): + # tests calculation of absolute step from rel_step + methods = ['2-point', '3-point', 'cs'] + + x0 = np.array([1e-5, 0, 1, 1e5]) + + EPS = np.finfo(np.float64).eps + relative_step = { + "2-point": EPS**0.5, + "3-point": EPS**(1/3), + "cs": EPS**0.5 + } + f0 = np.array(1.0) + + for method in methods: + rel_step = relative_step[method] + correct_step = np.array([rel_step, + rel_step * 1., + rel_step * 1., + rel_step * np.abs(x0[3])]) + + abs_step = _compute_absolute_step(None, x0, f0, method) + assert_allclose(abs_step, correct_step) + + sign_x0 = (-x0 >= 0).astype(float) * 2 - 1 + abs_step = _compute_absolute_step(None, -x0, f0, method) + assert_allclose(abs_step, sign_x0 * correct_step) + + # if a relative step is provided it should be used + rel_step = np.array([0.1, 1, 10, 100]) + correct_step = np.array([rel_step[0] * x0[0], + relative_step['2-point'], + rel_step[2] * 1., + rel_step[3] * np.abs(x0[3])]) + + abs_step = _compute_absolute_step(rel_step, x0, f0, '2-point') + assert_allclose(abs_step, correct_step) + + sign_x0 = (-x0 >= 0).astype(float) * 2 - 1 + abs_step = _compute_absolute_step(rel_step, -x0, f0, '2-point') + assert_allclose(abs_step, sign_x0 * correct_step) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__remove_redundancy.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__remove_redundancy.py new file mode 100644 index 0000000000000000000000000000000000000000..817282011699dea333042a4173f65c999a2925fc --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__remove_redundancy.py @@ -0,0 +1,228 @@ +""" +Unit test for Linear Programming via Simplex Algorithm. +""" + +# TODO: add tests for: +# https://github.com/scipy/scipy/issues/5400 +# https://github.com/scipy/scipy/issues/6690 + +import numpy as np +from numpy.testing import ( + assert_, + assert_allclose, + assert_equal) + +from .test_linprog import magic_square +from scipy.optimize._remove_redundancy import _remove_redundancy_svd +from scipy.optimize._remove_redundancy import _remove_redundancy_pivot_dense +from scipy.optimize._remove_redundancy import _remove_redundancy_pivot_sparse +from scipy.optimize._remove_redundancy import _remove_redundancy_id + +from scipy.sparse import csc_matrix + + +def setup_module(): + np.random.seed(2017) + + +def redundancy_removed(A, B): + """Checks whether a matrix contains only independent rows of another""" + for rowA in A: + # `rowA in B` is not a reliable check + for rowB in B: + if np.all(rowA == rowB): + break + else: + return False + return A.shape[0] == np.linalg.matrix_rank(A) == np.linalg.matrix_rank(B) + + +class RRCommonTests: + def test_no_redundancy(self): + m, n = 10, 10 + A0 = np.random.rand(m, n) + b0 = np.random.rand(m) + A1, b1, status, message = self.rr(A0, b0) + assert_allclose(A0, A1) + assert_allclose(b0, b1) + assert_equal(status, 0) + + def test_infeasible_zero_row(self): + A = np.eye(3) + A[1, :] = 0 + b = np.random.rand(3) + A1, b1, status, message = self.rr(A, b) + assert_equal(status, 2) + + def test_remove_zero_row(self): + A = np.eye(3) + A[1, :] = 0 + b = np.random.rand(3) + b[1] = 0 + A1, b1, status, message = self.rr(A, b) + assert_equal(status, 0) + assert_allclose(A1, A[[0, 2], :]) + assert_allclose(b1, b[[0, 2]]) + + def test_infeasible_m_gt_n(self): + m, n = 20, 10 + A0 = np.random.rand(m, n) + b0 = np.random.rand(m) + A1, b1, status, message = self.rr(A0, b0) + assert_equal(status, 2) + + def test_infeasible_m_eq_n(self): + m, n = 10, 10 + A0 = np.random.rand(m, n) + b0 = np.random.rand(m) + A0[-1, :] = 2 * A0[-2, :] + A1, b1, status, message = self.rr(A0, b0) + assert_equal(status, 2) + + def test_infeasible_m_lt_n(self): + m, n = 9, 10 + A0 = np.random.rand(m, n) + b0 = np.random.rand(m) + A0[-1, :] = np.arange(m - 1).dot(A0[:-1]) + A1, b1, status, message = self.rr(A0, b0) + assert_equal(status, 2) + + def test_m_gt_n(self): + np.random.seed(2032) + m, n = 20, 10 + A0 = np.random.rand(m, n) + b0 = np.random.rand(m) + x = np.linalg.solve(A0[:n, :], b0[:n]) + b0[n:] = A0[n:, :].dot(x) + A1, b1, status, message = self.rr(A0, b0) + assert_equal(status, 0) + assert_equal(A1.shape[0], n) + assert_equal(np.linalg.matrix_rank(A1), n) + + def test_m_gt_n_rank_deficient(self): + m, n = 20, 10 + A0 = np.zeros((m, n)) + A0[:, 0] = 1 + b0 = np.ones(m) + A1, b1, status, message = self.rr(A0, b0) + assert_equal(status, 0) + assert_allclose(A1, A0[0:1, :]) + assert_allclose(b1, b0[0]) + + def test_m_lt_n_rank_deficient(self): + m, n = 9, 10 + A0 = np.random.rand(m, n) + b0 = np.random.rand(m) + A0[-1, :] = np.arange(m - 1).dot(A0[:-1]) + b0[-1] = np.arange(m - 1).dot(b0[:-1]) + A1, b1, status, message = self.rr(A0, b0) + assert_equal(status, 0) + assert_equal(A1.shape[0], 8) + assert_equal(np.linalg.matrix_rank(A1), 8) + + def test_dense1(self): + A = np.ones((6, 6)) + A[0, :3] = 0 + A[1, 3:] = 0 + A[3:, ::2] = -1 + A[3, :2] = 0 + A[4, 2:] = 0 + b = np.zeros(A.shape[0]) + + A1, b1, status, message = self.rr(A, b) + assert_(redundancy_removed(A1, A)) + assert_equal(status, 0) + + def test_dense2(self): + A = np.eye(6) + A[-2, -1] = 1 + A[-1, :] = 1 + b = np.zeros(A.shape[0]) + A1, b1, status, message = self.rr(A, b) + assert_(redundancy_removed(A1, A)) + assert_equal(status, 0) + + def test_dense3(self): + A = np.eye(6) + A[-2, -1] = 1 + A[-1, :] = 1 + b = np.random.rand(A.shape[0]) + b[-1] = np.sum(b[:-1]) + A1, b1, status, message = self.rr(A, b) + assert_(redundancy_removed(A1, A)) + assert_equal(status, 0) + + def test_m_gt_n_sparse(self): + np.random.seed(2013) + m, n = 20, 5 + p = 0.1 + A = np.random.rand(m, n) + A[np.random.rand(m, n) > p] = 0 + rank = np.linalg.matrix_rank(A) + b = np.zeros(A.shape[0]) + A1, b1, status, message = self.rr(A, b) + assert_equal(status, 0) + assert_equal(A1.shape[0], rank) + assert_equal(np.linalg.matrix_rank(A1), rank) + + def test_m_lt_n_sparse(self): + np.random.seed(2017) + m, n = 20, 50 + p = 0.05 + A = np.random.rand(m, n) + A[np.random.rand(m, n) > p] = 0 + rank = np.linalg.matrix_rank(A) + b = np.zeros(A.shape[0]) + A1, b1, status, message = self.rr(A, b) + assert_equal(status, 0) + assert_equal(A1.shape[0], rank) + assert_equal(np.linalg.matrix_rank(A1), rank) + + def test_m_eq_n_sparse(self): + np.random.seed(2017) + m, n = 100, 100 + p = 0.01 + A = np.random.rand(m, n) + A[np.random.rand(m, n) > p] = 0 + rank = np.linalg.matrix_rank(A) + b = np.zeros(A.shape[0]) + A1, b1, status, message = self.rr(A, b) + assert_equal(status, 0) + assert_equal(A1.shape[0], rank) + assert_equal(np.linalg.matrix_rank(A1), rank) + + def test_magic_square(self): + A, b, c, numbers, _ = magic_square(3) + A1, b1, status, message = self.rr(A, b) + assert_equal(status, 0) + assert_equal(A1.shape[0], 23) + assert_equal(np.linalg.matrix_rank(A1), 23) + + def test_magic_square2(self): + A, b, c, numbers, _ = magic_square(4) + A1, b1, status, message = self.rr(A, b) + assert_equal(status, 0) + assert_equal(A1.shape[0], 39) + assert_equal(np.linalg.matrix_rank(A1), 39) + + +class TestRRSVD(RRCommonTests): + def rr(self, A, b): + return _remove_redundancy_svd(A, b) + + +class TestRRPivotDense(RRCommonTests): + def rr(self, A, b): + return _remove_redundancy_pivot_dense(A, b) + + +class TestRRID(RRCommonTests): + def rr(self, A, b): + return _remove_redundancy_id(A, b) + + +class TestRRPivotSparse(RRCommonTests): + def rr(self, A, b): + rr_res = _remove_redundancy_pivot_sparse(csc_matrix(A), b) + A1, b1, status, message = rr_res + return A1.toarray(), b1, status, message diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__root.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__root.py new file mode 100644 index 0000000000000000000000000000000000000000..1e2f45a10d3d976d02be18084d241adea8612b05 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__root.py @@ -0,0 +1,124 @@ +""" +Unit tests for optimization routines from _root.py. +""" +from numpy.testing import assert_, assert_equal +import pytest +from pytest import raises as assert_raises, warns as assert_warns +import numpy as np + +from scipy.optimize import root + + +class TestRoot: + def test_tol_parameter(self): + # Check that the minimize() tol= argument does something + def func(z): + x, y = z + return np.array([x**3 - 1, y**3 - 1]) + + def dfunc(z): + x, y = z + return np.array([[3*x**2, 0], [0, 3*y**2]]) + + for method in ['hybr', 'lm', 'broyden1', 'broyden2', 'anderson', + 'diagbroyden', 'krylov']: + if method in ('linearmixing', 'excitingmixing'): + # doesn't converge + continue + + if method in ('hybr', 'lm'): + jac = dfunc + else: + jac = None + + sol1 = root(func, [1.1,1.1], jac=jac, tol=1e-4, method=method) + sol2 = root(func, [1.1,1.1], jac=jac, tol=0.5, method=method) + msg = f"{method}: {func(sol1.x)} vs. {func(sol2.x)}" + assert_(sol1.success, msg) + assert_(sol2.success, msg) + assert_(abs(func(sol1.x)).max() < abs(func(sol2.x)).max(), + msg) + + def test_tol_norm(self): + + def norm(x): + return abs(x[0]) + + for method in ['excitingmixing', + 'diagbroyden', + 'linearmixing', + 'anderson', + 'broyden1', + 'broyden2', + 'krylov']: + + root(np.zeros_like, np.zeros(2), method=method, + options={"tol_norm": norm}) + + def test_minimize_scalar_coerce_args_param(self): + # GitHub issue #3503 + def func(z, f=1): + x, y = z + return np.array([x**3 - 1, y**3 - f]) + root(func, [1.1, 1.1], args=1.5) + + def test_f_size(self): + # gh8320 + # check that decreasing the size of the returned array raises an error + # and doesn't segfault + class fun: + def __init__(self): + self.count = 0 + + def __call__(self, x): + self.count += 1 + + if not (self.count % 5): + ret = x[0] + 0.5 * (x[0] - x[1]) ** 3 - 1.0 + else: + ret = ([x[0] + 0.5 * (x[0] - x[1]) ** 3 - 1.0, + 0.5 * (x[1] - x[0]) ** 3 + x[1]]) + + return ret + + F = fun() + with assert_raises(ValueError): + root(F, [0.1, 0.0], method='lm') + + @pytest.mark.thread_unsafe + def test_gh_10370(self): + # gh-10370 reported that passing both `args` and `jac` to `root` with + # `method='krylov'` caused a failure. Ensure that this is fixed whether + # the gradient is passed via `jac` or as a second output of `fun`. + def fun(x, ignored): + return [3*x[0] - 0.25*x[1]**2 + 10, 0.1*x[0]**2 + 5*x[1] - 2] + + def grad(x, ignored): + return [[3, 0.5 * x[1]], [0.2 * x[0], 5]] + + def fun_grad(x, ignored): + return fun(x, ignored), grad(x, ignored) + + x0 = np.zeros(2) + + ref = root(fun, x0, args=(1,), method='krylov') + message = 'Method krylov does not use the jacobian' + with assert_warns(RuntimeWarning, match=message): + res1 = root(fun, x0, args=(1,), method='krylov', jac=grad) + with assert_warns(RuntimeWarning, match=message): + res2 = root(fun_grad, x0, args=(1,), method='krylov', jac=True) + + assert_equal(res1.x, ref.x) + assert_equal(res2.x, ref.x) + assert res1.success is res2.success is ref.success is True + + @pytest.mark.parametrize("method", ["hybr", "lm", "broyden1", "broyden2", + "anderson", "linearmixing", + "diagbroyden", "excitingmixing", + "krylov", "df-sane"]) + def test_method_in_result(self, method): + def func(x): + return x - 1 + + res = root(func, x0=[1], method=method) + assert res.method == method diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__shgo.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__shgo.py new file mode 100644 index 0000000000000000000000000000000000000000..82efb74beee92eacc75f64c6c705374fa5ada322 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__shgo.py @@ -0,0 +1,1156 @@ +import logging +import sys + +import numpy as np +import time +from multiprocessing import Pool +from numpy.testing import assert_allclose, IS_PYPY +import pytest +from pytest import raises as assert_raises, warns +from scipy.optimize import (shgo, Bounds, minimize_scalar, minimize, rosen, + rosen_der, rosen_hess, NonlinearConstraint) +from scipy.optimize._constraints import new_constraint_to_old +from scipy.optimize._shgo import SHGO + + +class StructTestFunction: + def __init__(self, bounds, expected_x, expected_fun=None, + expected_xl=None, expected_funl=None): + self.bounds = bounds + self.expected_x = expected_x + self.expected_fun = expected_fun + self.expected_xl = expected_xl + self.expected_funl = expected_funl + + +def wrap_constraints(g): + cons = [] + if g is not None: + if not isinstance(g, (tuple, list)): + g = (g,) + else: + pass + for g in g: + cons.append({'type': 'ineq', + 'fun': g}) + cons = tuple(cons) + else: + cons = None + return cons + + +class StructTest1(StructTestFunction): + def f(self, x): + return x[0] ** 2 + x[1] ** 2 + + def g(x): + return -(np.sum(x, axis=0) - 6.0) + + cons = wrap_constraints(g) + + +test1_1 = StructTest1(bounds=[(-1, 6), (-1, 6)], + expected_x=[0, 0]) +test1_2 = StructTest1(bounds=[(0, 1), (0, 1)], + expected_x=[0, 0]) +test1_3 = StructTest1(bounds=[(None, None), (None, None)], + expected_x=[0, 0]) + + +class StructTest2(StructTestFunction): + """ + Scalar function with several minima to test all minimiser retrievals + """ + + def f(self, x): + return (x - 30) * np.sin(x) + + def g(x): + return 58 - np.sum(x, axis=0) + + cons = wrap_constraints(g) + + +test2_1 = StructTest2(bounds=[(0, 60)], + expected_x=[1.53567906], + expected_fun=-28.44677132, + # Important: test that funl return is in the correct + # order + expected_xl=np.array([[1.53567906], + [55.01782167], + [7.80894889], + [48.74797493], + [14.07445705], + [42.4913859], + [20.31743841], + [36.28607535], + [26.43039605], + [30.76371366]]), + + expected_funl=np.array([-28.44677132, -24.99785984, + -22.16855376, -18.72136195, + -15.89423937, -12.45154942, + -9.63133158, -6.20801301, + -3.43727232, -0.46353338]) + ) + +test2_2 = StructTest2(bounds=[(0, 4.5)], + expected_x=[1.53567906], + expected_fun=[-28.44677132], + expected_xl=np.array([[1.53567906]]), + expected_funl=np.array([-28.44677132]) + ) + + +class StructTest3(StructTestFunction): + """ + Hock and Schittkowski 18 problem (HS18). Hoch and Schittkowski (1981) + http://www.ai7.uni-bayreuth.de/test_problem_coll.pdf + Minimize: f = 0.01 * (x_1)**2 + (x_2)**2 + + Subject to: x_1 * x_2 - 25.0 >= 0, + (x_1)**2 + (x_2)**2 - 25.0 >= 0, + 2 <= x_1 <= 50, + 0 <= x_2 <= 50. + + Approx. Answer: + f([(250)**0.5 , (2.5)**0.5]) = 5.0 + + + """ + + # amended to test vectorisation of constraints + def f(self, x): + return 0.01 * (x[0]) ** 2 + (x[1]) ** 2 + + def g1(x): + return x[0] * x[1] - 25.0 + + def g2(x): + return x[0] ** 2 + x[1] ** 2 - 25.0 + + # g = (g1, g2) + # cons = wrap_constraints(g) + + def g(x): + return x[0] * x[1] - 25.0, x[0] ** 2 + x[1] ** 2 - 25.0 + + # this checks that shgo can be sent new-style constraints + __nlc = NonlinearConstraint(g, 0, np.inf) + cons = (__nlc,) + +test3_1 = StructTest3(bounds=[(2, 50), (0, 50)], + expected_x=[250 ** 0.5, 2.5 ** 0.5], + expected_fun=5.0 + ) + + +class StructTest4(StructTestFunction): + """ + Hock and Schittkowski 11 problem (HS11). Hoch and Schittkowski (1981) + + NOTE: Did not find in original reference to HS collection, refer to + Henderson (2015) problem 7 instead. 02.03.2016 + """ + + def f(self, x): + return ((x[0] - 10) ** 2 + 5 * (x[1] - 12) ** 2 + x[2] ** 4 + + 3 * (x[3] - 11) ** 2 + 10 * x[4] ** 6 + 7 * x[5] ** 2 + x[ + 6] ** 4 + - 4 * x[5] * x[6] - 10 * x[5] - 8 * x[6] + ) + + def g1(x): + return -(2 * x[0] ** 2 + 3 * x[1] ** 4 + x[2] + 4 * x[3] ** 2 + + 5 * x[4] - 127) + + def g2(x): + return -(7 * x[0] + 3 * x[1] + 10 * x[2] ** 2 + x[3] - x[4] - 282.0) + + def g3(x): + return -(23 * x[0] + x[1] ** 2 + 6 * x[5] ** 2 - 8 * x[6] - 196) + + def g4(x): + return -(4 * x[0] ** 2 + x[1] ** 2 - 3 * x[0] * x[1] + 2 * x[2] ** 2 + + 5 * x[5] - 11 * x[6]) + + g = (g1, g2, g3, g4) + + cons = wrap_constraints(g) + + +test4_1 = StructTest4(bounds=[(-10, 10), ] * 7, + expected_x=[2.330499, 1.951372, -0.4775414, + 4.365726, -0.6244870, 1.038131, 1.594227], + expected_fun=680.6300573 + ) + + +class StructTest5(StructTestFunction): + def f(self, x): + return ( + -(x[1] + 47.0)*np.sin(np.sqrt(abs(x[0]/2.0 + (x[1] + 47.0)))) + - x[0]*np.sin(np.sqrt(abs(x[0] - (x[1] + 47.0)))) + ) + + g = None + cons = wrap_constraints(g) + + +test5_1 = StructTest5(bounds=[(-512, 512), (-512, 512)], + expected_fun=[-959.64066272085051], + expected_x=[512., 404.23180542]) + + +class StructTestLJ(StructTestFunction): + """ + LennardJones objective function. Used to test symmetry constraints + settings. + """ + + def f(self, x, *args): + print(f'x = {x}') + self.N = args[0] + k = int(self.N / 3) + s = 0.0 + + for i in range(k - 1): + for j in range(i + 1, k): + a = 3 * i + b = 3 * j + xd = x[a] - x[b] + yd = x[a + 1] - x[b + 1] + zd = x[a + 2] - x[b + 2] + ed = xd * xd + yd * yd + zd * zd + ud = ed * ed * ed + if ed > 0.0: + s += (1.0 / ud - 2.0) / ud + + return s + + g = None + cons = wrap_constraints(g) + + +N = 6 +boundsLJ = list(zip([-4.0] * 6, [4.0] * 6)) + +testLJ = StructTestLJ(bounds=boundsLJ, + expected_fun=[-1.0], + expected_x=None, + # expected_x=[-2.71247337e-08, + # -2.71247337e-08, + # -2.50000222e+00, + # -2.71247337e-08, + # -2.71247337e-08, + # -1.50000222e+00] + ) + + +class StructTestS(StructTestFunction): + def f(self, x): + return ((x[0] - 0.5) ** 2 + (x[1] - 0.5) ** 2 + + (x[2] - 0.5) ** 2 + (x[3] - 0.5) ** 2) + + g = None + cons = wrap_constraints(g) + + +test_s = StructTestS(bounds=[(0, 2.0), ] * 4, + expected_fun=0.0, + expected_x=np.ones(4) - 0.5 + ) + + +class StructTestTable(StructTestFunction): + def f(self, x): + if x[0] == 3.0 and x[1] == 3.0: + return 50 + else: + return 100 + + g = None + cons = wrap_constraints(g) + + +test_table = StructTestTable(bounds=[(-10, 10), (-10, 10)], + expected_fun=[50], + expected_x=[3.0, 3.0]) + + +class StructTestInfeasible(StructTestFunction): + """ + Test function with no feasible domain. + """ + + def f(self, x, *args): + return x[0] ** 2 + x[1] ** 2 + + def g1(x): + return x[0] + x[1] - 1 + + def g2(x): + return -(x[0] + x[1] - 1) + + def g3(x): + return -x[0] + x[1] - 1 + + def g4(x): + return -(-x[0] + x[1] - 1) + + g = (g1, g2, g3, g4) + cons = wrap_constraints(g) + + +test_infeasible = StructTestInfeasible(bounds=[(2, 50), (-1, 1)], + expected_fun=None, + expected_x=None + ) + + +@pytest.mark.skip("Not a test") +def run_test(test, args=(), test_atol=1e-5, n=100, iters=None, + callback=None, minimizer_kwargs=None, options=None, + sampling_method='sobol', workers=1): + res = shgo(test.f, test.bounds, args=args, constraints=test.cons, + n=n, iters=iters, callback=callback, + minimizer_kwargs=minimizer_kwargs, options=options, + sampling_method=sampling_method, workers=workers) + + print(f'res = {res}') + logging.info(f'res = {res}') + if test.expected_x is not None: + np.testing.assert_allclose(res.x, test.expected_x, + rtol=test_atol, + atol=test_atol) + + # (Optional tests) + if test.expected_fun is not None: + np.testing.assert_allclose(res.fun, + test.expected_fun, + atol=test_atol) + + if test.expected_xl is not None: + np.testing.assert_allclose(res.xl, + test.expected_xl, + atol=test_atol) + + if test.expected_funl is not None: + np.testing.assert_allclose(res.funl, + test.expected_funl, + atol=test_atol) + return + + +# Base test functions: +class TestShgoSobolTestFunctions: + """ + Global optimisation tests with Sobol sampling: + """ + + # Sobol algorithm + def test_f1_1_sobol(self): + """Multivariate test function 1: + x[0]**2 + x[1]**2 with bounds=[(-1, 6), (-1, 6)]""" + run_test(test1_1) + + def test_f1_2_sobol(self): + """Multivariate test function 1: + x[0]**2 + x[1]**2 with bounds=[(0, 1), (0, 1)]""" + run_test(test1_2) + + def test_f1_3_sobol(self): + """Multivariate test function 1: + x[0]**2 + x[1]**2 with bounds=[(None, None),(None, None)]""" + options = {'disp': True} + run_test(test1_3, options=options) + + def test_f2_1_sobol(self): + """Univariate test function on + f(x) = (x - 30) * sin(x) with bounds=[(0, 60)]""" + run_test(test2_1) + + def test_f2_2_sobol(self): + """Univariate test function on + f(x) = (x - 30) * sin(x) bounds=[(0, 4.5)]""" + run_test(test2_2) + + def test_f3_sobol(self): + """NLP: Hock and Schittkowski problem 18""" + run_test(test3_1) + + @pytest.mark.slow + def test_f4_sobol(self): + """NLP: (High dimensional) Hock and Schittkowski 11 problem (HS11)""" + options = {'infty_constraints': False} + # run_test(test4_1, n=990, options=options) + run_test(test4_1, n=990 * 2, options=options) + + def test_f5_1_sobol(self): + """NLP: Eggholder, multimodal""" + # run_test(test5_1, n=30) + run_test(test5_1, n=60) + + def test_f5_2_sobol(self): + """NLP: Eggholder, multimodal""" + # run_test(test5_1, n=60, iters=5) + run_test(test5_1, n=60, iters=5) + + # def test_t911(self): + # """1D tabletop function""" + # run_test(test11_1) + + +class TestShgoSimplicialTestFunctions: + """ + Global optimisation tests with Simplicial sampling: + """ + + def test_f1_1_simplicial(self): + """Multivariate test function 1: + x[0]**2 + x[1]**2 with bounds=[(-1, 6), (-1, 6)]""" + run_test(test1_1, n=1, sampling_method='simplicial') + + def test_f1_2_simplicial(self): + """Multivariate test function 1: + x[0]**2 + x[1]**2 with bounds=[(0, 1), (0, 1)]""" + run_test(test1_2, n=1, sampling_method='simplicial') + + def test_f1_3_simplicial(self): + """Multivariate test function 1: x[0]**2 + x[1]**2 + with bounds=[(None, None),(None, None)]""" + run_test(test1_3, n=5, sampling_method='simplicial') + + def test_f2_1_simplicial(self): + """Univariate test function on + f(x) = (x - 30) * sin(x) with bounds=[(0, 60)]""" + options = {'minimize_every_iter': False} + run_test(test2_1, n=200, iters=7, options=options, + sampling_method='simplicial') + + def test_f2_2_simplicial(self): + """Univariate test function on + f(x) = (x - 30) * sin(x) bounds=[(0, 4.5)]""" + run_test(test2_2, n=1, sampling_method='simplicial') + + def test_f3_simplicial(self): + """NLP: Hock and Schittkowski problem 18""" + run_test(test3_1, n=1, sampling_method='simplicial') + + @pytest.mark.slow + def test_f4_simplicial(self): + """NLP: (High dimensional) Hock and Schittkowski 11 problem (HS11)""" + run_test(test4_1, n=1, sampling_method='simplicial') + + def test_lj_symmetry_old(self): + """LJ: Symmetry-constrained test function""" + options = {'symmetry': True, + 'disp': True} + args = (6,) # Number of atoms + run_test(testLJ, args=args, n=300, + options=options, iters=1, + sampling_method='simplicial') + + def test_f5_1_lj_symmetry(self): + """LJ: Symmetry constrained test function""" + options = {'symmetry': [0, ] * 6, + 'disp': True} + args = (6,) # No. of atoms + + run_test(testLJ, args=args, n=300, + options=options, iters=1, + sampling_method='simplicial') + + def test_f5_2_cons_symmetry(self): + """Symmetry constrained test function""" + options = {'symmetry': [0, 0], + 'disp': True} + + run_test(test1_1, n=200, + options=options, iters=1, + sampling_method='simplicial') + + @pytest.mark.fail_slow(10) + def test_f5_3_cons_symmetry(self): + """Asymmetrically constrained test function""" + options = {'symmetry': [0, 0, 0, 3], + 'disp': True} + + run_test(test_s, n=10000, + options=options, + iters=1, + sampling_method='simplicial') + + @pytest.mark.skip("Not a test") + def test_f0_min_variance(self): + """Return a minimum on a perfectly symmetric problem, based on + gh10429""" + avg = 0.5 # Given average value of x + cons = {'type': 'eq', 'fun': lambda x: np.mean(x) - avg} + + # Minimize the variance of x under the given constraint + res = shgo(np.var, bounds=6 * [(0, 1)], constraints=cons) + assert res.success + assert_allclose(res.fun, 0, atol=1e-15) + assert_allclose(res.x, 0.5) + + @pytest.mark.skip("Not a test") + def test_f0_min_variance_1D(self): + """Return a minimum on a perfectly symmetric 1D problem, based on + gh10538""" + + def fun(x): + return x * (x - 1.0) * (x - 0.5) + + bounds = [(0, 1)] + res = shgo(fun, bounds=bounds) + ref = minimize_scalar(fun, bounds=bounds[0]) + assert res.success + assert_allclose(res.fun, ref.fun) + assert_allclose(res.x, ref.x, rtol=1e-6) + +# Argument test functions +class TestShgoArguments: + def test_1_1_simpl_iter(self): + """Iterative simplicial sampling on TestFunction 1 (multivariate)""" + run_test(test1_2, n=None, iters=2, sampling_method='simplicial') + + def test_1_2_simpl_iter(self): + """Iterative simplicial on TestFunction 2 (univariate)""" + options = {'minimize_every_iter': False} + run_test(test2_1, n=None, iters=9, options=options, + sampling_method='simplicial') + + def test_2_1_sobol_iter(self): + """Iterative Sobol sampling on TestFunction 1 (multivariate)""" + run_test(test1_2, n=None, iters=1, sampling_method='sobol') + + def test_2_2_sobol_iter(self): + """Iterative Sobol sampling on TestFunction 2 (univariate)""" + res = shgo(test2_1.f, test2_1.bounds, constraints=test2_1.cons, + n=None, iters=1, sampling_method='sobol') + + np.testing.assert_allclose(res.x, test2_1.expected_x, rtol=1e-5, atol=1e-5) + np.testing.assert_allclose(res.fun, test2_1.expected_fun, atol=1e-5) + + def test_3_1_disp_simplicial(self): + """Iterative sampling on TestFunction 1 and 2 (multi and univariate) + """ + + def callback_func(x): + print("Local minimization callback test") + + for test in [test1_1, test2_1]: + shgo(test.f, test.bounds, iters=1, + sampling_method='simplicial', + callback=callback_func, options={'disp': True}) + shgo(test.f, test.bounds, n=1, sampling_method='simplicial', + callback=callback_func, options={'disp': True}) + + def test_3_2_disp_sobol(self): + """Iterative sampling on TestFunction 1 and 2 (multi and univariate)""" + + def callback_func(x): + print("Local minimization callback test") + + for test in [test1_1, test2_1]: + shgo(test.f, test.bounds, iters=1, sampling_method='sobol', + callback=callback_func, options={'disp': True}) + + shgo(test.f, test.bounds, n=1, sampling_method='simplicial', + callback=callback_func, options={'disp': True}) + + def test_args_gh14589(self): + """Using `args` used to cause `shgo` to fail; see #14589, #15986, + #16506""" + res = shgo(func=lambda x, y, z: x * z + y, bounds=[(0, 3)], args=(1, 2) + ) + ref = shgo(func=lambda x: 2 * x + 1, bounds=[(0, 3)]) + assert_allclose(res.fun, ref.fun) + assert_allclose(res.x, ref.x) + + @pytest.mark.slow + def test_4_1_known_f_min(self): + """Test known function minima stopping criteria""" + # Specify known function value + options = {'f_min': test4_1.expected_fun, + 'f_tol': 1e-6, + 'minimize_every_iter': True} + # TODO: Make default n higher for faster tests + run_test(test4_1, n=None, test_atol=1e-5, options=options, + sampling_method='simplicial') + + @pytest.mark.slow + def test_4_2_known_f_min(self): + """Test Global mode limiting local evaluations""" + options = { # Specify known function value + 'f_min': test4_1.expected_fun, + 'f_tol': 1e-6, + # Specify number of local iterations to perform + 'minimize_every_iter': True, + 'local_iter': 1} + + run_test(test4_1, n=None, test_atol=1e-5, options=options, + sampling_method='simplicial') + + def test_4_4_known_f_min(self): + """Test Global mode limiting local evaluations for 1D funcs""" + options = { # Specify known function value + 'f_min': test2_1.expected_fun, + 'f_tol': 1e-6, + # Specify number of local iterations to perform+ + 'minimize_every_iter': True, + 'local_iter': 1, + 'infty_constraints': False} + + res = shgo(test2_1.f, test2_1.bounds, constraints=test2_1.cons, + n=None, iters=None, options=options, + sampling_method='sobol') + np.testing.assert_allclose(res.x, test2_1.expected_x, rtol=1e-5, atol=1e-5) + + def test_5_1_simplicial_argless(self): + """Test Default simplicial sampling settings on TestFunction 1""" + res = shgo(test1_1.f, test1_1.bounds, constraints=test1_1.cons) + np.testing.assert_allclose(res.x, test1_1.expected_x, rtol=1e-5, atol=1e-5) + + def test_5_2_sobol_argless(self): + """Test Default sobol sampling settings on TestFunction 1""" + res = shgo(test1_1.f, test1_1.bounds, constraints=test1_1.cons, + sampling_method='sobol') + np.testing.assert_allclose(res.x, test1_1.expected_x, rtol=1e-5, atol=1e-5) + + def test_6_1_simplicial_max_iter(self): + """Test that maximum iteration option works on TestFunction 3""" + options = {'max_iter': 2} + res = shgo(test3_1.f, test3_1.bounds, constraints=test3_1.cons, + options=options, sampling_method='simplicial') + np.testing.assert_allclose(res.x, test3_1.expected_x, rtol=1e-5, atol=1e-5) + np.testing.assert_allclose(res.fun, test3_1.expected_fun, atol=1e-5) + + def test_6_2_simplicial_min_iter(self): + """Test that maximum iteration option works on TestFunction 3""" + options = {'min_iter': 2} + res = shgo(test3_1.f, test3_1.bounds, constraints=test3_1.cons, + options=options, sampling_method='simplicial') + np.testing.assert_allclose(res.x, test3_1.expected_x, rtol=1e-5, atol=1e-5) + np.testing.assert_allclose(res.fun, test3_1.expected_fun, atol=1e-5) + + def test_7_1_minkwargs(self): + """Test the minimizer_kwargs arguments for solvers with constraints""" + # Test solvers + for solver in ['COBYLA', 'COBYQA', 'SLSQP']: + # Note that passing global constraints to SLSQP is tested in other + # unittests which run test4_1 normally + minimizer_kwargs = {'method': solver, + 'constraints': test3_1.cons} + run_test(test3_1, n=100, test_atol=1e-3, + minimizer_kwargs=minimizer_kwargs, + sampling_method='sobol') + + def test_7_2_minkwargs(self): + """Test the minimizer_kwargs default inits""" + minimizer_kwargs = {'ftol': 1e-5} + options = {'disp': True} # For coverage purposes + SHGO(test3_1.f, test3_1.bounds, constraints=test3_1.cons[0], + minimizer_kwargs=minimizer_kwargs, options=options) + + def test_7_3_minkwargs(self): + """Test minimizer_kwargs arguments for solvers without constraints""" + for solver in ['Nelder-Mead', 'Powell', 'CG', 'BFGS', 'Newton-CG', + 'L-BFGS-B', 'TNC', 'dogleg', 'trust-ncg', 'trust-exact', + 'trust-krylov']: + def jac(x): + return np.array([2 * x[0], 2 * x[1]]).T + + def hess(x): + return np.array([[2, 0], [0, 2]]) + + minimizer_kwargs = {'method': solver, + 'jac': jac, + 'hess': hess} + logging.info(f"Solver = {solver}") + logging.info("=" * 100) + run_test(test1_1, n=100, test_atol=1e-3, + minimizer_kwargs=minimizer_kwargs, + sampling_method='sobol') + + def test_8_homology_group_diff(self): + options = {'minhgrd': 1, + 'minimize_every_iter': True} + + run_test(test1_1, n=None, iters=None, options=options, + sampling_method='simplicial') + + def test_9_cons_g(self): + """Test single function constraint passing""" + SHGO(test3_1.f, test3_1.bounds, constraints=test3_1.cons[0]) + + @pytest.mark.xfail(IS_PYPY and sys.platform == 'win32', + reason="Failing and fix in PyPy not planned (see gh-18632)") + def test_10_finite_time(self): + """Test single function constraint passing""" + options = {'maxtime': 1e-15} + + def f(x): + time.sleep(1e-14) + return 0.0 + + res = shgo(f, test1_1.bounds, iters=5, options=options) + # Assert that only 1 rather than 5 requested iterations ran: + assert res.nit == 1 + + def test_11_f_min_0(self): + """Test to cover the case where f_lowest == 0""" + options = {'f_min': 0.0, + 'disp': True} + res = shgo(test1_2.f, test1_2.bounds, n=10, iters=None, + options=options, sampling_method='sobol') + np.testing.assert_equal(0, res.x[0]) + np.testing.assert_equal(0, res.x[1]) + + # @nottest + @pytest.mark.skip(reason="no way of currently testing this") + def test_12_sobol_inf_cons(self): + """Test to cover the case where f_lowest == 0""" + # TODO: This test doesn't cover anything new, it is unknown what the + # original test was intended for as it was never complete. Delete or + # replace in the future. + options = {'maxtime': 1e-15, + 'f_min': 0.0} + res = shgo(test1_2.f, test1_2.bounds, n=1, iters=None, + options=options, sampling_method='sobol') + np.testing.assert_equal(0.0, res.fun) + + def test_13_high_sobol(self): + """Test init of high-dimensional sobol sequences""" + + def f(x): + return 0 + + bounds = [(None, None), ] * 41 + SHGOc = SHGO(f, bounds, sampling_method='sobol') + # SHGOc.sobol_points(2, 50) + SHGOc.sampling_function(2, 50) + + def test_14_local_iter(self): + """Test limited local iterations for a pseudo-global mode""" + options = {'local_iter': 4} + run_test(test5_1, n=60, options=options) + + def test_15_min_every_iter(self): + """Test minimize every iter options and cover function cache""" + options = {'minimize_every_iter': True} + run_test(test1_1, n=1, iters=7, options=options, + sampling_method='sobol') + + def test_16_disp_bounds_minimizer(self, capsys): + """Test disp=True with minimizers that do not support bounds """ + options = {'disp': True} + minimizer_kwargs = {'method': 'nelder-mead'} + run_test(test1_2, sampling_method='simplicial', + options=options, minimizer_kwargs=minimizer_kwargs) + + def test_17_custom_sampling(self): + """Test the functionality to add custom sampling methods to shgo""" + + def sample(n, d): + return np.random.uniform(size=(n, d)) + + run_test(test1_1, n=30, sampling_method=sample) + + def test_18_bounds_class(self): + # test that new and old bounds yield same result + def f(x): + return np.square(x).sum() + + lb = [-6., 1., -5.] + ub = [-1., 3., 5.] + bounds_old = list(zip(lb, ub)) + bounds_new = Bounds(lb, ub) + + res_old_bounds = shgo(f, bounds_old) + res_new_bounds = shgo(f, bounds_new) + + assert res_new_bounds.nfev == res_old_bounds.nfev + assert res_new_bounds.message == res_old_bounds.message + assert res_new_bounds.success == res_old_bounds.success + x_opt = np.array([-1., 1., 0.]) + np.testing.assert_allclose(res_new_bounds.x, x_opt) + np.testing.assert_allclose(res_new_bounds.x, res_old_bounds.x) + + @pytest.mark.fail_slow(10) + def test_19_parallelization(self): + """Test the functionality to add custom sampling methods to shgo""" + + with Pool(2) as p: + run_test(test1_1, n=30, workers=p.map) # Constrained + run_test(test1_1, n=30, workers=map) # Constrained + with Pool(2) as p: + run_test(test_s, n=30, workers=p.map) # Unconstrained + run_test(test_s, n=30, workers=map) # Unconstrained + + def test_20_constrained_args(self): + """Test that constraints can be passed to arguments""" + + def eggholder(x): + return ( + -(x[1] + 47.0)*np.sin(np.sqrt(abs(x[0] / 2.0 + (x[1] + 47.0)))) + - x[0]*np.sin(np.sqrt(abs(x[0] - (x[1] + 47.0)))) + ) + + def f(x): # (cattle-feed) + return 24.55 * x[0] + 26.75 * x[1] + 39 * x[2] + 40.50 * x[3] + + bounds = [(0, 1.0), ] * 4 + + def g1_modified(x, i): + return i * 2.3 * x[0] + i * 5.6 * x[1] + 11.1 * x[2] + 1.3 * x[ + 3] - 5 # >=0 + + def g2(x): + return ( + 12*x[0] + 11.9*x[1] + 41.8*x[2] + 52.1*x[3] - 21 + - 1.645*np.sqrt( + 0.28*x[0]**2 + 0.19*x[1]**2 + 20.5*x[2]**2 + 0.62*x[3]**2 + ) + ) # >=0 + + def h1(x): + return x[0] + x[1] + x[2] + x[3] - 1 # == 0 + + cons = ({'type': 'ineq', 'fun': g1_modified, "args": (0,)}, + {'type': 'ineq', 'fun': g2}, + {'type': 'eq', 'fun': h1}) + + shgo(f, bounds, n=300, iters=1, constraints=cons) + # using constrain with arguments AND sampling method sobol + shgo(f, bounds, n=300, iters=1, constraints=cons, + sampling_method='sobol') + + def test_21_1_jac_true(self): + """Test that shgo can handle objective functions that return the + gradient alongside the objective value. Fixes gh-13547""" + # previous + def func(x): + return np.sum(np.power(x, 2)), 2 * x + + shgo( + func, + bounds=[[-1, 1], [1, 2]], + n=100, iters=5, + sampling_method="sobol", + minimizer_kwargs={'method': 'SLSQP', 'jac': True} + ) + + # new + def func(x): + return np.sum(x ** 2), 2 * x + + bounds = [[-1, 1], [1, 2], [-1, 1], [1, 2], [0, 3]] + + res = shgo(func, bounds=bounds, sampling_method="sobol", + minimizer_kwargs={'method': 'SLSQP', 'jac': True}) + ref = minimize(func, x0=[1, 1, 1, 1, 1], bounds=bounds, + jac=True) + assert res.success + assert_allclose(res.fun, ref.fun) + assert_allclose(res.x, ref.x, atol=1e-15) + + @pytest.mark.parametrize('derivative', ['jac', 'hess', 'hessp']) + def test_21_2_derivative_options(self, derivative): + """shgo used to raise an error when passing `options` with 'jac' + # see gh-12963. check that this is resolved + """ + + def objective(x): + return 3 * x[0] * x[0] + 2 * x[0] + 5 + + def gradient(x): + return 6 * x[0] + 2 + + def hess(x): + return 6 + + def hessp(x, p): + return 6 * p + + derivative_funcs = {'jac': gradient, 'hess': hess, 'hessp': hessp} + options = {derivative: derivative_funcs[derivative]} + minimizer_kwargs = {'method': 'trust-constr'} + + bounds = [(-100, 100)] + res = shgo(objective, bounds, minimizer_kwargs=minimizer_kwargs, + options=options) + ref = minimize(objective, x0=[0], bounds=bounds, **minimizer_kwargs, + **options) + + assert res.success + np.testing.assert_allclose(res.fun, ref.fun) + np.testing.assert_allclose(res.x, ref.x) + + def test_21_3_hess_options_rosen(self): + """Ensure the Hessian gets passed correctly to the local minimizer + routine. Previous report gh-14533. + """ + bounds = [(0, 1.6), (0, 1.6), (0, 1.4), (0, 1.4), (0, 1.4)] + options = {'jac': rosen_der, 'hess': rosen_hess} + minimizer_kwargs = {'method': 'Newton-CG'} + res = shgo(rosen, bounds, minimizer_kwargs=minimizer_kwargs, + options=options) + ref = minimize(rosen, np.zeros(5), method='Newton-CG', + **options) + assert res.success + assert_allclose(res.fun, ref.fun) + assert_allclose(res.x, ref.x, atol=1e-15) + + def test_21_arg_tuple_sobol(self): + """shgo used to raise an error when passing `args` with Sobol sampling + # see gh-12114. check that this is resolved""" + + def fun(x, k): + return x[0] ** k + + constraints = ({'type': 'ineq', 'fun': lambda x: x[0] - 1}) + + bounds = [(0, 10)] + res = shgo(fun, bounds, args=(1,), constraints=constraints, + sampling_method='sobol') + ref = minimize(fun, np.zeros(1), bounds=bounds, args=(1,), + constraints=constraints) + assert res.success + assert_allclose(res.fun, ref.fun) + assert_allclose(res.x, ref.x) + + +# Failure test functions +class TestShgoFailures: + def test_1_maxiter(self): + """Test failure on insufficient iterations""" + options = {'maxiter': 2} + res = shgo(test4_1.f, test4_1.bounds, n=2, iters=None, + options=options, sampling_method='sobol') + + np.testing.assert_equal(False, res.success) + # np.testing.assert_equal(4, res.nfev) + np.testing.assert_equal(4, res.tnev) + + def test_2_sampling(self): + """Rejection of unknown sampling method""" + assert_raises(ValueError, shgo, test1_1.f, test1_1.bounds, + sampling_method='not_Sobol') + + def test_3_1_no_min_pool_sobol(self): + """Check that the routine stops when no minimiser is found + after maximum specified function evaluations""" + options = {'maxfev': 10, + # 'maxev': 10, + 'disp': True} + res = shgo(test_table.f, test_table.bounds, n=3, options=options, + sampling_method='sobol') + np.testing.assert_equal(False, res.success) + # np.testing.assert_equal(9, res.nfev) + np.testing.assert_equal(12, res.nfev) + + def test_3_2_no_min_pool_simplicial(self): + """Check that the routine stops when no minimiser is found + after maximum specified sampling evaluations""" + options = {'maxev': 10, + 'disp': True} + res = shgo(test_table.f, test_table.bounds, n=3, options=options, + sampling_method='simplicial') + np.testing.assert_equal(False, res.success) + + def test_4_1_bound_err(self): + """Specified bounds ub > lb""" + bounds = [(6, 3), (3, 5)] + assert_raises(ValueError, shgo, test1_1.f, bounds) + + def test_4_2_bound_err(self): + """Specified bounds are of the form (lb, ub)""" + bounds = [(3, 5, 5), (3, 5)] + assert_raises(ValueError, shgo, test1_1.f, bounds) + + def test_5_1_1_infeasible_sobol(self): + """Ensures the algorithm terminates on infeasible problems + after maxev is exceeded. Use infty constraints option""" + options = {'maxev': 100, + 'disp': True} + + res = shgo(test_infeasible.f, test_infeasible.bounds, + constraints=test_infeasible.cons, n=100, options=options, + sampling_method='sobol') + + np.testing.assert_equal(False, res.success) + + def test_5_1_2_infeasible_sobol(self): + """Ensures the algorithm terminates on infeasible problems + after maxev is exceeded. Do not use infty constraints option""" + options = {'maxev': 100, + 'disp': True, + 'infty_constraints': False} + + res = shgo(test_infeasible.f, test_infeasible.bounds, + constraints=test_infeasible.cons, n=100, options=options, + sampling_method='sobol') + + np.testing.assert_equal(False, res.success) + + def test_5_2_infeasible_simplicial(self): + """Ensures the algorithm terminates on infeasible problems + after maxev is exceeded.""" + options = {'maxev': 1000, + 'disp': False} + + res = shgo(test_infeasible.f, test_infeasible.bounds, + constraints=test_infeasible.cons, n=100, options=options, + sampling_method='simplicial') + + np.testing.assert_equal(False, res.success) + + @pytest.mark.thread_unsafe + def test_6_1_lower_known_f_min(self): + """Test Global mode limiting local evaluations with f* too high""" + options = { # Specify known function value + 'f_min': test2_1.expected_fun + 2.0, + 'f_tol': 1e-6, + # Specify number of local iterations to perform+ + 'minimize_every_iter': True, + 'local_iter': 1, + 'infty_constraints': False} + args = (test2_1.f, test2_1.bounds) + kwargs = {'constraints': test2_1.cons, + 'n': None, + 'iters': None, + 'options': options, + 'sampling_method': 'sobol' + } + warns(UserWarning, shgo, *args, **kwargs) + + def test(self): + from scipy.optimize import rosen, shgo + bounds = [(0, 2), (0, 2), (0, 2), (0, 2), (0, 2)] + + def fun(x): + fun.nfev += 1 + return rosen(x) + + fun.nfev = 0 + + result = shgo(fun, bounds) + print(result.x, result.fun, fun.nfev) # 50 + + +# Returns +class TestShgoReturns: + def test_1_nfev_simplicial(self): + bounds = [(0, 2), (0, 2), (0, 2), (0, 2), (0, 2)] + + def fun(x): + fun.nfev += 1 + return rosen(x) + + fun.nfev = 0 + + result = shgo(fun, bounds) + np.testing.assert_equal(fun.nfev, result.nfev) + + def test_1_nfev_sobol(self): + bounds = [(0, 2), (0, 2), (0, 2), (0, 2), (0, 2)] + + def fun(x): + fun.nfev += 1 + return rosen(x) + + fun.nfev = 0 + + result = shgo(fun, bounds, sampling_method='sobol') + np.testing.assert_equal(fun.nfev, result.nfev) + + +def test_vector_constraint(): + # gh15514 + def quad(x): + x = np.asarray(x) + return [np.sum(x ** 2)] + + nlc = NonlinearConstraint(quad, [2.2], [3]) + oldc = new_constraint_to_old(nlc, np.array([1.0, 1.0])) + + res = shgo(rosen, [(0, 10), (0, 10)], constraints=oldc, sampling_method='sobol') + assert np.all(np.sum((res.x)**2) >= 2.2) + assert np.all(np.sum((res.x) ** 2) <= 3.0) + assert res.success + + +@pytest.mark.filterwarnings("ignore:delta_grad") +def test_trust_constr(): + def quad(x): + x = np.asarray(x) + return [np.sum(x ** 2)] + + nlc = NonlinearConstraint(quad, [2.6], [3]) + minimizer_kwargs = {'method': 'trust-constr'} + # note that we don't supply the constraints in minimizer_kwargs, + # so if the final result obeys the constraints we know that shgo + # passed them on to 'trust-constr' + res = shgo( + rosen, + [(0, 10), (0, 10)], + constraints=nlc, + sampling_method='sobol', + minimizer_kwargs=minimizer_kwargs + ) + assert np.all(np.sum((res.x)**2) >= 2.6) + assert np.all(np.sum((res.x) ** 2) <= 3.0) + assert res.success + + +def test_equality_constraints(): + # gh16260 + bounds = [(0.9, 4.0)] * 2 # Constrain probabilities to 0 and 1. + + def faulty(x): + return x[0] + x[1] + + nlc = NonlinearConstraint(faulty, 3.9, 3.9) + res = shgo(rosen, bounds=bounds, constraints=nlc) + assert_allclose(np.sum(res.x), 3.9) + + def faulty(x): + return x[0] + x[1] - 3.9 + + constraints = {'type': 'eq', 'fun': faulty} + res = shgo(rosen, bounds=bounds, constraints=constraints) + assert_allclose(np.sum(res.x), 3.9) + + bounds = [(0, 1.0)] * 4 + # sum of variable should equal 1. + def faulty(x): + return x[0] + x[1] + x[2] + x[3] - 1 + + # options = {'minimize_every_iter': True, 'local_iter':10} + constraints = {'type': 'eq', 'fun': faulty} + res = shgo( + lambda x: - np.prod(x), + bounds=bounds, + constraints=constraints, + sampling_method='sobol' + ) + assert_allclose(np.sum(res.x), 1.0) + +def test_gh16971(): + def cons(x): + return np.sum(x**2) - 0 + + c = {'fun': cons, 'type': 'ineq'} + minimizer_kwargs = { + 'method': 'COBYLA', + 'options': {'rhobeg': 5, 'tol': 5e-1, 'catol': 0.05} + } + + s = SHGO( + rosen, [(0, 10)]*2, constraints=c, minimizer_kwargs=minimizer_kwargs + ) + + assert s.minimizer_kwargs['method'].lower() == 'cobyla' + assert s.minimizer_kwargs['options']['catol'] == 0.05 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__spectral.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__spectral.py new file mode 100644 index 0000000000000000000000000000000000000000..7b4dc52cc20caf0206fe53933d4dfc6d0fbb2c34 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test__spectral.py @@ -0,0 +1,226 @@ +import itertools + +import numpy as np +from numpy import exp +from numpy.testing import assert_, assert_equal + +from scipy.optimize import root + + +def test_performance(): + # Compare performance results to those listed in + # [Cheng & Li, IMA J. Num. An. 29, 814 (2008)] + # and + # [W. La Cruz, J.M. Martinez, M. Raydan, Math. Comp. 75, 1429 (2006)]. + # and those produced by dfsane.f from M. Raydan's website. + # + # Where the results disagree, the largest limits are taken. + + e_a = 1e-5 + e_r = 1e-4 + + table_1 = [ + dict(F=F_1, x0=x0_1, n=1000, nit=5, nfev=5), + dict(F=F_1, x0=x0_1, n=10000, nit=2, nfev=2), + dict(F=F_2, x0=x0_2, n=500, nit=11, nfev=11), + dict(F=F_2, x0=x0_2, n=2000, nit=11, nfev=11), + # dict(F=F_4, x0=x0_4, n=999, nit=243, nfev=1188) removed: + # too sensitive to rounding errors + # Results from dfsane.f; papers list nit=3, nfev=3 + dict(F=F_6, x0=x0_6, n=100, nit=6, nfev=6), + # Must have n%3==0, typo in papers? + dict(F=F_7, x0=x0_7, n=99, nit=23, nfev=29), + # Must have n%3==0, typo in papers? + dict(F=F_7, x0=x0_7, n=999, nit=23, nfev=29), + # Results from dfsane.f; papers list nit=nfev=6? + dict(F=F_9, x0=x0_9, n=100, nit=12, nfev=18), + dict(F=F_9, x0=x0_9, n=1000, nit=12, nfev=18), + # Results from dfsane.f; papers list nit=2, nfev=12 + dict(F=F_10, x0=x0_10, n=1000, nit=5, nfev=5), + ] + + # Check also scaling invariance + for xscale, yscale, line_search in itertools.product( + [1.0, 1e-10, 1e10], [1.0, 1e-10, 1e10], ['cruz', 'cheng'] + ): + for problem in table_1: + n = problem['n'] + def func(x, n): + return yscale * problem['F'](x / xscale, n) + args = (n,) + x0 = problem['x0'](n) * xscale + + fatol = np.sqrt(n) * e_a * yscale + e_r * np.linalg.norm(func(x0, n)) + + sigma_eps = 1e-10 * min(yscale/xscale, xscale/yscale) + sigma_0 = xscale/yscale + + with np.errstate(over='ignore'): + sol = root(func, x0, args=args, + options=dict(ftol=0, fatol=fatol, maxfev=problem['nfev'] + 1, + sigma_0=sigma_0, sigma_eps=sigma_eps, + line_search=line_search), + method='DF-SANE') + + err_msg = repr( + [xscale, yscale, line_search, problem, np.linalg.norm(func(sol.x, n)), + fatol, sol.success, sol.nit, sol.nfev] + ) + assert sol.success, err_msg + # nfev+1: dfsane.f doesn't count first eval + assert sol.nfev <= problem['nfev'] + 1, err_msg + assert sol.nit <= problem['nit'], err_msg + assert np.linalg.norm(func(sol.x, n)) <= fatol, err_msg + + +def test_complex(): + def func(z): + return z**2 - 1 + 2j + x0 = 2.0j + + ftol = 1e-4 + sol = root(func, x0, tol=ftol, method='DF-SANE') + + assert_(sol.success) + + f0 = np.linalg.norm(func(x0)) + fx = np.linalg.norm(func(sol.x)) + assert_(fx <= ftol*f0) + + +def test_linear_definite(): + # The DF-SANE paper proves convergence for "strongly isolated" + # solutions. + # + # For linear systems F(x) = A x - b = 0, with A positive or + # negative definite, the solution is strongly isolated. + + def check_solvability(A, b, line_search='cruz'): + def func(x): + return A.dot(x) - b + xp = np.linalg.solve(A, b) + eps = np.linalg.norm(func(xp)) * 1e3 + sol = root( + func, b, + options=dict(fatol=eps, ftol=0, maxfev=17523, line_search=line_search), + method='DF-SANE', + ) + assert_(sol.success) + assert_(np.linalg.norm(func(sol.x)) <= eps) + + n = 90 + + # Test linear pos.def. system + np.random.seed(1234) + A = np.arange(n*n).reshape(n, n) + A = A + n*n * np.diag(1 + np.arange(n)) + assert_(np.linalg.eigvals(A).min() > 0) + b = np.arange(n) * 1.0 + check_solvability(A, b, 'cruz') + check_solvability(A, b, 'cheng') + + # Test linear neg.def. system + check_solvability(-A, b, 'cruz') + check_solvability(-A, b, 'cheng') + + +def test_shape(): + def f(x, arg): + return x - arg + + for dt in [float, complex]: + x = np.zeros([2,2]) + arg = np.ones([2,2], dtype=dt) + + sol = root(f, x, args=(arg,), method='DF-SANE') + assert_(sol.success) + assert_equal(sol.x.shape, x.shape) + + +# Some of the test functions and initial guesses listed in +# [W. La Cruz, M. Raydan. Optimization Methods and Software, 18, 583 (2003)] + +def F_1(x, n): + g = np.zeros([n]) + i = np.arange(2, n+1) + g[0] = exp(x[0] - 1) - 1 + g[1:] = i*(exp(x[1:] - 1) - x[1:]) + return g + +def x0_1(n): + x0 = np.empty([n]) + x0.fill(n/(n-1)) + return x0 + +def F_2(x, n): + g = np.zeros([n]) + i = np.arange(2, n+1) + g[0] = exp(x[0]) - 1 + g[1:] = 0.1*i*(exp(x[1:]) + x[:-1] - 1) + return g + +def x0_2(n): + x0 = np.empty([n]) + x0.fill(1/n**2) + return x0 + + +def F_4(x, n): # skip name check + assert_equal(n % 3, 0) + g = np.zeros([n]) + # Note: the first line is typoed in some of the references; + # correct in original [Gasparo, Optimization Meth. 13, 79 (2000)] + g[::3] = 0.6 * x[::3] + 1.6 * x[1::3]**3 - 7.2 * x[1::3]**2 + 9.6 * x[1::3] - 4.8 + g[1::3] = (0.48 * x[::3] - 0.72 * x[1::3]**3 + 3.24 * x[1::3]**2 - 4.32 * x[1::3] + - x[2::3] + 0.2 * x[2::3]**3 + 2.16) + g[2::3] = 1.25 * x[2::3] - 0.25*x[2::3]**3 + return g + + +def x0_4(n): # skip name check + assert_equal(n % 3, 0) + x0 = np.array([-1, 1/2, -1] * (n//3)) + return x0 + +def F_6(x, n): + c = 0.9 + mu = (np.arange(1, n+1) - 0.5)/n + return x - 1/(1 - c/(2*n) * (mu[:,None]*x / (mu[:,None] + mu)).sum(axis=1)) + +def x0_6(n): + return np.ones([n]) + +def F_7(x, n): + assert_equal(n % 3, 0) + + def phi(t): + v = 0.5*t - 2 + v[t > -1] = ((-592*t**3 + 888*t**2 + 4551*t - 1924)/1998)[t > -1] + v[t >= 2] = (0.5*t + 2)[t >= 2] + return v + g = np.zeros([n]) + g[::3] = 1e4 * x[1::3]**2 - 1 + g[1::3] = exp(-x[::3]) + exp(-x[1::3]) - 1.0001 + g[2::3] = phi(x[2::3]) + return g + +def x0_7(n): + assert_equal(n % 3, 0) + return np.array([1e-3, 18, 1] * (n//3)) + +def F_9(x, n): + g = np.zeros([n]) + i = np.arange(2, n) + g[0] = x[0]**3/3 + x[1]**2/2 + g[1:-1] = -x[1:-1]**2/2 + i*x[1:-1]**3/3 + x[2:]**2/2 + g[-1] = -x[-1]**2/2 + n*x[-1]**3/3 + return g + +def x0_9(n): + return np.ones([n]) + +def F_10(x, n): + return np.log(1 + x) - x/n + +def x0_10(n): + return np.ones([n]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_bracket.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_bracket.py new file mode 100644 index 0000000000000000000000000000000000000000..f3a47fc005a2af6bbd02465634fdb72fa131f8f8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_bracket.py @@ -0,0 +1,906 @@ +import pytest + +import numpy as np + +from scipy.optimize._bracket import _ELIMITS +from scipy.optimize.elementwise import bracket_root, bracket_minimum +import scipy._lib._elementwise_iterative_method as eim +from scipy import stats +from scipy._lib._array_api_no_0d import (xp_assert_close, xp_assert_equal, + xp_assert_less, array_namespace) +from scipy._lib._array_api import xp_ravel +from scipy.conftest import array_api_compatible + + +# These tests were originally written for the private `optimize._bracket` +# interfaces, but now we want the tests to check the behavior of the public +# `optimize.elementwise` interfaces. Therefore, rather than importing +# `_bracket_root`/`_bracket_minimum` from `_bracket.py`, we import +# `bracket_root`/`bracket_minimum` from `optimize.elementwise` and wrap those +# functions to conform to the private interface. This may look a little strange, +# since it effectively just inverts the interface transformation done within the +# `bracket_root`/`bracket_minimum` functions, but it allows us to run the original, +# unmodified tests on the public interfaces, simplifying the PR that adds +# the public interfaces. We'll refactor this when we want to @parametrize the +# tests over multiple `method`s. +def _bracket_root(*args, **kwargs): + res = bracket_root(*args, **kwargs) + res.xl, res.xr = res.bracket + res.fl, res.fr = res.f_bracket + del res.bracket + del res.f_bracket + return res + + +def _bracket_minimum(*args, **kwargs): + res = bracket_minimum(*args, **kwargs) + res.xl, res.xm, res.xr = res.bracket + res.fl, res.fm, res.fr = res.f_bracket + del res.bracket + del res.f_bracket + return res + + +array_api_strict_skip_reason = 'Array API does not support fancy indexing assignment.' +jax_skip_reason = 'JAX arrays do not support item assignment.' + +@pytest.mark.skip_xp_backends('array_api_strict', reason=array_api_strict_skip_reason) +@pytest.mark.skip_xp_backends('jax.numpy', reason=jax_skip_reason) +@array_api_compatible +@pytest.mark.usefixtures("skip_xp_backends") +class TestBracketRoot: + @pytest.mark.parametrize("seed", (615655101, 3141866013, 238075752)) + @pytest.mark.parametrize("use_xmin", (False, True)) + @pytest.mark.parametrize("other_side", (False, True)) + @pytest.mark.parametrize("fix_one_side", (False, True)) + def test_nfev_expected(self, seed, use_xmin, other_side, fix_one_side, xp): + # Property-based test to confirm that _bracket_root is behaving as + # expected. The basic case is when root < a < b. + # The number of times bracket expands (per side) can be found by + # setting the expression for the left endpoint of the bracket to the + # root of f (x=0), solving for i, and rounding up. The corresponding + # lower and upper ends of the bracket are found by plugging this back + # into the expression for the ends of the bracket. + # `other_side=True` is the case that a < b < root + # Special cases like a < root < b are tested separately + rng = np.random.default_rng(seed) + xl0, d, factor = xp.asarray(rng.random(size=3) * [1e5, 10, 5]) + factor = 1 + factor # factor must be greater than 1 + xr0 = xl0 + d # xr0 must be greater than a in basic case + + def f(x): + f.count += 1 + return x # root is 0 + + if use_xmin: + xmin = xp.asarray(-rng.random()) + n = xp.ceil(xp.log(-(xl0 - xmin) / xmin) / xp.log(factor)) + l, u = xmin + (xl0 - xmin)*factor**-n, xmin + (xl0 - xmin)*factor**-(n - 1) + kwargs = dict(xl0=xl0, xr0=xr0, factor=factor, xmin=xmin) + else: + n = xp.ceil(xp.log(xr0/d) / xp.log(factor)) + l, u = xr0 - d*factor**n, xr0 - d*factor**(n-1) + kwargs = dict(xl0=xl0, xr0=xr0, factor=factor) + + if other_side: + kwargs['xl0'], kwargs['xr0'] = -kwargs['xr0'], -kwargs['xl0'] + l, u = -u, -l + if 'xmin' in kwargs: + kwargs['xmax'] = -kwargs.pop('xmin') + + if fix_one_side: + if other_side: + kwargs['xmin'] = -xr0 + else: + kwargs['xmax'] = xr0 + + f.count = 0 + res = _bracket_root(f, **kwargs) + + # Compare reported number of function evaluations `nfev` against + # reported `nit`, actual function call count `f.count`, and theoretical + # number of expansions `n`. + # When both sides are free, these get multiplied by 2 because function + # is evaluated on the left and the right each iteration. + # When one side is fixed, however, we add one: on the right side, the + # function gets evaluated once at b. + # Add 1 to `n` and `res.nit` because function evaluations occur at + # iterations *0*, 1, ..., `n`. Subtract 1 from `f.count` because + # function is called separately for left and right in iteration 0. + if not fix_one_side: + assert res.nfev == 2*(res.nit+1) == 2*(f.count-1) == 2*(n + 1) + else: + assert res.nfev == (res.nit+1)+1 == (f.count-1)+1 == (n+1)+1 + + # Compare reported bracket to theoretical bracket and reported function + # values to function evaluated at bracket. + bracket = xp.asarray([res.xl, res.xr]) + xp_assert_close(bracket, xp.asarray([l, u])) + f_bracket = xp.asarray([res.fl, res.fr]) + xp_assert_close(f_bracket, f(bracket)) + + # Check that bracket is valid and that status and success are correct + assert res.xr > res.xl + signs = xp.sign(f_bracket) + assert signs[0] == -signs[1] + assert res.status == 0 + assert res.success + + def f(self, q, p): + return stats._stats_py._SimpleNormal().cdf(q) - p + + @pytest.mark.parametrize('p', [0.6, np.linspace(0.05, 0.95, 10)]) + @pytest.mark.parametrize('xmin', [-5, None]) + @pytest.mark.parametrize('xmax', [5, None]) + @pytest.mark.parametrize('factor', [1.2, 2]) + def test_basic(self, p, xmin, xmax, factor, xp): + # Test basic functionality to bracket root (distribution PPF) + res = _bracket_root(self.f, xp.asarray(-0.01), 0.01, xmin=xmin, xmax=xmax, + factor=factor, args=(xp.asarray(p),)) + xp_assert_equal(-xp.sign(res.fl), xp.sign(res.fr)) + + @pytest.mark.parametrize('shape', [tuple(), (12,), (3, 4), (3, 2, 2)]) + def test_vectorization(self, shape, xp): + # Test for correct functionality, output shapes, and dtypes for various + # input shapes. + p = np.linspace(-0.05, 1.05, 12).reshape(shape) if shape else np.float64(0.6) + args = (p,) + maxiter = 10 + + @np.vectorize + def bracket_root_single(xl0, xr0, xmin, xmax, factor, p): + return _bracket_root(self.f, xl0, xr0, xmin=xmin, xmax=xmax, + factor=factor, args=(p,), + maxiter=maxiter) + + def f(*args, **kwargs): + f.f_evals += 1 + return self.f(*args, **kwargs) + f.f_evals = 0 + + rng = np.random.default_rng(2348234) + xl0 = -rng.random(size=shape) + xr0 = rng.random(size=shape) + xmin, xmax = 1e3*xl0, 1e3*xr0 + if shape: # make some elements un + i = rng.random(size=shape) > 0.5 + xmin[i], xmax[i] = -np.inf, np.inf + factor = rng.random(size=shape) + 1.5 + refs = bracket_root_single(xl0, xr0, xmin, xmax, factor, p).ravel() + xl0, xr0, xmin, xmax, factor = (xp.asarray(xl0), xp.asarray(xr0), + xp.asarray(xmin), xp.asarray(xmax), + xp.asarray(factor)) + args = tuple(map(xp.asarray, args)) + res = _bracket_root(f, xl0, xr0, xmin=xmin, xmax=xmax, factor=factor, + args=args, maxiter=maxiter) + + attrs = ['xl', 'xr', 'fl', 'fr', 'success', 'nfev', 'nit'] + for attr in attrs: + ref_attr = [xp.asarray(getattr(ref, attr)) for ref in refs] + res_attr = getattr(res, attr) + xp_assert_close(xp_ravel(res_attr, xp=xp), xp.stack(ref_attr)) + xp_assert_equal(res_attr.shape, shape) + + xp_test = array_namespace(xp.asarray(1.)) + assert res.success.dtype == xp_test.bool + if shape: + assert xp.all(res.success[1:-1]) + assert res.status.dtype == xp.int32 + assert res.nfev.dtype == xp.int32 + assert res.nit.dtype == xp.int32 + assert xp.max(res.nit) == f.f_evals - 2 + xp_assert_less(res.xl, res.xr) + xp_assert_close(res.fl, xp.asarray(self.f(res.xl, *args))) + xp_assert_close(res.fr, xp.asarray(self.f(res.xr, *args))) + + def test_flags(self, xp): + # Test cases that should produce different status flags; show that all + # can be produced simultaneously. + def f(xs, js): + funcs = [lambda x: x - 1.5, + lambda x: x - 1000, + lambda x: x - 1000, + lambda x: x * xp.nan, + lambda x: x] + + return [funcs[int(j)](x) for x, j in zip(xs, js)] + + args = (xp.arange(5, dtype=xp.int64),) + res = _bracket_root(f, + xl0=xp.asarray([-1., -1., -1., -1., 4.]), + xr0=xp.asarray([1, 1, 1, 1, -4]), + xmin=xp.asarray([-xp.inf, -1, -xp.inf, -xp.inf, 6]), + xmax=xp.asarray([xp.inf, 1, xp.inf, xp.inf, 2]), + args=args, maxiter=3) + + ref_flags = xp.asarray([eim._ECONVERGED, + _ELIMITS, + eim._ECONVERR, + eim._EVALUEERR, + eim._EINPUTERR], + dtype=xp.int32) + + xp_assert_equal(res.status, ref_flags) + + @pytest.mark.parametrize("root", (0.622, [0.622, 0.623])) + @pytest.mark.parametrize('xmin', [-5, None]) + @pytest.mark.parametrize('xmax', [5, None]) + @pytest.mark.parametrize("dtype", ("float16", "float32", "float64")) + def test_dtype(self, root, xmin, xmax, dtype, xp): + # Test that dtypes are preserved + dtype = getattr(xp, dtype) + xp_test = array_namespace(xp.asarray(1.)) + + xmin = xmin if xmin is None else xp.asarray(xmin, dtype=dtype) + xmax = xmax if xmax is None else xp.asarray(xmax, dtype=dtype) + root = xp.asarray(root, dtype=dtype) + def f(x, root): + return xp_test.astype((x - root) ** 3, dtype) + + bracket = xp.asarray([-0.01, 0.01], dtype=dtype) + res = _bracket_root(f, *bracket, xmin=xmin, xmax=xmax, args=(root,)) + assert xp.all(res.success) + assert res.xl.dtype == res.xr.dtype == dtype + assert res.fl.dtype == res.fr.dtype == dtype + + def test_input_validation(self, xp): + # Test input validation for appropriate error messages + + message = '`func` must be callable.' + with pytest.raises(ValueError, match=message): + _bracket_root(None, -4, 4) + + message = '...must be numeric and real.' + with pytest.raises(ValueError, match=message): + _bracket_root(lambda x: x, -4+1j, 4) + with pytest.raises(ValueError, match=message): + _bracket_root(lambda x: x, -4, 'hello') + with pytest.raises(ValueError, match=message): + _bracket_root(lambda x: x, -4, 4, xmin=np) + with pytest.raises(ValueError, match=message): + _bracket_root(lambda x: x, -4, 4, xmax=object()) + with pytest.raises(ValueError, match=message): + _bracket_root(lambda x: x, -4, 4, factor=sum) + + message = "All elements of `factor` must be greater than 1." + with pytest.raises(ValueError, match=message): + _bracket_root(lambda x: x, -4, 4, factor=0.5) + + message = "broadcast" + # raised by `xp.broadcast, but the traceback is readable IMO + with pytest.raises(Exception, match=message): + _bracket_root(lambda x: x, xp.asarray([-2, -3]), xp.asarray([3, 4, 5])) + # Consider making this give a more readable error message + # with pytest.raises(ValueError, match=message): + # _bracket_root(lambda x: [x[0], x[1], x[1]], [-3, -3], [5, 5]) + + message = '`maxiter` must be a non-negative integer.' + with pytest.raises(ValueError, match=message): + _bracket_root(lambda x: x, -4, 4, maxiter=1.5) + with pytest.raises(ValueError, match=message): + _bracket_root(lambda x: x, -4, 4, maxiter=-1) + with pytest.raises(ValueError, match=message): + _bracket_root(lambda x: x, -4, 4, maxiter="shrubbery") + + def test_special_cases(self, xp): + # Test edge cases and other special cases + xp_test = array_namespace(xp.asarray(1.)) + + # Test that integers are not passed to `f` + # (otherwise this would overflow) + def f(x): + assert xp_test.isdtype(x.dtype, "real floating") + return x ** 99 - 1 + + res = _bracket_root(f, xp.asarray(-7.), xp.asarray(5.)) + assert res.success + + # Test maxiter = 0. Should do nothing to bracket. + def f(x): + return x - 10 + + bracket = (xp.asarray(-3.), xp.asarray(5.)) + res = _bracket_root(f, *bracket, maxiter=0) + assert res.xl, res.xr == bracket + assert res.nit == 0 + assert res.nfev == 2 + assert res.status == -2 + + # Test scalar `args` (not in tuple) + def f(x, c): + return c*x - 1 + + res = _bracket_root(f, xp.asarray(-1.), xp.asarray(1.), + args=xp.asarray(3.)) + assert res.success + xp_assert_close(res.fl, f(res.xl, 3)) + + # Test other edge cases + + def f(x): + f.count += 1 + return x + + # 1. root lies within guess of bracket + f.count = 0 + _bracket_root(f, xp.asarray(-10), xp.asarray(20)) + assert f.count == 2 + + # 2. bracket endpoint hits root exactly + f.count = 0 + res = _bracket_root(f, xp.asarray(5.), xp.asarray(10.), + factor=2) + + assert res.nfev == 4 + xp_assert_close(res.xl, xp.asarray(0.), atol=1e-15) + xp_assert_close(res.xr, xp.asarray(5.), atol=1e-15) + + # 3. bracket limit hits root exactly + with np.errstate(over='ignore'): + res = _bracket_root(f, xp.asarray(5.), xp.asarray(10.), + xmin=0) + xp_assert_close(res.xl, xp.asarray(0.), atol=1e-15) + + with np.errstate(over='ignore'): + res = _bracket_root(f, xp.asarray(-10.), xp.asarray(-5.), + xmax=0) + xp_assert_close(res.xr, xp.asarray(0.), atol=1e-15) + + # 4. bracket not within min, max + with np.errstate(over='ignore'): + res = _bracket_root(f, xp.asarray(5.), xp.asarray(10.), + xmin=1) + assert not res.success + + def test_bug_fixes(self): + # 1. Bug in double sided bracket search. + # Happened in some cases where there are terminations on one side + # after corresponding searches on other side failed due to reaching the + # boundary. + + # https://github.com/scipy/scipy/pull/22560#discussion_r1962853839 + def f(x, p): + return np.exp(x) - p + + p = np.asarray([0.29, 0.35]) + res = _bracket_root(f, xl0=-1, xmin=-np.inf, xmax=0, args=(p, )) + + # https://github.com/scipy/scipy/pull/22560/files#r1962952517 + def f(x, p, c): + return np.exp(x*c) - p + + p = [0.32061201, 0.39175242, 0.40047535, 0.50527218, 0.55654373, + 0.11911647, 0.37507896, 0.66554191] + c = [1., -1., 1., 1., -1., 1., 1., 1.] + xl0 = [-7.63108551, 3.27840947, -8.36968526, -1.78124372, + 0.92201295, -2.48930123, -0.66733533, -0.44606749] + xr0 = [-6.63108551, 4.27840947, -7.36968526, -0.78124372, + 1.92201295, -1.48930123, 0., 0.] + xmin = [-np.inf, 0., -np.inf, -np.inf, 0., -np.inf, -np.inf, + -np.inf] + xmax = [0., np.inf, 0., 0., np.inf, 0., 0., 0.] + + res = _bracket_root(f, xl0=xl0, xr0=xr0, xmin=xmin, xmax=xmax, args=(p, c)) + + # 2. Default xl0 + 1 for xr0 exceeds xmax. + # https://github.com/scipy/scipy/pull/22560#discussion_r1962947434 + res = _bracket_root(lambda x: x + 0.25, xl0=-0.5, xmin=-np.inf, xmax=0) + assert res.success + + +@pytest.mark.skip_xp_backends('array_api_strict', reason=array_api_strict_skip_reason) +@pytest.mark.skip_xp_backends('jax.numpy', reason=jax_skip_reason) +@array_api_compatible +@pytest.mark.usefixtures("skip_xp_backends") +class TestBracketMinimum: + def init_f(self): + def f(x, a, b): + f.count += 1 + return (x - a)**2 + b + f.count = 0 + return f + + def assert_valid_bracket(self, result, xp): + assert xp.all( + (result.xl < result.xm) & (result.xm < result.xr) + ) + assert xp.all( + (result.fl >= result.fm) & (result.fr > result.fm) + | (result.fl > result.fm) & (result.fr > result.fm) + ) + + def get_kwargs( + self, *, xl0=None, xr0=None, factor=None, xmin=None, xmax=None, args=None + ): + names = ("xl0", "xr0", "xmin", "xmax", "factor", "args") + return { + name: val for name, val in zip(names, (xl0, xr0, xmin, xmax, factor, args)) + if val is not None + } + + @pytest.mark.parametrize( + "seed", + ( + 307448016549685229886351382450158984917, + 11650702770735516532954347931959000479, + 113767103358505514764278732330028568336, + ) + ) + @pytest.mark.parametrize("use_xmin", (False, True)) + @pytest.mark.parametrize("other_side", (False, True)) + def test_nfev_expected(self, seed, use_xmin, other_side, xp): + rng = np.random.default_rng(seed) + args = (xp.asarray(0.), xp.asarray(0.)) # f(x) = x^2 with minimum at 0 + # xl0, xm0, xr0 are chosen such that the initial bracket is to + # the right of the minimum, and the bracket will expand + # downhill towards zero. + xl0, d1, d2, factor = xp.asarray(rng.random(size=4) * [1e5, 10, 10, 5]) + xm0 = xl0 + d1 + xr0 = xm0 + d2 + # Factor should be greater than one. + factor += 1 + + if use_xmin: + xmin = xp.asarray(-rng.random() * 5, dtype=xp.float64) + n = int(xp.ceil(xp.log(-(xl0 - xmin) / xmin) / xp.log(factor))) + lower = xmin + (xl0 - xmin)*factor**-n + middle = xmin + (xl0 - xmin)*factor**-(n-1) + upper = xmin + (xl0 - xmin)*factor**-(n-2) if n > 1 else xm0 + # It may be the case the lower is below the minimum, but we still + # don't have a valid bracket. + if middle**2 > lower**2: + n += 1 + lower, middle, upper = ( + xmin + (xl0 - xmin)*factor**-n, lower, middle + ) + else: + xmin = None + n = int(xp.ceil(xp.log(xl0 / d1) / xp.log(factor))) + lower = xl0 - d1*factor**n + middle = xl0 - d1*factor**(n-1) if n > 1 else xl0 + upper = xl0 - d1*factor**(n-2) if n > 1 else xm0 + # It may be the case the lower is below the minimum, but we still + # don't have a valid bracket. + if middle**2 > lower**2: + n += 1 + lower, middle, upper = ( + xl0 - d1*factor**n, lower, middle + ) + f = self.init_f() + + xmax = None + if other_side: + xl0, xm0, xr0 = -xr0, -xm0, -xl0 + xmin, xmax = None, -xmin if xmin is not None else None + lower, middle, upper = -upper, -middle, -lower + + kwargs = self.get_kwargs( + xl0=xl0, xr0=xr0, xmin=xmin, xmax=xmax, factor=factor, args=args + ) + result = _bracket_minimum(f, xp.asarray(xm0), **kwargs) + + # Check that `nfev` and `nit` have the correct relationship + assert result.nfev == result.nit + 3 + # Check that `nfev` reports the correct number of function evaluations. + assert result.nfev == f.count + # Check that the number of iterations matches the theoretical value. + assert result.nit == n + + # Compare reported bracket to theoretical bracket and reported function + # values to function evaluated at bracket. + xp_assert_close(result.xl, lower) + xp_assert_close(result.xm, middle) + xp_assert_close(result.xr, upper) + xp_assert_close(result.fl, f(lower, *args)) + xp_assert_close(result.fm, f(middle, *args)) + xp_assert_close(result.fr, f(upper, *args)) + + self.assert_valid_bracket(result, xp) + assert result.status == 0 + assert result.success + + def test_flags(self, xp): + # Test cases that should produce different status flags; show that all + # can be produced simultaneously + def f(xs, js): + funcs = [lambda x: (x - 1.5)**2, + lambda x: x, + lambda x: x, + lambda x: xp.nan, + lambda x: x**2] + + return [funcs[j](x) for x, j in zip(xs, js)] + + args = (xp.arange(5, dtype=xp.int64),) + xl0 = xp.asarray([-1.0, -1.0, -1.0, -1.0, 6.0]) + xm0 = xp.asarray([0.0, 0.0, 0.0, 0.0, 4.0]) + xr0 = xp.asarray([1.0, 1.0, 1.0, 1.0, 2.0]) + xmin = xp.asarray([-xp.inf, -1.0, -xp.inf, -xp.inf, 8.0]) + + result = _bracket_minimum(f, xm0, xl0=xl0, xr0=xr0, xmin=xmin, + args=args, maxiter=3) + + reference_flags = xp.asarray([eim._ECONVERGED, _ELIMITS, + eim._ECONVERR, eim._EVALUEERR, + eim._EINPUTERR], dtype=xp.int32) + xp_assert_equal(result.status, reference_flags) + + @pytest.mark.parametrize("minimum", (0.622, [0.622, 0.623])) + @pytest.mark.parametrize("dtype", ("float16", "float32", "float64")) + @pytest.mark.parametrize("xmin", [-5, None]) + @pytest.mark.parametrize("xmax", [5, None]) + def test_dtypes(self, minimum, xmin, xmax, dtype, xp): + dtype = getattr(xp, dtype) + xp_test = array_namespace(xp.asarray(1.)) + xmin = xmin if xmin is None else xp.asarray(xmin, dtype=dtype) + xmax = xmax if xmax is None else xp.asarray(xmax, dtype=dtype) + minimum = xp.asarray(minimum, dtype=dtype) + + def f(x, minimum): + return xp_test.astype((x - minimum)**2, dtype) + + xl0, xm0, xr0 = [-0.01, 0.0, 0.01] + result = _bracket_minimum( + f, xp.asarray(xm0, dtype=dtype), xl0=xp.asarray(xl0, dtype=dtype), + xr0=xp.asarray(xr0, dtype=dtype), xmin=xmin, xmax=xmax, args=(minimum, ) + ) + assert xp.all(result.success) + assert result.xl.dtype == result.xm.dtype == result.xr.dtype == dtype + assert result.fl.dtype == result.fm.dtype == result.fr.dtype == dtype + + @pytest.mark.skip_xp_backends(np_only=True, reason="str/object arrays") + def test_input_validation(self, xp): + # Test input validation for appropriate error messages + + message = '`func` must be callable.' + with pytest.raises(ValueError, match=message): + _bracket_minimum(None, -4, xl0=4) + + message = '...must be numeric and real.' + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x**2, xp.asarray(4+1j)) + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x**2, xp.asarray(-4), xl0='hello') + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x**2, xp.asarray(-4), + xr0='farcical aquatic ceremony') + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x**2, xp.asarray(-4), xmin=np) + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x**2, xp.asarray(-4), xmax=object()) + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x**2, xp.asarray(-4), factor=sum) + + message = "All elements of `factor` must be greater than 1." + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x, xp.asarray(-4), factor=0.5) + + message = "shape mismatch: objects cannot be broadcast" + # raised by `xp.broadcast, but the traceback is readable IMO + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x**2, xp.asarray([-2, -3]), xl0=[-3, -4, -5]) + + message = '`maxiter` must be a non-negative integer.' + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x**2, xp.asarray(-4), xr0=4, maxiter=1.5) + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x**2, xp.asarray(-4), xr0=4, maxiter=-1) + with pytest.raises(ValueError, match=message): + _bracket_minimum(lambda x: x**2, xp.asarray(-4), xr0=4, maxiter="ekki") + + @pytest.mark.parametrize("xl0", [0.0, None]) + @pytest.mark.parametrize("xm0", (0.05, 0.1, 0.15)) + @pytest.mark.parametrize("xr0", (0.2, 0.4, 0.6, None)) + # Minimum is ``a`` for each tuple ``(a, b)`` below. Tests cases where minimum + # is within, or at varying distances to the left or right of the initial + # bracket. + @pytest.mark.parametrize( + "args", + ( + (1.2, 0), (-0.5, 0), (0.1, 0), (0.2, 0), (3.6, 0), (21.4, 0), + (121.6, 0), (5764.1, 0), (-6.4, 0), (-12.9, 0), (-146.2, 0) + ) + ) + def test_scalar_no_limits(self, xl0, xm0, xr0, args, xp): + f = self.init_f() + kwargs = self.get_kwargs(xl0=xl0, xr0=xr0, args=tuple(map(xp.asarray, args))) + result = _bracket_minimum(f, xp.asarray(xm0, dtype=xp.float64), **kwargs) + self.assert_valid_bracket(result, xp) + assert result.status == 0 + assert result.success + assert result.nfev == f.count + + @pytest.mark.parametrize( + # xmin is set at 0.0 in all cases. + "xl0,xm0,xr0,xmin", + ( + # Initial bracket at varying distances from the xmin. + (0.5, 0.75, 1.0, 0.0), + (1.0, 2.5, 4.0, 0.0), + (2.0, 4.0, 6.0, 0.0), + (12.0, 16.0, 20.0, 0.0), + # Test default initial left endpoint selection. It should not + # be below xmin. + (None, 0.75, 1.0, 0.0), + (None, 2.5, 4.0, 0.0), + (None, 4.0, 6.0, 0.0), + (None, 16.0, 20.0, 0.0), + ) + ) + @pytest.mark.parametrize( + "args", ( + (0.0, 0.0), # Minimum is directly at xmin. + (1e-300, 0.0), # Minimum is extremely close to xmin. + (1e-20, 0.0), # Minimum is very close to xmin. + # Minimum at varying distances from xmin. + (0.1, 0.0), + (0.2, 0.0), + (0.4, 0.0) + ) + ) + def test_scalar_with_limit_left(self, xl0, xm0, xr0, xmin, args, xp): + f = self.init_f() + kwargs = self.get_kwargs(xl0=xl0, xr0=xr0, xmin=xmin, + args=tuple(map(xp.asarray, args))) + result = _bracket_minimum(f, xp.asarray(xm0), **kwargs) + self.assert_valid_bracket(result, xp) + assert result.status == 0 + assert result.success + assert result.nfev == f.count + + @pytest.mark.parametrize( + #xmax is set to 1.0 in all cases. + "xl0,xm0,xr0,xmax", + ( + # Bracket at varying distances from xmax. + (0.2, 0.3, 0.4, 1.0), + (0.05, 0.075, 0.1, 1.0), + (-0.2, -0.1, 0.0, 1.0), + (-21.2, -17.7, -14.2, 1.0), + # Test default right endpoint selection. It should not exceed xmax. + (0.2, 0.3, None, 1.0), + (0.05, 0.075, None, 1.0), + (-0.2, -0.1, None, 1.0), + (-21.2, -17.7, None, 1.0), + ) + ) + @pytest.mark.parametrize( + "args", ( + (0.9999999999999999, 0.0), # Minimum very close to xmax. + # Minimum at varying distances from xmax. + (0.9, 0.0), + (0.7, 0.0), + (0.5, 0.0) + ) + ) + def test_scalar_with_limit_right(self, xl0, xm0, xr0, xmax, args, xp): + f = self.init_f() + args = tuple(xp.asarray(arg, dtype=xp.float64) for arg in args) + kwargs = self.get_kwargs(xl0=xl0, xr0=xr0, xmax=xmax, args=args) + result = _bracket_minimum(f, xp.asarray(xm0, dtype=xp.float64), **kwargs) + self.assert_valid_bracket(result, xp) + assert result.status == 0 + assert result.success + assert result.nfev == f.count + + @pytest.mark.parametrize( + "xl0,xm0,xr0,xmin,xmax,args", + ( + ( # Case 1: + # Initial bracket. + 0.2, + 0.3, + 0.4, + # Function slopes down to the right from the bracket to a minimum + # at 1.0. xmax is also at 1.0 + None, + 1.0, + (1.0, 0.0) + ), + ( # Case 2: + # Initial bracket. + 1.4, + 1.95, + 2.5, + # Function slopes down to the left from the bracket to a minimum at + # 0.3 with xmin set to 0.3. + 0.3, + None, + (0.3, 0.0) + ), + ( + # Case 3: + # Initial bracket. + 2.6, + 3.25, + 3.9, + # Function slopes down and to the right to a minimum at 99.4 with xmax + # at 99.4. Tests case where minimum is at xmax relatively further from + # the bracket. + None, + 99.4, + (99.4, 0) + ), + ( + # Case 4: + # Initial bracket. + 4, + 4.5, + 5, + # Function slopes down and to the left away from the bracket with a + # minimum at -26.3 with xmin set to -26.3. Tests case where minimum is + # at xmin relatively far from the bracket. + -26.3, + None, + (-26.3, 0) + ), + ( + # Case 5: + # Similar to Case 1 above, but tests default values of xl0 and xr0. + None, + 0.3, + None, + None, + 1.0, + (1.0, 0.0) + ), + ( # Case 6: + # Similar to Case 2 above, but tests default values of xl0 and xr0. + None, + 1.95, + None, + 0.3, + None, + (0.3, 0.0) + ), + ( + # Case 7: + # Similar to Case 3 above, but tests default values of xl0 and xr0. + None, + 3.25, + None, + None, + 99.4, + (99.4, 0) + ), + ( + # Case 8: + # Similar to Case 4 above, but tests default values of xl0 and xr0. + None, + 4.5, + None, + -26.3, + None, + (-26.3, 0) + ), + ) + ) + def test_minimum_at_boundary_point(self, xl0, xm0, xr0, xmin, xmax, args, xp): + f = self.init_f() + kwargs = self.get_kwargs(xr0=xr0, xmin=xmin, xmax=xmax, + args=tuple(map(xp.asarray, args))) + result = _bracket_minimum(f, xp.asarray(xm0), **kwargs) + assert result.status == -1 + assert args[0] in (result.xl, result.xr) + assert result.nfev == f.count + + @pytest.mark.parametrize('shape', [tuple(), (12, ), (3, 4), (3, 2, 2)]) + def test_vectorization(self, shape, xp): + # Test for correct functionality, output shapes, and dtypes for + # various input shapes. + a = np.linspace(-0.05, 1.05, 12).reshape(shape) if shape else 0.6 + args = (a, 0.) + maxiter = 10 + + @np.vectorize + def bracket_minimum_single(xm0, xl0, xr0, xmin, xmax, factor, a): + return _bracket_minimum(self.init_f(), xm0, xl0=xl0, xr0=xr0, xmin=xmin, + xmax=xmax, factor=factor, maxiter=maxiter, + args=(a, 0.0)) + + f = self.init_f() + + rng = np.random.default_rng(2348234) + xl0 = -rng.random(size=shape) + xr0 = rng.random(size=shape) + xm0 = xl0 + rng.random(size=shape) * (xr0 - xl0) + xmin, xmax = 1e3*xl0, 1e3*xr0 + if shape: # make some elements un + i = rng.random(size=shape) > 0.5 + xmin[i], xmax[i] = -np.inf, np.inf + factor = rng.random(size=shape) + 1.5 + refs = bracket_minimum_single(xm0, xl0, xr0, xmin, xmax, factor, a).ravel() + args = tuple(xp.asarray(arg, dtype=xp.float64) for arg in args) + res = _bracket_minimum(f, xp.asarray(xm0), xl0=xl0, xr0=xr0, xmin=xmin, + xmax=xmax, factor=factor, args=args, maxiter=maxiter) + + attrs = ['xl', 'xm', 'xr', 'fl', 'fm', 'fr', 'success', 'nfev', 'nit'] + for attr in attrs: + ref_attr = [xp.asarray(getattr(ref, attr)) for ref in refs] + res_attr = getattr(res, attr) + xp_assert_close(xp_ravel(res_attr, xp=xp), xp.stack(ref_attr)) + xp_assert_equal(res_attr.shape, shape) + + xp_test = array_namespace(xp.asarray(1.)) + assert res.success.dtype == xp_test.bool + if shape: + assert xp.all(res.success[1:-1]) + assert res.status.dtype == xp.int32 + assert res.nfev.dtype == xp.int32 + assert res.nit.dtype == xp.int32 + assert xp.max(res.nit) == f.count - 3 + self.assert_valid_bracket(res, xp) + xp_assert_close(res.fl, f(res.xl, *args)) + xp_assert_close(res.fm, f(res.xm, *args)) + xp_assert_close(res.fr, f(res.xr, *args)) + + def test_special_cases(self, xp): + # Test edge cases and other special cases. + xp_test = array_namespace(xp.asarray(1.)) + + # Test that integers are not passed to `f` + # (otherwise this would overflow) + def f(x): + assert xp_test.isdtype(x.dtype, "numeric") + return x ** 98 - 1 + + result = _bracket_minimum(f, xp.asarray(-7., dtype=xp.float64), xr0=5) + assert result.success + + # Test maxiter = 0. Should do nothing to bracket. + def f(x): + return x**2 - 10 + + xl0, xm0, xr0 = xp.asarray(-3.), xp.asarray(-1.), xp.asarray(2.) + result = _bracket_minimum(f, xm0, xl0=xl0, xr0=xr0, maxiter=0) + xp_assert_equal(result.xl, xl0) + xp_assert_equal(result.xm, xm0) + xp_assert_equal(result.xr, xr0) + + # Test scalar `args` (not in tuple) + def f(x, c): + return c*x**2 - 1 + + result = _bracket_minimum(f, xp.asarray(-1.), args=xp.asarray(3.)) + assert result.success + xp_assert_close(result.fl, f(result.xl, 3)) + + # Initial bracket is valid. + f = self.init_f() + xl0, xm0, xr0 = xp.asarray(-1.0), xp.asarray(-0.2), xp.asarray(1.0) + args = (xp.asarray(0.), xp.asarray(0.)) + result = _bracket_minimum(f, xm0, xl0=xl0, xr0=xr0, args=args) + assert f.count == 3 + + xp_assert_equal(result.xl, xl0) + xp_assert_equal(result.xm , xm0) + xp_assert_equal(result.xr, xr0) + xp_assert_equal(result.fl, f(xl0, *args)) + xp_assert_equal(result.fm, f(xm0, *args)) + xp_assert_equal(result.fr, f(xr0, *args)) + + def test_gh_20562_left(self, xp): + # Regression test for https://github.com/scipy/scipy/issues/20562 + # minimum of f in [xmin, xmax] is at xmin. + xmin, xmax = xp.asarray(0.21933608), xp.asarray(1.39713606) + + def f(x): + log_a, log_b = xp.log(xmin), xp.log(xmax) + return -((log_b - log_a)*x)**-1 + + result = _bracket_minimum(f, xp.asarray(0.5535723499480897), xmin=xmin, + xmax=xmax) + assert xmin == result.xl + + def test_gh_20562_right(self, xp): + # Regression test for https://github.com/scipy/scipy/issues/20562 + # minimum of f in [xmin, xmax] is at xmax. + xmin, xmax = xp.asarray(-1.39713606), xp.asarray(-0.21933608) + + def f(x): + log_a, log_b = xp.log(-xmax), xp.log(-xmin) + return ((log_b - log_a)*x)**-1 + + result = _bracket_minimum(f, xp.asarray(-0.5535723499480897), + xmin=xmin, xmax=xmax) + assert xmax == result.xr diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_chandrupatla.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_chandrupatla.py new file mode 100644 index 0000000000000000000000000000000000000000..bcc2ae1a70be2883a3d2346a1cb8d95d8ac027fa --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_chandrupatla.py @@ -0,0 +1,984 @@ +import math +import pytest +import numpy as np + +from scipy import stats, special +import scipy._lib._elementwise_iterative_method as eim +from scipy.conftest import array_api_compatible +from scipy._lib._array_api import array_namespace, is_cupy, is_numpy, xp_ravel, xp_size +from scipy._lib._array_api_no_0d import (xp_assert_close, xp_assert_equal, + xp_assert_less) + +from scipy.optimize.elementwise import find_minimum, find_root +from scipy.optimize._tstutils import _CHANDRUPATLA_TESTS + +from itertools import permutations +from .test_zeros import TestScalarRootFinders + + +def _vectorize(xp): + # xp-compatible version of np.vectorize + # assumes arguments are all arrays of the same shape + def decorator(f): + def wrapped(*arg_arrays): + shape = arg_arrays[0].shape + arg_arrays = [xp_ravel(arg_array, xp=xp) for arg_array in arg_arrays] + res = [] + for i in range(math.prod(shape)): + arg_scalars = [arg_array[i] for arg_array in arg_arrays] + res.append(f(*arg_scalars)) + return res + + return wrapped + + return decorator + + +# These tests were originally written for the private `optimize._chandrupatla` +# interfaces, but now we want the tests to check the behavior of the public +# `optimize.elementwise` interfaces. Therefore, rather than importing +# `_chandrupatla`/`_chandrupatla_minimize` from `_chandrupatla.py`, we import +# `find_root`/`find_minimum` from `optimize.elementwise` and wrap those +# functions to conform to the private interface. This may look a little strange, +# since it effectively just inverts the interface transformation done within the +# `find_root`/`find_minimum` functions, but it allows us to run the original, +# unmodified tests on the public interfaces, simplifying the PR that adds +# the public interfaces. We'll refactor this when we want to @parametrize the +# tests over multiple `method`s. +def _wrap_chandrupatla(func): + def _chandrupatla_wrapper(f, *bracket, **kwargs): + # avoid passing arguments to `find_minimum` to this function + tol_keys = {'xatol', 'xrtol', 'fatol', 'frtol'} + tolerances = {key: kwargs.pop(key) for key in tol_keys if key in kwargs} + _callback = kwargs.pop('callback', None) + if callable(_callback): + def callback(res): + if func == find_root: + res.xl, res.xr = res.bracket + res.fl, res.fr = res.f_bracket + else: + res.xl, res.xm, res.xr = res.bracket + res.fl, res.fm, res.fr = res.f_bracket + res.fun = res.f_x + del res.bracket + del res.f_bracket + del res.f_x + return _callback(res) + else: + callback = _callback + + res = func(f, bracket, tolerances=tolerances, callback=callback, **kwargs) + if func == find_root: + res.xl, res.xr = res.bracket + res.fl, res.fr = res.f_bracket + else: + res.xl, res.xm, res.xr = res.bracket + res.fl, res.fm, res.fr = res.f_bracket + res.fun = res.f_x + del res.bracket + del res.f_bracket + del res.f_x + return res + return _chandrupatla_wrapper + + +_chandrupatla_root = _wrap_chandrupatla(find_root) +_chandrupatla_minimize = _wrap_chandrupatla(find_minimum) + + +def f1(x): + return 100*(1 - x**3.)**2 + (1-x**2.) + 2*(1-x)**2. + + +def f2(x): + return 5 + (x - 2.)**6 + + +def f3(x): + xp = array_namespace(x) + return xp.exp(x) - 5*x + + +def f4(x): + return x**5. - 5*x**3. - 20.*x + 5. + + +def f5(x): + return 8*x**3 - 2*x**2 - 7*x + 3 + + +def _bracket_minimum(func, x1, x2): + phi = 1.61803398875 + maxiter = 100 + f1 = func(x1) + f2 = func(x2) + step = x2 - x1 + x1, x2, f1, f2, step = ((x2, x1, f2, f1, -step) if f2 > f1 + else (x1, x2, f1, f2, step)) + + for i in range(maxiter): + step *= phi + x3 = x2 + step + f3 = func(x3) + if f3 < f2: + x1, x2, f1, f2 = x2, x3, f2, f3 + else: + break + return x1, x2, x3, f1, f2, f3 + + +cases = [ + (f1, -1, 11), + (f1, -2, 13), + (f1, -4, 13), + (f1, -8, 15), + (f1, -16, 16), + (f1, -32, 19), + (f1, -64, 20), + (f1, -128, 21), + (f1, -256, 21), + (f1, -512, 19), + (f1, -1024, 24), + (f2, -1, 8), + (f2, -2, 6), + (f2, -4, 6), + (f2, -8, 7), + (f2, -16, 8), + (f2, -32, 8), + (f2, -64, 9), + (f2, -128, 11), + (f2, -256, 13), + (f2, -512, 12), + (f2, -1024, 13), + (f3, -1, 11), + (f3, -2, 11), + (f3, -4, 11), + (f3, -8, 10), + (f3, -16, 14), + (f3, -32, 12), + (f3, -64, 15), + (f3, -128, 18), + (f3, -256, 18), + (f3, -512, 19), + (f3, -1024, 19), + (f4, -0.05, 9), + (f4, -0.10, 11), + (f4, -0.15, 11), + (f4, -0.20, 11), + (f4, -0.25, 11), + (f4, -0.30, 9), + (f4, -0.35, 9), + (f4, -0.40, 9), + (f4, -0.45, 10), + (f4, -0.50, 10), + (f4, -0.55, 10), + (f5, -0.05, 6), + (f5, -0.10, 7), + (f5, -0.15, 8), + (f5, -0.20, 10), + (f5, -0.25, 9), + (f5, -0.30, 8), + (f5, -0.35, 7), + (f5, -0.40, 7), + (f5, -0.45, 9), + (f5, -0.50, 9), + (f5, -0.55, 8) +] + + +@array_api_compatible +@pytest.mark.usefixtures("skip_xp_backends") +@pytest.mark.skip_xp_backends('jax.numpy', + reason='JAX arrays do not support item assignment.') +@pytest.mark.skip_xp_backends('array_api_strict', + reason='Currently uses fancy indexing assignment.') +class TestChandrupatlaMinimize: + + def f(self, x, loc): + xp = array_namespace(x, loc) + res = -xp.exp(-1/2 * (x-loc)**2) / (2*xp.pi)**0.5 + return xp.asarray(res, dtype=x.dtype)[()] + + @pytest.mark.parametrize('dtype', ('float32', 'float64')) + @pytest.mark.parametrize('loc', [0.6, np.linspace(-1.05, 1.05, 10)]) + def test_basic(self, loc, xp, dtype): + # Find mode of normal distribution. Compare mode against location + # parameter and value of pdf at mode against expected pdf. + rtol = {'float32': 5e-3, 'float64': 5e-7}[dtype] + dtype = getattr(xp, dtype) + bracket = (xp.asarray(xi, dtype=dtype) for xi in (-5, 0, 5)) + loc = xp.asarray(loc, dtype=dtype) + fun = xp.broadcast_to(xp.asarray(-stats.norm.pdf(0), dtype=dtype), loc.shape) + + res = _chandrupatla_minimize(self.f, *bracket, args=(loc,)) + xp_assert_close(res.x, loc, rtol=rtol) + xp_assert_equal(res.fun, fun) + + @pytest.mark.parametrize('shape', [tuple(), (12,), (3, 4), (3, 2, 2)]) + def test_vectorization(self, shape, xp): + # Test for correct functionality, output shapes, and dtypes for various + # input shapes. + loc = xp.linspace(-0.05, 1.05, 12).reshape(shape) if shape else xp.asarray(0.6) + args = (loc,) + bracket = xp.asarray(-5.), xp.asarray(0.), xp.asarray(5.) + xp_test = array_namespace(loc) # need xp.stack + + @_vectorize(xp) + def chandrupatla_single(loc_single): + return _chandrupatla_minimize(self.f, *bracket, args=(loc_single,)) + + def f(*args, **kwargs): + f.f_evals += 1 + return self.f(*args, **kwargs) + f.f_evals = 0 + + res = _chandrupatla_minimize(f, *bracket, args=args) + refs = chandrupatla_single(loc) + + attrs = ['x', 'fun', 'success', 'status', 'nfev', 'nit', + 'xl', 'xm', 'xr', 'fl', 'fm', 'fr'] + for attr in attrs: + ref_attr = xp_test.stack([getattr(ref, attr) for ref in refs]) + res_attr = xp_ravel(getattr(res, attr)) + xp_assert_equal(res_attr, ref_attr) + assert getattr(res, attr).shape == shape + + xp_assert_equal(res.fun, self.f(res.x, *args)) + xp_assert_equal(res.fl, self.f(res.xl, *args)) + xp_assert_equal(res.fm, self.f(res.xm, *args)) + xp_assert_equal(res.fr, self.f(res.xr, *args)) + assert xp.max(res.nfev) == f.f_evals + assert xp.max(res.nit) == f.f_evals - 3 + + assert xp_test.isdtype(res.success.dtype, 'bool') + assert xp_test.isdtype(res.status.dtype, 'integral') + assert xp_test.isdtype(res.nfev.dtype, 'integral') + assert xp_test.isdtype(res.nit.dtype, 'integral') + + + def test_flags(self, xp): + # Test cases that should produce different status flags; show that all + # can be produced simultaneously. + def f(xs, js): + funcs = [lambda x: (x - 2.5) ** 2, + lambda x: x - 10, + lambda x: (x - 2.5) ** 4, + lambda x: xp.full_like(x, xp.asarray(xp.nan))] + res = [] + for i in range(xp_size(js)): + x = xs[i, ...] + j = int(xp_ravel(js)[i]) + res.append(funcs[j](x)) + return xp.stack(res) + + args = (xp.arange(4, dtype=xp.int64),) + bracket = (xp.asarray([0]*4, dtype=xp.float64), + xp.asarray([2]*4, dtype=xp.float64), + xp.asarray([np.pi]*4, dtype=xp.float64)) + res = _chandrupatla_minimize(f, *bracket, args=args, maxiter=10) + + ref_flags = xp.asarray([eim._ECONVERGED, eim._ESIGNERR, eim._ECONVERR, + eim._EVALUEERR], dtype=xp.int32) + xp_assert_equal(res.status, ref_flags) + + def test_convergence(self, xp): + # Test that the convergence tolerances behave as expected + rng = np.random.default_rng(2585255913088665241) + p = xp.asarray(rng.random(size=3)) + bracket = (xp.asarray(-5), xp.asarray(0), xp.asarray(5)) + args = (p,) + kwargs0 = dict(args=args, xatol=0, xrtol=0, fatol=0, frtol=0) + + kwargs = kwargs0.copy() + kwargs['xatol'] = 1e-3 + res1 = _chandrupatla_minimize(self.f, *bracket, **kwargs) + j1 = xp.abs(res1.xr - res1.xl) + tol = xp.asarray(4*kwargs['xatol'], dtype=p.dtype) + xp_assert_less(j1, xp.full((3,), tol, dtype=p.dtype)) + kwargs['xatol'] = 1e-6 + res2 = _chandrupatla_minimize(self.f, *bracket, **kwargs) + j2 = xp.abs(res2.xr - res2.xl) + tol = xp.asarray(4*kwargs['xatol'], dtype=p.dtype) + xp_assert_less(j2, xp.full((3,), tol, dtype=p.dtype)) + xp_assert_less(j2, j1) + + kwargs = kwargs0.copy() + kwargs['xrtol'] = 1e-3 + res1 = _chandrupatla_minimize(self.f, *bracket, **kwargs) + j1 = xp.abs(res1.xr - res1.xl) + tol = xp.asarray(4*kwargs['xrtol']*xp.abs(res1.x), dtype=p.dtype) + xp_assert_less(j1, tol) + kwargs['xrtol'] = 1e-6 + res2 = _chandrupatla_minimize(self.f, *bracket, **kwargs) + j2 = xp.abs(res2.xr - res2.xl) + tol = xp.asarray(4*kwargs['xrtol']*xp.abs(res2.x), dtype=p.dtype) + xp_assert_less(j2, tol) + xp_assert_less(j2, j1) + + kwargs = kwargs0.copy() + kwargs['fatol'] = 1e-3 + res1 = _chandrupatla_minimize(self.f, *bracket, **kwargs) + h1 = xp.abs(res1.fl - 2 * res1.fm + res1.fr) + tol = xp.asarray(2*kwargs['fatol'], dtype=p.dtype) + xp_assert_less(h1, xp.full((3,), tol, dtype=p.dtype)) + kwargs['fatol'] = 1e-6 + res2 = _chandrupatla_minimize(self.f, *bracket, **kwargs) + h2 = xp.abs(res2.fl - 2 * res2.fm + res2.fr) + tol = xp.asarray(2*kwargs['fatol'], dtype=p.dtype) + xp_assert_less(h2, xp.full((3,), tol, dtype=p.dtype)) + xp_assert_less(h2, h1) + + kwargs = kwargs0.copy() + kwargs['frtol'] = 1e-3 + res1 = _chandrupatla_minimize(self.f, *bracket, **kwargs) + h1 = xp.abs(res1.fl - 2 * res1.fm + res1.fr) + tol = xp.asarray(2*kwargs['frtol']*xp.abs(res1.fun), dtype=p.dtype) + xp_assert_less(h1, tol) + kwargs['frtol'] = 1e-6 + res2 = _chandrupatla_minimize(self.f, *bracket, **kwargs) + h2 = xp.abs(res2.fl - 2 * res2.fm + res2.fr) + tol = xp.asarray(2*kwargs['frtol']*abs(res2.fun), dtype=p.dtype) + xp_assert_less(h2, tol) + xp_assert_less(h2, h1) + + def test_maxiter_callback(self, xp): + # Test behavior of `maxiter` parameter and `callback` interface + loc = xp.asarray(0.612814) + bracket = (xp.asarray(-5), xp.asarray(0), xp.asarray(5)) + maxiter = 5 + + res = _chandrupatla_minimize(self.f, *bracket, args=(loc,), + maxiter=maxiter) + assert not xp.any(res.success) + assert xp.all(res.nfev == maxiter+3) + assert xp.all(res.nit == maxiter) + + def callback(res): + callback.iter += 1 + callback.res = res + assert hasattr(res, 'x') + if callback.iter == 0: + # callback is called once with initial bracket + assert (res.xl, res.xm, res.xr) == bracket + else: + changed_xr = (res.xl == callback.xl) & (res.xr != callback.xr) + changed_xl = (res.xl != callback.xl) & (res.xr == callback.xr) + assert xp.all(changed_xr | changed_xl) + + callback.xl = res.xl + callback.xr = res.xr + assert res.status == eim._EINPROGRESS + xp_assert_equal(self.f(res.xl, loc), res.fl) + xp_assert_equal(self.f(res.xm, loc), res.fm) + xp_assert_equal(self.f(res.xr, loc), res.fr) + xp_assert_equal(self.f(res.x, loc), res.fun) + if callback.iter == maxiter: + raise StopIteration + + callback.xl = xp.nan + callback.xr = xp.nan + callback.iter = -1 # callback called once before first iteration + callback.res = None + + res2 = _chandrupatla_minimize(self.f, *bracket, args=(loc,), + callback=callback) + + # terminating with callback is identical to terminating due to maxiter + # (except for `status`) + for key in res.keys(): + if key == 'status': + assert res[key] == eim._ECONVERR + # assert callback.res[key] == eim._EINPROGRESS + assert res2[key] == eim._ECALLBACK + else: + assert res2[key] == callback.res[key] == res[key] + + @pytest.mark.parametrize('case', cases) + def test_nit_expected(self, case, xp): + # Test that `_chandrupatla` implements Chandrupatla's algorithm: + # in all 55 test cases, the number of iterations performed + # matches the number reported in the original paper. + func, x1, nit = case + + # Find bracket using the algorithm in the paper + step = 0.2 + x2 = x1 + step + x1, x2, x3, f1, f2, f3 = _bracket_minimum(func, x1, x2) + + # Use tolerances from original paper + xatol = 0.0001 + fatol = 0.000001 + xrtol = 1e-16 + frtol = 1e-16 + + bracket = xp.asarray(x1), xp.asarray(x2), xp.asarray(x3, dtype=xp.float64) + res = _chandrupatla_minimize(func, *bracket, xatol=xatol, + fatol=fatol, xrtol=xrtol, frtol=frtol) + xp_assert_equal(res.nit, xp.asarray(nit, dtype=xp.int32)) + + @pytest.mark.parametrize("loc", (0.65, [0.65, 0.7])) + @pytest.mark.parametrize("dtype", ('float16', 'float32', 'float64')) + def test_dtype(self, loc, dtype, xp): + # Test that dtypes are preserved + dtype = getattr(xp, dtype) + + loc = xp.asarray(loc, dtype=dtype) + bracket = (xp.asarray(-3, dtype=dtype), + xp.asarray(1, dtype=dtype), + xp.asarray(5, dtype=dtype)) + + xp_test = array_namespace(loc) # need astype + def f(x, loc): + assert x.dtype == dtype + return xp_test.astype((x - loc)**2, dtype) + + res = _chandrupatla_minimize(f, *bracket, args=(loc,)) + assert res.x.dtype == dtype + xp_assert_close(res.x, loc, rtol=math.sqrt(xp.finfo(dtype).eps)) + + def test_input_validation(self, xp): + # Test input validation for appropriate error messages + + message = '`func` must be callable.' + bracket = xp.asarray(-4), xp.asarray(0), xp.asarray(4) + with pytest.raises(ValueError, match=message): + _chandrupatla_minimize(None, *bracket) + + message = 'Abscissae and function output must be real numbers.' + bracket = xp.asarray(-4 + 1j), xp.asarray(0), xp.asarray(4) + with pytest.raises(ValueError, match=message): + _chandrupatla_minimize(lambda x: x, *bracket) + + message = "...be broadcast..." + bracket = xp.asarray([-2, -3]), xp.asarray([0, 0]), xp.asarray([3, 4, 5]) + # raised by `np.broadcast, but the traceback is readable IMO + with pytest.raises((ValueError, RuntimeError), match=message): + _chandrupatla_minimize(lambda x: x, *bracket) + + message = "The shape of the array returned by `func` must be the same" + bracket = xp.asarray([-3, -3]), xp.asarray([0, 0]), xp.asarray([5, 5]) + with pytest.raises(ValueError, match=message): + _chandrupatla_minimize(lambda x: [x[0, ...], x[1, ...], x[1, ...]], + *bracket) + + message = 'Tolerances must be non-negative scalars.' + bracket = xp.asarray(-4), xp.asarray(0), xp.asarray(4) + with pytest.raises(ValueError, match=message): + _chandrupatla_minimize(lambda x: x, *bracket, xatol=-1) + with pytest.raises(ValueError, match=message): + _chandrupatla_minimize(lambda x: x, *bracket, xrtol=xp.nan) + with pytest.raises(ValueError, match=message): + _chandrupatla_minimize(lambda x: x, *bracket, fatol='ekki') + with pytest.raises(ValueError, match=message): + _chandrupatla_minimize(lambda x: x, *bracket, frtol=xp.nan) + + message = '`maxiter` must be a non-negative integer.' + with pytest.raises(ValueError, match=message): + _chandrupatla_minimize(lambda x: x, *bracket, maxiter=1.5) + with pytest.raises(ValueError, match=message): + _chandrupatla_minimize(lambda x: x, *bracket, maxiter=-1) + + message = '`callback` must be callable.' + with pytest.raises(ValueError, match=message): + _chandrupatla_minimize(lambda x: x, *bracket, callback='shrubbery') + + def test_bracket_order(self, xp): + # Confirm that order of points in bracket doesn't + xp_test = array_namespace(xp.asarray(1.)) # need `xp.newaxis` + loc = xp.linspace(-1, 1, 6)[:, xp_test.newaxis] + brackets = xp.asarray(list(permutations([-5, 0, 5]))).T + res = _chandrupatla_minimize(self.f, *brackets, args=(loc,)) + assert xp.all(xp.isclose(res.x, loc) | (res.fun == self.f(loc, loc))) + ref = res.x[:, 0] # all columns should be the same + xp_test = array_namespace(loc) # need `xp.broadcast_arrays + xp_assert_close(*xp_test.broadcast_arrays(res.x.T, ref), rtol=1e-15) + + def test_special_cases(self, xp): + # Test edge cases and other special cases + + # Test that integers are not passed to `f` + xp_test = array_namespace(xp.asarray(1.)) # need `xp.isdtype` + def f(x): + assert xp_test.isdtype(x.dtype, "real floating") + return (x - 1)**2 + + bracket = xp.asarray(-7), xp.asarray(0), xp.asarray(8) + with np.errstate(invalid='ignore'): + res = _chandrupatla_minimize(f, *bracket, fatol=0, frtol=0) + assert res.success + xp_assert_close(res.x, xp.asarray(1.), rtol=1e-3) + xp_assert_close(res.fun, xp.asarray(0.), atol=1e-200) + + # Test that if all elements of bracket equal minimizer, algorithm + # reports convergence + def f(x): + return (x-1)**2 + + bracket = xp.asarray(1), xp.asarray(1), xp.asarray(1) + res = _chandrupatla_minimize(f, *bracket) + assert res.success + xp_assert_equal(res.x, xp.asarray(1.)) + + # Test maxiter = 0. Should do nothing to bracket. + def f(x): + return (x-1)**2 + + bracket = xp.asarray(-3), xp.asarray(1.1), xp.asarray(5) + res = _chandrupatla_minimize(f, *bracket, maxiter=0) + assert res.xl, res.xr == bracket + assert res.nit == 0 + assert res.nfev == 3 + assert res.status == -2 + assert res.x == 1.1 # best so far + + # Test scalar `args` (not in tuple) + def f(x, c): + return (x-c)**2 - 1 + + bracket = xp.asarray(-1), xp.asarray(0), xp.asarray(1) + c = xp.asarray(1/3) + res = _chandrupatla_minimize(f, *bracket, args=(c,)) + xp_assert_close(res.x, c) + + # Test zero tolerances + def f(x): + return -xp.sin(x) + + bracket = xp.asarray(0), xp.asarray(1), xp.asarray(xp.pi) + res = _chandrupatla_minimize(f, *bracket, xatol=0, xrtol=0, fatol=0, frtol=0) + assert res.success + # found a minimum exactly (according to floating point arithmetic) + assert res.xl < res.xm < res.xr + assert f(res.xl) == f(res.xm) == f(res.xr) + + +@array_api_compatible +@pytest.mark.usefixtures("skip_xp_backends") +@pytest.mark.skip_xp_backends('array_api_strict', + reason='Currently uses fancy indexing assignment.') +@pytest.mark.skip_xp_backends('jax.numpy', + reason='JAX arrays do not support item assignment.') +@pytest.mark.skip_xp_backends('cupy', + reason='cupy/cupy#8391') +class TestChandrupatla(TestScalarRootFinders): + + def f(self, q, p): + return special.ndtr(q) - p + + @pytest.mark.parametrize('p', [0.6, np.linspace(-0.05, 1.05, 10)]) + def test_basic(self, p, xp): + # Invert distribution CDF and compare against distribution `ppf` + a, b = xp.asarray(-5.), xp.asarray(5.) + res = _chandrupatla_root(self.f, a, b, args=(xp.asarray(p),)) + ref = xp.asarray(stats.norm().ppf(p), dtype=xp.asarray(p).dtype) + xp_assert_close(res.x, ref) + + @pytest.mark.parametrize('shape', [tuple(), (12,), (3, 4), (3, 2, 2)]) + def test_vectorization(self, shape, xp): + # Test for correct functionality, output shapes, and dtypes for various + # input shapes. + p = (np.linspace(-0.05, 1.05, 12).reshape(shape) if shape + else np.float64(0.6)) + p_xp = xp.asarray(p) + args_xp = (p_xp,) + dtype = p_xp.dtype + xp_test = array_namespace(p_xp) # need xp.bool + + @np.vectorize + def chandrupatla_single(p): + return _chandrupatla_root(self.f, -5, 5, args=(p,)) + + def f(*args, **kwargs): + f.f_evals += 1 + return self.f(*args, **kwargs) + f.f_evals = 0 + + res = _chandrupatla_root(f, xp.asarray(-5.), xp.asarray(5.), args=args_xp) + refs = chandrupatla_single(p).ravel() + + ref_x = [ref.x for ref in refs] + ref_x = xp.reshape(xp.asarray(ref_x, dtype=dtype), shape) + xp_assert_close(res.x, ref_x) + + ref_fun = [ref.fun for ref in refs] + ref_fun = xp.reshape(xp.asarray(ref_fun, dtype=dtype), shape) + xp_assert_close(res.fun, ref_fun, atol=1e-15) + xp_assert_equal(res.fun, self.f(res.x, *args_xp)) + + ref_success = [bool(ref.success) for ref in refs] + ref_success = xp.reshape(xp.asarray(ref_success, dtype=xp_test.bool), shape) + xp_assert_equal(res.success, ref_success) + + ref_flag = [ref.status for ref in refs] + ref_flag = xp.reshape(xp.asarray(ref_flag, dtype=xp.int32), shape) + xp_assert_equal(res.status, ref_flag) + + ref_nfev = [ref.nfev for ref in refs] + ref_nfev = xp.reshape(xp.asarray(ref_nfev, dtype=xp.int32), shape) + if is_numpy(xp): + xp_assert_equal(res.nfev, ref_nfev) + assert xp.max(res.nfev) == f.f_evals + else: # different backend may lead to different nfev + assert res.nfev.shape == shape + assert res.nfev.dtype == xp.int32 + + ref_nit = [ref.nit for ref in refs] + ref_nit = xp.reshape(xp.asarray(ref_nit, dtype=xp.int32), shape) + if is_numpy(xp): + xp_assert_equal(res.nit, ref_nit) + assert xp.max(res.nit) == f.f_evals-2 + else: + assert res.nit.shape == shape + assert res.nit.dtype == xp.int32 + + ref_xl = [ref.xl for ref in refs] + ref_xl = xp.reshape(xp.asarray(ref_xl, dtype=dtype), shape) + xp_assert_close(res.xl, ref_xl) + + ref_xr = [ref.xr for ref in refs] + ref_xr = xp.reshape(xp.asarray(ref_xr, dtype=dtype), shape) + xp_assert_close(res.xr, ref_xr) + + xp_assert_less(res.xl, res.xr) + finite = xp.isfinite(res.x) + assert xp.all((res.x[finite] == res.xl[finite]) + | (res.x[finite] == res.xr[finite])) + + # PyTorch and CuPy don't solve to the same accuracy as NumPy - that's OK. + atol = 1e-15 if is_numpy(xp) else 1e-9 + + ref_fl = [ref.fl for ref in refs] + ref_fl = xp.reshape(xp.asarray(ref_fl, dtype=dtype), shape) + xp_assert_close(res.fl, ref_fl, atol=atol) + xp_assert_equal(res.fl, self.f(res.xl, *args_xp)) + + ref_fr = [ref.fr for ref in refs] + ref_fr = xp.reshape(xp.asarray(ref_fr, dtype=dtype), shape) + xp_assert_close(res.fr, ref_fr, atol=atol) + xp_assert_equal(res.fr, self.f(res.xr, *args_xp)) + + assert xp.all(xp.abs(res.fun[finite]) == + xp.minimum(xp.abs(res.fl[finite]), + xp.abs(res.fr[finite]))) + + def test_flags(self, xp): + # Test cases that should produce different status flags; show that all + # can be produced simultaneously. + def f(xs, js): + # Note that full_like and int(j) shouldn't really be required. CuPy + # is just really picky here, so I'm making it a special case to + # make sure the other backends work when the user is less careful. + assert js.dtype == xp.int64 + if is_cupy(xp): + funcs = [lambda x: x - 2.5, + lambda x: x - 10, + lambda x: (x - 0.1)**3, + lambda x: xp.full_like(x, xp.asarray(xp.nan))] + return [funcs[int(j)](x) for x, j in zip(xs, js)] + + funcs = [lambda x: x - 2.5, + lambda x: x - 10, + lambda x: (x - 0.1) ** 3, + lambda x: xp.nan] + return [funcs[j](x) for x, j in zip(xs, js)] + + args = (xp.arange(4, dtype=xp.int64),) + a, b = xp.asarray([0.]*4), xp.asarray([xp.pi]*4) + res = _chandrupatla_root(f, a, b, args=args, maxiter=2) + + ref_flags = xp.asarray([eim._ECONVERGED, + eim._ESIGNERR, + eim._ECONVERR, + eim._EVALUEERR], dtype=xp.int32) + xp_assert_equal(res.status, ref_flags) + + def test_convergence(self, xp): + # Test that the convergence tolerances behave as expected + rng = np.random.default_rng(2585255913088665241) + p = xp.asarray(rng.random(size=3)) + bracket = (-xp.asarray(5.), xp.asarray(5.)) + args = (p,) + kwargs0 = dict(args=args, xatol=0, xrtol=0, fatol=0, frtol=0) + + kwargs = kwargs0.copy() + kwargs['xatol'] = 1e-3 + res1 = _chandrupatla_root(self.f, *bracket, **kwargs) + xp_assert_less(res1.xr - res1.xl, xp.full_like(p, xp.asarray(1e-3))) + kwargs['xatol'] = 1e-6 + res2 = _chandrupatla_root(self.f, *bracket, **kwargs) + xp_assert_less(res2.xr - res2.xl, xp.full_like(p, xp.asarray(1e-6))) + xp_assert_less(res2.xr - res2.xl, res1.xr - res1.xl) + + kwargs = kwargs0.copy() + kwargs['xrtol'] = 1e-3 + res1 = _chandrupatla_root(self.f, *bracket, **kwargs) + xp_assert_less(res1.xr - res1.xl, 1e-3 * xp.abs(res1.x)) + kwargs['xrtol'] = 1e-6 + res2 = _chandrupatla_root(self.f, *bracket, **kwargs) + xp_assert_less(res2.xr - res2.xl, 1e-6 * xp.abs(res2.x)) + xp_assert_less(res2.xr - res2.xl, res1.xr - res1.xl) + + kwargs = kwargs0.copy() + kwargs['fatol'] = 1e-3 + res1 = _chandrupatla_root(self.f, *bracket, **kwargs) + xp_assert_less(xp.abs(res1.fun), xp.full_like(p, xp.asarray(1e-3))) + kwargs['fatol'] = 1e-6 + res2 = _chandrupatla_root(self.f, *bracket, **kwargs) + xp_assert_less(xp.abs(res2.fun), xp.full_like(p, xp.asarray(1e-6))) + xp_assert_less(xp.abs(res2.fun), xp.abs(res1.fun)) + + kwargs = kwargs0.copy() + kwargs['frtol'] = 1e-3 + x1, x2 = bracket + f0 = xp.minimum(xp.abs(self.f(x1, *args)), xp.abs(self.f(x2, *args))) + res1 = _chandrupatla_root(self.f, *bracket, **kwargs) + xp_assert_less(xp.abs(res1.fun), 1e-3*f0) + kwargs['frtol'] = 1e-6 + res2 = _chandrupatla_root(self.f, *bracket, **kwargs) + xp_assert_less(xp.abs(res2.fun), 1e-6*f0) + xp_assert_less(xp.abs(res2.fun), xp.abs(res1.fun)) + + def test_maxiter_callback(self, xp): + # Test behavior of `maxiter` parameter and `callback` interface + p = xp.asarray(0.612814) + bracket = (xp.asarray(-5.), xp.asarray(5.)) + maxiter = 5 + + def f(q, p): + res = special.ndtr(q) - p + f.x = q + f.fun = res + return res + f.x = None + f.fun = None + + res = _chandrupatla_root(f, *bracket, args=(p,), maxiter=maxiter) + assert not xp.any(res.success) + assert xp.all(res.nfev == maxiter+2) + assert xp.all(res.nit == maxiter) + + def callback(res): + callback.iter += 1 + callback.res = res + assert hasattr(res, 'x') + if callback.iter == 0: + # callback is called once with initial bracket + assert (res.xl, res.xr) == bracket + else: + changed = (((res.xl == callback.xl) & (res.xr != callback.xr)) + | ((res.xl != callback.xl) & (res.xr == callback.xr))) + assert xp.all(changed) + + callback.xl = res.xl + callback.xr = res.xr + assert res.status == eim._EINPROGRESS + xp_assert_equal(self.f(res.xl, p), res.fl) + xp_assert_equal(self.f(res.xr, p), res.fr) + xp_assert_equal(self.f(res.x, p), res.fun) + if callback.iter == maxiter: + raise StopIteration + callback.iter = -1 # callback called once before first iteration + callback.res = None + callback.xl = None + callback.xr = None + + res2 = _chandrupatla_root(f, *bracket, args=(p,), callback=callback) + + # terminating with callback is identical to terminating due to maxiter + # (except for `status`) + for key in res.keys(): + if key == 'status': + xp_assert_equal(res[key], xp.asarray(eim._ECONVERR, dtype=xp.int32)) + xp_assert_equal(res2[key], xp.asarray(eim._ECALLBACK, dtype=xp.int32)) + elif key.startswith('_'): + continue + else: + xp_assert_equal(res2[key], res[key]) + + @pytest.mark.parametrize('case', _CHANDRUPATLA_TESTS) + def test_nit_expected(self, case, xp): + # Test that `_chandrupatla` implements Chandrupatla's algorithm: + # in all 40 test cases, the number of iterations performed + # matches the number reported in the original paper. + f, bracket, root, nfeval, id = case + # Chandrupatla's criterion is equivalent to + # abs(x2-x1) < 4*abs(xmin)*xrtol + xatol, but we use the more standard + # abs(x2-x1) < abs(xmin)*xrtol + xatol. Therefore, set xrtol to 4x + # that used by Chandrupatla in tests. + bracket = (xp.asarray(bracket[0], dtype=xp.float64), + xp.asarray(bracket[1], dtype=xp.float64)) + root = xp.asarray(root, dtype=xp.float64) + + res = _chandrupatla_root(f, *bracket, xrtol=4e-10, xatol=1e-5) + xp_assert_close(res.fun, xp.asarray(f(root), dtype=xp.float64), + rtol=1e-8, atol=2e-3) + xp_assert_equal(res.nfev, xp.asarray(nfeval, dtype=xp.int32)) + + @pytest.mark.parametrize("root", (0.622, [0.622, 0.623])) + @pytest.mark.parametrize("dtype", ('float16', 'float32', 'float64')) + def test_dtype(self, root, dtype, xp): + # Test that dtypes are preserved + not_numpy = not is_numpy(xp) + if not_numpy and dtype == 'float16': + pytest.skip("`float16` dtype only supported for NumPy arrays.") + + dtype = getattr(xp, dtype, None) + if dtype is None: + pytest.skip(f"{xp} does not support {dtype}") + + def f(x, root): + res = (x - root) ** 3. + if is_numpy(xp): # NumPy does not preserve dtype + return xp.asarray(res, dtype=dtype) + return res + + a, b = xp.asarray(-3, dtype=dtype), xp.asarray(3, dtype=dtype) + root = xp.asarray(root, dtype=dtype) + res = _chandrupatla_root(f, a, b, args=(root,), xatol=1e-3) + try: + xp_assert_close(res.x, root, atol=1e-3) + except AssertionError: + assert res.x.dtype == dtype + xp.all(res.fun == 0) + + def test_input_validation(self, xp): + # Test input validation for appropriate error messages + + def func(x): + return x + + message = '`func` must be callable.' + with pytest.raises(ValueError, match=message): + bracket = xp.asarray(-4), xp.asarray(4) + _chandrupatla_root(None, *bracket) + + message = 'Abscissae and function output must be real numbers.' + with pytest.raises(ValueError, match=message): + bracket = xp.asarray(-4+1j), xp.asarray(4) + _chandrupatla_root(func, *bracket) + + # raised by `np.broadcast, but the traceback is readable IMO + message = "...not be broadcast..." # all messages include this part + with pytest.raises((ValueError, RuntimeError), match=message): + bracket = xp.asarray([-2, -3]), xp.asarray([3, 4, 5]) + _chandrupatla_root(func, *bracket) + + message = "The shape of the array returned by `func`..." + with pytest.raises(ValueError, match=message): + bracket = xp.asarray([-3, -3]), xp.asarray([5, 5]) + _chandrupatla_root(lambda x: [x[0], x[1], x[1]], *bracket) + + message = 'Tolerances must be non-negative scalars.' + bracket = xp.asarray(-4), xp.asarray(4) + with pytest.raises(ValueError, match=message): + _chandrupatla_root(func, *bracket, xatol=-1) + with pytest.raises(ValueError, match=message): + _chandrupatla_root(func, *bracket, xrtol=xp.nan) + with pytest.raises(ValueError, match=message): + _chandrupatla_root(func, *bracket, fatol='ekki') + with pytest.raises(ValueError, match=message): + _chandrupatla_root(func, *bracket, frtol=xp.nan) + + message = '`maxiter` must be a non-negative integer.' + with pytest.raises(ValueError, match=message): + _chandrupatla_root(func, *bracket, maxiter=1.5) + with pytest.raises(ValueError, match=message): + _chandrupatla_root(func, *bracket, maxiter=-1) + + message = '`callback` must be callable.' + with pytest.raises(ValueError, match=message): + _chandrupatla_root(func, *bracket, callback='shrubbery') + + def test_special_cases(self, xp): + # Test edge cases and other special cases + + # Test infinite function values + def f(x): + return 1 / x + 1 - 1 / (-x + 1) + + a, b = xp.asarray([0.1, 0., 0., 0.1]), xp.asarray([0.9, 1.0, 0.9, 1.0]) + + with np.errstate(divide='ignore', invalid='ignore'): + res = _chandrupatla_root(f, a, b) + + assert xp.all(res.success) + xp_assert_close(res.x[1:], xp.full((3,), res.x[0])) + + # Test that integers are not passed to `f` + # (otherwise this would overflow) + xp_test = array_namespace(a) # need isdtype + def f(x): + assert xp_test.isdtype(x.dtype, "real floating") + # this would overflow if x were an xp integer dtype + return x ** 31 - 1 + + # note that all inputs are integer type; result is automatically default float + res = _chandrupatla_root(f, xp.asarray(-7), xp.asarray(5)) + assert res.success + xp_assert_close(res.x, xp.asarray(1.)) + + # Test that if both ends of bracket equal root, algorithm reports + # convergence. + def f(x, root): + return x**2 - root + + root = xp.asarray([0, 1]) + res = _chandrupatla_root(f, xp.asarray(1), xp.asarray(1), args=(root,)) + xp_assert_equal(res.success, xp.asarray([False, True])) + xp_assert_equal(res.x, xp.asarray([xp.nan, 1.])) + + def f(x): + return 1/x + + with np.errstate(invalid='ignore'): + inf = xp.asarray(xp.inf) + res = _chandrupatla_root(f, inf, inf) + assert res.success + xp_assert_equal(res.x, xp.asarray(xp.inf)) + + # Test maxiter = 0. Should do nothing to bracket. + def f(x): + return x**3 - 1 + + a, b = xp.asarray(-3.), xp.asarray(5.) + res = _chandrupatla_root(f, a, b, maxiter=0) + xp_assert_equal(res.success, xp.asarray(False)) + xp_assert_equal(res.status, xp.asarray(-2, dtype=xp.int32)) + xp_assert_equal(res.nit, xp.asarray(0, dtype=xp.int32)) + xp_assert_equal(res.nfev, xp.asarray(2, dtype=xp.int32)) + xp_assert_equal(res.xl, a) + xp_assert_equal(res.xr, b) + # The `x` attribute is the one with the smaller function value + xp_assert_equal(res.x, a) + # Reverse bracket; check that this is still true + res = _chandrupatla_root(f, -b, -a, maxiter=0) + xp_assert_equal(res.x, -a) + + # Test maxiter = 1 + res = _chandrupatla_root(f, a, b, maxiter=1) + xp_assert_equal(res.success, xp.asarray(True)) + xp_assert_equal(res.status, xp.asarray(0, dtype=xp.int32)) + xp_assert_equal(res.nit, xp.asarray(1, dtype=xp.int32)) + xp_assert_equal(res.nfev, xp.asarray(3, dtype=xp.int32)) + xp_assert_close(res.x, xp.asarray(1.)) + + # Test scalar `args` (not in tuple) + def f(x, c): + return c*x - 1 + + res = _chandrupatla_root(f, xp.asarray(-1), xp.asarray(1), args=xp.asarray(3)) + xp_assert_close(res.x, xp.asarray(1/3)) + + # # TODO: Test zero tolerance + # # ~~What's going on here - why are iterations repeated?~~ + # # tl goes to zero when xatol=xrtol=0. When function is nearly linear, + # # this causes convergence issues. + # def f(x): + # return np.cos(x) + # + # res = _chandrupatla_root(f, 0, np.pi, xatol=0, xrtol=0) + # assert res.nit < 100 + # xp = np.nextafter(res.x, np.inf) + # xm = np.nextafter(res.x, -np.inf) + # assert np.abs(res.fun) < np.abs(f(xp)) + # assert np.abs(res.fun) < np.abs(f(xm)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_cobyla.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_cobyla.py new file mode 100644 index 0000000000000000000000000000000000000000..bd27eb9baa27433d1f801e8582dd2e8d29db3da2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_cobyla.py @@ -0,0 +1,166 @@ +import math + +import numpy as np +from numpy.testing import assert_allclose, assert_, assert_array_equal +import pytest + +from scipy.optimize import fmin_cobyla, minimize, Bounds + + +class TestCobyla: + def setup_method(self): + self.x0 = [4.95, 0.66] + self.solution = [math.sqrt(25 - (2.0/3)**2), 2.0/3] + self.opts = {'disp': False, 'rhobeg': 1, 'tol': 1e-5, + 'maxiter': 100} + + def fun(self, x): + return x[0]**2 + abs(x[1])**3 + + def con1(self, x): + return x[0]**2 + x[1]**2 - 25 + + def con2(self, x): + return -self.con1(x) + + @pytest.mark.xslow(True, reason='not slow, but noisy so only run rarely') + def test_simple(self, capfd): + # use disp=True as smoke test for gh-8118 + x = fmin_cobyla(self.fun, self.x0, [self.con1, self.con2], rhobeg=1, + rhoend=1e-5, maxfun=100, disp=True) + assert_allclose(x, self.solution, atol=1e-4) + + def test_minimize_simple(self): + class Callback: + def __init__(self): + self.n_calls = 0 + self.last_x = None + + def __call__(self, x): + self.n_calls += 1 + self.last_x = x + + callback = Callback() + + # Minimize with method='COBYLA' + cons = ({'type': 'ineq', 'fun': self.con1}, + {'type': 'ineq', 'fun': self.con2}) + sol = minimize(self.fun, self.x0, method='cobyla', constraints=cons, + callback=callback, options=self.opts) + assert_allclose(sol.x, self.solution, atol=1e-4) + assert_(sol.success, sol.message) + assert_(sol.maxcv < 1e-5, sol) + assert_(sol.nfev < 70, sol) + assert_(sol.fun < self.fun(self.solution) + 1e-3, sol) + assert_(sol.nfev == callback.n_calls, + "Callback is not called exactly once for every function eval.") + assert_array_equal( + sol.x, + callback.last_x, + "Last design vector sent to the callback is not equal to returned value.", + ) + + def test_minimize_constraint_violation(self): + rng = np.random.RandomState(1234) + pb = rng.rand(10, 10) + spread = rng.rand(10) + + def p(w): + return pb.dot(w) + + def f(w): + return -(w * spread).sum() + + def c1(w): + return 500 - abs(p(w)).sum() + + def c2(w): + return 5 - abs(p(w).sum()) + + def c3(w): + return 5 - abs(p(w)).max() + + cons = ({'type': 'ineq', 'fun': c1}, + {'type': 'ineq', 'fun': c2}, + {'type': 'ineq', 'fun': c3}) + w0 = np.zeros((10,)) + sol = minimize(f, w0, method='cobyla', constraints=cons, + options={'catol': 1e-6}) + assert_(sol.maxcv > 1e-6) + assert_(not sol.success) + + +def test_vector_constraints(): + # test that fmin_cobyla and minimize can take a combination + # of constraints, some returning a number and others an array + def fun(x): + return (x[0] - 1)**2 + (x[1] - 2.5)**2 + + def fmin(x): + return fun(x) - 1 + + def cons1(x): + a = np.array([[1, -2, 2], [-1, -2, 6], [-1, 2, 2]]) + return np.array([a[i, 0] * x[0] + a[i, 1] * x[1] + + a[i, 2] for i in range(len(a))]) + + def cons2(x): + return x # identity, acts as bounds x > 0 + + x0 = np.array([2, 0]) + cons_list = [fun, cons1, cons2] + + xsol = [1.4, 1.7] + fsol = 0.8 + + # testing fmin_cobyla + sol = fmin_cobyla(fun, x0, cons_list, rhoend=1e-5) + assert_allclose(sol, xsol, atol=1e-4) + + sol = fmin_cobyla(fun, x0, fmin, rhoend=1e-5) + assert_allclose(fun(sol), 1, atol=1e-4) + + # testing minimize + constraints = [{'type': 'ineq', 'fun': cons} for cons in cons_list] + sol = minimize(fun, x0, constraints=constraints, tol=1e-5) + assert_allclose(sol.x, xsol, atol=1e-4) + assert_(sol.success, sol.message) + assert_allclose(sol.fun, fsol, atol=1e-4) + + constraints = {'type': 'ineq', 'fun': fmin} + sol = minimize(fun, x0, constraints=constraints, tol=1e-5) + assert_allclose(sol.fun, 1, atol=1e-4) + + +class TestBounds: + # Test cobyla support for bounds (only when used via `minimize`) + # Invalid bounds is tested in + # test_optimize.TestOptimizeSimple.test_minimize_invalid_bounds + + def test_basic(self): + def f(x): + return np.sum(x**2) + + lb = [-1, None, 1, None, -0.5] + ub = [-0.5, -0.5, None, None, -0.5] + bounds = [(a, b) for a, b in zip(lb, ub)] + # these are converted to Bounds internally + + res = minimize(f, x0=[1, 2, 3, 4, 5], method='cobyla', bounds=bounds) + ref = [-0.5, -0.5, 1, 0, -0.5] + assert res.success + assert_allclose(res.x, ref, atol=1e-3) + + def test_unbounded(self): + def f(x): + return np.sum(x**2) + + bounds = Bounds([-np.inf, -np.inf], [np.inf, np.inf]) + res = minimize(f, x0=[1, 2], method='cobyla', bounds=bounds) + assert res.success + assert_allclose(res.x, 0, atol=1e-3) + + bounds = Bounds([1, -np.inf], [np.inf, np.inf]) + res = minimize(f, x0=[1, 2], method='cobyla', bounds=bounds) + assert res.success + assert_allclose(res.x, [1, 0], atol=1e-3) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_cobyqa.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_cobyqa.py new file mode 100644 index 0000000000000000000000000000000000000000..bf16af71625c4d476353407f9708c981915b098a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_cobyqa.py @@ -0,0 +1,252 @@ +import numpy as np +import pytest +import threading +from numpy.testing import assert_allclose, assert_equal + +from scipy.optimize import ( + Bounds, + LinearConstraint, + NonlinearConstraint, + OptimizeResult, + minimize, +) + + +class TestCOBYQA: + + def setup_method(self): + self.x0 = [4.95, 0.66] + self.options = {'maxfev': 100} + + @staticmethod + def fun(x, c=1.0): + return x[0]**2 + c * abs(x[1])**3 + + @staticmethod + def con(x): + return x[0]**2 + x[1]**2 - 25.0 + + def test_minimize_simple(self): + class Callback: + def __init__(self): + self.lock = threading.Lock() + self.n_calls = 0 + + def __call__(self, x): + assert isinstance(x, np.ndarray) + with self.lock: + self.n_calls += 1 + + class CallbackNewSyntax: + def __init__(self): + self.lock = threading.Lock() + self.n_calls = 0 + + def __call__(self, intermediate_result): + assert isinstance(intermediate_result, OptimizeResult) + with self.lock: + self.n_calls += 1 + + x0 = [4.95, 0.66] + callback = Callback() + callback_new_syntax = CallbackNewSyntax() + + # Minimize with method='cobyqa'. + constraints = NonlinearConstraint(self.con, 0.0, 0.0) + sol = minimize( + self.fun, + x0, + method='cobyqa', + constraints=constraints, + callback=callback, + options=self.options, + ) + sol_new = minimize( + self.fun, + x0, + method='cobyqa', + constraints=constraints, + callback=callback_new_syntax, + options=self.options, + ) + solution = [np.sqrt(25.0 - 4.0 / 9.0), 2.0 / 3.0] + assert_allclose(sol.x, solution, atol=1e-4) + assert sol.success, sol.message + assert sol.maxcv < 1e-8, sol + assert sol.nfev <= 100, sol + assert sol.fun < self.fun(solution) + 1e-3, sol + assert sol.nfev == callback.n_calls, \ + "Callback is not called exactly once for every function eval." + assert_equal(sol.x, sol_new.x) + assert sol_new.success, sol_new.message + assert sol.fun == sol_new.fun + assert sol.maxcv == sol_new.maxcv + assert sol.nfev == sol_new.nfev + assert sol.nit == sol_new.nit + assert sol_new.nfev == callback_new_syntax.n_calls, \ + "Callback is not called exactly once for every function eval." + + def test_minimize_bounds(self): + def fun_check_bounds(x): + assert np.all(bounds.lb <= x) and np.all(x <= bounds.ub) + return self.fun(x) + + # Case where the bounds are not active at the solution. + bounds = Bounds([4.5, 0.6], [5.0, 0.7]) + constraints = NonlinearConstraint(self.con, 0.0, 0.0) + sol = minimize( + fun_check_bounds, + self.x0, + method='cobyqa', + bounds=bounds, + constraints=constraints, + options=self.options, + ) + solution = [np.sqrt(25.0 - 4.0 / 9.0), 2.0 / 3.0] + assert_allclose(sol.x, solution, atol=1e-4) + assert sol.success, sol.message + assert sol.maxcv < 1e-8, sol + assert np.all(bounds.lb <= sol.x) and np.all(sol.x <= bounds.ub), sol + assert sol.nfev <= 100, sol + assert sol.fun < self.fun(solution) + 1e-3, sol + + # Case where the bounds are active at the solution. + bounds = Bounds([5.0, 0.6], [5.5, 0.65]) + sol = minimize( + fun_check_bounds, + self.x0, + method='cobyqa', + bounds=bounds, + constraints=constraints, + options=self.options, + ) + assert not sol.success, sol.message + assert sol.maxcv > 0.35, sol + assert np.all(bounds.lb <= sol.x) and np.all(sol.x <= bounds.ub), sol + assert sol.nfev <= 100, sol + + def test_minimize_linear_constraints(self): + constraints = LinearConstraint([1.0, 1.0], 1.0, 1.0) + sol = minimize( + self.fun, + self.x0, + method='cobyqa', + constraints=constraints, + options=self.options, + ) + solution = [(4 - np.sqrt(7)) / 3, (np.sqrt(7) - 1) / 3] + assert_allclose(sol.x, solution, atol=1e-4) + assert sol.success, sol.message + assert sol.maxcv < 1e-8, sol + assert sol.nfev <= 100, sol + assert sol.fun < self.fun(solution) + 1e-3, sol + + def test_minimize_args(self): + constraints = NonlinearConstraint(self.con, 0.0, 0.0) + sol = minimize( + self.fun, + self.x0, + args=(2.0,), + method='cobyqa', + constraints=constraints, + options=self.options, + ) + solution = [np.sqrt(25.0 - 4.0 / 36.0), 2.0 / 6.0] + assert_allclose(sol.x, solution, atol=1e-4) + assert sol.success, sol.message + assert sol.maxcv < 1e-8, sol + assert sol.nfev <= 100, sol + assert sol.fun < self.fun(solution, 2.0) + 1e-3, sol + + def test_minimize_array(self): + def fun_array(x, dim): + f = np.array(self.fun(x)) + return np.reshape(f, (1,) * dim) + + # The argument fun can return an array with a single element. + bounds = Bounds([4.5, 0.6], [5.0, 0.7]) + constraints = NonlinearConstraint(self.con, 0.0, 0.0) + sol = minimize( + self.fun, + self.x0, + method='cobyqa', + bounds=bounds, + constraints=constraints, + options=self.options, + ) + for dim in [0, 1, 2]: + sol_array = minimize( + fun_array, + self.x0, + args=(dim,), + method='cobyqa', + bounds=bounds, + constraints=constraints, + options=self.options, + ) + assert_equal(sol.x, sol_array.x) + assert sol_array.success, sol_array.message + assert sol.fun == sol_array.fun + assert sol.maxcv == sol_array.maxcv + assert sol.nfev == sol_array.nfev + assert sol.nit == sol_array.nit + + # The argument fun cannot return an array with more than one element. + with pytest.raises(TypeError): + minimize( + lambda x: np.array([self.fun(x), self.fun(x)]), + self.x0, + method='cobyqa', + bounds=bounds, + constraints=constraints, + options=self.options, + ) + + def test_minimize_maxfev(self): + constraints = NonlinearConstraint(self.con, 0.0, 0.0) + options = {'maxfev': 2} + sol = minimize( + self.fun, + self.x0, + method='cobyqa', + constraints=constraints, + options=options, + ) + assert not sol.success, sol.message + assert sol.nfev <= 2, sol + + def test_minimize_maxiter(self): + constraints = NonlinearConstraint(self.con, 0.0, 0.0) + options = {'maxiter': 2} + sol = minimize( + self.fun, + self.x0, + method='cobyqa', + constraints=constraints, + options=options, + ) + assert not sol.success, sol.message + assert sol.nit <= 2, sol + + def test_minimize_f_target(self): + constraints = NonlinearConstraint(self.con, 0.0, 0.0) + sol_ref = minimize( + self.fun, + self.x0, + method='cobyqa', + constraints=constraints, + options=self.options, + ) + options = dict(self.options) + options['f_target'] = sol_ref.fun + sol = minimize( + self.fun, + self.x0, + method='cobyqa', + constraints=constraints, + options=options, + ) + assert sol.success, sol.message + assert sol.maxcv < 1e-8, sol + assert sol.nfev <= sol_ref.nfev, sol + assert sol.fun <= sol_ref.fun, sol diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_constraint_conversion.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_constraint_conversion.py new file mode 100644 index 0000000000000000000000000000000000000000..c33183d5cb4e7ccaf4d755cbd2d8b28afaf46395 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_constraint_conversion.py @@ -0,0 +1,286 @@ +""" +Unit test for constraint conversion +""" + +import numpy as np +from numpy.testing import (assert_array_almost_equal, + assert_allclose, assert_warns, suppress_warnings) +import pytest +from scipy.optimize import (NonlinearConstraint, LinearConstraint, + OptimizeWarning, minimize, BFGS) +from .test_minimize_constrained import (Maratos, HyperbolicIneq, Rosenbrock, + IneqRosenbrock, EqIneqRosenbrock, + BoundedRosenbrock, Elec) + + +class TestOldToNew: + x0 = (2, 0) + bnds = ((0, None), (0, None)) + method = "trust-constr" + + def test_constraint_dictionary_1(self): + def fun(x): + return (x[0] - 1) ** 2 + (x[1] - 2.5) ** 2 + cons = ({'type': 'ineq', 'fun': lambda x: x[0] - 2 * x[1] + 2}, + {'type': 'ineq', 'fun': lambda x: -x[0] - 2 * x[1] + 6}, + {'type': 'ineq', 'fun': lambda x: -x[0] + 2 * x[1] + 2}) + + with suppress_warnings() as sup: + sup.filter(UserWarning, "delta_grad == 0.0") + res = minimize(fun, self.x0, method=self.method, + bounds=self.bnds, constraints=cons) + assert_allclose(res.x, [1.4, 1.7], rtol=1e-4) + assert_allclose(res.fun, 0.8, rtol=1e-4) + + def test_constraint_dictionary_2(self): + def fun(x): + return (x[0] - 1) ** 2 + (x[1] - 2.5) ** 2 + cons = {'type': 'eq', + 'fun': lambda x, p1, p2: p1*x[0] - p2*x[1], + 'args': (1, 1.1), + 'jac': lambda x, p1, p2: np.array([[p1, -p2]])} + with suppress_warnings() as sup: + sup.filter(UserWarning, "delta_grad == 0.0") + res = minimize(fun, self.x0, method=self.method, + bounds=self.bnds, constraints=cons) + assert_allclose(res.x, [1.7918552, 1.62895927]) + assert_allclose(res.fun, 1.3857466063348418) + + def test_constraint_dictionary_3(self): + def fun(x): + return (x[0] - 1) ** 2 + (x[1] - 2.5) ** 2 + cons = [{'type': 'ineq', 'fun': lambda x: x[0] - 2 * x[1] + 2}, + NonlinearConstraint(lambda x: x[0] - x[1], 0, 0)] + + with suppress_warnings() as sup: + sup.filter(UserWarning, "delta_grad == 0.0") + res = minimize(fun, self.x0, method=self.method, + bounds=self.bnds, constraints=cons) + assert_allclose(res.x, [1.75, 1.75], rtol=1e-4) + assert_allclose(res.fun, 1.125, rtol=1e-4) + + +class TestNewToOld: + @pytest.mark.fail_slow(2) + def test_multiple_constraint_objects(self, num_parallel_threads): + def fun(x): + return (x[0] - 1) ** 2 + (x[1] - 2.5) ** 2 + (x[2] - 0.75) ** 2 + x0 = [2, 0, 1] + coni = [] # only inequality constraints (can use cobyla) + methods = ["slsqp", "cobyla", "cobyqa", "trust-constr"] + + # mixed old and new + coni.append([{'type': 'ineq', 'fun': lambda x: x[0] - 2 * x[1] + 2}, + NonlinearConstraint(lambda x: x[0] - x[1], -1, 1)]) + + coni.append([LinearConstraint([1, -2, 0], -2, np.inf), + NonlinearConstraint(lambda x: x[0] - x[1], -1, 1)]) + + coni.append([NonlinearConstraint(lambda x: x[0] - 2 * x[1] + 2, 0, np.inf), + NonlinearConstraint(lambda x: x[0] - x[1], -1, 1)]) + + for con in coni: + funs = {} + for method in methods: + with suppress_warnings() as sup: + sup.filter(UserWarning) + result = minimize(fun, x0, method=method, constraints=con) + funs[method] = result.fun + assert_allclose(funs['slsqp'], funs['trust-constr'], rtol=1e-4) + assert_allclose(funs['cobyla'], funs['trust-constr'], rtol=1e-4) + if num_parallel_threads == 1: + assert_allclose(funs['cobyqa'], funs['trust-constr'], + rtol=1e-4) + + @pytest.mark.fail_slow(20) + def test_individual_constraint_objects(self, num_parallel_threads): + def fun(x): + return (x[0] - 1) ** 2 + (x[1] - 2.5) ** 2 + (x[2] - 0.75) ** 2 + x0 = [2, 0, 1] + + cone = [] # with equality constraints (can't use cobyla) + coni = [] # only inequality constraints (can use cobyla) + methods = ["slsqp", "cobyla", "cobyqa", "trust-constr"] + + # nonstandard data types for constraint equality bounds + cone.append(NonlinearConstraint(lambda x: x[0] - x[1], 1, 1)) + cone.append(NonlinearConstraint(lambda x: x[0] - x[1], [1.21], [1.21])) + cone.append(NonlinearConstraint(lambda x: x[0] - x[1], + 1.21, np.array([1.21]))) + + # multiple equalities + cone.append(NonlinearConstraint( + lambda x: [x[0] - x[1], x[1] - x[2]], + 1.21, 1.21)) # two same equalities + cone.append(NonlinearConstraint( + lambda x: [x[0] - x[1], x[1] - x[2]], + [1.21, 1.4], [1.21, 1.4])) # two different equalities + cone.append(NonlinearConstraint( + lambda x: [x[0] - x[1], x[1] - x[2]], + [1.21, 1.21], 1.21)) # equality specified two ways + cone.append(NonlinearConstraint( + lambda x: [x[0] - x[1], x[1] - x[2]], + [1.21, -np.inf], [1.21, np.inf])) # equality + unbounded + + # nonstandard data types for constraint inequality bounds + coni.append(NonlinearConstraint(lambda x: x[0] - x[1], 1.21, np.inf)) + coni.append(NonlinearConstraint(lambda x: x[0] - x[1], [1.21], np.inf)) + coni.append(NonlinearConstraint(lambda x: x[0] - x[1], + 1.21, np.array([np.inf]))) + coni.append(NonlinearConstraint(lambda x: x[0] - x[1], -np.inf, -3)) + coni.append(NonlinearConstraint(lambda x: x[0] - x[1], + np.array(-np.inf), -3)) + + # multiple inequalities/equalities + coni.append(NonlinearConstraint( + lambda x: [x[0] - x[1], x[1] - x[2]], + 1.21, np.inf)) # two same inequalities + cone.append(NonlinearConstraint( + lambda x: [x[0] - x[1], x[1] - x[2]], + [1.21, -np.inf], [1.21, 1.4])) # mixed equality/inequality + coni.append(NonlinearConstraint( + lambda x: [x[0] - x[1], x[1] - x[2]], + [1.1, .8], [1.2, 1.4])) # bounded above and below + coni.append(NonlinearConstraint( + lambda x: [x[0] - x[1], x[1] - x[2]], + [-1.2, -1.4], [-1.1, -.8])) # - bounded above and below + + # quick check of LinearConstraint class (very little new code to test) + cone.append(LinearConstraint([1, -1, 0], 1.21, 1.21)) + cone.append(LinearConstraint([[1, -1, 0], [0, 1, -1]], 1.21, 1.21)) + cone.append(LinearConstraint([[1, -1, 0], [0, 1, -1]], + [1.21, -np.inf], [1.21, 1.4])) + + for con in coni: + funs = {} + for method in methods: + with suppress_warnings() as sup: + sup.filter(UserWarning) + result = minimize(fun, x0, method=method, constraints=con) + funs[method] = result.fun + assert_allclose(funs['slsqp'], funs['trust-constr'], rtol=1e-3) + assert_allclose(funs['cobyla'], funs['trust-constr'], rtol=1e-3) + if num_parallel_threads == 1: + assert_allclose(funs['cobyqa'], funs['trust-constr'], + rtol=1e-3) + + for con in cone: + funs = {} + for method in [method for method in methods if method != 'cobyla']: + with suppress_warnings() as sup: + sup.filter(UserWarning) + result = minimize(fun, x0, method=method, constraints=con) + funs[method] = result.fun + assert_allclose(funs['slsqp'], funs['trust-constr'], rtol=1e-3) + if num_parallel_threads == 1: + assert_allclose(funs['cobyqa'], funs['trust-constr'], + rtol=1e-3) + + +class TestNewToOldSLSQP: + method = 'slsqp' + elec = Elec(n_electrons=2) + elec.x_opt = np.array([-0.58438468, 0.58438466, 0.73597047, + -0.73597044, 0.34180668, -0.34180667]) + brock = BoundedRosenbrock() + brock.x_opt = [0, 0] + list_of_problems = [Maratos(), + HyperbolicIneq(), + Rosenbrock(), + IneqRosenbrock(), + EqIneqRosenbrock(), + elec, + brock + ] + + def test_list_of_problems(self): + + for prob in self.list_of_problems: + + with suppress_warnings() as sup: + sup.filter(UserWarning) + result = minimize(prob.fun, prob.x0, + method=self.method, + bounds=prob.bounds, + constraints=prob.constr) + + assert_array_almost_equal(result.x, prob.x_opt, decimal=3) + + @pytest.mark.thread_unsafe + def test_warn_mixed_constraints(self): + # warns about inefficiency of mixed equality/inequality constraints + def fun(x): + return (x[0] - 1) ** 2 + (x[1] - 2.5) ** 2 + (x[2] - 0.75) ** 2 + cons = NonlinearConstraint(lambda x: [x[0]**2 - x[1], x[1] - x[2]], + [1.1, .8], [1.1, 1.4]) + bnds = ((0, None), (0, None), (0, None)) + with suppress_warnings() as sup: + sup.filter(UserWarning, "delta_grad == 0.0") + assert_warns(OptimizeWarning, minimize, fun, (2, 0, 1), + method=self.method, bounds=bnds, constraints=cons) + + @pytest.mark.thread_unsafe + def test_warn_ignored_options(self): + # warns about constraint options being ignored + def fun(x): + return (x[0] - 1) ** 2 + (x[1] - 2.5) ** 2 + (x[2] - 0.75) ** 2 + x0 = (2, 0, 1) + + if self.method == "slsqp": + bnds = ((0, None), (0, None), (0, None)) + else: + bnds = None + + cons = NonlinearConstraint(lambda x: x[0], 2, np.inf) + res = minimize(fun, x0, method=self.method, + bounds=bnds, constraints=cons) + # no warnings without constraint options + assert_allclose(res.fun, 1) + + cons = LinearConstraint([1, 0, 0], 2, np.inf) + res = minimize(fun, x0, method=self.method, + bounds=bnds, constraints=cons) + # no warnings without constraint options + assert_allclose(res.fun, 1) + + cons = [] + cons.append(NonlinearConstraint(lambda x: x[0]**2, 2, np.inf, + keep_feasible=True)) + cons.append(NonlinearConstraint(lambda x: x[0]**2, 2, np.inf, + hess=BFGS())) + cons.append(NonlinearConstraint(lambda x: x[0]**2, 2, np.inf, + finite_diff_jac_sparsity=42)) + cons.append(NonlinearConstraint(lambda x: x[0]**2, 2, np.inf, + finite_diff_rel_step=42)) + cons.append(LinearConstraint([1, 0, 0], 2, np.inf, + keep_feasible=True)) + for con in cons: + assert_warns(OptimizeWarning, minimize, fun, x0, + method=self.method, bounds=bnds, constraints=cons) + + +class TestNewToOldCobyla: + method = 'cobyla' + + list_of_problems = [ + Elec(n_electrons=2), + Elec(n_electrons=4), + ] + + @pytest.mark.slow + def test_list_of_problems(self): + + for prob in self.list_of_problems: + + with suppress_warnings() as sup: + sup.filter(UserWarning) + truth = minimize(prob.fun, prob.x0, + method='trust-constr', + bounds=prob.bounds, + constraints=prob.constr) + result = minimize(prob.fun, prob.x0, + method=self.method, + bounds=prob.bounds, + constraints=prob.constr) + + assert_allclose(result.fun, truth.fun, rtol=1e-3) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_constraints.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_constraints.py new file mode 100644 index 0000000000000000000000000000000000000000..4c4186ba7b6dd6f56b89e2f39add9eb16e6beccb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_constraints.py @@ -0,0 +1,255 @@ +import pytest +import numpy as np +from numpy.testing import TestCase, assert_array_equal +import scipy.sparse as sps +from scipy.optimize._constraints import ( + Bounds, LinearConstraint, NonlinearConstraint, PreparedConstraint, + new_bounds_to_old, old_bound_to_new, strict_bounds) + + +class TestStrictBounds(TestCase): + def test_scalarvalue_unique_enforce_feasibility(self): + m = 3 + lb = 2 + ub = 4 + enforce_feasibility = False + strict_lb, strict_ub = strict_bounds(lb, ub, + enforce_feasibility, + m) + assert_array_equal(strict_lb, [-np.inf, -np.inf, -np.inf]) + assert_array_equal(strict_ub, [np.inf, np.inf, np.inf]) + + enforce_feasibility = True + strict_lb, strict_ub = strict_bounds(lb, ub, + enforce_feasibility, + m) + assert_array_equal(strict_lb, [2, 2, 2]) + assert_array_equal(strict_ub, [4, 4, 4]) + + def test_vectorvalue_unique_enforce_feasibility(self): + m = 3 + lb = [1, 2, 3] + ub = [4, 5, 6] + enforce_feasibility = False + strict_lb, strict_ub = strict_bounds(lb, ub, + enforce_feasibility, + m) + assert_array_equal(strict_lb, [-np.inf, -np.inf, -np.inf]) + assert_array_equal(strict_ub, [np.inf, np.inf, np.inf]) + + enforce_feasibility = True + strict_lb, strict_ub = strict_bounds(lb, ub, + enforce_feasibility, + m) + assert_array_equal(strict_lb, [1, 2, 3]) + assert_array_equal(strict_ub, [4, 5, 6]) + + def test_scalarvalue_vector_enforce_feasibility(self): + m = 3 + lb = 2 + ub = 4 + enforce_feasibility = [False, True, False] + strict_lb, strict_ub = strict_bounds(lb, ub, + enforce_feasibility, + m) + assert_array_equal(strict_lb, [-np.inf, 2, -np.inf]) + assert_array_equal(strict_ub, [np.inf, 4, np.inf]) + + def test_vectorvalue_vector_enforce_feasibility(self): + m = 3 + lb = [1, 2, 3] + ub = [4, 6, np.inf] + enforce_feasibility = [True, False, True] + strict_lb, strict_ub = strict_bounds(lb, ub, + enforce_feasibility, + m) + assert_array_equal(strict_lb, [1, -np.inf, 3]) + assert_array_equal(strict_ub, [4, np.inf, np.inf]) + + +def test_prepare_constraint_infeasible_x0(): + lb = np.array([0, 20, 30]) + ub = np.array([0.5, np.inf, 70]) + x0 = np.array([1, 2, 3]) + enforce_feasibility = np.array([False, True, True], dtype=bool) + bounds = Bounds(lb, ub, enforce_feasibility) + pytest.raises(ValueError, PreparedConstraint, bounds, x0) + + pc = PreparedConstraint(Bounds(lb, ub), [1, 2, 3]) + assert (pc.violation([1, 2, 3]) > 0).any() + assert (pc.violation([0.25, 21, 31]) == 0).all() + + x0 = np.array([1, 2, 3, 4]) + A = np.array([[1, 2, 3, 4], [5, 0, 0, 6], [7, 0, 8, 0]]) + enforce_feasibility = np.array([True, True, True], dtype=bool) + linear = LinearConstraint(A, -np.inf, 0, enforce_feasibility) + pytest.raises(ValueError, PreparedConstraint, linear, x0) + + pc = PreparedConstraint(LinearConstraint(A, -np.inf, 0), + [1, 2, 3, 4]) + assert (pc.violation([1, 2, 3, 4]) > 0).any() + assert (pc.violation([-10, 2, -10, 4]) == 0).all() + + def fun(x): + return A.dot(x) + + def jac(x): + return A + + def hess(x, v): + return sps.csr_matrix((4, 4)) + + nonlinear = NonlinearConstraint(fun, -np.inf, 0, jac, hess, + enforce_feasibility) + pytest.raises(ValueError, PreparedConstraint, nonlinear, x0) + + pc = PreparedConstraint(nonlinear, [-10, 2, -10, 4]) + assert (pc.violation([1, 2, 3, 4]) > 0).any() + assert (pc.violation([-10, 2, -10, 4]) == 0).all() + + +def test_violation(): + def cons_f(x): + return np.array([x[0] ** 2 + x[1], x[0] ** 2 - x[1]]) + + nlc = NonlinearConstraint(cons_f, [-1, -0.8500], [2, 2]) + pc = PreparedConstraint(nlc, [0.5, 1]) + + assert_array_equal(pc.violation([0.5, 1]), [0., 0.]) + + np.testing.assert_almost_equal(pc.violation([0.5, 1.2]), [0., 0.1]) + + np.testing.assert_almost_equal(pc.violation([1.2, 1.2]), [0.64, 0]) + + np.testing.assert_almost_equal(pc.violation([0.1, -1.2]), [0.19, 0]) + + np.testing.assert_almost_equal(pc.violation([0.1, 2]), [0.01, 1.14]) + + +def test_new_bounds_to_old(): + lb = np.array([-np.inf, 2, 3]) + ub = np.array([3, np.inf, 10]) + + bounds = [(None, 3), (2, None), (3, 10)] + assert_array_equal(new_bounds_to_old(lb, ub, 3), bounds) + + bounds_single_lb = [(-1, 3), (-1, None), (-1, 10)] + assert_array_equal(new_bounds_to_old(-1, ub, 3), bounds_single_lb) + + bounds_no_lb = [(None, 3), (None, None), (None, 10)] + assert_array_equal(new_bounds_to_old(-np.inf, ub, 3), bounds_no_lb) + + bounds_single_ub = [(None, 20), (2, 20), (3, 20)] + assert_array_equal(new_bounds_to_old(lb, 20, 3), bounds_single_ub) + + bounds_no_ub = [(None, None), (2, None), (3, None)] + assert_array_equal(new_bounds_to_old(lb, np.inf, 3), bounds_no_ub) + + bounds_single_both = [(1, 2), (1, 2), (1, 2)] + assert_array_equal(new_bounds_to_old(1, 2, 3), bounds_single_both) + + bounds_no_both = [(None, None), (None, None), (None, None)] + assert_array_equal(new_bounds_to_old(-np.inf, np.inf, 3), bounds_no_both) + + +def test_old_bounds_to_new(): + bounds = ([1, 2], (None, 3), (-1, None)) + lb_true = np.array([1, -np.inf, -1]) + ub_true = np.array([2, 3, np.inf]) + + lb, ub = old_bound_to_new(bounds) + assert_array_equal(lb, lb_true) + assert_array_equal(ub, ub_true) + + bounds = [(-np.inf, np.inf), (np.array([1]), np.array([1]))] + lb, ub = old_bound_to_new(bounds) + + assert_array_equal(lb, [-np.inf, 1]) + assert_array_equal(ub, [np.inf, 1]) + + +class TestBounds: + def test_repr(self): + # so that eval works + from numpy import array, inf # noqa: F401 + for args in ( + (-1.0, 5.0), + (-1.0, np.inf, True), + (np.array([1.0, -np.inf]), np.array([2.0, np.inf])), + (np.array([1.0, -np.inf]), np.array([2.0, np.inf]), + np.array([True, False])), + ): + bounds = Bounds(*args) + bounds2 = eval(repr(Bounds(*args))) + assert_array_equal(bounds.lb, bounds2.lb) + assert_array_equal(bounds.ub, bounds2.ub) + assert_array_equal(bounds.keep_feasible, bounds2.keep_feasible) + + def test_array(self): + # gh13501 + b = Bounds(lb=[0.0, 0.0], ub=[1.0, 1.0]) + assert isinstance(b.lb, np.ndarray) + assert isinstance(b.ub, np.ndarray) + + def test_defaults(self): + b1 = Bounds() + b2 = Bounds(np.asarray(-np.inf), np.asarray(np.inf)) + assert b1.lb == b2.lb + assert b1.ub == b2.ub + + def test_input_validation(self): + message = "Lower and upper bounds must be dense arrays." + with pytest.raises(ValueError, match=message): + Bounds(sps.coo_array([1, 2]), [1, 2]) + with pytest.raises(ValueError, match=message): + Bounds([1, 2], sps.coo_array([1, 2])) + + message = "`keep_feasible` must be a dense array." + with pytest.raises(ValueError, match=message): + Bounds([1, 2], [1, 2], keep_feasible=sps.coo_array([True, True])) + + message = "`lb`, `ub`, and `keep_feasible` must be broadcastable." + with pytest.raises(ValueError, match=message): + Bounds([1, 2], [1, 2, 3]) + + def test_residual(self): + bounds = Bounds(-2, 4) + x0 = [-1, 2] + np.testing.assert_allclose(bounds.residual(x0), ([1, 4], [5, 2])) + + +class TestLinearConstraint: + def test_defaults(self): + A = np.eye(4) + lc = LinearConstraint(A) + lc2 = LinearConstraint(A, -np.inf, np.inf) + assert_array_equal(lc.lb, lc2.lb) + assert_array_equal(lc.ub, lc2.ub) + + def test_input_validation(self): + A = np.eye(4) + message = "`lb`, `ub`, and `keep_feasible` must be broadcastable" + with pytest.raises(ValueError, match=message): + LinearConstraint(A, [1, 2], [1, 2, 3]) + + message = "Constraint limits must be dense arrays" + with pytest.raises(ValueError, match=message): + LinearConstraint(A, sps.coo_array([1, 2]), [2, 3]) + with pytest.raises(ValueError, match=message): + LinearConstraint(A, [1, 2], sps.coo_array([2, 3])) + + message = "`keep_feasible` must be a dense array" + with pytest.raises(ValueError, match=message): + keep_feasible = sps.coo_array([True, True]) + LinearConstraint(A, [1, 2], [2, 3], keep_feasible=keep_feasible) + + A = np.empty((4, 3, 5)) + message = "`A` must have exactly two dimensions." + with pytest.raises(ValueError, match=message): + LinearConstraint(A) + + def test_residual(self): + A = np.eye(2) + lc = LinearConstraint(A, -2, 4) + x0 = [-1, 2] + np.testing.assert_allclose(lc.residual(x0), ([1, 4], [5, 2])) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_cython_optimize.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_cython_optimize.py new file mode 100644 index 0000000000000000000000000000000000000000..2f859c1143eb6b63c439fe278bfdd4fdaa15410f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_cython_optimize.py @@ -0,0 +1,92 @@ +""" +Test Cython optimize zeros API functions: ``bisect``, ``ridder``, ``brenth``, +and ``brentq`` in `scipy.optimize.cython_optimize`, by finding the roots of a +3rd order polynomial given a sequence of constant terms, ``a0``, and fixed 1st, +2nd, and 3rd order terms in ``args``. + +.. math:: + + f(x, a0, args) = ((args[2]*x + args[1])*x + args[0])*x + a0 + +The 3rd order polynomial function is written in Cython and called in a Python +wrapper named after the zero function. See the private ``_zeros`` Cython module +in `scipy.optimize.cython_optimze` for more information. +""" + +import numpy.testing as npt +from scipy.optimize.cython_optimize import _zeros + +# CONSTANTS +# Solve x**3 - A0 = 0 for A0 = [2.0, 2.1, ..., 2.9]. +# The ARGS have 3 elements just to show how this could be done for any cubic +# polynomial. +A0 = tuple(-2.0 - x/10.0 for x in range(10)) # constant term +ARGS = (0.0, 0.0, 1.0) # 1st, 2nd, and 3rd order terms +XLO, XHI = 0.0, 2.0 # first and second bounds of zeros functions +# absolute and relative tolerances and max iterations for zeros functions +XTOL, RTOL, MITR = 0.001, 0.001, 10 +EXPECTED = [(-a0) ** (1.0/3.0) for a0 in A0] +# = [1.2599210498948732, +# 1.2805791649874942, +# 1.300591446851387, +# 1.3200061217959123, +# 1.338865900164339, +# 1.3572088082974532, +# 1.375068867074141, +# 1.3924766500838337, +# 1.4094597464129783, +# 1.4260431471424087] + + +# test bisect +def test_bisect(): + npt.assert_allclose( + EXPECTED, + list( + _zeros.loop_example('bisect', A0, ARGS, XLO, XHI, XTOL, RTOL, MITR) + ), + rtol=RTOL, atol=XTOL + ) + + +# test ridder +def test_ridder(): + npt.assert_allclose( + EXPECTED, + list( + _zeros.loop_example('ridder', A0, ARGS, XLO, XHI, XTOL, RTOL, MITR) + ), + rtol=RTOL, atol=XTOL + ) + + +# test brenth +def test_brenth(): + npt.assert_allclose( + EXPECTED, + list( + _zeros.loop_example('brenth', A0, ARGS, XLO, XHI, XTOL, RTOL, MITR) + ), + rtol=RTOL, atol=XTOL + ) + + +# test brentq +def test_brentq(): + npt.assert_allclose( + EXPECTED, + list( + _zeros.loop_example('brentq', A0, ARGS, XLO, XHI, XTOL, RTOL, MITR) + ), + rtol=RTOL, atol=XTOL + ) + + +# test brentq with full output +def test_brentq_full_output(): + output = _zeros.full_output_example( + (A0[0],) + ARGS, XLO, XHI, XTOL, RTOL, MITR) + npt.assert_allclose(EXPECTED[0], output['root'], rtol=RTOL, atol=XTOL) + npt.assert_equal(6, output['iterations']) + npt.assert_equal(7, output['funcalls']) + npt.assert_equal(0, output['error_num']) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_differentiable_functions.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_differentiable_functions.py new file mode 100644 index 0000000000000000000000000000000000000000..cac12af0033c09c3fbf73c52a9c2cb52b84d8239 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_differentiable_functions.py @@ -0,0 +1,805 @@ +import pytest +import platform +import numpy as np +from numpy.testing import (TestCase, assert_array_almost_equal, + assert_array_equal, assert_, assert_allclose, + assert_equal) +from scipy._lib._gcutils import assert_deallocated +from scipy.sparse import csr_matrix +from scipy.sparse.linalg import LinearOperator +from scipy.optimize._differentiable_functions import (ScalarFunction, + VectorFunction, + LinearVectorFunction, + IdentityVectorFunction) +from scipy.optimize import rosen, rosen_der, rosen_hess +from scipy.optimize._hessian_update_strategy import BFGS + + +class ExScalarFunction: + + def __init__(self): + self.nfev = 0 + self.ngev = 0 + self.nhev = 0 + + def fun(self, x): + self.nfev += 1 + return 2*(x[0]**2 + x[1]**2 - 1) - x[0] + + def grad(self, x): + self.ngev += 1 + return np.array([4*x[0]-1, 4*x[1]]) + + def hess(self, x): + self.nhev += 1 + return 4*np.eye(2) + + +class TestScalarFunction(TestCase): + + def test_finite_difference_grad(self): + ex = ExScalarFunction() + nfev = 0 + ngev = 0 + + x0 = [1.0, 0.0] + analit = ScalarFunction(ex.fun, x0, (), ex.grad, + ex.hess, None, (-np.inf, np.inf)) + nfev += 1 + ngev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev, nfev) + approx = ScalarFunction(ex.fun, x0, (), '2-point', + ex.hess, None, (-np.inf, np.inf)) + nfev += 3 + ngev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_equal(analit.f, approx.f) + assert_array_almost_equal(analit.g, approx.g) + + x = [10, 0.3] + f_analit = analit.fun(x) + g_analit = analit.grad(x) + nfev += 1 + ngev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + f_approx = approx.fun(x) + g_approx = approx.grad(x) + nfev += 3 + ngev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_almost_equal(f_analit, f_approx) + assert_array_almost_equal(g_analit, g_approx) + + x = [2.0, 1.0] + g_analit = analit.grad(x) + ngev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + + g_approx = approx.grad(x) + nfev += 3 + ngev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_almost_equal(g_analit, g_approx) + + x = [2.5, 0.3] + f_analit = analit.fun(x) + g_analit = analit.grad(x) + nfev += 1 + ngev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + f_approx = approx.fun(x) + g_approx = approx.grad(x) + nfev += 3 + ngev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_almost_equal(f_analit, f_approx) + assert_array_almost_equal(g_analit, g_approx) + + x = [2, 0.3] + f_analit = analit.fun(x) + g_analit = analit.grad(x) + nfev += 1 + ngev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + f_approx = approx.fun(x) + g_approx = approx.grad(x) + nfev += 3 + ngev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_almost_equal(f_analit, f_approx) + assert_array_almost_equal(g_analit, g_approx) + + def test_fun_and_grad(self): + ex = ExScalarFunction() + + def fg_allclose(x, y): + assert_allclose(x[0], y[0]) + assert_allclose(x[1], y[1]) + + # with analytic gradient + x0 = [2.0, 0.3] + analit = ScalarFunction(ex.fun, x0, (), ex.grad, + ex.hess, None, (-np.inf, np.inf)) + + fg = ex.fun(x0), ex.grad(x0) + fg_allclose(analit.fun_and_grad(x0), fg) + assert analit.ngev == 1 + + x0[1] = 1. + fg = ex.fun(x0), ex.grad(x0) + fg_allclose(analit.fun_and_grad(x0), fg) + + # with finite difference gradient + x0 = [2.0, 0.3] + sf = ScalarFunction(ex.fun, x0, (), '3-point', + ex.hess, None, (-np.inf, np.inf)) + assert sf.ngev == 1 + fg = ex.fun(x0), ex.grad(x0) + fg_allclose(sf.fun_and_grad(x0), fg) + assert sf.ngev == 1 + + x0[1] = 1. + fg = ex.fun(x0), ex.grad(x0) + fg_allclose(sf.fun_and_grad(x0), fg) + + def test_finite_difference_hess_linear_operator(self): + ex = ExScalarFunction() + nfev = 0 + ngev = 0 + nhev = 0 + + x0 = [1.0, 0.0] + analit = ScalarFunction(ex.fun, x0, (), ex.grad, + ex.hess, None, (-np.inf, np.inf)) + nfev += 1 + ngev += 1 + nhev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev, ngev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev, nhev) + approx = ScalarFunction(ex.fun, x0, (), ex.grad, + '2-point', None, (-np.inf, np.inf)) + assert_(isinstance(approx.H, LinearOperator)) + for v in ([1.0, 2.0], [3.0, 4.0], [5.0, 2.0]): + assert_array_equal(analit.f, approx.f) + assert_array_almost_equal(analit.g, approx.g) + assert_array_almost_equal(analit.H.dot(v), approx.H.dot(v)) + nfev += 1 + ngev += 4 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + + x = [2.0, 1.0] + H_analit = analit.hess(x) + nhev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + H_approx = approx.hess(x) + assert_(isinstance(H_approx, LinearOperator)) + for v in ([1.0, 2.0], [3.0, 4.0], [5.0, 2.0]): + assert_array_almost_equal(H_analit.dot(v), H_approx.dot(v)) + ngev += 4 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + + x = [2.1, 1.2] + H_analit = analit.hess(x) + nhev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + H_approx = approx.hess(x) + assert_(isinstance(H_approx, LinearOperator)) + for v in ([1.0, 2.0], [3.0, 4.0], [5.0, 2.0]): + assert_array_almost_equal(H_analit.dot(v), H_approx.dot(v)) + ngev += 4 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + + x = [2.5, 0.3] + _ = analit.grad(x) + H_analit = analit.hess(x) + ngev += 1 + nhev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + _ = approx.grad(x) + H_approx = approx.hess(x) + assert_(isinstance(H_approx, LinearOperator)) + for v in ([1.0, 2.0], [3.0, 4.0], [5.0, 2.0]): + assert_array_almost_equal(H_analit.dot(v), H_approx.dot(v)) + ngev += 4 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + + x = [5.2, 2.3] + _ = analit.grad(x) + H_analit = analit.hess(x) + ngev += 1 + nhev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + _ = approx.grad(x) + H_approx = approx.hess(x) + assert_(isinstance(H_approx, LinearOperator)) + for v in ([1.0, 2.0], [3.0, 4.0], [5.0, 2.0]): + assert_array_almost_equal(H_analit.dot(v), H_approx.dot(v)) + ngev += 4 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.ngev, ngev) + assert_array_equal(analit.ngev+approx.ngev, ngev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + + @pytest.mark.thread_unsafe + def test_x_storage_overlap(self): + # Scalar_Function should not store references to arrays, it should + # store copies - this checks that updating an array in-place causes + # Scalar_Function.x to be updated. + + def f(x): + return np.sum(np.asarray(x) ** 2) + + x = np.array([1., 2., 3.]) + sf = ScalarFunction(f, x, (), '3-point', lambda x: x, None, (-np.inf, np.inf)) + + assert x is not sf.x + assert_equal(sf.fun(x), 14.0) + assert x is not sf.x + + x[0] = 0. + f1 = sf.fun(x) + assert_equal(f1, 13.0) + + x[0] = 1 + f2 = sf.fun(x) + assert_equal(f2, 14.0) + assert x is not sf.x + + # now test with a HessianUpdate strategy specified + hess = BFGS() + x = np.array([1., 2., 3.]) + sf = ScalarFunction(f, x, (), '3-point', hess, None, (-np.inf, np.inf)) + + assert x is not sf.x + assert_equal(sf.fun(x), 14.0) + assert x is not sf.x + + x[0] = 0. + f1 = sf.fun(x) + assert_equal(f1, 13.0) + + x[0] = 1 + f2 = sf.fun(x) + assert_equal(f2, 14.0) + assert x is not sf.x + + # gh13740 x is changed in user function + def ff(x): + x *= x # overwrite x + return np.sum(x) + + x = np.array([1., 2., 3.]) + sf = ScalarFunction( + ff, x, (), '3-point', lambda x: x, None, (-np.inf, np.inf) + ) + assert x is not sf.x + assert_equal(sf.fun(x), 14.0) + assert_equal(sf.x, np.array([1., 2., 3.])) + assert x is not sf.x + + def test_lowest_x(self): + # ScalarFunction should remember the lowest func(x) visited. + x0 = np.array([2, 3, 4]) + sf = ScalarFunction(rosen, x0, (), rosen_der, rosen_hess, + None, None) + sf.fun([1, 1, 1]) + sf.fun(x0) + sf.fun([1.01, 1, 1.0]) + sf.grad([1.01, 1, 1.0]) + assert_equal(sf._lowest_f, 0.0) + assert_equal(sf._lowest_x, [1.0, 1.0, 1.0]) + + sf = ScalarFunction(rosen, x0, (), '2-point', rosen_hess, + None, (-np.inf, np.inf)) + sf.fun([1, 1, 1]) + sf.fun(x0) + sf.fun([1.01, 1, 1.0]) + sf.grad([1.01, 1, 1.0]) + assert_equal(sf._lowest_f, 0.0) + assert_equal(sf._lowest_x, [1.0, 1.0, 1.0]) + + def test_float_size(self): + x0 = np.array([2, 3, 4]).astype(np.float32) + + # check that ScalarFunction/approx_derivative always send the correct + # float width + def rosen_(x): + assert x.dtype == np.float32 + return rosen(x) + + sf = ScalarFunction(rosen_, x0, (), '2-point', rosen_hess, + None, (-np.inf, np.inf)) + res = sf.fun(x0) + assert res.dtype == np.float32 + + +class ExVectorialFunction: + + def __init__(self): + self.nfev = 0 + self.njev = 0 + self.nhev = 0 + + def fun(self, x): + self.nfev += 1 + return np.array([2*(x[0]**2 + x[1]**2 - 1) - x[0], + 4*(x[0]**3 + x[1]**2 - 4) - 3*x[0]], dtype=x.dtype) + + def jac(self, x): + self.njev += 1 + return np.array([[4*x[0]-1, 4*x[1]], + [12*x[0]**2-3, 8*x[1]]], dtype=x.dtype) + + def hess(self, x, v): + self.nhev += 1 + return v[0]*4*np.eye(2) + v[1]*np.array([[24*x[0], 0], + [0, 8]]) + + +class TestVectorialFunction(TestCase): + + def test_finite_difference_jac(self): + ex = ExVectorialFunction() + nfev = 0 + njev = 0 + + x0 = [1.0, 0.0] + analit = VectorFunction(ex.fun, x0, ex.jac, ex.hess, None, None, + (-np.inf, np.inf), None) + nfev += 1 + njev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev, njev) + approx = VectorFunction(ex.fun, x0, '2-point', ex.hess, None, None, + (-np.inf, np.inf), None) + nfev += 3 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_equal(analit.f, approx.f) + assert_array_almost_equal(analit.J, approx.J) + + x = [10, 0.3] + f_analit = analit.fun(x) + J_analit = analit.jac(x) + nfev += 1 + njev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + f_approx = approx.fun(x) + J_approx = approx.jac(x) + nfev += 3 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_almost_equal(f_analit, f_approx) + assert_array_almost_equal(J_analit, J_approx, decimal=4) + + x = [2.0, 1.0] + J_analit = analit.jac(x) + njev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + J_approx = approx.jac(x) + nfev += 3 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_almost_equal(J_analit, J_approx) + + x = [2.5, 0.3] + f_analit = analit.fun(x) + J_analit = analit.jac(x) + nfev += 1 + njev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + f_approx = approx.fun(x) + J_approx = approx.jac(x) + nfev += 3 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_almost_equal(f_analit, f_approx) + assert_array_almost_equal(J_analit, J_approx) + + x = [2, 0.3] + f_analit = analit.fun(x) + J_analit = analit.jac(x) + nfev += 1 + njev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + f_approx = approx.fun(x) + J_approx = approx.jac(x) + nfev += 3 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_almost_equal(f_analit, f_approx) + assert_array_almost_equal(J_analit, J_approx) + + def test_finite_difference_hess_linear_operator(self): + ex = ExVectorialFunction() + nfev = 0 + njev = 0 + nhev = 0 + + x0 = [1.0, 0.0] + v0 = [1.0, 2.0] + analit = VectorFunction(ex.fun, x0, ex.jac, ex.hess, None, None, + (-np.inf, np.inf), None) + nfev += 1 + njev += 1 + nhev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev, njev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev, nhev) + approx = VectorFunction(ex.fun, x0, ex.jac, '2-point', None, None, + (-np.inf, np.inf), None) + assert_(isinstance(approx.H, LinearOperator)) + for p in ([1.0, 2.0], [3.0, 4.0], [5.0, 2.0]): + assert_array_equal(analit.f, approx.f) + assert_array_almost_equal(analit.J, approx.J) + assert_array_almost_equal(analit.H.dot(p), approx.H.dot(p)) + nfev += 1 + njev += 4 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + + x = [2.0, 1.0] + H_analit = analit.hess(x, v0) + nhev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + H_approx = approx.hess(x, v0) + assert_(isinstance(H_approx, LinearOperator)) + for p in ([1.0, 2.0], [3.0, 4.0], [5.0, 2.0]): + assert_array_almost_equal(H_analit.dot(p), H_approx.dot(p), + decimal=5) + njev += 4 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + + x = [2.1, 1.2] + v = [1.0, 1.0] + H_analit = analit.hess(x, v) + nhev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + H_approx = approx.hess(x, v) + assert_(isinstance(H_approx, LinearOperator)) + for v in ([1.0, 2.0], [3.0, 4.0], [5.0, 2.0]): + assert_array_almost_equal(H_analit.dot(v), H_approx.dot(v)) + njev += 4 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + + x = [2.5, 0.3] + _ = analit.jac(x) + H_analit = analit.hess(x, v0) + njev += 1 + nhev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + _ = approx.jac(x) + H_approx = approx.hess(x, v0) + assert_(isinstance(H_approx, LinearOperator)) + for v in ([1.0, 2.0], [3.0, 4.0], [5.0, 2.0]): + assert_array_almost_equal(H_analit.dot(v), H_approx.dot(v), decimal=4) + njev += 4 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + + x = [5.2, 2.3] + v = [2.3, 5.2] + _ = analit.jac(x) + H_analit = analit.hess(x, v) + njev += 1 + nhev += 1 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + _ = approx.jac(x) + H_approx = approx.hess(x, v) + assert_(isinstance(H_approx, LinearOperator)) + for v in ([1.0, 2.0], [3.0, 4.0], [5.0, 2.0]): + assert_array_almost_equal(H_analit.dot(v), H_approx.dot(v), decimal=4) + njev += 4 + assert_array_equal(ex.nfev, nfev) + assert_array_equal(analit.nfev+approx.nfev, nfev) + assert_array_equal(ex.njev, njev) + assert_array_equal(analit.njev+approx.njev, njev) + assert_array_equal(ex.nhev, nhev) + assert_array_equal(analit.nhev+approx.nhev, nhev) + + @pytest.mark.thread_unsafe + def test_x_storage_overlap(self): + # VectorFunction should not store references to arrays, it should + # store copies - this checks that updating an array in-place causes + # Scalar_Function.x to be updated. + ex = ExVectorialFunction() + x0 = np.array([1.0, 0.0]) + + vf = VectorFunction(ex.fun, x0, '3-point', ex.hess, None, None, + (-np.inf, np.inf), None) + + assert x0 is not vf.x + assert_equal(vf.fun(x0), ex.fun(x0)) + assert x0 is not vf.x + + x0[0] = 2. + assert_equal(vf.fun(x0), ex.fun(x0)) + assert x0 is not vf.x + + x0[0] = 1. + assert_equal(vf.fun(x0), ex.fun(x0)) + assert x0 is not vf.x + + # now test with a HessianUpdate strategy specified + hess = BFGS() + x0 = np.array([1.0, 0.0]) + vf = VectorFunction(ex.fun, x0, '3-point', hess, None, None, + (-np.inf, np.inf), None) + + with pytest.warns(UserWarning): + # filter UserWarning because ExVectorialFunction is linear and + # a quasi-Newton approximation is used for the Hessian. + assert x0 is not vf.x + assert_equal(vf.fun(x0), ex.fun(x0)) + assert x0 is not vf.x + + x0[0] = 2. + assert_equal(vf.fun(x0), ex.fun(x0)) + assert x0 is not vf.x + + x0[0] = 1. + assert_equal(vf.fun(x0), ex.fun(x0)) + assert x0 is not vf.x + + def test_float_size(self): + ex = ExVectorialFunction() + x0 = np.array([1.0, 0.0]).astype(np.float32) + + vf = VectorFunction(ex.fun, x0, ex.jac, ex.hess, None, None, + (-np.inf, np.inf), None) + + res = vf.fun(x0) + assert res.dtype == np.float32 + + res = vf.jac(x0) + assert res.dtype == np.float32 + + +def test_LinearVectorFunction(): + A_dense = np.array([ + [-1, 2, 0], + [0, 4, 2] + ]) + x0 = np.zeros(3) + A_sparse = csr_matrix(A_dense) + x = np.array([1, -1, 0]) + v = np.array([-1, 1]) + Ax = np.array([-3, -4]) + + f1 = LinearVectorFunction(A_dense, x0, None) + assert_(not f1.sparse_jacobian) + + f2 = LinearVectorFunction(A_dense, x0, True) + assert_(f2.sparse_jacobian) + + f3 = LinearVectorFunction(A_dense, x0, False) + assert_(not f3.sparse_jacobian) + + f4 = LinearVectorFunction(A_sparse, x0, None) + assert_(f4.sparse_jacobian) + + f5 = LinearVectorFunction(A_sparse, x0, True) + assert_(f5.sparse_jacobian) + + f6 = LinearVectorFunction(A_sparse, x0, False) + assert_(not f6.sparse_jacobian) + + assert_array_equal(f1.fun(x), Ax) + assert_array_equal(f2.fun(x), Ax) + assert_array_equal(f1.jac(x), A_dense) + assert_array_equal(f2.jac(x).toarray(), A_sparse.toarray()) + assert_array_equal(f1.hess(x, v).toarray(), np.zeros((3, 3))) + + +def test_LinearVectorFunction_memoization(): + A = np.array([[-1, 2, 0], [0, 4, 2]]) + x0 = np.array([1, 2, -1]) + fun = LinearVectorFunction(A, x0, False) + + assert_array_equal(x0, fun.x) + assert_array_equal(A.dot(x0), fun.f) + + x1 = np.array([-1, 3, 10]) + assert_array_equal(A, fun.jac(x1)) + assert_array_equal(x1, fun.x) + assert_array_equal(A.dot(x0), fun.f) + assert_array_equal(A.dot(x1), fun.fun(x1)) + assert_array_equal(A.dot(x1), fun.f) + + +def test_IdentityVectorFunction(): + x0 = np.zeros(3) + + f1 = IdentityVectorFunction(x0, None) + f2 = IdentityVectorFunction(x0, False) + f3 = IdentityVectorFunction(x0, True) + + assert_(f1.sparse_jacobian) + assert_(not f2.sparse_jacobian) + assert_(f3.sparse_jacobian) + + x = np.array([-1, 2, 1]) + v = np.array([-2, 3, 0]) + + assert_array_equal(f1.fun(x), x) + assert_array_equal(f2.fun(x), x) + + assert_array_equal(f1.jac(x).toarray(), np.eye(3)) + assert_array_equal(f2.jac(x), np.eye(3)) + + assert_array_equal(f1.hess(x, v).toarray(), np.zeros((3, 3))) + + +@pytest.mark.skipif( + platform.python_implementation() == "PyPy", + reason="assert_deallocate not available on PyPy" +) +def test_ScalarFunctionNoReferenceCycle(): + """Regression test for gh-20768.""" + ex = ExScalarFunction() + x0 = np.zeros(3) + with assert_deallocated(lambda: ScalarFunction(ex.fun, x0, (), ex.grad, + ex.hess, None, (-np.inf, np.inf))): + pass + + +@pytest.mark.skipif( + platform.python_implementation() == "PyPy", + reason="assert_deallocate not available on PyPy" +) +@pytest.mark.xfail(reason="TODO remove reference cycle from VectorFunction") +def test_VectorFunctionNoReferenceCycle(): + """Regression test for gh-20768.""" + ex = ExVectorialFunction() + x0 = [1.0, 0.0] + with assert_deallocated(lambda: VectorFunction(ex.fun, x0, ex.jac, + ex.hess, None, None, (-np.inf, np.inf), None)): + pass + + +@pytest.mark.skipif( + platform.python_implementation() == "PyPy", + reason="assert_deallocate not available on PyPy" +) +def test_LinearVectorFunctionNoReferenceCycle(): + """Regression test for gh-20768.""" + A_dense = np.array([ + [-1, 2, 0], + [0, 4, 2] + ]) + x0 = np.zeros(3) + A_sparse = csr_matrix(A_dense) + with assert_deallocated(lambda: LinearVectorFunction(A_sparse, x0, None)): + pass diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_direct.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_direct.py new file mode 100644 index 0000000000000000000000000000000000000000..835d3164c8d547599a507550dcc7629e7d327394 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_direct.py @@ -0,0 +1,321 @@ +""" +Unit test for DIRECT optimization algorithm. +""" +from numpy.testing import (assert_allclose, + assert_array_less) +import pytest +import numpy as np +from scipy.optimize import direct, Bounds +import threading + + +class TestDIRECT: + + def setup_method(self): + self.fun_calls = threading.local() + self.bounds_sphere = 4*[(-2, 3)] + self.optimum_sphere_pos = np.zeros((4, )) + self.optimum_sphere = 0.0 + self.bounds_stylinski_tang = Bounds([-4., -4.], [4., 4.]) + self.maxiter = 1000 + + # test functions + def sphere(self, x): + if not hasattr(self.fun_calls, 'c'): + self.fun_calls.c = 0 + self.fun_calls.c += 1 + return np.square(x).sum() + + def inv(self, x): + if np.sum(x) == 0: + raise ZeroDivisionError() + return 1/np.sum(x) + + def nan_fun(self, x): + return np.nan + + def inf_fun(self, x): + return np.inf + + def styblinski_tang(self, pos): + x, y = pos + return 0.5 * (x**4 - 16 * x**2 + 5 * x + y**4 - 16 * y**2 + 5 * y) + + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_direct(self, locally_biased): + res = direct(self.sphere, self.bounds_sphere, + locally_biased=locally_biased) + + # test accuracy + assert_allclose(res.x, self.optimum_sphere_pos, + rtol=1e-3, atol=1e-3) + assert_allclose(res.fun, self.optimum_sphere, atol=1e-5, rtol=1e-5) + + # test that result lies within bounds + _bounds = np.asarray(self.bounds_sphere) + assert_array_less(_bounds[:, 0], res.x) + assert_array_less(res.x, _bounds[:, 1]) + + # test number of function evaluations. Original DIRECT overshoots by + # up to 500 evaluations in last iteration + assert res.nfev <= 1000 * (len(self.bounds_sphere) + 1) + # test that number of function evaluations is correct + assert res.nfev == self.fun_calls.c + + # test that number of iterations is below supplied maximum + assert res.nit <= self.maxiter + + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_direct_callback(self, locally_biased): + # test that callback does not change the result + res = direct(self.sphere, self.bounds_sphere, + locally_biased=locally_biased) + + def callback(x): + x = 2*x + dummy = np.square(x) + print("DIRECT minimization algorithm callback test") + return dummy + + res_callback = direct(self.sphere, self.bounds_sphere, + locally_biased=locally_biased, + callback=callback) + + assert_allclose(res.x, res_callback.x) + + assert res.nit == res_callback.nit + assert res.nfev == res_callback.nfev + assert res.status == res_callback.status + assert res.success == res_callback.success + assert res.fun == res_callback.fun + assert_allclose(res.x, res_callback.x) + assert res.message == res_callback.message + + # test accuracy + assert_allclose(res_callback.x, self.optimum_sphere_pos, + rtol=1e-3, atol=1e-3) + assert_allclose(res_callback.fun, self.optimum_sphere, + atol=1e-5, rtol=1e-5) + + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_exception(self, locally_biased): + bounds = 4*[(-10, 10)] + with pytest.raises(ZeroDivisionError): + direct(self.inv, bounds=bounds, + locally_biased=locally_biased) + + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_nan(self, locally_biased): + bounds = 4*[(-10, 10)] + direct(self.nan_fun, bounds=bounds, + locally_biased=locally_biased) + + @pytest.mark.parametrize("len_tol", [1e-3, 1e-4]) + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_len_tol(self, len_tol, locally_biased): + bounds = 4*[(-10., 10.)] + res = direct(self.sphere, bounds=bounds, len_tol=len_tol, + vol_tol=1e-30, locally_biased=locally_biased) + assert res.status == 5 + assert res.success + assert_allclose(res.x, np.zeros((4, ))) + message = ("The side length measure of the hyperrectangle containing " + "the lowest function value found is below " + f"len_tol={len_tol}") + assert res.message == message + + @pytest.mark.parametrize("vol_tol", [1e-6, 1e-8]) + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_vol_tol(self, vol_tol, locally_biased): + bounds = 4*[(-10., 10.)] + res = direct(self.sphere, bounds=bounds, vol_tol=vol_tol, + len_tol=0., locally_biased=locally_biased) + assert res.status == 4 + assert res.success + assert_allclose(res.x, np.zeros((4, ))) + message = ("The volume of the hyperrectangle containing the lowest " + f"function value found is below vol_tol={vol_tol}") + assert res.message == message + + @pytest.mark.parametrize("f_min_rtol", [1e-3, 1e-5, 1e-7]) + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_f_min(self, f_min_rtol, locally_biased): + # test that desired function value is reached within + # relative tolerance of f_min_rtol + f_min = 1. + bounds = 4*[(-2., 10.)] + res = direct(self.sphere, bounds=bounds, f_min=f_min, + f_min_rtol=f_min_rtol, + locally_biased=locally_biased) + assert res.status == 3 + assert res.success + assert res.fun < f_min * (1. + f_min_rtol) + message = ("The best function value found is within a relative " + f"error={f_min_rtol} of the (known) global optimum f_min") + assert res.message == message + + def circle_with_args(self, x, a, b): + return np.square(x[0] - a) + np.square(x[1] - b).sum() + + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_f_circle_with_args(self, locally_biased): + bounds = 2*[(-2.0, 2.0)] + + res = direct(self.circle_with_args, bounds, args=(1, 1), maxfun=1250, + locally_biased=locally_biased) + assert_allclose(res.x, np.array([1., 1.]), rtol=1e-5) + + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_failure_maxfun(self, locally_biased): + # test that if optimization runs for the maximal number of + # evaluations, success = False is returned + + maxfun = 100 + result = direct(self.styblinski_tang, self.bounds_stylinski_tang, + maxfun=maxfun, locally_biased=locally_biased) + assert result.success is False + assert result.status == 1 + assert result.nfev >= maxfun + message = ("Number of function evaluations done is " + f"larger than maxfun={maxfun}") + assert result.message == message + + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_failure_maxiter(self, locally_biased): + # test that if optimization runs for the maximal number of + # iterations, success = False is returned + + maxiter = 10 + result = direct(self.styblinski_tang, self.bounds_stylinski_tang, + maxiter=maxiter, locally_biased=locally_biased) + assert result.success is False + assert result.status == 2 + assert result.nit >= maxiter + message = f"Number of iterations is larger than maxiter={maxiter}" + assert result.message == message + + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_bounds_variants(self, locally_biased): + # test that new and old bounds yield same result + + lb = [-6., 1., -5.] + ub = [-1., 3., 5.] + x_opt = np.array([-1., 1., 0.]) + bounds_old = list(zip(lb, ub)) + bounds_new = Bounds(lb, ub) + + res_old_bounds = direct(self.sphere, bounds_old, + locally_biased=locally_biased) + res_new_bounds = direct(self.sphere, bounds_new, + locally_biased=locally_biased) + + assert res_new_bounds.nfev == res_old_bounds.nfev + assert res_new_bounds.message == res_old_bounds.message + assert res_new_bounds.success == res_old_bounds.success + assert res_new_bounds.nit == res_old_bounds.nit + assert_allclose(res_new_bounds.x, res_old_bounds.x) + assert_allclose(res_new_bounds.x, x_opt, rtol=1e-2) + + @pytest.mark.parametrize("locally_biased", [True, False]) + @pytest.mark.parametrize("eps", [1e-5, 1e-4, 1e-3]) + def test_epsilon(self, eps, locally_biased): + result = direct(self.styblinski_tang, self.bounds_stylinski_tang, + eps=eps, vol_tol=1e-6, + locally_biased=locally_biased) + assert result.status == 4 + assert result.success + + @pytest.mark.xslow + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_no_segmentation_fault(self, locally_biased): + # test that an excessive number of function evaluations + # does not result in segmentation fault + bounds = [(-5., 20.)] * 100 + result = direct(self.sphere, bounds, maxfun=10000000, + maxiter=1000000, locally_biased=locally_biased) + assert result is not None + + @pytest.mark.parametrize("locally_biased", [True, False]) + def test_inf_fun(self, locally_biased): + # test that an objective value of infinity does not crash DIRECT + bounds = [(-5., 5.)] * 2 + result = direct(self.inf_fun, bounds, + locally_biased=locally_biased) + assert result is not None + + @pytest.mark.parametrize("len_tol", [-1, 2]) + def test_len_tol_validation(self, len_tol): + error_msg = "len_tol must be between 0 and 1." + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, self.bounds_stylinski_tang, + len_tol=len_tol) + + @pytest.mark.parametrize("vol_tol", [-1, 2]) + def test_vol_tol_validation(self, vol_tol): + error_msg = "vol_tol must be between 0 and 1." + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, self.bounds_stylinski_tang, + vol_tol=vol_tol) + + @pytest.mark.parametrize("f_min_rtol", [-1, 2]) + def test_fmin_rtol_validation(self, f_min_rtol): + error_msg = "f_min_rtol must be between 0 and 1." + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, self.bounds_stylinski_tang, + f_min_rtol=f_min_rtol, f_min=0.) + + @pytest.mark.parametrize("maxfun", [1.5, "string", (1, 2)]) + def test_maxfun_wrong_type(self, maxfun): + error_msg = "maxfun must be of type int." + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, self.bounds_stylinski_tang, + maxfun=maxfun) + + @pytest.mark.parametrize("maxiter", [1.5, "string", (1, 2)]) + def test_maxiter_wrong_type(self, maxiter): + error_msg = "maxiter must be of type int." + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, self.bounds_stylinski_tang, + maxiter=maxiter) + + def test_negative_maxiter(self): + error_msg = "maxiter must be > 0." + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, self.bounds_stylinski_tang, + maxiter=-1) + + def test_negative_maxfun(self): + error_msg = "maxfun must be > 0." + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, self.bounds_stylinski_tang, + maxfun=-1) + + @pytest.mark.parametrize("bounds", ["bounds", 2., 0]) + def test_invalid_bounds_type(self, bounds): + error_msg = ("bounds must be a sequence or " + "instance of Bounds class") + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, bounds) + + @pytest.mark.parametrize("bounds", + [Bounds([-1., -1], [-2, 1]), + Bounds([-np.nan, -1], [-2, np.nan]), + ] + ) + def test_incorrect_bounds(self, bounds): + error_msg = 'Bounds are not consistent min < max' + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, bounds) + + def test_inf_bounds(self): + error_msg = 'Bounds must not be inf.' + bounds = Bounds([-np.inf, -1], [-2, np.inf]) + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, bounds) + + @pytest.mark.parametrize("locally_biased", ["bias", [0, 0], 2.]) + def test_locally_biased_validation(self, locally_biased): + error_msg = 'locally_biased must be True or False.' + with pytest.raises(ValueError, match=error_msg): + direct(self.styblinski_tang, self.bounds_stylinski_tang, + locally_biased=locally_biased) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_extending.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_extending.py new file mode 100644 index 0000000000000000000000000000000000000000..279cac794e5f453fee52f19026f96eb530259485 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_extending.py @@ -0,0 +1,28 @@ +import os +import platform +import sysconfig + +import pytest + +from scipy._lib._testutils import IS_EDITABLE, _test_cython_extension, cython + + +@pytest.mark.fail_slow(40) +# essential per https://github.com/scipy/scipy/pull/20487#discussion_r1567057247 +@pytest.mark.skipif(IS_EDITABLE, + reason='Editable install cannot find .pxd headers.') +@pytest.mark.skipif((platform.system() == 'Windows' and + sysconfig.get_config_var('Py_GIL_DISABLED')), + reason='gh-22039') +@pytest.mark.skipif(platform.machine() in ["wasm32", "wasm64"], + reason="Can't start subprocess") +@pytest.mark.skipif(cython is None, reason="requires cython") +def test_cython(tmp_path): + srcdir = os.path.dirname(os.path.dirname(__file__)) + extensions, extensions_cpp = _test_cython_extension(tmp_path, srcdir) + # actually test the cython c-extensions + # From docstring for scipy.optimize.cython_optimize module + x = extensions.brentq_example() + assert x == 0.6999942848231314 + x = extensions_cpp.brentq_example() + assert x == 0.6999942848231314 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_hessian_update_strategy.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_hessian_update_strategy.py new file mode 100644 index 0000000000000000000000000000000000000000..2434e92434ff2ecc74ce2e8b8119632207fd0853 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_hessian_update_strategy.py @@ -0,0 +1,300 @@ +import re +from copy import deepcopy + +import numpy as np +import pytest +from numpy.linalg import norm +from numpy.testing import (TestCase, assert_array_almost_equal, + assert_array_equal, assert_array_less) +from scipy.optimize import (BFGS, SR1) + + +class Rosenbrock: + """Rosenbrock function. + + The following optimization problem: + minimize sum(100.0*(x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0) + """ + + def __init__(self, n=2, random_state=0): + rng = np.random.RandomState(random_state) + self.x0 = rng.uniform(-1, 1, n) + self.x_opt = np.ones(n) + + def fun(self, x): + x = np.asarray(x) + r = np.sum(100.0 * (x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0, + axis=0) + return r + + def grad(self, x): + x = np.asarray(x) + xm = x[1:-1] + xm_m1 = x[:-2] + xm_p1 = x[2:] + der = np.zeros_like(x) + der[1:-1] = (200 * (xm - xm_m1**2) - + 400 * (xm_p1 - xm**2) * xm - 2 * (1 - xm)) + der[0] = -400 * x[0] * (x[1] - x[0]**2) - 2 * (1 - x[0]) + der[-1] = 200 * (x[-1] - x[-2]**2) + return der + + def hess(self, x): + x = np.atleast_1d(x) + H = np.diag(-400 * x[:-1], 1) - np.diag(400 * x[:-1], -1) + diagonal = np.zeros(len(x), dtype=x.dtype) + diagonal[0] = 1200 * x[0]**2 - 400 * x[1] + 2 + diagonal[-1] = 200 + diagonal[1:-1] = 202 + 1200 * x[1:-1]**2 - 400 * x[2:] + H = H + np.diag(diagonal) + return H + + +class TestHessianUpdateStrategy(TestCase): + + + def test_hessian_initialization(self): + + ndims = 5 + symmetric_matrix = np.array([[43, 24, 33, 34, 49], + [24, 36, 44, 15, 44], + [33, 44, 37, 1, 30], + [34, 15, 1, 5, 46], + [49, 44, 30, 46, 22]]) + init_scales = ( + ('auto', np.eye(ndims)), + (2, np.eye(ndims) * 2), + (np.arange(1, ndims + 1) * np.eye(ndims), + np.arange(1, ndims + 1) * np.eye(ndims)), + (symmetric_matrix, symmetric_matrix),) + for approx_type in ['hess', 'inv_hess']: + for init_scale, true_matrix in init_scales: + # large min_{denominator,curvatur} makes them skip an update, + # so we can have our initial matrix + quasi_newton = (BFGS(init_scale=init_scale, + min_curvature=1e50, + exception_strategy='skip_update'), + SR1(init_scale=init_scale, + min_denominator=1e50)) + + for qn in quasi_newton: + qn.initialize(ndims, approx_type) + B = qn.get_matrix() + + assert_array_equal(B, np.eye(ndims)) + # don't test the auto init scale + if isinstance(init_scale, str) and init_scale == 'auto': + continue + + qn.update(np.ones(ndims) * 1e-5, np.arange(ndims) + 0.2) + B = qn.get_matrix() + assert_array_equal(B, true_matrix) + + # For this list of points, it is known + # that no exception occur during the + # Hessian update. Hence no update is + # skiped or damped. + + + def test_initialize_catch_illegal(self): + ndims = 3 + # no complex allowed + inits_msg_errtype = ((complex(3.14), + r"float\(\) argument must be a string or a " + r"(real )?number, not 'complex'", + TypeError), + + (np.array([3.2, 2.3, 1.2]).astype(np.complex128), + "init_scale contains complex elements, " + "must be real.", + TypeError), + + (np.array([[43, 24, 33], + [24, 36, 44, ], + [33, 44, 37, ]]).astype(np.complex128), + "init_scale contains complex elements, " + "must be real.", + TypeError), + + # not square + (np.array([[43, 55, 66]]), + re.escape( + "If init_scale is an array, it must have the " + "dimensions of the hess/inv_hess: (3, 3)." + " Got (1, 3)."), + ValueError), + + # not symmetric + (np.array([[43, 24, 33], + [24.1, 36, 44, ], + [33, 44, 37, ]]), + re.escape("If init_scale is an array, it must be" + " symmetric (passing scipy.linalg.issymmetric)" + " to be an approximation of a hess/inv_hess."), + ValueError), + ) + for approx_type in ['hess', 'inv_hess']: + for init_scale, message, errortype in inits_msg_errtype: + # large min_{denominator,curvatur} makes it skip an update, + # so we can retrieve our initial matrix + quasi_newton = (BFGS(init_scale=init_scale), + SR1(init_scale=init_scale)) + + for qn in quasi_newton: + qn.initialize(ndims, approx_type) + with pytest.raises(errortype, match=message): + qn.update(np.ones(ndims), np.arange(ndims)) + + def test_rosenbrock_with_no_exception(self): + # Define auxiliary problem + prob = Rosenbrock(n=5) + # Define iteration points + x_list = [[0.0976270, 0.4303787, 0.2055267, 0.0897663, -0.15269040], + [0.1847239, 0.0505757, 0.2123832, 0.0255081, 0.00083286], + [0.2142498, -0.0188480, 0.0503822, 0.0347033, 0.03323606], + [0.2071680, -0.0185071, 0.0341337, -0.0139298, 0.02881750], + [0.1533055, -0.0322935, 0.0280418, -0.0083592, 0.01503699], + [0.1382378, -0.0276671, 0.0266161, -0.0074060, 0.02801610], + [0.1651957, -0.0049124, 0.0269665, -0.0040025, 0.02138184], + [0.2354930, 0.0443711, 0.0173959, 0.0041872, 0.00794563], + [0.4168118, 0.1433867, 0.0111714, 0.0126265, -0.00658537], + [0.4681972, 0.2153273, 0.0225249, 0.0152704, -0.00463809], + [0.6023068, 0.3346815, 0.0731108, 0.0186618, -0.00371541], + [0.6415743, 0.3985468, 0.1324422, 0.0214160, -0.00062401], + [0.7503690, 0.5447616, 0.2804541, 0.0539851, 0.00242230], + [0.7452626, 0.5644594, 0.3324679, 0.0865153, 0.00454960], + [0.8059782, 0.6586838, 0.4229577, 0.1452990, 0.00976702], + [0.8549542, 0.7226562, 0.4991309, 0.2420093, 0.02772661], + [0.8571332, 0.7285741, 0.5279076, 0.2824549, 0.06030276], + [0.8835633, 0.7727077, 0.5957984, 0.3411303, 0.09652185], + [0.9071558, 0.8299587, 0.6771400, 0.4402896, 0.17469338], + [0.9190793, 0.8486480, 0.7163332, 0.5083780, 0.26107691], + [0.9371223, 0.8762177, 0.7653702, 0.5773109, 0.32181041], + [0.9554613, 0.9119893, 0.8282687, 0.6776178, 0.43162744], + [0.9545744, 0.9099264, 0.8270244, 0.6822220, 0.45237623], + [0.9688112, 0.9351710, 0.8730961, 0.7546601, 0.56622448], + [0.9743227, 0.9491953, 0.9005150, 0.8086497, 0.64505437], + [0.9807345, 0.9638853, 0.9283012, 0.8631675, 0.73812581], + [0.9886746, 0.9777760, 0.9558950, 0.9123417, 0.82726553], + [0.9899096, 0.9803828, 0.9615592, 0.9255600, 0.85822149], + [0.9969510, 0.9935441, 0.9864657, 0.9726775, 0.94358663], + [0.9979533, 0.9960274, 0.9921724, 0.9837415, 0.96626288], + [0.9995981, 0.9989171, 0.9974178, 0.9949954, 0.99023356], + [1.0002640, 1.0005088, 1.0010594, 1.0021161, 1.00386912], + [0.9998903, 0.9998459, 0.9997795, 0.9995484, 0.99916305], + [1.0000008, 0.9999905, 0.9999481, 0.9998903, 0.99978047], + [1.0000004, 0.9999983, 1.0000001, 1.0000031, 1.00000297], + [0.9999995, 1.0000003, 1.0000005, 1.0000001, 1.00000032], + [0.9999999, 0.9999997, 0.9999994, 0.9999989, 0.99999786], + [0.9999999, 0.9999999, 0.9999999, 0.9999999, 0.99999991]] + # Get iteration points + grad_list = [prob.grad(x) for x in x_list] + delta_x = [np.array(x_list[i+1])-np.array(x_list[i]) + for i in range(len(x_list)-1)] + delta_grad = [grad_list[i+1]-grad_list[i] + for i in range(len(grad_list)-1)] + # Check curvature condition + for s, y in zip(delta_x, delta_grad): + if np.dot(s, y) <= 0: + raise ArithmeticError() + # Define QuasiNewton update + for quasi_newton in (BFGS(init_scale=1, min_curvature=1e-4), + SR1(init_scale=1)): + hess = deepcopy(quasi_newton) + inv_hess = deepcopy(quasi_newton) + hess.initialize(len(x_list[0]), 'hess') + inv_hess.initialize(len(x_list[0]), 'inv_hess') + # Compare the hessian and its inverse + for s, y in zip(delta_x, delta_grad): + hess.update(s, y) + inv_hess.update(s, y) + B = hess.get_matrix() + H = inv_hess.get_matrix() + assert_array_almost_equal(np.linalg.inv(B), H, decimal=10) + B_true = prob.hess(x_list[len(delta_x)]) + assert_array_less(norm(B - B_true)/norm(B_true), 0.1) + + def test_SR1_skip_update(self): + # Define auxiliary problem + prob = Rosenbrock(n=5) + # Define iteration points + x_list = [[0.0976270, 0.4303787, 0.2055267, 0.0897663, -0.15269040], + [0.1847239, 0.0505757, 0.2123832, 0.0255081, 0.00083286], + [0.2142498, -0.0188480, 0.0503822, 0.0347033, 0.03323606], + [0.2071680, -0.0185071, 0.0341337, -0.0139298, 0.02881750], + [0.1533055, -0.0322935, 0.0280418, -0.0083592, 0.01503699], + [0.1382378, -0.0276671, 0.0266161, -0.0074060, 0.02801610], + [0.1651957, -0.0049124, 0.0269665, -0.0040025, 0.02138184], + [0.2354930, 0.0443711, 0.0173959, 0.0041872, 0.00794563], + [0.4168118, 0.1433867, 0.0111714, 0.0126265, -0.00658537], + [0.4681972, 0.2153273, 0.0225249, 0.0152704, -0.00463809], + [0.6023068, 0.3346815, 0.0731108, 0.0186618, -0.00371541], + [0.6415743, 0.3985468, 0.1324422, 0.0214160, -0.00062401], + [0.7503690, 0.5447616, 0.2804541, 0.0539851, 0.00242230], + [0.7452626, 0.5644594, 0.3324679, 0.0865153, 0.00454960], + [0.8059782, 0.6586838, 0.4229577, 0.1452990, 0.00976702], + [0.8549542, 0.7226562, 0.4991309, 0.2420093, 0.02772661], + [0.8571332, 0.7285741, 0.5279076, 0.2824549, 0.06030276], + [0.8835633, 0.7727077, 0.5957984, 0.3411303, 0.09652185], + [0.9071558, 0.8299587, 0.6771400, 0.4402896, 0.17469338]] + # Get iteration points + grad_list = [prob.grad(x) for x in x_list] + delta_x = [np.array(x_list[i+1])-np.array(x_list[i]) + for i in range(len(x_list)-1)] + delta_grad = [grad_list[i+1]-grad_list[i] + for i in range(len(grad_list)-1)] + hess = SR1(init_scale=1, min_denominator=1e-2) + hess.initialize(len(x_list[0]), 'hess') + # Compare the Hessian and its inverse + for i in range(len(delta_x)-1): + s = delta_x[i] + y = delta_grad[i] + hess.update(s, y) + # Test skip update + B = np.copy(hess.get_matrix()) + s = delta_x[17] + y = delta_grad[17] + hess.update(s, y) + B_updated = np.copy(hess.get_matrix()) + assert_array_equal(B, B_updated) + + def test_BFGS_skip_update(self): + # Define auxiliary problem + prob = Rosenbrock(n=5) + # Define iteration points + x_list = [[0.0976270, 0.4303787, 0.2055267, 0.0897663, -0.15269040], + [0.1847239, 0.0505757, 0.2123832, 0.0255081, 0.00083286], + [0.2142498, -0.0188480, 0.0503822, 0.0347033, 0.03323606], + [0.2071680, -0.0185071, 0.0341337, -0.0139298, 0.02881750], + [0.1533055, -0.0322935, 0.0280418, -0.0083592, 0.01503699], + [0.1382378, -0.0276671, 0.0266161, -0.0074060, 0.02801610], + [0.1651957, -0.0049124, 0.0269665, -0.0040025, 0.02138184]] + # Get iteration points + grad_list = [prob.grad(x) for x in x_list] + delta_x = [np.array(x_list[i+1])-np.array(x_list[i]) + for i in range(len(x_list)-1)] + delta_grad = [grad_list[i+1]-grad_list[i] + for i in range(len(grad_list)-1)] + hess = BFGS(init_scale=1, min_curvature=10) + hess.initialize(len(x_list[0]), 'hess') + # Compare the Hessian and its inverse + for i in range(len(delta_x)-1): + s = delta_x[i] + y = delta_grad[i] + hess.update(s, y) + # Test skip update + B = np.copy(hess.get_matrix()) + s = delta_x[5] + y = delta_grad[5] + hess.update(s, y) + B_updated = np.copy(hess.get_matrix()) + assert_array_equal(B, B_updated) + + +@pytest.mark.parametrize('strategy', [BFGS, SR1]) +@pytest.mark.parametrize('approx_type', ['hess', 'inv_hess']) +def test_matmul_equals_dot(strategy, approx_type): + H = strategy(init_scale=1) + H.initialize(2, approx_type) + v = np.array([1, 2]) + assert_array_equal(H @ v, H.dot(v)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_isotonic_regression.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_isotonic_regression.py new file mode 100644 index 0000000000000000000000000000000000000000..b49c56db5b4470c1e4e0f787df52c80eb055c120 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_isotonic_regression.py @@ -0,0 +1,167 @@ +import numpy as np +from numpy.testing import assert_allclose, assert_equal +import pytest + +from scipy.optimize._pava_pybind import pava +from scipy.optimize import isotonic_regression + + +class TestIsotonicRegression: + @pytest.mark.parametrize( + ("y", "w", "msg"), + [ + ([[0, 1]], None, + "array has incorrect number of dimensions: 2; expected 1"), + ([0, 1], [[1, 2]], + "Input arrays y and w must have one dimension of equal length"), + ([0, 1], [1], + "Input arrays y and w must have one dimension of equal length"), + (1, [1, 2], + "Input arrays y and w must have one dimension of equal length"), + ([1, 2], 1, + "Input arrays y and w must have one dimension of equal length"), + ([0, 1], [0, 1], + "Weights w must be strictly positive"), + ] + ) + def test_raise_error(self, y, w, msg): + with pytest.raises(ValueError, match=msg): + isotonic_regression(y=y, weights=w) + + def test_simple_pava(self): + # Test case of Busing 2020 + # https://doi.org/10.18637/jss.v102.c01 + y = np.array([8, 4, 8, 2, 2, 0, 8], dtype=np.float64) + w = np.ones_like(y) + r = np.full(shape=y.shape[0] + 1, fill_value=-1, dtype=np.intp) + pava(y, w, r) + assert_allclose(y, [4, 4, 4, 4, 4, 4, 8]) + # Only first 2 elements of w are changed. + assert_allclose(w, [6, 1, 1, 1, 1, 1, 1]) + # Only first 3 elements of r are changed. + assert_allclose(r, [0, 6, 7, -1, -1, -1, -1, -1]) + + @pytest.mark.parametrize("y_dtype", [np.float64, np.float32, np.int64, np.int32]) + @pytest.mark.parametrize("w_dtype", [np.float64, np.float32, np.int64, np.int32]) + @pytest.mark.parametrize("w", [None, "ones"]) + def test_simple_isotonic_regression(self, w, w_dtype, y_dtype): + # Test case of Busing 2020 + # https://doi.org/10.18637/jss.v102.c01 + y = np.array([8, 4, 8, 2, 2, 0, 8], dtype=y_dtype) + if w is not None: + w = np.ones_like(y, dtype=w_dtype) + res = isotonic_regression(y, weights=w) + assert res.x.dtype == np.float64 + assert res.weights.dtype == np.float64 + assert_allclose(res.x, [4, 4, 4, 4, 4, 4, 8]) + assert_allclose(res.weights, [6, 1]) + assert_allclose(res.blocks, [0, 6, 7]) + # Assert that y was not overwritten + assert_equal(y, np.array([8, 4, 8, 2, 2, 0, 8], dtype=np.float64)) + + @pytest.mark.parametrize("increasing", [True, False]) + def test_linspace(self, increasing): + n = 10 + y = np.linspace(0, 1, n) if increasing else np.linspace(1, 0, n) + res = isotonic_regression(y, increasing=increasing) + assert_allclose(res.x, y) + assert_allclose(res.blocks, np.arange(n + 1)) + + def test_weights(self): + w = np.array([1, 2, 5, 0.5, 0.5, 0.5, 1, 3]) + y = np.array([3, 2, 1, 10, 9, 8, 20, 10]) + res = isotonic_regression(y, weights=w) + assert_allclose(res.x, [12/8, 12/8, 12/8, 9, 9, 9, 50/4, 50/4]) + assert_allclose(res.weights, [8, 1.5, 4]) + assert_allclose(res.blocks, [0, 3, 6, 8]) + + # weights are like repeated observations, we repeat the 3rd element 5 + # times. + w2 = np.array([1, 2, 1, 1, 1, 1, 1, 0.5, 0.5, 0.5, 1, 3]) + y2 = np.array([3, 2, 1, 1, 1, 1, 1, 10, 9, 8, 20, 10]) + res2 = isotonic_regression(y2, weights=w2) + assert_allclose(np.diff(res2.x[0:7]), 0) + assert_allclose(res2.x[4:], res.x) + assert_allclose(res2.weights, res.weights) + assert_allclose(res2.blocks[1:] - 4, res.blocks[1:]) + + def test_against_R_monotone(self): + y = [0, 6, 8, 3, 5, 2, 1, 7, 9, 4] + res = isotonic_regression(y) + # R code + # library(monotone) + # options(digits=8) + # monotone(c(0, 6, 8, 3, 5, 2, 1, 7, 9, 4)) + x_R = [ + 0, 4.1666667, 4.1666667, 4.1666667, 4.1666667, 4.1666667, + 4.1666667, 6.6666667, 6.6666667, 6.6666667, + ] + assert_allclose(res.x, x_R) + assert_equal(res.blocks, [0, 1, 7, 10]) + + n = 100 + y = np.linspace(0, 1, num=n, endpoint=False) + y = 5 * y + np.sin(10 * y) + res = isotonic_regression(y) + # R code + # library(monotone) + # n <- 100 + # y <- 5 * ((1:n)-1)/n + sin(10 * ((1:n)-1)/n) + # options(digits=8) + # monotone(y) + x_R = [ + 0.00000000, 0.14983342, 0.29866933, 0.44552021, 0.58941834, 0.72942554, + 0.86464247, 0.99421769, 1.11735609, 1.23332691, 1.34147098, 1.44120736, + 1.53203909, 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, + 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, + 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, + 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, + 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, + 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, 1.57081100, + 1.57081100, 1.57081100, 1.57081100, 1.62418532, 1.71654534, 1.81773256, + 1.92723551, 2.04445967, 2.16873336, 2.29931446, 2.43539782, 2.57612334, + 2.72058450, 2.86783750, 3.01691060, 3.16681390, 3.31654920, 3.46511999, + 3.61154136, 3.75484992, 3.89411335, 4.02843976, 4.15698660, 4.27896904, + 4.39366786, 4.50043662, 4.59870810, 4.68799998, 4.76791967, 4.83816823, + 4.86564130, 4.86564130, 4.86564130, 4.86564130, 4.86564130, 4.86564130, + 4.86564130, 4.86564130, 4.86564130, 4.86564130, 4.86564130, 4.86564130, + 4.86564130, 4.86564130, 4.86564130, 4.86564130, 4.86564130, 4.86564130, + 4.86564130, 4.86564130, 4.86564130, 4.86564130, + ] + assert_allclose(res.x, x_R) + + # Test increasing + assert np.all(np.diff(res.x) >= 0) + + # Test balance property: sum(y) == sum(x) + assert_allclose(np.sum(res.x), np.sum(y)) + + # Reverse order + res_inv = isotonic_regression(-y, increasing=False) + assert_allclose(-res_inv.x, res.x) + assert_equal(res_inv.blocks, res.blocks) + + def test_readonly(self): + x = np.arange(3, dtype=float) + w = np.ones(3, dtype=float) + + x.flags.writeable = False + w.flags.writeable = False + + res = isotonic_regression(x, weights=w) + assert np.all(np.isfinite(res.x)) + assert np.all(np.isfinite(res.weights)) + assert np.all(np.isfinite(res.blocks)) + + def test_non_contiguous_arrays(self): + x = np.arange(10, dtype=float)[::3] + w = np.ones(10, dtype=float)[::3] + assert not x.flags.c_contiguous + assert not x.flags.f_contiguous + assert not w.flags.c_contiguous + assert not w.flags.f_contiguous + + res = isotonic_regression(x, weights=w) + assert np.all(np.isfinite(res.x)) + assert np.all(np.isfinite(res.weights)) + assert np.all(np.isfinite(res.blocks)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lbfgsb_hessinv.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lbfgsb_hessinv.py new file mode 100644 index 0000000000000000000000000000000000000000..8e4452cd61c5400c13f4f239055352bae754ad7e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lbfgsb_hessinv.py @@ -0,0 +1,43 @@ +import numpy as np +from numpy.testing import assert_allclose +import scipy.linalg +from scipy.optimize import minimize + + +def test_1(): + def f(x): + return x**4, 4*x**3 + + for gtol in [1e-8, 1e-12, 1e-20]: + for maxcor in range(20, 35): + result = minimize(fun=f, jac=True, method='L-BFGS-B', x0=20, + options={'gtol': gtol, 'maxcor': maxcor}) + + H1 = result.hess_inv(np.array([1])).reshape(1,1) + H2 = result.hess_inv.todense() + + assert_allclose(H1, H2) + + +def test_2(): + H0 = [[3, 0], [1, 2]] + + def f(x): + return np.dot(x, np.dot(scipy.linalg.inv(H0), x)) + + result1 = minimize(fun=f, method='L-BFGS-B', x0=[10, 20]) + result2 = minimize(fun=f, method='BFGS', x0=[10, 20]) + + H1 = result1.hess_inv.todense() + + H2 = np.vstack(( + result1.hess_inv(np.array([1, 0])), + result1.hess_inv(np.array([0, 1])))) + + assert_allclose( + result1.hess_inv(np.array([1, 0]).reshape(2,1)).reshape(-1), + result1.hess_inv(np.array([1, 0]))) + assert_allclose(H1, H2) + assert_allclose(H1, result2.hess_inv, rtol=1e-2, atol=0.03) + + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lbfgsb_setulb.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lbfgsb_setulb.py new file mode 100644 index 0000000000000000000000000000000000000000..ee47f45509c86968b9b551eb8740ad301fd37958 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lbfgsb_setulb.py @@ -0,0 +1,122 @@ +import numpy as np +from scipy.optimize import _lbfgsb, minimize + + +def objfun(x): + """simplified objective func to test lbfgsb bound violation""" + x0 = [0.8750000000000278, + 0.7500000000000153, + 0.9499999999999722, + 0.8214285714285992, + 0.6363636363636085] + x1 = [1.0, 0.0, 1.0, 0.0, 0.0] + x2 = [1.0, + 0.0, + 0.9889733043149325, + 0.0, + 0.026353554421041155] + x3 = [1.0, + 0.0, + 0.9889917442915558, + 0.0, + 0.020341986743231205] + + f0 = 5163.647901211178 + f1 = 5149.8181642072905 + f2 = 5149.379332309634 + f3 = 5149.374490771297 + + g0 = np.array([-0.5934820547965749, + 1.6251549718258351, + -71.99168459202559, + 5.346636965797545, + 37.10732723092604]) + g1 = np.array([-0.43295349282641515, + 1.008607936794592, + 18.223666726602975, + 31.927010036981997, + -19.667512518739386]) + g2 = np.array([-0.4699874455100256, + 0.9466285353668347, + -0.016874360242016825, + 48.44999161133457, + 5.819631620590712]) + g3 = np.array([-0.46970678696829116, + 0.9612719312174818, + 0.006129809488833699, + 48.43557729419473, + 6.005481418498221]) + + if np.allclose(x, x0): + f = f0 + g = g0 + elif np.allclose(x, x1): + f = f1 + g = g1 + elif np.allclose(x, x2): + f = f2 + g = g2 + elif np.allclose(x, x3): + f = f3 + g = g3 + else: + raise ValueError( + 'Simplified objective function not defined ' + 'at requested point') + return (np.copy(f), np.copy(g)) + + +def test_setulb_floatround(): + """test if setulb() violates bounds + + checks for violation due to floating point rounding error + """ + + n = 5 + m = 10 + factr = 1e7 + pgtol = 1e-5 + maxls = 20 + nbd = np.full(shape=(n,), fill_value=2, dtype=np.int32) + low_bnd = np.zeros(n, dtype=np.float64) + upper_bnd = np.ones(n, dtype=np.float64) + + x0 = np.array( + [0.8750000000000278, + 0.7500000000000153, + 0.9499999999999722, + 0.8214285714285992, + 0.6363636363636085]) + x = np.copy(x0) + + f = np.array(0.0, dtype=np.float64) + g = np.zeros(n, dtype=np.float64) + + wa = np.zeros(2*m*n + 5*n + 11*m*m + 8*m, dtype=np.float64) + iwa = np.zeros(3*n, dtype=np.int32) + task = np.zeros(2, dtype=np.int32) + ln_task = np.zeros(2, dtype=np.int32) + lsave = np.zeros(4, dtype=np.int32) + isave = np.zeros(44, dtype=np.int32) + dsave = np.zeros(29, dtype=np.float64) + + for n_iter in range(7): # 7 steps required to reproduce error + f, g = objfun(x) + + _lbfgsb.setulb(m, x, low_bnd, upper_bnd, nbd, f, g, factr, pgtol, wa, + iwa, task, lsave, isave, dsave, maxls, ln_task) + + assert (x <= upper_bnd).all() and (x >= low_bnd).all(), ( + "_lbfgsb.setulb() stepped to a point outside of the bounds") + + +def test_gh_issue18730(): + # issue 18730 reported that l-bfgs-b did not work with objectives + # returning single precision gradient arrays + def fun_single_precision(x): + x = x.astype(np.float32) + return np.sum(x**2), (2*x) + + res = minimize(fun_single_precision, x0=np.array([1., 1.]), jac=True, + method="l-bfgs-b") + np.testing.assert_allclose(res.fun, 0., atol=1e-15) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_least_squares.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_least_squares.py new file mode 100644 index 0000000000000000000000000000000000000000..d27d670a1aac01f7129d483e0a048a38dce35404 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_least_squares.py @@ -0,0 +1,874 @@ +from itertools import product + +import numpy as np +from numpy.linalg import norm +from numpy.testing import (assert_, assert_allclose, + assert_equal, suppress_warnings) +import pytest +from pytest import raises as assert_raises +from scipy.sparse import issparse, lil_matrix +from scipy.sparse.linalg import aslinearoperator + +from scipy.optimize import least_squares, Bounds +from scipy.optimize._lsq.least_squares import IMPLEMENTED_LOSSES +from scipy.optimize._lsq.common import EPS, make_strictly_feasible, CL_scaling_vector + + +def fun_trivial(x, a=0): + return (x - a)**2 + 5.0 + + +def jac_trivial(x, a=0.0): + return 2 * (x - a) + + +def fun_2d_trivial(x): + return np.array([x[0], x[1]]) + + +def jac_2d_trivial(x): + return np.identity(2) + + +def fun_rosenbrock(x): + return np.array([10 * (x[1] - x[0]**2), (1 - x[0])]) + + +def jac_rosenbrock(x): + return np.array([ + [-20 * x[0], 10], + [-1, 0] + ]) + + +def jac_rosenbrock_bad_dim(x): + return np.array([ + [-20 * x[0], 10], + [-1, 0], + [0.0, 0.0] + ]) + + +def fun_rosenbrock_cropped(x): + return fun_rosenbrock(x)[0] + + +def jac_rosenbrock_cropped(x): + return jac_rosenbrock(x)[0] + + +# When x is 1-D array, return is 2-D array. +def fun_wrong_dimensions(x): + return np.array([x, x**2, x**3]) + + +def jac_wrong_dimensions(x, a=0.0): + return np.atleast_3d(jac_trivial(x, a=a)) + + +def fun_bvp(x): + n = int(np.sqrt(x.shape[0])) + u = np.zeros((n + 2, n + 2)) + x = x.reshape((n, n)) + u[1:-1, 1:-1] = x + y = u[:-2, 1:-1] + u[2:, 1:-1] + u[1:-1, :-2] + u[1:-1, 2:] - 4 * x + x**3 + return y.ravel() + + +class BroydenTridiagonal: + def __init__(self, n=100, mode='sparse'): + rng = np.random.RandomState(0) + + self.n = n + + self.x0 = -np.ones(n) + self.lb = np.linspace(-2, -1.5, n) + self.ub = np.linspace(-0.8, 0.0, n) + + self.lb += 0.1 * rng.randn(n) + self.ub += 0.1 * rng.randn(n) + + self.x0 += 0.1 * rng.randn(n) + self.x0 = make_strictly_feasible(self.x0, self.lb, self.ub) + + if mode == 'sparse': + self.sparsity = lil_matrix((n, n), dtype=int) + i = np.arange(n) + self.sparsity[i, i] = 1 + i = np.arange(1, n) + self.sparsity[i, i - 1] = 1 + i = np.arange(n - 1) + self.sparsity[i, i + 1] = 1 + + self.jac = self._jac + elif mode == 'operator': + self.jac = lambda x: aslinearoperator(self._jac(x)) + elif mode == 'dense': + self.sparsity = None + self.jac = lambda x: self._jac(x).toarray() + else: + assert_(False) + + def fun(self, x): + f = (3 - x) * x + 1 + f[1:] -= x[:-1] + f[:-1] -= 2 * x[1:] + return f + + def _jac(self, x): + J = lil_matrix((self.n, self.n)) + i = np.arange(self.n) + J[i, i] = 3 - 2 * x + i = np.arange(1, self.n) + J[i, i - 1] = -1 + i = np.arange(self.n - 1) + J[i, i + 1] = -2 + return J + + +class ExponentialFittingProblem: + """Provide data and function for exponential fitting in the form + y = a + exp(b * x) + noise.""" + + def __init__(self, a, b, noise, n_outliers=1, x_range=(-1, 1), + n_points=11, random_seed=None): + rng = np.random.RandomState(random_seed) + self.m = n_points + self.n = 2 + + self.p0 = np.zeros(2) + self.x = np.linspace(x_range[0], x_range[1], n_points) + + self.y = a + np.exp(b * self.x) + self.y += noise * rng.randn(self.m) + + outliers = rng.randint(0, self.m, n_outliers) + self.y[outliers] += 50 * noise * rng.rand(n_outliers) + + self.p_opt = np.array([a, b]) + + def fun(self, p): + return p[0] + np.exp(p[1] * self.x) - self.y + + def jac(self, p): + J = np.empty((self.m, self.n)) + J[:, 0] = 1 + J[:, 1] = self.x * np.exp(p[1] * self.x) + return J + + +def cubic_soft_l1(z): + rho = np.empty((3, z.size)) + + t = 1 + z + rho[0] = 3 * (t**(1/3) - 1) + rho[1] = t ** (-2/3) + rho[2] = -2/3 * t**(-5/3) + + return rho + + +LOSSES = list(IMPLEMENTED_LOSSES.keys()) + [cubic_soft_l1] + + +class BaseMixin: + def test_basic(self): + # Test that the basic calling sequence works. + res = least_squares(fun_trivial, 2., method=self.method) + assert_allclose(res.x, 0, atol=1e-4) + assert_allclose(res.fun, fun_trivial(res.x)) + + def test_args_kwargs(self): + # Test that args and kwargs are passed correctly to the functions. + a = 3.0 + for jac in ['2-point', '3-point', 'cs', jac_trivial]: + with suppress_warnings() as sup: + sup.filter( + UserWarning, + "jac='(3-point|cs)' works equivalently to '2-point' for method='lm'" + ) + res = least_squares(fun_trivial, 2.0, jac, args=(a,), + method=self.method) + res1 = least_squares(fun_trivial, 2.0, jac, kwargs={'a': a}, + method=self.method) + + assert_allclose(res.x, a, rtol=1e-4) + assert_allclose(res1.x, a, rtol=1e-4) + + assert_raises(TypeError, least_squares, fun_trivial, 2.0, + args=(3, 4,), method=self.method) + assert_raises(TypeError, least_squares, fun_trivial, 2.0, + kwargs={'kaboom': 3}, method=self.method) + + def test_jac_options(self): + for jac in ['2-point', '3-point', 'cs', jac_trivial]: + with suppress_warnings() as sup: + sup.filter( + UserWarning, + "jac='(3-point|cs)' works equivalently to '2-point' for method='lm'" + ) + res = least_squares(fun_trivial, 2.0, jac, method=self.method) + assert_allclose(res.x, 0, atol=1e-4) + + assert_raises(ValueError, least_squares, fun_trivial, 2.0, jac='oops', + method=self.method) + + def test_nfev_options(self): + for max_nfev in [None, 20]: + res = least_squares(fun_trivial, 2.0, max_nfev=max_nfev, + method=self.method) + assert_allclose(res.x, 0, atol=1e-4) + + def test_x_scale_options(self): + for x_scale in [1.0, np.array([0.5]), 'jac']: + res = least_squares(fun_trivial, 2.0, x_scale=x_scale) + assert_allclose(res.x, 0) + assert_raises(ValueError, least_squares, fun_trivial, + 2.0, x_scale='auto', method=self.method) + assert_raises(ValueError, least_squares, fun_trivial, + 2.0, x_scale=-1.0, method=self.method) + assert_raises(ValueError, least_squares, fun_trivial, + 2.0, x_scale=None, method=self.method) + assert_raises(ValueError, least_squares, fun_trivial, + 2.0, x_scale=1.0+2.0j, method=self.method) + + def test_diff_step(self): + # res1 and res2 should be equivalent. + # res2 and res3 should be different. + res1 = least_squares(fun_trivial, 2.0, diff_step=1e-1, + method=self.method) + res2 = least_squares(fun_trivial, 2.0, diff_step=-1e-1, + method=self.method) + res3 = least_squares(fun_trivial, 2.0, + diff_step=None, method=self.method) + assert_allclose(res1.x, 0, atol=1e-4) + assert_allclose(res2.x, 0, atol=1e-4) + assert_allclose(res3.x, 0, atol=1e-4) + assert_equal(res1.x, res2.x) + assert_equal(res1.nfev, res2.nfev) + + def test_incorrect_options_usage(self): + assert_raises(TypeError, least_squares, fun_trivial, 2.0, + method=self.method, options={'no_such_option': 100}) + assert_raises(TypeError, least_squares, fun_trivial, 2.0, + method=self.method, options={'max_nfev': 100}) + + def test_full_result(self): + # MINPACK doesn't work very well with factor=100 on this problem, + # thus using low 'atol'. + res = least_squares(fun_trivial, 2.0, method=self.method) + assert_allclose(res.x, 0, atol=1e-4) + assert_allclose(res.cost, 12.5) + assert_allclose(res.fun, 5) + assert_allclose(res.jac, 0, atol=1e-4) + assert_allclose(res.grad, 0, atol=1e-2) + assert_allclose(res.optimality, 0, atol=1e-2) + assert_equal(res.active_mask, 0) + if self.method == 'lm': + assert_(res.nfev < 30) + assert_(res.njev is None) + else: + assert_(res.nfev < 10) + assert_(res.njev < 10) + assert_(res.status > 0) + assert_(res.success) + + def test_full_result_single_fev(self): + # MINPACK checks the number of nfev after the iteration, + # so it's hard to tell what he is going to compute. + if self.method == 'lm': + return + + res = least_squares(fun_trivial, 2.0, method=self.method, + max_nfev=1) + assert_equal(res.x, np.array([2])) + assert_equal(res.cost, 40.5) + assert_equal(res.fun, np.array([9])) + assert_equal(res.jac, np.array([[4]])) + assert_equal(res.grad, np.array([36])) + assert_equal(res.optimality, 36) + assert_equal(res.active_mask, np.array([0])) + assert_equal(res.nfev, 1) + assert_equal(res.njev, 1) + assert_equal(res.status, 0) + assert_equal(res.success, 0) + + def test_rosenbrock(self): + x0 = [-2, 1] + x_opt = [1, 1] + for jac, x_scale, tr_solver in product( + ['2-point', '3-point', 'cs', jac_rosenbrock], + [1.0, np.array([1.0, 0.2]), 'jac'], + ['exact', 'lsmr']): + with suppress_warnings() as sup: + sup.filter( + UserWarning, + "jac='(3-point|cs)' works equivalently to '2-point' for method='lm'" + ) + res = least_squares(fun_rosenbrock, x0, jac, x_scale=x_scale, + tr_solver=tr_solver, method=self.method) + assert_allclose(res.x, x_opt) + + def test_rosenbrock_cropped(self): + x0 = [-2, 1] + if self.method == 'lm': + assert_raises(ValueError, least_squares, fun_rosenbrock_cropped, + x0, method='lm') + else: + for jac, x_scale, tr_solver in product( + ['2-point', '3-point', 'cs', jac_rosenbrock_cropped], + [1.0, np.array([1.0, 0.2]), 'jac'], + ['exact', 'lsmr']): + res = least_squares( + fun_rosenbrock_cropped, x0, jac, x_scale=x_scale, + tr_solver=tr_solver, method=self.method) + assert_allclose(res.cost, 0, atol=1e-14) + + def test_fun_wrong_dimensions(self): + assert_raises(ValueError, least_squares, fun_wrong_dimensions, + 2.0, method=self.method) + + def test_jac_wrong_dimensions(self): + assert_raises(ValueError, least_squares, fun_trivial, + 2.0, jac_wrong_dimensions, method=self.method) + + def test_fun_and_jac_inconsistent_dimensions(self): + x0 = [1, 2] + assert_raises(ValueError, least_squares, fun_rosenbrock, x0, + jac_rosenbrock_bad_dim, method=self.method) + + def test_x0_multidimensional(self): + x0 = np.ones(4).reshape(2, 2) + assert_raises(ValueError, least_squares, fun_trivial, x0, + method=self.method) + + def test_x0_complex_scalar(self): + x0 = 2.0 + 0.0*1j + assert_raises(ValueError, least_squares, fun_trivial, x0, + method=self.method) + + def test_x0_complex_array(self): + x0 = [1.0, 2.0 + 0.0*1j] + assert_raises(ValueError, least_squares, fun_trivial, x0, + method=self.method) + + def test_bvp(self): + # This test was introduced with fix #5556. It turned out that + # dogbox solver had a bug with trust-region radius update, which + # could block its progress and create an infinite loop. And this + # discrete boundary value problem is the one which triggers it. + n = 10 + x0 = np.ones(n**2) + if self.method == 'lm': + max_nfev = 5000 # To account for Jacobian estimation. + else: + max_nfev = 100 + res = least_squares(fun_bvp, x0, ftol=1e-2, method=self.method, + max_nfev=max_nfev) + + assert_(res.nfev < max_nfev) + assert_(res.cost < 0.5) + + def test_error_raised_when_all_tolerances_below_eps(self): + # Test that all 0 tolerances are not allowed. + assert_raises(ValueError, least_squares, fun_trivial, 2.0, + method=self.method, ftol=None, xtol=None, gtol=None) + + def test_convergence_with_only_one_tolerance_enabled(self): + if self.method == 'lm': + return # should not do test + x0 = [-2, 1] + x_opt = [1, 1] + for ftol, xtol, gtol in [(1e-8, None, None), + (None, 1e-8, None), + (None, None, 1e-8)]: + res = least_squares(fun_rosenbrock, x0, jac=jac_rosenbrock, + ftol=ftol, gtol=gtol, xtol=xtol, + method=self.method) + assert_allclose(res.x, x_opt) + + +class BoundsMixin: + def test_inconsistent(self): + assert_raises(ValueError, least_squares, fun_trivial, 2.0, + bounds=(10.0, 0.0), method=self.method) + + def test_infeasible(self): + assert_raises(ValueError, least_squares, fun_trivial, 2.0, + bounds=(3., 4), method=self.method) + + def test_wrong_number(self): + assert_raises(ValueError, least_squares, fun_trivial, 2., + bounds=(1., 2, 3), method=self.method) + + def test_inconsistent_shape(self): + assert_raises(ValueError, least_squares, fun_trivial, 2.0, + bounds=(1.0, [2.0, 3.0]), method=self.method) + # 1-D array won't be broadcast + assert_raises(ValueError, least_squares, fun_rosenbrock, [1.0, 2.0], + bounds=([0.0], [3.0, 4.0]), method=self.method) + + def test_in_bounds(self): + for jac in ['2-point', '3-point', 'cs', jac_trivial]: + res = least_squares(fun_trivial, 2.0, jac=jac, + bounds=(-1.0, 3.0), method=self.method) + assert_allclose(res.x, 0.0, atol=1e-4) + assert_equal(res.active_mask, [0]) + assert_(-1 <= res.x <= 3) + res = least_squares(fun_trivial, 2.0, jac=jac, + bounds=(0.5, 3.0), method=self.method) + assert_allclose(res.x, 0.5, atol=1e-4) + assert_equal(res.active_mask, [-1]) + assert_(0.5 <= res.x <= 3) + + def test_bounds_shape(self): + def get_bounds_direct(lb, ub): + return lb, ub + + def get_bounds_instances(lb, ub): + return Bounds(lb, ub) + + for jac in ['2-point', '3-point', 'cs', jac_2d_trivial]: + for bounds_func in [get_bounds_direct, get_bounds_instances]: + x0 = [1.0, 1.0] + res = least_squares(fun_2d_trivial, x0, jac=jac) + assert_allclose(res.x, [0.0, 0.0]) + res = least_squares(fun_2d_trivial, x0, jac=jac, + bounds=bounds_func(0.5, [2.0, 2.0]), + method=self.method) + assert_allclose(res.x, [0.5, 0.5]) + res = least_squares(fun_2d_trivial, x0, jac=jac, + bounds=bounds_func([0.3, 0.2], 3.0), + method=self.method) + assert_allclose(res.x, [0.3, 0.2]) + res = least_squares( + fun_2d_trivial, x0, jac=jac, + bounds=bounds_func([-1, 0.5], [1.0, 3.0]), + method=self.method) + assert_allclose(res.x, [0.0, 0.5], atol=1e-5) + + def test_bounds_instances(self): + res = least_squares(fun_trivial, 0.5, bounds=Bounds()) + assert_allclose(res.x, 0.0, atol=1e-4) + + res = least_squares(fun_trivial, 3.0, bounds=Bounds(lb=1.0)) + assert_allclose(res.x, 1.0, atol=1e-4) + + res = least_squares(fun_trivial, 0.5, bounds=Bounds(lb=-1.0, ub=1.0)) + assert_allclose(res.x, 0.0, atol=1e-4) + + res = least_squares(fun_trivial, -3.0, bounds=Bounds(ub=-1.0)) + assert_allclose(res.x, -1.0, atol=1e-4) + + res = least_squares(fun_2d_trivial, [0.5, 0.5], + bounds=Bounds(lb=[-1.0, -1.0], ub=1.0)) + assert_allclose(res.x, [0.0, 0.0], atol=1e-5) + + res = least_squares(fun_2d_trivial, [0.5, 0.5], + bounds=Bounds(lb=[0.1, 0.1])) + assert_allclose(res.x, [0.1, 0.1], atol=1e-5) + + @pytest.mark.fail_slow(10) + def test_rosenbrock_bounds(self): + x0_1 = np.array([-2.0, 1.0]) + x0_2 = np.array([2.0, 2.0]) + x0_3 = np.array([-2.0, 2.0]) + x0_4 = np.array([0.0, 2.0]) + x0_5 = np.array([-1.2, 1.0]) + problems = [ + (x0_1, ([-np.inf, -1.5], np.inf)), + (x0_2, ([-np.inf, 1.5], np.inf)), + (x0_3, ([-np.inf, 1.5], np.inf)), + (x0_4, ([-np.inf, 1.5], [1.0, np.inf])), + (x0_2, ([1.0, 1.5], [3.0, 3.0])), + (x0_5, ([-50.0, 0.0], [0.5, 100])) + ] + for x0, bounds in problems: + for jac, x_scale, tr_solver in product( + ['2-point', '3-point', 'cs', jac_rosenbrock], + [1.0, [1.0, 0.5], 'jac'], + ['exact', 'lsmr']): + res = least_squares(fun_rosenbrock, x0, jac, bounds, + x_scale=x_scale, tr_solver=tr_solver, + method=self.method) + assert_allclose(res.optimality, 0.0, atol=1e-5) + + +class SparseMixin: + def test_exact_tr_solver(self): + p = BroydenTridiagonal() + assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac, + tr_solver='exact', method=self.method) + assert_raises(ValueError, least_squares, p.fun, p.x0, + tr_solver='exact', jac_sparsity=p.sparsity, + method=self.method) + + def test_equivalence(self): + sparse = BroydenTridiagonal(mode='sparse') + dense = BroydenTridiagonal(mode='dense') + res_sparse = least_squares( + sparse.fun, sparse.x0, jac=sparse.jac, + method=self.method) + res_dense = least_squares( + dense.fun, dense.x0, jac=sparse.jac, + method=self.method) + assert_equal(res_sparse.nfev, res_dense.nfev) + assert_allclose(res_sparse.x, res_dense.x, atol=1e-20) + assert_allclose(res_sparse.cost, 0, atol=1e-20) + assert_allclose(res_dense.cost, 0, atol=1e-20) + + def test_tr_options(self): + p = BroydenTridiagonal() + res = least_squares(p.fun, p.x0, p.jac, method=self.method, + tr_options={'btol': 1e-10}) + assert_allclose(res.cost, 0, atol=1e-20) + + def test_wrong_parameters(self): + p = BroydenTridiagonal() + assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac, + tr_solver='best', method=self.method) + assert_raises(TypeError, least_squares, p.fun, p.x0, p.jac, + tr_solver='lsmr', tr_options={'tol': 1e-10}) + + def test_solver_selection(self): + sparse = BroydenTridiagonal(mode='sparse') + dense = BroydenTridiagonal(mode='dense') + res_sparse = least_squares(sparse.fun, sparse.x0, jac=sparse.jac, + method=self.method) + res_dense = least_squares(dense.fun, dense.x0, jac=dense.jac, + method=self.method) + assert_allclose(res_sparse.cost, 0, atol=1e-20) + assert_allclose(res_dense.cost, 0, atol=1e-20) + assert_(issparse(res_sparse.jac)) + assert_(isinstance(res_dense.jac, np.ndarray)) + + def test_numerical_jac(self): + p = BroydenTridiagonal() + for jac in ['2-point', '3-point', 'cs']: + res_dense = least_squares(p.fun, p.x0, jac, method=self.method) + res_sparse = least_squares( + p.fun, p.x0, jac,method=self.method, + jac_sparsity=p.sparsity) + assert_equal(res_dense.nfev, res_sparse.nfev) + assert_allclose(res_dense.x, res_sparse.x, atol=1e-20) + assert_allclose(res_dense.cost, 0, atol=1e-20) + assert_allclose(res_sparse.cost, 0, atol=1e-20) + + @pytest.mark.fail_slow(10) + def test_with_bounds(self): + p = BroydenTridiagonal() + for jac, jac_sparsity in product( + [p.jac, '2-point', '3-point', 'cs'], [None, p.sparsity]): + res_1 = least_squares( + p.fun, p.x0, jac, bounds=(p.lb, np.inf), + method=self.method,jac_sparsity=jac_sparsity) + res_2 = least_squares( + p.fun, p.x0, jac, bounds=(-np.inf, p.ub), + method=self.method, jac_sparsity=jac_sparsity) + res_3 = least_squares( + p.fun, p.x0, jac, bounds=(p.lb, p.ub), + method=self.method, jac_sparsity=jac_sparsity) + assert_allclose(res_1.optimality, 0, atol=1e-10) + assert_allclose(res_2.optimality, 0, atol=1e-10) + assert_allclose(res_3.optimality, 0, atol=1e-10) + + def test_wrong_jac_sparsity(self): + p = BroydenTridiagonal() + sparsity = p.sparsity[:-1] + assert_raises(ValueError, least_squares, p.fun, p.x0, + jac_sparsity=sparsity, method=self.method) + + def test_linear_operator(self): + p = BroydenTridiagonal(mode='operator') + res = least_squares(p.fun, p.x0, p.jac, method=self.method) + assert_allclose(res.cost, 0.0, atol=1e-20) + assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac, + method=self.method, tr_solver='exact') + + def test_x_scale_jac_scale(self): + p = BroydenTridiagonal() + res = least_squares(p.fun, p.x0, p.jac, method=self.method, + x_scale='jac') + assert_allclose(res.cost, 0.0, atol=1e-20) + + p = BroydenTridiagonal(mode='operator') + assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac, + method=self.method, x_scale='jac') + + +class LossFunctionMixin: + def test_options(self): + for loss in LOSSES: + res = least_squares(fun_trivial, 2.0, loss=loss, + method=self.method) + assert_allclose(res.x, 0, atol=1e-15) + + assert_raises(ValueError, least_squares, fun_trivial, 2.0, + loss='hinge', method=self.method) + + def test_fun(self): + # Test that res.fun is actual residuals, and not modified by loss + # function stuff. + for loss in LOSSES: + res = least_squares(fun_trivial, 2.0, loss=loss, + method=self.method) + assert_equal(res.fun, fun_trivial(res.x)) + + def test_grad(self): + # Test that res.grad is true gradient of loss function at the + # solution. Use max_nfev = 1, to avoid reaching minimum. + x = np.array([2.0]) # res.x will be this. + + res = least_squares(fun_trivial, x, jac_trivial, loss='linear', + max_nfev=1, method=self.method) + assert_equal(res.grad, 2 * x * (x**2 + 5)) + + res = least_squares(fun_trivial, x, jac_trivial, loss='huber', + max_nfev=1, method=self.method) + assert_equal(res.grad, 2 * x) + + res = least_squares(fun_trivial, x, jac_trivial, loss='soft_l1', + max_nfev=1, method=self.method) + assert_allclose(res.grad, + 2 * x * (x**2 + 5) / (1 + (x**2 + 5)**2)**0.5) + + res = least_squares(fun_trivial, x, jac_trivial, loss='cauchy', + max_nfev=1, method=self.method) + assert_allclose(res.grad, 2 * x * (x**2 + 5) / (1 + (x**2 + 5)**2)) + + res = least_squares(fun_trivial, x, jac_trivial, loss='arctan', + max_nfev=1, method=self.method) + assert_allclose(res.grad, 2 * x * (x**2 + 5) / (1 + (x**2 + 5)**4)) + + res = least_squares(fun_trivial, x, jac_trivial, loss=cubic_soft_l1, + max_nfev=1, method=self.method) + assert_allclose(res.grad, + 2 * x * (x**2 + 5) / (1 + (x**2 + 5)**2)**(2/3)) + + def test_jac(self): + # Test that res.jac.T.dot(res.jac) gives Gauss-Newton approximation + # of Hessian. This approximation is computed by doubly differentiating + # the cost function and dropping the part containing second derivative + # of f. For a scalar function it is computed as + # H = (rho' + 2 * rho'' * f**2) * f'**2, if the expression inside the + # brackets is less than EPS it is replaced by EPS. Here, we check + # against the root of H. + + x = 2.0 # res.x will be this. + f = x**2 + 5 # res.fun will be this. + + res = least_squares(fun_trivial, x, jac_trivial, loss='linear', + max_nfev=1, method=self.method) + assert_equal(res.jac, 2 * x) + + # For `huber` loss the Jacobian correction is identically zero + # in outlier region, in such cases it is modified to be equal EPS**0.5. + res = least_squares(fun_trivial, x, jac_trivial, loss='huber', + max_nfev=1, method=self.method) + assert_equal(res.jac, 2 * x * EPS**0.5) + + # Now, let's apply `loss_scale` to turn the residual into an inlier. + # The loss function becomes linear. + res = least_squares(fun_trivial, x, jac_trivial, loss='huber', + f_scale=10, max_nfev=1) + assert_equal(res.jac, 2 * x) + + # 'soft_l1' always gives a positive scaling. + res = least_squares(fun_trivial, x, jac_trivial, loss='soft_l1', + max_nfev=1, method=self.method) + assert_allclose(res.jac, 2 * x * (1 + f**2)**-0.75) + + # For 'cauchy' the correction term turns out to be negative, and it + # replaced by EPS**0.5. + res = least_squares(fun_trivial, x, jac_trivial, loss='cauchy', + max_nfev=1, method=self.method) + assert_allclose(res.jac, 2 * x * EPS**0.5) + + # Now use scaling to turn the residual to inlier. + res = least_squares(fun_trivial, x, jac_trivial, loss='cauchy', + f_scale=10, max_nfev=1, method=self.method) + fs = f / 10 + assert_allclose(res.jac, 2 * x * (1 - fs**2)**0.5 / (1 + fs**2)) + + # 'arctan' gives an outlier. + res = least_squares(fun_trivial, x, jac_trivial, loss='arctan', + max_nfev=1, method=self.method) + assert_allclose(res.jac, 2 * x * EPS**0.5) + + # Turn to inlier. + res = least_squares(fun_trivial, x, jac_trivial, loss='arctan', + f_scale=20.0, max_nfev=1, method=self.method) + fs = f / 20 + assert_allclose(res.jac, 2 * x * (1 - 3 * fs**4)**0.5 / (1 + fs**4)) + + # cubic_soft_l1 will give an outlier. + res = least_squares(fun_trivial, x, jac_trivial, loss=cubic_soft_l1, + max_nfev=1) + assert_allclose(res.jac, 2 * x * EPS**0.5) + + # Turn to inlier. + res = least_squares(fun_trivial, x, jac_trivial, + loss=cubic_soft_l1, f_scale=6, max_nfev=1) + fs = f / 6 + assert_allclose(res.jac, + 2 * x * (1 - fs**2 / 3)**0.5 * (1 + fs**2)**(-5/6)) + + def test_robustness(self): + for noise in [0.1, 1.0]: + p = ExponentialFittingProblem(1, 0.1, noise, random_seed=0) + + for jac in ['2-point', '3-point', 'cs', p.jac]: + res_lsq = least_squares(p.fun, p.p0, jac=jac, + method=self.method) + assert_allclose(res_lsq.optimality, 0, atol=1e-2) + for loss in LOSSES: + if loss == 'linear': + continue + res_robust = least_squares( + p.fun, p.p0, jac=jac, loss=loss, f_scale=noise, + method=self.method) + assert_allclose(res_robust.optimality, 0, atol=1e-2) + assert_(norm(res_robust.x - p.p_opt) < + norm(res_lsq.x - p.p_opt)) + + +class TestDogbox(BaseMixin, BoundsMixin, SparseMixin, LossFunctionMixin): + method = 'dogbox' + + +class TestTRF(BaseMixin, BoundsMixin, SparseMixin, LossFunctionMixin): + method = 'trf' + + def test_lsmr_regularization(self): + p = BroydenTridiagonal() + for regularize in [True, False]: + res = least_squares(p.fun, p.x0, p.jac, method='trf', + tr_options={'regularize': regularize}) + assert_allclose(res.cost, 0, atol=1e-20) + + +class TestLM(BaseMixin): + method = 'lm' + + def test_bounds_not_supported(self): + assert_raises(ValueError, least_squares, fun_trivial, + 2.0, bounds=(-3.0, 3.0), method='lm') + + def test_m_less_n_not_supported(self): + x0 = [-2, 1] + assert_raises(ValueError, least_squares, fun_rosenbrock_cropped, x0, + method='lm') + + def test_sparse_not_supported(self): + p = BroydenTridiagonal() + assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac, + method='lm') + + def test_jac_sparsity_not_supported(self): + assert_raises(ValueError, least_squares, fun_trivial, 2.0, + jac_sparsity=[1], method='lm') + + def test_LinearOperator_not_supported(self): + p = BroydenTridiagonal(mode="operator") + assert_raises(ValueError, least_squares, p.fun, p.x0, p.jac, + method='lm') + + def test_loss(self): + res = least_squares(fun_trivial, 2.0, loss='linear', method='lm') + assert_allclose(res.x, 0.0, atol=1e-4) + + assert_raises(ValueError, least_squares, fun_trivial, 2.0, + method='lm', loss='huber') + + +def test_basic(): + # test that 'method' arg is really optional + res = least_squares(fun_trivial, 2.0) + assert_allclose(res.x, 0, atol=1e-10) + + +def test_small_tolerances_for_lm(): + for ftol, xtol, gtol in [(None, 1e-13, 1e-13), + (1e-13, None, 1e-13), + (1e-13, 1e-13, None)]: + assert_raises(ValueError, least_squares, fun_trivial, 2.0, xtol=xtol, + ftol=ftol, gtol=gtol, method='lm') + + +def test_fp32_gh12991(): + # checks that smaller FP sizes can be used in least_squares + # this is the minimum working example reported for gh12991 + rng = np.random.RandomState(1) + + x = np.linspace(0, 1, 100).astype("float32") + y = rng.random(100).astype("float32") + + def func(p, x): + return p[0] + p[1] * x + + def err(p, x, y): + return func(p, x) - y + + res = least_squares(err, [-1.0, -1.0], args=(x, y)) + # previously the initial jacobian calculated for this would be all 0 + # and the minimize would terminate immediately, with nfev=1, would + # report a successful minimization (it shouldn't have done), but be + # unchanged from the initial solution. + # It was terminating early because the underlying approx_derivative + # used a step size for FP64 when the working space was FP32. + assert res.nfev > 2 + assert_allclose(res.x, np.array([0.4082241, 0.15530563]), atol=5e-5) + + +def test_gh_18793_and_19351(): + answer = 1e-12 + initial_guess = 1.1e-12 + + def chi2(x): + return (x-answer)**2 + + gtol = 1e-15 + res = least_squares(chi2, x0=initial_guess, gtol=1e-15, bounds=(0, np.inf)) + # Original motivation: gh-18793 + # if we choose an initial condition that is close to the solution + # we shouldn't return an answer that is further away from the solution + + # Update: gh-19351 + # However this requirement does not go well with 'trf' algorithm logic. + # Some regressions were reported after the presumed fix. + # The returned solution is good as long as it satisfies the convergence + # conditions. + # Specifically in this case the scaled gradient will be sufficiently low. + + scaling, _ = CL_scaling_vector(res.x, res.grad, + np.atleast_1d(0), np.atleast_1d(np.inf)) + assert res.status == 1 # Converged by gradient + assert np.linalg.norm(res.grad * scaling, ord=np.inf) < gtol + + +def test_gh_19103(): + # Checks that least_squares trf method selects a strictly feasible point, + # and thus succeeds instead of failing, + # when the initial guess is reported exactly at a boundary point. + # This is a reduced example from gh191303 + + ydata = np.array([0.] * 66 + [ + 1., 0., 0., 0., 0., 0., 1., 1., 0., 0., 1., + 1., 1., 1., 0., 0., 0., 1., 0., 0., 2., 1., + 0., 3., 1., 6., 5., 0., 0., 2., 8., 4., 4., + 6., 9., 7., 2., 7., 8., 2., 13., 9., 8., 11., + 10., 13., 14., 19., 11., 15., 18., 26., 19., 32., 29., + 28., 36., 32., 35., 36., 43., 52., 32., 58., 56., 52., + 67., 53., 72., 88., 77., 95., 94., 84., 86., 101., 107., + 108., 118., 96., 115., 138., 137., + ]) + xdata = np.arange(0, ydata.size) * 0.1 + + def exponential_wrapped(params): + A, B, x0 = params + return A * np.exp(B * (xdata - x0)) - ydata + + x0 = [0.01, 1., 5.] + bounds = ((0.01, 0, 0), (np.inf, 10, 20.9)) + res = least_squares(exponential_wrapped, x0, method='trf', bounds=bounds) + assert res.success diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_linear_assignment.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_linear_assignment.py new file mode 100644 index 0000000000000000000000000000000000000000..d59792da9eef38e313eaa0bca70f873627f8d3cf --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_linear_assignment.py @@ -0,0 +1,116 @@ +# Author: Brian M. Clapper, G. Varoquaux, Lars Buitinck +# License: BSD + +from numpy.testing import assert_array_equal +import pytest + +import numpy as np + +from scipy.optimize import linear_sum_assignment +from scipy.sparse import random +from scipy.sparse._sputils import matrix +from scipy.sparse.csgraph import min_weight_full_bipartite_matching +from scipy.sparse.csgraph.tests.test_matching import ( + linear_sum_assignment_assertions, linear_sum_assignment_test_cases +) + + +def test_linear_sum_assignment_input_shape(): + with pytest.raises(ValueError, match="expected a matrix"): + linear_sum_assignment([1, 2, 3]) + + +def test_linear_sum_assignment_input_object(): + C = [[1, 2, 3], [4, 5, 6]] + assert_array_equal(linear_sum_assignment(C), + linear_sum_assignment(np.asarray(C))) + assert_array_equal(linear_sum_assignment(C), + linear_sum_assignment(matrix(C))) + + +def test_linear_sum_assignment_input_bool(): + I = np.identity(3) + assert_array_equal(linear_sum_assignment(I.astype(np.bool_)), + linear_sum_assignment(I)) + + +def test_linear_sum_assignment_input_string(): + I = np.identity(3) + with pytest.raises(TypeError, match="Cannot cast array data"): + linear_sum_assignment(I.astype(str)) + + +def test_linear_sum_assignment_input_nan(): + I = np.diag([np.nan, 1, 1]) + with pytest.raises(ValueError, match="contains invalid numeric entries"): + linear_sum_assignment(I) + + +def test_linear_sum_assignment_input_neginf(): + I = np.diag([1, -np.inf, 1]) + with pytest.raises(ValueError, match="contains invalid numeric entries"): + linear_sum_assignment(I) + + +def test_linear_sum_assignment_input_inf(): + I = np.identity(3) + I[:, 0] = np.inf + with pytest.raises(ValueError, match="cost matrix is infeasible"): + linear_sum_assignment(I) + + +def test_constant_cost_matrix(): + # Fixes #11602 + n = 8 + C = np.ones((n, n)) + row_ind, col_ind = linear_sum_assignment(C) + assert_array_equal(row_ind, np.arange(n)) + assert_array_equal(col_ind, np.arange(n)) + + +@pytest.mark.parametrize('num_rows,num_cols', [(0, 0), (2, 0), (0, 3)]) +def test_linear_sum_assignment_trivial_cost(num_rows, num_cols): + C = np.empty(shape=(num_cols, num_rows)) + row_ind, col_ind = linear_sum_assignment(C) + assert len(row_ind) == 0 + assert len(col_ind) == 0 + + +@pytest.mark.parametrize('sign,test_case', linear_sum_assignment_test_cases) +def test_linear_sum_assignment_small_inputs(sign, test_case): + linear_sum_assignment_assertions( + linear_sum_assignment, np.array, sign, test_case) + + +# Tests that combine scipy.optimize.linear_sum_assignment and +# scipy.sparse.csgraph.min_weight_full_bipartite_matching +def test_two_methods_give_same_result_on_many_sparse_inputs(): + # As opposed to the test above, here we do not spell out the expected + # output; only assert that the two methods give the same result. + # Concretely, the below tests 100 cases of size 100x100, out of which + # 36 are infeasible. + np.random.seed(1234) + for _ in range(100): + lsa_raises = False + mwfbm_raises = False + sparse = random(100, 100, density=0.06, + data_rvs=lambda size: np.random.randint(1, 100, size)) + # In csgraph, zeros correspond to missing edges, so we explicitly + # replace those with infinities + dense = np.full(sparse.shape, np.inf) + dense[sparse.row, sparse.col] = sparse.data + sparse = sparse.tocsr() + try: + row_ind, col_ind = linear_sum_assignment(dense) + lsa_cost = dense[row_ind, col_ind].sum() + except ValueError: + lsa_raises = True + try: + row_ind, col_ind = min_weight_full_bipartite_matching(sparse) + mwfbm_cost = sparse[row_ind, col_ind].sum() + except ValueError: + mwfbm_raises = True + # Ensure that if one method raises, so does the other one. + assert lsa_raises == mwfbm_raises + if not lsa_raises: + assert lsa_cost == mwfbm_cost diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_linesearch.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_linesearch.py new file mode 100644 index 0000000000000000000000000000000000000000..6eee0743d97665185c35cf144d66e00542925480 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_linesearch.py @@ -0,0 +1,328 @@ +""" +Tests for line search routines +""" +from numpy.testing import (assert_equal, assert_array_almost_equal, + assert_array_almost_equal_nulp, assert_warns, + suppress_warnings) +import scipy.optimize._linesearch as ls +from scipy.optimize._linesearch import LineSearchWarning +import numpy as np +import pytest +import threading + + +def assert_wolfe(s, phi, derphi, c1=1e-4, c2=0.9, err_msg=""): + """ + Check that strong Wolfe conditions apply + """ + phi1 = phi(s) + phi0 = phi(0) + derphi0 = derphi(0) + derphi1 = derphi(s) + msg = (f"s = {s}; phi(0) = {phi0}; phi(s) = {phi1}; phi'(0) = {derphi0};" + f" phi'(s) = {derphi1}; {err_msg}") + + assert phi1 <= phi0 + c1*s*derphi0, "Wolfe 1 failed: " + msg + assert abs(derphi1) <= abs(c2*derphi0), "Wolfe 2 failed: " + msg + + +def assert_armijo(s, phi, c1=1e-4, err_msg=""): + """ + Check that Armijo condition applies + """ + phi1 = phi(s) + phi0 = phi(0) + msg = f"s = {s}; phi(0) = {phi0}; phi(s) = {phi1}; {err_msg}" + assert phi1 <= (1 - c1*s)*phi0, msg + + +def assert_line_wolfe(x, p, s, f, fprime, **kw): + assert_wolfe(s, phi=lambda sp: f(x + p*sp), + derphi=lambda sp: np.dot(fprime(x + p*sp), p), **kw) + + +def assert_line_armijo(x, p, s, f, **kw): + assert_armijo(s, phi=lambda sp: f(x + p*sp), **kw) + + +def assert_fp_equal(x, y, err_msg="", nulp=50): + """Assert two arrays are equal, up to some floating-point rounding error""" + try: + assert_array_almost_equal_nulp(x, y, nulp) + except AssertionError as e: + raise AssertionError(f"{e}\n{err_msg}") from e + + +class TestLineSearch: + # -- scalar functions; must have dphi(0.) < 0 + def _scalar_func_1(self, s): # skip name check + if not hasattr(self.fcount, 'c'): + self.fcount.c = 0 + self.fcount.c += 1 + p = -s - s**3 + s**4 + dp = -1 - 3*s**2 + 4*s**3 + return p, dp + + def _scalar_func_2(self, s): # skip name check + if not hasattr(self.fcount, 'c'): + self.fcount.c = 0 + self.fcount.c += 1 + p = np.exp(-4*s) + s**2 + dp = -4*np.exp(-4*s) + 2*s + return p, dp + + def _scalar_func_3(self, s): # skip name check + if not hasattr(self.fcount, 'c'): + self.fcount.c = 0 + self.fcount.c += 1 + p = -np.sin(10*s) + dp = -10*np.cos(10*s) + return p, dp + + # -- n-d functions + + def _line_func_1(self, x): # skip name check + if not hasattr(self.fcount, 'c'): + self.fcount.c = 0 + self.fcount.c += 1 + f = np.dot(x, x) + df = 2*x + return f, df + + def _line_func_2(self, x): # skip name check + if not hasattr(self.fcount, 'c'): + self.fcount.c = 0 + self.fcount.c += 1 + f = np.dot(x, np.dot(self.A, x)) + 1 + df = np.dot(self.A + self.A.T, x) + return f, df + + # -- + + def setup_method(self): + self.scalar_funcs = [] + self.line_funcs = [] + self.N = 20 + self.fcount = threading.local() + + def bind_index(func, idx): + # Remember Python's closure semantics! + return lambda *a, **kw: func(*a, **kw)[idx] + + for name in sorted(dir(self)): + if name.startswith('_scalar_func_'): + value = getattr(self, name) + self.scalar_funcs.append( + (name, bind_index(value, 0), bind_index(value, 1))) + elif name.startswith('_line_func_'): + value = getattr(self, name) + self.line_funcs.append( + (name, bind_index(value, 0), bind_index(value, 1))) + + np.random.seed(1234) + self.A = np.random.randn(self.N, self.N) + + def scalar_iter(self): + for name, phi, derphi in self.scalar_funcs: + for old_phi0 in np.random.randn(3): + yield name, phi, derphi, old_phi0 + + def line_iter(self): + rng = np.random.RandomState(1234) + for name, f, fprime in self.line_funcs: + k = 0 + while k < 9: + x = rng.randn(self.N) + p = rng.randn(self.N) + if np.dot(p, fprime(x)) >= 0: + # always pick a descent direction + continue + k += 1 + old_fv = float(rng.randn()) + yield name, f, fprime, x, p, old_fv + + # -- Generic scalar searches + + def test_scalar_search_wolfe1(self): + c = 0 + for name, phi, derphi, old_phi0 in self.scalar_iter(): + c += 1 + s, phi1, phi0 = ls.scalar_search_wolfe1(phi, derphi, phi(0), + old_phi0, derphi(0)) + assert_fp_equal(phi0, phi(0), name) + assert_fp_equal(phi1, phi(s), name) + assert_wolfe(s, phi, derphi, err_msg=name) + + assert c > 3 # check that the iterator really works... + + def test_scalar_search_wolfe2(self): + for name, phi, derphi, old_phi0 in self.scalar_iter(): + s, phi1, phi0, derphi1 = ls.scalar_search_wolfe2( + phi, derphi, phi(0), old_phi0, derphi(0)) + assert_fp_equal(phi0, phi(0), name) + assert_fp_equal(phi1, phi(s), name) + if derphi1 is not None: + assert_fp_equal(derphi1, derphi(s), name) + assert_wolfe(s, phi, derphi, err_msg=f"{name} {old_phi0:g}") + + def test_scalar_search_wolfe2_with_low_amax(self): + def phi(alpha): + return (alpha - 5) ** 2 + + def derphi(alpha): + return 2 * (alpha - 5) + + alpha_star, _, _, derphi_star = ls.scalar_search_wolfe2(phi, derphi, amax=0.001) + assert alpha_star is None # Not converged + assert derphi_star is None # Not converged + + def test_scalar_search_wolfe2_regression(self): + # Regression test for gh-12157 + # This phi has its minimum at alpha=4/3 ~ 1.333. + def phi(alpha): + if alpha < 1: + return - 3*np.pi/2 * (alpha - 1) + else: + return np.cos(3*np.pi/2 * alpha - np.pi) + + def derphi(alpha): + if alpha < 1: + return - 3*np.pi/2 + else: + return - 3*np.pi/2 * np.sin(3*np.pi/2 * alpha - np.pi) + + s, _, _, _ = ls.scalar_search_wolfe2(phi, derphi) + # Without the fix in gh-13073, the scalar_search_wolfe2 + # returned s=2.0 instead. + assert s < 1.5 + + def test_scalar_search_armijo(self): + for name, phi, derphi, old_phi0 in self.scalar_iter(): + s, phi1 = ls.scalar_search_armijo(phi, phi(0), derphi(0)) + assert_fp_equal(phi1, phi(s), name) + assert_armijo(s, phi, err_msg=f"{name} {old_phi0:g}") + + # -- Generic line searches + + def test_line_search_wolfe1(self): + c = 0 + smax = 100 + for name, f, fprime, x, p, old_f in self.line_iter(): + f0 = f(x) + g0 = fprime(x) + self.fcount.c = 0 + s, fc, gc, fv, ofv, gv = ls.line_search_wolfe1(f, fprime, x, p, + g0, f0, old_f, + amax=smax) + assert_equal(self.fcount.c, fc+gc) + assert_fp_equal(ofv, f(x)) + if s is None: + continue + assert_fp_equal(fv, f(x + s*p)) + assert_array_almost_equal(gv, fprime(x + s*p), decimal=14) + if s < smax: + c += 1 + assert_line_wolfe(x, p, s, f, fprime, err_msg=name) + + assert c > 3 # check that the iterator really works... + + def test_line_search_wolfe2(self): + c = 0 + smax = 512 + for name, f, fprime, x, p, old_f in self.line_iter(): + f0 = f(x) + g0 = fprime(x) + self.fcount.c = 0 + with suppress_warnings() as sup: + sup.filter(LineSearchWarning, + "The line search algorithm could not find a solution") + sup.filter(LineSearchWarning, + "The line search algorithm did not converge") + s, fc, gc, fv, ofv, gv = ls.line_search_wolfe2(f, fprime, x, p, + g0, f0, old_f, + amax=smax) + assert_equal(self.fcount.c, fc+gc) + assert_fp_equal(ofv, f(x)) + assert_fp_equal(fv, f(x + s*p)) + if gv is not None: + assert_array_almost_equal(gv, fprime(x + s*p), decimal=14) + if s < smax: + c += 1 + assert_line_wolfe(x, p, s, f, fprime, err_msg=name) + assert c > 3 # check that the iterator really works... + + @pytest.mark.thread_unsafe + def test_line_search_wolfe2_bounds(self): + # See gh-7475 + + # For this f and p, starting at a point on axis 0, the strong Wolfe + # condition 2 is met if and only if the step length s satisfies + # |x + s| <= c2 * |x| + def f(x): + return np.dot(x, x) + def fp(x): + return 2 * x + p = np.array([1, 0]) + + # Smallest s satisfying strong Wolfe conditions for these arguments is 30 + x = -60 * p + c2 = 0.5 + + s, _, _, _, _, _ = ls.line_search_wolfe2(f, fp, x, p, amax=30, c2=c2) + assert_line_wolfe(x, p, s, f, fp) + + s, _, _, _, _, _ = assert_warns(LineSearchWarning, + ls.line_search_wolfe2, f, fp, x, p, + amax=29, c2=c2) + assert s is None + + # s=30 will only be tried on the 6th iteration, so this won't converge + assert_warns(LineSearchWarning, ls.line_search_wolfe2, f, fp, x, p, + c2=c2, maxiter=5) + + def test_line_search_armijo(self): + c = 0 + for name, f, fprime, x, p, old_f in self.line_iter(): + f0 = f(x) + g0 = fprime(x) + self.fcount.c = 0 + s, fc, fv = ls.line_search_armijo(f, x, p, g0, f0) + c += 1 + assert_equal(self.fcount.c, fc) + assert_fp_equal(fv, f(x + s*p)) + assert_line_armijo(x, p, s, f, err_msg=name) + assert c >= 9 + + # -- More specific tests + + def test_armijo_terminate_1(self): + # Armijo should evaluate the function only once if the trial step + # is already suitable + count = [0] + + def phi(s): + count[0] += 1 + return -s + 0.01*s**2 + s, phi1 = ls.scalar_search_armijo(phi, phi(0), -1, alpha0=1) + assert_equal(s, 1) + assert_equal(count[0], 2) + assert_armijo(s, phi) + + def test_wolfe_terminate(self): + # wolfe1 and wolfe2 should also evaluate the function only a few + # times if the trial step is already suitable + + def phi(s): + count[0] += 1 + return -s + 0.05*s**2 + + def derphi(s): + count[0] += 1 + return -1 + 0.05*2*s + + for func in [ls.scalar_search_wolfe1, ls.scalar_search_wolfe2]: + count = [0] + r = func(phi, derphi, phi(0), None, derphi(0)) + assert r[0] is not None, (r, func) + assert count[0] <= 2 + 2, (count, func) + assert_wolfe(r[0], phi, derphi, err_msg=str(func)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_linprog.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_linprog.py new file mode 100644 index 0000000000000000000000000000000000000000..4d18e68e394c31e3bc19f49e80c8e9adbc055193 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_linprog.py @@ -0,0 +1,2577 @@ +""" +Unit test for Linear Programming +""" +import sys +import platform + +import numpy as np +from numpy.testing import (assert_, assert_allclose, assert_equal, + assert_array_less, assert_warns, suppress_warnings) +from pytest import raises as assert_raises +from scipy.optimize import linprog, OptimizeWarning +from scipy.optimize._numdiff import approx_derivative +from scipy.sparse.linalg import MatrixRankWarning +from scipy.linalg import LinAlgWarning +from scipy._lib._util import VisibleDeprecationWarning +import scipy.sparse +import pytest + +has_umfpack = True +try: + from scikits.umfpack import UmfpackWarning +except ImportError: + has_umfpack = False + +has_cholmod = True +try: + import sksparse # noqa: F401 + from sksparse.cholmod import cholesky as cholmod # noqa: F401 +except ImportError: + has_cholmod = False + + +def _assert_iteration_limit_reached(res, maxiter): + assert_(not res.success, "Incorrectly reported success") + assert_(res.success < maxiter, "Incorrectly reported number of iterations") + assert_equal(res.status, 1, "Failed to report iteration limit reached") + + +def _assert_infeasible(res): + # res: linprog result object + assert_(not res.success, "incorrectly reported success") + assert_equal(res.status, 2, "failed to report infeasible status") + + +def _assert_unbounded(res): + # res: linprog result object + assert_(not res.success, "incorrectly reported success") + assert_equal(res.status, 3, "failed to report unbounded status") + + +def _assert_unable_to_find_basic_feasible_sol(res): + # res: linprog result object + + # The status may be either 2 or 4 depending on why the feasible solution + # could not be found. If the underlying problem is expected to not have a + # feasible solution, _assert_infeasible should be used. + assert_(not res.success, "incorrectly reported success") + assert_(res.status in (2, 4), "failed to report optimization failure") + + +def _assert_success(res, desired_fun=None, desired_x=None, + rtol=1e-8, atol=1e-8): + # res: linprog result object + # desired_fun: desired objective function value or None + # desired_x: desired solution or None + if not res.success: + msg = f"linprog status {res.status}, message: {res.message}" + raise AssertionError(msg) + + assert_equal(res.status, 0) + if desired_fun is not None: + assert_allclose(res.fun, desired_fun, + err_msg="converged to an unexpected objective value", + rtol=rtol, atol=atol) + if desired_x is not None: + assert_allclose(res.x, desired_x, + err_msg="converged to an unexpected solution", + rtol=rtol, atol=atol) + + +def magic_square(n): + """ + Generates a linear program for which integer solutions represent an + n x n magic square; binary decision variables represent the presence + (or absence) of an integer 1 to n^2 in each position of the square. + """ + + rng = np.random.RandomState(0) + M = n * (n**2 + 1) / 2 + + numbers = np.arange(n**4) // n**2 + 1 + + numbers = numbers.reshape(n**2, n, n) + + zeros = np.zeros((n**2, n, n)) + + A_list = [] + b_list = [] + + # Rule 1: use every number exactly once + for i in range(n**2): + A_row = zeros.copy() + A_row[i, :, :] = 1 + A_list.append(A_row.flatten()) + b_list.append(1) + + # Rule 2: Only one number per square + for i in range(n): + for j in range(n): + A_row = zeros.copy() + A_row[:, i, j] = 1 + A_list.append(A_row.flatten()) + b_list.append(1) + + # Rule 3: sum of rows is M + for i in range(n): + A_row = zeros.copy() + A_row[:, i, :] = numbers[:, i, :] + A_list.append(A_row.flatten()) + b_list.append(M) + + # Rule 4: sum of columns is M + for i in range(n): + A_row = zeros.copy() + A_row[:, :, i] = numbers[:, :, i] + A_list.append(A_row.flatten()) + b_list.append(M) + + # Rule 5: sum of diagonals is M + A_row = zeros.copy() + A_row[:, range(n), range(n)] = numbers[:, range(n), range(n)] + A_list.append(A_row.flatten()) + b_list.append(M) + A_row = zeros.copy() + A_row[:, range(n), range(-1, -n - 1, -1)] = \ + numbers[:, range(n), range(-1, -n - 1, -1)] + A_list.append(A_row.flatten()) + b_list.append(M) + + A = np.array(np.vstack(A_list), dtype=float) + b = np.array(b_list, dtype=float) + c = rng.rand(A.shape[1]) + + return A, b, c, numbers, M + + +def lpgen_2d(m, n): + """ -> A b c LP test: m*n vars, m+n constraints + row sums == n/m, col sums == 1 + https://gist.github.com/denis-bz/8647461 + """ + rng = np.random.RandomState(0) + c = - rng.exponential(size=(m, n)) + Arow = np.zeros((m, m * n)) + brow = np.zeros(m) + for j in range(m): + j1 = j + 1 + Arow[j, j * n:j1 * n] = 1 + brow[j] = n / m + + Acol = np.zeros((n, m * n)) + bcol = np.zeros(n) + for j in range(n): + j1 = j + 1 + Acol[j, j::n] = 1 + bcol[j] = 1 + + A = np.vstack((Arow, Acol)) + b = np.hstack((brow, bcol)) + + return A, b, c.ravel() + + +def very_random_gen(seed=0): + rng = np.random.RandomState(seed) + m_eq, m_ub, n = 10, 20, 50 + c = rng.rand(n)-0.5 + A_ub = rng.rand(m_ub, n)-0.5 + b_ub = rng.rand(m_ub)-0.5 + A_eq = rng.rand(m_eq, n)-0.5 + b_eq = rng.rand(m_eq)-0.5 + lb = -rng.rand(n) + ub = rng.rand(n) + lb[lb < -rng.rand()] = -np.inf + ub[ub > rng.rand()] = np.inf + bounds = np.vstack((lb, ub)).T + return c, A_ub, b_ub, A_eq, b_eq, bounds + + +def nontrivial_problem(): + c = [-1, 8, 4, -6] + A_ub = [[-7, -7, 6, 9], + [1, -1, -3, 0], + [10, -10, -7, 7], + [6, -1, 3, 4]] + b_ub = [-3, 6, -6, 6] + A_eq = [[-10, 1, 1, -8]] + b_eq = [-4] + x_star = [101 / 1391, 1462 / 1391, 0, 752 / 1391] + f_star = 7083 / 1391 + return c, A_ub, b_ub, A_eq, b_eq, x_star, f_star + + +def l1_regression_prob(seed=0, m=8, d=9, n=100): + ''' + Training data is {(x0, y0), (x1, y2), ..., (xn-1, yn-1)} + x in R^d + y in R + n: number of training samples + d: dimension of x, i.e. x in R^d + phi: feature map R^d -> R^m + m: dimension of feature space + ''' + rng = np.random.RandomState(seed) + phi = rng.normal(0, 1, size=(m, d)) # random feature mapping + w_true = rng.randn(m) + x = rng.normal(0, 1, size=(d, n)) # features + y = w_true @ (phi @ x) + rng.normal(0, 1e-5, size=n) # measurements + + # construct the problem + c = np.ones(m+n) + c[:m] = 0 + A_ub = scipy.sparse.lil_matrix((2*n, n+m)) + idx = 0 + for ii in range(n): + A_ub[idx, :m] = phi @ x[:, ii] + A_ub[idx, m+ii] = -1 + A_ub[idx+1, :m] = -1*phi @ x[:, ii] + A_ub[idx+1, m+ii] = -1 + idx += 2 + A_ub = A_ub.tocsc() + b_ub = np.zeros(2*n) + b_ub[0::2] = y + b_ub[1::2] = -y + bnds = [(None, None)]*m + [(0, None)]*n + return c, A_ub, b_ub, bnds + + +def generic_callback_test(self): + # Check that callback is as advertised + last_cb = {} + + def cb(res): + message = res.pop('message') + complete = res.pop('complete') + + assert_(res.pop('phase') in (1, 2)) + assert_(res.pop('status') in range(4)) + assert_(isinstance(res.pop('nit'), int)) + assert_(isinstance(complete, bool)) + assert_(isinstance(message, str)) + + last_cb['x'] = res['x'] + last_cb['fun'] = res['fun'] + last_cb['slack'] = res['slack'] + last_cb['con'] = res['con'] + + c = np.array([-3, -2]) + A_ub = [[2, 1], [1, 1], [1, 0]] + b_ub = [10, 8, 4] + res = linprog(c, A_ub=A_ub, b_ub=b_ub, callback=cb, method=self.method) + + _assert_success(res, desired_fun=-18.0, desired_x=[2, 6]) + assert_allclose(last_cb['fun'], res['fun']) + assert_allclose(last_cb['x'], res['x']) + assert_allclose(last_cb['con'], res['con']) + assert_allclose(last_cb['slack'], res['slack']) + + +@pytest.mark.thread_unsafe +def test_unknown_solvers_and_options(): + c = np.array([-3, -2]) + A_ub = [[2, 1], [1, 1], [1, 0]] + b_ub = [10, 8, 4] + + assert_raises(ValueError, linprog, + c, A_ub=A_ub, b_ub=b_ub, method='ekki-ekki-ekki') + assert_raises(ValueError, linprog, + c, A_ub=A_ub, b_ub=b_ub, method='highs-ekki') + message = "Unrecognized options detected: {'rr_method': 'ekki-ekki-ekki'}" + with pytest.warns(OptimizeWarning, match=message): + linprog(c, A_ub=A_ub, b_ub=b_ub, + options={"rr_method": 'ekki-ekki-ekki'}) + + +def test_choose_solver(): + # 'highs' chooses 'dual' + c = np.array([-3, -2]) + A_ub = [[2, 1], [1, 1], [1, 0]] + b_ub = [10, 8, 4] + + res = linprog(c, A_ub, b_ub, method='highs') + _assert_success(res, desired_fun=-18.0, desired_x=[2, 6]) + + +@pytest.mark.thread_unsafe +def test_deprecation(): + with pytest.warns(DeprecationWarning): + linprog(1, method='interior-point') + with pytest.warns(DeprecationWarning): + linprog(1, method='revised simplex') + with pytest.warns(DeprecationWarning): + linprog(1, method='simplex') + + +def test_highs_status_message(): + res = linprog(1, method='highs') + msg = "Optimization terminated successfully. (HiGHS Status 7:" + assert res.status == 0 + assert res.message.startswith(msg) + + A, b, c, numbers, M = magic_square(6) + bounds = [(0, 1)] * len(c) + integrality = [1] * len(c) + options = {"time_limit": 0.1} + res = linprog(c=c, A_eq=A, b_eq=b, bounds=bounds, method='highs', + options=options, integrality=integrality) + msg = "Time limit reached. (HiGHS Status 13:" + assert res.status == 1 + assert res.message.startswith(msg) + + options = {"maxiter": 10} + res = linprog(c=c, A_eq=A, b_eq=b, bounds=bounds, method='highs-ds', + options=options) + msg = "Iteration limit reached. (HiGHS Status 14:" + assert res.status == 1 + assert res.message.startswith(msg) + + res = linprog(1, bounds=(1, -1), method='highs') + msg = "The problem is infeasible. (HiGHS Status 8:" + assert res.status == 2 + assert res.message.startswith(msg) + + res = linprog(-1, method='highs') + msg = "The problem is unbounded. (HiGHS Status 10:" + assert res.status == 3 + assert res.message.startswith(msg) + + from scipy.optimize._linprog_highs import _highs_to_scipy_status_message + status, message = _highs_to_scipy_status_message(58, "Hello!") + msg = "The HiGHS status code was not recognized. (HiGHS Status 58:" + assert status == 4 + assert message.startswith(msg) + + status, message = _highs_to_scipy_status_message(None, None) + msg = "HiGHS did not provide a status code. (HiGHS Status None: None)" + assert status == 4 + assert message.startswith(msg) + + +def test_bug_17380(): + linprog([1, 1], A_ub=[[-1, 0]], b_ub=[-2.5], integrality=[1, 1]) + + +A_ub = None +b_ub = None +A_eq = None +b_eq = None +bounds = None + +################ +# Common Tests # +################ + + +class LinprogCommonTests: + """ + Base class for `linprog` tests. Generally, each test will be performed + once for every derived class of LinprogCommonTests, each of which will + typically change self.options and/or self.method. Effectively, these tests + are run for many combination of method (simplex, revised simplex, and + interior point) and options (such as pivoting rule or sparse treatment). + """ + + ################## + # Targeted Tests # + ################## + + def test_callback(self): + generic_callback_test(self) + + def test_disp(self): + # test that display option does not break anything. + A, b, c = lpgen_2d(20, 20) + res = linprog(c, A_ub=A, b_ub=b, method=self.method, + options={"disp": True}) + _assert_success(res, desired_fun=-64.049494229) + + def test_docstring_example(self): + # Example from linprog docstring. + c = [-1, 4] + A = [[-3, 1], [1, 2]] + b = [6, 4] + x0_bounds = (None, None) + x1_bounds = (-3, None) + res = linprog(c, A_ub=A, b_ub=b, bounds=(x0_bounds, x1_bounds), + options=self.options, method=self.method) + _assert_success(res, desired_fun=-22) + + def test_type_error(self): + # (presumably) checks that linprog recognizes type errors + # This is tested more carefully in test__linprog_clean_inputs.py + c = [1] + A_eq = [[1]] + b_eq = "hello" + assert_raises(TypeError, linprog, + c, A_eq=A_eq, b_eq=b_eq, + method=self.method, options=self.options) + + def test_aliasing_b_ub(self): + # (presumably) checks that linprog does not modify b_ub + # This is tested more carefully in test__linprog_clean_inputs.py + c = np.array([1.0]) + A_ub = np.array([[1.0]]) + b_ub_orig = np.array([3.0]) + b_ub = b_ub_orig.copy() + bounds = (-4.0, np.inf) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=-4, desired_x=[-4]) + assert_allclose(b_ub_orig, b_ub) + + def test_aliasing_b_eq(self): + # (presumably) checks that linprog does not modify b_eq + # This is tested more carefully in test__linprog_clean_inputs.py + c = np.array([1.0]) + A_eq = np.array([[1.0]]) + b_eq_orig = np.array([3.0]) + b_eq = b_eq_orig.copy() + bounds = (-4.0, np.inf) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=3, desired_x=[3]) + assert_allclose(b_eq_orig, b_eq) + + def test_non_ndarray_args(self): + # (presumably) checks that linprog accepts list in place of arrays + # This is tested more carefully in test__linprog_clean_inputs.py + c = [1.0] + A_ub = [[1.0]] + b_ub = [3.0] + A_eq = [[1.0]] + b_eq = [2.0] + bounds = (-1.0, 10.0) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=2, desired_x=[2]) + + @pytest.mark.thread_unsafe + def test_unknown_options(self): + c = np.array([-3, -2]) + A_ub = [[2, 1], [1, 1], [1, 0]] + b_ub = [10, 8, 4] + + def f(c, A_ub=None, b_ub=None, A_eq=None, + b_eq=None, bounds=None, options=None): + linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=options) + + o = {key: self.options[key] for key in self.options} + o['spam'] = 42 + + assert_warns(OptimizeWarning, f, + c, A_ub=A_ub, b_ub=b_ub, options=o) + + @pytest.mark.thread_unsafe + def test_integrality_without_highs(self): + # ensure that using `integrality` parameter without `method='highs'` + # raises warning and produces correct solution to relaxed problem + # source: https://en.wikipedia.org/wiki/Integer_programming#Example + A_ub = np.array([[-1, 1], [3, 2], [2, 3]]) + b_ub = np.array([1, 12, 12]) + c = -np.array([0, 1]) + + bounds = [(0, np.inf)] * len(c) + integrality = [1] * len(c) + + with np.testing.assert_warns(OptimizeWarning): + res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, bounds=bounds, + method=self.method, integrality=integrality) + + np.testing.assert_allclose(res.x, [1.8, 2.8]) + np.testing.assert_allclose(res.fun, -2.8) + + def test_invalid_inputs(self): + + def f(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None): + linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + + # Test ill-formatted bounds + assert_raises(ValueError, f, [1, 2, 3], bounds=[(1, 2), (3, 4)]) + with np.testing.suppress_warnings() as sup: + sup.filter(VisibleDeprecationWarning, "Creating an ndarray from ragged") + assert_raises(ValueError, f, [1, 2, 3], bounds=[(1, 2), (3, 4), (3, 4, 5)]) + assert_raises(ValueError, f, [1, 2, 3], bounds=[(1, -2), (1, 2)]) + + # Test other invalid inputs + assert_raises(ValueError, f, [1, 2], A_ub=[[1, 2]], b_ub=[1, 2]) + assert_raises(ValueError, f, [1, 2], A_ub=[[1]], b_ub=[1]) + assert_raises(ValueError, f, [1, 2], A_eq=[[1, 2]], b_eq=[1, 2]) + assert_raises(ValueError, f, [1, 2], A_eq=[[1]], b_eq=[1]) + assert_raises(ValueError, f, [1, 2], A_eq=[1], b_eq=1) + + # this last check doesn't make sense for sparse presolve + if ("_sparse_presolve" in self.options and + self.options["_sparse_presolve"]): + return + # there aren't 3-D sparse matrices + + assert_raises(ValueError, f, [1, 2], A_ub=np.zeros((1, 1, 3)), b_eq=1) + + def test_sparse_constraints(self): + # gh-13559: improve error message for sparse inputs when unsupported + def f(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None): + linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + + rng = np.random.RandomState(0) + m = 100 + n = 150 + A_eq = scipy.sparse.rand(m, n, 0.5) + x_valid = rng.randn(n) + c = rng.randn(n) + ub = x_valid + rng.rand(n) + lb = x_valid - rng.rand(n) + bounds = np.column_stack((lb, ub)) + b_eq = A_eq @ x_valid + + if self.method in {'simplex', 'revised simplex'}: + # simplex and revised simplex should raise error + with assert_raises(ValueError, match=f"Method '{self.method}' " + "does not support sparse constraint matrices."): + linprog(c=c, A_eq=A_eq, b_eq=b_eq, bounds=bounds, + method=self.method, options=self.options) + else: + # other methods should succeed + options = {**self.options} + if self.method in {'interior-point'}: + options['sparse'] = True + + res = linprog(c=c, A_eq=A_eq, b_eq=b_eq, bounds=bounds, + method=self.method, options=options) + assert res.success + + def test_maxiter(self): + # test iteration limit w/ Enzo example + c = [4, 8, 3, 0, 0, 0] + A = [ + [2, 5, 3, -1, 0, 0], + [3, 2.5, 8, 0, -1, 0], + [8, 10, 4, 0, 0, -1]] + b = [185, 155, 600] + np.random.seed(0) + maxiter = 3 + res = linprog(c, A_eq=A, b_eq=b, method=self.method, + options={"maxiter": maxiter}) + _assert_iteration_limit_reached(res, maxiter) + assert_equal(res.nit, maxiter) + + def test_bounds_fixed(self): + + # Test fixed bounds (upper equal to lower) + # If presolve option True, test if solution found in presolve (i.e. + # number of iterations is 0). + do_presolve = self.options.get('presolve', True) + + res = linprog([1], bounds=(1, 1), + method=self.method, options=self.options) + _assert_success(res, 1, 1) + if do_presolve: + assert_equal(res.nit, 0) + + res = linprog([1, 2, 3], bounds=[(5, 5), (-1, -1), (3, 3)], + method=self.method, options=self.options) + _assert_success(res, 12, [5, -1, 3]) + if do_presolve: + assert_equal(res.nit, 0) + + res = linprog([1, 1], bounds=[(1, 1), (1, 3)], + method=self.method, options=self.options) + _assert_success(res, 2, [1, 1]) + if do_presolve: + assert_equal(res.nit, 0) + + res = linprog([1, 1, 2], A_eq=[[1, 0, 0], [0, 1, 0]], b_eq=[1, 7], + bounds=[(-5, 5), (0, 10), (3.5, 3.5)], + method=self.method, options=self.options) + _assert_success(res, 15, [1, 7, 3.5]) + if do_presolve: + assert_equal(res.nit, 0) + + def test_bounds_infeasible(self): + + # Test ill-valued bounds (upper less than lower) + # If presolve option True, test if solution found in presolve (i.e. + # number of iterations is 0). + do_presolve = self.options.get('presolve', True) + + res = linprog([1], bounds=(1, -2), method=self.method, options=self.options) + _assert_infeasible(res) + if do_presolve: + assert_equal(res.nit, 0) + + res = linprog([1], bounds=[(1, -2)], method=self.method, options=self.options) + _assert_infeasible(res) + if do_presolve: + assert_equal(res.nit, 0) + + res = linprog([1, 2, 3], bounds=[(5, 0), (1, 2), (3, 4)], + method=self.method, options=self.options) + _assert_infeasible(res) + if do_presolve: + assert_equal(res.nit, 0) + + @pytest.mark.thread_unsafe + def test_bounds_infeasible_2(self): + + # Test ill-valued bounds (lower inf, upper -inf) + # If presolve option True, test if solution found in presolve (i.e. + # number of iterations is 0). + # For the simplex method, the cases do not result in an + # infeasible status, but in a RuntimeWarning. This is a + # consequence of having _presolve() take care of feasibility + # checks. See issue gh-11618. + do_presolve = self.options.get('presolve', True) + simplex_without_presolve = not do_presolve and self.method == 'simplex' + + c = [1, 2, 3] + bounds_1 = [(1, 2), (np.inf, np.inf), (3, 4)] + bounds_2 = [(1, 2), (-np.inf, -np.inf), (3, 4)] + + if simplex_without_presolve: + def g(c, bounds): + res = linprog(c, bounds=bounds, + method=self.method, options=self.options) + return res + + with pytest.warns(RuntimeWarning): + with pytest.raises(IndexError): + g(c, bounds=bounds_1) + + with pytest.warns(RuntimeWarning): + with pytest.raises(IndexError): + g(c, bounds=bounds_2) + else: + res = linprog(c=c, bounds=bounds_1, + method=self.method, options=self.options) + _assert_infeasible(res) + if do_presolve: + assert_equal(res.nit, 0) + res = linprog(c=c, bounds=bounds_2, + method=self.method, options=self.options) + _assert_infeasible(res) + if do_presolve: + assert_equal(res.nit, 0) + + def test_empty_constraint_1(self): + c = [-1, -2] + res = linprog(c, method=self.method, options=self.options) + _assert_unbounded(res) + + def test_empty_constraint_2(self): + c = [-1, 1, -1, 1] + bounds = [(0, np.inf), (-np.inf, 0), (-1, 1), (-1, 1)] + res = linprog(c, bounds=bounds, + method=self.method, options=self.options) + _assert_unbounded(res) + # Unboundedness detected in presolve requires no iterations + if self.options.get('presolve', True): + assert_equal(res.nit, 0) + + def test_empty_constraint_3(self): + c = [1, -1, 1, -1] + bounds = [(0, np.inf), (-np.inf, 0), (-1, 1), (-1, 1)] + res = linprog(c, bounds=bounds, + method=self.method, options=self.options) + _assert_success(res, desired_x=[0, 0, -1, 1], desired_fun=-2) + + def test_inequality_constraints(self): + # Minimize linear function subject to linear inequality constraints. + # http://www.dam.brown.edu/people/huiwang/classes/am121/Archive/simplex_121_c.pdf + c = np.array([3, 2]) * -1 # maximize + A_ub = [[2, 1], + [1, 1], + [1, 0]] + b_ub = [10, 8, 4] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=-18, desired_x=[2, 6]) + + def test_inequality_constraints2(self): + # Minimize linear function subject to linear inequality constraints. + # http://www.statslab.cam.ac.uk/~ff271/teaching/opt/notes/notes8.pdf + # (dead link) + c = [6, 3] + A_ub = [[0, 3], + [-1, -1], + [-2, 1]] + b_ub = [2, -1, -1] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=5, desired_x=[2 / 3, 1 / 3]) + + def test_bounds_simple(self): + c = [1, 2] + bounds = (1, 2) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_x=[1, 1]) + + bounds = [(1, 2), (1, 2)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_x=[1, 1]) + + def test_bounded_below_only_1(self): + c = np.array([1.0]) + A_eq = np.array([[1.0]]) + b_eq = np.array([3.0]) + bounds = (1.0, None) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=3, desired_x=[3]) + + def test_bounded_below_only_2(self): + c = np.ones(3) + A_eq = np.eye(3) + b_eq = np.array([1, 2, 3]) + bounds = (0.5, np.inf) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_x=b_eq, desired_fun=np.sum(b_eq)) + + def test_bounded_above_only_1(self): + c = np.array([1.0]) + A_eq = np.array([[1.0]]) + b_eq = np.array([3.0]) + bounds = (None, 10.0) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=3, desired_x=[3]) + + def test_bounded_above_only_2(self): + c = np.ones(3) + A_eq = np.eye(3) + b_eq = np.array([1, 2, 3]) + bounds = (-np.inf, 4) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_x=b_eq, desired_fun=np.sum(b_eq)) + + def test_bounds_infinity(self): + c = np.ones(3) + A_eq = np.eye(3) + b_eq = np.array([1, 2, 3]) + bounds = (-np.inf, np.inf) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_x=b_eq, desired_fun=np.sum(b_eq)) + + def test_bounds_mixed(self): + # Problem has one unbounded variable and + # another with a negative lower bound. + c = np.array([-1, 4]) * -1 # maximize + A_ub = np.array([[-3, 1], + [1, 2]], dtype=np.float64) + b_ub = [6, 4] + x0_bounds = (-np.inf, np.inf) + x1_bounds = (-3, np.inf) + bounds = (x0_bounds, x1_bounds) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=-80 / 7, desired_x=[-8 / 7, 18 / 7]) + + def test_bounds_equal_but_infeasible(self): + c = [-4, 1] + A_ub = [[7, -2], [0, 1], [2, -2]] + b_ub = [14, 0, 3] + bounds = [(2, 2), (0, None)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_infeasible(res) + + def test_bounds_equal_but_infeasible2(self): + c = [-4, 1] + A_eq = [[7, -2], [0, 1], [2, -2]] + b_eq = [14, 0, 3] + bounds = [(2, 2), (0, None)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_infeasible(res) + + def test_bounds_equal_no_presolve(self): + # There was a bug when a lower and upper bound were equal but + # presolve was not on to eliminate the variable. The bound + # was being converted to an equality constraint, but the bound + # was not eliminated, leading to issues in postprocessing. + c = [1, 2] + A_ub = [[1, 2], [1.1, 2.2]] + b_ub = [4, 8] + bounds = [(1, 2), (2, 2)] + + o = {key: self.options[key] for key in self.options} + o["presolve"] = False + + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=o) + _assert_infeasible(res) + + def test_zero_column_1(self): + m, n = 3, 4 + rng = np.random.RandomState(0) + c = rng.rand(n) + c[1] = 1 + A_eq = rng.rand(m, n) + A_eq[:, 1] = 0 + b_eq = rng.rand(m) + A_ub = [[1, 0, 1, 1]] + b_ub = 3 + bounds = [(-10, 10), (-10, 10), (-10, None), (None, None)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=-9.7087836730413404) + + def test_zero_column_2(self): + if self.method in {'highs-ds', 'highs-ipm'}: + # See upstream issue https://github.com/ERGO-Code/HiGHS/issues/648 + pytest.xfail() + + rng = np.random.RandomState(0) + m, n = 2, 4 + c = rng.rand(n) + c[1] = -1 + A_eq = rng.rand(m, n) + A_eq[:, 1] = 0 + b_eq = rng.rand(m) + + A_ub = rng.rand(m, n) + A_ub[:, 1] = 0 + b_ub = rng.rand(m) + bounds = (None, None) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_unbounded(res) + # Unboundedness detected in presolve + if self.options.get('presolve', True) and "highs" not in self.method: + # HiGHS detects unboundedness or infeasibility in presolve + # It needs an iteration of simplex to be sure of unboundedness + # Other solvers report that the problem is unbounded if feasible + assert_equal(res.nit, 0) + + def test_zero_row_1(self): + c = [1, 2, 3] + A_eq = [[0, 0, 0], [1, 1, 1], [0, 0, 0]] + b_eq = [0, 3, 0] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=3) + + def test_zero_row_2(self): + A_ub = [[0, 0, 0], [1, 1, 1], [0, 0, 0]] + b_ub = [0, 3, 0] + c = [1, 2, 3] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=0) + + def test_zero_row_3(self): + m, n = 2, 4 + rng = np.random.RandomState(1234) + c = rng.rand(n) + A_eq = rng.rand(m, n) + A_eq[0, :] = 0 + b_eq = rng.rand(m) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_infeasible(res) + + # Infeasibility detected in presolve + if self.options.get('presolve', True): + assert_equal(res.nit, 0) + + def test_zero_row_4(self): + m, n = 2, 4 + rng = np.random.RandomState(1234) + c = rng.rand(n) + A_ub = rng.rand(m, n) + A_ub[0, :] = 0 + b_ub = -rng.rand(m) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_infeasible(res) + + # Infeasibility detected in presolve + if self.options.get('presolve', True): + assert_equal(res.nit, 0) + + def test_singleton_row_eq_1(self): + c = [1, 1, 1, 2] + A_eq = [[1, 0, 0, 0], [0, 2, 0, 0], [1, 0, 0, 0], [1, 1, 1, 1]] + b_eq = [1, 2, 2, 4] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_infeasible(res) + + # Infeasibility detected in presolve + if self.options.get('presolve', True): + assert_equal(res.nit, 0) + + def test_singleton_row_eq_2(self): + c = [1, 1, 1, 2] + A_eq = [[1, 0, 0, 0], [0, 2, 0, 0], [1, 0, 0, 0], [1, 1, 1, 1]] + b_eq = [1, 2, 1, 4] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=4) + + def test_singleton_row_ub_1(self): + c = [1, 1, 1, 2] + A_ub = [[1, 0, 0, 0], [0, 2, 0, 0], [-1, 0, 0, 0], [1, 1, 1, 1]] + b_ub = [1, 2, -2, 4] + bounds = [(None, None), (0, None), (0, None), (0, None)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_infeasible(res) + + # Infeasibility detected in presolve + if self.options.get('presolve', True): + assert_equal(res.nit, 0) + + def test_singleton_row_ub_2(self): + c = [1, 1, 1, 2] + A_ub = [[1, 0, 0, 0], [0, 2, 0, 0], [-1, 0, 0, 0], [1, 1, 1, 1]] + b_ub = [1, 2, -0.5, 4] + bounds = [(None, None), (0, None), (0, None), (0, None)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=0.5) + + def test_infeasible(self): + # Test linprog response to an infeasible problem + c = [-1, -1] + A_ub = [[1, 0], + [0, 1], + [-1, -1]] + b_ub = [2, 2, -5] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_infeasible(res) + + def test_infeasible_inequality_bounds(self): + c = [1] + A_ub = [[2]] + b_ub = 4 + bounds = (5, 6) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_infeasible(res) + + # Infeasibility detected in presolve + if self.options.get('presolve', True): + assert_equal(res.nit, 0) + + def test_unbounded(self): + # Test linprog response to an unbounded problem + c = np.array([1, 1]) * -1 # maximize + A_ub = [[-1, 1], + [-1, -1]] + b_ub = [-1, -2] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_unbounded(res) + + def test_unbounded_below_no_presolve_corrected(self): + c = [1] + bounds = [(None, 1)] + + o = {key: self.options[key] for key in self.options} + o["presolve"] = False + + res = linprog(c=c, bounds=bounds, + method=self.method, + options=o) + if self.method == "revised simplex": + # Revised simplex has a special pathway for no constraints. + assert_equal(res.status, 5) + else: + _assert_unbounded(res) + + def test_unbounded_no_nontrivial_constraints_1(self): + """ + Test whether presolve pathway for detecting unboundedness after + constraint elimination is working. + """ + c = np.array([0, 0, 0, 1, -1, -1]) + A_ub = np.array([[1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, -1]]) + b_ub = np.array([2, -2, 0]) + bounds = [(None, None), (None, None), (None, None), + (-1, 1), (-1, 1), (0, None)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_unbounded(res) + if not self.method.lower().startswith("highs"): + assert_equal(res.x[-1], np.inf) + assert_equal(res.message[:36], + "The problem is (trivially) unbounded") + + def test_unbounded_no_nontrivial_constraints_2(self): + """ + Test whether presolve pathway for detecting unboundedness after + constraint elimination is working. + """ + c = np.array([0, 0, 0, 1, -1, 1]) + A_ub = np.array([[1, 0, 0, 0, 0, 0], + [0, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1]]) + b_ub = np.array([2, -2, 0]) + bounds = [(None, None), (None, None), (None, None), + (-1, 1), (-1, 1), (None, 0)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_unbounded(res) + if not self.method.lower().startswith("highs"): + assert_equal(res.x[-1], -np.inf) + assert_equal(res.message[:36], + "The problem is (trivially) unbounded") + + def test_cyclic_recovery(self): + # Test linprogs recovery from cycling using the Klee-Minty problem + # Klee-Minty https://www.math.ubc.ca/~israel/m340/kleemin3.pdf + c = np.array([100, 10, 1]) * -1 # maximize + A_ub = [[1, 0, 0], + [20, 1, 0], + [200, 20, 1]] + b_ub = [1, 100, 10000] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_x=[0, 0, 10000], atol=5e-6, rtol=1e-7) + + def test_cyclic_bland(self): + # Test the effect of Bland's rule on a cycling problem + c = np.array([-10, 57, 9, 24.]) + A_ub = np.array([[0.5, -5.5, -2.5, 9], + [0.5, -1.5, -0.5, 1], + [1, 0, 0, 0]]) + b_ub = [0, 0, 1] + + # copy the existing options dictionary but change maxiter + maxiter = 100 + o = {key: val for key, val in self.options.items()} + o['maxiter'] = maxiter + + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=o) + + if self.method == 'simplex' and not self.options.get('bland'): + # simplex cycles without Bland's rule + _assert_iteration_limit_reached(res, o['maxiter']) + else: + # other methods, including simplex with Bland's rule, succeed + _assert_success(res, desired_x=[1, 0, 1, 0]) + # note that revised simplex skips this test because it may or may not + # cycle depending on the initial basis + + def test_remove_redundancy_infeasibility(self): + # mostly a test of redundancy removal, which is carefully tested in + # test__remove_redundancy.py + m, n = 10, 10 + rng = np.random.RandomState(0) + c = rng.rand(n) + A_eq = rng.rand(m, n) + b_eq = rng.rand(m) + A_eq[-1, :] = 2 * A_eq[-2, :] + b_eq[-1] *= -1 + with suppress_warnings() as sup: + sup.filter(OptimizeWarning, "A_eq does not appear...") + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_infeasible(res) + + ################# + # General Tests # + ################# + + def test_nontrivial_problem(self): + # Problem involves all constraint types, + # negative resource limits, and rounding issues. + c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem() + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=f_star, desired_x=x_star) + + def test_lpgen_problem(self): + # Test linprog with a rather large problem (400 variables, + # 40 constraints) generated by https://gist.github.com/denis-bz/8647461 + A_ub, b_ub, c = lpgen_2d(20, 20) + + with suppress_warnings() as sup: + sup.filter(OptimizeWarning, "Solving system with option 'sym_pos'") + sup.filter(RuntimeWarning, "invalid value encountered") + sup.filter(LinAlgWarning) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=-64.049494229) + + def test_network_flow(self): + # A network flow problem with supply and demand at nodes + # and with costs along directed edges. + # https://www.princeton.edu/~rvdb/542/lectures/lec10.pdf + c = [2, 4, 9, 11, 4, 3, 8, 7, 0, 15, 16, 18] + n, p = -1, 1 + A_eq = [ + [n, n, p, 0, p, 0, 0, 0, 0, p, 0, 0], + [p, 0, 0, p, 0, p, 0, 0, 0, 0, 0, 0], + [0, 0, n, n, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, p, p, 0, 0, p, 0], + [0, 0, 0, 0, n, n, n, 0, p, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, n, n, 0, 0, p], + [0, 0, 0, 0, 0, 0, 0, 0, 0, n, n, n]] + b_eq = [0, 19, -16, 33, 0, 0, -36] + with suppress_warnings() as sup: + sup.filter(LinAlgWarning) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=755, atol=1e-6, rtol=1e-7) + + def test_network_flow_limited_capacity(self): + # A network flow problem with supply and demand at nodes + # and with costs and capacities along directed edges. + # http://blog.sommer-forst.de/2013/04/10/ + c = [2, 2, 1, 3, 1] + bounds = [ + [0, 4], + [0, 2], + [0, 2], + [0, 3], + [0, 5]] + n, p = -1, 1 + A_eq = [ + [n, n, 0, 0, 0], + [p, 0, n, n, 0], + [0, p, p, 0, n], + [0, 0, 0, p, p]] + b_eq = [-4, 0, 0, 4] + + with suppress_warnings() as sup: + # this is an UmfpackWarning but I had trouble importing it + if has_umfpack: + sup.filter(UmfpackWarning) + sup.filter(RuntimeWarning, "scipy.linalg.solve\nIll...") + sup.filter(OptimizeWarning, "A_eq does not appear...") + sup.filter(OptimizeWarning, "Solving system with option...") + sup.filter(LinAlgWarning) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=14) + + def test_simplex_algorithm_wikipedia_example(self): + # https://en.wikipedia.org/wiki/Simplex_algorithm#Example + c = [-2, -3, -4] + A_ub = [ + [3, 2, 1], + [2, 5, 3]] + b_ub = [10, 15] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=-20) + + def test_enzo_example(self): + # https://github.com/scipy/scipy/issues/1779 lp2.py + # + # Translated from Octave code at: + # http://www.ecs.shimane-u.ac.jp/~kyoshida/lpeng.htm + # and placed under MIT licence by Enzo Michelangeli + # with permission explicitly granted by the original author, + # Prof. Kazunobu Yoshida + c = [4, 8, 3, 0, 0, 0] + A_eq = [ + [2, 5, 3, -1, 0, 0], + [3, 2.5, 8, 0, -1, 0], + [8, 10, 4, 0, 0, -1]] + b_eq = [185, 155, 600] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=317.5, + desired_x=[66.25, 0, 17.5, 0, 183.75, 0], + atol=6e-6, rtol=1e-7) + + def test_enzo_example_b(self): + # rescued from https://github.com/scipy/scipy/pull/218 + c = [2.8, 6.3, 10.8, -2.8, -6.3, -10.8] + A_eq = [[-1, -1, -1, 0, 0, 0], + [0, 0, 0, 1, 1, 1], + [1, 0, 0, 1, 0, 0], + [0, 1, 0, 0, 1, 0], + [0, 0, 1, 0, 0, 1]] + b_eq = [-0.5, 0.4, 0.3, 0.3, 0.3] + + with suppress_warnings() as sup: + sup.filter(OptimizeWarning, "A_eq does not appear...") + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=-1.77, + desired_x=[0.3, 0.2, 0.0, 0.0, 0.1, 0.3]) + + def test_enzo_example_c_with_degeneracy(self): + # rescued from https://github.com/scipy/scipy/pull/218 + m = 20 + c = -np.ones(m) + tmp = 2 * np.pi * np.arange(1, m + 1) / (m + 1) + A_eq = np.vstack((np.cos(tmp) - 1, np.sin(tmp))) + b_eq = [0, 0] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=0, desired_x=np.zeros(m)) + + def test_enzo_example_c_with_unboundedness(self): + # rescued from https://github.com/scipy/scipy/pull/218 + m = 50 + c = -np.ones(m) + tmp = 2 * np.pi * np.arange(m) / (m + 1) + # This test relies on `cos(0) -1 == sin(0)`, so ensure that's true + # (SIMD code or -ffast-math may cause spurious failures otherwise) + row0 = np.cos(tmp) - 1 + row0[0] = 0.0 + row1 = np.sin(tmp) + row1[0] = 0.0 + A_eq = np.vstack((row0, row1)) + b_eq = [0, 0] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_unbounded(res) + + def test_enzo_example_c_with_infeasibility(self): + # rescued from https://github.com/scipy/scipy/pull/218 + m = 50 + c = -np.ones(m) + tmp = 2 * np.pi * np.arange(m) / (m + 1) + A_eq = np.vstack((np.cos(tmp) - 1, np.sin(tmp))) + b_eq = [1, 1] + + o = {key: self.options[key] for key in self.options} + o["presolve"] = False + + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=o) + _assert_infeasible(res) + + def test_basic_artificial_vars(self): + # Problem is chosen to test two phase simplex methods when at the end + # of phase 1 some artificial variables remain in the basis. + # Also, for `method='simplex'`, the row in the tableau corresponding + # with the artificial variables is not all zero. + c = np.array([-0.1, -0.07, 0.004, 0.004, 0.004, 0.004]) + A_ub = np.array([[1.0, 0, 0, 0, 0, 0], [-1.0, 0, 0, 0, 0, 0], + [0, -1.0, 0, 0, 0, 0], [0, 1.0, 0, 0, 0, 0], + [1.0, 1.0, 0, 0, 0, 0]]) + b_ub = np.array([3.0, 3.0, 3.0, 3.0, 20.0]) + A_eq = np.array([[1.0, 0, -1, 1, -1, 1], [0, -1.0, -1, 1, -1, 1]]) + b_eq = np.array([0, 0]) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=0, desired_x=np.zeros_like(c), + atol=2e-6) + + def test_optimize_result(self): + # check all fields in OptimizeResult + c, A_ub, b_ub, A_eq, b_eq, bounds = very_random_gen(0) + res = linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, + bounds=bounds, method=self.method, options=self.options) + assert_(res.success) + assert_(res.nit) + assert_(not res.status) + if 'highs' not in self.method: + # HiGHS status/message tested separately + assert_(res.message == "Optimization terminated successfully.") + assert_allclose(c @ res.x, res.fun) + assert_allclose(b_eq - A_eq @ res.x, res.con, atol=1e-11) + assert_allclose(b_ub - A_ub @ res.x, res.slack, atol=1e-11) + for key in ['eqlin', 'ineqlin', 'lower', 'upper']: + if key in res.keys(): + assert isinstance(res[key]['marginals'], np.ndarray) + assert isinstance(res[key]['residual'], np.ndarray) + + ################# + # Bug Fix Tests # + ################# + + def test_bug_5400(self): + # https://github.com/scipy/scipy/issues/5400 + bounds = [ + (0, None), + (0, 100), (0, 100), (0, 100), (0, 100), (0, 100), (0, 100), + (0, 900), (0, 900), (0, 900), (0, 900), (0, 900), (0, 900), + (0, None), (0, None), (0, None), (0, None), (0, None), (0, None)] + + f = 1 / 9 + g = -1e4 + h = -3.1 + A_ub = np.array([ + [1, -2.99, 0, 0, -3, 0, 0, 0, -1, -1, 0, -1, -1, 1, 1, 0, 0, 0, 0], + [1, 0, -2.9, h, 0, -3, 0, -1, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0, 0], + [1, 0, 0, h, 0, 0, -3, -1, -1, 0, -1, -1, 0, 0, 0, 0, 0, 1, 1], + [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], + [0, 1.99, -1, -1, 0, 0, 0, -1, f, f, 0, 0, 0, g, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 2, -1, -1, 0, 0, 0, -1, f, f, 0, g, 0, 0, 0, 0], + [0, -1, 1.9, 2.1, 0, 0, 0, f, -1, -1, 0, 0, 0, 0, 0, g, 0, 0, 0], + [0, 0, 0, 0, -1, 2, -1, 0, 0, 0, f, -1, f, 0, 0, 0, g, 0, 0], + [0, -1, -1, 2.1, 0, 0, 0, f, f, -1, 0, 0, 0, 0, 0, 0, 0, g, 0], + [0, 0, 0, 0, -1, -1, 2, 0, 0, 0, f, f, -1, 0, 0, 0, 0, 0, g]]) + + b_ub = np.array([ + 0.0, 0, 0, 100, 100, 100, 100, 100, 100, 900, 900, 900, 900, 900, + 900, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) + + c = np.array([-1.0, 1, 1, 1, 1, 1, 1, 1, 1, + 1, 1, 1, 1, 0, 0, 0, 0, 0, 0]) + with suppress_warnings() as sup: + sup.filter(OptimizeWarning, + "Solving system with option 'sym_pos'") + sup.filter(RuntimeWarning, "invalid value encountered") + sup.filter(LinAlgWarning) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=-106.63507541835018) + + def test_bug_6139(self): + # linprog(method='simplex') fails to find a basic feasible solution + # if phase 1 pseudo-objective function is outside the provided tol. + # https://github.com/scipy/scipy/issues/6139 + + # Note: This is not strictly a bug as the default tolerance determines + # if a result is "close enough" to zero and should not be expected + # to work for all cases. + + c = np.array([1, 1, 1]) + A_eq = np.array([[1., 0., 0.], [-1000., 0., - 1000.]]) + b_eq = np.array([5.00000000e+00, -1.00000000e+04]) + A_ub = -np.array([[0., 1000000., 1010000.]]) + b_ub = -np.array([10000000.]) + bounds = (None, None) + + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + + _assert_success(res, desired_fun=14.95, + desired_x=np.array([5, 4.95, 5])) + + def test_bug_6690(self): + # linprog simplex used to violate bound constraint despite reporting + # success. + # https://github.com/scipy/scipy/issues/6690 + + A_eq = np.array([[0, 0, 0, 0.93, 0, 0.65, 0, 0, 0.83, 0]]) + b_eq = np.array([0.9626]) + A_ub = np.array([ + [0, 0, 0, 1.18, 0, 0, 0, -0.2, 0, -0.22], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0.43, 0, 0, 0, 0, 0, 0], + [0, -1.22, -0.25, 0, 0, 0, -2.06, 0, 0, 1.37], + [0, 0, 0, 0, 0, 0, 0, -0.25, 0, 0] + ]) + b_ub = np.array([0.615, 0, 0.172, -0.869, -0.022]) + bounds = np.array([ + [-0.84, -0.97, 0.34, 0.4, -0.33, -0.74, 0.47, 0.09, -1.45, -0.73], + [0.37, 0.02, 2.86, 0.86, 1.18, 0.5, 1.76, 0.17, 0.32, -0.15] + ]).T + c = np.array([ + -1.64, 0.7, 1.8, -1.06, -1.16, 0.26, 2.13, 1.53, 0.66, 0.28 + ]) + + with suppress_warnings() as sup: + if has_umfpack: + sup.filter(UmfpackWarning) + sup.filter(OptimizeWarning, + "Solving system with option 'cholesky'") + sup.filter(OptimizeWarning, "Solving system with option 'sym_pos'") + sup.filter(RuntimeWarning, "invalid value encountered") + sup.filter(LinAlgWarning) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + + desired_fun = -1.19099999999 + desired_x = np.array([0.3700, -0.9700, 0.3400, 0.4000, 1.1800, + 0.5000, 0.4700, 0.0900, 0.3200, -0.7300]) + _assert_success(res, desired_fun=desired_fun, desired_x=desired_x) + + # Add small tol value to ensure arrays are less than or equal. + atol = 1e-6 + assert_array_less(bounds[:, 0] - atol, res.x) + assert_array_less(res.x, bounds[:, 1] + atol) + + def test_bug_7044(self): + # linprog simplex failed to "identify correct constraints" (?) + # leading to a non-optimal solution if A is rank-deficient. + # https://github.com/scipy/scipy/issues/7044 + + A_eq, b_eq, c, _, _ = magic_square(3) + with suppress_warnings() as sup: + sup.filter(OptimizeWarning, "A_eq does not appear...") + sup.filter(RuntimeWarning, "invalid value encountered") + sup.filter(LinAlgWarning) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + + desired_fun = 1.730550597 + _assert_success(res, desired_fun=desired_fun) + assert_allclose(A_eq.dot(res.x), b_eq) + assert_array_less(np.zeros(res.x.size) - 1e-5, res.x) + + def test_bug_7237(self): + # https://github.com/scipy/scipy/issues/7237 + # linprog simplex "explodes" when the pivot value is very + # close to zero. + + c = np.array([-1, 0, 0, 0, 0, 0, 0, 0, 0]) + A_ub = np.array([ + [1., -724., 911., -551., -555., -896., 478., -80., -293.], + [1., 566., 42., 937., 233., 883., 392., -909., 57.], + [1., -208., -894., 539., 321., 532., -924., 942., 55.], + [1., 857., -859., 83., 462., -265., -971., 826., 482.], + [1., 314., -424., 245., -424., 194., -443., -104., -429.], + [1., 540., 679., 361., 149., -827., 876., 633., 302.], + [0., -1., -0., -0., -0., -0., -0., -0., -0.], + [0., -0., -1., -0., -0., -0., -0., -0., -0.], + [0., -0., -0., -1., -0., -0., -0., -0., -0.], + [0., -0., -0., -0., -1., -0., -0., -0., -0.], + [0., -0., -0., -0., -0., -1., -0., -0., -0.], + [0., -0., -0., -0., -0., -0., -1., -0., -0.], + [0., -0., -0., -0., -0., -0., -0., -1., -0.], + [0., -0., -0., -0., -0., -0., -0., -0., -1.], + [0., 1., 0., 0., 0., 0., 0., 0., 0.], + [0., 0., 1., 0., 0., 0., 0., 0., 0.], + [0., 0., 0., 1., 0., 0., 0., 0., 0.], + [0., 0., 0., 0., 1., 0., 0., 0., 0.], + [0., 0., 0., 0., 0., 1., 0., 0., 0.], + [0., 0., 0., 0., 0., 0., 1., 0., 0.], + [0., 0., 0., 0., 0., 0., 0., 1., 0.], + [0., 0., 0., 0., 0., 0., 0., 0., 1.] + ]) + b_ub = np.array([ + 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., + 0., 0., 0., 1., 1., 1., 1., 1., 1., 1., 1.]) + A_eq = np.array([[0., 1., 1., 1., 1., 1., 1., 1., 1.]]) + b_eq = np.array([[1.]]) + bounds = [(None, None)] * 9 + + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=108.568535, atol=1e-6) + + def test_bug_8174(self): + # https://github.com/scipy/scipy/issues/8174 + # The simplex method sometimes "explodes" if the pivot value is very + # close to zero. + A_ub = np.array([ + [22714, 1008, 13380, -2713.5, -1116], + [-4986, -1092, -31220, 17386.5, 684], + [-4986, 0, 0, -2713.5, 0], + [22714, 0, 0, 17386.5, 0]]) + b_ub = np.zeros(A_ub.shape[0]) + c = -np.ones(A_ub.shape[1]) + bounds = [(0, 1)] * A_ub.shape[1] + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "invalid value encountered") + sup.filter(LinAlgWarning) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + + if self.options.get('tol', 1e-9) < 1e-10 and self.method == 'simplex': + _assert_unable_to_find_basic_feasible_sol(res) + else: + _assert_success(res, desired_fun=-2.0080717488789235, atol=1e-6) + + def test_bug_8174_2(self): + # Test supplementary example from issue 8174. + # https://github.com/scipy/scipy/issues/8174 + # https://stackoverflow.com/questions/47717012/linprog-in-scipy-optimize-checking-solution + c = np.array([1, 0, 0, 0, 0, 0, 0]) + A_ub = -np.identity(7) + b_ub = np.array([[-2], [-2], [-2], [-2], [-2], [-2], [-2]]) + A_eq = np.array([ + [1, 1, 1, 1, 1, 1, 0], + [0.3, 1.3, 0.9, 0, 0, 0, -1], + [0.3, 0, 0, 0, 0, 0, -2/3], + [0, 0.65, 0, 0, 0, 0, -1/15], + [0, 0, 0.3, 0, 0, 0, -1/15] + ]) + b_eq = np.array([[100], [0], [0], [0], [0]]) + + with suppress_warnings() as sup: + if has_umfpack: + sup.filter(UmfpackWarning) + sup.filter(OptimizeWarning, "A_eq does not appear...") + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_fun=43.3333333331385) + + def test_bug_8561(self): + # Test that pivot row is chosen correctly when using Bland's rule + # This was originally written for the simplex method with + # Bland's rule only, but it doesn't hurt to test all methods/options + # https://github.com/scipy/scipy/issues/8561 + c = np.array([7, 0, -4, 1.5, 1.5]) + A_ub = np.array([ + [4, 5.5, 1.5, 1.0, -3.5], + [1, -2.5, -2, 2.5, 0.5], + [3, -0.5, 4, -12.5, -7], + [-1, 4.5, 2, -3.5, -2], + [5.5, 2, -4.5, -1, 9.5]]) + b_ub = np.array([0, 0, 0, 0, 1]) + res = linprog(c, A_ub=A_ub, b_ub=b_ub, options=self.options, + method=self.method) + _assert_success(res, desired_x=[0, 0, 19, 16/3, 29/3]) + + def test_bug_8662(self): + # linprog simplex used to report incorrect optimal results + # https://github.com/scipy/scipy/issues/8662 + c = [-10, 10, 6, 3] + A_ub = [[8, -8, -4, 6], + [-8, 8, 4, -6], + [-4, 4, 8, -4], + [3, -3, -3, -10]] + b_ub = [9, -9, -9, -4] + bounds = [(0, None), (0, None), (0, None), (0, None)] + desired_fun = 36.0000000000 + + with suppress_warnings() as sup: + if has_umfpack: + sup.filter(UmfpackWarning) + sup.filter(RuntimeWarning, "invalid value encountered") + sup.filter(LinAlgWarning) + res1 = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + + # Set boundary condition as a constraint + A_ub.append([0, 0, -1, 0]) + b_ub.append(0) + bounds[2] = (None, None) + + with suppress_warnings() as sup: + if has_umfpack: + sup.filter(UmfpackWarning) + sup.filter(RuntimeWarning, "invalid value encountered") + sup.filter(LinAlgWarning) + res2 = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + rtol = 1e-5 + _assert_success(res1, desired_fun=desired_fun, rtol=rtol) + _assert_success(res2, desired_fun=desired_fun, rtol=rtol) + + def test_bug_8663(self): + # exposed a bug in presolve + # https://github.com/scipy/scipy/issues/8663 + c = [1, 5] + A_eq = [[0, -7]] + b_eq = [-6] + bounds = [(0, None), (None, None)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_x=[0, 6./7], desired_fun=5*6./7) + + def test_bug_8664(self): + # interior-point has trouble with this when presolve is off + # tested for interior-point with presolve off in TestLinprogIPSpecific + # https://github.com/scipy/scipy/issues/8664 + c = [4] + A_ub = [[2], [5]] + b_ub = [4, 4] + A_eq = [[0], [-8], [9]] + b_eq = [3, 2, 10] + with suppress_warnings() as sup: + sup.filter(RuntimeWarning) + sup.filter(OptimizeWarning, "Solving system with option...") + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_infeasible(res) + + def test_bug_8973(self): + """ + Test whether bug described at: + https://github.com/scipy/scipy/issues/8973 + was fixed. + """ + c = np.array([0, 0, 0, 1, -1]) + A_ub = np.array([[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]]) + b_ub = np.array([2, -2]) + bounds = [(None, None), (None, None), (None, None), (-1, 1), (-1, 1)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + # solution vector x is not unique + _assert_success(res, desired_fun=-2) + # HiGHS IPM had an issue where the following wasn't true! + assert_equal(c @ res.x, res.fun) + + def test_bug_8973_2(self): + """ + Additional test for: + https://github.com/scipy/scipy/issues/8973 + suggested in + https://github.com/scipy/scipy/pull/8985 + review by @antonior92 + """ + c = np.zeros(1) + A_ub = np.array([[1]]) + b_ub = np.array([-2]) + bounds = (None, None) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_x=[-2], desired_fun=0) + + def test_bug_10124(self): + """ + Test for linprog docstring problem + 'disp'=True caused revised simplex failure + """ + c = np.zeros(1) + A_ub = np.array([[1]]) + b_ub = np.array([-2]) + bounds = (None, None) + c = [-1, 4] + A_ub = [[-3, 1], [1, 2]] + b_ub = [6, 4] + bounds = [(None, None), (-3, None)] + o = {"disp": True} + o.update(self.options) + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=o) + _assert_success(res, desired_x=[10, -3], desired_fun=-22) + + def test_bug_10349(self): + """ + Test for redundancy removal tolerance issue + https://github.com/scipy/scipy/issues/10349 + """ + A_eq = np.array([[1, 1, 0, 0, 0, 0], + [0, 0, 1, 1, 0, 0], + [0, 0, 0, 0, 1, 1], + [1, 0, 1, 0, 0, 0], + [0, 0, 0, 1, 1, 0], + [0, 1, 0, 0, 0, 1]]) + b_eq = np.array([221, 210, 10, 141, 198, 102]) + c = np.concatenate((0, 1, np.zeros(4)), axis=None) + with suppress_warnings() as sup: + sup.filter(OptimizeWarning, "A_eq does not appear...") + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options) + _assert_success(res, desired_x=[129, 92, 12, 198, 0, 10], desired_fun=92) + + @pytest.mark.skipif(sys.platform == 'darwin', + reason=("Failing on some local macOS builds, " + "see gh-13846")) + def test_bug_10466(self): + """ + Test that autoscale fixes poorly-scaled problem + """ + c = [-8., -0., -8., -0., -8., -0., -0., -0., -0., -0., -0., -0., -0.] + A_eq = [[1., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.], + [0., 0., 1., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.], + [0., 0., 0., 0., 1., 1., 0., 0., 0., 0., 0., 0., 0.], + [1., 0., 1., 0., 1., 0., -1., 0., 0., 0., 0., 0., 0.], + [1., 0., 1., 0., 1., 0., 0., 1., 0., 0., 0., 0., 0.], + [1., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0.], + [1., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0.], + [1., 0., 1., 0., 1., 0., 0., 0., 0., 0., 1., 0., 0.], + [0., 0., 1., 0., 1., 0., 0., 0., 0., 0., 0., 1., 0.], + [0., 0., 1., 0., 1., 0., 0., 0., 0., 0., 0., 0., 1.]] + + b_eq = [3.14572800e+08, 4.19430400e+08, 5.24288000e+08, + 1.00663296e+09, 1.07374182e+09, 1.07374182e+09, + 1.07374182e+09, 1.07374182e+09, 1.07374182e+09, + 1.07374182e+09] + + o = {} + # HiGHS methods don't use autoscale option + if not self.method.startswith("highs"): + o = {"autoscale": True} + o.update(self.options) + + with suppress_warnings() as sup: + sup.filter(OptimizeWarning, "Solving system with option...") + if has_umfpack: + sup.filter(UmfpackWarning) + sup.filter(RuntimeWarning, "scipy.linalg.solve\nIll...") + sup.filter(RuntimeWarning, "divide by zero encountered...") + sup.filter(RuntimeWarning, "overflow encountered...") + sup.filter(RuntimeWarning, "invalid value encountered...") + sup.filter(LinAlgWarning, "Ill-conditioned matrix...") + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=o) + assert_allclose(res.fun, -8589934560) + + +######################### +# Method-specific Tests # +######################### + + +@pytest.mark.filterwarnings("ignore::DeprecationWarning") +class LinprogSimplexTests(LinprogCommonTests): + method = "simplex" + + +@pytest.mark.filterwarnings("ignore::DeprecationWarning") +class LinprogIPTests(LinprogCommonTests): + method = "interior-point" + + def test_bug_10466(self): + pytest.skip("Test is failing, but solver is deprecated.") + + +@pytest.mark.filterwarnings("ignore::DeprecationWarning") +class LinprogRSTests(LinprogCommonTests): + method = "revised simplex" + + # Revised simplex does not reliably solve these problems. + # Failure is intermittent due to the random choice of elements to complete + # the basis after phase 1 terminates. In any case, linprog exists + # gracefully, reporting numerical difficulties. I do not think this should + # prevent revised simplex from being merged, as it solves the problems + # most of the time and solves a broader range of problems than the existing + # simplex implementation. + # I believe that the root cause is the same for all three and that this + # same issue prevents revised simplex from solving many other problems + # reliably. Somehow the pivoting rule allows the algorithm to pivot into + # a singular basis. I haven't been able to find a reference that + # acknowledges this possibility, suggesting that there is a bug. On the + # other hand, the pivoting rule is quite simple, and I can't find a + # mistake, which suggests that this is a possibility with the pivoting + # rule. Hopefully, a better pivoting rule will fix the issue. + + def test_bug_5400(self): + pytest.skip("Intermittent failure acceptable.") + + def test_bug_8662(self): + pytest.skip("Intermittent failure acceptable.") + + def test_network_flow(self): + pytest.skip("Intermittent failure acceptable.") + + +class LinprogHiGHSTests(LinprogCommonTests): + def test_callback(self): + # this is the problem from test_callback + def cb(res): + return None + c = np.array([-3, -2]) + A_ub = [[2, 1], [1, 1], [1, 0]] + b_ub = [10, 8, 4] + assert_raises(NotImplementedError, linprog, c, A_ub=A_ub, b_ub=b_ub, + callback=cb, method=self.method) + res = linprog(c, A_ub=A_ub, b_ub=b_ub, method=self.method) + _assert_success(res, desired_fun=-18.0, desired_x=[2, 6]) + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize("options", + [{"maxiter": -1}, + {"disp": -1}, + {"presolve": -1}, + {"time_limit": -1}, + {"dual_feasibility_tolerance": -1}, + {"primal_feasibility_tolerance": -1}, + {"ipm_optimality_tolerance": -1}, + {"simplex_dual_edge_weight_strategy": "ekki"}, + ]) + def test_invalid_option_values(self, options): + def f(options): + linprog(1, method=self.method, options=options) + options.update(self.options) + assert_warns(OptimizeWarning, f, options=options) + + def test_crossover(self): + A_eq, b_eq, c, _, _ = magic_square(4) + bounds = (0, 1) + res = linprog(c, A_eq=A_eq, b_eq=b_eq, + bounds=bounds, method=self.method, options=self.options) + # there should be nonzero crossover iterations for IPM (only) + assert_equal(res.crossover_nit == 0, self.method != "highs-ipm") + + @pytest.mark.fail_slow(10) + def test_marginals(self): + # Ensure lagrange multipliers are correct by comparing the derivative + # w.r.t. b_ub/b_eq/ub/lb to the reported duals. + c, A_ub, b_ub, A_eq, b_eq, bounds = very_random_gen(seed=0) + res = linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, + bounds=bounds, method=self.method, options=self.options) + lb, ub = bounds.T + + # sensitivity w.r.t. b_ub + def f_bub(x): + return linprog(c, A_ub, x, A_eq, b_eq, bounds, + method=self.method).fun + + dfdbub = approx_derivative(f_bub, b_ub, method='3-point', f0=res.fun) + assert_allclose(res.ineqlin.marginals, dfdbub) + + # sensitivity w.r.t. b_eq + def f_beq(x): + return linprog(c, A_ub, b_ub, A_eq, x, bounds, + method=self.method).fun + + dfdbeq = approx_derivative(f_beq, b_eq, method='3-point', f0=res.fun) + assert_allclose(res.eqlin.marginals, dfdbeq) + + # sensitivity w.r.t. lb + def f_lb(x): + bounds = np.array([x, ub]).T + return linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method).fun + + with np.errstate(invalid='ignore'): + # approx_derivative has trouble where lb is infinite + dfdlb = approx_derivative(f_lb, lb, method='3-point', f0=res.fun) + dfdlb[~np.isfinite(lb)] = 0 + + assert_allclose(res.lower.marginals, dfdlb) + + # sensitivity w.r.t. ub + def f_ub(x): + bounds = np.array([lb, x]).T + return linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method).fun + + with np.errstate(invalid='ignore'): + dfdub = approx_derivative(f_ub, ub, method='3-point', f0=res.fun) + dfdub[~np.isfinite(ub)] = 0 + + assert_allclose(res.upper.marginals, dfdub) + + def test_dual_feasibility(self): + # Ensure solution is dual feasible using marginals + c, A_ub, b_ub, A_eq, b_eq, bounds = very_random_gen(seed=42) + res = linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, + bounds=bounds, method=self.method, options=self.options) + + # KKT dual feasibility equation from Theorem 1 from + # http://www.personal.psu.edu/cxg286/LPKKT.pdf + resid = (-c + A_ub.T @ res.ineqlin.marginals + + A_eq.T @ res.eqlin.marginals + + res.upper.marginals + + res.lower.marginals) + assert_allclose(resid, 0, atol=1e-12) + + def test_complementary_slackness(self): + # Ensure that the complementary slackness condition is satisfied. + c, A_ub, b_ub, A_eq, b_eq, bounds = very_random_gen(seed=42) + res = linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, + bounds=bounds, method=self.method, options=self.options) + + # KKT complementary slackness equation from Theorem 1 from + # http://www.personal.psu.edu/cxg286/LPKKT.pdf modified for + # non-zero RHS + assert np.allclose(res.ineqlin.marginals @ (b_ub - A_ub @ res.x), 0) + + @pytest.mark.xfail(reason='Upstream / Wrapper issue, see gh-20589') + def test_bug_20336(self): + """ + Test that `linprog` now solves a poorly-scaled problem + """ + boundaries = [(10000.0, 9010000.0), (0.0, None), (10000.0, None), + (0.0, 84.62623413258109), (10000.0, None), (10000.0, None), + (10000.0, None), (10000.0, None), (10000.0, None), + (10000.0, None), (10000.0, None), (10000.0, None), + (10000.0, None), (None, None), (None, None), (None, None), + (None, None), (None, None), (None, None), (None, None), + (None, None), (None, None), (None, None), (None, None), + (None, None), (None, None), (None, None), (None, None), + (None, None), (None, None), (None, None), (None, None), + (None, None)] + eq_entries = [-1.0, 1.0, -1.0, 1.0, -1.0, 1.0, -1.0, 1.0, -1.0, 1.0, -1.0, 1.0, + -1.0, 1.0, -1.0, 1.0, -1.0, 1.0, -1.0, 1.0, 1.0, 1.0, -1.0, 0.001, + -0.001, 3.7337777768059636e-10, 3.7337777768059636e-10, 1.0, -1.0, + 0.001, -0.001, 3.7337777768059636e-10, 3.7337777768059636e-10, + 1.0, -1.0, 0.001, -0.001, 3.7337777768059636e-10, + 3.7337777768059636e-10, 1.0, -1.0, 0.001, -0.001, + 3.7337777768059636e-10, 3.7337777768059636e-10, 1.0, -1.0, 0.001, + -0.001, 3.7337777768059636e-10, 3.7337777768059636e-10, 1.0, -1.0, + 0.001, -0.001, 3.7337777768059636e-10, 3.7337777768059636e-10, + 1.0, -1.0, 0.001, -0.001, 3.7337777768059636e-10, + 3.7337777768059636e-10, 1.0, -1.0, 0.001, -0.001, + 3.7337777768059636e-10, 3.7337777768059636e-10, 1.0, -1.0, 0.001, + -0.001, 3.7337777768059636e-10, 3.7337777768059636e-10, 1.0, + -1.0, 0.001, -0.001, 3.7337777768059636e-10, + 3.7337777768059636e-10, 1.0, -1.0] + eq_indizes = [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, + 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, + 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, + 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 26, 26, 26, 26, + 27, 27, 28, 28, 28, 28, 29, 29, 30, 30, 30, 30, 31, 31] + eq_vars = [15, 14, 17, 16, 19, 18, 21, 20, 23, 22, 25, 24, 27, 26, 29, 28, 31, + 30, 13, 1, 0, 32, 3, 14, 13, 4, 0, 4, 0, 32, 31, 2, 12, 2, 12, 16, + 15, 5, 4, 5, 4, 18, 17, 6, 5, 6, 5, 20, 19, 7, 6, 7, 6, 22, 21, 8, + 7, 8, 7, 24, 23, 9, 8, 9, 8, 26, 25, 10, 9, 10, 9, 28, 27, 11, 10, + 11, 10, 30, 29, 12, 11, 12, 11] + eq_values = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 9000000.0, 0.0, + 0.006587392118285457, -5032.197406716549, 0.0041860502789104696, + -7918.93439542944, 0.0063205763583549035, -5244.625751707402, + 0.006053760598424349, -5475.7793929428, 0.005786944838493795, + -5728.248403917573, 0.0055201290785632405, -6005.123623538355, + 0.005253313318632687, -6310.123825488683, 0.004986497558702133, + -6647.763714796453, 0.004719681798771578, -7023.578908071522, + 0.004452866038841024, -7444.431798646482] + coefficients = [0.0, 0.0, 0.0, -0.011816666666666668, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] + np_eq_entries = np.asarray(eq_entries, dtype=np.float64) + np_eq_indizes = np.asarray(eq_indizes, dtype=np.int32) + np_eq_vars = np.asarray(eq_vars, dtype=np.int32) + + a_eq= scipy.sparse.csr_array((np_eq_entries,(np_eq_indizes, np_eq_vars)), + shape=(32, 33)) + b_eq = np.asarray(eq_values, dtype=np.float64) + c = np.asarray(coefficients, dtype=np.float64) + + result = scipy.optimize.linprog(c, A_ub=None, b_ub=None, A_eq=a_eq, b_eq=b_eq, + bounds=boundaries) + assert result.status==0 + x = result.x + n_r_x = np.linalg.norm(a_eq @ x - b_eq) + n_r = np.linalg.norm(result.eqlin.residual) + assert_allclose(n_r, n_r_x) + + +################################ +# Simplex Option-Specific Tests# +################################ + + +class TestLinprogSimplexDefault(LinprogSimplexTests): + + def setup_method(self): + self.options = {} + + def test_bug_5400(self): + pytest.skip("Simplex fails on this problem.") + + def test_bug_7237_low_tol(self): + # Fails if the tolerance is too strict. Here, we test that + # even if the solution is wrong, the appropriate error is raised. + pytest.skip("Simplex fails on this problem.") + + @pytest.mark.thread_unsafe + def test_bug_8174_low_tol(self): + # Fails if the tolerance is too strict. Here, we test that + # even if the solution is wrong, the appropriate warning is issued. + self.options.update({'tol': 1e-12}) + with pytest.warns(OptimizeWarning): + super().test_bug_8174() + + +class TestLinprogSimplexBland(LinprogSimplexTests): + + def setup_method(self): + self.options = {'bland': True} + + def test_bug_5400(self): + pytest.skip("Simplex fails on this problem.") + + @pytest.mark.thread_unsafe + def test_bug_8174_low_tol(self): + # Fails if the tolerance is too strict. Here, we test that + # even if the solution is wrong, the appropriate error is raised. + self.options.update({'tol': 1e-12}) + with pytest.raises(AssertionError): + with pytest.warns(OptimizeWarning): + super().test_bug_8174() + + +class TestLinprogSimplexNoPresolve(LinprogSimplexTests): + + def setup_method(self): + self.options = {'presolve': False} + + is_32_bit = np.intp(0).itemsize < 8 + is_linux = sys.platform.startswith('linux') + + @pytest.mark.xfail( + condition=is_32_bit and is_linux, + reason='Fails with warning on 32-bit linux') + def test_bug_5400(self): + super().test_bug_5400() + + def test_bug_6139_low_tol(self): + # Linprog(method='simplex') fails to find a basic feasible solution + # if phase 1 pseudo-objective function is outside the provided tol. + # https://github.com/scipy/scipy/issues/6139 + # Without ``presolve`` eliminating such rows the result is incorrect. + self.options.update({'tol': 1e-12}) + with pytest.raises(AssertionError, match='linprog status 4'): + return super().test_bug_6139() + + def test_bug_7237_low_tol(self): + pytest.skip("Simplex fails on this problem.") + + @pytest.mark.thread_unsafe + def test_bug_8174_low_tol(self): + # Fails if the tolerance is too strict. Here, we test that + # even if the solution is wrong, the appropriate warning is issued. + self.options.update({'tol': 1e-12}) + with pytest.warns(OptimizeWarning): + super().test_bug_8174() + + def test_unbounded_no_nontrivial_constraints_1(self): + pytest.skip("Tests behavior specific to presolve") + + def test_unbounded_no_nontrivial_constraints_2(self): + pytest.skip("Tests behavior specific to presolve") + + +####################################### +# Interior-Point Option-Specific Tests# +####################################### + + +class TestLinprogIPDense(LinprogIPTests): + options = {"sparse": False} + + # see https://github.com/scipy/scipy/issues/20216 for skip reason + @pytest.mark.skipif( + sys.platform == 'darwin', + reason="Fails on some macOS builds for reason not relevant to test" + ) + def test_bug_6139(self): + super().test_bug_6139() + +if has_cholmod: + class TestLinprogIPSparseCholmod(LinprogIPTests): + options = {"sparse": True, "cholesky": True} + + +if has_umfpack: + class TestLinprogIPSparseUmfpack(LinprogIPTests): + options = {"sparse": True, "cholesky": False} + + def test_network_flow_limited_capacity(self): + pytest.skip("Failing due to numerical issues on some platforms.") + + +class TestLinprogIPSparse(LinprogIPTests): + options = {"sparse": True, "cholesky": False, "sym_pos": False} + + @pytest.mark.skipif( + sys.platform == 'darwin', + reason="Fails on macOS x86 Accelerate builds (gh-20510)" + ) + @pytest.mark.xfail_on_32bit("This test is sensitive to machine epsilon level " + "perturbations in linear system solution in " + "_linprog_ip._sym_solve.") + def test_bug_6139(self): + super().test_bug_6139() + + @pytest.mark.xfail(reason='Fails with ATLAS, see gh-7877') + def test_bug_6690(self): + # Test defined in base class, but can't mark as xfail there + super().test_bug_6690() + + def test_magic_square_sparse_no_presolve(self): + # test linprog with a problem with a rank-deficient A_eq matrix + A_eq, b_eq, c, _, _ = magic_square(3) + bounds = (0, 1) + + with suppress_warnings() as sup: + if has_umfpack: + sup.filter(UmfpackWarning) + sup.filter(MatrixRankWarning, "Matrix is exactly singular") + sup.filter(OptimizeWarning, "Solving system with option...") + + o = {key: self.options[key] for key in self.options} + o["presolve"] = False + + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=o) + _assert_success(res, desired_fun=1.730550597) + + def test_sparse_solve_options(self): + # checking that problem is solved with all column permutation options + A_eq, b_eq, c, _, _ = magic_square(3) + with suppress_warnings() as sup: + sup.filter(OptimizeWarning, "A_eq does not appear...") + sup.filter(OptimizeWarning, "Invalid permc_spec option") + o = {key: self.options[key] for key in self.options} + permc_specs = ('NATURAL', 'MMD_ATA', 'MMD_AT_PLUS_A', + 'COLAMD', 'ekki-ekki-ekki') + # 'ekki-ekki-ekki' raises warning about invalid permc_spec option + # and uses default + for permc_spec in permc_specs: + o["permc_spec"] = permc_spec + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=o) + _assert_success(res, desired_fun=1.730550597) + + +class TestLinprogIPSparsePresolve(LinprogIPTests): + options = {"sparse": True, "_sparse_presolve": True} + + @pytest.mark.skipif( + sys.platform == 'darwin', + reason="Fails on macOS x86 Accelerate builds (gh-20510)" + ) + @pytest.mark.xfail_on_32bit("This test is sensitive to machine epsilon level " + "perturbations in linear system solution in " + "_linprog_ip._sym_solve.") + def test_bug_6139(self): + super().test_bug_6139() + + def test_enzo_example_c_with_infeasibility(self): + pytest.skip('_sparse_presolve=True incompatible with presolve=False') + + @pytest.mark.xfail(reason='Fails with ATLAS, see gh-7877') + def test_bug_6690(self): + # Test defined in base class, but can't mark as xfail there + super().test_bug_6690() + + +@pytest.mark.filterwarnings("ignore::DeprecationWarning") +class TestLinprogIPSpecific: + method = "interior-point" + # the following tests don't need to be performed separately for + # sparse presolve, sparse after presolve, and dense + + def test_solver_select(self): + # check that default solver is selected as expected + if has_cholmod: + options = {'sparse': True, 'cholesky': True} + elif has_umfpack: + options = {'sparse': True, 'cholesky': False} + else: + options = {'sparse': True, 'cholesky': False, 'sym_pos': False} + A, b, c = lpgen_2d(20, 20) + res1 = linprog(c, A_ub=A, b_ub=b, method=self.method, options=options) + res2 = linprog(c, A_ub=A, b_ub=b, method=self.method) # default solver + assert_allclose(res1.fun, res2.fun, + err_msg="linprog default solver unexpected result", + rtol=2e-15, atol=1e-15) + + def test_unbounded_below_no_presolve_original(self): + # formerly caused segfault in TravisCI w/ "cholesky":True + c = [-1] + bounds = [(None, 1)] + res = linprog(c=c, bounds=bounds, + method=self.method, + options={"presolve": False, "cholesky": True}) + _assert_success(res, desired_fun=-1) + + def test_cholesky(self): + # use cholesky factorization and triangular solves + A, b, c = lpgen_2d(20, 20) + res = linprog(c, A_ub=A, b_ub=b, method=self.method, + options={"cholesky": True}) # only for dense + _assert_success(res, desired_fun=-64.049494229) + + def test_alternate_initial_point(self): + # use "improved" initial point + A, b, c = lpgen_2d(20, 20) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "scipy.linalg.solve\nIll...") + sup.filter(OptimizeWarning, "Solving system with option...") + sup.filter(LinAlgWarning, "Ill-conditioned matrix...") + res = linprog(c, A_ub=A, b_ub=b, method=self.method, + options={"ip": True, "disp": True}) + # ip code is independent of sparse/dense + _assert_success(res, desired_fun=-64.049494229) + + def test_bug_8664(self): + # interior-point has trouble with this when presolve is off + c = [4] + A_ub = [[2], [5]] + b_ub = [4, 4] + A_eq = [[0], [-8], [9]] + b_eq = [3, 2, 10] + with suppress_warnings() as sup: + sup.filter(RuntimeWarning) + sup.filter(OptimizeWarning, "Solving system with option...") + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options={"presolve": False}) + assert_(not res.success, "Incorrectly reported success") + + +######################################## +# Revised Simplex Option-Specific Tests# +######################################## + + +class TestLinprogRSCommon(LinprogRSTests): + options = {} + + def test_cyclic_bland(self): + pytest.skip("Intermittent failure acceptable.") + + def test_nontrivial_problem_with_guess(self): + c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem() + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options, x0=x_star) + _assert_success(res, desired_fun=f_star, desired_x=x_star) + assert_equal(res.nit, 0) + + def test_nontrivial_problem_with_unbounded_variables(self): + c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem() + bounds = [(None, None), (None, None), (0, None), (None, None)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options, x0=x_star) + _assert_success(res, desired_fun=f_star, desired_x=x_star) + assert_equal(res.nit, 0) + + def test_nontrivial_problem_with_bounded_variables(self): + c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem() + bounds = [(None, 1), (1, None), (0, None), (.4, .6)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options, x0=x_star) + _assert_success(res, desired_fun=f_star, desired_x=x_star) + assert_equal(res.nit, 0) + + def test_nontrivial_problem_with_negative_unbounded_variable(self): + c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem() + b_eq = [4] + x_star = np.array([-219/385, 582/385, 0, 4/10]) + f_star = 3951/385 + bounds = [(None, None), (1, None), (0, None), (.4, .6)] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options, x0=x_star) + _assert_success(res, desired_fun=f_star, desired_x=x_star) + assert_equal(res.nit, 0) + + def test_nontrivial_problem_with_bad_guess(self): + c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem() + bad_guess = [1, 2, 3, .5] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options, x0=bad_guess) + assert_equal(res.status, 6) + + def test_redundant_constraints_with_guess(self): + A, b, c, _, _ = magic_square(3) + p = np.random.rand(*c.shape) + with suppress_warnings() as sup: + sup.filter(OptimizeWarning, "A_eq does not appear...") + sup.filter(RuntimeWarning, "invalid value encountered") + sup.filter(LinAlgWarning) + res = linprog(c, A_eq=A, b_eq=b, method=self.method) + res2 = linprog(c, A_eq=A, b_eq=b, method=self.method, x0=res.x) + res3 = linprog(c + p, A_eq=A, b_eq=b, method=self.method, x0=res.x) + _assert_success(res2, desired_fun=1.730550597) + assert_equal(res2.nit, 0) + _assert_success(res3) + assert_(res3.nit < res.nit) # hot start reduces iterations + + +class TestLinprogRSBland(LinprogRSTests): + options = {"pivot": "bland"} + + +############################################ +# HiGHS-Simplex-Dual Option-Specific Tests # +############################################ + + +class TestLinprogHiGHSSimplexDual(LinprogHiGHSTests): + method = "highs-ds" + options = {} + + def test_lad_regression(self): + ''' + The scaled model should be optimal, i.e. not produce unscaled model + infeasible. See https://github.com/ERGO-Code/HiGHS/issues/494. + ''' + # Test to ensure gh-13610 is resolved (mismatch between HiGHS scaled + # and unscaled model statuses) + c, A_ub, b_ub, bnds = l1_regression_prob() + res = linprog(c, A_ub=A_ub, b_ub=b_ub, bounds=bnds, + method=self.method, options=self.options) + assert_equal(res.status, 0) + assert_(res.x is not None) + assert_(np.all(res.slack > -1e-6)) + assert_(np.all(res.x <= [np.inf if ub is None else ub + for lb, ub in bnds])) + assert_(np.all(res.x >= [-np.inf if lb is None else lb - 1e-7 + for lb, ub in bnds])) + + +################################### +# HiGHS-IPM Option-Specific Tests # +################################### + + +class TestLinprogHiGHSIPM(LinprogHiGHSTests): + method = "highs-ipm" + options = {} + + +################################### +# HiGHS-MIP Option-Specific Tests # +################################### + + +class TestLinprogHiGHSMIP: + method = "highs" + options = {} + + @pytest.mark.fail_slow(10) + @pytest.mark.xfail(condition=(sys.maxsize < 2 ** 32 and + platform.system() == "Linux"), + run=False, + reason="gh-16347") + def test_mip1(self): + # solve non-relaxed magic square problem (finally!) + # also check that values are all integers - they don't always + # come out of HiGHS that way + n = 4 + A, b, c, numbers, M = magic_square(n) + bounds = [(0, 1)] * len(c) + integrality = [1] * len(c) + + res = linprog(c=c*0, A_eq=A, b_eq=b, bounds=bounds, + method=self.method, integrality=integrality) + + s = (numbers.flatten() * res.x).reshape(n**2, n, n) + square = np.sum(s, axis=0) + np.testing.assert_allclose(square.sum(axis=0), M) + np.testing.assert_allclose(square.sum(axis=1), M) + np.testing.assert_allclose(np.diag(square).sum(), M) + np.testing.assert_allclose(np.diag(square[:, ::-1]).sum(), M) + + np.testing.assert_allclose(res.x, np.round(res.x), atol=1e-12) + + def test_mip2(self): + # solve MIP with inequality constraints and all integer constraints + # source: slide 5, + # https://www.cs.upc.edu/~erodri/webpage/cps/theory/lp/milp/slides.pdf + + # use all array inputs to test gh-16681 (integrality couldn't be array) + A_ub = np.array([[2, -2], [-8, 10]]) + b_ub = np.array([-1, 13]) + c = -np.array([1, 1]) + + bounds = np.array([(0, np.inf)] * len(c)) + integrality = np.ones_like(c) + + res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, bounds=bounds, + method=self.method, integrality=integrality) + + np.testing.assert_allclose(res.x, [1, 2]) + np.testing.assert_allclose(res.fun, -3) + + def test_mip3(self): + # solve MIP with inequality constraints and all integer constraints + # source: https://en.wikipedia.org/wiki/Integer_programming#Example + A_ub = np.array([[-1, 1], [3, 2], [2, 3]]) + b_ub = np.array([1, 12, 12]) + c = -np.array([0, 1]) + + bounds = [(0, np.inf)] * len(c) + integrality = [1] * len(c) + + res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, bounds=bounds, + method=self.method, integrality=integrality) + + np.testing.assert_allclose(res.fun, -2) + # two optimal solutions possible, just need one of them + assert np.allclose(res.x, [1, 2]) or np.allclose(res.x, [2, 2]) + + def test_mip4(self): + # solve MIP with inequality constraints and only one integer constraint + # source: https://www.mathworks.com/help/optim/ug/intlinprog.html + A_ub = np.array([[-1, -2], [-4, -1], [2, 1]]) + b_ub = np.array([14, -33, 20]) + c = np.array([8, 1]) + + bounds = [(0, np.inf)] * len(c) + integrality = [0, 1] + + res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, bounds=bounds, + method=self.method, integrality=integrality) + + np.testing.assert_allclose(res.x, [6.5, 7]) + np.testing.assert_allclose(res.fun, 59) + + def test_mip5(self): + # solve MIP with inequality and inequality constraints + # source: https://www.mathworks.com/help/optim/ug/intlinprog.html + A_ub = np.array([[1, 1, 1]]) + b_ub = np.array([7]) + A_eq = np.array([[4, 2, 1]]) + b_eq = np.array([12]) + c = np.array([-3, -2, -1]) + + bounds = [(0, np.inf), (0, np.inf), (0, 1)] + integrality = [0, 1, 0] + + res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, + bounds=bounds, method=self.method, + integrality=integrality) + + np.testing.assert_allclose(res.x, [0, 6, 0]) + np.testing.assert_allclose(res.fun, -12) + + # gh-16897: these fields were not present, ensure that they are now + assert res.get("mip_node_count", None) is not None + assert res.get("mip_dual_bound", None) is not None + assert res.get("mip_gap", None) is not None + + @pytest.mark.xslow + def test_mip6(self): + # solve a larger MIP with only equality constraints + # source: https://www.mathworks.com/help/optim/ug/intlinprog.html + A_eq = np.array([[22, 13, 26, 33, 21, 3, 14, 26], + [39, 16, 22, 28, 26, 30, 23, 24], + [18, 14, 29, 27, 30, 38, 26, 26], + [41, 26, 28, 36, 18, 38, 16, 26]]) + b_eq = np.array([7872, 10466, 11322, 12058]) + c = np.array([2, 10, 13, 17, 7, 5, 7, 3]) + + bounds = [(0, np.inf)]*8 + integrality = [1]*8 + + res = linprog(c=c, A_eq=A_eq, b_eq=b_eq, bounds=bounds, + method=self.method, integrality=integrality) + + np.testing.assert_allclose(res.fun, 1854) + + @pytest.mark.xslow + def test_mip_rel_gap_passdown(self): + # MIP taken from test_mip6, solved with different values of mip_rel_gap + # solve a larger MIP with only equality constraints + # source: https://www.mathworks.com/help/optim/ug/intlinprog.html + A_eq = np.array([[22, 13, 26, 33, 21, 3, 14, 26], + [39, 16, 22, 28, 26, 30, 23, 24], + [18, 14, 29, 27, 30, 38, 26, 26], + [41, 26, 28, 36, 18, 38, 16, 26]]) + b_eq = np.array([7872, 10466, 11322, 12058]) + c = np.array([2, 10, 13, 17, 7, 5, 7, 3]) + + bounds = [(0, np.inf)]*8 + integrality = [1]*8 + + mip_rel_gaps = [0.5, 0.25, 0.01, 0.001] + sol_mip_gaps = [] + for mip_rel_gap in mip_rel_gaps: + res = linprog(c=c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, + bounds=bounds, method=self.method, + integrality=integrality, + options={"mip_rel_gap": mip_rel_gap}) + final_mip_gap = res["mip_gap"] + # assert that the solution actually has mip_gap lower than the + # required mip_rel_gap supplied + assert final_mip_gap <= mip_rel_gap + sol_mip_gaps.append(final_mip_gap) + + # make sure that the mip_rel_gap parameter is actually doing something + # check that differences between solution gaps are declining + # monotonically with the mip_rel_gap parameter. np.diff does + # x[i+1] - x[i], so flip the array before differencing to get + # what should be a positive, monotone decreasing series of solution + # gaps + gap_diffs = np.diff(np.flip(sol_mip_gaps)) + assert np.all(gap_diffs >= 0) + assert not np.all(gap_diffs == 0) + + def test_semi_continuous(self): + # See issue #18106. This tests whether the solution is being + # checked correctly (status is 0) when integrality > 1: + # values are allowed to be 0 even if 0 is out of bounds. + + c = np.array([1., 1., -1, -1]) + bounds = np.array([[0.5, 1.5], [0.5, 1.5], [0.5, 1.5], [0.5, 1.5]]) + integrality = np.array([2, 3, 2, 3]) + + res = linprog(c, bounds=bounds, + integrality=integrality, method='highs') + + np.testing.assert_allclose(res.x, [0, 0, 1.5, 1]) + assert res.status == 0 + + def test_bug_20584(self): + """ + Test that when integrality is a list of all zeros, linprog gives the + same result as when it is an array of all zeros / integrality=None + """ + c = [1, 1] + A_ub = [[-1, 0]] + b_ub = [-2.5] + res1 = linprog(c, A_ub=A_ub, b_ub=b_ub, integrality=[0, 0]) + res2 = linprog(c, A_ub=A_ub, b_ub=b_ub, integrality=np.asarray([0, 0])) + res3 = linprog(c, A_ub=A_ub, b_ub=b_ub, integrality=None) + assert_equal(res1.x, res2.x) + assert_equal(res1.x, res3.x) + + +########################### +# Autoscale-Specific Tests# +########################### + + +@pytest.mark.filterwarnings("ignore::DeprecationWarning") +class AutoscaleTests: + options = {"autoscale": True} + + test_bug_6139 = LinprogCommonTests.test_bug_6139 + test_bug_6690 = LinprogCommonTests.test_bug_6690 + test_bug_7237 = LinprogCommonTests.test_bug_7237 + + +class TestAutoscaleIP(AutoscaleTests): + method = "interior-point" + + def test_bug_6139(self): + self.options['tol'] = 1e-10 + return AutoscaleTests.test_bug_6139(self) + + +class TestAutoscaleSimplex(AutoscaleTests): + method = "simplex" + + +class TestAutoscaleRS(AutoscaleTests): + method = "revised simplex" + + def test_nontrivial_problem_with_guess(self): + c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem() + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options, x0=x_star) + _assert_success(res, desired_fun=f_star, desired_x=x_star) + assert_equal(res.nit, 0) + + def test_nontrivial_problem_with_bad_guess(self): + c, A_ub, b_ub, A_eq, b_eq, x_star, f_star = nontrivial_problem() + bad_guess = [1, 2, 3, .5] + res = linprog(c, A_ub, b_ub, A_eq, b_eq, bounds, + method=self.method, options=self.options, x0=bad_guess) + assert_equal(res.status, 6) + + +########################### +# Redundancy Removal Tests# +########################### + + +@pytest.mark.filterwarnings("ignore::DeprecationWarning") +class RRTests: + method = "interior-point" + LCT = LinprogCommonTests + # these are a few of the existing tests that have redundancy + test_RR_infeasibility = LCT.test_remove_redundancy_infeasibility + test_bug_10349 = LCT.test_bug_10349 + test_bug_7044 = LCT.test_bug_7044 + test_NFLC = LCT.test_network_flow_limited_capacity + test_enzo_example_b = LCT.test_enzo_example_b + + +class TestRRSVD(RRTests): + options = {"rr_method": "SVD"} + + +class TestRRPivot(RRTests): + options = {"rr_method": "pivot"} + + +class TestRRID(RRTests): + options = {"rr_method": "ID"} diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lsq_common.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lsq_common.py new file mode 100644 index 0000000000000000000000000000000000000000..650deedce88b6babd8a3f2b62a5839f1a6cb966c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lsq_common.py @@ -0,0 +1,297 @@ +from numpy.testing import assert_, assert_allclose, assert_equal +from pytest import raises as assert_raises +import numpy as np + +from scipy.optimize._lsq.common import ( + step_size_to_bound, find_active_constraints, make_strictly_feasible, + CL_scaling_vector, intersect_trust_region, build_quadratic_1d, + minimize_quadratic_1d, evaluate_quadratic, reflective_transformation, + left_multiplied_operator, right_multiplied_operator) + + +class TestBounds: + def test_step_size_to_bounds(self): + lb = np.array([-1.0, 2.5, 10.0]) + ub = np.array([1.0, 5.0, 100.0]) + x = np.array([0.0, 2.5, 12.0]) + + s = np.array([0.1, 0.0, 0.0]) + step, hits = step_size_to_bound(x, s, lb, ub) + assert_equal(step, 10) + assert_equal(hits, [1, 0, 0]) + + s = np.array([0.01, 0.05, -1.0]) + step, hits = step_size_to_bound(x, s, lb, ub) + assert_equal(step, 2) + assert_equal(hits, [0, 0, -1]) + + s = np.array([10.0, -0.0001, 100.0]) + step, hits = step_size_to_bound(x, s, lb, ub) + assert_equal(step, np.array(-0)) + assert_equal(hits, [0, -1, 0]) + + s = np.array([1.0, 0.5, -2.0]) + step, hits = step_size_to_bound(x, s, lb, ub) + assert_equal(step, 1.0) + assert_equal(hits, [1, 0, -1]) + + s = np.zeros(3) + step, hits = step_size_to_bound(x, s, lb, ub) + assert_equal(step, np.inf) + assert_equal(hits, [0, 0, 0]) + + def test_find_active_constraints(self): + lb = np.array([0.0, -10.0, 1.0]) + ub = np.array([1.0, 0.0, 100.0]) + + x = np.array([0.5, -5.0, 2.0]) + active = find_active_constraints(x, lb, ub) + assert_equal(active, [0, 0, 0]) + + x = np.array([0.0, 0.0, 10.0]) + active = find_active_constraints(x, lb, ub) + assert_equal(active, [-1, 1, 0]) + + active = find_active_constraints(x, lb, ub, rtol=0) + assert_equal(active, [-1, 1, 0]) + + x = np.array([1e-9, -1e-8, 100 - 1e-9]) + active = find_active_constraints(x, lb, ub) + assert_equal(active, [0, 0, 1]) + + active = find_active_constraints(x, lb, ub, rtol=1.5e-9) + assert_equal(active, [-1, 0, 1]) + + lb = np.array([1.0, -np.inf, -np.inf]) + ub = np.array([np.inf, 10.0, np.inf]) + + x = np.ones(3) + active = find_active_constraints(x, lb, ub) + assert_equal(active, [-1, 0, 0]) + + # Handles out-of-bound cases. + x = np.array([0.0, 11.0, 0.0]) + active = find_active_constraints(x, lb, ub) + assert_equal(active, [-1, 1, 0]) + + active = find_active_constraints(x, lb, ub, rtol=0) + assert_equal(active, [-1, 1, 0]) + + def test_make_strictly_feasible(self): + lb = np.array([-0.5, -0.8, 2.0]) + ub = np.array([0.8, 1.0, 3.0]) + + x = np.array([-0.5, 0.0, 2 + 1e-10]) + + x_new = make_strictly_feasible(x, lb, ub, rstep=0) + assert_(x_new[0] > -0.5) + assert_equal(x_new[1:], x[1:]) + + x_new = make_strictly_feasible(x, lb, ub, rstep=1e-4) + assert_equal(x_new, [-0.5 + 1e-4, 0.0, 2 * (1 + 1e-4)]) + + x = np.array([-0.5, -1, 3.1]) + x_new = make_strictly_feasible(x, lb, ub) + assert_(np.all((x_new >= lb) & (x_new <= ub))) + + x_new = make_strictly_feasible(x, lb, ub, rstep=0) + assert_(np.all((x_new >= lb) & (x_new <= ub))) + + lb = np.array([-1, 100.0]) + ub = np.array([1, 100.0 + 1e-10]) + x = np.array([0, 100.0]) + x_new = make_strictly_feasible(x, lb, ub, rstep=1e-8) + assert_equal(x_new, [0, 100.0 + 0.5e-10]) + + def test_scaling_vector(self): + lb = np.array([-np.inf, -5.0, 1.0, -np.inf]) + ub = np.array([1.0, np.inf, 10.0, np.inf]) + x = np.array([0.5, 2.0, 5.0, 0.0]) + g = np.array([1.0, 0.1, -10.0, 0.0]) + v, dv = CL_scaling_vector(x, g, lb, ub) + assert_equal(v, [1.0, 7.0, 5.0, 1.0]) + assert_equal(dv, [0.0, 1.0, -1.0, 0.0]) + + +class TestQuadraticFunction: + def setup_method(self): + self.J = np.array([ + [0.1, 0.2], + [-1.0, 1.0], + [0.5, 0.2]]) + self.g = np.array([0.8, -2.0]) + self.diag = np.array([1.0, 2.0]) + + def test_build_quadratic_1d(self): + s = np.zeros(2) + a, b = build_quadratic_1d(self.J, self.g, s) + assert_equal(a, 0) + assert_equal(b, 0) + + a, b = build_quadratic_1d(self.J, self.g, s, diag=self.diag) + assert_equal(a, 0) + assert_equal(b, 0) + + s = np.array([1.0, -1.0]) + a, b = build_quadratic_1d(self.J, self.g, s) + assert_equal(a, 2.05) + assert_equal(b, 2.8) + + a, b = build_quadratic_1d(self.J, self.g, s, diag=self.diag) + assert_equal(a, 3.55) + assert_equal(b, 2.8) + + s0 = np.array([0.5, 0.5]) + a, b, c = build_quadratic_1d(self.J, self.g, s, diag=self.diag, s0=s0) + assert_equal(a, 3.55) + assert_allclose(b, 2.39) + assert_allclose(c, -0.1525) + + def test_minimize_quadratic_1d(self): + a = 5 + b = -1 + + t, y = minimize_quadratic_1d(a, b, 1, 2) + assert_equal(t, 1) + assert_allclose(y, a * t**2 + b * t, rtol=1e-15) + + t, y = minimize_quadratic_1d(a, b, -2, -1) + assert_equal(t, -1) + assert_allclose(y, a * t**2 + b * t, rtol=1e-15) + + t, y = minimize_quadratic_1d(a, b, -1, 1) + assert_equal(t, 0.1) + assert_allclose(y, a * t**2 + b * t, rtol=1e-15) + + c = 10 + t, y = minimize_quadratic_1d(a, b, -1, 1, c=c) + assert_equal(t, 0.1) + assert_allclose(y, a * t**2 + b * t + c, rtol=1e-15) + + t, y = minimize_quadratic_1d(a, b, -np.inf, np.inf, c=c) + assert_equal(t, 0.1) + assert_allclose(y, a * t ** 2 + b * t + c, rtol=1e-15) + + t, y = minimize_quadratic_1d(a, b, 0, np.inf, c=c) + assert_equal(t, 0.1) + assert_allclose(y, a * t ** 2 + b * t + c, rtol=1e-15) + + t, y = minimize_quadratic_1d(a, b, -np.inf, 0, c=c) + assert_equal(t, 0) + assert_allclose(y, a * t ** 2 + b * t + c, rtol=1e-15) + + a = -1 + b = 0.2 + t, y = minimize_quadratic_1d(a, b, -np.inf, np.inf) + assert_equal(y, -np.inf) + + t, y = minimize_quadratic_1d(a, b, 0, np.inf) + assert_equal(t, np.inf) + assert_equal(y, -np.inf) + + t, y = minimize_quadratic_1d(a, b, -np.inf, 0) + assert_equal(t, -np.inf) + assert_equal(y, -np.inf) + + def test_evaluate_quadratic(self): + s = np.array([1.0, -1.0]) + + value = evaluate_quadratic(self.J, self.g, s) + assert_equal(value, 4.85) + + value = evaluate_quadratic(self.J, self.g, s, diag=self.diag) + assert_equal(value, 6.35) + + s = np.array([[1.0, -1.0], + [1.0, 1.0], + [0.0, 0.0]]) + + values = evaluate_quadratic(self.J, self.g, s) + assert_allclose(values, [4.85, -0.91, 0.0]) + + values = evaluate_quadratic(self.J, self.g, s, diag=self.diag) + assert_allclose(values, [6.35, 0.59, 0.0]) + + +class TestTrustRegion: + def test_intersect(self): + Delta = 1.0 + + x = np.zeros(3) + s = np.array([1.0, 0.0, 0.0]) + t_neg, t_pos = intersect_trust_region(x, s, Delta) + assert_equal(t_neg, -1) + assert_equal(t_pos, 1) + + s = np.array([-1.0, 1.0, -1.0]) + t_neg, t_pos = intersect_trust_region(x, s, Delta) + assert_allclose(t_neg, -3**-0.5) + assert_allclose(t_pos, 3**-0.5) + + x = np.array([0.5, -0.5, 0]) + s = np.array([0, 0, 1.0]) + t_neg, t_pos = intersect_trust_region(x, s, Delta) + assert_allclose(t_neg, -2**-0.5) + assert_allclose(t_pos, 2**-0.5) + + x = np.ones(3) + assert_raises(ValueError, intersect_trust_region, x, s, Delta) + + x = np.zeros(3) + s = np.zeros(3) + assert_raises(ValueError, intersect_trust_region, x, s, Delta) + + +def test_reflective_transformation(): + lb = np.array([-1, -2], dtype=float) + ub = np.array([5, 3], dtype=float) + + y = np.array([0, 0]) + x, g = reflective_transformation(y, lb, ub) + assert_equal(x, y) + assert_equal(g, np.ones(2)) + + y = np.array([-4, 4], dtype=float) + + x, g = reflective_transformation(y, lb, np.array([np.inf, np.inf])) + assert_equal(x, [2, 4]) + assert_equal(g, [-1, 1]) + + x, g = reflective_transformation(y, np.array([-np.inf, -np.inf]), ub) + assert_equal(x, [-4, 2]) + assert_equal(g, [1, -1]) + + x, g = reflective_transformation(y, lb, ub) + assert_equal(x, [2, 2]) + assert_equal(g, [-1, -1]) + + lb = np.array([-np.inf, -2]) + ub = np.array([5, np.inf]) + y = np.array([10, 10], dtype=float) + x, g = reflective_transformation(y, lb, ub) + assert_equal(x, [0, 10]) + assert_equal(g, [-1, 1]) + + +def test_linear_operators(): + A = np.arange(6).reshape((3, 2)) + + d_left = np.array([-1, 2, 5]) + DA = np.diag(d_left).dot(A) + J_left = left_multiplied_operator(A, d_left) + + d_right = np.array([5, 10]) + AD = A.dot(np.diag(d_right)) + J_right = right_multiplied_operator(A, d_right) + + x = np.array([-2, 3]) + X = -2 * np.arange(2, 8).reshape((2, 3)) + xt = np.array([0, -2, 15]) + + assert_allclose(DA.dot(x), J_left.matvec(x)) + assert_allclose(DA.dot(X), J_left.matmat(X)) + assert_allclose(DA.T.dot(xt), J_left.rmatvec(xt)) + + assert_allclose(AD.dot(x), J_right.matvec(x)) + assert_allclose(AD.dot(X), J_right.matmat(X)) + assert_allclose(AD.T.dot(xt), J_right.rmatvec(xt)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lsq_linear.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lsq_linear.py new file mode 100644 index 0000000000000000000000000000000000000000..23032e99764ed2e90d7192c078e5cb4518e328fd --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_lsq_linear.py @@ -0,0 +1,287 @@ +import pytest + +import numpy as np +from numpy.linalg import lstsq +from numpy.testing import assert_allclose, assert_equal, assert_ + +from scipy.sparse import rand, coo_matrix +from scipy.sparse.linalg import aslinearoperator +from scipy.optimize import lsq_linear +from scipy.optimize._minimize import Bounds + + +A = np.array([ + [0.171, -0.057], + [-0.049, -0.248], + [-0.166, 0.054], +]) +b = np.array([0.074, 1.014, -0.383]) + + +class BaseMixin: + def setup_method(self): + self.rnd = np.random.RandomState(0) + + def test_dense_no_bounds(self): + for lsq_solver in self.lsq_solvers: + res = lsq_linear(A, b, method=self.method, lsq_solver=lsq_solver) + assert_allclose(res.x, lstsq(A, b, rcond=-1)[0]) + assert_allclose(res.x, res.unbounded_sol[0]) + + def test_dense_bounds(self): + # Solutions for comparison are taken from MATLAB. + lb = np.array([-1, -10]) + ub = np.array([1, 0]) + unbounded_sol = lstsq(A, b, rcond=-1)[0] + for lsq_solver in self.lsq_solvers: + res = lsq_linear(A, b, (lb, ub), method=self.method, + lsq_solver=lsq_solver) + assert_allclose(res.x, lstsq(A, b, rcond=-1)[0]) + assert_allclose(res.unbounded_sol[0], unbounded_sol) + + lb = np.array([0.0, -np.inf]) + for lsq_solver in self.lsq_solvers: + res = lsq_linear(A, b, (lb, np.inf), method=self.method, + lsq_solver=lsq_solver) + assert_allclose(res.x, np.array([0.0, -4.084174437334673]), + atol=1e-6) + assert_allclose(res.unbounded_sol[0], unbounded_sol) + + lb = np.array([-1, 0]) + for lsq_solver in self.lsq_solvers: + res = lsq_linear(A, b, (lb, np.inf), method=self.method, + lsq_solver=lsq_solver) + assert_allclose(res.x, np.array([0.448427311733504, 0]), + atol=1e-15) + assert_allclose(res.unbounded_sol[0], unbounded_sol) + + ub = np.array([np.inf, -5]) + for lsq_solver in self.lsq_solvers: + res = lsq_linear(A, b, (-np.inf, ub), method=self.method, + lsq_solver=lsq_solver) + assert_allclose(res.x, np.array([-0.105560998682388, -5])) + assert_allclose(res.unbounded_sol[0], unbounded_sol) + + ub = np.array([-1, np.inf]) + for lsq_solver in self.lsq_solvers: + res = lsq_linear(A, b, (-np.inf, ub), method=self.method, + lsq_solver=lsq_solver) + assert_allclose(res.x, np.array([-1, -4.181102129483254])) + assert_allclose(res.unbounded_sol[0], unbounded_sol) + + lb = np.array([0, -4]) + ub = np.array([1, 0]) + for lsq_solver in self.lsq_solvers: + res = lsq_linear(A, b, (lb, ub), method=self.method, + lsq_solver=lsq_solver) + assert_allclose(res.x, np.array([0.005236663400791, -4])) + assert_allclose(res.unbounded_sol[0], unbounded_sol) + + def test_bounds_variants(self): + x = np.array([1, 3]) + A = self.rnd.uniform(size=(2, 2)) + b = A@x + lb = np.array([1, 1]) + ub = np.array([2, 2]) + bounds_old = (lb, ub) + bounds_new = Bounds(lb, ub) + res_old = lsq_linear(A, b, bounds_old) + res_new = lsq_linear(A, b, bounds_new) + assert not np.allclose(res_new.x, res_new.unbounded_sol[0]) + assert_allclose(res_old.x, res_new.x) + + def test_np_matrix(self): + # gh-10711 + with np.testing.suppress_warnings() as sup: + sup.filter(PendingDeprecationWarning) + A = np.matrix([[20, -4, 0, 2, 3], [10, -2, 1, 0, -1]]) + k = np.array([20, 15]) + lsq_linear(A, k) + + def test_dense_rank_deficient(self): + A = np.array([[-0.307, -0.184]]) + b = np.array([0.773]) + lb = [-0.1, -0.1] + ub = [0.1, 0.1] + for lsq_solver in self.lsq_solvers: + res = lsq_linear(A, b, (lb, ub), method=self.method, + lsq_solver=lsq_solver) + assert_allclose(res.x, [-0.1, -0.1]) + assert_allclose(res.unbounded_sol[0], lstsq(A, b, rcond=-1)[0]) + + A = np.array([ + [0.334, 0.668], + [-0.516, -1.032], + [0.192, 0.384], + ]) + b = np.array([-1.436, 0.135, 0.909]) + lb = [0, -1] + ub = [1, -0.5] + for lsq_solver in self.lsq_solvers: + res = lsq_linear(A, b, (lb, ub), method=self.method, + lsq_solver=lsq_solver) + assert_allclose(res.optimality, 0, atol=1e-11) + assert_allclose(res.unbounded_sol[0], lstsq(A, b, rcond=-1)[0]) + + def test_full_result(self): + lb = np.array([0, -4]) + ub = np.array([1, 0]) + res = lsq_linear(A, b, (lb, ub), method=self.method) + + assert_allclose(res.x, [0.005236663400791, -4]) + assert_allclose(res.unbounded_sol[0], lstsq(A, b, rcond=-1)[0]) + + r = A.dot(res.x) - b + assert_allclose(res.cost, 0.5 * np.dot(r, r)) + assert_allclose(res.fun, r) + + assert_allclose(res.optimality, 0.0, atol=1e-12) + assert_equal(res.active_mask, [0, -1]) + assert_(res.nit < 15) + assert_(res.status == 1 or res.status == 3) + assert_(isinstance(res.message, str)) + assert_(res.success) + + # This is a test for issue #9982. + def test_almost_singular(self): + A = np.array( + [[0.8854232310355122, 0.0365312146937765, 0.0365312146836789], + [0.3742460132129041, 0.0130523214078376, 0.0130523214077873], + [0.9680633871281361, 0.0319366128718639, 0.0319366128718388]]) + + b = np.array( + [0.0055029366538097, 0.0026677442422208, 0.0066612514782381]) + + result = lsq_linear(A, b, method=self.method) + assert_(result.cost < 1.1e-8) + + @pytest.mark.xslow + def test_large_rank_deficient(self): + np.random.seed(0) + n, m = np.sort(np.random.randint(2, 1000, size=2)) + m *= 2 # make m >> n + A = 1.0 * np.random.randint(-99, 99, size=[m, n]) + b = 1.0 * np.random.randint(-99, 99, size=[m]) + bounds = 1.0 * np.sort(np.random.randint(-99, 99, size=(2, n)), axis=0) + bounds[1, :] += 1.0 # ensure up > lb + + # Make the A matrix strongly rank deficient by replicating some columns + w = np.random.choice(n, n) # Select random columns with duplicates + A = A[:, w] + + x_bvls = lsq_linear(A, b, bounds=bounds, method='bvls').x + x_trf = lsq_linear(A, b, bounds=bounds, method='trf').x + + cost_bvls = np.sum((A @ x_bvls - b)**2) + cost_trf = np.sum((A @ x_trf - b)**2) + + assert_(abs(cost_bvls - cost_trf) < cost_trf*1e-10) + + def test_convergence_small_matrix(self): + A = np.array([[49.0, 41.0, -32.0], + [-19.0, -32.0, -8.0], + [-13.0, 10.0, 69.0]]) + b = np.array([-41.0, -90.0, 47.0]) + bounds = np.array([[31.0, -44.0, 26.0], + [54.0, -32.0, 28.0]]) + + x_bvls = lsq_linear(A, b, bounds=bounds, method='bvls').x + x_trf = lsq_linear(A, b, bounds=bounds, method='trf').x + + cost_bvls = np.sum((A @ x_bvls - b)**2) + cost_trf = np.sum((A @ x_trf - b)**2) + + assert_(abs(cost_bvls - cost_trf) < cost_trf*1e-10) + + +class SparseMixin: + def test_sparse_and_LinearOperator(self): + m = 5000 + n = 1000 + rng = np.random.RandomState(0) + A = rand(m, n, random_state=rng) + b = rng.randn(m) + res = lsq_linear(A, b) + assert_allclose(res.optimality, 0, atol=1e-6) + + A = aslinearoperator(A) + res = lsq_linear(A, b) + assert_allclose(res.optimality, 0, atol=1e-6) + + @pytest.mark.fail_slow(10) + def test_sparse_bounds(self): + m = 5000 + n = 1000 + rng = np.random.RandomState(0) + A = rand(m, n, random_state=rng) + b = rng.randn(m) + lb = rng.randn(n) + ub = lb + 1 + res = lsq_linear(A, b, (lb, ub)) + assert_allclose(res.optimality, 0.0, atol=1e-6) + + res = lsq_linear(A, b, (lb, ub), lsmr_tol=1e-13, + lsmr_maxiter=1500) + assert_allclose(res.optimality, 0.0, atol=1e-6) + + res = lsq_linear(A, b, (lb, ub), lsmr_tol='auto') + assert_allclose(res.optimality, 0.0, atol=1e-6) + + def test_sparse_ill_conditioned(self): + # Sparse matrix with condition number of ~4 million + data = np.array([1., 1., 1., 1. + 1e-6, 1.]) + row = np.array([0, 0, 1, 2, 2]) + col = np.array([0, 2, 1, 0, 2]) + A = coo_matrix((data, (row, col)), shape=(3, 3)) + + # Get the exact solution + exact_sol = lsq_linear(A.toarray(), b, lsq_solver='exact') + + # Default lsmr arguments should not fully converge the solution + default_lsmr_sol = lsq_linear(A, b, lsq_solver='lsmr') + with pytest.raises(AssertionError, match=""): + assert_allclose(exact_sol.x, default_lsmr_sol.x) + + # By increasing the maximum lsmr iters, it will converge + conv_lsmr = lsq_linear(A, b, lsq_solver='lsmr', lsmr_maxiter=10) + assert_allclose(exact_sol.x, conv_lsmr.x) + + +class TestTRF(BaseMixin, SparseMixin): + method = 'trf' + lsq_solvers = ['exact', 'lsmr'] + + +class TestBVLS(BaseMixin): + method = 'bvls' + lsq_solvers = ['exact'] + + +class TestErrorChecking: + def test_option_lsmr_tol(self): + # Should work with a positive float, string equal to 'auto', or None + _ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_tol=1e-2) + _ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_tol='auto') + _ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_tol=None) + + # Should raise error with negative float, strings + # other than 'auto', and integers + err_message = "`lsmr_tol` must be None, 'auto', or positive float." + with pytest.raises(ValueError, match=err_message): + _ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_tol=-0.1) + with pytest.raises(ValueError, match=err_message): + _ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_tol='foo') + with pytest.raises(ValueError, match=err_message): + _ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_tol=1) + + def test_option_lsmr_maxiter(self): + # Should work with positive integers or None + _ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_maxiter=1) + _ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_maxiter=None) + + # Should raise error with 0 or negative max iter + err_message = "`lsmr_maxiter` must be None or positive integer." + with pytest.raises(ValueError, match=err_message): + _ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_maxiter=0) + with pytest.raises(ValueError, match=err_message): + _ = lsq_linear(A, b, lsq_solver='lsmr', lsmr_maxiter=-1) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_milp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_milp.py new file mode 100644 index 0000000000000000000000000000000000000000..165417fa1c0a9799056fd67a7218499c611a673e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_milp.py @@ -0,0 +1,459 @@ +""" +Unit test for Mixed Integer Linear Programming +""" +import re +import sys + +import numpy as np +from numpy.testing import assert_allclose, assert_array_equal +import pytest + +from .test_linprog import magic_square +from scipy.optimize import milp, Bounds, LinearConstraint +from scipy import sparse + + +_IS_32BIT = (sys.maxsize < 2**32) + +def test_milp_iv(): + + message = "`c` must be a dense array" + with pytest.raises(ValueError, match=message): + milp(sparse.coo_array([0, 0])) + + message = "`c` must be a one-dimensional array of finite numbers with" + with pytest.raises(ValueError, match=message): + milp(np.zeros((3, 4))) + with pytest.raises(ValueError, match=message): + milp([]) + with pytest.raises(ValueError, match=message): + milp(None) + + message = "`bounds` must be convertible into an instance of..." + with pytest.raises(ValueError, match=message): + milp(1, bounds=10) + + message = "`constraints` (or each element within `constraints`) must be" + with pytest.raises(ValueError, match=re.escape(message)): + milp(1, constraints=10) + with pytest.raises(ValueError, match=re.escape(message)): + milp(np.zeros(3), constraints=([[1, 2, 3]], [2, 3], [2, 3])) + with pytest.raises(ValueError, match=re.escape(message)): + milp(np.zeros(2), constraints=([[1, 2]], [2], sparse.coo_array([2]))) + + message = "The shape of `A` must be (len(b_l), len(c))." + with pytest.raises(ValueError, match=re.escape(message)): + milp(np.zeros(3), constraints=([[1, 2]], [2], [2])) + + message = "`integrality` must be a dense array" + with pytest.raises(ValueError, match=message): + milp([1, 2], integrality=sparse.coo_array([1, 2])) + + message = ("`integrality` must contain integers 0-3 and be broadcastable " + "to `c.shape`.") + with pytest.raises(ValueError, match=message): + milp([1, 2, 3], integrality=[1, 2]) + with pytest.raises(ValueError, match=message): + milp([1, 2, 3], integrality=[1, 5, 3]) + + message = "Lower and upper bounds must be dense arrays." + with pytest.raises(ValueError, match=message): + milp([1, 2, 3], bounds=([1, 2], sparse.coo_array([3, 4]))) + + message = "`lb`, `ub`, and `keep_feasible` must be broadcastable." + with pytest.raises(ValueError, match=message): + milp([1, 2, 3], bounds=([1, 2], [3, 4, 5])) + with pytest.raises(ValueError, match=message): + milp([1, 2, 3], bounds=([1, 2, 3], [4, 5])) + + message = "`bounds.lb` and `bounds.ub` must contain reals and..." + with pytest.raises(ValueError, match=message): + milp([1, 2, 3], bounds=([1, 2], [3, 4])) + with pytest.raises(ValueError, match=message): + milp([1, 2, 3], bounds=([1, 2, 3], ["3+4", 4, 5])) + with pytest.raises(ValueError, match=message): + milp([1, 2, 3], bounds=([1, 2, 3], [set(), 4, 5])) + + +@pytest.mark.xfail(run=False, + reason="Needs to be fixed in `_highs_wrapper`") +def test_milp_options(capsys): + # run=False now because of gh-16347 + message = "Unrecognized options detected: {'ekki'}..." + options = {'ekki': True} + with pytest.warns(RuntimeWarning, match=message): + milp(1, options=options) + + A, b, c, numbers, M = magic_square(3) + options = {"disp": True, "presolve": False, "time_limit": 0.05} + res = milp(c=c, constraints=(A, b, b), bounds=(0, 1), integrality=1, + options=options) + + captured = capsys.readouterr() + assert "Presolve is switched off" in captured.out + assert "Time Limit Reached" in captured.out + assert not res.success + + +def test_result(): + A, b, c, numbers, M = magic_square(3) + res = milp(c=c, constraints=(A, b, b), bounds=(0, 1), integrality=1) + assert res.status == 0 + assert res.success + msg = "Optimization terminated successfully. (HiGHS Status 7:" + assert res.message.startswith(msg) + assert isinstance(res.x, np.ndarray) + assert isinstance(res.fun, float) + assert isinstance(res.mip_node_count, int) + assert isinstance(res.mip_dual_bound, float) + assert isinstance(res.mip_gap, float) + + A, b, c, numbers, M = magic_square(6) + res = milp(c=c*0, constraints=(A, b, b), bounds=(0, 1), integrality=1, + options={'time_limit': 0.05}) + assert res.status == 1 + assert not res.success + msg = "Time limit reached. (HiGHS Status 13:" + assert res.message.startswith(msg) + assert (res.fun is res.mip_dual_bound is res.mip_gap + is res.mip_node_count is res.x is None) + + res = milp(1, bounds=(1, -1)) + assert res.status == 2 + assert not res.success + msg = "The problem is infeasible. (HiGHS Status 8:" + assert res.message.startswith(msg) + assert (res.fun is res.mip_dual_bound is res.mip_gap + is res.mip_node_count is res.x is None) + + res = milp(-1) + assert res.status == 3 + assert not res.success + msg = "The problem is unbounded. (HiGHS Status 10:" + assert res.message.startswith(msg) + assert (res.fun is res.mip_dual_bound is res.mip_gap + is res.mip_node_count is res.x is None) + + +def test_milp_optional_args(): + # check that arguments other than `c` are indeed optional + res = milp(1) + assert res.fun == 0 + assert_array_equal(res.x, [0]) + + +def test_milp_1(): + # solve magic square problem + n = 3 + A, b, c, numbers, M = magic_square(n) + A = sparse.csc_array(A) # confirm that sparse arrays are accepted + res = milp(c=c*0, constraints=(A, b, b), bounds=(0, 1), integrality=1) + + # check that solution is a magic square + x = np.round(res.x) + s = (numbers.flatten() * x).reshape(n**2, n, n) + square = np.sum(s, axis=0) + np.testing.assert_allclose(square.sum(axis=0), M) + np.testing.assert_allclose(square.sum(axis=1), M) + np.testing.assert_allclose(np.diag(square).sum(), M) + np.testing.assert_allclose(np.diag(square[:, ::-1]).sum(), M) + + +def test_milp_2(): + # solve MIP with inequality constraints and all integer constraints + # source: slide 5, + # https://www.cs.upc.edu/~erodri/webpage/cps/theory/lp/milp/slides.pdf + # also check that `milp` accepts all valid ways of specifying constraints + c = -np.ones(2) + A = [[-2, 2], [-8, 10]] + b_l = [1, -np.inf] + b_u = [np.inf, 13] + linear_constraint = LinearConstraint(A, b_l, b_u) + + # solve original problem + res1 = milp(c=c, constraints=(A, b_l, b_u), integrality=True) + res2 = milp(c=c, constraints=linear_constraint, integrality=True) + res3 = milp(c=c, constraints=[(A, b_l, b_u)], integrality=True) + res4 = milp(c=c, constraints=[linear_constraint], integrality=True) + res5 = milp(c=c, integrality=True, + constraints=[(A[:1], b_l[:1], b_u[:1]), + (A[1:], b_l[1:], b_u[1:])]) + res6 = milp(c=c, integrality=True, + constraints=[LinearConstraint(A[:1], b_l[:1], b_u[:1]), + LinearConstraint(A[1:], b_l[1:], b_u[1:])]) + res7 = milp(c=c, integrality=True, + constraints=[(A[:1], b_l[:1], b_u[:1]), + LinearConstraint(A[1:], b_l[1:], b_u[1:])]) + xs = np.array([res1.x, res2.x, res3.x, res4.x, res5.x, res6.x, res7.x]) + funs = np.array([res1.fun, res2.fun, res3.fun, + res4.fun, res5.fun, res6.fun, res7.fun]) + np.testing.assert_allclose(xs, np.broadcast_to([1, 2], xs.shape)) + np.testing.assert_allclose(funs, -3) + + # solve relaxed problem + res = milp(c=c, constraints=(A, b_l, b_u)) + np.testing.assert_allclose(res.x, [4, 4.5]) + np.testing.assert_allclose(res.fun, -8.5) + + +def test_milp_3(): + # solve MIP with inequality constraints and all integer constraints + # source: https://en.wikipedia.org/wiki/Integer_programming#Example + c = [0, -1] + A = [[-1, 1], [3, 2], [2, 3]] + b_u = [1, 12, 12] + b_l = np.full_like(b_u, -np.inf, dtype=np.float64) + constraints = LinearConstraint(A, b_l, b_u) + + integrality = np.ones_like(c) + + # solve original problem + res = milp(c=c, constraints=constraints, integrality=integrality) + assert_allclose(res.fun, -2) + # two optimal solutions possible, just need one of them + assert np.allclose(res.x, [1, 2]) or np.allclose(res.x, [2, 2]) + + # solve relaxed problem + res = milp(c=c, constraints=constraints) + assert_allclose(res.fun, -2.8) + assert_allclose(res.x, [1.8, 2.8]) + + +def test_milp_4(): + # solve MIP with inequality constraints and only one integer constraint + # source: https://www.mathworks.com/help/optim/ug/intlinprog.html + c = [8, 1] + integrality = [0, 1] + A = [[1, 2], [-4, -1], [2, 1]] + b_l = [-14, -np.inf, -np.inf] + b_u = [np.inf, -33, 20] + constraints = LinearConstraint(A, b_l, b_u) + bounds = Bounds(-np.inf, np.inf) + + res = milp(c, integrality=integrality, bounds=bounds, + constraints=constraints) + assert_allclose(res.fun, 59) + assert_allclose(res.x, [6.5, 7]) + + +def test_milp_5(): + # solve MIP with inequality and equality constraints + # source: https://www.mathworks.com/help/optim/ug/intlinprog.html + c = [-3, -2, -1] + integrality = [0, 0, 1] + lb = [0, 0, 0] + ub = [np.inf, np.inf, 1] + bounds = Bounds(lb, ub) + A = [[1, 1, 1], [4, 2, 1]] + b_l = [-np.inf, 12] + b_u = [7, 12] + constraints = LinearConstraint(A, b_l, b_u) + + res = milp(c, integrality=integrality, bounds=bounds, + constraints=constraints) + # there are multiple solutions + assert_allclose(res.fun, -12) + + +@pytest.mark.xslow +def test_milp_6(): + # solve a larger MIP with only equality constraints + # source: https://www.mathworks.com/help/optim/ug/intlinprog.html + integrality = 1 + A_eq = np.array([[22, 13, 26, 33, 21, 3, 14, 26], + [39, 16, 22, 28, 26, 30, 23, 24], + [18, 14, 29, 27, 30, 38, 26, 26], + [41, 26, 28, 36, 18, 38, 16, 26]]) + b_eq = np.array([7872, 10466, 11322, 12058]) + c = np.array([2, 10, 13, 17, 7, 5, 7, 3]) + + res = milp(c=c, constraints=(A_eq, b_eq, b_eq), integrality=integrality) + + np.testing.assert_allclose(res.fun, 1854) + + +def test_infeasible_prob_16609(): + # Ensure presolve does not mark trivially infeasible problems + # as Optimal -- see gh-16609 + c = [1.0, 0.0] + integrality = [0, 1] + + lb = [0, -np.inf] + ub = [np.inf, np.inf] + bounds = Bounds(lb, ub) + + A_eq = [[0.0, 1.0]] + b_eq = [0.5] + constraints = LinearConstraint(A_eq, b_eq, b_eq) + + res = milp(c, integrality=integrality, bounds=bounds, + constraints=constraints) + np.testing.assert_equal(res.status, 2) + + +_msg_time = "Time limit reached. (HiGHS Status 13:" +_msg_iter = "Iteration limit reached. (HiGHS Status 14:" + +@pytest.mark.thread_unsafe +# See https://github.com/scipy/scipy/pull/19255#issuecomment-1778438888 +@pytest.mark.xfail(reason="Often buggy, revisit with callbacks, gh-19255") +@pytest.mark.skipif(np.intp(0).itemsize < 8, + reason="Unhandled 32-bit GCC FP bug") +@pytest.mark.slow +@pytest.mark.parametrize(["options", "msg"], [({"time_limit": 0.1}, _msg_time), + ({"node_limit": 1}, _msg_iter)]) +def test_milp_timeout_16545(options, msg): + # Ensure solution is not thrown away if MILP solver times out + # -- see gh-16545 + rng = np.random.default_rng(5123833489170494244) + A = rng.integers(0, 5, size=(100, 100)) + b_lb = np.full(100, fill_value=-np.inf) + b_ub = np.full(100, fill_value=25) + constraints = LinearConstraint(A, b_lb, b_ub) + variable_lb = np.zeros(100) + variable_ub = np.ones(100) + variable_bounds = Bounds(variable_lb, variable_ub) + integrality = np.ones(100) + c_vector = -np.ones(100) + res = milp( + c_vector, + integrality=integrality, + bounds=variable_bounds, + constraints=constraints, + options=options, + ) + + assert res.message.startswith(msg) + assert res["x"] is not None + + # ensure solution is feasible + x = res["x"] + tol = 1e-8 # sometimes needed due to finite numerical precision + assert np.all(b_lb - tol <= A @ x) and np.all(A @ x <= b_ub + tol) + assert np.all(variable_lb - tol <= x) and np.all(x <= variable_ub + tol) + assert np.allclose(x, np.round(x)) + + +def test_three_constraints_16878(): + # `milp` failed when exactly three constraints were passed + # Ensure that this is no longer the case. + rng = np.random.default_rng(5123833489170494244) + A = rng.integers(0, 5, size=(6, 6)) + bl = np.full(6, fill_value=-np.inf) + bu = np.full(6, fill_value=10) + constraints = [LinearConstraint(A[:2], bl[:2], bu[:2]), + LinearConstraint(A[2:4], bl[2:4], bu[2:4]), + LinearConstraint(A[4:], bl[4:], bu[4:])] + constraints2 = [(A[:2], bl[:2], bu[:2]), + (A[2:4], bl[2:4], bu[2:4]), + (A[4:], bl[4:], bu[4:])] + lb = np.zeros(6) + ub = np.ones(6) + variable_bounds = Bounds(lb, ub) + c = -np.ones(6) + res1 = milp(c, bounds=variable_bounds, constraints=constraints) + res2 = milp(c, bounds=variable_bounds, constraints=constraints2) + ref = milp(c, bounds=variable_bounds, constraints=(A, bl, bu)) + assert res1.success and res2.success + assert_allclose(res1.x, ref.x) + assert_allclose(res2.x, ref.x) + + +@pytest.mark.xslow +def test_mip_rel_gap_passdown(): + # Solve problem with decreasing mip_gap to make sure mip_rel_gap decreases + # Adapted from test_linprog::TestLinprogHiGHSMIP::test_mip_rel_gap_passdown + # MIP taken from test_mip_6 above + A_eq = np.array([[22, 13, 26, 33, 21, 3, 14, 26], + [39, 16, 22, 28, 26, 30, 23, 24], + [18, 14, 29, 27, 30, 38, 26, 26], + [41, 26, 28, 36, 18, 38, 16, 26]]) + b_eq = np.array([7872, 10466, 11322, 12058]) + c = np.array([2, 10, 13, 17, 7, 5, 7, 3]) + + mip_rel_gaps = [0.25, 0.01, 0.001] + sol_mip_gaps = [] + for mip_rel_gap in mip_rel_gaps: + res = milp(c=c, bounds=(0, np.inf), constraints=(A_eq, b_eq, b_eq), + integrality=True, options={"mip_rel_gap": mip_rel_gap}) + # assert that the solution actually has mip_gap lower than the + # required mip_rel_gap supplied + assert res.mip_gap <= mip_rel_gap + # check that `res.mip_gap` is as defined in the documentation + assert res.mip_gap == (res.fun - res.mip_dual_bound)/res.fun + sol_mip_gaps.append(res.mip_gap) + + # make sure that the mip_rel_gap parameter is actually doing something + # check that differences between solution gaps are declining + # monotonically with the mip_rel_gap parameter. + assert np.all(np.diff(sol_mip_gaps) < 0) + +@pytest.mark.xfail(reason='Upstream / Wrapper issue, see gh-20116') +def test_large_numbers_gh20116(): + h = 10 ** 12 + A = np.array([[100.4534, h], [100.4534, -h]]) + b = np.array([h, 0]) + constraints = LinearConstraint(A=A, ub=b) + bounds = Bounds([0, 0], [1, 1]) + c = np.array([0, 0]) + res = milp(c=c, constraints=constraints, bounds=bounds, integrality=1) + assert res.status == 0 + assert np.all(A @ res.x < b) + + +def test_presolve_gh18907(): + from scipy.optimize import milp + import numpy as np + inf = np.inf + + # set up problem + c = np.array([-0.85850509, -0.82892676, -0.80026454, -0.63015535, -0.5099006, + -0.50077193, -0.4894404, -0.47285865, -0.39867774, -0.38069646, + -0.36733012, -0.36733012, -0.35820411, -0.31576141, -0.20626091, + -0.12466144, -0.10679516, -0.1061887, -0.1061887, -0.1061887, + -0., -0., -0., -0., 0., 0., 0., 0.]) + + A = np.array([[1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., + 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 0., 0., 0., 0.], + [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., + 1., 0., 0., 0., 0., 0., 1., 0., 0., 0., -25., -0., -0., -0.], + [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., + -1., 0., 0., 0., 0., 0., -1., 0., 0., 0., 2., 0., 0., 0.], + [0., 0., 0., 0., 1., 1., 1., 1., 0., 1., 0., 0., 0., 0., 0., + 0., 0., 0., 0., 0., 0., 0., 0., 0., -0., -25., -0., -0.], + [0., 0., 0., 0., -1., -1., -1., -1., 0., -1., 0., 0., 0., + 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 2., 0., 0.], + [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., + 0., 0., 1., 1., 1., 0., 0., 0., 0., -0., -0., -25., -0.], + [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., + 0., 0., -1., -1., -1., 0., 0., 0., 0., 0., 0., 2., 0.], + [1., 1., 1., 1., 0., 0., 0., 0., 1., 0., 1., 1., 1., 1., 0., + 1., 1., 0., 0., 0., 0., 1., 1., 1., -0., -0., -0., -25.], + [-1., -1., -1., -1., 0., 0., 0., 0., -1., 0., -1., -1., -1., -1., + 0., -1., -1., 0., 0., 0., 0., -1., -1., -1., 0., 0., 0., 2.]]) + bl = np.array([-inf, -inf, -inf, -inf, -inf, -inf, -inf, -inf, -inf]) + bu = np.array([100., 0., 0., 0., 0., 0., 0., 0., 0.]) + constraints = LinearConstraint(A, bl, bu) + integrality = 1 + bounds = (0, 1) + r1 = milp(c=c, constraints=constraints, integrality=integrality, bounds=bounds, + options={'presolve': True}) + r2 = milp(c=c, constraints=constraints, integrality=integrality, bounds=bounds, + options={'presolve': False}) + assert r1.status == r2.status + assert_allclose(r1.x, r2.x) + + # another example from the same issue + bounds = Bounds(lb=0, ub=1) + integrality = [1, 1, 0, 0] + c = [10, 9.52380952, -1000, -952.38095238] + A = [[1, 1, 0, 0], [0, 0, 1, 1], [200, 0, 0, 0], [0, 200, 0, 0], + [0, 0, 2000, 0], [0, 0, 0, 2000], [-1, 0, 1, 0], [-1, -1, 0, 1]] + ub = [1, 1, 200, 200, 1000, 1000, 0, 0] + constraints = LinearConstraint(A, ub=ub) + r1 = milp(c=c, constraints=constraints, bounds=bounds, + integrality=integrality, options={"presolve": False}) + r2 = milp(c=c, constraints=constraints, bounds=bounds, + integrality=integrality, options={"presolve": False}) + assert r1.status == r2.status + assert_allclose(r1.x, r2.x) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_minimize_constrained.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_minimize_constrained.py new file mode 100644 index 0000000000000000000000000000000000000000..cda21fd1dc2b24b128e47405e01353fdae41a75c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_minimize_constrained.py @@ -0,0 +1,845 @@ +import numpy as np +import pytest +from scipy.linalg import block_diag +from scipy.sparse import csc_matrix +from numpy.testing import (assert_array_almost_equal, + assert_array_less, assert_, + suppress_warnings) +from scipy.optimize import (NonlinearConstraint, + LinearConstraint, + Bounds, + minimize, + BFGS, + SR1, + rosen) + + +class Maratos: + """Problem 15.4 from Nocedal and Wright + + The following optimization problem: + minimize 2*(x[0]**2 + x[1]**2 - 1) - x[0] + Subject to: x[0]**2 + x[1]**2 - 1 = 0 + """ + + def __init__(self, degrees=60, constr_jac=None, constr_hess=None): + rads = degrees/180*np.pi + self.x0 = [np.cos(rads), np.sin(rads)] + self.x_opt = np.array([1.0, 0.0]) + self.constr_jac = constr_jac + self.constr_hess = constr_hess + self.bounds = None + + def fun(self, x): + return 2*(x[0]**2 + x[1]**2 - 1) - x[0] + + def grad(self, x): + return np.array([4*x[0]-1, 4*x[1]]) + + def hess(self, x): + return 4*np.eye(2) + + @property + def constr(self): + def fun(x): + return x[0]**2 + x[1]**2 + + if self.constr_jac is None: + def jac(x): + return [[2*x[0], 2*x[1]]] + else: + jac = self.constr_jac + + if self.constr_hess is None: + def hess(x, v): + return 2*v[0]*np.eye(2) + else: + hess = self.constr_hess + + return NonlinearConstraint(fun, 1, 1, jac, hess) + + +class MaratosTestArgs: + """Problem 15.4 from Nocedal and Wright + + The following optimization problem: + minimize 2*(x[0]**2 + x[1]**2 - 1) - x[0] + Subject to: x[0]**2 + x[1]**2 - 1 = 0 + """ + + def __init__(self, a, b, degrees=60, constr_jac=None, constr_hess=None): + rads = degrees/180*np.pi + self.x0 = [np.cos(rads), np.sin(rads)] + self.x_opt = np.array([1.0, 0.0]) + self.constr_jac = constr_jac + self.constr_hess = constr_hess + self.a = a + self.b = b + self.bounds = None + + def _test_args(self, a, b): + if self.a != a or self.b != b: + raise ValueError() + + def fun(self, x, a, b): + self._test_args(a, b) + return 2*(x[0]**2 + x[1]**2 - 1) - x[0] + + def grad(self, x, a, b): + self._test_args(a, b) + return np.array([4*x[0]-1, 4*x[1]]) + + def hess(self, x, a, b): + self._test_args(a, b) + return 4*np.eye(2) + + @property + def constr(self): + def fun(x): + return x[0]**2 + x[1]**2 + + if self.constr_jac is None: + def jac(x): + return [[4*x[0], 4*x[1]]] + else: + jac = self.constr_jac + + if self.constr_hess is None: + def hess(x, v): + return 2*v[0]*np.eye(2) + else: + hess = self.constr_hess + + return NonlinearConstraint(fun, 1, 1, jac, hess) + + +class MaratosGradInFunc: + """Problem 15.4 from Nocedal and Wright + + The following optimization problem: + minimize 2*(x[0]**2 + x[1]**2 - 1) - x[0] + Subject to: x[0]**2 + x[1]**2 - 1 = 0 + """ + + def __init__(self, degrees=60, constr_jac=None, constr_hess=None): + rads = degrees/180*np.pi + self.x0 = [np.cos(rads), np.sin(rads)] + self.x_opt = np.array([1.0, 0.0]) + self.constr_jac = constr_jac + self.constr_hess = constr_hess + self.bounds = None + + def fun(self, x): + return (2*(x[0]**2 + x[1]**2 - 1) - x[0], + np.array([4*x[0]-1, 4*x[1]])) + + @property + def grad(self): + return True + + def hess(self, x): + return 4*np.eye(2) + + @property + def constr(self): + def fun(x): + return x[0]**2 + x[1]**2 + + if self.constr_jac is None: + def jac(x): + return [[4*x[0], 4*x[1]]] + else: + jac = self.constr_jac + + if self.constr_hess is None: + def hess(x, v): + return 2*v[0]*np.eye(2) + else: + hess = self.constr_hess + + return NonlinearConstraint(fun, 1, 1, jac, hess) + + +class HyperbolicIneq: + """Problem 15.1 from Nocedal and Wright + + The following optimization problem: + minimize 1/2*(x[0] - 2)**2 + 1/2*(x[1] - 1/2)**2 + Subject to: 1/(x[0] + 1) - x[1] >= 1/4 + x[0] >= 0 + x[1] >= 0 + """ + def __init__(self, constr_jac=None, constr_hess=None): + self.x0 = [0, 0] + self.x_opt = [1.952823, 0.088659] + self.constr_jac = constr_jac + self.constr_hess = constr_hess + self.bounds = Bounds(0, np.inf) + + def fun(self, x): + return 1/2*(x[0] - 2)**2 + 1/2*(x[1] - 1/2)**2 + + def grad(self, x): + return [x[0] - 2, x[1] - 1/2] + + def hess(self, x): + return np.eye(2) + + @property + def constr(self): + def fun(x): + return 1/(x[0] + 1) - x[1] + + if self.constr_jac is None: + def jac(x): + return [[-1/(x[0] + 1)**2, -1]] + else: + jac = self.constr_jac + + if self.constr_hess is None: + def hess(x, v): + return 2*v[0]*np.array([[1/(x[0] + 1)**3, 0], + [0, 0]]) + else: + hess = self.constr_hess + + return NonlinearConstraint(fun, 0.25, np.inf, jac, hess) + + +class Rosenbrock: + """Rosenbrock function. + + The following optimization problem: + minimize sum(100.0*(x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0) + """ + + def __init__(self, n=2, random_state=0): + rng = np.random.RandomState(random_state) + self.x0 = rng.uniform(-1, 1, n) + self.x_opt = np.ones(n) + self.bounds = None + + def fun(self, x): + x = np.asarray(x) + r = np.sum(100.0 * (x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0, + axis=0) + return r + + def grad(self, x): + x = np.asarray(x) + xm = x[1:-1] + xm_m1 = x[:-2] + xm_p1 = x[2:] + der = np.zeros_like(x) + der[1:-1] = (200 * (xm - xm_m1**2) - + 400 * (xm_p1 - xm**2) * xm - 2 * (1 - xm)) + der[0] = -400 * x[0] * (x[1] - x[0]**2) - 2 * (1 - x[0]) + der[-1] = 200 * (x[-1] - x[-2]**2) + return der + + def hess(self, x): + x = np.atleast_1d(x) + H = np.diag(-400 * x[:-1], 1) - np.diag(400 * x[:-1], -1) + diagonal = np.zeros(len(x), dtype=x.dtype) + diagonal[0] = 1200 * x[0]**2 - 400 * x[1] + 2 + diagonal[-1] = 200 + diagonal[1:-1] = 202 + 1200 * x[1:-1]**2 - 400 * x[2:] + H = H + np.diag(diagonal) + return H + + @property + def constr(self): + return () + + +class IneqRosenbrock(Rosenbrock): + """Rosenbrock subject to inequality constraints. + + The following optimization problem: + minimize sum(100.0*(x[1] - x[0]**2)**2.0 + (1 - x[0])**2) + subject to: x[0] + 2 x[1] <= 1 + + Taken from matlab ``fmincon`` documentation. + """ + def __init__(self, random_state=0): + Rosenbrock.__init__(self, 2, random_state) + self.x0 = [-1, -0.5] + self.x_opt = [0.5022, 0.2489] + self.bounds = None + + @property + def constr(self): + A = [[1, 2]] + b = 1 + return LinearConstraint(A, -np.inf, b) + + +class BoundedRosenbrock(Rosenbrock): + """Rosenbrock subject to inequality constraints. + + The following optimization problem: + minimize sum(100.0*(x[1] - x[0]**2)**2.0 + (1 - x[0])**2) + subject to: -2 <= x[0] <= 0 + 0 <= x[1] <= 2 + + Taken from matlab ``fmincon`` documentation. + """ + def __init__(self, random_state=0): + Rosenbrock.__init__(self, 2, random_state) + self.x0 = [-0.2, 0.2] + self.x_opt = None + self.bounds = Bounds([-2, 0], [0, 2]) + + +class EqIneqRosenbrock(Rosenbrock): + """Rosenbrock subject to equality and inequality constraints. + + The following optimization problem: + minimize sum(100.0*(x[1] - x[0]**2)**2.0 + (1 - x[0])**2) + subject to: x[0] + 2 x[1] <= 1 + 2 x[0] + x[1] = 1 + + Taken from matlab ``fimincon`` documentation. + """ + def __init__(self, random_state=0): + Rosenbrock.__init__(self, 2, random_state) + self.x0 = [-1, -0.5] + self.x_opt = [0.41494, 0.17011] + self.bounds = None + + @property + def constr(self): + A_ineq = [[1, 2]] + b_ineq = 1 + A_eq = [[2, 1]] + b_eq = 1 + return (LinearConstraint(A_ineq, -np.inf, b_ineq), + LinearConstraint(A_eq, b_eq, b_eq)) + + +class Elec: + """Distribution of electrons on a sphere. + + Problem no 2 from COPS collection [2]_. Find + the equilibrium state distribution (of minimal + potential) of the electrons positioned on a + conducting sphere. + + References + ---------- + .. [1] E. D. Dolan, J. J. Mor\'{e}, and T. S. Munson, + "Benchmarking optimization software with COPS 3.0.", + Argonne National Lab., Argonne, IL (US), 2004. + """ + def __init__(self, n_electrons=200, random_state=0, + constr_jac=None, constr_hess=None): + self.n_electrons = n_electrons + self.rng = np.random.RandomState(random_state) + # Initial Guess + phi = self.rng.uniform(0, 2 * np.pi, self.n_electrons) + theta = self.rng.uniform(-np.pi, np.pi, self.n_electrons) + x = np.cos(theta) * np.cos(phi) + y = np.cos(theta) * np.sin(phi) + z = np.sin(theta) + self.x0 = np.hstack((x, y, z)) + self.x_opt = None + self.constr_jac = constr_jac + self.constr_hess = constr_hess + self.bounds = None + + def _get_cordinates(self, x): + x_coord = x[:self.n_electrons] + y_coord = x[self.n_electrons:2 * self.n_electrons] + z_coord = x[2 * self.n_electrons:] + return x_coord, y_coord, z_coord + + def _compute_coordinate_deltas(self, x): + x_coord, y_coord, z_coord = self._get_cordinates(x) + dx = x_coord[:, None] - x_coord + dy = y_coord[:, None] - y_coord + dz = z_coord[:, None] - z_coord + return dx, dy, dz + + def fun(self, x): + dx, dy, dz = self._compute_coordinate_deltas(x) + with np.errstate(divide='ignore'): + dm1 = (dx**2 + dy**2 + dz**2) ** -0.5 + dm1[np.diag_indices_from(dm1)] = 0 + return 0.5 * np.sum(dm1) + + def grad(self, x): + dx, dy, dz = self._compute_coordinate_deltas(x) + + with np.errstate(divide='ignore'): + dm3 = (dx**2 + dy**2 + dz**2) ** -1.5 + dm3[np.diag_indices_from(dm3)] = 0 + + grad_x = -np.sum(dx * dm3, axis=1) + grad_y = -np.sum(dy * dm3, axis=1) + grad_z = -np.sum(dz * dm3, axis=1) + + return np.hstack((grad_x, grad_y, grad_z)) + + def hess(self, x): + dx, dy, dz = self._compute_coordinate_deltas(x) + d = (dx**2 + dy**2 + dz**2) ** 0.5 + + with np.errstate(divide='ignore'): + dm3 = d ** -3 + dm5 = d ** -5 + + i = np.arange(self.n_electrons) + dm3[i, i] = 0 + dm5[i, i] = 0 + + Hxx = dm3 - 3 * dx**2 * dm5 + Hxx[i, i] = -np.sum(Hxx, axis=1) + + Hxy = -3 * dx * dy * dm5 + Hxy[i, i] = -np.sum(Hxy, axis=1) + + Hxz = -3 * dx * dz * dm5 + Hxz[i, i] = -np.sum(Hxz, axis=1) + + Hyy = dm3 - 3 * dy**2 * dm5 + Hyy[i, i] = -np.sum(Hyy, axis=1) + + Hyz = -3 * dy * dz * dm5 + Hyz[i, i] = -np.sum(Hyz, axis=1) + + Hzz = dm3 - 3 * dz**2 * dm5 + Hzz[i, i] = -np.sum(Hzz, axis=1) + + H = np.vstack(( + np.hstack((Hxx, Hxy, Hxz)), + np.hstack((Hxy, Hyy, Hyz)), + np.hstack((Hxz, Hyz, Hzz)) + )) + + return H + + @property + def constr(self): + def fun(x): + x_coord, y_coord, z_coord = self._get_cordinates(x) + return x_coord**2 + y_coord**2 + z_coord**2 - 1 + + if self.constr_jac is None: + def jac(x): + x_coord, y_coord, z_coord = self._get_cordinates(x) + Jx = 2 * np.diag(x_coord) + Jy = 2 * np.diag(y_coord) + Jz = 2 * np.diag(z_coord) + return csc_matrix(np.hstack((Jx, Jy, Jz))) + else: + jac = self.constr_jac + + if self.constr_hess is None: + def hess(x, v): + D = 2 * np.diag(v) + return block_diag(D, D, D) + else: + hess = self.constr_hess + + return NonlinearConstraint(fun, -np.inf, 0, jac, hess) + + +class TestTrustRegionConstr: + list_of_problems = [Maratos(), + Maratos(constr_hess='2-point'), + Maratos(constr_hess=SR1()), + Maratos(constr_jac='2-point', constr_hess=SR1()), + MaratosGradInFunc(), + HyperbolicIneq(), + HyperbolicIneq(constr_hess='3-point'), + HyperbolicIneq(constr_hess=BFGS()), + HyperbolicIneq(constr_jac='3-point', + constr_hess=BFGS()), + Rosenbrock(), + IneqRosenbrock(), + EqIneqRosenbrock(), + BoundedRosenbrock(), + Elec(n_electrons=2), + Elec(n_electrons=2, constr_hess='2-point'), + Elec(n_electrons=2, constr_hess=SR1()), + Elec(n_electrons=2, constr_jac='3-point', + constr_hess=SR1())] + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize('prob', list_of_problems) + @pytest.mark.parametrize('grad', ('prob.grad', '3-point', False)) + @pytest.mark.parametrize('hess', ("prob.hess", '3-point', SR1(), + BFGS(exception_strategy='damp_update'), + BFGS(exception_strategy='skip_update'))) + def test_list_of_problems(self, prob, grad, hess): + grad = prob.grad if grad == "prob.grad" else grad + hess = prob.hess if hess == "prob.hess" else hess + # Remove exceptions + if (grad in {'2-point', '3-point', 'cs', False} and + hess in {'2-point', '3-point', 'cs'}): + pytest.skip("Numerical Hessian needs analytical gradient") + if prob.grad is True and grad in {'3-point', False}: + pytest.skip("prob.grad incompatible with grad in {'3-point', False}") + sensitive = (isinstance(prob, BoundedRosenbrock) and grad == '3-point' + and isinstance(hess, BFGS)) + if sensitive: + pytest.xfail("Seems sensitive to initial conditions w/ Accelerate") + with suppress_warnings() as sup: + sup.filter(UserWarning, "delta_grad == 0.0") + result = minimize(prob.fun, prob.x0, + method='trust-constr', + jac=grad, hess=hess, + bounds=prob.bounds, + constraints=prob.constr) + + if prob.x_opt is not None: + assert_array_almost_equal(result.x, prob.x_opt, + decimal=5) + # gtol + if result.status == 1: + assert_array_less(result.optimality, 1e-8) + # xtol + if result.status == 2: + assert_array_less(result.tr_radius, 1e-8) + + if result.method == "tr_interior_point": + assert_array_less(result.barrier_parameter, 1e-8) + + # check for max iter + message = f"Invalid termination condition: {result.status}." + assert result.status not in {0, 3}, message + + + def test_default_jac_and_hess(self): + def fun(x): + return (x - 1) ** 2 + bounds = [(-2, 2)] + res = minimize(fun, x0=[-1.5], bounds=bounds, method='trust-constr') + assert_array_almost_equal(res.x, 1, decimal=5) + + def test_default_hess(self): + def fun(x): + return (x - 1) ** 2 + bounds = [(-2, 2)] + res = minimize(fun, x0=[-1.5], bounds=bounds, method='trust-constr', + jac='2-point') + assert_array_almost_equal(res.x, 1, decimal=5) + + def test_no_constraints(self): + prob = Rosenbrock() + result = minimize(prob.fun, prob.x0, + method='trust-constr', + jac=prob.grad, hess=prob.hess) + result1 = minimize(prob.fun, prob.x0, + method='L-BFGS-B', + jac='2-point') + + result2 = minimize(prob.fun, prob.x0, + method='L-BFGS-B', + jac='3-point') + assert_array_almost_equal(result.x, prob.x_opt, decimal=5) + assert_array_almost_equal(result1.x, prob.x_opt, decimal=5) + assert_array_almost_equal(result2.x, prob.x_opt, decimal=5) + + def test_hessp(self): + prob = Maratos() + + def hessp(x, p): + H = prob.hess(x) + return H.dot(p) + + result = minimize(prob.fun, prob.x0, + method='trust-constr', + jac=prob.grad, hessp=hessp, + bounds=prob.bounds, + constraints=prob.constr) + + if prob.x_opt is not None: + assert_array_almost_equal(result.x, prob.x_opt, decimal=2) + + # gtol + if result.status == 1: + assert_array_less(result.optimality, 1e-8) + # xtol + if result.status == 2: + assert_array_less(result.tr_radius, 1e-8) + + if result.method == "tr_interior_point": + assert_array_less(result.barrier_parameter, 1e-8) + # max iter + if result.status in (0, 3): + raise RuntimeError("Invalid termination condition.") + + def test_args(self): + prob = MaratosTestArgs("a", 234) + + result = minimize(prob.fun, prob.x0, ("a", 234), + method='trust-constr', + jac=prob.grad, hess=prob.hess, + bounds=prob.bounds, + constraints=prob.constr) + + if prob.x_opt is not None: + assert_array_almost_equal(result.x, prob.x_opt, decimal=2) + + # gtol + if result.status == 1: + assert_array_less(result.optimality, 1e-8) + # xtol + if result.status == 2: + assert_array_less(result.tr_radius, 1e-8) + if result.method == "tr_interior_point": + assert_array_less(result.barrier_parameter, 1e-8) + # max iter + if result.status in (0, 3): + raise RuntimeError("Invalid termination condition.") + + def test_raise_exception(self): + prob = Maratos() + message = "Whenever the gradient is estimated via finite-differences" + with pytest.raises(ValueError, match=message): + minimize(prob.fun, prob.x0, method='trust-constr', jac='2-point', + hess='2-point', constraints=prob.constr) + + def test_issue_9044(self): + # https://github.com/scipy/scipy/issues/9044 + # Test the returned `OptimizeResult` contains keys consistent with + # other solvers. + + def callback(x, info): + assert_('nit' in info) + assert_('niter' in info) + + result = minimize(lambda x: x**2, [0], jac=lambda x: 2*x, + hess=lambda x: 2, callback=callback, + method='trust-constr') + assert_(result.get('success')) + assert_(result.get('nit', -1) == 1) + + # Also check existence of the 'niter' attribute, for backward + # compatibility + assert_(result.get('niter', -1) == 1) + + def test_issue_15093(self): + # scipy docs define bounds as inclusive, so it shouldn't be + # an issue to set x0 on the bounds even if keep_feasible is + # True. Previously, trust-constr would treat bounds as + # exclusive. + + x0 = np.array([0., 0.5]) + + def obj(x): + x1 = x[0] + x2 = x[1] + return x1 ** 2 + x2 ** 2 + + bounds = Bounds(np.array([0., 0.]), np.array([1., 1.]), + keep_feasible=True) + + with suppress_warnings() as sup: + sup.filter(UserWarning, "delta_grad == 0.0") + result = minimize( + method='trust-constr', + fun=obj, + x0=x0, + bounds=bounds) + + assert result['success'] + +class TestEmptyConstraint: + """ + Here we minimize x^2+y^2 subject to x^2-y^2>1. + The actual minimum is at (0, 0) which fails the constraint. + Therefore we will find a minimum on the boundary at (+/-1, 0). + + When minimizing on the boundary, optimize uses a set of + constraints that removes the constraint that sets that + boundary. In our case, there's only one constraint, so + the result is an empty constraint. + + This tests that the empty constraint works. + """ + def test_empty_constraint(self): + + def function(x): + return x[0]**2 + x[1]**2 + + def functionjacobian(x): + return np.array([2.*x[0], 2.*x[1]]) + + def functionhvp(x, v): + return 2.*v + + def constraint(x): + return np.array([x[0]**2 - x[1]**2]) + + def constraintjacobian(x): + return np.array([[2*x[0], -2*x[1]]]) + + def constraintlcoh(x, v): + return np.array([[2., 0.], [0., -2.]]) * v[0] + + constraint = NonlinearConstraint(constraint, 1., np.inf, + constraintjacobian, constraintlcoh) + + startpoint = [1., 2.] + + bounds = Bounds([-np.inf, -np.inf], [np.inf, np.inf]) + + result = minimize( + function, + startpoint, + method='trust-constr', + jac=functionjacobian, + hessp=functionhvp, + constraints=[constraint], + bounds=bounds, + ) + + assert_array_almost_equal(abs(result.x), np.array([1, 0]), decimal=4) + + +def test_bug_11886(): + def opt(x): + return x[0]**2+x[1]**2 + + with np.testing.suppress_warnings() as sup: + sup.filter(PendingDeprecationWarning) + A = np.matrix(np.diag([1, 1])) + lin_cons = LinearConstraint(A, -1, np.inf) + # just checking that there are no errors + minimize(opt, 2*[1], constraints = lin_cons) + + +def test_gh11649(): + # trust - constr error when attempting to keep bound constrained solutions + # feasible. Algorithm attempts to go outside bounds when evaluating finite + # differences. (don't give objective an analytic gradient) + bnds = Bounds(lb=[-1, -1], ub=[1, 1], keep_feasible=True) + + def assert_inbounds(x): + assert np.all(x >= bnds.lb) + assert np.all(x <= bnds.ub) + + def obj(x): + assert_inbounds(x) + return np.exp(x[0])*(4*x[0]**2 + 2*x[1]**2 + 4*x[0]*x[1] + 2*x[1] + 1) + + def nce(x): + assert_inbounds(x) + return x[0]**2 + x[1] + + def nce_jac(x): + return np.array([2*x[0], 1]) + + def nci(x): + assert_inbounds(x) + return x[0]*x[1] + + x0 = np.array((0.99, -0.99)) + nlcs = [NonlinearConstraint(nci, -10, np.inf), + NonlinearConstraint(nce, 1, 1, jac=nce_jac)] + + res = minimize(fun=obj, x0=x0, method='trust-constr', + bounds=bnds, constraints=nlcs) + assert_inbounds(res.x) + assert nlcs[0].lb < nlcs[0].fun(res.x) < nlcs[0].ub + + +def test_gh20665_too_many_constraints(): + # gh-20665 reports a confusing error message when there are more equality + # constraints than variables. Check that the error message is improved. + message = "...more equality constraints than independent variables..." + with pytest.raises(ValueError, match=message): + x0 = np.ones((2,)) + A_eq, b_eq = np.arange(6).reshape((3, 2)), np.ones((3,)) + g = NonlinearConstraint(lambda x: A_eq @ x, lb=b_eq, ub=b_eq) + minimize(rosen, x0, method='trust-constr', constraints=[g]) + # no error with `SVDFactorization` + with np.testing.suppress_warnings() as sup: + sup.filter(UserWarning) + minimize(rosen, x0, method='trust-constr', constraints=[g], + options={'factorization_method': 'SVDFactorization'}) + +def test_issue_18882(): + def lsf(u): + u1, u2 = u + a, b = [3.0, 4.0] + return 1.0 + u1**2 / a**2 - u2**2 / b**2 + + def of(u): + return np.sum(u**2) + + with suppress_warnings() as sup: + sup.filter(UserWarning, "delta_grad == 0.0") + sup.filter(UserWarning, "Singular Jacobian matrix.") + res = minimize( + of, + [0.0, 0.0], + method="trust-constr", + constraints=NonlinearConstraint(lsf, 0, 0), + ) + assert (not res.success) and (res.constr_violation > 1e-8) + +class TestBoundedNelderMead: + + @pytest.mark.parametrize('bounds, x_opt', + [(Bounds(-np.inf, np.inf), Rosenbrock().x_opt), + (Bounds(-np.inf, -0.8), [-0.8, -0.8]), + (Bounds(3.0, np.inf), [3.0, 9.0]), + (Bounds([3.0, 1.0], [4.0, 5.0]), [3., 5.]), + ]) + def test_rosen_brock_with_bounds(self, bounds, x_opt): + prob = Rosenbrock() + with suppress_warnings() as sup: + sup.filter(UserWarning, "Initial guess is not within " + "the specified bounds") + result = minimize(prob.fun, [-10, -10], + method='Nelder-Mead', + bounds=bounds) + assert np.less_equal(bounds.lb, result.x).all() + assert np.less_equal(result.x, bounds.ub).all() + assert np.allclose(prob.fun(result.x), result.fun) + assert np.allclose(result.x, x_opt, atol=1.e-3) + + def test_equal_all_bounds(self): + prob = Rosenbrock() + bounds = Bounds([4.0, 5.0], [4.0, 5.0]) + with suppress_warnings() as sup: + sup.filter(UserWarning, "Initial guess is not within " + "the specified bounds") + result = minimize(prob.fun, [-10, 8], + method='Nelder-Mead', + bounds=bounds) + assert np.allclose(result.x, [4.0, 5.0]) + + def test_equal_one_bounds(self): + prob = Rosenbrock() + bounds = Bounds([4.0, 5.0], [4.0, 20.0]) + with suppress_warnings() as sup: + sup.filter(UserWarning, "Initial guess is not within " + "the specified bounds") + result = minimize(prob.fun, [-10, 8], + method='Nelder-Mead', + bounds=bounds) + assert np.allclose(result.x, [4.0, 16.0]) + + def test_invalid_bounds(self): + prob = Rosenbrock() + message = 'An upper bound is less than the corresponding lower bound.' + with pytest.raises(ValueError, match=message): + bounds = Bounds([-np.inf, 1.0], [4.0, -5.0]) + minimize(prob.fun, [-10, 3], + method='Nelder-Mead', + bounds=bounds) + + @pytest.mark.xfail(reason="Failing on Azure Linux and macOS builds, " + "see gh-13846") + def test_outside_bounds_warning(self): + prob = Rosenbrock() + message = "Initial guess is not within the specified bounds" + with pytest.warns(UserWarning, match=message): + bounds = Bounds([-np.inf, 1.0], [4.0, 5.0]) + minimize(prob.fun, [-10, 8], + method='Nelder-Mead', + bounds=bounds) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_minpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_minpack.py new file mode 100644 index 0000000000000000000000000000000000000000..ef107c5692c2d30a06b6943542696b95bc21e818 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_minpack.py @@ -0,0 +1,1194 @@ +""" +Unit tests for optimization routines from minpack.py. +""" +import warnings +import pytest +import threading + +from numpy.testing import (assert_, assert_almost_equal, assert_array_equal, + assert_array_almost_equal, assert_allclose, + assert_warns, suppress_warnings) +from pytest import raises as assert_raises +import numpy as np +from numpy import array, float64 +from multiprocessing.pool import ThreadPool + +from scipy import optimize, linalg +from scipy.special import lambertw +from scipy.optimize._minpack_py import leastsq, curve_fit, fixed_point +from scipy.optimize import OptimizeWarning +from scipy.optimize._minimize import Bounds + + +class ReturnShape: + """This class exists to create a callable that does not have a '__name__' attribute. + + __init__ takes the argument 'shape', which should be a tuple of ints. + When an instance is called with a single argument 'x', it returns numpy.ones(shape). + """ + + def __init__(self, shape): + self.shape = shape + + def __call__(self, x): + return np.ones(self.shape) + + +def dummy_func(x, shape): + """A function that returns an array of ones of the given shape. + `x` is ignored. + """ + return np.ones(shape) + + +def sequence_parallel(fs): + with ThreadPool(len(fs)) as pool: + return pool.map(lambda f: f(), fs) + + +# Function and Jacobian for tests of solvers for systems of nonlinear +# equations + + +def pressure_network(flow_rates, Qtot, k): + """Evaluate non-linear equation system representing + the pressures and flows in a system of n parallel pipes:: + + f_i = P_i - P_0, for i = 1..n + f_0 = sum(Q_i) - Qtot + + where Q_i is the flow rate in pipe i and P_i the pressure in that pipe. + Pressure is modeled as a P=kQ**2 where k is a valve coefficient and + Q is the flow rate. + + Parameters + ---------- + flow_rates : float + A 1-D array of n flow rates [kg/s]. + k : float + A 1-D array of n valve coefficients [1/kg m]. + Qtot : float + A scalar, the total input flow rate [kg/s]. + + Returns + ------- + F : float + A 1-D array, F[i] == f_i. + + """ + P = k * flow_rates**2 + F = np.hstack((P[1:] - P[0], flow_rates.sum() - Qtot)) + return F + + +def pressure_network_jacobian(flow_rates, Qtot, k): + """Return the jacobian of the equation system F(flow_rates) + computed by `pressure_network` with respect to + *flow_rates*. See `pressure_network` for the detailed + description of parameters. + + Returns + ------- + jac : float + *n* by *n* matrix ``df_i/dQ_i`` where ``n = len(flow_rates)`` + and *f_i* and *Q_i* are described in the doc for `pressure_network` + """ + n = len(flow_rates) + pdiff = np.diag(flow_rates[1:] * 2 * k[1:] - 2 * flow_rates[0] * k[0]) + + jac = np.empty((n, n)) + jac[:n-1, :n-1] = pdiff * 0 + jac[:n-1, n-1] = 0 + jac[n-1, :] = np.ones(n) + + return jac + + +def pressure_network_fun_and_grad(flow_rates, Qtot, k): + return (pressure_network(flow_rates, Qtot, k), + pressure_network_jacobian(flow_rates, Qtot, k)) + + +class TestFSolve: + def test_pressure_network_no_gradient(self): + # fsolve without gradient, equal pipes -> equal flows. + k = np.full(4, 0.5) + Qtot = 4 + initial_guess = array([2., 0., 2., 0.]) + final_flows, info, ier, mesg = optimize.fsolve( + pressure_network, initial_guess, args=(Qtot, k), + full_output=True) + assert_array_almost_equal(final_flows, np.ones(4)) + assert_(ier == 1, mesg) + + def test_pressure_network_with_gradient(self): + # fsolve with gradient, equal pipes -> equal flows + k = np.full(4, 0.5) + Qtot = 4 + initial_guess = array([2., 0., 2., 0.]) + final_flows = optimize.fsolve( + pressure_network, initial_guess, args=(Qtot, k), + fprime=pressure_network_jacobian) + assert_array_almost_equal(final_flows, np.ones(4)) + + def test_wrong_shape_func_callable(self): + func = ReturnShape(1) + # x0 is a list of two elements, but func will return an array with + # length 1, so this should result in a TypeError. + x0 = [1.5, 2.0] + assert_raises(TypeError, optimize.fsolve, func, x0) + + def test_wrong_shape_func_function(self): + # x0 is a list of two elements, but func will return an array with + # length 1, so this should result in a TypeError. + x0 = [1.5, 2.0] + assert_raises(TypeError, optimize.fsolve, dummy_func, x0, args=((1,),)) + + def test_wrong_shape_fprime_callable(self): + func = ReturnShape(1) + deriv_func = ReturnShape((2,2)) + assert_raises(TypeError, optimize.fsolve, func, x0=[0,1], fprime=deriv_func) + + def test_wrong_shape_fprime_function(self): + def func(x): + return dummy_func(x, (2,)) + def deriv_func(x): + return dummy_func(x, (3, 3)) + assert_raises(TypeError, optimize.fsolve, func, x0=[0,1], fprime=deriv_func) + + def test_func_can_raise(self): + def func(*args): + raise ValueError('I raised') + + with assert_raises(ValueError, match='I raised'): + optimize.fsolve(func, x0=[0]) + + def test_Dfun_can_raise(self): + def func(x): + return x - np.array([10]) + + def deriv_func(*args): + raise ValueError('I raised') + + with assert_raises(ValueError, match='I raised'): + optimize.fsolve(func, x0=[0], fprime=deriv_func) + + def test_float32(self): + def func(x): + return np.array([x[0] - 100, x[1] - 1000], dtype=np.float32) ** 2 + p = optimize.fsolve(func, np.array([1, 1], np.float32)) + assert_allclose(func(p), [0, 0], atol=1e-3) + + def test_reentrant_func(self): + def func(*args): + self.test_pressure_network_no_gradient() + return pressure_network(*args) + + # fsolve without gradient, equal pipes -> equal flows. + k = np.full(4, 0.5) + Qtot = 4 + initial_guess = array([2., 0., 2., 0.]) + final_flows, info, ier, mesg = optimize.fsolve( + func, initial_guess, args=(Qtot, k), + full_output=True) + assert_array_almost_equal(final_flows, np.ones(4)) + assert_(ier == 1, mesg) + + def test_reentrant_Dfunc(self): + def deriv_func(*args): + self.test_pressure_network_with_gradient() + return pressure_network_jacobian(*args) + + # fsolve with gradient, equal pipes -> equal flows + k = np.full(4, 0.5) + Qtot = 4 + initial_guess = array([2., 0., 2., 0.]) + final_flows = optimize.fsolve( + pressure_network, initial_guess, args=(Qtot, k), + fprime=deriv_func) + assert_array_almost_equal(final_flows, np.ones(4)) + + def test_concurrent_no_gradient(self): + v = sequence_parallel([self.test_pressure_network_no_gradient] * 10) + assert all([result is None for result in v]) + + def test_concurrent_with_gradient(self): + v = sequence_parallel([self.test_pressure_network_with_gradient] * 10) + assert all([result is None for result in v]) + + +class TestRootHybr: + def test_pressure_network_no_gradient(self): + # root/hybr without gradient, equal pipes -> equal flows + k = np.full(4, 0.5) + Qtot = 4 + initial_guess = array([2., 0., 2., 0.]) + final_flows = optimize.root(pressure_network, initial_guess, + method='hybr', args=(Qtot, k)).x + assert_array_almost_equal(final_flows, np.ones(4)) + + def test_pressure_network_with_gradient(self): + # root/hybr with gradient, equal pipes -> equal flows + k = np.full(4, 0.5) + Qtot = 4 + initial_guess = array([[2., 0., 2., 0.]]) + final_flows = optimize.root(pressure_network, initial_guess, + args=(Qtot, k), method='hybr', + jac=pressure_network_jacobian).x + assert_array_almost_equal(final_flows, np.ones(4)) + + def test_pressure_network_with_gradient_combined(self): + # root/hybr with gradient and function combined, equal pipes -> equal + # flows + k = np.full(4, 0.5) + Qtot = 4 + initial_guess = array([2., 0., 2., 0.]) + final_flows = optimize.root(pressure_network_fun_and_grad, + initial_guess, args=(Qtot, k), + method='hybr', jac=True).x + assert_array_almost_equal(final_flows, np.ones(4)) + + +class TestRootLM: + def test_pressure_network_no_gradient(self): + # root/lm without gradient, equal pipes -> equal flows + k = np.full(4, 0.5) + Qtot = 4 + initial_guess = array([2., 0., 2., 0.]) + final_flows = optimize.root(pressure_network, initial_guess, + method='lm', args=(Qtot, k)).x + assert_array_almost_equal(final_flows, np.ones(4)) + + +class TestNfev: + def setup_method(self): + self.nfev = threading.local() + + def zero_f(self, y): + if not hasattr(self.nfev, 'c'): + self.nfev.c = 0 + self.nfev.c += 1 + return y**2-3 + + @pytest.mark.parametrize('method', ['hybr', 'lm', 'broyden1', + 'broyden2', 'anderson', + 'linearmixing', 'diagbroyden', + 'excitingmixing', 'krylov', + 'df-sane']) + def test_root_nfev(self, method): + self.nfev.c = 0 + solution = optimize.root(self.zero_f, 100, method=method) + assert solution.nfev == self.nfev.c + + def test_fsolve_nfev(self): + self.nfev.c = 0 + x, info, ier, mesg = optimize.fsolve(self.zero_f, 100, full_output=True) + assert info['nfev'] == self.nfev.c + + +class TestLeastSq: + def setup_method(self): + x = np.linspace(0, 10, 40) + a,b,c = 3.1, 42, -304.2 + self.x = x + self.abc = a,b,c + y_true = a*x**2 + b*x + c + np.random.seed(0) + self.y_meas = y_true + 0.01*np.random.standard_normal(y_true.shape) + + def residuals(self, p, y, x): + a,b,c = p + err = y-(a*x**2 + b*x + c) + return err + + def residuals_jacobian(self, _p, _y, x): + return -np.vstack([x**2, x, np.ones_like(x)]).T + + def test_basic(self): + p0 = array([0,0,0]) + params_fit, ier = leastsq(self.residuals, p0, + args=(self.y_meas, self.x)) + assert_(ier in (1,2,3,4), 'solution not found (ier=%d)' % ier) + # low precision due to random + assert_array_almost_equal(params_fit, self.abc, decimal=2) + + def test_basic_with_gradient(self): + p0 = array([0,0,0]) + params_fit, ier = leastsq(self.residuals, p0, + args=(self.y_meas, self.x), + Dfun=self.residuals_jacobian) + assert_(ier in (1,2,3,4), 'solution not found (ier=%d)' % ier) + # low precision due to random + assert_array_almost_equal(params_fit, self.abc, decimal=2) + + def test_full_output(self): + p0 = array([[0,0,0]]) + full_output = leastsq(self.residuals, p0, + args=(self.y_meas, self.x), + full_output=True) + params_fit, cov_x, infodict, mesg, ier = full_output + assert_(ier in (1,2,3,4), f'solution not found: {mesg}') + + def test_input_untouched(self): + p0 = array([0,0,0],dtype=float64) + p0_copy = array(p0, copy=True) + full_output = leastsq(self.residuals, p0, + args=(self.y_meas, self.x), + full_output=True) + params_fit, cov_x, infodict, mesg, ier = full_output + assert_(ier in (1,2,3,4), f'solution not found: {mesg}') + assert_array_equal(p0, p0_copy) + + def test_wrong_shape_func_callable(self): + func = ReturnShape(1) + # x0 is a list of two elements, but func will return an array with + # length 1, so this should result in a TypeError. + x0 = [1.5, 2.0] + assert_raises(TypeError, optimize.leastsq, func, x0) + + def test_wrong_shape_func_function(self): + # x0 is a list of two elements, but func will return an array with + # length 1, so this should result in a TypeError. + x0 = [1.5, 2.0] + assert_raises(TypeError, optimize.leastsq, dummy_func, x0, args=((1,),)) + + def test_wrong_shape_Dfun_callable(self): + func = ReturnShape(1) + deriv_func = ReturnShape((2,2)) + assert_raises(TypeError, optimize.leastsq, func, x0=[0,1], Dfun=deriv_func) + + def test_wrong_shape_Dfun_function(self): + def func(x): + return dummy_func(x, (2,)) + def deriv_func(x): + return dummy_func(x, (3, 3)) + assert_raises(TypeError, optimize.leastsq, func, x0=[0,1], Dfun=deriv_func) + + def test_float32(self): + # Regression test for gh-1447 + def func(p,x,y): + q = p[0]*np.exp(-(x-p[1])**2/(2.0*p[2]**2))+p[3] + return q - y + + x = np.array([1.475,1.429,1.409,1.419,1.455,1.519,1.472, 1.368,1.286, + 1.231], dtype=np.float32) + y = np.array([0.0168,0.0193,0.0211,0.0202,0.0171,0.0151,0.0185,0.0258, + 0.034,0.0396], dtype=np.float32) + p0 = np.array([1.0,1.0,1.0,1.0]) + p1, success = optimize.leastsq(func, p0, args=(x,y)) + + assert_(success in [1,2,3,4]) + assert_((func(p1,x,y)**2).sum() < 1e-4 * (func(p0,x,y)**2).sum()) + + def test_func_can_raise(self): + def func(*args): + raise ValueError('I raised') + + with assert_raises(ValueError, match='I raised'): + optimize.leastsq(func, x0=[0]) + + def test_Dfun_can_raise(self): + def func(x): + return x - np.array([10]) + + def deriv_func(*args): + raise ValueError('I raised') + + with assert_raises(ValueError, match='I raised'): + optimize.leastsq(func, x0=[0], Dfun=deriv_func) + + def test_reentrant_func(self): + def func(*args): + self.test_basic() + return self.residuals(*args) + + p0 = array([0,0,0]) + params_fit, ier = leastsq(func, p0, + args=(self.y_meas, self.x)) + assert_(ier in (1,2,3,4), 'solution not found (ier=%d)' % ier) + # low precision due to random + assert_array_almost_equal(params_fit, self.abc, decimal=2) + + def test_reentrant_Dfun(self): + def deriv_func(*args): + self.test_basic() + return self.residuals_jacobian(*args) + + p0 = array([0,0,0]) + params_fit, ier = leastsq(self.residuals, p0, + args=(self.y_meas, self.x), + Dfun=deriv_func) + assert_(ier in (1,2,3,4), 'solution not found (ier=%d)' % ier) + # low precision due to random + assert_array_almost_equal(params_fit, self.abc, decimal=2) + + def test_concurrent_no_gradient(self): + v = sequence_parallel([self.test_basic] * 10) + assert all([result is None for result in v]) + + def test_concurrent_with_gradient(self): + v = sequence_parallel([self.test_basic_with_gradient] * 10) + assert all([result is None for result in v]) + + def test_func_input_output_length_check(self): + + def func(x): + return 2 * (x[0] - 3) ** 2 + 1 + + with assert_raises(TypeError, + match='Improper input: func input vector length N='): + optimize.leastsq(func, x0=[0, 1]) + + +class TestCurveFit: + def setup_method(self): + self.y = array([1.0, 3.2, 9.5, 13.7]) + self.x = array([1.0, 2.0, 3.0, 4.0]) + + def test_one_argument(self): + def func(x,a): + return x**a + popt, pcov = curve_fit(func, self.x, self.y) + assert_(len(popt) == 1) + assert_(pcov.shape == (1,1)) + assert_almost_equal(popt[0], 1.9149, decimal=4) + assert_almost_equal(pcov[0,0], 0.0016, decimal=4) + + # Test if we get the same with full_output. Regression test for #1415. + # Also test if check_finite can be turned off. + res = curve_fit(func, self.x, self.y, + full_output=1, check_finite=False) + (popt2, pcov2, infodict, errmsg, ier) = res + assert_array_almost_equal(popt, popt2) + + def test_two_argument(self): + def func(x, a, b): + return b*x**a + popt, pcov = curve_fit(func, self.x, self.y) + assert_(len(popt) == 2) + assert_(pcov.shape == (2,2)) + assert_array_almost_equal(popt, [1.7989, 1.1642], decimal=4) + assert_array_almost_equal(pcov, [[0.0852, -0.1260], [-0.1260, 0.1912]], + decimal=4) + + def test_func_is_classmethod(self): + class test_self: + """This class tests if curve_fit passes the correct number of + arguments when the model function is a class instance method. + """ + + def func(self, x, a, b): + return b * x**a + + test_self_inst = test_self() + popt, pcov = curve_fit(test_self_inst.func, self.x, self.y) + assert_(pcov.shape == (2,2)) + assert_array_almost_equal(popt, [1.7989, 1.1642], decimal=4) + assert_array_almost_equal(pcov, [[0.0852, -0.1260], [-0.1260, 0.1912]], + decimal=4) + + def test_regression_2639(self): + # This test fails if epsfcn in leastsq is too large. + x = [574.14200000000005, 574.154, 574.16499999999996, + 574.17700000000002, 574.18799999999999, 574.19899999999996, + 574.21100000000001, 574.22199999999998, 574.23400000000004, + 574.245] + y = [859.0, 997.0, 1699.0, 2604.0, 2013.0, 1964.0, 2435.0, + 1550.0, 949.0, 841.0] + guess = [574.1861428571428, 574.2155714285715, 1302.0, 1302.0, + 0.0035019999999983615, 859.0] + good = [5.74177150e+02, 5.74209188e+02, 1.74187044e+03, 1.58646166e+03, + 1.0068462e-02, 8.57450661e+02] + + def f_double_gauss(x, x0, x1, A0, A1, sigma, c): + return (A0*np.exp(-(x-x0)**2/(2.*sigma**2)) + + A1*np.exp(-(x-x1)**2/(2.*sigma**2)) + c) + popt, pcov = curve_fit(f_double_gauss, x, y, guess, maxfev=10000) + assert_allclose(popt, good, rtol=1e-5) + + def test_pcov(self): + xdata = np.array([0, 1, 2, 3, 4, 5]) + ydata = np.array([1, 1, 5, 7, 8, 12]) + sigma = np.array([1, 2, 1, 2, 1, 2]) + + def f(x, a, b): + return a*x + b + + for method in ['lm', 'trf', 'dogbox']: + popt, pcov = curve_fit(f, xdata, ydata, p0=[2, 0], sigma=sigma, + method=method) + perr_scaled = np.sqrt(np.diag(pcov)) + assert_allclose(perr_scaled, [0.20659803, 0.57204404], rtol=1e-3) + + popt, pcov = curve_fit(f, xdata, ydata, p0=[2, 0], sigma=3*sigma, + method=method) + perr_scaled = np.sqrt(np.diag(pcov)) + assert_allclose(perr_scaled, [0.20659803, 0.57204404], rtol=1e-3) + + popt, pcov = curve_fit(f, xdata, ydata, p0=[2, 0], sigma=sigma, + absolute_sigma=True, method=method) + perr = np.sqrt(np.diag(pcov)) + assert_allclose(perr, [0.30714756, 0.85045308], rtol=1e-3) + + popt, pcov = curve_fit(f, xdata, ydata, p0=[2, 0], sigma=3*sigma, + absolute_sigma=True, method=method) + perr = np.sqrt(np.diag(pcov)) + assert_allclose(perr, [3*0.30714756, 3*0.85045308], rtol=1e-3) + + # infinite variances + + def f_flat(x, a, b): + return a*x + + pcov_expected = np.array([np.inf]*4).reshape(2, 2) + + with suppress_warnings() as sup: + sup.filter(OptimizeWarning, + "Covariance of the parameters could not be estimated") + popt, pcov = curve_fit(f_flat, xdata, ydata, p0=[2, 0], sigma=sigma) + popt1, pcov1 = curve_fit(f, xdata[:2], ydata[:2], p0=[2, 0]) + + assert_(pcov.shape == (2, 2)) + assert_array_equal(pcov, pcov_expected) + + assert_(pcov1.shape == (2, 2)) + assert_array_equal(pcov1, pcov_expected) + + def test_array_like(self): + # Test sequence input. Regression test for gh-3037. + def f_linear(x, a, b): + return a*x + b + + x = [1, 2, 3, 4] + y = [3, 5, 7, 9] + assert_allclose(curve_fit(f_linear, x, y)[0], [2, 1], atol=1e-10) + + @pytest.mark.thread_unsafe + def test_indeterminate_covariance(self): + # Test that a warning is returned when pcov is indeterminate + xdata = np.array([1, 2, 3, 4, 5, 6]) + ydata = np.array([1, 2, 3, 4, 5.5, 6]) + assert_warns(OptimizeWarning, curve_fit, + lambda x, a, b: a*x, xdata, ydata) + + def test_NaN_handling(self): + # Test for correct handling of NaNs in input data: gh-3422 + + # create input with NaNs + xdata = np.array([1, np.nan, 3]) + ydata = np.array([1, 2, 3]) + + assert_raises(ValueError, curve_fit, + lambda x, a, b: a*x + b, xdata, ydata) + assert_raises(ValueError, curve_fit, + lambda x, a, b: a*x + b, ydata, xdata) + + assert_raises(ValueError, curve_fit, lambda x, a, b: a*x + b, + xdata, ydata, **{"check_finite": True}) + + @staticmethod + def _check_nan_policy(f, xdata_with_nan, xdata_without_nan, + ydata_with_nan, ydata_without_nan, method): + kwargs = {'f': f, 'xdata': xdata_with_nan, 'ydata': ydata_with_nan, + 'method': method, 'check_finite': False} + # propagate test + error_msg = ("`nan_policy='propagate'` is not supported " + "by this function.") + with assert_raises(ValueError, match=error_msg): + curve_fit(**kwargs, nan_policy="propagate", maxfev=2000) + + # raise test + with assert_raises(ValueError, match="The input contains nan"): + curve_fit(**kwargs, nan_policy="raise") + + # omit test + result_with_nan, _ = curve_fit(**kwargs, nan_policy="omit") + kwargs['xdata'] = xdata_without_nan + kwargs['ydata'] = ydata_without_nan + result_without_nan, _ = curve_fit(**kwargs) + assert_allclose(result_with_nan, result_without_nan) + + # not valid policy test + # check for argument names in any order + error_msg = (r"nan_policy must be one of \{(?:'raise'|'omit'|None)" + r"(?:, ?(?:'raise'|'omit'|None))*\}") + with assert_raises(ValueError, match=error_msg): + curve_fit(**kwargs, nan_policy="hi") + + @pytest.mark.parametrize('method', ["lm", "trf", "dogbox"]) + def test_nan_policy_1d(self, method): + def f(x, a, b): + return a*x + b + + xdata_with_nan = np.array([2, 3, np.nan, 4, 4, np.nan]) + ydata_with_nan = np.array([1, 2, 5, 3, np.nan, 7]) + xdata_without_nan = np.array([2, 3, 4]) + ydata_without_nan = np.array([1, 2, 3]) + + self._check_nan_policy(f, xdata_with_nan, xdata_without_nan, + ydata_with_nan, ydata_without_nan, method) + + @pytest.mark.parametrize('method', ["lm", "trf", "dogbox"]) + def test_nan_policy_2d(self, method): + def f(x, a, b): + x1 = x[0, :] + x2 = x[1, :] + return a*x1 + b + x2 + + xdata_with_nan = np.array([[2, 3, np.nan, 4, 4, np.nan, 5], + [2, 3, np.nan, np.nan, 4, np.nan, 7]]) + ydata_with_nan = np.array([1, 2, 5, 3, np.nan, 7, 10]) + xdata_without_nan = np.array([[2, 3, 5], [2, 3, 7]]) + ydata_without_nan = np.array([1, 2, 10]) + + self._check_nan_policy(f, xdata_with_nan, xdata_without_nan, + ydata_with_nan, ydata_without_nan, method) + + @pytest.mark.parametrize('n', [2, 3]) + @pytest.mark.parametrize('method', ["lm", "trf", "dogbox"]) + def test_nan_policy_2_3d(self, n, method): + def f(x, a, b): + x1 = x[..., 0, :].squeeze() + x2 = x[..., 1, :].squeeze() + return a*x1 + b + x2 + + xdata_with_nan = np.array([[[2, 3, np.nan, 4, 4, np.nan, 5], + [2, 3, np.nan, np.nan, 4, np.nan, 7]]]) + xdata_with_nan = xdata_with_nan.squeeze() if n == 2 else xdata_with_nan + ydata_with_nan = np.array([1, 2, 5, 3, np.nan, 7, 10]) + xdata_without_nan = np.array([[[2, 3, 5], [2, 3, 7]]]) + ydata_without_nan = np.array([1, 2, 10]) + + self._check_nan_policy(f, xdata_with_nan, xdata_without_nan, + ydata_with_nan, ydata_without_nan, method) + + def test_empty_inputs(self): + # Test both with and without bounds (regression test for gh-9864) + assert_raises(ValueError, curve_fit, lambda x, a: a*x, [], []) + assert_raises(ValueError, curve_fit, lambda x, a: a*x, [], [], + bounds=(1, 2)) + assert_raises(ValueError, curve_fit, lambda x, a: a*x, [1], []) + assert_raises(ValueError, curve_fit, lambda x, a: a*x, [2], [], + bounds=(1, 2)) + + def test_function_zero_params(self): + # Fit args is zero, so "Unable to determine number of fit parameters." + assert_raises(ValueError, curve_fit, lambda x: x, [1, 2], [3, 4]) + + def test_None_x(self): # Added in GH10196 + popt, pcov = curve_fit(lambda _, a: a * np.arange(10), + None, 2 * np.arange(10)) + assert_allclose(popt, [2.]) + + def test_method_argument(self): + def f(x, a, b): + return a * np.exp(-b*x) + + xdata = np.linspace(0, 1, 11) + ydata = f(xdata, 2., 2.) + + for method in ['trf', 'dogbox', 'lm', None]: + popt, pcov = curve_fit(f, xdata, ydata, method=method) + assert_allclose(popt, [2., 2.]) + + assert_raises(ValueError, curve_fit, f, xdata, ydata, method='unknown') + + def test_full_output(self): + def f(x, a, b): + return a * np.exp(-b * x) + + xdata = np.linspace(0, 1, 11) + ydata = f(xdata, 2., 2.) + + for method in ['trf', 'dogbox', 'lm', None]: + popt, pcov, infodict, errmsg, ier = curve_fit( + f, xdata, ydata, method=method, full_output=True) + assert_allclose(popt, [2., 2.]) + assert "nfev" in infodict + assert "fvec" in infodict + if method == 'lm' or method is None: + assert "fjac" in infodict + assert "ipvt" in infodict + assert "qtf" in infodict + assert isinstance(errmsg, str) + assert ier in (1, 2, 3, 4) + + def test_bounds(self): + def f(x, a, b): + return a * np.exp(-b*x) + + xdata = np.linspace(0, 1, 11) + ydata = f(xdata, 2., 2.) + + # The minimum w/out bounds is at [2., 2.], + # and with bounds it's at [1.5, smth]. + lb = [1., 0] + ub = [1.5, 3.] + + # Test that both variants of the bounds yield the same result + bounds = (lb, ub) + bounds_class = Bounds(lb, ub) + for method in [None, 'trf', 'dogbox']: + popt, pcov = curve_fit(f, xdata, ydata, bounds=bounds, + method=method) + assert_allclose(popt[0], 1.5) + + popt_class, pcov_class = curve_fit(f, xdata, ydata, + bounds=bounds_class, + method=method) + assert_allclose(popt_class, popt) + + # With bounds, the starting estimate is feasible. + popt, pcov = curve_fit(f, xdata, ydata, method='trf', + bounds=([0., 0], [0.6, np.inf])) + assert_allclose(popt[0], 0.6) + + # method='lm' doesn't support bounds. + assert_raises(ValueError, curve_fit, f, xdata, ydata, bounds=bounds, + method='lm') + + def test_bounds_p0(self): + # This test is for issue #5719. The problem was that an initial guess + # was ignored when 'trf' or 'dogbox' methods were invoked. + def f(x, a): + return np.sin(x + a) + + xdata = np.linspace(-2*np.pi, 2*np.pi, 40) + ydata = np.sin(xdata) + bounds = (-3 * np.pi, 3 * np.pi) + for method in ['trf', 'dogbox']: + popt_1, _ = curve_fit(f, xdata, ydata, p0=2.1*np.pi) + popt_2, _ = curve_fit(f, xdata, ydata, p0=2.1*np.pi, + bounds=bounds, method=method) + + # If the initial guess is ignored, then popt_2 would be close 0. + assert_allclose(popt_1, popt_2) + + def test_jac(self): + # Test that Jacobian callable is handled correctly and + # weighted if sigma is provided. + def f(x, a, b): + return a * np.exp(-b*x) + + def jac(x, a, b): + e = np.exp(-b*x) + return np.vstack((e, -a * x * e)).T + + xdata = np.linspace(0, 1, 11) + ydata = f(xdata, 2., 2.) + + # Test numerical options for least_squares backend. + for method in ['trf', 'dogbox']: + for scheme in ['2-point', '3-point', 'cs']: + popt, pcov = curve_fit(f, xdata, ydata, jac=scheme, + method=method) + assert_allclose(popt, [2, 2]) + + # Test the analytic option. + for method in ['lm', 'trf', 'dogbox']: + popt, pcov = curve_fit(f, xdata, ydata, method=method, jac=jac) + assert_allclose(popt, [2, 2]) + + # Now add an outlier and provide sigma. + ydata[5] = 100 + sigma = np.ones(xdata.shape[0]) + sigma[5] = 200 + for method in ['lm', 'trf', 'dogbox']: + popt, pcov = curve_fit(f, xdata, ydata, sigma=sigma, method=method, + jac=jac) + # Still the optimization process is influenced somehow, + # have to set rtol=1e-3. + assert_allclose(popt, [2, 2], rtol=1e-3) + + def test_maxfev_and_bounds(self): + # gh-6340: with no bounds, curve_fit accepts parameter maxfev (via leastsq) + # but with bounds, the parameter is `max_nfev` (via least_squares) + x = np.arange(0, 10) + y = 2*x + popt1, _ = curve_fit(lambda x,p: p*x, x, y, bounds=(0, 3), maxfev=100) + popt2, _ = curve_fit(lambda x,p: p*x, x, y, bounds=(0, 3), max_nfev=100) + + assert_allclose(popt1, 2, atol=1e-14) + assert_allclose(popt2, 2, atol=1e-14) + + @pytest.mark.parametrize("sigma_dim", [0, 1, 2]) + def test_curvefit_omitnan(self, sigma_dim): + def exponential(x, a, b): + return b * np.exp(a * x) + + rng = np.random.default_rng(578285731148908) + N = 100 + x = np.linspace(1, 10, N) + y = exponential(x, 0.2, 0.5) + + if (sigma_dim == 0): + sigma = 0.05 + y += rng.normal(0, sigma, N) + + elif (sigma_dim == 1): + sigma = x * 0.05 + y += rng.normal(0, sigma, N) + + elif (sigma_dim == 2): + # The covariance matrix must be symmetric positive-semidefinite + a = rng.normal(0, 2, (N, N)) + sigma = a @ a.T + y += rng.multivariate_normal(np.zeros_like(x), sigma) + else: + assert False, "The sigma must be a scalar, 1D array or 2D array." + + p0 = [0.1, 1.0] + + # Choose indices to place NaNs. + i_x = rng.integers(N, size=5) + i_y = rng.integers(N, size=5) + + # Add NaNs and compute result using `curve_fit` + x[i_x] = np.nan + y[i_y] = np.nan + res_opt, res_cov = curve_fit(exponential, x, y, p0=p0, sigma=sigma, + nan_policy="omit") + + # Manually remove elements that should be eliminated, and + # calculate reference using `curve_fit` + i_delete = np.unique(np.concatenate((i_x, i_y))) + x = np.delete(x, i_delete, axis=0) + y = np.delete(y, i_delete, axis=0) + + sigma = np.asarray(sigma) + if sigma.ndim == 1: + sigma = np.delete(sigma, i_delete) + elif sigma.ndim == 2: + sigma = np.delete(sigma, i_delete, axis=0) + sigma = np.delete(sigma, i_delete, axis=1) + ref_opt, ref_cov = curve_fit(exponential, x, y, p0=p0, sigma=sigma) + + assert_allclose(res_opt, ref_opt, atol=1e-14) + assert_allclose(res_cov, ref_cov, atol=1e-14) + + def test_curvefit_simplecovariance(self): + + def func(x, a, b): + return a * np.exp(-b*x) + + def jac(x, a, b): + e = np.exp(-b*x) + return np.vstack((e, -a * x * e)).T + + np.random.seed(0) + xdata = np.linspace(0, 4, 50) + y = func(xdata, 2.5, 1.3) + ydata = y + 0.2 * np.random.normal(size=len(xdata)) + + sigma = np.zeros(len(xdata)) + 0.2 + covar = np.diag(sigma**2) + + for jac1, jac2 in [(jac, jac), (None, None)]: + for absolute_sigma in [False, True]: + popt1, pcov1 = curve_fit(func, xdata, ydata, sigma=sigma, + jac=jac1, absolute_sigma=absolute_sigma) + popt2, pcov2 = curve_fit(func, xdata, ydata, sigma=covar, + jac=jac2, absolute_sigma=absolute_sigma) + + assert_allclose(popt1, popt2, atol=1e-14) + assert_allclose(pcov1, pcov2, atol=1e-14) + + def test_curvefit_covariance(self): + + def funcp(x, a, b): + rotn = np.array([[1./np.sqrt(2), -1./np.sqrt(2), 0], + [1./np.sqrt(2), 1./np.sqrt(2), 0], + [0, 0, 1.0]]) + return rotn.dot(a * np.exp(-b*x)) + + def jacp(x, a, b): + rotn = np.array([[1./np.sqrt(2), -1./np.sqrt(2), 0], + [1./np.sqrt(2), 1./np.sqrt(2), 0], + [0, 0, 1.0]]) + e = np.exp(-b*x) + return rotn.dot(np.vstack((e, -a * x * e)).T) + + def func(x, a, b): + return a * np.exp(-b*x) + + def jac(x, a, b): + e = np.exp(-b*x) + return np.vstack((e, -a * x * e)).T + + rng = np.random.RandomState(0) + xdata = np.arange(1, 4) + y = func(xdata, 2.5, 1.0) + ydata = y + 0.2 * rng.normal(size=len(xdata)) + sigma = np.zeros(len(xdata)) + 0.2 + covar = np.diag(sigma**2) + # Get a rotation matrix, and obtain ydatap = R ydata + # Chisq = ydata^T C^{-1} ydata + # = ydata^T R^T R C^{-1} R^T R ydata + # = ydatap^T Cp^{-1} ydatap + # Cp^{-1} = R C^{-1} R^T + # Cp = R C R^T, since R^-1 = R^T + rotn = np.array([[1./np.sqrt(2), -1./np.sqrt(2), 0], + [1./np.sqrt(2), 1./np.sqrt(2), 0], + [0, 0, 1.0]]) + ydatap = rotn.dot(ydata) + covarp = rotn.dot(covar).dot(rotn.T) + + for jac1, jac2 in [(jac, jacp), (None, None)]: + for absolute_sigma in [False, True]: + popt1, pcov1 = curve_fit(func, xdata, ydata, sigma=sigma, + jac=jac1, absolute_sigma=absolute_sigma) + popt2, pcov2 = curve_fit(funcp, xdata, ydatap, sigma=covarp, + jac=jac2, absolute_sigma=absolute_sigma) + + assert_allclose(popt1, popt2, rtol=1.2e-7, atol=1e-14) + assert_allclose(pcov1, pcov2, rtol=1.2e-7, atol=1e-14) + + @pytest.mark.parametrize("absolute_sigma", [False, True]) + def test_curvefit_scalar_sigma(self, absolute_sigma): + def func(x, a, b): + return a * x + b + + x, y = self.x, self.y + _, pcov1 = curve_fit(func, x, y, sigma=2, absolute_sigma=absolute_sigma) + # Explicitly building the sigma 1D array + _, pcov2 = curve_fit( + func, x, y, sigma=np.full_like(y, 2), absolute_sigma=absolute_sigma + ) + assert np.all(pcov1 == pcov2) + + def test_dtypes(self): + # regression test for gh-9581: curve_fit fails if x and y dtypes differ + x = np.arange(-3, 5) + y = 1.5*x + 3.0 + 0.5*np.sin(x) + + def func(x, a, b): + return a*x + b + + for method in ['lm', 'trf', 'dogbox']: + for dtx in [np.float32, np.float64]: + for dty in [np.float32, np.float64]: + x = x.astype(dtx) + y = y.astype(dty) + + with warnings.catch_warnings(): + warnings.simplefilter("error", OptimizeWarning) + p, cov = curve_fit(func, x, y, method=method) + + assert np.isfinite(cov).all() + assert not np.allclose(p, 1) # curve_fit's initial value + + def test_dtypes2(self): + # regression test for gh-7117: curve_fit fails if + # both inputs are float32 + def hyperbola(x, s_1, s_2, o_x, o_y, c): + b_2 = (s_1 + s_2) / 2 + b_1 = (s_2 - s_1) / 2 + return o_y + b_1*(x-o_x) + b_2*np.sqrt((x-o_x)**2 + c**2/4) + + min_fit = np.array([-3.0, 0.0, -2.0, -10.0, 0.0]) + max_fit = np.array([0.0, 3.0, 3.0, 0.0, 10.0]) + guess = np.array([-2.5/3.0, 4/3.0, 1.0, -4.0, 0.5]) + + params = [-2, .4, -1, -5, 9.5] + xdata = np.array([-32, -16, -8, 4, 4, 8, 16, 32]) + ydata = hyperbola(xdata, *params) + + # run optimization twice, with xdata being float32 and float64 + popt_64, _ = curve_fit(f=hyperbola, xdata=xdata, ydata=ydata, p0=guess, + bounds=(min_fit, max_fit)) + + xdata = xdata.astype(np.float32) + ydata = hyperbola(xdata, *params) + + popt_32, _ = curve_fit(f=hyperbola, xdata=xdata, ydata=ydata, p0=guess, + bounds=(min_fit, max_fit)) + + assert_allclose(popt_32, popt_64, atol=2e-5) + + def test_broadcast_y(self): + xdata = np.arange(10) + target = 4.7 * xdata ** 2 + 3.5 * xdata + np.random.rand(len(xdata)) + def fit_func(x, a, b): + return a * x ** 2 + b * x - target + for method in ['lm', 'trf', 'dogbox']: + popt0, pcov0 = curve_fit(fit_func, + xdata=xdata, + ydata=np.zeros_like(xdata), + method=method) + popt1, pcov1 = curve_fit(fit_func, + xdata=xdata, + ydata=0, + method=method) + assert_allclose(pcov0, pcov1) + + def test_args_in_kwargs(self): + # Ensure that `args` cannot be passed as keyword argument to `curve_fit` + + def func(x, a, b): + return a * x + b + + with assert_raises(ValueError): + curve_fit(func, + xdata=[1, 2, 3, 4], + ydata=[5, 9, 13, 17], + p0=[1], + args=(1,)) + + def test_data_point_number_validation(self): + def func(x, a, b, c, d, e): + return a * np.exp(-b * x) + c + d + e + + with assert_raises(TypeError, match="The number of func parameters="): + curve_fit(func, + xdata=[1, 2, 3, 4], + ydata=[5, 9, 13, 17]) + + @pytest.mark.filterwarnings('ignore::RuntimeWarning') + def test_gh4555(self): + # gh-4555 reported that covariance matrices returned by `leastsq` + # can have negative diagonal elements and eigenvalues. (In fact, + # they can also be asymmetric.) This shows up in the output of + # `scipy.optimize.curve_fit`. Check that it has been resolved.giit + def f(x, a, b, c, d, e): + return a*np.log(x + 1 + b) + c*np.log(x + 1 + d) + e + + rng = np.random.default_rng(408113519974467917) + n = 100 + x = np.arange(n) + y = np.linspace(2, 7, n) + rng.random(n) + p, cov = optimize.curve_fit(f, x, y, maxfev=100000) + assert np.all(np.diag(cov) > 0) + eigs = linalg.eigh(cov)[0] # separate line for debugging + # some platforms see a small negative eigevenvalue + assert np.all(eigs > -1e-2) + assert_allclose(cov, cov.T) + + def test_gh4555b(self): + # check that PR gh-17247 did not significantly change covariance matrix + # for simple cases + rng = np.random.default_rng(408113519974467917) + + def func(x, a, b, c): + return a * np.exp(-b * x) + c + + xdata = np.linspace(0, 4, 50) + y = func(xdata, 2.5, 1.3, 0.5) + y_noise = 0.2 * rng.normal(size=xdata.size) + ydata = y + y_noise + _, res = curve_fit(func, xdata, ydata) + # reference from commit 1d80a2f254380d2b45733258ca42eb6b55c8755b + ref = [[+0.0158972536486215, 0.0069207183284242, -0.0007474400714749], + [+0.0069207183284242, 0.0205057958128679, +0.0053997711275403], + [-0.0007474400714749, 0.0053997711275403, +0.0027833930320877]] + # Linux_Python_38_32bit_full fails with default tolerance + assert_allclose(res, ref, 2e-7) + + def test_gh13670(self): + # gh-13670 reported that `curve_fit` executes callables + # with the same values of the parameters at the beginning of + # optimization. Check that this has been resolved. + + rng = np.random.default_rng(8250058582555444926) + x = np.linspace(0, 3, 101) + y = 2 * x + 1 + rng.normal(size=101) * 0.5 + + def line(x, *p): + assert not np.all(line.last_p == p) + line.last_p = p + return x * p[0] + p[1] + + def jac(x, *p): + assert not np.all(jac.last_p == p) + jac.last_p = p + return np.array([x, np.ones_like(x)]).T + + line.last_p = None + jac.last_p = None + p0 = np.array([1.0, 5.0]) + curve_fit(line, x, y, p0, method='lm', jac=jac) + + @pytest.mark.parametrize('method', ['trf', 'dogbox']) + def test_gh20155_error_mentions_x0(self, method): + # `curve_fit` produced an error message that referred to an undocumented + # variable `x0`, which was really `p0`. Check that this is resolved. + def func(x,a): + return x**a + message = "Initial guess is outside of provided bounds" + with pytest.raises(ValueError, match=message): + curve_fit(func, self.x, self.y, p0=[1], bounds=(1000, 1001), + method=method) + + +class TestFixedPoint: + + def test_scalar_trivial(self): + # f(x) = 2x; fixed point should be x=0 + def func(x): + return 2.0*x + x0 = 1.0 + x = fixed_point(func, x0) + assert_almost_equal(x, 0.0) + + def test_scalar_basic1(self): + # f(x) = x**2; x0=1.05; fixed point should be x=1 + def func(x): + return x**2 + x0 = 1.05 + x = fixed_point(func, x0) + assert_almost_equal(x, 1.0) + + def test_scalar_basic2(self): + # f(x) = x**0.5; x0=1.05; fixed point should be x=1 + def func(x): + return x**0.5 + x0 = 1.05 + x = fixed_point(func, x0) + assert_almost_equal(x, 1.0) + + def test_array_trivial(self): + def func(x): + return 2.0*x + x0 = [0.3, 0.15] + with np.errstate(all='ignore'): + x = fixed_point(func, x0) + assert_almost_equal(x, [0.0, 0.0]) + + def test_array_basic1(self): + # f(x) = c * x**2; fixed point should be x=1/c + def func(x, c): + return c * x**2 + c = array([0.75, 1.0, 1.25]) + x0 = [1.1, 1.15, 0.9] + with np.errstate(all='ignore'): + x = fixed_point(func, x0, args=(c,)) + assert_almost_equal(x, 1.0/c) + + def test_array_basic2(self): + # f(x) = c * x**0.5; fixed point should be x=c**2 + def func(x, c): + return c * x**0.5 + c = array([0.75, 1.0, 1.25]) + x0 = [0.8, 1.1, 1.1] + x = fixed_point(func, x0, args=(c,)) + assert_almost_equal(x, c**2) + + def test_lambertw(self): + # python-list/2010-December/594592.html + xxroot = fixed_point(lambda xx: np.exp(-2.0*xx)/2.0, 1.0, + args=(), xtol=1e-12, maxiter=500) + assert_allclose(xxroot, np.exp(-2.0*xxroot)/2.0) + assert_allclose(xxroot, lambertw(1)/2) + + def test_no_acceleration(self): + # GitHub issue 5460 + ks = 2 + kl = 6 + m = 1.3 + n0 = 1.001 + i0 = ((m-1)/m)*(kl/ks/m)**(1/(m-1)) + + def func(n): + return np.log(kl/ks/n) / np.log(i0*n/(n - 1)) + 1 + + n = fixed_point(func, n0, method='iteration') + assert_allclose(n, m) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_nnls.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_nnls.py new file mode 100644 index 0000000000000000000000000000000000000000..67443dd6147bd7ee8898bac2a1a7993f6e56e799 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_nnls.py @@ -0,0 +1,429 @@ +import numpy as np +from numpy.testing import assert_allclose +from pytest import raises as assert_raises +from scipy.optimize import nnls + + +class TestNNLS: + def setup_method(self): + self.rng = np.random.default_rng(1685225766635251) + + def test_nnls(self): + a = np.arange(25.0).reshape(-1, 5) + x = np.arange(5.0) + y = a @ x + x, res = nnls(a, y) + assert res < 1e-7 + assert np.linalg.norm((a @ x) - y) < 1e-7 + + def test_nnls_tall(self): + a = self.rng.uniform(low=-10, high=10, size=[50, 10]) + x = np.abs(self.rng.uniform(low=-2, high=2, size=[10])) + x[::2] = 0 + b = a @ x + xact, rnorm = nnls(a, b, atol=500*np.linalg.norm(a, 1)*np.spacing(1.)) + assert_allclose(xact, x, rtol=0., atol=1e-10) + assert rnorm < 1e-12 + + def test_nnls_wide(self): + # If too wide then problem becomes too ill-conditioned ans starts + # emitting warnings, hence small m, n difference. + a = self.rng.uniform(low=-10, high=10, size=[100, 120]) + x = np.abs(self.rng.uniform(low=-2, high=2, size=[120])) + x[::2] = 0 + b = a @ x + xact, rnorm = nnls(a, b, atol=500*np.linalg.norm(a, 1)*np.spacing(1.)) + assert_allclose(xact, x, rtol=0., atol=1e-10) + assert rnorm < 1e-12 + + def test_maxiter(self): + # test that maxiter argument does stop iterations + a = self.rng.uniform(size=(5, 10)) + b = self.rng.uniform(size=5) + with assert_raises(RuntimeError): + nnls(a, b, maxiter=1) + + def test_nnls_inner_loop_case1(self): + # See gh-20168 + n = np.array( + [3, 2, 0, 1, 1, 1, 3, 8, 14, 16, 29, 23, 41, 47, 53, 57, 67, 76, + 103, 89, 97, 94, 85, 95, 78, 78, 78, 77, 73, 50, 50, 56, 68, 98, + 95, 112, 134, 145, 158, 172, 213, 234, 222, 215, 216, 216, 206, + 183, 135, 156, 110, 92, 63, 60, 52, 29, 20, 16, 12, 5, 5, 5, 1, 2, + 3, 0, 2]) + k = np.array( + [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., + 0., 0., 0., 0.7205812007860187, 0., 1.4411624015720375, + 0.7205812007860187, 2.882324803144075, 5.76464960628815, + 5.76464960628815, 12.249880413362318, 15.132205216506394, + 20.176273622008523, 27.382085629868712, 48.27894045266326, + 47.558359251877235, 68.45521407467177, 97.99904330689854, + 108.0871801179028, 135.46926574777152, 140.51333415327366, + 184.4687874012208, 171.49832578707245, 205.36564222401535, + 244.27702706646033, 214.01261663344755, 228.42424064916793, + 232.02714665309804, 205.36564222401535, 172.9394881886445, + 191.67459940908097, 162.1307701768542, 153.48379576742198, + 110.96950492104689, 103.04311171240067, 86.46974409432225, + 60.528820866025576, 43.234872047161126, 23.779179625938617, + 24.499760826724636, 17.29394881886445, 11.5292992125763, + 5.76464960628815, 5.044068405502131, 3.6029060039300935, 0., + 2.882324803144075, 0., 0., 0.]) + d = np.array( + [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., + 0., 0., 0., 0.003889242101538, 0., 0.007606268390096, 0., + 0.025457371599973, 0.036952882091577, 0., 0.08518359183449, + 0.048201126400243, 0.196234990022205, 0.144116240157247, + 0.171145134062442, 0., 0., 0.269555036538714, 0., 0., 0., + 0.010893241091872, 0., 0., 0., 0., 0., 0., 0., 0., + 0.048167058272886, 0.011238724891049, 0., 0., 0.055162603456078, + 0., 0., 0., 0., 0.027753339088588, 0., 0., 0., 0., 0., 0., 0., 0., + 0., 0.]) + # The following code sets up a system of equations such that + # $k_i-p_i*n_i$ is minimized for $p_i$ with weights $n_i$ and + # monotonicity constraints on $p_i$. This translates to a system of + # equations of the form $k_i - (d_1 + ... + d_i) * n_i$ and + # non-negativity constraints on the $d_i$. If $n_i$ is zero the + # system is modified such that $d_i - d_{i+1}$ is then minimized. + N = len(n) + A = np.diag(n) @ np.tril(np.ones((N, N))) + w = n ** 0.5 + + nz = (n == 0).nonzero()[0] + A[nz, nz] = 1 + A[nz, np.minimum(nz + 1, N - 1)] = -1 + w[nz] = 1 + k[nz] = 0 + W = np.diag(w) + + # Small perturbations can already make the infinite loop go away (just + # uncomment the next line) + # k = k + 1e-10 * np.random.normal(size=N) + dact, _ = nnls(W @ A, W @ k) + assert_allclose(dact, d, rtol=0., atol=1e-10) + + def test_nnls_inner_loop_case2(self): + # See gh-20168 + n = np.array( + [1, 0, 1, 2, 2, 2, 3, 3, 5, 4, 14, 14, 19, 26, 36, 42, 36, 64, 64, + 64, 81, 85, 85, 95, 95, 95, 75, 76, 69, 81, 62, 59, 68, 64, 71, 67, + 74, 78, 118, 135, 153, 159, 210, 195, 218, 243, 236, 215, 196, 175, + 185, 149, 144, 103, 104, 75, 56, 40, 32, 26, 17, 9, 12, 8, 2, 1, 1, + 1]) + k = np.array( + [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., + 0., 0., 0., 0., 0., 0.7064355064917867, 0., 0., 2.11930651947536, + 0.7064355064917867, 0., 3.5321775324589333, 7.064355064917867, + 11.302968103868587, 16.95445215580288, 20.486629688261814, + 20.486629688261814, 37.44108184406469, 55.808405012851146, + 78.41434122058831, 103.13958394780086, 105.965325973768, + 125.74552015553803, 149.057891869767, 176.60887662294667, + 197.09550631120848, 211.930651947536, 204.86629688261814, + 233.8301526487814, 221.1143135319292, 195.6826352982249, + 197.80194181770025, 191.4440222592742, 187.91184472681525, + 144.11284332432447, 131.39700420747232, 116.5618585711448, + 93.24948685691584, 89.01087381796512, 53.68909849337579, + 45.211872415474346, 31.083162285638615, 24.72524272721253, + 16.95445215580288, 9.890097090885014, 9.890097090885014, + 2.8257420259671466, 2.8257420259671466, 1.4128710129835733, + 0.7064355064917867, 1.4128710129835733]) + d = np.array( + [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., + 0., 0., 0., 0., 0., 0.0021916146355674473, 0., 0., + 0.011252740799789484, 0., 0., 0.037746623295934395, + 0.03602328132946222, 0.09509167709829734, 0.10505765870204821, + 0.01391037014274718, 0.0188296228752321, 0.20723559202324254, + 0.3056220879462608, 0.13304643490426477, 0., 0., 0., 0., 0., 0., + 0., 0., 0., 0., 0., 0.043185876949706214, 0.0037266261379722554, + 0., 0., 0., 0., 0., 0.094797899357143, 0., 0., 0., 0., 0., 0., 0., + 0., 0.23450935613672663, 0., 0., 0.07064355064917871]) + # The following code sets up a system of equations such that + # $k_i-p_i*n_i$ is minimized for $p_i$ with weights $n_i$ and + # monotonicity constraints on $p_i$. This translates to a system of + # equations of the form $k_i - (d_1 + ... + d_i) * n_i$ and + # non-negativity constraints on the $d_i$. If $n_i$ is zero the + # system is modified such that $d_i - d_{i+1}$ is then minimized. + N = len(n) + A = np.diag(n) @ np.tril(np.ones((N, N))) + w = n ** 0.5 + + nz = (n == 0).nonzero()[0] + A[nz, nz] = 1 + A[nz, np.minimum(nz + 1, N - 1)] = -1 + w[nz] = 1 + k[nz] = 0 + W = np.diag(w) + + dact, _ = nnls(W @ A, W @ k, atol=1e-7) + + p = np.cumsum(dact) + assert np.all(dact >= 0) + assert np.linalg.norm(k - n * p, ord=np.inf) < 28 + assert_allclose(dact, d, rtol=0., atol=1e-10) + + def test_nnls_gh20302(self): + # See gh-20302 + A = np.array( + [0.33408569134321575, 0.11136189711440525, 0.049140798007949286, + 0.03712063237146841, 0.055680948557202625, 0.16642814595936478, + 0.11095209730624318, 0.09791993030943345, 0.14793612974165757, + 0.44380838922497273, 0.11099502671044059, 0.11099502671044059, + 0.14693672599330593, 0.3329850801313218, 1.498432860590948, + 0.0832374225132955, 0.11098323001772734, 0.19589481249472837, + 0.5919105600945457, 3.5514633605672747, 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0.13739553134643406, 2128.794599579694, 56462810.11822766, + 2973783283306.8145, 1.0293367506254706e-05, 0.13533033372723272, + 2355.372854690074, 70176508.28667311, 4151852759764.441, + 9.678312586863569e-06, 0.14293577249119244, 2794.531827932675, + 93528671.31952812, 6215821967224.52, -1.174086323572049e-05, + 0.1429501325944908, 3139.4804810720925, 118031680.16618933, + -6466892421886.174, -2.1188265307407812e-05, 0.1477108290912869, + 3644.1133424610953, 153900132.62392554, -4828013117542.036, + -8.614483025123122e-05, 0.16037100755883044, 4444.386620899393, + 210846007.89660168, -1766340937974.433, 4.981445776141726e-05, + 0.16053420251962536, 4997.558254401547, 266327328.4755411, + 3862250287024.725, 1.8500019169456637e-05, 0.15448417164977674, + 5402.289867444643, 323399508.1475582, 12152445411933.408, + -5.647882376069748e-05, 0.1406372975946189, 5524.633133597753, + 371512945.9909363, -4162951345292.1514, 2.8048523486337994e-05, + 0.13183417571186926, 5817.462495763679, 439447252.3728975, + 9294740538175.03]).reshape(89, 5) + b = np.ones(89, dtype=np.float64) + sol, rnorm = nnls(A, b) + assert_allclose(sol, np.array([0.61124315, 8.22262829, 0., 0., 0.])) + assert_allclose(rnorm, 1.0556460808977297) + + def test_nnls_gh21021_ex1(self): + # Review examples used in gh-21021 + A = [[0.004734199143798789, -0.09661916455815653, -0.04308779048103441, + 0.4039475561867938, -0.27742598780954364, -0.20816924034369574, + -0.17264070902176, 0.05251808558963846], + [-0.030263548855047975, -0.30356483926431466, 0.18080406600591398, + -0.06892233941254086, -0.41837298885432317, 0.30245352819647003, + -0.19008975278116397, -0.00990809825429995], + [-0.2561747595787612, -0.04376282125249583, 0.4422181991706678, + -0.13720906318924858, -0.0069523811763796475, -0.059238287107464795, + 0.028663214369642594, 0.5415531284893763], + [0.2949336072968401, 0.33997647534935094, 0.38441519339815755, + -0.306001783010386, 0.18120773805949028, -0.36669767490747895, + -0.021539960590992304, -0.2784251712424615], + [0.5009075736232653, -0.20161970347571165, 0.08404512586550646, + 0.2520496489348788, 0.14812015101612894, -0.25823455803981266, + -0.1596872058396596, 0.5960141613922691] + ] + b = [18.036779281222124, -18.126530733870887, 13.535652034584029, + -2.6654275476795966, 9.166315328199575] + + # Obtained from matlab's lstnonneg + des_sol = np.array([0., 118.017802006619, 45.1996532316584, 102.62156313537, + 0., 55.8590204314398, 0., 29.7328833253434]) + sol, res = nnls(A, b) + assert_allclose(sol, des_sol) + assert np.abs(np.linalg.norm(A@sol - b) - res) < 5e-14 + + def test_nnls_gh21021_ex2(self): + A = np.array([ + [0.2508259992635229, -0.24031300195203256], + [0.510647748500133, 0.2872936081767836], + [0.8196387904102849, -0.03520620107046682], + [0.030739759120097084, -0.07768656359879388]]) + b = np.array([24.456141951303913, + 28.047143273432333, + 41.10526799545987, + -1.2078282698324068]) + + sol, res = nnls(A, b) + assert_allclose(sol, np.array([54.3047953202271, 0.0])) + assert np.abs(np.linalg.norm(A@sol - b) - res) < 5e-14 + + def test_nnls_gh21021_ex3(self): + A = np.array([ + [0.08247592017366788, 0.058398241636675674, -0.1031496693415968, + 0.03156983127072098, -0.029503680182026665], + [0.21463607509982277, -0.2164518969308173, -0.10816833396662294, + 0.12133867146012027, -0.15025010408668332], + [0.07251900316494089, -0.003044559315020767, 0.042682817961676424, + -0.018157525489298176, 0.11561953260568134], + [0.2328797918159187, -0.09112909645892767, 0.21348169727099078, + 0.00449447624089599, -0.16615256386885716], + [-0.02440856024843897, -0.20131427208575386, 0.030275781997161483, + -0.04560777213546784, 0.11007266012013553], + [-0.2928391429686263, -0.20437574856615687, -0.020892110811574407, + -0.10455040720819309, 0.05337267000160461], + [0.22041503019400316, 0.014262782992311842, 0.08274606359871121, + -0.17933172096518907, -0.11809690350702161], + [0.10440436007469953, 0.09171452270577712, 0.03942347724809893, + 0.11457669688231396, 0.07529747295631585], + [-0.052087576116032056, -0.15787717158077047, -0.08232202515883282, + -0.03194837933710708, -0.0546812506025729], + [-0.010388407673304468, 0.015174707581808923, 0.04764509565386281, + -0.1781221936030805, 0.10218894080536609], + [0.03272263140115928, -0.27576456949442574, 0.024897570959901753, + -0.1417129166632282, -0.03320796462136591], + [-0.12490006751823997, -0.03012003515442302, -0.051495264012509506, + 0.012070729698374614, 0.04811700123118234], + [0.15254854117990788, -0.051863547789218374, 0.058012914127346174, + -0.06717991061422621, -0.14514671564242257], + [0.12251250415395559, -0.17462495626695362, -0.025334728552179834, + 0.11425350676877533, 0.06183915953812639], + [0.19334259720491218, 0.2164301986218955, -0.018882278726614483, + 0.07950236716817938, -0.2220529357431092], + [-0.01822205701890852, 0.12630444976752267, -0.03118092027244001, + 0.02773743885242581, 0.06444433740044248], + [0.13344116850581977, -0.05142877469996826, 0.3385702016705455, + -0.25814970787123004, 0.2679034842977378], + [0.1309747058619377, 0.12090608957940627, -0.13957978654106512, + 0.17048819760322642, -0.241775259969348], + [0.28613102173467275, -0.47153463906732174, 0.20359970518269746, + -0.0962095202871843, -0.07703076550836387], + [0.2212788380372723, 0.02569245145758152, -0.021596152392209966, + 0.04610005150029433, -0.2024454395619734], + [-0.043225338359410316, 0.17816095186290315, -0.014709092962616079, + 0.06993970293287989, -0.09033722782555903], + [0.17747622942563512, -0.20991014784011458, 0.06265720409894943, + 0.0689704059061795, 0.024474319398401525], + [-0.1163880385601698, 0.29989570587630027, 0.033443765320984545, + 0.008470296514656, -0.0014457113271462002], + [0.024375314902718406, 0.05279830705548363, 0.02691082431023144, + 0.05265079368002343, 0.15542988147487913], + [-0.01855218360922308, -0.050265869142888164, 0.2567912677240452, + -0.2606428528561333, 0.25334396245022245]]) + + b = np.array([-7.876625373734849, -8.259856278691373, 3.2593082374900963, + 16.30170376973345, 2.311892943629045, -1.595345202555738, + 6.318582970536518, 3.0104212955340093, -6.286202915842167, + 3.6382333725029294, 1.9012066681249356, -3.932236581436514, + 4.4299317131740406, -1.9345885161292682, -1.4418721521970805, + -2.3810103256943926, 25.853603392922526, -10.658470311610483, + 15.547103681119214, -1.6491066136547277, -1.1232029689817422, + 4.7845749463206975, 2.553803732013229, 2.0549409701753705, + 19.60887153608244]) + + sol, res = nnls(A, b) + assert_allclose(sol, np.array([0.0, 0.0, 76.3611306173957, 0.0, 0.0]), + atol=5e-14) + assert np.abs(np.linalg.norm(A@sol - b) - res) < 5e-14 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_nonlin.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_nonlin.py new file mode 100644 index 0000000000000000000000000000000000000000..e5eb094c15902eca6e089ba3bbf6dfd8eb06970e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_nonlin.py @@ -0,0 +1,536 @@ +""" Unit tests for nonlinear solvers +Author: Ondrej Certik +May 2007 +""" +from numpy.testing import assert_ +import pytest +from functools import partial + +from scipy.optimize import _nonlin as nonlin, root +from scipy.sparse import csr_array +from numpy import diag, dot +from numpy.linalg import inv +import numpy as np +import scipy + +from .test_minpack import pressure_network + +SOLVERS = {'anderson': nonlin.anderson, + 'diagbroyden': nonlin.diagbroyden, + 'linearmixing': nonlin.linearmixing, + 'excitingmixing': nonlin.excitingmixing, + 'broyden1': nonlin.broyden1, + 'broyden2': nonlin.broyden2, + 'krylov': nonlin.newton_krylov} +MUST_WORK = {'anderson': nonlin.anderson, 'broyden1': nonlin.broyden1, + 'broyden2': nonlin.broyden2, 'krylov': nonlin.newton_krylov} + +# ---------------------------------------------------------------------------- +# Test problems +# ---------------------------------------------------------------------------- + + +def F(x): + x = np.asarray(x).T + d = diag([3, 2, 1.5, 1, 0.5]) + c = 0.01 + f = -d @ x - c * float(x.T @ x) * x + return f + + +F.xin = [1, 1, 1, 1, 1] +F.KNOWN_BAD = {} +F.JAC_KSP_BAD = {} +F.ROOT_JAC_KSP_BAD = {} + + +def F2(x): + return x + + +F2.xin = [1, 2, 3, 4, 5, 6] +F2.KNOWN_BAD = {'linearmixing': nonlin.linearmixing, + 'excitingmixing': nonlin.excitingmixing} +F2.JAC_KSP_BAD = {} +F2.ROOT_JAC_KSP_BAD = {} + + +def F2_lucky(x): + return x + + +F2_lucky.xin = [0, 0, 0, 0, 0, 0] +F2_lucky.KNOWN_BAD = {} +F2_lucky.JAC_KSP_BAD = {} +F2_lucky.ROOT_JAC_KSP_BAD = {} + + +def F3(x): + A = np.array([[-2, 1, 0.], [1, -2, 1], [0, 1, -2]]) + b = np.array([1, 2, 3.]) + return A @ x - b + + +F3.xin = [1, 2, 3] +F3.KNOWN_BAD = {} +F3.JAC_KSP_BAD = {} +F3.ROOT_JAC_KSP_BAD = {} + + +def F4_powell(x): + A = 1e4 + return [A*x[0]*x[1] - 1, np.exp(-x[0]) + np.exp(-x[1]) - (1 + 1/A)] + + +F4_powell.xin = [-1, -2] +F4_powell.KNOWN_BAD = {'linearmixing': nonlin.linearmixing, + 'excitingmixing': nonlin.excitingmixing, + 'diagbroyden': nonlin.diagbroyden} +# In the extreme case, it does not converge for nolinear problem solved by +# MINRES and root problem solved by GMRES/BiCGStab/CGS/MINRES/TFQMR when using +# Krylov method to approximate Jacobian +F4_powell.JAC_KSP_BAD = {'minres'} +F4_powell.ROOT_JAC_KSP_BAD = {'gmres', 'bicgstab', 'cgs', 'minres', 'tfqmr'} + + +def F5(x): + return pressure_network(x, 4, np.array([.5, .5, .5, .5])) + + +F5.xin = [2., 0, 2, 0] +F5.KNOWN_BAD = {'excitingmixing': nonlin.excitingmixing, + 'linearmixing': nonlin.linearmixing, + 'diagbroyden': nonlin.diagbroyden} +# In the extreme case, the Jacobian inversion yielded zero vector for nonlinear +# problem solved by CGS/MINRES and it does not converge for root problem solved +# by MINRES and when using Krylov method to approximate Jacobian +F5.JAC_KSP_BAD = {'cgs', 'minres'} +F5.ROOT_JAC_KSP_BAD = {'minres'} + + +def F6(x): + x1, x2 = x + J0 = np.array([[-4.256, 14.7], + [0.8394989, 0.59964207]]) + v = np.array([(x1 + 3) * (x2**5 - 7) + 3*6, + np.sin(x2 * np.exp(x1) - 1)]) + return -np.linalg.solve(J0, v) + + +F6.xin = [-0.5, 1.4] +F6.KNOWN_BAD = {'excitingmixing': nonlin.excitingmixing, + 'linearmixing': nonlin.linearmixing, + 'diagbroyden': nonlin.diagbroyden} +F6.JAC_KSP_BAD = {} +F6.ROOT_JAC_KSP_BAD = {} + + +# ---------------------------------------------------------------------------- +# Tests +# ---------------------------------------------------------------------------- + + +class TestNonlin: + """ + Check the Broyden methods for a few test problems. + + broyden1, broyden2, and newton_krylov must succeed for + all functions. Some of the others don't -- tests in KNOWN_BAD are skipped. + + """ + + def _check_nonlin_func(self, f, func, f_tol=1e-2): + # Test all methods mentioned in the class `KrylovJacobian` + if func == SOLVERS['krylov']: + for method in ['gmres', 'bicgstab', 'cgs', 'minres', 'tfqmr']: + if method in f.JAC_KSP_BAD: + continue + + x = func(f, f.xin, method=method, line_search=None, + f_tol=f_tol, maxiter=200, verbose=0) + assert_(np.absolute(f(x)).max() < f_tol) + + x = func(f, f.xin, f_tol=f_tol, maxiter=200, verbose=0) + assert_(np.absolute(f(x)).max() < f_tol) + + def _check_root(self, f, method, f_tol=1e-2): + # Test Krylov methods + if method == 'krylov': + for jac_method in ['gmres', 'bicgstab', 'cgs', 'minres', 'tfqmr']: + if jac_method in f.ROOT_JAC_KSP_BAD: + continue + + res = root(f, f.xin, method=method, + options={'ftol': f_tol, 'maxiter': 200, + 'disp': 0, + 'jac_options': {'method': jac_method}}) + assert_(np.absolute(res.fun).max() < f_tol) + + res = root(f, f.xin, method=method, + options={'ftol': f_tol, 'maxiter': 200, 'disp': 0}) + assert_(np.absolute(res.fun).max() < f_tol) + + @pytest.mark.xfail + def _check_func_fail(self, *a, **kw): + pass + + @pytest.mark.filterwarnings('ignore::DeprecationWarning') + def test_problem_nonlin(self): + for f in [F, F2, F2_lucky, F3, F4_powell, F5, F6]: + for func in SOLVERS.values(): + if func in f.KNOWN_BAD.values(): + if func in MUST_WORK.values(): + self._check_func_fail(f, func) + continue + self._check_nonlin_func(f, func) + + @pytest.mark.filterwarnings('ignore::DeprecationWarning') + @pytest.mark.parametrize("method", ['lgmres', 'gmres', 'bicgstab', 'cgs', + 'minres', 'tfqmr']) + def test_tol_norm_called(self, method): + # Check that supplying tol_norm keyword to nonlin_solve works + self._tol_norm_used = False + + def local_norm_func(x): + self._tol_norm_used = True + return np.absolute(x).max() + + nonlin.newton_krylov(F, F.xin, method=method, f_tol=1e-2, + maxiter=200, verbose=0, + tol_norm=local_norm_func) + assert_(self._tol_norm_used) + + @pytest.mark.filterwarnings('ignore::DeprecationWarning') + def test_problem_root(self): + for f in [F, F2, F2_lucky, F3, F4_powell, F5, F6]: + for meth in SOLVERS: + if meth in f.KNOWN_BAD: + if meth in MUST_WORK: + self._check_func_fail(f, meth) + continue + self._check_root(f, meth) + + def test_no_convergence(self): + def wont_converge(x): + return 1e3 + x + + with pytest.raises(scipy.optimize.NoConvergence): + nonlin.newton_krylov(wont_converge, xin=[0], maxiter=1) + + +class TestSecant: + """Check that some Jacobian approximations satisfy the secant condition""" + + xs = [np.array([1., 2., 3., 4., 5.]), + np.array([2., 3., 4., 5., 1.]), + np.array([3., 4., 5., 1., 2.]), + np.array([4., 5., 1., 2., 3.]), + np.array([9., 1., 9., 1., 3.]), + np.array([0., 1., 9., 1., 3.]), + np.array([5., 5., 7., 1., 1.]), + np.array([1., 2., 7., 5., 1.]),] + fs = [x**2 - 1 for x in xs] + + def _check_secant(self, jac_cls, npoints=1, **kw): + """ + Check that the given Jacobian approximation satisfies secant + conditions for last `npoints` points. + """ + jac = jac_cls(**kw) + jac.setup(self.xs[0], self.fs[0], None) + for j, (x, f) in enumerate(zip(self.xs[1:], self.fs[1:])): + jac.update(x, f) + + for k in range(min(npoints, j+1)): + dx = self.xs[j-k+1] - self.xs[j-k] + df = self.fs[j-k+1] - self.fs[j-k] + assert_(np.allclose(dx, jac.solve(df))) + + # Check that the `npoints` secant bound is strict + if j >= npoints: + dx = self.xs[j-npoints+1] - self.xs[j-npoints] + df = self.fs[j-npoints+1] - self.fs[j-npoints] + assert_(not np.allclose(dx, jac.solve(df))) + + def test_broyden1(self): + self._check_secant(nonlin.BroydenFirst) + + def test_broyden2(self): + self._check_secant(nonlin.BroydenSecond) + + def test_broyden1_update(self): + # Check that BroydenFirst update works as for a dense matrix + jac = nonlin.BroydenFirst(alpha=0.1) + jac.setup(self.xs[0], self.fs[0], None) + + B = np.identity(5) * (-1/0.1) + + for last_j, (x, f) in enumerate(zip(self.xs[1:], self.fs[1:])): + df = f - self.fs[last_j] + dx = x - self.xs[last_j] + B += (df - dot(B, dx))[:, None] * dx[None, :] / dot(dx, dx) + jac.update(x, f) + assert_(np.allclose(jac.todense(), B, rtol=1e-10, atol=1e-13)) + + def test_broyden2_update(self): + # Check that BroydenSecond update works as for a dense matrix + jac = nonlin.BroydenSecond(alpha=0.1) + jac.setup(self.xs[0], self.fs[0], None) + + H = np.identity(5) * (-0.1) + + for last_j, (x, f) in enumerate(zip(self.xs[1:], self.fs[1:])): + df = f - self.fs[last_j] + dx = x - self.xs[last_j] + H += (dx - dot(H, df))[:, None] * df[None, :] / dot(df, df) + jac.update(x, f) + assert_(np.allclose(jac.todense(), inv(H), rtol=1e-10, atol=1e-13)) + + def test_anderson(self): + # Anderson mixing (with w0=0) satisfies secant conditions + # for the last M iterates, see [Ey]_ + # + # .. [Ey] V. Eyert, J. Comp. Phys., 124, 271 (1996). + self._check_secant(nonlin.Anderson, M=3, w0=0, npoints=3) + + +class TestLinear: + """Solve a linear equation; + some methods find the exact solution in a finite number of steps""" + + def _check(self, jac, N, maxiter, complex=False, **kw): + np.random.seed(123) + + A = np.random.randn(N, N) + if complex: + A = A + 1j*np.random.randn(N, N) + b = np.random.randn(N) + if complex: + b = b + 1j*np.random.randn(N) + + def func(x): + return dot(A, x) - b + + sol = nonlin.nonlin_solve(func, np.zeros(N), jac, maxiter=maxiter, + f_tol=1e-6, line_search=None, verbose=0) + assert_(np.allclose(dot(A, sol), b, atol=1e-6)) + + def test_broyden1(self): + # Broyden methods solve linear systems exactly in 2*N steps + self._check(nonlin.BroydenFirst(alpha=1.0), 20, 41, False) + self._check(nonlin.BroydenFirst(alpha=1.0), 20, 41, True) + + def test_broyden2(self): + # Broyden methods solve linear systems exactly in 2*N steps + self._check(nonlin.BroydenSecond(alpha=1.0), 20, 41, False) + self._check(nonlin.BroydenSecond(alpha=1.0), 20, 41, True) + + def test_anderson(self): + # Anderson is rather similar to Broyden, if given enough storage space + self._check(nonlin.Anderson(M=50, alpha=1.0), 20, 29, False) + self._check(nonlin.Anderson(M=50, alpha=1.0), 20, 29, True) + + def test_krylov(self): + # Krylov methods solve linear systems exactly in N inner steps + self._check(nonlin.KrylovJacobian, 20, 2, False, inner_m=10) + self._check(nonlin.KrylovJacobian, 20, 2, True, inner_m=10) + + def _check_autojac(self, A, b): + def func(x): + return A.dot(x) - b + + def jac(v): + return A + + sol = nonlin.nonlin_solve(func, np.zeros(b.shape[0]), jac, maxiter=2, + f_tol=1e-6, line_search=None, verbose=0) + np.testing.assert_allclose(A @ sol, b, atol=1e-6) + # test jac input as array -- not a function + sol = nonlin.nonlin_solve(func, np.zeros(b.shape[0]), A, maxiter=2, + f_tol=1e-6, line_search=None, verbose=0) + np.testing.assert_allclose(A @ sol, b, atol=1e-6) + + def test_jac_sparse(self): + A = csr_array([[1, 2], [2, 1]]) + b = np.array([1, -1]) + self._check_autojac(A, b) + self._check_autojac((1 + 2j) * A, (2 + 2j) * b) + + def test_jac_ndarray(self): + A = np.array([[1, 2], [2, 1]]) + b = np.array([1, -1]) + self._check_autojac(A, b) + self._check_autojac((1 + 2j) * A, (2 + 2j) * b) + + +class TestJacobianDotSolve: + """ + Check that solve/dot methods in Jacobian approximations are consistent + """ + + def _func(self, x, A=None): + return x**2 - 1 + np.dot(A, x) + + def _check_dot(self, jac_cls, complex=False, tol=1e-6, **kw): + rng = np.random.RandomState(123) + + N = 7 + + def rand(*a): + q = rng.rand(*a) + if complex: + q = q + 1j*rng.rand(*a) + return q + + def assert_close(a, b, msg): + d = abs(a - b).max() + f = tol + abs(b).max()*tol + if d > f: + raise AssertionError(f'{msg}: err {d:g}') + + A = rand(N, N) + + # initialize + x0 = rng.rand(N) + jac = jac_cls(**kw) + jac.setup(x0, self._func(x0, A), partial(self._func, A=A)) + + # check consistency + for k in range(2*N): + v = rand(N) + + if hasattr(jac, '__array__'): + Jd = np.array(jac) + if hasattr(jac, 'solve'): + Gv = jac.solve(v) + Gv2 = np.linalg.solve(Jd, v) + assert_close(Gv, Gv2, 'solve vs array') + if hasattr(jac, 'rsolve'): + Gv = jac.rsolve(v) + Gv2 = np.linalg.solve(Jd.T.conj(), v) + assert_close(Gv, Gv2, 'rsolve vs array') + if hasattr(jac, 'matvec'): + Jv = jac.matvec(v) + Jv2 = np.dot(Jd, v) + assert_close(Jv, Jv2, 'dot vs array') + if hasattr(jac, 'rmatvec'): + Jv = jac.rmatvec(v) + Jv2 = np.dot(Jd.T.conj(), v) + assert_close(Jv, Jv2, 'rmatvec vs array') + + if hasattr(jac, 'matvec') and hasattr(jac, 'solve'): + Jv = jac.matvec(v) + Jv2 = jac.solve(jac.matvec(Jv)) + assert_close(Jv, Jv2, 'dot vs solve') + + if hasattr(jac, 'rmatvec') and hasattr(jac, 'rsolve'): + Jv = jac.rmatvec(v) + Jv2 = jac.rmatvec(jac.rsolve(Jv)) + assert_close(Jv, Jv2, 'rmatvec vs rsolve') + + x = rand(N) + jac.update(x, self._func(x, A)) + + def test_broyden1(self): + self._check_dot(nonlin.BroydenFirst, complex=False) + self._check_dot(nonlin.BroydenFirst, complex=True) + + def test_broyden2(self): + self._check_dot(nonlin.BroydenSecond, complex=False) + self._check_dot(nonlin.BroydenSecond, complex=True) + + def test_anderson(self): + self._check_dot(nonlin.Anderson, complex=False) + self._check_dot(nonlin.Anderson, complex=True) + + def test_diagbroyden(self): + self._check_dot(nonlin.DiagBroyden, complex=False) + self._check_dot(nonlin.DiagBroyden, complex=True) + + def test_linearmixing(self): + self._check_dot(nonlin.LinearMixing, complex=False) + self._check_dot(nonlin.LinearMixing, complex=True) + + def test_excitingmixing(self): + self._check_dot(nonlin.ExcitingMixing, complex=False) + self._check_dot(nonlin.ExcitingMixing, complex=True) + + @pytest.mark.thread_unsafe + def test_krylov(self): + self._check_dot(nonlin.KrylovJacobian, complex=False, tol=1e-3) + self._check_dot(nonlin.KrylovJacobian, complex=True, tol=1e-3) + + +class TestNonlinOldTests: + """ Test case for a simple constrained entropy maximization problem + (the machine translation example of Berger et al in + Computational Linguistics, vol 22, num 1, pp 39--72, 1996.) + """ + + def test_broyden1(self): + x = nonlin.broyden1(F, F.xin, iter=12, alpha=1) + assert_(nonlin.norm(x) < 1e-9) + assert_(nonlin.norm(F(x)) < 1e-9) + + def test_broyden2(self): + x = nonlin.broyden2(F, F.xin, iter=12, alpha=1) + assert_(nonlin.norm(x) < 1e-9) + assert_(nonlin.norm(F(x)) < 1e-9) + + def test_anderson(self): + x = nonlin.anderson(F, F.xin, iter=12, alpha=0.03, M=5) + assert_(nonlin.norm(x) < 0.33) + + def test_linearmixing(self): + x = nonlin.linearmixing(F, F.xin, iter=60, alpha=0.5) + assert_(nonlin.norm(x) < 1e-7) + assert_(nonlin.norm(F(x)) < 1e-7) + + def test_exciting(self): + x = nonlin.excitingmixing(F, F.xin, iter=20, alpha=0.5) + assert_(nonlin.norm(x) < 1e-5) + assert_(nonlin.norm(F(x)) < 1e-5) + + def test_diagbroyden(self): + x = nonlin.diagbroyden(F, F.xin, iter=11, alpha=1) + assert_(nonlin.norm(x) < 1e-8) + assert_(nonlin.norm(F(x)) < 1e-8) + + def test_root_broyden1(self): + res = root(F, F.xin, method='broyden1', + options={'nit': 12, 'jac_options': {'alpha': 1}}) + assert_(nonlin.norm(res.x) < 1e-9) + assert_(nonlin.norm(res.fun) < 1e-9) + + def test_root_broyden2(self): + res = root(F, F.xin, method='broyden2', + options={'nit': 12, 'jac_options': {'alpha': 1}}) + assert_(nonlin.norm(res.x) < 1e-9) + assert_(nonlin.norm(res.fun) < 1e-9) + + def test_root_anderson(self): + res = root(F, F.xin, method='anderson', + options={'nit': 12, + 'jac_options': {'alpha': 0.03, 'M': 5}}) + assert_(nonlin.norm(res.x) < 0.33) + + def test_root_linearmixing(self): + res = root(F, F.xin, method='linearmixing', + options={'nit': 60, + 'jac_options': {'alpha': 0.5}}) + assert_(nonlin.norm(res.x) < 1e-7) + assert_(nonlin.norm(res.fun) < 1e-7) + + def test_root_excitingmixing(self): + res = root(F, F.xin, method='excitingmixing', + options={'nit': 20, + 'jac_options': {'alpha': 0.5}}) + assert_(nonlin.norm(res.x) < 1e-5) + assert_(nonlin.norm(res.fun) < 1e-5) + + def test_root_diagbroyden(self): + res = root(F, F.xin, method='diagbroyden', + options={'nit': 11, + 'jac_options': {'alpha': 1}}) + assert_(nonlin.norm(res.x) < 1e-8) + assert_(nonlin.norm(res.fun) < 1e-8) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_optimize.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_optimize.py new file mode 100644 index 0000000000000000000000000000000000000000..913ef51f049386a06c61164a3fc07c15111ed212 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_optimize.py @@ -0,0 +1,3257 @@ +""" +Unit tests for optimization routines from optimize.py + +Authors: + Ed Schofield, Nov 2005 + Andrew Straw, April 2008 + +""" +import itertools +import platform +import threading +import numpy as np +from numpy.testing import (assert_allclose, assert_equal, + assert_almost_equal, + assert_no_warnings, assert_warns, + assert_array_less, suppress_warnings) +import pytest +from pytest import raises as assert_raises + +import scipy +from scipy import optimize +from scipy.optimize._minimize import Bounds, NonlinearConstraint +from scipy.optimize._minimize import (MINIMIZE_METHODS, + MINIMIZE_METHODS_NEW_CB, + MINIMIZE_SCALAR_METHODS) +from scipy.optimize._linprog import LINPROG_METHODS +from scipy.optimize._root import ROOT_METHODS +from scipy.optimize._root_scalar import ROOT_SCALAR_METHODS +from scipy.optimize._qap import QUADRATIC_ASSIGNMENT_METHODS +from scipy.optimize._differentiable_functions import ScalarFunction, FD_METHODS +from scipy.optimize._optimize import MemoizeJac, show_options, OptimizeResult +from scipy.optimize import rosen, rosen_der, rosen_hess + +from scipy.sparse import (coo_matrix, csc_matrix, csr_matrix, coo_array, + csr_array, csc_array) +from scipy.conftest import array_api_compatible +from scipy._lib._array_api_no_0d import xp_assert_equal, array_namespace + +skip_xp_backends = pytest.mark.skip_xp_backends + + +def test_check_grad(): + # Verify if check_grad is able to estimate the derivative of the + # expit (logistic sigmoid) function. + + def expit(x): + return 1 / (1 + np.exp(-x)) + + def der_expit(x): + return np.exp(-x) / (1 + np.exp(-x))**2 + + x0 = np.array([1.5]) + + r = optimize.check_grad(expit, der_expit, x0) + assert_almost_equal(r, 0) + # SPEC-007 leave one call with seed to check it still works + r = optimize.check_grad(expit, der_expit, x0, + direction='random', seed=1234) + assert_almost_equal(r, 0) + + r = optimize.check_grad(expit, der_expit, x0, epsilon=1e-6) + assert_almost_equal(r, 0) + r = optimize.check_grad(expit, der_expit, x0, epsilon=1e-6, + direction='random', rng=1234) + assert_almost_equal(r, 0) + + # Check if the epsilon parameter is being considered. + r = abs(optimize.check_grad(expit, der_expit, x0, epsilon=1e-1) - 0) + assert r > 1e-7 + r = abs(optimize.check_grad(expit, der_expit, x0, epsilon=1e-1, + direction='random', rng=1234) - 0) + assert r > 1e-7 + + def x_sinx(x): + return (x*np.sin(x)).sum() + + def der_x_sinx(x): + return np.sin(x) + x*np.cos(x) + + x0 = np.arange(0, 2, 0.2) + + r = optimize.check_grad(x_sinx, der_x_sinx, x0, + direction='random', rng=1234) + assert_almost_equal(r, 0) + + assert_raises(ValueError, optimize.check_grad, + x_sinx, der_x_sinx, x0, + direction='random_projection', rng=1234) + + # checking can be done for derivatives of vector valued functions + r = optimize.check_grad(himmelblau_grad, himmelblau_hess, himmelblau_x0, + direction='all', rng=1234) + assert r < 5e-7 + + +class CheckOptimize: + """ Base test case for a simple constrained entropy maximization problem + (the machine translation example of Berger et al in + Computational Linguistics, vol 22, num 1, pp 39--72, 1996.) + """ + + def setup_method(self): + self.F = np.array([[1, 1, 1], + [1, 1, 0], + [1, 0, 1], + [1, 0, 0], + [1, 0, 0]]) + self.K = np.array([1., 0.3, 0.5]) + self.startparams = np.zeros(3, np.float64) + self.solution = np.array([0., -0.524869316, 0.487525860]) + self.maxiter = 1000 + self.funccalls = threading.local() + self.gradcalls = threading.local() + self.trace = threading.local() + + def func(self, x): + if not hasattr(self.funccalls, 'c'): + self.funccalls.c = 0 + + if not hasattr(self.gradcalls, 'c'): + self.gradcalls.c = 0 + + self.funccalls.c += 1 + if self.funccalls.c > 6000: + raise RuntimeError("too many iterations in optimization routine") + log_pdot = np.dot(self.F, x) + logZ = np.log(sum(np.exp(log_pdot))) + f = logZ - np.dot(self.K, x) + if not hasattr(self.trace, 't'): + self.trace.t = [] + self.trace.t.append(np.copy(x)) + return f + + def grad(self, x): + if not hasattr(self.gradcalls, 'c'): + self.gradcalls.c = 0 + self.gradcalls.c += 1 + log_pdot = np.dot(self.F, x) + logZ = np.log(sum(np.exp(log_pdot))) + p = np.exp(log_pdot - logZ) + return np.dot(self.F.transpose(), p) - self.K + + def hess(self, x): + log_pdot = np.dot(self.F, x) + logZ = np.log(sum(np.exp(log_pdot))) + p = np.exp(log_pdot - logZ) + return np.dot(self.F.T, + np.dot(np.diag(p), self.F - np.dot(self.F.T, p))) + + def hessp(self, x, p): + return np.dot(self.hess(x), p) + + +class CheckOptimizeParameterized(CheckOptimize): + + def test_cg(self): + # conjugate gradient optimization routine + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': self.disp, + 'return_all': False} + res = optimize.minimize(self.func, self.startparams, args=(), + method='CG', jac=self.grad, + options=opts) + params, fopt, func_calls, grad_calls, warnflag = \ + res['x'], res['fun'], res['nfev'], res['njev'], res['status'] + else: + retval = optimize.fmin_cg(self.func, self.startparams, + self.grad, (), maxiter=self.maxiter, + full_output=True, disp=self.disp, + retall=False) + (params, fopt, func_calls, grad_calls, warnflag) = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + # Ensure that function call counts are 'known good'; these are from + # SciPy 0.7.0. Don't allow them to increase. + assert self.funccalls.c == 9, self.funccalls.c + assert self.gradcalls.c == 7, self.gradcalls.c + + # Ensure that the function behaves the same; this is from SciPy 0.7.0 + assert_allclose(self.trace.t[2:4], + [[0, -0.5, 0.5], + [0, -5.05700028e-01, 4.95985862e-01]], + atol=1e-14, rtol=1e-7) + + def test_cg_cornercase(self): + def f(r): + return 2.5 * (1 - np.exp(-1.5*(r - 0.5)))**2 + + # Check several initial guesses. (Too far away from the + # minimum, the function ends up in the flat region of exp.) + for x0 in np.linspace(-0.75, 3, 71): + sol = optimize.minimize(f, [x0], method='CG') + assert sol.success + assert_allclose(sol.x, [0.5], rtol=1e-5) + + def test_bfgs(self): + # Broyden-Fletcher-Goldfarb-Shanno optimization routine + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': self.disp, + 'return_all': False} + res = optimize.minimize(self.func, self.startparams, + jac=self.grad, method='BFGS', args=(), + options=opts) + + params, fopt, gopt, Hopt, func_calls, grad_calls, warnflag = ( + res['x'], res['fun'], res['jac'], res['hess_inv'], + res['nfev'], res['njev'], res['status']) + else: + retval = optimize.fmin_bfgs(self.func, self.startparams, self.grad, + args=(), maxiter=self.maxiter, + full_output=True, disp=self.disp, + retall=False) + (params, fopt, gopt, Hopt, + func_calls, grad_calls, warnflag) = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + # Ensure that function call counts are 'known good'; these are from + # SciPy 0.7.0. Don't allow them to increase. + assert self.funccalls.c == 10, self.funccalls.c + assert self.gradcalls.c == 8, self.gradcalls.c + + # Ensure that the function behaves the same; this is from SciPy 0.7.0 + assert_allclose(self.trace.t[6:8], + [[0, -5.25060743e-01, 4.87748473e-01], + [0, -5.24885582e-01, 4.87530347e-01]], + atol=1e-14, rtol=1e-7) + + def test_bfgs_hess_inv0_neg(self): + # Ensure that BFGS does not accept neg. def. initial inverse + # Hessian estimate. + with pytest.raises(ValueError, match="'hess_inv0' matrix isn't " + "positive definite."): + x0 = np.array([1.3, 0.7, 0.8, 1.9, 1.2]) + opts = {'disp': self.disp, 'hess_inv0': -np.eye(5)} + optimize.minimize(optimize.rosen, x0=x0, method='BFGS', args=(), + options=opts) + + def test_bfgs_hess_inv0_semipos(self): + # Ensure that BFGS does not accept semi pos. def. initial inverse + # Hessian estimate. + with pytest.raises(ValueError, match="'hess_inv0' matrix isn't " + "positive definite."): + x0 = np.array([1.3, 0.7, 0.8, 1.9, 1.2]) + hess_inv0 = np.eye(5) + hess_inv0[0, 0] = 0 + opts = {'disp': self.disp, 'hess_inv0': hess_inv0} + optimize.minimize(optimize.rosen, x0=x0, method='BFGS', args=(), + options=opts) + + def test_bfgs_hess_inv0_sanity(self): + # Ensure that BFGS handles `hess_inv0` parameter correctly. + fun = optimize.rosen + x0 = np.array([1.3, 0.7, 0.8, 1.9, 1.2]) + opts = {'disp': self.disp, 'hess_inv0': 1e-2 * np.eye(5)} + res = optimize.minimize(fun, x0=x0, method='BFGS', args=(), + options=opts) + res_true = optimize.minimize(fun, x0=x0, method='BFGS', args=(), + options={'disp': self.disp}) + assert_allclose(res.fun, res_true.fun, atol=1e-6) + + @pytest.mark.filterwarnings('ignore::UserWarning') + def test_bfgs_infinite(self): + # Test corner case where -Inf is the minimum. See gh-2019. + def func(x): + return -np.e ** (-x) + def fprime(x): + return -func(x) + x0 = [0] + with np.errstate(over='ignore'): + if self.use_wrapper: + opts = {'disp': self.disp} + x = optimize.minimize(func, x0, jac=fprime, method='BFGS', + args=(), options=opts)['x'] + else: + x = optimize.fmin_bfgs(func, x0, fprime, disp=self.disp) + assert not np.isfinite(func(x)) + + def test_bfgs_xrtol(self): + # test for #17345 to test xrtol parameter + x0 = [1.3, 0.7, 0.8, 1.9, 1.2] + res = optimize.minimize(optimize.rosen, + x0, method='bfgs', options={'xrtol': 1e-3}) + ref = optimize.minimize(optimize.rosen, + x0, method='bfgs', options={'gtol': 1e-3}) + assert res.nit != ref.nit + + def test_bfgs_c1(self): + # test for #18977 insufficiently low value of c1 leads to precision loss + # for poor starting parameters + x0 = [10.3, 20.7, 10.8, 1.9, -1.2] + res_c1_small = optimize.minimize(optimize.rosen, + x0, method='bfgs', options={'c1': 1e-8}) + res_c1_big = optimize.minimize(optimize.rosen, + x0, method='bfgs', options={'c1': 1e-1}) + + assert res_c1_small.nfev > res_c1_big.nfev + + def test_bfgs_c2(self): + # test that modification of c2 parameter + # results in different number of iterations + x0 = [1.3, 0.7, 0.8, 1.9, 1.2] + res_default = optimize.minimize(optimize.rosen, + x0, method='bfgs', options={'c2': .9}) + res_mod = optimize.minimize(optimize.rosen, + x0, method='bfgs', options={'c2': 1e-2}) + assert res_default.nit > res_mod.nit + + @pytest.mark.parametrize(["c1", "c2"], [[0.5, 2], + [-0.1, 0.1], + [0.2, 0.1]]) + def test_invalid_c1_c2(self, c1, c2): + with pytest.raises(ValueError, match="'c1' and 'c2'"): + x0 = [10.3, 20.7, 10.8, 1.9, -1.2] + optimize.minimize(optimize.rosen, x0, method='cg', + options={'c1': c1, 'c2': c2}) + + def test_powell(self): + # Powell (direction set) optimization routine + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': self.disp, + 'return_all': False} + res = optimize.minimize(self.func, self.startparams, args=(), + method='Powell', options=opts) + params, fopt, direc, numiter, func_calls, warnflag = ( + res['x'], res['fun'], res['direc'], res['nit'], + res['nfev'], res['status']) + else: + retval = optimize.fmin_powell(self.func, self.startparams, + args=(), maxiter=self.maxiter, + full_output=True, disp=self.disp, + retall=False) + (params, fopt, direc, numiter, func_calls, warnflag) = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + # params[0] does not affect the objective function + assert_allclose(params[1:], self.solution[1:], atol=5e-6) + + # Ensure that function call counts are 'known good'; these are from + # SciPy 0.7.0. Don't allow them to increase. + # + # However, some leeway must be added: the exact evaluation + # count is sensitive to numerical error, and floating-point + # computations are not bit-for-bit reproducible across + # machines, and when using e.g., MKL, data alignment + # etc., affect the rounding error. + # + assert self.funccalls.c <= 116 + 20, self.funccalls.c + assert self.gradcalls.c == 0, self.gradcalls.c + + @pytest.mark.xfail(reason="This part of test_powell fails on some " + "platforms, but the solution returned by powell is " + "still valid.") + def test_powell_gh14014(self): + # This part of test_powell started failing on some CI platforms; + # see gh-14014. Since the solution is still correct and the comments + # in test_powell suggest that small differences in the bits are known + # to change the "trace" of the solution, seems safe to xfail to get CI + # green now and investigate later. + + # Powell (direction set) optimization routine + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': self.disp, + 'return_all': False} + res = optimize.minimize(self.func, self.startparams, args=(), + method='Powell', options=opts) + params, fopt, direc, numiter, func_calls, warnflag = ( + res['x'], res['fun'], res['direc'], res['nit'], + res['nfev'], res['status']) + else: + retval = optimize.fmin_powell(self.func, self.startparams, + args=(), maxiter=self.maxiter, + full_output=True, disp=self.disp, + retall=False) + (params, fopt, direc, numiter, func_calls, warnflag) = retval + + # Ensure that the function behaves the same; this is from SciPy 0.7.0 + assert_allclose(self.trace[34:39], + [[0.72949016, -0.44156936, 0.47100962], + [0.72949016, -0.44156936, 0.48052496], + [1.45898031, -0.88313872, 0.95153458], + [0.72949016, -0.44156936, 0.47576729], + [1.72949016, -0.44156936, 0.47576729]], + atol=1e-14, rtol=1e-7) + + def test_powell_bounded(self): + # Powell (direction set) optimization routine + # same as test_powell above, but with bounds + bounds = [(-np.pi, np.pi) for _ in self.startparams] + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': self.disp, + 'return_all': False} + res = optimize.minimize(self.func, self.startparams, args=(), + bounds=bounds, + method='Powell', options=opts) + params, func_calls = (res['x'], res['nfev']) + + assert func_calls == self.funccalls.c + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6, rtol=1e-5) + + # The exact evaluation count is sensitive to numerical error, and + # floating-point computations are not bit-for-bit reproducible + # across machines, and when using e.g. MKL, data alignment etc. + # affect the rounding error. + # It takes 155 calls on my machine, but we can add the same +20 + # margin as is used in `test_powell` + assert self.funccalls.c <= 155 + 20 + assert self.gradcalls.c == 0 + + def test_neldermead(self): + # Nelder-Mead simplex algorithm + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': self.disp, + 'return_all': False} + res = optimize.minimize(self.func, self.startparams, args=(), + method='Nelder-mead', options=opts) + params, fopt, numiter, func_calls, warnflag = ( + res['x'], res['fun'], res['nit'], res['nfev'], + res['status']) + else: + retval = optimize.fmin(self.func, self.startparams, + args=(), maxiter=self.maxiter, + full_output=True, disp=self.disp, + retall=False) + (params, fopt, numiter, func_calls, warnflag) = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + # Ensure that function call counts are 'known good'; these are from + # SciPy 0.7.0. Don't allow them to increase. + assert self.funccalls.c == 167, self.funccalls.c + assert self.gradcalls.c == 0, self.gradcalls.c + + # Ensure that the function behaves the same; this is from SciPy 0.7.0 + assert_allclose(self.trace.t[76:78], + [[0.1928968, -0.62780447, 0.35166118], + [0.19572515, -0.63648426, 0.35838135]], + atol=1e-14, rtol=1e-7) + + def test_neldermead_initial_simplex(self): + # Nelder-Mead simplex algorithm + simplex = np.zeros((4, 3)) + simplex[...] = self.startparams + for j in range(3): + simplex[j+1, j] += 0.1 + + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': False, + 'return_all': True, 'initial_simplex': simplex} + res = optimize.minimize(self.func, self.startparams, args=(), + method='Nelder-mead', options=opts) + params, fopt, numiter, func_calls, warnflag = (res['x'], + res['fun'], + res['nit'], + res['nfev'], + res['status']) + assert_allclose(res['allvecs'][0], simplex[0]) + else: + retval = optimize.fmin(self.func, self.startparams, + args=(), maxiter=self.maxiter, + full_output=True, disp=False, retall=False, + initial_simplex=simplex) + + (params, fopt, numiter, func_calls, warnflag) = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + # Ensure that function call counts are 'known good'; these are from + # SciPy 0.17.0. Don't allow them to increase. + assert self.funccalls.c == 100, self.funccalls.c + assert self.gradcalls.c == 0, self.gradcalls.c + + # Ensure that the function behaves the same; this is from SciPy 0.15.0 + assert_allclose(self.trace.t[50:52], + [[0.14687474, -0.5103282, 0.48252111], + [0.14474003, -0.5282084, 0.48743951]], + atol=1e-14, rtol=1e-7) + + def test_neldermead_initial_simplex_bad(self): + # Check it fails with a bad simplices + bad_simplices = [] + + simplex = np.zeros((3, 2)) + simplex[...] = self.startparams[:2] + for j in range(2): + simplex[j+1, j] += 0.1 + bad_simplices.append(simplex) + + simplex = np.zeros((3, 3)) + bad_simplices.append(simplex) + + for simplex in bad_simplices: + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': False, + 'return_all': False, 'initial_simplex': simplex} + assert_raises(ValueError, + optimize.minimize, + self.func, + self.startparams, + args=(), + method='Nelder-mead', + options=opts) + else: + assert_raises(ValueError, optimize.fmin, + self.func, self.startparams, + args=(), maxiter=self.maxiter, + full_output=True, disp=False, retall=False, + initial_simplex=simplex) + + def test_neldermead_x0_ub(self): + # checks whether minimisation occurs correctly for entries where + # x0 == ub + # gh19991 + def quad(x): + return np.sum(x**2) + + res = optimize.minimize( + quad, + [1], + bounds=[(0, 1.)], + method='nelder-mead' + ) + assert_allclose(res.x, [0]) + + res = optimize.minimize( + quad, + [1, 2], + bounds=[(0, 1.), (1, 3.)], + method='nelder-mead' + ) + assert_allclose(res.x, [0, 1]) + + def test_ncg_negative_maxiter(self): + # Regression test for gh-8241 + opts = {'maxiter': -1} + result = optimize.minimize(self.func, self.startparams, + method='Newton-CG', jac=self.grad, + args=(), options=opts) + assert result.status == 1 + + def test_ncg_zero_xtol(self): + # Regression test for gh-20214 + def cosine(x): + return np.cos(x[0]) + + def jac(x): + return -np.sin(x[0]) + + x0 = [0.1] + xtol = 0 + result = optimize.minimize(cosine, + x0=x0, + jac=jac, + method="newton-cg", + options=dict(xtol=xtol)) + assert result.status == 0 + assert_almost_equal(result.x[0], np.pi) + + def test_ncg(self): + # line-search Newton conjugate gradient optimization routine + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': self.disp, + 'return_all': False} + retval = optimize.minimize(self.func, self.startparams, + method='Newton-CG', jac=self.grad, + args=(), options=opts)['x'] + else: + retval = optimize.fmin_ncg(self.func, self.startparams, self.grad, + args=(), maxiter=self.maxiter, + full_output=False, disp=self.disp, + retall=False) + + params = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + # Ensure that function call counts are 'known good'; these are from + # SciPy 0.7.0. Don't allow them to increase. + assert self.funccalls.c == 7, self.funccalls.c + assert self.gradcalls.c <= 22, self.gradcalls.c # 0.13.0 + # assert self.gradcalls <= 18, self.gradcalls # 0.9.0 + # assert self.gradcalls == 18, self.gradcalls # 0.8.0 + # assert self.gradcalls == 22, self.gradcalls # 0.7.0 + + # Ensure that the function behaves the same; this is from SciPy 0.7.0 + assert_allclose(self.trace.t[3:5], + [[-4.35700753e-07, -5.24869435e-01, 4.87527480e-01], + [-4.35700753e-07, -5.24869401e-01, 4.87527774e-01]], + atol=1e-6, rtol=1e-7) + + def test_ncg_hess(self): + # Newton conjugate gradient with Hessian + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': self.disp, + 'return_all': False} + retval = optimize.minimize(self.func, self.startparams, + method='Newton-CG', jac=self.grad, + hess=self.hess, + args=(), options=opts)['x'] + else: + retval = optimize.fmin_ncg(self.func, self.startparams, self.grad, + fhess=self.hess, + args=(), maxiter=self.maxiter, + full_output=False, disp=self.disp, + retall=False) + + params = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + # Ensure that function call counts are 'known good'; these are from + # SciPy 0.7.0. Don't allow them to increase. + assert self.funccalls.c <= 7, self.funccalls.c # gh10673 + assert self.gradcalls.c <= 18, self.gradcalls.c # 0.9.0 + # assert self.gradcalls == 18, self.gradcalls # 0.8.0 + # assert self.gradcalls == 22, self.gradcalls # 0.7.0 + + # Ensure that the function behaves the same; this is from SciPy 0.7.0 + assert_allclose(self.trace.t[3:5], + [[-4.35700753e-07, -5.24869435e-01, 4.87527480e-01], + [-4.35700753e-07, -5.24869401e-01, 4.87527774e-01]], + atol=1e-6, rtol=1e-7) + + def test_ncg_hessp(self): + # Newton conjugate gradient with Hessian times a vector p. + if self.use_wrapper: + opts = {'maxiter': self.maxiter, 'disp': self.disp, + 'return_all': False} + retval = optimize.minimize(self.func, self.startparams, + method='Newton-CG', jac=self.grad, + hessp=self.hessp, + args=(), options=opts)['x'] + else: + retval = optimize.fmin_ncg(self.func, self.startparams, self.grad, + fhess_p=self.hessp, + args=(), maxiter=self.maxiter, + full_output=False, disp=self.disp, + retall=False) + + params = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + # Ensure that function call counts are 'known good'; these are from + # SciPy 0.7.0. Don't allow them to increase. + assert self.funccalls.c <= 7, self.funccalls.c # gh10673 + assert self.gradcalls.c <= 18, self.gradcalls.c # 0.9.0 + # assert self.gradcalls == 18, self.gradcalls # 0.8.0 + # assert self.gradcalls == 22, self.gradcalls # 0.7.0 + + # Ensure that the function behaves the same; this is from SciPy 0.7.0 + assert_allclose(self.trace.t[3:5], + [[-4.35700753e-07, -5.24869435e-01, 4.87527480e-01], + [-4.35700753e-07, -5.24869401e-01, 4.87527774e-01]], + atol=1e-6, rtol=1e-7) + + def test_cobyqa(self): + # COBYQA method. + if self.use_wrapper: + res = optimize.minimize( + self.func, + self.startparams, + method='cobyqa', + options={'maxiter': self.maxiter, 'disp': self.disp}, + ) + assert_allclose(res.fun, self.func(self.solution), atol=1e-6) + + # Ensure that function call counts are 'known good'; these are from + # SciPy 1.14.0. Don't allow them to increase. The exact evaluation + # count is sensitive to numerical error and floating-point + # computations are not bit-for-bit reproducible across machines. It + # takes 45 calls on my machine, but we can add the same +20 margin + # as is used in `test_powell` + assert self.funccalls.c <= 45 + 20, self.funccalls.c + + +def test_maxfev_test(): + rng = np.random.default_rng(271707100830272976862395227613146332411) + + def cost(x): + return rng.random(1) * 1000 # never converged problem + + for imaxfev in [1, 10, 50]: + # "TNC" and "L-BFGS-B" also supports max function evaluation, but + # these may violate the limit because of evaluating gradients + # by numerical differentiation. See the discussion in PR #14805. + for method in ['Powell', 'Nelder-Mead']: + result = optimize.minimize(cost, rng.random(10), + method=method, + options={'maxfev': imaxfev}) + assert result["nfev"] == imaxfev + + +def test_wrap_scalar_function_with_validation(): + + def func_(x): + return x + + fcalls, func = optimize._optimize.\ + _wrap_scalar_function_maxfun_validation(func_, np.asarray(1), 5) + + for i in range(5): + func(np.asarray(i)) + assert fcalls[0] == i+1 + + msg = "Too many function calls" + with assert_raises(optimize._optimize._MaxFuncCallError, match=msg): + func(np.asarray(i)) # exceeded maximum function call + + fcalls, func = optimize._optimize.\ + _wrap_scalar_function_maxfun_validation(func_, np.asarray(1), 5) + + msg = "The user-provided objective function must return a scalar value." + with assert_raises(ValueError, match=msg): + func(np.array([1, 1])) + + +def test_obj_func_returns_scalar(): + match = ("The user-provided " + "objective function must " + "return a scalar value.") + with assert_raises(ValueError, match=match): + optimize.minimize(lambda x: x, np.array([1, 1]), method='BFGS') + + +def test_neldermead_iteration_num(): + x0 = np.array([1.3, 0.7, 0.8, 1.9, 1.2]) + res = optimize._minimize._minimize_neldermead(optimize.rosen, x0, + xatol=1e-8) + assert res.nit <= 339 + + +def test_neldermead_respect_fp(): + # Nelder-Mead should respect the fp type of the input + function + x0 = np.array([5.0, 4.0]).astype(np.float32) + def rosen_(x): + assert x.dtype == np.float32 + return optimize.rosen(x) + + optimize.minimize(rosen_, x0, method='Nelder-Mead') + + +def test_neldermead_xatol_fatol(): + # gh4484 + # test we can call with fatol, xatol specified + def func(x): + return x[0] ** 2 + x[1] ** 2 + + optimize._minimize._minimize_neldermead(func, [1, 1], maxiter=2, + xatol=1e-3, fatol=1e-3) + + +def test_neldermead_adaptive(): + def func(x): + return np.sum(x ** 2) + p0 = [0.15746215, 0.48087031, 0.44519198, 0.4223638, 0.61505159, + 0.32308456, 0.9692297, 0.4471682, 0.77411992, 0.80441652, + 0.35994957, 0.75487856, 0.99973421, 0.65063887, 0.09626474] + + res = optimize.minimize(func, p0, method='Nelder-Mead') + assert_equal(res.success, False) + + res = optimize.minimize(func, p0, method='Nelder-Mead', + options={'adaptive': True}) + assert_equal(res.success, True) + + +@pytest.mark.thread_unsafe +def test_bounded_powell_outsidebounds(): + # With the bounded Powell method if you start outside the bounds the final + # should still be within the bounds (provided that the user doesn't make a + # bad choice for the `direc` argument). + def func(x): + return np.sum(x ** 2) + bounds = (-1, 1), (-1, 1), (-1, 1) + x0 = [-4, .5, -.8] + + # we're starting outside the bounds, so we should get a warning + with assert_warns(optimize.OptimizeWarning): + res = optimize.minimize(func, x0, bounds=bounds, method="Powell") + assert_allclose(res.x, np.array([0.] * len(x0)), atol=1e-6) + assert_equal(res.success, True) + assert_equal(res.status, 0) + + # However, now if we change the `direc` argument such that the + # set of vectors does not span the parameter space, then we may + # not end up back within the bounds. Here we see that the first + # parameter cannot be updated! + direc = [[0, 0, 0], [0, 1, 0], [0, 0, 1]] + # we're starting outside the bounds, so we should get a warning + with assert_warns(optimize.OptimizeWarning): + res = optimize.minimize(func, x0, + bounds=bounds, method="Powell", + options={'direc': direc}) + assert_allclose(res.x, np.array([-4., 0, 0]), atol=1e-6) + assert_equal(res.success, False) + assert_equal(res.status, 4) + + +@pytest.mark.thread_unsafe +def test_bounded_powell_vs_powell(): + # here we test an example where the bounded Powell method + # will return a different result than the standard Powell + # method. + + # first we test a simple example where the minimum is at + # the origin and the minimum that is within the bounds is + # larger than the minimum at the origin. + def func(x): + return np.sum(x ** 2) + bounds = (-5, -1), (-10, -0.1), (1, 9.2), (-4, 7.6), (-15.9, -2) + x0 = [-2.1, -5.2, 1.9, 0, -2] + + options = {'ftol': 1e-10, 'xtol': 1e-10} + + res_powell = optimize.minimize(func, x0, method="Powell", options=options) + assert_allclose(res_powell.x, 0., atol=1e-6) + assert_allclose(res_powell.fun, 0., atol=1e-6) + + res_bounded_powell = optimize.minimize(func, x0, options=options, + bounds=bounds, + method="Powell") + p = np.array([-1, -0.1, 1, 0, -2]) + assert_allclose(res_bounded_powell.x, p, atol=1e-6) + assert_allclose(res_bounded_powell.fun, func(p), atol=1e-6) + + # now we test bounded Powell but with a mix of inf bounds. + bounds = (None, -1), (-np.inf, -.1), (1, np.inf), (-4, None), (-15.9, -2) + res_bounded_powell = optimize.minimize(func, x0, options=options, + bounds=bounds, + method="Powell") + p = np.array([-1, -0.1, 1, 0, -2]) + assert_allclose(res_bounded_powell.x, p, atol=1e-6) + assert_allclose(res_bounded_powell.fun, func(p), atol=1e-6) + + # next we test an example where the global minimum is within + # the bounds, but the bounded Powell method performs better + # than the standard Powell method. + def func(x): + t = np.sin(-x[0]) * np.cos(x[1]) * np.sin(-x[0] * x[1]) * np.cos(x[1]) + t -= np.cos(np.sin(x[1] * x[2]) * np.cos(x[2])) + return t**2 + + bounds = [(-2, 5)] * 3 + x0 = [-0.5, -0.5, -0.5] + + res_powell = optimize.minimize(func, x0, method="Powell") + res_bounded_powell = optimize.minimize(func, x0, + bounds=bounds, + method="Powell") + assert_allclose(res_powell.fun, 0.007136253919761627, atol=1e-6) + assert_allclose(res_bounded_powell.fun, 0, atol=1e-6) + + # next we test the previous example where the we provide Powell + # with (-inf, inf) bounds, and compare it to providing Powell + # with no bounds. They should end up the same. + bounds = [(-np.inf, np.inf)] * 3 + + res_bounded_powell = optimize.minimize(func, x0, + bounds=bounds, + method="Powell") + assert_allclose(res_powell.fun, res_bounded_powell.fun, atol=1e-6) + assert_allclose(res_powell.nfev, res_bounded_powell.nfev, atol=1e-6) + assert_allclose(res_powell.x, res_bounded_powell.x, atol=1e-6) + + # now test when x0 starts outside of the bounds. + x0 = [45.46254415, -26.52351498, 31.74830248] + bounds = [(-2, 5)] * 3 + # we're starting outside the bounds, so we should get a warning + with assert_warns(optimize.OptimizeWarning): + res_bounded_powell = optimize.minimize(func, x0, + bounds=bounds, + method="Powell") + assert_allclose(res_bounded_powell.fun, 0, atol=1e-6) + + +def test_onesided_bounded_powell_stability(): + # When the Powell method is bounded on only one side, a + # np.tan transform is done in order to convert it into a + # completely bounded problem. Here we do some simple tests + # of one-sided bounded Powell where the optimal solutions + # are large to test the stability of the transformation. + kwargs = {'method': 'Powell', + 'bounds': [(-np.inf, 1e6)] * 3, + 'options': {'ftol': 1e-8, 'xtol': 1e-8}} + x0 = [1, 1, 1] + + # df/dx is constant. + def f(x): + return -np.sum(x) + res = optimize.minimize(f, x0, **kwargs) + assert_allclose(res.fun, -3e6, atol=1e-4) + + # df/dx gets smaller and smaller. + def f(x): + return -np.abs(np.sum(x)) ** (0.1) * (1 if np.all(x > 0) else -1) + + res = optimize.minimize(f, x0, **kwargs) + assert_allclose(res.fun, -(3e6) ** (0.1)) + + # df/dx gets larger and larger. + def f(x): + return -np.abs(np.sum(x)) ** 10 * (1 if np.all(x > 0) else -1) + + res = optimize.minimize(f, x0, **kwargs) + assert_allclose(res.fun, -(3e6) ** 10, rtol=1e-7) + + # df/dx gets larger for some of the variables and smaller for others. + def f(x): + t = -np.abs(np.sum(x[:2])) ** 5 - np.abs(np.sum(x[2:])) ** (0.1) + t *= (1 if np.all(x > 0) else -1) + return t + + kwargs['bounds'] = [(-np.inf, 1e3)] * 3 + res = optimize.minimize(f, x0, **kwargs) + assert_allclose(res.fun, -(2e3) ** 5 - (1e6) ** (0.1), rtol=1e-7) + + +class TestOptimizeWrapperDisp(CheckOptimizeParameterized): + use_wrapper = True + disp = True + + +class TestOptimizeWrapperNoDisp(CheckOptimizeParameterized): + use_wrapper = True + disp = False + + +class TestOptimizeNoWrapperDisp(CheckOptimizeParameterized): + use_wrapper = False + disp = True + + +class TestOptimizeNoWrapperNoDisp(CheckOptimizeParameterized): + use_wrapper = False + disp = False + + +class TestOptimizeSimple(CheckOptimize): + + def test_bfgs_nan(self): + # Test corner case where nan is fed to optimizer. See gh-2067. + def func(x): + return x + def fprime(x): + return np.ones_like(x) + x0 = [np.nan] + with np.errstate(over='ignore', invalid='ignore'): + x = optimize.fmin_bfgs(func, x0, fprime, disp=False) + assert np.isnan(func(x)) + + def test_bfgs_nan_return(self): + # Test corner cases where fun returns NaN. See gh-4793. + + # First case: NaN from first call. + def func(x): + return np.nan + with np.errstate(invalid='ignore'): + result = optimize.minimize(func, 0) + + assert np.isnan(result['fun']) + assert result['success'] is False + + # Second case: NaN from second call. + def func(x): + return 0 if x == 0 else np.nan + def fprime(x): + return np.ones_like(x) # Steer away from zero. + with np.errstate(invalid='ignore'): + result = optimize.minimize(func, 0, jac=fprime) + + assert np.isnan(result['fun']) + assert result['success'] is False + + def test_bfgs_numerical_jacobian(self): + # BFGS with numerical Jacobian and a vector epsilon parameter. + # define the epsilon parameter using a random vector + epsilon = np.sqrt(np.spacing(1.)) * np.random.rand(len(self.solution)) + + params = optimize.fmin_bfgs(self.func, self.startparams, + epsilon=epsilon, args=(), + maxiter=self.maxiter, disp=False) + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + def test_finite_differences_jac(self): + methods = ['BFGS', 'CG', 'TNC'] + jacs = ['2-point', '3-point', None] + for method, jac in itertools.product(methods, jacs): + result = optimize.minimize(self.func, self.startparams, + method=method, jac=jac) + assert_allclose(self.func(result.x), self.func(self.solution), + atol=1e-6) + + def test_finite_differences_hess(self): + # test that all the methods that require hess can use finite-difference + # For Newton-CG, trust-ncg, trust-krylov the FD estimated hessian is + # wrapped in a hessp function + # dogleg, trust-exact actually require true hessians at the moment, so + # they're excluded. + methods = ['trust-constr', 'Newton-CG', 'trust-ncg', 'trust-krylov'] + hesses = FD_METHODS + (optimize.BFGS,) + for method, hess in itertools.product(methods, hesses): + if hess is optimize.BFGS: + hess = hess() + result = optimize.minimize(self.func, self.startparams, + method=method, jac=self.grad, + hess=hess) + assert result.success + + # check that the methods demand some sort of Hessian specification + # Newton-CG creates its own hessp, and trust-constr doesn't need a hess + # specified either + methods = ['trust-ncg', 'trust-krylov', 'dogleg', 'trust-exact'] + for method in methods: + with pytest.raises(ValueError): + optimize.minimize(self.func, self.startparams, + method=method, jac=self.grad, + hess=None) + + def test_bfgs_gh_2169(self): + def f(x): + if x < 0: + return 1.79769313e+308 + else: + return x + 1./x + xs = optimize.fmin_bfgs(f, [10.], disp=False) + assert_allclose(xs, 1.0, rtol=1e-4, atol=1e-4) + + def test_bfgs_double_evaluations(self): + # check BFGS does not evaluate twice in a row at same point + def f(x): + xp = x[0] + assert xp not in seen + seen.add(xp) + return 10*x**2, 20*x + + seen = set() + optimize.minimize(f, -100, method='bfgs', jac=True, tol=1e-7) + + def test_l_bfgs_b(self): + # limited-memory bound-constrained BFGS algorithm + retval = optimize.fmin_l_bfgs_b(self.func, self.startparams, + self.grad, args=(), + maxiter=self.maxiter) + + (params, fopt, d) = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + # Ensure that function call counts are 'known good'; these are from + # SciPy 0.7.0. Don't allow them to increase. + assert self.funccalls.c == 7, self.funccalls.c + assert self.gradcalls.c == 5, self.gradcalls.c + + # Ensure that the function behaves the same; this is from SciPy 0.7.0 + # test fixed in gh10673 + assert_allclose(self.trace.t[3:5], + [[8.117083e-16, -5.196198e-01, 4.897617e-01], + [0., -0.52489628, 0.48753042]], + atol=1e-14, rtol=1e-7) + + def test_l_bfgs_b_numjac(self): + # L-BFGS-B with numerical Jacobian + retval = optimize.fmin_l_bfgs_b(self.func, self.startparams, + approx_grad=True, + maxiter=self.maxiter) + + (params, fopt, d) = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + def test_l_bfgs_b_funjac(self): + # L-BFGS-B with combined objective function and Jacobian + def fun(x): + return self.func(x), self.grad(x) + + retval = optimize.fmin_l_bfgs_b(fun, self.startparams, + maxiter=self.maxiter) + + (params, fopt, d) = retval + + assert_allclose(self.func(params), self.func(self.solution), + atol=1e-6) + + def test_l_bfgs_b_maxiter(self): + # gh7854 + # Ensure that not more than maxiters are ever run. + class Callback: + def __init__(self): + self.nit = 0 + self.fun = None + self.x = None + + def __call__(self, x): + self.x = x + self.fun = optimize.rosen(x) + self.nit += 1 + + c = Callback() + res = optimize.minimize(optimize.rosen, [0., 0.], method='l-bfgs-b', + callback=c, options={'maxiter': 5}) + + assert_equal(res.nit, 5) + assert_almost_equal(res.x, c.x) + assert_almost_equal(res.fun, c.fun) + assert_equal(res.status, 1) + assert res.success is False + assert_equal(res.message, + 'STOP: TOTAL NO. OF ITERATIONS REACHED LIMIT') + + def test_minimize_l_bfgs_b(self): + # Minimize with L-BFGS-B method + opts = {'disp': False, 'maxiter': self.maxiter} + r = optimize.minimize(self.func, self.startparams, + method='L-BFGS-B', jac=self.grad, + options=opts) + assert_allclose(self.func(r.x), self.func(self.solution), + atol=1e-6) + assert self.gradcalls.c == r.njev + + self.funccalls.c = self.gradcalls.c = 0 + # approximate jacobian + ra = optimize.minimize(self.func, self.startparams, + method='L-BFGS-B', options=opts) + # check that function evaluations in approximate jacobian are counted + # assert_(ra.nfev > r.nfev) + assert self.funccalls.c == ra.nfev + assert_allclose(self.func(ra.x), self.func(self.solution), + atol=1e-6) + + self.funccalls.c = self.gradcalls.c = 0 + # approximate jacobian + ra = optimize.minimize(self.func, self.startparams, jac='3-point', + method='L-BFGS-B', options=opts) + assert self.funccalls.c == ra.nfev + assert_allclose(self.func(ra.x), self.func(self.solution), + atol=1e-6) + + def test_minimize_l_bfgs_b_ftol(self): + # Check that the `ftol` parameter in l_bfgs_b works as expected + v0 = None + for tol in [1e-1, 1e-4, 1e-7, 1e-10]: + opts = {'disp': False, 'maxiter': self.maxiter, 'ftol': tol} + sol = optimize.minimize(self.func, self.startparams, + method='L-BFGS-B', jac=self.grad, + options=opts) + v = self.func(sol.x) + + if v0 is None: + v0 = v + else: + assert v < v0 + + assert_allclose(v, self.func(self.solution), rtol=tol) + + def test_minimize_l_bfgs_maxls(self): + # check that the maxls is passed down to the Fortran routine + sol = optimize.minimize(optimize.rosen, np.array([-1.2, 1.0]), + method='L-BFGS-B', jac=optimize.rosen_der, + options={'disp': False, 'maxls': 1}) + assert not sol.success + + def test_minimize_l_bfgs_b_maxfun_interruption(self): + # gh-6162 + f = optimize.rosen + g = optimize.rosen_der + values = [] + x0 = np.full(7, 1000) + + def objfun(x): + value = f(x) + values.append(value) + return value + + # Look for an interesting test case. + # Request a maxfun that stops at a particularly bad function + # evaluation somewhere between 100 and 300 evaluations. + low, medium, high = 30, 100, 300 + optimize.fmin_l_bfgs_b(objfun, x0, fprime=g, maxfun=high) + v, k = max((y, i) for i, y in enumerate(values[medium:])) + maxfun = medium + k + # If the minimization strategy is reasonable, + # the minimize() result should not be worse than the best + # of the first 30 function evaluations. + target = min(values[:low]) + xmin, fmin, d = optimize.fmin_l_bfgs_b(f, x0, fprime=g, maxfun=maxfun) + assert_array_less(fmin, target) + + def test_custom(self): + # This function comes from the documentation example. + def custmin(fun, x0, args=(), maxfev=None, stepsize=0.1, + maxiter=100, callback=None, **options): + bestx = x0 + besty = fun(x0) + funcalls = 1 + niter = 0 + improved = True + stop = False + + while improved and not stop and niter < maxiter: + improved = False + niter += 1 + for dim in range(np.size(x0)): + for s in [bestx[dim] - stepsize, bestx[dim] + stepsize]: + testx = np.copy(bestx) + testx[dim] = s + testy = fun(testx, *args) + funcalls += 1 + if testy < besty: + besty = testy + bestx = testx + improved = True + if callback is not None: + callback(bestx) + if maxfev is not None and funcalls >= maxfev: + stop = True + break + + return optimize.OptimizeResult(fun=besty, x=bestx, nit=niter, + nfev=funcalls, success=(niter > 1)) + + x0 = [1.35, 0.9, 0.8, 1.1, 1.2] + res = optimize.minimize(optimize.rosen, x0, method=custmin, + options=dict(stepsize=0.05)) + assert_allclose(res.x, 1.0, rtol=1e-4, atol=1e-4) + + def test_gh10771(self): + # check that minimize passes bounds and constraints to a custom + # minimizer without altering them. + bounds = [(-2, 2), (0, 3)] + constraints = 'constraints' + + def custmin(fun, x0, **options): + assert options['bounds'] is bounds + assert options['constraints'] is constraints + return optimize.OptimizeResult() + + x0 = [1, 1] + optimize.minimize(optimize.rosen, x0, method=custmin, + bounds=bounds, constraints=constraints) + + def test_minimize_tol_parameter(self): + # Check that the minimize() tol= argument does something + def func(z): + x, y = z + return x**2*y**2 + x**4 + 1 + + def dfunc(z): + x, y = z + return np.array([2*x*y**2 + 4*x**3, 2*x**2*y]) + + for method in ['nelder-mead', 'powell', 'cg', 'bfgs', + 'newton-cg', 'l-bfgs-b', 'tnc', + 'cobyla', 'cobyqa', 'slsqp']: + if method in ('nelder-mead', 'powell', 'cobyla', 'cobyqa'): + jac = None + else: + jac = dfunc + + sol1 = optimize.minimize(func, [2, 2], jac=jac, tol=1e-10, + method=method) + sol2 = optimize.minimize(func, [2, 2], jac=jac, tol=1.0, + method=method) + assert func(sol1.x) < func(sol2.x), \ + f"{method}: {func(sol1.x)} vs. {func(sol2.x)}" + + @pytest.mark.fail_slow(10) + @pytest.mark.filterwarnings('ignore::UserWarning') + @pytest.mark.filterwarnings('ignore::RuntimeWarning') # See gh-18547 + @pytest.mark.parametrize('method', + ['fmin', 'fmin_powell', 'fmin_cg', 'fmin_bfgs', + 'fmin_ncg', 'fmin_l_bfgs_b', 'fmin_tnc', + 'fmin_slsqp'] + MINIMIZE_METHODS) + def test_minimize_callback_copies_array(self, method): + # Check that arrays passed to callbacks are not modified + # inplace by the optimizer afterward + + if method in ('fmin_tnc', 'fmin_l_bfgs_b'): + def func(x): + return optimize.rosen(x), optimize.rosen_der(x) + else: + func = optimize.rosen + jac = optimize.rosen_der + hess = optimize.rosen_hess + + x0 = np.zeros(10) + + # Set options + kwargs = {} + if method.startswith('fmin'): + routine = getattr(optimize, method) + if method == 'fmin_slsqp': + kwargs['iter'] = 5 + elif method == 'fmin_tnc': + kwargs['maxfun'] = 100 + elif method in ('fmin', 'fmin_powell'): + kwargs['maxiter'] = 3500 + else: + kwargs['maxiter'] = 5 + else: + def routine(*a, **kw): + kw['method'] = method + return optimize.minimize(*a, **kw) + + if method == 'tnc': + kwargs['options'] = dict(maxfun=100) + else: + kwargs['options'] = dict(maxiter=5) + + if method in ('fmin_ncg',): + kwargs['fprime'] = jac + elif method in ('newton-cg',): + kwargs['jac'] = jac + elif method in ('trust-krylov', 'trust-exact', 'trust-ncg', 'dogleg', + 'trust-constr'): + kwargs['jac'] = jac + kwargs['hess'] = hess + + # Run with callback + results = [] + + def callback(x, *args, **kwargs): + assert not isinstance(x, optimize.OptimizeResult) + results.append((x, np.copy(x))) + + routine(func, x0, callback=callback, **kwargs) + + # Check returned arrays coincide with their copies + # and have no memory overlap + assert len(results) > 2 + assert all(np.all(x == y) for x, y in results) + combinations = itertools.combinations(results, 2) + assert not any(np.may_share_memory(x[0], y[0]) for x, y in combinations) + + @pytest.mark.parametrize('method', ['nelder-mead', 'powell', 'cg', + 'bfgs', 'newton-cg', 'l-bfgs-b', + 'tnc', 'cobyla', 'cobyqa', 'slsqp']) + def test_no_increase(self, method): + # Check that the solver doesn't return a value worse than the + # initial point. + + def func(x): + return (x - 1)**2 + + def bad_grad(x): + # purposefully invalid gradient function, simulates a case + # where line searches start failing + return 2*(x - 1) * (-1) - 2 + + x0 = np.array([2.0]) + f0 = func(x0) + jac = bad_grad + options = dict(maxfun=20) if method == 'tnc' else dict(maxiter=20) + if method in ['nelder-mead', 'powell', 'cobyla', 'cobyqa']: + jac = None + sol = optimize.minimize(func, x0, jac=jac, method=method, + options=options) + assert_equal(func(sol.x), sol.fun) + + if method == 'slsqp': + pytest.xfail("SLSQP returns slightly worse") + assert func(sol.x) <= f0 + + def test_slsqp_respect_bounds(self): + # Regression test for gh-3108 + def f(x): + return sum((x - np.array([1., 2., 3., 4.]))**2) + + def cons(x): + a = np.array([[-1, -1, -1, -1], [-3, -3, -2, -1]]) + return np.concatenate([np.dot(a, x) + np.array([5, 10]), x]) + + x0 = np.array([0.5, 1., 1.5, 2.]) + res = optimize.minimize(f, x0, method='slsqp', + constraints={'type': 'ineq', 'fun': cons}) + assert_allclose(res.x, np.array([0., 2, 5, 8])/3, atol=1e-12) + + @pytest.mark.parametrize('method', ['Nelder-Mead', 'Powell', 'CG', 'BFGS', + 'Newton-CG', 'L-BFGS-B', 'SLSQP', + 'trust-constr', 'dogleg', 'trust-ncg', + 'trust-exact', 'trust-krylov', + 'cobyqa']) + def test_respect_maxiter(self, method): + # Check that the number of iterations equals max_iter, assuming + # convergence doesn't establish before + MAXITER = 4 + + x0 = np.zeros(10) + + sf = ScalarFunction(optimize.rosen, x0, (), optimize.rosen_der, + optimize.rosen_hess, None, None) + + # Set options + kwargs = {'method': method, 'options': dict(maxiter=MAXITER)} + + if method in ('Newton-CG',): + kwargs['jac'] = sf.grad + elif method in ('trust-krylov', 'trust-exact', 'trust-ncg', 'dogleg', + 'trust-constr'): + kwargs['jac'] = sf.grad + kwargs['hess'] = sf.hess + + sol = optimize.minimize(sf.fun, x0, **kwargs) + assert sol.nit == MAXITER + assert sol.nfev >= sf.nfev + if hasattr(sol, 'njev'): + assert sol.njev >= sf.ngev + + # method specific tests + if method == 'SLSQP': + assert sol.status == 9 # Iteration limit reached + elif method == 'cobyqa': + assert sol.status == 6 # Iteration limit reached + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize('method', ['Nelder-Mead', 'Powell', + 'fmin', 'fmin_powell']) + def test_runtime_warning(self, method): + x0 = np.zeros(10) + sf = ScalarFunction(optimize.rosen, x0, (), optimize.rosen_der, + optimize.rosen_hess, None, None) + options = {"maxiter": 1, "disp": True} + with pytest.warns(RuntimeWarning, + match=r'Maximum number of iterations'): + if method.startswith('fmin'): + routine = getattr(optimize, method) + routine(sf.fun, x0, **options) + else: + optimize.minimize(sf.fun, x0, method=method, options=options) + + def test_respect_maxiter_trust_constr_ineq_constraints(self): + # special case of minimization with trust-constr and inequality + # constraints to check maxiter limit is obeyed when using internal + # method 'tr_interior_point' + MAXITER = 4 + f = optimize.rosen + jac = optimize.rosen_der + hess = optimize.rosen_hess + + def fun(x): + return np.array([0.2 * x[0] - 0.4 * x[1] - 0.33 * x[2]]) + cons = ({'type': 'ineq', + 'fun': fun},) + + x0 = np.zeros(10) + sol = optimize.minimize(f, x0, constraints=cons, jac=jac, hess=hess, + method='trust-constr', + options=dict(maxiter=MAXITER)) + assert sol.nit == MAXITER + + def test_minimize_automethod(self): + def f(x): + return x**2 + + def cons(x): + return x - 2 + + x0 = np.array([10.]) + sol_0 = optimize.minimize(f, x0) + sol_1 = optimize.minimize(f, x0, constraints=[{'type': 'ineq', + 'fun': cons}]) + sol_2 = optimize.minimize(f, x0, bounds=[(5, 10)]) + sol_3 = optimize.minimize(f, x0, + constraints=[{'type': 'ineq', 'fun': cons}], + bounds=[(5, 10)]) + sol_4 = optimize.minimize(f, x0, + constraints=[{'type': 'ineq', 'fun': cons}], + bounds=[(1, 10)]) + for sol in [sol_0, sol_1, sol_2, sol_3, sol_4]: + assert sol.success + assert_allclose(sol_0.x, 0, atol=1e-7) + assert_allclose(sol_1.x, 2, atol=1e-7) + assert_allclose(sol_2.x, 5, atol=1e-7) + assert_allclose(sol_3.x, 5, atol=1e-7) + assert_allclose(sol_4.x, 2, atol=1e-7) + + def test_minimize_coerce_args_param(self): + # Regression test for gh-3503 + def Y(x, c): + return np.sum((x-c)**2) + + def dY_dx(x, c=None): + return 2*(x-c) + + c = np.array([3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]) + xinit = np.random.randn(len(c)) + optimize.minimize(Y, xinit, jac=dY_dx, args=(c), method="BFGS") + + def test_initial_step_scaling(self): + # Check that optimizer initial step is not huge even if the + # function and gradients are + + scales = [1e-50, 1, 1e50] + methods = ['CG', 'BFGS', 'L-BFGS-B', 'Newton-CG'] + + def f(x): + if first_step_size[0] is None and x[0] != x0[0]: + first_step_size[0] = abs(x[0] - x0[0]) + if abs(x).max() > 1e4: + raise AssertionError("Optimization stepped far away!") + return scale*(x[0] - 1)**2 + + def g(x): + return np.array([scale*(x[0] - 1)]) + + for scale, method in itertools.product(scales, methods): + if method in ('CG', 'BFGS'): + options = dict(gtol=scale*1e-8) + else: + options = dict() + + if scale < 1e-10 and method in ('L-BFGS-B', 'Newton-CG'): + # XXX: return initial point if they see small gradient + continue + + x0 = [-1.0] + first_step_size = [None] + res = optimize.minimize(f, x0, jac=g, method=method, + options=options) + + err_msg = f"{method} {scale}: {first_step_size}: {res}" + + assert res.success, err_msg + assert_allclose(res.x, [1.0], err_msg=err_msg) + assert res.nit <= 3, err_msg + + if scale > 1e-10: + if method in ('CG', 'BFGS'): + assert_allclose(first_step_size[0], 1.01, err_msg=err_msg) + else: + # Newton-CG and L-BFGS-B use different logic for the first + # step, but are both scaling invariant with step sizes ~ 1 + assert first_step_size[0] > 0.5 and first_step_size[0] < 3, err_msg + else: + # step size has upper bound of ||grad||, so line + # search makes many small steps + pass + + @pytest.mark.parametrize('method', ['nelder-mead', 'powell', 'cg', 'bfgs', + 'newton-cg', 'l-bfgs-b', 'tnc', + 'cobyla', 'cobyqa', 'slsqp', + 'trust-constr', 'dogleg', 'trust-ncg', + 'trust-exact', 'trust-krylov']) + def test_nan_values(self, method, num_parallel_threads): + if num_parallel_threads > 1 and method == 'cobyqa': + pytest.skip('COBYQA does not support concurrent execution') + + # Check nan values result to failed exit status + rng = np.random.RandomState(1234) + + count = [0] + + def func(x): + return np.nan + + def func2(x): + count[0] += 1 + if count[0] > 2: + return np.nan + else: + return rng.rand() + + def grad(x): + return np.array([1.0]) + + def hess(x): + return np.array([[1.0]]) + + x0 = np.array([1.0]) + + needs_grad = method in ('newton-cg', 'trust-krylov', 'trust-exact', + 'trust-ncg', 'dogleg') + needs_hess = method in ('trust-krylov', 'trust-exact', 'trust-ncg', + 'dogleg') + + funcs = [func, func2] + grads = [grad] if needs_grad else [grad, None] + hesss = [hess] if needs_hess else [hess, None] + options = dict(maxfun=20) if method == 'tnc' else dict(maxiter=20) + + with np.errstate(invalid='ignore'), suppress_warnings() as sup: + sup.filter(UserWarning, "delta_grad == 0.*") + sup.filter(RuntimeWarning, ".*does not use Hessian.*") + sup.filter(RuntimeWarning, ".*does not use gradient.*") + + for f, g, h in itertools.product(funcs, grads, hesss): + count = [0] + sol = optimize.minimize(f, x0, jac=g, hess=h, method=method, + options=options) + assert_equal(sol.success, False) + + @pytest.mark.parametrize('method', ['nelder-mead', 'cg', 'bfgs', + 'l-bfgs-b', 'tnc', + 'cobyla', 'cobyqa', 'slsqp', + 'trust-constr', 'dogleg', 'trust-ncg', + 'trust-exact', 'trust-krylov']) + def test_duplicate_evaluations(self, method): + # check that there are no duplicate evaluations for any methods + jac = hess = None + if method in ('newton-cg', 'trust-krylov', 'trust-exact', + 'trust-ncg', 'dogleg'): + jac = self.grad + if method in ('trust-krylov', 'trust-exact', 'trust-ncg', + 'dogleg'): + hess = self.hess + + with np.errstate(invalid='ignore'), suppress_warnings() as sup: + # for trust-constr + sup.filter(UserWarning, "delta_grad == 0.*") + optimize.minimize(self.func, self.startparams, + method=method, jac=jac, hess=hess) + + for i in range(1, len(self.trace.t)): + if np.array_equal(self.trace.t[i - 1], self.trace.t[i]): + raise RuntimeError( + f"Duplicate evaluations made by {method}") + + @pytest.mark.filterwarnings('ignore::RuntimeWarning') + @pytest.mark.parametrize('method', MINIMIZE_METHODS_NEW_CB) + @pytest.mark.parametrize('new_cb_interface', [0, 1, 2]) + def test_callback_stopiteration(self, method, new_cb_interface): + # Check that if callback raises StopIteration, optimization + # terminates with the same result as if iterations were limited + + def f(x): + f.flag = False # check that f isn't called after StopIteration + return optimize.rosen(x) + f.flag = False + + def g(x): + f.flag = False + return optimize.rosen_der(x) + + def h(x): + f.flag = False + return optimize.rosen_hess(x) + + maxiter = 5 + + if new_cb_interface == 1: + def callback_interface(*, intermediate_result): + assert intermediate_result.fun == f(intermediate_result.x) + callback() + elif new_cb_interface == 2: + class Callback: + def __call__(self, intermediate_result: OptimizeResult): + assert intermediate_result.fun == f(intermediate_result.x) + callback() + callback_interface = Callback() + else: + def callback_interface(xk, *args): # type: ignore[misc] + callback() + + def callback(): + callback.i += 1 + callback.flag = False + if callback.i == maxiter: + callback.flag = True + raise StopIteration() + callback.i = 0 + callback.flag = False + + kwargs = {'x0': [1.1]*5, 'method': method, + 'fun': f, 'jac': g, 'hess': h} + + res = optimize.minimize(**kwargs, callback=callback_interface) + if method == 'nelder-mead': + maxiter = maxiter + 1 # nelder-mead counts differently + if method == 'cobyqa': + ref = optimize.minimize(**kwargs, options={'maxfev': maxiter}) + assert res.nfev == ref.nfev == maxiter + else: + ref = optimize.minimize(**kwargs, options={'maxiter': maxiter}) + assert res.nit == ref.nit == maxiter + assert res.fun == ref.fun + assert_equal(res.x, ref.x) + assert res.status == (3 if method in [ + 'trust-constr', + 'cobyqa', + ] else 99) + + def test_ndim_error(self): + msg = "'x0' must only have one dimension." + with assert_raises(ValueError, match=msg): + optimize.minimize(lambda x: x, np.ones((2, 1))) + + @pytest.mark.parametrize('method', ('nelder-mead', 'l-bfgs-b', 'tnc', + 'powell', 'cobyla', 'cobyqa', + 'trust-constr')) + def test_minimize_invalid_bounds(self, method): + def f(x): + return np.sum(x**2) + + bounds = Bounds([1, 2], [3, 4]) + msg = 'The number of bounds is not compatible with the length of `x0`.' + with pytest.raises(ValueError, match=msg): + optimize.minimize(f, x0=[1, 2, 3], method=method, bounds=bounds) + + bounds = Bounds([1, 6, 1], [3, 4, 2]) + msg = 'An upper bound is less than the corresponding lower bound.' + with pytest.raises(ValueError, match=msg): + optimize.minimize(f, x0=[1, 2, 3], method=method, bounds=bounds) + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize('method', ['bfgs', 'cg', 'newton-cg', 'powell']) + def test_minimize_warnings_gh1953(self, method): + # test that minimize methods produce warnings rather than just using + # `print`; see gh-1953. + kwargs = {} if method=='powell' else {'jac': optimize.rosen_der} + warning_type = (RuntimeWarning if method=='powell' + else optimize.OptimizeWarning) + + options = {'disp': True, 'maxiter': 10} + with pytest.warns(warning_type, match='Maximum number'): + optimize.minimize(lambda x: optimize.rosen(x), [0, 0], + method=method, options=options, **kwargs) + + options['disp'] = False + optimize.minimize(lambda x: optimize.rosen(x), [0, 0], + method=method, options=options, **kwargs) + + +@pytest.mark.parametrize( + 'method', + ['l-bfgs-b', 'tnc', 'Powell', 'Nelder-Mead', 'cobyqa'] +) +def test_minimize_with_scalar(method): + # checks that minimize works with a scalar being provided to it. + def f(x): + return np.sum(x ** 2) + + res = optimize.minimize(f, 17, bounds=[(-100, 100)], method=method) + assert res.success + assert_allclose(res.x, [0.0], atol=1e-5) + + +class TestLBFGSBBounds: + def setup_method(self): + self.bounds = ((1, None), (None, None)) + self.solution = (1, 0) + + def fun(self, x, p=2.0): + return 1.0 / p * (x[0]**p + x[1]**p) + + def jac(self, x, p=2.0): + return x**(p - 1) + + def fj(self, x, p=2.0): + return self.fun(x, p), self.jac(x, p) + + def test_l_bfgs_b_bounds(self): + x, f, d = optimize.fmin_l_bfgs_b(self.fun, [0, -1], + fprime=self.jac, + bounds=self.bounds) + assert d['warnflag'] == 0, d['task'] + assert_allclose(x, self.solution, atol=1e-6) + + def test_l_bfgs_b_funjac(self): + # L-BFGS-B with fun and jac combined and extra arguments + x, f, d = optimize.fmin_l_bfgs_b(self.fj, [0, -1], args=(2.0, ), + bounds=self.bounds) + assert d['warnflag'] == 0, d['task'] + assert_allclose(x, self.solution, atol=1e-6) + + def test_minimize_l_bfgs_b_bounds(self): + # Minimize with method='L-BFGS-B' with bounds + res = optimize.minimize(self.fun, [0, -1], method='L-BFGS-B', + jac=self.jac, bounds=self.bounds) + assert res['success'], res['message'] + assert_allclose(res.x, self.solution, atol=1e-6) + + @pytest.mark.parametrize('bounds', [ + ([(10, 1), (1, 10)]), + ([(1, 10), (10, 1)]), + ([(10, 1), (10, 1)]) + ]) + def test_minimize_l_bfgs_b_incorrect_bounds(self, bounds): + with pytest.raises(ValueError, match='.*bound.*'): + optimize.minimize(self.fun, [0, -1], method='L-BFGS-B', + jac=self.jac, bounds=bounds) + + def test_minimize_l_bfgs_b_bounds_FD(self): + # test that initial starting value outside bounds doesn't raise + # an error (done with clipping). + # test all different finite differences combos, with and without args + + jacs = ['2-point', '3-point', None] + argss = [(2.,), ()] + for jac, args in itertools.product(jacs, argss): + res = optimize.minimize(self.fun, [0, -1], args=args, + method='L-BFGS-B', + jac=jac, bounds=self.bounds, + options={'finite_diff_rel_step': None}) + assert res['success'], res['message'] + assert_allclose(res.x, self.solution, atol=1e-6) + + +class TestOptimizeScalar: + def setup_method(self): + self.solution = 1.5 + + def fun(self, x, a=1.5): + """Objective function""" + return (x - a)**2 - 0.8 + + def test_brent(self): + x = optimize.brent(self.fun) + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.brent(self.fun, brack=(-3, -2)) + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.brent(self.fun, full_output=True) + assert_allclose(x[0], self.solution, atol=1e-6) + + x = optimize.brent(self.fun, brack=(-15, -1, 15)) + assert_allclose(x, self.solution, atol=1e-6) + + message = r"\(f\(xb\) < f\(xa\)\) and \(f\(xb\) < f\(xc\)\)" + with pytest.raises(ValueError, match=message): + optimize.brent(self.fun, brack=(-1, 0, 1)) + + message = r"\(xa < xb\) and \(xb < xc\)" + with pytest.raises(ValueError, match=message): + optimize.brent(self.fun, brack=(0, -1, 1)) + + @pytest.mark.filterwarnings('ignore::UserWarning') + def test_golden(self): + x = optimize.golden(self.fun) + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.golden(self.fun, brack=(-3, -2)) + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.golden(self.fun, full_output=True) + assert_allclose(x[0], self.solution, atol=1e-6) + + x = optimize.golden(self.fun, brack=(-15, -1, 15)) + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.golden(self.fun, tol=0) + assert_allclose(x, self.solution) + + maxiter_test_cases = [0, 1, 5] + for maxiter in maxiter_test_cases: + x0 = optimize.golden(self.fun, maxiter=0, full_output=True) + x = optimize.golden(self.fun, maxiter=maxiter, full_output=True) + nfev0, nfev = x0[2], x[2] + assert_equal(nfev - nfev0, maxiter) + + message = r"\(f\(xb\) < f\(xa\)\) and \(f\(xb\) < f\(xc\)\)" + with pytest.raises(ValueError, match=message): + optimize.golden(self.fun, brack=(-1, 0, 1)) + + message = r"\(xa < xb\) and \(xb < xc\)" + with pytest.raises(ValueError, match=message): + optimize.golden(self.fun, brack=(0, -1, 1)) + + def test_fminbound(self): + x = optimize.fminbound(self.fun, 0, 1) + assert_allclose(x, 1, atol=1e-4) + + x = optimize.fminbound(self.fun, 1, 5) + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.fminbound(self.fun, np.array([1]), np.array([5])) + assert_allclose(x, self.solution, atol=1e-6) + assert_raises(ValueError, optimize.fminbound, self.fun, 5, 1) + + def test_fminbound_scalar(self): + with pytest.raises(ValueError, match='.*must be finite scalars.*'): + optimize.fminbound(self.fun, np.zeros((1, 2)), 1) + + x = optimize.fminbound(self.fun, 1, np.array(5)) + assert_allclose(x, self.solution, atol=1e-6) + + def test_gh11207(self): + def fun(x): + return x**2 + optimize.fminbound(fun, 0, 0) + + def test_minimize_scalar(self): + # combine all tests above for the minimize_scalar wrapper + x = optimize.minimize_scalar(self.fun).x + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.minimize_scalar(self.fun, method='Brent') + assert x.success + + x = optimize.minimize_scalar(self.fun, method='Brent', + options=dict(maxiter=3)) + assert not x.success + + x = optimize.minimize_scalar(self.fun, bracket=(-3, -2), + args=(1.5, ), method='Brent').x + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.minimize_scalar(self.fun, method='Brent', + args=(1.5,)).x + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.minimize_scalar(self.fun, bracket=(-15, -1, 15), + args=(1.5, ), method='Brent').x + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.minimize_scalar(self.fun, bracket=(-3, -2), + args=(1.5, ), method='golden').x + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.minimize_scalar(self.fun, method='golden', + args=(1.5,)).x + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.minimize_scalar(self.fun, bracket=(-15, -1, 15), + args=(1.5, ), method='golden').x + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.minimize_scalar(self.fun, bounds=(0, 1), args=(1.5,), + method='Bounded').x + assert_allclose(x, 1, atol=1e-4) + + x = optimize.minimize_scalar(self.fun, bounds=(1, 5), args=(1.5, ), + method='bounded').x + assert_allclose(x, self.solution, atol=1e-6) + + x = optimize.minimize_scalar(self.fun, bounds=(np.array([1]), + np.array([5])), + args=(np.array([1.5]), ), + method='bounded').x + assert_allclose(x, self.solution, atol=1e-6) + + assert_raises(ValueError, optimize.minimize_scalar, self.fun, + bounds=(5, 1), method='bounded', args=(1.5, )) + + assert_raises(ValueError, optimize.minimize_scalar, self.fun, + bounds=(np.zeros(2), 1), method='bounded', args=(1.5, )) + + x = optimize.minimize_scalar(self.fun, bounds=(1, np.array(5)), + method='bounded').x + assert_allclose(x, self.solution, atol=1e-6) + + def test_minimize_scalar_custom(self): + # This function comes from the documentation example. + def custmin(fun, bracket, args=(), maxfev=None, stepsize=0.1, + maxiter=100, callback=None, **options): + bestx = (bracket[1] + bracket[0]) / 2.0 + besty = fun(bestx) + funcalls = 1 + niter = 0 + improved = True + stop = False + + while improved and not stop and niter < maxiter: + improved = False + niter += 1 + for testx in [bestx - stepsize, bestx + stepsize]: + testy = fun(testx, *args) + funcalls += 1 + if testy < besty: + besty = testy + bestx = testx + improved = True + if callback is not None: + callback(bestx) + if maxfev is not None and funcalls >= maxfev: + stop = True + break + + return optimize.OptimizeResult(fun=besty, x=bestx, nit=niter, + nfev=funcalls, success=(niter > 1)) + + res = optimize.minimize_scalar(self.fun, bracket=(0, 4), + method=custmin, + options=dict(stepsize=0.05)) + assert_allclose(res.x, self.solution, atol=1e-6) + + def test_minimize_scalar_coerce_args_param(self): + # Regression test for gh-3503 + optimize.minimize_scalar(self.fun, args=1.5) + + @pytest.mark.parametrize('method', ['brent', 'bounded', 'golden']) + def test_disp(self, method): + # test that all minimize_scalar methods accept a disp option. + for disp in [0, 1, 2, 3]: + optimize.minimize_scalar(self.fun, options={"disp": disp}) + + @pytest.mark.parametrize('method', ['brent', 'bounded', 'golden']) + def test_result_attributes(self, method): + kwargs = {"bounds": [-10, 10]} if method == 'bounded' else {} + result = optimize.minimize_scalar(self.fun, method=method, **kwargs) + assert hasattr(result, "x") + assert hasattr(result, "success") + assert hasattr(result, "message") + assert hasattr(result, "fun") + assert hasattr(result, "nfev") + assert hasattr(result, "nit") + + @pytest.mark.filterwarnings('ignore::UserWarning') + @pytest.mark.parametrize('method', ['brent', 'bounded', 'golden']) + def test_nan_values(self, method): + # Check nan values result to failed exit status + np.random.seed(1234) + + count = [0] + + def func(x): + count[0] += 1 + if count[0] > 4: + return np.nan + else: + return x**2 + 0.1 * np.sin(x) + + bracket = (-1, 0, 1) + bounds = (-1, 1) + + with np.errstate(invalid='ignore'), suppress_warnings() as sup: + sup.filter(UserWarning, "delta_grad == 0.*") + sup.filter(RuntimeWarning, ".*does not use Hessian.*") + sup.filter(RuntimeWarning, ".*does not use gradient.*") + + count = [0] + + kwargs = {"bounds": bounds} if method == 'bounded' else {} + sol = optimize.minimize_scalar(func, bracket=bracket, + **kwargs, method=method, + options=dict(maxiter=20)) + assert_equal(sol.success, False) + + def test_minimize_scalar_defaults_gh10911(self): + # Previously, bounds were silently ignored unless `method='bounds'` + # was chosen. See gh-10911. Check that this is no longer the case. + def f(x): + return x**2 + + res = optimize.minimize_scalar(f) + assert_allclose(res.x, 0, atol=1e-8) + + res = optimize.minimize_scalar(f, bounds=(1, 100), + options={'xatol': 1e-10}) + assert_allclose(res.x, 1) + + def test_minimize_non_finite_bounds_gh10911(self): + # Previously, minimize_scalar misbehaved with infinite bounds. + # See gh-10911. Check that it now raises an error, instead. + msg = "Optimization bounds must be finite scalars." + with pytest.raises(ValueError, match=msg): + optimize.minimize_scalar(np.sin, bounds=(1, np.inf)) + with pytest.raises(ValueError, match=msg): + optimize.minimize_scalar(np.sin, bounds=(np.nan, 1)) + + @pytest.mark.parametrize("method", ['brent', 'golden']) + def test_minimize_unbounded_method_with_bounds_gh10911(self, method): + # Previously, `bounds` were silently ignored when `method='brent'` or + # `method='golden'`. See gh-10911. Check that error is now raised. + msg = "Use of `bounds` is incompatible with..." + with pytest.raises(ValueError, match=msg): + optimize.minimize_scalar(np.sin, method=method, bounds=(1, 2)) + + @pytest.mark.filterwarnings('ignore::RuntimeWarning') + @pytest.mark.parametrize("method", MINIMIZE_SCALAR_METHODS) + @pytest.mark.parametrize("tol", [1, 1e-6]) + @pytest.mark.parametrize("fshape", [(), (1,), (1, 1)]) + def test_minimize_scalar_dimensionality_gh16196(self, method, tol, fshape): + # gh-16196 reported that the output shape of `minimize_scalar` was not + # consistent when an objective function returned an array. Check that + # `res.fun` and `res.x` are now consistent. + def f(x): + return np.array(x**4).reshape(fshape) + + a, b = -0.1, 0.2 + kwargs = (dict(bracket=(a, b)) if method != "bounded" + else dict(bounds=(a, b))) + kwargs.update(dict(method=method, tol=tol)) + + res = optimize.minimize_scalar(f, **kwargs) + assert res.x.shape == res.fun.shape == f(res.x).shape == fshape + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize('method', ['bounded', 'brent', 'golden']) + def test_minimize_scalar_warnings_gh1953(self, method): + # test that minimize_scalar methods produce warnings rather than just + # using `print`; see gh-1953. + def f(x): + return (x - 1)**2 + + kwargs = {} + kwd = 'bounds' if method == 'bounded' else 'bracket' + kwargs[kwd] = [-2, 10] + + options = {'disp': True, 'maxiter': 3} + with pytest.warns(optimize.OptimizeWarning, match='Maximum number'): + optimize.minimize_scalar(f, method=method, options=options, + **kwargs) + + options['disp'] = False + optimize.minimize_scalar(f, method=method, options=options, **kwargs) + + +class TestBracket: + + @pytest.mark.filterwarnings('ignore::RuntimeWarning') + def test_errors_and_status_false(self): + # Check that `bracket` raises the errors it is supposed to + def f(x): # gh-14858 + return x**2 if ((-1 < x) & (x < 1)) else 100.0 + + message = "The algorithm terminated without finding a valid bracket." + with pytest.raises(RuntimeError, match=message): + optimize.bracket(f, -1, 1) + with pytest.raises(RuntimeError, match=message): + optimize.bracket(f, -1, np.inf) + with pytest.raises(RuntimeError, match=message): + optimize.brent(f, brack=(-1, 1)) + with pytest.raises(RuntimeError, match=message): + optimize.golden(f, brack=(-1, 1)) + + def f(x): # gh-5899 + return -5 * x**5 + 4 * x**4 - 12 * x**3 + 11 * x**2 - 2 * x + 1 + + message = "No valid bracket was found before the iteration limit..." + with pytest.raises(RuntimeError, match=message): + optimize.bracket(f, -0.5, 0.5, maxiter=10) + + @pytest.mark.parametrize('method', ('brent', 'golden')) + def test_minimize_scalar_success_false(self, method): + # Check that status information from `bracket` gets to minimize_scalar + def f(x): # gh-14858 + return x**2 if ((-1 < x) & (x < 1)) else 100.0 + + message = "The algorithm terminated without finding a valid bracket." + + res = optimize.minimize_scalar(f, bracket=(-1, 1), method=method) + assert not res.success + assert message in res.message + assert res.nfev == 3 + assert res.nit == 0 + assert res.fun == 100 + + +def test_brent_negative_tolerance(): + assert_raises(ValueError, optimize.brent, np.cos, tol=-.01) + + +class TestNewtonCg: + def test_rosenbrock(self): + x0 = np.array([-1.2, 1.0]) + sol = optimize.minimize(optimize.rosen, x0, + jac=optimize.rosen_der, + hess=optimize.rosen_hess, + tol=1e-5, + method='Newton-CG') + assert sol.success, sol.message + assert_allclose(sol.x, np.array([1, 1]), rtol=1e-4) + + def test_himmelblau(self): + x0 = np.array(himmelblau_x0) + sol = optimize.minimize(himmelblau, + x0, + jac=himmelblau_grad, + hess=himmelblau_hess, + method='Newton-CG', + tol=1e-6) + assert sol.success, sol.message + assert_allclose(sol.x, himmelblau_xopt, rtol=1e-4) + assert_allclose(sol.fun, himmelblau_min, atol=1e-4) + + def test_finite_difference(self): + x0 = np.array([-1.2, 1.0]) + sol = optimize.minimize(optimize.rosen, x0, + jac=optimize.rosen_der, + hess='2-point', + tol=1e-5, + method='Newton-CG') + assert sol.success, sol.message + assert_allclose(sol.x, np.array([1, 1]), rtol=1e-4) + + def test_hessian_update_strategy(self): + x0 = np.array([-1.2, 1.0]) + sol = optimize.minimize(optimize.rosen, x0, + jac=optimize.rosen_der, + hess=optimize.BFGS(), + tol=1e-5, + method='Newton-CG') + assert sol.success, sol.message + assert_allclose(sol.x, np.array([1, 1]), rtol=1e-4) + + +def test_line_for_search(): + # _line_for_search is only used in _linesearch_powell, which is also + # tested below. Thus there are more tests of _line_for_search in the + # test_linesearch_powell_bounded function. + + line_for_search = optimize._optimize._line_for_search + # args are x0, alpha, lower_bound, upper_bound + # returns lmin, lmax + + lower_bound = np.array([-5.3, -1, -1.5, -3]) + upper_bound = np.array([1.9, 1, 2.8, 3]) + + # test when starting in the bounds + x0 = np.array([0., 0, 0, 0]) + # and when starting outside of the bounds + x1 = np.array([0., 2, -3, 0]) + + all_tests = ( + (x0, np.array([1., 0, 0, 0]), -5.3, 1.9), + (x0, np.array([0., 1, 0, 0]), -1, 1), + (x0, np.array([0., 0, 1, 0]), -1.5, 2.8), + (x0, np.array([0., 0, 0, 1]), -3, 3), + (x0, np.array([1., 1, 0, 0]), -1, 1), + (x0, np.array([1., 0, -1, 2]), -1.5, 1.5), + (x0, np.array([2., 0, -1, 2]), -1.5, 0.95), + (x1, np.array([1., 0, 0, 0]), -5.3, 1.9), + (x1, np.array([0., 1, 0, 0]), -3, -1), + (x1, np.array([0., 0, 1, 0]), 1.5, 5.8), + (x1, np.array([0., 0, 0, 1]), -3, 3), + (x1, np.array([1., 1, 0, 0]), -3, -1), + (x1, np.array([1., 0, -1, 0]), -5.3, -1.5), + ) + + for x, alpha, lmin, lmax in all_tests: + mi, ma = line_for_search(x, alpha, lower_bound, upper_bound) + assert_allclose(mi, lmin, atol=1e-6) + assert_allclose(ma, lmax, atol=1e-6) + + # now with infinite bounds + lower_bound = np.array([-np.inf, -1, -np.inf, -3]) + upper_bound = np.array([np.inf, 1, 2.8, np.inf]) + + all_tests = ( + (x0, np.array([1., 0, 0, 0]), -np.inf, np.inf), + (x0, np.array([0., 1, 0, 0]), -1, 1), + (x0, np.array([0., 0, 1, 0]), -np.inf, 2.8), + (x0, np.array([0., 0, 0, 1]), -3, np.inf), + (x0, np.array([1., 1, 0, 0]), -1, 1), + (x0, np.array([1., 0, -1, 2]), -1.5, np.inf), + (x1, np.array([1., 0, 0, 0]), -np.inf, np.inf), + (x1, np.array([0., 1, 0, 0]), -3, -1), + (x1, np.array([0., 0, 1, 0]), -np.inf, 5.8), + (x1, np.array([0., 0, 0, 1]), -3, np.inf), + (x1, np.array([1., 1, 0, 0]), -3, -1), + (x1, np.array([1., 0, -1, 0]), -5.8, np.inf), + ) + + for x, alpha, lmin, lmax in all_tests: + mi, ma = line_for_search(x, alpha, lower_bound, upper_bound) + assert_allclose(mi, lmin, atol=1e-6) + assert_allclose(ma, lmax, atol=1e-6) + + +def test_linesearch_powell(): + # helper function in optimize.py, not a public function. + linesearch_powell = optimize._optimize._linesearch_powell + # args are func, p, xi, fval, lower_bound=None, upper_bound=None, tol=1e-3 + # returns new_fval, p + direction, direction + def func(x): + return np.sum((x - np.array([-1.0, 2.0, 1.5, -0.4])) ** 2) + p0 = np.array([0., 0, 0, 0]) + fval = func(p0) + lower_bound = np.array([-np.inf] * 4) + upper_bound = np.array([np.inf] * 4) + + all_tests = ( + (np.array([1., 0, 0, 0]), -1), + (np.array([0., 1, 0, 0]), 2), + (np.array([0., 0, 1, 0]), 1.5), + (np.array([0., 0, 0, 1]), -.4), + (np.array([-1., 0, 1, 0]), 1.25), + (np.array([0., 0, 1, 1]), .55), + (np.array([2., 0, -1, 1]), -.65), + ) + + for xi, l in all_tests: + f, p, direction = linesearch_powell(func, p0, xi, + fval=fval, tol=1e-5) + assert_allclose(f, func(l * xi), atol=1e-6) + assert_allclose(p, l * xi, atol=1e-6) + assert_allclose(direction, l * xi, atol=1e-6) + + f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5, + lower_bound=lower_bound, + upper_bound=upper_bound, + fval=fval) + assert_allclose(f, func(l * xi), atol=1e-6) + assert_allclose(p, l * xi, atol=1e-6) + assert_allclose(direction, l * xi, atol=1e-6) + + +def test_linesearch_powell_bounded(): + # helper function in optimize.py, not a public function. + linesearch_powell = optimize._optimize._linesearch_powell + # args are func, p, xi, fval, lower_bound=None, upper_bound=None, tol=1e-3 + # returns new_fval, p+direction, direction + def func(x): + return np.sum((x - np.array([-1.0, 2.0, 1.5, -0.4])) ** 2) + p0 = np.array([0., 0, 0, 0]) + fval = func(p0) + + # first choose bounds such that the same tests from + # test_linesearch_powell should pass. + lower_bound = np.array([-2.]*4) + upper_bound = np.array([2.]*4) + + all_tests = ( + (np.array([1., 0, 0, 0]), -1), + (np.array([0., 1, 0, 0]), 2), + (np.array([0., 0, 1, 0]), 1.5), + (np.array([0., 0, 0, 1]), -.4), + (np.array([-1., 0, 1, 0]), 1.25), + (np.array([0., 0, 1, 1]), .55), + (np.array([2., 0, -1, 1]), -.65), + ) + + for xi, l in all_tests: + f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5, + lower_bound=lower_bound, + upper_bound=upper_bound, + fval=fval) + assert_allclose(f, func(l * xi), atol=1e-6) + assert_allclose(p, l * xi, atol=1e-6) + assert_allclose(direction, l * xi, atol=1e-6) + + # now choose bounds such that unbounded vs bounded gives different results + lower_bound = np.array([-.3]*3 + [-1]) + upper_bound = np.array([.45]*3 + [.9]) + + all_tests = ( + (np.array([1., 0, 0, 0]), -.3), + (np.array([0., 1, 0, 0]), .45), + (np.array([0., 0, 1, 0]), .45), + (np.array([0., 0, 0, 1]), -.4), + (np.array([-1., 0, 1, 0]), .3), + (np.array([0., 0, 1, 1]), .45), + (np.array([2., 0, -1, 1]), -.15), + ) + + for xi, l in all_tests: + f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5, + lower_bound=lower_bound, + upper_bound=upper_bound, + fval=fval) + assert_allclose(f, func(l * xi), atol=1e-6) + assert_allclose(p, l * xi, atol=1e-6) + assert_allclose(direction, l * xi, atol=1e-6) + + # now choose as above but start outside the bounds + p0 = np.array([-1., 0, 0, 2]) + fval = func(p0) + + all_tests = ( + (np.array([1., 0, 0, 0]), .7), + (np.array([0., 1, 0, 0]), .45), + (np.array([0., 0, 1, 0]), .45), + (np.array([0., 0, 0, 1]), -2.4), + ) + + for xi, l in all_tests: + f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5, + lower_bound=lower_bound, + upper_bound=upper_bound, + fval=fval) + assert_allclose(f, func(p0 + l * xi), atol=1e-6) + assert_allclose(p, p0 + l * xi, atol=1e-6) + assert_allclose(direction, l * xi, atol=1e-6) + + # now mix in inf + p0 = np.array([0., 0, 0, 0]) + fval = func(p0) + + # now choose bounds that mix inf + lower_bound = np.array([-.3, -np.inf, -np.inf, -1]) + upper_bound = np.array([np.inf, .45, np.inf, .9]) + + all_tests = ( + (np.array([1., 0, 0, 0]), -.3), + (np.array([0., 1, 0, 0]), .45), + (np.array([0., 0, 1, 0]), 1.5), + (np.array([0., 0, 0, 1]), -.4), + (np.array([-1., 0, 1, 0]), .3), + (np.array([0., 0, 1, 1]), .55), + (np.array([2., 0, -1, 1]), -.15), + ) + + for xi, l in all_tests: + f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5, + lower_bound=lower_bound, + upper_bound=upper_bound, + fval=fval) + assert_allclose(f, func(l * xi), atol=1e-6) + assert_allclose(p, l * xi, atol=1e-6) + assert_allclose(direction, l * xi, atol=1e-6) + + # now choose as above but start outside the bounds + p0 = np.array([-1., 0, 0, 2]) + fval = func(p0) + + all_tests = ( + (np.array([1., 0, 0, 0]), .7), + (np.array([0., 1, 0, 0]), .45), + (np.array([0., 0, 1, 0]), 1.5), + (np.array([0., 0, 0, 1]), -2.4), + ) + + for xi, l in all_tests: + f, p, direction = linesearch_powell(func, p0, xi, tol=1e-5, + lower_bound=lower_bound, + upper_bound=upper_bound, + fval=fval) + assert_allclose(f, func(p0 + l * xi), atol=1e-6) + assert_allclose(p, p0 + l * xi, atol=1e-6) + assert_allclose(direction, l * xi, atol=1e-6) + + +def test_powell_limits(): + # gh15342 - powell was going outside bounds for some function evaluations. + bounds = optimize.Bounds([0, 0], [0.6, 20]) + + def fun(x): + a, b = x + assert (x >= bounds.lb).all() and (x <= bounds.ub).all() + return a ** 2 + b ** 2 + + optimize.minimize(fun, x0=[0.6, 20], method='Powell', bounds=bounds) + + # Another test from the original report - gh-13411 + bounds = optimize.Bounds(lb=[0,], ub=[1,], keep_feasible=[True,]) + + def func(x): + assert x >= 0 and x <= 1 + return np.exp(x) + + optimize.minimize(fun=func, x0=[0.5], method='powell', bounds=bounds) + + +def test_powell_output(): + funs = [rosen, lambda x: np.array(rosen(x)), lambda x: np.array([rosen(x)])] + for fun in funs: + res = optimize.minimize(fun, x0=[0.6, 20], method='Powell') + assert np.isscalar(res.fun) + + +@array_api_compatible +class TestRosen: + def test_rosen(self, xp): + # integer input should be promoted to the default floating type + x = xp.asarray([1, 1, 1]) + xp_assert_equal(optimize.rosen(x), + xp.asarray(0.)) + + @skip_xp_backends('jax.numpy', + reasons=["JAX arrays do not support item assignment"]) + @pytest.mark.usefixtures("skip_xp_backends") + def test_rosen_der(self, xp): + x = xp.asarray([1, 1, 1, 1]) + xp_assert_equal(optimize.rosen_der(x), + xp.zeros_like(x, dtype=xp.asarray(1.).dtype)) + + @skip_xp_backends('jax.numpy', + reasons=["JAX arrays do not support item assignment"]) + @pytest.mark.usefixtures("skip_xp_backends") + def test_hess_prod(self, xp): + one = xp.asarray(1.) + xp_test = array_namespace(one) + # Compare rosen_hess(x) times p with rosen_hess_prod(x,p). See gh-1775. + x = xp.asarray([3, 4, 5]) + p = xp.asarray([2, 2, 2]) + hp = optimize.rosen_hess_prod(x, p) + p = xp_test.astype(p, one.dtype) + dothp = optimize.rosen_hess(x) @ p + xp_assert_equal(hp, dothp) + + +def himmelblau(p): + """ + R^2 -> R^1 test function for optimization. The function has four local + minima where himmelblau(xopt) == 0. + """ + x, y = p + a = x*x + y - 11 + b = x + y*y - 7 + return a*a + b*b + + +def himmelblau_grad(p): + x, y = p + return np.array([4*x**3 + 4*x*y - 42*x + 2*y**2 - 14, + 2*x**2 + 4*x*y + 4*y**3 - 26*y - 22]) + + +def himmelblau_hess(p): + x, y = p + return np.array([[12*x**2 + 4*y - 42, 4*x + 4*y], + [4*x + 4*y, 4*x + 12*y**2 - 26]]) + + +himmelblau_x0 = [-0.27, -0.9] +himmelblau_xopt = [3, 2] +himmelblau_min = 0.0 + + +def test_minimize_multiple_constraints(): + # Regression test for gh-4240. + def func(x): + return np.array([25 - 0.2 * x[0] - 0.4 * x[1] - 0.33 * x[2]]) + + def func1(x): + return np.array([x[1]]) + + def func2(x): + return np.array([x[2]]) + + cons = ({'type': 'ineq', 'fun': func}, + {'type': 'ineq', 'fun': func1}, + {'type': 'ineq', 'fun': func2}) + + def f(x): + return -1 * (x[0] + x[1] + x[2]) + + res = optimize.minimize(f, [0, 0, 0], method='SLSQP', constraints=cons) + assert_allclose(res.x, [125, 0, 0], atol=1e-10) + + +class TestOptimizeResultAttributes: + # Test that all minimizers return an OptimizeResult containing + # all the OptimizeResult attributes + def setup_method(self): + self.x0 = [5, 5] + self.func = optimize.rosen + self.jac = optimize.rosen_der + self.hess = optimize.rosen_hess + self.hessp = optimize.rosen_hess_prod + self.bounds = [(0., 10.), (0., 10.)] + + @pytest.mark.fail_slow(2) + def test_attributes_present(self): + attributes = ['nit', 'nfev', 'x', 'success', 'status', 'fun', + 'message'] + skip = {'cobyla': ['nit']} + for method in MINIMIZE_METHODS: + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, + ("Method .+ does not use (gradient|Hessian.*)" + " information")) + res = optimize.minimize(self.func, self.x0, method=method, + jac=self.jac, hess=self.hess, + hessp=self.hessp) + for attribute in attributes: + if method in skip and attribute in skip[method]: + continue + + assert hasattr(res, attribute) + assert attribute in dir(res) + + # gh13001, OptimizeResult.message should be a str + assert isinstance(res.message, str) + + +def f1(z, *params): + x, y = z + a, b, c, d, e, f, g, h, i, j, k, l, scale = params + return (a * x**2 + b * x * y + c * y**2 + d*x + e*y + f) + + +def f2(z, *params): + x, y = z + a, b, c, d, e, f, g, h, i, j, k, l, scale = params + return (-g*np.exp(-((x-h)**2 + (y-i)**2) / scale)) + + +def f3(z, *params): + x, y = z + a, b, c, d, e, f, g, h, i, j, k, l, scale = params + return (-j*np.exp(-((x-k)**2 + (y-l)**2) / scale)) + + +def brute_func(z, *params): + return f1(z, *params) + f2(z, *params) + f3(z, *params) + + +class TestBrute: + # Test the "brute force" method + def setup_method(self): + self.params = (2, 3, 7, 8, 9, 10, 44, -1, 2, 26, 1, -2, 0.5) + self.rranges = (slice(-4, 4, 0.25), slice(-4, 4, 0.25)) + self.solution = np.array([-1.05665192, 1.80834843]) + + def brute_func(self, z, *params): + # an instance method optimizing + return brute_func(z, *params) + + def test_brute(self): + # test fmin + resbrute = optimize.brute(brute_func, self.rranges, args=self.params, + full_output=True, finish=optimize.fmin) + assert_allclose(resbrute[0], self.solution, atol=1e-3) + assert_allclose(resbrute[1], brute_func(self.solution, *self.params), + atol=1e-3) + + # test minimize + resbrute = optimize.brute(brute_func, self.rranges, args=self.params, + full_output=True, + finish=optimize.minimize) + assert_allclose(resbrute[0], self.solution, atol=1e-3) + assert_allclose(resbrute[1], brute_func(self.solution, *self.params), + atol=1e-3) + + # test that brute can optimize an instance method (the other tests use + # a non-class based function + resbrute = optimize.brute(self.brute_func, self.rranges, + args=self.params, full_output=True, + finish=optimize.minimize) + assert_allclose(resbrute[0], self.solution, atol=1e-3) + + def test_1D(self): + # test that for a 1-D problem the test function is passed an array, + # not a scalar. + def f(x): + assert len(x.shape) == 1 + assert x.shape[0] == 1 + return x ** 2 + + optimize.brute(f, [(-1, 1)], Ns=3, finish=None) + + @pytest.mark.fail_slow(10) + def test_workers(self): + # check that parallel evaluation works + resbrute = optimize.brute(brute_func, self.rranges, args=self.params, + full_output=True, finish=None) + + resbrute1 = optimize.brute(brute_func, self.rranges, args=self.params, + full_output=True, finish=None, workers=2) + + assert_allclose(resbrute1[-1], resbrute[-1]) + assert_allclose(resbrute1[0], resbrute[0]) + + @pytest.mark.thread_unsafe + def test_runtime_warning(self, capsys): + rng = np.random.default_rng(1234) + + def func(z, *params): + return rng.random(1) * 1000 # never converged problem + + msg = "final optimization did not succeed.*|Maximum number of function eval.*" + with pytest.warns(RuntimeWarning, match=msg): + optimize.brute(func, self.rranges, args=self.params, disp=True) + + def test_coerce_args_param(self): + # optimize.brute should coerce non-iterable args to a tuple. + def f(x, *args): + return x ** args[0] + + resbrute = optimize.brute(f, (slice(-4, 4, .25),), args=2) + assert_allclose(resbrute, 0) + + +@pytest.mark.thread_unsafe +@pytest.mark.fail_slow(20) +def test_cobyla_threadsafe(): + + # Verify that cobyla is threadsafe. Will segfault if it is not. + + import concurrent.futures + import time + + def objective1(x): + time.sleep(0.1) + return x[0]**2 + + def objective2(x): + time.sleep(0.1) + return (x[0]-1)**2 + + min_method = "COBYLA" + + def minimizer1(): + return optimize.minimize(objective1, + [0.0], + method=min_method) + + def minimizer2(): + return optimize.minimize(objective2, + [0.0], + method=min_method) + + with concurrent.futures.ThreadPoolExecutor() as pool: + tasks = [] + tasks.append(pool.submit(minimizer1)) + tasks.append(pool.submit(minimizer2)) + for t in tasks: + t.result() + + +class TestIterationLimits: + # Tests that optimisation does not give up before trying requested + # number of iterations or evaluations. And that it does not succeed + # by exceeding the limits. + def setup_method(self): + self.funcalls = threading.local() + + def slow_func(self, v): + if not hasattr(self.funcalls, 'c'): + self.funcalls.c = 0 + self.funcalls.c += 1 + r, t = np.sqrt(v[0]**2+v[1]**2), np.arctan2(v[0], v[1]) + return np.sin(r*20 + t)+r*0.5 + + @pytest.mark.fail_slow(10) + def test_neldermead_limit(self): + self.check_limits("Nelder-Mead", 200) + + def test_powell_limit(self): + self.check_limits("powell", 1000) + + def check_limits(self, method, default_iters): + for start_v in [[0.1, 0.1], [1, 1], [2, 2]]: + for mfev in [50, 500, 5000]: + self.funcalls.c = 0 + res = optimize.minimize(self.slow_func, start_v, + method=method, + options={"maxfev": mfev}) + assert self.funcalls.c == res["nfev"] + if res["success"]: + assert res["nfev"] < mfev + else: + assert res["nfev"] >= mfev + for mit in [50, 500, 5000]: + res = optimize.minimize(self.slow_func, start_v, + method=method, + options={"maxiter": mit}) + if res["success"]: + assert res["nit"] <= mit + else: + assert res["nit"] >= mit + for mfev, mit in [[50, 50], [5000, 5000], [5000, np.inf]]: + self.funcalls.c = 0 + res = optimize.minimize(self.slow_func, start_v, + method=method, + options={"maxiter": mit, + "maxfev": mfev}) + assert self.funcalls.c == res["nfev"] + if res["success"]: + assert res["nfev"] < mfev and res["nit"] <= mit + else: + assert res["nfev"] >= mfev or res["nit"] >= mit + for mfev, mit in [[np.inf, None], [None, np.inf]]: + self.funcalls.c = 0 + res = optimize.minimize(self.slow_func, start_v, + method=method, + options={"maxiter": mit, + "maxfev": mfev}) + assert self.funcalls.c == res["nfev"] + if res["success"]: + if mfev is None: + assert res["nfev"] < default_iters*2 + else: + assert res["nit"] <= default_iters*2 + else: + assert (res["nfev"] >= default_iters*2 + or res["nit"] >= default_iters*2) + + +def test_result_x_shape_when_len_x_is_one(): + def fun(x): + return x * x + + def jac(x): + return 2. * x + + def hess(x): + return np.array([[2.]]) + + methods = ['Nelder-Mead', 'Powell', 'CG', 'BFGS', 'L-BFGS-B', 'TNC', + 'COBYLA', 'COBYQA', 'SLSQP'] + for method in methods: + res = optimize.minimize(fun, np.array([0.1]), method=method) + assert res.x.shape == (1,) + + # use jac + hess + methods = ['trust-constr', 'dogleg', 'trust-ncg', 'trust-exact', + 'trust-krylov', 'Newton-CG'] + for method in methods: + res = optimize.minimize(fun, np.array([0.1]), method=method, jac=jac, + hess=hess) + assert res.x.shape == (1,) + + +class FunctionWithGradient: + def __init__(self): + self.number_of_calls = threading.local() + + def __call__(self, x): + if not hasattr(self.number_of_calls, 'c'): + self.number_of_calls.c = 0 + self.number_of_calls.c += 1 + return np.sum(x**2), 2 * x + + +@pytest.fixture +def function_with_gradient(): + return FunctionWithGradient() + + +def test_memoize_jac_function_before_gradient(function_with_gradient): + memoized_function = MemoizeJac(function_with_gradient) + + x0 = np.array([1.0, 2.0]) + assert_allclose(memoized_function(x0), 5.0) + assert function_with_gradient.number_of_calls.c == 1 + + assert_allclose(memoized_function.derivative(x0), 2 * x0) + assert function_with_gradient.number_of_calls.c == 1, \ + "function is not recomputed " \ + "if gradient is requested after function value" + + assert_allclose( + memoized_function(2 * x0), 20.0, + err_msg="different input triggers new computation") + assert function_with_gradient.number_of_calls.c == 2, \ + "different input triggers new computation" + + +def test_memoize_jac_gradient_before_function(function_with_gradient): + memoized_function = MemoizeJac(function_with_gradient) + + x0 = np.array([1.0, 2.0]) + assert_allclose(memoized_function.derivative(x0), 2 * x0) + assert function_with_gradient.number_of_calls.c == 1 + + assert_allclose(memoized_function(x0), 5.0) + assert function_with_gradient.number_of_calls.c == 1, \ + "function is not recomputed " \ + "if function value is requested after gradient" + + assert_allclose( + memoized_function.derivative(2 * x0), 4 * x0, + err_msg="different input triggers new computation") + assert function_with_gradient.number_of_calls.c == 2, \ + "different input triggers new computation" + + +def test_memoize_jac_with_bfgs(function_with_gradient): + """ Tests that using MemoizedJac in combination with ScalarFunction + and BFGS does not lead to repeated function evaluations. + Tests changes made in response to GH11868. + """ + memoized_function = MemoizeJac(function_with_gradient) + jac = memoized_function.derivative + hess = optimize.BFGS() + + x0 = np.array([1.0, 0.5]) + scalar_function = ScalarFunction( + memoized_function, x0, (), jac, hess, None, None) + assert function_with_gradient.number_of_calls.c == 1 + + scalar_function.fun(x0 + 0.1) + assert function_with_gradient.number_of_calls.c == 2 + + scalar_function.fun(x0 + 0.2) + assert function_with_gradient.number_of_calls.c == 3 + + +def test_gh12696(): + # Test that optimize doesn't throw warning gh-12696 + with assert_no_warnings(): + optimize.fminbound( + lambda x: np.array([x**2]), -np.pi, np.pi, disp=False) + + +# --- Test minimize with equal upper and lower bounds --- # + +def setup_test_equal_bounds(): + + rng = np.random.RandomState(0) + x0 = rng.rand(4) + lb = np.array([0, 2, -1, -1.0]) + ub = np.array([3, 2, 2, -1.0]) + i_eb = (lb == ub) + + def check_x(x, check_size=True, check_values=True): + if check_size: + assert x.size == 4 + if check_values: + assert_allclose(x[i_eb], lb[i_eb]) + + def func(x): + check_x(x) + return optimize.rosen(x) + + def grad(x): + check_x(x) + return optimize.rosen_der(x) + + def callback(x, *args): + check_x(x) + + def constraint1(x): + check_x(x, check_values=False) + return x[0:1] - 1 + + def jacobian1(x): + check_x(x, check_values=False) + dc = np.zeros_like(x) + dc[0] = 1 + return dc + + def constraint2(x): + check_x(x, check_values=False) + return x[2:3] - 0.5 + + def jacobian2(x): + check_x(x, check_values=False) + dc = np.zeros_like(x) + dc[2] = 1 + return dc + + c1a = NonlinearConstraint(constraint1, -np.inf, 0) + c1b = NonlinearConstraint(constraint1, -np.inf, 0, jacobian1) + c2a = NonlinearConstraint(constraint2, -np.inf, 0) + c2b = NonlinearConstraint(constraint2, -np.inf, 0, jacobian2) + + # test using the three methods that accept bounds, use derivatives, and + # have some trouble when bounds fix variables + methods = ('L-BFGS-B', 'SLSQP', 'TNC') + + # test w/out gradient, w/ gradient, and w/ combined objective/gradient + kwds = ({"fun": func, "jac": False}, + {"fun": func, "jac": grad}, + {"fun": (lambda x: (func(x), grad(x))), + "jac": True}) + + # test with both old- and new-style bounds + bound_types = (lambda lb, ub: list(zip(lb, ub)), + Bounds) + + # Test for many combinations of constraints w/ and w/out jacobian + # Pairs in format: (test constraints, reference constraints) + # (always use analytical jacobian in reference) + constraints = ((None, None), ([], []), + (c1a, c1b), (c2b, c2b), + ([c1b], [c1b]), ([c2a], [c2b]), + ([c1a, c2a], [c1b, c2b]), + ([c1a, c2b], [c1b, c2b]), + ([c1b, c2b], [c1b, c2b])) + + # test with and without callback function + callbacks = (None, callback) + + data = {"methods": methods, "kwds": kwds, "bound_types": bound_types, + "constraints": constraints, "callbacks": callbacks, + "lb": lb, "ub": ub, "x0": x0, "i_eb": i_eb} + + return data + + +eb_data = setup_test_equal_bounds() + + +# This test is about handling fixed variables, not the accuracy of the solvers +@pytest.mark.xfail_on_32bit("Failures due to floating point issues, not logic") +@pytest.mark.xfail(scipy.show_config(mode='dicts')['Compilers']['fortran']['name'] == + "intel-llvm", + reason="Failures due to floating point issues, not logic") +@pytest.mark.parametrize('method', eb_data["methods"]) +@pytest.mark.parametrize('kwds', eb_data["kwds"]) +@pytest.mark.parametrize('bound_type', eb_data["bound_types"]) +@pytest.mark.parametrize('constraints', eb_data["constraints"]) +@pytest.mark.parametrize('callback', eb_data["callbacks"]) +def test_equal_bounds(method, kwds, bound_type, constraints, callback): + """ + Tests that minimizers still work if (bounds.lb == bounds.ub).any() + gh12502 - Divide by zero in Jacobian numerical differentiation when + equality bounds constraints are used + """ + # GH-15051; slightly more skips than necessary; hopefully fixed by GH-14882 + if (platform.machine() == 'aarch64' and method == "TNC" + and kwds["jac"] is False and callback is not None): + pytest.skip('Tolerance violation on aarch') + + lb, ub = eb_data["lb"], eb_data["ub"] + x0, i_eb = eb_data["x0"], eb_data["i_eb"] + + test_constraints, reference_constraints = constraints + if test_constraints and not method == 'SLSQP': + pytest.skip('Only SLSQP supports nonlinear constraints') + # reference constraints always have analytical jacobian + # if test constraints are not the same, we'll need finite differences + fd_needed = (test_constraints != reference_constraints) + + bounds = bound_type(lb, ub) # old- or new-style + + kwds.update({"x0": x0, "method": method, "bounds": bounds, + "constraints": test_constraints, "callback": callback}) + res = optimize.minimize(**kwds) + + expected = optimize.minimize(optimize.rosen, x0, method=method, + jac=optimize.rosen_der, bounds=bounds, + constraints=reference_constraints) + + # compare the output of a solution with FD vs that of an analytic grad + assert res.success + assert_allclose(res.fun, expected.fun, rtol=1.5e-6) + assert_allclose(res.x, expected.x, rtol=5e-4) + + if fd_needed or kwds['jac'] is False: + expected.jac[i_eb] = np.nan + assert res.jac.shape[0] == 4 + assert_allclose(res.jac[i_eb], expected.jac[i_eb], rtol=1e-6) + + if not (kwds['jac'] or test_constraints or isinstance(bounds, Bounds)): + # compare the output to an equivalent FD minimization that doesn't + # need factorization + def fun(x): + new_x = np.array([np.nan, 2, np.nan, -1]) + new_x[[0, 2]] = x + return optimize.rosen(new_x) + + fd_res = optimize.minimize(fun, + x0[[0, 2]], + method=method, + bounds=bounds[::2]) + assert_allclose(res.fun, fd_res.fun) + # TODO this test should really be equivalent to factorized version + # above, down to res.nfev. However, testing found that when TNC is + # called with or without a callback the output is different. The two + # should be the same! This indicates that the TNC callback may be + # mutating something when it shouldn't. + assert_allclose(res.x[[0, 2]], fd_res.x, rtol=2e-6) + + +@pytest.mark.parametrize('method', eb_data["methods"]) +def test_all_bounds_equal(method): + # this only tests methods that have parameters factored out when lb==ub + # it does not test other methods that work with bounds + def f(x, p1=1): + return np.linalg.norm(x) + p1 + + bounds = [(1, 1), (2, 2)] + x0 = (1.0, 3.0) + res = optimize.minimize(f, x0, bounds=bounds, method=method) + assert res.success + assert_allclose(res.fun, f([1.0, 2.0])) + assert res.nfev == 1 + assert res.message == 'All independent variables were fixed by bounds.' + + args = (2,) + res = optimize.minimize(f, x0, bounds=bounds, method=method, args=args) + assert res.success + assert_allclose(res.fun, f([1.0, 2.0], 2)) + + if method.upper() == 'SLSQP': + def con(x): + return np.sum(x) + nlc = NonlinearConstraint(con, -np.inf, 0.0) + res = optimize.minimize( + f, x0, bounds=bounds, method=method, constraints=[nlc] + ) + assert res.success is False + assert_allclose(res.fun, f([1.0, 2.0])) + assert res.nfev == 1 + message = "All independent variables were fixed by bounds, but" + assert res.message.startswith(message) + + nlc = NonlinearConstraint(con, -np.inf, 4) + res = optimize.minimize( + f, x0, bounds=bounds, method=method, constraints=[nlc] + ) + assert res.success is True + assert_allclose(res.fun, f([1.0, 2.0])) + assert res.nfev == 1 + message = "All independent variables were fixed by bounds at values" + assert res.message.startswith(message) + + +def test_eb_constraints(): + # make sure constraint functions aren't overwritten when equal bounds + # are employed, and a parameter is factored out. GH14859 + def f(x): + return x[0]**3 + x[1]**2 + x[2]*x[3] + + def cfun(x): + return x[0] + x[1] + x[2] + x[3] - 40 + + constraints = [{'type': 'ineq', 'fun': cfun}] + + bounds = [(0, 20)] * 4 + bounds[1] = (5, 5) + optimize.minimize( + f, + x0=[1, 2, 3, 4], + method='SLSQP', + bounds=bounds, + constraints=constraints, + ) + assert constraints[0]['fun'] == cfun + + +def test_show_options(): + solver_methods = { + 'minimize': MINIMIZE_METHODS, + 'minimize_scalar': MINIMIZE_SCALAR_METHODS, + 'root': ROOT_METHODS, + 'root_scalar': ROOT_SCALAR_METHODS, + 'linprog': LINPROG_METHODS, + 'quadratic_assignment': QUADRATIC_ASSIGNMENT_METHODS, + } + for solver, methods in solver_methods.items(): + for method in methods: + # testing that `show_options` works without error + show_options(solver, method) + + unknown_solver_method = { + 'minimize': "ekki", # unknown method + 'maximize': "cg", # unknown solver + 'maximize_scalar': "ekki", # unknown solver and method + } + for solver, method in unknown_solver_method.items(): + # testing that `show_options` raises ValueError + assert_raises(ValueError, show_options, solver, method) + + +def test_bounds_with_list(): + # gh13501. Bounds created with lists weren't working for Powell. + bounds = optimize.Bounds(lb=[5., 5.], ub=[10., 10.]) + optimize.minimize( + optimize.rosen, x0=np.array([9, 9]), method='Powell', bounds=bounds + ) + + +def test_x_overwritten_user_function(): + # if the user overwrites the x-array in the user function it's likely + # that the minimizer stops working properly. + # gh13740 + def fquad(x): + a = np.arange(np.size(x)) + x -= a + x *= x + return np.sum(x) + + def fquad_jac(x): + a = np.arange(np.size(x)) + x *= 2 + x -= 2 * a + return x + + def fquad_hess(x): + return np.eye(np.size(x)) * 2.0 + + meth_jac = [ + 'newton-cg', 'dogleg', 'trust-ncg', 'trust-exact', + 'trust-krylov', 'trust-constr' + ] + meth_hess = [ + 'dogleg', 'trust-ncg', 'trust-exact', 'trust-krylov', 'trust-constr' + ] + + x0 = np.ones(5) * 1.5 + + for meth in MINIMIZE_METHODS: + jac = None + hess = None + if meth in meth_jac: + jac = fquad_jac + if meth in meth_hess: + hess = fquad_hess + res = optimize.minimize(fquad, x0, method=meth, jac=jac, hess=hess) + assert_allclose(res.x, np.arange(np.size(x0)), atol=2e-4) + + +class TestGlobalOptimization: + + def test_optimize_result_attributes(self): + def func(x): + return x ** 2 + + # Note that `brute` solver does not return `OptimizeResult` + results = [optimize.basinhopping(func, x0=1), + optimize.differential_evolution(func, [(-4, 4)]), + optimize.shgo(func, [(-4, 4)]), + optimize.dual_annealing(func, [(-4, 4)]), + optimize.direct(func, [(-4, 4)]), + ] + + for result in results: + assert isinstance(result, optimize.OptimizeResult) + assert hasattr(result, "x") + assert hasattr(result, "success") + assert hasattr(result, "message") + assert hasattr(result, "fun") + assert hasattr(result, "nfev") + assert hasattr(result, "nit") + + +def test_approx_fprime(): + # check that approx_fprime (serviced by approx_derivative) works for + # jac and hess + g = optimize.approx_fprime(himmelblau_x0, himmelblau) + assert_allclose(g, himmelblau_grad(himmelblau_x0), rtol=5e-6) + + h = optimize.approx_fprime(himmelblau_x0, himmelblau_grad) + assert_allclose(h, himmelblau_hess(himmelblau_x0), rtol=5e-6) + + +def test_gh12594(): + # gh-12594 reported an error in `_linesearch_powell` and + # `_line_for_search` when `Bounds` was passed lists instead of arrays. + # Check that results are the same whether the inputs are lists or arrays. + + def f(x): + return x[0]**2 + (x[1] - 1)**2 + + bounds = Bounds(lb=[-10, -10], ub=[10, 10]) + res = optimize.minimize(f, x0=(0, 0), method='Powell', bounds=bounds) + bounds = Bounds(lb=np.array([-10, -10]), ub=np.array([10, 10])) + ref = optimize.minimize(f, x0=(0, 0), method='Powell', bounds=bounds) + + assert_allclose(res.fun, ref.fun) + assert_allclose(res.x, ref.x) + + +@pytest.mark.parametrize('method', ['Newton-CG', 'trust-constr']) +@pytest.mark.parametrize('sparse_type', [coo_matrix, csc_matrix, csr_matrix, + coo_array, csr_array, csc_array]) +def test_sparse_hessian(method, sparse_type): + # gh-8792 reported an error for minimization with `newton_cg` when `hess` + # returns a sparse matrix. Check that results are the same whether `hess` + # returns a dense or sparse matrix for optimization methods that accept + # sparse Hessian matrices. + + def sparse_rosen_hess(x): + return sparse_type(rosen_hess(x)) + + x0 = [2., 2.] + + res_sparse = optimize.minimize(rosen, x0, method=method, + jac=rosen_der, hess=sparse_rosen_hess) + res_dense = optimize.minimize(rosen, x0, method=method, + jac=rosen_der, hess=rosen_hess) + + assert_allclose(res_dense.fun, res_sparse.fun) + assert_allclose(res_dense.x, res_sparse.x) + assert res_dense.nfev == res_sparse.nfev + assert res_dense.njev == res_sparse.njev + assert res_dense.nhev == res_sparse.nhev diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_quadratic_assignment.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_quadratic_assignment.py new file mode 100644 index 0000000000000000000000000000000000000000..3f7f26158d8e7ddb33012db4dfc4030661aa75bb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_quadratic_assignment.py @@ -0,0 +1,455 @@ +import pytest +import numpy as np +from numpy.random import default_rng +from scipy.optimize import quadratic_assignment, OptimizeWarning +from scipy.optimize._qap import _calc_score as _score +from numpy.testing import assert_equal, assert_, assert_warns + + +################ +# Common Tests # +################ + +def chr12c(): + A = [ + [0, 90, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [90, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0], + [10, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0], + [0, 23, 0, 0, 0, 88, 0, 0, 0, 0, 0, 0], + [0, 0, 43, 0, 0, 0, 26, 0, 0, 0, 0, 0], + [0, 0, 0, 88, 0, 0, 0, 16, 0, 0, 0, 0], + [0, 0, 0, 0, 26, 0, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 16, 0, 0, 0, 96, 0, 0], + [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 29, 0], + [0, 0, 0, 0, 0, 0, 0, 96, 0, 0, 0, 37], + [0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 37, 0, 0], + ] + B = [ + [0, 36, 54, 26, 59, 72, 9, 34, 79, 17, 46, 95], + [36, 0, 73, 35, 90, 58, 30, 78, 35, 44, 79, 36], + [54, 73, 0, 21, 10, 97, 58, 66, 69, 61, 54, 63], + [26, 35, 21, 0, 93, 12, 46, 40, 37, 48, 68, 85], + [59, 90, 10, 93, 0, 64, 5, 29, 76, 16, 5, 76], + [72, 58, 97, 12, 64, 0, 96, 55, 38, 54, 0, 34], + [9, 30, 58, 46, 5, 96, 0, 83, 35, 11, 56, 37], + [34, 78, 66, 40, 29, 55, 83, 0, 44, 12, 15, 80], + [79, 35, 69, 37, 76, 38, 35, 44, 0, 64, 39, 33], + [17, 44, 61, 48, 16, 54, 11, 12, 64, 0, 70, 86], + [46, 79, 54, 68, 5, 0, 56, 15, 39, 70, 0, 18], + [95, 36, 63, 85, 76, 34, 37, 80, 33, 86, 18, 0], + ] + A, B = np.array(A), np.array(B) + n = A.shape[0] + + opt_perm = np.array([7, 5, 1, 3, 10, 4, 8, 6, 9, 11, 2, 12]) - [1] * n + + return A, B, opt_perm + + +@pytest.mark.filterwarnings("ignore:The NumPy global RNG was seeded by calling") +class QAPCommonTests: + """ + Base class for `quadratic_assignment` tests. + """ + # Test global optima of problem from Umeyama IVB + # https://pcl.sitehost.iu.edu/rgoldsto/papers/weighted%20graph%20match2.pdf + # Graph matching maximum is in the paper + # QAP minimum determined by brute force + def test_accuracy_1(self): + # besides testing accuracy, check that A and B can be lists + rng = np.random.default_rng(4358764578823597324) + + A = [[0, 3, 4, 2], + [0, 0, 1, 2], + [1, 0, 0, 1], + [0, 0, 1, 0]] + + B = [[0, 4, 2, 4], + [0, 0, 1, 0], + [0, 2, 0, 2], + [0, 1, 2, 0]] + + res = quadratic_assignment(A, B, method=self.method, + options={"rng": rng, "maximize": False}) + + assert_equal(res.fun, 10) + assert_equal(res.col_ind, np.array([1, 2, 3, 0])) + + res = quadratic_assignment(A, B, method=self.method, + options={"rng": rng, "maximize": True}) + + if self.method == 'faq': + # Global optimum is 40, but FAQ gets 37 + assert_equal(res.fun, 37) + assert_equal(res.col_ind, np.array([0, 2, 3, 1])) + else: + assert_equal(res.fun, 40) + assert_equal(res.col_ind, np.array([0, 3, 1, 2])) + + quadratic_assignment(A, B, method=self.method, + options={"rng": rng, "maximize": True}) + + # Test global optima of problem from Umeyama IIIB + # https://pcl.sitehost.iu.edu/rgoldsto/papers/weighted%20graph%20match2.pdf + # Graph matching maximum is in the paper + # QAP minimum determined by brute force + def test_accuracy_2(self): + rng = np.random.default_rng(4358764578823597324) + + A = np.array([[0, 5, 8, 6], + [5, 0, 5, 1], + [8, 5, 0, 2], + [6, 1, 2, 0]]) + + B = np.array([[0, 1, 8, 4], + [1, 0, 5, 2], + [8, 5, 0, 5], + [4, 2, 5, 0]]) + + res = quadratic_assignment(A, B, method=self.method, + options={"rng": rng, "maximize": False}) + + if self.method == 'faq': + # Global optimum is 176, but FAQ gets 178 + assert_equal(res.fun, 178) + assert_equal(res.col_ind, np.array([1, 0, 3, 2])) + else: + assert_equal(res.fun, 176) + assert_equal(res.col_ind, np.array([1, 2, 3, 0])) + + res = quadratic_assignment(A, B, method=self.method, + options={"rng": rng, "maximize": True}) + + assert_equal(res.fun, 286) + assert_equal(res.col_ind, np.array([2, 3, 0, 1])) + + def test_accuracy_3(self): + rng = np.random.default_rng(4358764578823597324) + A, B, opt_perm = chr12c() + + # basic minimization + res = quadratic_assignment(A, B, method=self.method, + options={"rng": rng}) + assert_(11156 <= res.fun < 21000) + assert_equal(res.fun, _score(A, B, res.col_ind)) + + # basic maximization + res = quadratic_assignment(A, B, method=self.method, + options={"rng": rng, 'maximize': True}) + assert_(74000 <= res.fun < 85000) + assert_equal(res.fun, _score(A, B, res.col_ind)) + + # check ofv with strictly partial match + seed_cost = np.array([4, 8, 10]) + seed = np.asarray([seed_cost, opt_perm[seed_cost]]).T + res = quadratic_assignment(A, B, method=self.method, + options={'partial_match': seed, "rng": rng}) + assert_(11156 <= res.fun < 21000) + assert_equal(res.col_ind[seed_cost], opt_perm[seed_cost]) + + # check performance when partial match is the global optimum + seed = np.asarray([np.arange(len(A)), opt_perm]).T + res = quadratic_assignment(A, B, method=self.method, + options={'partial_match': seed, "rng": rng}) + assert_equal(res.col_ind, seed[:, 1].T) + assert_equal(res.fun, 11156) + assert_equal(res.nit, 0) + + # check performance with zero sized matrix inputs + empty = np.empty((0, 0)) + res = quadratic_assignment(empty, empty, method=self.method, + options={"rng": rng}) + assert_equal(res.nit, 0) + assert_equal(res.fun, 0) + + @pytest.mark.thread_unsafe + def test_unknown_options(self): + A, B, opt_perm = chr12c() + + def f(): + quadratic_assignment(A, B, method=self.method, + options={"ekki-ekki": True}) + assert_warns(OptimizeWarning, f) + + @pytest.mark.thread_unsafe + def test_deprecation_future_warnings(self): + # May be removed after SPEC-7 transition is complete in SciPy 1.17 + A = np.arange(16).reshape((4, 4)) + B = np.arange(16).reshape((4, 4)) + + with pytest.warns(DeprecationWarning, match="Use of `RandomState`*"): + rng = np.random.RandomState(0) + quadratic_assignment(A, B, method=self.method, + options={"rng": rng, "maximize": False}) + + with pytest.warns(FutureWarning, match="The NumPy global RNG was seeded*"): + np.random.seed(0) + quadratic_assignment(A, B, method=self.method, + options={"maximize": False}) + + with pytest.warns(FutureWarning, match="The behavior when the rng option*"): + quadratic_assignment(A, B, method=self.method, + options={"rng": 0, "maximize": False}) + + +class TestFAQ(QAPCommonTests): + method = "faq" + + def test_options(self): + # cost and distance matrices of QAPLIB instance chr12c + rng = np.random.default_rng(4358764578823597324) + + A, B, opt_perm = chr12c() + n = len(A) + + # check that max_iter is obeying with low input value + res = quadratic_assignment(A, B, options={'maxiter': 5}) + assert_equal(res.nit, 5) + + # test with shuffle + res = quadratic_assignment(A, B, options={'shuffle_input': True}) + assert_(11156 <= res.fun < 21000) + + # test with randomized init + res = quadratic_assignment(A, B, options={'rng': rng, 'P0': "randomized"}) + assert_(11156 <= res.fun < 21000) + + # check with specified P0 + K = np.ones((n, n)) / float(n) + K = _doubly_stochastic(K) + res = quadratic_assignment(A, B, options={'P0': K}) + assert_(11156 <= res.fun < 21000) + + def test_specific_input_validation(self): + + A = np.identity(2) + B = A + + # method is implicitly faq + + # ValueError Checks: making sure single value parameters are of + # correct value + with pytest.raises(ValueError, match="Invalid 'P0' parameter"): + quadratic_assignment(A, B, options={'P0': "random"}) + with pytest.raises( + ValueError, match="'maxiter' must be a positive integer"): + quadratic_assignment(A, B, options={'maxiter': -1}) + with pytest.raises(ValueError, match="'tol' must be a positive float"): + quadratic_assignment(A, B, options={'tol': -1}) + + # TypeError Checks: making sure single value parameters are of + # correct type + with pytest.raises(TypeError): + quadratic_assignment(A, B, options={'maxiter': 1.5}) + + # test P0 matrix input + with pytest.raises( + ValueError, + match="`P0` matrix must have shape m' x m', where m'=n-m"): + quadratic_assignment( + np.identity(4), np.identity(4), + options={'P0': np.ones((3, 3))} + ) + + K = [[0.4, 0.2, 0.3], + [0.3, 0.6, 0.2], + [0.2, 0.2, 0.7]] + # matrix that isn't quite doubly stochastic + with pytest.raises( + ValueError, match="`P0` matrix must be doubly stochastic"): + quadratic_assignment( + np.identity(3), np.identity(3), options={'P0': K} + ) + + +class Test2opt(QAPCommonTests): + method = "2opt" + + def test_deterministic(self): + n = 20 + rng = default_rng(51982908) + A = rng.random(size=(n, n)) + B = rng.random(size=(n, n)) + res1 = quadratic_assignment(A, B, method=self.method, options={'rng': rng}) + + rng = default_rng(51982908) + A = rng.random(size=(n, n)) + B = rng.random(size=(n, n)) + res2 = quadratic_assignment(A, B, method=self.method, options={'rng': rng}) + + assert_equal(res1.nit, res2.nit) + + def test_partial_guess(self): + n = 5 + rng = np.random.default_rng(4358764578823597324) + + A = rng.random(size=(n, n)) + B = rng.random(size=(n, n)) + + res1 = quadratic_assignment(A, B, method=self.method, + options={'rng': rng}) + guess = np.array([np.arange(5), res1.col_ind]).T + res2 = quadratic_assignment(A, B, method=self.method, + options={'rng': rng, 'partial_guess': guess}) + fix = [2, 4] + match = np.array([np.arange(5)[fix], res1.col_ind[fix]]).T + res3 = quadratic_assignment(A, B, method=self.method, + options={'rng': rng, 'partial_guess': guess, + 'partial_match': match}) + assert_(res1.nit != n*(n+1)/2) + assert_equal(res2.nit, n*(n+1)/2) # tests each swap exactly once + assert_equal(res3.nit, (n-2)*(n-1)/2) # tests free swaps exactly once + + def test_specific_input_validation(self): + # can't have more seed nodes than cost/dist nodes + _rm = _range_matrix + with pytest.raises( + ValueError, + match="`partial_guess` can have only as many entries as"): + quadratic_assignment(np.identity(3), np.identity(3), + method=self.method, + options={'partial_guess': _rm(5, 2)}) + # test for only two seed columns + with pytest.raises( + ValueError, match="`partial_guess` must have two columns"): + quadratic_assignment( + np.identity(3), np.identity(3), method=self.method, + options={'partial_guess': _range_matrix(2, 3)} + ) + # test that seed has no more than two dimensions + with pytest.raises( + ValueError, match="`partial_guess` must have exactly two"): + quadratic_assignment( + np.identity(3), np.identity(3), method=self.method, + options={'partial_guess': np.random.rand(3, 2, 2)} + ) + # seeds cannot be negative valued + with pytest.raises( + ValueError, match="`partial_guess` must contain only pos"): + quadratic_assignment( + np.identity(3), np.identity(3), method=self.method, + options={'partial_guess': -1 * _range_matrix(2, 2)} + ) + # seeds can't have values greater than number of nodes + with pytest.raises( + ValueError, + match="`partial_guess` entries must be less than number"): + quadratic_assignment( + np.identity(5), np.identity(5), method=self.method, + options={'partial_guess': 2 * _range_matrix(4, 2)} + ) + # columns of seed matrix must be unique + with pytest.raises( + ValueError, + match="`partial_guess` column entries must be unique"): + quadratic_assignment( + np.identity(3), np.identity(3), method=self.method, + options={'partial_guess': np.ones((2, 2))} + ) + + +@pytest.mark.filterwarnings("ignore:The NumPy global RNG was seeded by calling") +class TestQAPOnce: + + # these don't need to be repeated for each method + def test_common_input_validation(self): + rng = default_rng(12349038) + # test that non square matrices return error + with pytest.raises(ValueError, match="`A` must be square"): + quadratic_assignment( + rng.random((3, 4)), + rng.random((3, 3)), + ) + with pytest.raises(ValueError, match="`B` must be square"): + quadratic_assignment( + rng.random((3, 3)), + rng.random((3, 4)), + ) + # test that cost and dist matrices have no more than two dimensions + with pytest.raises( + ValueError, match="`A` and `B` must have exactly two"): + quadratic_assignment( + rng.random((3, 3, 3)), + rng.random((3, 3, 3)), + ) + # test that cost and dist matrices of different sizes return error + with pytest.raises( + ValueError, + match="`A` and `B` matrices must be of equal size"): + quadratic_assignment( + rng.random((3, 3)), + rng.random((4, 4)), + ) + # can't have more seed nodes than cost/dist nodes + _rm = _range_matrix + with pytest.raises( + ValueError, + match="`partial_match` can have only as many seeds as"): + quadratic_assignment(np.identity(3), np.identity(3), + options={'partial_match': _rm(5, 2)}) + # test for only two seed columns + with pytest.raises( + ValueError, match="`partial_match` must have two columns"): + quadratic_assignment( + np.identity(3), np.identity(3), + options={'partial_match': _range_matrix(2, 3)} + ) + # test that seed has no more than two dimensions + with pytest.raises( + ValueError, match="`partial_match` must have exactly two"): + quadratic_assignment( + np.identity(3), np.identity(3), + options={'partial_match': np.random.rand(3, 2, 2)} + ) + # seeds cannot be negative valued + with pytest.raises( + ValueError, match="`partial_match` must contain only pos"): + quadratic_assignment( + np.identity(3), np.identity(3), + options={'partial_match': -1 * _range_matrix(2, 2)} + ) + # seeds can't have values greater than number of nodes + with pytest.raises( + ValueError, + match="`partial_match` entries must be less than number"): + quadratic_assignment( + np.identity(5), np.identity(5), + options={'partial_match': 2 * _range_matrix(4, 2)} + ) + # columns of seed matrix must be unique + with pytest.raises( + ValueError, + match="`partial_match` column entries must be unique"): + quadratic_assignment( + np.identity(3), np.identity(3), + options={'partial_match': np.ones((2, 2))} + ) + + +def _range_matrix(a, b): + mat = np.zeros((a, b)) + for i in range(b): + mat[:, i] = np.arange(a) + return mat + + +def _doubly_stochastic(P, tol=1e-3): + # cleaner implementation of btaba/sinkhorn_knopp + + max_iter = 1000 + c = 1 / P.sum(axis=0) + r = 1 / (P @ c) + P_eps = P + + for it in range(max_iter): + if ((np.abs(P_eps.sum(axis=1) - 1) < tol).all() and + (np.abs(P_eps.sum(axis=0) - 1) < tol).all()): + # All column/row sums ~= 1 within threshold + break + + c = 1 / (r @ P) + r = 1 / (P @ c) + P_eps = r[:, None] * P * c + + return P_eps diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_regression.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_regression.py new file mode 100644 index 0000000000000000000000000000000000000000..44916ba96293db19756b8222422e76945aa48ebb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_regression.py @@ -0,0 +1,40 @@ +"""Regression tests for optimize. + +""" +import numpy as np +from numpy.testing import assert_almost_equal +from pytest import raises as assert_raises + +import scipy.optimize + + +class TestRegression: + + def test_newton_x0_is_0(self): + # Regression test for gh-1601 + tgt = 1 + res = scipy.optimize.newton(lambda x: x - 1, 0) + assert_almost_equal(res, tgt) + + def test_newton_integers(self): + # Regression test for gh-1741 + root = scipy.optimize.newton(lambda x: x**2 - 1, x0=2, + fprime=lambda x: 2*x) + assert_almost_equal(root, 1.0) + + def test_lmdif_errmsg(self): + # This shouldn't cause a crash on Python 3 + class SomeError(Exception): + pass + counter = [0] + + def func(x): + counter[0] += 1 + if counter[0] < 3: + return x**2 - np.array([9, 10, 11]) + else: + raise SomeError() + assert_raises(SomeError, + scipy.optimize.leastsq, + func, [1, 2, 3]) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_slsqp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_slsqp.py new file mode 100644 index 0000000000000000000000000000000000000000..45216aa296b56a6a71b89c994e8fc360b826ba00 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_slsqp.py @@ -0,0 +1,613 @@ +""" +Unit test for SLSQP optimization. +""" +from numpy.testing import (assert_, assert_array_almost_equal, + assert_allclose, assert_equal) +from pytest import raises as assert_raises +import pytest +import numpy as np +import scipy + +from scipy.optimize import fmin_slsqp, minimize, Bounds, NonlinearConstraint + + +class MyCallBack: + """pass a custom callback function + + This makes sure it's being used. + """ + def __init__(self): + self.been_called = False + self.ncalls = 0 + + def __call__(self, x): + self.been_called = True + self.ncalls += 1 + + +class TestSLSQP: + """ + Test SLSQP algorithm using Example 14.4 from Numerical Methods for + Engineers by Steven Chapra and Raymond Canale. + This example maximizes the function f(x) = 2*x*y + 2*x - x**2 - 2*y**2, + which has a maximum at x=2, y=1. + """ + def setup_method(self): + self.opts = {'disp': False} + + def fun(self, d, sign=1.0): + """ + Arguments: + d - A list of two elements, where d[0] represents x and d[1] represents y + in the following equation. + sign - A multiplier for f. Since we want to optimize it, and the SciPy + optimizers can only minimize functions, we need to multiply it by + -1 to achieve the desired solution + Returns: + 2*x*y + 2*x - x**2 - 2*y**2 + + """ + x = d[0] + y = d[1] + return sign*(2*x*y + 2*x - x**2 - 2*y**2) + + def jac(self, d, sign=1.0): + """ + This is the derivative of fun, returning a NumPy array + representing df/dx and df/dy. + + """ + x = d[0] + y = d[1] + dfdx = sign*(-2*x + 2*y + 2) + dfdy = sign*(2*x - 4*y) + return np.array([dfdx, dfdy], float) + + def fun_and_jac(self, d, sign=1.0): + return self.fun(d, sign), self.jac(d, sign) + + def f_eqcon(self, x, sign=1.0): + """ Equality constraint """ + return np.array([x[0] - x[1]]) + + def fprime_eqcon(self, x, sign=1.0): + """ Equality constraint, derivative """ + return np.array([[1, -1]]) + + def f_eqcon_scalar(self, x, sign=1.0): + """ Scalar equality constraint """ + return self.f_eqcon(x, sign)[0] + + def fprime_eqcon_scalar(self, x, sign=1.0): + """ Scalar equality constraint, derivative """ + return self.fprime_eqcon(x, sign)[0].tolist() + + def f_ieqcon(self, x, sign=1.0): + """ Inequality constraint """ + return np.array([x[0] - x[1] - 1.0]) + + def fprime_ieqcon(self, x, sign=1.0): + """ Inequality constraint, derivative """ + return np.array([[1, -1]]) + + def f_ieqcon2(self, x): + """ Vector inequality constraint """ + return np.asarray(x) + + def fprime_ieqcon2(self, x): + """ Vector inequality constraint, derivative """ + return np.identity(x.shape[0]) + + # minimize + def test_minimize_unbounded_approximated(self): + # Minimize, method='SLSQP': unbounded, approximated jacobian. + jacs = [None, False, '2-point', '3-point'] + for jac in jacs: + res = minimize(self.fun, [-1.0, 1.0], args=(-1.0, ), + jac=jac, method='SLSQP', + options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [2, 1]) + + def test_minimize_unbounded_given(self): + # Minimize, method='SLSQP': unbounded, given Jacobian. + res = minimize(self.fun, [-1.0, 1.0], args=(-1.0, ), + jac=self.jac, method='SLSQP', options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [2, 1]) + + def test_minimize_bounded_approximated(self): + # Minimize, method='SLSQP': bounded, approximated jacobian. + jacs = [None, False, '2-point', '3-point'] + for jac in jacs: + with np.errstate(invalid='ignore'): + res = minimize(self.fun, [-1.0, 1.0], args=(-1.0, ), + jac=jac, + bounds=((2.5, None), (None, 0.5)), + method='SLSQP', options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [2.5, 0.5]) + assert_(2.5 <= res.x[0]) + assert_(res.x[1] <= 0.5) + + def test_minimize_unbounded_combined(self): + # Minimize, method='SLSQP': unbounded, combined function and Jacobian. + res = minimize(self.fun_and_jac, [-1.0, 1.0], args=(-1.0, ), + jac=True, method='SLSQP', options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [2, 1]) + + def test_minimize_equality_approximated(self): + # Minimize with method='SLSQP': equality constraint, approx. jacobian. + jacs = [None, False, '2-point', '3-point'] + for jac in jacs: + res = minimize(self.fun, [-1.0, 1.0], args=(-1.0, ), + jac=jac, + constraints={'type': 'eq', + 'fun': self.f_eqcon, + 'args': (-1.0, )}, + method='SLSQP', options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [1, 1]) + + def test_minimize_equality_given(self): + # Minimize with method='SLSQP': equality constraint, given Jacobian. + res = minimize(self.fun, [-1.0, 1.0], jac=self.jac, + method='SLSQP', args=(-1.0,), + constraints={'type': 'eq', 'fun':self.f_eqcon, + 'args': (-1.0, )}, + options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [1, 1]) + + def test_minimize_equality_given2(self): + # Minimize with method='SLSQP': equality constraint, given Jacobian + # for fun and const. + res = minimize(self.fun, [-1.0, 1.0], method='SLSQP', + jac=self.jac, args=(-1.0,), + constraints={'type': 'eq', + 'fun': self.f_eqcon, + 'args': (-1.0, ), + 'jac': self.fprime_eqcon}, + options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [1, 1]) + + def test_minimize_equality_given_cons_scalar(self): + # Minimize with method='SLSQP': scalar equality constraint, given + # Jacobian for fun and const. + res = minimize(self.fun, [-1.0, 1.0], method='SLSQP', + jac=self.jac, args=(-1.0,), + constraints={'type': 'eq', + 'fun': self.f_eqcon_scalar, + 'args': (-1.0, ), + 'jac': self.fprime_eqcon_scalar}, + options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [1, 1]) + + def test_minimize_inequality_given(self): + # Minimize with method='SLSQP': inequality constraint, given Jacobian. + res = minimize(self.fun, [-1.0, 1.0], method='SLSQP', + jac=self.jac, args=(-1.0, ), + constraints={'type': 'ineq', + 'fun': self.f_ieqcon, + 'args': (-1.0, )}, + options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [2, 1], atol=1e-3) + + def test_minimize_inequality_given_vector_constraints(self): + # Minimize with method='SLSQP': vector inequality constraint, given + # Jacobian. + res = minimize(self.fun, [-1.0, 1.0], jac=self.jac, + method='SLSQP', args=(-1.0,), + constraints={'type': 'ineq', + 'fun': self.f_ieqcon2, + 'jac': self.fprime_ieqcon2}, + options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [2, 1]) + + def test_minimize_bounded_constraint(self): + # when the constraint makes the solver go up against a parameter + # bound make sure that the numerical differentiation of the + # jacobian doesn't try to exceed that bound using a finite difference. + # gh11403 + def c(x): + assert 0 <= x[0] <= 1 and 0 <= x[1] <= 1, x + return x[0] ** 0.5 + x[1] + + def f(x): + assert 0 <= x[0] <= 1 and 0 <= x[1] <= 1, x + return -x[0] ** 2 + x[1] ** 2 + + cns = [NonlinearConstraint(c, 0, 1.5)] + x0 = np.asarray([0.9, 0.5]) + bnd = Bounds([0., 0.], [1.0, 1.0]) + minimize(f, x0, method='SLSQP', bounds=bnd, constraints=cns) + + def test_minimize_bound_equality_given2(self): + # Minimize with method='SLSQP': bounds, eq. const., given jac. for + # fun. and const. + res = minimize(self.fun, [-1.0, 1.0], method='SLSQP', + jac=self.jac, args=(-1.0, ), + bounds=[(-0.8, 1.), (-1, 0.8)], + constraints={'type': 'eq', + 'fun': self.f_eqcon, + 'args': (-1.0, ), + 'jac': self.fprime_eqcon}, + options=self.opts) + assert_(res['success'], res['message']) + assert_allclose(res.x, [0.8, 0.8], atol=1e-3) + assert_(-0.8 <= res.x[0] <= 1) + assert_(-1 <= res.x[1] <= 0.8) + + # fmin_slsqp + def test_unbounded_approximated(self): + # SLSQP: unbounded, approximated Jacobian. + res = fmin_slsqp(self.fun, [-1.0, 1.0], args=(-1.0, ), + iprint = 0, full_output = 1) + x, fx, its, imode, smode = res + assert_(imode == 0, imode) + assert_array_almost_equal(x, [2, 1]) + + def test_unbounded_given(self): + # SLSQP: unbounded, given Jacobian. + res = fmin_slsqp(self.fun, [-1.0, 1.0], args=(-1.0, ), + fprime = self.jac, iprint = 0, + full_output = 1) + x, fx, its, imode, smode = res + assert_(imode == 0, imode) + assert_array_almost_equal(x, [2, 1]) + + def test_equality_approximated(self): + # SLSQP: equality constraint, approximated Jacobian. + res = fmin_slsqp(self.fun,[-1.0,1.0], args=(-1.0,), + eqcons = [self.f_eqcon], + iprint = 0, full_output = 1) + x, fx, its, imode, smode = res + assert_(imode == 0, imode) + assert_array_almost_equal(x, [1, 1]) + + def test_equality_given(self): + # SLSQP: equality constraint, given Jacobian. + res = fmin_slsqp(self.fun, [-1.0, 1.0], + fprime=self.jac, args=(-1.0,), + eqcons = [self.f_eqcon], iprint = 0, + full_output = 1) + x, fx, its, imode, smode = res + assert_(imode == 0, imode) + assert_array_almost_equal(x, [1, 1]) + + def test_equality_given2(self): + # SLSQP: equality constraint, given Jacobian for fun and const. + res = fmin_slsqp(self.fun, [-1.0, 1.0], + fprime=self.jac, args=(-1.0,), + f_eqcons = self.f_eqcon, + fprime_eqcons = self.fprime_eqcon, + iprint = 0, + full_output = 1) + x, fx, its, imode, smode = res + assert_(imode == 0, imode) + assert_array_almost_equal(x, [1, 1]) + + def test_inequality_given(self): + # SLSQP: inequality constraint, given Jacobian. + res = fmin_slsqp(self.fun, [-1.0, 1.0], + fprime=self.jac, args=(-1.0, ), + ieqcons = [self.f_ieqcon], + iprint = 0, full_output = 1) + x, fx, its, imode, smode = res + assert_(imode == 0, imode) + assert_array_almost_equal(x, [2, 1], decimal=3) + + def test_bound_equality_given2(self): + # SLSQP: bounds, eq. const., given jac. for fun. and const. + res = fmin_slsqp(self.fun, [-1.0, 1.0], + fprime=self.jac, args=(-1.0, ), + bounds = [(-0.8, 1.), (-1, 0.8)], + f_eqcons = self.f_eqcon, + fprime_eqcons = self.fprime_eqcon, + iprint = 0, full_output = 1) + x, fx, its, imode, smode = res + assert_(imode == 0, imode) + assert_array_almost_equal(x, [0.8, 0.8], decimal=3) + assert_(-0.8 <= x[0] <= 1) + assert_(-1 <= x[1] <= 0.8) + + def test_scalar_constraints(self): + # Regression test for gh-2182 + x = fmin_slsqp(lambda z: z**2, [3.], + ieqcons=[lambda z: z[0] - 1], + iprint=0) + assert_array_almost_equal(x, [1.]) + + x = fmin_slsqp(lambda z: z**2, [3.], + f_ieqcons=lambda z: [z[0] - 1], + iprint=0) + assert_array_almost_equal(x, [1.]) + + def test_integer_bounds(self): + # This should not raise an exception + fmin_slsqp(lambda z: z**2 - 1, [0], bounds=[[0, 1]], iprint=0) + + def test_array_bounds(self): + # NumPy used to treat n-dimensional 1-element arrays as scalars + # in some cases. The handling of `bounds` by `fmin_slsqp` still + # supports this behavior. + bounds = [(-np.inf, np.inf), (np.array([2]), np.array([3]))] + x = fmin_slsqp(lambda z: np.sum(z**2 - 1), [2.5, 2.5], bounds=bounds, + iprint=0) + assert_array_almost_equal(x, [0, 2]) + + def test_obj_must_return_scalar(self): + # Regression test for Github Issue #5433 + # If objective function does not return a scalar, raises ValueError + with assert_raises(ValueError): + fmin_slsqp(lambda x: [0, 1], [1, 2, 3]) + + def test_obj_returns_scalar_in_list(self): + # Test for Github Issue #5433 and PR #6691 + # Objective function should be able to return length-1 Python list + # containing the scalar + fmin_slsqp(lambda x: [0], [1, 2, 3], iprint=0) + + def test_callback(self): + # Minimize, method='SLSQP': unbounded, approximated jacobian. Check for callback + callback = MyCallBack() + res = minimize(self.fun, [-1.0, 1.0], args=(-1.0, ), + method='SLSQP', callback=callback, options=self.opts) + assert_(res['success'], res['message']) + assert_(callback.been_called) + assert_equal(callback.ncalls, res['nit']) + + def test_inconsistent_linearization(self): + # SLSQP must be able to solve this problem, even if the + # linearized problem at the starting point is infeasible. + + # Linearized constraints are + # + # 2*x0[0]*x[0] >= 1 + # + # At x0 = [0, 1], the second constraint is clearly infeasible. + # This triggers a call with n2==1 in the LSQ subroutine. + x = [0, 1] + def f1(x): + return x[0] + x[1] - 2 + def f2(x): + return x[0] ** 2 - 1 + sol = minimize( + lambda x: x[0]**2 + x[1]**2, + x, + constraints=({'type':'eq','fun': f1}, + {'type':'ineq','fun': f2}), + bounds=((0,None), (0,None)), + method='SLSQP') + x = sol.x + + assert_allclose(f1(x), 0, atol=1e-8) + assert_(f2(x) >= -1e-8) + assert_(sol.success, sol) + + def test_regression_5743(self): + # SLSQP must not indicate success for this problem, + # which is infeasible. + x = [1, 2] + sol = minimize( + lambda x: x[0]**2 + x[1]**2, + x, + constraints=({'type':'eq','fun': lambda x: x[0]+x[1]-1}, + {'type':'ineq','fun': lambda x: x[0]-2}), + bounds=((0,None), (0,None)), + method='SLSQP') + assert_(not sol.success, sol) + + def test_gh_6676(self): + def func(x): + return (x[0] - 1)**2 + 2*(x[1] - 1)**2 + 0.5*(x[2] - 1)**2 + + sol = minimize(func, [0, 0, 0], method='SLSQP') + assert_(sol.jac.shape == (3,)) + + def test_invalid_bounds(self): + # Raise correct error when lower bound is greater than upper bound. + # See Github issue 6875. + bounds_list = [ + ((1, 2), (2, 1)), + ((2, 1), (1, 2)), + ((2, 1), (2, 1)), + ((np.inf, 0), (np.inf, 0)), + ((1, -np.inf), (0, 1)), + ] + for bounds in bounds_list: + with assert_raises(ValueError): + minimize(self.fun, [-1.0, 1.0], bounds=bounds, method='SLSQP') + + def test_bounds_clipping(self): + # + # SLSQP returns bogus results for initial guess out of bounds, gh-6859 + # + def f(x): + return (x[0] - 1)**2 + + sol = minimize(f, [10], method='slsqp', bounds=[(None, 0)]) + assert_(sol.success) + assert_allclose(sol.x, 0, atol=1e-10) + + sol = minimize(f, [-10], method='slsqp', bounds=[(2, None)]) + assert_(sol.success) + assert_allclose(sol.x, 2, atol=1e-10) + + sol = minimize(f, [-10], method='slsqp', bounds=[(None, 0)]) + assert_(sol.success) + assert_allclose(sol.x, 0, atol=1e-10) + + sol = minimize(f, [10], method='slsqp', bounds=[(2, None)]) + assert_(sol.success) + assert_allclose(sol.x, 2, atol=1e-10) + + sol = minimize(f, [-0.5], method='slsqp', bounds=[(-1, 0)]) + assert_(sol.success) + assert_allclose(sol.x, 0, atol=1e-10) + + sol = minimize(f, [10], method='slsqp', bounds=[(-1, 0)]) + assert_(sol.success) + assert_allclose(sol.x, 0, atol=1e-10) + + def test_infeasible_initial(self): + # Check SLSQP behavior with infeasible initial point + def f(x): + x, = x + return x*x - 2*x + 1 + + cons_u = [{'type': 'ineq', 'fun': lambda x: 0 - x}] + cons_l = [{'type': 'ineq', 'fun': lambda x: x - 2}] + cons_ul = [{'type': 'ineq', 'fun': lambda x: 0 - x}, + {'type': 'ineq', 'fun': lambda x: x + 1}] + + sol = minimize(f, [10], method='slsqp', constraints=cons_u) + assert_(sol.success) + assert_allclose(sol.x, 0, atol=1e-10) + + sol = minimize(f, [-10], method='slsqp', constraints=cons_l) + assert_(sol.success) + assert_allclose(sol.x, 2, atol=1e-10) + + sol = minimize(f, [-10], method='slsqp', constraints=cons_u) + assert_(sol.success) + assert_allclose(sol.x, 0, atol=1e-10) + + sol = minimize(f, [10], method='slsqp', constraints=cons_l) + assert_(sol.success) + assert_allclose(sol.x, 2, atol=1e-10) + + sol = minimize(f, [-0.5], method='slsqp', constraints=cons_ul) + assert_(sol.success) + assert_allclose(sol.x, 0, atol=1e-10) + + sol = minimize(f, [10], method='slsqp', constraints=cons_ul) + assert_(sol.success) + assert_allclose(sol.x, 0, atol=1e-10) + + @pytest.mark.xfail(scipy.show_config(mode='dicts')['Compilers']['fortran']['name'] + == "intel-llvm", + reason="Runtime warning due to floating point issues, not logic") + def test_inconsistent_inequalities(self): + # gh-7618 + + def cost(x): + return -1 * x[0] + 4 * x[1] + + def ineqcons1(x): + return x[1] - x[0] - 1 + + def ineqcons2(x): + return x[0] - x[1] + + # The inequalities are inconsistent, so no solution can exist: + # + # x1 >= x0 + 1 + # x0 >= x1 + + x0 = (1,5) + bounds = ((-5, 5), (-5, 5)) + cons = (dict(type='ineq', fun=ineqcons1), dict(type='ineq', fun=ineqcons2)) + res = minimize(cost, x0, method='SLSQP', bounds=bounds, constraints=cons) + + assert_(not res.success) + + def test_new_bounds_type(self): + def f(x): + return x[0] ** 2 + x[1] ** 2 + bounds = Bounds([1, 0], [np.inf, np.inf]) + sol = minimize(f, [0, 0], method='slsqp', bounds=bounds) + assert_(sol.success) + assert_allclose(sol.x, [1, 0]) + + def test_nested_minimization(self): + + class NestedProblem: + + def __init__(self): + self.F_outer_count = 0 + + def F_outer(self, x): + self.F_outer_count += 1 + if self.F_outer_count > 1000: + raise Exception("Nested minimization failed to terminate.") + inner_res = minimize(self.F_inner, (3, 4), method="SLSQP") + assert_(inner_res.success) + assert_allclose(inner_res.x, [1, 1]) + return x[0]**2 + x[1]**2 + x[2]**2 + + def F_inner(self, x): + return (x[0] - 1)**2 + (x[1] - 1)**2 + + def solve(self): + outer_res = minimize(self.F_outer, (5, 5, 5), method="SLSQP") + assert_(outer_res.success) + assert_allclose(outer_res.x, [0, 0, 0]) + + problem = NestedProblem() + problem.solve() + + def test_gh1758(self): + # the test suggested in gh1758 + # https://nlopt.readthedocs.io/en/latest/NLopt_Tutorial/ + # implement two equality constraints, in R^2. + def fun(x): + return np.sqrt(x[1]) + + def f_eqcon(x): + """ Equality constraint """ + return x[1] - (2 * x[0]) ** 3 + + def f_eqcon2(x): + """ Equality constraint """ + return x[1] - (-x[0] + 1) ** 3 + + c1 = {'type': 'eq', 'fun': f_eqcon} + c2 = {'type': 'eq', 'fun': f_eqcon2} + + res = minimize(fun, [8, 0.25], method='SLSQP', + constraints=[c1, c2], bounds=[(-0.5, 1), (0, 8)]) + + np.testing.assert_allclose(res.fun, 0.5443310539518) + np.testing.assert_allclose(res.x, [0.33333333, 0.2962963]) + assert res.success + + def test_gh9640(self): + np.random.seed(10) + cons = ({'type': 'ineq', 'fun': lambda x: -x[0] - x[1] - 3}, + {'type': 'ineq', 'fun': lambda x: x[1] + x[2] - 2}) + bnds = ((-2, 2), (-2, 2), (-2, 2)) + + def target(x): + return 1 + x0 = [-1.8869783504471584, -0.640096352696244, -0.8174212253407696] + res = minimize(target, x0, method='SLSQP', bounds=bnds, constraints=cons, + options={'disp':False, 'maxiter':10000}) + + # The problem is infeasible, so it cannot succeed + assert not res.success + + @pytest.mark.thread_unsafe + def test_parameters_stay_within_bounds(self): + # gh11403. For some problems the SLSQP Fortran code suggests a step + # outside one of the lower/upper bounds. When this happens + # approx_derivative complains because it's being asked to evaluate + # a gradient outside its domain. + np.random.seed(1) + bounds = Bounds(np.array([0.1]), np.array([1.0])) + n_inputs = len(bounds.lb) + x0 = np.array(bounds.lb + (bounds.ub - bounds.lb) * + np.random.random(n_inputs)) + + def f(x): + assert (x >= bounds.lb).all() + return np.linalg.norm(x) + + with pytest.warns(RuntimeWarning, match='x were outside bounds'): + res = minimize(f, x0, method='SLSQP', bounds=bounds) + assert res.success diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_tnc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_tnc.py new file mode 100644 index 0000000000000000000000000000000000000000..2cde9837bfd08e62916660a9750d833629b6b547 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_tnc.py @@ -0,0 +1,345 @@ +""" +Unit tests for TNC optimization routine from tnc.py +""" +import pytest +from numpy.testing import assert_allclose, assert_equal + +import numpy as np +from math import pow + +from scipy import optimize + + +class TestTnc: + """TNC non-linear optimization. + + These tests are taken from Prof. K. Schittkowski's test examples + for constrained non-linear programming. + + http://www.uni-bayreuth.de/departments/math/~kschittkowski/home.htm + + """ + def setup_method(self): + # options for minimize + self.opts = {'disp': False, 'maxfun': 200} + + # objective functions and Jacobian for each test + def f1(self, x, a=100.0): + return a * pow((x[1] - pow(x[0], 2)), 2) + pow(1.0 - x[0], 2) + + def g1(self, x, a=100.0): + dif = [0, 0] + dif[1] = 2 * a * (x[1] - pow(x[0], 2)) + dif[0] = -2.0 * (x[0] * (dif[1] - 1.0) + 1.0) + return dif + + def fg1(self, x, a=100.0): + return self.f1(x, a), self.g1(x, a) + + def f3(self, x): + return x[1] + pow(x[1] - x[0], 2) * 1.0e-5 + + def g3(self, x): + dif = [0, 0] + dif[0] = -2.0 * (x[1] - x[0]) * 1.0e-5 + dif[1] = 1.0 - dif[0] + return dif + + def fg3(self, x): + return self.f3(x), self.g3(x) + + def f4(self, x): + return pow(x[0] + 1.0, 3) / 3.0 + x[1] + + def g4(self, x): + dif = [0, 0] + dif[0] = pow(x[0] + 1.0, 2) + dif[1] = 1.0 + return dif + + def fg4(self, x): + return self.f4(x), self.g4(x) + + def f5(self, x): + return np.sin(x[0] + x[1]) + pow(x[0] - x[1], 2) - \ + 1.5 * x[0] + 2.5 * x[1] + 1.0 + + def g5(self, x): + dif = [0, 0] + v1 = np.cos(x[0] + x[1]) + v2 = 2.0*(x[0] - x[1]) + + dif[0] = v1 + v2 - 1.5 + dif[1] = v1 - v2 + 2.5 + return dif + + def fg5(self, x): + return self.f5(x), self.g5(x) + + def f38(self, x): + return (100.0 * pow(x[1] - pow(x[0], 2), 2) + + pow(1.0 - x[0], 2) + 90.0 * pow(x[3] - pow(x[2], 2), 2) + + pow(1.0 - x[2], 2) + 10.1 * (pow(x[1] - 1.0, 2) + + pow(x[3] - 1.0, 2)) + + 19.8 * (x[1] - 1.0) * (x[3] - 1.0)) * 1.0e-5 + + def g38(self, x): + dif = [0, 0, 0, 0] + dif[0] = (-400.0 * x[0] * (x[1] - pow(x[0], 2)) - + 2.0 * (1.0 - x[0])) * 1.0e-5 + dif[1] = (200.0 * (x[1] - pow(x[0], 2)) + 20.2 * (x[1] - 1.0) + + 19.8 * (x[3] - 1.0)) * 1.0e-5 + dif[2] = (- 360.0 * x[2] * (x[3] - pow(x[2], 2)) - + 2.0 * (1.0 - x[2])) * 1.0e-5 + dif[3] = (180.0 * (x[3] - pow(x[2], 2)) + 20.2 * (x[3] - 1.0) + + 19.8 * (x[1] - 1.0)) * 1.0e-5 + return dif + + def fg38(self, x): + return self.f38(x), self.g38(x) + + def f45(self, x): + return 2.0 - x[0] * x[1] * x[2] * x[3] * x[4] / 120.0 + + def g45(self, x): + dif = [0] * 5 + dif[0] = - x[1] * x[2] * x[3] * x[4] / 120.0 + dif[1] = - x[0] * x[2] * x[3] * x[4] / 120.0 + dif[2] = - x[0] * x[1] * x[3] * x[4] / 120.0 + dif[3] = - x[0] * x[1] * x[2] * x[4] / 120.0 + dif[4] = - x[0] * x[1] * x[2] * x[3] / 120.0 + return dif + + def fg45(self, x): + return self.f45(x), self.g45(x) + + # tests + # minimize with method=TNC + def test_minimize_tnc1(self): + x0, bnds = [-2, 1], ([-np.inf, None], [-1.5, None]) + xopt = [1, 1] + iterx = [] # to test callback + + res = optimize.minimize(self.f1, x0, method='TNC', jac=self.g1, + bounds=bnds, options=self.opts, + callback=iterx.append) + assert_allclose(res.fun, self.f1(xopt), atol=1e-8) + assert_equal(len(iterx), res.nit) + + def test_minimize_tnc1b(self): + x0, bnds = np.array([-2, 1]), ([-np.inf, None], [-1.5, None]) + xopt = [1, 1] + x = optimize.minimize(self.f1, x0, method='TNC', + bounds=bnds, options=self.opts).x + assert_allclose(self.f1(x), self.f1(xopt), atol=1e-4) + + def test_minimize_tnc1c(self): + x0, bnds = [-2, 1], ([-np.inf, None],[-1.5, None]) + xopt = [1, 1] + x = optimize.minimize(self.fg1, x0, method='TNC', + jac=True, bounds=bnds, + options=self.opts).x + assert_allclose(self.f1(x), self.f1(xopt), atol=1e-8) + + def test_minimize_tnc2(self): + x0, bnds = [-2, 1], ([-np.inf, None], [1.5, None]) + xopt = [-1.2210262419616387, 1.5] + x = optimize.minimize(self.f1, x0, method='TNC', + jac=self.g1, bounds=bnds, + options=self.opts).x + assert_allclose(self.f1(x), self.f1(xopt), atol=1e-8) + + def test_minimize_tnc3(self): + x0, bnds = [10, 1], ([-np.inf, None], [0.0, None]) + xopt = [0, 0] + x = optimize.minimize(self.f3, x0, method='TNC', + jac=self.g3, bounds=bnds, + options=self.opts).x + assert_allclose(self.f3(x), self.f3(xopt), atol=1e-8) + + def test_minimize_tnc4(self): + x0,bnds = [1.125, 0.125], [(1, None), (0, None)] + xopt = [1, 0] + x = optimize.minimize(self.f4, x0, method='TNC', + jac=self.g4, bounds=bnds, + options=self.opts).x + assert_allclose(self.f4(x), self.f4(xopt), atol=1e-8) + + def test_minimize_tnc5(self): + x0, bnds = [0, 0], [(-1.5, 4),(-3, 3)] + xopt = [-0.54719755119659763, -1.5471975511965976] + x = optimize.minimize(self.f5, x0, method='TNC', + jac=self.g5, bounds=bnds, + options=self.opts).x + assert_allclose(self.f5(x), self.f5(xopt), atol=1e-8) + + def test_minimize_tnc38(self): + x0, bnds = np.array([-3, -1, -3, -1]), [(-10, 10)]*4 + xopt = [1]*4 + x = optimize.minimize(self.f38, x0, method='TNC', + jac=self.g38, bounds=bnds, + options=self.opts).x + assert_allclose(self.f38(x), self.f38(xopt), atol=1e-8) + + def test_minimize_tnc45(self): + x0, bnds = [2] * 5, [(0, 1), (0, 2), (0, 3), (0, 4), (0, 5)] + xopt = [1, 2, 3, 4, 5] + x = optimize.minimize(self.f45, x0, method='TNC', + jac=self.g45, bounds=bnds, + options=self.opts).x + assert_allclose(self.f45(x), self.f45(xopt), atol=1e-8) + + # fmin_tnc + def test_tnc1(self): + fg, x, bounds = self.fg1, [-2, 1], ([-np.inf, None], [-1.5, None]) + xopt = [1, 1] + + x, nf, rc = optimize.fmin_tnc(fg, x, bounds=bounds, args=(100.0, ), + messages=optimize._tnc.MSG_NONE, + maxfun=200) + + assert_allclose(self.f1(x), self.f1(xopt), atol=1e-8, + err_msg="TNC failed with status: " + + optimize._tnc.RCSTRINGS[rc]) + + def test_tnc1b(self): + x, bounds = [-2, 1], ([-np.inf, None], [-1.5, None]) + xopt = [1, 1] + + x, nf, rc = optimize.fmin_tnc(self.f1, x, approx_grad=True, + bounds=bounds, + messages=optimize._tnc.MSG_NONE, + maxfun=200) + + assert_allclose(self.f1(x), self.f1(xopt), atol=1e-4, + err_msg="TNC failed with status: " + + optimize._tnc.RCSTRINGS[rc]) + + def test_tnc1c(self): + x, bounds = [-2, 1], ([-np.inf, None], [-1.5, None]) + xopt = [1, 1] + + x, nf, rc = optimize.fmin_tnc(self.f1, x, fprime=self.g1, + bounds=bounds, + messages=optimize._tnc.MSG_NONE, + maxfun=200) + + assert_allclose(self.f1(x), self.f1(xopt), atol=1e-8, + err_msg="TNC failed with status: " + + optimize._tnc.RCSTRINGS[rc]) + + def test_tnc2(self): + fg, x, bounds = self.fg1, [-2, 1], ([-np.inf, None], [1.5, None]) + xopt = [-1.2210262419616387, 1.5] + + x, nf, rc = optimize.fmin_tnc(fg, x, bounds=bounds, + messages=optimize._tnc.MSG_NONE, + maxfun=200) + + assert_allclose(self.f1(x), self.f1(xopt), atol=1e-8, + err_msg="TNC failed with status: " + + optimize._tnc.RCSTRINGS[rc]) + + def test_tnc3(self): + fg, x, bounds = self.fg3, [10, 1], ([-np.inf, None], [0.0, None]) + xopt = [0, 0] + + x, nf, rc = optimize.fmin_tnc(fg, x, bounds=bounds, + messages=optimize._tnc.MSG_NONE, + maxfun=200) + + assert_allclose(self.f3(x), self.f3(xopt), atol=1e-8, + err_msg="TNC failed with status: " + + optimize._tnc.RCSTRINGS[rc]) + + def test_tnc4(self): + fg, x, bounds = self.fg4, [1.125, 0.125], [(1, None), (0, None)] + xopt = [1, 0] + + x, nf, rc = optimize.fmin_tnc(fg, x, bounds=bounds, + messages=optimize._tnc.MSG_NONE, + maxfun=200) + + assert_allclose(self.f4(x), self.f4(xopt), atol=1e-8, + err_msg="TNC failed with status: " + + optimize._tnc.RCSTRINGS[rc]) + + def test_tnc5(self): + fg, x, bounds = self.fg5, [0, 0], [(-1.5, 4),(-3, 3)] + xopt = [-0.54719755119659763, -1.5471975511965976] + + x, nf, rc = optimize.fmin_tnc(fg, x, bounds=bounds, + messages=optimize._tnc.MSG_NONE, + maxfun=200) + + assert_allclose(self.f5(x), self.f5(xopt), atol=1e-8, + err_msg="TNC failed with status: " + + optimize._tnc.RCSTRINGS[rc]) + + def test_tnc38(self): + fg, x, bounds = self.fg38, np.array([-3, -1, -3, -1]), [(-10, 10)]*4 + xopt = [1]*4 + + x, nf, rc = optimize.fmin_tnc(fg, x, bounds=bounds, + messages=optimize._tnc.MSG_NONE, + maxfun=200) + + assert_allclose(self.f38(x), self.f38(xopt), atol=1e-8, + err_msg="TNC failed with status: " + + optimize._tnc.RCSTRINGS[rc]) + + def test_tnc45(self): + fg, x, bounds = self.fg45, [2] * 5, [(0, 1), (0, 2), (0, 3), + (0, 4), (0, 5)] + xopt = [1, 2, 3, 4, 5] + + x, nf, rc = optimize.fmin_tnc(fg, x, bounds=bounds, + messages=optimize._tnc.MSG_NONE, + maxfun=200) + + assert_allclose(self.f45(x), self.f45(xopt), atol=1e-8, + err_msg="TNC failed with status: " + + optimize._tnc.RCSTRINGS[rc]) + + def test_raising_exceptions(self): + # tnc was ported to cython from hand-crafted cpython code + # check that Exception handling works. + def myfunc(x): + raise RuntimeError("myfunc") + + def myfunc1(x): + return optimize.rosen(x) + + def callback(x): + raise ValueError("callback") + + with pytest.raises(RuntimeError): + optimize.minimize(myfunc, [0, 1], method="TNC") + + with pytest.raises(ValueError): + optimize.minimize( + myfunc1, [0, 1], method="TNC", callback=callback + ) + + def test_callback_shouldnt_affect_minimization(self): + # gh14879. The output of a TNC minimization was different depending + # on whether a callback was used or not. The two should be equivalent. + # The issue was that TNC was unscaling/scaling x, and this process was + # altering x in the process. Now the callback uses an unscaled + # temporary copy of x. + def callback(x): + pass + + fun = optimize.rosen + bounds = [(0, 10)] * 4 + x0 = [1, 2, 3, 4.] + res = optimize.minimize( + fun, x0, bounds=bounds, method="TNC", options={"maxfun": 1000} + ) + res2 = optimize.minimize( + fun, x0, bounds=bounds, method="TNC", options={"maxfun": 1000}, + callback=callback + ) + assert_allclose(res2.x, res.x) + assert_allclose(res2.fun, res.fun) + assert_equal(res2.nfev, res.nfev) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_trustregion.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_trustregion.py new file mode 100644 index 0000000000000000000000000000000000000000..0439f8125c565d70cbed8c6c41f762fdae06d4ce --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_trustregion.py @@ -0,0 +1,110 @@ +""" +Unit tests for trust-region optimization routines. + +""" +import pytest +import numpy as np +from numpy.testing import assert_, assert_equal, assert_allclose +from scipy.optimize import (minimize, rosen, rosen_der, rosen_hess, + rosen_hess_prod) + + +class Accumulator: + """ This is for testing callbacks.""" + def __init__(self): + self.count = 0 + self.accum = None + + def __call__(self, x): + self.count += 1 + if self.accum is None: + self.accum = np.array(x) + else: + self.accum += x + + +class TestTrustRegionSolvers: + + def setup_method(self): + self.x_opt = [1.0, 1.0] + self.easy_guess = [2.0, 2.0] + self.hard_guess = [-1.2, 1.0] + + def test_dogleg_accuracy(self): + # test the accuracy and the return_all option + x0 = self.hard_guess + r = minimize(rosen, x0, jac=rosen_der, hess=rosen_hess, tol=1e-8, + method='dogleg', options={'return_all': True},) + assert_allclose(x0, r['allvecs'][0]) + assert_allclose(r['x'], r['allvecs'][-1]) + assert_allclose(r['x'], self.x_opt) + + def test_dogleg_callback(self): + # test the callback mechanism and the maxiter and return_all options + accumulator = Accumulator() + maxiter = 5 + r = minimize(rosen, self.hard_guess, jac=rosen_der, hess=rosen_hess, + callback=accumulator, method='dogleg', + options={'return_all': True, 'maxiter': maxiter},) + assert_equal(accumulator.count, maxiter) + assert_equal(len(r['allvecs']), maxiter+1) + assert_allclose(r['x'], r['allvecs'][-1]) + assert_allclose(sum(r['allvecs'][1:]), accumulator.accum) + + @pytest.mark.thread_unsafe + def test_dogleg_user_warning(self): + with pytest.warns(RuntimeWarning, + match=r'Maximum number of iterations'): + minimize(rosen, self.hard_guess, jac=rosen_der, + hess=rosen_hess, method='dogleg', + options={'disp': True, 'maxiter': 1}, ) + + def test_solver_concordance(self): + # Assert that dogleg uses fewer iterations than ncg on the Rosenbrock + # test function, although this does not necessarily mean + # that dogleg is faster or better than ncg even for this function + # and especially not for other test functions. + f = rosen + g = rosen_der + h = rosen_hess + for x0 in (self.easy_guess, self.hard_guess): + r_dogleg = minimize(f, x0, jac=g, hess=h, tol=1e-8, + method='dogleg', options={'return_all': True}) + r_trust_ncg = minimize(f, x0, jac=g, hess=h, tol=1e-8, + method='trust-ncg', + options={'return_all': True}) + r_trust_krylov = minimize(f, x0, jac=g, hess=h, tol=1e-8, + method='trust-krylov', + options={'return_all': True}) + r_ncg = minimize(f, x0, jac=g, hess=h, tol=1e-8, + method='newton-cg', options={'return_all': True}) + r_iterative = minimize(f, x0, jac=g, hess=h, tol=1e-8, + method='trust-exact', + options={'return_all': True}) + assert_allclose(self.x_opt, r_dogleg['x']) + assert_allclose(self.x_opt, r_trust_ncg['x']) + assert_allclose(self.x_opt, r_trust_krylov['x']) + assert_allclose(self.x_opt, r_ncg['x']) + assert_allclose(self.x_opt, r_iterative['x']) + assert_(len(r_dogleg['allvecs']) < len(r_ncg['allvecs'])) + + def test_trust_ncg_hessp(self): + for x0 in (self.easy_guess, self.hard_guess, self.x_opt): + r = minimize(rosen, x0, jac=rosen_der, hessp=rosen_hess_prod, + tol=1e-8, method='trust-ncg') + assert_allclose(self.x_opt, r['x']) + + def test_trust_ncg_start_in_optimum(self): + r = minimize(rosen, x0=self.x_opt, jac=rosen_der, hess=rosen_hess, + tol=1e-8, method='trust-ncg') + assert_allclose(self.x_opt, r['x']) + + def test_trust_krylov_start_in_optimum(self): + r = minimize(rosen, x0=self.x_opt, jac=rosen_der, hess=rosen_hess, + tol=1e-8, method='trust-krylov') + assert_allclose(self.x_opt, r['x']) + + def test_trust_exact_start_in_optimum(self): + r = minimize(rosen, x0=self.x_opt, jac=rosen_der, hess=rosen_hess, + tol=1e-8, method='trust-exact') + assert_allclose(self.x_opt, r['x']) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_trustregion_exact.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_trustregion_exact.py new file mode 100644 index 0000000000000000000000000000000000000000..020b6d883a5fa59006c142df4fcb8c3e115c46bf --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_trustregion_exact.py @@ -0,0 +1,351 @@ +""" +Unit tests for trust-region iterative subproblem. + +""" +import pytest +import numpy as np +from scipy.optimize._trustregion_exact import ( + estimate_smallest_singular_value, + singular_leading_submatrix, + IterativeSubproblem) +from scipy.linalg import (svd, get_lapack_funcs, det, qr, norm) +from numpy.testing import (assert_array_equal, + assert_equal, assert_array_almost_equal) + + +def random_entry(n, min_eig, max_eig, case): + + # Generate random matrix + rand = np.random.uniform(-1, 1, (n, n)) + + # QR decomposition + Q, _, _ = qr(rand, pivoting='True') + + # Generate random eigenvalues + eigvalues = np.random.uniform(min_eig, max_eig, n) + eigvalues = np.sort(eigvalues)[::-1] + + # Generate matrix + Qaux = np.multiply(eigvalues, Q) + A = np.dot(Qaux, Q.T) + + # Generate gradient vector accordingly + # to the case is being tested. + if case == 'hard': + g = np.zeros(n) + g[:-1] = np.random.uniform(-1, 1, n-1) + g = np.dot(Q, g) + elif case == 'jac_equal_zero': + g = np.zeros(n) + else: + g = np.random.uniform(-1, 1, n) + + return A, g + + +class TestEstimateSmallestSingularValue: + + def test_for_ill_condiotioned_matrix(self): + + # Ill-conditioned triangular matrix + C = np.array([[1, 2, 3, 4], + [0, 0.05, 60, 7], + [0, 0, 0.8, 9], + [0, 0, 0, 10]]) + + # Get svd decomposition + U, s, Vt = svd(C) + + # Get smallest singular value and correspondent right singular vector. + smin_svd = s[-1] + zmin_svd = Vt[-1, :] + + # Estimate smallest singular value + smin, zmin = estimate_smallest_singular_value(C) + + # Check the estimation + assert_array_almost_equal(smin, smin_svd, decimal=8) + assert_array_almost_equal(abs(zmin), abs(zmin_svd), decimal=8) + + +class TestSingularLeadingSubmatrix: + + def test_for_already_singular_leading_submatrix(self): + + # Define test matrix A. + # Note that the leading 2x2 submatrix is singular. + A = np.array([[1, 2, 3], + [2, 4, 5], + [3, 5, 6]]) + + # Get Cholesky from lapack functions + cholesky, = get_lapack_funcs(('potrf',), (A,)) + + # Compute Cholesky Decomposition + c, k = cholesky(A, lower=False, overwrite_a=False, clean=True) + + delta, v = singular_leading_submatrix(A, c, k) + + A[k-1, k-1] += delta + + # Check if the leading submatrix is singular. + assert_array_almost_equal(det(A[:k, :k]), 0) + + # Check if `v` fulfil the specified properties + quadratic_term = np.dot(v, np.dot(A, v)) + assert_array_almost_equal(quadratic_term, 0) + + def test_for_simetric_indefinite_matrix(self): + + # Define test matrix A. + # Note that the leading 5x5 submatrix is indefinite. + A = np.asarray([[1, 2, 3, 7, 8], + [2, 5, 5, 9, 0], + [3, 5, 11, 1, 2], + [7, 9, 1, 7, 5], + [8, 0, 2, 5, 8]]) + + # Get Cholesky from lapack functions + cholesky, = get_lapack_funcs(('potrf',), (A,)) + + # Compute Cholesky Decomposition + c, k = cholesky(A, lower=False, overwrite_a=False, clean=True) + + delta, v = singular_leading_submatrix(A, c, k) + + A[k-1, k-1] += delta + + # Check if the leading submatrix is singular. + assert_array_almost_equal(det(A[:k, :k]), 0) + + # Check if `v` fulfil the specified properties + quadratic_term = np.dot(v, np.dot(A, v)) + assert_array_almost_equal(quadratic_term, 0) + + def test_for_first_element_equal_to_zero(self): + + # Define test matrix A. + # Note that the leading 2x2 submatrix is singular. + A = np.array([[0, 3, 11], + [3, 12, 5], + [11, 5, 6]]) + + # Get Cholesky from lapack functions + cholesky, = get_lapack_funcs(('potrf',), (A,)) + + # Compute Cholesky Decomposition + c, k = cholesky(A, lower=False, overwrite_a=False, clean=True) + + delta, v = singular_leading_submatrix(A, c, k) + + A[k-1, k-1] += delta + + # Check if the leading submatrix is singular + assert_array_almost_equal(det(A[:k, :k]), 0) + + # Check if `v` fulfil the specified properties + quadratic_term = np.dot(v, np.dot(A, v)) + assert_array_almost_equal(quadratic_term, 0) + + +class TestIterativeSubproblem: + + def test_for_the_easy_case(self): + + # `H` is chosen such that `g` is not orthogonal to the + # eigenvector associated with the smallest eigenvalue `s`. + H = [[10, 2, 3, 4], + [2, 1, 7, 1], + [3, 7, 1, 7], + [4, 1, 7, 2]] + g = [1, 1, 1, 1] + + # Trust Radius + trust_radius = 1 + + # Solve Subproblem + subprob = IterativeSubproblem(x=0, + fun=lambda x: 0, + jac=lambda x: np.array(g), + hess=lambda x: np.array(H), + k_easy=1e-10, + k_hard=1e-10) + p, hits_boundary = subprob.solve(trust_radius) + + assert_array_almost_equal(p, [0.00393332, -0.55260862, + 0.67065477, -0.49480341]) + assert_array_almost_equal(hits_boundary, True) + + def test_for_the_hard_case(self): + + # `H` is chosen such that `g` is orthogonal to the + # eigenvector associated with the smallest eigenvalue `s`. + H = [[10, 2, 3, 4], + [2, 1, 7, 1], + [3, 7, 1, 7], + [4, 1, 7, 2]] + g = [6.4852641521327437, 1, 1, 1] + s = -8.2151519874416614 + + # Trust Radius + trust_radius = 1 + + # Solve Subproblem + subprob = IterativeSubproblem(x=0, + fun=lambda x: 0, + jac=lambda x: np.array(g), + hess=lambda x: np.array(H), + k_easy=1e-10, + k_hard=1e-10) + p, hits_boundary = subprob.solve(trust_radius) + + assert_array_almost_equal(-s, subprob.lambda_current) + + def test_for_interior_convergence(self): + + H = [[1.812159, 0.82687265, 0.21838879, -0.52487006, 0.25436988], + [0.82687265, 2.66380283, 0.31508988, -0.40144163, 0.08811588], + [0.21838879, 0.31508988, 2.38020726, -0.3166346, 0.27363867], + [-0.52487006, -0.40144163, -0.3166346, 1.61927182, -0.42140166], + [0.25436988, 0.08811588, 0.27363867, -0.42140166, 1.33243101]] + + g = [0.75798952, 0.01421945, 0.33847612, 0.83725004, -0.47909534] + + # Solve Subproblem + subprob = IterativeSubproblem(x=0, + fun=lambda x: 0, + jac=lambda x: np.array(g), + hess=lambda x: np.array(H)) + p, hits_boundary = subprob.solve(1.1) + + assert_array_almost_equal(p, [-0.68585435, 0.1222621, -0.22090999, + -0.67005053, 0.31586769]) + assert_array_almost_equal(hits_boundary, False) + assert_array_almost_equal(subprob.lambda_current, 0) + assert_array_almost_equal(subprob.niter, 1) + + def test_for_jac_equal_zero(self): + + H = [[0.88547534, 2.90692271, 0.98440885, -0.78911503, -0.28035809], + [2.90692271, -0.04618819, 0.32867263, -0.83737945, 0.17116396], + [0.98440885, 0.32867263, -0.87355957, -0.06521957, -1.43030957], + [-0.78911503, -0.83737945, -0.06521957, -1.645709, -0.33887298], + [-0.28035809, 0.17116396, -1.43030957, -0.33887298, -1.68586978]] + + g = [0, 0, 0, 0, 0] + + # Solve Subproblem + subprob = IterativeSubproblem(x=0, + fun=lambda x: 0, + jac=lambda x: np.array(g), + hess=lambda x: np.array(H), + k_easy=1e-10, + k_hard=1e-10) + p, hits_boundary = subprob.solve(1.1) + + assert_array_almost_equal(p, [0.06910534, -0.01432721, + -0.65311947, -0.23815972, + -0.84954934]) + assert_array_almost_equal(hits_boundary, True) + + def test_for_jac_very_close_to_zero(self): + + H = [[0.88547534, 2.90692271, 0.98440885, -0.78911503, -0.28035809], + [2.90692271, -0.04618819, 0.32867263, -0.83737945, 0.17116396], + [0.98440885, 0.32867263, -0.87355957, -0.06521957, -1.43030957], + [-0.78911503, -0.83737945, -0.06521957, -1.645709, -0.33887298], + [-0.28035809, 0.17116396, -1.43030957, -0.33887298, -1.68586978]] + + g = [0, 0, 0, 0, 1e-15] + + # Solve Subproblem + subprob = IterativeSubproblem(x=0, + fun=lambda x: 0, + jac=lambda x: np.array(g), + hess=lambda x: np.array(H), + k_easy=1e-10, + k_hard=1e-10) + p, hits_boundary = subprob.solve(1.1) + + assert_array_almost_equal(p, [0.06910534, -0.01432721, + -0.65311947, -0.23815972, + -0.84954934]) + assert_array_almost_equal(hits_boundary, True) + + @pytest.mark.fail_slow(10) + def test_for_random_entries(self): + # Seed + np.random.seed(1) + + # Dimension + n = 5 + + for case in ('easy', 'hard', 'jac_equal_zero'): + + eig_limits = [(-20, -15), + (-10, -5), + (-10, 0), + (-5, 5), + (-10, 10), + (0, 10), + (5, 10), + (15, 20)] + + for min_eig, max_eig in eig_limits: + # Generate random symmetric matrix H with + # eigenvalues between min_eig and max_eig. + H, g = random_entry(n, min_eig, max_eig, case) + + # Trust radius + trust_radius_list = [0.1, 0.3, 0.6, 0.8, 1, 1.2, 3.3, 5.5, 10] + + for trust_radius in trust_radius_list: + # Solve subproblem with very high accuracy + subprob_ac = IterativeSubproblem(0, + lambda x: 0, + lambda x: g, + lambda x: H, + k_easy=1e-10, + k_hard=1e-10) + + p_ac, hits_boundary_ac = subprob_ac.solve(trust_radius) + + # Compute objective function value + J_ac = 1/2*np.dot(p_ac, np.dot(H, p_ac))+np.dot(g, p_ac) + + stop_criteria = [(0.1, 2), + (0.5, 1.1), + (0.9, 1.01)] + + for k_opt, k_trf in stop_criteria: + + # k_easy and k_hard computed in function + # of k_opt and k_trf accordingly to + # Conn, A. R., Gould, N. I., & Toint, P. L. (2000). + # "Trust region methods". Siam. p. 197. + k_easy = min(k_trf-1, + 1-np.sqrt(k_opt)) + k_hard = 1-k_opt + + # Solve subproblem + subprob = IterativeSubproblem(0, + lambda x: 0, + lambda x: g, + lambda x: H, + k_easy=k_easy, + k_hard=k_hard) + p, hits_boundary = subprob.solve(trust_radius) + + # Compute objective function value + J = 1/2*np.dot(p, np.dot(H, p))+np.dot(g, p) + + # Check if it respect k_trf + if hits_boundary: + assert_array_equal(np.abs(norm(p)-trust_radius) <= + (k_trf-1)*trust_radius, True) + else: + assert_equal(norm(p) <= trust_radius, True) + + # Check if it respect k_opt + assert_equal(J <= k_opt*J_ac, True) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_trustregion_krylov.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_trustregion_krylov.py new file mode 100644 index 0000000000000000000000000000000000000000..ee288c1b1348d280141bb62c09aba8fa67027142 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_trustregion_krylov.py @@ -0,0 +1,170 @@ +""" +Unit tests for Krylov space trust-region subproblem solver. + +""" +import pytest +import numpy as np +from scipy.optimize._trlib import (get_trlib_quadratic_subproblem) +from numpy.testing import (assert_, + assert_almost_equal, + assert_equal, assert_array_almost_equal) + +KrylovQP = get_trlib_quadratic_subproblem(tol_rel_i=1e-8, tol_rel_b=1e-6) +KrylovQP_disp = get_trlib_quadratic_subproblem(tol_rel_i=1e-8, tol_rel_b=1e-6, + disp=True) + +class TestKrylovQuadraticSubproblem: + + def test_for_the_easy_case(self): + + # `H` is chosen such that `g` is not orthogonal to the + # eigenvector associated with the smallest eigenvalue. + H = np.array([[1.0, 0.0, 4.0], + [0.0, 2.0, 0.0], + [4.0, 0.0, 3.0]]) + g = np.array([5.0, 0.0, 4.0]) + + # Trust Radius + trust_radius = 1.0 + + # Solve Subproblem + subprob = KrylovQP(x=0, + fun=lambda x: 0, + jac=lambda x: g, + hess=lambda x: None, + hessp=lambda x, y: H.dot(y)) + p, hits_boundary = subprob.solve(trust_radius) + + assert_array_almost_equal(p, np.array([-1.0, 0.0, 0.0])) + assert_equal(hits_boundary, True) + # check kkt satisfaction + assert_almost_equal( + np.linalg.norm(H.dot(p) + subprob.lam * p + g), + 0.0) + # check trust region constraint + assert_almost_equal(np.linalg.norm(p), trust_radius) + + trust_radius = 0.5 + p, hits_boundary = subprob.solve(trust_radius) + + assert_array_almost_equal(p, + np.array([-0.46125446, 0., -0.19298788])) + assert_equal(hits_boundary, True) + # check kkt satisfaction + assert_almost_equal( + np.linalg.norm(H.dot(p) + subprob.lam * p + g), + 0.0) + # check trust region constraint + assert_almost_equal(np.linalg.norm(p), trust_radius) + + def test_for_the_hard_case(self): + + # `H` is chosen such that `g` is orthogonal to the + # eigenvector associated with the smallest eigenvalue. + H = np.array([[1.0, 0.0, 4.0], + [0.0, 2.0, 0.0], + [4.0, 0.0, 3.0]]) + g = np.array([0.0, 2.0, 0.0]) + + # Trust Radius + trust_radius = 1.0 + + # Solve Subproblem + subprob = KrylovQP(x=0, + fun=lambda x: 0, + jac=lambda x: g, + hess=lambda x: None, + hessp=lambda x, y: H.dot(y)) + p, hits_boundary = subprob.solve(trust_radius) + + assert_array_almost_equal(p, np.array([0.0, -1.0, 0.0])) + # check kkt satisfaction + assert_almost_equal( + np.linalg.norm(H.dot(p) + subprob.lam * p + g), + 0.0) + # check trust region constraint + assert_almost_equal(np.linalg.norm(p), trust_radius) + + trust_radius = 0.5 + p, hits_boundary = subprob.solve(trust_radius) + + assert_array_almost_equal(p, np.array([0.0, -0.5, 0.0])) + # check kkt satisfaction + assert_almost_equal( + np.linalg.norm(H.dot(p) + subprob.lam * p + g), + 0.0) + # check trust region constraint + assert_almost_equal(np.linalg.norm(p), trust_radius) + + def test_for_interior_convergence(self): + + H = np.array([[1.812159, 0.82687265, 0.21838879, -0.52487006, 0.25436988], + [0.82687265, 2.66380283, 0.31508988, -0.40144163, 0.08811588], + [0.21838879, 0.31508988, 2.38020726, -0.3166346, 0.27363867], + [-0.52487006, -0.40144163, -0.3166346, 1.61927182, -0.42140166], + [0.25436988, 0.08811588, 0.27363867, -0.42140166, 1.33243101]]) + g = np.array([0.75798952, 0.01421945, 0.33847612, 0.83725004, -0.47909534]) + trust_radius = 1.1 + + # Solve Subproblem + subprob = KrylovQP(x=0, + fun=lambda x: 0, + jac=lambda x: g, + hess=lambda x: None, + hessp=lambda x, y: H.dot(y)) + p, hits_boundary = subprob.solve(trust_radius) + + # check kkt satisfaction + assert_almost_equal( + np.linalg.norm(H.dot(p) + subprob.lam * p + g), + 0.0) + + assert_array_almost_equal(p, [-0.68585435, 0.1222621, -0.22090999, + -0.67005053, 0.31586769]) + assert_array_almost_equal(hits_boundary, False) + + def test_for_very_close_to_zero(self): + + H = np.array([[0.88547534, 2.90692271, 0.98440885, -0.78911503, -0.28035809], + [2.90692271, -0.04618819, 0.32867263, -0.83737945, 0.17116396], + [0.98440885, 0.32867263, -0.87355957, -0.06521957, -1.43030957], + [-0.78911503, -0.83737945, -0.06521957, -1.645709, -0.33887298], + [-0.28035809, 0.17116396, -1.43030957, -0.33887298, -1.68586978]]) + g = np.array([0, 0, 0, 0, 1e-6]) + trust_radius = 1.1 + + # Solve Subproblem + subprob = KrylovQP(x=0, + fun=lambda x: 0, + jac=lambda x: g, + hess=lambda x: None, + hessp=lambda x, y: H.dot(y)) + p, hits_boundary = subprob.solve(trust_radius) + + # check kkt satisfaction + assert_almost_equal( + np.linalg.norm(H.dot(p) + subprob.lam * p + g), + 0.0) + # check trust region constraint + assert_almost_equal(np.linalg.norm(p), trust_radius) + + assert_array_almost_equal(p, [0.06910534, -0.01432721, + -0.65311947, -0.23815972, + -0.84954934]) + assert_array_almost_equal(hits_boundary, True) + + @pytest.mark.thread_unsafe + def test_disp(self, capsys): + H = -np.eye(5) + g = np.array([0, 0, 0, 0, 1e-6]) + trust_radius = 1.1 + + subprob = KrylovQP_disp(x=0, + fun=lambda x: 0, + jac=lambda x: g, + hess=lambda x: None, + hessp=lambda x, y: H.dot(y)) + p, hits_boundary = subprob.solve(trust_radius) + out, err = capsys.readouterr() + assert_(out.startswith(' TR Solving trust region problem'), repr(out)) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_zeros.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_zeros.py new file mode 100644 index 0000000000000000000000000000000000000000..99fedb181424a3669719f9d4703b739bb53fa8c4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tests/test_zeros.py @@ -0,0 +1,965 @@ +import pytest + +from functools import lru_cache + +from numpy.testing import (assert_warns, assert_, + assert_allclose, + assert_equal, + assert_array_equal, + suppress_warnings) +import numpy as np +from numpy import finfo, power, nan, isclose, sqrt, exp, sin, cos + +from scipy import optimize +from scipy.optimize import (_zeros_py as zeros, newton, root_scalar, + OptimizeResult) + +from scipy._lib._util import getfullargspec_no_self as _getfullargspec + +# Import testing parameters +from scipy.optimize._tstutils import get_tests, functions as tstutils_functions + +TOL = 4*np.finfo(float).eps # tolerance + +_FLOAT_EPS = finfo(float).eps + +bracket_methods = [zeros.bisect, zeros.ridder, zeros.brentq, zeros.brenth, + zeros.toms748] +gradient_methods = [zeros.newton] +all_methods = bracket_methods + gradient_methods + +# A few test functions used frequently: +# # A simple quadratic, (x-1)^2 - 1 +def f1(x): + return x ** 2 - 2 * x - 1 + + +def f1_1(x): + return 2 * x - 2 + + +def f1_2(x): + return 2.0 + 0 * x + + +def f1_and_p_and_pp(x): + return f1(x), f1_1(x), f1_2(x) + + +# Simple transcendental function +def f2(x): + return exp(x) - cos(x) + + +def f2_1(x): + return exp(x) + sin(x) + + +def f2_2(x): + return exp(x) + cos(x) + + +# lru cached function +@lru_cache +def f_lrucached(x): + return x + + +class TestScalarRootFinders: + # Basic tests for all scalar root finders + + xtol = 4 * np.finfo(float).eps + rtol = 4 * np.finfo(float).eps + + def _run_one_test(self, tc, method, sig_args_keys=None, + sig_kwargs_keys=None, **kwargs): + method_args = [] + for k in sig_args_keys or []: + if k not in tc: + # If a,b not present use x0, x1. Similarly for f and func + k = {'a': 'x0', 'b': 'x1', 'func': 'f'}.get(k, k) + method_args.append(tc[k]) + + method_kwargs = dict(**kwargs) + method_kwargs.update({'full_output': True, 'disp': False}) + for k in sig_kwargs_keys or []: + method_kwargs[k] = tc[k] + + root = tc.get('root') + func_args = tc.get('args', ()) + + try: + r, rr = method(*method_args, args=func_args, **method_kwargs) + return root, rr, tc + except Exception: + return root, zeros.RootResults(nan, -1, -1, zeros._EVALUEERR, method), tc + + def run_tests(self, tests, method, name, known_fail=None, **kwargs): + r"""Run test-cases using the specified method and the supplied signature. + + Extract the arguments for the method call from the test case + dictionary using the supplied keys for the method's signature.""" + # The methods have one of two base signatures: + # (f, a, b, **kwargs) # newton + # (func, x0, **kwargs) # bisect/brentq/... + + # FullArgSpec with args, varargs, varkw, defaults, ... + sig = _getfullargspec(method) + assert_(not sig.kwonlyargs) + nDefaults = len(sig.defaults) + nRequired = len(sig.args) - nDefaults + sig_args_keys = sig.args[:nRequired] + sig_kwargs_keys = [] + if name in ['secant', 'newton', 'halley']: + if name in ['newton', 'halley']: + sig_kwargs_keys.append('fprime') + if name in ['halley']: + sig_kwargs_keys.append('fprime2') + kwargs['tol'] = self.xtol + else: + kwargs['xtol'] = self.xtol + kwargs['rtol'] = self.rtol + + results = [list(self._run_one_test( + tc, method, sig_args_keys=sig_args_keys, + sig_kwargs_keys=sig_kwargs_keys, **kwargs)) for tc in tests] + # results= [[true root, full output, tc], ...] + + known_fail = known_fail or [] + notcvgd = [elt for elt in results if not elt[1].converged] + notcvgd = [elt for elt in notcvgd if elt[-1]['ID'] not in known_fail] + notcvged_IDS = [elt[-1]['ID'] for elt in notcvgd] + assert_equal([len(notcvged_IDS), notcvged_IDS], [0, []]) + + # The usable xtol and rtol depend on the test + tols = {'xtol': self.xtol, 'rtol': self.rtol} + tols.update(**kwargs) + rtol = tols['rtol'] + atol = tols.get('tol', tols['xtol']) + + cvgd = [elt for elt in results if elt[1].converged] + approx = [elt[1].root for elt in cvgd] + correct = [elt[0] for elt in cvgd] + # See if the root matches the reference value + notclose = [[a] + elt for a, c, elt in zip(approx, correct, cvgd) if + not isclose(a, c, rtol=rtol, atol=atol) + and elt[-1]['ID'] not in known_fail] + # If not, evaluate the function and see if is 0 at the purported root + fvs = [tc['f'](aroot, *tc.get('args', tuple())) + for aroot, c, fullout, tc in notclose] + notclose = [[fv] + elt for fv, elt in zip(fvs, notclose) if fv != 0] + assert_equal([notclose, len(notclose)], [[], 0]) + method_from_result = [result[1].method for result in results] + expected_method = [name for _ in results] + assert_equal(method_from_result, expected_method) + + def run_collection(self, collection, method, name, smoothness=None, + known_fail=None, **kwargs): + r"""Run a collection of tests using the specified method. + + The name is used to determine some optional arguments.""" + tests = get_tests(collection, smoothness=smoothness) + self.run_tests(tests, method, name, known_fail=known_fail, **kwargs) + + +class TestBracketMethods(TestScalarRootFinders): + @pytest.mark.parametrize('method', bracket_methods) + @pytest.mark.parametrize('function', tstutils_functions) + def test_basic_root_scalar(self, method, function): + # Tests bracketing root finders called via `root_scalar` on a small + # set of simple problems, each of which has a root at `x=1`. Checks for + # converged status and that the root was found. + a, b = .5, sqrt(3) + + r = root_scalar(function, method=method.__name__, bracket=[a, b], x0=a, + xtol=self.xtol, rtol=self.rtol) + assert r.converged + assert_allclose(r.root, 1.0, atol=self.xtol, rtol=self.rtol) + assert r.method == method.__name__ + + @pytest.mark.parametrize('method', bracket_methods) + @pytest.mark.parametrize('function', tstutils_functions) + def test_basic_individual(self, method, function): + # Tests individual bracketing root finders on a small set of simple + # problems, each of which has a root at `x=1`. Checks for converged + # status and that the root was found. + a, b = .5, sqrt(3) + root, r = method(function, a, b, xtol=self.xtol, rtol=self.rtol, + full_output=True) + + assert r.converged + assert_allclose(root, 1.0, atol=self.xtol, rtol=self.rtol) + + @pytest.mark.parametrize('method', bracket_methods) + @pytest.mark.parametrize('function', tstutils_functions) + def test_bracket_is_array(self, method, function): + # Test bracketing root finders called via `root_scalar` on a small set + # of simple problems, each of which has a root at `x=1`. Check that + # passing `bracket` as a `ndarray` is accepted and leads to finding the + # correct root. + a, b = .5, sqrt(3) + r = root_scalar(function, method=method.__name__, + bracket=np.array([a, b]), x0=a, xtol=self.xtol, + rtol=self.rtol) + assert r.converged + assert_allclose(r.root, 1.0, atol=self.xtol, rtol=self.rtol) + assert r.method == method.__name__ + + @pytest.mark.parametrize('method', bracket_methods) + def test_aps_collection(self, method): + self.run_collection('aps', method, method.__name__, smoothness=1) + + @pytest.mark.parametrize('method', [zeros.bisect, zeros.ridder, + zeros.toms748]) + def test_chandrupatla_collection(self, method): + known_fail = {'fun7.4'} if method == zeros.ridder else {} + self.run_collection('chandrupatla', method, method.__name__, + known_fail=known_fail) + + @pytest.mark.parametrize('method', bracket_methods) + def test_lru_cached_individual(self, method): + # check that https://github.com/scipy/scipy/issues/10846 is fixed + # (`root_scalar` failed when passed a function that was `@lru_cache`d) + a, b = -1, 1 + root, r = method(f_lrucached, a, b, full_output=True) + assert r.converged + assert_allclose(root, 0) + + +class TestNewton(TestScalarRootFinders): + def test_newton_collections(self): + known_fail = ['aps.13.00'] + known_fail += ['aps.12.05', 'aps.12.17'] # fails under Windows Py27 + for collection in ['aps', 'complex']: + self.run_collection(collection, zeros.newton, 'newton', + smoothness=2, known_fail=known_fail) + + def test_halley_collections(self): + known_fail = ['aps.12.06', 'aps.12.07', 'aps.12.08', 'aps.12.09', + 'aps.12.10', 'aps.12.11', 'aps.12.12', 'aps.12.13', + 'aps.12.14', 'aps.12.15', 'aps.12.16', 'aps.12.17', + 'aps.12.18', 'aps.13.00'] + for collection in ['aps', 'complex']: + self.run_collection(collection, zeros.newton, 'halley', + smoothness=2, known_fail=known_fail) + + def test_newton(self): + for f, f_1, f_2 in [(f1, f1_1, f1_2), (f2, f2_1, f2_2)]: + x = zeros.newton(f, 3, tol=1e-6) + assert_allclose(f(x), 0, atol=1e-6) + x = zeros.newton(f, 3, x1=5, tol=1e-6) # secant, x0 and x1 + assert_allclose(f(x), 0, atol=1e-6) + x = zeros.newton(f, 3, fprime=f_1, tol=1e-6) # newton + assert_allclose(f(x), 0, atol=1e-6) + x = zeros.newton(f, 3, fprime=f_1, fprime2=f_2, tol=1e-6) # halley + assert_allclose(f(x), 0, atol=1e-6) + + def test_newton_by_name(self): + r"""Invoke newton through root_scalar()""" + for f, f_1, f_2 in [(f1, f1_1, f1_2), (f2, f2_1, f2_2)]: + r = root_scalar(f, method='newton', x0=3, fprime=f_1, xtol=1e-6) + assert_allclose(f(r.root), 0, atol=1e-6) + for f, f_1, f_2 in [(f1, f1_1, f1_2), (f2, f2_1, f2_2)]: + r = root_scalar(f, method='newton', x0=3, xtol=1e-6) # without f' + assert_allclose(f(r.root), 0, atol=1e-6) + + def test_secant_by_name(self): + r"""Invoke secant through root_scalar()""" + for f, f_1, f_2 in [(f1, f1_1, f1_2), (f2, f2_1, f2_2)]: + r = root_scalar(f, method='secant', x0=3, x1=2, xtol=1e-6) + assert_allclose(f(r.root), 0, atol=1e-6) + r = root_scalar(f, method='secant', x0=3, x1=5, xtol=1e-6) + assert_allclose(f(r.root), 0, atol=1e-6) + for f, f_1, f_2 in [(f1, f1_1, f1_2), (f2, f2_1, f2_2)]: + r = root_scalar(f, method='secant', x0=3, xtol=1e-6) # without x1 + assert_allclose(f(r.root), 0, atol=1e-6) + + def test_halley_by_name(self): + r"""Invoke halley through root_scalar()""" + for f, f_1, f_2 in [(f1, f1_1, f1_2), (f2, f2_1, f2_2)]: + r = root_scalar(f, method='halley', x0=3, + fprime=f_1, fprime2=f_2, xtol=1e-6) + assert_allclose(f(r.root), 0, atol=1e-6) + + def test_root_scalar_fail(self): + message = 'fprime2 must be specified for halley' + with pytest.raises(ValueError, match=message): + root_scalar(f1, method='halley', fprime=f1_1, x0=3, xtol=1e-6) # no fprime2 + message = 'fprime must be specified for halley' + with pytest.raises(ValueError, match=message): + root_scalar(f1, method='halley', fprime2=f1_2, x0=3, xtol=1e-6) # no fprime + + def test_array_newton(self): + """test newton with array""" + + def f1(x, *a): + b = a[0] + x * a[3] + return a[1] - a[2] * (np.exp(b / a[5]) - 1.0) - b / a[4] - x + + def f1_1(x, *a): + b = a[3] / a[5] + return -a[2] * np.exp(a[0] / a[5] + x * b) * b - a[3] / a[4] - 1 + + def f1_2(x, *a): + b = a[3] / a[5] + return -a[2] * np.exp(a[0] / a[5] + x * b) * b**2 + + a0 = np.array([ + 5.32725221, 5.48673747, 5.49539973, + 5.36387202, 4.80237316, 1.43764452, + 5.23063958, 5.46094772, 5.50512718, + 5.42046290 + ]) + a1 = (np.sin(range(10)) + 1.0) * 7.0 + args = (a0, a1, 1e-09, 0.004, 10, 0.27456) + x0 = [7.0] * 10 + x = zeros.newton(f1, x0, f1_1, args) + x_expected = ( + 6.17264965, 11.7702805, 12.2219954, + 7.11017681, 1.18151293, 0.143707955, + 4.31928228, 10.5419107, 12.7552490, + 8.91225749 + ) + assert_allclose(x, x_expected) + # test halley's + x = zeros.newton(f1, x0, f1_1, args, fprime2=f1_2) + assert_allclose(x, x_expected) + # test secant + x = zeros.newton(f1, x0, args=args) + assert_allclose(x, x_expected) + + def test_array_newton_complex(self): + def f(x): + return x + 1+1j + + def fprime(x): + return 1.0 + + t = np.full(4, 1j) + x = zeros.newton(f, t, fprime=fprime) + assert_allclose(f(x), 0.) + + # should work even if x0 is not complex + t = np.ones(4) + x = zeros.newton(f, t, fprime=fprime) + assert_allclose(f(x), 0.) + + x = zeros.newton(f, t) + assert_allclose(f(x), 0.) + + def test_array_secant_active_zero_der(self): + """test secant doesn't continue to iterate zero derivatives""" + x = zeros.newton(lambda x, *a: x*x - a[0], x0=[4.123, 5], + args=[np.array([17, 25])]) + assert_allclose(x, (4.123105625617661, 5.0)) + + def test_array_newton_integers(self): + # test secant with float + x = zeros.newton(lambda y, z: z - y ** 2, [4.0] * 2, + args=([15.0, 17.0],)) + assert_allclose(x, (3.872983346207417, 4.123105625617661)) + # test integer becomes float + x = zeros.newton(lambda y, z: z - y ** 2, [4] * 2, args=([15, 17],)) + assert_allclose(x, (3.872983346207417, 4.123105625617661)) + + @pytest.mark.thread_unsafe + def test_array_newton_zero_der_failures(self): + # test derivative zero warning + assert_warns(RuntimeWarning, zeros.newton, + lambda y: y**2 - 2, [0., 0.], lambda y: 2 * y) + # test failures and zero_der + with pytest.warns(RuntimeWarning): + results = zeros.newton(lambda y: y**2 - 2, [0., 0.], + lambda y: 2*y, full_output=True) + assert_allclose(results.root, 0) + assert results.zero_der.all() + assert not results.converged.any() + + def test_newton_combined(self): + def f1(x): + return x ** 2 - 2 * x - 1 + def f1_1(x): + return 2 * x - 2 + def f1_2(x): + return 2.0 + 0 * x + + def f1_and_p_and_pp(x): + return x**2 - 2*x-1, 2*x-2, 2.0 + + sol0 = root_scalar(f1, method='newton', x0=3, fprime=f1_1) + sol = root_scalar(f1_and_p_and_pp, method='newton', x0=3, fprime=True) + assert_allclose(sol0.root, sol.root, atol=1e-8) + assert_equal(2*sol.function_calls, sol0.function_calls) + + sol0 = root_scalar(f1, method='halley', x0=3, fprime=f1_1, fprime2=f1_2) + sol = root_scalar(f1_and_p_and_pp, method='halley', x0=3, fprime2=True) + assert_allclose(sol0.root, sol.root, atol=1e-8) + assert_equal(3*sol.function_calls, sol0.function_calls) + + def test_newton_full_output(self, capsys): + # Test the full_output capability, both when converging and not. + # Use simple polynomials, to avoid hitting platform dependencies + # (e.g., exp & trig) in number of iterations + + x0 = 3 + expected_counts = [(6, 7), (5, 10), (3, 9)] + + for derivs in range(3): + kwargs = {'tol': 1e-6, 'full_output': True, } + for k, v in [['fprime', f1_1], ['fprime2', f1_2]][:derivs]: + kwargs[k] = v + + x, r = zeros.newton(f1, x0, disp=False, **kwargs) + assert_(r.converged) + assert_equal(x, r.root) + assert_equal((r.iterations, r.function_calls), expected_counts[derivs]) + if derivs == 0: + assert r.function_calls <= r.iterations + 1 + else: + assert_equal(r.function_calls, (derivs + 1) * r.iterations) + + # Now repeat, allowing one fewer iteration to force convergence failure + iters = r.iterations - 1 + x, r = zeros.newton(f1, x0, maxiter=iters, disp=False, **kwargs) + assert_(not r.converged) + assert_equal(x, r.root) + assert_equal(r.iterations, iters) + + if derivs == 1: + # Check that the correct Exception is raised and + # validate the start of the message. + msg = 'Failed to converge after %d iterations, value is .*' % (iters) + with pytest.raises(RuntimeError, match=msg): + x, r = zeros.newton(f1, x0, maxiter=iters, disp=True, **kwargs) + + @pytest.mark.thread_unsafe + def test_deriv_zero_warning(self): + def func(x): + return x ** 2 - 2.0 + def dfunc(x): + return 2 * x + assert_warns(RuntimeWarning, zeros.newton, func, 0.0, dfunc, disp=False) + with pytest.raises(RuntimeError, match='Derivative was zero'): + zeros.newton(func, 0.0, dfunc) + + def test_newton_does_not_modify_x0(self): + # https://github.com/scipy/scipy/issues/9964 + x0 = np.array([0.1, 3]) + x0_copy = x0.copy() # Copy to test for equality. + newton(np.sin, x0, np.cos) + assert_array_equal(x0, x0_copy) + + def test_gh17570_defaults(self): + # Previously, when fprime was not specified, root_scalar would default + # to secant. When x1 was not specified, secant failed. + # Check that without fprime, the default is secant if x1 is specified + # and newton otherwise. + # Also confirm that `x` is always a scalar (gh-21148) + def f(x): + assert np.isscalar(x) + return f1(x) + + res_newton_default = root_scalar(f, method='newton', x0=3, xtol=1e-6) + res_secant_default = root_scalar(f, method='secant', x0=3, x1=2, + xtol=1e-6) + # `newton` uses the secant method when `x1` and `x2` are specified + res_secant = newton(f, x0=3, x1=2, tol=1e-6, full_output=True)[1] + + # all three found a root + assert_allclose(f(res_newton_default.root), 0, atol=1e-6) + assert res_newton_default.root.shape == tuple() + assert_allclose(f(res_secant_default.root), 0, atol=1e-6) + assert res_secant_default.root.shape == tuple() + assert_allclose(f(res_secant.root), 0, atol=1e-6) + assert res_secant.root.shape == tuple() + + # Defaults are correct + assert (res_secant_default.root + == res_secant.root + != res_newton_default.iterations) + assert (res_secant_default.iterations + == res_secant_default.function_calls - 1 # true for secant + == res_secant.iterations + != res_newton_default.iterations + == res_newton_default.function_calls/2) # newton 2-point diff + + @pytest.mark.parametrize('kwargs', [dict(), {'method': 'newton'}]) + def test_args_gh19090(self, kwargs): + def f(x, a, b): + assert a == 3 + assert b == 1 + return (x ** a - b) + + res = optimize.root_scalar(f, x0=3, args=(3, 1), **kwargs) + assert res.converged + assert_allclose(res.root, 1) + + @pytest.mark.parametrize('method', ['secant', 'newton']) + def test_int_x0_gh19280(self, method): + # Originally, `newton` ensured that only floats were passed to the + # callable. This was inadvertently changed by gh-17669. Check that + # it has been changed back. + def f(x): + # an integer raised to a negative integer power would fail + return x**-2 - 2 + + res = optimize.root_scalar(f, x0=1, method=method) + assert res.converged + assert_allclose(abs(res.root), 2**-0.5) + assert res.root.dtype == np.dtype(np.float64) + + +def test_gh_5555(): + root = 0.1 + + def f(x): + return x - root + + methods = [zeros.bisect, zeros.ridder] + xtol = rtol = TOL + for method in methods: + res = method(f, -1e8, 1e7, xtol=xtol, rtol=rtol) + assert_allclose(root, res, atol=xtol, rtol=rtol, + err_msg=f'method {method.__name__}') + + +def test_gh_5557(): + # Show that without the changes in 5557 brentq and brenth might + # only achieve a tolerance of 2*(xtol + rtol*|res|). + + # f linearly interpolates (0, -0.1), (0.5, -0.1), and (1, + # 0.4). The important parts are that |f(0)| < |f(1)| (so that + # brent takes 0 as the initial guess), |f(0)| < atol (so that + # brent accepts 0 as the root), and that the exact root of f lies + # more than atol away from 0 (so that brent doesn't achieve the + # desired tolerance). + def f(x): + if x < 0.5: + return -0.1 + else: + return x - 0.6 + + atol = 0.51 + rtol = 4 * _FLOAT_EPS + methods = [zeros.brentq, zeros.brenth] + for method in methods: + res = method(f, 0, 1, xtol=atol, rtol=rtol) + assert_allclose(0.6, res, atol=atol, rtol=rtol) + + +def test_brent_underflow_in_root_bracketing(): + # Testing if an interval [a,b] brackets a zero of a function + # by checking f(a)*f(b) < 0 is not reliable when the product + # underflows/overflows. (reported in issue# 13737) + + underflow_scenario = (-450.0, -350.0, -400.0) + overflow_scenario = (350.0, 450.0, 400.0) + + for a, b, root in [underflow_scenario, overflow_scenario]: + c = np.exp(root) + for method in [zeros.brenth, zeros.brentq]: + res = method(lambda x: np.exp(x)-c, a, b) + assert_allclose(root, res) + + +class TestRootResults: + r = zeros.RootResults(root=1.0, iterations=44, function_calls=46, flag=0, + method="newton") + + def test_repr(self): + expected_repr = (" converged: True\n flag: converged" + "\n function_calls: 46\n iterations: 44\n" + " root: 1.0\n method: newton") + assert_equal(repr(self.r), expected_repr) + + def test_type(self): + assert isinstance(self.r, OptimizeResult) + + +def test_complex_halley(): + """Test Halley's works with complex roots""" + def f(x, *a): + return a[0] * x**2 + a[1] * x + a[2] + + def f_1(x, *a): + return 2 * a[0] * x + a[1] + + def f_2(x, *a): + retval = 2 * a[0] + try: + size = len(x) + except TypeError: + return retval + else: + return [retval] * size + + z = complex(1.0, 2.0) + coeffs = (2.0, 3.0, 4.0) + y = zeros.newton(f, z, args=coeffs, fprime=f_1, fprime2=f_2, tol=1e-6) + # (-0.75000000000000078+1.1989578808281789j) + assert_allclose(f(y, *coeffs), 0, atol=1e-6) + z = [z] * 10 + coeffs = (2.0, 3.0, 4.0) + y = zeros.newton(f, z, args=coeffs, fprime=f_1, fprime2=f_2, tol=1e-6) + assert_allclose(f(y, *coeffs), 0, atol=1e-6) + + +@pytest.mark.thread_unsafe +def test_zero_der_nz_dp(capsys): + """Test secant method with a non-zero dp, but an infinite newton step""" + # pick a symmetrical functions and choose a point on the side that with dx + # makes a secant that is a flat line with zero slope, EG: f = (x - 100)**2, + # which has a root at x = 100 and is symmetrical around the line x = 100 + # we have to pick a really big number so that it is consistently true + # now find a point on each side so that the secant has a zero slope + dx = np.finfo(float).eps ** 0.33 + # 100 - p0 = p1 - 100 = p0 * (1 + dx) + dx - 100 + # -> 200 = p0 * (2 + dx) + dx + p0 = (200.0 - dx) / (2.0 + dx) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "RMS of") + x = zeros.newton(lambda y: (y - 100.0)**2, x0=[p0] * 10) + assert_allclose(x, [100] * 10) + # test scalar cases too + p0 = (2.0 - 1e-4) / (2.0 + 1e-4) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "Tolerance of") + x = zeros.newton(lambda y: (y - 1.0) ** 2, x0=p0, disp=False) + assert_allclose(x, 1) + with pytest.raises(RuntimeError, match='Tolerance of'): + x = zeros.newton(lambda y: (y - 1.0) ** 2, x0=p0, disp=True) + p0 = (-2.0 + 1e-4) / (2.0 + 1e-4) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "Tolerance of") + x = zeros.newton(lambda y: (y + 1.0) ** 2, x0=p0, disp=False) + assert_allclose(x, -1) + with pytest.raises(RuntimeError, match='Tolerance of'): + x = zeros.newton(lambda y: (y + 1.0) ** 2, x0=p0, disp=True) + + +@pytest.mark.thread_unsafe +def test_array_newton_failures(): + """Test that array newton fails as expected""" + # p = 0.68 # [MPa] + # dp = -0.068 * 1e6 # [Pa] + # T = 323 # [K] + diameter = 0.10 # [m] + # L = 100 # [m] + roughness = 0.00015 # [m] + rho = 988.1 # [kg/m**3] + mu = 5.4790e-04 # [Pa*s] + u = 2.488 # [m/s] + reynolds_number = rho * u * diameter / mu # Reynolds number + + def colebrook_eqn(darcy_friction, re, dia): + return (1 / np.sqrt(darcy_friction) + + 2 * np.log10(roughness / 3.7 / dia + + 2.51 / re / np.sqrt(darcy_friction))) + + # only some failures + with pytest.warns(RuntimeWarning): + result = zeros.newton( + colebrook_eqn, x0=[0.01, 0.2, 0.02223, 0.3], maxiter=2, + args=[reynolds_number, diameter], full_output=True + ) + assert not result.converged.all() + # they all fail + with pytest.raises(RuntimeError): + result = zeros.newton( + colebrook_eqn, x0=[0.01] * 2, maxiter=2, + args=[reynolds_number, diameter], full_output=True + ) + + +# this test should **not** raise a RuntimeWarning +def test_gh8904_zeroder_at_root_fails(): + """Test that Newton or Halley don't warn if zero derivative at root""" + + # a function that has a zero derivative at it's root + def f_zeroder_root(x): + return x**3 - x**2 + + # should work with secant + r = zeros.newton(f_zeroder_root, x0=0) + assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol) + # test again with array + r = zeros.newton(f_zeroder_root, x0=[0]*10) + assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol) + + # 1st derivative + def fder(x): + return 3 * x**2 - 2 * x + + # 2nd derivative + def fder2(x): + return 6*x - 2 + + # should work with newton and halley + r = zeros.newton(f_zeroder_root, x0=0, fprime=fder) + assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol) + r = zeros.newton(f_zeroder_root, x0=0, fprime=fder, + fprime2=fder2) + assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol) + # test again with array + r = zeros.newton(f_zeroder_root, x0=[0]*10, fprime=fder) + assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol) + r = zeros.newton(f_zeroder_root, x0=[0]*10, fprime=fder, + fprime2=fder2) + assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol) + + # also test that if a root is found we do not raise RuntimeWarning even if + # the derivative is zero, EG: at x = 0.5, then fval = -0.125 and + # fder = -0.25 so the next guess is 0.5 - (-0.125/-0.5) = 0 which is the + # root, but if the solver continued with that guess, then it will calculate + # a zero derivative, so it should return the root w/o RuntimeWarning + r = zeros.newton(f_zeroder_root, x0=0.5, fprime=fder) + assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol) + # test again with array + r = zeros.newton(f_zeroder_root, x0=[0.5]*10, fprime=fder) + assert_allclose(r, 0, atol=zeros._xtol, rtol=zeros._rtol) + # doesn't apply to halley + + +def test_gh_8881(): + r"""Test that Halley's method realizes that the 2nd order adjustment + is too big and drops off to the 1st order adjustment.""" + n = 9 + + def f(x): + return power(x, 1.0/n) - power(n, 1.0/n) + + def fp(x): + return power(x, (1.0-n)/n)/n + + def fpp(x): + return power(x, (1.0-2*n)/n) * (1.0/n) * (1.0-n)/n + + x0 = 0.1 + # The root is at x=9. + # The function has positive slope, x0 < root. + # Newton succeeds in 8 iterations + rt, r = newton(f, x0, fprime=fp, full_output=True) + assert r.converged + # Before the Issue 8881/PR 8882, halley would send x in the wrong direction. + # Check that it now succeeds. + rt, r = newton(f, x0, fprime=fp, fprime2=fpp, full_output=True) + assert r.converged + + +def test_gh_9608_preserve_array_shape(): + """ + Test that shape is preserved for array inputs even if fprime or fprime2 is + scalar + """ + def f(x): + return x**2 + + def fp(x): + return 2 * x + + def fpp(x): + return 2 + + x0 = np.array([-2], dtype=np.float32) + rt, r = newton(f, x0, fprime=fp, fprime2=fpp, full_output=True) + assert r.converged + + x0_array = np.array([-2, -3], dtype=np.float32) + # This next invocation should fail + with pytest.raises(IndexError): + result = zeros.newton( + f, x0_array, fprime=fp, fprime2=fpp, full_output=True + ) + + def fpp_array(x): + return np.full(np.shape(x), 2, dtype=np.float32) + + result = zeros.newton( + f, x0_array, fprime=fp, fprime2=fpp_array, full_output=True + ) + assert result.converged.all() + + +@pytest.mark.parametrize( + "maximum_iterations,flag_expected", + [(10, zeros.CONVERR), (100, zeros.CONVERGED)]) +def test_gh9254_flag_if_maxiter_exceeded(maximum_iterations, flag_expected): + """ + Test that if the maximum iterations is exceeded that the flag is not + converged. + """ + result = zeros.brentq( + lambda x: ((1.2*x - 2.3)*x + 3.4)*x - 4.5, + -30, 30, (), 1e-6, 1e-6, maximum_iterations, + full_output=True, disp=False) + assert result[1].flag == flag_expected + if flag_expected == zeros.CONVERR: + # didn't converge because exceeded maximum iterations + assert result[1].iterations == maximum_iterations + elif flag_expected == zeros.CONVERGED: + # converged before maximum iterations + assert result[1].iterations < maximum_iterations + + +@pytest.mark.thread_unsafe +def test_gh9551_raise_error_if_disp_true(): + """Test that if disp is true then zero derivative raises RuntimeError""" + + def f(x): + return x*x + 1 + + def f_p(x): + return 2*x + + assert_warns(RuntimeWarning, zeros.newton, f, 1.0, f_p, disp=False) + with pytest.raises( + RuntimeError, + match=r'^Derivative was zero\. Failed to converge after \d+ iterations, ' + r'value is [+-]?\d*\.\d+\.$'): + zeros.newton(f, 1.0, f_p) + root = zeros.newton(f, complex(10.0, 10.0), f_p) + assert_allclose(root, complex(0.0, 1.0)) + + +@pytest.mark.parametrize('solver_name', + ['brentq', 'brenth', 'bisect', 'ridder', 'toms748']) +def test_gh3089_8394(solver_name): + # gh-3089 and gh-8394 reported that bracketing solvers returned incorrect + # results when they encountered NaNs. Check that this is resolved. + def f(x): + return np.nan + + solver = getattr(zeros, solver_name) + with pytest.raises(ValueError, match="The function value at x..."): + solver(f, 0, 1) + + +@pytest.mark.parametrize('method', + ['brentq', 'brenth', 'bisect', 'ridder', 'toms748']) +def test_gh18171(method): + # gh-3089 and gh-8394 reported that bracketing solvers returned incorrect + # results when they encountered NaNs. Check that `root_scalar` returns + # normally but indicates that convergence was unsuccessful. See gh-18171. + def f(x): + f._count += 1 + return np.nan + f._count = 0 + + res = root_scalar(f, bracket=(0, 1), method=method) + assert res.converged is False + assert res.flag.startswith("The function value at x") + assert res.function_calls == f._count + assert str(res.root) in res.flag + + +@pytest.mark.parametrize('solver_name', + ['brentq', 'brenth', 'bisect', 'ridder', 'toms748']) +@pytest.mark.parametrize('rs_interface', [True, False]) +def test_function_calls(solver_name, rs_interface): + # There do not appear to be checks that the bracketing solvers report the + # correct number of function evaluations. Check that this is the case. + solver = ((lambda f, a, b, **kwargs: root_scalar(f, bracket=(a, b))) + if rs_interface else getattr(zeros, solver_name)) + + def f(x): + f.calls += 1 + return x**2 - 1 + f.calls = 0 + + res = solver(f, 0, 10, full_output=True) + + if rs_interface: + assert res.function_calls == f.calls + else: + assert res[1].function_calls == f.calls + + +@pytest.mark.thread_unsafe +def test_gh_14486_converged_false(): + """Test that zero slope with secant method results in a converged=False""" + def lhs(x): + return x * np.exp(-x*x) - 0.07 + + with pytest.warns(RuntimeWarning, match='Tolerance of'): + res = root_scalar(lhs, method='secant', x0=-0.15, x1=1.0) + assert not res.converged + assert res.flag == 'convergence error' + + with pytest.warns(RuntimeWarning, match='Tolerance of'): + res = newton(lhs, x0=-0.15, x1=1.0, disp=False, full_output=True)[1] + assert not res.converged + assert res.flag == 'convergence error' + + +@pytest.mark.parametrize('solver_name', + ['brentq', 'brenth', 'bisect', 'ridder', 'toms748']) +@pytest.mark.parametrize('rs_interface', [True, False]) +def test_gh5584(solver_name, rs_interface): + # gh-5584 reported that an underflow can cause sign checks in the algorithm + # to fail. Check that this is resolved. + solver = ((lambda f, a, b, **kwargs: root_scalar(f, bracket=(a, b))) + if rs_interface else getattr(zeros, solver_name)) + + def f(x): + return 1e-200*x + + # Report failure when signs are the same + with pytest.raises(ValueError, match='...must have different signs'): + solver(f, -0.5, -0.4, full_output=True) + + # Solve successfully when signs are different + res = solver(f, -0.5, 0.4, full_output=True) + res = res if rs_interface else res[1] + assert res.converged + assert_allclose(res.root, 0, atol=1e-8) + + # Solve successfully when one side is negative zero + res = solver(f, -0.5, float('-0.0'), full_output=True) + res = res if rs_interface else res[1] + assert res.converged + assert_allclose(res.root, 0, atol=1e-8) + + +def test_gh13407(): + # gh-13407 reported that the message produced by `scipy.optimize.toms748` + # when `rtol < eps` is incorrect, and also that toms748 is unusual in + # accepting `rtol` as low as eps while other solvers raise at 4*eps. Check + # that the error message has been corrected and that `rtol=eps` can produce + # a lower function value than `rtol=4*eps`. + def f(x): + return x**3 - 2*x - 5 + + xtol = 1e-300 + eps = np.finfo(float).eps + x1 = zeros.toms748(f, 1e-10, 1e10, xtol=xtol, rtol=1*eps) + f1 = f(x1) + x4 = zeros.toms748(f, 1e-10, 1e10, xtol=xtol, rtol=4*eps) + f4 = f(x4) + assert f1 < f4 + + # using old-style syntax to get exactly the same message + message = fr"rtol too small \({eps/2:g} < {eps:g}\)" + with pytest.raises(ValueError, match=message): + zeros.toms748(f, 1e-10, 1e10, xtol=xtol, rtol=eps/2) + + +def test_newton_complex_gh10103(): + # gh-10103 reported a problem when `newton` is pass a Python complex x0, + # no `fprime` (secant method), and no `x1` (`x1` must be constructed). + # Check that this is resolved. + def f(z): + return z - 1 + res = newton(f, 1+1j) + assert_allclose(res, 1, atol=1e-12) + + res = root_scalar(f, x0=1+1j, x1=2+1.5j, method='secant') + assert_allclose(res.root, 1, atol=1e-12) + + +@pytest.mark.parametrize('method', all_methods) +def test_maxiter_int_check_gh10236(method): + # gh-10236 reported that the error message when `maxiter` is not an integer + # was difficult to interpret. Check that this was resolved (by gh-10907). + message = "'float' object cannot be interpreted as an integer" + with pytest.raises(TypeError, match=message): + method(f1, 0.0, 1.0, maxiter=72.45) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tnc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tnc.py new file mode 100644 index 0000000000000000000000000000000000000000..e0f66058bbcc501eb1303eb3075cb55705b93192 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/tnc.py @@ -0,0 +1,22 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'OptimizeResult', + 'fmin_tnc', + 'zeros', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="tnc", + private_modules=["_tnc"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/zeros.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/zeros.py new file mode 100644 index 0000000000000000000000000000000000000000..907d49d37fc1e7476e81a25dbbc0d3910cbbe004 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/optimize/zeros.py @@ -0,0 +1,26 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.optimize` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'RootResults', + 'bisect', + 'brenth', + 'brentq', + 'newton', + 'ridder', + 'toms748', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="optimize", module="zeros", + private_modules=["_zeros_py"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..a1eaa9243ff83caa8e8078adc0cfb268119a4c72 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/__init__.py @@ -0,0 +1,327 @@ +""" +======================================= +Signal processing (:mod:`scipy.signal`) +======================================= + +Convolution +=========== + +.. autosummary:: + :toctree: generated/ + + convolve -- N-D convolution. + correlate -- N-D correlation. + fftconvolve -- N-D convolution using the FFT. + oaconvolve -- N-D convolution using the overlap-add method. + convolve2d -- 2-D convolution (more options). + correlate2d -- 2-D correlation (more options). + sepfir2d -- Convolve with a 2-D separable FIR filter. + choose_conv_method -- Chooses faster of FFT and direct convolution methods. + correlation_lags -- Determines lag indices for 1D cross-correlation. + +B-splines +========= + +.. autosummary:: + :toctree: generated/ + + gauss_spline -- Gaussian approximation to the B-spline basis function. + cspline1d -- Coefficients for 1-D cubic (3rd order) B-spline. + qspline1d -- Coefficients for 1-D quadratic (2nd order) B-spline. + cspline2d -- Coefficients for 2-D cubic (3rd order) B-spline. + qspline2d -- Coefficients for 2-D quadratic (2nd order) B-spline. + cspline1d_eval -- Evaluate a cubic spline at the given points. + qspline1d_eval -- Evaluate a quadratic spline at the given points. + spline_filter -- Smoothing spline (cubic) filtering of a rank-2 array. + +Filtering +========= + +.. autosummary:: + :toctree: generated/ + + order_filter -- N-D order filter. + medfilt -- N-D median filter. + medfilt2d -- 2-D median filter (faster). + wiener -- N-D Wiener filter. + + symiirorder1 -- 2nd-order IIR filter (cascade of first-order systems). + symiirorder2 -- 4th-order IIR filter (cascade of second-order systems). + lfilter -- 1-D FIR and IIR digital linear filtering. + lfiltic -- Construct initial conditions for `lfilter`. + lfilter_zi -- Compute an initial state zi for the lfilter function that + -- corresponds to the steady state of the step response. + filtfilt -- A forward-backward filter. + savgol_filter -- Filter a signal using the Savitzky-Golay filter. + + deconvolve -- 1-D deconvolution using lfilter. + + sosfilt -- 1-D IIR digital linear filtering using + -- a second-order sections filter representation. + sosfilt_zi -- Compute an initial state zi for the sosfilt function that + -- corresponds to the steady state of the step response. + sosfiltfilt -- A forward-backward filter for second-order sections. + hilbert -- Compute 1-D analytic signal, using the Hilbert transform. + hilbert2 -- Compute 2-D analytic signal, using the Hilbert transform. + envelope -- Compute the envelope of a real- or complex-valued signal. + + decimate -- Downsample a signal. + detrend -- Remove linear and/or constant trends from data. + resample -- Resample using Fourier method. + resample_poly -- Resample using polyphase filtering method. + upfirdn -- Upsample, apply FIR filter, downsample. + +Filter design +============= + +.. autosummary:: + :toctree: generated/ + + bilinear -- Digital filter from an analog filter using + -- the bilinear transform. + bilinear_zpk -- Digital filter from an analog filter using + -- the bilinear transform. + findfreqs -- Find array of frequencies for computing filter response. + firls -- FIR filter design using least-squares error minimization. + firwin -- Windowed FIR filter design, with frequency response + -- defined as pass and stop bands. + firwin2 -- Windowed FIR filter design, with arbitrary frequency + -- response. + freqs -- Analog filter frequency response from TF coefficients. + freqs_zpk -- Analog filter frequency response from ZPK coefficients. + freqz -- Digital filter frequency response from TF coefficients. + freqz_sos -- Digital filter frequency response for SOS format filter. + freqz_zpk -- Digital filter frequency response from ZPK coefficients. + gammatone -- FIR and IIR gammatone filter design. + group_delay -- Digital filter group delay. + iirdesign -- IIR filter design given bands and gains. + iirfilter -- IIR filter design given order and critical frequencies. + kaiser_atten -- Compute the attenuation of a Kaiser FIR filter, given + -- the number of taps and the transition width at + -- discontinuities in the frequency response. + kaiser_beta -- Compute the Kaiser parameter beta, given the desired + -- FIR filter attenuation. + kaiserord -- Design a Kaiser window to limit ripple and width of + -- transition region. + minimum_phase -- Convert a linear phase FIR filter to minimum phase. + savgol_coeffs -- Compute the FIR filter coefficients for a Savitzky-Golay + -- filter. + remez -- Optimal FIR filter design. + + unique_roots -- Unique roots and their multiplicities. + residue -- Partial fraction expansion of b(s) / a(s). + residuez -- Partial fraction expansion of b(z) / a(z). + invres -- Inverse partial fraction expansion for analog filter. + invresz -- Inverse partial fraction expansion for digital filter. + BadCoefficients -- Warning on badly conditioned filter coefficients. + +Lower-level filter design functions: + +.. autosummary:: + :toctree: generated/ + + abcd_normalize -- Check state-space matrices and ensure they are rank-2. + band_stop_obj -- Band Stop Objective Function for order minimization. + besselap -- Return (z,p,k) for analog prototype of Bessel filter. + buttap -- Return (z,p,k) for analog prototype of Butterworth filter. + cheb1ap -- Return (z,p,k) for type I Chebyshev filter. + cheb2ap -- Return (z,p,k) for type II Chebyshev filter. + ellipap -- Return (z,p,k) for analog prototype of elliptic filter. + lp2bp -- Transform a lowpass filter prototype to a bandpass filter. + lp2bp_zpk -- Transform a lowpass filter prototype to a bandpass filter. + lp2bs -- Transform a lowpass filter prototype to a bandstop filter. + lp2bs_zpk -- Transform a lowpass filter prototype to a bandstop filter. + lp2hp -- Transform a lowpass filter prototype to a highpass filter. + lp2hp_zpk -- Transform a lowpass filter prototype to a highpass filter. + lp2lp -- Transform a lowpass filter prototype to a lowpass filter. + lp2lp_zpk -- Transform a lowpass filter prototype to a lowpass filter. + normalize -- Normalize polynomial representation of a transfer function. + + + +Matlab-style IIR filter design +============================== + +.. autosummary:: + :toctree: generated/ + + butter -- Butterworth + buttord + cheby1 -- Chebyshev Type I + cheb1ord + cheby2 -- Chebyshev Type II + cheb2ord + ellip -- Elliptic (Cauer) + ellipord + bessel -- Bessel (no order selection available -- try butterod) + iirnotch -- Design second-order IIR notch digital filter. + iirpeak -- Design second-order IIR peak (resonant) digital filter. + iircomb -- Design IIR comb filter. + +Continuous-time linear systems +============================== + +.. autosummary:: + :toctree: generated/ + + lti -- Continuous-time linear time invariant system base class. + StateSpace -- Linear time invariant system in state space form. + TransferFunction -- Linear time invariant system in transfer function form. + ZerosPolesGain -- Linear time invariant system in zeros, poles, gain form. + lsim -- Continuous-time simulation of output to linear system. + impulse -- Impulse response of linear, time-invariant (LTI) system. + step -- Step response of continuous-time LTI system. + freqresp -- Frequency response of a continuous-time LTI system. + bode -- Bode magnitude and phase data (continuous-time LTI). + +Discrete-time linear systems +============================ + +.. autosummary:: + :toctree: generated/ + + dlti -- Discrete-time linear time invariant system base class. + StateSpace -- Linear time invariant system in state space form. + TransferFunction -- Linear time invariant system in transfer function form. + ZerosPolesGain -- Linear time invariant system in zeros, poles, gain form. + dlsim -- Simulation of output to a discrete-time linear system. + dimpulse -- Impulse response of a discrete-time LTI system. + dstep -- Step response of a discrete-time LTI system. + dfreqresp -- Frequency response of a discrete-time LTI system. + dbode -- Bode magnitude and phase data (discrete-time LTI). + +LTI representations +=================== + +.. autosummary:: + :toctree: generated/ + + tf2zpk -- Transfer function to zero-pole-gain. + tf2sos -- Transfer function to second-order sections. + tf2ss -- Transfer function to state-space. + zpk2tf -- Zero-pole-gain to transfer function. + zpk2sos -- Zero-pole-gain to second-order sections. + zpk2ss -- Zero-pole-gain to state-space. + ss2tf -- State-pace to transfer function. + ss2zpk -- State-space to pole-zero-gain. + sos2zpk -- Second-order sections to zero-pole-gain. + sos2tf -- Second-order sections to transfer function. + cont2discrete -- Continuous-time to discrete-time LTI conversion. + place_poles -- Pole placement. + +Waveforms +========= + +.. autosummary:: + :toctree: generated/ + + chirp -- Frequency swept cosine signal, with several freq functions. + gausspulse -- Gaussian modulated sinusoid. + max_len_seq -- Maximum length sequence. + sawtooth -- Periodic sawtooth. + square -- Square wave. + sweep_poly -- Frequency swept cosine signal; freq is arbitrary polynomial. + unit_impulse -- Discrete unit impulse. + +Window functions +================ + +For window functions, see the `scipy.signal.windows` namespace. + +In the `scipy.signal` namespace, there is a convenience function to +obtain these windows by name: + +.. autosummary:: + :toctree: generated/ + + get_window -- Return a window of a given length and type. + +Peak finding +============ + +.. autosummary:: + :toctree: generated/ + + argrelmin -- Calculate the relative minima of data. + argrelmax -- Calculate the relative maxima of data. + argrelextrema -- Calculate the relative extrema of data. + find_peaks -- Find a subset of peaks inside a signal. + find_peaks_cwt -- Find peaks in a 1-D array with wavelet transformation. + peak_prominences -- Calculate the prominence of each peak in a signal. + peak_widths -- Calculate the width of each peak in a signal. + +Spectral analysis +================= + +.. autosummary:: + :toctree: generated/ + + periodogram -- Compute a (modified) periodogram. + welch -- Compute a periodogram using Welch's method. + csd -- Compute the cross spectral density, using Welch's method. + coherence -- Compute the magnitude squared coherence, using Welch's method. + spectrogram -- Compute the spectrogram (legacy). + lombscargle -- Computes the Lomb-Scargle periodogram. + vectorstrength -- Computes the vector strength. + ShortTimeFFT -- Interface for calculating the \ + :ref:`Short Time Fourier Transform ` and \ + its inverse. + stft -- Compute the Short Time Fourier Transform (legacy). + istft -- Compute the Inverse Short Time Fourier Transform (legacy). + check_COLA -- Check the COLA constraint for iSTFT reconstruction. + check_NOLA -- Check the NOLA constraint for iSTFT reconstruction. + +Chirp Z-transform and Zoom FFT +============================================ + +.. autosummary:: + :toctree: generated/ + + czt - Chirp z-transform convenience function + zoom_fft - Zoom FFT convenience function + CZT - Chirp z-transform function generator + ZoomFFT - Zoom FFT function generator + czt_points - Output the z-plane points sampled by a chirp z-transform + +The functions are simpler to use than the classes, but are less efficient when +using the same transform on many arrays of the same length, since they +repeatedly generate the same chirp signal with every call. In these cases, +use the classes to create a reusable function instead. + +""" + +from . import _sigtools, windows +from ._waveforms import * +from ._max_len_seq import max_len_seq +from ._upfirdn import upfirdn + +from ._spline import ( + sepfir2d +) + +from ._spline_filters import * +from ._filter_design import * +from ._fir_filter_design import * +from ._ltisys import * +from ._lti_conversion import * +from ._signaltools import * +from ._savitzky_golay import savgol_coeffs, savgol_filter +from ._spectral_py import * +from ._short_time_fft import * +from ._peak_finding import * +from ._czt import * +from .windows import get_window # keep this one in signal namespace + +# Deprecated namespaces, to be removed in v2.0.0 +from . import ( + bsplines, filter_design, fir_filter_design, lti_conversion, ltisys, + spectral, signaltools, waveforms, wavelets, spline +) + +__all__ = [ + s for s in dir() if not s.startswith("_") +] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_arraytools.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_arraytools.py new file mode 100644 index 0000000000000000000000000000000000000000..87ce75d8d892a64021da7abc5d149556c22cf983 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_arraytools.py @@ -0,0 +1,264 @@ +""" +Functions for acting on a axis of an array. +""" +import numpy as np + + +def axis_slice(a, start=None, stop=None, step=None, axis=-1): + """Take a slice along axis 'axis' from 'a'. + + Parameters + ---------- + a : numpy.ndarray + The array to be sliced. + start, stop, step : int or None + The slice parameters. + axis : int, optional + The axis of `a` to be sliced. + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal._arraytools import axis_slice + >>> a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) + >>> axis_slice(a, start=0, stop=1, axis=1) + array([[1], + [4], + [7]]) + >>> axis_slice(a, start=1, axis=0) + array([[4, 5, 6], + [7, 8, 9]]) + + Notes + ----- + The keyword arguments start, stop and step are used by calling + slice(start, stop, step). This implies axis_slice() does not + handle its arguments the exactly the same as indexing. To select + a single index k, for example, use + axis_slice(a, start=k, stop=k+1) + In this case, the length of the axis 'axis' in the result will + be 1; the trivial dimension is not removed. (Use numpy.squeeze() + to remove trivial axes.) + """ + a_slice = [slice(None)] * a.ndim + a_slice[axis] = slice(start, stop, step) + b = a[tuple(a_slice)] + return b + + +def axis_reverse(a, axis=-1): + """Reverse the 1-D slices of `a` along axis `axis`. + + Returns axis_slice(a, step=-1, axis=axis). + """ + return axis_slice(a, step=-1, axis=axis) + + +def odd_ext(x, n, axis=-1): + """ + Odd extension at the boundaries of an array + + Generate a new ndarray by making an odd extension of `x` along an axis. + + Parameters + ---------- + x : ndarray + The array to be extended. + n : int + The number of elements by which to extend `x` at each end of the axis. + axis : int, optional + The axis along which to extend `x`. Default is -1. + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal._arraytools import odd_ext + >>> a = np.array([[1, 2, 3, 4, 5], [0, 1, 4, 9, 16]]) + >>> odd_ext(a, 2) + array([[-1, 0, 1, 2, 3, 4, 5, 6, 7], + [-4, -1, 0, 1, 4, 9, 16, 23, 28]]) + + Odd extension is a "180 degree rotation" at the endpoints of the original + array: + + >>> t = np.linspace(0, 1.5, 100) + >>> a = 0.9 * np.sin(2 * np.pi * t**2) + >>> b = odd_ext(a, 40) + >>> import matplotlib.pyplot as plt + >>> plt.plot(np.arange(-40, 140), b, 'b', lw=1, label='odd extension') + >>> plt.plot(np.arange(100), a, 'r', lw=2, label='original') + >>> plt.legend(loc='best') + >>> plt.show() + """ + if n < 1: + return x + if n > x.shape[axis] - 1: + raise ValueError(("The extension length n (%d) is too big. " + + "It must not exceed x.shape[axis]-1, which is %d.") + % (n, x.shape[axis] - 1)) + left_end = axis_slice(x, start=0, stop=1, axis=axis) + left_ext = axis_slice(x, start=n, stop=0, step=-1, axis=axis) + right_end = axis_slice(x, start=-1, axis=axis) + right_ext = axis_slice(x, start=-2, stop=-(n + 2), step=-1, axis=axis) + ext = np.concatenate((2 * left_end - left_ext, + x, + 2 * right_end - right_ext), + axis=axis) + return ext + + +def even_ext(x, n, axis=-1): + """ + Even extension at the boundaries of an array + + Generate a new ndarray by making an even extension of `x` along an axis. + + Parameters + ---------- + x : ndarray + The array to be extended. + n : int + The number of elements by which to extend `x` at each end of the axis. + axis : int, optional + The axis along which to extend `x`. Default is -1. + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal._arraytools import even_ext + >>> a = np.array([[1, 2, 3, 4, 5], [0, 1, 4, 9, 16]]) + >>> even_ext(a, 2) + array([[ 3, 2, 1, 2, 3, 4, 5, 4, 3], + [ 4, 1, 0, 1, 4, 9, 16, 9, 4]]) + + Even extension is a "mirror image" at the boundaries of the original array: + + >>> t = np.linspace(0, 1.5, 100) + >>> a = 0.9 * np.sin(2 * np.pi * t**2) + >>> b = even_ext(a, 40) + >>> import matplotlib.pyplot as plt + >>> plt.plot(np.arange(-40, 140), b, 'b', lw=1, label='even extension') + >>> plt.plot(np.arange(100), a, 'r', lw=2, label='original') + >>> plt.legend(loc='best') + >>> plt.show() + """ + if n < 1: + return x + if n > x.shape[axis] - 1: + raise ValueError(("The extension length n (%d) is too big. " + + "It must not exceed x.shape[axis]-1, which is %d.") + % (n, x.shape[axis] - 1)) + left_ext = axis_slice(x, start=n, stop=0, step=-1, axis=axis) + right_ext = axis_slice(x, start=-2, stop=-(n + 2), step=-1, axis=axis) + ext = np.concatenate((left_ext, + x, + right_ext), + axis=axis) + return ext + + +def const_ext(x, n, axis=-1): + """ + Constant extension at the boundaries of an array + + Generate a new ndarray that is a constant extension of `x` along an axis. + + The extension repeats the values at the first and last element of + the axis. + + Parameters + ---------- + x : ndarray + The array to be extended. + n : int + The number of elements by which to extend `x` at each end of the axis. + axis : int, optional + The axis along which to extend `x`. Default is -1. + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal._arraytools import const_ext + >>> a = np.array([[1, 2, 3, 4, 5], [0, 1, 4, 9, 16]]) + >>> const_ext(a, 2) + array([[ 1, 1, 1, 2, 3, 4, 5, 5, 5], + [ 0, 0, 0, 1, 4, 9, 16, 16, 16]]) + + Constant extension continues with the same values as the endpoints of the + array: + + >>> t = np.linspace(0, 1.5, 100) + >>> a = 0.9 * np.sin(2 * np.pi * t**2) + >>> b = const_ext(a, 40) + >>> import matplotlib.pyplot as plt + >>> plt.plot(np.arange(-40, 140), b, 'b', lw=1, label='constant extension') + >>> plt.plot(np.arange(100), a, 'r', lw=2, label='original') + >>> plt.legend(loc='best') + >>> plt.show() + """ + if n < 1: + return x + left_end = axis_slice(x, start=0, stop=1, axis=axis) + ones_shape = [1] * x.ndim + ones_shape[axis] = n + ones = np.ones(ones_shape, dtype=x.dtype) + left_ext = ones * left_end + right_end = axis_slice(x, start=-1, axis=axis) + right_ext = ones * right_end + ext = np.concatenate((left_ext, + x, + right_ext), + axis=axis) + return ext + + +def zero_ext(x, n, axis=-1): + """ + Zero padding at the boundaries of an array + + Generate a new ndarray that is a zero-padded extension of `x` along + an axis. + + Parameters + ---------- + x : ndarray + The array to be extended. + n : int + The number of elements by which to extend `x` at each end of the + axis. + axis : int, optional + The axis along which to extend `x`. Default is -1. + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal._arraytools import zero_ext + >>> a = np.array([[1, 2, 3, 4, 5], [0, 1, 4, 9, 16]]) + >>> zero_ext(a, 2) + array([[ 0, 0, 1, 2, 3, 4, 5, 0, 0], + [ 0, 0, 0, 1, 4, 9, 16, 0, 0]]) + """ + if n < 1: + return x + zeros_shape = list(x.shape) + zeros_shape[axis] = n + zeros = np.zeros(zeros_shape, dtype=x.dtype) + ext = np.concatenate((zeros, x, zeros), axis=axis) + return ext + + +def _validate_fs(fs, allow_none=True): + """ + Check if the given sampling frequency is a scalar and raises an exception + otherwise. If allow_none is False, also raises an exception for none + sampling rates. Returns the sampling frequency as float or none if the + input is none. + """ + if fs is None: + if not allow_none: + raise ValueError("Sampling frequency can not be none.") + else: # should be float + if not np.isscalar(fs): + raise ValueError("Sampling frequency fs must be a single scalar.") + fs = float(fs) + return fs diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_czt.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_czt.py new file mode 100644 index 0000000000000000000000000000000000000000..c5e5715b460fb2719b68d4694474bc1efc0a9fa0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_czt.py @@ -0,0 +1,575 @@ +# This program is public domain +# Authors: Paul Kienzle, Nadav Horesh +""" +Chirp z-transform. + +We provide two interfaces to the chirp z-transform: an object interface +which precalculates part of the transform and can be applied efficiently +to many different data sets, and a functional interface which is applied +only to the given data set. + +Transforms +---------- + +CZT : callable (x, axis=-1) -> array + Define a chirp z-transform that can be applied to different signals. +ZoomFFT : callable (x, axis=-1) -> array + Define a Fourier transform on a range of frequencies. + +Functions +--------- + +czt : array + Compute the chirp z-transform for a signal. +zoom_fft : array + Compute the Fourier transform on a range of frequencies. +""" + +import cmath +import numbers +import numpy as np +from numpy import pi, arange +from scipy.fft import fft, ifft, next_fast_len + +__all__ = ['czt', 'zoom_fft', 'CZT', 'ZoomFFT', 'czt_points'] + + +def _validate_sizes(n, m): + if n < 1 or not isinstance(n, numbers.Integral): + raise ValueError('Invalid number of CZT data ' + f'points ({n}) specified. ' + 'n must be positive and integer type.') + + if m is None: + m = n + elif m < 1 or not isinstance(m, numbers.Integral): + raise ValueError('Invalid number of CZT output ' + f'points ({m}) specified. ' + 'm must be positive and integer type.') + + return m + + +def czt_points(m, w=None, a=1+0j): + """ + Return the points at which the chirp z-transform is computed. + + Parameters + ---------- + m : int + The number of points desired. + w : complex, optional + The ratio between points in each step. + Defaults to equally spaced points around the entire unit circle. + a : complex, optional + The starting point in the complex plane. Default is 1+0j. + + Returns + ------- + out : ndarray + The points in the Z plane at which `CZT` samples the z-transform, + when called with arguments `m`, `w`, and `a`, as complex numbers. + + See Also + -------- + CZT : Class that creates a callable chirp z-transform function. + czt : Convenience function for quickly calculating CZT. + + Examples + -------- + Plot the points of a 16-point FFT: + + >>> import numpy as np + >>> from scipy.signal import czt_points + >>> points = czt_points(16) + >>> import matplotlib.pyplot as plt + >>> plt.plot(points.real, points.imag, 'o') + >>> plt.gca().add_patch(plt.Circle((0,0), radius=1, fill=False, alpha=.3)) + >>> plt.axis('equal') + >>> plt.show() + + and a 91-point logarithmic spiral that crosses the unit circle: + + >>> m, w, a = 91, 0.995*np.exp(-1j*np.pi*.05), 0.8*np.exp(1j*np.pi/6) + >>> points = czt_points(m, w, a) + >>> plt.plot(points.real, points.imag, 'o') + >>> plt.gca().add_patch(plt.Circle((0,0), radius=1, fill=False, alpha=.3)) + >>> plt.axis('equal') + >>> plt.show() + """ + m = _validate_sizes(1, m) + + k = arange(m) + + a = 1.0 * a # at least float + + if w is None: + # Nothing specified, default to FFT + return a * np.exp(2j * pi * k / m) + else: + # w specified + w = 1.0 * w # at least float + return a * w**-k + + +class CZT: + """ + Create a callable chirp z-transform function. + + Transform to compute the frequency response around a spiral. + Objects of this class are callables which can compute the + chirp z-transform on their inputs. This object precalculates the constant + chirps used in the given transform. + + Parameters + ---------- + n : int + The size of the signal. + m : int, optional + The number of output points desired. Default is `n`. + w : complex, optional + The ratio between points in each step. This must be precise or the + accumulated error will degrade the tail of the output sequence. + Defaults to equally spaced points around the entire unit circle. + a : complex, optional + The starting point in the complex plane. Default is 1+0j. + + Returns + ------- + f : CZT + Callable object ``f(x, axis=-1)`` for computing the chirp z-transform + on `x`. + + See Also + -------- + czt : Convenience function for quickly calculating CZT. + ZoomFFT : Class that creates a callable partial FFT function. + + Notes + ----- + The defaults are chosen such that ``f(x)`` is equivalent to + ``fft.fft(x)`` and, if ``m > len(x)``, that ``f(x, m)`` is equivalent to + ``fft.fft(x, m)``. + + If `w` does not lie on the unit circle, then the transform will be + around a spiral with exponentially-increasing radius. Regardless, + angle will increase linearly. + + For transforms that do lie on the unit circle, accuracy is better when + using `ZoomFFT`, since any numerical error in `w` is + accumulated for long data lengths, drifting away from the unit circle. + + The chirp z-transform can be faster than an equivalent FFT with + zero padding. Try it with your own array sizes to see. + + However, the chirp z-transform is considerably less precise than the + equivalent zero-padded FFT. + + As this CZT is implemented using the Bluestein algorithm, it can compute + large prime-length Fourier transforms in O(N log N) time, rather than the + O(N**2) time required by the direct DFT calculation. (`scipy.fft` also + uses Bluestein's algorithm'.) + + (The name "chirp z-transform" comes from the use of a chirp in the + Bluestein algorithm. It does not decompose signals into chirps, like + other transforms with "chirp" in the name.) + + References + ---------- + .. [1] Leo I. Bluestein, "A linear filtering approach to the computation + of the discrete Fourier transform," Northeast Electronics Research + and Engineering Meeting Record 10, 218-219 (1968). + .. [2] Rabiner, Schafer, and Rader, "The chirp z-transform algorithm and + its application," Bell Syst. Tech. J. 48, 1249-1292 (1969). + + Examples + -------- + Compute multiple prime-length FFTs: + + >>> from scipy.signal import CZT + >>> import numpy as np + >>> a = np.random.rand(7) + >>> b = np.random.rand(7) + >>> c = np.random.rand(7) + >>> czt_7 = CZT(n=7) + >>> A = czt_7(a) + >>> B = czt_7(b) + >>> C = czt_7(c) + + Display the points at which the FFT is calculated: + + >>> czt_7.points() + array([ 1.00000000+0.j , 0.62348980+0.78183148j, + -0.22252093+0.97492791j, -0.90096887+0.43388374j, + -0.90096887-0.43388374j, -0.22252093-0.97492791j, + 0.62348980-0.78183148j]) + >>> import matplotlib.pyplot as plt + >>> plt.plot(czt_7.points().real, czt_7.points().imag, 'o') + >>> plt.gca().add_patch(plt.Circle((0,0), radius=1, fill=False, alpha=.3)) + >>> plt.axis('equal') + >>> plt.show() + """ + + def __init__(self, n, m=None, w=None, a=1+0j): + m = _validate_sizes(n, m) + + k = arange(max(m, n), dtype=np.min_scalar_type(-max(m, n)**2)) + + if w is None: + # Nothing specified, default to FFT-like + w = cmath.exp(-2j*pi/m) + wk2 = np.exp(-(1j * pi * ((k**2) % (2*m))) / m) + else: + # w specified + wk2 = w**(k**2/2.) + + a = 1.0 * a # at least float + + self.w, self.a = w, a + self.m, self.n = m, n + + nfft = next_fast_len(n + m - 1) + self._Awk2 = a**-k[:n] * wk2[:n] + self._nfft = nfft + self._Fwk2 = fft(1/np.hstack((wk2[n-1:0:-1], wk2[:m])), nfft) + self._wk2 = wk2[:m] + self._yidx = slice(n-1, n+m-1) + + def __call__(self, x, *, axis=-1): + """ + Calculate the chirp z-transform of a signal. + + Parameters + ---------- + x : array + The signal to transform. + axis : int, optional + Axis over which to compute the FFT. If not given, the last axis is + used. + + Returns + ------- + out : ndarray + An array of the same dimensions as `x`, but with the length of the + transformed axis set to `m`. + """ + x = np.asarray(x) + if x.shape[axis] != self.n: + raise ValueError(f"CZT defined for length {self.n}, not " + f"{x.shape[axis]}") + # Calculate transpose coordinates, to allow operation on any given axis + trnsp = np.arange(x.ndim) + trnsp[[axis, -1]] = [-1, axis] + x = x.transpose(*trnsp) + y = ifft(self._Fwk2 * fft(x*self._Awk2, self._nfft)) + y = y[..., self._yidx] * self._wk2 + return y.transpose(*trnsp) + + def points(self): + """ + Return the points at which the chirp z-transform is computed. + """ + return czt_points(self.m, self.w, self.a) + + +class ZoomFFT(CZT): + """ + Create a callable zoom FFT transform function. + + This is a specialization of the chirp z-transform (`CZT`) for a set of + equally-spaced frequencies around the unit circle, used to calculate a + section of the FFT more efficiently than calculating the entire FFT and + truncating. + + Parameters + ---------- + n : int + The size of the signal. + fn : array_like + A length-2 sequence [`f1`, `f2`] giving the frequency range, or a + scalar, for which the range [0, `fn`] is assumed. + m : int, optional + The number of points to evaluate. Default is `n`. + fs : float, optional + The sampling frequency. If ``fs=10`` represented 10 kHz, for example, + then `f1` and `f2` would also be given in kHz. + The default sampling frequency is 2, so `f1` and `f2` should be + in the range [0, 1] to keep the transform below the Nyquist + frequency. + endpoint : bool, optional + If True, `f2` is the last sample. Otherwise, it is not included. + Default is False. + + Returns + ------- + f : ZoomFFT + Callable object ``f(x, axis=-1)`` for computing the zoom FFT on `x`. + + See Also + -------- + zoom_fft : Convenience function for calculating a zoom FFT. + + Notes + ----- + The defaults are chosen such that ``f(x, 2)`` is equivalent to + ``fft.fft(x)`` and, if ``m > len(x)``, that ``f(x, 2, m)`` is equivalent to + ``fft.fft(x, m)``. + + Sampling frequency is 1/dt, the time step between samples in the + signal `x`. The unit circle corresponds to frequencies from 0 up + to the sampling frequency. The default sampling frequency of 2 + means that `f1`, `f2` values up to the Nyquist frequency are in the + range [0, 1). For `f1`, `f2` values expressed in radians, a sampling + frequency of 2*pi should be used. + + Remember that a zoom FFT can only interpolate the points of the existing + FFT. It cannot help to resolve two separate nearby frequencies. + Frequency resolution can only be increased by increasing acquisition + time. + + These functions are implemented using Bluestein's algorithm (as is + `scipy.fft`). [2]_ + + References + ---------- + .. [1] Steve Alan Shilling, "A study of the chirp z-transform and its + applications", pg 29 (1970) + https://krex.k-state.edu/dspace/bitstream/handle/2097/7844/LD2668R41972S43.pdf + .. [2] Leo I. Bluestein, "A linear filtering approach to the computation + of the discrete Fourier transform," Northeast Electronics Research + and Engineering Meeting Record 10, 218-219 (1968). + + Examples + -------- + To plot the transform results use something like the following: + + >>> import numpy as np + >>> from scipy.signal import ZoomFFT + >>> t = np.linspace(0, 1, 1021) + >>> x = np.cos(2*np.pi*15*t) + np.sin(2*np.pi*17*t) + >>> f1, f2 = 5, 27 + >>> transform = ZoomFFT(len(x), [f1, f2], len(x), fs=1021) + >>> X = transform(x) + >>> f = np.linspace(f1, f2, len(x)) + >>> import matplotlib.pyplot as plt + >>> plt.plot(f, 20*np.log10(np.abs(X))) + >>> plt.show() + """ + + def __init__(self, n, fn, m=None, *, fs=2, endpoint=False): + m = _validate_sizes(n, m) + + k = arange(max(m, n), dtype=np.min_scalar_type(-max(m, n)**2)) + + if np.size(fn) == 2: + f1, f2 = fn + elif np.size(fn) == 1: + f1, f2 = 0.0, fn + else: + raise ValueError('fn must be a scalar or 2-length sequence') + + self.f1, self.f2, self.fs = f1, f2, fs + + if endpoint: + scale = ((f2 - f1) * m) / (fs * (m - 1)) + else: + scale = (f2 - f1) / fs + a = cmath.exp(2j * pi * f1/fs) + wk2 = np.exp(-(1j * pi * scale * k**2) / m) + + self.w = cmath.exp(-2j*pi/m * scale) + self.a = a + self.m, self.n = m, n + + ak = np.exp(-2j * pi * f1/fs * k[:n]) + self._Awk2 = ak * wk2[:n] + + nfft = next_fast_len(n + m - 1) + self._nfft = nfft + self._Fwk2 = fft(1/np.hstack((wk2[n-1:0:-1], wk2[:m])), nfft) + self._wk2 = wk2[:m] + self._yidx = slice(n-1, n+m-1) + + +def czt(x, m=None, w=None, a=1+0j, *, axis=-1): + """ + Compute the frequency response around a spiral in the Z plane. + + Parameters + ---------- + x : array + The signal to transform. + m : int, optional + The number of output points desired. Default is the length of the + input data. + w : complex, optional + The ratio between points in each step. This must be precise or the + accumulated error will degrade the tail of the output sequence. + Defaults to equally spaced points around the entire unit circle. + a : complex, optional + The starting point in the complex plane. Default is 1+0j. + axis : int, optional + Axis over which to compute the FFT. If not given, the last axis is + used. + + Returns + ------- + out : ndarray + An array of the same dimensions as `x`, but with the length of the + transformed axis set to `m`. + + See Also + -------- + CZT : Class that creates a callable chirp z-transform function. + zoom_fft : Convenience function for partial FFT calculations. + + Notes + ----- + The defaults are chosen such that ``signal.czt(x)`` is equivalent to + ``fft.fft(x)`` and, if ``m > len(x)``, that ``signal.czt(x, m)`` is + equivalent to ``fft.fft(x, m)``. + + If the transform needs to be repeated, use `CZT` to construct a + specialized transform function which can be reused without + recomputing constants. + + An example application is in system identification, repeatedly evaluating + small slices of the z-transform of a system, around where a pole is + expected to exist, to refine the estimate of the pole's true location. [1]_ + + References + ---------- + .. [1] Steve Alan Shilling, "A study of the chirp z-transform and its + applications", pg 20 (1970) + https://krex.k-state.edu/dspace/bitstream/handle/2097/7844/LD2668R41972S43.pdf + + Examples + -------- + Generate a sinusoid: + + >>> import numpy as np + >>> f1, f2, fs = 8, 10, 200 # Hz + >>> t = np.linspace(0, 1, fs, endpoint=False) + >>> x = np.sin(2*np.pi*t*f2) + >>> import matplotlib.pyplot as plt + >>> plt.plot(t, x) + >>> plt.axis([0, 1, -1.1, 1.1]) + >>> plt.show() + + Its discrete Fourier transform has all of its energy in a single frequency + bin: + + >>> from scipy.fft import rfft, rfftfreq + >>> from scipy.signal import czt, czt_points + >>> plt.plot(rfftfreq(fs, 1/fs), abs(rfft(x))) + >>> plt.margins(0, 0.1) + >>> plt.show() + + However, if the sinusoid is logarithmically-decaying: + + >>> x = np.exp(-t*f1) * np.sin(2*np.pi*t*f2) + >>> plt.plot(t, x) + >>> plt.axis([0, 1, -1.1, 1.1]) + >>> plt.show() + + the DFT will have spectral leakage: + + >>> plt.plot(rfftfreq(fs, 1/fs), abs(rfft(x))) + >>> plt.margins(0, 0.1) + >>> plt.show() + + While the DFT always samples the z-transform around the unit circle, the + chirp z-transform allows us to sample the Z-transform along any + logarithmic spiral, such as a circle with radius smaller than unity: + + >>> M = fs // 2 # Just positive frequencies, like rfft + >>> a = np.exp(-f1/fs) # Starting point of the circle, radius < 1 + >>> w = np.exp(-1j*np.pi/M) # "Step size" of circle + >>> points = czt_points(M + 1, w, a) # M + 1 to include Nyquist + >>> plt.plot(points.real, points.imag, '.') + >>> plt.gca().add_patch(plt.Circle((0,0), radius=1, fill=False, alpha=.3)) + >>> plt.axis('equal'); plt.axis([-1.05, 1.05, -0.05, 1.05]) + >>> plt.show() + + With the correct radius, this transforms the decaying sinusoid (and others + with the same decay rate) without spectral leakage: + + >>> z_vals = czt(x, M + 1, w, a) # Include Nyquist for comparison to rfft + >>> freqs = np.angle(points)*fs/(2*np.pi) # angle = omega, radius = sigma + >>> plt.plot(freqs, abs(z_vals)) + >>> plt.margins(0, 0.1) + >>> plt.show() + """ + x = np.asarray(x) + transform = CZT(x.shape[axis], m=m, w=w, a=a) + return transform(x, axis=axis) + + +def zoom_fft(x, fn, m=None, *, fs=2, endpoint=False, axis=-1): + """ + Compute the DFT of `x` only for frequencies in range `fn`. + + Parameters + ---------- + x : array + The signal to transform. + fn : array_like + A length-2 sequence [`f1`, `f2`] giving the frequency range, or a + scalar, for which the range [0, `fn`] is assumed. + m : int, optional + The number of points to evaluate. The default is the length of `x`. + fs : float, optional + The sampling frequency. If ``fs=10`` represented 10 kHz, for example, + then `f1` and `f2` would also be given in kHz. + The default sampling frequency is 2, so `f1` and `f2` should be + in the range [0, 1] to keep the transform below the Nyquist + frequency. + endpoint : bool, optional + If True, `f2` is the last sample. Otherwise, it is not included. + Default is False. + axis : int, optional + Axis over which to compute the FFT. If not given, the last axis is + used. + + Returns + ------- + out : ndarray + The transformed signal. The Fourier transform will be calculated + at the points f1, f1+df, f1+2df, ..., f2, where df=(f2-f1)/m. + + See Also + -------- + ZoomFFT : Class that creates a callable partial FFT function. + + Notes + ----- + The defaults are chosen such that ``signal.zoom_fft(x, 2)`` is equivalent + to ``fft.fft(x)`` and, if ``m > len(x)``, that ``signal.zoom_fft(x, 2, m)`` + is equivalent to ``fft.fft(x, m)``. + + To graph the magnitude of the resulting transform, use:: + + plot(linspace(f1, f2, m, endpoint=False), abs(zoom_fft(x, [f1, f2], m))) + + If the transform needs to be repeated, use `ZoomFFT` to construct + a specialized transform function which can be reused without + recomputing constants. + + Examples + -------- + To plot the transform results use something like the following: + + >>> import numpy as np + >>> from scipy.signal import zoom_fft + >>> t = np.linspace(0, 1, 1021) + >>> x = np.cos(2*np.pi*15*t) + np.sin(2*np.pi*17*t) + >>> f1, f2 = 5, 27 + >>> X = zoom_fft(x, [f1, f2], len(x), fs=1021) + >>> f = np.linspace(f1, f2, len(x)) + >>> import matplotlib.pyplot as plt + >>> plt.plot(f, 20*np.log10(np.abs(X))) + >>> plt.show() + """ + x = np.asarray(x) + transform = ZoomFFT(x.shape[axis], fn, m=m, fs=fs, endpoint=endpoint) + return transform(x, axis=axis) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_filter_design.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_filter_design.py new file mode 100644 index 0000000000000000000000000000000000000000..0f177247c602cd529a064cc043d8052aa6cbc811 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_filter_design.py @@ -0,0 +1,5663 @@ +"""Filter design.""" +import math +import operator +import warnings + +import numpy as np +from numpy import (atleast_1d, poly, polyval, roots, real, asarray, + resize, pi, absolute, sqrt, tan, log10, + arcsinh, sin, exp, cosh, arccosh, ceil, conjugate, + zeros, sinh, append, concatenate, prod, ones, full, array, + mintypecode) +from numpy.polynomial.polynomial import polyval as npp_polyval +from numpy.polynomial.polynomial import polyvalfromroots + +from scipy import special, optimize, fft as sp_fft +from scipy.special import comb +from scipy._lib._util import float_factorial +from scipy.signal._arraytools import _validate_fs + + +__all__ = ['findfreqs', 'freqs', 'freqz', 'tf2zpk', 'zpk2tf', 'normalize', + 'lp2lp', 'lp2hp', 'lp2bp', 'lp2bs', 'bilinear', 'iirdesign', + 'iirfilter', 'butter', 'cheby1', 'cheby2', 'ellip', 'bessel', + 'band_stop_obj', 'buttord', 'cheb1ord', 'cheb2ord', 'ellipord', + 'buttap', 'cheb1ap', 'cheb2ap', 'ellipap', 'besselap', + 'BadCoefficients', 'freqs_zpk', 'freqz_zpk', + 'tf2sos', 'sos2tf', 'zpk2sos', 'sos2zpk', 'group_delay', + 'sosfreqz', 'freqz_sos', 'iirnotch', 'iirpeak', 'bilinear_zpk', + 'lp2lp_zpk', 'lp2hp_zpk', 'lp2bp_zpk', 'lp2bs_zpk', + 'gammatone', 'iircomb'] + + +class BadCoefficients(UserWarning): + """Warning about badly conditioned filter coefficients""" + pass + + +abs = absolute + + +def _is_int_type(x): + """ + Check if input is of a scalar integer type (so ``5`` and ``array(5)`` will + pass, while ``5.0`` and ``array([5])`` will fail. + """ + if np.ndim(x) != 0: + # Older versions of NumPy did not raise for np.array([1]).__index__() + # This is safe to remove when support for those versions is dropped + return False + try: + operator.index(x) + except TypeError: + return False + else: + return True + + +def findfreqs(num, den, N, kind='ba'): + """ + Find array of frequencies for computing the response of an analog filter. + + Parameters + ---------- + num, den : array_like, 1-D + The polynomial coefficients of the numerator and denominator of the + transfer function of the filter or LTI system, where the coefficients + are ordered from highest to lowest degree. Or, the roots of the + transfer function numerator and denominator (i.e., zeroes and poles). + N : int + The length of the array to be computed. + kind : str {'ba', 'zp'}, optional + Specifies whether the numerator and denominator are specified by their + polynomial coefficients ('ba'), or their roots ('zp'). + + Returns + ------- + w : (N,) ndarray + A 1-D array of frequencies, logarithmically spaced. + + Examples + -------- + Find a set of nine frequencies that span the "interesting part" of the + frequency response for the filter with the transfer function + + H(s) = s / (s^2 + 8s + 25) + + >>> from scipy import signal + >>> signal.findfreqs([1, 0], [1, 8, 25], N=9) + array([ 1.00000000e-02, 3.16227766e-02, 1.00000000e-01, + 3.16227766e-01, 1.00000000e+00, 3.16227766e+00, + 1.00000000e+01, 3.16227766e+01, 1.00000000e+02]) + """ + if kind == 'ba': + ep = atleast_1d(roots(den)) + 0j + tz = atleast_1d(roots(num)) + 0j + elif kind == 'zp': + ep = atleast_1d(den) + 0j + tz = atleast_1d(num) + 0j + else: + raise ValueError("input must be one of {'ba', 'zp'}") + + if len(ep) == 0: + ep = atleast_1d(-1000) + 0j + + ez = np.r_[ep[ep.imag >= 0], tz[(np.abs(tz) < 1e5) & (tz.imag >= 0)]] + + integ = np.abs(ez) < 1e-10 + hfreq = np.round(np.log10(np.max(3 * np.abs(ez.real + integ) + + 1.5 * ez.imag)) + 0.5) + lfreq = np.round(np.log10(0.1 * np.min(np.abs((ez + integ).real) + + 2 * ez.imag)) - 0.5) + + w = np.logspace(lfreq, hfreq, N) + return w + + +def freqs(b, a, worN=200, plot=None): + """ + Compute frequency response of analog filter. + + Given the M-order numerator `b` and N-order denominator `a` of an analog + filter, compute its frequency response:: + + b[0]*(jw)**M + b[1]*(jw)**(M-1) + ... + b[M] + H(w) = ---------------------------------------------- + a[0]*(jw)**N + a[1]*(jw)**(N-1) + ... + a[N] + + Parameters + ---------- + b : array_like + Numerator of a linear filter. + a : array_like + Denominator of a linear filter. + worN : {None, int, array_like}, optional + If None, then compute at 200 frequencies around the interesting parts + of the response curve (determined by pole-zero locations). If a single + integer, then compute at that many frequencies. Otherwise, compute the + response at the angular frequencies (e.g., rad/s) given in `worN`. + plot : callable, optional + A callable that takes two arguments. If given, the return parameters + `w` and `h` are passed to plot. Useful for plotting the frequency + response inside `freqs`. + + Returns + ------- + w : ndarray + The angular frequencies at which `h` was computed. + h : ndarray + The frequency response. + + See Also + -------- + freqz : Compute the frequency response of a digital filter. + + Notes + ----- + Using Matplotlib's "plot" function as the callable for `plot` produces + unexpected results, this plots the real part of the complex transfer + function, not the magnitude. Try ``lambda w, h: plot(w, abs(h))``. + + Examples + -------- + >>> from scipy.signal import freqs, iirfilter + >>> import numpy as np + + >>> b, a = iirfilter(4, [1, 10], 1, 60, analog=True, ftype='cheby1') + + >>> w, h = freqs(b, a, worN=np.logspace(-1, 2, 1000)) + + >>> import matplotlib.pyplot as plt + >>> plt.semilogx(w, 20 * np.log10(abs(h))) + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude response [dB]') + >>> plt.grid(True) + >>> plt.show() + + """ + if worN is None: + # For backwards compatibility + w = findfreqs(b, a, 200) + elif _is_int_type(worN): + w = findfreqs(b, a, worN) + else: + w = atleast_1d(worN) + + s = 1j * w + h = polyval(b, s) / polyval(a, s) + if plot is not None: + plot(w, h) + + return w, h + + +def freqs_zpk(z, p, k, worN=200): + """ + Compute frequency response of analog filter. + + Given the zeros `z`, poles `p`, and gain `k` of a filter, compute its + frequency response:: + + (jw-z[0]) * (jw-z[1]) * ... * (jw-z[-1]) + H(w) = k * ---------------------------------------- + (jw-p[0]) * (jw-p[1]) * ... * (jw-p[-1]) + + Parameters + ---------- + z : array_like + Zeroes of a linear filter + p : array_like + Poles of a linear filter + k : scalar + Gain of a linear filter + worN : {None, int, array_like}, optional + If None, then compute at 200 frequencies around the interesting parts + of the response curve (determined by pole-zero locations). If a single + integer, then compute at that many frequencies. Otherwise, compute the + response at the angular frequencies (e.g., rad/s) given in `worN`. + + Returns + ------- + w : ndarray + The angular frequencies at which `h` was computed. + h : ndarray + The frequency response. + + See Also + -------- + freqs : Compute the frequency response of an analog filter in TF form + freqz : Compute the frequency response of a digital filter in TF form + freqz_zpk : Compute the frequency response of a digital filter in ZPK form + + Notes + ----- + .. versionadded:: 0.19.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal import freqs_zpk, iirfilter + + >>> z, p, k = iirfilter(4, [1, 10], 1, 60, analog=True, ftype='cheby1', + ... output='zpk') + + >>> w, h = freqs_zpk(z, p, k, worN=np.logspace(-1, 2, 1000)) + + >>> import matplotlib.pyplot as plt + >>> plt.semilogx(w, 20 * np.log10(abs(h))) + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude response [dB]') + >>> plt.grid(True) + >>> plt.show() + + """ + k = np.asarray(k) + if k.size > 1: + raise ValueError('k must be a single scalar gain') + + if worN is None: + # For backwards compatibility + w = findfreqs(z, p, 200, kind='zp') + elif _is_int_type(worN): + w = findfreqs(z, p, worN, kind='zp') + else: + w = worN + + w = atleast_1d(w) + s = 1j * w + num = polyvalfromroots(s, z) + den = polyvalfromroots(s, p) + h = k * num/den + return w, h + + +def freqz(b, a=1, worN=512, whole=False, plot=None, fs=2*pi, + include_nyquist=False): + """ + Compute the frequency response of a digital filter. + + Given the M-order numerator `b` and N-order denominator `a` of a digital + filter, compute its frequency response:: + + jw -jw -jwM + jw B(e ) b[0] + b[1]e + ... + b[M]e + H(e ) = ------ = ----------------------------------- + jw -jw -jwN + A(e ) a[0] + a[1]e + ... + a[N]e + + Parameters + ---------- + b : array_like + Numerator of a linear filter. If `b` has dimension greater than 1, + it is assumed that the coefficients are stored in the first dimension, + and ``b.shape[1:]``, ``a.shape[1:]``, and the shape of the frequencies + array must be compatible for broadcasting. + a : array_like + Denominator of a linear filter. If `b` has dimension greater than 1, + it is assumed that the coefficients are stored in the first dimension, + and ``b.shape[1:]``, ``a.shape[1:]``, and the shape of the frequencies + array must be compatible for broadcasting. + worN : {None, int, array_like}, optional + If a single integer, then compute at that many frequencies (default is + N=512). This is a convenient alternative to:: + + np.linspace(0, fs if whole else fs/2, N, endpoint=include_nyquist) + + Using a number that is fast for FFT computations can result in + faster computations (see Notes). + + If an array_like, compute the response at the frequencies given. + These are in the same units as `fs`. + whole : bool, optional + Normally, frequencies are computed from 0 to the Nyquist frequency, + fs/2 (upper-half of unit-circle). If `whole` is True, compute + frequencies from 0 to fs. Ignored if worN is array_like. + plot : callable + A callable that takes two arguments. If given, the return parameters + `w` and `h` are passed to plot. Useful for plotting the frequency + response inside `freqz`. + fs : float, optional + The sampling frequency of the digital system. Defaults to 2*pi + radians/sample (so w is from 0 to pi). + + .. versionadded:: 1.2.0 + include_nyquist : bool, optional + If `whole` is False and `worN` is an integer, setting `include_nyquist` + to True will include the last frequency (Nyquist frequency) and is + otherwise ignored. + + .. versionadded:: 1.5.0 + + Returns + ------- + w : ndarray + The frequencies at which `h` was computed, in the same units as `fs`. + By default, `w` is normalized to the range [0, pi) (radians/sample). + h : ndarray + The frequency response, as complex numbers. + + See Also + -------- + freqz_zpk + freqz_sos + + Notes + ----- + Using Matplotlib's :func:`matplotlib.pyplot.plot` function as the callable + for `plot` produces unexpected results, as this plots the real part of the + complex transfer function, not the magnitude. + Try ``lambda w, h: plot(w, np.abs(h))``. + + A direct computation via (R)FFT is used to compute the frequency response + when the following conditions are met: + + 1. An integer value is given for `worN`. + 2. `worN` is fast to compute via FFT (i.e., + `next_fast_len(worN) ` equals `worN`). + 3. The denominator coefficients are a single value (``a.shape[0] == 1``). + 4. `worN` is at least as long as the numerator coefficients + (``worN >= b.shape[0]``). + 5. If ``b.ndim > 1``, then ``b.shape[-1] == 1``. + + For long FIR filters, the FFT approach can have lower error and be much + faster than the equivalent direct polynomial calculation. + + Examples + -------- + >>> from scipy import signal + >>> import numpy as np + >>> taps, f_c = 80, 1.0 # number of taps and cut-off frequency + >>> b = signal.firwin(taps, f_c, window=('kaiser', 8), fs=2*np.pi) + >>> w, h = signal.freqz(b) + + >>> import matplotlib.pyplot as plt + >>> fig, ax1 = plt.subplots(tight_layout=True) + >>> ax1.set_title(f"Frequency Response of {taps} tap FIR Filter" + + ... f"($f_c={f_c}$ rad/sample)") + >>> ax1.axvline(f_c, color='black', linestyle=':', linewidth=0.8) + >>> ax1.plot(w, 20 * np.log10(abs(h)), 'C0') + >>> ax1.set_ylabel("Amplitude in dB", color='C0') + >>> ax1.set(xlabel="Frequency in rad/sample", xlim=(0, np.pi)) + + >>> ax2 = ax1.twinx() + >>> phase = np.unwrap(np.angle(h)) + >>> ax2.plot(w, phase, 'C1') + >>> ax2.set_ylabel('Phase [rad]', color='C1') + >>> ax2.grid(True) + >>> ax2.axis('tight') + >>> plt.show() + + Broadcasting Examples + + Suppose we have two FIR filters whose coefficients are stored in the + rows of an array with shape (2, 25). For this demonstration, we'll + use random data: + + >>> rng = np.random.default_rng() + >>> b = rng.random((2, 25)) + + To compute the frequency response for these two filters with one call + to `freqz`, we must pass in ``b.T``, because `freqz` expects the first + axis to hold the coefficients. We must then extend the shape with a + trivial dimension of length 1 to allow broadcasting with the array + of frequencies. That is, we pass in ``b.T[..., np.newaxis]``, which has + shape (25, 2, 1): + + >>> w, h = signal.freqz(b.T[..., np.newaxis], worN=1024) + >>> w.shape + (1024,) + >>> h.shape + (2, 1024) + + Now, suppose we have two transfer functions, with the same numerator + coefficients ``b = [0.5, 0.5]``. The coefficients for the two denominators + are stored in the first dimension of the 2-D array `a`:: + + a = [ 1 1 ] + [ -0.25, -0.5 ] + + >>> b = np.array([0.5, 0.5]) + >>> a = np.array([[1, 1], [-0.25, -0.5]]) + + Only `a` is more than 1-D. To make it compatible for + broadcasting with the frequencies, we extend it with a trivial dimension + in the call to `freqz`: + + >>> w, h = signal.freqz(b, a[..., np.newaxis], worN=1024) + >>> w.shape + (1024,) + >>> h.shape + (2, 1024) + + """ + b = atleast_1d(b) + a = atleast_1d(a) + + fs = _validate_fs(fs, allow_none=False) + + if worN is None: + # For backwards compatibility + worN = 512 + + h = None + + if _is_int_type(worN): + N = operator.index(worN) + del worN + if N < 0: + raise ValueError(f'worN must be nonnegative, got {N}') + lastpoint = 2 * pi if whole else pi + # if include_nyquist is true and whole is false, w should + # include end point + w = np.linspace(0, lastpoint, N, + endpoint=include_nyquist and not whole) + n_fft = N if whole else 2 * (N - 1) if include_nyquist else 2 * N + if (a.size == 1 and (b.ndim == 1 or (b.shape[-1] == 1)) + and n_fft >= b.shape[0] + and n_fft > 0): # TODO: review threshold acc. to benchmark? + if np.isrealobj(b) and np.isrealobj(a): + fft_func = sp_fft.rfft + else: + fft_func = sp_fft.fft + h = fft_func(b, n=n_fft, axis=0)[:N] + h /= a + if fft_func is sp_fft.rfft and whole: + # exclude DC and maybe Nyquist (no need to use axis_reverse + # here because we can build reversal with the truncation) + stop = -1 if n_fft % 2 == 1 else -2 + h_flip = slice(stop, 0, -1) + h = np.concatenate((h, h[h_flip].conj())) + if b.ndim > 1: + # Last axis of h has length 1, so drop it. + h = h[..., 0] + # Move the first axis of h to the end. + h = np.moveaxis(h, 0, -1) + else: + w = atleast_1d(worN) + del worN + w = 2*pi*w/fs + + if h is None: # still need to compute using freqs w + zm1 = exp(-1j * w) + h = (npp_polyval(zm1, b, tensor=False) / + npp_polyval(zm1, a, tensor=False)) + + w = w*(fs/(2*pi)) + + if plot is not None: + plot(w, h) + + return w, h + + +def freqz_zpk(z, p, k, worN=512, whole=False, fs=2*pi): + r""" + Compute the frequency response of a digital filter in ZPK form. + + Given the Zeros, Poles and Gain of a digital filter, compute its frequency + response: + + :math:`H(z)=k \prod_i (z - Z[i]) / \prod_j (z - P[j])` + + where :math:`k` is the `gain`, :math:`Z` are the `zeros` and :math:`P` are + the `poles`. + + Parameters + ---------- + z : array_like + Zeroes of a linear filter + p : array_like + Poles of a linear filter + k : scalar + Gain of a linear filter + worN : {None, int, array_like}, optional + If a single integer, then compute at that many frequencies (default is + N=512). + + If an array_like, compute the response at the frequencies given. + These are in the same units as `fs`. + whole : bool, optional + Normally, frequencies are computed from 0 to the Nyquist frequency, + fs/2 (upper-half of unit-circle). If `whole` is True, compute + frequencies from 0 to fs. Ignored if w is array_like. + fs : float, optional + The sampling frequency of the digital system. Defaults to 2*pi + radians/sample (so w is from 0 to pi). + + .. versionadded:: 1.2.0 + + Returns + ------- + w : ndarray + The frequencies at which `h` was computed, in the same units as `fs`. + By default, `w` is normalized to the range [0, pi) (radians/sample). + h : ndarray + The frequency response, as complex numbers. + + See Also + -------- + freqs : Compute the frequency response of an analog filter in TF form + freqs_zpk : Compute the frequency response of an analog filter in ZPK form + freqz : Compute the frequency response of a digital filter in TF form + + Notes + ----- + .. versionadded:: 0.19.0 + + Examples + -------- + Design a 4th-order digital Butterworth filter with cut-off of 100 Hz in a + system with sample rate of 1000 Hz, and plot the frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> z, p, k = signal.butter(4, 100, output='zpk', fs=1000) + >>> w, h = signal.freqz_zpk(z, p, k, fs=1000) + + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> ax1 = fig.add_subplot(1, 1, 1) + >>> ax1.set_title('Digital filter frequency response') + + >>> ax1.plot(w, 20 * np.log10(abs(h)), 'b') + >>> ax1.set_ylabel('Amplitude [dB]', color='b') + >>> ax1.set_xlabel('Frequency [Hz]') + >>> ax1.grid(True) + + >>> ax2 = ax1.twinx() + >>> phase = np.unwrap(np.angle(h)) + >>> ax2.plot(w, phase, 'g') + >>> ax2.set_ylabel('Phase [rad]', color='g') + + >>> plt.axis('tight') + >>> plt.show() + + """ + z, p = map(atleast_1d, (z, p)) + + fs = _validate_fs(fs, allow_none=False) + + if whole: + lastpoint = 2 * pi + else: + lastpoint = pi + + if worN is None: + # For backwards compatibility + w = np.linspace(0, lastpoint, 512, endpoint=False) + elif _is_int_type(worN): + w = np.linspace(0, lastpoint, worN, endpoint=False) + else: + w = atleast_1d(worN) + w = 2*pi*w/fs + + zm1 = exp(1j * w) + h = k * polyvalfromroots(zm1, z) / polyvalfromroots(zm1, p) + + w = w*(fs/(2*pi)) + + return w, h + + +def group_delay(system, w=512, whole=False, fs=2*pi): + r"""Compute the group delay of a digital filter. + + The group delay measures by how many samples amplitude envelopes of + various spectral components of a signal are delayed by a filter. + It is formally defined as the derivative of continuous (unwrapped) phase:: + + d jw + D(w) = - -- arg H(e) + dw + + Parameters + ---------- + system : tuple of array_like (b, a) + Numerator and denominator coefficients of a filter transfer function. + w : {None, int, array_like}, optional + If a single integer, then compute at that many frequencies (default is + N=512). + + If an array_like, compute the delay at the frequencies given. These + are in the same units as `fs`. + whole : bool, optional + Normally, frequencies are computed from 0 to the Nyquist frequency, + fs/2 (upper-half of unit-circle). If `whole` is True, compute + frequencies from 0 to fs. Ignored if w is array_like. + fs : float, optional + The sampling frequency of the digital system. Defaults to 2*pi + radians/sample (so w is from 0 to pi). + + .. versionadded:: 1.2.0 + + Returns + ------- + w : ndarray + The frequencies at which group delay was computed, in the same units + as `fs`. By default, `w` is normalized to the range [0, pi) + (radians/sample). + gd : ndarray + The group delay. + + See Also + -------- + freqz : Frequency response of a digital filter + + Notes + ----- + The similar function in MATLAB is called `grpdelay`. + + If the transfer function :math:`H(z)` has zeros or poles on the unit + circle, the group delay at corresponding frequencies is undefined. + When such a case arises the warning is raised and the group delay + is set to 0 at those frequencies. + + For the details of numerical computation of the group delay refer to [1]_ or [2]_. + + .. versionadded:: 0.16.0 + + References + ---------- + .. [1] Richard G. Lyons, "Understanding Digital Signal Processing, + 3rd edition", p. 830. + .. [2] Julius O. Smith III, "Numerical Computation of Group Delay", + in "Introduction to Digital Filters with Audio Applications", + online book, 2007, + https://ccrma.stanford.edu/~jos/fp/Numerical_Computation_Group_Delay.html + + Examples + -------- + >>> from scipy import signal + >>> b, a = signal.iirdesign(0.1, 0.3, 5, 50, ftype='cheby1') + >>> w, gd = signal.group_delay((b, a)) + + >>> import matplotlib.pyplot as plt + >>> plt.title('Digital filter group delay') + >>> plt.plot(w, gd) + >>> plt.ylabel('Group delay [samples]') + >>> plt.xlabel('Frequency [rad/sample]') + >>> plt.show() + + """ + if w is None: + # For backwards compatibility + w = 512 + + fs = _validate_fs(fs, allow_none=False) + + if _is_int_type(w): + if whole: + w = np.linspace(0, 2 * pi, w, endpoint=False) + else: + w = np.linspace(0, pi, w, endpoint=False) + else: + w = np.atleast_1d(w) + w = 2*pi*w/fs + + b, a = map(np.atleast_1d, system) + c = np.convolve(b, conjugate(a[::-1])) + cr = c * np.arange(c.size) + z = np.exp(-1j * w) + num = np.polyval(cr[::-1], z) + den = np.polyval(c[::-1], z) + gd = np.real(num / den) - a.size + 1 + singular = ~np.isfinite(gd) + near_singular = np.absolute(den) < 10 * EPSILON + + if np.any(singular): + gd[singular] = 0 + warnings.warn( + "The group delay is singular at frequencies " + f"[{', '.join(f'{ws:.3f}' for ws in w[singular])}], setting to 0", + stacklevel=2 + ) + + elif np.any(near_singular): + warnings.warn( + "The filter's denominator is extremely small at frequencies " + f"[{', '.join(f'{ws:.3f}' for ws in w[near_singular])}], " + "around which a singularity may be present", + stacklevel=2 + ) + + w = w*(fs/(2*pi)) + + return w, gd + + +def _validate_sos(sos): + """Helper to validate a SOS input""" + sos = np.atleast_2d(sos) + if sos.ndim != 2: + raise ValueError('sos array must be 2D') + n_sections, m = sos.shape + if m != 6: + raise ValueError('sos array must be shape (n_sections, 6)') + if not (sos[:, 3] == 1).all(): + raise ValueError('sos[:, 3] should be all ones') + return sos, n_sections + + +def freqz_sos(sos, worN=512, whole=False, fs=2*pi): + r""" + Compute the frequency response of a digital filter in SOS format. + + Given `sos`, an array with shape (n, 6) of second order sections of + a digital filter, compute the frequency response of the system function:: + + B0(z) B1(z) B{n-1}(z) + H(z) = ----- * ----- * ... * --------- + A0(z) A1(z) A{n-1}(z) + + for z = exp(omega*1j), where B{k}(z) and A{k}(z) are numerator and + denominator of the transfer function of the k-th second order section. + + Parameters + ---------- + sos : array_like + Array of second-order filter coefficients, must have shape + ``(n_sections, 6)``. Each row corresponds to a second-order + section, with the first three columns providing the numerator + coefficients and the last three providing the denominator + coefficients. + worN : {None, int, array_like}, optional + If a single integer, then compute at that many frequencies (default is + N=512). Using a number that is fast for FFT computations can result + in faster computations (see Notes of `freqz`). + + If an array_like, compute the response at the frequencies given (must + be 1-D). These are in the same units as `fs`. + whole : bool, optional + Normally, frequencies are computed from 0 to the Nyquist frequency, + fs/2 (upper-half of unit-circle). If `whole` is True, compute + frequencies from 0 to fs. + fs : float, optional + The sampling frequency of the digital system. Defaults to 2*pi + radians/sample (so w is from 0 to pi). + + .. versionadded:: 1.2.0 + + Returns + ------- + w : ndarray + The frequencies at which `h` was computed, in the same units as `fs`. + By default, `w` is normalized to the range [0, pi) (radians/sample). + h : ndarray + The frequency response, as complex numbers. + + See Also + -------- + freqz, sosfilt + + Notes + ----- + .. versionadded:: 0.19.0 + + Examples + -------- + Design a 15th-order bandpass filter in SOS format. + + >>> from scipy import signal + >>> import numpy as np + >>> sos = signal.ellip(15, 0.5, 60, (0.2, 0.4), btype='bandpass', + ... output='sos') + + Compute the frequency response at 1500 points from DC to Nyquist. + + >>> w, h = signal.freqz_sos(sos, worN=1500) + + Plot the response. + + >>> import matplotlib.pyplot as plt + >>> plt.subplot(2, 1, 1) + >>> db = 20*np.log10(np.maximum(np.abs(h), 1e-5)) + >>> plt.plot(w/np.pi, db) + >>> plt.ylim(-75, 5) + >>> plt.grid(True) + >>> plt.yticks([0, -20, -40, -60]) + >>> plt.ylabel('Gain [dB]') + >>> plt.title('Frequency Response') + >>> plt.subplot(2, 1, 2) + >>> plt.plot(w/np.pi, np.angle(h)) + >>> plt.grid(True) + >>> plt.yticks([-np.pi, -0.5*np.pi, 0, 0.5*np.pi, np.pi], + ... [r'$-\pi$', r'$-\pi/2$', '0', r'$\pi/2$', r'$\pi$']) + >>> plt.ylabel('Phase [rad]') + >>> plt.xlabel('Normalized frequency (1.0 = Nyquist)') + >>> plt.show() + + If the same filter is implemented as a single transfer function, + numerical error corrupts the frequency response: + + >>> b, a = signal.ellip(15, 0.5, 60, (0.2, 0.4), btype='bandpass', + ... output='ba') + >>> w, h = signal.freqz(b, a, worN=1500) + >>> plt.subplot(2, 1, 1) + >>> db = 20*np.log10(np.maximum(np.abs(h), 1e-5)) + >>> plt.plot(w/np.pi, db) + >>> plt.ylim(-75, 5) + >>> plt.grid(True) + >>> plt.yticks([0, -20, -40, -60]) + >>> plt.ylabel('Gain [dB]') + >>> plt.title('Frequency Response') + >>> plt.subplot(2, 1, 2) + >>> plt.plot(w/np.pi, np.angle(h)) + >>> plt.grid(True) + >>> plt.yticks([-np.pi, -0.5*np.pi, 0, 0.5*np.pi, np.pi], + ... [r'$-\pi$', r'$-\pi/2$', '0', r'$\pi/2$', r'$\pi$']) + >>> plt.ylabel('Phase [rad]') + >>> plt.xlabel('Normalized frequency (1.0 = Nyquist)') + >>> plt.show() + + """ + fs = _validate_fs(fs, allow_none=False) + + sos, n_sections = _validate_sos(sos) + if n_sections == 0: + raise ValueError('Cannot compute frequencies with no sections') + h = 1. + for row in sos: + w, rowh = freqz(row[:3], row[3:], worN=worN, whole=whole, fs=fs) + h *= rowh + return w, h + + +def sosfreqz(*args, **kwargs): + """ + Compute the frequency response of a digital filter in SOS format. + + .. warning:: This function is an alias, provided for backward + compatibility. New code should use the function + :func:`scipy.signal.freqz_sos`. + """ + return freqz_sos(*args, **kwargs) + + +def _cplxreal(z, tol=None): + """ + Split into complex and real parts, combining conjugate pairs. + + The 1-D input vector `z` is split up into its complex (`zc`) and real (`zr`) + elements. Every complex element must be part of a complex-conjugate pair, + which are combined into a single number (with positive imaginary part) in + the output. Two complex numbers are considered a conjugate pair if their + real and imaginary parts differ in magnitude by less than ``tol * abs(z)``. + + Parameters + ---------- + z : array_like + Vector of complex numbers to be sorted and split + tol : float, optional + Relative tolerance for testing realness and conjugate equality. + Default is ``100 * spacing(1)`` of `z`'s data type (i.e., 2e-14 for + float64) + + Returns + ------- + zc : ndarray + Complex elements of `z`, with each pair represented by a single value + having positive imaginary part, sorted first by real part, and then + by magnitude of imaginary part. The pairs are averaged when combined + to reduce error. + zr : ndarray + Real elements of `z` (those having imaginary part less than + `tol` times their magnitude), sorted by value. + + Raises + ------ + ValueError + If there are any complex numbers in `z` for which a conjugate + cannot be found. + + See Also + -------- + _cplxpair + + Examples + -------- + >>> from scipy.signal._filter_design import _cplxreal + >>> a = [4, 3, 1, 2-2j, 2+2j, 2-1j, 2+1j, 2-1j, 2+1j, 1+1j, 1-1j] + >>> zc, zr = _cplxreal(a) + >>> print(zc) + [ 1.+1.j 2.+1.j 2.+1.j 2.+2.j] + >>> print(zr) + [ 1. 3. 4.] + """ + + z = atleast_1d(z) + if z.size == 0: + return z, z + elif z.ndim != 1: + raise ValueError('_cplxreal only accepts 1-D input') + + if tol is None: + # Get tolerance from dtype of input + tol = 100 * np.finfo((1.0 * z).dtype).eps + + # Sort by real part, magnitude of imaginary part (speed up further sorting) + z = z[np.lexsort((abs(z.imag), z.real))] + + # Split reals from conjugate pairs + real_indices = abs(z.imag) <= tol * abs(z) + zr = z[real_indices].real + + if len(zr) == len(z): + # Input is entirely real + return array([]), zr + + # Split positive and negative halves of conjugates + z = z[~real_indices] + zp = z[z.imag > 0] + zn = z[z.imag < 0] + + if len(zp) != len(zn): + raise ValueError('Array contains complex value with no matching ' + 'conjugate.') + + # Find runs of (approximately) the same real part + same_real = np.diff(zp.real) <= tol * abs(zp[:-1]) + diffs = np.diff(concatenate(([0], same_real, [0]))) + run_starts = np.nonzero(diffs > 0)[0] + run_stops = np.nonzero(diffs < 0)[0] + + # Sort each run by their imaginary parts + for i in range(len(run_starts)): + start = run_starts[i] + stop = run_stops[i] + 1 + for chunk in (zp[start:stop], zn[start:stop]): + chunk[...] = chunk[np.lexsort([abs(chunk.imag)])] + + # Check that negatives match positives + if any(abs(zp - zn.conj()) > tol * abs(zn)): + raise ValueError('Array contains complex value with no matching ' + 'conjugate.') + + # Average out numerical inaccuracy in real vs imag parts of pairs + zc = (zp + zn.conj()) / 2 + + return zc, zr + + +def _cplxpair(z, tol=None): + """ + Sort into pairs of complex conjugates. + + Complex conjugates in `z` are sorted by increasing real part. In each + pair, the number with negative imaginary part appears first. + + If pairs have identical real parts, they are sorted by increasing + imaginary magnitude. + + Two complex numbers are considered a conjugate pair if their real and + imaginary parts differ in magnitude by less than ``tol * abs(z)``. The + pairs are forced to be exact complex conjugates by averaging the positive + and negative values. + + Purely real numbers are also sorted, but placed after the complex + conjugate pairs. A number is considered real if its imaginary part is + smaller than `tol` times the magnitude of the number. + + Parameters + ---------- + z : array_like + 1-D input array to be sorted. + tol : float, optional + Relative tolerance for testing realness and conjugate equality. + Default is ``100 * spacing(1)`` of `z`'s data type (i.e., 2e-14 for + float64) + + Returns + ------- + y : ndarray + Complex conjugate pairs followed by real numbers. + + Raises + ------ + ValueError + If there are any complex numbers in `z` for which a conjugate + cannot be found. + + See Also + -------- + _cplxreal + + Examples + -------- + >>> from scipy.signal._filter_design import _cplxpair + >>> a = [4, 3, 1, 2-2j, 2+2j, 2-1j, 2+1j, 2-1j, 2+1j, 1+1j, 1-1j] + >>> z = _cplxpair(a) + >>> print(z) + [ 1.-1.j 1.+1.j 2.-1.j 2.+1.j 2.-1.j 2.+1.j 2.-2.j 2.+2.j 1.+0.j + 3.+0.j 4.+0.j] + """ + + z = atleast_1d(z) + if z.size == 0 or np.isrealobj(z): + return np.sort(z) + + if z.ndim != 1: + raise ValueError('z must be 1-D') + + zc, zr = _cplxreal(z, tol) + + # Interleave complex values and their conjugates, with negative imaginary + # parts first in each pair + zc = np.dstack((zc.conj(), zc)).flatten() + z = np.append(zc, zr) + return z + + +def tf2zpk(b, a): + r"""Return zero, pole, gain (z, p, k) representation from a numerator, + denominator representation of a linear filter. + + Parameters + ---------- + b : array_like + Numerator polynomial coefficients. + a : array_like + Denominator polynomial coefficients. + + Returns + ------- + z : ndarray + Zeros of the transfer function. + p : ndarray + Poles of the transfer function. + k : float + System gain. + + Notes + ----- + If some values of `b` are too close to 0, they are removed. In that case, + a BadCoefficients warning is emitted. + + The `b` and `a` arrays are interpreted as coefficients for positive, + descending powers of the transfer function variable. So the inputs + :math:`b = [b_0, b_1, ..., b_M]` and :math:`a =[a_0, a_1, ..., a_N]` + can represent an analog filter of the form: + + .. math:: + + H(s) = \frac + {b_0 s^M + b_1 s^{(M-1)} + \cdots + b_M} + {a_0 s^N + a_1 s^{(N-1)} + \cdots + a_N} + + or a discrete-time filter of the form: + + .. math:: + + H(z) = \frac + {b_0 z^M + b_1 z^{(M-1)} + \cdots + b_M} + {a_0 z^N + a_1 z^{(N-1)} + \cdots + a_N} + + This "positive powers" form is found more commonly in controls + engineering. If `M` and `N` are equal (which is true for all filters + generated by the bilinear transform), then this happens to be equivalent + to the "negative powers" discrete-time form preferred in DSP: + + .. math:: + + H(z) = \frac + {b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}} + {a_0 + a_1 z^{-1} + \cdots + a_N z^{-N}} + + Although this is true for common filters, remember that this is not true + in the general case. If `M` and `N` are not equal, the discrete-time + transfer function coefficients must first be converted to the "positive + powers" form before finding the poles and zeros. + + Examples + -------- + Find the zeroes, poles and gain of + a filter with the transfer function + + .. math:: + + H(s) = \frac{3s^2}{s^2 + 5s + 13} + + >>> from scipy.signal import tf2zpk + >>> tf2zpk([3, 0, 0], [1, 5, 13]) + ( array([ 0. , 0. ]), + array([ -2.5+2.59807621j , -2.5-2.59807621j]), + 3.0) + """ + b, a = normalize(b, a) + b = (b + 0.0) / a[0] + a = (a + 0.0) / a[0] + k = b[0] + b /= b[0] + z = roots(b) + p = roots(a) + return z, p, k + + +def zpk2tf(z, p, k): + r""" + Return polynomial transfer function representation from zeros and poles + + Parameters + ---------- + z : array_like + Zeros of the transfer function. + p : array_like + Poles of the transfer function. + k : float + System gain. + + Returns + ------- + b : ndarray + Numerator polynomial coefficients. + a : ndarray + Denominator polynomial coefficients. + + Examples + -------- + Find the polynomial representation of a transfer function H(s) + using its 'zpk' (Zero-Pole-Gain) representation. + + .. math:: + + H(z) = 5 \frac + { (s - 2)(s - 6) } + { (s - 1)(s - 8) } + + >>> from scipy.signal import zpk2tf + >>> z = [2, 6] + >>> p = [1, 8] + >>> k = 5 + >>> zpk2tf(z, p, k) + ( array([ 5., -40., 60.]), array([ 1., -9., 8.])) + """ + z = atleast_1d(z) + k = atleast_1d(k) + if len(z.shape) > 1: + temp = poly(z[0]) + b = np.empty((z.shape[0], z.shape[1] + 1), temp.dtype.char) + if len(k) == 1: + k = [k[0]] * z.shape[0] + for i in range(z.shape[0]): + b[i] = k[i] * poly(z[i]) + else: + b = k * poly(z) + a = atleast_1d(poly(p)) + + # Use real output if possible. Copied from np.poly, since + # we can't depend on a specific version of numpy. + if issubclass(b.dtype.type, np.complexfloating): + # if complex roots are all complex conjugates, the roots are real. + roots = np.asarray(z, complex) + pos_roots = np.compress(roots.imag > 0, roots) + neg_roots = np.conjugate(np.compress(roots.imag < 0, roots)) + if len(pos_roots) == len(neg_roots): + if np.all(np.sort_complex(neg_roots) == np.sort_complex(pos_roots)): + b = b.real.copy() + + if issubclass(a.dtype.type, np.complexfloating): + # if complex roots are all complex conjugates, the roots are real. + roots = np.asarray(p, complex) + pos_roots = np.compress(roots.imag > 0, roots) + neg_roots = np.conjugate(np.compress(roots.imag < 0, roots)) + if len(pos_roots) == len(neg_roots): + if np.all(np.sort_complex(neg_roots) == np.sort_complex(pos_roots)): + a = a.real.copy() + + return b, a + + +def tf2sos(b, a, pairing=None, *, analog=False): + r""" + Return second-order sections from transfer function representation + + Parameters + ---------- + b : array_like + Numerator polynomial coefficients. + a : array_like + Denominator polynomial coefficients. + pairing : {None, 'nearest', 'keep_odd', 'minimal'}, optional + The method to use to combine pairs of poles and zeros into sections. + See `zpk2sos` for information and restrictions on `pairing` and + `analog` arguments. + analog : bool, optional + If True, system is analog, otherwise discrete. + + .. versionadded:: 1.8.0 + + Returns + ------- + sos : ndarray + Array of second-order filter coefficients, with shape + ``(n_sections, 6)``. See `sosfilt` for the SOS filter format + specification. + + See Also + -------- + zpk2sos, sosfilt + + Notes + ----- + It is generally discouraged to convert from TF to SOS format, since doing + so usually will not improve numerical precision errors. Instead, consider + designing filters in ZPK format and converting directly to SOS. TF is + converted to SOS by first converting to ZPK format, then converting + ZPK to SOS. + + .. versionadded:: 0.16.0 + + Examples + -------- + Find the 'sos' (second-order sections) of the transfer function H(s) + using its polynomial representation. + + .. math:: + + H(s) = \frac{s^2 - 3.5s - 2}{s^4 + 3s^3 - 15s^2 - 19s + 30} + + >>> from scipy.signal import tf2sos + >>> tf2sos([1, -3.5, -2], [1, 3, -15, -19, 30], analog=True) + array([[ 0. , 0. , 1. , 1. , 2. , -15. ], + [ 1. , -3.5, -2. , 1. , 1. , -2. ]]) + """ + return zpk2sos(*tf2zpk(b, a), pairing=pairing, analog=analog) + + +def sos2tf(sos): + r""" + Return a single transfer function from a series of second-order sections + + Parameters + ---------- + sos : array_like + Array of second-order filter coefficients, must have shape + ``(n_sections, 6)``. See `sosfilt` for the SOS filter format + specification. + + Returns + ------- + b : ndarray + Numerator polynomial coefficients. + a : ndarray + Denominator polynomial coefficients. + + Notes + ----- + .. versionadded:: 0.16.0 + + Examples + -------- + Find the polynomial representation of an elliptic filter + using its 'sos' (second-order sections) format. + + >>> from scipy.signal import sos2tf + >>> from scipy import signal + >>> sos = signal.ellip(1, 0.001, 50, 0.1, output='sos') + >>> sos2tf(sos) + ( array([0.91256522, 0.91256522, 0. ]), + array([1. , 0.82513043, 0. ])) + """ + sos = np.asarray(sos) + result_type = sos.dtype + if result_type.kind in 'bui': + result_type = np.float64 + + b = np.array([1], dtype=result_type) + a = np.array([1], dtype=result_type) + n_sections = sos.shape[0] + for section in range(n_sections): + b = np.polymul(b, sos[section, :3]) + a = np.polymul(a, sos[section, 3:]) + return b, a + + +def sos2zpk(sos): + """ + Return zeros, poles, and gain of a series of second-order sections + + Parameters + ---------- + sos : array_like + Array of second-order filter coefficients, must have shape + ``(n_sections, 6)``. See `sosfilt` for the SOS filter format + specification. + + Returns + ------- + z : ndarray + Zeros of the transfer function. + p : ndarray + Poles of the transfer function. + k : float + System gain. + + Notes + ----- + The number of zeros and poles returned will be ``n_sections * 2`` + even if some of these are (effectively) zero. + + .. versionadded:: 0.16.0 + """ + sos = np.asarray(sos) + n_sections = sos.shape[0] + z = np.zeros(n_sections*2, np.complex128) + p = np.zeros(n_sections*2, np.complex128) + k = 1. + for section in range(n_sections): + zpk = tf2zpk(sos[section, :3], sos[section, 3:]) + z[2*section:2*section+len(zpk[0])] = zpk[0] + p[2*section:2*section+len(zpk[1])] = zpk[1] + k *= zpk[2] + return z, p, k + + +def _nearest_real_complex_idx(fro, to, which): + """Get the next closest real or complex element based on distance""" + assert which in ('real', 'complex', 'any') + order = np.argsort(np.abs(fro - to)) + if which == 'any': + return order[0] + else: + mask = np.isreal(fro[order]) + if which == 'complex': + mask = ~mask + return order[np.nonzero(mask)[0][0]] + + +def _single_zpksos(z, p, k): + """Create one second-order section from up to two zeros and poles""" + sos = np.zeros(6) + b, a = zpk2tf(z, p, k) + sos[3-len(b):3] = b + sos[6-len(a):6] = a + return sos + + +def zpk2sos(z, p, k, pairing=None, *, analog=False): + """Return second-order sections from zeros, poles, and gain of a system + + Parameters + ---------- + z : array_like + Zeros of the transfer function. + p : array_like + Poles of the transfer function. + k : float + System gain. + pairing : {None, 'nearest', 'keep_odd', 'minimal'}, optional + The method to use to combine pairs of poles and zeros into sections. + If analog is False and pairing is None, pairing is set to 'nearest'; + if analog is True, pairing must be 'minimal', and is set to that if + it is None. + analog : bool, optional + If True, system is analog, otherwise discrete. + + .. versionadded:: 1.8.0 + + Returns + ------- + sos : ndarray + Array of second-order filter coefficients, with shape + ``(n_sections, 6)``. See `sosfilt` for the SOS filter format + specification. + + See Also + -------- + sosfilt + + Notes + ----- + The algorithm used to convert ZPK to SOS format is designed to + minimize errors due to numerical precision issues. The pairing + algorithm attempts to minimize the peak gain of each biquadratic + section. This is done by pairing poles with the nearest zeros, starting + with the poles closest to the unit circle for discrete-time systems, and + poles closest to the imaginary axis for continuous-time systems. + + ``pairing='minimal'`` outputs may not be suitable for `sosfilt`, + and ``analog=True`` outputs will never be suitable for `sosfilt`. + + *Algorithms* + + The steps in the ``pairing='nearest'``, ``pairing='keep_odd'``, + and ``pairing='minimal'`` algorithms are mostly shared. The + ``'nearest'`` algorithm attempts to minimize the peak gain, while + ``'keep_odd'`` minimizes peak gain under the constraint that + odd-order systems should retain one section as first order. + ``'minimal'`` is similar to ``'keep_odd'``, but no additional + poles or zeros are introduced + + The algorithm steps are as follows: + + As a pre-processing step for ``pairing='nearest'``, + ``pairing='keep_odd'``, add poles or zeros to the origin as + necessary to obtain the same number of poles and zeros for + pairing. If ``pairing == 'nearest'`` and there are an odd number + of poles, add an additional pole and a zero at the origin. + + The following steps are then iterated over until no more poles or + zeros remain: + + 1. Take the (next remaining) pole (complex or real) closest to the + unit circle (or imaginary axis, for ``analog=True``) to + begin a new filter section. + + 2. If the pole is real and there are no other remaining real poles [#]_, + add the closest real zero to the section and leave it as a first + order section. Note that after this step we are guaranteed to be + left with an even number of real poles, complex poles, real zeros, + and complex zeros for subsequent pairing iterations. + + 3. Else: + + 1. If the pole is complex and the zero is the only remaining real + zero*, then pair the pole with the *next* closest zero + (guaranteed to be complex). This is necessary to ensure that + there will be a real zero remaining to eventually create a + first-order section (thus keeping the odd order). + + 2. Else pair the pole with the closest remaining zero (complex or + real). + + 3. Proceed to complete the second-order section by adding another + pole and zero to the current pole and zero in the section: + + 1. If the current pole and zero are both complex, add their + conjugates. + + 2. Else if the pole is complex and the zero is real, add the + conjugate pole and the next closest real zero. + + 3. Else if the pole is real and the zero is complex, add the + conjugate zero and the real pole closest to those zeros. + + 4. Else (we must have a real pole and real zero) add the next + real pole closest to the unit circle, and then add the real + zero closest to that pole. + + .. [#] This conditional can only be met for specific odd-order inputs + with the ``pairing = 'keep_odd'`` or ``'minimal'`` methods. + + .. versionadded:: 0.16.0 + + Examples + -------- + + Design a 6th order low-pass elliptic digital filter for a system with a + sampling rate of 8000 Hz that has a pass-band corner frequency of + 1000 Hz. The ripple in the pass-band should not exceed 0.087 dB, and + the attenuation in the stop-band should be at least 90 dB. + + In the following call to `ellip`, we could use ``output='sos'``, + but for this example, we'll use ``output='zpk'``, and then convert + to SOS format with `zpk2sos`: + + >>> from scipy import signal + >>> import numpy as np + >>> z, p, k = signal.ellip(6, 0.087, 90, 1000/(0.5*8000), output='zpk') + + Now convert to SOS format. + + >>> sos = signal.zpk2sos(z, p, k) + + The coefficients of the numerators of the sections: + + >>> sos[:, :3] + array([[0.0014152 , 0.00248677, 0.0014152 ], + [1. , 0.72976874, 1. ], + [1. , 0.17607852, 1. ]]) + + The symmetry in the coefficients occurs because all the zeros are on the + unit circle. + + The coefficients of the denominators of the sections: + + >>> sos[:, 3:] + array([[ 1. , -1.32544025, 0.46989976], + [ 1. , -1.26118294, 0.62625924], + [ 1. , -1.2570723 , 0.8619958 ]]) + + The next example shows the effect of the `pairing` option. We have a + system with three poles and three zeros, so the SOS array will have + shape (2, 6). The means there is, in effect, an extra pole and an extra + zero at the origin in the SOS representation. + + >>> z1 = np.array([-1, -0.5-0.5j, -0.5+0.5j]) + >>> p1 = np.array([0.75, 0.8+0.1j, 0.8-0.1j]) + + With ``pairing='nearest'`` (the default), we obtain + + >>> signal.zpk2sos(z1, p1, 1) + array([[ 1. , 1. , 0.5 , 1. , -0.75, 0. ], + [ 1. , 1. , 0. , 1. , -1.6 , 0.65]]) + + The first section has the zeros {-0.5-0.05j, -0.5+0.5j} and the poles + {0, 0.75}, and the second section has the zeros {-1, 0} and poles + {0.8+0.1j, 0.8-0.1j}. Note that the extra pole and zero at the origin + have been assigned to different sections. + + With ``pairing='keep_odd'``, we obtain: + + >>> signal.zpk2sos(z1, p1, 1, pairing='keep_odd') + array([[ 1. , 1. , 0. , 1. , -0.75, 0. ], + [ 1. , 1. , 0.5 , 1. , -1.6 , 0.65]]) + + The extra pole and zero at the origin are in the same section. + The first section is, in effect, a first-order section. + + With ``pairing='minimal'``, the first-order section doesn't have + the extra pole and zero at the origin: + + >>> signal.zpk2sos(z1, p1, 1, pairing='minimal') + array([[ 0. , 1. , 1. , 0. , 1. , -0.75], + [ 1. , 1. , 0.5 , 1. , -1.6 , 0.65]]) + + """ + # TODO in the near future: + # 1. Add SOS capability to `filtfilt`, `freqz`, etc. somehow (#3259). + # 2. Make `decimate` use `sosfilt` instead of `lfilter`. + # 3. Make sosfilt automatically simplify sections to first order + # when possible. Note this might make `sosfiltfilt` a bit harder (ICs). + # 4. Further optimizations of the section ordering / pole-zero pairing. + # See the wiki for other potential issues. + + if pairing is None: + pairing = 'minimal' if analog else 'nearest' + + valid_pairings = ['nearest', 'keep_odd', 'minimal'] + if pairing not in valid_pairings: + raise ValueError(f'pairing must be one of {valid_pairings}, not {pairing}') + + if analog and pairing != 'minimal': + raise ValueError('for analog zpk2sos conversion, ' + 'pairing must be "minimal"') + + if len(z) == len(p) == 0: + if not analog: + return np.array([[k, 0., 0., 1., 0., 0.]]) + else: + return np.array([[0., 0., k, 0., 0., 1.]]) + + if pairing != 'minimal': + # ensure we have the same number of poles and zeros, and make copies + p = np.concatenate((p, np.zeros(max(len(z) - len(p), 0)))) + z = np.concatenate((z, np.zeros(max(len(p) - len(z), 0)))) + n_sections = (max(len(p), len(z)) + 1) // 2 + + if len(p) % 2 == 1 and pairing == 'nearest': + p = np.concatenate((p, [0.])) + z = np.concatenate((z, [0.])) + assert len(p) == len(z) + else: + if len(p) < len(z): + raise ValueError('for analog zpk2sos conversion, ' + 'must have len(p)>=len(z)') + + n_sections = (len(p) + 1) // 2 + + # Ensure we have complex conjugate pairs + # (note that _cplxreal only gives us one element of each complex pair): + z = np.concatenate(_cplxreal(z)) + p = np.concatenate(_cplxreal(p)) + if not np.isreal(k): + raise ValueError('k must be real') + k = k.real + + if not analog: + # digital: "worst" is the closest to the unit circle + def idx_worst(p): + return np.argmin(np.abs(1 - np.abs(p))) + else: + # analog: "worst" is the closest to the imaginary axis + def idx_worst(p): + return np.argmin(np.abs(np.real(p))) + + sos = np.zeros((n_sections, 6)) + + # Construct the system, reversing order so the "worst" are last + for si in range(n_sections-1, -1, -1): + # Select the next "worst" pole + p1_idx = idx_worst(p) + p1 = p[p1_idx] + p = np.delete(p, p1_idx) + + # Pair that pole with a zero + + if np.isreal(p1) and np.isreal(p).sum() == 0: + # Special case (1): last remaining real pole + if pairing != 'minimal': + z1_idx = _nearest_real_complex_idx(z, p1, 'real') + z1 = z[z1_idx] + z = np.delete(z, z1_idx) + sos[si] = _single_zpksos([z1, 0], [p1, 0], 1) + elif len(z) > 0: + z1_idx = _nearest_real_complex_idx(z, p1, 'real') + z1 = z[z1_idx] + z = np.delete(z, z1_idx) + sos[si] = _single_zpksos([z1], [p1], 1) + else: + sos[si] = _single_zpksos([], [p1], 1) + + elif (len(p) + 1 == len(z) + and not np.isreal(p1) + and np.isreal(p).sum() == 1 + and np.isreal(z).sum() == 1): + + # Special case (2): there's one real pole and one real zero + # left, and an equal number of poles and zeros to pair up. + # We *must* pair with a complex zero + + z1_idx = _nearest_real_complex_idx(z, p1, 'complex') + z1 = z[z1_idx] + z = np.delete(z, z1_idx) + sos[si] = _single_zpksos([z1, z1.conj()], [p1, p1.conj()], 1) + + else: + if np.isreal(p1): + prealidx = np.flatnonzero(np.isreal(p)) + p2_idx = prealidx[idx_worst(p[prealidx])] + p2 = p[p2_idx] + p = np.delete(p, p2_idx) + else: + p2 = p1.conj() + + # find closest zero + if len(z) > 0: + z1_idx = _nearest_real_complex_idx(z, p1, 'any') + z1 = z[z1_idx] + z = np.delete(z, z1_idx) + + if not np.isreal(z1): + sos[si] = _single_zpksos([z1, z1.conj()], [p1, p2], 1) + else: + if len(z) > 0: + z2_idx = _nearest_real_complex_idx(z, p1, 'real') + z2 = z[z2_idx] + assert np.isreal(z2) + z = np.delete(z, z2_idx) + sos[si] = _single_zpksos([z1, z2], [p1, p2], 1) + else: + sos[si] = _single_zpksos([z1], [p1, p2], 1) + else: + # no more zeros + sos[si] = _single_zpksos([], [p1, p2], 1) + + assert len(p) == len(z) == 0 # we've consumed all poles and zeros + del p, z + + # put gain in first sos + sos[0][:3] *= k + return sos + + +def _align_nums(nums): + """Aligns the shapes of multiple numerators. + + Given an array of numerator coefficient arrays [[a_1, a_2,..., + a_n],..., [b_1, b_2,..., b_m]], this function pads shorter numerator + arrays with zero's so that all numerators have the same length. Such + alignment is necessary for functions like 'tf2ss', which needs the + alignment when dealing with SIMO transfer functions. + + Parameters + ---------- + nums: array_like + Numerator or list of numerators. Not necessarily with same length. + + Returns + ------- + nums: array + The numerator. If `nums` input was a list of numerators then a 2-D + array with padded zeros for shorter numerators is returned. Otherwise + returns ``np.asarray(nums)``. + """ + try: + # The statement can throw a ValueError if one + # of the numerators is a single digit and another + # is array-like e.g. if nums = [5, [1, 2, 3]] + nums = asarray(nums) + + if not np.issubdtype(nums.dtype, np.number): + raise ValueError("dtype of numerator is non-numeric") + + return nums + + except ValueError: + nums = [np.atleast_1d(num) for num in nums] + max_width = max(num.size for num in nums) + + # pre-allocate + aligned_nums = np.zeros((len(nums), max_width)) + + # Create numerators with padded zeros + for index, num in enumerate(nums): + aligned_nums[index, -num.size:] = num + + return aligned_nums + + +def normalize(b, a): + """Normalize numerator/denominator of a continuous-time transfer function. + + If values of `b` are too close to 0, they are removed. In that case, a + BadCoefficients warning is emitted. + + Parameters + ---------- + b: array_like + Numerator of the transfer function. Can be a 2-D array to normalize + multiple transfer functions. + a: array_like + Denominator of the transfer function. At most 1-D. + + Returns + ------- + num: array + The numerator of the normalized transfer function. At least a 1-D + array. A 2-D array if the input `num` is a 2-D array. + den: 1-D array + The denominator of the normalized transfer function. + + Notes + ----- + Coefficients for both the numerator and denominator should be specified in + descending exponent order (e.g., ``s^2 + 3s + 5`` would be represented as + ``[1, 3, 5]``). + + Examples + -------- + >>> from scipy.signal import normalize + + Normalize the coefficients of the transfer function + ``(3*s^2 - 2*s + 5) / (2*s^2 + 3*s + 1)``: + + >>> b = [3, -2, 5] + >>> a = [2, 3, 1] + >>> normalize(b, a) + (array([ 1.5, -1. , 2.5]), array([1. , 1.5, 0.5])) + + A warning is generated if, for example, the first coefficient of + `b` is 0. In the following example, the result is as expected: + + >>> import warnings + >>> with warnings.catch_warnings(record=True) as w: + ... num, den = normalize([0, 3, 6], [2, -5, 4]) + + >>> num + array([1.5, 3. ]) + >>> den + array([ 1. , -2.5, 2. ]) + + >>> print(w[0].message) + Badly conditioned filter coefficients (numerator): the results may be meaningless + + """ + num, den = b, a + + den = np.atleast_1d(den) + num = np.atleast_2d(_align_nums(num)) + + if den.ndim != 1: + raise ValueError("Denominator polynomial must be rank-1 array.") + if num.ndim > 2: + raise ValueError("Numerator polynomial must be rank-1 or" + " rank-2 array.") + if np.all(den == 0): + raise ValueError("Denominator must have at least on nonzero element.") + + # Trim leading zeros in denominator, leave at least one. + den = np.trim_zeros(den, 'f') + + # Normalize transfer function + num, den = num / den[0], den / den[0] + + # Count numerator columns that are all zero + leading_zeros = 0 + for col in num.T: + if np.allclose(col, 0, atol=1e-14): + leading_zeros += 1 + else: + break + + # Trim leading zeros of numerator + if leading_zeros > 0: + warnings.warn("Badly conditioned filter coefficients (numerator): the " + "results may be meaningless", + BadCoefficients, stacklevel=2) + # Make sure at least one column remains + if leading_zeros == num.shape[1]: + leading_zeros -= 1 + num = num[:, leading_zeros:] + + # Squeeze first dimension if singular + if num.shape[0] == 1: + num = num[0, :] + + return num, den + + +def lp2lp(b, a, wo=1.0): + r""" + Transform a lowpass filter prototype to a different frequency. + + Return an analog low-pass filter with cutoff frequency `wo` + from an analog low-pass filter prototype with unity cutoff frequency, in + transfer function ('ba') representation. + + Parameters + ---------- + b : array_like + Numerator polynomial coefficients. + a : array_like + Denominator polynomial coefficients. + wo : float + Desired cutoff, as angular frequency (e.g. rad/s). + Defaults to no change. + + Returns + ------- + b : array_like + Numerator polynomial coefficients of the transformed low-pass filter. + a : array_like + Denominator polynomial coefficients of the transformed low-pass filter. + + See Also + -------- + lp2hp, lp2bp, lp2bs, bilinear + lp2lp_zpk + + Notes + ----- + This is derived from the s-plane substitution + + .. math:: s \rightarrow \frac{s}{\omega_0} + + Examples + -------- + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> lp = signal.lti([1.0], [1.0, 1.0]) + >>> lp2 = signal.lti(*signal.lp2lp(lp.num, lp.den, 2)) + >>> w, mag_lp, p_lp = lp.bode() + >>> w, mag_lp2, p_lp2 = lp2.bode(w) + + >>> plt.plot(w, mag_lp, label='Lowpass') + >>> plt.plot(w, mag_lp2, label='Transformed Lowpass') + >>> plt.semilogx() + >>> plt.grid(True) + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.legend() + + """ + a, b = map(atleast_1d, (a, b)) + try: + wo = float(wo) + except TypeError: + wo = float(wo[0]) + d = len(a) + n = len(b) + M = max((d, n)) + pwo = pow(wo, np.arange(M - 1, -1, -1)) + start1 = max((n - d, 0)) + start2 = max((d - n, 0)) + b = b * pwo[start1] / pwo[start2:] + a = a * pwo[start1] / pwo[start1:] + return normalize(b, a) + + +def lp2hp(b, a, wo=1.0): + r""" + Transform a lowpass filter prototype to a highpass filter. + + Return an analog high-pass filter with cutoff frequency `wo` + from an analog low-pass filter prototype with unity cutoff frequency, in + transfer function ('ba') representation. + + Parameters + ---------- + b : array_like + Numerator polynomial coefficients. + a : array_like + Denominator polynomial coefficients. + wo : float + Desired cutoff, as angular frequency (e.g., rad/s). + Defaults to no change. + + Returns + ------- + b : array_like + Numerator polynomial coefficients of the transformed high-pass filter. + a : array_like + Denominator polynomial coefficients of the transformed high-pass filter. + + See Also + -------- + lp2lp, lp2bp, lp2bs, bilinear + lp2hp_zpk + + Notes + ----- + This is derived from the s-plane substitution + + .. math:: s \rightarrow \frac{\omega_0}{s} + + This maintains symmetry of the lowpass and highpass responses on a + logarithmic scale. + + Examples + -------- + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> lp = signal.lti([1.0], [1.0, 1.0]) + >>> hp = signal.lti(*signal.lp2hp(lp.num, lp.den)) + >>> w, mag_lp, p_lp = lp.bode() + >>> w, mag_hp, p_hp = hp.bode(w) + + >>> plt.plot(w, mag_lp, label='Lowpass') + >>> plt.plot(w, mag_hp, label='Highpass') + >>> plt.semilogx() + >>> plt.grid(True) + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.legend() + + """ + a, b = map(atleast_1d, (a, b)) + try: + wo = float(wo) + except TypeError: + wo = float(wo[0]) + d = len(a) + n = len(b) + if wo != 1: + pwo = pow(wo, np.arange(max((d, n)))) + else: + pwo = np.ones(max((d, n)), b.dtype.char) + if d >= n: + outa = a[::-1] * pwo + outb = resize(b, (d,)) + outb[n:] = 0.0 + outb[:n] = b[::-1] * pwo[:n] + else: + outb = b[::-1] * pwo + outa = resize(a, (n,)) + outa[d:] = 0.0 + outa[:d] = a[::-1] * pwo[:d] + + return normalize(outb, outa) + + +def lp2bp(b, a, wo=1.0, bw=1.0): + r""" + Transform a lowpass filter prototype to a bandpass filter. + + Return an analog band-pass filter with center frequency `wo` and + bandwidth `bw` from an analog low-pass filter prototype with unity + cutoff frequency, in transfer function ('ba') representation. + + Parameters + ---------- + b : array_like + Numerator polynomial coefficients. + a : array_like + Denominator polynomial coefficients. + wo : float + Desired passband center, as angular frequency (e.g., rad/s). + Defaults to no change. + bw : float + Desired passband width, as angular frequency (e.g., rad/s). + Defaults to 1. + + Returns + ------- + b : array_like + Numerator polynomial coefficients of the transformed band-pass filter. + a : array_like + Denominator polynomial coefficients of the transformed band-pass filter. + + See Also + -------- + lp2lp, lp2hp, lp2bs, bilinear + lp2bp_zpk + + Notes + ----- + This is derived from the s-plane substitution + + .. math:: s \rightarrow \frac{s^2 + {\omega_0}^2}{s \cdot \mathrm{BW}} + + This is the "wideband" transformation, producing a passband with + geometric (log frequency) symmetry about `wo`. + + Examples + -------- + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> lp = signal.lti([1.0], [1.0, 1.0]) + >>> bp = signal.lti(*signal.lp2bp(lp.num, lp.den)) + >>> w, mag_lp, p_lp = lp.bode() + >>> w, mag_bp, p_bp = bp.bode(w) + + >>> plt.plot(w, mag_lp, label='Lowpass') + >>> plt.plot(w, mag_bp, label='Bandpass') + >>> plt.semilogx() + >>> plt.grid(True) + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.legend() + """ + + a, b = map(atleast_1d, (a, b)) + D = len(a) - 1 + N = len(b) - 1 + artype = mintypecode((a, b)) + ma = max([N, D]) + Np = N + ma + Dp = D + ma + bprime = np.empty(Np + 1, artype) + aprime = np.empty(Dp + 1, artype) + wosq = wo * wo + for j in range(Np + 1): + val = 0.0 + for i in range(0, N + 1): + for k in range(0, i + 1): + if ma - i + 2 * k == j: + val += comb(i, k) * b[N - i] * (wosq) ** (i - k) / bw ** i + bprime[Np - j] = val + for j in range(Dp + 1): + val = 0.0 + for i in range(0, D + 1): + for k in range(0, i + 1): + if ma - i + 2 * k == j: + val += comb(i, k) * a[D - i] * (wosq) ** (i - k) / bw ** i + aprime[Dp - j] = val + + return normalize(bprime, aprime) + + +def lp2bs(b, a, wo=1.0, bw=1.0): + r""" + Transform a lowpass filter prototype to a bandstop filter. + + Return an analog band-stop filter with center frequency `wo` and + bandwidth `bw` from an analog low-pass filter prototype with unity + cutoff frequency, in transfer function ('ba') representation. + + Parameters + ---------- + b : array_like + Numerator polynomial coefficients. + a : array_like + Denominator polynomial coefficients. + wo : float + Desired stopband center, as angular frequency (e.g., rad/s). + Defaults to no change. + bw : float + Desired stopband width, as angular frequency (e.g., rad/s). + Defaults to 1. + + Returns + ------- + b : array_like + Numerator polynomial coefficients of the transformed band-stop filter. + a : array_like + Denominator polynomial coefficients of the transformed band-stop filter. + + See Also + -------- + lp2lp, lp2hp, lp2bp, bilinear + lp2bs_zpk + + Notes + ----- + This is derived from the s-plane substitution + + .. math:: s \rightarrow \frac{s \cdot \mathrm{BW}}{s^2 + {\omega_0}^2} + + This is the "wideband" transformation, producing a stopband with + geometric (log frequency) symmetry about `wo`. + + Examples + -------- + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> lp = signal.lti([1.0], [1.0, 1.5]) + >>> bs = signal.lti(*signal.lp2bs(lp.num, lp.den)) + >>> w, mag_lp, p_lp = lp.bode() + >>> w, mag_bs, p_bs = bs.bode(w) + >>> plt.plot(w, mag_lp, label='Lowpass') + >>> plt.plot(w, mag_bs, label='Bandstop') + >>> plt.semilogx() + >>> plt.grid(True) + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.legend() + """ + a, b = map(atleast_1d, (a, b)) + D = len(a) - 1 + N = len(b) - 1 + artype = mintypecode((a, b)) + M = max([N, D]) + Np = M + M + Dp = M + M + bprime = np.empty(Np + 1, artype) + aprime = np.empty(Dp + 1, artype) + wosq = wo * wo + for j in range(Np + 1): + val = 0.0 + for i in range(0, N + 1): + for k in range(0, M - i + 1): + if i + 2 * k == j: + val += (comb(M - i, k) * b[N - i] * + (wosq) ** (M - i - k) * bw ** i) + bprime[Np - j] = val + for j in range(Dp + 1): + val = 0.0 + for i in range(0, D + 1): + for k in range(0, M - i + 1): + if i + 2 * k == j: + val += (comb(M - i, k) * a[D - i] * + (wosq) ** (M - i - k) * bw ** i) + aprime[Dp - j] = val + + return normalize(bprime, aprime) + + +def bilinear(b, a, fs=1.0): + r""" + Return a digital IIR filter from an analog one using a bilinear transform. + + Transform a set of poles and zeros from the analog s-plane to the digital + z-plane using Tustin's method, which substitutes ``2*fs*(z-1) / (z+1)`` for + ``s``, maintaining the shape of the frequency response. + + Parameters + ---------- + b : array_like + Numerator of the analog filter transfer function. + a : array_like + Denominator of the analog filter transfer function. + fs : float + Sample rate, as ordinary frequency (e.g., hertz). No prewarping is + done in this function. + + Returns + ------- + b : ndarray + Numerator of the transformed digital filter transfer function. + a : ndarray + Denominator of the transformed digital filter transfer function. + + See Also + -------- + lp2lp, lp2hp, lp2bp, lp2bs + bilinear_zpk + + Examples + -------- + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> fs = 100 + >>> bf = 2 * np.pi * np.array([7, 13]) + >>> filts = signal.lti(*signal.butter(4, bf, btype='bandpass', + ... analog=True)) + >>> filtz = signal.lti(*signal.bilinear(filts.num, filts.den, fs)) + >>> wz, hz = signal.freqz(filtz.num, filtz.den) + >>> ws, hs = signal.freqs(filts.num, filts.den, worN=fs*wz) + + >>> plt.semilogx(wz*fs/(2*np.pi), 20*np.log10(np.abs(hz).clip(1e-15)), + ... label=r'$|H_z(e^{j \omega})|$') + >>> plt.semilogx(wz*fs/(2*np.pi), 20*np.log10(np.abs(hs).clip(1e-15)), + ... label=r'$|H(j \omega)|$') + >>> plt.legend() + >>> plt.xlabel('Frequency [Hz]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.grid(True) + """ + fs = _validate_fs(fs, allow_none=False) + a, b = map(atleast_1d, (a, b)) + D = len(a) - 1 + N = len(b) - 1 + artype = float + M = max([N, D]) + Np = M + Dp = M + bprime = np.empty(Np + 1, artype) + aprime = np.empty(Dp + 1, artype) + for j in range(Np + 1): + val = 0.0 + for i in range(N + 1): + for k in range(i + 1): + for l in range(M - i + 1): + if k + l == j: + val += (comb(i, k) * comb(M - i, l) * b[N - i] * + pow(2 * fs, i) * (-1) ** k) + bprime[j] = real(val) + for j in range(Dp + 1): + val = 0.0 + for i in range(D + 1): + for k in range(i + 1): + for l in range(M - i + 1): + if k + l == j: + val += (comb(i, k) * comb(M - i, l) * a[D - i] * + pow(2 * fs, i) * (-1) ** k) + aprime[j] = real(val) + + return normalize(bprime, aprime) + + +def _validate_gpass_gstop(gpass, gstop): + + if gpass <= 0.0: + raise ValueError("gpass should be larger than 0.0") + elif gstop <= 0.0: + raise ValueError("gstop should be larger than 0.0") + elif gpass > gstop: + raise ValueError("gpass should be smaller than gstop") + + +def iirdesign(wp, ws, gpass, gstop, analog=False, ftype='ellip', output='ba', + fs=None): + """Complete IIR digital and analog filter design. + + Given passband and stopband frequencies and gains, construct an analog or + digital IIR filter of minimum order for a given basic type. Return the + output in numerator, denominator ('ba'), pole-zero ('zpk') or second order + sections ('sos') form. + + Parameters + ---------- + wp, ws : float or array like, shape (2,) + Passband and stopband edge frequencies. Possible values are scalars + (for lowpass and highpass filters) or ranges (for bandpass and bandstop + filters). + For digital filters, these are in the same units as `fs`. By default, + `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1, + where 1 is the Nyquist frequency. For example: + + - Lowpass: wp = 0.2, ws = 0.3 + - Highpass: wp = 0.3, ws = 0.2 + - Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6] + - Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5] + + For analog filters, `wp` and `ws` are angular frequencies (e.g., rad/s). + Note, that for bandpass and bandstop filters passband must lie strictly + inside stopband or vice versa. Also note that the cutoff at the band edges + for IIR filters is defined as half-power, so -3dB, not half-amplitude (-6dB) + like for `scipy.signal.fiwin`. + gpass : float + The maximum loss in the passband (dB). + gstop : float + The minimum attenuation in the stopband (dB). + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. + ftype : str, optional + The type of IIR filter to design: + + - Butterworth : 'butter' + - Chebyshev I : 'cheby1' + - Chebyshev II : 'cheby2' + - Cauer/elliptic: 'ellip' + + output : {'ba', 'zpk', 'sos'}, optional + Filter form of the output: + + - second-order sections (recommended): 'sos' + - numerator/denominator (default) : 'ba' + - pole-zero : 'zpk' + + In general the second-order sections ('sos') form is + recommended because inferring the coefficients for the + numerator/denominator form ('ba') suffers from numerical + instabilities. For reasons of backward compatibility the default + form is the numerator/denominator form ('ba'), where the 'b' + and the 'a' in 'ba' refer to the commonly used names of the + coefficients used. + + Note: Using the second-order sections form ('sos') is sometimes + associated with additional computational costs: for + data-intense use cases it is therefore recommended to also + investigate the numerator/denominator form ('ba'). + + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + b, a : ndarray, ndarray + Numerator (`b`) and denominator (`a`) polynomials of the IIR filter. + Only returned if ``output='ba'``. + z, p, k : ndarray, ndarray, float + Zeros, poles, and system gain of the IIR filter transfer + function. Only returned if ``output='zpk'``. + sos : ndarray + Second-order sections representation of the IIR filter. + Only returned if ``output='sos'``. + + See Also + -------- + butter : Filter design using order and critical points + cheby1, cheby2, ellip, bessel + buttord : Find order and critical points from passband and stopband spec + cheb1ord, cheb2ord, ellipord + iirfilter : General filter design using order and critical frequencies + + Notes + ----- + The ``'sos'`` output parameter was added in 0.16.0. + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import matplotlib.ticker + + >>> wp = 0.2 + >>> ws = 0.3 + >>> gpass = 1 + >>> gstop = 40 + + >>> system = signal.iirdesign(wp, ws, gpass, gstop) + >>> w, h = signal.freqz(*system) + + >>> fig, ax1 = plt.subplots() + >>> ax1.set_title('Digital filter frequency response') + >>> ax1.plot(w, 20 * np.log10(abs(h)), 'b') + >>> ax1.set_ylabel('Amplitude [dB]', color='b') + >>> ax1.set_xlabel('Frequency [rad/sample]') + >>> ax1.grid(True) + >>> ax1.set_ylim([-120, 20]) + >>> ax2 = ax1.twinx() + >>> phase = np.unwrap(np.angle(h)) + >>> ax2.plot(w, phase, 'g') + >>> ax2.set_ylabel('Phase [rad]', color='g') + >>> ax2.grid(True) + >>> ax2.axis('tight') + >>> ax2.set_ylim([-6, 1]) + >>> nticks = 8 + >>> ax1.yaxis.set_major_locator(matplotlib.ticker.LinearLocator(nticks)) + >>> ax2.yaxis.set_major_locator(matplotlib.ticker.LinearLocator(nticks)) + + """ + try: + ordfunc = filter_dict[ftype][1] + except KeyError as e: + raise ValueError(f"Invalid IIR filter type: {ftype}") from e + except IndexError as e: + raise ValueError(f"{ftype} does not have order selection. " + "Use iirfilter function.") from e + + _validate_gpass_gstop(gpass, gstop) + + wp = atleast_1d(wp) + ws = atleast_1d(ws) + + fs = _validate_fs(fs, allow_none=True) + + if wp.shape[0] != ws.shape[0] or wp.shape not in [(1,), (2,)]: + raise ValueError("wp and ws must have one or two elements each, and " + f"the same shape, got {wp.shape} and {ws.shape}") + + if any(wp <= 0) or any(ws <= 0): + raise ValueError("Values for wp, ws must be greater than 0") + + if not analog: + if fs is None: + if any(wp >= 1) or any(ws >= 1): + raise ValueError("Values for wp, ws must be less than 1") + elif any(wp >= fs/2) or any(ws >= fs/2): + raise ValueError("Values for wp, ws must be less than fs/2 " + f"(fs={fs} -> fs/2={fs/2})") + + if wp.shape[0] == 2: + if not ((ws[0] < wp[0] and wp[1] < ws[1]) or + (wp[0] < ws[0] and ws[1] < wp[1])): + raise ValueError("Passband must lie strictly inside stopband " + "or vice versa") + + band_type = 2 * (len(wp) - 1) + band_type += 1 + if wp[0] >= ws[0]: + band_type += 1 + + btype = {1: 'lowpass', 2: 'highpass', + 3: 'bandstop', 4: 'bandpass'}[band_type] + + N, Wn = ordfunc(wp, ws, gpass, gstop, analog=analog, fs=fs) + return iirfilter(N, Wn, rp=gpass, rs=gstop, analog=analog, btype=btype, + ftype=ftype, output=output, fs=fs) + + +def iirfilter(N, Wn, rp=None, rs=None, btype='band', analog=False, + ftype='butter', output='ba', fs=None): + """ + IIR digital and analog filter design given order and critical points. + + Design an Nth-order digital or analog filter and return the filter + coefficients. + + Parameters + ---------- + N : int + The order of the filter. + Wn : array_like + A scalar or length-2 sequence giving the critical frequencies. + + For digital filters, `Wn` are in the same units as `fs`. By default, + `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1, + where 1 is the Nyquist frequency. (`Wn` is thus in + half-cycles / sample.) + + For analog filters, `Wn` is an angular frequency (e.g., rad/s). + + When Wn is a length-2 sequence, ``Wn[0]`` must be less than ``Wn[1]``. + rp : float, optional + For Chebyshev and elliptic filters, provides the maximum ripple + in the passband. (dB) + rs : float, optional + For Chebyshev and elliptic filters, provides the minimum attenuation + in the stop band. (dB) + btype : {'bandpass', 'lowpass', 'highpass', 'bandstop'}, optional + The type of filter. Default is 'bandpass'. + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. + ftype : str, optional + The type of IIR filter to design: + + - Butterworth : 'butter' + - Chebyshev I : 'cheby1' + - Chebyshev II : 'cheby2' + - Cauer/elliptic: 'ellip' + - Bessel/Thomson: 'bessel' + + output : {'ba', 'zpk', 'sos'}, optional + Filter form of the output: + + - second-order sections (recommended): 'sos' + - numerator/denominator (default) : 'ba' + - pole-zero : 'zpk' + + In general the second-order sections ('sos') form is + recommended because inferring the coefficients for the + numerator/denominator form ('ba') suffers from numerical + instabilities. For reasons of backward compatibility the default + form is the numerator/denominator form ('ba'), where the 'b' + and the 'a' in 'ba' refer to the commonly used names of the + coefficients used. + + Note: Using the second-order sections form ('sos') is sometimes + associated with additional computational costs: for + data-intense use cases it is therefore recommended to also + investigate the numerator/denominator form ('ba'). + + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + b, a : ndarray, ndarray + Numerator (`b`) and denominator (`a`) polynomials of the IIR filter. + Only returned if ``output='ba'``. + z, p, k : ndarray, ndarray, float + Zeros, poles, and system gain of the IIR filter transfer + function. Only returned if ``output='zpk'``. + sos : ndarray + Second-order sections representation of the IIR filter. + Only returned if ``output='sos'``. + + See Also + -------- + butter : Filter design using order and critical points + cheby1, cheby2, ellip, bessel + buttord : Find order and critical points from passband and stopband spec + cheb1ord, cheb2ord, ellipord + iirdesign : General filter design using passband and stopband spec + + Notes + ----- + The ``'sos'`` output parameter was added in 0.16.0. + + Examples + -------- + Generate a 17th-order Chebyshev II analog bandpass filter from 50 Hz to + 200 Hz and plot the frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> b, a = signal.iirfilter(17, [2*np.pi*50, 2*np.pi*200], rs=60, + ... btype='band', analog=True, ftype='cheby2') + >>> w, h = signal.freqs(b, a, 1000) + >>> fig = plt.figure() + >>> ax = fig.add_subplot(1, 1, 1) + >>> ax.semilogx(w / (2*np.pi), 20 * np.log10(np.maximum(abs(h), 1e-5))) + >>> ax.set_title('Chebyshev Type II bandpass frequency response') + >>> ax.set_xlabel('Frequency [Hz]') + >>> ax.set_ylabel('Amplitude [dB]') + >>> ax.axis((10, 1000, -100, 10)) + >>> ax.grid(which='both', axis='both') + >>> plt.show() + + Create a digital filter with the same properties, in a system with + sampling rate of 2000 Hz, and plot the frequency response. (Second-order + sections implementation is required to ensure stability of a filter of + this order): + + >>> sos = signal.iirfilter(17, [50, 200], rs=60, btype='band', + ... analog=False, ftype='cheby2', fs=2000, + ... output='sos') + >>> w, h = signal.freqz_sos(sos, 2000, fs=2000) + >>> fig = plt.figure() + >>> ax = fig.add_subplot(1, 1, 1) + >>> ax.semilogx(w, 20 * np.log10(np.maximum(abs(h), 1e-5))) + >>> ax.set_title('Chebyshev Type II bandpass frequency response') + >>> ax.set_xlabel('Frequency [Hz]') + >>> ax.set_ylabel('Amplitude [dB]') + >>> ax.axis((10, 1000, -100, 10)) + >>> ax.grid(which='both', axis='both') + >>> plt.show() + + """ + fs = _validate_fs(fs, allow_none=True) + ftype, btype, output = (x.lower() for x in (ftype, btype, output)) + Wn = asarray(Wn) + if fs is not None: + if analog: + raise ValueError("fs cannot be specified for an analog filter") + Wn = Wn / (fs/2) + + if np.any(Wn <= 0): + raise ValueError("filter critical frequencies must be greater than 0") + + if Wn.size > 1 and not Wn[0] < Wn[1]: + raise ValueError("Wn[0] must be less than Wn[1]") + + try: + btype = band_dict[btype] + except KeyError as e: + raise ValueError(f"'{btype}' is an invalid bandtype for filter.") from e + + try: + typefunc = filter_dict[ftype][0] + except KeyError as e: + raise ValueError(f"'{ftype}' is not a valid basic IIR filter.") from e + + if output not in ['ba', 'zpk', 'sos']: + raise ValueError(f"'{output}' is not a valid output form.") + + if rp is not None and rp < 0: + raise ValueError("passband ripple (rp) must be positive") + + if rs is not None and rs < 0: + raise ValueError("stopband attenuation (rs) must be positive") + + # Get analog lowpass prototype + if typefunc == buttap: + z, p, k = typefunc(N) + elif typefunc == besselap: + z, p, k = typefunc(N, norm=bessel_norms[ftype]) + elif typefunc == cheb1ap: + if rp is None: + raise ValueError("passband ripple (rp) must be provided to " + "design a Chebyshev I filter.") + z, p, k = typefunc(N, rp) + elif typefunc == cheb2ap: + if rs is None: + raise ValueError("stopband attenuation (rs) must be provided to " + "design an Chebyshev II filter.") + z, p, k = typefunc(N, rs) + elif typefunc == ellipap: + if rs is None or rp is None: + raise ValueError("Both rp and rs must be provided to design an " + "elliptic filter.") + z, p, k = typefunc(N, rp, rs) + else: + raise NotImplementedError(f"'{ftype}' not implemented in iirfilter.") + + # Pre-warp frequencies for digital filter design + if not analog: + if np.any(Wn <= 0) or np.any(Wn >= 1): + if fs is not None: + raise ValueError("Digital filter critical frequencies must " + f"be 0 < Wn < fs/2 (fs={fs} -> fs/2={fs/2})") + raise ValueError("Digital filter critical frequencies " + "must be 0 < Wn < 1") + fs = 2.0 + warped = 2 * fs * tan(pi * Wn / fs) + else: + warped = Wn + + # transform to lowpass, bandpass, highpass, or bandstop + if btype in ('lowpass', 'highpass'): + if np.size(Wn) != 1: + raise ValueError('Must specify a single critical frequency Wn ' + 'for lowpass or highpass filter') + + if btype == 'lowpass': + z, p, k = lp2lp_zpk(z, p, k, wo=warped) + elif btype == 'highpass': + z, p, k = lp2hp_zpk(z, p, k, wo=warped) + elif btype in ('bandpass', 'bandstop'): + try: + bw = warped[1] - warped[0] + wo = sqrt(warped[0] * warped[1]) + except IndexError as e: + raise ValueError('Wn must specify start and stop frequencies for ' + 'bandpass or bandstop filter') from e + + if btype == 'bandpass': + z, p, k = lp2bp_zpk(z, p, k, wo=wo, bw=bw) + elif btype == 'bandstop': + z, p, k = lp2bs_zpk(z, p, k, wo=wo, bw=bw) + else: + raise NotImplementedError(f"'{btype}' not implemented in iirfilter.") + + # Find discrete equivalent if necessary + if not analog: + z, p, k = bilinear_zpk(z, p, k, fs=fs) + + # Transform to proper out type (pole-zero, state-space, numer-denom) + if output == 'zpk': + return z, p, k + elif output == 'ba': + return zpk2tf(z, p, k) + elif output == 'sos': + return zpk2sos(z, p, k, analog=analog) + + +def _relative_degree(z, p): + """ + Return relative degree of transfer function from zeros and poles + """ + degree = len(p) - len(z) + if degree < 0: + raise ValueError("Improper transfer function. " + "Must have at least as many poles as zeros.") + else: + return degree + + +def bilinear_zpk(z, p, k, fs): + r""" + Return a digital IIR filter from an analog one using a bilinear transform. + + Transform a set of poles and zeros from the analog s-plane to the digital + z-plane using Tustin's method, which substitutes ``2*fs*(z-1) / (z+1)`` for + ``s``, maintaining the shape of the frequency response. + + Parameters + ---------- + z : array_like + Zeros of the analog filter transfer function. + p : array_like + Poles of the analog filter transfer function. + k : float + System gain of the analog filter transfer function. + fs : float + Sample rate, as ordinary frequency (e.g., hertz). No prewarping is + done in this function. + + Returns + ------- + z : ndarray + Zeros of the transformed digital filter transfer function. + p : ndarray + Poles of the transformed digital filter transfer function. + k : float + System gain of the transformed digital filter. + + See Also + -------- + lp2lp_zpk, lp2hp_zpk, lp2bp_zpk, lp2bs_zpk + bilinear + + Notes + ----- + .. versionadded:: 1.1.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> fs = 100 + >>> bf = 2 * np.pi * np.array([7, 13]) + >>> filts = signal.lti(*signal.butter(4, bf, btype='bandpass', analog=True, + ... output='zpk')) + >>> filtz = signal.lti(*signal.bilinear_zpk(filts.zeros, filts.poles, + ... filts.gain, fs)) + >>> wz, hz = signal.freqz_zpk(filtz.zeros, filtz.poles, filtz.gain) + >>> ws, hs = signal.freqs_zpk(filts.zeros, filts.poles, filts.gain, + ... worN=fs*wz) + >>> plt.semilogx(wz*fs/(2*np.pi), 20*np.log10(np.abs(hz).clip(1e-15)), + ... label=r'$|H_z(e^{j \omega})|$') + >>> plt.semilogx(wz*fs/(2*np.pi), 20*np.log10(np.abs(hs).clip(1e-15)), + ... label=r'$|H(j \omega)|$') + >>> plt.legend() + >>> plt.xlabel('Frequency [Hz]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.grid(True) + """ + z = atleast_1d(z) + p = atleast_1d(p) + + fs = _validate_fs(fs, allow_none=False) + + degree = _relative_degree(z, p) + + fs2 = 2.0*fs + + # Bilinear transform the poles and zeros + z_z = (fs2 + z) / (fs2 - z) + p_z = (fs2 + p) / (fs2 - p) + + # Any zeros that were at infinity get moved to the Nyquist frequency + z_z = append(z_z, -ones(degree)) + + # Compensate for gain change + k_z = k * real(prod(fs2 - z) / prod(fs2 - p)) + + return z_z, p_z, k_z + + +def lp2lp_zpk(z, p, k, wo=1.0): + r""" + Transform a lowpass filter prototype to a different frequency. + + Return an analog low-pass filter with cutoff frequency `wo` + from an analog low-pass filter prototype with unity cutoff frequency, + using zeros, poles, and gain ('zpk') representation. + + Parameters + ---------- + z : array_like + Zeros of the analog filter transfer function. + p : array_like + Poles of the analog filter transfer function. + k : float + System gain of the analog filter transfer function. + wo : float + Desired cutoff, as angular frequency (e.g., rad/s). + Defaults to no change. + + Returns + ------- + z : ndarray + Zeros of the transformed low-pass filter transfer function. + p : ndarray + Poles of the transformed low-pass filter transfer function. + k : float + System gain of the transformed low-pass filter. + + See Also + -------- + lp2hp_zpk, lp2bp_zpk, lp2bs_zpk, bilinear + lp2lp + + Notes + ----- + This is derived from the s-plane substitution + + .. math:: s \rightarrow \frac{s}{\omega_0} + + .. versionadded:: 1.1.0 + + Examples + -------- + Use the 'zpk' (Zero-Pole-Gain) representation of a lowpass filter to + transform it to a new 'zpk' representation associated with a cutoff frequency wo. + + >>> from scipy.signal import lp2lp_zpk + >>> z = [7, 2] + >>> p = [5, 13] + >>> k = 0.8 + >>> wo = 0.4 + >>> lp2lp_zpk(z, p, k, wo) + ( array([2.8, 0.8]), array([2. , 5.2]), 0.8) + """ + z = atleast_1d(z) + p = atleast_1d(p) + wo = float(wo) # Avoid int wraparound + + degree = _relative_degree(z, p) + + # Scale all points radially from origin to shift cutoff frequency + z_lp = wo * z + p_lp = wo * p + + # Each shifted pole decreases gain by wo, each shifted zero increases it. + # Cancel out the net change to keep overall gain the same + k_lp = k * wo**degree + + return z_lp, p_lp, k_lp + + +def lp2hp_zpk(z, p, k, wo=1.0): + r""" + Transform a lowpass filter prototype to a highpass filter. + + Return an analog high-pass filter with cutoff frequency `wo` + from an analog low-pass filter prototype with unity cutoff frequency, + using zeros, poles, and gain ('zpk') representation. + + Parameters + ---------- + z : array_like + Zeros of the analog filter transfer function. + p : array_like + Poles of the analog filter transfer function. + k : float + System gain of the analog filter transfer function. + wo : float + Desired cutoff, as angular frequency (e.g., rad/s). + Defaults to no change. + + Returns + ------- + z : ndarray + Zeros of the transformed high-pass filter transfer function. + p : ndarray + Poles of the transformed high-pass filter transfer function. + k : float + System gain of the transformed high-pass filter. + + See Also + -------- + lp2lp_zpk, lp2bp_zpk, lp2bs_zpk, bilinear + lp2hp + + Notes + ----- + This is derived from the s-plane substitution + + .. math:: s \rightarrow \frac{\omega_0}{s} + + This maintains symmetry of the lowpass and highpass responses on a + logarithmic scale. + + .. versionadded:: 1.1.0 + + Examples + -------- + Use the 'zpk' (Zero-Pole-Gain) representation of a lowpass filter to + transform it to a highpass filter with a cutoff frequency wo. + + >>> from scipy.signal import lp2hp_zpk + >>> z = [ -2 + 3j , -0.5 - 0.8j ] + >>> p = [ -1 , -4 ] + >>> k = 10 + >>> wo = 0.6 + >>> lp2hp_zpk(z, p, k, wo) + ( array([-0.09230769-0.13846154j, -0.33707865+0.53932584j]), + array([-0.6 , -0.15]), + 8.5) + """ + z = atleast_1d(z) + p = atleast_1d(p) + wo = float(wo) + + degree = _relative_degree(z, p) + + # Invert positions radially about unit circle to convert LPF to HPF + # Scale all points radially from origin to shift cutoff frequency + z_hp = wo / z + p_hp = wo / p + + # If lowpass had zeros at infinity, inverting moves them to origin. + z_hp = append(z_hp, zeros(degree)) + + # Cancel out gain change caused by inversion + k_hp = k * real(prod(-z) / prod(-p)) + + return z_hp, p_hp, k_hp + + +def lp2bp_zpk(z, p, k, wo=1.0, bw=1.0): + r""" + Transform a lowpass filter prototype to a bandpass filter. + + Return an analog band-pass filter with center frequency `wo` and + bandwidth `bw` from an analog low-pass filter prototype with unity + cutoff frequency, using zeros, poles, and gain ('zpk') representation. + + Parameters + ---------- + z : array_like + Zeros of the analog filter transfer function. + p : array_like + Poles of the analog filter transfer function. + k : float + System gain of the analog filter transfer function. + wo : float + Desired passband center, as angular frequency (e.g., rad/s). + Defaults to no change. + bw : float + Desired passband width, as angular frequency (e.g., rad/s). + Defaults to 1. + + Returns + ------- + z : ndarray + Zeros of the transformed band-pass filter transfer function. + p : ndarray + Poles of the transformed band-pass filter transfer function. + k : float + System gain of the transformed band-pass filter. + + See Also + -------- + lp2lp_zpk, lp2hp_zpk, lp2bs_zpk, bilinear + lp2bp + + Notes + ----- + This is derived from the s-plane substitution + + .. math:: s \rightarrow \frac{s^2 + {\omega_0}^2}{s \cdot \mathrm{BW}} + + This is the "wideband" transformation, producing a passband with + geometric (log frequency) symmetry about `wo`. + + .. versionadded:: 1.1.0 + + Examples + -------- + Use the 'zpk' (Zero-Pole-Gain) representation of a lowpass filter to + transform it to a bandpass filter with a center frequency wo and + bandwidth bw. + + >>> from scipy.signal import lp2bp_zpk + >>> z = [ 5 + 2j , 5 - 2j ] + >>> p = [ 7 , -16 ] + >>> k = 0.8 + >>> wo = 0.62 + >>> bw = 15 + >>> lp2bp_zpk(z, p, k, wo, bw) + ( array([7.49955815e+01+3.00017676e+01j, 7.49955815e+01-3.00017676e+01j, + 4.41850748e-03-1.76761126e-03j, 4.41850748e-03+1.76761126e-03j]), + array([1.04996339e+02+0.j, -1.60167736e-03+0.j, 3.66108003e-03+0.j, + -2.39998398e+02+0.j]), 0.8) + """ + z = atleast_1d(z) + p = atleast_1d(p) + wo = float(wo) + bw = float(bw) + + degree = _relative_degree(z, p) + + # Scale poles and zeros to desired bandwidth + z_lp = z * bw/2 + p_lp = p * bw/2 + + # Square root needs to produce complex result, not NaN + z_lp = z_lp.astype(complex) + p_lp = p_lp.astype(complex) + + # Duplicate poles and zeros and shift from baseband to +wo and -wo + z_bp = concatenate((z_lp + sqrt(z_lp**2 - wo**2), + z_lp - sqrt(z_lp**2 - wo**2))) + p_bp = concatenate((p_lp + sqrt(p_lp**2 - wo**2), + p_lp - sqrt(p_lp**2 - wo**2))) + + # Move degree zeros to origin, leaving degree zeros at infinity for BPF + z_bp = append(z_bp, zeros(degree)) + + # Cancel out gain change from frequency scaling + k_bp = k * bw**degree + + return z_bp, p_bp, k_bp + + +def lp2bs_zpk(z, p, k, wo=1.0, bw=1.0): + r""" + Transform a lowpass filter prototype to a bandstop filter. + + Return an analog band-stop filter with center frequency `wo` and + stopband width `bw` from an analog low-pass filter prototype with unity + cutoff frequency, using zeros, poles, and gain ('zpk') representation. + + Parameters + ---------- + z : array_like + Zeros of the analog filter transfer function. + p : array_like + Poles of the analog filter transfer function. + k : float + System gain of the analog filter transfer function. + wo : float + Desired stopband center, as angular frequency (e.g., rad/s). + Defaults to no change. + bw : float + Desired stopband width, as angular frequency (e.g., rad/s). + Defaults to 1. + + Returns + ------- + z : ndarray + Zeros of the transformed band-stop filter transfer function. + p : ndarray + Poles of the transformed band-stop filter transfer function. + k : float + System gain of the transformed band-stop filter. + + See Also + -------- + lp2lp_zpk, lp2hp_zpk, lp2bp_zpk, bilinear + lp2bs + + Notes + ----- + This is derived from the s-plane substitution + + .. math:: s \rightarrow \frac{s \cdot \mathrm{BW}}{s^2 + {\omega_0}^2} + + This is the "wideband" transformation, producing a stopband with + geometric (log frequency) symmetry about `wo`. + + .. versionadded:: 1.1.0 + + Examples + -------- + Transform a low-pass filter represented in 'zpk' (Zero-Pole-Gain) form + into a bandstop filter represented in 'zpk' form, with a center frequency wo and + bandwidth bw. + + >>> from scipy.signal import lp2bs_zpk + >>> z = [ ] + >>> p = [ 0.7 , -1 ] + >>> k = 9 + >>> wo = 0.5 + >>> bw = 10 + >>> lp2bs_zpk(z, p, k, wo, bw) + ( array([0.+0.5j, 0.+0.5j, 0.-0.5j, 0.-0.5j]), + array([14.2681928 +0.j, -0.02506281+0.j, 0.01752149+0.j, -9.97493719+0.j]), + -12.857142857142858) + """ + z = atleast_1d(z) + p = atleast_1d(p) + wo = float(wo) + bw = float(bw) + + degree = _relative_degree(z, p) + + # Invert to a highpass filter with desired bandwidth + z_hp = (bw/2) / z + p_hp = (bw/2) / p + + # Square root needs to produce complex result, not NaN + z_hp = z_hp.astype(complex) + p_hp = p_hp.astype(complex) + + # Duplicate poles and zeros and shift from baseband to +wo and -wo + z_bs = concatenate((z_hp + sqrt(z_hp**2 - wo**2), + z_hp - sqrt(z_hp**2 - wo**2))) + p_bs = concatenate((p_hp + sqrt(p_hp**2 - wo**2), + p_hp - sqrt(p_hp**2 - wo**2))) + + # Move any zeros that were at infinity to the center of the stopband + z_bs = append(z_bs, full(degree, +1j*wo)) + z_bs = append(z_bs, full(degree, -1j*wo)) + + # Cancel out gain change caused by inversion + k_bs = k * real(prod(-z) / prod(-p)) + + return z_bs, p_bs, k_bs + + +def butter(N, Wn, btype='low', analog=False, output='ba', fs=None): + """ + Butterworth digital and analog filter design. + + Design an Nth-order digital or analog Butterworth filter and return + the filter coefficients. + + Parameters + ---------- + N : int + The order of the filter. For 'bandpass' and 'bandstop' filters, + the resulting order of the final second-order sections ('sos') + matrix is ``2*N``, with `N` the number of biquad sections + of the desired system. + Wn : array_like + The critical frequency or frequencies. For lowpass and highpass + filters, Wn is a scalar; for bandpass and bandstop filters, + Wn is a length-2 sequence. + + For a Butterworth filter, this is the point at which the gain + drops to 1/sqrt(2) that of the passband (the "-3 dB point"). + + For digital filters, if `fs` is not specified, `Wn` units are + normalized from 0 to 1, where 1 is the Nyquist frequency (`Wn` is + thus in half cycles / sample and defined as 2*critical frequencies + / `fs`). If `fs` is specified, `Wn` is in the same units as `fs`. + + For analog filters, `Wn` is an angular frequency (e.g. rad/s). + btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional + The type of filter. Default is 'lowpass'. + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. + output : {'ba', 'zpk', 'sos'}, optional + Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or + second-order sections ('sos'). Default is 'ba' for backwards + compatibility, but 'sos' should be used for general-purpose filtering. + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + b, a : ndarray, ndarray + Numerator (`b`) and denominator (`a`) polynomials of the IIR filter. + Only returned if ``output='ba'``. + z, p, k : ndarray, ndarray, float + Zeros, poles, and system gain of the IIR filter transfer + function. Only returned if ``output='zpk'``. + sos : ndarray + Second-order sections representation of the IIR filter. + Only returned if ``output='sos'``. + + See Also + -------- + buttord, buttap + + Notes + ----- + The Butterworth filter has maximally flat frequency response in the + passband. + + The ``'sos'`` output parameter was added in 0.16.0. + + If the transfer function form ``[b, a]`` is requested, numerical + problems can occur since the conversion between roots and + the polynomial coefficients is a numerically sensitive operation, + even for N >= 4. It is recommended to work with the SOS + representation. + + .. warning:: + Designing high-order and narrowband IIR filters in TF form can + result in unstable or incorrect filtering due to floating point + numerical precision issues. Consider inspecting output filter + characteristics `freqz` or designing the filters with second-order + sections via ``output='sos'``. + + Examples + -------- + Design an analog filter and plot its frequency response, showing the + critical points: + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> b, a = signal.butter(4, 100, 'low', analog=True) + >>> w, h = signal.freqs(b, a) + >>> plt.semilogx(w, 20 * np.log10(abs(h))) + >>> plt.title('Butterworth filter frequency response') + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.margins(0, 0.1) + >>> plt.grid(which='both', axis='both') + >>> plt.axvline(100, color='green') # cutoff frequency + >>> plt.show() + + Generate a signal made up of 10 Hz and 20 Hz, sampled at 1 kHz + + >>> t = np.linspace(0, 1, 1000, False) # 1 second + >>> sig = np.sin(2*np.pi*10*t) + np.sin(2*np.pi*20*t) + >>> fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True) + >>> ax1.plot(t, sig) + >>> ax1.set_title('10 Hz and 20 Hz sinusoids') + >>> ax1.axis([0, 1, -2, 2]) + + Design a digital high-pass filter at 15 Hz to remove the 10 Hz tone, and + apply it to the signal. (It's recommended to use second-order sections + format when filtering, to avoid numerical error with transfer function + (``ba``) format): + + >>> sos = signal.butter(10, 15, 'hp', fs=1000, output='sos') + >>> filtered = signal.sosfilt(sos, sig) + >>> ax2.plot(t, filtered) + >>> ax2.set_title('After 15 Hz high-pass filter') + >>> ax2.axis([0, 1, -2, 2]) + >>> ax2.set_xlabel('Time [s]') + >>> plt.tight_layout() + >>> plt.show() + """ + return iirfilter(N, Wn, btype=btype, analog=analog, + output=output, ftype='butter', fs=fs) + + +def cheby1(N, rp, Wn, btype='low', analog=False, output='ba', fs=None): + """ + Chebyshev type I digital and analog filter design. + + Design an Nth-order digital or analog Chebyshev type I filter and + return the filter coefficients. + + Parameters + ---------- + N : int + The order of the filter. + rp : float + The maximum ripple allowed below unity gain in the passband. + Specified in decibels, as a positive number. + Wn : array_like + A scalar or length-2 sequence giving the critical frequencies. + For Type I filters, this is the point in the transition band at which + the gain first drops below -`rp`. + + For digital filters, `Wn` are in the same units as `fs`. By default, + `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1, + where 1 is the Nyquist frequency. (`Wn` is thus in + half-cycles / sample.) + + For analog filters, `Wn` is an angular frequency (e.g., rad/s). + btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional + The type of filter. Default is 'lowpass'. + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. + output : {'ba', 'zpk', 'sos'}, optional + Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or + second-order sections ('sos'). Default is 'ba' for backwards + compatibility, but 'sos' should be used for general-purpose filtering. + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + b, a : ndarray, ndarray + Numerator (`b`) and denominator (`a`) polynomials of the IIR filter. + Only returned if ``output='ba'``. + z, p, k : ndarray, ndarray, float + Zeros, poles, and system gain of the IIR filter transfer + function. Only returned if ``output='zpk'``. + sos : ndarray + Second-order sections representation of the IIR filter. + Only returned if ``output='sos'``. + + See Also + -------- + cheb1ord, cheb1ap + + Notes + ----- + The Chebyshev type I filter maximizes the rate of cutoff between the + frequency response's passband and stopband, at the expense of ripple in + the passband and increased ringing in the step response. + + Type I filters roll off faster than Type II (`cheby2`), but Type II + filters do not have any ripple in the passband. + + The equiripple passband has N maxima or minima (for example, a + 5th-order filter has 3 maxima and 2 minima). Consequently, the DC gain is + unity for odd-order filters, or -rp dB for even-order filters. + + The ``'sos'`` output parameter was added in 0.16.0. + + Examples + -------- + Design an analog filter and plot its frequency response, showing the + critical points: + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> b, a = signal.cheby1(4, 5, 100, 'low', analog=True) + >>> w, h = signal.freqs(b, a) + >>> plt.semilogx(w, 20 * np.log10(abs(h))) + >>> plt.title('Chebyshev Type I frequency response (rp=5)') + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.margins(0, 0.1) + >>> plt.grid(which='both', axis='both') + >>> plt.axvline(100, color='green') # cutoff frequency + >>> plt.axhline(-5, color='green') # rp + >>> plt.show() + + Generate a signal made up of 10 Hz and 20 Hz, sampled at 1 kHz + + >>> t = np.linspace(0, 1, 1000, False) # 1 second + >>> sig = np.sin(2*np.pi*10*t) + np.sin(2*np.pi*20*t) + >>> fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True) + >>> ax1.plot(t, sig) + >>> ax1.set_title('10 Hz and 20 Hz sinusoids') + >>> ax1.axis([0, 1, -2, 2]) + + Design a digital high-pass filter at 15 Hz to remove the 10 Hz tone, and + apply it to the signal. (It's recommended to use second-order sections + format when filtering, to avoid numerical error with transfer function + (``ba``) format): + + >>> sos = signal.cheby1(10, 1, 15, 'hp', fs=1000, output='sos') + >>> filtered = signal.sosfilt(sos, sig) + >>> ax2.plot(t, filtered) + >>> ax2.set_title('After 15 Hz high-pass filter') + >>> ax2.axis([0, 1, -2, 2]) + >>> ax2.set_xlabel('Time [s]') + >>> plt.tight_layout() + >>> plt.show() + """ + return iirfilter(N, Wn, rp=rp, btype=btype, analog=analog, + output=output, ftype='cheby1', fs=fs) + + +def cheby2(N, rs, Wn, btype='low', analog=False, output='ba', fs=None): + """ + Chebyshev type II digital and analog filter design. + + Design an Nth-order digital or analog Chebyshev type II filter and + return the filter coefficients. + + Parameters + ---------- + N : int + The order of the filter. + rs : float + The minimum attenuation required in the stop band. + Specified in decibels, as a positive number. + Wn : array_like + A scalar or length-2 sequence giving the critical frequencies. + For Type II filters, this is the point in the transition band at which + the gain first reaches -`rs`. + + For digital filters, `Wn` are in the same units as `fs`. By default, + `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1, + where 1 is the Nyquist frequency. (`Wn` is thus in + half-cycles / sample.) + + For analog filters, `Wn` is an angular frequency (e.g., rad/s). + btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional + The type of filter. Default is 'lowpass'. + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. + output : {'ba', 'zpk', 'sos'}, optional + Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or + second-order sections ('sos'). Default is 'ba' for backwards + compatibility, but 'sos' should be used for general-purpose filtering. + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + b, a : ndarray, ndarray + Numerator (`b`) and denominator (`a`) polynomials of the IIR filter. + Only returned if ``output='ba'``. + z, p, k : ndarray, ndarray, float + Zeros, poles, and system gain of the IIR filter transfer + function. Only returned if ``output='zpk'``. + sos : ndarray + Second-order sections representation of the IIR filter. + Only returned if ``output='sos'``. + + See Also + -------- + cheb2ord, cheb2ap + + Notes + ----- + The Chebyshev type II filter maximizes the rate of cutoff between the + frequency response's passband and stopband, at the expense of ripple in + the stopband and increased ringing in the step response. + + Type II filters do not roll off as fast as Type I (`cheby1`). + + The ``'sos'`` output parameter was added in 0.16.0. + + Examples + -------- + Design an analog filter and plot its frequency response, showing the + critical points: + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> b, a = signal.cheby2(4, 40, 100, 'low', analog=True) + >>> w, h = signal.freqs(b, a) + >>> plt.semilogx(w, 20 * np.log10(abs(h))) + >>> plt.title('Chebyshev Type II frequency response (rs=40)') + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.margins(0, 0.1) + >>> plt.grid(which='both', axis='both') + >>> plt.axvline(100, color='green') # cutoff frequency + >>> plt.axhline(-40, color='green') # rs + >>> plt.show() + + Generate a signal made up of 10 Hz and 20 Hz, sampled at 1 kHz + + >>> t = np.linspace(0, 1, 1000, False) # 1 second + >>> sig = np.sin(2*np.pi*10*t) + np.sin(2*np.pi*20*t) + >>> fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True) + >>> ax1.plot(t, sig) + >>> ax1.set_title('10 Hz and 20 Hz sinusoids') + >>> ax1.axis([0, 1, -2, 2]) + + Design a digital high-pass filter at 17 Hz to remove the 10 Hz tone, and + apply it to the signal. (It's recommended to use second-order sections + format when filtering, to avoid numerical error with transfer function + (``ba``) format): + + >>> sos = signal.cheby2(12, 20, 17, 'hp', fs=1000, output='sos') + >>> filtered = signal.sosfilt(sos, sig) + >>> ax2.plot(t, filtered) + >>> ax2.set_title('After 17 Hz high-pass filter') + >>> ax2.axis([0, 1, -2, 2]) + >>> ax2.set_xlabel('Time [s]') + >>> plt.show() + """ + return iirfilter(N, Wn, rs=rs, btype=btype, analog=analog, + output=output, ftype='cheby2', fs=fs) + + +def ellip(N, rp, rs, Wn, btype='low', analog=False, output='ba', fs=None): + """ + Elliptic (Cauer) digital and analog filter design. + + Design an Nth-order digital or analog elliptic filter and return + the filter coefficients. + + Parameters + ---------- + N : int + The order of the filter. + rp : float + The maximum ripple allowed below unity gain in the passband. + Specified in decibels, as a positive number. + rs : float + The minimum attenuation required in the stop band. + Specified in decibels, as a positive number. + Wn : array_like + A scalar or length-2 sequence giving the critical frequencies. + For elliptic filters, this is the point in the transition band at + which the gain first drops below -`rp`. + + For digital filters, `Wn` are in the same units as `fs`. By default, + `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1, + where 1 is the Nyquist frequency. (`Wn` is thus in + half-cycles / sample.) + + For analog filters, `Wn` is an angular frequency (e.g., rad/s). + btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional + The type of filter. Default is 'lowpass'. + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. + output : {'ba', 'zpk', 'sos'}, optional + Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or + second-order sections ('sos'). Default is 'ba' for backwards + compatibility, but 'sos' should be used for general-purpose filtering. + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + b, a : ndarray, ndarray + Numerator (`b`) and denominator (`a`) polynomials of the IIR filter. + Only returned if ``output='ba'``. + z, p, k : ndarray, ndarray, float + Zeros, poles, and system gain of the IIR filter transfer + function. Only returned if ``output='zpk'``. + sos : ndarray + Second-order sections representation of the IIR filter. + Only returned if ``output='sos'``. + + See Also + -------- + ellipord, ellipap + + Notes + ----- + Also known as Cauer or Zolotarev filters, the elliptical filter maximizes + the rate of transition between the frequency response's passband and + stopband, at the expense of ripple in both, and increased ringing in the + step response. + + As `rp` approaches 0, the elliptical filter becomes a Chebyshev + type II filter (`cheby2`). As `rs` approaches 0, it becomes a Chebyshev + type I filter (`cheby1`). As both approach 0, it becomes a Butterworth + filter (`butter`). + + The equiripple passband has N maxima or minima (for example, a + 5th-order filter has 3 maxima and 2 minima). Consequently, the DC gain is + unity for odd-order filters, or -rp dB for even-order filters. + + The ``'sos'`` output parameter was added in 0.16.0. + + Examples + -------- + Design an analog filter and plot its frequency response, showing the + critical points: + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> b, a = signal.ellip(4, 5, 40, 100, 'low', analog=True) + >>> w, h = signal.freqs(b, a) + >>> plt.semilogx(w, 20 * np.log10(abs(h))) + >>> plt.title('Elliptic filter frequency response (rp=5, rs=40)') + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.margins(0, 0.1) + >>> plt.grid(which='both', axis='both') + >>> plt.axvline(100, color='green') # cutoff frequency + >>> plt.axhline(-40, color='green') # rs + >>> plt.axhline(-5, color='green') # rp + >>> plt.show() + + Generate a signal made up of 10 Hz and 20 Hz, sampled at 1 kHz + + >>> t = np.linspace(0, 1, 1000, False) # 1 second + >>> sig = np.sin(2*np.pi*10*t) + np.sin(2*np.pi*20*t) + >>> fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True) + >>> ax1.plot(t, sig) + >>> ax1.set_title('10 Hz and 20 Hz sinusoids') + >>> ax1.axis([0, 1, -2, 2]) + + Design a digital high-pass filter at 17 Hz to remove the 10 Hz tone, and + apply it to the signal. (It's recommended to use second-order sections + format when filtering, to avoid numerical error with transfer function + (``ba``) format): + + >>> sos = signal.ellip(8, 1, 100, 17, 'hp', fs=1000, output='sos') + >>> filtered = signal.sosfilt(sos, sig) + >>> ax2.plot(t, filtered) + >>> ax2.set_title('After 17 Hz high-pass filter') + >>> ax2.axis([0, 1, -2, 2]) + >>> ax2.set_xlabel('Time [s]') + >>> plt.tight_layout() + >>> plt.show() + """ + return iirfilter(N, Wn, rs=rs, rp=rp, btype=btype, analog=analog, + output=output, ftype='elliptic', fs=fs) + + +def bessel(N, Wn, btype='low', analog=False, output='ba', norm='phase', + fs=None): + """ + Bessel/Thomson digital and analog filter design. + + Design an Nth-order digital or analog Bessel filter and return the + filter coefficients. + + Parameters + ---------- + N : int + The order of the filter. + Wn : array_like + A scalar or length-2 sequence giving the critical frequencies (defined + by the `norm` parameter). + For analog filters, `Wn` is an angular frequency (e.g., rad/s). + + For digital filters, `Wn` are in the same units as `fs`. By default, + `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1, + where 1 is the Nyquist frequency. (`Wn` is thus in + half-cycles / sample.) + btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional + The type of filter. Default is 'lowpass'. + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. (See Notes.) + output : {'ba', 'zpk', 'sos'}, optional + Type of output: numerator/denominator ('ba'), pole-zero ('zpk'), or + second-order sections ('sos'). Default is 'ba'. + norm : {'phase', 'delay', 'mag'}, optional + Critical frequency normalization: + + ``phase`` + The filter is normalized such that the phase response reaches its + midpoint at angular (e.g. rad/s) frequency `Wn`. This happens for + both low-pass and high-pass filters, so this is the + "phase-matched" case. + + The magnitude response asymptotes are the same as a Butterworth + filter of the same order with a cutoff of `Wn`. + + This is the default, and matches MATLAB's implementation. + + ``delay`` + The filter is normalized such that the group delay in the passband + is 1/`Wn` (e.g., seconds). This is the "natural" type obtained by + solving Bessel polynomials. + + ``mag`` + The filter is normalized such that the gain magnitude is -3 dB at + angular frequency `Wn`. + + .. versionadded:: 0.18.0 + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + b, a : ndarray, ndarray + Numerator (`b`) and denominator (`a`) polynomials of the IIR filter. + Only returned if ``output='ba'``. + z, p, k : ndarray, ndarray, float + Zeros, poles, and system gain of the IIR filter transfer + function. Only returned if ``output='zpk'``. + sos : ndarray + Second-order sections representation of the IIR filter. + Only returned if ``output='sos'``. + + Notes + ----- + Also known as a Thomson filter, the analog Bessel filter has maximally + flat group delay and maximally linear phase response, with very little + ringing in the step response. [1]_ + + The Bessel is inherently an analog filter. This function generates digital + Bessel filters using the bilinear transform, which does not preserve the + phase response of the analog filter. As such, it is only approximately + correct at frequencies below about fs/4. To get maximally-flat group + delay at higher frequencies, the analog Bessel filter must be transformed + using phase-preserving techniques. + + See `besselap` for implementation details and references. + + The ``'sos'`` output parameter was added in 0.16.0. + + References + ---------- + .. [1] Thomson, W.E., "Delay Networks having Maximally Flat Frequency + Characteristics", Proceedings of the Institution of Electrical + Engineers, Part III, November 1949, Vol. 96, No. 44, pp. 487-490. + + Examples + -------- + Plot the phase-normalized frequency response, showing the relationship + to the Butterworth's cutoff frequency (green): + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> b, a = signal.butter(4, 100, 'low', analog=True) + >>> w, h = signal.freqs(b, a) + >>> plt.semilogx(w, 20 * np.log10(np.abs(h)), color='silver', ls='dashed') + >>> b, a = signal.bessel(4, 100, 'low', analog=True, norm='phase') + >>> w, h = signal.freqs(b, a) + >>> plt.semilogx(w, 20 * np.log10(np.abs(h))) + >>> plt.title('Bessel filter magnitude response (with Butterworth)') + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.margins(0, 0.1) + >>> plt.grid(which='both', axis='both') + >>> plt.axvline(100, color='green') # cutoff frequency + >>> plt.show() + + and the phase midpoint: + + >>> plt.figure() + >>> plt.semilogx(w, np.unwrap(np.angle(h))) + >>> plt.axvline(100, color='green') # cutoff frequency + >>> plt.axhline(-np.pi, color='red') # phase midpoint + >>> plt.title('Bessel filter phase response') + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Phase [rad]') + >>> plt.margins(0, 0.1) + >>> plt.grid(which='both', axis='both') + >>> plt.show() + + Plot the magnitude-normalized frequency response, showing the -3 dB cutoff: + + >>> b, a = signal.bessel(3, 10, 'low', analog=True, norm='mag') + >>> w, h = signal.freqs(b, a) + >>> plt.semilogx(w, 20 * np.log10(np.abs(h))) + >>> plt.axhline(-3, color='red') # -3 dB magnitude + >>> plt.axvline(10, color='green') # cutoff frequency + >>> plt.title('Amplitude-normalized Bessel filter frequency response') + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.margins(0, 0.1) + >>> plt.grid(which='both', axis='both') + >>> plt.show() + + Plot the delay-normalized filter, showing the maximally-flat group delay + at 0.1 seconds: + + >>> b, a = signal.bessel(5, 1/0.1, 'low', analog=True, norm='delay') + >>> w, h = signal.freqs(b, a) + >>> plt.figure() + >>> plt.semilogx(w[1:], -np.diff(np.unwrap(np.angle(h)))/np.diff(w)) + >>> plt.axhline(0.1, color='red') # 0.1 seconds group delay + >>> plt.title('Bessel filter group delay') + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Group delay [s]') + >>> plt.margins(0, 0.1) + >>> plt.grid(which='both', axis='both') + >>> plt.show() + + """ + return iirfilter(N, Wn, btype=btype, analog=analog, + output=output, ftype='bessel_'+norm, fs=fs) + + +def maxflat(): + pass + + +def yulewalk(): + pass + + +def band_stop_obj(wp, ind, passb, stopb, gpass, gstop, type): + """ + Band Stop Objective Function for order minimization. + + Returns the non-integer order for an analog band stop filter. + + Parameters + ---------- + wp : scalar + Edge of passband `passb`. + ind : int, {0, 1} + Index specifying which `passb` edge to vary (0 or 1). + passb : ndarray + Two element sequence of fixed passband edges. + stopb : ndarray + Two element sequence of fixed stopband edges. + gstop : float + Amount of attenuation in stopband in dB. + gpass : float + Amount of ripple in the passband in dB. + type : {'butter', 'cheby', 'ellip'} + Type of filter. + + Returns + ------- + n : scalar + Filter order (possibly non-integer). + + Notes + ----- + Band-stop filters are used in applications where certain frequency + components need to be blocked while others are allowed; for instance, + removing noise at specific frequencies while allowing the desired signal + to pass through. The order of a filter often determines its complexity and + accuracy. Determining the right order can be a challenge. This function + aims to provide an appropriate order for an analog band stop filter. + + Examples + -------- + + >>> import numpy as np + >>> from scipy.signal import band_stop_obj + >>> wp = 2 + >>> ind = 1 + >>> passb = np.array([1, 3]) + >>> stopb = np.array([0.5, 4]) + >>> gstop = 30 + >>> gpass = 3 + >>> filter_type = 'butter' + >>> band_stop_obj(wp, ind, passb, stopb, gpass, gstop, filter_type) + np.float64(-2.758504160760643) + + """ + + _validate_gpass_gstop(gpass, gstop) + + passbC = passb.copy() + passbC[ind] = wp + nat = (stopb * (passbC[0] - passbC[1]) / + (stopb ** 2 - passbC[0] * passbC[1])) + nat = min(abs(nat)) + + if type == 'butter': + GSTOP = 10 ** (0.1 * abs(gstop)) + GPASS = 10 ** (0.1 * abs(gpass)) + n = (log10((GSTOP - 1.0) / (GPASS - 1.0)) / (2 * log10(nat))) + elif type == 'cheby': + GSTOP = 10 ** (0.1 * abs(gstop)) + GPASS = 10 ** (0.1 * abs(gpass)) + n = arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) / arccosh(nat) + elif type == 'ellip': + GSTOP = 10 ** (0.1 * gstop) + GPASS = 10 ** (0.1 * gpass) + arg1 = sqrt((GPASS - 1.0) / (GSTOP - 1.0)) + arg0 = 1.0 / nat + d0 = special.ellipk([arg0 ** 2, 1 - arg0 ** 2]) + d1 = special.ellipk([arg1 ** 2, 1 - arg1 ** 2]) + n = (d0[0] * d1[1] / (d0[1] * d1[0])) + else: + raise ValueError(f"Incorrect type: {type}") + return n + + +def _pre_warp(wp, ws, analog): + # Pre-warp frequencies for digital filter design + if not analog: + passb = np.tan(pi * wp / 2.0) + stopb = np.tan(pi * ws / 2.0) + else: + passb = wp * 1.0 + stopb = ws * 1.0 + return passb, stopb + + +def _validate_wp_ws(wp, ws, fs, analog): + wp = atleast_1d(wp) + ws = atleast_1d(ws) + if fs is not None: + if analog: + raise ValueError("fs cannot be specified for an analog filter") + wp = 2 * wp / fs + ws = 2 * ws / fs + + filter_type = 2 * (len(wp) - 1) + 1 + if wp[0] >= ws[0]: + filter_type += 1 + + return wp, ws, filter_type + + +def _find_nat_freq(stopb, passb, gpass, gstop, filter_type, filter_kind): + if filter_type == 1: # low + nat = stopb / passb + elif filter_type == 2: # high + nat = passb / stopb + elif filter_type == 3: # stop + + ### breakpoint() + + wp0 = optimize.fminbound(band_stop_obj, passb[0], stopb[0] - 1e-12, + args=(0, passb, stopb, gpass, gstop, + filter_kind), + disp=0) + passb[0] = wp0 + wp1 = optimize.fminbound(band_stop_obj, stopb[1] + 1e-12, passb[1], + args=(1, passb, stopb, gpass, gstop, + filter_kind), + disp=0) + passb[1] = wp1 + nat = ((stopb * (passb[0] - passb[1])) / + (stopb ** 2 - passb[0] * passb[1])) + elif filter_type == 4: # pass + nat = ((stopb ** 2 - passb[0] * passb[1]) / + (stopb * (passb[0] - passb[1]))) + else: + raise ValueError(f"should not happen: {filter_type =}.") + + nat = min(abs(nat)) + return nat, passb + + +def _postprocess_wn(WN, analog, fs): + wn = WN if analog else np.arctan(WN) * 2.0 / pi + if len(wn) == 1: + wn = wn[0] + if fs is not None: + wn = wn * fs / 2 + return wn + + +def buttord(wp, ws, gpass, gstop, analog=False, fs=None): + """Butterworth filter order selection. + + Return the order of the lowest order digital or analog Butterworth filter + that loses no more than `gpass` dB in the passband and has at least + `gstop` dB attenuation in the stopband. + + Parameters + ---------- + wp, ws : float + Passband and stopband edge frequencies. + + For digital filters, these are in the same units as `fs`. By default, + `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1, + where 1 is the Nyquist frequency. (`wp` and `ws` are thus in + half-cycles / sample.) For example: + + - Lowpass: wp = 0.2, ws = 0.3 + - Highpass: wp = 0.3, ws = 0.2 + - Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6] + - Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5] + + For analog filters, `wp` and `ws` are angular frequencies (e.g., rad/s). + gpass : float + The maximum loss in the passband (dB). + gstop : float + The minimum attenuation in the stopband (dB). + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + ord : int + The lowest order for a Butterworth filter which meets specs. + wn : ndarray or float + The Butterworth natural frequency (i.e. the "3dB frequency"). Should + be used with `butter` to give filter results. If `fs` is specified, + this is in the same units, and `fs` must also be passed to `butter`. + + See Also + -------- + butter : Filter design using order and critical points + cheb1ord : Find order and critical points from passband and stopband spec + cheb2ord, ellipord + iirfilter : General filter design using order and critical frequencies + iirdesign : General filter design using passband and stopband spec + + Examples + -------- + Design an analog bandpass filter with passband within 3 dB from 20 to + 50 rad/s, while rejecting at least -40 dB below 14 and above 60 rad/s. + Plot its frequency response, showing the passband and stopband + constraints in gray. + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> N, Wn = signal.buttord([20, 50], [14, 60], 3, 40, True) + >>> b, a = signal.butter(N, Wn, 'band', True) + >>> w, h = signal.freqs(b, a, np.logspace(1, 2, 500)) + >>> plt.semilogx(w, 20 * np.log10(abs(h))) + >>> plt.title('Butterworth bandpass filter fit to constraints') + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.grid(which='both', axis='both') + >>> plt.fill([1, 14, 14, 1], [-40, -40, 99, 99], '0.9', lw=0) # stop + >>> plt.fill([20, 20, 50, 50], [-99, -3, -3, -99], '0.9', lw=0) # pass + >>> plt.fill([60, 60, 1e9, 1e9], [99, -40, -40, 99], '0.9', lw=0) # stop + >>> plt.axis([10, 100, -60, 3]) + >>> plt.show() + + """ + _validate_gpass_gstop(gpass, gstop) + fs = _validate_fs(fs, allow_none=True) + wp, ws, filter_type = _validate_wp_ws(wp, ws, fs, analog) + passb, stopb = _pre_warp(wp, ws, analog) + nat, passb = _find_nat_freq(stopb, passb, gpass, gstop, filter_type, 'butter') + + GSTOP = 10 ** (0.1 * abs(gstop)) + GPASS = 10 ** (0.1 * abs(gpass)) + ord = int(ceil(log10((GSTOP - 1.0) / (GPASS - 1.0)) / (2 * log10(nat)))) + + # Find the Butterworth natural frequency WN (or the "3dB" frequency") + # to give exactly gpass at passb. + try: + W0 = (GPASS - 1.0) ** (-1.0 / (2.0 * ord)) + except ZeroDivisionError: + W0 = 1.0 + warnings.warn("Order is zero...check input parameters.", + RuntimeWarning, stacklevel=2) + + # now convert this frequency back from lowpass prototype + # to the original analog filter + + if filter_type == 1: # low + WN = W0 * passb + elif filter_type == 2: # high + WN = passb / W0 + elif filter_type == 3: # stop + WN = np.empty(2, float) + discr = sqrt((passb[1] - passb[0]) ** 2 + + 4 * W0 ** 2 * passb[0] * passb[1]) + WN[0] = ((passb[1] - passb[0]) + discr) / (2 * W0) + WN[1] = ((passb[1] - passb[0]) - discr) / (2 * W0) + WN = np.sort(abs(WN)) + elif filter_type == 4: # pass + W0 = np.array([-W0, W0], float) + WN = (-W0 * (passb[1] - passb[0]) / 2.0 + + sqrt(W0 ** 2 / 4.0 * (passb[1] - passb[0]) ** 2 + + passb[0] * passb[1])) + WN = np.sort(abs(WN)) + else: + raise ValueError(f"Bad type: {filter_type}") + + wn = _postprocess_wn(WN, analog, fs) + + return ord, wn + + +def cheb1ord(wp, ws, gpass, gstop, analog=False, fs=None): + """Chebyshev type I filter order selection. + + Return the order of the lowest order digital or analog Chebyshev Type I + filter that loses no more than `gpass` dB in the passband and has at + least `gstop` dB attenuation in the stopband. + + Parameters + ---------- + wp, ws : float + Passband and stopband edge frequencies. + + For digital filters, these are in the same units as `fs`. By default, + `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1, + where 1 is the Nyquist frequency. (`wp` and `ws` are thus in + half-cycles / sample.) For example: + + - Lowpass: wp = 0.2, ws = 0.3 + - Highpass: wp = 0.3, ws = 0.2 + - Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6] + - Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5] + + For analog filters, `wp` and `ws` are angular frequencies (e.g., rad/s). + gpass : float + The maximum loss in the passband (dB). + gstop : float + The minimum attenuation in the stopband (dB). + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + ord : int + The lowest order for a Chebyshev type I filter that meets specs. + wn : ndarray or float + The Chebyshev natural frequency (the "3dB frequency") for use with + `cheby1` to give filter results. If `fs` is specified, + this is in the same units, and `fs` must also be passed to `cheby1`. + + See Also + -------- + cheby1 : Filter design using order and critical points + buttord : Find order and critical points from passband and stopband spec + cheb2ord, ellipord + iirfilter : General filter design using order and critical frequencies + iirdesign : General filter design using passband and stopband spec + + Examples + -------- + Design a digital lowpass filter such that the passband is within 3 dB up + to 0.2*(fs/2), while rejecting at least -40 dB above 0.3*(fs/2). Plot its + frequency response, showing the passband and stopband constraints in gray. + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> N, Wn = signal.cheb1ord(0.2, 0.3, 3, 40) + >>> b, a = signal.cheby1(N, 3, Wn, 'low') + >>> w, h = signal.freqz(b, a) + >>> plt.semilogx(w / np.pi, 20 * np.log10(abs(h))) + >>> plt.title('Chebyshev I lowpass filter fit to constraints') + >>> plt.xlabel('Normalized frequency') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.grid(which='both', axis='both') + >>> plt.fill([.01, 0.2, 0.2, .01], [-3, -3, -99, -99], '0.9', lw=0) # stop + >>> plt.fill([0.3, 0.3, 2, 2], [ 9, -40, -40, 9], '0.9', lw=0) # pass + >>> plt.axis([0.08, 1, -60, 3]) + >>> plt.show() + + """ + fs = _validate_fs(fs, allow_none=True) + _validate_gpass_gstop(gpass, gstop) + wp, ws, filter_type = _validate_wp_ws(wp, ws, fs, analog) + passb, stopb = _pre_warp(wp, ws, analog) + nat, passb = _find_nat_freq(stopb, passb, gpass, gstop, filter_type, 'cheby') + + GSTOP = 10 ** (0.1 * abs(gstop)) + GPASS = 10 ** (0.1 * abs(gpass)) + v_pass_stop = np.arccosh(np.sqrt((GSTOP - 1.0) / (GPASS - 1.0))) + ord = int(ceil(v_pass_stop / np.arccosh(nat))) + + # Natural frequencies are just the passband edges + wn = _postprocess_wn(passb, analog, fs) + + return ord, wn + + +def cheb2ord(wp, ws, gpass, gstop, analog=False, fs=None): + """Chebyshev type II filter order selection. + + Return the order of the lowest order digital or analog Chebyshev Type II + filter that loses no more than `gpass` dB in the passband and has at least + `gstop` dB attenuation in the stopband. + + Parameters + ---------- + wp, ws : float + Passband and stopband edge frequencies. + + For digital filters, these are in the same units as `fs`. By default, + `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1, + where 1 is the Nyquist frequency. (`wp` and `ws` are thus in + half-cycles / sample.) For example: + + - Lowpass: wp = 0.2, ws = 0.3 + - Highpass: wp = 0.3, ws = 0.2 + - Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6] + - Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5] + + For analog filters, `wp` and `ws` are angular frequencies (e.g., rad/s). + gpass : float + The maximum loss in the passband (dB). + gstop : float + The minimum attenuation in the stopband (dB). + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + ord : int + The lowest order for a Chebyshev type II filter that meets specs. + wn : ndarray or float + The Chebyshev natural frequency (the "3dB frequency") for use with + `cheby2` to give filter results. If `fs` is specified, + this is in the same units, and `fs` must also be passed to `cheby2`. + + See Also + -------- + cheby2 : Filter design using order and critical points + buttord : Find order and critical points from passband and stopband spec + cheb1ord, ellipord + iirfilter : General filter design using order and critical frequencies + iirdesign : General filter design using passband and stopband spec + + Examples + -------- + Design a digital bandstop filter which rejects -60 dB from 0.2*(fs/2) to + 0.5*(fs/2), while staying within 3 dB below 0.1*(fs/2) or above + 0.6*(fs/2). Plot its frequency response, showing the passband and + stopband constraints in gray. + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> N, Wn = signal.cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 60) + >>> b, a = signal.cheby2(N, 60, Wn, 'stop') + >>> w, h = signal.freqz(b, a) + >>> plt.semilogx(w / np.pi, 20 * np.log10(abs(h))) + >>> plt.title('Chebyshev II bandstop filter fit to constraints') + >>> plt.xlabel('Normalized frequency') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.grid(which='both', axis='both') + >>> plt.fill([.01, .1, .1, .01], [-3, -3, -99, -99], '0.9', lw=0) # stop + >>> plt.fill([.2, .2, .5, .5], [ 9, -60, -60, 9], '0.9', lw=0) # pass + >>> plt.fill([.6, .6, 2, 2], [-99, -3, -3, -99], '0.9', lw=0) # stop + >>> plt.axis([0.06, 1, -80, 3]) + >>> plt.show() + + """ + fs = _validate_fs(fs, allow_none=True) + _validate_gpass_gstop(gpass, gstop) + wp, ws, filter_type = _validate_wp_ws(wp, ws, fs, analog) + passb, stopb = _pre_warp(wp, ws, analog) + nat, passb = _find_nat_freq(stopb, passb, gpass, gstop, filter_type, 'cheby') + + GSTOP = 10 ** (0.1 * abs(gstop)) + GPASS = 10 ** (0.1 * abs(gpass)) + v_pass_stop = np.arccosh(np.sqrt((GSTOP - 1.0) / (GPASS - 1.0))) + ord = int(ceil(v_pass_stop / arccosh(nat))) + + # Find frequency where analog response is -gpass dB. + # Then convert back from low-pass prototype to the original filter. + + new_freq = cosh(1.0 / ord * v_pass_stop) + new_freq = 1.0 / new_freq + + if filter_type == 1: + nat = passb / new_freq + elif filter_type == 2: + nat = passb * new_freq + elif filter_type == 3: + nat = np.empty(2, float) + nat[0] = (new_freq / 2.0 * (passb[0] - passb[1]) + + sqrt(new_freq ** 2 * (passb[1] - passb[0]) ** 2 / 4.0 + + passb[1] * passb[0])) + nat[1] = passb[1] * passb[0] / nat[0] + elif filter_type == 4: + nat = np.empty(2, float) + nat[0] = (1.0 / (2.0 * new_freq) * (passb[0] - passb[1]) + + sqrt((passb[1] - passb[0]) ** 2 / (4.0 * new_freq ** 2) + + passb[1] * passb[0])) + nat[1] = passb[0] * passb[1] / nat[0] + + wn = _postprocess_wn(nat, analog, fs) + + return ord, wn + + +_POW10_LOG10 = np.log(10) + + +def _pow10m1(x): + """10 ** x - 1 for x near 0""" + return np.expm1(_POW10_LOG10 * x) + + +def ellipord(wp, ws, gpass, gstop, analog=False, fs=None): + """Elliptic (Cauer) filter order selection. + + Return the order of the lowest order digital or analog elliptic filter + that loses no more than `gpass` dB in the passband and has at least + `gstop` dB attenuation in the stopband. + + Parameters + ---------- + wp, ws : float + Passband and stopband edge frequencies. + + For digital filters, these are in the same units as `fs`. By default, + `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1, + where 1 is the Nyquist frequency. (`wp` and `ws` are thus in + half-cycles / sample.) For example: + + - Lowpass: wp = 0.2, ws = 0.3 + - Highpass: wp = 0.3, ws = 0.2 + - Bandpass: wp = [0.2, 0.5], ws = [0.1, 0.6] + - Bandstop: wp = [0.1, 0.6], ws = [0.2, 0.5] + + For analog filters, `wp` and `ws` are angular frequencies (e.g., rad/s). + gpass : float + The maximum loss in the passband (dB). + gstop : float + The minimum attenuation in the stopband (dB). + analog : bool, optional + When True, return an analog filter, otherwise a digital filter is + returned. + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + ord : int + The lowest order for an Elliptic (Cauer) filter that meets specs. + wn : ndarray or float + The Chebyshev natural frequency (the "3dB frequency") for use with + `ellip` to give filter results. If `fs` is specified, + this is in the same units, and `fs` must also be passed to `ellip`. + + See Also + -------- + ellip : Filter design using order and critical points + buttord : Find order and critical points from passband and stopband spec + cheb1ord, cheb2ord + iirfilter : General filter design using order and critical frequencies + iirdesign : General filter design using passband and stopband spec + + Examples + -------- + Design an analog highpass filter such that the passband is within 3 dB + above 30 rad/s, while rejecting -60 dB at 10 rad/s. Plot its + frequency response, showing the passband and stopband constraints in gray. + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> N, Wn = signal.ellipord(30, 10, 3, 60, True) + >>> b, a = signal.ellip(N, 3, 60, Wn, 'high', True) + >>> w, h = signal.freqs(b, a, np.logspace(0, 3, 500)) + >>> plt.semilogx(w, 20 * np.log10(abs(h))) + >>> plt.title('Elliptical highpass filter fit to constraints') + >>> plt.xlabel('Frequency [rad/s]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.grid(which='both', axis='both') + >>> plt.fill([.1, 10, 10, .1], [1e4, 1e4, -60, -60], '0.9', lw=0) # stop + >>> plt.fill([30, 30, 1e9, 1e9], [-99, -3, -3, -99], '0.9', lw=0) # pass + >>> plt.axis([1, 300, -80, 3]) + >>> plt.show() + + """ + fs = _validate_fs(fs, allow_none=True) + _validate_gpass_gstop(gpass, gstop) + wp, ws, filter_type = _validate_wp_ws(wp, ws, fs, analog) + passb, stopb = _pre_warp(wp, ws, analog) + nat, passb = _find_nat_freq(stopb, passb, gpass, gstop, filter_type, 'ellip') + + arg1_sq = _pow10m1(0.1 * gpass) / _pow10m1(0.1 * gstop) + arg0 = 1.0 / nat + d0 = special.ellipk(arg0 ** 2), special.ellipkm1(arg0 ** 2) + d1 = special.ellipk(arg1_sq), special.ellipkm1(arg1_sq) + ord = int(ceil(d0[0] * d1[1] / (d0[1] * d1[0]))) + + wn = _postprocess_wn(passb, analog, fs) + + return ord, wn + + +def buttap(N): + """Return (z,p,k) for analog prototype of Nth-order Butterworth filter. + + The filter will have an angular (e.g., rad/s) cutoff frequency of 1. + + See Also + -------- + butter : Filter design function using this prototype + + """ + if abs(int(N)) != N: + raise ValueError("Filter order must be a nonnegative integer") + z = np.array([]) + m = np.arange(-N+1, N, 2) + # Middle value is 0 to ensure an exactly real pole + p = -np.exp(1j * pi * m / (2 * N)) + k = 1 + return z, p, k + + +def cheb1ap(N, rp): + """ + Return (z,p,k) for Nth-order Chebyshev type I analog lowpass filter. + + The returned filter prototype has `rp` decibels of ripple in the passband. + + The filter's angular (e.g. rad/s) cutoff frequency is normalized to 1, + defined as the point at which the gain first drops below ``-rp``. + + See Also + -------- + cheby1 : Filter design function using this prototype + + """ + if abs(int(N)) != N: + raise ValueError("Filter order must be a nonnegative integer") + elif N == 0: + # Avoid divide-by-zero error + # Even order filters have DC gain of -rp dB + return np.array([]), np.array([]), 10**(-rp/20) + z = np.array([]) + + # Ripple factor (epsilon) + eps = np.sqrt(10 ** (0.1 * rp) - 1.0) + mu = 1.0 / N * arcsinh(1 / eps) + + # Arrange poles in an ellipse on the left half of the S-plane + m = np.arange(-N+1, N, 2) + theta = pi * m / (2*N) + p = -sinh(mu + 1j*theta) + + k = np.prod(-p, axis=0).real + if N % 2 == 0: + k = k / sqrt(1 + eps * eps) + + return z, p, k + + +def cheb2ap(N, rs): + """ + Return (z,p,k) for Nth-order Chebyshev type II analog lowpass filter. + + The returned filter prototype has attenuation of at least ``rs`` decibels + in the stopband. + + The filter's angular (e.g. rad/s) cutoff frequency is normalized to 1, + defined as the point at which the attenuation first reaches ``rs``. + + See Also + -------- + cheby2 : Filter design function using this prototype + + """ + if abs(int(N)) != N: + raise ValueError("Filter order must be a nonnegative integer") + elif N == 0: + # Avoid divide-by-zero warning + return np.array([]), np.array([]), 1 + + # Ripple factor (epsilon) + de = 1.0 / sqrt(10 ** (0.1 * rs) - 1) + mu = arcsinh(1.0 / de) / N + + if N % 2: + m = np.concatenate((np.arange(-N+1, 0, 2), np.arange(2, N, 2))) + else: + m = np.arange(-N+1, N, 2) + + z = -conjugate(1j / sin(m * pi / (2.0 * N))) + + # Poles around the unit circle like Butterworth + p = -exp(1j * pi * np.arange(-N+1, N, 2) / (2 * N)) + # Warp into Chebyshev II + p = sinh(mu) * p.real + 1j * cosh(mu) * p.imag + p = 1.0 / p + + k = (np.prod(-p, axis=0) / np.prod(-z, axis=0)).real + return z, p, k + + +EPSILON = 2e-16 + +# number of terms in solving degree equation +_ELLIPDEG_MMAX = 7 + + +def _ellipdeg(n, m1): + """Solve degree equation using nomes + + Given n, m1, solve + n * K(m) / K'(m) = K1(m1) / K1'(m1) + for m + + See [1], Eq. (49) + + References + ---------- + .. [1] Orfanidis, "Lecture Notes on Elliptic Filter Design", + https://www.ece.rutgers.edu/~orfanidi/ece521/notes.pdf + """ + K1 = special.ellipk(m1) + K1p = special.ellipkm1(m1) + + q1 = np.exp(-np.pi * K1p / K1) + q = q1 ** (1/n) + + mnum = np.arange(_ELLIPDEG_MMAX + 1) + mden = np.arange(1, _ELLIPDEG_MMAX + 2) + + num = np.sum(q ** (mnum * (mnum+1))) + den = 1 + 2 * np.sum(q ** (mden**2)) + + return 16 * q * (num / den) ** 4 + + +# Maximum number of iterations in Landen transformation recursion +# sequence. 10 is conservative; unit tests pass with 4, Orfanidis +# (see _arc_jac_cn [1]) suggests 5. +_ARC_JAC_SN_MAXITER = 10 + + +def _arc_jac_sn(w, m): + """Inverse Jacobian elliptic sn + + Solve for z in w = sn(z, m) + + Parameters + ---------- + w : complex scalar + argument + + m : scalar + modulus; in interval [0, 1] + + + See [1], Eq. (56) + + References + ---------- + .. [1] Orfanidis, "Lecture Notes on Elliptic Filter Design", + https://www.ece.rutgers.edu/~orfanidi/ece521/notes.pdf + + """ + + def _complement(kx): + # (1-k**2) ** 0.5; the expression below + # works for small kx + return ((1 - kx) * (1 + kx)) ** 0.5 + + k = m ** 0.5 + + if k > 1: + return np.nan + elif k == 1: + return np.arctanh(w) + + ks = [k] + niter = 0 + while ks[-1] != 0: + k_ = ks[-1] + k_p = _complement(k_) + ks.append((1 - k_p) / (1 + k_p)) + niter += 1 + if niter > _ARC_JAC_SN_MAXITER: + raise ValueError('Landen transformation not converging') + + K = np.prod(1 + np.array(ks[1:])) * np.pi/2 + + wns = [w] + + for kn, knext in zip(ks[:-1], ks[1:]): + wn = wns[-1] + wnext = (2 * wn / + ((1 + knext) * (1 + _complement(kn * wn)))) + wns.append(wnext) + + u = 2 / np.pi * np.arcsin(wns[-1]) + + z = K * u + return z + + +def _arc_jac_sc1(w, m): + """Real inverse Jacobian sc, with complementary modulus + + Solve for z in w = sc(z, 1-m) + + w - real scalar + + m - modulus + + From [1], sc(z, m) = -i * sn(i * z, 1 - m) + + References + ---------- + # noqa: E501 + .. [1] https://functions.wolfram.com/EllipticFunctions/JacobiSC/introductions/JacobiPQs/ShowAll.html, + "Representations through other Jacobi functions" + + """ + + zcomplex = _arc_jac_sn(1j * w, m) + if abs(zcomplex.real) > 1e-14: + raise ValueError + + return zcomplex.imag + + +def ellipap(N, rp, rs): + """Return (z,p,k) of Nth-order elliptic analog lowpass filter. + + The filter is a normalized prototype that has `rp` decibels of ripple + in the passband and a stopband `rs` decibels down. + + The filter's angular (e.g., rad/s) cutoff frequency is normalized to 1, + defined as the point at which the gain first drops below ``-rp``. + + See Also + -------- + ellip : Filter design function using this prototype + + References + ---------- + .. [1] Lutovac, Tosic, and Evans, "Filter Design for Signal Processing", + Chapters 5 and 12. + + .. [2] Orfanidis, "Lecture Notes on Elliptic Filter Design", + https://www.ece.rutgers.edu/~orfanidi/ece521/notes.pdf + + """ + if abs(int(N)) != N: + raise ValueError("Filter order must be a nonnegative integer") + elif N == 0: + # Avoid divide-by-zero warning + # Even order filters have DC gain of -rp dB + return np.array([]), np.array([]), 10**(-rp/20) + elif N == 1: + p = -sqrt(1.0 / _pow10m1(0.1 * rp)) + k = -p + z = [] + return asarray(z), asarray(p), k + + eps_sq = _pow10m1(0.1 * rp) + + eps = np.sqrt(eps_sq) + ck1_sq = eps_sq / _pow10m1(0.1 * rs) + if ck1_sq == 0: + raise ValueError("Cannot design a filter with given rp and rs" + " specifications.") + + val = special.ellipk(ck1_sq), special.ellipkm1(ck1_sq) + + m = _ellipdeg(N, ck1_sq) + + capk = special.ellipk(m) + + j = np.arange(1 - N % 2, N, 2) + jj = len(j) + + [s, c, d, phi] = special.ellipj(j * capk / N, m * np.ones(jj)) + snew = np.compress(abs(s) > EPSILON, s, axis=-1) + z = 1.0 / (sqrt(m) * snew) + z = 1j * z + z = np.concatenate((z, conjugate(z))) + + r = _arc_jac_sc1(1. / eps, ck1_sq) + v0 = capk * r / (N * val[0]) + + [sv, cv, dv, phi] = special.ellipj(v0, 1 - m) + p = -(c * d * sv * cv + 1j * s * dv) / (1 - (d * sv) ** 2.0) + + if N % 2: + newp = np.compress( + abs(p.imag) > EPSILON * np.sqrt(np.sum(p * np.conjugate(p), axis=0).real), + p, axis=-1 + ) + p = np.concatenate((p, conjugate(newp))) + else: + p = np.concatenate((p, conjugate(p))) + + k = (np.prod(-p, axis=0) / np.prod(-z, axis=0)).real + if N % 2 == 0: + k = k / np.sqrt(1 + eps_sq) + + return z, p, k + + +# TODO: Make this a real public function scipy.misc.ff +def _falling_factorial(x, n): + r""" + Return the factorial of `x` to the `n` falling. + + This is defined as: + + .. math:: x^\underline n = (x)_n = x (x-1) \cdots (x-n+1) + + This can more efficiently calculate ratios of factorials, since: + + n!/m! == falling_factorial(n, n-m) + + where n >= m + + skipping the factors that cancel out + + the usual factorial n! == ff(n, n) + """ + val = 1 + for k in range(x - n + 1, x + 1): + val *= k + return val + + +def _bessel_poly(n, reverse=False): + """ + Return the coefficients of Bessel polynomial of degree `n` + + If `reverse` is true, a reverse Bessel polynomial is output. + + Output is a list of coefficients: + [1] = 1 + [1, 1] = 1*s + 1 + [1, 3, 3] = 1*s^2 + 3*s + 3 + [1, 6, 15, 15] = 1*s^3 + 6*s^2 + 15*s + 15 + [1, 10, 45, 105, 105] = 1*s^4 + 10*s^3 + 45*s^2 + 105*s + 105 + etc. + + Output is a Python list of arbitrary precision long ints, so n is only + limited by your hardware's memory. + + Sequence is http://oeis.org/A001498, and output can be confirmed to + match http://oeis.org/A001498/b001498.txt : + + >>> from scipy.signal._filter_design import _bessel_poly + >>> i = 0 + >>> for n in range(51): + ... for x in _bessel_poly(n, reverse=True): + ... print(i, x) + ... i += 1 + + """ + if abs(int(n)) != n: + raise ValueError("Polynomial order must be a nonnegative integer") + else: + n = int(n) # np.int32 doesn't work, for instance + + out = [] + for k in range(n + 1): + num = _falling_factorial(2*n - k, n) + den = 2**(n - k) * math.factorial(k) + out.append(num // den) + + if reverse: + return out[::-1] + else: + return out + + +def _campos_zeros(n): + """ + Return approximate zero locations of Bessel polynomials y_n(x) for order + `n` using polynomial fit (Campos-Calderon 2011) + """ + if n == 1: + return asarray([-1+0j]) + + s = npp_polyval(n, [0, 0, 2, 0, -3, 1]) + b3 = npp_polyval(n, [16, -8]) / s + b2 = npp_polyval(n, [-24, -12, 12]) / s + b1 = npp_polyval(n, [8, 24, -12, -2]) / s + b0 = npp_polyval(n, [0, -6, 0, 5, -1]) / s + + r = npp_polyval(n, [0, 0, 2, 1]) + a1 = npp_polyval(n, [-6, -6]) / r + a2 = 6 / r + + k = np.arange(1, n+1) + x = npp_polyval(k, [0, a1, a2]) + y = npp_polyval(k, [b0, b1, b2, b3]) + + return x + 1j*y + + +def _aberth(f, fp, x0, tol=1e-15, maxiter=50): + """ + Given a function `f`, its first derivative `fp`, and a set of initial + guesses `x0`, simultaneously find the roots of the polynomial using the + Aberth-Ehrlich method. + + ``len(x0)`` should equal the number of roots of `f`. + + (This is not a complete implementation of Bini's algorithm.) + """ + + N = len(x0) + + x = array(x0, complex) + beta = np.empty_like(x0) + + for iteration in range(maxiter): + alpha = -f(x) / fp(x) # Newton's method + + # Model "repulsion" between zeros + for k in range(N): + beta[k] = np.sum(1/(x[k] - x[k+1:])) + beta[k] += np.sum(1/(x[k] - x[:k])) + + x += alpha / (1 + alpha * beta) + + if not all(np.isfinite(x)): + raise RuntimeError('Root-finding calculation failed') + + # Mekwi: The iterative process can be stopped when |hn| has become + # less than the largest error one is willing to permit in the root. + if all(abs(alpha) <= tol): + break + else: + raise Exception('Zeros failed to converge') + + return x + + +def _bessel_zeros(N): + """ + Find zeros of ordinary Bessel polynomial of order `N`, by root-finding of + modified Bessel function of the second kind + """ + if N == 0: + return asarray([]) + + # Generate starting points + x0 = _campos_zeros(N) + + # Zeros are the same for exp(1/x)*K_{N+0.5}(1/x) and Nth-order ordinary + # Bessel polynomial y_N(x) + def f(x): + return special.kve(N+0.5, 1/x) + + # First derivative of above + def fp(x): + return (special.kve(N-0.5, 1/x)/(2*x**2) - + special.kve(N+0.5, 1/x)/(x**2) + + special.kve(N+1.5, 1/x)/(2*x**2)) + + # Starting points converge to true zeros + x = _aberth(f, fp, x0) + + # Improve precision using Newton's method on each + for i in range(len(x)): + x[i] = optimize.newton(f, x[i], fp, tol=1e-15) + + # Average complex conjugates to make them exactly symmetrical + x = np.mean((x, x[::-1].conj()), 0) + + # Zeros should sum to -1 + if abs(np.sum(x) + 1) > 1e-15: + raise RuntimeError('Generated zeros are inaccurate') + + return x + + +def _norm_factor(p, k): + """ + Numerically find frequency shift to apply to delay-normalized filter such + that -3 dB point is at 1 rad/sec. + + `p` is an array_like of polynomial poles + `k` is a float gain + + First 10 values are listed in "Bessel Scale Factors" table, + "Bessel Filters Polynomials, Poles and Circuit Elements 2003, C. Bond." + """ + p = asarray(p, dtype=complex) + + def G(w): + """ + Gain of filter + """ + return abs(k / prod(1j*w - p)) + + def cutoff(w): + """ + When gain = -3 dB, return 0 + """ + return G(w) - 1/np.sqrt(2) + + return optimize.newton(cutoff, 1.5) + + +def besselap(N, norm='phase'): + """ + Return (z,p,k) for analog prototype of an Nth-order Bessel filter. + + Parameters + ---------- + N : int + The order of the filter. + norm : {'phase', 'delay', 'mag'}, optional + Frequency normalization: + + ``phase`` + The filter is normalized such that the phase response reaches its + midpoint at an angular (e.g., rad/s) cutoff frequency of 1. This + happens for both low-pass and high-pass filters, so this is the + "phase-matched" case. [6]_ + + The magnitude response asymptotes are the same as a Butterworth + filter of the same order with a cutoff of `Wn`. + + This is the default, and matches MATLAB's implementation. + + ``delay`` + The filter is normalized such that the group delay in the passband + is 1 (e.g., 1 second). This is the "natural" type obtained by + solving Bessel polynomials + + ``mag`` + The filter is normalized such that the gain magnitude is -3 dB at + angular frequency 1. This is called "frequency normalization" by + Bond. [1]_ + + .. versionadded:: 0.18.0 + + Returns + ------- + z : ndarray + Zeros of the transfer function. Is always an empty array. + p : ndarray + Poles of the transfer function. + k : scalar + Gain of the transfer function. For phase-normalized, this is always 1. + + See Also + -------- + bessel : Filter design function using this prototype + + Notes + ----- + To find the pole locations, approximate starting points are generated [2]_ + for the zeros of the ordinary Bessel polynomial [3]_, then the + Aberth-Ehrlich method [4]_ [5]_ is used on the Kv(x) Bessel function to + calculate more accurate zeros, and these locations are then inverted about + the unit circle. + + References + ---------- + .. [1] C.R. Bond, "Bessel Filter Constants", + http://www.crbond.com/papers/bsf.pdf + .. [2] Campos and Calderon, "Approximate closed-form formulas for the + zeros of the Bessel Polynomials", :arXiv:`1105.0957`. + .. [3] Thomson, W.E., "Delay Networks having Maximally Flat Frequency + Characteristics", Proceedings of the Institution of Electrical + Engineers, Part III, November 1949, Vol. 96, No. 44, pp. 487-490. + .. [4] Aberth, "Iteration Methods for Finding all Zeros of a Polynomial + Simultaneously", Mathematics of Computation, Vol. 27, No. 122, + April 1973 + .. [5] Ehrlich, "A modified Newton method for polynomials", Communications + of the ACM, Vol. 10, Issue 2, pp. 107-108, Feb. 1967, + :DOI:`10.1145/363067.363115` + .. [6] Miller and Bohn, "A Bessel Filter Crossover, and Its Relation to + Others", RaneNote 147, 1998, + https://www.ranecommercial.com/legacy/note147.html + + """ + if abs(int(N)) != N: + raise ValueError("Filter order must be a nonnegative integer") + + N = int(N) # calculation below doesn't always fit in np.int64 + if N == 0: + p = [] + k = 1 + else: + # Find roots of reverse Bessel polynomial + p = 1/_bessel_zeros(N) + + a_last = _falling_factorial(2*N, N) // 2**N + + # Shift them to a different normalization if required + if norm in ('delay', 'mag'): + # Normalized for group delay of 1 + k = a_last + if norm == 'mag': + # -3 dB magnitude point is at 1 rad/sec + norm_factor = _norm_factor(p, k) + p /= norm_factor + k = norm_factor**-N * a_last + elif norm == 'phase': + # Phase-matched (1/2 max phase shift at 1 rad/sec) + # Asymptotes are same as Butterworth filter + p *= 10**(-math.log10(a_last)/N) + k = 1 + else: + raise ValueError('normalization not understood') + + return asarray([]), asarray(p, dtype=complex), float(k) + + +def iirnotch(w0, Q, fs=2.0): + """ + Design second-order IIR notch digital filter. + + A notch filter is a band-stop filter with a narrow bandwidth + (high quality factor). It rejects a narrow frequency band and + leaves the rest of the spectrum little changed. + + Parameters + ---------- + w0 : float + Frequency to remove from a signal. If `fs` is specified, this is in + the same units as `fs`. By default, it is a normalized scalar that must + satisfy ``0 < w0 < 1``, with ``w0 = 1`` corresponding to half of the + sampling frequency. + Q : float + Quality factor. Dimensionless parameter that characterizes + notch filter -3 dB bandwidth ``bw`` relative to its center + frequency, ``Q = w0/bw``. + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + b, a : ndarray, ndarray + Numerator (``b``) and denominator (``a``) polynomials + of the IIR filter. + + See Also + -------- + iirpeak + + Notes + ----- + .. versionadded:: 0.19.0 + + References + ---------- + .. [1] Sophocles J. Orfanidis, "Introduction To Signal Processing", + Prentice-Hall, 1996 + + Examples + -------- + Design and plot filter to remove the 60 Hz component from a + signal sampled at 200 Hz, using a quality factor Q = 30 + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> fs = 200.0 # Sample frequency (Hz) + >>> f0 = 60.0 # Frequency to be removed from signal (Hz) + >>> Q = 30.0 # Quality factor + >>> # Design notch filter + >>> b, a = signal.iirnotch(f0, Q, fs) + + >>> # Frequency response + >>> freq, h = signal.freqz(b, a, fs=fs) + >>> # Plot + >>> fig, ax = plt.subplots(2, 1, figsize=(8, 6)) + >>> ax[0].plot(freq, 20*np.log10(abs(h)), color='blue') + >>> ax[0].set_title("Frequency Response") + >>> ax[0].set_ylabel("Amplitude [dB]", color='blue') + >>> ax[0].set_xlim([0, 100]) + >>> ax[0].set_ylim([-25, 10]) + >>> ax[0].grid(True) + >>> ax[1].plot(freq, np.unwrap(np.angle(h))*180/np.pi, color='green') + >>> ax[1].set_ylabel("Phase [deg]", color='green') + >>> ax[1].set_xlabel("Frequency [Hz]") + >>> ax[1].set_xlim([0, 100]) + >>> ax[1].set_yticks([-90, -60, -30, 0, 30, 60, 90]) + >>> ax[1].set_ylim([-90, 90]) + >>> ax[1].grid(True) + >>> plt.show() + """ + + return _design_notch_peak_filter(w0, Q, "notch", fs) + + +def iirpeak(w0, Q, fs=2.0): + """ + Design second-order IIR peak (resonant) digital filter. + + A peak filter is a band-pass filter with a narrow bandwidth + (high quality factor). It rejects components outside a narrow + frequency band. + + Parameters + ---------- + w0 : float + Frequency to be retained in a signal. If `fs` is specified, this is in + the same units as `fs`. By default, it is a normalized scalar that must + satisfy ``0 < w0 < 1``, with ``w0 = 1`` corresponding to half of the + sampling frequency. + Q : float + Quality factor. Dimensionless parameter that characterizes + peak filter -3 dB bandwidth ``bw`` relative to its center + frequency, ``Q = w0/bw``. + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0 + + Returns + ------- + b, a : ndarray, ndarray + Numerator (``b``) and denominator (``a``) polynomials + of the IIR filter. + + See Also + -------- + iirnotch + + Notes + ----- + .. versionadded:: 0.19.0 + + References + ---------- + .. [1] Sophocles J. Orfanidis, "Introduction To Signal Processing", + Prentice-Hall, 1996 + + Examples + -------- + Design and plot filter to remove the frequencies other than the 300 Hz + component from a signal sampled at 1000 Hz, using a quality factor Q = 30 + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> fs = 1000.0 # Sample frequency (Hz) + >>> f0 = 300.0 # Frequency to be retained (Hz) + >>> Q = 30.0 # Quality factor + >>> # Design peak filter + >>> b, a = signal.iirpeak(f0, Q, fs) + + >>> # Frequency response + >>> freq, h = signal.freqz(b, a, fs=fs) + >>> # Plot + >>> fig, ax = plt.subplots(2, 1, figsize=(8, 6)) + >>> ax[0].plot(freq, 20*np.log10(np.maximum(abs(h), 1e-5)), color='blue') + >>> ax[0].set_title("Frequency Response") + >>> ax[0].set_ylabel("Amplitude [dB]", color='blue') + >>> ax[0].set_xlim([0, 500]) + >>> ax[0].set_ylim([-50, 10]) + >>> ax[0].grid(True) + >>> ax[1].plot(freq, np.unwrap(np.angle(h))*180/np.pi, color='green') + >>> ax[1].set_ylabel("Phase [deg]", color='green') + >>> ax[1].set_xlabel("Frequency [Hz]") + >>> ax[1].set_xlim([0, 500]) + >>> ax[1].set_yticks([-90, -60, -30, 0, 30, 60, 90]) + >>> ax[1].set_ylim([-90, 90]) + >>> ax[1].grid(True) + >>> plt.show() + """ + + return _design_notch_peak_filter(w0, Q, "peak", fs) + + +def _design_notch_peak_filter(w0, Q, ftype, fs=2.0): + """ + Design notch or peak digital filter. + + Parameters + ---------- + w0 : float + Normalized frequency to remove from a signal. If `fs` is specified, + this is in the same units as `fs`. By default, it is a normalized + scalar that must satisfy ``0 < w0 < 1``, with ``w0 = 1`` + corresponding to half of the sampling frequency. + Q : float + Quality factor. Dimensionless parameter that characterizes + notch filter -3 dB bandwidth ``bw`` relative to its center + frequency, ``Q = w0/bw``. + ftype : str + The type of IIR filter to design: + + - notch filter : ``notch`` + - peak filter : ``peak`` + fs : float, optional + The sampling frequency of the digital system. + + .. versionadded:: 1.2.0: + + Returns + ------- + b, a : ndarray, ndarray + Numerator (``b``) and denominator (``a``) polynomials + of the IIR filter. + """ + fs = _validate_fs(fs, allow_none=False) + + # Guarantee that the inputs are floats + w0 = float(w0) + Q = float(Q) + w0 = 2*w0/fs + + # Checks if w0 is within the range + if w0 > 1.0 or w0 < 0.0: + raise ValueError("w0 should be such that 0 < w0 < 1") + + # Get bandwidth + bw = w0/Q + + # Normalize inputs + bw = bw*np.pi + w0 = w0*np.pi + + if ftype not in ("notch", "peak"): + raise ValueError("Unknown ftype.") + + # Compute beta according to Eqs. 11.3.4 (p.575) and 11.3.19 (p.579) from + # reference [1]. Due to assuming a -3 dB attenuation value, i.e, assuming + # gb = 1 / np.sqrt(2), the following terms simplify to: + # (np.sqrt(1.0 - gb**2.0) / gb) = 1 + # (gb / np.sqrt(1.0 - gb**2.0)) = 1 + beta = np.tan(bw/2.0) + + # Compute gain: formula 11.3.6 (p.575) from reference [1] + gain = 1.0/(1.0+beta) + + # Compute numerator b and denominator a + # formulas 11.3.7 (p.575) and 11.3.21 (p.579) + # from reference [1] + if ftype == "notch": + b = gain*np.array([1.0, -2.0*np.cos(w0), 1.0]) + else: + b = (1.0-gain)*np.array([1.0, 0.0, -1.0]) + a = np.array([1.0, -2.0*gain*np.cos(w0), (2.0*gain-1.0)]) + + return b, a + + +def iircomb(w0, Q, ftype='notch', fs=2.0, *, pass_zero=False): + """ + Design IIR notching or peaking digital comb filter. + + A notching comb filter consists of regularly-spaced band-stop filters with + a narrow bandwidth (high quality factor). Each rejects a narrow frequency + band and leaves the rest of the spectrum little changed. + + A peaking comb filter consists of regularly-spaced band-pass filters with + a narrow bandwidth (high quality factor). Each rejects components outside + a narrow frequency band. + + Parameters + ---------- + w0 : float + The fundamental frequency of the comb filter (the spacing between its + peaks). This must evenly divide the sampling frequency. If `fs` is + specified, this is in the same units as `fs`. By default, it is + a normalized scalar that must satisfy ``0 < w0 < 1``, with + ``w0 = 1`` corresponding to half of the sampling frequency. + Q : float + Quality factor. Dimensionless parameter that characterizes + notch filter -3 dB bandwidth ``bw`` relative to its center + frequency, ``Q = w0/bw``. + ftype : {'notch', 'peak'} + The type of comb filter generated by the function. If 'notch', then + the Q factor applies to the notches. If 'peak', then the Q factor + applies to the peaks. Default is 'notch'. + fs : float, optional + The sampling frequency of the signal. Default is 2.0. + pass_zero : bool, optional + If False (default), the notches (nulls) of the filter are centered on + frequencies [0, w0, 2*w0, ...], and the peaks are centered on the + midpoints [w0/2, 3*w0/2, 5*w0/2, ...]. If True, the peaks are centered + on [0, w0, 2*w0, ...] (passing zero frequency) and vice versa. + + .. versionadded:: 1.9.0 + + Returns + ------- + b, a : ndarray, ndarray + Numerator (``b``) and denominator (``a``) polynomials + of the IIR filter. + + Raises + ------ + ValueError + If `w0` is less than or equal to 0 or greater than or equal to + ``fs/2``, if `fs` is not divisible by `w0`, if `ftype` + is not 'notch' or 'peak' + + See Also + -------- + iirnotch + iirpeak + + Notes + ----- + For implementation details, see [1]_. The TF implementation of the + comb filter is numerically stable even at higher orders due to the + use of a single repeated pole, which won't suffer from precision loss. + + References + ---------- + .. [1] Sophocles J. Orfanidis, "Introduction To Signal Processing", + Prentice-Hall, 1996, ch. 11, "Digital Filter Design" + + Examples + -------- + Design and plot notching comb filter at 20 Hz for a + signal sampled at 200 Hz, using quality factor Q = 30 + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> fs = 200.0 # Sample frequency (Hz) + >>> f0 = 20.0 # Frequency to be removed from signal (Hz) + >>> Q = 30.0 # Quality factor + >>> # Design notching comb filter + >>> b, a = signal.iircomb(f0, Q, ftype='notch', fs=fs) + + >>> # Frequency response + >>> freq, h = signal.freqz(b, a, fs=fs) + >>> response = abs(h) + >>> # To avoid divide by zero when graphing + >>> response[response == 0] = 1e-20 + >>> # Plot + >>> fig, ax = plt.subplots(2, 1, figsize=(8, 6), sharex=True) + >>> ax[0].plot(freq, 20*np.log10(abs(response)), color='blue') + >>> ax[0].set_title("Frequency Response") + >>> ax[0].set_ylabel("Amplitude [dB]", color='blue') + >>> ax[0].set_xlim([0, 100]) + >>> ax[0].set_ylim([-30, 10]) + >>> ax[0].grid(True) + >>> ax[1].plot(freq, (np.angle(h)*180/np.pi+180)%360 - 180, color='green') + >>> ax[1].set_ylabel("Phase [deg]", color='green') + >>> ax[1].set_xlabel("Frequency [Hz]") + >>> ax[1].set_xlim([0, 100]) + >>> ax[1].set_yticks([-90, -60, -30, 0, 30, 60, 90]) + >>> ax[1].set_ylim([-90, 90]) + >>> ax[1].grid(True) + >>> plt.show() + + Design and plot peaking comb filter at 250 Hz for a + signal sampled at 1000 Hz, using quality factor Q = 30 + + >>> fs = 1000.0 # Sample frequency (Hz) + >>> f0 = 250.0 # Frequency to be retained (Hz) + >>> Q = 30.0 # Quality factor + >>> # Design peaking filter + >>> b, a = signal.iircomb(f0, Q, ftype='peak', fs=fs, pass_zero=True) + + >>> # Frequency response + >>> freq, h = signal.freqz(b, a, fs=fs) + >>> response = abs(h) + >>> # To avoid divide by zero when graphing + >>> response[response == 0] = 1e-20 + >>> # Plot + >>> fig, ax = plt.subplots(2, 1, figsize=(8, 6), sharex=True) + >>> ax[0].plot(freq, 20*np.log10(np.maximum(abs(h), 1e-5)), color='blue') + >>> ax[0].set_title("Frequency Response") + >>> ax[0].set_ylabel("Amplitude [dB]", color='blue') + >>> ax[0].set_xlim([0, 500]) + >>> ax[0].set_ylim([-80, 10]) + >>> ax[0].grid(True) + >>> ax[1].plot(freq, (np.angle(h)*180/np.pi+180)%360 - 180, color='green') + >>> ax[1].set_ylabel("Phase [deg]", color='green') + >>> ax[1].set_xlabel("Frequency [Hz]") + >>> ax[1].set_xlim([0, 500]) + >>> ax[1].set_yticks([-90, -60, -30, 0, 30, 60, 90]) + >>> ax[1].set_ylim([-90, 90]) + >>> ax[1].grid(True) + >>> plt.show() + """ + + # Convert w0, Q, and fs to float + w0 = float(w0) + Q = float(Q) + fs = _validate_fs(fs, allow_none=False) + + # Check for invalid cutoff frequency or filter type + ftype = ftype.lower() + if not 0 < w0 < fs / 2: + raise ValueError(f"w0 must be between 0 and {fs / 2}" + f" (Nyquist), but given {w0}.") + if ftype not in ('notch', 'peak'): + raise ValueError('ftype must be either notch or peak.') + + # Compute the order of the filter + N = round(fs / w0) + + # Check for cutoff frequency divisibility + if abs(w0 - fs/N)/fs > 1e-14: + raise ValueError('fs must be divisible by w0.') + + # Compute frequency in radians and filter bandwidth + # Eq. 11.3.1 (p. 574) from reference [1] + w0 = (2 * np.pi * w0) / fs + w_delta = w0 / Q + + # Define base gain values depending on notch or peak filter + # Compute -3dB attenuation + # Eqs. 11.4.1 and 11.4.2 (p. 582) from reference [1] + if ftype == 'notch': + G0, G = 1, 0 + elif ftype == 'peak': + G0, G = 0, 1 + + # Compute beta according to Eq. 11.5.3 (p. 591) from reference [1]. Due to + # assuming a -3 dB attenuation value, i.e, assuming GB = 1 / np.sqrt(2), + # the following term simplifies to: + # np.sqrt((GB**2 - G0**2) / (G**2 - GB**2)) = 1 + beta = np.tan(N * w_delta / 4) + + # Compute filter coefficients + # Eq 11.5.1 (p. 590) variables a, b, c from reference [1] + ax = (1 - beta) / (1 + beta) + bx = (G0 + G * beta) / (1 + beta) + cx = (G0 - G * beta) / (1 + beta) + + # Last coefficients are negative to get peaking comb that passes zero or + # notching comb that doesn't. + negative_coef = ((ftype == 'peak' and pass_zero) or + (ftype == 'notch' and not pass_zero)) + + # Compute numerator coefficients + # Eq 11.5.1 (p. 590) or Eq 11.5.4 (p. 591) from reference [1] + # b - cz^-N or b + cz^-N + b = np.zeros(N + 1) + b[0] = bx + if negative_coef: + b[-1] = -cx + else: + b[-1] = +cx + + # Compute denominator coefficients + # Eq 11.5.1 (p. 590) or Eq 11.5.4 (p. 591) from reference [1] + # 1 - az^-N or 1 + az^-N + a = np.zeros(N + 1) + a[0] = 1 + if negative_coef: + a[-1] = -ax + else: + a[-1] = +ax + + return b, a + + +def _hz_to_erb(hz): + """ + Utility for converting from frequency (Hz) to the + Equivalent Rectangular Bandwidth (ERB) scale + ERB = frequency / EarQ + minBW + """ + EarQ = 9.26449 + minBW = 24.7 + return hz / EarQ + minBW + + +def gammatone(freq, ftype, order=None, numtaps=None, fs=None): + """ + Gammatone filter design. + + This function computes the coefficients of an FIR or IIR gammatone + digital filter [1]_. + + Parameters + ---------- + freq : float + Center frequency of the filter (expressed in the same units + as `fs`). + ftype : {'fir', 'iir'} + The type of filter the function generates. If 'fir', the function + will generate an Nth order FIR gammatone filter. If 'iir', the + function will generate an 8th order digital IIR filter, modeled as + as 4th order gammatone filter. + order : int, optional + The order of the filter. Only used when ``ftype='fir'``. + Default is 4 to model the human auditory system. Must be between + 0 and 24. + numtaps : int, optional + Length of the filter. Only used when ``ftype='fir'``. + Default is ``fs*0.015`` if `fs` is greater than 1000, + 15 if `fs` is less than or equal to 1000. + fs : float, optional + The sampling frequency of the signal. `freq` must be between + 0 and ``fs/2``. Default is 2. + + Returns + ------- + b, a : ndarray, ndarray + Numerator (``b``) and denominator (``a``) polynomials of the filter. + + Raises + ------ + ValueError + If `freq` is less than or equal to 0 or greater than or equal to + ``fs/2``, if `ftype` is not 'fir' or 'iir', if `order` is less than + or equal to 0 or greater than 24 when ``ftype='fir'`` + + See Also + -------- + firwin + iirfilter + + References + ---------- + .. [1] Slaney, Malcolm, "An Efficient Implementation of the + Patterson-Holdsworth Auditory Filter Bank", Apple Computer + Technical Report 35, 1993, pp.3-8, 34-39. + + Examples + -------- + 16-sample 4th order FIR Gammatone filter centered at 440 Hz + + >>> from scipy import signal + >>> signal.gammatone(440, 'fir', numtaps=16, fs=16000) + (array([ 0.00000000e+00, 2.22196719e-07, 1.64942101e-06, 4.99298227e-06, + 1.01993969e-05, 1.63125770e-05, 2.14648940e-05, 2.29947263e-05, + 1.76776931e-05, 2.04980537e-06, -2.72062858e-05, -7.28455299e-05, + -1.36651076e-04, -2.19066855e-04, -3.18905076e-04, -4.33156712e-04]), + [1.0]) + + IIR Gammatone filter centered at 440 Hz + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> fc, fs = 440, 16000 + >>> b, a = signal.gammatone(fc, 'iir', fs=fs) + >>> w, h = signal.freqz(b, a) + >>> plt.plot(w * fs / (2 * np.pi), 20 * np.log10(abs(h))) + >>> plt.xscale('log') + >>> plt.title('Gammatone filter frequency response') + >>> plt.xlabel('Frequency [Hz]') + >>> plt.ylabel('Amplitude [dB]') + >>> plt.margins(0, 0.1) + >>> plt.grid(which='both', axis='both') + >>> plt.axvline(fc, color='green') # cutoff frequency + >>> plt.show() + """ + # Converts freq to float + freq = float(freq) + + # Set sampling rate if not passed + if fs is None: + fs = 2 + fs = _validate_fs(fs, allow_none=False) + + # Check for invalid cutoff frequency or filter type + ftype = ftype.lower() + filter_types = ['fir', 'iir'] + if not 0 < freq < fs / 2: + raise ValueError(f"The frequency must be between 0 and {fs / 2}" + f" (Nyquist), but given {freq}.") + if ftype not in filter_types: + raise ValueError('ftype must be either fir or iir.') + + # Calculate FIR gammatone filter + if ftype == 'fir': + # Set order and numtaps if not passed + if order is None: + order = 4 + order = operator.index(order) + + if numtaps is None: + numtaps = max(int(fs * 0.015), 15) + numtaps = operator.index(numtaps) + + # Check for invalid order + if not 0 < order <= 24: + raise ValueError("Invalid order: order must be > 0 and <= 24.") + + # Gammatone impulse response settings + t = np.arange(numtaps) / fs + bw = 1.019 * _hz_to_erb(freq) + + # Calculate the FIR gammatone filter + b = (t ** (order - 1)) * np.exp(-2 * np.pi * bw * t) + b *= np.cos(2 * np.pi * freq * t) + + # Scale the FIR filter so the frequency response is 1 at cutoff + scale_factor = 2 * (2 * np.pi * bw) ** (order) + scale_factor /= float_factorial(order - 1) + scale_factor /= fs + b *= scale_factor + a = [1.0] + + # Calculate IIR gammatone filter + elif ftype == 'iir': + # Raise warning if order and/or numtaps is passed + if order is not None: + warnings.warn('order is not used for IIR gammatone filter.', stacklevel=2) + if numtaps is not None: + warnings.warn('numtaps is not used for IIR gammatone filter.', stacklevel=2) + + # Gammatone impulse response settings + T = 1./fs + bw = 2 * np.pi * 1.019 * _hz_to_erb(freq) + fr = 2 * freq * np.pi * T + bwT = bw * T + + # Calculate the gain to normalize the volume at the center frequency + g1 = -2 * np.exp(2j * fr) * T + g2 = 2 * np.exp(-(bwT) + 1j * fr) * T + g3 = np.sqrt(3 + 2 ** (3 / 2)) * np.sin(fr) + g4 = np.sqrt(3 - 2 ** (3 / 2)) * np.sin(fr) + g5 = np.exp(2j * fr) + + g = g1 + g2 * (np.cos(fr) - g4) + g *= (g1 + g2 * (np.cos(fr) + g4)) + g *= (g1 + g2 * (np.cos(fr) - g3)) + g *= (g1 + g2 * (np.cos(fr) + g3)) + g /= ((-2 / np.exp(2 * bwT) - 2 * g5 + 2 * (1 + g5) / np.exp(bwT)) ** 4) + g = np.abs(g) + + # Create empty filter coefficient lists + b = np.empty(5) + a = np.empty(9) + + # Calculate the numerator coefficients + b[0] = (T ** 4) / g + b[1] = -4 * T ** 4 * np.cos(fr) / np.exp(bw * T) / g + b[2] = 6 * T ** 4 * np.cos(2 * fr) / np.exp(2 * bw * T) / g + b[3] = -4 * T ** 4 * np.cos(3 * fr) / np.exp(3 * bw * T) / g + b[4] = T ** 4 * np.cos(4 * fr) / np.exp(4 * bw * T) / g + + # Calculate the denominator coefficients + a[0] = 1 + a[1] = -8 * np.cos(fr) / np.exp(bw * T) + a[2] = 4 * (4 + 3 * np.cos(2 * fr)) / np.exp(2 * bw * T) + a[3] = -8 * (6 * np.cos(fr) + np.cos(3 * fr)) + a[3] /= np.exp(3 * bw * T) + a[4] = 2 * (18 + 16 * np.cos(2 * fr) + np.cos(4 * fr)) + a[4] /= np.exp(4 * bw * T) + a[5] = -8 * (6 * np.cos(fr) + np.cos(3 * fr)) + a[5] /= np.exp(5 * bw * T) + a[6] = 4 * (4 + 3 * np.cos(2 * fr)) / np.exp(6 * bw * T) + a[7] = -8 * np.cos(fr) / np.exp(7 * bw * T) + a[8] = np.exp(-8 * bw * T) + + return b, a + + +filter_dict = {'butter': [buttap, buttord], + 'butterworth': [buttap, buttord], + + 'cauer': [ellipap, ellipord], + 'elliptic': [ellipap, ellipord], + 'ellip': [ellipap, ellipord], + + 'bessel': [besselap], + 'bessel_phase': [besselap], + 'bessel_delay': [besselap], + 'bessel_mag': [besselap], + + 'cheby1': [cheb1ap, cheb1ord], + 'chebyshev1': [cheb1ap, cheb1ord], + 'chebyshevi': [cheb1ap, cheb1ord], + + 'cheby2': [cheb2ap, cheb2ord], + 'chebyshev2': [cheb2ap, cheb2ord], + 'chebyshevii': [cheb2ap, cheb2ord], + } + +band_dict = {'band': 'bandpass', + 'bandpass': 'bandpass', + 'pass': 'bandpass', + 'bp': 'bandpass', + + 'bs': 'bandstop', + 'bandstop': 'bandstop', + 'bands': 'bandstop', + 'stop': 'bandstop', + + 'l': 'lowpass', + 'low': 'lowpass', + 'lowpass': 'lowpass', + 'lp': 'lowpass', + + 'high': 'highpass', + 'highpass': 'highpass', + 'h': 'highpass', + 'hp': 'highpass', + } + +bessel_norms = {'bessel': 'phase', + 'bessel_phase': 'phase', + 'bessel_delay': 'delay', + 'bessel_mag': 'mag'} diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_fir_filter_design.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_fir_filter_design.py new file mode 100644 index 0000000000000000000000000000000000000000..b46d1bcc72f5fb03965b66cf131c25c2ca2f5585 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_fir_filter_design.py @@ -0,0 +1,1286 @@ +"""Functions for FIR filter design.""" + +from math import ceil, log +import operator +import warnings +from typing import Literal + +import numpy as np +from numpy.fft import irfft, fft, ifft +from scipy.special import sinc +from scipy.linalg import (toeplitz, hankel, solve, LinAlgError, LinAlgWarning, + lstsq) +from scipy.signal._arraytools import _validate_fs + +from . import _sigtools + +__all__ = ['kaiser_beta', 'kaiser_atten', 'kaiserord', + 'firwin', 'firwin2', 'remez', 'firls', 'minimum_phase'] + + +# Some notes on function parameters: +# +# `cutoff` and `width` are given as numbers between 0 and 1. These are +# relative frequencies, expressed as a fraction of the Nyquist frequency. +# For example, if the Nyquist frequency is 2 KHz, then width=0.15 is a width +# of 300 Hz. +# +# The `order` of a FIR filter is one less than the number of taps. +# This is a potential source of confusion, so in the following code, +# we will always use the number of taps as the parameterization of +# the 'size' of the filter. The "number of taps" means the number +# of coefficients, which is the same as the length of the impulse +# response of the filter. + + +def kaiser_beta(a): + """Compute the Kaiser parameter `beta`, given the attenuation `a`. + + Parameters + ---------- + a : float + The desired attenuation in the stopband and maximum ripple in + the passband, in dB. This should be a *positive* number. + + Returns + ------- + beta : float + The `beta` parameter to be used in the formula for a Kaiser window. + + References + ---------- + Oppenheim, Schafer, "Discrete-Time Signal Processing", p.475-476. + + Examples + -------- + Suppose we want to design a lowpass filter, with 65 dB attenuation + in the stop band. The Kaiser window parameter to be used in the + window method is computed by ``kaiser_beta(65)``: + + >>> from scipy.signal import kaiser_beta + >>> kaiser_beta(65) + 6.20426 + + """ + if a > 50: + beta = 0.1102 * (a - 8.7) + elif a > 21: + beta = 0.5842 * (a - 21) ** 0.4 + 0.07886 * (a - 21) + else: + beta = 0.0 + return beta + + +def kaiser_atten(numtaps, width): + """Compute the attenuation of a Kaiser FIR filter. + + Given the number of taps `N` and the transition width `width`, compute the + attenuation `a` in dB, given by Kaiser's formula: + + a = 2.285 * (N - 1) * pi * width + 7.95 + + Parameters + ---------- + numtaps : int + The number of taps in the FIR filter. + width : float + The desired width of the transition region between passband and + stopband (or, in general, at any discontinuity) for the filter, + expressed as a fraction of the Nyquist frequency. + + Returns + ------- + a : float + The attenuation of the ripple, in dB. + + See Also + -------- + kaiserord, kaiser_beta + + Examples + -------- + Suppose we want to design a FIR filter using the Kaiser window method + that will have 211 taps and a transition width of 9 Hz for a signal that + is sampled at 480 Hz. Expressed as a fraction of the Nyquist frequency, + the width is 9/(0.5*480) = 0.0375. The approximate attenuation (in dB) + is computed as follows: + + >>> from scipy.signal import kaiser_atten + >>> kaiser_atten(211, 0.0375) + 64.48099630593983 + + """ + a = 2.285 * (numtaps - 1) * np.pi * width + 7.95 + return a + + +def kaiserord(ripple, width): + """ + Determine the filter window parameters for the Kaiser window method. + + The parameters returned by this function are generally used to create + a finite impulse response filter using the window method, with either + `firwin` or `firwin2`. + + Parameters + ---------- + ripple : float + Upper bound for the deviation (in dB) of the magnitude of the + filter's frequency response from that of the desired filter (not + including frequencies in any transition intervals). That is, if w + is the frequency expressed as a fraction of the Nyquist frequency, + A(w) is the actual frequency response of the filter and D(w) is the + desired frequency response, the design requirement is that:: + + abs(A(w) - D(w))) < 10**(-ripple/20) + + for 0 <= w <= 1 and w not in a transition interval. + width : float + Width of transition region, normalized so that 1 corresponds to pi + radians / sample. That is, the frequency is expressed as a fraction + of the Nyquist frequency. + + Returns + ------- + numtaps : int + The length of the Kaiser window. + beta : float + The beta parameter for the Kaiser window. + + See Also + -------- + kaiser_beta, kaiser_atten + + Notes + ----- + There are several ways to obtain the Kaiser window: + + - ``signal.windows.kaiser(numtaps, beta, sym=True)`` + - ``signal.get_window(beta, numtaps)`` + - ``signal.get_window(('kaiser', beta), numtaps)`` + + The empirical equations discovered by Kaiser are used. + + References + ---------- + Oppenheim, Schafer, "Discrete-Time Signal Processing", pp.475-476. + + Examples + -------- + We will use the Kaiser window method to design a lowpass FIR filter + for a signal that is sampled at 1000 Hz. + + We want at least 65 dB rejection in the stop band, and in the pass + band the gain should vary no more than 0.5%. + + We want a cutoff frequency of 175 Hz, with a transition between the + pass band and the stop band of 24 Hz. That is, in the band [0, 163], + the gain varies no more than 0.5%, and in the band [187, 500], the + signal is attenuated by at least 65 dB. + + >>> import numpy as np + >>> from scipy.signal import kaiserord, firwin, freqz + >>> import matplotlib.pyplot as plt + >>> fs = 1000.0 + >>> cutoff = 175 + >>> width = 24 + + The Kaiser method accepts just a single parameter to control the pass + band ripple and the stop band rejection, so we use the more restrictive + of the two. In this case, the pass band ripple is 0.005, or 46.02 dB, + so we will use 65 dB as the design parameter. + + Use `kaiserord` to determine the length of the filter and the + parameter for the Kaiser window. + + >>> numtaps, beta = kaiserord(65, width/(0.5*fs)) + >>> numtaps + 167 + >>> beta + 6.20426 + + Use `firwin` to create the FIR filter. + + >>> taps = firwin(numtaps, cutoff, window=('kaiser', beta), + ... scale=False, fs=fs) + + Compute the frequency response of the filter. ``w`` is the array of + frequencies, and ``h`` is the corresponding complex array of frequency + responses. + + >>> w, h = freqz(taps, worN=8000) + >>> w *= 0.5*fs/np.pi # Convert w to Hz. + + Compute the deviation of the magnitude of the filter's response from + that of the ideal lowpass filter. Values in the transition region are + set to ``nan``, so they won't appear in the plot. + + >>> ideal = w < cutoff # The "ideal" frequency response. + >>> deviation = np.abs(np.abs(h) - ideal) + >>> deviation[(w > cutoff - 0.5*width) & (w < cutoff + 0.5*width)] = np.nan + + Plot the deviation. A close look at the left end of the stop band shows + that the requirement for 65 dB attenuation is violated in the first lobe + by about 0.125 dB. This is not unusual for the Kaiser window method. + + >>> plt.plot(w, 20*np.log10(np.abs(deviation))) + >>> plt.xlim(0, 0.5*fs) + >>> plt.ylim(-90, -60) + >>> plt.grid(alpha=0.25) + >>> plt.axhline(-65, color='r', ls='--', alpha=0.3) + >>> plt.xlabel('Frequency (Hz)') + >>> plt.ylabel('Deviation from ideal (dB)') + >>> plt.title('Lowpass Filter Frequency Response') + >>> plt.show() + + """ + A = abs(ripple) # in case somebody is confused as to what's meant + if A < 8: + # Formula for N is not valid in this range. + raise ValueError("Requested maximum ripple attenuation " + f"{A:f} is too small for the Kaiser formula.") + beta = kaiser_beta(A) + + # Kaiser's formula (as given in Oppenheim and Schafer) is for the filter + # order, so we have to add 1 to get the number of taps. + numtaps = (A - 7.95) / 2.285 / (np.pi * width) + 1 + + return int(ceil(numtaps)), beta + + +def firwin(numtaps, cutoff, *, width=None, window='hamming', pass_zero=True, + scale=True, fs=None): + """ + FIR filter design using the window method. + + This function computes the coefficients of a finite impulse response + filter. The filter will have linear phase; it will be Type I if + `numtaps` is odd and Type II if `numtaps` is even. + + Type II filters always have zero response at the Nyquist frequency, so a + ValueError exception is raised if firwin is called with `numtaps` even and + having a passband whose right end is at the Nyquist frequency. + + Parameters + ---------- + numtaps : int + Length of the filter (number of coefficients, i.e. the filter + order + 1). `numtaps` must be odd if a passband includes the + Nyquist frequency. + cutoff : float or 1-D array_like + Cutoff frequency of filter (expressed in the same units as `fs`) + OR an array of cutoff frequencies (that is, band edges). In the + former case, as a float, the cutoff frequency should correspond + with the half-amplitude point, where the attenuation will be -6dB. + In the latter case, the frequencies in `cutoff` should be positive + and monotonically increasing between 0 and `fs/2`. The values 0 + and `fs/2` must not be included in `cutoff`. It should be noted + that this is different than the behavior of `scipy.signal.iirdesign`, + where the cutoff is the half-power point (-3dB). + width : float or None, optional + If `width` is not None, then assume it is the approximate width + of the transition region (expressed in the same units as `fs`) + for use in Kaiser FIR filter design. In this case, the `window` + argument is ignored. + window : string or tuple of string and parameter values, optional + Desired window to use. See `scipy.signal.get_window` for a list + of windows and required parameters. + pass_zero : {True, False, 'bandpass', 'lowpass', 'highpass', 'bandstop'}, optional + If True, the gain at the frequency 0 (i.e., the "DC gain") is 1. + If False, the DC gain is 0. Can also be a string argument for the + desired filter type (equivalent to ``btype`` in IIR design functions). + + .. versionadded:: 1.3.0 + Support for string arguments. + scale : bool, optional + Set to True to scale the coefficients so that the frequency + response is exactly unity at a certain frequency. + That frequency is either: + + - 0 (DC) if the first passband starts at 0 (i.e. pass_zero + is True) + - `fs/2` (the Nyquist frequency) if the first passband ends at + `fs/2` (i.e the filter is a single band highpass filter); + center of first passband otherwise + + fs : float, optional + The sampling frequency of the signal. Each frequency in `cutoff` + must be between 0 and ``fs/2``. Default is 2. + + Returns + ------- + h : (numtaps,) ndarray + Coefficients of length `numtaps` FIR filter. + + Raises + ------ + ValueError + If any value in `cutoff` is less than or equal to 0 or greater + than or equal to ``fs/2``, if the values in `cutoff` are not strictly + monotonically increasing, or if `numtaps` is even but a passband + includes the Nyquist frequency. + + See Also + -------- + firwin2 + firls + minimum_phase + remez + + Examples + -------- + Low-pass from 0 to f: + + >>> from scipy import signal + >>> numtaps = 3 + >>> f = 0.1 + >>> signal.firwin(numtaps, f) + array([ 0.06799017, 0.86401967, 0.06799017]) + + Use a specific window function: + + >>> signal.firwin(numtaps, f, window='nuttall') + array([ 3.56607041e-04, 9.99286786e-01, 3.56607041e-04]) + + High-pass ('stop' from 0 to f): + + >>> signal.firwin(numtaps, f, pass_zero=False) + array([-0.00859313, 0.98281375, -0.00859313]) + + Band-pass: + + >>> f1, f2 = 0.1, 0.2 + >>> signal.firwin(numtaps, [f1, f2], pass_zero=False) + array([ 0.06301614, 0.88770441, 0.06301614]) + + Band-stop: + + >>> signal.firwin(numtaps, [f1, f2]) + array([-0.00801395, 1.0160279 , -0.00801395]) + + Multi-band (passbands are [0, f1], [f2, f3] and [f4, 1]): + + >>> f3, f4 = 0.3, 0.4 + >>> signal.firwin(numtaps, [f1, f2, f3, f4]) + array([-0.01376344, 1.02752689, -0.01376344]) + + Multi-band (passbands are [f1, f2] and [f3,f4]): + + >>> signal.firwin(numtaps, [f1, f2, f3, f4], pass_zero=False) + array([ 0.04890915, 0.91284326, 0.04890915]) + + """ + # The major enhancements to this function added in November 2010 were + # developed by Tom Krauss (see ticket #902). + fs = _validate_fs(fs, allow_none=True) + fs = 2 if fs is None else fs + + nyq = 0.5 * fs + + cutoff = np.atleast_1d(cutoff) / float(nyq) + + # Check for invalid input. + if cutoff.ndim > 1: + raise ValueError("The cutoff argument must be at most " + "one-dimensional.") + if cutoff.size == 0: + raise ValueError("At least one cutoff frequency must be given.") + if cutoff.min() <= 0 or cutoff.max() >= 1: + raise ValueError("Invalid cutoff frequency: frequencies must be " + "greater than 0 and less than fs/2.") + if np.any(np.diff(cutoff) <= 0): + raise ValueError("Invalid cutoff frequencies: the frequencies " + "must be strictly increasing.") + + if width is not None: + # A width was given. Find the beta parameter of the Kaiser window + # and set `window`. This overrides the value of `window` passed in. + atten = kaiser_atten(numtaps, float(width) / nyq) + beta = kaiser_beta(atten) + window = ('kaiser', beta) + + if isinstance(pass_zero, str): + if pass_zero in ('bandstop', 'lowpass'): + if pass_zero == 'lowpass': + if cutoff.size != 1: + raise ValueError('cutoff must have one element if ' + f'pass_zero=="lowpass", got {cutoff.shape}') + elif cutoff.size <= 1: + raise ValueError('cutoff must have at least two elements if ' + f'pass_zero=="bandstop", got {cutoff.shape}') + pass_zero = True + elif pass_zero in ('bandpass', 'highpass'): + if pass_zero == 'highpass': + if cutoff.size != 1: + raise ValueError('cutoff must have one element if ' + f'pass_zero=="highpass", got {cutoff.shape}') + elif cutoff.size <= 1: + raise ValueError('cutoff must have at least two elements if ' + f'pass_zero=="bandpass", got {cutoff.shape}') + pass_zero = False + else: + raise ValueError('pass_zero must be True, False, "bandpass", ' + '"lowpass", "highpass", or "bandstop", got ' + f'{pass_zero}') + pass_zero = bool(operator.index(pass_zero)) # ensure bool-like + + pass_nyquist = bool(cutoff.size & 1) ^ pass_zero + if pass_nyquist and numtaps % 2 == 0: + raise ValueError("A filter with an even number of coefficients must " + "have zero response at the Nyquist frequency.") + + # Insert 0 and/or 1 at the ends of cutoff so that the length of cutoff + # is even, and each pair in cutoff corresponds to passband. + cutoff = np.hstack(([0.0] * pass_zero, cutoff, [1.0] * pass_nyquist)) + + # `bands` is a 2-D array; each row gives the left and right edges of + # a passband. + bands = cutoff.reshape(-1, 2) + + # Build up the coefficients. + alpha = 0.5 * (numtaps - 1) + m = np.arange(0, numtaps) - alpha + h = 0 + for left, right in bands: + h += right * sinc(right * m) + h -= left * sinc(left * m) + + # Get and apply the window function. + from .windows import get_window + win = get_window(window, numtaps, fftbins=False) + h *= win + + # Now handle scaling if desired. + if scale: + # Get the first passband. + left, right = bands[0] + if left == 0: + scale_frequency = 0.0 + elif right == 1: + scale_frequency = 1.0 + else: + scale_frequency = 0.5 * (left + right) + c = np.cos(np.pi * m * scale_frequency) + s = np.sum(h * c) + h /= s + + return h + + +# Original version of firwin2 from scipy ticket #457, submitted by "tash". +# +# Rewritten by Warren Weckesser, 2010. +def firwin2(numtaps, freq, gain, *, nfreqs=None, window='hamming', + antisymmetric=False, fs=None): + """ + FIR filter design using the window method. + + From the given frequencies `freq` and corresponding gains `gain`, + this function constructs an FIR filter with linear phase and + (approximately) the given frequency response. + + Parameters + ---------- + numtaps : int + The number of taps in the FIR filter. `numtaps` must be less than + `nfreqs`. + freq : array_like, 1-D + The frequency sampling points. Typically 0.0 to 1.0 with 1.0 being + Nyquist. The Nyquist frequency is half `fs`. + The values in `freq` must be nondecreasing. A value can be repeated + once to implement a discontinuity. The first value in `freq` must + be 0, and the last value must be ``fs/2``. Values 0 and ``fs/2`` must + not be repeated. + gain : array_like + The filter gains at the frequency sampling points. Certain + constraints to gain values, depending on the filter type, are applied, + see Notes for details. + nfreqs : int, optional + The size of the interpolation mesh used to construct the filter. + For most efficient behavior, this should be a power of 2 plus 1 + (e.g, 129, 257, etc). The default is one more than the smallest + power of 2 that is not less than `numtaps`. `nfreqs` must be greater + than `numtaps`. + window : string or (string, float) or float, or None, optional + Window function to use. Default is "hamming". See + `scipy.signal.get_window` for the complete list of possible values. + If None, no window function is applied. + antisymmetric : bool, optional + Whether resulting impulse response is symmetric/antisymmetric. + See Notes for more details. + fs : float, optional + The sampling frequency of the signal. Each frequency in `cutoff` + must be between 0 and ``fs/2``. Default is 2. + + Returns + ------- + taps : ndarray + The filter coefficients of the FIR filter, as a 1-D array of length + `numtaps`. + + See Also + -------- + firls + firwin + minimum_phase + remez + + Notes + ----- + From the given set of frequencies and gains, the desired response is + constructed in the frequency domain. The inverse FFT is applied to the + desired response to create the associated convolution kernel, and the + first `numtaps` coefficients of this kernel, scaled by `window`, are + returned. + + The FIR filter will have linear phase. The type of filter is determined by + the value of 'numtaps` and `antisymmetric` flag. + There are four possible combinations: + + - odd `numtaps`, `antisymmetric` is False, type I filter is produced + - even `numtaps`, `antisymmetric` is False, type II filter is produced + - odd `numtaps`, `antisymmetric` is True, type III filter is produced + - even `numtaps`, `antisymmetric` is True, type IV filter is produced + + Magnitude response of all but type I filters are subjects to following + constraints: + + - type II -- zero at the Nyquist frequency + - type III -- zero at zero and Nyquist frequencies + - type IV -- zero at zero frequency + + .. versionadded:: 0.9.0 + + References + ---------- + .. [1] Oppenheim, A. V. and Schafer, R. W., "Discrete-Time Signal + Processing", Prentice-Hall, Englewood Cliffs, New Jersey (1989). + (See, for example, Section 7.4.) + + .. [2] Smith, Steven W., "The Scientist and Engineer's Guide to Digital + Signal Processing", Ch. 17. http://www.dspguide.com/ch17/1.htm + + Examples + -------- + A lowpass FIR filter with a response that is 1 on [0.0, 0.5], and + that decreases linearly on [0.5, 1.0] from 1 to 0: + + >>> from scipy import signal + >>> taps = signal.firwin2(150, [0.0, 0.5, 1.0], [1.0, 1.0, 0.0]) + >>> print(taps[72:78]) + [-0.02286961 -0.06362756 0.57310236 0.57310236 -0.06362756 -0.02286961] + + """ + fs = _validate_fs(fs, allow_none=True) + fs = 2 if fs is None else fs + nyq = 0.5 * fs + + if len(freq) != len(gain): + raise ValueError('freq and gain must be of same length.') + + if nfreqs is not None and numtaps >= nfreqs: + raise ValueError(('ntaps must be less than nfreqs, but firwin2 was ' + 'called with ntaps=%d and nfreqs=%s') % + (numtaps, nfreqs)) + + if freq[0] != 0 or freq[-1] != nyq: + raise ValueError('freq must start with 0 and end with fs/2.') + d = np.diff(freq) + if (d < 0).any(): + raise ValueError('The values in freq must be nondecreasing.') + d2 = d[:-1] + d[1:] + if (d2 == 0).any(): + raise ValueError('A value in freq must not occur more than twice.') + if freq[1] == 0: + raise ValueError('Value 0 must not be repeated in freq') + if freq[-2] == nyq: + raise ValueError('Value fs/2 must not be repeated in freq') + + if antisymmetric: + if numtaps % 2 == 0: + ftype = 4 + else: + ftype = 3 + else: + if numtaps % 2 == 0: + ftype = 2 + else: + ftype = 1 + + if ftype == 2 and gain[-1] != 0.0: + raise ValueError("A Type II filter must have zero gain at the " + "Nyquist frequency.") + elif ftype == 3 and (gain[0] != 0.0 or gain[-1] != 0.0): + raise ValueError("A Type III filter must have zero gain at zero " + "and Nyquist frequencies.") + elif ftype == 4 and gain[0] != 0.0: + raise ValueError("A Type IV filter must have zero gain at zero " + "frequency.") + + if nfreqs is None: + nfreqs = 1 + 2 ** int(ceil(log(numtaps, 2))) + + if (d == 0).any(): + # Tweak any repeated values in freq so that interp works. + freq = np.array(freq, copy=True) + eps = np.finfo(float).eps * nyq + for k in range(len(freq) - 1): + if freq[k] == freq[k + 1]: + freq[k] = freq[k] - eps + freq[k + 1] = freq[k + 1] + eps + # Check if freq is strictly increasing after tweak + d = np.diff(freq) + if (d <= 0).any(): + raise ValueError("freq cannot contain numbers that are too close " + "(within eps * (fs/2): " + f"{eps}) to a repeated value") + + # Linearly interpolate the desired response on a uniform mesh `x`. + x = np.linspace(0.0, nyq, nfreqs) + fx = np.interp(x, freq, gain) + + # Adjust the phases of the coefficients so that the first `ntaps` of the + # inverse FFT are the desired filter coefficients. + shift = np.exp(-(numtaps - 1) / 2. * 1.j * np.pi * x / nyq) + if ftype > 2: + shift *= 1j + + fx2 = fx * shift + + # Use irfft to compute the inverse FFT. + out_full = irfft(fx2) + + if window is not None: + # Create the window to apply to the filter coefficients. + from .windows import get_window + wind = get_window(window, numtaps, fftbins=False) + else: + wind = 1 + + # Keep only the first `numtaps` coefficients in `out`, and multiply by + # the window. + out = out_full[:numtaps] * wind + + if ftype == 3: + out[out.size // 2] = 0.0 + + return out + + +def remez(numtaps, bands, desired, *, weight=None, type='bandpass', + maxiter=25, grid_density=16, fs=None): + """ + Calculate the minimax optimal filter using the Remez exchange algorithm. + + Calculate the filter-coefficients for the finite impulse response + (FIR) filter whose transfer function minimizes the maximum error + between the desired gain and the realized gain in the specified + frequency bands using the Remez exchange algorithm. + + Parameters + ---------- + numtaps : int + The desired number of taps in the filter. The number of taps is + the number of terms in the filter, or the filter order plus one. + bands : array_like + A monotonic sequence containing the band edges. + All elements must be non-negative and less than half the sampling + frequency as given by `fs`. + desired : array_like + A sequence half the size of bands containing the desired gain + in each of the specified bands. + weight : array_like, optional + A relative weighting to give to each band region. The length of + `weight` has to be half the length of `bands`. + type : {'bandpass', 'differentiator', 'hilbert'}, optional + The type of filter: + + * 'bandpass' : flat response in bands. This is the default. + + * 'differentiator' : frequency proportional response in bands. + + * 'hilbert' : filter with odd symmetry, that is, type III + (for even order) or type IV (for odd order) + linear phase filters. + + maxiter : int, optional + Maximum number of iterations of the algorithm. Default is 25. + grid_density : int, optional + Grid density. The dense grid used in `remez` is of size + ``(numtaps + 1) * grid_density``. Default is 16. + fs : float, optional + The sampling frequency of the signal. Default is 1. + + Returns + ------- + out : ndarray + A rank-1 array containing the coefficients of the optimal + (in a minimax sense) filter. + + See Also + -------- + firls + firwin + firwin2 + minimum_phase + + References + ---------- + .. [1] J. H. McClellan and T. W. Parks, "A unified approach to the + design of optimum FIR linear phase digital filters", + IEEE Trans. Circuit Theory, vol. CT-20, pp. 697-701, 1973. + .. [2] J. H. McClellan, T. W. Parks and L. R. Rabiner, "A Computer + Program for Designing Optimum FIR Linear Phase Digital + Filters", IEEE Trans. Audio Electroacoust., vol. AU-21, + pp. 506-525, 1973. + + Examples + -------- + In these examples, `remez` is used to design low-pass, high-pass, + band-pass and band-stop filters. The parameters that define each filter + are the filter order, the band boundaries, the transition widths of the + boundaries, the desired gains in each band, and the sampling frequency. + + We'll use a sample frequency of 22050 Hz in all the examples. In each + example, the desired gain in each band is either 0 (for a stop band) + or 1 (for a pass band). + + `freqz` is used to compute the frequency response of each filter, and + the utility function ``plot_response`` defined below is used to plot + the response. + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> fs = 22050 # Sample rate, Hz + + >>> def plot_response(w, h, title): + ... "Utility function to plot response functions" + ... fig = plt.figure() + ... ax = fig.add_subplot(111) + ... ax.plot(w, 20*np.log10(np.abs(h))) + ... ax.set_ylim(-40, 5) + ... ax.grid(True) + ... ax.set_xlabel('Frequency (Hz)') + ... ax.set_ylabel('Gain (dB)') + ... ax.set_title(title) + + The first example is a low-pass filter, with cutoff frequency 8 kHz. + The filter length is 325, and the transition width from pass to stop + is 100 Hz. + + >>> cutoff = 8000.0 # Desired cutoff frequency, Hz + >>> trans_width = 100 # Width of transition from pass to stop, Hz + >>> numtaps = 325 # Size of the FIR filter. + >>> taps = signal.remez(numtaps, [0, cutoff, cutoff + trans_width, 0.5*fs], + ... [1, 0], fs=fs) + >>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs) + >>> plot_response(w, h, "Low-pass Filter") + >>> plt.show() + + This example shows a high-pass filter: + + >>> cutoff = 2000.0 # Desired cutoff frequency, Hz + >>> trans_width = 250 # Width of transition from pass to stop, Hz + >>> numtaps = 125 # Size of the FIR filter. + >>> taps = signal.remez(numtaps, [0, cutoff - trans_width, cutoff, 0.5*fs], + ... [0, 1], fs=fs) + >>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs) + >>> plot_response(w, h, "High-pass Filter") + >>> plt.show() + + This example shows a band-pass filter with a pass-band from 2 kHz to + 5 kHz. The transition width is 260 Hz and the length of the filter + is 63, which is smaller than in the other examples: + + >>> band = [2000, 5000] # Desired pass band, Hz + >>> trans_width = 260 # Width of transition from pass to stop, Hz + >>> numtaps = 63 # Size of the FIR filter. + >>> edges = [0, band[0] - trans_width, band[0], band[1], + ... band[1] + trans_width, 0.5*fs] + >>> taps = signal.remez(numtaps, edges, [0, 1, 0], fs=fs) + >>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs) + >>> plot_response(w, h, "Band-pass Filter") + >>> plt.show() + + The low order leads to higher ripple and less steep transitions. + + The next example shows a band-stop filter. + + >>> band = [6000, 8000] # Desired stop band, Hz + >>> trans_width = 200 # Width of transition from pass to stop, Hz + >>> numtaps = 175 # Size of the FIR filter. + >>> edges = [0, band[0] - trans_width, band[0], band[1], + ... band[1] + trans_width, 0.5*fs] + >>> taps = signal.remez(numtaps, edges, [1, 0, 1], fs=fs) + >>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs) + >>> plot_response(w, h, "Band-stop Filter") + >>> plt.show() + + """ + fs = _validate_fs(fs, allow_none=True) + fs = 1.0 if fs is None else fs + + # Convert type + try: + tnum = {'bandpass': 1, 'differentiator': 2, 'hilbert': 3}[type] + except KeyError as e: + raise ValueError("Type must be 'bandpass', 'differentiator', " + "or 'hilbert'") from e + + # Convert weight + if weight is None: + weight = [1] * len(desired) + + bands = np.asarray(bands).copy() + return _sigtools._remez(numtaps, bands, desired, weight, tnum, fs, + maxiter, grid_density) + + +def firls(numtaps, bands, desired, *, weight=None, fs=None): + """ + FIR filter design using least-squares error minimization. + + Calculate the filter coefficients for the linear-phase finite + impulse response (FIR) filter which has the best approximation + to the desired frequency response described by `bands` and + `desired` in the least squares sense (i.e., the integral of the + weighted mean-squared error within the specified bands is + minimized). + + Parameters + ---------- + numtaps : int + The number of taps in the FIR filter. `numtaps` must be odd. + bands : array_like + A monotonic nondecreasing sequence containing the band edges in + Hz. All elements must be non-negative and less than or equal to + the Nyquist frequency given by `nyq`. The bands are specified as + frequency pairs, thus, if using a 1D array, its length must be + even, e.g., `np.array([0, 1, 2, 3, 4, 5])`. Alternatively, the + bands can be specified as an nx2 sized 2D array, where n is the + number of bands, e.g, `np.array([[0, 1], [2, 3], [4, 5]])`. + desired : array_like + A sequence the same size as `bands` containing the desired gain + at the start and end point of each band. + weight : array_like, optional + A relative weighting to give to each band region when solving + the least squares problem. `weight` has to be half the size of + `bands`. + fs : float, optional + The sampling frequency of the signal. Each frequency in `bands` + must be between 0 and ``fs/2`` (inclusive). Default is 2. + + Returns + ------- + coeffs : ndarray + Coefficients of the optimal (in a least squares sense) FIR filter. + + See Also + -------- + firwin + firwin2 + minimum_phase + remez + + Notes + ----- + This implementation follows the algorithm given in [1]_. + As noted there, least squares design has multiple advantages: + + 1. Optimal in a least-squares sense. + 2. Simple, non-iterative method. + 3. The general solution can obtained by solving a linear + system of equations. + 4. Allows the use of a frequency dependent weighting function. + + This function constructs a Type I linear phase FIR filter, which + contains an odd number of `coeffs` satisfying for :math:`n < numtaps`: + + .. math:: coeffs(n) = coeffs(numtaps - 1 - n) + + The odd number of coefficients and filter symmetry avoid boundary + conditions that could otherwise occur at the Nyquist and 0 frequencies + (e.g., for Type II, III, or IV variants). + + .. versionadded:: 0.18 + + References + ---------- + .. [1] Ivan Selesnick, Linear-Phase Fir Filter Design By Least Squares. + OpenStax CNX. Aug 9, 2005. + https://eeweb.engineering.nyu.edu/iselesni/EL713/firls/firls.pdf + + Examples + -------- + We want to construct a band-pass filter. Note that the behavior in the + frequency ranges between our stop bands and pass bands is unspecified, + and thus may overshoot depending on the parameters of our filter: + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> fig, axs = plt.subplots(2) + >>> fs = 10.0 # Hz + >>> desired = (0, 0, 1, 1, 0, 0) + >>> for bi, bands in enumerate(((0, 1, 2, 3, 4, 5), (0, 1, 2, 4, 4.5, 5))): + ... fir_firls = signal.firls(73, bands, desired, fs=fs) + ... fir_remez = signal.remez(73, bands, desired[::2], fs=fs) + ... fir_firwin2 = signal.firwin2(73, bands, desired, fs=fs) + ... hs = list() + ... ax = axs[bi] + ... for fir in (fir_firls, fir_remez, fir_firwin2): + ... freq, response = signal.freqz(fir) + ... hs.append(ax.semilogy(0.5*fs*freq/np.pi, np.abs(response))[0]) + ... for band, gains in zip(zip(bands[::2], bands[1::2]), + ... zip(desired[::2], desired[1::2])): + ... ax.semilogy(band, np.maximum(gains, 1e-7), 'k--', linewidth=2) + ... if bi == 0: + ... ax.legend(hs, ('firls', 'remez', 'firwin2'), + ... loc='lower center', frameon=False) + ... else: + ... ax.set_xlabel('Frequency (Hz)') + ... ax.grid(True) + ... ax.set(title='Band-pass %d-%d Hz' % bands[2:4], ylabel='Magnitude') + ... + >>> fig.tight_layout() + >>> plt.show() + + """ + fs = _validate_fs(fs, allow_none=True) + fs = 2 if fs is None else fs + nyq = 0.5 * fs + + numtaps = int(numtaps) + if numtaps % 2 == 0 or numtaps < 1: + raise ValueError("numtaps must be odd and >= 1") + M = (numtaps-1) // 2 + + # normalize bands 0->1 and make it 2 columns + nyq = float(nyq) + if nyq <= 0: + raise ValueError(f'nyq must be positive, got {nyq} <= 0.') + bands = np.asarray(bands).flatten() / nyq + if len(bands) % 2 != 0: + raise ValueError("bands must contain frequency pairs.") + if (bands < 0).any() or (bands > 1).any(): + raise ValueError("bands must be between 0 and 1 relative to Nyquist") + bands.shape = (-1, 2) + + # check remaining params + desired = np.asarray(desired).flatten() + if bands.size != desired.size: + raise ValueError( + f"desired must have one entry per frequency, got {desired.size} " + f"gains for {bands.size} frequencies." + ) + desired.shape = (-1, 2) + if (np.diff(bands) <= 0).any() or (np.diff(bands[:, 0]) < 0).any(): + raise ValueError("bands must be monotonically nondecreasing and have " + "width > 0.") + if (bands[:-1, 1] > bands[1:, 0]).any(): + raise ValueError("bands must not overlap.") + if (desired < 0).any(): + raise ValueError("desired must be non-negative.") + if weight is None: + weight = np.ones(len(desired)) + weight = np.asarray(weight).flatten() + if len(weight) != len(desired): + raise ValueError("weight must be the same size as the number of " + f"band pairs ({len(bands)}).") + if (weight < 0).any(): + raise ValueError("weight must be non-negative.") + + # Set up the linear matrix equation to be solved, Qa = b + + # We can express Q(k,n) = 0.5 Q1(k,n) + 0.5 Q2(k,n) + # where Q1(k,n)=q(k-n) and Q2(k,n)=q(k+n), i.e. a Toeplitz plus Hankel. + + # We omit the factor of 0.5 above, instead adding it during coefficient + # calculation. + + # We also omit the 1/π from both Q and b equations, as they cancel + # during solving. + + # We have that: + # q(n) = 1/π ∫W(ω)cos(nω)dω (over 0->π) + # Using our normalization ω=πf and with a constant weight W over each + # interval f1->f2 we get: + # q(n) = W∫cos(πnf)df (0->1) = Wf sin(πnf)/πnf + # integrated over each f1->f2 pair (i.e., value at f2 - value at f1). + n = np.arange(numtaps)[:, np.newaxis, np.newaxis] + q = np.dot(np.diff(np.sinc(bands * n) * bands, axis=2)[:, :, 0], weight) + + # Now we assemble our sum of Toeplitz and Hankel + Q1 = toeplitz(q[:M+1]) + Q2 = hankel(q[:M+1], q[M:]) + Q = Q1 + Q2 + + # Now for b(n) we have that: + # b(n) = 1/π ∫ W(ω)D(ω)cos(nω)dω (over 0->π) + # Using our normalization ω=πf and with a constant weight W over each + # interval and a linear term for D(ω) we get (over each f1->f2 interval): + # b(n) = W ∫ (mf+c)cos(πnf)df + # = f(mf+c)sin(πnf)/πnf + mf**2 cos(nπf)/(πnf)**2 + # integrated over each f1->f2 pair (i.e., value at f2 - value at f1). + n = n[:M + 1] # only need this many coefficients here + # Choose m and c such that we are at the start and end weights + m = (np.diff(desired, axis=1) / np.diff(bands, axis=1)) + c = desired[:, [0]] - bands[:, [0]] * m + b = bands * (m*bands + c) * np.sinc(bands * n) + # Use L'Hospital's rule here for cos(nπf)/(πnf)**2 @ n=0 + b[0] -= m * bands * bands / 2. + b[1:] += m * np.cos(n[1:] * np.pi * bands) / (np.pi * n[1:]) ** 2 + b = np.dot(np.diff(b, axis=2)[:, :, 0], weight) + + # Now we can solve the equation + try: # try the fast way + with warnings.catch_warnings(record=True) as w: + warnings.simplefilter('always') + a = solve(Q, b, assume_a="pos", check_finite=False) + for ww in w: + if (ww.category == LinAlgWarning and + str(ww.message).startswith('Ill-conditioned matrix')): + raise LinAlgError(str(ww.message)) + except LinAlgError: # in case Q is rank deficient + # This is faster than pinvh, even though we don't explicitly use + # the symmetry here. gelsy was faster than gelsd and gelss in + # some non-exhaustive tests. + a = lstsq(Q, b, lapack_driver='gelsy')[0] + + # make coefficients symmetric (linear phase) + coeffs = np.hstack((a[:0:-1], 2 * a[0], a[1:])) + return coeffs + + +def _dhtm(mag): + """Compute the modified 1-D discrete Hilbert transform + + Parameters + ---------- + mag : ndarray + The magnitude spectrum. Should be 1-D with an even length, and + preferably a fast length for FFT/IFFT. + """ + # Adapted based on code by Niranjan Damera-Venkata, + # Brian L. Evans and Shawn R. McCaslin (see refs for `minimum_phase`) + sig = np.zeros(len(mag)) + # Leave Nyquist and DC at 0, knowing np.abs(fftfreq(N)[midpt]) == 0.5 + midpt = len(mag) // 2 + sig[1:midpt] = 1 + sig[midpt+1:] = -1 + # eventually if we want to support complex filters, we will need a + # np.abs() on the mag inside the log, and should remove the .real + recon = ifft(mag * np.exp(fft(sig * ifft(np.log(mag))))).real + return recon + + +def minimum_phase(h: np.ndarray, + method: Literal['homomorphic', 'hilbert'] = 'homomorphic', + n_fft: int | None = None, *, half: bool = True) -> np.ndarray: + """Convert a linear-phase FIR filter to minimum phase + + Parameters + ---------- + h : array + Linear-phase FIR filter coefficients. + method : {'hilbert', 'homomorphic'} + The provided methods are: + + 'homomorphic' (default) + This method [4]_ [5]_ works best with filters with an + odd number of taps, and the resulting minimum phase filter + will have a magnitude response that approximates the square + root of the original filter's magnitude response using half + the number of taps when ``half=True`` (default), or the + original magnitude spectrum using the same number of taps + when ``half=False``. + + 'hilbert' + This method [1]_ is designed to be used with equiripple + filters (e.g., from `remez`) with unity or zero gain + regions. + + n_fft : int + The number of points to use for the FFT. Should be at least a + few times larger than the signal length (see Notes). + half : bool + If ``True``, create a filter that is half the length of the original, with a + magnitude spectrum that is the square root of the original. If ``False``, + create a filter that is the same length as the original, with a magnitude + spectrum that is designed to match the original (only supported when + ``method='homomorphic'``). + + .. versionadded:: 1.14.0 + + Returns + ------- + h_minimum : array + The minimum-phase version of the filter, with length + ``(len(h) + 1) // 2`` when ``half is True`` or ``len(h)`` otherwise. + + See Also + -------- + firwin + firwin2 + remez + + Notes + ----- + Both the Hilbert [1]_ or homomorphic [4]_ [5]_ methods require selection + of an FFT length to estimate the complex cepstrum of the filter. + + In the case of the Hilbert method, the deviation from the ideal + spectrum ``epsilon`` is related to the number of stopband zeros + ``n_stop`` and FFT length ``n_fft`` as:: + + epsilon = 2. * n_stop / n_fft + + For example, with 100 stopband zeros and a FFT length of 2048, + ``epsilon = 0.0976``. If we conservatively assume that the number of + stopband zeros is one less than the filter length, we can take the FFT + length to be the next power of 2 that satisfies ``epsilon=0.01`` as:: + + n_fft = 2 ** int(np.ceil(np.log2(2 * (len(h) - 1) / 0.01))) + + This gives reasonable results for both the Hilbert and homomorphic + methods, and gives the value used when ``n_fft=None``. + + Alternative implementations exist for creating minimum-phase filters, + including zero inversion [2]_ and spectral factorization [3]_ [4]_. + For more information, see `this DSPGuru page + `__. + + References + ---------- + .. [1] N. Damera-Venkata and B. L. Evans, "Optimal design of real and + complex minimum phase digital FIR filters," Acoustics, Speech, + and Signal Processing, 1999. Proceedings., 1999 IEEE International + Conference on, Phoenix, AZ, 1999, pp. 1145-1148 vol.3. + :doi:`10.1109/ICASSP.1999.756179` + .. [2] X. Chen and T. W. Parks, "Design of optimal minimum phase FIR + filters by direct factorization," Signal Processing, + vol. 10, no. 4, pp. 369-383, Jun. 1986. + .. [3] T. Saramaki, "Finite Impulse Response Filter Design," in + Handbook for Digital Signal Processing, chapter 4, + New York: Wiley-Interscience, 1993. + .. [4] J. S. Lim, Advanced Topics in Signal Processing. + Englewood Cliffs, N.J.: Prentice Hall, 1988. + .. [5] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, + "Discrete-Time Signal Processing," 3rd edition. + Upper Saddle River, N.J.: Pearson, 2009. + + Examples + -------- + Create an optimal linear-phase low-pass filter `h` with a transition band of + [0.2, 0.3] (assuming a Nyquist frequency of 1): + + >>> import numpy as np + >>> from scipy.signal import remez, minimum_phase, freqz, group_delay + >>> import matplotlib.pyplot as plt + >>> freq = [0, 0.2, 0.3, 1.0] + >>> desired = [1, 0] + >>> h_linear = remez(151, freq, desired, fs=2) + + Convert it to minimum phase: + + >>> h_hil = minimum_phase(h_linear, method='hilbert') + >>> h_hom = minimum_phase(h_linear, method='homomorphic') + >>> h_hom_full = minimum_phase(h_linear, method='homomorphic', half=False) + + Compare the impulse and frequency response of the four filters: + + >>> fig0, ax0 = plt.subplots(figsize=(6, 3), tight_layout=True) + >>> fig1, axs = plt.subplots(3, sharex='all', figsize=(6, 6), tight_layout=True) + >>> ax0.set_title("Impulse response") + >>> ax0.set(xlabel='Samples', ylabel='Amplitude', xlim=(0, len(h_linear) - 1)) + >>> axs[0].set_title("Frequency Response") + >>> axs[0].set(xlim=(0, .65), ylabel="Magnitude / dB") + >>> axs[1].set(ylabel="Phase / rad") + >>> axs[2].set(ylabel="Group Delay / samples", ylim=(-31, 81), + ... xlabel='Normalized Frequency (Nyqist frequency: 1)') + >>> for h, lb in ((h_linear, f'Linear ({len(h_linear)})'), + ... (h_hil, f'Min-Hilbert ({len(h_hil)})'), + ... (h_hom, f'Min-Homomorphic ({len(h_hom)})'), + ... (h_hom_full, f'Min-Homom. Full ({len(h_hom_full)})')): + ... w_H, H = freqz(h, fs=2) + ... w_gd, gd = group_delay((h, 1), fs=2) + ... + ... alpha = 1.0 if lb == 'linear' else 0.5 # full opacity for 'linear' line + ... ax0.plot(h, '.-', alpha=alpha, label=lb) + ... axs[0].plot(w_H, 20 * np.log10(np.abs(H)), alpha=alpha) + ... axs[1].plot(w_H, np.unwrap(np.angle(H)), alpha=alpha, label=lb) + ... axs[2].plot(w_gd, gd, alpha=alpha) + >>> ax0.grid(True) + >>> ax0.legend(title='Filter Phase (Order)') + >>> axs[1].legend(title='Filter Phase (Order)', loc='lower right') + >>> for ax_ in axs: # shade transition band: + ... ax_.axvspan(freq[1], freq[2], color='y', alpha=.25) + ... ax_.grid(True) + >>> plt.show() + + The impulse response and group delay plot depict the 75 sample delay of the linear + phase filter `h`. The phase should also be linear in the stop band--due to the small + magnitude, numeric noise dominates there. Furthermore, the plots show that the + minimum phase filters clearly show a reduced (negative) phase slope in the pass and + transition band. The plots also illustrate that the filter with parameters + ``method='homomorphic', half=False`` has same order and magnitude response as the + linear filter `h` whereas the other minimum phase filters have only half the order + and the square root of the magnitude response. + """ + h = np.asarray(h) + if np.iscomplexobj(h): + raise ValueError('Complex filters not supported') + if h.ndim != 1 or h.size <= 2: + raise ValueError('h must be 1-D and at least 2 samples long') + n_half = len(h) // 2 + if not np.allclose(h[-n_half:][::-1], h[:n_half]): + warnings.warn('h does not appear to by symmetric, conversion may fail', + RuntimeWarning, stacklevel=2) + if not isinstance(method, str) or method not in \ + ('homomorphic', 'hilbert',): + raise ValueError(f'method must be "homomorphic" or "hilbert", got {method!r}') + if method == "hilbert" and not half: + raise ValueError("`half=False` is only supported when `method='homomorphic'`") + if n_fft is None: + n_fft = 2 ** int(np.ceil(np.log2(2 * (len(h) - 1) / 0.01))) + n_fft = int(n_fft) + if n_fft < len(h): + raise ValueError(f'n_fft must be at least len(h)=={len(h)}') + if method == 'hilbert': + w = np.arange(n_fft) * (2 * np.pi / n_fft * n_half) + H = np.real(fft(h, n_fft) * np.exp(1j * w)) + dp = max(H) - 1 + ds = 0 - min(H) + S = 4. / (np.sqrt(1+dp+ds) + np.sqrt(1-dp+ds)) ** 2 + H += ds + H *= S + H = np.sqrt(H, out=H) + H += 1e-10 # ensure that the log does not explode + h_minimum = _dhtm(H) + else: # method == 'homomorphic' + # zero-pad; calculate the DFT + h_temp = np.abs(fft(h, n_fft)) + # take 0.25*log(|H|**2) = 0.5*log(|H|) + h_temp += 1e-7 * h_temp[h_temp > 0].min() # don't let log blow up + np.log(h_temp, out=h_temp) + if half: # halving of magnitude spectrum optional + h_temp *= 0.5 + # IDFT + h_temp = ifft(h_temp).real + # multiply pointwise by the homomorphic filter + # lmin[n] = 2u[n] - d[n] + # i.e., double the positive frequencies and zero out the negative ones; + # Oppenheim+Shafer 3rd ed p991 eq13.42b and p1004 fig13.7 + win = np.zeros(n_fft) + win[0] = 1 + stop = n_fft // 2 + win[1:stop] = 2 + if n_fft % 2: + win[stop] = 1 + h_temp *= win + h_temp = ifft(np.exp(fft(h_temp))) + h_minimum = h_temp.real + n_out = (n_half + len(h) % 2) if half else len(h) + return h_minimum[:n_out] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_lti_conversion.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_lti_conversion.py new file mode 100644 index 0000000000000000000000000000000000000000..52c6efbbfa53288934d12918566016db6c742ef1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_lti_conversion.py @@ -0,0 +1,533 @@ +""" +ltisys -- a collection of functions to convert linear time invariant systems +from one representation to another. +""" + +import numpy as np +from numpy import (r_, eye, atleast_2d, poly, dot, + asarray, zeros, array, outer) +from scipy import linalg + +from ._filter_design import tf2zpk, zpk2tf, normalize + + +__all__ = ['tf2ss', 'abcd_normalize', 'ss2tf', 'zpk2ss', 'ss2zpk', + 'cont2discrete'] + + +def tf2ss(num, den): + r"""Transfer function to state-space representation. + + Parameters + ---------- + num, den : array_like + Sequences representing the coefficients of the numerator and + denominator polynomials, in order of descending degree. The + denominator needs to be at least as long as the numerator. + + Returns + ------- + A, B, C, D : ndarray + State space representation of the system, in controller canonical + form. + + Examples + -------- + Convert the transfer function: + + .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1} + + >>> num = [1, 3, 3] + >>> den = [1, 2, 1] + + to the state-space representation: + + .. math:: + + \dot{\textbf{x}}(t) = + \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) + + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\ + + \textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) + + \begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t) + + >>> from scipy.signal import tf2ss + >>> A, B, C, D = tf2ss(num, den) + >>> A + array([[-2., -1.], + [ 1., 0.]]) + >>> B + array([[ 1.], + [ 0.]]) + >>> C + array([[ 1., 2.]]) + >>> D + array([[ 1.]]) + """ + # Controller canonical state-space representation. + # if M+1 = len(num) and K+1 = len(den) then we must have M <= K + # states are found by asserting that X(s) = U(s) / D(s) + # then Y(s) = N(s) * X(s) + # + # A, B, C, and D follow quite naturally. + # + num, den = normalize(num, den) # Strips zeros, checks arrays + nn = len(num.shape) + if nn == 1: + num = asarray([num], num.dtype) + M = num.shape[1] + K = len(den) + if M > K: + msg = "Improper transfer function. `num` is longer than `den`." + raise ValueError(msg) + if M == 0 or K == 0: # Null system + return (array([], float), array([], float), array([], float), + array([], float)) + + # pad numerator to have same number of columns has denominator + num = np.hstack((np.zeros((num.shape[0], K - M), dtype=num.dtype), num)) + + if num.shape[-1] > 0: + D = atleast_2d(num[:, 0]) + + else: + # We don't assign it an empty array because this system + # is not 'null'. It just doesn't have a non-zero D + # matrix. Thus, it should have a non-zero shape so that + # it can be operated on by functions like 'ss2tf' + D = array([[0]], float) + + if K == 1: + D = D.reshape(num.shape) + + return (zeros((1, 1)), zeros((1, D.shape[1])), + zeros((D.shape[0], 1)), D) + + frow = -array([den[1:]]) + A = r_[frow, eye(K - 2, K - 1)] + B = eye(K - 1, 1) + C = num[:, 1:] - outer(num[:, 0], den[1:]) + D = D.reshape((C.shape[0], B.shape[1])) + + return A, B, C, D + + +def _none_to_empty_2d(arg): + if arg is None: + return zeros((0, 0)) + else: + return arg + + +def _atleast_2d_or_none(arg): + if arg is not None: + return atleast_2d(arg) + + +def _shape_or_none(M): + if M is not None: + return M.shape + else: + return (None,) * 2 + + +def _choice_not_none(*args): + for arg in args: + if arg is not None: + return arg + + +def _restore(M, shape): + if M.shape == (0, 0): + return zeros(shape) + else: + if M.shape != shape: + raise ValueError("The input arrays have incompatible shapes.") + return M + + +def abcd_normalize(A=None, B=None, C=None, D=None): + """Check state-space matrices and ensure they are 2-D. + + If enough information on the system is provided, that is, enough + properly-shaped arrays are passed to the function, the missing ones + are built from this information, ensuring the correct number of + rows and columns. Otherwise a ValueError is raised. + + Parameters + ---------- + A, B, C, D : array_like, optional + State-space matrices. All of them are None (missing) by default. + See `ss2tf` for format. + + Returns + ------- + A, B, C, D : array + Properly shaped state-space matrices. + + Raises + ------ + ValueError + If not enough information on the system was provided. + + """ + A, B, C, D = map(_atleast_2d_or_none, (A, B, C, D)) + + MA, NA = _shape_or_none(A) + MB, NB = _shape_or_none(B) + MC, NC = _shape_or_none(C) + MD, ND = _shape_or_none(D) + + p = _choice_not_none(MA, MB, NC) + q = _choice_not_none(NB, ND) + r = _choice_not_none(MC, MD) + if p is None or q is None or r is None: + raise ValueError("Not enough information on the system.") + + A, B, C, D = map(_none_to_empty_2d, (A, B, C, D)) + A = _restore(A, (p, p)) + B = _restore(B, (p, q)) + C = _restore(C, (r, p)) + D = _restore(D, (r, q)) + + return A, B, C, D + + +def ss2tf(A, B, C, D, input=0): + r"""State-space to transfer function. + + A, B, C, D defines a linear state-space system with `p` inputs, + `q` outputs, and `n` state variables. + + Parameters + ---------- + A : array_like + State (or system) matrix of shape ``(n, n)`` + B : array_like + Input matrix of shape ``(n, p)`` + C : array_like + Output matrix of shape ``(q, n)`` + D : array_like + Feedthrough (or feedforward) matrix of shape ``(q, p)`` + input : int, optional + For multiple-input systems, the index of the input to use. + + Returns + ------- + num : 2-D ndarray + Numerator(s) of the resulting transfer function(s). `num` has one row + for each of the system's outputs. Each row is a sequence representation + of the numerator polynomial. + den : 1-D ndarray + Denominator of the resulting transfer function(s). `den` is a sequence + representation of the denominator polynomial. + + Examples + -------- + Convert the state-space representation: + + .. math:: + + \dot{\textbf{x}}(t) = + \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) + + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\ + + \textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) + + \begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t) + + >>> A = [[-2, -1], [1, 0]] + >>> B = [[1], [0]] # 2-D column vector + >>> C = [[1, 2]] # 2-D row vector + >>> D = 1 + + to the transfer function: + + .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1} + + >>> from scipy.signal import ss2tf + >>> ss2tf(A, B, C, D) + (array([[1., 3., 3.]]), array([ 1., 2., 1.])) + """ + # transfer function is C (sI - A)**(-1) B + D + + # Check consistency and make them all rank-2 arrays + A, B, C, D = abcd_normalize(A, B, C, D) + + nout, nin = D.shape + if input >= nin: + raise ValueError("System does not have the input specified.") + + # make SIMO from possibly MIMO system. + B = B[:, input:input + 1] + D = D[:, input:input + 1] + + try: + den = poly(A) + except ValueError: + den = 1 + + if (B.size == 0) and (C.size == 0): + num = np.ravel(D) + if (D.size == 0) and (A.size == 0): + den = [] + return num, den + + num_states = A.shape[0] + type_test = A[:, 0] + B[:, 0] + C[0, :] + D + 0.0 + num = np.empty((nout, num_states + 1), type_test.dtype) + for k in range(nout): + Ck = atleast_2d(C[k, :]) + num[k] = poly(A - dot(B, Ck)) + (D[k] - 1) * den + + return num, den + + +def zpk2ss(z, p, k): + """Zero-pole-gain representation to state-space representation + + Parameters + ---------- + z, p : sequence + Zeros and poles. + k : float + System gain. + + Returns + ------- + A, B, C, D : ndarray + State space representation of the system, in controller canonical + form. + + """ + return tf2ss(*zpk2tf(z, p, k)) + + +def ss2zpk(A, B, C, D, input=0): + """State-space representation to zero-pole-gain representation. + + A, B, C, D defines a linear state-space system with `p` inputs, + `q` outputs, and `n` state variables. + + Parameters + ---------- + A : array_like + State (or system) matrix of shape ``(n, n)`` + B : array_like + Input matrix of shape ``(n, p)`` + C : array_like + Output matrix of shape ``(q, n)`` + D : array_like + Feedthrough (or feedforward) matrix of shape ``(q, p)`` + input : int, optional + For multiple-input systems, the index of the input to use. + + Returns + ------- + z, p : sequence + Zeros and poles. + k : float + System gain. + + """ + return tf2zpk(*ss2tf(A, B, C, D, input=input)) + + +def cont2discrete(system, dt, method="zoh", alpha=None): + """ + Transform a continuous to a discrete state-space system. + + Parameters + ---------- + system : a tuple describing the system or an instance of `lti` + The following gives the number of elements in the tuple and + the interpretation: + + * 1: (instance of `lti`) + * 2: (num, den) + * 3: (zeros, poles, gain) + * 4: (A, B, C, D) + + dt : float + The discretization time step. + method : str, optional + Which method to use: + + * gbt: generalized bilinear transformation + * bilinear: Tustin's approximation ("gbt" with alpha=0.5) + * euler: Euler (or forward differencing) method ("gbt" with alpha=0) + * backward_diff: Backwards differencing ("gbt" with alpha=1.0) + * zoh: zero-order hold (default) + * foh: first-order hold (*versionadded: 1.3.0*) + * impulse: equivalent impulse response (*versionadded: 1.3.0*) + + alpha : float within [0, 1], optional + The generalized bilinear transformation weighting parameter, which + should only be specified with method="gbt", and is ignored otherwise + + Returns + ------- + sysd : tuple containing the discrete system + Based on the input type, the output will be of the form + + * (num, den, dt) for transfer function input + * (zeros, poles, gain, dt) for zeros-poles-gain input + * (A, B, C, D, dt) for state-space system input + + Notes + ----- + By default, the routine uses a Zero-Order Hold (zoh) method to perform + the transformation. Alternatively, a generalized bilinear transformation + may be used, which includes the common Tustin's bilinear approximation, + an Euler's method technique, or a backwards differencing technique. + + The Zero-Order Hold (zoh) method is based on [1]_, the generalized bilinear + approximation is based on [2]_ and [3]_, the First-Order Hold (foh) method + is based on [4]_. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Discretization#Discretization_of_linear_state_space_models + + .. [2] http://techteach.no/publications/discretetime_signals_systems/discrete.pdf + + .. [3] G. Zhang, X. Chen, and T. Chen, Digital redesign via the generalized + bilinear transformation, Int. J. Control, vol. 82, no. 4, pp. 741-754, + 2009. + (https://www.mypolyuweb.hk/~magzhang/Research/ZCC09_IJC.pdf) + + .. [4] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital control + of dynamic systems, 3rd ed. Menlo Park, Calif: Addison-Wesley, + pp. 204-206, 1998. + + Examples + -------- + We can transform a continuous state-space system to a discrete one: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.signal import cont2discrete, lti, dlti, dstep + + Define a continuous state-space system. + + >>> A = np.array([[0, 1],[-10., -3]]) + >>> B = np.array([[0],[10.]]) + >>> C = np.array([[1., 0]]) + >>> D = np.array([[0.]]) + >>> l_system = lti(A, B, C, D) + >>> t, x = l_system.step(T=np.linspace(0, 5, 100)) + >>> fig, ax = plt.subplots() + >>> ax.plot(t, x, label='Continuous', linewidth=3) + + Transform it to a discrete state-space system using several methods. + + >>> dt = 0.1 + >>> for method in ['zoh', 'bilinear', 'euler', 'backward_diff', 'foh', 'impulse']: + ... d_system = cont2discrete((A, B, C, D), dt, method=method) + ... s, x_d = dstep(d_system) + ... ax.step(s, np.squeeze(x_d), label=method, where='post') + >>> ax.axis([t[0], t[-1], x[0], 1.4]) + >>> ax.legend(loc='best') + >>> fig.tight_layout() + >>> plt.show() + + """ + if len(system) == 1: + return system.to_discrete() + if len(system) == 2: + sysd = cont2discrete(tf2ss(system[0], system[1]), dt, method=method, + alpha=alpha) + return ss2tf(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,) + elif len(system) == 3: + sysd = cont2discrete(zpk2ss(system[0], system[1], system[2]), dt, + method=method, alpha=alpha) + return ss2zpk(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,) + elif len(system) == 4: + a, b, c, d = system + else: + raise ValueError("First argument must either be a tuple of 2 (tf), " + "3 (zpk), or 4 (ss) arrays.") + + if method == 'gbt': + if alpha is None: + raise ValueError("Alpha parameter must be specified for the " + "generalized bilinear transform (gbt) method") + elif alpha < 0 or alpha > 1: + raise ValueError("Alpha parameter must be within the interval " + "[0,1] for the gbt method") + + if method == 'gbt': + # This parameter is used repeatedly - compute once here + ima = np.eye(a.shape[0]) - alpha*dt*a + ad = linalg.solve(ima, np.eye(a.shape[0]) + (1.0-alpha)*dt*a) + bd = linalg.solve(ima, dt*b) + + # Similarly solve for the output equation matrices + cd = linalg.solve(ima.transpose(), c.transpose()) + cd = cd.transpose() + dd = d + alpha*np.dot(c, bd) + + elif method == 'bilinear' or method == 'tustin': + return cont2discrete(system, dt, method="gbt", alpha=0.5) + + elif method == 'euler' or method == 'forward_diff': + return cont2discrete(system, dt, method="gbt", alpha=0.0) + + elif method == 'backward_diff': + return cont2discrete(system, dt, method="gbt", alpha=1.0) + + elif method == 'zoh': + # Build an exponential matrix + em_upper = np.hstack((a, b)) + + # Need to stack zeros under the a and b matrices + em_lower = np.hstack((np.zeros((b.shape[1], a.shape[0])), + np.zeros((b.shape[1], b.shape[1])))) + + em = np.vstack((em_upper, em_lower)) + ms = linalg.expm(dt * em) + + # Dispose of the lower rows + ms = ms[:a.shape[0], :] + + ad = ms[:, 0:a.shape[1]] + bd = ms[:, a.shape[1]:] + + cd = c + dd = d + + elif method == 'foh': + # Size parameters for convenience + n = a.shape[0] + m = b.shape[1] + + # Build an exponential matrix similar to 'zoh' method + em_upper = linalg.block_diag(np.block([a, b]) * dt, np.eye(m)) + em_lower = zeros((m, n + 2 * m)) + em = np.block([[em_upper], [em_lower]]) + + ms = linalg.expm(em) + + # Get the three blocks from upper rows + ms11 = ms[:n, 0:n] + ms12 = ms[:n, n:n + m] + ms13 = ms[:n, n + m:] + + ad = ms11 + bd = ms12 - ms13 + ms11 @ ms13 + cd = c + dd = d + c @ ms13 + + elif method == 'impulse': + if not np.allclose(d, 0): + raise ValueError("Impulse method is only applicable " + "to strictly proper systems") + + ad = linalg.expm(a * dt) + bd = ad @ b * dt + cd = c + dd = c @ b * dt + + else: + raise ValueError(f"Unknown transformation method '{method}'") + + return ad, bd, cd, dd, dt diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_ltisys.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_ltisys.py new file mode 100644 index 0000000000000000000000000000000000000000..3992797a09a3ceee4be1f052603fcd6593ae6274 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_ltisys.py @@ -0,0 +1,3519 @@ +""" +ltisys -- a collection of classes and functions for modeling linear +time invariant systems. +""" +# +# Author: Travis Oliphant 2001 +# +# Feb 2010: Warren Weckesser +# Rewrote lsim2 and added impulse2. +# Apr 2011: Jeffrey Armstrong +# Added dlsim, dstep, dimpulse, cont2discrete +# Aug 2013: Juan Luis Cano +# Rewrote abcd_normalize. +# Jan 2015: Irvin Probst irvin DOT probst AT ensta-bretagne DOT fr +# Added pole placement +# Mar 2015: Clancy Rowley +# Rewrote lsim +# May 2015: Felix Berkenkamp +# Split lti class into subclasses +# Merged discrete systems and added dlti + +import warnings + +# np.linalg.qr fails on some tests with LinAlgError: zgeqrf returns -7 +# use scipy's qr until this is solved + +from scipy.linalg import qr as s_qr +from scipy import linalg +from scipy.interpolate import make_interp_spline +from ._filter_design import (tf2zpk, zpk2tf, normalize, freqs, freqz, freqs_zpk, + freqz_zpk) +from ._lti_conversion import (tf2ss, abcd_normalize, ss2tf, zpk2ss, ss2zpk, + cont2discrete, _atleast_2d_or_none) + +import numpy as np +from numpy import (real, atleast_1d, squeeze, asarray, zeros, + dot, transpose, ones, linspace) +import copy + +__all__ = ['lti', 'dlti', 'TransferFunction', 'ZerosPolesGain', 'StateSpace', + 'lsim', 'impulse', 'step', 'bode', + 'freqresp', 'place_poles', 'dlsim', 'dstep', 'dimpulse', + 'dfreqresp', 'dbode'] + + +class LinearTimeInvariant: + def __new__(cls, *system, **kwargs): + """Create a new object, don't allow direct instances.""" + if cls is LinearTimeInvariant: + raise NotImplementedError('The LinearTimeInvariant class is not ' + 'meant to be used directly, use `lti` ' + 'or `dlti` instead.') + return super().__new__(cls) + + def __init__(self): + """ + Initialize the `lti` baseclass. + + The heavy lifting is done by the subclasses. + """ + super().__init__() + + self.inputs = None + self.outputs = None + self._dt = None + + @property + def dt(self): + """Return the sampling time of the system, `None` for `lti` systems.""" + return self._dt + + @property + def _dt_dict(self): + if self.dt is None: + return {} + else: + return {'dt': self.dt} + + @property + def zeros(self): + """Zeros of the system.""" + return self.to_zpk().zeros + + @property + def poles(self): + """Poles of the system.""" + return self.to_zpk().poles + + def _as_ss(self): + """Convert to `StateSpace` system, without copying. + + Returns + ------- + sys: StateSpace + The `StateSpace` system. If the class is already an instance of + `StateSpace` then this instance is returned. + """ + if isinstance(self, StateSpace): + return self + else: + return self.to_ss() + + def _as_zpk(self): + """Convert to `ZerosPolesGain` system, without copying. + + Returns + ------- + sys: ZerosPolesGain + The `ZerosPolesGain` system. If the class is already an instance of + `ZerosPolesGain` then this instance is returned. + """ + if isinstance(self, ZerosPolesGain): + return self + else: + return self.to_zpk() + + def _as_tf(self): + """Convert to `TransferFunction` system, without copying. + + Returns + ------- + sys: ZerosPolesGain + The `TransferFunction` system. If the class is already an instance of + `TransferFunction` then this instance is returned. + """ + if isinstance(self, TransferFunction): + return self + else: + return self.to_tf() + + +class lti(LinearTimeInvariant): + r""" + Continuous-time linear time invariant system base class. + + Parameters + ---------- + *system : arguments + The `lti` class can be instantiated with either 2, 3 or 4 arguments. + The following gives the number of arguments and the corresponding + continuous-time subclass that is created: + + * 2: `TransferFunction`: (numerator, denominator) + * 3: `ZerosPolesGain`: (zeros, poles, gain) + * 4: `StateSpace`: (A, B, C, D) + + Each argument can be an array or a sequence. + + See Also + -------- + ZerosPolesGain, StateSpace, TransferFunction, dlti + + Notes + ----- + `lti` instances do not exist directly. Instead, `lti` creates an instance + of one of its subclasses: `StateSpace`, `TransferFunction` or + `ZerosPolesGain`. + + If (numerator, denominator) is passed in for ``*system``, coefficients for + both the numerator and denominator should be specified in descending + exponent order (e.g., ``s^2 + 3s + 5`` would be represented as ``[1, 3, + 5]``). + + Changing the value of properties that are not directly part of the current + system representation (such as the `zeros` of a `StateSpace` system) is + very inefficient and may lead to numerical inaccuracies. It is better to + convert to the specific system representation first. For example, call + ``sys = sys.to_zpk()`` before accessing/changing the zeros, poles or gain. + + Examples + -------- + >>> from scipy import signal + + >>> signal.lti(1, 2, 3, 4) + StateSpaceContinuous( + array([[1]]), + array([[2]]), + array([[3]]), + array([[4]]), + dt: None + ) + + Construct the transfer function + :math:`H(s) = \frac{5(s - 1)(s - 2)}{(s - 3)(s - 4)}`: + + >>> signal.lti([1, 2], [3, 4], 5) + ZerosPolesGainContinuous( + array([1, 2]), + array([3, 4]), + 5, + dt: None + ) + + Construct the transfer function :math:`H(s) = \frac{3s + 4}{1s + 2}`: + + >>> signal.lti([3, 4], [1, 2]) + TransferFunctionContinuous( + array([3., 4.]), + array([1., 2.]), + dt: None + ) + + """ + def __new__(cls, *system): + """Create an instance of the appropriate subclass.""" + if cls is lti: + N = len(system) + if N == 2: + return TransferFunctionContinuous.__new__( + TransferFunctionContinuous, *system) + elif N == 3: + return ZerosPolesGainContinuous.__new__( + ZerosPolesGainContinuous, *system) + elif N == 4: + return StateSpaceContinuous.__new__(StateSpaceContinuous, + *system) + else: + raise ValueError("`system` needs to be an instance of `lti` " + "or have 2, 3 or 4 arguments.") + # __new__ was called from a subclass, let it call its own functions + return super().__new__(cls) + + def __init__(self, *system): + """ + Initialize the `lti` baseclass. + + The heavy lifting is done by the subclasses. + """ + super().__init__(*system) + + def impulse(self, X0=None, T=None, N=None): + """ + Return the impulse response of a continuous-time system. + See `impulse` for details. + """ + return impulse(self, X0=X0, T=T, N=N) + + def step(self, X0=None, T=None, N=None): + """ + Return the step response of a continuous-time system. + See `step` for details. + """ + return step(self, X0=X0, T=T, N=N) + + def output(self, U, T, X0=None): + """ + Return the response of a continuous-time system to input `U`. + See `lsim` for details. + """ + return lsim(self, U, T, X0=X0) + + def bode(self, w=None, n=100): + """ + Calculate Bode magnitude and phase data of a continuous-time system. + + Returns a 3-tuple containing arrays of frequencies [rad/s], magnitude + [dB] and phase [deg]. See `bode` for details. + + Examples + -------- + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> sys = signal.TransferFunction([1], [1, 1]) + >>> w, mag, phase = sys.bode() + + >>> plt.figure() + >>> plt.semilogx(w, mag) # Bode magnitude plot + >>> plt.figure() + >>> plt.semilogx(w, phase) # Bode phase plot + >>> plt.show() + + """ + return bode(self, w=w, n=n) + + def freqresp(self, w=None, n=10000): + """ + Calculate the frequency response of a continuous-time system. + + Returns a 2-tuple containing arrays of frequencies [rad/s] and + complex magnitude. + See `freqresp` for details. + """ + return freqresp(self, w=w, n=n) + + def to_discrete(self, dt, method='zoh', alpha=None): + """Return a discretized version of the current system. + + Parameters: See `cont2discrete` for details. + + Returns + ------- + sys: instance of `dlti` + """ + raise NotImplementedError('to_discrete is not implemented for this ' + 'system class.') + + +class dlti(LinearTimeInvariant): + r""" + Discrete-time linear time invariant system base class. + + Parameters + ---------- + *system: arguments + The `dlti` class can be instantiated with either 2, 3 or 4 arguments. + The following gives the number of arguments and the corresponding + discrete-time subclass that is created: + + * 2: `TransferFunction`: (numerator, denominator) + * 3: `ZerosPolesGain`: (zeros, poles, gain) + * 4: `StateSpace`: (A, B, C, D) + + Each argument can be an array or a sequence. + dt: float, optional + Sampling time [s] of the discrete-time systems. Defaults to ``True`` + (unspecified sampling time). Must be specified as a keyword argument, + for example, ``dt=0.1``. + + See Also + -------- + ZerosPolesGain, StateSpace, TransferFunction, lti + + Notes + ----- + `dlti` instances do not exist directly. Instead, `dlti` creates an instance + of one of its subclasses: `StateSpace`, `TransferFunction` or + `ZerosPolesGain`. + + Changing the value of properties that are not directly part of the current + system representation (such as the `zeros` of a `StateSpace` system) is + very inefficient and may lead to numerical inaccuracies. It is better to + convert to the specific system representation first. For example, call + ``sys = sys.to_zpk()`` before accessing/changing the zeros, poles or gain. + + If (numerator, denominator) is passed in for ``*system``, coefficients for + both the numerator and denominator should be specified in descending + exponent order (e.g., ``z^2 + 3z + 5`` would be represented as ``[1, 3, + 5]``). + + .. versionadded:: 0.18.0 + + Examples + -------- + >>> from scipy import signal + + >>> signal.dlti(1, 2, 3, 4) + StateSpaceDiscrete( + array([[1]]), + array([[2]]), + array([[3]]), + array([[4]]), + dt: True + ) + + >>> signal.dlti(1, 2, 3, 4, dt=0.1) + StateSpaceDiscrete( + array([[1]]), + array([[2]]), + array([[3]]), + array([[4]]), + dt: 0.1 + ) + + Construct the transfer function + :math:`H(z) = \frac{5(z - 1)(z - 2)}{(z - 3)(z - 4)}` with a sampling time + of 0.1 seconds: + + >>> signal.dlti([1, 2], [3, 4], 5, dt=0.1) + ZerosPolesGainDiscrete( + array([1, 2]), + array([3, 4]), + 5, + dt: 0.1 + ) + + Construct the transfer function :math:`H(z) = \frac{3z + 4}{1z + 2}` with + a sampling time of 0.1 seconds: + + >>> signal.dlti([3, 4], [1, 2], dt=0.1) + TransferFunctionDiscrete( + array([3., 4.]), + array([1., 2.]), + dt: 0.1 + ) + + """ + def __new__(cls, *system, **kwargs): + """Create an instance of the appropriate subclass.""" + if cls is dlti: + N = len(system) + if N == 2: + return TransferFunctionDiscrete.__new__( + TransferFunctionDiscrete, *system, **kwargs) + elif N == 3: + return ZerosPolesGainDiscrete.__new__(ZerosPolesGainDiscrete, + *system, **kwargs) + elif N == 4: + return StateSpaceDiscrete.__new__(StateSpaceDiscrete, *system, + **kwargs) + else: + raise ValueError("`system` needs to be an instance of `dlti` " + "or have 2, 3 or 4 arguments.") + # __new__ was called from a subclass, let it call its own functions + return super().__new__(cls) + + def __init__(self, *system, **kwargs): + """ + Initialize the `lti` baseclass. + + The heavy lifting is done by the subclasses. + """ + dt = kwargs.pop('dt', True) + super().__init__(*system, **kwargs) + + self.dt = dt + + @property + def dt(self): + """Return the sampling time of the system.""" + return self._dt + + @dt.setter + def dt(self, dt): + self._dt = dt + + def impulse(self, x0=None, t=None, n=None): + """ + Return the impulse response of the discrete-time `dlti` system. + See `dimpulse` for details. + """ + return dimpulse(self, x0=x0, t=t, n=n) + + def step(self, x0=None, t=None, n=None): + """ + Return the step response of the discrete-time `dlti` system. + See `dstep` for details. + """ + return dstep(self, x0=x0, t=t, n=n) + + def output(self, u, t, x0=None): + """ + Return the response of the discrete-time system to input `u`. + See `dlsim` for details. + """ + return dlsim(self, u, t, x0=x0) + + def bode(self, w=None, n=100): + r""" + Calculate Bode magnitude and phase data of a discrete-time system. + + Returns a 3-tuple containing arrays of frequencies [rad/s], magnitude + [dB] and phase [deg]. See `dbode` for details. + + Examples + -------- + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + Construct the transfer function :math:`H(z) = \frac{1}{z^2 + 2z + 3}` + with sampling time 0.5s: + + >>> sys = signal.TransferFunction([1], [1, 2, 3], dt=0.5) + + Equivalent: signal.dbode(sys) + + >>> w, mag, phase = sys.bode() + + >>> plt.figure() + >>> plt.semilogx(w, mag) # Bode magnitude plot + >>> plt.figure() + >>> plt.semilogx(w, phase) # Bode phase plot + >>> plt.show() + + """ + return dbode(self, w=w, n=n) + + def freqresp(self, w=None, n=10000, whole=False): + """ + Calculate the frequency response of a discrete-time system. + + Returns a 2-tuple containing arrays of frequencies [rad/s] and + complex magnitude. + See `dfreqresp` for details. + + """ + return dfreqresp(self, w=w, n=n, whole=whole) + + +class TransferFunction(LinearTimeInvariant): + r"""Linear Time Invariant system class in transfer function form. + + Represents the system as the continuous-time transfer function + :math:`H(s)=\sum_{i=0}^N b[N-i] s^i / \sum_{j=0}^M a[M-j] s^j` or the + discrete-time transfer function + :math:`H(z)=\sum_{i=0}^N b[N-i] z^i / \sum_{j=0}^M a[M-j] z^j`, where + :math:`b` are elements of the numerator `num`, :math:`a` are elements of + the denominator `den`, and ``N == len(b) - 1``, ``M == len(a) - 1``. + `TransferFunction` systems inherit additional + functionality from the `lti`, respectively the `dlti` classes, depending on + which system representation is used. + + Parameters + ---------- + *system: arguments + The `TransferFunction` class can be instantiated with 1 or 2 + arguments. The following gives the number of input arguments and their + interpretation: + + * 1: `lti` or `dlti` system: (`StateSpace`, `TransferFunction` or + `ZerosPolesGain`) + * 2: array_like: (numerator, denominator) + dt: float, optional + Sampling time [s] of the discrete-time systems. Defaults to `None` + (continuous-time). Must be specified as a keyword argument, for + example, ``dt=0.1``. + + See Also + -------- + ZerosPolesGain, StateSpace, lti, dlti + tf2ss, tf2zpk, tf2sos + + Notes + ----- + Changing the value of properties that are not part of the + `TransferFunction` system representation (such as the `A`, `B`, `C`, `D` + state-space matrices) is very inefficient and may lead to numerical + inaccuracies. It is better to convert to the specific system + representation first. For example, call ``sys = sys.to_ss()`` before + accessing/changing the A, B, C, D system matrices. + + If (numerator, denominator) is passed in for ``*system``, coefficients + for both the numerator and denominator should be specified in descending + exponent order (e.g. ``s^2 + 3s + 5`` or ``z^2 + 3z + 5`` would be + represented as ``[1, 3, 5]``) + + Examples + -------- + Construct the transfer function + :math:`H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}`: + + >>> from scipy import signal + + >>> num = [1, 3, 3] + >>> den = [1, 2, 1] + + >>> signal.TransferFunction(num, den) + TransferFunctionContinuous( + array([1., 3., 3.]), + array([1., 2., 1.]), + dt: None + ) + + Construct the transfer function + :math:`H(z) = \frac{z^2 + 3z + 3}{z^2 + 2z + 1}` with a sampling time of + 0.1 seconds: + + >>> signal.TransferFunction(num, den, dt=0.1) + TransferFunctionDiscrete( + array([1., 3., 3.]), + array([1., 2., 1.]), + dt: 0.1 + ) + + """ + def __new__(cls, *system, **kwargs): + """Handle object conversion if input is an instance of lti.""" + if len(system) == 1 and isinstance(system[0], LinearTimeInvariant): + return system[0].to_tf() + + # Choose whether to inherit from `lti` or from `dlti` + if cls is TransferFunction: + if kwargs.get('dt') is None: + return TransferFunctionContinuous.__new__( + TransferFunctionContinuous, + *system, + **kwargs) + else: + return TransferFunctionDiscrete.__new__( + TransferFunctionDiscrete, + *system, + **kwargs) + + # No special conversion needed + return super().__new__(cls) + + def __init__(self, *system, **kwargs): + """Initialize the state space LTI system.""" + # Conversion of lti instances is handled in __new__ + if isinstance(system[0], LinearTimeInvariant): + return + + # Remove system arguments, not needed by parents anymore + super().__init__(**kwargs) + + self._num = None + self._den = None + + self.num, self.den = normalize(*system) + + def __repr__(self): + """Return representation of the system's transfer function""" + return ( + f'{self.__class__.__name__}(\n' + f'{repr(self.num)},\n' + f'{repr(self.den)},\n' + f'dt: {repr(self.dt)}\n)' + ) + + @property + def num(self): + """Numerator of the `TransferFunction` system.""" + return self._num + + @num.setter + def num(self, num): + self._num = atleast_1d(num) + + # Update dimensions + if len(self.num.shape) > 1: + self.outputs, self.inputs = self.num.shape + else: + self.outputs = 1 + self.inputs = 1 + + @property + def den(self): + """Denominator of the `TransferFunction` system.""" + return self._den + + @den.setter + def den(self, den): + self._den = atleast_1d(den) + + def _copy(self, system): + """ + Copy the parameters of another `TransferFunction` object + + Parameters + ---------- + system : `TransferFunction` + The `StateSpace` system that is to be copied + + """ + self.num = system.num + self.den = system.den + + def to_tf(self): + """ + Return a copy of the current `TransferFunction` system. + + Returns + ------- + sys : instance of `TransferFunction` + The current system (copy) + + """ + return copy.deepcopy(self) + + def to_zpk(self): + """ + Convert system representation to `ZerosPolesGain`. + + Returns + ------- + sys : instance of `ZerosPolesGain` + Zeros, poles, gain representation of the current system + + """ + return ZerosPolesGain(*tf2zpk(self.num, self.den), + **self._dt_dict) + + def to_ss(self): + """ + Convert system representation to `StateSpace`. + + Returns + ------- + sys : instance of `StateSpace` + State space model of the current system + + """ + return StateSpace(*tf2ss(self.num, self.den), + **self._dt_dict) + + @staticmethod + def _z_to_zinv(num, den): + """Change a transfer function from the variable `z` to `z**-1`. + + Parameters + ---------- + num, den: 1d array_like + Sequences representing the coefficients of the numerator and + denominator polynomials, in order of descending degree of 'z'. + That is, ``5z**2 + 3z + 2`` is presented as ``[5, 3, 2]``. + + Returns + ------- + num, den: 1d array_like + Sequences representing the coefficients of the numerator and + denominator polynomials, in order of ascending degree of 'z**-1'. + That is, ``5 + 3 z**-1 + 2 z**-2`` is presented as ``[5, 3, 2]``. + """ + diff = len(num) - len(den) + if diff > 0: + den = np.hstack((np.zeros(diff), den)) + elif diff < 0: + num = np.hstack((np.zeros(-diff), num)) + return num, den + + @staticmethod + def _zinv_to_z(num, den): + """Change a transfer function from the variable `z` to `z**-1`. + + Parameters + ---------- + num, den: 1d array_like + Sequences representing the coefficients of the numerator and + denominator polynomials, in order of ascending degree of 'z**-1'. + That is, ``5 + 3 z**-1 + 2 z**-2`` is presented as ``[5, 3, 2]``. + + Returns + ------- + num, den: 1d array_like + Sequences representing the coefficients of the numerator and + denominator polynomials, in order of descending degree of 'z'. + That is, ``5z**2 + 3z + 2`` is presented as ``[5, 3, 2]``. + """ + diff = len(num) - len(den) + if diff > 0: + den = np.hstack((den, np.zeros(diff))) + elif diff < 0: + num = np.hstack((num, np.zeros(-diff))) + return num, den + + +class TransferFunctionContinuous(TransferFunction, lti): + r""" + Continuous-time Linear Time Invariant system in transfer function form. + + Represents the system as the transfer function + :math:`H(s)=\sum_{i=0}^N b[N-i] s^i / \sum_{j=0}^M a[M-j] s^j`, where + :math:`b` are elements of the numerator `num`, :math:`a` are elements of + the denominator `den`, and ``N == len(b) - 1``, ``M == len(a) - 1``. + Continuous-time `TransferFunction` systems inherit additional + functionality from the `lti` class. + + Parameters + ---------- + *system: arguments + The `TransferFunction` class can be instantiated with 1 or 2 + arguments. The following gives the number of input arguments and their + interpretation: + + * 1: `lti` system: (`StateSpace`, `TransferFunction` or + `ZerosPolesGain`) + * 2: array_like: (numerator, denominator) + + See Also + -------- + ZerosPolesGain, StateSpace, lti + tf2ss, tf2zpk, tf2sos + + Notes + ----- + Changing the value of properties that are not part of the + `TransferFunction` system representation (such as the `A`, `B`, `C`, `D` + state-space matrices) is very inefficient and may lead to numerical + inaccuracies. It is better to convert to the specific system + representation first. For example, call ``sys = sys.to_ss()`` before + accessing/changing the A, B, C, D system matrices. + + If (numerator, denominator) is passed in for ``*system``, coefficients + for both the numerator and denominator should be specified in descending + exponent order (e.g. ``s^2 + 3s + 5`` would be represented as + ``[1, 3, 5]``) + + Examples + -------- + Construct the transfer function + :math:`H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}`: + + >>> from scipy import signal + + >>> num = [1, 3, 3] + >>> den = [1, 2, 1] + + >>> signal.TransferFunction(num, den) + TransferFunctionContinuous( + array([ 1., 3., 3.]), + array([ 1., 2., 1.]), + dt: None + ) + + """ + + def to_discrete(self, dt, method='zoh', alpha=None): + """ + Returns the discretized `TransferFunction` system. + + Parameters: See `cont2discrete` for details. + + Returns + ------- + sys: instance of `dlti` and `StateSpace` + """ + return TransferFunction(*cont2discrete((self.num, self.den), + dt, + method=method, + alpha=alpha)[:-1], + dt=dt) + + +class TransferFunctionDiscrete(TransferFunction, dlti): + r""" + Discrete-time Linear Time Invariant system in transfer function form. + + Represents the system as the transfer function + :math:`H(z)=\sum_{i=0}^N b[N-i] z^i / \sum_{j=0}^M a[M-j] z^j`, where + :math:`b` are elements of the numerator `num`, :math:`a` are elements of + the denominator `den`, and ``N == len(b) - 1``, ``M == len(a) - 1``. + Discrete-time `TransferFunction` systems inherit additional functionality + from the `dlti` class. + + Parameters + ---------- + *system: arguments + The `TransferFunction` class can be instantiated with 1 or 2 + arguments. The following gives the number of input arguments and their + interpretation: + + * 1: `dlti` system: (`StateSpace`, `TransferFunction` or + `ZerosPolesGain`) + * 2: array_like: (numerator, denominator) + dt: float, optional + Sampling time [s] of the discrete-time systems. Defaults to `True` + (unspecified sampling time). Must be specified as a keyword argument, + for example, ``dt=0.1``. + + See Also + -------- + ZerosPolesGain, StateSpace, dlti + tf2ss, tf2zpk, tf2sos + + Notes + ----- + Changing the value of properties that are not part of the + `TransferFunction` system representation (such as the `A`, `B`, `C`, `D` + state-space matrices) is very inefficient and may lead to numerical + inaccuracies. + + If (numerator, denominator) is passed in for ``*system``, coefficients + for both the numerator and denominator should be specified in descending + exponent order (e.g., ``z^2 + 3z + 5`` would be represented as + ``[1, 3, 5]``). + + Examples + -------- + Construct the transfer function + :math:`H(z) = \frac{z^2 + 3z + 3}{z^2 + 2z + 1}` with a sampling time of + 0.5 seconds: + + >>> from scipy import signal + + >>> num = [1, 3, 3] + >>> den = [1, 2, 1] + + >>> signal.TransferFunction(num, den, dt=0.5) + TransferFunctionDiscrete( + array([ 1., 3., 3.]), + array([ 1., 2., 1.]), + dt: 0.5 + ) + + """ + pass + + +class ZerosPolesGain(LinearTimeInvariant): + r""" + Linear Time Invariant system class in zeros, poles, gain form. + + Represents the system as the continuous- or discrete-time transfer function + :math:`H(s)=k \prod_i (s - z[i]) / \prod_j (s - p[j])`, where :math:`k` is + the `gain`, :math:`z` are the `zeros` and :math:`p` are the `poles`. + `ZerosPolesGain` systems inherit additional functionality from the `lti`, + respectively the `dlti` classes, depending on which system representation + is used. + + Parameters + ---------- + *system : arguments + The `ZerosPolesGain` class can be instantiated with 1 or 3 + arguments. The following gives the number of input arguments and their + interpretation: + + * 1: `lti` or `dlti` system: (`StateSpace`, `TransferFunction` or + `ZerosPolesGain`) + * 3: array_like: (zeros, poles, gain) + dt: float, optional + Sampling time [s] of the discrete-time systems. Defaults to `None` + (continuous-time). Must be specified as a keyword argument, for + example, ``dt=0.1``. + + + See Also + -------- + TransferFunction, StateSpace, lti, dlti + zpk2ss, zpk2tf, zpk2sos + + Notes + ----- + Changing the value of properties that are not part of the + `ZerosPolesGain` system representation (such as the `A`, `B`, `C`, `D` + state-space matrices) is very inefficient and may lead to numerical + inaccuracies. It is better to convert to the specific system + representation first. For example, call ``sys = sys.to_ss()`` before + accessing/changing the A, B, C, D system matrices. + + Examples + -------- + Construct the transfer function + :math:`H(s) = \frac{5(s - 1)(s - 2)}{(s - 3)(s - 4)}`: + + >>> from scipy import signal + + >>> signal.ZerosPolesGain([1, 2], [3, 4], 5) + ZerosPolesGainContinuous( + array([1, 2]), + array([3, 4]), + 5, + dt: None + ) + + Construct the transfer function + :math:`H(z) = \frac{5(z - 1)(z - 2)}{(z - 3)(z - 4)}` with a sampling time + of 0.1 seconds: + + >>> signal.ZerosPolesGain([1, 2], [3, 4], 5, dt=0.1) + ZerosPolesGainDiscrete( + array([1, 2]), + array([3, 4]), + 5, + dt: 0.1 + ) + + """ + def __new__(cls, *system, **kwargs): + """Handle object conversion if input is an instance of `lti`""" + if len(system) == 1 and isinstance(system[0], LinearTimeInvariant): + return system[0].to_zpk() + + # Choose whether to inherit from `lti` or from `dlti` + if cls is ZerosPolesGain: + if kwargs.get('dt') is None: + return ZerosPolesGainContinuous.__new__( + ZerosPolesGainContinuous, + *system, + **kwargs) + else: + return ZerosPolesGainDiscrete.__new__( + ZerosPolesGainDiscrete, + *system, + **kwargs + ) + + # No special conversion needed + return super().__new__(cls) + + def __init__(self, *system, **kwargs): + """Initialize the zeros, poles, gain system.""" + # Conversion of lti instances is handled in __new__ + if isinstance(system[0], LinearTimeInvariant): + return + + super().__init__(**kwargs) + + self._zeros = None + self._poles = None + self._gain = None + + self.zeros, self.poles, self.gain = system + + def __repr__(self): + """Return representation of the `ZerosPolesGain` system.""" + return ( + f'{self.__class__.__name__}(\n' + f'{repr(self.zeros)},\n' + f'{repr(self.poles)},\n' + f'{repr(self.gain)},\n' + f'dt: {repr(self.dt)}\n)' + ) + + @property + def zeros(self): + """Zeros of the `ZerosPolesGain` system.""" + return self._zeros + + @zeros.setter + def zeros(self, zeros): + self._zeros = atleast_1d(zeros) + + # Update dimensions + if len(self.zeros.shape) > 1: + self.outputs, self.inputs = self.zeros.shape + else: + self.outputs = 1 + self.inputs = 1 + + @property + def poles(self): + """Poles of the `ZerosPolesGain` system.""" + return self._poles + + @poles.setter + def poles(self, poles): + self._poles = atleast_1d(poles) + + @property + def gain(self): + """Gain of the `ZerosPolesGain` system.""" + return self._gain + + @gain.setter + def gain(self, gain): + self._gain = gain + + def _copy(self, system): + """ + Copy the parameters of another `ZerosPolesGain` system. + + Parameters + ---------- + system : instance of `ZerosPolesGain` + The zeros, poles gain system that is to be copied + + """ + self.poles = system.poles + self.zeros = system.zeros + self.gain = system.gain + + def to_tf(self): + """ + Convert system representation to `TransferFunction`. + + Returns + ------- + sys : instance of `TransferFunction` + Transfer function of the current system + + """ + return TransferFunction(*zpk2tf(self.zeros, self.poles, self.gain), + **self._dt_dict) + + def to_zpk(self): + """ + Return a copy of the current 'ZerosPolesGain' system. + + Returns + ------- + sys : instance of `ZerosPolesGain` + The current system (copy) + + """ + return copy.deepcopy(self) + + def to_ss(self): + """ + Convert system representation to `StateSpace`. + + Returns + ------- + sys : instance of `StateSpace` + State space model of the current system + + """ + return StateSpace(*zpk2ss(self.zeros, self.poles, self.gain), + **self._dt_dict) + + +class ZerosPolesGainContinuous(ZerosPolesGain, lti): + r""" + Continuous-time Linear Time Invariant system in zeros, poles, gain form. + + Represents the system as the continuous time transfer function + :math:`H(s)=k \prod_i (s - z[i]) / \prod_j (s - p[j])`, where :math:`k` is + the `gain`, :math:`z` are the `zeros` and :math:`p` are the `poles`. + Continuous-time `ZerosPolesGain` systems inherit additional functionality + from the `lti` class. + + Parameters + ---------- + *system : arguments + The `ZerosPolesGain` class can be instantiated with 1 or 3 + arguments. The following gives the number of input arguments and their + interpretation: + + * 1: `lti` system: (`StateSpace`, `TransferFunction` or + `ZerosPolesGain`) + * 3: array_like: (zeros, poles, gain) + + See Also + -------- + TransferFunction, StateSpace, lti + zpk2ss, zpk2tf, zpk2sos + + Notes + ----- + Changing the value of properties that are not part of the + `ZerosPolesGain` system representation (such as the `A`, `B`, `C`, `D` + state-space matrices) is very inefficient and may lead to numerical + inaccuracies. It is better to convert to the specific system + representation first. For example, call ``sys = sys.to_ss()`` before + accessing/changing the A, B, C, D system matrices. + + Examples + -------- + Construct the transfer function + :math:`H(s)=\frac{5(s - 1)(s - 2)}{(s - 3)(s - 4)}`: + + >>> from scipy import signal + + >>> signal.ZerosPolesGain([1, 2], [3, 4], 5) + ZerosPolesGainContinuous( + array([1, 2]), + array([3, 4]), + 5, + dt: None + ) + + """ + + def to_discrete(self, dt, method='zoh', alpha=None): + """ + Returns the discretized `ZerosPolesGain` system. + + Parameters: See `cont2discrete` for details. + + Returns + ------- + sys: instance of `dlti` and `ZerosPolesGain` + """ + return ZerosPolesGain( + *cont2discrete((self.zeros, self.poles, self.gain), + dt, + method=method, + alpha=alpha)[:-1], + dt=dt) + + +class ZerosPolesGainDiscrete(ZerosPolesGain, dlti): + r""" + Discrete-time Linear Time Invariant system in zeros, poles, gain form. + + Represents the system as the discrete-time transfer function + :math:`H(z)=k \prod_i (z - q[i]) / \prod_j (z - p[j])`, where :math:`k` is + the `gain`, :math:`q` are the `zeros` and :math:`p` are the `poles`. + Discrete-time `ZerosPolesGain` systems inherit additional functionality + from the `dlti` class. + + Parameters + ---------- + *system : arguments + The `ZerosPolesGain` class can be instantiated with 1 or 3 + arguments. The following gives the number of input arguments and their + interpretation: + + * 1: `dlti` system: (`StateSpace`, `TransferFunction` or + `ZerosPolesGain`) + * 3: array_like: (zeros, poles, gain) + dt: float, optional + Sampling time [s] of the discrete-time systems. Defaults to `True` + (unspecified sampling time). Must be specified as a keyword argument, + for example, ``dt=0.1``. + + See Also + -------- + TransferFunction, StateSpace, dlti + zpk2ss, zpk2tf, zpk2sos + + Notes + ----- + Changing the value of properties that are not part of the + `ZerosPolesGain` system representation (such as the `A`, `B`, `C`, `D` + state-space matrices) is very inefficient and may lead to numerical + inaccuracies. It is better to convert to the specific system + representation first. For example, call ``sys = sys.to_ss()`` before + accessing/changing the A, B, C, D system matrices. + + Examples + -------- + Construct the transfer function + :math:`H(s) = \frac{5(s - 1)(s - 2)}{(s - 3)(s - 4)}`: + + >>> from scipy import signal + + >>> signal.ZerosPolesGain([1, 2], [3, 4], 5) + ZerosPolesGainContinuous( + array([1, 2]), + array([3, 4]), + 5, + dt: None + ) + + Construct the transfer function + :math:`H(z) = \frac{5(z - 1)(z - 2)}{(z - 3)(z - 4)}` with a sampling time + of 0.1 seconds: + + >>> signal.ZerosPolesGain([1, 2], [3, 4], 5, dt=0.1) + ZerosPolesGainDiscrete( + array([1, 2]), + array([3, 4]), + 5, + dt: 0.1 + ) + + """ + pass + + +class StateSpace(LinearTimeInvariant): + r""" + Linear Time Invariant system in state-space form. + + Represents the system as the continuous-time, first order differential + equation :math:`\dot{x} = A x + B u` or the discrete-time difference + equation :math:`x[k+1] = A x[k] + B u[k]`. `StateSpace` systems + inherit additional functionality from the `lti`, respectively the `dlti` + classes, depending on which system representation is used. + + Parameters + ---------- + *system: arguments + The `StateSpace` class can be instantiated with 1 or 4 arguments. + The following gives the number of input arguments and their + interpretation: + + * 1: `lti` or `dlti` system: (`StateSpace`, `TransferFunction` or + `ZerosPolesGain`) + * 4: array_like: (A, B, C, D) + dt: float, optional + Sampling time [s] of the discrete-time systems. Defaults to `None` + (continuous-time). Must be specified as a keyword argument, for + example, ``dt=0.1``. + + See Also + -------- + TransferFunction, ZerosPolesGain, lti, dlti + ss2zpk, ss2tf, zpk2sos + + Notes + ----- + Changing the value of properties that are not part of the + `StateSpace` system representation (such as `zeros` or `poles`) is very + inefficient and may lead to numerical inaccuracies. It is better to + convert to the specific system representation first. For example, call + ``sys = sys.to_zpk()`` before accessing/changing the zeros, poles or gain. + + Examples + -------- + >>> from scipy import signal + >>> import numpy as np + >>> a = np.array([[0, 1], [0, 0]]) + >>> b = np.array([[0], [1]]) + >>> c = np.array([[1, 0]]) + >>> d = np.array([[0]]) + + >>> sys = signal.StateSpace(a, b, c, d) + >>> print(sys) + StateSpaceContinuous( + array([[0, 1], + [0, 0]]), + array([[0], + [1]]), + array([[1, 0]]), + array([[0]]), + dt: None + ) + + >>> sys.to_discrete(0.1) + StateSpaceDiscrete( + array([[1. , 0.1], + [0. , 1. ]]), + array([[0.005], + [0.1 ]]), + array([[1, 0]]), + array([[0]]), + dt: 0.1 + ) + + >>> a = np.array([[1, 0.1], [0, 1]]) + >>> b = np.array([[0.005], [0.1]]) + + >>> signal.StateSpace(a, b, c, d, dt=0.1) + StateSpaceDiscrete( + array([[1. , 0.1], + [0. , 1. ]]), + array([[0.005], + [0.1 ]]), + array([[1, 0]]), + array([[0]]), + dt: 0.1 + ) + + """ + + # Override NumPy binary operations and ufuncs + __array_priority__ = 100.0 + __array_ufunc__ = None + + def __new__(cls, *system, **kwargs): + """Create new StateSpace object and settle inheritance.""" + # Handle object conversion if input is an instance of `lti` + if len(system) == 1 and isinstance(system[0], LinearTimeInvariant): + return system[0].to_ss() + + # Choose whether to inherit from `lti` or from `dlti` + if cls is StateSpace: + if kwargs.get('dt') is None: + return StateSpaceContinuous.__new__(StateSpaceContinuous, + *system, **kwargs) + else: + return StateSpaceDiscrete.__new__(StateSpaceDiscrete, + *system, **kwargs) + + # No special conversion needed + return super().__new__(cls) + + def __init__(self, *system, **kwargs): + """Initialize the state space lti/dlti system.""" + # Conversion of lti instances is handled in __new__ + if isinstance(system[0], LinearTimeInvariant): + return + + # Remove system arguments, not needed by parents anymore + super().__init__(**kwargs) + + self._A = None + self._B = None + self._C = None + self._D = None + + self.A, self.B, self.C, self.D = abcd_normalize(*system) + + def __repr__(self): + """Return representation of the `StateSpace` system.""" + return ( + f'{self.__class__.__name__}(\n' + f'{repr(self.A)},\n' + f'{repr(self.B)},\n' + f'{repr(self.C)},\n' + f'{repr(self.D)},\n' + f'dt: {repr(self.dt)}\n)' + ) + + def _check_binop_other(self, other): + return isinstance(other, (StateSpace, np.ndarray, float, complex, + np.number, int)) + + def __mul__(self, other): + """ + Post-multiply another system or a scalar + + Handles multiplication of systems in the sense of a frequency domain + multiplication. That means, given two systems E1(s) and E2(s), their + multiplication, H(s) = E1(s) * E2(s), means that applying H(s) to U(s) + is equivalent to first applying E2(s), and then E1(s). + + Notes + ----- + For SISO systems the order of system application does not matter. + However, for MIMO systems, where the two systems are matrices, the + order above ensures standard Matrix multiplication rules apply. + """ + if not self._check_binop_other(other): + return NotImplemented + + if isinstance(other, StateSpace): + # Disallow mix of discrete and continuous systems. + if type(other) is not type(self): + return NotImplemented + + if self.dt != other.dt: + raise TypeError('Cannot multiply systems with different `dt`.') + + n1 = self.A.shape[0] + n2 = other.A.shape[0] + + # Interconnection of systems + # x1' = A1 x1 + B1 u1 + # y1 = C1 x1 + D1 u1 + # x2' = A2 x2 + B2 y1 + # y2 = C2 x2 + D2 y1 + # + # Plugging in with u1 = y2 yields + # [x1'] [A1 B1*C2 ] [x1] [B1*D2] + # [x2'] = [0 A2 ] [x2] + [B2 ] u2 + # [x1] + # y2 = [C1 D1*C2] [x2] + D1*D2 u2 + a = np.vstack((np.hstack((self.A, np.dot(self.B, other.C))), + np.hstack((zeros((n2, n1)), other.A)))) + b = np.vstack((np.dot(self.B, other.D), other.B)) + c = np.hstack((self.C, np.dot(self.D, other.C))) + d = np.dot(self.D, other.D) + else: + # Assume that other is a scalar / matrix + # For post multiplication the input gets scaled + a = self.A + b = np.dot(self.B, other) + c = self.C + d = np.dot(self.D, other) + + common_dtype = np.result_type(a.dtype, b.dtype, c.dtype, d.dtype) + return StateSpace(np.asarray(a, dtype=common_dtype), + np.asarray(b, dtype=common_dtype), + np.asarray(c, dtype=common_dtype), + np.asarray(d, dtype=common_dtype), + **self._dt_dict) + + def __rmul__(self, other): + """Pre-multiply a scalar or matrix (but not StateSpace)""" + if not self._check_binop_other(other) or isinstance(other, StateSpace): + return NotImplemented + + # For pre-multiplication only the output gets scaled + a = self.A + b = self.B + c = np.dot(other, self.C) + d = np.dot(other, self.D) + + common_dtype = np.result_type(a.dtype, b.dtype, c.dtype, d.dtype) + return StateSpace(np.asarray(a, dtype=common_dtype), + np.asarray(b, dtype=common_dtype), + np.asarray(c, dtype=common_dtype), + np.asarray(d, dtype=common_dtype), + **self._dt_dict) + + def __neg__(self): + """Negate the system (equivalent to pre-multiplying by -1).""" + return StateSpace(self.A, self.B, -self.C, -self.D, **self._dt_dict) + + def __add__(self, other): + """ + Adds two systems in the sense of frequency domain addition. + """ + if not self._check_binop_other(other): + return NotImplemented + + if isinstance(other, StateSpace): + # Disallow mix of discrete and continuous systems. + if type(other) is not type(self): + raise TypeError(f'Cannot add {type(self)} and {type(other)}') + + if self.dt != other.dt: + raise TypeError('Cannot add systems with different `dt`.') + # Interconnection of systems + # x1' = A1 x1 + B1 u + # y1 = C1 x1 + D1 u + # x2' = A2 x2 + B2 u + # y2 = C2 x2 + D2 u + # y = y1 + y2 + # + # Plugging in yields + # [x1'] [A1 0 ] [x1] [B1] + # [x2'] = [0 A2] [x2] + [B2] u + # [x1] + # y = [C1 C2] [x2] + [D1 + D2] u + a = linalg.block_diag(self.A, other.A) + b = np.vstack((self.B, other.B)) + c = np.hstack((self.C, other.C)) + d = self.D + other.D + else: + other = np.atleast_2d(other) + if self.D.shape == other.shape: + # A scalar/matrix is really just a static system (A=0, B=0, C=0) + a = self.A + b = self.B + c = self.C + d = self.D + other + else: + raise ValueError("Cannot add systems with incompatible " + f"dimensions ({self.D.shape} and {other.shape})") + + common_dtype = np.result_type(a.dtype, b.dtype, c.dtype, d.dtype) + return StateSpace(np.asarray(a, dtype=common_dtype), + np.asarray(b, dtype=common_dtype), + np.asarray(c, dtype=common_dtype), + np.asarray(d, dtype=common_dtype), + **self._dt_dict) + + def __sub__(self, other): + if not self._check_binop_other(other): + return NotImplemented + + return self.__add__(-other) + + def __radd__(self, other): + if not self._check_binop_other(other): + return NotImplemented + + return self.__add__(other) + + def __rsub__(self, other): + if not self._check_binop_other(other): + return NotImplemented + + return (-self).__add__(other) + + def __truediv__(self, other): + """ + Divide by a scalar + """ + # Division by non-StateSpace scalars + if not self._check_binop_other(other) or isinstance(other, StateSpace): + return NotImplemented + + if isinstance(other, np.ndarray) and other.ndim > 0: + # It's ambiguous what this means, so disallow it + raise ValueError("Cannot divide StateSpace by non-scalar numpy arrays") + + return self.__mul__(1/other) + + @property + def A(self): + """State matrix of the `StateSpace` system.""" + return self._A + + @A.setter + def A(self, A): + self._A = _atleast_2d_or_none(A) + + @property + def B(self): + """Input matrix of the `StateSpace` system.""" + return self._B + + @B.setter + def B(self, B): + self._B = _atleast_2d_or_none(B) + self.inputs = self.B.shape[-1] + + @property + def C(self): + """Output matrix of the `StateSpace` system.""" + return self._C + + @C.setter + def C(self, C): + self._C = _atleast_2d_or_none(C) + self.outputs = self.C.shape[0] + + @property + def D(self): + """Feedthrough matrix of the `StateSpace` system.""" + return self._D + + @D.setter + def D(self, D): + self._D = _atleast_2d_or_none(D) + + def _copy(self, system): + """ + Copy the parameters of another `StateSpace` system. + + Parameters + ---------- + system : instance of `StateSpace` + The state-space system that is to be copied + + """ + self.A = system.A + self.B = system.B + self.C = system.C + self.D = system.D + + def to_tf(self, **kwargs): + """ + Convert system representation to `TransferFunction`. + + Parameters + ---------- + kwargs : dict, optional + Additional keywords passed to `ss2zpk` + + Returns + ------- + sys : instance of `TransferFunction` + Transfer function of the current system + + """ + return TransferFunction(*ss2tf(self._A, self._B, self._C, self._D, + **kwargs), **self._dt_dict) + + def to_zpk(self, **kwargs): + """ + Convert system representation to `ZerosPolesGain`. + + Parameters + ---------- + kwargs : dict, optional + Additional keywords passed to `ss2zpk` + + Returns + ------- + sys : instance of `ZerosPolesGain` + Zeros, poles, gain representation of the current system + + """ + return ZerosPolesGain(*ss2zpk(self._A, self._B, self._C, self._D, + **kwargs), **self._dt_dict) + + def to_ss(self): + """ + Return a copy of the current `StateSpace` system. + + Returns + ------- + sys : instance of `StateSpace` + The current system (copy) + + """ + return copy.deepcopy(self) + + +class StateSpaceContinuous(StateSpace, lti): + r""" + Continuous-time Linear Time Invariant system in state-space form. + + Represents the system as the continuous-time, first order differential + equation :math:`\dot{x} = A x + B u`. + Continuous-time `StateSpace` systems inherit additional functionality + from the `lti` class. + + Parameters + ---------- + *system: arguments + The `StateSpace` class can be instantiated with 1 or 3 arguments. + The following gives the number of input arguments and their + interpretation: + + * 1: `lti` system: (`StateSpace`, `TransferFunction` or + `ZerosPolesGain`) + * 4: array_like: (A, B, C, D) + + See Also + -------- + TransferFunction, ZerosPolesGain, lti + ss2zpk, ss2tf, zpk2sos + + Notes + ----- + Changing the value of properties that are not part of the + `StateSpace` system representation (such as `zeros` or `poles`) is very + inefficient and may lead to numerical inaccuracies. It is better to + convert to the specific system representation first. For example, call + ``sys = sys.to_zpk()`` before accessing/changing the zeros, poles or gain. + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + + >>> a = np.array([[0, 1], [0, 0]]) + >>> b = np.array([[0], [1]]) + >>> c = np.array([[1, 0]]) + >>> d = np.array([[0]]) + + >>> sys = signal.StateSpace(a, b, c, d) + >>> print(sys) + StateSpaceContinuous( + array([[0, 1], + [0, 0]]), + array([[0], + [1]]), + array([[1, 0]]), + array([[0]]), + dt: None + ) + + """ + + def to_discrete(self, dt, method='zoh', alpha=None): + """ + Returns the discretized `StateSpace` system. + + Parameters: See `cont2discrete` for details. + + Returns + ------- + sys: instance of `dlti` and `StateSpace` + """ + return StateSpace(*cont2discrete((self.A, self.B, self.C, self.D), + dt, + method=method, + alpha=alpha)[:-1], + dt=dt) + + +class StateSpaceDiscrete(StateSpace, dlti): + r""" + Discrete-time Linear Time Invariant system in state-space form. + + Represents the system as the discrete-time difference equation + :math:`x[k+1] = A x[k] + B u[k]`. + `StateSpace` systems inherit additional functionality from the `dlti` + class. + + Parameters + ---------- + *system: arguments + The `StateSpace` class can be instantiated with 1 or 3 arguments. + The following gives the number of input arguments and their + interpretation: + + * 1: `dlti` system: (`StateSpace`, `TransferFunction` or + `ZerosPolesGain`) + * 4: array_like: (A, B, C, D) + dt: float, optional + Sampling time [s] of the discrete-time systems. Defaults to `True` + (unspecified sampling time). Must be specified as a keyword argument, + for example, ``dt=0.1``. + + See Also + -------- + TransferFunction, ZerosPolesGain, dlti + ss2zpk, ss2tf, zpk2sos + + Notes + ----- + Changing the value of properties that are not part of the + `StateSpace` system representation (such as `zeros` or `poles`) is very + inefficient and may lead to numerical inaccuracies. It is better to + convert to the specific system representation first. For example, call + ``sys = sys.to_zpk()`` before accessing/changing the zeros, poles or gain. + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + + >>> a = np.array([[1, 0.1], [0, 1]]) + >>> b = np.array([[0.005], [0.1]]) + >>> c = np.array([[1, 0]]) + >>> d = np.array([[0]]) + + >>> signal.StateSpace(a, b, c, d, dt=0.1) + StateSpaceDiscrete( + array([[ 1. , 0.1], + [ 0. , 1. ]]), + array([[ 0.005], + [ 0.1 ]]), + array([[1, 0]]), + array([[0]]), + dt: 0.1 + ) + + """ + pass + + +def lsim(system, U, T, X0=None, interp=True): + """ + Simulate output of a continuous-time linear system. + + Parameters + ---------- + system : an instance of the LTI class or a tuple describing the system. + The following gives the number of elements in the tuple and + the interpretation: + + * 1: (instance of `lti`) + * 2: (num, den) + * 3: (zeros, poles, gain) + * 4: (A, B, C, D) + + U : array_like + An input array describing the input at each time `T` + (interpolation is assumed between given times). If there are + multiple inputs, then each column of the rank-2 array + represents an input. If U = 0 or None, a zero input is used. + T : array_like + The time steps at which the input is defined and at which the + output is desired. Must be nonnegative, increasing, and equally spaced. + X0 : array_like, optional + The initial conditions on the state vector (zero by default). + interp : bool, optional + Whether to use linear (True, the default) or zero-order-hold (False) + interpolation for the input array. + + Returns + ------- + T : 1D ndarray + Time values for the output. + yout : 1D ndarray + System response. + xout : ndarray + Time evolution of the state vector. + + Notes + ----- + If (num, den) is passed in for ``system``, coefficients for both the + numerator and denominator should be specified in descending exponent + order (e.g. ``s^2 + 3s + 5`` would be represented as ``[1, 3, 5]``). + + Examples + -------- + We'll use `lsim` to simulate an analog Bessel filter applied to + a signal. + + >>> import numpy as np + >>> from scipy.signal import bessel, lsim + >>> import matplotlib.pyplot as plt + + Create a low-pass Bessel filter with a cutoff of 12 Hz. + + >>> b, a = bessel(N=5, Wn=2*np.pi*12, btype='lowpass', analog=True) + + Generate data to which the filter is applied. + + >>> t = np.linspace(0, 1.25, 500, endpoint=False) + + The input signal is the sum of three sinusoidal curves, with + frequencies 4 Hz, 40 Hz, and 80 Hz. The filter should mostly + eliminate the 40 Hz and 80 Hz components, leaving just the 4 Hz signal. + + >>> u = (np.cos(2*np.pi*4*t) + 0.6*np.sin(2*np.pi*40*t) + + ... 0.5*np.cos(2*np.pi*80*t)) + + Simulate the filter with `lsim`. + + >>> tout, yout, xout = lsim((b, a), U=u, T=t) + + Plot the result. + + >>> plt.plot(t, u, 'r', alpha=0.5, linewidth=1, label='input') + >>> plt.plot(tout, yout, 'k', linewidth=1.5, label='output') + >>> plt.legend(loc='best', shadow=True, framealpha=1) + >>> plt.grid(alpha=0.3) + >>> plt.xlabel('t') + >>> plt.show() + + In a second example, we simulate a double integrator ``y'' = u``, with + a constant input ``u = 1``. We'll use the state space representation + of the integrator. + + >>> from scipy.signal import lti + >>> A = np.array([[0.0, 1.0], [0.0, 0.0]]) + >>> B = np.array([[0.0], [1.0]]) + >>> C = np.array([[1.0, 0.0]]) + >>> D = 0.0 + >>> system = lti(A, B, C, D) + + `t` and `u` define the time and input signal for the system to + be simulated. + + >>> t = np.linspace(0, 5, num=50) + >>> u = np.ones_like(t) + + Compute the simulation, and then plot `y`. As expected, the plot shows + the curve ``y = 0.5*t**2``. + + >>> tout, y, x = lsim(system, u, t) + >>> plt.plot(t, y) + >>> plt.grid(alpha=0.3) + >>> plt.xlabel('t') + >>> plt.show() + + """ + if isinstance(system, lti): + sys = system._as_ss() + elif isinstance(system, dlti): + raise AttributeError('lsim can only be used with continuous-time ' + 'systems.') + else: + sys = lti(*system)._as_ss() + T = atleast_1d(T) + if len(T.shape) != 1: + raise ValueError("T must be a rank-1 array.") + + A, B, C, D = map(np.asarray, (sys.A, sys.B, sys.C, sys.D)) + n_states = A.shape[0] + n_inputs = B.shape[1] + + n_steps = T.size + if X0 is None: + X0 = zeros(n_states, sys.A.dtype) + xout = np.empty((n_steps, n_states), sys.A.dtype) + + if T[0] == 0: + xout[0] = X0 + elif T[0] > 0: + # step forward to initial time, with zero input + xout[0] = dot(X0, linalg.expm(transpose(A) * T[0])) + else: + raise ValueError("Initial time must be nonnegative") + + no_input = (U is None or + (isinstance(U, (int, float)) and U == 0.) or + not np.any(U)) + + if n_steps == 1: + yout = squeeze(xout @ C.T) + if not no_input: + yout += squeeze(U @ D.T) + return T, yout, squeeze(xout) + + dt = T[1] - T[0] + if not np.allclose(np.diff(T), dt): + raise ValueError("Time steps are not equally spaced.") + + if no_input: + # Zero input: just use matrix exponential + # take transpose because state is a row vector + expAT_dt = linalg.expm(A.T * dt) + for i in range(1, n_steps): + xout[i] = xout[i-1] @ expAT_dt + yout = squeeze(xout @ C.T) + return T, yout, squeeze(xout) + + # Nonzero input + U = atleast_1d(U) + if U.ndim == 1: + U = U[:, np.newaxis] + + if U.shape[0] != n_steps: + raise ValueError("U must have the same number of rows " + "as elements in T.") + + if U.shape[1] != n_inputs: + raise ValueError("System does not define that many inputs.") + + if not interp: + # Zero-order hold + # Algorithm: to integrate from time 0 to time dt, we solve + # xdot = A x + B u, x(0) = x0 + # udot = 0, u(0) = u0. + # + # Solution is + # [ x(dt) ] [ A*dt B*dt ] [ x0 ] + # [ u(dt) ] = exp [ 0 0 ] [ u0 ] + M = np.vstack([np.hstack([A * dt, B * dt]), + np.zeros((n_inputs, n_states + n_inputs))]) + # transpose everything because the state and input are row vectors + expMT = linalg.expm(M.T) + Ad = expMT[:n_states, :n_states] + Bd = expMT[n_states:, :n_states] + for i in range(1, n_steps): + xout[i] = xout[i-1] @ Ad + U[i-1] @ Bd + else: + # Linear interpolation between steps + # Algorithm: to integrate from time 0 to time dt, with linear + # interpolation between inputs u(0) = u0 and u(dt) = u1, we solve + # xdot = A x + B u, x(0) = x0 + # udot = (u1 - u0) / dt, u(0) = u0. + # + # Solution is + # [ x(dt) ] [ A*dt B*dt 0 ] [ x0 ] + # [ u(dt) ] = exp [ 0 0 I ] [ u0 ] + # [u1 - u0] [ 0 0 0 ] [u1 - u0] + M = np.vstack([np.hstack([A * dt, B * dt, + np.zeros((n_states, n_inputs))]), + np.hstack([np.zeros((n_inputs, n_states + n_inputs)), + np.identity(n_inputs)]), + np.zeros((n_inputs, n_states + 2 * n_inputs))]) + expMT = linalg.expm(M.T) + Ad = expMT[:n_states, :n_states] + Bd1 = expMT[n_states+n_inputs:, :n_states] + Bd0 = expMT[n_states:n_states + n_inputs, :n_states] - Bd1 + for i in range(1, n_steps): + xout[i] = xout[i-1] @ Ad + U[i-1] @ Bd0 + U[i] @ Bd1 + + yout = squeeze(xout @ C.T) + squeeze(U @ D.T) + return T, yout, squeeze(xout) + + +def _default_response_times(A, n): + """Compute a reasonable set of time samples for the response time. + + This function is used by `impulse` and `step` to compute the response time + when the `T` argument to the function is None. + + Parameters + ---------- + A : array_like + The system matrix, which is square. + n : int + The number of time samples to generate. + + Returns + ------- + t : ndarray + The 1-D array of length `n` of time samples at which the response + is to be computed. + """ + # Create a reasonable time interval. + # TODO: This could use some more work. + # For example, what is expected when the system is unstable? + vals = linalg.eigvals(A) + r = min(abs(real(vals))) + if r == 0.0: + r = 1.0 + tc = 1.0 / r + t = linspace(0.0, 7 * tc, n) + return t + + +def impulse(system, X0=None, T=None, N=None): + """Impulse response of continuous-time system. + + Parameters + ---------- + system : an instance of the LTI class or a tuple of array_like + describing the system. + The following gives the number of elements in the tuple and + the interpretation: + + * 1 (instance of `lti`) + * 2 (num, den) + * 3 (zeros, poles, gain) + * 4 (A, B, C, D) + + X0 : array_like, optional + Initial state-vector. Defaults to zero. + T : array_like, optional + Time points. Computed if not given. + N : int, optional + The number of time points to compute (if `T` is not given). + + Returns + ------- + T : ndarray + A 1-D array of time points. + yout : ndarray + A 1-D array containing the impulse response of the system (except for + singularities at zero). + + Notes + ----- + If (num, den) is passed in for ``system``, coefficients for both the + numerator and denominator should be specified in descending exponent + order (e.g. ``s^2 + 3s + 5`` would be represented as ``[1, 3, 5]``). + + Examples + -------- + Compute the impulse response of a second order system with a repeated + root: ``x''(t) + 2*x'(t) + x(t) = u(t)`` + + >>> from scipy import signal + >>> system = ([1.0], [1.0, 2.0, 1.0]) + >>> t, y = signal.impulse(system) + >>> import matplotlib.pyplot as plt + >>> plt.plot(t, y) + + """ + if isinstance(system, lti): + sys = system._as_ss() + elif isinstance(system, dlti): + raise AttributeError('impulse can only be used with continuous-time ' + 'systems.') + else: + sys = lti(*system)._as_ss() + if X0 is None: + X = squeeze(sys.B) + else: + X = squeeze(sys.B + X0) + if N is None: + N = 100 + if T is None: + T = _default_response_times(sys.A, N) + else: + T = asarray(T) + + _, h, _ = lsim(sys, 0., T, X, interp=False) + return T, h + + +def step(system, X0=None, T=None, N=None): + """Step response of continuous-time system. + + Parameters + ---------- + system : an instance of the LTI class or a tuple of array_like + describing the system. + The following gives the number of elements in the tuple and + the interpretation: + + * 1 (instance of `lti`) + * 2 (num, den) + * 3 (zeros, poles, gain) + * 4 (A, B, C, D) + + X0 : array_like, optional + Initial state-vector (default is zero). + T : array_like, optional + Time points (computed if not given). + N : int, optional + Number of time points to compute if `T` is not given. + + Returns + ------- + T : 1D ndarray + Output time points. + yout : 1D ndarray + Step response of system. + + + Notes + ----- + If (num, den) is passed in for ``system``, coefficients for both the + numerator and denominator should be specified in descending exponent + order (e.g. ``s^2 + 3s + 5`` would be represented as ``[1, 3, 5]``). + + Examples + -------- + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> lti = signal.lti([1.0], [1.0, 1.0]) + >>> t, y = signal.step(lti) + >>> plt.plot(t, y) + >>> plt.xlabel('Time [s]') + >>> plt.ylabel('Amplitude') + >>> plt.title('Step response for 1. Order Lowpass') + >>> plt.grid() + + """ + if isinstance(system, lti): + sys = system._as_ss() + elif isinstance(system, dlti): + raise AttributeError('step can only be used with continuous-time ' + 'systems.') + else: + sys = lti(*system)._as_ss() + if N is None: + N = 100 + if T is None: + T = _default_response_times(sys.A, N) + else: + T = asarray(T) + U = ones(T.shape, sys.A.dtype) + vals = lsim(sys, U, T, X0=X0, interp=False) + return vals[0], vals[1] + + +def bode(system, w=None, n=100): + """ + Calculate Bode magnitude and phase data of a continuous-time system. + + Parameters + ---------- + system : an instance of the LTI class or a tuple describing the system. + The following gives the number of elements in the tuple and + the interpretation: + + * 1 (instance of `lti`) + * 2 (num, den) + * 3 (zeros, poles, gain) + * 4 (A, B, C, D) + + w : array_like, optional + Array of frequencies (in rad/s). Magnitude and phase data is calculated + for every value in this array. If not given a reasonable set will be + calculated. + n : int, optional + Number of frequency points to compute if `w` is not given. The `n` + frequencies are logarithmically spaced in an interval chosen to + include the influence of the poles and zeros of the system. + + Returns + ------- + w : 1D ndarray + Frequency array [rad/s] + mag : 1D ndarray + Magnitude array [dB] + phase : 1D ndarray + Phase array [deg] + + Notes + ----- + If (num, den) is passed in for ``system``, coefficients for both the + numerator and denominator should be specified in descending exponent + order (e.g. ``s^2 + 3s + 5`` would be represented as ``[1, 3, 5]``). + + .. versionadded:: 0.11.0 + + Examples + -------- + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> sys = signal.TransferFunction([1], [1, 1]) + >>> w, mag, phase = signal.bode(sys) + + >>> plt.figure() + >>> plt.semilogx(w, mag) # Bode magnitude plot + >>> plt.figure() + >>> plt.semilogx(w, phase) # Bode phase plot + >>> plt.show() + + """ + w, y = freqresp(system, w=w, n=n) + + mag = 20.0 * np.log10(abs(y)) + phase = np.unwrap(np.arctan2(y.imag, y.real)) * 180.0 / np.pi + + return w, mag, phase + + +def freqresp(system, w=None, n=10000): + r"""Calculate the frequency response of a continuous-time system. + + Parameters + ---------- + system : an instance of the `lti` class or a tuple describing the system. + The following gives the number of elements in the tuple and + the interpretation: + + * 1 (instance of `lti`) + * 2 (num, den) + * 3 (zeros, poles, gain) + * 4 (A, B, C, D) + + w : array_like, optional + Array of frequencies (in rad/s). Magnitude and phase data is + calculated for every value in this array. If not given, a reasonable + set will be calculated. + n : int, optional + Number of frequency points to compute if `w` is not given. The `n` + frequencies are logarithmically spaced in an interval chosen to + include the influence of the poles and zeros of the system. + + Returns + ------- + w : 1D ndarray + Frequency array [rad/s] + H : 1D ndarray + Array of complex magnitude values + + Notes + ----- + If (num, den) is passed in for ``system``, coefficients for both the + numerator and denominator should be specified in descending exponent + order (e.g. ``s^2 + 3s + 5`` would be represented as ``[1, 3, 5]``). + + Examples + -------- + Generating the Nyquist plot of a transfer function + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + Construct the transfer function :math:`H(s) = \frac{5}{(s-1)^3}`: + + >>> s1 = signal.ZerosPolesGain([], [1, 1, 1], [5]) + + >>> w, H = signal.freqresp(s1) + + >>> plt.figure() + >>> plt.plot(H.real, H.imag, "b") + >>> plt.plot(H.real, -H.imag, "r") + >>> plt.show() + """ + if isinstance(system, lti): + if isinstance(system, (TransferFunction, ZerosPolesGain)): + sys = system + else: + sys = system._as_zpk() + elif isinstance(system, dlti): + raise AttributeError('freqresp can only be used with continuous-time ' + 'systems.') + else: + sys = lti(*system)._as_zpk() + + if sys.inputs != 1 or sys.outputs != 1: + raise ValueError("freqresp() requires a SISO (single input, single " + "output) system.") + + if w is not None: + worN = w + else: + worN = n + + if isinstance(sys, TransferFunction): + # In the call to freqs(), sys.num.ravel() is used because there are + # cases where sys.num is a 2-D array with a single row. + w, h = freqs(sys.num.ravel(), sys.den, worN=worN) + + elif isinstance(sys, ZerosPolesGain): + w, h = freqs_zpk(sys.zeros, sys.poles, sys.gain, worN=worN) + + return w, h + + +# This class will be used by place_poles to return its results +# see https://code.activestate.com/recipes/52308/ +class Bunch: + def __init__(self, **kwds): + self.__dict__.update(kwds) + + +def _valid_inputs(A, B, poles, method, rtol, maxiter): + """ + Check the poles come in complex conjugate pairs + Check shapes of A, B and poles are compatible. + Check the method chosen is compatible with provided poles + Return update method to use and ordered poles + + """ + poles = np.asarray(poles) + if poles.ndim > 1: + raise ValueError("Poles must be a 1D array like.") + # Will raise ValueError if poles do not come in complex conjugates pairs + poles = _order_complex_poles(poles) + if A.ndim > 2: + raise ValueError("A must be a 2D array/matrix.") + if B.ndim > 2: + raise ValueError("B must be a 2D array/matrix") + if A.shape[0] != A.shape[1]: + raise ValueError("A must be square") + if len(poles) > A.shape[0]: + raise ValueError("maximum number of poles is %d but you asked for %d" % + (A.shape[0], len(poles))) + if len(poles) < A.shape[0]: + raise ValueError("number of poles is %d but you should provide %d" % + (len(poles), A.shape[0])) + r = np.linalg.matrix_rank(B) + for p in poles: + if sum(p == poles) > r: + raise ValueError("at least one of the requested pole is repeated " + "more than rank(B) times") + # Choose update method + update_loop = _YT_loop + if method not in ('KNV0','YT'): + raise ValueError("The method keyword must be one of 'YT' or 'KNV0'") + + if method == "KNV0": + update_loop = _KNV0_loop + if not all(np.isreal(poles)): + raise ValueError("Complex poles are not supported by KNV0") + + if maxiter < 1: + raise ValueError("maxiter must be at least equal to 1") + + # We do not check rtol <= 0 as the user can use a negative rtol to + # force maxiter iterations + if rtol > 1: + raise ValueError("rtol can not be greater than 1") + + return update_loop, poles + + +def _order_complex_poles(poles): + """ + Check we have complex conjugates pairs and reorder P according to YT, ie + real_poles, complex_i, conjugate complex_i, .... + The lexicographic sort on the complex poles is added to help the user to + compare sets of poles. + """ + ordered_poles = np.sort(poles[np.isreal(poles)]) + im_poles = [] + for p in np.sort(poles[np.imag(poles) < 0]): + if np.conj(p) in poles: + im_poles.extend((p, np.conj(p))) + + ordered_poles = np.hstack((ordered_poles, im_poles)) + + if poles.shape[0] != len(ordered_poles): + raise ValueError("Complex poles must come with their conjugates") + return ordered_poles + + +def _KNV0(B, ker_pole, transfer_matrix, j, poles): + """ + Algorithm "KNV0" Kautsky et Al. Robust pole + assignment in linear state feedback, Int journal of Control + 1985, vol 41 p 1129->1155 + https://la.epfl.ch/files/content/sites/la/files/ + users/105941/public/KautskyNicholsDooren + + """ + # Remove xj form the base + transfer_matrix_not_j = np.delete(transfer_matrix, j, axis=1) + # If we QR this matrix in full mode Q=Q0|Q1 + # then Q1 will be a single column orthogonal to + # Q0, that's what we are looking for ! + + # After merge of gh-4249 great speed improvements could be achieved + # using QR updates instead of full QR in the line below + + # To debug with numpy qr uncomment the line below + # Q, R = np.linalg.qr(transfer_matrix_not_j, mode="complete") + Q, R = s_qr(transfer_matrix_not_j, mode="full") + + mat_ker_pj = np.dot(ker_pole[j], ker_pole[j].T) + yj = np.dot(mat_ker_pj, Q[:, -1]) + + # If Q[:, -1] is "almost" orthogonal to ker_pole[j] its + # projection into ker_pole[j] will yield a vector + # close to 0. As we are looking for a vector in ker_pole[j] + # simply stick with transfer_matrix[:, j] (unless someone provides me with + # a better choice ?) + + if not np.allclose(yj, 0): + xj = yj/np.linalg.norm(yj) + transfer_matrix[:, j] = xj + + # KNV does not support complex poles, using YT technique the two lines + # below seem to work 9 out of 10 times but it is not reliable enough: + # transfer_matrix[:, j]=real(xj) + # transfer_matrix[:, j+1]=imag(xj) + + # Add this at the beginning of this function if you wish to test + # complex support: + # if ~np.isreal(P[j]) and (j>=B.shape[0]-1 or P[j]!=np.conj(P[j+1])): + # return + # Problems arise when imag(xj)=>0 I have no idea on how to fix this + + +def _YT_real(ker_pole, Q, transfer_matrix, i, j): + """ + Applies algorithm from YT section 6.1 page 19 related to real pairs + """ + # step 1 page 19 + u = Q[:, -2, np.newaxis] + v = Q[:, -1, np.newaxis] + + # step 2 page 19 + m = np.dot(np.dot(ker_pole[i].T, np.dot(u, v.T) - + np.dot(v, u.T)), ker_pole[j]) + + # step 3 page 19 + um, sm, vm = np.linalg.svd(m) + # mu1, mu2 two first columns of U => 2 first lines of U.T + mu1, mu2 = um.T[:2, :, np.newaxis] + # VM is V.T with numpy we want the first two lines of V.T + nu1, nu2 = vm[:2, :, np.newaxis] + + # what follows is a rough python translation of the formulas + # in section 6.2 page 20 (step 4) + transfer_matrix_j_mo_transfer_matrix_j = np.vstack(( + transfer_matrix[:, i, np.newaxis], + transfer_matrix[:, j, np.newaxis])) + + if not np.allclose(sm[0], sm[1]): + ker_pole_imo_mu1 = np.dot(ker_pole[i], mu1) + ker_pole_i_nu1 = np.dot(ker_pole[j], nu1) + ker_pole_mu_nu = np.vstack((ker_pole_imo_mu1, ker_pole_i_nu1)) + else: + ker_pole_ij = np.vstack(( + np.hstack((ker_pole[i], + np.zeros(ker_pole[i].shape))), + np.hstack((np.zeros(ker_pole[j].shape), + ker_pole[j])) + )) + mu_nu_matrix = np.vstack( + (np.hstack((mu1, mu2)), np.hstack((nu1, nu2))) + ) + ker_pole_mu_nu = np.dot(ker_pole_ij, mu_nu_matrix) + transfer_matrix_ij = np.dot(np.dot(ker_pole_mu_nu, ker_pole_mu_nu.T), + transfer_matrix_j_mo_transfer_matrix_j) + if not np.allclose(transfer_matrix_ij, 0): + transfer_matrix_ij = (np.sqrt(2)*transfer_matrix_ij / + np.linalg.norm(transfer_matrix_ij)) + transfer_matrix[:, i] = transfer_matrix_ij[ + :transfer_matrix[:, i].shape[0], 0 + ] + transfer_matrix[:, j] = transfer_matrix_ij[ + transfer_matrix[:, i].shape[0]:, 0 + ] + else: + # As in knv0 if transfer_matrix_j_mo_transfer_matrix_j is orthogonal to + # Vect{ker_pole_mu_nu} assign transfer_matrixi/transfer_matrix_j to + # ker_pole_mu_nu and iterate. As we are looking for a vector in + # Vect{Matker_pole_MU_NU} (see section 6.1 page 19) this might help + # (that's a guess, not a claim !) + transfer_matrix[:, i] = ker_pole_mu_nu[ + :transfer_matrix[:, i].shape[0], 0 + ] + transfer_matrix[:, j] = ker_pole_mu_nu[ + transfer_matrix[:, i].shape[0]:, 0 + ] + + +def _YT_complex(ker_pole, Q, transfer_matrix, i, j): + """ + Applies algorithm from YT section 6.2 page 20 related to complex pairs + """ + # step 1 page 20 + ur = np.sqrt(2)*Q[:, -2, np.newaxis] + ui = np.sqrt(2)*Q[:, -1, np.newaxis] + u = ur + 1j*ui + + # step 2 page 20 + ker_pole_ij = ker_pole[i] + m = np.dot(np.dot(np.conj(ker_pole_ij.T), np.dot(u, np.conj(u).T) - + np.dot(np.conj(u), u.T)), ker_pole_ij) + + # step 3 page 20 + e_val, e_vec = np.linalg.eig(m) + # sort eigenvalues according to their module + e_val_idx = np.argsort(np.abs(e_val)) + mu1 = e_vec[:, e_val_idx[-1], np.newaxis] + mu2 = e_vec[:, e_val_idx[-2], np.newaxis] + + # what follows is a rough python translation of the formulas + # in section 6.2 page 20 (step 4) + + # remember transfer_matrix_i has been split as + # transfer_matrix[i]=real(transfer_matrix_i) and + # transfer_matrix[j]=imag(transfer_matrix_i) + transfer_matrix_j_mo_transfer_matrix_j = ( + transfer_matrix[:, i, np.newaxis] + + 1j*transfer_matrix[:, j, np.newaxis] + ) + if not np.allclose(np.abs(e_val[e_val_idx[-1]]), + np.abs(e_val[e_val_idx[-2]])): + ker_pole_mu = np.dot(ker_pole_ij, mu1) + else: + mu1_mu2_matrix = np.hstack((mu1, mu2)) + ker_pole_mu = np.dot(ker_pole_ij, mu1_mu2_matrix) + transfer_matrix_i_j = np.dot(np.dot(ker_pole_mu, np.conj(ker_pole_mu.T)), + transfer_matrix_j_mo_transfer_matrix_j) + + if not np.allclose(transfer_matrix_i_j, 0): + transfer_matrix_i_j = (transfer_matrix_i_j / + np.linalg.norm(transfer_matrix_i_j)) + transfer_matrix[:, i] = np.real(transfer_matrix_i_j[:, 0]) + transfer_matrix[:, j] = np.imag(transfer_matrix_i_j[:, 0]) + else: + # same idea as in YT_real + transfer_matrix[:, i] = np.real(ker_pole_mu[:, 0]) + transfer_matrix[:, j] = np.imag(ker_pole_mu[:, 0]) + + +def _YT_loop(ker_pole, transfer_matrix, poles, B, maxiter, rtol): + """ + Algorithm "YT" Tits, Yang. Globally Convergent + Algorithms for Robust Pole Assignment by State Feedback + https://hdl.handle.net/1903/5598 + The poles P have to be sorted accordingly to section 6.2 page 20 + + """ + # The IEEE edition of the YT paper gives useful information on the + # optimal update order for the real poles in order to minimize the number + # of times we have to loop over all poles, see page 1442 + nb_real = poles[np.isreal(poles)].shape[0] + # hnb => Half Nb Real + hnb = nb_real // 2 + + # Stick to the indices in the paper and then remove one to get numpy array + # index it is a bit easier to link the code to the paper this way even if it + # is not very clean. The paper is unclear about what should be done when + # there is only one real pole => use KNV0 on this real pole seem to work + if nb_real > 0: + #update the biggest real pole with the smallest one + update_order = [[nb_real], [1]] + else: + update_order = [[],[]] + + r_comp = np.arange(nb_real+1, len(poles)+1, 2) + # step 1.a + r_p = np.arange(1, hnb+nb_real % 2) + update_order[0].extend(2*r_p) + update_order[1].extend(2*r_p+1) + # step 1.b + update_order[0].extend(r_comp) + update_order[1].extend(r_comp+1) + # step 1.c + r_p = np.arange(1, hnb+1) + update_order[0].extend(2*r_p-1) + update_order[1].extend(2*r_p) + # step 1.d + if hnb == 0 and np.isreal(poles[0]): + update_order[0].append(1) + update_order[1].append(1) + update_order[0].extend(r_comp) + update_order[1].extend(r_comp+1) + # step 2.a + r_j = np.arange(2, hnb+nb_real % 2) + for j in r_j: + for i in range(1, hnb+1): + update_order[0].append(i) + update_order[1].append(i+j) + # step 2.b + if hnb == 0 and np.isreal(poles[0]): + update_order[0].append(1) + update_order[1].append(1) + update_order[0].extend(r_comp) + update_order[1].extend(r_comp+1) + # step 2.c + r_j = np.arange(2, hnb+nb_real % 2) + for j in r_j: + for i in range(hnb+1, nb_real+1): + idx_1 = i+j + if idx_1 > nb_real: + idx_1 = i+j-nb_real + update_order[0].append(i) + update_order[1].append(idx_1) + # step 2.d + if hnb == 0 and np.isreal(poles[0]): + update_order[0].append(1) + update_order[1].append(1) + update_order[0].extend(r_comp) + update_order[1].extend(r_comp+1) + # step 3.a + for i in range(1, hnb+1): + update_order[0].append(i) + update_order[1].append(i+hnb) + # step 3.b + if hnb == 0 and np.isreal(poles[0]): + update_order[0].append(1) + update_order[1].append(1) + update_order[0].extend(r_comp) + update_order[1].extend(r_comp+1) + + update_order = np.array(update_order).T-1 + stop = False + nb_try = 0 + while nb_try < maxiter and not stop: + det_transfer_matrixb = np.abs(np.linalg.det(transfer_matrix)) + for i, j in update_order: + if i == j: + assert i == 0, "i!=0 for KNV call in YT" + assert np.isreal(poles[i]), "calling KNV on a complex pole" + _KNV0(B, ker_pole, transfer_matrix, i, poles) + else: + transfer_matrix_not_i_j = np.delete(transfer_matrix, (i, j), + axis=1) + # after merge of gh-4249 great speed improvements could be + # achieved using QR updates instead of full QR in the line below + + #to debug with numpy qr uncomment the line below + #Q, _ = np.linalg.qr(transfer_matrix_not_i_j, mode="complete") + Q, _ = s_qr(transfer_matrix_not_i_j, mode="full") + + if np.isreal(poles[i]): + assert np.isreal(poles[j]), "mixing real and complex " + \ + "in YT_real" + str(poles) + _YT_real(ker_pole, Q, transfer_matrix, i, j) + else: + assert ~np.isreal(poles[i]), "mixing real and complex " + \ + "in YT_real" + str(poles) + _YT_complex(ker_pole, Q, transfer_matrix, i, j) + + det_transfer_matrix = np.max((np.sqrt(np.spacing(1)), + np.abs(np.linalg.det(transfer_matrix)))) + cur_rtol = np.abs( + (det_transfer_matrix - + det_transfer_matrixb) / + det_transfer_matrix) + if cur_rtol < rtol and det_transfer_matrix > np.sqrt(np.spacing(1)): + # Convergence test from YT page 21 + stop = True + nb_try += 1 + return stop, cur_rtol, nb_try + + +def _KNV0_loop(ker_pole, transfer_matrix, poles, B, maxiter, rtol): + """ + Loop over all poles one by one and apply KNV method 0 algorithm + """ + # This method is useful only because we need to be able to call + # _KNV0 from YT without looping over all poles, otherwise it would + # have been fine to mix _KNV0_loop and _KNV0 in a single function + stop = False + nb_try = 0 + while nb_try < maxiter and not stop: + det_transfer_matrixb = np.abs(np.linalg.det(transfer_matrix)) + for j in range(B.shape[0]): + _KNV0(B, ker_pole, transfer_matrix, j, poles) + + det_transfer_matrix = np.max((np.sqrt(np.spacing(1)), + np.abs(np.linalg.det(transfer_matrix)))) + cur_rtol = np.abs((det_transfer_matrix - det_transfer_matrixb) / + det_transfer_matrix) + if cur_rtol < rtol and det_transfer_matrix > np.sqrt(np.spacing(1)): + # Convergence test from YT page 21 + stop = True + + nb_try += 1 + return stop, cur_rtol, nb_try + + +def place_poles(A, B, poles, method="YT", rtol=1e-3, maxiter=30): + """ + Compute K such that eigenvalues (A - dot(B, K))=poles. + + K is the gain matrix such as the plant described by the linear system + ``AX+BU`` will have its closed-loop poles, i.e the eigenvalues ``A - B*K``, + as close as possible to those asked for in poles. + + SISO, MISO and MIMO systems are supported. + + Parameters + ---------- + A, B : ndarray + State-space representation of linear system ``AX + BU``. + poles : array_like + Desired real poles and/or complex conjugates poles. + Complex poles are only supported with ``method="YT"`` (default). + method: {'YT', 'KNV0'}, optional + Which method to choose to find the gain matrix K. One of: + + - 'YT': Yang Tits + - 'KNV0': Kautsky, Nichols, Van Dooren update method 0 + + See References and Notes for details on the algorithms. + rtol: float, optional + After each iteration the determinant of the eigenvectors of + ``A - B*K`` is compared to its previous value, when the relative + error between these two values becomes lower than `rtol` the algorithm + stops. Default is 1e-3. + maxiter: int, optional + Maximum number of iterations to compute the gain matrix. + Default is 30. + + Returns + ------- + full_state_feedback : Bunch object + full_state_feedback is composed of: + gain_matrix : 1-D ndarray + The closed loop matrix K such as the eigenvalues of ``A-BK`` + are as close as possible to the requested poles. + computed_poles : 1-D ndarray + The poles corresponding to ``A-BK`` sorted as first the real + poles in increasing order, then the complex conjugates in + lexicographic order. + requested_poles : 1-D ndarray + The poles the algorithm was asked to place sorted as above, + they may differ from what was achieved. + X : 2-D ndarray + The transfer matrix such as ``X * diag(poles) = (A - B*K)*X`` + (see Notes) + rtol : float + The relative tolerance achieved on ``det(X)`` (see Notes). + `rtol` will be NaN if it is possible to solve the system + ``diag(poles) = (A - B*K)``, or 0 when the optimization + algorithms can't do anything i.e when ``B.shape[1] == 1``. + nb_iter : int + The number of iterations performed before converging. + `nb_iter` will be NaN if it is possible to solve the system + ``diag(poles) = (A - B*K)``, or 0 when the optimization + algorithms can't do anything i.e when ``B.shape[1] == 1``. + + Notes + ----- + The Tits and Yang (YT), [2]_ paper is an update of the original Kautsky et + al. (KNV) paper [1]_. KNV relies on rank-1 updates to find the transfer + matrix X such that ``X * diag(poles) = (A - B*K)*X``, whereas YT uses + rank-2 updates. This yields on average more robust solutions (see [2]_ + pp 21-22), furthermore the YT algorithm supports complex poles whereas KNV + does not in its original version. Only update method 0 proposed by KNV has + been implemented here, hence the name ``'KNV0'``. + + KNV extended to complex poles is used in Matlab's ``place`` function, YT is + distributed under a non-free licence by Slicot under the name ``robpole``. + It is unclear and undocumented how KNV0 has been extended to complex poles + (Tits and Yang claim on page 14 of their paper that their method can not be + used to extend KNV to complex poles), therefore only YT supports them in + this implementation. + + As the solution to the problem of pole placement is not unique for MIMO + systems, both methods start with a tentative transfer matrix which is + altered in various way to increase its determinant. Both methods have been + proven to converge to a stable solution, however depending on the way the + initial transfer matrix is chosen they will converge to different + solutions and therefore there is absolutely no guarantee that using + ``'KNV0'`` will yield results similar to Matlab's or any other + implementation of these algorithms. + + Using the default method ``'YT'`` should be fine in most cases; ``'KNV0'`` + is only provided because it is needed by ``'YT'`` in some specific cases. + Furthermore ``'YT'`` gives on average more robust results than ``'KNV0'`` + when ``abs(det(X))`` is used as a robustness indicator. + + [2]_ is available as a technical report on the following URL: + https://hdl.handle.net/1903/5598 + + References + ---------- + .. [1] J. Kautsky, N.K. Nichols and P. van Dooren, "Robust pole assignment + in linear state feedback", International Journal of Control, Vol. 41 + pp. 1129-1155, 1985. + .. [2] A.L. Tits and Y. Yang, "Globally convergent algorithms for robust + pole assignment by state feedback", IEEE Transactions on Automatic + Control, Vol. 41, pp. 1432-1452, 1996. + + Examples + -------- + A simple example demonstrating real pole placement using both KNV and YT + algorithms. This is example number 1 from section 4 of the reference KNV + publication ([1]_): + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> A = np.array([[ 1.380, -0.2077, 6.715, -5.676 ], + ... [-0.5814, -4.290, 0, 0.6750 ], + ... [ 1.067, 4.273, -6.654, 5.893 ], + ... [ 0.0480, 4.273, 1.343, -2.104 ]]) + >>> B = np.array([[ 0, 5.679 ], + ... [ 1.136, 1.136 ], + ... [ 0, 0, ], + ... [-3.146, 0 ]]) + >>> P = np.array([-0.2, -0.5, -5.0566, -8.6659]) + + Now compute K with KNV method 0, with the default YT method and with the YT + method while forcing 100 iterations of the algorithm and print some results + after each call. + + >>> fsf1 = signal.place_poles(A, B, P, method='KNV0') + >>> fsf1.gain_matrix + array([[ 0.20071427, -0.96665799, 0.24066128, -0.10279785], + [ 0.50587268, 0.57779091, 0.51795763, -0.41991442]]) + + >>> fsf2 = signal.place_poles(A, B, P) # uses YT method + >>> fsf2.computed_poles + array([-8.6659, -5.0566, -0.5 , -0.2 ]) + + >>> fsf3 = signal.place_poles(A, B, P, rtol=-1, maxiter=100) + >>> fsf3.X + array([[ 0.52072442+0.j, -0.08409372+0.j, -0.56847937+0.j, 0.74823657+0.j], + [-0.04977751+0.j, -0.80872954+0.j, 0.13566234+0.j, -0.29322906+0.j], + [-0.82266932+0.j, -0.19168026+0.j, -0.56348322+0.j, -0.43815060+0.j], + [ 0.22267347+0.j, 0.54967577+0.j, -0.58387806+0.j, -0.40271926+0.j]]) + + The absolute value of the determinant of X is a good indicator to check the + robustness of the results, both ``'KNV0'`` and ``'YT'`` aim at maximizing + it. Below a comparison of the robustness of the results above: + + >>> abs(np.linalg.det(fsf1.X)) < abs(np.linalg.det(fsf2.X)) + True + >>> abs(np.linalg.det(fsf2.X)) < abs(np.linalg.det(fsf3.X)) + True + + Now a simple example for complex poles: + + >>> A = np.array([[ 0, 7/3., 0, 0 ], + ... [ 0, 0, 0, 7/9. ], + ... [ 0, 0, 0, 0 ], + ... [ 0, 0, 0, 0 ]]) + >>> B = np.array([[ 0, 0 ], + ... [ 0, 0 ], + ... [ 1, 0 ], + ... [ 0, 1 ]]) + >>> P = np.array([-3, -1, -2-1j, -2+1j]) / 3. + >>> fsf = signal.place_poles(A, B, P, method='YT') + + We can plot the desired and computed poles in the complex plane: + + >>> t = np.linspace(0, 2*np.pi, 401) + >>> plt.plot(np.cos(t), np.sin(t), 'k--') # unit circle + >>> plt.plot(fsf.requested_poles.real, fsf.requested_poles.imag, + ... 'wo', label='Desired') + >>> plt.plot(fsf.computed_poles.real, fsf.computed_poles.imag, 'bx', + ... label='Placed') + >>> plt.grid() + >>> plt.axis('image') + >>> plt.axis([-1.1, 1.1, -1.1, 1.1]) + >>> plt.legend(bbox_to_anchor=(1.05, 1), loc=2, numpoints=1) + + """ + # Move away all the inputs checking, it only adds noise to the code + update_loop, poles = _valid_inputs(A, B, poles, method, rtol, maxiter) + + # The current value of the relative tolerance we achieved + cur_rtol = 0 + # The number of iterations needed before converging + nb_iter = 0 + + # Step A: QR decomposition of B page 1132 KN + # to debug with numpy qr uncomment the line below + # u, z = np.linalg.qr(B, mode="complete") + u, z = s_qr(B, mode="full") + rankB = np.linalg.matrix_rank(B) + u0 = u[:, :rankB] + u1 = u[:, rankB:] + z = z[:rankB, :] + + # If we can use the identity matrix as X the solution is obvious + if B.shape[0] == rankB: + # if B is square and full rank there is only one solution + # such as (A+BK)=inv(X)*diag(P)*X with X=eye(A.shape[0]) + # i.e K=inv(B)*(diag(P)-A) + # if B has as many lines as its rank (but not square) there are many + # solutions and we can choose one using least squares + # => use lstsq in both cases. + # In both cases the transfer matrix X will be eye(A.shape[0]) and I + # can hardly think of a better one so there is nothing to optimize + # + # for complex poles we use the following trick + # + # |a -b| has for eigenvalues a+b and a-b + # |b a| + # + # |a+bi 0| has the obvious eigenvalues a+bi and a-bi + # |0 a-bi| + # + # e.g solving the first one in R gives the solution + # for the second one in C + diag_poles = np.zeros(A.shape) + idx = 0 + while idx < poles.shape[0]: + p = poles[idx] + diag_poles[idx, idx] = np.real(p) + if ~np.isreal(p): + diag_poles[idx, idx+1] = -np.imag(p) + diag_poles[idx+1, idx+1] = np.real(p) + diag_poles[idx+1, idx] = np.imag(p) + idx += 1 # skip next one + idx += 1 + gain_matrix = np.linalg.lstsq(B, diag_poles-A, rcond=-1)[0] + transfer_matrix = np.eye(A.shape[0]) + cur_rtol = np.nan + nb_iter = np.nan + else: + # step A (p1144 KNV) and beginning of step F: decompose + # dot(U1.T, A-P[i]*I).T and build our set of transfer_matrix vectors + # in the same loop + ker_pole = [] + + # flag to skip the conjugate of a complex pole + skip_conjugate = False + # select orthonormal base ker_pole for each Pole and vectors for + # transfer_matrix + for j in range(B.shape[0]): + if skip_conjugate: + skip_conjugate = False + continue + pole_space_j = np.dot(u1.T, A-poles[j]*np.eye(B.shape[0])).T + + # after QR Q=Q0|Q1 + # only Q0 is used to reconstruct the qr'ed (dot Q, R) matrix. + # Q1 is orthogonal to Q0 and will be multiplied by the zeros in + # R when using mode "complete". In default mode Q1 and the zeros + # in R are not computed + + # To debug with numpy qr uncomment the line below + # Q, _ = np.linalg.qr(pole_space_j, mode="complete") + Q, _ = s_qr(pole_space_j, mode="full") + + ker_pole_j = Q[:, pole_space_j.shape[1]:] + + # We want to select one vector in ker_pole_j to build the transfer + # matrix, however qr returns sometimes vectors with zeros on the + # same line for each pole and this yields very long convergence + # times. + # Or some other times a set of vectors, one with zero imaginary + # part and one (or several) with imaginary parts. After trying + # many ways to select the best possible one (eg ditch vectors + # with zero imaginary part for complex poles) I ended up summing + # all vectors in ker_pole_j, this solves 100% of the problems and + # is a valid choice for transfer_matrix. + # This way for complex poles we are sure to have a non zero + # imaginary part that way, and the problem of lines full of zeros + # in transfer_matrix is solved too as when a vector from + # ker_pole_j has a zero the other one(s) when + # ker_pole_j.shape[1]>1) for sure won't have a zero there. + + transfer_matrix_j = np.sum(ker_pole_j, axis=1)[:, np.newaxis] + transfer_matrix_j = (transfer_matrix_j / + np.linalg.norm(transfer_matrix_j)) + if ~np.isreal(poles[j]): # complex pole + transfer_matrix_j = np.hstack([np.real(transfer_matrix_j), + np.imag(transfer_matrix_j)]) + ker_pole.extend([ker_pole_j, ker_pole_j]) + + # Skip next pole as it is the conjugate + skip_conjugate = True + else: # real pole, nothing to do + ker_pole.append(ker_pole_j) + + if j == 0: + transfer_matrix = transfer_matrix_j + else: + transfer_matrix = np.hstack((transfer_matrix, transfer_matrix_j)) + + if rankB > 1: # otherwise there is nothing we can optimize + stop, cur_rtol, nb_iter = update_loop(ker_pole, transfer_matrix, + poles, B, maxiter, rtol) + if not stop and rtol > 0: + # if rtol<=0 the user has probably done that on purpose, + # don't annoy them + err_msg = ( + "Convergence was not reached after maxiter iterations.\n" + f"You asked for a tolerance of {rtol}, we got {cur_rtol}." + ) + warnings.warn(err_msg, stacklevel=2) + + # reconstruct transfer_matrix to match complex conjugate pairs, + # ie transfer_matrix_j/transfer_matrix_j+1 are + # Re(Complex_pole), Im(Complex_pole) now and will be Re-Im/Re+Im after + transfer_matrix = transfer_matrix.astype(complex) + idx = 0 + while idx < poles.shape[0]-1: + if ~np.isreal(poles[idx]): + rel = transfer_matrix[:, idx].copy() + img = transfer_matrix[:, idx+1] + # rel will be an array referencing a column of transfer_matrix + # if we don't copy() it will changer after the next line and + # and the line after will not yield the correct value + transfer_matrix[:, idx] = rel-1j*img + transfer_matrix[:, idx+1] = rel+1j*img + idx += 1 # skip next one + idx += 1 + + try: + m = np.linalg.solve(transfer_matrix.T, np.dot(np.diag(poles), + transfer_matrix.T)).T + gain_matrix = np.linalg.solve(z, np.dot(u0.T, m-A)) + except np.linalg.LinAlgError as e: + raise ValueError("The poles you've chosen can't be placed. " + "Check the controllability matrix and try " + "another set of poles") from e + + # Beware: Kautsky solves A+BK but the usual form is A-BK + gain_matrix = -gain_matrix + # K still contains complex with ~=0j imaginary parts, get rid of them + gain_matrix = np.real(gain_matrix) + + full_state_feedback = Bunch() + full_state_feedback.gain_matrix = gain_matrix + full_state_feedback.computed_poles = _order_complex_poles( + np.linalg.eig(A - np.dot(B, gain_matrix))[0] + ) + full_state_feedback.requested_poles = poles + full_state_feedback.X = transfer_matrix + full_state_feedback.rtol = cur_rtol + full_state_feedback.nb_iter = nb_iter + + return full_state_feedback + + +def dlsim(system, u, t=None, x0=None): + """ + Simulate output of a discrete-time linear system. + + Parameters + ---------- + system : tuple of array_like or instance of `dlti` + A tuple describing the system. + The following gives the number of elements in the tuple and + the interpretation: + + * 1: (instance of `dlti`) + * 3: (num, den, dt) + * 4: (zeros, poles, gain, dt) + * 5: (A, B, C, D, dt) + + u : array_like + An input array describing the input at each time `t` (interpolation is + assumed between given times). If there are multiple inputs, then each + column of the rank-2 array represents an input. + t : array_like, optional + The time steps at which the input is defined. If `t` is given, it + must be the same length as `u`, and the final value in `t` determines + the number of steps returned in the output. + x0 : array_like, optional + The initial conditions on the state vector (zero by default). + + Returns + ------- + tout : ndarray + Time values for the output, as a 1-D array. + yout : ndarray + System response, as a 1-D array. + xout : ndarray, optional + Time-evolution of the state-vector. Only generated if the input is a + `StateSpace` system. + + See Also + -------- + lsim, dstep, dimpulse, cont2discrete + + Examples + -------- + A simple integrator transfer function with a discrete time step of 1.0 + could be implemented as: + + >>> import numpy as np + >>> from scipy import signal + >>> tf = ([1.0,], [1.0, -1.0], 1.0) + >>> t_in = [0.0, 1.0, 2.0, 3.0] + >>> u = np.asarray([0.0, 0.0, 1.0, 1.0]) + >>> t_out, y = signal.dlsim(tf, u, t=t_in) + >>> y.T + array([[ 0., 0., 0., 1.]]) + + """ + # Convert system to dlti-StateSpace + if isinstance(system, lti): + raise AttributeError('dlsim can only be used with discrete-time dlti ' + 'systems.') + elif not isinstance(system, dlti): + system = dlti(*system[:-1], dt=system[-1]) + + # Condition needed to ensure output remains compatible + is_ss_input = isinstance(system, StateSpace) + system = system._as_ss() + + u = np.atleast_1d(u) + + if u.ndim == 1: + u = np.atleast_2d(u).T + + if t is None: + out_samples = len(u) + stoptime = (out_samples - 1) * system.dt + else: + stoptime = t[-1] + out_samples = int(np.floor(stoptime / system.dt)) + 1 + + # Pre-build output arrays + xout = np.zeros((out_samples, system.A.shape[0])) + yout = np.zeros((out_samples, system.C.shape[0])) + tout = np.linspace(0.0, stoptime, num=out_samples) + + # Check initial condition + if x0 is None: + xout[0, :] = np.zeros((system.A.shape[1],)) + else: + xout[0, :] = np.asarray(x0) + + # Pre-interpolate inputs into the desired time steps + if t is None: + u_dt = u + else: + if len(u.shape) == 1: + u = u[:, np.newaxis] + + u_dt = make_interp_spline(t, u, k=1)(tout) + + # Simulate the system + for i in range(0, out_samples - 1): + xout[i+1, :] = (np.dot(system.A, xout[i, :]) + + np.dot(system.B, u_dt[i, :])) + yout[i, :] = (np.dot(system.C, xout[i, :]) + + np.dot(system.D, u_dt[i, :])) + + # Last point + yout[out_samples-1, :] = (np.dot(system.C, xout[out_samples-1, :]) + + np.dot(system.D, u_dt[out_samples-1, :])) + + if is_ss_input: + return tout, yout, xout + else: + return tout, yout + + +def dimpulse(system, x0=None, t=None, n=None): + """ + Impulse response of discrete-time system. + + Parameters + ---------- + system : tuple of array_like or instance of `dlti` + A tuple describing the system. + The following gives the number of elements in the tuple and + the interpretation: + + * 1: (instance of `dlti`) + * 3: (num, den, dt) + * 4: (zeros, poles, gain, dt) + * 5: (A, B, C, D, dt) + + x0 : array_like, optional + Initial state-vector. Defaults to zero. + t : array_like, optional + Time points. Computed if not given. + n : int, optional + The number of time points to compute (if `t` is not given). + + Returns + ------- + tout : ndarray + Time values for the output, as a 1-D array. + yout : tuple of ndarray + Impulse response of system. Each element of the tuple represents + the output of the system based on an impulse in each input. + + See Also + -------- + impulse, dstep, dlsim, cont2discrete + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> butter = signal.dlti(*signal.butter(3, 0.5)) + >>> t, y = signal.dimpulse(butter, n=25) + >>> plt.step(t, np.squeeze(y)) + >>> plt.grid() + >>> plt.xlabel('n [samples]') + >>> plt.ylabel('Amplitude') + + """ + # Convert system to dlti-StateSpace + if isinstance(system, dlti): + system = system._as_ss() + elif isinstance(system, lti): + raise AttributeError('dimpulse can only be used with discrete-time ' + 'dlti systems.') + else: + system = dlti(*system[:-1], dt=system[-1])._as_ss() + + # Default to 100 samples if unspecified + if n is None: + n = 100 + + # If time is not specified, use the number of samples + # and system dt + if t is None: + t = np.linspace(0, n * system.dt, n, endpoint=False) + else: + t = np.asarray(t) + + # For each input, implement a step change + yout = None + for i in range(0, system.inputs): + u = np.zeros((t.shape[0], system.inputs)) + u[0, i] = 1.0 + + one_output = dlsim(system, u, t=t, x0=x0) + + if yout is None: + yout = (one_output[1],) + else: + yout = yout + (one_output[1],) + + tout = one_output[0] + + return tout, yout + + +def dstep(system, x0=None, t=None, n=None): + """ + Step response of discrete-time system. + + Parameters + ---------- + system : tuple of array_like + A tuple describing the system. + The following gives the number of elements in the tuple and + the interpretation: + + * 1: (instance of `dlti`) + * 3: (num, den, dt) + * 4: (zeros, poles, gain, dt) + * 5: (A, B, C, D, dt) + + x0 : array_like, optional + Initial state-vector. Defaults to zero. + t : array_like, optional + Time points. Computed if not given. + n : int, optional + The number of time points to compute (if `t` is not given). + + Returns + ------- + tout : ndarray + Output time points, as a 1-D array. + yout : tuple of ndarray + Step response of system. Each element of the tuple represents + the output of the system based on a step response to each input. + + See Also + -------- + step, dimpulse, dlsim, cont2discrete + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> butter = signal.dlti(*signal.butter(3, 0.5)) + >>> t, y = signal.dstep(butter, n=25) + >>> plt.step(t, np.squeeze(y)) + >>> plt.grid() + >>> plt.xlabel('n [samples]') + >>> plt.ylabel('Amplitude') + """ + # Convert system to dlti-StateSpace + if isinstance(system, dlti): + system = system._as_ss() + elif isinstance(system, lti): + raise AttributeError('dstep can only be used with discrete-time dlti ' + 'systems.') + else: + system = dlti(*system[:-1], dt=system[-1])._as_ss() + + # Default to 100 samples if unspecified + if n is None: + n = 100 + + # If time is not specified, use the number of samples + # and system dt + if t is None: + t = np.linspace(0, n * system.dt, n, endpoint=False) + else: + t = np.asarray(t) + + # For each input, implement a step change + yout = None + for i in range(0, system.inputs): + u = np.zeros((t.shape[0], system.inputs)) + u[:, i] = np.ones((t.shape[0],)) + + one_output = dlsim(system, u, t=t, x0=x0) + + if yout is None: + yout = (one_output[1],) + else: + yout = yout + (one_output[1],) + + tout = one_output[0] + + return tout, yout + + +def dfreqresp(system, w=None, n=10000, whole=False): + r""" + Calculate the frequency response of a discrete-time system. + + Parameters + ---------- + system : an instance of the `dlti` class or a tuple describing the system. + The following gives the number of elements in the tuple and + the interpretation: + + * 1 (instance of `dlti`) + * 2 (numerator, denominator, dt) + * 3 (zeros, poles, gain, dt) + * 4 (A, B, C, D, dt) + + w : array_like, optional + Array of frequencies (in radians/sample). Magnitude and phase data is + calculated for every value in this array. If not given a reasonable + set will be calculated. + n : int, optional + Number of frequency points to compute if `w` is not given. The `n` + frequencies are logarithmically spaced in an interval chosen to + include the influence of the poles and zeros of the system. + whole : bool, optional + Normally, if 'w' is not given, frequencies are computed from 0 to the + Nyquist frequency, pi radians/sample (upper-half of unit-circle). If + `whole` is True, compute frequencies from 0 to 2*pi radians/sample. + + Returns + ------- + w : 1D ndarray + Frequency array [radians/sample] + H : 1D ndarray + Array of complex magnitude values + + Notes + ----- + If (num, den) is passed in for ``system``, coefficients for both the + numerator and denominator should be specified in descending exponent + order (e.g. ``z^2 + 3z + 5`` would be represented as ``[1, 3, 5]``). + + .. versionadded:: 0.18.0 + + Examples + -------- + Generating the Nyquist plot of a transfer function + + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + Construct the transfer function + :math:`H(z) = \frac{1}{z^2 + 2z + 3}` with a sampling time of 0.05 + seconds: + + >>> sys = signal.TransferFunction([1], [1, 2, 3], dt=0.05) + + >>> w, H = signal.dfreqresp(sys) + + >>> plt.figure() + >>> plt.plot(H.real, H.imag, "b") + >>> plt.plot(H.real, -H.imag, "r") + >>> plt.show() + + """ + if not isinstance(system, dlti): + if isinstance(system, lti): + raise AttributeError('dfreqresp can only be used with ' + 'discrete-time systems.') + + system = dlti(*system[:-1], dt=system[-1]) + + if isinstance(system, StateSpace): + # No SS->ZPK code exists right now, just SS->TF->ZPK + system = system._as_tf() + + if not isinstance(system, (TransferFunction, ZerosPolesGain)): + raise ValueError('Unknown system type') + + if system.inputs != 1 or system.outputs != 1: + raise ValueError("dfreqresp requires a SISO (single input, single " + "output) system.") + + if w is not None: + worN = w + else: + worN = n + + if isinstance(system, TransferFunction): + # Convert numerator and denominator from polynomials in the variable + # 'z' to polynomials in the variable 'z^-1', as freqz expects. + num, den = TransferFunction._z_to_zinv(system.num.ravel(), system.den) + w, h = freqz(num, den, worN=worN, whole=whole) + + elif isinstance(system, ZerosPolesGain): + w, h = freqz_zpk(system.zeros, system.poles, system.gain, worN=worN, + whole=whole) + + return w, h + + +def dbode(system, w=None, n=100): + r""" + Calculate Bode magnitude and phase data of a discrete-time system. + + Parameters + ---------- + system : + An instance of the LTI class `dlti` or a tuple describing the system. + The number of elements in the tuple determine the interpretation, i.e.: + + 1. ``(sys_dlti)``: Instance of LTI class `dlti`. Note that derived instances, + such as instances of `TransferFunction`, `ZerosPolesGain`, or `StateSpace`, + are allowed as well. + 2. ``(num, den, dt)``: Rational polynomial as described in `TransferFunction`. + The coefficients of the polynomials should be specified in descending + exponent order, e.g., z² + 3z + 5 would be represented as ``[1, 3, 5]``. + 3. ``(zeros, poles, gain, dt)``: Zeros, poles, gain form as described + in `ZerosPolesGain`. + 4. ``(A, B, C, D, dt)``: State-space form as described in `StateSpace`. + + w : array_like, optional + Array of frequencies normalized to the Nyquist frequency being π, i.e., + having unit radiant / sample. Magnitude and phase data is calculated for every + value in this array. If not given, a reasonable set will be calculated. + n : int, optional + Number of frequency points to compute if `w` is not given. The `n` + frequencies are logarithmically spaced in an interval chosen to + include the influence of the poles and zeros of the system. + + Returns + ------- + w : 1D ndarray + Array of frequencies normalized to the Nyquist frequency being ``np.pi/dt`` + with ``dt`` being the sampling interval of the `system` parameter. + The unit is rad/s assuming ``dt`` is in seconds. + mag : 1D ndarray + Magnitude array in dB + phase : 1D ndarray + Phase array in degrees + + Notes + ----- + This function is a convenience wrapper around `dfreqresp` for extracting + magnitude and phase from the calculated complex-valued amplitude of the + frequency response. + + .. versionadded:: 0.18.0 + + See Also + -------- + dfreqresp, dlti, TransferFunction, ZerosPolesGain, StateSpace + + + Examples + -------- + The following example shows how to create a Bode plot of a 5-th order + Butterworth lowpass filter with a corner frequency of 100 Hz: + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> from scipy import signal + ... + >>> T = 1e-4 # sampling interval in s + >>> f_c, o = 1e2, 5 # corner frequency in Hz (i.e., -3 dB value) and filter order + >>> bb, aa = signal.butter(o, f_c, 'lowpass', fs=1/T) + ... + >>> w, mag, phase = signal.dbode((bb, aa, T)) + >>> w /= 2*np.pi # convert unit of frequency into Hertz + ... + >>> fg, (ax0, ax1) = plt.subplots(2, 1, sharex='all', figsize=(5, 4), + ... tight_layout=True) + >>> ax0.set_title("Bode Plot of Butterworth Lowpass Filter " + + ... rf"($f_c={f_c:g}\,$Hz, order={o})") + >>> ax0.set_ylabel(r"Magnitude in dB") + >>> ax1.set(ylabel=r"Phase in Degrees", + ... xlabel="Frequency $f$ in Hertz", xlim=(w[1], w[-1])) + >>> ax0.semilogx(w, mag, 'C0-', label=r"$20\,\log_{10}|G(f)|$") # Magnitude plot + >>> ax1.semilogx(w, phase, 'C1-', label=r"$\angle G(f)$") # Phase plot + ... + >>> for ax_ in (ax0, ax1): + ... ax_.axvline(f_c, color='m', alpha=0.25, label=rf"${f_c=:g}\,$Hz") + ... ax_.grid(which='both', axis='x') # plot major & minor vertical grid lines + ... ax_.grid(which='major', axis='y') + ... ax_.legend() + >>> plt.show() + """ + w, y = dfreqresp(system, w=w, n=n) + + if isinstance(system, dlti): + dt = system.dt + else: + dt = system[-1] + + mag = 20.0 * np.log10(abs(y)) + phase = np.rad2deg(np.unwrap(np.angle(y))) + + return w / dt, mag, phase diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_max_len_seq.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_max_len_seq.py new file mode 100644 index 0000000000000000000000000000000000000000..4d64beaca86d1da50b563668679b6fc52c954ab0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_max_len_seq.py @@ -0,0 +1,139 @@ +# Author: Eric Larson +# 2014 + +"""Tools for MLS generation""" + +import numpy as np + +from ._max_len_seq_inner import _max_len_seq_inner + +__all__ = ['max_len_seq'] + + +# These are definitions of linear shift register taps for use in max_len_seq() +_mls_taps = {2: [1], 3: [2], 4: [3], 5: [3], 6: [5], 7: [6], 8: [7, 6, 1], + 9: [5], 10: [7], 11: [9], 12: [11, 10, 4], 13: [12, 11, 8], + 14: [13, 12, 2], 15: [14], 16: [15, 13, 4], 17: [14], + 18: [11], 19: [18, 17, 14], 20: [17], 21: [19], 22: [21], + 23: [18], 24: [23, 22, 17], 25: [22], 26: [25, 24, 20], + 27: [26, 25, 22], 28: [25], 29: [27], 30: [29, 28, 7], + 31: [28], 32: [31, 30, 10]} + +def max_len_seq(nbits, state=None, length=None, taps=None): + """ + Maximum length sequence (MLS) generator. + + Parameters + ---------- + nbits : int + Number of bits to use. Length of the resulting sequence will + be ``(2**nbits) - 1``. Note that generating long sequences + (e.g., greater than ``nbits == 16``) can take a long time. + state : array_like, optional + If array, must be of length ``nbits``, and will be cast to binary + (bool) representation. If None, a seed of ones will be used, + producing a repeatable representation. If ``state`` is all + zeros, an error is raised as this is invalid. Default: None. + length : int, optional + Number of samples to compute. If None, the entire length + ``(2**nbits) - 1`` is computed. + taps : array_like, optional + Polynomial taps to use (e.g., ``[7, 6, 1]`` for an 8-bit sequence). + If None, taps will be automatically selected (for up to + ``nbits == 32``). + + Returns + ------- + seq : array + Resulting MLS sequence of 0's and 1's. + state : array + The final state of the shift register. + + Notes + ----- + The algorithm for MLS generation is generically described in: + + https://en.wikipedia.org/wiki/Maximum_length_sequence + + The default values for taps are specifically taken from the first + option listed for each value of ``nbits`` in: + + https://web.archive.org/web/20181001062252/http://www.newwaveinstruments.com/resources/articles/m_sequence_linear_feedback_shift_register_lfsr.htm + + .. versionadded:: 0.15.0 + + Examples + -------- + MLS uses binary convention: + + >>> from scipy.signal import max_len_seq + >>> max_len_seq(4)[0] + array([1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0], dtype=int8) + + MLS has a white spectrum (except for DC): + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from numpy.fft import fft, ifft, fftshift, fftfreq + >>> seq = max_len_seq(6)[0]*2-1 # +1 and -1 + >>> spec = fft(seq) + >>> N = len(seq) + >>> plt.plot(fftshift(fftfreq(N)), fftshift(np.abs(spec)), '.-') + >>> plt.margins(0.1, 0.1) + >>> plt.grid(True) + >>> plt.show() + + Circular autocorrelation of MLS is an impulse: + + >>> acorrcirc = ifft(spec * np.conj(spec)).real + >>> plt.figure() + >>> plt.plot(np.arange(-N/2+1, N/2+1), fftshift(acorrcirc), '.-') + >>> plt.margins(0.1, 0.1) + >>> plt.grid(True) + >>> plt.show() + + Linear autocorrelation of MLS is approximately an impulse: + + >>> acorr = np.correlate(seq, seq, 'full') + >>> plt.figure() + >>> plt.plot(np.arange(-N+1, N), acorr, '.-') + >>> plt.margins(0.1, 0.1) + >>> plt.grid(True) + >>> plt.show() + + """ + taps_dtype = np.int32 if np.intp().itemsize == 4 else np.int64 + if taps is None: + if nbits not in _mls_taps: + known_taps = np.array(list(_mls_taps.keys())) + raise ValueError(f'nbits must be between {known_taps.min()} and ' + f'{known_taps.max()} if taps is None') + taps = np.array(_mls_taps[nbits], taps_dtype) + else: + taps = np.unique(np.array(taps, taps_dtype))[::-1] + if np.any(taps < 0) or np.any(taps > nbits) or taps.size < 1: + raise ValueError('taps must be non-empty with values between ' + 'zero and nbits (inclusive)') + taps = np.array(taps) # needed for Cython and Pythran + n_max = (2**nbits) - 1 + if length is None: + length = n_max + else: + length = int(length) + if length < 0: + raise ValueError('length must be greater than or equal to 0') + # We use int8 instead of bool here because NumPy arrays of bools + # don't seem to work nicely with Cython + if state is None: + state = np.ones(nbits, dtype=np.int8, order='c') + else: + # makes a copy if need be, ensuring it's 0's and 1's + state = np.array(state, dtype=bool, order='c').astype(np.int8) + if state.ndim != 1 or state.size != nbits: + raise ValueError('state must be a 1-D array of size nbits') + if np.all(state == 0): + raise ValueError('state must not be all zeros') + + seq = np.empty(length, dtype=np.int8, order='c') + state = _max_len_seq_inner(taps, state, nbits, length, seq) + return seq, state diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_max_len_seq_inner.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_max_len_seq_inner.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..4eb7c7b2a3b658919f14450c20cad1e612a327d7 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_max_len_seq_inner.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_peak_finding.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_peak_finding.py new file mode 100644 index 0000000000000000000000000000000000000000..ccbeca5b7a4839bbc28e9c1cfd1ebd1d028a82cf --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_peak_finding.py @@ -0,0 +1,1310 @@ +""" +Functions for identifying peaks in signals. +""" +import math +import numpy as np + +from scipy.signal._wavelets import _cwt, _ricker +from scipy.stats import scoreatpercentile + +from ._peak_finding_utils import ( + _local_maxima_1d, + _select_by_peak_distance, + _peak_prominences, + _peak_widths +) + + +__all__ = ['argrelmin', 'argrelmax', 'argrelextrema', 'peak_prominences', + 'peak_widths', 'find_peaks', 'find_peaks_cwt'] + + +def _boolrelextrema(data, comparator, axis=0, order=1, mode='clip'): + """ + Calculate the relative extrema of `data`. + + Relative extrema are calculated by finding locations where + ``comparator(data[n], data[n+1:n+order+1])`` is True. + + Parameters + ---------- + data : ndarray + Array in which to find the relative extrema. + comparator : callable + Function to use to compare two data points. + Should take two arrays as arguments. + axis : int, optional + Axis over which to select from `data`. Default is 0. + order : int, optional + How many points on each side to use for the comparison + to consider ``comparator(n,n+x)`` to be True. + mode : str, optional + How the edges of the vector are treated. 'wrap' (wrap around) or + 'clip' (treat overflow as the same as the last (or first) element). + Default 'clip'. See numpy.take. + + Returns + ------- + extrema : ndarray + Boolean array of the same shape as `data` that is True at an extrema, + False otherwise. + + See also + -------- + argrelmax, argrelmin + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal._peak_finding import _boolrelextrema + >>> testdata = np.array([1,2,3,2,1]) + >>> _boolrelextrema(testdata, np.greater, axis=0) + array([False, False, True, False, False], dtype=bool) + + """ + if (int(order) != order) or (order < 1): + raise ValueError('Order must be an int >= 1') + + datalen = data.shape[axis] + locs = np.arange(0, datalen) + + results = np.ones(data.shape, dtype=bool) + main = data.take(locs, axis=axis, mode=mode) + for shift in range(1, order + 1): + plus = data.take(locs + shift, axis=axis, mode=mode) + minus = data.take(locs - shift, axis=axis, mode=mode) + results &= comparator(main, plus) + results &= comparator(main, minus) + if ~results.any(): + return results + return results + + +def argrelmin(data, axis=0, order=1, mode='clip'): + """ + Calculate the relative minima of `data`. + + Parameters + ---------- + data : ndarray + Array in which to find the relative minima. + axis : int, optional + Axis over which to select from `data`. Default is 0. + order : int, optional + How many points on each side to use for the comparison + to consider ``comparator(n, n+x)`` to be True. + mode : str, optional + How the edges of the vector are treated. + Available options are 'wrap' (wrap around) or 'clip' (treat overflow + as the same as the last (or first) element). + Default 'clip'. See numpy.take. + + Returns + ------- + extrema : tuple of ndarrays + Indices of the minima in arrays of integers. ``extrema[k]`` is + the array of indices of axis `k` of `data`. Note that the + return value is a tuple even when `data` is 1-D. + + See Also + -------- + argrelextrema, argrelmax, find_peaks + + Notes + ----- + This function uses `argrelextrema` with np.less as comparator. Therefore, it + requires a strict inequality on both sides of a value to consider it a + minimum. This means flat minima (more than one sample wide) are not detected. + In case of 1-D `data` `find_peaks` can be used to detect all + local minima, including flat ones, by calling it with negated `data`. + + .. versionadded:: 0.11.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal import argrelmin + >>> x = np.array([2, 1, 2, 3, 2, 0, 1, 0]) + >>> argrelmin(x) + (array([1, 5]),) + >>> y = np.array([[1, 2, 1, 2], + ... [2, 2, 0, 0], + ... [5, 3, 4, 4]]) + ... + >>> argrelmin(y, axis=1) + (array([0, 2]), array([2, 1])) + + """ + return argrelextrema(data, np.less, axis, order, mode) + + +def argrelmax(data, axis=0, order=1, mode='clip'): + """ + Calculate the relative maxima of `data`. + + Parameters + ---------- + data : ndarray + Array in which to find the relative maxima. + axis : int, optional + Axis over which to select from `data`. Default is 0. + order : int, optional + How many points on each side to use for the comparison + to consider ``comparator(n, n+x)`` to be True. + mode : str, optional + How the edges of the vector are treated. + Available options are 'wrap' (wrap around) or 'clip' (treat overflow + as the same as the last (or first) element). + Default 'clip'. See `numpy.take`. + + Returns + ------- + extrema : tuple of ndarrays + Indices of the maxima in arrays of integers. ``extrema[k]`` is + the array of indices of axis `k` of `data`. Note that the + return value is a tuple even when `data` is 1-D. + + See Also + -------- + argrelextrema, argrelmin, find_peaks + + Notes + ----- + This function uses `argrelextrema` with np.greater as comparator. Therefore, + it requires a strict inequality on both sides of a value to consider it a + maximum. This means flat maxima (more than one sample wide) are not detected. + In case of 1-D `data` `find_peaks` can be used to detect all + local maxima, including flat ones. + + .. versionadded:: 0.11.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal import argrelmax + >>> x = np.array([2, 1, 2, 3, 2, 0, 1, 0]) + >>> argrelmax(x) + (array([3, 6]),) + >>> y = np.array([[1, 2, 1, 2], + ... [2, 2, 0, 0], + ... [5, 3, 4, 4]]) + ... + >>> argrelmax(y, axis=1) + (array([0]), array([1])) + """ + return argrelextrema(data, np.greater, axis, order, mode) + + +def argrelextrema(data, comparator, axis=0, order=1, mode='clip'): + """ + Calculate the relative extrema of `data`. + + Parameters + ---------- + data : ndarray + Array in which to find the relative extrema. + comparator : callable + Function to use to compare two data points. + Should take two arrays as arguments. + axis : int, optional + Axis over which to select from `data`. Default is 0. + order : int, optional + How many points on each side to use for the comparison + to consider ``comparator(n, n+x)`` to be True. + mode : str, optional + How the edges of the vector are treated. 'wrap' (wrap around) or + 'clip' (treat overflow as the same as the last (or first) element). + Default is 'clip'. See `numpy.take`. + + Returns + ------- + extrema : tuple of ndarrays + Indices of the maxima in arrays of integers. ``extrema[k]`` is + the array of indices of axis `k` of `data`. Note that the + return value is a tuple even when `data` is 1-D. + + See Also + -------- + argrelmin, argrelmax + + Notes + ----- + + .. versionadded:: 0.11.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal import argrelextrema + >>> x = np.array([2, 1, 2, 3, 2, 0, 1, 0]) + >>> argrelextrema(x, np.greater) + (array([3, 6]),) + >>> y = np.array([[1, 2, 1, 2], + ... [2, 2, 0, 0], + ... [5, 3, 4, 4]]) + ... + >>> argrelextrema(y, np.less, axis=1) + (array([0, 2]), array([2, 1])) + + """ + results = _boolrelextrema(data, comparator, + axis, order, mode) + return np.nonzero(results) + + +def _arg_x_as_expected(value): + """Ensure argument `x` is a 1-D C-contiguous array of dtype('float64'). + + Used in `find_peaks`, `peak_prominences` and `peak_widths` to make `x` + compatible with the signature of the wrapped Cython functions. + + Returns + ------- + value : ndarray + A 1-D C-contiguous array with dtype('float64'). + """ + value = np.asarray(value, order='C', dtype=np.float64) + if value.ndim != 1: + raise ValueError('`x` must be a 1-D array') + return value + + +def _arg_peaks_as_expected(value): + """Ensure argument `peaks` is a 1-D C-contiguous array of dtype('intp'). + + Used in `peak_prominences` and `peak_widths` to make `peaks` compatible + with the signature of the wrapped Cython functions. + + Returns + ------- + value : ndarray + A 1-D C-contiguous array with dtype('intp'). + """ + value = np.asarray(value) + if value.size == 0: + # Empty arrays default to np.float64 but are valid input + value = np.array([], dtype=np.intp) + try: + # Safely convert to C-contiguous array of type np.intp + value = value.astype(np.intp, order='C', casting='safe', + subok=False, copy=False) + except TypeError as e: + raise TypeError("cannot safely cast `peaks` to dtype('intp')") from e + if value.ndim != 1: + raise ValueError('`peaks` must be a 1-D array') + return value + + +def _arg_wlen_as_expected(value): + """Ensure argument `wlen` is of type `np.intp` and larger than 1. + + Used in `peak_prominences` and `peak_widths`. + + Returns + ------- + value : np.intp + The original `value` rounded up to an integer or -1 if `value` was + None. + """ + if value is None: + # _peak_prominences expects an intp; -1 signals that no value was + # supplied by the user + value = -1 + elif 1 < value: + # Round up to a positive integer + if isinstance(value, float): + value = math.ceil(value) + value = np.intp(value) + else: + raise ValueError(f'`wlen` must be larger than 1, was {value}') + return value + + +def peak_prominences(x, peaks, wlen=None): + """ + Calculate the prominence of each peak in a signal. + + The prominence of a peak measures how much a peak stands out from the + surrounding baseline of the signal and is defined as the vertical distance + between the peak and its lowest contour line. + + Parameters + ---------- + x : sequence + A signal with peaks. + peaks : sequence + Indices of peaks in `x`. + wlen : int, optional + A window length in samples that optionally limits the evaluated area for + each peak to a subset of `x`. The peak is always placed in the middle of + the window therefore the given length is rounded up to the next odd + integer. This parameter can speed up the calculation (see Notes). + + Returns + ------- + prominences : ndarray + The calculated prominences for each peak in `peaks`. + left_bases, right_bases : ndarray + The peaks' bases as indices in `x` to the left and right of each peak. + The higher base of each pair is a peak's lowest contour line. + + Raises + ------ + ValueError + If a value in `peaks` is an invalid index for `x`. + + Warns + ----- + PeakPropertyWarning + For indices in `peaks` that don't point to valid local maxima in `x`, + the returned prominence will be 0 and this warning is raised. This + also happens if `wlen` is smaller than the plateau size of a peak. + + Warnings + -------- + This function may return unexpected results for data containing NaNs. To + avoid this, NaNs should either be removed or replaced. + + See Also + -------- + find_peaks + Find peaks inside a signal based on peak properties. + peak_widths + Calculate the width of peaks. + + Notes + ----- + Strategy to compute a peak's prominence: + + 1. Extend a horizontal line from the current peak to the left and right + until the line either reaches the window border (see `wlen`) or + intersects the signal again at the slope of a higher peak. An + intersection with a peak of the same height is ignored. + 2. On each side find the minimal signal value within the interval defined + above. These points are the peak's bases. + 3. The higher one of the two bases marks the peak's lowest contour line. The + prominence can then be calculated as the vertical difference between the + peaks height itself and its lowest contour line. + + Searching for the peak's bases can be slow for large `x` with periodic + behavior because large chunks or even the full signal need to be evaluated + for the first algorithmic step. This evaluation area can be limited with the + parameter `wlen` which restricts the algorithm to a window around the + current peak and can shorten the calculation time if the window length is + short in relation to `x`. + However, this may stop the algorithm from finding the true global contour + line if the peak's true bases are outside this window. Instead, a higher + contour line is found within the restricted window leading to a smaller + calculated prominence. In practice, this is only relevant for the highest set + of peaks in `x`. This behavior may even be used intentionally to calculate + "local" prominences. + + .. versionadded:: 1.1.0 + + References + ---------- + .. [1] Wikipedia Article for Topographic Prominence: + https://en.wikipedia.org/wiki/Topographic_prominence + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal import find_peaks, peak_prominences + >>> import matplotlib.pyplot as plt + + Create a test signal with two overlaid harmonics + + >>> x = np.linspace(0, 6 * np.pi, 1000) + >>> x = np.sin(x) + 0.6 * np.sin(2.6 * x) + + Find all peaks and calculate prominences + + >>> peaks, _ = find_peaks(x) + >>> prominences = peak_prominences(x, peaks)[0] + >>> prominences + array([1.24159486, 0.47840168, 0.28470524, 3.10716793, 0.284603 , + 0.47822491, 2.48340261, 0.47822491]) + + Calculate the height of each peak's contour line and plot the results + + >>> contour_heights = x[peaks] - prominences + >>> plt.plot(x) + >>> plt.plot(peaks, x[peaks], "x") + >>> plt.vlines(x=peaks, ymin=contour_heights, ymax=x[peaks]) + >>> plt.show() + + Let's evaluate a second example that demonstrates several edge cases for + one peak at index 5. + + >>> x = np.array([0, 1, 0, 3, 1, 3, 0, 4, 0]) + >>> peaks = np.array([5]) + >>> plt.plot(x) + >>> plt.plot(peaks, x[peaks], "x") + >>> plt.show() + >>> peak_prominences(x, peaks) # -> (prominences, left_bases, right_bases) + (array([3.]), array([2]), array([6])) + + Note how the peak at index 3 of the same height is not considered as a + border while searching for the left base. Instead, two minima at 0 and 2 + are found in which case the one closer to the evaluated peak is always + chosen. On the right side, however, the base must be placed at 6 because the + higher peak represents the right border to the evaluated area. + + >>> peak_prominences(x, peaks, wlen=3.1) + (array([2.]), array([4]), array([6])) + + Here, we restricted the algorithm to a window from 3 to 7 (the length is 5 + samples because `wlen` was rounded up to the next odd integer). Thus, the + only two candidates in the evaluated area are the two neighboring samples + and a smaller prominence is calculated. + """ + x = _arg_x_as_expected(x) + peaks = _arg_peaks_as_expected(peaks) + wlen = _arg_wlen_as_expected(wlen) + return _peak_prominences(x, peaks, wlen) + + +def peak_widths(x, peaks, rel_height=0.5, prominence_data=None, wlen=None): + """ + Calculate the width of each peak in a signal. + + This function calculates the width of a peak in samples at a relative + distance to the peak's height and prominence. + + Parameters + ---------- + x : sequence + A signal with peaks. + peaks : sequence + Indices of peaks in `x`. + rel_height : float, optional + Chooses the relative height at which the peak width is measured as a + percentage of its prominence. 1.0 calculates the width of the peak at + its lowest contour line while 0.5 evaluates at half the prominence + height. Must be at least 0. See notes for further explanation. + prominence_data : tuple, optional + A tuple of three arrays matching the output of `peak_prominences` when + called with the same arguments `x` and `peaks`. This data are calculated + internally if not provided. + wlen : int, optional + A window length in samples passed to `peak_prominences` as an optional + argument for internal calculation of `prominence_data`. This argument + is ignored if `prominence_data` is given. + + Returns + ------- + widths : ndarray + The widths for each peak in samples. + width_heights : ndarray + The height of the contour lines at which the `widths` where evaluated. + left_ips, right_ips : ndarray + Interpolated positions of left and right intersection points of a + horizontal line at the respective evaluation height. + + Raises + ------ + ValueError + If `prominence_data` is supplied but doesn't satisfy the condition + ``0 <= left_base <= peak <= right_base < x.shape[0]`` for each peak, + has the wrong dtype, is not C-contiguous or does not have the same + shape. + + Warns + ----- + PeakPropertyWarning + Raised if any calculated width is 0. This may stem from the supplied + `prominence_data` or if `rel_height` is set to 0. + + Warnings + -------- + This function may return unexpected results for data containing NaNs. To + avoid this, NaNs should either be removed or replaced. + + See Also + -------- + find_peaks + Find peaks inside a signal based on peak properties. + peak_prominences + Calculate the prominence of peaks. + + Notes + ----- + The basic algorithm to calculate a peak's width is as follows: + + * Calculate the evaluation height :math:`h_{eval}` with the formula + :math:`h_{eval} = h_{Peak} - P \\cdot R`, where :math:`h_{Peak}` is the + height of the peak itself, :math:`P` is the peak's prominence and + :math:`R` a positive ratio specified with the argument `rel_height`. + * Draw a horizontal line at the evaluation height to both sides, starting at + the peak's current vertical position until the lines either intersect a + slope, the signal border or cross the vertical position of the peak's + base (see `peak_prominences` for an definition). For the first case, + intersection with the signal, the true intersection point is estimated + with linear interpolation. + * Calculate the width as the horizontal distance between the chosen + endpoints on both sides. As a consequence of this the maximal possible + width for each peak is the horizontal distance between its bases. + + As shown above to calculate a peak's width its prominence and bases must be + known. You can supply these yourself with the argument `prominence_data`. + Otherwise, they are internally calculated (see `peak_prominences`). + + .. versionadded:: 1.1.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal import chirp, find_peaks, peak_widths + >>> import matplotlib.pyplot as plt + + Create a test signal with two overlaid harmonics + + >>> x = np.linspace(0, 6 * np.pi, 1000) + >>> x = np.sin(x) + 0.6 * np.sin(2.6 * x) + + Find all peaks and calculate their widths at the relative height of 0.5 + (contour line at half the prominence height) and 1 (at the lowest contour + line at full prominence height). + + >>> peaks, _ = find_peaks(x) + >>> results_half = peak_widths(x, peaks, rel_height=0.5) + >>> results_half[0] # widths + array([ 64.25172825, 41.29465463, 35.46943289, 104.71586081, + 35.46729324, 41.30429622, 181.93835853, 45.37078546]) + >>> results_full = peak_widths(x, peaks, rel_height=1) + >>> results_full[0] # widths + array([181.9396084 , 72.99284945, 61.28657872, 373.84622694, + 61.78404617, 72.48822812, 253.09161876, 79.36860878]) + + Plot signal, peaks and contour lines at which the widths where calculated + + >>> plt.plot(x) + >>> plt.plot(peaks, x[peaks], "x") + >>> plt.hlines(*results_half[1:], color="C2") + >>> plt.hlines(*results_full[1:], color="C3") + >>> plt.show() + """ + x = _arg_x_as_expected(x) + peaks = _arg_peaks_as_expected(peaks) + if prominence_data is None: + # Calculate prominence if not supplied and use wlen if supplied. + wlen = _arg_wlen_as_expected(wlen) + prominence_data = _peak_prominences(x, peaks, wlen) + return _peak_widths(x, peaks, rel_height, *prominence_data) + + +def _unpack_condition_args(interval, x, peaks): + """ + Parse condition arguments for `find_peaks`. + + Parameters + ---------- + interval : number or ndarray or sequence + Either a number or ndarray or a 2-element sequence of the former. The + first value is always interpreted as `imin` and the second, if supplied, + as `imax`. + x : ndarray + The signal with `peaks`. + peaks : ndarray + An array with indices used to reduce `imin` and / or `imax` if those are + arrays. + + Returns + ------- + imin, imax : number or ndarray or None + Minimal and maximal value in `argument`. + + Raises + ------ + ValueError : + If interval border is given as array and its size does not match the size + of `x`. + + Notes + ----- + + .. versionadded:: 1.1.0 + """ + try: + imin, imax = interval + except (TypeError, ValueError): + imin, imax = (interval, None) + + # Reduce arrays if arrays + if isinstance(imin, np.ndarray): + if imin.size != x.size: + raise ValueError('array size of lower interval border must match x') + imin = imin[peaks] + if isinstance(imax, np.ndarray): + if imax.size != x.size: + raise ValueError('array size of upper interval border must match x') + imax = imax[peaks] + + return imin, imax + + +def _select_by_property(peak_properties, pmin, pmax): + """ + Evaluate where the generic property of peaks confirms to an interval. + + Parameters + ---------- + peak_properties : ndarray + An array with properties for each peak. + pmin : None or number or ndarray + Lower interval boundary for `peak_properties`. ``None`` is interpreted as + an open border. + pmax : None or number or ndarray + Upper interval boundary for `peak_properties`. ``None`` is interpreted as + an open border. + + Returns + ------- + keep : bool + A boolean mask evaluating to true where `peak_properties` confirms to the + interval. + + See Also + -------- + find_peaks + + Notes + ----- + + .. versionadded:: 1.1.0 + """ + keep = np.ones(peak_properties.size, dtype=bool) + if pmin is not None: + keep &= (pmin <= peak_properties) + if pmax is not None: + keep &= (peak_properties <= pmax) + return keep + + +def _select_by_peak_threshold(x, peaks, tmin, tmax): + """ + Evaluate which peaks fulfill the threshold condition. + + Parameters + ---------- + x : ndarray + A 1-D array which is indexable by `peaks`. + peaks : ndarray + Indices of peaks in `x`. + tmin, tmax : scalar or ndarray or None + Minimal and / or maximal required thresholds. If supplied as ndarrays + their size must match `peaks`. ``None`` is interpreted as an open + border. + + Returns + ------- + keep : bool + A boolean mask evaluating to true where `peaks` fulfill the threshold + condition. + left_thresholds, right_thresholds : ndarray + Array matching `peak` containing the thresholds of each peak on + both sides. + + Notes + ----- + + .. versionadded:: 1.1.0 + """ + # Stack thresholds on both sides to make min / max operations easier: + # tmin is compared with the smaller, and tmax with the greater threshold to + # each peak's side + stacked_thresholds = np.vstack([x[peaks] - x[peaks - 1], + x[peaks] - x[peaks + 1]]) + keep = np.ones(peaks.size, dtype=bool) + if tmin is not None: + min_thresholds = np.min(stacked_thresholds, axis=0) + keep &= (tmin <= min_thresholds) + if tmax is not None: + max_thresholds = np.max(stacked_thresholds, axis=0) + keep &= (max_thresholds <= tmax) + + return keep, stacked_thresholds[0], stacked_thresholds[1] + + +def find_peaks(x, height=None, threshold=None, distance=None, + prominence=None, width=None, wlen=None, rel_height=0.5, + plateau_size=None): + """ + Find peaks inside a signal based on peak properties. + + This function takes a 1-D array and finds all local maxima by + simple comparison of neighboring values. Optionally, a subset of these + peaks can be selected by specifying conditions for a peak's properties. + + Parameters + ---------- + x : sequence + A signal with peaks. + height : number or ndarray or sequence, optional + Required height of peaks. Either a number, ``None``, an array matching + `x` or a 2-element sequence of the former. The first element is + always interpreted as the minimal and the second, if supplied, as the + maximal required height. + threshold : number or ndarray or sequence, optional + Required threshold of peaks, the vertical distance to its neighboring + samples. Either a number, ``None``, an array matching `x` or a + 2-element sequence of the former. The first element is always + interpreted as the minimal and the second, if supplied, as the maximal + required threshold. + distance : number, optional + Required minimal horizontal distance (>= 1) in samples between + neighbouring peaks. Smaller peaks are removed first until the condition + is fulfilled for all remaining peaks. + prominence : number or ndarray or sequence, optional + Required prominence of peaks. Either a number, ``None``, an array + matching `x` or a 2-element sequence of the former. The first + element is always interpreted as the minimal and the second, if + supplied, as the maximal required prominence. + width : number or ndarray or sequence, optional + Required width of peaks in samples. Either a number, ``None``, an array + matching `x` or a 2-element sequence of the former. The first + element is always interpreted as the minimal and the second, if + supplied, as the maximal required width. + wlen : int, optional + Used for calculation of the peaks prominences, thus it is only used if + one of the arguments `prominence` or `width` is given. See argument + `wlen` in `peak_prominences` for a full description of its effects. + rel_height : float, optional + Used for calculation of the peaks width, thus it is only used if `width` + is given. See argument `rel_height` in `peak_widths` for a full + description of its effects. + plateau_size : number or ndarray or sequence, optional + Required size of the flat top of peaks in samples. Either a number, + ``None``, an array matching `x` or a 2-element sequence of the former. + The first element is always interpreted as the minimal and the second, + if supplied as the maximal required plateau size. + + .. versionadded:: 1.2.0 + + Returns + ------- + peaks : ndarray + Indices of peaks in `x` that satisfy all given conditions. + properties : dict + A dictionary containing properties of the returned peaks which were + calculated as intermediate results during evaluation of the specified + conditions: + + * 'peak_heights' + If `height` is given, the height of each peak in `x`. + * 'left_thresholds', 'right_thresholds' + If `threshold` is given, these keys contain a peaks vertical + distance to its neighbouring samples. + * 'prominences', 'right_bases', 'left_bases' + If `prominence` is given, these keys are accessible. See + `peak_prominences` for a description of their content. + * 'widths', 'width_heights', 'left_ips', 'right_ips' + If `width` is given, these keys are accessible. See `peak_widths` + for a description of their content. + * 'plateau_sizes', left_edges', 'right_edges' + If `plateau_size` is given, these keys are accessible and contain + the indices of a peak's edges (edges are still part of the + plateau) and the calculated plateau sizes. + + .. versionadded:: 1.2.0 + + To calculate and return properties without excluding peaks, provide the + open interval ``(None, None)`` as a value to the appropriate argument + (excluding `distance`). + + Warns + ----- + PeakPropertyWarning + Raised if a peak's properties have unexpected values (see + `peak_prominences` and `peak_widths`). + + Warnings + -------- + This function may return unexpected results for data containing NaNs. To + avoid this, NaNs should either be removed or replaced. + + See Also + -------- + find_peaks_cwt + Find peaks using the wavelet transformation. + peak_prominences + Directly calculate the prominence of peaks. + peak_widths + Directly calculate the width of peaks. + + Notes + ----- + In the context of this function, a peak or local maximum is defined as any + sample whose two direct neighbours have a smaller amplitude. For flat peaks + (more than one sample of equal amplitude wide) the index of the middle + sample is returned (rounded down in case the number of samples is even). + For noisy signals the peak locations can be off because the noise might + change the position of local maxima. In those cases consider smoothing the + signal before searching for peaks or use other peak finding and fitting + methods (like `find_peaks_cwt`). + + Some additional comments on specifying conditions: + + * Almost all conditions (excluding `distance`) can be given as half-open or + closed intervals, e.g., ``1`` or ``(1, None)`` defines the half-open + interval :math:`[1, \\infty]` while ``(None, 1)`` defines the interval + :math:`[-\\infty, 1]`. The open interval ``(None, None)`` can be specified + as well, which returns the matching properties without exclusion of peaks. + * The border is always included in the interval used to select valid peaks. + * For several conditions the interval borders can be specified with + arrays matching `x` in shape which enables dynamic constrains based on + the sample position. + * The conditions are evaluated in the following order: `plateau_size`, + `height`, `threshold`, `distance`, `prominence`, `width`. In most cases + this order is the fastest one because faster operations are applied first + to reduce the number of peaks that need to be evaluated later. + * While indices in `peaks` are guaranteed to be at least `distance` samples + apart, edges of flat peaks may be closer than the allowed `distance`. + * Use `wlen` to reduce the time it takes to evaluate the conditions for + `prominence` or `width` if `x` is large or has many local maxima + (see `peak_prominences`). + + .. versionadded:: 1.1.0 + + Examples + -------- + To demonstrate this function's usage we use a signal `x` supplied with + SciPy (see `scipy.datasets.electrocardiogram`). Let's find all peaks (local + maxima) in `x` whose amplitude lies above 0. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.datasets import electrocardiogram + >>> from scipy.signal import find_peaks + >>> x = electrocardiogram()[2000:4000] + >>> peaks, _ = find_peaks(x, height=0) + >>> plt.plot(x) + >>> plt.plot(peaks, x[peaks], "x") + >>> plt.plot(np.zeros_like(x), "--", color="gray") + >>> plt.show() + + We can select peaks below 0 with ``height=(None, 0)`` or use arrays matching + `x` in size to reflect a changing condition for different parts of the + signal. + + >>> border = np.sin(np.linspace(0, 3 * np.pi, x.size)) + >>> peaks, _ = find_peaks(x, height=(-border, border)) + >>> plt.plot(x) + >>> plt.plot(-border, "--", color="gray") + >>> plt.plot(border, ":", color="gray") + >>> plt.plot(peaks, x[peaks], "x") + >>> plt.show() + + Another useful condition for periodic signals can be given with the + `distance` argument. In this case, we can easily select the positions of + QRS complexes within the electrocardiogram (ECG) by demanding a distance of + at least 150 samples. + + >>> peaks, _ = find_peaks(x, distance=150) + >>> np.diff(peaks) + array([186, 180, 177, 171, 177, 169, 167, 164, 158, 162, 172]) + >>> plt.plot(x) + >>> plt.plot(peaks, x[peaks], "x") + >>> plt.show() + + Especially for noisy signals peaks can be easily grouped by their + prominence (see `peak_prominences`). E.g., we can select all peaks except + for the mentioned QRS complexes by limiting the allowed prominence to 0.6. + + >>> peaks, properties = find_peaks(x, prominence=(None, 0.6)) + >>> properties["prominences"].max() + 0.5049999999999999 + >>> plt.plot(x) + >>> plt.plot(peaks, x[peaks], "x") + >>> plt.show() + + And, finally, let's examine a different section of the ECG which contains + beat forms of different shape. To select only the atypical heart beats, we + combine two conditions: a minimal prominence of 1 and width of at least 20 + samples. + + >>> x = electrocardiogram()[17000:18000] + >>> peaks, properties = find_peaks(x, prominence=1, width=20) + >>> properties["prominences"], properties["widths"] + (array([1.495, 2.3 ]), array([36.93773946, 39.32723577])) + >>> plt.plot(x) + >>> plt.plot(peaks, x[peaks], "x") + >>> plt.vlines(x=peaks, ymin=x[peaks] - properties["prominences"], + ... ymax = x[peaks], color = "C1") + >>> plt.hlines(y=properties["width_heights"], xmin=properties["left_ips"], + ... xmax=properties["right_ips"], color = "C1") + >>> plt.show() + """ + # _argmaxima1d expects array of dtype 'float64' + x = _arg_x_as_expected(x) + if distance is not None and distance < 1: + raise ValueError('`distance` must be greater or equal to 1') + + peaks, left_edges, right_edges = _local_maxima_1d(x) + properties = {} + + if plateau_size is not None: + # Evaluate plateau size + plateau_sizes = right_edges - left_edges + 1 + pmin, pmax = _unpack_condition_args(plateau_size, x, peaks) + keep = _select_by_property(plateau_sizes, pmin, pmax) + peaks = peaks[keep] + properties["plateau_sizes"] = plateau_sizes + properties["left_edges"] = left_edges + properties["right_edges"] = right_edges + properties = {key: array[keep] for key, array in properties.items()} + + if height is not None: + # Evaluate height condition + peak_heights = x[peaks] + hmin, hmax = _unpack_condition_args(height, x, peaks) + keep = _select_by_property(peak_heights, hmin, hmax) + peaks = peaks[keep] + properties["peak_heights"] = peak_heights + properties = {key: array[keep] for key, array in properties.items()} + + if threshold is not None: + # Evaluate threshold condition + tmin, tmax = _unpack_condition_args(threshold, x, peaks) + keep, left_thresholds, right_thresholds = _select_by_peak_threshold( + x, peaks, tmin, tmax) + peaks = peaks[keep] + properties["left_thresholds"] = left_thresholds + properties["right_thresholds"] = right_thresholds + properties = {key: array[keep] for key, array in properties.items()} + + if distance is not None: + # Evaluate distance condition + keep = _select_by_peak_distance(peaks, x[peaks], distance) + peaks = peaks[keep] + properties = {key: array[keep] for key, array in properties.items()} + + if prominence is not None or width is not None: + # Calculate prominence (required for both conditions) + wlen = _arg_wlen_as_expected(wlen) + properties.update(zip( + ['prominences', 'left_bases', 'right_bases'], + _peak_prominences(x, peaks, wlen=wlen) + )) + + if prominence is not None: + # Evaluate prominence condition + pmin, pmax = _unpack_condition_args(prominence, x, peaks) + keep = _select_by_property(properties['prominences'], pmin, pmax) + peaks = peaks[keep] + properties = {key: array[keep] for key, array in properties.items()} + + if width is not None: + # Calculate widths + properties.update(zip( + ['widths', 'width_heights', 'left_ips', 'right_ips'], + _peak_widths(x, peaks, rel_height, properties['prominences'], + properties['left_bases'], properties['right_bases']) + )) + # Evaluate width condition + wmin, wmax = _unpack_condition_args(width, x, peaks) + keep = _select_by_property(properties['widths'], wmin, wmax) + peaks = peaks[keep] + properties = {key: array[keep] for key, array in properties.items()} + + return peaks, properties + + +def _identify_ridge_lines(matr, max_distances, gap_thresh): + """ + Identify ridges in the 2-D matrix. + + Expect that the width of the wavelet feature increases with increasing row + number. + + Parameters + ---------- + matr : 2-D ndarray + Matrix in which to identify ridge lines. + max_distances : 1-D sequence + At each row, a ridge line is only connected + if the relative max at row[n] is within + `max_distances`[n] from the relative max at row[n+1]. + gap_thresh : int + If a relative maximum is not found within `max_distances`, + there will be a gap. A ridge line is discontinued if + there are more than `gap_thresh` points without connecting + a new relative maximum. + + Returns + ------- + ridge_lines : tuple + Tuple of 2 1-D sequences. `ridge_lines`[ii][0] are the rows of the + ii-th ridge-line, `ridge_lines`[ii][1] are the columns. Empty if none + found. Each ridge-line will be sorted by row (increasing), but the + order of the ridge lines is not specified. + + References + ---------- + .. [1] Bioinformatics (2006) 22 (17): 2059-2065. + :doi:`10.1093/bioinformatics/btl355` + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal._peak_finding import _identify_ridge_lines + >>> rng = np.random.default_rng() + >>> data = rng.random((5,5)) + >>> max_dist = 3 + >>> max_distances = np.full(20, max_dist) + >>> ridge_lines = _identify_ridge_lines(data, max_distances, 1) + + Notes + ----- + This function is intended to be used in conjunction with `cwt` + as part of `find_peaks_cwt`. + + """ + if len(max_distances) < matr.shape[0]: + raise ValueError('Max_distances must have at least as many rows ' + 'as matr') + + all_max_cols = _boolrelextrema(matr, np.greater, axis=1, order=1) + # Highest row for which there are any relative maxima + has_relmax = np.nonzero(all_max_cols.any(axis=1))[0] + if len(has_relmax) == 0: + return [] + start_row = has_relmax[-1] + # Each ridge line is a 3-tuple: + # rows, cols,Gap number + ridge_lines = [[[start_row], + [col], + 0] for col in np.nonzero(all_max_cols[start_row])[0]] + final_lines = [] + rows = np.arange(start_row - 1, -1, -1) + cols = np.arange(0, matr.shape[1]) + for row in rows: + this_max_cols = cols[all_max_cols[row]] + + # Increment gap number of each line, + # set it to zero later if appropriate + for line in ridge_lines: + line[2] += 1 + + # XXX These should always be all_max_cols[row] + # But the order might be different. Might be an efficiency gain + # to make sure the order is the same and avoid this iteration + prev_ridge_cols = np.array([line[1][-1] for line in ridge_lines]) + # Look through every relative maximum found at current row + # Attempt to connect them with existing ridge lines. + for ind, col in enumerate(this_max_cols): + # If there is a previous ridge line within + # the max_distance to connect to, do so. + # Otherwise start a new one. + line = None + if len(prev_ridge_cols) > 0: + diffs = np.abs(col - prev_ridge_cols) + closest = np.argmin(diffs) + if diffs[closest] <= max_distances[row]: + line = ridge_lines[closest] + if line is not None: + # Found a point close enough, extend current ridge line + line[1].append(col) + line[0].append(row) + line[2] = 0 + else: + new_line = [[row], + [col], + 0] + ridge_lines.append(new_line) + + # Remove the ridge lines with gap_number too high + # XXX Modifying a list while iterating over it. + # Should be safe, since we iterate backwards, but + # still tacky. + for ind in range(len(ridge_lines) - 1, -1, -1): + line = ridge_lines[ind] + if line[2] > gap_thresh: + final_lines.append(line) + del ridge_lines[ind] + + out_lines = [] + for line in (final_lines + ridge_lines): + sortargs = np.array(np.argsort(line[0])) + rows, cols = np.zeros_like(sortargs), np.zeros_like(sortargs) + rows[sortargs] = line[0] + cols[sortargs] = line[1] + out_lines.append([rows, cols]) + + return out_lines + + +def _filter_ridge_lines(cwt, ridge_lines, window_size=None, min_length=None, + min_snr=1, noise_perc=10): + """ + Filter ridge lines according to prescribed criteria. Intended + to be used for finding relative maxima. + + Parameters + ---------- + cwt : 2-D ndarray + Continuous wavelet transform from which the `ridge_lines` were defined. + ridge_lines : 1-D sequence + Each element should contain 2 sequences, the rows and columns + of the ridge line (respectively). + window_size : int, optional + Size of window to use to calculate noise floor. + Default is ``cwt.shape[1] / 20``. + min_length : int, optional + Minimum length a ridge line needs to be acceptable. + Default is ``cwt.shape[0] / 4``, ie 1/4-th the number of widths. + min_snr : float, optional + Minimum SNR ratio. Default 1. The signal is the value of + the cwt matrix at the shortest length scale (``cwt[0, loc]``), the + noise is the `noise_perc`\\ th percentile of datapoints contained within a + window of `window_size` around ``cwt[0, loc]``. + noise_perc : float, optional + When calculating the noise floor, percentile of data points + examined below which to consider noise. Calculated using + scipy.stats.scoreatpercentile. + + References + ---------- + .. [1] Bioinformatics (2006) 22 (17): 2059-2065. + :doi:`10.1093/bioinformatics/btl355` + + """ + num_points = cwt.shape[1] + if min_length is None: + min_length = np.ceil(cwt.shape[0] / 4) + if window_size is None: + window_size = np.ceil(num_points / 20) + + window_size = int(window_size) + hf_window, odd = divmod(window_size, 2) + + # Filter based on SNR + row_one = cwt[0, :] + noises = np.empty_like(row_one) + for ind, val in enumerate(row_one): + window_start = max(ind - hf_window, 0) + window_end = min(ind + hf_window + odd, num_points) + noises[ind] = scoreatpercentile(row_one[window_start:window_end], + per=noise_perc) + + def filt_func(line): + if len(line[0]) < min_length: + return False + snr = abs(cwt[line[0][0], line[1][0]] / noises[line[1][0]]) + if snr < min_snr: + return False + return True + + return list(filter(filt_func, ridge_lines)) + + +def find_peaks_cwt(vector, widths, wavelet=None, max_distances=None, + gap_thresh=None, min_length=None, + min_snr=1, noise_perc=10, window_size=None): + """ + Find peaks in a 1-D array with wavelet transformation. + + The general approach is to smooth `vector` by convolving it with + `wavelet(width)` for each width in `widths`. Relative maxima which + appear at enough length scales, and with sufficiently high SNR, are + accepted. + + Parameters + ---------- + vector : ndarray + 1-D array in which to find the peaks. + widths : float or sequence + Single width or 1-D array-like of widths to use for calculating + the CWT matrix. In general, + this range should cover the expected width of peaks of interest. + wavelet : callable, optional + Should take two parameters and return a 1-D array to convolve + with `vector`. The first parameter determines the number of points + of the returned wavelet array, the second parameter is the scale + (`width`) of the wavelet. Should be normalized and symmetric. + Default is the ricker wavelet. + max_distances : ndarray, optional + At each row, a ridge line is only connected if the relative max at + row[n] is within ``max_distances[n]`` from the relative max at + ``row[n+1]``. Default value is ``widths/4``. + gap_thresh : float, optional + If a relative maximum is not found within `max_distances`, + there will be a gap. A ridge line is discontinued if there are more + than `gap_thresh` points without connecting a new relative maximum. + Default is the first value of the widths array i.e. widths[0]. + min_length : int, optional + Minimum length a ridge line needs to be acceptable. + Default is ``cwt.shape[0] / 4``, ie 1/4-th the number of widths. + min_snr : float, optional + Minimum SNR ratio. Default 1. The signal is the maximum CWT coefficient + on the largest ridge line. The noise is `noise_perc` th percentile of + datapoints contained within the same ridge line. + noise_perc : float, optional + When calculating the noise floor, percentile of data points + examined below which to consider noise. Calculated using + `stats.scoreatpercentile`. Default is 10. + window_size : int, optional + Size of window to use to calculate noise floor. + Default is ``cwt.shape[1] / 20``. + + Returns + ------- + peaks_indices : ndarray + Indices of the locations in the `vector` where peaks were found. + The list is sorted. + + See Also + -------- + find_peaks + Find peaks inside a signal based on peak properties. + + Notes + ----- + This approach was designed for finding sharp peaks among noisy data, + however with proper parameter selection it should function well for + different peak shapes. + + The algorithm is as follows: + 1. Perform a continuous wavelet transform on `vector`, for the supplied + `widths`. This is a convolution of `vector` with `wavelet(width)` for + each width in `widths`. See `cwt`. + 2. Identify "ridge lines" in the cwt matrix. These are relative maxima + at each row, connected across adjacent rows. See identify_ridge_lines + 3. Filter the ridge_lines using filter_ridge_lines. + + .. versionadded:: 0.11.0 + + References + ---------- + .. [1] Bioinformatics (2006) 22 (17): 2059-2065. + :doi:`10.1093/bioinformatics/btl355` + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> xs = np.arange(0, np.pi, 0.05) + >>> data = np.sin(xs) + >>> peakind = signal.find_peaks_cwt(data, np.arange(1,10)) + >>> peakind, xs[peakind], data[peakind] + ([32], array([ 1.6]), array([ 0.9995736])) + + """ + widths = np.atleast_1d(np.asarray(widths)) + + if gap_thresh is None: + gap_thresh = np.ceil(widths[0]) + if max_distances is None: + max_distances = widths / 4.0 + if wavelet is None: + wavelet = _ricker + + cwt_dat = _cwt(vector, wavelet, widths) + ridge_lines = _identify_ridge_lines(cwt_dat, max_distances, gap_thresh) + filtered = _filter_ridge_lines(cwt_dat, ridge_lines, min_length=min_length, + window_size=window_size, min_snr=min_snr, + noise_perc=noise_perc) + max_locs = np.asarray([x[1][0] for x in filtered]) + max_locs.sort() + + return max_locs diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_savitzky_golay.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_savitzky_golay.py new file mode 100644 index 0000000000000000000000000000000000000000..addcbe6951f8df093461c47848f8027dfdd406f2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_savitzky_golay.py @@ -0,0 +1,357 @@ +import numpy as np +from scipy.linalg import lstsq +from scipy._lib._util import float_factorial +from scipy.ndimage import convolve1d # type: ignore[attr-defined] +from ._arraytools import axis_slice + + +def savgol_coeffs(window_length, polyorder, deriv=0, delta=1.0, pos=None, + use="conv"): + """Compute the coefficients for a 1-D Savitzky-Golay FIR filter. + + Parameters + ---------- + window_length : int + The length of the filter window (i.e., the number of coefficients). + polyorder : int + The order of the polynomial used to fit the samples. + `polyorder` must be less than `window_length`. + deriv : int, optional + The order of the derivative to compute. This must be a + nonnegative integer. The default is 0, which means to filter + the data without differentiating. + delta : float, optional + The spacing of the samples to which the filter will be applied. + This is only used if deriv > 0. + pos : int or None, optional + If pos is not None, it specifies evaluation position within the + window. The default is the middle of the window. + use : str, optional + Either 'conv' or 'dot'. This argument chooses the order of the + coefficients. The default is 'conv', which means that the + coefficients are ordered to be used in a convolution. With + use='dot', the order is reversed, so the filter is applied by + dotting the coefficients with the data set. + + Returns + ------- + coeffs : 1-D ndarray + The filter coefficients. + + See Also + -------- + savgol_filter + + Notes + ----- + .. versionadded:: 0.14.0 + + References + ---------- + A. Savitzky, M. J. E. Golay, Smoothing and Differentiation of Data by + Simplified Least Squares Procedures. Analytical Chemistry, 1964, 36 (8), + pp 1627-1639. + Jianwen Luo, Kui Ying, and Jing Bai. 2005. Savitzky-Golay smoothing and + differentiation filter for even number data. Signal Process. + 85, 7 (July 2005), 1429-1434. + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal import savgol_coeffs + >>> savgol_coeffs(5, 2) + array([-0.08571429, 0.34285714, 0.48571429, 0.34285714, -0.08571429]) + >>> savgol_coeffs(5, 2, deriv=1) + array([ 2.00000000e-01, 1.00000000e-01, 2.07548111e-16, -1.00000000e-01, + -2.00000000e-01]) + + Note that use='dot' simply reverses the coefficients. + + >>> savgol_coeffs(5, 2, pos=3) + array([ 0.25714286, 0.37142857, 0.34285714, 0.17142857, -0.14285714]) + >>> savgol_coeffs(5, 2, pos=3, use='dot') + array([-0.14285714, 0.17142857, 0.34285714, 0.37142857, 0.25714286]) + >>> savgol_coeffs(4, 2, pos=3, deriv=1, use='dot') + array([0.45, -0.85, -0.65, 1.05]) + + `x` contains data from the parabola x = t**2, sampled at + t = -1, 0, 1, 2, 3. `c` holds the coefficients that will compute the + derivative at the last position. When dotted with `x` the result should + be 6. + + >>> x = np.array([1, 0, 1, 4, 9]) + >>> c = savgol_coeffs(5, 2, pos=4, deriv=1, use='dot') + >>> c.dot(x) + 6.0 + """ + + # An alternative method for finding the coefficients when deriv=0 is + # t = np.arange(window_length) + # unit = (t == pos).astype(int) + # coeffs = np.polyval(np.polyfit(t, unit, polyorder), t) + # The method implemented here is faster. + + # To recreate the table of sample coefficients shown in the chapter on + # the Savitzy-Golay filter in the Numerical Recipes book, use + # window_length = nL + nR + 1 + # pos = nL + 1 + # c = savgol_coeffs(window_length, M, pos=pos, use='dot') + + if polyorder >= window_length: + raise ValueError("polyorder must be less than window_length.") + + halflen, rem = divmod(window_length, 2) + + if pos is None: + if rem == 0: + pos = halflen - 0.5 + else: + pos = halflen + + if not (0 <= pos < window_length): + raise ValueError("pos must be nonnegative and less than " + "window_length.") + + if use not in ['conv', 'dot']: + raise ValueError("`use` must be 'conv' or 'dot'") + + if deriv > polyorder: + coeffs = np.zeros(window_length) + return coeffs + + # Form the design matrix A. The columns of A are powers of the integers + # from -pos to window_length - pos - 1. The powers (i.e., rows) range + # from 0 to polyorder. (That is, A is a vandermonde matrix, but not + # necessarily square.) + x = np.arange(-pos, window_length - pos, dtype=float) + + if use == "conv": + # Reverse so that result can be used in a convolution. + x = x[::-1] + + order = np.arange(polyorder + 1).reshape(-1, 1) + A = x ** order + + # y determines which order derivative is returned. + y = np.zeros(polyorder + 1) + # The coefficient assigned to y[deriv] scales the result to take into + # account the order of the derivative and the sample spacing. + y[deriv] = float_factorial(deriv) / (delta ** deriv) + + # Find the least-squares solution of A*c = y + coeffs, _, _, _ = lstsq(A, y) + + return coeffs + + +def _polyder(p, m): + """Differentiate polynomials represented with coefficients. + + p must be a 1-D or 2-D array. In the 2-D case, each column gives + the coefficients of a polynomial; the first row holds the coefficients + associated with the highest power. m must be a nonnegative integer. + (numpy.polyder doesn't handle the 2-D case.) + """ + + if m == 0: + result = p + else: + n = len(p) + if n <= m: + result = np.zeros_like(p[:1, ...]) + else: + dp = p[:-m].copy() + for k in range(m): + rng = np.arange(n - k - 1, m - k - 1, -1) + dp *= rng.reshape((n - m,) + (1,) * (p.ndim - 1)) + result = dp + return result + + +def _fit_edge(x, window_start, window_stop, interp_start, interp_stop, + axis, polyorder, deriv, delta, y): + """ + Given an N-d array `x` and the specification of a slice of `x` from + `window_start` to `window_stop` along `axis`, create an interpolating + polynomial of each 1-D slice, and evaluate that polynomial in the slice + from `interp_start` to `interp_stop`. Put the result into the + corresponding slice of `y`. + """ + + # Get the edge into a (window_length, -1) array. + x_edge = axis_slice(x, start=window_start, stop=window_stop, axis=axis) + if axis == 0 or axis == -x.ndim: + xx_edge = x_edge + swapped = False + else: + xx_edge = x_edge.swapaxes(axis, 0) + swapped = True + xx_edge = xx_edge.reshape(xx_edge.shape[0], -1) + + # Fit the edges. poly_coeffs has shape (polyorder + 1, -1), + # where '-1' is the same as in xx_edge. + poly_coeffs = np.polyfit(np.arange(0, window_stop - window_start), + xx_edge, polyorder) + + if deriv > 0: + poly_coeffs = _polyder(poly_coeffs, deriv) + + # Compute the interpolated values for the edge. + i = np.arange(interp_start - window_start, interp_stop - window_start) + values = np.polyval(poly_coeffs, i.reshape(-1, 1)) / (delta ** deriv) + + # Now put the values into the appropriate slice of y. + # First reshape values to match y. + shp = list(y.shape) + shp[0], shp[axis] = shp[axis], shp[0] + values = values.reshape(interp_stop - interp_start, *shp[1:]) + if swapped: + values = values.swapaxes(0, axis) + # Get a view of the data to be replaced by values. + y_edge = axis_slice(y, start=interp_start, stop=interp_stop, axis=axis) + y_edge[...] = values + + +def _fit_edges_polyfit(x, window_length, polyorder, deriv, delta, axis, y): + """ + Use polynomial interpolation of x at the low and high ends of the axis + to fill in the halflen values in y. + + This function just calls _fit_edge twice, once for each end of the axis. + """ + halflen = window_length // 2 + _fit_edge(x, 0, window_length, 0, halflen, axis, + polyorder, deriv, delta, y) + n = x.shape[axis] + _fit_edge(x, n - window_length, n, n - halflen, n, axis, + polyorder, deriv, delta, y) + + +def savgol_filter(x, window_length, polyorder, deriv=0, delta=1.0, + axis=-1, mode='interp', cval=0.0): + """ Apply a Savitzky-Golay filter to an array. + + This is a 1-D filter. If `x` has dimension greater than 1, `axis` + determines the axis along which the filter is applied. + + Parameters + ---------- + x : array_like + The data to be filtered. If `x` is not a single or double precision + floating point array, it will be converted to type ``numpy.float64`` + before filtering. + window_length : int + The length of the filter window (i.e., the number of coefficients). + If `mode` is 'interp', `window_length` must be less than or equal + to the size of `x`. + polyorder : int + The order of the polynomial used to fit the samples. + `polyorder` must be less than `window_length`. + deriv : int, optional + The order of the derivative to compute. This must be a + nonnegative integer. The default is 0, which means to filter + the data without differentiating. + delta : float, optional + The spacing of the samples to which the filter will be applied. + This is only used if deriv > 0. Default is 1.0. + axis : int, optional + The axis of the array `x` along which the filter is to be applied. + Default is -1. + mode : str, optional + Must be 'mirror', 'constant', 'nearest', 'wrap' or 'interp'. This + determines the type of extension to use for the padded signal to + which the filter is applied. When `mode` is 'constant', the padding + value is given by `cval`. See the Notes for more details on 'mirror', + 'constant', 'wrap', and 'nearest'. + When the 'interp' mode is selected (the default), no extension + is used. Instead, a degree `polyorder` polynomial is fit to the + last `window_length` values of the edges, and this polynomial is + used to evaluate the last `window_length // 2` output values. + cval : scalar, optional + Value to fill past the edges of the input if `mode` is 'constant'. + Default is 0.0. + + Returns + ------- + y : ndarray, same shape as `x` + The filtered data. + + See Also + -------- + savgol_coeffs + + Notes + ----- + Details on the `mode` options: + + 'mirror': + Repeats the values at the edges in reverse order. The value + closest to the edge is not included. + 'nearest': + The extension contains the nearest input value. + 'constant': + The extension contains the value given by the `cval` argument. + 'wrap': + The extension contains the values from the other end of the array. + + For example, if the input is [1, 2, 3, 4, 5, 6, 7, 8], and + `window_length` is 7, the following shows the extended data for + the various `mode` options (assuming `cval` is 0):: + + mode | Ext | Input | Ext + -----------+---------+------------------------+--------- + 'mirror' | 4 3 2 | 1 2 3 4 5 6 7 8 | 7 6 5 + 'nearest' | 1 1 1 | 1 2 3 4 5 6 7 8 | 8 8 8 + 'constant' | 0 0 0 | 1 2 3 4 5 6 7 8 | 0 0 0 + 'wrap' | 6 7 8 | 1 2 3 4 5 6 7 8 | 1 2 3 + + .. versionadded:: 0.14.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal import savgol_filter + >>> np.set_printoptions(precision=2) # For compact display. + >>> x = np.array([2, 2, 5, 2, 1, 0, 1, 4, 9]) + + Filter with a window length of 5 and a degree 2 polynomial. Use + the defaults for all other parameters. + + >>> savgol_filter(x, 5, 2) + array([1.66, 3.17, 3.54, 2.86, 0.66, 0.17, 1. , 4. , 9. ]) + + Note that the last five values in x are samples of a parabola, so + when mode='interp' (the default) is used with polyorder=2, the last + three values are unchanged. Compare that to, for example, + `mode='nearest'`: + + >>> savgol_filter(x, 5, 2, mode='nearest') + array([1.74, 3.03, 3.54, 2.86, 0.66, 0.17, 1. , 4.6 , 7.97]) + + """ + if mode not in ["mirror", "constant", "nearest", "interp", "wrap"]: + raise ValueError("mode must be 'mirror', 'constant', 'nearest' " + "'wrap' or 'interp'.") + + x = np.asarray(x) + # Ensure that x is either single or double precision floating point. + if x.dtype != np.float64 and x.dtype != np.float32: + x = x.astype(np.float64) + + coeffs = savgol_coeffs(window_length, polyorder, deriv=deriv, delta=delta) + + if mode == "interp": + if window_length > x.shape[axis]: + raise ValueError("If mode is 'interp', window_length must be less " + "than or equal to the size of x.") + + # Do not pad. Instead, for the elements within `window_length // 2` + # of the ends of the sequence, use the polynomial that is fitted to + # the last `window_length` elements. + y = convolve1d(x, coeffs, axis=axis, mode="constant") + _fit_edges_polyfit(x, window_length, polyorder, deriv, delta, axis, y) + else: + # Any mode other than 'interp' is passed on to ndimage.convolve1d. + y = convolve1d(x, coeffs, axis=axis, mode=mode, cval=cval) + + return y diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_short_time_fft.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_short_time_fft.py new file mode 100644 index 0000000000000000000000000000000000000000..6c87718dea8b777d23c77d6147b4ca6368204637 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_short_time_fft.py @@ -0,0 +1,1738 @@ +"""Implementation of an FFT-based Short-time Fourier Transform. """ + +# Implementation Notes for this file (as of 2023-07) +# -------------------------------------------------- +# * MyPy version 1.1.1 does not seem to support decorated property methods +# properly. Hence, applying ``@property`` to methods decorated with `@cache`` +# (as tried with the ``lower_border_end`` method) causes a mypy error when +# accessing it as an index (e.g., ``SFT.lower_border_end[0]``). +# * Since the method `stft` and `istft` have identical names as the legacy +# functions in the signal module, referencing them as HTML link in the +# docstrings has to be done by an explicit `~ShortTimeFFT.stft` instead of an +# ambiguous `stft` (The ``~`` hides the class / module name). +# * The HTML documentation currently renders each method/property on a separate +# page without reference to the parent class. Thus, a link to `ShortTimeFFT` +# was added to the "See Also" section of each method/property. These links +# can be removed, when SciPy updates ``pydata-sphinx-theme`` to >= 0.13.3 +# (currently 0.9). Consult Issue 18512 and PR 16660 for further details. +# + +# Provides typing union operator ``|`` in Python 3.9: +# Linter does not allow to import ``Generator`` from ``typing`` module: +from collections.abc import Generator, Callable +from functools import cache, lru_cache, partial +from typing import get_args, Literal + +import numpy as np + +import scipy.fft as fft_lib +from scipy.signal import detrend +from scipy.signal.windows import get_window + +__all__ = ['ShortTimeFFT'] + + +#: Allowed values for parameter `padding` of method `ShortTimeFFT.stft()`: +PAD_TYPE = Literal['zeros', 'edge', 'even', 'odd'] + +#: Allowed values for property `ShortTimeFFT.fft_mode`: +FFT_MODE_TYPE = Literal['twosided', 'centered', 'onesided', 'onesided2X'] + + +def _calc_dual_canonical_window(win: np.ndarray, hop: int) -> np.ndarray: + """Calculate canonical dual window for 1d window `win` and a time step + of `hop` samples. + + A ``ValueError`` is raised, if the inversion fails. + + This is a separate function not a method, since it is also used in the + class method ``ShortTimeFFT.from_dual()``. + """ + if hop > len(win): + raise ValueError(f"{hop=} is larger than window length of {len(win)}" + + " => STFT not invertible!") + if issubclass(win.dtype.type, np.integer): + raise ValueError("Parameter 'win' cannot be of integer type, but " + + f"{win.dtype=} => STFT not invertible!") + # The calculation of `relative_resolution` does not work for ints. + # Furthermore, `win / DD` casts the integers away, thus an implicit + # cast is avoided, which can always cause confusion when using 32-Bit + # floats. + + w2 = win.real**2 + win.imag**2 # win*win.conj() does not ensure w2 is real + DD = w2.copy() + for k_ in range(hop, len(win), hop): + DD[k_:] += w2[:-k_] + DD[:-k_] += w2[k_:] + + # check DD > 0: + relative_resolution = np.finfo(win.dtype).resolution * max(DD) + if not np.all(DD >= relative_resolution): + raise ValueError("Short-time Fourier Transform not invertible!") + + return win / DD + + +# noinspection PyShadowingNames +class ShortTimeFFT: + r"""Provide a parametrized discrete Short-time Fourier transform (stft) + and its inverse (istft). + + .. currentmodule:: scipy.signal.ShortTimeFFT + + The `~ShortTimeFFT.stft` calculates sequential FFTs by sliding a + window (`win`) over an input signal by `hop` increments. It can be used to + quantify the change of the spectrum over time. + + The `~ShortTimeFFT.stft` is represented by a complex-valued matrix S[q,p] + where the p-th column represents an FFT with the window centered at the + time t[p] = p * `delta_t` = p * `hop` * `T` where `T` is the sampling + interval of the input signal. The q-th row represents the values at the + frequency f[q] = q * `delta_f` with `delta_f` = 1 / (`mfft` * `T`) being + the bin width of the FFT. + + The inverse STFT `~ShortTimeFFT.istft` is calculated by reversing the steps + of the STFT: Take the IFFT of the p-th slice of S[q,p] and multiply the + result with the so-called dual window (see `dual_win`). Shift the result by + p * `delta_t` and add the result to previous shifted results to reconstruct + the signal. If only the dual window is known and the STFT is invertible, + `from_dual` can be used to instantiate this class. + + Due to the convention of time t = 0 being at the first sample of the input + signal, the STFT values typically have negative time slots. Hence, + negative indexes like `p_min` or `k_min` do not indicate counting + backwards from an array's end like in standard Python indexing but being + left of t = 0. + + More detailed information can be found in the :ref:`tutorial_stft` section + of the :ref:`user_guide`. + + Note that all parameters of the initializer, except `scale_to` (which uses + `scaling`) have identical named attributes. + + Parameters + ---------- + win : np.ndarray + The window must be a real- or complex-valued 1d array. + hop : int + The increment in samples, by which the window is shifted in each step. + fs : float + Sampling frequency of input signal and window. Its relation to the + sampling interval `T` is ``T = 1 / fs``. + fft_mode : 'twosided', 'centered', 'onesided', 'onesided2X' + Mode of FFT to be used (default 'onesided'). + See property `fft_mode` for details. + mfft: int | None + Length of the FFT used, if a zero padded FFT is desired. + If ``None`` (default), the length of the window `win` is used. + dual_win : np.ndarray | None + The dual window of `win`. If set to ``None``, it is calculated if + needed. + scale_to : 'magnitude', 'psd' | None + If not ``None`` (default) the window function is scaled, so each STFT + column represents either a 'magnitude' or a power spectral density + ('psd') spectrum. This parameter sets the property `scaling` to the + same value. See method `scale_to` for details. + phase_shift : int | None + If set, add a linear phase `phase_shift` / `mfft` * `f` to each + frequency `f`. The default value 0 ensures that there is no phase shift + on the zeroth slice (in which t=0 is centered). See property + `phase_shift` for more details. + + Examples + -------- + The following example shows the magnitude of the STFT of a sine with + varying frequency :math:`f_i(t)` (marked by a red dashed line in the plot): + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.signal import ShortTimeFFT + >>> from scipy.signal.windows import gaussian + ... + >>> T_x, N = 1 / 20, 1000 # 20 Hz sampling rate for 50 s signal + >>> t_x = np.arange(N) * T_x # time indexes for signal + >>> f_i = 1 * np.arctan((t_x - t_x[N // 2]) / 2) + 5 # varying frequency + >>> x = np.sin(2*np.pi*np.cumsum(f_i)*T_x) # the signal + + The utilized Gaussian window is 50 samples or 2.5 s long. The parameter + ``mfft=200`` in `ShortTimeFFT` causes the spectrum to be oversampled + by a factor of 4: + + >>> g_std = 8 # standard deviation for Gaussian window in samples + >>> w = gaussian(50, std=g_std, sym=True) # symmetric Gaussian window + >>> SFT = ShortTimeFFT(w, hop=10, fs=1/T_x, mfft=200, scale_to='magnitude') + >>> Sx = SFT.stft(x) # perform the STFT + + In the plot, the time extent of the signal `x` is marked by vertical dashed + lines. Note that the SFT produces values outside the time range of `x`. The + shaded areas on the left and the right indicate border effects caused + by the window slices in that area not fully being inside time range of + `x`: + + >>> fig1, ax1 = plt.subplots(figsize=(6., 4.)) # enlarge plot a bit + >>> t_lo, t_hi = SFT.extent(N)[:2] # time range of plot + >>> ax1.set_title(rf"STFT ({SFT.m_num*SFT.T:g}$\,s$ Gaussian window, " + + ... rf"$\sigma_t={g_std*SFT.T}\,$s)") + >>> ax1.set(xlabel=f"Time $t$ in seconds ({SFT.p_num(N)} slices, " + + ... rf"$\Delta t = {SFT.delta_t:g}\,$s)", + ... ylabel=f"Freq. $f$ in Hz ({SFT.f_pts} bins, " + + ... rf"$\Delta f = {SFT.delta_f:g}\,$Hz)", + ... xlim=(t_lo, t_hi)) + ... + >>> im1 = ax1.imshow(abs(Sx), origin='lower', aspect='auto', + ... extent=SFT.extent(N), cmap='viridis') + >>> ax1.plot(t_x, f_i, 'r--', alpha=.5, label='$f_i(t)$') + >>> fig1.colorbar(im1, label="Magnitude $|S_x(t, f)|$") + ... + >>> # Shade areas where window slices stick out to the side: + >>> for t0_, t1_ in [(t_lo, SFT.lower_border_end[0] * SFT.T), + ... (SFT.upper_border_begin(N)[0] * SFT.T, t_hi)]: + ... ax1.axvspan(t0_, t1_, color='w', linewidth=0, alpha=.2) + >>> for t_ in [0, N * SFT.T]: # mark signal borders with vertical line: + ... ax1.axvline(t_, color='y', linestyle='--', alpha=0.5) + >>> ax1.legend() + >>> fig1.tight_layout() + >>> plt.show() + + Reconstructing the signal with the `~ShortTimeFFT.istft` is + straightforward, but note that the length of `x1` should be specified, + since the SFT length increases in `hop` steps: + + >>> SFT.invertible # check if invertible + True + >>> x1 = SFT.istft(Sx, k1=N) + >>> np.allclose(x, x1) + True + + It is possible to calculate the SFT of signal parts: + + >>> N2 = SFT.nearest_k_p(N // 2) + >>> Sx0 = SFT.stft(x[:N2]) + >>> Sx1 = SFT.stft(x[N2:]) + + When assembling sequential STFT parts together, the overlap needs to be + considered: + + >>> p0_ub = SFT.upper_border_begin(N2)[1] - SFT.p_min + >>> p1_le = SFT.lower_border_end[1] - SFT.p_min + >>> Sx01 = np.hstack((Sx0[:, :p0_ub], + ... Sx0[:, p0_ub:] + Sx1[:, :p1_le], + ... Sx1[:, p1_le:])) + >>> np.allclose(Sx01, Sx) # Compare with SFT of complete signal + True + + It is also possible to calculate the `itsft` for signal parts: + + >>> y_p = SFT.istft(Sx, N//3, N//2) + >>> np.allclose(y_p, x[N//3:N//2]) + True + + """ + # immutable attributes (only have getters but no setters): + _win: np.ndarray # window + _dual_win: np.ndarray | None = None # canonical dual window + _hop: int # Step of STFT in number of samples + + # mutable attributes: + _fs: float # sampling frequency of input signal and window + _fft_mode: FFT_MODE_TYPE = 'onesided' # Mode of FFT to use + _mfft: int # length of FFT used - defaults to len(win) + _scaling: Literal['magnitude', 'psd'] | None = None # Scaling of _win + _phase_shift: int | None # amount to shift phase of FFT in samples + + # attributes for caching calculated values: + _fac_mag: float | None = None + _fac_psd: float | None = None + _lower_border_end: tuple[int, int] | None = None + + def __init__(self, win: np.ndarray, hop: int, fs: float, *, + fft_mode: FFT_MODE_TYPE = 'onesided', + mfft: int | None = None, + dual_win: np.ndarray | None = None, + scale_to: Literal['magnitude', 'psd'] | None = None, + phase_shift: int | None = 0): + if not (win.ndim == 1 and win.size > 0): + raise ValueError(f"Parameter win must be 1d, but {win.shape=}!") + if not all(np.isfinite(win)): + raise ValueError("Parameter win must have finite entries!") + if not (hop >= 1 and isinstance(hop, int)): + raise ValueError(f"Parameter {hop=} is not an integer >= 1!") + self._win, self._hop, self.fs = win, hop, fs + + self.mfft = len(win) if mfft is None else mfft + + if dual_win is not None: + if dual_win.shape != win.shape: + raise ValueError(f"{dual_win.shape=} must equal {win.shape=}!") + if not all(np.isfinite(dual_win)): + raise ValueError("Parameter dual_win must be a finite array!") + self._dual_win = dual_win # needs to be set before scaling + + if scale_to is not None: # needs to be set before fft_mode + self.scale_to(scale_to) + + self.fft_mode, self.phase_shift = fft_mode, phase_shift + + @classmethod + def from_dual(cls, dual_win: np.ndarray, hop: int, fs: float, *, + fft_mode: FFT_MODE_TYPE = 'onesided', + mfft: int | None = None, + scale_to: Literal['magnitude', 'psd'] | None = None, + phase_shift: int | None = 0): + r"""Instantiate a `ShortTimeFFT` by only providing a dual window. + + If an STFT is invertible, it is possible to calculate the window `win` + from a given dual window `dual_win`. All other parameters have the + same meaning as in the initializer of `ShortTimeFFT`. + + As explained in the :ref:`tutorial_stft` section of the + :ref:`user_guide`, an invertible STFT can be interpreted as series + expansion of time-shifted and frequency modulated dual windows. E.g., + the series coefficient S[q,p] belongs to the term, which shifted + `dual_win` by p * `delta_t` and multiplied it by + exp( 2 * j * pi * t * q * `delta_f`). + + + Examples + -------- + The following example discusses decomposing a signal into time- and + frequency-shifted Gaussians. A Gaussian with standard deviation of + one made up of 51 samples will be used: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.signal import ShortTimeFFT + >>> from scipy.signal.windows import gaussian + ... + >>> T, N = 0.1, 51 + >>> d_win = gaussian(N, std=1/T, sym=True) # symmetric Gaussian window + >>> t = T * (np.arange(N) - N//2) + ... + >>> fg1, ax1 = plt.subplots() + >>> ax1.set_title(r"Dual Window: Gaussian with $\sigma_t=1$") + >>> ax1.set(xlabel=f"Time $t$ in seconds ({N} samples, $T={T}$ s)", + ... xlim=(t[0], t[-1]), ylim=(0, 1.1*max(d_win))) + >>> ax1.plot(t, d_win, 'C0-') + + The following plot with the overlap of 41, 11 and 2 samples show how + the `hop` interval affects the shape of the window `win`: + + >>> fig2, axx = plt.subplots(3, 1, sharex='all') + ... + >>> axx[0].set_title(r"Windows for hop$\in\{10, 40, 49\}$") + >>> for c_, h_ in enumerate([10, 40, 49]): + ... SFT = ShortTimeFFT.from_dual(d_win, h_, 1/T) + ... axx[c_].plot(t + h_ * T, SFT.win, 'k--', alpha=.3, label=None) + ... axx[c_].plot(t - h_ * T, SFT.win, 'k:', alpha=.3, label=None) + ... axx[c_].plot(t, SFT.win, f'C{c_+1}', + ... label=r"$\Delta t=%0.1f\,$s" % SFT.delta_t) + ... axx[c_].set_ylim(0, 1.1*max(SFT.win)) + ... axx[c_].legend(loc='center') + >>> axx[-1].set(xlabel=f"Time $t$ in seconds ({N} samples, $T={T}$ s)", + ... xlim=(t[0], t[-1])) + >>> plt.show() + + Beside the window `win` centered at t = 0 the previous (t = -`delta_t`) + and following window (t = `delta_t`) are depicted. It can be seen that + for small `hop` intervals, the window is compact and smooth, having a + good time-frequency concentration in the STFT. For the large `hop` + interval of 4.9 s, the window has small values around t = 0, which are + not covered by the overlap of the adjacent windows, which could lead to + numeric inaccuracies. Furthermore, the peaky shape at the beginning and + the end of the window points to a higher bandwidth, resulting in a + poorer time-frequency resolution of the STFT. + Hence, the choice of the `hop` interval will be a compromise between + a time-frequency resolution and memory requirements demanded by small + `hop` sizes. + + See Also + -------- + from_window: Create instance by wrapping `get_window`. + ShortTimeFFT: Create instance using standard initializer. + """ + win = _calc_dual_canonical_window(dual_win, hop) + return cls(win=win, hop=hop, fs=fs, fft_mode=fft_mode, mfft=mfft, + dual_win=dual_win, scale_to=scale_to, + phase_shift=phase_shift) + + @classmethod + def from_window(cls, win_param: str | tuple | float, + fs: float, nperseg: int, noverlap: int, *, + symmetric_win: bool = False, + fft_mode: FFT_MODE_TYPE = 'onesided', + mfft: int | None = None, + scale_to: Literal['magnitude', 'psd'] | None = None, + phase_shift: int | None = 0): + """Instantiate `ShortTimeFFT` by using `get_window`. + + The method `get_window` is used to create a window of length + `nperseg`. The parameter names `noverlap`, and `nperseg` are used here, + since they more inline with other classical STFT libraries. + + Parameters + ---------- + win_param: Union[str, tuple, float], + Parameters passed to `get_window`. For windows with no parameters, + it may be a string (e.g., ``'hann'``), for parametrized windows a + tuple, (e.g., ``('gaussian', 2.)``) or a single float specifying + the shape parameter of a kaiser window (i.e. ``4.`` and + ``('kaiser', 4.)`` are equal. See `get_window` for more details. + fs : float + Sampling frequency of input signal. Its relation to the + sampling interval `T` is ``T = 1 / fs``. + nperseg: int + Window length in samples, which corresponds to the `m_num`. + noverlap: int + Window overlap in samples. It relates to the `hop` increment by + ``hop = npsereg - noverlap``. + symmetric_win: bool + If ``True`` then a symmetric window is generated, else a periodic + window is generated (default). Though symmetric windows seem for + most applications to be more sensible, the default of a periodic + windows was chosen to correspond to the default of `get_window`. + fft_mode : 'twosided', 'centered', 'onesided', 'onesided2X' + Mode of FFT to be used (default 'onesided'). + See property `fft_mode` for details. + mfft: int | None + Length of the FFT used, if a zero padded FFT is desired. + If ``None`` (default), the length of the window `win` is used. + scale_to : 'magnitude', 'psd' | None + If not ``None`` (default) the window function is scaled, so each + STFT column represents either a 'magnitude' or a power spectral + density ('psd') spectrum. This parameter sets the property + `scaling` to the same value. See method `scale_to` for details. + phase_shift : int | None + If set, add a linear phase `phase_shift` / `mfft` * `f` to each + frequency `f`. The default value 0 ensures that there is no phase + shift on the zeroth slice (in which t=0 is centered). See property + `phase_shift` for more details. + + Examples + -------- + The following instances ``SFT0`` and ``SFT1`` are equivalent: + + >>> from scipy.signal import ShortTimeFFT, get_window + >>> nperseg = 9 # window length + >>> w = get_window(('gaussian', 2.), nperseg) + >>> fs = 128 # sampling frequency + >>> hop = 3 # increment of STFT time slice + >>> SFT0 = ShortTimeFFT(w, hop, fs=fs) + >>> SFT1 = ShortTimeFFT.from_window(('gaussian', 2.), fs, nperseg, + ... noverlap=nperseg-hop) + + See Also + -------- + scipy.signal.get_window: Return a window of a given length and type. + from_dual: Create instance using dual window. + ShortTimeFFT: Create instance using standard initializer. + """ + win = get_window(win_param, nperseg, fftbins=not symmetric_win) + return cls(win, hop=nperseg-noverlap, fs=fs, fft_mode=fft_mode, + mfft=mfft, scale_to=scale_to, phase_shift=phase_shift) + + @property + def win(self) -> np.ndarray: + """Window function as real- or complex-valued 1d array. + + This attribute is read only, since `dual_win` depends on it. + + See Also + -------- + dual_win: Canonical dual window. + m_num: Number of samples in window `win`. + m_num_mid: Center index of window `win`. + mfft: Length of input for the FFT used - may be larger than `m_num`. + hop: ime increment in signal samples for sliding window. + win: Window function as real- or complex-valued 1d array. + ShortTimeFFT: Class this property belongs to. + """ + return self._win + + @property + def hop(self) -> int: + """Time increment in signal samples for sliding window. + + This attribute is read only, since `dual_win` depends on it. + + See Also + -------- + delta_t: Time increment of STFT (``hop*T``) + m_num: Number of samples in window `win`. + m_num_mid: Center index of window `win`. + mfft: Length of input for the FFT used - may be larger than `m_num`. + T: Sampling interval of input signal and of the window. + win: Window function as real- or complex-valued 1d array. + ShortTimeFFT: Class this property belongs to. + """ + return self._hop + + @property + def T(self) -> float: + """Sampling interval of input signal and of the window. + + A ``ValueError`` is raised if it is set to a non-positive value. + + See Also + -------- + delta_t: Time increment of STFT (``hop*T``) + hop: Time increment in signal samples for sliding window. + fs: Sampling frequency (being ``1/T``) + t: Times of STFT for an input signal with `n` samples. + ShortTimeFFT: Class this property belongs to. + """ + return 1 / self._fs + + @T.setter + def T(self, v: float): + """Sampling interval of input signal and of the window. + + A ``ValueError`` is raised if it is set to a non-positive value. + """ + if not (v > 0): + raise ValueError(f"Sampling interval T={v} must be positive!") + self._fs = 1 / v + + @property + def fs(self) -> float: + """Sampling frequency of input signal and of the window. + + The sampling frequency is the inverse of the sampling interval `T`. + A ``ValueError`` is raised if it is set to a non-positive value. + + See Also + -------- + delta_t: Time increment of STFT (``hop*T``) + hop: Time increment in signal samples for sliding window. + T: Sampling interval of input signal and of the window (``1/fs``). + ShortTimeFFT: Class this property belongs to. + """ + return self._fs + + @fs.setter + def fs(self, v: float): + """Sampling frequency of input signal and of the window. + + The sampling frequency is the inverse of the sampling interval `T`. + A ``ValueError`` is raised if it is set to a non-positive value. + """ + if not (v > 0): + raise ValueError(f"Sampling frequency fs={v} must be positive!") + self._fs = v + + @property + def fft_mode(self) -> FFT_MODE_TYPE: + """Mode of utilized FFT ('twosided', 'centered', 'onesided' or + 'onesided2X'). + + It can have the following values: + + 'twosided': + Two-sided FFT, where values for the negative frequencies are in + upper half of the array. Corresponds to :func:`~scipy.fft.fft()`. + 'centered': + Two-sided FFT with the values being ordered along monotonically + increasing frequencies. Corresponds to applying + :func:`~scipy.fft.fftshift()` to :func:`~scipy.fft.fft()`. + 'onesided': + Calculates only values for non-negative frequency values. + Corresponds to :func:`~scipy.fft.rfft()`. + 'onesided2X': + Like `onesided`, but the non-zero frequencies are doubled if + `scaling` is set to 'magnitude' or multiplied by ``sqrt(2)`` if + set to 'psd'. If `scaling` is ``None``, setting `fft_mode` to + `onesided2X` is not allowed. + If the FFT length `mfft` is even, the last FFT value is not paired, + and thus it is not scaled. + + Note that `onesided` and `onesided2X` do not work for complex-valued signals or + complex-valued windows. Furthermore, the frequency values can be obtained by + reading the `f` property, and the number of samples by accessing the `f_pts` + property. + + See Also + -------- + delta_f: Width of the frequency bins of the STFT. + f: Frequencies values of the STFT. + f_pts: Width of the frequency bins of the STFT. + onesided_fft: True if a one-sided FFT is used. + scaling: Normalization applied to the window function + ShortTimeFFT: Class this property belongs to. + """ + return self._fft_mode + + @fft_mode.setter + def fft_mode(self, t: FFT_MODE_TYPE): + """Set mode of FFT. + + Allowed values are 'twosided', 'centered', 'onesided', 'onesided2X'. + See the property `fft_mode` for more details. + """ + if t not in (fft_mode_types := get_args(FFT_MODE_TYPE)): + raise ValueError(f"fft_mode='{t}' not in {fft_mode_types}!") + + if t in {'onesided', 'onesided2X'} and np.iscomplexobj(self.win): + raise ValueError(f"One-sided spectra, i.e., fft_mode='{t}', " + + "are not allowed for complex-valued windows!") + + if t == 'onesided2X' and self.scaling is None: + raise ValueError(f"For scaling is None, fft_mode='{t}' is invalid!" + "Do scale_to('psd') or scale_to('magnitude')!") + self._fft_mode = t + + @property + def mfft(self) -> int: + """Length of input for the FFT used - may be larger than window + length `m_num`. + + If not set, `mfft` defaults to the window length `m_num`. + + See Also + -------- + f_pts: Number of points along the frequency axis. + f: Frequencies values of the STFT. + m_num: Number of samples in window `win`. + ShortTimeFFT: Class this property belongs to. + """ + return self._mfft + + @mfft.setter + def mfft(self, n_: int): + """Setter for the length of FFT utilized. + + See the property `mfft` for further details. + """ + if not (n_ >= self.m_num): + raise ValueError(f"Attribute mfft={n_} needs to be at least the " + + f"window length m_num={self.m_num}!") + self._mfft = n_ + + @property + def scaling(self) -> Literal['magnitude', 'psd'] | None: + """Normalization applied to the window function + ('magnitude', 'psd' or ``None``). + + If not ``None``, the FFTs can be either interpreted as a magnitude or + a power spectral density spectrum. + + The window function can be scaled by calling the `scale_to` method, + or it is set by the initializer parameter ``scale_to``. + + See Also + -------- + fac_magnitude: Scaling factor for to a magnitude spectrum. + fac_psd: Scaling factor for to a power spectral density spectrum. + fft_mode: Mode of utilized FFT + scale_to: Scale window to obtain 'magnitude' or 'psd' scaling. + ShortTimeFFT: Class this property belongs to. + """ + return self._scaling + + def scale_to(self, scaling: Literal['magnitude', 'psd']): + """Scale window to obtain 'magnitude' or 'psd' scaling for the STFT. + + The window of a 'magnitude' spectrum has an integral of one, i.e., unit + area for non-negative windows. This ensures that absolute the values of + spectrum does not change if the length of the window changes (given + the input signal is stationary). + + To represent the power spectral density ('psd') for varying length + windows the area of the absolute square of the window needs to be + unity. + + The `scaling` property shows the current scaling. The properties + `fac_magnitude` and `fac_psd` show the scaling factors required to + scale the STFT values to a magnitude or a psd spectrum. + + This method is called, if the initializer parameter `scale_to` is set. + + See Also + -------- + fac_magnitude: Scaling factor for to a magnitude spectrum. + fac_psd: Scaling factor for to a power spectral density spectrum. + fft_mode: Mode of utilized FFT + scaling: Normalization applied to the window function. + ShortTimeFFT: Class this method belongs to. + """ + if scaling not in (scaling_values := {'magnitude', 'psd'}): + raise ValueError(f"{scaling=} not in {scaling_values}!") + if self._scaling == scaling: # do nothing + return + + s_fac = self.fac_psd if scaling == 'psd' else self.fac_magnitude + self._win = self._win * s_fac + if self._dual_win is not None: + self._dual_win = self._dual_win / s_fac + self._fac_mag, self._fac_psd = None, None # reset scaling factors + self._scaling = scaling + + @property + def phase_shift(self) -> int | None: + """If set, add linear phase `phase_shift` / `mfft` * `f` to each FFT + slice of frequency `f`. + + Shifting (more precisely `rolling`) an `mfft`-point FFT input by + `phase_shift` samples results in a multiplication of the output by + ``np.exp(2j*np.pi*q*phase_shift/mfft)`` at the frequency q * `delta_f`. + + The default value 0 ensures that there is no phase shift on the + zeroth slice (in which t=0 is centered). + No phase shift (``phase_shift is None``) is equivalent to + ``phase_shift = -mfft//2``. In this case slices are not shifted + before calculating the FFT. + + The absolute value of `phase_shift` is limited to be less than `mfft`. + + See Also + -------- + delta_f: Width of the frequency bins of the STFT. + f: Frequencies values of the STFT. + mfft: Length of input for the FFT used + ShortTimeFFT: Class this property belongs to. + """ + return self._phase_shift + + @phase_shift.setter + def phase_shift(self, v: int | None): + """The absolute value of the phase shift needs to be less than mfft + samples. + + See the `phase_shift` getter method for more details. + """ + if v is None: + self._phase_shift = v + return + if not isinstance(v, int): + raise ValueError(f"phase_shift={v} has the unit samples. Hence " + + "it needs to be an int or it may be None!") + if not (-self.mfft < v < self.mfft): + raise ValueError("-mfft < phase_shift < mfft does not hold " + + f"for mfft={self.mfft}, phase_shift={v}!") + self._phase_shift = v + + def _x_slices(self, x: np.ndarray, k_off: int, p0: int, p1: int, + padding: PAD_TYPE) -> Generator[np.ndarray, None, None]: + """Generate signal slices along last axis of `x`. + + This method is only used by `stft_detrend`. The parameters are + described in `~ShortTimeFFT.stft`. + """ + if padding not in (padding_types := get_args(PAD_TYPE)): + raise ValueError(f"Parameter {padding=} not in {padding_types}!") + pad_kws: dict[str, dict] = { # possible keywords to pass to np.pad: + 'zeros': dict(mode='constant', constant_values=(0, 0)), + 'edge': dict(mode='edge'), + 'even': dict(mode='reflect', reflect_type='even'), + 'odd': dict(mode='reflect', reflect_type='odd'), + } # typing of pad_kws is needed to make mypy happy + + n, n1 = x.shape[-1], (p1 - p0) * self.hop + k0 = p0 * self.hop - self.m_num_mid + k_off # start sample + k1 = k0 + n1 + self.m_num # end sample + + i0, i1 = max(k0, 0), min(k1, n) # indexes to shorten x + # dimensions for padding x: + pad_width = [(0, 0)] * (x.ndim-1) + [(-min(k0, 0), max(k1 - n, 0))] + + x1 = np.pad(x[..., i0:i1], pad_width, **pad_kws[padding]) + for k_ in range(0, n1, self.hop): + yield x1[..., k_:k_ + self.m_num] + + def stft(self, x: np.ndarray, p0: int | None = None, + p1: int | None = None, *, k_offset: int = 0, + padding: PAD_TYPE = 'zeros', axis: int = -1) \ + -> np.ndarray: + """Perform the short-time Fourier transform. + + A two-dimensional matrix with ``p1-p0`` columns is calculated. + The `f_pts` rows represent value at the frequencies `f`. The q-th + column of the windowed FFT with the window `win` is centered at t[q]. + The columns represent the values at the frequencies `f`. + + Parameters + ---------- + x + The input signal as real or complex valued array. For complex values, the + property `fft_mode` must be set to 'twosided' or 'centered'. + p0 + The first element of the range of slices to calculate. If ``None`` + then it is set to :attr:`p_min`, which is the smallest possible + slice. + p1 + The end of the array. If ``None`` then `p_max(n)` is used. + k_offset + Index of first sample (t = 0) in `x`. + padding + Kind of values which are added, when the sliding window sticks out + on either the lower or upper end of the input `x`. Zeros are added + if the default 'zeros' is set. For 'edge' either the first or the + last value of `x` is used. 'even' pads by reflecting the + signal on the first or last sample and 'odd' additionally + multiplies it with -1. + axis + The axis of `x` over which to compute the STFT. + If not given, the last axis is used. + + Returns + ------- + S + A complex array is returned with the dimension always being larger + by one than of `x`. The last axis always represent the time slices + of the STFT. `axis` defines the frequency axis (default second to + last). E.g., for a one-dimensional `x`, a complex 2d array is + returned, with axis 0 representing frequency and axis 1 the time + slices. + + See Also + -------- + delta_f: Width of the frequency bins of the STFT. + delta_t: Time increment of STFT + f: Frequencies values of the STFT. + invertible: Check if STFT is invertible. + :meth:`~ShortTimeFFT.istft`: Inverse short-time Fourier transform. + p_range: Determine and validate slice index range. + stft_detrend: STFT with detrended segments. + t: Times of STFT for an input signal with `n` samples. + :class:`scipy.signal.ShortTimeFFT`: Class this method belongs to. + """ + return self.stft_detrend(x, None, p0, p1, k_offset=k_offset, + padding=padding, axis=axis) + + def stft_detrend(self, x: np.ndarray, + detr: Callable[[np.ndarray], np.ndarray] | Literal['linear', 'constant'] | None, # noqa: E501 + p0: int | None = None, p1: int | None = None, *, + k_offset: int = 0, padding: PAD_TYPE = 'zeros', + axis: int = -1) \ + -> np.ndarray: + """Short-time Fourier transform with a trend being subtracted from each + segment beforehand. + + If `detr` is set to 'constant', the mean is subtracted, if set to + "linear", the linear trend is removed. This is achieved by calling + :func:`scipy.signal.detrend`. If `detr` is a function, `detr` is + applied to each segment. + All other parameters have the same meaning as in `~ShortTimeFFT.stft`. + + Note that due to the detrending, the original signal cannot be + reconstructed by the `~ShortTimeFFT.istft`. + + See Also + -------- + invertible: Check if STFT is invertible. + :meth:`~ShortTimeFFT.istft`: Inverse short-time Fourier transform. + :meth:`~ShortTimeFFT.stft`: Short-time Fourier transform + (without detrending). + :class:`scipy.signal.ShortTimeFFT`: Class this method belongs to. + """ + if self.onesided_fft and np.iscomplexobj(x): + raise ValueError(f"Complex-valued `x` not allowed for {self.fft_mode=}'! " + "Set property `fft_mode` to 'twosided' or 'centered'.") + if isinstance(detr, str): + detr = partial(detrend, type=detr) + elif not (detr is None or callable(detr)): + raise ValueError(f"Parameter {detr=} is not a str, function or " + + "None!") + n = x.shape[axis] + if not (n >= (m2p := self.m_num-self.m_num_mid)): + e_str = f'{len(x)=}' if x.ndim == 1 else f'of {axis=} of {x.shape}' + raise ValueError(f"{e_str} must be >= ceil(m_num/2) = {m2p}!") + + if x.ndim > 1: # motivated by the NumPy broadcasting mechanisms: + x = np.moveaxis(x, axis, -1) + # determine slice index range: + p0, p1 = self.p_range(n, p0, p1) + S_shape_1d = (self.f_pts, p1 - p0) + S_shape = x.shape[:-1] + S_shape_1d if x.ndim > 1 else S_shape_1d + S = np.zeros(S_shape, dtype=complex) + for p_, x_ in enumerate(self._x_slices(x, k_offset, p0, p1, padding)): + if detr is not None: + x_ = detr(x_) + S[..., :, p_] = self._fft_func(x_ * self.win.conj()) + if x.ndim > 1: + return np.moveaxis(S, -2, axis if axis >= 0 else axis-1) + return S + + def spectrogram(self, x: np.ndarray, y: np.ndarray | None = None, + detr: Callable[[np.ndarray], np.ndarray] | Literal['linear', 'constant'] | None = None, # noqa: E501 + *, + p0: int | None = None, p1: int | None = None, + k_offset: int = 0, padding: PAD_TYPE = 'zeros', + axis: int = -1) \ + -> np.ndarray: + r"""Calculate spectrogram or cross-spectrogram. + + The spectrogram is the absolute square of the STFT, i.e., it is + ``abs(S[q,p])**2`` for given ``S[q,p]`` and thus is always + non-negative. + For two STFTs ``Sx[q,p], Sy[q,p]``, the cross-spectrogram is defined + as ``Sx[q,p] * np.conj(Sy[q,p])`` and is complex-valued. + This is a convenience function for calling `~ShortTimeFFT.stft` / + `stft_detrend`, hence all parameters are discussed there. If `y` is not + ``None`` it needs to have the same shape as `x`. + + Examples + -------- + The following example shows the spectrogram of a square wave with + varying frequency :math:`f_i(t)` (marked by a green dashed line in the + plot) sampled with 20 Hz: + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> from scipy.signal import square, ShortTimeFFT + >>> from scipy.signal.windows import gaussian + ... + >>> T_x, N = 1 / 20, 1000 # 20 Hz sampling rate for 50 s signal + >>> t_x = np.arange(N) * T_x # time indexes for signal + >>> f_i = 5e-3*(t_x - t_x[N // 3])**2 + 1 # varying frequency + >>> x = square(2*np.pi*np.cumsum(f_i)*T_x) # the signal + + The utilized Gaussian window is 50 samples or 2.5 s long. The + parameter ``mfft=800`` (oversampling factor 16) and the `hop` interval + of 2 in `ShortTimeFFT` was chosen to produce a sufficient number of + points: + + >>> g_std = 12 # standard deviation for Gaussian window in samples + >>> win = gaussian(50, std=g_std, sym=True) # symmetric Gaussian wind. + >>> SFT = ShortTimeFFT(win, hop=2, fs=1/T_x, mfft=800, scale_to='psd') + >>> Sx2 = SFT.spectrogram(x) # calculate absolute square of STFT + + The plot's colormap is logarithmically scaled as the power spectral + density is in dB. The time extent of the signal `x` is marked by + vertical dashed lines and the shaded areas mark the presence of border + effects: + + >>> fig1, ax1 = plt.subplots(figsize=(6., 4.)) # enlarge plot a bit + >>> t_lo, t_hi = SFT.extent(N)[:2] # time range of plot + >>> ax1.set_title(rf"Spectrogram ({SFT.m_num*SFT.T:g}$\,s$ Gaussian " + + ... rf"window, $\sigma_t={g_std*SFT.T:g}\,$s)") + >>> ax1.set(xlabel=f"Time $t$ in seconds ({SFT.p_num(N)} slices, " + + ... rf"$\Delta t = {SFT.delta_t:g}\,$s)", + ... ylabel=f"Freq. $f$ in Hz ({SFT.f_pts} bins, " + + ... rf"$\Delta f = {SFT.delta_f:g}\,$Hz)", + ... xlim=(t_lo, t_hi)) + >>> Sx_dB = 10 * np.log10(np.fmax(Sx2, 1e-4)) # limit range to -40 dB + >>> im1 = ax1.imshow(Sx_dB, origin='lower', aspect='auto', + ... extent=SFT.extent(N), cmap='magma') + >>> ax1.plot(t_x, f_i, 'g--', alpha=.5, label='$f_i(t)$') + >>> fig1.colorbar(im1, label='Power Spectral Density ' + + ... r"$20\,\log_{10}|S_x(t, f)|$ in dB") + ... + >>> # Shade areas where window slices stick out to the side: + >>> for t0_, t1_ in [(t_lo, SFT.lower_border_end[0] * SFT.T), + ... (SFT.upper_border_begin(N)[0] * SFT.T, t_hi)]: + ... ax1.axvspan(t0_, t1_, color='w', linewidth=0, alpha=.3) + >>> for t_ in [0, N * SFT.T]: # mark signal borders with vertical line + ... ax1.axvline(t_, color='c', linestyle='--', alpha=0.5) + >>> ax1.legend() + >>> fig1.tight_layout() + >>> plt.show() + + The logarithmic scaling reveals the odd harmonics of the square wave, + which are reflected at the Nyquist frequency of 10 Hz. This aliasing + is also the main source of the noise artifacts in the plot. + + + See Also + -------- + :meth:`~ShortTimeFFT.stft`: Perform the short-time Fourier transform. + stft_detrend: STFT with a trend subtracted from each segment. + :class:`scipy.signal.ShortTimeFFT`: Class this method belongs to. + """ + Sx = self.stft_detrend(x, detr, p0, p1, k_offset=k_offset, + padding=padding, axis=axis) + if y is None or y is x: # do spectrogram: + return Sx.real**2 + Sx.imag**2 + # Cross-spectrogram: + Sy = self.stft_detrend(y, detr, p0, p1, k_offset=k_offset, + padding=padding, axis=axis) + return Sx * Sy.conj() + + @property + def dual_win(self) -> np.ndarray: + """Canonical dual window. + + A STFT can be interpreted as the input signal being expressed as a + weighted sum of modulated and time-shifted dual windows. Note that for + a given window there exist many dual windows. The canonical window is + the one with the minimal energy (i.e., :math:`L_2` norm). + + `dual_win` has same length as `win`, namely `m_num` samples. + + If the dual window cannot be calculated a ``ValueError`` is raised. + This attribute is read only and calculated lazily. + + See Also + -------- + dual_win: Canonical dual window. + m_num: Number of samples in window `win`. + win: Window function as real- or complex-valued 1d array. + ShortTimeFFT: Class this property belongs to. + """ + if self._dual_win is None: + self._dual_win = _calc_dual_canonical_window(self.win, self.hop) + return self._dual_win + + @property + def invertible(self) -> bool: + """Check if STFT is invertible. + + This is achieved by trying to calculate the canonical dual window. + + See Also + -------- + :meth:`~ShortTimeFFT.istft`: Inverse short-time Fourier transform. + m_num: Number of samples in window `win` and `dual_win`. + dual_win: Canonical dual window. + win: Window for STFT. + ShortTimeFFT: Class this property belongs to. + """ + try: + return len(self.dual_win) > 0 # call self.dual_win() + except ValueError: + return False + + def istft(self, S: np.ndarray, k0: int = 0, k1: int | None = None, *, + f_axis: int = -2, t_axis: int = -1) \ + -> np.ndarray: + """Inverse short-time Fourier transform. + + It returns an array of dimension ``S.ndim - 1`` which is real + if `onesided_fft` is set, else complex. If the STFT is not + `invertible`, or the parameters are out of bounds a ``ValueError`` is + raised. + + Parameters + ---------- + S + A complex valued array where `f_axis` denotes the frequency + values and the `t-axis` dimension the temporal values of the + STFT values. + k0, k1 + The start and the end index of the reconstructed signal. The + default (``k0 = 0``, ``k1 = None``) assumes that the maximum length + signal should be reconstructed. + f_axis, t_axis + The axes in `S` denoting the frequency and the time dimension. + + Notes + ----- + It is required that `S` has `f_pts` entries along the `f_axis`. For + the `t_axis` it is assumed that the first entry corresponds to + `p_min` * `delta_t` (being <= 0). The length of `t_axis` needs to be + compatible with `k1`. I.e., ``S.shape[t_axis] >= self.p_max(k1)`` must + hold, if `k1` is not ``None``. Else `k1` is set to `k_max` with:: + + q_max = S.shape[t_range] + self.p_min + k_max = (q_max - 1) * self.hop + self.m_num - self.m_num_mid + + The :ref:`tutorial_stft` section of the :ref:`user_guide` discussed the + slicing behavior by means of an example. + + See Also + -------- + invertible: Check if STFT is invertible. + :meth:`~ShortTimeFFT.stft`: Perform Short-time Fourier transform. + :class:`scipy.signal.ShortTimeFFT`: Class this method belongs to. + """ + if f_axis == t_axis: + raise ValueError(f"{f_axis=} may not be equal to {t_axis=}!") + if S.shape[f_axis] != self.f_pts: + raise ValueError(f"{S.shape[f_axis]=} must be equal to " + + f"{self.f_pts=} ({S.shape=})!") + n_min = self.m_num-self.m_num_mid # minimum signal length + if not (S.shape[t_axis] >= (q_num := self.p_num(n_min))): + raise ValueError(f"{S.shape[t_axis]=} needs to have at least " + + f"{q_num} slices ({S.shape=})!") + if t_axis != S.ndim - 1 or f_axis != S.ndim - 2: + t_axis = S.ndim + t_axis if t_axis < 0 else t_axis + f_axis = S.ndim + f_axis if f_axis < 0 else f_axis + S = np.moveaxis(S, (f_axis, t_axis), (-2, -1)) + + q_max = S.shape[-1] + self.p_min + k_max = (q_max - 1) * self.hop + self.m_num - self.m_num_mid + + k1 = k_max if k1 is None else k1 + if not (self.k_min <= k0 < k1 <= k_max): + raise ValueError(f"({self.k_min=}) <= ({k0=}) < ({k1=}) <= " + + f"({k_max=}) is false!") + if not (num_pts := k1 - k0) >= n_min: + raise ValueError(f"({k1=}) - ({k0=}) = {num_pts} has to be at " + + f"least the half the window length {n_min}!") + + q0 = (k0 // self.hop + self.p_min if k0 >= 0 else # p_min always <= 0 + k0 // self.hop) + q1 = min(self.p_max(k1), q_max) + k_q0, k_q1 = self.nearest_k_p(k0), self.nearest_k_p(k1, left=False) + n_pts = k_q1 - k_q0 + self.m_num - self.m_num_mid + x = np.zeros(S.shape[:-2] + (n_pts,), + dtype=float if self.onesided_fft else complex) + for q_ in range(q0, q1): + xs = self._ifft_func(S[..., :, q_ - self.p_min]) * self.dual_win + i0 = q_ * self.hop - self.m_num_mid + i1 = min(i0 + self.m_num, n_pts+k0) + j0, j1 = 0, i1 - i0 + if i0 < k0: # xs sticks out to the left on x: + j0 += k0 - i0 + i0 = k0 + x[..., i0-k0:i1-k0] += xs[..., j0:j1] + x = x[..., :k1-k0] + if x.ndim > 1: + x = np.moveaxis(x, -1, f_axis if f_axis < x.ndim else t_axis) + return x + + @property + def fac_magnitude(self) -> float: + """Factor to multiply the STFT values by to scale each frequency slice + to a magnitude spectrum. + + It is 1 if attribute ``scaling == 'magnitude'``. + The window can be scaled to a magnitude spectrum by using the method + `scale_to`. + + See Also + -------- + fac_psd: Scaling factor for to a power spectral density spectrum. + scale_to: Scale window to obtain 'magnitude' or 'psd' scaling. + scaling: Normalization applied to the window function. + ShortTimeFFT: Class this property belongs to. + """ + if self.scaling == 'magnitude': + return 1 + if self._fac_mag is None: + self._fac_mag = 1 / abs(sum(self.win)) + return self._fac_mag + + @property + def fac_psd(self) -> float: + """Factor to multiply the STFT values by to scale each frequency slice + to a power spectral density (PSD). + + It is 1 if attribute ``scaling == 'psd'``. + The window can be scaled to a psd spectrum by using the method + `scale_to`. + + See Also + -------- + fac_magnitude: Scaling factor for to a magnitude spectrum. + scale_to: Scale window to obtain 'magnitude' or 'psd' scaling. + scaling: Normalization applied to the window function. + ShortTimeFFT: Class this property belongs to. + """ + if self.scaling == 'psd': + return 1 + if self._fac_psd is None: + self._fac_psd = 1 / np.sqrt( + sum(self.win.real**2+self.win.imag**2) / self.T) + return self._fac_psd + + @property + def m_num(self) -> int: + """Number of samples in window `win`. + + Note that the FFT can be oversampled by zero-padding. This is achieved + by setting the `mfft` property. + + See Also + -------- + m_num_mid: Center index of window `win`. + mfft: Length of input for the FFT used - may be larger than `m_num`. + hop: Time increment in signal samples for sliding window. + win: Window function as real- or complex-valued 1d array. + ShortTimeFFT: Class this property belongs to. + """ + return len(self.win) + + @property + def m_num_mid(self) -> int: + """Center index of window `win`. + + For odd `m_num`, ``(m_num - 1) / 2`` is returned and + for even `m_num` (per definition) ``m_num / 2`` is returned. + + See Also + -------- + m_num: Number of samples in window `win`. + mfft: Length of input for the FFT used - may be larger than `m_num`. + hop: ime increment in signal samples for sliding window. + win: Window function as real- or complex-valued 1d array. + ShortTimeFFT: Class this property belongs to. + """ + return self.m_num // 2 + + @cache + def _pre_padding(self) -> tuple[int, int]: + """Smallest signal index and slice index due to padding. + + Since, per convention, for time t=0, n,q is zero, the returned values + are negative or zero. + """ + w2 = self.win.real**2 + self.win.imag**2 + # move window to the left until the overlap with t >= 0 vanishes: + n0 = -self.m_num_mid + for q_, n_ in enumerate(range(n0, n0-self.m_num-1, -self.hop)): + n_next = n_ - self.hop + if n_next + self.m_num <= 0 or all(w2[n_next:] == 0): + return n_, -q_ + raise RuntimeError("This is code line should not have been reached!") + # If this case is reached, it probably means the first slice should be + # returned, i.e.: return n0, 0 + + @property + def k_min(self) -> int: + """The smallest possible signal index of the STFT. + + `k_min` is the index of the left-most non-zero value of the lowest + slice `p_min`. Since the zeroth slice is centered over the zeroth + sample of the input signal, `k_min` is never positive. + A detailed example is provided in the :ref:`tutorial_stft_sliding_win` + section of the :ref:`user_guide`. + + See Also + -------- + k_max: First sample index after signal end not touched by a time slice. + lower_border_end: Where pre-padding effects end. + p_min: The smallest possible slice index. + p_max: Index of first non-overlapping upper time slice. + p_num: Number of time slices, i.e., `p_max` - `p_min`. + p_range: Determine and validate slice index range. + upper_border_begin: Where post-padding effects start. + ShortTimeFFT: Class this property belongs to. + """ + return self._pre_padding()[0] + + @property + def p_min(self) -> int: + """The smallest possible slice index. + + `p_min` is the index of the left-most slice, where the window still + sticks into the signal, i.e., has non-zero part for t >= 0. + `k_min` is the smallest index where the window function of the slice + `p_min` is non-zero. + + Since, per convention the zeroth slice is centered at t=0, + `p_min` <= 0 always holds. + A detailed example is provided in the :ref:`tutorial_stft_sliding_win` + section of the :ref:`user_guide`. + + See Also + -------- + k_min: The smallest possible signal index. + k_max: First sample index after signal end not touched by a time slice. + p_max: Index of first non-overlapping upper time slice. + p_num: Number of time slices, i.e., `p_max` - `p_min`. + p_range: Determine and validate slice index range. + ShortTimeFFT: Class this property belongs to. + """ + return self._pre_padding()[1] + + @lru_cache(maxsize=256) + def _post_padding(self, n: int) -> tuple[int, int]: + """Largest signal index and slice index due to padding. + + Parameters + ---------- + n : int + Number of samples of input signal (must be ≥ half of the window length). + """ + if not (n >= (m2p := self.m_num - self.m_num_mid)): + raise ValueError(f"Parameter n must be >= ceil(m_num/2) = {m2p}!") + w2 = self.win.real**2 + self.win.imag**2 + # move window to the right until the overlap for t < t[n] vanishes: + q1 = n // self.hop # last slice index with t[p1] <= t[n] + k1 = q1 * self.hop - self.m_num_mid + for q_, k_ in enumerate(range(k1, n+self.m_num, self.hop), start=q1): + n_next = k_ + self.hop + if n_next >= n or all(w2[:n-n_next] == 0): + return k_ + self.m_num, q_ + 1 + raise RuntimeError("This is code line should not have been reached!") + # If this case is reached, it probably means the last slice should be + # returned, i.e.: return k1 + self.m_num - self.m_num_mid, q1 + 1 + + def k_max(self, n: int) -> int: + """First sample index after signal end not touched by a time slice. + + `k_max` - 1 is the largest sample index of the slice `p_max` for a + given input signal of `n` samples. + A detailed example is provided in the :ref:`tutorial_stft_sliding_win` + section of the :ref:`user_guide`. + + Parameters + ---------- + n : int + Number of samples of input signal (must be ≥ half of the window length). + + See Also + -------- + k_min: The smallest possible signal index. + p_min: The smallest possible slice index. + p_max: Index of first non-overlapping upper time slice. + p_num: Number of time slices, i.e., `p_max` - `p_min`. + p_range: Determine and validate slice index range. + ShortTimeFFT: Class this method belongs to. + """ + return self._post_padding(n)[0] + + def p_max(self, n: int) -> int: + """Index of first non-overlapping upper time slice for `n` sample + input. + + Note that center point t[p_max] = (p_max(n)-1) * `delta_t` is typically + larger than last time index t[n-1] == (`n`-1) * `T`. The upper border + of samples indexes covered by the window slices is given by `k_max`. + Furthermore, `p_max` does not denote the number of slices `p_num` since + `p_min` is typically less than zero. + A detailed example is provided in the :ref:`tutorial_stft_sliding_win` + section of the :ref:`user_guide`. + + See Also + -------- + k_min: The smallest possible signal index. + k_max: First sample index after signal end not touched by a time slice. + p_min: The smallest possible slice index. + p_num: Number of time slices, i.e., `p_max` - `p_min`. + p_range: Determine and validate slice index range. + ShortTimeFFT: Class this method belongs to. + """ + return self._post_padding(n)[1] + + def p_num(self, n: int) -> int: + """Number of time slices for an input signal with `n` samples. + + It is given by `p_num` = `p_max` - `p_min` with `p_min` typically + being negative. + A detailed example is provided in the :ref:`tutorial_stft_sliding_win` + section of the :ref:`user_guide`. + + See Also + -------- + k_min: The smallest possible signal index. + k_max: First sample index after signal end not touched by a time slice. + lower_border_end: Where pre-padding effects end. + p_min: The smallest possible slice index. + p_max: Index of first non-overlapping upper time slice. + p_range: Determine and validate slice index range. + upper_border_begin: Where post-padding effects start. + ShortTimeFFT: Class this method belongs to. + """ + return self.p_max(n) - self.p_min + + @property + def lower_border_end(self) -> tuple[int, int]: + """First signal index and first slice index unaffected by pre-padding. + + Describes the point where the window does not stick out to the left + of the signal domain. + A detailed example is provided in the :ref:`tutorial_stft_sliding_win` + section of the :ref:`user_guide`. + + See Also + -------- + k_min: The smallest possible signal index. + k_max: First sample index after signal end not touched by a time slice. + lower_border_end: Where pre-padding effects end. + p_min: The smallest possible slice index. + p_max: Index of first non-overlapping upper time slice. + p_num: Number of time slices, i.e., `p_max` - `p_min`. + p_range: Determine and validate slice index range. + upper_border_begin: Where post-padding effects start. + ShortTimeFFT: Class this property belongs to. + """ + # not using @cache decorator due to MyPy limitations + if self._lower_border_end is not None: + return self._lower_border_end + + # first non-zero element in self.win: + m0 = np.flatnonzero(self.win.real**2 + self.win.imag**2)[0] + + # move window to the right until does not stick out to the left: + k0 = -self.m_num_mid + m0 + for q_, k_ in enumerate(range(k0, self.hop + 1, self.hop)): + if k_ + self.hop >= 0: # next entry does not stick out anymore + self._lower_border_end = (k_ + self.m_num, q_ + 1) + return self._lower_border_end + self._lower_border_end = (0, max(self.p_min, 0)) # ends at first slice + return self._lower_border_end + + @lru_cache(maxsize=256) + def upper_border_begin(self, n: int) -> tuple[int, int]: + """First signal index and first slice index affected by post-padding. + + Describes the point where the window does begin stick out to the right + of the signal domain. + A detailed example is given :ref:`tutorial_stft_sliding_win` section + of the :ref:`user_guide`. + + Parameters + ---------- + n : int + Number of samples of input signal (must be ≥ half of the window length). + + Returns + ------- + k_ub : int + Lowest signal index, where a touching time slice sticks out past the + signal end. + p_ub : int + Lowest index of time slice of which the end sticks out past the signal end. + + See Also + -------- + k_min: The smallest possible signal index. + k_max: First sample index after signal end not touched by a time slice. + lower_border_end: Where pre-padding effects end. + p_min: The smallest possible slice index. + p_max: Index of first non-overlapping upper time slice. + p_num: Number of time slices, i.e., `p_max` - `p_min`. + p_range: Determine and validate slice index range. + ShortTimeFFT: Class this method belongs to. + """ + if not (n >= (m2p := self.m_num - self.m_num_mid)): + raise ValueError(f"Parameter n must be >= ceil(m_num/2) = {m2p}!") + w2 = self.win.real**2 + self.win.imag**2 + q2 = n // self.hop + 1 # first t[q] >= t[n] + q1 = max((n-self.m_num) // self.hop - 1, -1) + # move window left until does not stick out to the right: + for q_ in range(q2, q1, -1): + k_ = q_ * self.hop + (self.m_num - self.m_num_mid) + if k_ <= n or all(w2[n-k_:] == 0): + return (q_ + 1) * self.hop - self.m_num_mid, q_ + 1 + return 0, 0 # border starts at first slice + + @property + def delta_t(self) -> float: + """Time increment of STFT. + + The time increment `delta_t` = `T` * `hop` represents the sample + increment `hop` converted to time based on the sampling interval `T`. + + See Also + -------- + delta_f: Width of the frequency bins of the STFT. + hop: Hop size in signal samples for sliding window. + t: Times of STFT for an input signal with `n` samples. + T: Sampling interval of input signal and window `win`. + ShortTimeFFT: Class this property belongs to + """ + return self.T * self.hop + + def p_range(self, n: int, p0: int | None = None, + p1: int | None = None) -> tuple[int, int]: + """Determine and validate slice index range. + + Parameters + ---------- + n : int + Number of samples of input signal, assuming t[0] = 0. + p0 : int | None + First slice index. If 0 then the first slice is centered at t = 0. + If ``None`` then `p_min` is used. Note that p0 may be < 0 if + slices are left of t = 0. + p1 : int | None + End of interval (last value is p1-1). + If ``None`` then `p_max(n)` is used. + + + Returns + ------- + p0_ : int + The fist slice index + p1_ : int + End of interval (last value is p1-1). + + Notes + ----- + A ``ValueError`` is raised if ``p_min <= p0 < p1 <= p_max(n)`` does not + hold. + + See Also + -------- + k_min: The smallest possible signal index. + k_max: First sample index after signal end not touched by a time slice. + lower_border_end: Where pre-padding effects end. + p_min: The smallest possible slice index. + p_max: Index of first non-overlapping upper time slice. + p_num: Number of time slices, i.e., `p_max` - `p_min`. + upper_border_begin: Where post-padding effects start. + ShortTimeFFT: Class this property belongs to. + """ + p_max = self.p_max(n) # shorthand + p0_ = self.p_min if p0 is None else p0 + p1_ = p_max if p1 is None else p1 + if not (self.p_min <= p0_ < p1_ <= p_max): + raise ValueError(f"Invalid Parameter {p0=}, {p1=}, i.e., " + + f"{self.p_min=} <= p0 < p1 <= {p_max=} " + + f"does not hold for signal length {n=}!") + return p0_, p1_ + + @lru_cache(maxsize=1) + def t(self, n: int, p0: int | None = None, p1: int | None = None, + k_offset: int = 0) -> np.ndarray: + """Times of STFT for an input signal with `n` samples. + + Returns a 1d array with times of the `~ShortTimeFFT.stft` values with + the same parametrization. Note that the slices are + ``delta_t = hop * T`` time units apart. + + Parameters + ---------- + n + Number of sample of the input signal. + p0 + The first element of the range of slices to calculate. If ``None`` + then it is set to :attr:`p_min`, which is the smallest possible + slice. + p1 + The end of the array. If ``None`` then `p_max(n)` is used. + k_offset + Index of first sample (t = 0) in `x`. + + + See Also + -------- + delta_t: Time increment of STFT (``hop*T``) + hop: Time increment in signal samples for sliding window. + nearest_k_p: Nearest sample index k_p for which t[k_p] == t[p] holds. + T: Sampling interval of input signal and of the window (``1/fs``). + fs: Sampling frequency (being ``1/T``) + ShortTimeFFT: Class this method belongs to. + """ + p0, p1 = self.p_range(n, p0, p1) + return np.arange(p0, p1) * self.delta_t + k_offset * self.T + + def nearest_k_p(self, k: int, left: bool = True) -> int: + """Return nearest sample index k_p for which t[k_p] == t[p] holds. + + The nearest next smaller time sample p (where t[p] is the center + position of the window of the p-th slice) is p_k = k // `hop`. + If `hop` is a divisor of `k` than `k` is returned. + If `left` is set than p_k * `hop` is returned else (p_k+1) * `hop`. + + This method can be used to slice an input signal into chunks for + calculating the STFT and iSTFT incrementally. + + See Also + -------- + delta_t: Time increment of STFT (``hop*T``) + hop: Time increment in signal samples for sliding window. + T: Sampling interval of input signal and of the window (``1/fs``). + fs: Sampling frequency (being ``1/T``) + t: Times of STFT for an input signal with `n` samples. + ShortTimeFFT: Class this method belongs to. + """ + p_q, remainder = divmod(k, self.hop) + if remainder == 0: + return k + return p_q * self.hop if left else (p_q + 1) * self.hop + + @property + def delta_f(self) -> float: + """Width of the frequency bins of the STFT. + + Return the frequency interval `delta_f` = 1 / (`mfft` * `T`). + + See Also + -------- + delta_t: Time increment of STFT. + f_pts: Number of points along the frequency axis. + f: Frequencies values of the STFT. + mfft: Length of the input for FFT used. + T: Sampling interval. + t: Times of STFT for an input signal with `n` samples. + ShortTimeFFT: Class this property belongs to. + """ + return 1 / (self.mfft * self.T) + + @property + def f_pts(self) -> int: + """Number of points along the frequency axis. + + See Also + -------- + delta_f: Width of the frequency bins of the STFT. + f: Frequencies values of the STFT. + mfft: Length of the input for FFT used. + ShortTimeFFT: Class this property belongs to. + """ + return self.mfft // 2 + 1 if self.onesided_fft else self.mfft + + @property + def onesided_fft(self) -> bool: + """Return True if a one-sided FFT is used. + + Returns ``True`` if `fft_mode` is either 'onesided' or 'onesided2X'. + + See Also + -------- + fft_mode: Utilized FFT ('twosided', 'centered', 'onesided' or + 'onesided2X') + ShortTimeFFT: Class this property belongs to. + """ + return self.fft_mode in {'onesided', 'onesided2X'} + + @property + def f(self) -> np.ndarray: + """Frequencies values of the STFT. + + A 1d array of length `f_pts` with `delta_f` spaced entries is returned. + + See Also + -------- + delta_f: Width of the frequency bins of the STFT. + f_pts: Number of points along the frequency axis. + mfft: Length of the input for FFT used. + ShortTimeFFT: Class this property belongs to. + """ + if self.fft_mode in {'onesided', 'onesided2X'}: + return fft_lib.rfftfreq(self.mfft, self.T) + elif self.fft_mode == 'twosided': + return fft_lib.fftfreq(self.mfft, self.T) + elif self.fft_mode == 'centered': + return fft_lib.fftshift(fft_lib.fftfreq(self.mfft, self.T)) + # This should never happen but makes the Linters happy: + fft_modes = get_args(FFT_MODE_TYPE) + raise RuntimeError(f"{self.fft_mode=} not in {fft_modes}!") + + def _fft_func(self, x: np.ndarray) -> np.ndarray: + """FFT based on the `fft_mode`, `mfft`, `scaling` and `phase_shift` + attributes. + + For multidimensional arrays the transformation is carried out on the + last axis. + """ + if self.phase_shift is not None: + if x.shape[-1] < self.mfft: # zero pad if needed + z_shape = list(x.shape) + z_shape[-1] = self.mfft - x.shape[-1] + x = np.hstack((x, np.zeros(z_shape, dtype=x.dtype))) + p_s = (self.phase_shift + self.m_num_mid) % self.m_num + x = np.roll(x, -p_s, axis=-1) + + if self.fft_mode == 'twosided': + return fft_lib.fft(x, n=self.mfft, axis=-1) + if self.fft_mode == 'centered': + return fft_lib.fftshift(fft_lib.fft(x, self.mfft, axis=-1), axes=-1) + if self.fft_mode == 'onesided': + return fft_lib.rfft(x, n=self.mfft, axis=-1) + if self.fft_mode == 'onesided2X': + X = fft_lib.rfft(x, n=self.mfft, axis=-1) + # Either squared magnitude (psd) or magnitude is doubled: + fac = np.sqrt(2) if self.scaling == 'psd' else 2 + # For even input length, the last entry is unpaired: + X[..., 1: -1 if self.mfft % 2 == 0 else None] *= fac + return X + # This should never happen but makes the Linter happy: + fft_modes = get_args(FFT_MODE_TYPE) + raise RuntimeError(f"{self.fft_mode=} not in {fft_modes}!") + + def _ifft_func(self, X: np.ndarray) -> np.ndarray: + """Inverse to `_fft_func`. + + Returned is an array of length `m_num`. If the FFT is `onesided` + then a float array is returned else a complex array is returned. + For multidimensional arrays the transformation is carried out on the + last axis. + """ + if self.fft_mode == 'twosided': + x = fft_lib.ifft(X, n=self.mfft, axis=-1) + elif self.fft_mode == 'centered': + x = fft_lib.ifft(fft_lib.ifftshift(X, axes=-1), n=self.mfft, axis=-1) + elif self.fft_mode == 'onesided': + x = fft_lib.irfft(X, n=self.mfft, axis=-1) + elif self.fft_mode == 'onesided2X': + Xc = X.copy() # we do not want to modify function parameters + fac = np.sqrt(2) if self.scaling == 'psd' else 2 + # For even length X the last value is not paired with a negative + # value on the two-sided FFT: + q1 = -1 if self.mfft % 2 == 0 else None + Xc[..., 1:q1] /= fac + x = fft_lib.irfft(Xc, n=self.mfft, axis=-1) + else: # This should never happen but makes the Linter happy: + error_str = f"{self.fft_mode=} not in {get_args(FFT_MODE_TYPE)}!" + raise RuntimeError(error_str) + + if self.phase_shift is None: + return x[..., :self.m_num] + p_s = (self.phase_shift + self.m_num_mid) % self.m_num + return np.roll(x, p_s, axis=-1)[..., :self.m_num] + + def extent(self, n: int, axes_seq: Literal['tf', 'ft'] = 'tf', + center_bins: bool = False) -> tuple[float, float, float, float]: + """Return minimum and maximum values time-frequency values. + + A tuple with four floats ``(t0, t1, f0, f1)`` for 'tf' and + ``(f0, f1, t0, t1)`` for 'ft' is returned describing the corners + of the time-frequency domain of the `~ShortTimeFFT.stft`. + That tuple can be passed to `matplotlib.pyplot.imshow` as a parameter + with the same name. + + Parameters + ---------- + n : int + Number of samples in input signal. + axes_seq : {'tf', 'ft'} + Return time extent first and then frequency extent or vice-versa. + center_bins: bool + If set (default ``False``), the values of the time slots and + frequency bins are moved from the side the middle. This is useful, + when plotting the `~ShortTimeFFT.stft` values as step functions, + i.e., with no interpolation. + + See Also + -------- + :func:`matplotlib.pyplot.imshow`: Display data as an image. + :class:`scipy.signal.ShortTimeFFT`: Class this method belongs to. + + Examples + -------- + The following two plots illustrate the effect of the parameter `center_bins`: + The grid lines represent the three time and the four frequency values of the + STFT. + The left plot, where ``(t0, t1, f0, f1) = (0, 3, 0, 4)`` is passed as parameter + ``extent`` to `~matplotlib.pyplot.imshow`, shows the standard behavior of the + time and frequency values being at the lower edge of the corrsponding bin. + The right plot, with ``(t0, t1, f0, f1) = (-0.5, 2.5, -0.5, 3.5)``, shows that + the bins are centered over the respective values when passing + ``center_bins=True``. + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> from scipy.signal import ShortTimeFFT + ... + >>> n, m = 12, 6 + >>> SFT = ShortTimeFFT.from_window('hann', fs=m, nperseg=m, noverlap=0) + >>> Sxx = SFT.stft(np.cos(np.arange(n))) # produces a colorful plot + ... + >>> fig, axx = plt.subplots(1, 2, tight_layout=True, figsize=(6., 4.)) + >>> for ax_, center_bins in zip(axx, (False, True)): + ... ax_.imshow(abs(Sxx), origin='lower', interpolation=None, aspect='equal', + ... cmap='viridis', extent=SFT.extent(n, 'tf', center_bins)) + ... ax_.set_title(f"{center_bins=}") + ... ax_.set_xlabel(f"Time ({SFT.p_num(n)} points, Δt={SFT.delta_t})") + ... ax_.set_ylabel(f"Frequency ({SFT.f_pts} points, Δf={SFT.delta_f})") + ... ax_.set_xticks(SFT.t(n)) # vertical grid line are timestamps + ... ax_.set_yticks(SFT.f) # horizontal grid line are frequency values + ... ax_.grid(True) + >>> plt.show() + + Note that the step-like behavior with the constant colors is caused by passing + ``interpolation=None`` to `~matplotlib.pyplot.imshow`. + """ + if axes_seq not in ('tf', 'ft'): + raise ValueError(f"Parameter {axes_seq=} not in ['tf', 'ft']!") + + if self.onesided_fft: + q0, q1 = 0, self.f_pts + elif self.fft_mode == 'centered': + q0 = -(self.mfft // 2) + q1 = self.mfft // 2 if self.mfft % 2 == 0 else self.mfft // 2 + 1 + else: + raise ValueError(f"Attribute fft_mode={self.fft_mode} must be " + + "in ['centered', 'onesided', 'onesided2X']") + + p0, p1 = self.p_min, self.p_max(n) # shorthand + if center_bins: + t0, t1 = self.delta_t * (p0 - 0.5), self.delta_t * (p1 - 0.5) + f0, f1 = self.delta_f * (q0 - 0.5), self.delta_f * (q1 - 0.5) + else: + t0, t1 = self.delta_t * p0, self.delta_t * p1 + f0, f1 = self.delta_f * q0, self.delta_f * q1 + return (t0, t1, f0, f1) if axes_seq == 'tf' else (f0, f1, t0, t1) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_signaltools.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_signaltools.py new file mode 100644 index 0000000000000000000000000000000000000000..340e227760ecd0a4bbf7bf5a135b92acb123a4d3 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_signaltools.py @@ -0,0 +1,4989 @@ +# Author: Travis Oliphant +# 1999 -- 2002 + +from __future__ import annotations # Provides typing union operator `|` in Python 3.9 +import operator +import math +from math import prod as _prod +import timeit +import warnings +from typing import Literal + +from numpy._typing import ArrayLike + +from scipy.spatial import cKDTree +from . import _sigtools +from ._ltisys import dlti +from ._upfirdn import upfirdn, _output_len, _upfirdn_modes +from scipy import linalg, fft as sp_fft +from scipy import ndimage +from scipy.fft._helper import _init_nd_shape_and_axes +import numpy as np +from scipy.special import lambertw +from .windows import get_window +from ._arraytools import axis_slice, axis_reverse, odd_ext, even_ext, const_ext +from ._filter_design import cheby1, _validate_sos, zpk2sos +from ._fir_filter_design import firwin +from ._sosfilt import _sosfilt + + +__all__ = ['correlate', 'correlation_lags', 'correlate2d', + 'convolve', 'convolve2d', 'fftconvolve', 'oaconvolve', + 'order_filter', 'medfilt', 'medfilt2d', 'wiener', 'lfilter', + 'lfiltic', 'sosfilt', 'deconvolve', 'hilbert', 'hilbert2', 'envelope', + 'unique_roots', 'invres', 'invresz', 'residue', + 'residuez', 'resample', 'resample_poly', 'detrend', + 'lfilter_zi', 'sosfilt_zi', 'sosfiltfilt', 'choose_conv_method', + 'filtfilt', 'decimate', 'vectorstrength'] + + +_modedict = {'valid': 0, 'same': 1, 'full': 2} + +_boundarydict = {'fill': 0, 'pad': 0, 'wrap': 2, 'circular': 2, 'symm': 1, + 'symmetric': 1, 'reflect': 4} + + +def _valfrommode(mode): + try: + return _modedict[mode] + except KeyError as e: + raise ValueError("Acceptable mode flags are 'valid'," + " 'same', or 'full'.") from e + + +def _bvalfromboundary(boundary): + try: + return _boundarydict[boundary] << 2 + except KeyError as e: + raise ValueError("Acceptable boundary flags are 'fill', 'circular' " + "(or 'wrap'), and 'symmetric' (or 'symm').") from e + + +def _inputs_swap_needed(mode, shape1, shape2, axes=None): + """Determine if inputs arrays need to be swapped in `"valid"` mode. + + If in `"valid"` mode, returns whether or not the input arrays need to be + swapped depending on whether `shape1` is at least as large as `shape2` in + every calculated dimension. + + This is important for some of the correlation and convolution + implementations in this module, where the larger array input needs to come + before the smaller array input when operating in this mode. + + Note that if the mode provided is not 'valid', False is immediately + returned. + + """ + if mode != 'valid': + return False + + if not shape1: + return False + + if axes is None: + axes = range(len(shape1)) + + ok1 = all(shape1[i] >= shape2[i] for i in axes) + ok2 = all(shape2[i] >= shape1[i] for i in axes) + + if not (ok1 or ok2): + raise ValueError("For 'valid' mode, one must be at least " + "as large as the other in every dimension") + + return not ok1 + + +def _reject_objects(arr, name): + """Warn if arr.dtype is object or longdouble. + """ + dt = np.asarray(arr).dtype + if not (np.issubdtype(dt, np.integer) + or dt in [np.bool_, np.float16, np.float32, np.float64, + np.complex64, np.complex128] + ): + msg = ( + f"dtype={dt} is not supported by {name} and will raise an error in " + f"SciPy 1.17.0. Supported dtypes are: boolean, integer, `np.float16`," + f"`np.float32`, `np.float64`, `np.complex64`, `np.complex128`." + ) + warnings.warn(msg, category=DeprecationWarning, stacklevel=3) + + +def correlate(in1, in2, mode='full', method='auto'): + r""" + Cross-correlate two N-dimensional arrays. + + Cross-correlate `in1` and `in2`, with the output size determined by the + `mode` argument. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear cross-correlation + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + method : str {'auto', 'direct', 'fft'}, optional + A string indicating which method to use to calculate the correlation. + + ``direct`` + The correlation is determined directly from sums, the definition of + correlation. + ``fft`` + The Fast Fourier Transform is used to perform the correlation more + quickly (only available for numerical arrays.) + ``auto`` + Automatically chooses direct or Fourier method based on an estimate + of which is faster (default). See `convolve` Notes for more detail. + + .. versionadded:: 0.19.0 + + Returns + ------- + correlate : array + An N-dimensional array containing a subset of the discrete linear + cross-correlation of `in1` with `in2`. + + See Also + -------- + choose_conv_method : contains more documentation on `method`. + correlation_lags : calculates the lag / displacement indices array for 1D + cross-correlation. + + Notes + ----- + The correlation z of two d-dimensional arrays x and y is defined as:: + + z[...,k,...] = sum[..., i_l, ...] x[..., i_l,...] * conj(y[..., i_l - k,...]) + + This way, if x and y are 1-D arrays and ``z = correlate(x, y, 'full')`` + then + + .. math:: + + z[k] = (x * y)(k - N + 1) + = \sum_{l=0}^{||x||-1}x_l y_{l-k+N-1}^{*} + + for :math:`k = 0, 1, ..., ||x|| + ||y|| - 2` + + where :math:`||x||` is the length of ``x``, :math:`N = \max(||x||,||y||)`, + and :math:`y_m` is 0 when m is outside the range of y. + + ``method='fft'`` only works for numerical arrays as it relies on + `fftconvolve`. In certain cases (i.e., arrays of objects or when + rounding integers can lose precision), ``method='direct'`` is always used. + + When using "same" mode with even-length inputs, the outputs of `correlate` + and `correlate2d` differ: There is a 1-index offset between them. + + Examples + -------- + Implement a matched filter using cross-correlation, to recover a signal + that has passed through a noisy channel. + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + >>> sig = np.repeat([0., 1., 1., 0., 1., 0., 0., 1.], 128) + >>> sig_noise = sig + rng.standard_normal(len(sig)) + >>> corr = signal.correlate(sig_noise, np.ones(128), mode='same') / 128 + + >>> clock = np.arange(64, len(sig), 128) + >>> fig, (ax_orig, ax_noise, ax_corr) = plt.subplots(3, 1, sharex=True) + >>> ax_orig.plot(sig) + >>> ax_orig.plot(clock, sig[clock], 'ro') + >>> ax_orig.set_title('Original signal') + >>> ax_noise.plot(sig_noise) + >>> ax_noise.set_title('Signal with noise') + >>> ax_corr.plot(corr) + >>> ax_corr.plot(clock, corr[clock], 'ro') + >>> ax_corr.axhline(0.5, ls=':') + >>> ax_corr.set_title('Cross-correlated with rectangular pulse') + >>> ax_orig.margins(0, 0.1) + >>> fig.tight_layout() + >>> plt.show() + + Compute the cross-correlation of a noisy signal with the original signal. + + >>> x = np.arange(128) / 128 + >>> sig = np.sin(2 * np.pi * x) + >>> sig_noise = sig + rng.standard_normal(len(sig)) + >>> corr = signal.correlate(sig_noise, sig) + >>> lags = signal.correlation_lags(len(sig), len(sig_noise)) + >>> corr /= np.max(corr) + + >>> fig, (ax_orig, ax_noise, ax_corr) = plt.subplots(3, 1, figsize=(4.8, 4.8)) + >>> ax_orig.plot(sig) + >>> ax_orig.set_title('Original signal') + >>> ax_orig.set_xlabel('Sample Number') + >>> ax_noise.plot(sig_noise) + >>> ax_noise.set_title('Signal with noise') + >>> ax_noise.set_xlabel('Sample Number') + >>> ax_corr.plot(lags, corr) + >>> ax_corr.set_title('Cross-correlated signal') + >>> ax_corr.set_xlabel('Lag') + >>> ax_orig.margins(0, 0.1) + >>> ax_noise.margins(0, 0.1) + >>> ax_corr.margins(0, 0.1) + >>> fig.tight_layout() + >>> plt.show() + + """ + in1 = np.asarray(in1) + in2 = np.asarray(in2) + _reject_objects(in1, 'correlate') + _reject_objects(in2, 'correlate') + + if in1.ndim == in2.ndim == 0: + return in1 * in2.conj() + elif in1.ndim != in2.ndim: + raise ValueError("in1 and in2 should have the same dimensionality") + + # Don't use _valfrommode, since correlate should not accept numeric modes + try: + val = _modedict[mode] + except KeyError as e: + raise ValueError("Acceptable mode flags are 'valid'," + " 'same', or 'full'.") from e + + # this either calls fftconvolve or this function with method=='direct' + if method in ('fft', 'auto'): + return convolve(in1, _reverse_and_conj(in2), mode, method) + + elif method == 'direct': + # fastpath to faster numpy.correlate for 1d inputs when possible + if _np_conv_ok(in1, in2, mode): + return np.correlate(in1, in2, mode) + + # _correlateND is far slower when in2.size > in1.size, so swap them + # and then undo the effect afterward if mode == 'full'. Also, it fails + # with 'valid' mode if in2 is larger than in1, so swap those, too. + # Don't swap inputs for 'same' mode, since shape of in1 matters. + swapped_inputs = ((mode == 'full') and (in2.size > in1.size) or + _inputs_swap_needed(mode, in1.shape, in2.shape)) + + if swapped_inputs: + in1, in2 = in2, in1 + + if mode == 'valid': + ps = [i - j + 1 for i, j in zip(in1.shape, in2.shape)] + out = np.empty(ps, in1.dtype) + + z = _sigtools._correlateND(in1, in2, out, val) + + else: + ps = [i + j - 1 for i, j in zip(in1.shape, in2.shape)] + + # zero pad input + in1zpadded = np.zeros(ps, in1.dtype) + sc = tuple(slice(0, i) for i in in1.shape) + in1zpadded[sc] = in1.copy() + + if mode == 'full': + out = np.empty(ps, in1.dtype) + elif mode == 'same': + out = np.empty(in1.shape, in1.dtype) + + z = _sigtools._correlateND(in1zpadded, in2, out, val) + + if swapped_inputs: + # Reverse and conjugate to undo the effect of swapping inputs + z = _reverse_and_conj(z) + + return z + + else: + raise ValueError("Acceptable method flags are 'auto'," + " 'direct', or 'fft'.") + + +def correlation_lags(in1_len, in2_len, mode='full'): + r""" + Calculates the lag / displacement indices array for 1D cross-correlation. + + Parameters + ---------- + in1_len : int + First input size. + in2_len : int + Second input size. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output. + See the documentation `correlate` for more information. + + Returns + ------- + lags : array + Returns an array containing cross-correlation lag/displacement indices. + Indices can be indexed with the np.argmax of the correlation to return + the lag/displacement. + + See Also + -------- + correlate : Compute the N-dimensional cross-correlation. + + Notes + ----- + Cross-correlation for continuous functions :math:`f` and :math:`g` is + defined as: + + .. math:: + + \left ( f\star g \right )\left ( \tau \right ) + \triangleq \int_{t_0}^{t_0 +T} + \overline{f\left ( t \right )}g\left ( t+\tau \right )dt + + Where :math:`\tau` is defined as the displacement, also known as the lag. + + Cross correlation for discrete functions :math:`f` and :math:`g` is + defined as: + + .. math:: + \left ( f\star g \right )\left [ n \right ] + \triangleq \sum_{-\infty}^{\infty} + \overline{f\left [ m \right ]}g\left [ m+n \right ] + + Where :math:`n` is the lag. + + Examples + -------- + Cross-correlation of a signal with its time-delayed self. + + >>> import numpy as np + >>> from scipy import signal + >>> rng = np.random.default_rng() + >>> x = rng.standard_normal(1000) + >>> y = np.concatenate([rng.standard_normal(100), x]) + >>> correlation = signal.correlate(x, y, mode="full") + >>> lags = signal.correlation_lags(x.size, y.size, mode="full") + >>> lag = lags[np.argmax(correlation)] + """ + + # calculate lag ranges in different modes of operation + if mode == "full": + # the output is the full discrete linear convolution + # of the inputs. (Default) + lags = np.arange(-in2_len + 1, in1_len) + elif mode == "same": + # the output is the same size as `in1`, centered + # with respect to the 'full' output. + # calculate the full output + lags = np.arange(-in2_len + 1, in1_len) + # determine the midpoint in the full output + mid = lags.size // 2 + # determine lag_bound to be used with respect + # to the midpoint + lag_bound = in1_len // 2 + # calculate lag ranges for even and odd scenarios + if in1_len % 2 == 0: + lags = lags[(mid-lag_bound):(mid+lag_bound)] + else: + lags = lags[(mid-lag_bound):(mid+lag_bound)+1] + elif mode == "valid": + # the output consists only of those elements that do not + # rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + # must be at least as large as the other in every dimension. + + # the lag_bound will be either negative or positive + # this let's us infer how to present the lag range + lag_bound = in1_len - in2_len + if lag_bound >= 0: + lags = np.arange(lag_bound + 1) + else: + lags = np.arange(lag_bound, 1) + else: + raise ValueError(f"Mode {mode} is invalid") + return lags + + +def _centered(arr, newshape): + # Return the center newshape portion of the array. + newshape = np.asarray(newshape) + currshape = np.array(arr.shape) + startind = (currshape - newshape) // 2 + endind = startind + newshape + myslice = [slice(startind[k], endind[k]) for k in range(len(endind))] + return arr[tuple(myslice)] + + +def _init_freq_conv_axes(in1, in2, mode, axes, sorted_axes=False): + """Handle the axes argument for frequency-domain convolution. + + Returns the inputs and axes in a standard form, eliminating redundant axes, + swapping the inputs if necessary, and checking for various potential + errors. + + Parameters + ---------- + in1 : array + First input. + in2 : array + Second input. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output. + See the documentation `fftconvolve` for more information. + axes : list of ints + Axes over which to compute the FFTs. + sorted_axes : bool, optional + If `True`, sort the axes. + Default is `False`, do not sort. + + Returns + ------- + in1 : array + The first input, possible swapped with the second input. + in2 : array + The second input, possible swapped with the first input. + axes : list of ints + Axes over which to compute the FFTs. + + """ + s1 = in1.shape + s2 = in2.shape + noaxes = axes is None + + _, axes = _init_nd_shape_and_axes(in1, shape=None, axes=axes) + + if not noaxes and not len(axes): + raise ValueError("when provided, axes cannot be empty") + + # Axes of length 1 can rely on broadcasting rules for multiply, + # no fft needed. + axes = [a for a in axes if s1[a] != 1 and s2[a] != 1] + + if sorted_axes: + axes.sort() + + if not all(s1[a] == s2[a] or s1[a] == 1 or s2[a] == 1 + for a in range(in1.ndim) if a not in axes): + raise ValueError("incompatible shapes for in1 and in2:" + f" {s1} and {s2}") + + # Check that input sizes are compatible with 'valid' mode. + if _inputs_swap_needed(mode, s1, s2, axes=axes): + # Convolution is commutative; order doesn't have any effect on output. + in1, in2 = in2, in1 + + return in1, in2, axes + + +def _freq_domain_conv(in1, in2, axes, shape, calc_fast_len=False): + """Convolve two arrays in the frequency domain. + + This function implements only base the FFT-related operations. + Specifically, it converts the signals to the frequency domain, multiplies + them, then converts them back to the time domain. Calculations of axes, + shapes, convolution mode, etc. are implemented in higher level-functions, + such as `fftconvolve` and `oaconvolve`. Those functions should be used + instead of this one. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + axes : array_like of ints + Axes over which to compute the FFTs. + shape : array_like of ints + The sizes of the FFTs. + calc_fast_len : bool, optional + If `True`, set each value of `shape` to the next fast FFT length. + Default is `False`, use `axes` as-is. + + Returns + ------- + out : array + An N-dimensional array containing the discrete linear convolution of + `in1` with `in2`. + + """ + if not len(axes): + return in1 * in2 + + complex_result = (in1.dtype.kind == 'c' or in2.dtype.kind == 'c') + + if calc_fast_len: + # Speed up FFT by padding to optimal size. + fshape = [ + sp_fft.next_fast_len(shape[a], not complex_result) for a in axes] + else: + fshape = shape + + if not complex_result: + fft, ifft = sp_fft.rfftn, sp_fft.irfftn + else: + fft, ifft = sp_fft.fftn, sp_fft.ifftn + + sp1 = fft(in1, fshape, axes=axes) + sp2 = fft(in2, fshape, axes=axes) + + ret = ifft(sp1 * sp2, fshape, axes=axes) + + if calc_fast_len: + fslice = tuple([slice(sz) for sz in shape]) + ret = ret[fslice] + + return ret + + +def _apply_conv_mode(ret, s1, s2, mode, axes): + """Calculate the convolution result shape based on the `mode` argument. + + Returns the result sliced to the correct size for the given mode. + + Parameters + ---------- + ret : array + The result array, with the appropriate shape for the 'full' mode. + s1 : list of int + The shape of the first input. + s2 : list of int + The shape of the second input. + mode : str {'full', 'valid', 'same'} + A string indicating the size of the output. + See the documentation `fftconvolve` for more information. + axes : list of ints + Axes over which to compute the convolution. + + Returns + ------- + ret : array + A copy of `res`, sliced to the correct size for the given `mode`. + + """ + if mode == "full": + return ret.copy() + elif mode == "same": + return _centered(ret, s1).copy() + elif mode == "valid": + shape_valid = [ret.shape[a] if a not in axes else s1[a] - s2[a] + 1 + for a in range(ret.ndim)] + return _centered(ret, shape_valid).copy() + else: + raise ValueError("acceptable mode flags are 'valid'," + " 'same', or 'full'") + + +def fftconvolve(in1, in2, mode="full", axes=None): + """Convolve two N-dimensional arrays using FFT. + + Convolve `in1` and `in2` using the fast Fourier transform method, with + the output size determined by the `mode` argument. + + This is generally much faster than `convolve` for large arrays (n > ~500), + but can be slower when only a few output values are needed, and can only + output float arrays (int or object array inputs will be cast to float). + + As of v0.19, `convolve` automatically chooses this method or the direct + method based on an estimation of which is faster. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear convolution + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + axes : int or array_like of ints or None, optional + Axes over which to compute the convolution. + The default is over all axes. + + Returns + ------- + out : array + An N-dimensional array containing a subset of the discrete linear + convolution of `in1` with `in2`. + + See Also + -------- + convolve : Uses the direct convolution or FFT convolution algorithm + depending on which is faster. + oaconvolve : Uses the overlap-add method to do convolution, which is + generally faster when the input arrays are large and + significantly different in size. + + Examples + -------- + Autocorrelation of white noise is an impulse. + + >>> import numpy as np + >>> from scipy import signal + >>> rng = np.random.default_rng() + >>> sig = rng.standard_normal(1000) + >>> autocorr = signal.fftconvolve(sig, sig[::-1], mode='full') + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_orig, ax_mag) = plt.subplots(2, 1) + >>> ax_orig.plot(sig) + >>> ax_orig.set_title('White noise') + >>> ax_mag.plot(np.arange(-len(sig)+1,len(sig)), autocorr) + >>> ax_mag.set_title('Autocorrelation') + >>> fig.tight_layout() + >>> fig.show() + + Gaussian blur implemented using FFT convolution. Notice the dark borders + around the image, due to the zero-padding beyond its boundaries. + The `convolve2d` function allows for other types of image boundaries, + but is far slower. + + >>> from scipy import datasets + >>> face = datasets.face(gray=True) + >>> kernel = np.outer(signal.windows.gaussian(70, 8), + ... signal.windows.gaussian(70, 8)) + >>> blurred = signal.fftconvolve(face, kernel, mode='same') + + >>> fig, (ax_orig, ax_kernel, ax_blurred) = plt.subplots(3, 1, + ... figsize=(6, 15)) + >>> ax_orig.imshow(face, cmap='gray') + >>> ax_orig.set_title('Original') + >>> ax_orig.set_axis_off() + >>> ax_kernel.imshow(kernel, cmap='gray') + >>> ax_kernel.set_title('Gaussian kernel') + >>> ax_kernel.set_axis_off() + >>> ax_blurred.imshow(blurred, cmap='gray') + >>> ax_blurred.set_title('Blurred') + >>> ax_blurred.set_axis_off() + >>> fig.show() + + """ + in1 = np.asarray(in1) + in2 = np.asarray(in2) + + if in1.ndim == in2.ndim == 0: # scalar inputs + return in1 * in2 + elif in1.ndim != in2.ndim: + raise ValueError("in1 and in2 should have the same dimensionality") + elif in1.size == 0 or in2.size == 0: # empty arrays + return np.array([]) + + in1, in2, axes = _init_freq_conv_axes(in1, in2, mode, axes, + sorted_axes=False) + + s1 = in1.shape + s2 = in2.shape + + shape = [max((s1[i], s2[i])) if i not in axes else s1[i] + s2[i] - 1 + for i in range(in1.ndim)] + + ret = _freq_domain_conv(in1, in2, axes, shape, calc_fast_len=True) + + return _apply_conv_mode(ret, s1, s2, mode, axes) + + +def _calc_oa_lens(s1, s2): + """Calculate the optimal FFT lengths for overlap-add convolution. + + The calculation is done for a single dimension. + + Parameters + ---------- + s1 : int + Size of the dimension for the first array. + s2 : int + Size of the dimension for the second array. + + Returns + ------- + block_size : int + The size of the FFT blocks. + overlap : int + The amount of overlap between two blocks. + in1_step : int + The size of each step for the first array. + in2_step : int + The size of each step for the first array. + + """ + # Set up the arguments for the conventional FFT approach. + fallback = (s1+s2-1, None, s1, s2) + + # Use conventional FFT convolve if sizes are same. + if s1 == s2 or s1 == 1 or s2 == 1: + return fallback + + if s2 > s1: + s1, s2 = s2, s1 + swapped = True + else: + swapped = False + + # There cannot be a useful block size if s2 is more than half of s1. + if s2 >= s1/2: + return fallback + + # Derivation of optimal block length + # For original formula see: + # https://en.wikipedia.org/wiki/Overlap-add_method + # + # Formula: + # K = overlap = s2-1 + # N = block_size + # C = complexity + # e = exponential, exp(1) + # + # C = (N*(log2(N)+1))/(N-K) + # C = (N*log2(2N))/(N-K) + # C = N/(N-K) * log2(2N) + # C1 = N/(N-K) + # C2 = log2(2N) = ln(2N)/ln(2) + # + # dC1/dN = (1*(N-K)-N)/(N-K)^2 = -K/(N-K)^2 + # dC2/dN = 2/(2*N*ln(2)) = 1/(N*ln(2)) + # + # dC/dN = dC1/dN*C2 + dC2/dN*C1 + # dC/dN = -K*ln(2N)/(ln(2)*(N-K)^2) + N/(N*ln(2)*(N-K)) + # dC/dN = -K*ln(2N)/(ln(2)*(N-K)^2) + 1/(ln(2)*(N-K)) + # dC/dN = -K*ln(2N)/(ln(2)*(N-K)^2) + (N-K)/(ln(2)*(N-K)^2) + # dC/dN = (-K*ln(2N) + (N-K)/(ln(2)*(N-K)^2) + # dC/dN = (N - K*ln(2N) - K)/(ln(2)*(N-K)^2) + # + # Solve for minimum, where dC/dN = 0 + # 0 = (N - K*ln(2N) - K)/(ln(2)*(N-K)^2) + # 0 * ln(2)*(N-K)^2 = N - K*ln(2N) - K + # 0 = N - K*ln(2N) - K + # 0 = N - K*(ln(2N) + 1) + # 0 = N - K*ln(2Ne) + # N = K*ln(2Ne) + # N/K = ln(2Ne) + # + # e^(N/K) = e^ln(2Ne) + # e^(N/K) = 2Ne + # 1/e^(N/K) = 1/(2*N*e) + # e^(N/-K) = 1/(2*N*e) + # e^(N/-K) = K/N*1/(2*K*e) + # N/K*e^(N/-K) = 1/(2*e*K) + # N/-K*e^(N/-K) = -1/(2*e*K) + # + # Using Lambert W function + # https://en.wikipedia.org/wiki/Lambert_W_function + # x = W(y) It is the solution to y = x*e^x + # x = N/-K + # y = -1/(2*e*K) + # + # N/-K = W(-1/(2*e*K)) + # + # N = -K*W(-1/(2*e*K)) + overlap = s2-1 + opt_size = -overlap*lambertw(-1/(2*math.e*overlap), k=-1).real + block_size = sp_fft.next_fast_len(math.ceil(opt_size)) + + # Use conventional FFT convolve if there is only going to be one block. + if block_size >= s1: + return fallback + + if not swapped: + in1_step = block_size-s2+1 + in2_step = s2 + else: + in1_step = s2 + in2_step = block_size-s2+1 + + return block_size, overlap, in1_step, in2_step + + +def oaconvolve(in1, in2, mode="full", axes=None): + """Convolve two N-dimensional arrays using the overlap-add method. + + Convolve `in1` and `in2` using the overlap-add method, with + the output size determined by the `mode` argument. + + This is generally much faster than `convolve` for large arrays (n > ~500), + and generally much faster than `fftconvolve` when one array is much + larger than the other, but can be slower when only a few output values are + needed or when the arrays are very similar in shape, and can only + output float arrays (int or object array inputs will be cast to float). + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear convolution + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + axes : int or array_like of ints or None, optional + Axes over which to compute the convolution. + The default is over all axes. + + Returns + ------- + out : array + An N-dimensional array containing a subset of the discrete linear + convolution of `in1` with `in2`. + + See Also + -------- + convolve : Uses the direct convolution or FFT convolution algorithm + depending on which is faster. + fftconvolve : An implementation of convolution using FFT. + + Notes + ----- + .. versionadded:: 1.4.0 + + References + ---------- + .. [1] Wikipedia, "Overlap-add_method". + https://en.wikipedia.org/wiki/Overlap-add_method + .. [2] Richard G. Lyons. Understanding Digital Signal Processing, + Third Edition, 2011. Chapter 13.10. + ISBN 13: 978-0137-02741-5 + + Examples + -------- + Convolve a 100,000 sample signal with a 512-sample filter. + + >>> import numpy as np + >>> from scipy import signal + >>> rng = np.random.default_rng() + >>> sig = rng.standard_normal(100000) + >>> filt = signal.firwin(512, 0.01) + >>> fsig = signal.oaconvolve(sig, filt) + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_orig, ax_mag) = plt.subplots(2, 1) + >>> ax_orig.plot(sig) + >>> ax_orig.set_title('White noise') + >>> ax_mag.plot(fsig) + >>> ax_mag.set_title('Filtered noise') + >>> fig.tight_layout() + >>> fig.show() + + """ + in1 = np.asarray(in1) + in2 = np.asarray(in2) + + if in1.ndim == in2.ndim == 0: # scalar inputs + return in1 * in2 + elif in1.ndim != in2.ndim: + raise ValueError("in1 and in2 should have the same dimensionality") + elif in1.size == 0 or in2.size == 0: # empty arrays + return np.array([]) + elif in1.shape == in2.shape: # Equivalent to fftconvolve + return fftconvolve(in1, in2, mode=mode, axes=axes) + + in1, in2, axes = _init_freq_conv_axes(in1, in2, mode, axes, + sorted_axes=True) + + s1 = in1.shape + s2 = in2.shape + + if not axes: + ret = in1 * in2 + return _apply_conv_mode(ret, s1, s2, mode, axes) + + # Calculate this now since in1 is changed later + shape_final = [None if i not in axes else + s1[i] + s2[i] - 1 for i in range(in1.ndim)] + + # Calculate the block sizes for the output, steps, first and second inputs. + # It is simpler to calculate them all together than doing them in separate + # loops due to all the special cases that need to be handled. + optimal_sizes = ((-1, -1, s1[i], s2[i]) if i not in axes else + _calc_oa_lens(s1[i], s2[i]) for i in range(in1.ndim)) + block_size, overlaps, \ + in1_step, in2_step = zip(*optimal_sizes) + + # Fall back to fftconvolve if there is only one block in every dimension. + if in1_step == s1 and in2_step == s2: + return fftconvolve(in1, in2, mode=mode, axes=axes) + + # Figure out the number of steps and padding. + # This would get too complicated in a list comprehension. + nsteps1 = [] + nsteps2 = [] + pad_size1 = [] + pad_size2 = [] + for i in range(in1.ndim): + if i not in axes: + pad_size1 += [(0, 0)] + pad_size2 += [(0, 0)] + continue + + if s1[i] > in1_step[i]: + curnstep1 = math.ceil((s1[i]+1)/in1_step[i]) + if (block_size[i] - overlaps[i])*curnstep1 < shape_final[i]: + curnstep1 += 1 + + curpad1 = curnstep1*in1_step[i] - s1[i] + else: + curnstep1 = 1 + curpad1 = 0 + + if s2[i] > in2_step[i]: + curnstep2 = math.ceil((s2[i]+1)/in2_step[i]) + if (block_size[i] - overlaps[i])*curnstep2 < shape_final[i]: + curnstep2 += 1 + + curpad2 = curnstep2*in2_step[i] - s2[i] + else: + curnstep2 = 1 + curpad2 = 0 + + nsteps1 += [curnstep1] + nsteps2 += [curnstep2] + pad_size1 += [(0, curpad1)] + pad_size2 += [(0, curpad2)] + + # Pad the array to a size that can be reshaped to the desired shape + # if necessary. + if not all(curpad == (0, 0) for curpad in pad_size1): + in1 = np.pad(in1, pad_size1, mode='constant', constant_values=0) + + if not all(curpad == (0, 0) for curpad in pad_size2): + in2 = np.pad(in2, pad_size2, mode='constant', constant_values=0) + + # Reshape the overlap-add parts to input block sizes. + split_axes = [iax+i for i, iax in enumerate(axes)] + fft_axes = [iax+1 for iax in split_axes] + + # We need to put each new dimension before the corresponding dimension + # being reshaped in order to get the data in the right layout at the end. + reshape_size1 = list(in1_step) + reshape_size2 = list(in2_step) + for i, iax in enumerate(split_axes): + reshape_size1.insert(iax, nsteps1[i]) + reshape_size2.insert(iax, nsteps2[i]) + + in1 = in1.reshape(*reshape_size1) + in2 = in2.reshape(*reshape_size2) + + # Do the convolution. + fft_shape = [block_size[i] for i in axes] + ret = _freq_domain_conv(in1, in2, fft_axes, fft_shape, calc_fast_len=False) + + # Do the overlap-add. + for ax, ax_fft, ax_split in zip(axes, fft_axes, split_axes): + overlap = overlaps[ax] + if overlap is None: + continue + + ret, overpart = np.split(ret, [-overlap], ax_fft) + overpart = np.split(overpart, [-1], ax_split)[0] + + ret_overpart = np.split(ret, [overlap], ax_fft)[0] + ret_overpart = np.split(ret_overpart, [1], ax_split)[1] + ret_overpart += overpart + + # Reshape back to the correct dimensionality. + shape_ret = [ret.shape[i] if i not in fft_axes else + ret.shape[i]*ret.shape[i-1] + for i in range(ret.ndim) if i not in split_axes] + ret = ret.reshape(*shape_ret) + + # Slice to the correct size. + slice_final = tuple([slice(islice) for islice in shape_final]) + ret = ret[slice_final] + + return _apply_conv_mode(ret, s1, s2, mode, axes) + + +def _numeric_arrays(arrays, kinds='buifc'): + """ + See if a list of arrays are all numeric. + + Parameters + ---------- + arrays : array or list of arrays + arrays to check if numeric. + kinds : string-like + The dtypes of the arrays to be checked. If the dtype.kind of + the ndarrays are not in this string the function returns False and + otherwise returns True. + """ + if isinstance(arrays, np.ndarray): + return arrays.dtype.kind in kinds + for array_ in arrays: + if array_.dtype.kind not in kinds: + return False + return True + + +def _conv_ops(x_shape, h_shape, mode): + """ + Find the number of operations required for direct/fft methods of + convolution. The direct operations were recorded by making a dummy class to + record the number of operations by overriding ``__mul__`` and ``__add__``. + The FFT operations rely on the (well-known) computational complexity of the + FFT (and the implementation of ``_freq_domain_conv``). + + """ + if mode == "full": + out_shape = [n + k - 1 for n, k in zip(x_shape, h_shape)] + elif mode == "valid": + out_shape = [abs(n - k) + 1 for n, k in zip(x_shape, h_shape)] + elif mode == "same": + out_shape = x_shape + else: + raise ValueError("Acceptable mode flags are 'valid'," + f" 'same', or 'full', not mode={mode}") + + s1, s2 = x_shape, h_shape + if len(x_shape) == 1: + s1, s2 = s1[0], s2[0] + if mode == "full": + direct_ops = s1 * s2 + elif mode == "valid": + direct_ops = (s2 - s1 + 1) * s1 if s2 >= s1 else (s1 - s2 + 1) * s2 + elif mode == "same": + direct_ops = (s1 * s2 if s1 < s2 else + s1 * s2 - (s2 // 2) * ((s2 + 1) // 2)) + else: + if mode == "full": + direct_ops = min(_prod(s1), _prod(s2)) * _prod(out_shape) + elif mode == "valid": + direct_ops = min(_prod(s1), _prod(s2)) * _prod(out_shape) + elif mode == "same": + direct_ops = _prod(s1) * _prod(s2) + + full_out_shape = [n + k - 1 for n, k in zip(x_shape, h_shape)] + N = _prod(full_out_shape) + fft_ops = 3 * N * np.log(N) # 3 separate FFTs of size full_out_shape + return fft_ops, direct_ops + + +def _fftconv_faster(x, h, mode): + """ + See if using fftconvolve or convolve is faster. + + Parameters + ---------- + x : np.ndarray + Signal + h : np.ndarray + Kernel + mode : str + Mode passed to convolve + + Returns + ------- + fft_faster : bool + + Notes + ----- + See docstring of `choose_conv_method` for details on tuning hardware. + + See pull request 11031 for more detail: + https://github.com/scipy/scipy/pull/11031. + + """ + fft_ops, direct_ops = _conv_ops(x.shape, h.shape, mode) + offset = -1e-3 if x.ndim == 1 else -1e-4 + constants = { + "valid": (1.89095737e-9, 2.1364985e-10, offset), + "full": (1.7649070e-9, 2.1414831e-10, offset), + "same": (3.2646654e-9, 2.8478277e-10, offset) + if h.size <= x.size + else (3.21635404e-9, 1.1773253e-8, -1e-5), + } if x.ndim == 1 else { + "valid": (1.85927e-9, 2.11242e-8, offset), + "full": (1.99817e-9, 1.66174e-8, offset), + "same": (2.04735e-9, 1.55367e-8, offset), + } + O_fft, O_direct, O_offset = constants[mode] + return O_fft * fft_ops < O_direct * direct_ops + O_offset + + +def _reverse_and_conj(x): + """ + Reverse array `x` in all dimensions and perform the complex conjugate + """ + reverse = (slice(None, None, -1),) * x.ndim + return x[reverse].conj() + + +def _np_conv_ok(volume, kernel, mode): + """ + See if numpy supports convolution of `volume` and `kernel` (i.e. both are + 1D ndarrays and of the appropriate shape). NumPy's 'same' mode uses the + size of the larger input, while SciPy's uses the size of the first input. + + Invalid mode strings will return False and be caught by the calling func. + """ + if volume.ndim == kernel.ndim == 1: + if mode in ('full', 'valid'): + return True + elif mode == 'same': + return volume.size >= kernel.size + else: + return False + + +def _timeit_fast(stmt="pass", setup="pass", repeat=3): + """ + Returns the time the statement/function took, in seconds. + + Faster, less precise version of IPython's timeit. `stmt` can be a statement + written as a string or a callable. + + Will do only 1 loop (like IPython's timeit) with no repetitions + (unlike IPython) for very slow functions. For fast functions, only does + enough loops to take 5 ms, which seems to produce similar results (on + Windows at least), and avoids doing an extraneous cycle that isn't + measured. + + """ + timer = timeit.Timer(stmt, setup) + + # determine number of calls per rep so total time for 1 rep >= 5 ms + x = 0 + for p in range(0, 10): + number = 10**p + x = timer.timeit(number) # seconds + if x >= 5e-3 / 10: # 5 ms for final test, 1/10th that for this one + break + if x > 1: # second + # If it's macroscopic, don't bother with repetitions + best = x + else: + number *= 10 + r = timer.repeat(repeat, number) + best = min(r) + + sec = best / number + return sec + + +def choose_conv_method(in1, in2, mode='full', measure=False): + """ + Find the fastest convolution/correlation method. + + This primarily exists to be called during the ``method='auto'`` option in + `convolve` and `correlate`. It can also be used to determine the value of + ``method`` for many different convolutions of the same dtype/shape. + In addition, it supports timing the convolution to adapt the value of + ``method`` to a particular set of inputs and/or hardware. + + Parameters + ---------- + in1 : array_like + The first argument passed into the convolution function. + in2 : array_like + The second argument passed into the convolution function. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear convolution + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + measure : bool, optional + If True, run and time the convolution of `in1` and `in2` with both + methods and return the fastest. If False (default), predict the fastest + method using precomputed values. + + Returns + ------- + method : str + A string indicating which convolution method is fastest, either + 'direct' or 'fft' + times : dict, optional + A dictionary containing the times (in seconds) needed for each method. + This value is only returned if ``measure=True``. + + See Also + -------- + convolve + correlate + + Notes + ----- + Generally, this method is 99% accurate for 2D signals and 85% accurate + for 1D signals for randomly chosen input sizes. For precision, use + ``measure=True`` to find the fastest method by timing the convolution. + This can be used to avoid the minimal overhead of finding the fastest + ``method`` later, or to adapt the value of ``method`` to a particular set + of inputs. + + Experiments were run on an Amazon EC2 r5a.2xlarge machine to test this + function. These experiments measured the ratio between the time required + when using ``method='auto'`` and the time required for the fastest method + (i.e., ``ratio = time_auto / min(time_fft, time_direct)``). In these + experiments, we found: + + * There is a 95% chance of this ratio being less than 1.5 for 1D signals + and a 99% chance of being less than 2.5 for 2D signals. + * The ratio was always less than 2.5/5 for 1D/2D signals respectively. + * This function is most inaccurate for 1D convolutions that take between 1 + and 10 milliseconds with ``method='direct'``. A good proxy for this + (at least in our experiments) is ``1e6 <= in1.size * in2.size <= 1e7``. + + The 2D results almost certainly generalize to 3D/4D/etc because the + implementation is the same (the 1D implementation is different). + + All the numbers above are specific to the EC2 machine. However, we did find + that this function generalizes fairly decently across hardware. The speed + tests were of similar quality (and even slightly better) than the same + tests performed on the machine to tune this function's numbers (a mid-2014 + 15-inch MacBook Pro with 16GB RAM and a 2.5GHz Intel i7 processor). + + There are cases when `fftconvolve` supports the inputs but this function + returns `direct` (e.g., to protect against floating point integer + precision). + + .. versionadded:: 0.19 + + Examples + -------- + Estimate the fastest method for a given input: + + >>> import numpy as np + >>> from scipy import signal + >>> rng = np.random.default_rng() + >>> img = rng.random((32, 32)) + >>> filter = rng.random((8, 8)) + >>> method = signal.choose_conv_method(img, filter, mode='same') + >>> method + 'fft' + + This can then be applied to other arrays of the same dtype and shape: + + >>> img2 = rng.random((32, 32)) + >>> filter2 = rng.random((8, 8)) + >>> corr2 = signal.correlate(img2, filter2, mode='same', method=method) + >>> conv2 = signal.convolve(img2, filter2, mode='same', method=method) + + The output of this function (``method``) works with `correlate` and + `convolve`. + + """ + volume = np.asarray(in1) + kernel = np.asarray(in2) + + _reject_objects(volume, 'choose_conv_method') + _reject_objects(kernel, 'choose_conv_method') + + if measure: + times = {} + for method in ['fft', 'direct']: + times[method] = _timeit_fast(lambda: convolve(volume, kernel, + mode=mode, method=method)) + + chosen_method = 'fft' if times['fft'] < times['direct'] else 'direct' + return chosen_method, times + + # for integer input, + # catch when more precision required than float provides (representing an + # integer as float can lose precision in fftconvolve if larger than 2**52) + if any([_numeric_arrays([x], kinds='ui') for x in [volume, kernel]]): + max_value = int(np.abs(volume).max()) * int(np.abs(kernel).max()) + max_value *= int(min(volume.size, kernel.size)) + if max_value > 2**np.finfo('float').nmant - 1: + return 'direct' + + if _numeric_arrays([volume, kernel], kinds='b'): + return 'direct' + + if _numeric_arrays([volume, kernel]): + if _fftconv_faster(volume, kernel, mode): + return 'fft' + + return 'direct' + + +def convolve(in1, in2, mode='full', method='auto'): + """ + Convolve two N-dimensional arrays. + + Convolve `in1` and `in2`, with the output size determined by the + `mode` argument. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear convolution + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + method : str {'auto', 'direct', 'fft'}, optional + A string indicating which method to use to calculate the convolution. + + ``direct`` + The convolution is determined directly from sums, the definition of + convolution. + ``fft`` + The Fourier Transform is used to perform the convolution by calling + `fftconvolve`. + ``auto`` + Automatically chooses direct or Fourier method based on an estimate + of which is faster (default). See Notes for more detail. + + .. versionadded:: 0.19.0 + + Returns + ------- + convolve : array + An N-dimensional array containing a subset of the discrete linear + convolution of `in1` with `in2`. + + Warns + ----- + RuntimeWarning + Use of the FFT convolution on input containing NAN or INF will lead + to the entire output being NAN or INF. Use method='direct' when your + input contains NAN or INF values. + + See Also + -------- + numpy.polymul : performs polynomial multiplication (same operation, but + also accepts poly1d objects) + choose_conv_method : chooses the fastest appropriate convolution method + fftconvolve : Always uses the FFT method. + oaconvolve : Uses the overlap-add method to do convolution, which is + generally faster when the input arrays are large and + significantly different in size. + + Notes + ----- + By default, `convolve` and `correlate` use ``method='auto'``, which calls + `choose_conv_method` to choose the fastest method using pre-computed + values (`choose_conv_method` can also measure real-world timing with a + keyword argument). Because `fftconvolve` relies on floating point numbers, + there are certain constraints that may force ``method='direct'`` (more detail + in `choose_conv_method` docstring). + + Examples + -------- + Smooth a square pulse using a Hann window: + + >>> import numpy as np + >>> from scipy import signal + >>> sig = np.repeat([0., 1., 0.], 100) + >>> win = signal.windows.hann(50) + >>> filtered = signal.convolve(sig, win, mode='same') / sum(win) + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_orig, ax_win, ax_filt) = plt.subplots(3, 1, sharex=True) + >>> ax_orig.plot(sig) + >>> ax_orig.set_title('Original pulse') + >>> ax_orig.margins(0, 0.1) + >>> ax_win.plot(win) + >>> ax_win.set_title('Filter impulse response') + >>> ax_win.margins(0, 0.1) + >>> ax_filt.plot(filtered) + >>> ax_filt.set_title('Filtered signal') + >>> ax_filt.margins(0, 0.1) + >>> fig.tight_layout() + >>> fig.show() + + """ + volume = np.asarray(in1) + kernel = np.asarray(in2) + + _reject_objects(volume, 'correlate') + _reject_objects(kernel, 'correlate') + + if volume.ndim == kernel.ndim == 0: + return volume * kernel + elif volume.ndim != kernel.ndim: + raise ValueError("volume and kernel should have the same " + "dimensionality") + + if _inputs_swap_needed(mode, volume.shape, kernel.shape): + # Convolution is commutative; order doesn't have any effect on output + volume, kernel = kernel, volume + + if method == 'auto': + method = choose_conv_method(volume, kernel, mode=mode) + + if method == 'fft': + out = fftconvolve(volume, kernel, mode=mode) + result_type = np.result_type(volume, kernel) + if result_type.kind in {'u', 'i'}: + out = np.around(out) + + if np.isnan(out.flat[0]) or np.isinf(out.flat[0]): + warnings.warn("Use of fft convolution on input with NAN or inf" + " results in NAN or inf output. Consider using" + " method='direct' instead.", + category=RuntimeWarning, stacklevel=2) + + return out.astype(result_type) + elif method == 'direct': + # fastpath to faster numpy.convolve for 1d inputs when possible + if _np_conv_ok(volume, kernel, mode): + return np.convolve(volume, kernel, mode) + + return correlate(volume, _reverse_and_conj(kernel), mode, 'direct') + else: + raise ValueError("Acceptable method flags are 'auto'," + " 'direct', or 'fft'.") + + +def order_filter(a, domain, rank): + """ + Perform an order filter on an N-D array. + + Perform an order filter on the array in. The domain argument acts as a + mask centered over each pixel. The non-zero elements of domain are + used to select elements surrounding each input pixel which are placed + in a list. The list is sorted, and the output for that pixel is the + element corresponding to rank in the sorted list. + + Parameters + ---------- + a : ndarray + The N-dimensional input array. + domain : array_like + A mask array with the same number of dimensions as `a`. + Each dimension should have an odd number of elements. + rank : int + A non-negative integer which selects the element from the + sorted list (0 corresponds to the smallest element, 1 is the + next smallest element, etc.). + + Returns + ------- + out : ndarray + The results of the order filter in an array with the same + shape as `a`. + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> x = np.arange(25).reshape(5, 5) + >>> domain = np.identity(3) + >>> x + array([[ 0, 1, 2, 3, 4], + [ 5, 6, 7, 8, 9], + [10, 11, 12, 13, 14], + [15, 16, 17, 18, 19], + [20, 21, 22, 23, 24]]) + >>> signal.order_filter(x, domain, 0) + array([[ 0, 0, 0, 0, 0], + [ 0, 0, 1, 2, 0], + [ 0, 5, 6, 7, 0], + [ 0, 10, 11, 12, 0], + [ 0, 0, 0, 0, 0]]) + >>> signal.order_filter(x, domain, 2) + array([[ 6, 7, 8, 9, 4], + [ 11, 12, 13, 14, 9], + [ 16, 17, 18, 19, 14], + [ 21, 22, 23, 24, 19], + [ 20, 21, 22, 23, 24]]) + + """ + domain = np.asarray(domain) + for dimsize in domain.shape: + if (dimsize % 2) != 1: + raise ValueError("Each dimension of domain argument " + "should have an odd number of elements.") + + a = np.asarray(a) + if not (np.issubdtype(a.dtype, np.integer) + or a.dtype in [np.float32, np.float64]): + raise ValueError(f"dtype={a.dtype} is not supported by order_filter") + + result = ndimage.rank_filter(a, rank, footprint=domain, mode='constant') + return result + + +def medfilt(volume, kernel_size=None): + """ + Perform a median filter on an N-dimensional array. + + Apply a median filter to the input array using a local window-size + given by `kernel_size`. The array will automatically be zero-padded. + + Parameters + ---------- + volume : array_like + An N-dimensional input array. + kernel_size : array_like, optional + A scalar or an N-length list giving the size of the median filter + window in each dimension. Elements of `kernel_size` should be odd. + If `kernel_size` is a scalar, then this scalar is used as the size in + each dimension. Default size is 3 for each dimension. + + Returns + ------- + out : ndarray + An array the same size as input containing the median filtered + result. + + Warns + ----- + UserWarning + If array size is smaller than kernel size along any dimension + + See Also + -------- + scipy.ndimage.median_filter + scipy.signal.medfilt2d + + Notes + ----- + The more general function `scipy.ndimage.median_filter` has a more + efficient implementation of a median filter and therefore runs much faster. + + For 2-dimensional images with ``uint8``, ``float32`` or ``float64`` dtypes, + the specialised function `scipy.signal.medfilt2d` may be faster. + + """ + volume = np.atleast_1d(volume) + if not (np.issubdtype(volume.dtype, np.integer) + or volume.dtype in [np.float32, np.float64]): + raise ValueError(f"dtype={volume.dtype} is not supported by medfilt") + + if kernel_size is None: + kernel_size = [3] * volume.ndim + kernel_size = np.asarray(kernel_size) + if kernel_size.shape == (): + kernel_size = np.repeat(kernel_size.item(), volume.ndim) + + for k in range(volume.ndim): + if (kernel_size[k] % 2) != 1: + raise ValueError("Each element of kernel_size should be odd.") + if any(k > s for k, s in zip(kernel_size, volume.shape)): + warnings.warn('kernel_size exceeds volume extent: the volume will be ' + 'zero-padded.', + stacklevel=2) + + size = math.prod(kernel_size) + result = ndimage.rank_filter(volume, size // 2, size=kernel_size, + mode='constant') + + return result + + +def wiener(im, mysize=None, noise=None): + """ + Perform a Wiener filter on an N-dimensional array. + + Apply a Wiener filter to the N-dimensional array `im`. + + Parameters + ---------- + im : ndarray + An N-dimensional array. + mysize : int or array_like, optional + A scalar or an N-length list giving the size of the Wiener filter + window in each dimension. Elements of mysize should be odd. + If mysize is a scalar, then this scalar is used as the size + in each dimension. + noise : float, optional + The noise-power to use. If None, then noise is estimated as the + average of the local variance of the input. + + Returns + ------- + out : ndarray + Wiener filtered result with the same shape as `im`. + + Notes + ----- + This implementation is similar to wiener2 in Matlab/Octave. + For more details see [1]_ + + References + ---------- + .. [1] Lim, Jae S., Two-Dimensional Signal and Image Processing, + Englewood Cliffs, NJ, Prentice Hall, 1990, p. 548. + + Examples + -------- + >>> from scipy.datasets import face + >>> from scipy.signal import wiener + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> rng = np.random.default_rng() + >>> img = rng.random((40, 40)) #Create a random image + >>> filtered_img = wiener(img, (5, 5)) #Filter the image + >>> f, (plot1, plot2) = plt.subplots(1, 2) + >>> plot1.imshow(img) + >>> plot2.imshow(filtered_img) + >>> plt.show() + + """ + im = np.asarray(im) + if mysize is None: + mysize = [3] * im.ndim + mysize = np.asarray(mysize) + if mysize.shape == (): + mysize = np.repeat(mysize.item(), im.ndim) + + # Estimate the local mean + size = math.prod(mysize) + lMean = correlate(im, np.ones(mysize), 'same') / size + + # Estimate the local variance + lVar = (correlate(im ** 2, np.ones(mysize), 'same') / size - lMean ** 2) + + # Estimate the noise power if needed. + if noise is None: + noise = np.mean(np.ravel(lVar), axis=0) + + res = (im - lMean) + res *= (1 - noise / lVar) + res += lMean + out = np.where(lVar < noise, lMean, res) + + return out + + +def convolve2d(in1, in2, mode='full', boundary='fill', fillvalue=0): + """ + Convolve two 2-dimensional arrays. + + Convolve `in1` and `in2` with output size determined by `mode`, and + boundary conditions determined by `boundary` and `fillvalue`. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear convolution + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + boundary : str {'fill', 'wrap', 'symm'}, optional + A flag indicating how to handle boundaries: + + ``fill`` + pad input arrays with fillvalue. (default) + ``wrap`` + circular boundary conditions. + ``symm`` + symmetrical boundary conditions. + + fillvalue : scalar, optional + Value to fill pad input arrays with. Default is 0. + + Returns + ------- + out : ndarray + A 2-dimensional array containing a subset of the discrete linear + convolution of `in1` with `in2`. + + Examples + -------- + Compute the gradient of an image by 2D convolution with a complex Scharr + operator. (Horizontal operator is real, vertical is imaginary.) Use + symmetric boundary condition to avoid creating edges at the image + boundaries. + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy import datasets + >>> ascent = datasets.ascent() + >>> scharr = np.array([[ -3-3j, 0-10j, +3 -3j], + ... [-10+0j, 0+ 0j, +10 +0j], + ... [ -3+3j, 0+10j, +3 +3j]]) # Gx + j*Gy + >>> grad = signal.convolve2d(ascent, scharr, boundary='symm', mode='same') + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_orig, ax_mag, ax_ang) = plt.subplots(3, 1, figsize=(6, 15)) + >>> ax_orig.imshow(ascent, cmap='gray') + >>> ax_orig.set_title('Original') + >>> ax_orig.set_axis_off() + >>> ax_mag.imshow(np.absolute(grad), cmap='gray') + >>> ax_mag.set_title('Gradient magnitude') + >>> ax_mag.set_axis_off() + >>> ax_ang.imshow(np.angle(grad), cmap='hsv') # hsv is cyclic, like angles + >>> ax_ang.set_title('Gradient orientation') + >>> ax_ang.set_axis_off() + >>> fig.show() + + """ + in1 = np.asarray(in1) + in2 = np.asarray(in2) + + if not in1.ndim == in2.ndim == 2: + raise ValueError('convolve2d inputs must both be 2-D arrays') + + if _inputs_swap_needed(mode, in1.shape, in2.shape): + in1, in2 = in2, in1 + + val = _valfrommode(mode) + bval = _bvalfromboundary(boundary) + out = _sigtools._convolve2d(in1, in2, 1, val, bval, fillvalue) + return out + + +def correlate2d(in1, in2, mode='full', boundary='fill', fillvalue=0): + """ + Cross-correlate two 2-dimensional arrays. + + Cross correlate `in1` and `in2` with output size determined by `mode`, and + boundary conditions determined by `boundary` and `fillvalue`. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear cross-correlation + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + boundary : str {'fill', 'wrap', 'symm'}, optional + A flag indicating how to handle boundaries: + + ``fill`` + pad input arrays with fillvalue. (default) + ``wrap`` + circular boundary conditions. + ``symm`` + symmetrical boundary conditions. + + fillvalue : scalar, optional + Value to fill pad input arrays with. Default is 0. + + Returns + ------- + correlate2d : ndarray + A 2-dimensional array containing a subset of the discrete linear + cross-correlation of `in1` with `in2`. + + Notes + ----- + When using "same" mode with even-length inputs, the outputs of `correlate` + and `correlate2d` differ: There is a 1-index offset between them. + + Examples + -------- + Use 2D cross-correlation to find the location of a template in a noisy + image: + + >>> import numpy as np + >>> from scipy import signal, datasets, ndimage + >>> rng = np.random.default_rng() + >>> face = datasets.face(gray=True) - datasets.face(gray=True).mean() + >>> face = ndimage.zoom(face[30:500, 400:950], 0.5) # extract the face + >>> template = np.copy(face[135:165, 140:175]) # right eye + >>> template -= template.mean() + >>> face = face + rng.standard_normal(face.shape) * 50 # add noise + >>> corr = signal.correlate2d(face, template, boundary='symm', mode='same') + >>> y, x = np.unravel_index(np.argmax(corr), corr.shape) # find the match + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_orig, ax_template, ax_corr) = plt.subplots(3, 1, + ... figsize=(6, 15)) + >>> ax_orig.imshow(face, cmap='gray') + >>> ax_orig.set_title('Original') + >>> ax_orig.set_axis_off() + >>> ax_template.imshow(template, cmap='gray') + >>> ax_template.set_title('Template') + >>> ax_template.set_axis_off() + >>> ax_corr.imshow(corr, cmap='gray') + >>> ax_corr.set_title('Cross-correlation') + >>> ax_corr.set_axis_off() + >>> ax_orig.plot(x, y, 'ro') + >>> fig.show() + + """ + in1 = np.asarray(in1) + in2 = np.asarray(in2) + + if not in1.ndim == in2.ndim == 2: + raise ValueError('correlate2d inputs must both be 2-D arrays') + + swapped_inputs = _inputs_swap_needed(mode, in1.shape, in2.shape) + if swapped_inputs: + in1, in2 = in2, in1 + + val = _valfrommode(mode) + bval = _bvalfromboundary(boundary) + out = _sigtools._convolve2d(in1, in2.conj(), 0, val, bval, fillvalue) + + if swapped_inputs: + out = out[::-1, ::-1] + + return out + + +def medfilt2d(input, kernel_size=3): + """ + Median filter a 2-dimensional array. + + Apply a median filter to the `input` array using a local window-size + given by `kernel_size` (must be odd). The array is zero-padded + automatically. + + Parameters + ---------- + input : array_like + A 2-dimensional input array. + kernel_size : array_like, optional + A scalar or a list of length 2, giving the size of the + median filter window in each dimension. Elements of + `kernel_size` should be odd. If `kernel_size` is a scalar, + then this scalar is used as the size in each dimension. + Default is a kernel of size (3, 3). + + Returns + ------- + out : ndarray + An array the same size as input containing the median filtered + result. + + See Also + -------- + scipy.ndimage.median_filter + + Notes + ----- + This is faster than `medfilt` when the input dtype is ``uint8``, + ``float32``, or ``float64``; for other types, this falls back to + `medfilt`. In some situations, `scipy.ndimage.median_filter` may be + faster than this function. + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> x = np.arange(25).reshape(5, 5) + >>> x + array([[ 0, 1, 2, 3, 4], + [ 5, 6, 7, 8, 9], + [10, 11, 12, 13, 14], + [15, 16, 17, 18, 19], + [20, 21, 22, 23, 24]]) + + # Replaces i,j with the median out of 5*5 window + + >>> signal.medfilt2d(x, kernel_size=5) + array([[ 0, 0, 2, 0, 0], + [ 0, 3, 7, 4, 0], + [ 2, 8, 12, 9, 4], + [ 0, 8, 12, 9, 0], + [ 0, 0, 12, 0, 0]]) + + # Replaces i,j with the median out of default 3*3 window + + >>> signal.medfilt2d(x) + array([[ 0, 1, 2, 3, 0], + [ 1, 6, 7, 8, 4], + [ 6, 11, 12, 13, 9], + [11, 16, 17, 18, 14], + [ 0, 16, 17, 18, 0]]) + + # Replaces i,j with the median out of default 5*3 window + + >>> signal.medfilt2d(x, kernel_size=[5,3]) + array([[ 0, 1, 2, 3, 0], + [ 0, 6, 7, 8, 3], + [ 5, 11, 12, 13, 8], + [ 5, 11, 12, 13, 8], + [ 0, 11, 12, 13, 0]]) + + # Replaces i,j with the median out of default 3*5 window + + >>> signal.medfilt2d(x, kernel_size=[3,5]) + array([[ 0, 0, 2, 1, 0], + [ 1, 5, 7, 6, 3], + [ 6, 10, 12, 11, 8], + [11, 15, 17, 16, 13], + [ 0, 15, 17, 16, 0]]) + + # As seen in the examples, + # kernel numbers must be odd and not exceed original array dim + + """ + image = np.asarray(input) + + # checking dtype.type, rather than just dtype, is necessary for + # excluding np.longdouble with MS Visual C. + if image.dtype.type not in (np.ubyte, np.float32, np.float64): + return medfilt(image, kernel_size) + + if kernel_size is None: + kernel_size = [3] * 2 + kernel_size = np.asarray(kernel_size) + if kernel_size.shape == (): + kernel_size = np.repeat(kernel_size.item(), 2) + + for size in kernel_size: + if (size % 2) != 1: + raise ValueError("Each element of kernel_size should be odd.") + + return _sigtools._medfilt2d(image, kernel_size) + + +def lfilter(b, a, x, axis=-1, zi=None): + """ + Filter data along one-dimension with an IIR or FIR filter. + + Filter a data sequence, `x`, using a digital filter. This works for many + fundamental data types (including Object type). The filter is a direct + form II transposed implementation of the standard difference equation + (see Notes). + + The function `sosfilt` (and filter design using ``output='sos'``) should be + preferred over `lfilter` for most filtering tasks, as second-order sections + have fewer numerical problems. + + Parameters + ---------- + b : array_like + The numerator coefficient vector in a 1-D sequence. + a : array_like + The denominator coefficient vector in a 1-D sequence. If ``a[0]`` + is not 1, then both `a` and `b` are normalized by ``a[0]``. + x : array_like + An N-dimensional input array. + axis : int, optional + The axis of the input data array along which to apply the + linear filter. The filter is applied to each subarray along + this axis. Default is -1. + zi : array_like, optional + Initial conditions for the filter delays. It is a vector + (or array of vectors for an N-dimensional input) of length + ``max(len(a), len(b)) - 1``. If `zi` is None or is not given then + initial rest is assumed. See `lfiltic` for more information. + + Returns + ------- + y : array + The output of the digital filter. + zf : array, optional + If `zi` is None, this is not returned, otherwise, `zf` holds the + final filter delay values. + + See Also + -------- + lfiltic : Construct initial conditions for `lfilter`. + lfilter_zi : Compute initial state (steady state of step response) for + `lfilter`. + filtfilt : A forward-backward filter, to obtain a filter with zero phase. + savgol_filter : A Savitzky-Golay filter. + sosfilt: Filter data using cascaded second-order sections. + sosfiltfilt: A forward-backward filter using second-order sections. + + Notes + ----- + The filter function is implemented as a direct II transposed structure. + This means that the filter implements:: + + a[0]*y[n] = b[0]*x[n] + b[1]*x[n-1] + ... + b[M]*x[n-M] + - a[1]*y[n-1] - ... - a[N]*y[n-N] + + where `M` is the degree of the numerator, `N` is the degree of the + denominator, and `n` is the sample number. It is implemented using + the following difference equations (assuming M = N):: + + a[0]*y[n] = b[0] * x[n] + d[0][n-1] + d[0][n] = b[1] * x[n] - a[1] * y[n] + d[1][n-1] + d[1][n] = b[2] * x[n] - a[2] * y[n] + d[2][n-1] + ... + d[N-2][n] = b[N-1]*x[n] - a[N-1]*y[n] + d[N-1][n-1] + d[N-1][n] = b[N] * x[n] - a[N] * y[n] + + where `d` are the state variables. + + The rational transfer function describing this filter in the + z-transform domain is:: + + -1 -M + b[0] + b[1]z + ... + b[M] z + Y(z) = -------------------------------- X(z) + -1 -N + a[0] + a[1]z + ... + a[N] z + + Examples + -------- + Generate a noisy signal to be filtered: + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + >>> t = np.linspace(-1, 1, 201) + >>> x = (np.sin(2*np.pi*0.75*t*(1-t) + 2.1) + + ... 0.1*np.sin(2*np.pi*1.25*t + 1) + + ... 0.18*np.cos(2*np.pi*3.85*t)) + >>> xn = x + rng.standard_normal(len(t)) * 0.08 + + Create an order 3 lowpass butterworth filter: + + >>> b, a = signal.butter(3, 0.05) + + Apply the filter to xn. Use lfilter_zi to choose the initial condition of + the filter: + + >>> zi = signal.lfilter_zi(b, a) + >>> z, _ = signal.lfilter(b, a, xn, zi=zi*xn[0]) + + Apply the filter again, to have a result filtered at an order the same as + filtfilt: + + >>> z2, _ = signal.lfilter(b, a, z, zi=zi*z[0]) + + Use filtfilt to apply the filter: + + >>> y = signal.filtfilt(b, a, xn) + + Plot the original signal and the various filtered versions: + + >>> plt.figure + >>> plt.plot(t, xn, 'b', alpha=0.75) + >>> plt.plot(t, z, 'r--', t, z2, 'r', t, y, 'k') + >>> plt.legend(('noisy signal', 'lfilter, once', 'lfilter, twice', + ... 'filtfilt'), loc='best') + >>> plt.grid(True) + >>> plt.show() + + """ + b = np.atleast_1d(b) + a = np.atleast_1d(a) + + _reject_objects(x, 'lfilter') + _reject_objects(a, 'lfilter') + _reject_objects(b, 'lfilter') + + if len(a) == 1: + # This path only supports types fdgFDGO to mirror _linear_filter below. + # Any of b, a, x, or zi can set the dtype, but there is no default + # casting of other types; instead a NotImplementedError is raised. + b = np.asarray(b) + a = np.asarray(a) + if b.ndim != 1 and a.ndim != 1: + raise ValueError('object of too small depth for desired array') + x = _validate_x(x) + inputs = [b, a, x] + if zi is not None: + # _linear_filter does not broadcast zi, but does do expansion of + # singleton dims. + zi = np.asarray(zi) + if zi.ndim != x.ndim: + raise ValueError('object of too small depth for desired array') + expected_shape = list(x.shape) + expected_shape[axis] = b.shape[0] - 1 + expected_shape = tuple(expected_shape) + # check the trivial case where zi is the right shape first + if zi.shape != expected_shape: + strides = zi.ndim * [None] + if axis < 0: + axis += zi.ndim + for k in range(zi.ndim): + if k == axis and zi.shape[k] == expected_shape[k]: + strides[k] = zi.strides[k] + elif k != axis and zi.shape[k] == expected_shape[k]: + strides[k] = zi.strides[k] + elif k != axis and zi.shape[k] == 1: + strides[k] = 0 + else: + raise ValueError('Unexpected shape for zi: expected ' + f'{expected_shape}, found {zi.shape}.') + zi = np.lib.stride_tricks.as_strided(zi, expected_shape, + strides) + inputs.append(zi) + dtype = np.result_type(*inputs) + + if dtype.char not in 'fdgFDGO': + raise NotImplementedError(f"input type '{dtype}' not supported") + + b = np.array(b, dtype=dtype) + a = np.asarray(a, dtype=dtype) + b /= a[0] + x = np.asarray(x, dtype=dtype) + + out_full = np.apply_along_axis(lambda y: np.convolve(b, y), axis, x) + ind = out_full.ndim * [slice(None)] + if zi is not None: + ind[axis] = slice(zi.shape[axis]) + out_full[tuple(ind)] += zi + + ind[axis] = slice(out_full.shape[axis] - len(b) + 1) + out = out_full[tuple(ind)] + + if zi is None: + return out + else: + ind[axis] = slice(out_full.shape[axis] - len(b) + 1, None) + zf = out_full[tuple(ind)] + return out, zf + else: + if zi is None: + return _sigtools._linear_filter(b, a, x, axis) + else: + return _sigtools._linear_filter(b, a, x, axis, zi) + + +def lfiltic(b, a, y, x=None): + """ + Construct initial conditions for lfilter given input and output vectors. + + Given a linear filter (b, a) and initial conditions on the output `y` + and the input `x`, return the initial conditions on the state vector zi + which is used by `lfilter` to generate the output given the input. + + Parameters + ---------- + b : array_like + Linear filter term. + a : array_like + Linear filter term. + y : array_like + Initial conditions. + + If ``N = len(a) - 1``, then ``y = {y[-1], y[-2], ..., y[-N]}``. + + If `y` is too short, it is padded with zeros. + x : array_like, optional + Initial conditions. + + If ``M = len(b) - 1``, then ``x = {x[-1], x[-2], ..., x[-M]}``. + + If `x` is not given, its initial conditions are assumed zero. + + If `x` is too short, it is padded with zeros. + + Returns + ------- + zi : ndarray + The state vector ``zi = {z_0[-1], z_1[-1], ..., z_K-1[-1]}``, + where ``K = max(M, N)``. + + See Also + -------- + lfilter, lfilter_zi + + """ + N = np.size(a) - 1 + M = np.size(b) - 1 + K = max(M, N) + y = np.asarray(y) + + if x is None: + result_type = np.result_type(np.asarray(b), np.asarray(a), y) + if result_type.kind in 'bui': + result_type = np.float64 + x = np.zeros(M, dtype=result_type) + else: + x = np.asarray(x) + + result_type = np.result_type(np.asarray(b), np.asarray(a), y, x) + if result_type.kind in 'bui': + result_type = np.float64 + x = x.astype(result_type) + + L = np.size(x) + if L < M: + x = np.r_[x, np.zeros(M - L)] + + y = y.astype(result_type) + zi = np.zeros(K, result_type) + + L = np.size(y) + if L < N: + y = np.r_[y, np.zeros(N - L)] + + for m in range(M): + zi[m] = np.sum(b[m + 1:] * x[:M - m], axis=0) + + for m in range(N): + zi[m] -= np.sum(a[m + 1:] * y[:N - m], axis=0) + + return zi + + +def deconvolve(signal, divisor): + """Deconvolves ``divisor`` out of ``signal`` using inverse filtering. + + Returns the quotient and remainder such that + ``signal = convolve(divisor, quotient) + remainder`` + + Parameters + ---------- + signal : (N,) array_like + Signal data, typically a recorded signal + divisor : (N,) array_like + Divisor data, typically an impulse response or filter that was + applied to the original signal + + Returns + ------- + quotient : ndarray + Quotient, typically the recovered original signal + remainder : ndarray + Remainder + + See Also + -------- + numpy.polydiv : performs polynomial division (same operation, but + also accepts poly1d objects) + + Examples + -------- + Deconvolve a signal that's been filtered: + + >>> from scipy import signal + >>> original = [0, 1, 0, 0, 1, 1, 0, 0] + >>> impulse_response = [2, 1] + >>> recorded = signal.convolve(impulse_response, original) + >>> recorded + array([0, 2, 1, 0, 2, 3, 1, 0, 0]) + >>> recovered, remainder = signal.deconvolve(recorded, impulse_response) + >>> recovered + array([ 0., 1., 0., 0., 1., 1., 0., 0.]) + + """ + num = np.atleast_1d(signal) + den = np.atleast_1d(divisor) + if num.ndim > 1: + raise ValueError("signal must be 1-D.") + if den.ndim > 1: + raise ValueError("divisor must be 1-D.") + N = len(num) + D = len(den) + if D > N: + quot = [] + rem = num + else: + input = np.zeros(N - D + 1, float) + input[0] = 1 + quot = lfilter(num, den, input) + rem = num - convolve(den, quot, mode='full') + return quot, rem + + +def hilbert(x, N=None, axis=-1): + r"""FFT-based computation of the analytic signal. + + The analytic signal is calculated by filtering out the negative frequencies and + doubling the amplitudes of the positive frequencies in the FFT domain. + The imaginary part of the result is the hilbert transform of the real-valued input + signal. + + The transformation is done along the last axis by default. + + Parameters + ---------- + x : array_like + Signal data. Must be real. + N : int, optional + Number of Fourier components. Default: ``x.shape[axis]`` + axis : int, optional + Axis along which to do the transformation. Default: -1. + + Returns + ------- + xa : ndarray + Analytic signal of `x`, of each 1-D array along `axis` + + Notes + ----- + The analytic signal ``x_a(t)`` of a real-valued signal ``x(t)`` + can be expressed as [1]_ + + .. math:: x_a = F^{-1}(F(x) 2U) = x + i y\ , + + where `F` is the Fourier transform, `U` the unit step function, + and `y` the Hilbert transform of `x`. [2]_ + + In other words, the negative half of the frequency spectrum is zeroed + out, turning the real-valued signal into a complex-valued signal. The Hilbert + transformed signal can be obtained from ``np.imag(hilbert(x))``, and the + original signal from ``np.real(hilbert(x))``. + + References + ---------- + .. [1] Wikipedia, "Analytic signal". + https://en.wikipedia.org/wiki/Analytic_signal + .. [2] Wikipedia, "Hilbert Transform". + https://en.wikipedia.org/wiki/Hilbert_transform + .. [3] Leon Cohen, "Time-Frequency Analysis", 1995. Chapter 2. + .. [4] Alan V. Oppenheim, Ronald W. Schafer. Discrete-Time Signal + Processing, Third Edition, 2009. Chapter 12. + ISBN 13: 978-1292-02572-8 + + See Also + -------- + envelope: Compute envelope of a real- or complex-valued signal. + + Examples + -------- + In this example we use the Hilbert transform to determine the amplitude + envelope and instantaneous frequency of an amplitude-modulated signal. + + Let's create a chirp of which the frequency increases from 20 Hz to 100 Hz and + apply an amplitude modulation: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.signal import hilbert, chirp + ... + >>> duration, fs = 1, 400 # 1 s signal with sampling frequency of 400 Hz + >>> t = np.arange(int(fs*duration)) / fs # timestamps of samples + >>> signal = chirp(t, 20.0, t[-1], 100.0) + >>> signal *= (1.0 + 0.5 * np.sin(2.0*np.pi*3.0*t) ) + + The amplitude envelope is given by the magnitude of the analytic signal. The + instantaneous frequency can be obtained by differentiating the + instantaneous phase in respect to time. The instantaneous phase corresponds + to the phase angle of the analytic signal. + + >>> analytic_signal = hilbert(signal) + >>> amplitude_envelope = np.abs(analytic_signal) + >>> instantaneous_phase = np.unwrap(np.angle(analytic_signal)) + >>> instantaneous_frequency = np.diff(instantaneous_phase) / (2.0*np.pi) * fs + ... + >>> fig, (ax0, ax1) = plt.subplots(nrows=2, sharex='all', tight_layout=True) + >>> ax0.set_title("Amplitude-modulated Chirp Signal") + >>> ax0.set_ylabel("Amplitude") + >>> ax0.plot(t, signal, label='Signal') + >>> ax0.plot(t, amplitude_envelope, label='Envelope') + >>> ax0.legend() + >>> ax1.set(xlabel="Time in seconds", ylabel="Phase in rad", ylim=(0, 120)) + >>> ax1.plot(t[1:], instantaneous_frequency, 'C2-', label='Instantaneous Phase') + >>> ax1.legend() + >>> plt.show() + + """ + x = np.asarray(x) + if np.iscomplexobj(x): + raise ValueError("x must be real.") + if N is None: + N = x.shape[axis] + if N <= 0: + raise ValueError("N must be positive.") + + Xf = sp_fft.fft(x, N, axis=axis) + h = np.zeros(N, dtype=Xf.dtype) + if N % 2 == 0: + h[0] = h[N // 2] = 1 + h[1:N // 2] = 2 + else: + h[0] = 1 + h[1:(N + 1) // 2] = 2 + + if x.ndim > 1: + ind = [np.newaxis] * x.ndim + ind[axis] = slice(None) + h = h[tuple(ind)] + x = sp_fft.ifft(Xf * h, axis=axis) + return x + + +def hilbert2(x, N=None): + """ + Compute the '2-D' analytic signal of `x` + + Parameters + ---------- + x : array_like + 2-D signal data. + N : int or tuple of two ints, optional + Number of Fourier components. Default is ``x.shape`` + + Returns + ------- + xa : ndarray + Analytic signal of `x` taken along axes (0,1). + + References + ---------- + .. [1] Wikipedia, "Analytic signal", + https://en.wikipedia.org/wiki/Analytic_signal + + """ + x = np.atleast_2d(x) + if x.ndim > 2: + raise ValueError("x must be 2-D.") + if np.iscomplexobj(x): + raise ValueError("x must be real.") + if N is None: + N = x.shape + elif isinstance(N, int): + if N <= 0: + raise ValueError("N must be positive.") + N = (N, N) + elif len(N) != 2 or np.any(np.asarray(N) <= 0): + raise ValueError("When given as a tuple, N must hold exactly " + "two positive integers") + + Xf = sp_fft.fft2(x, N, axes=(0, 1)) + h1 = np.zeros(N[0], dtype=Xf.dtype) + h2 = np.zeros(N[1], dtype=Xf.dtype) + for h in (h1, h2): + N1 = h.shape[0] + if N1 % 2 == 0: + h[0] = h[N1 // 2] = 1 + h[1:N1 // 2] = 2 + else: + h[0] = 1 + h[1:(N1 + 1) // 2] = 2 + + h = h1[:, np.newaxis] * h2[np.newaxis, :] + k = x.ndim + while k > 2: + h = h[:, np.newaxis] + k -= 1 + x = sp_fft.ifft2(Xf * h, axes=(0, 1)) + return x + + +def envelope(z: np.ndarray, bp_in: tuple[int | None, int | None] = (1, None), *, + n_out: int | None = None, squared: bool = False, + residual: Literal['lowpass', 'all', None] = 'lowpass', + axis: int = -1) -> np.ndarray: + r"""Compute the envelope of a real- or complex-valued signal. + + Parameters + ---------- + z : ndarray + Real- or complex-valued input signal, which is assumed to be made up of ``n`` + samples and having sampling interval ``T``. `z` may also be a multidimensional + array with the time axis being defined by `axis`. + bp_in : tuple[int | None, int | None], optional + 2-tuple defining the frequency band ``bp_in[0]:bp_in[1]`` of the input filter. + The corner frequencies are specified as integer multiples of ``1/(n*T)`` with + ``-n//2 <= bp_in[0] < bp_in[1] <= (n+1)//2`` being the allowed frequency range. + ``None`` entries are replaced with ``-n//2`` or ``(n+1)//2`` respectively. The + default of ``(1, None)`` removes the mean value as well as the negative + frequency components. + n_out : int | None, optional + If not ``None`` the output will be resampled to `n_out` samples. The default + of ``None`` sets the output to the same length as the input `z`. + squared : bool, optional + If set, the square of the envelope is returned. The bandwidth of the squared + envelope is often smaller than the non-squared envelope bandwidth due to the + nonlinear nature of the utilized absolute value function. I.e., the embedded + square root function typically produces addiational harmonics. + The default is ``False``. + residual : Literal['lowpass', 'all', None], optional + This option determines what kind of residual, i.e., the signal part which the + input bandpass filter removes, is returned. ``'all'`` returns everything except + the contents of the frequency band ``bp_in[0]:bp_in[1]``, ``'lowpass'`` + returns the contents of the frequency band ``< bp_in[0]``. If ``None`` then + only the envelope is returned. Default: ``'lowpass'``. + axis : int, optional + Axis of `z` over which to compute the envelope. Default is last the axis. + + Returns + ------- + ndarray + If parameter `residual` is ``None`` then an array ``z_env`` with the same shape + as the input `z` is returned, containing its envelope. Otherwise, an array with + shape ``(2, *z.shape)``, containing the arrays ``z_env`` and ``z_res``, stacked + along the first axis, is returned. + It allows unpacking, i.e., ``z_env, z_res = envelope(z, residual='all')``. + The residual ``z_res`` contains the signal part which the input bandpass filter + removed, depending on the parameter `residual`. Note that for real-valued + signals, a real-valued residual is returned. Hence, the negative frequency + components of `bp_in` are ignored. + + Notes + ----- + Any complex-valued signal :math:`z(t)` can be described by a real-valued + instantaneous amplitude :math:`a(t)` and a real-valued instantaneous phase + :math:`\phi(t)`, i.e., :math:`z(t) = a(t) \exp\!\big(j \phi(t)\big)`. The + envelope is defined as the absolute value of the amplitude :math:`|a(t)| = |z(t)|`, + which is at the same time the absolute value of the signal. Hence, :math:`|a(t)|` + "envelopes" the class of all signals with amplitude :math:`a(t)` and arbitrary + phase :math:`\phi(t)`. + For real-valued signals, :math:`x(t) = a(t) \cos\!\big(\phi(t)\big)` is the + analogous formulation. Hence, :math:`|a(t)|` can be determined by converting + :math:`x(t)` into an analytic signal :math:`z_a(t)` by means of a Hilbert + transform, i.e., + :math:`z_a(t) = a(t) \cos\!\big(\phi(t)\big) + j a(t) \sin\!\big(\phi(t) \big)`, + which produces a complex-valued signal with the same envelope :math:`|a(t)|`. + + The implementation is based on computing the FFT of the input signal and then + performing the necessary operations in Fourier space. Hence, the typical FFT + caveats need to be taken into account: + + * The signal is assumed to be periodic. Discontinuities between signal start and + end can lead to unwanted results due to Gibbs phenomenon. + * The FFT is slow if the signal length is prime or very long. Also, the memory + demands are typically higher than a comparable FIR/IIR filter based + implementation. + * The frequency spacing ``1 / (n*T)`` for corner frequencies of the bandpass filter + corresponds to the frequencies produced by ``scipy.fft.fftfreq(len(z), T)``. + + If the envelope of a complex-valued signal `z` with no bandpass filtering is + desired, i.e., ``bp_in=(None, None)``, then the envelope corresponds to the + absolute value. Hence, it is more efficient to use ``np.abs(z)`` instead of this + function. + + Although computing the envelope based on the analytic signal [1]_ is the natural + method for real-valued signals, other methods are also frequently used. The most + popular alternative is probably the so-called "square-law" envelope detector and + its relatives [2]_. They do not always compute the correct result for all kinds of + signals, but are usually correct and typically computationally more efficient for + most kinds of narrowband signals. The definition for an envelope presented here is + common where instantaneous amplitude and phase are of interest (e.g., as described + in [3]_). There exist also other concepts, which rely on the general mathematical + idea of an envelope [4]_: A pragmatic approach is to determine all upper and lower + signal peaks and use a spline interpolation to determine the curves [5]_. + + + References + ---------- + .. [1] "Analytic Signal", Wikipedia, + https://en.wikipedia.org/wiki/Analytic_signal + .. [2] Lyons, Richard, "Digital envelope detection: The good, the bad, and the + ugly", IEEE Signal Processing Magazine 34.4 (2017): 183-187. + `PDF `__ + .. [3] T.G. Kincaid, "The complex representation of signals.", + TIS R67# MH5, General Electric Co. (1966). + `PDF `__ + .. [4] "Envelope (mathematics)", Wikipedia, + https://en.wikipedia.org/wiki/Envelope_(mathematics) + .. [5] Yang, Yanli. "A signal theoretic approach for envelope analysis of + real-valued signals." IEEE Access 5 (2017): 5623-5630. + `PDF `__ + + + See Also + -------- + hilbert: Compute analytic signal by means of Hilbert transform. + + + Examples + -------- + The following plot illustrates the envelope of a signal with variable frequency and + a low-frequency drift. To separate the drift from the envelope, a 4 Hz highpass + filter is used. The low-pass residuum of the input bandpass filter is utilized to + determine an asymmetric upper and lower bound to enclose the signal. Due to the + smoothness of the resulting envelope, it is down-sampled from 500 to 40 samples. + Note that the instantaneous amplitude ``a_x`` and the computed envelope ``x_env`` + are not perfectly identical. This is due to the signal not being perfectly periodic + as well as the existence of some spectral overlapping of ``x_carrier`` and + ``x_drift``. Hence, they cannot be completely separated by a bandpass filter. + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> from scipy.signal.windows import gaussian + >>> from scipy.signal import envelope + ... + >>> n, n_out = 500, 40 # number of signal samples and envelope samples + >>> T = 2 / n # sampling interval for 2 s duration + >>> t = np.arange(n) * T # time stamps + >>> a_x = gaussian(len(t), 0.4/T) # instantaneous amplitude + >>> phi_x = 30*np.pi*t + 35*np.cos(2*np.pi*0.25*t) # instantaneous phase + >>> x_carrier = a_x * np.cos(phi_x) + >>> x_drift = 0.3 * gaussian(len(t), 0.4/T) # drift + >>> x = x_carrier + x_drift + ... + >>> bp_in = (int(4 * (n*T)), None) # 4 Hz highpass input filter + >>> x_env, x_res = envelope(x, bp_in, n_out=n_out) + >>> t_out = np.arange(n_out) * (n / n_out) * T + ... + >>> fg0, ax0 = plt.subplots(1, 1, tight_layout=True) + >>> ax0.set_title(r"$4\,$Hz Highpass Envelope of Drifting Signal") + >>> ax0.set(xlabel="Time in seconds", xlim=(0, n*T), ylabel="Amplitude") + >>> ax0.plot(t, x, 'C0-', alpha=0.5, label="Signal") + >>> ax0.plot(t, x_drift, 'C2--', alpha=0.25, label="Drift") + >>> ax0.plot(t_out, x_res+x_env, 'C1.-', alpha=0.5, label="Envelope") + >>> ax0.plot(t_out, x_res-x_env, 'C1.-', alpha=0.5, label=None) + >>> ax0.grid(True) + >>> ax0.legend() + >>> plt.show() + + The second example provides a geometric envelope interpretation of complex-valued + signals: The following two plots show the complex-valued signal as a blue + 3d-trajectory and the envelope as an orange round tube with varying diameter, i.e., + as :math:`|a(t)| \exp(j\rho(t))`, with :math:`\rho(t)\in[-\pi,\pi]`. Also, the + projection into the 2d real and imaginary coordinate planes of trajectory and tube + is depicted. Every point of the complex-valued signal touches the tube's surface. + + The left plot shows an analytic signal, i.e, the phase difference between + imaginary and real part is always 90 degrees, resulting in a spiraling trajectory. + It can be seen that in this case the real part has also the expected envelope, + i.e., representing the absolute value of the instantaneous amplitude. + + The right plot shows the real part of that analytic signal being interpreted + as a complex-vauled signal, i.e., having zero imaginary part. There the resulting + envelope is not as smooth as in the analytic case and the instantaneous amplitude + in the real plane is not recovered. If ``z_re`` had been passed as a real-valued + signal, i.e., as ``z_re = z.real`` instead of ``z_re = z.real + 0j``, the result + would have been identical to the left plot. The reason for this is that real-valued + signals are interpreted as being the real part of a complex-valued analytic signal. + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> from scipy.signal.windows import gaussian + >>> from scipy.signal import envelope + ... + >>> n, T = 1000, 1/1000 # number of samples and sampling interval + >>> t = np.arange(n) * T # time stamps for 1 s duration + >>> f_c = 3 # Carrier frequency for signal + >>> z = gaussian(len(t), 0.3/T) * np.exp(2j*np.pi*f_c*t) # analytic signal + >>> z_re = z.real + 0j # complex signal with zero imaginary part + ... + >>> e_a, e_r = (envelope(z_, (None, None), residual=None) for z_ in (z, z_re)) + ... + >>> # Generate grids to visualize envelopes as 2d and 3d surfaces: + >>> E2d_t, E2_amp = np.meshgrid(t, [-1, 1]) + >>> E2d_1 = np.ones_like(E2_amp) + >>> E3d_t, E3d_phi = np.meshgrid(t, np.linspace(-np.pi, np.pi, 300)) + >>> ma = 1.8 # maximum axis values in real and imaginary direction + ... + >>> fg0 = plt.figure(figsize=(6.2, 4.)) + >>> ax00 = fg0.add_subplot(1, 2, 1, projection='3d') + >>> ax01 = fg0.add_subplot(1, 2, 2, projection='3d', sharex=ax00, + ... sharey=ax00, sharez=ax00) + >>> ax00.set_title("Analytic Signal") + >>> ax00.set(xlim=(0, 1), ylim=(-ma, ma), zlim=(-ma, ma)) + >>> ax01.set_title("Real-valued Signal") + >>> for z_, e_, ax_ in zip((z, z.real), (e_a, e_r), (ax00, ax01)): + ... ax_.set(xlabel="Time $t$", ylabel="Real Amp. $x(t)$", + ... zlabel="Imag. Amp. $y(t)$") + ... ax_.plot(t, z_.real, 'C0-', zs=-ma, zdir='z', alpha=0.5, label="Real") + ... ax_.plot_surface(E2d_t, e_*E2_amp, -ma*E2d_1, color='C1', alpha=0.25) + ... ax_.plot(t, z_.imag, 'C0-', zs=+ma, zdir='y', alpha=0.5, label="Imag.") + ... ax_.plot_surface(E2d_t, ma*E2d_1, e_*E2_amp, color='C1', alpha=0.25) + ... ax_.plot(t, z_.real, z_.imag, 'C0-', label="Signal") + ... ax_.plot_surface(E3d_t, e_*np.cos(E3d_phi), e_*np.sin(E3d_phi), + ... color='C1', alpha=0.5, shade=True, label="Envelope") + ... ax_.view_init(elev=22.7, azim=-114.3) + >>> fg0.subplots_adjust(left=0.08, right=0.97, wspace=0.15) + >>> plt.show() + """ + if not (-z.ndim <= axis < z.ndim): + raise ValueError(f"Invalid parameter {axis=} for {z.shape=}!") + if not (z.shape[axis] > 0): + raise ValueError(f"z.shape[axis] not > 0 for {z.shape=}, {axis=}!") + if len(bp_in) != 2 or not all((isinstance(b_, int) or b_ is None) for b_ in bp_in): + raise ValueError(f"{bp_in=} isn't a 2-tuple of type (int | None, int | None)!") + if not ((isinstance(n_out, int) and 0 < n_out) or n_out is None): + raise ValueError(f"{n_out=} is not a positive integer or None!") + if residual not in ('lowpass', 'all', None): + raise ValueError(f"{residual=} not in ['lowpass', 'all', None]!") + + n = z.shape[axis] # number of time samples of input + n_out = n if n_out is None else n_out + fak = n_out / n # scaling factor for resampling + + bp = slice(bp_in[0] if bp_in[0] is not None else -(n//2), + bp_in[1] if bp_in[1] is not None else (n+1)//2) + if not (-n//2 <= bp.start < bp.stop <= (n+1)//2): + raise ValueError("`-n//2 <= bp_in[0] < bp_in[1] <= (n+1)//2` does not hold " + + f"for n={z.shape[axis]=} and {bp_in=}!") + + # moving active axis to end allows to use `...` for indexing: + z = np.moveaxis(z, axis, -1) + + if np.iscomplexobj(z): + Z = sp_fft.fft(z) + else: # avoid calculating negative frequency bins for real signals: + Z = np.zeros_like(z, dtype=sp_fft.rfft(z.flat[:1]).dtype) + Z[..., :n//2 + 1] = sp_fft.rfft(z) + if bp.start > 0: # make signal analytic within bp_in band: + Z[..., bp] *= 2 + elif bp.stop > 0: + Z[..., 1:bp.stop] *= 2 + if not (bp.start <= 0 < bp.stop): # envelope is invariant to freq. shifts. + z_bb = sp_fft.ifft(Z[..., bp], n=n_out) * fak # baseband signal + else: + bp_shift = slice(bp.start + n//2, bp.stop + n//2) + z_bb = sp_fft.ifft(sp_fft.fftshift(Z, axes=-1)[..., bp_shift], n=n_out) * fak + + z_env = np.abs(z_bb) if not squared else z_bb.real ** 2 + z_bb.imag ** 2 + z_env = np.moveaxis(z_env, -1, axis) + + # Calculate the residual from the input bandpass filter: + if residual is None: + return z_env + if not (bp.start <= 0 < bp.stop): + Z[..., bp] = 0 + else: + Z[..., :bp.stop], Z[..., bp.start:] = 0, 0 + if residual == 'lowpass': + if bp.stop > 0: + Z[..., bp.stop:(n+1) // 2] = 0 + else: + Z[..., bp.start:], Z[..., 0:(n + 1) // 2] = 0, 0 + + z_res = fak * (sp_fft.ifft(Z, n=n_out) if np.iscomplexobj(z) else + sp_fft.irfft(Z, n=n_out)) + return np.stack((z_env, np.moveaxis(z_res, -1, axis)), axis=0) + +def _cmplx_sort(p): + """Sort roots based on magnitude. + + Parameters + ---------- + p : array_like + The roots to sort, as a 1-D array. + + Returns + ------- + p_sorted : ndarray + Sorted roots. + indx : ndarray + Array of indices needed to sort the input `p`. + + Examples + -------- + >>> from scipy import signal + >>> vals = [1, 4, 1+1.j, 3] + >>> p_sorted, indx = signal.cmplx_sort(vals) + >>> p_sorted + array([1.+0.j, 1.+1.j, 3.+0.j, 4.+0.j]) + >>> indx + array([0, 2, 3, 1]) + """ + p = np.asarray(p) + indx = np.argsort(abs(p)) + return np.take(p, indx, 0), indx + + +def unique_roots(p, tol=1e-3, rtype='min'): + """Determine unique roots and their multiplicities from a list of roots. + + Parameters + ---------- + p : array_like + The list of roots. + tol : float, optional + The tolerance for two roots to be considered equal in terms of + the distance between them. Default is 1e-3. Refer to Notes about + the details on roots grouping. + rtype : {'max', 'maximum', 'min', 'minimum', 'avg', 'mean'}, optional + How to determine the returned root if multiple roots are within + `tol` of each other. + + - 'max', 'maximum': pick the maximum of those roots + - 'min', 'minimum': pick the minimum of those roots + - 'avg', 'mean': take the average of those roots + + When finding minimum or maximum among complex roots they are compared + first by the real part and then by the imaginary part. + + Returns + ------- + unique : ndarray + The list of unique roots. + multiplicity : ndarray + The multiplicity of each root. + + Notes + ----- + If we have 3 roots ``a``, ``b`` and ``c``, such that ``a`` is close to + ``b`` and ``b`` is close to ``c`` (distance is less than `tol`), then it + doesn't necessarily mean that ``a`` is close to ``c``. It means that roots + grouping is not unique. In this function we use "greedy" grouping going + through the roots in the order they are given in the input `p`. + + This utility function is not specific to roots but can be used for any + sequence of values for which uniqueness and multiplicity has to be + determined. For a more general routine, see `numpy.unique`. + + Examples + -------- + >>> from scipy import signal + >>> vals = [0, 1.3, 1.31, 2.8, 1.25, 2.2, 10.3] + >>> uniq, mult = signal.unique_roots(vals, tol=2e-2, rtype='avg') + + Check which roots have multiplicity larger than 1: + + >>> uniq[mult > 1] + array([ 1.305]) + """ + if rtype in ['max', 'maximum']: + reduce = np.max + elif rtype in ['min', 'minimum']: + reduce = np.min + elif rtype in ['avg', 'mean']: + reduce = np.mean + else: + raise ValueError("`rtype` must be one of " + "{'max', 'maximum', 'min', 'minimum', 'avg', 'mean'}") + + p = np.asarray(p) + + points = np.empty((len(p), 2)) + points[:, 0] = np.real(p) + points[:, 1] = np.imag(p) + tree = cKDTree(points) + + p_unique = [] + p_multiplicity = [] + used = np.zeros(len(p), dtype=bool) + for i in range(len(p)): + if used[i]: + continue + + group = tree.query_ball_point(points[i], tol) + group = [x for x in group if not used[x]] + + p_unique.append(reduce(p[group])) + p_multiplicity.append(len(group)) + + used[group] = True + + return np.asarray(p_unique), np.asarray(p_multiplicity) + + +def invres(r, p, k, tol=1e-3, rtype='avg'): + """Compute b(s) and a(s) from partial fraction expansion. + + If `M` is the degree of numerator `b` and `N` the degree of denominator + `a`:: + + b(s) b[0] s**(M) + b[1] s**(M-1) + ... + b[M] + H(s) = ------ = ------------------------------------------ + a(s) a[0] s**(N) + a[1] s**(N-1) + ... + a[N] + + then the partial-fraction expansion H(s) is defined as:: + + r[0] r[1] r[-1] + = -------- + -------- + ... + --------- + k(s) + (s-p[0]) (s-p[1]) (s-p[-1]) + + If there are any repeated roots (closer together than `tol`), then H(s) + has terms like:: + + r[i] r[i+1] r[i+n-1] + -------- + ----------- + ... + ----------- + (s-p[i]) (s-p[i])**2 (s-p[i])**n + + This function is used for polynomials in positive powers of s or z, + such as analog filters or digital filters in controls engineering. For + negative powers of z (typical for digital filters in DSP), use `invresz`. + + Parameters + ---------- + r : array_like + Residues corresponding to the poles. For repeated poles, the residues + must be ordered to correspond to ascending by power fractions. + p : array_like + Poles. Equal poles must be adjacent. + k : array_like + Coefficients of the direct polynomial term. + tol : float, optional + The tolerance for two roots to be considered equal in terms of + the distance between them. Default is 1e-3. See `unique_roots` + for further details. + rtype : {'avg', 'min', 'max'}, optional + Method for computing a root to represent a group of identical roots. + Default is 'avg'. See `unique_roots` for further details. + + Returns + ------- + b : ndarray + Numerator polynomial coefficients. + a : ndarray + Denominator polynomial coefficients. + + See Also + -------- + residue, invresz, unique_roots + + """ + r = np.atleast_1d(r) + p = np.atleast_1d(p) + k = np.trim_zeros(np.atleast_1d(k), 'f') + + unique_poles, multiplicity = _group_poles(p, tol, rtype) + factors, denominator = _compute_factors(unique_poles, multiplicity, + include_powers=True) + + if len(k) == 0: + numerator = 0 + else: + numerator = np.polymul(k, denominator) + + for residue, factor in zip(r, factors): + numerator = np.polyadd(numerator, residue * factor) + + return numerator, denominator + + +def _compute_factors(roots, multiplicity, include_powers=False): + """Compute the total polynomial divided by factors for each root.""" + current = np.array([1]) + suffixes = [current] + for pole, mult in zip(roots[-1:0:-1], multiplicity[-1:0:-1]): + monomial = np.array([1, -pole]) + for _ in range(mult): + current = np.polymul(current, monomial) + suffixes.append(current) + suffixes = suffixes[::-1] + + factors = [] + current = np.array([1]) + for pole, mult, suffix in zip(roots, multiplicity, suffixes): + monomial = np.array([1, -pole]) + block = [] + for i in range(mult): + if i == 0 or include_powers: + block.append(np.polymul(current, suffix)) + current = np.polymul(current, monomial) + factors.extend(reversed(block)) + + return factors, current + + +def _compute_residues(poles, multiplicity, numerator): + denominator_factors, _ = _compute_factors(poles, multiplicity) + numerator = numerator.astype(poles.dtype) + + residues = [] + for pole, mult, factor in zip(poles, multiplicity, + denominator_factors): + if mult == 1: + residues.append(np.polyval(numerator, pole) / + np.polyval(factor, pole)) + else: + numer = numerator.copy() + monomial = np.array([1, -pole]) + factor, d = np.polydiv(factor, monomial) + + block = [] + for _ in range(mult): + numer, n = np.polydiv(numer, monomial) + r = n[0] / d[0] + numer = np.polysub(numer, r * factor) + block.append(r) + + residues.extend(reversed(block)) + + return np.asarray(residues) + + +def residue(b, a, tol=1e-3, rtype='avg'): + """Compute partial-fraction expansion of b(s) / a(s). + + If `M` is the degree of numerator `b` and `N` the degree of denominator + `a`:: + + b(s) b[0] s**(M) + b[1] s**(M-1) + ... + b[M] + H(s) = ------ = ------------------------------------------ + a(s) a[0] s**(N) + a[1] s**(N-1) + ... + a[N] + + then the partial-fraction expansion H(s) is defined as:: + + r[0] r[1] r[-1] + = -------- + -------- + ... + --------- + k(s) + (s-p[0]) (s-p[1]) (s-p[-1]) + + If there are any repeated roots (closer together than `tol`), then H(s) + has terms like:: + + r[i] r[i+1] r[i+n-1] + -------- + ----------- + ... + ----------- + (s-p[i]) (s-p[i])**2 (s-p[i])**n + + This function is used for polynomials in positive powers of s or z, + such as analog filters or digital filters in controls engineering. For + negative powers of z (typical for digital filters in DSP), use `residuez`. + + See Notes for details about the algorithm. + + Parameters + ---------- + b : array_like + Numerator polynomial coefficients. + a : array_like + Denominator polynomial coefficients. + tol : float, optional + The tolerance for two roots to be considered equal in terms of + the distance between them. Default is 1e-3. See `unique_roots` + for further details. + rtype : {'avg', 'min', 'max'}, optional + Method for computing a root to represent a group of identical roots. + Default is 'avg'. See `unique_roots` for further details. + + Returns + ------- + r : ndarray + Residues corresponding to the poles. For repeated poles, the residues + are ordered to correspond to ascending by power fractions. + p : ndarray + Poles ordered by magnitude in ascending order. + k : ndarray + Coefficients of the direct polynomial term. + + See Also + -------- + invres, residuez, numpy.poly, unique_roots + + Notes + ----- + The "deflation through subtraction" algorithm is used for + computations --- method 6 in [1]_. + + The form of partial fraction expansion depends on poles multiplicity in + the exact mathematical sense. However there is no way to exactly + determine multiplicity of roots of a polynomial in numerical computing. + Thus you should think of the result of `residue` with given `tol` as + partial fraction expansion computed for the denominator composed of the + computed poles with empirically determined multiplicity. The choice of + `tol` can drastically change the result if there are close poles. + + References + ---------- + .. [1] J. F. Mahoney, B. D. Sivazlian, "Partial fractions expansion: a + review of computational methodology and efficiency", Journal of + Computational and Applied Mathematics, Vol. 9, 1983. + """ + b = np.asarray(b) + a = np.asarray(a) + if (np.issubdtype(b.dtype, np.complexfloating) + or np.issubdtype(a.dtype, np.complexfloating)): + b = b.astype(complex) + a = a.astype(complex) + else: + b = b.astype(float) + a = a.astype(float) + + b = np.trim_zeros(np.atleast_1d(b), 'f') + a = np.trim_zeros(np.atleast_1d(a), 'f') + + if a.size == 0: + raise ValueError("Denominator `a` is zero.") + + poles = np.roots(a) + if b.size == 0: + return np.zeros(poles.shape), _cmplx_sort(poles)[0], np.array([]) + + if len(b) < len(a): + k = np.empty(0) + else: + k, b = np.polydiv(b, a) + + unique_poles, multiplicity = unique_roots(poles, tol=tol, rtype=rtype) + unique_poles, order = _cmplx_sort(unique_poles) + multiplicity = multiplicity[order] + + residues = _compute_residues(unique_poles, multiplicity, b) + + index = 0 + for pole, mult in zip(unique_poles, multiplicity): + poles[index:index + mult] = pole + index += mult + + return residues / a[0], poles, k + + +def residuez(b, a, tol=1e-3, rtype='avg'): + """Compute partial-fraction expansion of b(z) / a(z). + + If `M` is the degree of numerator `b` and `N` the degree of denominator + `a`:: + + b(z) b[0] + b[1] z**(-1) + ... + b[M] z**(-M) + H(z) = ------ = ------------------------------------------ + a(z) a[0] + a[1] z**(-1) + ... + a[N] z**(-N) + + then the partial-fraction expansion H(z) is defined as:: + + r[0] r[-1] + = --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ... + (1-p[0]z**(-1)) (1-p[-1]z**(-1)) + + If there are any repeated roots (closer than `tol`), then the partial + fraction expansion has terms like:: + + r[i] r[i+1] r[i+n-1] + -------------- + ------------------ + ... + ------------------ + (1-p[i]z**(-1)) (1-p[i]z**(-1))**2 (1-p[i]z**(-1))**n + + This function is used for polynomials in negative powers of z, + such as digital filters in DSP. For positive powers, use `residue`. + + See Notes of `residue` for details about the algorithm. + + Parameters + ---------- + b : array_like + Numerator polynomial coefficients. + a : array_like + Denominator polynomial coefficients. + tol : float, optional + The tolerance for two roots to be considered equal in terms of + the distance between them. Default is 1e-3. See `unique_roots` + for further details. + rtype : {'avg', 'min', 'max'}, optional + Method for computing a root to represent a group of identical roots. + Default is 'avg'. See `unique_roots` for further details. + + Returns + ------- + r : ndarray + Residues corresponding to the poles. For repeated poles, the residues + are ordered to correspond to ascending by power fractions. + p : ndarray + Poles ordered by magnitude in ascending order. + k : ndarray + Coefficients of the direct polynomial term. + + See Also + -------- + invresz, residue, unique_roots + """ + b = np.asarray(b) + a = np.asarray(a) + if (np.issubdtype(b.dtype, np.complexfloating) + or np.issubdtype(a.dtype, np.complexfloating)): + b = b.astype(complex) + a = a.astype(complex) + else: + b = b.astype(float) + a = a.astype(float) + + b = np.trim_zeros(np.atleast_1d(b), 'b') + a = np.trim_zeros(np.atleast_1d(a), 'b') + + if a.size == 0: + raise ValueError("Denominator `a` is zero.") + elif a[0] == 0: + raise ValueError("First coefficient of determinant `a` must be " + "non-zero.") + + poles = np.roots(a) + if b.size == 0: + return np.zeros(poles.shape), _cmplx_sort(poles)[0], np.array([]) + + b_rev = b[::-1] + a_rev = a[::-1] + + if len(b_rev) < len(a_rev): + k_rev = np.empty(0) + else: + k_rev, b_rev = np.polydiv(b_rev, a_rev) + + unique_poles, multiplicity = unique_roots(poles, tol=tol, rtype=rtype) + unique_poles, order = _cmplx_sort(unique_poles) + multiplicity = multiplicity[order] + + residues = _compute_residues(1 / unique_poles, multiplicity, b_rev) + + index = 0 + powers = np.empty(len(residues), dtype=int) + for pole, mult in zip(unique_poles, multiplicity): + poles[index:index + mult] = pole + powers[index:index + mult] = 1 + np.arange(mult) + index += mult + + residues *= (-poles) ** powers / a_rev[0] + + return residues, poles, k_rev[::-1] + + +def _group_poles(poles, tol, rtype): + if rtype in ['max', 'maximum']: + reduce = np.max + elif rtype in ['min', 'minimum']: + reduce = np.min + elif rtype in ['avg', 'mean']: + reduce = np.mean + else: + raise ValueError("`rtype` must be one of " + "{'max', 'maximum', 'min', 'minimum', 'avg', 'mean'}") + + unique = [] + multiplicity = [] + + pole = poles[0] + block = [pole] + for i in range(1, len(poles)): + if abs(poles[i] - pole) <= tol: + block.append(pole) + else: + unique.append(reduce(block)) + multiplicity.append(len(block)) + pole = poles[i] + block = [pole] + + unique.append(reduce(block)) + multiplicity.append(len(block)) + + return np.asarray(unique), np.asarray(multiplicity) + + +def invresz(r, p, k, tol=1e-3, rtype='avg'): + """Compute b(z) and a(z) from partial fraction expansion. + + If `M` is the degree of numerator `b` and `N` the degree of denominator + `a`:: + + b(z) b[0] + b[1] z**(-1) + ... + b[M] z**(-M) + H(z) = ------ = ------------------------------------------ + a(z) a[0] + a[1] z**(-1) + ... + a[N] z**(-N) + + then the partial-fraction expansion H(z) is defined as:: + + r[0] r[-1] + = --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ... + (1-p[0]z**(-1)) (1-p[-1]z**(-1)) + + If there are any repeated roots (closer than `tol`), then the partial + fraction expansion has terms like:: + + r[i] r[i+1] r[i+n-1] + -------------- + ------------------ + ... + ------------------ + (1-p[i]z**(-1)) (1-p[i]z**(-1))**2 (1-p[i]z**(-1))**n + + This function is used for polynomials in negative powers of z, + such as digital filters in DSP. For positive powers, use `invres`. + + Parameters + ---------- + r : array_like + Residues corresponding to the poles. For repeated poles, the residues + must be ordered to correspond to ascending by power fractions. + p : array_like + Poles. Equal poles must be adjacent. + k : array_like + Coefficients of the direct polynomial term. + tol : float, optional + The tolerance for two roots to be considered equal in terms of + the distance between them. Default is 1e-3. See `unique_roots` + for further details. + rtype : {'avg', 'min', 'max'}, optional + Method for computing a root to represent a group of identical roots. + Default is 'avg'. See `unique_roots` for further details. + + Returns + ------- + b : ndarray + Numerator polynomial coefficients. + a : ndarray + Denominator polynomial coefficients. + + See Also + -------- + residuez, unique_roots, invres + + """ + r = np.atleast_1d(r) + p = np.atleast_1d(p) + k = np.trim_zeros(np.atleast_1d(k), 'b') + + unique_poles, multiplicity = _group_poles(p, tol, rtype) + factors, denominator = _compute_factors(unique_poles, multiplicity, + include_powers=True) + + if len(k) == 0: + numerator = 0 + else: + numerator = np.polymul(k[::-1], denominator[::-1]) + + for residue, factor in zip(r, factors): + numerator = np.polyadd(numerator, residue * factor[::-1]) + + return numerator[::-1], denominator + + +def resample(x, num, t=None, axis=0, window=None, domain='time'): + """ + Resample `x` to `num` samples using Fourier method along the given axis. + + The resampled signal starts at the same value as `x` but is sampled + with a spacing of ``len(x) / num * (spacing of x)``. Because a + Fourier method is used, the signal is assumed to be periodic. + + Parameters + ---------- + x : array_like + The data to be resampled. + num : int + The number of samples in the resampled signal. + t : array_like, optional + If `t` is given, it is assumed to be the equally spaced sample + positions associated with the signal data in `x`. + axis : int, optional + The axis of `x` that is resampled. Default is 0. + window : array_like, callable, string, float, or tuple, optional + Specifies the window applied to the signal in the Fourier + domain. See below for details. + domain : string, optional + A string indicating the domain of the input `x`: + ``time`` Consider the input `x` as time-domain (Default), + ``freq`` Consider the input `x` as frequency-domain. + + Returns + ------- + resampled_x or (resampled_x, resampled_t) + Either the resampled array, or, if `t` was given, a tuple + containing the resampled array and the corresponding resampled + positions. + + See Also + -------- + decimate : Downsample the signal after applying an FIR or IIR filter. + resample_poly : Resample using polyphase filtering and an FIR filter. + + Notes + ----- + The argument `window` controls a Fourier-domain window that tapers + the Fourier spectrum before zero-padding to alleviate ringing in + the resampled values for sampled signals you didn't intend to be + interpreted as band-limited. + + If `window` is a function, then it is called with a vector of inputs + indicating the frequency bins (i.e. fftfreq(x.shape[axis]) ). + + If `window` is an array of the same length as `x.shape[axis]` it is + assumed to be the window to be applied directly in the Fourier + domain (with dc and low-frequency first). + + For any other type of `window`, the function `scipy.signal.get_window` + is called to generate the window. + + The first sample of the returned vector is the same as the first + sample of the input vector. The spacing between samples is changed + from ``dx`` to ``dx * len(x) / num``. + + If `t` is not None, then it is used solely to calculate the resampled + positions `resampled_t` + + As noted, `resample` uses FFT transformations, which can be very + slow if the number of input or output samples is large and prime; + see :func:`~scipy.fft.fft`. In such cases, it can be faster to first downsample + a signal of length ``n`` with :func:`~scipy.signal.resample_poly` by a factor of + ``n//num`` before using `resample`. Note that this approach changes the + characteristics of the antialiasing filter. + + Examples + -------- + Note that the end of the resampled data rises to meet the first + sample of the next cycle: + + >>> import numpy as np + >>> from scipy import signal + + >>> x = np.linspace(0, 10, 20, endpoint=False) + >>> y = np.cos(-x**2/6.0) + >>> f = signal.resample(y, 100) + >>> xnew = np.linspace(0, 10, 100, endpoint=False) + + >>> import matplotlib.pyplot as plt + >>> plt.plot(x, y, 'go-', xnew, f, '.-', 10, y[0], 'ro') + >>> plt.legend(['data', 'resampled'], loc='best') + >>> plt.show() + + Consider the following signal ``y`` where ``len(y)`` is a large prime number: + + >>> N = 55949 + >>> freq = 100 + >>> x = np.linspace(0, 1, N) + >>> y = np.cos(2 * np.pi * freq * x) + + Due to ``N`` being prime, + + >>> num = 5000 + >>> f = signal.resample(signal.resample_poly(y, 1, N // num), num) + + runs significantly faster than + + >>> f = signal.resample(y, num) + """ + + if domain not in ('time', 'freq'): + raise ValueError("Acceptable domain flags are 'time' or" + f" 'freq', not domain={domain}") + + x = np.asarray(x) + Nx = x.shape[axis] + + # Check if we can use faster real FFT + real_input = np.isrealobj(x) + + if domain == 'time': + # Forward transform + if real_input: + X = sp_fft.rfft(x, axis=axis) + else: # Full complex FFT + X = sp_fft.fft(x, axis=axis) + else: # domain == 'freq' + X = x + + # Apply window to spectrum + if window is not None: + if callable(window): + W = window(sp_fft.fftfreq(Nx)) + elif isinstance(window, np.ndarray): + if window.shape != (Nx,): + raise ValueError('window must have the same length as data') + W = window + else: + W = sp_fft.ifftshift(get_window(window, Nx)) + + newshape_W = [1] * x.ndim + newshape_W[axis] = X.shape[axis] + if real_input: + # Fold the window back on itself to mimic complex behavior + W_real = W.copy() + W_real[1:] += W_real[-1:0:-1] + W_real[1:] *= 0.5 + X *= W_real[:newshape_W[axis]].reshape(newshape_W) + else: + X *= W.reshape(newshape_W) + + # Copy each half of the original spectrum to the output spectrum, either + # truncating high frequencies (downsampling) or zero-padding them + # (upsampling) + + # Placeholder array for output spectrum + newshape = list(x.shape) + if real_input: + newshape[axis] = num // 2 + 1 + else: + newshape[axis] = num + Y = np.zeros(newshape, X.dtype) + + # Copy positive frequency components (and Nyquist, if present) + N = min(num, Nx) + nyq = N // 2 + 1 # Slice index that includes Nyquist if present + sl = [slice(None)] * x.ndim + sl[axis] = slice(0, nyq) + Y[tuple(sl)] = X[tuple(sl)] + if not real_input: + # Copy negative frequency components + if N > 2: # (slice expression doesn't collapse to empty array) + sl[axis] = slice(nyq - N, None) + Y[tuple(sl)] = X[tuple(sl)] + + # Split/join Nyquist component(s) if present + # So far we have set Y[+N/2]=X[+N/2] + if N % 2 == 0: + if num < Nx: # downsampling + if real_input: + sl[axis] = slice(N//2, N//2 + 1) + Y[tuple(sl)] *= 2. + else: + # select the component of Y at frequency +N/2, + # add the component of X at -N/2 + sl[axis] = slice(-N//2, -N//2 + 1) + Y[tuple(sl)] += X[tuple(sl)] + elif Nx < num: # upsampling + # select the component at frequency +N/2 and halve it + sl[axis] = slice(N//2, N//2 + 1) + Y[tuple(sl)] *= 0.5 + if not real_input: + temp = Y[tuple(sl)] + # set the component at -N/2 equal to the component at +N/2 + sl[axis] = slice(num-N//2, num-N//2 + 1) + Y[tuple(sl)] = temp + + # Inverse transform + if real_input: + y = sp_fft.irfft(Y, num, axis=axis) + else: + y = sp_fft.ifft(Y, axis=axis, overwrite_x=True) + + y *= (float(num) / float(Nx)) + + if t is None: + return y + else: + new_t = np.arange(0, num) * (t[1] - t[0]) * Nx / float(num) + t[0] + return y, new_t + + +def resample_poly(x, up, down, axis=0, window=('kaiser', 5.0), + padtype='constant', cval=None): + """ + Resample `x` along the given axis using polyphase filtering. + + The signal `x` is upsampled by the factor `up`, a zero-phase low-pass + FIR filter is applied, and then it is downsampled by the factor `down`. + The resulting sample rate is ``up / down`` times the original sample + rate. By default, values beyond the boundary of the signal are assumed + to be zero during the filtering step. + + Parameters + ---------- + x : array_like + The data to be resampled. + up : int + The upsampling factor. + down : int + The downsampling factor. + axis : int, optional + The axis of `x` that is resampled. Default is 0. + window : string, tuple, or array_like, optional + Desired window to use to design the low-pass filter, or the FIR filter + coefficients to employ. See below for details. + padtype : string, optional + `constant`, `line`, `mean`, `median`, `maximum`, `minimum` or any of + the other signal extension modes supported by `scipy.signal.upfirdn`. + Changes assumptions on values beyond the boundary. If `constant`, + assumed to be `cval` (default zero). If `line` assumed to continue a + linear trend defined by the first and last points. `mean`, `median`, + `maximum` and `minimum` work as in `np.pad` and assume that the values + beyond the boundary are the mean, median, maximum or minimum + respectively of the array along the axis. + + .. versionadded:: 1.4.0 + cval : float, optional + Value to use if `padtype='constant'`. Default is zero. + + .. versionadded:: 1.4.0 + + Returns + ------- + resampled_x : array + The resampled array. + + See Also + -------- + decimate : Downsample the signal after applying an FIR or IIR filter. + resample : Resample up or down using the FFT method. + + Notes + ----- + This polyphase method will likely be faster than the Fourier method + in `scipy.signal.resample` when the number of samples is large and + prime, or when the number of samples is large and `up` and `down` + share a large greatest common denominator. The length of the FIR + filter used will depend on ``max(up, down) // gcd(up, down)``, and + the number of operations during polyphase filtering will depend on + the filter length and `down` (see `scipy.signal.upfirdn` for details). + + The argument `window` specifies the FIR low-pass filter design. + + If `window` is an array_like it is assumed to be the FIR filter + coefficients. Note that the FIR filter is applied after the upsampling + step, so it should be designed to operate on a signal at a sampling + frequency higher than the original by a factor of `up//gcd(up, down)`. + This function's output will be centered with respect to this array, so it + is best to pass a symmetric filter with an odd number of samples if, as + is usually the case, a zero-phase filter is desired. + + For any other type of `window`, the functions `scipy.signal.get_window` + and `scipy.signal.firwin` are called to generate the appropriate filter + coefficients. + + The first sample of the returned vector is the same as the first + sample of the input vector. The spacing between samples is changed + from ``dx`` to ``dx * down / float(up)``. + + Examples + -------- + By default, the end of the resampled data rises to meet the first + sample of the next cycle for the FFT method, and gets closer to zero + for the polyphase method: + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> x = np.linspace(0, 10, 20, endpoint=False) + >>> y = np.cos(-x**2/6.0) + >>> f_fft = signal.resample(y, 100) + >>> f_poly = signal.resample_poly(y, 100, 20) + >>> xnew = np.linspace(0, 10, 100, endpoint=False) + + >>> plt.plot(xnew, f_fft, 'b.-', xnew, f_poly, 'r.-') + >>> plt.plot(x, y, 'ko-') + >>> plt.plot(10, y[0], 'bo', 10, 0., 'ro') # boundaries + >>> plt.legend(['resample', 'resamp_poly', 'data'], loc='best') + >>> plt.show() + + This default behaviour can be changed by using the padtype option: + + >>> N = 5 + >>> x = np.linspace(0, 1, N, endpoint=False) + >>> y = 2 + x**2 - 1.7*np.sin(x) + .2*np.cos(11*x) + >>> y2 = 1 + x**3 + 0.1*np.sin(x) + .1*np.cos(11*x) + >>> Y = np.stack([y, y2], axis=-1) + >>> up = 4 + >>> xr = np.linspace(0, 1, N*up, endpoint=False) + + >>> y2 = signal.resample_poly(Y, up, 1, padtype='constant') + >>> y3 = signal.resample_poly(Y, up, 1, padtype='mean') + >>> y4 = signal.resample_poly(Y, up, 1, padtype='line') + + >>> for i in [0,1]: + ... plt.figure() + ... plt.plot(xr, y4[:,i], 'g.', label='line') + ... plt.plot(xr, y3[:,i], 'y.', label='mean') + ... plt.plot(xr, y2[:,i], 'r.', label='constant') + ... plt.plot(x, Y[:,i], 'k-') + ... plt.legend() + >>> plt.show() + + """ + x = np.asarray(x) + if up != int(up): + raise ValueError("up must be an integer") + if down != int(down): + raise ValueError("down must be an integer") + up = int(up) + down = int(down) + if up < 1 or down < 1: + raise ValueError('up and down must be >= 1') + if cval is not None and padtype != 'constant': + raise ValueError('cval has no effect when padtype is ', padtype) + + # Determine our up and down factors + # Use a rational approximation to save computation time on really long + # signals + g_ = math.gcd(up, down) + up //= g_ + down //= g_ + if up == down == 1: + return x.copy() + n_in = x.shape[axis] + n_out = n_in * up + n_out = n_out // down + bool(n_out % down) + + if isinstance(window, (list | np.ndarray)): + window = np.array(window) # use array to force a copy (we modify it) + if window.ndim > 1: + raise ValueError('window must be 1-D') + half_len = (window.size - 1) // 2 + h = window + else: + # Design a linear-phase low-pass FIR filter + max_rate = max(up, down) + f_c = 1. / max_rate # cutoff of FIR filter (rel. to Nyquist) + half_len = 10 * max_rate # reasonable cutoff for sinc-like function + if np.issubdtype(x.dtype, np.complexfloating): + h = firwin(2 * half_len + 1, f_c, + window=window).astype(x.dtype) # match dtype of x + elif np.issubdtype(x.dtype, np.floating): + h = firwin(2 * half_len + 1, f_c, + window=window).astype(x.dtype) # match dtype of x + else: + h = firwin(2 * half_len + 1, f_c, + window=window) + h *= up + + # Zero-pad our filter to put the output samples at the center + n_pre_pad = (down - half_len % down) + n_post_pad = 0 + n_pre_remove = (half_len + n_pre_pad) // down + # We should rarely need to do this given our filter lengths... + while _output_len(len(h) + n_pre_pad + n_post_pad, n_in, + up, down) < n_out + n_pre_remove: + n_post_pad += 1 + h = np.concatenate((np.zeros(n_pre_pad, dtype=h.dtype), h, + np.zeros(n_post_pad, dtype=h.dtype))) + n_pre_remove_end = n_pre_remove + n_out + + # Remove background depending on the padtype option + funcs = {'mean': np.mean, 'median': np.median, + 'minimum': np.amin, 'maximum': np.amax} + upfirdn_kwargs = {'mode': 'constant', 'cval': 0} + if padtype in funcs: + background_values = funcs[padtype](x, axis=axis, keepdims=True) + elif padtype in _upfirdn_modes: + upfirdn_kwargs = {'mode': padtype} + if padtype == 'constant': + if cval is None: + cval = 0 + upfirdn_kwargs['cval'] = cval + else: + raise ValueError( + 'padtype must be one of: maximum, mean, median, minimum, ' + + ', '.join(_upfirdn_modes)) + + if padtype in funcs: + x = x - background_values + + # filter then remove excess + y = upfirdn(h, x, up, down, axis=axis, **upfirdn_kwargs) + keep = [slice(None), ]*x.ndim + keep[axis] = slice(n_pre_remove, n_pre_remove_end) + y_keep = y[tuple(keep)] + + # Add background back + if padtype in funcs: + y_keep += background_values + + return y_keep + + +def vectorstrength(events, period): + ''' + Determine the vector strength of the events corresponding to the given + period. + + The vector strength is a measure of phase synchrony, how well the + timing of the events is synchronized to a single period of a periodic + signal. + + If multiple periods are used, calculate the vector strength of each. + This is called the "resonating vector strength". + + Parameters + ---------- + events : 1D array_like + An array of time points containing the timing of the events. + period : float or array_like + The period of the signal that the events should synchronize to. + The period is in the same units as `events`. It can also be an array + of periods, in which case the outputs are arrays of the same length. + + Returns + ------- + strength : float or 1D array + The strength of the synchronization. 1.0 is perfect synchronization + and 0.0 is no synchronization. If `period` is an array, this is also + an array with each element containing the vector strength at the + corresponding period. + phase : float or array + The phase that the events are most strongly synchronized to in radians. + If `period` is an array, this is also an array with each element + containing the phase for the corresponding period. + + References + ---------- + van Hemmen, JL, Longtin, A, and Vollmayr, AN. Testing resonating vector + strength: Auditory system, electric fish, and noise. + Chaos 21, 047508 (2011); + :doi:`10.1063/1.3670512`. + van Hemmen, JL. Vector strength after Goldberg, Brown, and von Mises: + biological and mathematical perspectives. Biol Cybern. + 2013 Aug;107(4):385-96. :doi:`10.1007/s00422-013-0561-7`. + van Hemmen, JL and Vollmayr, AN. Resonating vector strength: what happens + when we vary the "probing" frequency while keeping the spike times + fixed. Biol Cybern. 2013 Aug;107(4):491-94. + :doi:`10.1007/s00422-013-0560-8`. + ''' + events = np.asarray(events) + period = np.asarray(period) + if events.ndim > 1: + raise ValueError('events cannot have dimensions more than 1') + if period.ndim > 1: + raise ValueError('period cannot have dimensions more than 1') + + # we need to know later if period was originally a scalar + scalarperiod = not period.ndim + + events = np.atleast_2d(events) + period = np.atleast_2d(period) + if (period <= 0).any(): + raise ValueError('periods must be positive') + + # this converts the times to vectors + vectors = np.exp(np.dot(2j*np.pi/period.T, events)) + + # the vector strength is just the magnitude of the mean of the vectors + # the vector phase is the angle of the mean of the vectors + vectormean = np.mean(vectors, axis=1) + strength = abs(vectormean) + phase = np.angle(vectormean) + + # if the original period was a scalar, return scalars + if scalarperiod: + strength = strength[0] + phase = phase[0] + return strength, phase + + +def detrend(data: np.ndarray, axis: int = -1, + type: Literal['linear', 'constant'] = 'linear', + bp: ArrayLike | int = 0, overwrite_data: bool = False) -> np.ndarray: + r"""Remove linear or constant trend along axis from data. + + Parameters + ---------- + data : array_like + The input data. + axis : int, optional + The axis along which to detrend the data. By default this is the + last axis (-1). + type : {'linear', 'constant'}, optional + The type of detrending. If ``type == 'linear'`` (default), + the result of a linear least-squares fit to `data` is subtracted + from `data`. + If ``type == 'constant'``, only the mean of `data` is subtracted. + bp : array_like of ints, optional + A sequence of break points. If given, an individual linear fit is + performed for each part of `data` between two break points. + Break points are specified as indices into `data`. This parameter + only has an effect when ``type == 'linear'``. + overwrite_data : bool, optional + If True, perform in place detrending and avoid a copy. Default is False + + Returns + ------- + ret : ndarray + The detrended input data. + + Notes + ----- + Detrending can be interpreted as subtracting a least squares fit polynomial: + Setting the parameter `type` to 'constant' corresponds to fitting a zeroth degree + polynomial, 'linear' to a first degree polynomial. Consult the example below. + + See Also + -------- + numpy.polynomial.polynomial.Polynomial.fit: Create least squares fit polynomial. + + + Examples + -------- + The following example detrends the function :math:`x(t) = \sin(\pi t) + 1/4`: + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> from scipy.signal import detrend + ... + >>> t = np.linspace(-0.5, 0.5, 21) + >>> x = np.sin(np.pi*t) + 1/4 + ... + >>> x_d_const = detrend(x, type='constant') + >>> x_d_linear = detrend(x, type='linear') + ... + >>> fig1, ax1 = plt.subplots() + >>> ax1.set_title(r"Detrending $x(t)=\sin(\pi t) + 1/4$") + >>> ax1.set(xlabel="t", ylabel="$x(t)$", xlim=(t[0], t[-1])) + >>> ax1.axhline(y=0, color='black', linewidth=.5) + >>> ax1.axvline(x=0, color='black', linewidth=.5) + >>> ax1.plot(t, x, 'C0.-', label="No detrending") + >>> ax1.plot(t, x_d_const, 'C1x-', label="type='constant'") + >>> ax1.plot(t, x_d_linear, 'C2+-', label="type='linear'") + >>> ax1.legend() + >>> plt.show() + + Alternatively, NumPy's `~numpy.polynomial.polynomial.Polynomial` can be used for + detrending as well: + + >>> pp0 = np.polynomial.Polynomial.fit(t, x, deg=0) # fit degree 0 polynomial + >>> np.allclose(x_d_const, x - pp0(t)) # compare with constant detrend + True + >>> pp1 = np.polynomial.Polynomial.fit(t, x, deg=1) # fit degree 1 polynomial + >>> np.allclose(x_d_linear, x - pp1(t)) # compare with linear detrend + True + + Note that `~numpy.polynomial.polynomial.Polynomial` also allows fitting higher + degree polynomials. Consult its documentation on how to extract the polynomial + coefficients. + """ + if type not in ['linear', 'l', 'constant', 'c']: + raise ValueError("Trend type must be 'linear' or 'constant'.") + data = np.asarray(data) + dtype = data.dtype.char + if dtype not in 'dfDF': + dtype = 'd' + if type in ['constant', 'c']: + ret = data - np.mean(data, axis, keepdims=True) + return ret + else: + dshape = data.shape + N = dshape[axis] + bp = np.sort(np.unique(np.concatenate(np.atleast_1d(0, bp, N)))) + if np.any(bp > N): + raise ValueError("Breakpoints must be less than length " + "of data along given axis.") + + # Restructure data so that axis is along first dimension and + # all other dimensions are collapsed into second dimension + rnk = len(dshape) + if axis < 0: + axis = axis + rnk + newdata = np.moveaxis(data, axis, 0) + newdata_shape = newdata.shape + newdata = newdata.reshape(N, -1) + + if not overwrite_data: + newdata = newdata.copy() # make sure we have a copy + if newdata.dtype.char not in 'dfDF': + newdata = newdata.astype(dtype) + +# Nreg = len(bp) - 1 + # Find leastsq fit and remove it for each piece + for m in range(len(bp) - 1): + Npts = bp[m + 1] - bp[m] + A = np.ones((Npts, 2), dtype) + A[:, 0] = np.arange(1, Npts + 1, dtype=dtype) / Npts + sl = slice(bp[m], bp[m + 1]) + coef, resids, rank, s = linalg.lstsq(A, newdata[sl]) + newdata[sl] = newdata[sl] - A @ coef + + # Put data back in original shape. + newdata = newdata.reshape(newdata_shape) + ret = np.moveaxis(newdata, 0, axis) + return ret + + +def lfilter_zi(b, a): + """ + Construct initial conditions for lfilter for step response steady-state. + + Compute an initial state `zi` for the `lfilter` function that corresponds + to the steady state of the step response. + + A typical use of this function is to set the initial state so that the + output of the filter starts at the same value as the first element of + the signal to be filtered. + + Parameters + ---------- + b, a : array_like (1-D) + The IIR filter coefficients. See `lfilter` for more + information. + + Returns + ------- + zi : 1-D ndarray + The initial state for the filter. + + See Also + -------- + lfilter, lfiltic, filtfilt + + Notes + ----- + A linear filter with order m has a state space representation (A, B, C, D), + for which the output y of the filter can be expressed as:: + + z(n+1) = A*z(n) + B*x(n) + y(n) = C*z(n) + D*x(n) + + where z(n) is a vector of length m, A has shape (m, m), B has shape + (m, 1), C has shape (1, m) and D has shape (1, 1) (assuming x(n) is + a scalar). lfilter_zi solves:: + + zi = A*zi + B + + In other words, it finds the initial condition for which the response + to an input of all ones is a constant. + + Given the filter coefficients `a` and `b`, the state space matrices + for the transposed direct form II implementation of the linear filter, + which is the implementation used by scipy.signal.lfilter, are:: + + A = scipy.linalg.companion(a).T + B = b[1:] - a[1:]*b[0] + + assuming ``a[0]`` is 1.0; if ``a[0]`` is not 1, `a` and `b` are first + divided by a[0]. + + Examples + -------- + The following code creates a lowpass Butterworth filter. Then it + applies that filter to an array whose values are all 1.0; the + output is also all 1.0, as expected for a lowpass filter. If the + `zi` argument of `lfilter` had not been given, the output would have + shown the transient signal. + + >>> from numpy import array, ones + >>> from scipy.signal import lfilter, lfilter_zi, butter + >>> b, a = butter(5, 0.25) + >>> zi = lfilter_zi(b, a) + >>> y, zo = lfilter(b, a, ones(10), zi=zi) + >>> y + array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]) + + Another example: + + >>> x = array([0.5, 0.5, 0.5, 0.0, 0.0, 0.0, 0.0]) + >>> y, zf = lfilter(b, a, x, zi=zi*x[0]) + >>> y + array([ 0.5 , 0.5 , 0.5 , 0.49836039, 0.48610528, + 0.44399389, 0.35505241]) + + Note that the `zi` argument to `lfilter` was computed using + `lfilter_zi` and scaled by ``x[0]``. Then the output `y` has no + transient until the input drops from 0.5 to 0.0. + + """ + + # FIXME: Can this function be replaced with an appropriate + # use of lfiltic? For example, when b,a = butter(N,Wn), + # lfiltic(b, a, y=numpy.ones_like(a), x=numpy.ones_like(b)). + # + + # We could use scipy.signal.normalize, but it uses warnings in + # cases where a ValueError is more appropriate, and it allows + # b to be 2D. + b = np.atleast_1d(b) + if b.ndim != 1: + raise ValueError("Numerator b must be 1-D.") + a = np.atleast_1d(a) + if a.ndim != 1: + raise ValueError("Denominator a must be 1-D.") + + while len(a) > 1 and a[0] == 0.0: + a = a[1:] + if a.size < 1: + raise ValueError("There must be at least one nonzero `a` coefficient.") + + if a[0] != 1.0: + # Normalize the coefficients so a[0] == 1. + b = b / a[0] + a = a / a[0] + + n = max(len(a), len(b)) + + # Pad a or b with zeros so they are the same length. + if len(a) < n: + a = np.r_[a, np.zeros(n - len(a), dtype=a.dtype)] + elif len(b) < n: + b = np.r_[b, np.zeros(n - len(b), dtype=b.dtype)] + + IminusA = np.eye(n - 1, dtype=np.result_type(a, b)) - linalg.companion(a).T + B = b[1:] - a[1:] * b[0] + # Solve zi = A*zi + B + zi = np.linalg.solve(IminusA, B) + + # For future reference: we could also use the following + # explicit formulas to solve the linear system: + # + # zi = np.zeros(n - 1) + # zi[0] = B.sum() / IminusA[:,0].sum() + # asum = 1.0 + # csum = 0.0 + # for k in range(1,n-1): + # asum += a[k] + # csum += b[k] - a[k]*b[0] + # zi[k] = asum*zi[0] - csum + + return zi + + +def sosfilt_zi(sos): + """ + Construct initial conditions for sosfilt for step response steady-state. + + Compute an initial state `zi` for the `sosfilt` function that corresponds + to the steady state of the step response. + + A typical use of this function is to set the initial state so that the + output of the filter starts at the same value as the first element of + the signal to be filtered. + + Parameters + ---------- + sos : array_like + Array of second-order filter coefficients, must have shape + ``(n_sections, 6)``. See `sosfilt` for the SOS filter format + specification. + + Returns + ------- + zi : ndarray + Initial conditions suitable for use with ``sosfilt``, shape + ``(n_sections, 2)``. + + See Also + -------- + sosfilt, zpk2sos + + Notes + ----- + .. versionadded:: 0.16.0 + + Examples + -------- + Filter a rectangular pulse that begins at time 0, with and without + the use of the `zi` argument of `scipy.signal.sosfilt`. + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> sos = signal.butter(9, 0.125, output='sos') + >>> zi = signal.sosfilt_zi(sos) + >>> x = (np.arange(250) < 100).astype(int) + >>> f1 = signal.sosfilt(sos, x) + >>> f2, zo = signal.sosfilt(sos, x, zi=zi) + + >>> plt.plot(x, 'k--', label='x') + >>> plt.plot(f1, 'b', alpha=0.5, linewidth=2, label='filtered') + >>> plt.plot(f2, 'g', alpha=0.25, linewidth=4, label='filtered with zi') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + sos = np.asarray(sos) + if sos.ndim != 2 or sos.shape[1] != 6: + raise ValueError('sos must be shape (n_sections, 6)') + + if sos.dtype.kind in 'bui': + sos = sos.astype(np.float64) + + n_sections = sos.shape[0] + zi = np.empty((n_sections, 2), dtype=sos.dtype) + scale = 1.0 + for section in range(n_sections): + b = sos[section, :3] + a = sos[section, 3:] + zi[section] = scale * lfilter_zi(b, a) + # If H(z) = B(z)/A(z) is this section's transfer function, then + # b.sum()/a.sum() is H(1), the gain at omega=0. That's the steady + # state value of this section's step response. + scale *= b.sum() / a.sum() + + return zi + + +def _filtfilt_gust(b, a, x, axis=-1, irlen=None): + """Forward-backward IIR filter that uses Gustafsson's method. + + Apply the IIR filter defined by ``(b,a)`` to `x` twice, first forward + then backward, using Gustafsson's initial conditions [1]_. + + Let ``y_fb`` be the result of filtering first forward and then backward, + and let ``y_bf`` be the result of filtering first backward then forward. + Gustafsson's method is to compute initial conditions for the forward + pass and the backward pass such that ``y_fb == y_bf``. + + Parameters + ---------- + b : scalar or 1-D ndarray + Numerator coefficients of the filter. + a : scalar or 1-D ndarray + Denominator coefficients of the filter. + x : ndarray + Data to be filtered. + axis : int, optional + Axis of `x` to be filtered. Default is -1. + irlen : int or None, optional + The length of the nonnegligible part of the impulse response. + If `irlen` is None, or if the length of the signal is less than + ``2 * irlen``, then no part of the impulse response is ignored. + + Returns + ------- + y : ndarray + The filtered data. + x0 : ndarray + Initial condition for the forward filter. + x1 : ndarray + Initial condition for the backward filter. + + Notes + ----- + Typically the return values `x0` and `x1` are not needed by the + caller. The intended use of these return values is in unit tests. + + References + ---------- + .. [1] F. Gustaffson. Determining the initial states in forward-backward + filtering. Transactions on Signal Processing, 46(4):988-992, 1996. + + """ + # In the comments, "Gustafsson's paper" and [1] refer to the + # paper referenced in the docstring. + + b = np.atleast_1d(b) + a = np.atleast_1d(a) + + order = max(len(b), len(a)) - 1 + if order == 0: + # The filter is just scalar multiplication, with no state. + scale = (b[0] / a[0])**2 + y = scale * x + return y, np.array([]), np.array([]) + + if axis != -1 or axis != x.ndim - 1: + # Move the axis containing the data to the end. + x = np.swapaxes(x, axis, x.ndim - 1) + + # n is the number of samples in the data to be filtered. + n = x.shape[-1] + + if irlen is None or n <= 2*irlen: + m = n + else: + m = irlen + + # Create Obs, the observability matrix (called O in the paper). + # This matrix can be interpreted as the operator that propagates + # an arbitrary initial state to the output, assuming the input is + # zero. + # In Gustafsson's paper, the forward and backward filters are not + # necessarily the same, so he has both O_f and O_b. We use the same + # filter in both directions, so we only need O. The same comment + # applies to S below. + Obs = np.zeros((m, order)) + zi = np.zeros(order) + zi[0] = 1 + Obs[:, 0] = lfilter(b, a, np.zeros(m), zi=zi)[0] + for k in range(1, order): + Obs[k:, k] = Obs[:-k, 0] + + # Obsr is O^R (Gustafsson's notation for row-reversed O) + Obsr = Obs[::-1] + + # Create S. S is the matrix that applies the filter to the reversed + # propagated initial conditions. That is, + # out = S.dot(zi) + # is the same as + # tmp, _ = lfilter(b, a, zeros(), zi=zi) # Propagate ICs. + # out = lfilter(b, a, tmp[::-1]) # Reverse and filter. + + # Equations (5) & (6) of [1] + S = lfilter(b, a, Obs[::-1], axis=0) + + # Sr is S^R (row-reversed S) + Sr = S[::-1] + + # M is [(S^R - O), (O^R - S)] + if m == n: + M = np.hstack((Sr - Obs, Obsr - S)) + else: + # Matrix described in section IV of [1]. + M = np.zeros((2*m, 2*order)) + M[:m, :order] = Sr - Obs + M[m:, order:] = Obsr - S + + # Naive forward-backward and backward-forward filters. + # These have large transients because the filters use zero initial + # conditions. + y_f = lfilter(b, a, x) + y_fb = lfilter(b, a, y_f[..., ::-1])[..., ::-1] + + y_b = lfilter(b, a, x[..., ::-1])[..., ::-1] + y_bf = lfilter(b, a, y_b) + + delta_y_bf_fb = y_bf - y_fb + if m == n: + delta = delta_y_bf_fb + else: + start_m = delta_y_bf_fb[..., :m] + end_m = delta_y_bf_fb[..., -m:] + delta = np.concatenate((start_m, end_m), axis=-1) + + # ic_opt holds the "optimal" initial conditions. + # The following code computes the result shown in the formula + # of the paper between equations (6) and (7). + if delta.ndim == 1: + ic_opt = linalg.lstsq(M, delta)[0] + else: + # Reshape delta so it can be used as an array of multiple + # right-hand-sides in linalg.lstsq. + delta2d = delta.reshape(-1, delta.shape[-1]).T + ic_opt0 = linalg.lstsq(M, delta2d)[0].T + ic_opt = ic_opt0.reshape(delta.shape[:-1] + (M.shape[-1],)) + + # Now compute the filtered signal using equation (7) of [1]. + # First, form [S^R, O^R] and call it W. + if m == n: + W = np.hstack((Sr, Obsr)) + else: + W = np.zeros((2*m, 2*order)) + W[:m, :order] = Sr + W[m:, order:] = Obsr + + # Equation (7) of [1] says + # Y_fb^opt = Y_fb^0 + W * [x_0^opt; x_{N-1}^opt] + # `wic` is (almost) the product on the right. + # W has shape (m, 2*order), and ic_opt has shape (..., 2*order), + # so we can't use W.dot(ic_opt). Instead, we dot ic_opt with W.T, + # so wic has shape (..., m). + wic = ic_opt.dot(W.T) + + # `wic` is "almost" the product of W and the optimal ICs in equation + # (7)--if we're using a truncated impulse response (m < n), `wic` + # contains only the adjustments required for the ends of the signal. + # Here we form y_opt, taking this into account if necessary. + y_opt = y_fb + if m == n: + y_opt += wic + else: + y_opt[..., :m] += wic[..., :m] + y_opt[..., -m:] += wic[..., -m:] + + x0 = ic_opt[..., :order] + x1 = ic_opt[..., -order:] + if axis != -1 or axis != x.ndim - 1: + # Restore the data axis to its original position. + x0 = np.swapaxes(x0, axis, x.ndim - 1) + x1 = np.swapaxes(x1, axis, x.ndim - 1) + y_opt = np.swapaxes(y_opt, axis, x.ndim - 1) + + return y_opt, x0, x1 + + +def filtfilt(b, a, x, axis=-1, padtype='odd', padlen=None, method='pad', + irlen=None): + """ + Apply a digital filter forward and backward to a signal. + + This function applies a linear digital filter twice, once forward and + once backwards. The combined filter has zero phase and a filter order + twice that of the original. + + The function provides options for handling the edges of the signal. + + The function `sosfiltfilt` (and filter design using ``output='sos'``) + should be preferred over `filtfilt` for most filtering tasks, as + second-order sections have fewer numerical problems. + + Parameters + ---------- + b : (N,) array_like + The numerator coefficient vector of the filter. + a : (N,) array_like + The denominator coefficient vector of the filter. If ``a[0]`` + is not 1, then both `a` and `b` are normalized by ``a[0]``. + x : array_like + The array of data to be filtered. + axis : int, optional + The axis of `x` to which the filter is applied. + Default is -1. + padtype : str or None, optional + Must be 'odd', 'even', 'constant', or None. This determines the + type of extension to use for the padded signal to which the filter + is applied. If `padtype` is None, no padding is used. The default + is 'odd'. + padlen : int or None, optional + The number of elements by which to extend `x` at both ends of + `axis` before applying the filter. This value must be less than + ``x.shape[axis] - 1``. ``padlen=0`` implies no padding. + The default value is ``3 * max(len(a), len(b))``. + method : str, optional + Determines the method for handling the edges of the signal, either + "pad" or "gust". When `method` is "pad", the signal is padded; the + type of padding is determined by `padtype` and `padlen`, and `irlen` + is ignored. When `method` is "gust", Gustafsson's method is used, + and `padtype` and `padlen` are ignored. + irlen : int or None, optional + When `method` is "gust", `irlen` specifies the length of the + impulse response of the filter. If `irlen` is None, no part + of the impulse response is ignored. For a long signal, specifying + `irlen` can significantly improve the performance of the filter. + + Returns + ------- + y : ndarray + The filtered output with the same shape as `x`. + + See Also + -------- + sosfiltfilt, lfilter_zi, lfilter, lfiltic, savgol_filter, sosfilt + + Notes + ----- + When `method` is "pad", the function pads the data along the given axis + in one of three ways: odd, even or constant. The odd and even extensions + have the corresponding symmetry about the end point of the data. The + constant extension extends the data with the values at the end points. On + both the forward and backward passes, the initial condition of the + filter is found by using `lfilter_zi` and scaling it by the end point of + the extended data. + + When `method` is "gust", Gustafsson's method [1]_ is used. Initial + conditions are chosen for the forward and backward passes so that the + forward-backward filter gives the same result as the backward-forward + filter. + + The option to use Gustaffson's method was added in scipy version 0.16.0. + + References + ---------- + .. [1] F. Gustaffson, "Determining the initial states in forward-backward + filtering", Transactions on Signal Processing, Vol. 46, pp. 988-992, + 1996. + + Examples + -------- + The examples will use several functions from `scipy.signal`. + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + First we create a one second signal that is the sum of two pure sine + waves, with frequencies 5 Hz and 250 Hz, sampled at 2000 Hz. + + >>> t = np.linspace(0, 1.0, 2001) + >>> xlow = np.sin(2 * np.pi * 5 * t) + >>> xhigh = np.sin(2 * np.pi * 250 * t) + >>> x = xlow + xhigh + + Now create a lowpass Butterworth filter with a cutoff of 0.125 times + the Nyquist frequency, or 125 Hz, and apply it to ``x`` with `filtfilt`. + The result should be approximately ``xlow``, with no phase shift. + + >>> b, a = signal.butter(8, 0.125) + >>> y = signal.filtfilt(b, a, x, padlen=150) + >>> np.abs(y - xlow).max() + 9.1086182074789912e-06 + + We get a fairly clean result for this artificial example because + the odd extension is exact, and with the moderately long padding, + the filter's transients have dissipated by the time the actual data + is reached. In general, transient effects at the edges are + unavoidable. + + The following example demonstrates the option ``method="gust"``. + + First, create a filter. + + >>> b, a = signal.ellip(4, 0.01, 120, 0.125) # Filter to be applied. + + `sig` is a random input signal to be filtered. + + >>> rng = np.random.default_rng() + >>> n = 60 + >>> sig = rng.standard_normal(n)**3 + 3*rng.standard_normal(n).cumsum() + + Apply `filtfilt` to `sig`, once using the Gustafsson method, and + once using padding, and plot the results for comparison. + + >>> fgust = signal.filtfilt(b, a, sig, method="gust") + >>> fpad = signal.filtfilt(b, a, sig, padlen=50) + >>> plt.plot(sig, 'k-', label='input') + >>> plt.plot(fgust, 'b-', linewidth=4, label='gust') + >>> plt.plot(fpad, 'c-', linewidth=1.5, label='pad') + >>> plt.legend(loc='best') + >>> plt.show() + + The `irlen` argument can be used to improve the performance + of Gustafsson's method. + + Estimate the impulse response length of the filter. + + >>> z, p, k = signal.tf2zpk(b, a) + >>> eps = 1e-9 + >>> r = np.max(np.abs(p)) + >>> approx_impulse_len = int(np.ceil(np.log(eps) / np.log(r))) + >>> approx_impulse_len + 137 + + Apply the filter to a longer signal, with and without the `irlen` + argument. The difference between `y1` and `y2` is small. For long + signals, using `irlen` gives a significant performance improvement. + + >>> x = rng.standard_normal(4000) + >>> y1 = signal.filtfilt(b, a, x, method='gust') + >>> y2 = signal.filtfilt(b, a, x, method='gust', irlen=approx_impulse_len) + >>> print(np.max(np.abs(y1 - y2))) + 2.875334415008979e-10 + + """ + b = np.atleast_1d(b) + a = np.atleast_1d(a) + x = np.asarray(x) + + if method not in ["pad", "gust"]: + raise ValueError("method must be 'pad' or 'gust'.") + + if method == "gust": + y, z1, z2 = _filtfilt_gust(b, a, x, axis=axis, irlen=irlen) + return y + + # method == "pad" + edge, ext = _validate_pad(padtype, padlen, x, axis, + ntaps=max(len(a), len(b))) + + # Get the steady state of the filter's step response. + zi = lfilter_zi(b, a) + + # Reshape zi and create x0 so that zi*x0 broadcasts + # to the correct value for the 'zi' keyword argument + # to lfilter. + zi_shape = [1] * x.ndim + zi_shape[axis] = zi.size + zi = np.reshape(zi, zi_shape) + x0 = axis_slice(ext, stop=1, axis=axis) + + # Forward filter. + (y, zf) = lfilter(b, a, ext, axis=axis, zi=zi * x0) + + # Backward filter. + # Create y0 so zi*y0 broadcasts appropriately. + y0 = axis_slice(y, start=-1, axis=axis) + (y, zf) = lfilter(b, a, axis_reverse(y, axis=axis), axis=axis, zi=zi * y0) + + # Reverse y. + y = axis_reverse(y, axis=axis) + + if edge > 0: + # Slice the actual signal from the extended signal. + y = axis_slice(y, start=edge, stop=-edge, axis=axis) + + return y + + +def _validate_pad(padtype, padlen, x, axis, ntaps): + """Helper to validate padding for filtfilt""" + if padtype not in ['even', 'odd', 'constant', None]: + raise ValueError(f"Unknown value '{padtype}' given to padtype. " + "padtype must be 'even', 'odd', 'constant', or None.") + + if padtype is None: + padlen = 0 + + if padlen is None: + # Original padding; preserved for backwards compatibility. + edge = ntaps * 3 + else: + edge = padlen + + # x's 'axis' dimension must be bigger than edge. + if x.shape[axis] <= edge: + raise ValueError("The length of the input vector x must be greater " + "than padlen, which is %d." % edge) + + if padtype is not None and edge > 0: + # Make an extension of length `edge` at each + # end of the input array. + if padtype == 'even': + ext = even_ext(x, edge, axis=axis) + elif padtype == 'odd': + ext = odd_ext(x, edge, axis=axis) + else: + ext = const_ext(x, edge, axis=axis) + else: + ext = x + return edge, ext + + +def _validate_x(x): + x = np.asarray(x) + if x.ndim == 0: + raise ValueError('x must be at least 1-D') + return x + + +def sosfilt(sos, x, axis=-1, zi=None): + """ + Filter data along one dimension using cascaded second-order sections. + + Filter a data sequence, `x`, using a digital IIR filter defined by + `sos`. + + Parameters + ---------- + sos : array_like + Array of second-order filter coefficients, must have shape + ``(n_sections, 6)``. Each row corresponds to a second-order + section, with the first three columns providing the numerator + coefficients and the last three providing the denominator + coefficients. + x : array_like + An N-dimensional input array. + axis : int, optional + The axis of the input data array along which to apply the + linear filter. The filter is applied to each subarray along + this axis. Default is -1. + zi : array_like, optional + Initial conditions for the cascaded filter delays. It is a (at + least 2D) vector of shape ``(n_sections, ..., 2, ...)``, where + ``..., 2, ...`` denotes the shape of `x`, but with ``x.shape[axis]`` + replaced by 2. If `zi` is None or is not given then initial rest + (i.e. all zeros) is assumed. + Note that these initial conditions are *not* the same as the initial + conditions given by `lfiltic` or `lfilter_zi`. + + Returns + ------- + y : ndarray + The output of the digital filter. + zf : ndarray, optional + If `zi` is None, this is not returned, otherwise, `zf` holds the + final filter delay values. + + See Also + -------- + zpk2sos, sos2zpk, sosfilt_zi, sosfiltfilt, freqz_sos + + Notes + ----- + The filter function is implemented as a series of second-order filters + with direct-form II transposed structure. It is designed to minimize + numerical precision errors for high-order filters. + + .. versionadded:: 0.16.0 + + Examples + -------- + Plot a 13th-order filter's impulse response using both `lfilter` and + `sosfilt`, showing the instability that results from trying to do a + 13th-order filter in a single stage (the numerical error pushes some poles + outside of the unit circle): + + >>> import matplotlib.pyplot as plt + >>> from scipy import signal + >>> b, a = signal.ellip(13, 0.009, 80, 0.05, output='ba') + >>> sos = signal.ellip(13, 0.009, 80, 0.05, output='sos') + >>> x = signal.unit_impulse(700) + >>> y_tf = signal.lfilter(b, a, x) + >>> y_sos = signal.sosfilt(sos, x) + >>> plt.plot(y_tf, 'r', label='TF') + >>> plt.plot(y_sos, 'k', label='SOS') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + _reject_objects(sos, 'sosfilt') + _reject_objects(x, 'sosfilt') + if zi is not None: + _reject_objects(zi, 'sosfilt') + + x = _validate_x(x) + sos, n_sections = _validate_sos(sos) + x_zi_shape = list(x.shape) + x_zi_shape[axis] = 2 + x_zi_shape = tuple([n_sections] + x_zi_shape) + inputs = [sos, x] + if zi is not None: + inputs.append(np.asarray(zi)) + dtype = np.result_type(*inputs) + if dtype.char not in 'fdgFDGO': + raise NotImplementedError(f"input type '{dtype}' not supported") + if zi is not None: + zi = np.array(zi, dtype) # make a copy so that we can operate in place + if zi.shape != x_zi_shape: + raise ValueError('Invalid zi shape. With axis=%r, an input with ' + 'shape %r, and an sos array with %d sections, zi ' + 'must have shape %r, got %r.' % + (axis, x.shape, n_sections, x_zi_shape, zi.shape)) + return_zi = True + else: + zi = np.zeros(x_zi_shape, dtype=dtype) + return_zi = False + axis = axis % x.ndim # make positive + x = np.moveaxis(x, axis, -1) + zi = np.moveaxis(zi, [0, axis + 1], [-2, -1]) + x_shape, zi_shape = x.shape, zi.shape + x = np.reshape(x, (-1, x.shape[-1])) + x = np.array(x, dtype, order='C') # make a copy, can modify in place + zi = np.ascontiguousarray(np.reshape(zi, (-1, n_sections, 2))) + sos = sos.astype(dtype, copy=False) + _sosfilt(sos, x, zi) + x.shape = x_shape + x = np.moveaxis(x, -1, axis) + if return_zi: + zi.shape = zi_shape + zi = np.moveaxis(zi, [-2, -1], [0, axis + 1]) + out = (x, zi) + else: + out = x + return out + + +def sosfiltfilt(sos, x, axis=-1, padtype='odd', padlen=None): + """ + A forward-backward digital filter using cascaded second-order sections. + + See `filtfilt` for more complete information about this method. + + Parameters + ---------- + sos : array_like + Array of second-order filter coefficients, must have shape + ``(n_sections, 6)``. Each row corresponds to a second-order + section, with the first three columns providing the numerator + coefficients and the last three providing the denominator + coefficients. + x : array_like + The array of data to be filtered. + axis : int, optional + The axis of `x` to which the filter is applied. + Default is -1. + padtype : str or None, optional + Must be 'odd', 'even', 'constant', or None. This determines the + type of extension to use for the padded signal to which the filter + is applied. If `padtype` is None, no padding is used. The default + is 'odd'. + padlen : int or None, optional + The number of elements by which to extend `x` at both ends of + `axis` before applying the filter. This value must be less than + ``x.shape[axis] - 1``. ``padlen=0`` implies no padding. + The default value is:: + + 3 * (2 * len(sos) + 1 - min((sos[:, 2] == 0).sum(), + (sos[:, 5] == 0).sum())) + + The extra subtraction at the end attempts to compensate for poles + and zeros at the origin (e.g. for odd-order filters) to yield + equivalent estimates of `padlen` to those of `filtfilt` for + second-order section filters built with `scipy.signal` functions. + + Returns + ------- + y : ndarray + The filtered output with the same shape as `x`. + + See Also + -------- + filtfilt, sosfilt, sosfilt_zi, freqz_sos + + Notes + ----- + .. versionadded:: 0.18.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal import sosfiltfilt, butter + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + Create an interesting signal to filter. + + >>> n = 201 + >>> t = np.linspace(0, 1, n) + >>> x = 1 + (t < 0.5) - 0.25*t**2 + 0.05*rng.standard_normal(n) + + Create a lowpass Butterworth filter, and use it to filter `x`. + + >>> sos = butter(4, 0.125, output='sos') + >>> y = sosfiltfilt(sos, x) + + For comparison, apply an 8th order filter using `sosfilt`. The filter + is initialized using the mean of the first four values of `x`. + + >>> from scipy.signal import sosfilt, sosfilt_zi + >>> sos8 = butter(8, 0.125, output='sos') + >>> zi = x[:4].mean() * sosfilt_zi(sos8) + >>> y2, zo = sosfilt(sos8, x, zi=zi) + + Plot the results. Note that the phase of `y` matches the input, while + `y2` has a significant phase delay. + + >>> plt.plot(t, x, alpha=0.5, label='x(t)') + >>> plt.plot(t, y, label='y(t)') + >>> plt.plot(t, y2, label='y2(t)') + >>> plt.legend(framealpha=1, shadow=True) + >>> plt.grid(alpha=0.25) + >>> plt.xlabel('t') + >>> plt.show() + + """ + sos, n_sections = _validate_sos(sos) + x = _validate_x(x) + + # `method` is "pad"... + ntaps = 2 * n_sections + 1 + ntaps -= min((sos[:, 2] == 0).sum(), (sos[:, 5] == 0).sum()) + edge, ext = _validate_pad(padtype, padlen, x, axis, + ntaps=ntaps) + + # These steps follow the same form as filtfilt with modifications + zi = sosfilt_zi(sos) # shape (n_sections, 2) --> (n_sections, ..., 2, ...) + zi_shape = [1] * x.ndim + zi_shape[axis] = 2 + zi.shape = [n_sections] + zi_shape + x_0 = axis_slice(ext, stop=1, axis=axis) + (y, zf) = sosfilt(sos, ext, axis=axis, zi=zi * x_0) + y_0 = axis_slice(y, start=-1, axis=axis) + (y, zf) = sosfilt(sos, axis_reverse(y, axis=axis), axis=axis, zi=zi * y_0) + y = axis_reverse(y, axis=axis) + if edge > 0: + y = axis_slice(y, start=edge, stop=-edge, axis=axis) + return y + + +def decimate(x, q, n=None, ftype='iir', axis=-1, zero_phase=True): + """ + Downsample the signal after applying an anti-aliasing filter. + + By default, an order 8 Chebyshev type I filter is used. A 30 point FIR + filter with Hamming window is used if `ftype` is 'fir'. + + Parameters + ---------- + x : array_like + The signal to be downsampled, as an N-dimensional array. + q : int + The downsampling factor. When using IIR downsampling, it is recommended + to call `decimate` multiple times for downsampling factors higher than + 13. + n : int, optional + The order of the filter (1 less than the length for 'fir'). Defaults to + 8 for 'iir' and 20 times the downsampling factor for 'fir'. + ftype : str {'iir', 'fir'} or ``dlti`` instance, optional + If 'iir' or 'fir', specifies the type of lowpass filter. If an instance + of an `dlti` object, uses that object to filter before downsampling. + axis : int, optional + The axis along which to decimate. + zero_phase : bool, optional + Prevent phase shift by filtering with `filtfilt` instead of `lfilter` + when using an IIR filter, and shifting the outputs back by the filter's + group delay when using an FIR filter. The default value of ``True`` is + recommended, since a phase shift is generally not desired. + + .. versionadded:: 0.18.0 + + Returns + ------- + y : ndarray + The down-sampled signal. + + See Also + -------- + resample : Resample up or down using the FFT method. + resample_poly : Resample using polyphase filtering and an FIR filter. + + Notes + ----- + The ``zero_phase`` keyword was added in 0.18.0. + The possibility to use instances of ``dlti`` as ``ftype`` was added in + 0.18.0. + + Examples + -------- + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + Define wave parameters. + + >>> wave_duration = 3 + >>> sample_rate = 100 + >>> freq = 2 + >>> q = 5 + + Calculate number of samples. + + >>> samples = wave_duration*sample_rate + >>> samples_decimated = int(samples/q) + + Create cosine wave. + + >>> x = np.linspace(0, wave_duration, samples, endpoint=False) + >>> y = np.cos(x*np.pi*freq*2) + + Decimate cosine wave. + + >>> ydem = signal.decimate(y, q) + >>> xnew = np.linspace(0, wave_duration, samples_decimated, endpoint=False) + + Plot original and decimated waves. + + >>> plt.plot(x, y, '.-', xnew, ydem, 'o-') + >>> plt.xlabel('Time, Seconds') + >>> plt.legend(['data', 'decimated'], loc='best') + >>> plt.show() + + """ + + x = np.asarray(x) + q = operator.index(q) + + if n is not None: + n = operator.index(n) + + result_type = x.dtype + if not np.issubdtype(result_type, np.inexact) \ + or result_type.type == np.float16: + # upcast integers and float16 to float64 + result_type = np.float64 + + if ftype == 'fir': + if n is None: + half_len = 10 * q # reasonable cutoff for our sinc-like function + n = 2 * half_len + b, a = firwin(n+1, 1. / q, window='hamming'), 1. + b = np.asarray(b, dtype=result_type) + a = np.asarray(a, dtype=result_type) + elif ftype == 'iir': + iir_use_sos = True + if n is None: + n = 8 + sos = cheby1(n, 0.05, 0.8 / q, output='sos') + sos = np.asarray(sos, dtype=result_type) + elif isinstance(ftype, dlti): + system = ftype._as_zpk() + if system.poles.shape[0] == 0: + # FIR + system = ftype._as_tf() + b, a = system.num, system.den + ftype = 'fir' + elif (any(np.iscomplex(system.poles)) + or any(np.iscomplex(system.poles)) + or np.iscomplex(system.gain)): + # sosfilt & sosfiltfilt don't handle complex coeffs + iir_use_sos = False + system = ftype._as_tf() + b, a = system.num, system.den + else: + iir_use_sos = True + sos = zpk2sos(system.zeros, system.poles, system.gain) + sos = np.asarray(sos, dtype=result_type) + else: + raise ValueError('invalid ftype') + + sl = [slice(None)] * x.ndim + + if ftype == 'fir': + b = b / a + if zero_phase: + y = resample_poly(x, 1, q, axis=axis, window=b) + else: + # upfirdn is generally faster than lfilter by a factor equal to the + # downsampling factor, since it only calculates the needed outputs + n_out = x.shape[axis] // q + bool(x.shape[axis] % q) + y = upfirdn(b, x, up=1, down=q, axis=axis) + sl[axis] = slice(None, n_out, None) + + else: # IIR case + if zero_phase: + if iir_use_sos: + y = sosfiltfilt(sos, x, axis=axis) + else: + y = filtfilt(b, a, x, axis=axis) + else: + if iir_use_sos: + y = sosfilt(sos, x, axis=axis) + else: + y = lfilter(b, a, x, axis=axis) + + sl[axis] = slice(None, None, q) + + return y[tuple(sl)] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_sigtools.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_sigtools.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..9697a8a7dda74e71dadf9522d0ba943d2946a581 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_sigtools.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spectral_py.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spectral_py.py new file mode 100644 index 0000000000000000000000000000000000000000..5151b2b335172a485b5d13408d35290812cadc43 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spectral_py.py @@ -0,0 +1,2291 @@ +"""Tools for spectral analysis. +""" +import numpy as np +import numpy.typing as npt +from scipy import fft as sp_fft +from . import _signaltools +from .windows import get_window +from ._arraytools import const_ext, even_ext, odd_ext, zero_ext +import warnings +from typing import Literal + + +__all__ = ['periodogram', 'welch', 'lombscargle', 'csd', 'coherence', + 'spectrogram', 'stft', 'istft', 'check_COLA', 'check_NOLA'] + + +def lombscargle( + x: npt.ArrayLike, + y: npt.ArrayLike, + freqs: npt.ArrayLike, + precenter: bool = False, + normalize: bool | Literal["power", "normalize", "amplitude"] = False, + *, + weights: npt.NDArray | None = None, + floating_mean: bool = False, +) -> npt.NDArray: + """ + Compute the generalized Lomb-Scargle periodogram. + + The Lomb-Scargle periodogram was developed by Lomb [1]_ and further + extended by Scargle [2]_ to find, and test the significance of weak + periodic signals with uneven temporal sampling. The algorithm used + here is based on a weighted least-squares fit of the form + ``y(ω) = a*cos(ω*x) + b*sin(ω*x) + c``, where the fit is calculated for + each frequency independently. This algorithm was developed by Zechmeister + and Kürster which improves the Lomb-Scargle periodogram by enabling + the weighting of individual samples and calculating an unknown y offset + (also called a "floating-mean" model) [3]_. For more details, and practical + considerations, see the excellent reference on the Lomb-Scargle periodogram [4]_. + + When *normalize* is False (or "power") (default) the computed periodogram + is unnormalized, it takes the value ``(A**2) * N/4`` for a harmonic + signal with amplitude A for sufficiently large N. Where N is the length of x or y. + + When *normalize* is True (or "normalize") the computed periodogram is normalized + by the residuals of the data around a constant reference model (at zero). + + When *normalize* is "amplitude" the computed periodogram is the complex + representation of the amplitude and phase. + + Input arrays should be 1-D of a real floating data type, which are converted into + float64 arrays before processing. + + Parameters + ---------- + x : array_like + Sample times. + y : array_like + Measurement values. Values are assumed to have a baseline of ``y = 0``. If + there is a possibility of a y offset, it is recommended to set `floating_mean` + to True. + freqs : array_like + Angular frequencies (e.g., having unit rad/s=2π/s for `x` having unit s) for + output periodogram. Frequencies are normally >= 0, as any peak at ``-freq`` will + also exist at ``+freq``. + precenter : bool, optional + Pre-center measurement values by subtracting the mean, if True. This is + a legacy parameter and unnecessary if `floating_mean` is True. + normalize : bool | str, optional + Compute normalized or complex (amplitude + phase) periodogram. + Valid options are: ``False``/``"power"``, ``True``/``"normalize"``, or + ``"amplitude"``. + weights : array_like, optional + Weights for each sample. Weights must be nonnegative. + floating_mean : bool, optional + Determines a y offset for each frequency independently, if True. + Else the y offset is assumed to be `0`. + + Returns + ------- + pgram : array_like + Lomb-Scargle periodogram. + + Raises + ------ + ValueError + If any of the input arrays x, y, freqs, or weights are not 1D, or if any are + zero length. Or, if the input arrays x, y, and weights do not have the same + shape as each other. + ValueError + If any weight is < 0, or the sum of the weights is <= 0. + ValueError + If the normalize parameter is not one of the allowed options. + + See Also + -------- + periodogram: Power spectral density using a periodogram + welch: Power spectral density by Welch's method + csd: Cross spectral density by Welch's method + + Notes + ----- + The algorithm used will not automatically account for any unknown y offset, unless + floating_mean is True. Therefore, for most use cases, if there is a possibility of + a y offset, it is recommended to set floating_mean to True. If precenter is True, + it performs the operation ``y -= y.mean()``. However, precenter is a legacy + parameter, and unnecessary when floating_mean is True. Furthermore, the mean + removed by precenter does not account for sample weights, nor will it correct for + any bias due to consistently missing observations at peaks and/or troughs. When the + normalize parameter is "amplitude", for any frequency in freqs that is below + ``(2*pi)/(x.max() - x.min())``, the predicted amplitude will tend towards infinity. + The concept of a "Nyquist frequency" limit (see Nyquist-Shannon sampling theorem) + is not generally applicable to unevenly sampled data. Therefore, with unevenly + sampled data, valid frequencies in freqs can often be much higher than expected. + + References + ---------- + .. [1] N.R. Lomb "Least-squares frequency analysis of unequally spaced + data", Astrophysics and Space Science, vol 39, pp. 447-462, 1976 + :doi:`10.1007/bf00648343` + + .. [2] J.D. Scargle "Studies in astronomical time series analysis. II - + Statistical aspects of spectral analysis of unevenly spaced data", + The Astrophysical Journal, vol 263, pp. 835-853, 1982 + :doi:`10.1086/160554` + + .. [3] M. Zechmeister and M. Kürster, "The generalised Lomb-Scargle periodogram. + A new formalism for the floating-mean and Keplerian periodograms," + Astronomy and Astrophysics, vol. 496, pp. 577-584, 2009 + :doi:`10.1051/0004-6361:200811296` + + .. [4] J.T. VanderPlas, "Understanding the Lomb-Scargle Periodogram," + The Astrophysical Journal Supplement Series, vol. 236, no. 1, p. 16, + May 2018 + :doi:`10.3847/1538-4365/aab766` + + + Examples + -------- + >>> import numpy as np + >>> rng = np.random.default_rng() + + First define some input parameters for the signal: + + >>> A = 2. # amplitude + >>> c = 2. # offset + >>> w0 = 1. # rad/sec + >>> nin = 150 + >>> nout = 1002 + + Randomly generate sample times: + + >>> x = rng.uniform(0, 10*np.pi, nin) + + Plot a sine wave for the selected times: + + >>> y = A * np.cos(w0*x) + c + + Define the array of frequencies for which to compute the periodogram: + + >>> w = np.linspace(0.25, 10, nout) + + Calculate Lomb-Scargle periodogram for each of the normalize options: + + >>> from scipy.signal import lombscargle + >>> pgram_power = lombscargle(x, y, w, normalize=False) + >>> pgram_norm = lombscargle(x, y, w, normalize=True) + >>> pgram_amp = lombscargle(x, y, w, normalize='amplitude') + ... + >>> pgram_power_f = lombscargle(x, y, w, normalize=False, floating_mean=True) + >>> pgram_norm_f = lombscargle(x, y, w, normalize=True, floating_mean=True) + >>> pgram_amp_f = lombscargle(x, y, w, normalize='amplitude', floating_mean=True) + + Now make a plot of the input data: + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_t, ax_p, ax_n, ax_a) = plt.subplots(4, 1, figsize=(5, 6)) + >>> ax_t.plot(x, y, 'b+') + >>> ax_t.set_xlabel('Time [s]') + >>> ax_t.set_ylabel('Amplitude') + + Then plot the periodogram for each of the normalize options, as well as with and + without floating_mean=True: + + >>> ax_p.plot(w, pgram_power, label='default') + >>> ax_p.plot(w, pgram_power_f, label='floating_mean=True') + >>> ax_p.set_xlabel('Angular frequency [rad/s]') + >>> ax_p.set_ylabel('Power') + >>> ax_p.legend(prop={'size': 7}) + ... + >>> ax_n.plot(w, pgram_norm, label='default') + >>> ax_n.plot(w, pgram_norm_f, label='floating_mean=True') + >>> ax_n.set_xlabel('Angular frequency [rad/s]') + >>> ax_n.set_ylabel('Normalized') + >>> ax_n.legend(prop={'size': 7}) + ... + >>> ax_a.plot(w, np.abs(pgram_amp), label='default') + >>> ax_a.plot(w, np.abs(pgram_amp_f), label='floating_mean=True') + >>> ax_a.set_xlabel('Angular frequency [rad/s]') + >>> ax_a.set_ylabel('Amplitude') + >>> ax_a.legend(prop={'size': 7}) + ... + >>> plt.tight_layout() + >>> plt.show() + + """ + + # if no weights are provided, assume all data points are equally important + if weights is None: + weights = np.ones_like(y, dtype=np.float64) + else: + # if provided, make sure weights is an array and cast to float64 + weights = np.asarray(weights, dtype=np.float64) + + # make sure other inputs are arrays and cast to float64 + # done before validation, in case they were not arrays + x = np.asarray(x, dtype=np.float64) + y = np.asarray(y, dtype=np.float64) + freqs = np.asarray(freqs, dtype=np.float64) + + # validate input shapes + if not (x.ndim == 1 and x.size > 0 and x.shape == y.shape == weights.shape): + raise ValueError("Parameters x, y, weights must be 1-D arrays of " + "equal non-zero length!") + if not (freqs.ndim == 1 and freqs.size > 0): + raise ValueError("Parameter freqs must be a 1-D array of non-zero length!") + + # validate weights + if not (np.all(weights >= 0) and np.sum(weights) > 0): + raise ValueError("Parameter weights must have only non-negative entries " + "which sum to a positive value!") + + # validate normalize parameter + if isinstance(normalize, bool): + # if bool, convert to str literal + normalize = "normalize" if normalize else "power" + + if normalize not in ["power", "normalize", "amplitude"]: + raise ValueError( + "Normalize must be: False (or 'power'), True (or 'normalize'), " + "or 'amplitude'." + ) + + # weight vector must sum to 1 + weights *= 1.0 / weights.sum() + + # if requested, perform precenter + if precenter: + y -= y.mean() + + # transform arrays + # row vector + freqs = freqs.reshape(1, -1) + # column vectors + x = x.reshape(-1, 1) + y = y.reshape(-1, 1) + weights = weights.reshape(-1, 1) + + # store frequent intermediates + weights_y = weights * y + freqst = freqs * x + coswt = np.cos(freqst) + sinwt = np.sin(freqst) + + Y = np.dot(weights.T, y) # Eq. 7 + CC = np.dot(weights.T, coswt * coswt) # Eq. 13 + SS = 1.0 - CC # trig identity: S^2 = 1 - C^2 Eq.14 + CS = np.dot(weights.T, coswt * sinwt) # Eq. 15 + + if floating_mean: + C = np.dot(weights.T, coswt) # Eq. 8 + S = np.dot(weights.T, sinwt) # Eq. 9 + CC -= C * C # Eq. 13 + SS -= S * S # Eq. 14 + CS -= C * S # Eq. 15 + + # calculate tau (phase offset to eliminate CS variable) + tau = 0.5 * np.arctan2(2.0 * CS, CC - SS) # Eq. 19 + freqst_tau = freqst - tau + + # coswt and sinwt are now offset by tau, which eliminates CS + coswt_tau = np.cos(freqst_tau) + sinwt_tau = np.sin(freqst_tau) + + YC = np.dot(weights_y.T, coswt_tau) # Eq. 11 + YS = np.dot(weights_y.T, sinwt_tau) # Eq. 12 + CC = np.dot(weights.T, coswt_tau * coswt_tau) # Eq. 13, CC range is [0.5, 1.0] + SS = 1.0 - CC # trig identity: S^2 = 1 - C^2 Eq. 14, SS range is [0.0, 0.5] + + if floating_mean: + C = np.dot(weights.T, coswt_tau) # Eq. 8 + S = np.dot(weights.T, sinwt_tau) # Eq. 9 + YC -= Y * C # Eq. 11 + YS -= Y * S # Eq. 12 + CC -= C * C # Eq. 13, CC range is now [0.0, 1.0] + SS -= S * S # Eq. 14, SS range is now [0.0, 0.5] + + # to prevent division by zero errors with a and b, as well as correcting for + # numerical precision errors that lead to CC or SS being approximately -0.0, + # make sure CC and SS are both > 0 + epsneg = np.finfo(dtype=y.dtype).epsneg + CC[CC < epsneg] = epsneg + SS[SS < epsneg] = epsneg + + # calculate a and b + # where: y(w) = a*cos(w) + b*sin(w) + c + a = YC / CC # Eq. A.4 and 6, eliminating CS + b = YS / SS # Eq. A.4 and 6, eliminating CS + # c = Y - a * C - b * S + + # store final value as power in A^2 (i.e., (y units)^2) + pgram = 2.0 * (a * YC + b * YS) + + # squeeze back to a vector + pgram = np.squeeze(pgram) + + if normalize == "power": # (default) + # return the legacy power units ((A**2) * N/4) + + pgram *= float(x.shape[0]) / 4.0 + + elif normalize == "normalize": + # return the normalized power (power at current frequency wrt the entire signal) + # range will be [0, 1] + + YY = np.dot(weights_y.T, y) # Eq. 10 + if floating_mean: + YY -= Y * Y # Eq. 10 + + pgram *= 0.5 / np.squeeze(YY) # Eq. 20 + + else: # normalize == "amplitude": + # return the complex representation of the best-fit amplitude and phase + + # squeeze back to vectors + a = np.squeeze(a) + b = np.squeeze(b) + tau = np.squeeze(tau) + + # calculate the complex representation, and correct for tau rotation + pgram = (a + 1j * b) * np.exp(1j * tau) + + return pgram + + +def periodogram(x, fs=1.0, window='boxcar', nfft=None, detrend='constant', + return_onesided=True, scaling='density', axis=-1): + """ + Estimate power spectral density using a periodogram. + + Parameters + ---------- + x : array_like + Time series of measurement values + fs : float, optional + Sampling frequency of the `x` time series. Defaults to 1.0. + window : str or tuple or array_like, optional + Desired window to use. If `window` is a string or tuple, it is + passed to `get_window` to generate the window values, which are + DFT-even by default. See `get_window` for a list of windows and + required parameters. If `window` is array_like it will be used + directly as the window and its length must be equal to the length + of the axis over which the periodogram is computed. Defaults + to 'boxcar'. + nfft : int, optional + Length of the FFT used. If `None` the length of `x` will be + used. + detrend : str or function or `False`, optional + Specifies how to detrend each segment. If `detrend` is a + string, it is passed as the `type` argument to the `detrend` + function. If it is a function, it takes a segment and returns a + detrended segment. If `detrend` is `False`, no detrending is + done. Defaults to 'constant'. + return_onesided : bool, optional + If `True`, return a one-sided spectrum for real data. If + `False` return a two-sided spectrum. Defaults to `True`, but for + complex data, a two-sided spectrum is always returned. + scaling : { 'density', 'spectrum' }, optional + Selects between computing the power spectral density ('density') + where `Pxx` has units of V**2/Hz and computing the squared magnitude + spectrum ('spectrum') where `Pxx` has units of V**2, if `x` + is measured in V and `fs` is measured in Hz. Defaults to + 'density' + axis : int, optional + Axis along which the periodogram is computed; the default is + over the last axis (i.e. ``axis=-1``). + + Returns + ------- + f : ndarray + Array of sample frequencies. + Pxx : ndarray + Power spectral density or power spectrum of `x`. + + See Also + -------- + welch: Estimate power spectral density using Welch's method + lombscargle: Lomb-Scargle periodogram for unevenly sampled data + + Notes + ----- + Consult the :ref:`tutorial_SpectralAnalysis` section of the :ref:`user_guide` + for a discussion of the scalings of the power spectral density and + the magnitude (squared) spectrum. + + .. versionadded:: 0.12.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + Generate a test signal, a 2 Vrms sine wave at 1234 Hz, corrupted by + 0.001 V**2/Hz of white noise sampled at 10 kHz. + + >>> fs = 10e3 + >>> N = 1e5 + >>> amp = 2*np.sqrt(2) + >>> freq = 1234.0 + >>> noise_power = 0.001 * fs / 2 + >>> time = np.arange(N) / fs + >>> x = amp*np.sin(2*np.pi*freq*time) + >>> x += rng.normal(scale=np.sqrt(noise_power), size=time.shape) + + Compute and plot the power spectral density. + + >>> f, Pxx_den = signal.periodogram(x, fs) + >>> plt.semilogy(f, Pxx_den) + >>> plt.ylim([1e-7, 1e2]) + >>> plt.xlabel('frequency [Hz]') + >>> plt.ylabel('PSD [V**2/Hz]') + >>> plt.show() + + If we average the last half of the spectral density, to exclude the + peak, we can recover the noise power on the signal. + + >>> np.mean(Pxx_den[25000:]) + 0.000985320699252543 + + Now compute and plot the power spectrum. + + >>> f, Pxx_spec = signal.periodogram(x, fs, 'flattop', scaling='spectrum') + >>> plt.figure() + >>> plt.semilogy(f, np.sqrt(Pxx_spec)) + >>> plt.ylim([1e-4, 1e1]) + >>> plt.xlabel('frequency [Hz]') + >>> plt.ylabel('Linear spectrum [V RMS]') + >>> plt.show() + + The peak height in the power spectrum is an estimate of the RMS + amplitude. + + >>> np.sqrt(Pxx_spec.max()) + 2.0077340678640727 + + """ + x = np.asarray(x) + + if x.size == 0: + return np.empty(x.shape), np.empty(x.shape) + + if window is None: + window = 'boxcar' + + if nfft is None: + nperseg = x.shape[axis] + elif nfft == x.shape[axis]: + nperseg = nfft + elif nfft > x.shape[axis]: + nperseg = x.shape[axis] + elif nfft < x.shape[axis]: + s = [np.s_[:]]*len(x.shape) + s[axis] = np.s_[:nfft] + x = x[tuple(s)] + nperseg = nfft + nfft = None + + if hasattr(window, 'size'): + if window.size != nperseg: + raise ValueError('the size of the window must be the same size ' + 'of the input on the specified axis') + + return welch(x, fs=fs, window=window, nperseg=nperseg, noverlap=0, + nfft=nfft, detrend=detrend, return_onesided=return_onesided, + scaling=scaling, axis=axis) + + +def welch(x, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None, + detrend='constant', return_onesided=True, scaling='density', + axis=-1, average='mean'): + r""" + Estimate power spectral density using Welch's method. + + Welch's method [1]_ computes an estimate of the power spectral + density by dividing the data into overlapping segments, computing a + modified periodogram for each segment and averaging the + periodograms. + + Parameters + ---------- + x : array_like + Time series of measurement values + fs : float, optional + Sampling frequency of the `x` time series. Defaults to 1.0. + window : str or tuple or array_like, optional + Desired window to use. If `window` is a string or tuple, it is + passed to `get_window` to generate the window values, which are + DFT-even by default. See `get_window` for a list of windows and + required parameters. If `window` is array_like it will be used + directly as the window and its length must be nperseg. Defaults + to a Hann window. + nperseg : int, optional + Length of each segment. Defaults to None, but if window is str or + tuple, is set to 256, and if window is array_like, is set to the + length of the window. + noverlap : int, optional + Number of points to overlap between segments. If `None`, + ``noverlap = nperseg // 2``. Defaults to `None`. + nfft : int, optional + Length of the FFT used, if a zero padded FFT is desired. If + `None`, the FFT length is `nperseg`. Defaults to `None`. + detrend : str or function or `False`, optional + Specifies how to detrend each segment. If `detrend` is a + string, it is passed as the `type` argument to the `detrend` + function. If it is a function, it takes a segment and returns a + detrended segment. If `detrend` is `False`, no detrending is + done. Defaults to 'constant'. + return_onesided : bool, optional + If `True`, return a one-sided spectrum for real data. If + `False` return a two-sided spectrum. Defaults to `True`, but for + complex data, a two-sided spectrum is always returned. + scaling : { 'density', 'spectrum' }, optional + Selects between computing the power spectral density ('density') + where `Pxx` has units of V**2/Hz and computing the squared magnitude + spectrum ('spectrum') where `Pxx` has units of V**2, if `x` + is measured in V and `fs` is measured in Hz. Defaults to + 'density' + axis : int, optional + Axis along which the periodogram is computed; the default is + over the last axis (i.e. ``axis=-1``). + average : { 'mean', 'median' }, optional + Method to use when averaging periodograms. Defaults to 'mean'. + + .. versionadded:: 1.2.0 + + Returns + ------- + f : ndarray + Array of sample frequencies. + Pxx : ndarray + Power spectral density or power spectrum of x. + + See Also + -------- + periodogram: Simple, optionally modified periodogram + lombscargle: Lomb-Scargle periodogram for unevenly sampled data + + Notes + ----- + An appropriate amount of overlap will depend on the choice of window + and on your requirements. For the default Hann window an overlap of + 50% is a reasonable trade off between accurately estimating the + signal power, while not over counting any of the data. Narrower + windows may require a larger overlap. + + If `noverlap` is 0, this method is equivalent to Bartlett's method + [2]_. + + Consult the :ref:`tutorial_SpectralAnalysis` section of the :ref:`user_guide` + for a discussion of the scalings of the power spectral density and + the (squared) magnitude spectrum. + + .. versionadded:: 0.12.0 + + References + ---------- + .. [1] P. Welch, "The use of the fast Fourier transform for the + estimation of power spectra: A method based on time averaging + over short, modified periodograms", IEEE Trans. Audio + Electroacoust. vol. 15, pp. 70-73, 1967. + .. [2] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra", + Biometrika, vol. 37, pp. 1-16, 1950. + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + Generate a test signal, a 2 Vrms sine wave at 1234 Hz, corrupted by + 0.001 V**2/Hz of white noise sampled at 10 kHz. + + >>> fs = 10e3 + >>> N = 1e5 + >>> amp = 2*np.sqrt(2) + >>> freq = 1234.0 + >>> noise_power = 0.001 * fs / 2 + >>> time = np.arange(N) / fs + >>> x = amp*np.sin(2*np.pi*freq*time) + >>> x += rng.normal(scale=np.sqrt(noise_power), size=time.shape) + + Compute and plot the power spectral density. + + >>> f, Pxx_den = signal.welch(x, fs, nperseg=1024) + >>> plt.semilogy(f, Pxx_den) + >>> plt.ylim([0.5e-3, 1]) + >>> plt.xlabel('frequency [Hz]') + >>> plt.ylabel('PSD [V**2/Hz]') + >>> plt.show() + + If we average the last half of the spectral density, to exclude the + peak, we can recover the noise power on the signal. + + >>> np.mean(Pxx_den[256:]) + 0.0009924865443739191 + + Now compute and plot the power spectrum. + + >>> f, Pxx_spec = signal.welch(x, fs, 'flattop', 1024, scaling='spectrum') + >>> plt.figure() + >>> plt.semilogy(f, np.sqrt(Pxx_spec)) + >>> plt.xlabel('frequency [Hz]') + >>> plt.ylabel('Linear spectrum [V RMS]') + >>> plt.show() + + The peak height in the power spectrum is an estimate of the RMS + amplitude. + + >>> np.sqrt(Pxx_spec.max()) + 2.0077340678640727 + + If we now introduce a discontinuity in the signal, by increasing the + amplitude of a small portion of the signal by 50, we can see the + corruption of the mean average power spectral density, but using a + median average better estimates the normal behaviour. + + >>> x[int(N//2):int(N//2)+10] *= 50. + >>> f, Pxx_den = signal.welch(x, fs, nperseg=1024) + >>> f_med, Pxx_den_med = signal.welch(x, fs, nperseg=1024, average='median') + >>> plt.semilogy(f, Pxx_den, label='mean') + >>> plt.semilogy(f_med, Pxx_den_med, label='median') + >>> plt.ylim([0.5e-3, 1]) + >>> plt.xlabel('frequency [Hz]') + >>> plt.ylabel('PSD [V**2/Hz]') + >>> plt.legend() + >>> plt.show() + + """ + freqs, Pxx = csd(x, x, fs=fs, window=window, nperseg=nperseg, + noverlap=noverlap, nfft=nfft, detrend=detrend, + return_onesided=return_onesided, scaling=scaling, + axis=axis, average=average) + + return freqs, Pxx.real + + +def csd(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None, + detrend='constant', return_onesided=True, scaling='density', + axis=-1, average='mean'): + r""" + Estimate the cross power spectral density, Pxy, using Welch's method. + + Parameters + ---------- + x : array_like + Time series of measurement values + y : array_like + Time series of measurement values + fs : float, optional + Sampling frequency of the `x` and `y` time series. Defaults + to 1.0. + window : str or tuple or array_like, optional + Desired window to use. If `window` is a string or tuple, it is + passed to `get_window` to generate the window values, which are + DFT-even by default. See `get_window` for a list of windows and + required parameters. If `window` is array_like it will be used + directly as the window and its length must be nperseg. Defaults + to a Hann window. + nperseg : int, optional + Length of each segment. Defaults to None, but if window is str or + tuple, is set to 256, and if window is array_like, is set to the + length of the window. + noverlap: int, optional + Number of points to overlap between segments. If `None`, + ``noverlap = nperseg // 2``. Defaults to `None`. + nfft : int, optional + Length of the FFT used, if a zero padded FFT is desired. If + `None`, the FFT length is `nperseg`. Defaults to `None`. + detrend : str or function or `False`, optional + Specifies how to detrend each segment. If `detrend` is a + string, it is passed as the `type` argument to the `detrend` + function. If it is a function, it takes a segment and returns a + detrended segment. If `detrend` is `False`, no detrending is + done. Defaults to 'constant'. + return_onesided : bool, optional + If `True`, return a one-sided spectrum for real data. If + `False` return a two-sided spectrum. Defaults to `True`, but for + complex data, a two-sided spectrum is always returned. + scaling : { 'density', 'spectrum' }, optional + Selects between computing the cross spectral density ('density') + where `Pxy` has units of V**2/Hz and computing the cross spectrum + ('spectrum') where `Pxy` has units of V**2, if `x` and `y` are + measured in V and `fs` is measured in Hz. Defaults to 'density' + axis : int, optional + Axis along which the CSD is computed for both inputs; the + default is over the last axis (i.e. ``axis=-1``). + average : { 'mean', 'median' }, optional + Method to use when averaging periodograms. If the spectrum is + complex, the average is computed separately for the real and + imaginary parts. Defaults to 'mean'. + + .. versionadded:: 1.2.0 + + Returns + ------- + f : ndarray + Array of sample frequencies. + Pxy : ndarray + Cross spectral density or cross power spectrum of x,y. + + See Also + -------- + periodogram: Simple, optionally modified periodogram + lombscargle: Lomb-Scargle periodogram for unevenly sampled data + welch: Power spectral density by Welch's method. [Equivalent to + csd(x,x)] + coherence: Magnitude squared coherence by Welch's method. + + Notes + ----- + By convention, Pxy is computed with the conjugate FFT of X + multiplied by the FFT of Y. + + If the input series differ in length, the shorter series will be + zero-padded to match. + + An appropriate amount of overlap will depend on the choice of window + and on your requirements. For the default Hann window an overlap of + 50% is a reasonable trade off between accurately estimating the + signal power, while not over counting any of the data. Narrower + windows may require a larger overlap. + + Consult the :ref:`tutorial_SpectralAnalysis` section of the :ref:`user_guide` + for a discussion of the scalings of a spectral density and an (amplitude) spectrum. + + .. versionadded:: 0.16.0 + + References + ---------- + .. [1] P. Welch, "The use of the fast Fourier transform for the + estimation of power spectra: A method based on time averaging + over short, modified periodograms", IEEE Trans. Audio + Electroacoust. vol. 15, pp. 70-73, 1967. + .. [2] Rabiner, Lawrence R., and B. Gold. "Theory and Application of + Digital Signal Processing" Prentice-Hall, pp. 414-419, 1975 + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + Generate two test signals with some common features. + + >>> fs = 10e3 + >>> N = 1e5 + >>> amp = 20 + >>> freq = 1234.0 + >>> noise_power = 0.001 * fs / 2 + >>> time = np.arange(N) / fs + >>> b, a = signal.butter(2, 0.25, 'low') + >>> x = rng.normal(scale=np.sqrt(noise_power), size=time.shape) + >>> y = signal.lfilter(b, a, x) + >>> x += amp*np.sin(2*np.pi*freq*time) + >>> y += rng.normal(scale=0.1*np.sqrt(noise_power), size=time.shape) + + Compute and plot the magnitude of the cross spectral density. + + >>> f, Pxy = signal.csd(x, y, fs, nperseg=1024) + >>> plt.semilogy(f, np.abs(Pxy)) + >>> plt.xlabel('frequency [Hz]') + >>> plt.ylabel('CSD [V**2/Hz]') + >>> plt.show() + + """ + freqs, _, Pxy = _spectral_helper(x, y, fs, window, nperseg, noverlap, + nfft, detrend, return_onesided, scaling, + axis, mode='psd') + + # Average over windows. + if len(Pxy.shape) >= 2 and Pxy.size > 0: + if Pxy.shape[-1] > 1: + if average == 'median': + # np.median must be passed real arrays for the desired result + bias = _median_bias(Pxy.shape[-1]) + if np.iscomplexobj(Pxy): + Pxy = (np.median(np.real(Pxy), axis=-1) + + 1j * np.median(np.imag(Pxy), axis=-1)) + else: + Pxy = np.median(Pxy, axis=-1) + Pxy /= bias + elif average == 'mean': + Pxy = Pxy.mean(axis=-1) + else: + raise ValueError(f'average must be "median" or "mean", got {average}') + else: + Pxy = np.reshape(Pxy, Pxy.shape[:-1]) + + return freqs, Pxy + + +def spectrogram(x, fs=1.0, window=('tukey', .25), nperseg=None, noverlap=None, + nfft=None, detrend='constant', return_onesided=True, + scaling='density', axis=-1, mode='psd'): + """Compute a spectrogram with consecutive Fourier transforms (legacy function). + + Spectrograms can be used as a way of visualizing the change of a + nonstationary signal's frequency content over time. + + .. legacy:: function + + :class:`ShortTimeFFT` is a newer STFT / ISTFT implementation with more + features also including a :meth:`~ShortTimeFFT.spectrogram` method. + A :ref:`comparison ` between the + implementations can be found in the :ref:`tutorial_stft` section of + the :ref:`user_guide`. + + Parameters + ---------- + x : array_like + Time series of measurement values + fs : float, optional + Sampling frequency of the `x` time series. Defaults to 1.0. + window : str or tuple or array_like, optional + Desired window to use. If `window` is a string or tuple, it is + passed to `get_window` to generate the window values, which are + DFT-even by default. See `get_window` for a list of windows and + required parameters. If `window` is array_like it will be used + directly as the window and its length must be nperseg. + Defaults to a Tukey window with shape parameter of 0.25. + nperseg : int, optional + Length of each segment. Defaults to None, but if window is str or + tuple, is set to 256, and if window is array_like, is set to the + length of the window. + noverlap : int, optional + Number of points to overlap between segments. If `None`, + ``noverlap = nperseg // 8``. Defaults to `None`. + nfft : int, optional + Length of the FFT used, if a zero padded FFT is desired. If + `None`, the FFT length is `nperseg`. Defaults to `None`. + detrend : str or function or `False`, optional + Specifies how to detrend each segment. If `detrend` is a + string, it is passed as the `type` argument to the `detrend` + function. If it is a function, it takes a segment and returns a + detrended segment. If `detrend` is `False`, no detrending is + done. Defaults to 'constant'. + return_onesided : bool, optional + If `True`, return a one-sided spectrum for real data. If + `False` return a two-sided spectrum. Defaults to `True`, but for + complex data, a two-sided spectrum is always returned. + scaling : { 'density', 'spectrum' }, optional + Selects between computing the power spectral density ('density') + where `Sxx` has units of V**2/Hz and computing the power + spectrum ('spectrum') where `Sxx` has units of V**2, if `x` + is measured in V and `fs` is measured in Hz. Defaults to + 'density'. + axis : int, optional + Axis along which the spectrogram is computed; the default is over + the last axis (i.e. ``axis=-1``). + mode : str, optional + Defines what kind of return values are expected. Options are + ['psd', 'complex', 'magnitude', 'angle', 'phase']. 'complex' is + equivalent to the output of `stft` with no padding or boundary + extension. 'magnitude' returns the absolute magnitude of the + STFT. 'angle' and 'phase' return the complex angle of the STFT, + with and without unwrapping, respectively. + + Returns + ------- + f : ndarray + Array of sample frequencies. + t : ndarray + Array of segment times. + Sxx : ndarray + Spectrogram of x. By default, the last axis of Sxx corresponds + to the segment times. + + See Also + -------- + periodogram: Simple, optionally modified periodogram + lombscargle: Lomb-Scargle periodogram for unevenly sampled data + welch: Power spectral density by Welch's method. + csd: Cross spectral density by Welch's method. + ShortTimeFFT: Newer STFT/ISTFT implementation providing more features, + which also includes a :meth:`~ShortTimeFFT.spectrogram` + method. + + Notes + ----- + An appropriate amount of overlap will depend on the choice of window + and on your requirements. In contrast to welch's method, where the + entire data stream is averaged over, one may wish to use a smaller + overlap (or perhaps none at all) when computing a spectrogram, to + maintain some statistical independence between individual segments. + It is for this reason that the default window is a Tukey window with + 1/8th of a window's length overlap at each end. + + + .. versionadded:: 0.16.0 + + References + ---------- + .. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck + "Discrete-Time Signal Processing", Prentice Hall, 1999. + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fftshift + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + Generate a test signal, a 2 Vrms sine wave whose frequency is slowly + modulated around 3kHz, corrupted by white noise of exponentially + decreasing magnitude sampled at 10 kHz. + + >>> fs = 10e3 + >>> N = 1e5 + >>> amp = 2 * np.sqrt(2) + >>> noise_power = 0.01 * fs / 2 + >>> time = np.arange(N) / float(fs) + >>> mod = 500*np.cos(2*np.pi*0.25*time) + >>> carrier = amp * np.sin(2*np.pi*3e3*time + mod) + >>> noise = rng.normal(scale=np.sqrt(noise_power), size=time.shape) + >>> noise *= np.exp(-time/5) + >>> x = carrier + noise + + Compute and plot the spectrogram. + + >>> f, t, Sxx = signal.spectrogram(x, fs) + >>> plt.pcolormesh(t, f, Sxx, shading='gouraud') + >>> plt.ylabel('Frequency [Hz]') + >>> plt.xlabel('Time [sec]') + >>> plt.show() + + Note, if using output that is not one sided, then use the following: + + >>> f, t, Sxx = signal.spectrogram(x, fs, return_onesided=False) + >>> plt.pcolormesh(t, fftshift(f), fftshift(Sxx, axes=0), shading='gouraud') + >>> plt.ylabel('Frequency [Hz]') + >>> plt.xlabel('Time [sec]') + >>> plt.show() + + """ + modelist = ['psd', 'complex', 'magnitude', 'angle', 'phase'] + if mode not in modelist: + raise ValueError(f'unknown value for mode {mode}, must be one of {modelist}') + + # need to set default for nperseg before setting default for noverlap below + window, nperseg = _triage_segments(window, nperseg, + input_length=x.shape[axis]) + + # Less overlap than welch, so samples are more statistically independent + if noverlap is None: + noverlap = nperseg // 8 + + if mode == 'psd': + freqs, time, Sxx = _spectral_helper(x, x, fs, window, nperseg, + noverlap, nfft, detrend, + return_onesided, scaling, axis, + mode='psd') + + else: + freqs, time, Sxx = _spectral_helper(x, x, fs, window, nperseg, + noverlap, nfft, detrend, + return_onesided, scaling, axis, + mode='stft') + + if mode == 'magnitude': + Sxx = np.abs(Sxx) + elif mode in ['angle', 'phase']: + Sxx = np.angle(Sxx) + if mode == 'phase': + # Sxx has one additional dimension for time strides + if axis < 0: + axis -= 1 + Sxx = np.unwrap(Sxx, axis=axis) + + # mode =='complex' is same as `stft`, doesn't need modification + + return freqs, time, Sxx + + +def check_COLA(window, nperseg, noverlap, tol=1e-10): + r"""Check whether the Constant OverLap Add (COLA) constraint is met. + + Parameters + ---------- + window : str or tuple or array_like + Desired window to use. If `window` is a string or tuple, it is + passed to `get_window` to generate the window values, which are + DFT-even by default. See `get_window` for a list of windows and + required parameters. If `window` is array_like it will be used + directly as the window and its length must be nperseg. + nperseg : int + Length of each segment. + noverlap : int + Number of points to overlap between segments. + tol : float, optional + The allowed variance of a bin's weighted sum from the median bin + sum. + + Returns + ------- + verdict : bool + `True` if chosen combination satisfies COLA within `tol`, + `False` otherwise + + See Also + -------- + check_NOLA: Check whether the Nonzero Overlap Add (NOLA) constraint is met + stft: Short Time Fourier Transform + istft: Inverse Short Time Fourier Transform + + Notes + ----- + In order to enable inversion of an STFT via the inverse STFT in + `istft`, it is sufficient that the signal windowing obeys the constraint of + "Constant OverLap Add" (COLA). This ensures that every point in the input + data is equally weighted, thereby avoiding aliasing and allowing full + reconstruction. + + Some examples of windows that satisfy COLA: + - Rectangular window at overlap of 0, 1/2, 2/3, 3/4, ... + - Bartlett window at overlap of 1/2, 3/4, 5/6, ... + - Hann window at 1/2, 2/3, 3/4, ... + - Any Blackman family window at 2/3 overlap + - Any window with ``noverlap = nperseg-1`` + + A very comprehensive list of other windows may be found in [2]_, + wherein the COLA condition is satisfied when the "Amplitude + Flatness" is unity. + + .. versionadded:: 0.19.0 + + References + ---------- + .. [1] Julius O. Smith III, "Spectral Audio Signal Processing", W3K + Publishing, 2011,ISBN 978-0-9745607-3-1. + .. [2] G. Heinzel, A. Ruediger and R. Schilling, "Spectrum and + spectral density estimation by the Discrete Fourier transform + (DFT), including a comprehensive list of window functions and + some new at-top windows", 2002, + http://hdl.handle.net/11858/00-001M-0000-0013-557A-5 + + Examples + -------- + >>> from scipy import signal + + Confirm COLA condition for rectangular window of 75% (3/4) overlap: + + >>> signal.check_COLA(signal.windows.boxcar(100), 100, 75) + True + + COLA is not true for 25% (1/4) overlap, though: + + >>> signal.check_COLA(signal.windows.boxcar(100), 100, 25) + False + + "Symmetrical" Hann window (for filter design) is not COLA: + + >>> signal.check_COLA(signal.windows.hann(120, sym=True), 120, 60) + False + + "Periodic" or "DFT-even" Hann window (for FFT analysis) is COLA for + overlap of 1/2, 2/3, 3/4, etc.: + + >>> signal.check_COLA(signal.windows.hann(120, sym=False), 120, 60) + True + + >>> signal.check_COLA(signal.windows.hann(120, sym=False), 120, 80) + True + + >>> signal.check_COLA(signal.windows.hann(120, sym=False), 120, 90) + True + + """ + nperseg = int(nperseg) + + if nperseg < 1: + raise ValueError('nperseg must be a positive integer') + + if noverlap >= nperseg: + raise ValueError('noverlap must be less than nperseg.') + noverlap = int(noverlap) + + if isinstance(window, str) or type(window) is tuple: + win = get_window(window, nperseg) + else: + win = np.asarray(window) + if len(win.shape) != 1: + raise ValueError('window must be 1-D') + if win.shape[0] != nperseg: + raise ValueError('window must have length of nperseg') + + step = nperseg - noverlap + binsums = sum(win[ii*step:(ii+1)*step] for ii in range(nperseg//step)) + + if nperseg % step != 0: + binsums[:nperseg % step] += win[-(nperseg % step):] + + deviation = binsums - np.median(binsums) + return np.max(np.abs(deviation)) < tol + + +def check_NOLA(window, nperseg, noverlap, tol=1e-10): + r"""Check whether the Nonzero Overlap Add (NOLA) constraint is met. + + Parameters + ---------- + window : str or tuple or array_like + Desired window to use. If `window` is a string or tuple, it is + passed to `get_window` to generate the window values, which are + DFT-even by default. See `get_window` for a list of windows and + required parameters. If `window` is array_like it will be used + directly as the window and its length must be nperseg. + nperseg : int + Length of each segment. + noverlap : int + Number of points to overlap between segments. + tol : float, optional + The allowed variance of a bin's weighted sum from the median bin + sum. + + Returns + ------- + verdict : bool + `True` if chosen combination satisfies the NOLA constraint within + `tol`, `False` otherwise + + See Also + -------- + check_COLA: Check whether the Constant OverLap Add (COLA) constraint is met + stft: Short Time Fourier Transform + istft: Inverse Short Time Fourier Transform + + Notes + ----- + In order to enable inversion of an STFT via the inverse STFT in + `istft`, the signal windowing must obey the constraint of "nonzero + overlap add" (NOLA): + + .. math:: \sum_{t}w^{2}[n-tH] \ne 0 + + for all :math:`n`, where :math:`w` is the window function, :math:`t` is the + frame index, and :math:`H` is the hop size (:math:`H` = `nperseg` - + `noverlap`). + + This ensures that the normalization factors in the denominator of the + overlap-add inversion equation are not zero. Only very pathological windows + will fail the NOLA constraint. + + .. versionadded:: 1.2.0 + + References + ---------- + .. [1] Julius O. Smith III, "Spectral Audio Signal Processing", W3K + Publishing, 2011,ISBN 978-0-9745607-3-1. + .. [2] G. Heinzel, A. Ruediger and R. Schilling, "Spectrum and + spectral density estimation by the Discrete Fourier transform + (DFT), including a comprehensive list of window functions and + some new at-top windows", 2002, + http://hdl.handle.net/11858/00-001M-0000-0013-557A-5 + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + + Confirm NOLA condition for rectangular window of 75% (3/4) overlap: + + >>> signal.check_NOLA(signal.windows.boxcar(100), 100, 75) + True + + NOLA is also true for 25% (1/4) overlap: + + >>> signal.check_NOLA(signal.windows.boxcar(100), 100, 25) + True + + "Symmetrical" Hann window (for filter design) is also NOLA: + + >>> signal.check_NOLA(signal.windows.hann(120, sym=True), 120, 60) + True + + As long as there is overlap, it takes quite a pathological window to fail + NOLA: + + >>> w = np.ones(64, dtype="float") + >>> w[::2] = 0 + >>> signal.check_NOLA(w, 64, 32) + False + + If there is not enough overlap, a window with zeros at the ends will not + work: + + >>> signal.check_NOLA(signal.windows.hann(64), 64, 0) + False + >>> signal.check_NOLA(signal.windows.hann(64), 64, 1) + False + >>> signal.check_NOLA(signal.windows.hann(64), 64, 2) + True + + """ + nperseg = int(nperseg) + + if nperseg < 1: + raise ValueError('nperseg must be a positive integer') + + if noverlap >= nperseg: + raise ValueError('noverlap must be less than nperseg') + if noverlap < 0: + raise ValueError('noverlap must be a nonnegative integer') + noverlap = int(noverlap) + + if isinstance(window, str) or type(window) is tuple: + win = get_window(window, nperseg) + else: + win = np.asarray(window) + if len(win.shape) != 1: + raise ValueError('window must be 1-D') + if win.shape[0] != nperseg: + raise ValueError('window must have length of nperseg') + + step = nperseg - noverlap + binsums = sum(win[ii*step:(ii+1)*step]**2 for ii in range(nperseg//step)) + + if nperseg % step != 0: + binsums[:nperseg % step] += win[-(nperseg % step):]**2 + + return np.min(binsums) > tol + + +def stft(x, fs=1.0, window='hann', nperseg=256, noverlap=None, nfft=None, + detrend=False, return_onesided=True, boundary='zeros', padded=True, + axis=-1, scaling='spectrum'): + r"""Compute the Short Time Fourier Transform (legacy function). + + STFTs can be used as a way of quantifying the change of a + nonstationary signal's frequency and phase content over time. + + .. legacy:: function + + `ShortTimeFFT` is a newer STFT / ISTFT implementation with more + features. A :ref:`comparison ` between the + implementations can be found in the :ref:`tutorial_stft` section of the + :ref:`user_guide`. + + Parameters + ---------- + x : array_like + Time series of measurement values + fs : float, optional + Sampling frequency of the `x` time series. Defaults to 1.0. + window : str or tuple or array_like, optional + Desired window to use. If `window` is a string or tuple, it is + passed to `get_window` to generate the window values, which are + DFT-even by default. See `get_window` for a list of windows and + required parameters. If `window` is array_like it will be used + directly as the window and its length must be nperseg. Defaults + to a Hann window. + nperseg : int, optional + Length of each segment. Defaults to 256. + noverlap : int, optional + Number of points to overlap between segments. If `None`, + ``noverlap = nperseg // 2``. Defaults to `None`. When + specified, the COLA constraint must be met (see Notes below). + nfft : int, optional + Length of the FFT used, if a zero padded FFT is desired. If + `None`, the FFT length is `nperseg`. Defaults to `None`. + detrend : str or function or `False`, optional + Specifies how to detrend each segment. If `detrend` is a + string, it is passed as the `type` argument to the `detrend` + function. If it is a function, it takes a segment and returns a + detrended segment. If `detrend` is `False`, no detrending is + done. Defaults to `False`. + return_onesided : bool, optional + If `True`, return a one-sided spectrum for real data. If + `False` return a two-sided spectrum. Defaults to `True`, but for + complex data, a two-sided spectrum is always returned. + boundary : str or None, optional + Specifies whether the input signal is extended at both ends, and + how to generate the new values, in order to center the first + windowed segment on the first input point. This has the benefit + of enabling reconstruction of the first input point when the + employed window function starts at zero. Valid options are + ``['even', 'odd', 'constant', 'zeros', None]``. Defaults to + 'zeros', for zero padding extension. I.e. ``[1, 2, 3, 4]`` is + extended to ``[0, 1, 2, 3, 4, 0]`` for ``nperseg=3``. + padded : bool, optional + Specifies whether the input signal is zero-padded at the end to + make the signal fit exactly into an integer number of window + segments, so that all of the signal is included in the output. + Defaults to `True`. Padding occurs after boundary extension, if + `boundary` is not `None`, and `padded` is `True`, as is the + default. + axis : int, optional + Axis along which the STFT is computed; the default is over the + last axis (i.e. ``axis=-1``). + scaling: {'spectrum', 'psd'} + The default 'spectrum' scaling allows each frequency line of `Zxx` to + be interpreted as a magnitude spectrum. The 'psd' option scales each + line to a power spectral density - it allows to calculate the signal's + energy by numerically integrating over ``abs(Zxx)**2``. + + .. versionadded:: 1.9.0 + + Returns + ------- + f : ndarray + Array of sample frequencies. + t : ndarray + Array of segment times. + Zxx : ndarray + STFT of `x`. By default, the last axis of `Zxx` corresponds + to the segment times. + + See Also + -------- + istft: Inverse Short Time Fourier Transform + ShortTimeFFT: Newer STFT/ISTFT implementation providing more features. + check_COLA: Check whether the Constant OverLap Add (COLA) constraint + is met + check_NOLA: Check whether the Nonzero Overlap Add (NOLA) constraint is met + welch: Power spectral density by Welch's method. + spectrogram: Spectrogram by Welch's method. + csd: Cross spectral density by Welch's method. + lombscargle: Lomb-Scargle periodogram for unevenly sampled data + + Notes + ----- + In order to enable inversion of an STFT via the inverse STFT in + `istft`, the signal windowing must obey the constraint of "Nonzero + OverLap Add" (NOLA), and the input signal must have complete + windowing coverage (i.e. ``(x.shape[axis] - nperseg) % + (nperseg-noverlap) == 0``). The `padded` argument may be used to + accomplish this. + + Given a time-domain signal :math:`x[n]`, a window :math:`w[n]`, and a hop + size :math:`H` = `nperseg - noverlap`, the windowed frame at time index + :math:`t` is given by + + .. math:: x_{t}[n]=x[n]w[n-tH] + + The overlap-add (OLA) reconstruction equation is given by + + .. math:: x[n]=\frac{\sum_{t}x_{t}[n]w[n-tH]}{\sum_{t}w^{2}[n-tH]} + + The NOLA constraint ensures that every normalization term that appears + in the denominator of the OLA reconstruction equation is nonzero. Whether a + choice of `window`, `nperseg`, and `noverlap` satisfy this constraint can + be tested with `check_NOLA`. + + + .. versionadded:: 0.19.0 + + References + ---------- + .. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck + "Discrete-Time Signal Processing", Prentice Hall, 1999. + .. [2] Daniel W. Griffin, Jae S. Lim "Signal Estimation from + Modified Short-Time Fourier Transform", IEEE 1984, + 10.1109/TASSP.1984.1164317 + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + Generate a test signal, a 2 Vrms sine wave whose frequency is slowly + modulated around 3kHz, corrupted by white noise of exponentially + decreasing magnitude sampled at 10 kHz. + + >>> fs = 10e3 + >>> N = 1e5 + >>> amp = 2 * np.sqrt(2) + >>> noise_power = 0.01 * fs / 2 + >>> time = np.arange(N) / float(fs) + >>> mod = 500*np.cos(2*np.pi*0.25*time) + >>> carrier = amp * np.sin(2*np.pi*3e3*time + mod) + >>> noise = rng.normal(scale=np.sqrt(noise_power), + ... size=time.shape) + >>> noise *= np.exp(-time/5) + >>> x = carrier + noise + + Compute and plot the STFT's magnitude. + + >>> f, t, Zxx = signal.stft(x, fs, nperseg=1000) + >>> plt.pcolormesh(t, f, np.abs(Zxx), vmin=0, vmax=amp, shading='gouraud') + >>> plt.title('STFT Magnitude') + >>> plt.ylabel('Frequency [Hz]') + >>> plt.xlabel('Time [sec]') + >>> plt.show() + + Compare the energy of the signal `x` with the energy of its STFT: + + >>> E_x = sum(x**2) / fs # Energy of x + >>> # Calculate a two-sided STFT with PSD scaling: + >>> f, t, Zxx = signal.stft(x, fs, nperseg=1000, return_onesided=False, + ... scaling='psd') + >>> # Integrate numerically over abs(Zxx)**2: + >>> df, dt = f[1] - f[0], t[1] - t[0] + >>> E_Zxx = sum(np.sum(Zxx.real**2 + Zxx.imag**2, axis=0) * df) * dt + >>> # The energy is the same, but the numerical errors are quite large: + >>> np.isclose(E_x, E_Zxx, rtol=1e-2) + True + + """ + if scaling == 'psd': + scaling = 'density' + elif scaling != 'spectrum': + raise ValueError(f"Parameter {scaling=} not in ['spectrum', 'psd']!") + + freqs, time, Zxx = _spectral_helper(x, x, fs, window, nperseg, noverlap, + nfft, detrend, return_onesided, + scaling=scaling, axis=axis, + mode='stft', boundary=boundary, + padded=padded) + + return freqs, time, Zxx + + +def istft(Zxx, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None, + input_onesided=True, boundary=True, time_axis=-1, freq_axis=-2, + scaling='spectrum'): + r"""Perform the inverse Short Time Fourier transform (legacy function). + + .. legacy:: function + + `ShortTimeFFT` is a newer STFT / ISTFT implementation with more + features. A :ref:`comparison ` between the + implementations can be found in the :ref:`tutorial_stft` section of the + :ref:`user_guide`. + + Parameters + ---------- + Zxx : array_like + STFT of the signal to be reconstructed. If a purely real array + is passed, it will be cast to a complex data type. + fs : float, optional + Sampling frequency of the time series. Defaults to 1.0. + window : str or tuple or array_like, optional + Desired window to use. If `window` is a string or tuple, it is + passed to `get_window` to generate the window values, which are + DFT-even by default. See `get_window` for a list of windows and + required parameters. If `window` is array_like it will be used + directly as the window and its length must be nperseg. Defaults + to a Hann window. Must match the window used to generate the + STFT for faithful inversion. + nperseg : int, optional + Number of data points corresponding to each STFT segment. This + parameter must be specified if the number of data points per + segment is odd, or if the STFT was padded via ``nfft > + nperseg``. If `None`, the value depends on the shape of + `Zxx` and `input_onesided`. If `input_onesided` is `True`, + ``nperseg=2*(Zxx.shape[freq_axis] - 1)``. Otherwise, + ``nperseg=Zxx.shape[freq_axis]``. Defaults to `None`. + noverlap : int, optional + Number of points to overlap between segments. If `None`, half + of the segment length. Defaults to `None`. When specified, the + COLA constraint must be met (see Notes below), and should match + the parameter used to generate the STFT. Defaults to `None`. + nfft : int, optional + Number of FFT points corresponding to each STFT segment. This + parameter must be specified if the STFT was padded via ``nfft > + nperseg``. If `None`, the default values are the same as for + `nperseg`, detailed above, with one exception: if + `input_onesided` is True and + ``nperseg==2*Zxx.shape[freq_axis] - 1``, `nfft` also takes on + that value. This case allows the proper inversion of an + odd-length unpadded STFT using ``nfft=None``. Defaults to + `None`. + input_onesided : bool, optional + If `True`, interpret the input array as one-sided FFTs, such + as is returned by `stft` with ``return_onesided=True`` and + `numpy.fft.rfft`. If `False`, interpret the input as a a + two-sided FFT. Defaults to `True`. + boundary : bool, optional + Specifies whether the input signal was extended at its + boundaries by supplying a non-`None` ``boundary`` argument to + `stft`. Defaults to `True`. + time_axis : int, optional + Where the time segments of the STFT is located; the default is + the last axis (i.e. ``axis=-1``). + freq_axis : int, optional + Where the frequency axis of the STFT is located; the default is + the penultimate axis (i.e. ``axis=-2``). + scaling: {'spectrum', 'psd'} + The default 'spectrum' scaling allows each frequency line of `Zxx` to + be interpreted as a magnitude spectrum. The 'psd' option scales each + line to a power spectral density - it allows to calculate the signal's + energy by numerically integrating over ``abs(Zxx)**2``. + + Returns + ------- + t : ndarray + Array of output data times. + x : ndarray + iSTFT of `Zxx`. + + See Also + -------- + stft: Short Time Fourier Transform + ShortTimeFFT: Newer STFT/ISTFT implementation providing more features. + check_COLA: Check whether the Constant OverLap Add (COLA) constraint + is met + check_NOLA: Check whether the Nonzero Overlap Add (NOLA) constraint is met + + Notes + ----- + In order to enable inversion of an STFT via the inverse STFT with + `istft`, the signal windowing must obey the constraint of "nonzero + overlap add" (NOLA): + + .. math:: \sum_{t}w^{2}[n-tH] \ne 0 + + This ensures that the normalization factors that appear in the denominator + of the overlap-add reconstruction equation + + .. math:: x[n]=\frac{\sum_{t}x_{t}[n]w[n-tH]}{\sum_{t}w^{2}[n-tH]} + + are not zero. The NOLA constraint can be checked with the `check_NOLA` + function. + + An STFT which has been modified (via masking or otherwise) is not + guaranteed to correspond to a exactly realizible signal. This + function implements the iSTFT via the least-squares estimation + algorithm detailed in [2]_, which produces a signal that minimizes + the mean squared error between the STFT of the returned signal and + the modified STFT. + + + .. versionadded:: 0.19.0 + + References + ---------- + .. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck + "Discrete-Time Signal Processing", Prentice Hall, 1999. + .. [2] Daniel W. Griffin, Jae S. Lim "Signal Estimation from + Modified Short-Time Fourier Transform", IEEE 1984, + 10.1109/TASSP.1984.1164317 + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + Generate a test signal, a 2 Vrms sine wave at 50Hz corrupted by + 0.001 V**2/Hz of white noise sampled at 1024 Hz. + + >>> fs = 1024 + >>> N = 10*fs + >>> nperseg = 512 + >>> amp = 2 * np.sqrt(2) + >>> noise_power = 0.001 * fs / 2 + >>> time = np.arange(N) / float(fs) + >>> carrier = amp * np.sin(2*np.pi*50*time) + >>> noise = rng.normal(scale=np.sqrt(noise_power), + ... size=time.shape) + >>> x = carrier + noise + + Compute the STFT, and plot its magnitude + + >>> f, t, Zxx = signal.stft(x, fs=fs, nperseg=nperseg) + >>> plt.figure() + >>> plt.pcolormesh(t, f, np.abs(Zxx), vmin=0, vmax=amp, shading='gouraud') + >>> plt.ylim([f[1], f[-1]]) + >>> plt.title('STFT Magnitude') + >>> plt.ylabel('Frequency [Hz]') + >>> plt.xlabel('Time [sec]') + >>> plt.yscale('log') + >>> plt.show() + + Zero the components that are 10% or less of the carrier magnitude, + then convert back to a time series via inverse STFT + + >>> Zxx = np.where(np.abs(Zxx) >= amp/10, Zxx, 0) + >>> _, xrec = signal.istft(Zxx, fs) + + Compare the cleaned signal with the original and true carrier signals. + + >>> plt.figure() + >>> plt.plot(time, x, time, xrec, time, carrier) + >>> plt.xlim([2, 2.1]) + >>> plt.xlabel('Time [sec]') + >>> plt.ylabel('Signal') + >>> plt.legend(['Carrier + Noise', 'Filtered via STFT', 'True Carrier']) + >>> plt.show() + + Note that the cleaned signal does not start as abruptly as the original, + since some of the coefficients of the transient were also removed: + + >>> plt.figure() + >>> plt.plot(time, x, time, xrec, time, carrier) + >>> plt.xlim([0, 0.1]) + >>> plt.xlabel('Time [sec]') + >>> plt.ylabel('Signal') + >>> plt.legend(['Carrier + Noise', 'Filtered via STFT', 'True Carrier']) + >>> plt.show() + + """ + # Make sure input is an ndarray of appropriate complex dtype + Zxx = np.asarray(Zxx) + 0j + freq_axis = int(freq_axis) + time_axis = int(time_axis) + + if Zxx.ndim < 2: + raise ValueError('Input stft must be at least 2d!') + + if freq_axis == time_axis: + raise ValueError('Must specify differing time and frequency axes!') + + nseg = Zxx.shape[time_axis] + + if input_onesided: + # Assume even segment length + n_default = 2*(Zxx.shape[freq_axis] - 1) + else: + n_default = Zxx.shape[freq_axis] + + # Check windowing parameters + if nperseg is None: + nperseg = n_default + else: + nperseg = int(nperseg) + if nperseg < 1: + raise ValueError('nperseg must be a positive integer') + + if nfft is None: + if (input_onesided) and (nperseg == n_default + 1): + # Odd nperseg, no FFT padding + nfft = nperseg + else: + nfft = n_default + elif nfft < nperseg: + raise ValueError('nfft must be greater than or equal to nperseg.') + else: + nfft = int(nfft) + + if noverlap is None: + noverlap = nperseg//2 + else: + noverlap = int(noverlap) + if noverlap >= nperseg: + raise ValueError('noverlap must be less than nperseg.') + nstep = nperseg - noverlap + + # Rearrange axes if necessary + if time_axis != Zxx.ndim-1 or freq_axis != Zxx.ndim-2: + # Turn negative indices to positive for the call to transpose + if freq_axis < 0: + freq_axis = Zxx.ndim + freq_axis + if time_axis < 0: + time_axis = Zxx.ndim + time_axis + zouter = list(range(Zxx.ndim)) + for ax in sorted([time_axis, freq_axis], reverse=True): + zouter.pop(ax) + Zxx = np.transpose(Zxx, zouter+[freq_axis, time_axis]) + + # Get window as array + if isinstance(window, str) or type(window) is tuple: + win = get_window(window, nperseg) + else: + win = np.asarray(window) + if len(win.shape) != 1: + raise ValueError('window must be 1-D') + if win.shape[0] != nperseg: + raise ValueError(f'window must have length of {nperseg}') + + ifunc = sp_fft.irfft if input_onesided else sp_fft.ifft + xsubs = ifunc(Zxx, axis=-2, n=nfft)[..., :nperseg, :] + + # Initialize output and normalization arrays + outputlength = nperseg + (nseg-1)*nstep + x = np.zeros(list(Zxx.shape[:-2])+[outputlength], dtype=xsubs.dtype) + norm = np.zeros(outputlength, dtype=xsubs.dtype) + + if np.result_type(win, xsubs) != xsubs.dtype: + win = win.astype(xsubs.dtype) + + if scaling == 'spectrum': + xsubs *= win.sum() + elif scaling == 'psd': + xsubs *= np.sqrt(fs * sum(win**2)) + else: + raise ValueError(f"Parameter {scaling=} not in ['spectrum', 'psd']!") + + # Construct the output from the ifft segments + # This loop could perhaps be vectorized/strided somehow... + for ii in range(nseg): + # Window the ifft + x[..., ii*nstep:ii*nstep+nperseg] += xsubs[..., ii] * win + norm[..., ii*nstep:ii*nstep+nperseg] += win**2 + + # Remove extension points + if boundary: + x = x[..., nperseg//2:-(nperseg//2)] + norm = norm[..., nperseg//2:-(nperseg//2)] + + # Divide out normalization where non-tiny + if np.sum(norm > 1e-10) != len(norm): + warnings.warn( + "NOLA condition failed, STFT may not be invertible." + + (" Possibly due to missing boundary" if not boundary else ""), + stacklevel=2 + ) + x /= np.where(norm > 1e-10, norm, 1.0) + + if input_onesided: + x = x.real + + # Put axes back + if x.ndim > 1: + if time_axis != Zxx.ndim-1: + if freq_axis < time_axis: + time_axis -= 1 + x = np.moveaxis(x, -1, time_axis) + + time = np.arange(x.shape[0])/float(fs) + return time, x + + +def coherence(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, + nfft=None, detrend='constant', axis=-1): + r""" + Estimate the magnitude squared coherence estimate, Cxy, of + discrete-time signals X and Y using Welch's method. + + ``Cxy = abs(Pxy)**2/(Pxx*Pyy)``, where `Pxx` and `Pyy` are power + spectral density estimates of X and Y, and `Pxy` is the cross + spectral density estimate of X and Y. + + Parameters + ---------- + x : array_like + Time series of measurement values + y : array_like + Time series of measurement values + fs : float, optional + Sampling frequency of the `x` and `y` time series. Defaults + to 1.0. + window : str or tuple or array_like, optional + Desired window to use. If `window` is a string or tuple, it is + passed to `get_window` to generate the window values, which are + DFT-even by default. See `get_window` for a list of windows and + required parameters. If `window` is array_like it will be used + directly as the window and its length must be nperseg. Defaults + to a Hann window. + nperseg : int, optional + Length of each segment. Defaults to None, but if window is str or + tuple, is set to 256, and if window is array_like, is set to the + length of the window. + noverlap: int, optional + Number of points to overlap between segments. If `None`, + ``noverlap = nperseg // 2``. Defaults to `None`. + nfft : int, optional + Length of the FFT used, if a zero padded FFT is desired. If + `None`, the FFT length is `nperseg`. Defaults to `None`. + detrend : str or function or `False`, optional + Specifies how to detrend each segment. If `detrend` is a + string, it is passed as the `type` argument to the `detrend` + function. If it is a function, it takes a segment and returns a + detrended segment. If `detrend` is `False`, no detrending is + done. Defaults to 'constant'. + axis : int, optional + Axis along which the coherence is computed for both inputs; the + default is over the last axis (i.e. ``axis=-1``). + + Returns + ------- + f : ndarray + Array of sample frequencies. + Cxy : ndarray + Magnitude squared coherence of x and y. + + See Also + -------- + periodogram: Simple, optionally modified periodogram + lombscargle: Lomb-Scargle periodogram for unevenly sampled data + welch: Power spectral density by Welch's method. + csd: Cross spectral density by Welch's method. + + Notes + ----- + An appropriate amount of overlap will depend on the choice of window + and on your requirements. For the default Hann window an overlap of + 50% is a reasonable trade off between accurately estimating the + signal power, while not over counting any of the data. Narrower + windows may require a larger overlap. + + .. versionadded:: 0.16.0 + + References + ---------- + .. [1] P. Welch, "The use of the fast Fourier transform for the + estimation of power spectra: A method based on time averaging + over short, modified periodograms", IEEE Trans. Audio + Electroacoust. vol. 15, pp. 70-73, 1967. + .. [2] Stoica, Petre, and Randolph Moses, "Spectral Analysis of + Signals" Prentice Hall, 2005 + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + Generate two test signals with some common features. + + >>> fs = 10e3 + >>> N = 1e5 + >>> amp = 20 + >>> freq = 1234.0 + >>> noise_power = 0.001 * fs / 2 + >>> time = np.arange(N) / fs + >>> b, a = signal.butter(2, 0.25, 'low') + >>> x = rng.normal(scale=np.sqrt(noise_power), size=time.shape) + >>> y = signal.lfilter(b, a, x) + >>> x += amp*np.sin(2*np.pi*freq*time) + >>> y += rng.normal(scale=0.1*np.sqrt(noise_power), size=time.shape) + + Compute and plot the coherence. + + >>> f, Cxy = signal.coherence(x, y, fs, nperseg=1024) + >>> plt.semilogy(f, Cxy) + >>> plt.xlabel('frequency [Hz]') + >>> plt.ylabel('Coherence') + >>> plt.show() + + """ + freqs, Pxx = welch(x, fs=fs, window=window, nperseg=nperseg, + noverlap=noverlap, nfft=nfft, detrend=detrend, + axis=axis) + _, Pyy = welch(y, fs=fs, window=window, nperseg=nperseg, noverlap=noverlap, + nfft=nfft, detrend=detrend, axis=axis) + _, Pxy = csd(x, y, fs=fs, window=window, nperseg=nperseg, + noverlap=noverlap, nfft=nfft, detrend=detrend, axis=axis) + + Cxy = np.abs(Pxy)**2 / Pxx / Pyy + + return freqs, Cxy + + +def _spectral_helper(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, + nfft=None, detrend='constant', return_onesided=True, + scaling='density', axis=-1, mode='psd', boundary=None, + padded=False): + """Calculate various forms of windowed FFTs for PSD, CSD, etc. + + This is a helper function that implements the commonality between + the stft, psd, csd, and spectrogram functions. It is not designed to + be called externally. The windows are not averaged over; the result + from each window is returned. + + Parameters + ---------- + x : array_like + Array or sequence containing the data to be analyzed. + y : array_like + Array or sequence containing the data to be analyzed. If this is + the same object in memory as `x` (i.e. ``_spectral_helper(x, + x, ...)``), the extra computations are spared. + fs : float, optional + Sampling frequency of the time series. Defaults to 1.0. + window : str or tuple or array_like, optional + Desired window to use. If `window` is a string or tuple, it is + passed to `get_window` to generate the window values, which are + DFT-even by default. See `get_window` for a list of windows and + required parameters. If `window` is array_like it will be used + directly as the window and its length must be nperseg. Defaults + to a Hann window. + nperseg : int, optional + Length of each segment. Defaults to None, but if window is str or + tuple, is set to 256, and if window is array_like, is set to the + length of the window. + noverlap : int, optional + Number of points to overlap between segments. If `None`, + ``noverlap = nperseg // 2``. Defaults to `None`. + nfft : int, optional + Length of the FFT used, if a zero padded FFT is desired. If + `None`, the FFT length is `nperseg`. Defaults to `None`. + detrend : str or function or `False`, optional + Specifies how to detrend each segment. If `detrend` is a + string, it is passed as the `type` argument to the `detrend` + function. If it is a function, it takes a segment and returns a + detrended segment. If `detrend` is `False`, no detrending is + done. Defaults to 'constant'. + return_onesided : bool, optional + If `True`, return a one-sided spectrum for real data. If + `False` return a two-sided spectrum. Defaults to `True`, but for + complex data, a two-sided spectrum is always returned. + scaling : { 'density', 'spectrum' }, optional + Selects between computing the cross spectral density ('density') + where `Pxy` has units of V**2/Hz and computing the cross + spectrum ('spectrum') where `Pxy` has units of V**2, if `x` + and `y` are measured in V and `fs` is measured in Hz. + Defaults to 'density' + axis : int, optional + Axis along which the FFTs are computed; the default is over the + last axis (i.e. ``axis=-1``). + mode: str {'psd', 'stft'}, optional + Defines what kind of return values are expected. Defaults to + 'psd'. + boundary : str or None, optional + Specifies whether the input signal is extended at both ends, and + how to generate the new values, in order to center the first + windowed segment on the first input point. This has the benefit + of enabling reconstruction of the first input point when the + employed window function starts at zero. Valid options are + ``['even', 'odd', 'constant', 'zeros', None]``. Defaults to + `None`. + padded : bool, optional + Specifies whether the input signal is zero-padded at the end to + make the signal fit exactly into an integer number of window + segments, so that all of the signal is included in the output. + Defaults to `False`. Padding occurs after boundary extension, if + `boundary` is not `None`, and `padded` is `True`. + + Returns + ------- + freqs : ndarray + Array of sample frequencies. + t : ndarray + Array of times corresponding to each data segment + result : ndarray + Array of output data, contents dependent on *mode* kwarg. + + Notes + ----- + Adapted from matplotlib.mlab + + .. versionadded:: 0.16.0 + """ + if mode not in ['psd', 'stft']: + raise ValueError(f"Unknown value for mode {mode}, must be one of: " + "{'psd', 'stft'}") + + boundary_funcs = {'even': even_ext, + 'odd': odd_ext, + 'constant': const_ext, + 'zeros': zero_ext, + None: None} + + if boundary not in boundary_funcs: + raise ValueError(f"Unknown boundary option '{boundary}', " + f"must be one of: {list(boundary_funcs.keys())}") + + # If x and y are the same object we can save ourselves some computation. + same_data = y is x + + if not same_data and mode != 'psd': + raise ValueError("x and y must be equal if mode is 'stft'") + + axis = int(axis) + + # Ensure we have np.arrays, get outdtype + x = np.asarray(x) + if not same_data: + y = np.asarray(y) + outdtype = np.result_type(x, y, np.complex64) + else: + outdtype = np.result_type(x, np.complex64) + + if not same_data: + # Check if we can broadcast the outer axes together + xouter = list(x.shape) + youter = list(y.shape) + xouter.pop(axis) + youter.pop(axis) + try: + outershape = np.broadcast(np.empty(xouter), np.empty(youter)).shape + except ValueError as e: + raise ValueError('x and y cannot be broadcast together.') from e + + if same_data: + if x.size == 0: + return np.empty(x.shape), np.empty(x.shape), np.empty(x.shape) + else: + if x.size == 0 or y.size == 0: + outshape = outershape + (min([x.shape[axis], y.shape[axis]]),) + emptyout = np.moveaxis(np.empty(outshape), -1, axis) + return emptyout, emptyout, emptyout + + if x.ndim > 1: + if axis != -1: + x = np.moveaxis(x, axis, -1) + if not same_data and y.ndim > 1: + y = np.moveaxis(y, axis, -1) + + # Check if x and y are the same length, zero-pad if necessary + if not same_data: + if x.shape[-1] != y.shape[-1]: + if x.shape[-1] < y.shape[-1]: + pad_shape = list(x.shape) + pad_shape[-1] = y.shape[-1] - x.shape[-1] + x = np.concatenate((x, np.zeros(pad_shape)), -1) + else: + pad_shape = list(y.shape) + pad_shape[-1] = x.shape[-1] - y.shape[-1] + y = np.concatenate((y, np.zeros(pad_shape)), -1) + + if nperseg is not None: # if specified by user + nperseg = int(nperseg) + if nperseg < 1: + raise ValueError('nperseg must be a positive integer') + + # parse window; if array like, then set nperseg = win.shape + win, nperseg = _triage_segments(window, nperseg, input_length=x.shape[-1]) + + if nfft is None: + nfft = nperseg + elif nfft < nperseg: + raise ValueError('nfft must be greater than or equal to nperseg.') + else: + nfft = int(nfft) + + if noverlap is None: + noverlap = nperseg//2 + else: + noverlap = int(noverlap) + if noverlap >= nperseg: + raise ValueError('noverlap must be less than nperseg.') + nstep = nperseg - noverlap + + # Padding occurs after boundary extension, so that the extended signal ends + # in zeros, instead of introducing an impulse at the end. + # I.e. if x = [..., 3, 2] + # extend then pad -> [..., 3, 2, 2, 3, 0, 0, 0] + # pad then extend -> [..., 3, 2, 0, 0, 0, 2, 3] + + if boundary is not None: + ext_func = boundary_funcs[boundary] + x = ext_func(x, nperseg//2, axis=-1) + if not same_data: + y = ext_func(y, nperseg//2, axis=-1) + + if padded: + # Pad to integer number of windowed segments + # I.e. make x.shape[-1] = nperseg + (nseg-1)*nstep, with integer nseg + nadd = (-(x.shape[-1]-nperseg) % nstep) % nperseg + zeros_shape = list(x.shape[:-1]) + [nadd] + x = np.concatenate((x, np.zeros(zeros_shape)), axis=-1) + if not same_data: + zeros_shape = list(y.shape[:-1]) + [nadd] + y = np.concatenate((y, np.zeros(zeros_shape)), axis=-1) + + # Handle detrending and window functions + if not detrend: + def detrend_func(d): + return d + elif not hasattr(detrend, '__call__'): + def detrend_func(d): + return _signaltools.detrend(d, type=detrend, axis=-1) + elif axis != -1: + # Wrap this function so that it receives a shape that it could + # reasonably expect to receive. + def detrend_func(d): + d = np.moveaxis(d, -1, axis) + d = detrend(d) + return np.moveaxis(d, axis, -1) + else: + detrend_func = detrend + + if np.result_type(win, np.complex64) != outdtype: + win = win.astype(outdtype) + + if scaling == 'density': + scale = 1.0 / (fs * (win*win).sum()) + elif scaling == 'spectrum': + scale = 1.0 / win.sum()**2 + else: + raise ValueError(f'Unknown scaling: {scaling!r}') + + if mode == 'stft': + scale = np.sqrt(scale) + + if return_onesided: + if np.iscomplexobj(x): + sides = 'twosided' + warnings.warn('Input data is complex, switching to return_onesided=False', + stacklevel=3) + else: + sides = 'onesided' + if not same_data: + if np.iscomplexobj(y): + sides = 'twosided' + warnings.warn('Input data is complex, switching to ' + 'return_onesided=False', + stacklevel=3) + else: + sides = 'twosided' + + if sides == 'twosided': + freqs = sp_fft.fftfreq(nfft, 1/fs) + elif sides == 'onesided': + freqs = sp_fft.rfftfreq(nfft, 1/fs) + + # Perform the windowed FFTs + result = _fft_helper(x, win, detrend_func, nperseg, noverlap, nfft, sides) + + if not same_data: + # All the same operations on the y data + result_y = _fft_helper(y, win, detrend_func, nperseg, noverlap, nfft, + sides) + result = np.conjugate(result) * result_y + elif mode == 'psd': + result = np.conjugate(result) * result + + result *= scale + if sides == 'onesided' and mode == 'psd': + if nfft % 2: + result[..., 1:] *= 2 + else: + # Last point is unpaired Nyquist freq point, don't double + result[..., 1:-1] *= 2 + + time = np.arange(nperseg/2, x.shape[-1] - nperseg/2 + 1, + nperseg - noverlap)/float(fs) + if boundary is not None: + time -= (nperseg/2) / fs + + result = result.astype(outdtype) + + # All imaginary parts are zero anyways + if same_data and mode != 'stft': + result = result.real + + # Output is going to have new last axis for time/window index, so a + # negative axis index shifts down one + if axis < 0: + axis -= 1 + + # Roll frequency axis back to axis where the data came from + result = np.moveaxis(result, -1, axis) + + return freqs, time, result + + +def _fft_helper(x, win, detrend_func, nperseg, noverlap, nfft, sides): + """ + Calculate windowed FFT, for internal use by + `scipy.signal._spectral_helper`. + + This is a helper function that does the main FFT calculation for + `_spectral helper`. All input validation is performed there, and the + data axis is assumed to be the last axis of x. It is not designed to + be called externally. The windows are not averaged over; the result + from each window is returned. + + Returns + ------- + result : ndarray + Array of FFT data + + Notes + ----- + Adapted from matplotlib.mlab + + .. versionadded:: 0.16.0 + """ + # Created sliding window view of array + if nperseg == 1 and noverlap == 0: + result = x[..., np.newaxis] + else: + step = nperseg - noverlap + result = np.lib.stride_tricks.sliding_window_view( + x, window_shape=nperseg, axis=-1, writeable=True + ) + result = result[..., 0::step, :] + + # Detrend each data segment individually + result = detrend_func(result) + + # Apply window by multiplication + result = win * result + + # Perform the fft. Acts on last axis by default. Zero-pads automatically + if sides == 'twosided': + func = sp_fft.fft + else: + result = result.real + func = sp_fft.rfft + result = func(result, n=nfft) + + return result + + +def _triage_segments(window, nperseg, input_length): + """ + Parses window and nperseg arguments for spectrogram and _spectral_helper. + This is a helper function, not meant to be called externally. + + Parameters + ---------- + window : string, tuple, or ndarray + If window is specified by a string or tuple and nperseg is not + specified, nperseg is set to the default of 256 and returns a window of + that length. + If instead the window is array_like and nperseg is not specified, then + nperseg is set to the length of the window. A ValueError is raised if + the user supplies both an array_like window and a value for nperseg but + nperseg does not equal the length of the window. + + nperseg : int + Length of each segment + + input_length: int + Length of input signal, i.e. x.shape[-1]. Used to test for errors. + + Returns + ------- + win : ndarray + window. If function was called with string or tuple than this will hold + the actual array used as a window. + + nperseg : int + Length of each segment. If window is str or tuple, nperseg is set to + 256. If window is array_like, nperseg is set to the length of the + window. + """ + # parse window; if array like, then set nperseg = win.shape + if isinstance(window, str) or isinstance(window, tuple): + # if nperseg not specified + if nperseg is None: + nperseg = 256 # then change to default + if nperseg > input_length: + warnings.warn(f'nperseg = {nperseg:d} is greater than input length ' + f' = {input_length:d}, using nperseg = {input_length:d}', + stacklevel=3) + nperseg = input_length + win = get_window(window, nperseg) + else: + win = np.asarray(window) + if len(win.shape) != 1: + raise ValueError('window must be 1-D') + if input_length < win.shape[-1]: + raise ValueError('window is longer than input signal') + if nperseg is None: + nperseg = win.shape[0] + elif nperseg is not None: + if nperseg != win.shape[0]: + raise ValueError("value specified for nperseg is different" + " from length of window") + return win, nperseg + + +def _median_bias(n): + """ + Returns the bias of the median of a set of periodograms relative to + the mean. + + See Appendix B from [1]_ for details. + + Parameters + ---------- + n : int + Numbers of periodograms being averaged. + + Returns + ------- + bias : float + Calculated bias. + + References + ---------- + .. [1] B. Allen, W.G. Anderson, P.R. Brady, D.A. Brown, J.D.E. Creighton. + "FINDCHIRP: an algorithm for detection of gravitational waves from + inspiraling compact binaries", Physical Review D 85, 2012, + :arxiv:`gr-qc/0509116` + """ + ii_2 = 2 * np.arange(1., (n-1) // 2 + 1) + return 1 + np.sum(1. / (ii_2 + 1) - 1. / ii_2) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spline.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spline.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..c776c84deedabf62573fdee3f6b0bea1089d0092 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spline.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spline.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spline.pyi new file mode 100644 index 0000000000000000000000000000000000000000..c4225577db7ea188a2add225ecec1fbec855de06 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spline.pyi @@ -0,0 +1,34 @@ + +import numpy as np +from numpy.typing import NDArray + +FloatingArray = NDArray[np.float32] | NDArray[np.float64] +ComplexArray = NDArray[np.complex64] | NDArray[np.complex128] +FloatingComplexArray = FloatingArray | ComplexArray + + +def symiirorder1_ic(signal: FloatingComplexArray, + c0: float, + z1: float, + precision: float) -> FloatingComplexArray: + ... + + +def symiirorder2_ic_fwd(signal: FloatingArray, + r: float, + omega: float, + precision: float) -> FloatingArray: + ... + + +def symiirorder2_ic_bwd(signal: FloatingArray, + r: float, + omega: float, + precision: float) -> FloatingArray: + ... + + +def sepfir2d(input: FloatingComplexArray, + hrow: FloatingComplexArray, + hcol: FloatingComplexArray) -> FloatingComplexArray: + ... diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spline_filters.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spline_filters.py new file mode 100644 index 0000000000000000000000000000000000000000..eb7884c5cc5544e9fc6d37476016e7a2d0e81d57 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_spline_filters.py @@ -0,0 +1,808 @@ +from numpy import (asarray, pi, zeros_like, + array, arctan2, tan, ones, arange, floor, + r_, atleast_1d, sqrt, exp, greater, cos, add, sin, + moveaxis, abs, arctan, complex64, float32) +import numpy as np + +from scipy._lib._util import normalize_axis_index + +# From splinemodule.c +from ._spline import sepfir2d, symiirorder1_ic, symiirorder2_ic_fwd, symiirorder2_ic_bwd +from ._signaltools import lfilter, sosfilt, lfiltic +from ._arraytools import axis_slice, axis_reverse + +from scipy.interpolate import BSpline + + +__all__ = ['spline_filter', 'gauss_spline', + 'cspline1d', 'qspline1d', 'qspline2d', 'cspline2d', + 'cspline1d_eval', 'qspline1d_eval', 'symiirorder1', 'symiirorder2'] + + +def spline_filter(Iin, lmbda=5.0): + """Smoothing spline (cubic) filtering of a rank-2 array. + + Filter an input data set, `Iin`, using a (cubic) smoothing spline of + fall-off `lmbda`. + + Parameters + ---------- + Iin : array_like + input data set + lmbda : float, optional + spline smoothing fall-off value, default is `5.0`. + + Returns + ------- + res : ndarray + filtered input data + + Examples + -------- + We can filter an multi dimensional signal (ex: 2D image) using cubic + B-spline filter: + + >>> import numpy as np + >>> from scipy.signal import spline_filter + >>> import matplotlib.pyplot as plt + >>> orig_img = np.eye(20) # create an image + >>> orig_img[10, :] = 1.0 + >>> sp_filter = spline_filter(orig_img, lmbda=0.1) + >>> f, ax = plt.subplots(1, 2, sharex=True) + >>> for ind, data in enumerate([[orig_img, "original image"], + ... [sp_filter, "spline filter"]]): + ... ax[ind].imshow(data[0], cmap='gray_r') + ... ax[ind].set_title(data[1]) + >>> plt.tight_layout() + >>> plt.show() + + """ + if Iin.dtype not in [np.float32, np.float64, np.complex64, np.complex128]: + raise TypeError(f"Invalid data type for Iin: {Iin.dtype = }") + + # XXX: note that complex-valued computations are done in single precision + # this is historic, and the root reason is unclear, + # see https://github.com/scipy/scipy/issues/9209 + # Attempting to work in complex double precision leads to symiirorder1 + # failing to converge for the boundary conditions. + intype = Iin.dtype + hcol = array([1.0, 4.0, 1.0], np.float32) / 6.0 + if intype == np.complex128: + Iin = Iin.astype(np.complex64) + + ck = cspline2d(Iin, lmbda) + out = sepfir2d(ck, hcol, hcol) + out = out.astype(intype) + return out + + +_splinefunc_cache = {} + + +def gauss_spline(x, n): + r"""Gaussian approximation to B-spline basis function of order n. + + Parameters + ---------- + x : array_like + a knot vector + n : int + The order of the spline. Must be non-negative, i.e., n >= 0 + + Returns + ------- + res : ndarray + B-spline basis function values approximated by a zero-mean Gaussian + function. + + Notes + ----- + The B-spline basis function can be approximated well by a zero-mean + Gaussian function with standard-deviation equal to :math:`\sigma=(n+1)/12` + for large `n` : + + .. math:: \frac{1}{\sqrt {2\pi\sigma^2}}exp(-\frac{x^2}{2\sigma}) + + References + ---------- + .. [1] Bouma H., Vilanova A., Bescos J.O., ter Haar Romeny B.M., Gerritsen + F.A. (2007) Fast and Accurate Gaussian Derivatives Based on B-Splines. In: + Sgallari F., Murli A., Paragios N. (eds) Scale Space and Variational + Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer + Science, vol 4485. Springer, Berlin, Heidelberg + .. [2] http://folk.uio.no/inf3330/scripting/doc/python/SciPy/tutorial/old/node24.html + + Examples + -------- + We can calculate B-Spline basis functions approximated by a gaussian + distribution: + + >>> import numpy as np + >>> from scipy.signal import gauss_spline + >>> knots = np.array([-1.0, 0.0, -1.0]) + >>> gauss_spline(knots, 3) + array([0.15418033, 0.6909883, 0.15418033]) # may vary + + """ + x = asarray(x) + signsq = (n + 1) / 12.0 + return 1 / sqrt(2 * pi * signsq) * exp(-x ** 2 / 2 / signsq) + + +def _cubic(x): + x = asarray(x, dtype=float) + b = BSpline.basis_element([-2, -1, 0, 1, 2], extrapolate=False) + out = b(x) + out[(x < -2) | (x > 2)] = 0 + return out + + +def _quadratic(x): + x = abs(asarray(x, dtype=float)) + b = BSpline.basis_element([-1.5, -0.5, 0.5, 1.5], extrapolate=False) + out = b(x) + out[(x < -1.5) | (x > 1.5)] = 0 + return out + + +def _coeff_smooth(lam): + xi = 1 - 96 * lam + 24 * lam * sqrt(3 + 144 * lam) + omeg = arctan2(sqrt(144 * lam - 1), sqrt(xi)) + rho = (24 * lam - 1 - sqrt(xi)) / (24 * lam) + rho = rho * sqrt((48 * lam + 24 * lam * sqrt(3 + 144 * lam)) / xi) + return rho, omeg + + +def _hc(k, cs, rho, omega): + return (cs / sin(omega) * (rho ** k) * sin(omega * (k + 1)) * + greater(k, -1)) + + +def _hs(k, cs, rho, omega): + c0 = (cs * cs * (1 + rho * rho) / (1 - rho * rho) / + (1 - 2 * rho * rho * cos(2 * omega) + rho ** 4)) + gamma = (1 - rho * rho) / (1 + rho * rho) / tan(omega) + ak = abs(k) + return c0 * rho ** ak * (cos(omega * ak) + gamma * sin(omega * ak)) + + +def _cubic_smooth_coeff(signal, lamb): + rho, omega = _coeff_smooth(lamb) + cs = 1 - 2 * rho * cos(omega) + rho * rho + K = len(signal) + k = arange(K) + + zi_2 = (_hc(0, cs, rho, omega) * signal[0] + + add.reduce(_hc(k + 1, cs, rho, omega) * signal)) + zi_1 = (_hc(0, cs, rho, omega) * signal[0] + + _hc(1, cs, rho, omega) * signal[1] + + add.reduce(_hc(k + 2, cs, rho, omega) * signal)) + + # Forward filter: + # for n in range(2, K): + # yp[n] = (cs * signal[n] + 2 * rho * cos(omega) * yp[n - 1] - + # rho * rho * yp[n - 2]) + zi = lfiltic(cs, r_[1, -2 * rho * cos(omega), rho * rho], r_[zi_1, zi_2]) + zi = zi.reshape(1, -1) + + sos = r_[cs, 0, 0, 1, -2 * rho * cos(omega), rho * rho] + sos = sos.reshape(1, -1) + + yp, _ = sosfilt(sos, signal[2:], zi=zi) + yp = r_[zi_2, zi_1, yp] + + # Reverse filter: + # for n in range(K - 3, -1, -1): + # y[n] = (cs * yp[n] + 2 * rho * cos(omega) * y[n + 1] - + # rho * rho * y[n + 2]) + + zi_2 = add.reduce((_hs(k, cs, rho, omega) + + _hs(k + 1, cs, rho, omega)) * signal[::-1]) + zi_1 = add.reduce((_hs(k - 1, cs, rho, omega) + + _hs(k + 2, cs, rho, omega)) * signal[::-1]) + + zi = lfiltic(cs, r_[1, -2 * rho * cos(omega), rho * rho], r_[zi_1, zi_2]) + zi = zi.reshape(1, -1) + y, _ = sosfilt(sos, yp[-3::-1], zi=zi) + y = r_[y[::-1], zi_1, zi_2] + return y + + +def _cubic_coeff(signal): + zi = -2 + sqrt(3) + K = len(signal) + powers = zi ** arange(K) + + if K == 1: + yplus = signal[0] + zi * add.reduce(powers * signal) + output = zi / (zi - 1) * yplus + return atleast_1d(output) + + # Forward filter: + # yplus[0] = signal[0] + zi * add.reduce(powers * signal) + # for k in range(1, K): + # yplus[k] = signal[k] + zi * yplus[k - 1] + + state = lfiltic(1, r_[1, -zi], atleast_1d(add.reduce(powers * signal))) + + b = ones(1) + a = r_[1, -zi] + yplus, _ = lfilter(b, a, signal, zi=state) + + # Reverse filter: + # output[K - 1] = zi / (zi - 1) * yplus[K - 1] + # for k in range(K - 2, -1, -1): + # output[k] = zi * (output[k + 1] - yplus[k]) + out_last = zi / (zi - 1) * yplus[K - 1] + state = lfiltic(-zi, r_[1, -zi], atleast_1d(out_last)) + + b = asarray([-zi]) + output, _ = lfilter(b, a, yplus[-2::-1], zi=state) + output = r_[output[::-1], out_last] + return output * 6.0 + + +def _quadratic_coeff(signal): + zi = -3 + 2 * sqrt(2.0) + K = len(signal) + powers = zi ** arange(K) + + if K == 1: + yplus = signal[0] + zi * add.reduce(powers * signal) + output = zi / (zi - 1) * yplus + return atleast_1d(output) + + # Forward filter: + # yplus[0] = signal[0] + zi * add.reduce(powers * signal) + # for k in range(1, K): + # yplus[k] = signal[k] + zi * yplus[k - 1] + + state = lfiltic(1, r_[1, -zi], atleast_1d(add.reduce(powers * signal))) + + b = ones(1) + a = r_[1, -zi] + yplus, _ = lfilter(b, a, signal, zi=state) + + # Reverse filter: + # output[K - 1] = zi / (zi - 1) * yplus[K - 1] + # for k in range(K - 2, -1, -1): + # output[k] = zi * (output[k + 1] - yplus[k]) + out_last = zi / (zi - 1) * yplus[K - 1] + state = lfiltic(-zi, r_[1, -zi], atleast_1d(out_last)) + + b = asarray([-zi]) + output, _ = lfilter(b, a, yplus[-2::-1], zi=state) + output = r_[output[::-1], out_last] + return output * 8.0 + + +def compute_root_from_lambda(lamb): + tmp = sqrt(3 + 144 * lamb) + xi = 1 - 96 * lamb + 24 * lamb * tmp + omega = arctan(sqrt((144 * lamb - 1.0) / xi)) + tmp2 = sqrt(xi) + r = ((24 * lamb - 1 - tmp2) / (24 * lamb) * + sqrt(48*lamb + 24 * lamb * tmp) / tmp2) + return r, omega + + +def cspline1d(signal, lamb=0.0): + """ + Compute cubic spline coefficients for rank-1 array. + + Find the cubic spline coefficients for a 1-D signal assuming + mirror-symmetric boundary conditions. To obtain the signal back from the + spline representation mirror-symmetric-convolve these coefficients with a + length 3 FIR window [1.0, 4.0, 1.0]/ 6.0 . + + Parameters + ---------- + signal : ndarray + A rank-1 array representing samples of a signal. + lamb : float, optional + Smoothing coefficient, default is 0.0. + + Returns + ------- + c : ndarray + Cubic spline coefficients. + + See Also + -------- + cspline1d_eval : Evaluate a cubic spline at the new set of points. + + Examples + -------- + We can filter a signal to reduce and smooth out high-frequency noise with + a cubic spline: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.signal import cspline1d, cspline1d_eval + >>> rng = np.random.default_rng() + >>> sig = np.repeat([0., 1., 0.], 100) + >>> sig += rng.standard_normal(len(sig))*0.05 # add noise + >>> time = np.linspace(0, len(sig)) + >>> filtered = cspline1d_eval(cspline1d(sig), time) + >>> plt.plot(sig, label="signal") + >>> plt.plot(time, filtered, label="filtered") + >>> plt.legend() + >>> plt.show() + + """ + if lamb != 0.0: + return _cubic_smooth_coeff(signal, lamb) + else: + return _cubic_coeff(signal) + + +def qspline1d(signal, lamb=0.0): + """Compute quadratic spline coefficients for rank-1 array. + + Parameters + ---------- + signal : ndarray + A rank-1 array representing samples of a signal. + lamb : float, optional + Smoothing coefficient (must be zero for now). + + Returns + ------- + c : ndarray + Quadratic spline coefficients. + + See Also + -------- + qspline1d_eval : Evaluate a quadratic spline at the new set of points. + + Notes + ----- + Find the quadratic spline coefficients for a 1-D signal assuming + mirror-symmetric boundary conditions. To obtain the signal back from the + spline representation mirror-symmetric-convolve these coefficients with a + length 3 FIR window [1.0, 6.0, 1.0]/ 8.0 . + + Examples + -------- + We can filter a signal to reduce and smooth out high-frequency noise with + a quadratic spline: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.signal import qspline1d, qspline1d_eval + >>> rng = np.random.default_rng() + >>> sig = np.repeat([0., 1., 0.], 100) + >>> sig += rng.standard_normal(len(sig))*0.05 # add noise + >>> time = np.linspace(0, len(sig)) + >>> filtered = qspline1d_eval(qspline1d(sig), time) + >>> plt.plot(sig, label="signal") + >>> plt.plot(time, filtered, label="filtered") + >>> plt.legend() + >>> plt.show() + + """ + if lamb != 0.0: + raise ValueError("Smoothing quadratic splines not supported yet.") + else: + return _quadratic_coeff(signal) + + +def collapse_2d(x, axis): + x = moveaxis(x, axis, -1) + x_shape = x.shape + x = x.reshape(-1, x.shape[-1]) + if not x.flags.c_contiguous: + x = x.copy() + return x, x_shape + + +def symiirorder_nd(func, input, *args, axis=-1, **kwargs): + axis = normalize_axis_index(axis, input.ndim) + input_shape = input.shape + input_ndim = input.ndim + if input.ndim > 1: + input, input_shape = collapse_2d(input, axis) + + out = func(input, *args, **kwargs) + + if input_ndim > 1: + out = out.reshape(input_shape) + out = moveaxis(out, -1, axis) + if not out.flags.c_contiguous: + out = out.copy() + return out + + +def qspline2d(signal, lamb=0.0, precision=-1.0): + """ + Coefficients for 2-D quadratic (2nd order) B-spline. + + Return the second-order B-spline coefficients over a regularly spaced + input grid for the two-dimensional input image. + + Parameters + ---------- + input : ndarray + The input signal. + lamb : float + Specifies the amount of smoothing in the transfer function. + precision : float + Specifies the precision for computing the infinite sum needed to apply + mirror-symmetric boundary conditions. + + Returns + ------- + output : ndarray + The filtered signal. + """ + if precision < 0.0 or precision >= 1.0: + if signal.dtype in [float32, complex64]: + precision = 1e-3 + else: + precision = 1e-6 + + if lamb > 0: + raise ValueError('lambda must be negative or zero') + + # normal quadratic spline + r = -3 + 2 * sqrt(2.0) + c0 = -r * 8.0 + z1 = r + + out = symiirorder_nd(symiirorder1, signal, c0, z1, precision, axis=-1) + out = symiirorder_nd(symiirorder1, out, c0, z1, precision, axis=0) + return out + + +def cspline2d(signal, lamb=0.0, precision=-1.0): + """ + Coefficients for 2-D cubic (3rd order) B-spline. + + Return the third-order B-spline coefficients over a regularly spaced + input grid for the two-dimensional input image. + + Parameters + ---------- + input : ndarray + The input signal. + lamb : float + Specifies the amount of smoothing in the transfer function. + precision : float + Specifies the precision for computing the infinite sum needed to apply + mirror-symmetric boundary conditions. + + Returns + ------- + output : ndarray + The filtered signal. + """ + if precision < 0.0 or precision >= 1.0: + if signal.dtype in [float32, complex64]: + precision = 1e-3 + else: + precision = 1e-6 + + if lamb <= 1 / 144.0: + # Normal cubic spline + r = -2 + sqrt(3.0) + out = symiirorder_nd( + symiirorder1, signal, -r * 6.0, r, precision=precision, axis=-1) + out = symiirorder_nd( + symiirorder1, out, -r * 6.0, r, precision=precision, axis=0) + return out + + r, omega = compute_root_from_lambda(lamb) + out = symiirorder_nd(symiirorder2, signal, r, omega, + precision=precision, axis=-1) + out = symiirorder_nd(symiirorder2, out, r, omega, + precision=precision, axis=0) + return out + + +def cspline1d_eval(cj, newx, dx=1.0, x0=0): + """Evaluate a cubic spline at the new set of points. + + `dx` is the old sample-spacing while `x0` was the old origin. In + other-words the old-sample points (knot-points) for which the `cj` + represent spline coefficients were at equally-spaced points of: + + oldx = x0 + j*dx j=0...N-1, with N=len(cj) + + Edges are handled using mirror-symmetric boundary conditions. + + Parameters + ---------- + cj : ndarray + cublic spline coefficients + newx : ndarray + New set of points. + dx : float, optional + Old sample-spacing, the default value is 1.0. + x0 : int, optional + Old origin, the default value is 0. + + Returns + ------- + res : ndarray + Evaluated a cubic spline points. + + See Also + -------- + cspline1d : Compute cubic spline coefficients for rank-1 array. + + Examples + -------- + We can filter a signal to reduce and smooth out high-frequency noise with + a cubic spline: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.signal import cspline1d, cspline1d_eval + >>> rng = np.random.default_rng() + >>> sig = np.repeat([0., 1., 0.], 100) + >>> sig += rng.standard_normal(len(sig))*0.05 # add noise + >>> time = np.linspace(0, len(sig)) + >>> filtered = cspline1d_eval(cspline1d(sig), time) + >>> plt.plot(sig, label="signal") + >>> plt.plot(time, filtered, label="filtered") + >>> plt.legend() + >>> plt.show() + + """ + newx = (asarray(newx) - x0) / float(dx) + res = zeros_like(newx, dtype=cj.dtype) + if res.size == 0: + return res + N = len(cj) + cond1 = newx < 0 + cond2 = newx > (N - 1) + cond3 = ~(cond1 | cond2) + # handle general mirror-symmetry + res[cond1] = cspline1d_eval(cj, -newx[cond1]) + res[cond2] = cspline1d_eval(cj, 2 * (N - 1) - newx[cond2]) + newx = newx[cond3] + if newx.size == 0: + return res + result = zeros_like(newx, dtype=cj.dtype) + jlower = floor(newx - 2).astype(int) + 1 + for i in range(4): + thisj = jlower + i + indj = thisj.clip(0, N - 1) # handle edge cases + result += cj[indj] * _cubic(newx - thisj) + res[cond3] = result + return res + + +def qspline1d_eval(cj, newx, dx=1.0, x0=0): + """Evaluate a quadratic spline at the new set of points. + + Parameters + ---------- + cj : ndarray + Quadratic spline coefficients + newx : ndarray + New set of points. + dx : float, optional + Old sample-spacing, the default value is 1.0. + x0 : int, optional + Old origin, the default value is 0. + + Returns + ------- + res : ndarray + Evaluated a quadratic spline points. + + See Also + -------- + qspline1d : Compute quadratic spline coefficients for rank-1 array. + + Notes + ----- + `dx` is the old sample-spacing while `x0` was the old origin. In + other-words the old-sample points (knot-points) for which the `cj` + represent spline coefficients were at equally-spaced points of:: + + oldx = x0 + j*dx j=0...N-1, with N=len(cj) + + Edges are handled using mirror-symmetric boundary conditions. + + Examples + -------- + We can filter a signal to reduce and smooth out high-frequency noise with + a quadratic spline: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.signal import qspline1d, qspline1d_eval + >>> rng = np.random.default_rng() + >>> sig = np.repeat([0., 1., 0.], 100) + >>> sig += rng.standard_normal(len(sig))*0.05 # add noise + >>> time = np.linspace(0, len(sig)) + >>> filtered = qspline1d_eval(qspline1d(sig), time) + >>> plt.plot(sig, label="signal") + >>> plt.plot(time, filtered, label="filtered") + >>> plt.legend() + >>> plt.show() + + """ + newx = (asarray(newx) - x0) / dx + res = zeros_like(newx) + if res.size == 0: + return res + N = len(cj) + cond1 = newx < 0 + cond2 = newx > (N - 1) + cond3 = ~(cond1 | cond2) + # handle general mirror-symmetry + res[cond1] = qspline1d_eval(cj, -newx[cond1]) + res[cond2] = qspline1d_eval(cj, 2 * (N - 1) - newx[cond2]) + newx = newx[cond3] + if newx.size == 0: + return res + result = zeros_like(newx) + jlower = floor(newx - 1.5).astype(int) + 1 + for i in range(3): + thisj = jlower + i + indj = thisj.clip(0, N - 1) # handle edge cases + result += cj[indj] * _quadratic(newx - thisj) + res[cond3] = result + return res + + +def symiirorder1(signal, c0, z1, precision=-1.0): + """ + Implement a smoothing IIR filter with mirror-symmetric boundary conditions + using a cascade of first-order sections. + + The second section uses a reversed sequence. This implements a system with + the following transfer function and mirror-symmetric boundary conditions:: + + c0 + H(z) = --------------------- + (1-z1/z) (1 - z1 z) + + The resulting signal will have mirror symmetric boundary conditions + as well. + + Parameters + ---------- + signal : ndarray + The input signal. If 2D, then the filter will be applied in a batched + fashion across the last axis. + c0, z1 : scalar + Parameters in the transfer function. + precision : + Specifies the precision for calculating initial conditions + of the recursive filter based on mirror-symmetric input. + + Returns + ------- + output : ndarray + The filtered signal. + """ + if np.abs(z1) >= 1: + raise ValueError('|z1| must be less than 1.0') + + if signal.ndim > 2: + raise ValueError('Input must be 1D or 2D') + + squeeze_dim = False + if signal.ndim == 1: + signal = signal[None, :] + squeeze_dim = True + + if np.issubdtype(signal.dtype, np.integer): + signal = signal.astype(np.promote_types(signal.dtype, np.float32)) + + y0 = symiirorder1_ic(signal, z1, precision) + + # Apply first the system 1 / (1 - z1 * z^-1) + b = np.ones(1, dtype=signal.dtype) + a = np.r_[1, -z1] + a = a.astype(signal.dtype) + + # Compute the initial state for lfilter. + zii = y0 * z1 + + y1, _ = lfilter(b, a, axis_slice(signal, 1), zi=zii) + y1 = np.c_[y0, y1] + + # Compute backward symmetric condition and apply the system + # c0 / (1 - z1 * z) + b = np.asarray([c0], dtype=signal.dtype) + out_last = -c0 / (z1 - 1.0) * axis_slice(y1, -1) + + # Compute the initial state for lfilter. + zii = out_last * z1 + + # Apply the system c0 / (1 - z1 * z) by reversing the output of the previous stage + out, _ = lfilter(b, a, axis_slice(y1, -2, step=-1), zi=zii) + out = np.c_[axis_reverse(out), out_last] + + if squeeze_dim: + out = out[0] + + return out + + +def symiirorder2(input, r, omega, precision=-1.0): + """ + Implement a smoothing IIR filter with mirror-symmetric boundary conditions + using a cascade of second-order sections. + + The second section uses a reversed sequence. This implements the following + transfer function:: + + cs^2 + H(z) = --------------------------------------- + (1 - a2/z - a3/z^2) (1 - a2 z - a3 z^2 ) + + where:: + + a2 = 2 * r * cos(omega) + a3 = - r ** 2 + cs = 1 - 2 * r * cos(omega) + r ** 2 + + Parameters + ---------- + input : ndarray + The input signal. + r, omega : float + Parameters in the transfer function. + precision : float + Specifies the precision for calculating initial conditions + of the recursive filter based on mirror-symmetric input. + + Returns + ------- + output : ndarray + The filtered signal. + """ + if r >= 1.0: + raise ValueError('r must be less than 1.0') + + if input.ndim > 2: + raise ValueError('Input must be 1D or 2D') + + if not input.flags.c_contiguous: + input = input.copy() + + squeeze_dim = False + if input.ndim == 1: + input = input[None, :] + squeeze_dim = True + + if np.issubdtype(input.dtype, np.integer): + input = input.astype(np.promote_types(input.dtype, np.float32)) + + rsq = r * r + a2 = 2 * r * np.cos(omega) + a3 = -rsq + cs = np.atleast_1d(1 - 2 * r * np.cos(omega) + rsq) + sos = np.atleast_2d(np.r_[cs, 0, 0, 1, -a2, -a3]).astype(input.dtype) + + # Find the starting (forward) conditions. + ic_fwd = symiirorder2_ic_fwd(input, r, omega, precision) + + # Apply first the system cs / (1 - a2 * z^-1 - a3 * z^-2) + # Compute the initial conditions in the form expected by sosfilt + # coef = np.asarray([[a3, a2], [0, a3]], dtype=input.dtype) + coef = np.r_[a3, a2, 0, a3].reshape(2, 2).astype(input.dtype) + zi = np.matmul(coef, ic_fwd[:, :, None])[:, :, 0] + + y_fwd, _ = sosfilt(sos, axis_slice(input, 2), zi=zi[None]) + y_fwd = np.c_[ic_fwd, y_fwd] + + # Then compute the symmetric backward starting conditions + ic_bwd = symiirorder2_ic_bwd(input, r, omega, precision) + + # Apply the system cs / (1 - a2 * z^1 - a3 * z^2) + # Compute the initial conditions in the form expected by sosfilt + zi = np.matmul(coef, ic_bwd[:, :, None])[:, :, 0] + y, _ = sosfilt(sos, axis_slice(y_fwd, -3, step=-1), zi=zi[None]) + out = np.c_[axis_reverse(y), axis_reverse(ic_bwd)] + + if squeeze_dim: + out = out[0] + + return out diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_upfirdn.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_upfirdn.py new file mode 100644 index 0000000000000000000000000000000000000000..d64cc142ff194b1404e380507289ddbaffab3359 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_upfirdn.py @@ -0,0 +1,216 @@ +# Code adapted from "upfirdn" python library with permission: +# +# Copyright (c) 2009, Motorola, Inc +# +# All Rights Reserved. +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions are +# met: +# +# * Redistributions of source code must retain the above copyright notice, +# this list of conditions and the following disclaimer. +# +# * Redistributions in binary form must reproduce the above copyright +# notice, this list of conditions and the following disclaimer in the +# documentation and/or other materials provided with the distribution. +# +# * Neither the name of Motorola nor the names of its contributors may be +# used to endorse or promote products derived from this software without +# specific prior written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +# IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, +# THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR +# PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR +# CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +# EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +# PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +# PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +# LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +import numpy as np + +from ._upfirdn_apply import _output_len, _apply, mode_enum + +__all__ = ['upfirdn', '_output_len'] + +_upfirdn_modes = [ + 'constant', 'wrap', 'edge', 'smooth', 'symmetric', 'reflect', + 'antisymmetric', 'antireflect', 'line', +] + + +def _pad_h(h, up): + """Store coefficients in a transposed, flipped arrangement. + + For example, suppose upRate is 3, and the + input number of coefficients is 10, represented as h[0], ..., h[9]. + + Then the internal buffer will look like this:: + + h[9], h[6], h[3], h[0], // flipped phase 0 coefs + 0, h[7], h[4], h[1], // flipped phase 1 coefs (zero-padded) + 0, h[8], h[5], h[2], // flipped phase 2 coefs (zero-padded) + + """ + h_padlen = len(h) + (-len(h) % up) + h_full = np.zeros(h_padlen, h.dtype) + h_full[:len(h)] = h + h_full = h_full.reshape(-1, up).T[:, ::-1].ravel() + return h_full + + +def _check_mode(mode): + mode = mode.lower() + enum = mode_enum(mode) + return enum + + +class _UpFIRDn: + """Helper for resampling.""" + + def __init__(self, h, x_dtype, up, down): + h = np.asarray(h) + if h.ndim != 1 or h.size == 0: + raise ValueError('h must be 1-D with non-zero length') + self._output_type = np.result_type(h.dtype, x_dtype, np.float32) + h = np.asarray(h, self._output_type) + self._up = int(up) + self._down = int(down) + if self._up < 1 or self._down < 1: + raise ValueError('Both up and down must be >= 1') + # This both transposes, and "flips" each phase for filtering + self._h_trans_flip = _pad_h(h, self._up) + self._h_trans_flip = np.ascontiguousarray(self._h_trans_flip) + self._h_len_orig = len(h) + + def apply_filter(self, x, axis=-1, mode='constant', cval=0): + """Apply the prepared filter to the specified axis of N-D signal x.""" + output_len = _output_len(self._h_len_orig, x.shape[axis], + self._up, self._down) + # Explicit use of np.int64 for output_shape dtype avoids OverflowError + # when allocating large array on platforms where intp is 32 bits. + output_shape = np.asarray(x.shape, dtype=np.int64) + output_shape[axis] = output_len + out = np.zeros(output_shape, dtype=self._output_type, order='C') + axis = axis % x.ndim + mode = _check_mode(mode) + _apply(np.asarray(x, self._output_type), + self._h_trans_flip, out, + self._up, self._down, axis, mode, cval) + return out + + +def upfirdn(h, x, up=1, down=1, axis=-1, mode='constant', cval=0): + """Upsample, FIR filter, and downsample. + + Parameters + ---------- + h : array_like + 1-D FIR (finite-impulse response) filter coefficients. + x : array_like + Input signal array. + up : int, optional + Upsampling rate. Default is 1. + down : int, optional + Downsampling rate. Default is 1. + axis : int, optional + The axis of the input data array along which to apply the + linear filter. The filter is applied to each subarray along + this axis. Default is -1. + mode : str, optional + The signal extension mode to use. The set + ``{"constant", "symmetric", "reflect", "edge", "wrap"}`` correspond to + modes provided by `numpy.pad`. ``"smooth"`` implements a smooth + extension by extending based on the slope of the last 2 points at each + end of the array. ``"antireflect"`` and ``"antisymmetric"`` are + anti-symmetric versions of ``"reflect"`` and ``"symmetric"``. The mode + `"line"` extends the signal based on a linear trend defined by the + first and last points along the ``axis``. + + .. versionadded:: 1.4.0 + cval : float, optional + The constant value to use when ``mode == "constant"``. + + .. versionadded:: 1.4.0 + + Returns + ------- + y : ndarray + The output signal array. Dimensions will be the same as `x` except + for along `axis`, which will change size according to the `h`, + `up`, and `down` parameters. + + Notes + ----- + The algorithm is an implementation of the block diagram shown on page 129 + of the Vaidyanathan text [1]_ (Figure 4.3-8d). + + The direct approach of upsampling by factor of P with zero insertion, + FIR filtering of length ``N``, and downsampling by factor of Q is + O(N*Q) per output sample. The polyphase implementation used here is + O(N/P). + + .. versionadded:: 0.18 + + References + ---------- + .. [1] P. P. Vaidyanathan, Multirate Systems and Filter Banks, + Prentice Hall, 1993. + + Examples + -------- + Simple operations: + + >>> import numpy as np + >>> from scipy.signal import upfirdn + >>> upfirdn([1, 1, 1], [1, 1, 1]) # FIR filter + array([ 1., 2., 3., 2., 1.]) + >>> upfirdn([1], [1, 2, 3], 3) # upsampling with zeros insertion + array([ 1., 0., 0., 2., 0., 0., 3.]) + >>> upfirdn([1, 1, 1], [1, 2, 3], 3) # upsampling with sample-and-hold + array([ 1., 1., 1., 2., 2., 2., 3., 3., 3.]) + >>> upfirdn([.5, 1, .5], [1, 1, 1], 2) # linear interpolation + array([ 0.5, 1. , 1. , 1. , 1. , 1. , 0.5]) + >>> upfirdn([1], np.arange(10), 1, 3) # decimation by 3 + array([ 0., 3., 6., 9.]) + >>> upfirdn([.5, 1, .5], np.arange(10), 2, 3) # linear interp, rate 2/3 + array([ 0. , 1. , 2.5, 4. , 5.5, 7. , 8.5]) + + Apply a single filter to multiple signals: + + >>> x = np.reshape(np.arange(8), (4, 2)) + >>> x + array([[0, 1], + [2, 3], + [4, 5], + [6, 7]]) + + Apply along the last dimension of ``x``: + + >>> h = [1, 1] + >>> upfirdn(h, x, 2) + array([[ 0., 0., 1., 1.], + [ 2., 2., 3., 3.], + [ 4., 4., 5., 5.], + [ 6., 6., 7., 7.]]) + + Apply along the 0th dimension of ``x``: + + >>> upfirdn(h, x, 2, axis=0) + array([[ 0., 1.], + [ 0., 1.], + [ 2., 3.], + [ 2., 3.], + [ 4., 5.], + [ 4., 5.], + [ 6., 7.], + [ 6., 7.]]) + """ + x = np.asarray(x) + ufd = _UpFIRDn(h, x.dtype, up, down) + # This is equivalent to (but faster than) using np.apply_along_axis + return ufd.apply_filter(x, axis, mode, cval) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_waveforms.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_waveforms.py new file mode 100644 index 0000000000000000000000000000000000000000..a6be46cfd38674ee8c3ae89c9762461440c1e620 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_waveforms.py @@ -0,0 +1,696 @@ +# Author: Travis Oliphant +# 2003 +# +# Feb. 2010: Updated by Warren Weckesser: +# Rewrote much of chirp() +# Added sweep_poly() +import numpy as np +from numpy import asarray, zeros, place, nan, mod, pi, extract, log, sqrt, \ + exp, cos, sin, polyval, polyint + + +__all__ = ['sawtooth', 'square', 'gausspulse', 'chirp', 'sweep_poly', + 'unit_impulse'] + + +def sawtooth(t, width=1): + """ + Return a periodic sawtooth or triangle waveform. + + The sawtooth waveform has a period ``2*pi``, rises from -1 to 1 on the + interval 0 to ``width*2*pi``, then drops from 1 to -1 on the interval + ``width*2*pi`` to ``2*pi``. `width` must be in the interval [0, 1]. + + Note that this is not band-limited. It produces an infinite number + of harmonics, which are aliased back and forth across the frequency + spectrum. + + Parameters + ---------- + t : array_like + Time. + width : array_like, optional + Width of the rising ramp as a proportion of the total cycle. + Default is 1, producing a rising ramp, while 0 produces a falling + ramp. `width` = 0.5 produces a triangle wave. + If an array, causes wave shape to change over time, and must be the + same length as t. + + Returns + ------- + y : ndarray + Output array containing the sawtooth waveform. + + Examples + -------- + A 5 Hz waveform sampled at 500 Hz for 1 second: + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> t = np.linspace(0, 1, 500) + >>> plt.plot(t, signal.sawtooth(2 * np.pi * 5 * t)) + + """ + t, w = asarray(t), asarray(width) + w = asarray(w + (t - t)) + t = asarray(t + (w - w)) + if t.dtype.char in ['fFdD']: + ytype = t.dtype.char + else: + ytype = 'd' + y = zeros(t.shape, ytype) + + # width must be between 0 and 1 inclusive + mask1 = (w > 1) | (w < 0) + place(y, mask1, nan) + + # take t modulo 2*pi + tmod = mod(t, 2 * pi) + + # on the interval 0 to width*2*pi function is + # tmod / (pi*w) - 1 + mask2 = (1 - mask1) & (tmod < w * 2 * pi) + tsub = extract(mask2, tmod) + wsub = extract(mask2, w) + place(y, mask2, tsub / (pi * wsub) - 1) + + # on the interval width*2*pi to 2*pi function is + # (pi*(w+1)-tmod) / (pi*(1-w)) + + mask3 = (1 - mask1) & (1 - mask2) + tsub = extract(mask3, tmod) + wsub = extract(mask3, w) + place(y, mask3, (pi * (wsub + 1) - tsub) / (pi * (1 - wsub))) + return y + + +def square(t, duty=0.5): + """ + Return a periodic square-wave waveform. + + The square wave has a period ``2*pi``, has value +1 from 0 to + ``2*pi*duty`` and -1 from ``2*pi*duty`` to ``2*pi``. `duty` must be in + the interval [0,1]. + + Note that this is not band-limited. It produces an infinite number + of harmonics, which are aliased back and forth across the frequency + spectrum. + + Parameters + ---------- + t : array_like + The input time array. + duty : array_like, optional + Duty cycle. Default is 0.5 (50% duty cycle). + If an array, causes wave shape to change over time, and must be the + same length as t. + + Returns + ------- + y : ndarray + Output array containing the square waveform. + + Examples + -------- + A 5 Hz waveform sampled at 500 Hz for 1 second: + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> t = np.linspace(0, 1, 500, endpoint=False) + >>> plt.plot(t, signal.square(2 * np.pi * 5 * t)) + >>> plt.ylim(-2, 2) + + A pulse-width modulated sine wave: + + >>> plt.figure() + >>> sig = np.sin(2 * np.pi * t) + >>> pwm = signal.square(2 * np.pi * 30 * t, duty=(sig + 1)/2) + >>> plt.subplot(2, 1, 1) + >>> plt.plot(t, sig) + >>> plt.subplot(2, 1, 2) + >>> plt.plot(t, pwm) + >>> plt.ylim(-1.5, 1.5) + + """ + t, w = asarray(t), asarray(duty) + w = asarray(w + (t - t)) + t = asarray(t + (w - w)) + if t.dtype.char in ['fFdD']: + ytype = t.dtype.char + else: + ytype = 'd' + + y = zeros(t.shape, ytype) + + # width must be between 0 and 1 inclusive + mask1 = (w > 1) | (w < 0) + place(y, mask1, nan) + + # on the interval 0 to duty*2*pi function is 1 + tmod = mod(t, 2 * pi) + mask2 = (1 - mask1) & (tmod < w * 2 * pi) + place(y, mask2, 1) + + # on the interval duty*2*pi to 2*pi function is + # (pi*(w+1)-tmod) / (pi*(1-w)) + mask3 = (1 - mask1) & (1 - mask2) + place(y, mask3, -1) + return y + + +def gausspulse(t, fc=1000, bw=0.5, bwr=-6, tpr=-60, retquad=False, + retenv=False): + """ + Return a Gaussian modulated sinusoid: + + ``exp(-a t^2) exp(1j*2*pi*fc*t).`` + + If `retquad` is True, then return the real and imaginary parts + (in-phase and quadrature). + If `retenv` is True, then return the envelope (unmodulated signal). + Otherwise, return the real part of the modulated sinusoid. + + Parameters + ---------- + t : ndarray or the string 'cutoff' + Input array. + fc : float, optional + Center frequency (e.g. Hz). Default is 1000. + bw : float, optional + Fractional bandwidth in frequency domain of pulse (e.g. Hz). + Default is 0.5. + bwr : float, optional + Reference level at which fractional bandwidth is calculated (dB). + Default is -6. + tpr : float, optional + If `t` is 'cutoff', then the function returns the cutoff + time for when the pulse amplitude falls below `tpr` (in dB). + Default is -60. + retquad : bool, optional + If True, return the quadrature (imaginary) as well as the real part + of the signal. Default is False. + retenv : bool, optional + If True, return the envelope of the signal. Default is False. + + Returns + ------- + yI : ndarray + Real part of signal. Always returned. + yQ : ndarray + Imaginary part of signal. Only returned if `retquad` is True. + yenv : ndarray + Envelope of signal. Only returned if `retenv` is True. + + Examples + -------- + Plot real component, imaginary component, and envelope for a 5 Hz pulse, + sampled at 100 Hz for 2 seconds: + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> t = np.linspace(-1, 1, 2 * 100, endpoint=False) + >>> i, q, e = signal.gausspulse(t, fc=5, retquad=True, retenv=True) + >>> plt.plot(t, i, t, q, t, e, '--') + + """ + if fc < 0: + raise ValueError(f"Center frequency (fc={fc:.2f}) must be >=0.") + if bw <= 0: + raise ValueError(f"Fractional bandwidth (bw={bw:.2f}) must be > 0.") + if bwr >= 0: + raise ValueError(f"Reference level for bandwidth (bwr={bwr:.2f}) " + "must be < 0 dB") + + # exp(-a t^2) <-> sqrt(pi/a) exp(-pi^2/a * f^2) = g(f) + + ref = pow(10.0, bwr / 20.0) + # fdel = fc*bw/2: g(fdel) = ref --- solve this for a + # + # pi^2/a * fc^2 * bw^2 /4=-log(ref) + a = -(pi * fc * bw) ** 2 / (4.0 * log(ref)) + + if isinstance(t, str): + if t == 'cutoff': # compute cut_off point + # Solve exp(-a tc**2) = tref for tc + # tc = sqrt(-log(tref) / a) where tref = 10^(tpr/20) + if tpr >= 0: + raise ValueError("Reference level for time cutoff must " + "be < 0 dB") + tref = pow(10.0, tpr / 20.0) + return sqrt(-log(tref) / a) + else: + raise ValueError("If `t` is a string, it must be 'cutoff'") + + yenv = exp(-a * t * t) + yI = yenv * cos(2 * pi * fc * t) + yQ = yenv * sin(2 * pi * fc * t) + if not retquad and not retenv: + return yI + if not retquad and retenv: + return yI, yenv + if retquad and not retenv: + return yI, yQ + if retquad and retenv: + return yI, yQ, yenv + + +def chirp(t, f0, t1, f1, method='linear', phi=0, vertex_zero=True, *, + complex=False): + r"""Frequency-swept cosine generator. + + In the following, 'Hz' should be interpreted as 'cycles per unit'; + there is no requirement here that the unit is one second. The + important distinction is that the units of rotation are cycles, not + radians. Likewise, `t` could be a measurement of space instead of time. + + Parameters + ---------- + t : array_like + Times at which to evaluate the waveform. + f0 : float + Frequency (e.g. Hz) at time t=0. + t1 : float + Time at which `f1` is specified. + f1 : float + Frequency (e.g. Hz) of the waveform at time `t1`. + method : {'linear', 'quadratic', 'logarithmic', 'hyperbolic'}, optional + Kind of frequency sweep. If not given, `linear` is assumed. See + Notes below for more details. + phi : float, optional + Phase offset, in degrees. Default is 0. + vertex_zero : bool, optional + This parameter is only used when `method` is 'quadratic'. + It determines whether the vertex of the parabola that is the graph + of the frequency is at t=0 or t=t1. + complex : bool, optional + This parameter creates a complex-valued analytic signal instead of a + real-valued signal. It allows the use of complex baseband (in communications + domain). Default is False. + + .. versionadded:: 1.15.0 + + Returns + ------- + y : ndarray + A numpy array containing the signal evaluated at `t` with the requested + time-varying frequency. More precisely, the function returns + ``exp(1j*phase + 1j*(pi/180)*phi) if complex else cos(phase + (pi/180)*phi)`` + where `phase` is the integral (from 0 to `t`) of ``2*pi*f(t)``. + The instantaneous frequency ``f(t)`` is defined below. + + See Also + -------- + sweep_poly + + Notes + ----- + There are four possible options for the parameter `method`, which have a (long) + standard form and some allowed abbreviations. The formulas for the instantaneous + frequency :math:`f(t)` of the generated signal are as follows: + + 1. Parameter `method` in ``('linear', 'lin', 'li')``: + + .. math:: + f(t) = f_0 + \beta\, t \quad\text{with}\quad + \beta = \frac{f_1 - f_0}{t_1} + + Frequency :math:`f(t)` varies linearly over time with a constant rate + :math:`\beta`. + + 2. Parameter `method` in ``('quadratic', 'quad', 'q')``: + + .. math:: + f(t) = + \begin{cases} + f_0 + \beta\, t^2 & \text{if vertex_zero is True,}\\ + f_1 + \beta\, (t_1 - t)^2 & \text{otherwise,} + \end{cases} + \quad\text{with}\quad + \beta = \frac{f_1 - f_0}{t_1^2} + + The graph of the frequency f(t) is a parabola through :math:`(0, f_0)` and + :math:`(t_1, f_1)`. By default, the vertex of the parabola is at + :math:`(0, f_0)`. If `vertex_zero` is ``False``, then the vertex is at + :math:`(t_1, f_1)`. + To use a more general quadratic function, or an arbitrary + polynomial, use the function `scipy.signal.sweep_poly`. + + 3. Parameter `method` in ``('logarithmic', 'log', 'lo')``: + + .. math:: + f(t) = f_0 \left(\frac{f_1}{f_0}\right)^{t/t_1} + + :math:`f_0` and :math:`f_1` must be nonzero and have the same sign. + This signal is also known as a geometric or exponential chirp. + + 4. Parameter `method` in ``('hyperbolic', 'hyp')``: + + .. math:: + f(t) = \frac{\alpha}{\beta\, t + \gamma} \quad\text{with}\quad + \alpha = f_0 f_1 t_1, \ \beta = f_0 - f_1, \ \gamma = f_1 t_1 + + :math:`f_0` and :math:`f_1` must be nonzero. + + + Examples + -------- + For the first example, a linear chirp ranging from 6 Hz to 1 Hz over 10 seconds is + plotted: + + >>> import numpy as np + >>> from matplotlib.pyplot import tight_layout + >>> from scipy.signal import chirp, square, ShortTimeFFT + >>> from scipy.signal.windows import gaussian + >>> import matplotlib.pyplot as plt + ... + >>> N, T = 1000, 0.01 # number of samples and sampling interval for 10 s signal + >>> t = np.arange(N) * T # timestamps + ... + >>> x_lin = chirp(t, f0=6, f1=1, t1=10, method='linear') + ... + >>> fg0, ax0 = plt.subplots() + >>> ax0.set_title(r"Linear Chirp from $f(0)=6\,$Hz to $f(10)=1\,$Hz") + >>> ax0.set(xlabel="Time $t$ in Seconds", ylabel=r"Amplitude $x_\text{lin}(t)$") + >>> ax0.plot(t, x_lin) + >>> plt.show() + + The following four plots each show the short-time Fourier transform of a chirp + ranging from 45 Hz to 5 Hz with different values for the parameter `method` + (and `vertex_zero`): + + >>> x_qu0 = chirp(t, f0=45, f1=5, t1=N*T, method='quadratic', vertex_zero=True) + >>> x_qu1 = chirp(t, f0=45, f1=5, t1=N*T, method='quadratic', vertex_zero=False) + >>> x_log = chirp(t, f0=45, f1=5, t1=N*T, method='logarithmic') + >>> x_hyp = chirp(t, f0=45, f1=5, t1=N*T, method='hyperbolic') + ... + >>> win = gaussian(50, std=12, sym=True) + >>> SFT = ShortTimeFFT(win, hop=2, fs=1/T, mfft=800, scale_to='magnitude') + >>> ts = ("'quadratic', vertex_zero=True", "'quadratic', vertex_zero=False", + ... "'logarithmic'", "'hyperbolic'") + >>> fg1, ax1s = plt.subplots(2, 2, sharex='all', sharey='all', + ... figsize=(6, 5), layout="constrained") + >>> for x_, ax_, t_ in zip([x_qu0, x_qu1, x_log, x_hyp], ax1s.ravel(), ts): + ... aSx = abs(SFT.stft(x_)) + ... im_ = ax_.imshow(aSx, origin='lower', aspect='auto', extent=SFT.extent(N), + ... cmap='plasma') + ... ax_.set_title(t_) + ... if t_ == "'hyperbolic'": + ... fg1.colorbar(im_, ax=ax1s, label='Magnitude $|S_z(t,f)|$') + >>> _ = fg1.supxlabel("Time $t$ in Seconds") # `_ =` is needed to pass doctests + >>> _ = fg1.supylabel("Frequency $f$ in Hertz") + >>> plt.show() + + Finally, the short-time Fourier transform of a complex-valued linear chirp + ranging from -30 Hz to 30 Hz is depicted: + + >>> z_lin = chirp(t, f0=-30, f1=30, t1=N*T, method="linear", complex=True) + >>> SFT.fft_mode = 'centered' # needed to work with complex signals + >>> aSz = abs(SFT.stft(z_lin)) + ... + >>> fg2, ax2 = plt.subplots() + >>> ax2.set_title(r"Linear Chirp from $-30\,$Hz to $30\,$Hz") + >>> ax2.set(xlabel="Time $t$ in Seconds", ylabel="Frequency $f$ in Hertz") + >>> im2 = ax2.imshow(aSz, origin='lower', aspect='auto', + ... extent=SFT.extent(N), cmap='viridis') + >>> fg2.colorbar(im2, label='Magnitude $|S_z(t,f)|$') + >>> plt.show() + + Note that using negative frequencies makes only sense with complex-valued signals. + Furthermore, the magnitude of the complex exponential function is one whereas the + magnitude of the real-valued cosine function is only 1/2. + """ + # 'phase' is computed in _chirp_phase, to make testing easier. + phase = _chirp_phase(t, f0, t1, f1, method, vertex_zero) + np.deg2rad(phi) + return np.exp(1j*phase) if complex else np.cos(phase) + + +def _chirp_phase(t, f0, t1, f1, method='linear', vertex_zero=True): + """ + Calculate the phase used by `chirp` to generate its output. + + See `chirp` for a description of the arguments. + + """ + t = asarray(t) + f0 = float(f0) + t1 = float(t1) + f1 = float(f1) + if method in ['linear', 'lin', 'li']: + beta = (f1 - f0) / t1 + phase = 2 * pi * (f0 * t + 0.5 * beta * t * t) + + elif method in ['quadratic', 'quad', 'q']: + beta = (f1 - f0) / (t1 ** 2) + if vertex_zero: + phase = 2 * pi * (f0 * t + beta * t ** 3 / 3) + else: + phase = 2 * pi * (f1 * t + beta * ((t1 - t) ** 3 - t1 ** 3) / 3) + + elif method in ['logarithmic', 'log', 'lo']: + if f0 * f1 <= 0.0: + raise ValueError("For a logarithmic chirp, f0 and f1 must be " + "nonzero and have the same sign.") + if f0 == f1: + phase = 2 * pi * f0 * t + else: + beta = t1 / log(f1 / f0) + phase = 2 * pi * beta * f0 * (pow(f1 / f0, t / t1) - 1.0) + + elif method in ['hyperbolic', 'hyp']: + if f0 == 0 or f1 == 0: + raise ValueError("For a hyperbolic chirp, f0 and f1 must be " + "nonzero.") + if f0 == f1: + # Degenerate case: constant frequency. + phase = 2 * pi * f0 * t + else: + # Singular point: the instantaneous frequency blows up + # when t == sing. + sing = -f1 * t1 / (f0 - f1) + phase = 2 * pi * (-sing * f0) * log(np.abs(1 - t/sing)) + + else: + raise ValueError("method must be 'linear', 'quadratic', 'logarithmic', " + f"or 'hyperbolic', but a value of {method!r} was given.") + + return phase + + +def sweep_poly(t, poly, phi=0): + """ + Frequency-swept cosine generator, with a time-dependent frequency. + + This function generates a sinusoidal function whose instantaneous + frequency varies with time. The frequency at time `t` is given by + the polynomial `poly`. + + Parameters + ---------- + t : ndarray + Times at which to evaluate the waveform. + poly : 1-D array_like or instance of numpy.poly1d + The desired frequency expressed as a polynomial. If `poly` is + a list or ndarray of length n, then the elements of `poly` are + the coefficients of the polynomial, and the instantaneous + frequency is + + ``f(t) = poly[0]*t**(n-1) + poly[1]*t**(n-2) + ... + poly[n-1]`` + + If `poly` is an instance of numpy.poly1d, then the + instantaneous frequency is + + ``f(t) = poly(t)`` + + phi : float, optional + Phase offset, in degrees, Default: 0. + + Returns + ------- + sweep_poly : ndarray + A numpy array containing the signal evaluated at `t` with the + requested time-varying frequency. More precisely, the function + returns ``cos(phase + (pi/180)*phi)``, where `phase` is the integral + (from 0 to t) of ``2 * pi * f(t)``; ``f(t)`` is defined above. + + See Also + -------- + chirp + + Notes + ----- + .. versionadded:: 0.8.0 + + If `poly` is a list or ndarray of length `n`, then the elements of + `poly` are the coefficients of the polynomial, and the instantaneous + frequency is: + + ``f(t) = poly[0]*t**(n-1) + poly[1]*t**(n-2) + ... + poly[n-1]`` + + If `poly` is an instance of `numpy.poly1d`, then the instantaneous + frequency is: + + ``f(t) = poly(t)`` + + Finally, the output `s` is: + + ``cos(phase + (pi/180)*phi)`` + + where `phase` is the integral from 0 to `t` of ``2 * pi * f(t)``, + ``f(t)`` as defined above. + + Examples + -------- + Compute the waveform with instantaneous frequency:: + + f(t) = 0.025*t**3 - 0.36*t**2 + 1.25*t + 2 + + over the interval 0 <= t <= 10. + + >>> import numpy as np + >>> from scipy.signal import sweep_poly + >>> p = np.poly1d([0.025, -0.36, 1.25, 2.0]) + >>> t = np.linspace(0, 10, 5001) + >>> w = sweep_poly(t, p) + + Plot it: + + >>> import matplotlib.pyplot as plt + >>> plt.subplot(2, 1, 1) + >>> plt.plot(t, w) + >>> plt.title("Sweep Poly\\nwith frequency " + + ... "$f(t) = 0.025t^3 - 0.36t^2 + 1.25t + 2$") + >>> plt.subplot(2, 1, 2) + >>> plt.plot(t, p(t), 'r', label='f(t)') + >>> plt.legend() + >>> plt.xlabel('t') + >>> plt.tight_layout() + >>> plt.show() + + """ + # 'phase' is computed in _sweep_poly_phase, to make testing easier. + phase = _sweep_poly_phase(t, poly) + # Convert to radians. + phi *= pi / 180 + return cos(phase + phi) + + +def _sweep_poly_phase(t, poly): + """ + Calculate the phase used by sweep_poly to generate its output. + + See `sweep_poly` for a description of the arguments. + + """ + # polyint handles lists, ndarrays and instances of poly1d automatically. + intpoly = polyint(poly) + phase = 2 * pi * polyval(intpoly, t) + return phase + + +def unit_impulse(shape, idx=None, dtype=float): + r""" + Unit impulse signal (discrete delta function) or unit basis vector. + + Parameters + ---------- + shape : int or tuple of int + Number of samples in the output (1-D), or a tuple that represents the + shape of the output (N-D). + idx : None or int or tuple of int or 'mid', optional + Index at which the value is 1. If None, defaults to the 0th element. + If ``idx='mid'``, the impulse will be centered at ``shape // 2`` in + all dimensions. If an int, the impulse will be at `idx` in all + dimensions. + dtype : data-type, optional + The desired data-type for the array, e.g., ``numpy.int8``. Default is + ``numpy.float64``. + + Returns + ------- + y : ndarray + Output array containing an impulse signal. + + Notes + ----- + In digital signal processing literature the unit impulse signal is often + represented by the Kronecker delta. [1]_ I.e., a signal :math:`u_k[n]`, + which is zero everywhere except being one at the :math:`k`-th sample, + can be expressed as + + .. math:: + + u_k[n] = \delta[n-k] \equiv \delta_{n,k}\ . + + Furthermore, the unit impulse is frequently interpreted as the discrete-time + version of the continuous-time Dirac distribution. [2]_ + + References + ---------- + .. [1] "Kronecker delta", *Wikipedia*, + https://en.wikipedia.org/wiki/Kronecker_delta#Digital_signal_processing + .. [2] "Dirac delta function" *Wikipedia*, + https://en.wikipedia.org/wiki/Dirac_delta_function#Relationship_to_the_Kronecker_delta + + .. versionadded:: 0.19.0 + + Examples + -------- + An impulse at the 0th element (:math:`\\delta[n]`): + + >>> from scipy import signal + >>> signal.unit_impulse(8) + array([ 1., 0., 0., 0., 0., 0., 0., 0.]) + + Impulse offset by 2 samples (:math:`\\delta[n-2]`): + + >>> signal.unit_impulse(7, 2) + array([ 0., 0., 1., 0., 0., 0., 0.]) + + 2-dimensional impulse, centered: + + >>> signal.unit_impulse((3, 3), 'mid') + array([[ 0., 0., 0.], + [ 0., 1., 0.], + [ 0., 0., 0.]]) + + Impulse at (2, 2), using broadcasting: + + >>> signal.unit_impulse((4, 4), 2) + array([[ 0., 0., 0., 0.], + [ 0., 0., 0., 0.], + [ 0., 0., 1., 0.], + [ 0., 0., 0., 0.]]) + + Plot the impulse response of a 4th-order Butterworth lowpass filter: + + >>> imp = signal.unit_impulse(100, 'mid') + >>> b, a = signal.butter(4, 0.2) + >>> response = signal.lfilter(b, a, imp) + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> plt.plot(np.arange(-50, 50), imp) + >>> plt.plot(np.arange(-50, 50), response) + >>> plt.margins(0.1, 0.1) + >>> plt.xlabel('Time [samples]') + >>> plt.ylabel('Amplitude') + >>> plt.grid(True) + >>> plt.show() + + """ + out = zeros(shape, dtype) + + shape = np.atleast_1d(shape) + + if idx is None: + idx = (0,) * len(shape) + elif idx == 'mid': + idx = tuple(shape // 2) + elif not hasattr(idx, "__iter__"): + idx = (idx,) * len(shape) + + out[idx] = 1 + return out diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_wavelets.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_wavelets.py new file mode 100644 index 0000000000000000000000000000000000000000..2b9f8fa32672e3f252f0f4ec4e387e0d474dc21e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/_wavelets.py @@ -0,0 +1,29 @@ +import numpy as np +from scipy.signal import convolve + + +def _ricker(points, a): + A = 2 / (np.sqrt(3 * a) * (np.pi**0.25)) + wsq = a**2 + vec = np.arange(0, points) - (points - 1.0) / 2 + xsq = vec**2 + mod = (1 - xsq / wsq) + gauss = np.exp(-xsq / (2 * wsq)) + total = A * mod * gauss + return total + + +def _cwt(data, wavelet, widths, dtype=None, **kwargs): + # Determine output type + if dtype is None: + if np.asarray(wavelet(1, widths[0], **kwargs)).dtype.char in 'FDG': + dtype = np.complex128 + else: + dtype = np.float64 + + output = np.empty((len(widths), len(data)), dtype=dtype) + for ind, width in enumerate(widths): + N = np.min([10 * width, len(data)]) + wavelet_data = np.conj(wavelet(N, width, **kwargs)[::-1]) + output[ind] = convolve(data, wavelet_data, mode='same') + return output diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/bsplines.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/bsplines.py new file mode 100644 index 0000000000000000000000000000000000000000..0328d45c107bda78cbdbd374148237ca09ac411d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/bsplines.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.signal` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'spline_filter', 'gauss_spline', + 'cspline1d', 'qspline1d', 'cspline1d_eval', 'qspline1d_eval', + 'cspline2d', 'sepfir2d' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="signal", module="bsplines", + private_modules=["_spline_filters"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/filter_design.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/filter_design.py new file mode 100644 index 0000000000000000000000000000000000000000..41dc230a7f24a7ac3209f821d8d0f9417130afbd --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/filter_design.py @@ -0,0 +1,28 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.signal` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'findfreqs', 'freqs', 'freqz', 'tf2zpk', 'zpk2tf', 'normalize', + 'lp2lp', 'lp2hp', 'lp2bp', 'lp2bs', 'bilinear', 'iirdesign', + 'iirfilter', 'butter', 'cheby1', 'cheby2', 'ellip', 'bessel', + 'band_stop_obj', 'buttord', 'cheb1ord', 'cheb2ord', 'ellipord', + 'buttap', 'cheb1ap', 'cheb2ap', 'ellipap', 'besselap', + 'BadCoefficients', 'freqs_zpk', 'freqz_zpk', + 'tf2sos', 'sos2tf', 'zpk2sos', 'sos2zpk', 'group_delay', + 'sosfreqz', 'freqz_sos', 'iirnotch', 'iirpeak', 'bilinear_zpk', + 'lp2lp_zpk', 'lp2hp_zpk', 'lp2bp_zpk', 'lp2bs_zpk', + 'gammatone', 'iircomb', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="signal", module="filter_design", + private_modules=["_filter_design"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/fir_filter_design.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/fir_filter_design.py new file mode 100644 index 0000000000000000000000000000000000000000..2214b82998bdefd2c6d6171cc952adf827269736 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/fir_filter_design.py @@ -0,0 +1,20 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.signal` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'kaiser_beta', 'kaiser_atten', 'kaiserord', + 'firwin', 'firwin2', 'remez', 'firls', 'minimum_phase', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="signal", module="fir_filter_design", + private_modules=["_fir_filter_design"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/lti_conversion.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/lti_conversion.py new file mode 100644 index 0000000000000000000000000000000000000000..7080990afc9e23e51e8a45aaa146b64c58dda3cf --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/lti_conversion.py @@ -0,0 +1,20 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.signal` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'tf2ss', 'abcd_normalize', 'ss2tf', 'zpk2ss', 'ss2zpk', + 'cont2discrete', 'tf2zpk', 'zpk2tf', 'normalize' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="signal", module="lti_conversion", + private_modules=["_lti_conversion"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/ltisys.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/ltisys.py new file mode 100644 index 0000000000000000000000000000000000000000..5123068de559f124bf444c12ef9824c3d14de64f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/ltisys.py @@ -0,0 +1,25 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.signal` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'lti', 'dlti', 'TransferFunction', 'ZerosPolesGain', 'StateSpace', + 'lsim', 'impulse', 'step', 'bode', + 'freqresp', 'place_poles', 'dlsim', 'dstep', 'dimpulse', + 'dfreqresp', 'dbode', + 'tf2zpk', 'zpk2tf', 'normalize', 'freqs', + 'freqz', 'freqs_zpk', 'freqz_zpk', 'tf2ss', 'abcd_normalize', + 'ss2tf', 'zpk2ss', 'ss2zpk', 'cont2discrete', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="signal", module="ltisys", + private_modules=["_ltisys"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/signaltools.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/signaltools.py new file mode 100644 index 0000000000000000000000000000000000000000..85d426f5fb2605c639fc6dbd1b4d0284a3f11e1b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/signaltools.py @@ -0,0 +1,27 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.signal` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'correlate', 'correlation_lags', 'correlate2d', + 'convolve', 'convolve2d', 'fftconvolve', 'oaconvolve', + 'order_filter', 'medfilt', 'medfilt2d', 'wiener', 'lfilter', + 'lfiltic', 'sosfilt', 'deconvolve', 'hilbert', 'hilbert2', + 'unique_roots', 'invres', 'invresz', 'residue', + 'residuez', 'resample', 'resample_poly', 'detrend', + 'lfilter_zi', 'sosfilt_zi', 'sosfiltfilt', 'choose_conv_method', + 'filtfilt', 'decimate', 'vectorstrength', + 'dlti', 'upfirdn', 'get_window', 'cheby1', 'firwin' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="signal", module="signaltools", + private_modules=["_signaltools"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/spectral.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/spectral.py new file mode 100644 index 0000000000000000000000000000000000000000..299ebed781b00a1f1f35e96c54f4c20d9bd9d0fc --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/spectral.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.signal` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'periodogram', 'welch', 'lombscargle', 'csd', 'coherence', + 'spectrogram', 'stft', 'istft', 'check_COLA', 'check_NOLA', + 'get_window', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="signal", module="spectral", + private_modules=["_spectral_py"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/spline.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/spline.py new file mode 100644 index 0000000000000000000000000000000000000000..7afd0d0a14beecd5bc4522050eaf3b195f1a3601 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/spline.py @@ -0,0 +1,25 @@ +# This file is not meant for public use and will be removed in the future +# versions of SciPy. Use the `scipy.signal` namespace for importing the +# functions included below. + +import warnings + +from . import _spline + +__all__ = ['sepfir2d'] # noqa: F822 + + +def __dir__(): + return __all__ + + +def __getattr__(name): + if name not in __all__: + raise AttributeError( + f"scipy.signal.spline is deprecated and has no attribute {name}. " + "Try looking in scipy.signal instead.") + + warnings.warn(f"Please use `{name}` from the `scipy.signal` namespace, " + "the `scipy.signal.spline` namespace is deprecated.", + category=DeprecationWarning, stacklevel=2) + return getattr(_spline, name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/_scipy_spectral_test_shim.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/_scipy_spectral_test_shim.py new file mode 100644 index 0000000000000000000000000000000000000000..42d3d830d0e39797832472b9e039259111cfccc8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/_scipy_spectral_test_shim.py @@ -0,0 +1,480 @@ +"""Helpers to utilize existing stft / istft tests for testing `ShortTimeFFT`. + +This module provides the functions stft_compare() and istft_compare(), which, +compares the output between the existing (i)stft() and the shortTimeFFT based +_(i)stft_wrapper() implementations in this module. + +For testing add the following imports to the file ``tests/test_spectral.py``:: + + from ._scipy_spectral_test_shim import stft_compare as stft + from ._scipy_spectral_test_shim import istft_compare as istft + +and remove the existing imports of stft and istft. + +The idea of these wrappers is not to provide a backward-compatible interface +but to demonstrate that the ShortTimeFFT implementation is at least as capable +as the existing one and delivers comparable results. Furthermore, the +wrappers highlight the different philosophies of the implementations, +especially in the border handling. +""" +import platform +from typing import cast, Literal + +import numpy as np +from numpy.testing import assert_allclose + +from scipy.signal import ShortTimeFFT +from scipy.signal import csd, get_window, stft, istft +from scipy.signal._arraytools import const_ext, even_ext, odd_ext, zero_ext +from scipy.signal._short_time_fft import FFT_MODE_TYPE +from scipy.signal._spectral_py import _spectral_helper, _triage_segments, \ + _median_bias + + +def _stft_wrapper(x, fs=1.0, window='hann', nperseg=256, noverlap=None, + nfft=None, detrend=False, return_onesided=True, + boundary='zeros', padded=True, axis=-1, scaling='spectrum'): + """Wrapper for the SciPy `stft()` function based on `ShortTimeFFT` for + unit testing. + + Handling the boundary and padding is where `ShortTimeFFT` and `stft()` + differ in behavior. Parts of `_spectral_helper()` were copied to mimic + the` stft()` behavior. + + This function is meant to be solely used by `stft_compare()`. + """ + if scaling not in ('psd', 'spectrum'): # same errors as in original stft: + raise ValueError(f"Parameter {scaling=} not in ['spectrum', 'psd']!") + + # The following lines are taken from the original _spectral_helper(): + boundary_funcs = {'even': even_ext, + 'odd': odd_ext, + 'constant': const_ext, + 'zeros': zero_ext, + None: None} + + if boundary not in boundary_funcs: + raise ValueError(f"Unknown boundary option '{boundary}', must be one" + + f" of: {list(boundary_funcs.keys())}") + if x.size == 0: + return np.empty(x.shape), np.empty(x.shape), np.empty(x.shape) + + if nperseg is not None: # if specified by user + nperseg = int(nperseg) + if nperseg < 1: + raise ValueError('nperseg must be a positive integer') + + # parse window; if array like, then set nperseg = win.shape + win, nperseg = _triage_segments(window, nperseg, + input_length=x.shape[axis]) + + if nfft is None: + nfft = nperseg + elif nfft < nperseg: + raise ValueError('nfft must be greater than or equal to nperseg.') + else: + nfft = int(nfft) + + if noverlap is None: + noverlap = nperseg//2 + else: + noverlap = int(noverlap) + if noverlap >= nperseg: + raise ValueError('noverlap must be less than nperseg.') + nstep = nperseg - noverlap + n = x.shape[axis] + + # Padding occurs after boundary extension, so that the extended signal ends + # in zeros, instead of introducing an impulse at the end. + # I.e. if x = [..., 3, 2] + # extend then pad -> [..., 3, 2, 2, 3, 0, 0, 0] + # pad then extend -> [..., 3, 2, 0, 0, 0, 2, 3] + + if boundary is not None: + ext_func = boundary_funcs[boundary] + # Extend by nperseg//2 in front and back: + x = ext_func(x, nperseg//2, axis=axis) + + if padded: + # Pad to integer number of windowed segments + # I.e make x.shape[-1] = nperseg + (nseg-1)*nstep, with integer nseg + x = np.moveaxis(x, axis, -1) + + # This is an edge case where shortTimeFFT returns one more time slice + # than the Scipy stft() shorten to remove last time slice: + if n % 2 == 1 and nperseg % 2 == 1 and noverlap % 2 == 1: + x = x[..., : -1] + + nadd = (-(x.shape[-1]-nperseg) % nstep) % nperseg + zeros_shape = list(x.shape[:-1]) + [nadd] + x = np.concatenate((x, np.zeros(zeros_shape)), axis=-1) + x = np.moveaxis(x, -1, axis) + + # ... end original _spectral_helper() code. + scale_to = {'spectrum': 'magnitude', 'psd': 'psd'}[scaling] + + if np.iscomplexobj(x) and return_onesided: + return_onesided = False + # using cast() to make mypy happy: + fft_mode = cast(FFT_MODE_TYPE, 'onesided' if return_onesided else 'twosided') + + ST = ShortTimeFFT(win, nstep, fs, fft_mode=fft_mode, mfft=nfft, + scale_to=scale_to, phase_shift=None) + + k_off = nperseg // 2 + p0 = 0 # ST.lower_border_end[1] + 1 + nn = x.shape[axis] if padded else n+k_off+1 + # number of frames akin to legacy stft computation + p1 = (x.shape[axis] - nperseg) // nstep + 1 + + detr = None if detrend is False else detrend + Sxx = ST.stft_detrend(x, detr, p0, p1, k_offset=k_off, axis=axis) + t = ST.t(nn, 0, p1 - p0, k_offset=0 if boundary is not None else k_off) + if x.dtype in (np.float32, np.complex64): + Sxx = Sxx.astype(np.complex64) + + return ST.f, t, Sxx + + +def _istft_wrapper(Zxx, fs=1.0, window='hann', nperseg=None, noverlap=None, + nfft=None, input_onesided=True, boundary=True, time_axis=-1, + freq_axis=-2, scaling='spectrum') -> \ + tuple[np.ndarray, np.ndarray, tuple[int, int]]: + """Wrapper for the SciPy `istft()` function based on `ShortTimeFFT` for + unit testing. + + Note that only option handling is implemented as far as to handle the unit + tests. E.g., the case ``nperseg=None`` is not handled. + + This function is meant to be solely used by `istft_compare()`. + """ + # *** Lines are taken from _spectral_py.istft() ***: + if Zxx.ndim < 2: + raise ValueError('Input stft must be at least 2d!') + + if freq_axis == time_axis: + raise ValueError('Must specify differing time and frequency axes!') + + nseg = Zxx.shape[time_axis] + + if input_onesided: + # Assume even segment length + n_default = 2*(Zxx.shape[freq_axis] - 1) + else: + n_default = Zxx.shape[freq_axis] + + # Check windowing parameters + if nperseg is None: + nperseg = n_default + else: + nperseg = int(nperseg) + if nperseg < 1: + raise ValueError('nperseg must be a positive integer') + + if nfft is None: + if input_onesided and (nperseg == n_default + 1): + # Odd nperseg, no FFT padding + nfft = nperseg + else: + nfft = n_default + elif nfft < nperseg: + raise ValueError('nfft must be greater than or equal to nperseg.') + else: + nfft = int(nfft) + + if noverlap is None: + noverlap = nperseg//2 + else: + noverlap = int(noverlap) + if noverlap >= nperseg: + raise ValueError('noverlap must be less than nperseg.') + nstep = nperseg - noverlap + + # Get window as array + if isinstance(window, str) or type(window) is tuple: + win = get_window(window, nperseg) + else: + win = np.asarray(window) + if len(win.shape) != 1: + raise ValueError('window must be 1-D') + if win.shape[0] != nperseg: + raise ValueError(f'window must have length of {nperseg}') + + outputlength = nperseg + (nseg-1)*nstep + # *** End block of: Taken from _spectral_py.istft() *** + + # Using cast() to make mypy happy: + fft_mode = cast(FFT_MODE_TYPE, 'onesided' if input_onesided else 'twosided') + scale_to = cast(Literal['magnitude', 'psd'], + {'spectrum': 'magnitude', 'psd': 'psd'}[scaling]) + + ST = ShortTimeFFT(win, nstep, fs, fft_mode=fft_mode, mfft=nfft, + scale_to=scale_to, phase_shift=None) + + if boundary: + j = nperseg if nperseg % 2 == 0 else nperseg - 1 + k0 = ST.k_min + nperseg // 2 + k1 = outputlength - j + k0 + else: + raise NotImplementedError("boundary=False does not make sense with" + + "ShortTimeFFT.istft()!") + + x = ST.istft(Zxx, k0=k0, k1=k1, f_axis=freq_axis, t_axis=time_axis) + t = np.arange(k1 - k0) * ST.T + k_hi = ST.upper_border_begin(k1 - k0)[0] + # using cast() to make mypy happy: + return t, x, (ST.lower_border_end[0], k_hi) + + +def _csd_wrapper(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, + nfft=None, detrend='constant', return_onesided=True, + scaling='density', axis=-1, average='mean'): + """Wrapper for the `csd()` function based on `ShortTimeFFT` for + unit testing. + """ + freqs, _, Pxy = _csd_test_shim(x, y, fs, window, nperseg, noverlap, nfft, + detrend, return_onesided, scaling, axis) + + # The following code is taken from csd(): + if len(Pxy.shape) >= 2 and Pxy.size > 0: + if Pxy.shape[-1] > 1: + if average == 'median': + # np.median must be passed real arrays for the desired result + bias = _median_bias(Pxy.shape[-1]) + if np.iscomplexobj(Pxy): + Pxy = (np.median(np.real(Pxy), axis=-1) + + 1j * np.median(np.imag(Pxy), axis=-1)) + else: + Pxy = np.median(Pxy, axis=-1) + Pxy /= bias + elif average == 'mean': + Pxy = Pxy.mean(axis=-1) + else: + raise ValueError(f'average must be "median" or "mean", got {average}') + else: + Pxy = np.reshape(Pxy, Pxy.shape[:-1]) + + return freqs, Pxy + + +def _csd_test_shim(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, + nfft=None, detrend='constant', return_onesided=True, + scaling='density', axis=-1): + """Compare output of _spectral_helper() and ShortTimeFFT, more + precisely _spect_helper_csd() for used in csd_wrapper(). + + The motivation of this function is to test if the ShortTimeFFT-based + wrapper `_spect_helper_csd()` returns the same values as `_spectral_helper`. + This function should only be usd by csd() in (unit) testing. + """ + freqs, t, Pxy = _spectral_helper(x, y, fs, window, nperseg, noverlap, nfft, + detrend, return_onesided, scaling, axis, + mode='psd') + freqs1, Pxy1 = _spect_helper_csd(x, y, fs, window, nperseg, noverlap, nfft, + detrend, return_onesided, scaling, axis) + + np.testing.assert_allclose(freqs1, freqs) + amax_Pxy = max(np.abs(Pxy).max(), 1) if Pxy.size else 1 + atol = np.finfo(Pxy.dtype).resolution * amax_Pxy # needed for large Pxy + # for c_ in range(Pxy.shape[-1]): + # np.testing.assert_allclose(Pxy1[:, c_], Pxy[:, c_], atol=atol) + np.testing.assert_allclose(Pxy1, Pxy, atol=atol) + return freqs, t, Pxy + + +def _spect_helper_csd(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, + nfft=None, detrend='constant', return_onesided=True, + scaling='density', axis=-1): + """Wrapper for replacing _spectral_helper() by using the ShortTimeFFT + for use by csd(). + + This function should be only used by _csd_test_shim() and is only useful + for testing the ShortTimeFFT implementation. + """ + + # The following lines are taken from the original _spectral_helper(): + same_data = y is x + axis = int(axis) + + # Ensure we have np.arrays, get outdtype + x = np.asarray(x) + if not same_data: + y = np.asarray(y) + # outdtype = np.result_type(x, y, np.complex64) + # else: + # outdtype = np.result_type(x, np.complex64) + + if not same_data: + # Check if we can broadcast the outer axes together + xouter = list(x.shape) + youter = list(y.shape) + xouter.pop(axis) + youter.pop(axis) + try: + outershape = np.broadcast(np.empty(xouter), np.empty(youter)).shape + except ValueError as e: + raise ValueError('x and y cannot be broadcast together.') from e + + if same_data: + if x.size == 0: + return np.empty(x.shape), np.empty(x.shape) + else: + if x.size == 0 or y.size == 0: + outshape = outershape + (min([x.shape[axis], y.shape[axis]]),) + emptyout = np.moveaxis(np.empty(outshape), -1, axis) + return emptyout, emptyout + + if nperseg is not None: # if specified by user + nperseg = int(nperseg) + if nperseg < 1: + raise ValueError('nperseg must be a positive integer') + + # parse window; if array like, then set nperseg = win.shape + n = x.shape[axis] if same_data else max(x.shape[axis], y.shape[axis]) + win, nperseg = _triage_segments(window, nperseg, input_length=n) + + if nfft is None: + nfft = nperseg + elif nfft < nperseg: + raise ValueError('nfft must be greater than or equal to nperseg.') + else: + nfft = int(nfft) + + if noverlap is None: + noverlap = nperseg // 2 + else: + noverlap = int(noverlap) + if noverlap >= nperseg: + raise ValueError('noverlap must be less than nperseg.') + nstep = nperseg - noverlap + + if np.iscomplexobj(x) and return_onesided: + return_onesided = False + + # using cast() to make mypy happy: + fft_mode = cast(FFT_MODE_TYPE, 'onesided' if return_onesided + else 'twosided') + scale = {'spectrum': 'magnitude', 'density': 'psd'}[scaling] + SFT = ShortTimeFFT(win, nstep, fs, fft_mode=fft_mode, mfft=nfft, + scale_to=scale, phase_shift=None) + + # _spectral_helper() calculates X.conj()*Y instead of X*Y.conj(): + Pxy = SFT.spectrogram(y, x, detr=None if detrend is False else detrend, + p0=0, p1=(n-noverlap)//SFT.hop, k_offset=nperseg//2, + axis=axis).conj() + # Note: + # 'onesided2X' scaling of ShortTimeFFT conflicts with the + # scaling='spectrum' parameter, since it doubles the squared magnitude, + # which in the view of the ShortTimeFFT implementation does not make sense. + # Hence, the doubling of the square is implemented here: + if return_onesided: + f_axis = Pxy.ndim - 1 + axis if axis < 0 else axis + Pxy = np.moveaxis(Pxy, f_axis, -1) + Pxy[..., 1:-1 if SFT.mfft % 2 == 0 else None] *= 2 + Pxy = np.moveaxis(Pxy, -1, f_axis) + + return SFT.f, Pxy + + +def stft_compare(x, fs=1.0, window='hann', nperseg=256, noverlap=None, + nfft=None, detrend=False, return_onesided=True, + boundary='zeros', padded=True, axis=-1, scaling='spectrum'): + """Assert that the results from the existing `stft()` and `_stft_wrapper()` + are close to each other. + + For comparing the STFT values an absolute tolerance of the floating point + resolution was added to circumvent problems with the following tests: + * For float32 the tolerances are much higher in + TestSTFT.test_roundtrip_float32()). + * The TestSTFT.test_roundtrip_scaling() has a high relative deviation. + Interestingly this did not appear in Scipy 1.9.1 but only in the current + development version. + """ + kw = dict(x=x, fs=fs, window=window, nperseg=nperseg, noverlap=noverlap, + nfft=nfft, detrend=detrend, return_onesided=return_onesided, + boundary=boundary, padded=padded, axis=axis, scaling=scaling) + f, t, Zxx = stft(**kw) + f_wrapper, t_wrapper, Zxx_wrapper = _stft_wrapper(**kw) + + e_msg_part = " of `stft_wrapper()` differ from `stft()`." + assert_allclose(f_wrapper, f, err_msg=f"Frequencies {e_msg_part}") + assert_allclose(t_wrapper, t, err_msg=f"Time slices {e_msg_part}") + + # Adapted tolerances to account for: + atol = np.finfo(Zxx.dtype).resolution * 2 + assert_allclose(Zxx_wrapper, Zxx, atol=atol, + err_msg=f"STFT values {e_msg_part}") + return f, t, Zxx + + +def istft_compare(Zxx, fs=1.0, window='hann', nperseg=None, noverlap=None, + nfft=None, input_onesided=True, boundary=True, time_axis=-1, + freq_axis=-2, scaling='spectrum'): + """Assert that the results from the existing `istft()` and + `_istft_wrapper()` are close to each other. + + Quirks: + * If ``boundary=False`` the comparison is skipped, since it does not + make sense with ShortTimeFFT.istft(). Only used in test + TestSTFT.test_roundtrip_boundary_extension(). + * If ShortTimeFFT.istft() decides the STFT is not invertible, the + comparison is skipped, since istft() only emits a warning and does not + return a correct result. Only used in + ShortTimeFFT.test_roundtrip_not_nola(). + * For comparing the signals an absolute tolerance of the floating point + resolution was added to account for the low accuracy of float32 (Occurs + only in TestSTFT.test_roundtrip_float32()). + """ + kw = dict(Zxx=Zxx, fs=fs, window=window, nperseg=nperseg, + noverlap=noverlap, nfft=nfft, input_onesided=input_onesided, + boundary=boundary, time_axis=time_axis, freq_axis=freq_axis, + scaling=scaling) + + t, x = istft(**kw) + if not boundary: # skip test_roundtrip_boundary_extension(): + return t, x # _istft_wrapper does() not implement this case + try: # if inversion fails, istft() only emits a warning: + t_wrapper, x_wrapper, (k_lo, k_hi) = _istft_wrapper(**kw) + except ValueError as v: # Do nothing if inversion fails: + if v.args[0] == "Short-time Fourier Transform not invertible!": + return t, x + raise v + + e_msg_part = " of `istft_wrapper()` differ from `istft()`" + assert_allclose(t, t_wrapper, err_msg=f"Sample times {e_msg_part}") + + # Adapted tolerances to account for resolution loss: + atol = np.finfo(x.dtype).resolution*2 # instead of default atol = 0 + rtol = 1e-7 # default for np.allclose() + + # Relax atol on 32-Bit platforms a bit to pass CI tests. + # - Not clear why there are discrepancies (in the FFT maybe?) + # - Not sure what changed on 'i686' since earlier on those test passed + if x.dtype == np.float32 and platform.machine() == 'i686': + # float32 gets only used by TestSTFT.test_roundtrip_float32() so + # we are using the tolerances from there to circumvent CI problems + atol, rtol = 1e-4, 1e-5 + elif platform.machine() in ('aarch64', 'i386', 'i686'): + atol = max(atol, 1e-12) # 2e-15 seems too tight for 32-Bit platforms + + assert_allclose(x_wrapper[k_lo:k_hi], x[k_lo:k_hi], atol=atol, rtol=rtol, + err_msg=f"Signal values {e_msg_part}") + return t, x + + +def csd_compare(x, y, fs=1.0, window='hann', nperseg=None, noverlap=None, + nfft=None, detrend='constant', return_onesided=True, + scaling='density', axis=-1, average='mean'): + """Assert that the results from the existing `csd()` and `_csd_wrapper()` + are close to each other. """ + kw = dict(x=x, y=y, fs=fs, window=window, nperseg=nperseg, + noverlap=noverlap, nfft=nfft, detrend=detrend, + return_onesided=return_onesided, scaling=scaling, axis=axis, + average=average) + freqs0, Pxy0 = csd(**kw) + freqs1, Pxy1 = _csd_wrapper(**kw) + + assert_allclose(freqs1, freqs0) + assert_allclose(Pxy1, Pxy0) + assert_allclose(freqs1, freqs0) + return freqs0, Pxy0 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/mpsig.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/mpsig.py new file mode 100644 index 0000000000000000000000000000000000000000..d129de74e5df00c22bc0b82c7d3f7b52483941f9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/mpsig.py @@ -0,0 +1,122 @@ +""" +Some signal functions implemented using mpmath. +""" + +try: + import mpmath +except ImportError: + mpmath = None + + +def _prod(seq): + """Returns the product of the elements in the sequence `seq`.""" + p = 1 + for elem in seq: + p *= elem + return p + + +def _relative_degree(z, p): + """ + Return relative degree of transfer function from zeros and poles. + + This is simply len(p) - len(z), which must be nonnegative. + A ValueError is raised if len(p) < len(z). + """ + degree = len(p) - len(z) + if degree < 0: + raise ValueError("Improper transfer function. " + "Must have at least as many poles as zeros.") + return degree + + +def _zpkbilinear(z, p, k, fs): + """Bilinear transformation to convert a filter from analog to digital.""" + + degree = _relative_degree(z, p) + + fs2 = 2*fs + + # Bilinear transform the poles and zeros + z_z = [(fs2 + z1) / (fs2 - z1) for z1 in z] + p_z = [(fs2 + p1) / (fs2 - p1) for p1 in p] + + # Any zeros that were at infinity get moved to the Nyquist frequency + z_z.extend([-1] * degree) + + # Compensate for gain change + numer = _prod(fs2 - z1 for z1 in z) + denom = _prod(fs2 - p1 for p1 in p) + k_z = k * numer / denom + + return z_z, p_z, k_z.real + + +def _zpklp2lp(z, p, k, wo=1): + """Transform a lowpass filter to a different cutoff frequency.""" + + degree = _relative_degree(z, p) + + # Scale all points radially from origin to shift cutoff frequency + z_lp = [wo * z1 for z1 in z] + p_lp = [wo * p1 for p1 in p] + + # Each shifted pole decreases gain by wo, each shifted zero increases it. + # Cancel out the net change to keep overall gain the same + k_lp = k * wo**degree + + return z_lp, p_lp, k_lp + + +def _butter_analog_poles(n): + """ + Poles of an analog Butterworth lowpass filter. + + This is the same calculation as scipy.signal.buttap(n) or + scipy.signal.butter(n, 1, analog=True, output='zpk'), but mpmath is used, + and only the poles are returned. + """ + poles = [-mpmath.exp(1j*mpmath.pi*k/(2*n)) for k in range(-n+1, n, 2)] + return poles + + +def butter_lp(n, Wn): + """ + Lowpass Butterworth digital filter design. + + This computes the same result as scipy.signal.butter(n, Wn, output='zpk'), + but it uses mpmath, and the results are returned in lists instead of NumPy + arrays. + """ + zeros = [] + poles = _butter_analog_poles(n) + k = 1 + fs = 2 + warped = 2 * fs * mpmath.tan(mpmath.pi * Wn / fs) + z, p, k = _zpklp2lp(zeros, poles, k, wo=warped) + z, p, k = _zpkbilinear(z, p, k, fs=fs) + return z, p, k + + +def zpkfreqz(z, p, k, worN=None): + """ + Frequency response of a filter in zpk format, using mpmath. + + This is the same calculation as scipy.signal.freqz, but the input is in + zpk format, the calculation is performed using mpath, and the results are + returned in lists instead of NumPy arrays. + """ + if worN is None or isinstance(worN, int): + N = worN or 512 + ws = [mpmath.pi * mpmath.mpf(j) / N for j in range(N)] + else: + ws = worN + + h = [] + for wk in ws: + zm1 = mpmath.exp(1j * wk) + numer = _prod([zm1 - t for t in z]) + denom = _prod([zm1 - t for t in p]) + hk = k * numer / denom + h.append(hk) + return ws, h diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_array_tools.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_array_tools.py new file mode 100644 index 0000000000000000000000000000000000000000..4bda9716e0bc4b6ad3ed0c3954147043b74c421a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_array_tools.py @@ -0,0 +1,111 @@ +import numpy as np + +from scipy._lib._array_api import xp_assert_equal +from pytest import raises as assert_raises + +from scipy.signal._arraytools import (axis_slice, axis_reverse, + odd_ext, even_ext, const_ext, zero_ext) + + +class TestArrayTools: + + def test_axis_slice(self): + a = np.arange(12).reshape(3, 4) + + s = axis_slice(a, start=0, stop=1, axis=0) + xp_assert_equal(s, a[0:1, :]) + + s = axis_slice(a, start=-1, axis=0) + xp_assert_equal(s, a[-1:, :]) + + s = axis_slice(a, start=0, stop=1, axis=1) + xp_assert_equal(s, a[:, 0:1]) + + s = axis_slice(a, start=-1, axis=1) + xp_assert_equal(s, a[:, -1:]) + + s = axis_slice(a, start=0, step=2, axis=0) + xp_assert_equal(s, a[::2, :]) + + s = axis_slice(a, start=0, step=2, axis=1) + xp_assert_equal(s, a[:, ::2]) + + def test_axis_reverse(self): + a = np.arange(12).reshape(3, 4) + + r = axis_reverse(a, axis=0) + xp_assert_equal(r, a[::-1, :]) + + r = axis_reverse(a, axis=1) + xp_assert_equal(r, a[:, ::-1]) + + def test_odd_ext(self): + a = np.array([[1, 2, 3, 4, 5], + [9, 8, 7, 6, 5]]) + + odd = odd_ext(a, 2, axis=1) + expected = np.array([[-1, 0, 1, 2, 3, 4, 5, 6, 7], + [11, 10, 9, 8, 7, 6, 5, 4, 3]]) + xp_assert_equal(odd, expected) + + odd = odd_ext(a, 1, axis=0) + expected = np.array([[-7, -4, -1, 2, 5], + [1, 2, 3, 4, 5], + [9, 8, 7, 6, 5], + [17, 14, 11, 8, 5]]) + xp_assert_equal(odd, expected) + + assert_raises(ValueError, odd_ext, a, 2, axis=0) + assert_raises(ValueError, odd_ext, a, 5, axis=1) + + def test_even_ext(self): + a = np.array([[1, 2, 3, 4, 5], + [9, 8, 7, 6, 5]]) + + even = even_ext(a, 2, axis=1) + expected = np.array([[3, 2, 1, 2, 3, 4, 5, 4, 3], + [7, 8, 9, 8, 7, 6, 5, 6, 7]]) + xp_assert_equal(even, expected) + + even = even_ext(a, 1, axis=0) + expected = np.array([[9, 8, 7, 6, 5], + [1, 2, 3, 4, 5], + [9, 8, 7, 6, 5], + [1, 2, 3, 4, 5]]) + xp_assert_equal(even, expected) + + assert_raises(ValueError, even_ext, a, 2, axis=0) + assert_raises(ValueError, even_ext, a, 5, axis=1) + + def test_const_ext(self): + a = np.array([[1, 2, 3, 4, 5], + [9, 8, 7, 6, 5]]) + + const = const_ext(a, 2, axis=1) + expected = np.array([[1, 1, 1, 2, 3, 4, 5, 5, 5], + [9, 9, 9, 8, 7, 6, 5, 5, 5]]) + xp_assert_equal(const, expected) + + const = const_ext(a, 1, axis=0) + expected = np.array([[1, 2, 3, 4, 5], + [1, 2, 3, 4, 5], + [9, 8, 7, 6, 5], + [9, 8, 7, 6, 5]]) + xp_assert_equal(const, expected) + + def test_zero_ext(self): + a = np.array([[1, 2, 3, 4, 5], + [9, 8, 7, 6, 5]]) + + zero = zero_ext(a, 2, axis=1) + expected = np.array([[0, 0, 1, 2, 3, 4, 5, 0, 0], + [0, 0, 9, 8, 7, 6, 5, 0, 0]]) + xp_assert_equal(zero, expected) + + zero = zero_ext(a, 1, axis=0) + expected = np.array([[0, 0, 0, 0, 0], + [1, 2, 3, 4, 5], + [9, 8, 7, 6, 5], + [0, 0, 0, 0, 0]]) + xp_assert_equal(zero, expected) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_bsplines.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_bsplines.py new file mode 100644 index 0000000000000000000000000000000000000000..9c7baf2d8d9f2f9a84452965c35b5262ca20105d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_bsplines.py @@ -0,0 +1,330 @@ +# pylint: disable=missing-docstring +import numpy as np + +from scipy._lib._array_api import ( + assert_almost_equal, xp_assert_close, xp_assert_equal +) +import pytest +from pytest import raises + +import scipy.signal._spline_filters as bsp +from scipy import signal + + +class TestBSplines: + """Test behaviors of B-splines. Some of the values tested against were + returned as of SciPy 1.1.0 and are included for regression testing + purposes. Others (at integer points) are compared to theoretical + expressions (cf. Unser, Aldroubi, Eden, IEEE TSP 1993, Table 1).""" + + def test_spline_filter(self): + rng = np.random.RandomState(12457) + # Test the type-error branch + raises(TypeError, bsp.spline_filter, np.asarray([0]), 0) + # Test the real branch + data_array_real = rng.rand(12, 12) + # make the magnitude exceed 1, and make some negative + data_array_real = 10*(1-2*data_array_real) + result_array_real = np.asarray( + [[-.463312621, 8.33391222, .697290949, 5.28390836, + 5.92066474, 6.59452137, 9.84406950, -8.78324188, + 7.20675750, -8.17222994, -4.38633345, 9.89917069], + [2.67755154, 6.24192170, -3.15730578, 9.87658581, + -9.96930425, 3.17194115, -4.50919947, 5.75423446, + 9.65979824, -8.29066885, .971416087, -2.38331897], + [-7.08868346, 4.89887705, -1.37062289, 7.70705838, + 2.51526461, 3.65885497, 5.16786604, -8.77715342e-03, + 4.10533325, 9.04761993, -.577960351, 9.86382519], + [-4.71444301, -1.68038985, 2.84695116, 1.14315938, + -3.17127091, 1.91830461, 7.13779687, -5.35737482, + -9.66586425, -9.87717456, 9.93160672, 4.71948144], + [9.49551194, -1.92958436, 6.25427993, -9.05582911, + 3.97562282, 7.68232426, -1.04514824, -5.86021443, + -8.43007451, 5.47528997, 2.06330736, -8.65968112], + [-8.91720100, 8.87065356, 3.76879937, 2.56222894, + -.828387146, 8.72288903, 6.42474741, -6.84576083, + 9.94724115, 6.90665380, -6.61084494, -9.44907391], + [9.25196790, -.774032030, 7.05371046, -2.73505725, + 2.53953305, -1.82889155, 2.95454824, -1.66362046, + 5.72478916, -3.10287679, 1.54017123, -7.87759020], + [-3.98464539, -2.44316992, -1.12708657, 1.01725672, + -8.89294671, -5.42145629, -6.16370321, 2.91775492, + 9.64132208, .702499998, -2.02622392, 1.56308431], + [-2.22050773, 7.89951554, 5.98970713, -7.35861835, + 5.45459283, -7.76427957, 3.67280490, -4.05521315, + 4.51967507, -3.22738749, -3.65080177, 3.05630155], + [-6.21240584, -.296796126, -8.34800163, 9.21564563, + -3.61958784, -4.77120006, -3.99454057, 1.05021988e-03, + -6.95982829, 6.04380797, 8.43181250, -2.71653339], + [1.19638037, 6.99718842e-02, 6.72020394, -2.13963198, + 3.75309875, -5.70076744, 5.92143551, -7.22150575, + -3.77114594, -1.11903194, -5.39151466, 3.06620093], + [9.86326886, 1.05134482, -7.75950607, -3.64429655, + 7.81848957, -9.02270373, 3.73399754, -4.71962549, + -7.71144306, 3.78263161, 6.46034818, -4.43444731]]) + xp_assert_close(bsp.spline_filter(data_array_real, 0), + result_array_real) + + def test_spline_filter_complex(self): + rng = np.random.RandomState(12457) + data_array_complex = rng.rand(7, 7) + rng.rand(7, 7)*1j + # make the magnitude exceed 1, and make some negative + data_array_complex = 10*(1+1j-2*data_array_complex) + result_array_complex = np.asarray( + [[-4.61489230e-01-1.92994022j, 8.33332443+6.25519943j, + 6.96300745e-01-9.05576038j, 5.28294849+3.97541356j, + 5.92165565+7.68240595j, 6.59493160-1.04542804j, + 9.84503460-5.85946894j], + [-8.78262329-8.4295969j, 7.20675516+5.47528982j, + -8.17223072+2.06330729j, -4.38633347-8.65968037j, + 9.89916801-8.91720295j, 2.67755103+8.8706522j, + 6.24192142+3.76879835j], + [-3.15627527+2.56303072j, 9.87658501-0.82838702j, + -9.96930313+8.72288895j, 3.17193985+6.42474651j, + -4.50919819-6.84576082j, 5.75423431+9.94723988j, + 9.65979767+6.90665293j], + [-8.28993416-6.61064005j, 9.71416473e-01-9.44907284j, + -2.38331890+9.25196648j, -7.08868170-0.77403212j, + 4.89887714+7.05371094j, -1.37062311-2.73505688j, + 7.70705748+2.5395329j], + [2.51528406-1.82964492j, 3.65885472+2.95454836j, + 5.16786575-1.66362023j, -8.77737999e-03+5.72478867j, + 4.10533333-3.10287571j, 9.04761887+1.54017115j, + -5.77960968e-01-7.87758923j], + [9.86398506-3.98528528j, -4.71444130-2.44316983j, + -1.68038976-1.12708664j, 2.84695053+1.01725709j, + 1.14315915-8.89294529j, -3.17127085-5.42145538j, + 1.91830420-6.16370344j], + [7.13875294+2.91851187j, -5.35737514+9.64132309j, + -9.66586399+0.70250005j, -9.87717438-2.0262239j, + 9.93160629+1.5630846j, 4.71948051-2.22050714j, + 9.49550819+7.8995142j]]) + # FIXME: for complex types, the computations are done in + # single precision (reason unclear). When this is changed, + # this test needs updating. + xp_assert_close(bsp.spline_filter(data_array_complex, 0), + result_array_complex, rtol=1e-6) + + def test_gauss_spline(self): + np.random.seed(12459) + assert_almost_equal(bsp.gauss_spline(0, 0), 1.381976597885342) + xp_assert_close(bsp.gauss_spline(np.asarray([1.]), 1), + np.asarray([0.04865217]), atol=1e-9 + ) + + def test_gauss_spline_list(self): + # regression test for gh-12152 (accept array_like) + knots = [-1.0, 0.0, -1.0] + assert_almost_equal(bsp.gauss_spline(knots, 3), + np.asarray([0.15418033, 0.6909883, 0.15418033]) + ) + + def test_cspline1d(self): + np.random.seed(12462) + xp_assert_equal(bsp.cspline1d(np.asarray([0])), [0.]) + c1d = np.asarray([1.21037185, 1.86293902, 2.98834059, 4.11660378, + 4.78893826]) + # test lamda != 0 + xp_assert_close(bsp.cspline1d(np.asarray([1., 2, 3, 4, 5]), 1), c1d) + c1d0 = np.asarray([0.78683946, 2.05333735, 2.99981113, 3.94741812, + 5.21051638]) + xp_assert_close(bsp.cspline1d(np.asarray([1., 2, 3, 4, 5])), c1d0) + + def test_qspline1d(self): + np.random.seed(12463) + xp_assert_equal(bsp.qspline1d(np.asarray([0])), [0.]) + # test lamda != 0 + raises(ValueError, bsp.qspline1d, np.asarray([1., 2, 3, 4, 5]), 1.) + raises(ValueError, bsp.qspline1d, np.asarray([1., 2, 3, 4, 5]), -1.) + q1d0 = np.asarray([0.85350007, 2.02441743, 2.99999534, 3.97561055, + 5.14634135]) + xp_assert_close(bsp.qspline1d(np.asarray([1., 2, 3, 4, 5])), q1d0) + + def test_cspline1d_eval(self): + np.random.seed(12464) + xp_assert_close(bsp.cspline1d_eval(np.asarray([0., 0]), [0.]), + np.asarray([0.]) + ) + xp_assert_equal(bsp.cspline1d_eval(np.asarray([1., 0, 1]), []), + np.asarray([]) + ) + x = [-3, -2, -1, 0, 1, 2, 3, 4, 5, 6] + dx = x[1] - x[0] + newx = [-6., -5.5, -5., -4.5, -4., -3.5, -3., -2.5, -2., -1.5, -1., + -0.5, 0., 0.5, 1., 1.5, 2., 2.5, 3., 3.5, 4., 4.5, 5., 5.5, 6., + 6.5, 7., 7.5, 8., 8.5, 9., 9.5, 10., 10.5, 11., 11.5, 12., + 12.5] + y = np.asarray([4.216, 6.864, 3.514, 6.203, 6.759, 7.433, 7.874, 5.879, + 1.396, 4.094]) + cj = bsp.cspline1d(y) + newy = np.asarray([6.203, 4.41570658, 3.514, 5.16924703, 6.864, 6.04643068, + 4.21600281, 6.04643068, 6.864, 5.16924703, 3.514, + 4.41570658, 6.203, 6.80717667, 6.759, 6.98971173, 7.433, + 7.79560142, 7.874, 7.41525761, 5.879, 3.18686814, 1.396, + 2.24889482, 4.094, 2.24889482, 1.396, 3.18686814, 5.879, + 7.41525761, 7.874, 7.79560142, 7.433, 6.98971173, 6.759, + 6.80717667, 6.203, 4.41570658]) + xp_assert_close(bsp.cspline1d_eval(cj, newx, dx=dx, x0=x[0]), newy) + + def test_qspline1d_eval(self): + np.random.seed(12465) + xp_assert_close(bsp.qspline1d_eval(np.asarray([0., 0]), [0.]), + np.asarray([0.]) + ) + xp_assert_equal(bsp.qspline1d_eval(np.asarray([1., 0, 1]), []), + np.asarray([]) + ) + x = [-3, -2, -1, 0, 1, 2, 3, 4, 5, 6] + dx = x[1]-x[0] + newx = [-6., -5.5, -5., -4.5, -4., -3.5, -3., -2.5, -2., -1.5, -1., + -0.5, 0., 0.5, 1., 1.5, 2., 2.5, 3., 3.5, 4., 4.5, 5., 5.5, 6., + 6.5, 7., 7.5, 8., 8.5, 9., 9.5, 10., 10.5, 11., 11.5, 12., + 12.5] + y = np.asarray([4.216, 6.864, 3.514, 6.203, 6.759, 7.433, 7.874, 5.879, + 1.396, 4.094]) + cj = bsp.qspline1d(y) + newy = np.asarray([6.203, 4.49418159, 3.514, 5.18390821, 6.864, 5.91436915, + 4.21600002, 5.91436915, 6.864, 5.18390821, 3.514, + 4.49418159, 6.203, 6.71900226, 6.759, 7.03980488, 7.433, + 7.81016848, 7.874, 7.32718426, 5.879, 3.23872593, 1.396, + 2.34046013, 4.094, 2.34046013, 1.396, 3.23872593, 5.879, + 7.32718426, 7.874, 7.81016848, 7.433, 7.03980488, 6.759, + 6.71900226, 6.203, 4.49418159]) + xp_assert_close(bsp.qspline1d_eval(cj, newx, dx=dx, x0=x[0]), newy) + + +# i/o dtypes with scipy 1.9.1, likely fixed by backwards compat +sepfir_dtype_map = {np.uint8: np.float32, int: np.float64, + np.float32: np.float32, float: float, + np.complex64: np.complex64, complex: complex} + +class TestSepfir2d: + def test_sepfir2d_invalid_filter(self): + filt = np.array([1.0, 2.0, 4.0, 2.0, 1.0]) + image = np.random.rand(7, 9) + # No error for odd lengths + signal.sepfir2d(image, filt, filt[2:]) + + # Row or column filter must be odd + with pytest.raises(ValueError, match="odd length"): + signal.sepfir2d(image, filt, filt[1:]) + with pytest.raises(ValueError, match="odd length"): + signal.sepfir2d(image, filt[1:], filt) + + # Filters must be 1-dimensional + with pytest.raises(ValueError, match="object too deep"): + signal.sepfir2d(image, filt.reshape(1, -1), filt) + with pytest.raises(ValueError, match="object too deep"): + signal.sepfir2d(image, filt, filt.reshape(1, -1)) + + def test_sepfir2d_invalid_image(self): + filt = np.array([1.0, 2.0, 4.0, 2.0, 1.0]) + image = np.random.rand(8, 8) + + # Image must be 2 dimensional + with pytest.raises(ValueError, match="object too deep"): + signal.sepfir2d(image.reshape(4, 4, 4), filt, filt) + + with pytest.raises(ValueError, match="object of too small depth"): + signal.sepfir2d(image[0], filt, filt) + + @pytest.mark.parametrize('dtyp', + [np.uint8, int, np.float32, float, np.complex64, complex] + ) + def test_simple(self, dtyp): + # test values on a paper-and-pencil example + a = np.array([[1, 2, 3, 3, 2, 1], + [1, 2, 3, 3, 2, 1], + [1, 2, 3, 3, 2, 1], + [1, 2, 3, 3, 2, 1]], dtype=dtyp) + h1 = [0.5, 1, 0.5] + h2 = [1] + result = signal.sepfir2d(a, h1, h2) + dt = sepfir_dtype_map[dtyp] + expected = np.asarray([[2.5, 4. , 5.5, 5.5, 4. , 2.5], + [2.5, 4. , 5.5, 5.5, 4. , 2.5], + [2.5, 4. , 5.5, 5.5, 4. , 2.5], + [2.5, 4. , 5.5, 5.5, 4. , 2.5]], dtype=dt) + xp_assert_close(result, expected, atol=1e-16) + + result = signal.sepfir2d(a, h2, h1) + expected = np.asarray([[2., 4., 6., 6., 4., 2.], + [2., 4., 6., 6., 4., 2.], + [2., 4., 6., 6., 4., 2.], + [2., 4., 6., 6., 4., 2.]], dtype=dt) + xp_assert_close(result, expected, atol=1e-16) + + @pytest.mark.parametrize('dtyp', + [np.uint8, int, np.float32, float, np.complex64, complex] + ) + def test_strided(self, dtyp): + a = np.array([[1, 2, 3, 3, 2, 1, 1, 2, 3], + [1, 2, 3, 3, 2, 1, 1, 2, 3], + [1, 2, 3, 3, 2, 1, 1, 2, 3], + [1, 2, 3, 3, 2, 1, 1, 2, 3]]) + h1, h2 = [0.5, 1, 0.5], [1] + result_strided = signal.sepfir2d(a[:, ::2], h1, h2) + result_contig = signal.sepfir2d(a[:, ::2].copy(), h1, h2) + xp_assert_close(result_strided, result_contig, atol=1e-15) + assert result_strided.dtype == result_contig.dtype + + @pytest.mark.xfail(reason="XXX: filt.size > image.shape: flaky") + def test_sepfir2d_strided_2(self): + # XXX: this test is flaky: fails on some reruns, with + # result[0, 1] and result[1, 1] being ~1e+224. + np.random.seed(1234) + filt = np.array([1.0, 2.0, 4.0, 2.0, 1.0, 3.0, 2.0]) + image = np.random.rand(4, 4) + + expected = np.asarray([[36.018162, 30.239061, 38.71187 , 43.878183], + [38.180999, 35.824583, 43.525247, 43.874945], + [43.269533, 40.834018, 46.757772, 44.276423], + [49.120928, 39.681844, 43.596067, 45.085854]]) + xp_assert_close(signal.sepfir2d(image, filt, filt[::3]), expected) + + @pytest.mark.xfail(reason="XXX: flaky. pointers OOB on some platforms") + @pytest.mark.parametrize('dtyp', + [np.uint8, int, np.float32, float, np.complex64, complex] + ) + def test_sepfir2d_strided_3(self, dtyp): + # NB: 'image' and 'filt' dtypes match here. Otherwise we can run into + # unsafe casting errors for many combinations. Historically, dtype handling + # in `sepfir2d` is a tad baroque; fixing it is an enhancement. + filt = np.array([1, 2, 4, 2, 1, 3, 2], dtype=dtyp) + image = np.asarray([[0, 3, 0, 1, 2], + [2, 2, 3, 3, 3], + [0, 1, 3, 0, 3], + [2, 3, 0, 1, 3], + [3, 3, 2, 1, 2]], dtype=dtyp) + + expected = [[123., 101., 91., 136., 127.], + [133., 125., 126., 152., 160.], + [136., 137., 150., 162., 177.], + [133., 124., 132., 148., 147.], + [173., 158., 152., 164., 141.]] + expected = np.asarray(expected) + result = signal.sepfir2d(image, filt, filt[::3]) + xp_assert_close(result, expected, atol=1e-15) + assert result.dtype == sepfir_dtype_map[dtyp] + + expected = [[22., 35., 41., 31., 47.], + [27., 39., 48., 47., 55.], + [33., 42., 49., 53., 59.], + [39., 44., 41., 36., 48.], + [67., 62., 47., 34., 46.]] + expected = np.asarray(expected) + result = signal.sepfir2d(image, filt[::3], filt[::3]) + xp_assert_close(result, expected, atol=1e-15) + assert result.dtype == sepfir_dtype_map[dtyp] + + +def test_cspline2d(): + np.random.seed(181819142) + image = np.random.rand(71, 73) + signal.cspline2d(image, 8.0) + + +def test_qspline2d(): + np.random.seed(181819143) + image = np.random.rand(71, 73) + signal.qspline2d(image) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_cont2discrete.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_cont2discrete.py new file mode 100644 index 0000000000000000000000000000000000000000..0b4008afd4728acb5b63a6626440b214f2d90f6f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_cont2discrete.py @@ -0,0 +1,417 @@ +import numpy as np +from scipy._lib._array_api import ( + assert_array_almost_equal, assert_almost_equal, xp_assert_close +) + +import pytest +from scipy.signal import cont2discrete as c2d +from scipy.signal import dlsim, ss2tf, ss2zpk, lsim, lti +from scipy.signal import tf2ss, impulse, dimpulse, step, dstep + +# Author: Jeffrey Armstrong +# March 29, 2011 + + +class TestC2D: + @pytest.mark.thread_unsafe # due to Cython fused types, see cython#6506 + def test_zoh(self): + ac = np.eye(2, dtype=np.float64) + bc = np.full((2, 1), 0.5, dtype=np.float64) + cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]]) + dc = np.array([[0.0], [0.0], [-0.33]]) + + ad_truth = 1.648721270700128 * np.eye(2) + bd_truth = np.full((2, 1), 0.324360635350064) + # c and d in discrete should be equal to their continuous counterparts + dt_requested = 0.5 + + ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, method='zoh') + + assert_array_almost_equal(ad_truth, ad) + assert_array_almost_equal(bd_truth, bd) + assert_array_almost_equal(cc, cd) + assert_array_almost_equal(dc, dd) + assert_almost_equal(dt_requested, dt) + + def test_foh(self): + ac = np.eye(2) + bc = np.full((2, 1), 0.5) + cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]]) + dc = np.array([[0.0], [0.0], [-0.33]]) + + # True values are verified with Matlab + ad_truth = 1.648721270700128 * np.eye(2) + bd_truth = np.full((2, 1), 0.420839287058789) + cd_truth = cc + dd_truth = np.array([[0.260262223725224], + [0.297442541400256], + [-0.144098411624840]]) + dt_requested = 0.5 + + ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, method='foh') + + assert_array_almost_equal(ad_truth, ad) + assert_array_almost_equal(bd_truth, bd) + assert_array_almost_equal(cd_truth, cd) + assert_array_almost_equal(dd_truth, dd) + assert_almost_equal(dt_requested, dt) + + def test_impulse(self): + ac = np.eye(2) + bc = np.full((2, 1), 0.5) + cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]]) + dc = np.array([[0.0], [0.0], [0.0]]) + + # True values are verified with Matlab + ad_truth = 1.648721270700128 * np.eye(2) + bd_truth = np.full((2, 1), 0.412180317675032) + cd_truth = cc + dd_truth = np.array([[0.4375], [0.5], [0.3125]]) + dt_requested = 0.5 + + ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, + method='impulse') + + assert_array_almost_equal(ad_truth, ad) + assert_array_almost_equal(bd_truth, bd) + assert_array_almost_equal(cd_truth, cd) + assert_array_almost_equal(dd_truth, dd) + assert_almost_equal(dt_requested, dt) + + def test_gbt(self): + ac = np.eye(2) + bc = np.full((2, 1), 0.5) + cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]]) + dc = np.array([[0.0], [0.0], [-0.33]]) + + dt_requested = 0.5 + alpha = 1.0 / 3.0 + + ad_truth = 1.6 * np.eye(2) + bd_truth = np.full((2, 1), 0.3) + cd_truth = np.array([[0.9, 1.2], + [1.2, 1.2], + [1.2, 0.3]]) + dd_truth = np.array([[0.175], + [0.2], + [-0.205]]) + + ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, + method='gbt', alpha=alpha) + + assert_array_almost_equal(ad_truth, ad) + assert_array_almost_equal(bd_truth, bd) + assert_array_almost_equal(cd_truth, cd) + assert_array_almost_equal(dd_truth, dd) + + def test_euler(self): + ac = np.eye(2) + bc = np.full((2, 1), 0.5) + cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]]) + dc = np.array([[0.0], [0.0], [-0.33]]) + + dt_requested = 0.5 + + ad_truth = 1.5 * np.eye(2) + bd_truth = np.full((2, 1), 0.25) + cd_truth = np.array([[0.75, 1.0], + [1.0, 1.0], + [1.0, 0.25]]) + dd_truth = dc + + ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, + method='euler') + + assert_array_almost_equal(ad_truth, ad) + assert_array_almost_equal(bd_truth, bd) + assert_array_almost_equal(cd_truth, cd) + assert_array_almost_equal(dd_truth, dd) + assert_almost_equal(dt_requested, dt) + + def test_backward_diff(self): + ac = np.eye(2) + bc = np.full((2, 1), 0.5) + cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]]) + dc = np.array([[0.0], [0.0], [-0.33]]) + + dt_requested = 0.5 + + ad_truth = 2.0 * np.eye(2) + bd_truth = np.full((2, 1), 0.5) + cd_truth = np.array([[1.5, 2.0], + [2.0, 2.0], + [2.0, 0.5]]) + dd_truth = np.array([[0.875], + [1.0], + [0.295]]) + + ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, + method='backward_diff') + + assert_array_almost_equal(ad_truth, ad) + assert_array_almost_equal(bd_truth, bd) + assert_array_almost_equal(cd_truth, cd) + assert_array_almost_equal(dd_truth, dd) + + def test_bilinear(self): + ac = np.eye(2) + bc = np.full((2, 1), 0.5) + cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]]) + dc = np.array([[0.0], [0.0], [-0.33]]) + + dt_requested = 0.5 + + ad_truth = (5.0 / 3.0) * np.eye(2) + bd_truth = np.full((2, 1), 1.0 / 3.0) + cd_truth = np.array([[1.0, 4.0 / 3.0], + [4.0 / 3.0, 4.0 / 3.0], + [4.0 / 3.0, 1.0 / 3.0]]) + dd_truth = np.array([[0.291666666666667], + [1.0 / 3.0], + [-0.121666666666667]]) + + ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, + method='bilinear') + + assert_array_almost_equal(ad_truth, ad) + assert_array_almost_equal(bd_truth, bd) + assert_array_almost_equal(cd_truth, cd) + assert_array_almost_equal(dd_truth, dd) + assert_almost_equal(dt_requested, dt) + + # Same continuous system again, but change sampling rate + + ad_truth = 1.4 * np.eye(2) + bd_truth = np.full((2, 1), 0.2) + cd_truth = np.array([[0.9, 1.2], [1.2, 1.2], [1.2, 0.3]]) + dd_truth = np.array([[0.175], [0.2], [-0.205]]) + + dt_requested = 1.0 / 3.0 + + ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, + method='bilinear') + + assert_array_almost_equal(ad_truth, ad) + assert_array_almost_equal(bd_truth, bd) + assert_array_almost_equal(cd_truth, cd) + assert_array_almost_equal(dd_truth, dd) + assert_almost_equal(dt_requested, dt) + + def test_transferfunction(self): + numc = np.array([0.25, 0.25, 0.5]) + denc = np.array([0.75, 0.75, 1.0]) + + numd = np.array([[1.0 / 3.0, -0.427419169438754, 0.221654141101125]]) + dend = np.array([1.0, -1.351394049721225, 0.606530659712634]) + + dt_requested = 0.5 + + num, den, dt = c2d((numc, denc), dt_requested, method='zoh') + + assert_array_almost_equal(numd, num) + assert_array_almost_equal(dend, den) + assert_almost_equal(dt_requested, dt) + + def test_zerospolesgain(self): + zeros_c = np.array([0.5, -0.5]) + poles_c = np.array([1.j / np.sqrt(2), -1.j / np.sqrt(2)]) + k_c = 1.0 + + zeros_d = [1.23371727305860, 0.735356894461267] + polls_d = [0.938148335039729 + 0.346233593780536j, + 0.938148335039729 - 0.346233593780536j] + k_d = 1.0 + + dt_requested = 0.5 + + zeros, poles, k, dt = c2d((zeros_c, poles_c, k_c), dt_requested, + method='zoh') + + assert_array_almost_equal(zeros_d, zeros) + assert_array_almost_equal(polls_d, poles) + assert_almost_equal(k_d, k) + assert_almost_equal(dt_requested, dt) + + def test_gbt_with_sio_tf_and_zpk(self): + """Test method='gbt' with alpha=0.25 for tf and zpk cases.""" + # State space coefficients for the continuous SIO system. + A = -1.0 + B = 1.0 + C = 1.0 + D = 0.5 + + # The continuous transfer function coefficients. + cnum, cden = ss2tf(A, B, C, D) + + # Continuous zpk representation + cz, cp, ck = ss2zpk(A, B, C, D) + + h = 1.0 + alpha = 0.25 + + # Explicit formulas, in the scalar case. + Ad = (1 + (1 - alpha) * h * A) / (1 - alpha * h * A) + Bd = h * B / (1 - alpha * h * A) + Cd = C / (1 - alpha * h * A) + Dd = D + alpha * C * Bd + + # Convert the explicit solution to tf + dnum, dden = ss2tf(Ad, Bd, Cd, Dd) + + # Compute the discrete tf using cont2discrete. + c2dnum, c2dden, dt = c2d((cnum, cden), h, method='gbt', alpha=alpha) + + xp_assert_close(dnum, c2dnum) + xp_assert_close(dden, c2dden) + + # Convert explicit solution to zpk. + dz, dp, dk = ss2zpk(Ad, Bd, Cd, Dd) + + # Compute the discrete zpk using cont2discrete. + c2dz, c2dp, c2dk, dt = c2d((cz, cp, ck), h, method='gbt', alpha=alpha) + + xp_assert_close(dz, c2dz) + xp_assert_close(dp, c2dp) + xp_assert_close(dk, c2dk) + + def test_discrete_approx(self): + """ + Test that the solution to the discrete approximation of a continuous + system actually approximates the solution to the continuous system. + This is an indirect test of the correctness of the implementation + of cont2discrete. + """ + + def u(t): + return np.sin(2.5 * t) + + a = np.array([[-0.01]]) + b = np.array([[1.0]]) + c = np.array([[1.0]]) + d = np.array([[0.2]]) + x0 = 1.0 + + t = np.linspace(0, 10.0, 101) + dt = t[1] - t[0] + u1 = u(t) + + # Use lsim to compute the solution to the continuous system. + t, yout, xout = lsim((a, b, c, d), T=t, U=u1, X0=x0) + + # Convert the continuous system to a discrete approximation. + dsys = c2d((a, b, c, d), dt, method='bilinear') + + # Use dlsim with the pairwise averaged input to compute the output + # of the discrete system. + u2 = 0.5 * (u1[:-1] + u1[1:]) + t2 = t[:-1] + td2, yd2, xd2 = dlsim(dsys, u=u2.reshape(-1, 1), t=t2, x0=x0) + + # ymid is the average of consecutive terms of the "exact" output + # computed by lsim2. This is what the discrete approximation + # actually approximates. + ymid = 0.5 * (yout[:-1] + yout[1:]) + + xp_assert_close(yd2.ravel(), ymid, rtol=1e-4) + + def test_simo_tf(self): + # See gh-5753 + tf = ([[1, 0], [1, 1]], [1, 1]) + num, den, dt = c2d(tf, 0.01) + + assert dt == 0.01 # sanity check + xp_assert_close(den, [1, -0.990404983], rtol=1e-3) + xp_assert_close(num, [[1, -1], [1, -0.99004983]], rtol=1e-3) + + def test_multioutput(self): + ts = 0.01 # time step + + tf = ([[1, -3], [1, 5]], [1, 1]) + num, den, dt = c2d(tf, ts) + + tf1 = (tf[0][0], tf[1]) + num1, den1, dt1 = c2d(tf1, ts) + + tf2 = (tf[0][1], tf[1]) + num2, den2, dt2 = c2d(tf2, ts) + + # Sanity checks + assert dt == dt1 + assert dt == dt2 + + # Check that we get the same results + xp_assert_close(num, np.vstack((num1, num2)), rtol=1e-13) + + # Single input, so the denominator should + # not be multidimensional like the numerator + xp_assert_close(den, den1, rtol=1e-13) + xp_assert_close(den, den2, rtol=1e-13) + +class TestC2dLti: + def test_c2d_ss(self): + # StateSpace + A = np.array([[-0.3, 0.1], [0.2, -0.7]]) + B = np.array([[0], [1]]) + C = np.array([[1, 0]]) + D = 0 + + A_res = np.array([[0.985136404135682, 0.004876671474795], + [0.009753342949590, 0.965629718236502]]) + B_res = np.array([[0.000122937599964], [0.049135527547844]]) + + sys_ssc = lti(A, B, C, D) + sys_ssd = sys_ssc.to_discrete(0.05) + + xp_assert_close(sys_ssd.A, A_res) + xp_assert_close(sys_ssd.B, B_res) + xp_assert_close(sys_ssd.C, C) + xp_assert_close(sys_ssd.D, np.zeros_like(sys_ssd.D)) + + def test_c2d_tf(self): + + sys = lti([0.5, 0.3], [1.0, 0.4]) + sys = sys.to_discrete(0.005) + + # Matlab results + num_res = np.array([0.5, -0.485149004980066]) + den_res = np.array([1.0, -0.980198673306755]) + + # Somehow a lot of numerical errors + xp_assert_close(sys.den, den_res, atol=0.02) + xp_assert_close(sys.num, num_res, atol=0.02) + + +class TestC2dInvariants: + # Some test cases for checking the invariances. + # Array of triplets: (system, sample time, number of samples) + cases = [ + (tf2ss([1, 1], [1, 1.5, 1]), 0.25, 10), + (tf2ss([1, 2], [1, 1.5, 3, 1]), 0.5, 10), + (tf2ss(0.1, [1, 1, 2, 1]), 0.5, 10), + ] + + # Check that systems discretized with the impulse-invariant + # method really hold the invariant + @pytest.mark.parametrize("sys,sample_time,samples_number", cases) + def test_impulse_invariant(self, sys, sample_time, samples_number): + time = np.arange(samples_number) * sample_time + _, yout_cont = impulse(sys, T=time) + _, yout_disc = dimpulse(c2d(sys, sample_time, method='impulse'), + n=len(time)) + xp_assert_close(sample_time * yout_cont.ravel(), yout_disc[0].ravel()) + + # Step invariant should hold for ZOH discretized systems + @pytest.mark.parametrize("sys,sample_time,samples_number", cases) + def test_step_invariant(self, sys, sample_time, samples_number): + time = np.arange(samples_number) * sample_time + _, yout_cont = step(sys, T=time) + _, yout_disc = dstep(c2d(sys, sample_time, method='zoh'), n=len(time)) + xp_assert_close(yout_cont.ravel(), yout_disc[0].ravel()) + + # Linear invariant should hold for FOH discretized systems + @pytest.mark.parametrize("sys,sample_time,samples_number", cases) + def test_linear_invariant(self, sys, sample_time, samples_number): + time = np.arange(samples_number) * sample_time + _, yout_cont, _ = lsim(sys, T=time, U=time) + _, yout_disc, _ = dlsim(c2d(sys, sample_time, method='foh'), u=time) + xp_assert_close(yout_cont.ravel(), yout_disc.ravel()) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_czt.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_czt.py new file mode 100644 index 0000000000000000000000000000000000000000..35087d99fec5057131f0735d43e0faa33a74ef82 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_czt.py @@ -0,0 +1,221 @@ +# This program is public domain +# Authors: Paul Kienzle, Nadav Horesh +''' +A unit test module for czt.py +''' +import pytest +from scipy._lib._array_api import xp_assert_close +from scipy.fft import fft +from scipy.signal import (czt, zoom_fft, czt_points, CZT, ZoomFFT) +import numpy as np + + +def check_czt(x): + # Check that czt is the equivalent of normal fft + y = fft(x) + y1 = czt(x) + xp_assert_close(y1, y, rtol=1e-13) + + # Check that interpolated czt is the equivalent of normal fft + y = fft(x, 100*len(x)) + y1 = czt(x, 100*len(x)) + xp_assert_close(y1, y, rtol=1e-12) + + +def check_zoom_fft(x): + # Check that zoom_fft is the equivalent of normal fft + y = fft(x) + y1 = zoom_fft(x, [0, 2-2./len(y)], endpoint=True) + xp_assert_close(y1, y, rtol=1e-11, atol=1e-14) + y1 = zoom_fft(x, [0, 2]) + xp_assert_close(y1, y, rtol=1e-11, atol=1e-14) + + # Test fn scalar + y1 = zoom_fft(x, 2-2./len(y), endpoint=True) + xp_assert_close(y1, y, rtol=1e-11, atol=1e-14) + y1 = zoom_fft(x, 2) + xp_assert_close(y1, y, rtol=1e-11, atol=1e-14) + + # Check that zoom_fft with oversampling is equivalent to zero padding + over = 10 + yover = fft(x, over*len(x)) + y2 = zoom_fft(x, [0, 2-2./len(yover)], m=len(yover), endpoint=True) + xp_assert_close(y2, yover, rtol=1e-12, atol=1e-10) + y2 = zoom_fft(x, [0, 2], m=len(yover)) + xp_assert_close(y2, yover, rtol=1e-12, atol=1e-10) + + # Check that zoom_fft works on a subrange + w = np.linspace(0, 2-2./len(x), len(x)) + f1, f2 = w[3], w[6] + y3 = zoom_fft(x, [f1, f2], m=3*over+1, endpoint=True) + idx3 = slice(3*over, 6*over+1) + xp_assert_close(y3, yover[idx3], rtol=1e-13) + + +def test_1D(): + # Test of 1D version of the transforms + + rng = np.random.RandomState(0) # Deterministic randomness + + # Random signals + lengths = rng.randint(8, 200, 20) + np.append(lengths, 1) + for length in lengths: + x = rng.random(length) + check_zoom_fft(x) + check_czt(x) + + # Gauss + t = np.linspace(-2, 2, 128) + x = np.exp(-t**2/0.01) + check_zoom_fft(x) + + # Linear + x = [1, 2, 3, 4, 5, 6, 7] + check_zoom_fft(x) + + # Check near powers of two + check_zoom_fft(range(126-31)) + check_zoom_fft(range(127-31)) + check_zoom_fft(range(128-31)) + check_zoom_fft(range(129-31)) + check_zoom_fft(range(130-31)) + + # Check transform on n-D array input + x = np.reshape(np.arange(3*2*28), (3, 2, 28)) + y1 = zoom_fft(x, [0, 2-2./28]) + y2 = zoom_fft(x[2, 0, :], [0, 2-2./28]) + xp_assert_close(y1[2, 0], y2, rtol=1e-13, atol=1e-12) + + y1 = zoom_fft(x, [0, 2], endpoint=False) + y2 = zoom_fft(x[2, 0, :], [0, 2], endpoint=False) + xp_assert_close(y1[2, 0], y2, rtol=1e-13, atol=1e-12) + + # Random (not a test condition) + x = rng.rand(101) + check_zoom_fft(x) + + # Spikes + t = np.linspace(0, 1, 128) + x = np.sin(2*np.pi*t*5)+np.sin(2*np.pi*t*13) + check_zoom_fft(x) + + # Sines + x = np.zeros(100, dtype=complex) + x[[1, 5, 21]] = 1 + check_zoom_fft(x) + + # Sines plus complex component + x += 1j*np.linspace(0, 0.5, x.shape[0]) + check_zoom_fft(x) + + +def test_large_prime_lengths(): + rng = np.random.RandomState(0) # Deterministic randomness + for N in (101, 1009, 10007): + x = rng.rand(N) + y = fft(x) + y1 = czt(x) + xp_assert_close(y, y1, rtol=1e-12) + + +@pytest.mark.slow +def test_czt_vs_fft(): + rng = np.random.RandomState(123) # Deterministic randomness + random_lengths = rng.exponential(100000, size=10).astype('int') + for n in random_lengths: + a = rng.randn(n) + xp_assert_close(czt(a), fft(a), rtol=1e-11) + + +def test_empty_input(): + with pytest.raises(ValueError, match='Invalid number of CZT'): + czt([]) + with pytest.raises(ValueError, match='Invalid number of CZT'): + zoom_fft([], 0.5) + + +def test_0_rank_input(): + with pytest.raises(IndexError, match='tuple index out of range'): + czt(5) + with pytest.raises(IndexError, match='tuple index out of range'): + zoom_fft(5, 0.5) + + +@pytest.mark.parametrize('impulse', ([0, 0, 1], [0, 0, 1, 0, 0], + np.concatenate((np.array([0, 0, 1]), + np.zeros(100))))) +@pytest.mark.parametrize('m', (1, 3, 5, 8, 101, 1021)) +@pytest.mark.parametrize('a', (1, 2, 0.5, 1.1)) +# Step that tests away from the unit circle, but not so far it explodes from +# numerical error +@pytest.mark.parametrize('w', (None, 0.98534 + 0.17055j)) +def test_czt_math(impulse, m, w, a): + # z-transform of an impulse is 1 everywhere + xp_assert_close(czt(impulse[2:], m=m, w=w, a=a), + np.ones(m, dtype=np.complex128), rtol=1e-10) + + # z-transform of a delayed impulse is z**-1 + xp_assert_close(czt(impulse[1:], m=m, w=w, a=a), + czt_points(m=m, w=w, a=a)**-1, rtol=1e-10) + + # z-transform of a 2-delayed impulse is z**-2 + xp_assert_close(czt(impulse, m=m, w=w, a=a), + czt_points(m=m, w=w, a=a)**-2, rtol=1e-10) + + +def test_int_args(): + # Integer argument `a` was producing all 0s + xp_assert_close(abs(czt([0, 1], m=10, a=2)), 0.5*np.ones(10), rtol=1e-15) + xp_assert_close(czt_points(11, w=2), + 1/(2**np.arange(11, dtype=np.complex128)), rtol=1e-30) + + +def test_czt_points(): + for N in (1, 2, 3, 8, 11, 100, 101, 10007): + xp_assert_close(czt_points(N), np.exp(2j*np.pi*np.arange(N)/N), + rtol=1e-30) + + xp_assert_close(czt_points(7, w=1), np.ones(7, dtype=np.complex128), rtol=1e-30) + xp_assert_close(czt_points(11, w=2.), + 1/(2**np.arange(11, dtype=np.complex128)), rtol=1e-30) + + func = CZT(12, m=11, w=2., a=1) + xp_assert_close(func.points(), 1/(2**np.arange(11)), rtol=1e-30) + + +@pytest.mark.parametrize('cls, args', [(CZT, (100,)), (ZoomFFT, (100, 0.2))]) +def test_CZT_size_mismatch(cls, args): + # Data size doesn't match function's expected size + myfunc = cls(*args) + with pytest.raises(ValueError, match='CZT defined for'): + myfunc(np.arange(5)) + + +def test_invalid_range(): + with pytest.raises(ValueError, match='2-length sequence'): + ZoomFFT(100, [1, 2, 3]) + + +@pytest.mark.parametrize('m', [0, -11, 5.5, 4.0]) +def test_czt_points_errors(m): + # Invalid number of points + with pytest.raises(ValueError, match='Invalid number of CZT'): + czt_points(m) + + +@pytest.mark.parametrize('size', [0, -5, 3.5, 4.0]) +def test_nonsense_size(size): + # Numpy and Scipy fft() give ValueError for 0 output size, so we do, too + with pytest.raises(ValueError, match='Invalid number of CZT'): + CZT(size, 3) + with pytest.raises(ValueError, match='Invalid number of CZT'): + ZoomFFT(size, 0.2, 3) + with pytest.raises(ValueError, match='Invalid number of CZT'): + CZT(3, size) + with pytest.raises(ValueError, match='Invalid number of CZT'): + ZoomFFT(3, 0.2, size) + with pytest.raises(ValueError, match='Invalid number of CZT'): + czt([1, 2, 3], size) + with pytest.raises(ValueError, match='Invalid number of CZT'): + zoom_fft([1, 2, 3], 0.2, size) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_dltisys.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_dltisys.py new file mode 100644 index 0000000000000000000000000000000000000000..872541543ba485f3e8a17735bee471875f92fd05 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_dltisys.py @@ -0,0 +1,599 @@ +# Author: Jeffrey Armstrong +# April 4, 2011 + +import numpy as np +from numpy.testing import suppress_warnings +from pytest import raises as assert_raises +from scipy._lib._array_api import ( + assert_array_almost_equal, assert_almost_equal, xp_assert_close, xp_assert_equal, +) + +from scipy.signal import (dlsim, dstep, dimpulse, tf2zpk, lti, dlti, + StateSpace, TransferFunction, ZerosPolesGain, + dfreqresp, dbode, BadCoefficients) + + +class TestDLTI: + + def test_dlsim(self): + + a = np.asarray([[0.9, 0.1], [-0.2, 0.9]]) + b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]]) + c = np.asarray([[0.1, 0.3]]) + d = np.asarray([[0.0, -0.1, 0.0]]) + dt = 0.5 + + # Create an input matrix with inputs down the columns (3 cols) and its + # respective time input vector + u = np.hstack((np.linspace(0, 4.0, num=5)[:, np.newaxis], + np.full((5, 1), 0.01), + np.full((5, 1), -0.002))) + t_in = np.linspace(0, 2.0, num=5) + + # Define the known result + yout_truth = np.array([[-0.001, + -0.00073, + 0.039446, + 0.0915387, + 0.13195948]]).T + xout_truth = np.asarray([[0, 0], + [0.0012, 0.0005], + [0.40233, 0.00071], + [1.163368, -0.079327], + [2.2402985, -0.3035679]]) + + tout, yout, xout = dlsim((a, b, c, d, dt), u, t_in) + + assert_array_almost_equal(yout_truth, yout) + assert_array_almost_equal(xout_truth, xout) + assert_array_almost_equal(t_in, tout) + + # Make sure input with single-dimension doesn't raise error + dlsim((1, 2, 3), 4) + + # Interpolated control - inputs should have different time steps + # than the discrete model uses internally + u_sparse = u[[0, 4], :] + t_sparse = np.asarray([0.0, 2.0]) + + tout, yout, xout = dlsim((a, b, c, d, dt), u_sparse, t_sparse) + + assert_array_almost_equal(yout_truth, yout) + assert_array_almost_equal(xout_truth, xout) + assert len(tout) == len(yout) + + # Transfer functions (assume dt = 0.5) + num = np.asarray([1.0, -0.1]) + den = np.asarray([0.3, 1.0, 0.2]) + yout_truth = np.array([[0.0, + 0.0, + 3.33333333333333, + -4.77777777777778, + 23.0370370370370]]).T + + # Assume use of the first column of the control input built earlier + tout, yout = dlsim((num, den, 0.5), u[:, 0], t_in) + + assert_array_almost_equal(yout, yout_truth) + assert_array_almost_equal(t_in, tout) + + # Retest the same with a 1-D input vector + uflat = np.asarray(u[:, 0]) + uflat = uflat.reshape((5,)) + tout, yout = dlsim((num, den, 0.5), uflat, t_in) + + assert_array_almost_equal(yout, yout_truth) + assert_array_almost_equal(t_in, tout) + + # zeros-poles-gain representation + zd = np.array([0.5, -0.5]) + pd = np.array([1.j / np.sqrt(2), -1.j / np.sqrt(2)]) + k = 1.0 + yout_truth = np.array([[0.0, 1.0, 2.0, 2.25, 2.5]]).T + + tout, yout = dlsim((zd, pd, k, 0.5), u[:, 0], t_in) + + assert_array_almost_equal(yout, yout_truth) + assert_array_almost_equal(t_in, tout) + + # Raise an error for continuous-time systems + system = lti([1], [1, 1]) + assert_raises(AttributeError, dlsim, system, u) + + def test_dstep(self): + + a = np.asarray([[0.9, 0.1], [-0.2, 0.9]]) + b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]]) + c = np.asarray([[0.1, 0.3]]) + d = np.asarray([[0.0, -0.1, 0.0]]) + dt = 0.5 + + # Because b.shape[1] == 3, dstep should result in a tuple of three + # result vectors + yout_step_truth = (np.asarray([0.0, 0.04, 0.052, 0.0404, 0.00956, + -0.036324, -0.093318, -0.15782348, + -0.226628324, -0.2969374948]), + np.asarray([-0.1, -0.075, -0.058, -0.04815, + -0.04453, -0.0461895, -0.0521812, + -0.061588875, -0.073549579, + -0.08727047595]), + np.asarray([0.0, -0.01, -0.013, -0.0101, -0.00239, + 0.009081, 0.0233295, 0.03945587, + 0.056657081, 0.0742343737])) + + tout, yout = dstep((a, b, c, d, dt), n=10) + + assert len(yout) == 3 + + for i in range(0, len(yout)): + assert yout[i].shape[0] == 10 + assert_array_almost_equal(yout[i].flatten(), yout_step_truth[i]) + + # Check that the other two inputs (tf, zpk) will work as well + tfin = ([1.0], [1.0, 1.0], 0.5) + yout_tfstep = np.asarray([0.0, 1.0, 0.0]) + tout, yout = dstep(tfin, n=3) + assert len(yout) == 1 + assert_array_almost_equal(yout[0].flatten(), yout_tfstep) + + zpkin = tf2zpk(tfin[0], tfin[1]) + (0.5,) + tout, yout = dstep(zpkin, n=3) + assert len(yout) == 1 + assert_array_almost_equal(yout[0].flatten(), yout_tfstep) + + # Raise an error for continuous-time systems + system = lti([1], [1, 1]) + assert_raises(AttributeError, dstep, system) + + def test_dimpulse(self): + + a = np.asarray([[0.9, 0.1], [-0.2, 0.9]]) + b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]]) + c = np.asarray([[0.1, 0.3]]) + d = np.asarray([[0.0, -0.1, 0.0]]) + dt = 0.5 + + # Because b.shape[1] == 3, dimpulse should result in a tuple of three + # result vectors + yout_imp_truth = (np.asarray([0.0, 0.04, 0.012, -0.0116, -0.03084, + -0.045884, -0.056994, -0.06450548, + -0.068804844, -0.0703091708]), + np.asarray([-0.1, 0.025, 0.017, 0.00985, 0.00362, + -0.0016595, -0.0059917, -0.009407675, + -0.011960704, -0.01372089695]), + np.asarray([0.0, -0.01, -0.003, 0.0029, 0.00771, + 0.011471, 0.0142485, 0.01612637, + 0.017201211, 0.0175772927])) + + tout, yout = dimpulse((a, b, c, d, dt), n=10) + + assert len(yout) == 3 + + for i in range(0, len(yout)): + assert yout[i].shape[0] == 10 + assert_array_almost_equal(yout[i].flatten(), yout_imp_truth[i]) + + # Check that the other two inputs (tf, zpk) will work as well + tfin = ([1.0], [1.0, 1.0], 0.5) + yout_tfimpulse = np.asarray([0.0, 1.0, -1.0]) + tout, yout = dimpulse(tfin, n=3) + assert len(yout) == 1 + assert_array_almost_equal(yout[0].flatten(), yout_tfimpulse) + + zpkin = tf2zpk(tfin[0], tfin[1]) + (0.5,) + tout, yout = dimpulse(zpkin, n=3) + assert len(yout) == 1 + assert_array_almost_equal(yout[0].flatten(), yout_tfimpulse) + + # Raise an error for continuous-time systems + system = lti([1], [1, 1]) + assert_raises(AttributeError, dimpulse, system) + + def test_dlsim_trivial(self): + a = np.array([[0.0]]) + b = np.array([[0.0]]) + c = np.array([[0.0]]) + d = np.array([[0.0]]) + n = 5 + u = np.zeros(n).reshape(-1, 1) + tout, yout, xout = dlsim((a, b, c, d, 1), u) + xp_assert_equal(tout, np.arange(float(n))) + xp_assert_equal(yout, np.zeros((n, 1))) + xp_assert_equal(xout, np.zeros((n, 1))) + + def test_dlsim_simple1d(self): + a = np.array([[0.5]]) + b = np.array([[0.0]]) + c = np.array([[1.0]]) + d = np.array([[0.0]]) + n = 5 + u = np.zeros(n).reshape(-1, 1) + tout, yout, xout = dlsim((a, b, c, d, 1), u, x0=1) + xp_assert_equal(tout, np.arange(float(n))) + expected = (0.5 ** np.arange(float(n))).reshape(-1, 1) + xp_assert_equal(yout, expected) + xp_assert_equal(xout, expected) + + def test_dlsim_simple2d(self): + lambda1 = 0.5 + lambda2 = 0.25 + a = np.array([[lambda1, 0.0], + [0.0, lambda2]]) + b = np.array([[0.0], + [0.0]]) + c = np.array([[1.0, 0.0], + [0.0, 1.0]]) + d = np.array([[0.0], + [0.0]]) + n = 5 + u = np.zeros(n).reshape(-1, 1) + tout, yout, xout = dlsim((a, b, c, d, 1), u, x0=1) + xp_assert_equal(tout, np.arange(float(n))) + # The analytical solution: + expected = (np.array([lambda1, lambda2]) ** + np.arange(float(n)).reshape(-1, 1)) + xp_assert_equal(yout, expected) + xp_assert_equal(xout, expected) + + def test_more_step_and_impulse(self): + lambda1 = 0.5 + lambda2 = 0.75 + a = np.array([[lambda1, 0.0], + [0.0, lambda2]]) + b = np.array([[1.0, 0.0], + [0.0, 1.0]]) + c = np.array([[1.0, 1.0]]) + d = np.array([[0.0, 0.0]]) + + n = 10 + + # Check a step response. + ts, ys = dstep((a, b, c, d, 1), n=n) + + # Create the exact step response. + stp0 = (1.0 / (1 - lambda1)) * (1.0 - lambda1 ** np.arange(n)) + stp1 = (1.0 / (1 - lambda2)) * (1.0 - lambda2 ** np.arange(n)) + + xp_assert_close(ys[0][:, 0], stp0) + xp_assert_close(ys[1][:, 0], stp1) + + # Check an impulse response with an initial condition. + x0 = np.array([1.0, 1.0]) + ti, yi = dimpulse((a, b, c, d, 1), n=n, x0=x0) + + # Create the exact impulse response. + imp = (np.array([lambda1, lambda2]) ** + np.arange(-1, n + 1).reshape(-1, 1)) + imp[0, :] = 0.0 + # Analytical solution to impulse response + y0 = imp[:n, 0] + np.dot(imp[1:n + 1, :], x0) + y1 = imp[:n, 1] + np.dot(imp[1:n + 1, :], x0) + + xp_assert_close(yi[0][:, 0], y0) + xp_assert_close(yi[1][:, 0], y1) + + # Check that dt=0.1, n=3 gives 3 time values. + system = ([1.0], [1.0, -0.5], 0.1) + t, (y,) = dstep(system, n=3) + xp_assert_close(t, [0, 0.1, 0.2]) + xp_assert_equal(y.T, [[0, 1.0, 1.5]]) + t, (y,) = dimpulse(system, n=3) + xp_assert_close(t, [0, 0.1, 0.2]) + xp_assert_equal(y.T, [[0, 1, 0.5]]) + + +class TestDlti: + def test_dlti_instantiation(self): + # Test that lti can be instantiated. + + dt = 0.05 + # TransferFunction + s = dlti([1], [-1], dt=dt) + assert isinstance(s, TransferFunction) + assert isinstance(s, dlti) + assert not isinstance(s, lti) + assert s.dt == dt + + # ZerosPolesGain + s = dlti(np.array([]), np.array([-1]), 1, dt=dt) + assert isinstance(s, ZerosPolesGain) + assert isinstance(s, dlti) + assert not isinstance(s, lti) + assert s.dt == dt + + # StateSpace + s = dlti([1], [-1], 1, 3, dt=dt) + assert isinstance(s, StateSpace) + assert isinstance(s, dlti) + assert not isinstance(s, lti) + assert s.dt == dt + + # Number of inputs + assert_raises(ValueError, dlti, 1) + assert_raises(ValueError, dlti, 1, 1, 1, 1, 1) + + +class TestStateSpaceDisc: + def test_initialization(self): + # Check that all initializations work + dt = 0.05 + StateSpace(1, 1, 1, 1, dt=dt) + StateSpace([1], [2], [3], [4], dt=dt) + StateSpace(np.array([[1, 2], [3, 4]]), np.array([[1], [2]]), + np.array([[1, 0]]), np.array([[0]]), dt=dt) + StateSpace(1, 1, 1, 1, dt=True) + + def test_conversion(self): + # Check the conversion functions + s = StateSpace(1, 2, 3, 4, dt=0.05) + assert isinstance(s.to_ss(), StateSpace) + assert isinstance(s.to_tf(), TransferFunction) + assert isinstance(s.to_zpk(), ZerosPolesGain) + + # Make sure copies work + assert StateSpace(s) is not s + assert s.to_ss() is not s + + def test_properties(self): + # Test setters/getters for cross class properties. + # This implicitly tests to_tf() and to_zpk() + + # Getters + s = StateSpace(1, 1, 1, 1, dt=0.05) + xp_assert_equal(s.poles, [1.]) + xp_assert_equal(s.zeros, [0.]) + + +class TestTransferFunction: + def test_initialization(self): + # Check that all initializations work + dt = 0.05 + TransferFunction(1, 1, dt=dt) + TransferFunction([1], [2], dt=dt) + TransferFunction(np.array([1]), np.array([2]), dt=dt) + TransferFunction(1, 1, dt=True) + + def test_conversion(self): + # Check the conversion functions + s = TransferFunction([1, 0], [1, -1], dt=0.05) + assert isinstance(s.to_ss(), StateSpace) + assert isinstance(s.to_tf(), TransferFunction) + assert isinstance(s.to_zpk(), ZerosPolesGain) + + # Make sure copies work + assert TransferFunction(s) is not s + assert s.to_tf() is not s + + def test_properties(self): + # Test setters/getters for cross class properties. + # This implicitly tests to_ss() and to_zpk() + + # Getters + s = TransferFunction([1, 0], [1, -1], dt=0.05) + xp_assert_equal(s.poles, [1.]) + xp_assert_equal(s.zeros, [0.]) + + +class TestZerosPolesGain: + def test_initialization(self): + # Check that all initializations work + dt = 0.05 + ZerosPolesGain(1, 1, 1, dt=dt) + ZerosPolesGain([1], [2], 1, dt=dt) + ZerosPolesGain(np.array([1]), np.array([2]), 1, dt=dt) + ZerosPolesGain(1, 1, 1, dt=True) + + def test_conversion(self): + # Check the conversion functions + s = ZerosPolesGain(1, 2, 3, dt=0.05) + assert isinstance(s.to_ss(), StateSpace) + assert isinstance(s.to_tf(), TransferFunction) + assert isinstance(s.to_zpk(), ZerosPolesGain) + + # Make sure copies work + assert ZerosPolesGain(s) is not s + assert s.to_zpk() is not s + + +class Test_dfreqresp: + + def test_manual(self): + # Test dfreqresp() real part calculation (manual sanity check). + # 1st order low-pass filter: H(z) = 1 / (z - 0.2), + system = TransferFunction(1, [1, -0.2], dt=0.1) + w = [0.1, 1, 10] + w, H = dfreqresp(system, w=w) + + # test real + expected_re = [1.2383, 0.4130, -0.7553] + assert_almost_equal(H.real, expected_re, decimal=4) + + # test imag + expected_im = [-0.1555, -1.0214, 0.3955] + assert_almost_equal(H.imag, expected_im, decimal=4) + + def test_auto(self): + # Test dfreqresp() real part calculation. + # 1st order low-pass filter: H(z) = 1 / (z - 0.2), + system = TransferFunction(1, [1, -0.2], dt=0.1) + w = [0.1, 1, 10, 100] + w, H = dfreqresp(system, w=w) + jw = np.exp(w * 1j) + y = np.polyval(system.num, jw) / np.polyval(system.den, jw) + + # test real + expected_re = y.real + assert_almost_equal(H.real, expected_re) + + # test imag + expected_im = y.imag + assert_almost_equal(H.imag, expected_im) + + def test_freq_range(self): + # Test that freqresp() finds a reasonable frequency range. + # 1st order low-pass filter: H(z) = 1 / (z - 0.2), + # Expected range is from 0.01 to 10. + system = TransferFunction(1, [1, -0.2], dt=0.1) + n = 10 + expected_w = np.linspace(0, np.pi, 10, endpoint=False) + w, H = dfreqresp(system, n=n) + assert_almost_equal(w, expected_w) + + def test_pole_one(self): + # Test that freqresp() doesn't fail on a system with a pole at 0. + # integrator, pole at zero: H(s) = 1 / s + system = TransferFunction([1], [1, -1], dt=0.1) + + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, message="divide by zero") + sup.filter(RuntimeWarning, message="invalid value encountered") + w, H = dfreqresp(system, n=2) + assert w[0] == 0. # a fail would give not-a-number + + def test_error(self): + # Raise an error for continuous-time systems + system = lti([1], [1, 1]) + assert_raises(AttributeError, dfreqresp, system) + + def test_from_state_space(self): + # H(z) = 2 / z^3 - 0.5 * z^2 + + system_TF = dlti([2], [1, -0.5, 0, 0]) + + A = np.array([[0.5, 0, 0], + [1, 0, 0], + [0, 1, 0]]) + B = np.array([[1, 0, 0]]).T + C = np.array([[0, 0, 2]]) + D = 0 + + system_SS = dlti(A, B, C, D) + w = 10.0**np.arange(-3,0,.5) + with suppress_warnings() as sup: + sup.filter(BadCoefficients) + w1, H1 = dfreqresp(system_TF, w=w) + w2, H2 = dfreqresp(system_SS, w=w) + + assert_almost_equal(H1, H2) + + def test_from_zpk(self): + # 1st order low-pass filter: H(s) = 0.3 / (z - 0.2), + system_ZPK = dlti([],[0.2],0.3) + system_TF = dlti(0.3, [1, -0.2]) + w = [0.1, 1, 10, 100] + w1, H1 = dfreqresp(system_ZPK, w=w) + w2, H2 = dfreqresp(system_TF, w=w) + assert_almost_equal(H1, H2) + + +class Test_bode: + + def test_manual(self): + # Test bode() magnitude calculation (manual sanity check). + # 1st order low-pass filter: H(s) = 0.3 / (z - 0.2), + dt = 0.1 + system = TransferFunction(0.3, [1, -0.2], dt=dt) + w = [0.1, 0.5, 1, np.pi] + w2, mag, phase = dbode(system, w=w) + + # Test mag + expected_mag = [-8.5329, -8.8396, -9.6162, -12.0412] + assert_almost_equal(mag, expected_mag, decimal=4) + + # Test phase + expected_phase = [-7.1575, -35.2814, -67.9809, -180.0000] + assert_almost_equal(phase, expected_phase, decimal=4) + + # Test frequency + xp_assert_equal(np.array(w) / dt, w2) + + def test_auto(self): + # Test bode() magnitude calculation. + # 1st order low-pass filter: H(s) = 0.3 / (z - 0.2), + system = TransferFunction(0.3, [1, -0.2], dt=0.1) + w = np.array([0.1, 0.5, 1, np.pi]) + w2, mag, phase = dbode(system, w=w) + jw = np.exp(w * 1j) + y = np.polyval(system.num, jw) / np.polyval(system.den, jw) + + # Test mag + expected_mag = 20.0 * np.log10(abs(y)) + assert_almost_equal(mag, expected_mag) + + # Test phase + expected_phase = np.rad2deg(np.angle(y)) + assert_almost_equal(phase, expected_phase) + + def test_range(self): + # Test that bode() finds a reasonable frequency range. + # 1st order low-pass filter: H(s) = 0.3 / (z - 0.2), + dt = 0.1 + system = TransferFunction(0.3, [1, -0.2], dt=0.1) + n = 10 + # Expected range is from 0.01 to 10. + expected_w = np.linspace(0, np.pi, n, endpoint=False) / dt + w, mag, phase = dbode(system, n=n) + assert_almost_equal(w, expected_w) + + def test_pole_one(self): + # Test that freqresp() doesn't fail on a system with a pole at 0. + # integrator, pole at zero: H(s) = 1 / s + system = TransferFunction([1], [1, -1], dt=0.1) + + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, message="divide by zero") + sup.filter(RuntimeWarning, message="invalid value encountered") + w, mag, phase = dbode(system, n=2) + assert w[0] == 0. # a fail would give not-a-number + + def test_imaginary(self): + # bode() should not fail on a system with pure imaginary poles. + # The test passes if bode doesn't raise an exception. + system = TransferFunction([1], [1, 0, 100], dt=0.1) + dbode(system, n=2) + + def test_error(self): + # Raise an error for continuous-time systems + system = lti([1], [1, 1]) + assert_raises(AttributeError, dbode, system) + + +class TestTransferFunctionZConversion: + """Test private conversions between 'z' and 'z**-1' polynomials.""" + + def test_full(self): + # Numerator and denominator same order + num = np.asarray([2.0, 3, 4]) + den = np.asarray([5.0, 6, 7]) + num2, den2 = TransferFunction._z_to_zinv(num, den) + xp_assert_equal(num, num2) + xp_assert_equal(den, den2) + + num2, den2 = TransferFunction._zinv_to_z(num, den) + xp_assert_equal(num, num2) + xp_assert_equal(den, den2) + + def test_numerator(self): + # Numerator lower order than denominator + num = np.asarray([2.0, 3]) + den = np.asarray([50, 6, 7]) + num2, den2 = TransferFunction._z_to_zinv(num, den) + xp_assert_equal([0.0, 2, 3], num2) + xp_assert_equal(den, den2) + + num2, den2 = TransferFunction._zinv_to_z(num, den) + xp_assert_equal([2.0, 3, 0], num2) + xp_assert_equal(den, den2) + + def test_denominator(self): + # Numerator higher order than denominator + num = np.asarray([2., 3, 4]) + den = np.asarray([5.0, 6]) + num2, den2 = TransferFunction._z_to_zinv(num, den) + xp_assert_equal(num, num2) + xp_assert_equal([0.0, 5, 6], den2) + + num2, den2 = TransferFunction._zinv_to_z(num, den) + xp_assert_equal(num, num2) + xp_assert_equal([5.0, 6, 0], den2) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_filter_design.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_filter_design.py new file mode 100644 index 0000000000000000000000000000000000000000..62613b5bb64ec2980dc8a306e7a2a997b5713d2c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_filter_design.py @@ -0,0 +1,4485 @@ +import warnings + +from scipy._lib import _pep440 +import numpy as np +from numpy.testing import ( + assert_array_almost_equal_nulp, assert_warns, suppress_warnings +) +import pytest +from pytest import raises as assert_raises +from scipy._lib._array_api import ( + xp_assert_close, xp_assert_equal, + assert_array_almost_equal, +) + +from numpy import array, spacing, sin, pi, sort, sqrt +from scipy.signal import (argrelextrema, BadCoefficients, bessel, besselap, bilinear, + buttap, butter, buttord, cheb1ap, cheb1ord, cheb2ap, + cheb2ord, cheby1, cheby2, ellip, ellipap, ellipord, + firwin, freqs_zpk, freqs, freqz, freqz_zpk, + gammatone, group_delay, iircomb, iirdesign, iirfilter, + iirnotch, iirpeak, lp2bp, lp2bs, lp2hp, lp2lp, normalize, + medfilt, order_filter, + sos2tf, sos2zpk, sosfreqz, freqz_sos, tf2sos, tf2zpk, zpk2sos, + zpk2tf, bilinear_zpk, lp2lp_zpk, lp2hp_zpk, lp2bp_zpk, + lp2bs_zpk) +from scipy.signal._filter_design import (_cplxreal, _cplxpair, _norm_factor, + _bessel_poly, _bessel_zeros) + +try: + import mpmath +except ImportError: + mpmath = None + + +def mpmath_check(min_ver): + return pytest.mark.skipif( + mpmath is None + or _pep440.parse(mpmath.__version__) < _pep440.Version(min_ver), + reason=f"mpmath version >= {min_ver} required", + ) + + +class TestCplxPair: + + def test_trivial_input(self): + assert _cplxpair([]).size == 0 + assert _cplxpair(1) == 1 + + def test_output_order(self): + xp_assert_close(_cplxpair([1+1j, 1-1j]), [1-1j, 1+1j]) + + a = [1+1j, 1+1j, 1, 1-1j, 1-1j, 2] + b = [1-1j, 1+1j, 1-1j, 1+1j, 1, 2] + xp_assert_close(_cplxpair(a), b) + + # points spaced around the unit circle + z = np.exp(2j*pi*array([4, 3, 5, 2, 6, 1, 0])/7) + z1 = np.copy(z) + np.random.shuffle(z) + xp_assert_close(_cplxpair(z), z1) + np.random.shuffle(z) + xp_assert_close(_cplxpair(z), z1) + np.random.shuffle(z) + xp_assert_close(_cplxpair(z), z1) + + # Should be able to pair up all the conjugates + x = np.random.rand(10000) + 1j * np.random.rand(10000) + y = x.conj() + z = np.random.rand(10000) + x = np.concatenate((x, y, z)) + np.random.shuffle(x) + c = _cplxpair(x) + + # Every other element of head should be conjugates: + xp_assert_close(c[0:20000:2], np.conj(c[1:20000:2])) + # Real parts of head should be in sorted order: + xp_assert_close(c[0:20000:2].real, np.sort(c[0:20000:2].real)) + # Tail should be sorted real numbers: + xp_assert_close(c[20000:], np.sort(c[20000:])) + + def test_real_integer_input(self): + xp_assert_equal(_cplxpair([2, 0, 1]), [0, 1, 2]) + + def test_tolerances(self): + eps = spacing(1) + xp_assert_close(_cplxpair([1j, -1j, 1+1j*eps], tol=2*eps), + [-1j, 1j, 1+1j*eps]) + + # sorting close to 0 + xp_assert_close(_cplxpair([-eps+1j, +eps-1j]), [-1j, +1j]) + xp_assert_close(_cplxpair([+eps+1j, -eps-1j]), [-1j, +1j]) + xp_assert_close(_cplxpair([+1j, -1j]), [-1j, +1j]) + + def test_unmatched_conjugates(self): + # 1+2j is unmatched + assert_raises(ValueError, _cplxpair, [1+3j, 1-3j, 1+2j]) + + # 1+2j and 1-3j are unmatched + assert_raises(ValueError, _cplxpair, [1+3j, 1-3j, 1+2j, 1-3j]) + + # 1+3j is unmatched + assert_raises(ValueError, _cplxpair, [1+3j, 1-3j, 1+3j]) + + # Not conjugates + assert_raises(ValueError, _cplxpair, [4+5j, 4+5j]) + assert_raises(ValueError, _cplxpair, [1-7j, 1-7j]) + + # No pairs + assert_raises(ValueError, _cplxpair, [1+3j]) + assert_raises(ValueError, _cplxpair, [1-3j]) + + +class TestCplxReal: + + def test_trivial_input(self): + assert all(x.size == 0 for x in _cplxreal([])) + + x = _cplxreal(1) + assert x[0].size == 0 + xp_assert_equal(x[1], np.asarray([1])) + + + def test_output_order(self): + zc, zr = _cplxreal(np.roots(array([1, 0, 0, 1]))) + xp_assert_close(np.append(zc, zr), [1/2 + 1j*sin(pi/3), -1]) + + eps = spacing(1) + + a = [0+1j, 0-1j, eps + 1j, eps - 1j, -eps + 1j, -eps - 1j, + 1, 4, 2, 3, 0, 0, + 2+3j, 2-3j, + 1-eps + 1j, 1+2j, 1-2j, 1+eps - 1j, # sorts out of order + 3+1j, 3+1j, 3+1j, 3-1j, 3-1j, 3-1j, + 2-3j, 2+3j] + zc, zr = _cplxreal(a) + xp_assert_close(zc, [1j, 1j, 1j, 1+1j, 1+2j, 2+3j, 2+3j, 3+1j, 3+1j, + 3+1j]) + xp_assert_close(zr, [0.0, 0, 1, 2, 3, 4]) + + z = array([1-eps + 1j, 1+2j, 1-2j, 1+eps - 1j, 1+eps+3j, 1-2*eps-3j, + 0+1j, 0-1j, 2+4j, 2-4j, 2+3j, 2-3j, 3+7j, 3-7j, 4-eps+1j, + 4+eps-2j, 4-1j, 4-eps+2j]) + + zc, zr = _cplxreal(z) + xp_assert_close(zc, [1j, 1+1j, 1+2j, 1+3j, 2+3j, 2+4j, 3+7j, 4+1j, + 4+2j]) + xp_assert_equal(zr, np.asarray([])) + + def test_unmatched_conjugates(self): + # 1+2j is unmatched + assert_raises(ValueError, _cplxreal, [1+3j, 1-3j, 1+2j]) + + # 1+2j and 1-3j are unmatched + assert_raises(ValueError, _cplxreal, [1+3j, 1-3j, 1+2j, 1-3j]) + + # 1+3j is unmatched + assert_raises(ValueError, _cplxreal, [1+3j, 1-3j, 1+3j]) + + # No pairs + assert_raises(ValueError, _cplxreal, [1+3j]) + assert_raises(ValueError, _cplxreal, [1-3j]) + + def test_real_integer_input(self): + zc, zr = _cplxreal([2, 0, 1, 4]) + xp_assert_equal(zc, []) + xp_assert_equal(zr, [0, 1, 2, 4]) + + +class TestTf2zpk: + + @pytest.mark.parametrize('dt', (np.float64, np.complex128)) + def test_simple(self, dt): + z_r = np.array([0.5, -0.5]) + p_r = np.array([1.j / np.sqrt(2), -1.j / np.sqrt(2)]) + # Sort the zeros/poles so that we don't fail the test if the order + # changes + z_r.sort() + p_r.sort() + b = np.poly(z_r).astype(dt) + a = np.poly(p_r).astype(dt) + + z, p, k = tf2zpk(b, a) + z.sort() + # The real part of `p` is ~0.0, so sort by imaginary part + p = p[np.argsort(p.imag)] + + assert_array_almost_equal(z, z_r) + assert_array_almost_equal(p, p_r) + assert_array_almost_equal(k, 1.) + assert k.dtype == dt + + def test_bad_filter(self): + # Regression test for #651: better handling of badly conditioned + # filter coefficients. + with suppress_warnings(): + warnings.simplefilter("error", BadCoefficients) + assert_raises(BadCoefficients, tf2zpk, [1e-15], [1.0, 1.0]) + + +class TestZpk2Tf: + + def test_identity(self): + """Test the identity transfer function.""" + z = [] + p = [] + k = 1. + b, a = zpk2tf(z, p, k) + b_r = np.array([1.]) # desired result + a_r = np.array([1.]) # desired result + # The test for the *type* of the return values is a regression + # test for ticket #1095. In the case p=[], zpk2tf used to + # return the scalar 1.0 instead of array([1.0]). + xp_assert_equal(b, b_r) + assert isinstance(b, np.ndarray) + xp_assert_equal(a, a_r) + assert isinstance(a, np.ndarray) + + +class TestSos2Zpk: + + def test_basic(self): + sos = [[1, 0, 1, 1, 0, -0.81], + [1, 0, 0, 1, 0, +0.49]] + z, p, k = sos2zpk(sos) + z2 = [1j, -1j, 0, 0] + p2 = [0.9, -0.9, 0.7j, -0.7j] + k2 = 1 + assert_array_almost_equal(sort(z), sort(z2), decimal=4) + assert_array_almost_equal(sort(p), sort(p2), decimal=4) + assert_array_almost_equal(k, k2) + + sos = [[1.00000, +0.61803, 1.0000, 1.00000, +0.60515, 0.95873], + [1.00000, -1.61803, 1.0000, 1.00000, -1.58430, 0.95873], + [1.00000, +1.00000, 0.0000, 1.00000, +0.97915, 0.00000]] + z, p, k = sos2zpk(sos) + z2 = [-0.3090 + 0.9511j, -0.3090 - 0.9511j, 0.8090 + 0.5878j, + 0.8090 - 0.5878j, -1.0000 + 0.0000j, 0] + p2 = [-0.3026 + 0.9312j, -0.3026 - 0.9312j, 0.7922 + 0.5755j, + 0.7922 - 0.5755j, -0.9791 + 0.0000j, 0] + k2 = 1 + assert_array_almost_equal(sort(z), sort(z2), decimal=4) + assert_array_almost_equal(sort(p), sort(p2), decimal=4) + + sos = array([[1, 2, 3, 1, 0.2, 0.3], + [4, 5, 6, 1, 0.4, 0.5]]) + z = array([-1 - 1.41421356237310j, -1 + 1.41421356237310j, + -0.625 - 1.05326872164704j, -0.625 + 1.05326872164704j]) + p = array([-0.2 - 0.678232998312527j, -0.2 + 0.678232998312527j, + -0.1 - 0.538516480713450j, -0.1 + 0.538516480713450j]) + k = 4 + z2, p2, k2 = sos2zpk(sos) + xp_assert_close(_cplxpair(z2), z) + xp_assert_close(_cplxpair(p2), p) + assert k2 == k + + @pytest.mark.thread_unsafe + def test_fewer_zeros(self): + """Test not the expected number of p/z (effectively at origin).""" + sos = butter(3, 0.1, output='sos') + z, p, k = sos2zpk(sos) + assert len(z) == 4 + assert len(p) == 4 + + sos = butter(12, [5., 30.], 'bandpass', fs=1200., analog=False, + output='sos') + with pytest.warns(BadCoefficients, match='Badly conditioned'): + z, p, k = sos2zpk(sos) + assert len(z) == 24 + assert len(p) == 24 + + +class TestSos2Tf: + + def test_basic(self): + sos = [[1, 1, 1, 1, 0, -1], + [-2, 3, 1, 1, 10, 1]] + b, a = sos2tf(sos) + assert_array_almost_equal(b, [-2, 1, 2, 4, 1]) + assert_array_almost_equal(a, [1, 10, 0, -10, -1]) + + +class TestTf2Sos: + + def test_basic(self): + num = [2, 16, 44, 56, 32] + den = [3, 3, -15, 18, -12] + sos = tf2sos(num, den) + sos2 = [[0.6667, 4.0000, 5.3333, 1.0000, +2.0000, -4.0000], + [1.0000, 2.0000, 2.0000, 1.0000, -1.0000, +1.0000]] + assert_array_almost_equal(sos, sos2, decimal=4) + + b = [1, -3, 11, -27, 18] + a = [16, 12, 2, -4, -1] + sos = tf2sos(b, a) + sos2 = [[0.0625, -0.1875, 0.1250, 1.0000, -0.2500, -0.1250], + [1.0000, +0.0000, 9.0000, 1.0000, +1.0000, +0.5000]] + # assert_array_almost_equal(sos, sos2, decimal=4) + + @pytest.mark.parametrize('b, a, analog, sos', + [([1], [1], False, [[1., 0., 0., 1., 0., 0.]]), + ([1], [1], True, [[0., 0., 1., 0., 0., 1.]]), + ([1], [1., 0., -1.01, 0, 0.01], False, + [[1., 0., 0., 1., 0., -0.01], + [1., 0., 0., 1., 0., -1]]), + ([1], [1., 0., -1.01, 0, 0.01], True, + [[0., 0., 1., 1., 0., -1], + [0., 0., 1., 1., 0., -0.01]])]) + def test_analog(self, b, a, analog, sos): + sos2 = tf2sos(b, a, analog=analog) + assert_array_almost_equal(sos, sos2, decimal=4) + + +class TestZpk2Sos: + + @pytest.mark.parametrize('dt', 'fdgFDG') + @pytest.mark.parametrize('pairing, analog', + [('nearest', False), + ('keep_odd', False), + ('minimal', False), + ('minimal', True)]) + def test_dtypes(self, dt, pairing, analog): + z = np.array([-1, -1]).astype(dt) + ct = dt.upper() # the poles have to be complex + p = np.array([0.57149 + 0.29360j, 0.57149 - 0.29360j]).astype(ct) + k = np.array(1).astype(dt) + sos = zpk2sos(z, p, k, pairing=pairing, analog=analog) + sos2 = [[1, 2, 1, 1, -1.14298, 0.41280]] # octave & MATLAB + assert_array_almost_equal(sos, sos2, decimal=4) + + def test_basic(self): + for pairing in ('nearest', 'keep_odd'): + # + # Cases that match octave + # + + z = [-1, -1] + p = [0.57149 + 0.29360j, 0.57149 - 0.29360j] + k = 1 + sos = zpk2sos(z, p, k, pairing=pairing) + sos2 = [[1, 2, 1, 1, -1.14298, 0.41280]] # octave & MATLAB + assert_array_almost_equal(sos, sos2, decimal=4) + + z = [1j, -1j] + p = [0.9, -0.9, 0.7j, -0.7j] + k = 1 + sos = zpk2sos(z, p, k, pairing=pairing) + sos2 = [[1, 0, 1, 1, 0, +0.49], + [1, 0, 0, 1, 0, -0.81]] # octave + # sos2 = [[0, 0, 1, 1, -0.9, 0], + # [1, 0, 1, 1, 0.9, 0]] # MATLAB + assert_array_almost_equal(sos, sos2, decimal=4) + + z = [] + p = [0.8, -0.5+0.25j, -0.5-0.25j] + k = 1. + sos = zpk2sos(z, p, k, pairing=pairing) + sos2 = [[1., 0., 0., 1., 1., 0.3125], + [1., 0., 0., 1., -0.8, 0.]] # octave, MATLAB fails + assert_array_almost_equal(sos, sos2, decimal=4) + + z = [1., 1., 0.9j, -0.9j] + p = [0.99+0.01j, 0.99-0.01j, 0.1+0.9j, 0.1-0.9j] + k = 1 + sos = zpk2sos(z, p, k, pairing=pairing) + sos2 = [[1, 0, 0.81, 1, -0.2, 0.82], + [1, -2, 1, 1, -1.98, 0.9802]] # octave + # sos2 = [[1, -2, 1, 1, -0.2, 0.82], + # [1, 0, 0.81, 1, -1.98, 0.9802]] # MATLAB + assert_array_almost_equal(sos, sos2, decimal=4) + + z = [0.9+0.1j, 0.9-0.1j, -0.9] + p = [0.75+0.25j, 0.75-0.25j, 0.9] + k = 1 + sos = zpk2sos(z, p, k, pairing=pairing) + if pairing == 'keep_odd': + sos2 = [[1, -1.8, 0.82, 1, -1.5, 0.625], + [1, 0.9, 0, 1, -0.9, 0]] # octave; MATLAB fails + assert_array_almost_equal(sos, sos2, decimal=4) + else: # pairing == 'nearest' + sos2 = [[1, 0.9, 0, 1, -1.5, 0.625], + [1, -1.8, 0.82, 1, -0.9, 0]] # our algorithm + assert_array_almost_equal(sos, sos2, decimal=4) + + # + # Cases that differ from octave: + # + + z = [-0.3090 + 0.9511j, -0.3090 - 0.9511j, 0.8090 + 0.5878j, + +0.8090 - 0.5878j, -1.0000 + 0.0000j] + p = [-0.3026 + 0.9312j, -0.3026 - 0.9312j, 0.7922 + 0.5755j, + +0.7922 - 0.5755j, -0.9791 + 0.0000j] + k = 1 + sos = zpk2sos(z, p, k, pairing=pairing) + # sos2 = [[1, 0.618, 1, 1, 0.6052, 0.95870], + # [1, -1.618, 1, 1, -1.5844, 0.95878], + # [1, 1, 0, 1, 0.9791, 0]] # octave, MATLAB fails + sos2 = [[1, 1, 0, 1, +0.97915, 0], + [1, 0.61803, 1, 1, +0.60515, 0.95873], + [1, -1.61803, 1, 1, -1.58430, 0.95873]] + assert_array_almost_equal(sos, sos2, decimal=4) + + z = [-1 - 1.4142j, -1 + 1.4142j, + -0.625 - 1.0533j, -0.625 + 1.0533j] + p = [-0.2 - 0.6782j, -0.2 + 0.6782j, + -0.1 - 0.5385j, -0.1 + 0.5385j] + k = 4 + sos = zpk2sos(z, p, k, pairing=pairing) + sos2 = [[4, 8, 12, 1, 0.2, 0.3], + [1, 1.25, 1.5, 1, 0.4, 0.5]] # MATLAB + # sos2 = [[4, 8, 12, 1, 0.4, 0.5], + # [1, 1.25, 1.5, 1, 0.2, 0.3]] # octave + xp_assert_close(sos, sos2, rtol=1e-4, atol=1e-4) + + z = [] + p = [0.2, -0.5+0.25j, -0.5-0.25j] + k = 1. + sos = zpk2sos(z, p, k, pairing=pairing) + sos2 = [[1., 0., 0., 1., -0.2, 0.], + [1., 0., 0., 1., 1., 0.3125]] + # sos2 = [[1., 0., 0., 1., 1., 0.3125], + # [1., 0., 0., 1., -0.2, 0]] # octave, MATLAB fails + assert_array_almost_equal(sos, sos2, decimal=4) + + # The next two examples are adapted from Leland B. Jackson, + # "Digital Filters and Signal Processing (1995) p.400: + # http://books.google.com/books?id=VZ8uabI1pNMC&lpg=PA400&ots=gRD9pi8Jua&dq=Pole%2Fzero%20pairing%20for%20minimum%20roundoff%20noise%20in%20BSF.&pg=PA400#v=onepage&q=Pole%2Fzero%20pairing%20for%20minimum%20roundoff%20noise%20in%20BSF.&f=false + + deg2rad = np.pi / 180. + k = 1. + + # first example + thetas = [22.5, 45, 77.5] + mags = [0.8, 0.6, 0.9] + z = np.array([np.exp(theta * deg2rad * 1j) for theta in thetas]) + z = np.concatenate((z, np.conj(z))) + p = np.array([mag * np.exp(theta * deg2rad * 1j) + for theta, mag in zip(thetas, mags)]) + p = np.concatenate((p, np.conj(p))) + sos = zpk2sos(z, p, k) + # sos2 = [[1, -0.43288, 1, 1, -0.38959, 0.81], # octave, + # [1, -1.41421, 1, 1, -0.84853, 0.36], # MATLAB fails + # [1, -1.84776, 1, 1, -1.47821, 0.64]] + # Note that pole-zero pairing matches, but ordering is different + sos2 = [[1, -1.41421, 1, 1, -0.84853, 0.36], + [1, -1.84776, 1, 1, -1.47821, 0.64], + [1, -0.43288, 1, 1, -0.38959, 0.81]] + assert_array_almost_equal(sos, sos2, decimal=4) + + # second example + z = np.array([np.exp(theta * deg2rad * 1j) + for theta in (85., 10.)]) + z = np.concatenate((z, np.conj(z), [1, -1])) + sos = zpk2sos(z, p, k) + + # sos2 = [[1, -0.17431, 1, 1, -0.38959, 0.81], # octave "wrong", + # [1, -1.96962, 1, 1, -0.84853, 0.36], # MATLAB fails + # [1, 0, -1, 1, -1.47821, 0.64000]] + # Our pole-zero pairing matches the text, Octave does not + sos2 = [[1, 0, -1, 1, -0.84853, 0.36], + [1, -1.96962, 1, 1, -1.47821, 0.64], + [1, -0.17431, 1, 1, -0.38959, 0.81]] + assert_array_almost_equal(sos, sos2, decimal=4) + + # these examples are taken from the doc string, and show the + # effect of the 'pairing' argument + @pytest.mark.parametrize('pairing, sos', + [('nearest', + np.array([[1., 1., 0.5, 1., -0.75, 0.], + [1., 1., 0., 1., -1.6, 0.65]])), + ('keep_odd', + np.array([[1., 1., 0, 1., -0.75, 0.], + [1., 1., 0.5, 1., -1.6, 0.65]])), + ('minimal', + np.array([[0., 1., 1., 0., 1., -0.75], + [1., 1., 0.5, 1., -1.6, 0.65]]))]) + def test_pairing(self, pairing, sos): + z1 = np.array([-1, -0.5-0.5j, -0.5+0.5j]) + p1 = np.array([0.75, 0.8+0.1j, 0.8-0.1j]) + sos2 = zpk2sos(z1, p1, 1, pairing=pairing) + assert_array_almost_equal(sos, sos2, decimal=4) + + @pytest.mark.parametrize('p, sos_dt', + [([-1, 1, -0.1, 0.1], + [[0., 0., 1., 1., 0., -0.01], + [0., 0., 1., 1., 0., -1]]), + ([-0.7071+0.7071j, -0.7071-0.7071j, -0.1j, 0.1j], + [[0., 0., 1., 1., 0., 0.01], + [0., 0., 1., 1., 1.4142, 1.]])]) + def test_analog(self, p, sos_dt): + # test `analog` argument + # for discrete time, poles closest to unit circle should appear last + # for cont. time, poles closest to imaginary axis should appear last + sos2_dt = zpk2sos([], p, 1, pairing='minimal', analog=False) + sos2_ct = zpk2sos([], p, 1, pairing='minimal', analog=True) + assert_array_almost_equal(sos_dt, sos2_dt, decimal=4) + assert_array_almost_equal(sos_dt[::-1], sos2_ct, decimal=4) + + def test_bad_args(self): + with pytest.raises(ValueError, match=r'pairing must be one of'): + zpk2sos([1], [2], 1, pairing='no_such_pairing') + + with pytest.raises(ValueError, match=r'.*pairing must be "minimal"'): + zpk2sos([1], [2], 1, pairing='keep_odd', analog=True) + + with pytest.raises(ValueError, + match=r'.*must have len\(p\)>=len\(z\)'): + zpk2sos([1, 1], [2], 1, analog=True) + + with pytest.raises(ValueError, match=r'k must be real'): + zpk2sos([1], [2], k=1j) + + +class TestFreqs: + + def test_basic(self): + _, h = freqs([1.0], [1.0], worN=8) + assert_array_almost_equal(h, np.ones(8)) + + def test_output(self): + # 1st order low-pass filter: H(s) = 1 / (s + 1) + w = [0.1, 1, 10, 100] + num = [1] + den = [1, 1] + w, H = freqs(num, den, worN=w) + s = w * 1j + expected = 1 / (s + 1) + assert_array_almost_equal(H.real, expected.real) + assert_array_almost_equal(H.imag, expected.imag) + + def test_freq_range(self): + # Test that freqresp() finds a reasonable frequency range. + # 1st order low-pass filter: H(s) = 1 / (s + 1) + # Expected range is from 0.01 to 10. + num = [1] + den = [1, 1] + n = 10 + expected_w = np.logspace(-2, 1, n) + w, H = freqs(num, den, worN=n) + assert_array_almost_equal(w, expected_w) + + def test_plot(self): + + def plot(w, h): + assert_array_almost_equal(h, np.ones(8)) + + assert_raises(ZeroDivisionError, freqs, [1.0], [1.0], worN=8, + plot=lambda w, h: 1 / 0) + freqs([1.0], [1.0], worN=8, plot=plot) + + def test_backward_compat(self): + # For backward compatibility, test if None act as a wrapper for default + w1, h1 = freqs([1.0], [1.0]) + w2, h2 = freqs([1.0], [1.0], None) + assert_array_almost_equal(w1, w2) + assert_array_almost_equal(h1, h2) + + def test_w_or_N_types(self): + # Measure at 8 equally-spaced points + for N in (8, np.int8(8), np.int16(8), np.int32(8), np.int64(8), + np.array(8)): + w, h = freqs([1.0], [1.0], worN=N) + assert len(w) == 8 + assert_array_almost_equal(h, np.ones(8)) + + # Measure at frequency 8 rad/sec + for w in (8.0, 8.0+0j): + w_out, h = freqs([1.0], [1.0], worN=w) + assert_array_almost_equal(w_out, [8]) + assert_array_almost_equal(h, [1]) + + +class TestFreqs_zpk: + + def test_basic(self): + _, h = freqs_zpk([1.0], [1.0], [1.0], worN=8) + assert_array_almost_equal(h, np.ones(8)) + + def test_output(self): + # 1st order low-pass filter: H(s) = 1 / (s + 1) + w = [0.1, 1, 10, 100] + z = [] + p = [-1] + k = 1 + w, H = freqs_zpk(z, p, k, worN=w) + s = w * 1j + expected = 1 / (s + 1) + assert_array_almost_equal(H.real, expected.real) + assert_array_almost_equal(H.imag, expected.imag) + + def test_freq_range(self): + # Test that freqresp() finds a reasonable frequency range. + # 1st order low-pass filter: H(s) = 1 / (s + 1) + # Expected range is from 0.01 to 10. + z = [] + p = [-1] + k = 1 + n = 10 + expected_w = np.logspace(-2, 1, n) + w, H = freqs_zpk(z, p, k, worN=n) + assert_array_almost_equal(w, expected_w) + + def test_vs_freqs(self): + b, a = cheby1(4, 5, 100, analog=True, output='ba') + z, p, k = cheby1(4, 5, 100, analog=True, output='zpk') + + w1, h1 = freqs(b, a) + w2, h2 = freqs_zpk(z, p, k) + xp_assert_close(w1, w2) + xp_assert_close(h1, h2, rtol=1e-6) + + def test_backward_compat(self): + # For backward compatibility, test if None act as a wrapper for default + w1, h1 = freqs_zpk([1.0], [1.0], [1.0]) + w2, h2 = freqs_zpk([1.0], [1.0], [1.0], None) + assert_array_almost_equal(w1, w2) + assert_array_almost_equal(h1, h2) + + def test_w_or_N_types(self): + # Measure at 8 equally-spaced points + for N in (8, np.int8(8), np.int16(8), np.int32(8), np.int64(8), + np.array(8)): + w, h = freqs_zpk([], [], 1, worN=N) + assert len(w) == 8 + assert_array_almost_equal(h, np.ones(8)) + + # Measure at frequency 8 rad/sec + for w in (8.0, 8.0+0j): + w_out, h = freqs_zpk([], [], 1, worN=w) + assert_array_almost_equal(w_out, [8]) + assert_array_almost_equal(h, [1]) + + +class TestFreqz: + + def test_ticket1441(self): + """Regression test for ticket 1441.""" + # Because freqz previously used arange instead of linspace, + # when N was large, it would return one more point than + # requested. + N = 100000 + w, h = freqz([1.0], worN=N) + assert w.shape == (N,) + + def test_basic(self): + w, h = freqz([1.0], worN=8) + assert_array_almost_equal(w, np.pi * np.arange(8) / 8.) + assert_array_almost_equal(h, np.ones(8)) + w, h = freqz([1.0], worN=9) + assert_array_almost_equal(w, np.pi * np.arange(9) / 9.) + assert_array_almost_equal(h, np.ones(9)) + + for a in [1, np.ones(2)]: + w, h = freqz(np.ones(2), a, worN=0) + assert w.shape == (0,) + assert h.shape == (0,) + assert h.dtype == np.dtype('complex128') + + t = np.linspace(0, 1, 4, endpoint=False) + for b, a, h_whole in zip( + ([1., 0, 0, 0], np.sin(2 * np.pi * t)), + ([1., 0, 0, 0], [0.5, 0, 0, 0]), + ([1., 1., 1., 1.], [0, -4j, 0, 4j])): + w, h = freqz(b, a, worN=4, whole=True) + expected_w = np.linspace(0, 2 * np.pi, 4, endpoint=False) + assert_array_almost_equal(w, expected_w) + assert_array_almost_equal(h, h_whole) + # simultaneously check int-like support + w, h = freqz(b, a, worN=np.int32(4), whole=True) + assert_array_almost_equal(w, expected_w) + assert_array_almost_equal(h, h_whole) + w, h = freqz(b, a, worN=w, whole=True) + assert_array_almost_equal(w, expected_w) + assert_array_almost_equal(h, h_whole) + + def test_basic_whole(self): + w, h = freqz([1.0], worN=8, whole=True) + assert_array_almost_equal(w, 2 * np.pi * np.arange(8.0) / 8) + assert_array_almost_equal(h, np.ones(8)) + + def test_plot(self): + + def plot(w, h): + assert_array_almost_equal(w, np.pi * np.arange(8.0) / 8) + assert_array_almost_equal(h, np.ones(8)) + + assert_raises(ZeroDivisionError, freqz, [1.0], worN=8, + plot=lambda w, h: 1 / 0) + freqz([1.0], worN=8, plot=plot) + + def test_fft_wrapping(self): + # Some simple real FIR filters + bs = list() # filters + as_ = list() + hs_whole = list() + hs_half = list() + # 3 taps + t = np.linspace(0, 1, 3, endpoint=False) + bs.append(np.sin(2 * np.pi * t)) + as_.append(3.) + hs_whole.append([0, -0.5j, 0.5j]) + hs_half.append([0, np.sqrt(1./12.), -0.5j]) + # 4 taps + t = np.linspace(0, 1, 4, endpoint=False) + bs.append(np.sin(2 * np.pi * t)) + as_.append(0.5) + hs_whole.append([0, -4j, 0, 4j]) + hs_half.append([0, np.sqrt(8), -4j, -np.sqrt(8)]) + del t + for ii, b in enumerate(bs): + # whole + a = as_[ii] + expected_w = np.linspace(0, 2 * np.pi, len(b), endpoint=False) + w, h = freqz(b, a, worN=expected_w, whole=True) # polyval + err_msg = f'b = {b}, a={a}' + assert_array_almost_equal(w, expected_w, err_msg=err_msg) + assert_array_almost_equal(h, hs_whole[ii], err_msg=err_msg) + w, h = freqz(b, a, worN=len(b), whole=True) # FFT + assert_array_almost_equal(w, expected_w, err_msg=err_msg) + assert_array_almost_equal(h, hs_whole[ii], err_msg=err_msg) + # non-whole + expected_w = np.linspace(0, np.pi, len(b), endpoint=False) + w, h = freqz(b, a, worN=expected_w, whole=False) # polyval + assert_array_almost_equal(w, expected_w, err_msg=err_msg) + assert_array_almost_equal(h, hs_half[ii], err_msg=err_msg) + w, h = freqz(b, a, worN=len(b), whole=False) # FFT + assert_array_almost_equal(w, expected_w, err_msg=err_msg) + assert_array_almost_equal(h, hs_half[ii], err_msg=err_msg) + + # some random FIR filters (real + complex) + # assume polyval is accurate + rng = np.random.RandomState(0) + for ii in range(2, 10): # number of taps + b = rng.randn(ii) + for kk in range(2): + a = rng.randn(1) if kk == 0 else rng.randn(3) + for jj in range(2): + if jj == 1: + b = b + rng.randn(ii) * 1j + # whole + expected_w = np.linspace(0, 2 * np.pi, ii, endpoint=False) + w, expected_h = freqz(b, a, worN=expected_w, whole=True) + assert_array_almost_equal(w, expected_w) + w, h = freqz(b, a, worN=ii, whole=True) + assert_array_almost_equal(w, expected_w) + assert_array_almost_equal(h, expected_h) + # half + expected_w = np.linspace(0, np.pi, ii, endpoint=False) + w, expected_h = freqz(b, a, worN=expected_w, whole=False) + assert_array_almost_equal(w, expected_w) + w, h = freqz(b, a, worN=ii, whole=False) + assert_array_almost_equal(w, expected_w) + assert_array_almost_equal(h, expected_h) + + def test_broadcasting1(self): + # Test broadcasting with worN an integer or a 1-D array, + # b and a are n-dimensional arrays. + np.random.seed(123) + b = np.random.rand(3, 5, 1) + a = np.random.rand(2, 1) + for whole in [False, True]: + # Test with worN being integers (one fast for FFT and one not), + # a 1-D array, and an empty array. + for worN in [16, 17, np.linspace(0, 1, 10), np.array([])]: + w, h = freqz(b, a, worN=worN, whole=whole) + for k in range(b.shape[1]): + bk = b[:, k, 0] + ak = a[:, 0] + ww, hh = freqz(bk, ak, worN=worN, whole=whole) + xp_assert_close(ww, w) + xp_assert_close(hh, h[k]) + + def test_broadcasting2(self): + # Test broadcasting with worN an integer or a 1-D array, + # b is an n-dimensional array, and a is left at the default value. + np.random.seed(123) + b = np.random.rand(3, 5, 1) + for whole in [False, True]: + for worN in [16, 17, np.linspace(0, 1, 10)]: + w, h = freqz(b, worN=worN, whole=whole) + for k in range(b.shape[1]): + bk = b[:, k, 0] + ww, hh = freqz(bk, worN=worN, whole=whole) + xp_assert_close(ww, w) + xp_assert_close(hh, h[k]) + + def test_broadcasting3(self): + # Test broadcasting where b.shape[-1] is the same length + # as worN, and a is left at the default value. + np.random.seed(123) + N = 16 + b = np.random.rand(3, N) + for whole in [False, True]: + for worN in [N, np.linspace(0, 1, N)]: + w, h = freqz(b, worN=worN, whole=whole) + assert w.size == N + for k in range(N): + bk = b[:, k] + ww, hh = freqz(bk, worN=w[k], whole=whole) + xp_assert_close(ww, np.asarray(w[k])[None]) + xp_assert_close(hh, np.asarray(h[k])[None]) + + def test_broadcasting4(self): + # Test broadcasting with worN a 2-D array. + np.random.seed(123) + b = np.random.rand(4, 2, 1, 1) + a = np.random.rand(5, 2, 1, 1) + for whole in [False, True]: + for worN in [np.random.rand(6, 7), np.empty((6, 0))]: + w, h = freqz(b, a, worN=worN, whole=whole) + xp_assert_close(w, worN, rtol=1e-14) + assert h.shape == (2,) + worN.shape + for k in range(2): + ww, hh = freqz(b[:, k, 0, 0], a[:, k, 0, 0], + worN=worN.ravel(), + whole=whole) + xp_assert_close(ww, worN.ravel(), rtol=1e-14) + xp_assert_close(hh, h[k, :, :].ravel()) + + def test_backward_compat(self): + # For backward compatibility, test if None act as a wrapper for default + w1, h1 = freqz([1.0], 1) + w2, h2 = freqz([1.0], 1, None) + assert_array_almost_equal(w1, w2) + assert_array_almost_equal(h1, h2) + + def test_fs_param(self): + fs = 900 + b = [0.039479155677484369, 0.11843746703245311, 0.11843746703245311, + 0.039479155677484369] + a = [1.0, -1.3199152021838287, 0.80341991081938424, + -0.16767146321568049] + + # N = None, whole=False + w1, h1 = freqz(b, a, fs=fs) + w2, h2 = freqz(b, a) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs/2, 512, endpoint=False)) + + # N = None, whole=True + w1, h1 = freqz(b, a, whole=True, fs=fs) + w2, h2 = freqz(b, a, whole=True) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs, 512, endpoint=False)) + + # N = 5, whole=False + w1, h1 = freqz(b, a, 5, fs=fs) + w2, h2 = freqz(b, a, 5) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs/2, 5, endpoint=False)) + + # N = 5, whole=True + w1, h1 = freqz(b, a, 5, whole=True, fs=fs) + w2, h2 = freqz(b, a, 5, whole=True) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs, 5, endpoint=False)) + + # w is an array_like + for w in ([123], (123,), np.array([123]), (50, 123, 230), + np.array([50, 123, 230])): + w1, h1 = freqz(b, a, w, fs=fs) + w2, h2 = freqz(b, a, 2*pi*np.array(w)/fs) + xp_assert_close(h1, h2) + xp_assert_close(w, w1, check_dtype=False) + + def test_w_or_N_types(self): + # Measure at 7 (polyval) or 8 (fft) equally-spaced points + for N in (7, np.int8(7), np.int16(7), np.int32(7), np.int64(7), + np.array(7), + 8, np.int8(8), np.int16(8), np.int32(8), np.int64(8), + np.array(8)): + + w, h = freqz([1.0], worN=N) + assert_array_almost_equal(w, np.pi * np.arange(N) / N) + assert_array_almost_equal(h, np.ones(N)) + + w, h = freqz([1.0], worN=N, fs=100) + assert_array_almost_equal(w, np.linspace(0, 50, N, endpoint=False)) + assert_array_almost_equal(h, np.ones(N)) + + # Measure at frequency 8 Hz + for w in (8.0, 8.0+0j): + # Only makes sense when fs is specified + w_out, h = freqz([1.0], worN=w, fs=100) + assert_array_almost_equal(w_out, [8]) + assert_array_almost_equal(h, [1]) + + def test_nyquist(self): + w, h = freqz([1.0], worN=8, include_nyquist=True) + assert_array_almost_equal(w, np.pi * np.arange(8) / 7.) + assert_array_almost_equal(h, np.ones(8)) + w, h = freqz([1.0], worN=9, include_nyquist=True) + assert_array_almost_equal(w, np.pi * np.arange(9) / 8.) + assert_array_almost_equal(h, np.ones(9)) + + for a in [1, np.ones(2)]: + w, h = freqz(np.ones(2), a, worN=0, include_nyquist=True) + assert w.shape == (0,) + assert h.shape == (0,) + assert h.dtype == np.dtype('complex128') + + w1, h1 = freqz([1.0], worN=8, whole = True, include_nyquist=True) + w2, h2 = freqz([1.0], worN=8, whole = True, include_nyquist=False) + assert_array_almost_equal(w1, w2) + assert_array_almost_equal(h1, h2) + + # https://github.com/scipy/scipy/issues/17289 + # https://github.com/scipy/scipy/issues/15273 + @pytest.mark.parametrize('whole,nyquist,worN', + [(False, False, 32), + (False, True, 32), + (True, False, 32), + (True, True, 32), + (False, False, 257), + (False, True, 257), + (True, False, 257), + (True, True, 257)]) + def test_17289(self, whole, nyquist, worN): + d = [0, 1] + w, Drfft = freqz(d, worN=32, whole=whole, include_nyquist=nyquist) + _, Dpoly = freqz(d, worN=w) + xp_assert_close(Drfft, Dpoly) + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + freqz([1.0], fs=np.array([10, 20])) + + with pytest.raises(ValueError, match="Sampling.*be none."): + freqz([1.0], fs=None) + + +class Testfreqz_sos: + + def test_freqz_sos_basic(self): + # Compare the results of freqz and freqz_sos for a low order + # Butterworth filter. + + N = 500 + + b, a = butter(4, 0.2) + sos = butter(4, 0.2, output='sos') + w, h = freqz(b, a, worN=N) + w2, h2 = freqz_sos(sos, worN=N) + xp_assert_equal(w2, w) + xp_assert_close(h2, h, rtol=1e-10, atol=1e-14) + + b, a = ellip(3, 1, 30, (0.2, 0.3), btype='bandpass') + sos = ellip(3, 1, 30, (0.2, 0.3), btype='bandpass', output='sos') + w, h = freqz(b, a, worN=N) + w2, h2 = freqz_sos(sos, worN=N) + xp_assert_equal(w2, w) + xp_assert_close(h2, h, rtol=1e-10, atol=1e-14) + # must have at least one section + assert_raises(ValueError, freqz_sos, sos[:0]) + + def test_backward_compat(self): + # For backward compatibility, test if None act as a wrapper for default + N = 500 + + sos = butter(4, 0.2, output='sos') + w1, h1 = freqz_sos(sos, worN=N) + w2, h2 = sosfreqz(sos, worN=N) + assert_array_almost_equal(w1, w2) + assert_array_almost_equal(h1, h2) + + def test_freqz_sos_design(self): + # Compare freqz_sos output against expected values for different + # filter types + + # from cheb2ord + N, Wn = cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 60) + sos = cheby2(N, 60, Wn, 'stop', output='sos') + w, h = freqz_sos(sos) + h = np.abs(h) + w /= np.pi + xp_assert_close(20 * np.log10(h[w <= 0.1]), np.asarray(0.), atol=3.01, + check_shape=False) + xp_assert_close(20 * np.log10(h[w >= 0.6]), np.asarray(0.), atol=3.01, + check_shape=False) + xp_assert_close(h[(w >= 0.2) & (w <= 0.5)], + np.asarray(0.), atol=1e-3, + check_shape=False) # <= -60 dB + + N, Wn = cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 150) + sos = cheby2(N, 150, Wn, 'stop', output='sos') + w, h = freqz_sos(sos) + dB = 20*np.log10(np.abs(h)) + w /= np.pi + xp_assert_close(dB[w <= 0.1], np.asarray(0.0), atol=3.01, check_shape=False) + xp_assert_close(dB[w >= 0.6], np.asarray(0.0), atol=3.01, check_shape=False) + assert np.all(dB[(w >= 0.2) & (w <= 0.5)] < -149.9) + + # from cheb1ord + N, Wn = cheb1ord(0.2, 0.3, 3, 40) + sos = cheby1(N, 3, Wn, 'low', output='sos') + w, h = freqz_sos(sos) + h = np.abs(h) + w /= np.pi + xp_assert_close(20 * np.log10(h[w <= 0.2]), np.asarray(0.0), atol=3.01, + check_shape=False) + xp_assert_close(h[w >= 0.3], np.asarray(0.0), atol=1e-2, + check_shape=False) # <= -40 dB + + N, Wn = cheb1ord(0.2, 0.3, 1, 150) + sos = cheby1(N, 1, Wn, 'low', output='sos') + w, h = freqz_sos(sos) + dB = 20*np.log10(np.abs(h)) + w /= np.pi + xp_assert_close(dB[w <= 0.2], np.asarray(0.0), atol=1.01, + check_shape=False) + assert np.all(dB[w >= 0.3] < -149.9) + + # adapted from ellipord + N, Wn = ellipord(0.3, 0.2, 3, 60) + sos = ellip(N, 0.3, 60, Wn, 'high', output='sos') + w, h = freqz_sos(sos) + h = np.abs(h) + w /= np.pi + xp_assert_close(20 * np.log10(h[w >= 0.3]), np.asarray(0.0), atol=3.01, + check_shape=False) + xp_assert_close(h[w <= 0.1], np.asarray(0.0), atol=1.5e-3, + check_shape=False) # <= -60 dB (approx) + + # adapted from buttord + N, Wn = buttord([0.2, 0.5], [0.14, 0.6], 3, 40) + sos = butter(N, Wn, 'band', output='sos') + w, h = freqz_sos(sos) + h = np.abs(h) + w /= np.pi + + h014 = h[w <= 0.14] + xp_assert_close(h014, np.zeros_like(h014), atol=1e-2) # <= -40 dB + h06 = h[w >= 0.6] + xp_assert_close(h06, np.zeros_like(h06), atol=1e-2) # <= -40 dB + h0205 = 20 * np.log10(h[(w >= 0.2) & (w <= 0.5)]) + xp_assert_close(h0205, np.zeros_like(h0205), atol=3.01) + + N, Wn = buttord([0.2, 0.5], [0.14, 0.6], 3, 100) + sos = butter(N, Wn, 'band', output='sos') + w, h = freqz_sos(sos) + dB = 20*np.log10(np.maximum(np.abs(h), 1e-10)) + w /= np.pi + + assert np.all(dB[(w > 0) & (w <= 0.14)] < -99.9) + assert np.all(dB[w >= 0.6] < -99.9) + db0205 = dB[(w >= 0.2) & (w <= 0.5)] + xp_assert_close(db0205, np.zeros_like(db0205), atol=3.01) + + def test_freqz_sos_design_ellip(self): + N, Wn = ellipord(0.3, 0.1, 3, 60) + sos = ellip(N, 0.3, 60, Wn, 'high', output='sos') + w, h = freqz_sos(sos) + h = np.abs(h) + w /= np.pi + + h03 = 20 * np.log10(h[w >= 0.3]) + xp_assert_close(h03, np.zeros_like(h03), atol=3.01) + h01 = h[w <= 0.1] + xp_assert_close(h01, np.zeros_like(h01), atol=1.5e-3) # <= -60 dB (approx) + + N, Wn = ellipord(0.3, 0.2, .5, 150) + sos = ellip(N, .5, 150, Wn, 'high', output='sos') + w, h = freqz_sos(sos) + dB = 20*np.log10(np.maximum(np.abs(h), 1e-10)) + w /= np.pi + + db03 = dB[w >= 0.3] + xp_assert_close(db03, np.zeros_like(db03), atol=.55) + # Allow some numerical slop in the upper bound -150, so this is + # a check that dB[w <= 0.2] is less than or almost equal to -150. + assert dB[w <= 0.2].max() < -150*(1 - 1e-12) + + @mpmath_check("0.10") + def test_freqz_sos_against_mp(self): + # Compare the result of freqz_sos applied to a high order Butterworth + # filter against the result computed using mpmath. (signal.freqz fails + # miserably with such high order filters.) + from . import mpsig + N = 500 + order = 25 + Wn = 0.15 + with mpmath.workdps(80): + z_mp, p_mp, k_mp = mpsig.butter_lp(order, Wn) + w_mp, h_mp = mpsig.zpkfreqz(z_mp, p_mp, k_mp, N) + w_mp = np.array([float(x) for x in w_mp]) + h_mp = np.array([complex(x) for x in h_mp]) + + sos = butter(order, Wn, output='sos') + w, h = freqz_sos(sos, worN=N) + xp_assert_close(w, w_mp, rtol=1e-12, atol=1e-14) + xp_assert_close(h, h_mp, rtol=1e-12, atol=1e-14) + + def test_fs_param(self): + fs = 900 + sos = [[0.03934683014103762, 0.07869366028207524, 0.03934683014103762, + 1.0, -0.37256600288916636, 0.0], + [1.0, 1.0, 0.0, 1.0, -0.9495739996946778, 0.45125966317124144]] + + # N = None, whole=False + w1, h1 = freqz_sos(sos, fs=fs) + w2, h2 = freqz_sos(sos) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs/2, 512, endpoint=False)) + + # N = None, whole=True + w1, h1 = freqz_sos(sos, whole=True, fs=fs) + w2, h2 = freqz_sos(sos, whole=True) + xp_assert_close(h1, h2, atol=1e-27) + xp_assert_close(w1, np.linspace(0, fs, 512, endpoint=False)) + + # N = 5, whole=False + w1, h1 = freqz_sos(sos, 5, fs=fs) + w2, h2 = freqz_sos(sos, 5) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs/2, 5, endpoint=False)) + + # N = 5, whole=True + w1, h1 = freqz_sos(sos, 5, whole=True, fs=fs) + w2, h2 = freqz_sos(sos, 5, whole=True) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs, 5, endpoint=False)) + + # w is an array_like + for w in ([123], (123,), np.array([123]), (50, 123, 230), + np.array([50, 123, 230])): + w1, h1 = freqz_sos(sos, w, fs=fs) + w2, h2 = freqz_sos(sos, 2*pi*np.array(w)/fs) + xp_assert_close(h1, h2) + xp_assert_close(w, w1, check_dtype=False) + + def test_w_or_N_types(self): + # Measure at 7 (polyval) or 8 (fft) equally-spaced points + for N in (7, np.int8(7), np.int16(7), np.int32(7), np.int64(7), + np.array(7), + 8, np.int8(8), np.int16(8), np.int32(8), np.int64(8), + np.array(8)): + + w, h = freqz_sos([1, 0, 0, 1, 0, 0], worN=N) + assert_array_almost_equal(w, np.pi * np.arange(N) / N) + assert_array_almost_equal(h, np.ones(N)) + + w, h = freqz_sos([1, 0, 0, 1, 0, 0], worN=N, fs=100) + assert_array_almost_equal(w, np.linspace(0, 50, N, endpoint=False)) + assert_array_almost_equal(h, np.ones(N)) + + # Measure at frequency 8 Hz + for w in (8.0, 8.0+0j): + # Only makes sense when fs is specified + w_out, h = freqz_sos([1, 0, 0, 1, 0, 0], worN=w, fs=100) + assert_array_almost_equal(w_out, [8]) + assert_array_almost_equal(h, [1]) + + def test_fs_validation(self): + sos = butter(4, 0.2, output='sos') + with pytest.raises(ValueError, match="Sampling.*single scalar"): + freqz_sos(sos, fs=np.array([10, 20])) + + +class TestFreqz_zpk: + + def test_ticket1441(self): + """Regression test for ticket 1441.""" + # Because freqz previously used arange instead of linspace, + # when N was large, it would return one more point than + # requested. + N = 100000 + w, h = freqz_zpk([0.5], [0.5], 1.0, worN=N) + assert w.shape == (N,) + + def test_basic(self): + w, h = freqz_zpk([0.5], [0.5], 1.0, worN=8) + assert_array_almost_equal(w, np.pi * np.arange(8.0) / 8) + assert_array_almost_equal(h, np.ones(8)) + + def test_basic_whole(self): + w, h = freqz_zpk([0.5], [0.5], 1.0, worN=8, whole=True) + assert_array_almost_equal(w, 2 * np.pi * np.arange(8.0) / 8) + assert_array_almost_equal(h, np.ones(8)) + + def test_vs_freqz(self): + b, a = cheby1(4, 5, 0.5, analog=False, output='ba') + z, p, k = cheby1(4, 5, 0.5, analog=False, output='zpk') + + w1, h1 = freqz(b, a) + w2, h2 = freqz_zpk(z, p, k) + xp_assert_close(w1, w2) + xp_assert_close(h1, h2, rtol=1e-6) + + def test_backward_compat(self): + # For backward compatibility, test if None act as a wrapper for default + w1, h1 = freqz_zpk([0.5], [0.5], 1.0) + w2, h2 = freqz_zpk([0.5], [0.5], 1.0, None) + assert_array_almost_equal(w1, w2) + assert_array_almost_equal(h1, h2) + + def test_fs_param(self): + fs = 900 + z = [-1, -1, -1] + p = [0.4747869998473389+0.4752230717749344j, 0.37256600288916636, + 0.4747869998473389-0.4752230717749344j] + k = 0.03934683014103762 + + # N = None, whole=False + w1, h1 = freqz_zpk(z, p, k, whole=False, fs=fs) + w2, h2 = freqz_zpk(z, p, k, whole=False) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs/2, 512, endpoint=False)) + + # N = None, whole=True + w1, h1 = freqz_zpk(z, p, k, whole=True, fs=fs) + w2, h2 = freqz_zpk(z, p, k, whole=True) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs, 512, endpoint=False)) + + # N = 5, whole=False + w1, h1 = freqz_zpk(z, p, k, 5, fs=fs) + w2, h2 = freqz_zpk(z, p, k, 5) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs/2, 5, endpoint=False)) + + # N = 5, whole=True + w1, h1 = freqz_zpk(z, p, k, 5, whole=True, fs=fs) + w2, h2 = freqz_zpk(z, p, k, 5, whole=True) + xp_assert_close(h1, h2) + xp_assert_close(w1, np.linspace(0, fs, 5, endpoint=False)) + + # w is an array_like + for w in ([123], (123,), np.array([123]), (50, 123, 230), + np.array([50, 123, 230])): + w1, h1 = freqz_zpk(z, p, k, w, fs=fs) + w2, h2 = freqz_zpk(z, p, k, 2*pi*np.array(w)/fs) + xp_assert_close(h1, h2) + xp_assert_close(w, w1, check_dtype=False) + + def test_w_or_N_types(self): + # Measure at 8 equally-spaced points + for N in (8, np.int8(8), np.int16(8), np.int32(8), np.int64(8), + np.array(8)): + + w, h = freqz_zpk([], [], 1, worN=N) + assert_array_almost_equal(w, np.pi * np.arange(8) / 8.) + assert_array_almost_equal(h, np.ones(8)) + + w, h = freqz_zpk([], [], 1, worN=N, fs=100) + assert_array_almost_equal(w, np.linspace(0, 50, 8, endpoint=False)) + assert_array_almost_equal(h, np.ones(8)) + + # Measure at frequency 8 Hz + for w in (8.0, 8.0+0j): + # Only makes sense when fs is specified + w_out, h = freqz_zpk([], [], 1, worN=w, fs=100) + assert_array_almost_equal(w_out, [8]) + assert_array_almost_equal(h, [1]) + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + freqz_zpk([1.0], [1.0], [1.0], fs=np.array([10, 20])) + + with pytest.raises(ValueError, match="Sampling.*be none."): + freqz_zpk([1.0], [1.0], [1.0], fs=None) + + +class TestNormalize: + + def test_allclose(self): + """Test for false positive on allclose in normalize() in + filter_design.py""" + # Test to make sure the allclose call within signal.normalize does not + # choose false positives. Then check against a known output from MATLAB + # to make sure the fix doesn't break anything. + + # These are the coefficients returned from + # `[b,a] = cheby1(8, 0.5, 0.048)' + # in MATLAB. There are at least 15 significant figures in each + # coefficient, so it makes sense to test for errors on the order of + # 1e-13 (this can always be relaxed if different platforms have + # different rounding errors) + b_matlab = np.array([2.150733144728282e-11, 1.720586515782626e-10, + 6.022052805239190e-10, 1.204410561047838e-09, + 1.505513201309798e-09, 1.204410561047838e-09, + 6.022052805239190e-10, 1.720586515782626e-10, + 2.150733144728282e-11]) + a_matlab = np.array([1.000000000000000e+00, -7.782402035027959e+00, + 2.654354569747454e+01, -5.182182531666387e+01, + 6.334127355102684e+01, -4.963358186631157e+01, + 2.434862182949389e+01, -6.836925348604676e+00, + 8.412934944449140e-01]) + + # This is the input to signal.normalize after passing through the + # equivalent steps in signal.iirfilter as was done for MATLAB + b_norm_in = np.array([1.5543135865293012e-06, 1.2434508692234413e-05, + 4.3520780422820447e-05, 8.7041560845640893e-05, + 1.0880195105705122e-04, 8.7041560845640975e-05, + 4.3520780422820447e-05, 1.2434508692234413e-05, + 1.5543135865293012e-06]) + a_norm_in = np.array([7.2269025909127173e+04, -5.6242661430467968e+05, + 1.9182761917308895e+06, -3.7451128364682454e+06, + 4.5776121393762771e+06, -3.5869706138592605e+06, + 1.7596511818472347e+06, -4.9409793515707983e+05, + 6.0799461347219651e+04]) + + b_output, a_output = normalize(b_norm_in, a_norm_in) + + # The test on b works for decimal=14 but the one for a does not. For + # the sake of consistency, both of these are decimal=13. If something + # breaks on another platform, it is probably fine to relax this lower. + assert_array_almost_equal(b_matlab, b_output, decimal=13) + assert_array_almost_equal(a_matlab, a_output, decimal=13) + + def test_errors(self): + """Test the error cases.""" + # all zero denominator + assert_raises(ValueError, normalize, [1, 2], 0) + + # denominator not 1 dimensional + assert_raises(ValueError, normalize, [1, 2], [[1]]) + + # numerator too many dimensions + assert_raises(ValueError, normalize, [[[1, 2]]], 1) + + +class TestLp2lp: + + def test_basic(self): + b = [1] + a = [1, np.sqrt(2), 1] + b_lp, a_lp = lp2lp(b, a, 0.38574256627112119) + assert_array_almost_equal(b_lp, [0.1488], decimal=4) + assert_array_almost_equal(a_lp, [1, 0.5455, 0.1488], decimal=4) + + +class TestLp2hp: + + def test_basic(self): + b = [0.25059432325190018] + a = [1, 0.59724041654134863, 0.92834805757524175, 0.25059432325190018] + b_hp, a_hp = lp2hp(b, a, 2*np.pi*5000) + xp_assert_close(b_hp, [1.0, 0, 0, 0]) + xp_assert_close(a_hp, [1, 1.1638e5, 2.3522e9, 1.2373e14], rtol=1e-4) + + +class TestLp2bp: + + def test_basic(self): + b = [1] + a = [1, 2, 2, 1] + b_bp, a_bp = lp2bp(b, a, 2*np.pi*4000, 2*np.pi*2000) + xp_assert_close(b_bp, [1.9844e12, 0, 0, 0], rtol=1e-6) + xp_assert_close(a_bp, [1, 2.5133e4, 2.2108e9, 3.3735e13, + 1.3965e18, 1.0028e22, 2.5202e26], rtol=1e-4) + + +class TestLp2bs: + + def test_basic(self): + b = [1] + a = [1, 1] + b_bs, a_bs = lp2bs(b, a, 0.41722257286366754, 0.18460575326152251) + assert_array_almost_equal(b_bs, [1, 0, 0.17407], decimal=5) + assert_array_almost_equal(a_bs, [1, 0.18461, 0.17407], decimal=5) + + +class TestBilinear: + + def test_basic(self): + b = [0.14879732743343033] + a = [1, 0.54552236880522209, 0.14879732743343033] + b_z, a_z = bilinear(b, a, 0.5) + assert_array_almost_equal(b_z, [0.087821, 0.17564, 0.087821], + decimal=5) + assert_array_almost_equal(a_z, [1, -1.0048, 0.35606], decimal=4) + + b = [1, 0, 0.17407467530697837] + a = [1, 0.18460575326152251, 0.17407467530697837] + b_z, a_z = bilinear(b, a, 0.5) + assert_array_almost_equal(b_z, [0.86413, -1.2158, 0.86413], + decimal=4) + assert_array_almost_equal(a_z, [1, -1.2158, 0.72826], + decimal=4) + + def test_fs_validation(self): + b = [0.14879732743343033] + a = [1, 0.54552236880522209, 0.14879732743343033] + with pytest.raises(ValueError, match="Sampling.*single scalar"): + bilinear(b, a, fs=np.array([10, 20])) + + with pytest.raises(ValueError, match="Sampling.*be none"): + bilinear(b, a, fs=None) + + +class TestLp2lp_zpk: + + def test_basic(self): + z = [] + p = [(-1+1j)/np.sqrt(2), (-1-1j)/np.sqrt(2)] + k = 1 + z_lp, p_lp, k_lp = lp2lp_zpk(z, p, k, 5) + xp_assert_equal(z_lp, []) + xp_assert_close(sort(p_lp), sort(p)*5) + xp_assert_close(k_lp, 25.) + + # Pseudo-Chebyshev with both poles and zeros + z = [-2j, +2j] + p = [-0.75, -0.5-0.5j, -0.5+0.5j] + k = 3 + z_lp, p_lp, k_lp = lp2lp_zpk(z, p, k, 20) + xp_assert_close(sort(z_lp), sort([-40j, +40j])) + xp_assert_close(sort(p_lp), sort([-15, -10-10j, -10+10j])) + xp_assert_close(k_lp, 60.) + + def test_fs_validation(self): + z = [-2j, +2j] + p = [-0.75, -0.5 - 0.5j, -0.5 + 0.5j] + k = 3 + + with pytest.raises(ValueError, match="Sampling.*single scalar"): + bilinear_zpk(z, p, k, fs=np.array([10, 20])) + + with pytest.raises(ValueError, match="Sampling.*be none"): + bilinear_zpk(z, p, k, fs=None) + + +class TestLp2hp_zpk: + + def test_basic(self): + z = [] + p = [(-1+1j)/np.sqrt(2), (-1-1j)/np.sqrt(2)] + k = 1 + + z_hp, p_hp, k_hp = lp2hp_zpk(z, p, k, 5) + xp_assert_equal(z_hp, np.asarray([0.0, 0.0])) + xp_assert_close(sort(p_hp), sort(p)*5) + xp_assert_close(k_hp, 1.0) + + z = [-2j, +2j] + p = [-0.75, -0.5-0.5j, -0.5+0.5j] + k = 3 + z_hp, p_hp, k_hp = lp2hp_zpk(z, p, k, 6) + xp_assert_close(sort(z_hp), sort([-3j, 0, +3j])) + xp_assert_close(sort(p_hp), sort([-8, -6-6j, -6+6j])) + xp_assert_close(k_hp, 32.0) + + +class TestLp2bp_zpk: + + def test_basic(self): + z = [-2j, +2j] + p = [-0.75, -0.5-0.5j, -0.5+0.5j] + k = 3 + z_bp, p_bp, k_bp = lp2bp_zpk(z, p, k, 15, 8) + xp_assert_close(sort(z_bp), sort([-25j, -9j, 0, +9j, +25j])) + xp_assert_close(sort(p_bp), sort([-3 + 6j*sqrt(6), + -3 - 6j*sqrt(6), + +2j+sqrt(-8j-225)-2, + -2j+sqrt(+8j-225)-2, + +2j-sqrt(-8j-225)-2, + -2j-sqrt(+8j-225)-2, ])) + xp_assert_close(k_bp, 24.0) + + +class TestLp2bs_zpk: + + def test_basic(self): + z = [-2j, +2j] + p = [-0.75, -0.5-0.5j, -0.5+0.5j] + k = 3 + + z_bs, p_bs, k_bs = lp2bs_zpk(z, p, k, 35, 12) + + xp_assert_close(sort(z_bs), sort([+35j, -35j, + +3j+sqrt(1234)*1j, + -3j+sqrt(1234)*1j, + +3j-sqrt(1234)*1j, + -3j-sqrt(1234)*1j])) + xp_assert_close(sort(p_bs), sort([+3j*sqrt(129) - 8, + -3j*sqrt(129) - 8, + (-6 + 6j) - sqrt(-1225 - 72j), + (-6 - 6j) - sqrt(-1225 + 72j), + (-6 + 6j) + sqrt(-1225 - 72j), + (-6 - 6j) + sqrt(-1225 + 72j), ])) + xp_assert_close(k_bs, 32.0) + + +class TestBilinear_zpk: + + def test_basic(self): + z = [-2j, +2j] + p = [-0.75, -0.5-0.5j, -0.5+0.5j] + k = 3 + + z_d, p_d, k_d = bilinear_zpk(z, p, k, 10) + + xp_assert_close(sort(z_d), sort([(20-2j)/(20+2j), (20+2j)/(20-2j), + -1])) + xp_assert_close(sort(p_d), sort([77/83, + (1j/2 + 39/2) / (41/2 - 1j/2), + (39/2 - 1j/2) / (1j/2 + 41/2), ])) + xp_assert_close(k_d, 9696/69803) + + +class TestPrototypeType: + + def test_output_type(self): + # Prototypes should consistently output arrays, not lists + # https://github.com/scipy/scipy/pull/441 + for func in (buttap, + besselap, + lambda N: cheb1ap(N, 1), + lambda N: cheb2ap(N, 20), + lambda N: ellipap(N, 1, 20)): + for N in range(7): + z, p, k = func(N) + assert isinstance(z, np.ndarray) + assert isinstance(p, np.ndarray) + + +def dB(x): + # Return magnitude in decibels, avoiding divide-by-zero warnings + # (and deal with some "not less-ordered" errors when -inf shows up) + return 20 * np.log10(np.maximum(np.abs(x), np.finfo(np.float64).tiny)) + + +class TestButtord: + + def test_lowpass(self): + wp = 0.2 + ws = 0.3 + rp = 3 + rs = 60 + N, Wn = buttord(wp, ws, rp, rs, False) + b, a = butter(N, Wn, 'lowpass', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp < dB(h[w <= wp])) + assert np.all(dB(h[ws <= w]) < -rs) + + assert N == 16 + xp_assert_close(Wn, + 2.0002776782743284e-01, rtol=1e-15) + + def test_highpass(self): + wp = 0.3 + ws = 0.2 + rp = 3 + rs = 70 + N, Wn = buttord(wp, ws, rp, rs, False) + b, a = butter(N, Wn, 'highpass', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp < dB(h[wp <= w])) + assert np.all(dB(h[w <= ws]) < -rs) + + assert N == 18 + xp_assert_close(Wn, + 2.9996603079132672e-01, rtol=1e-15) + + def test_bandpass(self): + wp = [0.2, 0.5] + ws = [0.1, 0.6] + rp = 3 + rs = 80 + N, Wn = buttord(wp, ws, rp, rs, False) + b, a = butter(N, Wn, 'bandpass', False) + w, h = freqz(b, a) + w /= np.pi + + assert np.all((-rp - 0.1) < dB(h[np.logical_and(wp[0] <= w, w <= wp[1])])) + + assert np.all(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]) < (-rs + 0.1)) + + assert N == 18 + xp_assert_close(Wn, [1.9998742411409134e-01, 5.0002139595676276e-01], + rtol=1e-15) + + def test_bandstop(self): + wp = [0.1, 0.6] + ws = [0.2, 0.5] + rp = 3 + rs = 90 + N, Wn = buttord(wp, ws, rp, rs, False) + b, a = butter(N, Wn, 'bandstop', False) + w, h = freqz(b, a) + w /= np.pi + + assert np.all(-rp < dB(h[np.logical_or(w <= wp[0], wp[1] <= w)])) + assert np.all(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]) < -rs) + + assert N == 20 + xp_assert_close(Wn, [1.4759432329294042e-01, 5.9997365985276407e-01], + rtol=1e-6) + + def test_analog(self): + wp = 200 + ws = 600 + rp = 3 + rs = 60 + N, Wn = buttord(wp, ws, rp, rs, True) + b, a = butter(N, Wn, 'lowpass', True) + w, h = freqs(b, a) + assert np.all(-rp < dB(h[w <= wp])) + assert np.all(dB(h[ws <= w]) < -rs) + + assert N == 7 + xp_assert_close(Wn, 2.0006785355671877e+02, rtol=1e-15) + + n, Wn = buttord(1, 550/450, 1, 26, analog=True) + assert n == 19 + xp_assert_close(Wn, 1.0361980524629517, rtol=1e-15) + + xp_assert_equal(buttord(1, 1.2, 1, 80, analog=True)[0], 55) + + def test_fs_param(self): + wp = [4410, 11025] + ws = [2205, 13230] + rp = 3 + rs = 80 + fs = 44100 + N, Wn = buttord(wp, ws, rp, rs, False, fs=fs) + b, a = butter(N, Wn, 'bandpass', False, fs=fs) + w, h = freqz(b, a, fs=fs) + assert np.all(-rp - 0.1 < dB(h[np.logical_and(wp[0] <= w, w <= wp[1])])) + assert np.all(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]) < -rs + 0.1) + + assert N == 18 + xp_assert_close(Wn, [4409.722701715714, 11025.47178084662], + rtol=1e-15) + + def test_invalid_input(self): + with pytest.raises(ValueError) as exc_info: + buttord([20, 50], [14, 60], 3, 2) + assert "gpass should be smaller than gstop" in str(exc_info.value) + + with pytest.raises(ValueError) as exc_info: + buttord([20, 50], [14, 60], -1, 2) + assert "gpass should be larger than 0.0" in str(exc_info.value) + + with pytest.raises(ValueError) as exc_info: + buttord([20, 50], [14, 60], 1, -2) + assert "gstop should be larger than 0.0" in str(exc_info.value) + + @pytest.mark.thread_unsafe + def test_runtime_warnings(self): + msg = "Order is zero.*|divide by zero encountered" + with pytest.warns(RuntimeWarning, match=msg): + buttord(0.0, 1.0, 3, 60) + + def test_ellip_butter(self): + # The purpose of the test is to compare to some known output from past + # scipy versions. The values to compare to are generated with scipy + # 1.9.1 (there is nothing special about this particular version though) + n, wn = buttord([0.1, 0.6], [0.2, 0.5], 3, 60) + assert n == 14 + + def test_fs_validation(self): + wp = 0.2 + ws = 0.3 + rp = 3 + rs = 60 + + with pytest.raises(ValueError, match="Sampling.*single scalar"): + buttord(wp, ws, rp, rs, False, fs=np.array([10, 20])) + + +class TestCheb1ord: + + def test_lowpass(self): + wp = 0.2 + ws = 0.3 + rp = 3 + rs = 60 + N, Wn = cheb1ord(wp, ws, rp, rs, False) + b, a = cheby1(N, rp, Wn, 'low', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[w <= wp])) + assert np.all(dB(h[ws <= w]) < -rs + 0.1) + + assert N == 8 + xp_assert_close(Wn, 0.2, rtol=1e-15) + + def test_highpass(self): + wp = 0.3 + ws = 0.2 + rp = 3 + rs = 70 + N, Wn = cheb1ord(wp, ws, rp, rs, False) + b, a = cheby1(N, rp, Wn, 'high', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[wp <= w])) + assert np.all(dB(h[w <= ws]) < -rs + 0.1) + + assert N == 9 + xp_assert_close(Wn, 0.3, rtol=1e-15) + + def test_bandpass(self): + wp = [0.2, 0.5] + ws = [0.1, 0.6] + rp = 3 + rs = 80 + N, Wn = cheb1ord(wp, ws, rp, rs, False) + b, a = cheby1(N, rp, Wn, 'band', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[np.logical_and(wp[0] <= w, w <= wp[1])])) + assert np.all(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]) < -rs + 0.1) + + assert N == 9 + xp_assert_close(Wn, [0.2, 0.5], rtol=1e-15) + + def test_bandstop(self): + wp = [0.1, 0.6] + ws = [0.2, 0.5] + rp = 3 + rs = 90 + N, Wn = cheb1ord(wp, ws, rp, rs, False) + b, a = cheby1(N, rp, Wn, 'stop', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[np.logical_or(w <= wp[0], wp[1] <= w)])) + assert np.all(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]) < -rs + 0.1) + + assert N == 10 + xp_assert_close(Wn, [0.14758232569947785, 0.6], rtol=1e-5) + + def test_analog(self): + wp = 700 + ws = 100 + rp = 3 + rs = 70 + N, Wn = cheb1ord(wp, ws, rp, rs, True) + b, a = cheby1(N, rp, Wn, 'high', True) + w, h = freqs(b, a) + assert np.all(-rp - 0.1 < dB(h[wp <= w])) + assert np.all(dB(h[w <= ws]) < -rs + 0.1) + + assert N == 4 + xp_assert_close(Wn, 700.0, rtol=1e-15) + + xp_assert_equal(cheb1ord(1, 1.2, 1, 80, analog=True)[0], 17) + + def test_fs_param(self): + wp = 4800 + ws = 7200 + rp = 3 + rs = 60 + fs = 48000 + N, Wn = cheb1ord(wp, ws, rp, rs, False, fs=fs) + b, a = cheby1(N, rp, Wn, 'low', False, fs=fs) + w, h = freqz(b, a, fs=fs) + assert np.all(-rp - 0.1 < dB(h[w <= wp])) + assert np.all(dB(h[ws <= w]) < -rs + 0.1) + + assert N == 8 + xp_assert_close(Wn, 4800.0, rtol=1e-15) + + def test_invalid_input(self): + with pytest.raises(ValueError) as exc_info: + cheb1ord(0.2, 0.3, 3, 2) + assert "gpass should be smaller than gstop" in str(exc_info.value) + + with pytest.raises(ValueError) as exc_info: + cheb1ord(0.2, 0.3, -1, 2) + assert "gpass should be larger than 0.0" in str(exc_info.value) + + with pytest.raises(ValueError) as exc_info: + cheb1ord(0.2, 0.3, 1, -2) + assert "gstop should be larger than 0.0" in str(exc_info.value) + + def test_ellip_cheb1(self): + # The purpose of the test is to compare to some known output from past + # scipy versions. The values to compare to are generated with scipy + # 1.9.1 (there is nothing special about this particular version though) + n, wn = cheb1ord([0.1, 0.6], [0.2, 0.5], 3, 60) + assert n == 7 + + n2, w2 = cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 60) + assert not (wn == w2).all() + + def test_fs_validation(self): + wp = 0.2 + ws = 0.3 + rp = 3 + rs = 60 + + with pytest.raises(ValueError, match="Sampling.*single scalar"): + cheb1ord(wp, ws, rp, rs, False, fs=np.array([10, 20])) + + +class TestCheb2ord: + + def test_lowpass(self): + wp = 0.2 + ws = 0.3 + rp = 3 + rs = 60 + N, Wn = cheb2ord(wp, ws, rp, rs, False) + b, a = cheby2(N, rs, Wn, 'lp', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[w <= wp])) + assert np.all(dB(h[ws <= w]) < -rs + 0.1) + + assert N == 8 + xp_assert_close(Wn, 0.28647639976553163, rtol=1e-15) + + def test_highpass(self): + wp = 0.3 + ws = 0.2 + rp = 3 + rs = 70 + N, Wn = cheb2ord(wp, ws, rp, rs, False) + b, a = cheby2(N, rs, Wn, 'hp', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[wp <= w])) + assert np.all(dB(h[w <= ws]) < -rs + 0.1) + + assert N == 9 + xp_assert_close(Wn, 0.20697492182903282, rtol=1e-15) + + def test_bandpass(self): + wp = [0.2, 0.5] + ws = [0.1, 0.6] + rp = 3 + rs = 80 + N, Wn = cheb2ord(wp, ws, rp, rs, False) + b, a = cheby2(N, rs, Wn, 'bp', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[np.logical_and(wp[0] <= w, w <= wp[1])])) + assert np.all(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]) < -rs + 0.1) + + assert N == 9 + xp_assert_close(Wn, [0.14876937565923479, 0.59748447842351482], + rtol=1e-15) + + def test_bandstop(self): + wp = [0.1, 0.6] + ws = [0.2, 0.5] + rp = 3 + rs = 90 + N, Wn = cheb2ord(wp, ws, rp, rs, False) + b, a = cheby2(N, rs, Wn, 'bs', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[np.logical_or(w <= wp[0], wp[1] <= w)])) + assert np.all(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]) < -rs + 0.1) + + assert N == 10 + xp_assert_close(Wn, [0.19926249974781743, 0.50125246585567362], + rtol=1e-6) + + def test_analog(self): + wp = [20, 50] + ws = [10, 60] + rp = 3 + rs = 80 + N, Wn = cheb2ord(wp, ws, rp, rs, True) + b, a = cheby2(N, rs, Wn, 'bp', True) + w, h = freqs(b, a) + assert np.all(-rp - 0.1 < dB(h[np.logical_and(wp[0] <= w, w <= wp[1])])) + assert np.all(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]) < -rs + 0.1) + + assert N == 11 + xp_assert_close(Wn, [1.673740595370124e+01, 5.974641487254268e+01], + rtol=1e-15) + + def test_fs_param(self): + wp = 150 + ws = 100 + rp = 3 + rs = 70 + fs = 1000 + N, Wn = cheb2ord(wp, ws, rp, rs, False, fs=fs) + b, a = cheby2(N, rs, Wn, 'hp', False, fs=fs) + w, h = freqz(b, a, fs=fs) + assert np.all(-rp - 0.1 < dB(h[wp <= w])) + assert np.all(dB(h[w <= ws]) < -rs + 0.1) + + assert N == 9 + xp_assert_close(Wn, 103.4874609145164, rtol=1e-15) + + def test_invalid_input(self): + with pytest.raises(ValueError) as exc_info: + cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 2) + assert "gpass should be smaller than gstop" in str(exc_info.value) + + with pytest.raises(ValueError) as exc_info: + cheb2ord([0.1, 0.6], [0.2, 0.5], -1, 2) + assert "gpass should be larger than 0.0" in str(exc_info.value) + + with pytest.raises(ValueError) as exc_info: + cheb2ord([0.1, 0.6], [0.2, 0.5], 1, -2) + assert "gstop should be larger than 0.0" in str(exc_info.value) + + def test_ellip_cheb2(self): + # The purpose of the test is to compare to some known output from past + # scipy versions. The values to compare to are generated with scipy + # 1.9.1 (there is nothing special about this particular version though) + n, wn = cheb2ord([0.1, 0.6], [0.2, 0.5], 3, 60) + assert n == 7 + + n1, w1 = cheb1ord([0.1, 0.6], [0.2, 0.5], 3, 60) + assert not (wn == w1).all() + + def test_fs_validation(self): + wp = 0.2 + ws = 0.3 + rp = 3 + rs = 60 + + with pytest.raises(ValueError, match="Sampling.*single scalar"): + cheb2ord(wp, ws, rp, rs, False, fs=np.array([10, 20])) + + +class TestEllipord: + + def test_lowpass(self): + wp = 0.2 + ws = 0.3 + rp = 3 + rs = 60 + N, Wn = ellipord(wp, ws, rp, rs, False) + b, a = ellip(N, rp, rs, Wn, 'lp', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[w <= wp])) + assert np.all(dB(h[ws <= w]) < -rs + 0.1) + + assert N == 5 + xp_assert_close(Wn, 0.2, rtol=1e-15) + + def test_lowpass_1000dB(self): + # failed when ellipkm1 wasn't used in ellipord and ellipap + wp = 0.2 + ws = 0.3 + rp = 3 + rs = 1000 + N, Wn = ellipord(wp, ws, rp, rs, False) + sos = ellip(N, rp, rs, Wn, 'lp', False, output='sos') + w, h = freqz_sos(sos) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[w <= wp])) + assert np.all(dB(h[ws <= w]) < -rs + 0.1) + + def test_highpass(self): + wp = 0.3 + ws = 0.2 + rp = 3 + rs = 70 + N, Wn = ellipord(wp, ws, rp, rs, False) + b, a = ellip(N, rp, rs, Wn, 'hp', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[wp <= w])) + assert np.all(dB(h[w <= ws]) < -rs + 0.1) + + assert N == 6 + xp_assert_close(Wn, 0.3, rtol=1e-15) + + def test_bandpass(self): + wp = [0.2, 0.5] + ws = [0.1, 0.6] + rp = 3 + rs = 80 + N, Wn = ellipord(wp, ws, rp, rs, False) + b, a = ellip(N, rp, rs, Wn, 'bp', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[np.logical_and(wp[0] <= w, w <= wp[1])])) + assert np.all(dB(h[np.logical_or(w <= ws[0], ws[1] <= w)]) < -rs + 0.1) + + assert N == 6 + xp_assert_close(Wn, [0.2, 0.5], rtol=1e-15) + + def test_bandstop(self): + wp = [0.1, 0.6] + ws = [0.2, 0.5] + rp = 3 + rs = 90 + N, Wn = ellipord(wp, ws, rp, rs, False) + b, a = ellip(N, rp, rs, Wn, 'bs', False) + w, h = freqz(b, a) + w /= np.pi + assert np.all(-rp - 0.1 < dB(h[np.logical_or(w <= wp[0], wp[1] <= w)])) + assert np.all(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]) < -rs + 0.1) + + assert N == 7 + xp_assert_close(Wn, [0.14758232794342988, 0.6], rtol=1e-5) + + def test_analog(self): + wp = [1000, 6000] + ws = [2000, 5000] + rp = 3 + rs = 90 + N, Wn = ellipord(wp, ws, rp, rs, True) + b, a = ellip(N, rp, rs, Wn, 'bs', True) + w, h = freqs(b, a) + assert np.all(-rp - 0.1 < dB(h[np.logical_or(w <= wp[0], wp[1] <= w)])) + assert np.all(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]) < -rs + 0.1) + + assert N == 8 + xp_assert_close(Wn, [1666.6666, 6000]) + + assert ellipord(1, 1.2, 1, 80, analog=True)[0] == 9 + + def test_fs_param(self): + wp = [400, 2400] + ws = [800, 2000] + rp = 3 + rs = 90 + fs = 8000 + N, Wn = ellipord(wp, ws, rp, rs, False, fs=fs) + b, a = ellip(N, rp, rs, Wn, 'bs', False, fs=fs) + w, h = freqz(b, a, fs=fs) + assert np.all(-rp - 0.1 < dB(h[np.logical_or(w <= wp[0], wp[1] <= w)])) + assert np.all(dB(h[np.logical_and(ws[0] <= w, w <= ws[1])]) < -rs + 0.1) + + assert N == 7 + xp_assert_close(Wn, [590.3293117737195, 2400], rtol=1e-5) + + def test_invalid_input(self): + with pytest.raises(ValueError) as exc_info: + ellipord(0.2, 0.5, 3, 2) + assert "gpass should be smaller than gstop" in str(exc_info.value) + + with pytest.raises(ValueError) as exc_info: + ellipord(0.2, 0.5, -1, 2) + assert "gpass should be larger than 0.0" in str(exc_info.value) + + with pytest.raises(ValueError) as exc_info: + ellipord(0.2, 0.5, 1, -2) + assert "gstop should be larger than 0.0" in str(exc_info.value) + + def test_ellip_butter(self): + # The purpose of the test is to compare to some known output from past + # scipy versions. The values to compare to are generated with scipy + # 1.9.1 (there is nothing special about this particular version though) + n, wn = ellipord([0.1, 0.6], [0.2, 0.5], 3, 60) + assert n == 5 + + def test_fs_validation(self): + wp = 0.2 + ws = 0.3 + rp = 3 + rs = 60 + + with pytest.raises(ValueError, match="Sampling.*single scalar"): + ellipord(wp, ws, rp, rs, False, fs=np.array([10, 20])) + + +class TestBessel: + + def test_degenerate(self): + for norm in ('delay', 'phase', 'mag'): + # 0-order filter is just a passthrough + b, a = bessel(0, 1, analog=True, norm=norm) + xp_assert_equal(b, np.asarray([1.0])) + xp_assert_equal(a, np.asarray([1.0])) + + # 1-order filter is same for all types + b, a = bessel(1, 1, analog=True, norm=norm) + xp_assert_close(b, np.asarray([1.0]), rtol=1e-15) + xp_assert_close(a, np.asarray([1.0, 1]), rtol=1e-15) + + z, p, k = bessel(1, 0.3, analog=True, output='zpk', norm=norm) + xp_assert_equal(z, np.asarray([])) + xp_assert_close(p, np.asarray([-0.3+0j]), rtol=1e-14) + xp_assert_close(k, 0.3, rtol=1e-14) + + def test_high_order(self): + # high even order, 'phase' + z, p, k = bessel(24, 100, analog=True, output='zpk') + z2 = [] + p2 = [ + -9.055312334014323e+01 + 4.844005815403969e+00j, + -8.983105162681878e+01 + 1.454056170018573e+01j, + -8.837357994162065e+01 + 2.426335240122282e+01j, + -8.615278316179575e+01 + 3.403202098404543e+01j, + -8.312326467067703e+01 + 4.386985940217900e+01j, + -7.921695461084202e+01 + 5.380628489700191e+01j, + -7.433392285433246e+01 + 6.388084216250878e+01j, + -6.832565803501586e+01 + 7.415032695116071e+01j, + -6.096221567378025e+01 + 8.470292433074425e+01j, + -5.185914574820616e+01 + 9.569048385258847e+01j, + -4.027853855197555e+01 + 1.074195196518679e+02j, + -2.433481337524861e+01 + 1.207298683731973e+02j, + ] + k2 = 9.999999999999989e+47 + xp_assert_equal(z, z2) + xp_assert_close(sorted(p, key=np.imag), + sorted(np.union1d(p2, np.conj(p2)), key=np.imag)) + xp_assert_close(k, k2, rtol=1e-14) + + # high odd order, 'phase' + z, p, k = bessel(23, 1000, analog=True, output='zpk') + z2 = [] + p2 = [ + -2.497697202208956e+02 + 1.202813187870698e+03j, + -4.126986617510172e+02 + 1.065328794475509e+03j, + -5.304922463809596e+02 + 9.439760364018479e+02j, + -9.027564978975828e+02 + 1.010534334242318e+02j, + -8.909283244406079e+02 + 2.023024699647598e+02j, + -8.709469394347836e+02 + 3.039581994804637e+02j, + -8.423805948131370e+02 + 4.062657947488952e+02j, + -8.045561642249877e+02 + 5.095305912401127e+02j, + -7.564660146766259e+02 + 6.141594859516342e+02j, + -6.965966033906477e+02 + 7.207341374730186e+02j, + -6.225903228776276e+02 + 8.301558302815096e+02j, + -9.066732476324988e+02] + k2 = 9.999999999999983e+68 + xp_assert_equal(z, z2) + xp_assert_close(sorted(p, key=np.imag), + sorted(np.union1d(p2, np.conj(p2)), key=np.imag)) + xp_assert_close(k, k2, rtol=1e-14) + + # high even order, 'delay' (Orchard 1965 "The Roots of the + # Maximally Flat-Delay Polynomials" Table 1) + z, p, k = bessel(31, 1, analog=True, output='zpk', norm='delay') + p2 = [-20.876706, + -20.826543 + 1.735732j, + -20.675502 + 3.473320j, + -20.421895 + 5.214702j, + -20.062802 + 6.961982j, + -19.593895 + 8.717546j, + -19.009148 + 10.484195j, + -18.300400 + 12.265351j, + -17.456663 + 14.065350j, + -16.463032 + 15.889910j, + -15.298849 + 17.746914j, + -13.934466 + 19.647827j, + -12.324914 + 21.610519j, + -10.395893 + 23.665701j, + - 8.005600 + 25.875019j, + - 4.792045 + 28.406037j, + ] + xp_assert_close(sorted(p, key=np.imag), + sorted(np.union1d(p2, np.conj(p2)), key=np.imag)) + + # high odd order, 'delay' + z, p, k = bessel(30, 1, analog=True, output='zpk', norm='delay') + p2 = [-20.201029 + 0.867750j, + -20.097257 + 2.604235j, + -19.888485 + 4.343721j, + -19.572188 + 6.088363j, + -19.144380 + 7.840570j, + -18.599342 + 9.603147j, + -17.929195 + 11.379494j, + -17.123228 + 13.173901j, + -16.166808 + 14.992008j, + -15.039580 + 16.841580j, + -13.712245 + 18.733902j, + -12.140295 + 20.686563j, + -10.250119 + 22.729808j, + - 7.901170 + 24.924391j, + - 4.734679 + 27.435615j, + ] + xp_assert_close(sorted(p, key=np.imag), + sorted(np.union1d(p2, np.conj(p2)), key=np.imag)) + + def test_refs(self): + # Compare to http://www.crbond.com/papers/bsf2.pdf + # "Delay Normalized Bessel Polynomial Coefficients" + bond_b = np.asarray([10395.0]) + bond_a = np.asarray([1.0, 21, 210, 1260, 4725, 10395, 10395]) + b, a = bessel(6, 1, norm='delay', analog=True) + xp_assert_close(b, bond_b) + xp_assert_close(a, bond_a) + + # "Delay Normalized Bessel Pole Locations" + bond_poles = { + 1: [-1.0000000000], + 2: [-1.5000000000 + 0.8660254038j], + 3: [-1.8389073227 + 1.7543809598j, -2.3221853546], + 4: [-2.1037893972 + 2.6574180419j, -2.8962106028 + 0.8672341289j], + 5: [-2.3246743032 + 3.5710229203j, -3.3519563992 + 1.7426614162j, + -3.6467385953], + 6: [-2.5159322478 + 4.4926729537j, -3.7357083563 + 2.6262723114j, + -4.2483593959 + 0.8675096732j], + 7: [-2.6856768789 + 5.4206941307j, -4.0701391636 + 3.5171740477j, + -4.7582905282 + 1.7392860611j, -4.9717868585], + 8: [-2.8389839489 + 6.3539112986j, -4.3682892172 + 4.4144425005j, + -5.2048407906 + 2.6161751526j, -5.5878860433 + 0.8676144454j], + 9: [-2.9792607982 + 7.2914636883j, -4.6384398872 + 5.3172716754j, + -5.6044218195 + 3.4981569179j, -6.1293679043 + 1.7378483835j, + -6.2970191817], + 10: [-3.1089162336 + 8.2326994591j, -4.8862195669 + 6.2249854825j, + -5.9675283286 + 4.3849471889j, -6.6152909655 + 2.6115679208j, + -6.9220449054 + 0.8676651955j] + } + + for N in range(1, 11): + p1 = np.sort(bond_poles[N]) + p2 = np.sort(np.concatenate(_cplxreal(besselap(N, 'delay')[1]))) + assert_array_almost_equal(p1, p2, decimal=10) + + # "Frequency Normalized Bessel Pole Locations" + bond_poles = { + 1: [-1.0000000000], + 2: [-1.1016013306 + 0.6360098248j], + 3: [-1.0474091610 + 0.9992644363j, -1.3226757999], + 4: [-0.9952087644 + 1.2571057395j, -1.3700678306 + 0.4102497175j], + 5: [-0.9576765486 + 1.4711243207j, -1.3808773259 + 0.7179095876j, + -1.5023162714], + 6: [-0.9306565229 + 1.6618632689j, -1.3818580976 + 0.9714718907j, + -1.5714904036 + 0.3208963742j], + 7: [-0.9098677806 + 1.8364513530j, -1.3789032168 + 1.1915667778j, + -1.6120387662 + 0.5892445069j, -1.6843681793], + 8: [-0.8928697188 + 1.9983258436j, -1.3738412176 + 1.3883565759j, + -1.6369394181 + 0.8227956251j, -1.7574084004 + 0.2728675751j], + 9: [-0.8783992762 + 2.1498005243j, -1.3675883098 + 1.5677337122j, + -1.6523964846 + 1.0313895670j, -1.8071705350 + 0.5123837306j, + -1.8566005012], + 10: [-0.8657569017 + 2.2926048310j, -1.3606922784 + 1.7335057427j, + -1.6618102414 + 1.2211002186j, -1.8421962445 + 0.7272575978j, + -1.9276196914 + 0.2416234710j] + } + + for N in range(1, 11): + p1 = np.sort(bond_poles[N]) + p2 = np.sort(np.concatenate(_cplxreal(besselap(N, 'mag')[1]))) + assert_array_almost_equal(p1, p2, decimal=10) + + # Compare to https://www.ranecommercial.com/legacy/note147.html + # "Table 1 - Bessel Crossovers of Second, Third, and Fourth-Order" + a = np.asarray([1, 1, 1/3]) + b2, a2 = bessel(2, 1, norm='delay', analog=True) + xp_assert_close(a[::-1], a2/b2) + + a = np.asarray([1, 1, 2/5, 1/15]) + b2, a2 = bessel(3, 1, norm='delay', analog=True) + xp_assert_close(a[::-1], a2/b2) + + a = np.asarray([1, 1, 9/21, 2/21, 1/105]) + b2, a2 = bessel(4, 1, norm='delay', analog=True) + xp_assert_close(a[::-1], a2/b2) + + a = np.asarray([1, np.sqrt(3), 1]) + b2, a2 = bessel(2, 1, norm='phase', analog=True) + xp_assert_close(a[::-1], a2/b2) + + # TODO: Why so inaccurate? Is reference flawed? + a = np.asarray([1, 2.481, 2.463, 1.018]) + b2, a2 = bessel(3, 1, norm='phase', analog=True) + assert_array_almost_equal(a[::-1], a2/b2, decimal=1) + + # TODO: Why so inaccurate? Is reference flawed? + a = np.asarray([1, 3.240, 4.5, 3.240, 1.050]) + b2, a2 = bessel(4, 1, norm='phase', analog=True) + assert_array_almost_equal(a[::-1], a2/b2, decimal=1) + + # Table of -3 dB factors: + N, scale = 2, np.asarray([1.272, 1.272], dtype=np.complex128) + scale2 = besselap(N, 'mag')[1] / besselap(N, 'phase')[1] + assert_array_almost_equal(scale2, scale, decimal=3) + + # TODO: Why so inaccurate? Is reference flawed? + N, scale = 3, np.asarray([1.413, 1.413, 1.413], dtype=np.complex128) + scale2 = besselap(N, 'mag')[1] / besselap(N, 'phase')[1] + assert_array_almost_equal(scale2, scale, decimal=2) + + # TODO: Why so inaccurate? Is reference flawed? + N, scale = 4, np.asarray([1.533]*4, dtype=np.complex128) + scale2 = besselap(N, 'mag')[1] / besselap(N, 'phase')[1] + assert_array_almost_equal(scale, scale2, decimal=1) + + def test_hardcoded(self): + # Compare to values from original hardcoded implementation + originals = { + 0: [], + 1: [-1], + 2: [-.8660254037844386467637229 + .4999999999999999999999996j], + 3: [-.9416000265332067855971980, + -.7456403858480766441810907 + .7113666249728352680992154j], + 4: [-.6572111716718829545787788 + .8301614350048733772399715j, + -.9047587967882449459642624 + .2709187330038746636700926j], + 5: [-.9264420773877602247196260, + -.8515536193688395541722677 + .4427174639443327209850002j, + -.5905759446119191779319432 + .9072067564574549539291747j], + 6: [-.9093906830472271808050953 + .1856964396793046769246397j, + -.7996541858328288520243325 + .5621717346937317988594118j, + -.5385526816693109683073792 + .9616876881954277199245657j], + 7: [-.9194871556490290014311619, + -.8800029341523374639772340 + .3216652762307739398381830j, + -.7527355434093214462291616 + .6504696305522550699212995j, + -.4966917256672316755024763 + 1.002508508454420401230220j], + 8: [-.9096831546652910216327629 + .1412437976671422927888150j, + -.8473250802359334320103023 + .4259017538272934994996429j, + -.7111381808485399250796172 + .7186517314108401705762571j, + -.4621740412532122027072175 + 1.034388681126901058116589j], + 9: [-.9154957797499037686769223, + -.8911217017079759323183848 + .2526580934582164192308115j, + -.8148021112269012975514135 + .5085815689631499483745341j, + -.6743622686854761980403401 + .7730546212691183706919682j, + -.4331415561553618854685942 + 1.060073670135929666774323j], + 10: [-.9091347320900502436826431 + .1139583137335511169927714j, + -.8688459641284764527921864 + .3430008233766309973110589j, + -.7837694413101441082655890 + .5759147538499947070009852j, + -.6417513866988316136190854 + .8175836167191017226233947j, + -.4083220732868861566219785 + 1.081274842819124562037210j], + 11: [-.9129067244518981934637318, + -.8963656705721166099815744 + .2080480375071031919692341j, + -.8453044014712962954184557 + .4178696917801248292797448j, + -.7546938934722303128102142 + .6319150050721846494520941j, + -.6126871554915194054182909 + .8547813893314764631518509j, + -.3868149510055090879155425 + 1.099117466763120928733632j], + 12: [-.9084478234140682638817772 + 95506365213450398415258360e-27j, + -.8802534342016826507901575 + .2871779503524226723615457j, + -.8217296939939077285792834 + .4810212115100676440620548j, + -.7276681615395159454547013 + .6792961178764694160048987j, + -.5866369321861477207528215 + .8863772751320727026622149j, + -.3679640085526312839425808 + 1.114373575641546257595657j], + 13: [-.9110914665984182781070663, + -.8991314665475196220910718 + .1768342956161043620980863j, + -.8625094198260548711573628 + .3547413731172988997754038j, + -.7987460692470972510394686 + .5350752120696801938272504j, + -.7026234675721275653944062 + .7199611890171304131266374j, + -.5631559842430199266325818 + .9135900338325109684927731j, + -.3512792323389821669401925 + 1.127591548317705678613239j], + 14: [-.9077932138396487614720659 + 82196399419401501888968130e-27j, + -.8869506674916445312089167 + .2470079178765333183201435j, + -.8441199160909851197897667 + .4131653825102692595237260j, + -.7766591387063623897344648 + .5819170677377608590492434j, + -.6794256425119233117869491 + .7552857305042033418417492j, + -.5418766775112297376541293 + .9373043683516919569183099j, + -.3363868224902037330610040 + 1.139172297839859991370924j], + 15: [-.9097482363849064167228581, + -.9006981694176978324932918 + .1537681197278439351298882j, + -.8731264620834984978337843 + .3082352470564267657715883j, + -.8256631452587146506294553 + .4642348752734325631275134j, + -.7556027168970728127850416 + .6229396358758267198938604j, + -.6579196593110998676999362 + .7862895503722515897065645j, + -.5224954069658330616875186 + .9581787261092526478889345j, + -.3229963059766444287113517 + 1.149416154583629539665297j], + 16: [-.9072099595087001356491337 + 72142113041117326028823950e-27j, + -.8911723070323647674780132 + .2167089659900576449410059j, + -.8584264231521330481755780 + .3621697271802065647661080j, + -.8074790293236003885306146 + .5092933751171800179676218j, + -.7356166304713115980927279 + .6591950877860393745845254j, + -.6379502514039066715773828 + .8137453537108761895522580j, + -.5047606444424766743309967 + .9767137477799090692947061j, + -.3108782755645387813283867 + 1.158552841199330479412225j], + 17: [-.9087141161336397432860029, + -.9016273850787285964692844 + .1360267995173024591237303j, + -.8801100704438627158492165 + .2725347156478803885651973j, + -.8433414495836129204455491 + .4100759282910021624185986j, + -.7897644147799708220288138 + .5493724405281088674296232j, + -.7166893842372349049842743 + .6914936286393609433305754j, + -.6193710717342144521602448 + .8382497252826992979368621j, + -.4884629337672704194973683 + .9932971956316781632345466j, + -.2998489459990082015466971 + 1.166761272925668786676672j], + 18: [-.9067004324162775554189031 + 64279241063930693839360680e-27j, + -.8939764278132455733032155 + .1930374640894758606940586j, + -.8681095503628830078317207 + .3224204925163257604931634j, + -.8281885016242836608829018 + .4529385697815916950149364j, + -.7726285030739558780127746 + .5852778162086640620016316j, + -.6987821445005273020051878 + .7204696509726630531663123j, + -.6020482668090644386627299 + .8602708961893664447167418j, + -.4734268069916151511140032 + 1.008234300314801077034158j, + -.2897592029880489845789953 + 1.174183010600059128532230j], + 19: [-.9078934217899404528985092, + -.9021937639390660668922536 + .1219568381872026517578164j, + -.8849290585034385274001112 + .2442590757549818229026280j, + -.8555768765618421591093993 + .3672925896399872304734923j, + -.8131725551578197705476160 + .4915365035562459055630005j, + -.7561260971541629355231897 + .6176483917970178919174173j, + -.6818424412912442033411634 + .7466272357947761283262338j, + -.5858613321217832644813602 + .8801817131014566284786759j, + -.4595043449730988600785456 + 1.021768776912671221830298j, + -.2804866851439370027628724 + 1.180931628453291873626003j], + 20: [-.9062570115576771146523497 + 57961780277849516990208850e-27j, + -.8959150941925768608568248 + .1740317175918705058595844j, + -.8749560316673332850673214 + .2905559296567908031706902j, + -.8427907479956670633544106 + .4078917326291934082132821j, + -.7984251191290606875799876 + .5264942388817132427317659j, + -.7402780309646768991232610 + .6469975237605228320268752j, + -.6658120544829934193890626 + .7703721701100763015154510j, + -.5707026806915714094398061 + .8982829066468255593407161j, + -.4465700698205149555701841 + 1.034097702560842962315411j, + -.2719299580251652601727704 + 1.187099379810885886139638j], + 21: [-.9072262653142957028884077, + -.9025428073192696303995083 + .1105252572789856480992275j, + -.8883808106664449854431605 + .2213069215084350419975358j, + -.8643915813643204553970169 + .3326258512522187083009453j, + -.8299435470674444100273463 + .4448177739407956609694059j, + -.7840287980408341576100581 + .5583186348022854707564856j, + -.7250839687106612822281339 + .6737426063024382240549898j, + -.6506315378609463397807996 + .7920349342629491368548074j, + -.5564766488918562465935297 + .9148198405846724121600860j, + -.4345168906815271799687308 + 1.045382255856986531461592j, + -.2640041595834031147954813 + 1.192762031948052470183960j], + 22: [-.9058702269930872551848625 + 52774908289999045189007100e-27j, + -.8972983138153530955952835 + .1584351912289865608659759j, + -.8799661455640176154025352 + .2644363039201535049656450j, + -.8534754036851687233084587 + .3710389319482319823405321j, + -.8171682088462720394344996 + .4785619492202780899653575j, + -.7700332930556816872932937 + .5874255426351153211965601j, + -.7105305456418785989070935 + .6982266265924524000098548j, + -.6362427683267827226840153 + .8118875040246347267248508j, + -.5430983056306302779658129 + .9299947824439872998916657j, + -.4232528745642628461715044 + 1.055755605227545931204656j, + -.2566376987939318038016012 + 1.197982433555213008346532j], + 23: [-.9066732476324988168207439, + -.9027564979912504609412993 + .1010534335314045013252480j, + -.8909283242471251458653994 + .2023024699381223418195228j, + -.8709469395587416239596874 + .3039581993950041588888925j, + -.8423805948021127057054288 + .4062657948237602726779246j, + -.8045561642053176205623187 + .5095305912227258268309528j, + -.7564660146829880581478138 + .6141594859476032127216463j, + -.6965966033912705387505040 + .7207341374753046970247055j, + -.6225903228771341778273152 + .8301558302812980678845563j, + -.5304922463810191698502226 + .9439760364018300083750242j, + -.4126986617510148836149955 + 1.065328794475513585531053j, + -.2497697202208956030229911 + 1.202813187870697831365338j], + 24: [-.9055312363372773709269407 + 48440066540478700874836350e-27j, + -.8983105104397872954053307 + .1454056133873610120105857j, + -.8837358034555706623131950 + .2426335234401383076544239j, + -.8615278304016353651120610 + .3403202112618624773397257j, + -.8312326466813240652679563 + .4386985933597305434577492j, + -.7921695462343492518845446 + .5380628490968016700338001j, + -.7433392285088529449175873 + .6388084216222567930378296j, + -.6832565803536521302816011 + .7415032695091650806797753j, + -.6096221567378335562589532 + .8470292433077202380020454j, + -.5185914574820317343536707 + .9569048385259054576937721j, + -.4027853855197518014786978 + 1.074195196518674765143729j, + -.2433481337524869675825448 + 1.207298683731972524975429j], + 25: [-.9062073871811708652496104, + -.9028833390228020537142561 + 93077131185102967450643820e-27j, + -.8928551459883548836774529 + .1863068969804300712287138j, + -.8759497989677857803656239 + .2798521321771408719327250j, + -.8518616886554019782346493 + .3738977875907595009446142j, + -.8201226043936880253962552 + .4686668574656966589020580j, + -.7800496278186497225905443 + .5644441210349710332887354j, + -.7306549271849967721596735 + .6616149647357748681460822j, + -.6704827128029559528610523 + .7607348858167839877987008j, + -.5972898661335557242320528 + .8626676330388028512598538j, + -.5073362861078468845461362 + .9689006305344868494672405j, + -.3934529878191079606023847 + 1.082433927173831581956863j, + -.2373280669322028974199184 + 1.211476658382565356579418j], + } + for N in originals: + p1 = sorted(np.union1d(originals[N], + np.conj(originals[N])), key=np.imag) + p2 = sorted(besselap(N)[1], key=np.imag) + xp_assert_close(p1, + p2, rtol=1e-14, check_dtype=False) + + def test_norm_phase(self): + # Test some orders and frequencies and see that they have the right + # phase at w0 + for N in (1, 2, 3, 4, 5, 51, 72): + for w0 in (1, 100): + b, a = bessel(N, w0, analog=True, norm='phase') + w = np.linspace(0, w0, 100) + w, h = freqs(b, a, w) + phase = np.unwrap(np.angle(h)) + xp_assert_close(phase[[0, -1]], (0, -N*pi/4), rtol=1e-1) + + def test_norm_mag(self): + # Test some orders and frequencies and see that they have the right + # mag at w0 + for N in (1, 2, 3, 4, 5, 51, 72): + for w0 in (1, 100): + b, a = bessel(N, w0, analog=True, norm='mag') + w = (0, w0) + w, h = freqs(b, a, w) + mag = abs(h) + xp_assert_close(mag, (1, 1/np.sqrt(2))) + + def test_norm_delay(self): + # Test some orders and frequencies and see that they have the right + # delay at DC + for N in (1, 2, 3, 4, 5, 51, 72): + for w0 in (1, 100): + b, a = bessel(N, w0, analog=True, norm='delay') + w = np.linspace(0, 10*w0, 1000) + w, h = freqs(b, a, w) + delay = -np.diff(np.unwrap(np.angle(h)))/np.diff(w) + xp_assert_close(delay[0], 1/w0, rtol=1e-4) + + def test_norm_factor(self): + mpmath_values = { + 1: 1.0, 2: 1.361654128716130520, 3: 1.755672368681210649, + 4: 2.113917674904215843, 5: 2.427410702152628137, + 6: 2.703395061202921876, 7: 2.951722147038722771, + 8: 3.179617237510651330, 9: 3.391693138911660101, + 10: 3.590980594569163482, 11: 3.779607416439620092, + 12: 3.959150821144285315, 13: 4.130825499383535980, + 14: 4.295593409533637564, 15: 4.454233021624377494, + 16: 4.607385465472647917, 17: 4.755586548961147727, + 18: 4.899289677284488007, 19: 5.038882681488207605, + 20: 5.174700441742707423, 21: 5.307034531360917274, + 22: 5.436140703250035999, 23: 5.562244783787878196, + 24: 5.685547371295963521, 25: 5.806227623775418541, + 50: 8.268963160013226298, 51: 8.352374541546012058, + } + for N in mpmath_values: + z, p, k = besselap(N, 'delay') + xp_assert_close(mpmath_values[N], _norm_factor(p, k), rtol=1e-13) + + def test_bessel_poly(self): + xp_assert_equal(_bessel_poly(5), [945, 945, 420, 105, 15, 1]) + xp_assert_equal(_bessel_poly(4, True), [1, 10, 45, 105, 105]) + + def test_bessel_zeros(self): + xp_assert_equal(_bessel_zeros(0), []) + + def test_invalid(self): + assert_raises(ValueError, besselap, 5, 'nonsense') + assert_raises(ValueError, besselap, -5) + assert_raises(ValueError, besselap, 3.2) + assert_raises(ValueError, _bessel_poly, -3) + assert_raises(ValueError, _bessel_poly, 3.3) + + @pytest.mark.fail_slow(10) + def test_fs_param(self): + for norm in ('phase', 'mag', 'delay'): + for fs in (900, 900.1, 1234.567): + for N in (0, 1, 2, 3, 10): + for fc in (100, 100.1, 432.12345): + for btype in ('lp', 'hp'): + ba1 = bessel(N, fc, btype, norm=norm, fs=fs) + ba2 = bessel(N, fc/(fs/2), btype, norm=norm) + for ba1_, ba2_ in zip(ba1, ba2): + xp_assert_close(ba1_, ba2_) + for fc in ((100, 200), (100.1, 200.2), (321.123, 432.123)): + for btype in ('bp', 'bs'): + ba1 = bessel(N, fc, btype, norm=norm, fs=fs) + for seq in (list, tuple, array): + fcnorm = seq([f/(fs/2) for f in fc]) + ba2 = bessel(N, fcnorm, btype, norm=norm) + for ba1_, ba2_ in zip(ba1, ba2): + xp_assert_close(ba1_, ba2_) + + +class TestButter: + + def test_degenerate(self): + # 0-order filter is just a passthrough + b, a = butter(0, 1, analog=True) + xp_assert_equal(b, np.asarray([1.0])) + xp_assert_equal(a, np.asarray([1.0])) + + # 1-order filter is same for all types + b, a = butter(1, 1, analog=True) + assert_array_almost_equal(b, [1]) + assert_array_almost_equal(a, [1, 1]) + + z, p, k = butter(1, 0.3, output='zpk') + xp_assert_equal(z, np.asarray([-1.0])) + xp_assert_close(p, [3.249196962329063e-01 + 0j], rtol=1e-14) + xp_assert_close(k, 3.375401518835469e-01, rtol=1e-14) + + def test_basic(self): + # analog s-plane + for N in range(25): + wn = 0.01 + z, p, k = butter(N, wn, 'low', analog=True, output='zpk') + assert_array_almost_equal([], z) + assert len(p) == N + # All poles should be at distance wn from origin + assert_array_almost_equal(abs(p), np.asarray(wn)) + assert all(np.real(p) <= 0) # No poles in right half of S-plane + assert_array_almost_equal(wn**N, k) + + # digital z-plane + for N in range(25): + wn = 0.01 + z, p, k = butter(N, wn, 'high', analog=False, output='zpk') + xp_assert_equal(np.ones(N), z) # All zeros exactly at DC + assert all(np.abs(p) <= 1) # No poles outside unit circle + + b1, a1 = butter(2, 1, analog=True) + assert_array_almost_equal(b1, [1]) + assert_array_almost_equal(a1, [1, np.sqrt(2), 1]) + + b2, a2 = butter(5, 1, analog=True) + assert_array_almost_equal(b2, [1]) + assert_array_almost_equal(a2, [1, 3.2361, 5.2361, + 5.2361, 3.2361, 1], decimal=4) + + b3, a3 = butter(10, 1, analog=True) + assert_array_almost_equal(b3, [1]) + assert_array_almost_equal(a3, [1, 6.3925, 20.4317, 42.8021, 64.8824, + 74.2334, 64.8824, 42.8021, 20.4317, + 6.3925, 1], decimal=4) + + b2, a2 = butter(19, 1.0441379169150726, analog=True) + assert_array_almost_equal(b2, [2.2720], decimal=4) + assert_array_almost_equal(a2, 1.0e+004 * np.array([ + 0.0001, 0.0013, 0.0080, 0.0335, 0.1045, 0.2570, + 0.5164, 0.8669, 1.2338, 1.5010, 1.5672, 1.4044, + 1.0759, 0.6986, 0.3791, 0.1681, 0.0588, 0.0153, + 0.0026, 0.0002]), decimal=0) + + b, a = butter(5, 0.4) + assert_array_almost_equal(b, [0.0219, 0.1097, 0.2194, + 0.2194, 0.1097, 0.0219], decimal=4) + assert_array_almost_equal(a, [1.0000, -0.9853, 0.9738, + -0.3864, 0.1112, -0.0113], decimal=4) + + def test_highpass(self): + # highpass, high even order + z, p, k = butter(28, 0.43, 'high', output='zpk') + z2 = np.ones(28) + p2 = [ + 2.068257195514592e-01 + 9.238294351481734e-01j, + 2.068257195514592e-01 - 9.238294351481734e-01j, + 1.874933103892023e-01 + 8.269455076775277e-01j, + 1.874933103892023e-01 - 8.269455076775277e-01j, + 1.717435567330153e-01 + 7.383078571194629e-01j, + 1.717435567330153e-01 - 7.383078571194629e-01j, + 1.588266870755982e-01 + 6.564623730651094e-01j, + 1.588266870755982e-01 - 6.564623730651094e-01j, + 1.481881532502603e-01 + 5.802343458081779e-01j, + 1.481881532502603e-01 - 5.802343458081779e-01j, + 1.394122576319697e-01 + 5.086609000582009e-01j, + 1.394122576319697e-01 - 5.086609000582009e-01j, + 1.321840881809715e-01 + 4.409411734716436e-01j, + 1.321840881809715e-01 - 4.409411734716436e-01j, + 1.262633413354405e-01 + 3.763990035551881e-01j, + 1.262633413354405e-01 - 3.763990035551881e-01j, + 1.214660449478046e-01 + 3.144545234797277e-01j, + 1.214660449478046e-01 - 3.144545234797277e-01j, + 1.104868766650320e-01 + 2.771505404367791e-02j, + 1.104868766650320e-01 - 2.771505404367791e-02j, + 1.111768629525075e-01 + 8.331369153155753e-02j, + 1.111768629525075e-01 - 8.331369153155753e-02j, + 1.125740630842972e-01 + 1.394219509611784e-01j, + 1.125740630842972e-01 - 1.394219509611784e-01j, + 1.147138487992747e-01 + 1.963932363793666e-01j, + 1.147138487992747e-01 - 1.963932363793666e-01j, + 1.176516491045901e-01 + 2.546021573417188e-01j, + 1.176516491045901e-01 - 2.546021573417188e-01j, + ] + k2 = 1.446671081817286e-06 + xp_assert_equal(z, z2) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag), rtol=1e-7) + xp_assert_close(k, k2, rtol=1e-10) + + # highpass, high odd order + z, p, k = butter(27, 0.56, 'high', output='zpk') + z2 = np.ones(27) + p2 = [ + -1.772572785680147e-01 + 9.276431102995948e-01j, + -1.772572785680147e-01 - 9.276431102995948e-01j, + -1.600766565322114e-01 + 8.264026279893268e-01j, + -1.600766565322114e-01 - 8.264026279893268e-01j, + -1.461948419016121e-01 + 7.341841939120078e-01j, + -1.461948419016121e-01 - 7.341841939120078e-01j, + -1.348975284762046e-01 + 6.493235066053785e-01j, + -1.348975284762046e-01 - 6.493235066053785e-01j, + -1.256628210712206e-01 + 5.704921366889227e-01j, + -1.256628210712206e-01 - 5.704921366889227e-01j, + -1.181038235962314e-01 + 4.966120551231630e-01j, + -1.181038235962314e-01 - 4.966120551231630e-01j, + -1.119304913239356e-01 + 4.267938916403775e-01j, + -1.119304913239356e-01 - 4.267938916403775e-01j, + -1.069237739782691e-01 + 3.602914879527338e-01j, + -1.069237739782691e-01 - 3.602914879527338e-01j, + -1.029178030691416e-01 + 2.964677964142126e-01j, + -1.029178030691416e-01 - 2.964677964142126e-01j, + -9.978747500816100e-02 + 2.347687643085738e-01j, + -9.978747500816100e-02 - 2.347687643085738e-01j, + -9.743974496324025e-02 + 1.747028739092479e-01j, + -9.743974496324025e-02 - 1.747028739092479e-01j, + -9.580754551625957e-02 + 1.158246860771989e-01j, + -9.580754551625957e-02 - 1.158246860771989e-01j, + -9.484562207782568e-02 + 5.772118357151691e-02j, + -9.484562207782568e-02 - 5.772118357151691e-02j, + -9.452783117928215e-02 + ] + k2 = 9.585686688851069e-09 + xp_assert_equal(z, z2) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag), rtol=1e-8) + xp_assert_close(k, k2) + + def test_bandpass(self): + z, p, k = butter(8, [0.25, 0.33], 'band', output='zpk') + z2 = [1, 1, 1, 1, 1, 1, 1, 1, + -1, -1, -1, -1, -1, -1, -1, -1] + p2 = [ + 4.979909925436156e-01 + 8.367609424799387e-01j, + 4.979909925436156e-01 - 8.367609424799387e-01j, + 4.913338722555539e-01 + 7.866774509868817e-01j, + 4.913338722555539e-01 - 7.866774509868817e-01j, + 5.035229361778706e-01 + 7.401147376726750e-01j, + 5.035229361778706e-01 - 7.401147376726750e-01j, + 5.307617160406101e-01 + 7.029184459442954e-01j, + 5.307617160406101e-01 - 7.029184459442954e-01j, + 5.680556159453138e-01 + 6.788228792952775e-01j, + 5.680556159453138e-01 - 6.788228792952775e-01j, + 6.100962560818854e-01 + 6.693849403338664e-01j, + 6.100962560818854e-01 - 6.693849403338664e-01j, + 6.904694312740631e-01 + 6.930501690145245e-01j, + 6.904694312740631e-01 - 6.930501690145245e-01j, + 6.521767004237027e-01 + 6.744414640183752e-01j, + 6.521767004237027e-01 - 6.744414640183752e-01j, + ] + k2 = 3.398854055800844e-08 + xp_assert_equal(z, z2, check_dtype=False) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag), rtol=1e-13) + xp_assert_close(k, k2, rtol=1e-13) + + # bandpass analog + z, p, k = butter(4, [90.5, 110.5], 'bp', analog=True, output='zpk') + z2 = np.zeros(4, dtype=z.dtype) + p2 = [ + -4.179137760733086e+00 + 1.095935899082837e+02j, + -4.179137760733086e+00 - 1.095935899082837e+02j, + -9.593598668443835e+00 + 1.034745398029734e+02j, + -9.593598668443835e+00 - 1.034745398029734e+02j, + -8.883991981781929e+00 + 9.582087115567160e+01j, + -8.883991981781929e+00 - 9.582087115567160e+01j, + -3.474530886568715e+00 + 9.111599925805801e+01j, + -3.474530886568715e+00 - 9.111599925805801e+01j, + ] + k2 = 1.600000000000001e+05 + xp_assert_equal(z, z2) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag)) + xp_assert_close(k, k2, rtol=1e-15) + + def test_bandstop(self): + z, p, k = butter(7, [0.45, 0.56], 'stop', output='zpk') + z2 = [-1.594474531383421e-02 + 9.998728744679880e-01j, + -1.594474531383421e-02 - 9.998728744679880e-01j, + -1.594474531383421e-02 + 9.998728744679880e-01j, + -1.594474531383421e-02 - 9.998728744679880e-01j, + -1.594474531383421e-02 + 9.998728744679880e-01j, + -1.594474531383421e-02 - 9.998728744679880e-01j, + -1.594474531383421e-02 + 9.998728744679880e-01j, + -1.594474531383421e-02 - 9.998728744679880e-01j, + -1.594474531383421e-02 + 9.998728744679880e-01j, + -1.594474531383421e-02 - 9.998728744679880e-01j, + -1.594474531383421e-02 + 9.998728744679880e-01j, + -1.594474531383421e-02 - 9.998728744679880e-01j, + -1.594474531383421e-02 + 9.998728744679880e-01j, + -1.594474531383421e-02 - 9.998728744679880e-01j] + p2 = [-1.766850742887729e-01 + 9.466951258673900e-01j, + -1.766850742887729e-01 - 9.466951258673900e-01j, + 1.467897662432886e-01 + 9.515917126462422e-01j, + 1.467897662432886e-01 - 9.515917126462422e-01j, + -1.370083529426906e-01 + 8.880376681273993e-01j, + -1.370083529426906e-01 - 8.880376681273993e-01j, + 1.086774544701390e-01 + 8.915240810704319e-01j, + 1.086774544701390e-01 - 8.915240810704319e-01j, + -7.982704457700891e-02 + 8.506056315273435e-01j, + -7.982704457700891e-02 - 8.506056315273435e-01j, + 5.238812787110331e-02 + 8.524011102699969e-01j, + 5.238812787110331e-02 - 8.524011102699969e-01j, + -1.357545000491310e-02 + 8.382287744986582e-01j, + -1.357545000491310e-02 - 8.382287744986582e-01j] + k2 = 4.577122512960063e-01 + xp_assert_close(sorted(z, key=np.imag), + sorted(z2, key=np.imag)) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag)) + xp_assert_close(k, k2, rtol=1e-14) + + def test_ba_output(self): + b, a = butter(4, [100, 300], 'bandpass', analog=True) + b2 = [1.6e+09, 0, 0, 0, 0] + a2 = [1.000000000000000e+00, 5.226251859505511e+02, + 2.565685424949238e+05, 6.794127417357160e+07, + 1.519411254969542e+10, 2.038238225207147e+12, + 2.309116882454312e+14, 1.411088002066486e+16, + 8.099999999999991e+17] + xp_assert_close(b, b2, rtol=1e-14) + xp_assert_close(a, a2, rtol=1e-14) + + def test_fs_param(self): + for fs in (900, 900.1, 1234.567): + for N in (0, 1, 2, 3, 10): + for fc in (100, 100.1, 432.12345): + for btype in ('lp', 'hp'): + ba1 = butter(N, fc, btype, fs=fs) + ba2 = butter(N, fc/(fs/2), btype) + for ba1_, ba2_ in zip(ba1, ba2): + xp_assert_close(ba1_, ba2_) + for fc in ((100, 200), (100.1, 200.2), (321.123, 432.123)): + for btype in ('bp', 'bs'): + ba1 = butter(N, fc, btype, fs=fs) + for seq in (list, tuple, array): + fcnorm = seq([f/(fs/2) for f in fc]) + ba2 = butter(N, fcnorm, btype) + for ba1_, ba2_ in zip(ba1, ba2): + xp_assert_close(ba1_, ba2_) + + +class TestCheby1: + + def test_degenerate(self): + # 0-order filter is just a passthrough + # Even-order filters have DC gain of -rp dB + b, a = cheby1(0, 10*np.log10(2), 1, analog=True) + assert_array_almost_equal(b, [1/np.sqrt(2)]) + xp_assert_equal(a, np.asarray([1.0])) + + # 1-order filter is same for all types + b, a = cheby1(1, 10*np.log10(2), 1, analog=True) + assert_array_almost_equal(b, [1]) + assert_array_almost_equal(a, [1, 1]) + + z, p, k = cheby1(1, 0.1, 0.3, output='zpk') + xp_assert_equal(z, np.asarray([-1.0])) + xp_assert_close(p, [-5.390126972799615e-01 + 0j], rtol=1e-14) + xp_assert_close(k, 7.695063486399808e-01, rtol=1e-14) + + def test_basic(self): + for N in range(25): + wn = 0.01 + z, p, k = cheby1(N, 1, wn, 'low', analog=True, output='zpk') + assert_array_almost_equal([], z) + assert len(p) == N + assert all(np.real(p) <= 0) # No poles in right half of S-plane + + for N in range(25): + wn = 0.01 + z, p, k = cheby1(N, 1, wn, 'high', analog=False, output='zpk') + xp_assert_equal(np.ones(N), z) # All zeros exactly at DC + assert all(np.abs(p) <= 1) # No poles outside unit circle + + # Same test as TestNormalize + b, a = cheby1(8, 0.5, 0.048) + assert_array_almost_equal(b, [ + 2.150733144728282e-11, 1.720586515782626e-10, + 6.022052805239190e-10, 1.204410561047838e-09, + 1.505513201309798e-09, 1.204410561047838e-09, + 6.022052805239190e-10, 1.720586515782626e-10, + 2.150733144728282e-11], decimal=14) + assert_array_almost_equal(a, [ + 1.000000000000000e+00, -7.782402035027959e+00, + 2.654354569747454e+01, -5.182182531666387e+01, + 6.334127355102684e+01, -4.963358186631157e+01, + 2.434862182949389e+01, -6.836925348604676e+00, + 8.412934944449140e-01], decimal=14) + + b, a = cheby1(4, 1, [0.4, 0.7], btype='band') + assert_array_almost_equal(b, [0.0084, 0, -0.0335, 0, 0.0502, 0, + -0.0335, 0, 0.0084], decimal=4) + assert_array_almost_equal(a, [1.0, 1.1191, 2.862, 2.2986, 3.4137, + 1.8653, 1.8982, 0.5676, 0.4103], + decimal=4) + + b2, a2 = cheby1(5, 3, 1, analog=True) + assert_array_almost_equal(b2, [0.0626], decimal=4) + assert_array_almost_equal(a2, [1, 0.5745, 1.4150, 0.5489, 0.4080, + 0.0626], decimal=4) + + b, a = cheby1(8, 0.5, 0.1) + assert_array_almost_equal(b, 1.0e-006 * np.array([ + 0.00703924326028, 0.05631394608227, 0.19709881128793, + 0.39419762257586, 0.49274702821983, 0.39419762257586, + 0.19709881128793, 0.05631394608227, 0.00703924326028]), + decimal=13) + assert_array_almost_equal(a, [ + 1.00000000000000, -7.44912258934158, 24.46749067762108, + -46.27560200466141, 55.11160187999928, -42.31640010161038, + 20.45543300484147, -5.69110270561444, 0.69770374759022], + decimal=13) + + b, a = cheby1(8, 0.5, 0.25) + assert_array_almost_equal(b, 1.0e-003 * np.array([ + 0.00895261138923, 0.07162089111382, 0.25067311889837, + 0.50134623779673, 0.62668279724591, 0.50134623779673, + 0.25067311889837, 0.07162089111382, 0.00895261138923]), + decimal=13) + assert_array_almost_equal(a, [1.00000000000000, -5.97529229188545, + 16.58122329202101, -27.71423273542923, + 30.39509758355313, -22.34729670426879, + 10.74509800434910, -3.08924633697497, + 0.40707685889802], decimal=13) + + def test_highpass(self): + # high even order + z, p, k = cheby1(24, 0.7, 0.2, 'high', output='zpk') + z2 = np.ones(24) + p2 = [-6.136558509657073e-01 + 2.700091504942893e-01j, + -6.136558509657073e-01 - 2.700091504942893e-01j, + -3.303348340927516e-01 + 6.659400861114254e-01j, + -3.303348340927516e-01 - 6.659400861114254e-01j, + 8.779713780557169e-03 + 8.223108447483040e-01j, + 8.779713780557169e-03 - 8.223108447483040e-01j, + 2.742361123006911e-01 + 8.356666951611864e-01j, + 2.742361123006911e-01 - 8.356666951611864e-01j, + 4.562984557158206e-01 + 7.954276912303594e-01j, + 4.562984557158206e-01 - 7.954276912303594e-01j, + 5.777335494123628e-01 + 7.435821817961783e-01j, + 5.777335494123628e-01 - 7.435821817961783e-01j, + 6.593260977749194e-01 + 6.955390907990932e-01j, + 6.593260977749194e-01 - 6.955390907990932e-01j, + 7.149590948466562e-01 + 6.559437858502012e-01j, + 7.149590948466562e-01 - 6.559437858502012e-01j, + 7.532432388188739e-01 + 6.256158042292060e-01j, + 7.532432388188739e-01 - 6.256158042292060e-01j, + 7.794365244268271e-01 + 6.042099234813333e-01j, + 7.794365244268271e-01 - 6.042099234813333e-01j, + 7.967253874772997e-01 + 5.911966597313203e-01j, + 7.967253874772997e-01 - 5.911966597313203e-01j, + 8.069756417293870e-01 + 5.862214589217275e-01j, + 8.069756417293870e-01 - 5.862214589217275e-01j] + k2 = 6.190427617192018e-04 + xp_assert_equal(z, z2) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag), rtol=1e-10) + xp_assert_close(k, k2, rtol=1e-10) + + # high odd order + z, p, k = cheby1(23, 0.8, 0.3, 'high', output='zpk') + z2 = np.ones(23) + p2 = [-7.676400532011010e-01, + -6.754621070166477e-01 + 3.970502605619561e-01j, + -6.754621070166477e-01 - 3.970502605619561e-01j, + -4.528880018446727e-01 + 6.844061483786332e-01j, + -4.528880018446727e-01 - 6.844061483786332e-01j, + -1.986009130216447e-01 + 8.382285942941594e-01j, + -1.986009130216447e-01 - 8.382285942941594e-01j, + 2.504673931532608e-02 + 8.958137635794080e-01j, + 2.504673931532608e-02 - 8.958137635794080e-01j, + 2.001089429976469e-01 + 9.010678290791480e-01j, + 2.001089429976469e-01 - 9.010678290791480e-01j, + 3.302410157191755e-01 + 8.835444665962544e-01j, + 3.302410157191755e-01 - 8.835444665962544e-01j, + 4.246662537333661e-01 + 8.594054226449009e-01j, + 4.246662537333661e-01 - 8.594054226449009e-01j, + 4.919620928120296e-01 + 8.366772762965786e-01j, + 4.919620928120296e-01 - 8.366772762965786e-01j, + 5.385746917494749e-01 + 8.191616180796720e-01j, + 5.385746917494749e-01 - 8.191616180796720e-01j, + 5.855636993537203e-01 + 8.060680937701062e-01j, + 5.855636993537203e-01 - 8.060680937701062e-01j, + 5.688812849391721e-01 + 8.086497795114683e-01j, + 5.688812849391721e-01 - 8.086497795114683e-01j] + k2 = 1.941697029206324e-05 + xp_assert_equal(z, z2) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag), rtol=1e-10) + xp_assert_close(k, k2, rtol=1e-10) + + z, p, k = cheby1(10, 1, 1000, 'high', analog=True, output='zpk') + z2 = np.zeros(10) + p2 = [-3.144743169501551e+03 + 3.511680029092744e+03j, + -3.144743169501551e+03 - 3.511680029092744e+03j, + -5.633065604514602e+02 + 2.023615191183945e+03j, + -5.633065604514602e+02 - 2.023615191183945e+03j, + -1.946412183352025e+02 + 1.372309454274755e+03j, + -1.946412183352025e+02 - 1.372309454274755e+03j, + -7.987162953085479e+01 + 1.105207708045358e+03j, + -7.987162953085479e+01 - 1.105207708045358e+03j, + -2.250315039031946e+01 + 1.001723931471477e+03j, + -2.250315039031946e+01 - 1.001723931471477e+03j] + k2 = 8.912509381337453e-01 + xp_assert_equal(z, z2) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag), rtol=1e-13) + xp_assert_close(k, k2, rtol=1e-15) + + def test_bandpass(self): + z, p, k = cheby1(8, 1, [0.3, 0.4], 'bp', output='zpk') + z2 = [1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1] + p2 = [3.077784854851463e-01 + 9.453307017592942e-01j, + 3.077784854851463e-01 - 9.453307017592942e-01j, + 3.280567400654425e-01 + 9.272377218689016e-01j, + 3.280567400654425e-01 - 9.272377218689016e-01j, + 3.677912763284301e-01 + 9.038008865279966e-01j, + 3.677912763284301e-01 - 9.038008865279966e-01j, + 4.194425632520948e-01 + 8.769407159656157e-01j, + 4.194425632520948e-01 - 8.769407159656157e-01j, + 4.740921994669189e-01 + 8.496508528630974e-01j, + 4.740921994669189e-01 - 8.496508528630974e-01j, + 5.234866481897429e-01 + 8.259608422808477e-01j, + 5.234866481897429e-01 - 8.259608422808477e-01j, + 5.844717632289875e-01 + 8.052901363500210e-01j, + 5.844717632289875e-01 - 8.052901363500210e-01j, + 5.615189063336070e-01 + 8.100667803850766e-01j, + 5.615189063336070e-01 - 8.100667803850766e-01j] + k2 = 5.007028718074307e-09 + xp_assert_equal(z, z2, check_dtype=False) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag), rtol=1e-13) + xp_assert_close(k, k2, rtol=1e-13) + + def test_bandstop(self): + z, p, k = cheby1(7, 1, [0.5, 0.6], 'stop', output='zpk') + z2 = [-1.583844403245361e-01 + 9.873775210440450e-01j, + -1.583844403245361e-01 - 9.873775210440450e-01j, + -1.583844403245361e-01 + 9.873775210440450e-01j, + -1.583844403245361e-01 - 9.873775210440450e-01j, + -1.583844403245361e-01 + 9.873775210440450e-01j, + -1.583844403245361e-01 - 9.873775210440450e-01j, + -1.583844403245361e-01 + 9.873775210440450e-01j, + -1.583844403245361e-01 - 9.873775210440450e-01j, + -1.583844403245361e-01 + 9.873775210440450e-01j, + -1.583844403245361e-01 - 9.873775210440450e-01j, + -1.583844403245361e-01 + 9.873775210440450e-01j, + -1.583844403245361e-01 - 9.873775210440450e-01j, + -1.583844403245361e-01 + 9.873775210440450e-01j, + -1.583844403245361e-01 - 9.873775210440450e-01j] + p2 = [-8.942974551472813e-02 + 3.482480481185926e-01j, + -8.942974551472813e-02 - 3.482480481185926e-01j, + 1.293775154041798e-01 + 8.753499858081858e-01j, + 1.293775154041798e-01 - 8.753499858081858e-01j, + 3.399741945062013e-02 + 9.690316022705607e-01j, + 3.399741945062013e-02 - 9.690316022705607e-01j, + 4.167225522796539e-04 + 9.927338161087488e-01j, + 4.167225522796539e-04 - 9.927338161087488e-01j, + -3.912966549550960e-01 + 8.046122859255742e-01j, + -3.912966549550960e-01 - 8.046122859255742e-01j, + -3.307805547127368e-01 + 9.133455018206508e-01j, + -3.307805547127368e-01 - 9.133455018206508e-01j, + -3.072658345097743e-01 + 9.443589759799366e-01j, + -3.072658345097743e-01 - 9.443589759799366e-01j] + k2 = 3.619438310405028e-01 + xp_assert_close(sorted(z, key=np.imag), + sorted(z2, key=np.imag), rtol=1e-13) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag), rtol=1e-13) + xp_assert_close(k, k2, rtol=0, atol=5e-16) + + def test_ba_output(self): + # with transfer function conversion, without digital conversion + b, a = cheby1(5, 0.9, [210, 310], 'stop', analog=True) + b2 = [1.000000000000006e+00, 0, + 3.255000000000020e+05, 0, + 4.238010000000026e+10, 0, + 2.758944510000017e+15, 0, + 8.980364380050052e+19, 0, + 1.169243442282517e+24 + ] + a2 = [1.000000000000000e+00, 4.630555945694342e+02, + 4.039266454794788e+05, 1.338060988610237e+08, + 5.844333551294591e+10, 1.357346371637638e+13, + 3.804661141892782e+15, 5.670715850340080e+17, + 1.114411200988328e+20, 8.316815934908471e+21, + 1.169243442282517e+24 + ] + xp_assert_close(b, b2, rtol=1e-14) + xp_assert_close(a, a2, rtol=1e-14) + + def test_fs_param(self): + for fs in (900, 900.1, 1234.567): + for N in (0, 1, 2, 3, 10): + for fc in (100, 100.1, 432.12345): + for btype in ('lp', 'hp'): + ba1 = cheby1(N, 1, fc, btype, fs=fs) + ba2 = cheby1(N, 1, fc/(fs/2), btype) + for ba1_, ba2_ in zip(ba1, ba2): + xp_assert_close(ba1_, ba2_) + for fc in ((100, 200), (100.1, 200.2), (321.123, 432.123)): + for btype in ('bp', 'bs'): + ba1 = cheby1(N, 1, fc, btype, fs=fs) + for seq in (list, tuple, array): + fcnorm = seq([f/(fs/2) for f in fc]) + ba2 = cheby1(N, 1, fcnorm, btype) + for ba1_, ba2_ in zip(ba1, ba2): + xp_assert_close(ba1_, ba2_) + +class TestCheby2: + + def test_degenerate(self): + # 0-order filter is just a passthrough + # Stopband ripple factor doesn't matter + b, a = cheby2(0, 123.456, 1, analog=True) + xp_assert_equal(b, np.asarray([1.0])) + xp_assert_equal(a, np.asarray([1.0])) + + # 1-order filter is same for all types + b, a = cheby2(1, 10*np.log10(2), 1, analog=True) + assert_array_almost_equal(b, [1]) + assert_array_almost_equal(a, [1, 1]) + + z, p, k = cheby2(1, 50, 0.3, output='zpk') + xp_assert_equal(z, np.asarray([-1], dtype=np.complex128)) + xp_assert_close(p, [9.967826460175649e-01 + 0j], rtol=1e-14) + xp_assert_close(k, 1.608676991217512e-03, rtol=1e-14) + + def test_basic(self): + for N in range(25): + wn = 0.01 + z, p, k = cheby2(N, 40, wn, 'low', analog=True, output='zpk') + assert len(p) == N + assert all(np.real(p) <= 0) # No poles in right half of S-plane + + for N in range(25): + wn = 0.01 + z, p, k = cheby2(N, 40, wn, 'high', analog=False, output='zpk') + assert all(np.abs(p) <= 1) # No poles outside unit circle + + B, A = cheby2(18, 100, 0.5) + assert_array_almost_equal(B, [ + 0.00167583914216, 0.01249479541868, 0.05282702120282, + 0.15939804265706, 0.37690207631117, 0.73227013789108, + 1.20191856962356, 1.69522872823393, 2.07598674519837, + 2.21972389625291, 2.07598674519838, 1.69522872823395, + 1.20191856962359, 0.73227013789110, 0.37690207631118, + 0.15939804265707, 0.05282702120282, 0.01249479541868, + 0.00167583914216], decimal=13) + assert_array_almost_equal(A, [ + 1.00000000000000, -0.27631970006174, 3.19751214254060, + -0.15685969461355, 4.13926117356269, 0.60689917820044, + 2.95082770636540, 0.89016501910416, 1.32135245849798, + 0.51502467236824, 0.38906643866660, 0.15367372690642, + 0.07255803834919, 0.02422454070134, 0.00756108751837, + 0.00179848550988, 0.00033713574499, 0.00004258794833, + 0.00000281030149], decimal=13) + + def test_highpass(self): + # high even order + z, p, k = cheby2(26, 60, 0.3, 'high', output='zpk') + z2 = [9.981088955489852e-01 + 6.147058341984388e-02j, + 9.981088955489852e-01 - 6.147058341984388e-02j, + 9.832702870387426e-01 + 1.821525257215483e-01j, + 9.832702870387426e-01 - 1.821525257215483e-01j, + 9.550760158089112e-01 + 2.963609353922882e-01j, + 9.550760158089112e-01 - 2.963609353922882e-01j, + 9.162054748821922e-01 + 4.007087817803773e-01j, + 9.162054748821922e-01 - 4.007087817803773e-01j, + 8.700619897368064e-01 + 4.929423232136168e-01j, + 8.700619897368064e-01 - 4.929423232136168e-01j, + 5.889791753434985e-01 + 8.081482110427953e-01j, + 5.889791753434985e-01 - 8.081482110427953e-01j, + 5.984900456570295e-01 + 8.011302423760501e-01j, + 5.984900456570295e-01 - 8.011302423760501e-01j, + 6.172880888914629e-01 + 7.867371958365343e-01j, + 6.172880888914629e-01 - 7.867371958365343e-01j, + 6.448899971038180e-01 + 7.642754030030161e-01j, + 6.448899971038180e-01 - 7.642754030030161e-01j, + 6.804845629637927e-01 + 7.327624168637228e-01j, + 6.804845629637927e-01 - 7.327624168637228e-01j, + 8.202619107108660e-01 + 5.719881098737678e-01j, + 8.202619107108660e-01 - 5.719881098737678e-01j, + 7.228410452536148e-01 + 6.910143437705678e-01j, + 7.228410452536148e-01 - 6.910143437705678e-01j, + 7.702121399578629e-01 + 6.377877856007792e-01j, + 7.702121399578629e-01 - 6.377877856007792e-01j] + p2 = [7.365546198286450e-01 + 4.842085129329526e-02j, + 7.365546198286450e-01 - 4.842085129329526e-02j, + 7.292038510962885e-01 + 1.442201672097581e-01j, + 7.292038510962885e-01 - 1.442201672097581e-01j, + 7.151293788040354e-01 + 2.369925800458584e-01j, + 7.151293788040354e-01 - 2.369925800458584e-01j, + 6.955051820787286e-01 + 3.250341363856910e-01j, + 6.955051820787286e-01 - 3.250341363856910e-01j, + 6.719122956045220e-01 + 4.070475750638047e-01j, + 6.719122956045220e-01 - 4.070475750638047e-01j, + 6.461722130611300e-01 + 4.821965916689270e-01j, + 6.461722130611300e-01 - 4.821965916689270e-01j, + 5.528045062872224e-01 + 8.162920513838372e-01j, + 5.528045062872224e-01 - 8.162920513838372e-01j, + 5.464847782492791e-01 + 7.869899955967304e-01j, + 5.464847782492791e-01 - 7.869899955967304e-01j, + 5.488033111260949e-01 + 7.520442354055579e-01j, + 5.488033111260949e-01 - 7.520442354055579e-01j, + 6.201874719022955e-01 + 5.500894392527353e-01j, + 6.201874719022955e-01 - 5.500894392527353e-01j, + 5.586478152536709e-01 + 7.112676877332921e-01j, + 5.586478152536709e-01 - 7.112676877332921e-01j, + 5.958145844148228e-01 + 6.107074340842115e-01j, + 5.958145844148228e-01 - 6.107074340842115e-01j, + 5.747812938519067e-01 + 6.643001536914696e-01j, + 5.747812938519067e-01 - 6.643001536914696e-01j] + k2 = 9.932997786497189e-02 + xp_assert_close(sorted(z, key=np.angle), + sorted(z2, key=np.angle), rtol=1e-13) + xp_assert_close(sorted(p, key=np.angle), + sorted(p2, key=np.angle), rtol=1e-12) + xp_assert_close(k, k2, rtol=1e-11) + + # high odd order + z, p, k = cheby2(25, 80, 0.5, 'high', output='zpk') + z2 = [9.690690376586687e-01 + 2.467897896011971e-01j, + 9.690690376586687e-01 - 2.467897896011971e-01j, + 9.999999999999492e-01, + 8.835111277191199e-01 + 4.684101698261429e-01j, + 8.835111277191199e-01 - 4.684101698261429e-01j, + 7.613142857900539e-01 + 6.483830335935022e-01j, + 7.613142857900539e-01 - 6.483830335935022e-01j, + 6.232625173626231e-01 + 7.820126817709752e-01j, + 6.232625173626231e-01 - 7.820126817709752e-01j, + 4.864456563413621e-01 + 8.737108351316745e-01j, + 4.864456563413621e-01 - 8.737108351316745e-01j, + 3.618368136816749e-01 + 9.322414495530347e-01j, + 3.618368136816749e-01 - 9.322414495530347e-01j, + 2.549486883466794e-01 + 9.669545833752675e-01j, + 2.549486883466794e-01 - 9.669545833752675e-01j, + 1.676175432109457e-01 + 9.858520980390212e-01j, + 1.676175432109457e-01 - 9.858520980390212e-01j, + 1.975218468277521e-03 + 9.999980492540941e-01j, + 1.975218468277521e-03 - 9.999980492540941e-01j, + 1.786959496651858e-02 + 9.998403260399917e-01j, + 1.786959496651858e-02 - 9.998403260399917e-01j, + 9.967933660557139e-02 + 9.950196127985684e-01j, + 9.967933660557139e-02 - 9.950196127985684e-01j, + 5.013970951219547e-02 + 9.987422137518890e-01j, + 5.013970951219547e-02 - 9.987422137518890e-01j] + p2 = [4.218866331906864e-01, + 4.120110200127552e-01 + 1.361290593621978e-01j, + 4.120110200127552e-01 - 1.361290593621978e-01j, + 3.835890113632530e-01 + 2.664910809911026e-01j, + 3.835890113632530e-01 - 2.664910809911026e-01j, + 3.399195570456499e-01 + 3.863983538639875e-01j, + 3.399195570456499e-01 - 3.863983538639875e-01j, + 2.855977834508353e-01 + 4.929444399540688e-01j, + 2.855977834508353e-01 - 4.929444399540688e-01j, + 2.255765441339322e-01 + 5.851631870205766e-01j, + 2.255765441339322e-01 - 5.851631870205766e-01j, + 1.644087535815792e-01 + 6.637356937277153e-01j, + 1.644087535815792e-01 - 6.637356937277153e-01j, + -7.293633845273095e-02 + 9.739218252516307e-01j, + -7.293633845273095e-02 - 9.739218252516307e-01j, + 1.058259206358626e-01 + 7.304739464862978e-01j, + 1.058259206358626e-01 - 7.304739464862978e-01j, + -5.703971947785402e-02 + 9.291057542169088e-01j, + -5.703971947785402e-02 - 9.291057542169088e-01j, + 5.263875132656864e-02 + 7.877974334424453e-01j, + 5.263875132656864e-02 - 7.877974334424453e-01j, + -3.007943405982616e-02 + 8.846331716180016e-01j, + -3.007943405982616e-02 - 8.846331716180016e-01j, + 6.857277464483946e-03 + 8.383275456264492e-01j, + 6.857277464483946e-03 - 8.383275456264492e-01j] + k2 = 6.507068761705037e-03 + xp_assert_close(sorted(z, key=np.angle), + sorted(z2, key=np.angle), rtol=1e-13) + xp_assert_close(sorted(p, key=np.angle), + sorted(p2, key=np.angle), rtol=1e-12) + xp_assert_close(k, k2, rtol=1e-11) + + def test_bandpass(self): + z, p, k = cheby2(9, 40, [0.07, 0.2], 'pass', output='zpk') + z2 = [-9.999999999999999e-01, + 3.676588029658514e-01 + 9.299607543341383e-01j, + 3.676588029658514e-01 - 9.299607543341383e-01j, + 7.009689684982283e-01 + 7.131917730894889e-01j, + 7.009689684982283e-01 - 7.131917730894889e-01j, + 7.815697973765858e-01 + 6.238178033919218e-01j, + 7.815697973765858e-01 - 6.238178033919218e-01j, + 8.063793628819866e-01 + 5.913986160941200e-01j, + 8.063793628819866e-01 - 5.913986160941200e-01j, + 1.000000000000001e+00, + 9.944493019920448e-01 + 1.052168511576739e-01j, + 9.944493019920448e-01 - 1.052168511576739e-01j, + 9.854674703367308e-01 + 1.698642543566085e-01j, + 9.854674703367308e-01 - 1.698642543566085e-01j, + 9.762751735919308e-01 + 2.165335665157851e-01j, + 9.762751735919308e-01 - 2.165335665157851e-01j, + 9.792277171575134e-01 + 2.027636011479496e-01j, + 9.792277171575134e-01 - 2.027636011479496e-01j] + p2 = [8.143803410489621e-01 + 5.411056063397541e-01j, + 8.143803410489621e-01 - 5.411056063397541e-01j, + 7.650769827887418e-01 + 5.195412242095543e-01j, + 7.650769827887418e-01 - 5.195412242095543e-01j, + 6.096241204063443e-01 + 3.568440484659796e-01j, + 6.096241204063443e-01 - 3.568440484659796e-01j, + 6.918192770246239e-01 + 4.770463577106911e-01j, + 6.918192770246239e-01 - 4.770463577106911e-01j, + 6.986241085779207e-01 + 1.146512226180060e-01j, + 6.986241085779207e-01 - 1.146512226180060e-01j, + 8.654645923909734e-01 + 1.604208797063147e-01j, + 8.654645923909734e-01 - 1.604208797063147e-01j, + 9.164831670444591e-01 + 1.969181049384918e-01j, + 9.164831670444591e-01 - 1.969181049384918e-01j, + 9.630425777594550e-01 + 2.317513360702271e-01j, + 9.630425777594550e-01 - 2.317513360702271e-01j, + 9.438104703725529e-01 + 2.193509900269860e-01j, + 9.438104703725529e-01 - 2.193509900269860e-01j] + k2 = 9.345352824659604e-03 + xp_assert_close(sorted(z, key=np.angle), + sorted(z2, key=np.angle), rtol=1e-13) + xp_assert_close(sorted(p, key=np.angle), + sorted(p2, key=np.angle), rtol=1e-13) + xp_assert_close(k, k2, rtol=1e-11) + + def test_bandstop(self): + z, p, k = cheby2(6, 55, [0.1, 0.9], 'stop', output='zpk') + z2 = [6.230544895101009e-01 + 7.821784343111114e-01j, + 6.230544895101009e-01 - 7.821784343111114e-01j, + 9.086608545660115e-01 + 4.175349702471991e-01j, + 9.086608545660115e-01 - 4.175349702471991e-01j, + 9.478129721465802e-01 + 3.188268649763867e-01j, + 9.478129721465802e-01 - 3.188268649763867e-01j, + -6.230544895100982e-01 + 7.821784343111109e-01j, + -6.230544895100982e-01 - 7.821784343111109e-01j, + -9.086608545660116e-01 + 4.175349702472088e-01j, + -9.086608545660116e-01 - 4.175349702472088e-01j, + -9.478129721465784e-01 + 3.188268649763897e-01j, + -9.478129721465784e-01 - 3.188268649763897e-01j] + p2 = [-9.464094036167638e-01 + 1.720048695084344e-01j, + -9.464094036167638e-01 - 1.720048695084344e-01j, + -8.715844103386737e-01 + 1.370665039509297e-01j, + -8.715844103386737e-01 - 1.370665039509297e-01j, + -8.078751204586425e-01 + 5.729329866682983e-02j, + -8.078751204586425e-01 - 5.729329866682983e-02j, + 9.464094036167665e-01 + 1.720048695084332e-01j, + 9.464094036167665e-01 - 1.720048695084332e-01j, + 8.078751204586447e-01 + 5.729329866683007e-02j, + 8.078751204586447e-01 - 5.729329866683007e-02j, + 8.715844103386721e-01 + 1.370665039509331e-01j, + 8.715844103386721e-01 - 1.370665039509331e-01j] + k2 = 2.917823332763358e-03 + xp_assert_close(sorted(z, key=np.angle), + sorted(z2, key=np.angle), rtol=1e-13) + xp_assert_close(sorted(p, key=np.angle), + sorted(p2, key=np.angle), rtol=1e-13) + xp_assert_close(k, k2, rtol=1e-11) + + def test_ba_output(self): + # with transfer function conversion, without digital conversion + b, a = cheby2(5, 20, [2010, 2100], 'stop', True) + b2 = [1.000000000000000e+00, 0, # Matlab: 6.683253076978249e-12, + 2.111512500000000e+07, 0, # Matlab: 1.134325604589552e-04, + 1.782966433781250e+14, 0, # Matlab: 7.216787944356781e+02, + 7.525901316990656e+20, 0, # Matlab: 2.039829265789886e+09, + 1.587960565565748e+27, 0, # Matlab: 2.161236218626134e+15, + 1.339913493808585e+33] + a2 = [1.000000000000000e+00, 1.849550755473371e+02, + 2.113222918998538e+07, 3.125114149732283e+09, + 1.785133457155609e+14, 1.979158697776348e+16, + 7.535048322653831e+20, 5.567966191263037e+22, + 1.589246884221346e+27, 5.871210648525566e+28, + 1.339913493808590e+33] + xp_assert_close(b, b2, rtol=1e-14) + xp_assert_close(a, a2, rtol=1e-14) + + def test_fs_param(self): + for fs in (900, 900.1, 1234.567): + for N in (0, 1, 2, 3, 10): + for fc in (100, 100.1, 432.12345): + for btype in ('lp', 'hp'): + ba1 = cheby2(N, 20, fc, btype, fs=fs) + ba2 = cheby2(N, 20, fc/(fs/2), btype) + for ba1_, ba2_ in zip(ba1, ba2): + xp_assert_close(ba1_, ba2_) + for fc in ((100, 200), (100.1, 200.2), (321.123, 432.123)): + for btype in ('bp', 'bs'): + ba1 = cheby2(N, 20, fc, btype, fs=fs) + for seq in (list, tuple, array): + fcnorm = seq([f/(fs/2) for f in fc]) + ba2 = cheby2(N, 20, fcnorm, btype) + for ba1_, ba2_ in zip(ba1, ba2): + xp_assert_close(ba1_, ba2_) + +class TestEllip: + + def test_degenerate(self): + # 0-order filter is just a passthrough + # Even-order filters have DC gain of -rp dB + # Stopband ripple factor doesn't matter + b, a = ellip(0, 10*np.log10(2), 123.456, 1, analog=True) + assert_array_almost_equal(b, [1/np.sqrt(2)]) + xp_assert_equal(a, np.asarray([1.0])) + + # 1-order filter is same for all types + b, a = ellip(1, 10*np.log10(2), 1, 1, analog=True) + assert_array_almost_equal(b, [1]) + assert_array_almost_equal(a, [1, 1]) + + z, p, k = ellip(1, 1, 55, 0.3, output='zpk') + xp_assert_close(z, [-9.999999999999998e-01], rtol=1e-14) + xp_assert_close(p, [-6.660721153525525e-04], rtol=1e-10) + xp_assert_close(k, 5.003330360576763e-01, rtol=1e-14) + + def test_basic(self): + for N in range(25): + wn = 0.01 + z, p, k = ellip(N, 1, 40, wn, 'low', analog=True, output='zpk') + assert len(p) == N + assert all(np.real(p) <= 0) # No poles in right half of S-plane + + for N in range(25): + wn = 0.01 + z, p, k = ellip(N, 1, 40, wn, 'high', analog=False, output='zpk') + assert all(np.abs(p) <= 1) # No poles outside unit circle + + b3, a3 = ellip(5, 3, 26, 1, analog=True) + assert_array_almost_equal(b3, [0.1420, 0, 0.3764, 0, + 0.2409], decimal=4) + assert_array_almost_equal(a3, [1, 0.5686, 1.8061, 0.8017, 0.8012, + 0.2409], decimal=4) + + b, a = ellip(3, 1, 60, [0.4, 0.7], 'stop') + assert_array_almost_equal(b, [0.3310, 0.3469, 1.1042, 0.7044, 1.1042, + 0.3469, 0.3310], decimal=4) + assert_array_almost_equal(a, [1.0000, 0.6973, 1.1441, 0.5878, 0.7323, + 0.1131, -0.0060], decimal=4) + + def test_highpass(self): + # high even order + z, p, k = ellip(24, 1, 80, 0.3, 'high', output='zpk') + z2 = [9.761875332501075e-01 + 2.169283290099910e-01j, + 9.761875332501075e-01 - 2.169283290099910e-01j, + 8.413503353963494e-01 + 5.404901600661900e-01j, + 8.413503353963494e-01 - 5.404901600661900e-01j, + 7.160082576305009e-01 + 6.980918098681732e-01j, + 7.160082576305009e-01 - 6.980918098681732e-01j, + 6.456533638965329e-01 + 7.636306264739803e-01j, + 6.456533638965329e-01 - 7.636306264739803e-01j, + 6.127321820971366e-01 + 7.902906256703928e-01j, + 6.127321820971366e-01 - 7.902906256703928e-01j, + 5.983607817490196e-01 + 8.012267936512676e-01j, + 5.983607817490196e-01 - 8.012267936512676e-01j, + 5.922577552594799e-01 + 8.057485658286990e-01j, + 5.922577552594799e-01 - 8.057485658286990e-01j, + 5.896952092563588e-01 + 8.076258788449631e-01j, + 5.896952092563588e-01 - 8.076258788449631e-01j, + 5.886248765538837e-01 + 8.084063054565607e-01j, + 5.886248765538837e-01 - 8.084063054565607e-01j, + 5.881802711123132e-01 + 8.087298490066037e-01j, + 5.881802711123132e-01 - 8.087298490066037e-01j, + 5.879995719101164e-01 + 8.088612386766461e-01j, + 5.879995719101164e-01 - 8.088612386766461e-01j, + 5.879354086709576e-01 + 8.089078780868164e-01j, + 5.879354086709576e-01 - 8.089078780868164e-01j] + p2 = [-3.184805259081650e-01 + 4.206951906775851e-01j, + -3.184805259081650e-01 - 4.206951906775851e-01j, + 1.417279173459985e-01 + 7.903955262836452e-01j, + 1.417279173459985e-01 - 7.903955262836452e-01j, + 4.042881216964651e-01 + 8.309042239116594e-01j, + 4.042881216964651e-01 - 8.309042239116594e-01j, + 5.128964442789670e-01 + 8.229563236799665e-01j, + 5.128964442789670e-01 - 8.229563236799665e-01j, + 5.569614712822724e-01 + 8.155957702908510e-01j, + 5.569614712822724e-01 - 8.155957702908510e-01j, + 5.750478870161392e-01 + 8.118633973883931e-01j, + 5.750478870161392e-01 - 8.118633973883931e-01j, + 5.825314018170804e-01 + 8.101960910679270e-01j, + 5.825314018170804e-01 - 8.101960910679270e-01j, + 5.856397379751872e-01 + 8.094825218722543e-01j, + 5.856397379751872e-01 - 8.094825218722543e-01j, + 5.869326035251949e-01 + 8.091827531557583e-01j, + 5.869326035251949e-01 - 8.091827531557583e-01j, + 5.874697218855733e-01 + 8.090593298213502e-01j, + 5.874697218855733e-01 - 8.090593298213502e-01j, + 5.876904783532237e-01 + 8.090127161018823e-01j, + 5.876904783532237e-01 - 8.090127161018823e-01j, + 5.877753105317594e-01 + 8.090050577978136e-01j, + 5.877753105317594e-01 - 8.090050577978136e-01j] + k2 = 4.918081266957108e-02 + xp_assert_close(sorted(z, key=np.angle), + sorted(z2, key=np.angle), rtol=1e-4) + xp_assert_close(sorted(p, key=np.angle), + sorted(p2, key=np.angle), rtol=1e-4) + xp_assert_close(k, k2, rtol=1e-3) + + # high odd order + z, p, k = ellip(23, 1, 70, 0.5, 'high', output='zpk') + z2 = [9.999999999998661e-01, + 6.603717261750994e-01 + 7.509388678638675e-01j, + 6.603717261750994e-01 - 7.509388678638675e-01j, + 2.788635267510325e-01 + 9.603307416968041e-01j, + 2.788635267510325e-01 - 9.603307416968041e-01j, + 1.070215532544218e-01 + 9.942567008268131e-01j, + 1.070215532544218e-01 - 9.942567008268131e-01j, + 4.049427369978163e-02 + 9.991797705105507e-01j, + 4.049427369978163e-02 - 9.991797705105507e-01j, + 1.531059368627931e-02 + 9.998827859909265e-01j, + 1.531059368627931e-02 - 9.998827859909265e-01j, + 5.808061438534933e-03 + 9.999831330689181e-01j, + 5.808061438534933e-03 - 9.999831330689181e-01j, + 2.224277847754599e-03 + 9.999975262909676e-01j, + 2.224277847754599e-03 - 9.999975262909676e-01j, + 8.731857107534554e-04 + 9.999996187732845e-01j, + 8.731857107534554e-04 - 9.999996187732845e-01j, + 3.649057346914968e-04 + 9.999999334218996e-01j, + 3.649057346914968e-04 - 9.999999334218996e-01j, + 1.765538109802615e-04 + 9.999999844143768e-01j, + 1.765538109802615e-04 - 9.999999844143768e-01j, + 1.143655290967426e-04 + 9.999999934602630e-01j, + 1.143655290967426e-04 - 9.999999934602630e-01j] + p2 = [-6.322017026545028e-01, + -4.648423756662754e-01 + 5.852407464440732e-01j, + -4.648423756662754e-01 - 5.852407464440732e-01j, + -2.249233374627773e-01 + 8.577853017985717e-01j, + -2.249233374627773e-01 - 8.577853017985717e-01j, + -9.234137570557621e-02 + 9.506548198678851e-01j, + -9.234137570557621e-02 - 9.506548198678851e-01j, + -3.585663561241373e-02 + 9.821494736043981e-01j, + -3.585663561241373e-02 - 9.821494736043981e-01j, + -1.363917242312723e-02 + 9.933844128330656e-01j, + -1.363917242312723e-02 - 9.933844128330656e-01j, + -5.131505238923029e-03 + 9.975221173308673e-01j, + -5.131505238923029e-03 - 9.975221173308673e-01j, + -1.904937999259502e-03 + 9.990680819857982e-01j, + -1.904937999259502e-03 - 9.990680819857982e-01j, + -6.859439885466834e-04 + 9.996492201426826e-01j, + -6.859439885466834e-04 - 9.996492201426826e-01j, + -2.269936267937089e-04 + 9.998686250679161e-01j, + -2.269936267937089e-04 - 9.998686250679161e-01j, + -5.687071588789117e-05 + 9.999527573294513e-01j, + -5.687071588789117e-05 - 9.999527573294513e-01j, + -6.948417068525226e-07 + 9.999882737700173e-01j, + -6.948417068525226e-07 - 9.999882737700173e-01j] + k2 = 1.220910020289434e-02 + xp_assert_close(sorted(z, key=np.angle), + sorted(z2, key=np.angle), rtol=1e-4) + xp_assert_close(sorted(p, key=np.angle), + sorted(p2, key=np.angle), rtol=1e-4) + xp_assert_close(k, k2, rtol=1e-3) + + def test_bandpass(self): + z, p, k = ellip(7, 1, 40, [0.07, 0.2], 'pass', output='zpk') + z2 = [-9.999999999999991e-01, + 6.856610961780020e-01 + 7.279209168501619e-01j, + 6.856610961780020e-01 - 7.279209168501619e-01j, + 7.850346167691289e-01 + 6.194518952058737e-01j, + 7.850346167691289e-01 - 6.194518952058737e-01j, + 7.999038743173071e-01 + 6.001281461922627e-01j, + 7.999038743173071e-01 - 6.001281461922627e-01j, + 9.999999999999999e-01, + 9.862938983554124e-01 + 1.649980183725925e-01j, + 9.862938983554124e-01 - 1.649980183725925e-01j, + 9.788558330548762e-01 + 2.045513580850601e-01j, + 9.788558330548762e-01 - 2.045513580850601e-01j, + 9.771155231720003e-01 + 2.127093189691258e-01j, + 9.771155231720003e-01 - 2.127093189691258e-01j] + p2 = [8.063992755498643e-01 + 5.858071374778874e-01j, + 8.063992755498643e-01 - 5.858071374778874e-01j, + 8.050395347071724e-01 + 5.639097428109795e-01j, + 8.050395347071724e-01 - 5.639097428109795e-01j, + 8.113124936559144e-01 + 4.855241143973142e-01j, + 8.113124936559144e-01 - 4.855241143973142e-01j, + 8.665595314082394e-01 + 3.334049560919331e-01j, + 8.665595314082394e-01 - 3.334049560919331e-01j, + 9.412369011968871e-01 + 2.457616651325908e-01j, + 9.412369011968871e-01 - 2.457616651325908e-01j, + 9.679465190411238e-01 + 2.228772501848216e-01j, + 9.679465190411238e-01 - 2.228772501848216e-01j, + 9.747235066273385e-01 + 2.178937926146544e-01j, + 9.747235066273385e-01 - 2.178937926146544e-01j] + k2 = 8.354782670263239e-03 + xp_assert_close(sorted(z, key=np.angle), + sorted(z2, key=np.angle), rtol=1e-4) + xp_assert_close(sorted(p, key=np.angle), + sorted(p2, key=np.angle), rtol=1e-4) + xp_assert_close(k, k2, rtol=1e-3) + + z, p, k = ellip(5, 1, 75, [90.5, 110.5], 'pass', True, 'zpk') + z2 = [-5.583607317695175e-14 + 1.433755965989225e+02j, + -5.583607317695175e-14 - 1.433755965989225e+02j, + 5.740106416459296e-14 + 1.261678754570291e+02j, + 5.740106416459296e-14 - 1.261678754570291e+02j, + -2.199676239638652e-14 + 6.974861996895196e+01j, + -2.199676239638652e-14 - 6.974861996895196e+01j, + -3.372595657044283e-14 + 7.926145989044531e+01j, + -3.372595657044283e-14 - 7.926145989044531e+01j, + 0] + p2 = [-8.814960004852743e-01 + 1.104124501436066e+02j, + -8.814960004852743e-01 - 1.104124501436066e+02j, + -2.477372459140184e+00 + 1.065638954516534e+02j, + -2.477372459140184e+00 - 1.065638954516534e+02j, + -3.072156842945799e+00 + 9.995404870405324e+01j, + -3.072156842945799e+00 - 9.995404870405324e+01j, + -2.180456023925693e+00 + 9.379206865455268e+01j, + -2.180456023925693e+00 - 9.379206865455268e+01j, + -7.230484977485752e-01 + 9.056598800801140e+01j, + -7.230484977485752e-01 - 9.056598800801140e+01j] + k2 = 3.774571622827070e-02 + xp_assert_close(sorted(z, key=np.imag), + sorted(z2, key=np.imag), rtol=1e-4) + xp_assert_close(sorted(p, key=np.imag), + sorted(p2, key=np.imag), rtol=1e-6) + xp_assert_close(k, k2, rtol=1e-3) + + def test_bandstop(self): + z, p, k = ellip(8, 1, 65, [0.2, 0.4], 'stop', output='zpk') + z2 = [3.528578094286510e-01 + 9.356769561794296e-01j, + 3.528578094286510e-01 - 9.356769561794296e-01j, + 3.769716042264783e-01 + 9.262248159096587e-01j, + 3.769716042264783e-01 - 9.262248159096587e-01j, + 4.406101783111199e-01 + 8.976985411420985e-01j, + 4.406101783111199e-01 - 8.976985411420985e-01j, + 5.539386470258847e-01 + 8.325574907062760e-01j, + 5.539386470258847e-01 - 8.325574907062760e-01j, + 6.748464963023645e-01 + 7.379581332490555e-01j, + 6.748464963023645e-01 - 7.379581332490555e-01j, + 7.489887970285254e-01 + 6.625826604475596e-01j, + 7.489887970285254e-01 - 6.625826604475596e-01j, + 7.913118471618432e-01 + 6.114127579150699e-01j, + 7.913118471618432e-01 - 6.114127579150699e-01j, + 7.806804740916381e-01 + 6.249303940216475e-01j, + 7.806804740916381e-01 - 6.249303940216475e-01j] + + p2 = [-1.025299146693730e-01 + 5.662682444754943e-01j, + -1.025299146693730e-01 - 5.662682444754943e-01j, + 1.698463595163031e-01 + 8.926678667070186e-01j, + 1.698463595163031e-01 - 8.926678667070186e-01j, + 2.750532687820631e-01 + 9.351020170094005e-01j, + 2.750532687820631e-01 - 9.351020170094005e-01j, + 3.070095178909486e-01 + 9.457373499553291e-01j, + 3.070095178909486e-01 - 9.457373499553291e-01j, + 7.695332312152288e-01 + 2.792567212705257e-01j, + 7.695332312152288e-01 - 2.792567212705257e-01j, + 8.083818999225620e-01 + 4.990723496863960e-01j, + 8.083818999225620e-01 - 4.990723496863960e-01j, + 8.066158014414928e-01 + 5.649811440393374e-01j, + 8.066158014414928e-01 - 5.649811440393374e-01j, + 8.062787978834571e-01 + 5.855780880424964e-01j, + 8.062787978834571e-01 - 5.855780880424964e-01j] + k2 = 2.068622545291259e-01 + xp_assert_close(sorted(z, key=np.angle), + sorted(z2, key=np.angle), rtol=1e-6) + xp_assert_close(sorted(p, key=np.angle), + sorted(p2, key=np.angle), rtol=1e-5) + xp_assert_close(k, k2, rtol=1e-5) + + def test_ba_output(self): + # with transfer function conversion, without digital conversion + b, a = ellip(5, 1, 40, [201, 240], 'stop', True) + b2 = [ + 1.000000000000000e+00, 0, # Matlab: 1.743506051190569e-13, + 2.426561778314366e+05, 0, # Matlab: 3.459426536825722e-08, + 2.348218683400168e+10, 0, # Matlab: 2.559179747299313e-03, + 1.132780692872241e+15, 0, # Matlab: 8.363229375535731e+01, + 2.724038554089566e+19, 0, # Matlab: 1.018700994113120e+06, + 2.612380874940186e+23 + ] + a2 = [ + 1.000000000000000e+00, 1.337266601804649e+02, + 2.486725353510667e+05, 2.628059713728125e+07, + 2.436169536928770e+10, 1.913554568577315e+12, + 1.175208184614438e+15, 6.115751452473410e+16, + 2.791577695211466e+19, 7.241811142725384e+20, + 2.612380874940182e+23 + ] + xp_assert_close(b, b2, rtol=1e-6) + xp_assert_close(a, a2, rtol=1e-4) + + def test_fs_param(self): + for fs in (900, 900.1, 1234.567): + for N in (0, 1, 2, 3, 10): + for fc in (100, 100.1, 432.12345): + for btype in ('lp', 'hp'): + ba1 = ellip(N, 1, 20, fc, btype, fs=fs) + ba2 = ellip(N, 1, 20, fc/(fs/2), btype) + for ba1_, ba2_ in zip(ba1, ba2): + xp_assert_close(ba1_, ba2_) + for fc in ((100, 200), (100.1, 200.2), (321.123, 432.123)): + for btype in ('bp', 'bs'): + ba1 = ellip(N, 1, 20, fc, btype, fs=fs) + for seq in (list, tuple, array): + fcnorm = seq([f/(fs/2) for f in fc]) + ba2 = ellip(N, 1, 20, fcnorm, btype) + for ba1_, ba2_ in zip(ba1, ba2): + xp_assert_close(ba1_, ba2_) + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + iirnotch(0.06, 30, fs=np.array([10, 20])) + + with pytest.raises(ValueError, match="Sampling.*be none"): + iirnotch(0.06, 30, fs=None) + + +def test_sos_consistency(): + # Consistency checks of output='sos' for the specialized IIR filter + # design functions. + design_funcs = [(bessel, (0.1,)), + (butter, (0.1,)), + (cheby1, (45.0, 0.1)), + (cheby2, (0.087, 0.1)), + (ellip, (0.087, 45, 0.1))] + for func, args in design_funcs: + name = func.__name__ + + b, a = func(2, *args, output='ba') + sos = func(2, *args, output='sos') + xp_assert_close(sos, [np.hstack((b, a))], err_msg=f"{name}(2,...)") + + zpk = func(3, *args, output='zpk') + sos = func(3, *args, output='sos') + xp_assert_close(sos, zpk2sos(*zpk), err_msg=f"{name}(3,...)") + + zpk = func(4, *args, output='zpk') + sos = func(4, *args, output='sos') + xp_assert_close(sos, zpk2sos(*zpk), err_msg=f"{name}(4,...)") + + +class TestIIRNotch: + + def test_ba_output(self): + # Compare coefficients with Matlab ones + # for the equivalent input: + b, a = iirnotch(0.06, 30) + b2 = [ + 9.9686824e-01, -1.9584219e+00, + 9.9686824e-01 + ] + a2 = [ + 1.0000000e+00, -1.9584219e+00, + 9.9373647e-01 + ] + + xp_assert_close(b, b2, rtol=1e-8) + xp_assert_close(a, a2, rtol=1e-8) + + def test_frequency_response(self): + # Get filter coefficients + b, a = iirnotch(0.3, 30) + + # Get frequency response + w, h = freqz(b, a, 1000) + + # Pick 5 point + p = [200, # w0 = 0.200 + 295, # w0 = 0.295 + 300, # w0 = 0.300 + 305, # w0 = 0.305 + 400] # w0 = 0.400 + + # Get frequency response correspondent to each of those points + hp = h[p] + + # Check if the frequency response fulfill the specifications: + # hp[0] and hp[4] correspond to frequencies distant from + # w0 = 0.3 and should be close to 1 + xp_assert_close(abs(hp[0]), np.asarray(1.), rtol=1e-2, check_0d=False) + xp_assert_close(abs(hp[4]), np.asarray(1.), rtol=1e-2, check_0d=False) + + # hp[1] and hp[3] correspond to frequencies approximately + # on the edges of the passband and should be close to -3dB + xp_assert_close(abs(hp[1]), 1/np.sqrt(2), rtol=1e-2) + xp_assert_close(abs(hp[3]), 1/np.sqrt(2), rtol=1e-2) + + # hp[2] correspond to the frequency that should be removed + # the frequency response should be very close to 0 + xp_assert_close(abs(hp[2]), np.asarray(0.0), atol=1e-10, check_0d=False) + + def test_errors(self): + # Exception should be raised if w0 > 1 or w0 <0 + assert_raises(ValueError, iirnotch, w0=2, Q=30) + assert_raises(ValueError, iirnotch, w0=-1, Q=30) + + # Exception should be raised if any of the parameters + # are not float (or cannot be converted to one) + assert_raises(ValueError, iirnotch, w0="blabla", Q=30) + assert_raises(TypeError, iirnotch, w0=-1, Q=[1, 2, 3]) + + def test_fs_param(self): + # Get filter coefficients + b, a = iirnotch(1500, 30, fs=10000) + + # Get frequency response + w, h = freqz(b, a, 1000, fs=10000) + + # Pick 5 point + p = [200, # w0 = 1000 + 295, # w0 = 1475 + 300, # w0 = 1500 + 305, # w0 = 1525 + 400] # w0 = 2000 + + # Get frequency response correspondent to each of those points + hp = h[p] + + # Check if the frequency response fulfill the specifications: + # hp[0] and hp[4] correspond to frequencies distant from + # w0 = 1500 and should be close to 1 + xp_assert_close(abs(hp[0]), np.ones_like(abs(hp[0])), rtol=1e-2, + check_0d=False) + xp_assert_close(abs(hp[4]), np.ones_like(abs(hp[4])), rtol=1e-2, + check_0d=False) + + # hp[1] and hp[3] correspond to frequencies approximately + # on the edges of the passband and should be close to -3dB + xp_assert_close(abs(hp[1]), 1/np.sqrt(2), rtol=1e-2) + xp_assert_close(abs(hp[3]), 1/np.sqrt(2), rtol=1e-2) + + # hp[2] correspond to the frequency that should be removed + # the frequency response should be very close to 0 + xp_assert_close(abs(hp[2]), np.asarray(0.0), atol=1e-10, check_0d=False) + + +class TestIIRPeak: + + def test_ba_output(self): + # Compare coefficients with Matlab ones + # for the equivalent input: + b, a = iirpeak(0.06, 30) + b2 = [ + 3.131764229e-03, 0, + -3.131764229e-03 + ] + a2 = [ + 1.0000000e+00, -1.958421917e+00, + 9.9373647e-01 + ] + xp_assert_close(b, b2, rtol=1e-8) + xp_assert_close(a, a2, rtol=1e-8) + + def test_frequency_response(self): + # Get filter coefficients + b, a = iirpeak(0.3, 30) + + # Get frequency response + w, h = freqz(b, a, 1000) + + # Pick 5 point + p = [30, # w0 = 0.030 + 295, # w0 = 0.295 + 300, # w0 = 0.300 + 305, # w0 = 0.305 + 800] # w0 = 0.800 + + # Get frequency response correspondent to each of those points + hp = h[p] + + # Check if the frequency response fulfill the specifications: + # hp[0] and hp[4] correspond to frequencies distant from + # w0 = 0.3 and should be close to 0 + xp_assert_close(abs(hp[0]), + np.zeros_like(abs(hp[0])), atol=1e-2, check_0d=False) + xp_assert_close(abs(hp[4]), + np.zeros_like(abs(hp[4])), atol=1e-2, check_0d=False) + + # hp[1] and hp[3] correspond to frequencies approximately + # on the edges of the passband and should be close to 10**(-3/20) + xp_assert_close(abs(hp[1]), 1/np.sqrt(2), rtol=1e-2) + xp_assert_close(abs(hp[3]), 1/np.sqrt(2), rtol=1e-2) + + # hp[2] correspond to the frequency that should be retained and + # the frequency response should be very close to 1 + xp_assert_close(abs(hp[2]), np.asarray(1.0), rtol=1e-10, check_0d=False) + + def test_errors(self): + # Exception should be raised if w0 > 1 or w0 <0 + assert_raises(ValueError, iirpeak, w0=2, Q=30) + assert_raises(ValueError, iirpeak, w0=-1, Q=30) + + # Exception should be raised if any of the parameters + # are not float (or cannot be converted to one) + assert_raises(ValueError, iirpeak, w0="blabla", Q=30) + assert_raises(TypeError, iirpeak, w0=-1, Q=[1, 2, 3]) + + def test_fs_param(self): + # Get filter coefficients + b, a = iirpeak(1200, 30, fs=8000) + + # Get frequency response + w, h = freqz(b, a, 1000, fs=8000) + + # Pick 5 point + p = [30, # w0 = 120 + 295, # w0 = 1180 + 300, # w0 = 1200 + 305, # w0 = 1220 + 800] # w0 = 3200 + + # Get frequency response correspondent to each of those points + hp = h[p] + + # Check if the frequency response fulfill the specifications: + # hp[0] and hp[4] correspond to frequencies distant from + # w0 = 1200 and should be close to 0 + xp_assert_close(abs(hp[0]), + np.zeros_like(abs(hp[0])), atol=1e-2, check_0d=False) + xp_assert_close(abs(hp[4]), + np.zeros_like(abs(hp[4])), atol=1e-2, check_0d=False) + + # hp[1] and hp[3] correspond to frequencies approximately + # on the edges of the passband and should be close to 10**(-3/20) + xp_assert_close(abs(hp[1]), 1/np.sqrt(2), rtol=1e-2) + xp_assert_close(abs(hp[3]), 1/np.sqrt(2), rtol=1e-2) + + # hp[2] correspond to the frequency that should be retained and + # the frequency response should be very close to 1 + xp_assert_close(abs(hp[2]), + np.ones_like(abs(hp[2])), rtol=1e-10, check_0d=False) + + +class TestIIRComb: + # Test erroneous input cases + def test_invalid_input(self): + # w0 is <= 0 or >= fs / 2 + fs = 1000 + for args in [(-fs, 30), (0, 35), (fs / 2, 40), (fs, 35)]: + with pytest.raises(ValueError, match='w0 must be between '): + iircomb(*args, fs=fs) + + # fs is not divisible by w0 + for args in [(120, 30), (157, 35)]: + with pytest.raises(ValueError, match='fs must be divisible '): + iircomb(*args, fs=fs) + + # https://github.com/scipy/scipy/issues/14043#issuecomment-1107349140 + # Previously, fs=44100, w0=49.999 was rejected, but fs=2, + # w0=49.999/int(44100/2) was accepted. Now it is rejected, too. + with pytest.raises(ValueError, match='fs must be divisible '): + iircomb(w0=49.999/int(44100/2), Q=30) + + with pytest.raises(ValueError, match='fs must be divisible '): + iircomb(w0=49.999, Q=30, fs=44100) + + # Filter type is not notch or peak + for args in [(0.2, 30, 'natch'), (0.5, 35, 'comb')]: + with pytest.raises(ValueError, match='ftype must be '): + iircomb(*args) + + # Verify that the filter's frequency response contains a + # notch at the cutoff frequency + @pytest.mark.parametrize('ftype', ('notch', 'peak')) + def test_frequency_response(self, ftype): + # Create a notching or peaking comb filter at 1000 Hz + b, a = iircomb(1000, 30, ftype=ftype, fs=10000) + + # Compute the frequency response + freqs, response = freqz(b, a, 1000, fs=10000) + + # Find the notch using argrelextrema + comb_points = argrelextrema(abs(response), np.less)[0] + + # Verify that the first notch sits at 1000 Hz + comb1 = comb_points[0] + xp_assert_close(freqs[comb1], np.asarray(1000.), check_0d=False) + + # Verify pass_zero parameter + @pytest.mark.parametrize('ftype,pass_zero,peak,notch', + [('peak', True, 123.45, 61.725), + ('peak', False, 61.725, 123.45), + ('peak', None, 61.725, 123.45), + ('notch', None, 61.725, 123.45), + ('notch', True, 123.45, 61.725), + ('notch', False, 61.725, 123.45)]) + def test_pass_zero(self, ftype, pass_zero, peak, notch): + # Create a notching or peaking comb filter + b, a = iircomb(123.45, 30, ftype=ftype, fs=1234.5, pass_zero=pass_zero) + + # Compute the frequency response + freqs, response = freqz(b, a, [peak, notch], fs=1234.5) + + # Verify that expected notches are notches and peaks are peaks + assert abs(response[0]) > 0.99 + assert abs(response[1]) < 1e-10 + + # All built-in IIR filters are real, so should have perfectly + # symmetrical poles and zeros. Then ba representation (using + # numpy.poly) will be purely real instead of having negligible + # imaginary parts. + def test_iir_symmetry(self): + b, a = iircomb(400, 30, fs=24000) + z, p, k = tf2zpk(b, a) + xp_assert_equal(sorted(z), sorted(z.conj())) + xp_assert_equal(sorted(p), sorted(p.conj())) + xp_assert_equal(k, np.real(k)) + + assert issubclass(b.dtype.type, np.floating) + assert issubclass(a.dtype.type, np.floating) + + # Verify filter coefficients with MATLAB's iircomb function + def test_ba_output(self): + b_notch, a_notch = iircomb(60, 35, ftype='notch', fs=600) + b_notch2 = [0.957020174408697, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, -0.957020174408697] + a_notch2 = [1.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, -0.914040348817395] + xp_assert_close(b_notch, b_notch2) + xp_assert_close(a_notch, a_notch2) + + b_peak, a_peak = iircomb(60, 35, ftype='peak', fs=600) + b_peak2 = [0.0429798255913026, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, -0.0429798255913026] + a_peak2 = [1.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.914040348817395] + xp_assert_close(b_peak, b_peak2) + xp_assert_close(a_peak, a_peak2) + + # Verify that https://github.com/scipy/scipy/issues/14043 is fixed + def test_nearest_divisor(self): + # Create a notching comb filter + b, a = iircomb(50/int(44100/2), 50.0, ftype='notch') + + # Compute the frequency response at an upper harmonic of 50 + freqs, response = freqz(b, a, [22000], fs=44100) + + # Before bug fix, this would produce N = 881, so that 22 kHz was ~0 dB. + # Now N = 882 correctly and 22 kHz should be a notch <-220 dB + assert abs(response[0]) < 1e-10 + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + iircomb(1000, 30, fs=np.array([10, 20])) + + with pytest.raises(ValueError, match="Sampling.*be none"): + iircomb(1000, 30, fs=None) + + +class TestIIRDesign: + + def test_exceptions(self): + with pytest.raises(ValueError, match="the same shape"): + iirdesign(0.2, [0.1, 0.3], 1, 40) + with pytest.raises(ValueError, match="the same shape"): + iirdesign(np.array([[0.3, 0.6], [0.3, 0.6]]), + np.array([[0.4, 0.5], [0.4, 0.5]]), 1, 40) + + # discrete filter with non-positive frequency + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign(0, 0.5, 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign(-0.1, 0.5, 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign(0.1, 0, 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign(0.1, -0.5, 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([0, 0.3], [0.1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([-0.1, 0.3], [0.1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([0.1, 0], [0.1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([0.1, -0.3], [0.1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([0.1, 0.3], [0, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([0.1, 0.3], [-0.1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([0.1, 0.3], [0.1, 0], 1, 40) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([0.1, 0.3], [0.1, -0.5], 1, 40) + + # analog filter with negative frequency + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign(-0.1, 0.5, 1, 40, analog=True) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign(0.1, -0.5, 1, 40, analog=True) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([-0.1, 0.3], [0.1, 0.5], 1, 40, analog=True) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([0.1, -0.3], [0.1, 0.5], 1, 40, analog=True) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([0.1, 0.3], [-0.1, 0.5], 1, 40, analog=True) + with pytest.raises(ValueError, match="must be greater than 0"): + iirdesign([0.1, 0.3], [0.1, -0.5], 1, 40, analog=True) + + # discrete filter with fs=None, freq > 1 + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign(1, 0.5, 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign(1.1, 0.5, 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign(0.1, 1, 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign(0.1, 1.5, 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign([1, 0.3], [0.1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign([1.1, 0.3], [0.1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign([0.1, 1], [0.1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign([0.1, 1.1], [0.1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign([0.1, 0.3], [1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign([0.1, 0.3], [1.1, 0.5], 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign([0.1, 0.3], [0.1, 1], 1, 40) + with pytest.raises(ValueError, match="must be less than 1"): + iirdesign([0.1, 0.3], [0.1, 1.5], 1, 40) + + # discrete filter with fs>2, wp, ws < fs/2 must pass + iirdesign(100, 500, 1, 40, fs=2000) + iirdesign(500, 100, 1, 40, fs=2000) + iirdesign([200, 400], [100, 500], 1, 40, fs=2000) + iirdesign([100, 500], [200, 400], 1, 40, fs=2000) + + # discrete filter with fs>2, freq > fs/2: this must raise + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign(1000, 400, 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign(1100, 500, 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign(100, 1000, 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign(100, 1100, 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign([1000, 400], [100, 500], 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign([1100, 400], [100, 500], 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign([200, 1000], [100, 500], 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign([200, 1100], [100, 500], 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign([200, 400], [1000, 500], 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign([200, 400], [1100, 500], 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign([200, 400], [100, 1000], 1, 40, fs=2000) + with pytest.raises(ValueError, match="must be less than fs/2"): + iirdesign([200, 400], [100, 1100], 1, 40, fs=2000) + + with pytest.raises(ValueError, match="strictly inside stopband"): + iirdesign([0.1, 0.4], [0.5, 0.6], 1, 40) + with pytest.raises(ValueError, match="strictly inside stopband"): + iirdesign([0.5, 0.6], [0.1, 0.4], 1, 40) + with pytest.raises(ValueError, match="strictly inside stopband"): + iirdesign([0.3, 0.6], [0.4, 0.7], 1, 40) + with pytest.raises(ValueError, match="strictly inside stopband"): + iirdesign([0.4, 0.7], [0.3, 0.6], 1, 40) + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + iirfilter(1, 1, btype="low", fs=np.array([10, 20])) + + +class TestIIRFilter: + + def test_symmetry(self): + # All built-in IIR filters are real, so should have perfectly + # symmetrical poles and zeros. Then ba representation (using + # numpy.poly) will be purely real instead of having negligible + # imaginary parts. + for N in np.arange(1, 26): + for ftype in ('butter', 'bessel', 'cheby1', 'cheby2', 'ellip'): + z, p, k = iirfilter(N, 1.1, 1, 20, 'low', analog=True, + ftype=ftype, output='zpk') + xp_assert_equal(sorted(z), + sorted(z.conj())) + xp_assert_equal(sorted(p), + sorted(p.conj())) + xp_assert_equal(k, np.real(k)) + + b, a = iirfilter(N, 1.1, 1, 20, 'low', analog=True, + ftype=ftype, output='ba') + assert issubclass(b.dtype.type, np.floating) + assert issubclass(a.dtype.type, np.floating) + + def test_int_inputs(self): + # Using integer frequency arguments and large N should not produce + # numpy integers that wraparound to negative numbers + k = iirfilter(24, 100, btype='low', analog=True, ftype='bessel', + output='zpk')[2] + k2 = 9.999999999999989e+47 + xp_assert_close(np.asarray(k), np.asarray(k2)) + # if fs is specified then the normalization of Wn to have + # 0 <= Wn <= 1 should not cause an integer overflow + # the following line should not raise an exception + iirfilter(20, [1000000000, 1100000000], btype='bp', + analog=False, fs=6250000000) + + def test_invalid_wn_size(self): + # low and high have 1 Wn, band and stop have 2 Wn + assert_raises(ValueError, iirfilter, 1, [0.1, 0.9], btype='low') + assert_raises(ValueError, iirfilter, 1, [0.2, 0.5], btype='high') + assert_raises(ValueError, iirfilter, 1, 0.2, btype='bp') + assert_raises(ValueError, iirfilter, 1, 400, btype='bs', analog=True) + + def test_invalid_wn_range(self): + # For digital filters, 0 <= Wn <= 1 + assert_raises(ValueError, iirfilter, 1, 2, btype='low') + assert_raises(ValueError, iirfilter, 1, [0.5, 1], btype='band') + assert_raises(ValueError, iirfilter, 1, [0., 0.5], btype='band') + assert_raises(ValueError, iirfilter, 1, -1, btype='high') + assert_raises(ValueError, iirfilter, 1, [1, 2], btype='band') + assert_raises(ValueError, iirfilter, 1, [10, 20], btype='stop') + + # analog=True with non-positive critical frequencies + with pytest.raises(ValueError, match="must be greater than 0"): + iirfilter(2, 0, btype='low', analog=True) + with pytest.raises(ValueError, match="must be greater than 0"): + iirfilter(2, -1, btype='low', analog=True) + with pytest.raises(ValueError, match="must be greater than 0"): + iirfilter(2, [0, 100], analog=True) + with pytest.raises(ValueError, match="must be greater than 0"): + iirfilter(2, [-1, 100], analog=True) + with pytest.raises(ValueError, match="must be greater than 0"): + iirfilter(2, [10, 0], analog=True) + with pytest.raises(ValueError, match="must be greater than 0"): + iirfilter(2, [10, -1], analog=True) + + def test_analog_sos(self): + # first order Butterworth filter with Wn = 1 has tf 1/(s+1) + sos = [[0., 0., 1., 0., 1., 1.]] + sos2 = iirfilter(N=1, Wn=1, btype='low', analog=True, output='sos') + assert_array_almost_equal(sos, sos2) + + def test_wn1_ge_wn0(self): + # gh-15773: should raise error if Wn[0] >= Wn[1] + with pytest.raises(ValueError, + match=r"Wn\[0\] must be less than Wn\[1\]"): + iirfilter(2, [0.5, 0.5]) + with pytest.raises(ValueError, + match=r"Wn\[0\] must be less than Wn\[1\]"): + iirfilter(2, [0.6, 0.5]) + + +class TestGroupDelay: + def test_identity_filter(self): + w, gd = group_delay((1, 1)) + assert_array_almost_equal(w, pi * np.arange(512) / 512) + assert_array_almost_equal(gd, np.zeros(512)) + w, gd = group_delay((1, 1), whole=True) + assert_array_almost_equal(w, 2 * pi * np.arange(512) / 512) + assert_array_almost_equal(gd, np.zeros(512)) + + def test_fir(self): + # Let's design linear phase FIR and check that the group delay + # is constant. + N = 100 + b = firwin(N + 1, 0.1) + w, gd = group_delay((b, 1)) + xp_assert_close(gd, np.ones_like(gd)*(0.5 * N)) + + def test_iir(self): + # Let's design Butterworth filter and test the group delay at + # some points against MATLAB answer. + b, a = butter(4, 0.1) + w = np.linspace(0, pi, num=10, endpoint=False) + w, gd = group_delay((b, a), w=w) + matlab_gd = np.array([8.249313898506037, 11.958947880907104, + 2.452325615326005, 1.048918665702008, + 0.611382575635897, 0.418293269460578, + 0.317932917836572, 0.261371844762525, + 0.229038045801298, 0.212185774208521]) + assert_array_almost_equal(gd, matlab_gd) + + @pytest.mark.thread_unsafe + def test_singular(self): + # Let's create a filter with zeros and poles on the unit circle and + # check if warnings are raised at those frequencies. + z1 = np.exp(1j * 0.1 * pi) + z2 = np.exp(1j * 0.25 * pi) + p1 = np.exp(1j * 0.5 * pi) + p2 = np.exp(1j * 0.8 * pi) + b = np.convolve([1, -z1], [1, -z2]) + a = np.convolve([1, -p1], [1, -p2]) + w = np.array([0.1 * pi, 0.25 * pi, -0.5 * pi, -0.8 * pi]) + + w, gd = assert_warns(UserWarning, group_delay, (b, a), w=w) + + def test_backward_compat(self): + # For backward compatibility, test if None act as a wrapper for default + w1, gd1 = group_delay((1, 1)) + w2, gd2 = group_delay((1, 1), None) + assert_array_almost_equal(w1, w2) + assert_array_almost_equal(gd1, gd2) + + def test_fs_param(self): + # Let's design Butterworth filter and test the group delay at + # some points against the normalized frequency answer. + b, a = butter(4, 4800, fs=96000) + w = np.linspace(0, 96000/2, num=10, endpoint=False) + w, gd = group_delay((b, a), w=w, fs=96000) + norm_gd = np.array([8.249313898506037, 11.958947880907104, + 2.452325615326005, 1.048918665702008, + 0.611382575635897, 0.418293269460578, + 0.317932917836572, 0.261371844762525, + 0.229038045801298, 0.212185774208521]) + assert_array_almost_equal(gd, norm_gd) + + def test_w_or_N_types(self): + # Measure at 8 equally-spaced points + for N in (8, np.int8(8), np.int16(8), np.int32(8), np.int64(8), + np.array(8)): + w, gd = group_delay((1, 1), N) + assert_array_almost_equal(w, pi * np.arange(8) / 8) + assert_array_almost_equal(gd, np.zeros(8)) + + # Measure at frequency 8 rad/sec + for w in (8.0, 8.0+0j): + w_out, gd = group_delay((1, 1), w) + assert_array_almost_equal(w_out, [8]) + assert_array_almost_equal(gd, [0]) + + def test_complex_coef(self): + # gh-19586: handle complex coef TFs + # + # for g(z) = (alpha*z+1)/(1+conjugate(alpha)), group delay is + # given by function below. + # + # def gd_expr(w, alpha): + # num = 1j*(abs(alpha)**2-1)*np.exp(1j*w) + # den = (alpha*np.exp(1j*w)+1)*(np.exp(1j*w)+np.conj(alpha)) + # return -np.imag(num/den) + + # arbitrary non-real alpha + alpha = -0.6143077933232609+0.3355978770229421j + # 8 points from from -pi to pi + wref = np.array([-3.141592653589793 , + -2.356194490192345 , + -1.5707963267948966, + -0.7853981633974483, + 0. , + 0.7853981633974483, + 1.5707963267948966, + 2.356194490192345 ]) + gdref = array([0.18759548150354619, + 0.17999770352712252, + 0.23598047471879877, + 0.46539443069907194, + 1.9511492420564165 , + 3.478129975138865 , + 0.6228594960517333 , + 0.27067831839471224]) + b = [alpha,1] + a = [1, np.conjugate(alpha)] + gdtest = group_delay((b,a), wref)[1] + # need nulp=14 for macOS arm64 wheel builds; added 2 for some + # robustness on other platforms. + assert_array_almost_equal_nulp(gdtest, gdref, nulp=16) + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + group_delay((1, 1), fs=np.array([10, 20])) + + with pytest.raises(ValueError, match="Sampling.*be none"): + group_delay((1, 1), fs=None) + + +class TestGammatone: + # Test erroneous input cases. + def test_invalid_input(self): + # Cutoff frequency is <= 0 or >= fs / 2. + fs = 16000 + for args in [(-fs, 'iir'), (0, 'fir'), (fs / 2, 'iir'), (fs, 'fir')]: + with pytest.raises(ValueError, match='The frequency must be ' + 'between '): + gammatone(*args, fs=fs) + + # Filter type is not fir or iir + for args in [(440, 'fie'), (220, 'it')]: + with pytest.raises(ValueError, match='ftype must be '): + gammatone(*args, fs=fs) + + # Order is <= 0 or > 24 for FIR filter. + for args in [(440, 'fir', -50), (220, 'fir', 0), (110, 'fir', 25), + (55, 'fir', 50)]: + with pytest.raises(ValueError, match='Invalid order: '): + gammatone(*args, numtaps=None, fs=fs) + + # Verify that the filter's frequency response is approximately + # 1 at the cutoff frequency. + def test_frequency_response(self): + fs = 16000 + ftypes = ['fir', 'iir'] + for ftype in ftypes: + # Create a gammatone filter centered at 1000 Hz. + b, a = gammatone(1000, ftype, fs=fs) + + # Calculate the frequency response. + freqs, response = freqz(b, a) + + # Determine peak magnitude of the response + # and corresponding frequency. + response_max = np.max(np.abs(response)) + freq_hz = freqs[np.argmax(np.abs(response))] / ((2 * np.pi) / fs) + + # Check that the peak magnitude is 1 and the frequency is 1000 Hz. + xp_assert_close(response_max, + np.ones_like(response_max), rtol=1e-2, check_0d=False) + xp_assert_close(freq_hz, + 1000*np.ones_like(freq_hz), rtol=1e-2, check_0d=False) + + # All built-in IIR filters are real, so should have perfectly + # symmetrical poles and zeros. Then ba representation (using + # numpy.poly) will be purely real instead of having negligible + # imaginary parts. + def test_iir_symmetry(self): + b, a = gammatone(440, 'iir', fs=24000) + z, p, k = tf2zpk(b, a) + xp_assert_equal(sorted(z), sorted(z.conj())) + xp_assert_equal(sorted(p), sorted(p.conj())) + xp_assert_equal(k, np.real(k)) + + assert issubclass(b.dtype.type, np.floating) + assert issubclass(a.dtype.type, np.floating) + + # Verify FIR filter coefficients with the paper's + # Mathematica implementation + def test_fir_ba_output(self): + b, _ = gammatone(15, 'fir', fs=1000) + b2 = [0.0, 2.2608075649884e-04, + 1.5077903981357e-03, 4.2033687753998e-03, + 8.1508962726503e-03, 1.2890059089154e-02, + 1.7833890391666e-02, 2.2392613558564e-02, + 2.6055195863104e-02, 2.8435872863284e-02, + 2.9293319149544e-02, 2.852976858014e-02, + 2.6176557156294e-02, 2.2371510270395e-02, + 1.7332485267759e-02] + xp_assert_close(b, b2) + + # Verify IIR filter coefficients with the paper's MATLAB implementation + def test_iir_ba_output(self): + b, a = gammatone(440, 'iir', fs=16000) + b2 = [1.31494461367464e-06, -5.03391196645395e-06, + 7.00649426000897e-06, -4.18951968419854e-06, + 9.02614910412011e-07] + a2 = [1.0, -7.65646235454218, + 25.7584699322366, -49.7319214483238, + 60.2667361289181, -46.9399590980486, + 22.9474798808461, -6.43799381299034, + 0.793651554625368] + xp_assert_close(b, b2) + xp_assert_close(a, a2) + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + gammatone(440, 'iir', fs=np.array([10, 20])) + + +class TestOrderFilter: + def test_doc_example(self): + x = np.arange(25).reshape(5, 5) + domain = np.identity(3) + + # minimum of elements 1,3,9 (zero-padded) on phone pad + # 7,5,3 on numpad + expected = np.array( + [[0., 0., 0., 0., 0.], + [0., 0., 1., 2., 0.], + [0., 5., 6., 7., 0.], + [0., 10., 11., 12., 0.], + [0., 0., 0., 0., 0.]], + ) + xp_assert_close(order_filter(x, domain, 0), expected, check_dtype=False) + + # maximum of elements 1,3,9 (zero-padded) on phone pad + # 7,5,3 on numpad + expected = np.array( + [[6., 7., 8., 9., 4.], + [11., 12., 13., 14., 9.], + [16., 17., 18., 19., 14.], + [21., 22., 23., 24., 19.], + [20., 21., 22., 23., 24.]], + ) + xp_assert_close(order_filter(x, domain, 2), expected, check_dtype=False) + + # and, just to complete the set, median of zero-padded elements + expected = np.array( + [[0, 1, 2, 3, 0], + [5, 6, 7, 8, 3], + [10, 11, 12, 13, 8], + [15, 16, 17, 18, 13], + [0, 15, 16, 17, 18]], + ) + xp_assert_close(order_filter(x, domain, 1), expected) + + def test_medfilt_order_filter(self): + x = np.arange(25).reshape(5, 5) + + # median of zero-padded elements 1,5,9 on phone pad + # 7,5,3 on numpad + expected = np.array( + [[0, 1, 2, 3, 0], + [1, 6, 7, 8, 4], + [6, 11, 12, 13, 9], + [11, 16, 17, 18, 14], + [0, 16, 17, 18, 0]], + ) + xp_assert_close(medfilt(x, 3), expected) + + xp_assert_close( + order_filter(x, np.ones((3, 3)), 4), + expected + ) + + def test_order_filter_asymmetric(self): + x = np.arange(25).reshape(5, 5) + domain = np.array( + [[1, 1, 0], + [0, 1, 0], + [0, 0, 0]], + ) + + expected = np.array( + [[0, 0, 0, 0, 0], + [0, 0, 1, 2, 3], + [0, 5, 6, 7, 8], + [0, 10, 11, 12, 13], + [0, 15, 16, 17, 18]] + ) + xp_assert_close(order_filter(x, domain, 0), expected) + + expected = np.array( + [[0, 0, 0, 0, 0], + [0, 1, 2, 3, 4], + [5, 6, 7, 8, 9], + [10, 11, 12, 13, 14], + [15, 16, 17, 18, 19]] + ) + xp_assert_close(order_filter(x, domain, 1), expected) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_fir_filter_design.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_fir_filter_design.py new file mode 100644 index 0000000000000000000000000000000000000000..cd2b60e63cdcb36e39c6775fa99394d123c2956b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_fir_filter_design.py @@ -0,0 +1,654 @@ +import numpy as np +from numpy.testing import assert_warns +from scipy._lib._array_api import ( + xp_assert_close, xp_assert_equal, + assert_almost_equal, assert_array_almost_equal, +) +from pytest import raises as assert_raises +import pytest + +from scipy.fft import fft +from scipy.special import sinc +from scipy.signal import (kaiser_beta, kaiser_atten, kaiserord, + firwin, firwin2, freqz, remez, firls, minimum_phase +) + + +def test_kaiser_beta(): + b = kaiser_beta(58.7) + assert_almost_equal(b, 0.1102 * 50.0) + b = kaiser_beta(22.0) + assert_almost_equal(b, 0.5842 + 0.07886) + b = kaiser_beta(21.0) + assert b == 0.0 + b = kaiser_beta(10.0) + assert b == 0.0 + + +def test_kaiser_atten(): + a = kaiser_atten(1, 1.0) + assert a == 7.95 + a = kaiser_atten(2, 1/np.pi) + assert a == 2.285 + 7.95 + + +def test_kaiserord(): + assert_raises(ValueError, kaiserord, 1.0, 1.0) + numtaps, beta = kaiserord(2.285 + 7.95 - 0.001, 1/np.pi) + assert (numtaps, beta) == (2, 0.0) + + +class TestFirwin: + + def check_response(self, h, expected_response, tol=.05): + N = len(h) + alpha = 0.5 * (N-1) + m = np.arange(0,N) - alpha # time indices of taps + for freq, expected in expected_response: + actual = abs(np.sum(h*np.exp(-1.j*np.pi*m*freq))) + mse = abs(actual-expected)**2 + assert mse < tol, f'response not as expected, mse={mse:g} > {tol:g}' + + def test_response(self): + N = 51 + f = .5 + # increase length just to try even/odd + h = firwin(N, f) # low-pass from 0 to f + self.check_response(h, [(.25,1), (.75,0)]) + + h = firwin(N+1, f, window='nuttall') # specific window + self.check_response(h, [(.25,1), (.75,0)]) + + h = firwin(N+2, f, pass_zero=False) # stop from 0 to f --> high-pass + self.check_response(h, [(.25,0), (.75,1)]) + + f1, f2, f3, f4 = .2, .4, .6, .8 + h = firwin(N+3, [f1, f2], pass_zero=False) # band-pass filter + self.check_response(h, [(.1,0), (.3,1), (.5,0)]) + + h = firwin(N+4, [f1, f2]) # band-stop filter + self.check_response(h, [(.1,1), (.3,0), (.5,1)]) + + h = firwin(N+5, [f1, f2, f3, f4], pass_zero=False, scale=False) + self.check_response(h, [(.1,0), (.3,1), (.5,0), (.7,1), (.9,0)]) + + h = firwin(N+6, [f1, f2, f3, f4]) # multiband filter + self.check_response(h, [(.1,1), (.3,0), (.5,1), (.7,0), (.9,1)]) + + h = firwin(N+7, 0.1, width=.03) # low-pass + self.check_response(h, [(.05,1), (.75,0)]) + + h = firwin(N+8, 0.1, pass_zero=False) # high-pass + self.check_response(h, [(.05,0), (.75,1)]) + + def mse(self, h, bands): + """Compute mean squared error versus ideal response across frequency + band. + h -- coefficients + bands -- list of (left, right) tuples relative to 1==Nyquist of + passbands + """ + w, H = freqz(h, worN=1024) + f = w/np.pi + passIndicator = np.zeros(len(w), bool) + for left, right in bands: + passIndicator |= (f >= left) & (f < right) + Hideal = np.where(passIndicator, 1, 0) + mse = np.mean(abs(abs(H)-Hideal)**2) + return mse + + def test_scaling(self): + """ + For one lowpass, bandpass, and highpass example filter, this test + checks two things: + - the mean squared error over the frequency domain of the unscaled + filter is smaller than the scaled filter (true for rectangular + window) + - the response of the scaled filter is exactly unity at the center + of the first passband + """ + N = 11 + cases = [ + ([.5], True, (0, 1)), + ([0.2, .6], False, (.4, 1)), + ([.5], False, (1, 1)), + ] + for cutoff, pass_zero, expected_response in cases: + h = firwin(N, cutoff, scale=False, pass_zero=pass_zero, window='ones') + hs = firwin(N, cutoff, scale=True, pass_zero=pass_zero, window='ones') + if len(cutoff) == 1: + if pass_zero: + cutoff = [0] + cutoff + else: + cutoff = cutoff + [1] + msg = 'least squares violation' + assert self.mse(h, [cutoff]) < self.mse(hs, [cutoff]), msg + self.check_response(hs, [expected_response], 1e-12) + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + firwin(51, .5, fs=np.array([10, 20])) + + +class TestFirWinMore: + """Different author, different style, different tests...""" + + def test_lowpass(self): + width = 0.04 + ntaps, beta = kaiserord(120, width) + kwargs = dict(cutoff=0.5, window=('kaiser', beta), scale=False) + taps = firwin(ntaps, **kwargs) + + # Check the symmetry of taps. + assert_array_almost_equal(taps[:ntaps//2], taps[ntaps:ntaps-ntaps//2-1:-1]) + + # Check the gain at a few samples where + # we know it should be approximately 0 or 1. + freq_samples = np.array([0.0, 0.25, 0.5-width/2, 0.5+width/2, 0.75, 1.0]) + freqs, response = freqz(taps, worN=np.pi*freq_samples) + assert_array_almost_equal(np.abs(response), + [1.0, 1.0, 1.0, 0.0, 0.0, 0.0], decimal=5) + + taps_str = firwin(ntaps, pass_zero='lowpass', **kwargs) + xp_assert_close(taps, taps_str) + + def test_highpass(self): + width = 0.04 + ntaps, beta = kaiserord(120, width) + + # Ensure that ntaps is odd. + ntaps |= 1 + + kwargs = dict(cutoff=0.5, window=('kaiser', beta), scale=False) + taps = firwin(ntaps, pass_zero=False, **kwargs) + + # Check the symmetry of taps. + assert_array_almost_equal(taps[:ntaps//2], taps[ntaps:ntaps-ntaps//2-1:-1]) + + # Check the gain at a few samples where + # we know it should be approximately 0 or 1. + freq_samples = np.array([0.0, 0.25, 0.5-width/2, 0.5+width/2, 0.75, 1.0]) + freqs, response = freqz(taps, worN=np.pi*freq_samples) + assert_array_almost_equal(np.abs(response), + [0.0, 0.0, 0.0, 1.0, 1.0, 1.0], decimal=5) + + taps_str = firwin(ntaps, pass_zero='highpass', **kwargs) + xp_assert_close(taps, taps_str) + + def test_bandpass(self): + width = 0.04 + ntaps, beta = kaiserord(120, width) + kwargs = dict(cutoff=[0.3, 0.7], window=('kaiser', beta), scale=False) + taps = firwin(ntaps, pass_zero=False, **kwargs) + + # Check the symmetry of taps. + assert_array_almost_equal(taps[:ntaps//2], taps[ntaps:ntaps-ntaps//2-1:-1]) + + # Check the gain at a few samples where + # we know it should be approximately 0 or 1. + freq_samples = np.array([0.0, 0.2, 0.3-width/2, 0.3+width/2, 0.5, + 0.7-width/2, 0.7+width/2, 0.8, 1.0]) + freqs, response = freqz(taps, worN=np.pi*freq_samples) + assert_array_almost_equal(np.abs(response), + [0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0], decimal=5) + + taps_str = firwin(ntaps, pass_zero='bandpass', **kwargs) + xp_assert_close(taps, taps_str) + + def test_bandstop_multi(self): + width = 0.04 + ntaps, beta = kaiserord(120, width) + kwargs = dict(cutoff=[0.2, 0.5, 0.8], window=('kaiser', beta), + scale=False) + taps = firwin(ntaps, **kwargs) + + # Check the symmetry of taps. + assert_array_almost_equal(taps[:ntaps//2], taps[ntaps:ntaps-ntaps//2-1:-1]) + + # Check the gain at a few samples where + # we know it should be approximately 0 or 1. + freq_samples = np.array([0.0, 0.1, 0.2-width/2, 0.2+width/2, 0.35, + 0.5-width/2, 0.5+width/2, 0.65, + 0.8-width/2, 0.8+width/2, 0.9, 1.0]) + freqs, response = freqz(taps, worN=np.pi*freq_samples) + assert_array_almost_equal(np.abs(response), + [1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0], + decimal=5) + + taps_str = firwin(ntaps, pass_zero='bandstop', **kwargs) + xp_assert_close(taps, taps_str) + + def test_fs_nyq(self): + """Test the fs and nyq keywords.""" + nyquist = 1000 + width = 40.0 + relative_width = width/nyquist + ntaps, beta = kaiserord(120, relative_width) + taps = firwin(ntaps, cutoff=[300, 700], window=('kaiser', beta), + pass_zero=False, scale=False, fs=2*nyquist) + + # Check the symmetry of taps. + assert_array_almost_equal(taps[:ntaps//2], taps[ntaps:ntaps-ntaps//2-1:-1]) + + # Check the gain at a few samples where + # we know it should be approximately 0 or 1. + freq_samples = np.array([0.0, 200, 300-width/2, 300+width/2, 500, + 700-width/2, 700+width/2, 800, 1000]) + freqs, response = freqz(taps, worN=np.pi*freq_samples/nyquist) + assert_array_almost_equal(np.abs(response), + [0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0], decimal=5) + + def test_bad_cutoff(self): + """Test that invalid cutoff argument raises ValueError.""" + # cutoff values must be greater than 0 and less than 1. + assert_raises(ValueError, firwin, 99, -0.5) + assert_raises(ValueError, firwin, 99, 1.5) + # Don't allow 0 or 1 in cutoff. + assert_raises(ValueError, firwin, 99, [0, 0.5]) + assert_raises(ValueError, firwin, 99, [0.5, 1]) + # cutoff values must be strictly increasing. + assert_raises(ValueError, firwin, 99, [0.1, 0.5, 0.2]) + assert_raises(ValueError, firwin, 99, [0.1, 0.5, 0.5]) + # Must have at least one cutoff value. + assert_raises(ValueError, firwin, 99, []) + # 2D array not allowed. + assert_raises(ValueError, firwin, 99, [[0.1, 0.2],[0.3, 0.4]]) + # cutoff values must be less than nyq. + assert_raises(ValueError, firwin, 99, 50.0, fs=80) + assert_raises(ValueError, firwin, 99, [10, 20, 30], fs=50) + + def test_even_highpass_raises_value_error(self): + """Test that attempt to create a highpass filter with an even number + of taps raises a ValueError exception.""" + assert_raises(ValueError, firwin, 40, 0.5, pass_zero=False) + assert_raises(ValueError, firwin, 40, [.25, 0.5]) + + def test_bad_pass_zero(self): + """Test degenerate pass_zero cases.""" + with assert_raises(ValueError, match='pass_zero must be'): + firwin(41, 0.5, pass_zero='foo') + with assert_raises(TypeError, match='cannot be interpreted'): + firwin(41, 0.5, pass_zero=1.) + for pass_zero in ('lowpass', 'highpass'): + with assert_raises(ValueError, match='cutoff must have one'): + firwin(41, [0.5, 0.6], pass_zero=pass_zero) + for pass_zero in ('bandpass', 'bandstop'): + with assert_raises(ValueError, match='must have at least two'): + firwin(41, [0.5], pass_zero=pass_zero) + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + firwin2(51, .5, 1, fs=np.array([10, 20])) + + +class TestFirwin2: + + def test_invalid_args(self): + # `freq` and `gain` have different lengths. + with assert_raises(ValueError, match='must be of same length'): + firwin2(50, [0, 0.5, 1], [0.0, 1.0]) + # `nfreqs` is less than `ntaps`. + with assert_raises(ValueError, match='ntaps must be less than nfreqs'): + firwin2(50, [0, 0.5, 1], [0.0, 1.0, 1.0], nfreqs=33) + # Decreasing value in `freq` + with assert_raises(ValueError, match='must be nondecreasing'): + firwin2(50, [0, 0.5, 0.4, 1.0], [0, .25, .5, 1.0]) + # Value in `freq` repeated more than once. + with assert_raises(ValueError, match='must not occur more than twice'): + firwin2(50, [0, .1, .1, .1, 1.0], [0.0, 0.5, 0.75, 1.0, 1.0]) + # `freq` does not start at 0.0. + with assert_raises(ValueError, match='start with 0'): + firwin2(50, [0.5, 1.0], [0.0, 1.0]) + # `freq` does not end at fs/2. + with assert_raises(ValueError, match='end with fs/2'): + firwin2(50, [0.0, 0.5], [0.0, 1.0]) + # Value 0 is repeated in `freq` + with assert_raises(ValueError, match='0 must not be repeated'): + firwin2(50, [0.0, 0.0, 0.5, 1.0], [1.0, 1.0, 0.0, 0.0]) + # Value fs/2 is repeated in `freq` + with assert_raises(ValueError, match='fs/2 must not be repeated'): + firwin2(50, [0.0, 0.5, 1.0, 1.0], [1.0, 1.0, 0.0, 0.0]) + # Value in `freq` that is too close to a repeated number + with assert_raises(ValueError, match='cannot contain numbers ' + 'that are too close'): + firwin2(50, [0.0, 0.5 - np.finfo(float).eps * 0.5, 0.5, 0.5, 1.0], + [1.0, 1.0, 1.0, 0.0, 0.0]) + + # Type II filter, but the gain at nyquist frequency is not zero. + with assert_raises(ValueError, match='Type II filter'): + firwin2(16, [0.0, 0.5, 1.0], [0.0, 1.0, 1.0]) + + # Type III filter, but the gains at nyquist and zero rate are not zero. + with assert_raises(ValueError, match='Type III filter'): + firwin2(17, [0.0, 0.5, 1.0], [0.0, 1.0, 1.0], antisymmetric=True) + with assert_raises(ValueError, match='Type III filter'): + firwin2(17, [0.0, 0.5, 1.0], [1.0, 1.0, 0.0], antisymmetric=True) + with assert_raises(ValueError, match='Type III filter'): + firwin2(17, [0.0, 0.5, 1.0], [1.0, 1.0, 1.0], antisymmetric=True) + + # Type IV filter, but the gain at zero rate is not zero. + with assert_raises(ValueError, match='Type IV filter'): + firwin2(16, [0.0, 0.5, 1.0], [1.0, 1.0, 0.0], antisymmetric=True) + + def test01(self): + width = 0.04 + beta = 12.0 + ntaps = 400 + # Filter is 1 from w=0 to w=0.5, then decreases linearly from 1 to 0 as w + # increases from w=0.5 to w=1 (w=1 is the Nyquist frequency). + freq = [0.0, 0.5, 1.0] + gain = [1.0, 1.0, 0.0] + taps = firwin2(ntaps, freq, gain, window=('kaiser', beta)) + freq_samples = np.array([0.0, 0.25, 0.5-width/2, 0.5+width/2, + 0.75, 1.0-width/2]) + freqs, response = freqz(taps, worN=np.pi*freq_samples) + assert_array_almost_equal(np.abs(response), + [1.0, 1.0, 1.0, 1.0-width, 0.5, width], decimal=5) + + def test02(self): + width = 0.04 + beta = 12.0 + # ntaps must be odd for positive gain at Nyquist. + ntaps = 401 + # An ideal highpass filter. + freq = [0.0, 0.5, 0.5, 1.0] + gain = [0.0, 0.0, 1.0, 1.0] + taps = firwin2(ntaps, freq, gain, window=('kaiser', beta)) + freq_samples = np.array([0.0, 0.25, 0.5-width, 0.5+width, 0.75, 1.0]) + freqs, response = freqz(taps, worN=np.pi*freq_samples) + assert_array_almost_equal(np.abs(response), + [0.0, 0.0, 0.0, 1.0, 1.0, 1.0], decimal=5) + + def test03(self): + width = 0.02 + ntaps, beta = kaiserord(120, width) + # ntaps must be odd for positive gain at Nyquist. + ntaps = int(ntaps) | 1 + freq = [0.0, 0.4, 0.4, 0.5, 0.5, 1.0] + gain = [1.0, 1.0, 0.0, 0.0, 1.0, 1.0] + taps = firwin2(ntaps, freq, gain, window=('kaiser', beta)) + freq_samples = np.array([0.0, 0.4-width, 0.4+width, 0.45, + 0.5-width, 0.5+width, 0.75, 1.0]) + freqs, response = freqz(taps, worN=np.pi*freq_samples) + assert_array_almost_equal(np.abs(response), + [1.0, 1.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0], decimal=5) + + def test04(self): + """Test firwin2 when window=None.""" + ntaps = 5 + # Ideal lowpass: gain is 1 on [0,0.5], and 0 on [0.5, 1.0] + freq = [0.0, 0.5, 0.5, 1.0] + gain = [1.0, 1.0, 0.0, 0.0] + taps = firwin2(ntaps, freq, gain, window=None, nfreqs=8193) + alpha = 0.5 * (ntaps - 1) + m = np.arange(0, ntaps) - alpha + h = 0.5 * sinc(0.5 * m) + assert_array_almost_equal(h, taps) + + def test05(self): + """Test firwin2 for calculating Type IV filters""" + ntaps = 1500 + + freq = [0.0, 1.0] + gain = [0.0, 1.0] + taps = firwin2(ntaps, freq, gain, window=None, antisymmetric=True) + assert_array_almost_equal(taps[: ntaps // 2], -taps[ntaps // 2:][::-1]) + + freqs, response = freqz(taps, worN=2048) + assert_array_almost_equal(abs(response), freqs / np.pi, decimal=4) + + def test06(self): + """Test firwin2 for calculating Type III filters""" + ntaps = 1501 + + freq = [0.0, 0.5, 0.55, 1.0] + gain = [0.0, 0.5, 0.0, 0.0] + taps = firwin2(ntaps, freq, gain, window=None, antisymmetric=True) + assert taps[ntaps // 2] == 0.0 + assert_array_almost_equal(taps[: ntaps // 2], -taps[ntaps // 2 + 1:][::-1]) + + freqs, response1 = freqz(taps, worN=2048) + response2 = np.interp(freqs / np.pi, freq, gain) + assert_array_almost_equal(abs(response1), response2, decimal=3) + + def test_fs_nyq(self): + taps1 = firwin2(80, [0.0, 0.5, 1.0], [1.0, 1.0, 0.0]) + taps2 = firwin2(80, [0.0, 30.0, 60.0], [1.0, 1.0, 0.0], fs=120.0) + assert_array_almost_equal(taps1, taps2) + + def test_tuple(self): + taps1 = firwin2(150, (0.0, 0.5, 0.5, 1.0), (1.0, 1.0, 0.0, 0.0)) + taps2 = firwin2(150, [0.0, 0.5, 0.5, 1.0], [1.0, 1.0, 0.0, 0.0]) + assert_array_almost_equal(taps1, taps2) + + def test_input_modyfication(self): + freq1 = np.array([0.0, 0.5, 0.5, 1.0]) + freq2 = np.array(freq1) + firwin2(80, freq1, [1.0, 1.0, 0.0, 0.0]) + xp_assert_equal(freq1, freq2) + + +class TestRemez: + + def test_bad_args(self): + assert_raises(ValueError, remez, 11, [0.1, 0.4], [1], type='pooka') + + def test_hilbert(self): + N = 11 # number of taps in the filter + a = 0.1 # width of the transition band + + # design an unity gain hilbert bandpass filter from w to 0.5-w + h = remez(11, [a, 0.5-a], [1], type='hilbert') + + # make sure the filter has correct # of taps + assert len(h) == N, "Number of Taps" + + # make sure it is type III (anti-symmetric tap coefficients) + assert_array_almost_equal(h[:(N-1)//2], -h[:-(N-1)//2-1:-1]) + + # Since the requested response is symmetric, all even coefficients + # should be zero (or in this case really small) + assert (abs(h[1::2]) < 1e-15).all(), "Even Coefficients Equal Zero" + + # now check the frequency response + w, H = freqz(h, 1) + f = w/2/np.pi + Hmag = abs(H) + + # should have a zero at 0 and pi (in this case close to zero) + assert (Hmag[[0, -1]] < 0.02).all(), "Zero at zero and pi" + + # check that the pass band is close to unity + idx = np.logical_and(f > a, f < 0.5-a) + assert (abs(Hmag[idx] - 1) < 0.015).all(), "Pass Band Close To Unity" + + def test_compare(self): + # test comparison to MATLAB + k = [0.024590270518440, -0.041314581814658, -0.075943803756711, + -0.003530911231040, 0.193140296954975, 0.373400753484939, + 0.373400753484939, 0.193140296954975, -0.003530911231040, + -0.075943803756711, -0.041314581814658, 0.024590270518440] + h = remez(12, [0, 0.3, 0.5, 1], [1, 0], fs=2.) + xp_assert_close(h, k) + + h = [-0.038976016082299, 0.018704846485491, -0.014644062687875, + 0.002879152556419, 0.016849978528150, -0.043276706138248, + 0.073641298245579, -0.103908158578635, 0.129770906801075, + -0.147163447297124, 0.153302248456347, -0.147163447297124, + 0.129770906801075, -0.103908158578635, 0.073641298245579, + -0.043276706138248, 0.016849978528150, 0.002879152556419, + -0.014644062687875, 0.018704846485491, -0.038976016082299] + xp_assert_close(remez(21, [0, 0.8, 0.9, 1], [0, 1], fs=2.), h) + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + remez(11, .1, 1, fs=np.array([10, 20])) + +class TestFirls: + + def test_bad_args(self): + # even numtaps + assert_raises(ValueError, firls, 10, [0.1, 0.2], [0, 0]) + # odd bands + assert_raises(ValueError, firls, 11, [0.1, 0.2, 0.4], [0, 0, 0]) + # len(bands) != len(desired) + assert_raises(ValueError, firls, 11, [0.1, 0.2, 0.3, 0.4], [0, 0, 0]) + # non-monotonic bands + assert_raises(ValueError, firls, 11, [0.2, 0.1], [0, 0]) + assert_raises(ValueError, firls, 11, [0.1, 0.2, 0.3, 0.3], [0] * 4) + assert_raises(ValueError, firls, 11, [0.3, 0.4, 0.1, 0.2], [0] * 4) + assert_raises(ValueError, firls, 11, [0.1, 0.3, 0.2, 0.4], [0] * 4) + # negative desired + assert_raises(ValueError, firls, 11, [0.1, 0.2], [-1, 1]) + # len(weight) != len(pairs) + assert_raises(ValueError, firls, 11, [0.1, 0.2], [0, 0], weight=[1, 2]) + # negative weight + assert_raises(ValueError, firls, 11, [0.1, 0.2], [0, 0], weight=[-1]) + + def test_firls(self): + N = 11 # number of taps in the filter + a = 0.1 # width of the transition band + + # design a halfband symmetric low-pass filter + h = firls(11, [0, a, 0.5-a, 0.5], [1, 1, 0, 0], fs=1.0) + + # make sure the filter has correct # of taps + assert h.shape[0] == N + + # make sure it is symmetric + midx = (N-1) // 2 + assert_array_almost_equal(h[:midx], h[:-midx-1:-1]) + + # make sure the center tap is 0.5 + assert_almost_equal(h[midx], 0.5) + + # For halfband symmetric, odd coefficients (except the center) + # should be zero (really small) + hodd = np.hstack((h[1:midx:2], h[-midx+1::2])) + assert_array_almost_equal(hodd, np.zeros_like(hodd)) + + # now check the frequency response + w, H = freqz(h, 1) + f = w/2/np.pi + Hmag = np.abs(H) + + # check that the pass band is close to unity + idx = np.logical_and(f > 0, f < a) + assert_array_almost_equal(Hmag[idx], np.ones_like(Hmag[idx]), decimal=3) + + # check that the stop band is close to zero + idx = np.logical_and(f > 0.5-a, f < 0.5) + assert_array_almost_equal(Hmag[idx], np.zeros_like(Hmag[idx]), decimal=3) + + def test_compare(self): + # compare to OCTAVE output + taps = firls(9, [0, 0.5, 0.55, 1], [1, 1, 0, 0], weight=[1, 2]) + # >> taps = firls(8, [0 0.5 0.55 1], [1 1 0 0], [1, 2]); + known_taps = [-6.26930101730182e-04, -1.03354450635036e-01, + -9.81576747564301e-03, 3.17271686090449e-01, + 5.11409425599933e-01, 3.17271686090449e-01, + -9.81576747564301e-03, -1.03354450635036e-01, + -6.26930101730182e-04] + xp_assert_close(taps, known_taps) + + # compare to MATLAB output + taps = firls(11, [0, 0.5, 0.5, 1], [1, 1, 0, 0], weight=[1, 2]) + # >> taps = firls(10, [0 0.5 0.5 1], [1 1 0 0], [1, 2]); + known_taps = [ + 0.058545300496815, -0.014233383714318, -0.104688258464392, + 0.012403323025279, 0.317930861136062, 0.488047220029700, + 0.317930861136062, 0.012403323025279, -0.104688258464392, + -0.014233383714318, 0.058545300496815] + xp_assert_close(taps, known_taps) + + # With linear changes: + taps = firls(7, (0, 1, 2, 3, 4, 5), [1, 0, 0, 1, 1, 0], fs=20) + # >> taps = firls(6, [0, 0.1, 0.2, 0.3, 0.4, 0.5], [1, 0, 0, 1, 1, 0]) + known_taps = [ + 1.156090832768218, -4.1385894727395849, 7.5288619164321826, + -8.5530572592947856, 7.5288619164321826, -4.1385894727395849, + 1.156090832768218] + xp_assert_close(taps, known_taps) + + def test_rank_deficient(self): + # solve() runs but warns (only sometimes, so here we don't use match) + x = firls(21, [0, 0.1, 0.9, 1], [1, 1, 0, 0]) + w, h = freqz(x, fs=2.) + absh2 = np.abs(h[:2]) + xp_assert_close(absh2, np.ones_like(absh2), atol=1e-5) + absh2 = np.abs(h[-2:]) + xp_assert_close(absh2, np.zeros_like(absh2), atol=1e-6, rtol=1e-7) + # switch to pinvh (tolerances could be higher with longer + # filters, but using shorter ones is faster computationally and + # the idea is the same) + x = firls(101, [0, 0.01, 0.99, 1], [1, 1, 0, 0]) + w, h = freqz(x, fs=2.) + mask = w < 0.01 + assert mask.sum() > 3 + habs = np.abs(h[mask]) + xp_assert_close(habs, np.ones_like(habs), atol=1e-4) + mask = w > 0.99 + assert mask.sum() > 3 + habs = np.abs(h[mask]) + xp_assert_close(habs, np.zeros_like(habs), atol=1e-4) + + def test_fs_validation(self): + with pytest.raises(ValueError, match="Sampling.*single scalar"): + firls(11, .1, 1, fs=np.array([10, 20])) + +class TestMinimumPhase: + @pytest.mark.thread_unsafe + def test_bad_args(self): + # not enough taps + assert_raises(ValueError, minimum_phase, [1.]) + assert_raises(ValueError, minimum_phase, [1., 1.]) + assert_raises(ValueError, minimum_phase, np.full(10, 1j)) + assert_raises(ValueError, minimum_phase, 'foo') + assert_raises(ValueError, minimum_phase, np.ones(10), n_fft=8) + assert_raises(ValueError, minimum_phase, np.ones(10), method='foo') + assert_warns(RuntimeWarning, minimum_phase, np.arange(3)) + with pytest.raises(ValueError, match="is only supported when"): + minimum_phase(np.ones(3), method='hilbert', half=False) + + def test_homomorphic(self): + # check that it can recover frequency responses of arbitrary + # linear-phase filters + + # for some cases we can get the actual filter back + h = [1, -1] + h_new = minimum_phase(np.convolve(h, h[::-1])) + xp_assert_close(h_new, np.asarray(h, dtype=np.float64), rtol=0.05) + + # but in general we only guarantee we get the magnitude back + rng = np.random.RandomState(0) + for n in (2, 3, 10, 11, 15, 16, 17, 20, 21, 100, 101): + h = rng.randn(n) + h_linear = np.convolve(h, h[::-1]) + h_new = minimum_phase(h_linear) + xp_assert_close(np.abs(fft(h_new)), np.abs(fft(h)), rtol=1e-4) + h_new = minimum_phase(h_linear, half=False) + assert len(h_linear) == len(h_new) + xp_assert_close(np.abs(fft(h_new)), np.abs(fft(h_linear)), rtol=1e-4) + + def test_hilbert(self): + # compare to MATLAB output of reference implementation + + # f=[0 0.3 0.5 1]; + # a=[1 1 0 0]; + # h=remez(11,f,a); + h = remez(12, [0, 0.3, 0.5, 1], [1, 0], fs=2.) + k = [0.349585548646686, 0.373552164395447, 0.326082685363438, + 0.077152207480935, -0.129943946349364, -0.059355880509749] + m = minimum_phase(h, 'hilbert') + xp_assert_close(m, k, rtol=5e-3) + + # f=[0 0.8 0.9 1]; + # a=[0 0 1 1]; + # h=remez(20,f,a); + h = remez(21, [0, 0.8, 0.9, 1], [0, 1], fs=2.) + k = [0.232486803906329, -0.133551833687071, 0.151871456867244, + -0.157957283165866, 0.151739294892963, -0.129293146705090, + 0.100787844523204, -0.065832656741252, 0.035361328741024, + -0.014977068692269, -0.158416139047557] + m = minimum_phase(h, 'hilbert', n_fft=2**19) + xp_assert_close(m, k, rtol=2e-3) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_ltisys.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_ltisys.py new file mode 100644 index 0000000000000000000000000000000000000000..826b39cb0e066b3a6198bbcf1a293f6e0497076b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_ltisys.py @@ -0,0 +1,1225 @@ +import warnings + +import numpy as np +from numpy.testing import suppress_warnings +import pytest +from pytest import raises as assert_raises +from scipy._lib._array_api import( + assert_almost_equal, xp_assert_equal, xp_assert_close +) + +from scipy.signal import (ss2tf, tf2ss, lti, + dlti, bode, freqresp, lsim, impulse, step, + abcd_normalize, place_poles, + TransferFunction, StateSpace, ZerosPolesGain) +from scipy.signal._filter_design import BadCoefficients +import scipy.linalg as linalg + + +def _assert_poles_close(P1,P2, rtol=1e-8, atol=1e-8): + """ + Check each pole in P1 is close to a pole in P2 with a 1e-8 + relative tolerance or 1e-8 absolute tolerance (useful for zero poles). + These tolerances are very strict but the systems tested are known to + accept these poles so we should not be far from what is requested. + """ + P2 = P2.copy() + for p1 in P1: + found = False + for p2_idx in range(P2.shape[0]): + if np.allclose([np.real(p1), np.imag(p1)], + [np.real(P2[p2_idx]), np.imag(P2[p2_idx])], + rtol, atol): + found = True + np.delete(P2, p2_idx) + break + if not found: + raise ValueError("Can't find pole " + str(p1) + " in " + str(P2)) + + +class TestPlacePoles: + + def _check(self, A, B, P, **kwargs): + """ + Perform the most common tests on the poles computed by place_poles + and return the Bunch object for further specific tests + """ + fsf = place_poles(A, B, P, **kwargs) + expected, _ = np.linalg.eig(A - np.dot(B, fsf.gain_matrix)) + _assert_poles_close(expected, fsf.requested_poles) + _assert_poles_close(expected, fsf.computed_poles) + _assert_poles_close(P,fsf.requested_poles) + return fsf + + def test_real(self): + # Test real pole placement using KNV and YT0 algorithm and example 1 in + # section 4 of the reference publication (see place_poles docstring) + A = np.array([1.380, -0.2077, 6.715, -5.676, -0.5814, -4.290, 0, + 0.6750, 1.067, 4.273, -6.654, 5.893, 0.0480, 4.273, + 1.343, -2.104]).reshape(4, 4) + B = np.array([0, 5.679, 1.136, 1.136, 0, 0, -3.146,0]).reshape(4, 2) + P = np.array([-0.2, -0.5, -5.0566, -8.6659]) + + # Check that both KNV and YT compute correct K matrix + self._check(A, B, P, method='KNV0') + self._check(A, B, P, method='YT') + + # Try to reach the specific case in _YT_real where two singular + # values are almost equal. This is to improve code coverage but I + # have no way to be sure this code is really reached + + # on some architectures this can lead to a RuntimeWarning invalid + # value in divide (see gh-7590), so suppress it for now + with np.errstate(invalid='ignore'): + self._check(A, B, (2,2,3,3)) + + def test_complex(self): + # Test complex pole placement on a linearized car model, taken from L. + # Jaulin, Automatique pour la robotique, Cours et Exercices, iSTE + # editions p 184/185 + A = np.array([[0, 7, 0, 0], + [0, 0, 0, 7/3.], + [0, 0, 0, 0], + [0, 0, 0, 0]]) + B = np.array([[0, 0], + [0, 0], + [1, 0], + [0, 1]]) + # Test complex poles on YT + P = np.array([-3, -1, -2-1j, -2+1j]) + # on macOS arm64 this can lead to a RuntimeWarning invalid + # value in divide, so suppress it for now + with np.errstate(divide='ignore', invalid='ignore'): + self._check(A, B, P) + + # Try to reach the specific case in _YT_complex where two singular + # values are almost equal. This is to improve code coverage but I + # have no way to be sure this code is really reached + + P = [0-1e-6j,0+1e-6j,-10,10] + with np.errstate(divide='ignore', invalid='ignore'): + self._check(A, B, P, maxiter=1000) + + # Try to reach the specific case in _YT_complex where the rank two + # update yields two null vectors. This test was found via Monte Carlo. + + A = np.array( + [-2148,-2902, -2267, -598, -1722, -1829, -165, -283, -2546, + -167, -754, -2285, -543, -1700, -584, -2978, -925, -1300, + -1583, -984, -386, -2650, -764, -897, -517, -1598, 2, -1709, + -291, -338, -153, -1804, -1106, -1168, -867, -2297] + ).reshape(6,6) + + B = np.array( + [-108, -374, -524, -1285, -1232, -161, -1204, -672, -637, + -15, -483, -23, -931, -780, -1245, -1129, -1290, -1502, + -952, -1374, -62, -964, -930, -939, -792, -756, -1437, + -491, -1543, -686] + ).reshape(6,5) + P = [-25.-29.j, -25.+29.j, 31.-42.j, 31.+42.j, 33.-41.j, 33.+41.j] + self._check(A, B, P) + + # Use a lot of poles to go through all cases for update_order + # in _YT_loop + + big_A = np.ones((11,11))-np.eye(11) + big_B = np.ones((11,10))-np.diag([1]*10,1)[:,1:] + big_A[:6,:6] = A + big_B[:6,:5] = B + + P = [-10,-20,-30,40,50,60,70,-20-5j,-20+5j,5+3j,5-3j] + with np.errstate(divide='ignore', invalid='ignore'): + self._check(big_A, big_B, P) + + #check with only complex poles and only real poles + P = [-10,-20,-30,-40,-50,-60,-70,-80,-90,-100] + self._check(big_A[:-1,:-1], big_B[:-1,:-1], P) + P = [-10+10j,-20+20j,-30+30j,-40+40j,-50+50j, + -10-10j,-20-20j,-30-30j,-40-40j,-50-50j] + self._check(big_A[:-1,:-1], big_B[:-1,:-1], P) + + # need a 5x5 array to ensure YT handles properly when there + # is only one real pole and several complex + A = np.array([0,7,0,0,0,0,0,7/3.,0,0,0,0,0,0,0,0, + 0,0,0,5,0,0,0,0,9]).reshape(5,5) + B = np.array([0,0,0,0,1,0,0,1,2,3]).reshape(5,2) + P = np.array([-2, -3+1j, -3-1j, -1+1j, -1-1j]) + with np.errstate(divide='ignore', invalid='ignore'): + place_poles(A, B, P) + + # same test with an odd number of real poles > 1 + # this is another specific case of YT + P = np.array([-2, -3, -4, -1+1j, -1-1j]) + with np.errstate(divide='ignore', invalid='ignore'): + self._check(A, B, P) + + def test_tricky_B(self): + # check we handle as we should the 1 column B matrices and + # n column B matrices (with n such as shape(A)=(n, n)) + A = np.array([1.380, -0.2077, 6.715, -5.676, -0.5814, -4.290, 0, + 0.6750, 1.067, 4.273, -6.654, 5.893, 0.0480, 4.273, + 1.343, -2.104]).reshape(4, 4) + B = np.array([0, 5.679, 1.136, 1.136, 0, 0, -3.146, 0, 1, 2, 3, 4, + 5, 6, 7, 8]).reshape(4, 4) + + # KNV or YT are not called here, it's a specific case with only + # one unique solution + P = np.array([-0.2, -0.5, -5.0566, -8.6659]) + fsf = self._check(A, B, P) + # rtol and nb_iter should be set to np.nan as the identity can be + # used as transfer matrix + assert np.isnan(fsf.rtol) + assert np.isnan(fsf.nb_iter) + + # check with complex poles too as they trigger a specific case in + # the specific case :-) + P = np.array((-2+1j,-2-1j,-3,-2)) + fsf = self._check(A, B, P) + assert np.isnan(fsf.rtol) + assert np.isnan(fsf.nb_iter) + + #now test with a B matrix with only one column (no optimisation) + B = B[:,0].reshape(4,1) + P = np.array((-2+1j,-2-1j,-3,-2)) + fsf = self._check(A, B, P) + + # we can't optimize anything, check they are set to 0 as expected + assert fsf.rtol == 0 + assert fsf.nb_iter == 0 + + @pytest.mark.thread_unsafe + def test_errors(self): + # Test input mistakes from user + A = np.array([0,7,0,0,0,0,0,7/3.,0,0,0,0,0,0,0,0]).reshape(4,4) + B = np.array([0,0,0,0,1,0,0,1]).reshape(4,2) + + #should fail as the method keyword is invalid + assert_raises(ValueError, place_poles, A, B, (-2.1,-2.2,-2.3,-2.4), + method="foo") + + #should fail as poles are not 1D array + assert_raises(ValueError, place_poles, A, B, + np.array((-2.1,-2.2,-2.3,-2.4)).reshape(4,1)) + + #should fail as A is not a 2D array + assert_raises(ValueError, place_poles, A[:,:,np.newaxis], B, + (-2.1,-2.2,-2.3,-2.4)) + + #should fail as B is not a 2D array + assert_raises(ValueError, place_poles, A, B[:,:,np.newaxis], + (-2.1,-2.2,-2.3,-2.4)) + + #should fail as there are too many poles + assert_raises(ValueError, place_poles, A, B, (-2.1,-2.2,-2.3,-2.4,-3)) + + #should fail as there are not enough poles + assert_raises(ValueError, place_poles, A, B, (-2.1,-2.2,-2.3)) + + #should fail as the rtol is greater than 1 + assert_raises(ValueError, place_poles, A, B, (-2.1,-2.2,-2.3,-2.4), + rtol=42) + + #should fail as maxiter is smaller than 1 + assert_raises(ValueError, place_poles, A, B, (-2.1,-2.2,-2.3,-2.4), + maxiter=-42) + + # should fail as ndim(B) is two + assert_raises(ValueError, place_poles, A, B, (-2,-2,-2,-2)) + + # uncontrollable system + assert_raises(ValueError, place_poles, np.ones((4,4)), + np.ones((4,2)), (1,2,3,4)) + + # Should not raise ValueError as the poles can be placed but should + # raise a warning as the convergence is not reached + with warnings.catch_warnings(record=True) as w: + warnings.simplefilter("always") + fsf = place_poles(A, B, (-1,-2,-3,-4), rtol=1e-16, maxiter=42) + assert len(w) == 1 + assert issubclass(w[-1].category, UserWarning) + assert ("Convergence was not reached after maxiter iterations" + in str(w[-1].message)) + assert fsf.nb_iter == 42 + + # should fail as a complex misses its conjugate + assert_raises(ValueError, place_poles, A, B, (-2+1j,-2-1j,-2+3j,-2)) + + # should fail as A is not square + assert_raises(ValueError, place_poles, A[:,:3], B, (-2,-3,-4,-5)) + + # should fail as B has not the same number of lines as A + assert_raises(ValueError, place_poles, A, B[:3,:], (-2,-3,-4,-5)) + + # should fail as KNV0 does not support complex poles + assert_raises(ValueError, place_poles, A, B, + (-2+1j,-2-1j,-2+3j,-2-3j), method="KNV0") + + +class TestSS2TF: + + def check_matrix_shapes(self, p, q, r): + ss2tf(np.zeros((p, p)), + np.zeros((p, q)), + np.zeros((r, p)), + np.zeros((r, q)), 0) + + def test_shapes(self): + # Each tuple holds: + # number of states, number of inputs, number of outputs + for p, q, r in [(3, 3, 3), (1, 3, 3), (1, 1, 1)]: + self.check_matrix_shapes(p, q, r) + + def test_basic(self): + # Test a round trip through tf2ss and ss2tf. + b = np.array([1.0, 3.0, 5.0]) + a = np.array([1.0, 2.0, 3.0]) + + A, B, C, D = tf2ss(b, a) + xp_assert_close(A, [[-2., -3], [1, 0]], rtol=1e-13) + xp_assert_close(B, [[1.], [0]], rtol=1e-13) + xp_assert_close(C, [[1., 2]], rtol=1e-13) + xp_assert_close(D, [[1.]], rtol=1e-14) + + bb, aa = ss2tf(A, B, C, D) + xp_assert_close(bb[0], b, rtol=1e-13) + xp_assert_close(aa, a, rtol=1e-13) + + def test_zero_order_round_trip(self): + # See gh-5760 + tf = (2, 1) + A, B, C, D = tf2ss(*tf) + xp_assert_close(A, [[0.]], rtol=1e-13) + xp_assert_close(B, [[0.]], rtol=1e-13) + xp_assert_close(C, [[0.]], rtol=1e-13) + xp_assert_close(D, [[2.]], rtol=1e-13) + + num, den = ss2tf(A, B, C, D) + xp_assert_close(num, [[2., 0]], rtol=1e-13) + xp_assert_close(den, [1., 0], rtol=1e-13) + + tf = ([[5], [2]], 1) + A, B, C, D = tf2ss(*tf) + xp_assert_close(A, [[0.]], rtol=1e-13) + xp_assert_close(B, [[0.]], rtol=1e-13) + xp_assert_close(C, [[0.], [0]], rtol=1e-13) + xp_assert_close(D, [[5.], [2]], rtol=1e-13) + + num, den = ss2tf(A, B, C, D) + xp_assert_close(num, [[5., 0], [2, 0]], rtol=1e-13) + xp_assert_close(den, [1., 0], rtol=1e-13) + + def test_simo_round_trip(self): + # See gh-5753 + tf = ([[1, 2], [1, 1]], [1, 2]) + A, B, C, D = tf2ss(*tf) + xp_assert_close(A, [[-2.]], rtol=1e-13) + xp_assert_close(B, [[1.]], rtol=1e-13) + xp_assert_close(C, [[0.], [-1.]], rtol=1e-13) + xp_assert_close(D, [[1.], [1.]], rtol=1e-13) + + num, den = ss2tf(A, B, C, D) + xp_assert_close(num, [[1., 2.], [1., 1.]], rtol=1e-13) + xp_assert_close(den, [1., 2.], rtol=1e-13) + + tf = ([[1, 0, 1], [1, 1, 1]], [1, 1, 1]) + A, B, C, D = tf2ss(*tf) + xp_assert_close(A, [[-1., -1.], [1., 0.]], rtol=1e-13) + xp_assert_close(B, [[1.], [0.]], rtol=1e-13) + xp_assert_close(C, [[-1., 0.], [0., 0.]], rtol=1e-13) + xp_assert_close(D, [[1.], [1.]], rtol=1e-13) + + num, den = ss2tf(A, B, C, D) + xp_assert_close(num, [[1., 0., 1.], [1., 1., 1.]], rtol=1e-13) + xp_assert_close(den, [1., 1., 1.], rtol=1e-13) + + tf = ([[1, 2, 3], [1, 2, 3]], [1, 2, 3, 4]) + A, B, C, D = tf2ss(*tf) + xp_assert_close(A, [[-2., -3, -4], [1, 0, 0], [0, 1, 0]], rtol=1e-13) + xp_assert_close(B, [[1.], [0], [0]], rtol=1e-13) + xp_assert_close(C, [[1., 2, 3], [1, 2, 3]], rtol=1e-13) + xp_assert_close(D, [[0.], [0]], rtol=1e-13) + + num, den = ss2tf(A, B, C, D) + xp_assert_close(num, [[0., 1, 2, 3], [0, 1, 2, 3]], rtol=1e-13) + xp_assert_close(den, [1., 2, 3, 4], rtol=1e-13) + + tf = (np.array([1, [2, 3]], dtype=object), [1, 6]) + A, B, C, D = tf2ss(*tf) + xp_assert_close(A, [[-6.]], rtol=1e-31) + xp_assert_close(B, [[1.]], rtol=1e-31) + xp_assert_close(C, [[1.], [-9]], rtol=1e-31) + xp_assert_close(D, [[0.], [2]], rtol=1e-31) + + num, den = ss2tf(A, B, C, D) + xp_assert_close(num, [[0., 1], [2, 3]], rtol=1e-13) + xp_assert_close(den, [1., 6], rtol=1e-13) + + tf = (np.array([[1, -3], [1, 2, 3]], dtype=object), [1, 6, 5]) + A, B, C, D = tf2ss(*tf) + xp_assert_close(A, [[-6., -5], [1, 0]], rtol=1e-13) + xp_assert_close(B, [[1.], [0]], rtol=1e-13) + xp_assert_close(C, [[1., -3], [-4, -2]], rtol=1e-13) + xp_assert_close(D, [[0.], [1]], rtol=1e-13) + + num, den = ss2tf(A, B, C, D) + xp_assert_close(num, [[0., 1, -3], [1, 2, 3]], rtol=1e-13) + xp_assert_close(den, [1., 6, 5], rtol=1e-13) + + def test_all_int_arrays(self): + A = [[0, 1, 0], [0, 0, 1], [-3, -4, -2]] + B = [[0], [0], [1]] + C = [[5, 1, 0]] + D = [[0]] + num, den = ss2tf(A, B, C, D) + xp_assert_close(num, [[0.0, 0.0, 1.0, 5.0]], rtol=1e-13, atol=1e-14) + xp_assert_close(den, [1.0, 2.0, 4.0, 3.0], rtol=1e-13) + + def test_multioutput(self): + # Regression test for gh-2669. + + # 4 states + A = np.array([[-1.0, 0.0, 1.0, 0.0], + [-1.0, 0.0, 2.0, 0.0], + [-4.0, 0.0, 3.0, 0.0], + [-8.0, 8.0, 0.0, 4.0]]) + + # 1 input + B = np.array([[0.3], + [0.0], + [7.0], + [0.0]]) + + # 3 outputs + C = np.array([[0.0, 1.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 1.0], + [8.0, 8.0, 0.0, 0.0]]) + + D = np.array([[0.0], + [0.0], + [1.0]]) + + # Get the transfer functions for all the outputs in one call. + b_all, a = ss2tf(A, B, C, D) + + # Get the transfer functions for each output separately. + b0, a0 = ss2tf(A, B, C[0], D[0]) + b1, a1 = ss2tf(A, B, C[1], D[1]) + b2, a2 = ss2tf(A, B, C[2], D[2]) + + # Check that we got the same results. + xp_assert_close(a0, a, rtol=1e-13) + xp_assert_close(a1, a, rtol=1e-13) + xp_assert_close(a2, a, rtol=1e-13) + xp_assert_close(b_all, np.vstack((b0, b1, b2)), rtol=1e-13, atol=1e-14) + + +class TestLsim: + digits_accuracy = 7 + + def lti_nowarn(self, *args): + with suppress_warnings() as sup: + sup.filter(BadCoefficients) + system = lti(*args) + return system + + def test_first_order(self): + # y' = -y + # exact solution is y(t) = exp(-t) + system = self.lti_nowarn(-1.,1.,1.,0.) + t = np.linspace(0,5) + u = np.zeros_like(t) + tout, y, x = lsim(system, u, t, X0=[1.0]) + expected_x = np.exp(-tout) + assert_almost_equal(x, expected_x) + assert_almost_equal(y, expected_x) + + def test_second_order(self): + t = np.linspace(0, 10, 1001) + u = np.zeros_like(t) + # Second order system with a repeated root: x''(t) + 2*x(t) + x(t) = 0. + # With initial conditions x(0)=1.0 and x'(t)=0.0, the exact solution + # is (1-t)*exp(-t). + system = self.lti_nowarn([1.0], [1.0, 2.0, 1.0]) + tout, y, x = lsim(system, u, t, X0=[1.0, 0.0]) + expected_x = (1.0 - tout) * np.exp(-tout) + assert_almost_equal(x[:, 0], expected_x) + + def test_integrator(self): + # integrator: y' = u + system = self.lti_nowarn(0., 1., 1., 0.) + t = np.linspace(0,5) + u = t + tout, y, x = lsim(system, u, t) + expected_x = 0.5 * tout**2 + assert_almost_equal(x, expected_x, decimal=self.digits_accuracy) + assert_almost_equal(y, expected_x, decimal=self.digits_accuracy) + + def test_two_states(self): + # A system with two state variables, two inputs, and one output. + A = np.array([[-1.0, 0.0], [0.0, -2.0]]) + B = np.array([[1.0, 0.0], [0.0, 1.0]]) + C = np.array([1.0, 0.0]) + D = np.zeros((1, 2)) + + system = self.lti_nowarn(A, B, C, D) + + t = np.linspace(0, 10.0, 21) + u = np.zeros((len(t), 2)) + tout, y, x = lsim(system, U=u, T=t, X0=[1.0, 1.0]) + expected_y = np.exp(-tout) + expected_x0 = np.exp(-tout) + expected_x1 = np.exp(-2.0 * tout) + assert_almost_equal(y, expected_y) + assert_almost_equal(x[:, 0], expected_x0) + assert_almost_equal(x[:, 1], expected_x1) + + def test_double_integrator(self): + # double integrator: y'' = 2u + A = np.array([[0., 1.], [0., 0.]]) + B = np.array([[0.], [1.]]) + C = np.array([[2., 0.]]) + system = self.lti_nowarn(A, B, C, 0.) + t = np.linspace(0,5) + u = np.ones_like(t) + tout, y, x = lsim(system, u, t) + expected_x = np.transpose(np.array([0.5 * tout**2, tout])) + expected_y = tout**2 + assert_almost_equal(x, expected_x, decimal=self.digits_accuracy) + assert_almost_equal(y, expected_y, decimal=self.digits_accuracy) + + def test_jordan_block(self): + # Non-diagonalizable A matrix + # x1' + x1 = x2 + # x2' + x2 = u + # y = x1 + # Exact solution with u = 0 is y(t) = t exp(-t) + A = np.array([[-1., 1.], [0., -1.]]) + B = np.array([[0.], [1.]]) + C = np.array([[1., 0.]]) + system = self.lti_nowarn(A, B, C, 0.) + t = np.linspace(0,5) + u = np.zeros_like(t) + tout, y, x = lsim(system, u, t, X0=[0.0, 1.0]) + expected_y = tout * np.exp(-tout) + assert_almost_equal(y, expected_y) + + def test_miso(self): + # A system with two state variables, two inputs, and one output. + A = np.array([[-1.0, 0.0], [0.0, -2.0]]) + B = np.array([[1.0, 0.0], [0.0, 1.0]]) + C = np.array([1.0, 0.0]) + D = np.zeros((1,2)) + system = self.lti_nowarn(A, B, C, D) + + t = np.linspace(0, 5.0, 101) + u = np.zeros((len(t), 2)) + tout, y, x = lsim(system, u, t, X0=[1.0, 1.0]) + expected_y = np.exp(-tout) + expected_x0 = np.exp(-tout) + expected_x1 = np.exp(-2.0*tout) + assert_almost_equal(y, expected_y) + assert_almost_equal(x[:,0], expected_x0) + assert_almost_equal(x[:,1], expected_x1) + + def test_nonzero_initial_time(self): + system = self.lti_nowarn(-1.,1.,1.,0.) + t = np.linspace(1,2) + u = np.zeros_like(t) + tout, y, x = lsim(system, u, t, X0=[1.0]) + expected_y = np.exp(-tout) + assert_almost_equal(y, expected_y) + + def test_nonequal_timesteps(self): + t = np.array([0.0, 1.0, 1.0, 3.0]) + u = np.array([0.0, 0.0, 1.0, 1.0]) + # Simple integrator: x'(t) = u(t) + system = ([1.0], [1.0, 0.0]) + with assert_raises(ValueError, + match="Time steps are not equally spaced."): + tout, y, x = lsim(system, u, t, X0=[1.0]) + + +class TestImpulse: + def test_first_order(self): + # First order system: x'(t) + x(t) = u(t) + # Exact impulse response is x(t) = exp(-t). + system = ([1.0], [1.0,1.0]) + tout, y = impulse(system) + expected_y = np.exp(-tout) + assert_almost_equal(y, expected_y) + + def test_first_order_fixed_time(self): + # Specify the desired time values for the output. + + # First order system: x'(t) + x(t) = u(t) + # Exact impulse response is x(t) = exp(-t). + system = ([1.0], [1.0,1.0]) + n = 21 + t = np.linspace(0, 2.0, n) + tout, y = impulse(system, T=t) + assert tout.shape == (n,) + assert_almost_equal(tout, t) + expected_y = np.exp(-t) + assert_almost_equal(y, expected_y) + + def test_first_order_initial(self): + # Specify an initial condition as a scalar. + + # First order system: x'(t) + x(t) = u(t), x(0)=3.0 + # Exact impulse response is x(t) = 4*exp(-t). + system = ([1.0], [1.0,1.0]) + tout, y = impulse(system, X0=3.0) + expected_y = 4.0 * np.exp(-tout) + assert_almost_equal(y, expected_y) + + def test_first_order_initial_list(self): + # Specify an initial condition as a list. + + # First order system: x'(t) + x(t) = u(t), x(0)=3.0 + # Exact impulse response is x(t) = 4*exp(-t). + system = ([1.0], [1.0,1.0]) + tout, y = impulse(system, X0=[3.0]) + expected_y = 4.0 * np.exp(-tout) + assert_almost_equal(y, expected_y) + + def test_integrator(self): + # Simple integrator: x'(t) = u(t) + system = ([1.0], [1.0,0.0]) + tout, y = impulse(system) + expected_y = np.ones_like(tout) + assert_almost_equal(y, expected_y) + + def test_second_order(self): + # Second order system with a repeated root: + # x''(t) + 2*x(t) + x(t) = u(t) + # The exact impulse response is t*exp(-t). + system = ([1.0], [1.0, 2.0, 1.0]) + tout, y = impulse(system) + expected_y = tout * np.exp(-tout) + assert_almost_equal(y, expected_y) + + def test_array_like(self): + # Test that function can accept sequences, scalars. + system = ([1.0], [1.0, 2.0, 1.0]) + # TODO: add meaningful test where X0 is a list + tout, y = impulse(system, X0=[3], T=[5, 6]) + tout, y = impulse(system, X0=[3], T=[5]) + + def test_array_like2(self): + system = ([1.0], [1.0, 2.0, 1.0]) + tout, y = impulse(system, X0=3, T=5) + + +class TestStep: + def test_first_order(self): + # First order system: x'(t) + x(t) = u(t) + # Exact step response is x(t) = 1 - exp(-t). + system = ([1.0], [1.0,1.0]) + tout, y = step(system) + expected_y = 1.0 - np.exp(-tout) + assert_almost_equal(y, expected_y) + + def test_first_order_fixed_time(self): + # Specify the desired time values for the output. + + # First order system: x'(t) + x(t) = u(t) + # Exact step response is x(t) = 1 - exp(-t). + system = ([1.0], [1.0,1.0]) + n = 21 + t = np.linspace(0, 2.0, n) + tout, y = step(system, T=t) + assert tout.shape == (n,) + assert_almost_equal(tout, t) + expected_y = 1 - np.exp(-t) + assert_almost_equal(y, expected_y) + + def test_first_order_initial(self): + # Specify an initial condition as a scalar. + + # First order system: x'(t) + x(t) = u(t), x(0)=3.0 + # Exact step response is x(t) = 1 + 2*exp(-t). + system = ([1.0], [1.0,1.0]) + tout, y = step(system, X0=3.0) + expected_y = 1 + 2.0*np.exp(-tout) + assert_almost_equal(y, expected_y) + + def test_first_order_initial_list(self): + # Specify an initial condition as a list. + + # First order system: x'(t) + x(t) = u(t), x(0)=3.0 + # Exact step response is x(t) = 1 + 2*exp(-t). + system = ([1.0], [1.0,1.0]) + tout, y = step(system, X0=[3.0]) + expected_y = 1 + 2.0*np.exp(-tout) + assert_almost_equal(y, expected_y) + + def test_integrator(self): + # Simple integrator: x'(t) = u(t) + # Exact step response is x(t) = t. + system = ([1.0],[1.0,0.0]) + tout, y = step(system) + expected_y = tout + assert_almost_equal(y, expected_y) + + def test_second_order(self): + # Second order system with a repeated root: + # x''(t) + 2*x(t) + x(t) = u(t) + # The exact step response is 1 - (1 + t)*exp(-t). + system = ([1.0], [1.0, 2.0, 1.0]) + tout, y = step(system) + expected_y = 1 - (1 + tout) * np.exp(-tout) + assert_almost_equal(y, expected_y) + + def test_array_like(self): + # Test that function can accept sequences, scalars. + system = ([1.0], [1.0, 2.0, 1.0]) + # TODO: add meaningful test where X0 is a list + tout, y = step(system, T=[5, 6]) + + def test_complex_input(self): + # Test that complex input doesn't raise an error. + # `step` doesn't seem to have been designed for complex input, but this + # works and may be used, so add regression test. See gh-2654. + step(([], [-1], 1+0j)) + + +class TestLti: + def test_lti_instantiation(self): + # Test that lti can be instantiated with sequences, scalars. + # See PR-225. + + # TransferFunction + s = lti([1], [-1]) + assert isinstance(s, TransferFunction) + assert isinstance(s, lti) + assert not isinstance(s, dlti) + assert s.dt is None + + # ZerosPolesGain + s = lti(np.array([]), np.array([-1]), 1) + assert isinstance(s, ZerosPolesGain) + assert isinstance(s, lti) + assert not isinstance(s, dlti) + assert s.dt is None + + # StateSpace + s = lti([], [-1], 1) + s = lti([1], [-1], 1, 3) + assert isinstance(s, StateSpace) + assert isinstance(s, lti) + assert not isinstance(s, dlti) + assert s.dt is None + + +class TestStateSpace: + def test_initialization(self): + # Check that all initializations work + StateSpace(1, 1, 1, 1) + StateSpace([1], [2], [3], [4]) + StateSpace(np.array([[1, 2], [3, 4]]), np.array([[1], [2]]), + np.array([[1, 0]]), np.array([[0]])) + + def test_conversion(self): + # Check the conversion functions + s = StateSpace(1, 2, 3, 4) + assert isinstance(s.to_ss(), StateSpace) + assert isinstance(s.to_tf(), TransferFunction) + assert isinstance(s.to_zpk(), ZerosPolesGain) + + # Make sure copies work + assert StateSpace(s) is not s + assert s.to_ss() is not s + + def test_properties(self): + # Test setters/getters for cross class properties. + # This implicitly tests to_tf() and to_zpk() + + # Getters + s = StateSpace(1, 1, 1, 1) + xp_assert_equal(s.poles, [1.]) + xp_assert_equal(s.zeros, [0.]) + assert s.dt is None + + def test_operators(self): + # Test +/-/* operators on systems + + class BadType: + pass + + s1 = StateSpace(np.array([[-0.5, 0.7], [0.3, -0.8]]), + np.array([[1], [0]]), + np.array([[1, 0]]), + np.array([[0]]), + ) + + s2 = StateSpace(np.array([[-0.2, -0.1], [0.4, -0.1]]), + np.array([[1], [0]]), + np.array([[1, 0]]), + np.array([[0]]) + ) + + s_discrete = s1.to_discrete(0.1) + s2_discrete = s2.to_discrete(0.2) + s3_discrete = s2.to_discrete(0.1) + + # Impulse response + t = np.linspace(0, 1, 100) + u = np.zeros_like(t) + u[0] = 1 + + # Test multiplication + for typ in (int, float, complex, np.float32, np.complex128, np.array): + xp_assert_close(lsim(typ(2) * s1, U=u, T=t)[1], + typ(2) * lsim(s1, U=u, T=t)[1]) + + xp_assert_close(lsim(s1 * typ(2), U=u, T=t)[1], + lsim(s1, U=u, T=t)[1] * typ(2)) + + xp_assert_close(lsim(s1 / typ(2), U=u, T=t)[1], + lsim(s1, U=u, T=t)[1] / typ(2)) + + with assert_raises(TypeError): + typ(2) / s1 + + xp_assert_close(lsim(s1 * 2, U=u, T=t)[1], + lsim(s1, U=2 * u, T=t)[1]) + + xp_assert_close(lsim(s1 * s2, U=u, T=t)[1], + lsim(s1, U=lsim(s2, U=u, T=t)[1], T=t)[1], + atol=1e-5) + + with assert_raises(TypeError): + s1 / s1 + + with assert_raises(TypeError): + s1 * s_discrete + + with assert_raises(TypeError): + # Check different discretization constants + s_discrete * s2_discrete + + with assert_raises(TypeError): + s1 * BadType() + + with assert_raises(TypeError): + BadType() * s1 + + with assert_raises(TypeError): + s1 / BadType() + + with assert_raises(TypeError): + BadType() / s1 + + # Test addition + xp_assert_close(lsim(s1 + 2, U=u, T=t)[1], + 2 * u + lsim(s1, U=u, T=t)[1]) + + # Check for dimension mismatch + with assert_raises(ValueError): + s1 + np.array([1, 2]) + + with assert_raises(ValueError): + np.array([1, 2]) + s1 + + with assert_raises(TypeError): + s1 + s_discrete + + with assert_raises(ValueError): + s1 / np.array([[1, 2], [3, 4]]) + + with assert_raises(TypeError): + # Check different discretization constants + s_discrete + s2_discrete + + with assert_raises(TypeError): + s1 + BadType() + + with assert_raises(TypeError): + BadType() + s1 + + xp_assert_close(lsim(s1 + s2, U=u, T=t)[1], + lsim(s1, U=u, T=t)[1] + lsim(s2, U=u, T=t)[1]) + + # Test subtraction + xp_assert_close(lsim(s1 - 2, U=u, T=t)[1], + -2 * u + lsim(s1, U=u, T=t)[1]) + + xp_assert_close(lsim(2 - s1, U=u, T=t)[1], + 2 * u + lsim(-s1, U=u, T=t)[1]) + + xp_assert_close(lsim(s1 - s2, U=u, T=t)[1], + lsim(s1, U=u, T=t)[1] - lsim(s2, U=u, T=t)[1]) + + with assert_raises(TypeError): + s1 - BadType() + + with assert_raises(TypeError): + BadType() - s1 + + s = s_discrete + s3_discrete + assert s.dt == 0.1 + + s = s_discrete * s3_discrete + assert s.dt == 0.1 + + s = 3 * s_discrete + assert s.dt == 0.1 + + s = -s_discrete + assert s.dt == 0.1 + +class TestTransferFunction: + def test_initialization(self): + # Check that all initializations work + TransferFunction(1, 1) + TransferFunction([1], [2]) + TransferFunction(np.array([1]), np.array([2])) + + def test_conversion(self): + # Check the conversion functions + s = TransferFunction([1, 0], [1, -1]) + assert isinstance(s.to_ss(), StateSpace) + assert isinstance(s.to_tf(), TransferFunction) + assert isinstance(s.to_zpk(), ZerosPolesGain) + + # Make sure copies work + assert TransferFunction(s) is not s + assert s.to_tf() is not s + + def test_properties(self): + # Test setters/getters for cross class properties. + # This implicitly tests to_ss() and to_zpk() + + # Getters + s = TransferFunction([1, 0], [1, -1]) + xp_assert_equal(s.poles, [1.]) + xp_assert_equal(s.zeros, [0.]) + + +class TestZerosPolesGain: + def test_initialization(self): + # Check that all initializations work + ZerosPolesGain(1, 1, 1) + ZerosPolesGain([1], [2], 1) + ZerosPolesGain(np.array([1]), np.array([2]), 1) + + def test_conversion(self): + #Check the conversion functions + s = ZerosPolesGain(1, 2, 3) + assert isinstance(s.to_ss(), StateSpace) + assert isinstance(s.to_tf(), TransferFunction) + assert isinstance(s.to_zpk(), ZerosPolesGain) + + # Make sure copies work + assert ZerosPolesGain(s) is not s + assert s.to_zpk() is not s + + +class Test_abcd_normalize: + def setup_method(self): + self.A = np.array([[1.0, 2.0], [3.0, 4.0]]) + self.B = np.array([[-1.0], [5.0]]) + self.C = np.array([[4.0, 5.0]]) + self.D = np.array([[2.5]]) + + def test_no_matrix_fails(self): + assert_raises(ValueError, abcd_normalize) + + def test_A_nosquare_fails(self): + assert_raises(ValueError, abcd_normalize, [1, -1], + self.B, self.C, self.D) + + def test_AB_mismatch_fails(self): + assert_raises(ValueError, abcd_normalize, self.A, [-1, 5], + self.C, self.D) + + def test_AC_mismatch_fails(self): + assert_raises(ValueError, abcd_normalize, self.A, self.B, + [[4.0], [5.0]], self.D) + + def test_CD_mismatch_fails(self): + assert_raises(ValueError, abcd_normalize, self.A, self.B, + self.C, [2.5, 0]) + + def test_BD_mismatch_fails(self): + assert_raises(ValueError, abcd_normalize, self.A, [-1, 5], + self.C, self.D) + + def test_normalized_matrices_unchanged(self): + A, B, C, D = abcd_normalize(self.A, self.B, self.C, self.D) + xp_assert_equal(A, self.A) + xp_assert_equal(B, self.B) + xp_assert_equal(C, self.C) + xp_assert_equal(D, self.D) + + def test_shapes(self): + A, B, C, D = abcd_normalize(self.A, self.B, [1, 0], 0) + xp_assert_equal(A.shape[0], A.shape[1]) + xp_assert_equal(A.shape[0], B.shape[0]) + xp_assert_equal(A.shape[0], C.shape[1]) + xp_assert_equal(C.shape[0], D.shape[0]) + xp_assert_equal(B.shape[1], D.shape[1]) + + def test_zero_dimension_is_not_none1(self): + B_ = np.zeros((2, 0)) + D_ = np.zeros((0, 0)) + A, B, C, D = abcd_normalize(A=self.A, B=B_, D=D_) + xp_assert_equal(A, self.A) + xp_assert_equal(B, B_) + xp_assert_equal(D, D_) + assert C.shape[0] == D_.shape[0] + assert C.shape[1] == self.A.shape[0] + + def test_zero_dimension_is_not_none2(self): + B_ = np.zeros((2, 0)) + C_ = np.zeros((0, 2)) + A, B, C, D = abcd_normalize(A=self.A, B=B_, C=C_) + xp_assert_equal(A, self.A) + xp_assert_equal(B, B_) + xp_assert_equal(C, C_) + assert D.shape[0] == C_.shape[0] + assert D.shape[1] == B_.shape[1] + + def test_missing_A(self): + A, B, C, D = abcd_normalize(B=self.B, C=self.C, D=self.D) + assert A.shape[0] == A.shape[1] + assert A.shape[0] == B.shape[0] + assert A.shape == (self.B.shape[0], self.B.shape[0]) + + def test_missing_B(self): + A, B, C, D = abcd_normalize(A=self.A, C=self.C, D=self.D) + assert B.shape[0] == A.shape[0] + assert B.shape[1] == D.shape[1] + assert B.shape == (self.A.shape[0], self.D.shape[1]) + + def test_missing_C(self): + A, B, C, D = abcd_normalize(A=self.A, B=self.B, D=self.D) + assert C.shape[0] == D.shape[0] + assert C.shape[1] == A.shape[0] + assert C.shape == (self.D.shape[0], self.A.shape[0]) + + def test_missing_D(self): + A, B, C, D = abcd_normalize(A=self.A, B=self.B, C=self.C) + assert D.shape[0] == C.shape[0] + assert D.shape[1] == B.shape[1] + assert D.shape == (self.C.shape[0], self.B.shape[1]) + + def test_missing_AB(self): + A, B, C, D = abcd_normalize(C=self.C, D=self.D) + assert A.shape[0] == A.shape[1] + assert A.shape[0] == B.shape[0] + assert B.shape[1] == D.shape[1] + assert A.shape == (self.C.shape[1], self.C.shape[1]) + assert B.shape == (self.C.shape[1], self.D.shape[1]) + + def test_missing_AC(self): + A, B, C, D = abcd_normalize(B=self.B, D=self.D) + assert A.shape[0] == A.shape[1] + assert A.shape[0] == B.shape[0] + assert C.shape[0] == D.shape[0] + assert C.shape[1] == A.shape[0] + assert A.shape == (self.B.shape[0], self.B.shape[0]) + assert C.shape == (self.D.shape[0], self.B.shape[0]) + + def test_missing_AD(self): + A, B, C, D = abcd_normalize(B=self.B, C=self.C) + assert A.shape[0] == A.shape[1] + assert A.shape[0] == B.shape[0] + assert D.shape[0] == C.shape[0] + assert D.shape[1] == B.shape[1] + assert A.shape == (self.B.shape[0], self.B.shape[0]) + assert D.shape == (self.C.shape[0], self.B.shape[1]) + + def test_missing_BC(self): + A, B, C, D = abcd_normalize(A=self.A, D=self.D) + assert B.shape[0] == A.shape[0] + assert B.shape[1] == D.shape[1] + assert C.shape[0] == D.shape[0] + assert C.shape[1], A.shape[0] + assert B.shape == (self.A.shape[0], self.D.shape[1]) + assert C.shape == (self.D.shape[0], self.A.shape[0]) + + def test_missing_ABC_fails(self): + assert_raises(ValueError, abcd_normalize, D=self.D) + + def test_missing_BD_fails(self): + assert_raises(ValueError, abcd_normalize, A=self.A, C=self.C) + + def test_missing_CD_fails(self): + assert_raises(ValueError, abcd_normalize, A=self.A, B=self.B) + + +class Test_bode: + + def test_01(self): + # Test bode() magnitude calculation (manual sanity check). + # 1st order low-pass filter: H(s) = 1 / (s + 1), + # cutoff: 1 rad/s, slope: -20 dB/decade + # H(s=0.1) ~= 0 dB + # H(s=1) ~= -3 dB + # H(s=10) ~= -20 dB + # H(s=100) ~= -40 dB + system = lti([1], [1, 1]) + w = [0.1, 1, 10, 100] + w, mag, phase = bode(system, w=w) + expected_mag = [0, -3, -20, -40] + assert_almost_equal(mag, expected_mag, decimal=1) + + def test_02(self): + # Test bode() phase calculation (manual sanity check). + # 1st order low-pass filter: H(s) = 1 / (s + 1), + # angle(H(s=0.1)) ~= -5.7 deg + # angle(H(s=1)) ~= -45 deg + # angle(H(s=10)) ~= -84.3 deg + system = lti([1], [1, 1]) + w = [0.1, 1, 10] + w, mag, phase = bode(system, w=w) + expected_phase = [-5.7, -45, -84.3] + assert_almost_equal(phase, expected_phase, decimal=1) + + def test_03(self): + # Test bode() magnitude calculation. + # 1st order low-pass filter: H(s) = 1 / (s + 1) + system = lti([1], [1, 1]) + w = [0.1, 1, 10, 100] + w, mag, phase = bode(system, w=w) + jw = w * 1j + y = np.polyval(system.num, jw) / np.polyval(system.den, jw) + expected_mag = 20.0 * np.log10(abs(y)) + assert_almost_equal(mag, expected_mag) + + def test_04(self): + # Test bode() phase calculation. + # 1st order low-pass filter: H(s) = 1 / (s + 1) + system = lti([1], [1, 1]) + w = [0.1, 1, 10, 100] + w, mag, phase = bode(system, w=w) + jw = w * 1j + y = np.polyval(system.num, jw) / np.polyval(system.den, jw) + expected_phase = np.arctan2(y.imag, y.real) * 180.0 / np.pi + assert_almost_equal(phase, expected_phase) + + def test_05(self): + # Test that bode() finds a reasonable frequency range. + # 1st order low-pass filter: H(s) = 1 / (s + 1) + system = lti([1], [1, 1]) + n = 10 + # Expected range is from 0.01 to 10. + expected_w = np.logspace(-2, 1, n) + w, mag, phase = bode(system, n=n) + assert_almost_equal(w, expected_w) + + def test_06(self): + # Test that bode() doesn't fail on a system with a pole at 0. + # integrator, pole at zero: H(s) = 1 / s + system = lti([1], [1, 0]) + w, mag, phase = bode(system, n=2) + assert w[0] == 0.01 # a fail would give not-a-number + + def test_07(self): + # bode() should not fail on a system with pure imaginary poles. + # The test passes if bode doesn't raise an exception. + system = lti([1], [1, 0, 100]) + w, mag, phase = bode(system, n=2) + + def test_08(self): + # Test that bode() return continuous phase, issues/2331. + system = lti([], [-10, -30, -40, -60, -70], 1) + w, mag, phase = system.bode(w=np.logspace(-3, 40, 100)) + assert_almost_equal(min(phase), -450, decimal=15) + + def test_from_state_space(self): + # Ensure that bode works with a system that was created from the + # state space representation matrices A, B, C, D. In this case, + # system.num will be a 2-D array with shape (1, n+1), where (n,n) + # is the shape of A. + # A Butterworth lowpass filter is used, so we know the exact + # frequency response. + a = np.array([1.0, 2.0, 2.0, 1.0]) + A = linalg.companion(a).T + B = np.array([[0.0], [0.0], [1.0]]) + C = np.array([[1.0, 0.0, 0.0]]) + D = np.array([[0.0]]) + with suppress_warnings() as sup: + sup.filter(BadCoefficients) + system = lti(A, B, C, D) + w, mag, phase = bode(system, n=100) + + expected_magnitude = 20 * np.log10(np.sqrt(1.0 / (1.0 + w**6))) + assert_almost_equal(mag, expected_magnitude) + + +class Test_freqresp: + + def test_output_manual(self): + # Test freqresp() output calculation (manual sanity check). + # 1st order low-pass filter: H(s) = 1 / (s + 1), + # re(H(s=0.1)) ~= 0.99 + # re(H(s=1)) ~= 0.5 + # re(H(s=10)) ~= 0.0099 + system = lti([1], [1, 1]) + w = [0.1, 1, 10] + w, H = freqresp(system, w=w) + expected_re = [0.99, 0.5, 0.0099] + expected_im = [-0.099, -0.5, -0.099] + assert_almost_equal(H.real, expected_re, decimal=1) + assert_almost_equal(H.imag, expected_im, decimal=1) + + def test_output(self): + # Test freqresp() output calculation. + # 1st order low-pass filter: H(s) = 1 / (s + 1) + system = lti([1], [1, 1]) + w = [0.1, 1, 10, 100] + w, H = freqresp(system, w=w) + s = w * 1j + expected = np.polyval(system.num, s) / np.polyval(system.den, s) + assert_almost_equal(H.real, expected.real) + assert_almost_equal(H.imag, expected.imag) + + def test_freq_range(self): + # Test that freqresp() finds a reasonable frequency range. + # 1st order low-pass filter: H(s) = 1 / (s + 1) + # Expected range is from 0.01 to 10. + system = lti([1], [1, 1]) + n = 10 + expected_w = np.logspace(-2, 1, n) + w, H = freqresp(system, n=n) + assert_almost_equal(w, expected_w) + + def test_pole_zero(self): + # Test that freqresp() doesn't fail on a system with a pole at 0. + # integrator, pole at zero: H(s) = 1 / s + system = lti([1], [1, 0]) + w, H = freqresp(system, n=2) + assert w[0] == 0.01 # a fail would give not-a-number + + def test_from_state_space(self): + # Ensure that freqresp works with a system that was created from the + # state space representation matrices A, B, C, D. In this case, + # system.num will be a 2-D array with shape (1, n+1), where (n,n) is + # the shape of A. + # A Butterworth lowpass filter is used, so we know the exact + # frequency response. + a = np.array([1.0, 2.0, 2.0, 1.0]) + A = linalg.companion(a).T + B = np.array([[0.0],[0.0],[1.0]]) + C = np.array([[1.0, 0.0, 0.0]]) + D = np.array([[0.0]]) + with suppress_warnings() as sup: + sup.filter(BadCoefficients) + system = lti(A, B, C, D) + w, H = freqresp(system, n=100) + s = w * 1j + expected = (1.0 / (1.0 + 2*s + 2*s**2 + s**3)) + assert_almost_equal(H.real, expected.real) + assert_almost_equal(H.imag, expected.imag) + + def test_from_zpk(self): + # 4th order low-pass filter: H(s) = 1 / (s + 1) + system = lti([],[-1]*4,[1]) + w = [0.1, 1, 10, 100] + w, H = freqresp(system, w=w) + s = w * 1j + expected = 1 / (s + 1)**4 + assert_almost_equal(H.real, expected.real) + assert_almost_equal(H.imag, expected.imag) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_max_len_seq.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_max_len_seq.py new file mode 100644 index 0000000000000000000000000000000000000000..7610b3f898571d10d75a64f00d900168c7142fbe --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_max_len_seq.py @@ -0,0 +1,71 @@ +import numpy as np +from pytest import raises as assert_raises +from scipy._lib._array_api import xp_assert_close, xp_assert_equal + +from numpy.fft import fft, ifft + +from scipy.signal import max_len_seq + + +class TestMLS: + + def test_mls_inputs(self): + # can't all be zero state + assert_raises(ValueError, max_len_seq, + 10, state=np.zeros(10)) + # wrong size state + assert_raises(ValueError, max_len_seq, 10, + state=np.ones(3)) + # wrong length + assert_raises(ValueError, max_len_seq, 10, length=-1) + xp_assert_equal(max_len_seq(10, length=0)[0], + np.asarray([], dtype=np.int8) + ) + # unknown taps + assert_raises(ValueError, max_len_seq, 64) + # bad taps + assert_raises(ValueError, max_len_seq, 10, taps=[-1, 1]) + + def test_mls_output(self): + # define some alternate working taps + alt_taps = {2: [1], 3: [2], 4: [3], 5: [4, 3, 2], 6: [5, 4, 1], 7: [4], + 8: [7, 5, 3]} + # assume the other bit levels work, too slow to test higher orders... + for nbits in range(2, 8): + for state in [None, np.round(np.random.rand(nbits))]: + for taps in [None, alt_taps[nbits]]: + if state is not None and np.all(state == 0): + state[0] = 1 # they can't all be zero + orig_m = max_len_seq(nbits, state=state, + taps=taps)[0] + m = 2. * orig_m - 1. # convert to +/- 1 representation + # First, make sure we got all 1's or -1 + err_msg = "mls had non binary terms" + xp_assert_equal(np.abs(m), np.ones_like(m), + err_msg=err_msg) + # Test via circular cross-correlation, which is just mult. + # in the frequency domain with one signal conjugated + tester = np.real(ifft(fft(m) * np.conj(fft(m)))) + out_len = 2**nbits - 1 + # impulse amplitude == test_len + err_msg = "mls impulse has incorrect value" + xp_assert_close(tester[0], + float(out_len), + err_msg=err_msg + ) + # steady-state is -1 + err_msg = "mls steady-state has incorrect value" + xp_assert_close(tester[1:], + np.full(out_len - 1, -1, dtype=tester.dtype), + err_msg=err_msg) + # let's do the split thing using a couple options + for n in (1, 2**(nbits - 1)): + m1, s1 = max_len_seq(nbits, state=state, taps=taps, + length=n) + m2, s2 = max_len_seq(nbits, state=s1, taps=taps, + length=1) + m3, s3 = max_len_seq(nbits, state=s2, taps=taps, + length=out_len - n - 1) + new_m = np.concatenate((m1, m2, m3)) + xp_assert_equal(orig_m, new_m) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_peak_finding.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_peak_finding.py new file mode 100644 index 0000000000000000000000000000000000000000..8de5a2379c2d43c0a10b5b3facd0b33be0778a36 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_peak_finding.py @@ -0,0 +1,915 @@ +import copy + +import numpy as np +import pytest +from pytest import raises, warns +from scipy._lib._array_api import xp_assert_close, xp_assert_equal + +from scipy.signal._peak_finding import ( + argrelmax, + argrelmin, + peak_prominences, + peak_widths, + _unpack_condition_args, + find_peaks, + find_peaks_cwt, + _identify_ridge_lines +) +from scipy.signal.windows import gaussian +from scipy.signal._peak_finding_utils import _local_maxima_1d, PeakPropertyWarning + + +def _gen_gaussians(center_locs, sigmas, total_length): + xdata = np.arange(0, total_length).astype(float) + out_data = np.zeros(total_length, dtype=float) + for ind, sigma in enumerate(sigmas): + tmp = (xdata - center_locs[ind]) / sigma + out_data += np.exp(-(tmp**2)) + return out_data + + +def _gen_gaussians_even(sigmas, total_length): + num_peaks = len(sigmas) + delta = total_length / (num_peaks + 1) + center_locs = np.linspace(delta, total_length - delta, num=num_peaks).astype(int) + out_data = _gen_gaussians(center_locs, sigmas, total_length) + return out_data, center_locs + + +def _gen_ridge_line(start_locs, max_locs, length, distances, gaps): + """ + Generate coordinates for a ridge line. + + Will be a series of coordinates, starting a start_loc (length 2). + The maximum distance between any adjacent columns will be + `max_distance`, the max distance between adjacent rows + will be `map_gap'. + + `max_locs` should be the size of the intended matrix. The + ending coordinates are guaranteed to be less than `max_locs`, + although they may not approach `max_locs` at all. + """ + + def keep_bounds(num, max_val): + out = max(num, 0) + out = min(out, max_val) + return out + + gaps = copy.deepcopy(gaps) + distances = copy.deepcopy(distances) + + locs = np.zeros([length, 2], dtype=int) + locs[0, :] = start_locs + total_length = max_locs[0] - start_locs[0] - sum(gaps) + if total_length < length: + raise ValueError('Cannot generate ridge line according to constraints') + dist_int = length / len(distances) - 1 + gap_int = length / len(gaps) - 1 + for ind in range(1, length): + nextcol = locs[ind - 1, 1] + nextrow = locs[ind - 1, 0] + 1 + if (ind % dist_int == 0) and (len(distances) > 0): + nextcol += ((-1)**ind)*distances.pop() + if (ind % gap_int == 0) and (len(gaps) > 0): + nextrow += gaps.pop() + nextrow = keep_bounds(nextrow, max_locs[0]) + nextcol = keep_bounds(nextcol, max_locs[1]) + locs[ind, :] = [nextrow, nextcol] + + return [locs[:, 0], locs[:, 1]] + + +class TestLocalMaxima1d: + + def test_empty(self): + """Test with empty signal.""" + x = np.array([], dtype=np.float64) + for array in _local_maxima_1d(x): + xp_assert_equal(array, np.array([]), check_dtype=False) + assert array.base is None + + def test_linear(self): + """Test with linear signal.""" + x = np.linspace(0, 100) + for array in _local_maxima_1d(x): + xp_assert_equal(array, np.array([], dtype=np.intp)) + assert array.base is None + + def test_simple(self): + """Test with simple signal.""" + x = np.linspace(-10, 10, 50) + x[2::3] += 1 + expected = np.arange(2, 50, 3, dtype=np.intp) + for array in _local_maxima_1d(x): + # For plateaus of size 1, the edges are identical with the + # midpoints + xp_assert_equal(array, expected, check_dtype=False) + assert array.base is None + + def test_flat_maxima(self): + """Test if flat maxima are detected correctly.""" + x = np.array([-1.3, 0, 1, 0, 2, 2, 0, 3, 3, 3, 2.99, 4, 4, 4, 4, -10, + -5, -5, -5, -5, -5, -10]) + midpoints, left_edges, right_edges = _local_maxima_1d(x) + xp_assert_equal(midpoints, np.array([2, 4, 8, 12, 18]), check_dtype=False) + xp_assert_equal(left_edges, np.array([2, 4, 7, 11, 16]), check_dtype=False) + xp_assert_equal(right_edges, np.array([2, 5, 9, 14, 20]), check_dtype=False) + + @pytest.mark.parametrize('x', [ + np.array([1., 0, 2]), + np.array([3., 3, 0, 4, 4]), + np.array([5., 5, 5, 0, 6, 6, 6]), + ]) + def test_signal_edges(self, x): + """Test if behavior on signal edges is correct.""" + for array in _local_maxima_1d(x): + xp_assert_equal(array, np.array([], dtype=np.intp)) + assert array.base is None + + def test_exceptions(self): + """Test input validation and raised exceptions.""" + with raises(ValueError, match="wrong number of dimensions"): + _local_maxima_1d(np.ones((1, 1))) + with raises(ValueError, match="expected 'const float64_t'"): + _local_maxima_1d(np.ones(1, dtype=int)) + with raises(TypeError, match="list"): + _local_maxima_1d([1., 2.]) + with raises(TypeError, match="'x' must not be None"): + _local_maxima_1d(None) + + +class TestRidgeLines: + + def test_empty(self): + test_matr = np.zeros([20, 100]) + lines = _identify_ridge_lines(test_matr, np.full(20, 2), 1) + assert len(lines) == 0 + + def test_minimal(self): + test_matr = np.zeros([20, 100]) + test_matr[0, 10] = 1 + lines = _identify_ridge_lines(test_matr, np.full(20, 2), 1) + assert len(lines) == 1 + + test_matr = np.zeros([20, 100]) + test_matr[0:2, 10] = 1 + lines = _identify_ridge_lines(test_matr, np.full(20, 2), 1) + assert len(lines) == 1 + + def test_single_pass(self): + distances = [0, 1, 2, 5] + gaps = [0, 1, 2, 0, 1] + test_matr = np.zeros([20, 50]) + 1e-12 + length = 12 + line = _gen_ridge_line([0, 25], test_matr.shape, length, distances, gaps) + test_matr[line[0], line[1]] = 1 + max_distances = np.full(20, max(distances)) + identified_lines = _identify_ridge_lines(test_matr, + max_distances, + max(gaps) + 1) + assert len(identified_lines) == 1 + for iline_, line_ in zip(identified_lines[0], line): + xp_assert_equal(iline_, line_, check_dtype=False) + + def test_single_bigdist(self): + distances = [0, 1, 2, 5] + gaps = [0, 1, 2, 4] + test_matr = np.zeros([20, 50]) + length = 12 + line = _gen_ridge_line([0, 25], test_matr.shape, length, distances, gaps) + test_matr[line[0], line[1]] = 1 + max_dist = 3 + max_distances = np.full(20, max_dist) + #This should get 2 lines, since the distance is too large + identified_lines = _identify_ridge_lines(test_matr, + max_distances, + max(gaps) + 1) + assert len(identified_lines) == 2 + + for iline in identified_lines: + adists = np.diff(iline[1]) + np.testing.assert_array_less(np.abs(adists), max_dist) + + agaps = np.diff(iline[0]) + np.testing.assert_array_less(np.abs(agaps), max(gaps) + 0.1) + + def test_single_biggap(self): + distances = [0, 1, 2, 5] + max_gap = 3 + gaps = [0, 4, 2, 1] + test_matr = np.zeros([20, 50]) + length = 12 + line = _gen_ridge_line([0, 25], test_matr.shape, length, distances, gaps) + test_matr[line[0], line[1]] = 1 + max_dist = 6 + max_distances = np.full(20, max_dist) + #This should get 2 lines, since the gap is too large + identified_lines = _identify_ridge_lines(test_matr, max_distances, max_gap) + assert len(identified_lines) == 2 + + for iline in identified_lines: + adists = np.diff(iline[1]) + np.testing.assert_array_less(np.abs(adists), max_dist) + + agaps = np.diff(iline[0]) + np.testing.assert_array_less(np.abs(agaps), max(gaps) + 0.1) + + def test_single_biggaps(self): + distances = [0] + max_gap = 1 + gaps = [3, 6] + test_matr = np.zeros([50, 50]) + length = 30 + line = _gen_ridge_line([0, 25], test_matr.shape, length, distances, gaps) + test_matr[line[0], line[1]] = 1 + max_dist = 1 + max_distances = np.full(50, max_dist) + #This should get 3 lines, since the gaps are too large + identified_lines = _identify_ridge_lines(test_matr, max_distances, max_gap) + assert len(identified_lines) == 3 + + for iline in identified_lines: + adists = np.diff(iline[1]) + np.testing.assert_array_less(np.abs(adists), max_dist) + + agaps = np.diff(iline[0]) + np.testing.assert_array_less(np.abs(agaps), max(gaps) + 0.1) + + +class TestArgrel: + + def test_empty(self): + # Regression test for gh-2832. + # When there are no relative extrema, make sure that + # the number of empty arrays returned matches the + # dimension of the input. + + empty_array = np.array([], dtype=int) + + z1 = np.zeros(5) + + i = argrelmin(z1) + xp_assert_equal(len(i), 1) + xp_assert_equal(i[0], empty_array, check_dtype=False) + + z2 = np.zeros((3, 5)) + + row, col = argrelmin(z2, axis=0) + xp_assert_equal(row, empty_array, check_dtype=False) + xp_assert_equal(col, empty_array, check_dtype=False) + + row, col = argrelmin(z2, axis=1) + xp_assert_equal(row, empty_array, check_dtype=False) + xp_assert_equal(col, empty_array, check_dtype=False) + + def test_basic(self): + # Note: the docstrings for the argrel{min,max,extrema} functions + # do not give a guarantee of the order of the indices, so we'll + # sort them before testing. + + x = np.array([[1, 2, 2, 3, 2], + [2, 1, 2, 2, 3], + [3, 2, 1, 2, 2], + [2, 3, 2, 1, 2], + [1, 2, 3, 2, 1]]) + + row, col = argrelmax(x, axis=0) + order = np.argsort(row) + xp_assert_equal(row[order], [1, 2, 3], check_dtype=False) + xp_assert_equal(col[order], [4, 0, 1], check_dtype=False) + + row, col = argrelmax(x, axis=1) + order = np.argsort(row) + xp_assert_equal(row[order], [0, 3, 4], check_dtype=False) + xp_assert_equal(col[order], [3, 1, 2], check_dtype=False) + + row, col = argrelmin(x, axis=0) + order = np.argsort(row) + xp_assert_equal(row[order], [1, 2, 3], check_dtype=False) + xp_assert_equal(col[order], [1, 2, 3], check_dtype=False) + + row, col = argrelmin(x, axis=1) + order = np.argsort(row) + xp_assert_equal(row[order], [1, 2, 3], check_dtype=False) + xp_assert_equal(col[order], [1, 2, 3], check_dtype=False) + + def test_highorder(self): + order = 2 + sigmas = [1.0, 2.0, 10.0, 5.0, 15.0] + test_data, act_locs = _gen_gaussians_even(sigmas, 500) + test_data[act_locs + order] = test_data[act_locs]*0.99999 + test_data[act_locs - order] = test_data[act_locs]*0.99999 + rel_max_locs = argrelmax(test_data, order=order, mode='clip')[0] + + assert len(rel_max_locs) == len(act_locs) + assert (rel_max_locs == act_locs).all() + + def test_2d_gaussians(self): + sigmas = [1.0, 2.0, 10.0] + test_data, act_locs = _gen_gaussians_even(sigmas, 100) + rot_factor = 20 + rot_range = np.arange(0, len(test_data)) - rot_factor + test_data_2 = np.vstack([test_data, test_data[rot_range]]) + rel_max_rows, rel_max_cols = argrelmax(test_data_2, axis=1, order=1) + + for rw in range(0, test_data_2.shape[0]): + inds = (rel_max_rows == rw) + + assert len(rel_max_cols[inds]) == len(act_locs) + assert (act_locs == (rel_max_cols[inds] - rot_factor*rw)).all() + + +class TestPeakProminences: + + def test_empty(self): + """ + Test if an empty array is returned if no peaks are provided. + """ + out = peak_prominences([1, 2, 3], []) + for arr, dtype in zip(out, [np.float64, np.intp, np.intp]): + assert arr.size == 0 + assert arr.dtype == dtype + + out = peak_prominences([], []) + for arr, dtype in zip(out, [np.float64, np.intp, np.intp]): + assert arr.size == 0 + assert arr.dtype == dtype + + def test_basic(self): + """ + Test if height of prominences is correctly calculated in signal with + rising baseline (peak widths are 1 sample). + """ + # Prepare basic signal + x = np.array([-1, 1.2, 1.2, 1, 3.2, 1.3, 2.88, 2.1]) + peaks = np.array([1, 2, 4, 6]) + lbases = np.array([0, 0, 0, 5]) + rbases = np.array([3, 3, 5, 7]) + proms = x[peaks] - np.max([x[lbases], x[rbases]], axis=0) + # Test if calculation matches handcrafted result + out = peak_prominences(x, peaks) + xp_assert_equal(out[0], proms, check_dtype=False) + xp_assert_equal(out[1], lbases, check_dtype=False) + xp_assert_equal(out[2], rbases, check_dtype=False) + + def test_edge_cases(self): + """ + Test edge cases. + """ + # Peaks have same height, prominence and bases + x = [0, 2, 1, 2, 1, 2, 0] + peaks = [1, 3, 5] + proms, lbases, rbases = peak_prominences(x, peaks) + xp_assert_equal(proms, np.asarray([2.0, 2, 2]), check_dtype=False) + xp_assert_equal(lbases, [0, 0, 0], check_dtype=False) + xp_assert_equal(rbases, [6, 6, 6], check_dtype=False) + + # Peaks have same height & prominence but different bases + x = [0, 1, 0, 1, 0, 1, 0] + peaks = np.array([1, 3, 5]) + proms, lbases, rbases = peak_prominences(x, peaks) + xp_assert_equal(proms, np.asarray([1.0, 1, 1])) + xp_assert_equal(lbases, peaks - 1, check_dtype=False) + xp_assert_equal(rbases, peaks + 1, check_dtype=False) + + def test_non_contiguous(self): + """ + Test with non-C-contiguous input arrays. + """ + x = np.repeat([-9, 9, 9, 0, 3, 1], 2) + peaks = np.repeat([1, 2, 4], 2) + proms, lbases, rbases = peak_prominences(x[::2], peaks[::2]) + xp_assert_equal(proms, np.asarray([9.0, 9, 2])) + xp_assert_equal(lbases, [0, 0, 3], check_dtype=False) + xp_assert_equal(rbases, [3, 3, 5], check_dtype=False) + + def test_wlen(self): + """ + Test if wlen actually shrinks the evaluation range correctly. + """ + x = [0, 1, 2, 3, 1, 0, -1] + peak = [3] + # Test rounding behavior of wlen + proms = peak_prominences(x, peak) + for prom, val in zip(proms, [3.0, 0, 6]): + assert prom == val + + for wlen, i in [(8, 0), (7, 0), (6, 0), (5, 1), (3.2, 1), (3, 2), (1.1, 2)]: + proms = peak_prominences(x, peak, wlen) + for prom, val in zip(proms, [3. - i, 0 + i, 6 - i]): + assert prom == val + + def test_exceptions(self): + """ + Verify that exceptions and warnings are raised. + """ + # x with dimension > 1 + with raises(ValueError, match='1-D array'): + peak_prominences([[0, 1, 1, 0]], [1, 2]) + # peaks with dimension > 1 + with raises(ValueError, match='1-D array'): + peak_prominences([0, 1, 1, 0], [[1, 2]]) + # x with dimension < 1 + with raises(ValueError, match='1-D array'): + peak_prominences(3, [0,]) + + # empty x with supplied + with raises(ValueError, match='not a valid index'): + peak_prominences([], [0]) + # invalid indices with non-empty x + for p in [-100, -1, 3, 1000]: + with raises(ValueError, match='not a valid index'): + peak_prominences([1, 0, 2], [p]) + + # peaks is not cast-able to np.intp + with raises(TypeError, match='cannot safely cast'): + peak_prominences([0, 1, 1, 0], [1.1, 2.3]) + + # wlen < 3 + with raises(ValueError, match='wlen'): + peak_prominences(np.arange(10), [3, 5], wlen=1) + + @pytest.mark.thread_unsafe + def test_warnings(self): + """ + Verify that appropriate warnings are raised. + """ + msg = "some peaks have a prominence of 0" + for p in [0, 1, 2]: + with warns(PeakPropertyWarning, match=msg): + peak_prominences([1, 0, 2], [p,]) + with warns(PeakPropertyWarning, match=msg): + peak_prominences([0, 1, 1, 1, 0], [2], wlen=2) + + +class TestPeakWidths: + + def test_empty(self): + """ + Test if an empty array is returned if no peaks are provided. + """ + widths = peak_widths([], [])[0] + assert isinstance(widths, np.ndarray) + assert widths.size == 0 + widths = peak_widths([1, 2, 3], [])[0] + assert isinstance(widths, np.ndarray) + assert widths.size == 0 + out = peak_widths([], []) + for arr in out: + assert isinstance(arr, np.ndarray) + assert arr.size == 0 + + @pytest.mark.filterwarnings("ignore:some peaks have a width of 0") + def test_basic(self): + """ + Test a simple use case with easy to verify results at different relative + heights. + """ + x = np.array([1, 0, 1, 2, 1, 0, -1]) + prominence = 2 + for rel_height, width_true, lip_true, rip_true in [ + (0., 0., 3., 3.), # raises warning + (0.25, 1., 2.5, 3.5), + (0.5, 2., 2., 4.), + (0.75, 3., 1.5, 4.5), + (1., 4., 1., 5.), + (2., 5., 1., 6.), + (3., 5., 1., 6.) + ]: + width_calc, height, lip_calc, rip_calc = peak_widths( + x, [3], rel_height) + xp_assert_close(width_calc, np.asarray([width_true])) + xp_assert_close(height, np.asarray([2 - rel_height * prominence])) + xp_assert_close(lip_calc, np.asarray([lip_true])) + xp_assert_close(rip_calc, np.asarray([rip_true])) + + def test_non_contiguous(self): + """ + Test with non-C-contiguous input arrays. + """ + x = np.repeat([0, 100, 50], 4) + peaks = np.repeat([1], 3) + result = peak_widths(x[::4], peaks[::3]) + xp_assert_equal(result, + np.asarray([[0.75], [75], [0.75], [1.5]]) + ) + + def test_exceptions(self): + """ + Verify that argument validation works as intended. + """ + with raises(ValueError, match='1-D array'): + # x with dimension > 1 + peak_widths(np.zeros((3, 4)), np.ones(3)) + with raises(ValueError, match='1-D array'): + # x with dimension < 1 + peak_widths(3, [0]) + with raises(ValueError, match='1-D array'): + # peaks with dimension > 1 + peak_widths(np.arange(10), np.ones((3, 2), dtype=np.intp)) + with raises(ValueError, match='1-D array'): + # peaks with dimension < 1 + peak_widths(np.arange(10), 3) + with raises(ValueError, match='not a valid index'): + # peak pos exceeds x.size + peak_widths(np.arange(10), [8, 11]) + with raises(ValueError, match='not a valid index'): + # empty x with peaks supplied + peak_widths([], [1, 2]) + with raises(TypeError, match='cannot safely cast'): + # peak cannot be safely cast to intp + peak_widths(np.arange(10), [1.1, 2.3]) + with raises(ValueError, match='rel_height'): + # rel_height is < 0 + peak_widths([0, 1, 0, 1, 0], [1, 3], rel_height=-1) + with raises(TypeError, match='None'): + # prominence data contains None + peak_widths([1, 2, 1], [1], prominence_data=(None, None, None)) + + @pytest.mark.thread_unsafe + def test_warnings(self): + """ + Verify that appropriate warnings are raised. + """ + msg = "some peaks have a width of 0" + with warns(PeakPropertyWarning, match=msg): + # Case: rel_height is 0 + peak_widths([0, 1, 0], [1], rel_height=0) + with warns(PeakPropertyWarning, match=msg): + # Case: prominence is 0 and bases are identical + peak_widths( + [0, 1, 1, 1, 0], [2], + prominence_data=(np.array([0.], np.float64), + np.array([2], np.intp), + np.array([2], np.intp)) + ) + + def test_mismatching_prominence_data(self): + """Test with mismatching peak and / or prominence data.""" + x = [0, 1, 0] + peak = [1] + for i, (prominences, left_bases, right_bases) in enumerate([ + ((1.,), (-1,), (2,)), # left base not in x + ((1.,), (0,), (3,)), # right base not in x + ((1.,), (2,), (0,)), # swapped bases same as peak + ((1., 1.), (0, 0), (2, 2)), # array shapes don't match peaks + ((1., 1.), (0,), (2,)), # arrays with different shapes + ((1.,), (0, 0), (2,)), # arrays with different shapes + ((1.,), (0,), (2, 2)) # arrays with different shapes + ]): + # Make sure input is matches output of signal.peak_prominences + prominence_data = (np.array(prominences, dtype=np.float64), + np.array(left_bases, dtype=np.intp), + np.array(right_bases, dtype=np.intp)) + # Test for correct exception + if i < 3: + match = "prominence data is invalid for peak" + else: + match = "arrays in `prominence_data` must have the same shape" + with raises(ValueError, match=match): + peak_widths(x, peak, prominence_data=prominence_data) + + @pytest.mark.filterwarnings("ignore:some peaks have a width of 0") + def test_intersection_rules(self): + """Test if x == eval_height counts as an intersection.""" + # Flatt peak with two possible intersection points if evaluated at 1 + x = [0, 1, 2, 1, 3, 3, 3, 1, 2, 1, 0] + # relative height is 0 -> width is 0 as well, raises warning + xp_assert_close(peak_widths(x, peaks=[5], rel_height=0), + [(0.,), (3.,), (5.,), (5.,)]) + # width_height == x counts as intersection -> nearest 1 is chosen + xp_assert_close(peak_widths(x, peaks=[5], rel_height=2/3), + [(4.,), (1.,), (3.,), (7.,)]) + + +def test_unpack_condition_args(): + """ + Verify parsing of condition arguments for `scipy.signal.find_peaks` function. + """ + x = np.arange(10) + amin_true = x + amax_true = amin_true + 10 + peaks = amin_true[1::2] + + # Test unpacking with None or interval + assert (None, None) == _unpack_condition_args((None, None), x, peaks) + assert (1, None) == _unpack_condition_args(1, x, peaks) + assert (1, None) == _unpack_condition_args((1, None), x, peaks) + assert (None, 2) == _unpack_condition_args((None, 2), x, peaks) + assert (3., 4.5) == _unpack_condition_args((3., 4.5), x, peaks) + + # Test if borders are correctly reduced with `peaks` + amin_calc, amax_calc = _unpack_condition_args((amin_true, amax_true), x, peaks) + xp_assert_equal(amin_calc, amin_true[peaks]) + xp_assert_equal(amax_calc, amax_true[peaks]) + + # Test raises if array borders don't match x + with raises(ValueError, match="array size of lower"): + _unpack_condition_args(amin_true, np.arange(11), peaks) + with raises(ValueError, match="array size of upper"): + _unpack_condition_args((None, amin_true), np.arange(11), peaks) + + +class TestFindPeaks: + + # Keys of optionally returned properties + property_keys = {'peak_heights', 'left_thresholds', 'right_thresholds', + 'prominences', 'left_bases', 'right_bases', 'widths', + 'width_heights', 'left_ips', 'right_ips'} + + def test_constant(self): + """ + Test behavior for signal without local maxima. + """ + open_interval = (None, None) + peaks, props = find_peaks(np.ones(10), + height=open_interval, threshold=open_interval, + prominence=open_interval, width=open_interval) + assert peaks.size == 0 + for key in self.property_keys: + assert props[key].size == 0 + + def test_plateau_size(self): + """ + Test plateau size condition for peaks. + """ + # Prepare signal with peaks with peak_height == plateau_size + plateau_sizes = np.array([1, 2, 3, 4, 8, 20, 111]) + x = np.zeros(plateau_sizes.size * 2 + 1) + x[1::2] = plateau_sizes + repeats = np.ones(x.size, dtype=int) + repeats[1::2] = x[1::2] + x = np.repeat(x, repeats) + + # Test full output + peaks, props = find_peaks(x, plateau_size=(None, None)) + xp_assert_equal(peaks, [1, 3, 7, 11, 18, 33, 100], check_dtype=False) + xp_assert_equal(props["plateau_sizes"], plateau_sizes, check_dtype=False) + xp_assert_equal(props["left_edges"], peaks - (plateau_sizes - 1) // 2, + check_dtype=False) + xp_assert_equal(props["right_edges"], peaks + plateau_sizes // 2, + check_dtype=False) + + # Test conditions + xp_assert_equal(find_peaks(x, plateau_size=4)[0], [11, 18, 33, 100], + check_dtype=False) + xp_assert_equal(find_peaks(x, plateau_size=(None, 3.5))[0], [1, 3, 7], + check_dtype=False) + xp_assert_equal(find_peaks(x, plateau_size=(5, 50))[0], [18, 33], + check_dtype=False) + + def test_height_condition(self): + """ + Test height condition for peaks. + """ + x = (0., 1/3, 0., 2.5, 0, 4., 0) + peaks, props = find_peaks(x, height=(None, None)) + xp_assert_equal(peaks, np.array([1, 3, 5]), check_dtype=False) + xp_assert_equal(props['peak_heights'], np.array([1/3, 2.5, 4.]), + check_dtype=False) + xp_assert_equal(find_peaks(x, height=0.5)[0], np.array([3, 5]), + check_dtype=False) + xp_assert_equal(find_peaks(x, height=(None, 3))[0], np.array([1, 3]), + check_dtype=False) + xp_assert_equal(find_peaks(x, height=(2, 3))[0], np.array([3]), + check_dtype=False) + + def test_threshold_condition(self): + """ + Test threshold condition for peaks. + """ + x = (0, 2, 1, 4, -1) + peaks, props = find_peaks(x, threshold=(None, None)) + xp_assert_equal(peaks, np.array([1, 3]), check_dtype=False) + xp_assert_equal(props['left_thresholds'], np.array([2.0, 3.0])) + xp_assert_equal(props['right_thresholds'], np.array([1.0, 5.0])) + xp_assert_equal(find_peaks(x, threshold=2)[0], np.array([3]), + check_dtype=False) + xp_assert_equal(find_peaks(x, threshold=3.5)[0], np.array([], dtype=int), + check_dtype=False) + xp_assert_equal(find_peaks(x, threshold=(None, 5))[0], np.array([1, 3]), + check_dtype=False) + xp_assert_equal(find_peaks(x, threshold=(None, 4))[0], np.array([1]), + check_dtype=False) + xp_assert_equal(find_peaks(x, threshold=(2, 4))[0], np.array([], dtype=int), + check_dtype=False) + + def test_distance_condition(self): + """ + Test distance condition for peaks. + """ + # Peaks of different height with constant distance 3 + peaks_all = np.arange(1, 21, 3) + x = np.zeros(21) + x[peaks_all] += np.linspace(1, 2, peaks_all.size) + + # Test if peaks with "minimal" distance are still selected (distance = 3) + xp_assert_equal(find_peaks(x, distance=3)[0], peaks_all, check_dtype=False) + + # Select every second peak (distance > 3) + peaks_subset = find_peaks(x, distance=3.0001)[0] + # Test if peaks_subset is subset of peaks_all + assert np.setdiff1d(peaks_subset, peaks_all, assume_unique=True).size == 0 + + # Test if every second peak was removed + dfs = np.diff(peaks_subset) + xp_assert_equal(dfs, 6*np.ones_like(dfs)) + + # Test priority of peak removal + x = [-2, 1, -1, 0, -3] + peaks_subset = find_peaks(x, distance=10)[0] # use distance > x size + assert peaks_subset.size == 1 and peaks_subset[0] == 1 + + def test_prominence_condition(self): + """ + Test prominence condition for peaks. + """ + x = np.linspace(0, 10, 100) + peaks_true = np.arange(1, 99, 2) + offset = np.linspace(1, 10, peaks_true.size) + x[peaks_true] += offset + prominences = x[peaks_true] - x[peaks_true + 1] + interval = (3, 9) + keep = np.nonzero( + (interval[0] <= prominences) & (prominences <= interval[1])) + + peaks_calc, properties = find_peaks(x, prominence=interval) + xp_assert_equal(peaks_calc, peaks_true[keep], check_dtype=False) + xp_assert_equal(properties['prominences'], prominences[keep], check_dtype=False) + xp_assert_equal(properties['left_bases'], + np.zeros_like(properties['left_bases'])) + xp_assert_equal(properties['right_bases'], peaks_true[keep] + 1, + check_dtype=False) + + def test_width_condition(self): + """ + Test width condition for peaks. + """ + x = np.array([1, 0, 1, 2, 1, 0, -1, 4, 0]) + peaks, props = find_peaks(x, width=(None, 2), rel_height=0.75) + assert peaks.size == 1 + xp_assert_equal(peaks, 7*np.ones_like(peaks)) + xp_assert_close(props['widths'], np.asarray([1.35])) + xp_assert_close(props['width_heights'], np.asarray([1.])) + xp_assert_close(props['left_ips'], np.asarray([6.4])) + xp_assert_close(props['right_ips'], np.asarray([7.75])) + + def test_properties(self): + """ + Test returned properties. + """ + open_interval = (None, None) + x = [0, 1, 0, 2, 1.5, 0, 3, 0, 5, 9] + peaks, props = find_peaks(x, + height=open_interval, threshold=open_interval, + prominence=open_interval, width=open_interval) + assert len(props) == len(self.property_keys) + for key in self.property_keys: + assert peaks.size == props[key].size + + def test_raises(self): + """ + Test exceptions raised by function. + """ + with raises(ValueError, match="1-D array"): + find_peaks(np.array(1)) + with raises(ValueError, match="1-D array"): + find_peaks(np.ones((2, 2))) + with raises(ValueError, match="distance"): + find_peaks(np.arange(10), distance=-1) + + @pytest.mark.filterwarnings("ignore:some peaks have a prominence of 0", + "ignore:some peaks have a width of 0") + def test_wlen_smaller_plateau(self): + """ + Test behavior of prominence and width calculation if the given window + length is smaller than a peak's plateau size. + + Regression test for gh-9110. + """ + peaks, props = find_peaks([0, 1, 1, 1, 0], prominence=(None, None), + width=(None, None), wlen=2) + xp_assert_equal(peaks, 2 * np.ones_like(peaks)) + xp_assert_equal(props["prominences"], np.zeros_like(props["prominences"])) + xp_assert_equal(props["widths"], np.zeros_like(props["widths"])) + xp_assert_equal(props["width_heights"], np.ones_like(props["width_heights"])) + for key in ("left_bases", "right_bases", "left_ips", "right_ips"): + xp_assert_equal(props[key], peaks, check_dtype=False) + + @pytest.mark.parametrize("kwargs", [ + {}, + {"distance": 3.0}, + {"prominence": (None, None)}, + {"width": (None, 2)}, + + ]) + def test_readonly_array(self, kwargs): + """ + Test readonly arrays are accepted. + """ + x = np.linspace(0, 10, 15) + x_readonly = x.copy() + x_readonly.flags.writeable = False + + peaks, _ = find_peaks(x) + peaks_readonly, _ = find_peaks(x_readonly, **kwargs) + + xp_assert_close(peaks, peaks_readonly) + + +class TestFindPeaksCwt: + + def test_find_peaks_exact(self): + """ + Generate a series of gaussians and attempt to find the peak locations. + """ + sigmas = [5.0, 3.0, 10.0, 20.0, 10.0, 50.0] + num_points = 500 + test_data, act_locs = _gen_gaussians_even(sigmas, num_points) + widths = np.arange(0.1, max(sigmas)) + found_locs = find_peaks_cwt(test_data, widths, gap_thresh=2, min_snr=0, + min_length=None) + xp_assert_equal(found_locs, act_locs, + check_dtype=False, + err_msg="Found maximum locations did not equal those expected" + ) + + def test_find_peaks_withnoise(self): + """ + Verify that peak locations are (approximately) found + for a series of gaussians with added noise. + """ + sigmas = [5.0, 3.0, 10.0, 20.0, 10.0, 50.0] + num_points = 500 + test_data, act_locs = _gen_gaussians_even(sigmas, num_points) + widths = np.arange(0.1, max(sigmas)) + noise_amp = 0.07 + np.random.seed(18181911) + test_data += (np.random.rand(num_points) - 0.5)*(2*noise_amp) + found_locs = find_peaks_cwt(test_data, widths, min_length=15, + gap_thresh=1, min_snr=noise_amp / 5) + + err_msg ='Different number of peaks found than expected' + assert len(found_locs) == len(act_locs), err_msg + diffs = np.abs(found_locs - act_locs) + max_diffs = np.array(sigmas) / 5 + np.testing.assert_array_less(diffs, max_diffs, 'Maximum location differed' + + f'by more than {max_diffs}') + + def test_find_peaks_nopeak(self): + """ + Verify that no peak is found in + data that's just noise. + """ + noise_amp = 1.0 + num_points = 100 + rng = np.random.RandomState(181819141) + test_data = (rng.rand(num_points) - 0.5)*(2*noise_amp) + widths = np.arange(10, 50) + found_locs = find_peaks_cwt(test_data, widths, min_snr=5, noise_perc=30) + assert len(found_locs) == 0 + + def test_find_peaks_with_non_default_wavelets(self): + x = gaussian(200, 2) + widths = np.array([1, 2, 3, 4]) + a = find_peaks_cwt(x, widths, wavelet=gaussian) + + xp_assert_equal(a, np.asarray([100]), check_dtype=False) + + def test_find_peaks_window_size(self): + """ + Verify that window_size is passed correctly to private function and + affects the result. + """ + sigmas = [2.0, 2.0] + num_points = 1000 + test_data, act_locs = _gen_gaussians_even(sigmas, num_points) + widths = np.arange(0.1, max(sigmas), 0.2) + noise_amp = 0.05 + rng = np.random.RandomState(18181911) + test_data += (rng.rand(num_points) - 0.5)*(2*noise_amp) + + # Possibly contrived negative region to throw off peak finding + # when window_size is too large + test_data[250:320] -= 1 + + found_locs = find_peaks_cwt(test_data, widths, gap_thresh=2, min_snr=3, + min_length=None, window_size=None) + with pytest.raises(AssertionError): + assert found_locs.size == act_locs.size + + found_locs = find_peaks_cwt(test_data, widths, gap_thresh=2, min_snr=3, + min_length=None, window_size=20) + assert found_locs.size == act_locs.size + + def test_find_peaks_with_one_width(self): + """ + Verify that the `width` argument + in `find_peaks_cwt` can be a float + """ + xs = np.arange(0, np.pi, 0.05) + test_data = np.sin(xs) + widths = 1 + found_locs = find_peaks_cwt(test_data, widths) + + np.testing.assert_equal(found_locs, 32) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_result_type.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_result_type.py new file mode 100644 index 0000000000000000000000000000000000000000..a2cadd325a7e36c877df8532ba957712831c2dad --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_result_type.py @@ -0,0 +1,51 @@ +# Regressions tests on result types of some signal functions + +import numpy as np + +from scipy.signal import (decimate, + lfilter_zi, + lfiltic, + sos2tf, + sosfilt_zi) + + +def test_decimate(): + ones_f32 = np.ones(32, dtype=np.float32) + assert decimate(ones_f32, 2).dtype == np.float32 + + ones_i64 = np.ones(32, dtype=np.int64) + assert decimate(ones_i64, 2).dtype == np.float64 + + +def test_lfilter_zi(): + b_f32 = np.array([1, 2, 3], dtype=np.float32) + a_f32 = np.array([4, 5, 6], dtype=np.float32) + assert lfilter_zi(b_f32, a_f32).dtype == np.float32 + + +def test_lfiltic(): + # this would return f32 when given a mix of f32 / f64 args + b_f32 = np.array([1, 2, 3], dtype=np.float32) + a_f32 = np.array([4, 5, 6], dtype=np.float32) + x_f32 = np.ones(32, dtype=np.float32) + + b_f64 = b_f32.astype(np.float64) + a_f64 = a_f32.astype(np.float64) + x_f64 = x_f32.astype(np.float64) + + assert lfiltic(b_f64, a_f32, x_f32).dtype == np.float64 + assert lfiltic(b_f32, a_f64, x_f32).dtype == np.float64 + assert lfiltic(b_f32, a_f32, x_f64).dtype == np.float64 + assert lfiltic(b_f32, a_f32, x_f32, x_f64).dtype == np.float64 + + +def test_sos2tf(): + sos_f32 = np.array([[4, 5, 6, 1, 2, 3]], dtype=np.float32) + b, a = sos2tf(sos_f32) + assert b.dtype == np.float32 + assert a.dtype == np.float32 + + +def test_sosfilt_zi(): + sos_f32 = np.array([[4, 5, 6, 1, 2, 3]], dtype=np.float32) + assert sosfilt_zi(sos_f32).dtype == np.float32 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_savitzky_golay.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_savitzky_golay.py new file mode 100644 index 0000000000000000000000000000000000000000..61d958e35d91b7d537e0bd3551b6cec3f45a4983 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_savitzky_golay.py @@ -0,0 +1,362 @@ +import pytest +import numpy as np +from numpy.testing import (assert_equal, + assert_array_equal, +) + +from scipy._lib._array_api import ( + assert_almost_equal, assert_array_almost_equal, xp_assert_close +) + +from scipy.ndimage import convolve1d # type: ignore[attr-defined] + +from scipy.signal import savgol_coeffs, savgol_filter +from scipy.signal._savitzky_golay import _polyder + + +def check_polyder(p, m, expected): + dp = _polyder(p, m) + assert_array_equal(dp, expected) + + +def test_polyder(): + cases = [ + ([5], 0, [5]), + ([5], 1, [0]), + ([3, 2, 1], 0, [3, 2, 1]), + ([3, 2, 1], 1, [6, 2]), + ([3, 2, 1], 2, [6]), + ([3, 2, 1], 3, [0]), + ([[3, 2, 1], [5, 6, 7]], 0, [[3, 2, 1], [5, 6, 7]]), + ([[3, 2, 1], [5, 6, 7]], 1, [[6, 2], [10, 6]]), + ([[3, 2, 1], [5, 6, 7]], 2, [[6], [10]]), + ([[3, 2, 1], [5, 6, 7]], 3, [[0], [0]]), + ] + for p, m, expected in cases: + check_polyder(np.array(p).T, m, np.array(expected).T) + + +#-------------------------------------------------------------------- +# savgol_coeffs tests +#-------------------------------------------------------------------- + +def alt_sg_coeffs(window_length, polyorder, pos): + """This is an alternative implementation of the SG coefficients. + + It uses numpy.polyfit and numpy.polyval. The results should be + equivalent to those of savgol_coeffs(), but this implementation + is slower. + + window_length should be odd. + + """ + if pos is None: + pos = window_length // 2 + t = np.arange(window_length) + unit = (t == pos).astype(int) + h = np.polyval(np.polyfit(t, unit, polyorder), t) + return h + + +def test_sg_coeffs_trivial(): + # Test a trivial case of savgol_coeffs: polyorder = window_length - 1 + h = savgol_coeffs(1, 0) + xp_assert_close(h, [1.0]) + + h = savgol_coeffs(3, 2) + xp_assert_close(h, [0.0, 1, 0], atol=1e-10) + + h = savgol_coeffs(5, 4) + xp_assert_close(h, [0.0, 0, 1, 0, 0], atol=1e-10) + + h = savgol_coeffs(5, 4, pos=1) + xp_assert_close(h, [0.0, 0, 0, 1, 0], atol=1e-10) + + h = savgol_coeffs(5, 4, pos=1, use='dot') + xp_assert_close(h, [0.0, 1, 0, 0, 0], atol=1e-10) + + +def compare_coeffs_to_alt(window_length, order): + # For the given window_length and order, compare the results + # of savgol_coeffs and alt_sg_coeffs for pos from 0 to window_length - 1. + # Also include pos=None. + for pos in [None] + list(range(window_length)): + h1 = savgol_coeffs(window_length, order, pos=pos, use='dot') + h2 = alt_sg_coeffs(window_length, order, pos=pos) + xp_assert_close(h1, h2, atol=1e-10, + err_msg=("window_length = %d, order = %d, pos = %s" % + (window_length, order, pos))) + + +def test_sg_coeffs_compare(): + # Compare savgol_coeffs() to alt_sg_coeffs(). + for window_length in range(1, 8, 2): + for order in range(window_length): + compare_coeffs_to_alt(window_length, order) + + +def test_sg_coeffs_exact(): + polyorder = 4 + window_length = 9 + halflen = window_length // 2 + + x = np.linspace(0, 21, 43) + delta = x[1] - x[0] + + # The data is a cubic polynomial. We'll use an order 4 + # SG filter, so the filtered values should equal the input data + # (except within half window_length of the edges). + y = 0.5 * x ** 3 - x + h = savgol_coeffs(window_length, polyorder) + y0 = convolve1d(y, h) + xp_assert_close(y0[halflen:-halflen], y[halflen:-halflen]) + + # Check the same input, but use deriv=1. dy is the exact result. + dy = 1.5 * x ** 2 - 1 + h = savgol_coeffs(window_length, polyorder, deriv=1, delta=delta) + y1 = convolve1d(y, h) + xp_assert_close(y1[halflen:-halflen], dy[halflen:-halflen]) + + # Check the same input, but use deriv=2. d2y is the exact result. + d2y = 3.0 * x + h = savgol_coeffs(window_length, polyorder, deriv=2, delta=delta) + y2 = convolve1d(y, h) + xp_assert_close(y2[halflen:-halflen], d2y[halflen:-halflen]) + + +def test_sg_coeffs_deriv(): + # The data in `x` is a sampled parabola, so using savgol_coeffs with an + # order 2 or higher polynomial should give exact results. + i = np.array([-2.0, 0.0, 2.0, 4.0, 6.0]) + x = i ** 2 / 4 + dx = i / 2 + d2x = np.full_like(i, 0.5) + for pos in range(x.size): + coeffs0 = savgol_coeffs(5, 3, pos=pos, delta=2.0, use='dot') + xp_assert_close(coeffs0.dot(x), x[pos], atol=1e-10) + coeffs1 = savgol_coeffs(5, 3, pos=pos, delta=2.0, use='dot', deriv=1) + xp_assert_close(coeffs1.dot(x), dx[pos], atol=1e-10) + coeffs2 = savgol_coeffs(5, 3, pos=pos, delta=2.0, use='dot', deriv=2) + xp_assert_close(coeffs2.dot(x), d2x[pos], atol=1e-10) + + +def test_sg_coeffs_deriv_gt_polyorder(): + """ + If deriv > polyorder, the coefficients should be all 0. + This is a regression test for a bug where, e.g., + savgol_coeffs(5, polyorder=1, deriv=2) + raised an error. + """ + coeffs = savgol_coeffs(5, polyorder=1, deriv=2) + assert_array_equal(coeffs, np.zeros(5)) + coeffs = savgol_coeffs(7, polyorder=4, deriv=6) + assert_array_equal(coeffs, np.zeros(7)) + + +def test_sg_coeffs_large(): + # Test that for large values of window_length and polyorder the array of + # coefficients returned is symmetric. The aim is to ensure that + # no potential numeric overflow occurs. + coeffs0 = savgol_coeffs(31, 9) + assert_array_almost_equal(coeffs0, coeffs0[::-1]) + coeffs1 = savgol_coeffs(31, 9, deriv=1) + assert_array_almost_equal(coeffs1, -coeffs1[::-1]) + +# -------------------------------------------------------------------- +# savgol_coeffs tests for even window length +# -------------------------------------------------------------------- + + +def test_sg_coeffs_even_window_length(): + # Simple case - deriv=0, polyorder=0, 1 + window_lengths = [4, 6, 8, 10, 12, 14, 16] + for length in window_lengths: + h_p_d = savgol_coeffs(length, 0, 0) + xp_assert_close(h_p_d, np.ones_like(h_p_d) / length) + + # Verify with closed forms + # deriv=1, polyorder=1, 2 + def h_p_d_closed_form_1(k, m): + return 6*(k - 0.5)/((2*m + 1)*m*(2*m - 1)) + + # deriv=2, polyorder=2 + def h_p_d_closed_form_2(k, m): + numer = 15*(-4*m**2 + 1 + 12*(k - 0.5)**2) + denom = 4*(2*m + 1)*(m + 1)*m*(m - 1)*(2*m - 1) + return numer/denom + + for length in window_lengths: + m = length//2 + expected_output = [h_p_d_closed_form_1(k, m) + for k in range(-m + 1, m + 1)][::-1] + actual_output = savgol_coeffs(length, 1, 1) + xp_assert_close(expected_output, actual_output) + actual_output = savgol_coeffs(length, 2, 1) + xp_assert_close(expected_output, actual_output) + + expected_output = [h_p_d_closed_form_2(k, m) + for k in range(-m + 1, m + 1)][::-1] + actual_output = savgol_coeffs(length, 2, 2) + xp_assert_close(expected_output, actual_output) + actual_output = savgol_coeffs(length, 3, 2) + xp_assert_close(expected_output, actual_output) + +#-------------------------------------------------------------------- +# savgol_filter tests +#-------------------------------------------------------------------- + + +def test_sg_filter_trivial(): + """ Test some trivial edge cases for savgol_filter().""" + x = np.array([1.0]) + y = savgol_filter(x, 1, 0) + assert_equal(y, [1.0]) + + # Input is a single value. With a window length of 3 and polyorder 1, + # the value in y is from the straight-line fit of (-1,0), (0,3) and + # (1, 0) at 0. This is just the average of the three values, hence 1.0. + x = np.array([3.0]) + y = savgol_filter(x, 3, 1, mode='constant') + assert_almost_equal(y, [1.0], decimal=15) + + x = np.array([3.0]) + y = savgol_filter(x, 3, 1, mode='nearest') + assert_almost_equal(y, [3.0], decimal=15) + + x = np.array([1.0] * 3) + y = savgol_filter(x, 3, 1, mode='wrap') + assert_almost_equal(y, [1.0, 1.0, 1.0], decimal=15) + + +def test_sg_filter_basic(): + # Some basic test cases for savgol_filter(). + x = np.array([1.0, 2.0, 1.0]) + y = savgol_filter(x, 3, 1, mode='constant') + xp_assert_close(y, [1.0, 4.0 / 3, 1.0]) + + y = savgol_filter(x, 3, 1, mode='mirror') + xp_assert_close(y, [5.0 / 3, 4.0 / 3, 5.0 / 3]) + + y = savgol_filter(x, 3, 1, mode='wrap') + xp_assert_close(y, [4.0 / 3, 4.0 / 3, 4.0 / 3]) + + +def test_sg_filter_2d(): + x = np.array([[1.0, 2.0, 1.0], + [2.0, 4.0, 2.0]]) + expected = np.array([[1.0, 4.0 / 3, 1.0], + [2.0, 8.0 / 3, 2.0]]) + y = savgol_filter(x, 3, 1, mode='constant') + xp_assert_close(y, expected) + + y = savgol_filter(x.T, 3, 1, mode='constant', axis=0) + xp_assert_close(y, expected.T) + + +def test_sg_filter_interp_edges(): + # Another test with low degree polynomial data, for which we can easily + # give the exact results. In this test, we use mode='interp', so + # savgol_filter should match the exact solution for the entire data set, + # including the edges. + t = np.linspace(-5, 5, 21) + delta = t[1] - t[0] + # Polynomial test data. + x = np.array([t, + 3 * t ** 2, + t ** 3 - t]) + dx = np.array([np.ones_like(t), + 6 * t, + 3 * t ** 2 - 1.0]) + d2x = np.array([np.zeros_like(t), + np.full_like(t, 6), + 6 * t]) + + window_length = 7 + + y = savgol_filter(x, window_length, 3, axis=-1, mode='interp') + xp_assert_close(y, x, atol=1e-12) + + y1 = savgol_filter(x, window_length, 3, axis=-1, mode='interp', + deriv=1, delta=delta) + xp_assert_close(y1, dx, atol=1e-12) + + y2 = savgol_filter(x, window_length, 3, axis=-1, mode='interp', + deriv=2, delta=delta) + xp_assert_close(y2, d2x, atol=1e-12) + + # Transpose everything, and test again with axis=0. + + x = x.T + dx = dx.T + d2x = d2x.T + + y = savgol_filter(x, window_length, 3, axis=0, mode='interp') + xp_assert_close(y, x, atol=1e-12) + + y1 = savgol_filter(x, window_length, 3, axis=0, mode='interp', + deriv=1, delta=delta) + xp_assert_close(y1, dx, atol=1e-12) + + y2 = savgol_filter(x, window_length, 3, axis=0, mode='interp', + deriv=2, delta=delta) + xp_assert_close(y2, d2x, atol=1e-12) + + +def test_sg_filter_interp_edges_3d(): + # Test mode='interp' with a 3-D array. + t = np.linspace(-5, 5, 21) + delta = t[1] - t[0] + x1 = np.array([t, -t]) + x2 = np.array([t ** 2, 3 * t ** 2 + 5]) + x3 = np.array([t ** 3, 2 * t ** 3 + t ** 2 - 0.5 * t]) + dx1 = np.array([np.ones_like(t), -np.ones_like(t)]) + dx2 = np.array([2 * t, 6 * t]) + dx3 = np.array([3 * t ** 2, 6 * t ** 2 + 2 * t - 0.5]) + + # z has shape (3, 2, 21) + z = np.array([x1, x2, x3]) + dz = np.array([dx1, dx2, dx3]) + + y = savgol_filter(z, 7, 3, axis=-1, mode='interp', delta=delta) + xp_assert_close(y, z, atol=1e-10) + + dy = savgol_filter(z, 7, 3, axis=-1, mode='interp', deriv=1, delta=delta) + xp_assert_close(dy, dz, atol=1e-10) + + # z has shape (3, 21, 2) + z = np.array([x1.T, x2.T, x3.T]) + dz = np.array([dx1.T, dx2.T, dx3.T]) + + y = savgol_filter(z, 7, 3, axis=1, mode='interp', delta=delta) + xp_assert_close(y, z, atol=1e-10) + + dy = savgol_filter(z, 7, 3, axis=1, mode='interp', deriv=1, delta=delta) + xp_assert_close(dy, dz, atol=1e-10) + + # z has shape (21, 3, 2) + z = z.swapaxes(0, 1).copy() + dz = dz.swapaxes(0, 1).copy() + + y = savgol_filter(z, 7, 3, axis=0, mode='interp', delta=delta) + xp_assert_close(y, z, atol=1e-10) + + dy = savgol_filter(z, 7, 3, axis=0, mode='interp', deriv=1, delta=delta) + xp_assert_close(dy, dz, atol=1e-10) + + +def test_sg_filter_valid_window_length_3d(): + """Tests that the window_length check is using the correct axis.""" + + x = np.ones((10, 20, 30)) + + savgol_filter(x, window_length=29, polyorder=3, mode='interp') + + with pytest.raises(ValueError, match='window_length must be less than'): + # window_length is more than x.shape[-1]. + savgol_filter(x, window_length=31, polyorder=3, mode='interp') + + savgol_filter(x, window_length=9, polyorder=3, axis=0, mode='interp') + + with pytest.raises(ValueError, match='window_length must be less than'): + # window_length is more than x.shape[0]. + savgol_filter(x, window_length=11, polyorder=3, axis=0, mode='interp') diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_short_time_fft.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_short_time_fft.py new file mode 100644 index 0000000000000000000000000000000000000000..df1ccc639f2041cfc8a6aec50190b34f72d25c84 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_short_time_fft.py @@ -0,0 +1,880 @@ +"""Unit tests for module `_short_time_fft`. + +This file's structure loosely groups the tests into the following sequential +categories: + +1. Test function `_calc_dual_canonical_window`. +2. Test for invalid parameters and exceptions in `ShortTimeFFT` (until the + `test_from_window` function). +3. Test algorithmic properties of STFT/ISTFT. Some tests were ported from + ``test_spectral.py``. + +Notes +----- +* Mypy 0.990 does interpret the line:: + + from scipy.stats import norm as normal_distribution + + incorrectly (but the code works), hence a ``type: ignore`` was appended. +""" +import math +from itertools import product +from typing import cast, get_args, Literal + +import numpy as np +import pytest +from scipy._lib._array_api import xp_assert_close, xp_assert_equal +from scipy.fft import fftshift +from scipy.stats import norm as normal_distribution # type: ignore +from scipy.signal import get_window, welch, stft, istft, spectrogram + +from scipy.signal._short_time_fft import FFT_MODE_TYPE, \ + _calc_dual_canonical_window, ShortTimeFFT, PAD_TYPE +from scipy.signal.windows import gaussian + + +def test__calc_dual_canonical_window_roundtrip(): + """Test dual window calculation with a round trip to verify duality. + + Note that this works only for canonical window pairs (having minimal + energy) like a Gaussian. + + The window is the same as in the example of `from ShortTimeFFT.from_dual`. + """ + win = gaussian(51, std=10, sym=True) + d_win = _calc_dual_canonical_window(win, 10) + win2 = _calc_dual_canonical_window(d_win, 10) + xp_assert_close(win2, win) + + +def test__calc_dual_canonical_window_exceptions(): + """Raise all exceptions in `_calc_dual_canonical_window`.""" + # Verify that calculation can fail: + with pytest.raises(ValueError, match="hop=5 is larger than window len.*"): + _calc_dual_canonical_window(np.ones(4), 5) + with pytest.raises(ValueError, match=".* Transform not invertible!"): + _calc_dual_canonical_window(np.array([.1, .2, .3, 0]), 4) + + # Verify that parameter `win` may not be integers: + with pytest.raises(ValueError, match="Parameter 'win' cannot be of int.*"): + _calc_dual_canonical_window(np.ones(4, dtype=int), 1) + + +def test_invalid_initializer_parameters(): + """Verify that exceptions get raised on invalid parameters when + instantiating ShortTimeFFT. """ + with pytest.raises(ValueError, match=r"Parameter win must be 1d, " + + r"but win.shape=\(2, 2\)!"): + ShortTimeFFT(np.ones((2, 2)), hop=4, fs=1) + with pytest.raises(ValueError, match="Parameter win must have " + + "finite entries"): + ShortTimeFFT(np.array([1, np.inf, 2, 3]), hop=4, fs=1) + with pytest.raises(ValueError, match="Parameter hop=0 is not " + + "an integer >= 1!"): + ShortTimeFFT(np.ones(4), hop=0, fs=1) + with pytest.raises(ValueError, match="Parameter hop=2.0 is not " + + "an integer >= 1!"): + # noinspection PyTypeChecker + ShortTimeFFT(np.ones(4), hop=2.0, fs=1) + with pytest.raises(ValueError, match=r"dual_win.shape=\(5,\) must equal " + + r"win.shape=\(4,\)!"): + ShortTimeFFT(np.ones(4), hop=2, fs=1, dual_win=np.ones(5)) + with pytest.raises(ValueError, match="Parameter dual_win must be " + + "a finite array!"): + ShortTimeFFT(np.ones(3), hop=2, fs=1, + dual_win=np.array([np.nan, 2, 3])) + + +def test_exceptions_properties_methods(): + """Verify that exceptions get raised when setting properties or calling + method of ShortTimeFFT to/with invalid values.""" + SFT = ShortTimeFFT(np.ones(8), hop=4, fs=1) + with pytest.raises(ValueError, match="Sampling interval T=-1 must be " + + "positive!"): + SFT.T = -1 + with pytest.raises(ValueError, match="Sampling frequency fs=-1 must be " + + "positive!"): + SFT.fs = -1 + with pytest.raises(ValueError, match="fft_mode='invalid_typ' not in " + + r"\('twosided', 'centered', " + + r"'onesided', 'onesided2X'\)!"): + SFT.fft_mode = 'invalid_typ' + with pytest.raises(ValueError, match="For scaling is None, " + + "fft_mode='onesided2X' is invalid.*"): + SFT.fft_mode = 'onesided2X' + with pytest.raises(ValueError, match="Attribute mfft=7 needs to be " + + "at least the window length.*"): + SFT.mfft = 7 + with pytest.raises(ValueError, match="scaling='invalid' not in.*"): + # noinspection PyTypeChecker + SFT.scale_to('invalid') + with pytest.raises(ValueError, match="phase_shift=3.0 has the unit .*"): + SFT.phase_shift = 3.0 + with pytest.raises(ValueError, match="-mfft < phase_shift < mfft " + + "does not hold.*"): + SFT.phase_shift = 2*SFT.mfft + with pytest.raises(ValueError, match="Parameter padding='invalid' not.*"): + # noinspection PyTypeChecker + g = SFT._x_slices(np.zeros(16), k_off=0, p0=0, p1=1, padding='invalid') + next(g) # execute generator + with pytest.raises(ValueError, match="Trend type must be 'linear' " + + "or 'constant'"): + # noinspection PyTypeChecker + SFT.stft_detrend(np.zeros(16), detr='invalid') + with pytest.raises(ValueError, match="Parameter detr=nan is not a str, " + + "function or None!"): + # noinspection PyTypeChecker + SFT.stft_detrend(np.zeros(16), detr=np.nan) + with pytest.raises(ValueError, match="Invalid Parameter p0=0, p1=200.*"): + SFT.p_range(100, 0, 200) + + with pytest.raises(ValueError, match="f_axis=0 may not be equal to " + + "t_axis=0!"): + SFT.istft(np.zeros((SFT.f_pts, 2)), t_axis=0, f_axis=0) + with pytest.raises(ValueError, match=r"S.shape\[f_axis\]=2 must be equal" + + " to self.f_pts=5.*"): + SFT.istft(np.zeros((2, 2))) + with pytest.raises(ValueError, match=r"S.shape\[t_axis\]=1 needs to have" + + " at least 2 slices.*"): + SFT.istft(np.zeros((SFT.f_pts, 1))) + with pytest.raises(ValueError, match=r".*\(k1=100\) <= \(k_max=12\) " + + "is false!$"): + SFT.istft(np.zeros((SFT.f_pts, 3)), k1=100) + with pytest.raises(ValueError, match=r"\(k1=1\) - \(k0=0\) = 1 has to " + + "be at least.* length 4!"): + SFT.istft(np.zeros((SFT.f_pts, 3)), k0=0, k1=1) + + with pytest.raises(ValueError, match=r"Parameter axes_seq='invalid' " + + r"not in \['tf', 'ft'\]!"): + # noinspection PyTypeChecker + SFT.extent(n=100, axes_seq='invalid') + with pytest.raises(ValueError, match="Attribute fft_mode=twosided must.*"): + SFT.fft_mode = 'twosided' + SFT.extent(n=100) + + +@pytest.mark.parametrize('m', ('onesided', 'onesided2X')) +def test_exceptions_fft_mode_complex_win(m: FFT_MODE_TYPE): + """Verify that one-sided spectra are not allowed with complex-valued + windows or with complex-valued signals. + + The reason being, the `rfft` function only accepts real-valued input. + """ + with pytest.raises(ValueError, + match=f"One-sided spectra, i.e., fft_mode='{m}'.*"): + ShortTimeFFT(np.ones(8)*1j, hop=4, fs=1, fft_mode=m) + + SFT = ShortTimeFFT(np.ones(8)*1j, hop=4, fs=1, fft_mode='twosided') + with pytest.raises(ValueError, + match=f"One-sided spectra, i.e., fft_mode='{m}'.*"): + SFT.fft_mode = m + + SFT = ShortTimeFFT(np.ones(8), hop=4, fs=1, scale_to='psd', fft_mode='onesided') + with pytest.raises(ValueError, match="Complex-valued `x` not allowed for self.*"): + SFT.stft(np.ones(8)*1j) + SFT.fft_mode = 'onesided2X' + with pytest.raises(ValueError, match="Complex-valued `x` not allowed for self.*"): + SFT.stft(np.ones(8)*1j) + + +def test_invalid_fft_mode_RuntimeError(): + """Ensure exception gets raised when property `fft_mode` is invalid. """ + SFT = ShortTimeFFT(np.ones(8), hop=4, fs=1) + SFT._fft_mode = 'invalid_typ' + + with pytest.raises(RuntimeError): + _ = SFT.f + with pytest.raises(RuntimeError): + SFT._fft_func(np.ones(8)) + with pytest.raises(RuntimeError): + SFT._ifft_func(np.ones(8)) + + +@pytest.mark.parametrize('win_params, Nx', [(('gaussian', 2.), 9), # in docstr + ('triang', 7), + (('kaiser', 4.0), 9), + (('exponential', None, 1.), 9), + (4.0, 9)]) +def test_from_window(win_params, Nx: int): + """Verify that `from_window()` handles parameters correctly. + + The window parameterizations are documented in the `get_window` docstring. + """ + w_sym, fs = get_window(win_params, Nx, fftbins=False), 16. + w_per = get_window(win_params, Nx, fftbins=True) + SFT0 = ShortTimeFFT(w_sym, hop=3, fs=fs, fft_mode='twosided', + scale_to='psd', phase_shift=1) + nperseg = len(w_sym) + noverlap = nperseg - SFT0.hop + SFT1 = ShortTimeFFT.from_window(win_params, fs, nperseg, noverlap, + symmetric_win=True, fft_mode='twosided', + scale_to='psd', phase_shift=1) + # periodic window: + SFT2 = ShortTimeFFT.from_window(win_params, fs, nperseg, noverlap, + symmetric_win=False, fft_mode='twosided', + scale_to='psd', phase_shift=1) + # Be informative when comparing instances: + xp_assert_equal(SFT1.win, SFT0.win) + xp_assert_close(SFT2.win, w_per / np.sqrt(sum(w_per**2) * fs)) + for n_ in ('hop', 'T', 'fft_mode', 'mfft', 'scaling', 'phase_shift'): + v0, v1, v2 = (getattr(SFT_, n_) for SFT_ in (SFT0, SFT1, SFT2)) + assert v1 == v0, f"SFT1.{n_}={v1} does not equal SFT0.{n_}={v0}" + assert v2 == v0, f"SFT2.{n_}={v2} does not equal SFT0.{n_}={v0}" + + +def test_dual_win_roundtrip(): + """Verify the duality of `win` and `dual_win`. + + Note that this test does not work for arbitrary windows, since dual windows + are not unique. It always works for invertible STFTs if the windows do not + overlap. + """ + # Non-standard values for keyword arguments (except for `scale_to`): + kw = dict(hop=4, fs=1, fft_mode='twosided', mfft=8, scale_to=None, + phase_shift=2) + SFT0 = ShortTimeFFT(np.ones(4), **kw) + SFT1 = ShortTimeFFT.from_dual(SFT0.dual_win, **kw) + xp_assert_close(SFT1.dual_win, SFT0.win) + + +@pytest.mark.parametrize('scale_to, fac_psd, fac_mag', + [(None, 0.25, 0.125), + ('magnitude', 2.0, 1), + ('psd', 1, 0.5)]) +def test_scaling(scale_to: Literal['magnitude', 'psd'], fac_psd, fac_mag): + """Verify scaling calculations. + + * Verify passing `scale_to`parameter to ``__init__(). + * Roundtrip while changing scaling factor. + """ + SFT = ShortTimeFFT(np.ones(4) * 2, hop=4, fs=1, scale_to=scale_to) + assert SFT.fac_psd == fac_psd + assert SFT.fac_magnitude == fac_mag + # increase coverage by accessing properties twice: + assert SFT.fac_psd == fac_psd + assert SFT.fac_magnitude == fac_mag + + x = np.fft.irfft([0, 0, 7, 0, 0, 0, 0]) # periodic signal + Sx = SFT.stft(x) + Sx_mag, Sx_psd = Sx * SFT.fac_magnitude, Sx * SFT.fac_psd + + SFT.scale_to('magnitude') + x_mag = SFT.istft(Sx_mag, k1=len(x)) + xp_assert_close(x_mag, x) + + SFT.scale_to('psd') + x_psd = SFT.istft(Sx_psd, k1=len(x)) + xp_assert_close(x_psd, x) + + +def test_scale_to(): + """Verify `scale_to()` method.""" + SFT = ShortTimeFFT(np.ones(4) * 2, hop=4, fs=1, scale_to=None) + + SFT.scale_to('magnitude') + assert SFT.scaling == 'magnitude' + assert SFT.fac_psd == 2.0 + assert SFT.fac_magnitude == 1 + + SFT.scale_to('psd') + assert SFT.scaling == 'psd' + assert SFT.fac_psd == 1 + assert SFT.fac_magnitude == 0.5 + + SFT.scale_to('psd') # needed for coverage + + for scale, s_fac in zip(('magnitude', 'psd'), (8, 4)): + SFT = ShortTimeFFT(np.ones(4) * 2, hop=4, fs=1, scale_to=None) + dual_win = SFT.dual_win.copy() + + SFT.scale_to(cast(Literal['magnitude', 'psd'], scale)) + xp_assert_close(SFT.dual_win, dual_win * s_fac) + + +def test_x_slices_padding(): + """Verify padding. + + The reference arrays were taken from the docstrings of `zero_ext`, + `const_ext`, `odd_ext()`, and `even_ext()` from the _array_tools module. + """ + SFT = ShortTimeFFT(np.ones(5), hop=4, fs=1) + x = np.array([[1, 2, 3, 4, 5], [0, 1, 4, 9, 16]], dtype=float) + d = {'zeros': [[[0, 0, 1, 2, 3], [0, 0, 0, 1, 4]], + [[3, 4, 5, 0, 0], [4, 9, 16, 0, 0]]], + 'edge': [[[1, 1, 1, 2, 3], [0, 0, 0, 1, 4]], + [[3, 4, 5, 5, 5], [4, 9, 16, 16, 16]]], + 'even': [[[3, 2, 1, 2, 3], [4, 1, 0, 1, 4]], + [[3, 4, 5, 4, 3], [4, 9, 16, 9, 4]]], + 'odd': [[[-1, 0, 1, 2, 3], [-4, -1, 0, 1, 4]], + [[3, 4, 5, 6, 7], [4, 9, 16, 23, 28]]]} + for p_, xx in d.items(): + gen = SFT._x_slices(np.array(x), 0, 0, 2, padding=cast(PAD_TYPE, p_)) + yy = np.array([y_.copy() for y_ in gen]) # due to inplace copying + xx = np.asarray(xx, dtype=np.float64) + xp_assert_equal(yy, xx, err_msg=f"Failed '{p_}' padding.") + + +def test_invertible(): + """Verify `invertible` property. """ + SFT = ShortTimeFFT(np.ones(8), hop=4, fs=1) + assert SFT.invertible + SFT = ShortTimeFFT(np.ones(8), hop=9, fs=1) + assert not SFT.invertible + + +def test_border_values(): + """Ensure that minimum and maximum values of slices are correct.""" + SFT = ShortTimeFFT(np.ones(8), hop=4, fs=1) + assert SFT.p_min == 0 + assert SFT.k_min == -4 + assert SFT.lower_border_end == (4, 1) + assert SFT.lower_border_end == (4, 1) # needed to test caching + assert SFT.p_max(10) == 4 + assert SFT.k_max(10) == 16 + assert SFT.upper_border_begin(10) == (4, 2) + # Raise exceptions: + with pytest.raises(ValueError, match="^Parameter n must be"): + SFT.upper_border_begin(3) + with pytest.raises(ValueError, match="^Parameter n must be"): + SFT._post_padding(3) + +def test_border_values_exotic(): + """Ensure that the border calculations are correct for windows with + zeros. """ + w = np.array([0, 0, 0, 0, 0, 0, 0, 1.]) + SFT = ShortTimeFFT(w, hop=1, fs=1) + assert SFT.lower_border_end == (0, 0) + + SFT = ShortTimeFFT(np.flip(w), hop=20, fs=1) + assert SFT.upper_border_begin(4) == (16, 1) + assert SFT.upper_border_begin(5) == (16, 1) + assert SFT.upper_border_begin(23) == (36, 2) + assert SFT.upper_border_begin(24) == (36, 2) + assert SFT.upper_border_begin(25) == (36, 2) + + SFT._hop = -1 # provoke unreachable line + with pytest.raises(RuntimeError): + _ = SFT.k_max(4) + with pytest.raises(RuntimeError): + _ = SFT.k_min + + +def test_t(): + """Verify that the times of the slices are correct. """ + SFT = ShortTimeFFT(np.ones(8), hop=4, fs=2) + assert SFT.T == 1/2 + assert SFT.fs == 2. + assert SFT.delta_t == 4 * 1/2 + t_stft = np.arange(0, SFT.p_max(10)) * SFT.delta_t + xp_assert_equal(SFT.t(10), t_stft) + xp_assert_equal(SFT.t(10, 1, 3), t_stft[1:3]) + SFT.T = 1/4 + assert SFT.T == 1/4 + assert SFT.fs == 4 + SFT.fs = 1/8 + assert SFT.fs == 1/8 + assert SFT.T == 8 + + +@pytest.mark.parametrize('fft_mode, f', + [('onesided', [0., 1., 2.]), + ('onesided2X', [0., 1., 2.]), + ('twosided', [0., 1., 2., -2., -1.]), + ('centered', [-2., -1., 0., 1., 2.])]) +def test_f(fft_mode: FFT_MODE_TYPE, f): + """Verify the frequency values property `f`.""" + SFT = ShortTimeFFT(np.ones(5), hop=4, fs=5, fft_mode=fft_mode, + scale_to='psd') + xp_assert_equal(SFT.f, f) + + +@pytest.mark.parametrize('n', [20, 21]) +@pytest.mark.parametrize('m', [5, 6]) +@pytest.mark.parametrize('fft_mode', ['onesided', 'centered']) +def test_extent(n, m, fft_mode: FFT_MODE_TYPE): + """Ensure that the `extent()` method is correct. """ + SFT = ShortTimeFFT(np.ones(m), hop=m, fs=m, fft_mode=fft_mode) + + t0 = SFT.t(n)[0] # first timestamp + t1 = SFT.t(n)[-1] + SFT.delta_t # last timestamp + 1 + t0c, t1c = t0 - SFT.delta_t / 2, t1 - SFT.delta_t / 2 # centered timestamps + + f0 = SFT.f[0] # first frequency + f1 = SFT.f[-1] + SFT.delta_f # last frequency + 1 + f0c, f1c = f0 - SFT.delta_f / 2, f1 - SFT.delta_f / 2 # centered frequencies + + assert SFT.extent(n, 'tf', False) == (t0, t1, f0, f1) + assert SFT.extent(n, 'ft', False) == (f0, f1, t0, t1) + assert SFT.extent(n, 'tf', True) == (t0c, t1c, f0c, f1c) + assert SFT.extent(n, 'ft', True) == (f0c, f1c, t0c, t1c) + + +def test_spectrogram(): + """Verify spectrogram and cross-spectrogram methods. """ + SFT = ShortTimeFFT(np.ones(8), hop=4, fs=1) + x, y = np.ones(10), np.arange(10) + X, Y = SFT.stft(x), SFT.stft(y) + xp_assert_close(SFT.spectrogram(x), X.real**2+X.imag**2) + xp_assert_close(SFT.spectrogram(x, y), X * Y.conj()) + + +@pytest.mark.parametrize('n', [8, 9]) +def test_fft_func_roundtrip(n: int): + """Test roundtrip `ifft_func(fft_func(x)) == x` for all permutations of + relevant parameters. """ + np.random.seed(2394795) + x0 = np.random.rand(n) + w, h_n = np.ones(n), 4 + + pp = dict( + fft_mode=get_args(FFT_MODE_TYPE), + mfft=[None, n, n+1, n+2], + scaling=[None, 'magnitude', 'psd'], + phase_shift=[None, -n+1, 0, n // 2, n-1]) + for f_typ, mfft, scaling, phase_shift in product(*pp.values()): + if f_typ == 'onesided2X' and scaling is None: + continue # this combination is forbidden + SFT = ShortTimeFFT(w, h_n, fs=n, fft_mode=f_typ, mfft=mfft, + scale_to=scaling, phase_shift=phase_shift) + X0 = SFT._fft_func(x0) + x1 = SFT._ifft_func(X0) + xp_assert_close(x0.astype(x1.dtype), x1, + err_msg="_fft_func() roundtrip failed for " + + f"{f_typ=}, {mfft=}, {scaling=}, {phase_shift=}") + + SFT = ShortTimeFFT(w, h_n, fs=1) + SFT._fft_mode = 'invalid_fft' # type: ignore + with pytest.raises(RuntimeError): + SFT._fft_func(x0) + with pytest.raises(RuntimeError): + SFT._ifft_func(x0) + + +@pytest.mark.parametrize('i', range(19)) +def test_impulse_roundtrip(i): + """Roundtrip for an impulse being at different positions `i`.""" + n = 19 + w, h_n = np.ones(8), 3 + x = np.zeros(n) + x[i] = 1 + + SFT = ShortTimeFFT(w, hop=h_n, fs=1, scale_to=None, phase_shift=None) + Sx = SFT.stft(x) + # test slicing the input signal into two parts: + n_q = SFT.nearest_k_p(n // 2) + Sx0 = SFT.stft(x[:n_q], padding='zeros') + Sx1 = SFT.stft(x[n_q:], padding='zeros') + q0_ub = SFT.upper_border_begin(n_q)[1] - SFT.p_min + q1_le = SFT.lower_border_end[1] - SFT.p_min + xp_assert_close(Sx0[:, :q0_ub], Sx[:, :q0_ub], err_msg=f"{i=}") + xp_assert_close(Sx1[:, q1_le:], Sx[:, q1_le-Sx1.shape[1]:], + err_msg=f"{i=}") + + Sx01 = np.hstack((Sx0[:, :q0_ub], + Sx0[:, q0_ub:] + Sx1[:, :q1_le], + Sx1[:, q1_le:])) + xp_assert_close(Sx, Sx01, atol=1e-8, err_msg=f"{i=}") + + y = SFT.istft(Sx, 0, n) + xp_assert_close(y, x, atol=1e-8, err_msg=f"{i=}") + y0 = SFT.istft(Sx, 0, n//2) + xp_assert_close(x[:n//2], y0, atol=1e-8, err_msg=f"{i=}") + y1 = SFT.istft(Sx, n // 2, n) + xp_assert_close(x[n // 2:], y1, atol=1e-8, err_msg=f"{i=}") + + +@pytest.mark.parametrize('hop', [1, 7, 8]) +def test_asymmetric_window_roundtrip(hop: int): + """An asymmetric window could uncover indexing problems. """ + np.random.seed(23371) + + w = np.arange(16) / 8 # must be of type float + w[len(w)//2:] = 1 + SFT = ShortTimeFFT(w, hop, fs=1) + + x = 10 * np.random.randn(64) + Sx = SFT.stft(x) + x1 = SFT.istft(Sx, k1=len(x)) + xp_assert_close(x1, x1, err_msg="Roundtrip for asymmetric window with " + + f" {hop=} failed!") + + +@pytest.mark.parametrize('m_num', [6, 7]) +def test_minimal_length_signal(m_num): + """Verify that the shortest allowed signal works. """ + SFT = ShortTimeFFT(np.ones(m_num), m_num//2, fs=1) + n = math.ceil(m_num/2) + x = np.ones(n) + Sx = SFT.stft(x) + x1 = SFT.istft(Sx, k1=n) + xp_assert_close(x1, x, err_msg=f"Roundtrip minimal length signal ({n=})" + + f" for {m_num} sample window failed!") + with pytest.raises(ValueError, match=rf"len\(x\)={n-1} must be >= ceil.*"): + SFT.stft(x[:-1]) + with pytest.raises(ValueError, match=rf"S.shape\[t_axis\]={Sx.shape[1]-1}" + f" needs to have at least {Sx.shape[1]} slices"): + SFT.istft(Sx[:, :-1], k1=n) + + +def test_tutorial_stft_sliding_win(): + """Verify example in "Sliding Windows" subsection from the "User Guide". + + In :ref:`tutorial_stft_sliding_win` (file ``signal.rst``) of the + :ref:`user_guide` the behavior the border behavior of + ``ShortTimeFFT(np.ones(6), 2, fs=1)`` with a 50 sample signal is discussed. + This test verifies the presented indexes. + """ + SFT = ShortTimeFFT(np.ones(6), 2, fs=1) + + # Lower border: + assert SFT.m_num_mid == 3, f"Slice middle is not 3 but {SFT.m_num_mid=}" + assert SFT.p_min == -1, f"Lowest slice {SFT.p_min=} is not -1" + assert SFT.k_min == -5, f"Lowest slice sample {SFT.p_min=} is not -5" + k_lb, p_lb = SFT.lower_border_end + assert p_lb == 2, f"First unaffected slice {p_lb=} is not 2" + assert k_lb == 5, f"First unaffected sample {k_lb=} is not 5" + + n = 50 # upper signal border + assert (p_max := SFT.p_max(n)) == 27, f"Last slice {p_max=} must be 27" + assert (k_max := SFT.k_max(n)) == 55, f"Last sample {k_max=} must be 55" + k_ub, p_ub = SFT.upper_border_begin(n) + assert p_ub == 24, f"First upper border slice {p_ub=} must be 24" + assert k_ub == 45, f"First upper border slice {k_ub=} must be 45" + + +def test_tutorial_stft_legacy_stft(): + """Verify STFT example in "Comparison with Legacy Implementation" from the + "User Guide". + + In :ref:`tutorial_stft_legacy_stft` (file ``signal.rst``) of the + :ref:`user_guide` the legacy and the new implementation are compared. + """ + fs, N = 200, 1001 # # 200 Hz sampling rate for 5 s signal + t_z = np.arange(N) / fs # time indexes for signal + z = np.exp(2j*np.pi * 70 * (t_z - 0.2 * t_z ** 2)) # complex-valued chirp + + nperseg, noverlap = 50, 40 + win = ('gaussian', 1e-2 * fs) # Gaussian with 0.01 s standard deviation + + # Legacy STFT: + f0_u, t0, Sz0_u = stft(z, fs, win, nperseg, noverlap, + return_onesided=False, scaling='spectrum') + Sz0 = fftshift(Sz0_u, axes=0) + + # New STFT: + SFT = ShortTimeFFT.from_window(win, fs, nperseg, noverlap, + fft_mode='centered', + scale_to='magnitude', phase_shift=None) + Sz1 = SFT.stft(z) + + xp_assert_close(Sz0, Sz1[:, 2:-1]) + + xp_assert_close((abs(Sz1[:, 1]).min(), abs(Sz1[:, 1]).max()), + (6.925060911593139e-07, 8.00271269218721e-07)) + + t0_r, z0_r = istft(Sz0_u, fs, win, nperseg, noverlap, input_onesided=False, + scaling='spectrum') + z1_r = SFT.istft(Sz1, k1=N) + assert len(z0_r) == N + 9 + xp_assert_close(z0_r[:N], z) + xp_assert_close(z1_r, z) + + # Spectrogram is just the absolute square of th STFT: + xp_assert_close(SFT.spectrogram(z), abs(Sz1) ** 2) + + +def test_tutorial_stft_legacy_spectrogram(): + """Verify spectrogram example in "Comparison with Legacy Implementation" + from the "User Guide". + + In :ref:`tutorial_stft_legacy_stft` (file ``signal.rst``) of the + :ref:`user_guide` the legacy and the new implementation are compared. + """ + fs, N = 200, 1001 # 200 Hz sampling rate for almost 5 s signal + t_z = np.arange(N) / fs # time indexes for signal + z = np.exp(2j*np.pi*70 * (t_z - 0.2*t_z**2)) # complex-valued sweep + + nperseg, noverlap = 50, 40 + win = ('gaussian', 1e-2 * fs) # Gaussian with 0.01 s standard dev. + + # Legacy spectrogram: + f2_u, t2, Sz2_u = spectrogram(z, fs, win, nperseg, noverlap, detrend=None, + return_onesided=False, scaling='spectrum', + mode='complex') + + f2, Sz2 = fftshift(f2_u), fftshift(Sz2_u, axes=0) + + # New STFT: + SFT = ShortTimeFFT.from_window(win, fs, nperseg, noverlap, + fft_mode='centered', scale_to='magnitude', + phase_shift=None) + Sz3 = SFT.stft(z, p0=0, p1=(N-noverlap) // SFT.hop, k_offset=nperseg // 2) + t3 = SFT.t(N, p0=0, p1=(N-noverlap) // SFT.hop, k_offset=nperseg // 2) + + xp_assert_close(t2, t3) + xp_assert_close(f2, SFT.f) + xp_assert_close(Sz2, Sz3) + + +def test_permute_axes(): + """Verify correctness of four-dimensional signal by permuting its + shape. """ + n = 25 + SFT = ShortTimeFFT(np.ones(8)/8, hop=3, fs=n) + x0 = np.arange(n, dtype=np.float64) + Sx0 = SFT.stft(x0) + Sx0 = Sx0.reshape((Sx0.shape[0], 1, 1, 1, Sx0.shape[-1])) + SxT = np.moveaxis(Sx0, (0, -1), (-1, 0)) + + atol = 2 * np.finfo(SFT.win.dtype).resolution + for i in range(4): + y = np.reshape(x0, np.roll((n, 1, 1, 1), i)) + Sy = SFT.stft(y, axis=i) + xp_assert_close(Sy, np.moveaxis(Sx0, 0, i)) + + yb0 = SFT.istft(Sy, k1=n, f_axis=i) + xp_assert_close(yb0, y, atol=atol) + # explicit t-axis parameter (for coverage): + yb1 = SFT.istft(Sy, k1=n, f_axis=i, t_axis=Sy.ndim-1) + xp_assert_close(yb1, y, atol=atol) + + SyT = np.moveaxis(Sy, (i, -1), (-1, i)) + xp_assert_close(SyT, np.moveaxis(SxT, 0, i)) + + ybT = SFT.istft(SyT, k1=n, t_axis=i, f_axis=-1) + xp_assert_close(ybT, y, atol=atol) + + +@pytest.mark.parametrize("fft_mode", + ('twosided', 'centered', 'onesided', 'onesided2X')) +def test_roundtrip_multidimensional(fft_mode: FFT_MODE_TYPE): + """Test roundtrip of a multidimensional input signal versus its components. + + This test can uncover potential problems with `fftshift()`. + """ + n = 9 + x = np.arange(4*n*2, dtype=np.float64).reshape(4, n, 2) + SFT = ShortTimeFFT(get_window('hann', 4), hop=2, fs=1, + scale_to='magnitude', fft_mode=fft_mode) + Sx = SFT.stft(x, axis=1) + y = SFT.istft(Sx, k1=n, f_axis=1, t_axis=-1) + xp_assert_close(y, x.astype(y.dtype), err_msg='Multidim. roundtrip failed!') + + for i, j in product(range(x.shape[0]), range(x.shape[2])): + y_ = SFT.istft(Sx[i, :, j, :], k1=n) + xp_assert_close(y_, x[i, :, j].astype(y_.dtype), + err_msg="Multidim. roundtrip for component " + + f"x[{i}, :, {j}] and {fft_mode=} failed!") + +@pytest.mark.parametrize("phase_shift", (0, 4, None)) +def test_roundtrip_two_dimensional(phase_shift: int|None): + """Test roundtrip of a 2 channel input signal with `mfft` set with different + values for `phase_shift` + + Tests for Issue https://github.com/scipy/scipy/issues/21671 + """ + n = 21 + SFT = ShortTimeFFT.from_window('hann', fs=1, nperseg=13, noverlap=7, + mfft=16, phase_shift=phase_shift) + x = np.arange(2*n, dtype=float).reshape(2, n) + Sx = SFT.stft(x) + y = SFT.istft(Sx, k1=n) + xp_assert_close(y, x, atol=2 * np.finfo(SFT.win.dtype).resolution, + err_msg='2-dim. roundtrip failed!') + + +@pytest.mark.parametrize('window, n, nperseg, noverlap', + [('boxcar', 100, 10, 0), # Test no overlap + ('boxcar', 100, 10, 9), # Test high overlap + ('bartlett', 101, 51, 26), # Test odd nperseg + ('hann', 1024, 256, 128), # Test defaults + (('tukey', 0.5), 1152, 256, 64), # Test Tukey + ('hann', 1024, 256, 255), # Test overlapped hann + ('boxcar', 100, 10, 3), # NOLA True, COLA False + ('bartlett', 101, 51, 37), # NOLA True, COLA False + ('hann', 1024, 256, 127), # NOLA True, COLA False + # NOLA True, COLA False: + (('tukey', 0.5), 1152, 256, 14), + ('hann', 1024, 256, 5)]) # NOLA True, COLA False +def test_roundtrip_windows(window, n: int, nperseg: int, noverlap: int): + """Roundtrip test adapted from `test_spectral.TestSTFT`. + + The parameters are taken from the methods test_roundtrip_real(), + test_roundtrip_nola_not_cola(), test_roundtrip_float32(), + test_roundtrip_complex(). + """ + np.random.seed(2394655) + + w = get_window(window, nperseg) + SFT = ShortTimeFFT(w, nperseg - noverlap, fs=1, fft_mode='twosided', + phase_shift=None) + + z = 10 * np.random.randn(n) + 10j * np.random.randn(n) + Sz = SFT.stft(z) + z1 = SFT.istft(Sz, k1=len(z)) + xp_assert_close(z, z1, err_msg="Roundtrip for complex values failed") + + x = 10 * np.random.randn(n) + Sx = SFT.stft(x) + x1 = SFT.istft(Sx, k1=len(z)) + xp_assert_close(x.astype(np.complex128), x1, + err_msg="Roundtrip for float values failed") + + x32 = x.astype(np.float32) + Sx32 = SFT.stft(x32) + x32_1 = SFT.istft(Sx32, k1=len(x32)) + x32_1_r = x32_1.real + xp_assert_close(x32, x32_1_r.astype(np.float32), + err_msg="Roundtrip for 32 Bit float values failed") + xp_assert_close(x32.imag, np.zeros_like(x32.imag), + err_msg="Roundtrip for 32 Bit float values failed") + + +@pytest.mark.parametrize('signal_type', ('real', 'complex')) +def test_roundtrip_complex_window(signal_type): + """Test roundtrip for complex-valued window function + + The purpose of this test is to check if the dual window is calculated + correctly for complex-valued windows. + """ + np.random.seed(1354654) + win = np.exp(2j*np.linspace(0, np.pi, 8)) + SFT = ShortTimeFFT(win, 3, fs=1, fft_mode='twosided') + + z = 10 * np.random.randn(11) + if signal_type == 'complex': + z = z + 2j * z + Sz = SFT.stft(z) + z1 = SFT.istft(Sz, k1=len(z)) + xp_assert_close(z.astype(np.complex128), z1, + err_msg="Roundtrip for complex-valued window failed") + + +def test_average_all_segments(): + """Compare `welch` function with stft mean. + + Ported from `TestSpectrogram.test_average_all_segments` from file + ``test__spectral.py``. + """ + x = np.random.randn(1024) + + fs = 1.0 + window = ('tukey', 0.25) + nperseg, noverlap = 16, 2 + fw, Pw = welch(x, fs, window, nperseg, noverlap) + SFT = ShortTimeFFT.from_window(window, fs, nperseg, noverlap, + fft_mode='onesided2X', scale_to='psd', + phase_shift=None) + # `welch` positions the window differently than the STFT: + P = SFT.spectrogram(x, detr='constant', p0=0, + p1=(len(x)-noverlap)//SFT.hop, k_offset=nperseg//2) + + xp_assert_close(SFT.f, fw) + xp_assert_close(np.mean(P, axis=-1), Pw) + + +@pytest.mark.parametrize('window, N, nperseg, noverlap, mfft', + # from test_roundtrip_padded_FFT: + [('hann', 1024, 256, 128, 512), + ('hann', 1024, 256, 128, 501), + ('boxcar', 100, 10, 0, 33), + (('tukey', 0.5), 1152, 256, 64, 1024), + # from test_roundtrip_padded_signal: + ('boxcar', 101, 10, 0, None), + ('hann', 1000, 256, 128, None), + # from test_roundtrip_boundary_extension: + ('boxcar', 100, 10, 0, None), + ('boxcar', 100, 10, 9, None)]) +@pytest.mark.parametrize('padding', get_args(PAD_TYPE)) +def test_stft_padding_roundtrip(window, N: int, nperseg: int, noverlap: int, + mfft: int, padding): + """Test the parameter 'padding' of `stft` with roundtrips. + + The STFT parametrizations were taken from the methods + `test_roundtrip_padded_FFT`, `test_roundtrip_padded_signal` and + `test_roundtrip_boundary_extension` from class `TestSTFT` in file + ``test_spectral.py``. Note that the ShortTimeFFT does not need the + concept of "boundary extension". + """ + x = normal_distribution.rvs(size=N, random_state=2909) # real signal + z = x * np.exp(1j * np.pi / 4) # complex signal + + SFT = ShortTimeFFT.from_window(window, 1, nperseg, noverlap, + fft_mode='twosided', mfft=mfft) + Sx = SFT.stft(x, padding=padding) + x1 = SFT.istft(Sx, k1=N) + xp_assert_close(x1, x.astype(np.complex128), + err_msg=f"Failed real roundtrip with '{padding}' padding") + + Sz = SFT.stft(z, padding=padding) + z1 = SFT.istft(Sz, k1=N) + xp_assert_close(z1, z, err_msg="Failed complex roundtrip with " + + f" '{padding}' padding") + + +@pytest.mark.parametrize('N_x', (128, 129, 255, 256, 1337)) # signal length +@pytest.mark.parametrize('w_size', (128, 256)) # window length +@pytest.mark.parametrize('t_step', (4, 64)) # SFT time hop +@pytest.mark.parametrize('f_c', (7., 23.)) # frequency of input sine +def test_energy_conservation(N_x: int, w_size: int, t_step: int, f_c: float): + """Test if a `psd`-scaled STFT conserves the L2 norm. + + This test is adapted from MNE-Python [1]_. Besides being battle-tested, + this test has the benefit of using non-standard window including + non-positive values and a 2d input signal. + + Since `ShortTimeFFT` requires the signal length `N_x` to be at least the + window length `w_size`, the parameter `N_x` was changed from + ``(127, 128, 255, 256, 1337)`` to ``(128, 129, 255, 256, 1337)`` to be + more useful. + + .. [1] File ``test_stft.py`` of MNE-Python + https://github.com/mne-tools/mne-python/blob/main/mne/time_frequency/tests/test_stft.py + """ + window = np.sin(np.arange(.5, w_size + .5) / w_size * np.pi) + SFT = ShortTimeFFT(window, t_step, fs=1000, fft_mode='onesided2X', + scale_to='psd') + atol = 2*np.finfo(window.dtype).resolution + N_x = max(N_x, w_size) # minimal sing + # Test with low frequency signal + t = np.arange(N_x).astype(np.float64) + x = np.sin(2 * np.pi * f_c * t * SFT.T) + x = np.array([x, x + 1.]) + X = SFT.stft(x) + xp = SFT.istft(X, k1=N_x) + + max_freq = SFT.f[np.argmax(np.sum(np.abs(X[0]) ** 2, axis=1))] + + assert X.shape[1] == SFT.f_pts + assert np.all(SFT.f >= 0.) + assert np.abs(max_freq - f_c) < 1. + xp_assert_close(x, xp, atol=atol) + + # check L2-norm squared (i.e., energy) conservation: + E_x = np.sum(x**2, axis=-1) * SFT.T # numerical integration + aX2 = X.real**2 + X.imag.real**2 + E_X = np.sum(np.sum(aX2, axis=-1) * SFT.delta_t, axis=-1) * SFT.delta_f + xp_assert_close(E_X, E_x, atol=atol) + + # Test with random signal + np.random.seed(2392795) + x = np.random.randn(2, N_x) + X = SFT.stft(x) + xp = SFT.istft(X, k1=N_x) + + assert X.shape[1] == SFT.f_pts + assert np.all(SFT.f >= 0.) + assert np.abs(max_freq - f_c) < 1. + xp_assert_close(x, xp, atol=atol) + + # check L2-norm squared (i.e., energy) conservation: + E_x = np.sum(x**2, axis=-1) * SFT.T # numeric integration + aX2 = X.real ** 2 + X.imag.real ** 2 + E_X = np.sum(np.sum(aX2, axis=-1) * SFT.delta_t, axis=-1) * SFT.delta_f + xp_assert_close(E_X, E_x, atol=atol) + + # Try with empty array + x = np.zeros((0, N_x)) + X = SFT.stft(x) + xp = SFT.istft(X, k1=N_x) + assert xp.shape == x.shape diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_signaltools.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_signaltools.py new file mode 100644 index 0000000000000000000000000000000000000000..7953fcb5ff7634f1c01d96920c1dd069cacf0c0c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_signaltools.py @@ -0,0 +1,3933 @@ +import sys + +from concurrent.futures import ThreadPoolExecutor, as_completed +from decimal import Decimal +from itertools import product +from math import gcd + +import pytest +from pytest import raises as assert_raises +from numpy.testing import ( + assert_equal, + assert_almost_equal, assert_array_equal, assert_array_almost_equal, + assert_allclose, assert_, assert_array_less, + suppress_warnings) +from numpy import array, arange +import numpy as np + +from scipy import fft as sp_fft +from scipy.ndimage import correlate1d +from scipy.optimize import fmin, linear_sum_assignment +from scipy import signal +from scipy.signal import ( + correlate, correlate2d, correlation_lags, convolve, convolve2d, + fftconvolve, oaconvolve, choose_conv_method, envelope, + hilbert, hilbert2, lfilter, lfilter_zi, filtfilt, butter, zpk2tf, zpk2sos, + invres, invresz, vectorstrength, lfiltic, tf2sos, sosfilt, sosfiltfilt, + sosfilt_zi, tf2zpk, BadCoefficients, detrend, unique_roots, residue, + residuez) +from scipy.signal.windows import hann +from scipy.signal._signaltools import (_filtfilt_gust, _compute_factors, + _group_poles) +from scipy.signal._upfirdn import _upfirdn_modes +from scipy._lib import _testutils +from scipy._lib._array_api import xp_assert_close +from scipy._lib._util import ComplexWarning, np_long, np_ulong + + +class _TestConvolve: + + def test_basic(self): + a = [3, 4, 5, 6, 5, 4] + b = [1, 2, 3] + c = convolve(a, b) + assert_array_equal(c, array([3, 10, 22, 28, 32, 32, 23, 12])) + + def test_same(self): + a = [3, 4, 5] + b = [1, 2, 3, 4] + c = convolve(a, b, mode="same") + assert_array_equal(c, array([10, 22, 34])) + + def test_same_eq(self): + a = [3, 4, 5] + b = [1, 2, 3] + c = convolve(a, b, mode="same") + assert_array_equal(c, array([10, 22, 22])) + + def test_complex(self): + x = array([1 + 1j, 2 + 1j, 3 + 1j]) + y = array([1 + 1j, 2 + 1j]) + z = convolve(x, y) + assert_array_equal(z, array([2j, 2 + 6j, 5 + 8j, 5 + 5j])) + + def test_zero_rank(self): + a = 1289 + b = 4567 + c = convolve(a, b) + assert_equal(c, a * b) + + def test_broadcastable(self): + a = np.arange(27).reshape(3, 3, 3) + b = np.arange(3) + for i in range(3): + b_shape = [1]*3 + b_shape[i] = 3 + x = convolve(a, b.reshape(b_shape), method='direct') + y = convolve(a, b.reshape(b_shape), method='fft') + assert_allclose(x, y) + + def test_single_element(self): + a = array([4967]) + b = array([3920]) + c = convolve(a, b) + assert_equal(c, a * b) + + def test_2d_arrays(self): + a = [[1, 2, 3], [3, 4, 5]] + b = [[2, 3, 4], [4, 5, 6]] + c = convolve(a, b) + d = array([[2, 7, 16, 17, 12], + [10, 30, 62, 58, 38], + [12, 31, 58, 49, 30]]) + assert_array_equal(c, d) + + def test_input_swapping(self): + small = arange(8).reshape(2, 2, 2) + big = 1j * arange(27).reshape(3, 3, 3) + big += arange(27)[::-1].reshape(3, 3, 3) + + out_array = array( + [[[0 + 0j, 26 + 0j, 25 + 1j, 24 + 2j], + [52 + 0j, 151 + 5j, 145 + 11j, 93 + 11j], + [46 + 6j, 133 + 23j, 127 + 29j, 81 + 23j], + [40 + 12j, 98 + 32j, 93 + 37j, 54 + 24j]], + + [[104 + 0j, 247 + 13j, 237 + 23j, 135 + 21j], + [282 + 30j, 632 + 96j, 604 + 124j, 330 + 86j], + [246 + 66j, 548 + 180j, 520 + 208j, 282 + 134j], + [142 + 66j, 307 + 161j, 289 + 179j, 153 + 107j]], + + [[68 + 36j, 157 + 103j, 147 + 113j, 81 + 75j], + [174 + 138j, 380 + 348j, 352 + 376j, 186 + 230j], + [138 + 174j, 296 + 432j, 268 + 460j, 138 + 278j], + [70 + 138j, 145 + 323j, 127 + 341j, 63 + 197j]], + + [[32 + 72j, 68 + 166j, 59 + 175j, 30 + 100j], + [68 + 192j, 139 + 433j, 117 + 455j, 57 + 255j], + [38 + 222j, 73 + 499j, 51 + 521j, 21 + 291j], + [12 + 144j, 20 + 318j, 7 + 331j, 0 + 182j]]]) + + assert_array_equal(convolve(small, big, 'full'), out_array) + assert_array_equal(convolve(big, small, 'full'), out_array) + assert_array_equal(convolve(small, big, 'same'), + out_array[1:3, 1:3, 1:3]) + assert_array_equal(convolve(big, small, 'same'), + out_array[0:3, 0:3, 0:3]) + assert_array_equal(convolve(small, big, 'valid'), + out_array[1:3, 1:3, 1:3]) + assert_array_equal(convolve(big, small, 'valid'), + out_array[1:3, 1:3, 1:3]) + + def test_invalid_params(self): + a = [3, 4, 5] + b = [1, 2, 3] + assert_raises(ValueError, convolve, a, b, mode='spam') + assert_raises(ValueError, convolve, a, b, mode='eggs', method='fft') + assert_raises(ValueError, convolve, a, b, mode='ham', method='direct') + assert_raises(ValueError, convolve, a, b, mode='full', method='bacon') + assert_raises(ValueError, convolve, a, b, mode='same', method='bacon') + + +class TestConvolve(_TestConvolve): + + def test_valid_mode2(self): + # See gh-5897 + a = [1, 2, 3, 6, 5, 3] + b = [2, 3, 4, 5, 3, 4, 2, 2, 1] + expected = [70, 78, 73, 65] + + out = convolve(a, b, 'valid') + assert_array_equal(out, expected) + + out = convolve(b, a, 'valid') + assert_array_equal(out, expected) + + a = [1 + 5j, 2 - 1j, 3 + 0j] + b = [2 - 3j, 1 + 0j] + expected = [2 - 3j, 8 - 10j] + + out = convolve(a, b, 'valid') + assert_array_equal(out, expected) + + out = convolve(b, a, 'valid') + assert_array_equal(out, expected) + + def test_same_mode(self): + a = [1, 2, 3, 3, 1, 2] + b = [1, 4, 3, 4, 5, 6, 7, 4, 3, 2, 1, 1, 3] + c = convolve(a, b, 'same') + d = array([57, 61, 63, 57, 45, 36]) + assert_array_equal(c, d) + + def test_invalid_shapes(self): + # By "invalid," we mean that no one + # array has dimensions that are all at + # least as large as the corresponding + # dimensions of the other array. This + # setup should throw a ValueError. + a = np.arange(1, 7).reshape((2, 3)) + b = np.arange(-6, 0).reshape((3, 2)) + + assert_raises(ValueError, convolve, *(a, b), **{'mode': 'valid'}) + assert_raises(ValueError, convolve, *(b, a), **{'mode': 'valid'}) + + def test_convolve_method(self, n=100): + # this types data structure was manually encoded instead of + # using custom filters on the soon-to-be-removed np.sctypes + types = {'uint16', 'uint64', 'int64', 'int32', + 'complex128', 'float64', 'float16', + 'complex64', 'float32', 'int16', + 'uint8', 'uint32', 'int8', 'bool'} + args = [(t1, t2, mode) for t1 in types for t2 in types + for mode in ['valid', 'full', 'same']] + + # These are random arrays, which means test is much stronger than + # convolving testing by convolving two np.ones arrays + rng = np.random.RandomState(42) + array_types = {'i': rng.choice([0, 1], size=n), + 'f': rng.randn(n)} + array_types['b'] = array_types['u'] = array_types['i'] + array_types['c'] = array_types['f'] + 0.5j*array_types['f'] + + for t1, t2, mode in args: + x1 = array_types[np.dtype(t1).kind].astype(t1) + x2 = array_types[np.dtype(t2).kind].astype(t2) + + results = {key: convolve(x1, x2, method=key, mode=mode) + for key in ['fft', 'direct']} + + assert_equal(results['fft'].dtype, results['direct'].dtype) + + if 'bool' in t1 and 'bool' in t2: + assert_equal(choose_conv_method(x1, x2), 'direct') + continue + + # Found by experiment. Found approx smallest value for (rtol, atol) + # threshold to have tests pass. + if any([t in {'complex64', 'float32'} for t in [t1, t2]]): + kwargs = {'rtol': 1.0e-4, 'atol': 1e-6} + elif 'float16' in [t1, t2]: + # atol is default for np.allclose + kwargs = {'rtol': 1e-3, 'atol': 1e-3} + else: + # defaults for np.allclose (different from assert_allclose) + kwargs = {'rtol': 1e-5, 'atol': 1e-8} + + assert_allclose(results['fft'], results['direct'], **kwargs) + + def test_convolve_method_large_input(self): + # This is really a test that convolving two large integers goes to the + # direct method even if they're in the fft method. + for n in [10, 20, 50, 51, 52, 53, 54, 60, 62]: + z = np.array([2**n], dtype=np.int64) + fft = convolve(z, z, method='fft') + direct = convolve(z, z, method='direct') + + # this is the case when integer precision gets to us + # issue #6076 has more detail, hopefully more tests after resolved + if n < 50: + assert_equal(fft, direct) + assert_equal(fft, 2**(2*n)) + assert_equal(direct, 2**(2*n)) + + def test_mismatched_dims(self): + # Input arrays should have the same number of dimensions + assert_raises(ValueError, convolve, [1], 2, method='direct') + assert_raises(ValueError, convolve, 1, [2], method='direct') + assert_raises(ValueError, convolve, [1], 2, method='fft') + assert_raises(ValueError, convolve, 1, [2], method='fft') + assert_raises(ValueError, convolve, [1], [[2]]) + assert_raises(ValueError, convolve, [3], 2) + + @pytest.mark.thread_unsafe + def test_dtype_deprecation(self): + # gh-21211 + a = np.asarray([1, 2, 3, 6, 5, 3], dtype=object) + b = np.asarray([2, 3, 4, 5, 3, 4, 2, 2, 1], dtype=object) + with pytest.deprecated_call(match="dtype=object is not supported"): + convolve(a, b) + + +class _TestConvolve2d: + + def test_2d_arrays(self): + a = [[1, 2, 3], [3, 4, 5]] + b = [[2, 3, 4], [4, 5, 6]] + d = array([[2, 7, 16, 17, 12], + [10, 30, 62, 58, 38], + [12, 31, 58, 49, 30]]) + e = convolve2d(a, b) + assert_array_equal(e, d) + + def test_valid_mode(self): + e = [[2, 3, 4, 5, 6, 7, 8], [4, 5, 6, 7, 8, 9, 10]] + f = [[1, 2, 3], [3, 4, 5]] + h = array([[62, 80, 98, 116, 134]]) + + g = convolve2d(e, f, 'valid') + assert_array_equal(g, h) + + # See gh-5897 + g = convolve2d(f, e, 'valid') + assert_array_equal(g, h) + + def test_valid_mode_complx(self): + e = [[2, 3, 4, 5, 6, 7, 8], [4, 5, 6, 7, 8, 9, 10]] + f = np.array([[1, 2, 3], [3, 4, 5]], dtype=complex) + 1j + h = array([[62.+24.j, 80.+30.j, 98.+36.j, 116.+42.j, 134.+48.j]]) + + g = convolve2d(e, f, 'valid') + assert_array_almost_equal(g, h) + + # See gh-5897 + g = convolve2d(f, e, 'valid') + assert_array_equal(g, h) + + def test_fillvalue(self): + a = [[1, 2, 3], [3, 4, 5]] + b = [[2, 3, 4], [4, 5, 6]] + fillval = 1 + c = convolve2d(a, b, 'full', 'fill', fillval) + d = array([[24, 26, 31, 34, 32], + [28, 40, 62, 64, 52], + [32, 46, 67, 62, 48]]) + assert_array_equal(c, d) + + def test_fillvalue_errors(self): + msg = "could not cast `fillvalue` directly to the output " + with np.testing.suppress_warnings() as sup: + sup.filter(ComplexWarning, "Casting complex values") + with assert_raises(ValueError, match=msg): + convolve2d([[1]], [[1, 2]], fillvalue=1j) + + msg = "`fillvalue` must be scalar or an array with " + with assert_raises(ValueError, match=msg): + convolve2d([[1]], [[1, 2]], fillvalue=[1, 2]) + + def test_fillvalue_empty(self): + # Check that fillvalue being empty raises an error: + assert_raises(ValueError, convolve2d, [[1]], [[1, 2]], + fillvalue=[]) + + def test_wrap_boundary(self): + a = [[1, 2, 3], [3, 4, 5]] + b = [[2, 3, 4], [4, 5, 6]] + c = convolve2d(a, b, 'full', 'wrap') + d = array([[80, 80, 74, 80, 80], + [68, 68, 62, 68, 68], + [80, 80, 74, 80, 80]]) + assert_array_equal(c, d) + + def test_sym_boundary(self): + a = [[1, 2, 3], [3, 4, 5]] + b = [[2, 3, 4], [4, 5, 6]] + c = convolve2d(a, b, 'full', 'symm') + d = array([[34, 30, 44, 62, 66], + [52, 48, 62, 80, 84], + [82, 78, 92, 110, 114]]) + assert_array_equal(c, d) + + @pytest.mark.parametrize('func', [convolve2d, correlate2d]) + @pytest.mark.parametrize('boundary, expected', + [('symm', [[37.0, 42.0, 44.0, 45.0]]), + ('wrap', [[43.0, 44.0, 42.0, 39.0]])]) + def test_same_with_boundary(self, func, boundary, expected): + # Test boundary='symm' and boundary='wrap' with a "long" kernel. + # The size of the kernel requires that the values in the "image" + # be extended more than once to handle the requested boundary method. + # This is a regression test for gh-8684 and gh-8814. + image = np.array([[2.0, -1.0, 3.0, 4.0]]) + kernel = np.ones((1, 21)) + result = func(image, kernel, mode='same', boundary=boundary) + # The expected results were calculated "by hand". Because the + # kernel is all ones, the same result is expected for convolve2d + # and correlate2d. + assert_array_equal(result, expected) + + def test_boundary_extension_same(self): + # Regression test for gh-12686. + # Use ndimage.convolve with appropriate arguments to create the + # expected result. + import scipy.ndimage as ndi + a = np.arange(1, 10*3+1, dtype=float).reshape(10, 3) + b = np.arange(1, 10*10+1, dtype=float).reshape(10, 10) + c = convolve2d(a, b, mode='same', boundary='wrap') + assert_array_equal(c, ndi.convolve(a, b, mode='wrap', origin=(-1, -1))) + + def test_boundary_extension_full(self): + # Regression test for gh-12686. + # Use ndimage.convolve with appropriate arguments to create the + # expected result. + import scipy.ndimage as ndi + a = np.arange(1, 3*3+1, dtype=float).reshape(3, 3) + b = np.arange(1, 6*6+1, dtype=float).reshape(6, 6) + c = convolve2d(a, b, mode='full', boundary='wrap') + apad = np.pad(a, ((3, 3), (3, 3)), 'wrap') + assert_array_equal(c, ndi.convolve(apad, b, mode='wrap')[:-1, :-1]) + + def test_invalid_shapes(self): + # By "invalid," we mean that no one + # array has dimensions that are all at + # least as large as the corresponding + # dimensions of the other array. This + # setup should throw a ValueError. + a = np.arange(1, 7).reshape((2, 3)) + b = np.arange(-6, 0).reshape((3, 2)) + + assert_raises(ValueError, convolve2d, *(a, b), **{'mode': 'valid'}) + assert_raises(ValueError, convolve2d, *(b, a), **{'mode': 'valid'}) + + +class TestConvolve2d(_TestConvolve2d): + + def test_same_mode(self): + e = [[1, 2, 3], [3, 4, 5]] + f = [[2, 3, 4, 5, 6, 7, 8], [4, 5, 6, 7, 8, 9, 10]] + g = convolve2d(e, f, 'same') + h = array([[22, 28, 34], + [80, 98, 116]]) + assert_array_equal(g, h) + + def test_valid_mode2(self): + # See gh-5897 + e = [[1, 2, 3], [3, 4, 5]] + f = [[2, 3, 4, 5, 6, 7, 8], [4, 5, 6, 7, 8, 9, 10]] + expected = [[62, 80, 98, 116, 134]] + + out = convolve2d(e, f, 'valid') + assert_array_equal(out, expected) + + out = convolve2d(f, e, 'valid') + assert_array_equal(out, expected) + + e = [[1 + 1j, 2 - 3j], [3 + 1j, 4 + 0j]] + f = [[2 - 1j, 3 + 2j, 4 + 0j], [4 - 0j, 5 + 1j, 6 - 3j]] + expected = [[27 - 1j, 46. + 2j]] + + out = convolve2d(e, f, 'valid') + assert_array_equal(out, expected) + + # See gh-5897 + out = convolve2d(f, e, 'valid') + assert_array_equal(out, expected) + + def test_consistency_convolve_funcs(self): + # Compare np.convolve, signal.convolve, signal.convolve2d + a = np.arange(5) + b = np.array([3.2, 1.4, 3]) + for mode in ['full', 'valid', 'same']: + assert_almost_equal(np.convolve(a, b, mode=mode), + signal.convolve(a, b, mode=mode)) + assert_almost_equal(np.squeeze( + signal.convolve2d([a], [b], mode=mode)), + signal.convolve(a, b, mode=mode)) + + def test_invalid_dims(self): + assert_raises(ValueError, convolve2d, 3, 4) + assert_raises(ValueError, convolve2d, [3], [4]) + assert_raises(ValueError, convolve2d, [[[3]]], [[[4]]]) + + @pytest.mark.slow + @pytest.mark.xfail_on_32bit("Can't create large array for test") + def test_large_array(self): + # Test indexing doesn't overflow an int (gh-10761) + n = 2**31 // (1000 * np.int64().itemsize) + _testutils.check_free_memory(2 * n * 1001 * np.int64().itemsize / 1e6) + + # Create a chequered pattern of 1s and 0s + a = np.zeros(1001 * n, dtype=np.int64) + a[::2] = 1 + a = np.lib.stride_tricks.as_strided(a, shape=(n, 1000), strides=(8008, 8)) + + count = signal.convolve2d(a, [[1, 1]]) + fails = np.where(count > 1) + assert fails[0].size == 0 + + +class TestFFTConvolve: + + @pytest.mark.parametrize('axes', ['', None, 0, [0], -1, [-1]]) + def test_real(self, axes): + a = array([1, 2, 3]) + expected = array([1, 4, 10, 12, 9.]) + + if axes == '': + out = fftconvolve(a, a) + else: + out = fftconvolve(a, a, axes=axes) + + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', [1, [1], -1, [-1]]) + def test_real_axes(self, axes): + a = array([1, 2, 3]) + expected = array([1, 4, 10, 12, 9.]) + + a = np.tile(a, [2, 1]) + expected = np.tile(expected, [2, 1]) + + out = fftconvolve(a, a, axes=axes) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', ['', None, 0, [0], -1, [-1]]) + def test_complex(self, axes): + a = array([1 + 1j, 2 + 2j, 3 + 3j]) + expected = array([0 + 2j, 0 + 8j, 0 + 20j, 0 + 24j, 0 + 18j]) + + if axes == '': + out = fftconvolve(a, a) + else: + out = fftconvolve(a, a, axes=axes) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', [1, [1], -1, [-1]]) + def test_complex_axes(self, axes): + a = array([1 + 1j, 2 + 2j, 3 + 3j]) + expected = array([0 + 2j, 0 + 8j, 0 + 20j, 0 + 24j, 0 + 18j]) + + a = np.tile(a, [2, 1]) + expected = np.tile(expected, [2, 1]) + + out = fftconvolve(a, a, axes=axes) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', ['', + None, + [0, 1], + [1, 0], + [0, -1], + [-1, 0], + [-2, 1], + [1, -2], + [-2, -1], + [-1, -2]]) + def test_2d_real_same(self, axes): + a = array([[1, 2, 3], + [4, 5, 6]]) + expected = array([[1, 4, 10, 12, 9], + [8, 26, 56, 54, 36], + [16, 40, 73, 60, 36]]) + + if axes == '': + out = fftconvolve(a, a) + else: + out = fftconvolve(a, a, axes=axes) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', [[1, 2], + [2, 1], + [1, -1], + [-1, 1], + [-2, 2], + [2, -2], + [-2, -1], + [-1, -2]]) + def test_2d_real_same_axes(self, axes): + a = array([[1, 2, 3], + [4, 5, 6]]) + expected = array([[1, 4, 10, 12, 9], + [8, 26, 56, 54, 36], + [16, 40, 73, 60, 36]]) + + a = np.tile(a, [2, 1, 1]) + expected = np.tile(expected, [2, 1, 1]) + + out = fftconvolve(a, a, axes=axes) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', ['', + None, + [0, 1], + [1, 0], + [0, -1], + [-1, 0], + [-2, 1], + [1, -2], + [-2, -1], + [-1, -2]]) + def test_2d_complex_same(self, axes): + a = array([[1 + 2j, 3 + 4j, 5 + 6j], + [2 + 1j, 4 + 3j, 6 + 5j]]) + expected = array([ + [-3 + 4j, -10 + 20j, -21 + 56j, -18 + 76j, -11 + 60j], + [10j, 44j, 118j, 156j, 122j], + [3 + 4j, 10 + 20j, 21 + 56j, 18 + 76j, 11 + 60j] + ]) + + if axes == '': + out = fftconvolve(a, a) + else: + out = fftconvolve(a, a, axes=axes) + + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', [[1, 2], + [2, 1], + [1, -1], + [-1, 1], + [-2, 2], + [2, -2], + [-2, -1], + [-1, -2]]) + def test_2d_complex_same_axes(self, axes): + a = array([[1 + 2j, 3 + 4j, 5 + 6j], + [2 + 1j, 4 + 3j, 6 + 5j]]) + expected = array([ + [-3 + 4j, -10 + 20j, -21 + 56j, -18 + 76j, -11 + 60j], + [10j, 44j, 118j, 156j, 122j], + [3 + 4j, 10 + 20j, 21 + 56j, 18 + 76j, 11 + 60j] + ]) + + a = np.tile(a, [2, 1, 1]) + expected = np.tile(expected, [2, 1, 1]) + + out = fftconvolve(a, a, axes=axes) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', ['', None, 0, [0], -1, [-1]]) + def test_real_same_mode(self, axes): + a = array([1, 2, 3]) + b = array([3, 3, 5, 6, 8, 7, 9, 0, 1]) + expected_1 = array([35., 41., 47.]) + expected_2 = array([9., 20., 25., 35., 41., 47., 39., 28., 2.]) + + if axes == '': + out = fftconvolve(a, b, 'same') + else: + out = fftconvolve(a, b, 'same', axes=axes) + assert_array_almost_equal(out, expected_1) + + if axes == '': + out = fftconvolve(b, a, 'same') + else: + out = fftconvolve(b, a, 'same', axes=axes) + assert_array_almost_equal(out, expected_2) + + @pytest.mark.parametrize('axes', [1, -1, [1], [-1]]) + def test_real_same_mode_axes(self, axes): + a = array([1, 2, 3]) + b = array([3, 3, 5, 6, 8, 7, 9, 0, 1]) + expected_1 = array([35., 41., 47.]) + expected_2 = array([9., 20., 25., 35., 41., 47., 39., 28., 2.]) + + a = np.tile(a, [2, 1]) + b = np.tile(b, [2, 1]) + expected_1 = np.tile(expected_1, [2, 1]) + expected_2 = np.tile(expected_2, [2, 1]) + + out = fftconvolve(a, b, 'same', axes=axes) + assert_array_almost_equal(out, expected_1) + + out = fftconvolve(b, a, 'same', axes=axes) + assert_array_almost_equal(out, expected_2) + + @pytest.mark.parametrize('axes', ['', None, 0, [0], -1, [-1]]) + def test_valid_mode_real(self, axes): + # See gh-5897 + a = array([3, 2, 1]) + b = array([3, 3, 5, 6, 8, 7, 9, 0, 1]) + expected = array([24., 31., 41., 43., 49., 25., 12.]) + + if axes == '': + out = fftconvolve(a, b, 'valid') + else: + out = fftconvolve(a, b, 'valid', axes=axes) + assert_array_almost_equal(out, expected) + + if axes == '': + out = fftconvolve(b, a, 'valid') + else: + out = fftconvolve(b, a, 'valid', axes=axes) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', [1, [1]]) + def test_valid_mode_real_axes(self, axes): + # See gh-5897 + a = array([3, 2, 1]) + b = array([3, 3, 5, 6, 8, 7, 9, 0, 1]) + expected = array([24., 31., 41., 43., 49., 25., 12.]) + + a = np.tile(a, [2, 1]) + b = np.tile(b, [2, 1]) + expected = np.tile(expected, [2, 1]) + + out = fftconvolve(a, b, 'valid', axes=axes) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', ['', None, 0, [0], -1, [-1]]) + def test_valid_mode_complex(self, axes): + a = array([3 - 1j, 2 + 7j, 1 + 0j]) + b = array([3 + 2j, 3 - 3j, 5 + 0j, 6 - 1j, 8 + 0j]) + expected = array([45. + 12.j, 30. + 23.j, 48 + 32.j]) + + if axes == '': + out = fftconvolve(a, b, 'valid') + else: + out = fftconvolve(a, b, 'valid', axes=axes) + assert_array_almost_equal(out, expected) + + if axes == '': + out = fftconvolve(b, a, 'valid') + else: + out = fftconvolve(b, a, 'valid', axes=axes) + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', [1, [1], -1, [-1]]) + def test_valid_mode_complex_axes(self, axes): + a = array([3 - 1j, 2 + 7j, 1 + 0j]) + b = array([3 + 2j, 3 - 3j, 5 + 0j, 6 - 1j, 8 + 0j]) + expected = array([45. + 12.j, 30. + 23.j, 48 + 32.j]) + + a = np.tile(a, [2, 1]) + b = np.tile(b, [2, 1]) + expected = np.tile(expected, [2, 1]) + + out = fftconvolve(a, b, 'valid', axes=axes) + assert_array_almost_equal(out, expected) + + out = fftconvolve(b, a, 'valid', axes=axes) + assert_array_almost_equal(out, expected) + + def test_valid_mode_ignore_nonaxes(self): + # See gh-5897 + a = array([3, 2, 1]) + b = array([3, 3, 5, 6, 8, 7, 9, 0, 1]) + expected = array([24., 31., 41., 43., 49., 25., 12.]) + + a = np.tile(a, [2, 1]) + b = np.tile(b, [1, 1]) + expected = np.tile(expected, [2, 1]) + + out = fftconvolve(a, b, 'valid', axes=1) + assert_array_almost_equal(out, expected) + + def test_empty(self): + # Regression test for #1745: crashes with 0-length input. + assert_(fftconvolve([], []).size == 0) + assert_(fftconvolve([5, 6], []).size == 0) + assert_(fftconvolve([], [7]).size == 0) + + def test_zero_rank(self): + a = array(4967) + b = array(3920) + out = fftconvolve(a, b) + assert_equal(out, a * b) + + def test_single_element(self): + a = array([4967]) + b = array([3920]) + out = fftconvolve(a, b) + assert_equal(out, a * b) + + @pytest.mark.parametrize('axes', ['', None, 0, [0], -1, [-1]]) + def test_random_data(self, axes): + np.random.seed(1234) + a = np.random.rand(1233) + 1j * np.random.rand(1233) + b = np.random.rand(1321) + 1j * np.random.rand(1321) + expected = np.convolve(a, b, 'full') + + if axes == '': + out = fftconvolve(a, b, 'full') + else: + out = fftconvolve(a, b, 'full', axes=axes) + assert_(np.allclose(out, expected, rtol=1e-10)) + + @pytest.mark.parametrize('axes', [1, [1], -1, [-1]]) + def test_random_data_axes(self, axes): + np.random.seed(1234) + a = np.random.rand(1233) + 1j * np.random.rand(1233) + b = np.random.rand(1321) + 1j * np.random.rand(1321) + expected = np.convolve(a, b, 'full') + + a = np.tile(a, [2, 1]) + b = np.tile(b, [2, 1]) + expected = np.tile(expected, [2, 1]) + + out = fftconvolve(a, b, 'full', axes=axes) + assert_(np.allclose(out, expected, rtol=1e-10)) + + @pytest.mark.parametrize('axes', [[1, 4], + [4, 1], + [1, -1], + [-1, 1], + [-4, 4], + [4, -4], + [-4, -1], + [-1, -4]]) + def test_random_data_multidim_axes(self, axes): + a_shape, b_shape = (123, 22), (132, 11) + np.random.seed(1234) + a = np.random.rand(*a_shape) + 1j * np.random.rand(*a_shape) + b = np.random.rand(*b_shape) + 1j * np.random.rand(*b_shape) + expected = convolve2d(a, b, 'full') + + a = a[:, :, None, None, None] + b = b[:, :, None, None, None] + expected = expected[:, :, None, None, None] + + a = np.moveaxis(a.swapaxes(0, 2), 1, 4) + b = np.moveaxis(b.swapaxes(0, 2), 1, 4) + expected = np.moveaxis(expected.swapaxes(0, 2), 1, 4) + + # use 1 for dimension 2 in a and 3 in b to test broadcasting + a = np.tile(a, [2, 1, 3, 1, 1]) + b = np.tile(b, [2, 1, 1, 4, 1]) + expected = np.tile(expected, [2, 1, 3, 4, 1]) + + out = fftconvolve(a, b, 'full', axes=axes) + assert_allclose(out, expected, rtol=1e-10, atol=1e-10) + + @pytest.mark.slow + @pytest.mark.parametrize( + 'n', + list(range(1, 100)) + + list(range(1000, 1500)) + + np.random.RandomState(1234).randint(1001, 10000, 5).tolist()) + def test_many_sizes(self, n): + a = np.random.rand(n) + 1j * np.random.rand(n) + b = np.random.rand(n) + 1j * np.random.rand(n) + expected = np.convolve(a, b, 'full') + + out = fftconvolve(a, b, 'full') + assert_allclose(out, expected, atol=1e-10) + + out = fftconvolve(a, b, 'full', axes=[0]) + assert_allclose(out, expected, atol=1e-10) + + @pytest.mark.thread_unsafe + def test_fft_nan(self): + n = 1000 + rng = np.random.default_rng(43876432987) + sig_nan = rng.standard_normal(n) + + for val in [np.nan, np.inf]: + sig_nan[100] = val + coeffs = signal.firwin(200, 0.2) + + msg = "Use of fft convolution.*|invalid value encountered.*" + with pytest.warns(RuntimeWarning, match=msg): + signal.convolve(sig_nan, coeffs, mode='same', method='fft') + +def fftconvolve_err(*args, **kwargs): + raise RuntimeError('Fell back to fftconvolve') + + +def gen_oa_shapes(sizes): + return [(a, b) for a, b in product(sizes, repeat=2) + if abs(a - b) > 3] + + +def gen_oa_shapes_2d(sizes): + shapes0 = gen_oa_shapes(sizes) + shapes1 = gen_oa_shapes(sizes) + shapes = [ishapes0+ishapes1 for ishapes0, ishapes1 in + zip(shapes0, shapes1)] + + modes = ['full', 'valid', 'same'] + return [ishapes+(imode,) for ishapes, imode in product(shapes, modes) + if imode != 'valid' or + (ishapes[0] > ishapes[1] and ishapes[2] > ishapes[3]) or + (ishapes[0] < ishapes[1] and ishapes[2] < ishapes[3])] + + +def gen_oa_shapes_eq(sizes): + return [(a, b) for a, b in product(sizes, repeat=2) + if a >= b] + + +class TestOAConvolve: + @pytest.mark.slow() + @pytest.mark.parametrize('shape_a_0, shape_b_0', + gen_oa_shapes_eq(list(range(100)) + + list(range(100, 1000, 23))) + ) + def test_real_manylens(self, shape_a_0, shape_b_0): + a = np.random.rand(shape_a_0) + b = np.random.rand(shape_b_0) + + expected = fftconvolve(a, b) + out = oaconvolve(a, b) + + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('shape_a_0, shape_b_0', + gen_oa_shapes([50, 47, 6, 4, 1])) + @pytest.mark.parametrize('is_complex', [True, False]) + @pytest.mark.parametrize('mode', ['full', 'valid', 'same']) + def test_1d_noaxes(self, shape_a_0, shape_b_0, + is_complex, mode, monkeypatch): + a = np.random.rand(shape_a_0) + b = np.random.rand(shape_b_0) + if is_complex: + a = a + 1j*np.random.rand(shape_a_0) + b = b + 1j*np.random.rand(shape_b_0) + + expected = fftconvolve(a, b, mode=mode) + + monkeypatch.setattr(signal._signaltools, 'fftconvolve', + fftconvolve_err) + out = oaconvolve(a, b, mode=mode) + + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', [0, 1]) + @pytest.mark.parametrize('shape_a_0, shape_b_0', + gen_oa_shapes([50, 47, 6, 4])) + @pytest.mark.parametrize('shape_a_extra', [1, 3]) + @pytest.mark.parametrize('shape_b_extra', [1, 3]) + @pytest.mark.parametrize('is_complex', [True, False]) + @pytest.mark.parametrize('mode', ['full', 'valid', 'same']) + def test_1d_axes(self, axes, shape_a_0, shape_b_0, + shape_a_extra, shape_b_extra, + is_complex, mode, monkeypatch): + ax_a = [shape_a_extra]*2 + ax_b = [shape_b_extra]*2 + ax_a[axes] = shape_a_0 + ax_b[axes] = shape_b_0 + + a = np.random.rand(*ax_a) + b = np.random.rand(*ax_b) + if is_complex: + a = a + 1j*np.random.rand(*ax_a) + b = b + 1j*np.random.rand(*ax_b) + + expected = fftconvolve(a, b, mode=mode, axes=axes) + + monkeypatch.setattr(signal._signaltools, 'fftconvolve', + fftconvolve_err) + out = oaconvolve(a, b, mode=mode, axes=axes) + + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('shape_a_0, shape_b_0, ' + 'shape_a_1, shape_b_1, mode', + gen_oa_shapes_2d([50, 47, 6, 4])) + @pytest.mark.parametrize('is_complex', [True, False]) + def test_2d_noaxes(self, shape_a_0, shape_b_0, + shape_a_1, shape_b_1, mode, + is_complex, monkeypatch): + a = np.random.rand(shape_a_0, shape_a_1) + b = np.random.rand(shape_b_0, shape_b_1) + if is_complex: + a = a + 1j*np.random.rand(shape_a_0, shape_a_1) + b = b + 1j*np.random.rand(shape_b_0, shape_b_1) + + expected = fftconvolve(a, b, mode=mode) + + monkeypatch.setattr(signal._signaltools, 'fftconvolve', + fftconvolve_err) + out = oaconvolve(a, b, mode=mode) + + assert_array_almost_equal(out, expected) + + @pytest.mark.parametrize('axes', [[0, 1], [0, 2], [1, 2]]) + @pytest.mark.parametrize('shape_a_0, shape_b_0, ' + 'shape_a_1, shape_b_1, mode', + gen_oa_shapes_2d([50, 47, 6, 4])) + @pytest.mark.parametrize('shape_a_extra', [1, 3]) + @pytest.mark.parametrize('shape_b_extra', [1, 3]) + @pytest.mark.parametrize('is_complex', [True, False]) + def test_2d_axes(self, axes, shape_a_0, shape_b_0, + shape_a_1, shape_b_1, mode, + shape_a_extra, shape_b_extra, + is_complex, monkeypatch): + ax_a = [shape_a_extra]*3 + ax_b = [shape_b_extra]*3 + ax_a[axes[0]] = shape_a_0 + ax_b[axes[0]] = shape_b_0 + ax_a[axes[1]] = shape_a_1 + ax_b[axes[1]] = shape_b_1 + + a = np.random.rand(*ax_a) + b = np.random.rand(*ax_b) + if is_complex: + a = a + 1j*np.random.rand(*ax_a) + b = b + 1j*np.random.rand(*ax_b) + + expected = fftconvolve(a, b, mode=mode, axes=axes) + + monkeypatch.setattr(signal._signaltools, 'fftconvolve', + fftconvolve_err) + out = oaconvolve(a, b, mode=mode, axes=axes) + + assert_array_almost_equal(out, expected) + + def test_empty(self): + # Regression test for #1745: crashes with 0-length input. + assert_(oaconvolve([], []).size == 0) + assert_(oaconvolve([5, 6], []).size == 0) + assert_(oaconvolve([], [7]).size == 0) + + def test_zero_rank(self): + a = array(4967) + b = array(3920) + out = oaconvolve(a, b) + assert_equal(out, a * b) + + def test_single_element(self): + a = array([4967]) + b = array([3920]) + out = oaconvolve(a, b) + assert_equal(out, a * b) + + +class TestAllFreqConvolves: + + @pytest.mark.parametrize('convapproach', + [fftconvolve, oaconvolve]) + def test_invalid_shapes(self, convapproach): + a = np.arange(1, 7).reshape((2, 3)) + b = np.arange(-6, 0).reshape((3, 2)) + with assert_raises(ValueError, + match="For 'valid' mode, one must be at least " + "as large as the other in every dimension"): + convapproach(a, b, mode='valid') + + @pytest.mark.parametrize('convapproach', + [fftconvolve, oaconvolve]) + def test_invalid_shapes_axes(self, convapproach): + a = np.zeros([5, 6, 2, 1]) + b = np.zeros([5, 6, 3, 1]) + with assert_raises(ValueError, + match=r"incompatible shapes for in1 and in2:" + r" \(5L?, 6L?, 2L?, 1L?\) and" + r" \(5L?, 6L?, 3L?, 1L?\)"): + convapproach(a, b, axes=[0, 1]) + + @pytest.mark.parametrize('a,b', + [([1], 2), + (1, [2]), + ([3], [[2]])]) + @pytest.mark.parametrize('convapproach', + [fftconvolve, oaconvolve]) + def test_mismatched_dims(self, a, b, convapproach): + with assert_raises(ValueError, + match="in1 and in2 should have the same" + " dimensionality"): + convapproach(a, b) + + @pytest.mark.parametrize('convapproach', + [fftconvolve, oaconvolve]) + def test_invalid_flags(self, convapproach): + with assert_raises(ValueError, + match="acceptable mode flags are 'valid'," + " 'same', or 'full'"): + convapproach([1], [2], mode='chips') + + with assert_raises(ValueError, + match="when provided, axes cannot be empty"): + convapproach([1], [2], axes=[]) + + with assert_raises(ValueError, match="axes must be a scalar or " + "iterable of integers"): + convapproach([1], [2], axes=[[1, 2], [3, 4]]) + + with assert_raises(ValueError, match="axes must be a scalar or " + "iterable of integers"): + convapproach([1], [2], axes=[1., 2., 3., 4.]) + + with assert_raises(ValueError, + match="axes exceeds dimensionality of input"): + convapproach([1], [2], axes=[1]) + + with assert_raises(ValueError, + match="axes exceeds dimensionality of input"): + convapproach([1], [2], axes=[-2]) + + with assert_raises(ValueError, + match="all axes must be unique"): + convapproach([1], [2], axes=[0, 0]) + + @pytest.mark.filterwarnings('ignore::DeprecationWarning') + @pytest.mark.parametrize('dtype', [np.longdouble, np.clongdouble]) + def test_longdtype_input(self, dtype): + x = np.random.random((27, 27)).astype(dtype) + y = np.random.random((4, 4)).astype(dtype) + if np.iscomplexobj(dtype()): + x += .1j + y -= .1j + + res = fftconvolve(x, y) + assert_allclose(res, convolve(x, y, method='direct')) + assert res.dtype == dtype + + +class TestMedFilt: + + IN = [[50, 50, 50, 50, 50, 92, 18, 27, 65, 46], + [50, 50, 50, 50, 50, 0, 72, 77, 68, 66], + [50, 50, 50, 50, 50, 46, 47, 19, 64, 77], + [50, 50, 50, 50, 50, 42, 15, 29, 95, 35], + [50, 50, 50, 50, 50, 46, 34, 9, 21, 66], + [70, 97, 28, 68, 78, 77, 61, 58, 71, 42], + [64, 53, 44, 29, 68, 32, 19, 68, 24, 84], + [3, 33, 53, 67, 1, 78, 74, 55, 12, 83], + [7, 11, 46, 70, 60, 47, 24, 43, 61, 26], + [32, 61, 88, 7, 39, 4, 92, 64, 45, 61]] + + OUT = [[0, 50, 50, 50, 42, 15, 15, 18, 27, 0], + [0, 50, 50, 50, 50, 42, 19, 21, 29, 0], + [50, 50, 50, 50, 50, 47, 34, 34, 46, 35], + [50, 50, 50, 50, 50, 50, 42, 47, 64, 42], + [50, 50, 50, 50, 50, 50, 46, 55, 64, 35], + [33, 50, 50, 50, 50, 47, 46, 43, 55, 26], + [32, 50, 50, 50, 50, 47, 46, 45, 55, 26], + [7, 46, 50, 50, 47, 46, 46, 43, 45, 21], + [0, 32, 33, 39, 32, 32, 43, 43, 43, 0], + [0, 7, 11, 7, 4, 4, 19, 19, 24, 0]] + + KERNEL_SIZE = [7,3] + + def test_basic(self): + d = signal.medfilt(self.IN, self.KERNEL_SIZE) + e = signal.medfilt2d(np.array(self.IN, float), self.KERNEL_SIZE) + assert_array_equal(d, self.OUT) + assert_array_equal(d, e) + + @pytest.mark.parametrize('dtype', [np.ubyte, np.byte, np.ushort, np.short, + np_ulong, np_long, np.ulonglong, np.ulonglong, + np.float32, np.float64]) + def test_types(self, dtype): + # volume input and output types match + in_typed = np.array(self.IN, dtype=dtype) + assert_equal(signal.medfilt(in_typed).dtype, dtype) + assert_equal(signal.medfilt2d(in_typed).dtype, dtype) + + @pytest.mark.parametrize('dtype', [np.bool_, np.complex64, np.complex128, + np.clongdouble, np.float16, np.object_, + "float96", "float128"]) + def test_invalid_dtypes(self, dtype): + # We can only test this on platforms that support a native type of float96 or + # float128; comparing to np.longdouble allows us to filter out non-native types + if (dtype in ["float96", "float128"] + and np.finfo(np.longdouble).dtype != dtype): + pytest.skip(f"Platform does not support {dtype}") + + in_typed = np.array(self.IN, dtype=dtype) + with pytest.raises(ValueError, match="not supported"): + signal.medfilt(in_typed) + + with pytest.raises(ValueError, match="not supported"): + signal.medfilt2d(in_typed) + + def test_none(self): + # gh-1651, trac #1124. Ensure this does not segfault. + msg = "dtype=object is not supported by medfilt" + with assert_raises(ValueError, match=msg): + signal.medfilt(None) + + def test_odd_strides(self): + # Avoid a regression with possible contiguous + # numpy arrays that have odd strides. The stride value below gets + # us into wrong memory if used (but it does not need to be used) + dummy = np.arange(10, dtype=np.float64) + a = dummy[5:6] + a.strides = 16 + assert_(signal.medfilt(a, 1) == 5.) + + @pytest.mark.parametrize("dtype", [np.ubyte, np.float32, np.float64]) + def test_medfilt2d_parallel(self, dtype): + in_typed = np.array(self.IN, dtype=dtype) + expected = np.array(self.OUT, dtype=dtype) + + # This is used to simplify the indexing calculations. + assert in_typed.shape == expected.shape + + # We'll do the calculation in four chunks. M1 and N1 are the dimensions + # of the first output chunk. We have to extend the input by half the + # kernel size to be able to calculate the full output chunk. + M1 = expected.shape[0] // 2 + N1 = expected.shape[1] // 2 + offM = self.KERNEL_SIZE[0] // 2 + 1 + offN = self.KERNEL_SIZE[1] // 2 + 1 + + def apply(chunk): + # in = slice of in_typed to use. + # sel = slice of output to crop it to the correct region. + # out = slice of output array to store in. + M, N = chunk + if M == 0: + Min = slice(0, M1 + offM) + Msel = slice(0, -offM) + Mout = slice(0, M1) + else: + Min = slice(M1 - offM, None) + Msel = slice(offM, None) + Mout = slice(M1, None) + if N == 0: + Nin = slice(0, N1 + offN) + Nsel = slice(0, -offN) + Nout = slice(0, N1) + else: + Nin = slice(N1 - offN, None) + Nsel = slice(offN, None) + Nout = slice(N1, None) + + # Do the calculation, but do not write to the output in the threads. + chunk_data = in_typed[Min, Nin] + med = signal.medfilt2d(chunk_data, self.KERNEL_SIZE) + return med[Msel, Nsel], Mout, Nout + + # Give each chunk to a different thread. + output = np.zeros_like(expected) + with ThreadPoolExecutor(max_workers=4) as pool: + chunks = {(0, 0), (0, 1), (1, 0), (1, 1)} + futures = {pool.submit(apply, chunk) for chunk in chunks} + + # Store each result in the output as it arrives. + for future in as_completed(futures): + data, Mslice, Nslice = future.result() + output[Mslice, Nslice] = data + + assert_array_equal(output, expected) + + +class TestWiener: + + def test_basic(self): + g = array([[5, 6, 4, 3], + [3, 5, 6, 2], + [2, 3, 5, 6], + [1, 6, 9, 7]], 'd') + h = array([[2.16374269, 3.2222222222, 2.8888888889, 1.6666666667], + [2.666666667, 4.33333333333, 4.44444444444, 2.8888888888], + [2.222222222, 4.4444444444, 5.4444444444, 4.801066874837], + [1.33333333333, 3.92735042735, 6.0712560386, 5.0404040404]]) + assert_array_almost_equal(signal.wiener(g), h, decimal=6) + assert_array_almost_equal(signal.wiener(g, mysize=3), h, decimal=6) + + +padtype_options = ["mean", "median", "minimum", "maximum", "line"] +padtype_options += _upfirdn_modes + + +class TestResample: + def test_basic(self): + # Some basic tests + + # Regression test for issue #3603. + # window.shape must equal to sig.shape[0] + sig = np.arange(128) + num = 256 + win = signal.get_window(('kaiser', 8.0), 160) + assert_raises(ValueError, signal.resample, sig, num, window=win) + + # Other degenerate conditions + assert_raises(ValueError, signal.resample_poly, sig, 'yo', 1) + assert_raises(ValueError, signal.resample_poly, sig, 1, 0) + assert_raises(ValueError, signal.resample_poly, sig, 2, 1, padtype='') + assert_raises(ValueError, signal.resample_poly, sig, 2, 1, + padtype='mean', cval=10) + + # test for issue #6505 - should not modify window.shape when axis ≠ 0 + sig2 = np.tile(np.arange(160), (2, 1)) + signal.resample(sig2, num, axis=-1, window=win) + assert_(win.shape == (160,)) + + @pytest.mark.parametrize('window', (None, 'hamming')) + @pytest.mark.parametrize('N', (20, 19)) + @pytest.mark.parametrize('num', (100, 101, 10, 11)) + def test_rfft(self, N, num, window): + # Make sure the speed up using rfft gives the same result as the normal + # way using fft + x = np.linspace(0, 10, N, endpoint=False) + y = np.cos(-x**2/6.0) + assert_allclose(signal.resample(y, num, window=window), + signal.resample(y + 0j, num, window=window).real) + + y = np.array([np.cos(-x**2/6.0), np.sin(-x**2/6.0)]) + y_complex = y + 0j + assert_allclose( + signal.resample(y, num, axis=1, window=window), + signal.resample(y_complex, num, axis=1, window=window).real, + atol=1e-9) + + def test_input_domain(self): + # Test if both input domain modes produce the same results. + tsig = np.arange(256) + 0j + fsig = sp_fft.fft(tsig) + num = 256 + assert_allclose( + signal.resample(fsig, num, domain='freq'), + signal.resample(tsig, num, domain='time'), + atol=1e-9) + + @pytest.mark.parametrize('nx', (1, 2, 3, 5, 8)) + @pytest.mark.parametrize('ny', (1, 2, 3, 5, 8)) + @pytest.mark.parametrize('dtype', ('float', 'complex')) + def test_dc(self, nx, ny, dtype): + x = np.array([1] * nx, dtype) + y = signal.resample(x, ny) + assert_allclose(y, [1] * ny) + + @pytest.mark.thread_unsafe # due to Cython fused types, see cython#6506 + @pytest.mark.parametrize('padtype', padtype_options) + def test_mutable_window(self, padtype): + # Test that a mutable window is not modified + impulse = np.zeros(3) + window = np.random.RandomState(0).randn(2) + window_orig = window.copy() + signal.resample_poly(impulse, 5, 1, window=window, padtype=padtype) + assert_array_equal(window, window_orig) + + @pytest.mark.parametrize('padtype', padtype_options) + def test_output_float32(self, padtype): + # Test that float32 inputs yield a float32 output + x = np.arange(10, dtype=np.float32) + h = np.array([1, 1, 1], dtype=np.float32) + y = signal.resample_poly(x, 1, 2, window=h, padtype=padtype) + assert y.dtype == np.float32 + + @pytest.mark.parametrize('padtype', padtype_options) + @pytest.mark.parametrize('dtype', [np.float32, np.float64]) + def test_output_match_dtype(self, padtype, dtype): + # Test that the dtype of x is preserved per issue #14733 + x = np.arange(10, dtype=dtype) + y = signal.resample_poly(x, 1, 2, padtype=padtype) + assert y.dtype == x.dtype + + @pytest.mark.parametrize( + "method, ext, padtype", + [("fft", False, None)] + + list( + product( + ["polyphase"], [False, True], padtype_options, + ) + ), + ) + def test_resample_methods(self, method, ext, padtype): + # Test resampling of sinusoids and random noise (1-sec) + rate = 100 + rates_to = [49, 50, 51, 99, 100, 101, 199, 200, 201] + + # Sinusoids, windowed to avoid edge artifacts + t = np.arange(rate) / float(rate) + freqs = np.array((1., 10., 40.))[:, np.newaxis] + x = np.sin(2 * np.pi * freqs * t) * hann(rate) + + for rate_to in rates_to: + t_to = np.arange(rate_to) / float(rate_to) + y_tos = np.sin(2 * np.pi * freqs * t_to) * hann(rate_to) + if method == 'fft': + y_resamps = signal.resample(x, rate_to, axis=-1) + else: + if ext and rate_to != rate: + # Match default window design + g = gcd(rate_to, rate) + up = rate_to // g + down = rate // g + max_rate = max(up, down) + f_c = 1. / max_rate + half_len = 10 * max_rate + window = signal.firwin(2 * half_len + 1, f_c, + window=('kaiser', 5.0)) + polyargs = {'window': window, 'padtype': padtype} + else: + polyargs = {'padtype': padtype} + + y_resamps = signal.resample_poly(x, rate_to, rate, axis=-1, + **polyargs) + + for y_to, y_resamp, freq in zip(y_tos, y_resamps, freqs): + if freq >= 0.5 * rate_to: + y_to.fill(0.) # mostly low-passed away + if padtype in ['minimum', 'maximum']: + assert_allclose(y_resamp, y_to, atol=3e-1) + else: + assert_allclose(y_resamp, y_to, atol=1e-3) + else: + assert_array_equal(y_to.shape, y_resamp.shape) + corr = np.corrcoef(y_to, y_resamp)[0, 1] + assert_(corr > 0.99, msg=(corr, rate, rate_to)) + + # Random data + rng = np.random.RandomState(0) + x = hann(rate) * np.cumsum(rng.randn(rate)) # low-pass, wind + for rate_to in rates_to: + # random data + t_to = np.arange(rate_to) / float(rate_to) + y_to = np.interp(t_to, t, x) + if method == 'fft': + y_resamp = signal.resample(x, rate_to) + else: + y_resamp = signal.resample_poly(x, rate_to, rate, + padtype=padtype) + assert_array_equal(y_to.shape, y_resamp.shape) + corr = np.corrcoef(y_to, y_resamp)[0, 1] + assert_(corr > 0.99, msg=corr) + + # More tests of fft method (Master 0.18.1 fails these) + if method == 'fft': + x1 = np.array([1.+0.j, 0.+0.j]) + y1_test = signal.resample(x1, 4) + # upsampling a complex array + y1_true = np.array([1.+0.j, 0.5+0.j, 0.+0.j, 0.5+0.j]) + assert_allclose(y1_test, y1_true, atol=1e-12) + x2 = np.array([1., 0.5, 0., 0.5]) + y2_test = signal.resample(x2, 2) # downsampling a real array + y2_true = np.array([1., 0.]) + assert_allclose(y2_test, y2_true, atol=1e-12) + + def test_poly_vs_filtfilt(self): + # Check that up=1.0 gives same answer as filtfilt + slicing + random_state = np.random.RandomState(17) + try_types = (int, np.float32, np.complex64, float, complex) + size = 10000 + down_factors = [2, 11, 79] + + for dtype in try_types: + x = random_state.randn(size).astype(dtype) + if dtype in (np.complex64, np.complex128): + x += 1j * random_state.randn(size) + + # resample_poly assumes zeros outside of signl, whereas filtfilt + # can only constant-pad. Make them equivalent: + x[0] = 0 + x[-1] = 0 + + for down in down_factors: + h = signal.firwin(31, 1. / down, window='hamming') + yf = filtfilt(h, 1.0, x, padtype='constant')[::down] + + # Need to pass convolved version of filter to resample_poly, + # since filtfilt does forward and backward, but resample_poly + # only goes forward + hc = convolve(h, h[::-1]) + y = signal.resample_poly(x, 1, down, window=hc) + assert_allclose(yf, y, atol=1e-7, rtol=1e-7) + + def test_correlate1d(self): + for down in [2, 4]: + for nx in range(1, 40, down): + for nweights in (32, 33): + x = np.random.random((nx,)) + weights = np.random.random((nweights,)) + y_g = correlate1d(x, weights[::-1], mode='constant') + y_s = signal.resample_poly( + x, up=1, down=down, window=weights) + assert_allclose(y_g[::down], y_s) + + @pytest.mark.parametrize('dtype', [np.int32, np.float32]) + def test_gh_15620(self, dtype): + data = np.array([0, 1, 2, 3, 2, 1, 0], dtype=dtype) + actual = signal.resample_poly(data, + up=2, + down=1, + padtype='smooth') + assert np.count_nonzero(actual) > 0 + + +class TestCSpline1DEval: + + def test_basic(self): + y = array([1, 2, 3, 4, 3, 2, 1, 2, 3.0]) + x = arange(len(y)) + dx = x[1] - x[0] + cj = signal.cspline1d(y) + + x2 = arange(len(y) * 10.0) / 10.0 + y2 = signal.cspline1d_eval(cj, x2, dx=dx, x0=x[0]) + + # make sure interpolated values are on knot points + assert_array_almost_equal(y2[::10], y, decimal=5) + + def test_complex(self): + # create some smoothly varying complex signal to interpolate + x = np.arange(2) + y = np.zeros(x.shape, dtype=np.complex64) + T = 10.0 + f = 1.0 / T + y = np.exp(2.0J * np.pi * f * x) + + # get the cspline transform + cy = signal.cspline1d(y) + + # determine new test x value and interpolate + xnew = np.array([0.5]) + ynew = signal.cspline1d_eval(cy, xnew) + + assert_equal(ynew.dtype, y.dtype) + +class TestOrderFilt: + + def test_basic(self): + assert_array_equal(signal.order_filter([1, 2, 3], [1, 0, 1], 1), + [2, 3, 2]) + + +class _TestLinearFilter: + + def generate(self, shape): + x = np.linspace(0, np.prod(shape) - 1, np.prod(shape)).reshape(shape) + return self.convert_dtype(x) + + def convert_dtype(self, arr): + if self.dtype == np.dtype('O'): + arr = np.asarray(arr) + out = np.empty(arr.shape, self.dtype) + iter = np.nditer([arr, out], ['refs_ok','zerosize_ok'], + [['readonly'],['writeonly']]) + for x, y in iter: + y[...] = self.type(x[()]) + return out + else: + return np.asarray(arr, dtype=self.dtype) + + def test_rank_1_IIR(self): + x = self.generate((6,)) + b = self.convert_dtype([1, -1]) + a = self.convert_dtype([0.5, -0.5]) + y_r = self.convert_dtype([0, 2, 4, 6, 8, 10.]) + assert_array_almost_equal(lfilter(b, a, x), y_r) + + def test_rank_1_FIR(self): + x = self.generate((6,)) + b = self.convert_dtype([1, 1]) + a = self.convert_dtype([1]) + y_r = self.convert_dtype([0, 1, 3, 5, 7, 9.]) + assert_array_almost_equal(lfilter(b, a, x), y_r) + + def test_rank_1_IIR_init_cond(self): + x = self.generate((6,)) + b = self.convert_dtype([1, 0, -1]) + a = self.convert_dtype([0.5, -0.5]) + zi = self.convert_dtype([1, 2]) + y_r = self.convert_dtype([1, 5, 9, 13, 17, 21]) + zf_r = self.convert_dtype([13, -10]) + y, zf = lfilter(b, a, x, zi=zi) + assert_array_almost_equal(y, y_r) + assert_array_almost_equal(zf, zf_r) + + def test_rank_1_FIR_init_cond(self): + x = self.generate((6,)) + b = self.convert_dtype([1, 1, 1]) + a = self.convert_dtype([1]) + zi = self.convert_dtype([1, 1]) + y_r = self.convert_dtype([1, 2, 3, 6, 9, 12.]) + zf_r = self.convert_dtype([9, 5]) + y, zf = lfilter(b, a, x, zi=zi) + assert_array_almost_equal(y, y_r) + assert_array_almost_equal(zf, zf_r) + + def test_rank_2_IIR_axis_0(self): + x = self.generate((4, 3)) + b = self.convert_dtype([1, -1]) + a = self.convert_dtype([0.5, 0.5]) + y_r2_a0 = self.convert_dtype([[0, 2, 4], [6, 4, 2], [0, 2, 4], + [6, 4, 2]]) + y = lfilter(b, a, x, axis=0) + assert_array_almost_equal(y_r2_a0, y) + + def test_rank_2_IIR_axis_1(self): + x = self.generate((4, 3)) + b = self.convert_dtype([1, -1]) + a = self.convert_dtype([0.5, 0.5]) + y_r2_a1 = self.convert_dtype([[0, 2, 0], [6, -4, 6], [12, -10, 12], + [18, -16, 18]]) + y = lfilter(b, a, x, axis=1) + assert_array_almost_equal(y_r2_a1, y) + + def test_rank_2_IIR_axis_0_init_cond(self): + x = self.generate((4, 3)) + b = self.convert_dtype([1, -1]) + a = self.convert_dtype([0.5, 0.5]) + zi = self.convert_dtype(np.ones((4,1))) + + y_r2_a0_1 = self.convert_dtype([[1, 1, 1], [7, -5, 7], [13, -11, 13], + [19, -17, 19]]) + zf_r = self.convert_dtype([-5, -17, -29, -41])[:, np.newaxis] + y, zf = lfilter(b, a, x, axis=1, zi=zi) + assert_array_almost_equal(y_r2_a0_1, y) + assert_array_almost_equal(zf, zf_r) + + def test_rank_2_IIR_axis_1_init_cond(self): + x = self.generate((4,3)) + b = self.convert_dtype([1, -1]) + a = self.convert_dtype([0.5, 0.5]) + zi = self.convert_dtype(np.ones((1,3))) + + y_r2_a0_0 = self.convert_dtype([[1, 3, 5], [5, 3, 1], + [1, 3, 5], [5, 3, 1]]) + zf_r = self.convert_dtype([[-23, -23, -23]]) + y, zf = lfilter(b, a, x, axis=0, zi=zi) + assert_array_almost_equal(y_r2_a0_0, y) + assert_array_almost_equal(zf, zf_r) + + def test_rank_3_IIR(self): + x = self.generate((4, 3, 2)) + b = self.convert_dtype([1, -1]) + a = self.convert_dtype([0.5, 0.5]) + + for axis in range(x.ndim): + y = lfilter(b, a, x, axis) + y_r = np.apply_along_axis(lambda w: lfilter(b, a, w), axis, x) + assert_array_almost_equal(y, y_r) + + def test_rank_3_IIR_init_cond(self): + x = self.generate((4, 3, 2)) + b = self.convert_dtype([1, -1]) + a = self.convert_dtype([0.5, 0.5]) + + for axis in range(x.ndim): + zi_shape = list(x.shape) + zi_shape[axis] = 1 + zi = self.convert_dtype(np.ones(zi_shape)) + zi1 = self.convert_dtype([1]) + y, zf = lfilter(b, a, x, axis, zi) + def lf0(w): + return lfilter(b, a, w, zi=zi1)[0] + def lf1(w): + return lfilter(b, a, w, zi=zi1)[1] + y_r = np.apply_along_axis(lf0, axis, x) + zf_r = np.apply_along_axis(lf1, axis, x) + assert_array_almost_equal(y, y_r) + assert_array_almost_equal(zf, zf_r) + + def test_rank_3_FIR(self): + x = self.generate((4, 3, 2)) + b = self.convert_dtype([1, 0, -1]) + a = self.convert_dtype([1]) + + for axis in range(x.ndim): + y = lfilter(b, a, x, axis) + y_r = np.apply_along_axis(lambda w: lfilter(b, a, w), axis, x) + assert_array_almost_equal(y, y_r) + + def test_rank_3_FIR_init_cond(self): + x = self.generate((4, 3, 2)) + b = self.convert_dtype([1, 0, -1]) + a = self.convert_dtype([1]) + + for axis in range(x.ndim): + zi_shape = list(x.shape) + zi_shape[axis] = 2 + zi = self.convert_dtype(np.ones(zi_shape)) + zi1 = self.convert_dtype([1, 1]) + y, zf = lfilter(b, a, x, axis, zi) + def lf0(w): + return lfilter(b, a, w, zi=zi1)[0] + def lf1(w): + return lfilter(b, a, w, zi=zi1)[1] + y_r = np.apply_along_axis(lf0, axis, x) + zf_r = np.apply_along_axis(lf1, axis, x) + assert_array_almost_equal(y, y_r) + assert_array_almost_equal(zf, zf_r) + + def test_zi_pseudobroadcast(self): + x = self.generate((4, 5, 20)) + b,a = signal.butter(8, 0.2, output='ba') + b = self.convert_dtype(b) + a = self.convert_dtype(a) + zi_size = b.shape[0] - 1 + + # lfilter requires x.ndim == zi.ndim exactly. However, zi can have + # length 1 dimensions. + zi_full = self.convert_dtype(np.ones((4, 5, zi_size))) + zi_sing = self.convert_dtype(np.ones((1, 1, zi_size))) + + y_full, zf_full = lfilter(b, a, x, zi=zi_full) + y_sing, zf_sing = lfilter(b, a, x, zi=zi_sing) + + assert_array_almost_equal(y_sing, y_full) + assert_array_almost_equal(zf_full, zf_sing) + + # lfilter does not prepend ones + assert_raises(ValueError, lfilter, b, a, x, -1, np.ones(zi_size)) + + def test_scalar_a(self): + # a can be a scalar. + x = self.generate(6) + b = self.convert_dtype([1, 0, -1]) + a = self.convert_dtype([1]) + y_r = self.convert_dtype([0, 1, 2, 2, 2, 2]) + + y = lfilter(b, a[0], x) + assert_array_almost_equal(y, y_r) + + def test_zi_some_singleton_dims(self): + # lfilter doesn't really broadcast (no prepending of 1's). But does + # do singleton expansion if x and zi have the same ndim. This was + # broken only if a subset of the axes were singletons (gh-4681). + x = self.convert_dtype(np.zeros((3,2,5), 'l')) + b = self.convert_dtype(np.ones(5, 'l')) + a = self.convert_dtype(np.array([1,0,0])) + zi = np.ones((3,1,4), 'l') + zi[1,:,:] *= 2 + zi[2,:,:] *= 3 + zi = self.convert_dtype(zi) + + zf_expected = self.convert_dtype(np.zeros((3,2,4), 'l')) + y_expected = np.zeros((3,2,5), 'l') + y_expected[:,:,:4] = [[[1]], [[2]], [[3]]] + y_expected = self.convert_dtype(y_expected) + + # IIR + y_iir, zf_iir = lfilter(b, a, x, -1, zi) + assert_array_almost_equal(y_iir, y_expected) + assert_array_almost_equal(zf_iir, zf_expected) + + # FIR + y_fir, zf_fir = lfilter(b, a[0], x, -1, zi) + assert_array_almost_equal(y_fir, y_expected) + assert_array_almost_equal(zf_fir, zf_expected) + + def base_bad_size_zi(self, b, a, x, axis, zi): + b = self.convert_dtype(b) + a = self.convert_dtype(a) + x = self.convert_dtype(x) + zi = self.convert_dtype(zi) + assert_raises(ValueError, lfilter, b, a, x, axis, zi) + + def test_bad_size_zi(self): + # rank 1 + x1 = np.arange(6) + self.base_bad_size_zi([1], [1], x1, -1, [1]) + self.base_bad_size_zi([1, 1], [1], x1, -1, [0, 1]) + self.base_bad_size_zi([1, 1], [1], x1, -1, [[0]]) + self.base_bad_size_zi([1, 1], [1], x1, -1, [0, 1, 2]) + self.base_bad_size_zi([1, 1, 1], [1], x1, -1, [[0]]) + self.base_bad_size_zi([1, 1, 1], [1], x1, -1, [0, 1, 2]) + self.base_bad_size_zi([1], [1, 1], x1, -1, [0, 1]) + self.base_bad_size_zi([1], [1, 1], x1, -1, [[0]]) + self.base_bad_size_zi([1], [1, 1], x1, -1, [0, 1, 2]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x1, -1, [0]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x1, -1, [[0], [1]]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x1, -1, [0, 1, 2]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x1, -1, [0, 1, 2, 3]) + self.base_bad_size_zi([1, 1], [1, 1, 1], x1, -1, [0]) + self.base_bad_size_zi([1, 1], [1, 1, 1], x1, -1, [[0], [1]]) + self.base_bad_size_zi([1, 1], [1, 1, 1], x1, -1, [0, 1, 2]) + self.base_bad_size_zi([1, 1], [1, 1, 1], x1, -1, [0, 1, 2, 3]) + + # rank 2 + x2 = np.arange(12).reshape((4,3)) + # for axis=0 zi.shape should == (max(len(a),len(b))-1, 3) + self.base_bad_size_zi([1], [1], x2, 0, [0]) + + # for each of these there are 5 cases tested (in this order): + # 1. not deep enough, right # elements + # 2. too deep, right # elements + # 3. right depth, right # elements, transposed + # 4. right depth, too few elements + # 5. right depth, too many elements + + self.base_bad_size_zi([1, 1], [1], x2, 0, [0,1,2]) + self.base_bad_size_zi([1, 1], [1], x2, 0, [[[0,1,2]]]) + self.base_bad_size_zi([1, 1], [1], x2, 0, [[0], [1], [2]]) + self.base_bad_size_zi([1, 1], [1], x2, 0, [[0,1]]) + self.base_bad_size_zi([1, 1], [1], x2, 0, [[0,1,2,3]]) + + self.base_bad_size_zi([1, 1, 1], [1], x2, 0, [0,1,2,3,4,5]) + self.base_bad_size_zi([1, 1, 1], [1], x2, 0, [[[0,1,2],[3,4,5]]]) + self.base_bad_size_zi([1, 1, 1], [1], x2, 0, [[0,1],[2,3],[4,5]]) + self.base_bad_size_zi([1, 1, 1], [1], x2, 0, [[0,1],[2,3]]) + self.base_bad_size_zi([1, 1, 1], [1], x2, 0, [[0,1,2,3],[4,5,6,7]]) + + self.base_bad_size_zi([1], [1, 1], x2, 0, [0,1,2]) + self.base_bad_size_zi([1], [1, 1], x2, 0, [[[0,1,2]]]) + self.base_bad_size_zi([1], [1, 1], x2, 0, [[0], [1], [2]]) + self.base_bad_size_zi([1], [1, 1], x2, 0, [[0,1]]) + self.base_bad_size_zi([1], [1, 1], x2, 0, [[0,1,2,3]]) + + self.base_bad_size_zi([1], [1, 1, 1], x2, 0, [0,1,2,3,4,5]) + self.base_bad_size_zi([1], [1, 1, 1], x2, 0, [[[0,1,2],[3,4,5]]]) + self.base_bad_size_zi([1], [1, 1, 1], x2, 0, [[0,1],[2,3],[4,5]]) + self.base_bad_size_zi([1], [1, 1, 1], x2, 0, [[0,1],[2,3]]) + self.base_bad_size_zi([1], [1, 1, 1], x2, 0, [[0,1,2,3],[4,5,6,7]]) + + self.base_bad_size_zi([1, 1, 1], [1, 1], x2, 0, [0,1,2,3,4,5]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x2, 0, [[[0,1,2],[3,4,5]]]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x2, 0, [[0,1],[2,3],[4,5]]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x2, 0, [[0,1],[2,3]]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x2, 0, [[0,1,2,3],[4,5,6,7]]) + + # for axis=1 zi.shape should == (4, max(len(a),len(b))-1) + self.base_bad_size_zi([1], [1], x2, 1, [0]) + + self.base_bad_size_zi([1, 1], [1], x2, 1, [0,1,2,3]) + self.base_bad_size_zi([1, 1], [1], x2, 1, [[[0],[1],[2],[3]]]) + self.base_bad_size_zi([1, 1], [1], x2, 1, [[0, 1, 2, 3]]) + self.base_bad_size_zi([1, 1], [1], x2, 1, [[0],[1],[2]]) + self.base_bad_size_zi([1, 1], [1], x2, 1, [[0],[1],[2],[3],[4]]) + + self.base_bad_size_zi([1, 1, 1], [1], x2, 1, [0,1,2,3,4,5,6,7]) + self.base_bad_size_zi([1, 1, 1], [1], x2, 1, [[[0,1],[2,3],[4,5],[6,7]]]) + self.base_bad_size_zi([1, 1, 1], [1], x2, 1, [[0,1,2,3],[4,5,6,7]]) + self.base_bad_size_zi([1, 1, 1], [1], x2, 1, [[0,1],[2,3],[4,5]]) + self.base_bad_size_zi([1, 1, 1], [1], x2, 1, [[0,1],[2,3],[4,5],[6,7],[8,9]]) + + self.base_bad_size_zi([1], [1, 1], x2, 1, [0,1,2,3]) + self.base_bad_size_zi([1], [1, 1], x2, 1, [[[0],[1],[2],[3]]]) + self.base_bad_size_zi([1], [1, 1], x2, 1, [[0, 1, 2, 3]]) + self.base_bad_size_zi([1], [1, 1], x2, 1, [[0],[1],[2]]) + self.base_bad_size_zi([1], [1, 1], x2, 1, [[0],[1],[2],[3],[4]]) + + self.base_bad_size_zi([1], [1, 1, 1], x2, 1, [0,1,2,3,4,5,6,7]) + self.base_bad_size_zi([1], [1, 1, 1], x2, 1, [[[0,1],[2,3],[4,5],[6,7]]]) + self.base_bad_size_zi([1], [1, 1, 1], x2, 1, [[0,1,2,3],[4,5,6,7]]) + self.base_bad_size_zi([1], [1, 1, 1], x2, 1, [[0,1],[2,3],[4,5]]) + self.base_bad_size_zi([1], [1, 1, 1], x2, 1, [[0,1],[2,3],[4,5],[6,7],[8,9]]) + + self.base_bad_size_zi([1, 1, 1], [1, 1], x2, 1, [0,1,2,3,4,5,6,7]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x2, 1, [[[0,1],[2,3],[4,5],[6,7]]]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x2, 1, [[0,1,2,3],[4,5,6,7]]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x2, 1, [[0,1],[2,3],[4,5]]) + self.base_bad_size_zi([1, 1, 1], [1, 1], x2, 1, [[0,1],[2,3],[4,5],[6,7],[8,9]]) + + def test_empty_zi(self): + # Regression test for #880: empty array for zi crashes. + x = self.generate((5,)) + a = self.convert_dtype([1]) + b = self.convert_dtype([1]) + zi = self.convert_dtype([]) + y, zf = lfilter(b, a, x, zi=zi) + assert_array_almost_equal(y, x) + assert_equal(zf.dtype, self.dtype) + assert_equal(zf.size, 0) + + def test_lfiltic_bad_zi(self): + # Regression test for #3699: bad initial conditions + a = self.convert_dtype([1]) + b = self.convert_dtype([1]) + # "y" sets the datatype of zi, so it truncates if int + zi = lfiltic(b, a, [1., 0]) + zi_1 = lfiltic(b, a, [1, 0]) + zi_2 = lfiltic(b, a, [True, False]) + assert_array_equal(zi, zi_1) + assert_array_equal(zi, zi_2) + + def test_short_x_FIR(self): + # regression test for #5116 + # x shorter than b, with non None zi fails + a = self.convert_dtype([1]) + b = self.convert_dtype([1, 0, -1]) + zi = self.convert_dtype([2, 7]) + x = self.convert_dtype([72]) + ye = self.convert_dtype([74]) + zfe = self.convert_dtype([7, -72]) + y, zf = lfilter(b, a, x, zi=zi) + assert_array_almost_equal(y, ye) + assert_array_almost_equal(zf, zfe) + + def test_short_x_IIR(self): + # regression test for #5116 + # x shorter than b, with non None zi fails + a = self.convert_dtype([1, 1]) + b = self.convert_dtype([1, 0, -1]) + zi = self.convert_dtype([2, 7]) + x = self.convert_dtype([72]) + ye = self.convert_dtype([74]) + zfe = self.convert_dtype([-67, -72]) + y, zf = lfilter(b, a, x, zi=zi) + assert_array_almost_equal(y, ye) + assert_array_almost_equal(zf, zfe) + + def test_do_not_modify_a_b_IIR(self): + x = self.generate((6,)) + b = self.convert_dtype([1, -1]) + b0 = b.copy() + a = self.convert_dtype([0.5, -0.5]) + a0 = a.copy() + y_r = self.convert_dtype([0, 2, 4, 6, 8, 10.]) + y_f = lfilter(b, a, x) + assert_array_almost_equal(y_f, y_r) + assert_equal(b, b0) + assert_equal(a, a0) + + def test_do_not_modify_a_b_FIR(self): + x = self.generate((6,)) + b = self.convert_dtype([1, 0, 1]) + b0 = b.copy() + a = self.convert_dtype([2]) + a0 = a.copy() + y_r = self.convert_dtype([0, 0.5, 1, 2, 3, 4.]) + y_f = lfilter(b, a, x) + assert_array_almost_equal(y_f, y_r) + assert_equal(b, b0) + assert_equal(a, a0) + + @pytest.mark.parametrize("a", [1.0, [1.0], np.array(1.0)]) + @pytest.mark.parametrize("b", [1.0, [1.0], np.array(1.0)]) + def test_scalar_input(self, a, b): + data = np.random.randn(10) + assert_allclose( + lfilter(np.array([1.0]), np.array([1.0]), data), + lfilter(b, a, data)) + + @pytest.mark.thread_unsafe + def test_dtype_deprecation(self): + # gh-21211 + a = np.asarray([1, 2, 3, 6, 5, 3], dtype=object) + b = np.asarray([2, 3, 4, 5, 3, 4, 2, 2, 1], dtype=object) + with pytest.deprecated_call(match="dtype=object is not supported"): + lfilter(a, b, [1, 2, 3, 4]) + + +class TestLinearFilterFloat32(_TestLinearFilter): + dtype = np.dtype('f') + + +class TestLinearFilterFloat64(_TestLinearFilter): + dtype = np.dtype('d') + + +@pytest.mark.filterwarnings('ignore::DeprecationWarning') +class TestLinearFilterFloatExtended(_TestLinearFilter): + dtype = np.dtype('g') + + +class TestLinearFilterComplex64(_TestLinearFilter): + dtype = np.dtype('F') + + +class TestLinearFilterComplex128(_TestLinearFilter): + dtype = np.dtype('D') + + +@pytest.mark.filterwarnings('ignore::DeprecationWarning') +class TestLinearFilterComplexExtended(_TestLinearFilter): + dtype = np.dtype('G') + + +@pytest.mark.filterwarnings('ignore::DeprecationWarning') +class TestLinearFilterDecimal(_TestLinearFilter): + dtype = np.dtype('O') + + def type(self, x): + return Decimal(str(x)) + + +@pytest.mark.filterwarnings('ignore::DeprecationWarning') +class TestLinearFilterObject(_TestLinearFilter): + dtype = np.dtype('O') + type = float + + +@pytest.mark.filterwarnings('ignore::DeprecationWarning') +def test_lfilter_bad_object(): + # lfilter: object arrays with non-numeric objects raise TypeError. + # Regression test for ticket #1452. + if hasattr(sys, 'abiflags') and 'd' in sys.abiflags: + pytest.skip('test is flaky when run with python3-dbg') + assert_raises(TypeError, lfilter, [1.0], [1.0], [1.0, None, 2.0]) + assert_raises(TypeError, lfilter, [1.0], [None], [1.0, 2.0, 3.0]) + assert_raises(TypeError, lfilter, [None], [1.0], [1.0, 2.0, 3.0]) + + +_pmf = pytest.mark.filterwarnings('ignore::DeprecationWarning') + +def test_lfilter_notimplemented_input(): + # Should not crash, gh-7991 + assert_raises(NotImplementedError, lfilter, [2,3], [4,5], [1,2,3,4,5]) + + +@pytest.mark.parametrize('dt', [np.ubyte, np.byte, np.ushort, np.short, + np_ulong, np_long, np.ulonglong, np.ulonglong, + np.float32, np.float64, + pytest.param(np.longdouble, marks=_pmf), + pytest.param(Decimal, marks=_pmf)] +) +class TestCorrelateReal: + def _setup_rank1(self, dt): + a = np.linspace(0, 3, 4).astype(dt) + b = np.linspace(1, 2, 2).astype(dt) + + y_r = np.array([0, 2, 5, 8, 3]).astype(dt) + return a, b, y_r + + def equal_tolerance(self, res_dt): + # default value of keyword + decimal = 6 + try: + dt_info = np.finfo(res_dt) + if hasattr(dt_info, 'resolution'): + decimal = int(-0.5*np.log10(dt_info.resolution)) + except Exception: + pass + return decimal + + def equal_tolerance_fft(self, res_dt): + # FFT implementations convert longdouble arguments down to + # double so don't expect better precision, see gh-9520 + if res_dt == np.longdouble: + return self.equal_tolerance(np.float64) + else: + return self.equal_tolerance(res_dt) + + def test_method(self, dt): + if dt == Decimal: + method = choose_conv_method([Decimal(4)], [Decimal(3)]) + assert_equal(method, 'direct') + else: + a, b, y_r = self._setup_rank3(dt) + y_fft = correlate(a, b, method='fft') + y_direct = correlate(a, b, method='direct') + + assert_array_almost_equal(y_r, + y_fft, + decimal=self.equal_tolerance_fft(y_fft.dtype),) + assert_array_almost_equal(y_r, + y_direct, + decimal=self.equal_tolerance(y_direct.dtype),) + assert_equal(y_fft.dtype, dt) + assert_equal(y_direct.dtype, dt) + + def test_rank1_valid(self, dt): + a, b, y_r = self._setup_rank1(dt) + y = correlate(a, b, 'valid') + assert_array_almost_equal(y, y_r[1:4]) + assert_equal(y.dtype, dt) + + # See gh-5897 + y = correlate(b, a, 'valid') + assert_array_almost_equal(y, y_r[1:4][::-1]) + assert_equal(y.dtype, dt) + + def test_rank1_same(self, dt): + a, b, y_r = self._setup_rank1(dt) + y = correlate(a, b, 'same') + assert_array_almost_equal(y, y_r[:-1]) + assert_equal(y.dtype, dt) + + def test_rank1_full(self, dt): + a, b, y_r = self._setup_rank1(dt) + y = correlate(a, b, 'full') + assert_array_almost_equal(y, y_r) + assert_equal(y.dtype, dt) + + def _setup_rank3(self, dt): + a = np.linspace(0, 39, 40).reshape((2, 4, 5), order='F').astype( + dt) + b = np.linspace(0, 23, 24).reshape((2, 3, 4), order='F').astype( + dt) + + y_r = array([[[0., 184., 504., 912., 1360., 888., 472., 160.], + [46., 432., 1062., 1840., 2672., 1698., 864., 266.], + [134., 736., 1662., 2768., 3920., 2418., 1168., 314.], + [260., 952., 1932., 3056., 4208., 2580., 1240., 332.], + [202., 664., 1290., 1984., 2688., 1590., 712., 150.], + [114., 344., 642., 960., 1280., 726., 296., 38.]], + + [[23., 400., 1035., 1832., 2696., 1737., 904., 293.], + [134., 920., 2166., 3680., 5280., 3306., 1640., 474.], + [325., 1544., 3369., 5512., 7720., 4683., 2192., 535.], + [571., 1964., 3891., 6064., 8272., 4989., 2324., 565.], + [434., 1360., 2586., 3920., 5264., 3054., 1312., 230.], + [241., 700., 1281., 1888., 2496., 1383., 532., 39.]], + + [[22., 214., 528., 916., 1332., 846., 430., 132.], + [86., 484., 1098., 1832., 2600., 1602., 772., 206.], + [188., 802., 1698., 2732., 3788., 2256., 1018., 218.], + [308., 1006., 1950., 2996., 4052., 2400., 1078., 230.], + [230., 692., 1290., 1928., 2568., 1458., 596., 78.], + [126., 354., 636., 924., 1212., 654., 234., 0.]]], + dtype=np.float64).astype(dt) + + return a, b, y_r + + def test_rank3_valid(self, dt): + a, b, y_r = self._setup_rank3(dt) + y = correlate(a, b, "valid") + assert_array_almost_equal(y, y_r[1:2, 2:4, 3:5]) + assert_equal(y.dtype, dt) + + # See gh-5897 + y = correlate(b, a, "valid") + assert_array_almost_equal(y, y_r[1:2, 2:4, 3:5][::-1, ::-1, ::-1]) + assert_equal(y.dtype, dt) + + def test_rank3_same(self, dt): + a, b, y_r = self._setup_rank3(dt) + y = correlate(a, b, "same") + assert_array_almost_equal(y, y_r[0:-1, 1:-1, 1:-2]) + assert_equal(y.dtype, dt) + + def test_rank3_all(self, dt): + a, b, y_r = self._setup_rank3(dt) + y = correlate(a, b) + assert_array_almost_equal(y, y_r) + assert_equal(y.dtype, dt) + + +class TestCorrelate: + # Tests that don't depend on dtype + + def test_invalid_shapes(self): + # By "invalid," we mean that no one + # array has dimensions that are all at + # least as large as the corresponding + # dimensions of the other array. This + # setup should throw a ValueError. + a = np.arange(1, 7).reshape((2, 3)) + b = np.arange(-6, 0).reshape((3, 2)) + + assert_raises(ValueError, correlate, *(a, b), **{'mode': 'valid'}) + assert_raises(ValueError, correlate, *(b, a), **{'mode': 'valid'}) + + def test_invalid_params(self): + a = [3, 4, 5] + b = [1, 2, 3] + assert_raises(ValueError, correlate, a, b, mode='spam') + assert_raises(ValueError, correlate, a, b, mode='eggs', method='fft') + assert_raises(ValueError, correlate, a, b, mode='ham', method='direct') + assert_raises(ValueError, correlate, a, b, mode='full', method='bacon') + assert_raises(ValueError, correlate, a, b, mode='same', method='bacon') + + def test_mismatched_dims(self): + # Input arrays should have the same number of dimensions + assert_raises(ValueError, correlate, [1], 2, method='direct') + assert_raises(ValueError, correlate, 1, [2], method='direct') + assert_raises(ValueError, correlate, [1], 2, method='fft') + assert_raises(ValueError, correlate, 1, [2], method='fft') + assert_raises(ValueError, correlate, [1], [[2]]) + assert_raises(ValueError, correlate, [3], 2) + + def test_numpy_fastpath(self): + a = [1, 2, 3] + b = [4, 5] + assert_allclose(correlate(a, b, mode='same'), [5, 14, 23]) + + a = [1, 2, 3] + b = [4, 5, 6] + assert_allclose(correlate(a, b, mode='same'), [17, 32, 23]) + assert_allclose(correlate(a, b, mode='full'), [6, 17, 32, 23, 12]) + assert_allclose(correlate(a, b, mode='valid'), [32]) + + @pytest.mark.thread_unsafe + def test_dtype_deprecation(self): + # gh-21211 + a = np.asarray([1, 2, 3, 6, 5, 3], dtype=object) + b = np.asarray([2, 3, 4, 5, 3, 4, 2, 2, 1], dtype=object) + with pytest.deprecated_call(match="dtype=object is not supported"): + correlate(a, b) + + +@pytest.mark.parametrize("mode", ["valid", "same", "full"]) +@pytest.mark.parametrize("behind", [True, False]) +@pytest.mark.parametrize("input_size", [100, 101, 1000, 1001, 10000, 10001]) +def test_correlation_lags(mode, behind, input_size): + # generate random data + rng = np.random.RandomState(0) + in1 = rng.standard_normal(input_size) + offset = int(input_size/10) + # generate offset version of array to correlate with + if behind: + # y is behind x + in2 = np.concatenate([rng.standard_normal(offset), in1]) + expected = -offset + else: + # y is ahead of x + in2 = in1[offset:] + expected = offset + # cross correlate, returning lag information + correlation = correlate(in1, in2, mode=mode) + lags = correlation_lags(in1.size, in2.size, mode=mode) + # identify the peak + lag_index = np.argmax(correlation) + # Check as expected + assert_equal(lags[lag_index], expected) + # Correlation and lags shape should match + assert_equal(lags.shape, correlation.shape) + + +def test_correlation_lags_invalid_mode(): + with pytest.raises(ValueError, match="Mode asdfgh is invalid"): + correlation_lags(100, 100, mode="asdfgh") + + +@pytest.mark.parametrize('dt', [np.csingle, np.cdouble, + pytest.param(np.clongdouble, marks=_pmf)]) +class TestCorrelateComplex: + # The decimal precision to be used for comparing results. + # This value will be passed as the 'decimal' keyword argument of + # assert_array_almost_equal(). + # Since correlate may chose to use FFT method which converts + # longdoubles to doubles internally don't expect better precision + # for longdouble than for double (see gh-9520). + + def decimal(self, dt): + if dt == np.clongdouble: + dt = np.cdouble + return int(2 * np.finfo(dt).precision / 3) + + def _setup_rank1(self, dt, mode): + np.random.seed(9) + a = np.random.randn(10).astype(dt) + a += 1j * np.random.randn(10).astype(dt) + b = np.random.randn(8).astype(dt) + b += 1j * np.random.randn(8).astype(dt) + + y_r = (correlate(a.real, b.real, mode=mode) + + correlate(a.imag, b.imag, mode=mode)).astype(dt) + y_r += 1j * (-correlate(a.real, b.imag, mode=mode) + + correlate(a.imag, b.real, mode=mode)) + return a, b, y_r + + def test_rank1_valid(self, dt): + a, b, y_r = self._setup_rank1(dt, 'valid') + y = correlate(a, b, 'valid') + assert_array_almost_equal(y, y_r, decimal=self.decimal(dt)) + assert_equal(y.dtype, dt) + + # See gh-5897 + y = correlate(b, a, 'valid') + assert_array_almost_equal(y, y_r[::-1].conj(), decimal=self.decimal(dt)) + assert_equal(y.dtype, dt) + + def test_rank1_same(self, dt): + a, b, y_r = self._setup_rank1(dt, 'same') + y = correlate(a, b, 'same') + assert_array_almost_equal(y, y_r, decimal=self.decimal(dt)) + assert_equal(y.dtype, dt) + + def test_rank1_full(self, dt): + a, b, y_r = self._setup_rank1(dt, 'full') + y = correlate(a, b, 'full') + assert_array_almost_equal(y, y_r, decimal=self.decimal(dt)) + assert_equal(y.dtype, dt) + + def test_swap_full(self, dt): + d = np.array([0.+0.j, 1.+1.j, 2.+2.j], dtype=dt) + k = np.array([1.+3.j, 2.+4.j, 3.+5.j, 4.+6.j], dtype=dt) + y = correlate(d, k) + assert_equal(y, [0.+0.j, 10.-2.j, 28.-6.j, 22.-6.j, 16.-6.j, 8.-4.j]) + + def test_swap_same(self, dt): + d = [0.+0.j, 1.+1.j, 2.+2.j] + k = [1.+3.j, 2.+4.j, 3.+5.j, 4.+6.j] + y = correlate(d, k, mode="same") + assert_equal(y, [10.-2.j, 28.-6.j, 22.-6.j]) + + def test_rank3(self, dt): + a = np.random.randn(10, 8, 6).astype(dt) + a += 1j * np.random.randn(10, 8, 6).astype(dt) + b = np.random.randn(8, 6, 4).astype(dt) + b += 1j * np.random.randn(8, 6, 4).astype(dt) + + y_r = (correlate(a.real, b.real) + + correlate(a.imag, b.imag)).astype(dt) + y_r += 1j * (-correlate(a.real, b.imag) + correlate(a.imag, b.real)) + + y = correlate(a, b, 'full') + assert_array_almost_equal(y, y_r, decimal=self.decimal(dt) - 1) + assert_equal(y.dtype, dt) + + def test_rank0(self, dt): + a = np.array(np.random.randn()).astype(dt) + a += 1j * np.array(np.random.randn()).astype(dt) + b = np.array(np.random.randn()).astype(dt) + b += 1j * np.array(np.random.randn()).astype(dt) + + y_r = (correlate(a.real, b.real) + + correlate(a.imag, b.imag)).astype(dt) + y_r += 1j * np.array(-correlate(a.real, b.imag) + + correlate(a.imag, b.real)) + + y = correlate(a, b, 'full') + assert_array_almost_equal(y, y_r, decimal=self.decimal(dt) - 1) + assert_equal(y.dtype, dt) + + assert_equal(correlate([1], [2j]), correlate(1, 2j)) + assert_equal(correlate([2j], [3j]), correlate(2j, 3j)) + assert_equal(correlate([3j], [4]), correlate(3j, 4)) + + +class TestCorrelate2d: + + def test_consistency_correlate_funcs(self): + # Compare np.correlate, signal.correlate, signal.correlate2d + a = np.arange(5) + b = np.array([3.2, 1.4, 3]) + for mode in ['full', 'valid', 'same']: + assert_almost_equal(np.correlate(a, b, mode=mode), + signal.correlate(a, b, mode=mode)) + assert_almost_equal(np.squeeze(signal.correlate2d([a], [b], + mode=mode)), + signal.correlate(a, b, mode=mode)) + + # See gh-5897 + if mode == 'valid': + assert_almost_equal(np.correlate(b, a, mode=mode), + signal.correlate(b, a, mode=mode)) + assert_almost_equal(np.squeeze(signal.correlate2d([b], [a], + mode=mode)), + signal.correlate(b, a, mode=mode)) + + def test_invalid_shapes(self): + # By "invalid," we mean that no one + # array has dimensions that are all at + # least as large as the corresponding + # dimensions of the other array. This + # setup should throw a ValueError. + a = np.arange(1, 7).reshape((2, 3)) + b = np.arange(-6, 0).reshape((3, 2)) + + assert_raises(ValueError, signal.correlate2d, *(a, b), **{'mode': 'valid'}) + assert_raises(ValueError, signal.correlate2d, *(b, a), **{'mode': 'valid'}) + + def test_complex_input(self): + assert_equal(signal.correlate2d([[1]], [[2j]]), -2j) + assert_equal(signal.correlate2d([[2j]], [[3j]]), 6) + assert_equal(signal.correlate2d([[3j]], [[4]]), 12j) + + +class TestLFilterZI: + + def test_basic(self): + a = np.array([1.0, -1.0, 0.5]) + b = np.array([1.0, 0.0, 2.0]) + zi_expected = np.array([5.0, -1.0]) + zi = lfilter_zi(b, a) + assert_array_almost_equal(zi, zi_expected) + + def test_scale_invariance(self): + # Regression test. There was a bug in which b was not correctly + # rescaled when a[0] was nonzero. + b = np.array([2, 8, 5]) + a = np.array([1, 1, 8]) + zi1 = lfilter_zi(b, a) + zi2 = lfilter_zi(2*b, 2*a) + assert_allclose(zi2, zi1, rtol=1e-12) + + @pytest.mark.parametrize('dtype', [np.float32, np.float64]) + def test_types(self, dtype): + b = np.zeros((8), dtype=dtype) + a = np.array([1], dtype=dtype) + assert_equal(np.real(signal.lfilter_zi(b, a)).dtype, dtype) + + +class TestFiltFilt: + filtfilt_kind = 'tf' + + def filtfilt(self, zpk, x, axis=-1, padtype='odd', padlen=None, + method='pad', irlen=None): + if self.filtfilt_kind == 'tf': + b, a = zpk2tf(*zpk) + return filtfilt(b, a, x, axis, padtype, padlen, method, irlen) + elif self.filtfilt_kind == 'sos': + sos = zpk2sos(*zpk) + return sosfiltfilt(sos, x, axis, padtype, padlen) + + def test_basic(self): + zpk = tf2zpk([1, 2, 3], [1, 2, 3]) + out = self.filtfilt(zpk, np.arange(12)) + assert_allclose(out, arange(12), atol=5.28e-11) + + def test_sine(self): + rate = 2000 + t = np.linspace(0, 1.0, rate + 1) + # A signal with low frequency and a high frequency. + xlow = np.sin(5 * 2 * np.pi * t) + xhigh = np.sin(250 * 2 * np.pi * t) + x = xlow + xhigh + + zpk = butter(8, 0.125, output='zpk') + # r is the magnitude of the largest pole. + r = np.abs(zpk[1]).max() + eps = 1e-5 + # n estimates the number of steps for the + # transient to decay by a factor of eps. + n = int(np.ceil(np.log(eps) / np.log(r))) + + # High order lowpass filter... + y = self.filtfilt(zpk, x, padlen=n) + # Result should be just xlow. + err = np.abs(y - xlow).max() + assert_(err < 1e-4) + + # A 2D case. + x2d = np.vstack([xlow, xlow + xhigh]) + y2d = self.filtfilt(zpk, x2d, padlen=n, axis=1) + assert_equal(y2d.shape, x2d.shape) + err = np.abs(y2d - xlow).max() + assert_(err < 1e-4) + + # Use the previous result to check the use of the axis keyword. + # (Regression test for ticket #1620) + y2dt = self.filtfilt(zpk, x2d.T, padlen=n, axis=0) + assert_equal(y2d, y2dt.T) + + def test_axis(self): + # Test the 'axis' keyword on a 3D array. + x = np.arange(10.0 * 11.0 * 12.0).reshape(10, 11, 12) + zpk = butter(3, 0.125, output='zpk') + y0 = self.filtfilt(zpk, x, padlen=0, axis=0) + y1 = self.filtfilt(zpk, np.swapaxes(x, 0, 1), padlen=0, axis=1) + assert_array_equal(y0, np.swapaxes(y1, 0, 1)) + y2 = self.filtfilt(zpk, np.swapaxes(x, 0, 2), padlen=0, axis=2) + assert_array_equal(y0, np.swapaxes(y2, 0, 2)) + + def test_acoeff(self): + if self.filtfilt_kind != 'tf': + return # only necessary for TF + # test for 'a' coefficient as single number + out = signal.filtfilt([.5, .5], 1, np.arange(10)) + assert_allclose(out, np.arange(10), rtol=1e-14, atol=1e-14) + + def test_gust_simple(self): + if self.filtfilt_kind != 'tf': + pytest.skip('gust only implemented for TF systems') + # The input array has length 2. The exact solution for this case + # was computed "by hand". + x = np.array([1.0, 2.0]) + b = np.array([0.5]) + a = np.array([1.0, -0.5]) + y, z1, z2 = _filtfilt_gust(b, a, x) + assert_allclose([z1[0], z2[0]], + [0.3*x[0] + 0.2*x[1], 0.2*x[0] + 0.3*x[1]]) + assert_allclose(y, [z1[0] + 0.25*z2[0] + 0.25*x[0] + 0.125*x[1], + 0.25*z1[0] + z2[0] + 0.125*x[0] + 0.25*x[1]]) + + def test_gust_scalars(self): + if self.filtfilt_kind != 'tf': + pytest.skip('gust only implemented for TF systems') + # The filter coefficients are both scalars, so the filter simply + # multiplies its input by b/a. When it is used in filtfilt, the + # factor is (b/a)**2. + x = np.arange(12) + b = 3.0 + a = 2.0 + y = filtfilt(b, a, x, method="gust") + expected = (b/a)**2 * x + assert_allclose(y, expected) + + +class TestSOSFiltFilt(TestFiltFilt): + filtfilt_kind = 'sos' + + def test_equivalence(self): + """Test equivalence between sosfiltfilt and filtfilt""" + x = np.random.RandomState(0).randn(1000) + for order in range(1, 6): + zpk = signal.butter(order, 0.35, output='zpk') + b, a = zpk2tf(*zpk) + sos = zpk2sos(*zpk) + y = filtfilt(b, a, x) + y_sos = sosfiltfilt(sos, x) + assert_allclose(y, y_sos, atol=1e-12, err_msg=f'order={order}') + + +def filtfilt_gust_opt(b, a, x): + """ + An alternative implementation of filtfilt with Gustafsson edges. + + This function computes the same result as + `scipy.signal._signaltools._filtfilt_gust`, but only 1-d arrays + are accepted. The problem is solved using `fmin` from `scipy.optimize`. + `_filtfilt_gust` is significantly faster than this implementation. + """ + def filtfilt_gust_opt_func(ics, b, a, x): + """Objective function used in filtfilt_gust_opt.""" + m = max(len(a), len(b)) - 1 + z0f = ics[:m] + z0b = ics[m:] + y_f = lfilter(b, a, x, zi=z0f)[0] + y_fb = lfilter(b, a, y_f[::-1], zi=z0b)[0][::-1] + + y_b = lfilter(b, a, x[::-1], zi=z0b)[0][::-1] + y_bf = lfilter(b, a, y_b, zi=z0f)[0] + value = np.sum((y_fb - y_bf)**2) + return value + + m = max(len(a), len(b)) - 1 + zi = lfilter_zi(b, a) + ics = np.concatenate((x[:m].mean()*zi, x[-m:].mean()*zi)) + result = fmin(filtfilt_gust_opt_func, ics, args=(b, a, x), + xtol=1e-10, ftol=1e-12, + maxfun=10000, maxiter=10000, + full_output=True, disp=False) + opt, fopt, niter, funcalls, warnflag = result + if warnflag > 0: + raise RuntimeError("minimization failed in filtfilt_gust_opt: " + "warnflag=%d" % warnflag) + z0f = opt[:m] + z0b = opt[m:] + + # Apply the forward-backward filter using the computed initial + # conditions. + y_b = lfilter(b, a, x[::-1], zi=z0b)[0][::-1] + y = lfilter(b, a, y_b, zi=z0f)[0] + + return y, z0f, z0b + + +def check_filtfilt_gust(b, a, shape, axis, irlen=None): + # Generate x, the data to be filtered. + np.random.seed(123) + x = np.random.randn(*shape) + + # Apply filtfilt to x. This is the main calculation to be checked. + y = filtfilt(b, a, x, axis=axis, method="gust", irlen=irlen) + + # Also call the private function so we can test the ICs. + yg, zg1, zg2 = _filtfilt_gust(b, a, x, axis=axis, irlen=irlen) + + # filtfilt_gust_opt is an independent implementation that gives the + # expected result, but it only handles 1-D arrays, so use some looping + # and reshaping shenanigans to create the expected output arrays. + xx = np.swapaxes(x, axis, -1) + out_shape = xx.shape[:-1] + yo = np.empty_like(xx) + m = max(len(a), len(b)) - 1 + zo1 = np.empty(out_shape + (m,)) + zo2 = np.empty(out_shape + (m,)) + for indx in product(*[range(d) for d in out_shape]): + yo[indx], zo1[indx], zo2[indx] = filtfilt_gust_opt(b, a, xx[indx]) + yo = np.swapaxes(yo, -1, axis) + zo1 = np.swapaxes(zo1, -1, axis) + zo2 = np.swapaxes(zo2, -1, axis) + + assert_allclose(y, yo, rtol=1e-8, atol=1e-9) + assert_allclose(yg, yo, rtol=1e-8, atol=1e-9) + assert_allclose(zg1, zo1, rtol=1e-8, atol=1e-9) + assert_allclose(zg2, zo2, rtol=1e-8, atol=1e-9) + + +@pytest.mark.fail_slow(10) +def test_choose_conv_method(): + for mode in ['valid', 'same', 'full']: + for ndim in [1, 2]: + n, k, true_method = 8, 6, 'direct' + x = np.random.randn(*((n,) * ndim)) + h = np.random.randn(*((k,) * ndim)) + + method = choose_conv_method(x, h, mode=mode) + assert_equal(method, true_method) + + method_try, times = choose_conv_method(x, h, mode=mode, measure=True) + assert_(method_try in {'fft', 'direct'}) + assert_(isinstance(times, dict)) + assert_('fft' in times.keys() and 'direct' in times.keys()) + + x = np.array([2**51], dtype=np.int64) + h = x.copy() + assert_equal(choose_conv_method(x, h, mode=mode), 'direct') + + +@pytest.mark.thread_unsafe +def test_choose_conv_dtype_deprecation(): + # gh-21211 + a = np.asarray([1, 2, 3, 6, 5, 3], dtype=object) + b = np.asarray([2, 3, 4, 5, 3, 4, 2, 2, 1], dtype=object) + with pytest.deprecated_call(match="dtype=object is not supported"): + choose_conv_method(a, b) + + +@pytest.mark.filterwarnings('ignore::DeprecationWarning') +def test_choose_conv_method_2(): + for mode in ['valid', 'same', 'full']: + x = [Decimal(3), Decimal(2)] + h = [Decimal(1), Decimal(4)] + assert_equal(choose_conv_method(x, h, mode=mode), 'direct') + + n = 10 + for not_fft_conv_supp in ["complex256", "complex192"]: + if hasattr(np, not_fft_conv_supp): + x = np.ones(n, dtype=not_fft_conv_supp) + h = x.copy() + assert_equal(choose_conv_method(x, h, mode=mode), 'direct') + + +@pytest.mark.fail_slow(10) +def test_filtfilt_gust(): + # Design a filter. + z, p, k = signal.ellip(3, 0.01, 120, 0.0875, output='zpk') + + # Find the approximate impulse response length of the filter. + eps = 1e-10 + r = np.max(np.abs(p)) + approx_impulse_len = int(np.ceil(np.log(eps) / np.log(r))) + + np.random.seed(123) + + b, a = zpk2tf(z, p, k) + for irlen in [None, approx_impulse_len]: + signal_len = 5 * approx_impulse_len + + # 1-d test case + check_filtfilt_gust(b, a, (signal_len,), 0, irlen) + + # 3-d test case; test each axis. + for axis in range(3): + shape = [2, 2, 2] + shape[axis] = signal_len + check_filtfilt_gust(b, a, shape, axis, irlen) + + # Test case with length less than 2*approx_impulse_len. + # In this case, `filtfilt_gust` should behave the same as if + # `irlen=None` was given. + length = 2*approx_impulse_len - 50 + check_filtfilt_gust(b, a, (length,), 0, approx_impulse_len) + + +class TestDecimate: + def test_bad_args(self): + x = np.arange(12) + assert_raises(TypeError, signal.decimate, x, q=0.5, n=1) + assert_raises(TypeError, signal.decimate, x, q=2, n=0.5) + + def test_basic_IIR(self): + x = np.arange(12) + y = signal.decimate(x, 2, n=1, ftype='iir', zero_phase=False).round() + assert_array_equal(y, x[::2]) + + def test_basic_FIR(self): + x = np.arange(12) + y = signal.decimate(x, 2, n=1, ftype='fir', zero_phase=False).round() + assert_array_equal(y, x[::2]) + + def test_shape(self): + # Regression test for ticket #1480. + z = np.zeros((30, 30)) + d0 = signal.decimate(z, 2, axis=0, zero_phase=False) + assert_equal(d0.shape, (15, 30)) + d1 = signal.decimate(z, 2, axis=1, zero_phase=False) + assert_equal(d1.shape, (30, 15)) + + def test_phaseshift_FIR(self): + with suppress_warnings() as sup: + sup.filter(BadCoefficients, "Badly conditioned filter") + self._test_phaseshift(method='fir', zero_phase=False) + + def test_zero_phase_FIR(self): + with suppress_warnings() as sup: + sup.filter(BadCoefficients, "Badly conditioned filter") + self._test_phaseshift(method='fir', zero_phase=True) + + def test_phaseshift_IIR(self): + self._test_phaseshift(method='iir', zero_phase=False) + + def test_zero_phase_IIR(self): + self._test_phaseshift(method='iir', zero_phase=True) + + def _test_phaseshift(self, method, zero_phase): + rate = 120 + rates_to = [15, 20, 30, 40] # q = 8, 6, 4, 3 + + t_tot = 100 # Need to let antialiasing filters settle + t = np.arange(rate*t_tot+1) / float(rate) + + # Sinusoids at 0.8*nyquist, windowed to avoid edge artifacts + freqs = np.array(rates_to) * 0.8 / 2 + d = (np.exp(1j * 2 * np.pi * freqs[:, np.newaxis] * t) + * signal.windows.tukey(t.size, 0.1)) + + for rate_to in rates_to: + q = rate // rate_to + t_to = np.arange(rate_to*t_tot+1) / float(rate_to) + d_tos = (np.exp(1j * 2 * np.pi * freqs[:, np.newaxis] * t_to) + * signal.windows.tukey(t_to.size, 0.1)) + + # Set up downsampling filters, match v0.17 defaults + if method == 'fir': + n = 30 + system = signal.dlti(signal.firwin(n + 1, 1. / q, + window='hamming'), 1.) + elif method == 'iir': + n = 8 + wc = 0.8*np.pi/q + system = signal.dlti(*signal.cheby1(n, 0.05, wc/np.pi)) + + # Calculate expected phase response, as unit complex vector + if zero_phase is False: + _, h_resps = signal.freqz(system.num, system.den, + freqs/rate*2*np.pi) + h_resps /= np.abs(h_resps) + else: + h_resps = np.ones_like(freqs) + + y_resamps = signal.decimate(d.real, q, n, ftype=system, + zero_phase=zero_phase) + + # Get phase from complex inner product, like CSD + h_resamps = np.sum(d_tos.conj() * y_resamps, axis=-1) + h_resamps /= np.abs(h_resamps) + subnyq = freqs < 0.5*rate_to + + # Complex vectors should be aligned, only compare below nyquist + assert_allclose(np.angle(h_resps.conj()*h_resamps)[subnyq], 0, + atol=1e-3, rtol=1e-3) + + def test_auto_n(self): + # Test that our value of n is a reasonable choice (depends on + # the downsampling factor) + sfreq = 100. + n = 1000 + t = np.arange(n) / sfreq + # will alias for decimations (>= 15) + x = np.sqrt(2. / n) * np.sin(2 * np.pi * (sfreq / 30.) * t) + assert_allclose(np.linalg.norm(x), 1., rtol=1e-3) + x_out = signal.decimate(x, 30, ftype='fir') + assert_array_less(np.linalg.norm(x_out), 0.01) + + def test_long_float32(self): + # regression: gh-15072. With 32-bit float and either lfilter + # or filtfilt, this is numerically unstable + x = signal.decimate(np.ones(10_000, dtype=np.float32), 10) + assert not any(np.isnan(x)) + + def test_float16_upcast(self): + # float16 must be upcast to float64 + x = signal.decimate(np.ones(100, dtype=np.float16), 10) + assert x.dtype.type == np.float64 + + def test_complex_iir_dlti(self): + # regression: gh-17845 + # centre frequency for filter [Hz] + fcentre = 50 + # filter passband width [Hz] + fwidth = 5 + # sample rate [Hz] + fs = 1e3 + + z, p, k = signal.butter(2, 2*np.pi*fwidth/2, output='zpk', fs=fs) + z = z.astype(complex) * np.exp(2j * np.pi * fcentre/fs) + p = p.astype(complex) * np.exp(2j * np.pi * fcentre/fs) + system = signal.dlti(z, p, k) + + t = np.arange(200) / fs + + # input + u = (np.exp(2j * np.pi * fcentre * t) + + 0.5 * np.exp(-2j * np.pi * fcentre * t)) + + ynzp = signal.decimate(u, 2, ftype=system, zero_phase=False) + ynzpref = signal.lfilter(*signal.zpk2tf(z, p, k), + u)[::2] + + assert_equal(ynzp, ynzpref) + + yzp = signal.decimate(u, 2, ftype=system, zero_phase=True) + yzpref = signal.filtfilt(*signal.zpk2tf(z, p, k), + u)[::2] + + assert_allclose(yzp, yzpref, rtol=1e-10, atol=1e-13) + + def test_complex_fir_dlti(self): + # centre frequency for filter [Hz] + fcentre = 50 + # filter passband width [Hz] + fwidth = 5 + # sample rate [Hz] + fs = 1e3 + numtaps = 20 + + # FIR filter about 0Hz + bbase = signal.firwin(numtaps, fwidth/2, fs=fs) + + # rotate these to desired frequency + zbase = np.roots(bbase) + zrot = zbase * np.exp(2j * np.pi * fcentre/fs) + # FIR filter about 50Hz, maintaining passband gain of 0dB + bz = bbase[0] * np.poly(zrot) + + system = signal.dlti(bz, 1) + + t = np.arange(200) / fs + + # input + u = (np.exp(2j * np.pi * fcentre * t) + + 0.5 * np.exp(-2j * np.pi * fcentre * t)) + + ynzp = signal.decimate(u, 2, ftype=system, zero_phase=False) + ynzpref = signal.upfirdn(bz, u, up=1, down=2)[:100] + + assert_equal(ynzp, ynzpref) + + yzp = signal.decimate(u, 2, ftype=system, zero_phase=True) + yzpref = signal.resample_poly(u, 1, 2, window=bz) + + assert_equal(yzp, yzpref) + + +class TestHilbert: + + def test_bad_args(self): + x = np.array([1.0 + 0.0j]) + assert_raises(ValueError, hilbert, x) + x = np.arange(8.0) + assert_raises(ValueError, hilbert, x, N=0) + + def test_hilbert_theoretical(self): + # test cases by Ariel Rokem + decimal = 14 + + pi = np.pi + t = np.arange(0, 2 * pi, pi / 256) + a0 = np.sin(t) + a1 = np.cos(t) + a2 = np.sin(2 * t) + a3 = np.cos(2 * t) + a = np.vstack([a0, a1, a2, a3]) + + h = hilbert(a) + h_abs = np.abs(h) + h_angle = np.angle(h) + h_real = np.real(h) + + # The real part should be equal to the original signals: + assert_almost_equal(h_real, a, decimal) + # The absolute value should be one everywhere, for this input: + assert_almost_equal(h_abs, np.ones(a.shape), decimal) + # For the 'slow' sine - the phase should go from -pi/2 to pi/2 in + # the first 256 bins: + assert_almost_equal(h_angle[0, :256], + np.arange(-pi / 2, pi / 2, pi / 256), + decimal) + # For the 'slow' cosine - the phase should go from 0 to pi in the + # same interval: + assert_almost_equal( + h_angle[1, :256], np.arange(0, pi, pi / 256), decimal) + # The 'fast' sine should make this phase transition in half the time: + assert_almost_equal(h_angle[2, :128], + np.arange(-pi / 2, pi / 2, pi / 128), + decimal) + # Ditto for the 'fast' cosine: + assert_almost_equal( + h_angle[3, :128], np.arange(0, pi, pi / 128), decimal) + + # The imaginary part of hilbert(cos(t)) = sin(t) Wikipedia + assert_almost_equal(h[1].imag, a0, decimal) + + def test_hilbert_axisN(self): + # tests for axis and N arguments + a = np.arange(18).reshape(3, 6) + # test axis + aa = hilbert(a, axis=-1) + assert_equal(hilbert(a.T, axis=0), aa.T) + # test 1d + assert_almost_equal(hilbert(a[0]), aa[0], 14) + + # test N + aan = hilbert(a, N=20, axis=-1) + assert_equal(aan.shape, [3, 20]) + assert_equal(hilbert(a.T, N=20, axis=0).shape, [20, 3]) + # the next test is just a regression test, + # no idea whether numbers make sense + a0hilb = np.array([0.000000000000000e+00 - 1.72015830311905j, + 1.000000000000000e+00 - 2.047794505137069j, + 1.999999999999999e+00 - 2.244055555687583j, + 3.000000000000000e+00 - 1.262750302935009j, + 4.000000000000000e+00 - 1.066489252384493j, + 5.000000000000000e+00 + 2.918022706971047j, + 8.881784197001253e-17 + 3.845658908989067j, + -9.444121133484362e-17 + 0.985044202202061j, + -1.776356839400251e-16 + 1.332257797702019j, + -3.996802888650564e-16 + 0.501905089898885j, + 1.332267629550188e-16 + 0.668696078880782j, + -1.192678053963799e-16 + 0.235487067862679j, + -1.776356839400251e-16 + 0.286439612812121j, + 3.108624468950438e-16 + 0.031676888064907j, + 1.332267629550188e-16 - 0.019275656884536j, + -2.360035624836702e-16 - 0.1652588660287j, + 0.000000000000000e+00 - 0.332049855010597j, + 3.552713678800501e-16 - 0.403810179797771j, + 8.881784197001253e-17 - 0.751023775297729j, + 9.444121133484362e-17 - 0.79252210110103j]) + assert_almost_equal(aan[0], a0hilb, 14, 'N regression') + + @pytest.mark.parametrize('dtype', [np.float32, np.float64]) + def test_hilbert_types(self, dtype): + in_typed = np.zeros(8, dtype=dtype) + assert_equal(np.real(signal.hilbert(in_typed)).dtype, dtype) + + +class TestHilbert2: + + def test_bad_args(self): + # x must be real. + x = np.array([[1.0 + 0.0j]]) + assert_raises(ValueError, hilbert2, x) + + # x must be rank 2. + x = np.arange(24).reshape(2, 3, 4) + assert_raises(ValueError, hilbert2, x) + + # Bad value for N. + x = np.arange(16).reshape(4, 4) + assert_raises(ValueError, hilbert2, x, N=0) + assert_raises(ValueError, hilbert2, x, N=(2, 0)) + assert_raises(ValueError, hilbert2, x, N=(2,)) + + @pytest.mark.parametrize('dtype', [np.float32, np.float64]) + def test_hilbert2_types(self, dtype): + in_typed = np.zeros((2, 32), dtype=dtype) + assert_equal(np.real(signal.hilbert2(in_typed)).dtype, dtype) + + +class TestEnvelope: + """Unit tests for function `._signaltools.envelope()`. """ + + @staticmethod + def assert_close(actual, desired, msg): + """Little helper to compare to arrays with proper tolerances""" + xp_assert_close(actual, desired, atol=1e-12, rtol=1e-12, err_msg=msg) + + def test_envelope_invalid_parameters(self): + """For `envelope()` Raise all exceptions that are used to verify function + parameters. """ + with pytest.raises(ValueError, + match=r"Invalid parameter axis=2 for z.shape=.*"): + envelope(np.ones(3), axis=2) + with pytest.raises(ValueError, + match=r"z.shape\[axis\] not > 0 for z.shape=.*"): + envelope(np.ones((3, 0)), axis=1) + for bp_in in [(0, 1, 2), (0, 2.), (None, 2.)]: + ts = ', '.join(map(str, bp_in)) + with pytest.raises(ValueError, + match=rf"bp_in=\({ts}\) isn't a 2-tuple of.*"): + # noinspection PyTypeChecker + envelope(np.ones(4), bp_in=bp_in) + with pytest.raises(ValueError, + match="n_out=10.0 is not a positive integer or.*"): + # noinspection PyTypeChecker + envelope(np.ones(4), n_out=10.) + for bp_in in [(-1, 3), (1, 1), (0, 10)]: + with pytest.raises(ValueError, + match=r"`-n//2 <= bp_in\[0\] < bp_in\[1\] <=.*"): + envelope(np.ones(4), bp_in=bp_in) + with pytest.raises(ValueError, match="residual='undefined' not in .*"): + # noinspection PyTypeChecker + envelope(np.ones(4), residual='undefined') + + def test_envelope_verify_parameters(self): + """Ensure that the various parametrizations produce compatible results. """ + Z, Zr_a = [4, 2, 2, 3, 0], [4, 0, 0, 6, 0, 0, 0, 0] + z = sp_fft.irfft(Z) + n = len(z) + + # the reference envelope: + ze2_0, zr_0 = envelope(z, (1, 3), residual='all', squared=True) + self.assert_close(sp_fft.rfft(ze2_0), np.array([4, 2, 0, 0, 0]).astype(complex), + msg="Envelope calculation error") + self.assert_close(sp_fft.rfft(zr_0), np.array([4, 0, 0, 3, 0]).astype(complex), + msg="Residual calculation error") + + ze_1, zr_1 = envelope(z, (1, 3), residual='all', squared=False) + self.assert_close(ze_1**2, ze2_0, + msg="Unsquared versus Squared envelope calculation error") + self.assert_close(zr_1, zr_0, + msg="Unsquared versus Squared residual calculation error") + + ze2_2, zr_2 = envelope(z, (1, 3), residual='all', squared=True, n_out=3*n) + self.assert_close(ze2_2[::3], ze2_0, + msg="3x up-sampled envelope calculation error") + self.assert_close(zr_2[::3], zr_0, + msg="3x up-sampled residual calculation error") + + ze2_3, zr_3 = envelope(z, (1, 3), residual='lowpass', squared=True) + self.assert_close(ze2_3, ze2_0, + msg="`residual='lowpass'` envelope calculation error") + self.assert_close(sp_fft.rfft(zr_3), np.array([4, 0, 0, 0, 0]).astype(complex), + msg="`residual='lowpass'` residual calculation error") + + ze2_4 = envelope(z, (1, 3), residual=None, squared=True) + self.assert_close(ze2_4, ze2_0, + msg="`residual=None` envelope calculation error") + + # compare complex analytic signal to real version + Z_a = np.copy(Z) + Z_a[1:] *= 2 + z_a = sp_fft.ifft(Z_a, n=n) # analytic signal of Z + self.assert_close(z_a.real, z, + msg="Reference analytic signal error") + ze2_a, zr_a = envelope(z_a, (1, 3), residual='all', squared=True) + self.assert_close(ze2_a, ze2_0.astype(complex), # dtypes must match + msg="Complex envelope calculation error") + self.assert_close(sp_fft.fft(zr_a), np.array(Zr_a).astype(complex), + msg="Complex residual calculation error") + + @pytest.mark.parametrize( + " Z, bp_in, Ze2_desired, Zr_desired", + [([1, 0, 2, 2, 0], (1, None), [4, 2, 0, 0, 0], [1, 0, 0, 0, 0]), + ([4, 0, 2, 0, 0], (0, None), [4, 0, 2, 0, 0], [0, 0, 0, 0, 0]), + ([4, 0, 0, 2, 0], (None, None), [4, 0, 0, 2, 0], [0, 0, 0, 0, 0]), + ([0, 0, 2, 2, 0], (1, 3), [2, 0, 0, 0, 0], [0, 0, 0, 2, 0]), + ([4, 0, 2, 2, 0], (-3, 3), [4, 0, 2, 0, 0], [0, 0, 0, 2, 0]), + ([4, 0, 3, 4, 0], (None, 1), [2, 0, 0, 0, 0], [0, 0, 3, 4, 0]), + ([4, 0, 3, 4, 0], (None, 0), [0, 0, 0, 0, 0], [4, 0, 3, 4, 0])]) + def test_envelope_real_signals(self, Z, bp_in, Ze2_desired, Zr_desired): + """Test envelope calculation with real-valued test signals. + + The comparisons are performed in the Fourier space, since it makes evaluating + the bandpass filter behavior straightforward. Note that also the squared + envelope can be easily calculated by hand, if one recalls that coefficients of + a complex-valued Fourier series representing the signal can be directly + determined by an FFT and that the absolute square of a Fourier series is again + a Fourier series. + """ + z = sp_fft.irfft(Z) + ze2, zr = envelope(z, bp_in, residual='all', squared=True) + ze2_lp, zr_lp = envelope(z, bp_in, residual='lowpass', squared=True) + Ze2, Zr, Ze2_lp, Zr_lp = (sp_fft.rfft(z_) for z_ in (ze2, zr, ze2_lp, zr_lp)) + + Ze2_desired = np.array(Ze2_desired).astype(complex) + Zr_desired = np.array(Zr_desired).astype(complex) + self.assert_close(Ze2, Ze2_desired, + msg="Envelope calculation error (residual='all')") + self.assert_close(Zr, Zr_desired, + msg="Residual calculation error (residual='all')") + + if bp_in[1] is not None: + Zr_desired[bp_in[1]:] = 0 + self.assert_close(Ze2_lp, Ze2_desired, + msg="Envelope calculation error (residual='lowpass')") + self.assert_close(Zr_lp, Zr_desired, + msg="Residual calculation error (residual='lowpass')") + + @pytest.mark.parametrize( + " Z, bp_in, Ze2_desired, Zr_desired", + [([0, 5, 0, 5, 0], (None, None), [5, 0, 10, 0, 5], [0, 0, 0, 0, 0]), + ([1, 5, 0, 5, 2], (-1, 2), [5, 0, 10, 0, 5], [1, 0, 0, 0, 2]), + ([1, 2, 6, 0, 6, 3], (-1, 2), [0, 6, 0, 12, 0, 6], [1, 2, 0, 0, 0, 3]) + ]) + def test_envelope_complex_signals(self, Z, bp_in, Ze2_desired, Zr_desired): + """Test envelope calculation with complex-valued test signals. + + We only need to test for the complex envelope here, since the ``Nones``s in the + bandpass filter were already tested in the previous test. + """ + z = sp_fft.ifft(sp_fft.ifftshift(Z)) + ze2, zr = envelope(z, bp_in, residual='all', squared=True) + Ze2, Zr = (sp_fft.fftshift(sp_fft.fft(z_)) for z_ in (ze2, zr)) + + self.assert_close(Ze2, np.array(Ze2_desired).astype(complex), + msg="Envelope calculation error") + self.assert_close(Zr, np.array(Zr_desired).astype(complex), + msg="Residual calculation error") + + def test_envelope_verify_axis_parameter(self): + """Test for multi-channel envelope calculations. """ + z = sp_fft.irfft([[1, 0, 2, 2, 0], [7, 0, 4, 4, 0]]) + Ze2_desired = np.array([[4, 2, 0, 0, 0], [16, 8, 0, 0, 0]], + dtype=complex) + Zr_desired = np.array([[1, 0, 0, 0, 0], [7, 0, 0, 0, 0]], dtype=complex) + + ze2, zr = envelope(z, squared=True, axis=1) + ye2T, yrT = envelope(z.T, squared=True, axis=0) + Ze2, Ye2, Zr, Yr = (sp_fft.rfft(z_) for z_ in (ze2, ye2T.T, zr, yrT.T)) + + self.assert_close(Ze2, Ze2_desired, msg="2d envelope calculation error") + self.assert_close(Zr, Zr_desired, msg="2d residual calculation error") + self.assert_close(Ye2, Ze2_desired, msg="Transposed 2d envelope calc. error") + self.assert_close(Yr, Zr_desired, msg="Transposed 2d residual calc. error") + + def test_envelope_verify_axis_parameter_complex(self): + """Test for multi-channel envelope calculations with complex values. """ + z = sp_fft.ifft(sp_fft.ifftshift([[1, 5, 0, 5, 2], [1, 10, 0, 10, 2]], axes=1)) + Ze2_des = np.array([[5, 0, 10, 0, 5], [20, 0, 40, 0, 20],], + dtype=complex) + Zr_des = np.array([[1, 0, 0, 0, 2], [1, 0, 0, 0, 2]], dtype=complex) + + kw = dict(bp_in=(-1, 2), residual='all', squared=True) + ze2, zr = envelope(z, axis=1, **kw) + ye2T, yrT = envelope(z.T, axis=0, **kw) + Ze2, Ye2, Zr, Yr = (sp_fft.fftshift(sp_fft.fft(z_), axes=1) + for z_ in (ze2, ye2T.T, zr, yrT.T)) + + self.assert_close(Ze2, Ze2_des, msg="2d envelope calculation error") + self.assert_close(Zr, Zr_des, msg="2d residual calculation error") + self.assert_close(Ye2, Ze2_des, msg="Transposed 2d envelope calc. error") + self.assert_close(Yr, Zr_des, msg="Transposed 2d residual calc. error") + + @pytest.mark.parametrize('X', [[4, 0, 0, 1, 2], [4, 0, 0, 2, 1, 2]]) + def test_compare_envelope_hilbert(self, X): + """Compare output of `envelope()` and `hilbert()`. """ + x = sp_fft.irfft(X) + e_hil = np.abs(hilbert(x)) + e_env = envelope(x, (None, None), residual=None) + self.assert_close(e_hil, e_env, msg="Hilbert-Envelope comparison error") + + +class TestPartialFractionExpansion: + @staticmethod + def assert_rp_almost_equal(r, p, r_true, p_true, decimal=7): + r_true = np.asarray(r_true) + p_true = np.asarray(p_true) + + distance = np.hypot(abs(p[:, None] - p_true), + abs(r[:, None] - r_true)) + + rows, cols = linear_sum_assignment(distance) + assert_almost_equal(p[rows], p_true[cols], decimal=decimal) + assert_almost_equal(r[rows], r_true[cols], decimal=decimal) + + def test_compute_factors(self): + factors, poly = _compute_factors([1, 2, 3], [3, 2, 1]) + assert_equal(len(factors), 3) + assert_almost_equal(factors[0], np.poly([2, 2, 3])) + assert_almost_equal(factors[1], np.poly([1, 1, 1, 3])) + assert_almost_equal(factors[2], np.poly([1, 1, 1, 2, 2])) + assert_almost_equal(poly, np.poly([1, 1, 1, 2, 2, 3])) + + factors, poly = _compute_factors([1, 2, 3], [3, 2, 1], + include_powers=True) + assert_equal(len(factors), 6) + assert_almost_equal(factors[0], np.poly([1, 1, 2, 2, 3])) + assert_almost_equal(factors[1], np.poly([1, 2, 2, 3])) + assert_almost_equal(factors[2], np.poly([2, 2, 3])) + assert_almost_equal(factors[3], np.poly([1, 1, 1, 2, 3])) + assert_almost_equal(factors[4], np.poly([1, 1, 1, 3])) + assert_almost_equal(factors[5], np.poly([1, 1, 1, 2, 2])) + assert_almost_equal(poly, np.poly([1, 1, 1, 2, 2, 3])) + + def test_group_poles(self): + unique, multiplicity = _group_poles( + [1.0, 1.001, 1.003, 2.0, 2.003, 3.0], 0.1, 'min') + assert_equal(unique, [1.0, 2.0, 3.0]) + assert_equal(multiplicity, [3, 2, 1]) + + def test_residue_general(self): + # Test are taken from issue #4464, note that poles in scipy are + # in increasing by absolute value order, opposite to MATLAB. + r, p, k = residue([5, 3, -2, 7], [-4, 0, 8, 3]) + assert_almost_equal(r, [1.3320, -0.6653, -1.4167], decimal=4) + assert_almost_equal(p, [-0.4093, -1.1644, 1.5737], decimal=4) + assert_almost_equal(k, [-1.2500], decimal=4) + + r, p, k = residue([-4, 8], [1, 6, 8]) + assert_almost_equal(r, [8, -12]) + assert_almost_equal(p, [-2, -4]) + assert_equal(k.size, 0) + + r, p, k = residue([4, 1], [1, -1, -2]) + assert_almost_equal(r, [1, 3]) + assert_almost_equal(p, [-1, 2]) + assert_equal(k.size, 0) + + r, p, k = residue([4, 3], [2, -3.4, 1.98, -0.406]) + self.assert_rp_almost_equal( + r, p, [-18.125 - 13.125j, -18.125 + 13.125j, 36.25], + [0.5 - 0.2j, 0.5 + 0.2j, 0.7]) + assert_equal(k.size, 0) + + r, p, k = residue([2, 1], [1, 5, 8, 4]) + self.assert_rp_almost_equal(r, p, [-1, 1, 3], [-1, -2, -2]) + assert_equal(k.size, 0) + + r, p, k = residue([3, -1.1, 0.88, -2.396, 1.348], + [1, -0.7, -0.14, 0.048]) + assert_almost_equal(r, [-3, 4, 1]) + assert_almost_equal(p, [0.2, -0.3, 0.8]) + assert_almost_equal(k, [3, 1]) + + r, p, k = residue([1], [1, 2, -3]) + assert_almost_equal(r, [0.25, -0.25]) + assert_almost_equal(p, [1, -3]) + assert_equal(k.size, 0) + + r, p, k = residue([1, 0, -5], [1, 0, 0, 0, -1]) + self.assert_rp_almost_equal(r, p, + [1, 1.5j, -1.5j, -1], [-1, -1j, 1j, 1]) + assert_equal(k.size, 0) + + r, p, k = residue([3, 8, 6], [1, 3, 3, 1]) + self.assert_rp_almost_equal(r, p, [1, 2, 3], [-1, -1, -1]) + assert_equal(k.size, 0) + + r, p, k = residue([3, -1], [1, -3, 2]) + assert_almost_equal(r, [-2, 5]) + assert_almost_equal(p, [1, 2]) + assert_equal(k.size, 0) + + r, p, k = residue([2, 3, -1], [1, -3, 2]) + assert_almost_equal(r, [-4, 13]) + assert_almost_equal(p, [1, 2]) + assert_almost_equal(k, [2]) + + r, p, k = residue([7, 2, 3, -1], [1, -3, 2]) + assert_almost_equal(r, [-11, 69]) + assert_almost_equal(p, [1, 2]) + assert_almost_equal(k, [7, 23]) + + r, p, k = residue([2, 3, -1], [1, -3, 4, -2]) + self.assert_rp_almost_equal(r, p, [4, -1 + 3.5j, -1 - 3.5j], + [1, 1 - 1j, 1 + 1j]) + assert_almost_equal(k.size, 0) + + def test_residue_leading_zeros(self): + # Leading zeros in numerator or denominator must not affect the answer. + r0, p0, k0 = residue([5, 3, -2, 7], [-4, 0, 8, 3]) + r1, p1, k1 = residue([0, 5, 3, -2, 7], [-4, 0, 8, 3]) + r2, p2, k2 = residue([5, 3, -2, 7], [0, -4, 0, 8, 3]) + r3, p3, k3 = residue([0, 0, 5, 3, -2, 7], [0, 0, 0, -4, 0, 8, 3]) + assert_almost_equal(r0, r1) + assert_almost_equal(r0, r2) + assert_almost_equal(r0, r3) + assert_almost_equal(p0, p1) + assert_almost_equal(p0, p2) + assert_almost_equal(p0, p3) + assert_almost_equal(k0, k1) + assert_almost_equal(k0, k2) + assert_almost_equal(k0, k3) + + def test_resiude_degenerate(self): + # Several tests for zero numerator and denominator. + r, p, k = residue([0, 0], [1, 6, 8]) + assert_almost_equal(r, [0, 0]) + assert_almost_equal(p, [-2, -4]) + assert_equal(k.size, 0) + + r, p, k = residue(0, 1) + assert_equal(r.size, 0) + assert_equal(p.size, 0) + assert_equal(k.size, 0) + + with pytest.raises(ValueError, match="Denominator `a` is zero."): + residue(1, 0) + + def test_residuez_general(self): + r, p, k = residuez([1, 6, 6, 2], [1, -(2 + 1j), (1 + 2j), -1j]) + self.assert_rp_almost_equal(r, p, [-2+2.5j, 7.5+7.5j, -4.5-12j], + [1j, 1, 1]) + assert_almost_equal(k, [2j]) + + r, p, k = residuez([1, 2, 1], [1, -1, 0.3561]) + self.assert_rp_almost_equal(r, p, + [-0.9041 - 5.9928j, -0.9041 + 5.9928j], + [0.5 + 0.3257j, 0.5 - 0.3257j], + decimal=4) + assert_almost_equal(k, [2.8082], decimal=4) + + r, p, k = residuez([1, -1], [1, -5, 6]) + assert_almost_equal(r, [-1, 2]) + assert_almost_equal(p, [2, 3]) + assert_equal(k.size, 0) + + r, p, k = residuez([2, 3, 4], [1, 3, 3, 1]) + self.assert_rp_almost_equal(r, p, [4, -5, 3], [-1, -1, -1]) + assert_equal(k.size, 0) + + r, p, k = residuez([1, -10, -4, 4], [2, -2, -4]) + assert_almost_equal(r, [0.5, -1.5]) + assert_almost_equal(p, [-1, 2]) + assert_almost_equal(k, [1.5, -1]) + + r, p, k = residuez([18], [18, 3, -4, -1]) + self.assert_rp_almost_equal(r, p, + [0.36, 0.24, 0.4], [0.5, -1/3, -1/3]) + assert_equal(k.size, 0) + + r, p, k = residuez([2, 3], np.polymul([1, -1/2], [1, 1/4])) + assert_almost_equal(r, [-10/3, 16/3]) + assert_almost_equal(p, [-0.25, 0.5]) + assert_equal(k.size, 0) + + r, p, k = residuez([1, -2, 1], [1, -1]) + assert_almost_equal(r, [0]) + assert_almost_equal(p, [1]) + assert_almost_equal(k, [1, -1]) + + r, p, k = residuez(1, [1, -1j]) + assert_almost_equal(r, [1]) + assert_almost_equal(p, [1j]) + assert_equal(k.size, 0) + + r, p, k = residuez(1, [1, -1, 0.25]) + assert_almost_equal(r, [0, 1]) + assert_almost_equal(p, [0.5, 0.5]) + assert_equal(k.size, 0) + + r, p, k = residuez(1, [1, -0.75, .125]) + assert_almost_equal(r, [-1, 2]) + assert_almost_equal(p, [0.25, 0.5]) + assert_equal(k.size, 0) + + r, p, k = residuez([1, 6, 2], [1, -2, 1]) + assert_almost_equal(r, [-10, 9]) + assert_almost_equal(p, [1, 1]) + assert_almost_equal(k, [2]) + + r, p, k = residuez([6, 2], [1, -2, 1]) + assert_almost_equal(r, [-2, 8]) + assert_almost_equal(p, [1, 1]) + assert_equal(k.size, 0) + + r, p, k = residuez([1, 6, 6, 2], [1, -2, 1]) + assert_almost_equal(r, [-24, 15]) + assert_almost_equal(p, [1, 1]) + assert_almost_equal(k, [10, 2]) + + r, p, k = residuez([1, 0, 1], [1, 0, 0, 0, 0, -1]) + self.assert_rp_almost_equal(r, p, + [0.2618 + 0.1902j, 0.2618 - 0.1902j, + 0.4, 0.0382 - 0.1176j, 0.0382 + 0.1176j], + [-0.8090 + 0.5878j, -0.8090 - 0.5878j, + 1.0, 0.3090 + 0.9511j, 0.3090 - 0.9511j], + decimal=4) + assert_equal(k.size, 0) + + def test_residuez_trailing_zeros(self): + # Trailing zeros in numerator or denominator must not affect the + # answer. + r0, p0, k0 = residuez([5, 3, -2, 7], [-4, 0, 8, 3]) + r1, p1, k1 = residuez([5, 3, -2, 7, 0], [-4, 0, 8, 3]) + r2, p2, k2 = residuez([5, 3, -2, 7], [-4, 0, 8, 3, 0]) + r3, p3, k3 = residuez([5, 3, -2, 7, 0, 0], [-4, 0, 8, 3, 0, 0, 0]) + assert_almost_equal(r0, r1) + assert_almost_equal(r0, r2) + assert_almost_equal(r0, r3) + assert_almost_equal(p0, p1) + assert_almost_equal(p0, p2) + assert_almost_equal(p0, p3) + assert_almost_equal(k0, k1) + assert_almost_equal(k0, k2) + assert_almost_equal(k0, k3) + + def test_residuez_degenerate(self): + r, p, k = residuez([0, 0], [1, 6, 8]) + assert_almost_equal(r, [0, 0]) + assert_almost_equal(p, [-2, -4]) + assert_equal(k.size, 0) + + r, p, k = residuez(0, 1) + assert_equal(r.size, 0) + assert_equal(p.size, 0) + assert_equal(k.size, 0) + + with pytest.raises(ValueError, match="Denominator `a` is zero."): + residuez(1, 0) + + with pytest.raises(ValueError, + match="First coefficient of determinant `a` must " + "be non-zero."): + residuez(1, [0, 1, 2, 3]) + + def test_inverse_unique_roots_different_rtypes(self): + # This test was inspired by GitHub issue 2496. + r = [3 / 10, -1 / 6, -2 / 15] + p = [0, -2, -5] + k = [] + b_expected = [0, 1, 3] + a_expected = [1, 7, 10, 0] + + # With the default tolerance, the rtype does not matter + # for this example. + for rtype in ('avg', 'mean', 'min', 'minimum', 'max', 'maximum'): + b, a = invres(r, p, k, rtype=rtype) + assert_allclose(b, b_expected) + assert_allclose(a, a_expected) + + b, a = invresz(r, p, k, rtype=rtype) + assert_allclose(b, b_expected) + assert_allclose(a, a_expected) + + def test_inverse_repeated_roots_different_rtypes(self): + r = [3 / 20, -7 / 36, -1 / 6, 2 / 45] + p = [0, -2, -2, -5] + k = [] + b_expected = [0, 0, 1, 3] + b_expected_z = [-1/6, -2/3, 11/6, 3] + a_expected = [1, 9, 24, 20, 0] + + for rtype in ('avg', 'mean', 'min', 'minimum', 'max', 'maximum'): + b, a = invres(r, p, k, rtype=rtype) + assert_allclose(b, b_expected, atol=1e-14) + assert_allclose(a, a_expected) + + b, a = invresz(r, p, k, rtype=rtype) + assert_allclose(b, b_expected_z, atol=1e-14) + assert_allclose(a, a_expected) + + def test_inverse_bad_rtype(self): + r = [3 / 20, -7 / 36, -1 / 6, 2 / 45] + p = [0, -2, -2, -5] + k = [] + with pytest.raises(ValueError, match="`rtype` must be one of"): + invres(r, p, k, rtype='median') + with pytest.raises(ValueError, match="`rtype` must be one of"): + invresz(r, p, k, rtype='median') + + def test_invresz_one_coefficient_bug(self): + # Regression test for issue in gh-4646. + r = [1] + p = [2] + k = [0] + b, a = invresz(r, p, k) + assert_allclose(b, [1.0]) + assert_allclose(a, [1.0, -2.0]) + + def test_invres(self): + b, a = invres([1], [1], []) + assert_almost_equal(b, [1]) + assert_almost_equal(a, [1, -1]) + + b, a = invres([1 - 1j, 2, 0.5 - 3j], [1, 0.5j, 1 + 1j], []) + assert_almost_equal(b, [3.5 - 4j, -8.5 + 0.25j, 3.5 + 3.25j]) + assert_almost_equal(a, [1, -2 - 1.5j, 0.5 + 2j, 0.5 - 0.5j]) + + b, a = invres([0.5, 1], [1 - 1j, 2 + 2j], [1, 2, 3]) + assert_almost_equal(b, [1, -1 - 1j, 1 - 2j, 0.5 - 3j, 10]) + assert_almost_equal(a, [1, -3 - 1j, 4]) + + b, a = invres([-1, 2, 1j, 3 - 1j, 4, -2], + [-1, 2 - 1j, 2 - 1j, 3, 3, 3], []) + assert_almost_equal(b, [4 - 1j, -28 + 16j, 40 - 62j, 100 + 24j, + -292 + 219j, 192 - 268j]) + assert_almost_equal(a, [1, -12 + 2j, 53 - 20j, -96 + 68j, 27 - 72j, + 108 - 54j, -81 + 108j]) + + b, a = invres([-1, 1j], [1, 1], [1, 2]) + assert_almost_equal(b, [1, 0, -4, 3 + 1j]) + assert_almost_equal(a, [1, -2, 1]) + + def test_invresz(self): + b, a = invresz([1], [1], []) + assert_almost_equal(b, [1]) + assert_almost_equal(a, [1, -1]) + + b, a = invresz([1 - 1j, 2, 0.5 - 3j], [1, 0.5j, 1 + 1j], []) + assert_almost_equal(b, [3.5 - 4j, -8.5 + 0.25j, 3.5 + 3.25j]) + assert_almost_equal(a, [1, -2 - 1.5j, 0.5 + 2j, 0.5 - 0.5j]) + + b, a = invresz([0.5, 1], [1 - 1j, 2 + 2j], [1, 2, 3]) + assert_almost_equal(b, [2.5, -3 - 1j, 1 - 2j, -1 - 3j, 12]) + assert_almost_equal(a, [1, -3 - 1j, 4]) + + b, a = invresz([-1, 2, 1j, 3 - 1j, 4, -2], + [-1, 2 - 1j, 2 - 1j, 3, 3, 3], []) + assert_almost_equal(b, [6, -50 + 11j, 100 - 72j, 80 + 58j, + -354 + 228j, 234 - 297j]) + assert_almost_equal(a, [1, -12 + 2j, 53 - 20j, -96 + 68j, 27 - 72j, + 108 - 54j, -81 + 108j]) + + b, a = invresz([-1, 1j], [1, 1], [1, 2]) + assert_almost_equal(b, [1j, 1, -3, 2]) + assert_almost_equal(a, [1, -2, 1]) + + def test_inverse_scalar_arguments(self): + b, a = invres(1, 1, 1) + assert_almost_equal(b, [1, 0]) + assert_almost_equal(a, [1, -1]) + + b, a = invresz(1, 1, 1) + assert_almost_equal(b, [2, -1]) + assert_almost_equal(a, [1, -1]) + + +class TestVectorstrength: + + def test_single_1dperiod(self): + events = np.array([.5]) + period = 5. + targ_strength = 1. + targ_phase = .1 + + strength, phase = vectorstrength(events, period) + + assert_equal(strength.ndim, 0) + assert_equal(phase.ndim, 0) + assert_almost_equal(strength, targ_strength) + assert_almost_equal(phase, 2 * np.pi * targ_phase) + + def test_single_2dperiod(self): + events = np.array([.5]) + period = [1, 2, 5.] + targ_strength = [1.] * 3 + targ_phase = np.array([.5, .25, .1]) + + strength, phase = vectorstrength(events, period) + + assert_equal(strength.ndim, 1) + assert_equal(phase.ndim, 1) + assert_array_almost_equal(strength, targ_strength) + assert_almost_equal(phase, 2 * np.pi * targ_phase) + + def test_equal_1dperiod(self): + events = np.array([.25, .25, .25, .25, .25, .25]) + period = 2 + targ_strength = 1. + targ_phase = .125 + + strength, phase = vectorstrength(events, period) + + assert_equal(strength.ndim, 0) + assert_equal(phase.ndim, 0) + assert_almost_equal(strength, targ_strength) + assert_almost_equal(phase, 2 * np.pi * targ_phase) + + def test_equal_2dperiod(self): + events = np.array([.25, .25, .25, .25, .25, .25]) + period = [1, 2, ] + targ_strength = [1.] * 2 + targ_phase = np.array([.25, .125]) + + strength, phase = vectorstrength(events, period) + + assert_equal(strength.ndim, 1) + assert_equal(phase.ndim, 1) + assert_almost_equal(strength, targ_strength) + assert_almost_equal(phase, 2 * np.pi * targ_phase) + + def test_spaced_1dperiod(self): + events = np.array([.1, 1.1, 2.1, 4.1, 10.1]) + period = 1 + targ_strength = 1. + targ_phase = .1 + + strength, phase = vectorstrength(events, period) + + assert_equal(strength.ndim, 0) + assert_equal(phase.ndim, 0) + assert_almost_equal(strength, targ_strength) + assert_almost_equal(phase, 2 * np.pi * targ_phase) + + def test_spaced_2dperiod(self): + events = np.array([.1, 1.1, 2.1, 4.1, 10.1]) + period = [1, .5] + targ_strength = [1.] * 2 + targ_phase = np.array([.1, .2]) + + strength, phase = vectorstrength(events, period) + + assert_equal(strength.ndim, 1) + assert_equal(phase.ndim, 1) + assert_almost_equal(strength, targ_strength) + assert_almost_equal(phase, 2 * np.pi * targ_phase) + + def test_partial_1dperiod(self): + events = np.array([.25, .5, .75]) + period = 1 + targ_strength = 1. / 3. + targ_phase = .5 + + strength, phase = vectorstrength(events, period) + + assert_equal(strength.ndim, 0) + assert_equal(phase.ndim, 0) + assert_almost_equal(strength, targ_strength) + assert_almost_equal(phase, 2 * np.pi * targ_phase) + + def test_partial_2dperiod(self): + events = np.array([.25, .5, .75]) + period = [1., 1., 1., 1.] + targ_strength = [1. / 3.] * 4 + targ_phase = np.array([.5, .5, .5, .5]) + + strength, phase = vectorstrength(events, period) + + assert_equal(strength.ndim, 1) + assert_equal(phase.ndim, 1) + assert_almost_equal(strength, targ_strength) + assert_almost_equal(phase, 2 * np.pi * targ_phase) + + def test_opposite_1dperiod(self): + events = np.array([0, .25, .5, .75]) + period = 1. + targ_strength = 0 + + strength, phase = vectorstrength(events, period) + + assert_equal(strength.ndim, 0) + assert_equal(phase.ndim, 0) + assert_almost_equal(strength, targ_strength) + + def test_opposite_2dperiod(self): + events = np.array([0, .25, .5, .75]) + period = [1.] * 10 + targ_strength = [0.] * 10 + + strength, phase = vectorstrength(events, period) + + assert_equal(strength.ndim, 1) + assert_equal(phase.ndim, 1) + assert_almost_equal(strength, targ_strength) + + def test_2d_events_ValueError(self): + events = np.array([[1, 2]]) + period = 1. + assert_raises(ValueError, vectorstrength, events, period) + + def test_2d_period_ValueError(self): + events = 1. + period = np.array([[1]]) + assert_raises(ValueError, vectorstrength, events, period) + + def test_zero_period_ValueError(self): + events = 1. + period = 0 + assert_raises(ValueError, vectorstrength, events, period) + + def test_negative_period_ValueError(self): + events = 1. + period = -1 + assert_raises(ValueError, vectorstrength, events, period) + + +def assert_allclose_cast(actual, desired, rtol=1e-7, atol=0): + """Wrap assert_allclose while casting object arrays.""" + if actual.dtype.kind == 'O': + dtype = np.array(actual.flat[0]).dtype + actual, desired = actual.astype(dtype), desired.astype(dtype) + assert_allclose(actual, desired, rtol, atol) + + +@pytest.mark.filterwarnings('ignore::DeprecationWarning') +@pytest.mark.parametrize('func', (sosfilt, lfilter)) +def test_nonnumeric_dtypes(func): + x = [Decimal(1), Decimal(2), Decimal(3)] + b = [Decimal(1), Decimal(2), Decimal(3)] + a = [Decimal(1), Decimal(2), Decimal(3)] + x = np.array(x) + assert x.dtype.kind == 'O' + desired = lfilter(np.array(b, float), np.array(a, float), x.astype(float)) + if func is sosfilt: + actual = sosfilt([b + a], x) + else: + actual = lfilter(b, a, x) + assert all(isinstance(x, Decimal) for x in actual) + assert_allclose(actual.astype(float), desired.astype(float)) + # Degenerate cases + if func is lfilter: + args = [1., 1.] + else: + args = [tf2sos(1., 1.)] + + with pytest.raises(ValueError, match='must be at least 1-D'): + func(*args, x=1.) + + +@pytest.mark.parametrize('dt', 'fdFD') +class TestSOSFilt: + + # The test_rank* tests are pulled from _TestLinearFilter + def test_rank1(self, dt): + x = np.linspace(0, 5, 6).astype(dt) + b = np.array([1, -1]).astype(dt) + a = np.array([0.5, -0.5]).astype(dt) + + # Test simple IIR + y_r = np.array([0, 2, 4, 6, 8, 10.]).astype(dt) + sos = tf2sos(b, a) + assert_array_almost_equal(sosfilt(tf2sos(b, a), x), y_r) + + # Test simple FIR + b = np.array([1, 1]).astype(dt) + # NOTE: This was changed (rel. to TestLinear...) to add a pole @zero: + a = np.array([1, 0]).astype(dt) + y_r = np.array([0, 1, 3, 5, 7, 9.]).astype(dt) + assert_array_almost_equal(sosfilt(tf2sos(b, a), x), y_r) + + b = [1, 1, 0] + a = [1, 0, 0] + x = np.ones(8) + sos = np.concatenate((b, a)) + sos.shape = (1, 6) + y = sosfilt(sos, x) + assert_allclose(y, [1, 2, 2, 2, 2, 2, 2, 2]) + + def test_rank2(self, dt): + shape = (4, 3) + x = np.linspace(0, np.prod(shape) - 1, np.prod(shape)).reshape(shape) + x = x.astype(dt) + + b = np.array([1, -1]).astype(dt) + a = np.array([0.5, 0.5]).astype(dt) + + y_r2_a0 = np.array([[0, 2, 4], [6, 4, 2], [0, 2, 4], [6, 4, 2]], + dtype=dt) + + y_r2_a1 = np.array([[0, 2, 0], [6, -4, 6], [12, -10, 12], + [18, -16, 18]], dtype=dt) + + y = sosfilt(tf2sos(b, a), x, axis=0) + assert_array_almost_equal(y_r2_a0, y) + + y = sosfilt(tf2sos(b, a), x, axis=1) + assert_array_almost_equal(y_r2_a1, y) + + def test_rank3(self, dt): + shape = (4, 3, 2) + x = np.linspace(0, np.prod(shape) - 1, np.prod(shape)).reshape(shape) + + b = np.array([1, -1]).astype(dt) + a = np.array([0.5, 0.5]).astype(dt) + + # Test last axis + y = sosfilt(tf2sos(b, a), x) + for i in range(x.shape[0]): + for j in range(x.shape[1]): + assert_array_almost_equal(y[i, j], lfilter(b, a, x[i, j])) + + def test_initial_conditions(self, dt): + b1, a1 = signal.butter(2, 0.25, 'low') + b2, a2 = signal.butter(2, 0.75, 'low') + b3, a3 = signal.butter(2, 0.75, 'low') + b = np.convolve(np.convolve(b1, b2), b3) + a = np.convolve(np.convolve(a1, a2), a3) + sos = np.array((np.r_[b1, a1], np.r_[b2, a2], np.r_[b3, a3])) + + x = np.random.rand(50).astype(dt) + + # Stopping filtering and continuing + y_true, zi = lfilter(b, a, x[:20], zi=np.zeros(6)) + y_true = np.r_[y_true, lfilter(b, a, x[20:], zi=zi)[0]] + assert_allclose_cast(y_true, lfilter(b, a, x)) + + y_sos, zi = sosfilt(sos, x[:20], zi=np.zeros((3, 2))) + y_sos = np.r_[y_sos, sosfilt(sos, x[20:], zi=zi)[0]] + assert_allclose_cast(y_true, y_sos) + + # Use a step function + zi = sosfilt_zi(sos) + x = np.ones(8, dt) + y, zf = sosfilt(sos, x, zi=zi) + + assert_allclose_cast(y, np.ones(8)) + assert_allclose_cast(zf, zi) + + # Initial condition shape matching + x.shape = (1, 1) + x.shape # 3D + assert_raises(ValueError, sosfilt, sos, x, zi=zi) + zi_nd = zi.copy() + zi_nd.shape = (zi.shape[0], 1, 1, zi.shape[-1]) + assert_raises(ValueError, sosfilt, sos, x, + zi=zi_nd[:, :, :, [0, 1, 1]]) + y, zf = sosfilt(sos, x, zi=zi_nd) + assert_allclose_cast(y[0, 0], np.ones(8)) + assert_allclose_cast(zf[:, 0, 0, :], zi) + + def test_initial_conditions_3d_axis1(self, dt): + # Test the use of zi when sosfilt is applied to axis 1 of a 3-d input. + + # Input array is x. + x = np.random.RandomState(159).randint(0, 5, size=(2, 15, 3)) + x = x.astype(dt) + + # Design a filter in ZPK format and convert to SOS + zpk = signal.butter(6, 0.35, output='zpk') + sos = zpk2sos(*zpk) + nsections = sos.shape[0] + + # Filter along this axis. + axis = 1 + + # Initial conditions, all zeros. + shp = list(x.shape) + shp[axis] = 2 + shp = [nsections] + shp + z0 = np.zeros(shp) + + # Apply the filter to x. + yf, zf = sosfilt(sos, x, axis=axis, zi=z0) + + # Apply the filter to x in two stages. + y1, z1 = sosfilt(sos, x[:, :5, :], axis=axis, zi=z0) + y2, z2 = sosfilt(sos, x[:, 5:, :], axis=axis, zi=z1) + + # y should equal yf, and z2 should equal zf. + y = np.concatenate((y1, y2), axis=axis) + assert_allclose_cast(y, yf, rtol=1e-10, atol=1e-13) + assert_allclose_cast(z2, zf, rtol=1e-10, atol=1e-13) + + # let's try the "step" initial condition + zi = sosfilt_zi(sos) + zi.shape = [nsections, 1, 2, 1] + zi = zi * x[:, 0:1, :] + y = sosfilt(sos, x, axis=axis, zi=zi)[0] + # check it against the TF form + b, a = zpk2tf(*zpk) + zi = lfilter_zi(b, a) + zi.shape = [1, zi.size, 1] + zi = zi * x[:, 0:1, :] + y_tf = lfilter(b, a, x, axis=axis, zi=zi)[0] + assert_allclose_cast(y, y_tf, rtol=1e-10, atol=1e-13) + + def test_bad_zi_shape(self, dt): + # The shape of zi is checked before using any values in the + # arguments, so np.empty is fine for creating the arguments. + x = np.empty((3, 15, 3), dt) + sos = np.zeros((4, 6)) + zi = np.empty((4, 3, 3, 2)) # Correct shape is (4, 3, 2, 3) + with pytest.raises(ValueError, match='should be all ones'): + sosfilt(sos, x, zi=zi, axis=1) + sos[:, 3] = 1. + with pytest.raises(ValueError, match='Invalid zi shape'): + sosfilt(sos, x, zi=zi, axis=1) + + def test_sosfilt_zi(self, dt): + sos = signal.butter(6, 0.2, output='sos') + zi = sosfilt_zi(sos) + + y, zf = sosfilt(sos, np.ones(40, dt), zi=zi) + assert_allclose_cast(zf, zi, rtol=1e-13) + + # Expected steady state value of the step response of this filter: + ss = np.prod(sos[:, :3].sum(axis=-1) / sos[:, 3:].sum(axis=-1)) + assert_allclose_cast(y, ss, rtol=1e-13) + + # zi as array-like + _, zf = sosfilt(sos, np.ones(40, dt), zi=zi.tolist()) + assert_allclose_cast(zf, zi, rtol=1e-13) + + @pytest.mark.thread_unsafe + def test_dtype_deprecation(self, dt): + # gh-21211 + sos = np.asarray([1, 2, 3, 1, 5, 3], dtype=object).reshape(1, 6) + x = np.asarray([2, 3, 4, 5, 3, 4, 2, 2, 1], dtype=object) + with pytest.deprecated_call(match="dtype=object is not supported"): + sosfilt(sos, x) + + +class TestDeconvolve: + + def test_basic(self): + # From docstring example + original = [0, 1, 0, 0, 1, 1, 0, 0] + impulse_response = [2, 1] + recorded = [0, 2, 1, 0, 2, 3, 1, 0, 0] + recovered, remainder = signal.deconvolve(recorded, impulse_response) + assert_allclose(recovered, original) + + def test_n_dimensional_signal(self): + recorded = [[0, 0], [0, 0]] + impulse_response = [0, 0] + with pytest.raises(ValueError, match="signal must be 1-D."): + quotient, remainder = signal.deconvolve(recorded, impulse_response) + + def test_n_dimensional_divisor(self): + recorded = [0, 0] + impulse_response = [[0, 0], [0, 0]] + with pytest.raises(ValueError, match="divisor must be 1-D."): + quotient, remainder = signal.deconvolve(recorded, impulse_response) + + +class TestDetrend: + + def test_basic(self): + detrended = detrend(array([1, 2, 3])) + detrended_exact = array([0, 0, 0]) + assert_array_almost_equal(detrended, detrended_exact) + + def test_copy(self): + x = array([1, 1.2, 1.5, 1.6, 2.4]) + copy_array = detrend(x, overwrite_data=False) + inplace = detrend(x, overwrite_data=True) + assert_array_almost_equal(copy_array, inplace) + + @pytest.mark.parametrize('kind', ['linear', 'constant']) + @pytest.mark.parametrize('axis', [0, 1, 2]) + def test_axis(self, axis, kind): + data = np.arange(5*6*7).reshape(5, 6, 7) + detrended = detrend(data, type=kind, axis=axis) + assert detrended.shape == data.shape + + def test_bp(self): + data = [0, 1, 2] + [5, 0, -5, -10] + detrended = detrend(data, type='linear', bp=3) + assert_allclose(detrended, 0, atol=1e-14) + + # repeat with ndim > 1 and axis + data = np.asarray(data)[None, :, None] + + detrended = detrend(data, type="linear", bp=3, axis=1) + assert_allclose(detrended, 0, atol=1e-14) + + # breakpoint index > shape[axis]: raises + with assert_raises(ValueError): + detrend(data, type="linear", bp=3) + + @pytest.mark.parametrize('bp', [np.array([0, 2]), [0, 2]]) + def test_detrend_array_bp(self, bp): + # regression test for https://github.com/scipy/scipy/issues/18675 + rng = np.random.RandomState(12345) + x = rng.rand(10) + # bp = np.array([0, 2]) + + res = detrend(x, bp=bp) + res_scipy_191 = np.array([-4.44089210e-16, -2.22044605e-16, + -1.11128506e-01, -1.69470553e-01, 1.14710683e-01, 6.35468419e-02, + 3.53533144e-01, -3.67877935e-02, -2.00417675e-02, -1.94362049e-01]) + + assert_allclose(res, res_scipy_191, atol=1e-14) + + +class TestUniqueRoots: + def test_real_no_repeat(self): + p = [-1.0, -0.5, 0.3, 1.2, 10.0] + unique, multiplicity = unique_roots(p) + assert_almost_equal(unique, p, decimal=15) + assert_equal(multiplicity, np.ones(len(p))) + + def test_real_repeat(self): + p = [-1.0, -0.95, -0.89, -0.8, 0.5, 1.0, 1.05] + + unique, multiplicity = unique_roots(p, tol=1e-1, rtype='min') + assert_almost_equal(unique, [-1.0, -0.89, 0.5, 1.0], decimal=15) + assert_equal(multiplicity, [2, 2, 1, 2]) + + unique, multiplicity = unique_roots(p, tol=1e-1, rtype='max') + assert_almost_equal(unique, [-0.95, -0.8, 0.5, 1.05], decimal=15) + assert_equal(multiplicity, [2, 2, 1, 2]) + + unique, multiplicity = unique_roots(p, tol=1e-1, rtype='avg') + assert_almost_equal(unique, [-0.975, -0.845, 0.5, 1.025], decimal=15) + assert_equal(multiplicity, [2, 2, 1, 2]) + + def test_complex_no_repeat(self): + p = [-1.0, 1.0j, 0.5 + 0.5j, -1.0 - 1.0j, 3.0 + 2.0j] + unique, multiplicity = unique_roots(p) + assert_almost_equal(unique, p, decimal=15) + assert_equal(multiplicity, np.ones(len(p))) + + def test_complex_repeat(self): + p = [-1.0, -1.0 + 0.05j, -0.95 + 0.15j, -0.90 + 0.15j, 0.0, + 0.5 + 0.5j, 0.45 + 0.55j] + + unique, multiplicity = unique_roots(p, tol=1e-1, rtype='min') + assert_almost_equal(unique, [-1.0, -0.95 + 0.15j, 0.0, 0.45 + 0.55j], + decimal=15) + assert_equal(multiplicity, [2, 2, 1, 2]) + + unique, multiplicity = unique_roots(p, tol=1e-1, rtype='max') + assert_almost_equal(unique, + [-1.0 + 0.05j, -0.90 + 0.15j, 0.0, 0.5 + 0.5j], + decimal=15) + assert_equal(multiplicity, [2, 2, 1, 2]) + + unique, multiplicity = unique_roots(p, tol=1e-1, rtype='avg') + assert_almost_equal( + unique, [-1.0 + 0.025j, -0.925 + 0.15j, 0.0, 0.475 + 0.525j], + decimal=15) + assert_equal(multiplicity, [2, 2, 1, 2]) + + def test_gh_4915(self): + p = np.roots(np.convolve(np.ones(5), np.ones(5))) + true_roots = [-(-1)**(1/5), (-1)**(4/5), -(-1)**(3/5), (-1)**(2/5)] + + unique, multiplicity = unique_roots(p) + unique = np.sort(unique) + + assert_almost_equal(np.sort(unique), true_roots, decimal=7) + assert_equal(multiplicity, [2, 2, 2, 2]) + + def test_complex_roots_extra(self): + unique, multiplicity = unique_roots([1.0, 1.0j, 1.0]) + assert_almost_equal(unique, [1.0, 1.0j], decimal=15) + assert_equal(multiplicity, [2, 1]) + + unique, multiplicity = unique_roots([1, 1 + 2e-9, 1e-9 + 1j], tol=0.1) + assert_almost_equal(unique, [1.0, 1e-9 + 1.0j], decimal=15) + assert_equal(multiplicity, [2, 1]) + + def test_single_unique_root(self): + p = np.random.rand(100) + 1j * np.random.rand(100) + unique, multiplicity = unique_roots(p, 2) + assert_almost_equal(unique, [np.min(p)], decimal=15) + assert_equal(multiplicity, [100]) + + +def test_gh_22684(): + actual = signal.resample_poly(np.arange(2000, dtype=np.complex64), 6, 4) + assert actual.dtype == np.complex64 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_spectral.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_spectral.py new file mode 100644 index 0000000000000000000000000000000000000000..12dac6300b9ef4b2eac0475f0b53517ab4867416 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_spectral.py @@ -0,0 +1,2059 @@ +import sys + +import numpy as np +from numpy.testing import (assert_, + assert_allclose, assert_array_equal, assert_equal, + assert_array_almost_equal_nulp, suppress_warnings) +import pytest +from pytest import raises as assert_raises + +from scipy import signal +from scipy.fft import fftfreq, rfftfreq, fft, irfft +from scipy.integrate import trapezoid +from scipy.signal import (periodogram, welch, lombscargle, coherence, + spectrogram, check_COLA, check_NOLA) +from scipy.signal.windows import hann +from scipy.signal._spectral_py import _spectral_helper + +# Compare ShortTimeFFT.stft() / ShortTimeFFT.istft() with stft() / istft(): +from scipy.signal.tests._scipy_spectral_test_shim import stft_compare as stft +from scipy.signal.tests._scipy_spectral_test_shim import istft_compare as istft +from scipy.signal.tests._scipy_spectral_test_shim import csd_compare as csd + + +class TestPeriodogram: + def test_real_onesided_even(self): + x = np.zeros(16) + x[0] = 1 + f, p = periodogram(x) + assert_allclose(f, np.linspace(0, 0.5, 9)) + q = np.ones(9) + q[0] = 0 + q[-1] /= 2.0 + q /= 8 + assert_allclose(p, q) + + def test_real_onesided_odd(self): + x = np.zeros(15) + x[0] = 1 + f, p = periodogram(x) + assert_allclose(f, np.arange(8.0)/15.0) + q = np.ones(8) + q[0] = 0 + q *= 2.0/15.0 + assert_allclose(p, q, atol=1e-15) + + def test_real_twosided(self): + x = np.zeros(16) + x[0] = 1 + f, p = periodogram(x, return_onesided=False) + assert_allclose(f, fftfreq(16, 1.0)) + q = np.full(16, 1/16.0) + q[0] = 0 + assert_allclose(p, q) + + def test_real_spectrum(self): + x = np.zeros(16) + x[0] = 1 + f, p = periodogram(x, scaling='spectrum') + g, q = periodogram(x, scaling='density') + assert_allclose(f, np.linspace(0, 0.5, 9)) + assert_allclose(p, q/16.0) + + def test_integer_even(self): + x = np.zeros(16, dtype=int) + x[0] = 1 + f, p = periodogram(x) + assert_allclose(f, np.linspace(0, 0.5, 9)) + q = np.ones(9) + q[0] = 0 + q[-1] /= 2.0 + q /= 8 + assert_allclose(p, q) + + def test_integer_odd(self): + x = np.zeros(15, dtype=int) + x[0] = 1 + f, p = periodogram(x) + assert_allclose(f, np.arange(8.0)/15.0) + q = np.ones(8) + q[0] = 0 + q *= 2.0/15.0 + assert_allclose(p, q, atol=1e-15) + + def test_integer_twosided(self): + x = np.zeros(16, dtype=int) + x[0] = 1 + f, p = periodogram(x, return_onesided=False) + assert_allclose(f, fftfreq(16, 1.0)) + q = np.full(16, 1/16.0) + q[0] = 0 + assert_allclose(p, q) + + def test_complex(self): + x = np.zeros(16, np.complex128) + x[0] = 1.0 + 2.0j + f, p = periodogram(x, return_onesided=False) + assert_allclose(f, fftfreq(16, 1.0)) + q = np.full(16, 5.0/16.0) + q[0] = 0 + assert_allclose(p, q) + + def test_unk_scaling(self): + assert_raises(ValueError, periodogram, np.zeros(4, np.complex128), + scaling='foo') + + @pytest.mark.skipif( + sys.maxsize <= 2**32, + reason="On some 32-bit tolerance issue" + ) + def test_nd_axis_m1(self): + x = np.zeros(20, dtype=np.float64) + x = x.reshape((2,1,10)) + x[:,:,0] = 1.0 + f, p = periodogram(x) + assert_array_equal(p.shape, (2, 1, 6)) + assert_array_almost_equal_nulp(p[0,0,:], p[1,0,:], 60) + f0, p0 = periodogram(x[0,0,:]) + assert_array_almost_equal_nulp(p0[np.newaxis,:], p[1,:], 60) + + @pytest.mark.skipif( + sys.maxsize <= 2**32, + reason="On some 32-bit tolerance issue" + ) + def test_nd_axis_0(self): + x = np.zeros(20, dtype=np.float64) + x = x.reshape((10,2,1)) + x[0,:,:] = 1.0 + f, p = periodogram(x, axis=0) + assert_array_equal(p.shape, (6,2,1)) + assert_array_almost_equal_nulp(p[:,0,0], p[:,1,0], 60) + f0, p0 = periodogram(x[:,0,0]) + assert_array_almost_equal_nulp(p0, p[:,1,0]) + + def test_window_external(self): + x = np.zeros(16) + x[0] = 1 + f, p = periodogram(x, 10, 'hann') + win = signal.get_window('hann', 16) + fe, pe = periodogram(x, 10, win) + assert_array_almost_equal_nulp(p, pe) + assert_array_almost_equal_nulp(f, fe) + win_err = signal.get_window('hann', 32) + assert_raises(ValueError, periodogram, x, + 10, win_err) # win longer than signal + + def test_padded_fft(self): + x = np.zeros(16) + x[0] = 1 + f, p = periodogram(x) + fp, pp = periodogram(x, nfft=32) + assert_allclose(f, fp[::2]) + assert_allclose(p, pp[::2]) + assert_array_equal(pp.shape, (17,)) + + def test_empty_input(self): + f, p = periodogram([]) + assert_array_equal(f.shape, (0,)) + assert_array_equal(p.shape, (0,)) + for shape in [(0,), (3,0), (0,5,2)]: + f, p = periodogram(np.empty(shape)) + assert_array_equal(f.shape, shape) + assert_array_equal(p.shape, shape) + + def test_empty_input_other_axis(self): + for shape in [(3,0), (0,5,2)]: + f, p = periodogram(np.empty(shape), axis=1) + assert_array_equal(f.shape, shape) + assert_array_equal(p.shape, shape) + + def test_short_nfft(self): + x = np.zeros(18) + x[0] = 1 + f, p = periodogram(x, nfft=16) + assert_allclose(f, np.linspace(0, 0.5, 9)) + q = np.ones(9) + q[0] = 0 + q[-1] /= 2.0 + q /= 8 + assert_allclose(p, q) + + def test_nfft_is_xshape(self): + x = np.zeros(16) + x[0] = 1 + f, p = periodogram(x, nfft=16) + assert_allclose(f, np.linspace(0, 0.5, 9)) + q = np.ones(9) + q[0] = 0 + q[-1] /= 2.0 + q /= 8 + assert_allclose(p, q) + + def test_real_onesided_even_32(self): + x = np.zeros(16, 'f') + x[0] = 1 + f, p = periodogram(x) + assert_allclose(f, np.linspace(0, 0.5, 9)) + q = np.ones(9, 'f') + q[0] = 0 + q[-1] /= 2.0 + q /= 8 + assert_allclose(p, q) + assert_(p.dtype == q.dtype) + + def test_real_onesided_odd_32(self): + x = np.zeros(15, 'f') + x[0] = 1 + f, p = periodogram(x) + assert_allclose(f, np.arange(8.0)/15.0) + q = np.ones(8, 'f') + q[0] = 0 + q *= 2.0/15.0 + assert_allclose(p, q, atol=1e-7) + assert_(p.dtype == q.dtype) + + def test_real_twosided_32(self): + x = np.zeros(16, 'f') + x[0] = 1 + f, p = periodogram(x, return_onesided=False) + assert_allclose(f, fftfreq(16, 1.0)) + q = np.full(16, 1/16.0, 'f') + q[0] = 0 + assert_allclose(p, q) + assert_(p.dtype == q.dtype) + + def test_complex_32(self): + x = np.zeros(16, 'F') + x[0] = 1.0 + 2.0j + f, p = periodogram(x, return_onesided=False) + assert_allclose(f, fftfreq(16, 1.0)) + q = np.full(16, 5.0/16.0, 'f') + q[0] = 0 + assert_allclose(p, q) + assert_(p.dtype == q.dtype) + + def test_shorter_window_error(self): + x = np.zeros(16) + x[0] = 1 + win = signal.get_window('hann', 10) + expected_msg = ('the size of the window must be the same size ' + 'of the input on the specified axis') + with assert_raises(ValueError, match=expected_msg): + periodogram(x, window=win) + + +class TestWelch: + def test_real_onesided_even(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=8) + assert_allclose(f, np.linspace(0, 0.5, 5)) + q = np.array([0.08333333, 0.15277778, 0.22222222, 0.22222222, + 0.11111111]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_real_onesided_odd(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=9) + assert_allclose(f, np.arange(5.0)/9.0) + q = np.array([0.12477455, 0.23430933, 0.17072113, 0.17072113, + 0.17072113]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_real_twosided(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=8, return_onesided=False) + assert_allclose(f, fftfreq(8, 1.0)) + q = np.array([0.08333333, 0.07638889, 0.11111111, 0.11111111, + 0.11111111, 0.11111111, 0.11111111, 0.07638889]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_real_spectrum(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=8, scaling='spectrum') + assert_allclose(f, np.linspace(0, 0.5, 5)) + q = np.array([0.015625, 0.02864583, 0.04166667, 0.04166667, + 0.02083333]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_integer_onesided_even(self): + x = np.zeros(16, dtype=int) + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=8) + assert_allclose(f, np.linspace(0, 0.5, 5)) + q = np.array([0.08333333, 0.15277778, 0.22222222, 0.22222222, + 0.11111111]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_integer_onesided_odd(self): + x = np.zeros(16, dtype=int) + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=9) + assert_allclose(f, np.arange(5.0)/9.0) + q = np.array([0.12477455, 0.23430933, 0.17072113, 0.17072113, + 0.17072113]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_integer_twosided(self): + x = np.zeros(16, dtype=int) + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=8, return_onesided=False) + assert_allclose(f, fftfreq(8, 1.0)) + q = np.array([0.08333333, 0.07638889, 0.11111111, 0.11111111, + 0.11111111, 0.11111111, 0.11111111, 0.07638889]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_complex(self): + x = np.zeros(16, np.complex128) + x[0] = 1.0 + 2.0j + x[8] = 1.0 + 2.0j + f, p = welch(x, nperseg=8, return_onesided=False) + assert_allclose(f, fftfreq(8, 1.0)) + q = np.array([0.41666667, 0.38194444, 0.55555556, 0.55555556, + 0.55555556, 0.55555556, 0.55555556, 0.38194444]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_unk_scaling(self): + assert_raises(ValueError, welch, np.zeros(4, np.complex128), + scaling='foo', nperseg=4) + + def test_detrend_linear(self): + x = np.arange(10, dtype=np.float64) + 0.04 + f, p = welch(x, nperseg=10, detrend='linear') + assert_allclose(p, np.zeros_like(p), atol=1e-15) + + def test_no_detrending(self): + x = np.arange(10, dtype=np.float64) + 0.04 + f1, p1 = welch(x, nperseg=10, detrend=False) + f2, p2 = welch(x, nperseg=10, detrend=lambda x: x) + assert_allclose(f1, f2, atol=1e-15) + assert_allclose(p1, p2, atol=1e-15) + + def test_detrend_external(self): + x = np.arange(10, dtype=np.float64) + 0.04 + f, p = welch(x, nperseg=10, + detrend=lambda seg: signal.detrend(seg, type='l')) + assert_allclose(p, np.zeros_like(p), atol=1e-15) + + def test_detrend_external_nd_m1(self): + x = np.arange(40, dtype=np.float64) + 0.04 + x = x.reshape((2,2,10)) + f, p = welch(x, nperseg=10, + detrend=lambda seg: signal.detrend(seg, type='l')) + assert_allclose(p, np.zeros_like(p), atol=1e-15) + + def test_detrend_external_nd_0(self): + x = np.arange(20, dtype=np.float64) + 0.04 + x = x.reshape((2,1,10)) + x = np.moveaxis(x, 2, 0) + f, p = welch(x, nperseg=10, axis=0, + detrend=lambda seg: signal.detrend(seg, axis=0, type='l')) + assert_allclose(p, np.zeros_like(p), atol=1e-15) + + def test_nd_axis_m1(self): + x = np.arange(20, dtype=np.float64) + 0.04 + x = x.reshape((2,1,10)) + f, p = welch(x, nperseg=10) + assert_array_equal(p.shape, (2, 1, 6)) + assert_allclose(p[0,0,:], p[1,0,:], atol=1e-13, rtol=1e-13) + f0, p0 = welch(x[0,0,:], nperseg=10) + assert_allclose(p0[np.newaxis,:], p[1,:], atol=1e-13, rtol=1e-13) + + def test_nd_axis_0(self): + x = np.arange(20, dtype=np.float64) + 0.04 + x = x.reshape((10,2,1)) + f, p = welch(x, nperseg=10, axis=0) + assert_array_equal(p.shape, (6,2,1)) + assert_allclose(p[:,0,0], p[:,1,0], atol=1e-13, rtol=1e-13) + f0, p0 = welch(x[:,0,0], nperseg=10) + assert_allclose(p0, p[:,1,0], atol=1e-13, rtol=1e-13) + + def test_window_external(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = welch(x, 10, 'hann', nperseg=8) + win = signal.get_window('hann', 8) + fe, pe = welch(x, 10, win, nperseg=None) + assert_array_almost_equal_nulp(p, pe) + assert_array_almost_equal_nulp(f, fe) + assert_array_equal(fe.shape, (5,)) # because win length used as nperseg + assert_array_equal(pe.shape, (5,)) + assert_raises(ValueError, welch, x, + 10, win, nperseg=4) # because nperseg != win.shape[-1] + win_err = signal.get_window('hann', 32) + assert_raises(ValueError, welch, x, + 10, win_err, nperseg=None) # win longer than signal + + def test_empty_input(self): + f, p = welch([]) + assert_array_equal(f.shape, (0,)) + assert_array_equal(p.shape, (0,)) + for shape in [(0,), (3,0), (0,5,2)]: + f, p = welch(np.empty(shape)) + assert_array_equal(f.shape, shape) + assert_array_equal(p.shape, shape) + + def test_empty_input_other_axis(self): + for shape in [(3,0), (0,5,2)]: + f, p = welch(np.empty(shape), axis=1) + assert_array_equal(f.shape, shape) + assert_array_equal(p.shape, shape) + + def test_short_data(self): + x = np.zeros(8) + x[0] = 1 + #for string-like window, input signal length < nperseg value gives + #UserWarning, sets nperseg to x.shape[-1] + with suppress_warnings() as sup: + msg = "nperseg = 256 is greater than input length = 8, using nperseg = 8" + sup.filter(UserWarning, msg) + f, p = welch(x,window='hann') # default nperseg + f1, p1 = welch(x,window='hann', nperseg=256) # user-specified nperseg + f2, p2 = welch(x, nperseg=8) # valid nperseg, doesn't give warning + assert_allclose(f, f2) + assert_allclose(p, p2) + assert_allclose(f1, f2) + assert_allclose(p1, p2) + + def test_window_long_or_nd(self): + assert_raises(ValueError, welch, np.zeros(4), 1, np.array([1,1,1,1,1])) + assert_raises(ValueError, welch, np.zeros(4), 1, + np.arange(6).reshape((2,3))) + + def test_nondefault_noverlap(self): + x = np.zeros(64) + x[::8] = 1 + f, p = welch(x, nperseg=16, noverlap=4) + q = np.array([0, 1./12., 1./3., 1./5., 1./3., 1./5., 1./3., 1./5., + 1./6.]) + assert_allclose(p, q, atol=1e-12) + + def test_bad_noverlap(self): + assert_raises(ValueError, welch, np.zeros(4), 1, 'hann', 2, 7) + + def test_nfft_too_short(self): + assert_raises(ValueError, welch, np.ones(12), nfft=3, nperseg=4) + + def test_real_onesided_even_32(self): + x = np.zeros(16, 'f') + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=8) + assert_allclose(f, np.linspace(0, 0.5, 5)) + q = np.array([0.08333333, 0.15277778, 0.22222222, 0.22222222, + 0.11111111], 'f') + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + assert_(p.dtype == q.dtype) + + def test_real_onesided_odd_32(self): + x = np.zeros(16, 'f') + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=9) + assert_allclose(f, np.arange(5.0)/9.0) + q = np.array([0.12477458, 0.23430935, 0.17072113, 0.17072116, + 0.17072113], 'f') + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + assert_(p.dtype == q.dtype) + + def test_real_twosided_32(self): + x = np.zeros(16, 'f') + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=8, return_onesided=False) + assert_allclose(f, fftfreq(8, 1.0)) + q = np.array([0.08333333, 0.07638889, 0.11111111, + 0.11111111, 0.11111111, 0.11111111, 0.11111111, + 0.07638889], 'f') + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + assert_(p.dtype == q.dtype) + + def test_complex_32(self): + x = np.zeros(16, 'F') + x[0] = 1.0 + 2.0j + x[8] = 1.0 + 2.0j + f, p = welch(x, nperseg=8, return_onesided=False) + assert_allclose(f, fftfreq(8, 1.0)) + q = np.array([0.41666666, 0.38194442, 0.55555552, 0.55555552, + 0.55555558, 0.55555552, 0.55555552, 0.38194442], 'f') + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + assert_(p.dtype == q.dtype, + f'dtype mismatch, {p.dtype}, {q.dtype}') + + def test_padded_freqs(self): + x = np.zeros(12) + + nfft = 24 + f = fftfreq(nfft, 1.0)[:nfft//2+1] + f[-1] *= -1 + fodd, _ = welch(x, nperseg=5, nfft=nfft) + feven, _ = welch(x, nperseg=6, nfft=nfft) + assert_allclose(f, fodd) + assert_allclose(f, feven) + + nfft = 25 + f = fftfreq(nfft, 1.0)[:(nfft + 1)//2] + fodd, _ = welch(x, nperseg=5, nfft=nfft) + feven, _ = welch(x, nperseg=6, nfft=nfft) + assert_allclose(f, fodd) + assert_allclose(f, feven) + + def test_window_correction(self): + A = 20 + fs = 1e4 + nperseg = int(fs//10) + fsig = 300 + ii = int(fsig*nperseg//fs) # Freq index of fsig + + tt = np.arange(fs)/fs + x = A*np.sin(2*np.pi*fsig*tt) + + for window in ['hann', 'bartlett', ('tukey', 0.1), 'flattop']: + _, p_spec = welch(x, fs=fs, nperseg=nperseg, window=window, + scaling='spectrum') + freq, p_dens = welch(x, fs=fs, nperseg=nperseg, window=window, + scaling='density') + + # Check peak height at signal frequency for 'spectrum' + assert_allclose(p_spec[ii], A**2/2.0) + # Check integrated spectrum RMS for 'density' + assert_allclose(np.sqrt(trapezoid(p_dens, freq)), A*np.sqrt(2)/2, + rtol=1e-3) + + def test_axis_rolling(self): + np.random.seed(1234) + + x_flat = np.random.randn(1024) + _, p_flat = welch(x_flat) + + for a in range(3): + newshape = [1,]*3 + newshape[a] = -1 + x = x_flat.reshape(newshape) + + _, p_plus = welch(x, axis=a) # Positive axis index + _, p_minus = welch(x, axis=a-x.ndim) # Negative axis index + + assert_equal(p_flat, p_plus.squeeze(), err_msg=a) + assert_equal(p_flat, p_minus.squeeze(), err_msg=a-x.ndim) + + def test_average(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = welch(x, nperseg=8, average='median') + assert_allclose(f, np.linspace(0, 0.5, 5)) + q = np.array([.1, .05, 0., 1.54074396e-33, 0.]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + assert_raises(ValueError, welch, x, nperseg=8, + average='unrecognised-average') + + +class TestCSD: + def test_pad_shorter_x(self): + x = np.zeros(8) + y = np.zeros(12) + + f = np.linspace(0, 0.5, 7) + c = np.zeros(7,dtype=np.complex128) + f1, c1 = csd(x, y, nperseg=12) + + assert_allclose(f, f1) + assert_allclose(c, c1) + + def test_pad_shorter_y(self): + x = np.zeros(12) + y = np.zeros(8) + + f = np.linspace(0, 0.5, 7) + c = np.zeros(7,dtype=np.complex128) + f1, c1 = csd(x, y, nperseg=12) + + assert_allclose(f, f1) + assert_allclose(c, c1) + + def test_real_onesided_even(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, nperseg=8) + assert_allclose(f, np.linspace(0, 0.5, 5)) + q = np.array([0.08333333, 0.15277778, 0.22222222, 0.22222222, + 0.11111111]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_real_onesided_odd(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, nperseg=9) + assert_allclose(f, np.arange(5.0)/9.0) + q = np.array([0.12477455, 0.23430933, 0.17072113, 0.17072113, + 0.17072113]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_real_twosided(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, nperseg=8, return_onesided=False) + assert_allclose(f, fftfreq(8, 1.0)) + q = np.array([0.08333333, 0.07638889, 0.11111111, 0.11111111, + 0.11111111, 0.11111111, 0.11111111, 0.07638889]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_real_spectrum(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, nperseg=8, scaling='spectrum') + assert_allclose(f, np.linspace(0, 0.5, 5)) + q = np.array([0.015625, 0.02864583, 0.04166667, 0.04166667, + 0.02083333]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_integer_onesided_even(self): + x = np.zeros(16, dtype=int) + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, nperseg=8) + assert_allclose(f, np.linspace(0, 0.5, 5)) + q = np.array([0.08333333, 0.15277778, 0.22222222, 0.22222222, + 0.11111111]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_integer_onesided_odd(self): + x = np.zeros(16, dtype=int) + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, nperseg=9) + assert_allclose(f, np.arange(5.0)/9.0) + q = np.array([0.12477455, 0.23430933, 0.17072113, 0.17072113, + 0.17072113]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_integer_twosided(self): + x = np.zeros(16, dtype=int) + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, nperseg=8, return_onesided=False) + assert_allclose(f, fftfreq(8, 1.0)) + q = np.array([0.08333333, 0.07638889, 0.11111111, 0.11111111, + 0.11111111, 0.11111111, 0.11111111, 0.07638889]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_complex(self): + x = np.zeros(16, np.complex128) + x[0] = 1.0 + 2.0j + x[8] = 1.0 + 2.0j + f, p = csd(x, x, nperseg=8, return_onesided=False) + assert_allclose(f, fftfreq(8, 1.0)) + q = np.array([0.41666667, 0.38194444, 0.55555556, 0.55555556, + 0.55555556, 0.55555556, 0.55555556, 0.38194444]) + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + + def test_unk_scaling(self): + assert_raises(ValueError, csd, np.zeros(4, np.complex128), + np.ones(4, np.complex128), scaling='foo', nperseg=4) + + def test_detrend_linear(self): + x = np.arange(10, dtype=np.float64) + 0.04 + f, p = csd(x, x, nperseg=10, detrend='linear') + assert_allclose(p, np.zeros_like(p), atol=1e-15) + + def test_no_detrending(self): + x = np.arange(10, dtype=np.float64) + 0.04 + f1, p1 = csd(x, x, nperseg=10, detrend=False) + f2, p2 = csd(x, x, nperseg=10, detrend=lambda x: x) + assert_allclose(f1, f2, atol=1e-15) + assert_allclose(p1, p2, atol=1e-15) + + def test_detrend_external(self): + x = np.arange(10, dtype=np.float64) + 0.04 + f, p = csd(x, x, nperseg=10, + detrend=lambda seg: signal.detrend(seg, type='l')) + assert_allclose(p, np.zeros_like(p), atol=1e-15) + + def test_detrend_external_nd_m1(self): + x = np.arange(40, dtype=np.float64) + 0.04 + x = x.reshape((2,2,10)) + f, p = csd(x, x, nperseg=10, + detrend=lambda seg: signal.detrend(seg, type='l')) + assert_allclose(p, np.zeros_like(p), atol=1e-15) + + def test_detrend_external_nd_0(self): + x = np.arange(20, dtype=np.float64) + 0.04 + x = x.reshape((2,1,10)) + x = np.moveaxis(x, 2, 0) + f, p = csd(x, x, nperseg=10, axis=0, + detrend=lambda seg: signal.detrend(seg, axis=0, type='l')) + assert_allclose(p, np.zeros_like(p), atol=1e-15) + + def test_nd_axis_m1(self): + x = np.arange(20, dtype=np.float64) + 0.04 + x = x.reshape((2,1,10)) + f, p = csd(x, x, nperseg=10) + assert_array_equal(p.shape, (2, 1, 6)) + assert_allclose(p[0,0,:], p[1,0,:], atol=1e-13, rtol=1e-13) + f0, p0 = csd(x[0,0,:], x[0,0,:], nperseg=10) + assert_allclose(p0[np.newaxis,:], p[1,:], atol=1e-13, rtol=1e-13) + + def test_nd_axis_0(self): + x = np.arange(20, dtype=np.float64) + 0.04 + x = x.reshape((10,2,1)) + f, p = csd(x, x, nperseg=10, axis=0) + assert_array_equal(p.shape, (6,2,1)) + assert_allclose(p[:,0,0], p[:,1,0], atol=1e-13, rtol=1e-13) + f0, p0 = csd(x[:,0,0], x[:,0,0], nperseg=10) + assert_allclose(p0, p[:,1,0], atol=1e-13, rtol=1e-13) + + def test_window_external(self): + x = np.zeros(16) + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, 10, 'hann', 8) + win = signal.get_window('hann', 8) + fe, pe = csd(x, x, 10, win, nperseg=None) + assert_array_almost_equal_nulp(p, pe) + assert_array_almost_equal_nulp(f, fe) + assert_array_equal(fe.shape, (5,)) # because win length used as nperseg + assert_array_equal(pe.shape, (5,)) + assert_raises(ValueError, csd, x, x, + 10, win, nperseg=256) # because nperseg != win.shape[-1] + win_err = signal.get_window('hann', 32) + assert_raises(ValueError, csd, x, x, + 10, win_err, nperseg=None) # because win longer than signal + + def test_empty_input(self): + f, p = csd([],np.zeros(10)) + assert_array_equal(f.shape, (0,)) + assert_array_equal(p.shape, (0,)) + + f, p = csd(np.zeros(10),[]) + assert_array_equal(f.shape, (0,)) + assert_array_equal(p.shape, (0,)) + + for shape in [(0,), (3,0), (0,5,2)]: + f, p = csd(np.empty(shape), np.empty(shape)) + assert_array_equal(f.shape, shape) + assert_array_equal(p.shape, shape) + + f, p = csd(np.ones(10), np.empty((5,0))) + assert_array_equal(f.shape, (5,0)) + assert_array_equal(p.shape, (5,0)) + + f, p = csd(np.empty((5,0)), np.ones(10)) + assert_array_equal(f.shape, (5,0)) + assert_array_equal(p.shape, (5,0)) + + def test_empty_input_other_axis(self): + for shape in [(3,0), (0,5,2)]: + f, p = csd(np.empty(shape), np.empty(shape), axis=1) + assert_array_equal(f.shape, shape) + assert_array_equal(p.shape, shape) + + f, p = csd(np.empty((10,10,3)), np.zeros((10,0,1)), axis=1) + assert_array_equal(f.shape, (10,0,3)) + assert_array_equal(p.shape, (10,0,3)) + + f, p = csd(np.empty((10,0,1)), np.zeros((10,10,3)), axis=1) + assert_array_equal(f.shape, (10,0,3)) + assert_array_equal(p.shape, (10,0,3)) + + def test_short_data(self): + x = np.zeros(8) + x[0] = 1 + + #for string-like window, input signal length < nperseg value gives + #UserWarning, sets nperseg to x.shape[-1] + with suppress_warnings() as sup: + msg = "nperseg = 256 is greater than input length = 8, using nperseg = 8" + sup.filter(UserWarning, msg) + f, p = csd(x, x, window='hann') # default nperseg + f1, p1 = csd(x, x, window='hann', nperseg=256) # user-specified nperseg + f2, p2 = csd(x, x, nperseg=8) # valid nperseg, doesn't give warning + assert_allclose(f, f2) + assert_allclose(p, p2) + assert_allclose(f1, f2) + assert_allclose(p1, p2) + + def test_window_long_or_nd(self): + assert_raises(ValueError, csd, np.zeros(4), np.ones(4), 1, + np.array([1,1,1,1,1])) + assert_raises(ValueError, csd, np.zeros(4), np.ones(4), 1, + np.arange(6).reshape((2,3))) + + def test_nondefault_noverlap(self): + x = np.zeros(64) + x[::8] = 1 + f, p = csd(x, x, nperseg=16, noverlap=4) + q = np.array([0, 1./12., 1./3., 1./5., 1./3., 1./5., 1./3., 1./5., + 1./6.]) + assert_allclose(p, q, atol=1e-12) + + def test_bad_noverlap(self): + assert_raises(ValueError, csd, np.zeros(4), np.ones(4), 1, 'hann', + 2, 7) + + def test_nfft_too_short(self): + assert_raises(ValueError, csd, np.ones(12), np.zeros(12), nfft=3, + nperseg=4) + + def test_real_onesided_even_32(self): + x = np.zeros(16, 'f') + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, nperseg=8) + assert_allclose(f, np.linspace(0, 0.5, 5)) + q = np.array([0.08333333, 0.15277778, 0.22222222, 0.22222222, + 0.11111111], 'f') + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + assert_(p.dtype == q.dtype) + + def test_real_onesided_odd_32(self): + x = np.zeros(16, 'f') + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, nperseg=9) + assert_allclose(f, np.arange(5.0)/9.0) + q = np.array([0.12477458, 0.23430935, 0.17072113, 0.17072116, + 0.17072113], 'f') + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + assert_(p.dtype == q.dtype) + + def test_real_twosided_32(self): + x = np.zeros(16, 'f') + x[0] = 1 + x[8] = 1 + f, p = csd(x, x, nperseg=8, return_onesided=False) + assert_allclose(f, fftfreq(8, 1.0)) + q = np.array([0.08333333, 0.07638889, 0.11111111, + 0.11111111, 0.11111111, 0.11111111, 0.11111111, + 0.07638889], 'f') + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + assert_(p.dtype == q.dtype) + + def test_complex_32(self): + x = np.zeros(16, 'F') + x[0] = 1.0 + 2.0j + x[8] = 1.0 + 2.0j + f, p = csd(x, x, nperseg=8, return_onesided=False) + assert_allclose(f, fftfreq(8, 1.0)) + q = np.array([0.41666666, 0.38194442, 0.55555552, 0.55555552, + 0.55555558, 0.55555552, 0.55555552, 0.38194442], 'f') + assert_allclose(p, q, atol=1e-7, rtol=1e-7) + assert_(p.dtype == q.dtype, + f'dtype mismatch, {p.dtype}, {q.dtype}') + + def test_padded_freqs(self): + x = np.zeros(12) + y = np.ones(12) + + nfft = 24 + f = fftfreq(nfft, 1.0)[:nfft//2+1] + f[-1] *= -1 + fodd, _ = csd(x, y, nperseg=5, nfft=nfft) + feven, _ = csd(x, y, nperseg=6, nfft=nfft) + assert_allclose(f, fodd) + assert_allclose(f, feven) + + nfft = 25 + f = fftfreq(nfft, 1.0)[:(nfft + 1)//2] + fodd, _ = csd(x, y, nperseg=5, nfft=nfft) + feven, _ = csd(x, y, nperseg=6, nfft=nfft) + assert_allclose(f, fodd) + assert_allclose(f, feven) + + def test_copied_data(self): + x = np.random.randn(64) + y = x.copy() + + _, p_same = csd(x, x, nperseg=8, average='mean', + return_onesided=False) + _, p_copied = csd(x, y, nperseg=8, average='mean', + return_onesided=False) + assert_allclose(p_same, p_copied) + + _, p_same = csd(x, x, nperseg=8, average='median', + return_onesided=False) + _, p_copied = csd(x, y, nperseg=8, average='median', + return_onesided=False) + assert_allclose(p_same, p_copied) + + +class TestCoherence: + def test_identical_input(self): + x = np.random.randn(20) + y = np.copy(x) # So `y is x` -> False + + f = np.linspace(0, 0.5, 6) + C = np.ones(6) + f1, C1 = coherence(x, y, nperseg=10) + + assert_allclose(f, f1) + assert_allclose(C, C1) + + def test_phase_shifted_input(self): + x = np.random.randn(20) + y = -x + + f = np.linspace(0, 0.5, 6) + C = np.ones(6) + f1, C1 = coherence(x, y, nperseg=10) + + assert_allclose(f, f1) + assert_allclose(C, C1) + + +class TestSpectrogram: + def test_average_all_segments(self): + x = np.random.randn(1024) + + fs = 1.0 + window = ('tukey', 0.25) + nperseg = 16 + noverlap = 2 + + f, _, P = spectrogram(x, fs, window, nperseg, noverlap) + fw, Pw = welch(x, fs, window, nperseg, noverlap) + assert_allclose(f, fw) + assert_allclose(np.mean(P, axis=-1), Pw) + + def test_window_external(self): + x = np.random.randn(1024) + + fs = 1.0 + window = ('tukey', 0.25) + nperseg = 16 + noverlap = 2 + f, _, P = spectrogram(x, fs, window, nperseg, noverlap) + + win = signal.get_window(('tukey', 0.25), 16) + fe, _, Pe = spectrogram(x, fs, win, nperseg=None, noverlap=2) + assert_array_equal(fe.shape, (9,)) # because win length used as nperseg + assert_array_equal(Pe.shape, (9,73)) + assert_raises(ValueError, spectrogram, x, + fs, win, nperseg=8) # because nperseg != win.shape[-1] + win_err = signal.get_window(('tukey', 0.25), 2048) + assert_raises(ValueError, spectrogram, x, + fs, win_err, nperseg=None) # win longer than signal + + def test_short_data(self): + x = np.random.randn(1024) + fs = 1.0 + + #for string-like window, input signal length < nperseg value gives + #UserWarning, sets nperseg to x.shape[-1] + f, _, p = spectrogram(x, fs, window=('tukey',0.25)) # default nperseg + with suppress_warnings() as sup: + sup.filter(UserWarning, + "nperseg = 1025 is greater than input length = 1024, " + "using nperseg = 1024",) + f1, _, p1 = spectrogram(x, fs, window=('tukey',0.25), + nperseg=1025) # user-specified nperseg + f2, _, p2 = spectrogram(x, fs, nperseg=256) # to compare w/default + f3, _, p3 = spectrogram(x, fs, nperseg=1024) # compare w/user-spec'd + assert_allclose(f, f2) + assert_allclose(p, p2) + assert_allclose(f1, f3) + assert_allclose(p1, p3) + +class TestLombscargle: + def test_frequency(self): + """Test if frequency location of peak corresponds to frequency of + generated input signal. + """ + + # Input parameters + ampl = 2. + w = 1. + phi = 0.5 * np.pi + nin = 100 + nout = 1000 + p = 0.7 # Fraction of points to select + + # Randomly select a fraction of an array with timesteps + rng = np.random.RandomState(2353425) + r = rng.rand(nin) + t = np.linspace(0.01*np.pi, 10.*np.pi, nin)[r >= p] + + # Plot a sine wave for the selected times + y = ampl * np.sin(w*t + phi) + + # Define the array of frequencies for which to compute the periodogram + f = np.linspace(0.01, 10., nout) + + # Calculate Lomb-Scargle periodogram + P = lombscargle(t, y, f) + + # Check if difference between found frequency maximum and input + # frequency is less than accuracy + delta = f[1] - f[0] + assert(w - f[np.argmax(P)] < (delta/2.)) + + # also, check that it works with weights + P = lombscargle(t, y, f, weights=np.ones_like(t, dtype=f.dtype)) + + # Check if difference between found frequency maximum and input + # frequency is less than accuracy + delta = f[1] - f[0] + assert(w - f[np.argmax(P)] < (delta/2.)) + + + def test_amplitude(self): + # Test if height of peak in unnormalized Lomb-Scargle periodogram + # corresponds to amplitude of the generated input signal. + + # Input parameters + ampl = 2. + w = 1. + phi = 0.5 * np.pi + nin = 1000 + nout = 1000 + p = 0.7 # Fraction of points to select + + # Randomly select a fraction of an array with timesteps + rng = np.random.RandomState(2353425) + r = rng.rand(nin) + t = np.linspace(0.01*np.pi, 10.*np.pi, nin)[r >= p] + + # Plot a sine wave for the selected times + y = ampl * np.sin(w*t + phi) + + # Define the array of frequencies for which to compute the periodogram + f = np.linspace(0.01, 10., nout) + + # Calculate Lomb-Scargle periodogram + pgram = lombscargle(t, y, f) + + # convert to the amplitude + pgram = np.sqrt(4.0 * pgram / t.shape[0]) + + # Check if amplitude is correct (this will not exactly match, due to + # numerical differences when data is removed) + assert_allclose(pgram[f==w], ampl, rtol=5e-2) + + def test_precenter(self): + # Test if precenter gives the same result as manually precentering + # (for a very simple offset) + + # Input parameters + ampl = 2. + w = 1. + phi = 0.5 * np.pi + nin = 100 + nout = 1000 + p = 0.7 # Fraction of points to select + offset = 0.15 # Offset to be subtracted in pre-centering + + # Randomly select a fraction of an array with timesteps + rng = np.random.RandomState(2353425) + r = rng.rand(nin) + t = np.linspace(0.01*np.pi, 10.*np.pi, nin)[r >= p] + + # Plot a sine wave for the selected times + y = ampl * np.sin(w*t + phi) + offset + + # Define the array of frequencies for which to compute the periodogram + f = np.linspace(0.01, 10., nout) + + # Calculate Lomb-Scargle periodogram + pgram = lombscargle(t, y, f, precenter=True) + pgram2 = lombscargle(t, y - y.mean(), f, precenter=False) + + # check if centering worked + assert_allclose(pgram, pgram2) + + # do this again, but with floating_mean=True + + # Calculate Lomb-Scargle periodogram + pgram = lombscargle(t, y, f, precenter=True, floating_mean=True) + pgram2 = lombscargle(t, y - y.mean(), f, precenter=False, floating_mean=True) + + # check if centering worked + assert_allclose(pgram, pgram2) + + def test_normalize(self): + # Test normalize option of Lomb-Scarge. + + # Input parameters + ampl = 2. + w = 1. + phi = 0.5 * np.pi + nin = 100 + nout = 1000 + p = 0.7 # Fraction of points to select + + # Randomly select a fraction of an array with timesteps + rng = np.random.RandomState(2353425) + r = rng.rand(nin) + t = np.linspace(0.01*np.pi, 10.*np.pi, nin)[r >= p] + + # Plot a sine wave for the selected times + y = ampl * np.sin(w*t + phi) + + # Define the array of frequencies for which to compute the periodogram + f = np.linspace(0.01, 10., nout) + + # Calculate Lomb-Scargle periodogram + pgram = lombscargle(t, y, f) + pgram2 = lombscargle(t, y, f, normalize=True) + + # Calculate the scale to convert from unnormalized to normalized + weights = np.ones_like(t)/float(t.shape[0]) + YY_hat = (weights * y * y).sum() + YY = YY_hat # correct formula for floating_mean=False + scale_to_use = 2/(YY*t.shape[0]) + + # check if normalization works as expected + assert_allclose(pgram * scale_to_use, pgram2) + assert_allclose(np.max(pgram2), 1.0) + + def test_wrong_shape(self): + + # different length t and y + t = np.linspace(0, 1, 1) + y = np.linspace(0, 1, 2) + f = np.linspace(0, 1, 3) + 0.1 + assert_raises(ValueError, lombscargle, t, y, f) + + # t is 2D, with both axes length > 1 + t = np.repeat(np.expand_dims(np.linspace(0, 1, 2), 1), 2, axis=1) + y = np.linspace(0, 1, 2) + f = np.linspace(0, 1, 3) + 0.1 + assert_raises(ValueError, lombscargle, t, y, f) + + # y is 2D, with both axes length > 1 + t = np.linspace(0, 1, 2) + y = np.repeat(np.expand_dims(np.linspace(0, 1, 2), 1), 2, axis=1) + f = np.linspace(0, 1, 3) + 0.1 + assert_raises(ValueError, lombscargle, t, y, f) + + # f is 2D, with both axes length > 1 + t = np.linspace(0, 1, 2) + y = np.linspace(0, 1, 2) + f = np.repeat(np.expand_dims(np.linspace(0, 1, 3), 1) + 0.1, 2, axis=1) + assert_raises(ValueError, lombscargle, t, y, f) + + # weights is 2D, with both axes length > 1 + t = np.linspace(0, 1, 2) + y = np.linspace(0, 1, 2) + f = np.linspace(0, 1, 3) + 0.1 + weights = np.repeat(np.expand_dims(np.linspace(0, 1, 2), 1), 2, axis=1) + assert_raises(ValueError, lombscargle, t, y, f, weights=weights) + + def test_lombscargle_atan_vs_atan2(self): + # https://github.com/scipy/scipy/issues/3787 + # This raised a ZeroDivisionError. + t = np.linspace(0, 10, 1000, endpoint=False) + y = np.sin(4*t) + f = np.linspace(0, 50, 500, endpoint=False) + 0.1 + lombscargle(t, y, f*2*np.pi) + + def test_wrong_shape_weights(self): + # Weights must be the same shape as t + + t = np.linspace(0, 1, 1) + y = np.linspace(0, 1, 1) + f = np.linspace(0, 1, 3) + 0.1 + weights = np.linspace(1, 2, 2) + assert_raises(ValueError, lombscargle, t, y, f, weights=weights) + + def test_zero_division_weights(self): + # Weights cannot sum to 0 + + t = np.zeros(1) + y = np.zeros(1) + f = np.ones(1) + weights = np.zeros(1) + assert_raises(ValueError, lombscargle, t, y, f, weights=weights) + + def test_normalize_parameter(self): + # Test the validity of the normalize parameter input + + # Input parameters + ampl = 2. + w = 1. + phi = 0 + nin = 100 + nout = 1000 + p = 0.7 # Fraction of points to select + + # Randomly select a fraction of an array with timesteps + rng = np.random.RandomState(2353425) + r = rng.rand(nin) + t = np.linspace(0.01*np.pi, 10.*np.pi, nin)[r >= p] + + # Plot a sine wave for the selected times + y = ampl * np.sin(w*t + phi) + + # Define the array of frequencies for which to compute the periodogram + f = np.linspace(0.01, 10., nout) + + # check each of the valid inputs + pgram_false = lombscargle(t, y, f, normalize=False) + pgram_true = lombscargle(t, y, f, normalize=True) + pgram_power = lombscargle(t, y, f, normalize='power') + pgram_norm = lombscargle(t, y, f, normalize='normalize') + pgram_amp = lombscargle(t, y, f, normalize='amplitude') + + # validate the results that should be the same + assert_allclose(pgram_false, pgram_power) + assert_allclose(pgram_true, pgram_norm) + + # validate that the power and norm outputs are proper wrt each other + weights = np.ones_like(y)/float(y.shape[0]) + YY_hat = (weights * y * y).sum() + YY = YY_hat # correct formula for floating_mean=False + assert_allclose(pgram_power * 2.0 / (float(t.shape[0]) * YY), pgram_norm) + + # validate that the amp output is correct for the given input + f_i = np.where(f==w)[0][0] + assert_allclose(np.abs(pgram_amp[f_i]), ampl) + + # check invalid inputs + # 1) a string that is not allowed + assert_raises(ValueError, lombscargle, t, y, f, normalize='lomb') + # 2) something besides a bool or str + assert_raises(ValueError, lombscargle, t, y, f, normalize=2) + + def test_offset_removal(self): + # Verify that the amplitude is the same, even with an offset + # must use floating_mean=True, otherwise it will not remove an offset + + # Input parameters + ampl = 2. + w = 1. + phi = 0.5 * np.pi + nin = 100 + nout = 1000 + p = 0.7 # Fraction of points to select + offset = 2.15 # Large offset + + # Randomly select a fraction of an array with timesteps + rng = np.random.RandomState(2353425) + r = rng.rand(nin) + t = np.linspace(0.01*np.pi, 10.*np.pi, nin)[r >= p] + + # Plot a sine wave for the selected times + y = ampl * np.sin(w*t + phi) + + # Define the array of frequencies for which to compute the periodogram + f = np.linspace(0.01, 10., nout) + + # Calculate Lomb-Scargle periodogram + pgram = lombscargle(t, y, f, floating_mean=True) + pgram_offset = lombscargle(t, y + offset, f, floating_mean=True) + + # check if offset removal works as expected + assert_allclose(pgram, pgram_offset) + + def test_floating_mean_false(self): + # Verify that when disabling the floating_mean, the calculations are correct + + # Input parameters + ampl = 2. + w = 1. + phi = 0 + nin = 1000 + nout = 1000 + p = 0.7 # Fraction of points to select + offset = 2 # Large offset + + # Randomly select a fraction of an array with timesteps + rng = np.random.RandomState(2353425) + r = rng.rand(nin) + t = np.linspace(0.01*np.pi, 10.*np.pi, nin)[r >= p] + + # Plot a cos wave for the selected times + y = ampl * np.cos(w*t + phi) + + # Define the array of frequencies for which to compute the periodogram + f = np.linspace(0.01, 10., nout) + + # Calculate Lomb-Scargle periodogram + pgram = lombscargle(t, y, f, normalize=True, floating_mean=False) + pgram_offset = lombscargle(t, y + offset, f, normalize=True, + floating_mean=False) + + # check if disabling floating_mean works as expected + # nearly-zero for no offset, exact value will change based on seed + assert(pgram[0] < 0.01) + # significant value with offset, exact value will change based on seed + assert(pgram_offset[0] > 0.5) + + def test_amplitude_is_correct(self): + # Verify that the amplitude is correct (when normalize='amplitude') + + # Input parameters + ampl = 2. + w = 1. + phi = 0.12 + nin = 100 + nout = 1000 + p = 0.7 # Fraction of points to select + offset = 2.15 # Large offset + + # Randomly select a fraction of an array with timesteps + rng = np.random.RandomState(2353425) + r = rng.rand(nin) + t = np.linspace(0.01*np.pi, 10.*np.pi, nin)[r >= p] + + # Plot a sine wave for the selected times + y = ampl * np.cos(w*t + phi) + offset + + # Define the array of frequencies for which to compute the periodogram + f = np.linspace(0.01, 10., nout) + + # Get the index of where the exact result should be + f_indx = np.where(f==w)[0][0] + + # Calculate Lomb-Scargle periodogram (amplitude + phase) + pgram = lombscargle(t, y, f, normalize='amplitude', floating_mean=True) + + # Check if amplitude is correct + assert_allclose(np.abs(pgram[f_indx]), ampl) + + # Check if phase is correct + # (phase angle is the negative of the phase offset) + assert_allclose(-np.angle(pgram[f_indx]), phi) + + def test_negative_weight(self): + # Test that a negative weight produces an error + + t = np.zeros(1) + y = np.zeros(1) + f = np.ones(1) + weights = -np.ones(1) + assert_raises(ValueError, lombscargle, t, y, f, weights=weights) + + def test_list_input(self): + # Test that input can be passsed in as lists and with a numerical issue + # https://github.com/scipy/scipy/issues/8787 + + t = [1.98201652e+09, 1.98201752e+09, 1.98201852e+09, 1.98201952e+09, + 1.98202052e+09, 1.98202152e+09, 1.98202252e+09, 1.98202352e+09, + 1.98202452e+09, 1.98202552e+09, 1.98202652e+09, 1.98202752e+09, + 1.98202852e+09, 1.98202952e+09, 1.98203052e+09, 1.98203152e+09, + 1.98203252e+09, 1.98203352e+09, 1.98203452e+09, 1.98203552e+09, + 1.98205452e+09, 1.98205552e+09, 1.98205652e+09, 1.98205752e+09, + 1.98205852e+09, 1.98205952e+09, 1.98206052e+09, 1.98206152e+09, + 1.98206252e+09, 1.98206352e+09, 1.98206452e+09, 1.98206552e+09, + 1.98206652e+09, 1.98206752e+09, 1.98206852e+09, 1.98206952e+09, + 1.98207052e+09, 1.98207152e+09, 1.98207252e+09, 1.98207352e+09, + 1.98209652e+09, 1.98209752e+09, 1.98209852e+09, 1.98209952e+09, + 1.98210052e+09, 1.98210152e+09, 1.98210252e+09, 1.98210352e+09, + 1.98210452e+09, 1.98210552e+09, 1.98210652e+09, 1.98210752e+09, + 1.98210852e+09, 1.98210952e+09, 1.98211052e+09, 1.98211152e+09, + 1.98211252e+09, 1.98211352e+09, 1.98211452e+09, 1.98211552e+09, + 1.98217252e+09, 1.98217352e+09, 1.98217452e+09, 1.98217552e+09, + 1.98217652e+09, 1.98217752e+09, 1.98217852e+09, 1.98217952e+09, + 1.98218052e+09, 1.98218152e+09, 1.98218252e+09, 1.98218352e+09, + 1.98218452e+09, 1.98218552e+09, 1.98218652e+09, 1.98218752e+09, + 1.98218852e+09, 1.98218952e+09, 1.98219052e+09, 1.98219152e+09, + 1.98219352e+09, 1.98219452e+09, 1.98219552e+09, 1.98219652e+09, + 1.98219752e+09, 1.98219852e+09, 1.98219952e+09, 1.98220052e+09, + 1.98220152e+09, 1.98220252e+09, 1.98220352e+09, 1.98220452e+09, + 1.98220552e+09, 1.98220652e+09, 1.98220752e+09, 1.98220852e+09, + 1.98220952e+09, 1.98221052e+09, 1.98221152e+09, 1.98221252e+09, + 1.98222752e+09, 1.98222852e+09, 1.98222952e+09, 1.98223052e+09, + 1.98223152e+09, 1.98223252e+09, 1.98223352e+09, 1.98223452e+09, + 1.98223552e+09, 1.98223652e+09, 1.98223752e+09, 1.98223852e+09, + 1.98223952e+09, 1.98224052e+09, 1.98224152e+09, 1.98224252e+09, + 1.98224352e+09, 1.98224452e+09, 1.98224552e+09, 1.98224652e+09, + 1.98224752e+09] + y = [2.97600000e+03, 3.18200000e+03, 3.74900000e+03, 4.53500000e+03, + 5.43300000e+03, 6.38000000e+03, 7.34000000e+03, 8.29200000e+03, + 9.21900000e+03, 1.01120000e+04, 1.09620000e+04, 1.17600000e+04, + 1.25010000e+04, 1.31790000e+04, 1.37900000e+04, 1.43290000e+04, + 1.47940000e+04, 1.51800000e+04, 1.54870000e+04, 1.57110000e+04, + 5.74200000e+03, 4.82300000e+03, 3.99100000e+03, 3.33600000e+03, + 2.99600000e+03, 3.08400000e+03, 3.56700000e+03, 4.30700000e+03, + 5.18200000e+03, 6.11900000e+03, 7.07900000e+03, 8.03400000e+03, + 8.97000000e+03, 9.87300000e+03, 1.07350000e+04, 1.15480000e+04, + 1.23050000e+04, 1.30010000e+04, 1.36300000e+04, 1.41890000e+04, + 6.00000000e+03, 5.06800000e+03, 4.20500000e+03, 3.49000000e+03, + 3.04900000e+03, 3.01600000e+03, 3.40400000e+03, 4.08800000e+03, + 4.93500000e+03, 5.86000000e+03, 6.81700000e+03, 7.77500000e+03, + 8.71800000e+03, 9.63100000e+03, 1.05050000e+04, 1.13320000e+04, + 1.21050000e+04, 1.28170000e+04, 1.34660000e+04, 1.40440000e+04, + 1.32730000e+04, 1.26040000e+04, 1.18720000e+04, 1.10820000e+04, + 1.02400000e+04, 9.35300000e+03, 8.43000000e+03, 7.48100000e+03, + 6.52100000e+03, 5.57000000e+03, 4.66200000e+03, 3.85400000e+03, + 3.24600000e+03, 2.97900000e+03, 3.14700000e+03, 3.68800000e+03, + 4.45900000e+03, 5.35000000e+03, 6.29400000e+03, 7.25400000e+03, + 9.13800000e+03, 1.00340000e+04, 1.08880000e+04, 1.16910000e+04, + 1.24370000e+04, 1.31210000e+04, 1.37380000e+04, 1.42840000e+04, + 1.47550000e+04, 1.51490000e+04, 1.54630000e+04, 1.56950000e+04, + 1.58430000e+04, 1.59070000e+04, 1.58860000e+04, 1.57800000e+04, + 1.55910000e+04, 1.53190000e+04, 1.49650000e+04, 1.45330000e+04, + 3.01000000e+03, 3.05900000e+03, 3.51200000e+03, 4.23400000e+03, + 5.10000000e+03, 6.03400000e+03, 6.99300000e+03, 7.95000000e+03, + 8.88800000e+03, 9.79400000e+03, 1.06600000e+04, 1.14770000e+04, + 1.22400000e+04, 1.29410000e+04, 1.35770000e+04, 1.41430000e+04, + 1.46350000e+04, 1.50500000e+04, 1.53850000e+04, 1.56400000e+04, + 1.58110000e+04] + + periods = np.linspace(400, 120, 1000) + angular_freq = 2 * np.pi / periods + + lombscargle(t, y, angular_freq, precenter=True, normalize=True) + + def test_zero_freq(self): + # Verify that function works when freqs includes 0 + # The value at f=0 will depend on the seed + + # Input parameters + ampl = 2. + w = 1. + phi = 0.12 + nin = 100 + nout = 1001 + p = 0.7 # Fraction of points to select + offset = 0 + + # Randomly select a fraction of an array with timesteps + rng = np.random.RandomState(2353425) + r = rng.rand(nin) + t = np.linspace(0.01*np.pi, 10.*np.pi, nin)[r >= p] + + # Plot a sine wave for the selected times + y = ampl * np.cos(w*t + phi) + offset + + # Define the array of frequencies for which to compute the periodogram + f = np.linspace(0, 10., nout) + + # Calculate Lomb-Scargle periodogram + pgram = lombscargle(t, y, f, normalize=True, floating_mean=True) + + # exact value will change based on seed + # testing to make sure it is very small + assert(pgram[0] < 1e-4) + + def test_simple_div_zero(self): + # these are bare-minimum examples that would, without the eps adjustments, + # cause division-by-zero errors + + # first, test with example that will cause first SS sum to be 0.0 + t = [t + 1 for t in range(0, 32)] + y = np.ones(len(t)) + freqs = [2.0*np.pi] * 2 # must have 2+ elements + lombscargle(t, y, freqs) + + # second, test with example that will cause first CC sum to be 0.0 + t = [t*4 + 1 for t in range(0, 32)] + y = np.ones(len(t)) + freqs = [np.pi/2.0] * 2 # must have 2+ elements + + lombscargle(t, y, freqs) + + +class TestSTFT: + @pytest.mark.thread_unsafe + def test_input_validation(self): + + def chk_VE(match): + """Assert for a ValueError matching regexp `match`. + + This little wrapper allows a more concise code layout. + """ + return pytest.raises(ValueError, match=match) + + # Checks for check_COLA(): + with chk_VE('nperseg must be a positive integer'): + check_COLA('hann', -10, 0) + with chk_VE('noverlap must be less than nperseg.'): + check_COLA('hann', 10, 20) + with chk_VE('window must be 1-D'): + check_COLA(np.ones((2, 2)), 10, 0) + with chk_VE('window must have length of nperseg'): + check_COLA(np.ones(20), 10, 0) + + # Checks for check_NOLA(): + with chk_VE('nperseg must be a positive integer'): + check_NOLA('hann', -10, 0) + with chk_VE('noverlap must be less than nperseg'): + check_NOLA('hann', 10, 20) + with chk_VE('window must be 1-D'): + check_NOLA(np.ones((2, 2)), 10, 0) + with chk_VE('window must have length of nperseg'): + check_NOLA(np.ones(20), 10, 0) + with chk_VE('noverlap must be a nonnegative integer'): + check_NOLA('hann', 64, -32) + + x = np.zeros(1024) + z = stft(x)[2] + + # Checks for stft(): + with chk_VE('window must be 1-D'): + stft(x, window=np.ones((2, 2))) + with chk_VE('value specified for nperseg is different ' + + 'from length of window'): + stft(x, window=np.ones(10), nperseg=256) + with chk_VE('nperseg must be a positive integer'): + stft(x, nperseg=-256) + with chk_VE('noverlap must be less than nperseg.'): + stft(x, nperseg=256, noverlap=1024) + with chk_VE('nfft must be greater than or equal to nperseg.'): + stft(x, nperseg=256, nfft=8) + + # Checks for istft(): + with chk_VE('Input stft must be at least 2d!'): + istft(x) + with chk_VE('window must be 1-D'): + istft(z, window=np.ones((2, 2))) + with chk_VE('window must have length of 256'): + istft(z, window=np.ones(10), nperseg=256) + with chk_VE('nperseg must be a positive integer'): + istft(z, nperseg=-256) + with chk_VE('noverlap must be less than nperseg.'): + istft(z, nperseg=256, noverlap=1024) + with chk_VE('nfft must be greater than or equal to nperseg.'): + istft(z, nperseg=256, nfft=8) + with pytest.warns(UserWarning, match="NOLA condition failed, " + + "STFT may not be invertible"): + istft(z, nperseg=256, noverlap=0, window='hann') + with chk_VE('Must specify differing time and frequency axes!'): + istft(z, time_axis=0, freq_axis=0) + + # Checks for _spectral_helper(): + with chk_VE("Unknown value for mode foo, must be one of: " + + r"\{'psd', 'stft'\}"): + _spectral_helper(x, x, mode='foo') + with chk_VE("x and y must be equal if mode is 'stft'"): + _spectral_helper(x[:512], x[512:], mode='stft') + with chk_VE("Unknown boundary option 'foo', must be one of: " + + r"\['even', 'odd', 'constant', 'zeros', None\]"): + _spectral_helper(x, x, boundary='foo') + + scaling = "not_valid" + with chk_VE(fr"Parameter {scaling=} not in \['spectrum', 'psd'\]!"): + stft(x, scaling=scaling) + with chk_VE(fr"Parameter {scaling=} not in \['spectrum', 'psd'\]!"): + istft(z, scaling=scaling) + + def test_check_COLA(self): + settings = [ + ('boxcar', 10, 0), + ('boxcar', 10, 9), + ('bartlett', 51, 26), + ('hann', 256, 128), + ('hann', 256, 192), + ('blackman', 300, 200), + (('tukey', 0.5), 256, 64), + ('hann', 256, 255), + ] + + for setting in settings: + msg = '{}, {}, {}'.format(*setting) + assert_equal(True, check_COLA(*setting), err_msg=msg) + + def test_check_NOLA(self): + settings_pass = [ + ('boxcar', 10, 0), + ('boxcar', 10, 9), + ('boxcar', 10, 7), + ('bartlett', 51, 26), + ('bartlett', 51, 10), + ('hann', 256, 128), + ('hann', 256, 192), + ('hann', 256, 37), + ('blackman', 300, 200), + ('blackman', 300, 123), + (('tukey', 0.5), 256, 64), + (('tukey', 0.5), 256, 38), + ('hann', 256, 255), + ('hann', 256, 39), + ] + for setting in settings_pass: + msg = '{}, {}, {}'.format(*setting) + assert_equal(True, check_NOLA(*setting), err_msg=msg) + + w_fail = np.ones(16) + w_fail[::2] = 0 + settings_fail = [ + (w_fail, len(w_fail), len(w_fail) // 2), + ('hann', 64, 0), + ] + for setting in settings_fail: + msg = '{}, {}, {}'.format(*setting) + assert_equal(False, check_NOLA(*setting), err_msg=msg) + + def test_average_all_segments(self): + rng = np.random.RandomState(1234) + x = rng.randn(1024) + + fs = 1.0 + window = 'hann' + nperseg = 16 + noverlap = 8 + + # Compare twosided, because onesided welch doubles non-DC terms to + # account for power at negative frequencies. stft doesn't do this, + # because it breaks invertibility. + f, _, Z = stft(x, fs, window, nperseg, noverlap, padded=False, + return_onesided=False, boundary=None) + fw, Pw = welch(x, fs, window, nperseg, noverlap, return_onesided=False, + scaling='spectrum', detrend=False) + + assert_allclose(f, fw) + assert_allclose(np.mean(np.abs(Z)**2, axis=-1), Pw) + + def test_permute_axes(self): + rng = np.random.RandomState(1234) + x = rng.randn(1024) + + fs = 1.0 + window = 'hann' + nperseg = 16 + noverlap = 8 + + f1, t1, Z1 = stft(x, fs, window, nperseg, noverlap) + f2, t2, Z2 = stft(x.reshape((-1, 1, 1)), fs, window, nperseg, noverlap, + axis=0) + + t3, x1 = istft(Z1, fs, window, nperseg, noverlap) + t4, x2 = istft(Z2.T, fs, window, nperseg, noverlap, time_axis=0, + freq_axis=-1) + + assert_allclose(f1, f2) + assert_allclose(t1, t2) + assert_allclose(t3, t4) + assert_allclose(Z1, Z2[:, 0, 0, :]) + assert_allclose(x1, x2[:, 0, 0]) + + @pytest.mark.parametrize('scaling', ['spectrum', 'psd']) + def test_roundtrip_real(self, scaling): + rng = np.random.RandomState(1234) + + settings = [ + ('boxcar', 100, 10, 0), # Test no overlap + ('boxcar', 100, 10, 9), # Test high overlap + ('bartlett', 101, 51, 26), # Test odd nperseg + ('hann', 1024, 256, 128), # Test defaults + (('tukey', 0.5), 1152, 256, 64), # Test Tukey + ('hann', 1024, 256, 255), # Test overlapped hann + ] + + for window, N, nperseg, noverlap in settings: + t = np.arange(N) + x = 10*rng.randn(t.size) + + _, _, zz = stft(x, nperseg=nperseg, noverlap=noverlap, + window=window, detrend=None, padded=False, + scaling=scaling) + + tr, xr = istft(zz, nperseg=nperseg, noverlap=noverlap, + window=window, scaling=scaling) + + msg = f'{window}, {noverlap}' + assert_allclose(t, tr, err_msg=msg) + assert_allclose(x, xr, err_msg=msg) + + @pytest.mark.thread_unsafe + def test_roundtrip_not_nola(self): + rng = np.random.RandomState(1234) + + w_fail = np.ones(16) + w_fail[::2] = 0 + settings = [ + (w_fail, 256, len(w_fail), len(w_fail) // 2), + ('hann', 256, 64, 0), + ] + + for window, N, nperseg, noverlap in settings: + msg = f'{window}, {N}, {nperseg}, {noverlap}' + assert not check_NOLA(window, nperseg, noverlap), msg + + t = np.arange(N) + x = 10 * rng.randn(t.size) + + _, _, zz = stft(x, nperseg=nperseg, noverlap=noverlap, + window=window, detrend=None, padded=True, + boundary='zeros') + with pytest.warns(UserWarning, match='NOLA'): + tr, xr = istft(zz, nperseg=nperseg, noverlap=noverlap, + window=window, boundary=True) + + assert np.allclose(t, tr[:len(t)]), msg + assert not np.allclose(x, xr[:len(x)]), msg + + def test_roundtrip_nola_not_cola(self): + rng = np.random.RandomState(1234) + + settings = [ + ('boxcar', 100, 10, 3), # NOLA True, COLA False + ('bartlett', 101, 51, 37), # NOLA True, COLA False + ('hann', 1024, 256, 127), # NOLA True, COLA False + (('tukey', 0.5), 1152, 256, 14), # NOLA True, COLA False + ('hann', 1024, 256, 5), # NOLA True, COLA False + ] + + for window, N, nperseg, noverlap in settings: + msg = f'{window}, {nperseg}, {noverlap}' + assert check_NOLA(window, nperseg, noverlap), msg + assert not check_COLA(window, nperseg, noverlap), msg + + t = np.arange(N) + x = 10 * rng.randn(t.size) + + _, _, zz = stft(x, nperseg=nperseg, noverlap=noverlap, + window=window, detrend=None, padded=True, + boundary='zeros') + + tr, xr = istft(zz, nperseg=nperseg, noverlap=noverlap, + window=window, boundary=True) + + msg = f'{window}, {noverlap}' + assert_allclose(t, tr[:len(t)], err_msg=msg) + assert_allclose(x, xr[:len(x)], err_msg=msg) + + def test_roundtrip_float32(self): + rng = np.random.RandomState(1234) + + settings = [('hann', 1024, 256, 128)] + + for window, N, nperseg, noverlap in settings: + t = np.arange(N) + x = 10*rng.randn(t.size) + x = x.astype(np.float32) + + _, _, zz = stft(x, nperseg=nperseg, noverlap=noverlap, + window=window, detrend=None, padded=False) + + tr, xr = istft(zz, nperseg=nperseg, noverlap=noverlap, + window=window) + + msg = f'{window}, {noverlap}' + assert_allclose(t, t, err_msg=msg) + assert_allclose(x, xr, err_msg=msg, rtol=1e-4, atol=1e-5) + assert_(x.dtype == xr.dtype) + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize('scaling', ['spectrum', 'psd']) + def test_roundtrip_complex(self, scaling): + rng = np.random.RandomState(1234) + + settings = [ + ('boxcar', 100, 10, 0), # Test no overlap + ('boxcar', 100, 10, 9), # Test high overlap + ('bartlett', 101, 51, 26), # Test odd nperseg + ('hann', 1024, 256, 128), # Test defaults + (('tukey', 0.5), 1152, 256, 64), # Test Tukey + ('hann', 1024, 256, 255), # Test overlapped hann + ] + + for window, N, nperseg, noverlap in settings: + t = np.arange(N) + x = 10*rng.randn(t.size) + 10j*rng.randn(t.size) + + _, _, zz = stft(x, nperseg=nperseg, noverlap=noverlap, + window=window, detrend=None, padded=False, + return_onesided=False, scaling=scaling) + + tr, xr = istft(zz, nperseg=nperseg, noverlap=noverlap, + window=window, input_onesided=False, + scaling=scaling) + + msg = f'{window}, {nperseg}, {noverlap}' + assert_allclose(t, tr, err_msg=msg) + assert_allclose(x, xr, err_msg=msg) + + # Check that asking for onesided switches to twosided + with suppress_warnings() as sup: + sup.filter(UserWarning, + "Input data is complex, switching to return_onesided=False") + _, _, zz = stft(x, nperseg=nperseg, noverlap=noverlap, + window=window, detrend=None, padded=False, + return_onesided=True, scaling=scaling) + + tr, xr = istft(zz, nperseg=nperseg, noverlap=noverlap, + window=window, input_onesided=False, scaling=scaling) + + msg = f'{window}, {nperseg}, {noverlap}' + assert_allclose(t, tr, err_msg=msg) + assert_allclose(x, xr, err_msg=msg) + + def test_roundtrip_boundary_extension(self): + rng = np.random.RandomState(1234) + + # Test against boxcar, since window is all ones, and thus can be fully + # recovered with no boundary extension + + settings = [ + ('boxcar', 100, 10, 0), # Test no overlap + ('boxcar', 100, 10, 9), # Test high overlap + ] + + for window, N, nperseg, noverlap in settings: + t = np.arange(N) + x = 10*rng.randn(t.size) + + _, _, zz = stft(x, nperseg=nperseg, noverlap=noverlap, + window=window, detrend=None, padded=True, + boundary=None) + + _, xr = istft(zz, noverlap=noverlap, window=window, boundary=False) + + for boundary in ['even', 'odd', 'constant', 'zeros']: + _, _, zz_ext = stft(x, nperseg=nperseg, noverlap=noverlap, + window=window, detrend=None, padded=True, + boundary=boundary) + + _, xr_ext = istft(zz_ext, noverlap=noverlap, window=window, + boundary=True) + + msg = f'{window}, {noverlap}, {boundary}' + assert_allclose(x, xr, err_msg=msg) + assert_allclose(x, xr_ext, err_msg=msg) + + def test_roundtrip_padded_signal(self): + rng = np.random.RandomState(1234) + + settings = [ + ('boxcar', 101, 10, 0), + ('hann', 1000, 256, 128), + ] + + for window, N, nperseg, noverlap in settings: + t = np.arange(N) + x = 10*rng.randn(t.size) + + _, _, zz = stft(x, nperseg=nperseg, noverlap=noverlap, + window=window, detrend=None, padded=True) + + tr, xr = istft(zz, noverlap=noverlap, window=window) + + msg = f'{window}, {noverlap}' + # Account for possible zero-padding at the end + assert_allclose(t, tr[:t.size], err_msg=msg) + assert_allclose(x, xr[:x.size], err_msg=msg) + + def test_roundtrip_padded_FFT(self): + rng = np.random.RandomState(1234) + + settings = [ + ('hann', 1024, 256, 128, 512), + ('hann', 1024, 256, 128, 501), + ('boxcar', 100, 10, 0, 33), + (('tukey', 0.5), 1152, 256, 64, 1024), + ] + + for window, N, nperseg, noverlap, nfft in settings: + t = np.arange(N) + x = 10*rng.randn(t.size) + xc = x*np.exp(1j*np.pi/4) + + # real signal + _, _, z = stft(x, nperseg=nperseg, noverlap=noverlap, nfft=nfft, + window=window, detrend=None, padded=True) + + # complex signal + _, _, zc = stft(xc, nperseg=nperseg, noverlap=noverlap, nfft=nfft, + window=window, detrend=None, padded=True, + return_onesided=False) + + tr, xr = istft(z, nperseg=nperseg, noverlap=noverlap, nfft=nfft, + window=window) + + tr, xcr = istft(zc, nperseg=nperseg, noverlap=noverlap, nfft=nfft, + window=window, input_onesided=False) + + msg = f'{window}, {noverlap}' + assert_allclose(t, tr, err_msg=msg) + assert_allclose(x, xr, err_msg=msg) + assert_allclose(xc, xcr, err_msg=msg) + + def test_axis_rolling(self): + rng = np.random.RandomState(1234) + + x_flat = rng.randn(1024) + _, _, z_flat = stft(x_flat) + + for a in range(3): + newshape = [1,]*3 + newshape[a] = -1 + x = x_flat.reshape(newshape) + + _, _, z_plus = stft(x, axis=a) # Positive axis index + _, _, z_minus = stft(x, axis=a-x.ndim) # Negative axis index + + assert_equal(z_flat, z_plus.squeeze(), err_msg=a) + assert_equal(z_flat, z_minus.squeeze(), err_msg=a-x.ndim) + + # z_flat has shape [n_freq, n_time] + + # Test vs. transpose + _, x_transpose_m = istft(z_flat.T, time_axis=-2, freq_axis=-1) + _, x_transpose_p = istft(z_flat.T, time_axis=0, freq_axis=1) + + assert_allclose(x_flat, x_transpose_m, err_msg='istft transpose minus') + assert_allclose(x_flat, x_transpose_p, err_msg='istft transpose plus') + + def test_roundtrip_scaling(self): + """Verify behavior of scaling parameter. """ + # Create 1024 sample cosine signal with amplitude 2: + X = np.zeros(513, dtype=complex) + X[256] = 1024 + x = np.fft.irfft(X) + power_x = sum(x**2) / len(x) # power of signal x is 2 + + # Calculate magnitude-scaled STFT: + Zs = stft(x, boundary='even', scaling='spectrum')[2] + + # Test round trip: + x1 = istft(Zs, boundary=True, scaling='spectrum')[1] + assert_allclose(x1, x) + + # For a Hann-windowed 256 sample length FFT, we expect a peak at + # frequency 64 (since it is 1/4 the length of X) with a height of 1 + # (half the amplitude). A Hann window of a perfectly centered sine has + # the magnitude [..., 0, 0, 0.5, 1, 0.5, 0, 0, ...]. + # Note that in this case the 'even' padding works for the beginning + # but not for the end of the STFT. + assert_allclose(abs(Zs[63, :-1]), 0.5) + assert_allclose(abs(Zs[64, :-1]), 1) + assert_allclose(abs(Zs[65, :-1]), 0.5) + # All other values should be zero: + Zs[63:66, :-1] = 0 + # Note since 'rtol' does not have influence here, atol needs to be set: + assert_allclose(Zs[:, :-1], 0, atol=np.finfo(Zs.dtype).resolution) + + # Calculate two-sided psd-scaled STFT: + # - using 'even' padding since signal is axis symmetric - this ensures + # stationary behavior on the boundaries + # - using the two-sided transform allows determining the spectral + # power by `sum(abs(Zp[:, k])**2) / len(f)` for the k-th time slot. + Zp = stft(x, return_onesided=False, boundary='even', scaling='psd')[2] + + # Calculate spectral power of Zd by summing over the frequency axis: + psd_Zp = np.sum(Zp.real**2 + Zp.imag**2, axis=0) / Zp.shape[0] + # Spectral power of Zp should be equal to the signal's power: + assert_allclose(psd_Zp, power_x) + + # Test round trip: + x1 = istft(Zp, input_onesided=False, boundary=True, scaling='psd')[1] + assert_allclose(x1, x) + + # The power of the one-sided psd-scaled STFT can be determined + # analogously (note that the two sides are not of equal shape): + Zp0 = stft(x, return_onesided=True, boundary='even', scaling='psd')[2] + + # Since x is real, its Fourier transform is conjugate symmetric, i.e., + # the missing 'second side' can be expressed through the 'first side': + Zp1 = np.conj(Zp0[-2:0:-1, :]) # 'second side' is conjugate reversed + assert_allclose(Zp[:129, :], Zp0) + assert_allclose(Zp[129:, :], Zp1) + + # Calculate the spectral power: + s2 = (np.sum(Zp0.real ** 2 + Zp0.imag ** 2, axis=0) + + np.sum(Zp1.real ** 2 + Zp1.imag ** 2, axis=0)) + psd_Zp01 = s2 / (Zp0.shape[0] + Zp1.shape[0]) + assert_allclose(psd_Zp01, power_x) + + # Test round trip: + x1 = istft(Zp0, input_onesided=True, boundary=True, scaling='psd')[1] + assert_allclose(x1, x) + + +class TestSampledSpectralRepresentations: + """Check energy/power relations from `Spectral Analysis` section in the user guide. + + A 32 sample cosine signal is used to compare the numerical to the expected results + stated in :ref:`tutorial_SpectralAnalysis` in + file ``doc/source/tutorial/signal.rst`` + """ + n: int = 32 #: number of samples + T: float = 1/16 #: sampling interval + a_ref: float = 3 #: amplitude of reference + l_a: int = 3 #: index in fft for defining frequency of test signal + + x_ref: np.ndarray #: reference signal + X_ref: np.ndarray #: two-sided FFT of x_ref + E_ref: float #: energy of signal + P_ref: float #: power of signal + + def setup_method(self): + """Create Cosine signal with amplitude a from spectrum. """ + f = rfftfreq(self.n, self.T) + X_ref = np.zeros_like(f) + self.l_a = 3 + X_ref[self.l_a] = self.a_ref/2 * self.n # set amplitude + self.x_ref = irfft(X_ref) + self.X_ref = fft(self.x_ref) + + # Closed form expression for continuous-time signal: + self.E_ref = self.tau * self.a_ref**2 / 2 # energy of signal + self.P_ref = self.a_ref**2 / 2 # power of signal + + @property + def tau(self) -> float: + """Duration of signal. """ + return self.n * self.T + + @property + def delta_f(self) -> float: + """Bin width """ + return 1 / (self.n * self.T) + + def test_reference_signal(self): + """Test energy and power formulas. """ + # Verify that amplitude is a: + assert_allclose(2*self.a_ref, np.ptp(self.x_ref), rtol=0.1) + # Verify that energy expression for sampled signal: + assert_allclose(self.T * sum(self.x_ref ** 2), self.E_ref) + + # Verify that spectral energy and power formulas are correct: + sum_X_ref_squared = sum(self.X_ref.real**2 + self.X_ref.imag**2) + assert_allclose(self.T/self.n * sum_X_ref_squared, self.E_ref) + assert_allclose(1/self.n**2 * sum_X_ref_squared, self.P_ref) + + def test_windowed_DFT(self): + """Verify spectral representations of windowed DFT. + + Furthermore, the scalings of `periodogram` and `welch` are verified. + """ + w = hann(self.n, sym=False) + c_amp, c_rms = abs(sum(w)), np.sqrt(sum(w.real**2 + w.imag**2)) + Xw = fft(self.x_ref*w) # unnormalized windowed DFT + + # Verify that the *spectrum* peak is consistent: + assert_allclose(self.tau * Xw[self.l_a] / c_amp, self.a_ref * self.tau / 2) + # Verify that the *amplitude spectrum* peak is consistent: + assert_allclose(Xw[self.l_a] / c_amp, self.a_ref/2) + + # Verify spectral power/energy equals signal's power/energy: + X_ESD = self.tau * self.T * abs(Xw / c_rms)**2 # Energy Spectral Density + X_PSD = self.T * abs(Xw / c_rms)**2 # Power Spectral Density + assert_allclose(self.delta_f * sum(X_ESD), self.E_ref) + assert_allclose(self.delta_f * sum(X_PSD), self.P_ref) + + # Verify scalings of periodogram: + kw = dict(fs=1/self.T, window=w, detrend=False, return_onesided=False) + _, P_mag = periodogram(self.x_ref, scaling='spectrum', **kw) + _, P_psd = periodogram(self.x_ref, scaling='density', **kw) + + # Verify that periodogram calculates a squared magnitude spectrum: + float_res = np.finfo(P_mag.dtype).resolution + assert_allclose(P_mag, abs(Xw/c_amp)**2, atol=float_res*max(P_mag)) + # Verify that periodogram calculates a PSD: + assert_allclose(P_psd, X_PSD, atol=float_res*max(P_psd)) + + # Ensure that scaling of welch is the same as of periodogram: + kw = dict(nperseg=len(self.x_ref), noverlap=0, **kw) + assert_allclose(welch(self.x_ref, scaling='spectrum', **kw)[1], P_mag, + atol=float_res*max(P_mag)) + assert_allclose(welch(self.x_ref, scaling='density', **kw)[1], P_psd, + atol=float_res*max(P_psd)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_splines.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_splines.py new file mode 100644 index 0000000000000000000000000000000000000000..e1be4083436f457582c7d8229198fd63ff0f9584 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_splines.py @@ -0,0 +1,362 @@ +# pylint: disable=missing-docstring +import numpy as np +import pytest +from scipy._lib._array_api import xp_assert_close + +from scipy.signal._spline import ( + symiirorder1_ic, symiirorder2_ic_fwd, symiirorder2_ic_bwd) +from scipy.signal import symiirorder1, symiirorder2 + + +def _compute_symiirorder2_bwd_hs(k, cs, rsq, omega): + cssq = cs * cs + k = np.abs(k) + rsupk = np.power(rsq, k / 2.0) + + c0 = (cssq * (1.0 + rsq) / (1.0 - rsq) / + (1 - 2 * rsq * np.cos(2 * omega) + rsq * rsq)) + gamma = (1.0 - rsq) / (1.0 + rsq) / np.tan(omega) + return c0 * rsupk * (np.cos(omega * k) + gamma * np.sin(omega * k)) + + +class TestSymIIR: + @pytest.mark.parametrize( + 'dtype', [np.float32, np.float64, np.complex64, np.complex128]) + @pytest.mark.parametrize('precision', [-1.0, 0.7, 0.5, 0.25, 0.0075]) + def test_symiir1_ic(self, dtype, precision): + c_precision = precision + if precision <= 0.0 or precision > 1.0: + if dtype in {np.float32, np.complex64}: + c_precision = 1e-6 + else: + c_precision = 1e-11 + + # Symmetrical initial conditions for a IIR filter of order 1 are: + # x[0] + z1 * \sum{k = 0}^{n - 1} x[k] * z1^k + + # Check the initial condition for a low-pass filter + # with coefficient b = 0.85 on a step signal. The initial condition is + # a geometric series: 1 + b * \sum_{k = 0}^{n - 1} u[k] b^k. + + # Finding the initial condition corresponds to + # 1. Computing the index n such that b**n < precision, which + # corresponds to ceil(log(precision) / log(b)) + # 2. Computing the geometric series until n, this can be computed + # using the partial sum formula: (1 - b**n) / (1 - b) + # This holds due to the input being a step signal. + b = 0.85 + n_exp = int(np.ceil(np.log(c_precision) / np.log(b))) + expected = np.asarray([[(1 - b ** n_exp) / (1 - b)]], dtype=dtype) + expected = 1 + b * expected + + # Create a step signal of size n + 1 + x = np.ones(n_exp + 1, dtype=dtype) + xp_assert_close(symiirorder1_ic(x, b, precision), expected, + atol=2e-6, rtol=2e-7) + + # Check the conditions for a exponential decreasing signal with base 2. + # Same conditions hold, as the product of 0.5^n * 0.85^n is + # still a geometric series + b_d = np.asarray(b, dtype=dtype) + expected = np.asarray( + [[(1 - (0.5 * b_d) ** n_exp) / (1 - (0.5 * b_d))]], dtype=dtype) + expected = 1 + b_d * expected + + # Create an exponential decreasing signal of size n + 1 + x = 2 ** -np.arange(n_exp + 1, dtype=dtype) + xp_assert_close(symiirorder1_ic(x, b, precision), expected, + atol=2e-6, rtol=2e-7) + + def test_symiir1_ic_fails(self): + # Test that symiirorder1_ic fails whenever \sum_{n = 1}^{n} b^n > eps + b = 0.85 + # Create a step signal of size 100 + x = np.ones(100, dtype=np.float64) + + # Compute the closed form for the geometrical series + precision = 1 / (1 - b) + pytest.raises(ValueError, symiirorder1_ic, x, b, precision) + + # Test that symiirorder1_ic fails when |z1| >= 1 + pytest.raises(ValueError, symiirorder1_ic, x, 1.0, -1) + pytest.raises(ValueError, symiirorder1_ic, x, 2.0, -1) + + @pytest.mark.parametrize( + 'dtype', [np.float32, np.float64, np.complex64, np.complex128]) + @pytest.mark.parametrize('precision', [-1.0, 0.7, 0.5, 0.25, 0.0075]) + def test_symiir1(self, dtype, precision): + c_precision = precision + if precision <= 0.0 or precision > 1.0: + if dtype in {np.float32, np.complex64}: + c_precision = 1e-6 + else: + c_precision = 1e-11 + + # Test for a low-pass filter with c0 = 0.15 and z1 = 0.85 + # using an unit step over 200 samples. + c0 = 0.15 + z1 = 0.85 + n = 200 + signal = np.ones(n, dtype=dtype) + + # Find the initial condition. See test_symiir1_ic for a detailed + # explanation + n_exp = int(np.ceil(np.log(c_precision) / np.log(z1))) + initial = np.asarray((1 - z1 ** n_exp) / (1 - z1), dtype=dtype) + initial = 1 + z1 * initial + + # Forward pass + # The transfer function for the system 1 / (1 - z1 * z^-1) when + # applied to an unit step with initial conditions y0 is + # 1 / (1 - z1 * z^-1) * (z^-1 / (1 - z^-1) + y0) + + # Solving the inverse Z-transform for the given expression yields: + # y[n] = y0 * z1**n * u[n] + + # -z1 / (1 - z1) * z1**(k - 1) * u[k - 1] + + # 1 / (1 - z1) * u[k - 1] + # d is the Kronecker delta function, and u is the unit step + + # y0 * z1**n * u[n] + pos = np.arange(n, dtype=dtype) + comp1 = initial * z1**pos + + # -z1 / (1 - z1) * z1**(k - 1) * u[k - 1] + comp2 = np.zeros(n, dtype=dtype) + comp2[1:] = -z1 / (1 - z1) * z1**pos[:-1] + + # 1 / (1 - z1) * u[k - 1] + comp3 = np.zeros(n, dtype=dtype) + comp3[1:] = 1 / (1 - z1) + + expected_fwd = comp1 + comp2 + comp3 + + # Reverse condition + sym_cond = -c0 / (z1 - 1.0) * expected_fwd[-1] + + # Backward pass + # The transfer function for the forward result is equivalent to + # the forward system times c0 / (1 - z1 * z). + + # Computing a closed form for the complete expression is difficult + # The result will be computed iteratively from the difference equation + exp_out = np.zeros(n, dtype=dtype) + exp_out[0] = sym_cond + + for i in range(1, n): + exp_out[i] = c0 * expected_fwd[n - 1 - i] + z1 * exp_out[i - 1] + + exp_out = exp_out[::-1] + + out = symiirorder1(signal, c0, z1, precision) + xp_assert_close(out, exp_out, atol=4e-6, rtol=6e-7) + + @pytest.mark.parametrize('dtype', ['float32', 'float64']) + def test_symiir1_values(self, dtype): + rng = np.random.RandomState(1234) + dtype = getattr(np, dtype) + s = rng.uniform(size=16).astype(dtype) + res = symiirorder1(s, 0.5, 0.1) + + # values from scipy 1.9.1 + exp_res = np.array([0.14387447, 0.35166047, 0.29735238, 0.46295986, 0.45174927, + 0.19982875, 0.20355805, 0.47378628, 0.57232247, 0.51597393, + 0.25935107, 0.31438554, 0.41096728, 0.4190693 , 0.25812255, + 0.33671467], dtype=res.dtype) + assert res.dtype == dtype + atol = {np.float64: 1e-15, np.float32: 1e-7}[dtype] + xp_assert_close(res, exp_res, atol=atol) + + s = s + 1j*s + res = symiirorder1(s, 0.5, 0.1) + assert res.dtype == np.complex64 if dtype == np.float32 else np.complex128 + xp_assert_close(res, exp_res + 1j*exp_res, atol=atol) + + @pytest.mark.parametrize( + 'dtype', ['float32', 'float64']) + @pytest.mark.parametrize('precision', [-1.0, 0.7, 0.5, 0.25, 0.0075]) + def test_symiir2_initial_fwd(self, dtype, precision): + dtype = getattr(np, dtype) + c_precision = precision + if precision <= 0.0 or precision > 1.0: + if dtype in {np.float32, np.complex64}: + c_precision = 1e-6 + else: + c_precision = 1e-11 + + # Compute the initial conditions for a order-two symmetrical low-pass + # filter with r = 0.5 and omega = pi / 3 for an unit step input. + r = np.asarray(0.5, dtype=dtype) + omega = np.asarray(np.pi / 3.0, dtype=dtype) + cs = 1 - 2 * r * np.cos(omega) + r**2 + + # The index n for the initial condition is bound from 0 to the + # first position where sin(omega * (n + 2)) = 0 => omega * (n + 2) = pi + # For omega = pi / 3, the maximum initial condition occurs when + # sqrt(3) / 2 * r**n < precision. + # => n = log(2 * sqrt(3) / 3 * precision) / log(r) + ub = np.ceil(np.log(c_precision / np.sin(omega)) / np.log(c_precision)) + lb = np.ceil(np.pi / omega) - 2 + n_exp = min(ub, lb) + + # The forward initial condition for a filter of order two is: + # \frac{cs}{\sin(\omega)} \sum_{n = 0}^{N - 1} { + # r^(n + 1) \sin{\omega(n + 2)}} + cs + # The closed expression for this sum is: + # s[n] = 2 * r * np.cos(omega) - + # r**2 - r**(n + 2) * np.sin(omega * (n + 3)) / np.sin(omega) + + # r**(n + 3) * np.sin(omega * (n + 2)) / np.sin(omega) + cs + fwd_initial_1 = ( + cs + + 2 * r * np.cos(omega) - + r**2 - + r**(n_exp + 2) * np.sin(omega * (n_exp + 3)) / np.sin(omega) + + r**(n_exp + 3) * np.sin(omega * (n_exp + 2)) / np.sin(omega)) + + # The second initial condition is given by + # s[n] = 1 / np.sin(omega) * ( + # r**2 * np.sin(3 * omega) - + # r**3 * np.sin(2 * omega) - + # r**(n + 3) * np.sin(omega * (n + 4)) + + # r**(n + 4) * np.sin(omega * (n + 3))) + ub = np.ceil(np.log(c_precision / np.sin(omega)) / np.log(c_precision)) + lb = np.ceil(np.pi / omega) - 3 + n_exp = min(ub, lb) + + fwd_initial_2 = ( + cs + cs * 2 * r * np.cos(omega) + + (r**2 * np.sin(3 * omega) - + r**3 * np.sin(2 * omega) - + r**(n_exp + 3) * np.sin(omega * (n_exp + 4)) + + r**(n_exp + 4) * np.sin(omega * (n_exp + 3))) / np.sin(omega)) + + expected = np.r_[fwd_initial_1, fwd_initial_2][None, :] + expected = expected.astype(dtype) + + n = 100 + signal = np.ones(n, dtype=dtype) + + out = symiirorder2_ic_fwd(signal, r, omega, precision) + xp_assert_close(out, expected, atol=4e-6, rtol=6e-7) + + @pytest.mark.parametrize( + 'dtype', [np.float32, np.float64]) + @pytest.mark.parametrize('precision', [-1.0, 0.7, 0.5, 0.25, 0.0075]) + def test_symiir2_initial_bwd(self, dtype, precision): + c_precision = precision + if precision <= 0.0 or precision > 1.0: + if dtype in {np.float32, np.complex64}: + c_precision = 1e-6 + else: + c_precision = 1e-11 + + r = np.asarray(0.5, dtype=dtype) + omega = np.asarray(np.pi / 3.0, dtype=dtype) + cs = 1 - 2 * r * np.cos(omega) + r * r + a2 = 2 * r * np.cos(omega) + a3 = -r * r + + n = 100 + signal = np.ones(n, dtype=dtype) + + # Compute initial forward conditions + ic = symiirorder2_ic_fwd(signal, r, omega, precision) + out = np.zeros(n + 2, dtype=dtype) + out[:2] = ic[0] + + # Apply the forward system cs / (1 - a2 * z^-1 - a3 * z^-2)) + for i in range(2, n + 2): + out[i] = cs * signal[i - 2] + a2 * out[i - 1] + a3 * out[i - 2] + + # Find the backward initial conditions + ic2 = np.zeros(2, dtype=dtype) + idx = np.arange(n) + + diff = (_compute_symiirorder2_bwd_hs(idx, cs, r * r, omega) + + _compute_symiirorder2_bwd_hs(idx + 1, cs, r * r, omega)) + ic2_0_all = np.cumsum(diff * out[:1:-1]) + pos = np.where(diff ** 2 < c_precision)[0] + ic2[0] = ic2_0_all[pos[0]] + + diff = (_compute_symiirorder2_bwd_hs(idx - 1, cs, r * r, omega) + + _compute_symiirorder2_bwd_hs(idx + 2, cs, r * r, omega)) + ic2_1_all = np.cumsum(diff * out[:1:-1]) + pos = np.where(diff ** 2 < c_precision)[0] + ic2[1] = ic2_1_all[pos[0]] + + out_ic = symiirorder2_ic_bwd(out, r, omega, precision)[0] + xp_assert_close(out_ic, ic2, atol=4e-6, rtol=6e-7) + + @pytest.mark.parametrize( + 'dtype', [np.float32, np.float64]) + @pytest.mark.parametrize('precision', [-1.0, 0.7, 0.5, 0.25, 0.0075]) + def test_symiir2(self, dtype, precision): + r = np.asarray(0.5, dtype=dtype) + omega = np.asarray(np.pi / 3.0, dtype=dtype) + cs = 1 - 2 * r * np.cos(omega) + r * r + a2 = 2 * r * np.cos(omega) + a3 = -r * r + + n = 100 + signal = np.ones(n, dtype=dtype) + + # Compute initial forward conditions + ic = symiirorder2_ic_fwd(signal, r, omega, precision) + out1 = np.zeros(n + 2, dtype=dtype) + out1[:2] = ic[0] + + # Apply the forward system cs / (1 - a2 * z^-1 - a3 * z^-2)) + for i in range(2, n + 2): + out1[i] = cs * signal[i - 2] + a2 * out1[i - 1] + a3 * out1[i - 2] + + # Find the backward initial conditions + ic2 = symiirorder2_ic_bwd(out1, r, omega, precision)[0] + + # Apply the system cs / (1 - a2 * z - a3 * z^2)) in backwards + exp = np.empty(n, dtype=dtype) + exp[-2:] = ic2[::-1] + + for i in range(n - 3, -1, -1): + exp[i] = cs * out1[i] + a2 * exp[i + 1] + a3 * exp[i + 2] + + out = symiirorder2(signal, r, omega, precision) + xp_assert_close(out, exp, atol=4e-6, rtol=6e-7) + + @pytest.mark.parametrize('dtyp', ['float32', 'float64']) + def test_symiir2_values(self, dtyp): + dtyp = getattr(np, dtyp) + rng = np.random.RandomState(1234) + s = rng.uniform(size=16).astype(dtyp) + res = symiirorder2(s, 0.1, 0.1, precision=1e-10) + + # values from scipy 1.9.1 + exp_res = np.array([0.26572609, 0.53408018, 0.51032696, 0.72115829, 0.69486885, + 0.3649055 , 0.37349478, 0.74165032, 0.89718521, 0.80582483, + 0.46758053, 0.51898709, 0.65025605, 0.65394321, 0.45273595, + 0.53539183], dtype=dtyp) + + assert res.dtype == dtyp + # The values in SciPy 1.14 agree with those in SciPy 1.9.1 to this + # accuracy only. Implementation differences are twofold: + # 1. boundary conditions are computed differently + # 2. the filter itself uses sosfilt instead of a hardcoded iteration + # The boundary conditions seem are tested separately (see + # test_symiir2_initial_{fwd,bwd} above, so the difference is likely + # due to a different way roundoff errors accumulate in the filter. + # In that respect, sosfilt is likely doing a better job. + xp_assert_close(res, exp_res, atol=2e-6) + + s = s + 1j*s + with pytest.raises(TypeError): + res = symiirorder2(s, 0.5, 0.1) + + def test_symiir1_integer_input(self): + s = np.where(np.arange(100) % 2, -1, 1) + expected = symiirorder1(s.astype(float), 0.5, 0.5) + out = symiirorder1(s, 0.5, 0.5) + xp_assert_close(out, expected) + + def test_symiir2_integer_input(self): + s = np.where(np.arange(100) % 2, -1, 1) + expected = symiirorder2(s.astype(float), 0.5, np.pi / 3.0) + out = symiirorder2(s, 0.5, np.pi / 3.0) + xp_assert_close(out, expected) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_upfirdn.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_upfirdn.py new file mode 100644 index 0000000000000000000000000000000000000000..0aaec38f06ba72a8bcd984d91b596e98eb3bbc54 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_upfirdn.py @@ -0,0 +1,288 @@ +# Code adapted from "upfirdn" python library with permission: +# +# Copyright (c) 2009, Motorola, Inc +# +# All Rights Reserved. +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions are +# met: +# +# * Redistributions of source code must retain the above copyright notice, +# this list of conditions and the following disclaimer. +# +# * Redistributions in binary form must reproduce the above copyright +# notice, this list of conditions and the following disclaimer in the +# documentation and/or other materials provided with the distribution. +# +# * Neither the name of Motorola nor the names of its contributors may be +# used to endorse or promote products derived from this software without +# specific prior written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +# IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, +# THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR +# PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR +# CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +# EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +# PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +# PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +# LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + + +import numpy as np +from itertools import product + +from scipy._lib._array_api import xp_assert_close +from pytest import raises as assert_raises +import pytest + +from scipy.signal import upfirdn, firwin +from scipy.signal._upfirdn import _output_len, _upfirdn_modes +from scipy.signal._upfirdn_apply import _pad_test + + +def upfirdn_naive(x, h, up=1, down=1): + """Naive upfirdn processing in Python. + + Note: arg order (x, h) differs to facilitate apply_along_axis use. + """ + h = np.asarray(h) + out = np.zeros(len(x) * up, x.dtype) + out[::up] = x + out = np.convolve(h, out)[::down][:_output_len(len(h), len(x), up, down)] + return out + + +class UpFIRDnCase: + """Test _UpFIRDn object""" + def __init__(self, up, down, h, x_dtype): + self.up = up + self.down = down + self.h = np.atleast_1d(h) + self.x_dtype = x_dtype + self.rng = np.random.RandomState(17) + + def __call__(self): + # tiny signal + self.scrub(np.ones(1, self.x_dtype)) + # ones + self.scrub(np.ones(10, self.x_dtype)) # ones + # randn + x = self.rng.randn(10).astype(self.x_dtype) + if self.x_dtype in (np.complex64, np.complex128): + x += 1j * self.rng.randn(10) + self.scrub(x) + # ramp + self.scrub(np.arange(10).astype(self.x_dtype)) + # 3D, random + size = (2, 3, 5) + x = self.rng.randn(*size).astype(self.x_dtype) + if self.x_dtype in (np.complex64, np.complex128): + x += 1j * self.rng.randn(*size) + for axis in range(len(size)): + self.scrub(x, axis=axis) + x = x[:, ::2, 1::3].T + for axis in range(len(size)): + self.scrub(x, axis=axis) + + def scrub(self, x, axis=-1): + yr = np.apply_along_axis(upfirdn_naive, axis, x, + self.h, self.up, self.down) + want_len = _output_len(len(self.h), x.shape[axis], self.up, self.down) + assert yr.shape[axis] == want_len + y = upfirdn(self.h, x, self.up, self.down, axis=axis) + assert y.shape[axis] == want_len + assert y.shape == yr.shape + dtypes = (self.h.dtype, x.dtype) + if all(d == np.complex64 for d in dtypes): + assert y.dtype == np.complex64 + elif np.complex64 in dtypes and np.float32 in dtypes: + assert y.dtype == np.complex64 + elif all(d == np.float32 for d in dtypes): + assert y.dtype == np.float32 + elif np.complex128 in dtypes or np.complex64 in dtypes: + assert y.dtype == np.complex128 + else: + assert y.dtype == np.float64 + xp_assert_close(yr.astype(y.dtype), y) + + +_UPFIRDN_TYPES = (int, np.float32, np.complex64, float, complex) + + +class TestUpfirdn: + + def test_valid_input(self): + assert_raises(ValueError, upfirdn, [1], [1], 1, 0) # up or down < 1 + assert_raises(ValueError, upfirdn, [], [1], 1, 1) # h.ndim != 1 + assert_raises(ValueError, upfirdn, [[1]], [1], 1, 1) + + @pytest.mark.parametrize('len_h', [1, 2, 3, 4, 5]) + @pytest.mark.parametrize('len_x', [1, 2, 3, 4, 5]) + def test_singleton(self, len_h, len_x): + # gh-9844: lengths producing expected outputs + h = np.zeros(len_h) + h[len_h // 2] = 1. # make h a delta + x = np.ones(len_x) + y = upfirdn(h, x, 1, 1) + want = np.pad(x, (len_h // 2, (len_h - 1) // 2), 'constant') + xp_assert_close(y, want) + + def test_shift_x(self): + # gh-9844: shifted x can change values? + y = upfirdn([1, 1], [1.], 1, 1) + xp_assert_close(y, np.asarray([1.0, 1.0])) # was [0, 1] in the issue + y = upfirdn([1, 1], [0., 1.], 1, 1) + xp_assert_close(y, np.asarray([0.0, 1.0, 1.0])) + + # A bunch of lengths/factors chosen because they exposed differences + # between the "old way" and new way of computing length, and then + # got `expected` from MATLAB + @pytest.mark.parametrize('len_h, len_x, up, down, expected', [ + (2, 2, 5, 2, [1, 0, 0, 0]), + (2, 3, 6, 3, [1, 0, 1, 0, 1]), + (2, 4, 4, 3, [1, 0, 0, 0, 1]), + (3, 2, 6, 2, [1, 0, 0, 1, 0]), + (4, 11, 3, 5, [1, 0, 0, 1, 0, 0, 1]), + ]) + def test_length_factors(self, len_h, len_x, up, down, expected): + # gh-9844: weird factors + h = np.zeros(len_h) + h[0] = 1. + x = np.ones(len_x) + y = upfirdn(h, x, up, down) + expected = np.asarray(expected, dtype=np.float64) + xp_assert_close(y, expected) + + @pytest.mark.parametrize('down, want_len', [ # lengths from MATLAB + (2, 5015), + (11, 912), + (79, 127), + ]) + def test_vs_convolve(self, down, want_len): + # Check that up=1.0 gives same answer as convolve + slicing + random_state = np.random.RandomState(17) + try_types = (int, np.float32, np.complex64, float, complex) + size = 10000 + + for dtype in try_types: + x = random_state.randn(size).astype(dtype) + if dtype in (np.complex64, np.complex128): + x += 1j * random_state.randn(size) + + h = firwin(31, 1. / down, window='hamming') + yl = upfirdn_naive(x, h, 1, down) + y = upfirdn(h, x, up=1, down=down) + assert y.shape == (want_len,) + assert yl.shape[0] == y.shape[0] + xp_assert_close(yl, y, atol=1e-7, rtol=1e-7) + + @pytest.mark.parametrize('x_dtype', _UPFIRDN_TYPES) + @pytest.mark.parametrize('h', (1., 1j)) + @pytest.mark.parametrize('up, down', [(1, 1), (2, 2), (3, 2), (2, 3)]) + def test_vs_naive_delta(self, x_dtype, h, up, down): + UpFIRDnCase(up, down, h, x_dtype)() + + @pytest.mark.parametrize('x_dtype', _UPFIRDN_TYPES) + @pytest.mark.parametrize('h_dtype', _UPFIRDN_TYPES) + @pytest.mark.parametrize('p_max, q_max', + list(product((10, 100), (10, 100)))) + def test_vs_naive(self, x_dtype, h_dtype, p_max, q_max): + tests = self._random_factors(p_max, q_max, h_dtype, x_dtype) + for test in tests: + test() + + def _random_factors(self, p_max, q_max, h_dtype, x_dtype): + n_rep = 3 + longest_h = 25 + random_state = np.random.RandomState(17) + tests = [] + + for _ in range(n_rep): + # Randomize the up/down factors somewhat + p_add = q_max if p_max > q_max else 1 + q_add = p_max if q_max > p_max else 1 + p = random_state.randint(p_max) + p_add + q = random_state.randint(q_max) + q_add + + # Generate random FIR coefficients + len_h = random_state.randint(longest_h) + 1 + h = np.atleast_1d(random_state.randint(len_h)) + h = h.astype(h_dtype) + if h_dtype is complex: + h += 1j * random_state.randint(len_h) + + tests.append(UpFIRDnCase(p, q, h, x_dtype)) + + return tests + + @pytest.mark.parametrize('mode', _upfirdn_modes) + def test_extensions(self, mode): + """Test vs. manually computed results for modes not in numpy's pad.""" + x = np.array([1, 2, 3, 1], dtype=float) + npre, npost = 6, 6 + y = _pad_test(x, npre=npre, npost=npost, mode=mode) + if mode == 'antisymmetric': + y_expected = np.asarray( + [3.0, 1, -1, -3, -2, -1, 1, 2, 3, 1, -1, -3, -2, -1, 1, 2]) + elif mode == 'antireflect': + y_expected = np.asarray( + [1.0, 2, 3, 1, -1, 0, 1, 2, 3, 1, -1, 0, 1, 2, 3, 1]) + elif mode == 'smooth': + y_expected = np.asarray( + [-5.0, -4, -3, -2, -1, 0, 1, 2, 3, 1, -1, -3, -5, -7, -9, -11]) + elif mode == "line": + lin_slope = (x[-1] - x[0]) / (len(x) - 1) + left = x[0] + np.arange(-npre, 0, 1) * lin_slope + right = x[-1] + np.arange(1, npost + 1) * lin_slope + y_expected = np.concatenate((left, x, right)) + else: + y_expected = np.pad(x, (npre, npost), mode=mode) + xp_assert_close(y, y_expected) + + @pytest.mark.parametrize( + 'size, h_len, mode, dtype', + product( + [8], + [4, 5, 26], # include cases with h_len > 2*size + _upfirdn_modes, + [np.float32, np.float64, np.complex64, np.complex128], + ) + ) + def test_modes(self, size, h_len, mode, dtype): + random_state = np.random.RandomState(5) + x = random_state.randn(size).astype(dtype) + if dtype in (np.complex64, np.complex128): + x += 1j * random_state.randn(size) + h = np.arange(1, 1 + h_len, dtype=x.real.dtype) + + y = upfirdn(h, x, up=1, down=1, mode=mode) + # expected result: pad the input, filter with zero padding, then crop + npad = h_len - 1 + if mode in ['antisymmetric', 'antireflect', 'smooth', 'line']: + # use _pad_test test function for modes not supported by np.pad. + xpad = _pad_test(x, npre=npad, npost=npad, mode=mode) + else: + xpad = np.pad(x, npad, mode=mode) + ypad = upfirdn(h, xpad, up=1, down=1, mode='constant') + y_expected = ypad[npad:-npad] + + atol = rtol = np.finfo(dtype).eps * 1e2 + xp_assert_close(y, y_expected, atol=atol, rtol=rtol) + + +def test_output_len_long_input(): + # Regression test for gh-17375. On Windows, a large enough input + # that should have been well within the capabilities of 64 bit integers + # would result in a 32 bit overflow because of a bug in Cython 0.29.32. + len_h = 1001 + in_len = 10**8 + up = 320 + down = 441 + out_len = _output_len(len_h, in_len, up, down) + # The expected value was computed "by hand" from the formula + # (((in_len - 1) * up + len_h) - 1) // down + 1 + assert out_len == 72562360 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_waveforms.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_waveforms.py new file mode 100644 index 0000000000000000000000000000000000000000..b30f7b9ceba80822f05ecfcabaad76fb87d5f335 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_waveforms.py @@ -0,0 +1,380 @@ +import numpy as np +from pytest import raises as assert_raises +from scipy._lib._array_api import ( + assert_almost_equal, xp_assert_equal, xp_assert_close +) + +import scipy.signal._waveforms as waveforms + + +# These chirp_* functions are the instantaneous frequencies of the signals +# returned by chirp(). + +def chirp_linear(t, f0, f1, t1): + f = f0 + (f1 - f0) * t / t1 + return f + + +def chirp_quadratic(t, f0, f1, t1, vertex_zero=True): + if vertex_zero: + f = f0 + (f1 - f0) * t**2 / t1**2 + else: + f = f1 - (f1 - f0) * (t1 - t)**2 / t1**2 + return f + + +def chirp_geometric(t, f0, f1, t1): + f = f0 * (f1/f0)**(t/t1) + return f + + +def chirp_hyperbolic(t, f0, f1, t1): + f = f0*f1*t1 / ((f0 - f1)*t + f1*t1) + return f + + +def compute_frequency(t, theta): + """ + Compute theta'(t)/(2*pi), where theta'(t) is the derivative of theta(t). + """ + # Assume theta and t are 1-D NumPy arrays. + # Assume that t is uniformly spaced. + dt = t[1] - t[0] + f = np.diff(theta)/(2*np.pi) / dt + tf = 0.5*(t[1:] + t[:-1]) + return tf, f + + +class TestChirp: + + def test_linear_at_zero(self): + w = waveforms.chirp(t=0, f0=1.0, f1=2.0, t1=1.0, method='linear') + assert_almost_equal(w, 1.0) + + def test_linear_freq_01(self): + method = 'linear' + f0 = 1.0 + f1 = 2.0 + t1 = 1.0 + t = np.linspace(0, t1, 100) + phase = waveforms._chirp_phase(t, f0, t1, f1, method) + tf, f = compute_frequency(t, phase) + abserr = np.max(np.abs(f - chirp_linear(tf, f0, f1, t1))) + assert abserr < 1e-6 + + def test_linear_freq_02(self): + method = 'linear' + f0 = 200.0 + f1 = 100.0 + t1 = 10.0 + t = np.linspace(0, t1, 100) + phase = waveforms._chirp_phase(t, f0, t1, f1, method) + tf, f = compute_frequency(t, phase) + abserr = np.max(np.abs(f - chirp_linear(tf, f0, f1, t1))) + assert abserr < 1e-6 + + def test_linear_complex_power(self): + method = 'linear' + f0 = 1.0 + f1 = 2.0 + t1 = 1.0 + t = np.linspace(0, t1, 100) + w_real = waveforms.chirp(t, f0, t1, f1, method, complex=False) + w_complex = waveforms.chirp(t, f0, t1, f1, method, complex=True) + w_pwr_r = np.var(w_real) + w_pwr_c = np.var(w_complex) + + # Making sure that power of the real part is not affected with + # complex conversion operation + err = w_pwr_r - np.real(w_pwr_c) + + assert(err < 1e-6) + + def test_linear_complex_at_zero(self): + w = waveforms.chirp(t=0, f0=-10.0, f1=1.0, t1=1.0, method='linear', + complex=True) + xp_assert_close(w, 1.0+0.0j) # dtype must match + + def test_quadratic_at_zero(self): + w = waveforms.chirp(t=0, f0=1.0, f1=2.0, t1=1.0, method='quadratic') + assert_almost_equal(w, 1.0) + + def test_quadratic_at_zero2(self): + w = waveforms.chirp(t=0, f0=1.0, f1=2.0, t1=1.0, method='quadratic', + vertex_zero=False) + assert_almost_equal(w, 1.0) + + def test_quadratic_complex_at_zero(self): + w = waveforms.chirp(t=0, f0=-1.0, f1=2.0, t1=1.0, method='quadratic', + complex=True) + xp_assert_close(w, 1.0+0j) + + def test_quadratic_freq_01(self): + method = 'quadratic' + f0 = 1.0 + f1 = 2.0 + t1 = 1.0 + t = np.linspace(0, t1, 2000) + phase = waveforms._chirp_phase(t, f0, t1, f1, method) + tf, f = compute_frequency(t, phase) + abserr = np.max(np.abs(f - chirp_quadratic(tf, f0, f1, t1))) + assert abserr < 1e-6 + + def test_quadratic_freq_02(self): + method = 'quadratic' + f0 = 20.0 + f1 = 10.0 + t1 = 10.0 + t = np.linspace(0, t1, 2000) + phase = waveforms._chirp_phase(t, f0, t1, f1, method) + tf, f = compute_frequency(t, phase) + abserr = np.max(np.abs(f - chirp_quadratic(tf, f0, f1, t1))) + assert abserr < 1e-6 + + def test_logarithmic_at_zero(self): + w = waveforms.chirp(t=0, f0=1.0, f1=2.0, t1=1.0, method='logarithmic') + assert_almost_equal(w, 1.0) + + def test_logarithmic_freq_01(self): + method = 'logarithmic' + f0 = 1.0 + f1 = 2.0 + t1 = 1.0 + t = np.linspace(0, t1, 10000) + phase = waveforms._chirp_phase(t, f0, t1, f1, method) + tf, f = compute_frequency(t, phase) + abserr = np.max(np.abs(f - chirp_geometric(tf, f0, f1, t1))) + assert abserr < 1e-6 + + def test_logarithmic_freq_02(self): + method = 'logarithmic' + f0 = 200.0 + f1 = 100.0 + t1 = 10.0 + t = np.linspace(0, t1, 10000) + phase = waveforms._chirp_phase(t, f0, t1, f1, method) + tf, f = compute_frequency(t, phase) + abserr = np.max(np.abs(f - chirp_geometric(tf, f0, f1, t1))) + assert abserr < 1e-6 + + def test_logarithmic_freq_03(self): + method = 'logarithmic' + f0 = 100.0 + f1 = 100.0 + t1 = 10.0 + t = np.linspace(0, t1, 10000) + phase = waveforms._chirp_phase(t, f0, t1, f1, method) + tf, f = compute_frequency(t, phase) + abserr = np.max(np.abs(f - chirp_geometric(tf, f0, f1, t1))) + assert abserr < 1e-6 + + def test_hyperbolic_at_zero(self): + w = waveforms.chirp(t=0, f0=10.0, f1=1.0, t1=1.0, method='hyperbolic') + assert_almost_equal(w, 1.0) + + def test_hyperbolic_freq_01(self): + method = 'hyperbolic' + t1 = 1.0 + t = np.linspace(0, t1, 10000) + # f0 f1 + cases = [[10.0, 1.0], + [1.0, 10.0], + [-10.0, -1.0], + [-1.0, -10.0]] + for f0, f1 in cases: + phase = waveforms._chirp_phase(t, f0, t1, f1, method) + tf, f = compute_frequency(t, phase) + expected = chirp_hyperbolic(tf, f0, f1, t1) + xp_assert_close(f, expected, atol=1e-7) + + def test_hyperbolic_zero_freq(self): + # f0=0 or f1=0 must raise a ValueError. + method = 'hyperbolic' + t1 = 1.0 + t = np.linspace(0, t1, 5) + assert_raises(ValueError, waveforms.chirp, t, 0, t1, 1, method) + assert_raises(ValueError, waveforms.chirp, t, 1, t1, 0, method) + + def test_unknown_method(self): + method = "foo" + f0 = 10.0 + f1 = 20.0 + t1 = 1.0 + t = np.linspace(0, t1, 10) + assert_raises(ValueError, waveforms.chirp, t, f0, t1, f1, method) + + def test_integer_t1(self): + f0 = 10.0 + f1 = 20.0 + t = np.linspace(-1, 1, 11) + t1 = 3.0 + float_result = waveforms.chirp(t, f0, t1, f1) + t1 = 3 + int_result = waveforms.chirp(t, f0, t1, f1) + err_msg = "Integer input 't1=3' gives wrong result" + xp_assert_equal(int_result, float_result, err_msg=err_msg) + + def test_integer_f0(self): + f1 = 20.0 + t1 = 3.0 + t = np.linspace(-1, 1, 11) + f0 = 10.0 + float_result = waveforms.chirp(t, f0, t1, f1) + f0 = 10 + int_result = waveforms.chirp(t, f0, t1, f1) + err_msg = "Integer input 'f0=10' gives wrong result" + xp_assert_equal(int_result, float_result, err_msg=err_msg) + + def test_integer_f1(self): + f0 = 10.0 + t1 = 3.0 + t = np.linspace(-1, 1, 11) + f1 = 20.0 + float_result = waveforms.chirp(t, f0, t1, f1) + f1 = 20 + int_result = waveforms.chirp(t, f0, t1, f1) + err_msg = "Integer input 'f1=20' gives wrong result" + xp_assert_equal(int_result, float_result, err_msg=err_msg) + + def test_integer_all(self): + f0 = 10 + t1 = 3 + f1 = 20 + t = np.linspace(-1, 1, 11) + float_result = waveforms.chirp(t, float(f0), float(t1), float(f1)) + int_result = waveforms.chirp(t, f0, t1, f1) + err_msg = "Integer input 'f0=10, t1=3, f1=20' gives wrong result" + xp_assert_equal(int_result, float_result, err_msg=err_msg) + + +class TestSweepPoly: + + def test_sweep_poly_quad1(self): + p = np.poly1d([1.0, 0.0, 1.0]) + t = np.linspace(0, 3.0, 10000) + phase = waveforms._sweep_poly_phase(t, p) + tf, f = compute_frequency(t, phase) + expected = p(tf) + abserr = np.max(np.abs(f - expected)) + assert abserr < 1e-6 + + def test_sweep_poly_const(self): + p = np.poly1d(2.0) + t = np.linspace(0, 3.0, 10000) + phase = waveforms._sweep_poly_phase(t, p) + tf, f = compute_frequency(t, phase) + expected = p(tf) + abserr = np.max(np.abs(f - expected)) + assert abserr < 1e-6 + + def test_sweep_poly_linear(self): + p = np.poly1d([-1.0, 10.0]) + t = np.linspace(0, 3.0, 10000) + phase = waveforms._sweep_poly_phase(t, p) + tf, f = compute_frequency(t, phase) + expected = p(tf) + abserr = np.max(np.abs(f - expected)) + assert abserr < 1e-6 + + def test_sweep_poly_quad2(self): + p = np.poly1d([1.0, 0.0, -2.0]) + t = np.linspace(0, 3.0, 10000) + phase = waveforms._sweep_poly_phase(t, p) + tf, f = compute_frequency(t, phase) + expected = p(tf) + abserr = np.max(np.abs(f - expected)) + assert abserr < 1e-6 + + def test_sweep_poly_cubic(self): + p = np.poly1d([2.0, 1.0, 0.0, -2.0]) + t = np.linspace(0, 2.0, 10000) + phase = waveforms._sweep_poly_phase(t, p) + tf, f = compute_frequency(t, phase) + expected = p(tf) + abserr = np.max(np.abs(f - expected)) + assert abserr < 1e-6 + + def test_sweep_poly_cubic2(self): + """Use an array of coefficients instead of a poly1d.""" + p = np.array([2.0, 1.0, 0.0, -2.0]) + t = np.linspace(0, 2.0, 10000) + phase = waveforms._sweep_poly_phase(t, p) + tf, f = compute_frequency(t, phase) + expected = np.poly1d(p)(tf) + abserr = np.max(np.abs(f - expected)) + assert abserr < 1e-6 + + def test_sweep_poly_cubic3(self): + """Use a list of coefficients instead of a poly1d.""" + p = [2.0, 1.0, 0.0, -2.0] + t = np.linspace(0, 2.0, 10000) + phase = waveforms._sweep_poly_phase(t, p) + tf, f = compute_frequency(t, phase) + expected = np.poly1d(p)(tf) + abserr = np.max(np.abs(f - expected)) + assert abserr < 1e-6 + + +class TestGaussPulse: + + def test_integer_fc(self): + float_result = waveforms.gausspulse('cutoff', fc=1000.0) + int_result = waveforms.gausspulse('cutoff', fc=1000) + err_msg = "Integer input 'fc=1000' gives wrong result" + xp_assert_equal(int_result, float_result, err_msg=err_msg) + + def test_integer_bw(self): + float_result = waveforms.gausspulse('cutoff', bw=1.0) + int_result = waveforms.gausspulse('cutoff', bw=1) + err_msg = "Integer input 'bw=1' gives wrong result" + xp_assert_equal(int_result, float_result, err_msg=err_msg) + + def test_integer_bwr(self): + float_result = waveforms.gausspulse('cutoff', bwr=-6.0) + int_result = waveforms.gausspulse('cutoff', bwr=-6) + err_msg = "Integer input 'bwr=-6' gives wrong result" + xp_assert_equal(int_result, float_result, err_msg=err_msg) + + def test_integer_tpr(self): + float_result = waveforms.gausspulse('cutoff', tpr=-60.0) + int_result = waveforms.gausspulse('cutoff', tpr=-60) + err_msg = "Integer input 'tpr=-60' gives wrong result" + xp_assert_equal(int_result, float_result, err_msg=err_msg) + + +class TestUnitImpulse: + + def test_no_index(self): + xp_assert_equal(waveforms.unit_impulse(7), + np.asarray([1.0, 0, 0, 0, 0, 0, 0])) + xp_assert_equal(waveforms.unit_impulse((3, 3)), + np.asarray([[1.0, 0, 0], [0, 0, 0], [0, 0, 0]])) + + def test_index(self): + xp_assert_equal(waveforms.unit_impulse(10, 3), + np.asarray([0.0, 0, 0, 1, 0, 0, 0, 0, 0, 0])) + xp_assert_equal(waveforms.unit_impulse((3, 3), (1, 1)), + np.asarray([[0.0, 0, 0], [0, 1, 0], [0, 0, 0]])) + + # Broadcasting + imp = waveforms.unit_impulse((4, 4), 2) + xp_assert_equal(imp, np.asarray([[0.0, 0, 0, 0], + [0.0, 0, 0, 0], + [0.0, 0, 1, 0], + [0.0, 0, 0, 0]])) + + def test_mid(self): + xp_assert_equal(waveforms.unit_impulse((3, 3), 'mid'), + np.asarray([[0.0, 0, 0], [0, 1, 0], [0, 0, 0]])) + xp_assert_equal(waveforms.unit_impulse(9, 'mid'), + np.asarray([0.0, 0, 0, 0, 1, 0, 0, 0, 0])) + + def test_dtype(self): + imp = waveforms.unit_impulse(7) + assert np.issubdtype(imp.dtype, np.floating) + + imp = waveforms.unit_impulse(5, 3, dtype=int) + assert np.issubdtype(imp.dtype, np.integer) + + imp = waveforms.unit_impulse((5, 2), (3, 1), dtype=complex) + assert np.issubdtype(imp.dtype, np.complexfloating) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_wavelets.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_wavelets.py new file mode 100644 index 0000000000000000000000000000000000000000..7a357d2eaf4a530930d612358b8ca69a18b5248e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_wavelets.py @@ -0,0 +1,59 @@ +import numpy as np +from numpy.testing import assert_array_equal, assert_array_almost_equal + +import scipy.signal._wavelets as wavelets + + +class TestWavelets: + def test_ricker(self): + w = wavelets._ricker(1.0, 1) + expected = 2 / (np.sqrt(3 * 1.0) * (np.pi ** 0.25)) + assert_array_equal(w, expected) + + lengths = [5, 11, 15, 51, 101] + for length in lengths: + w = wavelets._ricker(length, 1.0) + assert len(w) == length + max_loc = np.argmax(w) + assert max_loc == (length // 2) + + points = 100 + w = wavelets._ricker(points, 2.0) + half_vec = np.arange(0, points // 2) + # Wavelet should be symmetric + assert_array_almost_equal(w[half_vec], w[-(half_vec + 1)]) + + # Check zeros + aas = [5, 10, 15, 20, 30] + points = 99 + for a in aas: + w = wavelets._ricker(points, a) + vec = np.arange(0, points) - (points - 1.0) / 2 + exp_zero1 = np.argmin(np.abs(vec - a)) + exp_zero2 = np.argmin(np.abs(vec + a)) + assert_array_almost_equal(w[exp_zero1], 0) + assert_array_almost_equal(w[exp_zero2], 0) + + def test_cwt(self): + widths = [1.0] + def delta_wavelet(s, t): + return np.array([1]) + len_data = 100 + test_data = np.sin(np.pi * np.arange(0, len_data) / 10.0) + + # Test delta function input gives same data as output + cwt_dat = wavelets._cwt(test_data, delta_wavelet, widths) + assert cwt_dat.shape == (len(widths), len_data) + assert_array_almost_equal(test_data, cwt_dat.flatten()) + + # Check proper shape on output + widths = [1, 3, 4, 5, 10] + cwt_dat = wavelets._cwt(test_data, wavelets._ricker, widths) + assert cwt_dat.shape == (len(widths), len_data) + + widths = [len_data * 10] + # Note: this wavelet isn't defined quite right, but is fine for this test + def flat_wavelet(l, w): + return np.full(w, 1 / w) + cwt_dat = wavelets._cwt(test_data, flat_wavelet, widths) + assert_array_almost_equal(cwt_dat, np.mean(test_data)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_windows.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_windows.py new file mode 100644 index 0000000000000000000000000000000000000000..75c4da5327f0c1a806f72865687927bc7a380c6d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/tests/test_windows.py @@ -0,0 +1,846 @@ +import numpy as np +from numpy import array +from numpy.testing import (assert_array_almost_equal, assert_array_equal, + assert_allclose, + assert_equal, assert_, assert_array_less, + suppress_warnings) +from pytest import raises as assert_raises + +from scipy.fft import fft +from scipy.signal import windows, get_window, resample + + +window_funcs = [ + ('boxcar', ()), + ('triang', ()), + ('parzen', ()), + ('bohman', ()), + ('blackman', ()), + ('nuttall', ()), + ('blackmanharris', ()), + ('flattop', ()), + ('bartlett', ()), + ('barthann', ()), + ('hamming', ()), + ('kaiser', (1,)), + ('dpss', (2,)), + ('gaussian', (0.5,)), + ('general_gaussian', (1.5, 2)), + ('chebwin', (1,)), + ('cosine', ()), + ('hann', ()), + ('exponential', ()), + ('taylor', ()), + ('tukey', (0.5,)), + ('lanczos', ()), + ] + + +class TestBartHann: + + def test_basic(self): + assert_allclose(windows.barthann(6, sym=True), + [0, 0.35857354213752, 0.8794264578624801, + 0.8794264578624801, 0.3585735421375199, 0], + rtol=1e-15, atol=1e-15) + assert_allclose(windows.barthann(7), + [0, 0.27, 0.73, 1.0, 0.73, 0.27, 0], + rtol=1e-15, atol=1e-15) + assert_allclose(windows.barthann(6, False), + [0, 0.27, 0.73, 1.0, 0.73, 0.27], + rtol=1e-15, atol=1e-15) + + +class TestBartlett: + + def test_basic(self): + assert_allclose(windows.bartlett(6), [0, 0.4, 0.8, 0.8, 0.4, 0]) + assert_allclose(windows.bartlett(7), [0, 1/3, 2/3, 1.0, 2/3, 1/3, 0]) + assert_allclose(windows.bartlett(6, False), + [0, 1/3, 2/3, 1.0, 2/3, 1/3]) + + +class TestBlackman: + + def test_basic(self): + assert_allclose(windows.blackman(6, sym=False), + [0, 0.13, 0.63, 1.0, 0.63, 0.13], atol=1e-14) + assert_allclose(windows.blackman(7, sym=False), + [0, 0.09045342435412804, 0.4591829575459636, + 0.9203636180999081, 0.9203636180999081, + 0.4591829575459636, 0.09045342435412804], atol=1e-8) + assert_allclose(windows.blackman(6), + [0, 0.2007701432625305, 0.8492298567374694, + 0.8492298567374694, 0.2007701432625305, 0], + atol=1e-14) + assert_allclose(windows.blackman(7, True), + [0, 0.13, 0.63, 1.0, 0.63, 0.13, 0], atol=1e-14) + + +class TestBlackmanHarris: + + def test_basic(self): + assert_allclose(windows.blackmanharris(6, False), + [6.0e-05, 0.055645, 0.520575, 1.0, 0.520575, 0.055645]) + assert_allclose(windows.blackmanharris(7, sym=False), + [6.0e-05, 0.03339172347815117, 0.332833504298565, + 0.8893697722232837, 0.8893697722232838, + 0.3328335042985652, 0.03339172347815122]) + assert_allclose(windows.blackmanharris(6), + [6.0e-05, 0.1030114893456638, 0.7938335106543362, + 0.7938335106543364, 0.1030114893456638, 6.0e-05]) + assert_allclose(windows.blackmanharris(7, sym=True), + [6.0e-05, 0.055645, 0.520575, 1.0, 0.520575, 0.055645, + 6.0e-05]) + + +class TestTaylor: + + def test_normalized(self): + """Tests windows of small length that are normalized to 1. See the + documentation for the Taylor window for more information on + normalization. + """ + assert_allclose(windows.taylor(1, 2, 15), 1.0) + assert_allclose( + windows.taylor(5, 2, 15), + np.array([0.75803341, 0.90757699, 1.0, 0.90757699, 0.75803341]) + ) + assert_allclose( + windows.taylor(6, 2, 15), + np.array([ + 0.7504082, 0.86624416, 0.98208011, 0.98208011, 0.86624416, + 0.7504082 + ]) + ) + + def test_non_normalized(self): + """Test windows of small length that are not normalized to 1. See + the documentation for the Taylor window for more information on + normalization. + """ + assert_allclose( + windows.taylor(5, 2, 15, norm=False), + np.array([ + 0.87508054, 1.04771499, 1.15440894, 1.04771499, 0.87508054 + ]) + ) + assert_allclose( + windows.taylor(6, 2, 15, norm=False), + np.array([ + 0.86627793, 1.0, 1.13372207, 1.13372207, 1.0, 0.86627793 + ]) + ) + + def test_correctness(self): + """This test ensures the correctness of the implemented Taylor + Windowing function. A Taylor Window of 1024 points is created, its FFT + is taken, and the Peak Sidelobe Level (PSLL) and 3dB and 18dB bandwidth + are found and checked. + + A publication from Sandia National Laboratories was used as reference + for the correctness values [1]_. + + References + ----- + .. [1] Armin Doerry, "Catalog of Window Taper Functions for + Sidelobe Control", 2017. + https://www.researchgate.net/profile/Armin_Doerry/publication/316281181_Catalog_of_Window_Taper_Functions_for_Sidelobe_Control/links/58f92cb2a6fdccb121c9d54d/Catalog-of-Window-Taper-Functions-for-Sidelobe-Control.pdf + """ + M_win = 1024 + N_fft = 131072 + # Set norm=False for correctness as the values obtained from the + # scientific publication do not normalize the values. Normalizing + # changes the sidelobe level from the desired value. + w = windows.taylor(M_win, nbar=4, sll=35, norm=False, sym=False) + f = fft(w, N_fft) + spec = 20 * np.log10(np.abs(f / np.amax(f))) + + first_zero = np.argmax(np.diff(spec) > 0) + + PSLL = np.amax(spec[first_zero:-first_zero]) + + BW_3dB = 2*np.argmax(spec <= -3.0102999566398121) / N_fft * M_win + BW_18dB = 2*np.argmax(spec <= -18.061799739838872) / N_fft * M_win + + assert_allclose(PSLL, -35.1672, atol=1) + assert_allclose(BW_3dB, 1.1822, atol=0.1) + assert_allclose(BW_18dB, 2.6112, atol=0.1) + + +class TestBohman: + + def test_basic(self): + assert_allclose(windows.bohman(6), + [0, 0.1791238937062839, 0.8343114522576858, + 0.8343114522576858, 0.1791238937062838, 0]) + assert_allclose(windows.bohman(7, sym=True), + [0, 0.1089977810442293, 0.6089977810442293, 1.0, + 0.6089977810442295, 0.1089977810442293, 0]) + assert_allclose(windows.bohman(6, False), + [0, 0.1089977810442293, 0.6089977810442293, 1.0, + 0.6089977810442295, 0.1089977810442293]) + + +class TestBoxcar: + + def test_basic(self): + assert_allclose(windows.boxcar(6), [1, 1, 1, 1, 1, 1]) + assert_allclose(windows.boxcar(7), [1, 1, 1, 1, 1, 1, 1]) + assert_allclose(windows.boxcar(6, False), [1, 1, 1, 1, 1, 1]) + + +cheb_odd_true = array([0.200938, 0.107729, 0.134941, 0.165348, + 0.198891, 0.235450, 0.274846, 0.316836, + 0.361119, 0.407338, 0.455079, 0.503883, + 0.553248, 0.602637, 0.651489, 0.699227, + 0.745266, 0.789028, 0.829947, 0.867485, + 0.901138, 0.930448, 0.955010, 0.974482, + 0.988591, 0.997138, 1.000000, 0.997138, + 0.988591, 0.974482, 0.955010, 0.930448, + 0.901138, 0.867485, 0.829947, 0.789028, + 0.745266, 0.699227, 0.651489, 0.602637, + 0.553248, 0.503883, 0.455079, 0.407338, + 0.361119, 0.316836, 0.274846, 0.235450, + 0.198891, 0.165348, 0.134941, 0.107729, + 0.200938]) + +cheb_even_true = array([0.203894, 0.107279, 0.133904, + 0.163608, 0.196338, 0.231986, + 0.270385, 0.311313, 0.354493, + 0.399594, 0.446233, 0.493983, + 0.542378, 0.590916, 0.639071, + 0.686302, 0.732055, 0.775783, + 0.816944, 0.855021, 0.889525, + 0.920006, 0.946060, 0.967339, + 0.983557, 0.994494, 1.000000, + 1.000000, 0.994494, 0.983557, + 0.967339, 0.946060, 0.920006, + 0.889525, 0.855021, 0.816944, + 0.775783, 0.732055, 0.686302, + 0.639071, 0.590916, 0.542378, + 0.493983, 0.446233, 0.399594, + 0.354493, 0.311313, 0.270385, + 0.231986, 0.196338, 0.163608, + 0.133904, 0.107279, 0.203894]) + + +class TestChebWin: + + def test_basic(self): + with suppress_warnings() as sup: + sup.filter(UserWarning, "This window is not suitable") + assert_allclose(windows.chebwin(6, 100), + [0.1046401879356917, 0.5075781475823447, 1.0, 1.0, + 0.5075781475823447, 0.1046401879356917]) + assert_allclose(windows.chebwin(7, 100), + [0.05650405062850233, 0.316608530648474, + 0.7601208123539079, 1.0, 0.7601208123539079, + 0.316608530648474, 0.05650405062850233]) + assert_allclose(windows.chebwin(6, 10), + [1.0, 0.6071201674458373, 0.6808391469897297, + 0.6808391469897297, 0.6071201674458373, 1.0]) + assert_allclose(windows.chebwin(7, 10), + [1.0, 0.5190521247588651, 0.5864059018130382, + 0.6101519801307441, 0.5864059018130382, + 0.5190521247588651, 1.0]) + assert_allclose(windows.chebwin(6, 10, False), + [1.0, 0.5190521247588651, 0.5864059018130382, + 0.6101519801307441, 0.5864059018130382, + 0.5190521247588651]) + + def test_cheb_odd_high_attenuation(self): + with suppress_warnings() as sup: + sup.filter(UserWarning, "This window is not suitable") + cheb_odd = windows.chebwin(53, at=-40) + assert_array_almost_equal(cheb_odd, cheb_odd_true, decimal=4) + + def test_cheb_even_high_attenuation(self): + with suppress_warnings() as sup: + sup.filter(UserWarning, "This window is not suitable") + cheb_even = windows.chebwin(54, at=40) + assert_array_almost_equal(cheb_even, cheb_even_true, decimal=4) + + def test_cheb_odd_low_attenuation(self): + cheb_odd_low_at_true = array([1.000000, 0.519052, 0.586405, + 0.610151, 0.586405, 0.519052, + 1.000000]) + with suppress_warnings() as sup: + sup.filter(UserWarning, "This window is not suitable") + cheb_odd = windows.chebwin(7, at=10) + assert_array_almost_equal(cheb_odd, cheb_odd_low_at_true, decimal=4) + + def test_cheb_even_low_attenuation(self): + cheb_even_low_at_true = array([1.000000, 0.451924, 0.51027, + 0.541338, 0.541338, 0.51027, + 0.451924, 1.000000]) + with suppress_warnings() as sup: + sup.filter(UserWarning, "This window is not suitable") + cheb_even = windows.chebwin(8, at=-10) + assert_array_almost_equal(cheb_even, cheb_even_low_at_true, decimal=4) + + +exponential_data = { + (4, None, 0.2, False): + array([4.53999297624848542e-05, + 6.73794699908546700e-03, 1.00000000000000000e+00, + 6.73794699908546700e-03]), + (4, None, 0.2, True): array([0.00055308437014783, 0.0820849986238988, + 0.0820849986238988, 0.00055308437014783]), + (4, None, 1.0, False): array([0.1353352832366127, 0.36787944117144233, 1., + 0.36787944117144233]), + (4, None, 1.0, True): array([0.22313016014842982, 0.60653065971263342, + 0.60653065971263342, 0.22313016014842982]), + (4, 2, 0.2, False): + array([4.53999297624848542e-05, 6.73794699908546700e-03, + 1.00000000000000000e+00, 6.73794699908546700e-03]), + (4, 2, 0.2, True): None, + (4, 2, 1.0, False): array([0.1353352832366127, 0.36787944117144233, 1., + 0.36787944117144233]), + (4, 2, 1.0, True): None, + (5, None, 0.2, True): + array([4.53999297624848542e-05, + 6.73794699908546700e-03, 1.00000000000000000e+00, + 6.73794699908546700e-03, 4.53999297624848542e-05]), + (5, None, 1.0, True): array([0.1353352832366127, 0.36787944117144233, 1., + 0.36787944117144233, 0.1353352832366127]), + (5, 2, 0.2, True): None, + (5, 2, 1.0, True): None +} + + +def test_exponential(): + for k, v in exponential_data.items(): + if v is None: + assert_raises(ValueError, windows.exponential, *k) + else: + win = windows.exponential(*k) + assert_allclose(win, v, rtol=1e-14) + + +class TestFlatTop: + + def test_basic(self): + assert_allclose(windows.flattop(6, sym=False), + [-0.000421051, -0.051263156, 0.19821053, 1.0, + 0.19821053, -0.051263156]) + assert_allclose(windows.flattop(7, sym=False), + [-0.000421051, -0.03684078115492348, + 0.01070371671615342, 0.7808739149387698, + 0.7808739149387698, 0.01070371671615342, + -0.03684078115492348]) + assert_allclose(windows.flattop(6), + [-0.000421051, -0.0677142520762119, 0.6068721525762117, + 0.6068721525762117, -0.0677142520762119, + -0.000421051]) + assert_allclose(windows.flattop(7, True), + [-0.000421051, -0.051263156, 0.19821053, 1.0, + 0.19821053, -0.051263156, -0.000421051]) + + +class TestGaussian: + + def test_basic(self): + assert_allclose(windows.gaussian(6, 1.0), + [0.04393693362340742, 0.3246524673583497, + 0.8824969025845955, 0.8824969025845955, + 0.3246524673583497, 0.04393693362340742]) + assert_allclose(windows.gaussian(7, 1.2), + [0.04393693362340742, 0.2493522087772962, + 0.7066482778577162, 1.0, 0.7066482778577162, + 0.2493522087772962, 0.04393693362340742]) + assert_allclose(windows.gaussian(7, 3), + [0.6065306597126334, 0.8007374029168081, + 0.9459594689067654, 1.0, 0.9459594689067654, + 0.8007374029168081, 0.6065306597126334]) + assert_allclose(windows.gaussian(6, 3, False), + [0.6065306597126334, 0.8007374029168081, + 0.9459594689067654, 1.0, 0.9459594689067654, + 0.8007374029168081]) + + +class TestGeneralCosine: + + def test_basic(self): + assert_allclose(windows.general_cosine(5, [0.5, 0.3, 0.2]), + [0.4, 0.3, 1, 0.3, 0.4]) + assert_allclose(windows.general_cosine(4, [0.5, 0.3, 0.2], sym=False), + [0.4, 0.3, 1, 0.3]) + + +class TestGeneralHamming: + + def test_basic(self): + assert_allclose(windows.general_hamming(5, 0.7), + [0.4, 0.7, 1.0, 0.7, 0.4]) + assert_allclose(windows.general_hamming(5, 0.75, sym=False), + [0.5, 0.6727457514, 0.9522542486, + 0.9522542486, 0.6727457514]) + assert_allclose(windows.general_hamming(6, 0.75, sym=True), + [0.5, 0.6727457514, 0.9522542486, + 0.9522542486, 0.6727457514, 0.5]) + + +class TestHamming: + + def test_basic(self): + assert_allclose(windows.hamming(6, False), + [0.08, 0.31, 0.77, 1.0, 0.77, 0.31]) + assert_allclose(windows.hamming(7, sym=False), + [0.08, 0.2531946911449826, 0.6423596296199047, + 0.9544456792351128, 0.9544456792351128, + 0.6423596296199047, 0.2531946911449826]) + assert_allclose(windows.hamming(6), + [0.08, 0.3978521825875242, 0.9121478174124757, + 0.9121478174124757, 0.3978521825875242, 0.08]) + assert_allclose(windows.hamming(7, sym=True), + [0.08, 0.31, 0.77, 1.0, 0.77, 0.31, 0.08]) + + +class TestHann: + + def test_basic(self): + assert_allclose(windows.hann(6, sym=False), + [0, 0.25, 0.75, 1.0, 0.75, 0.25], + rtol=1e-15, atol=1e-15) + assert_allclose(windows.hann(7, sym=False), + [0, 0.1882550990706332, 0.6112604669781572, + 0.9504844339512095, 0.9504844339512095, + 0.6112604669781572, 0.1882550990706332], + rtol=1e-15, atol=1e-15) + assert_allclose(windows.hann(6, True), + [0, 0.3454915028125263, 0.9045084971874737, + 0.9045084971874737, 0.3454915028125263, 0], + rtol=1e-15, atol=1e-15) + assert_allclose(windows.hann(7), + [0, 0.25, 0.75, 1.0, 0.75, 0.25, 0], + rtol=1e-15, atol=1e-15) + + +class TestKaiser: + + def test_basic(self): + assert_allclose(windows.kaiser(6, 0.5), + [0.9403061933191572, 0.9782962393705389, + 0.9975765035372042, 0.9975765035372042, + 0.9782962393705389, 0.9403061933191572]) + assert_allclose(windows.kaiser(7, 0.5), + [0.9403061933191572, 0.9732402256999829, + 0.9932754654413773, 1.0, 0.9932754654413773, + 0.9732402256999829, 0.9403061933191572]) + assert_allclose(windows.kaiser(6, 2.7), + [0.2603047507678832, 0.6648106293528054, + 0.9582099802511439, 0.9582099802511439, + 0.6648106293528054, 0.2603047507678832]) + assert_allclose(windows.kaiser(7, 2.7), + [0.2603047507678832, 0.5985765418119844, + 0.8868495172060835, 1.0, 0.8868495172060835, + 0.5985765418119844, 0.2603047507678832]) + assert_allclose(windows.kaiser(6, 2.7, False), + [0.2603047507678832, 0.5985765418119844, + 0.8868495172060835, 1.0, 0.8868495172060835, + 0.5985765418119844]) + + +class TestKaiserBesselDerived: + + def test_basic(self): + M = 100 + w = windows.kaiser_bessel_derived(M, beta=4.0) + w2 = windows.get_window(('kaiser bessel derived', 4.0), + M, fftbins=False) + assert_allclose(w, w2) + + # Test for Princen-Bradley condition + assert_allclose(w[:M // 2] ** 2 + w[-M // 2:] ** 2, 1.) + + # Test actual values from other implementations + # M = 2: sqrt(2) / 2 + # M = 4: 0.518562710536, 0.855039598640 + # M = 6: 0.436168993154, 0.707106781187, 0.899864772847 + # Ref:https://github.com/scipy/scipy/pull/4747#issuecomment-172849418 + assert_allclose(windows.kaiser_bessel_derived(2, beta=np.pi / 2)[:1], + np.sqrt(2) / 2) + + assert_allclose(windows.kaiser_bessel_derived(4, beta=np.pi / 2)[:2], + [0.518562710536, 0.855039598640]) + + assert_allclose(windows.kaiser_bessel_derived(6, beta=np.pi / 2)[:3], + [0.436168993154, 0.707106781187, 0.899864772847]) + + def test_exceptions(self): + M = 100 + # Assert ValueError for odd window length + msg = ("Kaiser-Bessel Derived windows are only defined for even " + "number of points") + with assert_raises(ValueError, match=msg): + windows.kaiser_bessel_derived(M + 1, beta=4.) + + # Assert ValueError for non-symmetric setting + msg = ("Kaiser-Bessel Derived windows are only defined for " + "symmetric shapes") + with assert_raises(ValueError, match=msg): + windows.kaiser_bessel_derived(M + 1, beta=4., sym=False) + + +class TestNuttall: + + def test_basic(self): + assert_allclose(windows.nuttall(6, sym=False), + [0.0003628, 0.0613345, 0.5292298, 1.0, 0.5292298, + 0.0613345]) + assert_allclose(windows.nuttall(7, sym=False), + [0.0003628, 0.03777576895352025, 0.3427276199688195, + 0.8918518610776603, 0.8918518610776603, + 0.3427276199688196, 0.0377757689535203]) + assert_allclose(windows.nuttall(6), + [0.0003628, 0.1105152530498718, 0.7982580969501282, + 0.7982580969501283, 0.1105152530498719, 0.0003628]) + assert_allclose(windows.nuttall(7, True), + [0.0003628, 0.0613345, 0.5292298, 1.0, 0.5292298, + 0.0613345, 0.0003628]) + + +class TestParzen: + + def test_basic(self): + assert_allclose(windows.parzen(6), + [0.009259259259259254, 0.25, 0.8611111111111112, + 0.8611111111111112, 0.25, 0.009259259259259254]) + assert_allclose(windows.parzen(7, sym=True), + [0.00583090379008747, 0.1574344023323616, + 0.6501457725947521, 1.0, 0.6501457725947521, + 0.1574344023323616, 0.00583090379008747]) + assert_allclose(windows.parzen(6, False), + [0.00583090379008747, 0.1574344023323616, + 0.6501457725947521, 1.0, 0.6501457725947521, + 0.1574344023323616]) + + +class TestTriang: + + def test_basic(self): + + assert_allclose(windows.triang(6, True), + [1/6, 1/2, 5/6, 5/6, 1/2, 1/6]) + assert_allclose(windows.triang(7), + [1/4, 1/2, 3/4, 1, 3/4, 1/2, 1/4]) + assert_allclose(windows.triang(6, sym=False), + [1/4, 1/2, 3/4, 1, 3/4, 1/2]) + + +tukey_data = { + (4, 0.5, True): array([0.0, 1.0, 1.0, 0.0]), + (4, 0.9, True): array([0.0, 0.84312081893436686, + 0.84312081893436686, 0.0]), + (4, 1.0, True): array([0.0, 0.75, 0.75, 0.0]), + (4, 0.5, False): array([0.0, 1.0, 1.0, 1.0]), + (4, 0.9, False): array([0.0, 0.58682408883346526, + 1.0, 0.58682408883346526]), + (4, 1.0, False): array([0.0, 0.5, 1.0, 0.5]), + (5, 0.0, True): array([1.0, 1.0, 1.0, 1.0, 1.0]), + (5, 0.8, True): array([0.0, 0.69134171618254492, + 1.0, 0.69134171618254492, 0.0]), + (5, 1.0, True): array([0.0, 0.5, 1.0, 0.5, 0.0]), + + (6, 0): [1, 1, 1, 1, 1, 1], + (7, 0): [1, 1, 1, 1, 1, 1, 1], + (6, .25): [0, 1, 1, 1, 1, 0], + (7, .25): [0, 1, 1, 1, 1, 1, 0], + (6,): [0, 0.9045084971874737, 1.0, 1.0, 0.9045084971874735, 0], + (7,): [0, 0.75, 1.0, 1.0, 1.0, 0.75, 0], + (6, .75): [0, 0.5522642316338269, 1.0, 1.0, 0.5522642316338267, 0], + (7, .75): [0, 0.4131759111665348, 0.9698463103929542, 1.0, + 0.9698463103929542, 0.4131759111665347, 0], + (6, 1): [0, 0.3454915028125263, 0.9045084971874737, 0.9045084971874737, + 0.3454915028125263, 0], + (7, 1): [0, 0.25, 0.75, 1.0, 0.75, 0.25, 0], +} + + +class TestTukey: + + def test_basic(self): + # Test against hardcoded data + for k, v in tukey_data.items(): + if v is None: + assert_raises(ValueError, windows.tukey, *k) + else: + win = windows.tukey(*k) + assert_allclose(win, v, rtol=1e-15, atol=1e-15) + + def test_extremes(self): + # Test extremes of alpha correspond to boxcar and hann + tuk0 = windows.tukey(100, 0) + box0 = windows.boxcar(100) + assert_array_almost_equal(tuk0, box0) + + tuk1 = windows.tukey(100, 1) + han1 = windows.hann(100) + assert_array_almost_equal(tuk1, han1) + + +dpss_data = { + # All values from MATLAB: + # * taper[1] of (3, 1.4, 3) sign-flipped + # * taper[3] of (5, 1.5, 5) sign-flipped + (4, 0.1, 2): ([[0.497943898, 0.502047681, 0.502047681, 0.497943898], [0.670487993, 0.224601537, -0.224601537, -0.670487993]], [0.197961815, 0.002035474]), # noqa: E501 + (3, 1.4, 3): ([[0.410233151, 0.814504464, 0.410233151], [0.707106781, 0.0, -0.707106781], [0.575941629, -0.580157287, 0.575941629]], [0.999998093, 0.998067480, 0.801934426]), # noqa: E501 + (5, 1.5, 5): ([[0.1745071052, 0.4956749177, 0.669109327, 0.495674917, 0.174507105], [0.4399493348, 0.553574369, 0.0, -0.553574369, -0.439949334], [0.631452756, 0.073280238, -0.437943884, 0.073280238, 0.631452756], [0.553574369, -0.439949334, 0.0, 0.439949334, -0.553574369], [0.266110290, -0.498935248, 0.600414741, -0.498935248, 0.266110290147157]], [0.999728571, 0.983706916, 0.768457889, 0.234159338, 0.013947282907567]), # noqa: E501 + (100, 2, 4): ([[0.0030914414, 0.0041266922, 0.005315076, 0.006665149, 0.008184854, 0.0098814158, 0.011761239, 0.013829809, 0.016091597, 0.018549973, 0.02120712, 0.02406396, 0.027120092, 0.030373728, 0.033821651, 0.037459181, 0.041280145, 0.045276872, 0.049440192, 0.053759447, 0.058222524, 0.062815894, 0.067524661, 0.072332638, 0.077222418, 0.082175473, 0.087172252, 0.092192299, 0.097214376, 0.1022166, 0.10717657, 0.11207154, 0.11687856, 0.12157463, 0.12613686, 0.13054266, 0.13476986, 0.13879691, 0.14260302, 0.14616832, 0.14947401, 0.1525025, 0.15523755, 0.15766438, 0.15976981, 0.16154233, 0.16297223, 0.16405162, 0.16477455, 0.16513702, 0.16513702, 0.16477455, 0.16405162, 0.16297223, 0.16154233, 0.15976981, 0.15766438, 0.15523755, 0.1525025, 0.14947401, 0.14616832, 0.14260302, 0.13879691, 0.13476986, 0.13054266, 0.12613686, 0.12157463, 0.11687856, 0.11207154, 0.10717657, 0.1022166, 0.097214376, 0.092192299, 0.087172252, 0.082175473, 0.077222418, 0.072332638, 0.067524661, 0.062815894, 0.058222524, 0.053759447, 0.049440192, 0.045276872, 0.041280145, 0.037459181, 0.033821651, 0.030373728, 0.027120092, 0.02406396, 0.02120712, 0.018549973, 0.016091597, 0.013829809, 0.011761239, 0.0098814158, 0.008184854, 0.006665149, 0.005315076, 0.0041266922, 0.0030914414], [0.018064449, 0.022040342, 0.026325013, 0.030905288, 0.035764398, 0.040881982, 0.046234148, 0.051793558, 0.057529559, 0.063408356, 0.069393216, 0.075444716, 0.081521022, 0.087578202, 0.093570567, 0.099451049, 0.10517159, 0.11068356, 0.11593818, 0.12088699, 0.12548227, 0.12967752, 0.1334279, 0.13669069, 0.13942569, 0.1415957, 0.14316686, 0.14410905, 0.14439626, 0.14400686, 0.14292389, 0.1411353, 0.13863416, 0.13541876, 0.13149274, 0.12686516, 0.12155045, 0.1155684, 0.10894403, 0.10170748, 0.093893752, 0.08554251, 0.076697768, 0.067407559, 0.057723559, 0.04770068, 0.037396627, 0.026871428, 0.016186944, 0.0054063557, -0.0054063557, -0.016186944, -0.026871428, -0.037396627, -0.04770068, -0.057723559, -0.067407559, -0.076697768, -0.08554251, -0.093893752, -0.10170748, -0.10894403, -0.1155684, -0.12155045, -0.12686516, -0.13149274, -0.13541876, -0.13863416, -0.1411353, -0.14292389, -0.14400686, -0.14439626, -0.14410905, -0.14316686, -0.1415957, -0.13942569, -0.13669069, -0.1334279, -0.12967752, -0.12548227, -0.12088699, -0.11593818, -0.11068356, -0.10517159, -0.099451049, -0.093570567, -0.087578202, -0.081521022, -0.075444716, -0.069393216, -0.063408356, -0.057529559, -0.051793558, -0.046234148, -0.040881982, -0.035764398, -0.030905288, -0.026325013, -0.022040342, -0.018064449], [0.064817553, 0.072567801, 0.080292992, 0.087918235, 0.095367076, 0.10256232, 0.10942687, 0.1158846, 0.12186124, 0.12728523, 0.13208858, 0.13620771, 0.13958427, 0.14216587, 0.14390678, 0.14476863, 0.1447209, 0.14374148, 0.14181704, 0.13894336, 0.13512554, 0.13037812, 0.1247251, 0.11819984, 0.11084487, 0.10271159, 0.093859853, 0.084357497, 0.074279719, 0.063708406, 0.052731374, 0.041441525, 0.029935953, 0.018314987, 0.0066811877, -0.0048616765, -0.016209689, -0.027259848, -0.037911124, -0.048065512, -0.05762905, -0.066512804, -0.0746338, -0.081915903, -0.088290621, -0.09369783, -0.098086416, -0.10141482, -0.10365146, -0.10477512, -0.10477512, -0.10365146, -0.10141482, -0.098086416, -0.09369783, -0.088290621, -0.081915903, -0.0746338, -0.066512804, -0.05762905, -0.048065512, -0.037911124, -0.027259848, -0.016209689, -0.0048616765, 0.0066811877, 0.018314987, 0.029935953, 0.041441525, 0.052731374, 0.063708406, 0.074279719, 0.084357497, 0.093859853, 0.10271159, 0.11084487, 0.11819984, 0.1247251, 0.13037812, 0.13512554, 0.13894336, 0.14181704, 0.14374148, 0.1447209, 0.14476863, 0.14390678, 0.14216587, 0.13958427, 0.13620771, 0.13208858, 0.12728523, 0.12186124, 0.1158846, 0.10942687, 0.10256232, 0.095367076, 0.087918235, 0.080292992, 0.072567801, 0.064817553], [0.14985551, 0.15512305, 0.15931467, 0.16236806, 0.16423291, 0.16487165, 0.16426009, 0.1623879, 0.1592589, 0.15489114, 0.14931693, 0.14258255, 0.13474785, 0.1258857, 0.11608124, 0.10543095, 0.094041635, 0.082029213, 0.069517411, 0.056636348, 0.043521028, 0.030309756, 0.017142511, 0.0041592774, -0.0085016282, -0.020705223, -0.032321494, -0.043226982, -0.053306291, -0.062453515, -0.070573544, -0.077583253, -0.083412547, -0.088005244, -0.091319802, -0.093329861, -0.094024602, -0.093408915, -0.091503383, -0.08834406, -0.08398207, -0.078483012, -0.071926192, -0.064403681, -0.056019215, -0.046886954, -0.037130106, -0.026879442, -0.016271713, -0.005448, 0.005448, 0.016271713, 0.026879442, 0.037130106, 0.046886954, 0.056019215, 0.064403681, 0.071926192, 0.078483012, 0.08398207, 0.08834406, 0.091503383, 0.093408915, 0.094024602, 0.093329861, 0.091319802, 0.088005244, 0.083412547, 0.077583253, 0.070573544, 0.062453515, 0.053306291, 0.043226982, 0.032321494, 0.020705223, 0.0085016282, -0.0041592774, -0.017142511, -0.030309756, -0.043521028, -0.056636348, -0.069517411, -0.082029213, -0.094041635, -0.10543095, -0.11608124, -0.1258857, -0.13474785, -0.14258255, -0.14931693, -0.15489114, -0.1592589, -0.1623879, -0.16426009, -0.16487165, -0.16423291, -0.16236806, -0.15931467, -0.15512305, -0.14985551]], [0.999943140, 0.997571533, 0.959465463, 0.721862496]), # noqa: E501 +} + + +class TestDPSS: + + def test_basic(self): + # Test against hardcoded data + for k, v in dpss_data.items(): + win, ratios = windows.dpss(*k, return_ratios=True) + assert_allclose(win, v[0], atol=1e-7, err_msg=k) + assert_allclose(ratios, v[1], rtol=1e-5, atol=1e-7, err_msg=k) + + def test_unity(self): + # Test unity value handling (gh-2221) + for M in range(1, 21): + # corrected w/approximation (default) + win = windows.dpss(M, M / 2.1) + expected = M % 2 # one for odd, none for even + assert_equal(np.isclose(win, 1.).sum(), expected, + err_msg=f'{win}') + # corrected w/subsample delay (slower) + win_sub = windows.dpss(M, M / 2.1, norm='subsample') + if M > 2: + # @M=2 the subsample doesn't do anything + assert_equal(np.isclose(win_sub, 1.).sum(), expected, + err_msg=f'{win_sub}') + assert_allclose(win, win_sub, rtol=0.03) # within 3% + # not the same, l2-norm + win_2 = windows.dpss(M, M / 2.1, norm=2) + expected = 1 if M == 1 else 0 + assert_equal(np.isclose(win_2, 1.).sum(), expected, + err_msg=f'{win_2}') + + def test_extremes(self): + # Test extremes of alpha + lam = windows.dpss(31, 6, 4, return_ratios=True)[1] + assert_array_almost_equal(lam, 1.) + lam = windows.dpss(31, 7, 4, return_ratios=True)[1] + assert_array_almost_equal(lam, 1.) + lam = windows.dpss(31, 8, 4, return_ratios=True)[1] + assert_array_almost_equal(lam, 1.) + + def test_degenerate(self): + # Test failures + assert_raises(ValueError, windows.dpss, 4, 1.5, -1) # Bad Kmax + assert_raises(ValueError, windows.dpss, 4, 1.5, -5) + assert_raises(TypeError, windows.dpss, 4, 1.5, 1.1) + assert_raises(ValueError, windows.dpss, 3, 1.5, 3) # NW must be < N/2. + assert_raises(ValueError, windows.dpss, 3, -1, 3) # NW must be pos + assert_raises(ValueError, windows.dpss, 3, 0, 3) + assert_raises(ValueError, windows.dpss, -1, 1, 3) # negative M + + +class TestLanczos: + + def test_basic(self): + # Analytical results: + # sinc(x) = sinc(-x) + # sinc(pi) = 0, sinc(0) = 1 + # Hand computation on WolframAlpha: + # sinc(2 pi / 3) = 0.413496672 + # sinc(pi / 3) = 0.826993343 + # sinc(3 pi / 5) = 0.504551152 + # sinc(pi / 5) = 0.935489284 + assert_allclose(windows.lanczos(6, sym=False), + [0., 0.413496672, + 0.826993343, 1., 0.826993343, + 0.413496672], + atol=1e-9) + assert_allclose(windows.lanczos(6), + [0., 0.504551152, + 0.935489284, 0.935489284, + 0.504551152, 0.], + atol=1e-9) + assert_allclose(windows.lanczos(7, sym=True), + [0., 0.413496672, + 0.826993343, 1., 0.826993343, + 0.413496672, 0.], + atol=1e-9) + + def test_array_size(self): + for n in [0, 10, 11]: + assert_equal(len(windows.lanczos(n, sym=False)), n) + assert_equal(len(windows.lanczos(n, sym=True)), n) + + +class TestGetWindow: + + def test_boxcar(self): + w = windows.get_window('boxcar', 12) + assert_array_equal(w, np.ones_like(w)) + + # window is a tuple of len 1 + w = windows.get_window(('boxcar',), 16) + assert_array_equal(w, np.ones_like(w)) + + def test_cheb_odd(self): + with suppress_warnings() as sup: + sup.filter(UserWarning, "This window is not suitable") + w = windows.get_window(('chebwin', -40), 53, fftbins=False) + assert_array_almost_equal(w, cheb_odd_true, decimal=4) + + def test_cheb_even(self): + with suppress_warnings() as sup: + sup.filter(UserWarning, "This window is not suitable") + w = windows.get_window(('chebwin', 40), 54, fftbins=False) + assert_array_almost_equal(w, cheb_even_true, decimal=4) + + def test_dpss(self): + win1 = windows.get_window(('dpss', 3), 64, fftbins=False) + win2 = windows.dpss(64, 3) + assert_array_almost_equal(win1, win2, decimal=4) + + def test_kaiser_float(self): + win1 = windows.get_window(7.2, 64) + win2 = windows.kaiser(64, 7.2, False) + assert_allclose(win1, win2) + + def test_invalid_inputs(self): + # Window is not a float, tuple, or string + assert_raises(ValueError, windows.get_window, set('hann'), 8) + + # Unknown window type error + assert_raises(ValueError, windows.get_window, 'broken', 4) + + def test_array_as_window(self): + # GitHub issue 3603 + osfactor = 128 + sig = np.arange(128) + + win = windows.get_window(('kaiser', 8.0), osfactor // 2) + with assert_raises(ValueError, match='must have the same length'): + resample(sig, len(sig) * osfactor, window=win) + + def test_general_cosine(self): + assert_allclose(get_window(('general_cosine', [0.5, 0.3, 0.2]), 4), + [0.4, 0.3, 1, 0.3]) + assert_allclose(get_window(('general_cosine', [0.5, 0.3, 0.2]), 4, + fftbins=False), + [0.4, 0.55, 0.55, 0.4]) + + def test_general_hamming(self): + assert_allclose(get_window(('general_hamming', 0.7), 5), + [0.4, 0.6072949, 0.9427051, 0.9427051, 0.6072949]) + assert_allclose(get_window(('general_hamming', 0.7), 5, fftbins=False), + [0.4, 0.7, 1.0, 0.7, 0.4]) + + def test_lanczos(self): + assert_allclose(get_window('lanczos', 6), + [0., 0.413496672, 0.826993343, 1., 0.826993343, + 0.413496672], atol=1e-9) + assert_allclose(get_window('lanczos', 6, fftbins=False), + [0., 0.504551152, 0.935489284, 0.935489284, + 0.504551152, 0.], atol=1e-9) + assert_allclose(get_window('lanczos', 6), get_window('sinc', 6)) + + +def test_windowfunc_basics(): + for window_name, params in window_funcs: + window = getattr(windows, window_name) + with suppress_warnings() as sup: + sup.filter(UserWarning, "This window is not suitable") + # Check symmetry for odd and even lengths + w1 = window(8, *params, sym=True) + w2 = window(7, *params, sym=False) + assert_array_almost_equal(w1[:-1], w2) + + w1 = window(9, *params, sym=True) + w2 = window(8, *params, sym=False) + assert_array_almost_equal(w1[:-1], w2) + + # Check that functions run and output lengths are correct + assert_equal(len(window(6, *params, sym=True)), 6) + assert_equal(len(window(6, *params, sym=False)), 6) + assert_equal(len(window(7, *params, sym=True)), 7) + assert_equal(len(window(7, *params, sym=False)), 7) + + # Check invalid lengths + assert_raises(ValueError, window, 5.5, *params) + assert_raises(ValueError, window, -7, *params) + + # Check degenerate cases + assert_array_equal(window(0, *params, sym=True), []) + assert_array_equal(window(0, *params, sym=False), []) + assert_array_equal(window(1, *params, sym=True), [1]) + assert_array_equal(window(1, *params, sym=False), [1]) + + # Check dtype + assert_(window(0, *params, sym=True).dtype == 'float') + assert_(window(0, *params, sym=False).dtype == 'float') + assert_(window(1, *params, sym=True).dtype == 'float') + assert_(window(1, *params, sym=False).dtype == 'float') + assert_(window(6, *params, sym=True).dtype == 'float') + assert_(window(6, *params, sym=False).dtype == 'float') + + # Check normalization + assert_array_less(window(10, *params, sym=True), 1.01) + assert_array_less(window(10, *params, sym=False), 1.01) + assert_array_less(window(9, *params, sym=True), 1.01) + assert_array_less(window(9, *params, sym=False), 1.01) + + # Check that DFT-even spectrum is purely real for odd and even + assert_allclose(fft(window(10, *params, sym=False)).imag, + 0, atol=1e-14) + assert_allclose(fft(window(11, *params, sym=False)).imag, + 0, atol=1e-14) + + +def test_needs_params(): + for winstr in ['kaiser', 'ksr', 'kaiser_bessel_derived', 'kbd', + 'gaussian', 'gauss', 'gss', + 'general gaussian', 'general_gaussian', + 'general gauss', 'general_gauss', 'ggs', + 'dss', 'dpss', 'general cosine', 'general_cosine', + 'chebwin', 'cheb', 'general hamming', 'general_hamming', + ]: + assert_raises(ValueError, get_window, winstr, 7) + + +def test_not_needs_params(): + for winstr in ['barthann', + 'bartlett', + 'blackman', + 'blackmanharris', + 'bohman', + 'boxcar', + 'cosine', + 'flattop', + 'hamming', + 'nuttall', + 'parzen', + 'taylor', + 'exponential', + 'poisson', + 'tukey', + 'tuk', + 'triangle', + 'lanczos', + 'sinc', + ]: + win = get_window(winstr, 7) + assert_equal(len(win), 7) + + +def test_symmetric(): + + for win in [windows.lanczos]: + # Even sampling points + w = win(4096) + error = np.max(np.abs(w-np.flip(w))) + assert_equal(error, 0.0) + + # Odd sampling points + w = win(4097) + error = np.max(np.abs(w-np.flip(w))) + assert_equal(error, 0.0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/waveforms.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/waveforms.py new file mode 100644 index 0000000000000000000000000000000000000000..30e71348d04276a66470a4053d97cefc60f7136e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/waveforms.py @@ -0,0 +1,20 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.signal` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'sawtooth', 'square', 'gausspulse', 'chirp', 'sweep_poly', + 'unit_impulse', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="signal", module="waveforms", + private_modules=["_waveforms"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/wavelets.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/wavelets.py new file mode 100644 index 0000000000000000000000000000000000000000..fc897a2483536df7e995faaa29af621e25fe38c7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/wavelets.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.signal` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="signal", module="wavelets", + private_modules=["_wavelets"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/windows/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/windows/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..967a7c758f69c1c8002d886d78832904c402d2b3 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/windows/__init__.py @@ -0,0 +1,52 @@ +""" +Window functions (:mod:`scipy.signal.windows`) +============================================== + +The suite of window functions for filtering and spectral estimation. + +.. currentmodule:: scipy.signal.windows + +.. autosummary:: + :toctree: generated/ + + get_window -- Return a window of a given length and type. + + barthann -- Bartlett-Hann window + bartlett -- Bartlett window + blackman -- Blackman window + blackmanharris -- Minimum 4-term Blackman-Harris window + bohman -- Bohman window + boxcar -- Boxcar window + chebwin -- Dolph-Chebyshev window + cosine -- Cosine window + dpss -- Discrete prolate spheroidal sequences + exponential -- Exponential window + flattop -- Flat top window + gaussian -- Gaussian window + general_cosine -- Generalized Cosine window + general_gaussian -- Generalized Gaussian window + general_hamming -- Generalized Hamming window + hamming -- Hamming window + hann -- Hann window + kaiser -- Kaiser window + kaiser_bessel_derived -- Kaiser-Bessel derived window + lanczos -- Lanczos window also known as a sinc window + nuttall -- Nuttall's minimum 4-term Blackman-Harris window + parzen -- Parzen window + taylor -- Taylor window + triang -- Triangular window + tukey -- Tukey window + +""" + +from ._windows import * + +# Deprecated namespaces, to be removed in v2.0.0 +from . import windows + +__all__ = ['boxcar', 'triang', 'parzen', 'bohman', 'blackman', 'nuttall', + 'blackmanharris', 'flattop', 'bartlett', 'barthann', + 'hamming', 'kaiser', 'kaiser_bessel_derived', 'gaussian', + 'general_gaussian', 'general_cosine', 'general_hamming', + 'chebwin', 'cosine', 'hann', 'exponential', 'tukey', 'taylor', + 'get_window', 'dpss', 'lanczos'] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/windows/_windows.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/windows/_windows.py new file mode 100644 index 0000000000000000000000000000000000000000..e89c1aee6aba661f11a86cb2213904964d870782 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/windows/_windows.py @@ -0,0 +1,2374 @@ +"""The suite of window functions.""" + +import operator +import warnings + +import numpy as np +from scipy import linalg, special, fft as sp_fft + +__all__ = ['boxcar', 'triang', 'parzen', 'bohman', 'blackman', 'nuttall', + 'blackmanharris', 'flattop', 'bartlett', 'barthann', + 'hamming', 'kaiser', 'kaiser_bessel_derived', 'gaussian', + 'general_cosine', 'general_gaussian', 'general_hamming', + 'chebwin', 'cosine', 'hann', 'exponential', 'tukey', 'taylor', + 'dpss', 'get_window', 'lanczos'] + + +def _len_guards(M): + """Handle small or incorrect window lengths""" + if int(M) != M or M < 0: + raise ValueError('Window length M must be a non-negative integer') + return M <= 1 + + +def _extend(M, sym): + """Extend window by 1 sample if needed for DFT-even symmetry""" + if not sym: + return M + 1, True + else: + return M, False + + +def _truncate(w, needed): + """Truncate window by 1 sample if needed for DFT-even symmetry""" + if needed: + return w[:-1] + else: + return w + + +def general_cosine(M, a, sym=True): + r""" + Generic weighted sum of cosine terms window + + Parameters + ---------- + M : int + Number of points in the output window + a : array_like + Sequence of weighting coefficients. This uses the convention of being + centered on the origin, so these will typically all be positive + numbers, not alternating sign. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The array of window values. + + References + ---------- + .. [1] A. Nuttall, "Some windows with very good sidelobe behavior," IEEE + Transactions on Acoustics, Speech, and Signal Processing, vol. 29, + no. 1, pp. 84-91, Feb 1981. :doi:`10.1109/TASSP.1981.1163506`. + .. [2] Heinzel G. et al., "Spectrum and spectral density estimation by the + Discrete Fourier transform (DFT), including a comprehensive list of + window functions and some new flat-top windows", February 15, 2002 + https://holometer.fnal.gov/GH_FFT.pdf + + Examples + -------- + Heinzel describes a flat-top window named "HFT90D" with formula: [2]_ + + .. math:: w_j = 1 - 1.942604 \cos(z) + 1.340318 \cos(2z) + - 0.440811 \cos(3z) + 0.043097 \cos(4z) + + where + + .. math:: z = \frac{2 \pi j}{N}, j = 0...N - 1 + + Since this uses the convention of starting at the origin, to reproduce the + window, we need to convert every other coefficient to a positive number: + + >>> HFT90D = [1, 1.942604, 1.340318, 0.440811, 0.043097] + + The paper states that the highest sidelobe is at -90.2 dB. Reproduce + Figure 42 by plotting the window and its frequency response, and confirm + the sidelobe level in red: + + >>> import numpy as np + >>> from scipy.signal.windows import general_cosine + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = general_cosine(1000, HFT90D, sym=False) + >>> plt.plot(window) + >>> plt.title("HFT90D window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 10000) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = np.abs(fftshift(A / abs(A).max())) + >>> response = 20 * np.log10(np.maximum(response, 1e-10)) + >>> plt.plot(freq, response) + >>> plt.axis([-50/1000, 50/1000, -140, 0]) + >>> plt.title("Frequency response of the HFT90D window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + >>> plt.axhline(-90.2, color='red') + >>> plt.show() + """ + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + fac = np.linspace(-np.pi, np.pi, M) + w = np.zeros(M) + for k in range(len(a)): + w += a[k] * np.cos(k * fac) + + return _truncate(w, needs_trunc) + + +def boxcar(M, sym=True): + """Return a boxcar or rectangular window. + + Also known as a rectangular window or Dirichlet window, this is equivalent + to no window at all. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + Whether the window is symmetric. (Has no effect for boxcar.) + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1. + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.boxcar(51) + >>> plt.plot(window) + >>> plt.title("Boxcar window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the boxcar window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + w = np.ones(M, float) + + return _truncate(w, needs_trunc) + + +def triang(M, sym=True): + """Return a triangular window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + See Also + -------- + bartlett : A triangular window that touches zero + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.triang(51) + >>> plt.plot(window) + >>> plt.title("Triangular window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = np.abs(fftshift(A / abs(A).max())) + >>> response = 20 * np.log10(np.maximum(response, 1e-10)) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the triangular window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + n = np.arange(1, (M + 1) // 2 + 1) + if M % 2 == 0: + w = (2 * n - 1.0) / M + w = np.r_[w, w[::-1]] + else: + w = 2 * n / (M + 1.0) + w = np.r_[w, w[-2::-1]] + + return _truncate(w, needs_trunc) + + +def parzen(M, sym=True): + """Return a Parzen window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + References + ---------- + .. [1] E. Parzen, "Mathematical Considerations in the Estimation of + Spectra", Technometrics, Vol. 3, No. 2 (May, 1961), pp. 167-190 + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.parzen(51) + >>> plt.plot(window) + >>> plt.title("Parzen window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Parzen window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + n = np.arange(-(M - 1) / 2.0, (M - 1) / 2.0 + 0.5, 1.0) + na = np.extract(n < -(M - 1) / 4.0, n) + nb = np.extract(abs(n) <= (M - 1) / 4.0, n) + wa = 2 * (1 - np.abs(na) / (M / 2.0)) ** 3.0 + wb = (1 - 6 * (np.abs(nb) / (M / 2.0)) ** 2.0 + + 6 * (np.abs(nb) / (M / 2.0)) ** 3.0) + w = np.r_[wa, wb, wa[::-1]] + + return _truncate(w, needs_trunc) + + +def bohman(M, sym=True): + """Return a Bohman window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.bohman(51) + >>> plt.plot(window) + >>> plt.title("Bohman window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2047) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Bohman window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + fac = np.abs(np.linspace(-1, 1, M)[1:-1]) + w = (1 - fac) * np.cos(np.pi * fac) + 1.0 / np.pi * np.sin(np.pi * fac) + w = np.r_[0, w, 0] + + return _truncate(w, needs_trunc) + + +def blackman(M, sym=True): + r""" + Return a Blackman window. + + The Blackman window is a taper formed by using the first three terms of + a summation of cosines. It was designed to have close to the minimal + leakage possible. It is close to optimal, only slightly worse than a + Kaiser window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Notes + ----- + The Blackman window is defined as + + .. math:: w(n) = 0.42 - 0.5 \cos(2\pi n/M) + 0.08 \cos(4\pi n/M) + + The "exact Blackman" window was designed to null out the third and fourth + sidelobes, but has discontinuities at the boundaries, resulting in a + 6 dB/oct fall-off. This window is an approximation of the "exact" window, + which does not null the sidelobes as well, but is smooth at the edges, + improving the fall-off rate to 18 dB/oct. [3]_ + + Most references to the Blackman window come from the signal processing + literature, where it is used as one of many windowing functions for + smoothing values. It is also known as an apodization (which means + "removing the foot", i.e. smoothing discontinuities at the beginning + and end of the sampled signal) or tapering function. It is known as a + "near optimal" tapering function, almost as good (by some measures) + as the Kaiser window. + + References + ---------- + .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power + spectra, Dover Publications, New York. + .. [2] Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing. + Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471. + .. [3] Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic + Analysis with the Discrete Fourier Transform". Proceedings of the + IEEE 66 (1): 51-83. :doi:`10.1109/PROC.1978.10837`. + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.blackman(51) + >>> plt.plot(window) + >>> plt.title("Blackman window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = np.abs(fftshift(A / abs(A).max())) + >>> response = 20 * np.log10(np.maximum(response, 1e-10)) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Blackman window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + # Docstring adapted from NumPy's blackman function + return general_cosine(M, [0.42, 0.50, 0.08], sym) + + +def nuttall(M, sym=True): + """Return a minimum 4-term Blackman-Harris window according to Nuttall. + + This variation is called "Nuttall4c" by Heinzel. [2]_ + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + References + ---------- + .. [1] A. Nuttall, "Some windows with very good sidelobe behavior," IEEE + Transactions on Acoustics, Speech, and Signal Processing, vol. 29, + no. 1, pp. 84-91, Feb 1981. :doi:`10.1109/TASSP.1981.1163506`. + .. [2] Heinzel G. et al., "Spectrum and spectral density estimation by the + Discrete Fourier transform (DFT), including a comprehensive list of + window functions and some new flat-top windows", February 15, 2002 + https://holometer.fnal.gov/GH_FFT.pdf + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.nuttall(51) + >>> plt.plot(window) + >>> plt.title("Nuttall window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Nuttall window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + return general_cosine(M, [0.3635819, 0.4891775, 0.1365995, 0.0106411], sym) + + +def blackmanharris(M, sym=True): + """Return a minimum 4-term Blackman-Harris window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.blackmanharris(51) + >>> plt.plot(window) + >>> plt.title("Blackman-Harris window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Blackman-Harris window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + return general_cosine(M, [0.35875, 0.48829, 0.14128, 0.01168], sym) + + +def flattop(M, sym=True): + """Return a flat top window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Notes + ----- + Flat top windows are used for taking accurate measurements of signal + amplitude in the frequency domain, with minimal scalloping error from the + center of a frequency bin to its edges, compared to others. This is a + 5th-order cosine window, with the 5 terms optimized to make the main lobe + maximally flat. [1]_ + + References + ---------- + .. [1] D'Antona, Gabriele, and A. Ferrero, "Digital Signal Processing for + Measurement Systems", Springer Media, 2006, p. 70 + :doi:`10.1007/0-387-28666-7`. + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.flattop(51) + >>> plt.plot(window) + >>> plt.title("Flat top window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the flat top window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + a = [0.21557895, 0.41663158, 0.277263158, 0.083578947, 0.006947368] + return general_cosine(M, a, sym) + + +def bartlett(M, sym=True): + r""" + Return a Bartlett window. + + The Bartlett window is very similar to a triangular window, except + that the end points are at zero. It is often used in signal + processing for tapering a signal, without generating too much + ripple in the frequency domain. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The triangular window, with the first and last samples equal to zero + and the maximum value normalized to 1 (though the value 1 does not + appear if `M` is even and `sym` is True). + + See Also + -------- + triang : A triangular window that does not touch zero at the ends + + Notes + ----- + The Bartlett window is defined as + + .. math:: w(n) = \frac{2}{M-1} \left( + \frac{M-1}{2} - \left|n - \frac{M-1}{2}\right| + \right) + + Most references to the Bartlett window come from the signal + processing literature, where it is used as one of many windowing + functions for smoothing values. Note that convolution with this + window produces linear interpolation. It is also known as an + apodization (which means"removing the foot", i.e. smoothing + discontinuities at the beginning and end of the sampled signal) or + tapering function. The Fourier transform of the Bartlett is the product + of two sinc functions. + Note the excellent discussion in Kanasewich. [2]_ + + References + ---------- + .. [1] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra", + Biometrika 37, 1-16, 1950. + .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", + The University of Alberta Press, 1975, pp. 109-110. + .. [3] A.V. Oppenheim and R.W. Schafer, "Discrete-Time Signal + Processing", Prentice-Hall, 1999, pp. 468-471. + .. [4] Wikipedia, "Window function", + https://en.wikipedia.org/wiki/Window_function + .. [5] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, + "Numerical Recipes", Cambridge University Press, 1986, page 429. + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.bartlett(51) + >>> plt.plot(window) + >>> plt.title("Bartlett window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Bartlett window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + # Docstring adapted from NumPy's bartlett function + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + n = np.arange(0, M) + w = np.where(np.less_equal(n, (M - 1) / 2.0), + 2.0 * n / (M - 1), 2.0 - 2.0 * n / (M - 1)) + + return _truncate(w, needs_trunc) + + +def hann(M, sym=True): + r""" + Return a Hann window. + + The Hann window is a taper formed by using a raised cosine or sine-squared + with ends that touch zero. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Notes + ----- + The Hann window is defined as + + .. math:: w(n) = 0.5 - 0.5 \cos\left(\frac{2\pi{n}}{M-1}\right) + \qquad 0 \leq n \leq M-1 + + The window was named for Julius von Hann, an Austrian meteorologist. It is + also known as the Cosine Bell. It is sometimes erroneously referred to as + the "Hanning" window, from the use of "hann" as a verb in the original + paper and confusion with the very similar Hamming window. + + Most references to the Hann window come from the signal processing + literature, where it is used as one of many windowing functions for + smoothing values. It is also known as an apodization (which means + "removing the foot", i.e. smoothing discontinuities at the beginning + and end of the sampled signal) or tapering function. + + References + ---------- + .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power + spectra, Dover Publications, New York. + .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", + The University of Alberta Press, 1975, pp. 106-108. + .. [3] Wikipedia, "Window function", + https://en.wikipedia.org/wiki/Window_function + .. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, + "Numerical Recipes", Cambridge University Press, 1986, page 425. + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.hann(51) + >>> plt.plot(window) + >>> plt.title("Hann window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = np.abs(fftshift(A / abs(A).max())) + >>> response = 20 * np.log10(np.maximum(response, 1e-10)) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Hann window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + # Docstring adapted from NumPy's hanning function + return general_hamming(M, 0.5, sym) + + +def tukey(M, alpha=0.5, sym=True): + r"""Return a Tukey window, also known as a tapered cosine window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + alpha : float, optional + Shape parameter of the Tukey window, representing the fraction of the + window inside the cosine tapered region. + If zero, the Tukey window is equivalent to a rectangular window. + If one, the Tukey window is equivalent to a Hann window. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + References + ---------- + .. [1] Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic + Analysis with the Discrete Fourier Transform". Proceedings of the + IEEE 66 (1): 51-83. :doi:`10.1109/PROC.1978.10837` + .. [2] Wikipedia, "Window function", + https://en.wikipedia.org/wiki/Window_function#Tukey_window + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.tukey(51) + >>> plt.plot(window) + >>> plt.title("Tukey window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + >>> plt.ylim([0, 1.1]) + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Tukey window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + if _len_guards(M): + return np.ones(M) + + if alpha <= 0: + return np.ones(M, 'd') + elif alpha >= 1.0: + return hann(M, sym=sym) + + M, needs_trunc = _extend(M, sym) + + n = np.arange(0, M) + width = int(np.floor(alpha*(M-1)/2.0)) + n1 = n[0:width+1] + n2 = n[width+1:M-width-1] + n3 = n[M-width-1:] + + w1 = 0.5 * (1 + np.cos(np.pi * (-1 + 2.0*n1/alpha/(M-1)))) + w2 = np.ones(n2.shape) + w3 = 0.5 * (1 + np.cos(np.pi * (-2.0/alpha + 1 + 2.0*n3/alpha/(M-1)))) + + w = np.concatenate((w1, w2, w3)) + + return _truncate(w, needs_trunc) + + +def barthann(M, sym=True): + """Return a modified Bartlett-Hann window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.barthann(51) + >>> plt.plot(window) + >>> plt.title("Bartlett-Hann window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Bartlett-Hann window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + n = np.arange(0, M) + fac = np.abs(n / (M - 1.0) - 0.5) + w = 0.62 - 0.48 * fac + 0.38 * np.cos(2 * np.pi * fac) + + return _truncate(w, needs_trunc) + + +def general_hamming(M, alpha, sym=True): + r"""Return a generalized Hamming window. + + The generalized Hamming window is constructed by multiplying a rectangular + window by one period of a cosine function [1]_. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + alpha : float + The window coefficient, :math:`\alpha` + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + See Also + -------- + hamming, hann + + Notes + ----- + The generalized Hamming window is defined as + + .. math:: w(n) = \alpha - \left(1 - \alpha\right) + \cos\left(\frac{2\pi{n}}{M-1}\right) \qquad 0 \leq n \leq M-1 + + Both the common Hamming window and Hann window are special cases of the + generalized Hamming window with :math:`\alpha` = 0.54 and :math:`\alpha` = + 0.5, respectively [2]_. + + References + ---------- + .. [1] DSPRelated, "Generalized Hamming Window Family", + https://www.dsprelated.com/freebooks/sasp/Generalized_Hamming_Window_Family.html + .. [2] Wikipedia, "Window function", + https://en.wikipedia.org/wiki/Window_function + .. [3] Riccardo Piantanida ESA, "Sentinel-1 Level 1 Detailed Algorithm + Definition", + https://sentinel.esa.int/documents/247904/1877131/Sentinel-1-Level-1-Detailed-Algorithm-Definition + .. [4] Matthieu Bourbigot ESA, "Sentinel-1 Product Definition", + https://sentinel.esa.int/documents/247904/1877131/Sentinel-1-Product-Definition + + Examples + -------- + The Sentinel-1A/B Instrument Processing Facility uses generalized Hamming + windows in the processing of spaceborne Synthetic Aperture Radar (SAR) + data [3]_. The facility uses various values for the :math:`\alpha` + parameter based on operating mode of the SAR instrument. Some common + :math:`\alpha` values include 0.75, 0.7 and 0.52 [4]_. As an example, we + plot these different windows. + + >>> import numpy as np + >>> from scipy.signal.windows import general_hamming + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> fig1, spatial_plot = plt.subplots() + >>> spatial_plot.set_title("Generalized Hamming Windows") + >>> spatial_plot.set_ylabel("Amplitude") + >>> spatial_plot.set_xlabel("Sample") + + >>> fig2, freq_plot = plt.subplots() + >>> freq_plot.set_title("Frequency Responses") + >>> freq_plot.set_ylabel("Normalized magnitude [dB]") + >>> freq_plot.set_xlabel("Normalized frequency [cycles per sample]") + + >>> for alpha in [0.75, 0.7, 0.52]: + ... window = general_hamming(41, alpha) + ... spatial_plot.plot(window, label="{:.2f}".format(alpha)) + ... A = fft(window, 2048) / (len(window)/2.0) + ... freq = np.linspace(-0.5, 0.5, len(A)) + ... response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + ... freq_plot.plot(freq, response, label="{:.2f}".format(alpha)) + >>> freq_plot.legend(loc="upper right") + >>> spatial_plot.legend(loc="upper right") + + """ + return general_cosine(M, [alpha, 1. - alpha], sym) + + +def hamming(M, sym=True): + r"""Return a Hamming window. + + The Hamming window is a taper formed by using a raised cosine with + non-zero endpoints, optimized to minimize the nearest side lobe. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Notes + ----- + The Hamming window is defined as + + .. math:: w(n) = 0.54 - 0.46 \cos\left(\frac{2\pi{n}}{M-1}\right) + \qquad 0 \leq n \leq M-1 + + The Hamming was named for R. W. Hamming, an associate of J. W. Tukey and + is described in Blackman and Tukey. It was recommended for smoothing the + truncated autocovariance function in the time domain. + Most references to the Hamming window come from the signal processing + literature, where it is used as one of many windowing functions for + smoothing values. It is also known as an apodization (which means + "removing the foot", i.e. smoothing discontinuities at the beginning + and end of the sampled signal) or tapering function. + + References + ---------- + .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power + spectra, Dover Publications, New York. + .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The + University of Alberta Press, 1975, pp. 109-110. + .. [3] Wikipedia, "Window function", + https://en.wikipedia.org/wiki/Window_function + .. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, + "Numerical Recipes", Cambridge University Press, 1986, page 425. + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.hamming(51) + >>> plt.plot(window) + >>> plt.title("Hamming window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Hamming window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + # Docstring adapted from NumPy's hamming function + return general_hamming(M, 0.54, sym) + + +def kaiser(M, beta, sym=True): + r"""Return a Kaiser window. + + The Kaiser window is a taper formed by using a Bessel function. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + beta : float + Shape parameter, determines trade-off between main-lobe width and + side lobe level. As beta gets large, the window narrows. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Notes + ----- + The Kaiser window is defined as + + .. math:: w(n) = I_0\left( \beta \sqrt{1-\frac{4n^2}{(M-1)^2}} + \right)/I_0(\beta) + + with + + .. math:: \quad -\frac{M-1}{2} \leq n \leq \frac{M-1}{2}, + + where :math:`I_0` is the modified zeroth-order Bessel function. + + The Kaiser was named for Jim Kaiser, who discovered a simple approximation + to the DPSS window based on Bessel functions. + The Kaiser window is a very good approximation to the discrete prolate + spheroidal sequence, or Slepian window, which is the transform which + maximizes the energy in the main lobe of the window relative to total + energy. + + The Kaiser can approximate other windows by varying the beta parameter. + (Some literature uses alpha = beta/pi.) [4]_ + + ==== ======================= + beta Window shape + ==== ======================= + 0 Rectangular + 5 Similar to a Hamming + 6 Similar to a Hann + 8.6 Similar to a Blackman + ==== ======================= + + A beta value of 14 is probably a good starting point. Note that as beta + gets large, the window narrows, and so the number of samples needs to be + large enough to sample the increasingly narrow spike, otherwise NaNs will + be returned. + + Most references to the Kaiser window come from the signal processing + literature, where it is used as one of many windowing functions for + smoothing values. It is also known as an apodization (which means + "removing the foot", i.e. smoothing discontinuities at the beginning + and end of the sampled signal) or tapering function. + + References + ---------- + .. [1] J. F. Kaiser, "Digital Filters" - Ch 7 in "Systems analysis by + digital computer", Editors: F.F. Kuo and J.F. Kaiser, p 218-285. + John Wiley and Sons, New York, (1966). + .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The + University of Alberta Press, 1975, pp. 177-178. + .. [3] Wikipedia, "Window function", + https://en.wikipedia.org/wiki/Window_function + .. [4] F. J. Harris, "On the use of windows for harmonic analysis with the + discrete Fourier transform," Proceedings of the IEEE, vol. 66, + no. 1, pp. 51-83, Jan. 1978. :doi:`10.1109/PROC.1978.10837`. + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.kaiser(51, beta=14) + >>> plt.plot(window) + >>> plt.title(r"Kaiser window ($\beta$=14)") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title(r"Frequency response of the Kaiser window ($\beta$=14)") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + # Docstring adapted from NumPy's kaiser function + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + n = np.arange(0, M) + alpha = (M - 1) / 2.0 + w = (special.i0(beta * np.sqrt(1 - ((n - alpha) / alpha) ** 2.0)) / + special.i0(beta)) + + return _truncate(w, needs_trunc) + + +def kaiser_bessel_derived(M, beta, *, sym=True): + """Return a Kaiser-Bessel derived window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + Note that this window is only defined for an even + number of points. + beta : float + Kaiser window shape parameter. + sym : bool, optional + This parameter only exists to comply with the interface offered by + the other window functions and to be callable by `get_window`. + When True (default), generates a symmetric window, for use in filter + design. + + Returns + ------- + w : ndarray + The window, normalized to fulfil the Princen-Bradley condition. + + See Also + -------- + kaiser + + Notes + ----- + It is designed to be suitable for use with the modified discrete cosine + transform (MDCT) and is mainly used in audio signal processing and + audio coding. + + .. versionadded:: 1.9.0 + + References + ---------- + .. [1] Bosi, Marina, and Richard E. Goldberg. Introduction to Digital + Audio Coding and Standards. Dordrecht: Kluwer, 2003. + .. [2] Wikipedia, "Kaiser window", + https://en.wikipedia.org/wiki/Kaiser_window + + Examples + -------- + Plot the Kaiser-Bessel derived window based on the wikipedia + reference [2]_: + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> fig, ax = plt.subplots() + >>> N = 50 + >>> for alpha in [0.64, 2.55, 7.64, 31.83]: + ... ax.plot(signal.windows.kaiser_bessel_derived(2*N, np.pi*alpha), + ... label=f"{alpha=}") + >>> ax.grid(True) + >>> ax.set_title("Kaiser-Bessel derived window") + >>> ax.set_ylabel("Amplitude") + >>> ax.set_xlabel("Sample") + >>> ax.set_xticks([0, N, 2*N-1]) + >>> ax.set_xticklabels(["0", "N", "2N+1"]) # doctest: +SKIP + >>> ax.set_yticks([0.0, 0.2, 0.4, 0.6, 0.707, 0.8, 1.0]) + >>> fig.legend(loc="center") + >>> fig.tight_layout() + >>> fig.show() + """ + if not sym: + raise ValueError( + "Kaiser-Bessel Derived windows are only defined for symmetric " + "shapes" + ) + elif M < 1: + return np.array([]) + elif M % 2: + raise ValueError( + "Kaiser-Bessel Derived windows are only defined for even number " + "of points" + ) + + kaiser_window = kaiser(M // 2 + 1, beta) + csum = np.cumsum(kaiser_window) + half_window = np.sqrt(csum[:-1] / csum[-1]) + w = np.concatenate((half_window, half_window[::-1]), axis=0) + return w + + +def gaussian(M, std, sym=True): + r"""Return a Gaussian window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + std : float + The standard deviation, sigma. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Notes + ----- + The Gaussian window is defined as + + .. math:: w(n) = e^{ -\frac{1}{2}\left(\frac{n}{\sigma}\right)^2 } + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.gaussian(51, std=7) + >>> plt.plot(window) + >>> plt.title(r"Gaussian window ($\sigma$=7)") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title(r"Frequency response of the Gaussian window ($\sigma$=7)") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + n = np.arange(0, M) - (M - 1.0) / 2.0 + sig2 = 2 * std * std + w = np.exp(-n ** 2 / sig2) + + return _truncate(w, needs_trunc) + + +def general_gaussian(M, p, sig, sym=True): + r"""Return a window with a generalized Gaussian shape. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + p : float + Shape parameter. p = 1 is identical to `gaussian`, p = 0.5 is + the same shape as the Laplace distribution. + sig : float + The standard deviation, sigma. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Notes + ----- + The generalized Gaussian window is defined as + + .. math:: w(n) = e^{ -\frac{1}{2}\left|\frac{n}{\sigma}\right|^{2p} } + + the half-power point is at + + .. math:: (2 \log(2))^{1/(2 p)} \sigma + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.general_gaussian(51, p=1.5, sig=7) + >>> plt.plot(window) + >>> plt.title(r"Generalized Gaussian window (p=1.5, $\sigma$=7)") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title(r"Freq. resp. of the gen. Gaussian " + ... r"window (p=1.5, $\sigma$=7)") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + n = np.arange(0, M) - (M - 1.0) / 2.0 + w = np.exp(-0.5 * np.abs(n / sig) ** (2 * p)) + + return _truncate(w, needs_trunc) + + +# `chebwin` contributed by Kumar Appaiah. +def chebwin(M, at, sym=True): + r"""Return a Dolph-Chebyshev window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + at : float + Attenuation (in dB). + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value always normalized to 1 + + Notes + ----- + This window optimizes for the narrowest main lobe width for a given order + `M` and sidelobe equiripple attenuation `at`, using Chebyshev + polynomials. It was originally developed by Dolph to optimize the + directionality of radio antenna arrays. + + Unlike most windows, the Dolph-Chebyshev is defined in terms of its + frequency response: + + .. math:: W(k) = \frac + {\cos\{M \cos^{-1}[\beta \cos(\frac{\pi k}{M})]\}} + {\cosh[M \cosh^{-1}(\beta)]} + + where + + .. math:: \beta = \cosh \left [\frac{1}{M} + \cosh^{-1}(10^\frac{A}{20}) \right ] + + and 0 <= abs(k) <= M-1. A is the attenuation in decibels (`at`). + + The time domain window is then generated using the IFFT, so + power-of-two `M` are the fastest to generate, and prime number `M` are + the slowest. + + The equiripple condition in the frequency domain creates impulses in the + time domain, which appear at the ends of the window. + + References + ---------- + .. [1] C. Dolph, "A current distribution for broadside arrays which + optimizes the relationship between beam width and side-lobe level", + Proceedings of the IEEE, Vol. 34, Issue 6 + .. [2] Peter Lynch, "The Dolph-Chebyshev Window: A Simple Optimal Filter", + American Meteorological Society (April 1997) + http://mathsci.ucd.ie/~plynch/Publications/Dolph.pdf + .. [3] F. J. Harris, "On the use of windows for harmonic analysis with the + discrete Fourier transforms", Proceedings of the IEEE, Vol. 66, + No. 1, January 1978 + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.chebwin(51, at=100) + >>> plt.plot(window) + >>> plt.title("Dolph-Chebyshev window (100 dB)") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Dolph-Chebyshev window (100 dB)") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ + if np.abs(at) < 45: + warnings.warn("This window is not suitable for spectral analysis " + "for attenuation values lower than about 45dB because " + "the equivalent noise bandwidth of a Chebyshev window " + "does not grow monotonically with increasing sidelobe " + "attenuation when the attenuation is smaller than " + "about 45 dB.", + stacklevel=2) + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + # compute the parameter beta + order = M - 1.0 + beta = np.cosh(1.0 / order * np.arccosh(10 ** (np.abs(at) / 20.))) + k = np.r_[0:M] * 1.0 + x = beta * np.cos(np.pi * k / M) + # Find the window's DFT coefficients + # Use analytic definition of Chebyshev polynomial instead of expansion + # from scipy.special. Using the expansion in scipy.special leads to errors. + p = np.zeros(x.shape) + p[x > 1] = np.cosh(order * np.arccosh(x[x > 1])) + p[x < -1] = (2 * (M % 2) - 1) * np.cosh(order * np.arccosh(-x[x < -1])) + p[np.abs(x) <= 1] = np.cos(order * np.arccos(x[np.abs(x) <= 1])) + + # Appropriate IDFT and filling up + # depending on even/odd M + if M % 2: + w = np.real(sp_fft.fft(p)) + n = (M + 1) // 2 + w = w[:n] + w = np.concatenate((w[n - 1:0:-1], w)) + else: + p = p * np.exp(1.j * np.pi / M * np.r_[0:M]) + w = np.real(sp_fft.fft(p)) + n = M // 2 + 1 + w = np.concatenate((w[n - 1:0:-1], w[1:n])) + w = w / max(w) + + return _truncate(w, needs_trunc) + + +def cosine(M, sym=True): + """Return a window with a simple cosine shape. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Notes + ----- + + .. versionadded:: 0.13.0 + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.cosine(51) + >>> plt.plot(window) + >>> plt.title("Cosine window") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2047) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the cosine window") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + >>> plt.show() + + """ + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + w = np.sin(np.pi / M * (np.arange(0, M) + .5)) + + return _truncate(w, needs_trunc) + + +def exponential(M, center=None, tau=1., sym=True): + r"""Return an exponential (or Poisson) window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + center : float, optional + Parameter defining the center location of the window function. + The default value if not given is ``center = (M-1) / 2``. This + parameter must take its default value for symmetric windows. + tau : float, optional + Parameter defining the decay. For ``center = 0`` use + ``tau = -(M-1) / ln(x)`` if ``x`` is the fraction of the window + remaining at the end. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Notes + ----- + The Exponential window is defined as + + .. math:: w(n) = e^{-|n-center| / \tau} + + References + ---------- + .. [1] S. Gade and H. Herlufsen, "Windows to FFT analysis (Part I)", + Technical Review 3, Bruel & Kjaer, 1987. + + Examples + -------- + Plot the symmetric window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> M = 51 + >>> tau = 3.0 + >>> window = signal.windows.exponential(M, tau=tau) + >>> plt.plot(window) + >>> plt.title("Exponential Window (tau=3.0)") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -35, 0]) + >>> plt.title("Frequency response of the Exponential window (tau=3.0)") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + This function can also generate non-symmetric windows: + + >>> tau2 = -(M-1) / np.log(0.01) + >>> window2 = signal.windows.exponential(M, 0, tau2, False) + >>> plt.figure() + >>> plt.plot(window2) + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + """ + if sym and center is not None: + raise ValueError("If sym==True, center must be None.") + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + if center is None: + center = (M-1) / 2 + + n = np.arange(0, M) + w = np.exp(-np.abs(n-center) / tau) + + return _truncate(w, needs_trunc) + + +def taylor(M, nbar=4, sll=30, norm=True, sym=True): + """ + Return a Taylor window. + + The Taylor window taper function approximates the Dolph-Chebyshev window's + constant sidelobe level for a parameterized number of near-in sidelobes, + but then allows a taper beyond [2]_. + + The SAR (synthetic aperture radar) community commonly uses Taylor + weighting for image formation processing because it provides strong, + selectable sidelobe suppression with minimum broadening of the + mainlobe [1]_. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + nbar : int, optional + Number of nearly constant level sidelobes adjacent to the mainlobe. + sll : float, optional + Desired suppression of sidelobe level in decibels (dB) relative to the + DC gain of the mainlobe. This should be a positive number. + norm : bool, optional + When True (default), divides the window by the largest (middle) value + for odd-length windows or the value that would occur between the two + repeated middle values for even-length windows such that all values + are less than or equal to 1. When False the DC gain will remain at 1 + (0 dB) and the sidelobes will be `sll` dB down. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + out : array + The window. When `norm` is True (default), the maximum value is + normalized to 1 (though the value 1 does not appear if `M` is + even and `sym` is True). + + See Also + -------- + chebwin, kaiser, bartlett, blackman, hamming, hann + + References + ---------- + .. [1] W. Carrara, R. Goodman, and R. Majewski, "Spotlight Synthetic + Aperture Radar: Signal Processing Algorithms" Pages 512-513, + July 1995. + .. [2] Armin Doerry, "Catalog of Window Taper Functions for + Sidelobe Control", 2017. + https://www.researchgate.net/profile/Armin_Doerry/publication/316281181_Catalog_of_Window_Taper_Functions_for_Sidelobe_Control/links/58f92cb2a6fdccb121c9d54d/Catalog-of-Window-Taper-Functions-for-Sidelobe-Control.pdf + + Examples + -------- + Plot the window and its frequency response: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + + >>> window = signal.windows.taylor(51, nbar=20, sll=100, norm=False) + >>> plt.plot(window) + >>> plt.title("Taylor window (100 dB)") + >>> plt.ylabel("Amplitude") + >>> plt.xlabel("Sample") + + >>> plt.figure() + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> plt.plot(freq, response) + >>> plt.axis([-0.5, 0.5, -120, 0]) + >>> plt.title("Frequency response of the Taylor window (100 dB)") + >>> plt.ylabel("Normalized magnitude [dB]") + >>> plt.xlabel("Normalized frequency [cycles per sample]") + + """ # noqa: E501 + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + # Original text uses a negative sidelobe level parameter and then negates + # it in the calculation of B. To keep consistent with other methods we + # assume the sidelobe level parameter to be positive. + B = 10**(sll / 20) + A = np.arccosh(B) / np.pi + s2 = nbar**2 / (A**2 + (nbar - 0.5)**2) + ma = np.arange(1, nbar) + + Fm = np.empty(nbar-1) + signs = np.empty_like(ma) + signs[::2] = 1 + signs[1::2] = -1 + m2 = ma*ma + for mi, m in enumerate(ma): + numer = signs[mi] * np.prod(1 - m2[mi]/s2/(A**2 + (ma - 0.5)**2)) + denom = 2 * np.prod(1 - m2[mi]/m2[:mi]) * np.prod(1 - m2[mi]/m2[mi+1:]) + Fm[mi] = numer / denom + + def W(n): + return 1 + 2*np.dot(Fm, np.cos( + 2*np.pi*ma[:, np.newaxis]*(n-M/2.+0.5)/M)) + + w = W(np.arange(M)) + + # normalize (Note that this is not described in the original text [1]) + if norm: + scale = 1.0 / W((M - 1) / 2) + w *= scale + + return _truncate(w, needs_trunc) + + +def dpss(M, NW, Kmax=None, sym=True, norm=None, return_ratios=False): + """ + Compute the Discrete Prolate Spheroidal Sequences (DPSS). + + DPSS (or Slepian sequences) are often used in multitaper power spectral + density estimation (see [1]_). The first window in the sequence can be + used to maximize the energy concentration in the main lobe, and is also + called the Slepian window. + + Parameters + ---------- + M : int + Window length. + NW : float + Standardized half bandwidth corresponding to ``2*NW = BW/f0 = BW*M*dt`` + where ``dt`` is taken as 1. + Kmax : int | None, optional + Number of DPSS windows to return (orders ``0`` through ``Kmax-1``). + If None (default), return only a single window of shape ``(M,)`` + instead of an array of windows of shape ``(Kmax, M)``. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + norm : {2, 'approximate', 'subsample'} | None, optional + If 'approximate' or 'subsample', then the windows are normalized by the + maximum, and a correction scale-factor for even-length windows + is applied either using ``M**2/(M**2+NW)`` ("approximate") or + a FFT-based subsample shift ("subsample"), see Notes for details. + If None, then "approximate" is used when ``Kmax=None`` and 2 otherwise + (which uses the l2 norm). + return_ratios : bool, optional + If True, also return the concentration ratios in addition to the + windows. + + Returns + ------- + v : ndarray, shape (Kmax, M) or (M,) + The DPSS windows. Will be 1D if `Kmax` is None. + r : ndarray, shape (Kmax,) or float, optional + The concentration ratios for the windows. Only returned if + `return_ratios` evaluates to True. Will be 0D if `Kmax` is None. + + Notes + ----- + This computation uses the tridiagonal eigenvector formulation given + in [2]_. + + The default normalization for ``Kmax=None``, i.e. window-generation mode, + simply using the l-infinity norm would create a window with two unity + values, which creates slight normalization differences between even and odd + orders. The approximate correction of ``M**2/float(M**2+NW)`` for even + sample numbers is used to counteract this effect (see Examples below). + + For very long signals (e.g., 1e6 elements), it can be useful to compute + windows orders of magnitude shorter and use interpolation (e.g., + `scipy.interpolate.interp1d`) to obtain tapers of length `M`, + but this in general will not preserve orthogonality between the tapers. + + .. versionadded:: 1.1 + + References + ---------- + .. [1] Percival DB, Walden WT. Spectral Analysis for Physical Applications: + Multitaper and Conventional Univariate Techniques. + Cambridge University Press; 1993. + .. [2] Slepian, D. Prolate spheroidal wave functions, Fourier analysis, and + uncertainty V: The discrete case. Bell System Technical Journal, + Volume 57 (1978), 1371430. + .. [3] Kaiser, JF, Schafer RW. On the Use of the I0-Sinh Window for + Spectrum Analysis. IEEE Transactions on Acoustics, Speech and + Signal Processing. ASSP-28 (1): 105-107; 1980. + + Examples + -------- + We can compare the window to `kaiser`, which was invented as an alternative + that was easier to calculate [3]_ (example adapted from + `here `_): + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.signal import windows, freqz + >>> M = 51 + >>> fig, axes = plt.subplots(3, 2, figsize=(5, 7)) + >>> for ai, alpha in enumerate((1, 3, 5)): + ... win_dpss = windows.dpss(M, alpha) + ... beta = alpha*np.pi + ... win_kaiser = windows.kaiser(M, beta) + ... for win, c in ((win_dpss, 'k'), (win_kaiser, 'r')): + ... win /= win.sum() + ... axes[ai, 0].plot(win, color=c, lw=1.) + ... axes[ai, 0].set(xlim=[0, M-1], title=r'$\\alpha$ = %s' % alpha, + ... ylabel='Amplitude') + ... w, h = freqz(win) + ... axes[ai, 1].plot(w, 20 * np.log10(np.abs(h)), color=c, lw=1.) + ... axes[ai, 1].set(xlim=[0, np.pi], + ... title=r'$\\beta$ = %0.2f' % beta, + ... ylabel='Magnitude (dB)') + >>> for ax in axes.ravel(): + ... ax.grid(True) + >>> axes[2, 1].legend(['DPSS', 'Kaiser']) + >>> fig.tight_layout() + >>> plt.show() + + And here are examples of the first four windows, along with their + concentration ratios: + + >>> M = 512 + >>> NW = 2.5 + >>> win, eigvals = windows.dpss(M, NW, 4, return_ratios=True) + >>> fig, ax = plt.subplots(1) + >>> ax.plot(win.T, linewidth=1.) + >>> ax.set(xlim=[0, M-1], ylim=[-0.1, 0.1], xlabel='Samples', + ... title='DPSS, M=%d, NW=%0.1f' % (M, NW)) + >>> ax.legend(['win[%d] (%0.4f)' % (ii, ratio) + ... for ii, ratio in enumerate(eigvals)]) + >>> fig.tight_layout() + >>> plt.show() + + Using a standard :math:`l_{\\infty}` norm would produce two unity values + for even `M`, but only one unity value for odd `M`. This produces uneven + window power that can be counteracted by the approximate correction + ``M**2/float(M**2+NW)``, which can be selected by using + ``norm='approximate'`` (which is the same as ``norm=None`` when + ``Kmax=None``, as is the case here). Alternatively, the slower + ``norm='subsample'`` can be used, which uses subsample shifting in the + frequency domain (FFT) to compute the correction: + + >>> Ms = np.arange(1, 41) + >>> factors = (50, 20, 10, 5, 2.0001) + >>> energy = np.empty((3, len(Ms), len(factors))) + >>> for mi, M in enumerate(Ms): + ... for fi, factor in enumerate(factors): + ... NW = M / float(factor) + ... # Corrected using empirical approximation (default) + ... win = windows.dpss(M, NW) + ... energy[0, mi, fi] = np.sum(win ** 2) / np.sqrt(M) + ... # Corrected using subsample shifting + ... win = windows.dpss(M, NW, norm='subsample') + ... energy[1, mi, fi] = np.sum(win ** 2) / np.sqrt(M) + ... # Uncorrected (using l-infinity norm) + ... win /= win.max() + ... energy[2, mi, fi] = np.sum(win ** 2) / np.sqrt(M) + >>> fig, ax = plt.subplots(1) + >>> hs = ax.plot(Ms, energy[2], '-o', markersize=4, + ... markeredgecolor='none') + >>> leg = [hs[-1]] + >>> for hi, hh in enumerate(hs): + ... h1 = ax.plot(Ms, energy[0, :, hi], '-o', markersize=4, + ... color=hh.get_color(), markeredgecolor='none', + ... alpha=0.66) + ... h2 = ax.plot(Ms, energy[1, :, hi], '-o', markersize=4, + ... color=hh.get_color(), markeredgecolor='none', + ... alpha=0.33) + ... if hi == len(hs) - 1: + ... leg.insert(0, h1[0]) + ... leg.insert(0, h2[0]) + >>> ax.set(xlabel='M (samples)', ylabel=r'Power / $\\sqrt{M}$') + >>> ax.legend(leg, ['Uncorrected', r'Corrected: $\\frac{M^2}{M^2+NW}$', + ... 'Corrected (subsample)']) + >>> fig.tight_layout() + + """ + if _len_guards(M): + return np.ones(M) + if norm is None: + norm = 'approximate' if Kmax is None else 2 + known_norms = (2, 'approximate', 'subsample') + if norm not in known_norms: + raise ValueError(f'norm must be one of {known_norms}, got {norm}') + if Kmax is None: + singleton = True + Kmax = 1 + else: + singleton = False + Kmax = operator.index(Kmax) + if not 0 < Kmax <= M: + raise ValueError('Kmax must be greater than 0 and less than M') + if NW >= M/2.: + raise ValueError('NW must be less than M/2.') + if NW <= 0: + raise ValueError('NW must be positive') + M, needs_trunc = _extend(M, sym) + W = float(NW) / M + nidx = np.arange(M) + + # Here we want to set up an optimization problem to find a sequence + # whose energy is maximally concentrated within band [-W,W]. + # Thus, the measure lambda(T,W) is the ratio between the energy within + # that band, and the total energy. This leads to the eigen-system + # (A - (l1)I)v = 0, where the eigenvector corresponding to the largest + # eigenvalue is the sequence with maximally concentrated energy. The + # collection of eigenvectors of this system are called Slepian + # sequences, or discrete prolate spheroidal sequences (DPSS). Only the + # first K, K = 2NW/dt orders of DPSS will exhibit good spectral + # concentration + # [see https://en.wikipedia.org/wiki/Spectral_concentration_problem] + + # Here we set up an alternative symmetric tri-diagonal eigenvalue + # problem such that + # (B - (l2)I)v = 0, and v are our DPSS (but eigenvalues l2 != l1) + # the main diagonal = ([M-1-2*t]/2)**2 cos(2PIW), t=[0,1,2,...,M-1] + # and the first off-diagonal = t(M-t)/2, t=[1,2,...,M-1] + # [see Percival and Walden, 1993] + d = ((M - 1 - 2 * nidx) / 2.) ** 2 * np.cos(2 * np.pi * W) + e = nidx[1:] * (M - nidx[1:]) / 2. + + # only calculate the highest Kmax eigenvalues + w, windows = linalg.eigh_tridiagonal( + d, e, select='i', select_range=(M - Kmax, M - 1)) + w = w[::-1] + windows = windows[:, ::-1].T + + # By convention (Percival and Walden, 1993 pg 379) + # * symmetric tapers (k=0,2,4,...) should have a positive average. + fix_even = (windows[::2].sum(axis=1) < 0) + for i, f in enumerate(fix_even): + if f: + windows[2 * i] *= -1 + # * antisymmetric tapers should begin with a positive lobe + # (this depends on the definition of "lobe", here we'll take the first + # point above the numerical noise, which should be good enough for + # sufficiently smooth functions, and more robust than relying on an + # algorithm that uses max(abs(w)), which is susceptible to numerical + # noise problems) + thresh = max(1e-7, 1. / M) + for i, w in enumerate(windows[1::2]): + if w[w * w > thresh][0] < 0: + windows[2 * i + 1] *= -1 + + # Now find the eigenvalues of the original spectral concentration problem + # Use the autocorr sequence technique from Percival and Walden, 1993 pg 390 + if return_ratios: + dpss_rxx = _fftautocorr(windows) + r = 4 * W * np.sinc(2 * W * nidx) + r[0] = 2 * W + ratios = np.dot(dpss_rxx, r) + if singleton: + ratios = ratios[0] + # Deal with sym and Kmax=None + if norm != 2: + windows /= windows.max() + if M % 2 == 0: + if norm == 'approximate': + correction = M**2 / float(M**2 + NW) + else: + s = sp_fft.rfft(windows[0]) + shift = -(1 - 1./M) * np.arange(1, M//2 + 1) + s[1:] *= 2 * np.exp(-1j * np.pi * shift) + correction = M / s.real.sum() + windows *= correction + # else we're already l2 normed, so do nothing + if needs_trunc: + windows = windows[:, :-1] + if singleton: + windows = windows[0] + return (windows, ratios) if return_ratios else windows + + +def lanczos(M, *, sym=True): + r"""Return a Lanczos window also known as a sinc window. + + Parameters + ---------- + M : int + Number of points in the output window. If zero, an empty array + is returned. An exception is thrown when it is negative. + sym : bool, optional + When True (default), generates a symmetric window, for use in filter + design. + When False, generates a periodic window, for use in spectral analysis. + + Returns + ------- + w : ndarray + The window, with the maximum value normalized to 1 (though the value 1 + does not appear if `M` is even and `sym` is True). + + Notes + ----- + The Lanczos window is defined as + + .. math:: w(n) = sinc \left( \frac{2n}{M - 1} - 1 \right) + + where + + .. math:: sinc(x) = \frac{\sin(\pi x)}{\pi x} + + The Lanczos window has reduced Gibbs oscillations and is widely used for + filtering climate timeseries with good properties in the physical and + spectral domains. + + .. versionadded:: 1.10 + + References + ---------- + .. [1] Lanczos, C., and Teichmann, T. (1957). Applied analysis. + Physics Today, 10, 44. + .. [2] Duchon C. E. (1979) Lanczos Filtering in One and Two Dimensions. + Journal of Applied Meteorology, Vol 18, pp 1016-1022. + .. [3] Thomson, R. E. and Emery, W. J. (2014) Data Analysis Methods in + Physical Oceanography (Third Edition), Elsevier, pp 593-637. + .. [4] Wikipedia, "Window function", + http://en.wikipedia.org/wiki/Window_function + + Examples + -------- + Plot the window + + >>> import numpy as np + >>> from scipy.signal.windows import lanczos + >>> from scipy.fft import fft, fftshift + >>> import matplotlib.pyplot as plt + >>> fig, ax = plt.subplots(1) + >>> window = lanczos(51) + >>> ax.plot(window) + >>> ax.set_title("Lanczos window") + >>> ax.set_ylabel("Amplitude") + >>> ax.set_xlabel("Sample") + >>> fig.tight_layout() + >>> plt.show() + + and its frequency response: + + >>> fig, ax = plt.subplots(1) + >>> A = fft(window, 2048) / (len(window)/2.0) + >>> freq = np.linspace(-0.5, 0.5, len(A)) + >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) + >>> ax.plot(freq, response) + >>> ax.set_xlim(-0.5, 0.5) + >>> ax.set_ylim(-120, 0) + >>> ax.set_title("Frequency response of the lanczos window") + >>> ax.set_ylabel("Normalized magnitude [dB]") + >>> ax.set_xlabel("Normalized frequency [cycles per sample]") + >>> fig.tight_layout() + >>> plt.show() + """ + if _len_guards(M): + return np.ones(M) + M, needs_trunc = _extend(M, sym) + + # To make sure that the window is symmetric, we concatenate the right hand + # half of the window and the flipped one which is the left hand half of + # the window. + def _calc_right_side_lanczos(n, m): + return np.sinc(2. * np.arange(n, m) / (m - 1) - 1.0) + + if M % 2 == 0: + wh = _calc_right_side_lanczos(M/2, M) + w = np.r_[np.flip(wh), wh] + else: + wh = _calc_right_side_lanczos((M+1)/2, M) + w = np.r_[np.flip(wh), 1.0, wh] + + return _truncate(w, needs_trunc) + + +def _fftautocorr(x): + """Compute the autocorrelation of a real array and crop the result.""" + N = x.shape[-1] + use_N = sp_fft.next_fast_len(2*N-1) + x_fft = sp_fft.rfft(x, use_N, axis=-1) + cxy = sp_fft.irfft(x_fft * x_fft.conj(), n=use_N)[:, :N] + # Or equivalently (but in most cases slower): + # cxy = np.array([np.convolve(xx, yy[::-1], mode='full') + # for xx, yy in zip(x, x)])[:, N-1:2*N-1] + return cxy + + +_win_equiv_raw = { + ('barthann', 'brthan', 'bth'): (barthann, False), + ('bartlett', 'bart', 'brt'): (bartlett, False), + ('blackman', 'black', 'blk'): (blackman, False), + ('blackmanharris', 'blackharr', 'bkh'): (blackmanharris, False), + ('bohman', 'bman', 'bmn'): (bohman, False), + ('boxcar', 'box', 'ones', + 'rect', 'rectangular'): (boxcar, False), + ('chebwin', 'cheb'): (chebwin, True), + ('cosine', 'halfcosine'): (cosine, False), + ('dpss',): (dpss, True), + ('exponential', 'poisson'): (exponential, False), + ('flattop', 'flat', 'flt'): (flattop, False), + ('gaussian', 'gauss', 'gss'): (gaussian, True), + ('general cosine', 'general_cosine'): (general_cosine, True), + ('general gaussian', 'general_gaussian', + 'general gauss', 'general_gauss', 'ggs'): (general_gaussian, True), + ('general hamming', 'general_hamming'): (general_hamming, True), + ('hamming', 'hamm', 'ham'): (hamming, False), + ('hann', 'han'): (hann, False), + ('kaiser', 'ksr'): (kaiser, True), + ('kaiser bessel derived', 'kbd'): (kaiser_bessel_derived, True), + ('lanczos', 'sinc'): (lanczos, False), + ('nuttall', 'nutl', 'nut'): (nuttall, False), + ('parzen', 'parz', 'par'): (parzen, False), + ('taylor', 'taylorwin'): (taylor, False), + ('triangle', 'triang', 'tri'): (triang, False), + ('tukey', 'tuk'): (tukey, False), +} + +# Fill dict with all valid window name strings +_win_equiv = {} +for k, v in _win_equiv_raw.items(): + for key in k: + _win_equiv[key] = v[0] + +# Keep track of which windows need additional parameters +_needs_param = set() +for k, v in _win_equiv_raw.items(): + if v[1]: + _needs_param.update(k) + + +def get_window(window, Nx, fftbins=True): + """ + Return a window of a given length and type. + + Parameters + ---------- + window : string, float, or tuple + The type of window to create. See below for more details. + Nx : int + The number of samples in the window. + fftbins : bool, optional + If True (default), create a "periodic" window, ready to use with + `ifftshift` and be multiplied by the result of an FFT (see also + :func:`~scipy.fft.fftfreq`). + If False, create a "symmetric" window, for use in filter design. + + Returns + ------- + get_window : ndarray + Returns a window of length `Nx` and type `window` + + Notes + ----- + Window types: + + - `~scipy.signal.windows.boxcar` + - `~scipy.signal.windows.triang` + - `~scipy.signal.windows.blackman` + - `~scipy.signal.windows.hamming` + - `~scipy.signal.windows.hann` + - `~scipy.signal.windows.bartlett` + - `~scipy.signal.windows.flattop` + - `~scipy.signal.windows.parzen` + - `~scipy.signal.windows.bohman` + - `~scipy.signal.windows.blackmanharris` + - `~scipy.signal.windows.nuttall` + - `~scipy.signal.windows.barthann` + - `~scipy.signal.windows.cosine` + - `~scipy.signal.windows.exponential` + - `~scipy.signal.windows.tukey` + - `~scipy.signal.windows.taylor` + - `~scipy.signal.windows.lanczos` + - `~scipy.signal.windows.kaiser` (needs beta) + - `~scipy.signal.windows.kaiser_bessel_derived` (needs beta) + - `~scipy.signal.windows.gaussian` (needs standard deviation) + - `~scipy.signal.windows.general_cosine` (needs weighting coefficients) + - `~scipy.signal.windows.general_gaussian` (needs power, width) + - `~scipy.signal.windows.general_hamming` (needs window coefficient) + - `~scipy.signal.windows.dpss` (needs normalized half-bandwidth) + - `~scipy.signal.windows.chebwin` (needs attenuation) + + + If the window requires no parameters, then `window` can be a string. + + If the window requires parameters, then `window` must be a tuple + with the first argument the string name of the window, and the next + arguments the needed parameters. + + If `window` is a floating point number, it is interpreted as the beta + parameter of the `~scipy.signal.windows.kaiser` window. + + Each of the window types listed above is also the name of + a function that can be called directly to create a window of + that type. + + Examples + -------- + >>> from scipy import signal + >>> signal.get_window('triang', 7) + array([ 0.125, 0.375, 0.625, 0.875, 0.875, 0.625, 0.375]) + >>> signal.get_window(('kaiser', 4.0), 9) + array([ 0.08848053, 0.29425961, 0.56437221, 0.82160913, 0.97885093, + 0.97885093, 0.82160913, 0.56437221, 0.29425961]) + >>> signal.get_window(('exponential', None, 1.), 9) + array([ 0.011109 , 0.03019738, 0.082085 , 0.22313016, 0.60653066, + 0.60653066, 0.22313016, 0.082085 , 0.03019738]) + >>> signal.get_window(4.0, 9) + array([ 0.08848053, 0.29425961, 0.56437221, 0.82160913, 0.97885093, + 0.97885093, 0.82160913, 0.56437221, 0.29425961]) + + """ + sym = not fftbins + try: + beta = float(window) + except (TypeError, ValueError) as e: + args = () + if isinstance(window, tuple): + winstr = window[0] + if len(window) > 1: + args = window[1:] + elif isinstance(window, str): + if window in _needs_param: + raise ValueError("The '" + window + "' window needs one or " + "more parameters -- pass a tuple.") from e + else: + winstr = window + else: + raise ValueError( + f"{str(type(window))} as window type is not supported.") from e + + try: + winfunc = _win_equiv[winstr] + except KeyError as e: + raise ValueError("Unknown window type.") from e + + if winfunc is dpss: + params = (Nx,) + args + (None,) + else: + params = (Nx,) + args + else: + winfunc = kaiser + params = (Nx, beta) + + return winfunc(*params, sym=sym) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/windows/windows.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/windows/windows.py new file mode 100644 index 0000000000000000000000000000000000000000..6858f71aceeb29ca6110864d01fb250e8c8ce403 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/signal/windows/windows.py @@ -0,0 +1,23 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.signal.windows` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'boxcar', 'triang', 'parzen', 'bohman', 'blackman', 'nuttall', + 'blackmanharris', 'flattop', 'bartlett', 'barthann', + 'hamming', 'kaiser', 'gaussian', 'general_cosine', + 'general_gaussian', 'general_hamming', 'chebwin', 'cosine', + 'hann', 'exponential', 'tukey', 'taylor', 'dpss', 'get_window', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="signal.windows", module="windows", + private_modules=["_windows"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..18fe7e011db102f57a8263d1db343818715aeeee --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/__init__.py @@ -0,0 +1,331 @@ +""" +=================================== +Sparse arrays (:mod:`scipy.sparse`) +=================================== + +.. currentmodule:: scipy.sparse + +.. toctree:: + :hidden: + + sparse.csgraph + sparse.linalg + sparse.migration_to_sparray + +SciPy 2-D sparse array package for numeric data. + +.. note:: + + This package is switching to an array interface, compatible with + NumPy arrays, from the older matrix interface. We recommend that + you use the array objects (`bsr_array`, `coo_array`, etc.) for + all new work. + + When using the array interface, please note that: + + - ``x * y`` no longer performs matrix multiplication, but + element-wise multiplication (just like with NumPy arrays). To + make code work with both arrays and matrices, use ``x @ y`` for + matrix multiplication. + - Operations such as ``sum``, that used to produce dense matrices, now + produce arrays, whose multiplication behavior differs similarly. + - Sparse arrays use array style *slicing* operations, returning scalars, + 1D, or 2D sparse arrays. If you need 2D results, use an appropriate index. + E.g. ``A[:, i, None]`` or ``A[:, [i]]``. + + The construction utilities (`eye`, `kron`, `random`, `diags`, etc.) + have appropriate replacements (see :ref:`sparse-construction-functions`). + + For more information see + :ref:`Migration from spmatrix to sparray `. + + +Submodules +========== + +.. autosummary:: + + csgraph - Compressed sparse graph routines + linalg - Sparse linear algebra routines + + +Sparse array classes +==================== + +.. autosummary:: + :toctree: generated/ + + bsr_array - Block Sparse Row array + coo_array - A sparse array in COOrdinate format + csc_array - Compressed Sparse Column array + csr_array - Compressed Sparse Row array + dia_array - Sparse array with DIAgonal storage + dok_array - Dictionary Of Keys based sparse array + lil_array - Row-based list of lists sparse array + sparray - Sparse array base class + +.. _sparse-construction-functions: + +Building sparse arrays +---------------------- + +.. autosummary:: + :toctree: generated/ + + diags_array - Return a sparse array from diagonals + eye_array - Sparse MxN array whose k-th diagonal is all ones + random_array - Random values in a given shape array + block_array - Build a sparse array from sub-blocks + +.. _combining-arrays: + +Combining arrays +---------------- + +.. autosummary:: + :toctree: generated/ + + kron - Kronecker product of two sparse arrays + kronsum - Kronecker sum of sparse arrays + block_diag - Build a block diagonal sparse array + tril - Lower triangular portion of a sparse array + triu - Upper triangular portion of a sparse array + hstack - Stack sparse arrays horizontally (column wise) + vstack - Stack sparse arrays vertically (row wise) + +Sparse tools +------------ + +.. autosummary:: + :toctree: generated/ + + save_npz - Save a sparse array to a file using ``.npz`` format. + load_npz - Load a sparse array from a file using ``.npz`` format. + find - Return the indices and values of the nonzero elements + get_index_dtype - determine a good dtype for index arrays. + safely_cast_index_arrays - cast index array dtype or raise if shape too big + +Identifying sparse arrays +------------------------- + +.. autosummary:: + :toctree: generated/ + + issparse - Check if the argument is a sparse object (array or matrix). + + +Sparse matrix classes +===================== + +.. autosummary:: + :toctree: generated/ + + bsr_matrix - Block Sparse Row matrix + coo_matrix - A sparse matrix in COOrdinate format + csc_matrix - Compressed Sparse Column matrix + csr_matrix - Compressed Sparse Row matrix + dia_matrix - Sparse matrix with DIAgonal storage + dok_matrix - Dictionary Of Keys based sparse matrix + lil_matrix - Row-based list of lists sparse matrix + spmatrix - Sparse matrix base class + +Building sparse matrices +------------------------ + +.. autosummary:: + :toctree: generated/ + + eye - Sparse MxN matrix whose k-th diagonal is all ones + identity - Identity matrix in sparse matrix format + diags - Return a sparse matrix from diagonals + spdiags - Return a sparse matrix from diagonals + bmat - Build a sparse matrix from sparse sub-blocks + random - Random values in a given shape matrix + rand - Random values in a given shape matrix (old interface) + +**Combining matrices use the same functions as for** :ref:`combining-arrays`. + +Identifying sparse matrices +--------------------------- + +.. autosummary:: + :toctree: generated/ + + issparse + isspmatrix + isspmatrix_csc + isspmatrix_csr + isspmatrix_bsr + isspmatrix_lil + isspmatrix_dok + isspmatrix_coo + isspmatrix_dia + + +Warnings +======== + +.. autosummary:: + :toctree: generated/ + + SparseEfficiencyWarning + SparseWarning + + +Usage information +================= + +There are seven available sparse array types: + + 1. csc_array: Compressed Sparse Column format + 2. csr_array: Compressed Sparse Row format + 3. bsr_array: Block Sparse Row format + 4. lil_array: List of Lists format + 5. dok_array: Dictionary of Keys format + 6. coo_array: COOrdinate format (aka IJV, triplet format) + 7. dia_array: DIAgonal format + +To construct an array efficiently, use any of `coo_array`, +`dok_array` or `lil_array`. `dok_array` and `lil_array` +support basic slicing and fancy indexing with a similar syntax +to NumPy arrays. The COO format does not support indexing (yet) +but can also be used to efficiently construct arrays using coord +and value info. + +Despite their similarity to NumPy arrays, it is **strongly discouraged** +to use NumPy functions directly on these arrays because NumPy typically +treats them as generic Python objects rather than arrays, leading to +unexpected (and incorrect) results. If you do want to apply a NumPy +function to these arrays, first check if SciPy has its own implementation +for the given sparse array class, or **convert the sparse array to +a NumPy array** (e.g., using the `toarray` method of the class) +before applying the method. + +All conversions among the CSR, CSC, and COO formats are efficient, +linear-time operations. + +To perform manipulations such as multiplication or inversion, first +convert the array to either CSC or CSR format. The `lil_array` +format is row-based, so conversion to CSR is efficient, whereas +conversion to CSC is less so. + +Matrix vector product +--------------------- + +To do a vector product between a 2D sparse array and a vector use +the matmul operator (i.e., ``@``) which performs a dot product (like the +``dot`` method): + +>>> import numpy as np +>>> from scipy.sparse import csr_array +>>> A = csr_array([[1, 2, 0], [0, 0, 3], [4, 0, 5]]) +>>> v = np.array([1, 0, -1]) +>>> A @ v +array([ 1, -3, -1], dtype=int64) + +The CSR format is especially suitable for fast matrix vector products. + +Example 1 +--------- + +Construct a 1000x1000 `lil_array` and add some values to it: + +>>> from scipy.sparse import lil_array +>>> from scipy.sparse.linalg import spsolve +>>> from numpy.linalg import solve, norm +>>> from numpy.random import rand + +>>> A = lil_array((1000, 1000)) +>>> A[0, :100] = rand(100) +>>> A.setdiag(rand(1000)) + +Now convert it to CSR format and solve A x = b for x: + +>>> A = A.tocsr() +>>> b = rand(1000) +>>> x = spsolve(A, b) + +Convert it to a dense array and solve, and check that the result +is the same: + +>>> x_ = solve(A.toarray(), b) + +Now we can compute norm of the error with: + +>>> err = norm(x-x_) +>>> err < 1e-10 +True + +It should be small :) + + +Example 2 +--------- + +Construct an array in COO format: + +>>> from scipy import sparse +>>> from numpy import array +>>> I = array([0,3,1,0]) +>>> J = array([0,3,1,2]) +>>> V = array([4,5,7,9]) +>>> A = sparse.coo_array((V,(I,J)),shape=(4,4)) + +Notice that the indices do not need to be sorted. + +Duplicate (i,j) entries are summed when converting to CSR or CSC. + +>>> I = array([0,0,1,3,1,0,0]) +>>> J = array([0,2,1,3,1,0,0]) +>>> V = array([1,1,1,1,1,1,1]) +>>> B = sparse.coo_array((V,(I,J)),shape=(4,4)).tocsr() + +This is useful for constructing finite-element stiffness and mass matrices. + +Further details +--------------- + +CSR column indices are not necessarily sorted. Likewise for CSC row +indices. Use the ``.sorted_indices()`` and ``.sort_indices()`` methods when +sorted indices are required (e.g., when passing data to other libraries). + +""" + +# Original code by Travis Oliphant. +# Modified and extended by Ed Schofield, Robert Cimrman, +# Nathan Bell, and Jake Vanderplas. + +import warnings as _warnings + +from ._base import * +from ._csr import * +from ._csc import * +from ._lil import * +from ._dok import * +from ._coo import * +from ._dia import * +from ._bsr import * +from ._construct import * +from ._extract import * +from ._matrix import spmatrix +from ._matrix_io import * +from ._sputils import get_index_dtype, safely_cast_index_arrays + +# For backward compatibility with v0.19. +from . import csgraph + +# Deprecated namespaces, to be removed in v2.0.0 +from . import ( + base, bsr, compressed, construct, coo, csc, csr, data, dia, dok, extract, + lil, sparsetools, sputils +) + +__all__ = [s for s in dir() if not s.startswith('_')] + +# Filter PendingDeprecationWarning for np.matrix introduced with numpy 1.15 +msg = 'the matrix subclass is not the recommended way' +_warnings.filterwarnings('ignore', message=msg) + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_base.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_base.py new file mode 100644 index 0000000000000000000000000000000000000000..926e191013c7cacebab7a8e025a017358ba82f50 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_base.py @@ -0,0 +1,1448 @@ +"""Base class for sparse matrices""" + +import numpy as np + +from ._sputils import (asmatrix, check_reshape_kwargs, check_shape, + get_sum_dtype, isdense, isscalarlike, + matrix, validateaxis, getdtype) + +from ._matrix import spmatrix + +__all__ = ['isspmatrix', 'issparse', 'sparray', + 'SparseWarning', 'SparseEfficiencyWarning'] + + +class SparseWarning(Warning): + pass + + +class SparseFormatWarning(SparseWarning): + pass + + +class SparseEfficiencyWarning(SparseWarning): + pass + + +# The formats that we might potentially understand. +_formats = {'csc': [0, "Compressed Sparse Column"], + 'csr': [1, "Compressed Sparse Row"], + 'dok': [2, "Dictionary Of Keys"], + 'lil': [3, "List of Lists"], + 'dod': [4, "Dictionary of Dictionaries"], + 'sss': [5, "Symmetric Sparse Skyline"], + 'coo': [6, "COOrdinate"], + 'lba': [7, "Linpack BAnded"], + 'egd': [8, "Ellpack-itpack Generalized Diagonal"], + 'dia': [9, "DIAgonal"], + 'bsr': [10, "Block Sparse Row"], + 'msr': [11, "Modified compressed Sparse Row"], + 'bsc': [12, "Block Sparse Column"], + 'msc': [13, "Modified compressed Sparse Column"], + 'ssk': [14, "Symmetric SKyline"], + 'nsk': [15, "Nonsymmetric SKyline"], + 'jad': [16, "JAgged Diagonal"], + 'uss': [17, "Unsymmetric Sparse Skyline"], + 'vbr': [18, "Variable Block Row"], + 'und': [19, "Undefined"] + } + + +# These univariate ufuncs preserve zeros. +_ufuncs_with_fixed_point_at_zero = frozenset([ + np.sin, np.tan, np.arcsin, np.arctan, np.sinh, np.tanh, np.arcsinh, + np.arctanh, np.rint, np.sign, np.expm1, np.log1p, np.deg2rad, + np.rad2deg, np.floor, np.ceil, np.trunc, np.sqrt]) + + +MAXPRINT = 50 + + +class _spbase: + """ This class provides a base class for all sparse arrays. It + cannot be instantiated. Most of the work is provided by subclasses. + """ + + __array_priority__ = 10.1 + _format = 'und' # undefined + _allow_nd = (2,) + + @property + def ndim(self) -> int: + return len(self._shape) + + @property + def _shape_as_2d(self): + s = self._shape + return (1, s[-1]) if len(s) == 1 else s + + @property + def _bsr_container(self): + from ._bsr import bsr_array + return bsr_array + + @property + def _coo_container(self): + from ._coo import coo_array + return coo_array + + @property + def _csc_container(self): + from ._csc import csc_array + return csc_array + + @property + def _csr_container(self): + from ._csr import csr_array + return csr_array + + @property + def _dia_container(self): + from ._dia import dia_array + return dia_array + + @property + def _dok_container(self): + from ._dok import dok_array + return dok_array + + @property + def _lil_container(self): + from ._lil import lil_array + return lil_array + + def __init__(self, arg1, *, maxprint=None): + self._shape = None + if self.__class__.__name__ == '_spbase': + raise ValueError("This class is not intended" + " to be instantiated directly.") + if isinstance(self, sparray) and np.isscalar(arg1): + raise ValueError( + "scipy sparse array classes do not support instantiation from a scalar" + ) + self.maxprint = MAXPRINT if maxprint is None else maxprint + + @property + def shape(self): + return self._shape + + def reshape(self, *args, **kwargs): + """reshape(self, shape, order='C', copy=False) + + Gives a new shape to a sparse array/matrix without changing its data. + + Parameters + ---------- + shape : length-2 tuple of ints + The new shape should be compatible with the original shape. + order : {'C', 'F'}, optional + Read the elements using this index order. 'C' means to read and + write the elements using C-like index order; e.g., read entire first + row, then second row, etc. 'F' means to read and write the elements + using Fortran-like index order; e.g., read entire first column, then + second column, etc. + copy : bool, optional + Indicates whether or not attributes of self should be copied + whenever possible. The degree to which attributes are copied varies + depending on the type of sparse array being used. + + Returns + ------- + reshaped : sparse array/matrix + A sparse array/matrix with the given `shape`, not necessarily of the same + format as the current object. + + See Also + -------- + numpy.reshape : NumPy's implementation of 'reshape' for ndarrays + """ + # If the shape already matches, don't bother doing an actual reshape + # Otherwise, the default is to convert to COO and use its reshape + # Don't restrict ndim on this first call. That happens in constructor + shape = check_shape(args, self.shape, allow_nd=range(1, 65)) + order, copy = check_reshape_kwargs(kwargs) + if shape == self.shape: + if copy: + return self.copy() + else: + return self + + return self.tocoo(copy=copy).reshape(shape, order=order, copy=False) + + def resize(self, shape): + """Resize the array/matrix in-place to dimensions given by ``shape`` + + Any elements that lie within the new shape will remain at the same + indices, while non-zero elements lying outside the new shape are + removed. + + Parameters + ---------- + shape : (int, int) + number of rows and columns in the new array/matrix + + Notes + ----- + The semantics are not identical to `numpy.ndarray.resize` or + `numpy.resize`. Here, the same data will be maintained at each index + before and after reshape, if that index is within the new bounds. In + numpy, resizing maintains contiguity of the array, moving elements + around in the logical array but not within a flattened representation. + + We give no guarantees about whether the underlying data attributes + (arrays, etc.) will be modified in place or replaced with new objects. + """ + # As an inplace operation, this requires implementation in each format. + raise NotImplementedError( + f'{type(self).__name__}.resize is not implemented') + + def astype(self, dtype, casting='unsafe', copy=True): + """Cast the array/matrix elements to a specified type. + + Parameters + ---------- + dtype : string or numpy dtype + Typecode or data-type to which to cast the data. + casting : {'no', 'equiv', 'safe', 'same_kind', 'unsafe'}, optional + Controls what kind of data casting may occur. + Defaults to 'unsafe' for backwards compatibility. + 'no' means the data types should not be cast at all. + 'equiv' means only byte-order changes are allowed. + 'safe' means only casts which can preserve values are allowed. + 'same_kind' means only safe casts or casts within a kind, + like float64 to float32, are allowed. + 'unsafe' means any data conversions may be done. + copy : bool, optional + If `copy` is `False`, the result might share some memory with this + array/matrix. If `copy` is `True`, it is guaranteed that the result and + this array/matrix do not share any memory. + """ + + dtype = getdtype(dtype) + if self.dtype != dtype: + return self.tocsr().astype( + dtype, casting=casting, copy=copy).asformat(self.format) + elif copy: + return self.copy() + else: + return self + + @classmethod + def _ascontainer(cls, X, **kwargs): + if issubclass(cls, sparray): + return np.asarray(X, **kwargs) + else: + return asmatrix(X, **kwargs) + + @classmethod + def _container(cls, X, **kwargs): + if issubclass(cls, sparray): + return np.array(X, **kwargs) + else: + return matrix(X, **kwargs) + + def _asfptype(self): + """Upcast array to a floating point format (if necessary)""" + + fp_types = ['f', 'd', 'F', 'D'] + + if self.dtype.char in fp_types: + return self + else: + for fp_type in fp_types: + if self.dtype <= np.dtype(fp_type): + return self.astype(fp_type) + + raise TypeError( + f'cannot upcast [{self.dtype.name}] to a floating point format' + ) + + def __iter__(self): + for r in range(self.shape[0]): + yield self[r] + + def _getmaxprint(self): + """Maximum number of elements to display when printed.""" + return self.maxprint + + def count_nonzero(self, axis=None): + """Number of non-zero entries, equivalent to + + np.count_nonzero(a.toarray(), axis=axis) + + Unlike the nnz property, which return the number of stored + entries (the length of the data attribute), this method counts the + actual number of non-zero entries in data. + + Duplicate entries are summed before counting. + + Parameters + ---------- + axis : {-2, -1, 0, 1, None} optional + Count nonzeros for the whole array, or along a specified axis. + + .. versionadded:: 1.15.0 + + Returns + ------- + numpy array + A reduced array (no axis `axis`) holding the number of nonzero values + for each of the indices of the nonaxis dimensions. + + Notes + ----- + If you want to count nonzero and explicit zero stored values (e.g. nnz) + along an axis, two fast idioms are provided by `numpy` functions for the + common CSR, CSC, COO formats. + + For the major axis in CSR (rows) and CSC (cols) use `np.diff`: + + >>> import numpy as np + >>> import scipy as sp + >>> A = sp.sparse.csr_array([[4, 5, 0], [7, 0, 0]]) + >>> major_axis_stored_values = np.diff(A.indptr) # -> np.array([2, 1]) + + For the minor axis in CSR (cols) and CSC (rows) use `numpy.bincount` with + minlength ``A.shape[1]`` for CSR and ``A.shape[0]`` for CSC: + + >>> csr_minor_stored_values = np.bincount(A.indices, minlength=A.shape[1]) + + For COO, use the minor axis approach for either `axis`: + + >>> A = A.tocoo() + >>> coo_axis0_stored_values = np.bincount(A.coords[0], minlength=A.shape[1]) + >>> coo_axis1_stored_values = np.bincount(A.coords[1], minlength=A.shape[0]) + + Examples + -------- + + >>> A = sp.sparse.csr_array([[4, 5, 0], [7, 0, 0]]) + >>> A.count_nonzero(axis=0) + array([2, 1, 0]) + """ + clsname = self.__class__.__name__ + raise NotImplementedError(f"count_nonzero not implemented for {clsname}.") + + def _getnnz(self, axis=None): + """Number of stored values, including explicit zeros. + + Parameters + ---------- + axis : {-2, -1, 0, 1, None} optional + Report stored values for the whole array, or along a specified axis. + + See also + -------- + count_nonzero : Number of non-zero entries + """ + clsname = self.__class__.__name__ + raise NotImplementedError(f"getnnz not implemented for {clsname}.") + + @property + def nnz(self) -> int: + """Number of stored values, including explicit zeros. + + See also + -------- + count_nonzero : Number of non-zero entries + """ + return self._getnnz() + + @property + def size(self) -> int: + """Number of stored values. + + See also + -------- + count_nonzero : Number of non-zero values. + """ + return self._getnnz() + + @property + def format(self) -> str: + """Format string for matrix.""" + return self._format + + @property + def T(self): + """Transpose.""" + return self.transpose() + + @property + def real(self): + return self._real() + + @property + def imag(self): + return self._imag() + + def __repr__(self): + _, format_name = _formats[self.format] + sparse_cls = 'array' if isinstance(self, sparray) else 'matrix' + return ( + f"<{format_name} sparse {sparse_cls} of dtype '{self.dtype}'\n" + f"\twith {self.nnz} stored elements and shape {self.shape}>" + ) + + def __str__(self): + maxprint = self._getmaxprint() + + A = self.tocoo() + + # helper function, outputs "(i,j) v" + def tostr(coords, data): + pairs = zip(zip(*(c.tolist() for c in coords)), data) + return '\n'.join(f' {idx}\t{val}' for idx, val in pairs) + + out = repr(self) + if self.nnz == 0: + return out + + out += '\n Coords\tValues\n' + if self.nnz > maxprint: + half = maxprint // 2 + out += tostr(tuple(c[:half] for c in A.coords), A.data[:half]) + out += "\n :\t:\n" + half = maxprint - half + out += tostr(tuple(c[-half:] for c in A.coords), A.data[-half:]) + else: + out += tostr(A.coords, A.data) + + return out + + def __bool__(self): # Simple -- other ideas? + if self.shape == (1, 1): + return self.nnz != 0 + else: + raise ValueError("The truth value of an array with more than one " + "element is ambiguous. Use a.any() or a.all().") + __nonzero__ = __bool__ + + # What should len(sparse) return? For consistency with dense matrices, + # perhaps it should be the number of rows? But for some uses the number of + # non-zeros is more important. For now, raise an exception! + def __len__(self): + raise TypeError("sparse array length is ambiguous; use getnnz()" + " or shape[0]") + + def asformat(self, format, copy=False): + """Return this array/matrix in the passed format. + + Parameters + ---------- + format : {str, None} + The desired sparse format ("csr", "csc", "lil", "dok", "array", ...) + or None for no conversion. + copy : bool, optional + If True, the result is guaranteed to not share data with self. + + Returns + ------- + A : This array/matrix in the passed format. + """ + if format is None or format == self.format: + if copy: + return self.copy() + else: + return self + else: + try: + convert_method = getattr(self, 'to' + format) + except AttributeError as e: + raise ValueError(f'Format {format} is unknown.') from e + + # Forward the copy kwarg, if it's accepted. + try: + return convert_method(copy=copy) + except TypeError: + return convert_method() + + ################################################################### + # NOTE: All arithmetic operations use csr_matrix by default. + # Therefore a new sparse array format just needs to define a + # .tocsr() method to provide arithmetic support. Any of these + # methods can be overridden for efficiency. + #################################################################### + + def multiply(self, other): + """Point-wise multiplication by another array/matrix.""" + if isscalarlike(other): + return self._mul_scalar(other) + return self.tocsr().multiply(other) + + def maximum(self, other): + """Element-wise maximum between this and another array/matrix.""" + return self.tocsr().maximum(other) + + def minimum(self, other): + """Element-wise minimum between this and another array/matrix.""" + return self.tocsr().minimum(other) + + def dot(self, other): + """Ordinary dot product + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csr_array + >>> A = csr_array([[1, 2, 0], [0, 0, 3], [4, 0, 5]]) + >>> v = np.array([1, 0, -1]) + >>> A.dot(v) + array([ 1, -3, -1], dtype=int64) + + """ + if np.isscalar(other): + return self * other + else: + return self @ other + + def power(self, n, dtype=None): + """Element-wise power.""" + return self.tocsr().power(n, dtype=dtype) + + def _broadcast_to(self, shape, copy=False): + if self.shape == shape: + return self.copy() if copy else self + else: + return self.tocsr()._broadcast_to(shape, copy) + + def __eq__(self, other): + return self.tocsr().__eq__(other) + + def __ne__(self, other): + return self.tocsr().__ne__(other) + + def __lt__(self, other): + return self.tocsr().__lt__(other) + + def __gt__(self, other): + return self.tocsr().__gt__(other) + + def __le__(self, other): + return self.tocsr().__le__(other) + + def __ge__(self, other): + return self.tocsr().__ge__(other) + + def __abs__(self): + return abs(self.tocsr()) + + def __round__(self, ndigits=0): + return round(self.tocsr(), ndigits=ndigits) + + def _add_sparse(self, other): + return self.tocsr()._add_sparse(other) + + def _add_dense(self, other): + return self.tocoo()._add_dense(other) + + def _sub_sparse(self, other): + return self.tocsr()._sub_sparse(other) + + def _sub_dense(self, other): + return self.todense() - other + + def _rsub_dense(self, other): + # note: this can't be replaced by other + (-self) for unsigned types + return other - self.todense() + + def __add__(self, other): # self + other + if isscalarlike(other): + if other == 0: + return self.copy() + # Now we would add this scalar to every element. + raise NotImplementedError('adding a nonzero scalar to a ' + 'sparse array is not supported') + elif issparse(other): + if other.shape != self.shape: + raise ValueError("inconsistent shapes") + return self._add_sparse(other) + elif isdense(other): + other = np.broadcast_to(other, self.shape) + return self._add_dense(other) + else: + return NotImplemented + + def __radd__(self,other): # other + self + return self.__add__(other) + + def __sub__(self, other): # self - other + if isscalarlike(other): + if other == 0: + return self.copy() + raise NotImplementedError('subtracting a nonzero scalar from a ' + 'sparse array is not supported') + elif issparse(other): + if other.shape != self.shape: + raise ValueError("inconsistent shapes") + return self._sub_sparse(other) + elif isdense(other): + other = np.broadcast_to(other, self.shape) + return self._sub_dense(other) + else: + return NotImplemented + + def __rsub__(self,other): # other - self + if isscalarlike(other): + if other == 0: + return -self.copy() + raise NotImplementedError('subtracting a sparse array from a ' + 'nonzero scalar is not supported') + elif isdense(other): + other = np.broadcast_to(other, self.shape) + return self._rsub_dense(other) + else: + return NotImplemented + + def _matmul_dispatch(self, other): + """np.array-like matmul & `np.matrix`-like mul, i.e. `dot` or `NotImplemented` + + interpret other and call one of the following + self._mul_scalar() + self._matmul_vector() + self._matmul_multivector() + self._matmul_sparse() + """ + # This method has to be different from `__matmul__` because it is also + # called by sparse matrix classes. + + # Currently matrix multiplication is only supported + # for 2D arrays. Hence we unpacked and use only the + # two last axes' lengths. + M, N = self._shape_as_2d + + if other.__class__ is np.ndarray: + # Fast path for the most common case + if other.shape == (N,): + return self._matmul_vector(other) + elif other.shape == (N, 1): + result = self._matmul_vector(other.ravel()) + if self.ndim == 1: + return result.reshape(1) + return result.reshape(M, 1) + elif other.ndim == 2 and other.shape[0] == N: + return self._matmul_multivector(other) + + if isscalarlike(other): + # scalar value + return self._mul_scalar(other) + + err_prefix = "matmul: dimension mismatch with signature" + if issparse(other): + if N != other.shape[0]: + raise ValueError( + f"{err_prefix} (n,k={N}),(k={other.shape[0]},m)->(n,m)" + ) + return self._matmul_sparse(other) + + # If it's a list or whatever, treat it like an array + other_a = np.asanyarray(other) + + if other_a.ndim == 0 and other_a.dtype == np.object_: + # Not interpretable as an array; return NotImplemented so that + # other's __rmatmul__ can kick in if that's implemented. + return NotImplemented + + try: + other.shape + except AttributeError: + other = other_a + + if other.ndim == 1 or other.ndim == 2 and other.shape[1] == 1: + # dense row or column vector + if other.shape[0] != N: + raise ValueError( + f"{err_prefix} (n,k={N}),(k={other.shape[0]},1?)->(n,1?)" + ) + + result = self._matmul_vector(np.ravel(other)) + + if isinstance(other, np.matrix): + result = self._ascontainer(result) + + if other.ndim == 2 and other.shape[1] == 1: + # If 'other' was an (nx1) column vector, reshape the result + if self.ndim == 1: + result = result.reshape(1) + else: + result = result.reshape(-1, 1) + + return result + + elif other.ndim == 2: + ## + # dense 2D array or matrix ("multivector") + + if other.shape[0] != N: + raise ValueError( + f"{err_prefix} (n,k={N}),(k={other.shape[0]},m)->(n,m)" + ) + + result = self._matmul_multivector(np.asarray(other)) + + if isinstance(other, np.matrix): + result = self._ascontainer(result) + + return result + + else: + raise ValueError('could not interpret dimensions') + + def __mul__(self, other): + return self.multiply(other) + + def __rmul__(self, other): # other * self + return self.multiply(other) + + # by default, use CSR for __mul__ handlers + def _mul_scalar(self, other): + return self.tocsr()._mul_scalar(other) + + def _matmul_vector(self, other): + return self.tocsr()._matmul_vector(other) + + def _matmul_multivector(self, other): + return self.tocsr()._matmul_multivector(other) + + def _matmul_sparse(self, other): + return self.tocsr()._matmul_sparse(other) + + def _rmatmul_dispatch(self, other): + if isscalarlike(other): + return self._mul_scalar(other) + else: + # Don't use asarray unless we have to + try: + tr = other.transpose() + except AttributeError: + tr = np.asarray(other).transpose() + ret = self.transpose()._matmul_dispatch(tr) + if ret is NotImplemented: + return NotImplemented + return ret.transpose() + + ####################### + # matmul (@) operator # + ####################### + + def __matmul__(self, other): + if isscalarlike(other): + raise ValueError("Scalar operands are not allowed, " + "use '*' instead") + return self._matmul_dispatch(other) + + def __rmatmul__(self, other): + if isscalarlike(other): + raise ValueError("Scalar operands are not allowed, " + "use '*' instead") + return self._rmatmul_dispatch(other) + + #################### + # Other Arithmetic # + #################### + + def _divide(self, other, true_divide=False, rdivide=False): + if isscalarlike(other): + if rdivide: + if true_divide: + return np.true_divide(other, self.todense()) + else: + return np.divide(other, self.todense()) + + if true_divide and np.can_cast(self.dtype, np.float64): + return self.astype(np.float64)._mul_scalar(1./other) + else: + r = self._mul_scalar(1./other) + + scalar_dtype = np.asarray(other).dtype + if (np.issubdtype(self.dtype, np.integer) and + np.issubdtype(scalar_dtype, np.integer)): + return r.astype(self.dtype) + else: + return r + + elif isdense(other): + if not rdivide: + if true_divide: + recip = np.true_divide(1., other) + else: + recip = np.divide(1., other) + return self.multiply(recip) + else: + if true_divide: + return np.true_divide(other, self.todense()) + else: + return np.divide(other, self.todense()) + elif issparse(other): + if rdivide: + return other._divide(self, true_divide, rdivide=False) + + self_csr = self.tocsr() + if true_divide and np.can_cast(self.dtype, np.float64): + return self_csr.astype(np.float64)._divide_sparse(other) + else: + return self_csr._divide_sparse(other) + else: + return NotImplemented + + def __truediv__(self, other): + return self._divide(other, true_divide=True) + + def __div__(self, other): + # Always do true division + return self._divide(other, true_divide=True) + + def __rtruediv__(self, other): + # Implementing this as the inverse would be too magical -- bail out + return NotImplemented + + def __rdiv__(self, other): + # Implementing this as the inverse would be too magical -- bail out + return NotImplemented + + def __neg__(self): + return -self.tocsr() + + def __iadd__(self, other): + return NotImplemented + + def __isub__(self, other): + return NotImplemented + + def __imul__(self, other): + return NotImplemented + + def __idiv__(self, other): + return self.__itruediv__(other) + + def __itruediv__(self, other): + return NotImplemented + + def __pow__(self, *args, **kwargs): + return self.power(*args, **kwargs) + + def transpose(self, axes=None, copy=False): + """ + Reverses the dimensions of the sparse array/matrix. + + Parameters + ---------- + axes : None, optional + This argument is in the signature *solely* for NumPy + compatibility reasons. Do not pass in anything except + for the default value. + copy : bool, optional + Indicates whether or not attributes of `self` should be + copied whenever possible. The degree to which attributes + are copied varies depending on the type of sparse array/matrix + being used. + + Returns + ------- + p : `self` with the dimensions reversed. + + Notes + ----- + If `self` is a `csr_array` or a `csc_array`, then this will return a + `csc_array` or a `csr_array`, respectively. + + See Also + -------- + numpy.transpose : NumPy's implementation of 'transpose' for ndarrays + """ + return self.tocsr(copy=copy).transpose(axes=axes, copy=False) + + def conjugate(self, copy=True): + """Element-wise complex conjugation. + + If the array/matrix is of non-complex data type and `copy` is False, + this method does nothing and the data is not copied. + + Parameters + ---------- + copy : bool, optional + If True, the result is guaranteed to not share data with self. + + Returns + ------- + A : The element-wise complex conjugate. + + """ + if np.issubdtype(self.dtype, np.complexfloating): + return self.tocsr(copy=copy).conjugate(copy=False) + elif copy: + return self.copy() + else: + return self + + def conj(self, copy=True): + return self.conjugate(copy=copy) + + conj.__doc__ = conjugate.__doc__ + + def _real(self): + return self.tocsr()._real() + + def _imag(self): + return self.tocsr()._imag() + + def nonzero(self): + """Nonzero indices of the array/matrix. + + Returns a tuple of arrays (row,col) containing the indices + of the non-zero elements of the array. + + Examples + -------- + >>> from scipy.sparse import csr_array + >>> A = csr_array([[1, 2, 0], [0, 0, 3], [4, 0, 5]]) + >>> A.nonzero() + (array([0, 0, 1, 2, 2], dtype=int32), array([0, 1, 2, 0, 2], dtype=int32)) + + """ + + # convert to COOrdinate format + A = self.tocoo() + nz_mask = A.data != 0 + return tuple(idx[nz_mask] for idx in A.coords) + + def _getcol(self, j): + """Returns a copy of column j of the array, as an (m x 1) sparse + array (column vector). + """ + if self.ndim == 1: + raise ValueError("getcol not provided for 1d arrays. Use indexing A[j]") + # Subclasses should override this method for efficiency. + # Post-multiply by a (n x 1) column vector 'a' containing all zeros + # except for a_j = 1 + N = self.shape[-1] + if j < 0: + j += N + if j < 0 or j >= N: + raise IndexError("index out of bounds") + col_selector = self._csc_container(([1], [[j], [0]]), + shape=(N, 1), dtype=self.dtype) + result = self @ col_selector + return result + + def _getrow(self, i): + """Returns a copy of row i of the array, as a (1 x n) sparse + array (row vector). + """ + if self.ndim == 1: + raise ValueError("getrow not meaningful for a 1d array") + # Subclasses should override this method for efficiency. + # Pre-multiply by a (1 x m) row vector 'a' containing all zeros + # except for a_i = 1 + M = self.shape[0] + if i < 0: + i += M + if i < 0 or i >= M: + raise IndexError("index out of bounds") + row_selector = self._csr_container(([1], [[0], [i]]), + shape=(1, M), dtype=self.dtype) + return row_selector @ self + + # The following dunder methods cannot be implemented. + # + # def __array__(self): + # # Sparse matrices rely on NumPy wrapping them in object arrays under + # # the hood to make unary ufuncs work on them. So we cannot raise + # # TypeError here - which would be handy to not give users object + # # arrays they probably don't want (they're looking for `.toarray()`). + # # + # # Conversion with `toarray()` would also break things because of the + # # behavior discussed above, plus we want to avoid densification by + # # accident because that can too easily blow up memory. + # + # def __array_ufunc__(self): + # # We cannot implement __array_ufunc__ due to mismatching semantics. + # # See gh-7707 and gh-7349 for details. + # + # def __array_function__(self): + # # We cannot implement __array_function__ due to mismatching semantics. + # # See gh-10362 for details. + + def todense(self, order=None, out=None): + """ + Return a dense representation of this sparse array. + + Parameters + ---------- + order : {'C', 'F'}, optional + Whether to store multi-dimensional data in C (row-major) + or Fortran (column-major) order in memory. The default + is 'None', which provides no ordering guarantees. + Cannot be specified in conjunction with the `out` + argument. + + out : ndarray, 2-D, optional + If specified, uses this array as the output buffer + instead of allocating a new array to return. The + provided array must have the same shape and dtype as + the sparse array on which you are calling the method. + + Returns + ------- + arr : ndarray, 2-D + An array with the same shape and containing the same + data represented by the sparse array, with the requested + memory order. If `out` was passed, the same object is + returned after being modified in-place to contain the + appropriate values. + """ + return self._ascontainer(self.toarray(order=order, out=out)) + + def toarray(self, order=None, out=None): + """ + Return a dense ndarray representation of this sparse array/matrix. + + Parameters + ---------- + order : {'C', 'F'}, optional + Whether to store multidimensional data in C (row-major) + or Fortran (column-major) order in memory. The default + is 'None', which provides no ordering guarantees. + Cannot be specified in conjunction with the `out` + argument. + + out : ndarray, 2-D, optional + If specified, uses this array as the output buffer + instead of allocating a new array to return. The provided + array must have the same shape and dtype as the sparse + array/matrix on which you are calling the method. For most + sparse types, `out` is required to be memory contiguous + (either C or Fortran ordered). + + Returns + ------- + arr : ndarray, 2-D + An array with the same shape and containing the same + data represented by the sparse array/matrix, with the requested + memory order. If `out` was passed, the same object is + returned after being modified in-place to contain the + appropriate values. + """ + return self.tocoo(copy=False).toarray(order=order, out=out) + + # Any sparse array format deriving from _spbase must define one of + # tocsr or tocoo. The other conversion methods may be implemented for + # efficiency, but are not required. + def tocsr(self, copy=False): + """Convert this array/matrix to Compressed Sparse Row format. + + With copy=False, the data/indices may be shared between this array/matrix and + the resultant csr_array/matrix. + """ + return self.tocoo(copy=copy).tocsr(copy=False) + + def todok(self, copy=False): + """Convert this array/matrix to Dictionary Of Keys format. + + With copy=False, the data/indices may be shared between this array/matrix and + the resultant dok_array/matrix. + """ + return self.tocoo(copy=copy).todok(copy=False) + + def tocoo(self, copy=False): + """Convert this array/matrix to COOrdinate format. + + With copy=False, the data/indices may be shared between this array/matrix and + the resultant coo_array/matrix. + """ + return self.tocsr(copy=False).tocoo(copy=copy) + + def tolil(self, copy=False): + """Convert this array/matrix to List of Lists format. + + With copy=False, the data/indices may be shared between this array/matrix and + the resultant lil_array/matrix. + """ + return self.tocsr(copy=False).tolil(copy=copy) + + def todia(self, copy=False): + """Convert this array/matrix to sparse DIAgonal format. + + With copy=False, the data/indices may be shared between this array/matrix and + the resultant dia_array/matrix. + """ + return self.tocoo(copy=copy).todia(copy=False) + + def tobsr(self, blocksize=None, copy=False): + """Convert this array/matrix to Block Sparse Row format. + + With copy=False, the data/indices may be shared between this array/matrix and + the resultant bsr_array/matrix. + + When blocksize=(R, C) is provided, it will be used for construction of + the bsr_array/matrix. + """ + return self.tocsr(copy=False).tobsr(blocksize=blocksize, copy=copy) + + def tocsc(self, copy=False): + """Convert this array/matrix to Compressed Sparse Column format. + + With copy=False, the data/indices may be shared between this array/matrix and + the resultant csc_array/matrix. + """ + return self.tocsr(copy=copy).tocsc(copy=False) + + def copy(self): + """Returns a copy of this array/matrix. + + No data/indices will be shared between the returned value and current + array/matrix. + """ + return self.__class__(self, copy=True) + + def sum(self, axis=None, dtype=None, out=None): + """ + Sum the array/matrix elements over a given axis. + + Parameters + ---------- + axis : {-2, -1, 0, 1, None} optional + Axis along which the sum is computed. The default is to + compute the sum of all the array/matrix elements, returning a scalar + (i.e., `axis` = `None`). + dtype : dtype, optional + The type of the returned array/matrix and of the accumulator in which + the elements are summed. The dtype of `a` is used by default + unless `a` has an integer dtype of less precision than the default + platform integer. In that case, if `a` is signed then the platform + integer is used while if `a` is unsigned then an unsigned integer + of the same precision as the platform integer is used. + + .. versionadded:: 0.18.0 + + out : np.matrix, optional + Alternative output matrix in which to place the result. It must + have the same shape as the expected output, but the type of the + output values will be cast if necessary. + + .. versionadded:: 0.18.0 + + Returns + ------- + sum_along_axis : np.matrix + A matrix with the same shape as `self`, with the specified + axis removed. + + See Also + -------- + numpy.matrix.sum : NumPy's implementation of 'sum' for matrices + + """ + validateaxis(axis) + + # Mimic numpy's casting. + res_dtype = get_sum_dtype(self.dtype) + + if self.ndim == 1: + if axis not in (None, -1, 0): + raise ValueError("axis must be None, -1 or 0") + res = self @ np.ones(self.shape, dtype=res_dtype) + return res.sum(dtype=dtype, out=out) + + # We use multiplication by a matrix of ones to achieve this. + # For some sparse array formats more efficient methods are + # possible -- these should override this function. + M, N = self.shape + + if axis is None: + # sum over rows and columns + return ( + self @ self._ascontainer(np.ones((N, 1), dtype=res_dtype)) + ).sum(dtype=dtype, out=out) + + if axis < 0: + axis += 2 + + # axis = 0 or 1 now + if axis == 0: + # sum over columns + ret = self._ascontainer( + np.ones((1, M), dtype=res_dtype) + ) @ self + else: + # sum over rows + ret = self @ self._ascontainer( + np.ones((N, 1), dtype=res_dtype) + ) + + return ret.sum(axis=axis, dtype=dtype, out=out) + + def mean(self, axis=None, dtype=None, out=None): + """ + Compute the arithmetic mean along the specified axis. + + Returns the average of the array/matrix elements. The average is taken + over all elements in the array/matrix by default, otherwise over the + specified axis. `float64` intermediate and return values are used + for integer inputs. + + Parameters + ---------- + axis : {-2, -1, 0, 1, None} optional + Axis along which the mean is computed. The default is to compute + the mean of all elements in the array/matrix (i.e., `axis` = `None`). + dtype : data-type, optional + Type to use in computing the mean. For integer inputs, the default + is `float64`; for floating point inputs, it is the same as the + input dtype. + + .. versionadded:: 0.18.0 + + out : np.matrix, optional + Alternative output matrix in which to place the result. It must + have the same shape as the expected output, but the type of the + output values will be cast if necessary. + + .. versionadded:: 0.18.0 + + Returns + ------- + m : np.matrix + + See Also + -------- + numpy.matrix.mean : NumPy's implementation of 'mean' for matrices + + """ + validateaxis(axis) + + res_dtype = self.dtype.type + integral = (np.issubdtype(self.dtype, np.integer) or + np.issubdtype(self.dtype, np.bool_)) + + # output dtype + if dtype is None: + if integral: + res_dtype = np.float64 + else: + res_dtype = np.dtype(dtype).type + + # intermediate dtype for summation + inter_dtype = np.float64 if integral else res_dtype + inter_self = self.astype(inter_dtype) + + if self.ndim == 1: + if axis not in (None, -1, 0): + raise ValueError("axis must be None, -1 or 0") + res = inter_self / self.shape[-1] + return res.sum(dtype=res_dtype, out=out) + + if axis is None: + return (inter_self / (self.shape[0] * self.shape[1]))\ + .sum(dtype=res_dtype, out=out) + + if axis < 0: + axis += 2 + + # axis = 0 or 1 now + if axis == 0: + return (inter_self * (1.0 / self.shape[0])).sum( + axis=0, dtype=res_dtype, out=out) + else: + return (inter_self * (1.0 / self.shape[1])).sum( + axis=1, dtype=res_dtype, out=out) + + def diagonal(self, k=0): + """Returns the kth diagonal of the array/matrix. + + Parameters + ---------- + k : int, optional + Which diagonal to get, corresponding to elements a[i, i+k]. + Default: 0 (the main diagonal). + + .. versionadded:: 1.0 + + See also + -------- + numpy.diagonal : Equivalent numpy function. + + Examples + -------- + >>> from scipy.sparse import csr_array + >>> A = csr_array([[1, 2, 0], [0, 0, 3], [4, 0, 5]]) + >>> A.diagonal() + array([1, 0, 5]) + >>> A.diagonal(k=1) + array([2, 3]) + """ + return self.tocsr().diagonal(k=k) + + def trace(self, offset=0): + """Returns the sum along diagonals of the sparse array/matrix. + + Parameters + ---------- + offset : int, optional + Which diagonal to get, corresponding to elements a[i, i+offset]. + Default: 0 (the main diagonal). + + """ + return self.diagonal(k=offset).sum() + + def setdiag(self, values, k=0): + """ + Set diagonal or off-diagonal elements of the array/matrix. + + Parameters + ---------- + values : array_like + New values of the diagonal elements. + + Values may have any length. If the diagonal is longer than values, + then the remaining diagonal entries will not be set. If values are + longer than the diagonal, then the remaining values are ignored. + + If a scalar value is given, all of the diagonal is set to it. + + k : int, optional + Which off-diagonal to set, corresponding to elements a[i,i+k]. + Default: 0 (the main diagonal). + + """ + M, N = self.shape + if (k > 0 and k >= N) or (k < 0 and -k >= M): + raise ValueError("k exceeds array dimensions") + self._setdiag(np.asarray(values), k) + + def _setdiag(self, values, k): + """This part of the implementation gets overridden by the + different formats. + """ + M, N = self.shape + if k < 0: + if values.ndim == 0: + # broadcast + max_index = min(M+k, N) + for i in range(max_index): + self[i - k, i] = values + else: + max_index = min(M+k, N, len(values)) + if max_index <= 0: + return + for i, v in enumerate(values[:max_index]): + self[i - k, i] = v + else: + if values.ndim == 0: + # broadcast + max_index = min(M, N-k) + for i in range(max_index): + self[i, i + k] = values + else: + max_index = min(M, N-k, len(values)) + if max_index <= 0: + return + for i, v in enumerate(values[:max_index]): + self[i, i + k] = v + + def _process_toarray_args(self, order, out): + if out is not None: + if order is not None: + raise ValueError('order cannot be specified if out ' + 'is not None') + if out.shape != self.shape or out.dtype != self.dtype: + raise ValueError('out array must be same dtype and shape as ' + 'sparse array') + out[...] = 0. + return out + else: + return np.zeros(self.shape, dtype=self.dtype, order=order) + + def _get_index_dtype(self, arrays=(), maxval=None, check_contents=False): + """ + Determine index dtype for array. + + This wraps _sputils.get_index_dtype, providing compatibility for both + array and matrix API sparse matrices. Matrix API sparse matrices would + attempt to downcast the indices - which can be computationally + expensive and undesirable for users. The array API changes this + behaviour. + + See discussion: https://github.com/scipy/scipy/issues/16774 + + The get_index_dtype import is due to implementation details of the test + suite. It allows the decorator ``with_64bit_maxval_limit`` to mock a + lower int32 max value for checks on the matrix API's downcasting + behaviour. + """ + from ._sputils import get_index_dtype + + # Don't check contents for array API + return get_index_dtype(arrays, + maxval, + (check_contents and not isinstance(self, sparray))) + + +class sparray: + """A namespace class to separate sparray from spmatrix""" + + +sparray.__doc__ = _spbase.__doc__ + + +def issparse(x): + """Is `x` of a sparse array or sparse matrix type? + + Parameters + ---------- + x + object to check for being a sparse array or sparse matrix + + Returns + ------- + bool + True if `x` is a sparse array or a sparse matrix, False otherwise + + Notes + ----- + Use `isinstance(x, sp.sparse.sparray)` to check between an array or matrix. + Use `a.format` to check the sparse format, e.g. `a.format == 'csr'`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csr_array, csr_matrix, issparse + >>> issparse(csr_matrix([[5]])) + True + >>> issparse(csr_array([[5]])) + True + >>> issparse(np.array([[5]])) + False + >>> issparse(5) + False + """ + return isinstance(x, _spbase) + + +def isspmatrix(x): + """Is `x` of a sparse matrix type? + + Parameters + ---------- + x + object to check for being a sparse matrix + + Returns + ------- + bool + True if `x` is a sparse matrix, False otherwise + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csr_array, csr_matrix, isspmatrix + >>> isspmatrix(csr_matrix([[5]])) + True + >>> isspmatrix(csr_array([[5]])) + False + >>> isspmatrix(np.array([[5]])) + False + >>> isspmatrix(5) + False + """ + return isinstance(x, spmatrix) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_bsr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_bsr.py new file mode 100644 index 0000000000000000000000000000000000000000..14cf85f8ccf8d7d2b4c63d19938b215e54a9736b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_bsr.py @@ -0,0 +1,877 @@ +"""Compressed Block Sparse Row format""" + +__docformat__ = "restructuredtext en" + +__all__ = ['bsr_array', 'bsr_matrix', 'isspmatrix_bsr'] + +from warnings import warn + +import numpy as np + +from scipy._lib._util import copy_if_needed +from ._matrix import spmatrix +from ._data import _data_matrix, _minmax_mixin +from ._compressed import _cs_matrix +from ._base import issparse, _formats, _spbase, sparray +from ._sputils import (isshape, getdtype, getdata, to_native, upcast, + check_shape) +from . import _sparsetools +from ._sparsetools import (bsr_matvec, bsr_matvecs, csr_matmat_maxnnz, + bsr_matmat, bsr_transpose, bsr_sort_indices, + bsr_tocsr) + + +class _bsr_base(_cs_matrix, _minmax_mixin): + _format = 'bsr' + + def __init__(self, arg1, shape=None, dtype=None, copy=False, + blocksize=None, *, maxprint=None): + _data_matrix.__init__(self, arg1, maxprint=maxprint) + + if issparse(arg1): + if arg1.format == self.format and copy: + arg1 = arg1.copy() + else: + arg1 = arg1.tobsr(blocksize=blocksize) + self.indptr, self.indices, self.data, self._shape = ( + arg1.indptr, arg1.indices, arg1.data, arg1._shape + ) + + elif isinstance(arg1,tuple): + if isshape(arg1): + # it's a tuple of matrix dimensions (M,N) + self._shape = check_shape(arg1) + M,N = self.shape + # process blocksize + if blocksize is None: + blocksize = (1,1) + else: + if not isshape(blocksize): + raise ValueError(f'invalid blocksize={blocksize}') + blocksize = tuple(blocksize) + self.data = np.zeros((0,) + blocksize, getdtype(dtype, default=float)) + + R,C = blocksize + if (M % R) != 0 or (N % C) != 0: + raise ValueError('shape must be multiple of blocksize') + + # Select index dtype large enough to pass array and + # scalar parameters to sparsetools + idx_dtype = self._get_index_dtype(maxval=max(M//R, N//C, R, C)) + self.indices = np.zeros(0, dtype=idx_dtype) + self.indptr = np.zeros(M//R + 1, dtype=idx_dtype) + + elif len(arg1) == 2: + # (data,(row,col)) format + coo = self._coo_container(arg1, dtype=dtype, shape=shape) + bsr = coo.tobsr(blocksize=blocksize) + self.indptr, self.indices, self.data, self._shape = ( + bsr.indptr, bsr.indices, bsr.data, bsr._shape + ) + + elif len(arg1) == 3: + # (data,indices,indptr) format + (data, indices, indptr) = arg1 + + # Select index dtype large enough to pass array and + # scalar parameters to sparsetools + maxval = 1 + if shape is not None: + maxval = max(shape) + if blocksize is not None: + maxval = max(maxval, max(blocksize)) + idx_dtype = self._get_index_dtype((indices, indptr), maxval=maxval, + check_contents=True) + if not copy: + copy = copy_if_needed + self.indices = np.array(indices, copy=copy, dtype=idx_dtype) + self.indptr = np.array(indptr, copy=copy, dtype=idx_dtype) + self.data = getdata(data, copy=copy, dtype=dtype) + if self.data.ndim != 3: + raise ValueError( + f'BSR data must be 3-dimensional, got shape={self.data.shape}' + ) + if blocksize is not None: + if not isshape(blocksize): + raise ValueError(f'invalid blocksize={blocksize}') + if tuple(blocksize) != self.data.shape[1:]: + raise ValueError( + f'mismatching blocksize={blocksize}' + f' vs {self.data.shape[1:]}' + ) + else: + raise ValueError('unrecognized bsr_array constructor usage') + else: + # must be dense + try: + arg1 = np.asarray(arg1) + except Exception as e: + raise ValueError("unrecognized form for " + f"{self.format}_matrix constructor") from e + if isinstance(self, sparray) and arg1.ndim != 2: + raise ValueError(f"BSR arrays don't support {arg1.ndim}D input. Use 2D") + arg1 = self._coo_container(arg1, dtype=dtype).tobsr(blocksize=blocksize) + self.indptr, self.indices, self.data, self._shape = ( + arg1.indptr, arg1.indices, arg1.data, arg1._shape + ) + + if shape is not None: + self._shape = check_shape(shape) + else: + if self.shape is None: + # shape not already set, try to infer dimensions + try: + M = len(self.indptr) - 1 + N = self.indices.max() + 1 + except Exception as e: + raise ValueError('unable to infer matrix dimensions') from e + else: + R,C = self.blocksize + self._shape = check_shape((M*R,N*C)) + + if self.shape is None: + if shape is None: + # TODO infer shape here + raise ValueError('need to infer shape') + else: + self._shape = check_shape(shape) + + if dtype is not None: + self.data = self.data.astype(getdtype(dtype, self.data), copy=False) + + self.check_format(full_check=False) + + def check_format(self, full_check=True): + """Check whether the array/matrix respects the BSR format. + + Parameters + ---------- + full_check : bool, optional + If `True`, run rigorous check, scanning arrays for valid values. + Note that activating those check might copy arrays for casting, + modifying indices and index pointers' inplace. + If `False`, run basic checks on attributes. O(1) operations. + Default is `True`. + """ + M,N = self.shape + R,C = self.blocksize + + # index arrays should have integer data types + if self.indptr.dtype.kind != 'i': + warn(f"indptr array has non-integer dtype ({self.indptr.dtype.name})", + stacklevel=2) + if self.indices.dtype.kind != 'i': + warn(f"indices array has non-integer dtype ({self.indices.dtype.name})", + stacklevel=2) + + # check array shapes + if self.indices.ndim != 1 or self.indptr.ndim != 1: + raise ValueError("indices, and indptr should be 1-D") + if self.data.ndim != 3: + raise ValueError("data should be 3-D") + + # check index pointer + if (len(self.indptr) != M//R + 1): + raise ValueError("index pointer size (%d) should be (%d)" % + (len(self.indptr), M//R + 1)) + if (self.indptr[0] != 0): + raise ValueError("index pointer should start with 0") + + # check index and data arrays + if (len(self.indices) != len(self.data)): + raise ValueError("indices and data should have the same size") + if (self.indptr[-1] > len(self.indices)): + raise ValueError("Last value of index pointer should be less than " + "the size of index and data arrays") + + self.prune() + + if full_check: + # check format validity (more expensive) + if self.nnz > 0: + if self.indices.max() >= N//C: + raise ValueError("column index values must be < %d (now max %d)" + % (N//C, self.indices.max())) + if self.indices.min() < 0: + raise ValueError("column index values must be >= 0") + if np.diff(self.indptr).min() < 0: + raise ValueError("index pointer values must form a " + "non-decreasing sequence") + + idx_dtype = self._get_index_dtype((self.indices, self.indptr)) + self.indptr = np.asarray(self.indptr, dtype=idx_dtype) + self.indices = np.asarray(self.indices, dtype=idx_dtype) + self.data = to_native(self.data) + # if not self.has_sorted_indices(): + # warn('Indices were not in sorted order. Sorting indices.') + # self.sort_indices(check_first=False) + + @property + def blocksize(self) -> tuple: + """Block size of the matrix.""" + return self.data.shape[1:] + + def _getnnz(self, axis=None): + if axis is not None: + raise NotImplementedError("_getnnz over an axis is not implemented " + "for BSR format") + R, C = self.blocksize + return int(self.indptr[-1]) * R * C + + _getnnz.__doc__ = _spbase._getnnz.__doc__ + + def count_nonzero(self, axis=None): + if axis is not None: + raise NotImplementedError( + "count_nonzero over axis is not implemented for BSR format." + ) + return np.count_nonzero(self._deduped_data()) + + count_nonzero.__doc__ = _spbase.count_nonzero.__doc__ + + def __repr__(self): + _, fmt = _formats[self.format] + sparse_cls = 'array' if isinstance(self, sparray) else 'matrix' + b = 'x'.join(str(x) for x in self.blocksize) + return ( + f"<{fmt} sparse {sparse_cls} of dtype '{self.dtype}'\n" + f"\twith {self.nnz} stored elements (blocksize={b}) and shape {self.shape}>" + ) + + def diagonal(self, k=0): + rows, cols = self.shape + if k <= -rows or k >= cols: + return np.empty(0, dtype=self.data.dtype) + R, C = self.blocksize + y = np.zeros(min(rows + min(k, 0), cols - max(k, 0)), + dtype=upcast(self.dtype)) + _sparsetools.bsr_diagonal(k, rows // R, cols // C, R, C, + self.indptr, self.indices, + np.ravel(self.data), y) + return y + + diagonal.__doc__ = _spbase.diagonal.__doc__ + + ########################## + # NotImplemented methods # + ########################## + + def __getitem__(self,key): + raise NotImplementedError + + def __setitem__(self,key,val): + raise NotImplementedError + + ###################### + # Arithmetic methods # + ###################### + + def _add_dense(self, other): + return self.tocoo(copy=False)._add_dense(other) + + def _matmul_vector(self, other): + M,N = self.shape + R,C = self.blocksize + + result = np.zeros(self.shape[0], dtype=upcast(self.dtype, other.dtype)) + + bsr_matvec(M//R, N//C, R, C, + self.indptr, self.indices, self.data.ravel(), + other, result) + + return result + + def _matmul_multivector(self,other): + R,C = self.blocksize + M,N = self.shape + n_vecs = other.shape[1] # number of column vectors + + result = np.zeros((M,n_vecs), dtype=upcast(self.dtype,other.dtype)) + + bsr_matvecs(M//R, N//C, n_vecs, R, C, + self.indptr, self.indices, self.data.ravel(), + other.ravel(), result.ravel()) + + return result + + def _matmul_sparse(self, other): + M, K1 = self.shape + K2, N = other.shape + + R,n = self.blocksize + + # convert to this format + if other.format == "bsr": + C = other.blocksize[1] + else: + C = 1 + + if other.format == "csr" and n == 1: + other = other.tobsr(blocksize=(n,C), copy=False) # lightweight conversion + else: + other = other.tobsr(blocksize=(n,C)) + + idx_dtype = self._get_index_dtype((self.indptr, self.indices, + other.indptr, other.indices)) + + bnnz = csr_matmat_maxnnz(M//R, N//C, + self.indptr.astype(idx_dtype), + self.indices.astype(idx_dtype), + other.indptr.astype(idx_dtype), + other.indices.astype(idx_dtype)) + + idx_dtype = self._get_index_dtype((self.indptr, self.indices, + other.indptr, other.indices), + maxval=bnnz) + indptr = np.empty(self.indptr.shape, dtype=idx_dtype) + indices = np.empty(bnnz, dtype=idx_dtype) + data = np.empty(R*C*bnnz, dtype=upcast(self.dtype,other.dtype)) + + bsr_matmat(bnnz, M//R, N//C, R, C, n, + self.indptr.astype(idx_dtype), + self.indices.astype(idx_dtype), + np.ravel(self.data), + other.indptr.astype(idx_dtype), + other.indices.astype(idx_dtype), + np.ravel(other.data), + indptr, + indices, + data) + + data = data.reshape(-1,R,C) + + # TODO eliminate zeros + + return self._bsr_container( + (data, indices, indptr), shape=(M, N), blocksize=(R, C) + ) + + ###################### + # Conversion methods # + ###################### + + def tobsr(self, blocksize=None, copy=False): + """Convert this array/matrix into Block Sparse Row Format. + + With copy=False, the data/indices may be shared between this + array/matrix and the resultant bsr_array/bsr_matrix. + + If blocksize=(R, C) is provided, it will be used for determining + block size of the bsr_array/bsr_matrix. + """ + if blocksize not in [None, self.blocksize]: + return self.tocsr().tobsr(blocksize=blocksize) + if copy: + return self.copy() + else: + return self + + def tocsr(self, copy=False): + M, N = self.shape + R, C = self.blocksize + nnz = self.nnz + idx_dtype = self._get_index_dtype((self.indptr, self.indices), + maxval=max(nnz, N)) + indptr = np.empty(M + 1, dtype=idx_dtype) + indices = np.empty(nnz, dtype=idx_dtype) + data = np.empty(nnz, dtype=upcast(self.dtype)) + + bsr_tocsr(M // R, # n_brow + N // C, # n_bcol + R, C, + self.indptr.astype(idx_dtype, copy=False), + self.indices.astype(idx_dtype, copy=False), + self.data, + indptr, + indices, + data) + return self._csr_container((data, indices, indptr), shape=self.shape) + + tocsr.__doc__ = _spbase.tocsr.__doc__ + + def tocsc(self, copy=False): + return self.tocsr(copy=False).tocsc(copy=copy) + + tocsc.__doc__ = _spbase.tocsc.__doc__ + + def tocoo(self, copy=True): + """Convert this array/matrix to COOrdinate format. + + When copy=False the data array will be shared between + this array/matrix and the resultant coo_array/coo_matrix. + """ + + M,N = self.shape + R,C = self.blocksize + + indptr_diff = np.diff(self.indptr) + if indptr_diff.dtype.itemsize > np.dtype(np.intp).itemsize: + # Check for potential overflow + indptr_diff_limited = indptr_diff.astype(np.intp) + if np.any(indptr_diff_limited != indptr_diff): + raise ValueError("Matrix too big to convert") + indptr_diff = indptr_diff_limited + + idx_dtype = self._get_index_dtype(maxval=max(M, N)) + row = (R * np.arange(M//R, dtype=idx_dtype)).repeat(indptr_diff) + row = row.repeat(R*C).reshape(-1,R,C) + row += np.tile(np.arange(R, dtype=idx_dtype).reshape(-1,1), (1,C)) + row = row.reshape(-1) + + col = ((C * self.indices).astype(idx_dtype, copy=False) + .repeat(R*C).reshape(-1,R,C)) + col += np.tile(np.arange(C, dtype=idx_dtype), (R,1)) + col = col.reshape(-1) + + data = self.data.reshape(-1) + + if copy: + data = data.copy() + + return self._coo_container( + (data, (row, col)), shape=self.shape + ) + + def toarray(self, order=None, out=None): + return self.tocoo(copy=False).toarray(order=order, out=out) + + toarray.__doc__ = _spbase.toarray.__doc__ + + def transpose(self, axes=None, copy=False): + if axes is not None and axes != (1, 0): + raise ValueError("Sparse matrices do not support " + "an 'axes' parameter because swapping " + "dimensions is the only logical permutation.") + + R, C = self.blocksize + M, N = self.shape + NBLK = self.nnz//(R*C) + + if self.nnz == 0: + return self._bsr_container((N, M), blocksize=(C, R), + dtype=self.dtype, copy=copy) + + indptr = np.empty(N//C + 1, dtype=self.indptr.dtype) + indices = np.empty(NBLK, dtype=self.indices.dtype) + data = np.empty((NBLK, C, R), dtype=self.data.dtype) + + bsr_transpose(M//R, N//C, R, C, + self.indptr, self.indices, self.data.ravel(), + indptr, indices, data.ravel()) + + return self._bsr_container((data, indices, indptr), + shape=(N, M), copy=copy) + + transpose.__doc__ = _spbase.transpose.__doc__ + + ############################################################## + # methods that examine or modify the internal data structure # + ############################################################## + + def eliminate_zeros(self): + """Remove zero elements in-place.""" + + if not self.nnz: + return # nothing to do + + R,C = self.blocksize + M,N = self.shape + + mask = (self.data != 0).reshape(-1,R*C).sum(axis=1) # nonzero blocks + + nonzero_blocks = mask.nonzero()[0] + + self.data[:len(nonzero_blocks)] = self.data[nonzero_blocks] + + # modifies self.indptr and self.indices *in place* + _sparsetools.csr_eliminate_zeros(M//R, N//C, self.indptr, + self.indices, mask) + self.prune() + + def sum_duplicates(self): + """Eliminate duplicate array/matrix entries by adding them together + + The is an *in place* operation + """ + if self.has_canonical_format: + return + self.sort_indices() + R, C = self.blocksize + M, N = self.shape + + # port of _sparsetools.csr_sum_duplicates + n_row = M // R + nnz = 0 + row_end = 0 + for i in range(n_row): + jj = row_end + row_end = self.indptr[i+1] + while jj < row_end: + j = self.indices[jj] + x = self.data[jj] + jj += 1 + while jj < row_end and self.indices[jj] == j: + x += self.data[jj] + jj += 1 + self.indices[nnz] = j + self.data[nnz] = x + nnz += 1 + self.indptr[i+1] = nnz + + self.prune() # nnz may have changed + self.has_canonical_format = True + + def sort_indices(self): + """Sort the indices of this array/matrix *in place* + """ + if self.has_sorted_indices: + return + + R,C = self.blocksize + M,N = self.shape + + bsr_sort_indices(M//R, N//C, R, C, self.indptr, self.indices, self.data.ravel()) + + self.has_sorted_indices = True + + def prune(self): + """Remove empty space after all non-zero elements. + """ + + R,C = self.blocksize + M,N = self.shape + + if len(self.indptr) != M//R + 1: + raise ValueError("index pointer has invalid length") + + bnnz = self.indptr[-1] + + if len(self.indices) < bnnz: + raise ValueError("indices array has too few elements") + if len(self.data) < bnnz: + raise ValueError("data array has too few elements") + + self.data = self.data[:bnnz] + self.indices = self.indices[:bnnz] + + # utility functions + def _binopt(self, other, op, in_shape=None, out_shape=None): + """Apply the binary operation fn to two sparse matrices.""" + + # Ideally we'd take the GCDs of the blocksize dimensions + # and explode self and other to match. + other = self.__class__(other, blocksize=self.blocksize) + + # e.g. bsr_plus_bsr, etc. + fn = getattr(_sparsetools, self.format + op + self.format) + + R,C = self.blocksize + + max_bnnz = len(self.data) + len(other.data) + idx_dtype = self._get_index_dtype((self.indptr, self.indices, + other.indptr, other.indices), + maxval=max_bnnz) + indptr = np.empty(self.indptr.shape, dtype=idx_dtype) + indices = np.empty(max_bnnz, dtype=idx_dtype) + + bool_ops = ['_ne_', '_lt_', '_gt_', '_le_', '_ge_'] + if op in bool_ops: + data = np.empty(R*C*max_bnnz, dtype=np.bool_) + else: + data = np.empty(R*C*max_bnnz, dtype=upcast(self.dtype,other.dtype)) + + fn(self.shape[0]//R, self.shape[1]//C, R, C, + self.indptr.astype(idx_dtype), + self.indices.astype(idx_dtype), + self.data, + other.indptr.astype(idx_dtype), + other.indices.astype(idx_dtype), + np.ravel(other.data), + indptr, + indices, + data) + + actual_bnnz = indptr[-1] + indices = indices[:actual_bnnz] + data = data[:R*C*actual_bnnz] + + if actual_bnnz < max_bnnz/2: + indices = indices.copy() + data = data.copy() + + data = data.reshape(-1,R,C) + + return self.__class__((data, indices, indptr), shape=self.shape) + + # needed by _data_matrix + def _with_data(self,data,copy=True): + """Returns a matrix with the same sparsity structure as self, + but with different data. By default the structure arrays + (i.e. .indptr and .indices) are copied. + """ + if copy: + return self.__class__((data,self.indices.copy(),self.indptr.copy()), + shape=self.shape,dtype=data.dtype) + else: + return self.__class__((data,self.indices,self.indptr), + shape=self.shape,dtype=data.dtype) + +# # these functions are used by the parent class +# # to remove redundancy between bsc_matrix and bsr_matrix +# def _swap(self,x): +# """swap the members of x if this is a column-oriented matrix +# """ +# return (x[0],x[1]) + + def _broadcast_to(self, shape, copy=False): + return _spbase._broadcast_to(self, shape, copy) + + +def isspmatrix_bsr(x): + """Is `x` of a bsr_matrix type? + + Parameters + ---------- + x + object to check for being a bsr matrix + + Returns + ------- + bool + True if `x` is a bsr matrix, False otherwise + + Examples + -------- + >>> from scipy.sparse import bsr_array, bsr_matrix, csr_matrix, isspmatrix_bsr + >>> isspmatrix_bsr(bsr_matrix([[5]])) + True + >>> isspmatrix_bsr(bsr_array([[5]])) + False + >>> isspmatrix_bsr(csr_matrix([[5]])) + False + """ + return isinstance(x, bsr_matrix) + + +# This namespace class separates array from matrix with isinstance +class bsr_array(_bsr_base, sparray): + """ + Block Sparse Row format sparse array. + + This can be instantiated in several ways: + bsr_array(D, [blocksize=(R,C)]) + where D is a 2-D ndarray. + + bsr_array(S, [blocksize=(R,C)]) + with another sparse array or matrix S (equivalent to S.tobsr()) + + bsr_array((M, N), [blocksize=(R,C), dtype]) + to construct an empty sparse array with shape (M, N) + dtype is optional, defaulting to dtype='d'. + + bsr_array((data, ij), [blocksize=(R,C), shape=(M, N)]) + where ``data`` and ``ij`` satisfy ``a[ij[0, k], ij[1, k]] = data[k]`` + + bsr_array((data, indices, indptr), [shape=(M, N)]) + is the standard BSR representation where the block column + indices for row i are stored in ``indices[indptr[i]:indptr[i+1]]`` + and their corresponding block values are stored in + ``data[ indptr[i]: indptr[i+1] ]``. If the shape parameter is not + supplied, the array dimensions are inferred from the index arrays. + + Attributes + ---------- + dtype : dtype + Data type of the array + shape : 2-tuple + Shape of the array + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + BSR format data array of the array + indices + BSR format index array of the array + indptr + BSR format index pointer array of the array + blocksize + Block size + has_sorted_indices : bool + Whether indices are sorted + has_canonical_format : bool + T + + Notes + ----- + Sparse arrays can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + **Summary of BSR format** + + The Block Sparse Row (BSR) format is very similar to the Compressed + Sparse Row (CSR) format. BSR is appropriate for sparse matrices with dense + sub matrices like the last example below. Such sparse block matrices often + arise in vector-valued finite element discretizations. In such cases, BSR is + considerably more efficient than CSR and CSC for many sparse arithmetic + operations. + + **Blocksize** + + The blocksize (R,C) must evenly divide the shape of the sparse array (M,N). + That is, R and C must satisfy the relationship ``M % R = 0`` and + ``N % C = 0``. + + If no blocksize is specified, a simple heuristic is applied to determine + an appropriate blocksize. + + **Canonical Format** + + In canonical format, there are no duplicate blocks and indices are sorted + per row. + + **Limitations** + + Block Sparse Row format sparse arrays do not support slicing. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import bsr_array + >>> bsr_array((3, 4), dtype=np.int8).toarray() + array([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=int8) + + >>> row = np.array([0, 0, 1, 2, 2, 2]) + >>> col = np.array([0, 2, 2, 0, 1, 2]) + >>> data = np.array([1, 2, 3 ,4, 5, 6]) + >>> bsr_array((data, (row, col)), shape=(3, 3)).toarray() + array([[1, 0, 2], + [0, 0, 3], + [4, 5, 6]]) + + >>> indptr = np.array([0, 2, 3, 6]) + >>> indices = np.array([0, 2, 2, 0, 1, 2]) + >>> data = np.array([1, 2, 3, 4, 5, 6]).repeat(4).reshape(6, 2, 2) + >>> bsr_array((data,indices,indptr), shape=(6, 6)).toarray() + array([[1, 1, 0, 0, 2, 2], + [1, 1, 0, 0, 2, 2], + [0, 0, 0, 0, 3, 3], + [0, 0, 0, 0, 3, 3], + [4, 4, 5, 5, 6, 6], + [4, 4, 5, 5, 6, 6]]) + + """ + + +class bsr_matrix(spmatrix, _bsr_base): + """ + Block Sparse Row format sparse matrix. + + This can be instantiated in several ways: + bsr_matrix(D, [blocksize=(R,C)]) + where D is a 2-D ndarray. + + bsr_matrix(S, [blocksize=(R,C)]) + with another sparse array or matrix S (equivalent to S.tobsr()) + + bsr_matrix((M, N), [blocksize=(R,C), dtype]) + to construct an empty sparse matrix with shape (M, N) + dtype is optional, defaulting to dtype='d'. + + bsr_matrix((data, ij), [blocksize=(R,C), shape=(M, N)]) + where ``data`` and ``ij`` satisfy ``a[ij[0, k], ij[1, k]] = data[k]`` + + bsr_matrix((data, indices, indptr), [shape=(M, N)]) + is the standard BSR representation where the block column + indices for row i are stored in ``indices[indptr[i]:indptr[i+1]]`` + and their corresponding block values are stored in + ``data[ indptr[i]: indptr[i+1] ]``. If the shape parameter is not + supplied, the matrix dimensions are inferred from the index arrays. + + Attributes + ---------- + dtype : dtype + Data type of the matrix + shape : 2-tuple + Shape of the matrix + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + BSR format data array of the matrix + indices + BSR format index array of the matrix + indptr + BSR format index pointer array of the matrix + blocksize + Block size + has_sorted_indices : bool + Whether indices are sorted + has_canonical_format : bool + T + + Notes + ----- + Sparse matrices can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + **Summary of BSR format** + + The Block Sparse Row (BSR) format is very similar to the Compressed + Sparse Row (CSR) format. BSR is appropriate for sparse matrices with dense + sub matrices like the last example below. Such sparse block matrices often + arise in vector-valued finite element discretizations. In such cases, BSR is + considerably more efficient than CSR and CSC for many sparse arithmetic + operations. + + **Blocksize** + + The blocksize (R,C) must evenly divide the shape of the sparse matrix (M,N). + That is, R and C must satisfy the relationship ``M % R = 0`` and + ``N % C = 0``. + + If no blocksize is specified, a simple heuristic is applied to determine + an appropriate blocksize. + + **Canonical Format** + + In canonical format, there are no duplicate blocks and indices are sorted + per row. + + **Limitations** + + Block Sparse Row format sparse matrices do not support slicing. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import bsr_matrix + >>> bsr_matrix((3, 4), dtype=np.int8).toarray() + array([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=int8) + + >>> row = np.array([0, 0, 1, 2, 2, 2]) + >>> col = np.array([0, 2, 2, 0, 1, 2]) + >>> data = np.array([1, 2, 3 ,4, 5, 6]) + >>> bsr_matrix((data, (row, col)), shape=(3, 3)).toarray() + array([[1, 0, 2], + [0, 0, 3], + [4, 5, 6]]) + + >>> indptr = np.array([0, 2, 3, 6]) + >>> indices = np.array([0, 2, 2, 0, 1, 2]) + >>> data = np.array([1, 2, 3, 4, 5, 6]).repeat(4).reshape(6, 2, 2) + >>> bsr_matrix((data,indices,indptr), shape=(6, 6)).toarray() + array([[1, 1, 0, 0, 2, 2], + [1, 1, 0, 0, 2, 2], + [0, 0, 0, 0, 3, 3], + [0, 0, 0, 0, 3, 3], + [4, 4, 5, 5, 6, 6], + [4, 4, 5, 5, 6, 6]]) + + """ + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_compressed.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_compressed.py new file mode 100644 index 0000000000000000000000000000000000000000..e5f43c16bd924ac6bd70cfd4634dc3afcd298391 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_compressed.py @@ -0,0 +1,1500 @@ +"""Base class for sparse matrix formats using compressed storage.""" +__all__ = [] + +from warnings import warn +import itertools +import operator + +import numpy as np +from scipy._lib._util import _prune_array, copy_if_needed + +from ._base import _spbase, issparse, sparray, SparseEfficiencyWarning +from ._data import _data_matrix, _minmax_mixin +from . import _sparsetools +from ._sparsetools import (get_csr_submatrix, csr_sample_offsets, csr_todense, + csr_sample_values, csr_row_index, csr_row_slice, + csr_column_index1, csr_column_index2) +from ._index import IndexMixin +from ._sputils import (upcast, upcast_char, to_native, isdense, isshape, + getdtype, isscalarlike, isintlike, downcast_intp_index, + get_sum_dtype, check_shape, get_index_dtype, broadcast_shapes, + is_pydata_spmatrix) + + +class _cs_matrix(_data_matrix, _minmax_mixin, IndexMixin): + """ + base array/matrix class for compressed row- and column-oriented arrays/matrices + """ + + def __init__(self, arg1, shape=None, dtype=None, copy=False, *, maxprint=None): + _data_matrix.__init__(self, arg1, maxprint=maxprint) + + if issparse(arg1): + if arg1.format == self.format and copy: + arg1 = arg1.copy() + else: + arg1 = arg1.asformat(self.format) + self.indptr, self.indices, self.data, self._shape = ( + arg1.indptr, arg1.indices, arg1.data, arg1._shape + ) + + elif isinstance(arg1, tuple): + if isshape(arg1, allow_nd=self._allow_nd): + # It's a tuple of matrix dimensions (M, N) + # create empty matrix + self._shape = check_shape(arg1, allow_nd=self._allow_nd) + M, N = self._swap(self._shape_as_2d) + # Select index dtype large enough to pass array and + # scalar parameters to sparsetools + idx_dtype = self._get_index_dtype(maxval=max(self.shape)) + self.data = np.zeros(0, getdtype(dtype, default=float)) + self.indices = np.zeros(0, idx_dtype) + self.indptr = np.zeros(M + 1, dtype=idx_dtype) + else: + if len(arg1) == 2: + # (data, ij) format + coo = self._coo_container(arg1, shape=shape, dtype=dtype) + arrays = coo._coo_to_compressed(self._swap) + self.indptr, self.indices, self.data, self._shape = arrays + self.sum_duplicates() + elif len(arg1) == 3: + # (data, indices, indptr) format + (data, indices, indptr) = arg1 + + # Select index dtype large enough to pass array and + # scalar parameters to sparsetools + maxval = None + if shape is not None and 0 not in shape: + maxval = max(shape) + idx_dtype = self._get_index_dtype((indices, indptr), + maxval=maxval, + check_contents=True) + + if not copy: + copy = copy_if_needed + self.indices = np.array(indices, copy=copy, dtype=idx_dtype) + self.indptr = np.array(indptr, copy=copy, dtype=idx_dtype) + self.data = np.array(data, copy=copy, dtype=dtype) + else: + raise ValueError(f"unrecognized {self.__class__.__name__} " + f"constructor input: {arg1}") + + else: + # must be dense + try: + arg1 = np.asarray(arg1) + except Exception as e: + raise ValueError(f"unrecognized {self.__class__.__name__} " + f"constructor input: {arg1}") from e + if isinstance(self, sparray) and arg1.ndim != 2 and self.format == "csc": + raise ValueError(f"CSC arrays don't support {arg1.ndim}D input. Use 2D") + if arg1.ndim > 2: + raise ValueError(f"CSR arrays don't yet support {arg1.ndim}D.") + + coo = self._coo_container(arg1, dtype=dtype) + arrays = coo._coo_to_compressed(self._swap) + self.indptr, self.indices, self.data, self._shape = arrays + + # Read matrix dimensions given, if any + if shape is not None: + self._shape = check_shape(shape, allow_nd=self._allow_nd) + elif self.shape is None: + # shape not already set, try to infer dimensions + try: + M = len(self.indptr) - 1 + N = self.indices.max() + 1 + except Exception as e: + raise ValueError('unable to infer matrix dimensions') from e + + self._shape = check_shape(self._swap((M, N)), allow_nd=self._allow_nd) + + if dtype is not None: + newdtype = getdtype(dtype) + self.data = self.data.astype(newdtype, copy=False) + + self.check_format(full_check=False) + + def _getnnz(self, axis=None): + if axis is None: + return int(self.indptr[-1]) + elif self.ndim == 1: + if axis in (0, -1): + return int(self.indptr[-1]) + raise ValueError('axis out of bounds') + else: + if axis < 0: + axis += 2 + axis, _ = self._swap((axis, 1 - axis)) + _, N = self._swap(self.shape) + if axis == 0: + return np.bincount(downcast_intp_index(self.indices), minlength=N) + elif axis == 1: + return np.diff(self.indptr) + raise ValueError('axis out of bounds') + + _getnnz.__doc__ = _spbase._getnnz.__doc__ + + def count_nonzero(self, axis=None): + self.sum_duplicates() + if axis is None: + return np.count_nonzero(self.data) + + if self.ndim == 1: + if axis not in (0, -1): + raise ValueError('axis out of bounds') + return np.count_nonzero(self.data) + + if axis < 0: + axis += 2 + axis, _ = self._swap((axis, 1 - axis)) + if axis == 0: + _, N = self._swap(self.shape) + mask = self.data != 0 + idx = self.indices if mask.all() else self.indices[mask] + return np.bincount(downcast_intp_index(idx), minlength=N) + elif axis == 1: + if self.data.all(): + return np.diff(self.indptr) + pairs = itertools.pairwise(self.indptr) + return np.array([np.count_nonzero(self.data[i:j]) for i, j in pairs]) + else: + raise ValueError('axis out of bounds') + + count_nonzero.__doc__ = _spbase.count_nonzero.__doc__ + + def check_format(self, full_check=True): + """Check whether the array/matrix respects the CSR or CSC format. + + Parameters + ---------- + full_check : bool, optional + If `True`, run rigorous check, scanning arrays for valid values. + Note that activating those check might copy arrays for casting, + modifying indices and index pointers' inplace. + If `False`, run basic checks on attributes. O(1) operations. + Default is `True`. + """ + # index arrays should have integer data types + if self.indptr.dtype.kind != 'i': + warn(f"indptr array has non-integer dtype ({self.indptr.dtype.name})", + stacklevel=3) + if self.indices.dtype.kind != 'i': + warn(f"indices array has non-integer dtype ({self.indices.dtype.name})", + stacklevel=3) + + # check array shapes + for x in [self.data.ndim, self.indices.ndim, self.indptr.ndim]: + if x != 1: + raise ValueError('data, indices, and indptr should be 1-D') + + # check index pointer. Use _swap to determine proper bounds + M, N = self._swap(self._shape_as_2d) + + if (len(self.indptr) != M + 1): + raise ValueError(f"index pointer size {len(self.indptr)} should be {M + 1}") + if (self.indptr[0] != 0): + raise ValueError("index pointer should start with 0") + + # check index and data arrays + if (len(self.indices) != len(self.data)): + raise ValueError("indices and data should have the same size") + if (self.indptr[-1] > len(self.indices)): + raise ValueError("Last value of index pointer should be less than " + "the size of index and data arrays") + + self.prune() + + if full_check: + # check format validity (more expensive) + if self.nnz > 0: + if self.indices.max() >= N: + raise ValueError(f"indices must be < {N}") + if self.indices.min() < 0: + raise ValueError("indices must be >= 0") + if np.diff(self.indptr).min() < 0: + raise ValueError("indptr must be a non-decreasing sequence") + + idx_dtype = self._get_index_dtype((self.indptr, self.indices)) + self.indptr = np.asarray(self.indptr, dtype=idx_dtype) + self.indices = np.asarray(self.indices, dtype=idx_dtype) + self.data = to_native(self.data) + + # if not self.has_sorted_indices(): + # warn('Indices were not in sorted order. Sorting indices.') + # self.sort_indices() + # assert(self.has_sorted_indices()) + # TODO check for duplicates? + + ####################### + # Boolean comparisons # + ####################### + + def _scalar_binopt(self, other, op): + """Scalar version of self._binopt, for cases in which no new nonzeros + are added. Produces a new sparse array in canonical form. + """ + self.sum_duplicates() + res = self._with_data(op(self.data, other), copy=True) + res.eliminate_zeros() + return res + + def __eq__(self, other): + # Scalar other. + if isscalarlike(other): + if np.isnan(other): + return self.__class__(self.shape, dtype=np.bool_) + + if other == 0: + warn("Comparing a sparse matrix with 0 using == is inefficient" + ", try using != instead.", SparseEfficiencyWarning, + stacklevel=3) + all_true = self.__class__(np.ones(self.shape, dtype=np.bool_)) + inv = self._scalar_binopt(other, operator.ne) + return all_true - inv + else: + return self._scalar_binopt(other, operator.eq) + # Dense other. + elif isdense(other): + return self.todense() == other + # Pydata sparse other. + elif is_pydata_spmatrix(other): + return NotImplemented + # Sparse other. + elif issparse(other): + warn("Comparing sparse matrices using == is inefficient, try using" + " != instead.", SparseEfficiencyWarning, stacklevel=3) + # TODO sparse broadcasting + if self.shape != other.shape: + return False + elif self.format != other.format: + other = other.asformat(self.format) + res = self._binopt(other, '_ne_') + all_true = self.__class__(np.ones(self.shape, dtype=np.bool_)) + return all_true - res + else: + return NotImplemented + + def __ne__(self, other): + # Scalar other. + if isscalarlike(other): + if np.isnan(other): + warn("Comparing a sparse matrix with nan using != is" + " inefficient", SparseEfficiencyWarning, stacklevel=3) + all_true = self.__class__(np.ones(self.shape, dtype=np.bool_)) + return all_true + elif other != 0: + warn("Comparing a sparse matrix with a nonzero scalar using !=" + " is inefficient, try using == instead.", + SparseEfficiencyWarning, stacklevel=3) + all_true = self.__class__(np.ones(self.shape), dtype=np.bool_) + inv = self._scalar_binopt(other, operator.eq) + return all_true - inv + else: + return self._scalar_binopt(other, operator.ne) + # Dense other. + elif isdense(other): + return self.todense() != other + # Pydata sparse other. + elif is_pydata_spmatrix(other): + return NotImplemented + # Sparse other. + elif issparse(other): + # TODO sparse broadcasting + if self.shape != other.shape: + return True + elif self.format != other.format: + other = other.asformat(self.format) + return self._binopt(other, '_ne_') + else: + return NotImplemented + + def _inequality(self, other, op, op_name, bad_scalar_msg): + # Scalar other. + if isscalarlike(other): + if 0 == other and op_name in ('_le_', '_ge_'): + raise NotImplementedError(" >= and <= don't work with 0.") + elif op(0, other): + warn(bad_scalar_msg, SparseEfficiencyWarning, stacklevel=3) + other_arr = np.empty(self.shape, dtype=np.result_type(other)) + other_arr.fill(other) + other_arr = self.__class__(other_arr) + return self._binopt(other_arr, op_name) + else: + return self._scalar_binopt(other, op) + # Dense other. + elif isdense(other): + return op(self.todense(), other) + # Sparse other. + elif issparse(other): + # TODO sparse broadcasting + if self.shape != other.shape: + raise ValueError("inconsistent shapes") + elif self.format != other.format: + other = other.asformat(self.format) + if op_name not in ('_ge_', '_le_'): + return self._binopt(other, op_name) + + warn("Comparing sparse matrices using >= and <= is inefficient, " + "using <, >, or !=, instead.", + SparseEfficiencyWarning, stacklevel=3) + all_true = self.__class__(np.ones(self.shape, dtype=np.bool_)) + res = self._binopt(other, '_gt_' if op_name == '_le_' else '_lt_') + return all_true - res + else: + return NotImplemented + + def __lt__(self, other): + return self._inequality(other, operator.lt, '_lt_', + "Comparing a sparse matrix with a scalar " + "greater than zero using < is inefficient, " + "try using >= instead.") + + def __gt__(self, other): + return self._inequality(other, operator.gt, '_gt_', + "Comparing a sparse matrix with a scalar " + "less than zero using > is inefficient, " + "try using <= instead.") + + def __le__(self, other): + return self._inequality(other, operator.le, '_le_', + "Comparing a sparse matrix with a scalar " + "greater than zero using <= is inefficient, " + "try using > instead.") + + def __ge__(self, other): + return self._inequality(other, operator.ge, '_ge_', + "Comparing a sparse matrix with a scalar " + "less than zero using >= is inefficient, " + "try using < instead.") + + ################################# + # Arithmetic operator overrides # + ################################# + + def _add_dense(self, other): + if other.shape != self.shape: + raise ValueError(f'Incompatible shapes ({self.shape} and {other.shape})') + dtype = upcast_char(self.dtype.char, other.dtype.char) + order = self._swap('CF')[0] + result = np.array(other, dtype=dtype, order=order, copy=True) + y = result if result.flags.c_contiguous else result.T + M, N = self._swap(self._shape_as_2d) + csr_todense(M, N, self.indptr, self.indices, self.data, y) + return self._container(result, copy=False) + + def _add_sparse(self, other): + return self._binopt(other, '_plus_') + + def _sub_sparse(self, other): + return self._binopt(other, '_minus_') + + def multiply(self, other): + """Point-wise multiplication by array/matrix, vector, or scalar.""" + # Scalar multiplication. + if isscalarlike(other): + return self._mul_scalar(other) + # Sparse matrix or vector. + if issparse(other): + if self.shape == other.shape: + other = self.__class__(other) + return self._binopt(other, '_elmul_') + # Single element. + if other.shape == (1, 1): + result = self._mul_scalar(other.toarray()[0, 0]) + if self.ndim == 1: + return result.reshape((1, self.shape[0])) + return result + if other.shape == (1,): + return self._mul_scalar(other.toarray()[0]) + if self.shape in ((1,), (1, 1)): + return other._mul_scalar(self.data.sum()) + + # broadcast. treat 1d like a row + sM, sN = self._shape_as_2d + oM, oN = other._shape_as_2d + # A row times a column. + if sM == 1 and oN == 1: + return other._matmul_sparse(self.reshape(sM, sN).tocsc()) + if sN == 1 and oM == 1: + return self._matmul_sparse(other.reshape(oM, oN).tocsc()) + + is_array = isinstance(self, sparray) + # Other is a row. + if oM == 1 and sN == oN: + new_other = _make_diagonal_csr(other.toarray().ravel(), is_array) + result = self._matmul_sparse(new_other) + return result if self.ndim == 2 else result.reshape((1, oN)) + # self is a row. + if sM == 1 and sN == oN: + copy = _make_diagonal_csr(self.toarray().ravel(), is_array) + return other._matmul_sparse(copy) + + # Other is a column. + if oN == 1 and sM == oM: + new_other = _make_diagonal_csr(other.toarray().ravel(), is_array) + return new_other._matmul_sparse(self) + # self is a column. + if sN == 1 and sM == oM: + new_self = _make_diagonal_csr(self.toarray().ravel(), is_array) + return new_self._matmul_sparse(other) + raise ValueError("inconsistent shapes") + + # Assume other is a dense matrix/array, which produces a single-item + # object array if other isn't convertible to ndarray. + other = np.asanyarray(other) + + if other.ndim > 2: + return np.multiply(self.toarray(), other) + # Single element / wrapped object. + if other.size == 1: + if other.dtype == np.object_: + # 'other' not convertible to ndarray. + return NotImplemented + bshape = broadcast_shapes(self.shape, other.shape) + return self._mul_scalar(other.flat[0]).reshape(bshape) + # Fast case for trivial sparse matrix. + if self.shape in ((1,), (1, 1)): + bshape = broadcast_shapes(self.shape, other.shape) + return np.multiply(self.data.sum(), other).reshape(bshape) + + ret = self.tocoo() + # Matching shapes. + if self.shape == other.shape: + data = np.multiply(ret.data, other[ret.coords]) + ret.data = data.view(np.ndarray).ravel() + return ret + + # convert other to 2d + other2d = np.atleast_2d(other) + # Sparse row vector times... + if self.shape[0] == 1 or self.ndim == 1: + if other2d.shape[1] == 1: # Dense column vector. + data = np.multiply(ret.data, other2d) + elif other2d.shape[1] == self.shape[-1]: # Dense 2d matrix. + data = np.multiply(ret.data, other2d[:, ret.col]) + else: + raise ValueError("inconsistent shapes") + idx_dtype = self._get_index_dtype(ret.col, + maxval=ret.nnz * other2d.shape[0]) + row = np.repeat(np.arange(other2d.shape[0], dtype=idx_dtype), ret.nnz) + col = np.tile(ret.col.astype(idx_dtype, copy=False), other2d.shape[0]) + return self._coo_container( + (data.view(np.ndarray).ravel(), (row, col)), + shape=(other2d.shape[0], self.shape[-1]), + copy=False + ) + # Sparse column vector times... + if self.shape[1] == 1: + if other2d.shape[0] == 1: # Dense row vector. + data = np.multiply(ret.data[:, None], other2d) + elif other2d.shape[0] == self.shape[0]: # Dense 2d array. + data = np.multiply(ret.data[:, None], other2d[ret.row]) + else: + raise ValueError("inconsistent shapes") + idx_dtype = self._get_index_dtype(ret.row, + maxval=ret.nnz * other2d.shape[1]) + row = np.repeat(ret.row.astype(idx_dtype, copy=False), other2d.shape[1]) + col = np.tile(np.arange(other2d.shape[1], dtype=idx_dtype), ret.nnz) + return self._coo_container( + (data.view(np.ndarray).ravel(), (row, col)), + shape=(self.shape[0], other2d.shape[1]), + copy=False + ) + # Sparse matrix times dense row vector. + if other2d.shape[0] == 1 and self.shape[1] == other2d.shape[1]: + data = np.multiply(ret.data, other2d[:, ret.col].ravel()) + # Sparse matrix times dense column vector. + elif other2d.shape[1] == 1 and self.shape[0] == other2d.shape[0]: + data = np.multiply(ret.data, other2d[ret.row].ravel()) + else: + raise ValueError("inconsistent shapes") + ret.data = data.view(np.ndarray).ravel() + return ret + + ########################### + # Multiplication handlers # + ########################### + + def _matmul_vector(self, other): + M, N = self._shape_as_2d + + # output array + result = np.zeros(M, dtype=upcast_char(self.dtype.char, other.dtype.char)) + + # csr_matvec or csc_matvec + fn = getattr(_sparsetools, self.format + '_matvec') + fn(M, N, self.indptr, self.indices, self.data, other, result) + + return result[0] if self.ndim == 1 else result + + def _matmul_multivector(self, other): + M, N = self._shape_as_2d + n_vecs = other.shape[-1] # number of column vectors + + result = np.zeros((M, n_vecs), + dtype=upcast_char(self.dtype.char, other.dtype.char)) + + # csr_matvecs or csc_matvecs + fn = getattr(_sparsetools, self.format + '_matvecs') + fn(M, N, n_vecs, self.indptr, self.indices, self.data, + other.ravel(), result.ravel()) + + if self.ndim == 1: + return result.reshape((n_vecs,)) + return result + + def _matmul_sparse(self, other): + M, K1 = self._shape_as_2d + # if other is 1d, treat as a **column** + o_ndim = other.ndim + if o_ndim == 1: + # convert 1d array to a 2d column when on the right of @ + other = other.reshape((1, other.shape[0])).T # Note: converts to CSC + K2, N = other._shape if other.ndim == 2 else (other.shape[0], 1) + + # find new_shape: (M, N), (M,), (N,) or () + new_shape = () + if self.ndim == 2: + new_shape += (M,) + if o_ndim == 2: + new_shape += (N,) + faux_shape = (M if self.ndim == 2 else 1, N if o_ndim == 2 else 1) + + major_dim = self._swap((M, N))[0] + other = self.__class__(other) # convert to this format + + idx_dtype = self._get_index_dtype((self.indptr, self.indices, + other.indptr, other.indices)) + + fn = getattr(_sparsetools, self.format + '_matmat_maxnnz') + nnz = fn(M, N, + np.asarray(self.indptr, dtype=idx_dtype), + np.asarray(self.indices, dtype=idx_dtype), + np.asarray(other.indptr, dtype=idx_dtype), + np.asarray(other.indices, dtype=idx_dtype)) + if nnz == 0: + if new_shape == (): + return np.array(0, dtype=upcast(self.dtype, other.dtype)) + return self.__class__(new_shape, dtype=upcast(self.dtype, other.dtype)) + + idx_dtype = self._get_index_dtype((self.indptr, self.indices, + other.indptr, other.indices), + maxval=nnz) + + indptr = np.empty(major_dim + 1, dtype=idx_dtype) + indices = np.empty(nnz, dtype=idx_dtype) + data = np.empty(nnz, dtype=upcast(self.dtype, other.dtype)) + + fn = getattr(_sparsetools, self.format + '_matmat') + fn(M, N, np.asarray(self.indptr, dtype=idx_dtype), + np.asarray(self.indices, dtype=idx_dtype), + self.data, + np.asarray(other.indptr, dtype=idx_dtype), + np.asarray(other.indices, dtype=idx_dtype), + other.data, + indptr, indices, data) + + if new_shape == (): + return np.array(data[0]) + res = self.__class__((data, indices, indptr), shape=faux_shape) + if faux_shape != new_shape: + if res.format != 'csr': + res = res.tocsr() + res = res.reshape(new_shape) + return res + + def diagonal(self, k=0): + rows, cols = self.shape + if k <= -rows or k >= cols: + return np.empty(0, dtype=self.data.dtype) + fn = getattr(_sparsetools, self.format + "_diagonal") + y = np.empty(min(rows + min(k, 0), cols - max(k, 0)), + dtype=upcast(self.dtype)) + fn(k, self.shape[0], self.shape[1], self.indptr, self.indices, + self.data, y) + return y + + diagonal.__doc__ = _spbase.diagonal.__doc__ + + ##################### + # Other binary ops # + ##################### + + def _maximum_minimum(self, other, npop, op_name, dense_check): + if isscalarlike(other): + if dense_check(other): + warn("Taking maximum (minimum) with > 0 (< 0) number results" + " to a dense matrix.", SparseEfficiencyWarning, + stacklevel=3) + other_arr = np.empty(self.shape, dtype=np.asarray(other).dtype) + other_arr.fill(other) + other_arr = self.__class__(other_arr) + return self._binopt(other_arr, op_name) + else: + self.sum_duplicates() + new_data = npop(self.data, np.asarray(other)) + mat = self.__class__((new_data, self.indices, self.indptr), + dtype=new_data.dtype, shape=self.shape) + return mat + elif isdense(other): + return npop(self.todense(), other) + elif issparse(other): + return self._binopt(other, op_name) + else: + raise ValueError("Operands not compatible.") + + def maximum(self, other): + return self._maximum_minimum(other, np.maximum, + '_maximum_', lambda x: np.asarray(x) > 0) + + maximum.__doc__ = _spbase.maximum.__doc__ + + def minimum(self, other): + return self._maximum_minimum(other, np.minimum, + '_minimum_', lambda x: np.asarray(x) < 0) + + minimum.__doc__ = _spbase.minimum.__doc__ + + ##################### + # Reduce operations # + ##################### + + def sum(self, axis=None, dtype=None, out=None): + """Sum the array/matrix over the given axis. If the axis is None, sum + over both rows and columns, returning a scalar. + """ + # The _spbase base class already does axis=0 and axis=1 efficiently + # so we only do the case axis=None here + if (self.ndim == 2 and not hasattr(self, 'blocksize') and + axis in self._swap(((1, -1), (0, -2)))[0]): + # faster than multiplication for large minor axis in CSC/CSR + res_dtype = get_sum_dtype(self.dtype) + ret = np.zeros(len(self.indptr) - 1, dtype=res_dtype) + + major_index, value = self._minor_reduce(np.add) + ret[major_index] = value + ret = self._ascontainer(ret) + if axis % 2 == 1: + ret = ret.T + + return ret.sum(axis=(), dtype=dtype, out=out) + else: + # _spbase handles the situations when axis is in {None, -2, -1, 0, 1} + return _spbase.sum(self, axis=axis, dtype=dtype, out=out) + + sum.__doc__ = _spbase.sum.__doc__ + + def _minor_reduce(self, ufunc, data=None): + """Reduce nonzeros with a ufunc over the minor axis when non-empty + + Can be applied to a function of self.data by supplying data parameter. + + Warning: this does not call sum_duplicates() + + Returns + ------- + major_index : array of ints + Major indices where nonzero + + value : array of self.dtype + Reduce result for nonzeros in each major_index + """ + if data is None: + data = self.data + major_index = np.flatnonzero(np.diff(self.indptr)) + value = ufunc.reduceat(data, + downcast_intp_index(self.indptr[major_index])) + return major_index, value + + ####################### + # Getting and Setting # + ####################### + + def _get_intXint(self, row, col): + M, N = self._swap(self.shape) + major, minor = self._swap((row, col)) + indptr, indices, data = get_csr_submatrix( + M, N, self.indptr, self.indices, self.data, + major, major + 1, minor, minor + 1) + return data.sum(dtype=self.dtype) + + def _get_sliceXslice(self, row, col): + major, minor = self._swap((row, col)) + if major.step in (1, None) and minor.step in (1, None): + return self._get_submatrix(major, minor, copy=True) + return self._major_slice(major)._minor_slice(minor) + + def _get_arrayXarray(self, row, col): + # inner indexing + idx_dtype = self.indices.dtype + M, N = self._swap(self.shape) + major, minor = self._swap((row, col)) + major = np.asarray(major, dtype=idx_dtype) + minor = np.asarray(minor, dtype=idx_dtype) + + val = np.empty(major.size, dtype=self.dtype) + csr_sample_values(M, N, self.indptr, self.indices, self.data, + major.size, major.ravel(), minor.ravel(), val) + if major.ndim == 1: + return self._ascontainer(val) + return self.__class__(val.reshape(major.shape)) + + def _get_columnXarray(self, row, col): + # outer indexing + major, minor = self._swap((row, col)) + return self._major_index_fancy(major)._minor_index_fancy(minor) + + def _major_index_fancy(self, idx): + """Index along the major axis where idx is an array of ints. + """ + idx_dtype = self._get_index_dtype((self.indptr, self.indices)) + indices = np.asarray(idx, dtype=idx_dtype).ravel() + + N = self._swap(self._shape_as_2d)[1] + M = len(indices) + new_shape = self._swap((M, N)) if self.ndim == 2 else (M,) + if M == 0: + return self.__class__(new_shape, dtype=self.dtype) + + row_nnz = (self.indptr[indices + 1] - self.indptr[indices]).astype(idx_dtype) + res_indptr = np.zeros(M + 1, dtype=idx_dtype) + np.cumsum(row_nnz, out=res_indptr[1:]) + + nnz = res_indptr[-1] + res_indices = np.empty(nnz, dtype=idx_dtype) + res_data = np.empty(nnz, dtype=self.dtype) + csr_row_index( + M, + indices, + self.indptr.astype(idx_dtype, copy=False), + self.indices.astype(idx_dtype, copy=False), + self.data, + res_indices, + res_data + ) + + return self.__class__((res_data, res_indices, res_indptr), + shape=new_shape, copy=False) + + def _major_slice(self, idx, copy=False): + """Index along the major axis where idx is a slice object. + """ + if idx == slice(None): + return self.copy() if copy else self + + M, N = self._swap(self._shape_as_2d) + start, stop, step = idx.indices(M) + M = len(range(start, stop, step)) + new_shape = self._swap((M, N)) if self.ndim == 2 else (M,) + if M == 0: + return self.__class__(new_shape, dtype=self.dtype) + + # Work out what slices are needed for `row_nnz` + # start,stop can be -1, only if step is negative + start0, stop0 = start, stop + if stop == -1 and start >= 0: + stop0 = None + start1, stop1 = start + 1, stop + 1 + + row_nnz = self.indptr[start1:stop1:step] - \ + self.indptr[start0:stop0:step] + idx_dtype = self.indices.dtype + res_indptr = np.zeros(M+1, dtype=idx_dtype) + np.cumsum(row_nnz, out=res_indptr[1:]) + + if step == 1: + all_idx = slice(self.indptr[start], self.indptr[stop]) + res_indices = np.array(self.indices[all_idx], copy=copy) + res_data = np.array(self.data[all_idx], copy=copy) + else: + nnz = res_indptr[-1] + res_indices = np.empty(nnz, dtype=idx_dtype) + res_data = np.empty(nnz, dtype=self.dtype) + csr_row_slice(start, stop, step, self.indptr, self.indices, + self.data, res_indices, res_data) + + return self.__class__((res_data, res_indices, res_indptr), + shape=new_shape, copy=False) + + def _minor_index_fancy(self, idx): + """Index along the minor axis where idx is an array of ints. + """ + idx_dtype = self._get_index_dtype((self.indices, self.indptr)) + indices = self.indices.astype(idx_dtype, copy=False) + indptr = self.indptr.astype(idx_dtype, copy=False) + + idx = np.asarray(idx, dtype=idx_dtype).ravel() + + M, N = self._swap(self._shape_as_2d) + k = len(idx) + new_shape = self._swap((M, k)) if self.ndim == 2 else (k,) + if k == 0: + return self.__class__(new_shape, dtype=self.dtype) + + # pass 1: count idx entries and compute new indptr + col_offsets = np.zeros(N, dtype=idx_dtype) + res_indptr = np.empty_like(self.indptr, dtype=idx_dtype) + csr_column_index1( + k, + idx, + M, + N, + indptr, + indices, + col_offsets, + res_indptr, + ) + + # pass 2: copy indices/data for selected idxs + col_order = np.argsort(idx).astype(idx_dtype, copy=False) + nnz = res_indptr[-1] + res_indices = np.empty(nnz, dtype=idx_dtype) + res_data = np.empty(nnz, dtype=self.dtype) + csr_column_index2(col_order, col_offsets, len(self.indices), + indices, self.data, res_indices, res_data) + return self.__class__((res_data, res_indices, res_indptr), + shape=new_shape, copy=False) + + def _minor_slice(self, idx, copy=False): + """Index along the minor axis where idx is a slice object. + """ + if idx == slice(None): + return self.copy() if copy else self + + M, N = self._swap(self._shape_as_2d) + start, stop, step = idx.indices(N) + N = len(range(start, stop, step)) + if N == 0: + return self.__class__(self._swap((M, N)), dtype=self.dtype) + if step == 1: + return self._get_submatrix(minor=idx, copy=copy) + # TODO: don't fall back to fancy indexing here + return self._minor_index_fancy(np.arange(start, stop, step)) + + def _get_submatrix(self, major=None, minor=None, copy=False): + """Return a submatrix of this matrix. + + major, minor: None, int, or slice with step 1 + """ + M, N = self._swap(self._shape_as_2d) + i0, i1 = _process_slice(major, M) + j0, j1 = _process_slice(minor, N) + + if i0 == 0 and j0 == 0 and i1 == M and j1 == N: + return self.copy() if copy else self + + indptr, indices, data = get_csr_submatrix( + M, N, self.indptr, self.indices, self.data, i0, i1, j0, j1) + + shape = self._swap((i1 - i0, j1 - j0)) + if self.ndim == 1: + shape = (shape[1],) + return self.__class__((data, indices, indptr), shape=shape, + dtype=self.dtype, copy=False) + + def _set_intXint(self, row, col, x): + i, j = self._swap((row, col)) + self._set_many(i, j, x) + + def _set_arrayXarray(self, row, col, x): + i, j = self._swap((row, col)) + self._set_many(i, j, x) + + def _set_arrayXarray_sparse(self, row, col, x): + # clear entries that will be overwritten + self._zero_many(*self._swap((row, col))) + + M, N = row.shape # matches col.shape + broadcast_row = M != 1 and x.shape[0] == 1 + broadcast_col = N != 1 and x.shape[1] == 1 + r, c = x.row, x.col + + x = np.asarray(x.data, dtype=self.dtype) + if x.size == 0: + return + + if broadcast_row: + r = np.repeat(np.arange(M), len(r)) + c = np.tile(c, M) + x = np.tile(x, M) + if broadcast_col: + r = np.repeat(r, N) + c = np.tile(np.arange(N), len(c)) + x = np.repeat(x, N) + # only assign entries in the new sparsity structure + i, j = self._swap((row[r, c], col[r, c])) + self._set_many(i, j, x) + + def _setdiag(self, values, k): + if 0 in self.shape: + return + if self.ndim == 1: + raise NotImplementedError('diagonals cant be set in 1d arrays') + + M, N = self.shape + broadcast = (values.ndim == 0) + + if k < 0: + if broadcast: + max_index = min(M + k, N) + else: + max_index = min(M + k, N, len(values)) + i = np.arange(-k, max_index - k, dtype=self.indices.dtype) + j = np.arange(max_index, dtype=self.indices.dtype) + + else: + if broadcast: + max_index = min(M, N - k) + else: + max_index = min(M, N - k, len(values)) + i = np.arange(max_index, dtype=self.indices.dtype) + j = np.arange(k, k + max_index, dtype=self.indices.dtype) + + if not broadcast: + values = values[:len(i)] + + x = np.atleast_1d(np.asarray(values, dtype=self.dtype)).ravel() + if x.squeeze().shape != i.squeeze().shape: + x = np.broadcast_to(x, i.shape) + if x.size == 0: + return + + M, N = self._swap((M, N)) + i, j = self._swap((i, j)) + n_samples = x.size + offsets = np.empty(n_samples, dtype=self.indices.dtype) + ret = csr_sample_offsets(M, N, self.indptr, self.indices, n_samples, + i, j, offsets) + if ret == 1: + # rinse and repeat + self.sum_duplicates() + csr_sample_offsets(M, N, self.indptr, self.indices, n_samples, + i, j, offsets) + if -1 not in offsets: + # only affects existing non-zero cells + self.data[offsets] = x + return + + mask = (offsets >= 0) + # Boundary between csc and convert to coo + # The value 0.001 is justified in gh-19962#issuecomment-1920499678 + if self.nnz - mask.sum() < self.nnz * 0.001: + # replace existing entries + self.data[offsets[mask]] = x[mask] + # create new entries + mask = ~mask + i = i[mask] + j = j[mask] + self._insert_many(i, j, x[mask]) + else: + # convert to coo for _set_diag + coo = self.tocoo() + coo._setdiag(values, k) + arrays = coo._coo_to_compressed(self._swap) + self.indptr, self.indices, self.data, _ = arrays + + def _prepare_indices(self, i, j): + M, N = self._swap(self._shape_as_2d) + + def check_bounds(indices, bound): + idx = indices.max() + if idx >= bound: + raise IndexError(f'index ({idx}) out of range (>= {bound})') + idx = indices.min() + if idx < -bound: + raise IndexError(f'index ({idx}) out of range (< -{bound})') + + i = np.atleast_1d(np.asarray(i, dtype=self.indices.dtype)).ravel() + j = np.atleast_1d(np.asarray(j, dtype=self.indices.dtype)).ravel() + check_bounds(i, M) + check_bounds(j, N) + return i, j, M, N + + def _set_many(self, i, j, x): + """Sets value at each (i, j) to x + + Here (i,j) index major and minor respectively, and must not contain + duplicate entries. + """ + i, j, M, N = self._prepare_indices(i, j) + x = np.atleast_1d(np.asarray(x, dtype=self.dtype)).ravel() + + n_samples = x.size + offsets = np.empty(n_samples, dtype=self.indices.dtype) + ret = csr_sample_offsets(M, N, self.indptr, self.indices, n_samples, + i, j, offsets) + if ret == 1: + # rinse and repeat + self.sum_duplicates() + csr_sample_offsets(M, N, self.indptr, self.indices, n_samples, + i, j, offsets) + + if -1 not in offsets: + # only affects existing non-zero cells + self.data[offsets] = x + return + + else: + warn(f"Changing the sparsity structure of a {self.__class__.__name__} is" + " expensive. lil and dok are more efficient.", + SparseEfficiencyWarning, stacklevel=3) + # replace where possible + mask = offsets > -1 + self.data[offsets[mask]] = x[mask] + # only insertions remain + mask = ~mask + i = i[mask] + i[i < 0] += M + j = j[mask] + j[j < 0] += N + self._insert_many(i, j, x[mask]) + + def _zero_many(self, i, j): + """Sets value at each (i, j) to zero, preserving sparsity structure. + + Here (i,j) index major and minor respectively. + """ + i, j, M, N = self._prepare_indices(i, j) + + n_samples = len(i) + offsets = np.empty(n_samples, dtype=self.indices.dtype) + ret = csr_sample_offsets(M, N, self.indptr, self.indices, n_samples, + i, j, offsets) + if ret == 1: + # rinse and repeat + self.sum_duplicates() + csr_sample_offsets(M, N, self.indptr, self.indices, n_samples, + i, j, offsets) + + # only assign zeros to the existing sparsity structure + self.data[offsets[offsets > -1]] = 0 + + def _insert_many(self, i, j, x): + """Inserts new nonzero at each (i, j) with value x + + Here (i,j) index major and minor respectively. + i, j and x must be non-empty, 1d arrays. + Inserts each major group (e.g. all entries per row) at a time. + Maintains has_sorted_indices property. + Modifies i, j, x in place. + """ + order = np.argsort(i, kind='mergesort') # stable for duplicates + i = i.take(order, mode='clip') + j = j.take(order, mode='clip') + x = x.take(order, mode='clip') + + do_sort = self.has_sorted_indices + + # Update index data type + idx_dtype = self._get_index_dtype((self.indices, self.indptr), + maxval=(self.indptr[-1] + x.size)) + self.indptr = np.asarray(self.indptr, dtype=idx_dtype) + self.indices = np.asarray(self.indices, dtype=idx_dtype) + i = np.asarray(i, dtype=idx_dtype) + j = np.asarray(j, dtype=idx_dtype) + + # Collate old and new in chunks by major index + indices_parts = [] + data_parts = [] + ui, ui_indptr = np.unique(i, return_index=True) + ui_indptr = np.append(ui_indptr, len(j)) + new_nnzs = np.diff(ui_indptr) + prev = 0 + for c, (ii, js, je) in enumerate(zip(ui, ui_indptr, ui_indptr[1:])): + # old entries + start = self.indptr[prev] + stop = self.indptr[ii] + indices_parts.append(self.indices[start:stop]) + data_parts.append(self.data[start:stop]) + + # handle duplicate j: keep last setting + uj, uj_indptr = np.unique(j[js:je][::-1], return_index=True) + if len(uj) == je - js: + indices_parts.append(j[js:je]) + data_parts.append(x[js:je]) + else: + indices_parts.append(j[js:je][::-1][uj_indptr]) + data_parts.append(x[js:je][::-1][uj_indptr]) + new_nnzs[c] = len(uj) + + prev = ii + + # remaining old entries + start = self.indptr[ii] + indices_parts.append(self.indices[start:]) + data_parts.append(self.data[start:]) + + # update attributes + self.indices = np.concatenate(indices_parts) + self.data = np.concatenate(data_parts) + nnzs = np.empty(self.indptr.shape, dtype=idx_dtype) + nnzs[0] = idx_dtype(0) + indptr_diff = np.diff(self.indptr) + indptr_diff[ui] += new_nnzs + nnzs[1:] = indptr_diff + self.indptr = np.cumsum(nnzs, out=nnzs) + + if do_sort: + # TODO: only sort where necessary + self.has_sorted_indices = False + self.sort_indices() + + self.check_format(full_check=False) + + ###################### + # Conversion methods # + ###################### + + def tocoo(self, copy=True): + if self.ndim == 1: + csr = self.tocsr() + return self._coo_container((csr.data, (csr.indices,)), csr.shape, copy=copy) + major_dim, minor_dim = self._swap(self.shape) + minor_indices = self.indices + major_indices = np.empty(len(minor_indices), dtype=self.indices.dtype) + _sparsetools.expandptr(major_dim, self.indptr, major_indices) + coords = self._swap((major_indices, minor_indices)) + + return self._coo_container( + (self.data, coords), self.shape, copy=copy, dtype=self.dtype + ) + + tocoo.__doc__ = _spbase.tocoo.__doc__ + + def toarray(self, order=None, out=None): + if out is None and order is None: + order = self._swap('cf')[0] + out = self._process_toarray_args(order, out) + if not (out.flags.c_contiguous or out.flags.f_contiguous): + raise ValueError('Output array must be C or F contiguous') + # align ideal order with output array order + if out.flags.c_contiguous: + x = self.tocsr() + y = out + else: + x = self.tocsc() + y = out.T + M, N = x._swap(x._shape_as_2d) + csr_todense(M, N, x.indptr, x.indices, x.data, y) + return out + + toarray.__doc__ = _spbase.toarray.__doc__ + + ############################################################## + # methods that examine or modify the internal data structure # + ############################################################## + + def eliminate_zeros(self): + """Remove zero entries from the array/matrix + + This is an *in place* operation. + """ + M, N = self._swap(self._shape_as_2d) + _sparsetools.csr_eliminate_zeros(M, N, self.indptr, self.indices, self.data) + self.prune() # nnz may have changed + + @property + def has_canonical_format(self) -> bool: + """Whether the array/matrix has sorted indices and no duplicates + + Returns + - True: if the above applies + - False: otherwise + + has_canonical_format implies has_sorted_indices, so if the latter flag + is False, so will the former be; if the former is found True, the + latter flag is also set. + """ + # first check to see if result was cached + if not getattr(self, '_has_sorted_indices', True): + # not sorted => not canonical + self._has_canonical_format = False + elif not hasattr(self, '_has_canonical_format'): + self.has_canonical_format = bool( + _sparsetools.csr_has_canonical_format( + len(self.indptr) - 1, self.indptr, self.indices) + ) + return self._has_canonical_format + + @has_canonical_format.setter + def has_canonical_format(self, val: bool): + self._has_canonical_format = bool(val) + if val: + self.has_sorted_indices = True + + def sum_duplicates(self): + """Eliminate duplicate entries by adding them together + + This is an *in place* operation. + """ + if self.has_canonical_format: + return + self.sort_indices() + + M, N = self._swap(self._shape_as_2d) + _sparsetools.csr_sum_duplicates(M, N, self.indptr, self.indices, self.data) + + self.prune() # nnz may have changed + self.has_canonical_format = True + + @property + def has_sorted_indices(self) -> bool: + """Whether the indices are sorted + + Returns + - True: if the indices of the array/matrix are in sorted order + - False: otherwise + """ + # first check to see if result was cached + if not hasattr(self, '_has_sorted_indices'): + self._has_sorted_indices = bool( + _sparsetools.csr_has_sorted_indices( + len(self.indptr) - 1, self.indptr, self.indices) + ) + return self._has_sorted_indices + + @has_sorted_indices.setter + def has_sorted_indices(self, val: bool): + self._has_sorted_indices = bool(val) + + + def sorted_indices(self): + """Return a copy of this array/matrix with sorted indices + """ + A = self.copy() + A.sort_indices() + return A + + # an alternative that has linear complexity is the following + # although the previous option is typically faster + # return self.toother().toother() + + def sort_indices(self): + """Sort the indices of this array/matrix *in place* + """ + + if not self.has_sorted_indices: + _sparsetools.csr_sort_indices(len(self.indptr) - 1, self.indptr, + self.indices, self.data) + self.has_sorted_indices = True + + def prune(self): + """Remove empty space after all non-zero elements. + """ + major_dim = self._swap(self._shape_as_2d)[0] + + if len(self.indptr) != major_dim + 1: + raise ValueError('index pointer has invalid length') + if len(self.indices) < self.nnz: + raise ValueError('indices array has fewer than nnz elements') + if len(self.data) < self.nnz: + raise ValueError('data array has fewer than nnz elements') + + self.indices = _prune_array(self.indices[:self.nnz]) + self.data = _prune_array(self.data[:self.nnz]) + + def resize(self, *shape): + shape = check_shape(shape, allow_nd=self._allow_nd) + + if hasattr(self, 'blocksize'): + bm, bn = self.blocksize + new_M, rm = divmod(shape[0], bm) + new_N, rn = divmod(shape[1], bn) + if rm or rn: + raise ValueError(f"shape must be divisible into {self.blocksize}" + f" blocks. Got {shape}") + M, N = self.shape[0] // bm, self.shape[1] // bn + else: + new_M, new_N = self._swap(shape if len(shape)>1 else (1, shape[0])) + M, N = self._swap(self._shape_as_2d) + + if new_M < M: + self.indices = self.indices[:self.indptr[new_M]] + self.data = self.data[:self.indptr[new_M]] + self.indptr = self.indptr[:new_M + 1] + elif new_M > M: + self.indptr = np.resize(self.indptr, new_M + 1) + self.indptr[M + 1:].fill(self.indptr[M]) + + if new_N < N: + mask = self.indices < new_N + if not np.all(mask): + self.indices = self.indices[mask] + self.data = self.data[mask] + major_index, val = self._minor_reduce(np.add, mask) + self.indptr.fill(0) + self.indptr[1:][major_index] = val + np.cumsum(self.indptr, out=self.indptr) + + self._shape = shape + + resize.__doc__ = _spbase.resize.__doc__ + + ################### + # utility methods # + ################### + + # needed by _data_matrix + def _with_data(self, data, copy=True): + """Returns a matrix with the same sparsity structure as self, + but with different data. By default the structure arrays + (i.e. .indptr and .indices) are copied. + """ + if copy: + return self.__class__((data, self.indices.copy(), + self.indptr.copy()), + shape=self.shape, + dtype=data.dtype) + else: + return self.__class__((data, self.indices, self.indptr), + shape=self.shape, dtype=data.dtype) + + def _binopt(self, other, op): + """apply the binary operation fn to two sparse matrices.""" + other = self.__class__(other) + + # e.g. csr_plus_csr, csr_minus_csr, etc. + fn = getattr(_sparsetools, self.format + op + self.format) + + maxnnz = self.nnz + other.nnz + idx_dtype = self._get_index_dtype((self.indptr, self.indices, + other.indptr, other.indices), + maxval=maxnnz) + indptr = np.empty(self.indptr.shape, dtype=idx_dtype) + indices = np.empty(maxnnz, dtype=idx_dtype) + + bool_ops = ['_ne_', '_lt_', '_gt_', '_le_', '_ge_'] + if op in bool_ops: + data = np.empty(maxnnz, dtype=np.bool_) + else: + data = np.empty(maxnnz, dtype=upcast(self.dtype, other.dtype)) + + M, N = self._shape_as_2d + fn(M, N, + np.asarray(self.indptr, dtype=idx_dtype), + np.asarray(self.indices, dtype=idx_dtype), + self.data, + np.asarray(other.indptr, dtype=idx_dtype), + np.asarray(other.indices, dtype=idx_dtype), + other.data, + indptr, indices, data) + + A = self.__class__((data, indices, indptr), shape=self.shape) + A.prune() + + return A + + def _divide_sparse(self, other): + """ + Divide this matrix by a second sparse matrix. + """ + if other.shape != self.shape: + raise ValueError('inconsistent shapes') + + r = self._binopt(other, '_eldiv_') + + if np.issubdtype(r.dtype, np.inexact): + # Eldiv leaves entries outside the combined sparsity + # pattern empty, so they must be filled manually. + # Everything outside of other's sparsity is NaN, and everything + # inside it is either zero or defined by eldiv. + out = np.empty(self.shape, dtype=self.dtype) + out.fill(np.nan) + coords = other.nonzero() + if self.ndim == 1: + coords = (coords[-1],) + out[coords] = 0 + r = r.tocoo() + out[r.coords] = r.data + return self._container(out) + else: + # integers types go with nan <-> 0 + out = r + return out + + def _broadcast_to(self, shape, copy=False): + if self.shape == shape: + return self.copy() if copy else self + + shape = check_shape(shape, allow_nd=(self._allow_nd)) + + if broadcast_shapes(self.shape, shape) != shape: + raise ValueError("cannot be broadcast") + + if len(self.shape) == 1 and len(shape) == 1: + self.sum_duplicates() + if self.nnz == 0: # array has no non zero elements + return self.__class__(shape, dtype=self.dtype, copy=False) + + N = shape[0] + data = np.full(N, self.data[0]) + indices = np.arange(0,N) + indptr = np.array([0, N]) + return self._csr_container((data, indices, indptr), shape=shape, copy=False) + + # treat 1D as a 2D row + old_shape = self._shape_as_2d + + if len(shape) != 2: + ndim = len(shape) + raise ValueError(f'CSR/CSC broadcast_to cannot have shape >2D. Got {ndim}D') + + if self.nnz == 0: # array has no non zero elements + return self.__class__(shape, dtype=self.dtype, copy=False) + + self.sum_duplicates() + M, N = self._swap(shape) + oM, oN = self._swap(old_shape) + if all(s == 1 for s in old_shape): + # Broadcast a single element to the entire shape + data = np.full(M * N, self.data[0]) + indices = np.tile(np.arange(N), M) + indptr = np.arange(0, len(data) + 1, N) + elif oM == 1 and oN == N: + # Broadcast row-wise (columns for CSC) + data = np.tile(self.data, M) + indices = np.tile(self.indices, M) + indptr = np.arange(0, len(data) + 1, len(self.data)) + elif oN == 1 and oM == M: + # Broadcast column-wise (rows for CSC) + data = np.repeat(self.data, N) + indices = np.tile(np.arange(N), len(self.data)) + indptr = self.indptr * N + return self.__class__((data, indices, indptr), shape=shape, copy=False) + + +def _make_diagonal_csr(data, is_array=False): + """build diagonal csc_array/csr_array => self._csr_container + + Parameter `data` should be a raveled numpy array holding the + values on the diagonal of the resulting sparse matrix. + """ + from ._csr import csr_array, csr_matrix + csr_array = csr_array if is_array else csr_matrix + + N = len(data) + idx_dtype = get_index_dtype(maxval=N) + indptr = np.arange(N + 1, dtype=idx_dtype) + indices = indptr[:-1] + + return csr_array((data, indices, indptr), shape=(N, N)) + + +def _process_slice(sl, num): + if sl is None: + i0, i1 = 0, num + elif isinstance(sl, slice): + i0, i1, stride = sl.indices(num) + if stride != 1: + raise ValueError('slicing with step != 1 not supported') + i0 = min(i0, i1) # give an empty slice when i0 > i1 + elif isintlike(sl): + if sl < 0: + sl += num + i0, i1 = sl, sl + 1 + if i0 < 0 or i1 > num: + raise IndexError(f'index out of bounds: 0 <= {i0} < {i1} <= {num}') + else: + raise TypeError('expected slice or scalar') + + return i0, i1 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_construct.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_construct.py new file mode 100644 index 0000000000000000000000000000000000000000..f483976badb771d777c9dba0eecfbeb94d2e76fd --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_construct.py @@ -0,0 +1,1402 @@ +"""Functions to construct sparse matrices and arrays +""" + +__docformat__ = "restructuredtext en" + +__all__ = ['spdiags', 'eye', 'identity', 'kron', 'kronsum', + 'hstack', 'vstack', 'bmat', 'rand', 'random', 'diags', 'block_diag', + 'diags_array', 'block_array', 'eye_array', 'random_array'] + +import numbers +import math +import numpy as np + +from scipy._lib._util import check_random_state, rng_integers, _transition_to_rng +from ._sputils import upcast, get_index_dtype, isscalarlike + +from ._sparsetools import csr_hstack +from ._bsr import bsr_matrix, bsr_array +from ._coo import coo_matrix, coo_array +from ._csc import csc_matrix, csc_array +from ._csr import csr_matrix, csr_array +from ._dia import dia_matrix, dia_array + +from ._base import issparse, sparray + + +def spdiags(data, diags, m=None, n=None, format=None): + """ + Return a sparse matrix from diagonals. + + Parameters + ---------- + data : array_like + Matrix diagonals stored row-wise + diags : sequence of int or an int + Diagonals to set: + + * k = 0 the main diagonal + * k > 0 the kth upper diagonal + * k < 0 the kth lower diagonal + m, n : int, tuple, optional + Shape of the result. If `n` is None and `m` is a given tuple, + the shape is this tuple. If omitted, the matrix is square and + its shape is len(data[0]). + format : str, optional + Format of the result. By default (format=None) an appropriate sparse + matrix format is returned. This choice is subject to change. + + .. warning:: + + This function returns a sparse matrix -- not a sparse array. + You are encouraged to use ``dia_array`` to take advantage + of the sparse array functionality. + + Notes + ----- + This function can be replaced by an equivalent call to ``dia_matrix`` + as:: + + dia_matrix((data, diags), shape=(m, n)).asformat(format) + + See Also + -------- + diags_array : more convenient form of this function + diags : matrix version of diags_array + dia_matrix : the sparse DIAgonal format. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import spdiags + >>> data = np.array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]]) + >>> diags = np.array([0, -1, 2]) + >>> spdiags(data, diags, 4, 4).toarray() + array([[1, 0, 3, 0], + [1, 2, 0, 4], + [0, 2, 3, 0], + [0, 0, 3, 4]]) + + """ + if m is None and n is None: + m = n = len(data[0]) + elif n is None: + m, n = m + return dia_matrix((data, diags), shape=(m, n)).asformat(format) + + +def diags_array(diagonals, /, *, offsets=0, shape=None, format=None, dtype=None): + """ + Construct a sparse array from diagonals. + + Parameters + ---------- + diagonals : sequence of array_like + Sequence of arrays containing the array diagonals, + corresponding to `offsets`. + offsets : sequence of int or an int, optional + Diagonals to set (repeated offsets are not allowed): + - k = 0 the main diagonal (default) + - k > 0 the kth upper diagonal + - k < 0 the kth lower diagonal + shape : tuple of int, optional + Shape of the result. If omitted, a square array large enough + to contain the diagonals is returned. + format : {"dia", "csr", "csc", "lil", ...}, optional + Matrix format of the result. By default (format=None) an + appropriate sparse array format is returned. This choice is + subject to change. + dtype : dtype, optional + Data type of the array. + + Notes + ----- + Repeated diagonal offsets are disallowed. + + The result from `diags_array` is the sparse equivalent of:: + + np.diag(diagonals[0], offsets[0]) + + ... + + np.diag(diagonals[k], offsets[k]) + + ``diags_array`` differs from `dia_array` in the way it handles off-diagonals. + Specifically, `dia_array` assumes the data input includes padding + (ignored values) at the start/end of the rows for positive/negative + offset, while ``diags_array` assumes the input data has no padding. + Each value in the input ``diagonals`` is used. + + .. versionadded:: 1.11 + + Examples + -------- + >>> from scipy.sparse import diags_array + >>> diagonals = [[1, 2, 3, 4], [1, 2, 3], [1, 2]] + >>> diags_array(diagonals, offsets=[0, -1, 2]).toarray() + array([[1., 0., 1., 0.], + [1., 2., 0., 2.], + [0., 2., 3., 0.], + [0., 0., 3., 4.]]) + + Broadcasting of scalars is supported (but shape needs to be + specified): + + >>> diags_array([1, -2, 1], offsets=[-1, 0, 1], shape=(4, 4)).toarray() + array([[-2., 1., 0., 0.], + [ 1., -2., 1., 0.], + [ 0., 1., -2., 1.], + [ 0., 0., 1., -2.]]) + + + If only one diagonal is wanted (as in `numpy.diag`), the following + works as well: + + >>> diags_array([1, 2, 3], offsets=1).toarray() + array([[ 0., 1., 0., 0.], + [ 0., 0., 2., 0.], + [ 0., 0., 0., 3.], + [ 0., 0., 0., 0.]]) + + """ + # if offsets is not a sequence, assume that there's only one diagonal + if isscalarlike(offsets): + # now check that there's actually only one diagonal + if len(diagonals) == 0 or isscalarlike(diagonals[0]): + diagonals = [np.atleast_1d(diagonals)] + else: + raise ValueError("Different number of diagonals and offsets.") + else: + diagonals = list(map(np.atleast_1d, diagonals)) + + offsets = np.atleast_1d(offsets) + + # Basic check + if len(diagonals) != len(offsets): + raise ValueError("Different number of diagonals and offsets.") + + # Determine shape, if omitted + if shape is None: + m = len(diagonals[0]) + abs(int(offsets[0])) + shape = (m, m) + + # Determine data type, if omitted + if dtype is None: + dtype = np.common_type(*diagonals) + + # Construct data array + m, n = shape + + M = max([min(m + offset, n - offset) + max(0, offset) + for offset in offsets]) + M = max(0, M) + data_arr = np.zeros((len(offsets), M), dtype=dtype) + + K = min(m, n) + + for j, diagonal in enumerate(diagonals): + offset = offsets[j] + k = max(0, offset) + length = min(m + offset, n - offset, K) + if length < 0: + raise ValueError(f"Offset {offset} (index {j}) out of bounds") + try: + data_arr[j, k:k+length] = diagonal[...,:length] + except ValueError as e: + if len(diagonal) != length and len(diagonal) != 1: + raise ValueError( + f"Diagonal length (index {j}: {len(diagonal)} at" + f" offset {offset}) does not agree with array size ({m}, {n})." + ) from e + raise + + return dia_array((data_arr, offsets), shape=(m, n)).asformat(format) + + +def diags(diagonals, offsets=0, shape=None, format=None, dtype=None): + """ + Construct a sparse matrix from diagonals. + + .. warning:: + + This function returns a sparse matrix -- not a sparse array. + You are encouraged to use ``diags_array`` to take advantage + of the sparse array functionality. + + Parameters + ---------- + diagonals : sequence of array_like + Sequence of arrays containing the matrix diagonals, + corresponding to `offsets`. + offsets : sequence of int or an int, optional + Diagonals to set (repeated offsets are not allowed): + - k = 0 the main diagonal (default) + - k > 0 the kth upper diagonal + - k < 0 the kth lower diagonal + shape : tuple of int, optional + Shape of the result. If omitted, a square matrix large enough + to contain the diagonals is returned. + format : {"dia", "csr", "csc", "lil", ...}, optional + Matrix format of the result. By default (format=None) an + appropriate sparse matrix format is returned. This choice is + subject to change. + dtype : dtype, optional + Data type of the matrix. + + See Also + -------- + spdiags : construct matrix from diagonals + diags_array : construct sparse array instead of sparse matrix + + Notes + ----- + Repeated diagonal offsets are disallowed. + + The result from `diags` is the sparse equivalent of:: + + np.diag(diagonals[0], offsets[0]) + + ... + + np.diag(diagonals[k], offsets[k]) + + ``diags`` differs from ``dia_matrix`` in the way it handles off-diagonals. + Specifically, `dia_matrix` assumes the data input includes padding + (ignored values) at the start/end of the rows for positive/negative + offset, while ``diags` assumes the input data has no padding. + Each value in the input ``diagonals`` is used. + + .. versionadded:: 0.11 + + Examples + -------- + >>> from scipy.sparse import diags + >>> diagonals = [[1, 2, 3, 4], [1, 2, 3], [1, 2]] + >>> diags(diagonals, [0, -1, 2]).toarray() + array([[1., 0., 1., 0.], + [1., 2., 0., 2.], + [0., 2., 3., 0.], + [0., 0., 3., 4.]]) + + Broadcasting of scalars is supported (but shape needs to be + specified): + + >>> diags([1, -2, 1], [-1, 0, 1], shape=(4, 4)).toarray() + array([[-2., 1., 0., 0.], + [ 1., -2., 1., 0.], + [ 0., 1., -2., 1.], + [ 0., 0., 1., -2.]]) + + + If only one diagonal is wanted (as in `numpy.diag`), the following + works as well: + + >>> diags([1, 2, 3], 1).toarray() + array([[ 0., 1., 0., 0.], + [ 0., 0., 2., 0.], + [ 0., 0., 0., 3.], + [ 0., 0., 0., 0.]]) + + """ + A = diags_array(diagonals, offsets=offsets, shape=shape, dtype=dtype) + return dia_matrix(A).asformat(format) + + +def identity(n, dtype='d', format=None): + """Identity matrix in sparse format + + Returns an identity matrix with shape (n,n) using a given + sparse format and dtype. This differs from `eye_array` in + that it has a square shape with ones only on the main diagonal. + It is thus the multiplicative identity. `eye_array` allows + rectangular shapes and the diagonal can be offset from the main one. + + .. warning:: + + This function returns a sparse matrix -- not a sparse array. + You are encouraged to use ``eye_array`` to take advantage + of the sparse array functionality. + + Parameters + ---------- + n : int + Shape of the identity matrix. + dtype : dtype, optional + Data type of the matrix + format : str, optional + Sparse format of the result, e.g., format="csr", etc. + + Examples + -------- + >>> import scipy as sp + >>> sp.sparse.identity(3).toarray() + array([[ 1., 0., 0.], + [ 0., 1., 0.], + [ 0., 0., 1.]]) + >>> sp.sparse.identity(3, dtype='int8', format='dia') + + >>> sp.sparse.eye_array(3, dtype='int8', format='dia') + + + """ + return eye(n, n, dtype=dtype, format=format) + + +def eye_array(m, n=None, *, k=0, dtype=float, format=None): + """Identity matrix in sparse array format + + Return a sparse array with ones on diagonal. + Specifically a sparse array (m x n) where the kth diagonal + is all ones and everything else is zeros. + + Parameters + ---------- + m : int + Number of rows requested. + n : int, optional + Number of columns. Default: `m`. + k : int, optional + Diagonal to place ones on. Default: 0 (main diagonal). + dtype : dtype, optional + Data type of the array + format : str, optional (default: "dia") + Sparse format of the result, e.g., format="csr", etc. + + Examples + -------- + >>> import numpy as np + >>> import scipy as sp + >>> sp.sparse.eye_array(3).toarray() + array([[ 1., 0., 0.], + [ 0., 1., 0.], + [ 0., 0., 1.]]) + >>> sp.sparse.eye_array(3, dtype=np.int8) + + + """ + # TODO: delete next 15 lines [combine with _eye()] once spmatrix removed + return _eye(m, n, k, dtype, format) + + +def _eye(m, n, k, dtype, format, as_sparray=True): + if as_sparray: + csr_sparse = csr_array + csc_sparse = csc_array + coo_sparse = coo_array + diags_sparse = diags_array + else: + csr_sparse = csr_matrix + csc_sparse = csc_matrix + coo_sparse = coo_matrix + diags_sparse = diags + + if n is None: + n = m + m, n = int(m), int(n) + + if m == n and k == 0: + # fast branch for special formats + if format in ['csr', 'csc']: + idx_dtype = get_index_dtype(maxval=n) + indptr = np.arange(n+1, dtype=idx_dtype) + indices = np.arange(n, dtype=idx_dtype) + data = np.ones(n, dtype=dtype) + cls = {'csr': csr_sparse, 'csc': csc_sparse}[format] + return cls((data, indices, indptr), (n, n)) + + elif format == 'coo': + idx_dtype = get_index_dtype(maxval=n) + row = np.arange(n, dtype=idx_dtype) + col = np.arange(n, dtype=idx_dtype) + data = np.ones(n, dtype=dtype) + return coo_sparse((data, (row, col)), (n, n)) + + data = np.ones((1, max(0, min(m + k, n))), dtype=dtype) + return diags_sparse(data, offsets=[k], shape=(m, n), dtype=dtype).asformat(format) + + +def eye(m, n=None, k=0, dtype=float, format=None): + """Sparse matrix with ones on diagonal + + Returns a sparse matrix (m x n) where the kth diagonal + is all ones and everything else is zeros. + + Parameters + ---------- + m : int + Number of rows in the matrix. + n : int, optional + Number of columns. Default: `m`. + k : int, optional + Diagonal to place ones on. Default: 0 (main diagonal). + dtype : dtype, optional + Data type of the matrix. + format : str, optional + Sparse format of the result, e.g., format="csr", etc. + + .. warning:: + + This function returns a sparse matrix -- not a sparse array. + You are encouraged to use ``eye_array`` to take advantage + of the sparse array functionality. + + Examples + -------- + >>> import numpy as np + >>> import scipy as sp + >>> sp.sparse.eye(3).toarray() + array([[ 1., 0., 0.], + [ 0., 1., 0.], + [ 0., 0., 1.]]) + >>> sp.sparse.eye(3, dtype=np.int8) + + + """ + return _eye(m, n, k, dtype, format, False) + + +def kron(A, B, format=None): + """kronecker product of sparse matrices A and B + + Parameters + ---------- + A : sparse or dense matrix + first matrix of the product + B : sparse or dense matrix + second matrix of the product + format : str, optional (default: 'bsr' or 'coo') + format of the result (e.g. "csr") + If None, choose 'bsr' for relatively dense array and 'coo' for others + + Returns + ------- + kronecker product in a sparse format. + Returns a sparse matrix unless either A or B is a + sparse array in which case returns a sparse array. + + Examples + -------- + >>> import numpy as np + >>> import scipy as sp + >>> A = sp.sparse.csr_array(np.array([[0, 2], [5, 0]])) + >>> B = sp.sparse.csr_array(np.array([[1, 2], [3, 4]])) + >>> sp.sparse.kron(A, B).toarray() + array([[ 0, 0, 2, 4], + [ 0, 0, 6, 8], + [ 5, 10, 0, 0], + [15, 20, 0, 0]]) + + >>> sp.sparse.kron(A, [[1, 2], [3, 4]]).toarray() + array([[ 0, 0, 2, 4], + [ 0, 0, 6, 8], + [ 5, 10, 0, 0], + [15, 20, 0, 0]]) + + """ + # TODO: delete next 10 lines and replace _sparse with _array when spmatrix removed + if isinstance(A, sparray) or isinstance(B, sparray): + # convert to local variables + bsr_sparse = bsr_array + csr_sparse = csr_array + coo_sparse = coo_array + else: # use spmatrix + bsr_sparse = bsr_matrix + csr_sparse = csr_matrix + coo_sparse = coo_matrix + + B = coo_sparse(B) + if B.ndim != 2: + raise ValueError(f"kron requires 2D input arrays. `B` is {B.ndim}D.") + + # B is fairly dense, use BSR + if (format is None or format == "bsr") and 2*B.nnz >= B.shape[0] * B.shape[1]: + A = csr_sparse(A,copy=True) + if A.ndim != 2: + raise ValueError(f"kron requires 2D input arrays. `A` is {A.ndim}D.") + output_shape = (A.shape[0]*B.shape[0], A.shape[1]*B.shape[1]) + + if A.nnz == 0 or B.nnz == 0: + # kronecker product is the zero matrix + return coo_sparse(output_shape).asformat(format) + + B = B.toarray() + data = A.data.repeat(B.size).reshape(-1,B.shape[0],B.shape[1]) + data = data * B + + return bsr_sparse((data,A.indices,A.indptr), shape=output_shape) + else: + # use COO + A = coo_sparse(A) + if A.ndim != 2: + raise ValueError(f"kron requires 2D input arrays. `A` is {A.ndim}D.") + output_shape = (A.shape[0]*B.shape[0], A.shape[1]*B.shape[1]) + + if A.nnz == 0 or B.nnz == 0: + # kronecker product is the zero matrix + return coo_sparse(output_shape).asformat(format) + + # expand entries of a into blocks + idx_dtype = get_index_dtype(A.coords, maxval=max(output_shape)) + row = np.asarray(A.row, dtype=idx_dtype).repeat(B.nnz) + col = np.asarray(A.col, dtype=idx_dtype).repeat(B.nnz) + data = A.data.repeat(B.nnz) + + row *= B.shape[0] + col *= B.shape[1] + + # increment block indices + row,col = row.reshape(-1,B.nnz),col.reshape(-1,B.nnz) + row += B.row + col += B.col + row,col = row.reshape(-1),col.reshape(-1) + + # compute block entries + data = data.reshape(-1,B.nnz) * B.data + data = data.reshape(-1) + + return coo_sparse((data,(row,col)), shape=output_shape).asformat(format) + + +def kronsum(A, B, format=None): + """kronecker sum of square sparse matrices A and B + + Kronecker sum of two sparse matrices is a sum of two Kronecker + products kron(I_n,A) + kron(B,I_m) where A has shape (m,m) + and B has shape (n,n) and I_m and I_n are identity matrices + of shape (m,m) and (n,n), respectively. + + Parameters + ---------- + A + square matrix + B + square matrix + format : str + format of the result (e.g. "csr") + + Returns + ------- + kronecker sum in a sparse matrix format + + """ + # TODO: delete next 8 lines and replace _sparse with _array when spmatrix removed + if isinstance(A, sparray) or isinstance(B, sparray): + # convert to local variables + coo_sparse = coo_array + identity_sparse = eye_array + else: + coo_sparse = coo_matrix + identity_sparse = identity + + A = coo_sparse(A) + B = coo_sparse(B) + + if A.ndim != 2: + raise ValueError(f"kronsum requires 2D inputs. `A` is {A.ndim}D.") + if B.ndim != 2: + raise ValueError(f"kronsum requires 2D inputs. `B` is {B.ndim}D.") + if A.shape[0] != A.shape[1]: + raise ValueError('A is not square') + if B.shape[0] != B.shape[1]: + raise ValueError('B is not square') + + dtype = upcast(A.dtype, B.dtype) + + I_n = identity_sparse(A.shape[0], dtype=dtype) + I_m = identity_sparse(B.shape[0], dtype=dtype) + L = kron(I_m, A, format='coo') + R = kron(B, I_n, format='coo') + + return (L + R).asformat(format) + + +def _compressed_sparse_stack(blocks, axis, return_spmatrix): + """ + Stacking fast path for CSR/CSC matrices or arrays + (i) vstack for CSR, (ii) hstack for CSC. + """ + other_axis = 1 if axis == 0 else 0 + data = np.concatenate([b.data for b in blocks]) + constant_dim = blocks[0]._shape_as_2d[other_axis] + idx_dtype = get_index_dtype(arrays=[b.indptr for b in blocks], + maxval=max(data.size, constant_dim)) + indices = np.empty(data.size, dtype=idx_dtype) + indptr = np.empty(sum(b._shape_as_2d[axis] for b in blocks) + 1, dtype=idx_dtype) + last_indptr = idx_dtype(0) + sum_dim = 0 + sum_indices = 0 + for b in blocks: + if b._shape_as_2d[other_axis] != constant_dim: + raise ValueError(f'incompatible dimensions for axis {other_axis}') + indices[sum_indices:sum_indices+b.indices.size] = b.indices + sum_indices += b.indices.size + idxs = slice(sum_dim, sum_dim + b._shape_as_2d[axis]) + indptr[idxs] = b.indptr[:-1] + indptr[idxs] += last_indptr + sum_dim += b._shape_as_2d[axis] + last_indptr += b.indptr[-1] + indptr[-1] = last_indptr + # TODO remove this if-structure when sparse matrices removed + if return_spmatrix: + if axis == 0: + return csr_matrix((data, indices, indptr), + shape=(sum_dim, constant_dim)) + else: + return csc_matrix((data, indices, indptr), + shape=(constant_dim, sum_dim)) + + if axis == 0: + return csr_array((data, indices, indptr), + shape=(sum_dim, constant_dim)) + else: + return csc_array((data, indices, indptr), + shape=(constant_dim, sum_dim)) + + +def _stack_along_minor_axis(blocks, axis): + """ + Stacking fast path for CSR/CSC matrices along the minor axis + (i) hstack for CSR, (ii) vstack for CSC. + """ + n_blocks = len(blocks) + if n_blocks == 0: + raise ValueError('Missing block matrices') + + if n_blocks == 1: + return blocks[0] + + # check for incompatible dimensions + other_axis = 1 if axis == 0 else 0 + other_axis_dims = {b._shape_as_2d[other_axis] for b in blocks} + if len(other_axis_dims) > 1: + raise ValueError(f'Mismatching dimensions along axis {other_axis}: ' + f'{other_axis_dims}') + constant_dim, = other_axis_dims + + # Do the stacking + indptr_list = [b.indptr for b in blocks] + data_cat = np.concatenate([b.data for b in blocks]) + + # Need to check if any indices/indptr, would be too large post- + # concatenation for np.int32: + # - The max value of indices is the output array's stacking-axis length - 1 + # - The max value in indptr is the number of non-zero entries. This is + # exceedingly unlikely to require int64, but is checked out of an + # abundance of caution. + sum_dim = sum(b._shape_as_2d[axis] for b in blocks) + nnz = sum(len(b.indices) for b in blocks) + idx_dtype = get_index_dtype(indptr_list, maxval=max(sum_dim - 1, nnz)) + stack_dim_cat = np.array([b._shape_as_2d[axis] for b in blocks], dtype=idx_dtype) + if data_cat.size > 0: + indptr_cat = np.concatenate(indptr_list, dtype=idx_dtype) + indices_cat = np.concatenate([b.indices for b in blocks], dtype=idx_dtype) + indptr = np.empty(constant_dim + 1, dtype=idx_dtype) + indices = np.empty_like(indices_cat) + data = np.empty_like(data_cat) + csr_hstack(n_blocks, constant_dim, stack_dim_cat, + indptr_cat, indices_cat, data_cat, + indptr, indices, data) + else: + indptr = np.zeros(constant_dim + 1, dtype=idx_dtype) + indices = np.empty(0, dtype=idx_dtype) + data = np.empty(0, dtype=data_cat.dtype) + + if axis == 0: + return blocks[0]._csc_container((data, indices, indptr), + shape=(sum_dim, constant_dim)) + else: + return blocks[0]._csr_container((data, indices, indptr), + shape=(constant_dim, sum_dim)) + + +def hstack(blocks, format=None, dtype=None): + """ + Stack sparse matrices horizontally (column wise) + + Parameters + ---------- + blocks + sequence of sparse matrices with compatible shapes + format : str + sparse format of the result (e.g., "csr") + by default an appropriate sparse matrix format is returned. + This choice is subject to change. + dtype : dtype, optional + The data-type of the output matrix. If not given, the dtype is + determined from that of `blocks`. + + Returns + ------- + new_array : sparse matrix or array + If any block in blocks is a sparse array, return a sparse array. + Otherwise return a sparse matrix. + + If you want a sparse array built from blocks that are not sparse + arrays, use ``block(hstack(blocks))`` or convert one block + e.g. ``blocks[0] = csr_array(blocks[0])``. + + See Also + -------- + vstack : stack sparse matrices vertically (row wise) + + Examples + -------- + >>> from scipy.sparse import coo_matrix, hstack + >>> A = coo_matrix([[1, 2], [3, 4]]) + >>> B = coo_matrix([[5], [6]]) + >>> hstack([A,B]).toarray() + array([[1, 2, 5], + [3, 4, 6]]) + + """ + blocks = np.asarray(blocks, dtype='object') + if any(isinstance(b, sparray) for b in blocks.flat): + return _block([blocks], format, dtype) + else: + return _block([blocks], format, dtype, return_spmatrix=True) + + +def vstack(blocks, format=None, dtype=None): + """ + Stack sparse arrays vertically (row wise) + + Parameters + ---------- + blocks + sequence of sparse arrays with compatible shapes + format : str, optional + sparse format of the result (e.g., "csr") + by default an appropriate sparse array format is returned. + This choice is subject to change. + dtype : dtype, optional + The data-type of the output array. If not given, the dtype is + determined from that of `blocks`. + + Returns + ------- + new_array : sparse matrix or array + If any block in blocks is a sparse array, return a sparse array. + Otherwise return a sparse matrix. + + If you want a sparse array built from blocks that are not sparse + arrays, use ``block(vstack(blocks))`` or convert one block + e.g. `blocks[0] = csr_array(blocks[0])`. + + See Also + -------- + hstack : stack sparse matrices horizontally (column wise) + + Examples + -------- + >>> from scipy.sparse import coo_array, vstack + >>> A = coo_array([[1, 2], [3, 4]]) + >>> B = coo_array([[5, 6]]) + >>> vstack([A, B]).toarray() + array([[1, 2], + [3, 4], + [5, 6]]) + + """ + blocks = np.asarray(blocks, dtype='object') + if any(isinstance(b, sparray) for b in blocks.flat): + return _block([[b] for b in blocks], format, dtype) + else: + return _block([[b] for b in blocks], format, dtype, return_spmatrix=True) + + +def bmat(blocks, format=None, dtype=None): + """ + Build a sparse array or matrix from sparse sub-blocks + + Note: `block_array` is preferred over `bmat`. They are the same function + except that `bmat` can return a deprecated sparse matrix. + `bmat` returns a coo_matrix if none of the inputs are a sparse array. + + .. warning:: + + This function returns a sparse matrix -- not a sparse array. + You are encouraged to use ``block_array`` to take advantage + of the sparse array functionality. + + Parameters + ---------- + blocks : array_like + Grid of sparse matrices with compatible shapes. + An entry of None implies an all-zero matrix. + format : {'bsr', 'coo', 'csc', 'csr', 'dia', 'dok', 'lil'}, optional + The sparse format of the result (e.g. "csr"). By default an + appropriate sparse matrix format is returned. + This choice is subject to change. + dtype : dtype, optional + The data-type of the output matrix. If not given, the dtype is + determined from that of `blocks`. + + Returns + ------- + bmat : sparse matrix or array + If any block in blocks is a sparse array, return a sparse array. + Otherwise return a sparse matrix. + + If you want a sparse array built from blocks that are not sparse + arrays, use ``block_array()``. + + See Also + -------- + block_array + + Examples + -------- + >>> from scipy.sparse import coo_array, bmat + >>> A = coo_array([[1, 2], [3, 4]]) + >>> B = coo_array([[5], [6]]) + >>> C = coo_array([[7]]) + >>> bmat([[A, B], [None, C]]).toarray() + array([[1, 2, 5], + [3, 4, 6], + [0, 0, 7]]) + + >>> bmat([[A, None], [None, C]]).toarray() + array([[1, 2, 0], + [3, 4, 0], + [0, 0, 7]]) + + """ + blocks = np.asarray(blocks, dtype='object') + if any(isinstance(b, sparray) for b in blocks.flat): + return _block(blocks, format, dtype) + else: + return _block(blocks, format, dtype, return_spmatrix=True) + + +def block_array(blocks, *, format=None, dtype=None): + """ + Build a sparse array from sparse sub-blocks + + Parameters + ---------- + blocks : array_like + Grid of sparse arrays with compatible shapes. + An entry of None implies an all-zero array. + format : {'bsr', 'coo', 'csc', 'csr', 'dia', 'dok', 'lil'}, optional + The sparse format of the result (e.g. "csr"). By default an + appropriate sparse array format is returned. + This choice is subject to change. + dtype : dtype, optional + The data-type of the output array. If not given, the dtype is + determined from that of `blocks`. + + Returns + ------- + block : sparse array + + See Also + -------- + block_diag : specify blocks along the main diagonals + diags : specify (possibly offset) diagonals + + Examples + -------- + >>> from scipy.sparse import coo_array, block_array + >>> A = coo_array([[1, 2], [3, 4]]) + >>> B = coo_array([[5], [6]]) + >>> C = coo_array([[7]]) + >>> block_array([[A, B], [None, C]]).toarray() + array([[1, 2, 5], + [3, 4, 6], + [0, 0, 7]]) + + >>> block_array([[A, None], [None, C]]).toarray() + array([[1, 2, 0], + [3, 4, 0], + [0, 0, 7]]) + + """ + return _block(blocks, format, dtype) + + +def _block(blocks, format, dtype, return_spmatrix=False): + blocks = np.asarray(blocks, dtype='object') + + if blocks.ndim != 2: + raise ValueError('blocks must be 2-D') + + M,N = blocks.shape + + # check for fast path cases + if (format in (None, 'csr') and + all(issparse(b) and b.format == 'csr' for b in blocks.flat) + ): + if N > 1: + # stack along columns (axis 1): must have shape (M, 1) + blocks = [[_stack_along_minor_axis(blocks[b, :], 1)] for b in range(M)] + blocks = np.asarray(blocks, dtype='object') + + # stack along rows (axis 0): + A = _compressed_sparse_stack(blocks[:, 0], 0, return_spmatrix) + if dtype is not None: + A = A.astype(dtype, copy=False) + return A + elif (format in (None, 'csc') and + all(issparse(b) and b.format == 'csc' for b in blocks.flat) + ): + if M > 1: + # stack along rows (axis 0): must have shape (1, N) + blocks = [[_stack_along_minor_axis(blocks[:, b], 0) for b in range(N)]] + blocks = np.asarray(blocks, dtype='object') + + # stack along columns (axis 1): + A = _compressed_sparse_stack(blocks[0, :], 1, return_spmatrix) + if dtype is not None: + A = A.astype(dtype, copy=False) + return A + + block_mask = np.zeros(blocks.shape, dtype=bool) + brow_lengths = np.zeros(M, dtype=np.int64) + bcol_lengths = np.zeros(N, dtype=np.int64) + + # convert everything to COO format + for i in range(M): + for j in range(N): + if blocks[i,j] is not None: + A = coo_array(blocks[i,j]) + blocks[i,j] = A + block_mask[i,j] = True + + if brow_lengths[i] == 0: + brow_lengths[i] = A._shape_as_2d[0] + elif brow_lengths[i] != A._shape_as_2d[0]: + msg = (f'blocks[{i},:] has incompatible row dimensions. ' + f'Got blocks[{i},{j}].shape[0] == {A._shape_as_2d[0]}, ' + f'expected {brow_lengths[i]}.') + raise ValueError(msg) + + if bcol_lengths[j] == 0: + bcol_lengths[j] = A._shape_as_2d[1] + elif bcol_lengths[j] != A._shape_as_2d[1]: + msg = (f'blocks[:,{j}] has incompatible column ' + f'dimensions. ' + f'Got blocks[{i},{j}].shape[1] == {A._shape_as_2d[1]}, ' + f'expected {bcol_lengths[j]}.') + raise ValueError(msg) + + nnz = sum(block.nnz for block in blocks[block_mask]) + if dtype is None: + all_dtypes = [blk.dtype for blk in blocks[block_mask]] + dtype = upcast(*all_dtypes) if all_dtypes else None + + row_offsets = np.append(0, np.cumsum(brow_lengths)) + col_offsets = np.append(0, np.cumsum(bcol_lengths)) + + shape = (row_offsets[-1], col_offsets[-1]) + + data = np.empty(nnz, dtype=dtype) + idx_dtype = get_index_dtype([b.coords[0] for b in blocks[block_mask]], + maxval=max(shape)) + row = np.empty(nnz, dtype=idx_dtype) + col = np.empty(nnz, dtype=idx_dtype) + + nnz = 0 + ii, jj = np.nonzero(block_mask) + for i, j in zip(ii, jj): + B = blocks[i, j] + idx = slice(nnz, nnz + B.nnz) + data[idx] = B.data + np.add(B.row, row_offsets[i], out=row[idx], dtype=idx_dtype) + np.add(B.col, col_offsets[j], out=col[idx], dtype=idx_dtype) + nnz += B.nnz + + if return_spmatrix: + return coo_matrix((data, (row, col)), shape=shape).asformat(format) + return coo_array((data, (row, col)), shape=shape).asformat(format) + + +def block_diag(mats, format=None, dtype=None): + """ + Build a block diagonal sparse matrix or array from provided matrices. + + Parameters + ---------- + mats : sequence of matrices or arrays + Input matrices or arrays. + format : str, optional + The sparse format of the result (e.g., "csr"). If not given, the result + is returned in "coo" format. + dtype : dtype specifier, optional + The data-type of the output. If not given, the dtype is + determined from that of `blocks`. + + Returns + ------- + res : sparse matrix or array + If at least one input is a sparse array, the output is a sparse array. + Otherwise the output is a sparse matrix. + + Notes + ----- + + .. versionadded:: 0.11.0 + + See Also + -------- + block_array + diags_array + + Examples + -------- + >>> from scipy.sparse import coo_array, block_diag + >>> A = coo_array([[1, 2], [3, 4]]) + >>> B = coo_array([[5], [6]]) + >>> C = coo_array([[7]]) + >>> block_diag((A, B, C)).toarray() + array([[1, 2, 0, 0], + [3, 4, 0, 0], + [0, 0, 5, 0], + [0, 0, 6, 0], + [0, 0, 0, 7]]) + + """ + if any(isinstance(a, sparray) for a in mats): + container = coo_array + else: + container = coo_matrix + + row = [] + col = [] + data = [] + idx_arrays = [] # track idx_dtype of incoming sparse arrays + r_idx = 0 + c_idx = 0 + for a in mats: + if isinstance(a, (list | numbers.Number)): + a = coo_array(np.atleast_2d(a)) + if issparse(a): + a = a.tocoo() + if not idx_arrays and a.coords[0].dtype == np.int64: + idx_arrays.append(a.coords[0]) + nrows, ncols = a._shape_as_2d + row.append(a.row + r_idx) + col.append(a.col + c_idx) + data.append(a.data) + else: + nrows, ncols = a.shape + a_row, a_col = np.divmod(np.arange(nrows*ncols), ncols) + row.append(a_row + r_idx) + col.append(a_col + c_idx) + data.append(a.ravel()) + r_idx += nrows + c_idx += ncols + idx_dtype = get_index_dtype(idx_arrays, maxval=max(r_idx, c_idx)) + row = np.concatenate(row, dtype=idx_dtype) + col = np.concatenate(col, dtype=idx_dtype) + data = np.concatenate(data) + new_shape = (r_idx, c_idx) + + return container((data, (row, col)), shape=new_shape, dtype=dtype).asformat(format) + + +@_transition_to_rng("random_state") +def random_array(shape, *, density=0.01, format='coo', dtype=None, + rng=None, data_sampler=None): + """Return a sparse array of uniformly random numbers in [0, 1) + + Returns a sparse array with the given shape and density + where values are generated uniformly randomly in the range [0, 1). + + Parameters + ---------- + shape : int or tuple of ints + shape of the array + density : real, optional (default: 0.01) + density of the generated matrix: density equal to one means a full + matrix, density of 0 means a matrix with no non-zero items. + format : str, optional (default: 'coo') + sparse matrix format. + dtype : dtype, optional (default: np.float64) + type of the returned matrix values. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + + This random state will be used for sampling `indices` (the sparsity + structure), and by default for the data values too (see `data_sampler`). + data_sampler : callable, optional (default depends on dtype) + Sampler of random data values with keyword arg `size`. + This function should take a single keyword argument `size` specifying + the length of its returned ndarray. It is used to generate the nonzero + values in the matrix after the locations of those values are chosen. + By default, uniform [0, 1) random values are used unless `dtype` is + an integer (default uniform integers from that dtype) or + complex (default uniform over the unit square in the complex plane). + For these, the `rng` is used e.g. ``rng.uniform(size=size)``. + + Returns + ------- + res : sparse array + + Examples + -------- + + Passing a ``np.random.Generator`` instance for better performance: + + >>> import numpy as np + >>> import scipy as sp + >>> rng = np.random.default_rng() + + Default sampling uniformly from [0, 1): + + >>> S = sp.sparse.random_array((3, 4), density=0.25, rng=rng) + + Providing a sampler for the values: + + >>> rvs = sp.stats.poisson(25, loc=10).rvs + >>> S = sp.sparse.random_array((3, 4), density=0.25, + ... rng=rng, data_sampler=rvs) + >>> S.toarray() + array([[ 36., 0., 33., 0.], # random + [ 0., 0., 0., 0.], + [ 0., 0., 36., 0.]]) + + Building a custom distribution. + This example builds a squared normal from np.random: + + >>> def np_normal_squared(size=None, rng=rng): + ... return rng.standard_normal(size) ** 2 + >>> S = sp.sparse.random_array((3, 4), density=0.25, rng=rng, + ... data_sampler=np_normal_squared) + + Or we can build it from sp.stats style rvs functions: + + >>> def sp_stats_normal_squared(size=None, rng=rng): + ... std_normal = sp.stats.distributions.norm_gen().rvs + ... return std_normal(size=size, random_state=rng) ** 2 + >>> S = sp.sparse.random_array((3, 4), density=0.25, rng=rng, + ... data_sampler=sp_stats_normal_squared) + + Or we can subclass sp.stats rv_continuous or rv_discrete: + + >>> class NormalSquared(sp.stats.rv_continuous): + ... def _rvs(self, size=None, random_state=rng): + ... return rng.standard_normal(size) ** 2 + >>> X = NormalSquared() + >>> Y = X().rvs + >>> S = sp.sparse.random_array((3, 4), density=0.25, + ... rng=rng, data_sampler=Y) + """ + data, ind = _random(shape, density, format, dtype, rng, data_sampler) + + # downcast, if safe, before calling coo_constructor + idx_dtype = get_index_dtype(maxval=max(shape)) + ind = tuple(np.asarray(co, dtype=idx_dtype) for co in ind) + return coo_array((data, ind), shape=shape).asformat(format) + + +def _random(shape, density=0.01, format=None, dtype=None, + rng=None, data_sampler=None): + if density < 0 or density > 1: + raise ValueError("density expected to be 0 <= density <= 1") + + tot_prod = math.prod(shape) # use `math` for when prod is >= 2**64 + + # Number of non zero values + size = int(round(density * tot_prod)) + + rng = check_random_state(rng) + + if data_sampler is None: + if np.issubdtype(dtype, np.integer): + def data_sampler(size): + return rng_integers(rng, + np.iinfo(dtype).min, + np.iinfo(dtype).max, + size, + dtype=dtype) + elif np.issubdtype(dtype, np.complexfloating): + def data_sampler(size): + return (rng.uniform(size=size) + + rng.uniform(size=size) * 1j) + else: + data_sampler = rng.uniform + + idx_dtype = get_index_dtype(maxval=max(shape)) + # rng.choice uses int64 if first arg is an int + if tot_prod <= np.iinfo(np.int64).max: + raveled_ind = rng.choice(tot_prod, size=size, replace=False) + ind = np.unravel_index(raveled_ind, shape=shape, order='F') + ind = tuple(np.asarray(co, idx_dtype) for co in ind) + else: + # for ravel indices bigger than dtype max, use sets to remove duplicates + ndim = len(shape) + seen = set() + while len(seen) < size: + dsize = size - len(seen) + seen.update(map(tuple, rng_integers(rng, shape, size=(dsize, ndim)))) + ind = tuple(np.array(list(seen), dtype=idx_dtype).T) + + # size kwarg allows eg data_sampler=partial(np.random.poisson, lam=5) + vals = data_sampler(size=size).astype(dtype, copy=False) + return vals, ind + + +@_transition_to_rng("random_state", position_num=5) +def random(m, n, density=0.01, format='coo', dtype=None, + rng=None, data_rvs=None): + """Generate a sparse matrix of the given shape and density with randomly + distributed values. + + .. warning:: + + This function returns a sparse matrix -- not a sparse array. + You are encouraged to use ``random_array`` to take advantage of the + sparse array functionality. + + Parameters + ---------- + m, n : int + shape of the matrix + density : real, optional + density of the generated matrix: density equal to one means a full + matrix, density of 0 means a matrix with no non-zero items. + format : str, optional + sparse matrix format. + dtype : dtype, optional + type of the returned matrix values. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + + This random state will be used for sampling the sparsity structure, but + not necessarily for sampling the values of the structurally nonzero + entries of the matrix. + data_rvs : callable, optional + Samples a requested number of random values. + This function should take a single argument specifying the length + of the ndarray that it will return. The structurally nonzero entries + of the sparse random matrix will be taken from the array sampled + by this function. By default, uniform [0, 1) random values will be + sampled using the same random state as is used for sampling + the sparsity structure. + + Returns + ------- + res : sparse matrix + + See Also + -------- + random_array : constructs sparse arrays instead of sparse matrices + + Examples + -------- + + Passing a ``np.random.Generator`` instance for better performance: + + >>> import scipy as sp + >>> import numpy as np + >>> rng = np.random.default_rng() + >>> S = sp.sparse.random(3, 4, density=0.25, rng=rng) + + Providing a sampler for the values: + + >>> rvs = sp.stats.poisson(25, loc=10).rvs + >>> S = sp.sparse.random(3, 4, density=0.25, rng=rng, data_rvs=rvs) + >>> S.toarray() + array([[ 36., 0., 33., 0.], # random + [ 0., 0., 0., 0.], + [ 0., 0., 36., 0.]]) + + Building a custom distribution. + This example builds a squared normal from np.random: + + >>> def np_normal_squared(size=None, rng=rng): + ... return rng.standard_normal(size) ** 2 + >>> S = sp.sparse.random(3, 4, density=0.25, rng=rng, + ... data_rvs=np_normal_squared) + + Or we can build it from sp.stats style rvs functions: + + >>> def sp_stats_normal_squared(size=None, rng=rng): + ... std_normal = sp.stats.distributions.norm_gen().rvs + ... return std_normal(size=size, random_state=rng) ** 2 + >>> S = sp.sparse.random(3, 4, density=0.25, rng=rng, + ... data_rvs=sp_stats_normal_squared) + + Or we can subclass sp.stats rv_continuous or rv_discrete: + + >>> class NormalSquared(sp.stats.rv_continuous): + ... def _rvs(self, size=None, random_state=rng): + ... return rng.standard_normal(size) ** 2 + >>> X = NormalSquared() + >>> Y = X() # get a frozen version of the distribution + >>> S = sp.sparse.random(3, 4, density=0.25, rng=rng, data_rvs=Y.rvs) + """ + if n is None: + n = m + m, n = int(m), int(n) + # make keyword syntax work for data_rvs e.g. data_rvs(size=7) + if data_rvs is not None: + def data_rvs_kw(size): + return data_rvs(size) + else: + data_rvs_kw = None + vals, ind = _random((m, n), density, format, dtype, rng, data_rvs_kw) + return coo_matrix((vals, ind), shape=(m, n)).asformat(format) + + +@_transition_to_rng("random_state", position_num=5) +def rand(m, n, density=0.01, format="coo", dtype=None, rng=None): + """Generate a sparse matrix of the given shape and density with uniformly + distributed values. + + .. warning:: + + This function returns a sparse matrix -- not a sparse array. + You are encouraged to use ``random_array`` to take advantage + of the sparse array functionality. + + Parameters + ---------- + m, n : int + shape of the matrix + density : real, optional + density of the generated matrix: density equal to one means a full + matrix, density of 0 means a matrix with no non-zero items. + format : str, optional + sparse matrix format. + dtype : dtype, optional + type of the returned matrix values. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + + Returns + ------- + res : sparse matrix + + Notes + ----- + Only float types are supported for now. + + See Also + -------- + random : Similar function allowing a custom random data sampler + random_array : Similar to random() but returns a sparse array + + Examples + -------- + >>> from scipy.sparse import rand + >>> matrix = rand(3, 4, density=0.25, format="csr", rng=42) + >>> matrix + + >>> matrix.toarray() + array([[0.05641158, 0. , 0. , 0.65088847], # random + [0. , 0. , 0. , 0.14286682], + [0. , 0. , 0. , 0. ]]) + + """ + return random(m, n, density, format, dtype, rng) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_coo.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_coo.py new file mode 100644 index 0000000000000000000000000000000000000000..3b1d577f90e5cf4d585aede3f30c6caf5b8ae059 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_coo.py @@ -0,0 +1,1647 @@ +""" A sparse matrix in COOrdinate or 'triplet' format""" + +__docformat__ = "restructuredtext en" + +__all__ = ['coo_array', 'coo_matrix', 'isspmatrix_coo'] + +import math +from warnings import warn + +import numpy as np + +from .._lib._util import copy_if_needed +from ._matrix import spmatrix +from ._sparsetools import (coo_tocsr, coo_todense, coo_todense_nd, + coo_matvec, coo_matvec_nd, coo_matmat_dense, + coo_matmat_dense_nd) +from ._base import issparse, SparseEfficiencyWarning, _spbase, sparray +from ._data import _data_matrix, _minmax_mixin +from ._sputils import (upcast_char, to_native, isshape, getdtype, + getdata, downcast_intp_index, get_index_dtype, + check_shape, check_reshape_kwargs, isscalarlike, isdense) + +import operator + + +class _coo_base(_data_matrix, _minmax_mixin): + _format = 'coo' + _allow_nd = range(1, 65) + + def __init__(self, arg1, shape=None, dtype=None, copy=False, *, maxprint=None): + _data_matrix.__init__(self, arg1, maxprint=maxprint) + if not copy: + copy = copy_if_needed + + if isinstance(arg1, tuple): + if isshape(arg1, allow_nd=self._allow_nd): + self._shape = check_shape(arg1, allow_nd=self._allow_nd) + idx_dtype = self._get_index_dtype(maxval=max(self._shape)) + data_dtype = getdtype(dtype, default=float) + self.coords = tuple(np.array([], dtype=idx_dtype) + for _ in range(len(self._shape))) + self.data = np.array([], dtype=data_dtype) + self.has_canonical_format = True + else: + try: + obj, coords = arg1 + except (TypeError, ValueError) as e: + raise TypeError('invalid input format') from e + + if shape is None: + if any(len(idx) == 0 for idx in coords): + raise ValueError('cannot infer dimensions from zero ' + 'sized index arrays') + shape = tuple(operator.index(np.max(idx)) + 1 + for idx in coords) + self._shape = check_shape(shape, allow_nd=self._allow_nd) + idx_dtype = self._get_index_dtype(coords, + maxval=max(self.shape), + check_contents=True) + self.coords = tuple(np.array(idx, copy=copy, dtype=idx_dtype) + for idx in coords) + self.data = getdata(obj, copy=copy, dtype=dtype) + self.has_canonical_format = False + else: + if issparse(arg1): + if arg1.format == self.format and copy: + self.coords = tuple(idx.copy() for idx in arg1.coords) + self.data = arg1.data.astype(getdtype(dtype, arg1)) # copy=True + self._shape = check_shape(arg1.shape, allow_nd=self._allow_nd) + self.has_canonical_format = arg1.has_canonical_format + else: + coo = arg1.tocoo() + self.coords = tuple(coo.coords) + self.data = coo.data.astype(getdtype(dtype, coo), copy=False) + self._shape = check_shape(coo.shape, allow_nd=self._allow_nd) + self.has_canonical_format = False + else: + # dense argument + M = np.asarray(arg1) + if not isinstance(self, sparray): + M = np.atleast_2d(M) + if M.ndim != 2: + raise TypeError(f'expected 2D array or matrix, not {M.ndim}D') + + self._shape = check_shape(M.shape, allow_nd=self._allow_nd) + if shape is not None: + if check_shape(shape, allow_nd=self._allow_nd) != self._shape: + message = f'inconsistent shapes: {shape} != {self._shape}' + raise ValueError(message) + + index_dtype = self._get_index_dtype(maxval=max(self._shape)) + coords = M.nonzero() + self.coords = tuple(idx.astype(index_dtype, copy=False) + for idx in coords) + self.data = getdata(M[coords], copy=copy, dtype=dtype) + self.has_canonical_format = True + + if len(self._shape) > 2: + self.coords = tuple(idx.astype(np.int64, copy=False) for idx in self.coords) + + self._check() + + @property + def row(self): + if self.ndim > 1: + return self.coords[-2] + result = np.zeros_like(self.col) + result.setflags(write=False) + return result + + + @row.setter + def row(self, new_row): + if self.ndim < 2: + raise ValueError('cannot set row attribute of a 1-dimensional sparse array') + new_row = np.asarray(new_row, dtype=self.coords[-2].dtype) + self.coords = self.coords[:-2] + (new_row,) + self.coords[-1:] + + @property + def col(self): + return self.coords[-1] + + @col.setter + def col(self, new_col): + new_col = np.asarray(new_col, dtype=self.coords[-1].dtype) + self.coords = self.coords[:-1] + (new_col,) + + def reshape(self, *args, **kwargs): + shape = check_shape(args, self.shape, allow_nd=self._allow_nd) + order, copy = check_reshape_kwargs(kwargs) + + # Return early if reshape is not required + if shape == self.shape: + if copy: + return self.copy() + else: + return self + + # When reducing the number of dimensions, we need to be careful about + # index overflow. This is why we can't simply call + # `np.ravel_multi_index()` followed by `np.unravel_index()` here. + flat_coords = _ravel_coords(self.coords, self.shape, order=order) + if len(shape) == 2: + if order == 'C': + new_coords = divmod(flat_coords, shape[1]) + else: + new_coords = divmod(flat_coords, shape[0])[::-1] + else: + new_coords = np.unravel_index(flat_coords, shape, order=order) + + idx_dtype = self._get_index_dtype(self.coords, maxval=max(shape)) + new_coords = tuple(np.asarray(co, dtype=idx_dtype) for co in new_coords) + + # Handle copy here rather than passing on to the constructor so that no + # copy will be made of `new_coords` regardless. + if copy: + new_data = self.data.copy() + else: + new_data = self.data + + return self.__class__((new_data, new_coords), shape=shape, copy=False) + + reshape.__doc__ = _spbase.reshape.__doc__ + + def _getnnz(self, axis=None): + if axis is None or (axis == 0 and self.ndim == 1): + nnz = len(self.data) + if any(len(idx) != nnz for idx in self.coords): + raise ValueError('all index and data arrays must have the ' + 'same length') + + if self.data.ndim != 1 or any(idx.ndim != 1 for idx in self.coords): + raise ValueError('coordinates and data arrays must be 1-D') + + return int(nnz) + + if axis < 0: + axis += self.ndim + if axis >= self.ndim: + raise ValueError('axis out of bounds') + + return np.bincount(downcast_intp_index(self.coords[1 - axis]), + minlength=self.shape[1 - axis]) + + _getnnz.__doc__ = _spbase._getnnz.__doc__ + + def count_nonzero(self, axis=None): + self.sum_duplicates() + if axis is None: + return np.count_nonzero(self.data) + + if axis < 0: + axis += self.ndim + if axis < 0 or axis >= self.ndim: + raise ValueError('axis out of bounds') + mask = self.data != 0 + coord = self.coords[1 - axis][mask] + return np.bincount(downcast_intp_index(coord), minlength=self.shape[1 - axis]) + + count_nonzero.__doc__ = _spbase.count_nonzero.__doc__ + + def _check(self): + """ Checks data structure for consistency """ + if self.ndim != len(self.coords): + raise ValueError('mismatching number of index arrays for shape; ' + f'got {len(self.coords)}, expected {self.ndim}') + + # index arrays should have integer data types + for i, idx in enumerate(self.coords): + if idx.dtype.kind != 'i': + warn(f'index array {i} has non-integer dtype ({idx.dtype.name})', + stacklevel=3) + + idx_dtype = self._get_index_dtype(self.coords, maxval=max(self.shape)) + self.coords = tuple(np.asarray(idx, dtype=idx_dtype) + for idx in self.coords) + self.data = to_native(self.data) + + if self.nnz > 0: + for i, idx in enumerate(self.coords): + if idx.max() >= self.shape[i]: + raise ValueError(f'axis {i} index {idx.max()} exceeds ' + f'matrix dimension {self.shape[i]}') + if idx.min() < 0: + raise ValueError(f'negative axis {i} index: {idx.min()}') + + def transpose(self, axes=None, copy=False): + if axes is None: + axes = range(self.ndim)[::-1] + elif isinstance(self, sparray): + if not hasattr(axes, "__len__") or len(axes) != self.ndim: + raise ValueError("axes don't match matrix dimensions") + if len(set(axes)) != self.ndim: + raise ValueError("repeated axis in transpose") + elif axes != (1, 0): + raise ValueError("Sparse matrices do not support an 'axes' " + "parameter because swapping dimensions is the " + "only logical permutation.") + + permuted_shape = tuple(self._shape[i] for i in axes) + permuted_coords = tuple(self.coords[i] for i in axes) + return self.__class__((self.data, permuted_coords), + shape=permuted_shape, copy=copy) + + transpose.__doc__ = _spbase.transpose.__doc__ + + def resize(self, *shape) -> None: + shape = check_shape(shape, allow_nd=self._allow_nd) + if self.ndim > 2: + raise ValueError("only 1-D or 2-D input accepted") + if len(shape) > 2: + raise ValueError("shape argument must be 1-D or 2-D") + # Check for added dimensions. + if len(shape) > self.ndim: + flat_coords = _ravel_coords(self.coords, self.shape) + max_size = math.prod(shape) + self.coords = np.unravel_index(flat_coords[:max_size], shape) + self.data = self.data[:max_size] + self._shape = shape + return + + # Check for removed dimensions. + if len(shape) < self.ndim: + tmp_shape = ( + self._shape[:len(shape) - 1] # Original shape without last axis + + (-1,) # Last axis is used to flatten the array + + (1,) * (self.ndim - len(shape)) # Pad with ones + ) + tmp = self.reshape(tmp_shape) + self.coords = tmp.coords[:len(shape)] + self._shape = tmp.shape[:len(shape)] + + # Handle truncation of existing dimensions. + is_truncating = any(old > new for old, new in zip(self.shape, shape)) + if is_truncating: + mask = np.logical_and.reduce([ + idx < size for idx, size in zip(self.coords, shape) + ]) + if not mask.all(): + self.coords = tuple(idx[mask] for idx in self.coords) + self.data = self.data[mask] + + self._shape = shape + + resize.__doc__ = _spbase.resize.__doc__ + + def toarray(self, order=None, out=None): + B = self._process_toarray_args(order, out) + fortran = int(B.flags.f_contiguous) + if not fortran and not B.flags.c_contiguous: + raise ValueError("Output array must be C or F contiguous") + # This handles both 0D and 1D cases correctly regardless of the + # original shape. + if self.ndim == 1: + coo_todense_nd(np.array([1]), self.nnz, self.ndim, + self.coords[0], self.data, B.ravel('A'), fortran) + elif self.ndim == 2: + M, N = self.shape + coo_todense(M, N, self.nnz, self.row, self.col, self.data, + B.ravel('A'), fortran) + else: + if fortran: + strides = np.append(1, np.cumprod(self.shape[:-1])) + else: + strides = np.append(np.cumprod(self.shape[1:][::-1])[::-1], 1) + coords = np.concatenate(self.coords) + coo_todense_nd(strides, self.nnz, self.ndim, + coords, self.data, B.ravel('A'), fortran) + # Note: reshape() doesn't copy here, but does return a new array (view). + return B.reshape(self.shape) + + toarray.__doc__ = _spbase.toarray.__doc__ + + def tocsc(self, copy=False): + """Convert this array/matrix to Compressed Sparse Column format + + Duplicate entries will be summed together. + + Examples + -------- + >>> from numpy import array + >>> from scipy.sparse import coo_array + >>> row = array([0, 0, 1, 3, 1, 0, 0]) + >>> col = array([0, 2, 1, 3, 1, 0, 0]) + >>> data = array([1, 1, 1, 1, 1, 1, 1]) + >>> A = coo_array((data, (row, col)), shape=(4, 4)).tocsc() + >>> A.toarray() + array([[3, 0, 1, 0], + [0, 2, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 1]]) + + """ + if self.ndim != 2: + raise ValueError(f'Cannot convert. CSC format must be 2D. Got {self.ndim}D') + if self.nnz == 0: + return self._csc_container(self.shape, dtype=self.dtype) + else: + from ._csc import csc_array + indptr, indices, data, shape = self._coo_to_compressed(csc_array._swap) + + x = self._csc_container((data, indices, indptr), shape=shape) + if not self.has_canonical_format: + x.sum_duplicates() + return x + + def tocsr(self, copy=False): + """Convert this array/matrix to Compressed Sparse Row format + + Duplicate entries will be summed together. + + Examples + -------- + >>> from numpy import array + >>> from scipy.sparse import coo_array + >>> row = array([0, 0, 1, 3, 1, 0, 0]) + >>> col = array([0, 2, 1, 3, 1, 0, 0]) + >>> data = array([1, 1, 1, 1, 1, 1, 1]) + >>> A = coo_array((data, (row, col)), shape=(4, 4)).tocsr() + >>> A.toarray() + array([[3, 0, 1, 0], + [0, 2, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 1]]) + + """ + if self.ndim > 2: + raise ValueError(f'Cannot convert. CSR must be 1D or 2D. Got {self.ndim}D') + if self.nnz == 0: + return self._csr_container(self.shape, dtype=self.dtype) + else: + from ._csr import csr_array + arrays = self._coo_to_compressed(csr_array._swap, copy=copy) + indptr, indices, data, shape = arrays + + x = self._csr_container((data, indices, indptr), shape=self.shape) + if not self.has_canonical_format: + x.sum_duplicates() + return x + + def _coo_to_compressed(self, swap, copy=False): + """convert (shape, coords, data) to (indptr, indices, data, shape)""" + M, N = swap(self._shape_as_2d) + # convert idx_dtype intc to int32 for pythran. + # tested in scipy/optimize/tests/test__numdiff.py::test_group_columns + idx_dtype = self._get_index_dtype(self.coords, maxval=max(self.nnz, N)) + + if self.ndim == 1: + indices = self.coords[0].copy() if copy else self.coords[0] + nnz = len(indices) + indptr = np.array([0, nnz], dtype=idx_dtype) + data = self.data.copy() if copy else self.data + return indptr, indices, data, self.shape + + # ndim == 2 + major, minor = swap(self.coords) + nnz = len(major) + major = major.astype(idx_dtype, copy=False) + minor = minor.astype(idx_dtype, copy=False) + + indptr = np.empty(M + 1, dtype=idx_dtype) + indices = np.empty_like(minor, dtype=idx_dtype) + data = np.empty_like(self.data, dtype=self.dtype) + + coo_tocsr(M, N, nnz, major, minor, self.data, indptr, indices, data) + return indptr, indices, data, self.shape + + def tocoo(self, copy=False): + if copy: + return self.copy() + else: + return self + + tocoo.__doc__ = _spbase.tocoo.__doc__ + + def todia(self, copy=False): + if self.ndim != 2: + raise ValueError(f'Cannot convert. DIA format must be 2D. Got {self.ndim}D') + self.sum_duplicates() + ks = self.col - self.row # the diagonal for each nonzero + diags, diag_idx = np.unique(ks, return_inverse=True) + + if len(diags) > 100: + # probably undesired, should todia() have a maxdiags parameter? + warn(f"Constructing a DIA matrix with {len(diags)} diagonals " + "is inefficient", + SparseEfficiencyWarning, stacklevel=2) + + #initialize and fill in data array + if self.data.size == 0: + data = np.zeros((0, 0), dtype=self.dtype) + else: + data = np.zeros((len(diags), self.col.max()+1), dtype=self.dtype) + data[diag_idx, self.col] = self.data + + return self._dia_container((data, diags), shape=self.shape) + + todia.__doc__ = _spbase.todia.__doc__ + + def todok(self, copy=False): + if self.ndim > 2: + raise ValueError(f'Cannot convert. DOK must be 1D or 2D. Got {self.ndim}D') + self.sum_duplicates() + dok = self._dok_container(self.shape, dtype=self.dtype) + # ensure that 1d coordinates are not tuples + if self.ndim == 1: + coords = self.coords[0] + else: + coords = zip(*self.coords) + + dok._dict = dict(zip(coords, self.data)) + return dok + + todok.__doc__ = _spbase.todok.__doc__ + + def diagonal(self, k=0): + if self.ndim != 2: + raise ValueError("diagonal requires two dimensions") + rows, cols = self.shape + if k <= -rows or k >= cols: + return np.empty(0, dtype=self.data.dtype) + diag = np.zeros(min(rows + min(k, 0), cols - max(k, 0)), + dtype=self.dtype) + diag_mask = (self.row + k) == self.col + + if self.has_canonical_format: + row = self.row[diag_mask] + data = self.data[diag_mask] + else: + inds = tuple(idx[diag_mask] for idx in self.coords) + (row, _), data = self._sum_duplicates(inds, self.data[diag_mask]) + diag[row + min(k, 0)] = data + + return diag + + diagonal.__doc__ = _data_matrix.diagonal.__doc__ + + def _setdiag(self, values, k): + if self.ndim != 2: + raise ValueError("setting a diagonal requires two dimensions") + M, N = self.shape + if values.ndim and not len(values): + return + idx_dtype = self.row.dtype + + # Determine which triples to keep and where to put the new ones. + full_keep = self.col - self.row != k + if k < 0: + max_index = min(M+k, N) + if values.ndim: + max_index = min(max_index, len(values)) + keep = np.logical_or(full_keep, self.col >= max_index) + new_row = np.arange(-k, -k + max_index, dtype=idx_dtype) + new_col = np.arange(max_index, dtype=idx_dtype) + else: + max_index = min(M, N-k) + if values.ndim: + max_index = min(max_index, len(values)) + keep = np.logical_or(full_keep, self.row >= max_index) + new_row = np.arange(max_index, dtype=idx_dtype) + new_col = np.arange(k, k + max_index, dtype=idx_dtype) + + # Define the array of data consisting of the entries to be added. + if values.ndim: + new_data = values[:max_index] + else: + new_data = np.empty(max_index, dtype=self.dtype) + new_data[:] = values + + # Update the internal structure. + self.coords = (np.concatenate((self.row[keep], new_row)), + np.concatenate((self.col[keep], new_col))) + self.data = np.concatenate((self.data[keep], new_data)) + self.has_canonical_format = False + + # needed by _data_matrix + def _with_data(self, data, copy=True): + """Returns a matrix with the same sparsity structure as self, + but with different data. By default the index arrays are copied. + """ + if copy: + coords = tuple(idx.copy() for idx in self.coords) + else: + coords = self.coords + return self.__class__((data, coords), shape=self.shape, dtype=data.dtype) + + def sum_duplicates(self) -> None: + """Eliminate duplicate entries by adding them together + + This is an *in place* operation + """ + if self.has_canonical_format: + return + summed = self._sum_duplicates(self.coords, self.data) + self.coords, self.data = summed + self.has_canonical_format = True + + def _sum_duplicates(self, coords, data): + # Assumes coords not in canonical format. + if len(data) == 0: + return coords, data + # Sort coords w.r.t. rows, then cols. This corresponds to C-order, + # which we rely on for argmin/argmax to return the first index in the + # same way that numpy does (in the case of ties). + order = np.lexsort(coords[::-1]) + coords = tuple(idx[order] for idx in coords) + data = data[order] + unique_mask = np.logical_or.reduce([ + idx[1:] != idx[:-1] for idx in coords + ]) + unique_mask = np.append(True, unique_mask) + coords = tuple(idx[unique_mask] for idx in coords) + unique_inds, = np.nonzero(unique_mask) + data = np.add.reduceat(data, downcast_intp_index(unique_inds), dtype=self.dtype) + return coords, data + + def eliminate_zeros(self): + """Remove zero entries from the array/matrix + + This is an *in place* operation + """ + mask = self.data != 0 + self.data = self.data[mask] + self.coords = tuple(idx[mask] for idx in self.coords) + + ####################### + # Arithmetic handlers # + ####################### + + def _add_dense(self, other): + if other.shape != self.shape: + raise ValueError(f'Incompatible shapes ({self.shape} and {other.shape})') + dtype = upcast_char(self.dtype.char, other.dtype.char) + result = np.array(other, dtype=dtype, copy=True) + fortran = int(result.flags.f_contiguous) + if self.ndim == 1: + coo_todense_nd(np.array([1]), self.nnz, self.ndim, + self.coords[0], self.data, result.ravel('A'), fortran) + elif self.ndim == 2: + M, N = self._shape_as_2d + coo_todense(M, N, self.nnz, self.row, self.col, self.data, + result.ravel('A'), fortran) + else: + if fortran: + strides = np.append(1, np.cumprod(self.shape[:-1])) + else: + strides = np.append(np.cumprod(self.shape[1:][::-1])[::-1], 1) + coords = np.concatenate(self.coords) + coo_todense_nd(strides, self.nnz, self.ndim, + coords, self.data, result.ravel('A'), fortran) + return self._container(result, copy=False) + + + def _add_sparse(self, other): + if self.ndim < 3: + return self.tocsr()._add_sparse(other) + + if other.shape != self.shape: + raise ValueError(f'Incompatible shapes ({self.shape} and {other.shape})') + other = self.__class__(other) + new_data = np.concatenate((self.data, other.data)) + new_coords = tuple(np.concatenate((self.coords, other.coords), axis=1)) + A = self.__class__((new_data, new_coords), shape=self.shape) + return A + + + def _sub_sparse(self, other): + if self.ndim < 3: + return self.tocsr()._sub_sparse(other) + + if other.shape != self.shape: + raise ValueError(f'Incompatible shapes ({self.shape} and {other.shape})') + other = self.__class__(other) + new_data = np.concatenate((self.data, -other.data)) + new_coords = tuple(np.concatenate((self.coords, other.coords), axis=1)) + A = coo_array((new_data, new_coords), shape=self.shape) + return A + + + def _matmul_vector(self, other): + if self.ndim > 2: + result = np.zeros(math.prod(self.shape[:-1]), + dtype=upcast_char(self.dtype.char, other.dtype.char)) + shape = np.array(self.shape) + strides = np.append(np.cumprod(shape[:-1][::-1])[::-1][1:], 1) + coords = np.concatenate(self.coords) + coo_matvec_nd(self.nnz, len(self.shape), strides, coords, self.data, + other, result) + + result = result.reshape(self.shape[:-1]) + return result + + # self.ndim <= 2 + result_shape = self.shape[0] if self.ndim > 1 else 1 + result = np.zeros(result_shape, + dtype=upcast_char(self.dtype.char, other.dtype.char)) + if self.ndim == 2: + col = self.col + row = self.row + elif self.ndim == 1: + col = self.coords[0] + row = np.zeros_like(col) + else: + raise NotImplementedError( + f"coo_matvec not implemented for ndim={self.ndim}") + + coo_matvec(self.nnz, row, col, self.data, other, result) + # Array semantics return a scalar here, not a single-element array. + if isinstance(self, sparray) and result_shape == 1: + return result[0] + return result + + + def _rmatmul_dispatch(self, other): + if isscalarlike(other): + return self._mul_scalar(other) + else: + # Don't use asarray unless we have to + try: + o_ndim = other.ndim + except AttributeError: + other = np.asarray(other) + o_ndim = other.ndim + perm = tuple(range(o_ndim)[:-2]) + tuple(range(o_ndim)[-2:][::-1]) + tr = other.transpose(perm) + + s_ndim = self.ndim + perm = tuple(range(s_ndim)[:-2]) + tuple(range(s_ndim)[-2:][::-1]) + ret = self.transpose(perm)._matmul_dispatch(tr) + if ret is NotImplemented: + return NotImplemented + + if s_ndim == 1 or o_ndim == 1: + perm = range(ret.ndim) + else: + perm = tuple(range(ret.ndim)[:-2]) + tuple(range(ret.ndim)[-2:][::-1]) + return ret.transpose(perm) + + + def _matmul_dispatch(self, other): + if isscalarlike(other): + return self.multiply(other) + + if not (issparse(other) or isdense(other)): + # If it's a list or whatever, treat it like an array + other_a = np.asanyarray(other) + + if other_a.ndim == 0 and other_a.dtype == np.object_: + # Not interpretable as an array; return NotImplemented so that + # other's __rmatmul__ can kick in if that's implemented. + return NotImplemented + + try: + other.shape + except AttributeError: + other = other_a + + if self.ndim < 3 and other.ndim < 3: + return _spbase._matmul_dispatch(self, other) + + N = self.shape[-1] + err_prefix = "matmul: dimension mismatch with signature" + if other.__class__ is np.ndarray: + if other.shape == (N,): + return self._matmul_vector(other) + if other.shape == (N, 1): + result = self._matmul_vector(other.ravel()) + return result.reshape(*self.shape[:-1], 1) + if other.ndim == 1: + msg = f"{err_prefix} (n,k={N}),(k={other.shape[0]},)->(n,)" + raise ValueError(msg) + if other.shape[-2] == N: + # check for batch dimensions compatibility + batch_shape_A = self.shape[:-2] + batch_shape_B = other.shape[:-2] + if batch_shape_A != batch_shape_B: + try: + # This will raise an error if the shapes are not broadcastable + np.broadcast_shapes(batch_shape_A, batch_shape_B) + except ValueError: + raise ValueError("Batch dimensions are not broadcastable") + + return self._matmul_multivector(other) + else: + raise ValueError( + f"{err_prefix} (n,..,k={N}),(k={other.shape[-2]},..,m)->(n,..,m)" + ) + + + if isscalarlike(other): + # scalar value + return self._mul_scalar(other) + + if issparse(other): + self_is_1d = self.ndim == 1 + other_is_1d = other.ndim == 1 + + # reshape to 2-D if self or other is 1-D + if self_is_1d: + self = self.reshape(self._shape_as_2d) # prepend 1 to shape + + if other_is_1d: + other = other.reshape((other.shape[0], 1)) # append 1 to shape + + # Check if the inner dimensions match for matrix multiplication + if N != other.shape[-2]: + raise ValueError( + f"{err_prefix} (n,..,k={N}),(k={other.shape[-2]},..,m)->(n,..,m)" + ) + + # If A or B has more than 2 dimensions, check for + # batch dimensions compatibility + if self.ndim > 2 or other.ndim > 2: + batch_shape_A = self.shape[:-2] + batch_shape_B = other.shape[:-2] + if batch_shape_A != batch_shape_B: + try: + # This will raise an error if the shapes are not broadcastable + np.broadcast_shapes(batch_shape_A, batch_shape_B) + except ValueError: + raise ValueError("Batch dimensions are not broadcastable") + + result = self._matmul_sparse(other) + + # reshape back if a or b were originally 1-D + if self_is_1d: + # if self was originally 1-D, reshape result accordingly + result = result.reshape(tuple(result.shape[:-2]) + + tuple(result.shape[-1:])) + if other_is_1d: + result = result.reshape(result.shape[:-1]) + return result + + + def _matmul_multivector(self, other): + result_dtype = upcast_char(self.dtype.char, other.dtype.char) + if self.ndim >= 3 or other.ndim >= 3: + # if self has shape (N,), reshape to (1,N) + if self.ndim == 1: + result = self.reshape(1, self.shape[0])._matmul_multivector(other) + return result.reshape(tuple(other.shape[:-2]) + tuple(other.shape[-1:])) + + broadcast_shape = np.broadcast_shapes(self.shape[:-2], other.shape[:-2]) + self_shape = broadcast_shape + self.shape[-2:] + other_shape = broadcast_shape + other.shape[-2:] + + self = self._broadcast_to(self_shape) + other = np.broadcast_to(other, other_shape) + result_shape = broadcast_shape + self.shape[-2:-1] + other.shape[-1:] + result = np.zeros(result_shape, dtype=result_dtype) + coo_matmat_dense_nd(self.nnz, len(self.shape), other.shape[-1], + np.array(other_shape), np.array(result_shape), + np.concatenate(self.coords), + self.data, other.ravel('C'), result) + return result + + if self.ndim == 2: + result_shape = (self.shape[0], other.shape[1]) + col = self.col + row = self.row + elif self.ndim == 1: + result_shape = (other.shape[1],) + col = self.coords[0] + row = np.zeros_like(col) + result = np.zeros(result_shape, dtype=result_dtype) + coo_matmat_dense(self.nnz, other.shape[-1], row, col, + self.data, other.ravel('C'), result) + return result.view(type=type(other)) + + + def dot(self, other): + """Return the dot product of two arrays. + + Strictly speaking a dot product involves two vectors. + But in the sense that an array with ndim >= 1 is a collection + of vectors, the function computes the collection of dot products + between each vector in the first array with each vector in the + second array. The axis upon which the sum of products is performed + is the last axis of the first array and the second to last axis of + the second array. If the second array is 1-D, the last axis is used. + + Thus, if both arrays are 1-D, the inner product is returned. + If both are 2-D, we have matrix multiplication. If `other` is 1-D, + the sum product is taken along the last axis of each array. If + `other` is N-D for N>=2, the sum product is over the last axis of + the first array and the second-to-last axis of the second array. + + Parameters + ---------- + other : array_like (dense or sparse) + Second array + + Returns + ------- + output : array (sparse or dense) + The dot product of this array with `other`. + It will be dense/sparse if `other` is dense/sparse. + + Examples + -------- + + >>> import numpy as np + >>> from scipy.sparse import coo_array + >>> A = coo_array([[1, 2, 0], [0, 0, 3], [4, 0, 5]]) + >>> v = np.array([1, 0, -1]) + >>> A.dot(v) + array([ 1, -3, -1], dtype=int64) + + For 2-D arrays it is the matrix product: + + >>> A = coo_array([[1, 0], [0, 1]]) + >>> B = coo_array([[4, 1], [2, 2]]) + >>> A.dot(B).toarray() + array([[4, 1], + [2, 2]]) + + For 3-D arrays the shape extends unused axes by other unused axes. + + >>> A = coo_array(np.arange(3*4*5*6)).reshape((3,4,5,6)) + >>> B = coo_array(np.arange(3*4*5*6)).reshape((5,4,6,3)) + >>> A.dot(B).shape + (3, 4, 5, 5, 4, 3) + """ + if not (issparse(other) or isdense(other) or isscalarlike(other)): + # If it's a list or whatever, treat it like an array + o_array = np.asanyarray(other) + + if o_array.ndim == 0 and o_array.dtype == np.object_: + raise TypeError(f"dot argument not supported type: '{type(other)}'") + try: + other.shape + except AttributeError: + other = o_array + + if self.ndim < 3 and (np.isscalar(other) or other.ndim<3): + return _spbase.dot(self, other) + # Handle scalar multiplication + if np.isscalar(other): + return self * other + if isdense(other): + return self._dense_dot(other) + elif other.format != "coo": + raise TypeError("input must be a COO matrix/array") + elif self.ndim == 1 and other.ndim == 1: + # Handle inner product of vectors (1-D arrays) + if self.shape[0] != other.shape[0]: + raise ValueError(f"shapes {self.shape} and {other.shape}" + " are not aligned for inner product") + return self @ other + elif self.ndim == 2 and other.ndim == 2: + # Handle matrix multiplication (2-D arrays) + if self.shape[1] != other.shape[0]: + raise ValueError(f"shapes {self.shape} and {other.shape}" + " are not aligned for matmul") + return self @ other + else: + return self._sparse_dot(other) + + + def _sparse_dot(self, other): + self_is_1d = self.ndim == 1 + other_is_1d = other.ndim == 1 + + # reshape to 2-D if self or other is 1-D + if self_is_1d: + self = self.reshape(self._shape_as_2d) # prepend 1 to shape + if other_is_1d: + other = other.reshape((other.shape[0], 1)) # append 1 to shape + + if self.shape[-1] != other.shape[-2]: + raise ValueError(f"shapes {self.shape} and {other.shape}" + " are not aligned for n-D dot") + + # Prepare the tensors for dot operation + # Ravel non-reduced axes coordinates + self_raveled_coords = _ravel_non_reduced_axes(self.coords, + self.shape, [self.ndim-1]) + other_raveled_coords = _ravel_non_reduced_axes(other.coords, + other.shape, [other.ndim-2]) + + # Get the shape of the non-reduced axes + self_nonreduced_shape = self.shape[:-1] + other_nonreduced_shape = other.shape[:-2] + other.shape[-1:] + + # Create 2D coords arrays + ravel_coords_shape_self = (math.prod(self_nonreduced_shape), self.shape[-1]) + ravel_coords_shape_other = (other.shape[-2], math.prod(other_nonreduced_shape)) + + self_2d_coords = (self_raveled_coords, self.coords[-1]) + other_2d_coords = (other.coords[-2], other_raveled_coords) + + self_2d = coo_array((self.data, self_2d_coords), ravel_coords_shape_self) + other_2d = coo_array((other.data, other_2d_coords), ravel_coords_shape_other) + + prod = (self_2d @ other_2d).tocoo() # routes via 2-D CSR + + # Combine the shapes of the non-reduced axes + combined_shape = self_nonreduced_shape + other_nonreduced_shape + + # Unravel the 2D coordinates to get multi-dimensional coordinates + shapes = (self_nonreduced_shape, other_nonreduced_shape) + prod_coords = [] + for c, s in zip(prod.coords, shapes): + prod_coords.extend(np.unravel_index(c, s)) + + prod_arr = coo_array((prod.data, prod_coords), combined_shape) + + # reshape back if a or b were originally 1-D + # TODO: Move this logic before computation of prod_coords for efficiency + if self_is_1d: + prod_arr = prod_arr.reshape(combined_shape[1:]) + if other_is_1d: + prod_arr = prod_arr.reshape(combined_shape[:-1]) + + return prod_arr + + def _dense_dot(self, other): + self_is_1d = self.ndim == 1 + other_is_1d = other.ndim == 1 + + # reshape to 2-D if self or other is 1-D + if self_is_1d: + self = self.reshape(self._shape_as_2d) # prepend 1 to shape + if other_is_1d: + other = other.reshape((other.shape[0], 1)) # append 1 to shape + + if self.shape[-1] != other.shape[-2]: + raise ValueError(f"shapes {self.shape} and {other.shape}" + " are not aligned for n-D dot") + + new_shape_self = ( + self.shape[:-1] + (1,) * (len(other.shape) - 1) + self.shape[-1:] + ) + new_shape_other = (1,) * (len(self.shape) - 1) + other.shape + + result_shape = self.shape[:-1] + other.shape[:-2] + other.shape[-1:] + result = self.reshape(new_shape_self) @ other.reshape(new_shape_other) + prod_arr = result.reshape(result_shape) + + # reshape back if a or b were originally 1-D + if self_is_1d: + prod_arr = prod_arr.reshape(result_shape[1:]) + if other_is_1d: + prod_arr = prod_arr.reshape(result_shape[:-1]) + + return prod_arr + + def tensordot(self, other, axes=2): + """Return the tensordot product with another array along the given axes. + + The tensordot differs from dot and matmul in that any axis can be + chosen for each of the first and second array and the sum of the + products is computed just like for matrix multiplication, only not + just for the rows of the first times the columns of the second. It + takes the dot product of the collection of vectors along the specified + axes. Here we can even take the sum of the products along two or even + more axes if desired. So, tensordot is a dot product computation + applied to arrays of any dimension >= 1. It is like matmul but over + arbitrary axes for each matrix. + + Given two tensors, `a` and `b`, and the desired axes specified as a + 2-tuple/list/array containing two sequences of axis numbers, + ``(a_axes, b_axes)``, sum the products of `a`'s and `b`'s elements + (components) over the axes specified by ``a_axes`` and ``b_axes``. + The `axes` input can be a single non-negative integer, ``N``; + if it is, then the last ``N`` dimensions of `a` and the first + ``N`` dimensions of `b` are summed over. + + Parameters + ---------- + a, b : array_like + Tensors to "dot". + + axes : int or (2,) array_like + * integer_like + If an int N, sum over the last N axes of `a` and the first N axes + of `b` in order. The sizes of the corresponding axes must match. + * (2,) array_like + A 2-tuple of sequences of axes to be summed over, the first applying + to `a`, the second to `b`. The sequences must be the same length. + The shape of the corresponding axes must match between `a` and `b`. + + Returns + ------- + output : coo_array + The tensor dot product of this array with `other`. + It will be dense/sparse if `other` is dense/sparse. + + See Also + -------- + dot + + Examples + -------- + >>> import numpy as np + >>> import scipy.sparse + >>> A = scipy.sparse.coo_array([[[2, 3], [0, 0]], [[0, 1], [0, 5]]]) + >>> A.shape + (2, 2, 2) + + Integer axes N are shorthand for (range(-N, 0), range(0, N)): + + >>> A.tensordot(A, axes=1).toarray() + array([[[[ 4, 9], + [ 0, 15]], + + [[ 0, 0], + [ 0, 0]]], + + + [[[ 0, 1], + [ 0, 5]], + + [[ 0, 5], + [ 0, 25]]]]) + >>> A.tensordot(A, axes=2).toarray() + array([[ 4, 6], + [ 0, 25]]) + >>> A.tensordot(A, axes=3) + array(39) + + Using tuple for axes: + + >>> a = scipy.sparse.coo_array(np.arange(60).reshape(3,4,5)) + >>> b = np.arange(24).reshape(4,3,2) + >>> c = a.tensordot(b, axes=([1,0],[0,1])) + >>> c.shape + (5, 2) + >>> c + array([[4400, 4730], + [4532, 4874], + [4664, 5018], + [4796, 5162], + [4928, 5306]]) + + """ + if not isdense(other) and not issparse(other): + # If it's a list or whatever, treat it like an array + other_array = np.asanyarray(other) + + if other_array.ndim == 0 and other_array.dtype == np.object_: + raise TypeError(f"tensordot arg not supported type: '{type(other)}'") + try: + other.shape + except AttributeError: + other = other_array + + axes_self, axes_other = _process_axes(self.ndim, other.ndim, axes) + + # Check for shape compatibility along specified axes + if any(self.shape[ax] != other.shape[bx] + for ax, bx in zip(axes_self, axes_other)): + raise ValueError("sizes of the corresponding axes must match") + + if isdense(other): + return self._dense_tensordot(other, axes_self, axes_other) + else: + return self._sparse_tensordot(other, axes_self, axes_other) + + + def _sparse_tensordot(self, other, axes_self, axes_other): + ndim_self = len(self.shape) + ndim_other = len(other.shape) + + # Prepare the tensors for tensordot operation + # Ravel non-reduced axes coordinates + self_non_red_coords = _ravel_non_reduced_axes(self.coords, self.shape, + axes_self) + self_reduced_coords = np.ravel_multi_index( + [self.coords[ax] for ax in axes_self], [self.shape[ax] for ax in axes_self]) + other_non_red_coords = _ravel_non_reduced_axes(other.coords, other.shape, + axes_other) + other_reduced_coords = np.ravel_multi_index( + [other.coords[a] for a in axes_other], [other.shape[a] for a in axes_other] + ) + # Get the shape of the non-reduced axes + self_nonreduced_shape = tuple(self.shape[ax] for ax in range(ndim_self) + if ax not in axes_self) + other_nonreduced_shape = tuple(other.shape[ax] for ax in range(ndim_other) + if ax not in axes_other) + + # Create 2D coords arrays + ravel_coords_shape_self = (math.prod(self_nonreduced_shape), + math.prod([self.shape[ax] for ax in axes_self])) + ravel_coords_shape_other = (math.prod([other.shape[ax] for ax in axes_other]), + math.prod(other_nonreduced_shape)) + + self_2d_coords = (self_non_red_coords, self_reduced_coords) + other_2d_coords = (other_reduced_coords, other_non_red_coords) + + self_2d = coo_array((self.data, self_2d_coords), ravel_coords_shape_self) + other_2d = coo_array((other.data, other_2d_coords), ravel_coords_shape_other) + + # Perform matrix multiplication (routed via 2-D CSR) + prod = (self_2d @ other_2d).tocoo() + + # Combine the shapes of the non-contracted axes + combined_shape = self_nonreduced_shape + other_nonreduced_shape + + # Unravel the 2D coordinates to get multi-dimensional coordinates + coords = [] + for c, s in zip(prod.coords, (self_nonreduced_shape, other_nonreduced_shape)): + if s: + coords.extend(np.unravel_index(c, s)) + + if coords == []: # if result is scalar + return sum(prod.data) + + # Construct the resulting COO array with combined coordinates and shape + return coo_array((prod.data, coords), shape=combined_shape) + + + def _dense_tensordot(self, other, axes_self, axes_other): + ndim_self = len(self.shape) + ndim_other = len(other.shape) + + non_reduced_axes_self = [ax for ax in range(ndim_self) if ax not in axes_self] + reduced_shape_self = [self.shape[s] for s in axes_self] + non_reduced_shape_self = [self.shape[s] for s in non_reduced_axes_self] + + non_reduced_axes_other = [ax for ax in range(ndim_other) + if ax not in axes_other] + reduced_shape_other = [other.shape[s] for s in axes_other] + non_reduced_shape_other = [other.shape[s] for s in non_reduced_axes_other] + + permute_self = non_reduced_axes_self + axes_self + permute_other = ( + non_reduced_axes_other[:-1] + axes_other + non_reduced_axes_other[-1:] + ) + self = self.transpose(permute_self) + other = np.transpose(other, permute_other) + + reshape_self = (*non_reduced_shape_self, math.prod(reduced_shape_self)) + reshape_other = (*non_reduced_shape_other[:-1], math.prod(reduced_shape_other), + *non_reduced_shape_other[-1:]) + + return self.reshape(reshape_self).dot(other.reshape(reshape_other)) + + + def _matmul_sparse(self, other): + """ + Perform sparse-sparse matrix multiplication for two n-D COO arrays. + The method converts input n-D arrays to 2-D block array format, + uses csr_matmat to multiply them, and then converts the + result back to n-D COO array. + + Parameters: + self (COO): The first n-D sparse array in COO format. + other (COO): The second n-D sparse array in COO format. + + Returns: + prod (COO): The resulting n-D sparse array after multiplication. + """ + if self.ndim < 3 and other.ndim < 3: + return _spbase._matmul_sparse(self, other) + + # Get the shapes of self and other + self_shape = self.shape + other_shape = other.shape + + # Determine the new shape to broadcast self and other + broadcast_shape = np.broadcast_shapes(self_shape[:-2], other_shape[:-2]) + self_new_shape = tuple(broadcast_shape) + self_shape[-2:] + other_new_shape = tuple(broadcast_shape) + other_shape[-2:] + + self_broadcasted = self._broadcast_to(self_new_shape) + other_broadcasted = other._broadcast_to(other_new_shape) + + # Convert n-D COO arrays to 2-D block diagonal arrays + self_block_diag = _block_diag(self_broadcasted) + other_block_diag = _block_diag(other_broadcasted) + + # Use csr_matmat to perform sparse matrix multiplication + prod_block_diag = (self_block_diag @ other_block_diag).tocoo() + + # Convert the 2-D block diagonal array back to n-D + return _extract_block_diag( + prod_block_diag, + shape=(*broadcast_shape, self.shape[-2], other.shape[-1]), + ) + + + def _broadcast_to(self, new_shape, copy=False): + if self.shape == new_shape: + return self.copy() if copy else self + + old_shape = self.shape + + # Check if the new shape is compatible for broadcasting + if len(new_shape) < len(old_shape): + raise ValueError("New shape must have at least as many dimensions" + " as the current shape") + + # Add leading ones to shape to ensure same length as `new_shape` + shape = (1,) * (len(new_shape) - len(old_shape)) + tuple(old_shape) + + # Ensure the old shape can be broadcast to the new shape + if any((o != 1 and o != n) for o, n in zip(shape, new_shape)): + raise ValueError(f"current shape {old_shape} cannot be " + "broadcast to new shape {new_shape}") + + # Reshape the COO array to match the new dimensions + self = self.reshape(shape) + + idx_dtype = get_index_dtype(self.coords, maxval=max(new_shape)) + coords = self.coords + new_data = self.data + new_coords = coords[-1:] # Copy last coordinate to start + cum_repeat = 1 # Cumulative repeat factor for broadcasting + + if shape[-1] != new_shape[-1]: # broadcasting the n-th (col) dimension + repeat_count = new_shape[-1] + cum_repeat *= repeat_count + new_data = np.tile(new_data, repeat_count) + new_dim = np.repeat(np.arange(0, repeat_count, dtype=idx_dtype), self.nnz) + new_coords = (new_dim,) + + for i in range(-2, -(len(shape)+1), -1): + if shape[i] != new_shape[i]: + repeat_count = new_shape[i] # number of times to repeat data, coords + cum_repeat *= repeat_count # update cumulative repeat factor + nnz = len(new_data) # Number of non-zero elements so far + + # Tile data and coordinates to match the new repeat count + new_data = np.tile(new_data, repeat_count) + new_coords = tuple(np.tile(new_coords[i+1:], repeat_count)) + + # Create new dimensions and stack them + new_dim = np.repeat(np.arange(0, repeat_count, dtype=idx_dtype), nnz) + new_coords = (new_dim,) + new_coords + else: + # If no broadcasting needed, tile the coordinates + new_dim = np.tile(coords[i], cum_repeat) + new_coords = (new_dim,) + new_coords + + return coo_array((new_data, new_coords), new_shape) + + +def _block_diag(self): + """ + Converts an N-D COO array into a 2-D COO array in block diagonal form. + + Parameters: + self (coo_array): An N-Dimensional COO sparse array. + + Returns: + coo_array: A 2-Dimensional COO sparse array in block diagonal form. + """ + if self.ndim<2: + raise ValueError("array must have atleast dim=2") + num_blocks = math.prod(self.shape[:-2]) + n_col = self.shape[-1] + n_row = self.shape[-2] + res_arr = self.reshape((num_blocks, n_row, n_col)) + new_coords = ( + res_arr.coords[1] + res_arr.coords[0] * res_arr.shape[1], + res_arr.coords[2] + res_arr.coords[0] * res_arr.shape[2], + ) + + new_shape = (num_blocks * n_row, num_blocks * n_col) + return coo_array((self.data, tuple(new_coords)), shape=new_shape) + + +def _extract_block_diag(self, shape): + n_row, n_col = shape[-2], shape[-1] + + # Extract data and coordinates from the block diagonal COO array + data = self.data + row, col = self.row, self.col + + # Initialize new coordinates array + new_coords = np.empty((len(shape), self.nnz), dtype=int) + + # Calculate within-block indices + new_coords[-2] = row % n_row + new_coords[-1] = col % n_col + + # Calculate coordinates for higher dimensions + temp_block_idx = row // n_row + for i in range(len(shape) - 3, -1, -1): + size = shape[i] + new_coords[i] = temp_block_idx % size + temp_block_idx = temp_block_idx // size + + # Create the new COO array with the original n-D shape + return coo_array((data, tuple(new_coords)), shape=shape) + + +def _process_axes(ndim_a, ndim_b, axes): + if isinstance(axes, int): + if axes < 1 or axes > min(ndim_a, ndim_b): + raise ValueError("axes integer is out of bounds for input arrays") + axes_a = list(range(ndim_a - axes, ndim_a)) + axes_b = list(range(axes)) + elif isinstance(axes, (tuple, list)): + if len(axes) != 2: + raise ValueError("axes must be a tuple/list of length 2") + axes_a, axes_b = axes + if len(axes_a) != len(axes_b): + raise ValueError("axes lists/tuples must be of the same length") + if any(ax >= ndim_a or ax < -ndim_a for ax in axes_a) or \ + any(bx >= ndim_b or bx < -ndim_b for bx in axes_b): + raise ValueError("axes indices are out of bounds for input arrays") + else: + raise TypeError("axes must be an integer or a tuple/list of integers") + + axes_a = [axis + ndim_a if axis < 0 else axis for axis in axes_a] + axes_b = [axis + ndim_b if axis < 0 else axis for axis in axes_b] + return axes_a, axes_b + + +def _ravel_non_reduced_axes(coords, shape, axes): + ndim = len(shape) + non_reduced_axes = [ax for ax in range(ndim) if ax not in axes] + + if not non_reduced_axes: + # Return an array with one row + return np.zeros_like(coords[0]) + + # Extract the shape of the non-reduced axes + non_reduced_shape = [shape[ax] for ax in non_reduced_axes] + + # Extract the coordinates of the non-reduced axes + non_reduced_coords = tuple(coords[idx] for idx in non_reduced_axes) + + # Ravel the coordinates into 1D + return np.ravel_multi_index(non_reduced_coords, non_reduced_shape) + + +def _ravel_coords(coords, shape, order='C'): + """Like np.ravel_multi_index, but avoids some overflow issues.""" + if len(coords) == 1: + return coords[0] + # Handle overflow as in https://github.com/scipy/scipy/pull/9132 + if len(coords) == 2: + nrows, ncols = shape + row, col = coords + if order == 'C': + maxval = (ncols * max(0, nrows - 1) + max(0, ncols - 1)) + idx_dtype = get_index_dtype(maxval=maxval) + return np.multiply(ncols, row, dtype=idx_dtype) + col + elif order == 'F': + maxval = (nrows * max(0, ncols - 1) + max(0, nrows - 1)) + idx_dtype = get_index_dtype(maxval=maxval) + return np.multiply(nrows, col, dtype=idx_dtype) + row + else: + raise ValueError("'order' must be 'C' or 'F'") + return np.ravel_multi_index(coords, shape, order=order) + + +def isspmatrix_coo(x): + """Is `x` of coo_matrix type? + + Parameters + ---------- + x + object to check for being a coo matrix + + Returns + ------- + bool + True if `x` is a coo matrix, False otherwise + + Examples + -------- + >>> from scipy.sparse import coo_array, coo_matrix, csr_matrix, isspmatrix_coo + >>> isspmatrix_coo(coo_matrix([[5]])) + True + >>> isspmatrix_coo(coo_array([[5]])) + False + >>> isspmatrix_coo(csr_matrix([[5]])) + False + """ + return isinstance(x, coo_matrix) + + +# This namespace class separates array from matrix with isinstance +class coo_array(_coo_base, sparray): + """ + A sparse array in COOrdinate format. + + Also known as the 'ijv' or 'triplet' format. + + This can be instantiated in several ways: + coo_array(D) + where D is an ndarray + + coo_array(S) + with another sparse array or matrix S (equivalent to S.tocoo()) + + coo_array(shape, [dtype]) + to construct an empty sparse array with shape `shape` + dtype is optional, defaulting to dtype='d'. + + coo_array((data, coords), [shape]) + to construct from existing data and index arrays: + 1. data[:] the entries of the sparse array, in any order + 2. coords[i][:] the axis-i coordinates of the data entries + + Where ``A[coords] = data``, and coords is a tuple of index arrays. + When shape is not specified, it is inferred from the index arrays. + + Attributes + ---------- + dtype : dtype + Data type of the sparse array + shape : tuple of integers + Shape of the sparse array + ndim : int + Number of dimensions of the sparse array + nnz + size + data + COO format data array of the sparse array + coords + COO format tuple of index arrays + has_canonical_format : bool + Whether the matrix has sorted coordinates and no duplicates + format + T + + Notes + ----- + + Sparse arrays can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + Advantages of the COO format + - facilitates fast conversion among sparse formats + - permits duplicate entries (see example) + - very fast conversion to and from CSR/CSC formats + + Disadvantages of the COO format + - does not directly support: + + arithmetic operations + + slicing + + Intended Usage + - COO is a fast format for constructing sparse arrays + - Once a COO array has been constructed, convert to CSR or + CSC format for fast arithmetic and matrix vector operations + - By default when converting to CSR or CSC format, duplicate (i,j) + entries will be summed together. This facilitates efficient + construction of finite element matrices and the like. (see example) + + Canonical format + - Entries and coordinates sorted by row, then column. + - There are no duplicate entries (i.e. duplicate (i,j) locations) + - Data arrays MAY have explicit zeros. + + Examples + -------- + + >>> # Constructing an empty sparse array + >>> import numpy as np + >>> from scipy.sparse import coo_array + >>> coo_array((3, 4), dtype=np.int8).toarray() + array([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=int8) + + >>> # Constructing a sparse array using ijv format + >>> row = np.array([0, 3, 1, 0]) + >>> col = np.array([0, 3, 1, 2]) + >>> data = np.array([4, 5, 7, 9]) + >>> coo_array((data, (row, col)), shape=(4, 4)).toarray() + array([[4, 0, 9, 0], + [0, 7, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 5]]) + + >>> # Constructing a sparse array with duplicate coordinates + >>> row = np.array([0, 0, 1, 3, 1, 0, 0]) + >>> col = np.array([0, 2, 1, 3, 1, 0, 0]) + >>> data = np.array([1, 1, 1, 1, 1, 1, 1]) + >>> coo = coo_array((data, (row, col)), shape=(4, 4)) + >>> # Duplicate coordinates are maintained until implicitly or explicitly summed + >>> np.max(coo.data) + 1 + >>> coo.toarray() + array([[3, 0, 1, 0], + [0, 2, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 1]]) + + """ + + +class coo_matrix(spmatrix, _coo_base): + """ + A sparse matrix in COOrdinate format. + + Also known as the 'ijv' or 'triplet' format. + + This can be instantiated in several ways: + coo_matrix(D) + where D is a 2-D ndarray + + coo_matrix(S) + with another sparse array or matrix S (equivalent to S.tocoo()) + + coo_matrix((M, N), [dtype]) + to construct an empty matrix with shape (M, N) + dtype is optional, defaulting to dtype='d'. + + coo_matrix((data, (i, j)), [shape=(M, N)]) + to construct from three arrays: + 1. data[:] the entries of the matrix, in any order + 2. i[:] the row indices of the matrix entries + 3. j[:] the column indices of the matrix entries + + Where ``A[i[k], j[k]] = data[k]``. When shape is not + specified, it is inferred from the index arrays + + Attributes + ---------- + dtype : dtype + Data type of the matrix + shape : 2-tuple + Shape of the matrix + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + COO format data array of the matrix + row + COO format row index array of the matrix + col + COO format column index array of the matrix + has_canonical_format : bool + Whether the matrix has sorted indices and no duplicates + format + T + + Notes + ----- + + Sparse matrices can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + Advantages of the COO format + - facilitates fast conversion among sparse formats + - permits duplicate entries (see example) + - very fast conversion to and from CSR/CSC formats + + Disadvantages of the COO format + - does not directly support: + + arithmetic operations + + slicing + + Intended Usage + - COO is a fast format for constructing sparse matrices + - Once a COO matrix has been constructed, convert to CSR or + CSC format for fast arithmetic and matrix vector operations + - By default when converting to CSR or CSC format, duplicate (i,j) + entries will be summed together. This facilitates efficient + construction of finite element matrices and the like. (see example) + + Canonical format + - Entries and coordinates sorted by row, then column. + - There are no duplicate entries (i.e. duplicate (i,j) locations) + - Data arrays MAY have explicit zeros. + + Examples + -------- + + >>> # Constructing an empty matrix + >>> import numpy as np + >>> from scipy.sparse import coo_matrix + >>> coo_matrix((3, 4), dtype=np.int8).toarray() + array([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=int8) + + >>> # Constructing a matrix using ijv format + >>> row = np.array([0, 3, 1, 0]) + >>> col = np.array([0, 3, 1, 2]) + >>> data = np.array([4, 5, 7, 9]) + >>> coo_matrix((data, (row, col)), shape=(4, 4)).toarray() + array([[4, 0, 9, 0], + [0, 7, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 5]]) + + >>> # Constructing a matrix with duplicate coordinates + >>> row = np.array([0, 0, 1, 3, 1, 0, 0]) + >>> col = np.array([0, 2, 1, 3, 1, 0, 0]) + >>> data = np.array([1, 1, 1, 1, 1, 1, 1]) + >>> coo = coo_matrix((data, (row, col)), shape=(4, 4)) + >>> # Duplicate coordinates are maintained until implicitly or explicitly summed + >>> np.max(coo.data) + 1 + >>> coo.toarray() + array([[3, 0, 1, 0], + [0, 2, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 1]]) + + """ + + def __setstate__(self, state): + if 'coords' not in state: + # For retro-compatibility with the previous attributes + # storing nnz coordinates for 2D COO matrix. + state['coords'] = (state.pop('row'), state.pop('col')) + self.__dict__.update(state) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_csc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_csc.py new file mode 100644 index 0000000000000000000000000000000000000000..9a03f50f6a29c71a838e6a9c93d61c20114bb432 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_csc.py @@ -0,0 +1,367 @@ +"""Compressed Sparse Column matrix format""" +__docformat__ = "restructuredtext en" + +__all__ = ['csc_array', 'csc_matrix', 'isspmatrix_csc'] + + +import numpy as np + +from ._matrix import spmatrix +from ._base import _spbase, sparray +from ._sparsetools import csc_tocsr, expandptr +from ._sputils import upcast + +from ._compressed import _cs_matrix + + +class _csc_base(_cs_matrix): + _format = 'csc' + + def transpose(self, axes=None, copy=False): + if axes is not None and axes != (1, 0): + raise ValueError("Sparse arrays/matrices do not support " + "an 'axes' parameter because swapping " + "dimensions is the only logical permutation.") + + M, N = self.shape + + return self._csr_container((self.data, self.indices, + self.indptr), (N, M), copy=copy) + + transpose.__doc__ = _spbase.transpose.__doc__ + + def __iter__(self): + yield from self.tocsr() + + def tocsc(self, copy=False): + if copy: + return self.copy() + else: + return self + + tocsc.__doc__ = _spbase.tocsc.__doc__ + + def tocsr(self, copy=False): + M,N = self.shape + idx_dtype = self._get_index_dtype((self.indptr, self.indices), + maxval=max(self.nnz, N)) + indptr = np.empty(M + 1, dtype=idx_dtype) + indices = np.empty(self.nnz, dtype=idx_dtype) + data = np.empty(self.nnz, dtype=upcast(self.dtype)) + + csc_tocsr(M, N, + self.indptr.astype(idx_dtype), + self.indices.astype(idx_dtype), + self.data, + indptr, + indices, + data) + + A = self._csr_container( + (data, indices, indptr), + shape=self.shape, copy=False + ) + A.has_sorted_indices = True + return A + + tocsr.__doc__ = _spbase.tocsr.__doc__ + + def nonzero(self): + # CSC can't use _cs_matrix's .nonzero method because it + # returns the indices sorted for self transposed. + + # Get row and col indices, from _cs_matrix.tocoo + major_dim, minor_dim = self._swap(self.shape) + minor_indices = self.indices + major_indices = np.empty(len(minor_indices), dtype=self.indices.dtype) + expandptr(major_dim, self.indptr, major_indices) + row, col = self._swap((major_indices, minor_indices)) + + # Remove explicit zeros + nz_mask = self.data != 0 + row = row[nz_mask] + col = col[nz_mask] + + # Sort them to be in C-style order + ind = np.argsort(row, kind='mergesort') + row = row[ind] + col = col[ind] + + return row, col + + nonzero.__doc__ = _cs_matrix.nonzero.__doc__ + + def _getrow(self, i): + """Returns a copy of row i of the matrix, as a (1 x n) + CSR matrix (row vector). + """ + M, N = self.shape + i = int(i) + if i < 0: + i += M + if i < 0 or i >= M: + raise IndexError('index (%d) out of range' % i) + return self._get_submatrix(minor=i).tocsr() + + def _getcol(self, i): + """Returns a copy of column i of the matrix, as a (m x 1) + CSC matrix (column vector). + """ + M, N = self.shape + i = int(i) + if i < 0: + i += N + if i < 0 or i >= N: + raise IndexError('index (%d) out of range' % i) + return self._get_submatrix(major=i, copy=True) + + def _get_intXarray(self, row, col): + return self._major_index_fancy(col)._get_submatrix(minor=row) + + def _get_intXslice(self, row, col): + if col.step in (1, None): + return self._get_submatrix(major=col, minor=row, copy=True) + return self._major_slice(col)._get_submatrix(minor=row) + + def _get_sliceXint(self, row, col): + if row.step in (1, None): + return self._get_submatrix(major=col, minor=row, copy=True) + return self._get_submatrix(major=col)._minor_slice(row) + + def _get_sliceXarray(self, row, col): + return self._major_index_fancy(col)._minor_slice(row) + + def _get_arrayXint(self, row, col): + res = self._get_submatrix(major=col)._minor_index_fancy(row) + if row.ndim > 1: + return res.reshape(row.shape) + return res + + def _get_arrayXslice(self, row, col): + return self._major_slice(col)._minor_index_fancy(row) + + # these functions are used by the parent class (_cs_matrix) + # to remove redundancy between csc_array and csr_matrix + @staticmethod + def _swap(x): + """swap the members of x if this is a column-oriented matrix + """ + return x[1], x[0] + + +def isspmatrix_csc(x): + """Is `x` of csc_matrix type? + + Parameters + ---------- + x + object to check for being a csc matrix + + Returns + ------- + bool + True if `x` is a csc matrix, False otherwise + + Examples + -------- + >>> from scipy.sparse import csc_array, csc_matrix, coo_matrix, isspmatrix_csc + >>> isspmatrix_csc(csc_matrix([[5]])) + True + >>> isspmatrix_csc(csc_array([[5]])) + False + >>> isspmatrix_csc(coo_matrix([[5]])) + False + """ + return isinstance(x, csc_matrix) + + +# This namespace class separates array from matrix with isinstance +class csc_array(_csc_base, sparray): + """ + Compressed Sparse Column array. + + This can be instantiated in several ways: + csc_array(D) + where D is a 2-D ndarray + + csc_array(S) + with another sparse array or matrix S (equivalent to S.tocsc()) + + csc_array((M, N), [dtype]) + to construct an empty array with shape (M, N) + dtype is optional, defaulting to dtype='d'. + + csc_array((data, (row_ind, col_ind)), [shape=(M, N)]) + where ``data``, ``row_ind`` and ``col_ind`` satisfy the + relationship ``a[row_ind[k], col_ind[k]] = data[k]``. + + csc_array((data, indices, indptr), [shape=(M, N)]) + is the standard CSC representation where the row indices for + column i are stored in ``indices[indptr[i]:indptr[i+1]]`` + and their corresponding values are stored in + ``data[indptr[i]:indptr[i+1]]``. If the shape parameter is + not supplied, the array dimensions are inferred from + the index arrays. + + Attributes + ---------- + dtype : dtype + Data type of the array + shape : 2-tuple + Shape of the array + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + CSC format data array of the array + indices + CSC format index array of the array + indptr + CSC format index pointer array of the array + has_sorted_indices + has_canonical_format + T + + Notes + ----- + + Sparse arrays can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + Advantages of the CSC format + - efficient arithmetic operations CSC + CSC, CSC * CSC, etc. + - efficient column slicing + - fast matrix vector products (CSR, BSR may be faster) + + Disadvantages of the CSC format + - slow row slicing operations (consider CSR) + - changes to the sparsity structure are expensive (consider LIL or DOK) + + Canonical format + - Within each column, indices are sorted by row. + - There are no duplicate entries. + + Examples + -------- + + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> csc_array((3, 4), dtype=np.int8).toarray() + array([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=int8) + + >>> row = np.array([0, 2, 2, 0, 1, 2]) + >>> col = np.array([0, 0, 1, 2, 2, 2]) + >>> data = np.array([1, 2, 3, 4, 5, 6]) + >>> csc_array((data, (row, col)), shape=(3, 3)).toarray() + array([[1, 0, 4], + [0, 0, 5], + [2, 3, 6]]) + + >>> indptr = np.array([0, 2, 3, 6]) + >>> indices = np.array([0, 2, 2, 0, 1, 2]) + >>> data = np.array([1, 2, 3, 4, 5, 6]) + >>> csc_array((data, indices, indptr), shape=(3, 3)).toarray() + array([[1, 0, 4], + [0, 0, 5], + [2, 3, 6]]) + + """ + + +class csc_matrix(spmatrix, _csc_base): + """ + Compressed Sparse Column matrix. + + This can be instantiated in several ways: + csc_matrix(D) + where D is a 2-D ndarray + + csc_matrix(S) + with another sparse array or matrix S (equivalent to S.tocsc()) + + csc_matrix((M, N), [dtype]) + to construct an empty matrix with shape (M, N) + dtype is optional, defaulting to dtype='d'. + + csc_matrix((data, (row_ind, col_ind)), [shape=(M, N)]) + where ``data``, ``row_ind`` and ``col_ind`` satisfy the + relationship ``a[row_ind[k], col_ind[k]] = data[k]``. + + csc_matrix((data, indices, indptr), [shape=(M, N)]) + is the standard CSC representation where the row indices for + column i are stored in ``indices[indptr[i]:indptr[i+1]]`` + and their corresponding values are stored in + ``data[indptr[i]:indptr[i+1]]``. If the shape parameter is + not supplied, the matrix dimensions are inferred from + the index arrays. + + Attributes + ---------- + dtype : dtype + Data type of the matrix + shape : 2-tuple + Shape of the matrix + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + CSC format data array of the matrix + indices + CSC format index array of the matrix + indptr + CSC format index pointer array of the matrix + has_sorted_indices + has_canonical_format + T + + Notes + ----- + + Sparse matrices can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + Advantages of the CSC format + - efficient arithmetic operations CSC + CSC, CSC * CSC, etc. + - efficient column slicing + - fast matrix vector products (CSR, BSR may be faster) + + Disadvantages of the CSC format + - slow row slicing operations (consider CSR) + - changes to the sparsity structure are expensive (consider LIL or DOK) + + Canonical format + - Within each column, indices are sorted by row. + - There are no duplicate entries. + + Examples + -------- + + >>> import numpy as np + >>> from scipy.sparse import csc_matrix + >>> csc_matrix((3, 4), dtype=np.int8).toarray() + array([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=int8) + + >>> row = np.array([0, 2, 2, 0, 1, 2]) + >>> col = np.array([0, 0, 1, 2, 2, 2]) + >>> data = np.array([1, 2, 3, 4, 5, 6]) + >>> csc_matrix((data, (row, col)), shape=(3, 3)).toarray() + array([[1, 0, 4], + [0, 0, 5], + [2, 3, 6]]) + + >>> indptr = np.array([0, 2, 3, 6]) + >>> indices = np.array([0, 2, 2, 0, 1, 2]) + >>> data = np.array([1, 2, 3, 4, 5, 6]) + >>> csc_matrix((data, indices, indptr), shape=(3, 3)).toarray() + array([[1, 0, 4], + [0, 0, 5], + [2, 3, 6]]) + + """ + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_csr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_csr.py new file mode 100644 index 0000000000000000000000000000000000000000..52ce35cdb1fbf75c26a08b4e13b93d236566ab0b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_csr.py @@ -0,0 +1,558 @@ +"""Compressed Sparse Row matrix format""" + +__docformat__ = "restructuredtext en" + +__all__ = ['csr_array', 'csr_matrix', 'isspmatrix_csr'] + +import numpy as np + +from ._matrix import spmatrix +from ._base import _spbase, sparray +from ._sparsetools import (csr_tocsc, csr_tobsr, csr_count_blocks, + get_csr_submatrix, csr_sample_values) +from ._sputils import upcast + +from ._compressed import _cs_matrix + + +class _csr_base(_cs_matrix): + _format = 'csr' + _allow_nd = (1, 2) + + def transpose(self, axes=None, copy=False): + if axes is not None and axes != (1, 0): + raise ValueError("Sparse arrays/matrices do not support " + "an 'axes' parameter because swapping " + "dimensions is the only logical permutation.") + + if self.ndim == 1: + return self.copy() if copy else self + M, N = self.shape + return self._csc_container((self.data, self.indices, + self.indptr), shape=(N, M), copy=copy) + + transpose.__doc__ = _spbase.transpose.__doc__ + + def tolil(self, copy=False): + if self.ndim != 2: + raise ValueError("Cannot convert a 1d sparse array to lil format") + lil = self._lil_container(self.shape, dtype=self.dtype) + + self.sum_duplicates() + ptr,ind,dat = self.indptr,self.indices,self.data + rows, data = lil.rows, lil.data + + for n in range(self.shape[0]): + start = ptr[n] + end = ptr[n+1] + rows[n] = ind[start:end].tolist() + data[n] = dat[start:end].tolist() + + return lil + + tolil.__doc__ = _spbase.tolil.__doc__ + + def tocsr(self, copy=False): + if copy: + return self.copy() + else: + return self + + tocsr.__doc__ = _spbase.tocsr.__doc__ + + def tocsc(self, copy=False): + if self.ndim != 2: + raise ValueError("Cannot convert a 1d sparse array to csc format") + M, N = self.shape + idx_dtype = self._get_index_dtype((self.indptr, self.indices), + maxval=max(self.nnz, M)) + indptr = np.empty(N + 1, dtype=idx_dtype) + indices = np.empty(self.nnz, dtype=idx_dtype) + data = np.empty(self.nnz, dtype=upcast(self.dtype)) + + csr_tocsc(M, N, + self.indptr.astype(idx_dtype), + self.indices.astype(idx_dtype), + self.data, + indptr, + indices, + data) + + A = self._csc_container((data, indices, indptr), shape=self.shape) + A.has_sorted_indices = True + return A + + tocsc.__doc__ = _spbase.tocsc.__doc__ + + def tobsr(self, blocksize=None, copy=True): + if self.ndim != 2: + raise ValueError("Cannot convert a 1d sparse array to bsr format") + if blocksize is None: + from ._spfuncs import estimate_blocksize + return self.tobsr(blocksize=estimate_blocksize(self)) + + elif blocksize == (1,1): + arg1 = (self.data.reshape(-1,1,1),self.indices,self.indptr) + return self._bsr_container(arg1, shape=self.shape, copy=copy) + + else: + R,C = blocksize + M,N = self.shape + + if R < 1 or C < 1 or M % R != 0 or N % C != 0: + raise ValueError(f'invalid blocksize {blocksize}') + + blks = csr_count_blocks(M,N,R,C,self.indptr,self.indices) + + idx_dtype = self._get_index_dtype((self.indptr, self.indices), + maxval=max(N//C, blks)) + indptr = np.empty(M//R+1, dtype=idx_dtype) + indices = np.empty(blks, dtype=idx_dtype) + data = np.zeros((blks,R,C), dtype=self.dtype) + + csr_tobsr(M, N, R, C, + self.indptr.astype(idx_dtype), + self.indices.astype(idx_dtype), + self.data, + indptr, indices, data.ravel()) + + return self._bsr_container( + (data, indices, indptr), shape=self.shape + ) + + tobsr.__doc__ = _spbase.tobsr.__doc__ + + # these functions are used by the parent class (_cs_matrix) + # to remove redundancy between csc_matrix and csr_array + @staticmethod + def _swap(x): + """swap the members of x if this is a column-oriented matrix + """ + return x + + def __iter__(self): + if self.ndim == 1: + zero = self.dtype.type(0) + u = 0 + for v, d in zip(self.indices, self.data): + for _ in range(v - u): + yield zero + yield d + u = v + 1 + for _ in range(self.shape[0] - u): + yield zero + return + + indptr = np.zeros(2, dtype=self.indptr.dtype) + # return 1d (sparray) or 2drow (spmatrix) + shape = self.shape[1:] if isinstance(self, sparray) else (1, self.shape[1]) + i0 = 0 + for i1 in self.indptr[1:]: + indptr[1] = i1 - i0 + indices = self.indices[i0:i1] + data = self.data[i0:i1] + yield self.__class__((data, indices, indptr), shape=shape, copy=True) + i0 = i1 + + def _getrow(self, i): + """Returns a copy of row i of the matrix, as a (1 x n) + CSR matrix (row vector). + """ + if self.ndim == 1: + if i not in (0, -1): + raise IndexError(f'index ({i}) out of range') + return self.reshape((1, self.shape[0]), copy=True) + + M, N = self.shape + i = int(i) + if i < 0: + i += M + if i < 0 or i >= M: + raise IndexError('index (%d) out of range' % i) + indptr, indices, data = get_csr_submatrix( + M, N, self.indptr, self.indices, self.data, i, i + 1, 0, N) + return self.__class__((data, indices, indptr), shape=(1, N), + dtype=self.dtype, copy=False) + + def _getcol(self, i): + """Returns a copy of column i. A (m x 1) sparse array (column vector). + """ + if self.ndim == 1: + raise ValueError("getcol not provided for 1d arrays. Use indexing A[j]") + M, N = self.shape + i = int(i) + if i < 0: + i += N + if i < 0 or i >= N: + raise IndexError('index (%d) out of range' % i) + indptr, indices, data = get_csr_submatrix( + M, N, self.indptr, self.indices, self.data, 0, M, i, i + 1) + return self.__class__((data, indices, indptr), shape=(M, 1), + dtype=self.dtype, copy=False) + + def _get_int(self, idx): + spot = np.flatnonzero(self.indices == idx) + if spot.size: + return self.data[spot[0]] + return self.data.dtype.type(0) + + def _get_slice(self, idx): + if idx == slice(None): + return self.copy() + if idx.step in (1, None): + ret = self._get_submatrix(0, idx, copy=True) + return ret.reshape(ret.shape[-1]) + return self._minor_slice(idx) + + def _get_array(self, idx): + idx_dtype = self._get_index_dtype(self.indices) + idx = np.asarray(idx, dtype=idx_dtype) + if idx.size == 0: + return self.__class__([], dtype=self.dtype) + + M, N = 1, self.shape[0] + row = np.zeros_like(idx, dtype=idx_dtype) + col = np.asarray(idx, dtype=idx_dtype) + val = np.empty(row.size, dtype=self.dtype) + csr_sample_values(M, N, self.indptr, self.indices, self.data, + row.size, row, col, val) + + new_shape = col.shape if col.shape[0] > 1 else (col.shape[0],) + return self.__class__(val.reshape(new_shape)) + + def _get_intXarray(self, row, col): + return self._getrow(row)._minor_index_fancy(col) + + def _get_intXslice(self, row, col): + if col.step in (1, None): + return self._get_submatrix(row, col, copy=True) + # TODO: uncomment this once it's faster: + # return self._getrow(row)._minor_slice(col) + + M, N = self.shape + start, stop, stride = col.indices(N) + + ii, jj = self.indptr[row:row+2] + row_indices = self.indices[ii:jj] + row_data = self.data[ii:jj] + + if stride > 0: + ind = (row_indices >= start) & (row_indices < stop) + else: + ind = (row_indices <= start) & (row_indices > stop) + + if abs(stride) > 1: + ind &= (row_indices - start) % stride == 0 + + row_indices = (row_indices[ind] - start) // stride + row_data = row_data[ind] + row_indptr = np.array([0, len(row_indices)]) + + if stride < 0: + row_data = row_data[::-1] + row_indices = abs(row_indices[::-1]) + + shape = (1, max(0, int(np.ceil(float(stop - start) / stride)))) + return self.__class__((row_data, row_indices, row_indptr), shape=shape, + dtype=self.dtype, copy=False) + + def _get_sliceXint(self, row, col): + if row.step in (1, None): + return self._get_submatrix(row, col, copy=True) + return self._major_slice(row)._get_submatrix(minor=col) + + def _get_sliceXarray(self, row, col): + return self._major_slice(row)._minor_index_fancy(col) + + def _get_arrayXint(self, row, col): + res = self._major_index_fancy(row)._get_submatrix(minor=col) + if row.ndim > 1: + return res.reshape(row.shape) + return res + + def _get_arrayXslice(self, row, col): + if col.step not in (1, None): + col = np.arange(*col.indices(self.shape[1])) + return self._get_arrayXarray(row, col) + return self._major_index_fancy(row)._get_submatrix(minor=col) + + def _set_int(self, idx, x): + self._set_many(0, idx, x) + + def _set_array(self, idx, x): + x = np.broadcast_to(x, idx.shape) + self._set_many(np.zeros_like(idx), idx, x) + + +def isspmatrix_csr(x): + """Is `x` of csr_matrix type? + + Parameters + ---------- + x + object to check for being a csr matrix + + Returns + ------- + bool + True if `x` is a csr matrix, False otherwise + + Examples + -------- + >>> from scipy.sparse import csr_array, csr_matrix, coo_matrix, isspmatrix_csr + >>> isspmatrix_csr(csr_matrix([[5]])) + True + >>> isspmatrix_csr(csr_array([[5]])) + False + >>> isspmatrix_csr(coo_matrix([[5]])) + False + """ + return isinstance(x, csr_matrix) + + +# This namespace class separates array from matrix with isinstance +class csr_array(_csr_base, sparray): + """ + Compressed Sparse Row array. + + This can be instantiated in several ways: + csr_array(D) + where D is a 2-D ndarray + + csr_array(S) + with another sparse array or matrix S (equivalent to S.tocsr()) + + csr_array((M, N), [dtype]) + to construct an empty array with shape (M, N) + dtype is optional, defaulting to dtype='d'. + + csr_array((data, (row_ind, col_ind)), [shape=(M, N)]) + where ``data``, ``row_ind`` and ``col_ind`` satisfy the + relationship ``a[row_ind[k], col_ind[k]] = data[k]``. + + csr_array((data, indices, indptr), [shape=(M, N)]) + is the standard CSR representation where the column indices for + row i are stored in ``indices[indptr[i]:indptr[i+1]]`` and their + corresponding values are stored in ``data[indptr[i]:indptr[i+1]]``. + If the shape parameter is not supplied, the array dimensions + are inferred from the index arrays. + + Attributes + ---------- + dtype : dtype + Data type of the array + shape : 2-tuple + Shape of the array + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + CSR format data array of the array + indices + CSR format index array of the array + indptr + CSR format index pointer array of the array + has_sorted_indices + has_canonical_format + T + + Notes + ----- + + Sparse arrays can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + Advantages of the CSR format + - efficient arithmetic operations CSR + CSR, CSR * CSR, etc. + - efficient row slicing + - fast matrix vector products + + Disadvantages of the CSR format + - slow column slicing operations (consider CSC) + - changes to the sparsity structure are expensive (consider LIL or DOK) + + Canonical Format + - Within each row, indices are sorted by column. + - There are no duplicate entries. + + Examples + -------- + + >>> import numpy as np + >>> from scipy.sparse import csr_array + >>> csr_array((3, 4), dtype=np.int8).toarray() + array([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=int8) + + >>> row = np.array([0, 0, 1, 2, 2, 2]) + >>> col = np.array([0, 2, 2, 0, 1, 2]) + >>> data = np.array([1, 2, 3, 4, 5, 6]) + >>> csr_array((data, (row, col)), shape=(3, 3)).toarray() + array([[1, 0, 2], + [0, 0, 3], + [4, 5, 6]]) + + >>> indptr = np.array([0, 2, 3, 6]) + >>> indices = np.array([0, 2, 2, 0, 1, 2]) + >>> data = np.array([1, 2, 3, 4, 5, 6]) + >>> csr_array((data, indices, indptr), shape=(3, 3)).toarray() + array([[1, 0, 2], + [0, 0, 3], + [4, 5, 6]]) + + Duplicate entries are summed together: + + >>> row = np.array([0, 1, 2, 0]) + >>> col = np.array([0, 1, 1, 0]) + >>> data = np.array([1, 2, 4, 8]) + >>> csr_array((data, (row, col)), shape=(3, 3)).toarray() + array([[9, 0, 0], + [0, 2, 0], + [0, 4, 0]]) + + As an example of how to construct a CSR array incrementally, + the following snippet builds a term-document array from texts: + + >>> docs = [["hello", "world", "hello"], ["goodbye", "cruel", "world"]] + >>> indptr = [0] + >>> indices = [] + >>> data = [] + >>> vocabulary = {} + >>> for d in docs: + ... for term in d: + ... index = vocabulary.setdefault(term, len(vocabulary)) + ... indices.append(index) + ... data.append(1) + ... indptr.append(len(indices)) + ... + >>> csr_array((data, indices, indptr), dtype=int).toarray() + array([[2, 1, 0, 0], + [0, 1, 1, 1]]) + + """ + + +class csr_matrix(spmatrix, _csr_base): + """ + Compressed Sparse Row matrix. + + This can be instantiated in several ways: + csr_matrix(D) + where D is a 2-D ndarray + + csr_matrix(S) + with another sparse array or matrix S (equivalent to S.tocsr()) + + csr_matrix((M, N), [dtype]) + to construct an empty matrix with shape (M, N) + dtype is optional, defaulting to dtype='d'. + + csr_matrix((data, (row_ind, col_ind)), [shape=(M, N)]) + where ``data``, ``row_ind`` and ``col_ind`` satisfy the + relationship ``a[row_ind[k], col_ind[k]] = data[k]``. + + csr_matrix((data, indices, indptr), [shape=(M, N)]) + is the standard CSR representation where the column indices for + row i are stored in ``indices[indptr[i]:indptr[i+1]]`` and their + corresponding values are stored in ``data[indptr[i]:indptr[i+1]]``. + If the shape parameter is not supplied, the matrix dimensions + are inferred from the index arrays. + + Attributes + ---------- + dtype : dtype + Data type of the matrix + shape : 2-tuple + Shape of the matrix + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + CSR format data array of the matrix + indices + CSR format index array of the matrix + indptr + CSR format index pointer array of the matrix + has_sorted_indices + has_canonical_format + T + + Notes + ----- + + Sparse matrices can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + Advantages of the CSR format + - efficient arithmetic operations CSR + CSR, CSR * CSR, etc. + - efficient row slicing + - fast matrix vector products + + Disadvantages of the CSR format + - slow column slicing operations (consider CSC) + - changes to the sparsity structure are expensive (consider LIL or DOK) + + Canonical Format + - Within each row, indices are sorted by column. + - There are no duplicate entries. + + Examples + -------- + + >>> import numpy as np + >>> from scipy.sparse import csr_matrix + >>> csr_matrix((3, 4), dtype=np.int8).toarray() + array([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=int8) + + >>> row = np.array([0, 0, 1, 2, 2, 2]) + >>> col = np.array([0, 2, 2, 0, 1, 2]) + >>> data = np.array([1, 2, 3, 4, 5, 6]) + >>> csr_matrix((data, (row, col)), shape=(3, 3)).toarray() + array([[1, 0, 2], + [0, 0, 3], + [4, 5, 6]]) + + >>> indptr = np.array([0, 2, 3, 6]) + >>> indices = np.array([0, 2, 2, 0, 1, 2]) + >>> data = np.array([1, 2, 3, 4, 5, 6]) + >>> csr_matrix((data, indices, indptr), shape=(3, 3)).toarray() + array([[1, 0, 2], + [0, 0, 3], + [4, 5, 6]]) + + Duplicate entries are summed together: + + >>> row = np.array([0, 1, 2, 0]) + >>> col = np.array([0, 1, 1, 0]) + >>> data = np.array([1, 2, 4, 8]) + >>> csr_matrix((data, (row, col)), shape=(3, 3)).toarray() + array([[9, 0, 0], + [0, 2, 0], + [0, 4, 0]]) + + As an example of how to construct a CSR matrix incrementally, + the following snippet builds a term-document matrix from texts: + + >>> docs = [["hello", "world", "hello"], ["goodbye", "cruel", "world"]] + >>> indptr = [0] + >>> indices = [] + >>> data = [] + >>> vocabulary = {} + >>> for d in docs: + ... for term in d: + ... index = vocabulary.setdefault(term, len(vocabulary)) + ... indices.append(index) + ... data.append(1) + ... indptr.append(len(indices)) + ... + >>> csr_matrix((data, indices, indptr), dtype=int).toarray() + array([[2, 1, 0, 0], + [0, 1, 1, 1]]) + + """ + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_data.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_data.py new file mode 100644 index 0000000000000000000000000000000000000000..585820b10a65271b52b81d140e82130eb4177979 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_data.py @@ -0,0 +1,569 @@ +"""Base class for sparse matrice with a .data attribute + + subclasses must provide a _with_data() method that + creates a new matrix with the same sparsity pattern + as self but with a different data array + +""" + +import math +import numpy as np + +from ._base import _spbase, sparray, _ufuncs_with_fixed_point_at_zero +from ._sputils import isscalarlike, validateaxis + +__all__ = [] + + +# TODO implement all relevant operations +# use .data.__methods__() instead of /=, *=, etc. +class _data_matrix(_spbase): + def __init__(self, arg1, *, maxprint=None): + _spbase.__init__(self, arg1, maxprint=maxprint) + + @property + def dtype(self): + return self.data.dtype + + @dtype.setter + def dtype(self, newtype): + self.data.dtype = newtype + + def _deduped_data(self): + if hasattr(self, 'sum_duplicates'): + self.sum_duplicates() + return self.data + + def __abs__(self): + return self._with_data(abs(self._deduped_data())) + + def __round__(self, ndigits=0): + return self._with_data(np.around(self._deduped_data(), decimals=ndigits)) + + def _real(self): + return self._with_data(self.data.real) + + def _imag(self): + return self._with_data(self.data.imag) + + def __neg__(self): + if self.dtype.kind == 'b': + raise NotImplementedError('negating a boolean sparse array is not ' + 'supported') + return self._with_data(-self.data) + + def __imul__(self, other): # self *= other + if isscalarlike(other): + self.data *= other + return self + return NotImplemented + + def __itruediv__(self, other): # self /= other + if isscalarlike(other): + recip = 1.0 / other + self.data *= recip + return self + else: + return NotImplemented + + def astype(self, dtype, casting='unsafe', copy=True): + dtype = np.dtype(dtype) + if self.dtype != dtype: + matrix = self._with_data( + self.data.astype(dtype, casting=casting, copy=True), + copy=True + ) + return matrix._with_data(matrix._deduped_data(), copy=False) + elif copy: + return self.copy() + else: + return self + + astype.__doc__ = _spbase.astype.__doc__ + + def conjugate(self, copy=True): + if np.issubdtype(self.dtype, np.complexfloating): + return self._with_data(self.data.conjugate(), copy=copy) + elif copy: + return self.copy() + else: + return self + + conjugate.__doc__ = _spbase.conjugate.__doc__ + + def copy(self): + return self._with_data(self.data.copy(), copy=True) + + copy.__doc__ = _spbase.copy.__doc__ + + def power(self, n, dtype=None): + """ + This function performs element-wise power. + + Parameters + ---------- + n : scalar + n is a non-zero scalar (nonzero avoids dense ones creation) + If zero power is desired, special case it to use `np.ones` + + dtype : If dtype is not specified, the current dtype will be preserved. + + Raises + ------ + NotImplementedError : if n is a zero scalar + If zero power is desired, special case it to use + ``np.ones(A.shape, dtype=A.dtype)`` + """ + if not isscalarlike(n): + raise NotImplementedError("input is not scalar") + if not n: + raise NotImplementedError( + "zero power is not supported as it would densify the matrix.\n" + "Use `np.ones(A.shape, dtype=A.dtype)` for this case." + ) + + data = self._deduped_data() + if dtype is not None: + data = data.astype(dtype) + return self._with_data(data ** n) + + ########################### + # Multiplication handlers # + ########################### + + def _mul_scalar(self, other): + return self._with_data(self.data * other) + + +# Add the numpy unary ufuncs for which func(0) = 0 to _data_matrix. +for npfunc in _ufuncs_with_fixed_point_at_zero: + name = npfunc.__name__ + + def _create_method(op): + def method(self): + result = op(self._deduped_data()) + return self._with_data(result, copy=True) + + method.__doc__ = (f"Element-wise {name}.\n\n" + f"See `numpy.{name}` for more information.") + method.__name__ = name + + return method + + setattr(_data_matrix, name, _create_method(npfunc)) + + +def _find_missing_index(ind, n): + for k, a in enumerate(ind): + if k != a: + return k + + k += 1 + if k < n: + return k + else: + return -1 + + +class _minmax_mixin: + """Mixin for min and max methods. + + These are not implemented for dia_matrix, hence the separate class. + """ + + def _min_or_max_axis(self, axis, min_or_max, explicit): + N = self.shape[axis] + if N == 0: + raise ValueError("zero-size array to reduction operation") + M = self.shape[1 - axis] + idx_dtype = self._get_index_dtype(maxval=M) + + mat = self.tocsc() if axis == 0 else self.tocsr() + mat.sum_duplicates() + + major_index, value = mat._minor_reduce(min_or_max) + if not explicit: + not_full = np.diff(mat.indptr)[major_index] < N + value[not_full] = min_or_max(value[not_full], 0) + + mask = value != 0 + major_index = np.compress(mask, major_index).astype(idx_dtype, copy=False) + value = np.compress(mask, value) + + if isinstance(self, sparray): + coords = (major_index,) + shape = (M,) + return self._coo_container((value, coords), shape=shape, dtype=self.dtype) + + if axis == 0: + return self._coo_container( + (value, (np.zeros(len(value), dtype=idx_dtype), major_index)), + dtype=self.dtype, shape=(1, M) + ) + else: + return self._coo_container( + (value, (major_index, np.zeros(len(value), dtype=idx_dtype))), + dtype=self.dtype, shape=(M, 1) + ) + + def _min_or_max(self, axis, out, min_or_max, explicit): + if out is not None: + raise ValueError("Sparse arrays do not support an 'out' parameter.") + + validateaxis(axis) + if self.ndim == 1: + if axis not in (None, 0, -1): + raise ValueError("axis out of range") + axis = None # avoid calling special axis case. no impact on 1d + + if axis is None: + if 0 in self.shape: + raise ValueError("zero-size array to reduction operation") + + zero = self.dtype.type(0) + if self.nnz == 0: + return zero + m = min_or_max.reduce(self._deduped_data().ravel()) + if self.nnz != math.prod(self.shape) and not explicit: + m = min_or_max(zero, m) + return m + + if axis < 0: + axis += 2 + + if (axis == 0) or (axis == 1): + return self._min_or_max_axis(axis, min_or_max, explicit) + else: + raise ValueError("axis out of range") + + def _arg_min_or_max_axis(self, axis, argmin_or_argmax, compare, explicit): + if self.shape[axis] == 0: + raise ValueError("Cannot apply the operation along a zero-sized dimension.") + + if axis < 0: + axis += 2 + + zero = self.dtype.type(0) + + mat = self.tocsc() if axis == 0 else self.tocsr() + mat.sum_duplicates() + + ret_size, line_size = mat._swap(mat.shape) + ret = np.zeros(ret_size, dtype=int) + + nz_lines, = np.nonzero(np.diff(mat.indptr)) + for i in nz_lines: + p, q = mat.indptr[i:i + 2] + data = mat.data[p:q] + indices = mat.indices[p:q] + extreme_index = argmin_or_argmax(data) + extreme_value = data[extreme_index] + if explicit: + if q - p > 0: + ret[i] = indices[extreme_index] + else: + if compare(extreme_value, zero) or q - p == line_size: + ret[i] = indices[extreme_index] + else: + zero_ind = _find_missing_index(indices, line_size) + if extreme_value == zero: + ret[i] = min(extreme_index, zero_ind) + else: + ret[i] = zero_ind + + if isinstance(self, sparray): + return ret + + if axis == 1: + ret = ret.reshape(-1, 1) + + return self._ascontainer(ret) + + def _arg_min_or_max(self, axis, out, argmin_or_argmax, compare, explicit): + if out is not None: + raise ValueError("Sparse types do not support an 'out' parameter.") + + validateaxis(axis) + + if self.ndim == 1: + if axis not in (None, 0, -1): + raise ValueError("axis out of range") + axis = None # avoid calling special axis case. no impact on 1d + + if axis is not None: + return self._arg_min_or_max_axis(axis, argmin_or_argmax, compare, explicit) + + if 0 in self.shape: + raise ValueError("Cannot apply the operation to an empty matrix.") + + if self.nnz == 0: + if explicit: + raise ValueError("Cannot apply the operation to zero matrix " + "when explicit=True.") + return 0 + + zero = self.dtype.type(0) + mat = self.tocoo() + # Convert to canonical form: no duplicates, sorted indices. + mat.sum_duplicates() + extreme_index = argmin_or_argmax(mat.data) + if explicit: + return extreme_index + extreme_value = mat.data[extreme_index] + num_col = mat.shape[-1] + + # If the min value is less than zero, or max is greater than zero, + # then we do not need to worry about implicit zeros. + if compare(extreme_value, zero): + # cast to Python int to avoid overflow and RuntimeError + return int(mat.row[extreme_index]) * num_col + int(mat.col[extreme_index]) + + # Cheap test for the rare case where we have no implicit zeros. + size = math.prod(self.shape) + if size == mat.nnz: + return int(mat.row[extreme_index]) * num_col + int(mat.col[extreme_index]) + + # At this stage, any implicit zero could be the min or max value. + # After sum_duplicates(), the `row` and `col` arrays are guaranteed to + # be sorted in C-order, which means the linearized indices are sorted. + linear_indices = mat.row * num_col + mat.col + first_implicit_zero_index = _find_missing_index(linear_indices, size) + if extreme_value == zero: + return min(first_implicit_zero_index, extreme_index) + return first_implicit_zero_index + + def max(self, axis=None, out=None, *, explicit=False): + """Return the maximum of the array/matrix or maximum along an axis. + + By default, all elements are taken into account, not just the non-zero ones. + But with `explicit` set, only the stored elements are considered. + + Parameters + ---------- + axis : {-2, -1, 0, 1, None} optional + Axis along which the sum is computed. The default is to + compute the maximum over all elements, returning + a scalar (i.e., `axis` = `None`). + + out : None, optional + This argument is in the signature *solely* for NumPy + compatibility reasons. Do not pass in anything except + for the default value, as this argument is not used. + + explicit : {False, True} optional (default: False) + When set to True, only the stored elements will be considered. + If a row/column is empty, the sparse.coo_array returned + has no stored element (i.e. an implicit zero) for that row/column. + + .. versionadded:: 1.15.0 + + Returns + ------- + amax : coo_array or scalar + Maximum of `a`. If `axis` is None, the result is a scalar value. + If `axis` is given, the result is a sparse.coo_array of dimension + ``a.ndim - 1``. + + See Also + -------- + min : The minimum value of a sparse array/matrix along a given axis. + numpy.max : NumPy's implementation of 'max' + + """ + return self._min_or_max(axis, out, np.maximum, explicit) + + def min(self, axis=None, out=None, *, explicit=False): + """Return the minimum of the array/matrix or maximum along an axis. + + By default, all elements are taken into account, not just the non-zero ones. + But with `explicit` set, only the stored elements are considered. + + Parameters + ---------- + axis : {-2, -1, 0, 1, None} optional + Axis along which the sum is computed. The default is to + compute the minimum over all elements, returning + a scalar (i.e., `axis` = `None`). + + out : None, optional + This argument is in the signature *solely* for NumPy + compatibility reasons. Do not pass in anything except for + the default value, as this argument is not used. + + explicit : {False, True} optional (default: False) + When set to True, only the stored elements will be considered. + If a row/column is empty, the sparse.coo_array returned + has no stored element (i.e. an implicit zero) for that row/column. + + .. versionadded:: 1.15.0 + + Returns + ------- + amin : coo_matrix or scalar + Minimum of `a`. If `axis` is None, the result is a scalar value. + If `axis` is given, the result is a sparse.coo_array of dimension + ``a.ndim - 1``. + + See Also + -------- + max : The maximum value of a sparse array/matrix along a given axis. + numpy.min : NumPy's implementation of 'min' + + """ + return self._min_or_max(axis, out, np.minimum, explicit) + + def nanmax(self, axis=None, out=None, *, explicit=False): + """Return the maximum, ignoring any Nans, along an axis. + + Return the maximum, ignoring any Nans, of the array/matrix along an axis. + By default this takes all elements into account, but with `explicit` set, + only stored elements are considered. + + .. versionadded:: 1.11.0 + + Parameters + ---------- + axis : {-2, -1, 0, 1, None} optional + Axis along which the maximum is computed. The default is to + compute the maximum over all elements, returning + a scalar (i.e., `axis` = `None`). + + out : None, optional + This argument is in the signature *solely* for NumPy + compatibility reasons. Do not pass in anything except + for the default value, as this argument is not used. + + explicit : {False, True} optional (default: False) + When set to True, only the stored elements will be considered. + If a row/column is empty, the sparse.coo_array returned + has no stored element (i.e. an implicit zero) for that row/column. + + .. versionadded:: 1.15.0 + + Returns + ------- + amax : coo_array or scalar + Maximum of `a`. If `axis` is None, the result is a scalar value. + If `axis` is given, the result is a sparse.coo_array of dimension + ``a.ndim - 1``. + + See Also + -------- + nanmin : The minimum value of a sparse array/matrix along a given axis, + ignoring NaNs. + max : The maximum value of a sparse array/matrix along a given axis, + propagating NaNs. + numpy.nanmax : NumPy's implementation of 'nanmax'. + + """ + return self._min_or_max(axis, out, np.fmax, explicit) + + def nanmin(self, axis=None, out=None, *, explicit=False): + """Return the minimum, ignoring any Nans, along an axis. + + Return the minimum, ignoring any Nans, of the array/matrix along an axis. + By default this takes all elements into account, but with `explicit` set, + only stored elements are considered. + + .. versionadded:: 1.11.0 + + Parameters + ---------- + axis : {-2, -1, 0, 1, None} optional + Axis along which the minimum is computed. The default is to + compute the minimum over all elements, returning + a scalar (i.e., `axis` = `None`). + + out : None, optional + This argument is in the signature *solely* for NumPy + compatibility reasons. Do not pass in anything except for + the default value, as this argument is not used. + + explicit : {False, True} optional (default: False) + When set to True, only the stored elements will be considered. + If a row/column is empty, the sparse.coo_array returned + has no stored element (i.e. an implicit zero) for that row/column. + + .. versionadded:: 1.15.0 + + Returns + ------- + amin : coo_array or scalar + Minimum of `a`. If `axis` is None, the result is a scalar value. + If `axis` is given, the result is a sparse.coo_array of dimension + ``a.ndim - 1``. + + See Also + -------- + nanmax : The maximum value of a sparse array/matrix along a given axis, + ignoring NaNs. + min : The minimum value of a sparse array/matrix along a given axis, + propagating NaNs. + numpy.nanmin : NumPy's implementation of 'nanmin'. + + """ + return self._min_or_max(axis, out, np.fmin, explicit) + + def argmax(self, axis=None, out=None, *, explicit=False): + """Return indices of maximum elements along an axis. + + By default, implicit zero elements are taken into account. If there are + several minimum values, the index of the first occurrence is returned. + If `explicit` is set, only explicitly stored elements will be considered. + + Parameters + ---------- + axis : {-2, -1, 0, 1, None}, optional + Axis along which the argmax is computed. If None (default), index + of the maximum element in the flatten data is returned. + + out : None, optional + This argument is in the signature *solely* for NumPy + compatibility reasons. Do not pass in anything except for + the default value, as this argument is not used. + + explicit : {False, True} optional (default: False) + When set to True, only explicitly stored elements will be considered. + If axis is not None and a row/column has no stored elements, argmax + is undefined, so the index ``0`` is returned for that row/column. + + .. versionadded:: 1.15.0 + + Returns + ------- + ind : numpy.matrix or int + Indices of maximum elements. If matrix, its size along `axis` is 1. + """ + return self._arg_min_or_max(axis, out, np.argmax, np.greater, explicit) + + def argmin(self, axis=None, out=None, *, explicit=False): + """Return indices of minimum elements along an axis. + + By default, implicit zero elements are taken into account. If there are + several minimum values, the index of the first occurrence is returned. + If `explicit` is set, only explicitly stored elements will be considered. + + Parameters + ---------- + axis : {-2, -1, 0, 1, None}, optional + Axis along which the argmin is computed. If None (default), index + of the minimum element in the flatten data is returned. + + out : None, optional + This argument is in the signature *solely* for NumPy + compatibility reasons. Do not pass in anything except for + the default value, as this argument is not used. + + explicit : {False, True} optional (default: False) + When set to True, only explicitly stored elements will be considered. + If axis is not None and a row/column has no stored elements, argmin + is undefined, so the index ``0`` is returned for that row/column. + + .. versionadded:: 1.15.0 + + Returns + ------- + ind : numpy.matrix or int + Indices of minimum elements. If matrix, its size along `axis` is 1. + """ + return self._arg_min_or_max(axis, out, np.argmin, np.less, explicit) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_dia.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_dia.py new file mode 100644 index 0000000000000000000000000000000000000000..c2944e080b6abc1705d3f597dda7478335b2bcf3 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_dia.py @@ -0,0 +1,590 @@ +"""Sparse DIAgonal format""" + +__docformat__ = "restructuredtext en" + +__all__ = ['dia_array', 'dia_matrix', 'isspmatrix_dia'] + +import numpy as np + +from .._lib._util import copy_if_needed +from ._matrix import spmatrix +from ._base import issparse, _formats, _spbase, sparray +from ._data import _data_matrix +from ._sputils import ( + isshape, upcast_char, getdtype, get_sum_dtype, validateaxis, check_shape +) +from ._sparsetools import dia_matvec + + +class _dia_base(_data_matrix): + _format = 'dia' + + def __init__(self, arg1, shape=None, dtype=None, copy=False, *, maxprint=None): + _data_matrix.__init__(self, arg1, maxprint=maxprint) + + if issparse(arg1): + if arg1.format == "dia": + if copy: + arg1 = arg1.copy() + self.data = arg1.data + self.offsets = arg1.offsets + self._shape = check_shape(arg1.shape) + else: + if arg1.format == self.format and copy: + A = arg1.copy() + else: + A = arg1.todia() + self.data = A.data + self.offsets = A.offsets + self._shape = check_shape(A.shape) + elif isinstance(arg1, tuple): + if isshape(arg1): + # It's a tuple of matrix dimensions (M, N) + # create empty matrix + self._shape = check_shape(arg1) + self.data = np.zeros((0,0), getdtype(dtype, default=float)) + idx_dtype = self._get_index_dtype(maxval=max(self.shape)) + self.offsets = np.zeros((0), dtype=idx_dtype) + else: + try: + # Try interpreting it as (data, offsets) + data, offsets = arg1 + except Exception as e: + message = 'unrecognized form for dia_array constructor' + raise ValueError(message) from e + else: + if shape is None: + raise ValueError('expected a shape argument') + if not copy: + copy = copy_if_needed + self.data = np.atleast_2d(np.array(arg1[0], dtype=dtype, copy=copy)) + offsets = np.array(arg1[1], + dtype=self._get_index_dtype(maxval=max(shape)), + copy=copy) + self.offsets = np.atleast_1d(offsets) + self._shape = check_shape(shape) + else: + # must be dense, convert to COO first, then to DIA + try: + arg1 = np.asarray(arg1) + except Exception as e: + raise ValueError("unrecognized form for " + f"{self.format}_matrix constructor") from e + if isinstance(self, sparray) and arg1.ndim != 2: + raise ValueError(f"DIA arrays don't support {arg1.ndim}D input. Use 2D") + A = self._coo_container(arg1, dtype=dtype, shape=shape).todia() + self.data = A.data + self.offsets = A.offsets + self._shape = check_shape(A.shape) + + if dtype is not None: + newdtype = getdtype(dtype) + self.data = self.data.astype(newdtype) + + # check format + if self.offsets.ndim != 1: + raise ValueError('offsets array must have rank 1') + + if self.data.ndim != 2: + raise ValueError('data array must have rank 2') + + if self.data.shape[0] != len(self.offsets): + raise ValueError('number of diagonals (%d) ' + 'does not match the number of offsets (%d)' + % (self.data.shape[0], len(self.offsets))) + + if len(np.unique(self.offsets)) != len(self.offsets): + raise ValueError('offset array contains duplicate values') + + def __repr__(self): + _, fmt = _formats[self.format] + sparse_cls = 'array' if isinstance(self, sparray) else 'matrix' + d = self.data.shape[0] + return ( + f"<{fmt} sparse {sparse_cls} of dtype '{self.dtype}'\n" + f"\twith {self.nnz} stored elements ({d} diagonals) and shape {self.shape}>" + ) + + def _data_mask(self): + """Returns a mask of the same shape as self.data, where + mask[i,j] is True when data[i,j] corresponds to a stored element.""" + num_rows, num_cols = self.shape + offset_inds = np.arange(self.data.shape[1]) + row = offset_inds - self.offsets[:,None] + mask = (row >= 0) + mask &= (row < num_rows) + mask &= (offset_inds < num_cols) + return mask + + def count_nonzero(self, axis=None): + if axis is not None: + raise NotImplementedError( + "count_nonzero over an axis is not implemented for DIA format" + ) + mask = self._data_mask() + return np.count_nonzero(self.data[mask]) + + count_nonzero.__doc__ = _spbase.count_nonzero.__doc__ + + def _getnnz(self, axis=None): + if axis is not None: + raise NotImplementedError("_getnnz over an axis is not implemented " + "for DIA format") + M,N = self.shape + nnz = 0 + for k in self.offsets: + if k > 0: + nnz += min(M,N-k) + else: + nnz += min(M+k,N) + return int(nnz) + + _getnnz.__doc__ = _spbase._getnnz.__doc__ + + def sum(self, axis=None, dtype=None, out=None): + validateaxis(axis) + + if axis is not None and axis < 0: + axis += 2 + + res_dtype = get_sum_dtype(self.dtype) + num_rows, num_cols = self.shape + ret = None + + if axis == 0: + mask = self._data_mask() + x = (self.data * mask).sum(axis=0) + if x.shape[0] == num_cols: + res = x + else: + res = np.zeros(num_cols, dtype=x.dtype) + res[:x.shape[0]] = x + ret = self._ascontainer(res, dtype=res_dtype) + + else: + row_sums = np.zeros((num_rows, 1), dtype=res_dtype) + one = np.ones(num_cols, dtype=res_dtype) + dia_matvec(num_rows, num_cols, len(self.offsets), + self.data.shape[1], self.offsets, self.data, one, row_sums) + + row_sums = self._ascontainer(row_sums) + + if axis is None: + return row_sums.sum(dtype=dtype, out=out) + + ret = self._ascontainer(row_sums.sum(axis=axis)) + + return ret.sum(axis=(), dtype=dtype, out=out) + + sum.__doc__ = _spbase.sum.__doc__ + + def _add_sparse(self, other): + # If other is not DIA format, let them handle us instead. + if not isinstance(other, _dia_base): + return other._add_sparse(self) + + # Fast path for exact equality of the sparsity structure. + if np.array_equal(self.offsets, other.offsets): + return self._with_data(self.data + other.data) + + # Find the union of the offsets (which will be sorted and unique). + new_offsets = np.union1d(self.offsets, other.offsets) + self_idx = np.searchsorted(new_offsets, self.offsets) + other_idx = np.searchsorted(new_offsets, other.offsets) + + self_d = self.data.shape[1] + other_d = other.data.shape[1] + # Fast path for a sparsity structure where the final offsets are a + # permutation of the existing offsets and the diagonal lengths match. + if self_d == other_d and len(new_offsets) == len(self.offsets): + new_data = self.data[_invert_index(self_idx)] + new_data[other_idx, :] += other.data + elif self_d == other_d and len(new_offsets) == len(other.offsets): + new_data = other.data[_invert_index(other_idx)] + new_data[self_idx, :] += self.data + else: + # Maximum diagonal length of the result. + d = min(self.shape[0] + new_offsets[-1], self.shape[1]) + + # Add all diagonals to a freshly-allocated data array. + new_data = np.zeros( + (len(new_offsets), d), + dtype=np.result_type(self.data, other.data), + ) + new_data[self_idx, :self_d] += self.data[:, :d] + new_data[other_idx, :other_d] += other.data[:, :d] + return self._dia_container((new_data, new_offsets), shape=self.shape) + + def _mul_scalar(self, other): + return self._with_data(self.data * other) + + def _matmul_vector(self, other): + x = other + + y = np.zeros(self.shape[0], dtype=upcast_char(self.dtype.char, + x.dtype.char)) + + L = self.data.shape[1] + + M,N = self.shape + + dia_matvec(M,N, len(self.offsets), L, self.offsets, self.data, + x.ravel(), y.ravel()) + + return y + + def _setdiag(self, values, k=0): + M, N = self.shape + + if values.ndim == 0: + # broadcast + values_n = np.inf + else: + values_n = len(values) + + if k < 0: + n = min(M + k, N, values_n) + min_index = 0 + max_index = n + else: + n = min(M, N - k, values_n) + min_index = k + max_index = k + n + + if values.ndim != 0: + # allow also longer sequences + values = values[:n] + + data_rows, data_cols = self.data.shape + if k in self.offsets: + if max_index > data_cols: + data = np.zeros((data_rows, max_index), dtype=self.data.dtype) + data[:, :data_cols] = self.data + self.data = data + self.data[self.offsets == k, min_index:max_index] = values + else: + self.offsets = np.append(self.offsets, self.offsets.dtype.type(k)) + m = max(max_index, data_cols) + data = np.zeros((data_rows + 1, m), dtype=self.data.dtype) + data[:-1, :data_cols] = self.data + data[-1, min_index:max_index] = values + self.data = data + + def todia(self, copy=False): + if copy: + return self.copy() + else: + return self + + todia.__doc__ = _spbase.todia.__doc__ + + def transpose(self, axes=None, copy=False): + if axes is not None and axes != (1, 0): + raise ValueError("Sparse arrays/matrices do not support " + "an 'axes' parameter because swapping " + "dimensions is the only logical permutation.") + + num_rows, num_cols = self.shape + max_dim = max(self.shape) + + # flip diagonal offsets + offsets = -self.offsets + + # re-align the data matrix + r = np.arange(len(offsets), dtype=np.intc)[:, None] + c = np.arange(num_rows, dtype=np.intc) - (offsets % max_dim)[:, None] + pad_amount = max(0, max_dim-self.data.shape[1]) + data = np.hstack((self.data, np.zeros((self.data.shape[0], pad_amount), + dtype=self.data.dtype))) + data = data[r, c] + return self._dia_container((data, offsets), shape=( + num_cols, num_rows), copy=copy) + + transpose.__doc__ = _spbase.transpose.__doc__ + + def diagonal(self, k=0): + rows, cols = self.shape + if k <= -rows or k >= cols: + return np.empty(0, dtype=self.data.dtype) + idx, = np.nonzero(self.offsets == k) + first_col = max(0, k) + last_col = min(rows + k, cols) + result_size = last_col - first_col + if idx.size == 0: + return np.zeros(result_size, dtype=self.data.dtype) + result = self.data[idx[0], first_col:last_col] + padding = result_size - len(result) + if padding > 0: + result = np.pad(result, (0, padding), mode='constant') + return result + + diagonal.__doc__ = _spbase.diagonal.__doc__ + + def tocsc(self, copy=False): + if self.nnz == 0: + return self._csc_container(self.shape, dtype=self.dtype) + + num_rows, num_cols = self.shape + num_offsets, offset_len = self.data.shape + offset_inds = np.arange(offset_len) + + row = offset_inds - self.offsets[:,None] + mask = (row >= 0) + mask &= (row < num_rows) + mask &= (offset_inds < num_cols) + mask &= (self.data != 0) + + idx_dtype = self._get_index_dtype(maxval=max(self.shape)) + indptr = np.zeros(num_cols + 1, dtype=idx_dtype) + indptr[1:offset_len+1] = np.cumsum(mask.sum(axis=0)[:num_cols]) + if offset_len < num_cols: + indptr[offset_len+1:] = indptr[offset_len] + indices = row.T[mask.T].astype(idx_dtype, copy=False) + data = self.data.T[mask.T] + return self._csc_container((data, indices, indptr), shape=self.shape, + dtype=self.dtype) + + tocsc.__doc__ = _spbase.tocsc.__doc__ + + def tocoo(self, copy=False): + num_rows, num_cols = self.shape + num_offsets, offset_len = self.data.shape + offset_inds = np.arange(offset_len) + + row = offset_inds - self.offsets[:,None] + mask = (row >= 0) + mask &= (row < num_rows) + mask &= (offset_inds < num_cols) + mask &= (self.data != 0) + row = row[mask] + col = np.tile(offset_inds, num_offsets)[mask.ravel()] + idx_dtype = self._get_index_dtype( + arrays=(self.offsets,), maxval=max(self.shape) + ) + row = row.astype(idx_dtype, copy=False) + col = col.astype(idx_dtype, copy=False) + data = self.data[mask] + # Note: this cannot set has_canonical_format=True, because despite the + # lack of duplicates, we do not generate sorted indices. + return self._coo_container( + (data, (row, col)), shape=self.shape, dtype=self.dtype, copy=False + ) + + tocoo.__doc__ = _spbase.tocoo.__doc__ + + # needed by _data_matrix + def _with_data(self, data, copy=True): + """Returns a matrix with the same sparsity structure as self, + but with different data. By default the structure arrays are copied. + """ + if copy: + return self._dia_container( + (data, self.offsets.copy()), shape=self.shape + ) + else: + return self._dia_container( + (data, self.offsets), shape=self.shape + ) + + def resize(self, *shape): + shape = check_shape(shape) + M, N = shape + # we do not need to handle the case of expanding N + self.data = self.data[:, :N] + + if (M > self.shape[0] and + np.any(self.offsets + self.shape[0] < self.data.shape[1])): + # explicitly clear values that were previously hidden + mask = (self.offsets[:, None] + self.shape[0] <= + np.arange(self.data.shape[1])) + self.data[mask] = 0 + + self._shape = shape + + resize.__doc__ = _spbase.resize.__doc__ + + +def _invert_index(idx): + """Helper function to invert an index array.""" + inv = np.zeros_like(idx) + inv[idx] = np.arange(len(idx)) + return inv + + +def isspmatrix_dia(x): + """Is `x` of dia_matrix type? + + Parameters + ---------- + x + object to check for being a dia matrix + + Returns + ------- + bool + True if `x` is a dia matrix, False otherwise + + Examples + -------- + >>> from scipy.sparse import dia_array, dia_matrix, coo_matrix, isspmatrix_dia + >>> isspmatrix_dia(dia_matrix([[5]])) + True + >>> isspmatrix_dia(dia_array([[5]])) + False + >>> isspmatrix_dia(coo_matrix([[5]])) + False + """ + return isinstance(x, dia_matrix) + + +# This namespace class separates array from matrix with isinstance +class dia_array(_dia_base, sparray): + """ + Sparse array with DIAgonal storage. + + This can be instantiated in several ways: + dia_array(D) + where D is a 2-D ndarray + + dia_array(S) + with another sparse array or matrix S (equivalent to S.todia()) + + dia_array((M, N), [dtype]) + to construct an empty array with shape (M, N), + dtype is optional, defaulting to dtype='d'. + + dia_array((data, offsets), shape=(M, N)) + where the ``data[k,:]`` stores the diagonal entries for + diagonal ``offsets[k]`` (See example below) + + Attributes + ---------- + dtype : dtype + Data type of the array + shape : 2-tuple + Shape of the array + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + DIA format data array of the array + offsets + DIA format offset array of the array + T + + Notes + ----- + + Sparse arrays can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + Sparse arrays with DIAgonal storage do not support slicing. + + Examples + -------- + + >>> import numpy as np + >>> from scipy.sparse import dia_array + >>> dia_array((3, 4), dtype=np.int8).toarray() + array([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=int8) + + >>> data = np.array([[1, 2, 3, 4]]).repeat(3, axis=0) + >>> offsets = np.array([0, -1, 2]) + >>> dia_array((data, offsets), shape=(4, 4)).toarray() + array([[1, 0, 3, 0], + [1, 2, 0, 4], + [0, 2, 3, 0], + [0, 0, 3, 4]]) + + >>> from scipy.sparse import dia_array + >>> n = 10 + >>> ex = np.ones(n) + >>> data = np.array([ex, 2 * ex, ex]) + >>> offsets = np.array([-1, 0, 1]) + >>> dia_array((data, offsets), shape=(n, n)).toarray() + array([[2., 1., 0., ..., 0., 0., 0.], + [1., 2., 1., ..., 0., 0., 0.], + [0., 1., 2., ..., 0., 0., 0.], + ..., + [0., 0., 0., ..., 2., 1., 0.], + [0., 0., 0., ..., 1., 2., 1.], + [0., 0., 0., ..., 0., 1., 2.]]) + """ + + +class dia_matrix(spmatrix, _dia_base): + """ + Sparse matrix with DIAgonal storage. + + This can be instantiated in several ways: + dia_matrix(D) + where D is a 2-D ndarray + + dia_matrix(S) + with another sparse array or matrix S (equivalent to S.todia()) + + dia_matrix((M, N), [dtype]) + to construct an empty matrix with shape (M, N), + dtype is optional, defaulting to dtype='d'. + + dia_matrix((data, offsets), shape=(M, N)) + where the ``data[k,:]`` stores the diagonal entries for + diagonal ``offsets[k]`` (See example below) + + Attributes + ---------- + dtype : dtype + Data type of the matrix + shape : 2-tuple + Shape of the matrix + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + DIA format data array of the matrix + offsets + DIA format offset array of the matrix + T + + Notes + ----- + + Sparse matrices can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + Sparse matrices with DIAgonal storage do not support slicing. + + Examples + -------- + + >>> import numpy as np + >>> from scipy.sparse import dia_matrix + >>> dia_matrix((3, 4), dtype=np.int8).toarray() + array([[0, 0, 0, 0], + [0, 0, 0, 0], + [0, 0, 0, 0]], dtype=int8) + + >>> data = np.array([[1, 2, 3, 4]]).repeat(3, axis=0) + >>> offsets = np.array([0, -1, 2]) + >>> dia_matrix((data, offsets), shape=(4, 4)).toarray() + array([[1, 0, 3, 0], + [1, 2, 0, 4], + [0, 2, 3, 0], + [0, 0, 3, 4]]) + + >>> from scipy.sparse import dia_matrix + >>> n = 10 + >>> ex = np.ones(n) + >>> data = np.array([ex, 2 * ex, ex]) + >>> offsets = np.array([-1, 0, 1]) + >>> dia_matrix((data, offsets), shape=(n, n)).toarray() + array([[2., 1., 0., ..., 0., 0., 0.], + [1., 2., 1., ..., 0., 0., 0.], + [0., 1., 2., ..., 0., 0., 0.], + ..., + [0., 0., 0., ..., 2., 1., 0.], + [0., 0., 0., ..., 1., 2., 1.], + [0., 0., 0., ..., 0., 1., 2.]]) + """ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_dok.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_dok.py new file mode 100644 index 0000000000000000000000000000000000000000..c9814a9e8d0b2d5a67185faae9311f4216cc7d13 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_dok.py @@ -0,0 +1,692 @@ +"""Dictionary Of Keys based matrix""" + +__docformat__ = "restructuredtext en" + +__all__ = ['dok_array', 'dok_matrix', 'isspmatrix_dok'] + +import itertools +from warnings import warn +import numpy as np + +from ._matrix import spmatrix +from ._base import _spbase, sparray, issparse +from ._index import IndexMixin +from ._sputils import (isdense, getdtype, isshape, isintlike, isscalarlike, + upcast, upcast_scalar, check_shape) + + +class _dok_base(_spbase, IndexMixin, dict): + _format = 'dok' + _allow_nd = (1, 2) + + def __init__(self, arg1, shape=None, dtype=None, copy=False, *, maxprint=None): + _spbase.__init__(self, arg1, maxprint=maxprint) + + if isinstance(arg1, tuple) and isshape(arg1, allow_nd=self._allow_nd): + self._shape = check_shape(arg1, allow_nd=self._allow_nd) + self._dict = {} + self.dtype = getdtype(dtype, default=float) + elif issparse(arg1): # Sparse ctor + if arg1.format == self.format: + arg1 = arg1.copy() if copy else arg1 + else: + arg1 = arg1.todok() + + if dtype is not None: + arg1 = arg1.astype(dtype, copy=False) + + self._dict = arg1._dict + self._shape = check_shape(arg1.shape, allow_nd=self._allow_nd) + self.dtype = getdtype(arg1.dtype) + else: # Dense ctor + try: + arg1 = np.asarray(arg1) + except Exception as e: + raise TypeError('Invalid input format.') from e + + if arg1.ndim > 2: + raise ValueError(f"DOK arrays don't yet support {arg1.ndim}D input.") + + if arg1.ndim == 1: + if dtype is not None: + arg1 = arg1.astype(dtype) + self._dict = {i: v for i, v in enumerate(arg1) if v != 0} + self.dtype = getdtype(arg1.dtype) + else: + d = self._coo_container(arg1, shape=shape, dtype=dtype).todok() + self._dict = d._dict + self.dtype = getdtype(d.dtype) + self._shape = check_shape(arg1.shape, allow_nd=self._allow_nd) + + def update(self, val): + # Prevent direct usage of update + raise NotImplementedError("Direct update to DOK sparse format is not allowed.") + + def _getnnz(self, axis=None): + if axis is not None: + raise NotImplementedError( + "_getnnz over an axis is not implemented for DOK format." + ) + return len(self._dict) + + def count_nonzero(self, axis=None): + if axis is not None: + raise NotImplementedError( + "count_nonzero over an axis is not implemented for DOK format." + ) + return sum(x != 0 for x in self.values()) + + _getnnz.__doc__ = _spbase._getnnz.__doc__ + count_nonzero.__doc__ = _spbase.count_nonzero.__doc__ + + def __len__(self): + return len(self._dict) + + def __contains__(self, key): + return key in self._dict + + def setdefault(self, key, default=None, /): + return self._dict.setdefault(key, default) + + def __delitem__(self, key, /): + del self._dict[key] + + def clear(self): + return self._dict.clear() + + def pop(self, /, *args): + return self._dict.pop(*args) + + def __reversed__(self): + raise TypeError("reversed is not defined for dok_array type") + + def __or__(self, other): + type_names = f"{type(self).__name__} and {type(other).__name__}" + raise TypeError(f"unsupported operand type for |: {type_names}") + + def __ror__(self, other): + type_names = f"{type(self).__name__} and {type(other).__name__}" + raise TypeError(f"unsupported operand type for |: {type_names}") + + def __ior__(self, other): + type_names = f"{type(self).__name__} and {type(other).__name__}" + raise TypeError(f"unsupported operand type for |: {type_names}") + + def popitem(self): + return self._dict.popitem() + + def items(self): + return self._dict.items() + + def keys(self): + return self._dict.keys() + + def values(self): + return self._dict.values() + + def get(self, key, default=0.0): + """This provides dict.get method functionality with type checking""" + if key in self._dict: + return self._dict[key] + if isintlike(key) and self.ndim == 1: + key = (key,) + if self.ndim != len(key): + raise IndexError(f'Index {key} length needs to match self.shape') + try: + for i in key: + assert isintlike(i) + except (AssertionError, TypeError, ValueError) as e: + raise IndexError('Index must be or consist of integers.') from e + key = tuple(i + M if i < 0 else i for i, M in zip(key, self.shape)) + if any(i < 0 or i >= M for i, M in zip(key, self.shape)): + raise IndexError('Index out of bounds.') + if self.ndim == 1: + key = key[0] + return self._dict.get(key, default) + + # 1D get methods + def _get_int(self, idx): + return self._dict.get(idx, self.dtype.type(0)) + + def _get_slice(self, idx): + i_range = range(*idx.indices(self.shape[0])) + return self._get_array(list(i_range)) + + def _get_array(self, idx): + idx = np.asarray(idx) + if idx.ndim == 0: + val = self._dict.get(int(idx), self.dtype.type(0)) + return np.array(val, stype=self.dtype) + new_dok = self._dok_container(idx.shape, dtype=self.dtype) + dok_vals = [self._dict.get(i, 0) for i in idx.ravel()] + if dok_vals: + if len(idx.shape) == 1: + for i, v in enumerate(dok_vals): + if v: + new_dok._dict[i] = v + else: + new_idx = np.unravel_index(np.arange(len(dok_vals)), idx.shape) + new_idx = new_idx[0] if len(new_idx) == 1 else zip(*new_idx) + for i, v in zip(new_idx, dok_vals, strict=True): + if v: + new_dok._dict[i] = v + return new_dok + + # 2D get methods + def _get_intXint(self, row, col): + return self._dict.get((row, col), self.dtype.type(0)) + + def _get_intXslice(self, row, col): + return self._get_sliceXslice(slice(row, row + 1), col) + + def _get_sliceXint(self, row, col): + return self._get_sliceXslice(row, slice(col, col + 1)) + + def _get_sliceXslice(self, row, col): + row_start, row_stop, row_step = row.indices(self.shape[0]) + col_start, col_stop, col_step = col.indices(self.shape[1]) + row_range = range(row_start, row_stop, row_step) + col_range = range(col_start, col_stop, col_step) + shape = (len(row_range), len(col_range)) + # Switch paths only when advantageous + # (count the iterations in the loops, adjust for complexity) + if len(self) >= 2 * shape[0] * shape[1]: + # O(nr*nc) path: loop over + return self._get_columnXarray(row_range, col_range) + # O(nnz) path: loop over entries of self + newdok = self._dok_container(shape, dtype=self.dtype) + for key in self.keys(): + i, ri = divmod(int(key[0]) - row_start, row_step) + if ri != 0 or i < 0 or i >= shape[0]: + continue + j, rj = divmod(int(key[1]) - col_start, col_step) + if rj != 0 or j < 0 or j >= shape[1]: + continue + newdok._dict[i, j] = self._dict[key] + return newdok + + def _get_intXarray(self, row, col): + return self._get_columnXarray([row], col.ravel()) + + def _get_arrayXint(self, row, col): + res = self._get_columnXarray(row.ravel(), [col]) + if row.ndim > 1: + return res.reshape(row.shape) + return res + + def _get_sliceXarray(self, row, col): + row = list(range(*row.indices(self.shape[0]))) + return self._get_columnXarray(row, col) + + def _get_arrayXslice(self, row, col): + col = list(range(*col.indices(self.shape[1]))) + return self._get_columnXarray(row, col) + + def _get_columnXarray(self, row, col): + # outer indexing + newdok = self._dok_container((len(row), len(col)), dtype=self.dtype) + + for i, r in enumerate(row): + for j, c in enumerate(col): + v = self._dict.get((r, c), 0) + if v: + newdok._dict[i, j] = v + return newdok + + def _get_arrayXarray(self, row, col): + # inner indexing + i, j = map(np.atleast_2d, np.broadcast_arrays(row, col)) + newdok = self._dok_container(i.shape, dtype=self.dtype) + + for key in itertools.product(range(i.shape[0]), range(i.shape[1])): + v = self._dict.get((i[key], j[key]), 0) + if v: + newdok._dict[key] = v + return newdok + + # 1D set methods + def _set_int(self, idx, x): + if x: + self._dict[idx] = x + elif idx in self._dict: + del self._dict[idx] + + def _set_array(self, idx, x): + idx_set = idx.ravel() + x_set = x.ravel() + if len(idx_set) != len(x_set): + if len(x_set) == 1: + x_set = np.full(len(idx_set), x_set[0], dtype=self.dtype) + else: + raise ValueError("Need len(index)==len(data) or len(data)==1") + for i, v in zip(idx_set, x_set): + if v: + self._dict[i] = v + elif i in self._dict: + del self._dict[i] + + # 2D set methods + def _set_intXint(self, row, col, x): + key = (row, col) + if x: + self._dict[key] = x + elif key in self._dict: + del self._dict[key] + + def _set_arrayXarray(self, row, col, x): + row = list(map(int, row.ravel())) + col = list(map(int, col.ravel())) + x = x.ravel() + self._dict.update(zip(zip(row, col), x)) + + for i in np.nonzero(x == 0)[0]: + key = (row[i], col[i]) + if self._dict[key] == 0: + # may have been superseded by later update + del self._dict[key] + + def __add__(self, other): + if isscalarlike(other): + res_dtype = upcast_scalar(self.dtype, other) + new = self._dok_container(self.shape, dtype=res_dtype) + # Add this scalar to each element. + for key in itertools.product(*[range(d) for d in self.shape]): + aij = self._dict.get(key, 0) + other + if aij: + new[key] = aij + elif issparse(other): + if other.shape != self.shape: + raise ValueError("Matrix dimensions are not equal.") + res_dtype = upcast(self.dtype, other.dtype) + new = self._dok_container(self.shape, dtype=res_dtype) + new._dict = self._dict.copy() + if other.format == "dok": + o_items = other.items() + else: + other = other.tocoo() + if self.ndim == 1: + o_items = zip(other.coords[0], other.data) + else: + o_items = zip(zip(*other.coords), other.data) + with np.errstate(over='ignore'): + new._dict.update((k, new[k] + v) for k, v in o_items) + elif isdense(other): + new = self.todense() + other + else: + return NotImplemented + return new + + def __radd__(self, other): + return self + other # addition is commutative + + def __neg__(self): + if self.dtype.kind == 'b': + raise NotImplementedError( + 'Negating a sparse boolean matrix is not supported.' + ) + new = self._dok_container(self.shape, dtype=self.dtype) + new._dict.update((k, -v) for k, v in self.items()) + return new + + def _mul_scalar(self, other): + res_dtype = upcast_scalar(self.dtype, other) + # Multiply this scalar by every element. + new = self._dok_container(self.shape, dtype=res_dtype) + new._dict.update(((k, v * other) for k, v in self.items())) + return new + + def _matmul_vector(self, other): + res_dtype = upcast(self.dtype, other.dtype) + + # vector @ vector + if self.ndim == 1: + if issparse(other): + if other.format == "dok": + keys = self.keys() & other.keys() + else: + keys = self.keys() & other.tocoo().coords[0] + return res_dtype(sum(self._dict[k] * other._dict[k] for k in keys)) + elif isdense(other): + return res_dtype(sum(other[k] * v for k, v in self.items())) + else: + return NotImplemented + + # matrix @ vector + result = np.zeros(self.shape[0], dtype=res_dtype) + for (i, j), v in self.items(): + result[i] += v * other[j] + return result + + def _matmul_multivector(self, other): + result_dtype = upcast(self.dtype, other.dtype) + # vector @ multivector + if self.ndim == 1: + # works for other 1d or 2d + return sum(v * other[j] for j, v in self._dict.items()) + + # matrix @ multivector + M = self.shape[0] + new_shape = (M,) if other.ndim == 1 else (M, other.shape[1]) + result = np.zeros(new_shape, dtype=result_dtype) + for (i, j), v in self.items(): + result[i] += v * other[j] + return result + + def __imul__(self, other): + if isscalarlike(other): + self._dict.update((k, v * other) for k, v in self.items()) + return self + return NotImplemented + + def __truediv__(self, other): + if isscalarlike(other): + res_dtype = upcast_scalar(self.dtype, other) + new = self._dok_container(self.shape, dtype=res_dtype) + new._dict.update(((k, v / other) for k, v in self.items())) + return new + return self.tocsr() / other + + def __itruediv__(self, other): + if isscalarlike(other): + self._dict.update((k, v / other) for k, v in self.items()) + return self + return NotImplemented + + def __reduce__(self): + # this approach is necessary because __setstate__ is called after + # __setitem__ upon unpickling and since __init__ is not called there + # is no shape attribute hence it is not possible to unpickle it. + return dict.__reduce__(self) + + def diagonal(self, k=0): + if self.ndim == 2: + return super().diagonal(k) + raise ValueError("diagonal requires two dimensions") + + def transpose(self, axes=None, copy=False): + if self.ndim == 1: + return self.copy() + + if axes is not None and axes != (1, 0): + raise ValueError( + "Sparse arrays/matrices do not support " + "an 'axes' parameter because swapping " + "dimensions is the only logical permutation." + ) + + M, N = self.shape + new = self._dok_container((N, M), dtype=self.dtype, copy=copy) + new._dict.update((((right, left), val) for (left, right), val in self.items())) + return new + + transpose.__doc__ = _spbase.transpose.__doc__ + + def conjtransp(self): + """DEPRECATED: Return the conjugate transpose. + + .. deprecated:: 1.14.0 + + `conjtransp` is deprecated and will be removed in v1.16.0. + Use ``.T.conj()`` instead. + """ + msg = ("`conjtransp` is deprecated and will be removed in v1.16.0. " + "Use `.T.conj()` instead.") + warn(msg, DeprecationWarning, stacklevel=2) + + if self.ndim == 1: + new = self.tocoo() + new.data = new.data.conjugate() + return new + + M, N = self.shape + new = self._dok_container((N, M), dtype=self.dtype) + new._dict = {(right, left): np.conj(val) for (left, right), val in self.items()} + return new + + def copy(self): + new = self._dok_container(self.shape, dtype=self.dtype) + new._dict.update(self._dict) + return new + + copy.__doc__ = _spbase.copy.__doc__ + + @classmethod + def fromkeys(cls, iterable, value=1, /): + tmp = dict.fromkeys(iterable, value) + if isinstance(next(iter(tmp)), tuple): + shape = tuple(max(idx) + 1 for idx in zip(*tmp)) + else: + shape = (max(tmp) + 1,) + result = cls(shape, dtype=type(value)) + result._dict = tmp + return result + + def tocoo(self, copy=False): + nnz = self.nnz + if nnz == 0: + return self._coo_container(self.shape, dtype=self.dtype) + + idx_dtype = self._get_index_dtype(maxval=max(self.shape)) + data = np.fromiter(self.values(), dtype=self.dtype, count=nnz) + # handle 1d keys specially b/c not a tuple + inds = zip(*self.keys()) if self.ndim > 1 else (self.keys(),) + coords = tuple(np.fromiter(ix, dtype=idx_dtype, count=nnz) for ix in inds) + A = self._coo_container((data, coords), shape=self.shape, dtype=self.dtype) + A.has_canonical_format = True + return A + + tocoo.__doc__ = _spbase.tocoo.__doc__ + + def todok(self, copy=False): + if copy: + return self.copy() + return self + + todok.__doc__ = _spbase.todok.__doc__ + + def tocsc(self, copy=False): + if self.ndim == 1: + raise NotImplementedError("tocsr() not valid for 1d sparse array") + return self.tocoo(copy=False).tocsc(copy=copy) + + tocsc.__doc__ = _spbase.tocsc.__doc__ + + def resize(self, *shape): + shape = check_shape(shape, allow_nd=self._allow_nd) + if len(shape) != len(self.shape): + # TODO implement resize across dimensions + raise NotImplementedError + + if self.ndim == 1: + newN = shape[-1] + for i in list(self._dict): + if i >= newN: + del self._dict[i] + self._shape = shape + return + + newM, newN = shape + M, N = self.shape + if newM < M or newN < N: + # Remove all elements outside new dimensions + for i, j in list(self.keys()): + if i >= newM or j >= newN: + del self._dict[i, j] + self._shape = shape + + resize.__doc__ = _spbase.resize.__doc__ + + # Added for 1d to avoid `tocsr` from _base.py + def astype(self, dtype, casting='unsafe', copy=True): + dtype = np.dtype(dtype) + if self.dtype != dtype: + result = self._dok_container(self.shape, dtype=dtype) + data = np.array(list(self._dict.values()), dtype=dtype) + result._dict = dict(zip(self._dict, data)) + return result + elif copy: + return self.copy() + return self + + +def isspmatrix_dok(x): + """Is `x` of dok_array type? + + Parameters + ---------- + x + object to check for being a dok matrix + + Returns + ------- + bool + True if `x` is a dok matrix, False otherwise + + Examples + -------- + >>> from scipy.sparse import dok_array, dok_matrix, coo_matrix, isspmatrix_dok + >>> isspmatrix_dok(dok_matrix([[5]])) + True + >>> isspmatrix_dok(dok_array([[5]])) + False + >>> isspmatrix_dok(coo_matrix([[5]])) + False + """ + return isinstance(x, dok_matrix) + + +# This namespace class separates array from matrix with isinstance +class dok_array(_dok_base, sparray): + """ + Dictionary Of Keys based sparse array. + + This is an efficient structure for constructing sparse + arrays incrementally. + + This can be instantiated in several ways: + dok_array(D) + where D is a 2-D ndarray + + dok_array(S) + with another sparse array or matrix S (equivalent to S.todok()) + + dok_array((M,N), [dtype]) + create the array with initial shape (M,N) + dtype is optional, defaulting to dtype='d' + + Attributes + ---------- + dtype : dtype + Data type of the array + shape : 2-tuple + Shape of the array + ndim : int + Number of dimensions (this is always 2) + nnz + Number of nonzero elements + size + T + + Notes + ----- + + Sparse arrays can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + - Allows for efficient O(1) access of individual elements. + - Duplicates are not allowed. + - Can be efficiently converted to a coo_array once constructed. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import dok_array + >>> S = dok_array((5, 5), dtype=np.float32) + >>> for i in range(5): + ... for j in range(5): + ... S[i, j] = i + j # Update element + + """ + + +class dok_matrix(spmatrix, _dok_base): + """ + Dictionary Of Keys based sparse matrix. + + This is an efficient structure for constructing sparse + matrices incrementally. + + This can be instantiated in several ways: + dok_matrix(D) + where D is a 2-D ndarray + + dok_matrix(S) + with another sparse array or matrix S (equivalent to S.todok()) + + dok_matrix((M,N), [dtype]) + create the matrix with initial shape (M,N) + dtype is optional, defaulting to dtype='d' + + Attributes + ---------- + dtype : dtype + Data type of the matrix + shape : 2-tuple + Shape of the matrix + ndim : int + Number of dimensions (this is always 2) + nnz + Number of nonzero elements + size + T + + Notes + ----- + + Sparse matrices can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + - Allows for efficient O(1) access of individual elements. + - Duplicates are not allowed. + - Can be efficiently converted to a coo_matrix once constructed. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import dok_matrix + >>> S = dok_matrix((5, 5), dtype=np.float32) + >>> for i in range(5): + ... for j in range(5): + ... S[i, j] = i + j # Update element + + """ + + def set_shape(self, shape): + new_matrix = self.reshape(shape, copy=False).asformat(self.format) + self.__dict__ = new_matrix.__dict__ + + def get_shape(self): + """Get shape of a sparse matrix.""" + return self._shape + + shape = property(fget=get_shape, fset=set_shape) + + def __reversed__(self): + return self._dict.__reversed__() + + def __or__(self, other): + if isinstance(other, _dok_base): + return self._dict | other._dict + return self._dict | other + + def __ror__(self, other): + if isinstance(other, _dok_base): + return self._dict | other._dict + return self._dict | other + + def __ior__(self, other): + if isinstance(other, _dok_base): + self._dict |= other._dict + else: + self._dict |= other + return self diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_extract.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_extract.py new file mode 100644 index 0000000000000000000000000000000000000000..0ee1a88575926efa1d5a921edbd3d88696157dc2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_extract.py @@ -0,0 +1,178 @@ +"""Functions to extract parts of sparse matrices +""" + +__docformat__ = "restructuredtext en" + +__all__ = ['find', 'tril', 'triu'] + + +from ._coo import coo_matrix, coo_array +from ._base import sparray + + +def find(A): + """Return the indices and values of the nonzero elements of a matrix + + Parameters + ---------- + A : dense or sparse array or matrix + Matrix whose nonzero elements are desired. + + Returns + ------- + (I,J,V) : tuple of arrays + I,J, and V contain the row indices, column indices, and values + of the nonzero entries. + + + Examples + -------- + >>> from scipy.sparse import csr_array, find + >>> A = csr_array([[7.0, 8.0, 0],[0, 0, 9.0]]) + >>> find(A) + (array([0, 0, 1], dtype=int32), + array([0, 1, 2], dtype=int32), + array([ 7., 8., 9.])) + + """ + + A = coo_array(A, copy=True) + A.sum_duplicates() + # remove explicit zeros + nz_mask = A.data != 0 + return A.row[nz_mask], A.col[nz_mask], A.data[nz_mask] + + +def tril(A, k=0, format=None): + """Return the lower triangular portion of a sparse array or matrix + + Returns the elements on or below the k-th diagonal of A. + - k = 0 corresponds to the main diagonal + - k > 0 is above the main diagonal + - k < 0 is below the main diagonal + + Parameters + ---------- + A : dense or sparse array or matrix + Matrix whose lower trianglar portion is desired. + k : integer : optional + The top-most diagonal of the lower triangle. + format : string + Sparse format of the result, e.g. format="csr", etc. + + Returns + ------- + L : sparse matrix + Lower triangular portion of A in sparse format. + + See Also + -------- + triu : upper triangle in sparse format + + Examples + -------- + >>> from scipy.sparse import csr_array, tril + >>> A = csr_array([[1, 2, 0, 0, 3], [4, 5, 0, 6, 7], [0, 0, 8, 9, 0]], + ... dtype='int32') + >>> A.toarray() + array([[1, 2, 0, 0, 3], + [4, 5, 0, 6, 7], + [0, 0, 8, 9, 0]], dtype=int32) + >>> tril(A).toarray() + array([[1, 0, 0, 0, 0], + [4, 5, 0, 0, 0], + [0, 0, 8, 0, 0]], dtype=int32) + >>> tril(A).nnz + 4 + >>> tril(A, k=1).toarray() + array([[1, 2, 0, 0, 0], + [4, 5, 0, 0, 0], + [0, 0, 8, 9, 0]], dtype=int32) + >>> tril(A, k=-1).toarray() + array([[0, 0, 0, 0, 0], + [4, 0, 0, 0, 0], + [0, 0, 0, 0, 0]], dtype=int32) + >>> tril(A, format='csc') + + + """ + coo_sparse = coo_array if isinstance(A, sparray) else coo_matrix + + # convert to COOrdinate format where things are easy + A = coo_sparse(A, copy=False) + mask = A.row + k >= A.col + + row = A.row[mask] + col = A.col[mask] + data = A.data[mask] + new_coo = coo_sparse((data, (row, col)), shape=A.shape, dtype=A.dtype) + return new_coo.asformat(format) + + +def triu(A, k=0, format=None): + """Return the upper triangular portion of a sparse array or matrix + + Returns the elements on or above the k-th diagonal of A. + - k = 0 corresponds to the main diagonal + - k > 0 is above the main diagonal + - k < 0 is below the main diagonal + + Parameters + ---------- + A : dense or sparse array or matrix + Matrix whose upper trianglar portion is desired. + k : integer : optional + The bottom-most diagonal of the upper triangle. + format : string + Sparse format of the result, e.g. format="csr", etc. + + Returns + ------- + L : sparse array or matrix + Upper triangular portion of A in sparse format. + Sparse array if A is a sparse array, otherwise matrix. + + See Also + -------- + tril : lower triangle in sparse format + + Examples + -------- + >>> from scipy.sparse import csr_array, triu + >>> A = csr_array([[1, 2, 0, 0, 3], [4, 5, 0, 6, 7], [0, 0, 8, 9, 0]], + ... dtype='int32') + >>> A.toarray() + array([[1, 2, 0, 0, 3], + [4, 5, 0, 6, 7], + [0, 0, 8, 9, 0]], dtype=int32) + >>> triu(A).toarray() + array([[1, 2, 0, 0, 3], + [0, 5, 0, 6, 7], + [0, 0, 8, 9, 0]], dtype=int32) + >>> triu(A).nnz + 8 + >>> triu(A, k=1).toarray() + array([[0, 2, 0, 0, 3], + [0, 0, 0, 6, 7], + [0, 0, 0, 9, 0]], dtype=int32) + >>> triu(A, k=-1).toarray() + array([[1, 2, 0, 0, 3], + [4, 5, 0, 6, 7], + [0, 0, 8, 9, 0]], dtype=int32) + >>> triu(A, format='csc') + + + """ + coo_sparse = coo_array if isinstance(A, sparray) else coo_matrix + + # convert to COOrdinate format where things are easy + A = coo_sparse(A, copy=False) + mask = A.row + k <= A.col + + row = A.row[mask] + col = A.col[mask] + data = A.data[mask] + new_coo = coo_sparse((data, (row, col)), shape=A.shape, dtype=A.dtype) + return new_coo.asformat(format) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_index.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_index.py new file mode 100644 index 0000000000000000000000000000000000000000..451df8b242c73b35036878feaf1f7ad302dc1f91 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_index.py @@ -0,0 +1,444 @@ +"""Indexing mixin for sparse array/matrix classes. +""" +import numpy as np +from ._sputils import isintlike +from ._base import sparray, issparse + +INT_TYPES = (int, np.integer) + + +def _broadcast_arrays(a, b): + """ + Same as np.broadcast_arrays(a, b) but old writeability rules. + + NumPy >= 1.17.0 transitions broadcast_arrays to return + read-only arrays. Set writeability explicitly to avoid warnings. + Retain the old writeability rules, as our Cython code assumes + the old behavior. + """ + x, y = np.broadcast_arrays(a, b) + x.flags.writeable = a.flags.writeable + y.flags.writeable = b.flags.writeable + return x, y + + +class IndexMixin: + """ + This class provides common dispatching and validation logic for indexing. + """ + def __getitem__(self, key): + index, new_shape = self._validate_indices(key) + + # 1D array + if len(index) == 1: + idx = index[0] + if isinstance(idx, np.ndarray): + if idx.shape == (): + idx = idx.item() + if isinstance(idx, INT_TYPES): + res = self._get_int(idx) + elif isinstance(idx, slice): + res = self._get_slice(idx) + else: # assume array idx + res = self._get_array(idx) + + # package the result and return + if not isinstance(self, sparray): + return res + # handle np.newaxis in idx when result would otherwise be a scalar + if res.shape == () and new_shape != (): + if len(new_shape) == 1: + return self.__class__([res], shape=new_shape, dtype=self.dtype) + if len(new_shape) == 2: + return self.__class__([[res]], shape=new_shape, dtype=self.dtype) + return res.reshape(new_shape) + + # 2D array + row, col = index + + # Dispatch to specialized methods. + if isinstance(row, INT_TYPES): + if isinstance(col, INT_TYPES): + res = self._get_intXint(row, col) + elif isinstance(col, slice): + res = self._get_intXslice(row, col) + elif col.ndim == 1: + res = self._get_intXarray(row, col) + elif col.ndim == 2: + res = self._get_intXarray(row, col) + else: + raise IndexError('index results in >2 dimensions') + elif isinstance(row, slice): + if isinstance(col, INT_TYPES): + res = self._get_sliceXint(row, col) + elif isinstance(col, slice): + if row == slice(None) and row == col: + res = self.copy() + else: + res = self._get_sliceXslice(row, col) + elif col.ndim == 1: + res = self._get_sliceXarray(row, col) + else: + raise IndexError('index results in >2 dimensions') + else: + if isinstance(col, INT_TYPES): + res = self._get_arrayXint(row, col) + elif isinstance(col, slice): + res = self._get_arrayXslice(row, col) + # arrayXarray preprocess + elif (row.ndim == 2 and row.shape[1] == 1 + and (col.ndim == 1 or col.shape[0] == 1)): + # outer indexing + res = self._get_columnXarray(row[:, 0], col.ravel()) + else: + # inner indexing + row, col = _broadcast_arrays(row, col) + if row.shape != col.shape: + raise IndexError('number of row and column indices differ') + if row.size == 0: + res = self.__class__(np.atleast_2d(row).shape, dtype=self.dtype) + else: + res = self._get_arrayXarray(row, col) + + # handle spmatrix (must be 2d, dont let 1d new_shape start reshape) + if not isinstance(self, sparray): + if new_shape == () or (len(new_shape) == 1 and res.ndim != 0): + # res handles cases not inflated by None + return res + if len(new_shape) == 1: + # shape inflated to 1D by None in index. Make 2D + new_shape = (1,) + new_shape + # reshape if needed (when None changes shape, e.g. A[1,:,None]) + return res if new_shape == res.shape else res.reshape(new_shape) + + # package the result and return + if res.shape != new_shape: + # handle formats that support indexing but not 1D (lil for now) + if self.format == "lil" and len(new_shape) != 2: + if res.shape == (): + return self._coo_container([res], shape = new_shape) + return res.tocoo().reshape(new_shape) + return res.reshape(new_shape) + return res + + def __setitem__(self, key, x): + index, _ = self._validate_indices(key) + + # 1D array + if len(index) == 1: + idx = index[0] + + if issparse(x): + x = x.toarray() + else: + x = np.asarray(x, dtype=self.dtype) + + if isinstance(idx, INT_TYPES): + if x.size != 1: + raise ValueError('Trying to assign a sequence to an item') + self._set_int(idx, x.flat[0]) + return + + if isinstance(idx, slice): + # check for simple case of slice that gives 1 item + # Note: Python `range` does not use lots of memory + idx_range = range(*idx.indices(self.shape[0])) + N = len(idx_range) + if N == 1 and x.size == 1: + self._set_int(idx_range[0], x.flat[0]) + return + idx = np.arange(*idx.indices(self.shape[0])) + idx_shape = idx.shape + else: + idx_shape = idx.squeeze().shape + # broadcast scalar to full 1d + if x.squeeze().shape != idx_shape: + x = np.broadcast_to(x, idx.shape) + if x.size != 0: + self._set_array(idx, x) + return + + # 2D array + row, col = index + + if isinstance(row, INT_TYPES) and isinstance(col, INT_TYPES): + x = np.asarray(x, dtype=self.dtype) + if x.size != 1: + raise ValueError('Trying to assign a sequence to an item') + self._set_intXint(row, col, x.flat[0]) + return + + if isinstance(row, slice): + row = np.arange(*row.indices(self.shape[0]))[:, None] + else: + row = np.atleast_1d(row) + + if isinstance(col, slice): + col = np.arange(*col.indices(self.shape[1]))[None, :] + if row.ndim == 1: + row = row[:, None] + else: + col = np.atleast_1d(col) + + i, j = _broadcast_arrays(row, col) + if i.shape != j.shape: + raise IndexError('number of row and column indices differ') + + if issparse(x): + if 0 in x.shape: + return + if i.ndim == 1: + # Inner indexing, so treat them like row vectors. + i = i[None] + j = j[None] + x = x.tocoo(copy=False).reshape(x._shape_as_2d, copy=True) + broadcast_row = x.shape[0] == 1 and i.shape[0] != 1 + broadcast_col = x.shape[1] == 1 and i.shape[1] != 1 + if not ((broadcast_row or x.shape[0] == i.shape[0]) and + (broadcast_col or x.shape[1] == i.shape[1])): + raise ValueError('shape mismatch in assignment') + x.sum_duplicates() + self._set_arrayXarray_sparse(i, j, x) + else: + # Make x and i into the same shape + x = np.asarray(x, dtype=self.dtype) + if x.squeeze().shape != i.squeeze().shape: + x = np.broadcast_to(x, i.shape) + if x.size == 0: + return + x = x.reshape(i.shape) + self._set_arrayXarray(i, j, x) + + def _validate_indices(self, key): + """Returns two tuples: (index tuple, requested shape tuple)""" + # single ellipsis + if key is Ellipsis: + return (slice(None),) * self.ndim, self.shape + + if not isinstance(key, tuple): + key = [key] + + ellps_pos = None + index_1st = [] + prelim_ndim = 0 + for i, idx in enumerate(key): + if idx is Ellipsis: + if ellps_pos is not None: + raise IndexError('an index can only have a single ellipsis') + ellps_pos = i + elif idx is None: + index_1st.append(idx) + elif isinstance(idx, slice) or isintlike(idx): + index_1st.append(idx) + prelim_ndim += 1 + elif (ix := _compatible_boolean_index(idx, self.ndim)) is not None: + index_1st.append(ix) + prelim_ndim += ix.ndim + elif issparse(idx): + # TODO: make sparse matrix indexing work for sparray + raise IndexError( + 'Indexing with sparse matrices is not supported ' + 'except boolean indexing where matrix and index ' + 'are equal shapes.') + else: # dense array + index_1st.append(np.asarray(idx)) + prelim_ndim += 1 + ellip_slices = (self.ndim - prelim_ndim) * [slice(None)] + if ellip_slices: + if ellps_pos is None: + index_1st.extend(ellip_slices) + else: + index_1st = index_1st[:ellps_pos] + ellip_slices + index_1st[ellps_pos:] + + # second pass (have processed ellipsis and preprocessed arrays) + idx_shape = [] + index_ndim = 0 + index = [] + array_indices = [] + for i, idx in enumerate(index_1st): + if idx is None: + idx_shape.append(1) + elif isinstance(idx, slice): + index.append(idx) + Ms = self._shape[index_ndim] + len_slice = len(range(*idx.indices(Ms))) + idx_shape.append(len_slice) + index_ndim += 1 + elif isintlike(idx): + N = self._shape[index_ndim] + if not (-N <= idx < N): + raise IndexError(f'index ({idx}) out of range') + idx = int(idx + N if idx < 0 else idx) + index.append(idx) + index_ndim += 1 + # bool array (checked in first pass) + elif idx.dtype.kind == 'b': + ix = idx + tmp_ndim = index_ndim + ix.ndim + mid_shape = self._shape[index_ndim:tmp_ndim] + if ix.shape != mid_shape: + raise IndexError( + f"bool index {i} has shape {mid_shape} instead of {ix.shape}" + ) + index.extend(ix.nonzero()) + array_indices.extend(range(index_ndim, tmp_ndim)) + index_ndim = tmp_ndim + else: # dense array + N = self._shape[index_ndim] + idx = self._asindices(idx, N) + index.append(idx) + array_indices.append(index_ndim) + index_ndim += 1 + if index_ndim > self.ndim: + raise IndexError( + f'invalid index ndim. Array is {self.ndim}D. Index needs {index_ndim}D' + ) + if len(array_indices) > 1: + idx_arrays = _broadcast_arrays(*(index[i] for i in array_indices)) + if any(idx_arrays[0].shape != ix.shape for ix in idx_arrays[1:]): + shapes = " ".join(str(ix.shape) for ix in idx_arrays) + msg = (f'shape mismatch: indexing arrays could not be broadcast ' + f'together with shapes {shapes}') + raise IndexError(msg) + # TODO: handle this for nD (adjacent arrays stay, separated move to start) + idx_shape = list(idx_arrays[0].shape) + idx_shape + elif len(array_indices) == 1: + arr_index = array_indices[0] + arr_shape = list(index[arr_index].shape) + idx_shape = idx_shape[:arr_index] + arr_shape + idx_shape[arr_index:] + if (ndim := len(idx_shape)) > 2: + raise IndexError(f'Only 1D or 2D arrays allowed. Index makes {ndim}D') + return tuple(index), tuple(idx_shape) + + def _asindices(self, idx, length): + """Convert `idx` to a valid index for an axis with a given length. + + Subclasses that need special validation can override this method. + """ + try: + x = np.asarray(idx) + except (ValueError, TypeError, MemoryError) as e: + raise IndexError('invalid index') from e + + if x.ndim not in (1, 2): + raise IndexError('Index dimension must be 1 or 2') + + if x.size == 0: + return x + + # Check bounds + max_indx = x.max() + if max_indx >= length: + raise IndexError('index (%d) out of range' % max_indx) + + min_indx = x.min() + if min_indx < 0: + if min_indx < -length: + raise IndexError('index (%d) out of range' % min_indx) + if x is idx or not x.flags.owndata: + x = x.copy() + x[x < 0] += length + return x + + def _getrow(self, i): + """Return a copy of row i of the matrix, as a (1 x n) row vector. + """ + M, N = self.shape + i = int(i) + if i < -M or i >= M: + raise IndexError('index (%d) out of range' % i) + if i < 0: + i += M + return self._get_intXslice(i, slice(None)) + + def _getcol(self, i): + """Return a copy of column i of the matrix, as a (m x 1) column vector. + """ + M, N = self.shape + i = int(i) + if i < -N or i >= N: + raise IndexError('index (%d) out of range' % i) + if i < 0: + i += N + return self._get_sliceXint(slice(None), i) + + def _get_int(self, idx): + raise NotImplementedError() + + def _get_slice(self, idx): + raise NotImplementedError() + + def _get_array(self, idx): + raise NotImplementedError() + + def _get_intXint(self, row, col): + raise NotImplementedError() + + def _get_intXarray(self, row, col): + raise NotImplementedError() + + def _get_intXslice(self, row, col): + raise NotImplementedError() + + def _get_sliceXint(self, row, col): + raise NotImplementedError() + + def _get_sliceXslice(self, row, col): + raise NotImplementedError() + + def _get_sliceXarray(self, row, col): + raise NotImplementedError() + + def _get_arrayXint(self, row, col): + raise NotImplementedError() + + def _get_arrayXslice(self, row, col): + raise NotImplementedError() + + def _get_columnXarray(self, row, col): + raise NotImplementedError() + + def _get_arrayXarray(self, row, col): + raise NotImplementedError() + + def _set_int(self, idx, x): + raise NotImplementedError() + + def _set_array(self, idx, x): + raise NotImplementedError() + + def _set_intXint(self, row, col, x): + raise NotImplementedError() + + def _set_arrayXarray(self, row, col, x): + raise NotImplementedError() + + def _set_arrayXarray_sparse(self, row, col, x): + # Fall back to densifying x + x = np.asarray(x.toarray(), dtype=self.dtype) + x, _ = _broadcast_arrays(x, row) + self._set_arrayXarray(row, col, x) + + +def _compatible_boolean_index(idx, desired_ndim): + """Check for boolean array or array-like. peek before asarray for array-like""" + # use attribute ndim to indicate a compatible array and check dtype + # if not, look at 1st element as quick rejection of bool, else slower asanyarray + if not hasattr(idx, 'ndim'): + # is first element boolean? + try: + ix = next(iter(idx), None) + for _ in range(desired_ndim): + if isinstance(ix, bool): + break + ix = next(iter(ix), None) + else: + return None + except TypeError: + return None + # since first is boolean, construct array and check all elements + idx = np.asanyarray(idx) + + if idx.dtype.kind == 'b': + return idx + return None diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_lil.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_lil.py new file mode 100644 index 0000000000000000000000000000000000000000..479472445cd8bdc36374b3d8bc18e6f9df123306 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_lil.py @@ -0,0 +1,632 @@ +"""List of Lists sparse matrix class +""" + +__docformat__ = "restructuredtext en" + +__all__ = ['lil_array', 'lil_matrix', 'isspmatrix_lil'] + +from bisect import bisect_left + +import numpy as np + +from ._matrix import spmatrix +from ._base import _spbase, sparray, issparse +from ._index import IndexMixin, INT_TYPES, _broadcast_arrays +from ._sputils import (getdtype, isshape, isscalarlike, upcast_scalar, + check_shape, check_reshape_kwargs) +from . import _csparsetools + + +class _lil_base(_spbase, IndexMixin): + _format = 'lil' + + def __init__(self, arg1, shape=None, dtype=None, copy=False, *, maxprint=None): + _spbase.__init__(self, arg1, maxprint=maxprint) + self.dtype = getdtype(dtype, arg1, default=float) + + # First get the shape + if issparse(arg1): + if arg1.format == "lil" and copy: + A = arg1.copy() + else: + A = arg1.tolil() + + if dtype is not None: + newdtype = getdtype(dtype) + A = A.astype(newdtype, copy=False) + + self._shape = check_shape(A.shape) + self.dtype = A.dtype + self.rows = A.rows + self.data = A.data + elif isinstance(arg1,tuple): + if isshape(arg1): + if shape is not None: + raise ValueError('invalid use of shape parameter') + M, N = arg1 + self._shape = check_shape((M, N)) + self.rows = np.empty((M,), dtype=object) + self.data = np.empty((M,), dtype=object) + for i in range(M): + self.rows[i] = [] + self.data[i] = [] + else: + raise TypeError('unrecognized lil_array constructor usage') + else: + # assume A is dense + try: + A = self._ascontainer(arg1) + except TypeError as e: + raise TypeError('unsupported matrix type') from e + if isinstance(self, sparray) and A.ndim != 2: + raise ValueError(f"LIL arrays don't support {A.ndim}D input. Use 2D") + A = self._csr_container(A, dtype=dtype).tolil() + + self._shape = check_shape(A.shape) + self.dtype = getdtype(A.dtype) + self.rows = A.rows + self.data = A.data + + def __iadd__(self,other): + self[:,:] = self + other + return self + + def __isub__(self,other): + self[:,:] = self - other + return self + + def __imul__(self,other): + if isscalarlike(other): + self[:,:] = self * other + return self + else: + return NotImplemented + + def __itruediv__(self,other): + if isscalarlike(other): + self[:,:] = self / other + return self + else: + return NotImplemented + + # Whenever the dimensions change, empty lists should be created for each + # row + + def _getnnz(self, axis=None): + if axis is None: + return sum([len(rowvals) for rowvals in self.data]) + if axis < 0: + axis += 2 + if axis == 0: + out = np.zeros(self.shape[1], dtype=np.intp) + for row in self.rows: + out[row] += 1 + return out + elif axis == 1: + return np.array([len(rowvals) for rowvals in self.data], dtype=np.intp) + else: + raise ValueError('axis out of bounds') + + _getnnz.__doc__ = _spbase._getnnz.__doc__ + + def count_nonzero(self, axis=None): + if axis is None: + return sum(np.count_nonzero(rowvals) for rowvals in self.data) + + if axis < 0: + axis += 2 + if axis == 0: + out = np.zeros(self.shape[1], dtype=np.intp) + for row, data in zip(self.rows, self.data): + mask = [c for c, d in zip(row, data) if d != 0] + out[mask] += 1 + return out + elif axis == 1: + return np.array( + [np.count_nonzero(rowvals) for rowvals in self.data], dtype=np.intp, + ) + else: + raise ValueError('axis out of bounds') + + count_nonzero.__doc__ = _spbase.count_nonzero.__doc__ + + def getrowview(self, i): + """Returns a view of the 'i'th row (without copying). + """ + new = self._lil_container((1, self.shape[1]), dtype=self.dtype) + new.rows[0] = self.rows[i] + new.data[0] = self.data[i] + return new + + def getrow(self, i): + """Returns a copy of the 'i'th row. + """ + M, N = self.shape + if i < 0: + i += M + if i < 0 or i >= M: + raise IndexError('row index out of bounds') + new = self._lil_container((1, N), dtype=self.dtype) + new.rows[0] = self.rows[i][:] + new.data[0] = self.data[i][:] + return new + + def __getitem__(self, key): + # Fast path for simple (int, int) indexing. + if (isinstance(key, tuple) and len(key) == 2 and + isinstance(key[0], INT_TYPES) and + isinstance(key[1], INT_TYPES)): + # lil_get1 handles validation for us. + return self._get_intXint(*key) + # Everything else takes the normal path. + return IndexMixin.__getitem__(self, key) + + def _asindices(self, idx, N): + # LIL routines handle bounds-checking for us, so don't do it here. + try: + x = np.asarray(idx) + except (ValueError, TypeError, MemoryError) as e: + raise IndexError('invalid index') from e + if x.ndim not in (1, 2): + raise IndexError('Index dimension must be <= 2') + return x + + def _get_intXint(self, row, col): + v = _csparsetools.lil_get1(self.shape[0], self.shape[1], self.rows, + self.data, row, col) + return self.dtype.type(v) + + def _get_sliceXint(self, row, col): + row = range(*row.indices(self.shape[0])) + return self._get_row_ranges(row, slice(col, col+1)) + + def _get_arrayXint(self, row, col): + res = self._get_row_ranges(row.ravel(), slice(col, col+1)) + if row.ndim > 1: + return res.reshape(row.shape) + return res + + def _get_intXslice(self, row, col): + return self._get_row_ranges((row,), col) + + def _get_sliceXslice(self, row, col): + row = range(*row.indices(self.shape[0])) + return self._get_row_ranges(row, col) + + def _get_arrayXslice(self, row, col): + return self._get_row_ranges(row, col) + + def _get_intXarray(self, row, col): + row = np.array(row, dtype=col.dtype, ndmin=1) + return self._get_columnXarray(row, col) + + def _get_sliceXarray(self, row, col): + row = np.arange(*row.indices(self.shape[0])) + return self._get_columnXarray(row, col) + + def _get_columnXarray(self, row, col): + # outer indexing + row, col = _broadcast_arrays(row[:,None], col) + return self._get_arrayXarray(row, col) + + def _get_arrayXarray(self, row, col): + # inner indexing + i, j = map(np.atleast_2d, _prepare_index_for_memoryview(row, col)) + new = self._lil_container(i.shape, dtype=self.dtype) + _csparsetools.lil_fancy_get(self.shape[0], self.shape[1], + self.rows, self.data, + new.rows, new.data, + i, j) + return new + + def _get_row_ranges(self, rows, col_slice): + """ + Fast path for indexing in the case where column index is slice. + + This gains performance improvement over brute force by more + efficient skipping of zeros, by accessing the elements + column-wise in order. + + Parameters + ---------- + rows : sequence or range + Rows indexed. If range, must be within valid bounds. + col_slice : slice + Columns indexed + + """ + j_start, j_stop, j_stride = col_slice.indices(self.shape[1]) + col_range = range(j_start, j_stop, j_stride) + nj = len(col_range) + new = self._lil_container((len(rows), nj), dtype=self.dtype) + + _csparsetools.lil_get_row_ranges(self.shape[0], self.shape[1], + self.rows, self.data, + new.rows, new.data, + rows, + j_start, j_stop, j_stride, nj) + + return new + + def _set_intXint(self, row, col, x): + _csparsetools.lil_insert(self.shape[0], self.shape[1], self.rows, + self.data, row, col, x) + + def _set_arrayXarray(self, row, col, x): + i, j, x = map(np.atleast_2d, _prepare_index_for_memoryview(row, col, x)) + _csparsetools.lil_fancy_set(self.shape[0], self.shape[1], + self.rows, self.data, + i, j, x) + + def _set_arrayXarray_sparse(self, row, col, x): + # Fall back to densifying x + x = np.asarray(x.toarray(), dtype=self.dtype) + x, _ = _broadcast_arrays(x, row) + self._set_arrayXarray(row, col, x) + + def __setitem__(self, key, x): + if isinstance(key, tuple) and len(key) == 2: + row, col = key + # Fast path for simple (int, int) indexing. + if isinstance(row, INT_TYPES) and isinstance(col, INT_TYPES): + x = self.dtype.type(x) + if x.size > 1: + raise ValueError("Trying to assign a sequence to an item") + return self._set_intXint(row, col, x) + # Fast path for full-matrix sparse assignment. + if (isinstance(row, slice) and isinstance(col, slice) and + row == slice(None) and col == slice(None) and + issparse(x) and x.shape == self.shape): + x = self._lil_container(x, dtype=self.dtype) + self.rows = x.rows + self.data = x.data + return + # Everything else takes the normal path. + IndexMixin.__setitem__(self, key, x) + + def _mul_scalar(self, other): + if other == 0: + # Multiply by zero: return the zero matrix + new = self._lil_container(self.shape, dtype=self.dtype) + else: + res_dtype = upcast_scalar(self.dtype, other) + + new = self.copy() + new = new.astype(res_dtype) + # Multiply this scalar by every element. + for j, rowvals in enumerate(new.data): + new.data[j] = [val*other for val in rowvals] + return new + + def __truediv__(self, other): # self / other + if isscalarlike(other): + new = self.copy() + new.dtype = np.result_type(self, other) + # Divide every element by this scalar + for j, rowvals in enumerate(new.data): + new.data[j] = [val/other for val in rowvals] + return new + else: + return self.tocsr() / other + + def copy(self): + M, N = self.shape + new = self._lil_container(self.shape, dtype=self.dtype) + # This is ~14x faster than calling deepcopy() on rows and data. + _csparsetools.lil_get_row_ranges(M, N, self.rows, self.data, + new.rows, new.data, range(M), + 0, N, 1, N) + return new + + copy.__doc__ = _spbase.copy.__doc__ + + def reshape(self, *args, **kwargs): + shape = check_shape(args, self.shape) + order, copy = check_reshape_kwargs(kwargs) + + # Return early if reshape is not required + if shape == self.shape: + if copy: + return self.copy() + else: + return self + + new = self._lil_container(shape, dtype=self.dtype) + + if order == 'C': + ncols = self.shape[1] + for i, row in enumerate(self.rows): + for col, j in enumerate(row): + new_r, new_c = np.unravel_index(i * ncols + j, shape) + new[new_r, new_c] = self[i, j] + elif order == 'F': + nrows = self.shape[0] + for i, row in enumerate(self.rows): + for col, j in enumerate(row): + new_r, new_c = np.unravel_index(i + j * nrows, shape, order) + new[new_r, new_c] = self[i, j] + else: + raise ValueError("'order' must be 'C' or 'F'") + + return new + + reshape.__doc__ = _spbase.reshape.__doc__ + + def resize(self, *shape): + shape = check_shape(shape) + new_M, new_N = shape + M, N = self.shape + + if new_M < M: + self.rows = self.rows[:new_M] + self.data = self.data[:new_M] + elif new_M > M: + self.rows = np.resize(self.rows, new_M) + self.data = np.resize(self.data, new_M) + for i in range(M, new_M): + self.rows[i] = [] + self.data[i] = [] + + if new_N < N: + for row, data in zip(self.rows, self.data): + trunc = bisect_left(row, new_N) + del row[trunc:] + del data[trunc:] + + self._shape = shape + + resize.__doc__ = _spbase.resize.__doc__ + + def toarray(self, order=None, out=None): + d = self._process_toarray_args(order, out) + for i, row in enumerate(self.rows): + for pos, j in enumerate(row): + d[i, j] = self.data[i][pos] + return d + + toarray.__doc__ = _spbase.toarray.__doc__ + + def transpose(self, axes=None, copy=False): + return self.tocsr(copy=copy).transpose(axes=axes, copy=False).tolil(copy=False) + + transpose.__doc__ = _spbase.transpose.__doc__ + + def tolil(self, copy=False): + if copy: + return self.copy() + else: + return self + + tolil.__doc__ = _spbase.tolil.__doc__ + + def tocsr(self, copy=False): + M, N = self.shape + if M == 0 or N == 0: + return self._csr_container((M, N), dtype=self.dtype) + + # construct indptr array + if M*N <= np.iinfo(np.int32).max: + # fast path: it is known that 64-bit indexing will not be needed. + idx_dtype = np.int32 + indptr = np.empty(M + 1, dtype=idx_dtype) + indptr[0] = 0 + _csparsetools.lil_get_lengths(self.rows, indptr[1:]) + np.cumsum(indptr, out=indptr) + nnz = indptr[-1] + else: + idx_dtype = self._get_index_dtype(maxval=N) + lengths = np.empty(M, dtype=idx_dtype) + _csparsetools.lil_get_lengths(self.rows, lengths) + nnz = lengths.sum(dtype=np.int64) + idx_dtype = self._get_index_dtype(maxval=max(N, nnz)) + indptr = np.empty(M + 1, dtype=idx_dtype) + indptr[0] = 0 + np.cumsum(lengths, dtype=idx_dtype, out=indptr[1:]) + + indices = np.empty(nnz, dtype=idx_dtype) + data = np.empty(nnz, dtype=self.dtype) + _csparsetools.lil_flatten_to_array(self.rows, indices) + _csparsetools.lil_flatten_to_array(self.data, data) + + # init csr matrix + return self._csr_container((data, indices, indptr), shape=self.shape) + + tocsr.__doc__ = _spbase.tocsr.__doc__ + + +def _prepare_index_for_memoryview(i, j, x=None): + """ + Convert index and data arrays to form suitable for passing to the + Cython fancy getset routines. + + The conversions are necessary since to (i) ensure the integer + index arrays are in one of the accepted types, and (ii) to ensure + the arrays are writable so that Cython memoryview support doesn't + choke on them. + + Parameters + ---------- + i, j + Index arrays + x : optional + Data arrays + + Returns + ------- + i, j, x + Re-formatted arrays (x is omitted, if input was None) + + """ + if i.dtype > j.dtype: + j = j.astype(i.dtype) + elif i.dtype < j.dtype: + i = i.astype(j.dtype) + + if not i.flags.writeable or i.dtype not in (np.int32, np.int64): + i = i.astype(np.intp) + if not j.flags.writeable or j.dtype not in (np.int32, np.int64): + j = j.astype(np.intp) + + if x is not None: + if not x.flags.writeable: + x = x.copy() + return i, j, x + else: + return i, j + + +def isspmatrix_lil(x): + """Is `x` of lil_matrix type? + + Parameters + ---------- + x + object to check for being a lil matrix + + Returns + ------- + bool + True if `x` is a lil matrix, False otherwise + + Examples + -------- + >>> from scipy.sparse import lil_array, lil_matrix, coo_matrix, isspmatrix_lil + >>> isspmatrix_lil(lil_matrix([[5]])) + True + >>> isspmatrix_lil(lil_array([[5]])) + False + >>> isspmatrix_lil(coo_matrix([[5]])) + False + """ + return isinstance(x, lil_matrix) + + +# This namespace class separates array from matrix with isinstance +class lil_array(_lil_base, sparray): + """ + Row-based LIst of Lists sparse array. + + This is a structure for constructing sparse arrays incrementally. + Note that inserting a single item can take linear time in the worst case; + to construct the array efficiently, make sure the items are pre-sorted by + index, per row. + + This can be instantiated in several ways: + lil_array(D) + where D is a 2-D ndarray + + lil_array(S) + with another sparse array or matrix S (equivalent to S.tolil()) + + lil_array((M, N), [dtype]) + to construct an empty array with shape (M, N) + dtype is optional, defaulting to dtype='d'. + + Attributes + ---------- + dtype : dtype + Data type of the array + shape : 2-tuple + Shape of the array + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + LIL format data array of the array + rows + LIL format row index array of the array + T + + Notes + ----- + Sparse arrays can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + Advantages of the LIL format + - supports flexible slicing + - changes to the array sparsity structure are efficient + + Disadvantages of the LIL format + - arithmetic operations LIL + LIL are slow (consider CSR or CSC) + - slow column slicing (consider CSC) + - slow matrix vector products (consider CSR or CSC) + + Intended Usage + - LIL is a convenient format for constructing sparse arrays + - once an array has been constructed, convert to CSR or + CSC format for fast arithmetic and matrix vector operations + - consider using the COO format when constructing large arrays + + Data Structure + - An array (``self.rows``) of rows, each of which is a sorted + list of column indices of non-zero elements. + - The corresponding nonzero values are stored in similar + fashion in ``self.data``. + + """ + + +class lil_matrix(spmatrix, _lil_base): + """ + Row-based LIst of Lists sparse matrix. + + This is a structure for constructing sparse matrices incrementally. + Note that inserting a single item can take linear time in the worst case; + to construct the matrix efficiently, make sure the items are pre-sorted by + index, per row. + + This can be instantiated in several ways: + lil_matrix(D) + where D is a 2-D ndarray + + lil_matrix(S) + with another sparse array or matrix S (equivalent to S.tolil()) + + lil_matrix((M, N), [dtype]) + to construct an empty matrix with shape (M, N) + dtype is optional, defaulting to dtype='d'. + + Attributes + ---------- + dtype : dtype + Data type of the matrix + shape : 2-tuple + Shape of the matrix + ndim : int + Number of dimensions (this is always 2) + nnz + size + data + LIL format data array of the matrix + rows + LIL format row index array of the matrix + T + + Notes + ----- + Sparse matrices can be used in arithmetic operations: they support + addition, subtraction, multiplication, division, and matrix power. + + Advantages of the LIL format + - supports flexible slicing + - changes to the matrix sparsity structure are efficient + + Disadvantages of the LIL format + - arithmetic operations LIL + LIL are slow (consider CSR or CSC) + - slow column slicing (consider CSC) + - slow matrix vector products (consider CSR or CSC) + + Intended Usage + - LIL is a convenient format for constructing sparse matrices + - once a matrix has been constructed, convert to CSR or + CSC format for fast arithmetic and matrix vector operations + - consider using the COO format when constructing large matrices + + Data Structure + - An array (``self.rows``) of rows, each of which is a sorted + list of column indices of non-zero elements. + - The corresponding nonzero values are stored in similar + fashion in ``self.data``. + + """ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_matrix.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_matrix.py new file mode 100644 index 0000000000000000000000000000000000000000..351660ba389ea7f2adf2576e350e840783d894fa --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_matrix.py @@ -0,0 +1,146 @@ +class spmatrix: + """This class provides a base class for all sparse matrix classes. + + It cannot be instantiated. Most of the work is provided by subclasses. + """ + _allow_nd = (2,) + + @property + def _bsr_container(self): + from ._bsr import bsr_matrix + return bsr_matrix + + @property + def _coo_container(self): + from ._coo import coo_matrix + return coo_matrix + + @property + def _csc_container(self): + from ._csc import csc_matrix + return csc_matrix + + @property + def _csr_container(self): + from ._csr import csr_matrix + return csr_matrix + + @property + def _dia_container(self): + from ._dia import dia_matrix + return dia_matrix + + @property + def _dok_container(self): + from ._dok import dok_matrix + return dok_matrix + + @property + def _lil_container(self): + from ._lil import lil_matrix + return lil_matrix + + # Restore matrix multiplication + def __mul__(self, other): + return self._matmul_dispatch(other) + + def __rmul__(self, other): + return self._rmatmul_dispatch(other) + + # Restore matrix power + def __pow__(self, power): + from .linalg import matrix_power + + return matrix_power(self, power) + + ## Backward compatibility + + def set_shape(self, shape): + """Set the shape of the matrix in-place""" + # Make sure copy is False since this is in place + # Make sure format is unchanged because we are doing a __dict__ swap + new_self = self.reshape(shape, copy=False).asformat(self.format) + self.__dict__ = new_self.__dict__ + + def get_shape(self): + """Get the shape of the matrix""" + return self._shape + + shape = property(fget=get_shape, fset=set_shape, + doc="Shape of the matrix") + + def asfptype(self): + """Upcast matrix to a floating point format (if necessary)""" + return self._asfptype() + + def getmaxprint(self): + """Maximum number of elements to display when printed.""" + return self._getmaxprint() + + def getformat(self): + """Matrix storage format""" + return self.format + + def getnnz(self, axis=None): + """Number of stored values, including explicit zeros. + + Parameters + ---------- + axis : None, 0, or 1 + Select between the number of values across the whole array, in + each column, or in each row. + """ + return self._getnnz(axis=axis) + + def getH(self): + """Return the Hermitian transpose of this matrix. + + See Also + -------- + numpy.matrix.getH : NumPy's implementation of `getH` for matrices + """ + return self.conjugate().transpose() + + def getcol(self, j): + """Returns a copy of column j of the matrix, as an (m x 1) sparse + matrix (column vector). + """ + return self._getcol(j) + + def getrow(self, i): + """Returns a copy of row i of the matrix, as a (1 x n) sparse + matrix (row vector). + """ + return self._getrow(i) + + def todense(self, order=None, out=None): + """ + Return a dense representation of this sparse matrix. + + Parameters + ---------- + order : {'C', 'F'}, optional + Whether to store multi-dimensional data in C (row-major) + or Fortran (column-major) order in memory. The default + is 'None', which provides no ordering guarantees. + Cannot be specified in conjunction with the `out` + argument. + + out : ndarray, 2-D, optional + If specified, uses this array (or `numpy.matrix`) as the + output buffer instead of allocating a new array to + return. The provided array must have the same shape and + dtype as the sparse matrix on which you are calling the + method. + + Returns + ------- + arr : numpy.matrix, 2-D + A NumPy matrix object with the same shape and containing + the same data represented by the sparse matrix, with the + requested memory order. If `out` was passed and was an + array (rather than a `numpy.matrix`), it will be filled + with the appropriate values and returned wrapped in a + `numpy.matrix` object that shares the same memory. + """ + return super().todense(order, out) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_matrix_io.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_matrix_io.py new file mode 100644 index 0000000000000000000000000000000000000000..5b7f533926fd415a379cb08420b4a65a14baeb43 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_matrix_io.py @@ -0,0 +1,167 @@ +import numpy as np +import scipy as sp + +__all__ = ['save_npz', 'load_npz'] + + +# Make loading safe vs. malicious input +PICKLE_KWARGS = dict(allow_pickle=False) + + +def save_npz(file, matrix, compressed=True): + """ Save a sparse matrix or array to a file using ``.npz`` format. + + Parameters + ---------- + file : str or file-like object + Either the file name (string) or an open file (file-like object) + where the data will be saved. If file is a string, the ``.npz`` + extension will be appended to the file name if it is not already + there. + matrix: spmatrix or sparray + The sparse matrix or array to save. + Supported formats: ``csc``, ``csr``, ``bsr``, ``dia`` or ``coo``. + compressed : bool, optional + Allow compressing the file. Default: True + + See Also + -------- + scipy.sparse.load_npz: Load a sparse matrix from a file using ``.npz`` format. + numpy.savez: Save several arrays into a ``.npz`` archive. + numpy.savez_compressed : Save several arrays into a compressed ``.npz`` archive. + + Examples + -------- + Store sparse matrix to disk, and load it again: + + >>> import numpy as np + >>> import scipy as sp + >>> sparse_matrix = sp.sparse.csc_matrix([[0, 0, 3], [4, 0, 0]]) + >>> sparse_matrix + + >>> sparse_matrix.toarray() + array([[0, 0, 3], + [4, 0, 0]], dtype=int64) + + >>> sp.sparse.save_npz('/tmp/sparse_matrix.npz', sparse_matrix) + >>> sparse_matrix = sp.sparse.load_npz('/tmp/sparse_matrix.npz') + + >>> sparse_matrix + + >>> sparse_matrix.toarray() + array([[0, 0, 3], + [4, 0, 0]], dtype=int64) + """ + arrays_dict = {} + if matrix.format in ('csc', 'csr', 'bsr'): + arrays_dict.update(indices=matrix.indices, indptr=matrix.indptr) + elif matrix.format == 'dia': + arrays_dict.update(offsets=matrix.offsets) + elif matrix.format == 'coo': + arrays_dict.update(row=matrix.row, col=matrix.col) + else: + msg = f'Save is not implemented for sparse matrix of format {matrix.format}.' + raise NotImplementedError(msg) + arrays_dict.update( + format=matrix.format.encode('ascii'), + shape=matrix.shape, + data=matrix.data + ) + if isinstance(matrix, sp.sparse.sparray): + arrays_dict.update(_is_array=True) + if compressed: + np.savez_compressed(file, **arrays_dict) + else: + np.savez(file, **arrays_dict) + + +def load_npz(file): + """ Load a sparse array/matrix from a file using ``.npz`` format. + + Parameters + ---------- + file : str or file-like object + Either the file name (string) or an open file (file-like object) + where the data will be loaded. + + Returns + ------- + result : csc_array, csr_array, bsr_array, dia_array or coo_array + A sparse array/matrix containing the loaded data. + + Raises + ------ + OSError + If the input file does not exist or cannot be read. + + See Also + -------- + scipy.sparse.save_npz: Save a sparse array/matrix to a file using ``.npz`` format. + numpy.load: Load several arrays from a ``.npz`` archive. + + Examples + -------- + Store sparse array/matrix to disk, and load it again: + + >>> import numpy as np + >>> import scipy as sp + >>> sparse_array = sp.sparse.csc_array([[0, 0, 3], [4, 0, 0]]) + >>> sparse_array + + >>> sparse_array.toarray() + array([[0, 0, 3], + [4, 0, 0]], dtype=int64) + + >>> sp.sparse.save_npz('/tmp/sparse_array.npz', sparse_array) + >>> sparse_array = sp.sparse.load_npz('/tmp/sparse_array.npz') + + >>> sparse_array + + >>> sparse_array.toarray() + array([[0, 0, 3], + [4, 0, 0]], dtype=int64) + + In this example we force the result to be csr_array from csr_matrix + >>> sparse_matrix = sp.sparse.csc_matrix([[0, 0, 3], [4, 0, 0]]) + >>> sp.sparse.save_npz('/tmp/sparse_matrix.npz', sparse_matrix) + >>> tmp = sp.sparse.load_npz('/tmp/sparse_matrix.npz') + >>> sparse_array = sp.sparse.csr_array(tmp) + """ + with np.load(file, **PICKLE_KWARGS) as loaded: + sparse_format = loaded.get('format') + if sparse_format is None: + raise ValueError(f'The file {file} does not contain ' + f'a sparse array or matrix.') + sparse_format = sparse_format.item() + + if not isinstance(sparse_format, str): + # Play safe with Python 2 vs 3 backward compatibility; + # files saved with SciPy < 1.0.0 may contain unicode or bytes. + sparse_format = sparse_format.decode('ascii') + + if loaded.get('_is_array'): + sparse_type = sparse_format + '_array' + else: + sparse_type = sparse_format + '_matrix' + + try: + cls = getattr(sp.sparse, f'{sparse_type}') + except AttributeError as e: + raise ValueError(f'Unknown format "{sparse_type}"') from e + + if sparse_format in ('csc', 'csr', 'bsr'): + return cls((loaded['data'], loaded['indices'], loaded['indptr']), + shape=loaded['shape']) + elif sparse_format == 'dia': + return cls((loaded['data'], loaded['offsets']), + shape=loaded['shape']) + elif sparse_format == 'coo': + return cls((loaded['data'], (loaded['row'], loaded['col'])), + shape=loaded['shape']) + else: + raise NotImplementedError(f'Load is not implemented for ' + f'sparse matrix of format {sparse_format}.') diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_spfuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_spfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..8e9b0abcede6387e74538baf839a303c6cc1b6be --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_spfuncs.py @@ -0,0 +1,76 @@ +""" Functions that operate on sparse matrices +""" + +__all__ = ['count_blocks','estimate_blocksize'] + +from ._base import issparse +from ._csr import csr_array +from ._sparsetools import csr_count_blocks + + +def estimate_blocksize(A,efficiency=0.7): + """Attempt to determine the blocksize of a sparse matrix + + Returns a blocksize=(r,c) such that + - A.nnz / A.tobsr( (r,c) ).nnz > efficiency + """ + if not (issparse(A) and A.format in ("csc", "csr")): + A = csr_array(A) + + if A.nnz == 0: + return (1,1) + + if not 0 < efficiency < 1.0: + raise ValueError('efficiency must satisfy 0.0 < efficiency < 1.0') + + high_efficiency = (1.0 + efficiency) / 2.0 + nnz = float(A.nnz) + M,N = A.shape + + if M % 2 == 0 and N % 2 == 0: + e22 = nnz / (4 * count_blocks(A,(2,2))) + else: + e22 = 0.0 + + if M % 3 == 0 and N % 3 == 0: + e33 = nnz / (9 * count_blocks(A,(3,3))) + else: + e33 = 0.0 + + if e22 > high_efficiency and e33 > high_efficiency: + e66 = nnz / (36 * count_blocks(A,(6,6))) + if e66 > efficiency: + return (6,6) + else: + return (3,3) + else: + if M % 4 == 0 and N % 4 == 0: + e44 = nnz / (16 * count_blocks(A,(4,4))) + else: + e44 = 0.0 + + if e44 > efficiency: + return (4,4) + elif e33 > efficiency: + return (3,3) + elif e22 > efficiency: + return (2,2) + else: + return (1,1) + + +def count_blocks(A,blocksize): + """For a given blocksize=(r,c) count the number of occupied + blocks in a sparse matrix A + """ + r,c = blocksize + if r < 1 or c < 1: + raise ValueError('r and c must be positive') + + if issparse(A): + if A.format == "csr": + M,N = A.shape + return csr_count_blocks(M,N,r,c,A.indptr,A.indices) + elif A.format == "csc": + return count_blocks(A.T,(c,r)) + return count_blocks(csr_array(A),blocksize) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_sputils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_sputils.py new file mode 100644 index 0000000000000000000000000000000000000000..4fb5b5fc6c2e37d0c61dd3a3aa0aa382c7486a03 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/_sputils.py @@ -0,0 +1,617 @@ +""" Utility functions for sparse matrix module +""" + +import sys +from typing import Any, Literal, Union +import operator +import numpy as np +from math import prod +import scipy.sparse as sp +from scipy._lib._util import np_long, np_ulong + + +__all__ = ['upcast', 'getdtype', 'getdata', 'isscalarlike', 'isintlike', + 'isshape', 'issequence', 'isdense', 'ismatrix', 'get_sum_dtype', + 'broadcast_shapes'] + +supported_dtypes = [np.bool_, np.byte, np.ubyte, np.short, np.ushort, np.intc, + np.uintc, np_long, np_ulong, np.longlong, np.ulonglong, + np.float32, np.float64, np.longdouble, + np.complex64, np.complex128, np.clongdouble] + +_upcast_memo = {} + + +def upcast(*args): + """Returns the nearest supported sparse dtype for the + combination of one or more types. + + upcast(t0, t1, ..., tn) -> T where T is a supported dtype + + Examples + -------- + >>> from scipy.sparse._sputils import upcast + >>> upcast('int32') + + >>> upcast('bool') + + >>> upcast('int32','float32') + + >>> upcast('bool',complex,float) + + + """ + + t = _upcast_memo.get(hash(args)) + if t is not None: + return t + + upcast = np.result_type(*args) + + for t in supported_dtypes: + if np.can_cast(upcast, t): + _upcast_memo[hash(args)] = t + return t + + raise TypeError(f'no supported conversion for types: {args!r}') + + +def upcast_char(*args): + """Same as `upcast` but taking dtype.char as input (faster).""" + t = _upcast_memo.get(args) + if t is not None: + return t + t = upcast(*map(np.dtype, args)) + _upcast_memo[args] = t + return t + + +def upcast_scalar(dtype, scalar): + """Determine data type for binary operation between an array of + type `dtype` and a scalar. + """ + return (np.array([0], dtype=dtype) * scalar).dtype + + +def downcast_intp_index(arr): + """ + Down-cast index array to np.intp dtype if it is of a larger dtype. + + Raise an error if the array contains a value that is too large for + intp. + """ + if arr.dtype.itemsize > np.dtype(np.intp).itemsize: + if arr.size == 0: + return arr.astype(np.intp) + maxval = arr.max() + minval = arr.min() + if maxval > np.iinfo(np.intp).max or minval < np.iinfo(np.intp).min: + raise ValueError("Cannot deal with arrays with indices larger " + "than the machine maximum address size " + "(e.g. 64-bit indices on 32-bit machine).") + return arr.astype(np.intp) + return arr + + +def to_native(A): + """ + Ensure that the data type of the NumPy array `A` has native byte order. + + `A` must be a NumPy array. If the data type of `A` does not have native + byte order, a copy of `A` with a native byte order is returned. Otherwise + `A` is returned. + """ + dt = A.dtype + if dt.isnative: + # Don't call `asarray()` if A is already native, to avoid unnecessarily + # creating a view of the input array. + return A + return np.asarray(A, dtype=dt.newbyteorder('native')) + + +def getdtype(dtype, a=None, default=None): + """Form a supported numpy dtype based on input arguments. + + Returns a valid ``numpy.dtype`` from `dtype` if not None, + or else ``a.dtype`` if possible, or else the given `default` + if not None, or else raise a ``TypeError``. + + The resulting ``dtype`` must be in ``supported_dtypes``: + bool_, int8, uint8, int16, uint16, int32, uint32, + int64, uint64, longlong, ulonglong, float32, float64, + longdouble, complex64, complex128, clongdouble + """ + if dtype is None: + try: + newdtype = a.dtype + except AttributeError as e: + if default is not None: + newdtype = np.dtype(default) + else: + raise TypeError("could not interpret data type") from e + else: + newdtype = np.dtype(dtype) + + if newdtype not in supported_dtypes: + supported_dtypes_fmt = ", ".join(t.__name__ for t in supported_dtypes) + raise ValueError(f"scipy.sparse does not support dtype {newdtype.name}. " + f"The only supported types are: {supported_dtypes_fmt}.") + return newdtype + + +def getdata(obj, dtype=None, copy=False) -> np.ndarray: + """ + This is a wrapper of `np.array(obj, dtype=dtype, copy=copy)` + that will generate a warning if the result is an object array. + """ + data = np.array(obj, dtype=dtype, copy=copy) + # Defer to getdtype for checking that the dtype is OK. + # This is called for the validation only; we don't need the return value. + getdtype(data.dtype) + return data + + +def safely_cast_index_arrays(A, idx_dtype=np.int32, msg=""): + """Safely cast sparse array indices to `idx_dtype`. + + Check the shape of `A` to determine if it is safe to cast its index + arrays to dtype `idx_dtype`. If any dimension in shape is larger than + fits in the dtype, casting is unsafe so raise ``ValueError``. + If safe, cast the index arrays to `idx_dtype` and return the result + without changing the input `A`. The caller can assign results to `A` + attributes if desired or use the recast index arrays directly. + + Unless downcasting is needed, the original index arrays are returned. + You can test e.g. ``A.indptr is new_indptr`` to see if downcasting occurred. + + .. versionadded:: 1.15.0 + + Parameters + ---------- + A : sparse array or matrix + The array for which index arrays should be downcast. + idx_dtype : dtype + Desired dtype. Should be an integer dtype (default: ``np.int32``). + Most of scipy.sparse uses either int64 or int32. + msg : string, optional + A string to be added to the end of the ValueError message + if the array shape is too big to fit in `idx_dtype`. + The error message is ``f" values too large for {msg}"`` + It should indicate why the downcasting is needed, e.g. "SuperLU", + and defaults to f"dtype {idx_dtype}". + + Returns + ------- + idx_arrays : ndarray or tuple of ndarrays + Based on ``A.format``, index arrays are returned after casting to `idx_dtype`. + For CSC/CSR, returns ``(indices, indptr)``. + For COO, returns ``coords``. + For DIA, returns ``offsets``. + For BSR, returns ``(indices, indptr)``. + + Raises + ------ + ValueError + If the array has shape that would not fit in the new dtype, or if + the sparse format does not use index arrays. + + Examples + -------- + >>> import numpy as np + >>> from scipy import sparse + >>> data = [3] + >>> coords = (np.array([3]), np.array([1])) # Note: int64 arrays + >>> A = sparse.coo_array((data, coords)) + >>> A.coords[0].dtype + dtype('int64') + + >>> # rescast after construction, raising exception if shape too big + >>> coords = sparse.safely_cast_index_arrays(A, np.int32) + >>> A.coords[0] is coords[0] # False if casting is needed + False + >>> A.coords = coords # set the index dtype of A + >>> A.coords[0].dtype + dtype('int32') + """ + if not msg: + msg = f"dtype {idx_dtype}" + # check for safe downcasting + max_value = np.iinfo(idx_dtype).max + + if A.format in ("csc", "csr"): + # indptr[-1] is max b/c indptr always sorted + if A.indptr[-1] > max_value: + raise ValueError(f"indptr values too large for {msg}") + + # check shape vs dtype + if max(*A.shape) > max_value: + if (A.indices > max_value).any(): + raise ValueError(f"indices values too large for {msg}") + + indices = A.indices.astype(idx_dtype, copy=False) + indptr = A.indptr.astype(idx_dtype, copy=False) + return indices, indptr + + elif A.format == "coo": + if max(*A.shape) > max_value: + if any((co > max_value).any() for co in A.coords): + raise ValueError(f"coords values too large for {msg}") + return tuple(co.astype(idx_dtype, copy=False) for co in A.coords) + + elif A.format == "dia": + if max(*A.shape) > max_value: + if (A.offsets > max_value).any(): + raise ValueError(f"offsets values too large for {msg}") + offsets = A.offsets.astype(idx_dtype, copy=False) + return offsets + + elif A.format == 'bsr': + R, C = A.blocksize + if A.indptr[-1] * R > max_value: + raise ValueError("indptr values too large for {msg}") + if max(*A.shape) > max_value: + if (A.indices * C > max_value).any(): + raise ValueError(f"indices values too large for {msg}") + indices = A.indices.astype(idx_dtype, copy=False) + indptr = A.indptr.astype(idx_dtype, copy=False) + return indices, indptr + + else: + raise TypeError(f'Format {A.format} is not associated with index arrays. ' + 'DOK and LIL have dict and list, not array.') + + +def get_index_dtype(arrays=(), maxval=None, check_contents=False): + """ + Based on input (integer) arrays `a`, determine a suitable index data + type that can hold the data in the arrays. + + Parameters + ---------- + arrays : tuple of array_like + Input arrays whose types/contents to check + maxval : float, optional + Maximum value needed + check_contents : bool, optional + Whether to check the values in the arrays and not just their types. + Default: False (check only the types) + + Returns + ------- + dtype : dtype + Suitable index data type (int32 or int64) + + Examples + -------- + >>> import numpy as np + >>> from scipy import sparse + >>> # select index dtype based on shape + >>> shape = (3, 3) + >>> idx_dtype = sparse.get_index_dtype(maxval=max(shape)) + >>> data = [1.1, 3.0, 1.5] + >>> indices = np.array([0, 1, 0], dtype=idx_dtype) + >>> indptr = np.array([0, 2, 3, 3], dtype=idx_dtype) + >>> A = sparse.csr_array((data, indices, indptr), shape=shape) + >>> A.indptr.dtype + dtype('int32') + + >>> # select based on larger of existing arrays and shape + >>> shape = (3, 3) + >>> idx_dtype = sparse.get_index_dtype(A.indptr, maxval=max(shape)) + >>> idx_dtype + + """ + # not using intc directly due to misinteractions with pythran + if np.intc().itemsize != 4: + return np.int64 + + int32min = np.int32(np.iinfo(np.int32).min) + int32max = np.int32(np.iinfo(np.int32).max) + + if maxval is not None: + maxval = np.int64(maxval) + if maxval > int32max: + return np.int64 + + if isinstance(arrays, np.ndarray): + arrays = (arrays,) + + for arr in arrays: + arr = np.asarray(arr) + if not np.can_cast(arr.dtype, np.int32): + if check_contents: + if arr.size == 0: + # a bigger type not needed + continue + elif np.issubdtype(arr.dtype, np.integer): + maxval = arr.max() + minval = arr.min() + if minval >= int32min and maxval <= int32max: + # a bigger type not needed + continue + return np.int64 + return np.int32 + + +def get_sum_dtype(dtype: np.dtype) -> np.dtype: + """Mimic numpy's casting for np.sum""" + if dtype.kind == 'u' and np.can_cast(dtype, np.uint): + return np.uint + if np.can_cast(dtype, np.int_): + return np.int_ + return dtype + + +def isscalarlike(x) -> bool: + """Is x either a scalar, an array scalar, or a 0-dim array?""" + return np.isscalar(x) or (isdense(x) and x.ndim == 0) + + +def isintlike(x) -> bool: + """Is x appropriate as an index into a sparse matrix? Returns True + if it can be cast safely to a machine int. + """ + # Fast-path check to eliminate non-scalar values. operator.index would + # catch this case too, but the exception catching is slow. + if np.ndim(x) != 0: + return False + try: + operator.index(x) + except (TypeError, ValueError): + try: + loose_int = bool(int(x) == x) + except (TypeError, ValueError): + return False + if loose_int: + msg = "Inexact indices into sparse matrices are not allowed" + raise ValueError(msg) + return loose_int + return True + + +def isshape(x, nonneg=False, *, allow_nd=(2,)) -> bool: + """Is x a valid tuple of dimensions? + + If nonneg, also checks that the dimensions are non-negative. + Shapes of length in the tuple allow_nd are allowed. + """ + ndim = len(x) + if ndim not in allow_nd: + return False + + for d in x: + if not isintlike(d): + return False + if nonneg and d < 0: + return False + return True + + +def issequence(t) -> bool: + return ((isinstance(t, list | tuple) and + (len(t) == 0 or np.isscalar(t[0]))) or + (isinstance(t, np.ndarray) and (t.ndim == 1))) + + +def ismatrix(t) -> bool: + return ((isinstance(t, list | tuple) and + len(t) > 0 and issequence(t[0])) or + (isinstance(t, np.ndarray) and t.ndim == 2)) + + +def isdense(x) -> bool: + return isinstance(x, np.ndarray) + + +def validateaxis(axis) -> None: + if axis is None: + return + axis_type = type(axis) + + # In NumPy, you can pass in tuples for 'axis', but they are + # not very useful for sparse matrices given their limited + # dimensions, so let's make it explicit that they are not + # allowed to be passed in + if isinstance(axis, tuple): + raise TypeError("Tuples are not accepted for the 'axis' parameter. " + "Please pass in one of the following: " + "{-2, -1, 0, 1, None}.") + + # If not a tuple, check that the provided axis is actually + # an integer and raise a TypeError similar to NumPy's + if not np.issubdtype(np.dtype(axis_type), np.integer): + raise TypeError(f"axis must be an integer, not {axis_type.__name__}") + + if not (-2 <= axis <= 1): + raise ValueError("axis out of range") + + +def check_shape(args, current_shape=None, *, allow_nd=(2,)) -> tuple[int, ...]: + """Imitate numpy.matrix handling of shape arguments + + Parameters + ---------- + args : array_like + Data structures providing information about the shape of the sparse array. + current_shape : tuple, optional + The current shape of the sparse array or matrix. + If None (default), the current shape will be inferred from args. + allow_nd : tuple of ints, optional default: (2,) + If shape does not have a length in the tuple allow_nd an error is raised. + + Returns + ------- + new_shape: tuple + The new shape after validation. + """ + if len(args) == 0: + raise TypeError("function missing 1 required positional argument: 'shape'") + if len(args) == 1: + try: + shape_iter = iter(args[0]) + except TypeError: + new_shape = (operator.index(args[0]), ) + else: + new_shape = tuple(operator.index(arg) for arg in shape_iter) + else: + new_shape = tuple(operator.index(arg) for arg in args) + + if current_shape is None: + if len(new_shape) not in allow_nd: + raise ValueError(f'shape must have length in {allow_nd}. Got {new_shape=}') + if any(d < 0 for d in new_shape): + raise ValueError("'shape' elements cannot be negative") + else: + # Check the current size only if needed + current_size = prod(current_shape) + + # Check for negatives + negative_indexes = [i for i, x in enumerate(new_shape) if x < 0] + if not negative_indexes: + new_size = prod(new_shape) + if new_size != current_size: + raise ValueError(f'cannot reshape array of size {current_size}' + f' into shape {new_shape}') + elif len(negative_indexes) == 1: + skip = negative_indexes[0] + specified = prod(new_shape[:skip] + new_shape[skip+1:]) + unspecified, remainder = divmod(current_size, specified) + if remainder != 0: + err_shape = tuple('newshape' if x < 0 else x for x in new_shape) + raise ValueError(f'cannot reshape array of size {current_size}' + f' into shape {err_shape}') + new_shape = new_shape[:skip] + (unspecified,) + new_shape[skip+1:] + else: + raise ValueError('can only specify one unknown dimension') + + if len(new_shape) not in allow_nd: + raise ValueError(f'shape must have length in {allow_nd}. Got {new_shape=}') + + return new_shape + + +def broadcast_shapes(*shapes): + """Check if shapes can be broadcast and return resulting shape + + This is similar to the NumPy ``broadcast_shapes`` function but + does not check memory consequences of the resulting dense matrix. + + Parameters + ---------- + *shapes : tuple of shape tuples + The tuple of shapes to be considered for broadcasting. + Shapes should be tuples of non-negative integers. + + Returns + ------- + new_shape : tuple of integers + The shape that results from broadcasting th input shapes. + """ + if not shapes: + return () + shapes = [shp if isinstance(shp, (tuple, list)) else (shp,) for shp in shapes] + big_shp = max(shapes, key=len) + out = list(big_shp) + for shp in shapes: + if shp is big_shp: + continue + for i, x in enumerate(shp, start=-len(shp)): + if x != 1 and x != out[i]: + if out[i] != 1: + raise ValueError("shapes cannot be broadcast to a single shape.") + out[i] = x + return (*out,) + + +def check_reshape_kwargs(kwargs): + """Unpack keyword arguments for reshape function. + + This is useful because keyword arguments after star arguments are not + allowed in Python 2, but star keyword arguments are. This function unpacks + 'order' and 'copy' from the star keyword arguments (with defaults) and + throws an error for any remaining. + """ + + order = kwargs.pop('order', 'C') + copy = kwargs.pop('copy', False) + if kwargs: # Some unused kwargs remain + raise TypeError("reshape() got unexpected keywords arguments: " + f"{', '.join(kwargs.keys())}") + return order, copy + + +def is_pydata_spmatrix(m) -> bool: + """ + Check whether object is pydata/sparse matrix, avoiding importing the module. + """ + base_cls = getattr(sys.modules.get('sparse'), 'SparseArray', None) + return base_cls is not None and isinstance(m, base_cls) + + +def convert_pydata_sparse_to_scipy( + arg: Any, + target_format: None | Literal["csc", "csr"] = None, + accept_fv: Any = None, +) -> Union[Any, "sp.spmatrix"]: + """ + Convert a pydata/sparse array to scipy sparse matrix, + pass through anything else. + """ + if is_pydata_spmatrix(arg): + # The `accept_fv` keyword is new in PyData Sparse 0.15.4 (May 2024), + # remove the `except` once the minimum supported version is >=0.15.4 + try: + arg = arg.to_scipy_sparse(accept_fv=accept_fv) + except TypeError: + arg = arg.to_scipy_sparse() + if target_format is not None: + arg = arg.asformat(target_format) + elif arg.format not in ("csc", "csr"): + arg = arg.tocsc() + return arg + + +############################################################################### +# Wrappers for NumPy types that are deprecated + +# Numpy versions of these functions raise deprecation warnings, the +# ones below do not. + +def matrix(*args, **kwargs): + return np.array(*args, **kwargs).view(np.matrix) + + +def asmatrix(data, dtype=None): + if isinstance(data, np.matrix) and (dtype is None or data.dtype == dtype): + return data + return np.asarray(data, dtype=dtype).view(np.matrix) + +############################################################################### + + +def _todata(s) -> np.ndarray: + """Access nonzero values, possibly after summing duplicates. + + Parameters + ---------- + s : sparse array + Input sparse array. + + Returns + ------- + data: ndarray + Nonzero values of the array, with shape (s.nnz,) + + """ + if isinstance(s, sp._data._data_matrix): + return s._deduped_data() + + if isinstance(s, sp.dok_array): + return np.fromiter(s.values(), dtype=s.dtype, count=s.nnz) + + if isinstance(s, sp.lil_array): + data = np.empty(s.nnz, dtype=s.dtype) + sp._csparsetools.lil_flatten_to_array(s.data, data) + return data + + return s.tocoo()._deduped_data() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/base.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/base.py new file mode 100644 index 0000000000000000000000000000000000000000..d0a427e4570e07cc71e9e45bf98c7cf61798125b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/base.py @@ -0,0 +1,33 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'MAXPRINT', + 'SparseEfficiencyWarning', + 'SparseFormatWarning', + 'SparseWarning', + 'asmatrix', + 'check_reshape_kwargs', + 'check_shape', + 'get_sum_dtype', + 'isdense', + 'isscalarlike', + 'issparse', + 'isspmatrix', + 'spmatrix', + 'validateaxis', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="base", + private_modules=["_base"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/bsr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/bsr.py new file mode 100644 index 0000000000000000000000000000000000000000..c686301a78fc3e2221600eb06035a5cb12898cdb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/bsr.py @@ -0,0 +1,36 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'bsr_matmat', + 'bsr_matrix', + 'bsr_matvec', + 'bsr_matvecs', + 'bsr_sort_indices', + 'bsr_tocsr', + 'bsr_transpose', + 'check_shape', + 'csr_matmat_maxnnz', + 'getdata', + 'getdtype', + 'isshape', + 'isspmatrix_bsr', + 'spmatrix', + 'to_native', + 'upcast', + 'warn', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="bsr", + private_modules=["_bsr"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/compressed.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/compressed.py new file mode 100644 index 0000000000000000000000000000000000000000..e6dc8a73e5ab527cfe0b73d558dae25047cfb98b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/compressed.py @@ -0,0 +1,43 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'IndexMixin', + 'SparseEfficiencyWarning', + 'check_shape', + 'csr_column_index1', + 'csr_column_index2', + 'csr_row_index', + 'csr_row_slice', + 'csr_sample_offsets', + 'csr_sample_values', + 'csr_todense', + 'downcast_intp_index', + 'get_csr_submatrix', + 'get_sum_dtype', + 'getdtype', + 'is_pydata_spmatrix', + 'isdense', + 'isintlike', + 'isscalarlike', + 'isshape', + 'operator', + 'to_native', + 'upcast', + 'upcast_char', + 'warn', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="compressed", + private_modules=["_compressed"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/construct.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/construct.py new file mode 100644 index 0000000000000000000000000000000000000000..c3d34d2fd38887877980727bceaaa215129bf283 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/construct.py @@ -0,0 +1,44 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'block_diag', + 'bmat', + 'bsr_matrix', + 'check_random_state', + 'coo_matrix', + 'csc_matrix', + 'csr_hstack', + 'csr_matrix', + 'dia_matrix', + 'diags', + 'eye', + 'get_index_dtype', + 'hstack', + 'identity', + 'isscalarlike', + 'issparse', + 'kron', + 'kronsum', + 'numbers', + 'rand', + 'random', + 'rng_integers', + 'spdiags', + 'upcast', + 'vstack', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="construct", + private_modules=["_construct"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/coo.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/coo.py new file mode 100644 index 0000000000000000000000000000000000000000..bda2da3d09a676ab79739331a21ba26102bb90ae --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/coo.py @@ -0,0 +1,37 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'SparseEfficiencyWarning', + 'check_reshape_kwargs', + 'check_shape', + 'coo_matrix', + 'coo_matvec', + 'coo_tocsr', + 'coo_todense', + 'downcast_intp_index', + 'getdata', + 'getdtype', + 'isshape', + 'isspmatrix_coo', + 'operator', + 'spmatrix', + 'to_native', + 'upcast', + 'upcast_char', + 'warn', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="coo", + private_modules=["_coo"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csc.py new file mode 100644 index 0000000000000000000000000000000000000000..d140b841e0724155f8602a4215836e2c8a7fad72 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csc.py @@ -0,0 +1,25 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'csc_matrix', + 'csc_tocsr', + 'expandptr', + 'isspmatrix_csc', + 'spmatrix', + 'upcast', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="csc", + private_modules=["_csc"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..00ab19af4748147d748fccb51a3710d5c711f4b4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/__init__.py @@ -0,0 +1,210 @@ +r""" +Compressed sparse graph routines (:mod:`scipy.sparse.csgraph`) +============================================================== + +.. currentmodule:: scipy.sparse.csgraph + +Fast graph algorithms based on sparse matrix representations. + +Contents +-------- + +.. autosummary:: + :toctree: generated/ + + connected_components -- determine connected components of a graph + laplacian -- compute the laplacian of a graph + shortest_path -- compute the shortest path between points on a positive graph + dijkstra -- use Dijkstra's algorithm for shortest path + floyd_warshall -- use the Floyd-Warshall algorithm for shortest path + bellman_ford -- use the Bellman-Ford algorithm for shortest path + johnson -- use Johnson's algorithm for shortest path + yen -- use Yen's algorithm for K-shortest paths between to nodes. + breadth_first_order -- compute a breadth-first order of nodes + depth_first_order -- compute a depth-first order of nodes + breadth_first_tree -- construct the breadth-first tree from a given node + depth_first_tree -- construct a depth-first tree from a given node + minimum_spanning_tree -- construct the minimum spanning tree of a graph + reverse_cuthill_mckee -- compute permutation for reverse Cuthill-McKee ordering + maximum_flow -- solve the maximum flow problem for a graph + maximum_bipartite_matching -- compute a maximum matching of a bipartite graph + min_weight_full_bipartite_matching - compute a minimum weight full matching of a bipartite graph + structural_rank -- compute the structural rank of a graph + NegativeCycleError + +.. autosummary:: + :toctree: generated/ + + construct_dist_matrix + csgraph_from_dense + csgraph_from_masked + csgraph_masked_from_dense + csgraph_to_dense + csgraph_to_masked + reconstruct_path + +Graph Representations +--------------------- +This module uses graphs which are stored in a matrix format. A +graph with N nodes can be represented by an (N x N) adjacency matrix G. +If there is a connection from node i to node j, then G[i, j] = w, where +w is the weight of the connection. For nodes i and j which are +not connected, the value depends on the representation: + +- for dense array representations, non-edges are represented by + G[i, j] = 0, infinity, or NaN. + +- for dense masked representations (of type np.ma.MaskedArray), non-edges + are represented by masked values. This can be useful when graphs with + zero-weight edges are desired. + +- for sparse array representations, non-edges are represented by + non-entries in the matrix. This sort of sparse representation also + allows for edges with zero weights. + +As a concrete example, imagine that you would like to represent the following +undirected graph:: + + G + + (0) + / \ + 1 2 + / \ + (2) (1) + +This graph has three nodes, where node 0 and 1 are connected by an edge of +weight 2, and nodes 0 and 2 are connected by an edge of weight 1. +We can construct the dense, masked, and sparse representations as follows, +keeping in mind that an undirected graph is represented by a symmetric matrix:: + + >>> import numpy as np + >>> G_dense = np.array([[0, 2, 1], + ... [2, 0, 0], + ... [1, 0, 0]]) + >>> G_masked = np.ma.masked_values(G_dense, 0) + >>> from scipy.sparse import csr_array + >>> G_sparse = csr_array(G_dense) + +This becomes more difficult when zero edges are significant. For example, +consider the situation when we slightly modify the above graph:: + + G2 + + (0) + / \ + 0 2 + / \ + (2) (1) + +This is identical to the previous graph, except nodes 0 and 2 are connected +by an edge of zero weight. In this case, the dense representation above +leads to ambiguities: how can non-edges be represented if zero is a meaningful +value? In this case, either a masked or sparse representation must be used +to eliminate the ambiguity:: + + >>> import numpy as np + >>> G2_data = np.array([[np.inf, 2, 0 ], + ... [2, np.inf, np.inf], + ... [0, np.inf, np.inf]]) + >>> G2_masked = np.ma.masked_invalid(G2_data) + >>> from scipy.sparse.csgraph import csgraph_from_dense + >>> # G2_sparse = csr_array(G2_data) would give the wrong result + >>> G2_sparse = csgraph_from_dense(G2_data, null_value=np.inf) + >>> G2_sparse.data + array([ 2., 0., 2., 0.]) + +Here we have used a utility routine from the csgraph submodule in order to +convert the dense representation to a sparse representation which can be +understood by the algorithms in submodule. By viewing the data array, we +can see that the zero values are explicitly encoded in the graph. + +Directed vs. undirected +^^^^^^^^^^^^^^^^^^^^^^^ +Matrices may represent either directed or undirected graphs. This is +specified throughout the csgraph module by a boolean keyword. Graphs are +assumed to be directed by default. In a directed graph, traversal from node +i to node j can be accomplished over the edge G[i, j], but not the edge +G[j, i]. Consider the following dense graph:: + + >>> import numpy as np + >>> G_dense = np.array([[0, 1, 0], + ... [2, 0, 3], + ... [0, 4, 0]]) + +When ``directed=True`` we get the graph:: + + ---1--> ---3--> + (0) (1) (2) + <--2--- <--4--- + +In a non-directed graph, traversal from node i to node j can be +accomplished over either G[i, j] or G[j, i]. If both edges are not null, +and the two have unequal weights, then the smaller of the two is used. + +So for the same graph, when ``directed=False`` we get the graph:: + + (0)--1--(1)--3--(2) + +Note that a symmetric matrix will represent an undirected graph, regardless +of whether the 'directed' keyword is set to True or False. In this case, +using ``directed=True`` generally leads to more efficient computation. + +The routines in this module accept as input either scipy.sparse representations +(csr, csc, or lil format), masked representations, or dense representations +with non-edges indicated by zeros, infinities, and NaN entries. +""" # noqa: E501 + +__docformat__ = "restructuredtext en" + +__all__ = ['connected_components', + 'laplacian', + 'shortest_path', + 'floyd_warshall', + 'dijkstra', + 'bellman_ford', + 'johnson', + 'yen', + 'breadth_first_order', + 'depth_first_order', + 'breadth_first_tree', + 'depth_first_tree', + 'minimum_spanning_tree', + 'reverse_cuthill_mckee', + 'maximum_flow', + 'maximum_bipartite_matching', + 'min_weight_full_bipartite_matching', + 'structural_rank', + 'construct_dist_matrix', + 'reconstruct_path', + 'csgraph_masked_from_dense', + 'csgraph_from_dense', + 'csgraph_from_masked', + 'csgraph_to_dense', + 'csgraph_to_masked', + 'NegativeCycleError'] + +from ._laplacian import laplacian +from ._shortest_path import ( + shortest_path, floyd_warshall, dijkstra, bellman_ford, johnson, yen, + NegativeCycleError +) +from ._traversal import ( + breadth_first_order, depth_first_order, breadth_first_tree, + depth_first_tree, connected_components +) +from ._min_spanning_tree import minimum_spanning_tree +from ._flow import maximum_flow +from ._matching import ( + maximum_bipartite_matching, min_weight_full_bipartite_matching +) +from ._reordering import reverse_cuthill_mckee, structural_rank +from ._tools import ( + construct_dist_matrix, reconstruct_path, csgraph_from_dense, + csgraph_to_dense, csgraph_masked_from_dense, csgraph_from_masked, + csgraph_to_masked +) + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/_laplacian.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/_laplacian.py new file mode 100644 index 0000000000000000000000000000000000000000..e5529a0662a3f9db006bc5411664908f10d8fe23 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/_laplacian.py @@ -0,0 +1,563 @@ +""" +Laplacian of a compressed-sparse graph +""" + +import numpy as np +from scipy.sparse import issparse +from scipy.sparse.linalg import LinearOperator +from scipy.sparse._sputils import convert_pydata_sparse_to_scipy, is_pydata_spmatrix + + +############################################################################### +# Graph laplacian +def laplacian( + csgraph, + normed=False, + return_diag=False, + use_out_degree=False, + *, + copy=True, + form="array", + dtype=None, + symmetrized=False, +): + """ + Return the Laplacian of a directed graph. + + Parameters + ---------- + csgraph : array_like or sparse array or matrix, 2 dimensions + compressed-sparse graph, with shape (N, N). + normed : bool, optional + If True, then compute symmetrically normalized Laplacian. + Default: False. + return_diag : bool, optional + If True, then also return an array related to vertex degrees. + Default: False. + use_out_degree : bool, optional + If True, then use out-degree instead of in-degree. + This distinction matters only if the graph is asymmetric. + Default: False. + copy: bool, optional + If False, then change `csgraph` in place if possible, + avoiding doubling the memory use. + Default: True, for backward compatibility. + form: 'array', or 'function', or 'lo' + Determines the format of the output Laplacian: + + * 'array' is a numpy array; + * 'function' is a pointer to evaluating the Laplacian-vector + or Laplacian-matrix product; + * 'lo' results in the format of the `LinearOperator`. + + Choosing 'function' or 'lo' always avoids doubling + the memory use, ignoring `copy` value. + Default: 'array', for backward compatibility. + dtype: None or one of numeric numpy dtypes, optional + The dtype of the output. If ``dtype=None``, the dtype of the + output matches the dtype of the input csgraph, except for + the case ``normed=True`` and integer-like csgraph, where + the output dtype is 'float' allowing accurate normalization, + but dramatically increasing the memory use. + Default: None, for backward compatibility. + symmetrized: bool, optional + If True, then the output Laplacian is symmetric/Hermitian. + The symmetrization is done by ``csgraph + csgraph.T.conj`` + without dividing by 2 to preserve integer dtypes if possible + prior to the construction of the Laplacian. + The symmetrization will increase the memory footprint of + sparse matrices unless the sparsity pattern is symmetric or + `form` is 'function' or 'lo'. + Default: False, for backward compatibility. + + Returns + ------- + lap : ndarray, or sparse array or matrix, or `LinearOperator` + The N x N Laplacian of csgraph. It will be a NumPy array (dense) + if the input was dense, or a sparse array otherwise, or + the format of a function or `LinearOperator` if + `form` equals 'function' or 'lo', respectively. + diag : ndarray, optional + The length-N main diagonal of the Laplacian matrix. + For the normalized Laplacian, this is the array of square roots + of vertex degrees or 1 if the degree is zero. + + Notes + ----- + The Laplacian matrix of a graph is sometimes referred to as the + "Kirchhoff matrix" or just the "Laplacian", and is useful in many + parts of spectral graph theory. + In particular, the eigen-decomposition of the Laplacian can give + insight into many properties of the graph, e.g., + is commonly used for spectral data embedding and clustering. + + The constructed Laplacian doubles the memory use if ``copy=True`` and + ``form="array"`` which is the default. + Choosing ``copy=False`` has no effect unless ``form="array"`` + or the matrix is sparse in the ``coo`` format, or dense array, except + for the integer input with ``normed=True`` that forces the float output. + + Sparse input is reformatted into ``coo`` if ``form="array"``, + which is the default. + + If the input adjacency matrix is not symmetric, the Laplacian is + also non-symmetric unless ``symmetrized=True`` is used. + + Diagonal entries of the input adjacency matrix are ignored and + replaced with zeros for the purpose of normalization where ``normed=True``. + The normalization uses the inverse square roots of row-sums of the input + adjacency matrix, and thus may fail if the row-sums contain + negative or complex with a non-zero imaginary part values. + + The normalization is symmetric, making the normalized Laplacian also + symmetric if the input csgraph was symmetric. + + References + ---------- + .. [1] Laplacian matrix. https://en.wikipedia.org/wiki/Laplacian_matrix + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csgraph + + Our first illustration is the symmetric graph + + >>> G = np.arange(4) * np.arange(4)[:, np.newaxis] + >>> G + array([[0, 0, 0, 0], + [0, 1, 2, 3], + [0, 2, 4, 6], + [0, 3, 6, 9]]) + + and its symmetric Laplacian matrix + + >>> csgraph.laplacian(G) + array([[ 0, 0, 0, 0], + [ 0, 5, -2, -3], + [ 0, -2, 8, -6], + [ 0, -3, -6, 9]]) + + The non-symmetric graph + + >>> G = np.arange(9).reshape(3, 3) + >>> G + array([[0, 1, 2], + [3, 4, 5], + [6, 7, 8]]) + + has different row- and column sums, resulting in two varieties + of the Laplacian matrix, using an in-degree, which is the default + + >>> L_in_degree = csgraph.laplacian(G) + >>> L_in_degree + array([[ 9, -1, -2], + [-3, 8, -5], + [-6, -7, 7]]) + + or alternatively an out-degree + + >>> L_out_degree = csgraph.laplacian(G, use_out_degree=True) + >>> L_out_degree + array([[ 3, -1, -2], + [-3, 8, -5], + [-6, -7, 13]]) + + Constructing a symmetric Laplacian matrix, one can add the two as + + >>> L_in_degree + L_out_degree.T + array([[ 12, -4, -8], + [ -4, 16, -12], + [ -8, -12, 20]]) + + or use the ``symmetrized=True`` option + + >>> csgraph.laplacian(G, symmetrized=True) + array([[ 12, -4, -8], + [ -4, 16, -12], + [ -8, -12, 20]]) + + that is equivalent to symmetrizing the original graph + + >>> csgraph.laplacian(G + G.T) + array([[ 12, -4, -8], + [ -4, 16, -12], + [ -8, -12, 20]]) + + The goal of normalization is to make the non-zero diagonal entries + of the Laplacian matrix to be all unit, also scaling off-diagonal + entries correspondingly. The normalization can be done manually, e.g., + + >>> G = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]]) + >>> L, d = csgraph.laplacian(G, return_diag=True) + >>> L + array([[ 2, -1, -1], + [-1, 2, -1], + [-1, -1, 2]]) + >>> d + array([2, 2, 2]) + >>> scaling = np.sqrt(d) + >>> scaling + array([1.41421356, 1.41421356, 1.41421356]) + >>> (1/scaling)*L*(1/scaling) + array([[ 1. , -0.5, -0.5], + [-0.5, 1. , -0.5], + [-0.5, -0.5, 1. ]]) + + Or using ``normed=True`` option + + >>> L, d = csgraph.laplacian(G, return_diag=True, normed=True) + >>> L + array([[ 1. , -0.5, -0.5], + [-0.5, 1. , -0.5], + [-0.5, -0.5, 1. ]]) + + which now instead of the diagonal returns the scaling coefficients + + >>> d + array([1.41421356, 1.41421356, 1.41421356]) + + Zero scaling coefficients are substituted with 1s, where scaling + has thus no effect, e.g., + + >>> G = np.array([[0, 0, 0], [0, 0, 1], [0, 1, 0]]) + >>> G + array([[0, 0, 0], + [0, 0, 1], + [0, 1, 0]]) + >>> L, d = csgraph.laplacian(G, return_diag=True, normed=True) + >>> L + array([[ 0., -0., -0.], + [-0., 1., -1.], + [-0., -1., 1.]]) + >>> d + array([1., 1., 1.]) + + Only the symmetric normalization is implemented, resulting + in a symmetric Laplacian matrix if and only if its graph is symmetric + and has all non-negative degrees, like in the examples above. + + The output Laplacian matrix is by default a dense array or a sparse + array or matrix inferring its class, shape, format, and dtype from + the input graph matrix: + + >>> G = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]]).astype(np.float32) + >>> G + array([[0., 1., 1.], + [1., 0., 1.], + [1., 1., 0.]], dtype=float32) + >>> csgraph.laplacian(G) + array([[ 2., -1., -1.], + [-1., 2., -1.], + [-1., -1., 2.]], dtype=float32) + + but can alternatively be generated matrix-free as a LinearOperator: + + >>> L = csgraph.laplacian(G, form="lo") + >>> L + <3x3 _CustomLinearOperator with dtype=float32> + >>> L(np.eye(3)) + array([[ 2., -1., -1.], + [-1., 2., -1.], + [-1., -1., 2.]]) + + or as a lambda-function: + + >>> L = csgraph.laplacian(G, form="function") + >>> L + . at 0x0000012AE6F5A598> + >>> L(np.eye(3)) + array([[ 2., -1., -1.], + [-1., 2., -1.], + [-1., -1., 2.]]) + + The Laplacian matrix is used for + spectral data clustering and embedding + as well as for spectral graph partitioning. + Our final example illustrates the latter + for a noisy directed linear graph. + + >>> from scipy.sparse import diags_array, random_array + >>> from scipy.sparse.linalg import lobpcg + + Create a directed linear graph with ``N=35`` vertices + using a sparse adjacency matrix ``G``: + + >>> N = 35 + >>> G = diags_array(np.ones(N - 1), offsets=1, format="csr") + + Fix a random seed ``rng`` and add a random sparse noise to the graph ``G``: + + >>> rng = np.random.default_rng() + >>> G += 1e-2 * random_array((N, N), density=0.1, rng=rng) + + Set initial approximations for eigenvectors: + + >>> X = rng.random((N, 2)) + + The constant vector of ones is always a trivial eigenvector + of the non-normalized Laplacian to be filtered out: + + >>> Y = np.ones((N, 1)) + + Alternating (1) the sign of the graph weights allows determining + labels for spectral max- and min- cuts in a single loop. + Since the graph is undirected, the option ``symmetrized=True`` + must be used in the construction of the Laplacian. + The option ``normed=True`` cannot be used in (2) for the negative weights + here as the symmetric normalization evaluates square roots. + The option ``form="lo"`` in (2) is matrix-free, i.e., guarantees + a fixed memory footprint and read-only access to the graph. + Calling the eigenvalue solver ``lobpcg`` (3) computes the Fiedler vector + that determines the labels as the signs of its components in (5). + Since the sign in an eigenvector is not deterministic and can flip, + we fix the sign of the first component to be always +1 in (4). + + >>> for cut in ["max", "min"]: + ... G = -G # 1. + ... L = csgraph.laplacian(G, symmetrized=True, form="lo") # 2. + ... _, eves = lobpcg(L, X, Y=Y, largest=False, tol=1e-2) # 3. + ... eves *= np.sign(eves[0, 0]) # 4. + ... print(cut + "-cut labels:\\n", 1 * (eves[:, 0]>0)) # 5. + max-cut labels: + [1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1] + min-cut labels: + [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] + + As anticipated for a (slightly noisy) linear graph, + the max-cut strips all the edges of the graph coloring all + odd vertices into one color and all even vertices into another one, + while the balanced min-cut partitions the graph + in the middle by deleting a single edge. + Both determined partitions are optimal. + """ + is_pydata_sparse = is_pydata_spmatrix(csgraph) + if is_pydata_sparse: + pydata_sparse_cls = csgraph.__class__ + csgraph = convert_pydata_sparse_to_scipy(csgraph) + if csgraph.ndim != 2 or csgraph.shape[0] != csgraph.shape[1]: + raise ValueError('csgraph must be a square matrix or array') + + if normed and ( + np.issubdtype(csgraph.dtype, np.signedinteger) + or np.issubdtype(csgraph.dtype, np.uint) + ): + csgraph = csgraph.astype(np.float64) + + if form == "array": + create_lap = ( + _laplacian_sparse if issparse(csgraph) else _laplacian_dense + ) + else: + create_lap = ( + _laplacian_sparse_flo + if issparse(csgraph) + else _laplacian_dense_flo + ) + + degree_axis = 1 if use_out_degree else 0 + + lap, d = create_lap( + csgraph, + normed=normed, + axis=degree_axis, + copy=copy, + form=form, + dtype=dtype, + symmetrized=symmetrized, + ) + if is_pydata_sparse: + lap = pydata_sparse_cls.from_scipy_sparse(lap) + if return_diag: + return lap, d + return lap + + +def _setdiag_dense(m, d): + step = len(d) + 1 + m.flat[::step] = d + + +def _laplace(m, d): + return lambda v: v * d[:, np.newaxis] - m @ v + + +def _laplace_normed(m, d, nd): + laplace = _laplace(m, d) + return lambda v: nd[:, np.newaxis] * laplace(v * nd[:, np.newaxis]) + + +def _laplace_sym(m, d): + return ( + lambda v: v * d[:, np.newaxis] + - m @ v + - np.transpose(np.conjugate(np.transpose(np.conjugate(v)) @ m)) + ) + + +def _laplace_normed_sym(m, d, nd): + laplace_sym = _laplace_sym(m, d) + return lambda v: nd[:, np.newaxis] * laplace_sym(v * nd[:, np.newaxis]) + + +def _linearoperator(mv, shape, dtype): + return LinearOperator(matvec=mv, matmat=mv, shape=shape, dtype=dtype) + + +def _laplacian_sparse_flo(graph, normed, axis, copy, form, dtype, symmetrized): + # The keyword argument `copy` is unused and has no effect here. + del copy + + if dtype is None: + dtype = graph.dtype + + graph_sum = np.asarray(graph.sum(axis=axis)).ravel() + graph_diagonal = graph.diagonal() + diag = graph_sum - graph_diagonal + if symmetrized: + graph_sum += np.asarray(graph.sum(axis=1 - axis)).ravel() + diag = graph_sum - graph_diagonal - graph_diagonal + + if normed: + isolated_node_mask = diag == 0 + w = np.where(isolated_node_mask, 1, np.sqrt(diag)) + if symmetrized: + md = _laplace_normed_sym(graph, graph_sum, 1.0 / w) + else: + md = _laplace_normed(graph, graph_sum, 1.0 / w) + if form == "function": + return md, w.astype(dtype, copy=False) + elif form == "lo": + m = _linearoperator(md, shape=graph.shape, dtype=dtype) + return m, w.astype(dtype, copy=False) + else: + raise ValueError(f"Invalid form: {form!r}") + else: + if symmetrized: + md = _laplace_sym(graph, graph_sum) + else: + md = _laplace(graph, graph_sum) + if form == "function": + return md, diag.astype(dtype, copy=False) + elif form == "lo": + m = _linearoperator(md, shape=graph.shape, dtype=dtype) + return m, diag.astype(dtype, copy=False) + else: + raise ValueError(f"Invalid form: {form!r}") + + +def _laplacian_sparse(graph, normed, axis, copy, form, dtype, symmetrized): + # The keyword argument `form` is unused and has no effect here. + del form + + if dtype is None: + dtype = graph.dtype + + needs_copy = False + if graph.format in ('lil', 'dok'): + m = graph.tocoo() + else: + m = graph + if copy: + needs_copy = True + + if symmetrized: + m += m.T.conj() + + w = np.asarray(m.sum(axis=axis)).ravel() - m.diagonal() + if normed: + m = m.tocoo(copy=needs_copy) + isolated_node_mask = (w == 0) + w = np.where(isolated_node_mask, 1, np.sqrt(w)) + m.data /= w[m.row] + m.data /= w[m.col] + m.data *= -1 + m.setdiag(1 - isolated_node_mask) + else: + if m.format == 'dia': + m = m.copy() + else: + m = m.tocoo(copy=needs_copy) + m.data *= -1 + m.setdiag(w) + + return m.astype(dtype, copy=False), w.astype(dtype) + + +def _laplacian_dense_flo(graph, normed, axis, copy, form, dtype, symmetrized): + + if copy: + m = np.array(graph) + else: + m = np.asarray(graph) + + if dtype is None: + dtype = m.dtype + + graph_sum = m.sum(axis=axis) + graph_diagonal = m.diagonal() + diag = graph_sum - graph_diagonal + if symmetrized: + graph_sum += m.sum(axis=1 - axis) + diag = graph_sum - graph_diagonal - graph_diagonal + + if normed: + isolated_node_mask = diag == 0 + w = np.where(isolated_node_mask, 1, np.sqrt(diag)) + if symmetrized: + md = _laplace_normed_sym(m, graph_sum, 1.0 / w) + else: + md = _laplace_normed(m, graph_sum, 1.0 / w) + if form == "function": + return md, w.astype(dtype, copy=False) + elif form == "lo": + m = _linearoperator(md, shape=graph.shape, dtype=dtype) + return m, w.astype(dtype, copy=False) + else: + raise ValueError(f"Invalid form: {form!r}") + else: + if symmetrized: + md = _laplace_sym(m, graph_sum) + else: + md = _laplace(m, graph_sum) + if form == "function": + return md, diag.astype(dtype, copy=False) + elif form == "lo": + m = _linearoperator(md, shape=graph.shape, dtype=dtype) + return m, diag.astype(dtype, copy=False) + else: + raise ValueError(f"Invalid form: {form!r}") + + +def _laplacian_dense(graph, normed, axis, copy, form, dtype, symmetrized): + + if form != "array": + raise ValueError(f'{form!r} must be "array"') + + if dtype is None: + dtype = graph.dtype + + if copy: + m = np.array(graph) + else: + m = np.asarray(graph) + + if dtype is None: + dtype = m.dtype + + if symmetrized: + m += m.T.conj() + np.fill_diagonal(m, 0) + w = m.sum(axis=axis) + if normed: + isolated_node_mask = (w == 0) + w = np.where(isolated_node_mask, 1, np.sqrt(w)) + m /= w + m /= w[:, np.newaxis] + m *= -1 + _setdiag_dense(m, 1 - isolated_node_mask) + else: + m *= -1 + _setdiag_dense(m, w) + + return m.astype(dtype, copy=False), w.astype(dtype, copy=False) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/_validation.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/_validation.py new file mode 100644 index 0000000000000000000000000000000000000000..6eb9ce811b73e751ebb1cd6b226b73f7bcfe7ceb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/_validation.py @@ -0,0 +1,66 @@ +import numpy as np +from scipy.sparse import issparse +from scipy.sparse._sputils import convert_pydata_sparse_to_scipy +from scipy.sparse.csgraph._tools import ( + csgraph_to_dense, csgraph_from_dense, + csgraph_masked_from_dense, csgraph_from_masked +) + +DTYPE = np.float64 + + +def validate_graph(csgraph, directed, dtype=DTYPE, + csr_output=True, dense_output=True, + copy_if_dense=False, copy_if_sparse=False, + null_value_in=0, null_value_out=np.inf, + infinity_null=True, nan_null=True): + """Routine for validation and conversion of csgraph inputs""" + if not (csr_output or dense_output): + raise ValueError("Internal: dense or csr output must be true") + + accept_fv = [null_value_in] + if infinity_null: + accept_fv.append(np.inf) + if nan_null: + accept_fv.append(np.nan) + csgraph = convert_pydata_sparse_to_scipy(csgraph, accept_fv=accept_fv) + + # if undirected and csc storage, then transposing in-place + # is quicker than later converting to csr. + if (not directed) and issparse(csgraph) and csgraph.format == "csc": + csgraph = csgraph.T + + if issparse(csgraph): + if csr_output: + csgraph = csgraph.tocsr(copy=copy_if_sparse).astype(DTYPE, copy=False) + else: + csgraph = csgraph_to_dense(csgraph, null_value=null_value_out) + elif np.ma.isMaskedArray(csgraph): + if dense_output: + mask = csgraph.mask + csgraph = np.array(csgraph.data, dtype=DTYPE, copy=copy_if_dense) + csgraph[mask] = null_value_out + else: + csgraph = csgraph_from_masked(csgraph) + else: + if dense_output: + csgraph = csgraph_masked_from_dense(csgraph, + copy=copy_if_dense, + null_value=null_value_in, + nan_null=nan_null, + infinity_null=infinity_null) + mask = csgraph.mask + csgraph = np.asarray(csgraph.data, dtype=DTYPE) + csgraph[mask] = null_value_out + else: + csgraph = csgraph_from_dense(csgraph, null_value=null_value_in, + infinity_null=infinity_null, + nan_null=nan_null) + + if csgraph.ndim != 2: + raise ValueError("compressed-sparse graph must be 2-D") + + if csgraph.shape[0] != csgraph.shape[1]: + raise ValueError("compressed-sparse graph must be shape (N, N)") + + return csgraph diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_connected_components.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_connected_components.py new file mode 100644 index 0000000000000000000000000000000000000000..0b190a24deb9f2818893a120f8ea376fbfb8d6fe --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_connected_components.py @@ -0,0 +1,119 @@ +import numpy as np +from numpy.testing import assert_equal, assert_array_almost_equal +from scipy.sparse import csgraph, csr_array + + +def test_weak_connections(): + Xde = np.array([[0, 1, 0], + [0, 0, 0], + [0, 0, 0]]) + + Xsp = csgraph.csgraph_from_dense(Xde, null_value=0) + + for X in Xsp, Xde: + n_components, labels =\ + csgraph.connected_components(X, directed=True, + connection='weak') + + assert_equal(n_components, 2) + assert_array_almost_equal(labels, [0, 0, 1]) + + +def test_strong_connections(): + X1de = np.array([[0, 1, 0], + [0, 0, 0], + [0, 0, 0]]) + X2de = X1de + X1de.T + + X1sp = csgraph.csgraph_from_dense(X1de, null_value=0) + X2sp = csgraph.csgraph_from_dense(X2de, null_value=0) + + for X in X1sp, X1de: + n_components, labels =\ + csgraph.connected_components(X, directed=True, + connection='strong') + + assert_equal(n_components, 3) + labels.sort() + assert_array_almost_equal(labels, [0, 1, 2]) + + for X in X2sp, X2de: + n_components, labels =\ + csgraph.connected_components(X, directed=True, + connection='strong') + + assert_equal(n_components, 2) + labels.sort() + assert_array_almost_equal(labels, [0, 0, 1]) + + +def test_strong_connections2(): + X = np.array([[0, 0, 0, 0, 0, 0], + [1, 0, 1, 0, 0, 0], + [0, 0, 0, 1, 0, 0], + [0, 0, 1, 0, 1, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 1, 0]]) + n_components, labels =\ + csgraph.connected_components(X, directed=True, + connection='strong') + assert_equal(n_components, 5) + labels.sort() + assert_array_almost_equal(labels, [0, 1, 2, 2, 3, 4]) + + +def test_weak_connections2(): + X = np.array([[0, 0, 0, 0, 0, 0], + [1, 0, 0, 0, 0, 0], + [0, 0, 0, 1, 0, 0], + [0, 0, 1, 0, 1, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 1, 0]]) + n_components, labels =\ + csgraph.connected_components(X, directed=True, + connection='weak') + assert_equal(n_components, 2) + labels.sort() + assert_array_almost_equal(labels, [0, 0, 1, 1, 1, 1]) + + +def test_ticket1876(): + # Regression test: this failed in the original implementation + # There should be two strongly-connected components; previously gave one + g = np.array([[0, 1, 1, 0], + [1, 0, 0, 1], + [0, 0, 0, 1], + [0, 0, 1, 0]]) + n_components, labels = csgraph.connected_components(g, connection='strong') + + assert_equal(n_components, 2) + assert_equal(labels[0], labels[1]) + assert_equal(labels[2], labels[3]) + + +def test_fully_connected_graph(): + # Fully connected dense matrices raised an exception. + # https://github.com/scipy/scipy/issues/3818 + g = np.ones((4, 4)) + n_components, labels = csgraph.connected_components(g) + assert_equal(n_components, 1) + + +def test_int64_indices_undirected(): + # See https://github.com/scipy/scipy/issues/18716 + g = csr_array(([1], np.array([[0], [1]], dtype=np.int64)), shape=(2, 2)) + assert g.indices.dtype == np.int64 + n, labels = csgraph.connected_components(g, directed=False) + assert n == 1 + assert_array_almost_equal(labels, [0, 0]) + + +def test_int64_indices_directed(): + # See https://github.com/scipy/scipy/issues/18716 + g = csr_array(([1], np.array([[0], [1]], dtype=np.int64)), shape=(2, 2)) + assert g.indices.dtype == np.int64 + n, labels = csgraph.connected_components(g, directed=True, + connection='strong') + assert n == 2 + assert_array_almost_equal(labels, [1, 0]) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_conversions.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_conversions.py new file mode 100644 index 0000000000000000000000000000000000000000..65f141e5b371367018a6e9985f8325850d8972da --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_conversions.py @@ -0,0 +1,61 @@ +import numpy as np +from numpy.testing import assert_array_almost_equal +from scipy.sparse import csr_array +from scipy.sparse.csgraph import csgraph_from_dense, csgraph_to_dense + + +def test_csgraph_from_dense(): + np.random.seed(1234) + G = np.random.random((10, 10)) + some_nulls = (G < 0.4) + all_nulls = (G < 0.8) + + for null_value in [0, np.nan, np.inf]: + G[all_nulls] = null_value + with np.errstate(invalid="ignore"): + G_csr = csgraph_from_dense(G, null_value=0) + + G[all_nulls] = 0 + assert_array_almost_equal(G, G_csr.toarray()) + + for null_value in [np.nan, np.inf]: + G[all_nulls] = 0 + G[some_nulls] = null_value + with np.errstate(invalid="ignore"): + G_csr = csgraph_from_dense(G, null_value=0) + + G[all_nulls] = 0 + assert_array_almost_equal(G, G_csr.toarray()) + + +def test_csgraph_to_dense(): + np.random.seed(1234) + G = np.random.random((10, 10)) + nulls = (G < 0.8) + G[nulls] = np.inf + + G_csr = csgraph_from_dense(G) + + for null_value in [0, 10, -np.inf, np.inf]: + G[nulls] = null_value + assert_array_almost_equal(G, csgraph_to_dense(G_csr, null_value)) + + +def test_multiple_edges(): + # create a random square matrix with an even number of elements + np.random.seed(1234) + X = np.random.random((10, 10)) + Xcsr = csr_array(X) + + # now double-up every other column + Xcsr.indices[::2] = Xcsr.indices[1::2] + + # normal sparse toarray() will sum the duplicated edges + Xdense = Xcsr.toarray() + assert_array_almost_equal(Xdense[:, 1::2], + X[:, ::2] + X[:, 1::2]) + + # csgraph_to_dense chooses the minimum of each duplicated edge + Xdense = csgraph_to_dense(Xcsr) + assert_array_almost_equal(Xdense[:, 1::2], + np.minimum(X[:, ::2], X[:, 1::2])) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_flow.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_flow.py new file mode 100644 index 0000000000000000000000000000000000000000..c92eb985a1145c4b7c1777f0449bb423402f6d66 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_flow.py @@ -0,0 +1,209 @@ +import numpy as np +from numpy.testing import assert_array_equal +import pytest + +from scipy.sparse import csr_array, csc_array, csr_matrix +from scipy.sparse.csgraph import maximum_flow +from scipy.sparse.csgraph._flow import ( + _add_reverse_edges, _make_edge_pointers, _make_tails +) + +methods = ['edmonds_karp', 'dinic'] + +def test_raises_on_dense_input(): + with pytest.raises(TypeError): + graph = np.array([[0, 1], [0, 0]]) + maximum_flow(graph, 0, 1) + maximum_flow(graph, 0, 1, method='edmonds_karp') + + +def test_raises_on_csc_input(): + with pytest.raises(TypeError): + graph = csc_array([[0, 1], [0, 0]]) + maximum_flow(graph, 0, 1) + maximum_flow(graph, 0, 1, method='edmonds_karp') + + +def test_raises_on_floating_point_input(): + with pytest.raises(ValueError): + graph = csr_array([[0, 1.5], [0, 0]], dtype=np.float64) + maximum_flow(graph, 0, 1) + maximum_flow(graph, 0, 1, method='edmonds_karp') + + +def test_raises_on_non_square_input(): + with pytest.raises(ValueError): + graph = csr_array([[0, 1, 2], [2, 1, 0]]) + maximum_flow(graph, 0, 1) + + +def test_raises_when_source_is_sink(): + with pytest.raises(ValueError): + graph = csr_array([[0, 1], [0, 0]]) + maximum_flow(graph, 0, 0) + maximum_flow(graph, 0, 0, method='edmonds_karp') + + +@pytest.mark.parametrize('method', methods) +@pytest.mark.parametrize('source', [-1, 2, 3]) +def test_raises_when_source_is_out_of_bounds(source, method): + with pytest.raises(ValueError): + graph = csr_array([[0, 1], [0, 0]]) + maximum_flow(graph, source, 1, method=method) + + +@pytest.mark.parametrize('method', methods) +@pytest.mark.parametrize('sink', [-1, 2, 3]) +def test_raises_when_sink_is_out_of_bounds(sink, method): + with pytest.raises(ValueError): + graph = csr_array([[0, 1], [0, 0]]) + maximum_flow(graph, 0, sink, method=method) + + +@pytest.mark.parametrize('method', methods) +def test_simple_graph(method): + # This graph looks as follows: + # (0) --5--> (1) + graph = csr_array([[0, 5], [0, 0]]) + res = maximum_flow(graph, 0, 1, method=method) + assert res.flow_value == 5 + expected_flow = np.array([[0, 5], [-5, 0]]) + assert_array_equal(res.flow.toarray(), expected_flow) + + +@pytest.mark.parametrize('method', methods) +def test_return_type(method): + graph = csr_array([[0, 5], [0, 0]]) + assert isinstance(maximum_flow(graph, 0, 1, method=method).flow, csr_array) + graph = csr_matrix([[0, 5], [0, 0]]) + assert isinstance(maximum_flow(graph, 0, 1, method=method).flow, csr_matrix) + + +@pytest.mark.parametrize('method', methods) +def test_bottle_neck_graph(method): + # This graph cannot use the full capacity between 0 and 1: + # (0) --5--> (1) --3--> (2) + graph = csr_array([[0, 5, 0], [0, 0, 3], [0, 0, 0]]) + res = maximum_flow(graph, 0, 2, method=method) + assert res.flow_value == 3 + expected_flow = np.array([[0, 3, 0], [-3, 0, 3], [0, -3, 0]]) + assert_array_equal(res.flow.toarray(), expected_flow) + + +@pytest.mark.parametrize('method', methods) +def test_backwards_flow(method): + # This example causes backwards flow between vertices 3 and 4, + # and so this test ensures that we handle that accordingly. See + # https://stackoverflow.com/q/38843963/5085211 + # for more information. + graph = csr_array([[0, 10, 0, 0, 10, 0, 0, 0], + [0, 0, 10, 0, 0, 0, 0, 0], + [0, 0, 0, 10, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 10], + [0, 0, 0, 10, 0, 10, 0, 0], + [0, 0, 0, 0, 0, 0, 10, 0], + [0, 0, 0, 0, 0, 0, 0, 10], + [0, 0, 0, 0, 0, 0, 0, 0]]) + res = maximum_flow(graph, 0, 7, method=method) + assert res.flow_value == 20 + expected_flow = np.array([[0, 10, 0, 0, 10, 0, 0, 0], + [-10, 0, 10, 0, 0, 0, 0, 0], + [0, -10, 0, 10, 0, 0, 0, 0], + [0, 0, -10, 0, 0, 0, 0, 10], + [-10, 0, 0, 0, 0, 10, 0, 0], + [0, 0, 0, 0, -10, 0, 10, 0], + [0, 0, 0, 0, 0, -10, 0, 10], + [0, 0, 0, -10, 0, 0, -10, 0]]) + assert_array_equal(res.flow.toarray(), expected_flow) + + +@pytest.mark.parametrize('method', methods) +def test_example_from_clrs_chapter_26_1(method): + # See page 659 in CLRS second edition, but note that the maximum flow + # we find is slightly different than the one in CLRS; we push a flow of + # 12 to v_1 instead of v_2. + graph = csr_array([[0, 16, 13, 0, 0, 0], + [0, 0, 10, 12, 0, 0], + [0, 4, 0, 0, 14, 0], + [0, 0, 9, 0, 0, 20], + [0, 0, 0, 7, 0, 4], + [0, 0, 0, 0, 0, 0]]) + res = maximum_flow(graph, 0, 5, method=method) + assert res.flow_value == 23 + expected_flow = np.array([[0, 12, 11, 0, 0, 0], + [-12, 0, 0, 12, 0, 0], + [-11, 0, 0, 0, 11, 0], + [0, -12, 0, 0, -7, 19], + [0, 0, -11, 7, 0, 4], + [0, 0, 0, -19, -4, 0]]) + assert_array_equal(res.flow.toarray(), expected_flow) + + +@pytest.mark.parametrize('method', methods) +def test_disconnected_graph(method): + # This tests the following disconnected graph: + # (0) --5--> (1) (2) --3--> (3) + graph = csr_array([[0, 5, 0, 0], + [0, 0, 0, 0], + [0, 0, 9, 3], + [0, 0, 0, 0]]) + res = maximum_flow(graph, 0, 3, method=method) + assert res.flow_value == 0 + expected_flow = np.zeros((4, 4), dtype=np.int32) + assert_array_equal(res.flow.toarray(), expected_flow) + + +@pytest.mark.parametrize('method', methods) +def test_add_reverse_edges_large_graph(method): + # Regression test for https://github.com/scipy/scipy/issues/14385 + n = 100_000 + indices = np.arange(1, n) + indptr = np.array(list(range(n)) + [n - 1]) + data = np.ones(n - 1, dtype=np.int32) + graph = csr_array((data, indices, indptr), shape=(n, n)) + res = maximum_flow(graph, 0, n - 1, method=method) + assert res.flow_value == 1 + expected_flow = graph - graph.transpose() + assert_array_equal(res.flow.data, expected_flow.data) + assert_array_equal(res.flow.indices, expected_flow.indices) + assert_array_equal(res.flow.indptr, expected_flow.indptr) + + +@pytest.mark.parametrize("a,b_data_expected", [ + ([[]], []), + ([[0], [0]], []), + ([[1, 0, 2], [0, 0, 0], [0, 3, 0]], [1, 2, 0, 0, 3]), + ([[9, 8, 7], [4, 5, 6], [0, 0, 0]], [9, 8, 7, 4, 5, 6, 0, 0])]) +def test_add_reverse_edges(a, b_data_expected): + """Test that the reversal of the edges of the input graph works + as expected. + """ + a = csr_array(a, dtype=np.int32, shape=(len(a), len(a))) + b = _add_reverse_edges(a) + assert_array_equal(b.data, b_data_expected) + + +@pytest.mark.parametrize("a,expected", [ + ([[]], []), + ([[0]], []), + ([[1]], [0]), + ([[0, 1], [10, 0]], [1, 0]), + ([[1, 0, 2], [0, 0, 3], [4, 5, 0]], [0, 3, 4, 1, 2]) +]) +def test_make_edge_pointers(a, expected): + a = csr_array(a, dtype=np.int32) + rev_edge_ptr = _make_edge_pointers(a) + assert_array_equal(rev_edge_ptr, expected) + + +@pytest.mark.parametrize("a,expected", [ + ([[]], []), + ([[0]], []), + ([[1]], [0]), + ([[0, 1], [10, 0]], [0, 1]), + ([[1, 0, 2], [0, 0, 3], [4, 5, 0]], [0, 0, 1, 2, 2]) +]) +def test_make_tails(a, expected): + a = csr_array(a, dtype=np.int32) + tails = _make_tails(a) + assert_array_equal(tails, expected) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_graph_laplacian.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_graph_laplacian.py new file mode 100644 index 0000000000000000000000000000000000000000..0ed5e2edf92ef0cacc819fcbd06bc6d4e195cb44 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_graph_laplacian.py @@ -0,0 +1,368 @@ +import pytest +import numpy as np +from numpy.testing import assert_allclose +from pytest import raises as assert_raises +from scipy import sparse + +from scipy.sparse import csgraph +from scipy._lib._util import np_long, np_ulong + + +def check_int_type(mat): + return np.issubdtype(mat.dtype, np.signedinteger) or np.issubdtype( + mat.dtype, np_ulong + ) + + +def test_laplacian_value_error(): + for t in int, float, complex: + for m in ([1, 1], + [[[1]]], + [[1, 2, 3], [4, 5, 6]], + [[1, 2], [3, 4], [5, 5]]): + A = np.array(m, dtype=t) + assert_raises(ValueError, csgraph.laplacian, A) + + +def _explicit_laplacian(x, normed=False): + if sparse.issparse(x): + x = x.toarray() + x = np.asarray(x) + y = -1.0 * x + for j in range(y.shape[0]): + y[j,j] = x[j,j+1:].sum() + x[j,:j].sum() + if normed: + d = np.diag(y).copy() + d[d == 0] = 1.0 + y /= d[:,None]**.5 + y /= d[None,:]**.5 + return y + + +def _check_symmetric_graph_laplacian(mat, normed, copy=True): + if not hasattr(mat, 'shape'): + mat = eval(mat, dict(np=np, sparse=sparse)) + + if sparse.issparse(mat): + sp_mat = mat + mat = sp_mat.toarray() + else: + sp_mat = sparse.csr_array(mat) + + mat_copy = np.copy(mat) + sp_mat_copy = sparse.csr_array(sp_mat, copy=True) + + n_nodes = mat.shape[0] + explicit_laplacian = _explicit_laplacian(mat, normed=normed) + laplacian = csgraph.laplacian(mat, normed=normed, copy=copy) + sp_laplacian = csgraph.laplacian(sp_mat, normed=normed, + copy=copy) + + if copy: + assert_allclose(mat, mat_copy) + _assert_allclose_sparse(sp_mat, sp_mat_copy) + else: + if not (normed and check_int_type(mat)): + assert_allclose(laplacian, mat) + if sp_mat.format == 'coo': + _assert_allclose_sparse(sp_laplacian, sp_mat) + + assert_allclose(laplacian, sp_laplacian.toarray()) + + for tested in [laplacian, sp_laplacian.toarray()]: + if not normed: + assert_allclose(tested.sum(axis=0), np.zeros(n_nodes)) + assert_allclose(tested.T, tested) + assert_allclose(tested, explicit_laplacian) + + +def test_symmetric_graph_laplacian(): + symmetric_mats = ( + 'np.arange(10) * np.arange(10)[:, np.newaxis]', + 'np.ones((7, 7))', + 'np.eye(19)', + 'sparse.diags([1, 1], [-1, 1], shape=(4, 4))', + 'sparse.diags([1, 1], [-1, 1], shape=(4, 4)).toarray()', + 'sparse.diags([1, 1], [-1, 1], shape=(4, 4)).todense()', + 'np.vander(np.arange(4)) + np.vander(np.arange(4)).T' + ) + for mat in symmetric_mats: + for normed in True, False: + for copy in True, False: + _check_symmetric_graph_laplacian(mat, normed, copy) + + +def _assert_allclose_sparse(a, b, **kwargs): + # helper function that can deal with sparse matrices + if sparse.issparse(a): + a = a.toarray() + if sparse.issparse(b): + b = b.toarray() + assert_allclose(a, b, **kwargs) + + +def _check_laplacian_dtype_none( + A, desired_L, desired_d, normed, use_out_degree, copy, dtype, arr_type +): + mat = arr_type(A, dtype=dtype) + L, d = csgraph.laplacian( + mat, + normed=normed, + return_diag=True, + use_out_degree=use_out_degree, + copy=copy, + dtype=None, + ) + if normed and check_int_type(mat): + assert L.dtype == np.float64 + assert d.dtype == np.float64 + _assert_allclose_sparse(L, desired_L, atol=1e-12) + _assert_allclose_sparse(d, desired_d, atol=1e-12) + else: + assert L.dtype == dtype + assert d.dtype == dtype + desired_L = np.asarray(desired_L).astype(dtype) + desired_d = np.asarray(desired_d).astype(dtype) + _assert_allclose_sparse(L, desired_L, atol=1e-12) + _assert_allclose_sparse(d, desired_d, atol=1e-12) + + if not copy: + if not (normed and check_int_type(mat)): + if type(mat) is np.ndarray: + assert_allclose(L, mat) + elif mat.format == "coo": + _assert_allclose_sparse(L, mat) + + +def _check_laplacian_dtype( + A, desired_L, desired_d, normed, use_out_degree, copy, dtype, arr_type +): + mat = arr_type(A, dtype=dtype) + L, d = csgraph.laplacian( + mat, + normed=normed, + return_diag=True, + use_out_degree=use_out_degree, + copy=copy, + dtype=dtype, + ) + assert L.dtype == dtype + assert d.dtype == dtype + desired_L = np.asarray(desired_L).astype(dtype) + desired_d = np.asarray(desired_d).astype(dtype) + _assert_allclose_sparse(L, desired_L, atol=1e-12) + _assert_allclose_sparse(d, desired_d, atol=1e-12) + + if not copy: + if not (normed and check_int_type(mat)): + if type(mat) is np.ndarray: + assert_allclose(L, mat) + elif mat.format == 'coo': + _assert_allclose_sparse(L, mat) + + +INT_DTYPES = (np.intc, np_long, np.longlong) +REAL_DTYPES = (np.float32, np.float64, np.longdouble) +COMPLEX_DTYPES = (np.complex64, np.complex128, np.clongdouble) +DTYPES = INT_DTYPES + REAL_DTYPES + COMPLEX_DTYPES + + +@pytest.mark.parametrize("dtype", DTYPES) +@pytest.mark.parametrize("arr_type", [np.array, + sparse.csr_matrix, + sparse.coo_matrix, + sparse.csr_array, + sparse.coo_array]) +@pytest.mark.parametrize("copy", [True, False]) +@pytest.mark.parametrize("normed", [True, False]) +@pytest.mark.parametrize("use_out_degree", [True, False]) +def test_asymmetric_laplacian(use_out_degree, normed, + copy, dtype, arr_type): + # adjacency matrix + A = [[0, 1, 0], + [4, 2, 0], + [0, 0, 0]] + A = arr_type(np.array(A), dtype=dtype) + A_copy = A.copy() + + if not normed and use_out_degree: + # Laplacian matrix using out-degree + L = [[1, -1, 0], + [-4, 4, 0], + [0, 0, 0]] + d = [1, 4, 0] + + if normed and use_out_degree: + # normalized Laplacian matrix using out-degree + L = [[1, -0.5, 0], + [-2, 1, 0], + [0, 0, 0]] + d = [1, 2, 1] + + if not normed and not use_out_degree: + # Laplacian matrix using in-degree + L = [[4, -1, 0], + [-4, 1, 0], + [0, 0, 0]] + d = [4, 1, 0] + + if normed and not use_out_degree: + # normalized Laplacian matrix using in-degree + L = [[1, -0.5, 0], + [-2, 1, 0], + [0, 0, 0]] + d = [2, 1, 1] + + _check_laplacian_dtype_none( + A, + L, + d, + normed=normed, + use_out_degree=use_out_degree, + copy=copy, + dtype=dtype, + arr_type=arr_type, + ) + + _check_laplacian_dtype( + A_copy, + L, + d, + normed=normed, + use_out_degree=use_out_degree, + copy=copy, + dtype=dtype, + arr_type=arr_type, + ) + + +@pytest.mark.parametrize("fmt", ['csr', 'csc', 'coo', 'lil', + 'dok', 'dia', 'bsr']) +@pytest.mark.parametrize("normed", [True, False]) +@pytest.mark.parametrize("copy", [True, False]) +def test_sparse_formats(fmt, normed, copy): + mat = sparse.diags_array([1, 1], offsets=[-1, 1], shape=(4, 4), format=fmt) + _check_symmetric_graph_laplacian(mat, normed, copy) + + +@pytest.mark.parametrize( + "arr_type", [np.asarray, + sparse.csr_matrix, + sparse.coo_matrix, + sparse.csr_array, + sparse.coo_array] +) +@pytest.mark.parametrize("form", ["array", "function", "lo"]) +def test_laplacian_symmetrized(arr_type, form): + # adjacency matrix + n = 3 + mat = arr_type(np.arange(n * n).reshape(n, n)) + L_in, d_in = csgraph.laplacian( + mat, + return_diag=True, + form=form, + ) + L_out, d_out = csgraph.laplacian( + mat, + return_diag=True, + use_out_degree=True, + form=form, + ) + Ls, ds = csgraph.laplacian( + mat, + return_diag=True, + symmetrized=True, + form=form, + ) + Ls_normed, ds_normed = csgraph.laplacian( + mat, + return_diag=True, + symmetrized=True, + normed=True, + form=form, + ) + mat += mat.T + Lss, dss = csgraph.laplacian(mat, return_diag=True, form=form) + Lss_normed, dss_normed = csgraph.laplacian( + mat, + return_diag=True, + normed=True, + form=form, + ) + + assert_allclose(ds, d_in + d_out) + assert_allclose(ds, dss) + assert_allclose(ds_normed, dss_normed) + + d = {} + for L in ["L_in", "L_out", "Ls", "Ls_normed", "Lss", "Lss_normed"]: + if form == "array": + d[L] = eval(L) + else: + d[L] = eval(L)(np.eye(n, dtype=mat.dtype)) + + _assert_allclose_sparse(d["Ls"], d["L_in"] + d["L_out"].T) + _assert_allclose_sparse(d["Ls"], d["Lss"]) + _assert_allclose_sparse(d["Ls_normed"], d["Lss_normed"]) + + +@pytest.mark.parametrize( + "arr_type", [np.asarray, + sparse.csr_matrix, + sparse.coo_matrix, + sparse.csr_array, + sparse.coo_array] +) +@pytest.mark.parametrize("dtype", DTYPES) +@pytest.mark.parametrize("normed", [True, False]) +@pytest.mark.parametrize("symmetrized", [True, False]) +@pytest.mark.parametrize("use_out_degree", [True, False]) +@pytest.mark.parametrize("form", ["function", "lo"]) +def test_format(dtype, arr_type, normed, symmetrized, use_out_degree, form): + n = 3 + mat = [[0, 1, 0], [4, 2, 0], [0, 0, 0]] + mat = arr_type(np.array(mat), dtype=dtype) + Lo, do = csgraph.laplacian( + mat, + return_diag=True, + normed=normed, + symmetrized=symmetrized, + use_out_degree=use_out_degree, + dtype=dtype, + ) + La, da = csgraph.laplacian( + mat, + return_diag=True, + normed=normed, + symmetrized=symmetrized, + use_out_degree=use_out_degree, + dtype=dtype, + form="array", + ) + assert_allclose(do, da) + _assert_allclose_sparse(Lo, La) + + L, d = csgraph.laplacian( + mat, + return_diag=True, + normed=normed, + symmetrized=symmetrized, + use_out_degree=use_out_degree, + dtype=dtype, + form=form, + ) + assert_allclose(d, do) + assert d.dtype == dtype + Lm = L(np.eye(n, dtype=mat.dtype)).astype(dtype) + _assert_allclose_sparse(Lm, Lo, rtol=2e-7, atol=2e-7) + x = np.arange(6).reshape(3, 2) + if not (normed and dtype in INT_DTYPES): + assert_allclose(L(x), Lo @ x) + else: + # Normalized Lo is casted to integer, but L() is not + pass + + +def test_format_error_message(): + with pytest.raises(ValueError, match="Invalid form: 'toto'"): + _ = csgraph.laplacian(np.eye(1), form='toto') diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_matching.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_matching.py new file mode 100644 index 0000000000000000000000000000000000000000..8477861d3e563c43379c615ec0913f47a4349abd --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_matching.py @@ -0,0 +1,295 @@ +from itertools import product + +import numpy as np +from numpy.testing import assert_array_equal, assert_equal +import pytest + +from scipy.sparse import csr_array, diags_array +from scipy.sparse.csgraph import ( + maximum_bipartite_matching, min_weight_full_bipartite_matching +) + + +def test_maximum_bipartite_matching_raises_on_dense_input(): + with pytest.raises(TypeError): + graph = np.array([[0, 1], [0, 0]]) + maximum_bipartite_matching(graph) + + +def test_maximum_bipartite_matching_empty_graph(): + graph = csr_array((0, 0)) + x = maximum_bipartite_matching(graph, perm_type='row') + y = maximum_bipartite_matching(graph, perm_type='column') + expected_matching = np.array([]) + assert_array_equal(expected_matching, x) + assert_array_equal(expected_matching, y) + + +def test_maximum_bipartite_matching_empty_left_partition(): + graph = csr_array((2, 0)) + x = maximum_bipartite_matching(graph, perm_type='row') + y = maximum_bipartite_matching(graph, perm_type='column') + assert_array_equal(np.array([]), x) + assert_array_equal(np.array([-1, -1]), y) + + +def test_maximum_bipartite_matching_empty_right_partition(): + graph = csr_array((0, 3)) + x = maximum_bipartite_matching(graph, perm_type='row') + y = maximum_bipartite_matching(graph, perm_type='column') + assert_array_equal(np.array([-1, -1, -1]), x) + assert_array_equal(np.array([]), y) + + +def test_maximum_bipartite_matching_graph_with_no_edges(): + graph = csr_array((2, 2)) + x = maximum_bipartite_matching(graph, perm_type='row') + y = maximum_bipartite_matching(graph, perm_type='column') + assert_array_equal(np.array([-1, -1]), x) + assert_array_equal(np.array([-1, -1]), y) + + +def test_maximum_bipartite_matching_graph_that_causes_augmentation(): + # In this graph, column 1 is initially assigned to row 1, but it should be + # reassigned to make room for row 2. + graph = csr_array([[1, 1], [1, 0]]) + x = maximum_bipartite_matching(graph, perm_type='column') + y = maximum_bipartite_matching(graph, perm_type='row') + expected_matching = np.array([1, 0]) + assert_array_equal(expected_matching, x) + assert_array_equal(expected_matching, y) + + +def test_maximum_bipartite_matching_graph_with_more_rows_than_columns(): + graph = csr_array([[1, 1], [1, 0], [0, 1]]) + x = maximum_bipartite_matching(graph, perm_type='column') + y = maximum_bipartite_matching(graph, perm_type='row') + assert_array_equal(np.array([0, -1, 1]), x) + assert_array_equal(np.array([0, 2]), y) + + +def test_maximum_bipartite_matching_graph_with_more_columns_than_rows(): + graph = csr_array([[1, 1, 0], [0, 0, 1]]) + x = maximum_bipartite_matching(graph, perm_type='column') + y = maximum_bipartite_matching(graph, perm_type='row') + assert_array_equal(np.array([0, 2]), x) + assert_array_equal(np.array([0, -1, 1]), y) + + +def test_maximum_bipartite_matching_explicit_zeros_count_as_edges(): + data = [0, 0] + indices = [1, 0] + indptr = [0, 1, 2] + graph = csr_array((data, indices, indptr), shape=(2, 2)) + x = maximum_bipartite_matching(graph, perm_type='row') + y = maximum_bipartite_matching(graph, perm_type='column') + expected_matching = np.array([1, 0]) + assert_array_equal(expected_matching, x) + assert_array_equal(expected_matching, y) + + +def test_maximum_bipartite_matching_feasibility_of_result(): + # This is a regression test for GitHub issue #11458 + data = np.ones(50, dtype=int) + indices = [11, 12, 19, 22, 23, 5, 22, 3, 8, 10, 5, 6, 11, 12, 13, 5, 13, + 14, 20, 22, 3, 15, 3, 13, 14, 11, 12, 19, 22, 23, 5, 22, 3, 8, + 10, 5, 6, 11, 12, 13, 5, 13, 14, 20, 22, 3, 15, 3, 13, 14] + indptr = [0, 5, 7, 10, 10, 15, 20, 22, 22, 23, 25, 30, 32, 35, 35, 40, 45, + 47, 47, 48, 50] + graph = csr_array((data, indices, indptr), shape=(20, 25)) + x = maximum_bipartite_matching(graph, perm_type='row') + y = maximum_bipartite_matching(graph, perm_type='column') + assert (x != -1).sum() == 13 + assert (y != -1).sum() == 13 + # Ensure that each element of the matching is in fact an edge in the graph. + for u, v in zip(range(graph.shape[0]), y): + if v != -1: + assert graph[u, v] + for u, v in zip(x, range(graph.shape[1])): + if u != -1: + assert graph[u, v] + + +def test_matching_large_random_graph_with_one_edge_incident_to_each_vertex(): + np.random.seed(42) + A = diags_array(np.ones(25), offsets=0, format='csr') + rand_perm = np.random.permutation(25) + rand_perm2 = np.random.permutation(25) + + Rrow = np.arange(25) + Rcol = rand_perm + Rdata = np.ones(25, dtype=int) + Rmat = csr_array((Rdata, (Rrow, Rcol))) + + Crow = rand_perm2 + Ccol = np.arange(25) + Cdata = np.ones(25, dtype=int) + Cmat = csr_array((Cdata, (Crow, Ccol))) + # Randomly permute identity matrix + B = Rmat @ A @ Cmat + + # Row permute + perm = maximum_bipartite_matching(B, perm_type='row') + Rrow = np.arange(25) + Rcol = perm + Rdata = np.ones(25, dtype=int) + Rmat = csr_array((Rdata, (Rrow, Rcol))) + C1 = Rmat @ B + + # Column permute + perm2 = maximum_bipartite_matching(B, perm_type='column') + Crow = perm2 + Ccol = np.arange(25) + Cdata = np.ones(25, dtype=int) + Cmat = csr_array((Cdata, (Crow, Ccol))) + C2 = B @ Cmat + + # Should get identity matrix back + assert_equal(any(C1.diagonal() == 0), False) + assert_equal(any(C2.diagonal() == 0), False) + + +@pytest.mark.parametrize('num_rows,num_cols', [(0, 0), (2, 0), (0, 3)]) +def test_min_weight_full_matching_trivial_graph(num_rows, num_cols): + biadjacency = csr_array((num_cols, num_rows)) + row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency) + assert len(row_ind) == 0 + assert len(col_ind) == 0 + + +@pytest.mark.parametrize('biadjacency', + [ + [[1, 1, 1], [1, 0, 0], [1, 0, 0]], + [[1, 1, 1], [0, 0, 1], [0, 0, 1]], + [[1, 0, 0, 1], [1, 1, 0, 1], [0, 0, 0, 0]], + [[1, 0, 0], [2, 0, 0]], + [[0, 1, 0], [0, 2, 0]], + [[1, 0], [2, 0], [5, 0]] + ]) +def test_min_weight_full_matching_infeasible_problems(biadjacency): + with pytest.raises(ValueError): + min_weight_full_bipartite_matching(csr_array(biadjacency)) + + +def test_min_weight_full_matching_large_infeasible(): + # Regression test for GitHub issue #17269 + a = np.asarray([ + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.001, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.001, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.001, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.001, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.001, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.001, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.001, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.001, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.001], + [0.0, 0.11687445, 0.0, 0.0, 0.01319788, 0.07509257, 0.0, + 0.0, 0.0, 0.74228317, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.81087935, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.8408466, 0.0, 0.0, 0.0, 0.0, 0.01194389, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.82994211, 0.0, 0.0, 0.0, 0.11468516, 0.0, 0.0, 0.0, + 0.11173505, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.0, 0.0], + [0.18796507, 0.0, 0.04002318, 0.0, 0.0, 0.0, 0.0, 0.0, 0.75883335, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.71545464, 0.0, 0.0, 0.0, 0.0, 0.0, 0.02748488, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.78470564, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.14829198, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.10870609, 0.0, 0.0, 0.0, 0.8918677, 0.0, 0.0, 0.0, 0.06306644, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, + 0.63844085, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7442354, 0.0, 0.0, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.09850549, 0.0, 0.0, 0.18638258, + 0.2769244, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.73182464, 0.0, 0.0, 0.46443561, + 0.38589284, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.29510278, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.09666032, 0.0, + 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] + ]) + with pytest.raises(ValueError, match='no full matching exists'): + min_weight_full_bipartite_matching(csr_array(a)) + + +@pytest.mark.thread_unsafe +def test_explicit_zero_causes_warning(): + with pytest.warns(UserWarning): + biadjacency = csr_array(((2, 0, 3), (0, 1, 1), (0, 2, 3))) + min_weight_full_bipartite_matching(biadjacency) + + +# General test for linear sum assignment solvers to make it possible to rely +# on the same tests for scipy.optimize.linear_sum_assignment. +def linear_sum_assignment_assertions( + solver, array_type, sign, test_case +): + cost_matrix, expected_cost = test_case + maximize = sign == -1 + cost_matrix = sign * array_type(cost_matrix) + expected_cost = sign * np.array(expected_cost) + + row_ind, col_ind = solver(cost_matrix, maximize=maximize) + assert_array_equal(row_ind, np.sort(row_ind)) + assert_array_equal(expected_cost, + np.array(cost_matrix[row_ind, col_ind]).flatten()) + + cost_matrix = cost_matrix.T + row_ind, col_ind = solver(cost_matrix, maximize=maximize) + assert_array_equal(row_ind, np.sort(row_ind)) + assert_array_equal(np.sort(expected_cost), + np.sort(np.array( + cost_matrix[row_ind, col_ind])).flatten()) + + +linear_sum_assignment_test_cases = product( + [-1, 1], + [ + # Square + ([[400, 150, 400], + [400, 450, 600], + [300, 225, 300]], + [150, 400, 300]), + + # Rectangular variant + ([[400, 150, 400, 1], + [400, 450, 600, 2], + [300, 225, 300, 3]], + [150, 2, 300]), + + ([[10, 10, 8], + [9, 8, 1], + [9, 7, 4]], + [10, 1, 7]), + + # Square + ([[10, 10, 8, 11], + [9, 8, 1, 1], + [9, 7, 4, 10]], + [10, 1, 4]), + + # Rectangular variant + ([[10, float("inf"), float("inf")], + [float("inf"), float("inf"), 1], + [float("inf"), 7, float("inf")]], + [10, 1, 7]) + ]) + + +@pytest.mark.parametrize('sign,test_case', linear_sum_assignment_test_cases) +def test_min_weight_full_matching_small_inputs(sign, test_case): + linear_sum_assignment_assertions( + min_weight_full_bipartite_matching, csr_array, sign, test_case) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_pydata_sparse.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_pydata_sparse.py new file mode 100644 index 0000000000000000000000000000000000000000..1476c29a3ba97869c6c38be4d717641a494c1183 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_pydata_sparse.py @@ -0,0 +1,194 @@ +import pytest + +import numpy as np +import scipy.sparse as sp +import scipy.sparse.csgraph as spgraph +from scipy._lib import _pep440 + +from numpy.testing import assert_equal + +try: + import sparse +except Exception: + sparse = None + +pytestmark = pytest.mark.skipif(sparse is None, + reason="pydata/sparse not installed") + + +msg = "pydata/sparse (0.15.1) does not implement necessary operations" + + +sparse_params = (pytest.param("COO"), + pytest.param("DOK", marks=[pytest.mark.xfail(reason=msg)])) + + +def check_sparse_version(min_ver): + if sparse is None: + return pytest.mark.skip(reason="sparse is not installed") + return pytest.mark.skipif( + _pep440.parse(sparse.__version__) < _pep440.Version(min_ver), + reason=f"sparse version >= {min_ver} required" + ) + + +@pytest.fixture(params=sparse_params) +def sparse_cls(request): + return getattr(sparse, request.param) + + +@pytest.fixture +def graphs(sparse_cls): + graph = [ + [0, 1, 1, 0, 0], + [0, 0, 1, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 1], + [0, 0, 0, 0, 0], + ] + A_dense = np.array(graph) + A_sparse = sparse_cls(A_dense) + return A_dense, A_sparse + + +@pytest.mark.parametrize( + "func", + [ + spgraph.shortest_path, + spgraph.dijkstra, + spgraph.floyd_warshall, + spgraph.bellman_ford, + spgraph.johnson, + spgraph.reverse_cuthill_mckee, + spgraph.maximum_bipartite_matching, + spgraph.structural_rank, + ] +) +def test_csgraph_equiv(func, graphs): + A_dense, A_sparse = graphs + actual = func(A_sparse) + desired = func(sp.csc_array(A_dense)) + assert_equal(actual, desired) + + +def test_connected_components(graphs): + A_dense, A_sparse = graphs + func = spgraph.connected_components + + actual_comp, actual_labels = func(A_sparse) + desired_comp, desired_labels, = func(sp.csc_array(A_dense)) + + assert actual_comp == desired_comp + assert_equal(actual_labels, desired_labels) + + +def test_laplacian(graphs): + A_dense, A_sparse = graphs + sparse_cls = type(A_sparse) + func = spgraph.laplacian + + actual = func(A_sparse) + desired = func(sp.csc_array(A_dense)) + + assert isinstance(actual, sparse_cls) + + assert_equal(actual.todense(), desired.todense()) + + +@pytest.mark.parametrize( + "func", [spgraph.breadth_first_order, spgraph.depth_first_order] +) +def test_order_search(graphs, func): + A_dense, A_sparse = graphs + + actual = func(A_sparse, 0) + desired = func(sp.csc_array(A_dense), 0) + + assert_equal(actual, desired) + + +@pytest.mark.parametrize( + "func", [spgraph.breadth_first_tree, spgraph.depth_first_tree] +) +def test_tree_search(graphs, func): + A_dense, A_sparse = graphs + sparse_cls = type(A_sparse) + + actual = func(A_sparse, 0) + desired = func(sp.csc_array(A_dense), 0) + + assert isinstance(actual, sparse_cls) + + assert_equal(actual.todense(), desired.todense()) + + +def test_minimum_spanning_tree(graphs): + A_dense, A_sparse = graphs + sparse_cls = type(A_sparse) + func = spgraph.minimum_spanning_tree + + actual = func(A_sparse) + desired = func(sp.csc_array(A_dense)) + + assert isinstance(actual, sparse_cls) + + assert_equal(actual.todense(), desired.todense()) + + +def test_maximum_flow(graphs): + A_dense, A_sparse = graphs + sparse_cls = type(A_sparse) + func = spgraph.maximum_flow + + actual = func(A_sparse, 0, 2) + desired = func(sp.csr_array(A_dense), 0, 2) + + assert actual.flow_value == desired.flow_value + assert isinstance(actual.flow, sparse_cls) + + assert_equal(actual.flow.todense(), desired.flow.todense()) + + +def test_min_weight_full_bipartite_matching(graphs): + A_dense, A_sparse = graphs + func = spgraph.min_weight_full_bipartite_matching + + actual = func(A_sparse[0:2, 1:3]) + desired = func(sp.csc_array(A_dense)[0:2, 1:3]) + + assert_equal(actual, desired) + + +@check_sparse_version("0.15.4") +@pytest.mark.parametrize( + "func", + [ + spgraph.shortest_path, + spgraph.dijkstra, + spgraph.floyd_warshall, + spgraph.bellman_ford, + spgraph.johnson, + spgraph.minimum_spanning_tree, + ] +) +@pytest.mark.parametrize( + "fill_value, comp_func", + [(np.inf, np.isposinf), (np.nan, np.isnan)], +) +def test_nonzero_fill_value(graphs, func, fill_value, comp_func): + A_dense, A_sparse = graphs + A_sparse = A_sparse.astype(float) + A_sparse.fill_value = fill_value + sparse_cls = type(A_sparse) + + actual = func(A_sparse) + desired = func(sp.csc_array(A_dense)) + + if func == spgraph.minimum_spanning_tree: + assert isinstance(actual, sparse_cls) + assert comp_func(actual.fill_value) + actual = actual.todense() + actual[comp_func(actual)] = 0.0 + assert_equal(actual, desired.todense()) + else: + assert_equal(actual, desired) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_reordering.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_reordering.py new file mode 100644 index 0000000000000000000000000000000000000000..add76cdc29c39c079f386a6ca73075bb90b0a6e8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_reordering.py @@ -0,0 +1,70 @@ +import numpy as np +from numpy.testing import assert_equal +from scipy.sparse.csgraph import reverse_cuthill_mckee, structural_rank +from scipy.sparse import csc_array, csr_array, coo_array + + +def test_graph_reverse_cuthill_mckee(): + A = np.array([[1, 0, 0, 0, 1, 0, 0, 0], + [0, 1, 1, 0, 0, 1, 0, 1], + [0, 1, 1, 0, 1, 0, 0, 0], + [0, 0, 0, 1, 0, 0, 1, 0], + [1, 0, 1, 0, 1, 0, 0, 0], + [0, 1, 0, 0, 0, 1, 0, 1], + [0, 0, 0, 1, 0, 0, 1, 0], + [0, 1, 0, 0, 0, 1, 0, 1]], dtype=int) + + graph = csr_array(A) + perm = reverse_cuthill_mckee(graph) + correct_perm = np.array([6, 3, 7, 5, 1, 2, 4, 0]) + assert_equal(perm, correct_perm) + + # Test int64 indices input + graph.indices = graph.indices.astype('int64') + graph.indptr = graph.indptr.astype('int64') + perm = reverse_cuthill_mckee(graph, True) + assert_equal(perm, correct_perm) + + +def test_graph_reverse_cuthill_mckee_ordering(): + data = np.ones(63,dtype=int) + rows = np.array([0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, + 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, + 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, + 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, + 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, + 14, 15, 15, 15, 15, 15]) + cols = np.array([0, 2, 5, 8, 10, 1, 3, 9, 11, 0, 2, + 7, 10, 1, 3, 11, 4, 6, 12, 14, 0, 7, 13, + 15, 4, 6, 14, 2, 5, 7, 15, 0, 8, 10, 13, + 1, 9, 11, 0, 2, 8, 10, 15, 1, 3, 9, 11, + 4, 12, 14, 5, 8, 13, 15, 4, 6, 12, 14, + 5, 7, 10, 13, 15]) + graph = csr_array((data, (rows,cols))) + perm = reverse_cuthill_mckee(graph) + correct_perm = np.array([12, 14, 4, 6, 10, 8, 2, 15, + 0, 13, 7, 5, 9, 11, 1, 3]) + assert_equal(perm, correct_perm) + + +def test_graph_structural_rank(): + # Test square matrix #1 + A = csc_array([[1, 1, 0], + [1, 0, 1], + [0, 1, 0]]) + assert_equal(structural_rank(A), 3) + + # Test square matrix #2 + rows = np.array([0,0,0,0,0,1,1,2,2,3,3,3,3,3,3,4,4,5,5,6,6,7,7]) + cols = np.array([0,1,2,3,4,2,5,2,6,0,1,3,5,6,7,4,5,5,6,2,6,2,4]) + data = np.ones_like(rows) + B = coo_array((data,(rows,cols)), shape=(8,8)) + assert_equal(structural_rank(B), 6) + + #Test non-square matrix + C = csc_array([[1, 0, 2, 0], + [2, 0, 4, 0]]) + assert_equal(structural_rank(C), 2) + + #Test tall matrix + assert_equal(structural_rank(C.T), 2) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_shortest_path.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_shortest_path.py new file mode 100644 index 0000000000000000000000000000000000000000..ba50f760e750ce1dbdec3624c2ab985fca1af7d1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_shortest_path.py @@ -0,0 +1,484 @@ +from io import StringIO +import warnings +import numpy as np +from numpy.testing import assert_array_almost_equal, assert_array_equal, assert_allclose +from pytest import raises as assert_raises +from scipy.sparse.csgraph import (shortest_path, dijkstra, johnson, + bellman_ford, construct_dist_matrix, yen, + NegativeCycleError) +import scipy.sparse +from scipy.io import mmread +import pytest + +directed_G = np.array([[0, 3, 3, 0, 0], + [0, 0, 0, 2, 4], + [0, 0, 0, 0, 0], + [1, 0, 0, 0, 0], + [2, 0, 0, 2, 0]], dtype=float) + +undirected_G = np.array([[0, 3, 3, 1, 2], + [3, 0, 0, 2, 4], + [3, 0, 0, 0, 0], + [1, 2, 0, 0, 2], + [2, 4, 0, 2, 0]], dtype=float) + +unweighted_G = (directed_G > 0).astype(float) + +directed_SP = [[0, 3, 3, 5, 7], + [3, 0, 6, 2, 4], + [np.inf, np.inf, 0, np.inf, np.inf], + [1, 4, 4, 0, 8], + [2, 5, 5, 2, 0]] + +directed_2SP_0_to_3 = [[-9999, 0, -9999, 1, -9999], + [-9999, 0, -9999, 4, 1]] + +directed_sparse_zero_G = scipy.sparse.csr_array( + ( + [0, 1, 2, 3, 1], + ([0, 1, 2, 3, 4], [1, 2, 0, 4, 3]), + ), + shape=(5, 5), +) + +directed_sparse_zero_SP = [[0, 0, 1, np.inf, np.inf], + [3, 0, 1, np.inf, np.inf], + [2, 2, 0, np.inf, np.inf], + [np.inf, np.inf, np.inf, 0, 3], + [np.inf, np.inf, np.inf, 1, 0]] + +undirected_sparse_zero_G = scipy.sparse.csr_array( + ( + [0, 0, 1, 1, 2, 2, 1, 1], + ([0, 1, 1, 2, 2, 0, 3, 4], [1, 0, 2, 1, 0, 2, 4, 3]) + ), + shape=(5, 5), +) + +undirected_sparse_zero_SP = [[0, 0, 1, np.inf, np.inf], + [0, 0, 1, np.inf, np.inf], + [1, 1, 0, np.inf, np.inf], + [np.inf, np.inf, np.inf, 0, 1], + [np.inf, np.inf, np.inf, 1, 0]] + +directed_pred = np.array([[-9999, 0, 0, 1, 1], + [3, -9999, 0, 1, 1], + [-9999, -9999, -9999, -9999, -9999], + [3, 0, 0, -9999, 1], + [4, 0, 0, 4, -9999]], dtype=float) + +undirected_SP = np.array([[0, 3, 3, 1, 2], + [3, 0, 6, 2, 4], + [3, 6, 0, 4, 5], + [1, 2, 4, 0, 2], + [2, 4, 5, 2, 0]], dtype=float) + +undirected_SP_limit_2 = np.array([[0, np.inf, np.inf, 1, 2], + [np.inf, 0, np.inf, 2, np.inf], + [np.inf, np.inf, 0, np.inf, np.inf], + [1, 2, np.inf, 0, 2], + [2, np.inf, np.inf, 2, 0]], dtype=float) + +undirected_SP_limit_0 = np.ones((5, 5), dtype=float) - np.eye(5) +undirected_SP_limit_0[undirected_SP_limit_0 > 0] = np.inf + +undirected_pred = np.array([[-9999, 0, 0, 0, 0], + [1, -9999, 0, 1, 1], + [2, 0, -9999, 0, 0], + [3, 3, 0, -9999, 3], + [4, 4, 0, 4, -9999]], dtype=float) + +directed_negative_weighted_G = np.array([[0, 0, 0], + [-1, 0, 0], + [0, -1, 0]], dtype=float) + +directed_negative_weighted_SP = np.array([[0, np.inf, np.inf], + [-1, 0, np.inf], + [-2, -1, 0]], dtype=float) + +methods = ['auto', 'FW', 'D', 'BF', 'J'] + + +def test_dijkstra_limit(): + limits = [0, 2, np.inf] + results = [undirected_SP_limit_0, + undirected_SP_limit_2, + undirected_SP] + + def check(limit, result): + SP = dijkstra(undirected_G, directed=False, limit=limit) + assert_array_almost_equal(SP, result) + + for limit, result in zip(limits, results): + check(limit, result) + + +def test_directed(): + def check(method): + SP = shortest_path(directed_G, method=method, directed=True, + overwrite=False) + assert_array_almost_equal(SP, directed_SP) + + for method in methods: + check(method) + + +def test_undirected(): + def check(method, directed_in): + if directed_in: + SP1 = shortest_path(directed_G, method=method, directed=False, + overwrite=False) + assert_array_almost_equal(SP1, undirected_SP) + else: + SP2 = shortest_path(undirected_G, method=method, directed=True, + overwrite=False) + assert_array_almost_equal(SP2, undirected_SP) + + for method in methods: + for directed_in in (True, False): + check(method, directed_in) + + +def test_directed_sparse_zero(): + # test directed sparse graph with zero-weight edge and two connected components + def check(method): + SP = shortest_path(directed_sparse_zero_G, method=method, directed=True, + overwrite=False) + assert_array_almost_equal(SP, directed_sparse_zero_SP) + + for method in methods: + check(method) + + +def test_undirected_sparse_zero(): + def check(method, directed_in): + if directed_in: + SP1 = shortest_path(directed_sparse_zero_G, method=method, directed=False, + overwrite=False) + assert_array_almost_equal(SP1, undirected_sparse_zero_SP) + else: + SP2 = shortest_path(undirected_sparse_zero_G, method=method, directed=True, + overwrite=False) + assert_array_almost_equal(SP2, undirected_sparse_zero_SP) + + for method in methods: + for directed_in in (True, False): + check(method, directed_in) + + +@pytest.mark.parametrize('directed, SP_ans', + ((True, directed_SP), + (False, undirected_SP))) +@pytest.mark.parametrize('indices', ([0, 2, 4], [0, 4], [3, 4], [0, 0])) +def test_dijkstra_indices_min_only(directed, SP_ans, indices): + SP_ans = np.array(SP_ans) + indices = np.array(indices, dtype=np.int64) + min_ind_ans = indices[np.argmin(SP_ans[indices, :], axis=0)] + min_d_ans = np.zeros(SP_ans.shape[0], SP_ans.dtype) + for k in range(SP_ans.shape[0]): + min_d_ans[k] = SP_ans[min_ind_ans[k], k] + min_ind_ans[np.isinf(min_d_ans)] = -9999 + + SP, pred, sources = dijkstra(directed_G, + directed=directed, + indices=indices, + min_only=True, + return_predecessors=True) + assert_array_almost_equal(SP, min_d_ans) + assert_array_equal(min_ind_ans, sources) + SP = dijkstra(directed_G, + directed=directed, + indices=indices, + min_only=True, + return_predecessors=False) + assert_array_almost_equal(SP, min_d_ans) + + +@pytest.mark.parametrize('n', (10, 100, 1000)) +def test_dijkstra_min_only_random(n): + rng = np.random.default_rng(7345782358920239234) + data = scipy.sparse.random_array((n, n), density=0.5, format='lil', + rng=rng, dtype=np.float64) + data.setdiag(np.zeros(n, dtype=np.bool_)) + # choose some random vertices + v = np.arange(n) + rng.shuffle(v) + indices = v[:int(n*.1)] + ds, pred, sources = dijkstra(data, + directed=True, + indices=indices, + min_only=True, + return_predecessors=True) + for k in range(n): + p = pred[k] + s = sources[k] + while p != -9999: + assert sources[p] == s + p = pred[p] + + +def test_dijkstra_random(): + # reproduces the hang observed in gh-17782 + n = 10 + indices = [0, 4, 4, 5, 7, 9, 0, 6, 2, 3, 7, 9, 1, 2, 9, 2, 5, 6] + indptr = [0, 0, 2, 5, 6, 7, 8, 12, 15, 18, 18] + data = [0.33629, 0.40458, 0.47493, 0.42757, 0.11497, 0.91653, 0.69084, + 0.64979, 0.62555, 0.743, 0.01724, 0.99945, 0.31095, 0.15557, + 0.02439, 0.65814, 0.23478, 0.24072] + graph = scipy.sparse.csr_array((data, indices, indptr), shape=(n, n)) + dijkstra(graph, directed=True, return_predecessors=True) + + +def test_gh_17782_segfault(): + text = """%%MatrixMarket matrix coordinate real general + 84 84 22 + 2 1 4.699999809265137e+00 + 6 14 1.199999973177910e-01 + 9 6 1.199999973177910e-01 + 10 16 2.012000083923340e+01 + 11 10 1.422000026702881e+01 + 12 1 9.645999908447266e+01 + 13 18 2.012000083923340e+01 + 14 13 4.679999828338623e+00 + 15 11 1.199999973177910e-01 + 16 12 1.199999973177910e-01 + 18 15 1.199999973177910e-01 + 32 2 2.299999952316284e+00 + 33 20 6.000000000000000e+00 + 33 32 5.000000000000000e+00 + 36 9 3.720000028610229e+00 + 36 37 3.720000028610229e+00 + 36 38 3.720000028610229e+00 + 37 44 8.159999847412109e+00 + 38 32 7.903999328613281e+01 + 43 20 2.400000000000000e+01 + 43 33 4.000000000000000e+00 + 44 43 6.028000259399414e+01 + """ + data = mmread(StringIO(text), spmatrix=False) + dijkstra(data, directed=True, return_predecessors=True) + + +def test_shortest_path_indices(): + indices = np.arange(4) + + def check(func, indshape): + outshape = indshape + (5,) + SP = func(directed_G, directed=False, + indices=indices.reshape(indshape)) + assert_array_almost_equal(SP, undirected_SP[indices].reshape(outshape)) + + for indshape in [(4,), (4, 1), (2, 2)]: + for func in (dijkstra, bellman_ford, johnson, shortest_path): + check(func, indshape) + + assert_raises(ValueError, shortest_path, directed_G, method='FW', + indices=indices) + + +def test_predecessors(): + SP_res = {True: directed_SP, + False: undirected_SP} + pred_res = {True: directed_pred, + False: undirected_pred} + + def check(method, directed): + SP, pred = shortest_path(directed_G, method, directed=directed, + overwrite=False, + return_predecessors=True) + assert_array_almost_equal(SP, SP_res[directed]) + assert_array_almost_equal(pred, pred_res[directed]) + + for method in methods: + for directed in (True, False): + check(method, directed) + + +def test_construct_shortest_path(): + def check(method, directed): + SP1, pred = shortest_path(directed_G, + directed=directed, + overwrite=False, + return_predecessors=True) + SP2 = construct_dist_matrix(directed_G, pred, directed=directed) + assert_array_almost_equal(SP1, SP2) + + for method in methods: + for directed in (True, False): + check(method, directed) + + +def test_unweighted_path(): + def check(method, directed): + SP1 = shortest_path(directed_G, + directed=directed, + overwrite=False, + unweighted=True) + SP2 = shortest_path(unweighted_G, + directed=directed, + overwrite=False, + unweighted=False) + assert_array_almost_equal(SP1, SP2) + + for method in methods: + for directed in (True, False): + check(method, directed) + + +def test_negative_cycles(): + # create a small graph with a negative cycle + graph = np.ones([5, 5]) + graph.flat[::6] = 0 + graph[1, 2] = -2 + + def check(method, directed): + assert_raises(NegativeCycleError, shortest_path, graph, method, + directed) + + for directed in (True, False): + for method in ['FW', 'J', 'BF']: + check(method, directed) + + assert_raises(NegativeCycleError, yen, graph, 0, 1, 1, + directed=directed) + + +@pytest.mark.parametrize("method", ['FW', 'J', 'BF']) +def test_negative_weights(method): + SP = shortest_path(directed_negative_weighted_G, method, directed=True) + assert_allclose(SP, directed_negative_weighted_SP, atol=1e-10) + + +def test_masked_input(): + np.ma.masked_equal(directed_G, 0) + + def check(method): + SP = shortest_path(directed_G, method=method, directed=True, + overwrite=False) + assert_array_almost_equal(SP, directed_SP) + + for method in methods: + check(method) + + +def test_overwrite(): + G = np.array([[0, 3, 3, 1, 2], + [3, 0, 0, 2, 4], + [3, 0, 0, 0, 0], + [1, 2, 0, 0, 2], + [2, 4, 0, 2, 0]], dtype=float) + foo = G.copy() + shortest_path(foo, overwrite=False) + assert_array_equal(foo, G) + + +@pytest.mark.parametrize('method', methods) +def test_buffer(method): + # Smoke test that sparse matrices with read-only buffers (e.g., those from + # joblib workers) do not cause:: + # + # ValueError: buffer source array is read-only + # + G = scipy.sparse.csr_array([[1.]]) + G.data.flags['WRITEABLE'] = False + shortest_path(G, method=method) + + +def test_NaN_warnings(): + with warnings.catch_warnings(record=True) as record: + shortest_path(np.array([[0, 1], [np.nan, 0]])) + for r in record: + assert r.category is not RuntimeWarning + + +def test_sparse_matrices(): + # Test that using lil,csr and csc sparse matrix do not cause error + G_dense = np.array([[0, 3, 0, 0, 0], + [0, 0, -1, 0, 0], + [0, 0, 0, 2, 0], + [0, 0, 0, 0, 4], + [0, 0, 0, 0, 0]], dtype=float) + SP = shortest_path(G_dense) + G_csr = scipy.sparse.csr_array(G_dense) + G_csc = scipy.sparse.csc_array(G_dense) + G_lil = scipy.sparse.lil_array(G_dense) + assert_array_almost_equal(SP, shortest_path(G_csr)) + assert_array_almost_equal(SP, shortest_path(G_csc)) + assert_array_almost_equal(SP, shortest_path(G_lil)) + + +def test_yen_directed(): + distances, predecessors = yen( + directed_G, + source=0, + sink=3, + K=2, + return_predecessors=True + ) + assert_allclose(distances, [5., 9.]) + assert_allclose(predecessors, directed_2SP_0_to_3) + + +def test_yen_undirected(): + distances = yen( + undirected_G, + source=0, + sink=3, + K=4, + ) + assert_allclose(distances, [1., 4., 5., 8.]) + +def test_yen_unweighted(): + # Ask for more paths than there are, verify only the available paths are returned + distances, predecessors = yen( + directed_G, + source=0, + sink=3, + K=4, + unweighted=True, + return_predecessors=True, + ) + assert_allclose(distances, [2., 3.]) + assert_allclose(predecessors, directed_2SP_0_to_3) + +def test_yen_no_paths(): + distances = yen( + directed_G, + source=2, + sink=3, + K=1, + ) + assert distances.size == 0 + +def test_yen_negative_weights(): + distances = yen( + directed_negative_weighted_G, + source=2, + sink=0, + K=1, + ) + assert_allclose(distances, [-2.]) + + +@pytest.mark.parametrize("min_only", (True, False)) +@pytest.mark.parametrize("directed", (True, False)) +@pytest.mark.parametrize("return_predecessors", (True, False)) +@pytest.mark.parametrize("index_dtype", (np.int32, np.int64)) +@pytest.mark.parametrize("indices", (None, [1])) +def test_20904(min_only, directed, return_predecessors, index_dtype, indices): + """Test two failures from gh-20904: int32 and indices-as-None.""" + adj_mat = scipy.sparse.eye_array(4, format="csr") + adj_mat = scipy.sparse.csr_array( + ( + adj_mat.data, + adj_mat.indices.astype(index_dtype), + adj_mat.indptr.astype(index_dtype), + ), + ) + dijkstra( + adj_mat, + directed, + indices=indices, + min_only=min_only, + return_predecessors=return_predecessors, + ) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_spanning_tree.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_spanning_tree.py new file mode 100644 index 0000000000000000000000000000000000000000..3237a14584d42022184a54f174b809a2b06d16ed --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_spanning_tree.py @@ -0,0 +1,66 @@ +"""Test the minimum spanning tree function""" +import numpy as np +from numpy.testing import assert_ +import numpy.testing as npt +from scipy.sparse import csr_array +from scipy.sparse.csgraph import minimum_spanning_tree + + +def test_minimum_spanning_tree(): + + # Create a graph with two connected components. + graph = [[0,1,0,0,0], + [1,0,0,0,0], + [0,0,0,8,5], + [0,0,8,0,1], + [0,0,5,1,0]] + graph = np.asarray(graph) + + # Create the expected spanning tree. + expected = [[0,1,0,0,0], + [0,0,0,0,0], + [0,0,0,0,5], + [0,0,0,0,1], + [0,0,0,0,0]] + expected = np.asarray(expected) + + # Ensure minimum spanning tree code gives this expected output. + csgraph = csr_array(graph) + mintree = minimum_spanning_tree(csgraph) + mintree_array = mintree.toarray() + npt.assert_array_equal(mintree_array, expected, + 'Incorrect spanning tree found.') + + # Ensure that the original graph was not modified. + npt.assert_array_equal(csgraph.toarray(), graph, + 'Original graph was modified.') + + # Now let the algorithm modify the csgraph in place. + mintree = minimum_spanning_tree(csgraph, overwrite=True) + npt.assert_array_equal(mintree.toarray(), expected, + 'Graph was not properly modified to contain MST.') + + np.random.seed(1234) + for N in (5, 10, 15, 20): + + # Create a random graph. + graph = 3 + np.random.random((N, N)) + csgraph = csr_array(graph) + + # The spanning tree has at most N - 1 edges. + mintree = minimum_spanning_tree(csgraph) + assert_(mintree.nnz < N) + + # Set the sub diagonal to 1 to create a known spanning tree. + idx = np.arange(N-1) + graph[idx,idx+1] = 1 + csgraph = csr_array(graph) + mintree = minimum_spanning_tree(csgraph) + + # We expect to see this pattern in the spanning tree and otherwise + # have this zero. + expected = np.zeros((N, N)) + expected[idx, idx+1] = 1 + + npt.assert_array_equal(mintree.toarray(), expected, + 'Incorrect spanning tree found.') diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_traversal.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_traversal.py new file mode 100644 index 0000000000000000000000000000000000000000..aef6def21b61ff6b8d8bfb990a3a69136349d7c5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csgraph/tests/test_traversal.py @@ -0,0 +1,148 @@ +import numpy as np +import pytest +from numpy.testing import assert_array_almost_equal +from scipy.sparse import csr_array, csr_matrix, coo_array, coo_matrix +from scipy.sparse.csgraph import (breadth_first_tree, depth_first_tree, + csgraph_to_dense, csgraph_from_dense, csgraph_masked_from_dense) + + +def test_graph_breadth_first(): + csgraph = np.array([[0, 1, 2, 0, 0], + [1, 0, 0, 0, 3], + [2, 0, 0, 7, 0], + [0, 0, 7, 0, 1], + [0, 3, 0, 1, 0]]) + csgraph = csgraph_from_dense(csgraph, null_value=0) + + bfirst = np.array([[0, 1, 2, 0, 0], + [0, 0, 0, 0, 3], + [0, 0, 0, 7, 0], + [0, 0, 0, 0, 0], + [0, 0, 0, 0, 0]]) + + for directed in [True, False]: + bfirst_test = breadth_first_tree(csgraph, 0, directed) + assert_array_almost_equal(csgraph_to_dense(bfirst_test), + bfirst) + + +def test_graph_depth_first(): + csgraph = np.array([[0, 1, 2, 0, 0], + [1, 0, 0, 0, 3], + [2, 0, 0, 7, 0], + [0, 0, 7, 0, 1], + [0, 3, 0, 1, 0]]) + csgraph = csgraph_from_dense(csgraph, null_value=0) + + dfirst = np.array([[0, 1, 0, 0, 0], + [0, 0, 0, 0, 3], + [0, 0, 0, 0, 0], + [0, 0, 7, 0, 0], + [0, 0, 0, 1, 0]]) + + for directed in [True, False]: + dfirst_test = depth_first_tree(csgraph, 0, directed) + assert_array_almost_equal(csgraph_to_dense(dfirst_test), dfirst) + + +def test_return_type(): + from .._laplacian import laplacian + from .._min_spanning_tree import minimum_spanning_tree + + np_csgraph = np.array([[0, 1, 2, 0, 0], + [1, 0, 0, 0, 3], + [2, 0, 0, 7, 0], + [0, 0, 7, 0, 1], + [0, 3, 0, 1, 0]]) + csgraph = csr_array(np_csgraph) + assert isinstance(laplacian(csgraph), coo_array) + assert isinstance(minimum_spanning_tree(csgraph), csr_array) + for directed in [True, False]: + assert isinstance(depth_first_tree(csgraph, 0, directed), csr_array) + assert isinstance(breadth_first_tree(csgraph, 0, directed), csr_array) + + csgraph = csgraph_from_dense(np_csgraph, null_value=0) + assert isinstance(csgraph, csr_array) + assert isinstance(laplacian(csgraph), coo_array) + assert isinstance(minimum_spanning_tree(csgraph), csr_array) + for directed in [True, False]: + assert isinstance(depth_first_tree(csgraph, 0, directed), csr_array) + assert isinstance(breadth_first_tree(csgraph, 0, directed), csr_array) + + csgraph = csgraph_masked_from_dense(np_csgraph, null_value=0) + assert isinstance(csgraph, np.ma.MaskedArray) + assert csgraph._baseclass is np.ndarray + # laplacian doesnt work with masked arrays so not here + assert isinstance(minimum_spanning_tree(csgraph), csr_array) + for directed in [True, False]: + assert isinstance(depth_first_tree(csgraph, 0, directed), csr_array) + assert isinstance(breadth_first_tree(csgraph, 0, directed), csr_array) + + # start of testing with matrix/spmatrix types + with np.testing.suppress_warnings() as sup: + sup.filter(DeprecationWarning, "the matrix subclass.*") + sup.filter(PendingDeprecationWarning, "the matrix subclass.*") + + nm_csgraph = np.matrix([[0, 1, 2, 0, 0], + [1, 0, 0, 0, 3], + [2, 0, 0, 7, 0], + [0, 0, 7, 0, 1], + [0, 3, 0, 1, 0]]) + + csgraph = csr_matrix(nm_csgraph) + assert isinstance(laplacian(csgraph), coo_matrix) + assert isinstance(minimum_spanning_tree(csgraph), csr_matrix) + for directed in [True, False]: + assert isinstance(depth_first_tree(csgraph, 0, directed), csr_matrix) + assert isinstance(breadth_first_tree(csgraph, 0, directed), csr_matrix) + + csgraph = csgraph_from_dense(nm_csgraph, null_value=0) + assert isinstance(csgraph, csr_matrix) + assert isinstance(laplacian(csgraph), coo_matrix) + assert isinstance(minimum_spanning_tree(csgraph), csr_matrix) + for directed in [True, False]: + assert isinstance(depth_first_tree(csgraph, 0, directed), csr_matrix) + assert isinstance(breadth_first_tree(csgraph, 0, directed), csr_matrix) + + mm_csgraph = csgraph_masked_from_dense(nm_csgraph, null_value=0) + assert isinstance(mm_csgraph, np.ma.MaskedArray) + # laplacian doesnt work with masked arrays so not here + assert isinstance(minimum_spanning_tree(csgraph), csr_matrix) + for directed in [True, False]: + assert isinstance(depth_first_tree(csgraph, 0, directed), csr_matrix) + assert isinstance(breadth_first_tree(csgraph, 0, directed), csr_matrix) + # end of testing with matrix/spmatrix types + + +def test_graph_breadth_first_trivial_graph(): + csgraph = np.array([[0]]) + csgraph = csgraph_from_dense(csgraph, null_value=0) + + bfirst = np.array([[0]]) + + for directed in [True, False]: + bfirst_test = breadth_first_tree(csgraph, 0, directed) + assert_array_almost_equal(csgraph_to_dense(bfirst_test), bfirst) + + +def test_graph_depth_first_trivial_graph(): + csgraph = np.array([[0]]) + csgraph = csgraph_from_dense(csgraph, null_value=0) + + bfirst = np.array([[0]]) + + for directed in [True, False]: + bfirst_test = depth_first_tree(csgraph, 0, directed) + assert_array_almost_equal(csgraph_to_dense(bfirst_test), + bfirst) + + +@pytest.mark.parametrize('directed', [True, False]) +@pytest.mark.parametrize('tree_func', [breadth_first_tree, depth_first_tree]) +def test_int64_indices(tree_func, directed): + # See https://github.com/scipy/scipy/issues/18716 + g = csr_array(([1], np.array([[0], [1]], dtype=np.int64)), shape=(2, 2)) + assert g.indices.dtype == np.int64 + tree = tree_func(g, 0, directed=directed) + assert_array_almost_equal(csgraph_to_dense(tree), [[0, 1], [0, 0]]) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csr.py new file mode 100644 index 0000000000000000000000000000000000000000..86bb1e072ebe4480e9dcb01f2d36f7387872b898 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/csr.py @@ -0,0 +1,27 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'csr_count_blocks', + 'csr_matrix', + 'csr_tobsr', + 'csr_tocsc', + 'get_csr_submatrix', + 'isspmatrix_csr', + 'spmatrix', + 'upcast', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="csr", + private_modules=["_csr"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/data.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/data.py new file mode 100644 index 0000000000000000000000000000000000000000..a9958bcda6dd35ac0779514d79b7f1c494c1b01a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/data.py @@ -0,0 +1,23 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'isscalarlike', + 'name', + 'npfunc', + 'validateaxis', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="data", + private_modules=["_data"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/dia.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/dia.py new file mode 100644 index 0000000000000000000000000000000000000000..f79abd39f114b23df8ceb6eafb7fcc1c07218dcb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/dia.py @@ -0,0 +1,29 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'check_shape', + 'dia_matrix', + 'dia_matvec', + 'get_sum_dtype', + 'getdtype', + 'isshape', + 'isspmatrix_dia', + 'spmatrix', + 'upcast_char', + 'validateaxis', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="dia", + private_modules=["_dia"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/dok.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/dok.py new file mode 100644 index 0000000000000000000000000000000000000000..847824456eaa3145d5ecb078e30251875168775b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/dok.py @@ -0,0 +1,32 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'IndexMixin', + 'check_shape', + 'dok_matrix', + 'getdtype', + 'isdense', + 'isintlike', + 'isscalarlike', + 'isshape', + 'isspmatrix_dok', + 'itertools', + 'spmatrix', + 'upcast', + 'upcast_scalar', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="dok", + private_modules=["_dok"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/extract.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/extract.py new file mode 100644 index 0000000000000000000000000000000000000000..be5e161b6f99e57e2b2a6b3d4f1ef6427c07658d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/extract.py @@ -0,0 +1,23 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'coo_matrix', + 'find', + 'tril', + 'triu', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="extract", + private_modules=["_extract"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/lil.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/lil.py new file mode 100644 index 0000000000000000000000000000000000000000..5f7bf8eb03bb36a1b2fa77c5fc0840e532ab64fd --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/lil.py @@ -0,0 +1,22 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'isspmatrix_lil', + 'lil_array', + 'lil_matrix', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="lil", + private_modules=["_lil"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..ae19314d48b3689a556b2a795689a3a1b75458da --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/__init__.py @@ -0,0 +1,148 @@ +""" +Sparse linear algebra (:mod:`scipy.sparse.linalg`) +================================================== + +.. currentmodule:: scipy.sparse.linalg + +Abstract linear operators +------------------------- + +.. autosummary:: + :toctree: generated/ + + LinearOperator -- abstract representation of a linear operator + aslinearoperator -- convert an object to an abstract linear operator + +Matrix Operations +----------------- + +.. autosummary:: + :toctree: generated/ + + inv -- compute the sparse matrix inverse + expm -- compute the sparse matrix exponential + expm_multiply -- compute the product of a matrix exponential and a matrix + matrix_power -- compute the matrix power by raising a matrix to an exponent + +Matrix norms +------------ + +.. autosummary:: + :toctree: generated/ + + norm -- Norm of a sparse matrix + onenormest -- Estimate the 1-norm of a sparse matrix + +Solving linear problems +----------------------- + +Direct methods for linear equation systems: + +.. autosummary:: + :toctree: generated/ + + spsolve -- Solve the sparse linear system Ax=b + spsolve_triangular -- Solve sparse linear system Ax=b for a triangular A. + is_sptriangular -- Check if sparse A is triangular. + spbandwidth -- Find the bandwidth of a sparse matrix. + factorized -- Pre-factorize matrix to a function solving a linear system + MatrixRankWarning -- Warning on exactly singular matrices + use_solver -- Select direct solver to use + +Iterative methods for linear equation systems: + +.. autosummary:: + :toctree: generated/ + + bicg -- Use BIConjugate Gradient iteration to solve Ax = b + bicgstab -- Use BIConjugate Gradient STABilized iteration to solve Ax = b + cg -- Use Conjugate Gradient iteration to solve Ax = b + cgs -- Use Conjugate Gradient Squared iteration to solve Ax = b + gmres -- Use Generalized Minimal RESidual iteration to solve Ax = b + lgmres -- Solve a matrix equation using the LGMRES algorithm + minres -- Use MINimum RESidual iteration to solve Ax = b + qmr -- Use Quasi-Minimal Residual iteration to solve Ax = b + gcrotmk -- Solve a matrix equation using the GCROT(m,k) algorithm + tfqmr -- Use Transpose-Free Quasi-Minimal Residual iteration to solve Ax = b + +Iterative methods for least-squares problems: + +.. autosummary:: + :toctree: generated/ + + lsqr -- Find the least-squares solution to a sparse linear equation system + lsmr -- Find the least-squares solution to a sparse linear equation system + +Matrix factorizations +--------------------- + +Eigenvalue problems: + +.. autosummary:: + :toctree: generated/ + + eigs -- Find k eigenvalues and eigenvectors of the square matrix A + eigsh -- Find k eigenvalues and eigenvectors of a symmetric matrix + lobpcg -- Solve symmetric partial eigenproblems with optional preconditioning + +Singular values problems: + +.. autosummary:: + :toctree: generated/ + + svds -- Compute k singular values/vectors for a sparse matrix + +The `svds` function supports the following solvers: + +.. toctree:: + + sparse.linalg.svds-arpack + sparse.linalg.svds-lobpcg + sparse.linalg.svds-propack + +Complete or incomplete LU factorizations + +.. autosummary:: + :toctree: generated/ + + splu -- Compute a LU decomposition for a sparse matrix + spilu -- Compute an incomplete LU decomposition for a sparse matrix + SuperLU -- Object representing an LU factorization + +Sparse arrays with structure +---------------------------- + +.. autosummary:: + :toctree: generated/ + + LaplacianNd -- Laplacian on a uniform rectangular grid in ``N`` dimensions + +Exceptions +---------- + +.. autosummary:: + :toctree: generated/ + + ArpackNoConvergence + ArpackError + +""" + +from ._isolve import * +from ._dsolve import * +from ._interface import * +from ._eigen import * +from ._matfuncs import * +from ._onenormest import * +from ._norm import * +from ._expm_multiply import * +from ._special_sparse_arrays import * + +# Deprecated namespaces, to be removed in v2.0.0 +from . import isolve, dsolve, interface, eigen, matfuncs + +__all__ = [s for s in dir() if not s.startswith('_')] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..90005e3af863d2f7913e6fc4ea8de85962a3efdf --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/__init__.py @@ -0,0 +1,71 @@ +""" +Linear Solvers +============== + +The default solver is SuperLU (included in the scipy distribution), +which can solve real or complex linear systems in both single and +double precisions. It is automatically replaced by UMFPACK, if +available. Note that UMFPACK works in double precision only, so +switch it off by:: + + >>> from scipy.sparse.linalg import spsolve, use_solver + >>> use_solver(useUmfpack=False) + +to solve in the single precision. See also use_solver documentation. + +Example session:: + + >>> from scipy.sparse import csc_array, dia_array + >>> from numpy import array + >>> + >>> print("Inverting a sparse linear system:") + >>> print("The sparse matrix (constructed from diagonals):") + >>> a = dia_array(([[1, 2, 3, 4, 5], [6, 5, 8, 9, 10]], [0, 1]), shape=(5, 5)) + >>> b = array([1, 2, 3, 4, 5]) + >>> print("Solve: single precision complex:") + >>> use_solver( useUmfpack = False ) + >>> a = a.astype('F') + >>> x = spsolve(a, b) + >>> print(x) + >>> print("Error: ", a@x-b) + >>> + >>> print("Solve: double precision complex:") + >>> use_solver( useUmfpack = True ) + >>> a = a.astype('D') + >>> x = spsolve(a, b) + >>> print(x) + >>> print("Error: ", a@x-b) + >>> + >>> print("Solve: double precision:") + >>> a = a.astype('d') + >>> x = spsolve(a, b) + >>> print(x) + >>> print("Error: ", a@x-b) + >>> + >>> print("Solve: single precision:") + >>> use_solver( useUmfpack = False ) + >>> a = a.astype('f') + >>> x = spsolve(a, b.astype('f')) + >>> print(x) + >>> print("Error: ", a@x-b) + +""" + +#import umfpack +#__doc__ = '\n\n'.join( (__doc__, umfpack.__doc__) ) +#del umfpack + +from .linsolve import * +from ._superlu import SuperLU +from . import _add_newdocs +from . import linsolve + +__all__ = [ + 'MatrixRankWarning', 'SuperLU', 'factorized', + 'spilu', 'splu', 'spsolve', 'is_sptriangular', + 'spsolve_triangular', 'use_solver', 'spbandwidth', +] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/_add_newdocs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/_add_newdocs.py new file mode 100644 index 0000000000000000000000000000000000000000..cec34dca456b36a1f77e64f853fc1d821f5215eb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/_add_newdocs.py @@ -0,0 +1,147 @@ +from numpy.lib import add_newdoc + +add_newdoc('scipy.sparse.linalg._dsolve._superlu', 'SuperLU', + """ + LU factorization of a sparse matrix. + + Factorization is represented as:: + + Pr @ A @ Pc = L @ U + + To construct these `SuperLU` objects, call the `splu` and `spilu` + functions. + + Attributes + ---------- + shape + nnz + perm_c + perm_r + L + U + + Methods + ------- + solve + + Notes + ----- + + .. versionadded:: 0.14.0 + + Examples + -------- + The LU decomposition can be used to solve matrix equations. Consider: + + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import splu + >>> A = csc_array([[1,2,0,4], [1,0,0,1], [1,0,2,1], [2,2,1,0.]]) + + This can be solved for a given right-hand side: + + >>> lu = splu(A) + >>> b = np.array([1, 2, 3, 4]) + >>> x = lu.solve(b) + >>> A.dot(x) + array([ 1., 2., 3., 4.]) + + The ``lu`` object also contains an explicit representation of the + decomposition. The permutations are represented as mappings of + indices: + + >>> lu.perm_r + array([2, 1, 3, 0], dtype=int32) # may vary + >>> lu.perm_c + array([0, 1, 3, 2], dtype=int32) # may vary + + The L and U factors are sparse matrices in CSC format: + + >>> lu.L.toarray() + array([[ 1. , 0. , 0. , 0. ], # may vary + [ 0.5, 1. , 0. , 0. ], + [ 0.5, -1. , 1. , 0. ], + [ 0.5, 1. , 0. , 1. ]]) + >>> lu.U.toarray() + array([[ 2. , 2. , 0. , 1. ], # may vary + [ 0. , -1. , 1. , -0.5], + [ 0. , 0. , 5. , -1. ], + [ 0. , 0. , 0. , 2. ]]) + + The permutation matrices can be constructed: + + >>> Pr = csc_array((np.ones(4), (lu.perm_r, np.arange(4)))) + >>> Pc = csc_array((np.ones(4), (np.arange(4), lu.perm_c))) + + We can reassemble the original matrix: + + >>> (Pr.T @ (lu.L @ lu.U) @ Pc.T).toarray() + array([[ 1., 2., 0., 4.], + [ 1., 0., 0., 1.], + [ 1., 0., 2., 1.], + [ 2., 2., 1., 0.]]) + """) + +add_newdoc('scipy.sparse.linalg._dsolve._superlu', 'SuperLU', ('solve', + """ + solve(rhs[, trans]) + + Solves linear system of equations with one or several right-hand sides. + + Parameters + ---------- + rhs : ndarray, shape (n,) or (n, k) + Right hand side(s) of equation + trans : {'N', 'T', 'H'}, optional + Type of system to solve:: + + 'N': A @ x == rhs (default) + 'T': A^T @ x == rhs + 'H': A^H @ x == rhs + + i.e., normal, transposed, and hermitian conjugate. + + Returns + ------- + x : ndarray, shape ``rhs.shape`` + Solution vector(s) + """)) + +add_newdoc('scipy.sparse.linalg._dsolve._superlu', 'SuperLU', ('L', + """ + Lower triangular factor with unit diagonal as a + `scipy.sparse.csc_array`. + + .. versionadded:: 0.14.0 + """)) + +add_newdoc('scipy.sparse.linalg._dsolve._superlu', 'SuperLU', ('U', + """ + Upper triangular factor as a `scipy.sparse.csc_array`. + + .. versionadded:: 0.14.0 + """)) + +add_newdoc('scipy.sparse.linalg._dsolve._superlu', 'SuperLU', ('shape', + """ + Shape of the original matrix as a tuple of ints. + """)) + +add_newdoc('scipy.sparse.linalg._dsolve._superlu', 'SuperLU', ('nnz', + """ + Number of nonzero elements in the matrix. + """)) + +add_newdoc('scipy.sparse.linalg._dsolve._superlu', 'SuperLU', ('perm_c', + """ + Permutation Pc represented as an array of indices. + + See the `SuperLU` docstring for details. + """)) + +add_newdoc('scipy.sparse.linalg._dsolve._superlu', 'SuperLU', ('perm_r', + """ + Permutation Pr represented as an array of indices. + + See the `SuperLU` docstring for details. + """)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/linsolve.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/linsolve.py new file mode 100644 index 0000000000000000000000000000000000000000..192da969d89300d1c616f8f795d91ee4bf65ff8e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/linsolve.py @@ -0,0 +1,873 @@ +from warnings import warn, catch_warnings, simplefilter + +import numpy as np +from numpy import asarray +from scipy.sparse import (issparse, SparseEfficiencyWarning, + csr_array, csc_array, eye_array, diags_array) +from scipy.sparse._sputils import (is_pydata_spmatrix, convert_pydata_sparse_to_scipy, + get_index_dtype, safely_cast_index_arrays) +from scipy.linalg import LinAlgError +import copy +import threading + +from . import _superlu + +noScikit = False +try: + import scikits.umfpack as umfpack +except ImportError: + noScikit = True + +useUmfpack = threading.local() + + +__all__ = ['use_solver', 'spsolve', 'splu', 'spilu', 'factorized', + 'MatrixRankWarning', 'spsolve_triangular', 'is_sptriangular', 'spbandwidth'] + + +class MatrixRankWarning(UserWarning): + pass + + +def use_solver(**kwargs): + """ + Select default sparse direct solver to be used. + + Parameters + ---------- + useUmfpack : bool, optional + Use UMFPACK [1]_, [2]_, [3]_, [4]_. over SuperLU. Has effect only + if ``scikits.umfpack`` is installed. Default: True + assumeSortedIndices : bool, optional + Allow UMFPACK to skip the step of sorting indices for a CSR/CSC matrix. + Has effect only if useUmfpack is True and ``scikits.umfpack`` is + installed. Default: False + + Notes + ----- + The default sparse solver is UMFPACK when available + (``scikits.umfpack`` is installed). This can be changed by passing + useUmfpack = False, which then causes the always present SuperLU + based solver to be used. + + UMFPACK requires a CSR/CSC matrix to have sorted column/row indices. If + sure that the matrix fulfills this, pass ``assumeSortedIndices=True`` + to gain some speed. + + References + ---------- + .. [1] T. A. Davis, Algorithm 832: UMFPACK - an unsymmetric-pattern + multifrontal method with a column pre-ordering strategy, ACM + Trans. on Mathematical Software, 30(2), 2004, pp. 196--199. + https://dl.acm.org/doi/abs/10.1145/992200.992206 + + .. [2] T. A. Davis, A column pre-ordering strategy for the + unsymmetric-pattern multifrontal method, ACM Trans. + on Mathematical Software, 30(2), 2004, pp. 165--195. + https://dl.acm.org/doi/abs/10.1145/992200.992205 + + .. [3] T. A. Davis and I. S. Duff, A combined unifrontal/multifrontal + method for unsymmetric sparse matrices, ACM Trans. on + Mathematical Software, 25(1), 1999, pp. 1--19. + https://doi.org/10.1145/305658.287640 + + .. [4] T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal + method for sparse LU factorization, SIAM J. Matrix Analysis and + Computations, 18(1), 1997, pp. 140--158. + https://doi.org/10.1137/S0895479894246905T. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse.linalg import use_solver, spsolve + >>> from scipy.sparse import csc_array + >>> R = np.random.randn(5, 5) + >>> A = csc_array(R) + >>> b = np.random.randn(5) + >>> use_solver(useUmfpack=False) # enforce superLU over UMFPACK + >>> x = spsolve(A, b) + >>> np.allclose(A.dot(x), b) + True + >>> use_solver(useUmfpack=True) # reset umfPack usage to default + """ + global useUmfpack + if 'useUmfpack' in kwargs: + useUmfpack.u = kwargs['useUmfpack'] + if useUmfpack.u and 'assumeSortedIndices' in kwargs: + umfpack.configure(assumeSortedIndices=kwargs['assumeSortedIndices']) + +def _get_umf_family(A): + """Get umfpack family string given the sparse matrix dtype.""" + _families = { + (np.float64, np.int32): 'di', + (np.complex128, np.int32): 'zi', + (np.float64, np.int64): 'dl', + (np.complex128, np.int64): 'zl' + } + + # A.dtype.name can only be "float64" or + # "complex128" in control flow + f_type = getattr(np, A.dtype.name) + # control flow may allow for more index + # types to get through here + i_type = getattr(np, A.indices.dtype.name) + + try: + family = _families[(f_type, i_type)] + + except KeyError as e: + msg = ('only float64 or complex128 matrices with int32 or int64 ' + f'indices are supported! (got: matrix: {f_type}, indices: {i_type})') + raise ValueError(msg) from e + + # See gh-8278. Considered converting only if + # A.shape[0]*A.shape[1] > np.iinfo(np.int32).max, + # but that didn't always fix the issue. + family = family[0] + "l" + A_new = copy.copy(A) + A_new.indptr = np.asarray(A.indptr, dtype=np.int64) + A_new.indices = np.asarray(A.indices, dtype=np.int64) + + return family, A_new + +def spsolve(A, b, permc_spec=None, use_umfpack=True): + """Solve the sparse linear system Ax=b, where b may be a vector or a matrix. + + Parameters + ---------- + A : ndarray or sparse array or matrix + The square matrix A will be converted into CSC or CSR form + b : ndarray or sparse array or matrix + The matrix or vector representing the right hand side of the equation. + If a vector, b.shape must be (n,) or (n, 1). + permc_spec : str, optional + How to permute the columns of the matrix for sparsity preservation. + (default: 'COLAMD') + + - ``NATURAL``: natural ordering. + - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. + - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. + - ``COLAMD``: approximate minimum degree column ordering [1]_, [2]_. + + use_umfpack : bool, optional + if True (default) then use UMFPACK for the solution [3]_, [4]_, [5]_, + [6]_ . This is only referenced if b is a vector and + ``scikits.umfpack`` is installed. + + Returns + ------- + x : ndarray or sparse array or matrix + the solution of the sparse linear equation. + If b is a vector, then x is a vector of size A.shape[1] + If b is a matrix, then x is a matrix of size (A.shape[1], b.shape[1]) + + Notes + ----- + For solving the matrix expression AX = B, this solver assumes the resulting + matrix X is sparse, as is often the case for very sparse inputs. If the + resulting X is dense, the construction of this sparse result will be + relatively expensive. In that case, consider converting A to a dense + matrix and using scipy.linalg.solve or its variants. + + References + ---------- + .. [1] T. A. Davis, J. R. Gilbert, S. Larimore, E. Ng, Algorithm 836: + COLAMD, an approximate column minimum degree ordering algorithm, + ACM Trans. on Mathematical Software, 30(3), 2004, pp. 377--380. + :doi:`10.1145/1024074.1024080` + + .. [2] T. A. Davis, J. R. Gilbert, S. Larimore, E. Ng, A column approximate + minimum degree ordering algorithm, ACM Trans. on Mathematical + Software, 30(3), 2004, pp. 353--376. :doi:`10.1145/1024074.1024079` + + .. [3] T. A. Davis, Algorithm 832: UMFPACK - an unsymmetric-pattern + multifrontal method with a column pre-ordering strategy, ACM + Trans. on Mathematical Software, 30(2), 2004, pp. 196--199. + https://dl.acm.org/doi/abs/10.1145/992200.992206 + + .. [4] T. A. Davis, A column pre-ordering strategy for the + unsymmetric-pattern multifrontal method, ACM Trans. + on Mathematical Software, 30(2), 2004, pp. 165--195. + https://dl.acm.org/doi/abs/10.1145/992200.992205 + + .. [5] T. A. Davis and I. S. Duff, A combined unifrontal/multifrontal + method for unsymmetric sparse matrices, ACM Trans. on + Mathematical Software, 25(1), 1999, pp. 1--19. + https://doi.org/10.1145/305658.287640 + + .. [6] T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal + method for sparse LU factorization, SIAM J. Matrix Analysis and + Computations, 18(1), 1997, pp. 140--158. + https://doi.org/10.1137/S0895479894246905T. + + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import spsolve + >>> A = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) + >>> B = csc_array([[2, 0], [-1, 0], [2, 0]], dtype=float) + >>> x = spsolve(A, B) + >>> np.allclose(A.dot(x).toarray(), B.toarray()) + True + """ + is_pydata_sparse = is_pydata_spmatrix(b) + pydata_sparse_cls = b.__class__ if is_pydata_sparse else None + A = convert_pydata_sparse_to_scipy(A) + b = convert_pydata_sparse_to_scipy(b) + + if not (issparse(A) and A.format in ("csc", "csr")): + A = csc_array(A) + warn('spsolve requires A be CSC or CSR matrix format', + SparseEfficiencyWarning, stacklevel=2) + + # b is a vector only if b have shape (n,) or (n, 1) + b_is_sparse = issparse(b) + if not b_is_sparse: + b = asarray(b) + b_is_vector = ((b.ndim == 1) or (b.ndim == 2 and b.shape[1] == 1)) + + # sum duplicates for non-canonical format + A.sum_duplicates() + A = A._asfptype() # upcast to a floating point format + result_dtype = np.promote_types(A.dtype, b.dtype) + if A.dtype != result_dtype: + A = A.astype(result_dtype) + if b.dtype != result_dtype: + b = b.astype(result_dtype) + + # validate input shapes + M, N = A.shape + if (M != N): + raise ValueError(f"matrix must be square (has shape {(M, N)})") + + if M != b.shape[0]: + raise ValueError(f"matrix - rhs dimension mismatch ({A.shape} - {b.shape[0]})") + + if not hasattr(useUmfpack, 'u'): + useUmfpack.u = not noScikit + + use_umfpack = use_umfpack and useUmfpack.u + + if b_is_vector and use_umfpack: + if b_is_sparse: + b_vec = b.toarray() + else: + b_vec = b + b_vec = asarray(b_vec, dtype=A.dtype).ravel() + + if noScikit: + raise RuntimeError('Scikits.umfpack not installed.') + + if A.dtype.char not in 'dD': + raise ValueError("convert matrix data to double, please, using" + " .astype(), or set linsolve.useUmfpack.u = False") + + umf_family, A = _get_umf_family(A) + umf = umfpack.UmfpackContext(umf_family) + x = umf.linsolve(umfpack.UMFPACK_A, A, b_vec, + autoTranspose=True) + else: + if b_is_vector and b_is_sparse: + b = b.toarray() + b_is_sparse = False + + if not b_is_sparse: + if A.format == "csc": + flag = 1 # CSC format + else: + flag = 0 # CSR format + + indices = A.indices.astype(np.intc, copy=False) + indptr = A.indptr.astype(np.intc, copy=False) + options = dict(ColPerm=permc_spec) + x, info = _superlu.gssv(N, A.nnz, A.data, indices, indptr, + b, flag, options=options) + if info != 0: + warn("Matrix is exactly singular", MatrixRankWarning, stacklevel=2) + x.fill(np.nan) + if b_is_vector: + x = x.ravel() + else: + # b is sparse + Afactsolve = factorized(A) + + if not (b.format == "csc" or is_pydata_spmatrix(b)): + warn('spsolve is more efficient when sparse b ' + 'is in the CSC matrix format', + SparseEfficiencyWarning, stacklevel=2) + b = csc_array(b) + + # Create a sparse output matrix by repeatedly applying + # the sparse factorization to solve columns of b. + data_segs = [] + row_segs = [] + col_segs = [] + for j in range(b.shape[1]): + bj = b[:, j].toarray().ravel() + xj = Afactsolve(bj) + w = np.flatnonzero(xj) + segment_length = w.shape[0] + row_segs.append(w) + col_segs.append(np.full(segment_length, j, dtype=int)) + data_segs.append(np.asarray(xj[w], dtype=A.dtype)) + sparse_data = np.concatenate(data_segs) + idx_dtype = get_index_dtype(maxval=max(b.shape)) + sparse_row = np.concatenate(row_segs, dtype=idx_dtype) + sparse_col = np.concatenate(col_segs, dtype=idx_dtype) + x = A.__class__((sparse_data, (sparse_row, sparse_col)), + shape=b.shape, dtype=A.dtype) + + if is_pydata_sparse: + x = pydata_sparse_cls.from_scipy_sparse(x) + + return x + + +def splu(A, permc_spec=None, diag_pivot_thresh=None, + relax=None, panel_size=None, options=None): + """ + Compute the LU decomposition of a sparse, square matrix. + + Parameters + ---------- + A : sparse array or matrix + Sparse array to factorize. Most efficient when provided in CSC + format. Other formats will be converted to CSC before factorization. + permc_spec : str, optional + How to permute the columns of the matrix for sparsity preservation. + (default: 'COLAMD') + + - ``NATURAL``: natural ordering. + - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. + - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. + - ``COLAMD``: approximate minimum degree column ordering + + diag_pivot_thresh : float, optional + Threshold used for a diagonal entry to be an acceptable pivot. + See SuperLU user's guide for details [1]_ + relax : int, optional + Expert option for customizing the degree of relaxing supernodes. + See SuperLU user's guide for details [1]_ + panel_size : int, optional + Expert option for customizing the panel size. + See SuperLU user's guide for details [1]_ + options : dict, optional + Dictionary containing additional expert options to SuperLU. + See SuperLU user guide [1]_ (section 2.4 on the 'Options' argument) + for more details. For example, you can specify + ``options=dict(Equil=False, IterRefine='SINGLE'))`` + to turn equilibration off and perform a single iterative refinement. + + Returns + ------- + invA : scipy.sparse.linalg.SuperLU + Object, which has a ``solve`` method. + + See also + -------- + spilu : incomplete LU decomposition + + Notes + ----- + This function uses the SuperLU library. + + References + ---------- + .. [1] SuperLU https://portal.nersc.gov/project/sparse/superlu/ + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import splu + >>> A = csc_array([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float) + >>> B = splu(A) + >>> x = np.array([1., 2., 3.], dtype=float) + >>> B.solve(x) + array([ 1. , -3. , -1.5]) + >>> A.dot(B.solve(x)) + array([ 1., 2., 3.]) + >>> B.solve(A.dot(x)) + array([ 1., 2., 3.]) + """ + + if is_pydata_spmatrix(A): + A_cls = type(A) + def csc_construct_func(*a, cls=A_cls): + return cls.from_scipy_sparse(csc_array(*a)) + A = A.to_scipy_sparse().tocsc() + else: + csc_construct_func = csc_array + + if not (issparse(A) and A.format == "csc"): + A = csc_array(A) + warn('splu converted its input to CSC format', + SparseEfficiencyWarning, stacklevel=2) + + # sum duplicates for non-canonical format + A.sum_duplicates() + A = A._asfptype() # upcast to a floating point format + + M, N = A.shape + if (M != N): + raise ValueError("can only factor square matrices") # is this true? + + indices, indptr = safely_cast_index_arrays(A, np.intc, "SuperLU") + + _options = dict(DiagPivotThresh=diag_pivot_thresh, ColPerm=permc_spec, + PanelSize=panel_size, Relax=relax) + if options is not None: + _options.update(options) + + # Ensure that no column permutations are applied + if (_options["ColPerm"] == "NATURAL"): + _options["SymmetricMode"] = True + + return _superlu.gstrf(N, A.nnz, A.data, indices, indptr, + csc_construct_func=csc_construct_func, + ilu=False, options=_options) + + +def spilu(A, drop_tol=None, fill_factor=None, drop_rule=None, permc_spec=None, + diag_pivot_thresh=None, relax=None, panel_size=None, options=None): + """ + Compute an incomplete LU decomposition for a sparse, square matrix. + + The resulting object is an approximation to the inverse of `A`. + + Parameters + ---------- + A : (N, N) array_like + Sparse array to factorize. Most efficient when provided in CSC format. + Other formats will be converted to CSC before factorization. + drop_tol : float, optional + Drop tolerance (0 <= tol <= 1) for an incomplete LU decomposition. + (default: 1e-4) + fill_factor : float, optional + Specifies the fill ratio upper bound (>= 1.0) for ILU. (default: 10) + drop_rule : str, optional + Comma-separated string of drop rules to use. + Available rules: ``basic``, ``prows``, ``column``, ``area``, + ``secondary``, ``dynamic``, ``interp``. (Default: ``basic,area``) + + See SuperLU documentation for details. + + Remaining other options + Same as for `splu` + + Returns + ------- + invA_approx : scipy.sparse.linalg.SuperLU + Object, which has a ``solve`` method. + + See also + -------- + splu : complete LU decomposition + + Notes + ----- + To improve the better approximation to the inverse, you may need to + increase `fill_factor` AND decrease `drop_tol`. + + This function uses the SuperLU library. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import spilu + >>> A = csc_array([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float) + >>> B = spilu(A) + >>> x = np.array([1., 2., 3.], dtype=float) + >>> B.solve(x) + array([ 1. , -3. , -1.5]) + >>> A.dot(B.solve(x)) + array([ 1., 2., 3.]) + >>> B.solve(A.dot(x)) + array([ 1., 2., 3.]) + """ + + if is_pydata_spmatrix(A): + A_cls = type(A) + def csc_construct_func(*a, cls=A_cls): + return cls.from_scipy_sparse(csc_array(*a)) + A = A.to_scipy_sparse().tocsc() + else: + csc_construct_func = csc_array + + if not (issparse(A) and A.format == "csc"): + A = csc_array(A) + warn('spilu converted its input to CSC format', + SparseEfficiencyWarning, stacklevel=2) + + # sum duplicates for non-canonical format + A.sum_duplicates() + A = A._asfptype() # upcast to a floating point format + + M, N = A.shape + if (M != N): + raise ValueError("can only factor square matrices") # is this true? + + indices, indptr = safely_cast_index_arrays(A, np.intc, "SuperLU") + + _options = dict(ILU_DropRule=drop_rule, ILU_DropTol=drop_tol, + ILU_FillFactor=fill_factor, + DiagPivotThresh=diag_pivot_thresh, ColPerm=permc_spec, + PanelSize=panel_size, Relax=relax) + if options is not None: + _options.update(options) + + # Ensure that no column permutations are applied + if (_options["ColPerm"] == "NATURAL"): + _options["SymmetricMode"] = True + + return _superlu.gstrf(N, A.nnz, A.data, indices, indptr, + csc_construct_func=csc_construct_func, + ilu=True, options=_options) + + +def factorized(A): + """ + Return a function for solving a sparse linear system, with A pre-factorized. + + Parameters + ---------- + A : (N, N) array_like + Input. A in CSC format is most efficient. A CSR format matrix will + be converted to CSC before factorization. + + Returns + ------- + solve : callable + To solve the linear system of equations given in `A`, the `solve` + callable should be passed an ndarray of shape (N,). + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse.linalg import factorized + >>> from scipy.sparse import csc_array + >>> A = np.array([[ 3. , 2. , -1. ], + ... [ 2. , -2. , 4. ], + ... [-1. , 0.5, -1. ]]) + >>> solve = factorized(csc_array(A)) # Makes LU decomposition. + >>> rhs1 = np.array([1, -2, 0]) + >>> solve(rhs1) # Uses the LU factors. + array([ 1., -2., -2.]) + + """ + if is_pydata_spmatrix(A): + A = A.to_scipy_sparse().tocsc() + + if not hasattr(useUmfpack, 'u'): + useUmfpack.u = not noScikit + + if useUmfpack.u: + if noScikit: + raise RuntimeError('Scikits.umfpack not installed.') + + if not (issparse(A) and A.format == "csc"): + A = csc_array(A) + warn('splu converted its input to CSC format', + SparseEfficiencyWarning, stacklevel=2) + + A = A._asfptype() # upcast to a floating point format + + if A.dtype.char not in 'dD': + raise ValueError("convert matrix data to double, please, using" + " .astype(), or set linsolve.useUmfpack.u = False") + + umf_family, A = _get_umf_family(A) + umf = umfpack.UmfpackContext(umf_family) + + # Make LU decomposition. + umf.numeric(A) + + def solve(b): + with np.errstate(divide="ignore", invalid="ignore"): + # Ignoring warnings with numpy >= 1.23.0, see gh-16523 + result = umf.solve(umfpack.UMFPACK_A, A, b, autoTranspose=True) + + return result + + return solve + else: + return splu(A).solve + + +def spsolve_triangular(A, b, lower=True, overwrite_A=False, overwrite_b=False, + unit_diagonal=False): + """ + Solve the equation ``A x = b`` for `x`, assuming A is a triangular matrix. + + Parameters + ---------- + A : (M, M) sparse array or matrix + A sparse square triangular matrix. Should be in CSR or CSC format. + b : (M,) or (M, N) array_like + Right-hand side matrix in ``A x = b`` + lower : bool, optional + Whether `A` is a lower or upper triangular matrix. + Default is lower triangular matrix. + overwrite_A : bool, optional + Allow changing `A`. + Enabling gives a performance gain. Default is False. + overwrite_b : bool, optional + Allow overwriting data in `b`. + Enabling gives a performance gain. Default is False. + If `overwrite_b` is True, it should be ensured that + `b` has an appropriate dtype to be able to store the result. + unit_diagonal : bool, optional + If True, diagonal elements of `a` are assumed to be 1. + + .. versionadded:: 1.4.0 + + Returns + ------- + x : (M,) or (M, N) ndarray + Solution to the system ``A x = b``. Shape of return matches shape + of `b`. + + Raises + ------ + LinAlgError + If `A` is singular or not triangular. + ValueError + If shape of `A` or shape of `b` do not match the requirements. + + Notes + ----- + .. versionadded:: 0.19.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import spsolve_triangular + >>> A = csc_array([[3, 0, 0], [1, -1, 0], [2, 0, 1]], dtype=float) + >>> B = np.array([[2, 0], [-1, 0], [2, 0]], dtype=float) + >>> x = spsolve_triangular(A, B) + >>> np.allclose(A.dot(x), B) + True + """ + + if is_pydata_spmatrix(A): + A = A.to_scipy_sparse().tocsc() + + trans = "N" + if issparse(A) and A.format == "csr": + A = A.T + trans = "T" + lower = not lower + + if not (issparse(A) and A.format == "csc"): + warn('CSC or CSR matrix format is required. Converting to CSC matrix.', + SparseEfficiencyWarning, stacklevel=2) + A = csc_array(A) + elif not overwrite_A: + A = A.copy() + + + M, N = A.shape + if M != N: + raise ValueError( + f'A must be a square matrix but its shape is {A.shape}.') + + if unit_diagonal: + with catch_warnings(): + simplefilter('ignore', SparseEfficiencyWarning) + A.setdiag(1) + else: + diag = A.diagonal() + if np.any(diag == 0): + raise LinAlgError( + 'A is singular: zero entry on diagonal.') + invdiag = 1/diag + if trans == "N": + A = A @ diags_array(invdiag) + else: + A = (A.T @ diags_array(invdiag)).T + + # sum duplicates for non-canonical format + A.sum_duplicates() + + b = np.asanyarray(b) + + if b.ndim not in [1, 2]: + raise ValueError( + f'b must have 1 or 2 dims but its shape is {b.shape}.') + if M != b.shape[0]: + raise ValueError( + 'The size of the dimensions of A must be equal to ' + 'the size of the first dimension of b but the shape of A is ' + f'{A.shape} and the shape of b is {b.shape}.' + ) + + result_dtype = np.promote_types(np.promote_types(A.dtype, np.float32), b.dtype) + if A.dtype != result_dtype: + A = A.astype(result_dtype) + if b.dtype != result_dtype: + b = b.astype(result_dtype) + elif not overwrite_b: + b = b.copy() + + if lower: + L = A + U = csc_array((N, N), dtype=result_dtype) + else: + L = eye_array(N, dtype=result_dtype, format='csc') + U = A + U.setdiag(0) + + x, info = _superlu.gstrs(trans, + N, L.nnz, L.data, L.indices, L.indptr, + N, U.nnz, U.data, U.indices, U.indptr, + b) + if info: + raise LinAlgError('A is singular.') + + if not unit_diagonal: + invdiag = invdiag.reshape(-1, *([1] * (len(x.shape) - 1))) + x = x * invdiag + + return x + + +def is_sptriangular(A): + """Returns 2-tuple indicating lower/upper triangular structure for sparse ``A`` + + Checks for triangular structure in ``A``. The result is summarized in + two boolean values ``lower`` and ``upper`` to designate whether ``A`` is + lower triangular or upper triangular respectively. Diagonal ``A`` will + result in both being True. Non-triangular structure results in False for both. + + Only the sparse structure is used here. Values are not checked for zeros. + + This function will convert a copy of ``A`` to CSC format if it is not already + CSR or CSC format. So it may be more efficient to convert it yourself if you + have other uses for the CSR/CSC version. + + If ``A`` is not square, the portions outside the upper left square of the + matrix do not affect its triangular structure. You probably want to work + with the square portion of the matrix, though it is not requred here. + + Parameters + ---------- + A : SciPy sparse array or matrix + A sparse matrix preferrably in CSR or CSC format. + + Returns + ------- + lower, upper : 2-tuple of bool + + .. versionadded:: 1.15.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array, eye_array + >>> from scipy.sparse.linalg import is_sptriangular + >>> A = csc_array([[3, 0, 0], [1, -1, 0], [2, 0, 1]], dtype=float) + >>> is_sptriangular(A) + (True, False) + >>> D = eye_array(3, format='csr') + >>> is_sptriangular(D) + (True, True) + """ + if not (issparse(A) and A.format in ("csc", "csr", "coo", "dia", "dok", "lil")): + warn('is_sptriangular needs sparse and not BSR format. Converting to CSR.', + SparseEfficiencyWarning, stacklevel=2) + A = csr_array(A) + + # bsr is better off converting to csr + if A.format == "dia": + return A.offsets.max() <= 0, A.offsets.min() >= 0 + elif A.format == "coo": + rows, cols = A.coords + return (cols <= rows).all(), (cols >= rows).all() + elif A.format == "dok": + return all(c <= r for r, c in A.keys()), all(c >= r for r, c in A.keys()) + elif A.format == "lil": + lower = all(col <= row for row, cols in enumerate(A.rows) for col in cols) + upper = all(col >= row for row, cols in enumerate(A.rows) for col in cols) + return lower, upper + # format in ("csc", "csr") + indptr, indices = A.indptr, A.indices + N = len(indptr) - 1 + + lower, upper = True, True + # check middle, 1st, last col (treat as CSC and switch at end if CSR) + for col in [N // 2, 0, -1]: + rows = indices[indptr[col]:indptr[col + 1]] + upper = upper and (col >= rows).all() + lower = lower and (col <= rows).all() + if not upper and not lower: + return False, False + # check all cols + cols = np.repeat(np.arange(N), np.diff(indptr)) + rows = indices + upper = upper and (cols >= rows).all() + lower = lower and (cols <= rows).all() + if A.format == 'csr': + return upper, lower + return lower, upper + + +def spbandwidth(A): + """Return the lower and upper bandwidth of a 2D numeric array. + + Computes the lower and upper limits on the bandwidth of the + sparse 2D array ``A``. The result is summarized as a 2-tuple + of positive integers ``(lo, hi)``. A zero denotes no sub/super + diagonal entries on that side (tringular). The maximum value + for ``lo``(``hi``) is one less than the number of rows(cols). + + Only the sparse structure is used here. Values are not checked for zeros. + + Parameters + ---------- + A : SciPy sparse array or matrix + A sparse matrix preferrably in CSR or CSC format. + + Returns + ------- + below, above : 2-tuple of int + The distance to the farthest non-zero diagonal below/above the + main diagonal. + + .. versionadded:: 1.15.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse.linalg import spbandwidth + >>> from scipy.sparse import csc_array, eye_array + >>> A = csc_array([[3, 0, 0], [1, -1, 0], [2, 0, 1]], dtype=float) + >>> spbandwidth(A) + (2, 0) + >>> D = eye_array(3, format='csr') + >>> spbandwidth(D) + (0, 0) + """ + if not (issparse(A) and A.format in ("csc", "csr", "coo", "dia", "dok")): + warn('spbandwidth needs sparse format not LIL and BSR. Converting to CSR.', + SparseEfficiencyWarning, stacklevel=2) + A = csr_array(A) + + # bsr and lil are better off converting to csr + if A.format == "dia": + return max(0, -A.offsets.min().item()), max(0, A.offsets.max().item()) + if A.format in ("csc", "csr"): + indptr, indices = A.indptr, A.indices + N = len(indptr) - 1 + gap = np.repeat(np.arange(N), np.diff(indptr)) - indices + if A.format == 'csr': + gap = -gap + elif A.format == "coo": + gap = A.coords[1] - A.coords[0] + elif A.format == "dok": + gap = [(c - r) for r, c in A.keys()] + [0] + return -min(gap), max(gap) + return max(-np.min(gap).item(), 0), max(np.max(gap).item(), 0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/tests/test_linsolve.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/tests/test_linsolve.py new file mode 100644 index 0000000000000000000000000000000000000000..c9c5bfed1b5458a8c43ee2234087f164f2a37a80 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/tests/test_linsolve.py @@ -0,0 +1,921 @@ +import sys +import threading + +import numpy as np +from numpy import array, finfo, arange, eye, all, unique, ones, dot +import numpy.random as random +from numpy.testing import ( + assert_array_almost_equal, assert_almost_equal, + assert_equal, assert_array_equal, assert_, assert_allclose, + assert_warns, suppress_warnings) +import pytest +from pytest import raises as assert_raises + +import scipy.linalg +from scipy.linalg import norm, inv +from scipy.sparse import (dia_array, SparseEfficiencyWarning, csc_array, + csr_array, eye_array, issparse, dok_array, lil_array, bsr_array, kron) +from scipy.sparse.linalg import SuperLU +from scipy.sparse.linalg._dsolve import (spsolve, use_solver, splu, spilu, + MatrixRankWarning, _superlu, spsolve_triangular, factorized, + is_sptriangular, spbandwidth) +import scipy.sparse + +from scipy._lib._testutils import check_free_memory +from scipy._lib._util import ComplexWarning + + +sup_sparse_efficiency = suppress_warnings() +sup_sparse_efficiency.filter(SparseEfficiencyWarning) + +# scikits.umfpack is not a SciPy dependency but it is optionally used in +# dsolve, so check whether it's available +try: + import scikits.umfpack as umfpack + has_umfpack = True +except ImportError: + has_umfpack = False + +def toarray(a): + if issparse(a): + return a.toarray() + else: + return a + + +def setup_bug_8278(): + N = 2 ** 6 + h = 1/N + Ah1D = dia_array(([-1, 2, -1], [-1, 0, 1]), shape=(N-1, N-1))/(h**2) + eyeN = eye_array(N - 1) + A = (kron(eyeN, kron(eyeN, Ah1D)) + + kron(eyeN, kron(Ah1D, eyeN)) + + kron(Ah1D, kron(eyeN, eyeN))) + b = np.random.rand((N-1)**3) + return A, b + + +class TestFactorized: + def setup_method(self): + n = 5 + d = arange(n) + 1 + self.n = n + self.A = dia_array(((d, 2*d, d[::-1]), (-3, 0, 5)), shape=(n,n)).tocsc() + random.seed(1234) + + def _check_singular(self): + A = csc_array((5,5), dtype='d') + b = ones(5) + assert_array_almost_equal(0. * b, factorized(A)(b)) + + def _check_non_singular(self): + # Make a diagonal dominant, to make sure it is not singular + n = 5 + a = csc_array(random.rand(n, n)) + b = ones(n) + + expected = splu(a).solve(b) + assert_array_almost_equal(factorized(a)(b), expected) + + def test_singular_without_umfpack(self): + use_solver(useUmfpack=False) + with assert_raises(RuntimeError, match="Factor is exactly singular"): + self._check_singular() + + @pytest.mark.skipif(not has_umfpack, reason="umfpack not available") + def test_singular_with_umfpack(self): + use_solver(useUmfpack=True) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "divide by zero encountered in double_scalars") + assert_warns(umfpack.UmfpackWarning, self._check_singular) + + def test_non_singular_without_umfpack(self): + use_solver(useUmfpack=False) + self._check_non_singular() + + @pytest.mark.skipif(not has_umfpack, reason="umfpack not available") + def test_non_singular_with_umfpack(self): + use_solver(useUmfpack=True) + self._check_non_singular() + + def test_cannot_factorize_nonsquare_matrix_without_umfpack(self): + use_solver(useUmfpack=False) + msg = "can only factor square matrices" + with assert_raises(ValueError, match=msg): + factorized(self.A[:, :4]) + + @pytest.mark.skipif(not has_umfpack, reason="umfpack not available") + def test_factorizes_nonsquare_matrix_with_umfpack(self): + use_solver(useUmfpack=True) + # does not raise + factorized(self.A[:,:4]) + + def test_call_with_incorrectly_sized_matrix_without_umfpack(self): + use_solver(useUmfpack=False) + solve = factorized(self.A) + b = random.rand(4) + B = random.rand(4, 3) + BB = random.rand(self.n, 3, 9) + + with assert_raises(ValueError, match="is of incompatible size"): + solve(b) + with assert_raises(ValueError, match="is of incompatible size"): + solve(B) + with assert_raises(ValueError, + match="object too deep for desired array"): + solve(BB) + + @pytest.mark.skipif(not has_umfpack, reason="umfpack not available") + def test_call_with_incorrectly_sized_matrix_with_umfpack(self): + use_solver(useUmfpack=True) + solve = factorized(self.A) + b = random.rand(4) + B = random.rand(4, 3) + BB = random.rand(self.n, 3, 9) + + # does not raise + solve(b) + msg = "object too deep for desired array" + with assert_raises(ValueError, match=msg): + solve(B) + with assert_raises(ValueError, match=msg): + solve(BB) + + def test_call_with_cast_to_complex_without_umfpack(self): + use_solver(useUmfpack=False) + solve = factorized(self.A) + b = random.rand(4) + for t in [np.complex64, np.complex128]: + with assert_raises(TypeError, match="Cannot cast array data"): + solve(b.astype(t)) + + @pytest.mark.skipif(not has_umfpack, reason="umfpack not available") + def test_call_with_cast_to_complex_with_umfpack(self): + use_solver(useUmfpack=True) + solve = factorized(self.A) + b = random.rand(4) + for t in [np.complex64, np.complex128]: + assert_warns(ComplexWarning, solve, b.astype(t)) + + @pytest.mark.skipif(not has_umfpack, reason="umfpack not available") + def test_assume_sorted_indices_flag(self): + # a sparse matrix with unsorted indices + unsorted_inds = np.array([2, 0, 1, 0]) + data = np.array([10, 16, 5, 0.4]) + indptr = np.array([0, 1, 2, 4]) + A = csc_array((data, unsorted_inds, indptr), (3, 3)) + b = ones(3) + + # should raise when incorrectly assuming indices are sorted + use_solver(useUmfpack=True, assumeSortedIndices=True) + with assert_raises(RuntimeError, + match="UMFPACK_ERROR_invalid_matrix"): + factorized(A) + + # should sort indices and succeed when not assuming indices are sorted + use_solver(useUmfpack=True, assumeSortedIndices=False) + expected = splu(A.copy()).solve(b) + + assert_equal(A.has_sorted_indices, 0) + assert_array_almost_equal(factorized(A)(b), expected) + + @pytest.mark.slow + @pytest.mark.skipif(not has_umfpack, reason="umfpack not available") + def test_bug_8278(self): + check_free_memory(8000) + use_solver(useUmfpack=True) + A, b = setup_bug_8278() + A = A.tocsc() + f = factorized(A) + x = f(b) + assert_array_almost_equal(A @ x, b) + + +class TestLinsolve: + def setup_method(self): + use_solver(useUmfpack=False) + + def test_singular(self): + A = csc_array((5,5), dtype='d') + b = array([1, 2, 3, 4, 5],dtype='d') + with suppress_warnings() as sup: + sup.filter(MatrixRankWarning, "Matrix is exactly singular") + x = spsolve(A, b) + assert_(not np.isfinite(x).any()) + + def test_singular_gh_3312(self): + # "Bad" test case that leads SuperLU to call LAPACK with invalid + # arguments. Check that it fails moderately gracefully. + ij = np.array([(17, 0), (17, 6), (17, 12), (10, 13)], dtype=np.int32) + v = np.array([0.284213, 0.94933781, 0.15767017, 0.38797296]) + A = csc_array((v, ij.T), shape=(20, 20)) + b = np.arange(20) + + try: + # should either raise a runtime error or return value + # appropriate for singular input (which yields the warning) + with suppress_warnings() as sup: + sup.filter(MatrixRankWarning, "Matrix is exactly singular") + x = spsolve(A, b) + assert not np.isfinite(x).any() + except RuntimeError: + pass + + @pytest.mark.parametrize('format', ['csc', 'csr']) + @pytest.mark.parametrize('idx_dtype', [np.int32, np.int64]) + def test_twodiags(self, format: str, idx_dtype: np.dtype): + A = dia_array(([[1, 2, 3, 4, 5], [6, 5, 8, 9, 10]], [0, 1]), + shape=(5, 5)).asformat(format) + b = array([1, 2, 3, 4, 5]) + + # condition number of A + cond_A = norm(A.toarray(), 2) * norm(inv(A.toarray()), 2) + + for t in ['f','d','F','D']: + eps = finfo(t).eps # floating point epsilon + b = b.astype(t) + Asp = A.astype(t) + Asp.indices = Asp.indices.astype(idx_dtype, copy=False) + Asp.indptr = Asp.indptr.astype(idx_dtype, copy=False) + + x = spsolve(Asp, b) + assert_(norm(b - Asp@x) < 10 * cond_A * eps) + + def test_bvector_smoketest(self): + Adense = array([[0., 1., 1.], + [1., 0., 1.], + [0., 0., 1.]]) + As = csc_array(Adense) + random.seed(1234) + x = random.randn(3) + b = As@x + x2 = spsolve(As, b) + + assert_array_almost_equal(x, x2) + + def test_bmatrix_smoketest(self): + Adense = array([[0., 1., 1.], + [1., 0., 1.], + [0., 0., 1.]]) + As = csc_array(Adense) + random.seed(1234) + x = random.randn(3, 4) + Bdense = As.dot(x) + Bs = csc_array(Bdense) + x2 = spsolve(As, Bs) + assert_array_almost_equal(x, x2.toarray()) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + def test_non_square(self): + # A is not square. + A = ones((3, 4)) + b = ones((4, 1)) + assert_raises(ValueError, spsolve, A, b) + # A2 and b2 have incompatible shapes. + A2 = csc_array(eye(3)) + b2 = array([1.0, 2.0]) + assert_raises(ValueError, spsolve, A2, b2) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + def test_example_comparison(self): + row = array([0,0,1,2,2,2]) + col = array([0,2,2,0,1,2]) + data = array([1,2,3,-4,5,6]) + sM = csr_array((data,(row,col)), shape=(3,3), dtype=float) + M = sM.toarray() + + row = array([0,0,1,1,0,0]) + col = array([0,2,1,1,0,0]) + data = array([1,1,1,1,1,1]) + sN = csr_array((data, (row,col)), shape=(3,3), dtype=float) + N = sN.toarray() + + sX = spsolve(sM, sN) + X = scipy.linalg.solve(M, N) + + assert_array_almost_equal(X, sX.toarray()) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + @pytest.mark.skipif(not has_umfpack, reason="umfpack not available") + def test_shape_compatibility(self): + use_solver(useUmfpack=True) + A = csc_array([[1., 0], [0, 2]]) + bs = [ + [1, 6], + array([1, 6]), + [[1], [6]], + array([[1], [6]]), + csc_array([[1], [6]]), + csr_array([[1], [6]]), + dok_array([[1], [6]]), + bsr_array([[1], [6]]), + array([[1., 2., 3.], [6., 8., 10.]]), + csc_array([[1., 2., 3.], [6., 8., 10.]]), + csr_array([[1., 2., 3.], [6., 8., 10.]]), + dok_array([[1., 2., 3.], [6., 8., 10.]]), + bsr_array([[1., 2., 3.], [6., 8., 10.]]), + ] + + for b in bs: + x = np.linalg.solve(A.toarray(), toarray(b)) + for spmattype in [csc_array, csr_array, dok_array, lil_array]: + x1 = spsolve(spmattype(A), b, use_umfpack=True) + x2 = spsolve(spmattype(A), b, use_umfpack=False) + + # check solution + if x.ndim == 2 and x.shape[1] == 1: + # interprets also these as "vectors" + x = x.ravel() + + assert_array_almost_equal(toarray(x1), x, + err_msg=repr((b, spmattype, 1))) + assert_array_almost_equal(toarray(x2), x, + err_msg=repr((b, spmattype, 2))) + + # dense vs. sparse output ("vectors" are always dense) + if issparse(b) and x.ndim > 1: + assert_(issparse(x1), repr((b, spmattype, 1))) + assert_(issparse(x2), repr((b, spmattype, 2))) + else: + assert_(isinstance(x1, np.ndarray), repr((b, spmattype, 1))) + assert_(isinstance(x2, np.ndarray), repr((b, spmattype, 2))) + + # check output shape + if x.ndim == 1: + # "vector" + assert_equal(x1.shape, (A.shape[1],)) + assert_equal(x2.shape, (A.shape[1],)) + else: + # "matrix" + assert_equal(x1.shape, x.shape) + assert_equal(x2.shape, x.shape) + + A = csc_array((3, 3)) + b = csc_array((1, 3)) + assert_raises(ValueError, spsolve, A, b) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + def test_ndarray_support(self): + A = array([[1., 2.], [2., 0.]]) + x = array([[1., 1.], [0.5, -0.5]]) + b = array([[2., 0.], [2., 2.]]) + + assert_array_almost_equal(x, spsolve(A, b)) + + def test_gssv_badinput(self): + N = 10 + d = arange(N) + 1.0 + A = dia_array(((d, 2*d, d[::-1]), (-3, 0, 5)), shape=(N, N)) + + for container in (csc_array, csr_array): + A = container(A) + b = np.arange(N) + + def not_c_contig(x): + return x.repeat(2)[::2] + + def not_1dim(x): + return x[:,None] + + def bad_type(x): + return x.astype(bool) + + def too_short(x): + return x[:-1] + + badops = [not_c_contig, not_1dim, bad_type, too_short] + + for badop in badops: + msg = f"{container!r} {badop!r}" + # Not C-contiguous + assert_raises((ValueError, TypeError), _superlu.gssv, + N, A.nnz, badop(A.data), A.indices, A.indptr, + b, int(A.format == 'csc'), err_msg=msg) + assert_raises((ValueError, TypeError), _superlu.gssv, + N, A.nnz, A.data, badop(A.indices), A.indptr, + b, int(A.format == 'csc'), err_msg=msg) + assert_raises((ValueError, TypeError), _superlu.gssv, + N, A.nnz, A.data, A.indices, badop(A.indptr), + b, int(A.format == 'csc'), err_msg=msg) + + def test_sparsity_preservation(self): + ident = csc_array([ + [1, 0, 0], + [0, 1, 0], + [0, 0, 1]]) + b = csc_array([ + [0, 1], + [1, 0], + [0, 0]]) + x = spsolve(ident, b) + assert_equal(ident.nnz, 3) + assert_equal(b.nnz, 2) + assert_equal(x.nnz, 2) + assert_allclose(x.toarray(), b.toarray(), atol=1e-12, rtol=1e-12) + + def test_dtype_cast(self): + A_real = scipy.sparse.csr_array([[1, 2, 0], + [0, 0, 3], + [4, 0, 5]]) + A_complex = scipy.sparse.csr_array([[1, 2, 0], + [0, 0, 3], + [4, 0, 5 + 1j]]) + b_real = np.array([1,1,1]) + b_complex = np.array([1,1,1]) + 1j*np.array([1,1,1]) + x = spsolve(A_real, b_real) + assert_(np.issubdtype(x.dtype, np.floating)) + x = spsolve(A_real, b_complex) + assert_(np.issubdtype(x.dtype, np.complexfloating)) + x = spsolve(A_complex, b_real) + assert_(np.issubdtype(x.dtype, np.complexfloating)) + x = spsolve(A_complex, b_complex) + assert_(np.issubdtype(x.dtype, np.complexfloating)) + + @pytest.mark.slow + @pytest.mark.skipif(not has_umfpack, reason="umfpack not available") + def test_bug_8278(self): + check_free_memory(8000) + use_solver(useUmfpack=True) + A, b = setup_bug_8278() + x = spsolve(A, b) + assert_array_almost_equal(A @ x, b) + + +class TestSplu: + def setup_method(self): + use_solver(useUmfpack=False) + n = 40 + d = arange(n) + 1 + self.n = n + self.A = dia_array(((d, 2*d, d[::-1]), (-3, 0, 5)), shape=(n, n)).tocsc() + random.seed(1234) + + def _smoketest(self, spxlu, check, dtype, idx_dtype): + if np.issubdtype(dtype, np.complexfloating): + A = self.A + 1j*self.A.T + else: + A = self.A + + A = A.astype(dtype) + A.indices = A.indices.astype(idx_dtype, copy=False) + A.indptr = A.indptr.astype(idx_dtype, copy=False) + lu = spxlu(A) + + rng = random.RandomState(1234) + + # Input shapes + for k in [None, 1, 2, self.n, self.n+2]: + msg = f"k={k!r}" + + if k is None: + b = rng.rand(self.n) + else: + b = rng.rand(self.n, k) + + if np.issubdtype(dtype, np.complexfloating): + b = b + 1j*rng.rand(*b.shape) + b = b.astype(dtype) + + x = lu.solve(b) + check(A, b, x, msg) + + x = lu.solve(b, 'T') + check(A.T, b, x, msg) + + x = lu.solve(b, 'H') + check(A.T.conj(), b, x, msg) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + def test_splu_smoketest(self): + self._internal_test_splu_smoketest() + + def _internal_test_splu_smoketest(self): + # Check that splu works at all + def check(A, b, x, msg=""): + eps = np.finfo(A.dtype).eps + r = A @ x + assert_(abs(r - b).max() < 1e3*eps, msg) + + for dtype in [np.float32, np.float64, np.complex64, np.complex128]: + for idx_dtype in [np.int32, np.int64]: + self._smoketest(splu, check, dtype, idx_dtype) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + def test_spilu_smoketest(self): + self._internal_test_spilu_smoketest() + + def _internal_test_spilu_smoketest(self): + errors = [] + + def check(A, b, x, msg=""): + r = A @ x + err = abs(r - b).max() + assert_(err < 1e-2, msg) + if b.dtype in (np.float64, np.complex128): + errors.append(err) + + for dtype in [np.float32, np.float64, np.complex64, np.complex128]: + for idx_dtype in [np.int32, np.int64]: + self._smoketest(spilu, check, dtype, idx_dtype) + + assert_(max(errors) > 1e-5) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + def test_spilu_drop_rule(self): + # Test passing in the drop_rule argument to spilu. + A = eye_array(2) + + rules = [ + b'basic,area'.decode('ascii'), # unicode + b'basic,area', # ascii + [b'basic', b'area'.decode('ascii')] + ] + for rule in rules: + # Argument should be accepted + assert_(isinstance(spilu(A, drop_rule=rule), SuperLU)) + + def test_splu_nnz0(self): + A = csc_array((5,5), dtype='d') + assert_raises(RuntimeError, splu, A) + + def test_spilu_nnz0(self): + A = csc_array((5,5), dtype='d') + assert_raises(RuntimeError, spilu, A) + + def test_splu_basic(self): + # Test basic splu functionality. + n = 30 + rng = random.RandomState(12) + a = rng.rand(n, n) + a[a < 0.95] = 0 + # First test with a singular matrix + a[:, 0] = 0 + a_ = csc_array(a) + # Matrix is exactly singular + assert_raises(RuntimeError, splu, a_) + + # Make a diagonal dominant, to make sure it is not singular + a += 4*eye(n) + a_ = csc_array(a) + lu = splu(a_) + b = ones(n) + x = lu.solve(b) + assert_almost_equal(dot(a, x), b) + + def test_splu_perm(self): + # Test the permutation vectors exposed by splu. + n = 30 + a = random.random((n, n)) + a[a < 0.95] = 0 + # Make a diagonal dominant, to make sure it is not singular + a += 4*eye(n) + a_ = csc_array(a) + lu = splu(a_) + # Check that the permutation indices do belong to [0, n-1]. + for perm in (lu.perm_r, lu.perm_c): + assert_(all(perm > -1)) + assert_(all(perm < n)) + assert_equal(len(unique(perm)), len(perm)) + + # Now make a symmetric, and test that the two permutation vectors are + # the same + # Note: a += a.T relies on undefined behavior. + a = a + a.T + a_ = csc_array(a) + lu = splu(a_) + assert_array_equal(lu.perm_r, lu.perm_c) + + @pytest.mark.parametrize("splu_fun, rtol", [(splu, 1e-7), (spilu, 1e-1)]) + def test_natural_permc(self, splu_fun, rtol): + # Test that the "NATURAL" permc_spec does not permute the matrix + rng = np.random.RandomState(42) + n = 500 + p = 0.01 + A = scipy.sparse.random(n, n, p, random_state=rng) + x = rng.rand(n) + # Make A diagonal dominant to make sure it is not singular + A += (n+1)*scipy.sparse.eye_array(n) + A_ = csc_array(A) + b = A_ @ x + + # without permc_spec, permutation is not identity + lu = splu_fun(A_) + assert_(np.any(lu.perm_c != np.arange(n))) + + # with permc_spec="NATURAL", permutation is identity + lu = splu_fun(A_, permc_spec="NATURAL") + assert_array_equal(lu.perm_c, np.arange(n)) + + # Also, lu decomposition is valid + x2 = lu.solve(b) + assert_allclose(x, x2, rtol=rtol) + + @pytest.mark.skipif(not hasattr(sys, 'getrefcount'), reason="no sys.getrefcount") + def test_lu_refcount(self): + # Test that we are keeping track of the reference count with splu. + n = 30 + a = random.random((n, n)) + a[a < 0.95] = 0 + # Make a diagonal dominant, to make sure it is not singular + a += 4*eye(n) + a_ = csc_array(a) + lu = splu(a_) + + # And now test that we don't have a refcount bug + rc = sys.getrefcount(lu) + for attr in ('perm_r', 'perm_c'): + perm = getattr(lu, attr) + assert_equal(sys.getrefcount(lu), rc + 1) + del perm + assert_equal(sys.getrefcount(lu), rc) + + def test_bad_inputs(self): + A = self.A.tocsc() + + assert_raises(ValueError, splu, A[:,:4]) + assert_raises(ValueError, spilu, A[:,:4]) + + for lu in [splu(A), spilu(A)]: + b = random.rand(42) + B = random.rand(42, 3) + BB = random.rand(self.n, 3, 9) + assert_raises(ValueError, lu.solve, b) + assert_raises(ValueError, lu.solve, B) + assert_raises(ValueError, lu.solve, BB) + assert_raises(TypeError, lu.solve, + b.astype(np.complex64)) + assert_raises(TypeError, lu.solve, + b.astype(np.complex128)) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + def test_superlu_dlamch_i386_nan(self): + # SuperLU 4.3 calls some functions returning floats without + # declaring them. On i386@linux call convention, this fails to + # clear floating point registers after call. As a result, NaN + # can appear in the next floating point operation made. + # + # Here's a test case that triggered the issue. + n = 8 + d = np.arange(n) + 1 + A = dia_array(((d, 2*d, d[::-1]), (-3, 0, 5)), shape=(n, n)) + A = A.astype(np.float32) + spilu(A) + A = A + 1j*A + B = A.toarray() + assert_(not np.isnan(B).any()) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + def test_lu_attr(self): + + def check(dtype, complex_2=False): + A = self.A.astype(dtype) + + if complex_2: + A = A + 1j*A.T + + n = A.shape[0] + lu = splu(A) + + # Check that the decomposition is as advertised + + Pc = np.zeros((n, n)) + Pc[np.arange(n), lu.perm_c] = 1 + + Pr = np.zeros((n, n)) + Pr[lu.perm_r, np.arange(n)] = 1 + + Ad = A.toarray() + lhs = Pr.dot(Ad).dot(Pc) + rhs = (lu.L @ lu.U).toarray() + + eps = np.finfo(dtype).eps + + assert_allclose(lhs, rhs, atol=100*eps) + + check(np.float32) + check(np.float64) + check(np.complex64) + check(np.complex128) + check(np.complex64, True) + check(np.complex128, True) + + @pytest.mark.thread_unsafe + @pytest.mark.slow + @sup_sparse_efficiency + def test_threads_parallel(self): + oks = [] + + def worker(): + try: + self.test_splu_basic() + self._internal_test_splu_smoketest() + self._internal_test_spilu_smoketest() + oks.append(True) + except Exception: + pass + + threads = [threading.Thread(target=worker) + for k in range(20)] + for t in threads: + t.start() + for t in threads: + t.join() + + assert_equal(len(oks), 20) + + @pytest.mark.thread_unsafe + def test_singular_matrix(self): + # Test that SuperLU does not print to stdout when a singular matrix is + # passed. See gh-20993. + A = eye_array(10, format='csr') + A[-1, -1] = 0 + b = np.zeros(10) + with pytest.warns(MatrixRankWarning): + res = spsolve(A, b) + assert np.isnan(res).all() + + +class TestGstrsErrors: + def setup_method(self): + self.A = array([[1.0,2.0,3.0],[4.0,5.0,6.0],[7.0,8.0,9.0]], dtype=np.float64) + self.b = np.array([[1.0],[2.0],[3.0]], dtype=np.float64) + + def test_trans(self): + L = scipy.sparse.tril(self.A, format='csc') + U = scipy.sparse.triu(self.A, k=1, format='csc') + with assert_raises(ValueError, match="trans must be N, T, or H"): + _superlu.gstrs('X', L.shape[0], L.nnz, L.data, L.indices, L.indptr, + U.shape[0], U.nnz, U.data, U.indices, U.indptr, self.b) + + def test_shape_LU(self): + L = scipy.sparse.tril(self.A[0:2,0:2], format='csc') + U = scipy.sparse.triu(self.A, k=1, format='csc') + with assert_raises(ValueError, match="L and U must have the same dimension"): + _superlu.gstrs('N', L.shape[0], L.nnz, L.data, L.indices, L.indptr, + U.shape[0], U.nnz, U.data, U.indices, U.indptr, self.b) + + def test_shape_b(self): + L = scipy.sparse.tril(self.A, format='csc') + U = scipy.sparse.triu(self.A, k=1, format='csc') + with assert_raises(ValueError, match="right hand side array has invalid shape"): + _superlu.gstrs('N', L.shape[0], L.nnz, L.data, L.indices, L.indptr, + U.shape[0], U.nnz, U.data, U.indices, U.indptr, + self.b[0:2]) + + def test_types_differ(self): + L = scipy.sparse.tril(self.A.astype(np.float32), format='csc') + U = scipy.sparse.triu(self.A, k=1, format='csc') + with assert_raises(TypeError, match="nzvals types of L and U differ"): + _superlu.gstrs('N', L.shape[0], L.nnz, L.data, L.indices, L.indptr, + U.shape[0], U.nnz, U.data, U.indices, U.indptr, self.b) + + def test_types_unsupported(self): + L = scipy.sparse.tril(self.A.astype(np.uint8), format='csc') + U = scipy.sparse.triu(self.A.astype(np.uint8), k=1, format='csc') + with assert_raises(TypeError, match="nzvals is not of a type supported"): + _superlu.gstrs('N', L.shape[0], L.nnz, L.data, L.indices, L.indptr, + U.shape[0], U.nnz, U.data, U.indices, U.indptr, + self.b.astype(np.uint8)) + +class TestSpsolveTriangular: + def setup_method(self): + use_solver(useUmfpack=False) + + @pytest.mark.parametrize("fmt",["csr","csc"]) + def test_zero_diagonal(self,fmt): + n = 5 + rng = np.random.default_rng(43876432987) + A = rng.standard_normal((n, n)) + b = np.arange(n) + A = scipy.sparse.tril(A, k=0, format=fmt) + + x = spsolve_triangular(A, b, unit_diagonal=True, lower=True) + + A.setdiag(1) + assert_allclose(A.dot(x), b) + + # Regression test from gh-15199 + A = np.array([[0, 0, 0], [1, 0, 0], [1, 1, 0]], dtype=np.float64) + b = np.array([1., 2., 3.]) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "CSC or CSR matrix format is") + spsolve_triangular(A, b, unit_diagonal=True) + + @pytest.mark.parametrize("fmt",["csr","csc"]) + def test_singular(self,fmt): + n = 5 + if fmt == "csr": + A = csr_array((n, n)) + else: + A = csc_array((n, n)) + b = np.arange(n) + for lower in (True, False): + assert_raises(scipy.linalg.LinAlgError, + spsolve_triangular, A, b, lower=lower) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + def test_bad_shape(self): + # A is not square. + A = np.zeros((3, 4)) + b = ones((4, 1)) + assert_raises(ValueError, spsolve_triangular, A, b) + # A2 and b2 have incompatible shapes. + A2 = csr_array(eye(3)) + b2 = array([1.0, 2.0]) + assert_raises(ValueError, spsolve_triangular, A2, b2) + + @pytest.mark.thread_unsafe + @sup_sparse_efficiency + def test_input_types(self): + A = array([[1., 0.], [1., 2.]]) + b = array([[2., 0.], [2., 2.]]) + for matrix_type in (array, csc_array, csr_array): + x = spsolve_triangular(matrix_type(A), b, lower=True) + assert_array_almost_equal(A.dot(x), b) + + @pytest.mark.thread_unsafe + @pytest.mark.slow + @sup_sparse_efficiency + @pytest.mark.parametrize("n", [10, 10**2, 10**3]) + @pytest.mark.parametrize("m", [1, 10]) + @pytest.mark.parametrize("lower", [True, False]) + @pytest.mark.parametrize("format", ["csr", "csc"]) + @pytest.mark.parametrize("unit_diagonal", [False, True]) + @pytest.mark.parametrize("choice_of_A", ["real", "complex"]) + @pytest.mark.parametrize("choice_of_b", ["floats", "ints", "complexints"]) + def test_random(self, n, m, lower, format, unit_diagonal, choice_of_A, choice_of_b): + def random_triangle_matrix(n, lower=True, format="csr", choice_of_A="real"): + if choice_of_A == "real": + dtype = np.float64 + elif choice_of_A == "complex": + dtype = np.complex128 + else: + raise ValueError("choice_of_A must be 'real' or 'complex'.") + rng = np.random.default_rng(789002319) + rvs = rng.random + A = scipy.sparse.random(n, n, density=0.1, format='lil', dtype=dtype, + random_state=rng, data_rvs=rvs) + if lower: + A = scipy.sparse.tril(A, format="lil") + else: + A = scipy.sparse.triu(A, format="lil") + for i in range(n): + A[i, i] = np.random.rand() + 1 + if format == "csc": + A = A.tocsc(copy=False) + else: + A = A.tocsr(copy=False) + return A + + np.random.seed(1234) + A = random_triangle_matrix(n, lower=lower) + if choice_of_b == "floats": + b = np.random.rand(n, m) + elif choice_of_b == "ints": + b = np.random.randint(-9, 9, (n, m)) + elif choice_of_b == "complexints": + b = np.random.randint(-9, 9, (n, m)) + np.random.randint(-9, 9, (n, m)) * 1j + else: + raise ValueError( + "choice_of_b must be 'floats', 'ints', or 'complexints'.") + x = spsolve_triangular(A, b, lower=lower, unit_diagonal=unit_diagonal) + if unit_diagonal: + A.setdiag(1) + assert_allclose(A.dot(x), b, atol=1.5e-6) + + +@pytest.mark.thread_unsafe +@sup_sparse_efficiency +@pytest.mark.parametrize("nnz", [10, 10**2, 10**3]) +@pytest.mark.parametrize("fmt", ["csr", "csc", "coo", "dia", "dok", "lil"]) +def test_is_sptriangular_and_spbandwidth(nnz, fmt): + rng = np.random.default_rng(42) + + N = nnz // 2 + dens = 0.1 + A = scipy.sparse.random_array((N, N), density=dens, format="csr", rng=rng) + A[1, 3] = A[3, 1] = 22 # ensure not upper or lower + A = A.asformat(fmt) + AU = scipy.sparse.triu(A, format=fmt) + AL = scipy.sparse.tril(A, format=fmt) + D = 0.1 * scipy.sparse.eye_array(N, format=fmt) + + assert is_sptriangular(A) == (False, False) + assert is_sptriangular(AL) == (True, False) + assert is_sptriangular(AU) == (False, True) + assert is_sptriangular(D) == (True, True) + + assert spbandwidth(A) == scipy.linalg.bandwidth(A.toarray()) + assert spbandwidth(AU) == scipy.linalg.bandwidth(AU.toarray()) + assert spbandwidth(AL) == scipy.linalg.bandwidth(AL.toarray()) + assert spbandwidth(D) == scipy.linalg.bandwidth(D.toarray()) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..25278d34ecd3353d409a25f7a94797902fe6ef93 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/__init__.py @@ -0,0 +1,22 @@ +""" +Sparse Eigenvalue Solvers +------------------------- + +The submodules of sparse.linalg._eigen: + 1. lobpcg: Locally Optimal Block Preconditioned Conjugate Gradient Method + +""" +from .arpack import * +from .lobpcg import * +from ._svds import svds + +from . import arpack + +__all__ = [ + 'ArpackError', 'ArpackNoConvergence', + 'eigs', 'eigsh', 'lobpcg', 'svds' +] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/_svds.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/_svds.py new file mode 100644 index 0000000000000000000000000000000000000000..ce57e841f9f163edd3945f248a653c94acd2a93c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/_svds.py @@ -0,0 +1,540 @@ +import math +import numpy as np + +from .arpack import _arpack # type: ignore[attr-defined] +from . import eigsh + +from scipy._lib._util import check_random_state, _transition_to_rng +from scipy.sparse.linalg._interface import LinearOperator, aslinearoperator +from scipy.sparse.linalg._eigen.lobpcg import lobpcg # type: ignore[no-redef] +from scipy.sparse.linalg._svdp import _svdp +from scipy.linalg import svd + +arpack_int = _arpack.timing.nbx.dtype +__all__ = ['svds'] + + +def _herm(x): + return x.T.conj() + + +def _iv(A, k, ncv, tol, which, v0, maxiter, + return_singular, solver, rng): + + # input validation/standardization for `solver` + # out of order because it's needed for other parameters + solver = str(solver).lower() + solvers = {"arpack", "lobpcg", "propack"} + if solver not in solvers: + raise ValueError(f"solver must be one of {solvers}.") + + # input validation/standardization for `A` + A = aslinearoperator(A) # this takes care of some input validation + if not np.issubdtype(A.dtype, np.number): + message = "`A` must be of numeric data type." + raise ValueError(message) + if math.prod(A.shape) == 0: + message = "`A` must not be empty." + raise ValueError(message) + + # input validation/standardization for `k` + kmax = min(A.shape) if solver == 'propack' else min(A.shape) - 1 + if int(k) != k or not (0 < k <= kmax): + message = "`k` must be an integer satisfying `0 < k < min(A.shape)`." + raise ValueError(message) + k = int(k) + + # input validation/standardization for `ncv` + if solver == "arpack" and ncv is not None: + if int(ncv) != ncv or not (k < ncv < min(A.shape)): + message = ("`ncv` must be an integer satisfying " + "`k < ncv < min(A.shape)`.") + raise ValueError(message) + ncv = int(ncv) + + # input validation/standardization for `tol` + if tol < 0 or not np.isfinite(tol): + message = "`tol` must be a non-negative floating point value." + raise ValueError(message) + tol = float(tol) + + # input validation/standardization for `which` + which = str(which).upper() + whichs = {'LM', 'SM'} + if which not in whichs: + raise ValueError(f"`which` must be in {whichs}.") + + # input validation/standardization for `v0` + if v0 is not None: + v0 = np.atleast_1d(v0) + if not (np.issubdtype(v0.dtype, np.complexfloating) + or np.issubdtype(v0.dtype, np.floating)): + message = ("`v0` must be of floating or complex floating " + "data type.") + raise ValueError(message) + + shape = (A.shape[0],) if solver == 'propack' else (min(A.shape),) + if v0.shape != shape: + message = f"`v0` must have shape {shape}." + raise ValueError(message) + + # input validation/standardization for `maxiter` + if maxiter is not None and (int(maxiter) != maxiter or maxiter <= 0): + message = "`maxiter` must be a positive integer." + raise ValueError(message) + maxiter = int(maxiter) if maxiter is not None else maxiter + + # input validation/standardization for `return_singular_vectors` + # not going to be flexible with this; too complicated for little gain + rs_options = {True, False, "vh", "u"} + if return_singular not in rs_options: + raise ValueError(f"`return_singular_vectors` must be in {rs_options}.") + + rng = check_random_state(rng) + + return (A, k, ncv, tol, which, v0, maxiter, + return_singular, solver, rng) + + +@_transition_to_rng("random_state", position_num=9) +def svds(A, k=6, ncv=None, tol=0, which='LM', v0=None, + maxiter=None, return_singular_vectors=True, + solver='arpack', rng=None, options=None): + """ + Partial singular value decomposition of a sparse matrix. + + Compute the largest or smallest `k` singular values and corresponding + singular vectors of a sparse matrix `A`. The order in which the singular + values are returned is not guaranteed. + + In the descriptions below, let ``M, N = A.shape``. + + Parameters + ---------- + A : ndarray, sparse matrix, or LinearOperator + Matrix to decompose of a floating point numeric dtype. + k : int, default: 6 + Number of singular values and singular vectors to compute. + Must satisfy ``1 <= k <= kmax``, where ``kmax=min(M, N)`` for + ``solver='propack'`` and ``kmax=min(M, N) - 1`` otherwise. + ncv : int, optional + When ``solver='arpack'``, this is the number of Lanczos vectors + generated. See :ref:`'arpack' ` for details. + When ``solver='lobpcg'`` or ``solver='propack'``, this parameter is + ignored. + tol : float, optional + Tolerance for singular values. Zero (default) means machine precision. + which : {'LM', 'SM'} + Which `k` singular values to find: either the largest magnitude ('LM') + or smallest magnitude ('SM') singular values. + v0 : ndarray, optional + The starting vector for iteration; see method-specific + documentation (:ref:`'arpack' `, + :ref:`'lobpcg' `), or + :ref:`'propack' ` for details. + maxiter : int, optional + Maximum number of iterations; see method-specific + documentation (:ref:`'arpack' `, + :ref:`'lobpcg' `), or + :ref:`'propack' ` for details. + return_singular_vectors : {True, False, "u", "vh"} + Singular values are always computed and returned; this parameter + controls the computation and return of singular vectors. + + - ``True``: return singular vectors. + - ``False``: do not return singular vectors. + - ``"u"``: if ``M <= N``, compute only the left singular vectors and + return ``None`` for the right singular vectors. Otherwise, compute + all singular vectors. + - ``"vh"``: if ``M > N``, compute only the right singular vectors and + return ``None`` for the left singular vectors. Otherwise, compute + all singular vectors. + + If ``solver='propack'``, the option is respected regardless of the + matrix shape. + + solver : {'arpack', 'propack', 'lobpcg'}, optional + The solver used. + :ref:`'arpack' `, + :ref:`'lobpcg' `, and + :ref:`'propack' ` are supported. + Default: `'arpack'`. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + options : dict, optional + A dictionary of solver-specific options. No solver-specific options + are currently supported; this parameter is reserved for future use. + + Returns + ------- + u : ndarray, shape=(M, k) + Unitary matrix having left singular vectors as columns. + s : ndarray, shape=(k,) + The singular values. + vh : ndarray, shape=(k, N) + Unitary matrix having right singular vectors as rows. + + Notes + ----- + This is a naive implementation using ARPACK or LOBPCG as an eigensolver + on the matrix ``A.conj().T @ A`` or ``A @ A.conj().T``, depending on + which one is smaller size, followed by the Rayleigh-Ritz method + as postprocessing; see + Using the normal matrix, in Rayleigh-Ritz method, (2022, Nov. 19), + Wikipedia, https://w.wiki/4zms. + + Alternatively, the PROPACK solver can be called. + + Choices of the input matrix `A` numeric dtype may be limited. + Only ``solver="lobpcg"`` supports all floating point dtypes + real: 'np.float32', 'np.float64', 'np.longdouble' and + complex: 'np.complex64', 'np.complex128', 'np.clongdouble'. + The ``solver="arpack"`` supports only + 'np.float32', 'np.float64', and 'np.complex128'. + + Examples + -------- + Construct a matrix `A` from singular values and vectors. + + >>> import numpy as np + >>> from scipy import sparse, linalg, stats + >>> from scipy.sparse.linalg import svds, aslinearoperator, LinearOperator + + Construct a dense matrix `A` from singular values and vectors. + + >>> rng = np.random.default_rng(258265244568965474821194062361901728911) + >>> orthogonal = stats.ortho_group.rvs(10, random_state=rng) + >>> s = [1e-3, 1, 2, 3, 4] # non-zero singular values + >>> u = orthogonal[:, :5] # left singular vectors + >>> vT = orthogonal[:, 5:].T # right singular vectors + >>> A = u @ np.diag(s) @ vT + + With only four singular values/vectors, the SVD approximates the original + matrix. + + >>> u4, s4, vT4 = svds(A, k=4) + >>> A4 = u4 @ np.diag(s4) @ vT4 + >>> np.allclose(A4, A, atol=1e-3) + True + + With all five non-zero singular values/vectors, we can reproduce + the original matrix more accurately. + + >>> u5, s5, vT5 = svds(A, k=5) + >>> A5 = u5 @ np.diag(s5) @ vT5 + >>> np.allclose(A5, A) + True + + The singular values match the expected singular values. + + >>> np.allclose(s5, s) + True + + Since the singular values are not close to each other in this example, + every singular vector matches as expected up to a difference in sign. + + >>> (np.allclose(np.abs(u5), np.abs(u)) and + ... np.allclose(np.abs(vT5), np.abs(vT))) + True + + The singular vectors are also orthogonal. + + >>> (np.allclose(u5.T @ u5, np.eye(5)) and + ... np.allclose(vT5 @ vT5.T, np.eye(5))) + True + + If there are (nearly) multiple singular values, the corresponding + individual singular vectors may be unstable, but the whole invariant + subspace containing all such singular vectors is computed accurately + as can be measured by angles between subspaces via 'subspace_angles'. + + >>> rng = np.random.default_rng(178686584221410808734965903901790843963) + >>> s = [1, 1 + 1e-6] # non-zero singular values + >>> u, _ = np.linalg.qr(rng.standard_normal((99, 2))) + >>> v, _ = np.linalg.qr(rng.standard_normal((99, 2))) + >>> vT = v.T + >>> A = u @ np.diag(s) @ vT + >>> A = A.astype(np.float32) + >>> u2, s2, vT2 = svds(A, k=2, rng=rng) + >>> np.allclose(s2, s) + True + + The angles between the individual exact and computed singular vectors + may not be so small. To check use: + + >>> (linalg.subspace_angles(u2[:, :1], u[:, :1]) + + ... linalg.subspace_angles(u2[:, 1:], u[:, 1:])) + array([0.06562513]) # may vary + >>> (linalg.subspace_angles(vT2[:1, :].T, vT[:1, :].T) + + ... linalg.subspace_angles(vT2[1:, :].T, vT[1:, :].T)) + array([0.06562507]) # may vary + + As opposed to the angles between the 2-dimensional invariant subspaces + that these vectors span, which are small for rights singular vectors + + >>> linalg.subspace_angles(u2, u).sum() < 1e-6 + True + + as well as for left singular vectors. + + >>> linalg.subspace_angles(vT2.T, vT.T).sum() < 1e-6 + True + + The next example follows that of 'sklearn.decomposition.TruncatedSVD'. + + >>> rng = np.random.default_rng(0) + >>> X_dense = rng.random(size=(100, 100)) + >>> X_dense[:, 2 * np.arange(50)] = 0 + >>> X = sparse.csr_array(X_dense) + >>> _, singular_values, _ = svds(X, k=5, rng=rng) + >>> print(singular_values) + [ 4.3221... 4.4043... 4.4907... 4.5858... 35.4549...] + + The function can be called without the transpose of the input matrix + ever explicitly constructed. + + >>> rng = np.random.default_rng(102524723947864966825913730119128190974) + >>> G = sparse.random_array((8, 9), density=0.5, rng=rng) + >>> Glo = aslinearoperator(G) + >>> _, singular_values_svds, _ = svds(Glo, k=5, rng=rng) + >>> _, singular_values_svd, _ = linalg.svd(G.toarray()) + >>> np.allclose(singular_values_svds, singular_values_svd[-4::-1]) + True + + The most memory efficient scenario is where neither + the original matrix, nor its transpose, is explicitly constructed. + Our example computes the smallest singular values and vectors + of 'LinearOperator' constructed from the numpy function 'np.diff' used + column-wise to be consistent with 'LinearOperator' operating on columns. + + >>> diff0 = lambda a: np.diff(a, axis=0) + + Let us create the matrix from 'diff0' to be used for validation only. + + >>> n = 5 # The dimension of the space. + >>> M_from_diff0 = diff0(np.eye(n)) + >>> print(M_from_diff0.astype(int)) + [[-1 1 0 0 0] + [ 0 -1 1 0 0] + [ 0 0 -1 1 0] + [ 0 0 0 -1 1]] + + The matrix 'M_from_diff0' is bi-diagonal and could be alternatively + created directly by + + >>> M = - np.eye(n - 1, n, dtype=int) + >>> np.fill_diagonal(M[:,1:], 1) + >>> np.allclose(M, M_from_diff0) + True + + Its transpose + + >>> print(M.T) + [[-1 0 0 0] + [ 1 -1 0 0] + [ 0 1 -1 0] + [ 0 0 1 -1] + [ 0 0 0 1]] + + can be viewed as the incidence matrix; see + Incidence matrix, (2022, Nov. 19), Wikipedia, https://w.wiki/5YXU, + of a linear graph with 5 vertices and 4 edges. The 5x5 normal matrix + ``M.T @ M`` thus is + + >>> print(M.T @ M) + [[ 1 -1 0 0 0] + [-1 2 -1 0 0] + [ 0 -1 2 -1 0] + [ 0 0 -1 2 -1] + [ 0 0 0 -1 1]] + + the graph Laplacian, while the actually used in 'svds' smaller size + 4x4 normal matrix ``M @ M.T`` + + >>> print(M @ M.T) + [[ 2 -1 0 0] + [-1 2 -1 0] + [ 0 -1 2 -1] + [ 0 0 -1 2]] + + is the so-called edge-based Laplacian; see + Symmetric Laplacian via the incidence matrix, in Laplacian matrix, + (2022, Nov. 19), Wikipedia, https://w.wiki/5YXW. + + The 'LinearOperator' setup needs the options 'rmatvec' and 'rmatmat' + of multiplication by the matrix transpose ``M.T``, but we want to be + matrix-free to save memory, so knowing how ``M.T`` looks like, we + manually construct the following function to be + used in ``rmatmat=diff0t``. + + >>> def diff0t(a): + ... if a.ndim == 1: + ... a = a[:,np.newaxis] # Turn 1D into 2D array + ... d = np.zeros((a.shape[0] + 1, a.shape[1]), dtype=a.dtype) + ... d[0, :] = - a[0, :] + ... d[1:-1, :] = a[0:-1, :] - a[1:, :] + ... d[-1, :] = a[-1, :] + ... return d + + We check that our function 'diff0t' for the matrix transpose is valid. + + >>> np.allclose(M.T, diff0t(np.eye(n-1))) + True + + Now we setup our matrix-free 'LinearOperator' called 'diff0_func_aslo' + and for validation the matrix-based 'diff0_matrix_aslo'. + + >>> def diff0_func_aslo_def(n): + ... return LinearOperator(matvec=diff0, + ... matmat=diff0, + ... rmatvec=diff0t, + ... rmatmat=diff0t, + ... shape=(n - 1, n)) + >>> diff0_func_aslo = diff0_func_aslo_def(n) + >>> diff0_matrix_aslo = aslinearoperator(M_from_diff0) + + And validate both the matrix and its transpose in 'LinearOperator'. + + >>> np.allclose(diff0_func_aslo(np.eye(n)), + ... diff0_matrix_aslo(np.eye(n))) + True + >>> np.allclose(diff0_func_aslo.T(np.eye(n-1)), + ... diff0_matrix_aslo.T(np.eye(n-1))) + True + + Having the 'LinearOperator' setup validated, we run the solver. + + >>> n = 100 + >>> diff0_func_aslo = diff0_func_aslo_def(n) + >>> u, s, vT = svds(diff0_func_aslo, k=3, which='SM') + + The singular values squared and the singular vectors are known + explicitly; see + Pure Dirichlet boundary conditions, in + Eigenvalues and eigenvectors of the second derivative, + (2022, Nov. 19), Wikipedia, https://w.wiki/5YX6, + since 'diff' corresponds to first + derivative, and its smaller size n-1 x n-1 normal matrix + ``M @ M.T`` represent the discrete second derivative with the Dirichlet + boundary conditions. We use these analytic expressions for validation. + + >>> se = 2. * np.sin(np.pi * np.arange(1, 4) / (2. * n)) + >>> ue = np.sqrt(2 / n) * np.sin(np.pi * np.outer(np.arange(1, n), + ... np.arange(1, 4)) / n) + >>> np.allclose(s, se, atol=1e-3) + True + >>> np.allclose(np.abs(u), np.abs(ue), atol=1e-6) + True + + """ + args = _iv(A, k, ncv, tol, which, v0, maxiter, return_singular_vectors, + solver, rng) + (A, k, ncv, tol, which, v0, maxiter, + return_singular_vectors, solver, rng) = args + + largest = (which == 'LM') + n, m = A.shape + + if n >= m: + X_dot = A.matvec + X_matmat = A.matmat + XH_dot = A.rmatvec + XH_mat = A.rmatmat + transpose = False + else: + X_dot = A.rmatvec + X_matmat = A.rmatmat + XH_dot = A.matvec + XH_mat = A.matmat + transpose = True + + dtype = getattr(A, 'dtype', None) + if dtype is None: + dtype = A.dot(np.zeros([m, 1])).dtype + + def matvec_XH_X(x): + return XH_dot(X_dot(x)) + + def matmat_XH_X(x): + return XH_mat(X_matmat(x)) + + XH_X = LinearOperator(matvec=matvec_XH_X, dtype=A.dtype, + matmat=matmat_XH_X, + shape=(min(A.shape), min(A.shape))) + + # Get a low rank approximation of the implicitly defined gramian matrix. + # This is not a stable way to approach the problem. + if solver == 'lobpcg': + + if k == 1 and v0 is not None: + X = np.reshape(v0, (-1, 1)) + else: + X = rng.standard_normal(size=(min(A.shape), k)) + + _, eigvec = lobpcg(XH_X, X, tol=tol ** 2, maxiter=maxiter, + largest=largest) + + elif solver == 'propack': + jobu = return_singular_vectors in {True, 'u'} + jobv = return_singular_vectors in {True, 'vh'} + irl_mode = (which == 'SM') + res = _svdp(A, k=k, tol=tol**2, which=which, maxiter=None, + compute_u=jobu, compute_v=jobv, irl_mode=irl_mode, + kmax=maxiter, v0=v0, rng=rng) + + u, s, vh, _ = res # but we'll ignore bnd, the last output + + # PROPACK order appears to be largest first. `svds` output order is not + # guaranteed, according to documentation, but for ARPACK and LOBPCG + # they actually are ordered smallest to largest, so reverse for + # consistency. + s = s[::-1] + u = u[:, ::-1] + vh = vh[::-1] + + u = u if jobu else None + vh = vh if jobv else None + + if return_singular_vectors: + return u, s, vh + else: + return s + + elif solver == 'arpack' or solver is None: + if v0 is None: + v0 = rng.standard_normal(size=(min(A.shape),)) + _, eigvec = eigsh(XH_X, k=k, tol=tol ** 2, maxiter=maxiter, + ncv=ncv, which=which, v0=v0) + # arpack do not guarantee exactly orthonormal eigenvectors + # for clustered eigenvalues, especially in complex arithmetic + eigvec, _ = np.linalg.qr(eigvec) + + # the eigenvectors eigvec must be orthonomal here; see gh-16712 + Av = X_matmat(eigvec) + if not return_singular_vectors: + s = svd(Av, compute_uv=False, overwrite_a=True) + return s[::-1] + + # compute the left singular vectors of X and update the right ones + # accordingly + u, s, vh = svd(Av, full_matrices=False, overwrite_a=True) + u = u[:, ::-1] + s = s[::-1] + vh = vh[::-1] + + jobu = return_singular_vectors in {True, 'u'} + jobv = return_singular_vectors in {True, 'vh'} + + if transpose: + u_tmp = eigvec @ _herm(vh) if jobu else None + vh = _herm(u) if jobv else None + u = u_tmp + else: + if not jobu: + u = None + vh = vh @ _herm(eigvec) if jobv else None + + return u, s, vh diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/_svds_doc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/_svds_doc.py new file mode 100644 index 0000000000000000000000000000000000000000..90b85876d2a69dd3c2efa56445e46c93bf0eaa9e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/_svds_doc.py @@ -0,0 +1,382 @@ +def _svds_arpack_doc(A, k=6, ncv=None, tol=0, which='LM', v0=None, + maxiter=None, return_singular_vectors=True, + solver='arpack', rng=None): + """ + Partial singular value decomposition of a sparse matrix using ARPACK. + + Compute the largest or smallest `k` singular values and corresponding + singular vectors of a sparse matrix `A`. The order in which the singular + values are returned is not guaranteed. + + In the descriptions below, let ``M, N = A.shape``. + + Parameters + ---------- + A : sparse matrix or LinearOperator + Matrix to decompose. + k : int, optional + Number of singular values and singular vectors to compute. + Must satisfy ``1 <= k <= min(M, N) - 1``. + Default is 6. + ncv : int, optional + The number of Lanczos vectors generated. + The default is ``min(n, max(2*k + 1, 20))``. + If specified, must satisfy ``k + 1 < ncv < min(M, N)``; ``ncv > 2*k`` + is recommended. + tol : float, optional + Tolerance for singular values. Zero (default) means machine precision. + which : {'LM', 'SM'} + Which `k` singular values to find: either the largest magnitude ('LM') + or smallest magnitude ('SM') singular values. + v0 : ndarray, optional + The starting vector for iteration: + an (approximate) left singular vector if ``N > M`` and a right singular + vector otherwise. Must be of length ``min(M, N)``. + Default: random + maxiter : int, optional + Maximum number of Arnoldi update iterations allowed; + default is ``min(M, N) * 10``. + return_singular_vectors : {True, False, "u", "vh"} + Singular values are always computed and returned; this parameter + controls the computation and return of singular vectors. + + - ``True``: return singular vectors. + - ``False``: do not return singular vectors. + - ``"u"``: if ``M <= N``, compute only the left singular vectors and + return ``None`` for the right singular vectors. Otherwise, compute + all singular vectors. + - ``"vh"``: if ``M > N``, compute only the right singular vectors and + return ``None`` for the left singular vectors. Otherwise, compute + all singular vectors. + + solver : {'arpack', 'propack', 'lobpcg'}, optional + This is the solver-specific documentation for ``solver='arpack'``. + :ref:`'lobpcg' ` and + :ref:`'propack' ` + are also supported. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + options : dict, optional + A dictionary of solver-specific options. No solver-specific options + are currently supported; this parameter is reserved for future use. + + Returns + ------- + u : ndarray, shape=(M, k) + Unitary matrix having left singular vectors as columns. + s : ndarray, shape=(k,) + The singular values. + vh : ndarray, shape=(k, N) + Unitary matrix having right singular vectors as rows. + + Notes + ----- + This is a naive implementation using ARPACK as an eigensolver + on ``A.conj().T @ A`` or ``A @ A.conj().T``, depending on which one is more + efficient. + + Examples + -------- + Construct a matrix ``A`` from singular values and vectors. + + >>> import numpy as np + >>> from scipy.stats import ortho_group + >>> from scipy.sparse import csc_array, diags_array + >>> from scipy.sparse.linalg import svds + >>> rng = np.random.default_rng() + >>> orthogonal = csc_array(ortho_group.rvs(10, random_state=rng)) + >>> s = [0.0001, 0.001, 3, 4, 5] # singular values + >>> u = orthogonal[:, :5] # left singular vectors + >>> vT = orthogonal[:, 5:].T # right singular vectors + >>> A = u @ diags_array(s) @ vT + + With only three singular values/vectors, the SVD approximates the original + matrix. + + >>> u2, s2, vT2 = svds(A, k=3, solver='arpack') + >>> A2 = u2 @ np.diag(s2) @ vT2 + >>> np.allclose(A2, A.toarray(), atol=1e-3) + True + + With all five singular values/vectors, we can reproduce the original + matrix. + + >>> u3, s3, vT3 = svds(A, k=5, solver='arpack') + >>> A3 = u3 @ np.diag(s3) @ vT3 + >>> np.allclose(A3, A.toarray()) + True + + The singular values match the expected singular values, and the singular + vectors are as expected up to a difference in sign. + + >>> (np.allclose(s3, s) and + ... np.allclose(np.abs(u3), np.abs(u.toarray())) and + ... np.allclose(np.abs(vT3), np.abs(vT.toarray()))) + True + + The singular vectors are also orthogonal. + + >>> (np.allclose(u3.T @ u3, np.eye(5)) and + ... np.allclose(vT3 @ vT3.T, np.eye(5))) + True + """ + pass + + +def _svds_lobpcg_doc(A, k=6, ncv=None, tol=0, which='LM', v0=None, + maxiter=None, return_singular_vectors=True, + solver='lobpcg', rng=None): + """ + Partial singular value decomposition of a sparse matrix using LOBPCG. + + Compute the largest or smallest `k` singular values and corresponding + singular vectors of a sparse matrix `A`. The order in which the singular + values are returned is not guaranteed. + + In the descriptions below, let ``M, N = A.shape``. + + Parameters + ---------- + A : sparse matrix or LinearOperator + Matrix to decompose. + k : int, default: 6 + Number of singular values and singular vectors to compute. + Must satisfy ``1 <= k <= min(M, N) - 1``. + ncv : int, optional + Ignored. + tol : float, optional + Tolerance for singular values. Zero (default) means machine precision. + which : {'LM', 'SM'} + Which `k` singular values to find: either the largest magnitude ('LM') + or smallest magnitude ('SM') singular values. + v0 : ndarray, optional + If `k` is 1, the starting vector for iteration: + an (approximate) left singular vector if ``N > M`` and a right singular + vector otherwise. Must be of length ``min(M, N)``. + Ignored otherwise. + Default: random + maxiter : int, default: 20 + Maximum number of iterations. + return_singular_vectors : {True, False, "u", "vh"} + Singular values are always computed and returned; this parameter + controls the computation and return of singular vectors. + + - ``True``: return singular vectors. + - ``False``: do not return singular vectors. + - ``"u"``: if ``M <= N``, compute only the left singular vectors and + return ``None`` for the right singular vectors. Otherwise, compute + all singular vectors. + - ``"vh"``: if ``M > N``, compute only the right singular vectors and + return ``None`` for the left singular vectors. Otherwise, compute + all singular vectors. + + solver : {'arpack', 'propack', 'lobpcg'}, optional + This is the solver-specific documentation for ``solver='lobpcg'``. + :ref:`'arpack' ` and + :ref:`'propack' ` + are also supported. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + options : dict, optional + A dictionary of solver-specific options. No solver-specific options + are currently supported; this parameter is reserved for future use. + + Returns + ------- + u : ndarray, shape=(M, k) + Unitary matrix having left singular vectors as columns. + s : ndarray, shape=(k,) + The singular values. + vh : ndarray, shape=(k, N) + Unitary matrix having right singular vectors as rows. + + Notes + ----- + This is a naive implementation using LOBPCG as an eigensolver + on ``A.conj().T @ A`` or ``A @ A.conj().T``, depending on which one is more + efficient. + + Examples + -------- + Construct a matrix ``A`` from singular values and vectors. + + >>> import numpy as np + >>> from scipy.stats import ortho_group + >>> from scipy.sparse import csc_array, diags_array + >>> from scipy.sparse.linalg import svds + >>> rng = np.random.default_rng() + >>> orthogonal = csc_array(ortho_group.rvs(10, random_state=rng)) + >>> s = [0.0001, 0.001, 3, 4, 5] # singular values + >>> u = orthogonal[:, :5] # left singular vectors + >>> vT = orthogonal[:, 5:].T # right singular vectors + >>> A = u @ diags_array(s) @ vT + + With only three singular values/vectors, the SVD approximates the original + matrix. + + >>> u2, s2, vT2 = svds(A, k=3, solver='lobpcg') + >>> A2 = u2 @ np.diag(s2) @ vT2 + >>> np.allclose(A2, A.toarray(), atol=1e-3) + True + + With all five singular values/vectors, we can reproduce the original + matrix. + + >>> u3, s3, vT3 = svds(A, k=5, solver='lobpcg') + >>> A3 = u3 @ np.diag(s3) @ vT3 + >>> np.allclose(A3, A.toarray()) + True + + The singular values match the expected singular values, and the singular + vectors are as expected up to a difference in sign. + + >>> (np.allclose(s3, s) and + ... np.allclose(np.abs(u3), np.abs(u.todense())) and + ... np.allclose(np.abs(vT3), np.abs(vT.todense()))) + True + + The singular vectors are also orthogonal. + + >>> (np.allclose(u3.T @ u3, np.eye(5)) and + ... np.allclose(vT3 @ vT3.T, np.eye(5))) + True + + """ + pass + + +def _svds_propack_doc(A, k=6, ncv=None, tol=0, which='LM', v0=None, + maxiter=None, return_singular_vectors=True, + solver='propack', rng=None): + """ + Partial singular value decomposition of a sparse matrix using PROPACK. + + Compute the largest or smallest `k` singular values and corresponding + singular vectors of a sparse matrix `A`. The order in which the singular + values are returned is not guaranteed. + + In the descriptions below, let ``M, N = A.shape``. + + Parameters + ---------- + A : sparse matrix or LinearOperator + Matrix to decompose. If `A` is a ``LinearOperator`` + object, it must define both ``matvec`` and ``rmatvec`` methods. + k : int, default: 6 + Number of singular values and singular vectors to compute. + Must satisfy ``1 <= k <= min(M, N)``. + ncv : int, optional + Ignored. + tol : float, optional + The desired relative accuracy for computed singular values. + Zero (default) means machine precision. + which : {'LM', 'SM'} + Which `k` singular values to find: either the largest magnitude ('LM') + or smallest magnitude ('SM') singular values. Note that choosing + ``which='SM'`` will force the ``irl`` option to be set ``True``. + v0 : ndarray, optional + Starting vector for iterations: must be of length ``A.shape[0]``. + If not specified, PROPACK will generate a starting vector. + maxiter : int, optional + Maximum number of iterations / maximal dimension of the Krylov + subspace. Default is ``10 * k``. + return_singular_vectors : {True, False, "u", "vh"} + Singular values are always computed and returned; this parameter + controls the computation and return of singular vectors. + + - ``True``: return singular vectors. + - ``False``: do not return singular vectors. + - ``"u"``: compute only the left singular vectors; return ``None`` for + the right singular vectors. + - ``"vh"``: compute only the right singular vectors; return ``None`` + for the left singular vectors. + + solver : {'arpack', 'propack', 'lobpcg'}, optional + This is the solver-specific documentation for ``solver='propack'``. + :ref:`'arpack' ` and + :ref:`'lobpcg' ` + are also supported. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + options : dict, optional + A dictionary of solver-specific options. No solver-specific options + are currently supported; this parameter is reserved for future use. + + Returns + ------- + u : ndarray, shape=(M, k) + Unitary matrix having left singular vectors as columns. + s : ndarray, shape=(k,) + The singular values. + vh : ndarray, shape=(k, N) + Unitary matrix having right singular vectors as rows. + + Notes + ----- + This is an interface to the Fortran library PROPACK [1]_. + The current default is to run with IRL mode disabled unless seeking the + smallest singular values/vectors (``which='SM'``). + + References + ---------- + + .. [1] Larsen, Rasmus Munk. "PROPACK-Software for large and sparse SVD + calculations." Available online. URL + http://sun.stanford.edu/~rmunk/PROPACK (2004): 2008-2009. + + Examples + -------- + Construct a matrix ``A`` from singular values and vectors. + + >>> import numpy as np + >>> from scipy.stats import ortho_group + >>> from scipy.sparse import csc_array, diags_array + >>> from scipy.sparse.linalg import svds + >>> rng = np.random.default_rng() + >>> orthogonal = csc_array(ortho_group.rvs(10, random_state=rng)) + >>> s = [0.0001, 0.001, 3, 4, 5] # singular values + >>> u = orthogonal[:, :5] # left singular vectors + >>> vT = orthogonal[:, 5:].T # right singular vectors + >>> A = u @ diags_array(s) @ vT + + With only three singular values/vectors, the SVD approximates the original + matrix. + + >>> u2, s2, vT2 = svds(A, k=3, solver='propack') + >>> A2 = u2 @ np.diag(s2) @ vT2 + >>> np.allclose(A2, A.todense(), atol=1e-3) + True + + With all five singular values/vectors, we can reproduce the original + matrix. + + >>> u3, s3, vT3 = svds(A, k=5, solver='propack') + >>> A3 = u3 @ np.diag(s3) @ vT3 + >>> np.allclose(A3, A.todense()) + True + + The singular values match the expected singular values, and the singular + vectors are as expected up to a difference in sign. + + >>> (np.allclose(s3, s) and + ... np.allclose(np.abs(u3), np.abs(u.toarray())) and + ... np.allclose(np.abs(vT3), np.abs(vT.toarray()))) + True + + The singular vectors are also orthogonal. + + >>> (np.allclose(u3.T @ u3, np.eye(5)) and + ... np.allclose(vT3 @ vT3.T, np.eye(5))) + True + + """ + pass diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/COPYING b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/COPYING new file mode 100644 index 0000000000000000000000000000000000000000..e87667e1b8c178e53c6a7c6268ebc09ab4b0476c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/COPYING @@ -0,0 +1,45 @@ + +BSD Software License + +Pertains to ARPACK and P_ARPACK + +Copyright (c) 1996-2008 Rice University. +Developed by D.C. Sorensen, R.B. Lehoucq, C. Yang, and K. Maschhoff. +All rights reserved. + +Arpack has been renamed to arpack-ng. + +Copyright (c) 2001-2011 - Scilab Enterprises +Updated by Allan Cornet, Sylvestre Ledru. + +Copyright (c) 2010 - Jordi Gutiérrez Hermoso (Octave patch) + +Copyright (c) 2007 - Sébastien Fabbro (gentoo patch) + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + +- Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + +- Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer listed + in this license in the documentation and/or other materials + provided with the distribution. + +- Neither the name of the copyright holders nor the names of its + contributors may be used to endorse or promote products derived from + this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR +A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT +OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, +SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT +LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, +DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY +THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT +(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE +OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..679b94480d7ff5a11e037ffb758f2214c6e5097f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/__init__.py @@ -0,0 +1,20 @@ +""" +Eigenvalue solver using iterative methods. + +Find k eigenvectors and eigenvalues of a matrix A using the +Arnoldi/Lanczos iterative methods from ARPACK [1]_,[2]_. + +These methods are most useful for large sparse matrices. + + - eigs(A,k) + - eigsh(A,k) + +References +---------- +.. [1] ARPACK Software, http://www.caam.rice.edu/software/ARPACK/ +.. [2] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK USERS GUIDE: + Solution of Large Scale Eigenvalue Problems by Implicitly Restarted + Arnoldi Methods. SIAM, Philadelphia, PA, 1998. + +""" +from .arpack import * diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py new file mode 100644 index 0000000000000000000000000000000000000000..623b605b2da51b462c88b97c8083bc1a135cc161 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py @@ -0,0 +1,1700 @@ +""" +Find a few eigenvectors and eigenvalues of a matrix. + + +Uses ARPACK: https://github.com/opencollab/arpack-ng + +""" +# Wrapper implementation notes +# +# ARPACK Entry Points +# ------------------- +# The entry points to ARPACK are +# - (s,d)seupd : single and double precision symmetric matrix +# - (s,d,c,z)neupd: single,double,complex,double complex general matrix +# This wrapper puts the *neupd (general matrix) interfaces in eigs() +# and the *seupd (symmetric matrix) in eigsh(). +# There is no specialized interface for complex Hermitian matrices. +# To find eigenvalues of a complex Hermitian matrix you +# may use eigsh(), but eigsh() will simply call eigs() +# and return the real part of the eigenvalues thus obtained. + +# Number of eigenvalues returned and complex eigenvalues +# ------------------------------------------------------ +# The ARPACK nonsymmetric real and double interface (s,d)naupd return +# eigenvalues and eigenvectors in real (float,double) arrays. +# Since the eigenvalues and eigenvectors are, in general, complex +# ARPACK puts the real and imaginary parts in consecutive entries +# in real-valued arrays. This wrapper puts the real entries +# into complex data types and attempts to return the requested eigenvalues +# and eigenvectors. + + +# Solver modes +# ------------ +# ARPACK and handle shifted and shift-inverse computations +# for eigenvalues by providing a shift (sigma) and a solver. + +import numpy as np +import warnings +from scipy.sparse.linalg._interface import aslinearoperator, LinearOperator +from scipy.sparse import eye, issparse +from scipy.linalg import eig, eigh, lu_factor, lu_solve +from scipy.sparse._sputils import ( + convert_pydata_sparse_to_scipy, isdense, is_pydata_spmatrix, +) +from scipy.sparse.linalg import gmres, splu +from scipy._lib._util import _aligned_zeros +from scipy._lib._threadsafety import ReentrancyLock +from . import _arpack +arpack_int = _arpack.timing.nbx.dtype + +__docformat__ = "restructuredtext en" + +__all__ = ['eigs', 'eigsh', 'ArpackError', 'ArpackNoConvergence'] + + +_type_conv = {'f': 's', 'd': 'd', 'F': 'c', 'D': 'z'} +_ndigits = {'f': 5, 'd': 12, 'F': 5, 'D': 12} + +DNAUPD_ERRORS = { + 0: "Normal exit.", + 1: "Maximum number of iterations taken. " + "All possible eigenvalues of OP has been found. IPARAM(5) " + "returns the number of wanted converged Ritz values.", + 2: "No longer an informational error. Deprecated starting " + "with release 2 of ARPACK.", + 3: "No shifts could be applied during a cycle of the " + "Implicitly restarted Arnoldi iteration. One possibility " + "is to increase the size of NCV relative to NEV. ", + -1: "N must be positive.", + -2: "NEV must be positive.", + -3: "NCV-NEV >= 2 and less than or equal to N.", + -4: "The maximum number of Arnoldi update iterations allowed " + "must be greater than zero.", + -5: " WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'", + -6: "BMAT must be one of 'I' or 'G'.", + -7: "Length of private work array WORKL is not sufficient.", + -8: "Error return from LAPACK eigenvalue calculation;", + -9: "Starting vector is zero.", + -10: "IPARAM(7) must be 1,2,3,4.", + -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.", + -12: "IPARAM(1) must be equal to 0 or 1.", + -13: "NEV and WHICH = 'BE' are incompatible.", + -9999: "Could not build an Arnoldi factorization. " + "IPARAM(5) returns the size of the current Arnoldi " + "factorization. The user is advised to check that " + "enough workspace and array storage has been allocated." +} + +SNAUPD_ERRORS = DNAUPD_ERRORS + +ZNAUPD_ERRORS = DNAUPD_ERRORS.copy() +ZNAUPD_ERRORS[-10] = "IPARAM(7) must be 1,2,3." + +CNAUPD_ERRORS = ZNAUPD_ERRORS + +DSAUPD_ERRORS = { + 0: "Normal exit.", + 1: "Maximum number of iterations taken. " + "All possible eigenvalues of OP has been found.", + 2: "No longer an informational error. Deprecated starting with " + "release 2 of ARPACK.", + 3: "No shifts could be applied during a cycle of the Implicitly " + "restarted Arnoldi iteration. One possibility is to increase " + "the size of NCV relative to NEV. ", + -1: "N must be positive.", + -2: "NEV must be positive.", + -3: "NCV must be greater than NEV and less than or equal to N.", + -4: "The maximum number of Arnoldi update iterations allowed " + "must be greater than zero.", + -5: "WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.", + -6: "BMAT must be one of 'I' or 'G'.", + -7: "Length of private work array WORKL is not sufficient.", + -8: "Error return from trid. eigenvalue calculation; " + "Informational error from LAPACK routine dsteqr .", + -9: "Starting vector is zero.", + -10: "IPARAM(7) must be 1,2,3,4,5.", + -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.", + -12: "IPARAM(1) must be equal to 0 or 1.", + -13: "NEV and WHICH = 'BE' are incompatible. ", + -9999: "Could not build an Arnoldi factorization. " + "IPARAM(5) returns the size of the current Arnoldi " + "factorization. The user is advised to check that " + "enough workspace and array storage has been allocated.", +} + +SSAUPD_ERRORS = DSAUPD_ERRORS + +DNEUPD_ERRORS = { + 0: "Normal exit.", + 1: "The Schur form computed by LAPACK routine dlahqr " + "could not be reordered by LAPACK routine dtrsen. " + "Re-enter subroutine dneupd with IPARAM(5)NCV and " + "increase the size of the arrays DR and DI to have " + "dimension at least dimension NCV and allocate at least NCV " + "columns for Z. NOTE: Not necessary if Z and V share " + "the same space. Please notify the authors if this error" + "occurs.", + -1: "N must be positive.", + -2: "NEV must be positive.", + -3: "NCV-NEV >= 2 and less than or equal to N.", + -5: "WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'", + -6: "BMAT must be one of 'I' or 'G'.", + -7: "Length of private work WORKL array is not sufficient.", + -8: "Error return from calculation of a real Schur form. " + "Informational error from LAPACK routine dlahqr .", + -9: "Error return from calculation of eigenvectors. " + "Informational error from LAPACK routine dtrevc.", + -10: "IPARAM(7) must be 1,2,3,4.", + -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.", + -12: "HOWMNY = 'S' not yet implemented", + -13: "HOWMNY must be one of 'A' or 'P' if RVEC = .true.", + -14: "DNAUPD did not find any eigenvalues to sufficient " + "accuracy.", + -15: "DNEUPD got a different count of the number of converged " + "Ritz values than DNAUPD got. This indicates the user " + "probably made an error in passing data from DNAUPD to " + "DNEUPD or that the data was modified before entering " + "DNEUPD", +} + +SNEUPD_ERRORS = DNEUPD_ERRORS.copy() +SNEUPD_ERRORS[1] = ("The Schur form computed by LAPACK routine slahqr " + "could not be reordered by LAPACK routine strsen . " + "Re-enter subroutine dneupd with IPARAM(5)=NCV and " + "increase the size of the arrays DR and DI to have " + "dimension at least dimension NCV and allocate at least " + "NCV columns for Z. NOTE: Not necessary if Z and V share " + "the same space. Please notify the authors if this error " + "occurs.") +SNEUPD_ERRORS[-14] = ("SNAUPD did not find any eigenvalues to sufficient " + "accuracy.") +SNEUPD_ERRORS[-15] = ("SNEUPD got a different count of the number of " + "converged Ritz values than SNAUPD got. This indicates " + "the user probably made an error in passing data from " + "SNAUPD to SNEUPD or that the data was modified before " + "entering SNEUPD") + +ZNEUPD_ERRORS = {0: "Normal exit.", + 1: "The Schur form computed by LAPACK routine csheqr " + "could not be reordered by LAPACK routine ztrsen. " + "Re-enter subroutine zneupd with IPARAM(5)=NCV and " + "increase the size of the array D to have " + "dimension at least dimension NCV and allocate at least " + "NCV columns for Z. NOTE: Not necessary if Z and V share " + "the same space. Please notify the authors if this error " + "occurs.", + -1: "N must be positive.", + -2: "NEV must be positive.", + -3: "NCV-NEV >= 1 and less than or equal to N.", + -5: "WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'", + -6: "BMAT must be one of 'I' or 'G'.", + -7: "Length of private work WORKL array is not sufficient.", + -8: "Error return from LAPACK eigenvalue calculation. " + "This should never happened.", + -9: "Error return from calculation of eigenvectors. " + "Informational error from LAPACK routine ztrevc.", + -10: "IPARAM(7) must be 1,2,3", + -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.", + -12: "HOWMNY = 'S' not yet implemented", + -13: "HOWMNY must be one of 'A' or 'P' if RVEC = .true.", + -14: "ZNAUPD did not find any eigenvalues to sufficient " + "accuracy.", + -15: "ZNEUPD got a different count of the number of " + "converged Ritz values than ZNAUPD got. This " + "indicates the user probably made an error in passing " + "data from ZNAUPD to ZNEUPD or that the data was " + "modified before entering ZNEUPD" + } + +CNEUPD_ERRORS = ZNEUPD_ERRORS.copy() +CNEUPD_ERRORS[-14] = ("CNAUPD did not find any eigenvalues to sufficient " + "accuracy.") +CNEUPD_ERRORS[-15] = ("CNEUPD got a different count of the number of " + "converged Ritz values than CNAUPD got. This indicates " + "the user probably made an error in passing data from " + "CNAUPD to CNEUPD or that the data was modified before " + "entering CNEUPD") + +DSEUPD_ERRORS = { + 0: "Normal exit.", + -1: "N must be positive.", + -2: "NEV must be positive.", + -3: "NCV must be greater than NEV and less than or equal to N.", + -5: "WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.", + -6: "BMAT must be one of 'I' or 'G'.", + -7: "Length of private work WORKL array is not sufficient.", + -8: ("Error return from trid. eigenvalue calculation; " + "Information error from LAPACK routine dsteqr."), + -9: "Starting vector is zero.", + -10: "IPARAM(7) must be 1,2,3,4,5.", + -11: "IPARAM(7) = 1 and BMAT = 'G' are incompatible.", + -12: "NEV and WHICH = 'BE' are incompatible.", + -14: "DSAUPD did not find any eigenvalues to sufficient accuracy.", + -15: "HOWMNY must be one of 'A' or 'S' if RVEC = .true.", + -16: "HOWMNY = 'S' not yet implemented", + -17: ("DSEUPD got a different count of the number of converged " + "Ritz values than DSAUPD got. This indicates the user " + "probably made an error in passing data from DSAUPD to " + "DSEUPD or that the data was modified before entering " + "DSEUPD.") +} + +SSEUPD_ERRORS = DSEUPD_ERRORS.copy() +SSEUPD_ERRORS[-14] = ("SSAUPD did not find any eigenvalues " + "to sufficient accuracy.") +SSEUPD_ERRORS[-17] = ("SSEUPD got a different count of the number of " + "converged " + "Ritz values than SSAUPD got. This indicates the user " + "probably made an error in passing data from SSAUPD to " + "SSEUPD or that the data was modified before entering " + "SSEUPD.") + +_SAUPD_ERRORS = {'d': DSAUPD_ERRORS, + 's': SSAUPD_ERRORS} +_NAUPD_ERRORS = {'d': DNAUPD_ERRORS, + 's': SNAUPD_ERRORS, + 'z': ZNAUPD_ERRORS, + 'c': CNAUPD_ERRORS} +_SEUPD_ERRORS = {'d': DSEUPD_ERRORS, + 's': SSEUPD_ERRORS} +_NEUPD_ERRORS = {'d': DNEUPD_ERRORS, + 's': SNEUPD_ERRORS, + 'z': ZNEUPD_ERRORS, + 'c': CNEUPD_ERRORS} + +# accepted values of parameter WHICH in _SEUPD +_SEUPD_WHICH = ['LM', 'SM', 'LA', 'SA', 'BE'] + +# accepted values of parameter WHICH in _NAUPD +_NEUPD_WHICH = ['LM', 'SM', 'LR', 'SR', 'LI', 'SI'] + + +class ArpackError(RuntimeError): + """ + ARPACK error + """ + + def __init__(self, info, infodict=None): + if infodict is None: + infodict = _NAUPD_ERRORS + + msg = infodict.get(info, "Unknown error") + super().__init__(f"ARPACK error {info}: {msg}") + + +class ArpackNoConvergence(ArpackError): + """ + ARPACK iteration did not converge + + Attributes + ---------- + eigenvalues : ndarray + Partial result. Converged eigenvalues. + eigenvectors : ndarray + Partial result. Converged eigenvectors. + + """ + + def __init__(self, msg, eigenvalues, eigenvectors): + ArpackError.__init__(self, -1, {-1: msg}) + self.eigenvalues = eigenvalues + self.eigenvectors = eigenvectors + + +def choose_ncv(k): + """ + Choose number of lanczos vectors based on target number + of singular/eigen values and vectors to compute, k. + """ + return max(2 * k + 1, 20) + + +class _ArpackParams: + def __init__(self, n, k, tp, mode=1, sigma=None, + ncv=None, v0=None, maxiter=None, which="LM", tol=0): + if k <= 0: + raise ValueError("k must be positive, k=%d" % k) + + if maxiter is None: + maxiter = n * 10 + if maxiter <= 0: + raise ValueError("maxiter must be positive, maxiter=%d" % maxiter) + + if tp not in 'fdFD': + # Use `float64` libraries from integer dtypes. + if np.can_cast(tp, 'd'): + tp = 'd' + else: + raise ValueError("matrix type must be 'f', 'd', 'F', or 'D'") + + if v0 is not None: + # ARPACK overwrites its initial resid, make a copy + self.resid = np.array(v0, copy=True) + info = 1 + else: + # ARPACK will use a random initial vector. + self.resid = np.zeros(n, tp) + info = 0 + + if sigma is None: + #sigma not used + self.sigma = 0 + else: + self.sigma = sigma + + if ncv is None: + ncv = choose_ncv(k) + ncv = min(ncv, n) + + self.v = np.zeros((n, ncv), tp) # holds Ritz vectors + self.iparam = np.zeros(11, arpack_int) + + # set solver mode and parameters + ishfts = 1 + self.mode = mode + self.iparam[0] = ishfts + self.iparam[2] = maxiter + self.iparam[3] = 1 + self.iparam[6] = mode + + self.n = n + self.tol = tol + self.k = k + self.maxiter = maxiter + self.ncv = ncv + self.which = which + self.tp = tp + self.info = info + + self.converged = False + self.ido = 0 + + def _raise_no_convergence(self): + msg = "No convergence (%d iterations, %d/%d eigenvectors converged)" + k_ok = self.iparam[4] + num_iter = self.iparam[2] + try: + ev, vec = self.extract(True) + except ArpackError as err: + msg = f"{msg} [{err}]" + ev = np.zeros((0,)) + vec = np.zeros((self.n, 0)) + k_ok = 0 + raise ArpackNoConvergence(msg % (num_iter, k_ok, self.k), ev, vec) + + +class _SymmetricArpackParams(_ArpackParams): + def __init__(self, n, k, tp, matvec, mode=1, M_matvec=None, + Minv_matvec=None, sigma=None, + ncv=None, v0=None, maxiter=None, which="LM", tol=0): + # The following modes are supported: + # mode = 1: + # Solve the standard eigenvalue problem: + # A*x = lambda*x : + # A - symmetric + # Arguments should be + # matvec = left multiplication by A + # M_matvec = None [not used] + # Minv_matvec = None [not used] + # + # mode = 2: + # Solve the general eigenvalue problem: + # A*x = lambda*M*x + # A - symmetric + # M - symmetric positive definite + # Arguments should be + # matvec = left multiplication by A + # M_matvec = left multiplication by M + # Minv_matvec = left multiplication by M^-1 + # + # mode = 3: + # Solve the general eigenvalue problem in shift-invert mode: + # A*x = lambda*M*x + # A - symmetric + # M - symmetric positive semi-definite + # Arguments should be + # matvec = None [not used] + # M_matvec = left multiplication by M + # or None, if M is the identity + # Minv_matvec = left multiplication by [A-sigma*M]^-1 + # + # mode = 4: + # Solve the general eigenvalue problem in Buckling mode: + # A*x = lambda*AG*x + # A - symmetric positive semi-definite + # AG - symmetric indefinite + # Arguments should be + # matvec = left multiplication by A + # M_matvec = None [not used] + # Minv_matvec = left multiplication by [A-sigma*AG]^-1 + # + # mode = 5: + # Solve the general eigenvalue problem in Cayley-transformed mode: + # A*x = lambda*M*x + # A - symmetric + # M - symmetric positive semi-definite + # Arguments should be + # matvec = left multiplication by A + # M_matvec = left multiplication by M + # or None, if M is the identity + # Minv_matvec = left multiplication by [A-sigma*M]^-1 + if mode == 1: + if matvec is None: + raise ValueError("matvec must be specified for mode=1") + if M_matvec is not None: + raise ValueError("M_matvec cannot be specified for mode=1") + if Minv_matvec is not None: + raise ValueError("Minv_matvec cannot be specified for mode=1") + + self.OP = matvec + self.B = lambda x: x + self.bmat = 'I' + elif mode == 2: + if matvec is None: + raise ValueError("matvec must be specified for mode=2") + if M_matvec is None: + raise ValueError("M_matvec must be specified for mode=2") + if Minv_matvec is None: + raise ValueError("Minv_matvec must be specified for mode=2") + + self.OP = lambda x: Minv_matvec(matvec(x)) + self.OPa = Minv_matvec + self.OPb = matvec + self.B = M_matvec + self.bmat = 'G' + elif mode == 3: + if matvec is not None: + raise ValueError("matvec must not be specified for mode=3") + if Minv_matvec is None: + raise ValueError("Minv_matvec must be specified for mode=3") + + if M_matvec is None: + self.OP = Minv_matvec + self.OPa = Minv_matvec + self.B = lambda x: x + self.bmat = 'I' + else: + self.OP = lambda x: Minv_matvec(M_matvec(x)) + self.OPa = Minv_matvec + self.B = M_matvec + self.bmat = 'G' + elif mode == 4: + if matvec is None: + raise ValueError("matvec must be specified for mode=4") + if M_matvec is not None: + raise ValueError("M_matvec must not be specified for mode=4") + if Minv_matvec is None: + raise ValueError("Minv_matvec must be specified for mode=4") + self.OPa = Minv_matvec + self.OP = lambda x: self.OPa(matvec(x)) + self.B = matvec + self.bmat = 'G' + elif mode == 5: + if matvec is None: + raise ValueError("matvec must be specified for mode=5") + if Minv_matvec is None: + raise ValueError("Minv_matvec must be specified for mode=5") + + self.OPa = Minv_matvec + self.A_matvec = matvec + + if M_matvec is None: + self.OP = lambda x: Minv_matvec(matvec(x) + sigma * x) + self.B = lambda x: x + self.bmat = 'I' + else: + self.OP = lambda x: Minv_matvec(matvec(x) + + sigma * M_matvec(x)) + self.B = M_matvec + self.bmat = 'G' + else: + raise ValueError("mode=%i not implemented" % mode) + + if which not in _SEUPD_WHICH: + raise ValueError(f"which must be one of {' '.join(_SEUPD_WHICH)}") + if k >= n: + raise ValueError("k must be less than ndim(A), k=%d" % k) + + _ArpackParams.__init__(self, n, k, tp, mode, sigma, + ncv, v0, maxiter, which, tol) + + if self.ncv > n or self.ncv <= k: + raise ValueError(f"ncv must be k= n - 1: + raise ValueError("k must be less than ndim(A)-1, k=%d" % k) + + _ArpackParams.__init__(self, n, k, tp, mode, sigma, + ncv, v0, maxiter, which, tol) + + if self.ncv > n or self.ncv <= k + 1: + raise ValueError(f"ncv must be k+1 k, so we'll + # throw out this case. + nreturned -= 1 + i += 1 + + else: + # real matrix, mode 3 or 4, imag(sigma) is nonzero: + # see remark 3 in neupd.f + # Build complex eigenvalues from real and imaginary parts + i = 0 + while i <= k: + if abs(d[i].imag) == 0: + d[i] = np.dot(zr[:, i], self.matvec(zr[:, i])) + else: + if i < k: + z[:, i] = zr[:, i] + 1.0j * zr[:, i + 1] + z[:, i + 1] = z[:, i].conjugate() + d[i] = ((np.dot(zr[:, i], + self.matvec(zr[:, i])) + + np.dot(zr[:, i + 1], + self.matvec(zr[:, i + 1]))) + + 1j * (np.dot(zr[:, i], + self.matvec(zr[:, i + 1])) + - np.dot(zr[:, i + 1], + self.matvec(zr[:, i])))) + d[i + 1] = d[i].conj() + i += 1 + else: + #last eigenvalue is complex: the imaginary part of + # the eigenvector has not been returned + #this can only happen if nreturned > k, so we'll + # throw out this case. + nreturned -= 1 + i += 1 + + # Now we have k+1 possible eigenvalues and eigenvectors + # Return the ones specified by the keyword "which" + + if nreturned <= k: + # we got less or equal as many eigenvalues we wanted + d = d[:nreturned] + z = z[:, :nreturned] + else: + # we got one extra eigenvalue (likely a cc pair, but which?) + if self.mode in (1, 2): + rd = d + elif self.mode in (3, 4): + rd = 1 / (d - self.sigma) + + if self.which in ['LR', 'SR']: + ind = np.argsort(rd.real) + elif self.which in ['LI', 'SI']: + # for LI,SI ARPACK returns largest,smallest + # abs(imaginary) (complex pairs come together) + ind = np.argsort(abs(rd.imag)) + else: + ind = np.argsort(abs(rd)) + + if self.which in ['LR', 'LM', 'LI']: + ind = ind[-k:][::-1] + elif self.which in ['SR', 'SM', 'SI']: + ind = ind[:k] + + d = d[ind] + z = z[:, ind] + else: + # complex is so much simpler... + d, z, ierr =\ + self._arpack_extract(return_eigenvectors, + howmny, sselect, self.sigma, workev, + self.bmat, self.which, k, self.tol, self.resid, + self.v, self.iparam, self.ipntr, + self.workd, self.workl, self.rwork, ierr) + + if ierr != 0: + raise ArpackError(ierr, infodict=self.extract_infodict) + + k_ok = self.iparam[4] + d = d[:k_ok] + z = z[:, :k_ok] + + if return_eigenvectors: + return d, z + else: + return d + +class SpLuInv(LinearOperator): + """ + SpLuInv: + helper class to repeatedly solve M*x=b + using a sparse LU-decomposition of M + """ + + def __init__(self, M): + self.M_lu = splu(M) + self.shape = M.shape + self.dtype = M.dtype + self.isreal = not np.issubdtype(self.dtype, np.complexfloating) + + def _matvec(self, x): + # careful here: splu.solve will throw away imaginary + # part of x if M is real + x = np.asarray(x) + if self.isreal and np.issubdtype(x.dtype, np.complexfloating): + return (self.M_lu.solve(np.real(x).astype(self.dtype)) + + 1j * self.M_lu.solve(np.imag(x).astype(self.dtype))) + else: + return self.M_lu.solve(x.astype(self.dtype)) + + +class LuInv(LinearOperator): + """ + LuInv: + helper class to repeatedly solve M*x=b + using an LU-decomposition of M + """ + + def __init__(self, M): + self.M_lu = lu_factor(M) + self.shape = M.shape + self.dtype = M.dtype + + def _matvec(self, x): + return lu_solve(self.M_lu, x) + + +def gmres_loose(A, b, tol): + """ + gmres with looser termination condition. + """ + b = np.asarray(b) + min_tol = 1000 * np.sqrt(b.size) * np.finfo(b.dtype).eps + return gmres(A, b, rtol=max(tol, min_tol), atol=0) + + +class IterInv(LinearOperator): + """ + IterInv: + helper class to repeatedly solve M*x=b + using an iterative method. + """ + + def __init__(self, M, ifunc=gmres_loose, tol=0): + self.M = M + if hasattr(M, 'dtype'): + self.dtype = M.dtype + else: + x = np.zeros(M.shape[1]) + self.dtype = (M * x).dtype + self.shape = M.shape + + if tol <= 0: + # when tol=0, ARPACK uses machine tolerance as calculated + # by LAPACK's _LAMCH function. We should match this + tol = 2 * np.finfo(self.dtype).eps + self.ifunc = ifunc + self.tol = tol + + def _matvec(self, x): + b, info = self.ifunc(self.M, x, tol=self.tol) + if info != 0: + raise ValueError("Error in inverting M: function " + "%s did not converge (info = %i)." + % (self.ifunc.__name__, info)) + return b + + +class IterOpInv(LinearOperator): + """ + IterOpInv: + helper class to repeatedly solve [A-sigma*M]*x = b + using an iterative method + """ + + def __init__(self, A, M, sigma, ifunc=gmres_loose, tol=0): + self.A = A + self.M = M + self.sigma = sigma + + def mult_func(x): + return A.matvec(x) - sigma * M.matvec(x) + + def mult_func_M_None(x): + return A.matvec(x) - sigma * x + + x = np.zeros(A.shape[1]) + if M is None: + dtype = mult_func_M_None(x).dtype + self.OP = LinearOperator(self.A.shape, + mult_func_M_None, + dtype=dtype) + else: + dtype = mult_func(x).dtype + self.OP = LinearOperator(self.A.shape, + mult_func, + dtype=dtype) + self.shape = A.shape + + if tol <= 0: + # when tol=0, ARPACK uses machine tolerance as calculated + # by LAPACK's _LAMCH function. We should match this + tol = 2 * np.finfo(self.OP.dtype).eps + self.ifunc = ifunc + self.tol = tol + + def _matvec(self, x): + b, info = self.ifunc(self.OP, x, tol=self.tol) + if info != 0: + raise ValueError("Error in inverting [A-sigma*M]: function " + "%s did not converge (info = %i)." + % (self.ifunc.__name__, info)) + return b + + @property + def dtype(self): + return self.OP.dtype + + +def _fast_spmatrix_to_csc(A, hermitian=False): + """Convert sparse matrix to CSC (by transposing, if possible)""" + if (A.format == "csr" and hermitian + and not np.issubdtype(A.dtype, np.complexfloating)): + return A.T + elif is_pydata_spmatrix(A): + # No need to convert + return A + else: + return A.tocsc() + + +def get_inv_matvec(M, hermitian=False, tol=0): + if isdense(M): + return LuInv(M).matvec + elif issparse(M) or is_pydata_spmatrix(M): + M = _fast_spmatrix_to_csc(M, hermitian=hermitian) + return SpLuInv(M).matvec + else: + return IterInv(M, tol=tol).matvec + + +def get_OPinv_matvec(A, M, sigma, hermitian=False, tol=0): + if sigma == 0: + return get_inv_matvec(A, hermitian=hermitian, tol=tol) + + if M is None: + #M is the identity matrix + if isdense(A): + if (np.issubdtype(A.dtype, np.complexfloating) + or np.imag(sigma) == 0): + A = np.copy(A) + else: + A = A + 0j + A.flat[::A.shape[1] + 1] -= sigma + return LuInv(A).matvec + elif issparse(A) or is_pydata_spmatrix(A): + A = A - sigma * eye(A.shape[0]) + A = _fast_spmatrix_to_csc(A, hermitian=hermitian) + return SpLuInv(A).matvec + else: + return IterOpInv(aslinearoperator(A), + M, sigma, tol=tol).matvec + else: + if ((not isdense(A) and not issparse(A) and not is_pydata_spmatrix(A)) or + (not isdense(M) and not issparse(M) and not is_pydata_spmatrix(A))): + return IterOpInv(aslinearoperator(A), + aslinearoperator(M), + sigma, tol=tol).matvec + elif isdense(A) or isdense(M): + return LuInv(A - sigma * M).matvec + else: + OP = A - sigma * M + OP = _fast_spmatrix_to_csc(OP, hermitian=hermitian) + return SpLuInv(OP).matvec + + +# ARPACK is not threadsafe or reentrant (SAVE variables), so we need a +# lock and a re-entering check. +_ARPACK_LOCK = ReentrancyLock("Nested calls to eigs/eighs not allowed: " + "ARPACK is not re-entrant") + + +def eigs(A, k=6, M=None, sigma=None, which='LM', v0=None, + ncv=None, maxiter=None, tol=0, return_eigenvectors=True, + Minv=None, OPinv=None, OPpart=None): + """ + Find k eigenvalues and eigenvectors of the square matrix A. + + Solves ``A @ x[i] = w[i] * x[i]``, the standard eigenvalue problem + for w[i] eigenvalues with corresponding eigenvectors x[i]. + + If M is specified, solves ``A @ x[i] = w[i] * M @ x[i]``, the + generalized eigenvalue problem for w[i] eigenvalues + with corresponding eigenvectors x[i] + + Parameters + ---------- + A : ndarray, sparse matrix or LinearOperator + An array, sparse matrix, or LinearOperator representing + the operation ``A @ x``, where A is a real or complex square matrix. + k : int, optional + The number of eigenvalues and eigenvectors desired. + `k` must be smaller than N-1. It is not possible to compute all + eigenvectors of a matrix. + M : ndarray, sparse matrix or LinearOperator, optional + An array, sparse matrix, or LinearOperator representing + the operation M@x for the generalized eigenvalue problem + + A @ x = w * M @ x. + + M must represent a real symmetric matrix if A is real, and must + represent a complex Hermitian matrix if A is complex. For best + results, the data type of M should be the same as that of A. + Additionally: + + If `sigma` is None, M is positive definite + + If sigma is specified, M is positive semi-definite + + If sigma is None, eigs requires an operator to compute the solution + of the linear equation ``M @ x = b``. This is done internally via a + (sparse) LU decomposition for an explicit matrix M, or via an + iterative solver for a general linear operator. Alternatively, + the user can supply the matrix or operator Minv, which gives + ``x = Minv @ b = M^-1 @ b``. + sigma : real or complex, optional + Find eigenvalues near sigma using shift-invert mode. This requires + an operator to compute the solution of the linear system + ``[A - sigma * M] @ x = b``, where M is the identity matrix if + unspecified. This is computed internally via a (sparse) LU + decomposition for explicit matrices A & M, or via an iterative + solver if either A or M is a general linear operator. + Alternatively, the user can supply the matrix or operator OPinv, + which gives ``x = OPinv @ b = [A - sigma * M]^-1 @ b``. + For a real matrix A, shift-invert can either be done in imaginary + mode or real mode, specified by the parameter OPpart ('r' or 'i'). + Note that when sigma is specified, the keyword 'which' (below) + refers to the shifted eigenvalues ``w'[i]`` where: + + If A is real and OPpart == 'r' (default), + ``w'[i] = 1/2 * [1/(w[i]-sigma) + 1/(w[i]-conj(sigma))]``. + + If A is real and OPpart == 'i', + ``w'[i] = 1/2i * [1/(w[i]-sigma) - 1/(w[i]-conj(sigma))]``. + + If A is complex, ``w'[i] = 1/(w[i]-sigma)``. + + v0 : ndarray, optional + Starting vector for iteration. + Default: random + ncv : int, optional + The number of Lanczos vectors generated + `ncv` must be greater than `k`; it is recommended that ``ncv > 2*k``. + Default: ``min(n, max(2*k + 1, 20))`` + which : str, ['LM' | 'SM' | 'LR' | 'SR' | 'LI' | 'SI'], optional + Which `k` eigenvectors and eigenvalues to find: + + 'LM' : largest magnitude + + 'SM' : smallest magnitude + + 'LR' : largest real part + + 'SR' : smallest real part + + 'LI' : largest imaginary part + + 'SI' : smallest imaginary part + + When sigma != None, 'which' refers to the shifted eigenvalues w'[i] + (see discussion in 'sigma', above). ARPACK is generally better + at finding large values than small values. If small eigenvalues are + desired, consider using shift-invert mode for better performance. + maxiter : int, optional + Maximum number of Arnoldi update iterations allowed + Default: ``n*10`` + tol : float, optional + Relative accuracy for eigenvalues (stopping criterion) + The default value of 0 implies machine precision. + return_eigenvectors : bool, optional + Return eigenvectors (True) in addition to eigenvalues + Minv : ndarray, sparse matrix or LinearOperator, optional + See notes in M, above. + OPinv : ndarray, sparse matrix or LinearOperator, optional + See notes in sigma, above. + OPpart : {'r' or 'i'}, optional + See notes in sigma, above + + Returns + ------- + w : ndarray + Array of k eigenvalues. + v : ndarray + An array of `k` eigenvectors. + ``v[:, i]`` is the eigenvector corresponding to the eigenvalue w[i]. + + Raises + ------ + ArpackNoConvergence + When the requested convergence is not obtained. + The currently converged eigenvalues and eigenvectors can be found + as ``eigenvalues`` and ``eigenvectors`` attributes of the exception + object. + + See Also + -------- + eigsh : eigenvalues and eigenvectors for symmetric matrix A + svds : singular value decomposition for a matrix A + + Notes + ----- + This function is a wrapper to the ARPACK [1]_ SNEUPD, DNEUPD, CNEUPD, + ZNEUPD, functions which use the Implicitly Restarted Arnoldi Method to + find the eigenvalues and eigenvectors [2]_. + + References + ---------- + .. [1] ARPACK Software, https://github.com/opencollab/arpack-ng + .. [2] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK USERS GUIDE: + Solution of Large Scale Eigenvalue Problems by Implicitly Restarted + Arnoldi Methods. SIAM, Philadelphia, PA, 1998. + + Examples + -------- + Find 6 eigenvectors of the identity matrix: + + >>> import numpy as np + >>> from scipy.sparse.linalg import eigs + >>> id = np.eye(13) + >>> vals, vecs = eigs(id, k=6) + >>> vals + array([ 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j]) + >>> vecs.shape + (13, 6) + + """ + A = convert_pydata_sparse_to_scipy(A) + M = convert_pydata_sparse_to_scipy(M) + if A.shape[0] != A.shape[1]: + raise ValueError(f'expected square matrix (shape={A.shape})') + if M is not None: + if M.shape != A.shape: + raise ValueError(f'wrong M dimensions {M.shape}, should be {A.shape}') + if np.dtype(M.dtype).char.lower() != np.dtype(A.dtype).char.lower(): + warnings.warn('M does not have the same type precision as A. ' + 'This may adversely affect ARPACK convergence', + stacklevel=2) + + n = A.shape[0] + + if k <= 0: + raise ValueError("k=%d must be greater than 0." % k) + + if k >= n - 1: + warnings.warn("k >= N - 1 for N * N square matrix. " + "Attempting to use scipy.linalg.eig instead.", + RuntimeWarning, stacklevel=2) + + if issparse(A): + raise TypeError("Cannot use scipy.linalg.eig for sparse A with " + "k >= N - 1. Use scipy.linalg.eig(A.toarray()) or" + " reduce k.") + if isinstance(A, LinearOperator): + raise TypeError("Cannot use scipy.linalg.eig for LinearOperator " + "A with k >= N - 1.") + if isinstance(M, LinearOperator): + raise TypeError("Cannot use scipy.linalg.eig for LinearOperator " + "M with k >= N - 1.") + + return eig(A, b=M, right=return_eigenvectors) + + if sigma is None: + matvec = aslinearoperator(A).matvec + + if OPinv is not None: + raise ValueError("OPinv should not be specified " + "with sigma = None.") + if OPpart is not None: + raise ValueError("OPpart should not be specified with " + "sigma = None or complex A") + + if M is None: + #standard eigenvalue problem + mode = 1 + M_matvec = None + Minv_matvec = None + if Minv is not None: + raise ValueError("Minv should not be " + "specified with M = None.") + else: + #general eigenvalue problem + mode = 2 + if Minv is None: + Minv_matvec = get_inv_matvec(M, hermitian=True, tol=tol) + else: + Minv = aslinearoperator(Minv) + Minv_matvec = Minv.matvec + M_matvec = aslinearoperator(M).matvec + else: + #sigma is not None: shift-invert mode + if np.issubdtype(A.dtype, np.complexfloating): + if OPpart is not None: + raise ValueError("OPpart should not be specified " + "with sigma=None or complex A") + mode = 3 + elif OPpart is None or OPpart.lower() == 'r': + mode = 3 + elif OPpart.lower() == 'i': + if np.imag(sigma) == 0: + raise ValueError("OPpart cannot be 'i' if sigma is real") + mode = 4 + else: + raise ValueError("OPpart must be one of ('r','i')") + + matvec = aslinearoperator(A).matvec + if Minv is not None: + raise ValueError("Minv should not be specified when sigma is") + if OPinv is None: + Minv_matvec = get_OPinv_matvec(A, M, sigma, + hermitian=False, tol=tol) + else: + OPinv = aslinearoperator(OPinv) + Minv_matvec = OPinv.matvec + if M is None: + M_matvec = None + else: + M_matvec = aslinearoperator(M).matvec + + params = _UnsymmetricArpackParams(n, k, A.dtype.char, matvec, mode, + M_matvec, Minv_matvec, sigma, + ncv, v0, maxiter, which, tol) + + with _ARPACK_LOCK: + while not params.converged: + params.iterate() + + return params.extract(return_eigenvectors) + + +def eigsh(A, k=6, M=None, sigma=None, which='LM', v0=None, + ncv=None, maxiter=None, tol=0, return_eigenvectors=True, + Minv=None, OPinv=None, mode='normal'): + """ + Find k eigenvalues and eigenvectors of the real symmetric square matrix + or complex Hermitian matrix A. + + Solves ``A @ x[i] = w[i] * x[i]``, the standard eigenvalue problem for + w[i] eigenvalues with corresponding eigenvectors x[i]. + + If M is specified, solves ``A @ x[i] = w[i] * M @ x[i]``, the + generalized eigenvalue problem for w[i] eigenvalues + with corresponding eigenvectors x[i]. + + Note that there is no specialized routine for the case when A is a complex + Hermitian matrix. In this case, ``eigsh()`` will call ``eigs()`` and return the + real parts of the eigenvalues thus obtained. + + Parameters + ---------- + A : ndarray, sparse matrix or LinearOperator + A square operator representing the operation ``A @ x``, where ``A`` is + real symmetric or complex Hermitian. For buckling mode (see below) + ``A`` must additionally be positive-definite. + k : int, optional + The number of eigenvalues and eigenvectors desired. + `k` must be smaller than N. It is not possible to compute all + eigenvectors of a matrix. + + Returns + ------- + w : array + Array of k eigenvalues. + v : array + An array representing the `k` eigenvectors. The column ``v[:, i]`` is + the eigenvector corresponding to the eigenvalue ``w[i]``. + + Other Parameters + ---------------- + M : An N x N matrix, array, sparse matrix, or linear operator representing + the operation ``M @ x`` for the generalized eigenvalue problem + + A @ x = w * M @ x. + + M must represent a real symmetric matrix if A is real, and must + represent a complex Hermitian matrix if A is complex. For best + results, the data type of M should be the same as that of A. + Additionally: + + If sigma is None, M is symmetric positive definite. + + If sigma is specified, M is symmetric positive semi-definite. + + In buckling mode, M is symmetric indefinite. + + If sigma is None, eigsh requires an operator to compute the solution + of the linear equation ``M @ x = b``. This is done internally via a + (sparse) LU decomposition for an explicit matrix M, or via an + iterative solver for a general linear operator. Alternatively, + the user can supply the matrix or operator Minv, which gives + ``x = Minv @ b = M^-1 @ b``. + sigma : real + Find eigenvalues near sigma using shift-invert mode. This requires + an operator to compute the solution of the linear system + ``[A - sigma * M] x = b``, where M is the identity matrix if + unspecified. This is computed internally via a (sparse) LU + decomposition for explicit matrices A & M, or via an iterative + solver if either A or M is a general linear operator. + Alternatively, the user can supply the matrix or operator OPinv, + which gives ``x = OPinv @ b = [A - sigma * M]^-1 @ b``. + Note that when sigma is specified, the keyword 'which' refers to + the shifted eigenvalues ``w'[i]`` where: + + if mode == 'normal', ``w'[i] = 1 / (w[i] - sigma)``. + + if mode == 'cayley', ``w'[i] = (w[i] + sigma) / (w[i] - sigma)``. + + if mode == 'buckling', ``w'[i] = w[i] / (w[i] - sigma)``. + + (see further discussion in 'mode' below) + v0 : ndarray, optional + Starting vector for iteration. + Default: random + ncv : int, optional + The number of Lanczos vectors generated ncv must be greater than k and + smaller than n; it is recommended that ``ncv > 2*k``. + Default: ``min(n, max(2*k + 1, 20))`` + which : str ['LM' | 'SM' | 'LA' | 'SA' | 'BE'] + If A is a complex Hermitian matrix, 'BE' is invalid. + Which `k` eigenvectors and eigenvalues to find: + + 'LM' : Largest (in magnitude) eigenvalues. + + 'SM' : Smallest (in magnitude) eigenvalues. + + 'LA' : Largest (algebraic) eigenvalues. + + 'SA' : Smallest (algebraic) eigenvalues. + + 'BE' : Half (k/2) from each end of the spectrum. + + When k is odd, return one more (k/2+1) from the high end. + When sigma != None, 'which' refers to the shifted eigenvalues ``w'[i]`` + (see discussion in 'sigma', above). ARPACK is generally better + at finding large values than small values. If small eigenvalues are + desired, consider using shift-invert mode for better performance. + maxiter : int, optional + Maximum number of Arnoldi update iterations allowed. + Default: ``n*10`` + tol : float + Relative accuracy for eigenvalues (stopping criterion). + The default value of 0 implies machine precision. + Minv : N x N matrix, array, sparse matrix, or LinearOperator + See notes in M, above. + OPinv : N x N matrix, array, sparse matrix, or LinearOperator + See notes in sigma, above. + return_eigenvectors : bool + Return eigenvectors (True) in addition to eigenvalues. + This value determines the order in which eigenvalues are sorted. + The sort order is also dependent on the `which` variable. + + For which = 'LM' or 'SA': + If `return_eigenvectors` is True, eigenvalues are sorted by + algebraic value. + + If `return_eigenvectors` is False, eigenvalues are sorted by + absolute value. + + For which = 'BE' or 'LA': + eigenvalues are always sorted by algebraic value. + + For which = 'SM': + If `return_eigenvectors` is True, eigenvalues are sorted by + algebraic value. + + If `return_eigenvectors` is False, eigenvalues are sorted by + decreasing absolute value. + + mode : string ['normal' | 'buckling' | 'cayley'] + Specify strategy to use for shift-invert mode. This argument applies + only for real-valued A and sigma != None. For shift-invert mode, + ARPACK internally solves the eigenvalue problem + ``OP @ x'[i] = w'[i] * B @ x'[i]`` + and transforms the resulting Ritz vectors x'[i] and Ritz values w'[i] + into the desired eigenvectors and eigenvalues of the problem + ``A @ x[i] = w[i] * M @ x[i]``. + The modes are as follows: + + 'normal' : + OP = [A - sigma * M]^-1 @ M, + B = M, + w'[i] = 1 / (w[i] - sigma) + + 'buckling' : + OP = [A - sigma * M]^-1 @ A, + B = A, + w'[i] = w[i] / (w[i] - sigma) + + 'cayley' : + OP = [A - sigma * M]^-1 @ [A + sigma * M], + B = M, + w'[i] = (w[i] + sigma) / (w[i] - sigma) + + The choice of mode will affect which eigenvalues are selected by + the keyword 'which', and can also impact the stability of + convergence (see [2] for a discussion). + + Raises + ------ + ArpackNoConvergence + When the requested convergence is not obtained. + + The currently converged eigenvalues and eigenvectors can be found + as ``eigenvalues`` and ``eigenvectors`` attributes of the exception + object. + + See Also + -------- + eigs : eigenvalues and eigenvectors for a general (nonsymmetric) matrix A + svds : singular value decomposition for a matrix A + + Notes + ----- + This function is a wrapper to the ARPACK [1]_ SSEUPD and DSEUPD + functions which use the Implicitly Restarted Lanczos Method to + find the eigenvalues and eigenvectors [2]_. + + References + ---------- + .. [1] ARPACK Software, https://github.com/opencollab/arpack-ng + .. [2] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK USERS GUIDE: + Solution of Large Scale Eigenvalue Problems by Implicitly Restarted + Arnoldi Methods. SIAM, Philadelphia, PA, 1998. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse.linalg import eigsh + >>> identity = np.eye(13) + >>> eigenvalues, eigenvectors = eigsh(identity, k=6) + >>> eigenvalues + array([1., 1., 1., 1., 1., 1.]) + >>> eigenvectors.shape + (13, 6) + + """ + # complex Hermitian matrices should be solved with eigs + if np.issubdtype(A.dtype, np.complexfloating): + if mode != 'normal': + raise ValueError(f"mode={mode} cannot be used with complex matrix A") + if which == 'BE': + raise ValueError("which='BE' cannot be used with complex matrix A") + elif which == 'LA': + which = 'LR' + elif which == 'SA': + which = 'SR' + ret = eigs(A, k, M=M, sigma=sigma, which=which, v0=v0, + ncv=ncv, maxiter=maxiter, tol=tol, + return_eigenvectors=return_eigenvectors, Minv=Minv, + OPinv=OPinv) + + if return_eigenvectors: + return ret[0].real, ret[1] + else: + return ret.real + + if A.shape[0] != A.shape[1]: + raise ValueError(f'expected square matrix (shape={A.shape})') + if M is not None: + if M.shape != A.shape: + raise ValueError(f'wrong M dimensions {M.shape}, should be {A.shape}') + if np.dtype(M.dtype).char.lower() != np.dtype(A.dtype).char.lower(): + warnings.warn('M does not have the same type precision as A. ' + 'This may adversely affect ARPACK convergence', + stacklevel=2) + + n = A.shape[0] + + if k <= 0: + raise ValueError("k must be greater than 0.") + + if k >= n: + warnings.warn("k >= N for N * N square matrix. " + "Attempting to use scipy.linalg.eigh instead.", + RuntimeWarning, stacklevel=2) + + if issparse(A): + raise TypeError("Cannot use scipy.linalg.eigh for sparse A with " + "k >= N. Use scipy.linalg.eigh(A.toarray()) or" + " reduce k.") + if isinstance(A, LinearOperator): + raise TypeError("Cannot use scipy.linalg.eigh for LinearOperator " + "A with k >= N.") + if isinstance(M, LinearOperator): + raise TypeError("Cannot use scipy.linalg.eigh for LinearOperator " + "M with k >= N.") + + return eigh(A, b=M, eigvals_only=not return_eigenvectors) + + if sigma is None: + A = aslinearoperator(A) + matvec = A.matvec + + if OPinv is not None: + raise ValueError("OPinv should not be specified " + "with sigma = None.") + if M is None: + #standard eigenvalue problem + mode = 1 + M_matvec = None + Minv_matvec = None + if Minv is not None: + raise ValueError("Minv should not be " + "specified with M = None.") + else: + #general eigenvalue problem + mode = 2 + if Minv is None: + Minv_matvec = get_inv_matvec(M, hermitian=True, tol=tol) + else: + Minv = aslinearoperator(Minv) + Minv_matvec = Minv.matvec + M_matvec = aslinearoperator(M).matvec + else: + # sigma is not None: shift-invert mode + if Minv is not None: + raise ValueError("Minv should not be specified when sigma is") + + # normal mode + if mode == 'normal': + mode = 3 + matvec = None + if OPinv is None: + Minv_matvec = get_OPinv_matvec(A, M, sigma, + hermitian=True, tol=tol) + else: + OPinv = aslinearoperator(OPinv) + Minv_matvec = OPinv.matvec + if M is None: + M_matvec = None + else: + M = aslinearoperator(M) + M_matvec = M.matvec + + # buckling mode + elif mode == 'buckling': + mode = 4 + if OPinv is None: + Minv_matvec = get_OPinv_matvec(A, M, sigma, + hermitian=True, tol=tol) + else: + Minv_matvec = aslinearoperator(OPinv).matvec + matvec = aslinearoperator(A).matvec + M_matvec = None + + # cayley-transform mode + elif mode == 'cayley': + mode = 5 + matvec = aslinearoperator(A).matvec + if OPinv is None: + Minv_matvec = get_OPinv_matvec(A, M, sigma, + hermitian=True, tol=tol) + else: + Minv_matvec = aslinearoperator(OPinv).matvec + if M is None: + M_matvec = None + else: + M_matvec = aslinearoperator(M).matvec + + # unrecognized mode + else: + raise ValueError(f"unrecognized mode '{mode}'") + + params = _SymmetricArpackParams(n, k, A.dtype.char, matvec, mode, + M_matvec, Minv_matvec, sigma, + ncv, v0, maxiter, which, tol) + + with _ARPACK_LOCK: + while not params.converged: + params.iterate() + + return params.extract(return_eigenvectors) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/tests/test_arpack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/tests/test_arpack.py new file mode 100644 index 0000000000000000000000000000000000000000..e962798a9ffce0b380c17cdf1f9deb4d6fa83159 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/arpack/tests/test_arpack.py @@ -0,0 +1,717 @@ +__usage__ = """ +To run tests locally: + python tests/test_arpack.py [-l] [-v] + +""" + +import threading +import itertools + +import numpy as np + +from numpy.testing import assert_allclose, assert_equal, suppress_warnings +from pytest import raises as assert_raises +import pytest + +from numpy import dot, conj, random +from scipy.linalg import eig, eigh +from scipy.sparse import csc_array, csr_array, diags_array, random_array +from scipy.sparse.linalg import LinearOperator, aslinearoperator +from scipy.sparse.linalg._eigen.arpack import (eigs, eigsh, arpack, + ArpackNoConvergence) + + +from scipy._lib._gcutils import assert_deallocated, IS_PYPY + + +# precision for tests +_ndigits = {'f': 3, 'd': 11, 'F': 3, 'D': 11} + + +def _get_test_tolerance(type_char, mattype=None, D_type=None, which=None): + """ + Return tolerance values suitable for a given test: + + Parameters + ---------- + type_char : {'f', 'd', 'F', 'D'} + Data type in ARPACK eigenvalue problem + mattype : {csr_array, aslinearoperator, asarray}, optional + Linear operator type + + Returns + ------- + tol + Tolerance to pass to the ARPACK routine + rtol + Relative tolerance for outputs + atol + Absolute tolerance for outputs + + """ + + rtol = {'f': 3000 * np.finfo(np.float32).eps, + 'F': 3000 * np.finfo(np.float32).eps, + 'd': 2000 * np.finfo(np.float64).eps, + 'D': 2000 * np.finfo(np.float64).eps}[type_char] + atol = rtol + tol = 0 + + if mattype is aslinearoperator and type_char in ('f', 'F'): + # iterative methods in single precision: worse errors + # also: bump ARPACK tolerance so that the iterative method converges + tol = 30 * np.finfo(np.float32).eps + rtol *= 5 + + if ( + isinstance(mattype, type) and issubclass(mattype, csr_array) + and type_char in ('f', 'F') + ): + # sparse in single precision: worse errors + rtol *= 5 + + if ( + which in ('LM', 'SM', 'LA') + and D_type.name == "gen-hermitian-Mc" + ): + if type_char == 'F': + # missing case 1, 2, and more, from PR 14798 + rtol *= 5 + + if type_char == 'D': + # missing more cases, from PR 14798 + rtol *= 10 + atol *= 10 + + return tol, rtol, atol + + +def generate_matrix(N, complex_=False, hermitian=False, + pos_definite=False, sparse=False, rng=None): + M = rng.random((N, N)) + if complex_: + M = M + 1j * rng.random((N, N)) + + if hermitian: + if pos_definite: + if sparse: + i = np.arange(N) + j = rng.randint(N, size=N-2) + i, j = np.meshgrid(i, j) + M[i, j] = 0 + M = np.dot(M.conj(), M.T) + else: + M = np.dot(M.conj(), M.T) + if sparse: + i = rng.randint(N, size=N * N // 4) + j = rng.randint(N, size=N * N // 4) + ind = np.nonzero(i == j) + j[ind] = (j[ind] + 1) % N + M[i, j] = 0 + M[j, i] = 0 + else: + if sparse: + i = rng.randint(N, size=N * N // 2) + j = rng.randint(N, size=N * N // 2) + M[i, j] = 0 + return M + + +def generate_matrix_symmetric(N, pos_definite=False, sparse=False, rng=None): + M = rng.random((N, N)) + + M = 0.5 * (M + M.T) # Make M symmetric + + if pos_definite: + Id = N * np.eye(N) + if sparse: + M = csr_array(M) + M += Id + else: + if sparse: + M = csr_array(M) + + return M + + +def assert_allclose_cc(actual, desired, **kw): + """Almost equal or complex conjugates almost equal""" + try: + assert_allclose(actual, desired, **kw) + except AssertionError: + assert_allclose(actual, conj(desired), **kw) + + +def argsort_which(eigenvalues, typ, k, which, + sigma=None, OPpart=None, mode=None): + """Return sorted indices of eigenvalues using the "which" keyword + from eigs and eigsh""" + if sigma is None: + reval = np.round(eigenvalues, decimals=_ndigits[typ]) + else: + if mode is None or mode == 'normal': + if OPpart is None: + reval = 1. / (eigenvalues - sigma) + elif OPpart == 'r': + reval = 0.5 * (1. / (eigenvalues - sigma) + + 1. / (eigenvalues - np.conj(sigma))) + elif OPpart == 'i': + reval = -0.5j * (1. / (eigenvalues - sigma) + - 1. / (eigenvalues - np.conj(sigma))) + elif mode == 'cayley': + reval = (eigenvalues + sigma) / (eigenvalues - sigma) + elif mode == 'buckling': + reval = eigenvalues / (eigenvalues - sigma) + else: + raise ValueError(f"mode='{mode}' not recognized") + + reval = np.round(reval, decimals=_ndigits[typ]) + + if which in ['LM', 'SM']: + ind = np.argsort(abs(reval)) + elif which in ['LR', 'SR', 'LA', 'SA', 'BE']: + ind = np.argsort(np.real(reval)) + elif which in ['LI', 'SI']: + # for LI,SI ARPACK returns largest,smallest abs(imaginary) why? + if typ.islower(): + ind = np.argsort(abs(np.imag(reval))) + else: + ind = np.argsort(np.imag(reval)) + else: + raise ValueError(f"which='{which}' is unrecognized") + + if which in ['LM', 'LA', 'LR', 'LI']: + return ind[-k:] + elif which in ['SM', 'SA', 'SR', 'SI']: + return ind[:k] + elif which == 'BE': + return np.concatenate((ind[:k//2], ind[k//2-k:])) + + +def eval_evec(symmetric, d, typ, k, which, v0=None, sigma=None, + mattype=np.asarray, OPpart=None, mode='normal'): + general = ('bmat' in d) + + if symmetric: + eigs_func = eigsh + else: + eigs_func = eigs + + if general: + err = (f"error for {eigs_func.__name__}:general, typ={typ}, which={which}, " + f"sigma={sigma}, mattype={mattype.__name__}," + f" OPpart={OPpart}, mode={mode}") + else: + err = (f"error for {eigs_func.__name__}:standard, typ={typ}, which={which}, " + f"sigma={sigma}, mattype={mattype.__name__}, " + f"OPpart={OPpart}, mode={mode}") + + a = d['mat'].astype(typ) + ac = mattype(a) + + if general: + b = d['bmat'].astype(typ) + bc = mattype(b) + + # get exact eigenvalues + exact_eval = d['eval'].astype(typ.upper()) + ind = argsort_which(exact_eval, typ, k, which, + sigma, OPpart, mode) + exact_eval = exact_eval[ind] + + # compute arpack eigenvalues + kwargs = dict(which=which, v0=v0, sigma=sigma) + if eigs_func is eigsh: + kwargs['mode'] = mode + else: + kwargs['OPpart'] = OPpart + + # compute suitable tolerances + kwargs['tol'], rtol, atol = _get_test_tolerance(typ, mattype, d, which) + # on rare occasions, ARPACK routines return results that are proper + # eigenvalues and -vectors, but not necessarily the ones requested in + # the parameter which. This is inherent to the Krylov methods, and + # should not be treated as a failure. If such a rare situation + # occurs, the calculation is tried again (but at most a few times). + ntries = 0 + while ntries < 5: + # solve + if general: + try: + eigenvalues, evec = eigs_func(ac, k, bc, **kwargs) + except ArpackNoConvergence: + kwargs['maxiter'] = 20*a.shape[0] + eigenvalues, evec = eigs_func(ac, k, bc, **kwargs) + else: + try: + eigenvalues, evec = eigs_func(ac, k, **kwargs) + except ArpackNoConvergence: + kwargs['maxiter'] = 20*a.shape[0] + eigenvalues, evec = eigs_func(ac, k, **kwargs) + + ind = argsort_which(eigenvalues, typ, k, which, + sigma, OPpart, mode) + eigenvalues = eigenvalues[ind] + evec = evec[:, ind] + + try: + # check eigenvalues + assert_allclose_cc(eigenvalues, exact_eval, rtol=rtol, atol=atol, + err_msg=err) + check_evecs = True + except AssertionError: + check_evecs = False + ntries += 1 + + if check_evecs: + # check eigenvectors + LHS = np.dot(a, evec) + if general: + RHS = eigenvalues * np.dot(b, evec) + else: + RHS = eigenvalues * evec + + assert_allclose(LHS, RHS, rtol=rtol, atol=atol, err_msg=err) + break + + # check eigenvalues + assert_allclose_cc(eigenvalues, exact_eval, rtol=rtol, atol=atol, err_msg=err) + + +class DictWithRepr(dict): + def __init__(self, name): + self.name = name + + def __repr__(self): + return f"<{self.name}>" + + +class SymmetricParams: + def __init__(self): + self.eigs = eigsh + self.which = ['LM', 'SM', 'LA', 'SA', 'BE'] + self.mattypes = [csr_array, aslinearoperator, np.asarray] + self.sigmas_modes = {None: ['normal'], + 0.5: ['normal', 'buckling', 'cayley']} + + # generate matrices + # these should all be float32 so that the eigenvalues + # are the same in float32 and float64 + N = 6 + rng = np.random.RandomState(2300) + Ar = generate_matrix(N, hermitian=True, + pos_definite=True, + rng=rng).astype('f').astype('d') + M = generate_matrix(N, hermitian=True, + pos_definite=True, + rng=rng).astype('f').astype('d') + Ac = generate_matrix(N, hermitian=True, pos_definite=True, + complex_=True, rng=rng).astype('F').astype('D') + Mc = generate_matrix(N, hermitian=True, pos_definite=True, + complex_=True, rng=rng).astype('F').astype('D') + v0 = rng.random(N) + + # standard symmetric problem + SS = DictWithRepr("std-symmetric") + SS['mat'] = Ar + SS['v0'] = v0 + SS['eval'] = eigh(SS['mat'], eigvals_only=True) + + # general symmetric problem + GS = DictWithRepr("gen-symmetric") + GS['mat'] = Ar + GS['bmat'] = M + GS['v0'] = v0 + GS['eval'] = eigh(GS['mat'], GS['bmat'], eigvals_only=True) + + # standard hermitian problem + SH = DictWithRepr("std-hermitian") + SH['mat'] = Ac + SH['v0'] = v0 + SH['eval'] = eigh(SH['mat'], eigvals_only=True) + + # general hermitian problem + GH = DictWithRepr("gen-hermitian") + GH['mat'] = Ac + GH['bmat'] = M + GH['v0'] = v0 + GH['eval'] = eigh(GH['mat'], GH['bmat'], eigvals_only=True) + + # general hermitian problem with hermitian M + GHc = DictWithRepr("gen-hermitian-Mc") + GHc['mat'] = Ac + GHc['bmat'] = Mc + GHc['v0'] = v0 + GHc['eval'] = eigh(GHc['mat'], GHc['bmat'], eigvals_only=True) + + self.real_test_cases = [SS, GS] + self.complex_test_cases = [SH, GH, GHc] + + +class NonSymmetricParams: + def __init__(self): + self.eigs = eigs + self.which = ['LM', 'LR', 'LI'] # , 'SM', 'LR', 'SR', 'LI', 'SI'] + self.mattypes = [csr_array, aslinearoperator, np.asarray] + self.sigmas_OPparts = {None: [None], + 0.1: ['r'], + 0.1 + 0.1j: ['r', 'i']} + + # generate matrices + # these should all be float32 so that the eigenvalues + # are the same in float32 and float64 + N = 6 + rng = np.random.RandomState(2300) + Ar = generate_matrix(N, rng=rng).astype('f').astype('d') + M = generate_matrix(N, hermitian=True, + pos_definite=True, rng=rng).astype('f').astype('d') + Ac = generate_matrix(N, complex_=True, rng=rng).astype('F').astype('D') + v0 = rng.random(N) + + # standard real nonsymmetric problem + SNR = DictWithRepr("std-real-nonsym") + SNR['mat'] = Ar + SNR['v0'] = v0 + SNR['eval'] = eig(SNR['mat'], left=False, right=False) + + # general real nonsymmetric problem + GNR = DictWithRepr("gen-real-nonsym") + GNR['mat'] = Ar + GNR['bmat'] = M + GNR['v0'] = v0 + GNR['eval'] = eig(GNR['mat'], GNR['bmat'], left=False, right=False) + + # standard complex nonsymmetric problem + SNC = DictWithRepr("std-cmplx-nonsym") + SNC['mat'] = Ac + SNC['v0'] = v0 + SNC['eval'] = eig(SNC['mat'], left=False, right=False) + + # general complex nonsymmetric problem + GNC = DictWithRepr("gen-cmplx-nonsym") + GNC['mat'] = Ac + GNC['bmat'] = M + GNC['v0'] = v0 + GNC['eval'] = eig(GNC['mat'], GNC['bmat'], left=False, right=False) + + self.real_test_cases = [SNR, GNR] + self.complex_test_cases = [SNC, GNC] + + +@pytest.mark.iterations(1) +@pytest.mark.thread_unsafe +def test_symmetric_modes(num_parallel_threads): + assert num_parallel_threads == 1 + params = SymmetricParams() + k = 2 + symmetric = True + for D in params.real_test_cases: + for typ in 'fd': + for which in params.which: + for mattype in params.mattypes: + for (sigma, modes) in params.sigmas_modes.items(): + for mode in modes: + eval_evec(symmetric, D, typ, k, which, + None, sigma, mattype, None, mode) + + +def test_hermitian_modes(): + params = SymmetricParams() + k = 2 + symmetric = True + for D in params.complex_test_cases: + for typ in 'FD': + for which in params.which: + if which == 'BE': + continue # BE invalid for complex + for mattype in params.mattypes: + for sigma in params.sigmas_modes: + eval_evec(symmetric, D, typ, k, which, + None, sigma, mattype) + + +def test_symmetric_starting_vector(): + params = SymmetricParams() + symmetric = True + for k in [1, 2, 3, 4, 5]: + for D in params.real_test_cases: + for typ in 'fd': + v0 = random.rand(len(D['v0'])).astype(typ) + eval_evec(symmetric, D, typ, k, 'LM', v0) + + +def test_symmetric_no_convergence(): + rng = np.random.RandomState(1234) + m = generate_matrix(30, hermitian=True, pos_definite=True, rng=rng) + tol, rtol, atol = _get_test_tolerance('d') + try: + w, v = eigsh(m, 4, which='LM', v0=m[:, 0], maxiter=5, tol=tol, ncv=9) + raise AssertionError("Spurious no-error exit") + except ArpackNoConvergence as err: + k = len(err.eigenvalues) + if k <= 0: + raise AssertionError("Spurious no-eigenvalues-found case") from err + w, v = err.eigenvalues, err.eigenvectors + assert_allclose(dot(m, v), w * v, rtol=rtol, atol=atol) + + +def test_real_nonsymmetric_modes(): + params = NonSymmetricParams() + k = 2 + symmetric = False + for D in params.real_test_cases: + for typ in 'fd': + for which in params.which: + for mattype in params.mattypes: + for sigma, OPparts in params.sigmas_OPparts.items(): + for OPpart in OPparts: + eval_evec(symmetric, D, typ, k, which, + None, sigma, mattype, OPpart) + + +def test_complex_nonsymmetric_modes(): + params = NonSymmetricParams() + k = 2 + symmetric = False + for D in params.complex_test_cases: + for typ in 'DF': + for which in params.which: + for mattype in params.mattypes: + for sigma in params.sigmas_OPparts: + eval_evec(symmetric, D, typ, k, which, + None, sigma, mattype) + + +def test_standard_nonsymmetric_starting_vector(): + params = NonSymmetricParams() + sigma = None + symmetric = False + for k in [1, 2, 3, 4]: + for d in params.complex_test_cases: + for typ in 'FD': + A = d['mat'] + n = A.shape[0] + v0 = random.rand(n).astype(typ) + eval_evec(symmetric, d, typ, k, "LM", v0, sigma) + + +def test_general_nonsymmetric_starting_vector(): + params = NonSymmetricParams() + sigma = None + symmetric = False + for k in [1, 2, 3, 4]: + for d in params.complex_test_cases: + for typ in 'FD': + A = d['mat'] + n = A.shape[0] + v0 = random.rand(n).astype(typ) + eval_evec(symmetric, d, typ, k, "LM", v0, sigma) + + +def test_standard_nonsymmetric_no_convergence(): + rng = np.random.RandomState(1234) + m = generate_matrix(30, complex_=True, rng=rng) + tol, rtol, atol = _get_test_tolerance('d') + try: + w, v = eigs(m, 4, which='LM', v0=m[:, 0], maxiter=5, tol=tol) + raise AssertionError("Spurious no-error exit") + except ArpackNoConvergence as err: + k = len(err.eigenvalues) + if k <= 0: + raise AssertionError("Spurious no-eigenvalues-found case") from err + w, v = err.eigenvalues, err.eigenvectors + for ww, vv in zip(w, v.T): + assert_allclose(dot(m, vv), ww * vv, rtol=rtol, atol=atol) + + +def test_eigen_bad_shapes(): + # A is not square. + A = csc_array(np.zeros((2, 3))) + assert_raises(ValueError, eigs, A) + + +def test_eigen_bad_kwargs(): + # Test eigen on wrong keyword argument + A = csc_array(np.zeros((8, 8))) + assert_raises(ValueError, eigs, A, which='XX') + + +def test_ticket_1459_arpack_crash(): + for dtype in [np.float32, np.float64]: + # This test does not seem to catch the issue for float32, + # but we made the same fix there, just to be sure + + N = 6 + k = 2 + + np.random.seed(2301) + A = np.random.random((N, N)).astype(dtype) + v0 = np.array([-0.71063568258907849895, -0.83185111795729227424, + -0.34365925382227402451, 0.46122533684552280420, + -0.58001341115969040629, -0.78844877570084292984e-01], + dtype=dtype) + + # Should not crash: + evals, evecs = eigs(A, k, v0=v0) + + +@pytest.mark.skipif(IS_PYPY, reason="Test not meaningful on PyPy") +def test_linearoperator_deallocation(): + # Check that the linear operators used by the Arpack wrappers are + # deallocatable by reference counting -- they are big objects, so + # Python's cyclic GC may not collect them fast enough before + # running out of memory if eigs/eigsh are called in a tight loop. + + M_d = np.eye(10) + M_s = csc_array(M_d) + M_o = aslinearoperator(M_d) + + with assert_deallocated(lambda: arpack.SpLuInv(M_s)): + pass + with assert_deallocated(lambda: arpack.LuInv(M_d)): + pass + with assert_deallocated(lambda: arpack.IterInv(M_s)): + pass + with assert_deallocated(lambda: arpack.IterOpInv(M_o, None, 0.3)): + pass + with assert_deallocated(lambda: arpack.IterOpInv(M_o, M_o, 0.3)): + pass + + +@pytest.mark.thread_unsafe +def test_parallel_threads(): + results = [] + v0 = np.random.rand(50) + + def worker(): + x = diags_array([1, -2, 1], offsets=[-1, 0, 1], shape=(50, 50)) + w, v = eigs(x, k=3, v0=v0) + results.append(w) + + w, v = eigsh(x, k=3, v0=v0) + results.append(w) + + threads = [threading.Thread(target=worker) for k in range(10)] + for t in threads: + t.start() + for t in threads: + t.join() + + worker() + + for r in results: + assert_allclose(r, results[-1]) + + +def test_reentering(): + # Just some linear operator that calls eigs recursively + def A_matvec(x): + x = diags_array([1, -2, 1], offsets=[-1, 0, 1], shape=(50, 50)) + w, v = eigs(x, k=1) + return v / w[0] + A = LinearOperator(matvec=A_matvec, dtype=float, shape=(50, 50)) + + # The Fortran code is not reentrant, so this fails (gracefully, not crashing) + assert_raises(RuntimeError, eigs, A, k=1) + assert_raises(RuntimeError, eigsh, A, k=1) + + +def test_regression_arpackng_1315(): + # Check that issue arpack-ng/#1315 is not present. + # Adapted from arpack-ng/TESTS/bug_1315_single.c + # If this fails, then the installed ARPACK library is faulty. + + for dtype in [np.float32, np.float64]: + np.random.seed(1234) + + w0 = np.arange(1, 1000+1).astype(dtype) + A = diags_array([w0], offsets=[0], shape=(1000, 1000)) + + v0 = np.random.rand(1000).astype(dtype) + w, v = eigs(A, k=9, ncv=2*9+1, which="LM", v0=v0) + + assert_allclose(np.sort(w), np.sort(w0[-9:]), + rtol=1e-4) + + +def test_eigs_for_k_greater(): + # Test eigs() for k beyond limits. + rng = np.random.RandomState(1234) + A_sparse = diags_array([1, -2, 1], offsets=[-1, 0, 1], shape=(4, 4)) # sparse + A = generate_matrix(4, sparse=False, rng=rng) + M_dense = rng.random((4, 4)) + M_sparse = generate_matrix(4, sparse=True, rng=rng) + M_linop = aslinearoperator(M_dense) + eig_tuple1 = eig(A, b=M_dense) + eig_tuple2 = eig(A, b=M_sparse) + + with suppress_warnings() as sup: + sup.filter(RuntimeWarning) + + assert_equal(eigs(A, M=M_dense, k=3), eig_tuple1) + assert_equal(eigs(A, M=M_dense, k=4), eig_tuple1) + assert_equal(eigs(A, M=M_dense, k=5), eig_tuple1) + assert_equal(eigs(A, M=M_sparse, k=5), eig_tuple2) + + # M as LinearOperator + assert_raises(TypeError, eigs, A, M=M_linop, k=3) + + # Test 'A' for different types + assert_raises(TypeError, eigs, aslinearoperator(A), k=3) + assert_raises(TypeError, eigs, A_sparse, k=3) + + +def test_eigsh_for_k_greater(): + # Test eigsh() for k beyond limits. + rng = np.random.RandomState(1234) + A_sparse = diags_array([1, -2, 1], offsets=[-1, 0, 1], shape=(4, 4)) # sparse + A = generate_matrix(4, sparse=False, rng=rng) + M_dense = generate_matrix_symmetric(4, pos_definite=True, rng=rng) + M_sparse = generate_matrix_symmetric( + 4, pos_definite=True, sparse=True, rng=rng) + M_linop = aslinearoperator(M_dense) + eig_tuple1 = eigh(A, b=M_dense) + eig_tuple2 = eigh(A, b=M_sparse) + + with suppress_warnings() as sup: + sup.filter(RuntimeWarning) + + assert_equal(eigsh(A, M=M_dense, k=4), eig_tuple1) + assert_equal(eigsh(A, M=M_dense, k=5), eig_tuple1) + assert_equal(eigsh(A, M=M_sparse, k=5), eig_tuple2) + + # M as LinearOperator + assert_raises(TypeError, eigsh, A, M=M_linop, k=4) + + # Test 'A' for different types + assert_raises(TypeError, eigsh, aslinearoperator(A), k=4) + assert_raises(TypeError, eigsh, A_sparse, M=M_dense, k=4) + + +def test_real_eigs_real_k_subset(): + rng = np.random.default_rng(2) + + n = 10 + A = random_array(shape=(n, n), density=0.5, rng=rng) + A.data *= 2 + A.data -= 1 + A += A.T # make symmetric to test real eigenvalues + + v0 = np.ones(n) + + whichs = ['LM', 'SM', 'LR', 'SR', 'LI', 'SI'] + dtypes = [np.float32, np.float64] + + for which, sigma, dtype in itertools.product(whichs, [None, 0, 5], dtypes): + prev_w = np.array([], dtype=dtype) + eps = np.finfo(dtype).eps + for k in range(1, 9): + w, z = eigs(A.astype(dtype), k=k, which=which, sigma=sigma, + v0=v0.astype(dtype), tol=0) + assert_allclose(np.linalg.norm(A.dot(z) - z * w), 0, atol=np.sqrt(eps)) + + # Check that the set of eigenvalues for `k` is a subset of that for `k+1` + dist = abs(prev_w[:,None] - w).min(axis=1) + assert_allclose(dist, 0, atol=np.sqrt(eps)) + + prev_w = w diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..6ab5330361a6bcc2a8403f9b3788aedae750d57f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/__init__.py @@ -0,0 +1,16 @@ +""" +Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG) + +LOBPCG is a preconditioned eigensolver for large symmetric positive definite +(SPD) generalized eigenproblems. + +Call the function lobpcg - see help for lobpcg.lobpcg. + +""" +from .lobpcg import * + +__all__ = [s for s in dir() if not s.startswith('_')] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/lobpcg.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/lobpcg.py new file mode 100644 index 0000000000000000000000000000000000000000..3bace8c76ccd11be7bb25f02806382a60ac5e9c7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/lobpcg.py @@ -0,0 +1,1110 @@ +""" +Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG). + +References +---------- +.. [1] A. V. Knyazev (2001), + Toward the Optimal Preconditioned Eigensolver: Locally Optimal + Block Preconditioned Conjugate Gradient Method. + SIAM Journal on Scientific Computing 23, no. 2, + pp. 517-541. :doi:`10.1137/S1064827500366124` + +.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov (2007), + Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) + in hypre and PETSc. :arxiv:`0705.2626` + +.. [3] A. V. Knyazev's C and MATLAB implementations: + https://github.com/lobpcg/blopex +""" + +import warnings +import numpy as np +from scipy.linalg import (inv, eigh, cho_factor, cho_solve, + cholesky, LinAlgError) +from scipy.sparse.linalg import LinearOperator +from scipy.sparse import issparse + +__all__ = ["lobpcg"] + + +def _report_nonhermitian(M, name): + """ + Report if `M` is not a Hermitian matrix given its type. + """ + from scipy.linalg import norm + + md = M - M.T.conj() + nmd = norm(md, 1) + tol = 10 * np.finfo(M.dtype).eps + tol = max(tol, tol * norm(M, 1)) + if nmd > tol: + warnings.warn( + f"Matrix {name} of the type {M.dtype} is not Hermitian: " + f"condition: {nmd} < {tol} fails.", + UserWarning, stacklevel=4 + ) + +def _as2d(ar): + """ + If the input array is 2D return it, if it is 1D, append a dimension, + making it a column vector. + """ + if ar.ndim == 2: + return ar + else: # Assume 1! + aux = np.asarray(ar) + aux.shape = (ar.shape[0], 1) + return aux + + +def _makeMatMat(m): + if m is None: + return None + elif callable(m): + return lambda v: m(v) + else: + return lambda v: m @ v + + +def _matmul_inplace(x, y, verbosityLevel=0): + """Perform 'np.matmul' in-place if possible. + + If some sufficient conditions for inplace matmul are met, do so. + Otherwise try inplace update and fall back to overwrite if that fails. + """ + if x.flags["CARRAY"] and x.shape[1] == y.shape[1] and x.dtype == y.dtype: + # conditions where we can guarantee that inplace updates will work; + # i.e. x is not a view/slice, x & y have compatible dtypes, and the + # shape of the result of x @ y matches the shape of x. + np.matmul(x, y, out=x) + else: + # ideally, we'd have an exhaustive list of conditions above when + # inplace updates are possible; since we don't, we opportunistically + # try if it works, and fall back to overwriting if necessary + try: + np.matmul(x, y, out=x) + except Exception: + if verbosityLevel: + warnings.warn( + "Inplace update of x = x @ y failed, " + "x needs to be overwritten.", + UserWarning, stacklevel=3 + ) + x = x @ y + return x + + +def _applyConstraints(blockVectorV, factYBY, blockVectorBY, blockVectorY): + """Changes blockVectorV in-place.""" + YBV = blockVectorBY.T.conj() @ blockVectorV + tmp = cho_solve(factYBY, YBV) + blockVectorV -= blockVectorY @ tmp + + +def _b_orthonormalize(B, blockVectorV, blockVectorBV=None, + verbosityLevel=0): + """in-place B-orthonormalize the given block vector using Cholesky.""" + if blockVectorBV is None: + if B is None: + blockVectorBV = blockVectorV + else: + try: + blockVectorBV = B(blockVectorV) + except Exception as e: + if verbosityLevel: + warnings.warn( + f"Secondary MatMul call failed with error\n" + f"{e}\n", + UserWarning, stacklevel=3 + ) + return None, None, None + if blockVectorBV.shape != blockVectorV.shape: + raise ValueError( + f"The shape {blockVectorV.shape} " + f"of the orthogonalized matrix not preserved\n" + f"and changed to {blockVectorBV.shape} " + f"after multiplying by the secondary matrix.\n" + ) + + VBV = blockVectorV.T.conj() @ blockVectorBV + try: + # VBV is a Cholesky factor from now on... + VBV = cholesky(VBV, overwrite_a=True) + VBV = inv(VBV, overwrite_a=True) + blockVectorV = _matmul_inplace( + blockVectorV, VBV, + verbosityLevel=verbosityLevel + ) + if B is not None: + blockVectorBV = _matmul_inplace( + blockVectorBV, VBV, + verbosityLevel=verbosityLevel + ) + return blockVectorV, blockVectorBV, VBV + except LinAlgError: + if verbosityLevel: + warnings.warn( + "Cholesky has failed.", + UserWarning, stacklevel=3 + ) + return None, None, None + + +def _get_indx(_lambda, num, largest): + """Get `num` indices into `_lambda` depending on `largest` option.""" + ii = np.argsort(_lambda) + if largest: + ii = ii[:-num - 1:-1] + else: + ii = ii[:num] + + return ii + + +def _handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel): + if verbosityLevel: + _report_nonhermitian(gramA, "gramA") + _report_nonhermitian(gramB, "gramB") + + +def lobpcg( + A, + X, + B=None, + M=None, + Y=None, + tol=None, + maxiter=None, + largest=True, + verbosityLevel=0, + retLambdaHistory=False, + retResidualNormsHistory=False, + restartControl=20, +): + """Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG). + + LOBPCG is a preconditioned eigensolver for large real symmetric and complex + Hermitian definite generalized eigenproblems. + + Parameters + ---------- + A : {sparse matrix, ndarray, LinearOperator, callable object} + The Hermitian linear operator of the problem, usually given by a + sparse matrix. Often called the "stiffness matrix". + X : ndarray, float32 or float64 + Initial approximation to the ``k`` eigenvectors (non-sparse). + If `A` has ``shape=(n,n)`` then `X` must have ``shape=(n,k)``. + B : {sparse matrix, ndarray, LinearOperator, callable object} + Optional. By default ``B = None``, which is equivalent to identity. + The right hand side operator in a generalized eigenproblem if present. + Often called the "mass matrix". Must be Hermitian positive definite. + M : {sparse matrix, ndarray, LinearOperator, callable object} + Optional. By default ``M = None``, which is equivalent to identity. + Preconditioner aiming to accelerate convergence. + Y : ndarray, float32 or float64, default: None + An ``n-by-sizeY`` ndarray of constraints with ``sizeY < n``. + The iterations will be performed in the ``B``-orthogonal complement + of the column-space of `Y`. `Y` must be full rank if present. + tol : scalar, optional + The default is ``tol=n*sqrt(eps)``. + Solver tolerance for the stopping criterion. + maxiter : int, default: 20 + Maximum number of iterations. + largest : bool, default: True + When True, solve for the largest eigenvalues, otherwise the smallest. + verbosityLevel : int, optional + By default ``verbosityLevel=0`` no output. + Controls the solver standard/screen output. + retLambdaHistory : bool, default: False + Whether to return iterative eigenvalue history. + retResidualNormsHistory : bool, default: False + Whether to return iterative history of residual norms. + restartControl : int, optional. + Iterations restart if the residuals jump ``2**restartControl`` times + compared to the smallest recorded in ``retResidualNormsHistory``. + The default is ``restartControl=20``, making the restarts rare for + backward compatibility. + + Returns + ------- + lambda : ndarray of the shape ``(k, )``. + Array of ``k`` approximate eigenvalues. + v : ndarray of the same shape as ``X.shape``. + An array of ``k`` approximate eigenvectors. + lambdaHistory : ndarray, optional. + The eigenvalue history, if `retLambdaHistory` is ``True``. + ResidualNormsHistory : ndarray, optional. + The history of residual norms, if `retResidualNormsHistory` + is ``True``. + + Notes + ----- + The iterative loop runs ``maxit=maxiter`` (20 if ``maxit=None``) + iterations at most and finishes earlier if the tolerance is met. + Breaking backward compatibility with the previous version, LOBPCG + now returns the block of iterative vectors with the best accuracy rather + than the last one iterated, as a cure for possible divergence. + + If ``X.dtype == np.float32`` and user-provided operations/multiplications + by `A`, `B`, and `M` all preserve the ``np.float32`` data type, + all the calculations and the output are in ``np.float32``. + + The size of the iteration history output equals to the number of the best + (limited by `maxit`) iterations plus 3: initial, final, and postprocessing. + + If both `retLambdaHistory` and `retResidualNormsHistory` are ``True``, + the return tuple has the following format + ``(lambda, V, lambda history, residual norms history)``. + + In the following ``n`` denotes the matrix size and ``k`` the number + of required eigenvalues (smallest or largest). + + The LOBPCG code internally solves eigenproblems of the size ``3k`` on every + iteration by calling the dense eigensolver `eigh`, so if ``k`` is not + small enough compared to ``n``, it makes no sense to call the LOBPCG code. + Moreover, if one calls the LOBPCG algorithm for ``5k > n``, it would likely + break internally, so the code calls the standard function `eigh` instead. + It is not that ``n`` should be large for the LOBPCG to work, but rather the + ratio ``n / k`` should be large. It you call LOBPCG with ``k=1`` + and ``n=10``, it works though ``n`` is small. The method is intended + for extremely large ``n / k``. + + The convergence speed depends basically on three factors: + + 1. Quality of the initial approximations `X` to the seeking eigenvectors. + Randomly distributed around the origin vectors work well if no better + choice is known. + + 2. Relative separation of the desired eigenvalues from the rest + of the eigenvalues. One can vary ``k`` to improve the separation. + + 3. Proper preconditioning to shrink the spectral spread. + For example, a rod vibration test problem (under tests + directory) is ill-conditioned for large ``n``, so convergence will be + slow, unless efficient preconditioning is used. For this specific + problem, a good simple preconditioner function would be a linear solve + for `A`, which is easy to code since `A` is tridiagonal. + + References + ---------- + .. [1] A. V. Knyazev (2001), + Toward the Optimal Preconditioned Eigensolver: Locally Optimal + Block Preconditioned Conjugate Gradient Method. + SIAM Journal on Scientific Computing 23, no. 2, + pp. 517-541. :doi:`10.1137/S1064827500366124` + + .. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov + (2007), Block Locally Optimal Preconditioned Eigenvalue Xolvers + (BLOPEX) in hypre and PETSc. :arxiv:`0705.2626` + + .. [3] A. V. Knyazev's C and MATLAB implementations: + https://github.com/lobpcg/blopex + + Examples + -------- + Our first example is minimalistic - find the largest eigenvalue of + a diagonal matrix by solving the non-generalized eigenvalue problem + ``A x = lambda x`` without constraints or preconditioning. + + >>> import numpy as np + >>> from scipy.sparse import spdiags + >>> from scipy.sparse.linalg import LinearOperator, aslinearoperator + >>> from scipy.sparse.linalg import lobpcg + + The square matrix size is + + >>> n = 100 + + and its diagonal entries are 1, ..., 100 defined by + + >>> vals = np.arange(1, n + 1).astype(np.int16) + + The first mandatory input parameter in this test is + the sparse diagonal matrix `A` + of the eigenvalue problem ``A x = lambda x`` to solve. + + >>> A = spdiags(vals, 0, n, n) + >>> A = A.astype(np.int16) + >>> A.toarray() + array([[ 1, 0, 0, ..., 0, 0, 0], + [ 0, 2, 0, ..., 0, 0, 0], + [ 0, 0, 3, ..., 0, 0, 0], + ..., + [ 0, 0, 0, ..., 98, 0, 0], + [ 0, 0, 0, ..., 0, 99, 0], + [ 0, 0, 0, ..., 0, 0, 100]], shape=(100, 100), dtype=int16) + + The second mandatory input parameter `X` is a 2D array with the + row dimension determining the number of requested eigenvalues. + `X` is an initial guess for targeted eigenvectors. + `X` must have linearly independent columns. + If no initial approximations available, randomly oriented vectors + commonly work best, e.g., with components normally distributed + around zero or uniformly distributed on the interval [-1 1]. + Setting the initial approximations to dtype ``np.float32`` + forces all iterative values to dtype ``np.float32`` speeding up + the run while still allowing accurate eigenvalue computations. + + >>> k = 1 + >>> rng = np.random.default_rng() + >>> X = rng.normal(size=(n, k)) + >>> X = X.astype(np.float32) + + >>> eigenvalues, _ = lobpcg(A, X, maxiter=60) + >>> eigenvalues + array([100.], dtype=float32) + + `lobpcg` needs only access the matrix product with `A` rather + then the matrix itself. Since the matrix `A` is diagonal in + this example, one can write a function of the matrix product + ``A @ X`` using the diagonal values ``vals`` only, e.g., by + element-wise multiplication with broadcasting in the lambda-function + + >>> A_lambda = lambda X: vals[:, np.newaxis] * X + + or the regular function + + >>> def A_matmat(X): + ... return vals[:, np.newaxis] * X + + and use the handle to one of these callables as an input + + >>> eigenvalues, _ = lobpcg(A_lambda, X, maxiter=60) + >>> eigenvalues + array([100.], dtype=float32) + >>> eigenvalues, _ = lobpcg(A_matmat, X, maxiter=60) + >>> eigenvalues + array([100.], dtype=float32) + + The traditional callable `LinearOperator` is no longer + necessary but still supported as the input to `lobpcg`. + Specifying ``matmat=A_matmat`` explicitly improves performance. + + >>> A_lo = LinearOperator((n, n), matvec=A_matmat, matmat=A_matmat, dtype=np.int16) + >>> eigenvalues, _ = lobpcg(A_lo, X, maxiter=80) + >>> eigenvalues + array([100.], dtype=float32) + + The least efficient callable option is `aslinearoperator`: + + >>> eigenvalues, _ = lobpcg(aslinearoperator(A), X, maxiter=80) + >>> eigenvalues + array([100.], dtype=float32) + + We now switch to computing the three smallest eigenvalues specifying + + >>> k = 3 + >>> X = np.random.default_rng().normal(size=(n, k)) + + and ``largest=False`` parameter + + >>> eigenvalues, _ = lobpcg(A, X, largest=False, maxiter=90) + >>> print(eigenvalues) + [1. 2. 3.] + + The next example illustrates computing 3 smallest eigenvalues of + the same matrix `A` given by the function handle ``A_matmat`` but + with constraints and preconditioning. + + Constraints - an optional input parameter is a 2D array comprising + of column vectors that the eigenvectors must be orthogonal to + + >>> Y = np.eye(n, 3) + + The preconditioner acts as the inverse of `A` in this example, but + in the reduced precision ``np.float32`` even though the initial `X` + and thus all iterates and the output are in full ``np.float64``. + + >>> inv_vals = 1./vals + >>> inv_vals = inv_vals.astype(np.float32) + >>> M = lambda X: inv_vals[:, np.newaxis] * X + + Let us now solve the eigenvalue problem for the matrix `A` first + without preconditioning requesting 80 iterations + + >>> eigenvalues, _ = lobpcg(A_matmat, X, Y=Y, largest=False, maxiter=80) + >>> eigenvalues + array([4., 5., 6.]) + >>> eigenvalues.dtype + dtype('float64') + + With preconditioning we need only 20 iterations from the same `X` + + >>> eigenvalues, _ = lobpcg(A_matmat, X, Y=Y, M=M, largest=False, maxiter=20) + >>> eigenvalues + array([4., 5., 6.]) + + Note that the vectors passed in `Y` are the eigenvectors of the 3 + smallest eigenvalues. The results returned above are orthogonal to those. + + The primary matrix `A` may be indefinite, e.g., after shifting + ``vals`` by 50 from 1, ..., 100 to -49, ..., 50, we still can compute + the 3 smallest or largest eigenvalues. + + >>> vals = vals - 50 + >>> X = rng.normal(size=(n, k)) + >>> eigenvalues, _ = lobpcg(A_matmat, X, largest=False, maxiter=99) + >>> eigenvalues + array([-49., -48., -47.]) + >>> eigenvalues, _ = lobpcg(A_matmat, X, largest=True, maxiter=99) + >>> eigenvalues + array([50., 49., 48.]) + + """ + blockVectorX = X + bestblockVectorX = blockVectorX + blockVectorY = Y + residualTolerance = tol + if maxiter is None: + maxiter = 20 + + bestIterationNumber = maxiter + + sizeY = 0 + if blockVectorY is not None: + if len(blockVectorY.shape) != 2: + warnings.warn( + f"Expected rank-2 array for argument Y, instead got " + f"{len(blockVectorY.shape)}, " + f"so ignore it and use no constraints.", + UserWarning, stacklevel=2 + ) + blockVectorY = None + else: + sizeY = blockVectorY.shape[1] + + # Block size. + if blockVectorX is None: + raise ValueError("The mandatory initial matrix X cannot be None") + if len(blockVectorX.shape) != 2: + raise ValueError("expected rank-2 array for argument X") + + n, sizeX = blockVectorX.shape + + # Data type of iterates, determined by X, must be inexact + if not np.issubdtype(blockVectorX.dtype, np.inexact): + warnings.warn( + f"Data type for argument X is {blockVectorX.dtype}, " + f"which is not inexact, so casted to np.float32.", + UserWarning, stacklevel=2 + ) + blockVectorX = np.asarray(blockVectorX, dtype=np.float32) + + if retLambdaHistory: + lambdaHistory = np.zeros((maxiter + 3, sizeX), + dtype=blockVectorX.dtype) + if retResidualNormsHistory: + residualNormsHistory = np.zeros((maxiter + 3, sizeX), + dtype=blockVectorX.dtype) + + if verbosityLevel: + aux = "Solving " + if B is None: + aux += "standard" + else: + aux += "generalized" + aux += " eigenvalue problem with" + if M is None: + aux += "out" + aux += " preconditioning\n\n" + aux += "matrix size %d\n" % n + aux += "block size %d\n\n" % sizeX + if blockVectorY is None: + aux += "No constraints\n\n" + else: + if sizeY > 1: + aux += "%d constraints\n\n" % sizeY + else: + aux += "%d constraint\n\n" % sizeY + print(aux) + + if (n - sizeY) < (5 * sizeX): + warnings.warn( + f"The problem size {n} minus the constraints size {sizeY} " + f"is too small relative to the block size {sizeX}. " + f"Using a dense eigensolver instead of LOBPCG iterations." + f"No output of the history of the iterations.", + UserWarning, stacklevel=2 + ) + + sizeX = min(sizeX, n) + + if blockVectorY is not None: + raise NotImplementedError( + "The dense eigensolver does not support constraints." + ) + + # Define the closed range of indices of eigenvalues to return. + if largest: + eigvals = (n - sizeX, n - 1) + else: + eigvals = (0, sizeX - 1) + + try: + if isinstance(A, LinearOperator): + A = A(np.eye(n, dtype=int)) + elif callable(A): + A = A(np.eye(n, dtype=int)) + if A.shape != (n, n): + raise ValueError( + f"The shape {A.shape} of the primary matrix\n" + f"defined by a callable object is wrong.\n" + ) + elif issparse(A): + A = A.toarray() + else: + A = np.asarray(A) + except Exception as e: + raise Exception( + f"Primary MatMul call failed with error\n" + f"{e}\n") + + if B is not None: + try: + if isinstance(B, LinearOperator): + B = B(np.eye(n, dtype=int)) + elif callable(B): + B = B(np.eye(n, dtype=int)) + if B.shape != (n, n): + raise ValueError( + f"The shape {B.shape} of the secondary matrix\n" + f"defined by a callable object is wrong.\n" + ) + elif issparse(B): + B = B.toarray() + else: + B = np.asarray(B) + except Exception as e: + raise Exception( + f"Secondary MatMul call failed with error\n" + f"{e}\n") + + try: + vals, vecs = eigh(A, + B, + subset_by_index=eigvals, + check_finite=False) + if largest: + # Reverse order to be compatible with eigs() in 'LM' mode. + vals = vals[::-1] + vecs = vecs[:, ::-1] + + return vals, vecs + except Exception as e: + raise Exception( + f"Dense eigensolver failed with error\n" + f"{e}\n" + ) + + if (residualTolerance is None) or (residualTolerance <= 0.0): + residualTolerance = np.sqrt(np.finfo(blockVectorX.dtype).eps) * n + + A = _makeMatMat(A) + B = _makeMatMat(B) + M = _makeMatMat(M) + + # Apply constraints to X. + if blockVectorY is not None: + + if B is not None: + blockVectorBY = B(blockVectorY) + if blockVectorBY.shape != blockVectorY.shape: + raise ValueError( + f"The shape {blockVectorY.shape} " + f"of the constraint not preserved\n" + f"and changed to {blockVectorBY.shape} " + f"after multiplying by the secondary matrix.\n" + ) + else: + blockVectorBY = blockVectorY + + # gramYBY is a dense array. + gramYBY = blockVectorY.T.conj() @ blockVectorBY + try: + # gramYBY is a Cholesky factor from now on... + gramYBY = cho_factor(gramYBY, overwrite_a=True) + except LinAlgError as e: + raise ValueError("Linearly dependent constraints") from e + + _applyConstraints(blockVectorX, gramYBY, blockVectorBY, blockVectorY) + + ## + # B-orthonormalize X. + blockVectorX, blockVectorBX, _ = _b_orthonormalize( + B, blockVectorX, verbosityLevel=verbosityLevel) + if blockVectorX is None: + raise ValueError("Linearly dependent initial approximations") + + ## + # Compute the initial Ritz vectors: solve the eigenproblem. + blockVectorAX = A(blockVectorX) + if blockVectorAX.shape != blockVectorX.shape: + raise ValueError( + f"The shape {blockVectorX.shape} " + f"of the initial approximations not preserved\n" + f"and changed to {blockVectorAX.shape} " + f"after multiplying by the primary matrix.\n" + ) + + gramXAX = blockVectorX.T.conj() @ blockVectorAX + + _lambda, eigBlockVector = eigh(gramXAX, check_finite=False) + ii = _get_indx(_lambda, sizeX, largest) + _lambda = _lambda[ii] + if retLambdaHistory: + lambdaHistory[0, :] = _lambda + + eigBlockVector = np.asarray(eigBlockVector[:, ii]) + blockVectorX = _matmul_inplace( + blockVectorX, eigBlockVector, + verbosityLevel=verbosityLevel + ) + blockVectorAX = _matmul_inplace( + blockVectorAX, eigBlockVector, + verbosityLevel=verbosityLevel + ) + if B is not None: + blockVectorBX = _matmul_inplace( + blockVectorBX, eigBlockVector, + verbosityLevel=verbosityLevel + ) + + ## + # Active index set. + activeMask = np.ones((sizeX,), dtype=bool) + + ## + # Main iteration loop. + + blockVectorP = None # set during iteration + blockVectorAP = None + blockVectorBP = None + + smallestResidualNorm = np.abs(np.finfo(blockVectorX.dtype).max) + + iterationNumber = -1 + restart = True + forcedRestart = False + explicitGramFlag = False + while iterationNumber < maxiter: + iterationNumber += 1 + + if B is not None: + aux = blockVectorBX * _lambda[np.newaxis, :] + else: + aux = blockVectorX * _lambda[np.newaxis, :] + + blockVectorR = blockVectorAX - aux + + aux = np.sum(blockVectorR.conj() * blockVectorR, 0) + residualNorms = np.sqrt(np.abs(aux)) + if retResidualNormsHistory: + residualNormsHistory[iterationNumber, :] = residualNorms + residualNorm = np.sum(np.abs(residualNorms)) / sizeX + + if residualNorm < smallestResidualNorm: + smallestResidualNorm = residualNorm + bestIterationNumber = iterationNumber + bestblockVectorX = blockVectorX + elif residualNorm > 2**restartControl * smallestResidualNorm: + forcedRestart = True + blockVectorAX = A(blockVectorX) + if blockVectorAX.shape != blockVectorX.shape: + raise ValueError( + f"The shape {blockVectorX.shape} " + f"of the restarted iterate not preserved\n" + f"and changed to {blockVectorAX.shape} " + f"after multiplying by the primary matrix.\n" + ) + if B is not None: + blockVectorBX = B(blockVectorX) + if blockVectorBX.shape != blockVectorX.shape: + raise ValueError( + f"The shape {blockVectorX.shape} " + f"of the restarted iterate not preserved\n" + f"and changed to {blockVectorBX.shape} " + f"after multiplying by the secondary matrix.\n" + ) + + ii = np.where(residualNorms > residualTolerance, True, False) + activeMask = activeMask & ii + currentBlockSize = activeMask.sum() + + if verbosityLevel: + print(f"iteration {iterationNumber}") + print(f"current block size: {currentBlockSize}") + print(f"eigenvalue(s):\n{_lambda}") + print(f"residual norm(s):\n{residualNorms}") + + if currentBlockSize == 0: + break + + activeBlockVectorR = _as2d(blockVectorR[:, activeMask]) + + if iterationNumber > 0: + activeBlockVectorP = _as2d(blockVectorP[:, activeMask]) + activeBlockVectorAP = _as2d(blockVectorAP[:, activeMask]) + if B is not None: + activeBlockVectorBP = _as2d(blockVectorBP[:, activeMask]) + + if M is not None: + # Apply preconditioner T to the active residuals. + activeBlockVectorR = M(activeBlockVectorR) + + ## + # Apply constraints to the preconditioned residuals. + if blockVectorY is not None: + _applyConstraints(activeBlockVectorR, + gramYBY, + blockVectorBY, + blockVectorY) + + ## + # B-orthogonalize the preconditioned residuals to X. + if B is not None: + activeBlockVectorR = activeBlockVectorR - ( + blockVectorX @ + (blockVectorBX.T.conj() @ activeBlockVectorR) + ) + else: + activeBlockVectorR = activeBlockVectorR - ( + blockVectorX @ + (blockVectorX.T.conj() @ activeBlockVectorR) + ) + + ## + # B-orthonormalize the preconditioned residuals. + aux = _b_orthonormalize( + B, activeBlockVectorR, verbosityLevel=verbosityLevel) + activeBlockVectorR, activeBlockVectorBR, _ = aux + + if activeBlockVectorR is None: + warnings.warn( + f"Failed at iteration {iterationNumber} with accuracies " + f"{residualNorms}\n not reaching the requested " + f"tolerance {residualTolerance}.", + UserWarning, stacklevel=2 + ) + break + activeBlockVectorAR = A(activeBlockVectorR) + + if iterationNumber > 0: + if B is not None: + aux = _b_orthonormalize( + B, activeBlockVectorP, activeBlockVectorBP, + verbosityLevel=verbosityLevel + ) + activeBlockVectorP, activeBlockVectorBP, invR = aux + else: + aux = _b_orthonormalize(B, activeBlockVectorP, + verbosityLevel=verbosityLevel) + activeBlockVectorP, _, invR = aux + # Function _b_orthonormalize returns None if Cholesky fails + if activeBlockVectorP is not None: + activeBlockVectorAP = _matmul_inplace( + activeBlockVectorAP, invR, + verbosityLevel=verbosityLevel + ) + restart = forcedRestart + else: + restart = True + + ## + # Perform the Rayleigh Ritz Procedure: + # Compute symmetric Gram matrices: + + if activeBlockVectorAR.dtype == "float32": + myeps = 1 + else: + myeps = np.sqrt(np.finfo(activeBlockVectorR.dtype).eps) + + if residualNorms.max() > myeps and not explicitGramFlag: + explicitGramFlag = False + else: + # Once explicitGramFlag, forever explicitGramFlag. + explicitGramFlag = True + + # Shared memory assignments to simplify the code + if B is None: + blockVectorBX = blockVectorX + activeBlockVectorBR = activeBlockVectorR + if not restart: + activeBlockVectorBP = activeBlockVectorP + + # Common submatrices: + gramXAR = np.dot(blockVectorX.T.conj(), activeBlockVectorAR) + gramRAR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR) + + gramDtype = activeBlockVectorAR.dtype + if explicitGramFlag: + gramRAR = (gramRAR + gramRAR.T.conj()) / 2 + gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX) + gramXAX = (gramXAX + gramXAX.T.conj()) / 2 + gramXBX = np.dot(blockVectorX.T.conj(), blockVectorBX) + gramRBR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBR) + gramXBR = np.dot(blockVectorX.T.conj(), activeBlockVectorBR) + else: + gramXAX = np.diag(_lambda).astype(gramDtype) + gramXBX = np.eye(sizeX, dtype=gramDtype) + gramRBR = np.eye(currentBlockSize, dtype=gramDtype) + gramXBR = np.zeros((sizeX, currentBlockSize), dtype=gramDtype) + + if not restart: + gramXAP = np.dot(blockVectorX.T.conj(), activeBlockVectorAP) + gramRAP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP) + gramPAP = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP) + gramXBP = np.dot(blockVectorX.T.conj(), activeBlockVectorBP) + gramRBP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBP) + if explicitGramFlag: + gramPAP = (gramPAP + gramPAP.T.conj()) / 2 + gramPBP = np.dot(activeBlockVectorP.T.conj(), + activeBlockVectorBP) + else: + gramPBP = np.eye(currentBlockSize, dtype=gramDtype) + + gramA = np.block( + [ + [gramXAX, gramXAR, gramXAP], + [gramXAR.T.conj(), gramRAR, gramRAP], + [gramXAP.T.conj(), gramRAP.T.conj(), gramPAP], + ] + ) + gramB = np.block( + [ + [gramXBX, gramXBR, gramXBP], + [gramXBR.T.conj(), gramRBR, gramRBP], + [gramXBP.T.conj(), gramRBP.T.conj(), gramPBP], + ] + ) + + _handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel) + + try: + _lambda, eigBlockVector = eigh(gramA, + gramB, + check_finite=False) + except LinAlgError as e: + # raise ValueError("eigh failed in lobpcg iterations") from e + if verbosityLevel: + warnings.warn( + f"eigh failed at iteration {iterationNumber} \n" + f"with error {e} causing a restart.\n", + UserWarning, stacklevel=2 + ) + # try again after dropping the direction vectors P from RR + restart = True + + if restart: + gramA = np.block([[gramXAX, gramXAR], [gramXAR.T.conj(), gramRAR]]) + gramB = np.block([[gramXBX, gramXBR], [gramXBR.T.conj(), gramRBR]]) + + _handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel) + + try: + _lambda, eigBlockVector = eigh(gramA, + gramB, + check_finite=False) + except LinAlgError as e: + # raise ValueError("eigh failed in lobpcg iterations") from e + warnings.warn( + f"eigh failed at iteration {iterationNumber} with error\n" + f"{e}\n", + UserWarning, stacklevel=2 + ) + break + + ii = _get_indx(_lambda, sizeX, largest) + _lambda = _lambda[ii] + eigBlockVector = eigBlockVector[:, ii] + if retLambdaHistory: + lambdaHistory[iterationNumber + 1, :] = _lambda + + # Compute Ritz vectors. + if B is not None: + if not restart: + eigBlockVectorX = eigBlockVector[:sizeX] + eigBlockVectorR = eigBlockVector[sizeX: + sizeX + currentBlockSize] + eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize:] + + pp = np.dot(activeBlockVectorR, eigBlockVectorR) + pp += np.dot(activeBlockVectorP, eigBlockVectorP) + + app = np.dot(activeBlockVectorAR, eigBlockVectorR) + app += np.dot(activeBlockVectorAP, eigBlockVectorP) + + bpp = np.dot(activeBlockVectorBR, eigBlockVectorR) + bpp += np.dot(activeBlockVectorBP, eigBlockVectorP) + else: + eigBlockVectorX = eigBlockVector[:sizeX] + eigBlockVectorR = eigBlockVector[sizeX:] + + pp = np.dot(activeBlockVectorR, eigBlockVectorR) + app = np.dot(activeBlockVectorAR, eigBlockVectorR) + bpp = np.dot(activeBlockVectorBR, eigBlockVectorR) + + blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp + blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app + blockVectorBX = np.dot(blockVectorBX, eigBlockVectorX) + bpp + + blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp + + else: + if not restart: + eigBlockVectorX = eigBlockVector[:sizeX] + eigBlockVectorR = eigBlockVector[sizeX: + sizeX + currentBlockSize] + eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize:] + + pp = np.dot(activeBlockVectorR, eigBlockVectorR) + pp += np.dot(activeBlockVectorP, eigBlockVectorP) + + app = np.dot(activeBlockVectorAR, eigBlockVectorR) + app += np.dot(activeBlockVectorAP, eigBlockVectorP) + else: + eigBlockVectorX = eigBlockVector[:sizeX] + eigBlockVectorR = eigBlockVector[sizeX:] + + pp = np.dot(activeBlockVectorR, eigBlockVectorR) + app = np.dot(activeBlockVectorAR, eigBlockVectorR) + + blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp + blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app + + blockVectorP, blockVectorAP = pp, app + + if B is not None: + aux = blockVectorBX * _lambda[np.newaxis, :] + else: + aux = blockVectorX * _lambda[np.newaxis, :] + + blockVectorR = blockVectorAX - aux + + aux = np.sum(blockVectorR.conj() * blockVectorR, 0) + residualNorms = np.sqrt(np.abs(aux)) + # Use old lambda in case of early loop exit. + if retLambdaHistory: + lambdaHistory[iterationNumber + 1, :] = _lambda + if retResidualNormsHistory: + residualNormsHistory[iterationNumber + 1, :] = residualNorms + residualNorm = np.sum(np.abs(residualNorms)) / sizeX + if residualNorm < smallestResidualNorm: + smallestResidualNorm = residualNorm + bestIterationNumber = iterationNumber + 1 + bestblockVectorX = blockVectorX + + if np.max(np.abs(residualNorms)) > residualTolerance: + warnings.warn( + f"Exited at iteration {iterationNumber} with accuracies \n" + f"{residualNorms}\n" + f"not reaching the requested tolerance {residualTolerance}.\n" + f"Use iteration {bestIterationNumber} instead with accuracy \n" + f"{smallestResidualNorm}.\n", + UserWarning, stacklevel=2 + ) + + if verbosityLevel: + print(f"Final iterative eigenvalue(s):\n{_lambda}") + print(f"Final iterative residual norm(s):\n{residualNorms}") + + blockVectorX = bestblockVectorX + # Making eigenvectors "exactly" satisfy the blockVectorY constrains + if blockVectorY is not None: + _applyConstraints(blockVectorX, + gramYBY, + blockVectorBY, + blockVectorY) + + # Making eigenvectors "exactly" othonormalized by final "exact" RR + blockVectorAX = A(blockVectorX) + if blockVectorAX.shape != blockVectorX.shape: + raise ValueError( + f"The shape {blockVectorX.shape} " + f"of the postprocessing iterate not preserved\n" + f"and changed to {blockVectorAX.shape} " + f"after multiplying by the primary matrix.\n" + ) + gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX) + + blockVectorBX = blockVectorX + if B is not None: + blockVectorBX = B(blockVectorX) + if blockVectorBX.shape != blockVectorX.shape: + raise ValueError( + f"The shape {blockVectorX.shape} " + f"of the postprocessing iterate not preserved\n" + f"and changed to {blockVectorBX.shape} " + f"after multiplying by the secondary matrix.\n" + ) + + gramXBX = np.dot(blockVectorX.T.conj(), blockVectorBX) + _handle_gramA_gramB_verbosity(gramXAX, gramXBX, verbosityLevel) + gramXAX = (gramXAX + gramXAX.T.conj()) / 2 + gramXBX = (gramXBX + gramXBX.T.conj()) / 2 + try: + _lambda, eigBlockVector = eigh(gramXAX, + gramXBX, + check_finite=False) + except LinAlgError as e: + raise ValueError("eigh has failed in lobpcg postprocessing") from e + + ii = _get_indx(_lambda, sizeX, largest) + _lambda = _lambda[ii] + eigBlockVector = np.asarray(eigBlockVector[:, ii]) + + blockVectorX = np.dot(blockVectorX, eigBlockVector) + blockVectorAX = np.dot(blockVectorAX, eigBlockVector) + + if B is not None: + blockVectorBX = np.dot(blockVectorBX, eigBlockVector) + aux = blockVectorBX * _lambda[np.newaxis, :] + else: + aux = blockVectorX * _lambda[np.newaxis, :] + + blockVectorR = blockVectorAX - aux + + aux = np.sum(blockVectorR.conj() * blockVectorR, 0) + residualNorms = np.sqrt(np.abs(aux)) + + if retLambdaHistory: + lambdaHistory[bestIterationNumber + 1, :] = _lambda + if retResidualNormsHistory: + residualNormsHistory[bestIterationNumber + 1, :] = residualNorms + + if retLambdaHistory: + lambdaHistory = lambdaHistory[ + : bestIterationNumber + 2, :] + if retResidualNormsHistory: + residualNormsHistory = residualNormsHistory[ + : bestIterationNumber + 2, :] + + if np.max(np.abs(residualNorms)) > residualTolerance: + warnings.warn( + f"Exited postprocessing with accuracies \n" + f"{residualNorms}\n" + f"not reaching the requested tolerance {residualTolerance}.", + UserWarning, stacklevel=2 + ) + + if verbosityLevel: + print(f"Final postprocessing eigenvalue(s):\n{_lambda}") + print(f"Final residual norm(s):\n{residualNorms}") + + if retLambdaHistory: + lambdaHistory = np.vsplit(lambdaHistory, np.shape(lambdaHistory)[0]) + lambdaHistory = [np.squeeze(i) for i in lambdaHistory] + if retResidualNormsHistory: + residualNormsHistory = np.vsplit(residualNormsHistory, + np.shape(residualNormsHistory)[0]) + residualNormsHistory = [np.squeeze(i) for i in residualNormsHistory] + + if retLambdaHistory: + if retResidualNormsHistory: + return _lambda, blockVectorX, lambdaHistory, residualNormsHistory + else: + return _lambda, blockVectorX, lambdaHistory + else: + if retResidualNormsHistory: + return _lambda, blockVectorX, residualNormsHistory + else: + return _lambda, blockVectorX diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/tests/test_lobpcg.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/tests/test_lobpcg.py new file mode 100644 index 0000000000000000000000000000000000000000..850aa0c1b3f5d0c47c1ba66d518b8a7059c85e95 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/lobpcg/tests/test_lobpcg.py @@ -0,0 +1,725 @@ +""" Test functions for the sparse.linalg._eigen.lobpcg module +""" + +import itertools +import platform +import sys +import pytest +import numpy as np +from numpy import ones, r_, diag +from numpy.testing import (assert_almost_equal, assert_equal, + assert_allclose, assert_array_less) + +from scipy import sparse +from scipy.linalg import (eigh, toeplitz, + cholesky_banded, cho_solve_banded) +from scipy.sparse import dia_array, eye_array, csr_array +from scipy.sparse.linalg import eigsh, LinearOperator +from scipy.sparse.linalg._eigen.lobpcg import lobpcg +from scipy.sparse.linalg._eigen.lobpcg.lobpcg import _b_orthonormalize +from scipy._lib._util import np_long, np_ulong +from scipy.sparse.linalg._special_sparse_arrays import (Sakurai, + MikotaPair) + +_IS_32BIT = (sys.maxsize < 2**32) + +INT_DTYPES = (np.intc, np_long, np.longlong, np.uintc, np_ulong, np.ulonglong) +# np.half is unsupported on many test systems so excluded +REAL_DTYPES = (np.float32, np.float64, np.longdouble) +COMPLEX_DTYPES = (np.complex64, np.complex128, np.clongdouble) +INEXACTDTYPES = REAL_DTYPES + COMPLEX_DTYPES +ALLDTYPES = INT_DTYPES + INEXACTDTYPES + + +def sign_align(A, B): + """Align signs of columns of A match those of B: column-wise remove + sign of A by multiplying with its sign then multiply in sign of B. + """ + return np.array([col_A * np.sign(col_A[0]) * np.sign(col_B[0]) + for col_A, col_B in zip(A.T, B.T)]).T + +def ElasticRod(n): + """Build the matrices for the generalized eigenvalue problem of the + fixed-free elastic rod vibration model. + """ + L = 1.0 + le = L/n + rho = 7.85e3 + S = 1.e-4 + E = 2.1e11 + mass = rho*S*le/6. + k = E*S/le + A = k*(diag(r_[2.*ones(n-1), 1])-diag(ones(n-1), 1)-diag(ones(n-1), -1)) + B = mass*(diag(r_[4.*ones(n-1), 2])+diag(ones(n-1), 1)+diag(ones(n-1), -1)) + return A, B + + +@pytest.mark.filterwarnings("ignore:The problem size") +@pytest.mark.parametrize("n", [10, 20]) +@pytest.mark.filterwarnings("ignore:Exited at iteration") +@pytest.mark.filterwarnings("ignore:Exited postprocessing") +def test_ElasticRod(n): + """Check eigh vs. lobpcg consistency for elastic rod model. + """ + A, B = ElasticRod(n) + m = 2 + rnd = np.random.RandomState(0) + X = rnd.standard_normal((n, m)) + eigvals, _ = lobpcg(A, X, B=B, tol=1e-2, maxiter=50, largest=False) + eigvals.sort() + w, _ = eigh(A, b=B) + w.sort() + assert_almost_equal(w[:int(m/2)], eigvals[:int(m/2)], decimal=2) + + +@pytest.mark.parametrize("n", [50]) +@pytest.mark.parametrize("m", [1, 2, 10]) +@pytest.mark.filterwarnings("ignore:Casting complex values to real") +@pytest.mark.parametrize("Vdtype", INEXACTDTYPES) +@pytest.mark.parametrize("Bdtype", ALLDTYPES) +@pytest.mark.parametrize("BVdtype", INEXACTDTYPES) +def test_b_orthonormalize(n, m, Vdtype, Bdtype, BVdtype): + """Test B-orthonormalization by Cholesky with callable 'B'. + The function '_b_orthonormalize' is key in LOBPCG but may + lead to numerical instabilities. The input vectors are often + badly scaled, so the function needs scale-invariant Cholesky; + see https://netlib.org/lapack/lawnspdf/lawn14.pdf. + """ + rnd = np.random.RandomState(0) + X = rnd.standard_normal((n, m)).astype(Vdtype) + Xcopy = np.copy(X) + vals = np.arange(1, n+1, dtype=float) + B = dia_array(([vals], [0]), shape=(n, n)).astype(Bdtype) + BX = B @ X + BX = BX.astype(BVdtype) + is_all_complex = (np.issubdtype(Vdtype, np.complexfloating) and + np.issubdtype(BVdtype, np.complexfloating)) + is_all_notcomplex = (not np.issubdtype(Vdtype, np.complexfloating) and + not np.issubdtype(Bdtype, np.complexfloating) and + not np.issubdtype(BVdtype, np.complexfloating)) + + # All complex or all not complex can calculate in-place + check_inplace = is_all_complex or is_all_notcomplex + # np.longdouble tol cannot be achieved on most systems + atol = m * n * max(np.finfo(Vdtype).eps, + np.finfo(BVdtype).eps, + np.finfo(np.float64).eps) + + Xo, BXo, _ = _b_orthonormalize(lambda v: B @ v, X, BX) + if check_inplace: + # Check in-place + assert_equal(X, Xo) + assert_equal(id(X), id(Xo)) + assert_equal(BX, BXo) + assert_equal(id(BX), id(BXo)) + # Check BXo + assert_allclose(B @ Xo, BXo, atol=atol, rtol=atol) + # Check B-orthonormality + assert_allclose(Xo.T.conj() @ B @ Xo, np.identity(m), + atol=atol, rtol=atol) + # Repeat without BX in outputs + X = np.copy(Xcopy) + Xo1, BXo1, _ = _b_orthonormalize(lambda v: B @ v, X) + assert_allclose(Xo, Xo1, atol=atol, rtol=atol) + assert_allclose(BXo, BXo1, atol=atol, rtol=atol) + if check_inplace: + # Check in-place. + assert_equal(X, Xo1) + assert_equal(id(X), id(Xo1)) + # Check BXo1 + assert_allclose(B @ Xo1, BXo1, atol=atol, rtol=atol) + + # Introduce column-scaling in X + scaling = 1.0 / np.geomspace(10, 1e10, num=m) + X = Xcopy * scaling + X = X.astype(Vdtype) + BX = B @ X + BX = BX.astype(BVdtype) + # Check scaling-invariance of Cholesky-based orthonormalization + Xo1, BXo1, _ = _b_orthonormalize(lambda v: B @ v, X, BX) + # The output should be the same, up the signs of the columns + Xo1 = sign_align(Xo1, Xo) + assert_allclose(Xo, Xo1, atol=atol, rtol=atol) + BXo1 = sign_align(BXo1, BXo) + assert_allclose(BXo, BXo1, atol=atol, rtol=atol) + + +@pytest.mark.thread_unsafe +@pytest.mark.filterwarnings("ignore:Exited at iteration 0") +@pytest.mark.filterwarnings("ignore:Exited postprocessing") +def test_nonhermitian_warning(capsys): + """Check the warning of a Ritz matrix being not Hermitian + by feeding a non-Hermitian input matrix. + Also check stdout since verbosityLevel=1 and lack of stderr. + """ + n = 10 + X = np.arange(n * 2).reshape(n, 2).astype(np.float32) + A = np.arange(n * n).reshape(n, n).astype(np.float32) + with pytest.warns(UserWarning, match="Matrix gramA"): + _, _ = lobpcg(A, X, verbosityLevel=1, maxiter=0) + out, err = capsys.readouterr() # Capture output + assert out.startswith("Solving standard eigenvalue") # Test stdout + assert err == '' # Test empty stderr + # Make the matrix symmetric and the UserWarning disappears. + A += A.T + _, _ = lobpcg(A, X, verbosityLevel=1, maxiter=0) + out, err = capsys.readouterr() # Capture output + assert out.startswith("Solving standard eigenvalue") # Test stdout + assert err == '' # Test empty stderr + + +def test_regression(): + """Check the eigenvalue of the identity matrix is one. + """ + # https://mail.python.org/pipermail/scipy-user/2010-October/026944.html + n = 10 + X = np.ones((n, 1)) + A = np.identity(n) + w, _ = lobpcg(A, X) + assert_allclose(w, [1]) + + +@pytest.mark.filterwarnings("ignore:The problem size") +@pytest.mark.parametrize('n, m, m_excluded', [(30, 4, 3), (4, 2, 0)]) +def test_diagonal(n, m, m_excluded): + """Test ``m - m_excluded`` eigenvalues and eigenvectors of + diagonal matrices of the size ``n`` varying matrix formats: + dense array, spare matrix, and ``LinearOperator`` for both + matrixes in the generalized eigenvalue problem ``Av = cBv`` + and for the preconditioner. + """ + rnd = np.random.RandomState(0) + + # Define the generalized eigenvalue problem Av = cBv + # where (c, v) is a generalized eigenpair, + # A is the diagonal matrix whose entries are 1,...n, + # B is the identity matrix. + vals = np.arange(1, n+1, dtype=float) + A_s = dia_array(([vals], [0]), shape=(n, n)) + A_a = A_s.toarray() + + def A_f(x): + return A_s @ x + + A_lo = LinearOperator(matvec=A_f, + matmat=A_f, + shape=(n, n), dtype=float) + + B_a = eye_array(n) + B_s = csr_array(B_a) + + def B_f(x): + return B_a @ x + + B_lo = LinearOperator(matvec=B_f, + matmat=B_f, + shape=(n, n), dtype=float) + + # Let the preconditioner M be the inverse of A. + M_s = dia_array(([1./vals], [0]), shape=(n, n)) + M_a = M_s.toarray() + + def M_f(x): + return M_s @ x + + M_lo = LinearOperator(matvec=M_f, + matmat=M_f, + shape=(n, n), dtype=float) + + # Pick random initial vectors. + X = rnd.normal(size=(n, m)) + + # Require that the returned eigenvectors be in the orthogonal complement + # of the first few standard basis vectors. + if m_excluded > 0: + Y = np.eye(n, m_excluded) + else: + Y = None + + for A in [A_a, A_s, A_lo]: + for B in [B_a, B_s, B_lo]: + for M in [M_a, M_s, M_lo]: + eigvals, vecs = lobpcg(A, X, B, M=M, Y=Y, + maxiter=40, largest=False) + + assert_allclose(eigvals, np.arange(1+m_excluded, + 1+m_excluded+m)) + _check_eigen(A, eigvals, vecs, rtol=1e-3, atol=1e-3) + + +def _check_eigen(M, w, V, rtol=1e-8, atol=1e-14): + """Check if the eigenvalue residual is small. + """ + mult_wV = np.multiply(w, V) + dot_MV = M.dot(V) + assert_allclose(mult_wV, dot_MV, rtol=rtol, atol=atol) + + +def _check_fiedler(n, p): + """Check the Fiedler vector computation. + """ + # This is not necessarily the recommended way to find the Fiedler vector. + col = np.zeros(n) + col[1] = 1 + A = toeplitz(col) + D = np.diag(A.sum(axis=1)) + L = D - A + # Compute the full eigendecomposition using tricks, e.g. + # http://www.cs.yale.edu/homes/spielman/561/2009/lect02-09.pdf + tmp = np.pi * np.arange(n) / n + analytic_w = 2 * (1 - np.cos(tmp)) + analytic_V = np.cos(np.outer(np.arange(n) + 1/2, tmp)) + _check_eigen(L, analytic_w, analytic_V) + # Compute the full eigendecomposition using eigh. + eigh_w, eigh_V = eigh(L) + _check_eigen(L, eigh_w, eigh_V) + # Check that the first eigenvalue is near zero and that the rest agree. + assert_array_less(np.abs([eigh_w[0], analytic_w[0]]), 1e-14) + assert_allclose(eigh_w[1:], analytic_w[1:]) + + # Check small lobpcg eigenvalues. + X = analytic_V[:, :p] + lobpcg_w, lobpcg_V = lobpcg(L, X, largest=False) + assert_equal(lobpcg_w.shape, (p,)) + assert_equal(lobpcg_V.shape, (n, p)) + _check_eigen(L, lobpcg_w, lobpcg_V) + assert_array_less(np.abs(np.min(lobpcg_w)), 1e-14) + assert_allclose(np.sort(lobpcg_w)[1:], analytic_w[1:p]) + + # Check large lobpcg eigenvalues. + X = analytic_V[:, -p:] + lobpcg_w, lobpcg_V = lobpcg(L, X, largest=True) + assert_equal(lobpcg_w.shape, (p,)) + assert_equal(lobpcg_V.shape, (n, p)) + _check_eigen(L, lobpcg_w, lobpcg_V) + assert_allclose(np.sort(lobpcg_w), analytic_w[-p:]) + + # Look for the Fiedler vector using good but not exactly correct guesses. + fiedler_guess = np.concatenate((np.ones(n//2), -np.ones(n-n//2))) + X = np.vstack((np.ones(n), fiedler_guess)).T + lobpcg_w, _ = lobpcg(L, X, largest=False) + # Mathematically, the smaller eigenvalue should be zero + # and the larger should be the algebraic connectivity. + lobpcg_w = np.sort(lobpcg_w) + assert_allclose(lobpcg_w, analytic_w[:2], atol=1e-14) + + +@pytest.mark.thread_unsafe +def test_fiedler_small_8(): + """Check the dense workaround path for small matrices. + """ + # This triggers the dense path because 8 < 2*5. + with pytest.warns(UserWarning, match="The problem size"): + _check_fiedler(8, 2) + + +def test_fiedler_large_12(): + """Check the dense workaround path avoided for non-small matrices. + """ + # This does not trigger the dense path, because 2*5 <= 12. + _check_fiedler(12, 2) + + +@pytest.mark.filterwarnings("ignore:Failed at iteration") +@pytest.mark.filterwarnings("ignore:Exited at iteration") +@pytest.mark.filterwarnings("ignore:Exited postprocessing") +def test_failure_to_run_iterations(): + """Check that the code exits gracefully without breaking. Issue #10974. + The code may or not issue a warning, filtered out. Issue #15935, #17954. + """ + rnd = np.random.RandomState(0) + X = rnd.standard_normal((100, 10)) + A = X @ X.T + Q = rnd.standard_normal((X.shape[0], 4)) + eigenvalues, _ = lobpcg(A, Q, maxiter=40, tol=1e-12) + assert np.max(eigenvalues) > 0 + + +@pytest.mark.thread_unsafe +def test_failure_to_run_iterations_nonsymmetric(): + """Check that the code exists gracefully without breaking + if the matrix in not symmetric. + """ + A = np.zeros((10, 10)) + A[0, 1] = 1 + Q = np.ones((10, 1)) + msg = "Exited at iteration 2|Exited postprocessing with accuracies.*" + with pytest.warns(UserWarning, match=msg): + eigenvalues, _ = lobpcg(A, Q, maxiter=20) + assert np.max(eigenvalues) > 0 + + +@pytest.mark.filterwarnings("ignore:The problem size") +def test_hermitian(): + """Check complex-value Hermitian cases. + """ + rnd = np.random.RandomState(0) + + sizes = [3, 12] + ks = [1, 2] + gens = [True, False] + + for s, k, gen, dh, dx, db in ( + itertools.product(sizes, ks, gens, gens, gens, gens) + ): + H = rnd.random((s, s)) + 1.j * rnd.random((s, s)) + H = 10 * np.eye(s) + H + H.T.conj() + H = H.astype(np.complex128) if dh else H.astype(np.complex64) + + X = rnd.standard_normal((s, k)) + X = X + 1.j * rnd.standard_normal((s, k)) + X = X.astype(np.complex128) if dx else X.astype(np.complex64) + + if not gen: + B = np.eye(s) + w, v = lobpcg(H, X, maxiter=99, verbosityLevel=0) + # Also test mixing complex H with real B. + wb, _ = lobpcg(H, X, B, maxiter=99, verbosityLevel=0) + assert_allclose(w, wb, rtol=1e-6) + w0, _ = eigh(H) + else: + B = rnd.random((s, s)) + 1.j * rnd.random((s, s)) + B = 10 * np.eye(s) + B.dot(B.T.conj()) + B = B.astype(np.complex128) if db else B.astype(np.complex64) + w, v = lobpcg(H, X, B, maxiter=99, verbosityLevel=0) + w0, _ = eigh(H, B) + + for wx, vx in zip(w, v.T): + # Check eigenvector + assert_allclose(np.linalg.norm(H.dot(vx) - B.dot(vx) * wx) + / np.linalg.norm(H.dot(vx)), + 0, atol=5e-2, rtol=0) + + # Compare eigenvalues + j = np.argmin(abs(w0 - wx)) + assert_allclose(wx, w0[j], rtol=1e-4) + + +# The n=5 case tests the alternative small matrix code path that uses eigh(). +@pytest.mark.filterwarnings("ignore:The problem size") +@pytest.mark.parametrize('n, atol', [(20, 1e-3), (5, 1e-8)]) +def test_eigsh_consistency(n, atol): + """Check eigsh vs. lobpcg consistency. + """ + vals = np.arange(1, n+1, dtype=np.float64) + A = dia_array((vals, 0), shape=(n, n)) + rnd = np.random.RandomState(0) + X = rnd.standard_normal((n, 2)) + lvals, lvecs = lobpcg(A, X, largest=True, maxiter=100) + vals, _ = eigsh(A, k=2) + + _check_eigen(A, lvals, lvecs, atol=atol, rtol=0) + assert_allclose(np.sort(vals), np.sort(lvals), atol=1e-14) + + +@pytest.mark.thread_unsafe +def test_verbosity(): + """Check that nonzero verbosity level code runs. + """ + rnd = np.random.RandomState(0) + X = rnd.standard_normal((10, 10)) + A = X @ X.T + Q = rnd.standard_normal((X.shape[0], 1)) + msg = "Exited at iteration.*|Exited postprocessing with accuracies.*" + with pytest.warns(UserWarning, match=msg): + _, _ = lobpcg(A, Q, maxiter=3, verbosityLevel=9) + + +@pytest.mark.xfail(_IS_32BIT and sys.platform == 'win32', + reason="tolerance violation on windows") +@pytest.mark.xfail(platform.machine() == 'ppc64le', + reason="fails on ppc64le") +@pytest.mark.filterwarnings("ignore:Exited postprocessing") +def test_tolerance_float32(): + """Check lobpcg for attainable tolerance in float32. + """ + rnd = np.random.RandomState(0) + n = 50 + m = 3 + vals = -np.arange(1, n + 1) + A = dia_array(([vals], [0]), shape=(n, n)) + A = A.astype(np.float32) + X = rnd.standard_normal((n, m)) + X = X.astype(np.float32) + eigvals, _ = lobpcg(A, X, tol=1.25e-5, maxiter=50, verbosityLevel=0) + assert_allclose(eigvals, -np.arange(1, 1 + m), atol=2e-5, rtol=1e-5) + + +@pytest.mark.parametrize("vdtype", INEXACTDTYPES) +@pytest.mark.parametrize("mdtype", ALLDTYPES) +@pytest.mark.parametrize("arr_type", [np.array, + sparse.csr_array, + sparse.coo_array]) +def test_dtypes(vdtype, mdtype, arr_type): + """Test lobpcg in various dtypes. + """ + rnd = np.random.RandomState(0) + n = 12 + m = 2 + A = arr_type(np.diag(np.arange(1, n + 1)).astype(mdtype)) + X = rnd.random((n, m)) + X = X.astype(vdtype) + eigvals, eigvecs = lobpcg(A, X, tol=1e-2, largest=False) + assert_allclose(eigvals, np.arange(1, 1 + m), atol=1e-1) + # eigenvectors must be nearly real in any case + assert_allclose(np.sum(np.abs(eigvecs - eigvecs.conj())), 0, atol=1e-2) + + +@pytest.mark.thread_unsafe +@pytest.mark.filterwarnings("ignore:Exited at iteration") +@pytest.mark.filterwarnings("ignore:Exited postprocessing") +def test_inplace_warning(): + """Check lobpcg gives a warning in '_b_orthonormalize' + that in-place orthogonalization is impossible due to dtype mismatch. + """ + rnd = np.random.RandomState(0) + n = 6 + m = 1 + vals = -np.arange(1, n + 1) + A = dia_array(([vals], [0]), shape=(n, n)) + A = A.astype(np.cdouble) + X = rnd.standard_normal((n, m)) + with pytest.warns(UserWarning, match="Inplace update"): + eigvals, _ = lobpcg(A, X, maxiter=2, verbosityLevel=1) + + +@pytest.mark.thread_unsafe +def test_maxit(): + """Check lobpcg if maxit=maxiter runs maxiter iterations and + if maxit=None runs 20 iterations (the default) + by checking the size of the iteration history output, which should + be the number of iterations plus 3 (initial, final, and postprocessing) + typically when maxiter is small and the choice of the best is passive. + """ + rnd = np.random.RandomState(0) + n = 50 + m = 4 + vals = -np.arange(1, n + 1) + A = dia_array(([vals], [0]), shape=(n, n)) + A = A.astype(np.float32) + X = rnd.standard_normal((n, m)) + X = X.astype(np.float64) + msg = "Exited at iteration.*|Exited postprocessing with accuracies.*" + for maxiter in range(1, 4): + with pytest.warns(UserWarning, match=msg): + _, _, l_h, r_h = lobpcg(A, X, tol=1e-8, maxiter=maxiter, + retLambdaHistory=True, + retResidualNormsHistory=True) + assert_allclose(np.shape(l_h)[0], maxiter+3) + assert_allclose(np.shape(r_h)[0], maxiter+3) + with pytest.warns(UserWarning, match=msg): + l, _, l_h, r_h = lobpcg(A, X, tol=1e-8, + retLambdaHistory=True, + retResidualNormsHistory=True) + assert_allclose(np.shape(l_h)[0], 20+3) + assert_allclose(np.shape(r_h)[0], 20+3) + # Check that eigenvalue output is the last one in history + assert_allclose(l, l_h[-1]) + # Make sure that both history outputs are lists + assert isinstance(l_h, list) + assert isinstance(r_h, list) + # Make sure that both history lists are arrays-like + assert_allclose(np.shape(l_h), np.shape(np.asarray(l_h))) + assert_allclose(np.shape(r_h), np.shape(np.asarray(r_h))) + + +@pytest.mark.xslow +@pytest.mark.filterwarnings("ignore:Exited at iteration") +@pytest.mark.filterwarnings("ignore:Exited postprocessing") +def test_sakurai(): + """Check lobpcg and eighs accuracy for the Sakurai example + already used in `benchmarks/benchmarks/sparse_linalg_lobpcg.py`. + """ + n = 50 + tol = 100 * n * n * n* np.finfo(float).eps + sakurai_obj = Sakurai(n, dtype='int') + A = sakurai_obj + m = 3 + ee = sakurai_obj.eigenvalues(3) + rng = np.random.default_rng(0) + X = rng.normal(size=(n, m)) + el, _ = lobpcg(A, X, tol=1e-9, maxiter=5000, largest=False) + accuracy = max(abs(ee - el) / ee) + assert_allclose(accuracy, 0., atol=tol) + a_l = LinearOperator((n, n), matvec=A, matmat=A, dtype='float64') + ea, _ = eigsh(a_l, k=m, which='SA', tol=1e-9, maxiter=15000, + v0 = rng.normal(size=(n, 1))) + accuracy = max(abs(ee - ea) / ee) + assert_allclose(accuracy, 0., atol=tol) + + +@pytest.mark.parametrize("n", [500, 1000]) +@pytest.mark.filterwarnings("ignore:Exited at iteration") +@pytest.mark.filterwarnings("ignore:Exited postprocessing") +def test_sakurai_inverse(n): + """Check lobpcg and eighs accuracy for the sakurai_inverse example + already used in `benchmarks/benchmarks/sparse_linalg_lobpcg.py`. + """ + def a(x): + return cho_solve_banded((c, False), x) + tol = 100 * n * n * n* np.finfo(float).eps + sakurai_obj = Sakurai(n) + A = sakurai_obj.tobanded().astype(np.float64) + m = 3 + ee = sakurai_obj.eigenvalues(3) + rng = np.random.default_rng(0) + X = rng.normal(size=(n, m)) + c = cholesky_banded(A) + el, _ = lobpcg(a, X, tol=1e-9, maxiter=8) + accuracy = max(abs(ee - 1. / el) / ee) + assert_allclose(accuracy, 0., atol=tol) + a_l = LinearOperator((n, n), matvec=a, matmat=a, dtype='float64') + ea, _ = eigsh(a_l, k=m, which='LA', tol=1e-9, maxiter=8, + v0 = rng.normal(size=(n, 1))) + accuracy = max(abs(ee - np.sort(1. / ea)) / ee) + assert_allclose(accuracy, 0., atol=tol) + + +@pytest.mark.filterwarnings("ignore:The problem size") +@pytest.mark.parametrize("n", [10, 20, 128, 256, 512, 1024, 2048]) +@pytest.mark.filterwarnings("ignore:Exited at iteration") +@pytest.mark.filterwarnings("ignore:Exited postprocessing") +def test_MikotaPair(n): + """Check lobpcg and eighs accuracy for the Mikota example + already used in `benchmarks/benchmarks/sparse_linalg_lobpcg.py`. + """ + def a(x): + return cho_solve_banded((c, False), x) + mik = MikotaPair(n) + mik_k = mik.k + mik_m = mik.m + Ac = mik_k + Bc = mik_m + Ab = mik_k.tobanded() + eigenvalues = mik.eigenvalues + if n == 10: + m = 3 # lobpcg calls eigh + elif n == 20: + m = 2 + else: + m = 10 + ee = eigenvalues(m) + tol = 100 * m * n * n * np.finfo(float).eps + rng = np.random.default_rng(0) + X = rng.normal(size=(n, m)) + c = cholesky_banded(Ab.astype(np.float32)) + el, _ = lobpcg(Ac, X, Bc, M=a, tol=1e-4, + maxiter=40, largest=False) + accuracy = max(abs(ee - el) / ee) + assert_allclose(accuracy, 0., atol=tol) + B = LinearOperator((n, n), matvec=Bc, matmat=Bc, dtype='float64') + A = LinearOperator((n, n), matvec=Ac, matmat=Ac, dtype='float64') + c = cholesky_banded(Ab) + a_l = LinearOperator((n, n), matvec=a, matmat=a, dtype='float64') + ea, _ = eigsh(B, k=m, M=A, Minv=a_l, which='LA', tol=1e-4, maxiter=50, + v0 = rng.normal(size=(n, 1))) + accuracy = max(abs(ee - np.sort(1./ea)) / ee) + assert_allclose(accuracy, 0., atol=tol) + + +@pytest.mark.slow +@pytest.mark.parametrize("n", [15]) +@pytest.mark.parametrize("m", [1, 2]) +@pytest.mark.filterwarnings("ignore:Exited at iteration") +@pytest.mark.filterwarnings("ignore:Exited postprocessing") +def test_diagonal_data_types(n, m): + """Check lobpcg for diagonal matrices for all matrix types. + Constraints are imposed, so a dense eigensolver eig cannot run. + """ + rnd = np.random.RandomState(0) + # Define the generalized eigenvalue problem Av = cBv + # where (c, v) is a generalized eigenpair, + # and where we choose A and B to be diagonal. + vals = np.arange(1, n + 1) + + list_sparse_format = ['bsr', 'coo', 'csc', 'csr', 'dia', 'dok', 'lil'] + for s_f_i, s_f in enumerate(list_sparse_format): + + As64 = dia_array(([vals * vals], [0]), shape=(n, n)).asformat(s_f) + As32 = As64.astype(np.float32) + Af64 = As64.toarray() + Af32 = Af64.astype(np.float32) + + def As32f(x): + return As32 @ x + As32LO = LinearOperator(matvec=As32f, + matmat=As32f, + shape=(n, n), + dtype=As32.dtype) + + listA = [Af64, As64, Af32, As32, As32f, As32LO, lambda v: As32 @ v] + + Bs64 = dia_array(([vals], [0]), shape=(n, n)).asformat(s_f) + Bf64 = Bs64.toarray() + Bs32 = Bs64.astype(np.float32) + + def Bs32f(x): + return Bs32 @ x + Bs32LO = LinearOperator(matvec=Bs32f, + matmat=Bs32f, + shape=(n, n), + dtype=Bs32.dtype) + listB = [Bf64, Bs64, Bs32, Bs32f, Bs32LO, lambda v: Bs32 @ v] + + # Define the preconditioner function as LinearOperator. + Ms64 = dia_array(([1./vals], [0]), shape=(n, n)).asformat(s_f) + + def Ms64precond(x): + return Ms64 @ x + Ms64precondLO = LinearOperator(matvec=Ms64precond, + matmat=Ms64precond, + shape=(n, n), + dtype=Ms64.dtype) + Mf64 = Ms64.toarray() + + def Mf64precond(x): + return Mf64 @ x + Mf64precondLO = LinearOperator(matvec=Mf64precond, + matmat=Mf64precond, + shape=(n, n), + dtype=Mf64.dtype) + Ms32 = Ms64.astype(np.float32) + + def Ms32precond(x): + return Ms32 @ x + Ms32precondLO = LinearOperator(matvec=Ms32precond, + matmat=Ms32precond, + shape=(n, n), + dtype=Ms32.dtype) + Mf32 = Ms32.toarray() + + def Mf32precond(x): + return Mf32 @ x + Mf32precondLO = LinearOperator(matvec=Mf32precond, + matmat=Mf32precond, + shape=(n, n), + dtype=Mf32.dtype) + listM = [None, Ms64, Ms64precondLO, Mf64precondLO, Ms64precond, + Ms32, Ms32precondLO, Mf32precondLO, Ms32precond] + + # Setup matrix of the initial approximation to the eigenvectors + # (cannot be sparse array). + Xf64 = rnd.random((n, m)) + Xf32 = Xf64.astype(np.float32) + listX = [Xf64, Xf32] + + # Require that the returned eigenvectors be in the orthogonal complement + # of the first few standard basis vectors (cannot be sparse array). + m_excluded = 3 + Yf64 = np.eye(n, m_excluded, dtype=float) + Yf32 = np.eye(n, m_excluded, dtype=np.float32) + listY = [Yf64, Yf32] + + tests = list(itertools.product(listA, listB, listM, listX, listY)) + + for A, B, M, X, Y in tests: + # This is one of the slower tests because there are >1,000 configs + # to test here. Flip a biased coin to decide whether to run each + # test to get decent coverage in less time. + if rnd.random() < 0.98: + continue # too many tests + eigvals, _ = lobpcg(A, X, B=B, M=M, Y=Y, tol=1e-4, + maxiter=100, largest=False) + assert_allclose(eigvals, + np.arange(1 + m_excluded, 1 + m_excluded + m), + atol=1e-5) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/tests/test_svds.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/tests/test_svds.py new file mode 100644 index 0000000000000000000000000000000000000000..7fddf19af26811364ca8551021244e20c6da2239 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_eigen/tests/test_svds.py @@ -0,0 +1,886 @@ +import re +import copy +import numpy as np + +from numpy.testing import assert_allclose, assert_equal, assert_array_equal +import pytest + +from scipy.linalg import svd, null_space +from scipy.sparse import csc_array, issparse, dia_array, random_array +from scipy.sparse.linalg import LinearOperator, aslinearoperator +from scipy.sparse.linalg import svds +from scipy.sparse.linalg._eigen.arpack import ArpackNoConvergence + + +# --- Helper Functions / Classes --- + + +def sorted_svd(m, k, which='LM'): + # Compute svd of a dense matrix m, and return singular vectors/values + # sorted. + if issparse(m): + m = m.toarray() + u, s, vh = svd(m) + if which == 'LM': + ii = np.argsort(s)[-k:] + elif which == 'SM': + ii = np.argsort(s)[:k] + else: + raise ValueError(f"unknown which={which!r}") + + return u[:, ii], s[ii], vh[ii] + + +def _check_svds(A, k, u, s, vh, which="LM", check_usvh_A=False, + check_svd=True, atol=1e-10, rtol=1e-7): + n, m = A.shape + + # Check shapes. + assert_equal(u.shape, (n, k)) + assert_equal(s.shape, (k,)) + assert_equal(vh.shape, (k, m)) + + # Check that the original matrix can be reconstituted. + A_rebuilt = (u*s).dot(vh) + assert_equal(A_rebuilt.shape, A.shape) + if check_usvh_A: + assert_allclose(A_rebuilt, A, atol=atol, rtol=rtol) + + # Check that u is a semi-orthogonal matrix. + uh_u = np.dot(u.T.conj(), u) + assert_equal(uh_u.shape, (k, k)) + assert_allclose(uh_u, np.identity(k), atol=atol, rtol=rtol) + + # Check that vh is a semi-orthogonal matrix. + vh_v = np.dot(vh, vh.T.conj()) + assert_equal(vh_v.shape, (k, k)) + assert_allclose(vh_v, np.identity(k), atol=atol, rtol=rtol) + + # Check that scipy.sparse.linalg.svds ~ scipy.linalg.svd + if check_svd: + u2, s2, vh2 = sorted_svd(A, k, which) + assert_allclose(np.abs(u), np.abs(u2), atol=atol, rtol=rtol) + assert_allclose(s, s2, atol=atol, rtol=rtol) + assert_allclose(np.abs(vh), np.abs(vh2), atol=atol, rtol=rtol) + + +def _check_svds_n(A, k, u, s, vh, which="LM", check_res=True, + check_svd=True, atol=1e-10, rtol=1e-7): + n, m = A.shape + + # Check shapes. + assert_equal(u.shape, (n, k)) + assert_equal(s.shape, (k,)) + assert_equal(vh.shape, (k, m)) + + # Check that u is a semi-orthogonal matrix. + uh_u = np.dot(u.T.conj(), u) + assert_equal(uh_u.shape, (k, k)) + error = np.sum(np.abs(uh_u - np.identity(k))) / (k * k) + assert_allclose(error, 0.0, atol=atol, rtol=rtol) + + # Check that vh is a semi-orthogonal matrix. + vh_v = np.dot(vh, vh.T.conj()) + assert_equal(vh_v.shape, (k, k)) + error = np.sum(np.abs(vh_v - np.identity(k))) / (k * k) + assert_allclose(error, 0.0, atol=atol, rtol=rtol) + + # Check residuals + if check_res: + ru = A.T.conj() @ u - vh.T.conj() * s + rus = np.sum(np.abs(ru)) / (n * k) + rvh = A @ vh.T.conj() - u * s + rvhs = np.sum(np.abs(rvh)) / (m * k) + assert_allclose(rus, 0.0, atol=atol, rtol=rtol) + assert_allclose(rvhs, 0.0, atol=atol, rtol=rtol) + + # Check that scipy.sparse.linalg.svds ~ scipy.linalg.svd + if check_svd: + u2, s2, vh2 = sorted_svd(A, k, which) + assert_allclose(s, s2, atol=atol, rtol=rtol) + A_rebuilt_svd = (u2*s2).dot(vh2) + A_rebuilt = (u*s).dot(vh) + assert_equal(A_rebuilt.shape, A.shape) + error = np.sum(np.abs(A_rebuilt_svd - A_rebuilt)) / (k * k) + assert_allclose(error, 0.0, atol=atol, rtol=rtol) + + +class CheckingLinearOperator(LinearOperator): + def __init__(self, A): + self.A = A + self.dtype = A.dtype + self.shape = A.shape + + def _matvec(self, x): + assert_equal(max(x.shape), np.size(x)) + return self.A.dot(x) + + def _rmatvec(self, x): + assert_equal(max(x.shape), np.size(x)) + return self.A.T.conjugate().dot(x) + + +# --- Test Input Validation --- +# Tests input validation on parameters `k` and `which`. +# Needs better input validation checks for all other parameters. + +class SVDSCommonTests: + + solver = None + + # some of these IV tests could run only once, say with solver=None + + _A_empty_msg = "`A` must not be empty." + _A_dtype_msg = "`A` must be of numeric data type" + _A_type_msg = "type not understood" + _A_ndim_msg = "array must have ndim <= 2" + _A_validation_inputs = [ + (np.asarray([[]]), ValueError, _A_empty_msg), + (np.array([['a', 'b'], ['c', 'd']], dtype='object'), ValueError, _A_dtype_msg), + ("hi", TypeError, _A_type_msg), + (np.asarray([[[1., 2.], [3., 4.]]]), ValueError, _A_ndim_msg)] + + @pytest.mark.parametrize("args", _A_validation_inputs) + def test_svds_input_validation_A(self, args): + A, error_type, message = args + with pytest.raises(error_type, match=message): + svds(A, k=1, solver=self.solver, rng=0) + + @pytest.mark.parametrize("which", ["LM", "SM"]) + def test_svds_int_A(self, which): + A = np.asarray([[1, 2], [3, 4]]) + if self.solver == 'lobpcg': + with pytest.warns(UserWarning, match="The problem size"): + res = svds(A, k=1, which=which, solver=self.solver, rng=0) + else: + res = svds(A, k=1, which=which, solver=self.solver, rng=0) + _check_svds(A, 1, *res, which=which, atol=8e-10) + + def test_svds_diff0_docstring_example(self): + def diff0(a): + return np.diff(a, axis=0) + def diff0t(a): + if a.ndim == 1: + a = a[:,np.newaxis] # Turn 1D into 2D array + d = np.zeros((a.shape[0] + 1, a.shape[1]), dtype=a.dtype) + d[0, :] = - a[0, :] + d[1:-1, :] = a[0:-1, :] - a[1:, :] + d[-1, :] = a[-1, :] + return d + def diff0_func_aslo_def(n): + return LinearOperator(matvec=diff0, + matmat=diff0, + rmatvec=diff0t, + rmatmat=diff0t, + shape=(n - 1, n)) + n = 100 + diff0_func_aslo = diff0_func_aslo_def(n) + # preserve a use of legacy keyword `random_state` during SPEC 7 transition + u, s, _ = svds(diff0_func_aslo, k=3, which='SM', random_state=0) + se = 2. * np.sin(np.pi * np.arange(1, 4) / (2. * n)) + ue = np.sqrt(2 / n) * np.sin(np.pi * np.outer(np.arange(1, n), + np.arange(1, 4)) / n) + assert_allclose(s, se, atol=1e-3) + assert_allclose(np.abs(u), np.abs(ue), atol=1e-6) + + @pytest.mark.parametrize("k", [-1, 0, 3, 4, 5, 1.5, "1"]) + def test_svds_input_validation_k_1(self, k): + rng = np.random.default_rng(0) + A = rng.random((4, 3)) + + # propack can do complete SVD + if self.solver == 'propack' and k == 3: + res = svds(A, k=k, solver=self.solver, rng=0) + _check_svds(A, k, *res, check_usvh_A=True, check_svd=True) + return + + message = ("`k` must be an integer satisfying") + with pytest.raises(ValueError, match=message): + svds(A, k=k, solver=self.solver, rng=0) + + def test_svds_input_validation_k_2(self): + # I think the stack trace is reasonable when `k` can't be converted + # to an int. + message = "int() argument must be a" + with pytest.raises(TypeError, match=re.escape(message)): + svds(np.eye(10), k=[], solver=self.solver, rng=0) + + message = "invalid literal for int()" + with pytest.raises(ValueError, match=message): + svds(np.eye(10), k="hi", solver=self.solver, rng=0) + + @pytest.mark.parametrize("tol", (-1, np.inf, np.nan)) + def test_svds_input_validation_tol_1(self, tol): + message = "`tol` must be a non-negative floating point value." + with pytest.raises(ValueError, match=message): + svds(np.eye(10), tol=tol, solver=self.solver, rng=0) + + @pytest.mark.parametrize("tol", ([], 'hi')) + def test_svds_input_validation_tol_2(self, tol): + # I think the stack trace is reasonable here + message = "'<' not supported between instances" + with pytest.raises(TypeError, match=message): + svds(np.eye(10), tol=tol, solver=self.solver, rng=0) + + @pytest.mark.parametrize("which", ('LA', 'SA', 'ekki', 0)) + def test_svds_input_validation_which(self, which): + # Regression test for a github issue. + # https://github.com/scipy/scipy/issues/4590 + # Function was not checking for eigenvalue type and unintended + # values could be returned. + with pytest.raises(ValueError, match="`which` must be in"): + svds(np.eye(10), which=which, solver=self.solver, rng=0) + + @pytest.mark.parametrize("transpose", (True, False)) + @pytest.mark.parametrize("n", range(4, 9)) + def test_svds_input_validation_v0_1(self, transpose, n): + rng = np.random.default_rng(0) + A = rng.random((5, 7)) + v0 = rng.random(n) + if transpose: + A = A.T + k = 2 + message = "`v0` must have shape" + + required_length = (A.shape[0] if self.solver == 'propack' + else min(A.shape)) + if n != required_length: + with pytest.raises(ValueError, match=message): + svds(A, k=k, v0=v0, solver=self.solver, rng=0) + + def test_svds_input_validation_v0_2(self): + A = np.ones((10, 10)) + v0 = np.ones((1, 10)) + message = "`v0` must have shape" + with pytest.raises(ValueError, match=message): + svds(A, k=1, v0=v0, solver=self.solver, rng=0) + + @pytest.mark.parametrize("v0", ("hi", 1, np.ones(10, dtype=int))) + def test_svds_input_validation_v0_3(self, v0): + A = np.ones((10, 10)) + message = "`v0` must be of floating or complex floating data type." + with pytest.raises(ValueError, match=message): + svds(A, k=1, v0=v0, solver=self.solver, rng=0) + + @pytest.mark.parametrize("maxiter", (-1, 0, 5.5)) + def test_svds_input_validation_maxiter_1(self, maxiter): + message = ("`maxiter` must be a positive integer.") + with pytest.raises(ValueError, match=message): + svds(np.eye(10), maxiter=maxiter, solver=self.solver, rng=0) + + def test_svds_input_validation_maxiter_2(self): + # I think the stack trace is reasonable when `k` can't be converted + # to an int. + message = "int() argument must be a" + with pytest.raises(TypeError, match=re.escape(message)): + svds(np.eye(10), maxiter=[], solver=self.solver, rng=0) + + message = "invalid literal for int()" + with pytest.raises(ValueError, match=message): + svds(np.eye(10), maxiter="hi", solver=self.solver, rng=0) + + @pytest.mark.parametrize("rsv", ('ekki', 10)) + def test_svds_input_validation_return_singular_vectors(self, rsv): + message = "`return_singular_vectors` must be in" + with pytest.raises(ValueError, match=message): + svds(np.eye(10), return_singular_vectors=rsv, solver=self.solver, rng=0) + + # --- Test Parameters --- + @pytest.mark.thread_unsafe + @pytest.mark.parametrize("k", [3, 5]) + @pytest.mark.parametrize("which", ["LM", "SM"]) + def test_svds_parameter_k_which(self, k, which): + # check that the `k` parameter sets the number of eigenvalues/ + # eigenvectors returned. + # Also check that the `which` parameter sets whether the largest or + # smallest eigenvalues are returned + rng = np.random.default_rng(0) + A = rng.random((10, 10)) + if self.solver == 'lobpcg': + with pytest.warns(UserWarning, match="The problem size"): + res = svds(A, k=k, which=which, solver=self.solver, rng=0) + else: + res = svds(A, k=k, which=which, solver=self.solver, rng=0) + _check_svds(A, k, *res, which=which, atol=1e-9, rtol=2e-13) + + @pytest.mark.filterwarnings("ignore:Exited", + reason="Ignore LOBPCG early exit.") + # loop instead of parametrize for simplicity + def test_svds_parameter_tol(self): + # check the effect of the `tol` parameter on solver accuracy by solving + # the same problem with varying `tol` and comparing the eigenvalues + # against ground truth computed + n = 100 # matrix size + k = 3 # number of eigenvalues to check + + # generate a random, sparse-ish matrix + # effect isn't apparent for matrices that are too small + rng = np.random.default_rng(0) + A = rng.random((n, n)) + A[A > .1] = 0 + A = A @ A.T + + _, s, _ = svd(A) # calculate ground truth + + # calculate the error as a function of `tol` + A = csc_array(A) + + def err(tol): + _, s2, _ = svds(A, k=k, v0=np.ones(n), maxiter=1000, + solver=self.solver, tol=tol, rng=0) + return np.linalg.norm((s2 - s[k-1::-1])/s[k-1::-1]) + + tols = [1e-4, 1e-2, 1e0] # tolerance levels to check + # for 'arpack' and 'propack', accuracies make discrete steps + accuracies = {'propack': [1e-12, 1e-6, 1e-4], + 'arpack': [2.5e-15, 1e-10, 1e-10], + 'lobpcg': [2e-12, 4e-2, 2]} + + for tol, accuracy in zip(tols, accuracies[self.solver]): + error = err(tol) + assert error < accuracy + + def test_svd_v0(self): + # check that the `v0` parameter affects the solution + n = 100 + k = 1 + # If k != 1, LOBPCG needs more initial vectors, which are generated + # with rng, so it does not pass w/ k >= 2. + # For some other values of `n`, the AssertionErrors are not raised + # with different v0s, which is reasonable. + + rng = np.random.default_rng(0) + A = rng.random((n, n)) + + # with the same v0, solutions are the same, and they are accurate + # v0 takes precedence over rng + v0a = rng.random(n) + res1a = svds(A, k, v0=v0a, solver=self.solver, rng=0) + res2a = svds(A, k, v0=v0a, solver=self.solver, rng=1) + for idx in range(3): + assert_allclose(res1a[idx], res2a[idx], rtol=1e-15, atol=2e-16) + _check_svds(A, k, *res1a) + + # with the same v0, solutions are the same, and they are accurate + v0b = rng.random(n) + res1b = svds(A, k, v0=v0b, solver=self.solver, rng=2) + res2b = svds(A, k, v0=v0b, solver=self.solver, rng=3) + for idx in range(3): + assert_allclose(res1b[idx], res2b[idx], rtol=1e-15, atol=2e-16) + _check_svds(A, k, *res1b) + + # with different v0, solutions can be numerically different + message = "Arrays are not equal" + with pytest.raises(AssertionError, match=message): + assert_equal(res1a, res1b) + + def test_svd_rng(self): + # check that the `rng` parameter affects the solution + # Admittedly, `n` and `k` are chosen so that all solver pass all + # these checks. That's a tall order, since LOBPCG doesn't want to + # achieve the desired accuracy and ARPACK often returns the same + # singular values/vectors for different v0. + n = 100 + k = 1 + + rng = np.random.default_rng(0) + A = rng.random((n, n)) + + # with the same rng, solutions are the same and accurate + res1a = svds(A, k, solver=self.solver, rng=0) + res2a = svds(A, k, solver=self.solver, rng=0) + for idx in range(3): + assert_allclose(res1a[idx], res2a[idx], rtol=1e-15, atol=2e-16) + _check_svds(A, k, *res1a) + + # with the same rng, solutions are the same and accurate + res1b = svds(A, k, solver=self.solver, rng=1) + res2b = svds(A, k, solver=self.solver, rng=1) + for idx in range(3): + assert_allclose(res1b[idx], res2b[idx], rtol=1e-15, atol=2e-16) + _check_svds(A, k, *res1b) + + # with different rng, solutions can be numerically different + message = "Arrays are not equal" + with pytest.raises(AssertionError, match=message): + assert_equal(res1a, res1b) + + def test_svd_rng_2(self): + n = 100 + k = 1 + + rng = np.random.default_rng(234981) + A = rng.random((n, n)) + rng_2 = copy.deepcopy(rng) + + # with the same rng, solutions are the same and accurate + res1a = svds(A, k, solver=self.solver, rng=rng) + res2a = svds(A, k, solver=self.solver, rng=rng_2) + for idx in range(3): + assert_allclose(res1a[idx], res2a[idx], rtol=1e-15, atol=2e-16) + _check_svds(A, k, *res1a) + + @pytest.mark.filterwarnings("ignore:Exited", + reason="Ignore LOBPCG early exit.") + def test_svd_rng_3(self): + n = 100 + k = 5 + + rng1 = np.random.default_rng(0) + rng2 = np.random.default_rng(234832) + A = rng1.random((n, n)) + + # rng in different state produces accurate - but not + # not necessarily identical - results + res1a = svds(A, k, solver=self.solver, rng=rng1, maxiter=1000) + res2a = svds(A, k, solver=self.solver, rng=rng2, maxiter=1000) + _check_svds(A, k, *res1a, atol=2e-7) + _check_svds(A, k, *res2a, atol=2e-7) + + message = "Arrays are not equal" + with pytest.raises(AssertionError, match=message): + assert_equal(res1a, res2a) + + @pytest.mark.thread_unsafe + @pytest.mark.filterwarnings("ignore:Exited postprocessing") + def test_svd_maxiter(self): + # check that maxiter works as expected: should not return accurate + # solution after 1 iteration, but should with default `maxiter` + A = np.diag(np.arange(9)).astype(np.float64) + k = 1 + u, s, vh = sorted_svd(A, k) + # Use default maxiter by default + maxiter = None + + if self.solver == 'arpack': + message = "ARPACK error -1: No convergence" + with pytest.raises(ArpackNoConvergence, match=message): + svds(A, k, ncv=3, maxiter=1, solver=self.solver, rng=0) + elif self.solver == 'lobpcg': + # Set maxiter higher so test passes without changing + # default and breaking backward compatibility (gh-20221) + maxiter = 30 + with pytest.warns(UserWarning, match="Exited at iteration"): + svds(A, k, maxiter=1, solver=self.solver, rng=0) + elif self.solver == 'propack': + message = "k=1 singular triplets did not converge within" + with pytest.raises(np.linalg.LinAlgError, match=message): + svds(A, k, maxiter=1, solver=self.solver, rng=0) + + ud, sd, vhd = svds(A, k, solver=self.solver, maxiter=maxiter, rng=0) + _check_svds(A, k, ud, sd, vhd, atol=1e-8) + assert_allclose(np.abs(ud), np.abs(u), atol=1e-8) + assert_allclose(np.abs(vhd), np.abs(vh), atol=1e-8) + assert_allclose(np.abs(sd), np.abs(s), atol=1e-9) + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize("rsv", (True, False, 'u', 'vh')) + @pytest.mark.parametrize("shape", ((5, 7), (6, 6), (7, 5))) + def test_svd_return_singular_vectors(self, rsv, shape): + # check that the return_singular_vectors parameter works as expected + rng = np.random.default_rng(0) + A = rng.random(shape) + k = 2 + M, N = shape + u, s, vh = sorted_svd(A, k) + + respect_u = True if self.solver == 'propack' else M <= N + respect_vh = True if self.solver == 'propack' else M > N + + if self.solver == 'lobpcg': + with pytest.warns(UserWarning, match="The problem size"): + if rsv is False: + s2 = svds(A, k, return_singular_vectors=rsv, + solver=self.solver, rng=rng) + assert_allclose(s2, s) + elif rsv == 'u' and respect_u: + u2, s2, vh2 = svds(A, k, return_singular_vectors=rsv, + solver=self.solver, rng=rng) + assert_allclose(np.abs(u2), np.abs(u)) + assert_allclose(s2, s) + assert vh2 is None + elif rsv == 'vh' and respect_vh: + u2, s2, vh2 = svds(A, k, return_singular_vectors=rsv, + solver=self.solver, rng=rng) + assert u2 is None + assert_allclose(s2, s) + assert_allclose(np.abs(vh2), np.abs(vh)) + else: + u2, s2, vh2 = svds(A, k, return_singular_vectors=rsv, + solver=self.solver, rng=rng) + if u2 is not None: + assert_allclose(np.abs(u2), np.abs(u)) + assert_allclose(s2, s) + if vh2 is not None: + assert_allclose(np.abs(vh2), np.abs(vh)) + else: + if rsv is False: + s2 = svds(A, k, return_singular_vectors=rsv, + solver=self.solver, rng=rng) + assert_allclose(s2, s) + elif rsv == 'u' and respect_u: + u2, s2, vh2 = svds(A, k, return_singular_vectors=rsv, + solver=self.solver, rng=rng) + assert_allclose(np.abs(u2), np.abs(u)) + assert_allclose(s2, s) + assert vh2 is None + elif rsv == 'vh' and respect_vh: + u2, s2, vh2 = svds(A, k, return_singular_vectors=rsv, + solver=self.solver, rng=rng) + assert u2 is None + assert_allclose(s2, s) + assert_allclose(np.abs(vh2), np.abs(vh)) + else: + u2, s2, vh2 = svds(A, k, return_singular_vectors=rsv, + solver=self.solver, rng=rng) + if u2 is not None: + assert_allclose(np.abs(u2), np.abs(u)) + assert_allclose(s2, s) + if vh2 is not None: + assert_allclose(np.abs(vh2), np.abs(vh)) + + # --- Test Basic Functionality --- + # Tests the accuracy of each solver for real and complex matrices provided + # as list, dense array, sparse matrix, and LinearOperator. + + A1 = [[1, 2, 3], [3, 4, 3], [1 + 1j, 0, 2], [0, 0, 1]] + A2 = [[1, 2, 3, 8 + 5j], [3 - 2j, 4, 3, 5], [1, 0, 2, 3], [0, 0, 1, 0]] + + @pytest.mark.thread_unsafe + @pytest.mark.filterwarnings("ignore:k >= N - 1", + reason="needed to demonstrate #16725") + @pytest.mark.parametrize('A', (A1, A2)) + @pytest.mark.parametrize('k', range(1, 5)) + # PROPACK fails a lot if @pytest.mark.parametrize('which', ("SM", "LM")) + @pytest.mark.parametrize('real', (True, False)) + @pytest.mark.parametrize('transpose', (False, True)) + # In gh-14299, it was suggested the `svds` should _not_ work with lists + @pytest.mark.parametrize('lo_type', (np.asarray, csc_array, + aslinearoperator)) + def test_svd_simple(self, A, k, real, transpose, lo_type): + + A = np.asarray(A) + A = np.real(A) if real else A + A = A.T if transpose else A + A2 = lo_type(A) + + # could check for the appropriate errors, but that is tested above + if k > min(A.shape): + pytest.skip("`k` cannot be greater than `min(A.shape)`") + if self.solver != 'propack' and k >= min(A.shape): + pytest.skip("Only PROPACK supports complete SVD") + if self.solver == 'arpack' and not real and k == min(A.shape) - 1: + pytest.skip("#16725") + + atol = 3e-10 + if self.solver == 'propack': + atol = 3e-9 # otherwise test fails on Linux aarch64 (see gh-19855) + + if self.solver == 'lobpcg': + with pytest.warns(UserWarning, match="The problem size"): + u, s, vh = svds(A2, k, solver=self.solver, rng=0) + else: + u, s, vh = svds(A2, k, solver=self.solver, rng=0) + _check_svds(A, k, u, s, vh, atol=atol) + + @pytest.mark.thread_unsafe + def test_svd_linop(self): + solver = self.solver + + nmks = [(6, 7, 3), + (9, 5, 4), + (10, 8, 5)] + + def reorder(args): + U, s, VH = args + j = np.argsort(s) + return U[:, j], s[j], VH[j, :] + + for n, m, k in nmks: + # Test svds on a LinearOperator. + A = np.random.RandomState(52).randn(n, m) + L = CheckingLinearOperator(A) + + if solver == 'propack': + v0 = np.ones(n) + else: + v0 = np.ones(min(A.shape)) + if solver == 'lobpcg': + with pytest.warns(UserWarning, match="The problem size"): + U1, s1, VH1 = reorder(svds(A, k, v0=v0, solver=solver, rng=0)) + U2, s2, VH2 = reorder(svds(L, k, v0=v0, solver=solver, rng=0)) + else: + U1, s1, VH1 = reorder(svds(A, k, v0=v0, solver=solver, rng=0)) + U2, s2, VH2 = reorder(svds(L, k, v0=v0, solver=solver, rng=0)) + + assert_allclose(np.abs(U1), np.abs(U2)) + assert_allclose(s1, s2) + assert_allclose(np.abs(VH1), np.abs(VH2)) + assert_allclose(np.dot(U1, np.dot(np.diag(s1), VH1)), + np.dot(U2, np.dot(np.diag(s2), VH2))) + + # Try again with which="SM". + A = np.random.RandomState(1909).randn(n, m) + L = CheckingLinearOperator(A) + + # TODO: arpack crashes when v0=v0, which="SM" + kwargs = {'v0': v0} if solver not in {None, 'arpack'} else {} + if self.solver == 'lobpcg': + with pytest.warns(UserWarning, match="The problem size"): + U1, s1, VH1 = reorder(svds(A, k, which="SM", solver=solver, + rng=0, **kwargs)) + U2, s2, VH2 = reorder(svds(L, k, which="SM", solver=solver, + rng=0, **kwargs)) + else: + U1, s1, VH1 = reorder(svds(A, k, which="SM", solver=solver, + rng=0, **kwargs)) + U2, s2, VH2 = reorder(svds(L, k, which="SM", solver=solver, + rng=0, **kwargs)) + + assert_allclose(np.abs(U1), np.abs(U2)) + assert_allclose(s1 + 1, s2 + 1) + assert_allclose(np.abs(VH1), np.abs(VH2)) + assert_allclose(np.dot(U1, np.dot(np.diag(s1), VH1)), + np.dot(U2, np.dot(np.diag(s2), VH2))) + + if k < min(n, m) - 1: + # Complex input and explicit which="LM". + for (dt, eps) in [(complex, 1e-7), (np.complex64, 3e-3)]: + rng = np.random.RandomState(1648) + A = (rng.randn(n, m) + 1j * rng.randn(n, m)).astype(dt) + L = CheckingLinearOperator(A) + + if self.solver == 'lobpcg': + with pytest.warns(UserWarning, + match="The problem size"): + U1, s1, VH1 = reorder(svds(A, k, which="LM", + solver=solver, rng=0)) + U2, s2, VH2 = reorder(svds(L, k, which="LM", + solver=solver, rng=0)) + else: + U1, s1, VH1 = reorder(svds(A, k, which="LM", + solver=solver, rng=0)) + U2, s2, VH2 = reorder(svds(L, k, which="LM", + solver=solver, rng=0)) + + assert_allclose(np.abs(U1), np.abs(U2), rtol=eps) + assert_allclose(s1, s2, rtol=eps) + assert_allclose(np.abs(VH1), np.abs(VH2), rtol=eps) + assert_allclose(np.dot(U1, np.dot(np.diag(s1), VH1)), + np.dot(U2, np.dot(np.diag(s2), VH2)), + rtol=eps) + + SHAPES = ((100, 100), (100, 101), (101, 100)) + + @pytest.mark.filterwarnings("ignore:Exited at iteration") + @pytest.mark.filterwarnings("ignore:Exited postprocessing") + @pytest.mark.parametrize("shape", SHAPES) + # ARPACK supports only dtype float, complex, or np.float32 + @pytest.mark.parametrize("dtype", (float, complex, np.float32)) + def test_small_sigma_sparse(self, shape, dtype): + # https://github.com/scipy/scipy/pull/11829 + solver = self.solver + # 2do: PROPACK fails orthogonality of singular vectors + # if dtype == complex and self.solver == 'propack': + # pytest.skip("PROPACK unsupported for complex dtype") + rng = np.random.default_rng(0) + k = 5 + (m, n) = shape + S = random_array(shape=(m, n), density=0.1, rng=rng) + if dtype is complex: + S = + 1j * random_array(shape=(m, n), density=0.1, rng=rng) + e = np.ones(m) + e[0:5] *= 1e1 ** np.arange(-5, 0, 1) + S = dia_array((e, 0), shape=(m, m)) @ S + S = S.astype(dtype) + u, s, vh = svds(S, k, which='SM', solver=solver, maxiter=1000, rng=0) + c_svd = False # partial SVD can be different from full SVD + _check_svds_n(S, k, u, s, vh, which="SM", check_svd=c_svd, atol=2e-1) + + # --- Test Edge Cases --- + # Checks a few edge cases. + @pytest.mark.thread_unsafe + @pytest.mark.parametrize("shape", ((6, 5), (5, 5), (5, 6))) + @pytest.mark.parametrize("dtype", (float, complex)) + def test_svd_LM_ones_matrix(self, shape, dtype): + # Check that svds can deal with matrix_rank less than k in LM mode. + k = 3 + n, m = shape + A = np.ones((n, m), dtype=dtype) + + if self.solver == 'lobpcg': + with pytest.warns(UserWarning, match="The problem size"): + U, s, VH = svds(A, k, solver=self.solver, rng=0) + else: + U, s, VH = svds(A, k, solver=self.solver, rng=0) + + _check_svds(A, k, U, s, VH, check_usvh_A=True, check_svd=False) + + # Check that the largest singular value is near sqrt(n*m) + # and the other singular values have been forced to zero. + assert_allclose(np.max(s), np.sqrt(n*m)) + s = np.array(sorted(s)[:-1]) + 1 + z = np.ones_like(s) + assert_allclose(s, z) + + @pytest.mark.thread_unsafe + @pytest.mark.filterwarnings("ignore:k >= N - 1", + reason="needed to demonstrate #16725") + @pytest.mark.parametrize("shape", ((3, 4), (4, 4), (4, 3), (4, 2))) + @pytest.mark.parametrize("dtype", (float, complex)) + def test_zero_matrix(self, shape, dtype): + # Check that svds can deal with matrices containing only zeros; + # see https://github.com/scipy/scipy/issues/3452/ + # shape = (4, 2) is included because it is the particular case + # reported in the issue + k = 1 + n, m = shape + A = np.zeros((n, m), dtype=dtype) + + if (self.solver == 'arpack'): + pytest.skip('See gh-21110.') + + if (self.solver == 'arpack' and dtype is complex + and k == min(A.shape) - 1): + pytest.skip("#16725") + + if self.solver == 'propack': + pytest.skip("PROPACK failures unrelated to PR #16712") + + if self.solver == 'lobpcg': + with pytest.warns(UserWarning, match="The problem size"): + U, s, VH = svds(A, k, solver=self.solver, rng=0) + else: + U, s, VH = svds(A, k, solver=self.solver, rng=0) + + # Check some generic properties of svd. + _check_svds(A, k, U, s, VH, check_usvh_A=True, check_svd=False) + + # Check that the singular values are zero. + assert_array_equal(s, 0) + + @pytest.mark.parametrize("shape", ((20, 20), (20, 21), (21, 20))) + # ARPACK supports only dtype float, complex, or np.float32 + @pytest.mark.parametrize("dtype", (float, complex, np.float32)) + @pytest.mark.filterwarnings("ignore:Exited", + reason="Ignore LOBPCG early exit.") + def test_small_sigma(self, shape, dtype): + rng = np.random.default_rng(179847540) + A = rng.random(shape).astype(dtype) + u, _, vh = svd(A, full_matrices=False) + if dtype == np.float32: + e = 10.0 + else: + e = 100.0 + t = e**(-np.arange(len(vh))).astype(dtype) + A = (u*t).dot(vh) + k = 4 + u, s, vh = svds(A, k, solver=self.solver, maxiter=100, rng=0) + t = np.sum(s > 0) + assert_equal(t, k) + # LOBPCG needs larger atol and rtol to pass + _check_svds_n(A, k, u, s, vh, atol=1e-3, rtol=1e0, check_svd=False) + + # ARPACK supports only dtype float, complex, or np.float32 + @pytest.mark.filterwarnings("ignore:The problem size") + @pytest.mark.parametrize("dtype", (float, complex, np.float32)) + def test_small_sigma2(self, dtype): + rng = np.random.default_rng(179847540) + # create a 10x10 singular matrix with a 4-dim null space + dim = 4 + size = 10 + x = rng.random((size, size-dim)) + y = x[:, :dim] * rng.random(dim) + mat = np.hstack((x, y)) + mat = mat.astype(dtype) + + nz = null_space(mat) + assert_equal(nz.shape[1], dim) + + # Tolerances atol and rtol adjusted to pass np.float32 + # Use non-sparse svd + u, s, vh = svd(mat) + # Singular values are 0: + assert_allclose(s[-dim:], 0, atol=1e-6, rtol=1e0) + # Smallest right singular vectors in null space: + assert_allclose(mat @ vh[-dim:, :].T, 0, atol=1e-6, rtol=1e0) + + # Smallest singular values should be 0 + sp_mat = csc_array(mat) + su, ss, svh = svds(sp_mat, k=dim, which='SM', solver=self.solver, rng=0) + # Smallest dim singular values are 0: + assert_allclose(ss, 0, atol=1e-5, rtol=1e0) + # Smallest singular vectors via svds in null space: + n, m = mat.shape + if n < m: # else the assert fails with some libraries unclear why + assert_allclose(sp_mat.transpose() @ su, 0, atol=1e-5, rtol=1e0) + assert_allclose(sp_mat @ svh.T, 0, atol=1e-5, rtol=1e0) + +# --- Perform tests with each solver --- + + +class Test_SVDS_once: + @pytest.mark.parametrize("solver", ['ekki', object]) + def test_svds_input_validation_solver(self, solver): + message = "solver must be one of" + with pytest.raises(ValueError, match=message): + svds(np.ones((3, 4)), k=2, solver=solver, rng=0) + + +class Test_SVDS_ARPACK(SVDSCommonTests): + + def setup_method(self): + self.solver = 'arpack' + + @pytest.mark.parametrize("ncv", list(range(-1, 8)) + [4.5, "5"]) + def test_svds_input_validation_ncv_1(self, ncv): + rng = np.random.default_rng(0) + A = rng.random((6, 7)) + k = 3 + if ncv in {4, 5}: + u, s, vh = svds(A, k=k, ncv=ncv, solver=self.solver, rng=0) + # partial decomposition, so don't check that u@diag(s)@vh=A; + # do check that scipy.sparse.linalg.svds ~ scipy.linalg.svd + _check_svds(A, k, u, s, vh) + else: + message = ("`ncv` must be an integer satisfying") + with pytest.raises(ValueError, match=message): + svds(A, k=k, ncv=ncv, solver=self.solver, rng=0) + + def test_svds_input_validation_ncv_2(self): + # I think the stack trace is reasonable when `ncv` can't be converted + # to an int. + message = "int() argument must be a" + with pytest.raises(TypeError, match=re.escape(message)): + svds(np.eye(10), ncv=[], solver=self.solver, rng=0) + + message = "invalid literal for int()" + with pytest.raises(ValueError, match=message): + svds(np.eye(10), ncv="hi", solver=self.solver, rng=0) + + # I can't see a robust relationship between `ncv` and relevant outputs + # (e.g. accuracy, time), so no test of the parameter. + + +class Test_SVDS_LOBPCG(SVDSCommonTests): + + def setup_method(self): + self.solver = 'lobpcg' + + +class Test_SVDS_PROPACK(SVDSCommonTests): + + def setup_method(self): + self.solver = 'propack' + + def test_svd_LM_ones_matrix(self): + message = ("PROPACK does not return orthonormal singular vectors " + "associated with zero singular values.") + # There are some other issues with this matrix of all ones, e.g. + # `which='sm'` and `k=1` returns the largest singular value + pytest.xfail(message) + + def test_svd_LM_zeros_matrix(self): + message = ("PROPACK does not return orthonormal singular vectors " + "associated with zero singular values.") + pytest.xfail(message) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_expm_multiply.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_expm_multiply.py new file mode 100644 index 0000000000000000000000000000000000000000..8f5a7a8508a6a9d102a836469d8fb76ddc9534b4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_expm_multiply.py @@ -0,0 +1,816 @@ +"""Compute the action of the matrix exponential.""" +from warnings import warn + +import numpy as np + +import scipy.linalg +import scipy.sparse.linalg +from scipy.linalg._decomp_qr import qr +from scipy.sparse._sputils import is_pydata_spmatrix +from scipy.sparse.linalg import aslinearoperator +from scipy.sparse.linalg._interface import IdentityOperator +from scipy.sparse.linalg._onenormest import onenormest + +__all__ = ['expm_multiply'] + + +def _exact_inf_norm(A): + # A compatibility function which should eventually disappear. + if scipy.sparse.issparse(A): + return max(abs(A).sum(axis=1).flat) + elif is_pydata_spmatrix(A): + return max(abs(A).sum(axis=1)) + else: + return np.linalg.norm(A, np.inf) + + +def _exact_1_norm(A): + # A compatibility function which should eventually disappear. + if scipy.sparse.issparse(A): + return max(abs(A).sum(axis=0).flat) + elif is_pydata_spmatrix(A): + return max(abs(A).sum(axis=0)) + else: + return np.linalg.norm(A, 1) + + +def _trace(A): + # A compatibility function which should eventually disappear. + if is_pydata_spmatrix(A): + return A.to_scipy_sparse().trace() + else: + return A.trace() + + +def traceest(A, m3, seed=None): + """Estimate `np.trace(A)` using `3*m3` matrix-vector products. + + The result is not deterministic. + + Parameters + ---------- + A : LinearOperator + Linear operator whose trace will be estimated. Has to be square. + m3 : int + Number of matrix-vector products divided by 3 used to estimate the + trace. + seed : optional + Seed for `numpy.random.default_rng`. + Can be provided to obtain deterministic results. + + Returns + ------- + trace : LinearOperator.dtype + Estimate of the trace + + Notes + ----- + This is the Hutch++ algorithm given in [1]_. + + References + ---------- + .. [1] Meyer, Raphael A., Cameron Musco, Christopher Musco, and David P. + Woodruff. "Hutch++: Optimal Stochastic Trace Estimation." In Symposium + on Simplicity in Algorithms (SOSA), pp. 142-155. Society for Industrial + and Applied Mathematics, 2021 + https://doi.org/10.1137/1.9781611976496.16 + + """ + rng = np.random.default_rng(seed) + if len(A.shape) != 2 or A.shape[-1] != A.shape[-2]: + raise ValueError("Expected A to be like a square matrix.") + n = A.shape[-1] + S = rng.choice([-1.0, +1.0], [n, m3]) + Q, _ = qr(A.matmat(S), overwrite_a=True, mode='economic') + trQAQ = np.trace(Q.conj().T @ A.matmat(Q)) + G = rng.choice([-1, +1], [n, m3]) + right = G - Q@(Q.conj().T @ G) + trGAG = np.trace(right.conj().T @ A.matmat(right)) + return trQAQ + trGAG/m3 + + +def _ident_like(A): + # A compatibility function which should eventually disappear. + if scipy.sparse.issparse(A): + # Creates a sparse matrix in dia format + out = scipy.sparse.eye(A.shape[0], A.shape[1], dtype=A.dtype) + if scipy.sparse.issparse(A): + return out.asformat(A.format) + return scipy.sparse.dia_array(out).asformat(A.format) + elif is_pydata_spmatrix(A): + import sparse + return sparse.eye(A.shape[0], A.shape[1], dtype=A.dtype) + elif isinstance(A, scipy.sparse.linalg.LinearOperator): + return IdentityOperator(A.shape, dtype=A.dtype) + else: + return np.eye(A.shape[0], A.shape[1], dtype=A.dtype) + + +def expm_multiply(A, B, start=None, stop=None, num=None, + endpoint=None, traceA=None): + """ + Compute the action of the matrix exponential of A on B. + + Parameters + ---------- + A : transposable linear operator + The operator whose exponential is of interest. + B : ndarray, sparse array + The matrix or vector to be multiplied by the matrix exponential of A. + start : scalar, optional + The starting time point of the sequence. + stop : scalar, optional + The end time point of the sequence, unless `endpoint` is set to False. + In that case, the sequence consists of all but the last of ``num + 1`` + evenly spaced time points, so that `stop` is excluded. + Note that the step size changes when `endpoint` is False. + num : int, optional + Number of time points to use. + endpoint : bool, optional + If True, `stop` is the last time point. Otherwise, it is not included. + traceA : scalar, optional + Trace of `A`. If not given the trace is estimated for linear operators, + or calculated exactly for sparse matrices. It is used to precondition + `A`, thus an approximate trace is acceptable. + For linear operators, `traceA` should be provided to ensure performance + as the estimation is not guaranteed to be reliable for all cases. + + .. versionadded:: 1.9.0 + + Returns + ------- + expm_A_B : ndarray + The result of the action :math:`e^{t_k A} B`. + + Warns + ----- + UserWarning + If `A` is a linear operator and ``traceA=None`` (default). + + Notes + ----- + The optional arguments defining the sequence of evenly spaced time points + are compatible with the arguments of `numpy.linspace`. + + The output ndarray shape is somewhat complicated so I explain it here. + The ndim of the output could be either 1, 2, or 3. + It would be 1 if you are computing the expm action on a single vector + at a single time point. + It would be 2 if you are computing the expm action on a vector + at multiple time points, or if you are computing the expm action + on a matrix at a single time point. + It would be 3 if you want the action on a matrix with multiple + columns at multiple time points. + If multiple time points are requested, expm_A_B[0] will always + be the action of the expm at the first time point, + regardless of whether the action is on a vector or a matrix. + + References + ---------- + .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2011) + "Computing the Action of the Matrix Exponential, + with an Application to Exponential Integrators." + SIAM Journal on Scientific Computing, + 33 (2). pp. 488-511. ISSN 1064-8275 + http://eprints.ma.man.ac.uk/1591/ + + .. [2] Nicholas J. Higham and Awad H. Al-Mohy (2010) + "Computing Matrix Functions." + Acta Numerica, + 19. 159-208. ISSN 0962-4929 + http://eprints.ma.man.ac.uk/1451/ + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import expm, expm_multiply + >>> A = csc_array([[1, 0], [0, 1]]) + >>> A.toarray() + array([[1, 0], + [0, 1]], dtype=int64) + >>> B = np.array([np.exp(-1.), np.exp(-2.)]) + >>> B + array([ 0.36787944, 0.13533528]) + >>> expm_multiply(A, B, start=1, stop=2, num=3, endpoint=True) + array([[ 1. , 0.36787944], + [ 1.64872127, 0.60653066], + [ 2.71828183, 1. ]]) + >>> expm(A).dot(B) # Verify 1st timestep + array([ 1. , 0.36787944]) + >>> expm(1.5*A).dot(B) # Verify 2nd timestep + array([ 1.64872127, 0.60653066]) + >>> expm(2*A).dot(B) # Verify 3rd timestep + array([ 2.71828183, 1. ]) + """ + if all(arg is None for arg in (start, stop, num, endpoint)): + X = _expm_multiply_simple(A, B, traceA=traceA) + else: + X, status = _expm_multiply_interval(A, B, start, stop, num, + endpoint, traceA=traceA) + return X + + +def _expm_multiply_simple(A, B, t=1.0, traceA=None, balance=False): + """ + Compute the action of the matrix exponential at a single time point. + + Parameters + ---------- + A : transposable linear operator + The operator whose exponential is of interest. + B : ndarray + The matrix to be multiplied by the matrix exponential of A. + t : float + A time point. + traceA : scalar, optional + Trace of `A`. If not given the trace is estimated for linear operators, + or calculated exactly for sparse matrices. It is used to precondition + `A`, thus an approximate trace is acceptable + balance : bool + Indicates whether or not to apply balancing. + + Returns + ------- + F : ndarray + :math:`e^{t A} B` + + Notes + ----- + This is algorithm (3.2) in Al-Mohy and Higham (2011). + + """ + if balance: + raise NotImplementedError + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected A to be like a square matrix') + if A.shape[1] != B.shape[0]: + raise ValueError(f'shapes of matrices A {A.shape} and B {B.shape}' + ' are incompatible') + ident = _ident_like(A) + is_linear_operator = isinstance(A, scipy.sparse.linalg.LinearOperator) + n = A.shape[0] + if len(B.shape) == 1: + n0 = 1 + elif len(B.shape) == 2: + n0 = B.shape[1] + else: + raise ValueError('expected B to be like a matrix or a vector') + u_d = 2**-53 + tol = u_d + if traceA is None: + if is_linear_operator: + warn("Trace of LinearOperator not available, it will be estimated." + " Provide `traceA` to ensure performance.", stacklevel=3) + # m3=1 is bit arbitrary choice, a more accurate trace (larger m3) might + # speed up exponential calculation, but trace estimation is more costly + traceA = traceest(A, m3=1) if is_linear_operator else _trace(A) + mu = traceA / float(n) + A = A - mu * ident + A_1_norm = onenormest(A) if is_linear_operator else _exact_1_norm(A) + if t*A_1_norm == 0: + m_star, s = 0, 1 + else: + ell = 2 + norm_info = LazyOperatorNormInfo(t*A, A_1_norm=t*A_1_norm, ell=ell) + m_star, s = _fragment_3_1(norm_info, n0, tol, ell=ell) + return _expm_multiply_simple_core(A, B, t, mu, m_star, s, tol, balance) + + +def _expm_multiply_simple_core(A, B, t, mu, m_star, s, tol=None, balance=False): + """ + A helper function. + """ + if balance: + raise NotImplementedError + if tol is None: + u_d = 2 ** -53 + tol = u_d + F = B + eta = np.exp(t*mu / float(s)) + for i in range(s): + c1 = _exact_inf_norm(B) + for j in range(m_star): + coeff = t / float(s*(j+1)) + B = coeff * A.dot(B) + c2 = _exact_inf_norm(B) + F = F + B + if c1 + c2 <= tol * _exact_inf_norm(F): + break + c1 = c2 + F = eta * F + B = F + return F + + +# This table helps to compute bounds. +# They seem to have been difficult to calculate, involving symbolic +# manipulation of equations, followed by numerical root finding. +_theta = { + # The first 30 values are from table A.3 of Computing Matrix Functions. + 1: 2.29e-16, + 2: 2.58e-8, + 3: 1.39e-5, + 4: 3.40e-4, + 5: 2.40e-3, + 6: 9.07e-3, + 7: 2.38e-2, + 8: 5.00e-2, + 9: 8.96e-2, + 10: 1.44e-1, + # 11 + 11: 2.14e-1, + 12: 3.00e-1, + 13: 4.00e-1, + 14: 5.14e-1, + 15: 6.41e-1, + 16: 7.81e-1, + 17: 9.31e-1, + 18: 1.09, + 19: 1.26, + 20: 1.44, + # 21 + 21: 1.62, + 22: 1.82, + 23: 2.01, + 24: 2.22, + 25: 2.43, + 26: 2.64, + 27: 2.86, + 28: 3.08, + 29: 3.31, + 30: 3.54, + # The rest are from table 3.1 of + # Computing the Action of the Matrix Exponential. + 35: 4.7, + 40: 6.0, + 45: 7.2, + 50: 8.5, + 55: 9.9, + } + + +def _onenormest_matrix_power(A, p, + t=2, itmax=5, compute_v=False, compute_w=False): + """ + Efficiently estimate the 1-norm of A^p. + + Parameters + ---------- + A : ndarray + Matrix whose 1-norm of a power is to be computed. + p : int + Non-negative integer power. + t : int, optional + A positive parameter controlling the tradeoff between + accuracy versus time and memory usage. + Larger values take longer and use more memory + but give more accurate output. + itmax : int, optional + Use at most this many iterations. + compute_v : bool, optional + Request a norm-maximizing linear operator input vector if True. + compute_w : bool, optional + Request a norm-maximizing linear operator output vector if True. + + Returns + ------- + est : float + An underestimate of the 1-norm of the sparse matrix. + v : ndarray, optional + The vector such that ||Av||_1 == est*||v||_1. + It can be thought of as an input to the linear operator + that gives an output with particularly large norm. + w : ndarray, optional + The vector Av which has relatively large 1-norm. + It can be thought of as an output of the linear operator + that is relatively large in norm compared to the input. + + """ + #XXX Eventually turn this into an API function in the _onenormest module, + #XXX and remove its underscore, + #XXX but wait until expm_multiply goes into scipy. + from scipy.sparse.linalg._onenormest import onenormest + return onenormest(aslinearoperator(A) ** p) + +class LazyOperatorNormInfo: + """ + Information about an operator is lazily computed. + + The information includes the exact 1-norm of the operator, + in addition to estimates of 1-norms of powers of the operator. + This uses the notation of Computing the Action (2011). + This class is specialized enough to probably not be of general interest + outside of this module. + + """ + + def __init__(self, A, A_1_norm=None, ell=2, scale=1): + """ + Provide the operator and some norm-related information. + + Parameters + ---------- + A : linear operator + The operator of interest. + A_1_norm : float, optional + The exact 1-norm of A. + ell : int, optional + A technical parameter controlling norm estimation quality. + scale : int, optional + If specified, return the norms of scale*A instead of A. + + """ + self._A = A + self._A_1_norm = A_1_norm + self._ell = ell + self._d = {} + self._scale = scale + + def set_scale(self,scale): + """ + Set the scale parameter. + """ + self._scale = scale + + def onenorm(self): + """ + Compute the exact 1-norm. + """ + if self._A_1_norm is None: + self._A_1_norm = _exact_1_norm(self._A) + return self._scale*self._A_1_norm + + def d(self, p): + """ + Lazily estimate :math:`d_p(A) ~= || A^p ||^(1/p)` + where :math:`||.||` is the 1-norm. + """ + if p not in self._d: + est = _onenormest_matrix_power(self._A, p, self._ell) + self._d[p] = est ** (1.0 / p) + return self._scale*self._d[p] + + def alpha(self, p): + """ + Lazily compute max(d(p), d(p+1)). + """ + return max(self.d(p), self.d(p+1)) + +def _compute_cost_div_m(m, p, norm_info): + """ + A helper function for computing bounds. + + This is equation (3.10). + It measures cost in terms of the number of required matrix products. + + Parameters + ---------- + m : int + A valid key of _theta. + p : int + A matrix power. + norm_info : LazyOperatorNormInfo + Information about 1-norms of related operators. + + Returns + ------- + cost_div_m : int + Required number of matrix products divided by m. + + """ + return int(np.ceil(norm_info.alpha(p) / _theta[m])) + + +def _compute_p_max(m_max): + """ + Compute the largest positive integer p such that p*(p-1) <= m_max + 1. + + Do this in a slightly dumb way, but safe and not too slow. + + Parameters + ---------- + m_max : int + A count related to bounds. + + """ + sqrt_m_max = np.sqrt(m_max) + p_low = int(np.floor(sqrt_m_max)) + p_high = int(np.ceil(sqrt_m_max + 1)) + return max(p for p in range(p_low, p_high+1) if p*(p-1) <= m_max + 1) + + +def _fragment_3_1(norm_info, n0, tol, m_max=55, ell=2): + """ + A helper function for the _expm_multiply_* functions. + + Parameters + ---------- + norm_info : LazyOperatorNormInfo + Information about norms of certain linear operators of interest. + n0 : int + Number of columns in the _expm_multiply_* B matrix. + tol : float + Expected to be + :math:`2^{-24}` for single precision or + :math:`2^{-53}` for double precision. + m_max : int + A value related to a bound. + ell : int + The number of columns used in the 1-norm approximation. + This is usually taken to be small, maybe between 1 and 5. + + Returns + ------- + best_m : int + Related to bounds for error control. + best_s : int + Amount of scaling. + + Notes + ----- + This is code fragment (3.1) in Al-Mohy and Higham (2011). + The discussion of default values for m_max and ell + is given between the definitions of equation (3.11) + and the definition of equation (3.12). + + """ + if ell < 1: + raise ValueError('expected ell to be a positive integer') + best_m = None + best_s = None + if _condition_3_13(norm_info.onenorm(), n0, m_max, ell): + for m, theta in _theta.items(): + s = int(np.ceil(norm_info.onenorm() / theta)) + if best_m is None or m * s < best_m * best_s: + best_m = m + best_s = s + else: + # Equation (3.11). + for p in range(2, _compute_p_max(m_max) + 1): + for m in range(p*(p-1)-1, m_max+1): + if m in _theta: + s = _compute_cost_div_m(m, p, norm_info) + if best_m is None or m * s < best_m * best_s: + best_m = m + best_s = s + best_s = max(best_s, 1) + return best_m, best_s + + +def _condition_3_13(A_1_norm, n0, m_max, ell): + """ + A helper function for the _expm_multiply_* functions. + + Parameters + ---------- + A_1_norm : float + The precomputed 1-norm of A. + n0 : int + Number of columns in the _expm_multiply_* B matrix. + m_max : int + A value related to a bound. + ell : int + The number of columns used in the 1-norm approximation. + This is usually taken to be small, maybe between 1 and 5. + + Returns + ------- + value : bool + Indicates whether or not the condition has been met. + + Notes + ----- + This is condition (3.13) in Al-Mohy and Higham (2011). + + """ + + # This is the rhs of equation (3.12). + p_max = _compute_p_max(m_max) + a = 2 * ell * p_max * (p_max + 3) + + # Evaluate the condition (3.13). + b = _theta[m_max] / float(n0 * m_max) + return A_1_norm <= a * b + + +def _expm_multiply_interval(A, B, start=None, stop=None, num=None, + endpoint=None, traceA=None, balance=False, + status_only=False): + """ + Compute the action of the matrix exponential at multiple time points. + + Parameters + ---------- + A : transposable linear operator + The operator whose exponential is of interest. + B : ndarray + The matrix to be multiplied by the matrix exponential of A. + start : scalar, optional + The starting time point of the sequence. + stop : scalar, optional + The end time point of the sequence, unless `endpoint` is set to False. + In that case, the sequence consists of all but the last of ``num + 1`` + evenly spaced time points, so that `stop` is excluded. + Note that the step size changes when `endpoint` is False. + num : int, optional + Number of time points to use. + traceA : scalar, optional + Trace of `A`. If not given the trace is estimated for linear operators, + or calculated exactly for sparse matrices. It is used to precondition + `A`, thus an approximate trace is acceptable + endpoint : bool, optional + If True, `stop` is the last time point. Otherwise, it is not included. + balance : bool + Indicates whether or not to apply balancing. + status_only : bool + A flag that is set to True for some debugging and testing operations. + + Returns + ------- + F : ndarray + :math:`e^{t_k A} B` + status : int + An integer status for testing and debugging. + + Notes + ----- + This is algorithm (5.2) in Al-Mohy and Higham (2011). + + There seems to be a typo, where line 15 of the algorithm should be + moved to line 6.5 (between lines 6 and 7). + + """ + if balance: + raise NotImplementedError + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected A to be like a square matrix') + if A.shape[1] != B.shape[0]: + raise ValueError(f'shapes of matrices A {A.shape} and B {B.shape}' + ' are incompatible') + ident = _ident_like(A) + is_linear_operator = isinstance(A, scipy.sparse.linalg.LinearOperator) + n = A.shape[0] + if len(B.shape) == 1: + n0 = 1 + elif len(B.shape) == 2: + n0 = B.shape[1] + else: + raise ValueError('expected B to be like a matrix or a vector') + u_d = 2**-53 + tol = u_d + if traceA is None: + if is_linear_operator: + warn("Trace of LinearOperator not available, it will be estimated." + " Provide `traceA` to ensure performance.", stacklevel=3) + # m3=5 is bit arbitrary choice, a more accurate trace (larger m3) might + # speed up exponential calculation, but trace estimation is also costly + # an educated guess would need to consider the number of time points + traceA = traceest(A, m3=5) if is_linear_operator else _trace(A) + mu = traceA / float(n) + + # Get the linspace samples, attempting to preserve the linspace defaults. + linspace_kwargs = {'retstep': True} + if num is not None: + linspace_kwargs['num'] = num + if endpoint is not None: + linspace_kwargs['endpoint'] = endpoint + samples, step = np.linspace(start, stop, **linspace_kwargs) + + # Convert the linspace output to the notation used by the publication. + nsamples = len(samples) + if nsamples < 2: + raise ValueError('at least two time points are required') + q = nsamples - 1 + h = step + t_0 = samples[0] + t_q = samples[q] + + # Define the output ndarray. + # Use an ndim=3 shape, such that the last two indices + # are the ones that may be involved in level 3 BLAS operations. + X_shape = (nsamples,) + B.shape + X = np.empty(X_shape, dtype=np.result_type(A.dtype, B.dtype, float)) + t = t_q - t_0 + A = A - mu * ident + A_1_norm = onenormest(A) if is_linear_operator else _exact_1_norm(A) + ell = 2 + norm_info = LazyOperatorNormInfo(t*A, A_1_norm=t*A_1_norm, ell=ell) + if t*A_1_norm == 0: + m_star, s = 0, 1 + else: + m_star, s = _fragment_3_1(norm_info, n0, tol, ell=ell) + + # Compute the expm action up to the initial time point. + action_t0 = _expm_multiply_simple_core(A, B, t_0, mu, m_star, s) + if scipy.sparse.issparse(action_t0): + action_t0 = action_t0.toarray() + elif is_pydata_spmatrix(action_t0): + action_t0 = action_t0.todense() + X[0] = action_t0 + + # Compute the expm action at the rest of the time points. + if q <= s: + if status_only: + return 0 + else: + return _expm_multiply_interval_core_0(A, X, + h, mu, q, norm_info, tol, ell,n0) + elif not (q % s): + if status_only: + return 1 + else: + return _expm_multiply_interval_core_1(A, X, + h, mu, m_star, s, q, tol) + elif (q % s): + if status_only: + return 2 + else: + return _expm_multiply_interval_core_2(A, X, + h, mu, m_star, s, q, tol) + else: + raise Exception('internal error') + + +def _expm_multiply_interval_core_0(A, X, h, mu, q, norm_info, tol, ell, n0): + """ + A helper function, for the case q <= s. + """ + + # Compute the new values of m_star and s which should be applied + # over intervals of size t/q + if norm_info.onenorm() == 0: + m_star, s = 0, 1 + else: + norm_info.set_scale(1./q) + m_star, s = _fragment_3_1(norm_info, n0, tol, ell=ell) + norm_info.set_scale(1) + + for k in range(q): + X[k+1] = _expm_multiply_simple_core(A, X[k], h, mu, m_star, s) + return X, 0 + + +def _expm_multiply_interval_core_1(A, X, h, mu, m_star, s, q, tol): + """ + A helper function, for the case q > s and q % s == 0. + """ + d = q // s + input_shape = X.shape[1:] + K_shape = (m_star + 1, ) + input_shape + K = np.empty(K_shape, dtype=X.dtype) + for i in range(s): + Z = X[i*d] + K[0] = Z + high_p = 0 + for k in range(1, d+1): + F = K[0] + c1 = _exact_inf_norm(F) + for p in range(1, m_star+1): + if p > high_p: + K[p] = h * A.dot(K[p-1]) / float(p) + coeff = float(pow(k, p)) + F = F + coeff * K[p] + inf_norm_K_p_1 = _exact_inf_norm(K[p]) + c2 = coeff * inf_norm_K_p_1 + if c1 + c2 <= tol * _exact_inf_norm(F): + break + c1 = c2 + X[k + i*d] = np.exp(k*h*mu) * F + return X, 1 + + +def _expm_multiply_interval_core_2(A, X, h, mu, m_star, s, q, tol): + """ + A helper function, for the case q > s and q % s > 0. + """ + d = q // s + j = q // d + r = q - d * j + input_shape = X.shape[1:] + K_shape = (m_star + 1, ) + input_shape + K = np.empty(K_shape, dtype=X.dtype) + for i in range(j + 1): + Z = X[i*d] + K[0] = Z + high_p = 0 + if i < j: + effective_d = d + else: + effective_d = r + for k in range(1, effective_d+1): + F = K[0] + c1 = _exact_inf_norm(F) + for p in range(1, m_star+1): + if p == high_p + 1: + K[p] = h * A.dot(K[p-1]) / float(p) + high_p = p + coeff = float(pow(k, p)) + F = F + coeff * K[p] + inf_norm_K_p_1 = _exact_inf_norm(K[p]) + c2 = coeff * inf_norm_K_p_1 + if c1 + c2 <= tol * _exact_inf_norm(F): + break + c1 = c2 + X[k + i*d] = np.exp(k*h*mu) * F + return X, 2 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_interface.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_interface.py new file mode 100644 index 0000000000000000000000000000000000000000..510c4ff254eb561fd93da5ca3bda66c2dc92770f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_interface.py @@ -0,0 +1,921 @@ +"""Abstract linear algebra library. + +This module defines a class hierarchy that implements a kind of "lazy" +matrix representation, called the ``LinearOperator``. It can be used to do +linear algebra with extremely large sparse or structured matrices, without +representing those explicitly in memory. Such matrices can be added, +multiplied, transposed, etc. + +As a motivating example, suppose you want have a matrix where almost all of +the elements have the value one. The standard sparse matrix representation +skips the storage of zeros, but not ones. By contrast, a LinearOperator is +able to represent such matrices efficiently. First, we need a compact way to +represent an all-ones matrix:: + + >>> import numpy as np + >>> from scipy.sparse.linalg._interface import LinearOperator + >>> class Ones(LinearOperator): + ... def __init__(self, shape): + ... super().__init__(dtype=None, shape=shape) + ... def _matvec(self, x): + ... return np.repeat(x.sum(), self.shape[0]) + +Instances of this class emulate ``np.ones(shape)``, but using a constant +amount of storage, independent of ``shape``. The ``_matvec`` method specifies +how this linear operator multiplies with (operates on) a vector. We can now +add this operator to a sparse matrix that stores only offsets from one:: + + >>> from scipy.sparse.linalg._interface import aslinearoperator + >>> from scipy.sparse import csr_array + >>> offsets = csr_array([[1, 0, 2], [0, -1, 0], [0, 0, 3]]) + >>> A = aslinearoperator(offsets) + Ones(offsets.shape) + >>> A.dot([1, 2, 3]) + array([13, 4, 15]) + +The result is the same as that given by its dense, explicitly-stored +counterpart:: + + >>> (np.ones(A.shape, A.dtype) + offsets.toarray()).dot([1, 2, 3]) + array([13, 4, 15]) + +Several algorithms in the ``scipy.sparse`` library are able to operate on +``LinearOperator`` instances. +""" + +import warnings + +import numpy as np + +from scipy.sparse import issparse +from scipy.sparse._sputils import isshape, isintlike, asmatrix, is_pydata_spmatrix + +__all__ = ['LinearOperator', 'aslinearoperator'] + + +class LinearOperator: + """Common interface for performing matrix vector products + + Many iterative methods (e.g. cg, gmres) do not need to know the + individual entries of a matrix to solve a linear system A@x=b. + Such solvers only require the computation of matrix vector + products, A@v where v is a dense vector. This class serves as + an abstract interface between iterative solvers and matrix-like + objects. + + To construct a concrete LinearOperator, either pass appropriate + callables to the constructor of this class, or subclass it. + + A subclass must implement either one of the methods ``_matvec`` + and ``_matmat``, and the attributes/properties ``shape`` (pair of + integers) and ``dtype`` (may be None). It may call the ``__init__`` + on this class to have these attributes validated. Implementing + ``_matvec`` automatically implements ``_matmat`` (using a naive + algorithm) and vice-versa. + + Optionally, a subclass may implement ``_rmatvec`` or ``_adjoint`` + to implement the Hermitian adjoint (conjugate transpose). As with + ``_matvec`` and ``_matmat``, implementing either ``_rmatvec`` or + ``_adjoint`` implements the other automatically. Implementing + ``_adjoint`` is preferable; ``_rmatvec`` is mostly there for + backwards compatibility. + + Parameters + ---------- + shape : tuple + Matrix dimensions (M, N). + matvec : callable f(v) + Returns returns A @ v. + rmatvec : callable f(v) + Returns A^H @ v, where A^H is the conjugate transpose of A. + matmat : callable f(V) + Returns A @ V, where V is a dense matrix with dimensions (N, K). + dtype : dtype + Data type of the matrix. + rmatmat : callable f(V) + Returns A^H @ V, where V is a dense matrix with dimensions (M, K). + + Attributes + ---------- + args : tuple + For linear operators describing products etc. of other linear + operators, the operands of the binary operation. + ndim : int + Number of dimensions (this is always 2) + + See Also + -------- + aslinearoperator : Construct LinearOperators + + Notes + ----- + The user-defined matvec() function must properly handle the case + where v has shape (N,) as well as the (N,1) case. The shape of + the return type is handled internally by LinearOperator. + + It is highly recommended to explicitly specify the `dtype`, otherwise + it is determined automatically at the cost of a single matvec application + on `int8` zero vector using the promoted `dtype` of the output. + Python `int` could be difficult to automatically cast to numpy integers + in the definition of the `matvec` so the determination may be inaccurate. + It is assumed that `matmat`, `rmatvec`, and `rmatmat` would result in + the same dtype of the output given an `int8` input as `matvec`. + + LinearOperator instances can also be multiplied, added with each + other and exponentiated, all lazily: the result of these operations + is always a new, composite LinearOperator, that defers linear + operations to the original operators and combines the results. + + More details regarding how to subclass a LinearOperator and several + examples of concrete LinearOperator instances can be found in the + external project `PyLops `_. + + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse.linalg import LinearOperator + >>> def mv(v): + ... return np.array([2*v[0], 3*v[1]]) + ... + >>> A = LinearOperator((2,2), matvec=mv) + >>> A + <2x2 _CustomLinearOperator with dtype=int8> + >>> A.matvec(np.ones(2)) + array([ 2., 3.]) + >>> A @ np.ones(2) + array([ 2., 3.]) + + """ + + ndim = 2 + # Necessary for right matmul with numpy arrays. + __array_ufunc__ = None + + def __new__(cls, *args, **kwargs): + if cls is LinearOperator: + # Operate as _CustomLinearOperator factory. + return super().__new__(_CustomLinearOperator) + else: + obj = super().__new__(cls) + + if (type(obj)._matvec == LinearOperator._matvec + and type(obj)._matmat == LinearOperator._matmat): + warnings.warn("LinearOperator subclass should implement" + " at least one of _matvec and _matmat.", + category=RuntimeWarning, stacklevel=2) + + return obj + + def __init__(self, dtype, shape): + """Initialize this LinearOperator. + + To be called by subclasses. ``dtype`` may be None; ``shape`` should + be convertible to a length-2 tuple. + """ + if dtype is not None: + dtype = np.dtype(dtype) + + shape = tuple(shape) + if not isshape(shape): + raise ValueError(f"invalid shape {shape!r} (must be 2-d)") + + self.dtype = dtype + self.shape = shape + + def _init_dtype(self): + """Determine the dtype by executing `matvec` on an `int8` test vector. + + In `np.promote_types` hierarchy, the type `int8` is the smallest, + so we call `matvec` on `int8` and use the promoted dtype of the output + to set the default `dtype` of the `LinearOperator`. + We assume that `matmat`, `rmatvec`, and `rmatmat` would result in + the same dtype of the output given an `int8` input as `matvec`. + + Called from subclasses at the end of the __init__ routine. + """ + if self.dtype is None: + v = np.zeros(self.shape[-1], dtype=np.int8) + try: + matvec_v = np.asarray(self.matvec(v)) + except OverflowError: + # Python large `int` promoted to `np.int64`or `np.int32` + self.dtype = np.dtype(int) + else: + self.dtype = matvec_v.dtype + + def _matmat(self, X): + """Default matrix-matrix multiplication handler. + + Falls back on the user-defined _matvec method, so defining that will + define matrix multiplication (though in a very suboptimal way). + """ + + return np.hstack([self.matvec(col.reshape(-1,1)) for col in X.T]) + + def _matvec(self, x): + """Default matrix-vector multiplication handler. + + If self is a linear operator of shape (M, N), then this method will + be called on a shape (N,) or (N, 1) ndarray, and should return a + shape (M,) or (M, 1) ndarray. + + This default implementation falls back on _matmat, so defining that + will define matrix-vector multiplication as well. + """ + return self.matmat(x.reshape(-1, 1)) + + def matvec(self, x): + """Matrix-vector multiplication. + + Performs the operation y=A@x where A is an MxN linear + operator and x is a column vector or 1-d array. + + Parameters + ---------- + x : {matrix, ndarray} + An array with shape (N,) or (N,1). + + Returns + ------- + y : {matrix, ndarray} + A matrix or ndarray with shape (M,) or (M,1) depending + on the type and shape of the x argument. + + Notes + ----- + This matvec wraps the user-specified matvec routine or overridden + _matvec method to ensure that y has the correct shape and type. + + """ + + x = np.asanyarray(x) + + M,N = self.shape + + if x.shape != (N,) and x.shape != (N,1): + raise ValueError('dimension mismatch') + + y = self._matvec(x) + + if isinstance(x, np.matrix): + y = asmatrix(y) + else: + y = np.asarray(y) + + if x.ndim == 1: + y = y.reshape(M) + elif x.ndim == 2: + y = y.reshape(M,1) + else: + raise ValueError('invalid shape returned by user-defined matvec()') + + return y + + def rmatvec(self, x): + """Adjoint matrix-vector multiplication. + + Performs the operation y = A^H @ x where A is an MxN linear + operator and x is a column vector or 1-d array. + + Parameters + ---------- + x : {matrix, ndarray} + An array with shape (M,) or (M,1). + + Returns + ------- + y : {matrix, ndarray} + A matrix or ndarray with shape (N,) or (N,1) depending + on the type and shape of the x argument. + + Notes + ----- + This rmatvec wraps the user-specified rmatvec routine or overridden + _rmatvec method to ensure that y has the correct shape and type. + + """ + + x = np.asanyarray(x) + + M,N = self.shape + + if x.shape != (M,) and x.shape != (M,1): + raise ValueError('dimension mismatch') + + y = self._rmatvec(x) + + if isinstance(x, np.matrix): + y = asmatrix(y) + else: + y = np.asarray(y) + + if x.ndim == 1: + y = y.reshape(N) + elif x.ndim == 2: + y = y.reshape(N,1) + else: + raise ValueError('invalid shape returned by user-defined rmatvec()') + + return y + + def _rmatvec(self, x): + """Default implementation of _rmatvec; defers to adjoint.""" + if type(self)._adjoint == LinearOperator._adjoint: + # _adjoint not overridden, prevent infinite recursion + if (hasattr(self, "_rmatmat") + and type(self)._rmatmat != LinearOperator._rmatmat): + # Try to use _rmatmat as a fallback + return self._rmatmat(x.reshape(-1, 1)).reshape(-1) + raise NotImplementedError + else: + return self.H.matvec(x) + + def matmat(self, X): + """Matrix-matrix multiplication. + + Performs the operation y=A@X where A is an MxN linear + operator and X dense N*K matrix or ndarray. + + Parameters + ---------- + X : {matrix, ndarray} + An array with shape (N,K). + + Returns + ------- + Y : {matrix, ndarray} + A matrix or ndarray with shape (M,K) depending on + the type of the X argument. + + Notes + ----- + This matmat wraps any user-specified matmat routine or overridden + _matmat method to ensure that y has the correct type. + + """ + if not (issparse(X) or is_pydata_spmatrix(X)): + X = np.asanyarray(X) + + if X.ndim != 2: + raise ValueError(f'expected 2-d ndarray or matrix, not {X.ndim}-d') + + if X.shape[0] != self.shape[1]: + raise ValueError(f'dimension mismatch: {self.shape}, {X.shape}') + + try: + Y = self._matmat(X) + except Exception as e: + if issparse(X) or is_pydata_spmatrix(X): + raise TypeError( + "Unable to multiply a LinearOperator with a sparse matrix." + " Wrap the matrix in aslinearoperator first." + ) from e + raise + + if isinstance(Y, np.matrix): + Y = asmatrix(Y) + + return Y + + def rmatmat(self, X): + """Adjoint matrix-matrix multiplication. + + Performs the operation y = A^H @ x where A is an MxN linear + operator and x is a column vector or 1-d array, or 2-d array. + The default implementation defers to the adjoint. + + Parameters + ---------- + X : {matrix, ndarray} + A matrix or 2D array. + + Returns + ------- + Y : {matrix, ndarray} + A matrix or 2D array depending on the type of the input. + + Notes + ----- + This rmatmat wraps the user-specified rmatmat routine. + + """ + if not (issparse(X) or is_pydata_spmatrix(X)): + X = np.asanyarray(X) + + if X.ndim != 2: + raise ValueError('expected 2-d ndarray or matrix, not %d-d' + % X.ndim) + + if X.shape[0] != self.shape[0]: + raise ValueError(f'dimension mismatch: {self.shape}, {X.shape}') + + try: + Y = self._rmatmat(X) + except Exception as e: + if issparse(X) or is_pydata_spmatrix(X): + raise TypeError( + "Unable to multiply a LinearOperator with a sparse matrix." + " Wrap the matrix in aslinearoperator() first." + ) from e + raise + + if isinstance(Y, np.matrix): + Y = asmatrix(Y) + return Y + + def _rmatmat(self, X): + """Default implementation of _rmatmat defers to rmatvec or adjoint.""" + if type(self)._adjoint == LinearOperator._adjoint: + return np.hstack([self.rmatvec(col.reshape(-1, 1)) for col in X.T]) + else: + return self.H.matmat(X) + + def __call__(self, x): + return self@x + + def __mul__(self, x): + return self.dot(x) + + def __truediv__(self, other): + if not np.isscalar(other): + raise ValueError("Can only divide a linear operator by a scalar.") + + return _ScaledLinearOperator(self, 1.0/other) + + def dot(self, x): + """Matrix-matrix or matrix-vector multiplication. + + Parameters + ---------- + x : array_like + 1-d or 2-d array, representing a vector or matrix. + + Returns + ------- + Ax : array + 1-d or 2-d array (depending on the shape of x) that represents + the result of applying this linear operator on x. + + """ + if isinstance(x, LinearOperator): + return _ProductLinearOperator(self, x) + elif np.isscalar(x): + return _ScaledLinearOperator(self, x) + else: + if not issparse(x) and not is_pydata_spmatrix(x): + # Sparse matrices shouldn't be converted to numpy arrays. + x = np.asarray(x) + + if x.ndim == 1 or x.ndim == 2 and x.shape[1] == 1: + return self.matvec(x) + elif x.ndim == 2: + return self.matmat(x) + else: + raise ValueError(f'expected 1-d or 2-d array or matrix, got {x!r}') + + def __matmul__(self, other): + if np.isscalar(other): + raise ValueError("Scalar operands are not allowed, " + "use '*' instead") + return self.__mul__(other) + + def __rmatmul__(self, other): + if np.isscalar(other): + raise ValueError("Scalar operands are not allowed, " + "use '*' instead") + return self.__rmul__(other) + + def __rmul__(self, x): + if np.isscalar(x): + return _ScaledLinearOperator(self, x) + else: + return self._rdot(x) + + def _rdot(self, x): + """Matrix-matrix or matrix-vector multiplication from the right. + + Parameters + ---------- + x : array_like + 1-d or 2-d array, representing a vector or matrix. + + Returns + ------- + xA : array + 1-d or 2-d array (depending on the shape of x) that represents + the result of applying this linear operator on x from the right. + + Notes + ----- + This is copied from dot to implement right multiplication. + """ + if isinstance(x, LinearOperator): + return _ProductLinearOperator(x, self) + elif np.isscalar(x): + return _ScaledLinearOperator(self, x) + else: + if not issparse(x) and not is_pydata_spmatrix(x): + # Sparse matrices shouldn't be converted to numpy arrays. + x = np.asarray(x) + + # We use transpose instead of rmatvec/rmatmat to avoid + # unnecessary complex conjugation if possible. + if x.ndim == 1 or x.ndim == 2 and x.shape[0] == 1: + return self.T.matvec(x.T).T + elif x.ndim == 2: + return self.T.matmat(x.T).T + else: + raise ValueError(f'expected 1-d or 2-d array or matrix, got {x!r}') + + def __pow__(self, p): + if np.isscalar(p): + return _PowerLinearOperator(self, p) + else: + return NotImplemented + + def __add__(self, x): + if isinstance(x, LinearOperator): + return _SumLinearOperator(self, x) + else: + return NotImplemented + + def __neg__(self): + return _ScaledLinearOperator(self, -1) + + def __sub__(self, x): + return self.__add__(-x) + + def __repr__(self): + M,N = self.shape + if self.dtype is None: + dt = 'unspecified dtype' + else: + dt = 'dtype=' + str(self.dtype) + + return '<%dx%d %s with %s>' % (M, N, self.__class__.__name__, dt) + + def adjoint(self): + """Hermitian adjoint. + + Returns the Hermitian adjoint of self, aka the Hermitian + conjugate or Hermitian transpose. For a complex matrix, the + Hermitian adjoint is equal to the conjugate transpose. + + Can be abbreviated self.H instead of self.adjoint(). + + Returns + ------- + A_H : LinearOperator + Hermitian adjoint of self. + """ + return self._adjoint() + + H = property(adjoint) + + def transpose(self): + """Transpose this linear operator. + + Returns a LinearOperator that represents the transpose of this one. + Can be abbreviated self.T instead of self.transpose(). + """ + return self._transpose() + + T = property(transpose) + + def _adjoint(self): + """Default implementation of _adjoint; defers to rmatvec.""" + return _AdjointLinearOperator(self) + + def _transpose(self): + """ Default implementation of _transpose; defers to rmatvec + conj""" + return _TransposedLinearOperator(self) + + +class _CustomLinearOperator(LinearOperator): + """Linear operator defined in terms of user-specified operations.""" + + def __init__(self, shape, matvec, rmatvec=None, matmat=None, + dtype=None, rmatmat=None): + super().__init__(dtype, shape) + + self.args = () + + self.__matvec_impl = matvec + self.__rmatvec_impl = rmatvec + self.__rmatmat_impl = rmatmat + self.__matmat_impl = matmat + + self._init_dtype() + + def _matmat(self, X): + if self.__matmat_impl is not None: + return self.__matmat_impl(X) + else: + return super()._matmat(X) + + def _matvec(self, x): + return self.__matvec_impl(x) + + def _rmatvec(self, x): + func = self.__rmatvec_impl + if func is None: + raise NotImplementedError("rmatvec is not defined") + return self.__rmatvec_impl(x) + + def _rmatmat(self, X): + if self.__rmatmat_impl is not None: + return self.__rmatmat_impl(X) + else: + return super()._rmatmat(X) + + def _adjoint(self): + return _CustomLinearOperator(shape=(self.shape[1], self.shape[0]), + matvec=self.__rmatvec_impl, + rmatvec=self.__matvec_impl, + matmat=self.__rmatmat_impl, + rmatmat=self.__matmat_impl, + dtype=self.dtype) + + +class _AdjointLinearOperator(LinearOperator): + """Adjoint of arbitrary Linear Operator""" + + def __init__(self, A): + shape = (A.shape[1], A.shape[0]) + super().__init__(dtype=A.dtype, shape=shape) + self.A = A + self.args = (A,) + + def _matvec(self, x): + return self.A._rmatvec(x) + + def _rmatvec(self, x): + return self.A._matvec(x) + + def _matmat(self, x): + return self.A._rmatmat(x) + + def _rmatmat(self, x): + return self.A._matmat(x) + +class _TransposedLinearOperator(LinearOperator): + """Transposition of arbitrary Linear Operator""" + + def __init__(self, A): + shape = (A.shape[1], A.shape[0]) + super().__init__(dtype=A.dtype, shape=shape) + self.A = A + self.args = (A,) + + def _matvec(self, x): + # NB. np.conj works also on sparse matrices + return np.conj(self.A._rmatvec(np.conj(x))) + + def _rmatvec(self, x): + return np.conj(self.A._matvec(np.conj(x))) + + def _matmat(self, x): + # NB. np.conj works also on sparse matrices + return np.conj(self.A._rmatmat(np.conj(x))) + + def _rmatmat(self, x): + return np.conj(self.A._matmat(np.conj(x))) + +def _get_dtype(operators, dtypes=None): + if dtypes is None: + dtypes = [] + for obj in operators: + if obj is not None and hasattr(obj, 'dtype'): + dtypes.append(obj.dtype) + return np.result_type(*dtypes) + + +class _SumLinearOperator(LinearOperator): + def __init__(self, A, B): + if not isinstance(A, LinearOperator) or \ + not isinstance(B, LinearOperator): + raise ValueError('both operands have to be a LinearOperator') + if A.shape != B.shape: + raise ValueError(f'cannot add {A} and {B}: shape mismatch') + self.args = (A, B) + super().__init__(_get_dtype([A, B]), A.shape) + + def _matvec(self, x): + return self.args[0].matvec(x) + self.args[1].matvec(x) + + def _rmatvec(self, x): + return self.args[0].rmatvec(x) + self.args[1].rmatvec(x) + + def _rmatmat(self, x): + return self.args[0].rmatmat(x) + self.args[1].rmatmat(x) + + def _matmat(self, x): + return self.args[0].matmat(x) + self.args[1].matmat(x) + + def _adjoint(self): + A, B = self.args + return A.H + B.H + + +class _ProductLinearOperator(LinearOperator): + def __init__(self, A, B): + if not isinstance(A, LinearOperator) or \ + not isinstance(B, LinearOperator): + raise ValueError('both operands have to be a LinearOperator') + if A.shape[1] != B.shape[0]: + raise ValueError(f'cannot multiply {A} and {B}: shape mismatch') + super().__init__(_get_dtype([A, B]), + (A.shape[0], B.shape[1])) + self.args = (A, B) + + def _matvec(self, x): + return self.args[0].matvec(self.args[1].matvec(x)) + + def _rmatvec(self, x): + return self.args[1].rmatvec(self.args[0].rmatvec(x)) + + def _rmatmat(self, x): + return self.args[1].rmatmat(self.args[0].rmatmat(x)) + + def _matmat(self, x): + return self.args[0].matmat(self.args[1].matmat(x)) + + def _adjoint(self): + A, B = self.args + return B.H @ A.H + + +class _ScaledLinearOperator(LinearOperator): + def __init__(self, A, alpha): + if not isinstance(A, LinearOperator): + raise ValueError('LinearOperator expected as A') + if not np.isscalar(alpha): + raise ValueError('scalar expected as alpha') + if isinstance(A, _ScaledLinearOperator): + A, alpha_original = A.args + # Avoid in-place multiplication so that we don't accidentally mutate + # the original prefactor. + alpha = alpha * alpha_original + + dtype = _get_dtype([A], [type(alpha)]) + super().__init__(dtype, A.shape) + self.args = (A, alpha) + # Note: args[1] is alpha (a scalar), so use `*` below, not `@` + + def _matvec(self, x): + return self.args[1] * self.args[0].matvec(x) + + def _rmatvec(self, x): + return np.conj(self.args[1]) * self.args[0].rmatvec(x) + + def _rmatmat(self, x): + return np.conj(self.args[1]) * self.args[0].rmatmat(x) + + def _matmat(self, x): + return self.args[1] * self.args[0].matmat(x) + + def _adjoint(self): + A, alpha = self.args + return A.H * np.conj(alpha) + + +class _PowerLinearOperator(LinearOperator): + def __init__(self, A, p): + if not isinstance(A, LinearOperator): + raise ValueError('LinearOperator expected as A') + if A.shape[0] != A.shape[1]: + raise ValueError(f'square LinearOperator expected, got {A!r}') + if not isintlike(p) or p < 0: + raise ValueError('non-negative integer expected as p') + + super().__init__(_get_dtype([A]), A.shape) + self.args = (A, p) + + def _power(self, fun, x): + res = np.array(x, copy=True) + for i in range(self.args[1]): + res = fun(res) + return res + + def _matvec(self, x): + return self._power(self.args[0].matvec, x) + + def _rmatvec(self, x): + return self._power(self.args[0].rmatvec, x) + + def _rmatmat(self, x): + return self._power(self.args[0].rmatmat, x) + + def _matmat(self, x): + return self._power(self.args[0].matmat, x) + + def _adjoint(self): + A, p = self.args + return A.H ** p + + +class MatrixLinearOperator(LinearOperator): + def __init__(self, A): + super().__init__(A.dtype, A.shape) + self.A = A + self.__adj = None + self.args = (A,) + + def _matmat(self, X): + return self.A.dot(X) + + def _adjoint(self): + if self.__adj is None: + self.__adj = _AdjointMatrixOperator(self.A) + return self.__adj + + +class _AdjointMatrixOperator(MatrixLinearOperator): + def __init__(self, adjoint_array): + self.A = adjoint_array.T.conj() + self.args = (adjoint_array,) + self.shape = adjoint_array.shape[1], adjoint_array.shape[0] + + @property + def dtype(self): + return self.args[0].dtype + + def _adjoint(self): + return MatrixLinearOperator(self.args[0]) + + +class IdentityOperator(LinearOperator): + def __init__(self, shape, dtype=None): + super().__init__(dtype, shape) + + def _matvec(self, x): + return x + + def _rmatvec(self, x): + return x + + def _rmatmat(self, x): + return x + + def _matmat(self, x): + return x + + def _adjoint(self): + return self + + +def aslinearoperator(A): + """Return A as a LinearOperator. + + 'A' may be any of the following types: + - ndarray + - matrix + - sparse array (e.g. csr_array, lil_array, etc.) + - LinearOperator + - An object with .shape and .matvec attributes + + See the LinearOperator documentation for additional information. + + Notes + ----- + If 'A' has no .dtype attribute, the data type is determined by calling + :func:`LinearOperator.matvec()` - set the .dtype attribute to prevent this + call upon the linear operator creation. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse.linalg import aslinearoperator + >>> M = np.array([[1,2,3],[4,5,6]], dtype=np.int32) + >>> aslinearoperator(M) + <2x3 MatrixLinearOperator with dtype=int32> + """ + if isinstance(A, LinearOperator): + return A + + elif isinstance(A, np.ndarray) or isinstance(A, np.matrix): + if A.ndim > 2: + raise ValueError('array must have ndim <= 2') + A = np.atleast_2d(np.asarray(A)) + return MatrixLinearOperator(A) + + elif issparse(A) or is_pydata_spmatrix(A): + return MatrixLinearOperator(A) + + else: + if hasattr(A, 'shape') and hasattr(A, 'matvec'): + rmatvec = None + rmatmat = None + dtype = None + + if hasattr(A, 'rmatvec'): + rmatvec = A.rmatvec + if hasattr(A, 'rmatmat'): + rmatmat = A.rmatmat + if hasattr(A, 'dtype'): + dtype = A.dtype + return LinearOperator(A.shape, A.matvec, rmatvec=rmatvec, + rmatmat=rmatmat, dtype=dtype) + + else: + raise TypeError('type not understood') diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..3b57274542928e79c234bb6955849a90be21990e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/__init__.py @@ -0,0 +1,20 @@ +"Iterative Solvers for Sparse Linear Systems" + +#from info import __doc__ +from .iterative import * +from .minres import minres +from .lgmres import lgmres +from .lsqr import lsqr +from .lsmr import lsmr +from ._gcrotmk import gcrotmk +from .tfqmr import tfqmr + +__all__ = [ + 'bicg', 'bicgstab', 'cg', 'cgs', 'gcrotmk', 'gmres', + 'lgmres', 'lsmr', 'lsqr', + 'minres', 'qmr', 'tfqmr' +] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/_gcrotmk.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/_gcrotmk.py new file mode 100644 index 0000000000000000000000000000000000000000..e5c1fe9daee2c5a955dd71de3e82f33e3dc02179 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/_gcrotmk.py @@ -0,0 +1,503 @@ +# Copyright (C) 2015, Pauli Virtanen +# Distributed under the same license as SciPy. + +import numpy as np +from numpy.linalg import LinAlgError +from scipy.linalg import (get_blas_funcs, qr, solve, svd, qr_insert, lstsq) +from .iterative import _get_atol_rtol +from scipy.sparse.linalg._isolve.utils import make_system + + +__all__ = ['gcrotmk'] + + +def _fgmres(matvec, v0, m, atol, lpsolve=None, rpsolve=None, cs=(), outer_v=(), + prepend_outer_v=False): + """ + FGMRES Arnoldi process, with optional projection or augmentation + + Parameters + ---------- + matvec : callable + Operation A*x + v0 : ndarray + Initial vector, normalized to nrm2(v0) == 1 + m : int + Number of GMRES rounds + atol : float + Absolute tolerance for early exit + lpsolve : callable + Left preconditioner L + rpsolve : callable + Right preconditioner R + cs : list of (ndarray, ndarray) + Columns of matrices C and U in GCROT + outer_v : list of ndarrays + Augmentation vectors in LGMRES + prepend_outer_v : bool, optional + Whether augmentation vectors come before or after + Krylov iterates + + Raises + ------ + LinAlgError + If nans encountered + + Returns + ------- + Q, R : ndarray + QR decomposition of the upper Hessenberg H=QR + B : ndarray + Projections corresponding to matrix C + vs : list of ndarray + Columns of matrix V + zs : list of ndarray + Columns of matrix Z + y : ndarray + Solution to ||H y - e_1||_2 = min! + res : float + The final (preconditioned) residual norm + + """ + + if lpsolve is None: + def lpsolve(x): + return x + if rpsolve is None: + def rpsolve(x): + return x + + axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (v0,)) + + vs = [v0] + zs = [] + y = None + res = np.nan + + m = m + len(outer_v) + + # Orthogonal projection coefficients + B = np.zeros((len(cs), m), dtype=v0.dtype) + + # H is stored in QR factorized form + Q = np.ones((1, 1), dtype=v0.dtype) + R = np.zeros((1, 0), dtype=v0.dtype) + + eps = np.finfo(v0.dtype).eps + + breakdown = False + + # FGMRES Arnoldi process + for j in range(m): + # L A Z = C B + V H + + if prepend_outer_v and j < len(outer_v): + z, w = outer_v[j] + elif prepend_outer_v and j == len(outer_v): + z = rpsolve(v0) + w = None + elif not prepend_outer_v and j >= m - len(outer_v): + z, w = outer_v[j - (m - len(outer_v))] + else: + z = rpsolve(vs[-1]) + w = None + + if w is None: + w = lpsolve(matvec(z)) + else: + # w is clobbered below + w = w.copy() + + w_norm = nrm2(w) + + # GCROT projection: L A -> (1 - C C^H) L A + # i.e. orthogonalize against C + for i, c in enumerate(cs): + alpha = dot(c, w) + B[i,j] = alpha + w = axpy(c, w, c.shape[0], -alpha) # w -= alpha*c + + # Orthogonalize against V + hcur = np.zeros(j+2, dtype=Q.dtype) + for i, v in enumerate(vs): + alpha = dot(v, w) + hcur[i] = alpha + w = axpy(v, w, v.shape[0], -alpha) # w -= alpha*v + hcur[i+1] = nrm2(w) + + with np.errstate(over='ignore', divide='ignore'): + # Careful with denormals + alpha = 1/hcur[-1] + + if np.isfinite(alpha): + w = scal(alpha, w) + + if not (hcur[-1] > eps * w_norm): + # w essentially in the span of previous vectors, + # or we have nans. Bail out after updating the QR + # solution. + breakdown = True + + vs.append(w) + zs.append(z) + + # Arnoldi LSQ problem + + # Add new column to H=Q@R, padding other columns with zeros + Q2 = np.zeros((j+2, j+2), dtype=Q.dtype, order='F') + Q2[:j+1,:j+1] = Q + Q2[j+1,j+1] = 1 + + R2 = np.zeros((j+2, j), dtype=R.dtype, order='F') + R2[:j+1,:] = R + + Q, R = qr_insert(Q2, R2, hcur, j, which='col', + overwrite_qru=True, check_finite=False) + + # Transformed least squares problem + # || Q R y - inner_res_0 * e_1 ||_2 = min! + # Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0] + + # Residual is immediately known + res = abs(Q[0,-1]) + + # Check for termination + if res < atol or breakdown: + break + + if not np.isfinite(R[j,j]): + # nans encountered, bail out + raise LinAlgError() + + # -- Get the LSQ problem solution + + # The problem is triangular, but the condition number may be + # bad (or in case of breakdown the last diagonal entry may be + # zero), so use lstsq instead of trtrs. + y, _, _, _, = lstsq(R[:j+1,:j+1], Q[0,:j+1].conj()) + + B = B[:,:j+1] + + return Q, R, B, vs, zs, y, res + + +def gcrotmk(A, b, x0=None, *, rtol=1e-5, atol=0., maxiter=1000, M=None, callback=None, + m=20, k=None, CU=None, discard_C=False, truncate='oldest'): + """ + Solve a matrix equation using flexible GCROT(m,k) algorithm. + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + The real or complex N-by-N matrix of the linear system. + Alternatively, `A` can be a linear operator which can + produce ``Ax`` using, e.g., + `LinearOperator`. + b : ndarray + Right hand side of the linear system. Has shape (N,) or (N,1). + x0 : ndarray + Starting guess for the solution. + rtol, atol : float, optional + Parameters for the convergence test. For convergence, + ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. + The default is ``rtol=1e-5`` and ``atol=0.0``. + maxiter : int, optional + Maximum number of iterations. Iteration will stop after maxiter + steps even if the specified tolerance has not been achieved. The + default is ``1000``. + M : {sparse array, ndarray, LinearOperator}, optional + Preconditioner for `A`. The preconditioner should approximate the + inverse of `A`. gcrotmk is a 'flexible' algorithm and the preconditioner + can vary from iteration to iteration. Effective preconditioning + dramatically improves the rate of convergence, which implies that + fewer iterations are needed to reach a given error tolerance. + callback : function, optional + User-supplied function to call after each iteration. It is called + as ``callback(xk)``, where ``xk`` is the current solution vector. + m : int, optional + Number of inner FGMRES iterations per each outer iteration. + Default: 20 + k : int, optional + Number of vectors to carry between inner FGMRES iterations. + According to [2]_, good values are around `m`. + Default: `m` + CU : list of tuples, optional + List of tuples ``(c, u)`` which contain the columns of the matrices + C and U in the GCROT(m,k) algorithm. For details, see [2]_. + The list given and vectors contained in it are modified in-place. + If not given, start from empty matrices. The ``c`` elements in the + tuples can be ``None``, in which case the vectors are recomputed + via ``c = A u`` on start and orthogonalized as described in [3]_. + discard_C : bool, optional + Discard the C-vectors at the end. Useful if recycling Krylov subspaces + for different linear systems. + truncate : {'oldest', 'smallest'}, optional + Truncation scheme to use. Drop: oldest vectors, or vectors with + smallest singular values using the scheme discussed in [1,2]. + See [2]_ for detailed comparison. + Default: 'oldest' + + Returns + ------- + x : ndarray + The solution found. + info : int + Provides convergence information: + + * 0 : successful exit + * >0 : convergence to tolerance not achieved, number of iterations + + References + ---------- + .. [1] E. de Sturler, ''Truncation strategies for optimal Krylov subspace + methods'', SIAM J. Numer. Anal. 36, 864 (1999). + .. [2] J.E. Hicken and D.W. Zingg, ''A simplified and flexible variant + of GCROT for solving nonsymmetric linear systems'', + SIAM J. Sci. Comput. 32, 172 (2010). + .. [3] M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti, + ''Recycling Krylov subspaces for sequences of linear systems'', + SIAM J. Sci. Comput. 28, 1651 (2006). + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import gcrotmk + >>> R = np.random.randn(5, 5) + >>> A = csc_array(R) + >>> b = np.random.randn(5) + >>> x, exit_code = gcrotmk(A, b, atol=1e-5) + >>> print(exit_code) + 0 + >>> np.allclose(A.dot(x), b) + True + + """ + A,M,x,b,postprocess = make_system(A,M,x0,b) + + if not np.isfinite(b).all(): + raise ValueError("RHS must contain only finite numbers") + + if truncate not in ('oldest', 'smallest'): + raise ValueError(f"Invalid value for 'truncate': {truncate!r}") + + matvec = A.matvec + psolve = M.matvec + + if CU is None: + CU = [] + + if k is None: + k = m + + axpy, dot, scal = None, None, None + + if x0 is None: + r = b.copy() + else: + r = b - matvec(x) + + axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (x, r)) + + b_norm = nrm2(b) + + # we call this to get the right atol/rtol and raise errors as necessary + atol, rtol = _get_atol_rtol('gcrotmk', b_norm, atol, rtol) + + if b_norm == 0: + x = b + return (postprocess(x), 0) + + if discard_C: + CU[:] = [(None, u) for c, u in CU] + + # Reorthogonalize old vectors + if CU: + # Sort already existing vectors to the front + CU.sort(key=lambda cu: cu[0] is not None) + + # Fill-in missing ones + C = np.empty((A.shape[0], len(CU)), dtype=r.dtype, order='F') + us = [] + j = 0 + while CU: + # More memory-efficient: throw away old vectors as we go + c, u = CU.pop(0) + if c is None: + c = matvec(u) + C[:,j] = c + j += 1 + us.append(u) + + # Orthogonalize + Q, R, P = qr(C, overwrite_a=True, mode='economic', pivoting=True) + del C + + # C := Q + cs = list(Q.T) + + # U := U P R^-1, back-substitution + new_us = [] + for j in range(len(cs)): + u = us[P[j]] + for i in range(j): + u = axpy(us[P[i]], u, u.shape[0], -R[i,j]) + if abs(R[j,j]) < 1e-12 * abs(R[0,0]): + # discard rest of the vectors + break + u = scal(1.0/R[j,j], u) + new_us.append(u) + + # Form the new CU lists + CU[:] = list(zip(cs, new_us))[::-1] + + if CU: + axpy, dot = get_blas_funcs(['axpy', 'dot'], (r,)) + + # Solve first the projection operation with respect to the CU + # vectors. This corresponds to modifying the initial guess to + # be + # + # x' = x + U y + # y = argmin_y || b - A (x + U y) ||^2 + # + # The solution is y = C^H (b - A x) + for c, u in CU: + yc = dot(c, r) + x = axpy(u, x, x.shape[0], yc) + r = axpy(c, r, r.shape[0], -yc) + + # GCROT main iteration + for j_outer in range(maxiter): + # -- callback + if callback is not None: + callback(x) + + beta = nrm2(r) + + # -- check stopping condition + beta_tol = max(atol, rtol * b_norm) + + if beta <= beta_tol and (j_outer > 0 or CU): + # recompute residual to avoid rounding error + r = b - matvec(x) + beta = nrm2(r) + + if beta <= beta_tol: + j_outer = -1 + break + + ml = m + max(k - len(CU), 0) + + cs = [c for c, u in CU] + + try: + Q, R, B, vs, zs, y, pres = _fgmres(matvec, + r/beta, + ml, + rpsolve=psolve, + atol=max(atol, rtol*b_norm)/beta, + cs=cs) + y *= beta + except LinAlgError: + # Floating point over/underflow, non-finite result from + # matmul etc. -- report failure. + break + + # + # At this point, + # + # [A U, A Z] = [C, V] G; G = [ I B ] + # [ 0 H ] + # + # where [C, V] has orthonormal columns, and r = beta v_0. Moreover, + # + # || b - A (x + Z y + U q) ||_2 = || r - C B y - V H y - C q ||_2 = min! + # + # from which y = argmin_y || beta e_1 - H y ||_2, and q = -B y + # + + # + # GCROT(m,k) update + # + + # Define new outer vectors + + # ux := (Z - U B) y + ux = zs[0]*y[0] + for z, yc in zip(zs[1:], y[1:]): + ux = axpy(z, ux, ux.shape[0], yc) # ux += z*yc + by = B.dot(y) + for cu, byc in zip(CU, by): + c, u = cu + ux = axpy(u, ux, ux.shape[0], -byc) # ux -= u*byc + + # cx := V H y + with np.errstate(invalid="ignore"): + hy = Q.dot(R.dot(y)) + cx = vs[0] * hy[0] + for v, hyc in zip(vs[1:], hy[1:]): + cx = axpy(v, cx, cx.shape[0], hyc) # cx += v*hyc + + # Normalize cx, maintaining cx = A ux + # This new cx is orthogonal to the previous C, by construction + try: + alpha = 1/nrm2(cx) + if not np.isfinite(alpha): + raise FloatingPointError() + except (FloatingPointError, ZeroDivisionError): + # Cannot update, so skip it + continue + + cx = scal(alpha, cx) + ux = scal(alpha, ux) + + # Update residual and solution + gamma = dot(cx, r) + r = axpy(cx, r, r.shape[0], -gamma) # r -= gamma*cx + x = axpy(ux, x, x.shape[0], gamma) # x += gamma*ux + + # Truncate CU + if truncate == 'oldest': + while len(CU) >= k and CU: + del CU[0] + elif truncate == 'smallest': + if len(CU) >= k and CU: + # cf. [1,2] + D = solve(R[:-1,:].T, B.T).T + W, sigma, V = svd(D) + + # C := C W[:,:k-1], U := U W[:,:k-1] + new_CU = [] + for j, w in enumerate(W[:,:k-1].T): + c, u = CU[0] + c = c * w[0] + u = u * w[0] + for cup, wp in zip(CU[1:], w[1:]): + cp, up = cup + c = axpy(cp, c, c.shape[0], wp) + u = axpy(up, u, u.shape[0], wp) + + # Reorthogonalize at the same time; not necessary + # in exact arithmetic, but floating point error + # tends to accumulate here + for cp, up in new_CU: + alpha = dot(cp, c) + c = axpy(cp, c, c.shape[0], -alpha) + u = axpy(up, u, u.shape[0], -alpha) + alpha = nrm2(c) + c = scal(1.0/alpha, c) + u = scal(1.0/alpha, u) + + new_CU.append((c, u)) + CU[:] = new_CU + + # Add new vector to CU + CU.append((cx, ux)) + + # Include the solution vector to the span + CU.append((None, x.copy())) + if discard_C: + CU[:] = [(None, uz) for cz, uz in CU] + + return postprocess(x), j_outer + 1 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/iterative.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/iterative.py new file mode 100644 index 0000000000000000000000000000000000000000..4b91ef8fe4b37191e76710670eccd9557a397964 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/iterative.py @@ -0,0 +1,1045 @@ +import warnings +import numpy as np +from scipy.sparse.linalg._interface import LinearOperator +from .utils import make_system +from scipy.linalg import get_lapack_funcs + +__all__ = ['bicg', 'bicgstab', 'cg', 'cgs', 'gmres', 'qmr'] + + +def _get_atol_rtol(name, b_norm, atol=0., rtol=1e-5): + """ + A helper function to handle tolerance normalization + """ + if atol == 'legacy' or atol is None or atol < 0: + msg = (f"'scipy.sparse.linalg.{name}' called with invalid `atol`={atol}; " + "if set, `atol` must be a real, non-negative number.") + raise ValueError(msg) + + atol = max(float(atol), float(rtol) * float(b_norm)) + + return atol, rtol + + +def bicg(A, b, x0=None, *, rtol=1e-5, atol=0., maxiter=None, M=None, callback=None): + """Use BIConjugate Gradient iteration to solve ``Ax = b``. + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + The real or complex N-by-N matrix of the linear system. + Alternatively, `A` can be a linear operator which can + produce ``Ax`` and ``A^T x`` using, e.g., + ``scipy.sparse.linalg.LinearOperator``. + b : ndarray + Right hand side of the linear system. Has shape (N,) or (N,1). + x0 : ndarray + Starting guess for the solution. + rtol, atol : float, optional + Parameters for the convergence test. For convergence, + ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. + The default is ``atol=0.`` and ``rtol=1e-5``. + maxiter : integer + Maximum number of iterations. Iteration will stop after maxiter + steps even if the specified tolerance has not been achieved. + M : {sparse array, ndarray, LinearOperator} + Preconditioner for `A`. It should approximate the + inverse of `A` (see Notes). Effective preconditioning dramatically improves the + rate of convergence, which implies that fewer iterations are needed + to reach a given error tolerance. + callback : function + User-supplied function to call after each iteration. It is called + as ``callback(xk)``, where ``xk`` is the current solution vector. + + Returns + ------- + x : ndarray + The converged solution. + info : integer + Provides convergence information: + 0 : successful exit + >0 : convergence to tolerance not achieved, number of iterations + <0 : parameter breakdown + + Notes + ----- + The preconditioner `M` should be a matrix such that ``M @ A`` has a smaller + condition number than `A`, see [1]_ . + + References + ---------- + .. [1] "Preconditioner", Wikipedia, + https://en.wikipedia.org/wiki/Preconditioner + .. [2] "Biconjugate gradient method", Wikipedia, + https://en.wikipedia.org/wiki/Biconjugate_gradient_method + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import bicg + >>> A = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1.]]) + >>> b = np.array([2., 4., -1.]) + >>> x, exitCode = bicg(A, b, atol=1e-5) + >>> print(exitCode) # 0 indicates successful convergence + 0 + >>> np.allclose(A.dot(x), b) + True + """ + A, M, x, b, postprocess = make_system(A, M, x0, b) + bnrm2 = np.linalg.norm(b) + + atol, _ = _get_atol_rtol('bicg', bnrm2, atol, rtol) + + if bnrm2 == 0: + return postprocess(b), 0 + + n = len(b) + dotprod = np.vdot if np.iscomplexobj(x) else np.dot + + if maxiter is None: + maxiter = n*10 + + matvec, rmatvec = A.matvec, A.rmatvec + psolve, rpsolve = M.matvec, M.rmatvec + + rhotol = np.finfo(x.dtype.char).eps**2 + + # Dummy values to initialize vars, silence linter warnings + rho_prev, p, ptilde = None, None, None + + r = b - matvec(x) if x.any() else b.copy() + rtilde = r.copy() + + for iteration in range(maxiter): + if np.linalg.norm(r) < atol: # Are we done? + return postprocess(x), 0 + + z = psolve(r) + ztilde = rpsolve(rtilde) + # order matters in this dot product + rho_cur = dotprod(rtilde, z) + + if np.abs(rho_cur) < rhotol: # Breakdown case + return postprocess, -10 + + if iteration > 0: + beta = rho_cur / rho_prev + p *= beta + p += z + ptilde *= beta.conj() + ptilde += ztilde + else: # First spin + p = z.copy() + ptilde = ztilde.copy() + + q = matvec(p) + qtilde = rmatvec(ptilde) + rv = dotprod(ptilde, q) + + if rv == 0: + return postprocess(x), -11 + + alpha = rho_cur / rv + x += alpha*p + r -= alpha*q + rtilde -= alpha.conj()*qtilde + rho_prev = rho_cur + + if callback: + callback(x) + + else: # for loop exhausted + # Return incomplete progress + return postprocess(x), maxiter + + +def bicgstab(A, b, x0=None, *, rtol=1e-5, atol=0., maxiter=None, M=None, + callback=None): + """Use BIConjugate Gradient STABilized iteration to solve ``Ax = b``. + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + The real or complex N-by-N matrix of the linear system. + Alternatively, `A` can be a linear operator which can + produce ``Ax`` and ``A^T x`` using, e.g., + ``scipy.sparse.linalg.LinearOperator``. + b : ndarray + Right hand side of the linear system. Has shape (N,) or (N,1). + x0 : ndarray + Starting guess for the solution. + rtol, atol : float, optional + Parameters for the convergence test. For convergence, + ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. + The default is ``atol=0.`` and ``rtol=1e-5``. + maxiter : integer + Maximum number of iterations. Iteration will stop after maxiter + steps even if the specified tolerance has not been achieved. + M : {sparse array, ndarray, LinearOperator} + Preconditioner for `A`. It should approximate the + inverse of `A` (see Notes). Effective preconditioning dramatically improves the + rate of convergence, which implies that fewer iterations are needed + to reach a given error tolerance. + callback : function + User-supplied function to call after each iteration. It is called + as ``callback(xk)``, where ``xk`` is the current solution vector. + + Returns + ------- + x : ndarray + The converged solution. + info : integer + Provides convergence information: + 0 : successful exit + >0 : convergence to tolerance not achieved, number of iterations + <0 : parameter breakdown + + Notes + ----- + The preconditioner `M` should be a matrix such that ``M @ A`` has a smaller + condition number than `A`, see [1]_ . + + References + ---------- + .. [1] "Preconditioner", Wikipedia, + https://en.wikipedia.org/wiki/Preconditioner + .. [2] "Biconjugate gradient stabilized method", + Wikipedia, https://en.wikipedia.org/wiki/Biconjugate_gradient_stabilized_method + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import bicgstab + >>> R = np.array([[4, 2, 0, 1], + ... [3, 0, 0, 2], + ... [0, 1, 1, 1], + ... [0, 2, 1, 0]]) + >>> A = csc_array(R) + >>> b = np.array([-1, -0.5, -1, 2]) + >>> x, exit_code = bicgstab(A, b, atol=1e-5) + >>> print(exit_code) # 0 indicates successful convergence + 0 + >>> np.allclose(A.dot(x), b) + True + """ + A, M, x, b, postprocess = make_system(A, M, x0, b) + bnrm2 = np.linalg.norm(b) + + atol, _ = _get_atol_rtol('bicgstab', bnrm2, atol, rtol) + + if bnrm2 == 0: + return postprocess(b), 0 + + n = len(b) + + dotprod = np.vdot if np.iscomplexobj(x) else np.dot + + if maxiter is None: + maxiter = n*10 + + matvec = A.matvec + psolve = M.matvec + + # These values make no sense but coming from original Fortran code + # sqrt might have been meant instead. + rhotol = np.finfo(x.dtype.char).eps**2 + omegatol = rhotol + + # Dummy values to initialize vars, silence linter warnings + rho_prev, omega, alpha, p, v = None, None, None, None, None + + r = b - matvec(x) if x.any() else b.copy() + rtilde = r.copy() + + for iteration in range(maxiter): + if np.linalg.norm(r) < atol: # Are we done? + return postprocess(x), 0 + + rho = dotprod(rtilde, r) + if np.abs(rho) < rhotol: # rho breakdown + return postprocess(x), -10 + + if iteration > 0: + if np.abs(omega) < omegatol: # omega breakdown + return postprocess(x), -11 + + beta = (rho / rho_prev) * (alpha / omega) + p -= omega*v + p *= beta + p += r + else: # First spin + s = np.empty_like(r) + p = r.copy() + + phat = psolve(p) + v = matvec(phat) + rv = dotprod(rtilde, v) + if rv == 0: + return postprocess(x), -11 + alpha = rho / rv + r -= alpha*v + s[:] = r[:] + + if np.linalg.norm(s) < atol: + x += alpha*phat + return postprocess(x), 0 + + shat = psolve(s) + t = matvec(shat) + omega = dotprod(t, s) / dotprod(t, t) + x += alpha*phat + x += omega*shat + r -= omega*t + rho_prev = rho + + if callback: + callback(x) + + else: # for loop exhausted + # Return incomplete progress + return postprocess(x), maxiter + + +def cg(A, b, x0=None, *, rtol=1e-5, atol=0., maxiter=None, M=None, callback=None): + """Use Conjugate Gradient iteration to solve ``Ax = b``. + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + The real or complex N-by-N matrix of the linear system. + `A` must represent a hermitian, positive definite matrix. + Alternatively, `A` can be a linear operator which can + produce ``Ax`` using, e.g., + ``scipy.sparse.linalg.LinearOperator``. + b : ndarray + Right hand side of the linear system. Has shape (N,) or (N,1). + x0 : ndarray + Starting guess for the solution. + rtol, atol : float, optional + Parameters for the convergence test. For convergence, + ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. + The default is ``atol=0.`` and ``rtol=1e-5``. + maxiter : integer + Maximum number of iterations. Iteration will stop after maxiter + steps even if the specified tolerance has not been achieved. + M : {sparse array, ndarray, LinearOperator} + Preconditioner for `A`. `M` must represent a hermitian, positive definite + matrix. It should approximate the inverse of `A` (see Notes). + Effective preconditioning dramatically improves the + rate of convergence, which implies that fewer iterations are needed + to reach a given error tolerance. + callback : function + User-supplied function to call after each iteration. It is called + as ``callback(xk)``, where ``xk`` is the current solution vector. + + Returns + ------- + x : ndarray + The converged solution. + info : integer + Provides convergence information: + 0 : successful exit + >0 : convergence to tolerance not achieved, number of iterations + + Notes + ----- + The preconditioner `M` should be a matrix such that ``M @ A`` has a smaller + condition number than `A`, see [2]_. + + References + ---------- + .. [1] "Conjugate Gradient Method, Wikipedia, + https://en.wikipedia.org/wiki/Conjugate_gradient_method + .. [2] "Preconditioner", + Wikipedia, https://en.wikipedia.org/wiki/Preconditioner + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import cg + >>> P = np.array([[4, 0, 1, 0], + ... [0, 5, 0, 0], + ... [1, 0, 3, 2], + ... [0, 0, 2, 4]]) + >>> A = csc_array(P) + >>> b = np.array([-1, -0.5, -1, 2]) + >>> x, exit_code = cg(A, b, atol=1e-5) + >>> print(exit_code) # 0 indicates successful convergence + 0 + >>> np.allclose(A.dot(x), b) + True + """ + A, M, x, b, postprocess = make_system(A, M, x0, b) + bnrm2 = np.linalg.norm(b) + + atol, _ = _get_atol_rtol('cg', bnrm2, atol, rtol) + + if bnrm2 == 0: + return postprocess(b), 0 + + n = len(b) + + if maxiter is None: + maxiter = n*10 + + dotprod = np.vdot if np.iscomplexobj(x) else np.dot + + matvec = A.matvec + psolve = M.matvec + r = b - matvec(x) if x.any() else b.copy() + + # Dummy value to initialize var, silences warnings + rho_prev, p = None, None + + for iteration in range(maxiter): + if np.linalg.norm(r) < atol: # Are we done? + return postprocess(x), 0 + + z = psolve(r) + rho_cur = dotprod(r, z) + if iteration > 0: + beta = rho_cur / rho_prev + p *= beta + p += z + else: # First spin + p = np.empty_like(r) + p[:] = z[:] + + q = matvec(p) + alpha = rho_cur / dotprod(p, q) + x += alpha*p + r -= alpha*q + rho_prev = rho_cur + + if callback: + callback(x) + + else: # for loop exhausted + # Return incomplete progress + return postprocess(x), maxiter + + +def cgs(A, b, x0=None, *, rtol=1e-5, atol=0., maxiter=None, M=None, callback=None): + """Use Conjugate Gradient Squared iteration to solve ``Ax = b``. + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + The real-valued N-by-N matrix of the linear system. + Alternatively, `A` can be a linear operator which can + produce ``Ax`` using, e.g., + ``scipy.sparse.linalg.LinearOperator``. + b : ndarray + Right hand side of the linear system. Has shape (N,) or (N,1). + x0 : ndarray + Starting guess for the solution. + rtol, atol : float, optional + Parameters for the convergence test. For convergence, + ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. + The default is ``atol=0.`` and ``rtol=1e-5``. + maxiter : integer + Maximum number of iterations. Iteration will stop after maxiter + steps even if the specified tolerance has not been achieved. + M : {sparse array, ndarray, LinearOperator} + Preconditioner for ``A``. It should approximate the + inverse of `A` (see Notes). Effective preconditioning dramatically improves the + rate of convergence, which implies that fewer iterations are needed + to reach a given error tolerance. + callback : function + User-supplied function to call after each iteration. It is called + as ``callback(xk)``, where ``xk`` is the current solution vector. + + Returns + ------- + x : ndarray + The converged solution. + info : integer + Provides convergence information: + 0 : successful exit + >0 : convergence to tolerance not achieved, number of iterations + <0 : parameter breakdown + + Notes + ----- + The preconditioner `M` should be a matrix such that ``M @ A`` has a smaller + condition number than `A`, see [1]_. + + References + ---------- + .. [1] "Preconditioner", Wikipedia, + https://en.wikipedia.org/wiki/Preconditioner + .. [2] "Conjugate gradient squared", Wikipedia, + https://en.wikipedia.org/wiki/Conjugate_gradient_squared_method + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import cgs + >>> R = np.array([[4, 2, 0, 1], + ... [3, 0, 0, 2], + ... [0, 1, 1, 1], + ... [0, 2, 1, 0]]) + >>> A = csc_array(R) + >>> b = np.array([-1, -0.5, -1, 2]) + >>> x, exit_code = cgs(A, b) + >>> print(exit_code) # 0 indicates successful convergence + 0 + >>> np.allclose(A.dot(x), b) + True + """ + A, M, x, b, postprocess = make_system(A, M, x0, b) + bnrm2 = np.linalg.norm(b) + + atol, _ = _get_atol_rtol('cgs', bnrm2, atol, rtol) + + if bnrm2 == 0: + return postprocess(b), 0 + + n = len(b) + + dotprod = np.vdot if np.iscomplexobj(x) else np.dot + + if maxiter is None: + maxiter = n*10 + + matvec = A.matvec + psolve = M.matvec + + rhotol = np.finfo(x.dtype.char).eps**2 + + r = b - matvec(x) if x.any() else b.copy() + + rtilde = r.copy() + bnorm = np.linalg.norm(b) + if bnorm == 0: + bnorm = 1 + + # Dummy values to initialize vars, silence linter warnings + rho_prev, p, u, q = None, None, None, None + + for iteration in range(maxiter): + rnorm = np.linalg.norm(r) + if rnorm < atol: # Are we done? + return postprocess(x), 0 + + rho_cur = dotprod(rtilde, r) + if np.abs(rho_cur) < rhotol: # Breakdown case + return postprocess, -10 + + if iteration > 0: + beta = rho_cur / rho_prev + + # u = r + beta * q + # p = u + beta * (q + beta * p); + u[:] = r[:] + u += beta*q + + p *= beta + p += q + p *= beta + p += u + + else: # First spin + p = r.copy() + u = r.copy() + q = np.empty_like(r) + + phat = psolve(p) + vhat = matvec(phat) + rv = dotprod(rtilde, vhat) + + if rv == 0: # Dot product breakdown + return postprocess(x), -11 + + alpha = rho_cur / rv + q[:] = u[:] + q -= alpha*vhat + uhat = psolve(u + q) + x += alpha*uhat + + # Due to numerical error build-up the actual residual is computed + # instead of the following two lines that were in the original + # FORTRAN templates, still using a single matvec. + + # qhat = matvec(uhat) + # r -= alpha*qhat + r = b - matvec(x) + + rho_prev = rho_cur + + if callback: + callback(x) + + else: # for loop exhausted + # Return incomplete progress + return postprocess(x), maxiter + + +def gmres(A, b, x0=None, *, rtol=1e-5, atol=0., restart=None, maxiter=None, M=None, + callback=None, callback_type=None): + """ + Use Generalized Minimal RESidual iteration to solve ``Ax = b``. + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + The real or complex N-by-N matrix of the linear system. + Alternatively, `A` can be a linear operator which can + produce ``Ax`` using, e.g., + ``scipy.sparse.linalg.LinearOperator``. + b : ndarray + Right hand side of the linear system. Has shape (N,) or (N,1). + x0 : ndarray + Starting guess for the solution (a vector of zeros by default). + atol, rtol : float + Parameters for the convergence test. For convergence, + ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. + The default is ``atol=0.`` and ``rtol=1e-5``. + restart : int, optional + Number of iterations between restarts. Larger values increase + iteration cost, but may be necessary for convergence. + If omitted, ``min(20, n)`` is used. + maxiter : int, optional + Maximum number of iterations (restart cycles). Iteration will stop + after maxiter steps even if the specified tolerance has not been + achieved. See `callback_type`. + M : {sparse array, ndarray, LinearOperator} + Inverse of the preconditioner of `A`. `M` should approximate the + inverse of `A` and be easy to solve for (see Notes). Effective + preconditioning dramatically improves the rate of convergence, + which implies that fewer iterations are needed to reach a given + error tolerance. By default, no preconditioner is used. + In this implementation, left preconditioning is used, + and the preconditioned residual is minimized. However, the final + convergence is tested with respect to the ``b - A @ x`` residual. + callback : function + User-supplied function to call after each iteration. It is called + as ``callback(args)``, where ``args`` are selected by `callback_type`. + callback_type : {'x', 'pr_norm', 'legacy'}, optional + Callback function argument requested: + - ``x``: current iterate (ndarray), called on every restart + - ``pr_norm``: relative (preconditioned) residual norm (float), + called on every inner iteration + - ``legacy`` (default): same as ``pr_norm``, but also changes the + meaning of `maxiter` to count inner iterations instead of restart + cycles. + + This keyword has no effect if `callback` is not set. + + Returns + ------- + x : ndarray + The converged solution. + info : int + Provides convergence information: + 0 : successful exit + >0 : convergence to tolerance not achieved, number of iterations + + See Also + -------- + LinearOperator + + Notes + ----- + A preconditioner, P, is chosen such that P is close to A but easy to solve + for. The preconditioner parameter required by this routine is + ``M = P^-1``. The inverse should preferably not be calculated + explicitly. Rather, use the following template to produce M:: + + # Construct a linear operator that computes P^-1 @ x. + import scipy.sparse.linalg as spla + M_x = lambda x: spla.spsolve(P, x) + M = spla.LinearOperator((n, n), M_x) + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import gmres + >>> A = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) + >>> b = np.array([2, 4, -1], dtype=float) + >>> x, exitCode = gmres(A, b, atol=1e-5) + >>> print(exitCode) # 0 indicates successful convergence + 0 + >>> np.allclose(A.dot(x), b) + True + """ + if callback is not None and callback_type is None: + # Warn about 'callback_type' semantic changes. + # Probably should be removed only in far future, Scipy 2.0 or so. + msg = ("scipy.sparse.linalg.gmres called without specifying " + "`callback_type`. The default value will be changed in" + " a future release. For compatibility, specify a value " + "for `callback_type` explicitly, e.g., " + "``gmres(..., callback_type='pr_norm')``, or to retain the " + "old behavior ``gmres(..., callback_type='legacy')``" + ) + warnings.warn(msg, category=DeprecationWarning, stacklevel=3) + + if callback_type is None: + callback_type = 'legacy' + + if callback_type not in ('x', 'pr_norm', 'legacy'): + raise ValueError(f"Unknown callback_type: {callback_type!r}") + + if callback is None: + callback_type = None + + A, M, x, b, postprocess = make_system(A, M, x0, b) + matvec = A.matvec + psolve = M.matvec + n = len(b) + bnrm2 = np.linalg.norm(b) + + atol, _ = _get_atol_rtol('gmres', bnrm2, atol, rtol) + + if bnrm2 == 0: + return postprocess(b), 0 + + eps = np.finfo(x.dtype.char).eps + + dotprod = np.vdot if np.iscomplexobj(x) else np.dot + + if maxiter is None: + maxiter = n*10 + + if restart is None: + restart = 20 + restart = min(restart, n) + + Mb_nrm2 = np.linalg.norm(psolve(b)) + + # ==================================================== + # =========== Tolerance control from gh-8400 ========= + # ==================================================== + # Tolerance passed to GMRESREVCOM applies to the inner + # iteration and deals with the left-preconditioned + # residual. + ptol_max_factor = 1. + ptol = Mb_nrm2 * min(ptol_max_factor, atol / bnrm2) + presid = 0. + # ==================================================== + lartg = get_lapack_funcs('lartg', dtype=x.dtype) + + # allocate internal variables + v = np.empty([restart+1, n], dtype=x.dtype) + h = np.zeros([restart, restart+1], dtype=x.dtype) + givens = np.zeros([restart, 2], dtype=x.dtype) + + # legacy iteration count + inner_iter = 0 + + for iteration in range(maxiter): + if iteration == 0: + r = b - matvec(x) if x.any() else b.copy() + if np.linalg.norm(r) < atol: # Are we done? + return postprocess(x), 0 + + v[0, :] = psolve(r) + tmp = np.linalg.norm(v[0, :]) + v[0, :] *= (1 / tmp) + # RHS of the Hessenberg problem + S = np.zeros(restart+1, dtype=x.dtype) + S[0] = tmp + + breakdown = False + for col in range(restart): + av = matvec(v[col, :]) + w = psolve(av) + + # Modified Gram-Schmidt + h0 = np.linalg.norm(w) + for k in range(col+1): + tmp = dotprod(v[k, :], w) + h[col, k] = tmp + w -= tmp*v[k, :] + + h1 = np.linalg.norm(w) + h[col, col + 1] = h1 + v[col + 1, :] = w[:] + + # Exact solution indicator + if h1 <= eps*h0: + h[col, col + 1] = 0 + breakdown = True + else: + v[col + 1, :] *= (1 / h1) + + # apply past Givens rotations to current h column + for k in range(col): + c, s = givens[k, 0], givens[k, 1] + n0, n1 = h[col, [k, k+1]] + h[col, [k, k + 1]] = [c*n0 + s*n1, -s.conj()*n0 + c*n1] + + # get and apply current rotation to h and S + c, s, mag = lartg(h[col, col], h[col, col+1]) + givens[col, :] = [c, s] + h[col, [col, col+1]] = mag, 0 + + # S[col+1] component is always 0 + tmp = -np.conjugate(s)*S[col] + S[[col, col + 1]] = [c*S[col], tmp] + presid = np.abs(tmp) + inner_iter += 1 + + if callback_type in ('legacy', 'pr_norm'): + callback(presid / bnrm2) + # Legacy behavior + if callback_type == 'legacy' and inner_iter == maxiter: + break + if presid <= ptol or breakdown: + break + + # Solve h(col, col) upper triangular system and allow pseudo-solve + # singular cases as in (but without the f2py copies): + # y = trsv(h[:col+1, :col+1].T, S[:col+1]) + + if h[col, col] == 0: + S[col] = 0 + + y = np.zeros([col+1], dtype=x.dtype) + y[:] = S[:col+1] + for k in range(col, 0, -1): + if y[k] != 0: + y[k] /= h[k, k] + tmp = y[k] + y[:k] -= tmp*h[k, :k] + if y[0] != 0: + y[0] /= h[0, 0] + + x += y @ v[:col+1, :] + + r = b - matvec(x) + rnorm = np.linalg.norm(r) + + # Legacy exit + if callback_type == 'legacy' and inner_iter == maxiter: + return postprocess(x), 0 if rnorm <= atol else maxiter + + if callback_type == 'x': + callback(x) + + if rnorm <= atol: + break + elif breakdown: + # Reached breakdown (= exact solution), but the external + # tolerance check failed. Bail out with failure. + break + elif presid <= ptol: + # Inner loop passed but outer didn't + ptol_max_factor = max(eps, 0.25 * ptol_max_factor) + else: + ptol_max_factor = min(1.0, 1.5 * ptol_max_factor) + + ptol = presid * min(ptol_max_factor, atol / rnorm) + + info = 0 if (rnorm <= atol) else maxiter + return postprocess(x), info + + +def qmr(A, b, x0=None, *, rtol=1e-5, atol=0., maxiter=None, M1=None, M2=None, + callback=None): + """Use Quasi-Minimal Residual iteration to solve ``Ax = b``. + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + The real-valued N-by-N matrix of the linear system. + Alternatively, ``A`` can be a linear operator which can + produce ``Ax`` and ``A^T x`` using, e.g., + ``scipy.sparse.linalg.LinearOperator``. + b : ndarray + Right hand side of the linear system. Has shape (N,) or (N,1). + x0 : ndarray + Starting guess for the solution. + atol, rtol : float, optional + Parameters for the convergence test. For convergence, + ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. + The default is ``atol=0.`` and ``rtol=1e-5``. + maxiter : integer + Maximum number of iterations. Iteration will stop after maxiter + steps even if the specified tolerance has not been achieved. + M1 : {sparse array, ndarray, LinearOperator} + Left preconditioner for A. + M2 : {sparse array, ndarray, LinearOperator} + Right preconditioner for A. Used together with the left + preconditioner M1. The matrix M1@A@M2 should have better + conditioned than A alone. + callback : function + User-supplied function to call after each iteration. It is called + as callback(xk), where xk is the current solution vector. + + Returns + ------- + x : ndarray + The converged solution. + info : integer + Provides convergence information: + 0 : successful exit + >0 : convergence to tolerance not achieved, number of iterations + <0 : parameter breakdown + + See Also + -------- + LinearOperator + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import qmr + >>> A = csc_array([[3., 2., 0.], [1., -1., 0.], [0., 5., 1.]]) + >>> b = np.array([2., 4., -1.]) + >>> x, exitCode = qmr(A, b, atol=1e-5) + >>> print(exitCode) # 0 indicates successful convergence + 0 + >>> np.allclose(A.dot(x), b) + True + """ + A_ = A + A, M, x, b, postprocess = make_system(A, None, x0, b) + bnrm2 = np.linalg.norm(b) + + atol, _ = _get_atol_rtol('qmr', bnrm2, atol, rtol) + + if bnrm2 == 0: + return postprocess(b), 0 + + if M1 is None and M2 is None: + if hasattr(A_, 'psolve'): + def left_psolve(b): + return A_.psolve(b, 'left') + + def right_psolve(b): + return A_.psolve(b, 'right') + + def left_rpsolve(b): + return A_.rpsolve(b, 'left') + + def right_rpsolve(b): + return A_.rpsolve(b, 'right') + M1 = LinearOperator(A.shape, + matvec=left_psolve, + rmatvec=left_rpsolve) + M2 = LinearOperator(A.shape, + matvec=right_psolve, + rmatvec=right_rpsolve) + else: + def id(b): + return b + M1 = LinearOperator(A.shape, matvec=id, rmatvec=id) + M2 = LinearOperator(A.shape, matvec=id, rmatvec=id) + + n = len(b) + if maxiter is None: + maxiter = n*10 + + dotprod = np.vdot if np.iscomplexobj(x) else np.dot + + rhotol = np.finfo(x.dtype.char).eps + betatol = rhotol + gammatol = rhotol + deltatol = rhotol + epsilontol = rhotol + xitol = rhotol + + r = b - A.matvec(x) if x.any() else b.copy() + + vtilde = r.copy() + y = M1.matvec(vtilde) + rho = np.linalg.norm(y) + wtilde = r.copy() + z = M2.rmatvec(wtilde) + xi = np.linalg.norm(z) + gamma, eta, theta = 1, -1, 0 + v = np.empty_like(vtilde) + w = np.empty_like(wtilde) + + # Dummy values to initialize vars, silence linter warnings + epsilon, q, d, p, s = None, None, None, None, None + + for iteration in range(maxiter): + if np.linalg.norm(r) < atol: # Are we done? + return postprocess(x), 0 + if np.abs(rho) < rhotol: # rho breakdown + return postprocess(x), -10 + if np.abs(xi) < xitol: # xi breakdown + return postprocess(x), -15 + + v[:] = vtilde[:] + v *= (1 / rho) + y *= (1 / rho) + w[:] = wtilde[:] + w *= (1 / xi) + z *= (1 / xi) + delta = dotprod(z, y) + + if np.abs(delta) < deltatol: # delta breakdown + return postprocess(x), -13 + + ytilde = M2.matvec(y) + ztilde = M1.rmatvec(z) + + if iteration > 0: + ytilde -= (xi * delta / epsilon) * p + p[:] = ytilde[:] + ztilde -= (rho * (delta / epsilon).conj()) * q + q[:] = ztilde[:] + else: # First spin + p = ytilde.copy() + q = ztilde.copy() + + ptilde = A.matvec(p) + epsilon = dotprod(q, ptilde) + if np.abs(epsilon) < epsilontol: # epsilon breakdown + return postprocess(x), -14 + + beta = epsilon / delta + if np.abs(beta) < betatol: # beta breakdown + return postprocess(x), -11 + + vtilde[:] = ptilde[:] + vtilde -= beta*v + y = M1.matvec(vtilde) + + rho_prev = rho + rho = np.linalg.norm(y) + wtilde[:] = w[:] + wtilde *= - beta.conj() + wtilde += A.rmatvec(q) + z = M2.rmatvec(wtilde) + xi = np.linalg.norm(z) + gamma_prev = gamma + theta_prev = theta + theta = rho / (gamma_prev * np.abs(beta)) + gamma = 1 / np.sqrt(1 + theta**2) + + if np.abs(gamma) < gammatol: # gamma breakdown + return postprocess(x), -12 + + eta *= -(rho_prev / beta) * (gamma / gamma_prev)**2 + + if iteration > 0: + d *= (theta_prev * gamma) ** 2 + d += eta*p + s *= (theta_prev * gamma) ** 2 + s += eta*ptilde + else: + d = p.copy() + d *= eta + s = ptilde.copy() + s *= eta + + x += d + r -= s + + if callback: + callback(x) + + else: # for loop exhausted + # Return incomplete progress + return postprocess(x), maxiter diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/lgmres.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/lgmres.py new file mode 100644 index 0000000000000000000000000000000000000000..ce368e81a07f282d091cd9bc8281c98e720a206b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/lgmres.py @@ -0,0 +1,230 @@ +# Copyright (C) 2009, Pauli Virtanen +# Distributed under the same license as SciPy. + +import numpy as np +from numpy.linalg import LinAlgError +from scipy.linalg import get_blas_funcs +from .iterative import _get_atol_rtol +from .utils import make_system + +from ._gcrotmk import _fgmres + +__all__ = ['lgmres'] + + +def lgmres(A, b, x0=None, *, rtol=1e-5, atol=0., maxiter=1000, M=None, callback=None, + inner_m=30, outer_k=3, outer_v=None, store_outer_Av=True, + prepend_outer_v=False): + """ + Solve a matrix equation using the LGMRES algorithm. + + The LGMRES algorithm [1]_ [2]_ is designed to avoid some problems + in the convergence in restarted GMRES, and often converges in fewer + iterations. + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + The real or complex N-by-N matrix of the linear system. + Alternatively, ``A`` can be a linear operator which can + produce ``Ax`` using, e.g., + ``scipy.sparse.linalg.LinearOperator``. + b : ndarray + Right hand side of the linear system. Has shape (N,) or (N,1). + x0 : ndarray + Starting guess for the solution. + rtol, atol : float, optional + Parameters for the convergence test. For convergence, + ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. + The default is ``rtol=1e-5``, the default for ``atol`` is ``0.0``. + maxiter : int, optional + Maximum number of iterations. Iteration will stop after maxiter + steps even if the specified tolerance has not been achieved. + M : {sparse array, ndarray, LinearOperator}, optional + Preconditioner for A. The preconditioner should approximate the + inverse of A. Effective preconditioning dramatically improves the + rate of convergence, which implies that fewer iterations are needed + to reach a given error tolerance. + callback : function, optional + User-supplied function to call after each iteration. It is called + as callback(xk), where xk is the current solution vector. + inner_m : int, optional + Number of inner GMRES iterations per each outer iteration. + outer_k : int, optional + Number of vectors to carry between inner GMRES iterations. + According to [1]_, good values are in the range of 1...3. + However, note that if you want to use the additional vectors to + accelerate solving multiple similar problems, larger values may + be beneficial. + outer_v : list of tuples, optional + List containing tuples ``(v, Av)`` of vectors and corresponding + matrix-vector products, used to augment the Krylov subspace, and + carried between inner GMRES iterations. The element ``Av`` can + be `None` if the matrix-vector product should be re-evaluated. + This parameter is modified in-place by `lgmres`, and can be used + to pass "guess" vectors in and out of the algorithm when solving + similar problems. + store_outer_Av : bool, optional + Whether LGMRES should store also A@v in addition to vectors `v` + in the `outer_v` list. Default is True. + prepend_outer_v : bool, optional + Whether to put outer_v augmentation vectors before Krylov iterates. + In standard LGMRES, prepend_outer_v=False. + + Returns + ------- + x : ndarray + The converged solution. + info : int + Provides convergence information: + + - 0 : successful exit + - >0 : convergence to tolerance not achieved, number of iterations + - <0 : illegal input or breakdown + + Notes + ----- + The LGMRES algorithm [1]_ [2]_ is designed to avoid the + slowing of convergence in restarted GMRES, due to alternating + residual vectors. Typically, it often outperforms GMRES(m) of + comparable memory requirements by some measure, or at least is not + much worse. + + Another advantage in this algorithm is that you can supply it with + 'guess' vectors in the `outer_v` argument that augment the Krylov + subspace. If the solution lies close to the span of these vectors, + the algorithm converges faster. This can be useful if several very + similar matrices need to be inverted one after another, such as in + Newton-Krylov iteration where the Jacobian matrix often changes + little in the nonlinear steps. + + References + ---------- + .. [1] A.H. Baker and E.R. Jessup and T. Manteuffel, "A Technique for + Accelerating the Convergence of Restarted GMRES", SIAM J. Matrix + Anal. Appl. 26, 962 (2005). + .. [2] A.H. Baker, "On Improving the Performance of the Linear Solver + restarted GMRES", PhD thesis, University of Colorado (2003). + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import lgmres + >>> A = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) + >>> b = np.array([2, 4, -1], dtype=float) + >>> x, exitCode = lgmres(A, b, atol=1e-5) + >>> print(exitCode) # 0 indicates successful convergence + 0 + >>> np.allclose(A.dot(x), b) + True + """ + A,M,x,b,postprocess = make_system(A,M,x0,b) + + if not np.isfinite(b).all(): + raise ValueError("RHS must contain only finite numbers") + + matvec = A.matvec + psolve = M.matvec + + if outer_v is None: + outer_v = [] + + axpy, dot, scal = None, None, None + nrm2 = get_blas_funcs('nrm2', [b]) + + b_norm = nrm2(b) + + # we call this to get the right atol/rtol and raise errors as necessary + atol, rtol = _get_atol_rtol('lgmres', b_norm, atol, rtol) + + if b_norm == 0: + x = b + return (postprocess(x), 0) + + ptol_max_factor = 1.0 + + for k_outer in range(maxiter): + r_outer = matvec(x) - b + + # -- callback + if callback is not None: + callback(x) + + # -- determine input type routines + if axpy is None: + if np.iscomplexobj(r_outer) and not np.iscomplexobj(x): + x = x.astype(r_outer.dtype) + axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], + (x, r_outer)) + + # -- check stopping condition + r_norm = nrm2(r_outer) + if r_norm <= max(atol, rtol * b_norm): + break + + # -- inner LGMRES iteration + v0 = -psolve(r_outer) + inner_res_0 = nrm2(v0) + + if inner_res_0 == 0: + rnorm = nrm2(r_outer) + raise RuntimeError("Preconditioner returned a zero vector; " + f"|v| ~ {rnorm:.1g}, |M v| = 0") + + v0 = scal(1.0/inner_res_0, v0) + + ptol = min(ptol_max_factor, max(atol, rtol*b_norm)/r_norm) + + try: + Q, R, B, vs, zs, y, pres = _fgmres(matvec, + v0, + inner_m, + lpsolve=psolve, + atol=ptol, + outer_v=outer_v, + prepend_outer_v=prepend_outer_v) + y *= inner_res_0 + if not np.isfinite(y).all(): + # Overflow etc. in computation. There's no way to + # recover from this, so we have to bail out. + raise LinAlgError() + except LinAlgError: + # Floating point over/underflow, non-finite result from + # matmul etc. -- report failure. + return postprocess(x), k_outer + 1 + + # Inner loop tolerance control + if pres > ptol: + ptol_max_factor = min(1.0, 1.5 * ptol_max_factor) + else: + ptol_max_factor = max(1e-16, 0.25 * ptol_max_factor) + + # -- GMRES terminated: eval solution + dx = zs[0]*y[0] + for w, yc in zip(zs[1:], y[1:]): + dx = axpy(w, dx, dx.shape[0], yc) # dx += w*yc + + # -- Store LGMRES augmentation vectors + nx = nrm2(dx) + if nx > 0: + if store_outer_Av: + q = Q.dot(R.dot(y)) + ax = vs[0]*q[0] + for v, qc in zip(vs[1:], q[1:]): + ax = axpy(v, ax, ax.shape[0], qc) + outer_v.append((dx/nx, ax/nx)) + else: + outer_v.append((dx/nx, None)) + + # -- Retain only a finite number of augmentation vectors + while len(outer_v) > outer_k: + del outer_v[0] + + # -- Apply step + x += dx + else: + # didn't converge ... + return postprocess(x), maxiter + + return postprocess(x), 0 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/lsmr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/lsmr.py new file mode 100644 index 0000000000000000000000000000000000000000..97eb734aa64c3145044a81e97b0e1b8df9506ce2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/lsmr.py @@ -0,0 +1,486 @@ +""" +Copyright (C) 2010 David Fong and Michael Saunders + +LSMR uses an iterative method. + +07 Jun 2010: Documentation updated +03 Jun 2010: First release version in Python + +David Chin-lung Fong clfong@stanford.edu +Institute for Computational and Mathematical Engineering +Stanford University + +Michael Saunders saunders@stanford.edu +Systems Optimization Laboratory +Dept of MS&E, Stanford University. + +""" + +__all__ = ['lsmr'] + +from numpy import zeros, inf, atleast_1d, result_type +from numpy.linalg import norm +from math import sqrt +from scipy.sparse.linalg._interface import aslinearoperator + +from scipy.sparse.linalg._isolve.lsqr import _sym_ortho + + +def lsmr(A, b, damp=0.0, atol=1e-6, btol=1e-6, conlim=1e8, + maxiter=None, show=False, x0=None): + """Iterative solver for least-squares problems. + + lsmr solves the system of linear equations ``Ax = b``. If the system + is inconsistent, it solves the least-squares problem ``min ||b - Ax||_2``. + ``A`` is a rectangular matrix of dimension m-by-n, where all cases are + allowed: m = n, m > n, or m < n. ``b`` is a vector of length m. + The matrix A may be dense or sparse (usually sparse). + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + Matrix A in the linear system. + Alternatively, ``A`` can be a linear operator which can + produce ``Ax`` and ``A^H x`` using, e.g., + ``scipy.sparse.linalg.LinearOperator``. + b : array_like, shape (m,) + Vector ``b`` in the linear system. + damp : float + Damping factor for regularized least-squares. `lsmr` solves + the regularized least-squares problem:: + + min ||(b) - ( A )x|| + ||(0) (damp*I) ||_2 + + where damp is a scalar. If damp is None or 0, the system + is solved without regularization. Default is 0. + atol, btol : float, optional + Stopping tolerances. `lsmr` continues iterations until a + certain backward error estimate is smaller than some quantity + depending on atol and btol. Let ``r = b - Ax`` be the + residual vector for the current approximate solution ``x``. + If ``Ax = b`` seems to be consistent, `lsmr` terminates + when ``norm(r) <= atol * norm(A) * norm(x) + btol * norm(b)``. + Otherwise, `lsmr` terminates when ``norm(A^H r) <= + atol * norm(A) * norm(r)``. If both tolerances are 1.0e-6 (default), + the final ``norm(r)`` should be accurate to about 6 + digits. (The final ``x`` will usually have fewer correct digits, + depending on ``cond(A)`` and the size of LAMBDA.) If `atol` + or `btol` is None, a default value of 1.0e-6 will be used. + Ideally, they should be estimates of the relative error in the + entries of ``A`` and ``b`` respectively. For example, if the entries + of ``A`` have 7 correct digits, set ``atol = 1e-7``. This prevents + the algorithm from doing unnecessary work beyond the + uncertainty of the input data. + conlim : float, optional + `lsmr` terminates if an estimate of ``cond(A)`` exceeds + `conlim`. For compatible systems ``Ax = b``, conlim could be + as large as 1.0e+12 (say). For least-squares problems, + `conlim` should be less than 1.0e+8. If `conlim` is None, the + default value is 1e+8. Maximum precision can be obtained by + setting ``atol = btol = conlim = 0``, but the number of + iterations may then be excessive. Default is 1e8. + maxiter : int, optional + `lsmr` terminates if the number of iterations reaches + `maxiter`. The default is ``maxiter = min(m, n)``. For + ill-conditioned systems, a larger value of `maxiter` may be + needed. Default is False. + show : bool, optional + Print iterations logs if ``show=True``. Default is False. + x0 : array_like, shape (n,), optional + Initial guess of ``x``, if None zeros are used. Default is None. + + .. versionadded:: 1.0.0 + + Returns + ------- + x : ndarray of float + Least-square solution returned. + istop : int + istop gives the reason for stopping:: + + istop = 0 means x=0 is a solution. If x0 was given, then x=x0 is a + solution. + = 1 means x is an approximate solution to A@x = B, + according to atol and btol. + = 2 means x approximately solves the least-squares problem + according to atol. + = 3 means COND(A) seems to be greater than CONLIM. + = 4 is the same as 1 with atol = btol = eps (machine + precision) + = 5 is the same as 2 with atol = eps. + = 6 is the same as 3 with CONLIM = 1/eps. + = 7 means ITN reached maxiter before the other stopping + conditions were satisfied. + + itn : int + Number of iterations used. + normr : float + ``norm(b-Ax)`` + normar : float + ``norm(A^H (b - Ax))`` + norma : float + ``norm(A)`` + conda : float + Condition number of A. + normx : float + ``norm(x)`` + + Notes + ----- + + .. versionadded:: 0.11.0 + + References + ---------- + .. [1] D. C.-L. Fong and M. A. Saunders, + "LSMR: An iterative algorithm for sparse least-squares problems", + SIAM J. Sci. Comput., vol. 33, pp. 2950-2971, 2011. + :arxiv:`1006.0758` + .. [2] LSMR Software, https://web.stanford.edu/group/SOL/software/lsmr/ + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import lsmr + >>> A = csc_array([[1., 0.], [1., 1.], [0., 1.]], dtype=float) + + The first example has the trivial solution ``[0, 0]`` + + >>> b = np.array([0., 0., 0.], dtype=float) + >>> x, istop, itn, normr = lsmr(A, b)[:4] + >>> istop + 0 + >>> x + array([0., 0.]) + + The stopping code ``istop=0`` returned indicates that a vector of zeros was + found as a solution. The returned solution `x` indeed contains + ``[0., 0.]``. The next example has a non-trivial solution: + + >>> b = np.array([1., 0., -1.], dtype=float) + >>> x, istop, itn, normr = lsmr(A, b)[:4] + >>> istop + 1 + >>> x + array([ 1., -1.]) + >>> itn + 1 + >>> normr + 4.440892098500627e-16 + + As indicated by ``istop=1``, `lsmr` found a solution obeying the tolerance + limits. The given solution ``[1., -1.]`` obviously solves the equation. The + remaining return values include information about the number of iterations + (`itn=1`) and the remaining difference of left and right side of the solved + equation. + The final example demonstrates the behavior in the case where there is no + solution for the equation: + + >>> b = np.array([1., 0.01, -1.], dtype=float) + >>> x, istop, itn, normr = lsmr(A, b)[:4] + >>> istop + 2 + >>> x + array([ 1.00333333, -0.99666667]) + >>> A.dot(x)-b + array([ 0.00333333, -0.00333333, 0.00333333]) + >>> normr + 0.005773502691896255 + + `istop` indicates that the system is inconsistent and thus `x` is rather an + approximate solution to the corresponding least-squares problem. `normr` + contains the minimal distance that was found. + """ + + A = aslinearoperator(A) + b = atleast_1d(b) + if b.ndim > 1: + b = b.squeeze() + + msg = ('The exact solution is x = 0, or x = x0, if x0 was given ', + 'Ax - b is small enough, given atol, btol ', + 'The least-squares solution is good enough, given atol ', + 'The estimate of cond(Abar) has exceeded conlim ', + 'Ax - b is small enough for this machine ', + 'The least-squares solution is good enough for this machine', + 'Cond(Abar) seems to be too large for this machine ', + 'The iteration limit has been reached ') + + hdg1 = ' itn x(1) norm r norm Ar' + hdg2 = ' compatible LS norm A cond A' + pfreq = 20 # print frequency (for repeating the heading) + pcount = 0 # print counter + + m, n = A.shape + + # stores the num of singular values + minDim = min([m, n]) + + if maxiter is None: + maxiter = minDim + + if x0 is None: + dtype = result_type(A, b, float) + else: + dtype = result_type(A, b, x0, float) + + if show: + print(' ') + print('LSMR Least-squares solution of Ax = b\n') + print(f'The matrix A has {m} rows and {n} columns') + print(f'damp = {damp:20.14e}\n') + print(f'atol = {atol:8.2e} conlim = {conlim:8.2e}\n') + print(f'btol = {btol:8.2e} maxiter = {maxiter:8g}\n') + + u = b + normb = norm(b) + if x0 is None: + x = zeros(n, dtype) + beta = normb.copy() + else: + x = atleast_1d(x0.copy()) + u = u - A.matvec(x) + beta = norm(u) + + if beta > 0: + u = (1 / beta) * u + v = A.rmatvec(u) + alpha = norm(v) + else: + v = zeros(n, dtype) + alpha = 0 + + if alpha > 0: + v = (1 / alpha) * v + + # Initialize variables for 1st iteration. + + itn = 0 + zetabar = alpha * beta + alphabar = alpha + rho = 1 + rhobar = 1 + cbar = 1 + sbar = 0 + + h = v.copy() + hbar = zeros(n, dtype) + + # Initialize variables for estimation of ||r||. + + betadd = beta + betad = 0 + rhodold = 1 + tautildeold = 0 + thetatilde = 0 + zeta = 0 + d = 0 + + # Initialize variables for estimation of ||A|| and cond(A) + + normA2 = alpha * alpha + maxrbar = 0 + minrbar = 1e+100 + normA = sqrt(normA2) + condA = 1 + normx = 0 + + # Items for use in stopping rules, normb set earlier + istop = 0 + ctol = 0 + if conlim > 0: + ctol = 1 / conlim + normr = beta + + # Reverse the order here from the original matlab code because + # there was an error on return when arnorm==0 + normar = alpha * beta + if normar == 0: + if show: + print(msg[0]) + return x, istop, itn, normr, normar, normA, condA, normx + + if normb == 0: + x[()] = 0 + return x, istop, itn, normr, normar, normA, condA, normx + + if show: + print(' ') + print(hdg1, hdg2) + test1 = 1 + test2 = alpha / beta + str1 = f'{itn:6g} {x[0]:12.5e}' + str2 = f' {normr:10.3e} {normar:10.3e}' + str3 = f' {test1:8.1e} {test2:8.1e}' + print(''.join([str1, str2, str3])) + + # Main iteration loop. + while itn < maxiter: + itn = itn + 1 + + # Perform the next step of the bidiagonalization to obtain the + # next beta, u, alpha, v. These satisfy the relations + # beta*u = A@v - alpha*u, + # alpha*v = A'@u - beta*v. + + u *= -alpha + u += A.matvec(v) + beta = norm(u) + + if beta > 0: + u *= (1 / beta) + v *= -beta + v += A.rmatvec(u) + alpha = norm(v) + if alpha > 0: + v *= (1 / alpha) + + # At this point, beta = beta_{k+1}, alpha = alpha_{k+1}. + + # Construct rotation Qhat_{k,2k+1}. + + chat, shat, alphahat = _sym_ortho(alphabar, damp) + + # Use a plane rotation (Q_i) to turn B_i to R_i + + rhoold = rho + c, s, rho = _sym_ortho(alphahat, beta) + thetanew = s*alpha + alphabar = c*alpha + + # Use a plane rotation (Qbar_i) to turn R_i^T to R_i^bar + + rhobarold = rhobar + zetaold = zeta + thetabar = sbar * rho + rhotemp = cbar * rho + cbar, sbar, rhobar = _sym_ortho(cbar * rho, thetanew) + zeta = cbar * zetabar + zetabar = - sbar * zetabar + + # Update h, h_hat, x. + + hbar *= - (thetabar * rho / (rhoold * rhobarold)) + hbar += h + x += (zeta / (rho * rhobar)) * hbar + h *= - (thetanew / rho) + h += v + + # Estimate of ||r||. + + # Apply rotation Qhat_{k,2k+1}. + betaacute = chat * betadd + betacheck = -shat * betadd + + # Apply rotation Q_{k,k+1}. + betahat = c * betaacute + betadd = -s * betaacute + + # Apply rotation Qtilde_{k-1}. + # betad = betad_{k-1} here. + + thetatildeold = thetatilde + ctildeold, stildeold, rhotildeold = _sym_ortho(rhodold, thetabar) + thetatilde = stildeold * rhobar + rhodold = ctildeold * rhobar + betad = - stildeold * betad + ctildeold * betahat + + # betad = betad_k here. + # rhodold = rhod_k here. + + tautildeold = (zetaold - thetatildeold * tautildeold) / rhotildeold + taud = (zeta - thetatilde * tautildeold) / rhodold + d = d + betacheck * betacheck + normr = sqrt(d + (betad - taud)**2 + betadd * betadd) + + # Estimate ||A||. + normA2 = normA2 + beta * beta + normA = sqrt(normA2) + normA2 = normA2 + alpha * alpha + + # Estimate cond(A). + maxrbar = max(maxrbar, rhobarold) + if itn > 1: + minrbar = min(minrbar, rhobarold) + condA = max(maxrbar, rhotemp) / min(minrbar, rhotemp) + + # Test for convergence. + + # Compute norms for convergence testing. + normar = abs(zetabar) + normx = norm(x) + + # Now use these norms to estimate certain other quantities, + # some of which will be small near a solution. + + test1 = normr / normb + if (normA * normr) != 0: + test2 = normar / (normA * normr) + else: + test2 = inf + test3 = 1 / condA + t1 = test1 / (1 + normA * normx / normb) + rtol = btol + atol * normA * normx / normb + + # The following tests guard against extremely small values of + # atol, btol or ctol. (The user may have set any or all of + # the parameters atol, btol, conlim to 0.) + # The effect is equivalent to the normAl tests using + # atol = eps, btol = eps, conlim = 1/eps. + + if itn >= maxiter: + istop = 7 + if 1 + test3 <= 1: + istop = 6 + if 1 + test2 <= 1: + istop = 5 + if 1 + t1 <= 1: + istop = 4 + + # Allow for tolerances set by the user. + + if test3 <= ctol: + istop = 3 + if test2 <= atol: + istop = 2 + if test1 <= rtol: + istop = 1 + + # See if it is time to print something. + + if show: + if (n <= 40) or (itn <= 10) or (itn >= maxiter - 10) or \ + (itn % 10 == 0) or (test3 <= 1.1 * ctol) or \ + (test2 <= 1.1 * atol) or (test1 <= 1.1 * rtol) or \ + (istop != 0): + + if pcount >= pfreq: + pcount = 0 + print(' ') + print(hdg1, hdg2) + pcount = pcount + 1 + str1 = f'{itn:6g} {x[0]:12.5e}' + str2 = f' {normr:10.3e} {normar:10.3e}' + str3 = f' {test1:8.1e} {test2:8.1e}' + str4 = f' {normA:8.1e} {condA:8.1e}' + print(''.join([str1, str2, str3, str4])) + + if istop > 0: + break + + # Print the stopping condition. + + if show: + print(' ') + print('LSMR finished') + print(msg[istop]) + print(f'istop ={istop:8g} normr ={normr:8.1e}') + print(f' normA ={normA:8.1e} normAr ={normar:8.1e}') + print(f'itn ={itn:8g} condA ={condA:8.1e}') + print(f' normx ={normx:8.1e}') + print(str1, str2) + print(str3, str4) + + return x, istop, itn, normr, normar, normA, condA, normx diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/lsqr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/lsqr.py new file mode 100644 index 0000000000000000000000000000000000000000..3e490a0769e6d20303b1e5fdd111237b72d3abc9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/lsqr.py @@ -0,0 +1,589 @@ +"""Sparse Equations and Least Squares. + +The original Fortran code was written by C. C. Paige and M. A. Saunders as +described in + +C. C. Paige and M. A. Saunders, LSQR: An algorithm for sparse linear +equations and sparse least squares, TOMS 8(1), 43--71 (1982). + +C. C. Paige and M. A. Saunders, Algorithm 583; LSQR: Sparse linear +equations and least-squares problems, TOMS 8(2), 195--209 (1982). + +It is licensed under the following BSD license: + +Copyright (c) 2006, Systems Optimization Laboratory +All rights reserved. + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + * Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + * Redistributions in binary form must reproduce the above + copyright notice, this list of conditions and the following + disclaimer in the documentation and/or other materials provided + with the distribution. + + * Neither the name of Stanford University nor the names of its + contributors may be used to endorse or promote products derived + from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR +A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT +OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, +SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT +LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, +DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY +THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT +(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE +OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +The Fortran code was translated to Python for use in CVXOPT by Jeffery +Kline with contributions by Mridul Aanjaneya and Bob Myhill. + +Adapted for SciPy by Stefan van der Walt. + +""" + +__all__ = ['lsqr'] + +import numpy as np +from math import sqrt +from scipy.sparse.linalg._interface import aslinearoperator +from scipy.sparse._sputils import convert_pydata_sparse_to_scipy + +eps = np.finfo(np.float64).eps + + +def _sym_ortho(a, b): + """ + Stable implementation of Givens rotation. + + Notes + ----- + The routine 'SymOrtho' was added for numerical stability. This is + recommended by S.-C. Choi in [1]_. It removes the unpleasant potential of + ``1/eps`` in some important places (see, for example text following + "Compute the next plane rotation Qk" in minres.py). + + References + ---------- + .. [1] S.-C. Choi, "Iterative Methods for Singular Linear Equations + and Least-Squares Problems", Dissertation, + http://www.stanford.edu/group/SOL/dissertations/sou-cheng-choi-thesis.pdf + + """ + if b == 0: + return np.sign(a), 0, abs(a) + elif a == 0: + return 0, np.sign(b), abs(b) + elif abs(b) > abs(a): + tau = a / b + s = np.sign(b) / sqrt(1 + tau * tau) + c = s * tau + r = b / s + else: + tau = b / a + c = np.sign(a) / sqrt(1+tau*tau) + s = c * tau + r = a / c + return c, s, r + + +def lsqr(A, b, damp=0.0, atol=1e-6, btol=1e-6, conlim=1e8, + iter_lim=None, show=False, calc_var=False, x0=None): + """Find the least-squares solution to a large, sparse, linear system + of equations. + + The function solves ``Ax = b`` or ``min ||Ax - b||^2`` or + ``min ||Ax - b||^2 + d^2 ||x - x0||^2``. + + The matrix A may be square or rectangular (over-determined or + under-determined), and may have any rank. + + :: + + 1. Unsymmetric equations -- solve Ax = b + + 2. Linear least squares -- solve Ax = b + in the least-squares sense + + 3. Damped least squares -- solve ( A )*x = ( b ) + ( damp*I ) ( damp*x0 ) + in the least-squares sense + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + Representation of an m-by-n matrix. + Alternatively, ``A`` can be a linear operator which can + produce ``Ax`` and ``A^T x`` using, e.g., + ``scipy.sparse.linalg.LinearOperator``. + b : array_like, shape (m,) + Right-hand side vector ``b``. + damp : float + Damping coefficient. Default is 0. + atol, btol : float, optional + Stopping tolerances. `lsqr` continues iterations until a + certain backward error estimate is smaller than some quantity + depending on atol and btol. Let ``r = b - Ax`` be the + residual vector for the current approximate solution ``x``. + If ``Ax = b`` seems to be consistent, `lsqr` terminates + when ``norm(r) <= atol * norm(A) * norm(x) + btol * norm(b)``. + Otherwise, `lsqr` terminates when ``norm(A^H r) <= + atol * norm(A) * norm(r)``. If both tolerances are 1.0e-6 (default), + the final ``norm(r)`` should be accurate to about 6 + digits. (The final ``x`` will usually have fewer correct digits, + depending on ``cond(A)`` and the size of LAMBDA.) If `atol` + or `btol` is None, a default value of 1.0e-6 will be used. + Ideally, they should be estimates of the relative error in the + entries of ``A`` and ``b`` respectively. For example, if the entries + of ``A`` have 7 correct digits, set ``atol = 1e-7``. This prevents + the algorithm from doing unnecessary work beyond the + uncertainty of the input data. + conlim : float, optional + Another stopping tolerance. lsqr terminates if an estimate of + ``cond(A)`` exceeds `conlim`. For compatible systems ``Ax = + b``, `conlim` could be as large as 1.0e+12 (say). For + least-squares problems, conlim should be less than 1.0e+8. + Maximum precision can be obtained by setting ``atol = btol = + conlim = zero``, but the number of iterations may then be + excessive. Default is 1e8. + iter_lim : int, optional + Explicit limitation on number of iterations (for safety). + show : bool, optional + Display an iteration log. Default is False. + calc_var : bool, optional + Whether to estimate diagonals of ``(A'A + damp^2*I)^{-1}``. + x0 : array_like, shape (n,), optional + Initial guess of x, if None zeros are used. Default is None. + + .. versionadded:: 1.0.0 + + Returns + ------- + x : ndarray of float + The final solution. + istop : int + Gives the reason for termination. + 1 means x is an approximate solution to Ax = b. + 2 means x approximately solves the least-squares problem. + itn : int + Iteration number upon termination. + r1norm : float + ``norm(r)``, where ``r = b - Ax``. + r2norm : float + ``sqrt( norm(r)^2 + damp^2 * norm(x - x0)^2 )``. Equal to `r1norm` + if ``damp == 0``. + anorm : float + Estimate of Frobenius norm of ``Abar = [[A]; [damp*I]]``. + acond : float + Estimate of ``cond(Abar)``. + arnorm : float + Estimate of ``norm(A'@r - damp^2*(x - x0))``. + xnorm : float + ``norm(x)`` + var : ndarray of float + If ``calc_var`` is True, estimates all diagonals of + ``(A'A)^{-1}`` (if ``damp == 0``) or more generally ``(A'A + + damp^2*I)^{-1}``. This is well defined if A has full column + rank or ``damp > 0``. (Not sure what var means if ``rank(A) + < n`` and ``damp = 0.``) + + Notes + ----- + LSQR uses an iterative method to approximate the solution. The + number of iterations required to reach a certain accuracy depends + strongly on the scaling of the problem. Poor scaling of the rows + or columns of A should therefore be avoided where possible. + + For example, in problem 1 the solution is unaltered by + row-scaling. If a row of A is very small or large compared to + the other rows of A, the corresponding row of ( A b ) should be + scaled up or down. + + In problems 1 and 2, the solution x is easily recovered + following column-scaling. Unless better information is known, + the nonzero columns of A should be scaled so that they all have + the same Euclidean norm (e.g., 1.0). + + In problem 3, there is no freedom to re-scale if damp is + nonzero. However, the value of damp should be assigned only + after attention has been paid to the scaling of A. + + The parameter damp is intended to help regularize + ill-conditioned systems, by preventing the true solution from + being very large. Another aid to regularization is provided by + the parameter acond, which may be used to terminate iterations + before the computed solution becomes very large. + + If some initial estimate ``x0`` is known and if ``damp == 0``, + one could proceed as follows: + + 1. Compute a residual vector ``r0 = b - A@x0``. + 2. Use LSQR to solve the system ``A@dx = r0``. + 3. Add the correction dx to obtain a final solution ``x = x0 + dx``. + + This requires that ``x0`` be available before and after the call + to LSQR. To judge the benefits, suppose LSQR takes k1 iterations + to solve A@x = b and k2 iterations to solve A@dx = r0. + If x0 is "good", norm(r0) will be smaller than norm(b). + If the same stopping tolerances atol and btol are used for each + system, k1 and k2 will be similar, but the final solution x0 + dx + should be more accurate. The only way to reduce the total work + is to use a larger stopping tolerance for the second system. + If some value btol is suitable for A@x = b, the larger value + btol*norm(b)/norm(r0) should be suitable for A@dx = r0. + + Preconditioning is another way to reduce the number of iterations. + If it is possible to solve a related system ``M@x = b`` + efficiently, where M approximates A in some helpful way (e.g. M - + A has low rank or its elements are small relative to those of A), + LSQR may converge more rapidly on the system ``A@M(inverse)@z = + b``, after which x can be recovered by solving M@x = z. + + If A is symmetric, LSQR should not be used! + + Alternatives are the symmetric conjugate-gradient method (cg) + and/or SYMMLQ. SYMMLQ is an implementation of symmetric cg that + applies to any symmetric A and will converge more rapidly than + LSQR. If A is positive definite, there are other implementations + of symmetric cg that require slightly less work per iteration than + SYMMLQ (but will take the same number of iterations). + + References + ---------- + .. [1] C. C. Paige and M. A. Saunders (1982a). + "LSQR: An algorithm for sparse linear equations and + sparse least squares", ACM TOMS 8(1), 43-71. + .. [2] C. C. Paige and M. A. Saunders (1982b). + "Algorithm 583. LSQR: Sparse linear equations and least + squares problems", ACM TOMS 8(2), 195-209. + .. [3] M. A. Saunders (1995). "Solution of sparse rectangular + systems using LSQR and CRAIG", BIT 35, 588-604. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import lsqr + >>> A = csc_array([[1., 0.], [1., 1.], [0., 1.]], dtype=float) + + The first example has the trivial solution ``[0, 0]`` + + >>> b = np.array([0., 0., 0.], dtype=float) + >>> x, istop, itn, normr = lsqr(A, b)[:4] + >>> istop + 0 + >>> x + array([ 0., 0.]) + + The stopping code ``istop=0`` returned indicates that a vector of zeros was + found as a solution. The returned solution `x` indeed contains + ``[0., 0.]``. The next example has a non-trivial solution: + + >>> b = np.array([1., 0., -1.], dtype=float) + >>> x, istop, itn, r1norm = lsqr(A, b)[:4] + >>> istop + 1 + >>> x + array([ 1., -1.]) + >>> itn + 1 + >>> r1norm + 4.440892098500627e-16 + + As indicated by ``istop=1``, `lsqr` found a solution obeying the tolerance + limits. The given solution ``[1., -1.]`` obviously solves the equation. The + remaining return values include information about the number of iterations + (`itn=1`) and the remaining difference of left and right side of the solved + equation. + The final example demonstrates the behavior in the case where there is no + solution for the equation: + + >>> b = np.array([1., 0.01, -1.], dtype=float) + >>> x, istop, itn, r1norm = lsqr(A, b)[:4] + >>> istop + 2 + >>> x + array([ 1.00333333, -0.99666667]) + >>> A.dot(x)-b + array([ 0.00333333, -0.00333333, 0.00333333]) + >>> r1norm + 0.005773502691896255 + + `istop` indicates that the system is inconsistent and thus `x` is rather an + approximate solution to the corresponding least-squares problem. `r1norm` + contains the norm of the minimal residual that was found. + """ + A = convert_pydata_sparse_to_scipy(A) + A = aslinearoperator(A) + b = np.atleast_1d(b) + if b.ndim > 1: + b = b.squeeze() + + m, n = A.shape + if iter_lim is None: + iter_lim = 2 * n + var = np.zeros(n) + + msg = ('The exact solution is x = 0 ', + 'Ax - b is small enough, given atol, btol ', + 'The least-squares solution is good enough, given atol ', + 'The estimate of cond(Abar) has exceeded conlim ', + 'Ax - b is small enough for this machine ', + 'The least-squares solution is good enough for this machine', + 'Cond(Abar) seems to be too large for this machine ', + 'The iteration limit has been reached ') + + if show: + print(' ') + print('LSQR Least-squares solution of Ax = b') + str1 = f'The matrix A has {m} rows and {n} columns' + str2 = f'damp = {damp:20.14e} calc_var = {calc_var:8g}' + str3 = f'atol = {atol:8.2e} conlim = {conlim:8.2e}' + str4 = f'btol = {btol:8.2e} iter_lim = {iter_lim:8g}' + print(str1) + print(str2) + print(str3) + print(str4) + + itn = 0 + istop = 0 + ctol = 0 + if conlim > 0: + ctol = 1/conlim + anorm = 0 + acond = 0 + dampsq = damp**2 + ddnorm = 0 + res2 = 0 + xnorm = 0 + xxnorm = 0 + z = 0 + cs2 = -1 + sn2 = 0 + + # Set up the first vectors u and v for the bidiagonalization. + # These satisfy beta*u = b - A@x, alfa*v = A'@u. + u = b + bnorm = np.linalg.norm(b) + + if x0 is None: + x = np.zeros(n) + beta = bnorm.copy() + else: + x = np.asarray(x0) + u = u - A.matvec(x) + beta = np.linalg.norm(u) + + if beta > 0: + u = (1/beta) * u + v = A.rmatvec(u) + alfa = np.linalg.norm(v) + else: + v = x.copy() + alfa = 0 + + if alfa > 0: + v = (1/alfa) * v + w = v.copy() + + rhobar = alfa + phibar = beta + rnorm = beta + r1norm = rnorm + r2norm = rnorm + + # Reverse the order here from the original matlab code because + # there was an error on return when arnorm==0 + arnorm = alfa * beta + if arnorm == 0: + if show: + print(msg[0]) + return x, istop, itn, r1norm, r2norm, anorm, acond, arnorm, xnorm, var + + head1 = ' Itn x[0] r1norm r2norm ' + head2 = ' Compatible LS Norm A Cond A' + + if show: + print(' ') + print(head1, head2) + test1 = 1 + test2 = alfa / beta + str1 = f'{itn:6g} {x[0]:12.5e}' + str2 = f' {r1norm:10.3e} {r2norm:10.3e}' + str3 = f' {test1:8.1e} {test2:8.1e}' + print(str1, str2, str3) + + # Main iteration loop. + while itn < iter_lim: + itn = itn + 1 + # Perform the next step of the bidiagonalization to obtain the + # next beta, u, alfa, v. These satisfy the relations + # beta*u = a@v - alfa*u, + # alfa*v = A'@u - beta*v. + u = A.matvec(v) - alfa * u + beta = np.linalg.norm(u) + + if beta > 0: + u = (1/beta) * u + anorm = sqrt(anorm**2 + alfa**2 + beta**2 + dampsq) + v = A.rmatvec(u) - beta * v + alfa = np.linalg.norm(v) + if alfa > 0: + v = (1 / alfa) * v + + # Use a plane rotation to eliminate the damping parameter. + # This alters the diagonal (rhobar) of the lower-bidiagonal matrix. + if damp > 0: + rhobar1 = sqrt(rhobar**2 + dampsq) + cs1 = rhobar / rhobar1 + sn1 = damp / rhobar1 + psi = sn1 * phibar + phibar = cs1 * phibar + else: + # cs1 = 1 and sn1 = 0 + rhobar1 = rhobar + psi = 0. + + # Use a plane rotation to eliminate the subdiagonal element (beta) + # of the lower-bidiagonal matrix, giving an upper-bidiagonal matrix. + cs, sn, rho = _sym_ortho(rhobar1, beta) + + theta = sn * alfa + rhobar = -cs * alfa + phi = cs * phibar + phibar = sn * phibar + tau = sn * phi + + # Update x and w. + t1 = phi / rho + t2 = -theta / rho + dk = (1 / rho) * w + + x = x + t1 * w + w = v + t2 * w + ddnorm = ddnorm + np.linalg.norm(dk)**2 + + if calc_var: + var = var + dk**2 + + # Use a plane rotation on the right to eliminate the + # super-diagonal element (theta) of the upper-bidiagonal matrix. + # Then use the result to estimate norm(x). + delta = sn2 * rho + gambar = -cs2 * rho + rhs = phi - delta * z + zbar = rhs / gambar + xnorm = sqrt(xxnorm + zbar**2) + gamma = sqrt(gambar**2 + theta**2) + cs2 = gambar / gamma + sn2 = theta / gamma + z = rhs / gamma + xxnorm = xxnorm + z**2 + + # Test for convergence. + # First, estimate the condition of the matrix Abar, + # and the norms of rbar and Abar'rbar. + acond = anorm * sqrt(ddnorm) + res1 = phibar**2 + res2 = res2 + psi**2 + rnorm = sqrt(res1 + res2) + arnorm = alfa * abs(tau) + + # Distinguish between + # r1norm = ||b - Ax|| and + # r2norm = rnorm in current code + # = sqrt(r1norm^2 + damp^2*||x - x0||^2). + # Estimate r1norm from + # r1norm = sqrt(r2norm^2 - damp^2*||x - x0||^2). + # Although there is cancellation, it might be accurate enough. + if damp > 0: + r1sq = rnorm**2 - dampsq * xxnorm + r1norm = sqrt(abs(r1sq)) + if r1sq < 0: + r1norm = -r1norm + else: + r1norm = rnorm + r2norm = rnorm + + # Now use these norms to estimate certain other quantities, + # some of which will be small near a solution. + test1 = rnorm / bnorm + test2 = arnorm / (anorm * rnorm + eps) + test3 = 1 / (acond + eps) + t1 = test1 / (1 + anorm * xnorm / bnorm) + rtol = btol + atol * anorm * xnorm / bnorm + + # The following tests guard against extremely small values of + # atol, btol or ctol. (The user may have set any or all of + # the parameters atol, btol, conlim to 0.) + # The effect is equivalent to the normal tests using + # atol = eps, btol = eps, conlim = 1/eps. + if itn >= iter_lim: + istop = 7 + if 1 + test3 <= 1: + istop = 6 + if 1 + test2 <= 1: + istop = 5 + if 1 + t1 <= 1: + istop = 4 + + # Allow for tolerances set by the user. + if test3 <= ctol: + istop = 3 + if test2 <= atol: + istop = 2 + if test1 <= rtol: + istop = 1 + + if show: + # See if it is time to print something. + prnt = False + if n <= 40: + prnt = True + if itn <= 10: + prnt = True + if itn >= iter_lim-10: + prnt = True + # if itn%10 == 0: prnt = True + if test3 <= 2*ctol: + prnt = True + if test2 <= 10*atol: + prnt = True + if test1 <= 10*rtol: + prnt = True + if istop != 0: + prnt = True + + if prnt: + str1 = f'{itn:6g} {x[0]:12.5e}' + str2 = f' {r1norm:10.3e} {r2norm:10.3e}' + str3 = f' {test1:8.1e} {test2:8.1e}' + str4 = f' {anorm:8.1e} {acond:8.1e}' + print(str1, str2, str3, str4) + + if istop != 0: + break + + # End of iteration loop. + # Print the stopping condition. + if show: + print(' ') + print('LSQR finished') + print(msg[istop]) + print(' ') + str1 = f'istop ={istop:8g} r1norm ={r1norm:8.1e}' + str2 = f'anorm ={anorm:8.1e} arnorm ={arnorm:8.1e}' + str3 = f'itn ={itn:8g} r2norm ={r2norm:8.1e}' + str4 = f'acond ={acond:8.1e} xnorm ={xnorm:8.1e}' + print(str1 + ' ' + str2) + print(str3 + ' ' + str4) + print(' ') + + return x, istop, itn, r1norm, r2norm, anorm, acond, arnorm, xnorm, var diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/minres.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/minres.py new file mode 100644 index 0000000000000000000000000000000000000000..719d4eed991f15dda61da2c01f28d7f2244fc97d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/minres.py @@ -0,0 +1,372 @@ +from numpy import inner, zeros, inf, finfo +from numpy.linalg import norm +from math import sqrt + +from .utils import make_system + +__all__ = ['minres'] + + +def minres(A, b, x0=None, *, rtol=1e-5, shift=0.0, maxiter=None, + M=None, callback=None, show=False, check=False): + """ + Use MINimum RESidual iteration to solve Ax=b + + MINRES minimizes norm(Ax - b) for a real symmetric matrix A. Unlike + the Conjugate Gradient method, A can be indefinite or singular. + + If shift != 0 then the method solves (A - shift*I)x = b + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + The real symmetric N-by-N matrix of the linear system + Alternatively, ``A`` can be a linear operator which can + produce ``Ax`` using, e.g., + ``scipy.sparse.linalg.LinearOperator``. + b : ndarray + Right hand side of the linear system. Has shape (N,) or (N,1). + + Returns + ------- + x : ndarray + The converged solution. + info : integer + Provides convergence information: + 0 : successful exit + >0 : convergence to tolerance not achieved, number of iterations + <0 : illegal input or breakdown + + Other Parameters + ---------------- + x0 : ndarray + Starting guess for the solution. + shift : float + Value to apply to the system ``(A - shift * I)x = b``. Default is 0. + rtol : float + Tolerance to achieve. The algorithm terminates when the relative + residual is below ``rtol``. + maxiter : integer + Maximum number of iterations. Iteration will stop after maxiter + steps even if the specified tolerance has not been achieved. + M : {sparse array, ndarray, LinearOperator} + Preconditioner for A. The preconditioner should approximate the + inverse of A. Effective preconditioning dramatically improves the + rate of convergence, which implies that fewer iterations are needed + to reach a given error tolerance. + callback : function + User-supplied function to call after each iteration. It is called + as callback(xk), where xk is the current solution vector. + show : bool + If ``True``, print out a summary and metrics related to the solution + during iterations. Default is ``False``. + check : bool + If ``True``, run additional input validation to check that `A` and + `M` (if specified) are symmetric. Default is ``False``. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import minres + >>> A = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) + >>> A = A + A.T + >>> b = np.array([2, 4, -1], dtype=float) + >>> x, exitCode = minres(A, b) + >>> print(exitCode) # 0 indicates successful convergence + 0 + >>> np.allclose(A.dot(x), b) + True + + References + ---------- + Solution of sparse indefinite systems of linear equations, + C. C. Paige and M. A. Saunders (1975), + SIAM J. Numer. Anal. 12(4), pp. 617-629. + https://web.stanford.edu/group/SOL/software/minres/ + + This file is a translation of the following MATLAB implementation: + https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip + + """ + A, M, x, b, postprocess = make_system(A, M, x0, b) + + matvec = A.matvec + psolve = M.matvec + + first = 'Enter minres. ' + last = 'Exit minres. ' + + n = A.shape[0] + + if maxiter is None: + maxiter = 5 * n + + msg = [' beta2 = 0. If M = I, b and x are eigenvectors ', # -1 + ' beta1 = 0. The exact solution is x0 ', # 0 + ' A solution to Ax = b was found, given rtol ', # 1 + ' A least-squares solution was found, given rtol ', # 2 + ' Reasonable accuracy achieved, given eps ', # 3 + ' x has converged to an eigenvector ', # 4 + ' acond has exceeded 0.1/eps ', # 5 + ' The iteration limit was reached ', # 6 + ' A does not define a symmetric matrix ', # 7 + ' M does not define a symmetric matrix ', # 8 + ' M does not define a pos-def preconditioner '] # 9 + + if show: + print(first + 'Solution of symmetric Ax = b') + print(first + f'n = {n:3g} shift = {shift:23.14e}') + print(first + f'itnlim = {maxiter:3g} rtol = {rtol:11.2e}') + print() + + istop = 0 + itn = 0 + Anorm = 0 + Acond = 0 + rnorm = 0 + ynorm = 0 + + xtype = x.dtype + + eps = finfo(xtype).eps + + # Set up y and v for the first Lanczos vector v1. + # y = beta1 P' v1, where P = C**(-1). + # v is really P' v1. + + if x0 is None: + r1 = b.copy() + else: + r1 = b - A@x + y = psolve(r1) + + beta1 = inner(r1, y) + + if beta1 < 0: + raise ValueError('indefinite preconditioner') + elif beta1 == 0: + return (postprocess(x), 0) + + bnorm = norm(b) + if bnorm == 0: + x = b + return (postprocess(x), 0) + + beta1 = sqrt(beta1) + + if check: + # are these too strict? + + # see if A is symmetric + w = matvec(y) + r2 = matvec(w) + s = inner(w,w) + t = inner(y,r2) + z = abs(s - t) + epsa = (s + eps) * eps**(1.0/3.0) + if z > epsa: + raise ValueError('non-symmetric matrix') + + # see if M is symmetric + r2 = psolve(y) + s = inner(y,y) + t = inner(r1,r2) + z = abs(s - t) + epsa = (s + eps) * eps**(1.0/3.0) + if z > epsa: + raise ValueError('non-symmetric preconditioner') + + # Initialize other quantities + oldb = 0 + beta = beta1 + dbar = 0 + epsln = 0 + qrnorm = beta1 + phibar = beta1 + rhs1 = beta1 + rhs2 = 0 + tnorm2 = 0 + gmax = 0 + gmin = finfo(xtype).max + cs = -1 + sn = 0 + w = zeros(n, dtype=xtype) + w2 = zeros(n, dtype=xtype) + r2 = r1 + + if show: + print() + print() + print(' Itn x(1) Compatible LS norm(A) cond(A) gbar/|A|') + + while itn < maxiter: + itn += 1 + + s = 1.0/beta + v = s*y + + y = matvec(v) + y = y - shift * v + + if itn >= 2: + y = y - (beta/oldb)*r1 + + alfa = inner(v,y) + y = y - (alfa/beta)*r2 + r1 = r2 + r2 = y + y = psolve(r2) + oldb = beta + beta = inner(r2,y) + if beta < 0: + raise ValueError('non-symmetric matrix') + beta = sqrt(beta) + tnorm2 += alfa**2 + oldb**2 + beta**2 + + if itn == 1: + if beta/beta1 <= 10*eps: + istop = -1 # Terminate later + + # Apply previous rotation Qk-1 to get + # [deltak epslnk+1] = [cs sn][dbark 0 ] + # [gbar k dbar k+1] [sn -cs][alfak betak+1]. + + oldeps = epsln + delta = cs * dbar + sn * alfa # delta1 = 0 deltak + gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k + epsln = sn * beta # epsln2 = 0 epslnk+1 + dbar = - cs * beta # dbar 2 = beta2 dbar k+1 + root = norm([gbar, dbar]) + Arnorm = phibar * root + + # Compute the next plane rotation Qk + + gamma = norm([gbar, beta]) # gammak + gamma = max(gamma, eps) + cs = gbar / gamma # ck + sn = beta / gamma # sk + phi = cs * phibar # phik + phibar = sn * phibar # phibark+1 + + # Update x. + + denom = 1.0/gamma + w1 = w2 + w2 = w + w = (v - oldeps*w1 - delta*w2) * denom + x = x + phi*w + + # Go round again. + + gmax = max(gmax, gamma) + gmin = min(gmin, gamma) + z = rhs1 / gamma + rhs1 = rhs2 - delta*z + rhs2 = - epsln*z + + # Estimate various norms and test for convergence. + + Anorm = sqrt(tnorm2) + ynorm = norm(x) + epsa = Anorm * eps + epsx = Anorm * ynorm * eps + epsr = Anorm * ynorm * rtol + diag = gbar + + if diag == 0: + diag = epsa + + qrnorm = phibar + rnorm = qrnorm + if ynorm == 0 or Anorm == 0: + test1 = inf + else: + test1 = rnorm / (Anorm*ynorm) # ||r|| / (||A|| ||x||) + if Anorm == 0: + test2 = inf + else: + test2 = root / Anorm # ||Ar|| / (||A|| ||r||) + + # Estimate cond(A). + # In this version we look at the diagonals of R in the + # factorization of the lower Hessenberg matrix, Q @ H = R, + # where H is the tridiagonal matrix from Lanczos with one + # extra row, beta(k+1) e_k^T. + + Acond = gmax/gmin + + # See if any of the stopping criteria are satisfied. + # In rare cases, istop is already -1 from above (Abar = const*I). + + if istop == 0: + t1 = 1 + test1 # These tests work if rtol < eps + t2 = 1 + test2 + if t2 <= 1: + istop = 2 + if t1 <= 1: + istop = 1 + + if itn >= maxiter: + istop = 6 + if Acond >= 0.1/eps: + istop = 4 + if epsx >= beta1: + istop = 3 + # if rnorm <= epsx : istop = 2 + # if rnorm <= epsr : istop = 1 + if test2 <= rtol: + istop = 2 + if test1 <= rtol: + istop = 1 + + # See if it is time to print something. + + prnt = False + if n <= 40: + prnt = True + if itn <= 10: + prnt = True + if itn >= maxiter-10: + prnt = True + if itn % 10 == 0: + prnt = True + if qrnorm <= 10*epsx: + prnt = True + if qrnorm <= 10*epsr: + prnt = True + if Acond <= 1e-2/eps: + prnt = True + if istop != 0: + prnt = True + + if show and prnt: + str1 = f'{itn:6g} {x[0]:12.5e} {test1:10.3e}' + str2 = f' {test2:10.3e}' + str3 = f' {Anorm:8.1e} {Acond:8.1e} {gbar/Anorm:8.1e}' + + print(str1 + str2 + str3) + + if itn % 10 == 0: + print() + + if callback is not None: + callback(x) + + if istop != 0: + break # TODO check this + + if show: + print() + print(last + f' istop = {istop:3g} itn ={itn:5g}') + print(last + f' Anorm = {Anorm:12.4e} Acond = {Acond:12.4e}') + print(last + f' rnorm = {rnorm:12.4e} ynorm = {ynorm:12.4e}') + print(last + f' Arnorm = {Arnorm:12.4e}') + print(last + msg[istop+1]) + + if istop == 6: + info = maxiter + else: + info = 0 + + return (postprocess(x),info) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_gcrotmk.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_gcrotmk.py new file mode 100644 index 0000000000000000000000000000000000000000..6523f0b344e86bdcb1e520a61cbaed0a28e77e70 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_gcrotmk.py @@ -0,0 +1,183 @@ +#!/usr/bin/env python +"""Tests for the linalg._isolve.gcrotmk module +""" + +import threading +from numpy.testing import (assert_, assert_allclose, assert_equal, + suppress_warnings) + +import numpy as np +from numpy import zeros, array, allclose +from scipy.linalg import norm +from scipy.sparse import csr_array, eye_array, random_array + +from scipy.sparse.linalg._interface import LinearOperator +from scipy.sparse.linalg import splu +from scipy.sparse.linalg._isolve import gcrotmk, gmres + + +Am = csr_array(array([[-2,1,0,0,0,9], + [1,-2,1,0,5,0], + [0,1,-2,1,0,0], + [0,0,1,-2,1,0], + [0,3,0,1,-2,1], + [1,0,0,0,1,-2]])) +b = array([1,2,3,4,5,6]) +count = threading.local() # [0] +niter = threading.local() # [0] + + +def matvec(v): + if not hasattr(count, 'c'): + count.c = [0] + count.c[0] += 1 + return Am@v + + +def cb(v): + if not hasattr(niter, 'n'): + niter.n = [0] + niter.n[0] += 1 + + +A = LinearOperator(matvec=matvec, shape=Am.shape, dtype=Am.dtype) + + +def do_solve(**kw): + if not hasattr(niter, 'n'): + niter.n = [0] + + if not hasattr(count, 'c'): + count.c = [0] + + count.c[0] = 0 + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + x0, flag = gcrotmk(A, b, x0=zeros(A.shape[0]), rtol=1e-14, **kw) + count_0 = count.c[0] + assert_(allclose(A@x0, b, rtol=1e-12, atol=1e-12), norm(A@x0-b)) + return x0, count_0 + + +class TestGCROTMK: + def test_preconditioner(self): + # Check that preconditioning works + pc = splu(Am.tocsc()) + M = LinearOperator(matvec=pc.solve, shape=A.shape, dtype=A.dtype) + + x0, count_0 = do_solve() + niter.n[0] = 0 + x1, count_1 = do_solve(M=M, callback=cb) + + assert_equal(count_1, 3) + assert count_1 < count_0/2 + assert allclose(x1, x0, rtol=1e-14) + assert niter.n[0] < 3 + + def test_arnoldi(self): + rng = np.random.default_rng(1) + + A = eye_array(2000) + random_array((2000, 2000), density=5e-4, rng=rng) + b = rng.random(2000) + + # The inner arnoldi should be equivalent to gmres + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + x0, flag0 = gcrotmk(A, b, x0=zeros(A.shape[0]), m=10, k=0, maxiter=1) + x1, flag1 = gmres(A, b, x0=zeros(A.shape[0]), restart=10, maxiter=1) + + assert_equal(flag0, 1) + assert_equal(flag1, 1) + assert np.linalg.norm(A.dot(x0) - b) > 1e-4 + + assert_allclose(x0, x1) + + def test_cornercase(self): + np.random.seed(1234) + + # Rounding error may prevent convergence with tol=0 --- ensure + # that the return values in this case are correct, and no + # exceptions are raised + + for n in [3, 5, 10, 100]: + A = 2*eye_array(n) + + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + b = np.ones(n) + x, info = gcrotmk(A, b, maxiter=10) + assert_equal(info, 0) + assert_allclose(A.dot(x) - b, 0, atol=1e-14) + + x, info = gcrotmk(A, b, rtol=0, maxiter=10) + if info == 0: + assert_allclose(A.dot(x) - b, 0, atol=1e-14) + + b = np.random.rand(n) + x, info = gcrotmk(A, b, maxiter=10) + assert_equal(info, 0) + assert_allclose(A.dot(x) - b, 0, atol=1e-14) + + x, info = gcrotmk(A, b, rtol=0, maxiter=10) + if info == 0: + assert_allclose(A.dot(x) - b, 0, atol=1e-14) + + def test_nans(self): + A = eye_array(3, format='lil') + A[1,1] = np.nan + b = np.ones(3) + + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + x, info = gcrotmk(A, b, rtol=0, maxiter=10) + assert_equal(info, 1) + + def test_truncate(self): + np.random.seed(1234) + A = np.random.rand(30, 30) + np.eye(30) + b = np.random.rand(30) + + for truncate in ['oldest', 'smallest']: + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + x, info = gcrotmk(A, b, m=10, k=10, truncate=truncate, + rtol=1e-4, maxiter=200) + assert_equal(info, 0) + assert_allclose(A.dot(x) - b, 0, atol=1e-3) + + def test_CU(self): + for discard_C in (True, False): + # Check that C,U behave as expected + CU = [] + x0, count_0 = do_solve(CU=CU, discard_C=discard_C) + assert_(len(CU) > 0) + assert_(len(CU) <= 6) + + if discard_C: + for c, u in CU: + assert_(c is None) + + # should converge immediately + x1, count_1 = do_solve(CU=CU, discard_C=discard_C) + if discard_C: + assert_equal(count_1, 2 + len(CU)) + else: + assert_equal(count_1, 3) + assert_(count_1 <= count_0/2) + assert_allclose(x1, x0, atol=1e-14) + + def test_denormals(self): + # Check that no warnings are emitted if the matrix contains + # numbers for which 1/x has no float representation, and that + # the solver behaves properly. + A = np.array([[1, 2], [3, 4]], dtype=float) + A *= 100 * np.nextafter(0, 1) + + b = np.array([1, 1]) + + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + xp, info = gcrotmk(A, b) + + if info == 0: + assert_allclose(A.dot(xp), b) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_iterative.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_iterative.py new file mode 100644 index 0000000000000000000000000000000000000000..7daff5fc854a0a53fea80f7e5d7fde8736ceb2e7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_iterative.py @@ -0,0 +1,809 @@ +""" Test functions for the sparse.linalg._isolve module +""" + +import itertools +import platform +import pytest + +import numpy as np +from numpy.testing import assert_array_equal, assert_allclose +from numpy import zeros, arange, array, ones, eye, iscomplexobj +from numpy.linalg import norm + +from scipy.sparse import dia_array, csr_array, kronsum + +from scipy.sparse.linalg import LinearOperator, aslinearoperator +from scipy.sparse.linalg._isolve import (bicg, bicgstab, cg, cgs, + gcrotmk, gmres, lgmres, + minres, qmr, tfqmr) + +# TODO check that method preserve shape and type +# TODO test both preconditioner methods + + +# list of all solvers under test +_SOLVERS = [bicg, bicgstab, cg, cgs, gcrotmk, gmres, lgmres, + minres, qmr, tfqmr] + +CB_TYPE_FILTER = ".*called without specifying `callback_type`.*" + + +# create parametrized fixture for easy reuse in tests +@pytest.fixture(params=_SOLVERS, scope="session") +def solver(request): + """ + Fixture for all solvers in scipy.sparse.linalg._isolve + """ + return request.param + + +class Case: + def __init__(self, name, A, b=None, skip=None, nonconvergence=None): + self.name = name + self.A = A + if b is None: + self.b = arange(A.shape[0], dtype=float) + else: + self.b = b + if skip is None: + self.skip = [] + else: + self.skip = skip + if nonconvergence is None: + self.nonconvergence = [] + else: + self.nonconvergence = nonconvergence + + +class SingleTest: + def __init__(self, A, b, solver, casename, convergence=True): + self.A = A + self.b = b + self.solver = solver + self.name = casename + '-' + solver.__name__ + self.convergence = convergence + + def __repr__(self): + return f"<{self.name}>" + + +class IterativeParams: + def __init__(self): + sym_solvers = [minres, cg] + posdef_solvers = [cg] + real_solvers = [minres] + + # list of Cases + self.cases = [] + + # Symmetric and Positive Definite + N = 40 + data = ones((3, N)) + data[0, :] = 2 + data[1, :] = -1 + data[2, :] = -1 + Poisson1D = dia_array((data, [0, -1, 1]), shape=(N, N)).tocsr() + self.cases.append(Case("poisson1d", Poisson1D)) + # note: minres fails for single precision + self.cases.append(Case("poisson1d-F", Poisson1D.astype('f'), + skip=[minres])) + + # Symmetric and Negative Definite + self.cases.append(Case("neg-poisson1d", -Poisson1D, + skip=posdef_solvers)) + # note: minres fails for single precision + self.cases.append(Case("neg-poisson1d-F", (-Poisson1D).astype('f'), + skip=posdef_solvers + [minres])) + + # 2-dimensional Poisson equations + Poisson2D = kronsum(Poisson1D, Poisson1D) + # note: minres fails for 2-d poisson problem, + # it will be fixed in the future PR + self.cases.append(Case("poisson2d", Poisson2D, skip=[minres])) + # note: minres fails for single precision + self.cases.append(Case("poisson2d-F", Poisson2D.astype('f'), + skip=[minres])) + + # Symmetric and Indefinite + data = array([[6, -5, 2, 7, -1, 10, 4, -3, -8, 9]], dtype='d') + RandDiag = dia_array((data, [0]), shape=(10, 10)).tocsr() + self.cases.append(Case("rand-diag", RandDiag, skip=posdef_solvers)) + self.cases.append(Case("rand-diag-F", RandDiag.astype('f'), + skip=posdef_solvers)) + + # Random real-valued + rng = np.random.RandomState(1234) + data = rng.rand(4, 4) + self.cases.append(Case("rand", data, + skip=posdef_solvers + sym_solvers)) + self.cases.append(Case("rand-F", data.astype('f'), + skip=posdef_solvers + sym_solvers)) + + # Random symmetric real-valued + rng = np.random.RandomState(1234) + data = rng.rand(4, 4) + data = data + data.T + self.cases.append(Case("rand-sym", data, skip=posdef_solvers)) + self.cases.append(Case("rand-sym-F", data.astype('f'), + skip=posdef_solvers)) + + # Random pos-def symmetric real + np.random.seed(1234) + data = np.random.rand(9, 9) + data = np.dot(data.conj(), data.T) + self.cases.append(Case("rand-sym-pd", data)) + # note: minres fails for single precision + self.cases.append(Case("rand-sym-pd-F", data.astype('f'), + skip=[minres])) + + # Random complex-valued + rng = np.random.RandomState(1234) + data = rng.rand(4, 4) + 1j * rng.rand(4, 4) + skip_cmplx = posdef_solvers + sym_solvers + real_solvers + self.cases.append(Case("rand-cmplx", data, skip=skip_cmplx)) + self.cases.append(Case("rand-cmplx-F", data.astype('F'), + skip=skip_cmplx)) + + # Random hermitian complex-valued + rng = np.random.RandomState(1234) + data = rng.rand(4, 4) + 1j * rng.rand(4, 4) + data = data + data.T.conj() + self.cases.append(Case("rand-cmplx-herm", data, + skip=posdef_solvers + real_solvers)) + self.cases.append(Case("rand-cmplx-herm-F", data.astype('F'), + skip=posdef_solvers + real_solvers)) + + # Random pos-def hermitian complex-valued + rng = np.random.RandomState(1234) + data = rng.rand(9, 9) + 1j * rng.rand(9, 9) + data = np.dot(data.conj(), data.T) + self.cases.append(Case("rand-cmplx-sym-pd", data, skip=real_solvers)) + self.cases.append(Case("rand-cmplx-sym-pd-F", data.astype('F'), + skip=real_solvers)) + + # Non-symmetric and Positive Definite + # + # cgs, qmr, bicg and tfqmr fail to converge on this one + # -- algorithmic limitation apparently + data = ones((2, 10)) + data[0, :] = 2 + data[1, :] = -1 + A = dia_array((data, [0, -1]), shape=(10, 10)).tocsr() + self.cases.append(Case("nonsymposdef", A, + skip=sym_solvers + [cgs, qmr, bicg, tfqmr])) + self.cases.append(Case("nonsymposdef-F", A.astype('F'), + skip=sym_solvers + [cgs, qmr, bicg, tfqmr])) + + # Symmetric, non-pd, hitting cgs/bicg/bicgstab/qmr/tfqmr breakdown + A = np.array([[0, 0, 0, 0, 0, 1, -1, -0, -0, -0, -0], + [0, 0, 0, 0, 0, 2, -0, -1, -0, -0, -0], + [0, 0, 0, 0, 0, 2, -0, -0, -1, -0, -0], + [0, 0, 0, 0, 0, 2, -0, -0, -0, -1, -0], + [0, 0, 0, 0, 0, 1, -0, -0, -0, -0, -1], + [1, 2, 2, 2, 1, 0, -0, -0, -0, -0, -0], + [-1, 0, 0, 0, 0, 0, -1, -0, -0, -0, -0], + [0, -1, 0, 0, 0, 0, -0, -1, -0, -0, -0], + [0, 0, -1, 0, 0, 0, -0, -0, -1, -0, -0], + [0, 0, 0, -1, 0, 0, -0, -0, -0, -1, -0], + [0, 0, 0, 0, -1, 0, -0, -0, -0, -0, -1]], dtype=float) + b = np.array([0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], dtype=float) + assert (A == A.T).all() + self.cases.append(Case("sym-nonpd", A, b, + skip=posdef_solvers, + nonconvergence=[cgs, bicg, bicgstab, qmr, tfqmr] + ) + ) + + def generate_tests(self): + # generate test cases with skips applied + tests = [] + for case in self.cases: + for solver in _SOLVERS: + if (solver in case.skip): + continue + if solver in case.nonconvergence: + tests += [SingleTest(case.A, case.b, solver, case.name, + convergence=False)] + else: + tests += [SingleTest(case.A, case.b, solver, case.name)] + return tests + + +cases = IterativeParams().generate_tests() + + +@pytest.fixture(params=cases, ids=[x.name for x in cases], scope="module") +def case(request): + """ + Fixture for all cases in IterativeParams + """ + return request.param + +@pytest.mark.thread_unsafe +def test_maxiter(case): + if not case.convergence: + pytest.skip("Solver - Breakdown case, see gh-8829") + A = case.A + rtol = 1e-12 + + b = case.b + x0 = 0 * b + + residuals = [] + + def callback(x): + if x.ndim == 0: + residuals.append(norm(b - case.A * x)) + else: + residuals.append(norm(b - case.A @ x)) + + if case.solver == gmres: + with pytest.warns(DeprecationWarning, match=CB_TYPE_FILTER): + x, info = case.solver(A, b, x0=x0, rtol=rtol, maxiter=1, callback=callback) + else: + x, info = case.solver(A, b, x0=x0, rtol=rtol, maxiter=1, callback=callback) + + assert len(residuals) == 1 + assert info == 1 + + +def test_convergence(case): + A = case.A + + if A.dtype.char in "dD": + rtol = 1e-8 + else: + rtol = 1e-2 + + b = case.b + x0 = 0 * b + + x, info = case.solver(A, b, x0=x0, rtol=rtol) + + assert_array_equal(x0, 0 * b) # ensure that x0 is not overwritten + if case.convergence: + assert info == 0 + assert norm(A @ x - b) <= norm(b) * rtol + else: + assert info != 0 + assert norm(A @ x - b) <= norm(b) + + +def test_precond_dummy(case): + if not case.convergence: + pytest.skip("Solver - Breakdown case, see gh-8829") + + rtol = 1e-8 + + def identity(b, which=None): + """trivial preconditioner""" + return b + + A = case.A + + M, N = A.shape + # Ensure the diagonal elements of A are non-zero before calculating + # 1.0/A.diagonal() + diagOfA = A.diagonal() + if np.count_nonzero(diagOfA) == len(diagOfA): + dia_array(([1.0 / diagOfA], [0]), shape=(M, N)) + + b = case.b + x0 = 0 * b + + precond = LinearOperator(A.shape, identity, rmatvec=identity) + + if case.solver is qmr: + x, info = case.solver(A, b, M1=precond, M2=precond, x0=x0, rtol=rtol) + else: + x, info = case.solver(A, b, M=precond, x0=x0, rtol=rtol) + assert info == 0 + assert norm(A @ x - b) <= norm(b) * rtol + + A = aslinearoperator(A) + A.psolve = identity + A.rpsolve = identity + + x, info = case.solver(A, b, x0=x0, rtol=rtol) + assert info == 0 + assert norm(A @ x - b) <= norm(b) * rtol + + +# Specific test for poisson1d and poisson2d cases +@pytest.mark.fail_slow(10) +@pytest.mark.parametrize('case', [x for x in IterativeParams().cases + if x.name in ('poisson1d', 'poisson2d')], + ids=['poisson1d', 'poisson2d']) +def test_precond_inverse(case): + for solver in _SOLVERS: + if solver in case.skip or solver is qmr: + continue + + rtol = 1e-8 + + def inverse(b, which=None): + """inverse preconditioner""" + A = case.A + if not isinstance(A, np.ndarray): + A = A.toarray() + return np.linalg.solve(A, b) + + def rinverse(b, which=None): + """inverse preconditioner""" + A = case.A + if not isinstance(A, np.ndarray): + A = A.toarray() + return np.linalg.solve(A.T, b) + + matvec_count = [0] + + def matvec(b): + matvec_count[0] += 1 + return case.A @ b + + def rmatvec(b): + matvec_count[0] += 1 + return case.A.T @ b + + b = case.b + x0 = 0 * b + + A = LinearOperator(case.A.shape, matvec, rmatvec=rmatvec) + precond = LinearOperator(case.A.shape, inverse, rmatvec=rinverse) + + # Solve with preconditioner + matvec_count = [0] + x, info = solver(A, b, M=precond, x0=x0, rtol=rtol) + + assert info == 0 + assert norm(case.A @ x - b) <= norm(b) * rtol + + # Solution should be nearly instant + assert matvec_count[0] <= 3 + + +def test_atol(solver): + # TODO: minres / tfqmr. It didn't historically use absolute tolerances, so + # fixing it is less urgent. + if solver in (minres, tfqmr): + pytest.skip("TODO: Add atol to minres/tfqmr") + + # Historically this is tested as below, all pass but for some reason + # gcrotmk is over-sensitive to difference between random.seed/rng.random + # Hence tol lower bound is changed from -10 to -9 + # np.random.seed(1234) + # A = np.random.rand(10, 10) + # A = A @ A.T + 10 * np.eye(10) + # b = 1e3*np.random.rand(10) + + rng = np.random.default_rng(168441431005389) + A = rng.uniform(size=[10, 10]) + A = A @ A.T + 10*np.eye(10) + b = 1e3 * rng.uniform(size=10) + + b_norm = np.linalg.norm(b) + + tols = np.r_[0, np.logspace(-9, 2, 7), np.inf] + + # Check effect of badly scaled preconditioners + M0 = rng.standard_normal(size=(10, 10)) + M0 = M0 @ M0.T + Ms = [None, 1e-6 * M0, 1e6 * M0] + + for M, rtol, atol in itertools.product(Ms, tols, tols): + if rtol == 0 and atol == 0: + continue + + if solver is qmr: + if M is not None: + M = aslinearoperator(M) + M2 = aslinearoperator(np.eye(10)) + else: + M2 = None + x, info = solver(A, b, M1=M, M2=M2, rtol=rtol, atol=atol) + else: + x, info = solver(A, b, M=M, rtol=rtol, atol=atol) + + assert info == 0 + residual = A @ x - b + err = np.linalg.norm(residual) + atol2 = rtol * b_norm + # Added 1.00025 fudge factor because of `err` exceeding `atol` just + # very slightly on s390x (see gh-17839) + assert err <= 1.00025 * max(atol, atol2) + + +def test_zero_rhs(solver): + rng = np.random.default_rng(1684414984100503) + A = rng.random(size=[10, 10]) + A = A @ A.T + 10 * np.eye(10) + + b = np.zeros(10) + tols = np.r_[np.logspace(-10, 2, 7)] + + for tol in tols: + x, info = solver(A, b, rtol=tol) + assert info == 0 + assert_allclose(x, 0., atol=1e-15) + + x, info = solver(A, b, rtol=tol, x0=ones(10)) + assert info == 0 + assert_allclose(x, 0., atol=tol) + + if solver is not minres: + x, info = solver(A, b, rtol=tol, atol=0, x0=ones(10)) + if info == 0: + assert_allclose(x, 0) + + x, info = solver(A, b, rtol=tol, atol=tol) + assert info == 0 + assert_allclose(x, 0, atol=1e-300) + + x, info = solver(A, b, rtol=tol, atol=0) + assert info == 0 + assert_allclose(x, 0, atol=1e-300) + + +@pytest.mark.xfail(reason="see gh-18697") +def test_maxiter_worsening(solver): + if solver not in (gmres, lgmres, qmr): + # these were skipped from the very beginning, see gh-9201; gh-14160 + pytest.skip("Solver breakdown case") + # Check error does not grow (boundlessly) with increasing maxiter. + # This can occur due to the solvers hitting close to breakdown, + # which they should detect and halt as necessary. + # cf. gh-9100 + if (solver is lgmres and + platform.machine() not in ['x86_64' 'x86', 'aarch64', 'arm64']): + # see gh-17839 + pytest.xfail(reason="fails on at least ppc64le, ppc64 and riscv64") + + # Singular matrix, rhs numerically not in range + A = np.array([[-0.1112795288033378, 0, 0, 0.16127952880333685], + [0, -0.13627952880333782 + 6.283185307179586j, 0, 0], + [0, 0, -0.13627952880333782 - 6.283185307179586j, 0], + [0.1112795288033368, 0j, 0j, -0.16127952880333785]]) + v = np.ones(4) + best_error = np.inf + + # Unable to match the Fortran code tolerance levels with this example + # Original tolerance values + + # slack_tol = 7 if platform.machine() == 'aarch64' else 5 + slack_tol = 9 + + for maxiter in range(1, 20): + x, info = solver(A, v, maxiter=maxiter, rtol=1e-8, atol=0) + + if info == 0: + assert norm(A @ x - v) <= 1e-8 * norm(v) + + error = np.linalg.norm(A @ x - v) + best_error = min(best_error, error) + + # Check with slack + assert error <= slack_tol * best_error + + +def test_x0_working(solver): + # Easy problem + rng = np.random.default_rng(1685363802304750) + n = 10 + A = rng.random(size=[n, n]) + A = A @ A.T + b = rng.random(n) + x0 = rng.random(n) + + if solver is minres: + kw = dict(rtol=1e-6) + else: + kw = dict(atol=0, rtol=1e-6) + + x, info = solver(A, b, **kw) + assert info == 0 + assert norm(A @ x - b) <= 1e-6 * norm(b) + + x, info = solver(A, b, x0=x0, **kw) + assert info == 0 + assert norm(A @ x - b) <= 4.5e-6*norm(b) + + +def test_x0_equals_Mb(case): + if (case.solver is bicgstab) and (case.name == 'nonsymposdef-bicgstab'): + pytest.skip("Solver fails due to numerical noise " + "on some architectures (see gh-15533).") + if case.solver is tfqmr: + pytest.skip("Solver does not support x0='Mb'") + + A = case.A + b = case.b + x0 = 'Mb' + rtol = 1e-8 + x, info = case.solver(A, b, x0=x0, rtol=rtol) + + assert_array_equal(x0, 'Mb') # ensure that x0 is not overwritten + assert info == 0 + assert norm(A @ x - b) <= rtol * norm(b) + + +@pytest.mark.parametrize('solver', _SOLVERS) +def test_x0_solves_problem_exactly(solver): + # See gh-19948 + mat = np.eye(2) + rhs = np.array([-1., -1.]) + + sol, info = solver(mat, rhs, x0=rhs) + assert_allclose(sol, rhs) + assert info == 0 + + +# Specific tfqmr test +@pytest.mark.thread_unsafe +@pytest.mark.parametrize('case', IterativeParams().cases) +def test_show(case, capsys): + def cb(x): + pass + + x, info = tfqmr(case.A, case.b, callback=cb, show=True) + out, err = capsys.readouterr() + + if case.name == "sym-nonpd": + # no logs for some reason + exp = "" + elif case.name in ("nonsymposdef", "nonsymposdef-F"): + # Asymmetric and Positive Definite + exp = "TFQMR: Linear solve not converged due to reach MAXIT iterations" + else: # all other cases + exp = "TFQMR: Linear solve converged due to reach TOL iterations" + + assert out.startswith(exp) + assert err == "" + + +def test_positional_error(solver): + # from test_x0_working + rng = np.random.default_rng(1685363802304750) + n = 10 + A = rng.random(size=[n, n]) + A = A @ A.T + b = rng.random(n) + x0 = rng.random(n) + with pytest.raises(TypeError): + solver(A, b, x0, 1e-5) + + +@pytest.mark.parametrize("atol", ["legacy", None, -1]) +def test_invalid_atol(solver, atol): + if solver == minres: + pytest.skip("minres has no `atol` argument") + # from test_x0_working + rng = np.random.default_rng(1685363802304750) + n = 10 + A = rng.random(size=[n, n]) + A = A @ A.T + b = rng.random(n) + x0 = rng.random(n) + with pytest.raises(ValueError): + solver(A, b, x0, atol=atol) + + +class TestQMR: + @pytest.mark.filterwarnings('ignore::scipy.sparse.SparseEfficiencyWarning') + def test_leftright_precond(self): + """Check that QMR works with left and right preconditioners""" + + from scipy.sparse.linalg._dsolve import splu + from scipy.sparse.linalg._interface import LinearOperator + + n = 100 + + dat = ones(n) + A = dia_array(([-2 * dat, 4 * dat, -dat], [-1, 0, 1]), shape=(n, n)) + b = arange(n, dtype='d') + + L = dia_array(([-dat / 2, dat], [-1, 0]), shape=(n, n)) + U = dia_array(([4 * dat, -dat], [0, 1]), shape=(n, n)) + L_solver = splu(L) + U_solver = splu(U) + + def L_solve(b): + return L_solver.solve(b) + + def U_solve(b): + return U_solver.solve(b) + + def LT_solve(b): + return L_solver.solve(b, 'T') + + def UT_solve(b): + return U_solver.solve(b, 'T') + + M1 = LinearOperator((n, n), matvec=L_solve, rmatvec=LT_solve) + M2 = LinearOperator((n, n), matvec=U_solve, rmatvec=UT_solve) + + rtol = 1e-8 + x, info = qmr(A, b, rtol=rtol, maxiter=15, M1=M1, M2=M2) + + assert info == 0 + assert norm(A @ x - b) <= rtol * norm(b) + + +class TestGMRES: + def test_basic(self): + A = np.vander(np.arange(10) + 1)[:, ::-1] + b = np.zeros(10) + b[0] = 1 + + x_gm, err = gmres(A, b, restart=5, maxiter=1) + + assert_allclose(x_gm[0], 0.359, rtol=1e-2) + + @pytest.mark.filterwarnings(f"ignore:{CB_TYPE_FILTER}:DeprecationWarning") + def test_callback(self): + + def store_residual(r, rvec): + rvec[rvec.nonzero()[0].max() + 1] = r + + # Define, A,b + A = csr_array(array([[-2, 1, 0, 0, 0, 0], + [1, -2, 1, 0, 0, 0], + [0, 1, -2, 1, 0, 0], + [0, 0, 1, -2, 1, 0], + [0, 0, 0, 1, -2, 1], + [0, 0, 0, 0, 1, -2]])) + b = ones((A.shape[0],)) + maxiter = 1 + rvec = zeros(maxiter + 1) + rvec[0] = 1.0 + + def callback(r): + return store_residual(r, rvec) + + x, flag = gmres(A, b, x0=zeros(A.shape[0]), rtol=1e-16, + maxiter=maxiter, callback=callback) + + # Expected output from SciPy 1.0.0 + assert_allclose(rvec, array([1.0, 0.81649658092772603]), rtol=1e-10) + + # Test preconditioned callback + M = 1e-3 * np.eye(A.shape[0]) + rvec = zeros(maxiter + 1) + rvec[0] = 1.0 + x, flag = gmres(A, b, M=M, rtol=1e-16, maxiter=maxiter, + callback=callback) + + # Expected output from SciPy 1.0.0 + # (callback has preconditioned residual!) + assert_allclose(rvec, array([1.0, 1e-3 * 0.81649658092772603]), + rtol=1e-10) + + def test_abi(self): + # Check we don't segfault on gmres with complex argument + A = eye(2) + b = ones(2) + r_x, r_info = gmres(A, b) + r_x = r_x.astype(complex) + x, info = gmres(A.astype(complex), b.astype(complex)) + + assert iscomplexobj(x) + assert_allclose(r_x, x) + assert r_info == info + + @pytest.mark.fail_slow(10) + def test_atol_legacy(self): + + A = eye(2) + b = ones(2) + x, info = gmres(A, b, rtol=1e-5) + assert np.linalg.norm(A @ x - b) <= 1e-5 * np.linalg.norm(b) + assert_allclose(x, b, atol=0, rtol=1e-8) + + rndm = np.random.RandomState(12345) + A = rndm.rand(30, 30) + b = 1e-6 * ones(30) + x, info = gmres(A, b, rtol=1e-7, restart=20) + assert np.linalg.norm(A @ x - b) > 1e-7 + + A = eye(2) + b = 1e-10 * ones(2) + x, info = gmres(A, b, rtol=1e-8, atol=0) + assert np.linalg.norm(A @ x - b) <= 1e-8 * np.linalg.norm(b) + + def test_defective_precond_breakdown(self): + # Breakdown due to defective preconditioner + M = np.eye(3) + M[2, 2] = 0 + + b = np.array([0, 1, 1]) + x = np.array([1, 0, 0]) + A = np.diag([2, 3, 4]) + + x, info = gmres(A, b, x0=x, M=M, rtol=1e-15, atol=0) + + # Should not return nans, nor terminate with false success + assert not np.isnan(x).any() + if info == 0: + assert np.linalg.norm(A @ x - b) <= 1e-15 * np.linalg.norm(b) + + # The solution should be OK outside null space of M + assert_allclose(M @ (A @ x), M @ b) + + def test_defective_matrix_breakdown(self): + # Breakdown due to defective matrix + A = np.array([[0, 1, 0], [1, 0, 0], [0, 0, 0]]) + b = np.array([1, 0, 1]) + rtol = 1e-8 + x, info = gmres(A, b, rtol=rtol, atol=0) + + # Should not return nans, nor terminate with false success + assert not np.isnan(x).any() + if info == 0: + assert np.linalg.norm(A @ x - b) <= rtol * np.linalg.norm(b) + + # The solution should be OK outside null space of A + assert_allclose(A @ (A @ x), A @ b) + + @pytest.mark.filterwarnings(f"ignore:{CB_TYPE_FILTER}:DeprecationWarning") + def test_callback_type(self): + # The legacy callback type changes meaning of 'maxiter' + np.random.seed(1) + A = np.random.rand(20, 20) + b = np.random.rand(20) + + cb_count = [0] + + def pr_norm_cb(r): + cb_count[0] += 1 + assert isinstance(r, float) + + def x_cb(x): + cb_count[0] += 1 + assert isinstance(x, np.ndarray) + + # 2 iterations is not enough to solve the problem + cb_count = [0] + x, info = gmres(A, b, rtol=1e-6, atol=0, callback=pr_norm_cb, + maxiter=2, restart=50) + assert info == 2 + assert cb_count[0] == 2 + + # With `callback_type` specified, no warning should be raised + cb_count = [0] + x, info = gmres(A, b, rtol=1e-6, atol=0, callback=pr_norm_cb, + maxiter=2, restart=50, callback_type='legacy') + assert info == 2 + assert cb_count[0] == 2 + + # 2 restart cycles is enough to solve the problem + cb_count = [0] + x, info = gmres(A, b, rtol=1e-6, atol=0, callback=pr_norm_cb, + maxiter=2, restart=50, callback_type='pr_norm') + assert info == 0 + assert cb_count[0] > 2 + + # 2 restart cycles is enough to solve the problem + cb_count = [0] + x, info = gmres(A, b, rtol=1e-6, atol=0, callback=x_cb, maxiter=2, + restart=50, callback_type='x') + assert info == 0 + assert cb_count[0] == 1 + + def test_callback_x_monotonic(self): + # Check that callback_type='x' gives monotonic norm decrease + rng = np.random.RandomState(1) + A = rng.rand(20, 20) + np.eye(20) + b = rng.rand(20) + + prev_r = [np.inf] + count = [0] + + def x_cb(x): + r = np.linalg.norm(A @ x - b) + assert r <= prev_r[0] + prev_r[0] = r + count[0] += 1 + + x, info = gmres(A, b, rtol=1e-6, atol=0, callback=x_cb, maxiter=20, + restart=10, callback_type='x') + assert info == 20 + assert count[0] == 20 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_lgmres.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_lgmres.py new file mode 100644 index 0000000000000000000000000000000000000000..a33fc9bc13b36faef48f04388b76ed33eac775d6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_lgmres.py @@ -0,0 +1,225 @@ +"""Tests for the linalg._isolve.lgmres module +""" + +import threading +from numpy.testing import (assert_, assert_allclose, assert_equal, + suppress_warnings) + +import pytest +from platform import python_implementation + +import numpy as np +from numpy import zeros, array, allclose +from scipy.linalg import norm +from scipy.sparse import csr_array, eye_array, random_array + +from scipy.sparse.linalg._interface import LinearOperator +from scipy.sparse.linalg import splu +from scipy.sparse.linalg._isolve import lgmres, gmres + + +Am = csr_array(array([[-2, 1, 0, 0, 0, 9], + [1, -2, 1, 0, 5, 0], + [0, 1, -2, 1, 0, 0], + [0, 0, 1, -2, 1, 0], + [0, 3, 0, 1, -2, 1], + [1, 0, 0, 0, 1, -2]])) +b = array([1, 2, 3, 4, 5, 6]) +count = threading.local() # [0] +niter = threading.local() # [0] + + +def matvec(v): + if not hasattr(count, 'c'): + count.c = [0] + count.c[0] += 1 + return Am@v + + +def cb(v): + if not hasattr(niter, 'n'): + niter.n = [0] + niter.n[0] += 1 + + +A = LinearOperator(matvec=matvec, shape=Am.shape, dtype=Am.dtype) + + +def do_solve(**kw): + if not hasattr(niter, 'n'): + niter.n = [0] + if not hasattr(count, 'c'): + count.c = [0] + count.c[0] = 0 + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + x0, flag = lgmres(A, b, x0=zeros(A.shape[0]), + inner_m=6, rtol=1e-14, **kw) + count_0 = count.c[0] + assert_(allclose(A@x0, b, rtol=1e-12, atol=1e-12), norm(A@x0-b)) + return x0, count_0 + + +class TestLGMRES: + def test_preconditioner(self): + # Check that preconditioning works + pc = splu(Am.tocsc()) + M = LinearOperator(matvec=pc.solve, shape=A.shape, dtype=A.dtype) + + x0, count_0 = do_solve() + niter.n[0] = 0 + x1, count_1 = do_solve(M=M, callback=cb) + + assert count_1 == 3 + assert count_1 < count_0/2 + assert allclose(x1, x0, rtol=1e-14) + assert niter.n[0] < 3 + + def test_outer_v(self): + # Check that the augmentation vectors behave as expected + + outer_v = [] + x0, count_0 = do_solve(outer_k=6, outer_v=outer_v) + assert_(len(outer_v) > 0) + assert_(len(outer_v) <= 6) + + x1, count_1 = do_solve(outer_k=6, outer_v=outer_v, + prepend_outer_v=True) + assert_(count_1 == 2, count_1) + assert_(count_1 < count_0/2) + assert_(allclose(x1, x0, rtol=1e-14)) + + # --- + + outer_v = [] + x0, count_0 = do_solve(outer_k=6, outer_v=outer_v, + store_outer_Av=False) + assert_(array([v[1] is None for v in outer_v]).all()) + assert_(len(outer_v) > 0) + assert_(len(outer_v) <= 6) + + x1, count_1 = do_solve(outer_k=6, outer_v=outer_v, + prepend_outer_v=True) + assert_(count_1 == 3, count_1) + assert_(count_1 < count_0/2) + assert_(allclose(x1, x0, rtol=1e-14)) + + @pytest.mark.skipif(python_implementation() == 'PyPy', + reason="Fails on PyPy CI runs. See #9507") + def test_arnoldi(self): + rng = np.random.default_rng(123) + + A = eye_array(2000) + random_array((2000, 2000), density=5e-4, rng=rng) + b = rng.random(2000) + + # The inner arnoldi should be equivalent to gmres + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + x0, flag0 = lgmres(A, b, x0=zeros(A.shape[0]), inner_m=10, maxiter=1) + x1, flag1 = gmres(A, b, x0=zeros(A.shape[0]), restart=10, maxiter=1) + + assert_equal(flag0, 1) + assert_equal(flag1, 1) + norm = np.linalg.norm(A.dot(x0) - b) + assert_(norm > 1e-4) + assert_allclose(x0, x1) + + def test_cornercase(self): + rng = np.random.RandomState(1234) + + # Rounding error may prevent convergence with tol=0 --- ensure + # that the return values in this case are correct, and no + # exceptions are raised + + for n in [3, 5, 10, 100]: + A = 2*eye_array(n) + + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + + b = np.ones(n) + x, info = lgmres(A, b, maxiter=10) + assert_equal(info, 0) + assert_allclose(A.dot(x) - b, 0, atol=1e-14) + + x, info = lgmres(A, b, rtol=0, maxiter=10) + if info == 0: + assert_allclose(A.dot(x) - b, 0, atol=1e-14) + + b = rng.rand(n) + x, info = lgmres(A, b, maxiter=10) + assert_equal(info, 0) + assert_allclose(A.dot(x) - b, 0, atol=1e-14) + + x, info = lgmres(A, b, rtol=0, maxiter=10) + if info == 0: + assert_allclose(A.dot(x) - b, 0, atol=1e-14) + + def test_nans(self): + A = eye_array(3, format='lil') + A[1, 1] = np.nan + b = np.ones(3) + + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + x, info = lgmres(A, b, rtol=0, maxiter=10) + assert_equal(info, 1) + + def test_breakdown_with_outer_v(self): + A = np.array([[1, 2], [3, 4]], dtype=float) + b = np.array([1, 2]) + + x = np.linalg.solve(A, b) + v0 = np.array([1, 0]) + + # The inner iteration should converge to the correct solution, + # since it's in the outer vector list + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + xp, info = lgmres(A, b, outer_v=[(v0, None), (x, None)], maxiter=1) + + assert_allclose(xp, x, atol=1e-12) + + def test_breakdown_underdetermined(self): + # Should find LSQ solution in the Krylov span in one inner + # iteration, despite solver breakdown from nilpotent A. + A = np.array([[0, 1, 1, 1], + [0, 0, 1, 1], + [0, 0, 0, 1], + [0, 0, 0, 0]], dtype=float) + + bs = [ + np.array([1, 1, 1, 1]), + np.array([1, 1, 1, 0]), + np.array([1, 1, 0, 0]), + np.array([1, 0, 0, 0]), + ] + + for b in bs: + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + xp, info = lgmres(A, b, maxiter=1) + resp = np.linalg.norm(A.dot(xp) - b) + + K = np.c_[b, A.dot(b), A.dot(A.dot(b)), A.dot(A.dot(A.dot(b)))] + y, _, _, _ = np.linalg.lstsq(A.dot(K), b, rcond=-1) + x = K.dot(y) + res = np.linalg.norm(A.dot(x) - b) + + assert_allclose(resp, res, err_msg=repr(b)) + + def test_denormals(self): + # Check that no warnings are emitted if the matrix contains + # numbers for which 1/x has no float representation, and that + # the solver behaves properly. + A = np.array([[1, 2], [3, 4]], dtype=float) + A *= 100 * np.nextafter(0, 1) + + b = np.array([1, 1]) + + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, ".*called without specifying.*") + xp, info = lgmres(A, b) + + if info == 0: + assert_allclose(A.dot(xp), b) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_lsmr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_lsmr.py new file mode 100644 index 0000000000000000000000000000000000000000..0d1196b23e0b7a24513850116ee134193709619e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_lsmr.py @@ -0,0 +1,185 @@ +""" +Copyright (C) 2010 David Fong and Michael Saunders +Distributed under the same license as SciPy + +Testing Code for LSMR. + +03 Jun 2010: First version release with lsmr.py + +David Chin-lung Fong clfong@stanford.edu +Institute for Computational and Mathematical Engineering +Stanford University + +Michael Saunders saunders@stanford.edu +Systems Optimization Laboratory +Dept of MS&E, Stanford University. + +""" + +from numpy import array, arange, eye, zeros, ones, transpose, hstack +from numpy.linalg import norm +from numpy.testing import assert_allclose +import pytest +from scipy.sparse import coo_array +from scipy.sparse.linalg._interface import aslinearoperator +from scipy.sparse.linalg import lsmr +from .test_lsqr import G, b + + +class TestLSMR: + def setup_method(self): + self.n = 10 + self.m = 10 + + def assertCompatibleSystem(self, A, xtrue): + Afun = aslinearoperator(A) + b = Afun.matvec(xtrue) + x = lsmr(A, b)[0] + assert norm(x - xtrue) == pytest.approx(0, abs=1e-5) + + def testIdentityACase1(self): + A = eye(self.n) + xtrue = zeros((self.n, 1)) + self.assertCompatibleSystem(A, xtrue) + + def testIdentityACase2(self): + A = eye(self.n) + xtrue = ones((self.n,1)) + self.assertCompatibleSystem(A, xtrue) + + def testIdentityACase3(self): + A = eye(self.n) + xtrue = transpose(arange(self.n,0,-1)) + self.assertCompatibleSystem(A, xtrue) + + def testBidiagonalA(self): + A = lowerBidiagonalMatrix(20,self.n) + xtrue = transpose(arange(self.n,0,-1)) + self.assertCompatibleSystem(A,xtrue) + + def testScalarB(self): + A = array([[1.0, 2.0]]) + b = 3.0 + x = lsmr(A, b)[0] + assert norm(A.dot(x) - b) == pytest.approx(0) + + def testComplexX(self): + A = eye(self.n) + xtrue = transpose(arange(self.n, 0, -1) * (1 + 1j)) + self.assertCompatibleSystem(A, xtrue) + + def testComplexX0(self): + A = 4 * eye(self.n) + ones((self.n, self.n)) + xtrue = transpose(arange(self.n, 0, -1)) + b = aslinearoperator(A).matvec(xtrue) + x0 = zeros(self.n, dtype=complex) + x = lsmr(A, b, x0=x0)[0] + assert norm(x - xtrue) == pytest.approx(0, abs=1e-5) + + def testComplexA(self): + A = 4 * eye(self.n) + 1j * ones((self.n, self.n)) + xtrue = transpose(arange(self.n, 0, -1).astype(complex)) + self.assertCompatibleSystem(A, xtrue) + + def testComplexB(self): + A = 4 * eye(self.n) + ones((self.n, self.n)) + xtrue = transpose(arange(self.n, 0, -1) * (1 + 1j)) + b = aslinearoperator(A).matvec(xtrue) + x = lsmr(A, b)[0] + assert norm(x - xtrue) == pytest.approx(0, abs=1e-5) + + def testColumnB(self): + A = eye(self.n) + b = ones((self.n, 1)) + x = lsmr(A, b)[0] + assert norm(A.dot(x) - b.ravel()) == pytest.approx(0) + + def testInitialization(self): + # Test that the default setting is not modified + x_ref, _, itn_ref, normr_ref, *_ = lsmr(G, b) + assert_allclose(norm(b - G@x_ref), normr_ref, atol=1e-6) + + # Test passing zeros yields similar result + x0 = zeros(b.shape) + x = lsmr(G, b, x0=x0)[0] + assert_allclose(x, x_ref) + + # Test warm-start with single iteration + x0 = lsmr(G, b, maxiter=1)[0] + + x, _, itn, normr, *_ = lsmr(G, b, x0=x0) + assert_allclose(norm(b - G@x), normr, atol=1e-6) + + # NOTE(gh-12139): This doesn't always converge to the same value as + # ref because error estimates will be slightly different when calculated + # from zeros vs x0 as a result only compare norm and itn (not x). + + # x generally converges 1 iteration faster because it started at x0. + # itn == itn_ref means that lsmr(x0) took an extra iteration see above. + # -1 is technically possible but is rare (1 in 100000) so it's more + # likely to be an error elsewhere. + assert itn - itn_ref in (0, 1) + + # If an extra iteration is performed normr may be 0, while normr_ref + # may be much larger. + assert normr < normr_ref * (1 + 1e-6) + + +class TestLSMRReturns: + def setup_method(self): + self.n = 10 + self.A = lowerBidiagonalMatrix(20, self.n) + self.xtrue = transpose(arange(self.n, 0, -1)) + self.Afun = aslinearoperator(self.A) + self.b = self.Afun.matvec(self.xtrue) + self.x0 = ones(self.n) + self.x00 = self.x0.copy() + self.returnValues = lsmr(self.A, self.b) + self.returnValuesX0 = lsmr(self.A, self.b, x0=self.x0) + + def test_unchanged_x0(self): + x, istop, itn, normr, normar, normA, condA, normx = self.returnValuesX0 + assert_allclose(self.x00, self.x0) + + def testNormr(self): + x, istop, itn, normr, normar, normA, condA, normx = self.returnValues + assert norm(self.b - self.Afun.matvec(x)) == pytest.approx(normr) + + def testNormar(self): + x, istop, itn, normr, normar, normA, condA, normx = self.returnValues + assert (norm(self.Afun.rmatvec(self.b - self.Afun.matvec(x))) + == pytest.approx(normar)) + + def testNormx(self): + x, istop, itn, normr, normar, normA, condA, normx = self.returnValues + assert norm(x) == pytest.approx(normx) + + +def lowerBidiagonalMatrix(m, n): + # This is a simple example for testing LSMR. + # It uses the leading m*n submatrix from + # A = [ 1 + # 1 2 + # 2 3 + # 3 4 + # ... + # n ] + # suitably padded by zeros. + # + # 04 Jun 2010: First version for distribution with lsmr.py + if m <= n: + row = hstack((arange(m, dtype=int), + arange(1, m, dtype=int))) + col = hstack((arange(m, dtype=int), + arange(m-1, dtype=int))) + data = hstack((arange(1, m+1, dtype=float), + arange(1,m, dtype=float))) + return coo_array((data, (row, col)), shape=(m,n)) + else: + row = hstack((arange(n, dtype=int), + arange(1, n+1, dtype=int))) + col = hstack((arange(n, dtype=int), + arange(n, dtype=int))) + data = hstack((arange(1, n+1, dtype=float), + arange(1,n+1, dtype=float))) + return coo_array((data,(row, col)), shape=(m,n)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_lsqr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_lsqr.py new file mode 100644 index 0000000000000000000000000000000000000000..d77048af48a6b4495d23c9bc9a3b2d71466bade6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_lsqr.py @@ -0,0 +1,120 @@ +import numpy as np +from numpy.testing import assert_allclose, assert_array_equal, assert_equal +import pytest +import scipy.sparse +import scipy.sparse.linalg +from scipy.sparse.linalg import lsqr + +# Set up a test problem +n = 35 +G = np.eye(n) +normal = np.random.normal +norm = np.linalg.norm + +for jj in range(5): + gg = normal(size=n) + hh = gg * gg.T + G += (hh + hh.T) * 0.5 + G += normal(size=n) * normal(size=n) + +b = normal(size=n) + +# tolerance for atol/btol keywords of lsqr() +tol = 2e-10 +# tolerances for testing the results of the lsqr() call with assert_allclose +# These tolerances are a bit fragile - see discussion in gh-15301. +atol_test = 4e-10 +rtol_test = 2e-8 +show = False +maxit = None + + +def test_lsqr_basic(): + b_copy = b.copy() + xo, *_ = lsqr(G, b, show=show, atol=tol, btol=tol, iter_lim=maxit) + assert_array_equal(b_copy, b) + + svx = np.linalg.solve(G, b) + assert_allclose(xo, svx, atol=atol_test, rtol=rtol_test) + + # Now the same but with damp > 0. + # This is equivalent to solving the extended system: + # ( G ) @ x = ( b ) + # ( damp*I ) ( 0 ) + damp = 1.5 + xo, *_ = lsqr( + G, b, damp=damp, show=show, atol=tol, btol=tol, iter_lim=maxit) + + Gext = np.r_[G, damp * np.eye(G.shape[1])] + bext = np.r_[b, np.zeros(G.shape[1])] + svx, *_ = np.linalg.lstsq(Gext, bext, rcond=None) + assert_allclose(xo, svx, atol=atol_test, rtol=rtol_test) + + +def test_gh_2466(): + row = np.array([0, 0]) + col = np.array([0, 1]) + val = np.array([1, -1]) + A = scipy.sparse.coo_array((val, (row, col)), shape=(1, 2)) + b = np.asarray([4]) + lsqr(A, b) + + +def test_well_conditioned_problems(): + # Test that sparse the lsqr solver returns the right solution + # on various problems with different random seeds. + # This is a non-regression test for a potential ZeroDivisionError + # raised when computing the `test2` & `test3` convergence conditions. + n = 10 + A_sparse = scipy.sparse.eye_array(n, n) + A_dense = A_sparse.toarray() + + with np.errstate(invalid='raise'): + for seed in range(30): + rng = np.random.RandomState(seed + 10) + beta = rng.rand(n) + beta[beta == 0] = 0.00001 # ensure that all the betas are not null + b = A_sparse @ beta[:, np.newaxis] + output = lsqr(A_sparse, b, show=show) + + # Check that the termination condition corresponds to an approximate + # solution to Ax = b + assert_equal(output[1], 1) + solution = output[0] + + # Check that we recover the ground truth solution + assert_allclose(solution, beta) + + # Sanity check: compare to the dense array solver + reference_solution = np.linalg.solve(A_dense, b).ravel() + assert_allclose(solution, reference_solution) + + +def test_b_shapes(): + # Test b being a scalar. + A = np.array([[1.0, 2.0]]) + b = 3.0 + x = lsqr(A, b)[0] + assert norm(A.dot(x) - b) == pytest.approx(0) + + # Test b being a column vector. + A = np.eye(10) + b = np.ones((10, 1)) + x = lsqr(A, b)[0] + assert norm(A.dot(x) - b.ravel()) == pytest.approx(0) + + +def test_initialization(): + # Test the default setting is the same as zeros + b_copy = b.copy() + x_ref = lsqr(G, b, show=show, atol=tol, btol=tol, iter_lim=maxit) + x0 = np.zeros(x_ref[0].shape) + x = lsqr(G, b, show=show, atol=tol, btol=tol, iter_lim=maxit, x0=x0) + assert_array_equal(b_copy, b) + assert_allclose(x_ref[0], x[0]) + + # Test warm-start with single iteration + x0 = lsqr(G, b, show=show, atol=tol, btol=tol, iter_lim=1)[0] + x = lsqr(G, b, show=show, atol=tol, btol=tol, iter_lim=maxit, x0=x0) + assert_allclose(x_ref[0], x[0]) + assert_array_equal(b_copy, b) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_minres.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_minres.py new file mode 100644 index 0000000000000000000000000000000000000000..cae169e14f0306b195bca73ce86547391851aab6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_minres.py @@ -0,0 +1,97 @@ +import numpy as np +from numpy.linalg import norm +from numpy.testing import assert_equal, assert_allclose, assert_ +from scipy.sparse.linalg._isolve import minres + +from pytest import raises as assert_raises + + +def get_sample_problem(): + # A random 10 x 10 symmetric matrix + rng = np.random.RandomState(1234) + matrix = rng.rand(10, 10) + matrix = matrix + matrix.T + # A random vector of length 10 + vector = rng.rand(10) + return matrix, vector + + +def test_singular(): + A, b = get_sample_problem() + A[0, ] = 0 + b[0] = 0 + xp, info = minres(A, b) + assert_equal(info, 0) + assert norm(A @ xp - b) <= 1e-5 * norm(b) + + +def test_x0_is_used_by(): + A, b = get_sample_problem() + # Random x0 to feed minres + rng = np.random.RandomState(12345) + x0 = rng.rand(10) + trace = [] + + def trace_iterates(xk): + trace.append(xk) + minres(A, b, x0=x0, callback=trace_iterates) + trace_with_x0 = trace + + trace = [] + minres(A, b, callback=trace_iterates) + assert_(not np.array_equal(trace_with_x0[0], trace[0])) + + +def test_shift(): + A, b = get_sample_problem() + shift = 0.5 + shifted_A = A - shift * np.eye(10) + x1, info1 = minres(A, b, shift=shift) + x2, info2 = minres(shifted_A, b) + assert_equal(info1, 0) + assert_allclose(x1, x2, rtol=1e-5) + + +def test_asymmetric_fail(): + """Asymmetric matrix should raise `ValueError` when check=True""" + A, b = get_sample_problem() + A[1, 2] = 1 + A[2, 1] = 2 + with assert_raises(ValueError): + xp, info = minres(A, b, check=True) + + +def test_minres_non_default_x0(): + rng = np.random.RandomState(1234) + rtol = 1e-6 + a = rng.randn(5, 5) + a = np.dot(a, a.T) + b = rng.randn(5) + c = rng.randn(5) + x = minres(a, b, x0=c, rtol=rtol)[0] + assert norm(a @ x - b) <= rtol * norm(b) + + +def test_minres_precond_non_default_x0(): + rng = np.random.RandomState(12345) + rtol = 1e-6 + a = rng.randn(5, 5) + a = np.dot(a, a.T) + b = rng.randn(5) + c = rng.randn(5) + m = rng.randn(5, 5) + m = np.dot(m, m.T) + x = minres(a, b, M=m, x0=c, rtol=rtol)[0] + assert norm(a @ x - b) <= rtol * norm(b) + + +def test_minres_precond_exact_x0(): + rng = np.random.RandomState(1234) + rtol = 1e-6 + a = np.eye(10) + b = np.ones(10) + c = np.ones(10) + m = rng.randn(10, 10) + m = np.dot(m, m.T) + x = minres(a, b, M=m, x0=c, rtol=rtol)[0] + assert norm(a @ x - b) <= rtol * norm(b) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_utils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_utils.py new file mode 100644 index 0000000000000000000000000000000000000000..bb62e2223d515288bd5aa6c5a70028279c4f6d30 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tests/test_utils.py @@ -0,0 +1,9 @@ +import numpy as np +from pytest import raises as assert_raises + +import scipy.sparse.linalg._isolve.utils as utils + + +def test_make_system_bad_shape(): + assert_raises(ValueError, + utils.make_system, np.zeros((5,3)), None, np.zeros(4), np.zeros(4)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tfqmr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tfqmr.py new file mode 100644 index 0000000000000000000000000000000000000000..efec0302d53f107d8ffb3fcfe82f65cfa37ada5f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/tfqmr.py @@ -0,0 +1,179 @@ +import numpy as np +from .iterative import _get_atol_rtol +from .utils import make_system + + +__all__ = ['tfqmr'] + + +def tfqmr(A, b, x0=None, *, rtol=1e-5, atol=0., maxiter=None, M=None, + callback=None, show=False): + """ + Use Transpose-Free Quasi-Minimal Residual iteration to solve ``Ax = b``. + + Parameters + ---------- + A : {sparse array, ndarray, LinearOperator} + The real or complex N-by-N matrix of the linear system. + Alternatively, `A` can be a linear operator which can + produce ``Ax`` using, e.g., + `scipy.sparse.linalg.LinearOperator`. + b : {ndarray} + Right hand side of the linear system. Has shape (N,) or (N,1). + x0 : {ndarray} + Starting guess for the solution. + rtol, atol : float, optional + Parameters for the convergence test. For convergence, + ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. + The default is ``rtol=1e-5``, the default for ``atol`` is ``0.0``. + maxiter : int, optional + Maximum number of iterations. Iteration will stop after maxiter + steps even if the specified tolerance has not been achieved. + Default is ``min(10000, ndofs * 10)``, where ``ndofs = A.shape[0]``. + M : {sparse array, ndarray, LinearOperator} + Inverse of the preconditioner of A. M should approximate the + inverse of A and be easy to solve for (see Notes). Effective + preconditioning dramatically improves the rate of convergence, + which implies that fewer iterations are needed to reach a given + error tolerance. By default, no preconditioner is used. + callback : function, optional + User-supplied function to call after each iteration. It is called + as ``callback(xk)``, where ``xk`` is the current solution vector. + show : bool, optional + Specify ``show = True`` to show the convergence, ``show = False`` is + to close the output of the convergence. + Default is `False`. + + Returns + ------- + x : ndarray + The converged solution. + info : int + Provides convergence information: + + - 0 : successful exit + - >0 : convergence to tolerance not achieved, number of iterations + - <0 : illegal input or breakdown + + Notes + ----- + The Transpose-Free QMR algorithm is derived from the CGS algorithm. + However, unlike CGS, the convergence curves for the TFQMR method is + smoothed by computing a quasi minimization of the residual norm. The + implementation supports left preconditioner, and the "residual norm" + to compute in convergence criterion is actually an upper bound on the + actual residual norm ``||b - Axk||``. + + References + ---------- + .. [1] R. W. Freund, A Transpose-Free Quasi-Minimal Residual Algorithm for + Non-Hermitian Linear Systems, SIAM J. Sci. Comput., 14(2), 470-482, + 1993. + .. [2] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, + SIAM, Philadelphia, 2003. + .. [3] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, + number 16 in Frontiers in Applied Mathematics, SIAM, Philadelphia, + 1995. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import tfqmr + >>> A = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) + >>> b = np.array([2, 4, -1], dtype=float) + >>> x, exitCode = tfqmr(A, b, atol=0.0) + >>> print(exitCode) # 0 indicates successful convergence + 0 + >>> np.allclose(A.dot(x), b) + True + """ + + # Check data type + dtype = A.dtype + if np.issubdtype(dtype, np.int64): + dtype = float + A = A.astype(dtype) + if np.issubdtype(b.dtype, np.int64): + b = b.astype(dtype) + + A, M, x, b, postprocess = make_system(A, M, x0, b) + + # Check if the R.H.S is a zero vector + if np.linalg.norm(b) == 0.: + x = b.copy() + return (postprocess(x), 0) + + ndofs = A.shape[0] + if maxiter is None: + maxiter = min(10000, ndofs * 10) + + if x0 is None: + r = b.copy() + else: + r = b - A.matvec(x) + u = r + w = r.copy() + # Take rstar as b - Ax0, that is rstar := r = b - Ax0 mathematically + rstar = r + v = M.matvec(A.matvec(r)) + uhat = v + d = theta = eta = 0. + # at this point we know rstar == r, so rho is always real + rho = np.inner(rstar.conjugate(), r).real + rhoLast = rho + r0norm = np.sqrt(rho) + tau = r0norm + if r0norm == 0: + return (postprocess(x), 0) + + # we call this to get the right atol and raise errors as necessary + atol, _ = _get_atol_rtol('tfqmr', r0norm, atol, rtol) + + for iter in range(maxiter): + even = iter % 2 == 0 + if (even): + vtrstar = np.inner(rstar.conjugate(), v) + # Check breakdown + if vtrstar == 0.: + return (postprocess(x), -1) + alpha = rho / vtrstar + uNext = u - alpha * v # [1]-(5.6) + w -= alpha * uhat # [1]-(5.8) + d = u + (theta**2 / alpha) * eta * d # [1]-(5.5) + # [1]-(5.2) + theta = np.linalg.norm(w) / tau + c = np.sqrt(1. / (1 + theta**2)) + tau *= theta * c + # Calculate step and direction [1]-(5.4) + eta = (c**2) * alpha + z = M.matvec(d) + x += eta * z + + if callback is not None: + callback(x) + + # Convergence criterion + if tau * np.sqrt(iter+1) < atol: + if (show): + print("TFQMR: Linear solve converged due to reach TOL " + f"iterations {iter+1}") + return (postprocess(x), 0) + + if (not even): + # [1]-(5.7) + rho = np.inner(rstar.conjugate(), w) + beta = rho / rhoLast + u = w + beta * u + v = beta * uhat + (beta**2) * v + uhat = M.matvec(A.matvec(u)) + v += uhat + else: + uhat = M.matvec(A.matvec(uNext)) + u = uNext + rhoLast = rho + + if (show): + print("TFQMR: Linear solve not converged due to reach MAXIT " + f"iterations {iter+1}") + return (postprocess(x), maxiter) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/utils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/utils.py new file mode 100644 index 0000000000000000000000000000000000000000..80f37fc1cf63fa0352fd93d62be758f87c065db5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/utils.py @@ -0,0 +1,127 @@ +__docformat__ = "restructuredtext en" + +__all__ = [] + + +from numpy import asanyarray, asarray, array, zeros + +from scipy.sparse.linalg._interface import aslinearoperator, LinearOperator, \ + IdentityOperator + +_coerce_rules = {('f','f'):'f', ('f','d'):'d', ('f','F'):'F', + ('f','D'):'D', ('d','f'):'d', ('d','d'):'d', + ('d','F'):'D', ('d','D'):'D', ('F','f'):'F', + ('F','d'):'D', ('F','F'):'F', ('F','D'):'D', + ('D','f'):'D', ('D','d'):'D', ('D','F'):'D', + ('D','D'):'D'} + + +def coerce(x,y): + if x not in 'fdFD': + x = 'd' + if y not in 'fdFD': + y = 'd' + return _coerce_rules[x,y] + + +def id(x): + return x + + +def make_system(A, M, x0, b): + """Make a linear system Ax=b + + Parameters + ---------- + A : LinearOperator + sparse or dense matrix (or any valid input to aslinearoperator) + M : {LinearOperator, Nones} + preconditioner + sparse or dense matrix (or any valid input to aslinearoperator) + x0 : {array_like, str, None} + initial guess to iterative method. + ``x0 = 'Mb'`` means using the nonzero initial guess ``M @ b``. + Default is `None`, which means using the zero initial guess. + b : array_like + right hand side + + Returns + ------- + (A, M, x, b, postprocess) + A : LinearOperator + matrix of the linear system + M : LinearOperator + preconditioner + x : rank 1 ndarray + initial guess + b : rank 1 ndarray + right hand side + postprocess : function + converts the solution vector to the appropriate + type and dimensions (e.g. (N,1) matrix) + + """ + A_ = A + A = aslinearoperator(A) + + if A.shape[0] != A.shape[1]: + raise ValueError(f'expected square matrix, but got shape={(A.shape,)}') + + N = A.shape[0] + + b = asanyarray(b) + + if not (b.shape == (N,1) or b.shape == (N,)): + raise ValueError(f'shapes of A {A.shape} and b {b.shape} are ' + 'incompatible') + + if b.dtype.char not in 'fdFD': + b = b.astype('d') # upcast non-FP types to double + + def postprocess(x): + return x + + if hasattr(A,'dtype'): + xtype = A.dtype.char + else: + xtype = A.matvec(b).dtype.char + xtype = coerce(xtype, b.dtype.char) + + b = asarray(b,dtype=xtype) # make b the same type as x + b = b.ravel() + + # process preconditioner + if M is None: + if hasattr(A_,'psolve'): + psolve = A_.psolve + else: + psolve = id + if hasattr(A_,'rpsolve'): + rpsolve = A_.rpsolve + else: + rpsolve = id + if psolve is id and rpsolve is id: + M = IdentityOperator(shape=A.shape, dtype=A.dtype) + else: + M = LinearOperator(A.shape, matvec=psolve, rmatvec=rpsolve, + dtype=A.dtype) + else: + M = aslinearoperator(M) + if A.shape != M.shape: + raise ValueError('matrix and preconditioner have different shapes') + + # set initial guess + if x0 is None: + x = zeros(N, dtype=xtype) + elif isinstance(x0, str): + if x0 == 'Mb': # use nonzero initial guess ``M @ b`` + bCopy = b.copy() + x = M.matvec(bCopy) + else: + x = array(x0, dtype=xtype) + if not (x.shape == (N, 1) or x.shape == (N,)): + raise ValueError(f'shapes of A {A.shape} and ' + f'x0 {x.shape} are incompatible') + x = x.ravel() + + return A, M, x, b, postprocess diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_matfuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_matfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..5dff48df52d8b95eb12c64c0117ac57101d9f031 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_matfuncs.py @@ -0,0 +1,940 @@ +""" +Sparse matrix functions +""" + +# +# Authors: Travis Oliphant, March 2002 +# Anthony Scopatz, August 2012 (Sparse Updates) +# Jake Vanderplas, August 2012 (Sparse Updates) +# + +__all__ = ['expm', 'inv', 'matrix_power'] + +import numpy as np +from scipy.linalg._basic import solve, solve_triangular + +from scipy.sparse._base import issparse +from scipy.sparse.linalg import spsolve +from scipy.sparse._sputils import is_pydata_spmatrix, isintlike + +import scipy.sparse +import scipy.sparse.linalg +from scipy.sparse.linalg._interface import LinearOperator +from scipy.sparse._construct import eye_array + +from ._expm_multiply import _ident_like, _exact_1_norm as _onenorm + + +UPPER_TRIANGULAR = 'upper_triangular' + + +def inv(A): + """ + Compute the inverse of a sparse arrays + + Parameters + ---------- + A : (M, M) sparse arrays + square matrix to be inverted + + Returns + ------- + Ainv : (M, M) sparse arrays + inverse of `A` + + Notes + ----- + This computes the sparse inverse of `A`. If the inverse of `A` is expected + to be non-sparse, it will likely be faster to convert `A` to dense and use + `scipy.linalg.inv`. + + Examples + -------- + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import inv + >>> A = csc_array([[1., 0.], [1., 2.]]) + >>> Ainv = inv(A) + >>> Ainv + + >>> A.dot(Ainv) + + >>> A.dot(Ainv).toarray() + array([[ 1., 0.], + [ 0., 1.]]) + + .. versionadded:: 0.12.0 + + """ + # Check input + if not (issparse(A) or is_pydata_spmatrix(A)): + raise TypeError('Input must be a sparse arrays') + + # Use sparse direct solver to solve "AX = I" accurately + I = _ident_like(A) + Ainv = spsolve(A, I) + return Ainv + + +def _onenorm_matrix_power_nnm(A, p): + """ + Compute the 1-norm of a non-negative integer power of a non-negative matrix. + + Parameters + ---------- + A : a square ndarray or matrix or sparse arrays + Input matrix with non-negative entries. + p : non-negative integer + The power to which the matrix is to be raised. + + Returns + ------- + out : float + The 1-norm of the matrix power p of A. + + """ + # Check input + if int(p) != p or p < 0: + raise ValueError('expected non-negative integer p') + p = int(p) + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected A to be like a square matrix') + + # Explicitly make a column vector so that this works when A is a + # numpy matrix (in addition to ndarray and sparse arrays). + v = np.ones((A.shape[0], 1), dtype=float) + M = A.T + for i in range(p): + v = M.dot(v) + return np.max(v) + + +def _is_upper_triangular(A): + # This function could possibly be of wider interest. + if issparse(A): + lower_part = scipy.sparse.tril(A, -1) + # Check structural upper triangularity, + # then coincidental upper triangularity if needed. + return lower_part.nnz == 0 or lower_part.count_nonzero() == 0 + elif is_pydata_spmatrix(A): + import sparse + lower_part = sparse.tril(A, -1) + return lower_part.nnz == 0 + else: + return not np.tril(A, -1).any() + + +def _smart_matrix_product(A, B, alpha=None, structure=None): + """ + A matrix product that knows about sparse and structured matrices. + + Parameters + ---------- + A : 2d ndarray + First matrix. + B : 2d ndarray + Second matrix. + alpha : float + The matrix product will be scaled by this constant. + structure : str, optional + A string describing the structure of both matrices `A` and `B`. + Only `upper_triangular` is currently supported. + + Returns + ------- + M : 2d ndarray + Matrix product of A and B. + + """ + if len(A.shape) != 2: + raise ValueError('expected A to be a rectangular matrix') + if len(B.shape) != 2: + raise ValueError('expected B to be a rectangular matrix') + f = None + if structure == UPPER_TRIANGULAR: + if (not issparse(A) and not issparse(B) + and not is_pydata_spmatrix(A) and not is_pydata_spmatrix(B)): + f, = scipy.linalg.get_blas_funcs(('trmm',), (A, B)) + if f is not None: + if alpha is None: + alpha = 1. + out = f(alpha, A, B) + else: + if alpha is None: + out = A.dot(B) + else: + out = alpha * A.dot(B) + return out + + +class MatrixPowerOperator(LinearOperator): + + def __init__(self, A, p, structure=None): + if A.ndim != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected A to be like a square matrix') + if p < 0: + raise ValueError('expected p to be a non-negative integer') + self._A = A + self._p = p + self._structure = structure + self.dtype = A.dtype + self.ndim = A.ndim + self.shape = A.shape + + def _matvec(self, x): + for i in range(self._p): + x = self._A.dot(x) + return x + + def _rmatvec(self, x): + A_T = self._A.T + x = x.ravel() + for i in range(self._p): + x = A_T.dot(x) + return x + + def _matmat(self, X): + for i in range(self._p): + X = _smart_matrix_product(self._A, X, structure=self._structure) + return X + + @property + def T(self): + return MatrixPowerOperator(self._A.T, self._p) + + +class ProductOperator(LinearOperator): + """ + For now, this is limited to products of multiple square matrices. + """ + + def __init__(self, *args, **kwargs): + self._structure = kwargs.get('structure', None) + for A in args: + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError( + 'For now, the ProductOperator implementation is ' + 'limited to the product of multiple square matrices.') + if args: + n = args[0].shape[0] + for A in args: + for d in A.shape: + if d != n: + raise ValueError( + 'The square matrices of the ProductOperator ' + 'must all have the same shape.') + self.shape = (n, n) + self.ndim = len(self.shape) + self.dtype = np.result_type(*[x.dtype for x in args]) + self._operator_sequence = args + + def _matvec(self, x): + for A in reversed(self._operator_sequence): + x = A.dot(x) + return x + + def _rmatvec(self, x): + x = x.ravel() + for A in self._operator_sequence: + x = A.T.dot(x) + return x + + def _matmat(self, X): + for A in reversed(self._operator_sequence): + X = _smart_matrix_product(A, X, structure=self._structure) + return X + + @property + def T(self): + T_args = [A.T for A in reversed(self._operator_sequence)] + return ProductOperator(*T_args) + + +def _onenormest_matrix_power(A, p, + t=2, itmax=5, compute_v=False, compute_w=False, structure=None): + """ + Efficiently estimate the 1-norm of A^p. + + Parameters + ---------- + A : ndarray + Matrix whose 1-norm of a power is to be computed. + p : int + Non-negative integer power. + t : int, optional + A positive parameter controlling the tradeoff between + accuracy versus time and memory usage. + Larger values take longer and use more memory + but give more accurate output. + itmax : int, optional + Use at most this many iterations. + compute_v : bool, optional + Request a norm-maximizing linear operator input vector if True. + compute_w : bool, optional + Request a norm-maximizing linear operator output vector if True. + + Returns + ------- + est : float + An underestimate of the 1-norm of the sparse arrays. + v : ndarray, optional + The vector such that ||Av||_1 == est*||v||_1. + It can be thought of as an input to the linear operator + that gives an output with particularly large norm. + w : ndarray, optional + The vector Av which has relatively large 1-norm. + It can be thought of as an output of the linear operator + that is relatively large in norm compared to the input. + + """ + return scipy.sparse.linalg.onenormest( + MatrixPowerOperator(A, p, structure=structure)) + + +def _onenormest_product(operator_seq, + t=2, itmax=5, compute_v=False, compute_w=False, structure=None): + """ + Efficiently estimate the 1-norm of the matrix product of the args. + + Parameters + ---------- + operator_seq : linear operator sequence + Matrices whose 1-norm of product is to be computed. + t : int, optional + A positive parameter controlling the tradeoff between + accuracy versus time and memory usage. + Larger values take longer and use more memory + but give more accurate output. + itmax : int, optional + Use at most this many iterations. + compute_v : bool, optional + Request a norm-maximizing linear operator input vector if True. + compute_w : bool, optional + Request a norm-maximizing linear operator output vector if True. + structure : str, optional + A string describing the structure of all operators. + Only `upper_triangular` is currently supported. + + Returns + ------- + est : float + An underestimate of the 1-norm of the sparse arrays. + v : ndarray, optional + The vector such that ||Av||_1 == est*||v||_1. + It can be thought of as an input to the linear operator + that gives an output with particularly large norm. + w : ndarray, optional + The vector Av which has relatively large 1-norm. + It can be thought of as an output of the linear operator + that is relatively large in norm compared to the input. + + """ + return scipy.sparse.linalg.onenormest( + ProductOperator(*operator_seq, structure=structure)) + + +class _ExpmPadeHelper: + """ + Help lazily evaluate a matrix exponential. + + The idea is to not do more work than we need for high expm precision, + so we lazily compute matrix powers and store or precompute + other properties of the matrix. + + """ + + def __init__(self, A, structure=None, use_exact_onenorm=False): + """ + Initialize the object. + + Parameters + ---------- + A : a dense or sparse square numpy matrix or ndarray + The matrix to be exponentiated. + structure : str, optional + A string describing the structure of matrix `A`. + Only `upper_triangular` is currently supported. + use_exact_onenorm : bool, optional + If True then only the exact one-norm of matrix powers and products + will be used. Otherwise, the one-norm of powers and products + may initially be estimated. + """ + self.A = A + self._A2 = None + self._A4 = None + self._A6 = None + self._A8 = None + self._A10 = None + self._d4_exact = None + self._d6_exact = None + self._d8_exact = None + self._d10_exact = None + self._d4_approx = None + self._d6_approx = None + self._d8_approx = None + self._d10_approx = None + self.ident = _ident_like(A) + self.structure = structure + self.use_exact_onenorm = use_exact_onenorm + + @property + def A2(self): + if self._A2 is None: + self._A2 = _smart_matrix_product( + self.A, self.A, structure=self.structure) + return self._A2 + + @property + def A4(self): + if self._A4 is None: + self._A4 = _smart_matrix_product( + self.A2, self.A2, structure=self.structure) + return self._A4 + + @property + def A6(self): + if self._A6 is None: + self._A6 = _smart_matrix_product( + self.A4, self.A2, structure=self.structure) + return self._A6 + + @property + def A8(self): + if self._A8 is None: + self._A8 = _smart_matrix_product( + self.A6, self.A2, structure=self.structure) + return self._A8 + + @property + def A10(self): + if self._A10 is None: + self._A10 = _smart_matrix_product( + self.A4, self.A6, structure=self.structure) + return self._A10 + + @property + def d4_tight(self): + if self._d4_exact is None: + self._d4_exact = _onenorm(self.A4)**(1/4.) + return self._d4_exact + + @property + def d6_tight(self): + if self._d6_exact is None: + self._d6_exact = _onenorm(self.A6)**(1/6.) + return self._d6_exact + + @property + def d8_tight(self): + if self._d8_exact is None: + self._d8_exact = _onenorm(self.A8)**(1/8.) + return self._d8_exact + + @property + def d10_tight(self): + if self._d10_exact is None: + self._d10_exact = _onenorm(self.A10)**(1/10.) + return self._d10_exact + + @property + def d4_loose(self): + if self.use_exact_onenorm: + return self.d4_tight + if self._d4_exact is not None: + return self._d4_exact + else: + if self._d4_approx is None: + self._d4_approx = _onenormest_matrix_power(self.A2, 2, + structure=self.structure)**(1/4.) + return self._d4_approx + + @property + def d6_loose(self): + if self.use_exact_onenorm: + return self.d6_tight + if self._d6_exact is not None: + return self._d6_exact + else: + if self._d6_approx is None: + self._d6_approx = _onenormest_matrix_power(self.A2, 3, + structure=self.structure)**(1/6.) + return self._d6_approx + + @property + def d8_loose(self): + if self.use_exact_onenorm: + return self.d8_tight + if self._d8_exact is not None: + return self._d8_exact + else: + if self._d8_approx is None: + self._d8_approx = _onenormest_matrix_power(self.A4, 2, + structure=self.structure)**(1/8.) + return self._d8_approx + + @property + def d10_loose(self): + if self.use_exact_onenorm: + return self.d10_tight + if self._d10_exact is not None: + return self._d10_exact + else: + if self._d10_approx is None: + self._d10_approx = _onenormest_product((self.A4, self.A6), + structure=self.structure)**(1/10.) + return self._d10_approx + + def pade3(self): + b = (120., 60., 12., 1.) + U = _smart_matrix_product(self.A, + b[3]*self.A2 + b[1]*self.ident, + structure=self.structure) + V = b[2]*self.A2 + b[0]*self.ident + return U, V + + def pade5(self): + b = (30240., 15120., 3360., 420., 30., 1.) + U = _smart_matrix_product(self.A, + b[5]*self.A4 + b[3]*self.A2 + b[1]*self.ident, + structure=self.structure) + V = b[4]*self.A4 + b[2]*self.A2 + b[0]*self.ident + return U, V + + def pade7(self): + b = (17297280., 8648640., 1995840., 277200., 25200., 1512., 56., 1.) + U = _smart_matrix_product(self.A, + b[7]*self.A6 + b[5]*self.A4 + b[3]*self.A2 + b[1]*self.ident, + structure=self.structure) + V = b[6]*self.A6 + b[4]*self.A4 + b[2]*self.A2 + b[0]*self.ident + return U, V + + def pade9(self): + b = (17643225600., 8821612800., 2075673600., 302702400., 30270240., + 2162160., 110880., 3960., 90., 1.) + U = _smart_matrix_product(self.A, + (b[9]*self.A8 + b[7]*self.A6 + b[5]*self.A4 + + b[3]*self.A2 + b[1]*self.ident), + structure=self.structure) + V = (b[8]*self.A8 + b[6]*self.A6 + b[4]*self.A4 + + b[2]*self.A2 + b[0]*self.ident) + return U, V + + def pade13_scaled(self, s): + b = (64764752532480000., 32382376266240000., 7771770303897600., + 1187353796428800., 129060195264000., 10559470521600., + 670442572800., 33522128640., 1323241920., 40840800., 960960., + 16380., 182., 1.) + B = self.A * 2**-s + B2 = self.A2 * 2**(-2*s) + B4 = self.A4 * 2**(-4*s) + B6 = self.A6 * 2**(-6*s) + U2 = _smart_matrix_product(B6, + b[13]*B6 + b[11]*B4 + b[9]*B2, + structure=self.structure) + U = _smart_matrix_product(B, + (U2 + b[7]*B6 + b[5]*B4 + + b[3]*B2 + b[1]*self.ident), + structure=self.structure) + V2 = _smart_matrix_product(B6, + b[12]*B6 + b[10]*B4 + b[8]*B2, + structure=self.structure) + V = V2 + b[6]*B6 + b[4]*B4 + b[2]*B2 + b[0]*self.ident + return U, V + + +def expm(A): + """ + Compute the matrix exponential using Pade approximation. + + Parameters + ---------- + A : (M,M) array_like or sparse array + 2D Array or Matrix (sparse or dense) to be exponentiated + + Returns + ------- + expA : (M,M) ndarray + Matrix exponential of `A` + + Notes + ----- + This is algorithm (6.1) which is a simplification of algorithm (5.1). + + .. versionadded:: 0.12.0 + + References + ---------- + .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2009) + "A New Scaling and Squaring Algorithm for the Matrix Exponential." + SIAM Journal on Matrix Analysis and Applications. + 31 (3). pp. 970-989. ISSN 1095-7162 + + Examples + -------- + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import expm + >>> A = csc_array([[1, 0, 0], [0, 2, 0], [0, 0, 3]]) + >>> A.toarray() + array([[1, 0, 0], + [0, 2, 0], + [0, 0, 3]], dtype=int64) + >>> Aexp = expm(A) + >>> Aexp + + >>> Aexp.toarray() + array([[ 2.71828183, 0. , 0. ], + [ 0. , 7.3890561 , 0. ], + [ 0. , 0. , 20.08553692]]) + """ + return _expm(A, use_exact_onenorm='auto') + + +def _expm(A, use_exact_onenorm): + # Core of expm, separated to allow testing exact and approximate + # algorithms. + + # Avoid indiscriminate asarray() to allow sparse or other strange arrays. + if isinstance(A, (list, tuple, np.matrix)): + A = np.asarray(A) + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected a square matrix') + + # gracefully handle size-0 input, + # carefully handling sparse scenario + if A.shape == (0, 0): + out = np.zeros([0, 0], dtype=A.dtype) + if issparse(A) or is_pydata_spmatrix(A): + return A.__class__(out) + return out + + # Trivial case + if A.shape == (1, 1): + out = [[np.exp(A[0, 0])]] + + # Avoid indiscriminate casting to ndarray to + # allow for sparse or other strange arrays + if issparse(A) or is_pydata_spmatrix(A): + return A.__class__(out) + + return np.array(out) + + # Ensure input is of float type, to avoid integer overflows etc. + if ((isinstance(A, np.ndarray) or issparse(A) or is_pydata_spmatrix(A)) + and not np.issubdtype(A.dtype, np.inexact)): + A = A.astype(float) + + # Detect upper triangularity. + structure = UPPER_TRIANGULAR if _is_upper_triangular(A) else None + + if use_exact_onenorm == "auto": + # Hardcode a matrix order threshold for exact vs. estimated one-norms. + use_exact_onenorm = A.shape[0] < 200 + + # Track functions of A to help compute the matrix exponential. + h = _ExpmPadeHelper( + A, structure=structure, use_exact_onenorm=use_exact_onenorm) + + # Try Pade order 3. + eta_1 = max(h.d4_loose, h.d6_loose) + if eta_1 < 1.495585217958292e-002 and _ell(h.A, 3) == 0: + U, V = h.pade3() + return _solve_P_Q(U, V, structure=structure) + + # Try Pade order 5. + eta_2 = max(h.d4_tight, h.d6_loose) + if eta_2 < 2.539398330063230e-001 and _ell(h.A, 5) == 0: + U, V = h.pade5() + return _solve_P_Q(U, V, structure=structure) + + # Try Pade orders 7 and 9. + eta_3 = max(h.d6_tight, h.d8_loose) + if eta_3 < 9.504178996162932e-001 and _ell(h.A, 7) == 0: + U, V = h.pade7() + return _solve_P_Q(U, V, structure=structure) + if eta_3 < 2.097847961257068e+000 and _ell(h.A, 9) == 0: + U, V = h.pade9() + return _solve_P_Q(U, V, structure=structure) + + # Use Pade order 13. + eta_4 = max(h.d8_loose, h.d10_loose) + eta_5 = min(eta_3, eta_4) + theta_13 = 4.25 + + # Choose smallest s>=0 such that 2**(-s) eta_5 <= theta_13 + if eta_5 == 0: + # Nilpotent special case + s = 0 + else: + s = max(int(np.ceil(np.log2(eta_5 / theta_13))), 0) + s = s + _ell(2**-s * h.A, 13) + U, V = h.pade13_scaled(s) + X = _solve_P_Q(U, V, structure=structure) + if structure == UPPER_TRIANGULAR: + # Invoke Code Fragment 2.1. + X = _fragment_2_1(X, h.A, s) + else: + # X = r_13(A)^(2^s) by repeated squaring. + for i in range(s): + X = X.dot(X) + return X + + +def _solve_P_Q(U, V, structure=None): + """ + A helper function for expm_2009. + + Parameters + ---------- + U : ndarray + Pade numerator. + V : ndarray + Pade denominator. + structure : str, optional + A string describing the structure of both matrices `U` and `V`. + Only `upper_triangular` is currently supported. + + Notes + ----- + The `structure` argument is inspired by similar args + for theano and cvxopt functions. + + """ + P = U + V + Q = -U + V + if issparse(U) or is_pydata_spmatrix(U): + return spsolve(Q, P) + elif structure is None: + return solve(Q, P) + elif structure == UPPER_TRIANGULAR: + return solve_triangular(Q, P) + else: + raise ValueError('unsupported matrix structure: ' + str(structure)) + + +def _exp_sinch(a, x): + """ + Stably evaluate exp(a)*sinh(x)/x + + Notes + ----- + The strategy of falling back to a sixth order Taylor expansion + was suggested by the Spallation Neutron Source docs + which was found on the internet by google search. + http://www.ornl.gov/~t6p/resources/xal/javadoc/gov/sns/tools/math/ElementaryFunction.html + The details of the cutoff point and the Horner-like evaluation + was picked without reference to anything in particular. + + Note that sinch is not currently implemented in scipy.special, + whereas the "engineer's" definition of sinc is implemented. + The implementation of sinc involves a scaling factor of pi + that distinguishes it from the "mathematician's" version of sinc. + + """ + + # If x is small then use sixth order Taylor expansion. + # How small is small? I am using the point where the relative error + # of the approximation is less than 1e-14. + # If x is large then directly evaluate sinh(x) / x. + if abs(x) < 0.0135: + x2 = x*x + return np.exp(a) * (1 + (x2/6.)*(1 + (x2/20.)*(1 + (x2/42.)))) + else: + return (np.exp(a + x) - np.exp(a - x)) / (2*x) + + +def _eq_10_42(lam_1, lam_2, t_12): + """ + Equation (10.42) of Functions of Matrices: Theory and Computation. + + Notes + ----- + This is a helper function for _fragment_2_1 of expm_2009. + Equation (10.42) is on page 251 in the section on Schur algorithms. + In particular, section 10.4.3 explains the Schur-Parlett algorithm. + expm([[lam_1, t_12], [0, lam_1]) + = + [[exp(lam_1), t_12*exp((lam_1 + lam_2)/2)*sinch((lam_1 - lam_2)/2)], + [0, exp(lam_2)] + """ + + # The plain formula t_12 * (exp(lam_2) - exp(lam_2)) / (lam_2 - lam_1) + # apparently suffers from cancellation, according to Higham's textbook. + # A nice implementation of sinch, defined as sinh(x)/x, + # will apparently work around the cancellation. + a = 0.5 * (lam_1 + lam_2) + b = 0.5 * (lam_1 - lam_2) + return t_12 * _exp_sinch(a, b) + + +def _fragment_2_1(X, T, s): + """ + A helper function for expm_2009. + + Notes + ----- + The argument X is modified in-place, but this modification is not the same + as the returned value of the function. + This function also takes pains to do things in ways that are compatible + with sparse arrays, for example by avoiding fancy indexing + and by using methods of the matrices whenever possible instead of + using functions of the numpy or scipy libraries themselves. + + """ + # Form X = r_m(2^-s T) + # Replace diag(X) by exp(2^-s diag(T)). + n = X.shape[0] + diag_T = np.ravel(T.diagonal().copy()) + + # Replace diag(X) by exp(2^-s diag(T)). + scale = 2 ** -s + exp_diag = np.exp(scale * diag_T) + for k in range(n): + X[k, k] = exp_diag[k] + + for i in range(s-1, -1, -1): + X = X.dot(X) + + # Replace diag(X) by exp(2^-i diag(T)). + scale = 2 ** -i + exp_diag = np.exp(scale * diag_T) + for k in range(n): + X[k, k] = exp_diag[k] + + # Replace (first) superdiagonal of X by explicit formula + # for superdiagonal of exp(2^-i T) from Eq (10.42) of + # the author's 2008 textbook + # Functions of Matrices: Theory and Computation. + for k in range(n-1): + lam_1 = scale * diag_T[k] + lam_2 = scale * diag_T[k+1] + t_12 = scale * T[k, k+1] + value = _eq_10_42(lam_1, lam_2, t_12) + X[k, k+1] = value + + # Return the updated X matrix. + return X + + +def _ell(A, m): + """ + A helper function for expm_2009. + + Parameters + ---------- + A : linear operator + A linear operator whose norm of power we care about. + m : int + The power of the linear operator + + Returns + ------- + value : int + A value related to a bound. + + """ + if len(A.shape) != 2 or A.shape[0] != A.shape[1]: + raise ValueError('expected A to be like a square matrix') + + # The c_i are explained in (2.2) and (2.6) of the 2005 expm paper. + # They are coefficients of terms of a generating function series expansion. + c_i = {3: 100800., + 5: 10059033600., + 7: 4487938430976000., + 9: 5914384781877411840000., + 13: 113250775606021113483283660800000000. + } + abs_c_recip = c_i[m] + + # This is explained after Eq. (1.2) of the 2009 expm paper. + # It is the "unit roundoff" of IEEE double precision arithmetic. + u = 2**-53 + + # Compute the one-norm of matrix power p of abs(A). + A_abs_onenorm = _onenorm_matrix_power_nnm(abs(A), 2*m + 1) + + # Treat zero norm as a special case. + if not A_abs_onenorm: + return 0 + + alpha = A_abs_onenorm / (_onenorm(A) * abs_c_recip) + log2_alpha_div_u = np.log2(alpha/u) + value = int(np.ceil(log2_alpha_div_u / (2 * m))) + return max(value, 0) + +def matrix_power(A, power): + """ + Raise a square matrix to the integer power, `power`. + + For non-negative integers, ``A**power`` is computed using repeated + matrix multiplications. Negative integers are not supported. + + Parameters + ---------- + A : (M, M) square sparse array or matrix + sparse array that will be raised to power `power` + power : int + Exponent used to raise sparse array `A` + + Returns + ------- + A**power : (M, M) sparse array or matrix + The output matrix will be the same shape as A, and will preserve + the class of A, but the format of the output may be changed. + + Notes + ----- + This uses a recursive implementation of the matrix power. For computing + the matrix power using a reasonably large `power`, this may be less efficient + than computing the product directly, using A @ A @ ... @ A. + This is contingent upon the number of nonzero entries in the matrix. + + .. versionadded:: 1.12.0 + + Examples + -------- + >>> from scipy import sparse + >>> A = sparse.csc_array([[0,1,0],[1,0,1],[0,1,0]]) + >>> A.todense() + array([[0, 1, 0], + [1, 0, 1], + [0, 1, 0]]) + >>> (A @ A).todense() + array([[1, 0, 1], + [0, 2, 0], + [1, 0, 1]]) + >>> A2 = sparse.linalg.matrix_power(A, 2) + >>> A2.todense() + array([[1, 0, 1], + [0, 2, 0], + [1, 0, 1]]) + >>> A4 = sparse.linalg.matrix_power(A, 4) + >>> A4.todense() + array([[2, 0, 2], + [0, 4, 0], + [2, 0, 2]]) + + """ + M, N = A.shape + if M != N: + raise TypeError('sparse matrix is not square') + + if isintlike(power): + power = int(power) + if power < 0: + raise ValueError('exponent must be >= 0') + + if power == 0: + return eye_array(M, dtype=A.dtype) + + if power == 1: + return A.copy() + + tmp = matrix_power(A, power // 2) + if power % 2: + return A @ tmp @ tmp + else: + return tmp @ tmp + else: + raise ValueError("exponent must be an integer") diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_norm.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_norm.py new file mode 100644 index 0000000000000000000000000000000000000000..821ed02bb1b1d7c2047d5e5dcb3049cc2bf8ad02 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_norm.py @@ -0,0 +1,195 @@ +"""Sparse matrix norms. + +""" +import numpy as np +from scipy.sparse import issparse +from scipy.sparse.linalg import svds +from scipy.sparse._sputils import convert_pydata_sparse_to_scipy +import scipy.sparse as sp + +from numpy import sqrt, abs + +__all__ = ['norm'] + + +def _sparse_frobenius_norm(x): + data = sp._sputils._todata(x) + return np.linalg.norm(data) + + +def norm(x, ord=None, axis=None): + """ + Norm of a sparse matrix + + This function is able to return one of seven different matrix norms, + depending on the value of the ``ord`` parameter. + + Parameters + ---------- + x : a sparse array + Input sparse array. + ord : {non-zero int, inf, -inf, 'fro'}, optional + Order of the norm (see table under ``Notes``). inf means numpy's + `inf` object. + axis : {int, 2-tuple of ints, None}, optional + If `axis` is an integer, it specifies the axis of `x` along which to + compute the vector norms. If `axis` is a 2-tuple, it specifies the + axes that hold 2-D matrices, and the matrix norms of these matrices + are computed. If `axis` is None then either a vector norm (when `x` + is 1-D) or a matrix norm (when `x` is 2-D) is returned. + + Returns + ------- + n : float or ndarray + + Notes + ----- + Some of the ord are not implemented because some associated functions like, + _multi_svd_norm, are not yet available for sparse array. + + This docstring is modified based on numpy.linalg.norm. + https://github.com/numpy/numpy/blob/main/numpy/linalg/linalg.py + + The following norms can be calculated: + + ===== ============================ + ord norm for sparse arrays + ===== ============================ + None Frobenius norm + 'fro' Frobenius norm + inf max(sum(abs(x), axis=1)) + -inf min(sum(abs(x), axis=1)) + 0 abs(x).sum(axis=axis) + 1 max(sum(abs(x), axis=0)) + -1 min(sum(abs(x), axis=0)) + 2 Spectral norm (the largest singular value) + -2 Not implemented + other Not implemented + ===== ============================ + + The Frobenius norm is given by [1]_: + + :math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}` + + References + ---------- + .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*, + Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15 + + Examples + -------- + >>> from scipy.sparse import csr_array, diags_array + >>> import numpy as np + >>> from scipy.sparse.linalg import norm + >>> a = np.arange(9) - 4 + >>> a + array([-4, -3, -2, -1, 0, 1, 2, 3, 4]) + >>> b = a.reshape((3, 3)) + >>> b + array([[-4, -3, -2], + [-1, 0, 1], + [ 2, 3, 4]]) + + >>> b = csr_array(b) + >>> norm(b) + 7.745966692414834 + >>> norm(b, 'fro') + 7.745966692414834 + >>> norm(b, np.inf) + 9 + >>> norm(b, -np.inf) + 2 + >>> norm(b, 1) + 7 + >>> norm(b, -1) + 6 + + The matrix 2-norm or the spectral norm is the largest singular + value, computed approximately and with limitations. + + >>> b = diags_array([-1, 1], [0, 1], shape=(9, 10)) + >>> norm(b, 2) + 1.9753... + """ + x = convert_pydata_sparse_to_scipy(x, target_format="csr") + if not issparse(x): + raise TypeError("input is not sparse. use numpy.linalg.norm") + + # Check the default case first and handle it immediately. + if axis is None and ord in (None, 'fro', 'f'): + return _sparse_frobenius_norm(x) + + # Some norms require functions that are not implemented for all types. + x = x.tocsr() + + if axis is None: + axis = tuple(range(x.ndim)) + elif not isinstance(axis, tuple): + msg = "'axis' must be None, an integer or a tuple of integers" + try: + int_axis = int(axis) + except TypeError as e: + raise TypeError(msg) from e + if axis != int_axis: + raise TypeError(msg) + axis = (int_axis,) + + nd = x.ndim + if len(axis) == 2: + row_axis, col_axis = axis + if not (-nd <= row_axis < nd and -nd <= col_axis < nd): + message = f'Invalid axis {axis!r} for an array with shape {x.shape!r}' + raise ValueError(message) + if row_axis % nd == col_axis % nd: + raise ValueError('Duplicate axes given.') + if ord == 2: + # Only solver="lobpcg" supports all numpy dtypes + _, s, _ = svds(x, k=1, solver="lobpcg") + return s[0] + elif ord == -2: + raise NotImplementedError + #return _multi_svd_norm(x, row_axis, col_axis, amin) + elif ord == 1: + return abs(x).sum(axis=row_axis).max() + elif ord == np.inf: + return abs(x).sum(axis=col_axis).max() + elif ord == -1: + return abs(x).sum(axis=row_axis).min() + elif ord == -np.inf: + return abs(x).sum(axis=col_axis).min() + elif ord in (None, 'f', 'fro'): + # The axis order does not matter for this norm. + return _sparse_frobenius_norm(x) + else: + raise ValueError("Invalid norm order for matrices.") + elif len(axis) == 1: + a, = axis + if not (-nd <= a < nd): + message = f'Invalid axis {axis!r} for an array with shape {x.shape!r}' + raise ValueError(message) + if ord == np.inf: + M = abs(x).max(axis=a) + elif ord == -np.inf: + M = abs(x).min(axis=a) + elif ord == 0: + # Zero norm + M = (x != 0).sum(axis=a) + elif ord == 1: + # special case for speedup + M = abs(x).sum(axis=a) + elif ord in (2, None): + M = sqrt(abs(x).power(2).sum(axis=a)) + else: + try: + ord + 1 + except TypeError as e: + raise ValueError('Invalid norm order for vectors.') from e + M = np.power(abs(x).power(ord).sum(axis=a), 1 / ord) + if hasattr(M, 'toarray'): + return M.toarray().ravel() + elif hasattr(M, 'A'): + return M.A.ravel() + else: + return M.ravel() + else: + raise ValueError("Improper number of dimensions to norm.") diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_onenormest.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_onenormest.py new file mode 100644 index 0000000000000000000000000000000000000000..a9e806ab6fbd5a2910de823a1046bce225d60a13 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_onenormest.py @@ -0,0 +1,467 @@ +"""Sparse block 1-norm estimator. +""" + +import numpy as np +from scipy.sparse.linalg import aslinearoperator + + +__all__ = ['onenormest'] + + +def onenormest(A, t=2, itmax=5, compute_v=False, compute_w=False): + """ + Compute a lower bound of the 1-norm of a sparse array. + + Parameters + ---------- + A : ndarray or other linear operator + A linear operator that can be transposed and that can + produce matrix products. + t : int, optional + A positive parameter controlling the tradeoff between + accuracy versus time and memory usage. + Larger values take longer and use more memory + but give more accurate output. + itmax : int, optional + Use at most this many iterations. + compute_v : bool, optional + Request a norm-maximizing linear operator input vector if True. + compute_w : bool, optional + Request a norm-maximizing linear operator output vector if True. + + Returns + ------- + est : float + An underestimate of the 1-norm of the sparse array. + v : ndarray, optional + The vector such that ||Av||_1 == est*||v||_1. + It can be thought of as an input to the linear operator + that gives an output with particularly large norm. + w : ndarray, optional + The vector Av which has relatively large 1-norm. + It can be thought of as an output of the linear operator + that is relatively large in norm compared to the input. + + Notes + ----- + This is algorithm 2.4 of [1]. + + In [2] it is described as follows. + "This algorithm typically requires the evaluation of + about 4t matrix-vector products and almost invariably + produces a norm estimate (which is, in fact, a lower + bound on the norm) correct to within a factor 3." + + .. versionadded:: 0.13.0 + + References + ---------- + .. [1] Nicholas J. Higham and Francoise Tisseur (2000), + "A Block Algorithm for Matrix 1-Norm Estimation, + with an Application to 1-Norm Pseudospectra." + SIAM J. Matrix Anal. Appl. Vol. 21, No. 4, pp. 1185-1201. + + .. [2] Awad H. Al-Mohy and Nicholas J. Higham (2009), + "A new scaling and squaring algorithm for the matrix exponential." + SIAM J. Matrix Anal. Appl. Vol. 31, No. 3, pp. 970-989. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse import csc_array + >>> from scipy.sparse.linalg import onenormest + >>> A = csc_array([[1., 0., 0.], [5., 8., 2.], [0., -1., 0.]], dtype=float) + >>> A.toarray() + array([[ 1., 0., 0.], + [ 5., 8., 2.], + [ 0., -1., 0.]]) + >>> onenormest(A) + 9.0 + >>> np.linalg.norm(A.toarray(), ord=1) + 9.0 + """ + + # Check the input. + A = aslinearoperator(A) + if A.shape[0] != A.shape[1]: + raise ValueError('expected the operator to act like a square matrix') + + # If the operator size is small compared to t, + # then it is easier to compute the exact norm. + # Otherwise estimate the norm. + n = A.shape[1] + if t >= n: + A_explicit = np.asarray(aslinearoperator(A).matmat(np.identity(n))) + if A_explicit.shape != (n, n): + raise Exception('internal error: ', + 'unexpected shape ' + str(A_explicit.shape)) + col_abs_sums = abs(A_explicit).sum(axis=0) + if col_abs_sums.shape != (n, ): + raise Exception('internal error: ', + 'unexpected shape ' + str(col_abs_sums.shape)) + argmax_j = np.argmax(col_abs_sums) + v = elementary_vector(n, argmax_j) + w = A_explicit[:, argmax_j] + est = col_abs_sums[argmax_j] + else: + est, v, w, nmults, nresamples = _onenormest_core(A, A.H, t, itmax) + + # Report the norm estimate along with some certificates of the estimate. + if compute_v or compute_w: + result = (est,) + if compute_v: + result += (v,) + if compute_w: + result += (w,) + return result + else: + return est + + +def _blocked_elementwise(func): + """ + Decorator for an elementwise function, to apply it blockwise along + first dimension, to avoid excessive memory usage in temporaries. + """ + block_size = 2**20 + + def wrapper(x): + if x.shape[0] < block_size: + return func(x) + else: + y0 = func(x[:block_size]) + y = np.zeros((x.shape[0],) + y0.shape[1:], dtype=y0.dtype) + y[:block_size] = y0 + del y0 + for j in range(block_size, x.shape[0], block_size): + y[j:j+block_size] = func(x[j:j+block_size]) + return y + return wrapper + + +@_blocked_elementwise +def sign_round_up(X): + """ + This should do the right thing for both real and complex matrices. + + From Higham and Tisseur: + "Everything in this section remains valid for complex matrices + provided that sign(A) is redefined as the matrix (aij / |aij|) + (and sign(0) = 1) transposes are replaced by conjugate transposes." + + """ + Y = X.copy() + Y[Y == 0] = 1 + Y /= np.abs(Y) + return Y + + +@_blocked_elementwise +def _max_abs_axis1(X): + return np.max(np.abs(X), axis=1) + + +def _sum_abs_axis0(X): + block_size = 2**20 + r = None + for j in range(0, X.shape[0], block_size): + y = np.sum(np.abs(X[j:j+block_size]), axis=0) + if r is None: + r = y + else: + r += y + return r + + +def elementary_vector(n, i): + v = np.zeros(n, dtype=float) + v[i] = 1 + return v + + +def vectors_are_parallel(v, w): + # Columns are considered parallel when they are equal or negative. + # Entries are required to be in {-1, 1}, + # which guarantees that the magnitudes of the vectors are identical. + if v.ndim != 1 or v.shape != w.shape: + raise ValueError('expected conformant vectors with entries in {-1,1}') + n = v.shape[0] + return np.dot(v, w) == n + + +def every_col_of_X_is_parallel_to_a_col_of_Y(X, Y): + for v in X.T: + if not any(vectors_are_parallel(v, w) for w in Y.T): + return False + return True + + +def column_needs_resampling(i, X, Y=None): + # column i of X needs resampling if either + # it is parallel to a previous column of X or + # it is parallel to a column of Y + n, t = X.shape + v = X[:, i] + if any(vectors_are_parallel(v, X[:, j]) for j in range(i)): + return True + if Y is not None: + if any(vectors_are_parallel(v, w) for w in Y.T): + return True + return False + + +def resample_column(i, X): + X[:, i] = np.random.randint(0, 2, size=X.shape[0])*2 - 1 + + +def less_than_or_close(a, b): + return np.allclose(a, b) or (a < b) + + +def _algorithm_2_2(A, AT, t): + """ + This is Algorithm 2.2. + + Parameters + ---------- + A : ndarray or other linear operator + A linear operator that can produce matrix products. + AT : ndarray or other linear operator + The transpose of A. + t : int, optional + A positive parameter controlling the tradeoff between + accuracy versus time and memory usage. + + Returns + ------- + g : sequence + A non-negative decreasing vector + such that g[j] is a lower bound for the 1-norm + of the column of A of jth largest 1-norm. + The first entry of this vector is therefore a lower bound + on the 1-norm of the linear operator A. + This sequence has length t. + ind : sequence + The ith entry of ind is the index of the column A whose 1-norm + is given by g[i]. + This sequence of indices has length t, and its entries are + chosen from range(n), possibly with repetition, + where n is the order of the operator A. + + Notes + ----- + This algorithm is mainly for testing. + It uses the 'ind' array in a way that is similar to + its usage in algorithm 2.4. This algorithm 2.2 may be easier to test, + so it gives a chance of uncovering bugs related to indexing + which could have propagated less noticeably to algorithm 2.4. + + """ + A_linear_operator = aslinearoperator(A) + AT_linear_operator = aslinearoperator(AT) + n = A_linear_operator.shape[0] + + # Initialize the X block with columns of unit 1-norm. + X = np.ones((n, t)) + if t > 1: + X[:, 1:] = np.random.randint(0, 2, size=(n, t-1))*2 - 1 + X /= float(n) + + # Iteratively improve the lower bounds. + # Track extra things, to assert invariants for debugging. + g_prev = None + h_prev = None + k = 1 + ind = range(t) + while True: + Y = np.asarray(A_linear_operator.matmat(X)) + g = _sum_abs_axis0(Y) + best_j = np.argmax(g) + g.sort() + g = g[::-1] + S = sign_round_up(Y) + Z = np.asarray(AT_linear_operator.matmat(S)) + h = _max_abs_axis1(Z) + + # If this algorithm runs for fewer than two iterations, + # then its return values do not have the properties indicated + # in the description of the algorithm. + # In particular, the entries of g are not 1-norms of any + # column of A until the second iteration. + # Therefore we will require the algorithm to run for at least + # two iterations, even though this requirement is not stated + # in the description of the algorithm. + if k >= 2: + if less_than_or_close(max(h), np.dot(Z[:, best_j], X[:, best_j])): + break + ind = np.argsort(h)[::-1][:t] + h = h[ind] + for j in range(t): + X[:, j] = elementary_vector(n, ind[j]) + + # Check invariant (2.2). + if k >= 2: + if not less_than_or_close(g_prev[0], h_prev[0]): + raise Exception('invariant (2.2) is violated') + if not less_than_or_close(h_prev[0], g[0]): + raise Exception('invariant (2.2) is violated') + + # Check invariant (2.3). + if k >= 3: + for j in range(t): + if not less_than_or_close(g[j], g_prev[j]): + raise Exception('invariant (2.3) is violated') + + # Update for the next iteration. + g_prev = g + h_prev = h + k += 1 + + # Return the lower bounds and the corresponding column indices. + return g, ind + + +def _onenormest_core(A, AT, t, itmax): + """ + Compute a lower bound of the 1-norm of a sparse array. + + Parameters + ---------- + A : ndarray or other linear operator + A linear operator that can produce matrix products. + AT : ndarray or other linear operator + The transpose of A. + t : int, optional + A positive parameter controlling the tradeoff between + accuracy versus time and memory usage. + itmax : int, optional + Use at most this many iterations. + + Returns + ------- + est : float + An underestimate of the 1-norm of the sparse array. + v : ndarray, optional + The vector such that ||Av||_1 == est*||v||_1. + It can be thought of as an input to the linear operator + that gives an output with particularly large norm. + w : ndarray, optional + The vector Av which has relatively large 1-norm. + It can be thought of as an output of the linear operator + that is relatively large in norm compared to the input. + nmults : int, optional + The number of matrix products that were computed. + nresamples : int, optional + The number of times a parallel column was observed, + necessitating a re-randomization of the column. + + Notes + ----- + This is algorithm 2.4. + + """ + # This function is a more or less direct translation + # of Algorithm 2.4 from the Higham and Tisseur (2000) paper. + A_linear_operator = aslinearoperator(A) + AT_linear_operator = aslinearoperator(AT) + if itmax < 2: + raise ValueError('at least two iterations are required') + if t < 1: + raise ValueError('at least one column is required') + n = A.shape[0] + if t >= n: + raise ValueError('t should be smaller than the order of A') + # Track the number of big*small matrix multiplications + # and the number of resamplings. + nmults = 0 + nresamples = 0 + # "We now explain our choice of starting matrix. We take the first + # column of X to be the vector of 1s [...] This has the advantage that + # for a matrix with nonnegative elements the algorithm converges + # with an exact estimate on the second iteration, and such matrices + # arise in applications [...]" + X = np.ones((n, t), dtype=float) + # "The remaining columns are chosen as rand{-1,1}, + # with a check for and correction of parallel columns, + # exactly as for S in the body of the algorithm." + if t > 1: + for i in range(1, t): + # These are technically initial samples, not resamples, + # so the resampling count is not incremented. + resample_column(i, X) + for i in range(t): + while column_needs_resampling(i, X): + resample_column(i, X) + nresamples += 1 + # "Choose starting matrix X with columns of unit 1-norm." + X /= float(n) + # "indices of used unit vectors e_j" + ind_hist = np.zeros(0, dtype=np.intp) + est_old = 0 + S = np.zeros((n, t), dtype=float) + k = 1 + ind = None + while True: + Y = np.asarray(A_linear_operator.matmat(X)) + nmults += 1 + mags = _sum_abs_axis0(Y) + est = np.max(mags) + best_j = np.argmax(mags) + if est > est_old or k == 2: + if k >= 2: + ind_best = ind[best_j] + w = Y[:, best_j] + # (1) + if k >= 2 and est <= est_old: + est = est_old + break + est_old = est + S_old = S + if k > itmax: + break + S = sign_round_up(Y) + del Y + # (2) + if every_col_of_X_is_parallel_to_a_col_of_Y(S, S_old): + break + if t > 1: + # "Ensure that no column of S is parallel to another column of S + # or to a column of S_old by replacing columns of S by rand{-1,1}." + for i in range(t): + while column_needs_resampling(i, S, S_old): + resample_column(i, S) + nresamples += 1 + del S_old + # (3) + Z = np.asarray(AT_linear_operator.matmat(S)) + nmults += 1 + h = _max_abs_axis1(Z) + del Z + # (4) + if k >= 2 and max(h) == h[ind_best]: + break + # "Sort h so that h_first >= ... >= h_last + # and re-order ind correspondingly." + # + # Later on, we will need at most t+len(ind_hist) largest + # entries, so drop the rest + ind = np.argsort(h)[::-1][:t+len(ind_hist)].copy() + del h + if t > 1: + # (5) + # Break if the most promising t vectors have been visited already. + if np.isin(ind[:t], ind_hist).all(): + break + # Put the most promising unvisited vectors at the front of the list + # and put the visited vectors at the end of the list. + # Preserve the order of the indices induced by the ordering of h. + seen = np.isin(ind, ind_hist) + ind = np.concatenate((ind[~seen], ind[seen])) + for j in range(t): + X[:, j] = elementary_vector(n, ind[j]) + + new_ind = ind[:t][~np.isin(ind[:t], ind_hist)] + ind_hist = np.concatenate((ind_hist, new_ind)) + k += 1 + v = elementary_vector(n, ind_best) + return est, v, w, nmults, nresamples diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_special_sparse_arrays.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_special_sparse_arrays.py new file mode 100644 index 0000000000000000000000000000000000000000..9d7415e1ec9a7e03b94dd8893935d46294b4b215 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_special_sparse_arrays.py @@ -0,0 +1,948 @@ +import numpy as np +from scipy.sparse.linalg import LinearOperator +from scipy.sparse import kron, eye, dia_array + +__all__ = ['LaplacianNd'] +# Sakurai and Mikota classes are intended for tests and benchmarks +# and explicitly not included in the public API of this module. + + +class LaplacianNd(LinearOperator): + """ + The grid Laplacian in ``N`` dimensions and its eigenvalues/eigenvectors. + + Construct Laplacian on a uniform rectangular grid in `N` dimensions + and output its eigenvalues and eigenvectors. + The Laplacian ``L`` is square, negative definite, real symmetric array + with signed integer entries and zeros otherwise. + + Parameters + ---------- + grid_shape : tuple + A tuple of integers of length ``N`` (corresponding to the dimension of + the Lapacian), where each entry gives the size of that dimension. The + Laplacian matrix is square of the size ``np.prod(grid_shape)``. + boundary_conditions : {'neumann', 'dirichlet', 'periodic'}, optional + The type of the boundary conditions on the boundaries of the grid. + Valid values are ``'dirichlet'`` or ``'neumann'``(default) or + ``'periodic'``. + dtype : dtype + Numerical type of the array. Default is ``np.int8``. + + Methods + ------- + toarray() + Construct a dense array from Laplacian data + tosparse() + Construct a sparse array from Laplacian data + eigenvalues(m=None) + Construct a 1D array of `m` largest (smallest in absolute value) + eigenvalues of the Laplacian matrix in ascending order. + eigenvectors(m=None): + Construct the array with columns made of `m` eigenvectors (``float``) + of the ``Nd`` Laplacian corresponding to the `m` ordered eigenvalues. + + .. versionadded:: 1.12.0 + + Notes + ----- + Compared to the MATLAB/Octave implementation [1] of 1-, 2-, and 3-D + Laplacian, this code allows the arbitrary N-D case and the matrix-free + callable option, but is currently limited to pure Dirichlet, Neumann or + Periodic boundary conditions only. + + The Laplacian matrix of a graph (`scipy.sparse.csgraph.laplacian`) of a + rectangular grid corresponds to the negative Laplacian with the Neumann + conditions, i.e., ``boundary_conditions = 'neumann'``. + + All eigenvalues and eigenvectors of the discrete Laplacian operator for + an ``N``-dimensional regular grid of shape `grid_shape` with the grid + step size ``h=1`` are analytically known [2]. + + References + ---------- + .. [1] https://github.com/lobpcg/blopex/blob/master/blopex_\ +tools/matlab/laplacian/laplacian.m + .. [2] "Eigenvalues and eigenvectors of the second derivative", Wikipedia + https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors_\ +of_the_second_derivative + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse.linalg import LaplacianNd + >>> from scipy.sparse import diags, csgraph + >>> from scipy.linalg import eigvalsh + + The one-dimensional Laplacian demonstrated below for pure Neumann boundary + conditions on a regular grid with ``n=6`` grid points is exactly the + negative graph Laplacian for the undirected linear graph with ``n`` + vertices using the sparse adjacency matrix ``G`` represented by the + famous tri-diagonal matrix: + + >>> n = 6 + >>> G = diags(np.ones(n - 1), 1, format='csr') + >>> Lf = csgraph.laplacian(G, symmetrized=True, form='function') + >>> grid_shape = (n, ) + >>> lap = LaplacianNd(grid_shape, boundary_conditions='neumann') + >>> np.array_equal(lap.matmat(np.eye(n)), -Lf(np.eye(n))) + True + + Since all matrix entries of the Laplacian are integers, ``'int8'`` is + the default dtype for storing matrix representations. + + >>> lap.tosparse() + + >>> lap.toarray() + array([[-1, 1, 0, 0, 0, 0], + [ 1, -2, 1, 0, 0, 0], + [ 0, 1, -2, 1, 0, 0], + [ 0, 0, 1, -2, 1, 0], + [ 0, 0, 0, 1, -2, 1], + [ 0, 0, 0, 0, 1, -1]], dtype=int8) + >>> np.array_equal(lap.matmat(np.eye(n)), lap.toarray()) + True + >>> np.array_equal(lap.tosparse().toarray(), lap.toarray()) + True + + Any number of extreme eigenvalues and/or eigenvectors can be computed. + + >>> lap = LaplacianNd(grid_shape, boundary_conditions='periodic') + >>> lap.eigenvalues() + array([-4., -3., -3., -1., -1., 0.]) + >>> lap.eigenvalues()[-2:] + array([-1., 0.]) + >>> lap.eigenvalues(2) + array([-1., 0.]) + >>> lap.eigenvectors(1) + array([[0.40824829], + [0.40824829], + [0.40824829], + [0.40824829], + [0.40824829], + [0.40824829]]) + >>> lap.eigenvectors(2) + array([[ 0.5 , 0.40824829], + [ 0. , 0.40824829], + [-0.5 , 0.40824829], + [-0.5 , 0.40824829], + [ 0. , 0.40824829], + [ 0.5 , 0.40824829]]) + >>> lap.eigenvectors() + array([[ 0.40824829, 0.28867513, 0.28867513, 0.5 , 0.5 , + 0.40824829], + [-0.40824829, -0.57735027, -0.57735027, 0. , 0. , + 0.40824829], + [ 0.40824829, 0.28867513, 0.28867513, -0.5 , -0.5 , + 0.40824829], + [-0.40824829, 0.28867513, 0.28867513, -0.5 , -0.5 , + 0.40824829], + [ 0.40824829, -0.57735027, -0.57735027, 0. , 0. , + 0.40824829], + [-0.40824829, 0.28867513, 0.28867513, 0.5 , 0.5 , + 0.40824829]]) + + The two-dimensional Laplacian is illustrated on a regular grid with + ``grid_shape = (2, 3)`` points in each dimension. + + >>> grid_shape = (2, 3) + >>> n = np.prod(grid_shape) + + Numeration of grid points is as follows: + + >>> np.arange(n).reshape(grid_shape + (-1,)) + array([[[0], + [1], + [2]], + + [[3], + [4], + [5]]]) + + Each of the boundary conditions ``'dirichlet'``, ``'periodic'``, and + ``'neumann'`` is illustrated separately; with ``'dirichlet'`` + + >>> lap = LaplacianNd(grid_shape, boundary_conditions='dirichlet') + >>> lap.tosparse() + + >>> lap.toarray() + array([[-4, 1, 0, 1, 0, 0], + [ 1, -4, 1, 0, 1, 0], + [ 0, 1, -4, 0, 0, 1], + [ 1, 0, 0, -4, 1, 0], + [ 0, 1, 0, 1, -4, 1], + [ 0, 0, 1, 0, 1, -4]], dtype=int8) + >>> np.array_equal(lap.matmat(np.eye(n)), lap.toarray()) + True + >>> np.array_equal(lap.tosparse().toarray(), lap.toarray()) + True + >>> lap.eigenvalues() + array([-6.41421356, -5. , -4.41421356, -3.58578644, -3. , + -1.58578644]) + >>> eigvals = eigvalsh(lap.toarray().astype(np.float64)) + >>> np.allclose(lap.eigenvalues(), eigvals) + True + >>> np.allclose(lap.toarray() @ lap.eigenvectors(), + ... lap.eigenvectors() @ np.diag(lap.eigenvalues())) + True + + with ``'periodic'`` + + >>> lap = LaplacianNd(grid_shape, boundary_conditions='periodic') + >>> lap.tosparse() + + >>> lap.toarray() + array([[-4, 1, 1, 2, 0, 0], + [ 1, -4, 1, 0, 2, 0], + [ 1, 1, -4, 0, 0, 2], + [ 2, 0, 0, -4, 1, 1], + [ 0, 2, 0, 1, -4, 1], + [ 0, 0, 2, 1, 1, -4]], dtype=int8) + >>> np.array_equal(lap.matmat(np.eye(n)), lap.toarray()) + True + >>> np.array_equal(lap.tosparse().toarray(), lap.toarray()) + True + >>> lap.eigenvalues() + array([-7., -7., -4., -3., -3., 0.]) + >>> eigvals = eigvalsh(lap.toarray().astype(np.float64)) + >>> np.allclose(lap.eigenvalues(), eigvals) + True + >>> np.allclose(lap.toarray() @ lap.eigenvectors(), + ... lap.eigenvectors() @ np.diag(lap.eigenvalues())) + True + + and with ``'neumann'`` + + >>> lap = LaplacianNd(grid_shape, boundary_conditions='neumann') + >>> lap.tosparse() + + >>> lap.toarray() + array([[-2, 1, 0, 1, 0, 0], + [ 1, -3, 1, 0, 1, 0], + [ 0, 1, -2, 0, 0, 1], + [ 1, 0, 0, -2, 1, 0], + [ 0, 1, 0, 1, -3, 1], + [ 0, 0, 1, 0, 1, -2]], dtype=int8) + >>> np.array_equal(lap.matmat(np.eye(n)), lap.toarray()) + True + >>> np.array_equal(lap.tosparse().toarray(), lap.toarray()) + True + >>> lap.eigenvalues() + array([-5., -3., -3., -2., -1., 0.]) + >>> eigvals = eigvalsh(lap.toarray().astype(np.float64)) + >>> np.allclose(lap.eigenvalues(), eigvals) + True + >>> np.allclose(lap.toarray() @ lap.eigenvectors(), + ... lap.eigenvectors() @ np.diag(lap.eigenvalues())) + True + + """ + + def __init__(self, grid_shape, *, + boundary_conditions='neumann', + dtype=np.int8): + + if boundary_conditions not in ('dirichlet', 'neumann', 'periodic'): + raise ValueError( + f"Unknown value {boundary_conditions!r} is given for " + "'boundary_conditions' parameter. The valid options are " + "'dirichlet', 'periodic', and 'neumann' (default)." + ) + + self.grid_shape = grid_shape + self.boundary_conditions = boundary_conditions + # LaplacianNd folds all dimensions in `grid_shape` into a single one + N = np.prod(grid_shape) + super().__init__(dtype=dtype, shape=(N, N)) + + def _eigenvalue_ordering(self, m): + """Compute `m` largest eigenvalues in each of the ``N`` directions, + i.e., up to ``m * N`` total, order them and return `m` largest. + """ + grid_shape = self.grid_shape + if m is None: + indices = np.indices(grid_shape) + Leig = np.zeros(grid_shape) + else: + grid_shape_min = min(grid_shape, + tuple(np.ones_like(grid_shape) * m)) + indices = np.indices(grid_shape_min) + Leig = np.zeros(grid_shape_min) + + for j, n in zip(indices, grid_shape): + if self.boundary_conditions == 'dirichlet': + Leig += -4 * np.sin(np.pi * (j + 1) / (2 * (n + 1))) ** 2 + elif self.boundary_conditions == 'neumann': + Leig += -4 * np.sin(np.pi * j / (2 * n)) ** 2 + else: # boundary_conditions == 'periodic' + Leig += -4 * np.sin(np.pi * np.floor((j + 1) / 2) / n) ** 2 + + Leig_ravel = Leig.ravel() + ind = np.argsort(Leig_ravel) + eigenvalues = Leig_ravel[ind] + if m is not None: + eigenvalues = eigenvalues[-m:] + ind = ind[-m:] + + return eigenvalues, ind + + def eigenvalues(self, m=None): + """Return the requested number of eigenvalues. + + Parameters + ---------- + m : int, optional + The positive number of smallest eigenvalues to return. + If not provided, then all eigenvalues will be returned. + + Returns + ------- + eigenvalues : float array + The requested `m` smallest or all eigenvalues, in ascending order. + """ + eigenvalues, _ = self._eigenvalue_ordering(m) + return eigenvalues + + def _ev1d(self, j, n): + """Return 1 eigenvector in 1d with index `j` + and number of grid points `n` where ``j < n``. + """ + if self.boundary_conditions == 'dirichlet': + i = np.pi * (np.arange(n) + 1) / (n + 1) + ev = np.sqrt(2. / (n + 1.)) * np.sin(i * (j + 1)) + elif self.boundary_conditions == 'neumann': + i = np.pi * (np.arange(n) + 0.5) / n + ev = np.sqrt((1. if j == 0 else 2.) / n) * np.cos(i * j) + else: # boundary_conditions == 'periodic' + if j == 0: + ev = np.sqrt(1. / n) * np.ones(n) + elif j + 1 == n and n % 2 == 0: + ev = np.sqrt(1. / n) * np.tile([1, -1], n//2) + else: + i = 2. * np.pi * (np.arange(n) + 0.5) / n + ev = np.sqrt(2. / n) * np.cos(i * np.floor((j + 1) / 2)) + # make small values exact zeros correcting round-off errors + # due to symmetry of eigenvectors the exact 0. is correct + ev[np.abs(ev) < np.finfo(np.float64).eps] = 0. + return ev + + def _one_eve(self, k): + """Return 1 eigenvector in Nd with multi-index `j` + as a tensor product of the corresponding 1d eigenvectors. + """ + phi = [self._ev1d(j, n) for j, n in zip(k, self.grid_shape)] + result = phi[0] + for phi in phi[1:]: + result = np.tensordot(result, phi, axes=0) + return np.asarray(result).ravel() + + def eigenvectors(self, m=None): + """Return the requested number of eigenvectors for ordered eigenvalues. + + Parameters + ---------- + m : int, optional + The positive number of eigenvectors to return. If not provided, + then all eigenvectors will be returned. + + Returns + ------- + eigenvectors : float array + An array with columns made of the requested `m` or all eigenvectors. + The columns are ordered according to the `m` ordered eigenvalues. + """ + _, ind = self._eigenvalue_ordering(m) + if m is None: + grid_shape_min = self.grid_shape + else: + grid_shape_min = min(self.grid_shape, + tuple(np.ones_like(self.grid_shape) * m)) + + N_indices = np.unravel_index(ind, grid_shape_min) + N_indices = [tuple(x) for x in zip(*N_indices)] + eigenvectors_list = [self._one_eve(k) for k in N_indices] + return np.column_stack(eigenvectors_list) + + def toarray(self): + """ + Converts the Laplacian data to a dense array. + + Returns + ------- + L : ndarray + The shape is ``(N, N)`` where ``N = np.prod(grid_shape)``. + + """ + grid_shape = self.grid_shape + n = np.prod(grid_shape) + L = np.zeros([n, n], dtype=np.int8) + # Scratch arrays + L_i = np.empty_like(L) + Ltemp = np.empty_like(L) + + for ind, dim in enumerate(grid_shape): + # Start zeroing out L_i + L_i[:] = 0 + # Allocate the top left corner with the kernel of L_i + # Einsum returns writable view of arrays + np.einsum("ii->i", L_i[:dim, :dim])[:] = -2 + np.einsum("ii->i", L_i[: dim - 1, 1:dim])[:] = 1 + np.einsum("ii->i", L_i[1:dim, : dim - 1])[:] = 1 + + if self.boundary_conditions == 'neumann': + L_i[0, 0] = -1 + L_i[dim - 1, dim - 1] = -1 + elif self.boundary_conditions == 'periodic': + if dim > 1: + L_i[0, dim - 1] += 1 + L_i[dim - 1, 0] += 1 + else: + L_i[0, 0] += 1 + + # kron is too slow for large matrices hence the next two tricks + # 1- kron(eye, mat) is block_diag(mat, mat, ...) + # 2- kron(mat, eye) can be performed by 4d stride trick + + # 1- + new_dim = dim + # for block_diag we tile the top left portion on the diagonal + if ind > 0: + tiles = np.prod(grid_shape[:ind]) + for j in range(1, tiles): + L_i[j*dim:(j+1)*dim, j*dim:(j+1)*dim] = L_i[:dim, :dim] + new_dim += dim + # 2- + # we need the keep L_i, but reset the array + Ltemp[:new_dim, :new_dim] = L_i[:new_dim, :new_dim] + tiles = int(np.prod(grid_shape[ind+1:])) + # Zero out the top left, the rest is already 0 + L_i[:new_dim, :new_dim] = 0 + idx = [x for x in range(tiles)] + L_i.reshape( + (new_dim, tiles, + new_dim, tiles) + )[:, idx, :, idx] = Ltemp[:new_dim, :new_dim] + + L += L_i + + return L.astype(self.dtype) + + def tosparse(self): + """ + Constructs a sparse array from the Laplacian data. The returned sparse + array format is dependent on the selected boundary conditions. + + Returns + ------- + L : scipy.sparse.sparray + The shape is ``(N, N)`` where ``N = np.prod(grid_shape)``. + + """ + N = len(self.grid_shape) + p = np.prod(self.grid_shape) + L = dia_array((p, p), dtype=np.int8) + + for i in range(N): + dim = self.grid_shape[i] + data = np.ones([3, dim], dtype=np.int8) + data[1, :] *= -2 + + if self.boundary_conditions == 'neumann': + data[1, 0] = -1 + data[1, -1] = -1 + + L_i = dia_array((data, [-1, 0, 1]), shape=(dim, dim), + dtype=np.int8 + ) + + if self.boundary_conditions == 'periodic': + t = dia_array((dim, dim), dtype=np.int8) + t.setdiag([1], k=-dim+1) + t.setdiag([1], k=dim-1) + L_i += t + + for j in range(i): + L_i = kron(eye(self.grid_shape[j], dtype=np.int8), L_i) + for j in range(i + 1, N): + L_i = kron(L_i, eye(self.grid_shape[j], dtype=np.int8)) + L += L_i + return L.astype(self.dtype) + + def _matvec(self, x): + grid_shape = self.grid_shape + N = len(grid_shape) + X = x.reshape(grid_shape + (-1,)) + Y = -2 * N * X + for i in range(N): + Y += np.roll(X, 1, axis=i) + Y += np.roll(X, -1, axis=i) + if self.boundary_conditions in ('neumann', 'dirichlet'): + Y[(slice(None),)*i + (0,) + (slice(None),)*(N-i-1) + ] -= np.roll(X, 1, axis=i)[ + (slice(None),) * i + (0,) + (slice(None),) * (N-i-1) + ] + Y[ + (slice(None),) * i + (-1,) + (slice(None),) * (N-i-1) + ] -= np.roll(X, -1, axis=i)[ + (slice(None),) * i + (-1,) + (slice(None),) * (N-i-1) + ] + + if self.boundary_conditions == 'neumann': + Y[ + (slice(None),) * i + (0,) + (slice(None),) * (N-i-1) + ] += np.roll(X, 0, axis=i)[ + (slice(None),) * i + (0,) + (slice(None),) * (N-i-1) + ] + Y[ + (slice(None),) * i + (-1,) + (slice(None),) * (N-i-1) + ] += np.roll(X, 0, axis=i)[ + (slice(None),) * i + (-1,) + (slice(None),) * (N-i-1) + ] + + return Y.reshape(-1, X.shape[-1]) + + def _matmat(self, x): + return self._matvec(x) + + def _adjoint(self): + return self + + def _transpose(self): + return self + + +class Sakurai(LinearOperator): + """ + Construct a Sakurai matrix in various formats and its eigenvalues. + + Constructs the "Sakurai" matrix motivated by reference [1]_: + square real symmetric positive definite and 5-diagonal + with the main diagonal ``[5, 6, 6, ..., 6, 6, 5], the ``+1`` and ``-1`` + diagonals filled with ``-4``, and the ``+2`` and ``-2`` diagonals + made of ``1``. Its eigenvalues are analytically known to be + ``16. * np.power(np.cos(0.5 * k * np.pi / (n + 1)), 4)``. + The matrix gets ill-conditioned with its size growing. + It is useful for testing and benchmarking sparse eigenvalue solvers + especially those taking advantage of its banded 5-diagonal structure. + See the notes below for details. + + Parameters + ---------- + n : int + The size of the matrix. + dtype : dtype + Numerical type of the array. Default is ``np.int8``. + + Methods + ------- + toarray() + Construct a dense array from Laplacian data + tosparse() + Construct a sparse array from Laplacian data + tobanded() + The Sakurai matrix in the format for banded symmetric matrices, + i.e., (3, n) ndarray with 3 upper diagonals + placing the main diagonal at the bottom. + eigenvalues + All eigenvalues of the Sakurai matrix ordered ascending. + + Notes + ----- + Reference [1]_ introduces a generalized eigenproblem for the matrix pair + `A` and `B` where `A` is the identity so we turn it into an eigenproblem + just for the matrix `B` that this function outputs in various formats + together with its eigenvalues. + + .. versionadded:: 1.12.0 + + References + ---------- + .. [1] T. Sakurai, H. Tadano, Y. Inadomi, and U. Nagashima, + "A moment-based method for large-scale generalized + eigenvalue problems", + Appl. Num. Anal. Comp. Math. Vol. 1 No. 2 (2004). + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse.linalg._special_sparse_arrays import Sakurai + >>> from scipy.linalg import eig_banded + >>> n = 6 + >>> sak = Sakurai(n) + + Since all matrix entries are small integers, ``'int8'`` is + the default dtype for storing matrix representations. + + >>> sak.toarray() + array([[ 5, -4, 1, 0, 0, 0], + [-4, 6, -4, 1, 0, 0], + [ 1, -4, 6, -4, 1, 0], + [ 0, 1, -4, 6, -4, 1], + [ 0, 0, 1, -4, 6, -4], + [ 0, 0, 0, 1, -4, 5]], dtype=int8) + >>> sak.tobanded() + array([[ 1, 1, 1, 1, 1, 1], + [-4, -4, -4, -4, -4, -4], + [ 5, 6, 6, 6, 6, 5]], dtype=int8) + >>> sak.tosparse() + + >>> np.array_equal(sak.dot(np.eye(n)), sak.tosparse().toarray()) + True + >>> sak.eigenvalues() + array([0.03922866, 0.56703972, 2.41789479, 5.97822974, + 10.54287655, 14.45473055]) + >>> sak.eigenvalues(2) + array([0.03922866, 0.56703972]) + + The banded form can be used in scipy functions for banded matrices, e.g., + + >>> e = eig_banded(sak.tobanded(), eigvals_only=True) + >>> np.allclose(sak.eigenvalues, e, atol= n * n * n * np.finfo(float).eps) + True + + """ + def __init__(self, n, dtype=np.int8): + self.n = n + self.dtype = dtype + shape = (n, n) + super().__init__(dtype, shape) + + def eigenvalues(self, m=None): + """Return the requested number of eigenvalues. + + Parameters + ---------- + m : int, optional + The positive number of smallest eigenvalues to return. + If not provided, then all eigenvalues will be returned. + + Returns + ------- + eigenvalues : `np.float64` array + The requested `m` smallest or all eigenvalues, in ascending order. + """ + if m is None: + m = self.n + k = np.arange(self.n + 1 -m, self.n + 1) + return np.flip(16. * np.power(np.cos(0.5 * k * np.pi / (self.n + 1)), 4)) + + def tobanded(self): + """ + Construct the Sakurai matrix as a banded array. + """ + d0 = np.r_[5, 6 * np.ones(self.n - 2, dtype=self.dtype), 5] + d1 = -4 * np.ones(self.n, dtype=self.dtype) + d2 = np.ones(self.n, dtype=self.dtype) + return np.array([d2, d1, d0]).astype(self.dtype) + + def tosparse(self): + """ + Construct the Sakurai matrix is a sparse format. + """ + from scipy.sparse import spdiags + d = self.tobanded() + # the banded format has the main diagonal at the bottom + # `spdiags` has no `dtype` parameter so inherits dtype from banded + return spdiags([d[0], d[1], d[2], d[1], d[0]], [-2, -1, 0, 1, 2], + self.n, self.n) + + def toarray(self): + return self.tosparse().toarray() + + def _matvec(self, x): + """ + Construct matrix-free callable banded-matrix-vector multiplication by + the Sakurai matrix without constructing or storing the matrix itself + using the knowledge of its entries and the 5-diagonal format. + """ + x = x.reshape(self.n, -1) + result_dtype = np.promote_types(x.dtype, self.dtype) + sx = np.zeros_like(x, dtype=result_dtype) + sx[0, :] = 5 * x[0, :] - 4 * x[1, :] + x[2, :] + sx[-1, :] = 5 * x[-1, :] - 4 * x[-2, :] + x[-3, :] + sx[1: -1, :] = (6 * x[1: -1, :] - 4 * (x[:-2, :] + x[2:, :]) + + np.pad(x[:-3, :], ((1, 0), (0, 0))) + + np.pad(x[3:, :], ((0, 1), (0, 0)))) + return sx + + def _matmat(self, x): + """ + Construct matrix-free callable matrix-matrix multiplication by + the Sakurai matrix without constructing or storing the matrix itself + by reusing the ``_matvec(x)`` that supports both 1D and 2D arrays ``x``. + """ + return self._matvec(x) + + def _adjoint(self): + return self + + def _transpose(self): + return self + + +class MikotaM(LinearOperator): + """ + Construct a mass matrix in various formats of Mikota pair. + + The mass matrix `M` is square real diagonal + positive definite with entries that are reciprocal to integers. + + Parameters + ---------- + shape : tuple of int + The shape of the matrix. + dtype : dtype + Numerical type of the array. Default is ``np.float64``. + + Methods + ------- + toarray() + Construct a dense array from Mikota data + tosparse() + Construct a sparse array from Mikota data + tobanded() + The format for banded symmetric matrices, + i.e., (1, n) ndarray with the main diagonal. + """ + def __init__(self, shape, dtype=np.float64): + self.shape = shape + self.dtype = dtype + super().__init__(dtype, shape) + + def _diag(self): + # The matrix is constructed from its diagonal 1 / [1, ..., N+1]; + # compute in a function to avoid duplicated code & storage footprint + return (1. / np.arange(1, self.shape[0] + 1)).astype(self.dtype) + + def tobanded(self): + return self._diag() + + def tosparse(self): + from scipy.sparse import diags + return diags([self._diag()], [0], shape=self.shape, dtype=self.dtype) + + def toarray(self): + return np.diag(self._diag()).astype(self.dtype) + + def _matvec(self, x): + """ + Construct matrix-free callable banded-matrix-vector multiplication by + the Mikota mass matrix without constructing or storing the matrix itself + using the knowledge of its entries and the diagonal format. + """ + x = x.reshape(self.shape[0], -1) + return self._diag()[:, np.newaxis] * x + + def _matmat(self, x): + """ + Construct matrix-free callable matrix-matrix multiplication by + the Mikota mass matrix without constructing or storing the matrix itself + by reusing the ``_matvec(x)`` that supports both 1D and 2D arrays ``x``. + """ + return self._matvec(x) + + def _adjoint(self): + return self + + def _transpose(self): + return self + + +class MikotaK(LinearOperator): + """ + Construct a stiffness matrix in various formats of Mikota pair. + + The stiffness matrix `K` is square real tri-diagonal symmetric + positive definite with integer entries. + + Parameters + ---------- + shape : tuple of int + The shape of the matrix. + dtype : dtype + Numerical type of the array. Default is ``np.int32``. + + Methods + ------- + toarray() + Construct a dense array from Mikota data + tosparse() + Construct a sparse array from Mikota data + tobanded() + The format for banded symmetric matrices, + i.e., (2, n) ndarray with 2 upper diagonals + placing the main diagonal at the bottom. + """ + def __init__(self, shape, dtype=np.int32): + self.shape = shape + self.dtype = dtype + super().__init__(dtype, shape) + # The matrix is constructed from its diagonals; + # we precompute these to avoid duplicating the computation + n = shape[0] + self._diag0 = np.arange(2 * n - 1, 0, -2, dtype=self.dtype) + self._diag1 = - np.arange(n - 1, 0, -1, dtype=self.dtype) + + def tobanded(self): + return np.array([np.pad(self._diag1, (1, 0), 'constant'), self._diag0]) + + def tosparse(self): + from scipy.sparse import diags + return diags([self._diag1, self._diag0, self._diag1], [-1, 0, 1], + shape=self.shape, dtype=self.dtype) + + def toarray(self): + return self.tosparse().toarray() + + def _matvec(self, x): + """ + Construct matrix-free callable banded-matrix-vector multiplication by + the Mikota stiffness matrix without constructing or storing the matrix + itself using the knowledge of its entries and the 3-diagonal format. + """ + x = x.reshape(self.shape[0], -1) + result_dtype = np.promote_types(x.dtype, self.dtype) + kx = np.zeros_like(x, dtype=result_dtype) + d1 = self._diag1 + d0 = self._diag0 + kx[0, :] = d0[0] * x[0, :] + d1[0] * x[1, :] + kx[-1, :] = d1[-1] * x[-2, :] + d0[-1] * x[-1, :] + kx[1: -1, :] = (d1[:-1, None] * x[: -2, :] + + d0[1: -1, None] * x[1: -1, :] + + d1[1:, None] * x[2:, :]) + return kx + + def _matmat(self, x): + """ + Construct matrix-free callable matrix-matrix multiplication by + the Stiffness mass matrix without constructing or storing the matrix itself + by reusing the ``_matvec(x)`` that supports both 1D and 2D arrays ``x``. + """ + return self._matvec(x) + + def _adjoint(self): + return self + + def _transpose(self): + return self + + +class MikotaPair: + """ + Construct the Mikota pair of matrices in various formats and + eigenvalues of the generalized eigenproblem with them. + + The Mikota pair of matrices [1, 2]_ models a vibration problem + of a linear mass-spring system with the ends attached where + the stiffness of the springs and the masses increase along + the system length such that vibration frequencies are subsequent + integers 1, 2, ..., `n` where `n` is the number of the masses. Thus, + eigenvalues of the generalized eigenvalue problem for + the matrix pair `K` and `M` where `K` is the system stiffness matrix + and `M` is the system mass matrix are the squares of the integers, + i.e., 1, 4, 9, ..., ``n * n``. + + The stiffness matrix `K` is square real tri-diagonal symmetric + positive definite. The mass matrix `M` is diagonal with diagonal + entries 1, 1/2, 1/3, ...., ``1/n``. Both matrices get + ill-conditioned with `n` growing. + + Parameters + ---------- + n : int + The size of the matrices of the Mikota pair. + dtype : dtype + Numerical type of the array. Default is ``np.float64``. + + Attributes + ---------- + eigenvalues : 1D ndarray, ``np.uint64`` + All eigenvalues of the Mikota pair ordered ascending. + + Methods + ------- + MikotaK() + A `LinearOperator` custom object for the stiffness matrix. + MikotaM() + A `LinearOperator` custom object for the mass matrix. + + .. versionadded:: 1.12.0 + + References + ---------- + .. [1] J. Mikota, "Frequency tuning of chain structure multibody oscillators + to place the natural frequencies at omega1 and N-1 integer multiples + omega2,..., omegaN", Z. Angew. Math. Mech. 81 (2001), S2, S201-S202. + Appl. Num. Anal. Comp. Math. Vol. 1 No. 2 (2004). + .. [2] Peter C. Muller and Metin Gurgoze, + "Natural frequencies of a multi-degree-of-freedom vibration system", + Proc. Appl. Math. Mech. 6, 319-320 (2006). + http://dx.doi.org/10.1002/pamm.200610141. + + Examples + -------- + >>> import numpy as np + >>> from scipy.sparse.linalg._special_sparse_arrays import MikotaPair + >>> n = 6 + >>> mik = MikotaPair(n) + >>> mik_k = mik.k + >>> mik_m = mik.m + >>> mik_k.toarray() + array([[11., -5., 0., 0., 0., 0.], + [-5., 9., -4., 0., 0., 0.], + [ 0., -4., 7., -3., 0., 0.], + [ 0., 0., -3., 5., -2., 0.], + [ 0., 0., 0., -2., 3., -1.], + [ 0., 0., 0., 0., -1., 1.]]) + >>> mik_k.tobanded() + array([[ 0., -5., -4., -3., -2., -1.], + [11., 9., 7., 5., 3., 1.]]) + >>> mik_m.tobanded() + array([1. , 0.5 , 0.33333333, 0.25 , 0.2 , + 0.16666667]) + >>> mik_k.tosparse() + + >>> mik_m.tosparse() + + >>> np.array_equal(mik_k(np.eye(n)), mik_k.toarray()) + True + >>> np.array_equal(mik_m(np.eye(n)), mik_m.toarray()) + True + >>> mik.eigenvalues() + array([ 1, 4, 9, 16, 25, 36]) + >>> mik.eigenvalues(2) + array([ 1, 4]) + + """ + def __init__(self, n, dtype=np.float64): + self.n = n + self.dtype = dtype + self.shape = (n, n) + self.m = MikotaM(self.shape, self.dtype) + self.k = MikotaK(self.shape, self.dtype) + + def eigenvalues(self, m=None): + """Return the requested number of eigenvalues. + + Parameters + ---------- + m : int, optional + The positive number of smallest eigenvalues to return. + If not provided, then all eigenvalues will be returned. + + Returns + ------- + eigenvalues : `np.uint64` array + The requested `m` smallest or all eigenvalues, in ascending order. + """ + if m is None: + m = self.n + arange_plus1 = np.arange(1, m + 1, dtype=np.uint64) + return arange_plus1 * arange_plus1 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_svdp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_svdp.py new file mode 100644 index 0000000000000000000000000000000000000000..fd64c6d0c3069eceb5a347111f1ac77bea8ea942 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/_svdp.py @@ -0,0 +1,309 @@ +""" +Python wrapper for PROPACK +-------------------------- + +PROPACK is a collection of Fortran routines for iterative computation +of partial SVDs of large matrices or linear operators. + +Based on BSD licensed pypropack project: + http://github.com/jakevdp/pypropack + Author: Jake Vanderplas + +PROPACK source is BSD licensed, and available at + http://soi.stanford.edu/~rmunk/PROPACK/ +""" + +__all__ = ['_svdp'] + +import numpy as np + +from scipy.sparse.linalg import aslinearoperator +from scipy.linalg import LinAlgError + +from ._propack import _spropack # type: ignore[attr-defined] +from ._propack import _dpropack # type: ignore[attr-defined] +from ._propack import _cpropack # type: ignore[attr-defined] +from ._propack import _zpropack # type: ignore[attr-defined] + + +_lansvd_dict = { + 'f': _spropack.slansvd, + 'd': _dpropack.dlansvd, + 'F': _cpropack.clansvd, + 'D': _zpropack.zlansvd, +} + + +_lansvd_irl_dict = { + 'f': _spropack.slansvd_irl, + 'd': _dpropack.dlansvd_irl, + 'F': _cpropack.clansvd_irl, + 'D': _zpropack.zlansvd_irl, +} + +_which_converter = { + 'LM': 'L', + 'SM': 'S', +} + + +class _AProd: + """ + Wrapper class for linear operator + + The call signature of the __call__ method matches the callback of + the PROPACK routines. + """ + def __init__(self, A): + try: + self.A = aslinearoperator(A) + except TypeError: + self.A = aslinearoperator(np.asarray(A)) + + def __call__(self, transa, m, n, x, y, sparm, iparm): + if transa == 'n': + y[:] = self.A.matvec(x) + else: + y[:] = self.A.rmatvec(x) + + @property + def shape(self): + return self.A.shape + + @property + def dtype(self): + try: + return self.A.dtype + except AttributeError: + return self.A.matvec(np.zeros(self.A.shape[1])).dtype + + +def _svdp(A, k, which='LM', irl_mode=True, kmax=None, + compute_u=True, compute_v=True, v0=None, full_output=False, tol=0, + delta=None, eta=None, anorm=0, cgs=False, elr=True, + min_relgap=0.002, shifts=None, maxiter=None, rng=None): + """ + Compute the singular value decomposition of a linear operator using PROPACK + + Parameters + ---------- + A : array_like, sparse matrix, or LinearOperator + Operator for which SVD will be computed. If `A` is a LinearOperator + object, it must define both ``matvec`` and ``rmatvec`` methods. + k : int + Number of singular values/vectors to compute + which : {"LM", "SM"} + Which singular triplets to compute: + - 'LM': compute triplets corresponding to the `k` largest singular + values + - 'SM': compute triplets corresponding to the `k` smallest singular + values + `which='SM'` requires `irl_mode=True`. Computes largest singular + values by default. + irl_mode : bool, optional + If `True`, then compute SVD using IRL (implicitly restarted Lanczos) + mode. Default is `True`. + kmax : int, optional + Maximal number of iterations / maximal dimension of the Krylov + subspace. Default is ``10 * k``. + compute_u : bool, optional + If `True` (default) then compute left singular vectors, `u`. + compute_v : bool, optional + If `True` (default) then compute right singular vectors, `v`. + tol : float, optional + The desired relative accuracy for computed singular values. + If not specified, it will be set based on machine precision. + v0 : array_like, optional + Starting vector for iterations: must be of length ``A.shape[0]``. + If not specified, PROPACK will generate a starting vector. + full_output : bool, optional + If `True`, then return sigma_bound. Default is `False`. + delta : float, optional + Level of orthogonality to maintain between Lanczos vectors. + Default is set based on machine precision. + eta : float, optional + Orthogonality cutoff. During reorthogonalization, vectors with + component larger than `eta` along the Lanczos vector will be purged. + Default is set based on machine precision. + anorm : float, optional + Estimate of ``||A||``. Default is ``0``. + cgs : bool, optional + If `True`, reorthogonalization is done using classical Gram-Schmidt. + If `False` (default), it is done using modified Gram-Schmidt. + elr : bool, optional + If `True` (default), then extended local orthogonality is enforced + when obtaining singular vectors. + min_relgap : float, optional + The smallest relative gap allowed between any shift in IRL mode. + Default is ``0.001``. Accessed only if ``irl_mode=True``. + shifts : int, optional + Number of shifts per restart in IRL mode. Default is determined + to satisfy ``k <= min(kmax-shifts, m, n)``. Must be + >= 0, but choosing 0 might lead to performance degradation. + Accessed only if ``irl_mode=True``. + maxiter : int, optional + Maximum number of restarts in IRL mode. Default is ``1000``. + Accessed only if ``irl_mode=True``. + rng : `numpy.random.Generator`, optional + Pseudorandom number generator state. When `rng` is None, a new + `numpy.random.Generator` is created using entropy from the + operating system. Types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + + Returns + ------- + u : ndarray + The `k` largest (``which="LM"``) or smallest (``which="SM"``) left + singular vectors, ``shape == (A.shape[0], 3)``, returned only if + ``compute_u=True``. + sigma : ndarray + The top `k` singular values, ``shape == (k,)`` + vt : ndarray + The `k` largest (``which="LM"``) or smallest (``which="SM"``) right + singular vectors, ``shape == (3, A.shape[1])``, returned only if + ``compute_v=True``. + sigma_bound : ndarray + the error bounds on the singular values sigma, returned only if + ``full_output=True``. + + """ + if rng is None: + raise ValueError("`rng` must be a normalized numpy.random.Generator instance") + + which = which.upper() + if which not in {'LM', 'SM'}: + raise ValueError("`which` must be either 'LM' or 'SM'") + if not irl_mode and which == 'SM': + raise ValueError("`which`='SM' requires irl_mode=True") + + aprod = _AProd(A) + typ = aprod.dtype.char + + try: + lansvd_irl = _lansvd_irl_dict[typ] + lansvd = _lansvd_dict[typ] + except KeyError: + # work with non-supported types using native system precision + if np.iscomplexobj(np.empty(0, dtype=typ)): + typ = np.dtype(complex).char + else: + typ = np.dtype(float).char + lansvd_irl = _lansvd_irl_dict[typ] + lansvd = _lansvd_dict[typ] + + m, n = aprod.shape + if (k < 1) or (k > min(m, n)): + raise ValueError("k must be positive and not greater than m or n") + + if kmax is None: + kmax = 10*k + if maxiter is None: + maxiter = 1000 + + # guard against unnecessarily large kmax + kmax = min(m + 1, n + 1, kmax) + if kmax < k: + raise ValueError( + "kmax must be greater than or equal to k, " + f"but kmax ({kmax}) < k ({k})") + + # convert python args to fortran args + jobu = 'y' if compute_u else 'n' + jobv = 'y' if compute_v else 'n' + + # these will be the output arrays + u = np.zeros((m, kmax + 1), order='F', dtype=typ) + v = np.zeros((n, kmax), order='F', dtype=typ) + + # Specify the starting vector. if v0 is all zero, PROPACK will generate + # a random starting vector: the random seed cannot be controlled in that + # case, so we'll instead use numpy to generate a random vector + if v0 is None: + u[:, 0] = rng.uniform(size=m) + if np.iscomplexobj(np.empty(0, dtype=typ)): # complex type + u[:, 0] += 1j * rng.uniform(size=m) + else: + try: + u[:, 0] = v0 + except ValueError: + raise ValueError(f"v0 must be of length {m}") + + # process options for the fit + if delta is None: + delta = np.sqrt(np.finfo(typ).eps) + if eta is None: + eta = np.finfo(typ).eps ** 0.75 + + if irl_mode: + doption = np.array((delta, eta, anorm, min_relgap), dtype=typ.lower()) + + # validate or find default shifts + if shifts is None: + shifts = kmax - k + if k > min(kmax - shifts, m, n): + raise ValueError('shifts must satisfy ' + 'k <= min(kmax-shifts, m, n)!') + elif shifts < 0: + raise ValueError('shifts must be >= 0!') + + else: + doption = np.array((delta, eta, anorm), dtype=typ.lower()) + + ioption = np.array((int(bool(cgs)), int(bool(elr))), dtype='i') + + # If computing `u` or `v` (left and right singular vectors, + # respectively), `blocksize` controls how large a fraction of the + # work is done via fast BLAS level 3 operations. A larger blocksize + # may lead to faster computation at the expense of greater memory + # consumption. `blocksize` must be ``>= 1``. Choosing blocksize + # of 16, but docs don't specify; it's almost surely a + # power of 2. + blocksize = 16 + + # Determine lwork & liwork: + # the required lengths are specified in the PROPACK documentation + if compute_u or compute_v: + lwork = m + n + 9*kmax + 5*kmax*kmax + 4 + max( + 3*kmax*kmax + 4*kmax + 4, + blocksize*max(m, n)) + liwork = 8*kmax + else: + lwork = m + n + 9*kmax + 2*kmax*kmax + 4 + max(m + n, 4*kmax + 4) + liwork = 2*kmax + 1 + work = np.empty(lwork, dtype=typ.lower()) + iwork = np.empty(liwork, dtype=np.int32) + + # dummy arguments: these are passed to aprod, and not used in this wrapper + dparm = np.empty(1, dtype=typ.lower()) + iparm = np.empty(1, dtype=np.int32) + + if typ.isupper(): + # PROPACK documentation is unclear on the required length of zwork. + # Use the same length Julia's wrapper uses + # see https://github.com/JuliaSmoothOptimizers/PROPACK.jl/ + zwork = np.empty(m + n + 32*m, dtype=typ) + works = work, zwork, iwork + else: + works = work, iwork + + if irl_mode: + u, sigma, bnd, v, info = lansvd_irl(_which_converter[which], jobu, + jobv, m, n, shifts, k, maxiter, + aprod, u, v, tol, *works, doption, + ioption, dparm, iparm) + else: + u, sigma, bnd, v, info = lansvd(jobu, jobv, m, n, k, aprod, u, v, tol, + *works, doption, ioption, dparm, iparm) + + if info > 0: + raise LinAlgError( + f"An invariant subspace of dimension {info} was found.") + elif info < 0: + raise LinAlgError( + f"k={k} singular triplets did not converge within " + f"kmax={kmax} iterations") + + # info == 0: The K largest (or smallest) singular triplets were computed + # successfully! + + return u[:, :k], sigma, v[:, :k].conj().T, bnd diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/dsolve.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/dsolve.py new file mode 100644 index 0000000000000000000000000000000000000000..45139f6b280d047386652577d9e1c8d2aaeb7033 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/dsolve.py @@ -0,0 +1,22 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'MatrixRankWarning', 'SuperLU', 'factorized', + 'spilu', 'splu', 'spsolve', + 'spsolve_triangular', 'use_solver', 'test' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse.linalg", module="dsolve", + private_modules=["_dsolve"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/eigen.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/eigen.py new file mode 100644 index 0000000000000000000000000000000000000000..588986d6650aad334e6a9a682ed76cef94295298 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/eigen.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'ArpackError', 'ArpackNoConvergence', 'ArpackError', + 'eigs', 'eigsh', 'lobpcg', 'svds', 'test' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse.linalg", module="eigen", + private_modules=["_eigen"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/interface.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/interface.py new file mode 100644 index 0000000000000000000000000000000000000000..24f40f185b1328b16e7e239e5a165cc6b1ed4317 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/interface.py @@ -0,0 +1,20 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'LinearOperator', 'aslinearoperator', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse.linalg", module="interface", + private_modules=["_interface"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/isolve.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/isolve.py new file mode 100644 index 0000000000000000000000000000000000000000..e032ddd9c673be3bc8790adad3bdae1839127050 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/isolve.py @@ -0,0 +1,22 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'bicg', 'bicgstab', 'cg', 'cgs', 'gcrotmk', 'gmres', + 'lgmres', 'lsmr', 'lsqr', + 'minres', 'qmr', 'tfqmr', 'test' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse.linalg", module="isolve", + private_modules=["_isolve"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/matfuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/matfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..8ed877ff1aa6f5a5466ce94729b9225dcce37b36 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/matfuncs.py @@ -0,0 +1,18 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse.linalg` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = ["expm", "inv", "spsolve", "LinearOperator"] # noqa: F822 + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse.linalg", module="matfuncs", + private_modules=["_matfuncs"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_expm_multiply.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_expm_multiply.py new file mode 100644 index 0000000000000000000000000000000000000000..b6d3661e99580906501dac66c9960d8d4671137f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_expm_multiply.py @@ -0,0 +1,367 @@ +"""Test functions for the sparse.linalg._expm_multiply module.""" +from functools import partial +from itertools import product + +import numpy as np +import pytest +from numpy.testing import (assert_allclose, assert_, assert_equal, + suppress_warnings) +from scipy.sparse import SparseEfficiencyWarning +import scipy.sparse +from scipy.sparse.linalg import aslinearoperator +import scipy.linalg +from scipy.sparse.linalg import expm as sp_expm +from scipy.sparse.linalg._expm_multiply import (_theta, _compute_p_max, + _onenormest_matrix_power, expm_multiply, _expm_multiply_simple, + _expm_multiply_interval) +from scipy._lib._util import np_long + + +IMPRECISE = {np.single, np.csingle} +REAL_DTYPES = (np.intc, np_long, np.longlong, + np.float32, np.float64, np.longdouble) +COMPLEX_DTYPES = (np.complex64, np.complex128, np.clongdouble) +DTYPES = REAL_DTYPES + COMPLEX_DTYPES + + +def estimated(func): + """If trace is estimated, it should warn. + + We warn that estimation of trace might impact performance. + All result have to be correct nevertheless! + + """ + def wrapped(*args, **kwds): + with pytest.warns(UserWarning, + match="Trace of LinearOperator not available"): + return func(*args, **kwds) + return wrapped + + +def less_than_or_close(a, b): + return np.allclose(a, b) or (a < b) + + +class TestExpmActionSimple: + """ + These tests do not consider the case of multiple time steps in one call. + """ + + def test_theta_monotonicity(self): + pairs = sorted(_theta.items()) + for (m_a, theta_a), (m_b, theta_b) in zip(pairs[:-1], pairs[1:]): + assert_(theta_a < theta_b) + + def test_p_max_default(self): + m_max = 55 + expected_p_max = 8 + observed_p_max = _compute_p_max(m_max) + assert_equal(observed_p_max, expected_p_max) + + def test_p_max_range(self): + for m_max in range(1, 55+1): + p_max = _compute_p_max(m_max) + assert_(p_max*(p_max - 1) <= m_max + 1) + p_too_big = p_max + 1 + assert_(p_too_big*(p_too_big - 1) > m_max + 1) + + def test_onenormest_matrix_power(self): + rng = np.random.RandomState(1234) + n = 40 + nsamples = 10 + for i in range(nsamples): + A = scipy.linalg.inv(rng.randn(n, n)) + for p in range(4): + if not p: + M = np.identity(n) + else: + M = np.dot(M, A) + estimated = _onenormest_matrix_power(A, p) + exact = np.linalg.norm(M, 1) + assert_(less_than_or_close(estimated, exact)) + assert_(less_than_or_close(exact, 3*estimated)) + + @pytest.mark.thread_unsafe + def test_expm_multiply(self): + np.random.seed(1234) + n = 40 + k = 3 + nsamples = 10 + for i in range(nsamples): + A = scipy.linalg.inv(np.random.randn(n, n)) + B = np.random.randn(n, k) + observed = expm_multiply(A, B) + expected = np.dot(sp_expm(A), B) + assert_allclose(observed, expected) + observed = estimated(expm_multiply)(aslinearoperator(A), B) + assert_allclose(observed, expected) + traceA = np.trace(A) + observed = expm_multiply(aslinearoperator(A), B, traceA=traceA) + assert_allclose(observed, expected) + + @pytest.mark.thread_unsafe + def test_matrix_vector_multiply(self): + np.random.seed(1234) + n = 40 + nsamples = 10 + for i in range(nsamples): + A = scipy.linalg.inv(np.random.randn(n, n)) + v = np.random.randn(n) + observed = expm_multiply(A, v) + expected = np.dot(sp_expm(A), v) + assert_allclose(observed, expected) + observed = estimated(expm_multiply)(aslinearoperator(A), v) + assert_allclose(observed, expected) + + @pytest.mark.thread_unsafe + def test_scaled_expm_multiply(self): + np.random.seed(1234) + n = 40 + k = 3 + nsamples = 10 + for i, t in product(range(nsamples), [0.2, 1.0, 1.5]): + with np.errstate(invalid='ignore'): + A = scipy.linalg.inv(np.random.randn(n, n)) + B = np.random.randn(n, k) + observed = _expm_multiply_simple(A, B, t=t) + expected = np.dot(sp_expm(t*A), B) + assert_allclose(observed, expected) + observed = estimated(_expm_multiply_simple)( + aslinearoperator(A), B, t=t + ) + assert_allclose(observed, expected) + + @pytest.mark.thread_unsafe + def test_scaled_expm_multiply_single_timepoint(self): + np.random.seed(1234) + t = 0.1 + n = 5 + k = 2 + A = np.random.randn(n, n) + B = np.random.randn(n, k) + observed = _expm_multiply_simple(A, B, t=t) + expected = sp_expm(t*A).dot(B) + assert_allclose(observed, expected) + observed = estimated(_expm_multiply_simple)( + aslinearoperator(A), B, t=t + ) + assert_allclose(observed, expected) + + @pytest.mark.thread_unsafe + def test_sparse_expm_multiply(self): + rng = np.random.default_rng(1234) + n = 40 + k = 3 + nsamples = 10 + for i in range(nsamples): + A = scipy.sparse.random_array((n, n), density=0.05, rng=rng) + B = rng.standard_normal((n, k)) + observed = expm_multiply(A, B) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, + "splu converted its input to CSC format") + sup.filter(SparseEfficiencyWarning, + "spsolve is more efficient when sparse b is in the" + " CSC matrix format") + expected = sp_expm(A).dot(B) + assert_allclose(observed, expected) + observed = estimated(expm_multiply)(aslinearoperator(A), B) + assert_allclose(observed, expected) + + @pytest.mark.thread_unsafe + def test_complex(self): + A = np.array([ + [1j, 1j], + [0, 1j]], dtype=complex) + B = np.array([1j, 1j]) + observed = expm_multiply(A, B) + expected = np.array([ + 1j * np.exp(1j) + 1j * (1j*np.cos(1) - np.sin(1)), + 1j * np.exp(1j)], dtype=complex) + assert_allclose(observed, expected) + observed = estimated(expm_multiply)(aslinearoperator(A), B) + assert_allclose(observed, expected) + + +class TestExpmActionInterval: + + @pytest.mark.fail_slow(20) + def test_sparse_expm_multiply_interval(self): + rng = np.random.default_rng(1234) + start = 0.1 + stop = 3.2 + n = 40 + k = 3 + endpoint = True + for num in (14, 13, 2): + A = scipy.sparse.random_array((n, n), density=0.05, rng=rng) + B = rng.standard_normal((n, k)) + v = rng.standard_normal((n,)) + for target in (B, v): + X = expm_multiply(A, target, start=start, stop=stop, + num=num, endpoint=endpoint) + samples = np.linspace(start=start, stop=stop, + num=num, endpoint=endpoint) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, + "splu converted its input to CSC format") + sup.filter(SparseEfficiencyWarning, + "spsolve is more efficient when sparse b is in" + " the CSC matrix format") + for solution, t in zip(X, samples): + assert_allclose(solution, sp_expm(t*A).dot(target)) + + @pytest.mark.thread_unsafe + @pytest.mark.fail_slow(20) + def test_expm_multiply_interval_vector(self): + np.random.seed(1234) + interval = {'start': 0.1, 'stop': 3.2, 'endpoint': True} + for num, n in product([14, 13, 2], [1, 2, 5, 20, 40]): + A = scipy.linalg.inv(np.random.randn(n, n)) + v = np.random.randn(n) + samples = np.linspace(num=num, **interval) + X = expm_multiply(A, v, num=num, **interval) + for solution, t in zip(X, samples): + assert_allclose(solution, sp_expm(t*A).dot(v)) + # test for linear operator with unknown trace -> estimate trace + Xguess = estimated(expm_multiply)(aslinearoperator(A), v, + num=num, **interval) + # test for linear operator with given trace + Xgiven = expm_multiply(aslinearoperator(A), v, num=num, **interval, + traceA=np.trace(A)) + # test robustness for linear operator with wrong trace + Xwrong = expm_multiply(aslinearoperator(A), v, num=num, **interval, + traceA=np.trace(A)*5) + for sol_guess, sol_given, sol_wrong, t in zip(Xguess, Xgiven, + Xwrong, samples): + correct = sp_expm(t*A).dot(v) + assert_allclose(sol_guess, correct) + assert_allclose(sol_given, correct) + assert_allclose(sol_wrong, correct) + + @pytest.mark.thread_unsafe + @pytest.mark.fail_slow(20) + def test_expm_multiply_interval_matrix(self): + np.random.seed(1234) + interval = {'start': 0.1, 'stop': 3.2, 'endpoint': True} + for num, n, k in product([14, 13, 2], [1, 2, 5, 20, 40], [1, 2]): + A = scipy.linalg.inv(np.random.randn(n, n)) + B = np.random.randn(n, k) + samples = np.linspace(num=num, **interval) + X = expm_multiply(A, B, num=num, **interval) + for solution, t in zip(X, samples): + assert_allclose(solution, sp_expm(t*A).dot(B)) + X = estimated(expm_multiply)(aslinearoperator(A), B, num=num, + **interval) + for solution, t in zip(X, samples): + assert_allclose(solution, sp_expm(t*A).dot(B)) + + def test_sparse_expm_multiply_interval_dtypes(self): + # Test A & B int + A = scipy.sparse.diags_array(np.arange(5),format='csr', dtype=int) + B = np.ones(5, dtype=int) + Aexpm = scipy.sparse.diags_array(np.exp(np.arange(5)),format='csr') + BI = np.identity(5, dtype=int) + BI_sparse = scipy.sparse.csr_array(BI) + assert_allclose(expm_multiply(A,B,0,1)[-1], Aexpm.dot(B)) + assert_allclose(np.diag(expm_multiply(A, BI_sparse, 0, 1)[-1]), Aexpm.dot(B)) + + # Test A complex, B int + A = scipy.sparse.diags_array(-1j*np.arange(5),format='csr', dtype=complex) + B = np.ones(5, dtype=int) + Aexpm = scipy.sparse.diags_array(np.exp(-1j*np.arange(5)),format='csr') + assert_allclose(expm_multiply(A,B,0,1)[-1], Aexpm.dot(B)) + assert_allclose(np.diag(expm_multiply(A, BI_sparse, 0, 1)[-1]), Aexpm.dot(B)) + + # Test A int, B complex + A = scipy.sparse.diags_array(np.arange(5),format='csr', dtype=int) + B = np.full(5, 1j, dtype=complex) + Aexpm = scipy.sparse.diags_array(np.exp(np.arange(5)),format='csr') + assert_allclose(expm_multiply(A,B,0,1)[-1], Aexpm.dot(B)) + BI = np.identity(5, dtype=complex)*1j + assert_allclose( + np.diag(expm_multiply(A, scipy.sparse.csr_array(BI), 0, 1)[-1]), + Aexpm.dot(B) + ) + + def test_expm_multiply_interval_status_0(self): + self._help_test_specific_expm_interval_status(0) + + def test_expm_multiply_interval_status_1(self): + self._help_test_specific_expm_interval_status(1) + + def test_expm_multiply_interval_status_2(self): + self._help_test_specific_expm_interval_status(2) + + def _help_test_specific_expm_interval_status(self, target_status): + rng = np.random.RandomState(1234) + start = 0.1 + stop = 3.2 + num = 13 + endpoint = True + n = 5 + k = 2 + nrepeats = 10 + nsuccesses = 0 + for num in [14, 13, 2] * nrepeats: + A = rng.randn(n, n) + B = rng.randn(n, k) + status = _expm_multiply_interval(A, B, + start=start, stop=stop, num=num, endpoint=endpoint, + status_only=True) + if status == target_status: + X, status = _expm_multiply_interval(A, B, + start=start, stop=stop, num=num, endpoint=endpoint, + status_only=False) + assert_equal(X.shape, (num, n, k)) + samples = np.linspace(start=start, stop=stop, + num=num, endpoint=endpoint) + for solution, t in zip(X, samples): + assert_allclose(solution, sp_expm(t*A).dot(B)) + nsuccesses += 1 + if not nsuccesses: + msg = 'failed to find a status-' + str(target_status) + ' interval' + raise Exception(msg) + + +@pytest.mark.thread_unsafe +@pytest.mark.parametrize("dtype_a", DTYPES) +@pytest.mark.parametrize("dtype_b", DTYPES) +@pytest.mark.parametrize("b_is_matrix", [False, True]) +def test_expm_multiply_dtype(dtype_a, dtype_b, b_is_matrix): + """Make sure `expm_multiply` handles all numerical dtypes correctly.""" + assert_allclose_ = (partial(assert_allclose, rtol=1.8e-3, atol=1e-5) + if {dtype_a, dtype_b} & IMPRECISE else assert_allclose) + rng = np.random.default_rng(1234) + # test data + n = 7 + b_shape = (n, 3) if b_is_matrix else (n, ) + if dtype_a in REAL_DTYPES: + A = scipy.linalg.inv(rng.random([n, n])).astype(dtype_a) + else: + A = scipy.linalg.inv( + rng.random([n, n]) + 1j*rng.random([n, n]) + ).astype(dtype_a) + if dtype_b in REAL_DTYPES: + B = (2*rng.random(b_shape)).astype(dtype_b) + else: + B = (rng.random(b_shape) + 1j*rng.random(b_shape)).astype(dtype_b) + + # single application + sol_mat = expm_multiply(A, B) + sol_op = estimated(expm_multiply)(aslinearoperator(A), B) + direct_sol = np.dot(sp_expm(A), B) + assert_allclose_(sol_mat, direct_sol) + assert_allclose_(sol_op, direct_sol) + sol_op = expm_multiply(aslinearoperator(A), B, traceA=np.trace(A)) + assert_allclose_(sol_op, direct_sol) + + # for time points + interval = {'start': 0.1, 'stop': 3.2, 'num': 13, 'endpoint': True} + samples = np.linspace(**interval) + X_mat = expm_multiply(A, B, **interval) + X_op = estimated(expm_multiply)(aslinearoperator(A), B, **interval) + for sol_mat, sol_op, t in zip(X_mat, X_op, samples): + direct_sol = sp_expm(t*A).dot(B) + assert_allclose_(sol_mat, direct_sol) + assert_allclose_(sol_op, direct_sol) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_interface.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_interface.py new file mode 100644 index 0000000000000000000000000000000000000000..13bbcf16dbfc391a7e1b6fab666221d4c21b4c64 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_interface.py @@ -0,0 +1,561 @@ +"""Test functions for the sparse.linalg._interface module +""" + +from functools import partial +from itertools import product +import operator +import pytest +from pytest import raises as assert_raises, warns +from numpy.testing import assert_, assert_equal + +import numpy as np +import scipy.sparse as sparse + +import scipy.sparse.linalg._interface as interface +from scipy.sparse._sputils import matrix +from scipy._lib._gcutils import assert_deallocated, IS_PYPY + + +class TestLinearOperator: + def setup_method(self): + self.A = np.array([[1,2,3], + [4,5,6]]) + self.B = np.array([[1,2], + [3,4], + [5,6]]) + self.C = np.array([[1,2], + [3,4]]) + + def test_matvec(self): + def get_matvecs(A): + return [{ + 'shape': A.shape, + 'matvec': lambda x: np.dot(A, x).reshape(A.shape[0]), + 'rmatvec': lambda x: np.dot(A.T.conj(), + x).reshape(A.shape[1]) + }, + { + 'shape': A.shape, + 'matvec': lambda x: np.dot(A, x), + 'rmatvec': lambda x: np.dot(A.T.conj(), x), + 'rmatmat': lambda x: np.dot(A.T.conj(), x), + 'matmat': lambda x: np.dot(A, x) + }] + + for matvecs in get_matvecs(self.A): + A = interface.LinearOperator(**matvecs) + + assert_(A.args == ()) + + assert_equal(A.matvec(np.array([1,2,3])), [14,32]) + assert_equal(A.matvec(np.array([[1],[2],[3]])), [[14],[32]]) + assert_equal(A @ np.array([1,2,3]), [14,32]) + assert_equal(A @ np.array([[1],[2],[3]]), [[14],[32]]) + assert_equal(A.dot(np.array([1,2,3])), [14,32]) + assert_equal(A.dot(np.array([[1],[2],[3]])), [[14],[32]]) + + assert_equal(A.matvec(matrix([[1],[2],[3]])), [[14],[32]]) + assert_equal(A @ matrix([[1],[2],[3]]), [[14],[32]]) + assert_equal(A.dot(matrix([[1],[2],[3]])), [[14],[32]]) + + assert_equal((2*A)@[1,1,1], [12,30]) + assert_equal((2 * A).rmatvec([1, 1]), [10, 14, 18]) + assert_equal((2*A).H.matvec([1,1]), [10, 14, 18]) + assert_equal((2*A).adjoint().matvec([1,1]), [10, 14, 18]) + assert_equal((2*A)@[[1],[1],[1]], [[12],[30]]) + assert_equal((2 * A).matmat([[1], [1], [1]]), [[12], [30]]) + assert_equal((A*2)@[1,1,1], [12,30]) + assert_equal((A*2)@[[1],[1],[1]], [[12],[30]]) + assert_equal((2j*A)@[1,1,1], [12j,30j]) + assert_equal((A+A)@[1,1,1], [12, 30]) + assert_equal((A + A).rmatvec([1, 1]), [10, 14, 18]) + assert_equal((A+A).H.matvec([1,1]), [10, 14, 18]) + assert_equal((A+A).adjoint().matvec([1,1]), [10, 14, 18]) + assert_equal((A+A)@[[1],[1],[1]], [[12], [30]]) + assert_equal((A+A).matmat([[1],[1],[1]]), [[12], [30]]) + assert_equal((-A)@[1,1,1], [-6,-15]) + assert_equal((-A)@[[1],[1],[1]], [[-6],[-15]]) + assert_equal((A-A)@[1,1,1], [0,0]) + assert_equal((A - A) @ [[1], [1], [1]], [[0], [0]]) + + X = np.array([[1, 2], [3, 4]]) + # A_asarray = np.array([[1, 2, 3], [4, 5, 6]]) + assert_equal((2 * A).rmatmat(X), np.dot((2 * self.A).T, X)) + assert_equal((A * 2).rmatmat(X), np.dot((self.A * 2).T, X)) + assert_equal((2j * A).rmatmat(X), + np.dot((2j * self.A).T.conj(), X)) + assert_equal((A * 2j).rmatmat(X), + np.dot((self.A * 2j).T.conj(), X)) + assert_equal((A + A).rmatmat(X), + np.dot((self.A + self.A).T, X)) + assert_equal((A + 2j * A).rmatmat(X), + np.dot((self.A + 2j * self.A).T.conj(), X)) + assert_equal((-A).rmatmat(X), np.dot((-self.A).T, X)) + assert_equal((A - A).rmatmat(X), + np.dot((self.A - self.A).T, X)) + assert_equal((2j * A).rmatmat(2j * X), + np.dot((2j * self.A).T.conj(), 2j * X)) + + z = A+A + assert_(len(z.args) == 2 and z.args[0] is A and z.args[1] is A) + z = 2*A + assert_(len(z.args) == 2 and z.args[0] is A and z.args[1] == 2) + + assert_(isinstance(A.matvec([1, 2, 3]), np.ndarray)) + assert_(isinstance(A.matvec(np.array([[1],[2],[3]])), np.ndarray)) + assert_(isinstance(A @ np.array([1,2,3]), np.ndarray)) + assert_(isinstance(A @ np.array([[1],[2],[3]]), np.ndarray)) + assert_(isinstance(A.dot(np.array([1,2,3])), np.ndarray)) + assert_(isinstance(A.dot(np.array([[1],[2],[3]])), np.ndarray)) + + assert_(isinstance(A.matvec(matrix([[1],[2],[3]])), np.ndarray)) + assert_(isinstance(A @ matrix([[1],[2],[3]]), np.ndarray)) + assert_(isinstance(A.dot(matrix([[1],[2],[3]])), np.ndarray)) + + assert_(isinstance(2*A, interface._ScaledLinearOperator)) + assert_(isinstance(2j*A, interface._ScaledLinearOperator)) + assert_(isinstance(A+A, interface._SumLinearOperator)) + assert_(isinstance(-A, interface._ScaledLinearOperator)) + assert_(isinstance(A-A, interface._SumLinearOperator)) + assert_(isinstance(A/2, interface._ScaledLinearOperator)) + assert_(isinstance(A/2j, interface._ScaledLinearOperator)) + assert_(((A * 3) / 3).args[0] is A) # check for simplification + + # Test that prefactor is of _ScaledLinearOperator is not mutated + # when the operator is multiplied by a number + result = A @ np.array([1, 2, 3]) + B = A * 3 + C = A / 5 + assert_equal(A @ np.array([1, 2, 3]), result) + + assert_((2j*A).dtype == np.complex128) + + # Test division by non-scalar + msg = "Can only divide a linear operator by a scalar." + with assert_raises(ValueError, match=msg): + A / np.array([1, 2]) + + assert_raises(ValueError, A.matvec, np.array([1,2])) + assert_raises(ValueError, A.matvec, np.array([1,2,3,4])) + assert_raises(ValueError, A.matvec, np.array([[1],[2]])) + assert_raises(ValueError, A.matvec, np.array([[1],[2],[3],[4]])) + + assert_raises(ValueError, lambda: A@A) + assert_raises(ValueError, lambda: A**2) + + for matvecsA, matvecsB in product(get_matvecs(self.A), + get_matvecs(self.B)): + A = interface.LinearOperator(**matvecsA) + B = interface.LinearOperator(**matvecsB) + # AtimesB = np.array([[22, 28], [49, 64]]) + AtimesB = self.A.dot(self.B) + X = np.array([[1, 2], [3, 4]]) + + assert_equal((A @ B).rmatmat(X), np.dot((AtimesB).T, X)) + assert_equal((2j * A @ B).rmatmat(X), + np.dot((2j * AtimesB).T.conj(), X)) + + assert_equal((A@B)@[1,1], [50,113]) + assert_equal((A@B)@[[1],[1]], [[50],[113]]) + assert_equal((A@B).matmat([[1],[1]]), [[50],[113]]) + + assert_equal((A @ B).rmatvec([1, 1]), [71, 92]) + assert_equal((A @ B).H.matvec([1, 1]), [71, 92]) + assert_equal((A @ B).adjoint().matvec([1, 1]), [71, 92]) + + assert_(isinstance(A@B, interface._ProductLinearOperator)) + + assert_raises(ValueError, lambda: A+B) + assert_raises(ValueError, lambda: A**2) + + z = A@B + assert_(len(z.args) == 2 and z.args[0] is A and z.args[1] is B) + + for matvecsC in get_matvecs(self.C): + C = interface.LinearOperator(**matvecsC) + X = np.array([[1, 2], [3, 4]]) + + assert_equal(C.rmatmat(X), np.dot((self.C).T, X)) + assert_equal((C**2).rmatmat(X), + np.dot((np.dot(self.C, self.C)).T, X)) + + assert_equal((C**2)@[1,1], [17,37]) + assert_equal((C**2).rmatvec([1, 1]), [22, 32]) + assert_equal((C**2).H.matvec([1, 1]), [22, 32]) + assert_equal((C**2).adjoint().matvec([1, 1]), [22, 32]) + assert_equal((C**2).matmat([[1],[1]]), [[17],[37]]) + + assert_(isinstance(C**2, interface._PowerLinearOperator)) + + def test_matmul(self): + D = {'shape': self.A.shape, + 'matvec': lambda x: np.dot(self.A, x).reshape(self.A.shape[0]), + 'rmatvec': lambda x: np.dot(self.A.T.conj(), + x).reshape(self.A.shape[1]), + 'rmatmat': lambda x: np.dot(self.A.T.conj(), x), + 'matmat': lambda x: np.dot(self.A, x)} + A = interface.LinearOperator(**D) + B = np.array([[1 + 1j, 2, 3], + [4, 5, 6], + [7, 8, 9]]) + b = B[0] + + assert_equal(operator.matmul(A, b), A * b) + assert_equal(operator.matmul(A, b.reshape(-1, 1)), A * b.reshape(-1, 1)) + assert_equal(operator.matmul(A, B), A @ B) + assert_equal(operator.matmul(b, A.H), b * A.H) + assert_equal(operator.matmul(b, A.adjoint()), b * A.adjoint()) + assert_equal(operator.matmul(b.reshape(1, -1), A.H), b.reshape(1, -1) * A.H) + assert_equal(operator.matmul(b.reshape(1, -1), A.adjoint()), + b.reshape(1, -1) * A.adjoint()) + assert_equal(operator.matmul(B, A.H), B @ A.H) + assert_equal(operator.matmul(B, A.adjoint()), B @ A.adjoint()) + assert_raises(ValueError, operator.matmul, A, 2) + assert_raises(ValueError, operator.matmul, 2, A) + + +class TestAsLinearOperator: + def setup_method(self): + self.cases = [] + + def make_cases(original, dtype): + cases = [] + + cases.append((matrix(original, dtype=dtype), original)) + cases.append((np.array(original, dtype=dtype), original)) + cases.append((sparse.csr_array(original, dtype=dtype), original)) + + # Test default implementations of _adjoint and _rmatvec, which + # refer to each other. + def mv(x, dtype): + y = original.dot(x) + if len(x.shape) == 2: + y = y.reshape(-1, 1) + return y + + def rmv(x, dtype): + return original.T.conj().dot(x) + + class BaseMatlike(interface.LinearOperator): + args = () + + def __init__(self, dtype): + self.dtype = np.dtype(dtype) + self.shape = original.shape + + def _matvec(self, x): + return mv(x, self.dtype) + + class HasRmatvec(BaseMatlike): + args = () + + def _rmatvec(self,x): + return rmv(x, self.dtype) + + class HasAdjoint(BaseMatlike): + args = () + + def _adjoint(self): + shape = self.shape[1], self.shape[0] + matvec = partial(rmv, dtype=self.dtype) + rmatvec = partial(mv, dtype=self.dtype) + return interface.LinearOperator(matvec=matvec, + rmatvec=rmatvec, + dtype=self.dtype, + shape=shape) + + class HasRmatmat(HasRmatvec): + def _matmat(self, x): + return original.dot(x) + + def _rmatmat(self, x): + return original.T.conj().dot(x) + + cases.append((HasRmatvec(dtype), original)) + cases.append((HasAdjoint(dtype), original)) + cases.append((HasRmatmat(dtype), original)) + return cases + + original = np.array([[1,2,3], [4,5,6]]) + self.cases += make_cases(original, np.int32) + self.cases += make_cases(original, np.float32) + self.cases += make_cases(original, np.float64) + self.cases += [(interface.aslinearoperator(M).T, A.T) + for M, A in make_cases(original.T, np.float64)] + self.cases += [(interface.aslinearoperator(M).H, A.T.conj()) + for M, A in make_cases(original.T, np.float64)] + self.cases += [(interface.aslinearoperator(M).adjoint(), A.T.conj()) + for M, A in make_cases(original.T, np.float64)] + + original = np.array([[1, 2j, 3j], [4j, 5j, 6]]) + self.cases += make_cases(original, np.complex128) + self.cases += [(interface.aslinearoperator(M).T, A.T) + for M, A in make_cases(original.T, np.complex128)] + self.cases += [(interface.aslinearoperator(M).H, A.T.conj()) + for M, A in make_cases(original.T, np.complex128)] + self.cases += [(interface.aslinearoperator(M).adjoint(), A.T.conj()) + for M, A in make_cases(original.T, np.complex128)] + + def test_basic(self): + + for M, A_array in self.cases: + A = interface.aslinearoperator(M) + M,N = A.shape + + xs = [np.array([1, 2, 3]), + np.array([[1], [2], [3]])] + ys = [np.array([1, 2]), np.array([[1], [2]])] + + if A.dtype == np.complex128: + xs += [np.array([1, 2j, 3j]), + np.array([[1], [2j], [3j]])] + ys += [np.array([1, 2j]), np.array([[1], [2j]])] + + x2 = np.array([[1, 4], [2, 5], [3, 6]]) + + for x in xs: + assert_equal(A.matvec(x), A_array.dot(x)) + assert_equal(A @ x, A_array.dot(x)) + + assert_equal(A.matmat(x2), A_array.dot(x2)) + assert_equal(A @ x2, A_array.dot(x2)) + + for y in ys: + assert_equal(A.rmatvec(y), A_array.T.conj().dot(y)) + assert_equal(A.T.matvec(y), A_array.T.dot(y)) + assert_equal(A.H.matvec(y), A_array.T.conj().dot(y)) + assert_equal(A.adjoint().matvec(y), A_array.T.conj().dot(y)) + + for y in ys: + if y.ndim < 2: + continue + assert_equal(A.rmatmat(y), A_array.T.conj().dot(y)) + assert_equal(A.T.matmat(y), A_array.T.dot(y)) + assert_equal(A.H.matmat(y), A_array.T.conj().dot(y)) + assert_equal(A.adjoint().matmat(y), A_array.T.conj().dot(y)) + + if hasattr(M,'dtype'): + assert_equal(A.dtype, M.dtype) + + assert_(hasattr(A, 'args')) + + def test_dot(self): + + for M, A_array in self.cases: + A = interface.aslinearoperator(M) + M,N = A.shape + + x0 = np.array([1, 2, 3]) + x1 = np.array([[1], [2], [3]]) + x2 = np.array([[1, 4], [2, 5], [3, 6]]) + + assert_equal(A.dot(x0), A_array.dot(x0)) + assert_equal(A.dot(x1), A_array.dot(x1)) + assert_equal(A.dot(x2), A_array.dot(x2)) + + +def test_repr(): + A = interface.LinearOperator(shape=(1, 1), matvec=lambda x: 1) + repr_A = repr(A) + assert_('unspecified dtype' not in repr_A, repr_A) + + +def test_identity(): + ident = interface.IdentityOperator((3, 3)) + assert_equal(ident @ [1, 2, 3], [1, 2, 3]) + assert_equal(ident.dot(np.arange(9).reshape(3, 3)).ravel(), np.arange(9)) + + assert_raises(ValueError, ident.matvec, [1, 2, 3, 4]) + + +def test_attributes(): + A = interface.aslinearoperator(np.arange(16).reshape(4, 4)) + + def always_four_ones(x): + x = np.asarray(x) + assert_(x.shape == (3,) or x.shape == (3, 1)) + return np.ones(4) + + B = interface.LinearOperator(shape=(4, 3), matvec=always_four_ones) + + ops = [A, B, A * B, A @ B, A.H, A.adjoint(), A + A, B + B, A**4] + for op in ops: + assert_(hasattr(op, "dtype")) + assert_(hasattr(op, "shape")) + assert_(hasattr(op, "_matvec")) + +def matvec(x): + """ Needed for test_pickle as local functions are not pickleable """ + return np.zeros(3) + +def test_pickle(): + import pickle + + for protocol in range(pickle.HIGHEST_PROTOCOL + 1): + A = interface.LinearOperator((3, 3), matvec) + s = pickle.dumps(A, protocol=protocol) + B = pickle.loads(s) + + for k in A.__dict__: + assert_equal(getattr(A, k), getattr(B, k)) + + +@pytest.mark.thread_unsafe +def test_inheritance(): + class Empty(interface.LinearOperator): + pass + + with warns(RuntimeWarning, match="should implement at least"): + assert_raises(TypeError, Empty) + + class Identity(interface.LinearOperator): + def __init__(self, n): + super().__init__(dtype=None, shape=(n, n)) + + def _matvec(self, x): + return x + + id3 = Identity(3) + assert_equal(id3.matvec([1, 2, 3]), [1, 2, 3]) + assert_raises(NotImplementedError, id3.rmatvec, [4, 5, 6]) + + class MatmatOnly(interface.LinearOperator): + def __init__(self, A): + super().__init__(A.dtype, A.shape) + self.A = A + + def _matmat(self, x): + return self.A.dot(x) + + mm = MatmatOnly(np.random.randn(5, 3)) + assert_equal(mm.matvec(np.random.randn(3)).shape, (5,)) + +def test_dtypes_of_operator_sum(): + # gh-6078 + + mat_complex = np.random.rand(2,2) + 1j * np.random.rand(2,2) + mat_real = np.random.rand(2,2) + + complex_operator = interface.aslinearoperator(mat_complex) + real_operator = interface.aslinearoperator(mat_real) + + sum_complex = complex_operator + complex_operator + sum_real = real_operator + real_operator + + assert_equal(sum_real.dtype, np.float64) + assert_equal(sum_complex.dtype, np.complex128) + +def test_no_double_init(): + call_count = [0] + + def matvec(v): + call_count[0] += 1 + return v + + # It should call matvec exactly once (in order to determine the + # operator dtype) + interface.LinearOperator((2, 2), matvec=matvec) + assert_equal(call_count[0], 1) + +INT_DTYPES = (np.int8, np.int16, np.int32, np.int64) +REAL_DTYPES = (np.float32, np.float64, np.longdouble) +COMPLEX_DTYPES = (np.complex64, np.complex128, np.clongdouble) +INEXACTDTYPES = REAL_DTYPES + COMPLEX_DTYPES +ALLDTYPES = INT_DTYPES + INEXACTDTYPES + + +@pytest.mark.parametrize("test_dtype", ALLDTYPES) +def test_determine_lo_dtype_from_matvec(test_dtype): + # gh-19209 + scalar = np.array(1, dtype=test_dtype) + def mv(v): + return np.array([scalar * v[0], v[1]]) + + lo = interface.LinearOperator((2, 2), matvec=mv) + assert lo.dtype == np.dtype(test_dtype) + +def test_determine_lo_dtype_for_int(): + # gh-19209 + # test Python int larger than int8 max cast to some int + def mv(v): + return np.array([128 * v[0], v[1]]) + + lo = interface.LinearOperator((2, 2), matvec=mv) + assert lo.dtype in INT_DTYPES + +def test_adjoint_conjugate(): + X = np.array([[1j]]) + A = interface.aslinearoperator(X) + + B = 1j * A + Y = 1j * X + + v = np.array([1]) + + assert_equal(B.dot(v), Y.dot(v)) + assert_equal(B.H.dot(v), Y.T.conj().dot(v)) + assert_equal(B.adjoint().dot(v), Y.T.conj().dot(v)) + +def test_ndim(): + X = np.array([[1]]) + A = interface.aslinearoperator(X) + assert_equal(A.ndim, 2) + +def test_transpose_noconjugate(): + X = np.array([[1j]]) + A = interface.aslinearoperator(X) + + B = 1j * A + Y = 1j * X + + v = np.array([1]) + + assert_equal(B.dot(v), Y.dot(v)) + assert_equal(B.T.dot(v), Y.T.dot(v)) + +def test_transpose_multiplication(): + class MyMatrix(interface.LinearOperator): + def __init__(self, A): + super().__init__(A.dtype, A.shape) + self.A = A + def _matmat(self, other): return self.A @ other + def _rmatmat(self, other): return self.A.T @ other + + A = MyMatrix(np.array([[1, 2], [3, 4]])) + X = np.array([1, 2]) + B = np.array([[10, 20], [30, 40]]) + X2 = X.reshape(-1, 1) + Y = np.array([[1, 2], [3, 4]]) + + assert_equal(A @ B, Y @ B) + assert_equal(B.T @ A, B.T @ Y) + assert_equal(A.T @ B, Y.T @ B) + assert_equal(A @ X, Y @ X) + assert_equal(X.T @ A, X.T @ Y) + assert_equal(A.T @ X, Y.T @ X) + assert_equal(A @ X2, Y @ X2) + assert_equal(X2.T @ A, X2.T @ Y) + assert_equal(A.T @ X2, Y.T @ X2) + +def test_sparse_matmat_exception(): + A = interface.LinearOperator((2, 2), matvec=lambda x: x) + B = sparse.eye_array(2) + msg = "Unable to multiply a LinearOperator with a sparse matrix." + with assert_raises(TypeError, match=msg): + A @ B + with assert_raises(TypeError, match=msg): + B @ A + with assert_raises(ValueError): + A @ np.identity(4) + with assert_raises(ValueError): + np.identity(4) @ A + + +@pytest.mark.skipif(IS_PYPY, reason="Test not meaningful on PyPy") +def test_MatrixLinearOperator_refcycle(): + # gh-10634 + # Test that MatrixLinearOperator can be automatically garbage collected + A = np.eye(2) + with assert_deallocated(interface.MatrixLinearOperator, A) as op: + op.adjoint() + del op diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_matfuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_matfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..8a468c39cad30f6f852cc6df64b98588fa70b5ff --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_matfuncs.py @@ -0,0 +1,592 @@ +# +# Created by: Pearu Peterson, March 2002 +# +""" Test functions for scipy.linalg._matfuncs module + +""" +import math + +import numpy as np +from numpy import array, eye, exp, random +from numpy.testing import ( + assert_allclose, assert_, assert_array_almost_equal, assert_equal, + assert_array_almost_equal_nulp, suppress_warnings) + +from scipy.sparse import csc_array, SparseEfficiencyWarning +from scipy.sparse._construct import eye_array +from scipy.sparse.linalg._matfuncs import (expm, _expm, + ProductOperator, MatrixPowerOperator, + _onenorm_matrix_power_nnm, matrix_power) +from scipy.sparse._sputils import matrix +from scipy.linalg import logm +from scipy.special import factorial, binom +import scipy.sparse +import scipy.sparse.linalg + + +def _burkardt_13_power(n, p): + """ + A helper function for testing matrix functions. + + Parameters + ---------- + n : integer greater than 1 + Order of the square matrix to be returned. + p : non-negative integer + Power of the matrix. + + Returns + ------- + out : ndarray representing a square matrix + A Forsythe matrix of order n, raised to the power p. + + """ + # Input validation. + if n != int(n) or n < 2: + raise ValueError('n must be an integer greater than 1') + n = int(n) + if p != int(p) or p < 0: + raise ValueError('p must be a non-negative integer') + p = int(p) + + # Construct the matrix explicitly. + a, b = divmod(p, n) + large = np.power(10.0, -n*a) + small = large * np.power(10.0, -n) + return np.diag([large]*(n-b), b) + np.diag([small]*b, b-n) + + +def test_onenorm_matrix_power_nnm(): + np.random.seed(1234) + for n in range(1, 5): + for p in range(5): + M = np.random.random((n, n)) + Mp = np.linalg.matrix_power(M, p) + observed = _onenorm_matrix_power_nnm(M, p) + expected = np.linalg.norm(Mp, 1) + assert_allclose(observed, expected) + +def test_matrix_power(): + np.random.seed(1234) + row, col = np.random.randint(0, 4, size=(2, 6)) + data = np.random.random(size=(6,)) + Amat = csc_array((data, (row, col)), shape=(4, 4)) + A = csc_array((data, (row, col)), shape=(4, 4)) + Adense = A.toarray() + for power in (2, 5, 6): + Apow = matrix_power(A, power).toarray() + Amat_pow = matrix_power(Amat, power).toarray() + Adense_pow = np.linalg.matrix_power(Adense, power) + assert_allclose(Apow, Adense_pow) + assert_allclose(Apow, Amat_pow) + + +class TestExpM: + def test_zero_ndarray(self): + a = array([[0.,0],[0,0]]) + assert_array_almost_equal(expm(a),[[1,0],[0,1]]) + + def test_zero_sparse(self): + a = csc_array([[0.,0],[0,0]]) + assert_array_almost_equal(expm(a).toarray(),[[1,0],[0,1]]) + + def test_zero_matrix(self): + a = matrix([[0.,0],[0,0]]) + assert_array_almost_equal(expm(a),[[1,0],[0,1]]) + + def test_misc_types(self): + A = expm(np.array([[1]])) + assert_allclose(expm(((1,),)), A) + assert_allclose(expm([[1]]), A) + assert_allclose(expm(matrix([[1]])), A) + assert_allclose(expm(np.array([[1]])), A) + assert_allclose(expm(csc_array([[1]])).toarray(), A) + B = expm(np.array([[1j]])) + assert_allclose(expm(((1j,),)), B) + assert_allclose(expm([[1j]]), B) + assert_allclose(expm(matrix([[1j]])), B) + assert_allclose(expm(csc_array([[1j]])).toarray(), B) + + def test_bidiagonal_sparse(self): + A = csc_array([ + [1, 3, 0], + [0, 1, 5], + [0, 0, 2]], dtype=float) + e1 = math.exp(1) + e2 = math.exp(2) + expected = np.array([ + [e1, 3*e1, 15*(e2 - 2*e1)], + [0, e1, 5*(e2 - e1)], + [0, 0, e2]], dtype=float) + observed = expm(A).toarray() + assert_array_almost_equal(observed, expected) + + def test_padecases_dtype_float(self): + for dtype in [np.float32, np.float64]: + for scale in [1e-2, 1e-1, 5e-1, 1, 10]: + A = scale * eye(3, dtype=dtype) + observed = expm(A) + expected = exp(scale, dtype=dtype) * eye(3, dtype=dtype) + assert_array_almost_equal_nulp(observed, expected, nulp=100) + + def test_padecases_dtype_complex(self): + for dtype in [np.complex64, np.complex128]: + for scale in [1e-2, 1e-1, 5e-1, 1, 10]: + A = scale * eye(3, dtype=dtype) + observed = expm(A) + expected = exp(scale, dtype=dtype) * eye(3, dtype=dtype) + assert_array_almost_equal_nulp(observed, expected, nulp=100) + + def test_padecases_dtype_sparse_float(self): + # float32 and complex64 lead to errors in spsolve/UMFpack + dtype = np.float64 + for scale in [1e-2, 1e-1, 5e-1, 1, 10]: + a = scale * eye_array(3, 3, dtype=dtype, format='csc') + e = exp(scale, dtype=dtype) * eye(3, dtype=dtype) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + exact_onenorm = _expm(a, use_exact_onenorm=True).toarray() + inexact_onenorm = _expm(a, use_exact_onenorm=False).toarray() + assert_array_almost_equal_nulp(exact_onenorm, e, nulp=100) + assert_array_almost_equal_nulp(inexact_onenorm, e, nulp=100) + + def test_padecases_dtype_sparse_complex(self): + # float32 and complex64 lead to errors in spsolve/UMFpack + dtype = np.complex128 + for scale in [1e-2, 1e-1, 5e-1, 1, 10]: + a = scale * eye_array(3, 3, dtype=dtype, format='csc') + e = exp(scale) * eye(3, dtype=dtype) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + assert_array_almost_equal_nulp(expm(a).toarray(), e, nulp=100) + + def test_logm_consistency(self): + random.seed(1234) + for dtype in [np.float64, np.complex128]: + for n in range(1, 10): + for scale in [1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1, 1e2]: + # make logm(A) be of a given scale + A = (eye(n) + random.rand(n, n) * scale).astype(dtype) + if np.iscomplexobj(A): + A = A + 1j * random.rand(n, n) * scale + assert_array_almost_equal(expm(logm(A)), A) + + def test_integer_matrix(self): + Q = np.array([ + [-3, 1, 1, 1], + [1, -3, 1, 1], + [1, 1, -3, 1], + [1, 1, 1, -3]]) + assert_allclose(expm(Q), expm(1.0 * Q)) + + def test_integer_matrix_2(self): + # Check for integer overflows + Q = np.array([[-500, 500, 0, 0], + [0, -550, 360, 190], + [0, 630, -630, 0], + [0, 0, 0, 0]], dtype=np.int16) + assert_allclose(expm(Q), expm(1.0 * Q)) + + Q = csc_array(Q) + assert_allclose(expm(Q).toarray(), expm(1.0 * Q).toarray()) + + def test_triangularity_perturbation(self): + # Experiment (1) of + # Awad H. Al-Mohy and Nicholas J. Higham (2012) + # Improved Inverse Scaling and Squaring Algorithms + # for the Matrix Logarithm. + A = np.array([ + [3.2346e-1, 3e4, 3e4, 3e4], + [0, 3.0089e-1, 3e4, 3e4], + [0, 0, 3.221e-1, 3e4], + [0, 0, 0, 3.0744e-1]], + dtype=float) + A_logm = np.array([ + [-1.12867982029050462e+00, 9.61418377142025565e+04, + -4.52485573953179264e+09, 2.92496941103871812e+14], + [0.00000000000000000e+00, -1.20101052953082288e+00, + 9.63469687211303099e+04, -4.68104828911105442e+09], + [0.00000000000000000e+00, 0.00000000000000000e+00, + -1.13289322264498393e+00, 9.53249183094775653e+04], + [0.00000000000000000e+00, 0.00000000000000000e+00, + 0.00000000000000000e+00, -1.17947533272554850e+00]], + dtype=float) + assert_allclose(expm(A_logm), A, rtol=1e-4) + + # Perturb the upper triangular matrix by tiny amounts, + # so that it becomes technically not upper triangular. + random.seed(1234) + tiny = 1e-17 + A_logm_perturbed = A_logm.copy() + A_logm_perturbed[1, 0] = tiny + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "Ill-conditioned.*") + A_expm_logm_perturbed = expm(A_logm_perturbed) + rtol = 1e-4 + atol = 100 * tiny + assert_(not np.allclose(A_expm_logm_perturbed, A, rtol=rtol, atol=atol)) + + def test_burkardt_1(self): + # This matrix is diagonal. + # The calculation of the matrix exponential is simple. + # + # This is the first of a series of matrix exponential tests + # collected by John Burkardt from the following sources. + # + # Alan Laub, + # Review of "Linear System Theory" by Joao Hespanha, + # SIAM Review, + # Volume 52, Number 4, December 2010, pages 779--781. + # + # Cleve Moler and Charles Van Loan, + # Nineteen Dubious Ways to Compute the Exponential of a Matrix, + # Twenty-Five Years Later, + # SIAM Review, + # Volume 45, Number 1, March 2003, pages 3--49. + # + # Cleve Moler, + # Cleve's Corner: A Balancing Act for the Matrix Exponential, + # 23 July 2012. + # + # Robert Ward, + # Numerical computation of the matrix exponential + # with accuracy estimate, + # SIAM Journal on Numerical Analysis, + # Volume 14, Number 4, September 1977, pages 600--610. + exp1 = np.exp(1) + exp2 = np.exp(2) + A = np.array([ + [1, 0], + [0, 2], + ], dtype=float) + desired = np.array([ + [exp1, 0], + [0, exp2], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_2(self): + # This matrix is symmetric. + # The calculation of the matrix exponential is straightforward. + A = np.array([ + [1, 3], + [3, 2], + ], dtype=float) + desired = np.array([ + [39.322809708033859, 46.166301438885753], + [46.166301438885768, 54.711576854329110], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_3(self): + # This example is due to Laub. + # This matrix is ill-suited for the Taylor series approach. + # As powers of A are computed, the entries blow up too quickly. + exp1 = np.exp(1) + exp39 = np.exp(39) + A = np.array([ + [0, 1], + [-39, -40], + ], dtype=float) + desired = np.array([ + [ + 39/(38*exp1) - 1/(38*exp39), + -np.expm1(-38) / (38*exp1)], + [ + 39*np.expm1(-38) / (38*exp1), + -1/(38*exp1) + 39/(38*exp39)], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_4(self): + # This example is due to Moler and Van Loan. + # The example will cause problems for the series summation approach, + # as well as for diagonal Pade approximations. + A = np.array([ + [-49, 24], + [-64, 31], + ], dtype=float) + U = np.array([[3, 1], [4, 2]], dtype=float) + V = np.array([[1, -1/2], [-2, 3/2]], dtype=float) + w = np.array([-17, -1], dtype=float) + desired = np.dot(U * np.exp(w), V) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_5(self): + # This example is due to Moler and Van Loan. + # This matrix is strictly upper triangular + # All powers of A are zero beyond some (low) limit. + # This example will cause problems for Pade approximations. + A = np.array([ + [0, 6, 0, 0], + [0, 0, 6, 0], + [0, 0, 0, 6], + [0, 0, 0, 0], + ], dtype=float) + desired = np.array([ + [1, 6, 18, 36], + [0, 1, 6, 18], + [0, 0, 1, 6], + [0, 0, 0, 1], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_6(self): + # This example is due to Moler and Van Loan. + # This matrix does not have a complete set of eigenvectors. + # That means the eigenvector approach will fail. + exp1 = np.exp(1) + A = np.array([ + [1, 1], + [0, 1], + ], dtype=float) + desired = np.array([ + [exp1, exp1], + [0, exp1], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_7(self): + # This example is due to Moler and Van Loan. + # This matrix is very close to example 5. + # Mathematically, it has a complete set of eigenvectors. + # Numerically, however, the calculation will be suspect. + exp1 = np.exp(1) + eps = np.spacing(1) + A = np.array([ + [1 + eps, 1], + [0, 1 - eps], + ], dtype=float) + desired = np.array([ + [exp1, exp1], + [0, exp1], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_8(self): + # This matrix was an example in Wikipedia. + exp4 = np.exp(4) + exp16 = np.exp(16) + A = np.array([ + [21, 17, 6], + [-5, -1, -6], + [4, 4, 16], + ], dtype=float) + desired = np.array([ + [13*exp16 - exp4, 13*exp16 - 5*exp4, 2*exp16 - 2*exp4], + [-9*exp16 + exp4, -9*exp16 + 5*exp4, -2*exp16 + 2*exp4], + [16*exp16, 16*exp16, 4*exp16], + ], dtype=float) * 0.25 + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_9(self): + # This matrix is due to the NAG Library. + # It is an example for function F01ECF. + A = np.array([ + [1, 2, 2, 2], + [3, 1, 1, 2], + [3, 2, 1, 2], + [3, 3, 3, 1], + ], dtype=float) + desired = np.array([ + [740.7038, 610.8500, 542.2743, 549.1753], + [731.2510, 603.5524, 535.0884, 542.2743], + [823.7630, 679.4257, 603.5524, 610.8500], + [998.4355, 823.7630, 731.2510, 740.7038], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_10(self): + # This is Ward's example #1. + # It is defective and nonderogatory. + A = np.array([ + [4, 2, 0], + [1, 4, 1], + [1, 1, 4], + ], dtype=float) + assert_allclose(sorted(scipy.linalg.eigvals(A)), (3, 3, 6)) + desired = np.array([ + [147.8666224463699, 183.7651386463682, 71.79703239999647], + [127.7810855231823, 183.7651386463682, 91.88256932318415], + [127.7810855231824, 163.6796017231806, 111.9681062463718], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_11(self): + # This is Ward's example #2. + # It is a symmetric matrix. + A = np.array([ + [29.87942128909879, 0.7815750847907159, -2.289519314033932], + [0.7815750847907159, 25.72656945571064, 8.680737820540137], + [-2.289519314033932, 8.680737820540137, 34.39400925519054], + ], dtype=float) + assert_allclose(scipy.linalg.eigvalsh(A), (20, 30, 40)) + desired = np.array([ + [ + 5.496313853692378E+15, + -1.823188097200898E+16, + -3.047577080858001E+16], + [ + -1.823188097200899E+16, + 6.060522870222108E+16, + 1.012918429302482E+17], + [ + -3.047577080858001E+16, + 1.012918429302482E+17, + 1.692944112408493E+17], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_12(self): + # This is Ward's example #3. + # Ward's algorithm has difficulty estimating the accuracy + # of its results. + A = np.array([ + [-131, 19, 18], + [-390, 56, 54], + [-387, 57, 52], + ], dtype=float) + assert_allclose(sorted(scipy.linalg.eigvals(A)), (-20, -2, -1)) + desired = np.array([ + [-1.509644158793135, 0.3678794391096522, 0.1353352811751005], + [-5.632570799891469, 1.471517758499875, 0.4060058435250609], + [-4.934938326088363, 1.103638317328798, 0.5413411267617766], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_burkardt_13(self): + # This is Ward's example #4. + # This is a version of the Forsythe matrix. + # The eigenvector problem is badly conditioned. + # Ward's algorithm has difficulty estimating the accuracy + # of its results for this problem. + # + # Check the construction of one instance of this family of matrices. + A4_actual = _burkardt_13_power(4, 1) + A4_desired = [[0, 1, 0, 0], + [0, 0, 1, 0], + [0, 0, 0, 1], + [1e-4, 0, 0, 0]] + assert_allclose(A4_actual, A4_desired) + # Check the expm for a few instances. + for n in (2, 3, 4, 10): + # Approximate expm using Taylor series. + # This works well for this matrix family + # because each matrix in the summation, + # even before dividing by the factorial, + # is entrywise positive with max entry 10**(-floor(p/n)*n). + k = max(1, int(np.ceil(16/n))) + desired = np.zeros((n, n), dtype=float) + for p in range(n*k): + Ap = _burkardt_13_power(n, p) + assert_equal(np.min(Ap), 0) + assert_allclose(np.max(Ap), np.power(10, -np.floor(p/n)*n)) + desired += Ap / factorial(p) + actual = expm(_burkardt_13_power(n, 1)) + assert_allclose(actual, desired) + + def test_burkardt_14(self): + # This is Moler's example. + # This badly scaled matrix caused problems for MATLAB's expm(). + A = np.array([ + [0, 1e-8, 0], + [-(2e10 + 4e8/6.), -3, 2e10], + [200./3., 0, -200./3.], + ], dtype=float) + desired = np.array([ + [0.446849468283175, 1.54044157383952e-09, 0.462811453558774], + [-5743067.77947947, -0.0152830038686819, -4526542.71278401], + [0.447722977849494, 1.54270484519591e-09, 0.463480648837651], + ], dtype=float) + actual = expm(A) + assert_allclose(actual, desired) + + def test_pascal(self): + # Test pascal triangle. + # Nilpotent exponential, used to trigger a failure (gh-8029) + + for scale in [1.0, 1e-3, 1e-6]: + for n in range(0, 80, 3): + sc = scale ** np.arange(n, -1, -1) + if np.any(sc < 1e-300): + break + + A = np.diag(np.arange(1, n + 1), -1) * scale + B = expm(A) + + got = B + expected = binom(np.arange(n + 1)[:,None], + np.arange(n + 1)[None,:]) * sc[None,:] / sc[:,None] + atol = 1e-13 * abs(expected).max() + assert_allclose(got, expected, atol=atol) + + def test_matrix_input(self): + # Large np.matrix inputs should work, gh-5546 + A = np.zeros((200, 200)) + A[-1,0] = 1 + B0 = expm(A) + with suppress_warnings() as sup: + sup.filter(DeprecationWarning, "the matrix subclass.*") + sup.filter(PendingDeprecationWarning, "the matrix subclass.*") + B = expm(np.matrix(A)) + assert_allclose(B, B0) + + def test_exp_sinch_overflow(self): + # Check overflow in intermediate steps is fixed (gh-11839) + L = np.array([[1.0, -0.5, -0.5, 0.0, 0.0, 0.0, 0.0], + [0.0, 1.0, 0.0, -0.5, -0.5, 0.0, 0.0], + [0.0, 0.0, 1.0, 0.0, 0.0, -0.5, -0.5], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], + [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]) + + E0 = expm(-L) + E1 = expm(-2**11 * L) + E2 = E0 + for j in range(11): + E2 = E2 @ E2 + + assert_allclose(E1, E2) + + +class TestOperators: + + def test_product_operator(self): + random.seed(1234) + n = 5 + k = 2 + nsamples = 10 + for i in range(nsamples): + A = np.random.randn(n, n) + B = np.random.randn(n, n) + C = np.random.randn(n, n) + D = np.random.randn(n, k) + op = ProductOperator(A, B, C) + assert_allclose(op.matmat(D), A.dot(B).dot(C).dot(D)) + assert_allclose(op.T.matmat(D), (A.dot(B).dot(C)).T.dot(D)) + + def test_matrix_power_operator(self): + random.seed(1234) + n = 5 + k = 2 + p = 3 + nsamples = 10 + for i in range(nsamples): + A = np.random.randn(n, n) + B = np.random.randn(n, k) + op = MatrixPowerOperator(A, p) + assert_allclose(op.matmat(B), np.linalg.matrix_power(A, p).dot(B)) + assert_allclose(op.T.matmat(B), np.linalg.matrix_power(A, p).T.dot(B)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_norm.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_norm.py new file mode 100644 index 0000000000000000000000000000000000000000..7350f7f61d8d32b24930e063d705031b4e94a419 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_norm.py @@ -0,0 +1,154 @@ +"""Test functions for the sparse.linalg.norm module +""" + +import pytest +import numpy as np +from numpy.linalg import norm as npnorm +from numpy.testing import assert_allclose, assert_equal +from pytest import raises as assert_raises + +import scipy.sparse +from scipy.sparse.linalg import norm as spnorm + + +# https://github.com/scipy/scipy/issues/16031 +# https://github.com/scipy/scipy/issues/21690 +def test_sparray_norm(): + row = np.array([0, 0, 1, 1]) + col = np.array([0, 1, 2, 3]) + data = np.array([4, 5, 7, 9]) + test_arr = scipy.sparse.coo_array((data, (row, col)), shape=(2, 4)) + test_mat = scipy.sparse.coo_matrix((data, (row, col)), shape=(2, 4)) + for ord in (1, np.inf, None): + for ax in [0, 1, None, (0, 1), (1, 0)]: + for A in (test_arr, test_mat): + expected = npnorm(A.toarray(), ord=ord, axis=ax) + actual = spnorm(A, ord=ord, axis=ax) + assert hasattr(actual, "dtype") + assert_equal(actual, expected) + # test 1d array and 1d-like (column) matrix + test_arr_1d = scipy.sparse.coo_array((data, (col,)), shape=(4,)) + test_mat_col = scipy.sparse.coo_matrix((data, (col, [0, 0, 0, 0])), shape=(4, 1)) + for ord in (1, np.inf, None): + for ax in [0, None]: + for A in (test_arr_1d, test_mat_col): + expected = npnorm(A.toarray(), ord=ord, axis=ax) + assert_equal(spnorm(A, ord=ord, axis=ax), expected) + + +class TestNorm: + def setup_method(self): + a = np.arange(9) - 4 + b = a.reshape((3, 3)) + self.b = scipy.sparse.csr_array(b) + + @pytest.mark.thread_unsafe + def test_matrix_norm(self): + + # Frobenius norm is the default + assert_allclose(spnorm(self.b), 7.745966692414834) + assert_allclose(spnorm(self.b, 'fro'), 7.745966692414834) + + assert_allclose(spnorm(self.b, np.inf), 9) + assert_allclose(spnorm(self.b, -np.inf), 2) + assert_allclose(spnorm(self.b, 1), 7) + assert_allclose(spnorm(self.b, -1), 6) + # Only floating or complex floating dtype supported by svds. + with pytest.warns(UserWarning, match="The problem size"): + assert_allclose(spnorm(self.b.astype(np.float64), 2), + 7.348469228349534) + + # _multi_svd_norm is not implemented for sparse array + assert_raises(NotImplementedError, spnorm, self.b, -2) + + def test_matrix_norm_axis(self): + for m, axis in ((self.b, None), (self.b, (0, 1)), (self.b.T, (1, 0))): + assert_allclose(spnorm(m, axis=axis), 7.745966692414834) + assert_allclose(spnorm(m, 'fro', axis=axis), 7.745966692414834) + assert_allclose(spnorm(m, np.inf, axis=axis), 9) + assert_allclose(spnorm(m, -np.inf, axis=axis), 2) + assert_allclose(spnorm(m, 1, axis=axis), 7) + assert_allclose(spnorm(m, -1, axis=axis), 6) + + def test_vector_norm(self): + v = [4.5825756949558398, 4.2426406871192848, 4.5825756949558398] + for m, a in (self.b, 0), (self.b.T, 1): + for axis in a, (a, ), a-2, (a-2, ): + assert_allclose(spnorm(m, 1, axis=axis), [7, 6, 7]) + assert_allclose(spnorm(m, np.inf, axis=axis), [4, 3, 4]) + assert_allclose(spnorm(m, axis=axis), v) + assert_allclose(spnorm(m, ord=2, axis=axis), v) + assert_allclose(spnorm(m, ord=None, axis=axis), v) + + def test_norm_exceptions(self): + m = self.b + assert_raises(TypeError, spnorm, m, None, 1.5) + assert_raises(TypeError, spnorm, m, None, [2]) + assert_raises(ValueError, spnorm, m, None, ()) + assert_raises(ValueError, spnorm, m, None, (0, 1, 2)) + assert_raises(ValueError, spnorm, m, None, (0, 0)) + assert_raises(ValueError, spnorm, m, None, (0, 2)) + assert_raises(ValueError, spnorm, m, None, (-3, 0)) + assert_raises(ValueError, spnorm, m, None, 2) + assert_raises(ValueError, spnorm, m, None, -3) + assert_raises(ValueError, spnorm, m, 'plate_of_shrimp', 0) + assert_raises(ValueError, spnorm, m, 'plate_of_shrimp', (0, 1)) + + +class TestVsNumpyNorm: + _sparse_types = ( + scipy.sparse.bsr_array, + scipy.sparse.coo_array, + scipy.sparse.csc_array, + scipy.sparse.csr_array, + scipy.sparse.dia_array, + scipy.sparse.dok_array, + scipy.sparse.lil_array, + ) + _test_matrices = ( + (np.arange(9) - 4).reshape((3, 3)), + [ + [1, 2, 3], + [-1, 1, 4]], + [ + [1, 0, 3], + [-1, 1, 4j]], + ) + + def test_sparse_matrix_norms(self): + for sparse_type in self._sparse_types: + for M in self._test_matrices: + S = sparse_type(M) + assert_allclose(spnorm(S), npnorm(M)) + assert_allclose(spnorm(S, 'fro'), npnorm(M, 'fro')) + assert_allclose(spnorm(S, np.inf), npnorm(M, np.inf)) + assert_allclose(spnorm(S, -np.inf), npnorm(M, -np.inf)) + assert_allclose(spnorm(S, 1), npnorm(M, 1)) + assert_allclose(spnorm(S, -1), npnorm(M, -1)) + + def test_sparse_matrix_norms_with_axis(self): + for sparse_type in self._sparse_types: + for M in self._test_matrices: + S = sparse_type(M) + for axis in None, (0, 1), (1, 0): + assert_allclose(spnorm(S, axis=axis), npnorm(M, axis=axis)) + for ord in 'fro', np.inf, -np.inf, 1, -1: + assert_allclose(spnorm(S, ord, axis=axis), + npnorm(M, ord, axis=axis)) + # Some numpy matrix norms are allergic to negative axes. + for axis in (-2, -1), (-1, -2), (1, -2): + assert_allclose(spnorm(S, axis=axis), npnorm(M, axis=axis)) + assert_allclose(spnorm(S, 'f', axis=axis), + npnorm(M, 'f', axis=axis)) + assert_allclose(spnorm(S, 'fro', axis=axis), + npnorm(M, 'fro', axis=axis)) + + def test_sparse_vector_norms(self): + for sparse_type in self._sparse_types: + for M in self._test_matrices: + S = sparse_type(M) + for axis in (0, 1, -1, -2, (0, ), (1, ), (-1, ), (-2, )): + assert_allclose(spnorm(S, axis=axis), npnorm(M, axis=axis)) + for ord in None, 2, np.inf, -np.inf, 1, 0.5, 0.42: + assert_allclose(spnorm(S, ord, axis=axis), + npnorm(M, ord, axis=axis)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_onenormest.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_onenormest.py new file mode 100644 index 0000000000000000000000000000000000000000..d306e8177568d8db31189a5c67b5862e37e8a0fc --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_onenormest.py @@ -0,0 +1,252 @@ +"""Test functions for the sparse.linalg._onenormest module +""" + +import numpy as np +from numpy.testing import assert_allclose, assert_equal, assert_ +import pytest +import scipy.linalg +import scipy.sparse.linalg +from scipy.sparse.linalg._onenormest import _onenormest_core, _algorithm_2_2 + + +class MatrixProductOperator(scipy.sparse.linalg.LinearOperator): + """ + This is purely for onenormest testing. + """ + + def __init__(self, A, B): + if A.ndim != 2 or B.ndim != 2: + raise ValueError('expected ndarrays representing matrices') + if A.shape[1] != B.shape[0]: + raise ValueError('incompatible shapes') + self.A = A + self.B = B + self.ndim = 2 + self.shape = (A.shape[0], B.shape[1]) + + def _matvec(self, x): + return np.dot(self.A, np.dot(self.B, x)) + + def _rmatvec(self, x): + return np.dot(np.dot(x, self.A), self.B) + + def _matmat(self, X): + return np.dot(self.A, np.dot(self.B, X)) + + @property + def T(self): + return MatrixProductOperator(self.B.T, self.A.T) + + +class TestOnenormest: + + @pytest.mark.xslow + def test_onenormest_table_3_t_2(self): + # This will take multiple seconds if your computer is slow like mine. + # It is stochastic, so the tolerance could be too strict. + np.random.seed(1234) + t = 2 + n = 100 + itmax = 5 + nsamples = 5000 + observed = [] + expected = [] + nmult_list = [] + nresample_list = [] + for i in range(nsamples): + A = scipy.linalg.inv(np.random.randn(n, n)) + est, v, w, nmults, nresamples = _onenormest_core(A, A.T, t, itmax) + observed.append(est) + expected.append(scipy.linalg.norm(A, 1)) + nmult_list.append(nmults) + nresample_list.append(nresamples) + observed = np.array(observed, dtype=float) + expected = np.array(expected, dtype=float) + relative_errors = np.abs(observed - expected) / expected + + # check the mean underestimation ratio + underestimation_ratio = observed / expected + assert_(0.99 < np.mean(underestimation_ratio) < 1.0) + + # check the max and mean required column resamples + assert_equal(np.max(nresample_list), 2) + assert_(0.05 < np.mean(nresample_list) < 0.2) + + # check the proportion of norms computed exactly correctly + nexact = np.count_nonzero(relative_errors < 1e-14) + proportion_exact = nexact / float(nsamples) + assert_(0.9 < proportion_exact < 0.95) + + # check the average number of matrix*vector multiplications + assert_(3.5 < np.mean(nmult_list) < 4.5) + + @pytest.mark.xslow + def test_onenormest_table_4_t_7(self): + # This will take multiple seconds if your computer is slow like mine. + # It is stochastic, so the tolerance could be too strict. + np.random.seed(1234) + t = 7 + n = 100 + itmax = 5 + nsamples = 5000 + observed = [] + expected = [] + nmult_list = [] + nresample_list = [] + for i in range(nsamples): + A = np.random.randint(-1, 2, size=(n, n)) + est, v, w, nmults, nresamples = _onenormest_core(A, A.T, t, itmax) + observed.append(est) + expected.append(scipy.linalg.norm(A, 1)) + nmult_list.append(nmults) + nresample_list.append(nresamples) + observed = np.array(observed, dtype=float) + expected = np.array(expected, dtype=float) + relative_errors = np.abs(observed - expected) / expected + + # check the mean underestimation ratio + underestimation_ratio = observed / expected + assert_(0.90 < np.mean(underestimation_ratio) < 0.99) + + # check the required column resamples + assert_equal(np.max(nresample_list), 0) + + # check the proportion of norms computed exactly correctly + nexact = np.count_nonzero(relative_errors < 1e-14) + proportion_exact = nexact / float(nsamples) + assert_(0.15 < proportion_exact < 0.25) + + # check the average number of matrix*vector multiplications + assert_(3.5 < np.mean(nmult_list) < 4.5) + + def test_onenormest_table_5_t_1(self): + # "note that there is no randomness and hence only one estimate for t=1" + t = 1 + n = 100 + itmax = 5 + alpha = 1 - 1e-6 + A = -scipy.linalg.inv(np.identity(n) + alpha*np.eye(n, k=1)) + first_col = np.array([1] + [0]*(n-1)) + first_row = np.array([(-alpha)**i for i in range(n)]) + B = -scipy.linalg.toeplitz(first_col, first_row) + assert_allclose(A, B) + est, v, w, nmults, nresamples = _onenormest_core(B, B.T, t, itmax) + exact_value = scipy.linalg.norm(B, 1) + underest_ratio = est / exact_value + assert_allclose(underest_ratio, 0.05, rtol=1e-4) + assert_equal(nmults, 11) + assert_equal(nresamples, 0) + # check the non-underscored version of onenormest + est_plain = scipy.sparse.linalg.onenormest(B, t=t, itmax=itmax) + assert_allclose(est, est_plain) + + @pytest.mark.xslow + def test_onenormest_table_6_t_2(self): + #TODO this test seems to give estimates that match the table, + #TODO even though no attempt has been made to deal with + #TODO complex numbers in the one-norm estimation. + # This will take multiple seconds if your computer is slow like mine. + # It is stochastic, so the tolerance could be too strict. + np.random.seed(1234) + t = 2 + n = 100 + itmax = 5 + nsamples = 5000 + observed = [] + expected = [] + nmult_list = [] + nresample_list = [] + for i in range(nsamples): + A_inv = np.random.rand(n, n) + 1j * np.random.rand(n, n) + A = scipy.linalg.inv(A_inv) + est, v, w, nmults, nresamples = _onenormest_core(A, A.T, t, itmax) + observed.append(est) + expected.append(scipy.linalg.norm(A, 1)) + nmult_list.append(nmults) + nresample_list.append(nresamples) + observed = np.array(observed, dtype=float) + expected = np.array(expected, dtype=float) + relative_errors = np.abs(observed - expected) / expected + + # check the mean underestimation ratio + underestimation_ratio = observed / expected + underestimation_ratio_mean = np.mean(underestimation_ratio) + assert_(0.90 < underestimation_ratio_mean < 0.99) + + # check the required column resamples + max_nresamples = np.max(nresample_list) + assert_equal(max_nresamples, 0) + + # check the proportion of norms computed exactly correctly + nexact = np.count_nonzero(relative_errors < 1e-14) + proportion_exact = nexact / float(nsamples) + assert_(0.7 < proportion_exact < 0.8) + + # check the average number of matrix*vector multiplications + mean_nmult = np.mean(nmult_list) + assert_(4 < mean_nmult < 5) + + def _help_product_norm_slow(self, A, B): + # for profiling + C = np.dot(A, B) + return scipy.linalg.norm(C, 1) + + def _help_product_norm_fast(self, A, B): + # for profiling + t = 2 + itmax = 5 + D = MatrixProductOperator(A, B) + est, v, w, nmults, nresamples = _onenormest_core(D, D.T, t, itmax) + return est + + @pytest.mark.slow + def test_onenormest_linear_operator(self): + # Define a matrix through its product A B. + # Depending on the shapes of A and B, + # it could be easy to multiply this product by a small matrix, + # but it could be annoying to look at all of + # the entries of the product explicitly. + np.random.seed(1234) + n = 6000 + k = 3 + A = np.random.randn(n, k) + B = np.random.randn(k, n) + fast_estimate = self._help_product_norm_fast(A, B) + exact_value = self._help_product_norm_slow(A, B) + assert_(fast_estimate <= exact_value <= 3*fast_estimate, + f'fast: {fast_estimate:g}\nexact:{exact_value:g}') + + def test_returns(self): + np.random.seed(1234) + A = scipy.sparse.rand(50, 50, 0.1) + + s0 = scipy.linalg.norm(A.toarray(), 1) + s1, v = scipy.sparse.linalg.onenormest(A, compute_v=True) + s2, w = scipy.sparse.linalg.onenormest(A, compute_w=True) + s3, v2, w2 = scipy.sparse.linalg.onenormest(A, compute_w=True, compute_v=True) + + assert_allclose(s1, s0, rtol=1e-9) + assert_allclose(np.linalg.norm(A.dot(v), 1), s0*np.linalg.norm(v, 1), rtol=1e-9) + assert_allclose(A.dot(v), w, rtol=1e-9) + + +class TestAlgorithm_2_2: + + @pytest.mark.thread_unsafe + def test_randn_inv(self): + rng = np.random.RandomState(1234) + n = 20 + nsamples = 100 + for i in range(nsamples): + + # Choose integer t uniformly between 1 and 3 inclusive. + t = rng.randint(1, 4) + + # Choose n uniformly between 10 and 40 inclusive. + n = rng.randint(10, 41) + + # Sample the inverse of a matrix with random normal entries. + A = scipy.linalg.inv(rng.randn(n, n)) + + # Compute the 1-norm bounds. + g, ind = _algorithm_2_2(A, A.T, t) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_propack.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_propack.py new file mode 100644 index 0000000000000000000000000000000000000000..f13905de08470a410b5aa83d33b6e29ec87e905c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_propack.py @@ -0,0 +1,165 @@ +import os +import pytest + +import numpy as np +from numpy.testing import assert_allclose +from pytest import raises as assert_raises +from scipy.sparse.linalg._svdp import _svdp +from scipy.sparse import csr_array, csc_array + + +# dtype_flavour to tolerance +TOLS = { + np.float32: 1e-4, + np.float64: 1e-8, + np.complex64: 1e-4, + np.complex128: 1e-8, +} + + +def is_complex_type(dtype): + return np.dtype(dtype).kind == "c" + + +_dtypes = [] +for dtype_flavour in TOLS.keys(): + marks = [] + if is_complex_type(dtype_flavour): + marks = [pytest.mark.slow] + _dtypes.append(pytest.param(dtype_flavour, marks=marks, + id=dtype_flavour.__name__)) +_dtypes = tuple(_dtypes) # type: ignore[assignment] + + +def generate_matrix(constructor, n, m, f, + dtype=float, rseed=0, **kwargs): + """Generate a random sparse array""" + rng = np.random.RandomState(rseed) + if is_complex_type(dtype): + M = (- 5 + 10 * rng.rand(n, m) + - 5j + 10j * rng.rand(n, m)).astype(dtype) + else: + M = (-5 + 10 * rng.rand(n, m)).astype(dtype) + M[M.real > 10 * f - 5] = 0 + return constructor(M, **kwargs) + + +def assert_orthogonal(u1, u2, rtol, atol): + """Check that the first k rows of u1 and u2 are orthogonal""" + A = abs(np.dot(u1.conj().T, u2)) + assert_allclose(A, np.eye(u1.shape[1], u2.shape[1]), rtol=rtol, atol=atol) + + +def check_svdp(n, m, constructor, dtype, k, irl_mode, which, f=0.8): + tol = TOLS[dtype] + + M = generate_matrix(np.asarray, n, m, f, dtype) + Msp = constructor(M) + + u1, sigma1, vt1 = np.linalg.svd(M, full_matrices=False) + u2, sigma2, vt2, _ = _svdp(Msp, k=k, which=which, irl_mode=irl_mode, + tol=tol, rng=np.random.default_rng(0)) + + # check the which + if which.upper() == 'SM': + u1 = np.roll(u1, k, 1) + vt1 = np.roll(vt1, k, 0) + sigma1 = np.roll(sigma1, k) + + # check that singular values agree + assert_allclose(sigma1[:k], sigma2, rtol=tol, atol=tol) + + # check that singular vectors are orthogonal + assert_orthogonal(u1, u2, rtol=tol, atol=tol) + assert_orthogonal(vt1.T, vt2.T, rtol=tol, atol=tol) + + +@pytest.mark.parametrize('ctor', (np.array, csr_array, csc_array)) +@pytest.mark.parametrize('dtype', _dtypes) +@pytest.mark.parametrize('irl', (True, False)) +@pytest.mark.parametrize('which', ('LM', 'SM')) +def test_svdp(ctor, dtype, irl, which): + np.random.seed(0) + n, m, k = 10, 20, 3 + if which == 'SM' and not irl: + message = "`which`='SM' requires irl_mode=True" + with assert_raises(ValueError, match=message): + check_svdp(n, m, ctor, dtype, k, irl, which) + else: + check_svdp(n, m, ctor, dtype, k, irl, which) + + +@pytest.mark.xslow +@pytest.mark.parametrize('dtype', _dtypes) +@pytest.mark.parametrize('irl', (False, True)) +def test_examples(dtype, irl): + # Note: atol for complex64 bumped from 1e-4 to 1e-3 due to test failures + # with BLIS, Netlib, and MKL+AVX512 - see + # https://github.com/conda-forge/scipy-feedstock/pull/198#issuecomment-999180432 + atol = { + np.float32: 1.3e-4, + np.float64: 1e-9, + np.complex64: 1e-3, + np.complex128: 1e-9, + }[dtype] + + path_prefix = os.path.dirname(__file__) + # Test matrices from `illc1850.coord` and `mhd1280b.cua` distributed with + # PROPACK 2.1: http://sun.stanford.edu/~rmunk/PROPACK/ + relative_path = "propack_test_data.npz" + filename = os.path.join(path_prefix, relative_path) + with np.load(filename, allow_pickle=True) as data: + if is_complex_type(dtype): + A = data['A_complex'].item().astype(dtype) + else: + A = data['A_real'].item().astype(dtype) + + k = 200 + u, s, vh, _ = _svdp(A, k, irl_mode=irl, rng=np.random.default_rng(0)) + + # complex example matrix has many repeated singular values, so check only + # beginning non-repeated singular vectors to avoid permutations + sv_check = 27 if is_complex_type(dtype) else k + u = u[:, :sv_check] + vh = vh[:sv_check, :] + s = s[:sv_check] + + # Check orthogonality of singular vectors + assert_allclose(np.eye(u.shape[1]), u.conj().T @ u, atol=atol) + assert_allclose(np.eye(vh.shape[0]), vh @ vh.conj().T, atol=atol) + + # Ensure the norm of the difference between the np.linalg.svd and + # PROPACK reconstructed matrices is small + u3, s3, vh3 = np.linalg.svd(A.todense()) + u3 = u3[:, :sv_check] + s3 = s3[:sv_check] + vh3 = vh3[:sv_check, :] + A3 = u3 @ np.diag(s3) @ vh3 + recon = u @ np.diag(s) @ vh + assert_allclose(np.linalg.norm(A3 - recon), 0, atol=atol) + + +@pytest.mark.parametrize('shifts', (None, -10, 0, 1, 10, 70)) +@pytest.mark.parametrize('dtype', _dtypes[:2]) +def test_shifts(shifts, dtype): + rng = np.random.default_rng(0) + n, k = 70, 10 + A = rng.random((n, n)) + if shifts is not None and ((shifts < 0) or (k > min(n-1-shifts, n))): + with pytest.raises(ValueError): + _svdp(A, k, shifts=shifts, kmax=5*k, irl_mode=True, rng=rng) + else: + _svdp(A, k, shifts=shifts, kmax=5*k, irl_mode=True, rng=rng) + + +@pytest.mark.slow +@pytest.mark.xfail() +def test_shifts_accuracy(): + rng = np.random.default_rng(0) + n, k = 70, 10 + A = rng.random((n, n)).astype(np.float64) + u1, s1, vt1, _ = _svdp(A, k, shifts=None, which='SM', irl_mode=True, rng=rng) + u2, s2, vt2, _ = _svdp(A, k, shifts=32, which='SM', irl_mode=True, rng=rng) + # shifts <= 32 doesn't agree with shifts > 32 + # Does agree when which='LM' instead of 'SM' + assert_allclose(s1, s2) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_pydata_sparse.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_pydata_sparse.py new file mode 100644 index 0000000000000000000000000000000000000000..f6b855271063ee769c70095c62dd8a09bd05bd95 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_pydata_sparse.py @@ -0,0 +1,258 @@ +import pytest + +import numpy as np +import scipy.sparse as sp +import scipy.sparse.linalg as splin + +from numpy.testing import assert_allclose, assert_equal + +try: + import sparse +except Exception: + sparse = None + +pytestmark = pytest.mark.skipif(sparse is None, + reason="pydata/sparse not installed") + + +msg = "pydata/sparse (0.15.1) does not implement necessary operations" + + +sparse_params = (pytest.param("COO"), + pytest.param("DOK", marks=[pytest.mark.xfail(reason=msg)])) + +scipy_sparse_classes = [ + sp.bsr_array, + sp.csr_array, + sp.coo_array, + sp.csc_array, + sp.dia_array, + sp.dok_array +] + + +@pytest.fixture(params=sparse_params) +def sparse_cls(request): + return getattr(sparse, request.param) + + +@pytest.fixture(params=scipy_sparse_classes) +def sp_sparse_cls(request): + return request.param + + +@pytest.fixture +def same_matrix(sparse_cls, sp_sparse_cls): + np.random.seed(1234) + A_dense = np.random.rand(9, 9) + return sp_sparse_cls(A_dense), sparse_cls(A_dense) + + +@pytest.fixture +def matrices(sparse_cls): + np.random.seed(1234) + A_dense = np.random.rand(9, 9) + A_dense = A_dense @ A_dense.T + A_sparse = sparse_cls(A_dense) + b = np.random.rand(9) + return A_dense, A_sparse, b + + +def test_isolve_gmres(matrices): + # Several of the iterative solvers use the same + # isolve.utils.make_system wrapper code, so test just one of them. + A_dense, A_sparse, b = matrices + x, info = splin.gmres(A_sparse, b, atol=1e-15) + assert info == 0 + assert isinstance(x, np.ndarray) + assert_allclose(A_sparse @ x, b) + + +def test_lsmr(matrices): + A_dense, A_sparse, b = matrices + res0 = splin.lsmr(A_dense, b) + res = splin.lsmr(A_sparse, b) + assert_allclose(res[0], res0[0], atol=1e-3) + + +# test issue 17012 +def test_lsmr_output_shape(): + x = splin.lsmr(A=np.ones((10, 1)), b=np.zeros(10), x0=np.ones(1))[0] + assert_equal(x.shape, (1,)) + + +def test_lsqr(matrices): + A_dense, A_sparse, b = matrices + res0 = splin.lsqr(A_dense, b) + res = splin.lsqr(A_sparse, b) + assert_allclose(res[0], res0[0], atol=1e-5) + + +def test_eigs(matrices): + A_dense, A_sparse, v0 = matrices + + M_dense = np.diag(v0**2) + M_sparse = A_sparse.__class__(M_dense) + + w_dense, v_dense = splin.eigs(A_dense, k=3, v0=v0) + w, v = splin.eigs(A_sparse, k=3, v0=v0) + + assert_allclose(w, w_dense) + assert_allclose(v, v_dense) + + for M in [M_sparse, M_dense]: + w_dense, v_dense = splin.eigs(A_dense, M=M_dense, k=3, v0=v0) + w, v = splin.eigs(A_sparse, M=M, k=3, v0=v0) + + assert_allclose(w, w_dense) + assert_allclose(v, v_dense) + + w_dense, v_dense = splin.eigsh(A_dense, M=M_dense, k=3, v0=v0) + w, v = splin.eigsh(A_sparse, M=M, k=3, v0=v0) + + assert_allclose(w, w_dense) + assert_allclose(v, v_dense) + + +def test_svds(matrices): + A_dense, A_sparse, v0 = matrices + + u0, s0, vt0 = splin.svds(A_dense, k=2, v0=v0) + u, s, vt = splin.svds(A_sparse, k=2, v0=v0) + + assert_allclose(s, s0) + assert_allclose(np.abs(u), np.abs(u0)) + assert_allclose(np.abs(vt), np.abs(vt0)) + + +def test_lobpcg(matrices): + A_dense, A_sparse, x = matrices + X = x[:,None] + + w_dense, v_dense = splin.lobpcg(A_dense, X) + w, v = splin.lobpcg(A_sparse, X) + + assert_allclose(w, w_dense) + assert_allclose(v, v_dense) + + +def test_spsolve(matrices): + A_dense, A_sparse, b = matrices + b2 = np.random.rand(len(b), 3) + + x0 = splin.spsolve(sp.csc_array(A_dense), b) + x = splin.spsolve(A_sparse, b) + assert isinstance(x, np.ndarray) + assert_allclose(x, x0) + + x0 = splin.spsolve(sp.csc_array(A_dense), b) + x = splin.spsolve(A_sparse, b, use_umfpack=True) + assert isinstance(x, np.ndarray) + assert_allclose(x, x0) + + x0 = splin.spsolve(sp.csc_array(A_dense), b2) + x = splin.spsolve(A_sparse, b2) + assert isinstance(x, np.ndarray) + assert_allclose(x, x0) + + x0 = splin.spsolve(sp.csc_array(A_dense), + sp.csc_array(A_dense)) + x = splin.spsolve(A_sparse, A_sparse) + assert isinstance(x, type(A_sparse)) + assert_allclose(x.todense(), x0.todense()) + + +def test_splu(matrices): + A_dense, A_sparse, b = matrices + n = len(b) + sparse_cls = type(A_sparse) + + lu = splin.splu(A_sparse) + + assert isinstance(lu.L, sparse_cls) + assert isinstance(lu.U, sparse_cls) + + _Pr_scipy = sp.csc_array((np.ones(n), (lu.perm_r, np.arange(n)))) + _Pc_scipy = sp.csc_array((np.ones(n), (np.arange(n), lu.perm_c))) + Pr = sparse_cls.from_scipy_sparse(_Pr_scipy) + Pc = sparse_cls.from_scipy_sparse(_Pc_scipy) + A2 = Pr.T @ lu.L @ lu.U @ Pc.T + + assert_allclose(A2.todense(), A_sparse.todense()) + + z = lu.solve(A_sparse.todense()) + assert_allclose(z, np.eye(n), atol=1e-10) + + +def test_spilu(matrices): + A_dense, A_sparse, b = matrices + sparse_cls = type(A_sparse) + + lu = splin.spilu(A_sparse) + + assert isinstance(lu.L, sparse_cls) + assert isinstance(lu.U, sparse_cls) + + z = lu.solve(A_sparse.todense()) + assert_allclose(z, np.eye(len(b)), atol=1e-3) + + +def test_spsolve_triangular(matrices): + A_dense, A_sparse, b = matrices + A_sparse = sparse.tril(A_sparse) + + x = splin.spsolve_triangular(A_sparse, b) + assert_allclose(A_sparse @ x, b) + + +def test_onenormest(matrices): + A_dense, A_sparse, b = matrices + est0 = splin.onenormest(A_dense) + est = splin.onenormest(A_sparse) + assert_allclose(est, est0) + + +def test_norm(matrices): + A_dense, A_sparse, b = matrices + norm0 = splin.norm(sp.csr_array(A_dense)) + norm = splin.norm(A_sparse) + assert_allclose(norm, norm0) + + +def test_inv(matrices): + A_dense, A_sparse, b = matrices + x0 = splin.inv(sp.csc_array(A_dense)) + x = splin.inv(A_sparse) + assert_allclose(x.todense(), x0.todense()) + + +def test_expm(matrices): + A_dense, A_sparse, b = matrices + x0 = splin.expm(sp.csc_array(A_dense)) + x = splin.expm(A_sparse) + assert_allclose(x.todense(), x0.todense()) + + +def test_expm_multiply(matrices): + A_dense, A_sparse, b = matrices + x0 = splin.expm_multiply(A_dense, b) + x = splin.expm_multiply(A_sparse, b) + assert_allclose(x, x0) + + x0 = splin.expm_multiply(A_dense, A_dense) + x = splin.expm_multiply(A_sparse, A_sparse) + assert_allclose(x.todense(), x0) + + +def test_eq(same_matrix): + sp_sparse, pd_sparse = same_matrix + # temporary splint until pydata sparse support sparray equality + sp_sparse = sp.coo_matrix(sp_sparse).asformat(sp_sparse.format) + assert (sp_sparse == pd_sparse).all() + + +def test_ne(same_matrix): + sp_sparse, pd_sparse = same_matrix + # temporary splint until pydata sparse support sparray equality + sp_sparse = sp.coo_matrix(sp_sparse).asformat(sp_sparse.format) + assert not (sp_sparse != pd_sparse).any() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_special_sparse_arrays.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_special_sparse_arrays.py new file mode 100644 index 0000000000000000000000000000000000000000..d9d1c4001af6697233380edf0047409a41847834 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/linalg/tests/test_special_sparse_arrays.py @@ -0,0 +1,337 @@ +import pytest +import numpy as np +from numpy.testing import assert_array_equal, assert_allclose + +from scipy.sparse import diags, csgraph +from scipy.linalg import eigh + +from scipy.sparse.linalg import LaplacianNd +from scipy.sparse.linalg._special_sparse_arrays import Sakurai +from scipy.sparse.linalg._special_sparse_arrays import MikotaPair + +INT_DTYPES = [np.int8, np.int16, np.int32, np.int64] +REAL_DTYPES = [np.float32, np.float64] +COMPLEX_DTYPES = [np.complex64, np.complex128] +ALLDTYPES = INT_DTYPES + REAL_DTYPES + COMPLEX_DTYPES + + +class TestLaplacianNd: + """ + LaplacianNd tests + """ + + @pytest.mark.parametrize('bc', ['neumann', 'dirichlet', 'periodic']) + def test_1d_specific_shape(self, bc): + lap = LaplacianNd(grid_shape=(6, ), boundary_conditions=bc) + lapa = lap.toarray() + if bc == 'neumann': + a = np.array( + [ + [-1, 1, 0, 0, 0, 0], + [1, -2, 1, 0, 0, 0], + [0, 1, -2, 1, 0, 0], + [0, 0, 1, -2, 1, 0], + [0, 0, 0, 1, -2, 1], + [0, 0, 0, 0, 1, -1], + ] + ) + elif bc == 'dirichlet': + a = np.array( + [ + [-2, 1, 0, 0, 0, 0], + [1, -2, 1, 0, 0, 0], + [0, 1, -2, 1, 0, 0], + [0, 0, 1, -2, 1, 0], + [0, 0, 0, 1, -2, 1], + [0, 0, 0, 0, 1, -2], + ] + ) + else: + a = np.array( + [ + [-2, 1, 0, 0, 0, 1], + [1, -2, 1, 0, 0, 0], + [0, 1, -2, 1, 0, 0], + [0, 0, 1, -2, 1, 0], + [0, 0, 0, 1, -2, 1], + [1, 0, 0, 0, 1, -2], + ] + ) + assert_array_equal(a, lapa) + + def test_1d_with_graph_laplacian(self): + n = 6 + G = diags(np.ones(n - 1), 1, format='dia') + Lf = csgraph.laplacian(G, symmetrized=True, form='function') + La = csgraph.laplacian(G, symmetrized=True, form='array') + grid_shape = (n,) + bc = 'neumann' + lap = LaplacianNd(grid_shape, boundary_conditions=bc) + assert_array_equal(lap(np.eye(n)), -Lf(np.eye(n))) + assert_array_equal(lap.toarray(), -La.toarray()) + # https://github.com/numpy/numpy/issues/24351 + assert_array_equal(lap.tosparse().toarray(), -La.toarray()) + + @pytest.mark.parametrize('grid_shape', [(6, ), (2, 3), (2, 3, 4)]) + @pytest.mark.parametrize('bc', ['neumann', 'dirichlet', 'periodic']) + def test_eigenvalues(self, grid_shape, bc): + lap = LaplacianNd(grid_shape, boundary_conditions=bc, dtype=np.float64) + L = lap.toarray() + eigvals = eigh(L, eigvals_only=True) + n = np.prod(grid_shape) + eigenvalues = lap.eigenvalues() + dtype = eigenvalues.dtype + atol = n * n * np.finfo(dtype).eps + # test the default ``m = None`` + assert_allclose(eigenvalues, eigvals, atol=atol) + # test every ``m > 0`` + for m in np.arange(1, n + 1): + assert_array_equal(lap.eigenvalues(m), eigenvalues[-m:]) + + @pytest.mark.parametrize('grid_shape', [(6, ), (2, 3), (2, 3, 4)]) + @pytest.mark.parametrize('bc', ['neumann', 'dirichlet', 'periodic']) + def test_eigenvectors(self, grid_shape, bc): + lap = LaplacianNd(grid_shape, boundary_conditions=bc, dtype=np.float64) + n = np.prod(grid_shape) + eigenvalues = lap.eigenvalues() + eigenvectors = lap.eigenvectors() + dtype = eigenvectors.dtype + atol = n * n * max(np.finfo(dtype).eps, np.finfo(np.double).eps) + # test the default ``m = None`` every individual eigenvector + for i in np.arange(n): + r = lap.toarray() @ eigenvectors[:, i] - eigenvectors[:, i] * eigenvalues[i] + assert_allclose(r, np.zeros_like(r), atol=atol) + # test every ``m > 0`` + for m in np.arange(1, n + 1): + e = lap.eigenvalues(m) + ev = lap.eigenvectors(m) + r = lap.toarray() @ ev - ev @ np.diag(e) + assert_allclose(r, np.zeros_like(r), atol=atol) + + @pytest.mark.parametrize('grid_shape', [(6, ), (2, 3), (2, 3, 4)]) + @pytest.mark.parametrize('bc', ['neumann', 'dirichlet', 'periodic']) + def test_toarray_tosparse_consistency(self, grid_shape, bc): + lap = LaplacianNd(grid_shape, boundary_conditions=bc) + n = np.prod(grid_shape) + assert_array_equal(lap.toarray(), lap(np.eye(n))) + assert_array_equal(lap.tosparse().toarray(), lap.toarray()) + + @pytest.mark.parametrize('dtype', ALLDTYPES) + @pytest.mark.parametrize('grid_shape', [(6, ), (2, 3), (2, 3, 4)]) + @pytest.mark.parametrize('bc', ['neumann', 'dirichlet', 'periodic']) + def test_linearoperator_shape_dtype(self, grid_shape, bc, dtype): + lap = LaplacianNd(grid_shape, boundary_conditions=bc, dtype=dtype) + n = np.prod(grid_shape) + assert lap.shape == (n, n) + assert lap.dtype == dtype + assert_array_equal( + LaplacianNd( + grid_shape, boundary_conditions=bc, dtype=dtype + ).toarray(), + LaplacianNd(grid_shape, boundary_conditions=bc) + .toarray() + .astype(dtype), + ) + assert_array_equal( + LaplacianNd(grid_shape, boundary_conditions=bc, dtype=dtype) + .tosparse() + .toarray(), + LaplacianNd(grid_shape, boundary_conditions=bc) + .tosparse() + .toarray() + .astype(dtype), + ) + + @pytest.mark.parametrize('dtype', ALLDTYPES) + @pytest.mark.parametrize('grid_shape', [(6, ), (2, 3), (2, 3, 4)]) + @pytest.mark.parametrize('bc', ['neumann', 'dirichlet', 'periodic']) + def test_dot(self, grid_shape, bc, dtype): + """ Test the dot-product for type preservation and consistency. + """ + lap = LaplacianNd(grid_shape, boundary_conditions=bc) + n = np.prod(grid_shape) + x0 = np.arange(n) + x1 = x0.reshape((-1, 1)) + x2 = np.arange(2 * n).reshape((n, 2)) + input_set = [x0, x1, x2] + for x in input_set: + y = lap.dot(x.astype(dtype)) + assert x.shape == y.shape + assert y.dtype == dtype + if x.ndim == 2: + yy = lap.toarray() @ x.astype(dtype) + assert yy.dtype == dtype + np.array_equal(y, yy) + + def test_boundary_conditions_value_error(self): + with pytest.raises(ValueError, match="Unknown value 'robin'"): + LaplacianNd(grid_shape=(6, ), boundary_conditions='robin') + + +class TestSakurai: + """ + Sakurai tests + """ + + def test_specific_shape(self): + sak = Sakurai(6) + assert_array_equal(sak.toarray(), sak(np.eye(6))) + a = np.array( + [ + [ 5, -4, 1, 0, 0, 0], + [-4, 6, -4, 1, 0, 0], + [ 1, -4, 6, -4, 1, 0], + [ 0, 1, -4, 6, -4, 1], + [ 0, 0, 1, -4, 6, -4], + [ 0, 0, 0, 1, -4, 5] + ] + ) + + np.array_equal(a, sak.toarray()) + np.array_equal(sak.tosparse().toarray(), sak.toarray()) + ab = np.array( + [ + [ 1, 1, 1, 1, 1, 1], + [-4, -4, -4, -4, -4, -4], + [ 5, 6, 6, 6, 6, 5] + ] + ) + np.array_equal(ab, sak.tobanded()) + e = np.array( + [0.03922866, 0.56703972, 2.41789479, 5.97822974, + 10.54287655, 14.45473055] + ) + np.array_equal(e, sak.eigenvalues()) + np.array_equal(e[:2], sak.eigenvalues(2)) + + # `Sakurai` default `dtype` is `np.int8` as its entries are small integers + @pytest.mark.parametrize('dtype', ALLDTYPES) + def test_linearoperator_shape_dtype(self, dtype): + n = 7 + sak = Sakurai(n, dtype=dtype) + assert sak.shape == (n, n) + assert sak.dtype == dtype + assert_array_equal(sak.toarray(), Sakurai(n).toarray().astype(dtype)) + assert_array_equal(sak.tosparse().toarray(), + Sakurai(n).tosparse().toarray().astype(dtype)) + + @pytest.mark.parametrize('dtype', ALLDTYPES) + @pytest.mark.parametrize('argument_dtype', ALLDTYPES) + def test_dot(self, dtype, argument_dtype): + """ Test the dot-product for type preservation and consistency. + """ + result_dtype = np.promote_types(argument_dtype, dtype) + n = 5 + sak = Sakurai(n) + x0 = np.arange(n) + x1 = x0.reshape((-1, 1)) + x2 = np.arange(2 * n).reshape((n, 2)) + input_set = [x0, x1, x2] + for x in input_set: + y = sak.dot(x.astype(argument_dtype)) + assert x.shape == y.shape + assert np.can_cast(y.dtype, result_dtype) + if x.ndim == 2: + ya = sak.toarray() @ x.astype(argument_dtype) + np.array_equal(y, ya) + assert np.can_cast(ya.dtype, result_dtype) + ys = sak.tosparse() @ x.astype(argument_dtype) + np.array_equal(y, ys) + assert np.can_cast(ys.dtype, result_dtype) + +class TestMikotaPair: + """ + MikotaPair tests + """ + # both MikotaPair `LinearOperator`s share the same dtype + # while `MikotaK` `dtype` can be as small as its default `np.int32` + # since its entries are integers, the `MikotaM` involves inverses + # so its smallest still accurate `dtype` is `np.float32` + tested_types = REAL_DTYPES + COMPLEX_DTYPES + + def test_specific_shape(self): + n = 6 + mik = MikotaPair(n) + mik_k = mik.k + mik_m = mik.m + assert_array_equal(mik_k.toarray(), mik_k(np.eye(n))) + assert_array_equal(mik_m.toarray(), mik_m(np.eye(n))) + + k = np.array( + [ + [11, -5, 0, 0, 0, 0], + [-5, 9, -4, 0, 0, 0], + [ 0, -4, 7, -3, 0, 0], + [ 0, 0, -3, 5, -2, 0], + [ 0, 0, 0, -2, 3, -1], + [ 0, 0, 0, 0, -1, 1] + ] + ) + np.array_equal(k, mik_k.toarray()) + np.array_equal(mik_k.tosparse().toarray(), k) + kb = np.array( + [ + [ 0, -5, -4, -3, -2, -1], + [11, 9, 7, 5, 3, 1] + ] + ) + np.array_equal(kb, mik_k.tobanded()) + + minv = np.arange(1, n + 1) + np.array_equal(np.diag(1. / minv), mik_m.toarray()) + np.array_equal(mik_m.tosparse().toarray(), mik_m.toarray()) + np.array_equal(1. / minv, mik_m.tobanded()) + + e = np.array([ 1, 4, 9, 16, 25, 36]) + np.array_equal(e, mik.eigenvalues()) + np.array_equal(e[:2], mik.eigenvalues(2)) + + @pytest.mark.parametrize('dtype', tested_types) + def test_linearoperator_shape_dtype(self, dtype): + n = 7 + mik = MikotaPair(n, dtype=dtype) + mik_k = mik.k + mik_m = mik.m + assert mik_k.shape == (n, n) + assert mik_k.dtype == dtype + assert mik_m.shape == (n, n) + assert mik_m.dtype == dtype + mik_default_dtype = MikotaPair(n) + mikd_k = mik_default_dtype.k + mikd_m = mik_default_dtype.m + assert mikd_k.shape == (n, n) + assert mikd_k.dtype == np.float64 + assert mikd_m.shape == (n, n) + assert mikd_m.dtype == np.float64 + assert_array_equal(mik_k.toarray(), + mikd_k.toarray().astype(dtype)) + assert_array_equal(mik_k.tosparse().toarray(), + mikd_k.tosparse().toarray().astype(dtype)) + + @pytest.mark.parametrize('dtype', tested_types) + @pytest.mark.parametrize('argument_dtype', ALLDTYPES) + def test_dot(self, dtype, argument_dtype): + """ Test the dot-product for type preservation and consistency. + """ + result_dtype = np.promote_types(argument_dtype, dtype) + n = 5 + mik = MikotaPair(n, dtype=dtype) + mik_k = mik.k + mik_m = mik.m + x0 = np.arange(n) + x1 = x0.reshape((-1, 1)) + x2 = np.arange(2 * n).reshape((n, 2)) + lo_set = [mik_k, mik_m] + input_set = [x0, x1, x2] + for lo in lo_set: + for x in input_set: + y = lo.dot(x.astype(argument_dtype)) + assert x.shape == y.shape + assert np.can_cast(y.dtype, result_dtype) + if x.ndim == 2: + ya = lo.toarray() @ x.astype(argument_dtype) + np.array_equal(y, ya) + assert np.can_cast(ya.dtype, result_dtype) + ys = lo.tosparse() @ x.astype(argument_dtype) + np.array_equal(y, ys) + assert np.can_cast(ys.dtype, result_dtype) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/sparsetools.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/sparsetools.py new file mode 100644 index 0000000000000000000000000000000000000000..404e431d89d479520d2198ae73b9eab7b23a80f7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/sparsetools.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="sparsetools", + private_modules=["_sparsetools"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/spfuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/spfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..911969e414d4a1d3888900ad8392b5fc2177c850 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/spfuncs.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="spfuncs", + private_modules=["_spfuncs"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/sputils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/sputils.py new file mode 100644 index 0000000000000000000000000000000000000000..4ddd27a43889609b0642bd7579e13c8e3c460a8b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/sputils.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.sparse` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="sparse", module="sputils", + private_modules=["_sputils"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_arithmetic1d.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_arithmetic1d.py new file mode 100644 index 0000000000000000000000000000000000000000..3d5d2ee2f1bc2c3fef03c71fc0206cd06f3f8618 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_arithmetic1d.py @@ -0,0 +1,338 @@ +"""Test of 1D arithmetic operations""" + +import pytest + +import numpy as np +from numpy.testing import assert_equal, assert_allclose + +from scipy.sparse import coo_array, csr_array +from scipy.sparse._sputils import isscalarlike + + +spcreators = [coo_array, csr_array] +math_dtypes = [np.int64, np.float64, np.complex128] + + +def toarray(a): + if isinstance(a, np.ndarray) or isscalarlike(a): + return a + return a.toarray() + +@pytest.fixture +def dat1d(): + return np.array([3, 0, 1, 0], 'd') + + +@pytest.fixture +def datsp_math_dtypes(dat1d): + dat_dtypes = {dtype: dat1d.astype(dtype) for dtype in math_dtypes} + return { + sp: [(dtype, dat, sp(dat)) for dtype, dat in dat_dtypes.items()] + for sp in spcreators + } + + +@pytest.mark.parametrize("spcreator", spcreators) +class TestArithmetic1D: + def test_empty_arithmetic(self, spcreator): + shape = (5,) + for mytype in [ + np.dtype('int32'), + np.dtype('float32'), + np.dtype('float64'), + np.dtype('complex64'), + np.dtype('complex128'), + ]: + a = spcreator(shape, dtype=mytype) + b = a + a + c = 2 * a + assert isinstance(a @ a.tocsr(), np.ndarray) + assert isinstance(a @ a.tocoo(), np.ndarray) + for m in [a, b, c]: + assert m @ m == a.toarray() @ a.toarray() + assert m.dtype == mytype + assert toarray(m).dtype == mytype + + def test_abs(self, spcreator): + A = np.array([-1, 0, 17, 0, -5, 0, 1, -4, 0, 0, 0, 0], 'd') + assert_equal(abs(A), abs(spcreator(A)).toarray()) + + def test_round(self, spcreator): + A = np.array([-1.35, 0.56, 17.25, -5.98], 'd') + Asp = spcreator(A) + assert_equal(np.around(A, decimals=1), round(Asp, ndigits=1).toarray()) + + def test_elementwise_power(self, spcreator): + A = np.array([-4, -3, -2, -1, 0, 1, 2, 3, 4], 'd') + Asp = spcreator(A) + assert_equal(np.power(A, 2), Asp.power(2).toarray()) + + # element-wise power function needs a scalar power + with pytest.raises(NotImplementedError, match='input is not scalar'): + spcreator(A).power(A) + + def test_real(self, spcreator): + D = np.array([1 + 3j, 2 - 4j]) + A = spcreator(D) + assert_equal(A.real.toarray(), D.real) + + def test_imag(self, spcreator): + D = np.array([1 + 3j, 2 - 4j]) + A = spcreator(D) + assert_equal(A.imag.toarray(), D.imag) + + def test_mul_scalar(self, spcreator, datsp_math_dtypes): + for dtype, dat, datsp in datsp_math_dtypes[spcreator]: + assert_equal(dat * 2, (datsp * 2).toarray()) + assert_equal(dat * 17.3, (datsp * 17.3).toarray()) + + def test_rmul_scalar(self, spcreator, datsp_math_dtypes): + for dtype, dat, datsp in datsp_math_dtypes[spcreator]: + assert_equal(2 * dat, (2 * datsp).toarray()) + assert_equal(17.3 * dat, (17.3 * datsp).toarray()) + + def test_sub(self, spcreator, datsp_math_dtypes): + for dtype, dat, datsp in datsp_math_dtypes[spcreator]: + if dtype == np.dtype('bool'): + # boolean array subtraction deprecated in 1.9.0 + continue + + assert_equal((datsp - datsp).toarray(), np.zeros(4)) + assert_equal((datsp - 0).toarray(), dat) + + A = spcreator([1, -4, 0, 2], dtype='d') + assert_equal((datsp - A).toarray(), dat - A.toarray()) + assert_equal((A - datsp).toarray(), A.toarray() - dat) + + # test broadcasting + assert_equal(datsp.toarray() - dat[0], dat - dat[0]) + + def test_add0(self, spcreator, datsp_math_dtypes): + for dtype, dat, datsp in datsp_math_dtypes[spcreator]: + # Adding 0 to a sparse matrix + assert_equal((datsp + 0).toarray(), dat) + # use sum (which takes 0 as a starting value) + sumS = sum([k * datsp for k in range(1, 3)]) + sumD = sum([k * dat for k in range(1, 3)]) + assert_allclose(sumS.toarray(), sumD) + + def test_elementwise_multiply(self, spcreator): + # real/real + A = np.array([4, 0, 9]) + B = np.array([0, 7, -1]) + Asp = spcreator(A) + Bsp = spcreator(B) + assert_allclose(Asp.multiply(Bsp).toarray(), A * B) # sparse/sparse + assert_allclose(Asp.multiply(B).toarray(), A * B) # sparse/dense + + # complex/complex + C = np.array([1 - 2j, 0 + 5j, -1 + 0j]) + D = np.array([5 + 2j, 7 - 3j, -2 + 1j]) + Csp = spcreator(C) + Dsp = spcreator(D) + assert_allclose(Csp.multiply(Dsp).toarray(), C * D) # sparse/sparse + assert_allclose(Csp.multiply(D).toarray(), C * D) # sparse/dense + + # real/complex + assert_allclose(Asp.multiply(Dsp).toarray(), A * D) # sparse/sparse + assert_allclose(Asp.multiply(D).toarray(), A * D) # sparse/dense + + def test_elementwise_multiply_broadcast(self, spcreator): + A = np.array([4]) + B = np.array([[-9]]) + C = np.array([1, -1, 0]) + D = np.array([[7, 9, -9]]) + E = np.array([[3], [2], [1]]) + F = np.array([[8, 6, 3], [-4, 3, 2], [6, 6, 6]]) + G = [1, 2, 3] + H = np.ones((3, 4)) + J = H.T + K = np.array([[0]]) + L = np.array([[[1, 2], [0, 1]]]) + + # Some arrays can't be cast as spmatrices (A, C, L) so leave + # them out. + Asp = spcreator(A) + Csp = spcreator(C) + Gsp = spcreator(G) + # 2d arrays + Bsp = spcreator(B) + Dsp = spcreator(D) + Esp = spcreator(E) + Fsp = spcreator(F) + Hsp = spcreator(H) + Hspp = spcreator(H[0, None]) + Jsp = spcreator(J) + Jspp = spcreator(J[:, 0, None]) + Ksp = spcreator(K) + + matrices = [A, B, C, D, E, F, G, H, J, K, L] + spmatrices = [Asp, Bsp, Csp, Dsp, Esp, Fsp, Gsp, Hsp, Hspp, Jsp, Jspp, Ksp] + sp1dmatrices = [Asp, Csp, Gsp] + + # sparse/sparse + for i in sp1dmatrices: + for j in spmatrices: + try: + dense_mult = i.toarray() * j.toarray() + except ValueError: + with pytest.raises(ValueError, match='inconsistent shapes'): + i.multiply(j) + continue + sp_mult = i.multiply(j) + assert_allclose(sp_mult.toarray(), dense_mult) + + # sparse/dense + for i in sp1dmatrices: + for j in matrices: + try: + dense_mult = i.toarray() * j + except TypeError: + continue + except ValueError: + matchme = 'broadcast together|inconsistent shapes' + with pytest.raises(ValueError, match=matchme): + i.multiply(j) + continue + sp_mult = i.multiply(j) + assert_allclose(toarray(sp_mult), dense_mult) + + def test_elementwise_divide(self, spcreator, dat1d): + datsp = spcreator(dat1d) + expected = np.array([1, np.nan, 1, np.nan]) + actual = datsp / datsp + # need assert_array_equal to handle nan values + np.testing.assert_array_equal(actual, expected) + + denom = spcreator([1, 0, 0, 4], dtype='d') + expected = [3, np.nan, np.inf, 0] + np.testing.assert_array_equal(datsp / denom, expected) + + # complex + A = np.array([1 - 2j, 0 + 5j, -1 + 0j]) + B = np.array([5 + 2j, 7 - 3j, -2 + 1j]) + Asp = spcreator(A) + Bsp = spcreator(B) + assert_allclose(Asp / Bsp, A / B) + + # integer + A = np.array([1, 2, 3]) + B = np.array([0, 1, 2]) + Asp = spcreator(A) + Bsp = spcreator(B) + with np.errstate(divide='ignore'): + assert_equal(Asp / Bsp, A / B) + + # mismatching sparsity patterns + A = np.array([0, 1]) + B = np.array([1, 0]) + Asp = spcreator(A) + Bsp = spcreator(B) + with np.errstate(divide='ignore', invalid='ignore'): + assert_equal(Asp / Bsp, A / B) + + def test_pow(self, spcreator): + A = np.array([1, 0, 2, 0]) + B = spcreator(A) + + # unusual exponents + with pytest.raises(ValueError, match='negative integer powers'): + B**-1 + with pytest.raises(NotImplementedError, match='zero power'): + B**0 + + for exponent in [1, 2, 3, 2.2]: + ret_sp = B**exponent + ret_np = A**exponent + assert_equal(ret_sp.toarray(), ret_np) + assert_equal(ret_sp.dtype, ret_np.dtype) + + def test_dot_scalar(self, spcreator, dat1d): + A = spcreator(dat1d) + scalar = 10 + actual = A.dot(scalar) + expected = A * scalar + + assert_allclose(actual.toarray(), expected.toarray()) + + def test_matmul(self, spcreator): + Msp = spcreator([2, 0, 3.0]) + B = spcreator(np.array([[0, 1], [1, 0], [0, 2]], 'd')) + col = np.array([[1, 2, 3]]).T + + # check sparse @ dense 2d column + assert_allclose(Msp @ col, Msp.toarray() @ col) + + # check sparse1d @ sparse2d, sparse1d @ dense2d, dense1d @ sparse2d + assert_allclose((Msp @ B).toarray(), (Msp @ B).toarray()) + assert_allclose(Msp.toarray() @ B, (Msp @ B).toarray()) + assert_allclose(Msp @ B.toarray(), (Msp @ B).toarray()) + + # check sparse1d @ dense1d, sparse1d @ sparse1d + V = np.array([0, 0, 1]) + assert_allclose(Msp @ V, Msp.toarray() @ V) + + Vsp = spcreator(V) + Msp_Vsp = Msp @ Vsp + assert isinstance(Msp_Vsp, np.ndarray) + assert Msp_Vsp.shape == () + + # output is 0-dim ndarray + assert_allclose(np.array(3), Msp_Vsp) + assert_allclose(np.array(3), Msp.toarray() @ Vsp) + assert_allclose(np.array(3), Msp @ Vsp.toarray()) + assert_allclose(np.array(3), Msp.toarray() @ Vsp.toarray()) + + # check error on matrix-scalar + with pytest.raises(ValueError, match='Scalar operands are not allowed'): + Msp @ 1 + with pytest.raises(ValueError, match='Scalar operands are not allowed'): + 1 @ Msp + + def test_sub_dense(self, spcreator, datsp_math_dtypes): + # subtracting a dense matrix to/from a sparse matrix + for dtype, dat, datsp in datsp_math_dtypes[spcreator]: + if dtype == np.dtype('bool'): + # boolean array subtraction deprecated in 1.9.0 + continue + + # Manually add to avoid upcasting from scalar + # multiplication. + sum1 = (dat + dat + dat) - datsp + assert_equal(sum1, dat + dat) + sum2 = (datsp + datsp + datsp) - dat + assert_equal(sum2, dat + dat) + + def test_size_zero_matrix_arithmetic(self, spcreator): + # Test basic matrix arithmetic with shapes like 0, (1, 0), (0, 3), etc. + mat = np.array([]) + a = mat.reshape(0) + d = mat.reshape((1, 0)) + f = np.ones([5, 5]) + + asp = spcreator(a) + dsp = spcreator(d) + # bad shape for addition + with pytest.raises(ValueError, match='inconsistent shapes'): + asp.__add__(dsp) + + # matrix product. + assert_equal(asp.dot(asp), np.dot(a, a)) + + # bad matrix products + with pytest.raises(ValueError, match='dimension mismatch'): + asp.dot(f) + + # elemente-wise multiplication + assert_equal(asp.multiply(asp).toarray(), np.multiply(a, a)) + + assert_equal(asp.multiply(a).toarray(), np.multiply(a, a)) + + assert_equal(asp.multiply(6).toarray(), np.multiply(a, 6)) + + # bad element-wise multiplication + with pytest.raises(ValueError, match='inconsistent shapes'): + asp.multiply(f) + + # Addition + assert_equal(asp.__add__(asp).toarray(), a.__add__(a)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_array_api.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_array_api.py new file mode 100644 index 0000000000000000000000000000000000000000..96ca22939df371c0d11b06fb06f2920c705f4578 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_array_api.py @@ -0,0 +1,561 @@ +import pytest +import numpy as np +import numpy.testing as npt +import scipy.sparse +import scipy.sparse.linalg as spla + + +sparray_types = ('bsr', 'coo', 'csc', 'csr', 'dia', 'dok', 'lil') + +sparray_classes = [ + getattr(scipy.sparse, f'{T}_array') for T in sparray_types +] + +A = np.array([ + [0, 1, 2, 0], + [2, 0, 0, 3], + [1, 4, 0, 0] +]) + +B = np.array([ + [0, 1], + [2, 0] +]) + +X = np.array([ + [1, 0, 0, 1], + [2, 1, 2, 0], + [0, 2, 1, 0], + [0, 0, 1, 2] +], dtype=float) + + +sparrays = [sparray(A) for sparray in sparray_classes] +square_sparrays = [sparray(B) for sparray in sparray_classes] +eig_sparrays = [sparray(X) for sparray in sparray_classes] + +parametrize_sparrays = pytest.mark.parametrize( + "A", sparrays, ids=sparray_types +) +parametrize_square_sparrays = pytest.mark.parametrize( + "B", square_sparrays, ids=sparray_types +) +parametrize_eig_sparrays = pytest.mark.parametrize( + "X", eig_sparrays, ids=sparray_types +) + + +@parametrize_sparrays +def test_sum(A): + assert not isinstance(A.sum(axis=0), np.matrix), \ + "Expected array, got matrix" + assert A.sum(axis=0).shape == (4,) + assert A.sum(axis=1).shape == (3,) + + +@parametrize_sparrays +def test_mean(A): + assert not isinstance(A.mean(axis=1), np.matrix), \ + "Expected array, got matrix" + + +@parametrize_sparrays +def test_min_max(A): + # Some formats don't support min/max operations, so we skip those here. + if hasattr(A, 'min'): + assert not isinstance(A.min(axis=1), np.matrix), \ + "Expected array, got matrix" + if hasattr(A, 'max'): + assert not isinstance(A.max(axis=1), np.matrix), \ + "Expected array, got matrix" + if hasattr(A, 'argmin'): + assert not isinstance(A.argmin(axis=1), np.matrix), \ + "Expected array, got matrix" + if hasattr(A, 'argmax'): + assert not isinstance(A.argmax(axis=1), np.matrix), \ + "Expected array, got matrix" + + +@parametrize_sparrays +def test_todense(A): + assert not isinstance(A.todense(), np.matrix), \ + "Expected array, got matrix" + + +@parametrize_sparrays +def test_indexing(A): + if A.__class__.__name__[:3] in ('dia', 'coo', 'bsr'): + return + + all_res = ( + A[1, :], + A[:, 1], + A[1, [1, 2]], + A[[1, 2], 1], + A[[0]], + A[:, [1, 2]], + A[[1, 2], :], + A[1, [[1, 2]]], + A[[[1, 2]], 1], + ) + + for res in all_res: + assert isinstance(res, scipy.sparse.sparray), \ + f"Expected sparse array, got {res._class__.__name__}" + + +@parametrize_sparrays +def test_dense_addition(A): + X = np.random.random(A.shape) + assert not isinstance(A + X, np.matrix), "Expected array, got matrix" + + +@parametrize_sparrays +def test_sparse_addition(A): + assert isinstance((A + A), scipy.sparse.sparray), "Expected array, got matrix" + + +@parametrize_sparrays +def test_elementwise_mul(A): + assert np.all((A * A).todense() == A.power(2).todense()) + + +@parametrize_sparrays +def test_elementwise_rmul(A): + with pytest.raises(TypeError): + None * A + + with pytest.raises(ValueError): + np.eye(3) * scipy.sparse.csr_array(np.arange(6).reshape(2, 3)) + + assert np.all((2 * A) == (A.todense() * 2)) + + assert np.all((A.todense() * A) == (A.todense() ** 2)) + + +@parametrize_sparrays +def test_matmul(A): + assert np.all((A @ A.T).todense() == A.dot(A.T).todense()) + + +@parametrize_sparrays +def test_power_operator(A): + assert isinstance((A**2), scipy.sparse.sparray), "Expected array, got matrix" + + # https://github.com/scipy/scipy/issues/15948 + npt.assert_equal((A**2).todense(), (A.todense())**2) + + # power of zero is all ones (dense) so helpful msg exception + with pytest.raises(NotImplementedError, match="zero power"): + A**0 + + +@parametrize_sparrays +def test_sparse_divide(A): + assert isinstance(A / A, np.ndarray) + +@parametrize_sparrays +@pytest.mark.thread_unsafe +def test_sparse_dense_divide(A): + with pytest.warns(RuntimeWarning): + assert isinstance((A / A.todense()), scipy.sparse.sparray) + +@parametrize_sparrays +def test_dense_divide(A): + assert isinstance((A / 2), scipy.sparse.sparray), "Expected array, got matrix" + + +@parametrize_sparrays +def test_no_A_attr(A): + with pytest.raises(AttributeError): + A.A + + +@parametrize_sparrays +def test_no_H_attr(A): + with pytest.raises(AttributeError): + A.H + + +@parametrize_sparrays +def test_getrow_getcol(A): + assert isinstance(A._getcol(0), scipy.sparse.sparray) + assert isinstance(A._getrow(0), scipy.sparse.sparray) + + +# -- linalg -- + +@parametrize_sparrays +def test_as_linearoperator(A): + L = spla.aslinearoperator(A) + npt.assert_allclose(L * [1, 2, 3, 4], A @ [1, 2, 3, 4]) + + +@parametrize_square_sparrays +def test_inv(B): + if B.__class__.__name__[:3] != 'csc': + return + + C = spla.inv(B) + + assert isinstance(C, scipy.sparse.sparray) + npt.assert_allclose(C.todense(), np.linalg.inv(B.todense())) + + +@parametrize_square_sparrays +def test_expm(B): + if B.__class__.__name__[:3] != 'csc': + return + + Bmat = scipy.sparse.csc_matrix(B) + + C = spla.expm(B) + + assert isinstance(C, scipy.sparse.sparray) + npt.assert_allclose( + C.todense(), + spla.expm(Bmat).todense() + ) + + +@parametrize_square_sparrays +def test_expm_multiply(B): + if B.__class__.__name__[:3] != 'csc': + return + + npt.assert_allclose( + spla.expm_multiply(B, np.array([1, 2])), + spla.expm(B) @ [1, 2] + ) + + +@parametrize_sparrays +def test_norm(A): + C = spla.norm(A) + npt.assert_allclose(C, np.linalg.norm(A.todense())) + + +@parametrize_square_sparrays +def test_onenormest(B): + C = spla.onenormest(B) + npt.assert_allclose(C, np.linalg.norm(B.todense(), 1)) + + +@parametrize_square_sparrays +def test_spsolve(B): + if B.__class__.__name__[:3] not in ('csc', 'csr'): + return + + npt.assert_allclose( + spla.spsolve(B, [1, 2]), + np.linalg.solve(B.todense(), [1, 2]) + ) + + +@pytest.mark.parametrize("fmt",["csr","csc"]) +def test_spsolve_triangular(fmt): + arr = [ + [1, 0, 0, 0], + [2, 1, 0, 0], + [3, 2, 1, 0], + [4, 3, 2, 1], + ] + if fmt == "csr": + X = scipy.sparse.csr_array(arr) + else: + X = scipy.sparse.csc_array(arr) + spla.spsolve_triangular(X, [1, 2, 3, 4]) + + +@parametrize_square_sparrays +def test_factorized(B): + if B.__class__.__name__[:3] != 'csc': + return + + LU = spla.factorized(B) + npt.assert_allclose( + LU(np.array([1, 2])), + np.linalg.solve(B.todense(), [1, 2]) + ) + + +@parametrize_square_sparrays +@pytest.mark.parametrize( + "solver", + ["bicg", "bicgstab", "cg", "cgs", "gmres", "lgmres", "minres", "qmr", + "gcrotmk", "tfqmr"] +) +def test_solvers(B, solver): + if solver == "minres": + kwargs = {} + else: + kwargs = {'atol': 1e-5} + + x, info = getattr(spla, solver)(B, np.array([1, 2]), **kwargs) + assert info >= 0 # no errors, even if perhaps did not converge fully + npt.assert_allclose(x, [1, 1], atol=1e-1) + + +@parametrize_sparrays +@pytest.mark.parametrize( + "solver", + ["lsqr", "lsmr"] +) +def test_lstsqr(A, solver): + x, *_ = getattr(spla, solver)(A, [1, 2, 3]) + npt.assert_allclose(A @ x, [1, 2, 3]) + + +@parametrize_eig_sparrays +def test_eigs(X): + e, v = spla.eigs(X, k=1) + npt.assert_allclose( + X @ v, + e[0] * v + ) + + +@parametrize_eig_sparrays +def test_eigsh(X): + X = X + X.T + e, v = spla.eigsh(X, k=1) + npt.assert_allclose( + X @ v, + e[0] * v + ) + + +@parametrize_eig_sparrays +def test_svds(X): + u, s, vh = spla.svds(X, k=3) + u2, s2, vh2 = np.linalg.svd(X.todense()) + s = np.sort(s) + s2 = np.sort(s2[:3]) + npt.assert_allclose(s, s2, atol=1e-3) + + +def test_splu(): + X = scipy.sparse.csc_array([ + [1, 0, 0, 0], + [2, 1, 0, 0], + [3, 2, 1, 0], + [4, 3, 2, 1], + ]) + LU = spla.splu(X) + npt.assert_allclose( + LU.solve(np.array([1, 2, 3, 4])), + np.asarray([1, 0, 0, 0], dtype=np.float64), + rtol=1e-14, atol=3e-16 + ) + + +def test_spilu(): + X = scipy.sparse.csc_array([ + [1, 0, 0, 0], + [2, 1, 0, 0], + [3, 2, 1, 0], + [4, 3, 2, 1], + ]) + LU = spla.spilu(X) + npt.assert_allclose( + LU.solve(np.array([1, 2, 3, 4])), + np.asarray([1, 0, 0, 0], dtype=np.float64), + rtol=1e-14, atol=3e-16 + ) + + +@pytest.mark.parametrize( + "cls,indices_attrs", + [ + ( + scipy.sparse.csr_array, + ["indices", "indptr"], + ), + ( + scipy.sparse.csc_array, + ["indices", "indptr"], + ), + ( + scipy.sparse.coo_array, + ["row", "col"], + ), + ] +) +@pytest.mark.parametrize("expected_dtype", [np.int64, np.int32]) +def test_index_dtype_compressed(cls, indices_attrs, expected_dtype): + input_array = scipy.sparse.coo_array(np.arange(9).reshape(3, 3)) + coo_tuple = ( + input_array.data, + ( + input_array.row.astype(expected_dtype), + input_array.col.astype(expected_dtype), + ) + ) + + result = cls(coo_tuple) + for attr in indices_attrs: + assert getattr(result, attr).dtype == expected_dtype + + result = cls(coo_tuple, shape=(3, 3)) + for attr in indices_attrs: + assert getattr(result, attr).dtype == expected_dtype + + if issubclass(cls, scipy.sparse._compressed._cs_matrix): + input_array_csr = input_array.tocsr() + csr_tuple = ( + input_array_csr.data, + input_array_csr.indices.astype(expected_dtype), + input_array_csr.indptr.astype(expected_dtype), + ) + + result = cls(csr_tuple) + for attr in indices_attrs: + assert getattr(result, attr).dtype == expected_dtype + + result = cls(csr_tuple, shape=(3, 3)) + for attr in indices_attrs: + assert getattr(result, attr).dtype == expected_dtype + + +def test_default_is_matrix_diags(): + m = scipy.sparse.diags([0, 1, 2]) + assert not isinstance(m, scipy.sparse.sparray) + + +def test_default_is_matrix_eye(): + m = scipy.sparse.eye(3) + assert not isinstance(m, scipy.sparse.sparray) + + +def test_default_is_matrix_spdiags(): + m = scipy.sparse.spdiags([1, 2, 3], 0, 3, 3) + assert not isinstance(m, scipy.sparse.sparray) + + +def test_default_is_matrix_identity(): + m = scipy.sparse.identity(3) + assert not isinstance(m, scipy.sparse.sparray) + + +def test_default_is_matrix_kron_dense(): + m = scipy.sparse.kron( + np.array([[1, 2], [3, 4]]), np.array([[4, 3], [2, 1]]) + ) + assert not isinstance(m, scipy.sparse.sparray) + + +def test_default_is_matrix_kron_sparse(): + m = scipy.sparse.kron( + np.array([[1, 2], [3, 4]]), np.array([[1, 0], [0, 0]]) + ) + assert not isinstance(m, scipy.sparse.sparray) + + +def test_default_is_matrix_kronsum(): + m = scipy.sparse.kronsum( + np.array([[1, 0], [0, 1]]), np.array([[0, 1], [1, 0]]) + ) + assert not isinstance(m, scipy.sparse.sparray) + + +def test_default_is_matrix_random(): + m = scipy.sparse.random(3, 3) + assert not isinstance(m, scipy.sparse.sparray) + + +def test_default_is_matrix_rand(): + m = scipy.sparse.rand(3, 3) + assert not isinstance(m, scipy.sparse.sparray) + + +@pytest.mark.parametrize("fn", (scipy.sparse.hstack, scipy.sparse.vstack)) +def test_default_is_matrix_stacks(fn): + """Same idea as `test_default_construction_fn_matrices`, but for the + stacking creation functions.""" + A = scipy.sparse.coo_matrix(np.eye(2)) + B = scipy.sparse.coo_matrix([[0, 1], [1, 0]]) + m = fn([A, B]) + assert not isinstance(m, scipy.sparse.sparray) + + +def test_blocks_default_construction_fn_matrices(): + """Same idea as `test_default_construction_fn_matrices`, but for the block + creation function""" + A = scipy.sparse.coo_matrix(np.eye(2)) + B = scipy.sparse.coo_matrix([[2], [0]]) + C = scipy.sparse.coo_matrix([[3]]) + + # block diag + m = scipy.sparse.block_diag((A, B, C)) + assert not isinstance(m, scipy.sparse.sparray) + + # bmat + m = scipy.sparse.bmat([[A, None], [None, C]]) + assert not isinstance(m, scipy.sparse.sparray) + + +def test_format_property(): + for fmt in sparray_types: + arr_cls = getattr(scipy.sparse, f"{fmt}_array") + M = arr_cls([[1, 2]]) + assert M.format == fmt + assert M._format == fmt + with pytest.raises(AttributeError): + M.format = "qqq" + + +def test_issparse(): + m = scipy.sparse.eye(3) + a = scipy.sparse.csr_array(m) + assert not isinstance(m, scipy.sparse.sparray) + assert isinstance(a, scipy.sparse.sparray) + + # Both sparse arrays and sparse matrices should be sparse + assert scipy.sparse.issparse(a) + assert scipy.sparse.issparse(m) + + # ndarray and array_likes are not sparse + assert not scipy.sparse.issparse(a.todense()) + assert not scipy.sparse.issparse(m.todense()) + + +def test_isspmatrix(): + m = scipy.sparse.eye(3) + a = scipy.sparse.csr_array(m) + assert not isinstance(m, scipy.sparse.sparray) + assert isinstance(a, scipy.sparse.sparray) + + # Should only be true for sparse matrices, not sparse arrays + assert not scipy.sparse.isspmatrix(a) + assert scipy.sparse.isspmatrix(m) + + # ndarray and array_likes are not sparse + assert not scipy.sparse.isspmatrix(a.todense()) + assert not scipy.sparse.isspmatrix(m.todense()) + + +@pytest.mark.parametrize( + ("fmt", "fn"), + ( + ("bsr", scipy.sparse.isspmatrix_bsr), + ("coo", scipy.sparse.isspmatrix_coo), + ("csc", scipy.sparse.isspmatrix_csc), + ("csr", scipy.sparse.isspmatrix_csr), + ("dia", scipy.sparse.isspmatrix_dia), + ("dok", scipy.sparse.isspmatrix_dok), + ("lil", scipy.sparse.isspmatrix_lil), + ), +) +def test_isspmatrix_format(fmt, fn): + m = scipy.sparse.eye(3, format=fmt) + a = scipy.sparse.csr_array(m).asformat(fmt) + assert not isinstance(m, scipy.sparse.sparray) + assert isinstance(a, scipy.sparse.sparray) + + # Should only be true for sparse matrices, not sparse arrays + assert not fn(a) + assert fn(m) + + # ndarray and array_likes are not sparse + assert not fn(a.todense()) + assert not fn(m.todense()) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_base.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_base.py new file mode 100644 index 0000000000000000000000000000000000000000..5f7ead8b134dc2e09bb0cef8441cb2b75a5aee8b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_base.py @@ -0,0 +1,5695 @@ +# +# Authors: Travis Oliphant, Ed Schofield, Robert Cimrman, Nathan Bell, and others + +""" Test functions for sparse matrices. Each class in the "Matrix class +based tests" section become subclasses of the classes in the "Generic +tests" section. This is done by the functions in the "Tailored base +class for generic tests" section. + +""" + + +import contextlib +import functools +import operator +import platform +import itertools +import sys + +import pytest +from pytest import raises as assert_raises + +import numpy as np +from numpy import (arange, zeros, array, dot, asarray, + vstack, ndarray, transpose, diag, kron, inf, conjugate, + int8) + +import random +from numpy.testing import (assert_equal, assert_array_equal, + assert_array_almost_equal, assert_almost_equal, assert_, + assert_allclose, suppress_warnings) + +import scipy.linalg + +import scipy.sparse as sparse +from scipy.sparse import (csc_matrix, csr_matrix, dok_matrix, + coo_matrix, lil_matrix, dia_matrix, bsr_matrix, + csc_array, csr_array, dok_array, + coo_array, lil_array, dia_array, bsr_array, + eye, issparse, SparseEfficiencyWarning, sparray) +from scipy.sparse._base import _formats +from scipy.sparse._sputils import (supported_dtypes, isscalarlike, + get_index_dtype, asmatrix, matrix) +from scipy.sparse.linalg import splu, expm, inv + +from scipy._lib.decorator import decorator +from scipy._lib._util import ComplexWarning + +IS_COLAB = ('google.colab' in sys.modules) + + +def assert_in(member, collection, msg=None): + message = msg if msg is not None else f"{member!r} not found in {collection!r}" + assert_(member in collection, msg=message) + + +def assert_array_equal_dtype(x, y, **kwargs): + assert_(x.dtype == y.dtype) + assert_array_equal(x, y, **kwargs) + + +NON_ARRAY_BACKED_FORMATS = frozenset(['dok']) + +def sparse_may_share_memory(A, B): + # Checks if A and B have any numpy array sharing memory. + + def _underlying_arrays(x): + # Given any object (e.g. a sparse array), returns all numpy arrays + # stored in any attribute. + + arrays = [] + for a in x.__dict__.values(): + if isinstance(a, np.ndarray | np.generic): + arrays.append(a) + return arrays + + for a in _underlying_arrays(A): + for b in _underlying_arrays(B): + if np.may_share_memory(a, b): + return True + return False + + +sup_complex = suppress_warnings() +sup_complex.filter(ComplexWarning) + + +def with_64bit_maxval_limit(maxval_limit=None, random=False, fixed_dtype=None, + downcast_maxval=None, assert_32bit=False): + """ + Monkeypatch the maxval threshold at which scipy.sparse switches to + 64-bit index arrays, or make it (pseudo-)random. + + """ + if maxval_limit is None: + maxval_limit = np.int64(10) + else: + # Ensure we use numpy scalars rather than Python scalars (matters for + # NEP 50 casting rule changes) + maxval_limit = np.int64(maxval_limit) + + if assert_32bit: + def new_get_index_dtype(arrays=(), maxval=None, check_contents=False): + tp = get_index_dtype(arrays, maxval, check_contents) + assert_equal(np.iinfo(tp).max, np.iinfo(np.int32).max) + assert_(tp == np.int32 or tp == np.intc) + return tp + elif fixed_dtype is not None: + def new_get_index_dtype(arrays=(), maxval=None, check_contents=False): + return fixed_dtype + elif random: + counter = np.random.RandomState(seed=1234) + + def new_get_index_dtype(arrays=(), maxval=None, check_contents=False): + return (np.int32, np.int64)[counter.randint(2)] + else: + def new_get_index_dtype(arrays=(), maxval=None, check_contents=False): + dtype = np.int32 + if maxval is not None: + if maxval > maxval_limit: + dtype = np.int64 + for arr in arrays: + arr = np.asarray(arr) + if arr.dtype > np.int32: + if check_contents: + if arr.size == 0: + # a bigger type not needed + continue + elif np.issubdtype(arr.dtype, np.integer): + maxval = arr.max() + minval = arr.min() + if minval >= -maxval_limit and maxval <= maxval_limit: + # a bigger type not needed + continue + dtype = np.int64 + return dtype + + if downcast_maxval is not None: + def new_downcast_intp_index(arr): + if arr.max() > downcast_maxval: + raise AssertionError("downcast limited") + return arr.astype(np.intp) + + @decorator + def deco(func, *a, **kw): + backup = [] + modules = [scipy.sparse._bsr, scipy.sparse._coo, scipy.sparse._csc, + scipy.sparse._csr, scipy.sparse._dia, scipy.sparse._dok, + scipy.sparse._lil, scipy.sparse._sputils, + scipy.sparse._compressed, scipy.sparse._construct] + try: + for mod in modules: + backup.append((mod, 'get_index_dtype', + getattr(mod, 'get_index_dtype', None))) + setattr(mod, 'get_index_dtype', new_get_index_dtype) + if downcast_maxval is not None: + backup.append((mod, 'downcast_intp_index', + getattr(mod, 'downcast_intp_index', None))) + setattr(mod, 'downcast_intp_index', new_downcast_intp_index) + return func(*a, **kw) + finally: + for mod, name, oldfunc in backup: + if oldfunc is not None: + setattr(mod, name, oldfunc) + + return deco + + +def toarray(a): + if isinstance(a, np.ndarray) or isscalarlike(a): + return a + return a.toarray() + + +class BinopTester: + # Custom type to test binary operations on sparse matrices. + + def __add__(self, mat): + return "matrix on the right" + + def __mul__(self, mat): + return "matrix on the right" + + def __sub__(self, mat): + return "matrix on the right" + + def __radd__(self, mat): + return "matrix on the left" + + def __rmul__(self, mat): + return "matrix on the left" + + def __rsub__(self, mat): + return "matrix on the left" + + def __matmul__(self, mat): + return "matrix on the right" + + def __rmatmul__(self, mat): + return "matrix on the left" + +class BinopTester_with_shape: + # Custom type to test binary operations on sparse matrices + # with object which has shape attribute. + def __init__(self,shape): + self._shape = shape + + def shape(self): + return self._shape + + def ndim(self): + return len(self._shape) + + def __add__(self, mat): + return "matrix on the right" + + def __mul__(self, mat): + return "matrix on the right" + + def __sub__(self, mat): + return "matrix on the right" + + def __radd__(self, mat): + return "matrix on the left" + + def __rmul__(self, mat): + return "matrix on the left" + + def __rsub__(self, mat): + return "matrix on the left" + + def __matmul__(self, mat): + return "matrix on the right" + + def __rmatmul__(self, mat): + return "matrix on the left" + +class ComparisonTester: + # Custom type to test comparison operations on sparse matrices. + def __eq__(self, other): + return "eq" + + def __ne__(self, other): + return "ne" + + def __lt__(self, other): + return "lt" + + def __le__(self, other): + return "le" + + def __gt__(self, other): + return "gt" + + def __ge__(self, other): + return "ge" + + +#------------------------------------------------------------------------------ +# Generic tests +#------------------------------------------------------------------------------ + + +class _MatrixMixin: + """mixin to easily allow tests of both sparray and spmatrix""" + bsr_container = bsr_matrix + coo_container = coo_matrix + csc_container = csc_matrix + csr_container = csr_matrix + dia_container = dia_matrix + dok_container = dok_matrix + lil_container = lil_matrix + asdense = staticmethod(asmatrix) + + def test_getrow(self): + assert_array_equal(self.datsp.getrow(1).toarray(), self.dat[[1], :]) + assert_array_equal(self.datsp.getrow(-1).toarray(), self.dat[[-1], :]) + + def test_getcol(self): + assert_array_equal(self.datsp.getcol(1).toarray(), self.dat[:, [1]]) + assert_array_equal(self.datsp.getcol(-1).toarray(), self.dat[:, [-1]]) + + def test_asfptype(self): + A = self.spcreator(arange(6,dtype='int32').reshape(2,3)) + + assert_equal(A.asfptype().dtype, np.dtype('float64')) + assert_equal(A.asfptype().format, A.format) + assert_equal(A.astype('int16').asfptype().dtype, np.dtype('float32')) + assert_equal(A.astype('complex128').asfptype().dtype, np.dtype('complex128')) + + B = A.asfptype() + C = B.asfptype() + assert_(B is C) + + +# TODO test prune +# TODO test has_sorted_indices +class _TestCommon: + """test common functionality shared by all sparse formats""" + math_dtypes = supported_dtypes + + bsr_container = bsr_array + coo_container = coo_array + csc_container = csc_array + csr_container = csr_array + dia_container = dia_array + dok_container = dok_array + lil_container = lil_array + asdense = array + + @classmethod + def init_class(cls): + # Canonical data. + cls.dat = array([[1, 0, 0, 2], [3, 0, 1, 0], [0, 2, 0, 0]], 'd') + cls.datsp = cls.spcreator(cls.dat) + + # Some sparse and dense matrices with data for every supported dtype. + # This set union is a workaround for numpy#6295, which means that + # two np.int64 dtypes don't hash to the same value. + cls.checked_dtypes = set(supported_dtypes).union(cls.math_dtypes) + cls.dat_dtypes = {} + cls.datsp_dtypes = {} + for dtype in cls.checked_dtypes: + cls.dat_dtypes[dtype] = cls.dat.astype(dtype) + cls.datsp_dtypes[dtype] = cls.spcreator(cls.dat.astype(dtype)) + + # Check that the original data is equivalent to the + # corresponding dat_dtypes & datsp_dtypes. + assert_equal(cls.dat, cls.dat_dtypes[np.float64]) + assert_equal(cls.datsp.toarray(), + cls.datsp_dtypes[np.float64].toarray()) + + cls.is_array_test = isinstance(cls.datsp, sparray) + + def test_bool(self): + def check(dtype): + datsp = self.datsp_dtypes[dtype] + + assert_raises(ValueError, bool, datsp) + assert_(self.spcreator([[1]])) + assert_(not self.spcreator([[0]])) + + if isinstance(self, TestDOK): + pytest.skip("Cannot create a rank <= 2 DOK matrix.") + for dtype in self.checked_dtypes: + check(dtype) + + def test_bool_rollover(self): + # bool's underlying dtype is 1 byte, check that it does not + # rollover True -> False at 256. + dat = array([[True, False]]) + datsp = self.spcreator(dat) + + for _ in range(10): + datsp = datsp + datsp + dat = dat + dat + assert_array_equal(dat, datsp.toarray()) + + def test_eq(self): + sup = suppress_warnings() + sup.filter(SparseEfficiencyWarning) + + @sup + @sup_complex + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + dat2 = dat.copy() + dat2[:,0] = 0 + datsp2 = self.spcreator(dat2) + datbsr = self.bsr_container(dat) + datcsr = self.csr_container(dat) + datcsc = self.csc_container(dat) + datlil = self.lil_container(dat) + + # sparse/sparse + assert_array_equal_dtype(dat == dat2, (datsp == datsp2).toarray()) + # mix sparse types + assert_array_equal_dtype(dat == dat2, (datbsr == datsp2).toarray()) + assert_array_equal_dtype(dat == dat2, (datcsr == datsp2).toarray()) + assert_array_equal_dtype(dat == dat2, (datcsc == datsp2).toarray()) + assert_array_equal_dtype(dat == dat2, (datlil == datsp2).toarray()) + # sparse/dense + assert_array_equal_dtype(dat == datsp2, datsp2 == dat) + # sparse/scalar + assert_array_equal_dtype(dat == 0, (datsp == 0).toarray()) + assert_array_equal_dtype(dat == 1, (datsp == 1).toarray()) + assert_array_equal_dtype(dat == np.nan, + (datsp == np.nan).toarray()) + + if self.datsp.format not in ['bsr', 'csc', 'csr']: + pytest.skip("Bool comparisons only implemented for BSR, CSC, and CSR.") + for dtype in self.checked_dtypes: + check(dtype) + + def test_ne(self): + sup = suppress_warnings() + sup.filter(SparseEfficiencyWarning) + + @sup + @sup_complex + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + dat2 = dat.copy() + dat2[:,0] = 0 + datsp2 = self.spcreator(dat2) + datbsr = self.bsr_container(dat) + datcsc = self.csc_container(dat) + datcsr = self.csr_container(dat) + datlil = self.lil_container(dat) + + # sparse/sparse + assert_array_equal_dtype(dat != dat2, (datsp != datsp2).toarray()) + # mix sparse types + assert_array_equal_dtype(dat != dat2, (datbsr != datsp2).toarray()) + assert_array_equal_dtype(dat != dat2, (datcsc != datsp2).toarray()) + assert_array_equal_dtype(dat != dat2, (datcsr != datsp2).toarray()) + assert_array_equal_dtype(dat != dat2, (datlil != datsp2).toarray()) + # sparse/dense + assert_array_equal_dtype(dat != datsp2, datsp2 != dat) + # sparse/scalar + assert_array_equal_dtype(dat != 0, (datsp != 0).toarray()) + assert_array_equal_dtype(dat != 1, (datsp != 1).toarray()) + assert_array_equal_dtype(0 != dat, (0 != datsp).toarray()) + assert_array_equal_dtype(1 != dat, (1 != datsp).toarray()) + assert_array_equal_dtype(dat != np.nan, + (datsp != np.nan).toarray()) + + if self.datsp.format not in ['bsr', 'csc', 'csr']: + pytest.skip("Bool comparisons only implemented for BSR, CSC, and CSR.") + for dtype in self.checked_dtypes: + check(dtype) + + def test_lt(self): + sup = suppress_warnings() + sup.filter(SparseEfficiencyWarning) + + @sup + @sup_complex + def check(dtype): + # data + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + dat2 = dat.copy() + dat2[:,0] = 0 + datsp2 = self.spcreator(dat2) + datcomplex = dat.astype(complex) + datcomplex[:,0] = 1 + 1j + datspcomplex = self.spcreator(datcomplex) + datbsr = self.bsr_container(dat) + datcsc = self.csc_container(dat) + datcsr = self.csr_container(dat) + datlil = self.lil_container(dat) + + # sparse/sparse + assert_array_equal_dtype(dat < dat2, (datsp < datsp2).toarray()) + assert_array_equal_dtype(datcomplex < dat2, + (datspcomplex < datsp2).toarray()) + # mix sparse types + assert_array_equal_dtype(dat < dat2, (datbsr < datsp2).toarray()) + assert_array_equal_dtype(dat < dat2, (datcsc < datsp2).toarray()) + assert_array_equal_dtype(dat < dat2, (datcsr < datsp2).toarray()) + assert_array_equal_dtype(dat < dat2, (datlil < datsp2).toarray()) + + assert_array_equal_dtype(dat2 < dat, (datsp2 < datbsr).toarray()) + assert_array_equal_dtype(dat2 < dat, (datsp2 < datcsc).toarray()) + assert_array_equal_dtype(dat2 < dat, (datsp2 < datcsr).toarray()) + assert_array_equal_dtype(dat2 < dat, (datsp2 < datlil).toarray()) + # sparse/dense + assert_array_equal_dtype(dat < dat2, datsp < dat2) + assert_array_equal_dtype(datcomplex < dat2, datspcomplex < dat2) + # sparse/scalar + for val in [2, 1, 0, -1, -2]: + val = np.int64(val) # avoid Python scalar (due to NEP 50 changes) + assert_array_equal_dtype((datsp < val).toarray(), dat < val) + assert_array_equal_dtype((val < datsp).toarray(), val < dat) + + with np.errstate(invalid='ignore'): + assert_array_equal_dtype((datsp < np.nan).toarray(), + dat < np.nan) + + # data + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + dat2 = dat.copy() + dat2[:,0] = 0 + datsp2 = self.spcreator(dat2) + + # dense rhs + assert_array_equal_dtype(dat < datsp2, datsp < dat2) + + if self.datsp.format not in ['bsr', 'csc', 'csr']: + pytest.skip("Bool comparisons only implemented for BSR, CSC, and CSR.") + for dtype in self.checked_dtypes: + check(dtype) + + def test_gt(self): + sup = suppress_warnings() + sup.filter(SparseEfficiencyWarning) + + @sup + @sup_complex + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + dat2 = dat.copy() + dat2[:,0] = 0 + datsp2 = self.spcreator(dat2) + datcomplex = dat.astype(complex) + datcomplex[:,0] = 1 + 1j + datspcomplex = self.spcreator(datcomplex) + datbsr = self.bsr_container(dat) + datcsc = self.csc_container(dat) + datcsr = self.csr_container(dat) + datlil = self.lil_container(dat) + + # sparse/sparse + assert_array_equal_dtype(dat > dat2, (datsp > datsp2).toarray()) + assert_array_equal_dtype(datcomplex > dat2, + (datspcomplex > datsp2).toarray()) + # mix sparse types + assert_array_equal_dtype(dat > dat2, (datbsr > datsp2).toarray()) + assert_array_equal_dtype(dat > dat2, (datcsc > datsp2).toarray()) + assert_array_equal_dtype(dat > dat2, (datcsr > datsp2).toarray()) + assert_array_equal_dtype(dat > dat2, (datlil > datsp2).toarray()) + + assert_array_equal_dtype(dat2 > dat, (datsp2 > datbsr).toarray()) + assert_array_equal_dtype(dat2 > dat, (datsp2 > datcsc).toarray()) + assert_array_equal_dtype(dat2 > dat, (datsp2 > datcsr).toarray()) + assert_array_equal_dtype(dat2 > dat, (datsp2 > datlil).toarray()) + # sparse/dense + assert_array_equal_dtype(dat > dat2, datsp > dat2) + assert_array_equal_dtype(datcomplex > dat2, datspcomplex > dat2) + # sparse/scalar + for val in [2, 1, 0, -1, -2]: + val = np.int64(val) # avoid Python scalar (due to NEP 50 changes) + assert_array_equal_dtype((datsp > val).toarray(), dat > val) + assert_array_equal_dtype((val > datsp).toarray(), val > dat) + + with np.errstate(invalid='ignore'): + assert_array_equal_dtype((datsp > np.nan).toarray(), + dat > np.nan) + + # data + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + dat2 = dat.copy() + dat2[:,0] = 0 + datsp2 = self.spcreator(dat2) + + # dense rhs + assert_array_equal_dtype(dat > datsp2, datsp > dat2) + + if self.datsp.format not in ['bsr', 'csc', 'csr']: + pytest.skip("Bool comparisons only implemented for BSR, CSC, and CSR.") + for dtype in self.checked_dtypes: + check(dtype) + + def test_le(self): + sup = suppress_warnings() + sup.filter(SparseEfficiencyWarning) + + @sup + @sup_complex + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + dat2 = dat.copy() + dat2[:,0] = 0 + datsp2 = self.spcreator(dat2) + datcomplex = dat.astype(complex) + datcomplex[:,0] = 1 + 1j + datspcomplex = self.spcreator(datcomplex) + datbsr = self.bsr_container(dat) + datcsc = self.csc_container(dat) + datcsr = self.csr_container(dat) + datlil = self.lil_container(dat) + + # sparse/sparse + assert_array_equal_dtype(dat <= dat2, (datsp <= datsp2).toarray()) + assert_array_equal_dtype(datcomplex <= dat2, + (datspcomplex <= datsp2).toarray()) + # mix sparse types + assert_array_equal_dtype((datbsr <= datsp2).toarray(), dat <= dat2) + assert_array_equal_dtype((datcsc <= datsp2).toarray(), dat <= dat2) + assert_array_equal_dtype((datcsr <= datsp2).toarray(), dat <= dat2) + assert_array_equal_dtype((datlil <= datsp2).toarray(), dat <= dat2) + + assert_array_equal_dtype((datsp2 <= datbsr).toarray(), dat2 <= dat) + assert_array_equal_dtype((datsp2 <= datcsc).toarray(), dat2 <= dat) + assert_array_equal_dtype((datsp2 <= datcsr).toarray(), dat2 <= dat) + assert_array_equal_dtype((datsp2 <= datlil).toarray(), dat2 <= dat) + # sparse/dense + assert_array_equal_dtype(datsp <= dat2, dat <= dat2) + assert_array_equal_dtype(datspcomplex <= dat2, datcomplex <= dat2) + # sparse/scalar + for val in [2, 1, -1, -2]: + val = np.int64(val) # avoid Python scalar (due to NEP 50 changes) + assert_array_equal_dtype((datsp <= val).toarray(), dat <= val) + assert_array_equal_dtype((val <= datsp).toarray(), val <= dat) + + # data + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + dat2 = dat.copy() + dat2[:,0] = 0 + datsp2 = self.spcreator(dat2) + + # dense rhs + assert_array_equal_dtype(dat <= datsp2, datsp <= dat2) + + if self.datsp.format not in ['bsr', 'csc', 'csr']: + pytest.skip("Bool comparisons only implemented for BSR, CSC, and CSR.") + for dtype in self.checked_dtypes: + check(dtype) + + def test_ge(self): + sup = suppress_warnings() + sup.filter(SparseEfficiencyWarning) + + @sup + @sup_complex + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + dat2 = dat.copy() + dat2[:,0] = 0 + datsp2 = self.spcreator(dat2) + datcomplex = dat.astype(complex) + datcomplex[:,0] = 1 + 1j + datspcomplex = self.spcreator(datcomplex) + datbsr = self.bsr_container(dat) + datcsc = self.csc_container(dat) + datcsr = self.csr_container(dat) + datlil = self.lil_container(dat) + + # sparse/sparse + assert_array_equal_dtype(dat >= dat2, (datsp >= datsp2).toarray()) + assert_array_equal_dtype(datcomplex >= dat2, + (datspcomplex >= datsp2).toarray()) + # mix sparse types + assert_array_equal_dtype((datbsr >= datsp2).toarray(), dat >= dat2) + assert_array_equal_dtype((datcsc >= datsp2).toarray(), dat >= dat2) + assert_array_equal_dtype((datcsr >= datsp2).toarray(), dat >= dat2) + assert_array_equal_dtype((datlil >= datsp2).toarray(), dat >= dat2) + + assert_array_equal_dtype((datsp2 >= datbsr).toarray(), dat2 >= dat) + assert_array_equal_dtype((datsp2 >= datcsc).toarray(), dat2 >= dat) + assert_array_equal_dtype((datsp2 >= datcsr).toarray(), dat2 >= dat) + assert_array_equal_dtype((datsp2 >= datlil).toarray(), dat2 >= dat) + # sparse/dense + assert_array_equal_dtype(datsp >= dat2, dat >= dat2) + assert_array_equal_dtype(datspcomplex >= dat2, datcomplex >= dat2) + # sparse/scalar + for val in [2, 1, -1, -2]: + val = np.int64(val) # avoid Python scalar (due to NEP 50 changes) + assert_array_equal_dtype((datsp >= val).toarray(), dat >= val) + assert_array_equal_dtype((val >= datsp).toarray(), val >= dat) + + # dense data + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + dat2 = dat.copy() + dat2[:,0] = 0 + datsp2 = self.spcreator(dat2) + + # dense rhs + assert_array_equal_dtype(dat >= datsp2, datsp >= dat2) + + if self.datsp.format not in ['bsr', 'csc', 'csr']: + pytest.skip("Bool comparisons only implemented for BSR, CSC, and CSR.") + for dtype in self.checked_dtypes: + check(dtype) + + def test_empty(self): + # create empty matrices + assert_equal(self.spcreator((3, 3)).toarray(), zeros((3, 3))) + assert_equal(self.spcreator((3, 3)).nnz, 0) + assert_equal(self.spcreator((3, 3)).count_nonzero(), 0) + if self.datsp.format in ["coo", "csr", "csc", "lil"]: + assert_equal(self.spcreator((3, 3)).count_nonzero(axis=0), array([0, 0, 0])) + + def test_count_nonzero(self): + axis_support = self.datsp.format in ["coo", "csr", "csc", "lil"] + axes = [None, 0, 1, -1, -2] if axis_support else [None] + + for A in (self.datsp, self.datsp.T): + for ax in axes: + expected = np.count_nonzero(A.toarray(), axis=ax) + assert_equal(A.count_nonzero(axis=ax), expected) + + if not axis_support: + with assert_raises(NotImplementedError, match="not implemented .* format"): + self.datsp.count_nonzero(axis=0) + + def test_invalid_shapes(self): + assert_raises(ValueError, self.spcreator, (-1,3)) + assert_raises(ValueError, self.spcreator, (3,-1)) + assert_raises(ValueError, self.spcreator, (-1,-1)) + + def test_repr(self): + datsp = self.spcreator([[1, 0, 0], [0, 0, 0], [0, 0, -2]]) + extra = ( + "(1 diagonals) " if datsp.format == "dia" + else "(blocksize=1x1) " if datsp.format == "bsr" + else "" + ) + _, fmt = _formats[datsp.format] + sparse_cls = "array" if self.is_array_test else "matrix" + expected = ( + f"<{fmt} sparse {sparse_cls} of dtype '{datsp.dtype}'\n" + f"\twith {datsp.nnz} stored elements {extra}and shape {datsp.shape}>" + ) + assert repr(datsp) == expected + + def test_str_maxprint(self): + datsp = self.spcreator(np.arange(75).reshape(5, 15)) + assert datsp.maxprint == 50 + assert len(str(datsp).split('\n')) == 51 + 3 + + dat = np.arange(15).reshape(5,3) + datsp = self.spcreator(dat) + # format dia reports nnz=15, but we want 14 + nnz_small = 14 if datsp.format == 'dia' else datsp.nnz + datsp_mp6 = self.spcreator(dat, maxprint=6) + + assert len(str(datsp).split('\n')) == nnz_small + 3 + assert len(str(datsp_mp6).split('\n')) == 6 + 4 + + # Check parameter `maxprint` is keyword only + datsp = self.spcreator(dat, shape=(5, 3), dtype='i', copy=False, maxprint=4) + datsp = self.spcreator(dat, (5, 3), 'i', False, maxprint=4) + with pytest.raises(TypeError, match="positional argument|unpack non-iterable"): + self.spcreator(dat, (5, 3), 'i', False, 4) + + def test_str(self): + datsp = self.spcreator([[1, 0, 0], [0, 0, 0], [0, 0, -2]]) + if datsp.nnz != 2: + return + extra = ( + "(1 diagonals) " if datsp.format == "dia" + else "(blocksize=1x1) " if datsp.format == "bsr" + else "" + ) + _, fmt = _formats[datsp.format] + sparse_cls = "array" if self.is_array_test else "matrix" + expected = ( + f"<{fmt} sparse {sparse_cls} of dtype '{datsp.dtype}'\n" + f"\twith {datsp.nnz} stored elements {extra}and shape {datsp.shape}>" + "\n Coords\tValues" + "\n (0, 0)\t1" + "\n (2, 2)\t-2" + ) + assert str(datsp) == expected + + def test_empty_arithmetic(self): + # Test manipulating empty matrices. Fails in SciPy SVN <= r1768 + shape = (5, 5) + for mytype in [np.dtype('int32'), np.dtype('float32'), + np.dtype('float64'), np.dtype('complex64'), + np.dtype('complex128')]: + a = self.spcreator(shape, dtype=mytype) + b = a + a + c = 2 * a + d = a @ a.tocsc() + e = a @ a.tocsr() + f = a @ a.tocoo() + for m in [a,b,c,d,e,f]: + assert_equal(m.toarray(), a.toarray()@a.toarray()) + # These fail in all revisions <= r1768: + assert_equal(m.dtype,mytype) + assert_equal(m.toarray().dtype,mytype) + + def test_abs(self): + A = array([[-1, 0, 17], [0, -5, 0], [1, -4, 0], [0, 0, 0]], 'd') + assert_equal(abs(A), abs(self.spcreator(A)).toarray()) + + def test_round(self): + decimal = 1 + A = array([[-1.35, 0.56], [17.25, -5.98]], 'd') + assert_equal(np.around(A, decimals=decimal), + round(self.spcreator(A), ndigits=decimal).toarray()) + + def test_elementwise_power(self): + A = array([[-4, -3, -2], [-1, 0, 1], [2, 3, 4]], 'd') + assert_equal(np.power(A, 2), self.spcreator(A).power(2).toarray()) + + #it's element-wise power function, input has to be a scalar + assert_raises(NotImplementedError, self.spcreator(A).power, A) + + def test_neg(self): + A = array([[-1, 0, 17], [0, -5, 0], [1, -4, 0], [0, 0, 0]], 'd') + assert_equal(-A, (-self.spcreator(A)).toarray()) + + # see gh-5843 + A = array([[True, False, False], [False, False, True]]) + assert_raises(NotImplementedError, self.spcreator(A).__neg__) + + def test_real(self): + D = array([[1 + 3j, 2 - 4j]]) + A = self.spcreator(D) + assert_equal(A.real.toarray(), D.real) + + def test_imag(self): + D = array([[1 + 3j, 2 - 4j]]) + A = self.spcreator(D) + assert_equal(A.imag.toarray(), D.imag) + + def test_diagonal(self): + # Does the matrix's .diagonal() method work? + mats = [] + mats.append([[1,0,2]]) + mats.append([[1],[0],[2]]) + mats.append([[0,1],[0,2],[0,3]]) + mats.append([[0,0,1],[0,0,2],[0,3,0]]) + mats.append([[1,0],[0,0]]) + + mats.append(kron(mats[0],[[1,2]])) + mats.append(kron(mats[0],[[1],[2]])) + mats.append(kron(mats[1],[[1,2],[3,4]])) + mats.append(kron(mats[2],[[1,2],[3,4]])) + mats.append(kron(mats[3],[[1,2],[3,4]])) + mats.append(kron(mats[3],[[1,2,3,4]])) + + for m in mats: + rows, cols = array(m).shape + sparse_mat = self.spcreator(m) + for k in range(-rows-1, cols+2): + assert_equal(sparse_mat.diagonal(k=k), diag(m, k=k)) + # Test for k beyond boundaries(issue #11949) + assert_equal(sparse_mat.diagonal(k=10), diag(m, k=10)) + assert_equal(sparse_mat.diagonal(k=-99), diag(m, k=-99)) + + # Test all-zero matrix. + assert_equal(self.spcreator((40, 16130)).diagonal(), np.zeros(40)) + # Test empty matrix + # https://github.com/scipy/scipy/issues/11949 + assert_equal(self.spcreator((0, 0)).diagonal(), np.empty(0)) + assert_equal(self.spcreator((15, 0)).diagonal(), np.empty(0)) + assert_equal(self.spcreator((0, 5)).diagonal(10), np.empty(0)) + + def test_trace(self): + # For square matrix + A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) + B = self.spcreator(A) + for k in range(-2, 3): + assert_equal(A.trace(offset=k), B.trace(offset=k)) + + # For rectangular matrix + A = np.array([[1, 2, 3], [4, 5, 6]]) + B = self.spcreator(A) + for k in range(-1, 3): + assert_equal(A.trace(offset=k), B.trace(offset=k)) + + def test_reshape(self): + x = self.spcreator([[1, 0, 7], [0, 0, 0], [0, 3, 0], [0, 0, 5]]) + for order in ['C', 'F']: + for s in [(12, 1), (1, 12)]: + assert_array_equal(x.reshape(s, order=order).toarray(), + x.toarray().reshape(s, order=order)) + + # This example is taken from the stackoverflow answer at + # https://stackoverflow.com/q/16511879 + x = self.spcreator([[0, 10, 0, 0], [0, 0, 0, 0], [0, 20, 30, 40]]) + y = x.reshape((2, 6)) # Default order is 'C' + desired = [[0, 10, 0, 0, 0, 0], [0, 0, 0, 20, 30, 40]] + assert_array_equal(y.toarray(), desired) + + # Reshape with negative indexes + y = x.reshape((2, -1)) + assert_array_equal(y.toarray(), desired) + y = x.reshape((-1, 6)) + assert_array_equal(y.toarray(), desired) + assert_raises(ValueError, x.reshape, (-1, -1)) + + # Reshape with star args + y = x.reshape(2, 6) + assert_array_equal(y.toarray(), desired) + assert_raises(TypeError, x.reshape, 2, 6, not_an_arg=1) + + # Reshape with same size is noop unless copy=True + y = x.reshape((3, 4)) + assert_(y is x) + y = x.reshape((3, 4), copy=True) + assert_(y is not x) + + # Ensure reshape did not alter original size + assert_array_equal(x.shape, (3, 4)) + + if self.is_array_test: + with assert_raises(AttributeError, match="has no setter|n't set attribute"): + x.shape = (2, 6) + else: # spmatrix test + # Reshape in place + x.shape = (2, 6) + assert_array_equal(x.toarray(), desired) + + # Reshape to bad ndim + assert_raises(ValueError, x.reshape, (x.size,)) + assert_raises(ValueError, x.reshape, (1, x.size, 1)) + + @pytest.mark.slow + def test_setdiag_comprehensive(self): + def dense_setdiag(a, v, k): + v = np.asarray(v) + if k >= 0: + n = min(a.shape[0], a.shape[1] - k) + if v.ndim != 0: + n = min(n, len(v)) + v = v[:n] + i = np.arange(0, n) + j = np.arange(k, k + n) + a[i,j] = v + elif k < 0: + dense_setdiag(a.T, v, -k) + + def check_setdiag(a, b, k): + # Check setting diagonal using a scalar, a vector of + # correct length, and too short or too long vectors + for r in [-1, len(np.diag(a, k)), 2, 30]: + if r < 0: + v = np.random.choice(range(1, 20)) + else: + v = np.random.randint(1, 20, size=r) + + dense_setdiag(a, v, k) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structu") + b.setdiag(v, k) + + # check that dense_setdiag worked + d = np.diag(a, k) + if np.asarray(v).ndim == 0: + assert_array_equal(d, v, err_msg="{msg} {r}") + else: + n = min(len(d), len(v)) + assert_array_equal(d[:n], v[:n], err_msg="{msg} {r}") + # check that sparse setdiag worked + assert_array_equal(b.toarray(), a, err_msg="{msg} {r}") + + # comprehensive test + np.random.seed(1234) + shapes = [(0,5), (5,0), (1,5), (5,1), (5,5)] + for dtype in [np.int8, np.float64]: + for m,n in shapes: + ks = np.arange(-m+1, n-1) + for k in ks: + a = np.zeros((m, n), dtype=dtype) + b = self.spcreator((m, n), dtype=dtype) + + check_setdiag(a, b, k) + + # check overwriting etc + for k2 in np.random.choice(ks, size=min(len(ks), 5)): + check_setdiag(a, b, k2) + + def test_setdiag(self): + # simple test cases + m = self.spcreator(np.eye(3)) + m2 = self.spcreator((4, 4)) + values = [3, 2, 1] + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + assert_raises(ValueError, m.setdiag, values, k=4) + m.setdiag(values) + assert_array_equal(m.diagonal(), values) + m.setdiag(values, k=1) + assert_array_equal(m.toarray(), np.array([[3, 3, 0], + [0, 2, 2], + [0, 0, 1]])) + m.setdiag(values, k=-2) + assert_array_equal(m.toarray(), np.array([[3, 3, 0], + [0, 2, 2], + [3, 0, 1]])) + m.setdiag((9,), k=2) + assert_array_equal(m.toarray()[0,2], 9) + m.setdiag((9,), k=-2) + assert_array_equal(m.toarray()[2,0], 9) + # test short values on an empty matrix + m2.setdiag([1], k=2) + assert_array_equal(m2.toarray()[0], [0, 0, 1, 0]) + # test overwriting that same diagonal + m2.setdiag([1, 1], k=2) + assert_array_equal(m2.toarray()[:2], [[0, 0, 1, 0], + [0, 0, 0, 1]]) + + def test_nonzero(self): + A = array([[1, 0, 1],[0, 1, 1],[0, 0, 1]]) + Asp = self.spcreator(A) + + A_nz = {tuple(ij) for ij in transpose(A.nonzero())} + Asp_nz = {tuple(ij) for ij in transpose(Asp.nonzero())} + + assert_equal(A_nz, Asp_nz) + + def test_numpy_nonzero(self): + # See gh-5987 + A = array([[1, 0, 1], [0, 1, 1], [0, 0, 1]]) + Asp = self.spcreator(A) + + A_nz = {tuple(ij) for ij in transpose(np.nonzero(A))} + Asp_nz = {tuple(ij) for ij in transpose(np.nonzero(Asp))} + + assert_equal(A_nz, Asp_nz) + + def test_sum(self): + np.random.seed(1234) + dat_1 = np.array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + dat_2 = np.random.rand(5, 5) + dat_3 = np.array([[]]) + dat_4 = np.zeros((40, 40)) + dat_5 = sparse.rand(5, 5, density=1e-2).toarray() + matrices = [dat_1, dat_2, dat_3, dat_4, dat_5] + + def check(dtype, j): + dat = self.asdense(matrices[j], dtype=dtype) + datsp = self.spcreator(dat, dtype=dtype) + with np.errstate(over='ignore'): + assert_array_almost_equal(dat.sum(), datsp.sum()) + assert_equal(dat.sum().dtype, datsp.sum().dtype) + assert_(np.isscalar(datsp.sum(axis=None))) + assert_array_almost_equal(dat.sum(axis=None), + datsp.sum(axis=None)) + assert_equal(dat.sum(axis=None).dtype, + datsp.sum(axis=None).dtype) + assert_array_almost_equal(dat.sum(axis=0), datsp.sum(axis=0)) + assert_equal(dat.sum(axis=0).dtype, datsp.sum(axis=0).dtype) + assert_array_almost_equal(dat.sum(axis=1), datsp.sum(axis=1)) + assert_equal(dat.sum(axis=1).dtype, datsp.sum(axis=1).dtype) + assert_array_almost_equal(dat.sum(axis=-2), datsp.sum(axis=-2)) + assert_equal(dat.sum(axis=-2).dtype, datsp.sum(axis=-2).dtype) + assert_array_almost_equal(dat.sum(axis=-1), datsp.sum(axis=-1)) + assert_equal(dat.sum(axis=-1).dtype, datsp.sum(axis=-1).dtype) + + for dtype in self.checked_dtypes: + for j in range(len(matrices)): + check(dtype, j) + + def test_sum_invalid_params(self): + out = np.zeros((1, 3)) + dat = array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + datsp = self.spcreator(dat) + + assert_raises(ValueError, datsp.sum, axis=3) + assert_raises(TypeError, datsp.sum, axis=(0, 1)) + assert_raises(TypeError, datsp.sum, axis=1.5) + assert_raises(ValueError, datsp.sum, axis=1, out=out) + + def test_sum_dtype(self): + dat = array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + datsp = self.spcreator(dat) + + def check(dtype): + dat_sum = dat.sum(dtype=dtype) + datsp_sum = datsp.sum(dtype=dtype) + + assert_array_almost_equal(dat_sum, datsp_sum) + assert_equal(dat_sum.dtype, datsp_sum.dtype) + + for dtype in self.checked_dtypes: + check(dtype) + + def test_sum_out(self): + keep = not self.is_array_test + dat = array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + datsp = self.spcreator(dat) + + dat_out = array(0) if self.is_array_test else array([[0]]) + datsp_out = array(0) if self.is_array_test else matrix([[0]]) + + dat.sum(out=dat_out, keepdims=keep) + datsp.sum(out=datsp_out) + assert_array_almost_equal(dat_out, datsp_out) + + dat_out = np.zeros((3,)) if self.is_array_test else np.zeros((3, 1)) + datsp_out = np.zeros((3,)) if self.is_array_test else matrix(np.zeros((3, 1))) + + dat.sum(axis=1, out=dat_out, keepdims=keep) + datsp.sum(axis=1, out=datsp_out) + assert_array_almost_equal(dat_out, datsp_out) + + # check that wrong shape out parameter raises + with assert_raises(ValueError, match="output parameter.*wrong.*dimension"): + datsp.sum(out=array([0])) + with assert_raises(ValueError, match="output parameter.*wrong.*dimension"): + datsp.sum(out=array([[0]] if self.is_array_test else 0)) + + def test_numpy_sum(self): + # See gh-5987 + dat = array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + datsp = self.spcreator(dat) + + dat_sum = np.sum(dat) + datsp_sum = np.sum(datsp) + + assert_array_almost_equal(dat_sum, datsp_sum) + assert_equal(dat_sum.dtype, datsp_sum.dtype) + + def test_mean(self): + keep = not self.is_array_test + def check(dtype): + dat = array([[0, 1, 2], + [3, 4, 5], + [6, 7, 9]], dtype=dtype) + datsp = self.spcreator(dat, dtype=dtype) + + assert_array_almost_equal(dat.mean(), datsp.mean()) + assert_equal(dat.mean().dtype, datsp.mean().dtype) + assert_(np.isscalar(datsp.mean(axis=None))) + assert_array_almost_equal( + dat.mean(axis=None, keepdims=keep), datsp.mean(axis=None) + ) + assert_equal(dat.mean(axis=None).dtype, datsp.mean(axis=None).dtype) + assert_array_almost_equal( + dat.mean(axis=0, keepdims=keep), datsp.mean(axis=0) + ) + assert_equal(dat.mean(axis=0).dtype, datsp.mean(axis=0).dtype) + assert_array_almost_equal( + dat.mean(axis=1, keepdims=keep), datsp.mean(axis=1) + ) + assert_equal(dat.mean(axis=1).dtype, datsp.mean(axis=1).dtype) + assert_array_almost_equal( + dat.mean(axis=-2, keepdims=keep), datsp.mean(axis=-2) + ) + assert_equal(dat.mean(axis=-2).dtype, datsp.mean(axis=-2).dtype) + assert_array_almost_equal( + dat.mean(axis=-1, keepdims=keep), datsp.mean(axis=-1) + ) + assert_equal(dat.mean(axis=-1).dtype, datsp.mean(axis=-1).dtype) + + for dtype in self.checked_dtypes: + check(dtype) + + def test_mean_invalid_params(self): + out = self.asdense(np.zeros((1, 3))) + dat = array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + datsp = self.spcreator(dat) + + assert_raises(ValueError, datsp.mean, axis=3) + assert_raises(TypeError, datsp.mean, axis=(0, 1)) + assert_raises(TypeError, datsp.mean, axis=1.5) + assert_raises(ValueError, datsp.mean, axis=1, out=out) + + def test_mean_dtype(self): + dat = array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + datsp = self.spcreator(dat) + + def check(dtype): + dat_mean = dat.mean(dtype=dtype) + datsp_mean = datsp.mean(dtype=dtype) + + assert_array_almost_equal(dat_mean, datsp_mean) + assert_equal(dat_mean.dtype, datsp_mean.dtype) + + for dtype in self.checked_dtypes: + check(dtype) + + def test_mean_out(self): + keep = not self.is_array_test + dat = array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + datsp = self.spcreator(dat) + + dat_out = array(0) if self.is_array_test else array([[0]]) + datsp_out = array(0) if self.is_array_test else matrix([[0]]) + + dat.mean(out=dat_out, keepdims=keep) + datsp.mean(out=datsp_out) + assert_array_almost_equal(dat_out, datsp_out) + + dat_out = np.zeros((3,)) if self.is_array_test else np.zeros((3, 1)) + datsp_out = np.zeros((3,)) if self.is_array_test else matrix(np.zeros((3, 1))) + + dat.mean(axis=1, out=dat_out, keepdims=keep) + datsp.mean(axis=1, out=datsp_out) + assert_array_almost_equal(dat_out, datsp_out) + + # check that wrong shape out parameter raises + with assert_raises(ValueError, match="output parameter.*wrong.*dimension"): + datsp.mean(out=array([0])) + with assert_raises(ValueError, match="output parameter.*wrong.*dimension"): + datsp.mean(out=array([[0]] if self.is_array_test else 0)) + + def test_numpy_mean(self): + # See gh-5987 + dat = array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + datsp = self.spcreator(dat) + + dat_mean = np.mean(dat) + datsp_mean = np.mean(datsp) + + assert_array_almost_equal(dat_mean, datsp_mean) + assert_equal(dat_mean.dtype, datsp_mean.dtype) + + def test_expm(self): + M = array([[1, 0, 2], [0, 0, 3], [-4, 5, 6]], float) + sM = self.spcreator(M, shape=(3,3), dtype=float) + Mexp = scipy.linalg.expm(M) + + N = array([[3., 0., 1.], [0., 2., 0.], [0., 0., 0.]]) + sN = self.spcreator(N, shape=(3,3), dtype=float) + Nexp = scipy.linalg.expm(N) + + with suppress_warnings() as sup: + sup.filter( + SparseEfficiencyWarning, + "splu converted its input to CSC format", + ) + sup.filter( + SparseEfficiencyWarning, + "spsolve is more efficient when sparse b is in the CSC matrix format", + ) + sup.filter( + SparseEfficiencyWarning, + "spsolve requires A be CSC or CSR matrix format", + ) + sMexp = expm(sM).toarray() + sNexp = expm(sN).toarray() + + assert_array_almost_equal((sMexp - Mexp), zeros((3, 3))) + assert_array_almost_equal((sNexp - Nexp), zeros((3, 3))) + + def test_inv(self): + def check(dtype): + M = array([[1, 0, 2], [0, 0, 3], [-4, 5, 6]], dtype) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, + "spsolve requires A be CSC or CSR matrix format",) + sup.filter(SparseEfficiencyWarning, + "spsolve is more efficient when sparse b " + "is in the CSC matrix format",) + sup.filter(SparseEfficiencyWarning, + "splu converted its input to CSC format",) + sM = self.spcreator(M, shape=(3,3), dtype=dtype) + sMinv = inv(sM) + assert_array_almost_equal(sMinv.dot(sM).toarray(), np.eye(3)) + assert_raises(TypeError, inv, M) + for dtype in [float]: + check(dtype) + + @sup_complex + def test_from_array(self): + A = array([[1,0,0],[2,3,4],[0,5,0],[0,0,0]]) + assert_array_equal(self.spcreator(A).toarray(), A) + + A = array([[1.0 + 3j, 0, 0], + [0, 2.0 + 5, 0], + [0, 0, 0]]) + assert_array_equal(self.spcreator(A).toarray(), A) + assert_array_equal(self.spcreator(A, dtype='int16').toarray(),A.astype('int16')) + + @sup_complex + def test_from_matrix(self): + A = self.asdense([[1, 0, 0], [2, 3, 4], [0, 5, 0], [0, 0, 0]]) + assert_array_equal(self.spcreator(A).todense(), A) + + A = self.asdense([[1.0 + 3j, 0, 0], + [0, 2.0 + 5, 0], + [0, 0, 0]]) + assert_array_equal(self.spcreator(A).todense(), A) + assert_array_equal( + self.spcreator(A, dtype='int16').todense(), A.astype('int16') + ) + + @sup_complex + def test_from_list(self): + A = [[1,0,0],[2,3,4],[0,5,0],[0,0,0]] + assert_array_equal(self.spcreator(A).toarray(), A) + + A = [[1.0 + 3j, 0, 0], + [0, 2.0 + 5, 0], + [0, 0, 0]] + assert_array_equal(self.spcreator(A).toarray(), array(A)) + assert_array_equal( + self.spcreator(A, dtype='int16').toarray(), array(A).astype('int16') + ) + + @sup_complex + def test_from_sparse(self): + D = array([[1,0,0],[2,3,4],[0,5,0],[0,0,0]]) + S = self.csr_container(D) + assert_array_equal(self.spcreator(S).toarray(), D) + S = self.spcreator(D) + assert_array_equal(self.spcreator(S).toarray(), D) + + D = array([[1.0 + 3j, 0, 0], + [0, 2.0 + 5, 0], + [0, 0, 0]]) + S = self.csr_container(D) + assert_array_equal(self.spcreator(S).toarray(), D) + assert_array_equal(self.spcreator(S, dtype='int16').toarray(), + D.astype('int16')) + S = self.spcreator(D) + assert_array_equal(self.spcreator(S).toarray(), D) + assert_array_equal(self.spcreator(S, dtype='int16').toarray(), + D.astype('int16')) + + # def test_array(self): + # """test array(A) where A is in sparse format""" + # assert_equal( array(self.datsp), self.dat ) + + def test_todense(self): + # Check C- or F-contiguous (default). + chk = self.datsp.todense() + assert isinstance(chk, np.ndarray if self.is_array_test else np.matrix) + assert_array_equal(chk, self.dat) + assert_(chk.flags.c_contiguous != chk.flags.f_contiguous) + # Check C-contiguous (with arg). + chk = self.datsp.todense(order='C') + assert_array_equal(chk, self.dat) + assert_(chk.flags.c_contiguous) + assert_(not chk.flags.f_contiguous) + # Check F-contiguous (with arg). + chk = self.datsp.todense(order='F') + assert_array_equal(chk, self.dat) + assert_(not chk.flags.c_contiguous) + assert_(chk.flags.f_contiguous) + # Check with out argument (array). + out = np.zeros(self.datsp.shape, dtype=self.datsp.dtype) + chk = self.datsp.todense(out=out) + assert_array_equal(self.dat, out) + assert_array_equal(self.dat, chk) + assert np.may_share_memory(chk, out) + # Check with out array (matrix). + out = self.asdense(np.zeros(self.datsp.shape, dtype=self.datsp.dtype)) + chk = self.datsp.todense(out=out) + assert_array_equal(self.dat, out) + assert_array_equal(self.dat, chk) + assert np.may_share_memory(chk, out) + a = array([[1.,2.,3.]]) + dense_dot_dense = a @ self.dat + check = a @ self.datsp.todense() + assert_array_equal(dense_dot_dense, check) + b = array([[1.,2.,3.,4.]]).T + dense_dot_dense = self.dat @ b + check2 = self.datsp.todense() @ b + assert_array_equal(dense_dot_dense, check2) + # Check bool data works. + spbool = self.spcreator(self.dat, dtype=bool) + matbool = self.dat.astype(bool) + assert_array_equal(spbool.todense(), matbool) + + def test_toarray(self): + # Check C- or F-contiguous (default). + dat = asarray(self.dat) + chk = self.datsp.toarray() + assert_array_equal(chk, dat) + assert_(chk.flags.c_contiguous != chk.flags.f_contiguous) + # Check C-contiguous (with arg). + chk = self.datsp.toarray(order='C') + assert_array_equal(chk, dat) + assert_(chk.flags.c_contiguous) + assert_(not chk.flags.f_contiguous) + # Check F-contiguous (with arg). + chk = self.datsp.toarray(order='F') + assert_array_equal(chk, dat) + assert_(not chk.flags.c_contiguous) + assert_(chk.flags.f_contiguous) + # Check with output arg. + out = np.zeros(self.datsp.shape, dtype=self.datsp.dtype) + self.datsp.toarray(out=out) + assert_array_equal(chk, dat) + # Check that things are fine when we don't initialize with zeros. + out[...] = 1. + self.datsp.toarray(out=out) + assert_array_equal(chk, dat) + a = array([1.,2.,3.]) + dense_dot_dense = dot(a, dat) + check = dot(a, self.datsp.toarray()) + assert_array_equal(dense_dot_dense, check) + b = array([1.,2.,3.,4.]) + dense_dot_dense = dot(dat, b) + check2 = dot(self.datsp.toarray(), b) + assert_array_equal(dense_dot_dense, check2) + # Check bool data works. + spbool = self.spcreator(self.dat, dtype=bool) + arrbool = dat.astype(bool) + assert_array_equal(spbool.toarray(), arrbool) + + @sup_complex + def test_astype(self): + D = array([[2.0 + 3j, 0, 0], + [0, 4.0 + 5j, 0], + [0, 0, 0]]) + S = self.spcreator(D) + + for x in supported_dtypes: + # Check correctly casted + D_casted = D.astype(x) + for copy in (True, False): + S_casted = S.astype(x, copy=copy) + assert_equal(S_casted.dtype, D_casted.dtype) # correct type + assert_equal(S_casted.toarray(), D_casted) # correct values + assert_equal(S_casted.format, S.format) # format preserved + # Check correctly copied + assert_(S_casted.astype(x, copy=False) is S_casted) + S_copied = S_casted.astype(x, copy=True) + assert_(S_copied is not S_casted) + + def check_equal_but_not_same_array_attribute(attribute): + a = getattr(S_casted, attribute) + b = getattr(S_copied, attribute) + assert_array_equal(a, b) + assert_(a is not b) + i = (0,) * b.ndim + b_i = b[i] + b[i] = not b[i] + assert_(a[i] != b[i]) + b[i] = b_i + + if S_casted.format in ('csr', 'csc', 'bsr'): + for attribute in ('indices', 'indptr', 'data'): + check_equal_but_not_same_array_attribute(attribute) + elif S_casted.format == 'coo': + for attribute in ('row', 'col', 'data'): + check_equal_but_not_same_array_attribute(attribute) + elif S_casted.format == 'dia': + for attribute in ('offsets', 'data'): + check_equal_but_not_same_array_attribute(attribute) + + @sup_complex + def test_astype_immutable(self): + D = array([[2.0 + 3j, 0, 0], + [0, 4.0 + 5j, 0], + [0, 0, 0]]) + S = self.spcreator(D) + if hasattr(S, 'data'): + S.data.flags.writeable = False + if S.format in ('csr', 'csc', 'bsr'): + S.indptr.flags.writeable = False + S.indices.flags.writeable = False + for x in supported_dtypes: + D_casted = D.astype(x) + S_casted = S.astype(x) + assert_equal(S_casted.dtype, D_casted.dtype) + + def test_mul_scalar(self): + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + assert_array_equal(dat*2, (datsp*2).toarray()) + assert_array_equal(dat*17.3, (datsp*17.3).toarray()) + + for dtype in self.math_dtypes: + check(dtype) + + def test_rmul_scalar(self): + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + assert_array_equal(2*dat, (2*datsp).toarray()) + assert_array_equal(17.3*dat, (17.3*datsp).toarray()) + + for dtype in self.math_dtypes: + check(dtype) + + # GitHub issue #15210 + def test_rmul_scalar_type_error(self): + datsp = self.datsp_dtypes[np.float64] + with assert_raises(TypeError): + None * datsp + + def test_add(self): + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + a = dat.copy() + a[0,2] = 2.0 + b = datsp + c = b + a + assert_array_equal(c, b.toarray() + a) + + c = b + b.tocsr() + assert_array_equal(c.toarray(), + b.toarray() + b.toarray()) + + # test broadcasting + c = b + a[0] + assert_array_equal(c, b.toarray() + a[0]) + + for dtype in self.math_dtypes: + check(dtype) + + def test_radd(self): + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + a = dat.copy() + a[0,2] = 2.0 + b = datsp + c = a + b + assert_array_equal(c, a + b.toarray()) + + for dtype in self.math_dtypes: + check(dtype) + + def test_sub(self): + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + assert_array_equal((datsp - datsp).toarray(), np.zeros((3, 4))) + assert_array_equal((datsp - 0).toarray(), dat) + + A = self.spcreator( + np.array([[1, 0, 0, 4], [-1, 0, 0, 0], [0, 8, 0, -5]], 'd') + ) + assert_array_equal((datsp - A).toarray(), dat - A.toarray()) + assert_array_equal((A - datsp).toarray(), A.toarray() - dat) + + # test broadcasting + assert_array_equal(datsp - dat[0], dat - dat[0]) + + for dtype in self.math_dtypes: + if dtype == np.dtype('bool'): + # boolean array subtraction deprecated in 1.9.0 + continue + + check(dtype) + + def test_rsub(self): + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + assert_array_equal((dat - datsp),[[0,0,0,0],[0,0,0,0],[0,0,0,0]]) + assert_array_equal((datsp - dat),[[0,0,0,0],[0,0,0,0],[0,0,0,0]]) + assert_array_equal((0 - datsp).toarray(), -dat) + + A = self.spcreator([[1,0,0,4],[-1,0,0,0],[0,8,0,-5]],dtype='d') + assert_array_equal((dat - A), dat - A.toarray()) + assert_array_equal((A - dat), A.toarray() - dat) + assert_array_equal(A.toarray() - datsp, A.toarray() - dat) + assert_array_equal(datsp - A.toarray(), dat - A.toarray()) + + # test broadcasting + assert_array_equal(dat[0] - datsp, dat[0] - dat) + + for dtype in self.math_dtypes: + if dtype == np.dtype('bool'): + # boolean array subtraction deprecated in 1.9.0 + continue + + check(dtype) + + def test_add0(self): + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + # Adding 0 to a sparse matrix + assert_array_equal((datsp + 0).toarray(), dat) + # use sum (which takes 0 as a starting value) + sumS = sum([k * datsp for k in range(1, 3)]) + sumD = sum([k * dat for k in range(1, 3)]) + assert_almost_equal(sumS.toarray(), sumD) + + for dtype in self.math_dtypes: + check(dtype) + + def test_elementwise_multiply(self): + # real/real + A = array([[4,0,9],[2,-3,5]]) + B = array([[0,7,0],[0,-4,0]]) + Asp = self.spcreator(A) + Bsp = self.spcreator(B) + assert_almost_equal(Asp.multiply(Bsp).toarray(), A*B) # sparse/sparse + assert_almost_equal(Asp.multiply(B).toarray(), A*B) # sparse/dense + + # complex/complex + C = array([[1-2j,0+5j,-1+0j],[4-3j,-3+6j,5]]) + D = array([[5+2j,7-3j,-2+1j],[0-1j,-4+2j,9]]) + Csp = self.spcreator(C) + Dsp = self.spcreator(D) + assert_almost_equal(Csp.multiply(Dsp).toarray(), C*D) # sparse/sparse + assert_almost_equal(Csp.multiply(D).toarray(), C*D) # sparse/dense + + # real/complex + assert_almost_equal(Asp.multiply(Dsp).toarray(), A*D) # sparse/sparse + assert_almost_equal(Asp.multiply(D).toarray(), A*D) # sparse/dense + + def test_elementwise_multiply_broadcast(self): + A = array([4]) + B = array([[-9]]) + C = array([1,-1,0]) + D = array([[7,9,-9]]) + E = array([[3],[2],[1]]) + F = array([[8,6,3],[-4,3,2],[6,6,6]]) + G = [1, 2, 3] + H = np.ones((3, 4)) + J = H.T + K = array([[0]]) + L = array([[[1,2],[0,1]]]) + + # Some arrays can't be cast as spmatrices (A,C,L) so leave + # them out. + Bsp = self.spcreator(B) + Dsp = self.spcreator(D) + Esp = self.spcreator(E) + Fsp = self.spcreator(F) + Hsp = self.spcreator(H) + Hspp = self.spcreator(H[0,None]) + Jsp = self.spcreator(J) + Jspp = self.spcreator(J[:,0,None]) + Ksp = self.spcreator(K) + + matrices = [A, B, C, D, E, F, G, H, J, K, L] + spmatrices = [Bsp, Dsp, Esp, Fsp, Hsp, Hspp, Jsp, Jspp, Ksp] + + # sparse/sparse + for i in spmatrices: + for j in spmatrices: + try: + dense_mult = i.toarray() * j.toarray() + except ValueError: + assert_raises(ValueError, i.multiply, j) + continue + sp_mult = i.multiply(j) + assert_almost_equal(sp_mult.toarray(), dense_mult) + + # sparse/dense + for i in spmatrices: + for j in matrices: + try: + dense_mult = i.toarray() * j + except TypeError: + continue + except ValueError: + assert_raises(ValueError, i.multiply, j) + continue + sp_mult = i.multiply(j) + if issparse(sp_mult): + assert_almost_equal(sp_mult.toarray(), dense_mult) + else: + assert_almost_equal(sp_mult, dense_mult) + + def test_elementwise_divide(self): + expected = [[1,np.nan,np.nan,1], + [1,np.nan,1,np.nan], + [np.nan,1,np.nan,np.nan]] + assert_array_equal(toarray(self.datsp / self.datsp), expected) + + denom = self.spcreator([[1,0,0,4],[-1,0,0,0],[0,8,0,-5]],dtype='d') + expected = [[1,np.nan,np.nan,0.5], + [-3,np.nan,inf,np.nan], + [np.nan,0.25,np.nan,0]] + assert_array_equal(toarray(self.datsp / denom), expected) + + # complex + A = array([[1-2j,0+5j,-1+0j],[4-3j,-3+6j,5]]) + B = array([[5+2j,7-3j,-2+1j],[0-1j,-4+2j,9]]) + Asp = self.spcreator(A) + Bsp = self.spcreator(B) + assert_almost_equal(toarray(Asp / Bsp), A/B) + + # integer + A = array([[1,2,3],[-3,2,1]]) + B = array([[0,1,2],[0,-2,3]]) + Asp = self.spcreator(A) + Bsp = self.spcreator(B) + with np.errstate(divide='ignore'): + assert_array_equal(toarray(Asp / Bsp), A / B) + + # mismatching sparsity patterns + A = array([[0,1],[1,0]]) + B = array([[1,0],[1,0]]) + Asp = self.spcreator(A) + Bsp = self.spcreator(B) + with np.errstate(divide='ignore', invalid='ignore'): + assert_array_equal(np.array(toarray(Asp / Bsp)), A / B) + + def test_pow(self): + A = array([[1, 0, 2, 0], [0, 3, 4, 0], [0, 5, 0, 0], [0, 6, 7, 8]]) + B = self.spcreator(A) + + if self.is_array_test: # sparrays use element-wise power + # Todo: Add 1+3j to tested exponent list when np1.24 is no longer supported + # Complex exponents of 0 (our implicit fill value) change in numpy-1.25 + # from `(nan+nanj)` to `0`. Old value makes array element-wise result + # dense and is hard to check for without any `isnan` method. + # So while untested here, element-wise complex exponents work with np>=1.25. + # for exponent in [1, 2, 2.2, 3, 1+3j]: + for exponent in [1, 2, 2.2, 3]: + ret_sp = B**exponent + ret_np = A**exponent + assert_array_equal(ret_sp.toarray(), ret_np) + assert_equal(ret_sp.dtype, ret_np.dtype) + + # invalid exponents + assert_raises(NotImplementedError, B.__pow__, 0) + assert_raises(ValueError, B.__pow__, -1) + + # nonsquare matrix + B = self.spcreator(A[:3,:]) + assert_equal((B**1).toarray(), B.toarray()) + else: # test sparse matrix. spmatrices use matrix multiplicative power + for exponent in [0,1,2,3]: + ret_sp = B**exponent + ret_np = np.linalg.matrix_power(A, exponent) + assert_array_equal(ret_sp.toarray(), ret_np) + assert_equal(ret_sp.dtype, ret_np.dtype) + + # invalid exponents + for exponent in [-1, 2.2, 1 + 3j]: + assert_raises(ValueError, B.__pow__, exponent) + + # nonsquare matrix + B = self.spcreator(A[:3,:]) + assert_raises(TypeError, B.__pow__, 1) + + def test_rmatvec(self): + M = self.spcreator([[3,0,0],[0,1,0],[2,0,3.0],[2,3,0]]) + assert_array_almost_equal([1,2,3,4] @ M, dot([1,2,3,4], M.toarray())) + row = array([[1,2,3,4]]) + assert_array_almost_equal(row @ M, row @ M.toarray()) + + def test_small_multiplication(self): + # test that A*x works for x with shape () (1,) (1,1) and (1,0) + A = self.spcreator([[1],[2],[3]]) + + assert_(issparse(A * array(1))) + assert_equal((A * array(1)).toarray(), [[1], [2], [3]]) + + assert_equal(A @ array([1]), array([1, 2, 3])) + assert_equal(A @ array([[1]]), array([[1], [2], [3]])) + assert_equal(A @ np.ones((1, 1)), array([[1], [2], [3]])) + assert_equal(A @ np.ones((1, 0)), np.ones((3, 0))) + + def test_star_vs_at_sign_for_sparray_and_spmatrix(self): + # test that * is matmul for spmatrix and mul for sparray + A = np.array([[1], [2], [3]]) + Asp = self.spcreator(A) + + if self.is_array_test: + assert_array_almost_equal((Asp * np.ones((3, 1))).toarray(), A) + assert_array_almost_equal((Asp * array([[1]])).toarray(), A) + else: + assert_equal(Asp * array([1]), array([1, 2, 3])) + assert_equal(Asp * array([[1]]), array([[1], [2], [3]])) + assert_equal(Asp * np.ones((1, 0)), np.ones((3, 0))) + + def test_binop_custom_type(self): + # Non-regression test: previously, binary operations would raise + # NotImplementedError instead of returning NotImplemented + # (https://docs.python.org/library/constants.html#NotImplemented) + # so overloading Custom + matrix etc. didn't work. + A = self.spcreator([[1], [2], [3]]) + B = BinopTester() + assert_equal(A + B, "matrix on the left") + assert_equal(A - B, "matrix on the left") + assert_equal(A * B, "matrix on the left") + assert_equal(B + A, "matrix on the right") + assert_equal(B - A, "matrix on the right") + assert_equal(B * A, "matrix on the right") + + assert_equal(A @ B, "matrix on the left") + assert_equal(B @ A, "matrix on the right") + + def test_binop_custom_type_with_shape(self): + A = self.spcreator([[1], [2], [3]]) + B = BinopTester_with_shape((3,1)) + assert_equal(A + B, "matrix on the left") + assert_equal(A - B, "matrix on the left") + assert_equal(A * B, "matrix on the left") + assert_equal(B + A, "matrix on the right") + assert_equal(B - A, "matrix on the right") + assert_equal(B * A, "matrix on the right") + + assert_equal(A @ B, "matrix on the left") + assert_equal(B @ A, "matrix on the right") + + def test_mul_custom_type(self): + class Custom: + def __init__(self, scalar): + self.scalar = scalar + + def __rmul__(self, other): + return other * self.scalar + + scalar = 2 + A = self.spcreator([[1],[2],[3]]) + c = Custom(scalar) + A_scalar = A * scalar + A_c = A * c + assert_array_equal_dtype(A_scalar.toarray(), A_c.toarray()) + assert_equal(A_scalar.format, A_c.format) + + def test_comparisons_custom_type(self): + A = self.spcreator([[1], [2], [3]]) + B = ComparisonTester() + assert_equal(A == B, "eq") + assert_equal(A != B, "ne") + assert_equal(A > B, "lt") + assert_equal(A >= B, "le") + assert_equal(A < B, "gt") + assert_equal(A <= B, "ge") + + def test_dot_scalar(self): + M = self.spcreator(array([[3,0,0],[0,1,0],[2,0,3.0],[2,3,0]])) + scalar = 10 + actual = M.dot(scalar) + expected = M * scalar + + assert_allclose(actual.toarray(), expected.toarray()) + + def test_matmul(self): + M = self.spcreator(array([[3,0,0],[0,1,0],[2,0,3.0],[2,3,0]])) + B = self.spcreator(array([[0,1],[1,0],[0,2]],'d')) + col = array([[1,2,3]]).T + + matmul = operator.matmul + # check matrix-vector + assert_array_almost_equal(matmul(M, col), M.toarray() @ col) + + # check matrix-matrix + assert_array_almost_equal(matmul(M, B).toarray(), (M @ B).toarray()) + assert_array_almost_equal(matmul(M.toarray(), B), (M @ B).toarray()) + assert_array_almost_equal(matmul(M, B.toarray()), (M @ B).toarray()) + + # check error on matrix-scalar + assert_raises(ValueError, matmul, M, 1) + assert_raises(ValueError, matmul, 1, M) + + def test_matvec(self): + M = self.spcreator([[3,0,0],[0,1,0],[2,0,3.0],[2,3,0]]) + col = array([[1,2,3]]).T + + assert_array_almost_equal(M @ col, M.toarray() @ col) + + # check result dimensions (ticket #514) + assert_equal((M @ array([1,2,3])).shape,(4,)) + assert_equal((M @ array([[1],[2],[3]])).shape,(4,1)) + assert_equal((M @ matrix([[1],[2],[3]])).shape,(4,1)) + + # check result type + assert_(isinstance(M @ array([1,2,3]), ndarray)) + matrix_or_array = ndarray if self.is_array_test else np.matrix + assert_(isinstance(M @ matrix([1,2,3]).T, matrix_or_array)) + + # ensure exception is raised for improper dimensions + bad_vecs = [array([1,2]), array([1,2,3,4]), array([[1],[2]]), + matrix([1,2,3]), matrix([[1],[2]])] + for x in bad_vecs: + assert_raises(ValueError, M.__matmul__, x) + + # The current relationship between sparse matrix products and array + # products is as follows: + assert_almost_equal(M@array([1,2,3]), dot(M.toarray(),[1,2,3])) + assert_almost_equal(M@[[1],[2],[3]], np.atleast_2d(dot(M.toarray(),[1,2,3])).T) + # Note that the result of M * x is dense if x has a singleton dimension. + + # Currently M.matvec(asarray(col)) is rank-1, whereas M.matvec(col) + # is rank-2. Is this desirable? + + def test_matmat_sparse(self): + a = matrix([[3,0,0],[0,1,0],[2,0,3.0],[2,3,0]]) + a2 = array([[3,0,0],[0,1,0],[2,0,3.0],[2,3,0]]) + b = matrix([[0,1],[1,0],[0,2]],'d') + asp = self.spcreator(a) + bsp = self.spcreator(b) + assert_array_almost_equal((asp @ bsp).toarray(), a @ b) + assert_array_almost_equal(asp @ b, a @ b) + assert_array_almost_equal(a @ bsp, a @ b) + assert_array_almost_equal(a2 @ bsp, a @ b) + + # Now try performing cross-type multiplication: + csp = bsp.tocsc() + c = b + want = a @ c + assert_array_almost_equal((asp @ csp).toarray(), want) + assert_array_almost_equal(asp @ c, want) + + assert_array_almost_equal(a @ csp, want) + assert_array_almost_equal(a2 @ csp, want) + csp = bsp.tocsr() + assert_array_almost_equal((asp @ csp).toarray(), want) + assert_array_almost_equal(asp @ c, want) + + assert_array_almost_equal(a @ csp, want) + assert_array_almost_equal(a2 @ csp, want) + csp = bsp.tocoo() + assert_array_almost_equal((asp @ csp).toarray(), want) + assert_array_almost_equal(asp @ c, want) + + assert_array_almost_equal(a @ csp, want) + assert_array_almost_equal(a2 @ csp, want) + + # Test provided by Andy Fraser, 2006-03-26 + L = 30 + frac = .3 + random.seed(0) # make runs repeatable + A = zeros((L,2)) + for i in range(L): + for j in range(2): + r = random.random() + if r < frac: + A[i,j] = r/frac + + A = self.spcreator(A) + B = A @ A.T + assert_array_almost_equal(B.toarray(), A.toarray() @ A.T.toarray()) + assert_array_almost_equal(B.toarray(), A.toarray() @ A.toarray().T) + + # check dimension mismatch 2x2 times 3x2 + A = self.spcreator([[1,2],[3,4]]) + B = self.spcreator([[1,2],[3,4],[5,6]]) + assert_raises(ValueError, A.__matmul__, B) + if self.is_array_test: + assert_raises(ValueError, A.__mul__, B) + + def test_matmat_dense(self): + a = [[3,0,0],[0,1,0],[2,0,3.0],[2,3,0]] + asp = self.spcreator(a) + + # check both array and matrix types + bs = [array([[1,2],[3,4],[5,6]]), matrix([[1,2],[3,4],[5,6]])] + + for b in bs: + result = asp @ b + assert_(isinstance(result, ndarray if self.is_array_test else type(b))) + assert_equal(result.shape, (4,2)) + assert_equal(result, dot(a,b)) + + def test_sparse_format_conversions(self): + A = sparse.kron([[1,0,2],[0,3,4],[5,0,0]], [[1,2],[0,3]]) + D = A.toarray() + A = self.spcreator(A) + + for format in ['bsr','coo','csc','csr','dia','dok','lil']: + a = A.asformat(format) + assert_equal(a.format,format) + assert_array_equal(a.toarray(), D) + + b = self.spcreator(D+3j).asformat(format) + assert_equal(b.format,format) + assert_array_equal(b.toarray(), D+3j) + + c = self.spcreator(D).asformat(format) + assert_equal(c.format,format) + assert_array_equal(c.toarray(), D) + + for format in ['array', 'dense']: + a = A.asformat(format) + assert_array_equal(a, D) + + b = self.spcreator(D+3j).asformat(format) + assert_array_equal(b, D+3j) + + def test_tobsr(self): + x = array([[1,0,2,0],[0,0,0,0],[0,0,4,5]]) + y = array([[0,1,2],[3,0,5]]) + A = kron(x,y) + Asp = self.spcreator(A) + for format in ['bsr']: + fn = getattr(Asp, 'to' + format) + + for X in [1, 2, 3, 6]: + for Y in [1, 2, 3, 4, 6, 12]: + assert_equal(fn(blocksize=(X, Y)).toarray(), A) + + def test_transpose(self): + dat_1 = self.dat + dat_2 = np.array([[]]) + matrices = [dat_1, dat_2] + + def check(dtype, j): + dat = array(matrices[j], dtype=dtype) + datsp = self.spcreator(dat) + + a = datsp.transpose() + b = dat.transpose() + + assert_array_equal(a.toarray(), b) + assert_array_equal(a.transpose().toarray(), dat) + assert_array_equal(datsp.transpose(axes=(1, 0)).toarray(), b) + assert_equal(a.dtype, b.dtype) + + # See gh-5987 + empty = self.spcreator((3, 4)) + assert_array_equal(np.transpose(empty).toarray(), + np.transpose(zeros((3, 4)))) + assert_array_equal(empty.T.toarray(), zeros((4, 3))) + assert_raises(ValueError, empty.transpose, axes=0) + + for dtype in self.checked_dtypes: + for j in range(len(matrices)): + check(dtype, j) + + def test_add_dense(self): + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + # adding a dense matrix to a sparse matrix + sum1 = dat + datsp + assert_array_equal(sum1, dat + dat) + sum2 = datsp + dat + assert_array_equal(sum2, dat + dat) + + for dtype in self.math_dtypes: + check(dtype) + + def test_sub_dense(self): + # subtracting a dense matrix to/from a sparse matrix + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + # Behavior is different for bool. + if dat.dtype == bool: + sum1 = dat - datsp + assert_array_equal(sum1, dat - dat) + sum2 = datsp - dat + assert_array_equal(sum2, dat - dat) + else: + # Manually add to avoid upcasting from scalar + # multiplication. + sum1 = (dat + dat + dat) - datsp + assert_array_equal(sum1, dat + dat) + sum2 = (datsp + datsp + datsp) - dat + assert_array_equal(sum2, dat + dat) + + for dtype in self.math_dtypes: + if dtype == np.dtype('bool'): + # boolean array subtraction deprecated in 1.9.0 + continue + + check(dtype) + + def test_maximum_minimum(self): + A_dense = np.array([[1, 0, 3], [0, 4, 5], [0, 0, 0]]) + B_dense = np.array([[1, 1, 2], [0, 3, 6], [1, -1, 0]]) + + A_dense_cpx = np.array([[1, 0, 3], [0, 4+2j, 5], [0, 1j, -1j]]) + + def check(dtype, dtype2, btype): + if np.issubdtype(dtype, np.complexfloating): + A = self.spcreator(A_dense_cpx.astype(dtype)) + else: + A = self.spcreator(A_dense.astype(dtype)) + if btype == 'scalar': + B = dtype2.type(1) + elif btype == 'scalar2': + B = dtype2.type(-1) + elif btype == 'dense': + B = B_dense.astype(dtype2) + elif btype == 'sparse': + B = self.spcreator(B_dense.astype(dtype2)) + else: + raise ValueError() + + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, + "Taking maximum .minimum. with > 0 .< 0. number " + "results to a dense matrix") + + max_s = A.maximum(B) + min_s = A.minimum(B) + + max_d = np.maximum(toarray(A), toarray(B)) + assert_array_equal(toarray(max_s), max_d) + assert_equal(max_s.dtype, max_d.dtype) + + min_d = np.minimum(toarray(A), toarray(B)) + assert_array_equal(toarray(min_s), min_d) + assert_equal(min_s.dtype, min_d.dtype) + + for dtype in self.math_dtypes: + for dtype2 in [np.int8, np.float64, np.complex128]: + for btype in ['scalar', 'scalar2', 'dense', 'sparse']: + check(np.dtype(dtype), np.dtype(dtype2), btype) + + def test_copy(self): + # Check whether the copy=True and copy=False keywords work + A = self.datsp + + # check that copy preserves format + assert_equal(A.copy().format, A.format) + assert_equal(A.__class__(A,copy=True).format, A.format) + assert_equal(A.__class__(A,copy=False).format, A.format) + + assert_equal(A.copy().toarray(), A.toarray()) + assert_equal(A.__class__(A, copy=True).toarray(), A.toarray()) + assert_equal(A.__class__(A, copy=False).toarray(), A.toarray()) + + # check that XXX_array.toXXX() works + toself = getattr(A,'to' + A.format) + assert_(toself() is A) + assert_(toself(copy=False) is A) + assert_equal(toself(copy=True).format, A.format) + assert_equal(toself(copy=True).toarray(), A.toarray()) + + # check whether the data is copied? + assert_(not sparse_may_share_memory(A.copy(), A)) + + # test that __iter__ is compatible with NumPy matrix + def test_iterator(self): + B = self.asdense(np.arange(50).reshape(5, 10)) + A = self.spcreator(B) + + for x, y in zip(A, B): + assert_equal(x.toarray(), y) + + def test_size_zero_matrix_arithmetic(self): + # Test basic matrix arithmetic with shapes like (0,0), (10,0), + # (0, 3), etc. + mat = array([]) + a = mat.reshape((0, 0)) + b = mat.reshape((0, 1)) + c = mat.reshape((0, 5)) + d = mat.reshape((1, 0)) + e = mat.reshape((5, 0)) + f = np.ones([5, 5]) + + asp = self.spcreator(a) + bsp = self.spcreator(b) + csp = self.spcreator(c) + dsp = self.spcreator(d) + esp = self.spcreator(e) + fsp = self.spcreator(f) + + # matrix product. + assert_array_equal(asp.dot(asp).toarray(), np.dot(a, a)) + assert_array_equal(bsp.dot(dsp).toarray(), np.dot(b, d)) + assert_array_equal(dsp.dot(bsp).toarray(), np.dot(d, b)) + assert_array_equal(csp.dot(esp).toarray(), np.dot(c, e)) + assert_array_equal(csp.dot(fsp).toarray(), np.dot(c, f)) + assert_array_equal(esp.dot(csp).toarray(), np.dot(e, c)) + assert_array_equal(dsp.dot(csp).toarray(), np.dot(d, c)) + assert_array_equal(fsp.dot(esp).toarray(), np.dot(f, e)) + + # bad matrix products + assert_raises(ValueError, dsp.dot, e) + assert_raises(ValueError, asp.dot, d) + + # elemente-wise multiplication + assert_array_equal(asp.multiply(asp).toarray(), np.multiply(a, a)) + assert_array_equal(bsp.multiply(bsp).toarray(), np.multiply(b, b)) + assert_array_equal(dsp.multiply(dsp).toarray(), np.multiply(d, d)) + + assert_array_equal(asp.multiply(a).toarray(), np.multiply(a, a)) + assert_array_equal(bsp.multiply(b).toarray(), np.multiply(b, b)) + assert_array_equal(dsp.multiply(d).toarray(), np.multiply(d, d)) + + assert_array_equal(asp.multiply(6).toarray(), np.multiply(a, 6)) + assert_array_equal(bsp.multiply(6).toarray(), np.multiply(b, 6)) + assert_array_equal(dsp.multiply(6).toarray(), np.multiply(d, 6)) + + # bad element-wise multiplication + assert_raises(ValueError, asp.multiply, c) + assert_raises(ValueError, esp.multiply, c) + + # Addition + assert_array_equal(asp.__add__(asp).toarray(), a.__add__(a)) + assert_array_equal(bsp.__add__(bsp).toarray(), b.__add__(b)) + assert_array_equal(dsp.__add__(dsp).toarray(), d.__add__(d)) + + # bad addition + assert_raises(ValueError, asp.__add__, dsp) + assert_raises(ValueError, bsp.__add__, asp) + + def test_size_zero_conversions(self): + mat = array([]) + a = mat.reshape((0, 0)) + b = mat.reshape((0, 5)) + c = mat.reshape((5, 0)) + + for m in [a, b, c]: + spm = self.spcreator(m) + assert_array_equal(spm.tocoo().toarray(), m) + assert_array_equal(spm.tocsr().toarray(), m) + assert_array_equal(spm.tocsc().toarray(), m) + assert_array_equal(spm.tolil().toarray(), m) + assert_array_equal(spm.todok().toarray(), m) + assert_array_equal(spm.tobsr().toarray(), m) + + def test_dtype_check(self): + a = np.array([[3.5, 0, 1.1], [0, 0, 0]], dtype=np.float16) + with assert_raises(ValueError, match="does not support dtype"): + self.spcreator(a) + + A32 = self.spcreator(a.astype(np.float32)) + with assert_raises(ValueError, match="does not support dtype"): + self.spcreator(A32, dtype=np.float16) + + def test_pickle(self): + import pickle + sup = suppress_warnings() + sup.filter(SparseEfficiencyWarning) + + @sup + def check(): + datsp = self.datsp.copy() + for protocol in range(pickle.HIGHEST_PROTOCOL): + sploaded = pickle.loads(pickle.dumps(datsp, protocol=protocol)) + assert_equal(datsp.shape, sploaded.shape) + assert_array_equal(datsp.toarray(), sploaded.toarray()) + assert_equal(datsp.format, sploaded.format) + # Hacky check for class member equality. This assumes that + # all instance variables are one of: + # 1. Plain numpy ndarrays + # 2. Tuples of ndarrays + # 3. Types that support equality comparison with == + for key, val in datsp.__dict__.items(): + if isinstance(val, np.ndarray): + assert_array_equal(val, sploaded.__dict__[key]) + elif (isinstance(val, tuple) and val + and isinstance(val[0], np.ndarray)): + assert_array_equal(val, sploaded.__dict__[key]) + else: + assert_(val == sploaded.__dict__[key]) + check() + + def test_unary_ufunc_overrides(self): + def check(name): + if name == "sign": + pytest.skip("sign conflicts with comparison op " + "support on Numpy") + if self.datsp.format in ["dok", "lil"]: + pytest.skip("Unary ops not implemented for dok/lil") + ufunc = getattr(np, name) + + X = self.spcreator(np.arange(20).reshape(4, 5) / 20.) + X0 = ufunc(X.toarray()) + + X2 = ufunc(X) + assert_array_equal(X2.toarray(), X0) + + for name in ["sin", "tan", "arcsin", "arctan", "sinh", "tanh", + "arcsinh", "arctanh", "rint", "sign", "expm1", "log1p", + "deg2rad", "rad2deg", "floor", "ceil", "trunc", "sqrt", + "abs"]: + check(name) + + def test_resize(self): + # resize(shape) resizes the matrix in-place + D = np.array([[1, 0, 3, 4], + [2, 0, 0, 0], + [3, 0, 0, 0]]) + S = self.spcreator(D) + assert_(S.resize((3, 2)) is None) + assert_array_equal(S.toarray(), [[1, 0], + [2, 0], + [3, 0]]) + S.resize((2, 2)) + assert_array_equal(S.toarray(), [[1, 0], + [2, 0]]) + S.resize((3, 2)) + assert_array_equal(S.toarray(), [[1, 0], + [2, 0], + [0, 0]]) + S.resize((3, 3)) + assert_array_equal(S.toarray(), [[1, 0, 0], + [2, 0, 0], + [0, 0, 0]]) + # test no-op + S.resize((3, 3)) + assert_array_equal(S.toarray(), [[1, 0, 0], + [2, 0, 0], + [0, 0, 0]]) + + # test *args + S.resize(3, 2) + assert_array_equal(S.toarray(), [[1, 0], + [2, 0], + [0, 0]]) + + if self.is_array_test and S.format in ["coo", "csr"]: + S.resize(1) + else: + assert_raises((ValueError, NotImplementedError, IndexError), S.resize, 1) + + for bad_shape in [(-1, 2), (2, -1), (1, 2, 3)]: + assert_raises(ValueError, S.resize, bad_shape) + + def test_constructor1_base(self): + A = self.datsp + + self_format = A.format + + C = A.__class__(A, copy=False) + assert_array_equal_dtype(A.toarray(), C.toarray()) + if self_format not in NON_ARRAY_BACKED_FORMATS: + assert_(sparse_may_share_memory(A, C)) + + C = A.__class__(A, dtype=A.dtype, copy=False) + assert_array_equal_dtype(A.toarray(), C.toarray()) + if self_format not in NON_ARRAY_BACKED_FORMATS: + assert_(sparse_may_share_memory(A, C)) + + C = A.__class__(A, dtype=np.float32, copy=False) + assert_array_equal(A.toarray(), C.toarray()) + + C = A.__class__(A, copy=True) + assert_array_equal_dtype(A.toarray(), C.toarray()) + assert_(not sparse_may_share_memory(A, C)) + + for other_format in ['csr', 'csc', 'coo', 'dia', 'dok', 'lil']: + if other_format == self_format: + continue + B = A.asformat(other_format) + C = A.__class__(B, copy=False) + assert_array_equal_dtype(A.toarray(), C.toarray()) + + C = A.__class__(B, copy=True) + assert_array_equal_dtype(A.toarray(), C.toarray()) + assert_(not sparse_may_share_memory(B, C)) + + +class _TestInplaceArithmetic: + def test_inplace_dense(self): + a = np.ones((3, 4)) + b = self.spcreator(a) + + x = a.copy() + y = a.copy() + x += a + y += b + assert_array_equal(x, y) + + x = a.copy() + y = a.copy() + x -= a + y -= b + assert_array_equal(x, y) + + if self.is_array_test: + # Elementwise multiply from sparray.__rmul__ + x = a.copy() + y = a.copy() + with assert_raises(ValueError, match="inconsistent shapes"): + x *= b.T + x = x * a + y *= b + assert_array_equal(x, y.toarray()) + else: + # Matrix multiply from spmatrix.__rmul__ + x = a.copy() + y = a.copy() + with assert_raises(ValueError, match="dimension mismatch"): + x *= b + x = x.dot(a.T) + y *= b.T + assert_array_equal(x, y) + + # Matrix multiply from __rmatmul__ + y = a.copy() + # skip this test if numpy doesn't support __imatmul__ yet. + # move out of the try/except once numpy 1.24 is no longer supported. + try: + y @= b.T + except TypeError: + pass + else: + x = a.copy() + y = a.copy() + with assert_raises(ValueError, match="dimension mismatch"): + x @= b + x = x.dot(a.T) + y @= b.T + assert_array_equal(x, y) + + # Floor division is not supported + with assert_raises(TypeError, match="unsupported operand"): + x //= b + + def test_imul_scalar(self): + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + # Avoid implicit casting. + if np.can_cast(int, dtype, casting='same_kind'): + a = datsp.copy() + a *= 2 + b = dat.copy() + b *= 2 + assert_array_equal(b, a.toarray()) + + if np.can_cast(float, dtype, casting='same_kind'): + a = datsp.copy() + a *= 17.3 + b = dat.copy() + b *= 17.3 + assert_array_equal(b, a.toarray()) + + for dtype in self.math_dtypes: + check(dtype) + + def test_idiv_scalar(self): + def check(dtype): + dat = self.dat_dtypes[dtype] + datsp = self.datsp_dtypes[dtype] + + if np.can_cast(int, dtype, casting='same_kind'): + a = datsp.copy() + a /= 2 + b = dat.copy() + b /= 2 + assert_array_equal(b, a.toarray()) + + if np.can_cast(float, dtype, casting='same_kind'): + a = datsp.copy() + a /= 17.3 + b = dat.copy() + b /= 17.3 + assert_array_equal(b, a.toarray()) + + for dtype in self.math_dtypes: + # /= should only be used with float dtypes to avoid implicit + # casting. + if not np.can_cast(dtype, np.dtype(int)): + check(dtype) + + def test_inplace_success(self): + # Inplace ops should work even if a specialized version is not + # implemented, falling back to x = x y + a = self.spcreator(np.eye(5)) + b = self.spcreator(np.eye(5)) + bp = self.spcreator(np.eye(5)) + + b += a + bp = bp + a + assert_allclose(b.toarray(), bp.toarray()) + + if self.is_array_test: + b *= a + bp = bp * a + assert_allclose(b.toarray(), bp.toarray()) + + b @= a + bp = bp @ a + assert_allclose(b.toarray(), bp.toarray()) + + b -= a + bp = bp - a + assert_allclose(b.toarray(), bp.toarray()) + + with assert_raises(TypeError, match="unsupported operand"): + a //= b + + +class _TestGetSet: + def test_getelement(self): + def check(dtype): + D = array([[1,0,0], + [4,3,0], + [0,2,0], + [0,0,0]], dtype=dtype) + A = self.spcreator(D) + + M,N = D.shape + + for i in range(-M, M): + for j in range(-N, N): + assert_equal(A[i,j], D[i,j]) + + assert_equal(type(A[1,1]), dtype) + + for ij in [(0,3),(-1,3),(4,0),(4,3),(4,-1), (1, 2, 3)]: + assert_raises((IndexError, TypeError), A.__getitem__, ij) + + for dtype in supported_dtypes: + check(np.dtype(dtype)) + + def test_setelement(self): + def check(dtype): + A = self.spcreator((3,4), dtype=dtype) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + A[0, 0] = dtype.type(0) # bug 870 + A[1, 2] = dtype.type(4.0) + A[0, 1] = dtype.type(3) + A[2, 0] = dtype.type(2.0) + A[0,-1] = dtype.type(8) + A[-1,-2] = dtype.type(7) + A[0, 1] = dtype.type(5) + + if dtype != np.bool_: + assert_array_equal( + A.toarray(), + [ + [0, 5, 0, 8], + [0, 0, 4, 0], + [2, 0, 7, 0] + ] + ) + + for ij in [(0,4),(-1,4),(3,0),(3,4),(3,-1)]: + assert_raises(IndexError, A.__setitem__, ij, 123.0) + + for v in [[1,2,3], array([1,2,3])]: + assert_raises(ValueError, A.__setitem__, (0,0), v) + + if (not np.issubdtype(dtype, np.complexfloating) and + dtype != np.bool_): + for v in [3j]: + assert_raises(TypeError, A.__setitem__, (0,0), v) + + for dtype in supported_dtypes: + check(np.dtype(dtype)) + + def test_negative_index_assignment(self): + # Regression test for GitHub issue 4428. + + def check(dtype): + A = self.spcreator((3, 10), dtype=dtype) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + A[0, -4] = 1 + assert_equal(A[0, -4], 1) + + for dtype in self.math_dtypes: + check(np.dtype(dtype)) + + def test_scalar_assign_2(self): + n, m = (5, 10) + + def _test_set(i, j, nitems): + msg = f"{i!r} ; {j!r} ; {nitems!r}" + A = self.spcreator((n, m)) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + A[i, j] = 1 + assert_almost_equal(A.sum(), nitems, err_msg=msg) + assert_almost_equal(A[i, j], 1, err_msg=msg) + + # [i,j] + for i, j in [(2, 3), (-1, 8), (-1, -2), (array(-1), -2), (-1, array(-2)), + (array(-1), array(-2))]: + _test_set(i, j, 1) + + def test_index_scalar_assign(self): + A = self.spcreator((5, 5)) + B = np.zeros((5, 5)) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + for C in [A, B]: + C[0,1] = 1 + C[3,0] = 4 + C[3,0] = 9 + assert_array_equal(A.toarray(), B) + + +@pytest.mark.thread_unsafe +class _TestSolve: + def test_solve(self): + # Test whether the lu_solve command segfaults, as reported by Nils + # Wagner for a 64-bit machine, 02 March 2005 (EJS) + n = 20 + np.random.seed(0) # make tests repeatable + A = zeros((n,n), dtype=complex) + x = np.random.rand(n) + y = np.random.rand(n-1)+1j*np.random.rand(n-1) + r = np.random.rand(n) + for i in range(len(x)): + A[i,i] = x[i] + for i in range(len(y)): + A[i,i+1] = y[i] + A[i+1,i] = conjugate(y[i]) + A = self.spcreator(A) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, + "splu converted its input to CSC format") + x = splu(A).solve(r) + assert_almost_equal(A @ x,r) + + +class _TestSlicing: + def test_dtype_preservation(self): + assert_equal(self.spcreator((1,10), dtype=np.int16)[0,1:5].dtype, np.int16) + assert_equal(self.spcreator((1,10), dtype=np.int32)[0,1:5].dtype, np.int32) + assert_equal(self.spcreator((1,10), dtype=np.float32)[0,1:5].dtype, np.float32) + assert_equal(self.spcreator((1,10), dtype=np.float64)[0,1:5].dtype, np.float64) + + def test_dtype_preservation_empty_slice(self): + # This should be parametrized with pytest, but something in the parent + # class creation used in this file breaks pytest.mark.parametrize. + for dt in [np.int16, np.int32, np.float32, np.float64]: + A = self.spcreator((3, 2), dtype=dt) + assert_equal(A[:, 0:0:2].dtype, dt) + assert_equal(A[0:0:2, :].dtype, dt) + assert_equal(A[0, 0:0:2].dtype, dt) + assert_equal(A[0:0:2, 0].dtype, dt) + + def test_get_horiz_slice(self): + B = self.asdense(arange(50.).reshape(5,10)) + A = self.spcreator(B) + r0, r1, r2 = (0, 1, 2) if self.is_array_test else ([0], [1], [2]) + assert_array_equal(B[r1, :], A[1, :].toarray()) + assert_array_equal(B[r1, 2:5], A[1, 2:5].toarray()) + + C = self.asdense([[1, 2, 1], [4, 0, 6], [0, 0, 0], [0, 0, 1]]) + D = self.spcreator(C) + assert_array_equal(C[r1, 1:3], D[1, 1:3].toarray()) + + # Now test slicing when a row contains only zeros + E = self.asdense([[1, 2, 1], [4, 0, 0], [0, 0, 0], [0, 0, 1]]) + F = self.spcreator(E) + assert_array_equal(E[r1, 1:3], F[1, 1:3].toarray()) + assert_array_equal(E[r2, -2:], F[2, -2:].toarray()) + + # The following should raise exceptions: + assert_raises(IndexError, A.__getitem__, (slice(None), 11)) + assert_raises(IndexError, A.__getitem__, (6, slice(3, 7))) + + def test_get_vert_slice(self): + B = arange(50.).reshape(5, 10) + A = self.spcreator(B) + c0, c1, c2 = (0, 1, 2) if self.is_array_test else ([0], [1], [2]) + assert_array_equal(B[2:5, c0], A[2:5, 0].toarray()) + assert_array_equal(B[:, c1], A[:, 1].toarray()) + + C = array([[1, 2, 1], [4, 0, 6], [0, 0, 0], [0, 0, 1]]) + D = self.spcreator(C) + assert_array_equal(C[1:3, c1], D[1:3, 1].toarray()) + assert_array_equal(C[:, c2], D[:, 2].toarray()) + + # Now test slicing when a column contains only zeros + E = array([[1, 0, 1], [4, 0, 0], [0, 0, 0], [0, 0, 1]]) + F = self.spcreator(E) + assert_array_equal(E[:, c1], F[:, 1].toarray()) + assert_array_equal(E[-2:, c2], F[-2:, 2].toarray()) + + # The following should raise exceptions: + assert_raises(IndexError, A.__getitem__, (slice(None), 11)) + assert_raises(IndexError, A.__getitem__, (6, slice(3, 7))) + + def test_get_slices(self): + B = arange(50.).reshape(5, 10) + A = self.spcreator(B) + assert_array_equal(A[2:5, 0:3].toarray(), B[2:5, 0:3]) + assert_array_equal(A[1:, :-1].toarray(), B[1:, :-1]) + assert_array_equal(A[:-1, 1:].toarray(), B[:-1, 1:]) + + # Now test slicing when a column contains only zeros + E = array([[1, 0, 1], [4, 0, 0], [0, 0, 0], [0, 0, 1]]) + F = self.spcreator(E) + assert_array_equal(E[1:2, 1:2], F[1:2, 1:2].toarray()) + assert_array_equal(E[:, 1:], F[:, 1:].toarray()) + + def test_non_unit_stride_2d_indexing(self): + # Regression test -- used to silently ignore the stride. + v0 = np.random.rand(50, 50) + try: + v = self.spcreator(v0)[0:25:2, 2:30:3] + except ValueError: + # if unsupported + raise pytest.skip("feature not implemented") + + assert_array_equal(v.toarray(), v0[0:25:2, 2:30:3]) + + def test_slicing_2(self): + B = self.asdense(arange(50).reshape(5,10)) + A = self.spcreator(B) + + # [i,j] + assert_equal(A[2,3], B[2,3]) + assert_equal(A[-1,8], B[-1,8]) + assert_equal(A[-1,-2],B[-1,-2]) + assert_equal(A[array(-1),-2],B[-1,-2]) + assert_equal(A[-1,array(-2)],B[-1,-2]) + assert_equal(A[array(-1),array(-2)],B[-1,-2]) + + # [i,1:2] + assert_equal(A[2, :].toarray(), B[2, :]) + assert_equal(A[2, 5:-2].toarray(), B[2, 5:-2]) + assert_equal(A[array(2), 5:-2].toarray(), B[2, 5:-2]) + + # [1:2,j] + assert_equal(A[:, 2].toarray(), B[:, 2]) + assert_equal(A[3:4, 9].toarray(), B[3:4, 9]) + assert_equal(A[1:4, -5].toarray(), B[1:4, -5]) + assert_equal(A[2:-1, 3].toarray(), B[2:-1, 3]) + assert_equal(A[2:-1, array(3)].toarray(), B[2:-1, 3]) + + # [1:2,1:2] + assert_equal(A[1:2, 1:2].toarray(), B[1:2, 1:2]) + assert_equal(A[4:, 3:].toarray(), B[4:, 3:]) + assert_equal(A[:4, :5].toarray(), B[:4, :5]) + assert_equal(A[2:-1, :5].toarray(), B[2:-1, :5]) + + # [i] + assert_equal(A[1, :].toarray(), B[1, :]) + assert_equal(A[-2, :].toarray(), B[-2, :]) + assert_equal(A[array(-2), :].toarray(), B[-2, :]) + + # [1:2] + assert_equal(A[1:4].toarray(), B[1:4]) + assert_equal(A[1:-2].toarray(), B[1:-2]) + + # Check bug reported by Robert Cimrman: + # http://thread.gmane.org/gmane.comp.python.scientific.devel/7986 (dead link) + s = slice(int8(2),int8(4),None) + assert_equal(A[s, :].toarray(), B[2:4, :]) + assert_equal(A[:, s].toarray(), B[:, 2:4]) + + @pytest.mark.fail_slow(2) + def test_slicing_3(self): + B = self.asdense(arange(50).reshape(5,10)) + A = self.spcreator(B) + + s_ = np.s_ + slices = [s_[:2], s_[1:2], s_[3:], s_[3::2], + s_[15:20], s_[3:2], + s_[8:3:-1], s_[4::-2], s_[:5:-1], + 0, 1, s_[:], s_[1:5], -1, -2, -5, + array(-1), np.int8(-3)] + + def check_1(a): + x = A[a] + y = B[a] + if y.shape == (): + assert_equal(x, y, repr(a)) + else: + if x.size == 0 and y.size == 0: + pass + else: + assert_array_equal(x.toarray(), y, repr(a)) + + for j, a in enumerate(slices): + check_1(a) + + def check_2(a, b): + # Indexing np.matrix with 0-d arrays seems to be broken, + # as they seem not to be treated as scalars. + # https://github.com/numpy/numpy/issues/3110 + if isinstance(a, np.ndarray): + ai = int(a) + else: + ai = a + if isinstance(b, np.ndarray): + bi = int(b) + else: + bi = b + + x = A[a, b] + y = B[ai, bi] + + if y.shape == (): + assert_equal(x, y, repr((a, b))) + else: + if x.size == 0 and y.size == 0: + pass + else: + assert_array_equal(x.toarray(), y, repr((a, b))) + + for i, a in enumerate(slices): + for j, b in enumerate(slices): + check_2(a, b) + + # Check out of bounds etc. systematically + extra_slices = [] + for a, b, c in itertools.product(*([(None, 0, 1, 2, 5, 15, + -1, -2, 5, -15)]*3)): + if c == 0: + continue + extra_slices.append(slice(a, b, c)) + + for a in extra_slices: + check_2(a, a) + check_2(a, -2) + check_2(-2, a) + + def test_None_slicing(self): + B = self.asdense(arange(50).reshape(5,10)) + A = self.spcreator(B) + + assert A[1, 2].ndim == 0 + assert A[None, 1, 2:4].shape == (1, 2) + assert A[None, 1, 2, None].shape == (1, 1) + + # see gh-22458 + assert A[None, 1].shape == (1, 10) + assert A[1, None].shape == (1, 10) + assert A[None, 1, :].shape == (1, 10) + assert A[1, None, :].shape == (1, 10) + assert A[1, :, None].shape == (10, 1) + + assert A[None, 1:3, 2].shape == B[None, 1:3, 2].shape == (1, 2) + assert A[1:3, None, 2].shape == B[1:3, None, 2].shape == (2, 1) + assert A[1:3, 2, None].shape == B[1:3, 2, None].shape == (2, 1) + assert A[None, 1, 2:4].shape == B[None, 1, 2:4].shape == (1, 2) + assert A[1, None, 2:4].shape == B[1, None, 2:4].shape == (1, 2) + assert A[1, 2:4, None].shape == B[1, 2:4, None].shape == (2, 1) + + # different for spmatrix + if self.is_array_test: + assert A[1:3, 2].shape == B[1:3, 2].shape == (2,) + assert A[1, 2:4].shape == B[1, 2:4].shape == (2,) + assert A[None, 1, 2].shape == B[None, 1, 2].shape == (1,) + assert A[1, None, 2].shape == B[1, None, 2].shape == (1,) + assert A[1, 2, None].shape == B[1, 2, None].shape == (1,) + else: + assert A[1, 2:4].shape == B[1, 2:4].shape == (1, 2) + assert A[1:3, 2].shape == B[1:3, 2].shape == (2, 1) + assert A[None, 1, 2].shape == B[None, 1, 2].shape == (1, 1) + assert A[1, None, 2].shape == B[1, None, 2].shape == (1, 1) + assert A[1, 2, None].shape == B[1, 2, None].shape == (1, 1) + + def test_ellipsis_slicing(self): + b = self.asdense(arange(50).reshape(5,10)) + a = self.spcreator(b) + + assert_array_equal(a[...].toarray(), b[...]) + assert_array_equal(a[...,].toarray(), b[...,]) + + assert_array_equal(a[4, ...].toarray(), b[4, ...]) + assert_array_equal(a[..., 4].toarray(), b[..., 4]) + assert_array_equal(a[..., 5].toarray(), b[..., 5]) + with pytest.raises(IndexError, match='index .5. out of range'): + a[5, ...] + with pytest.raises(IndexError, match='index .10. out of range'): + a[..., 10] + with pytest.raises(IndexError, match='index .5. out of range'): + a.T[..., 5] + + assert_array_equal(a[1:, ...].toarray(), b[1:, ...]) + assert_array_equal(a[..., 1:].toarray(), b[..., 1:]) + assert_array_equal(a[:2, ...].toarray(), b[:2, ...]) + assert_array_equal(a[..., :2].toarray(), b[..., :2]) + + # check slice limit outside range + assert_array_equal(a[:5, ...].toarray(), b[:5, ...]) + assert_array_equal(a[..., :5].toarray(), b[..., :5]) + assert_array_equal(a[5:, ...].toarray(), b[5:, ...]) + assert_array_equal(a[..., 5:].toarray(), b[..., 5:]) + assert_array_equal(a[10:, ...].toarray(), b[10:, ...]) + assert_array_equal(a[..., 10:].toarray(), b[..., 10:]) + + # ellipsis should be ignored + assert_array_equal(a[1:, 1, ...].toarray(), b[1:, 1, ...]) + assert_array_equal(a[1, ..., 1:].toarray(), b[1, ..., 1:]) + assert_array_equal(a[..., 1, 1:].toarray(), b[1, ..., 1:]) + assert_array_equal(a[:2, 1, ...].toarray(), b[:2, 1, ...]) + assert_array_equal(a[1, ..., :2].toarray(), b[1, ..., :2]) + assert_array_equal(a[..., 1, :2].toarray(), b[1, ..., :2]) + # These return ints + assert_equal(a[1, 1, ...], b[1, 1, ...]) + assert_equal(a[1, ..., 1], b[1, ..., 1]) + + def test_ellipsis_fancy_bool(self): + numpy_a = self.asdense(arange(50).reshape(5, 10)) + a = self.spcreator(numpy_a) + + ix5 = [True, False, True, False, True] + ix10 = [False] * 5 + ix5 # same number of True values as ix5 + ix10_6True = ix5 + ix5 # not same number of True values as ix5 + full_ix = [ix10] * 5 + + assert_array_equal(toarray(a[full_ix, ...]), numpy_a[full_ix, ...]) + assert_array_equal(toarray(a[..., full_ix]), numpy_a[..., full_ix]) + + assert_array_equal(toarray(a[ix5, ...]), numpy_a[ix5, ...]) + assert_array_equal(toarray(a[..., ix10]), numpy_a[..., ix10]) + + assert_array_equal(toarray(a[ix5, ..., ix10]), numpy_a[ix5, ..., ix10]) + assert_array_equal(toarray(a[..., ix5, ix10]), numpy_a[..., ix5, ix10]) + assert_array_equal(toarray(a[ix5, ix10, ...]), numpy_a[ix5, ix10, ...]) + + with assert_raises(ValueError, match="shape mismatch"): + a[ix5, ix10_6True] + + def test_ellipsis_fancy_slicing(self): + b = self.asdense(arange(50).reshape(5, 10)) + a = self.spcreator(b) + + assert_array_equal(a[[4], ...].toarray(), b[[4], ...]) + assert_array_equal(a[[2, 4], ...].toarray(), b[[2, 4], ...]) + assert_array_equal(a[..., [4]].toarray(), b[..., [4]]) + assert_array_equal(a[..., [2, 4]].toarray(), b[..., [2, 4]]) + + assert_array_equal(a[[4], 1, ...].toarray(), b[[4], 1, ...]) + assert_array_equal(a[[2, 4], 1, ...].toarray(), b[[2, 4], 1, ...]) + assert_array_equal(a[[4], ..., 1].toarray(), b[[4], ..., 1]) + assert_array_equal(a[..., [4], 1].toarray(), b[..., [4], 1]) + # fancy index gives dense + assert_array_equal(toarray(a[[2, 4], ..., [2, 4]]), b[[2, 4], ..., [2, 4]]) + assert_array_equal(toarray(a[..., [2, 4], [2, 4]]), b[..., [2, 4], [2, 4]]) + + def test_multiple_ellipsis_slicing(self): + a = self.spcreator(arange(6).reshape(3, 2)) + + with pytest.raises(IndexError, + match='an index can only have a single ellipsis'): + a[..., ...] + with pytest.raises(IndexError, + match='an index can only have a single ellipsis'): + a[..., 1, ...] + + +class _TestSlicingAssign: + def test_slice_scalar_assign(self): + A = self.spcreator((5, 5)) + B = np.zeros((5, 5)) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + for C in [A, B]: + C[0:1,1] = 1 + C[3:0,0] = 4 + C[3:4,0] = 9 + C[0,4:] = 1 + C[3::-1,4:] = 9 + assert_array_equal(A.toarray(), B) + + def test_slice_assign_2(self): + n, m = (5, 10) + + def _test_set(i, j): + msg = f"i={i!r}; j={j!r}" + A = self.spcreator((n, m)) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + A[i, j] = 1 + B = np.zeros((n, m)) + B[i, j] = 1 + assert_array_almost_equal(A.toarray(), B, err_msg=msg) + # [i,1:2] + for i, j in [(2, slice(3)), (2, slice(None, 10, 4)), (2, slice(5, -2)), + (array(2), slice(5, -2))]: + _test_set(i, j) + + def test_self_self_assignment(self): + # Tests whether a row of one sparse array can be assigned to another. + B = self.spcreator((4,3)) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + B[0,0] = 2 + B[1,2] = 7 + B[2,1] = 3 + B[3,0] = 10 + + A = B / 10 + B[0,:] = A[0,:] + assert_array_equal(A[0,:].toarray(), B[0,:].toarray()) + + A = B / 10 + B[:,:] = A[:1,:1] + assert_array_equal(np.zeros((4,3)) + A[0,0], B.toarray()) + + A = B / 10 + B[:-1,0] = A[None,0,:].T + assert_array_equal(A[0,:].toarray().T, B[:-1,0].toarray()) + + def test_slice_assignment(self): + B = self.spcreator((4,3)) + expected = array([[10,0,0], + [0,0,6], + [0,14,0], + [0,0,0]]) + block = [[1,0],[0,4]] + + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + B[0,0] = 5 + B[1,2] = 3 + B[2,1] = 7 + B[:,:] = B+B + assert_array_equal(B.toarray(), expected) + + B[:2,:2] = self.csc_container(array(block)) + assert_array_equal(B.toarray()[:2, :2], block) + + def test_sparsity_modifying_assignment(self): + B = self.spcreator((4,3)) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + B[0,0] = 5 + B[1,2] = 3 + B[2,1] = 7 + B[3,0] = 10 + B[:3] = self.csr_container(np.eye(3)) + + expected = array([[1,0,0],[0,1,0],[0,0,1],[10,0,0]]) + assert_array_equal(B.toarray(), expected) + + def test_set_slice(self): + A = self.spcreator((5,10)) + B = array(zeros((5, 10), float)) + s_ = np.s_ + slices = [s_[:2], s_[1:2], s_[3:], s_[3::2], + s_[8:3:-1], s_[4::-2], s_[:5:-1], + 0, 1, s_[:], s_[1:5], -1, -2, -5, + array(-1), np.int8(-3)] + + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + for j, a in enumerate(slices): + A[a] = j + B[a] = j + assert_array_equal(A.toarray(), B, repr(a)) + + for i, a in enumerate(slices): + for j, b in enumerate(slices): + A[a,b] = 10*i + 1000*(j+1) + B[a,b] = 10*i + 1000*(j+1) + assert_array_equal(A.toarray(), B, repr((a, b))) + + A[0, 1:10:2] = range(1, 10, 2) + B[0, 1:10:2] = range(1, 10, 2) + assert_array_equal(A.toarray(), B) + A[1:5:2, 0] = np.arange(1, 5, 2)[:, None] + B[1:5:2, 0] = np.arange(1, 5, 2)[:] + assert_array_equal(A.toarray(), B) + + # The next commands should raise exceptions + assert_raises(ValueError, A.__setitem__, (0, 0), list(range(100))) + assert_raises(ValueError, A.__setitem__, (0, 0), arange(100)) + assert_raises(ValueError, A.__setitem__, (0, slice(None)), + list(range(100))) + assert_raises(ValueError, A.__setitem__, (slice(None), 1), + list(range(100))) + assert_raises(ValueError, A.__setitem__, (slice(None), 1), A.copy()) + assert_raises(ValueError, A.__setitem__, + ([[1, 2, 3], [0, 3, 4]], [1, 2, 3]), [1, 2, 3, 4]) + assert_raises(ValueError, A.__setitem__, + ([[1, 2, 3], [0, 3, 4], [4, 1, 3]], + [[1, 2, 4], [0, 1, 3]]), [2, 3, 4]) + assert_raises(ValueError, A.__setitem__, (slice(4), 0), + [[1, 2], [3, 4]]) + + def test_assign_empty(self): + A = self.spcreator(np.ones((2, 3))) + B = self.spcreator((1, 2)) + A[1, :2] = B + assert_array_equal(A.toarray(), [[1, 1, 1], [0, 0, 1]]) + + def test_assign_1d_slice(self): + A = self.spcreator(np.ones((3, 3))) + x = np.zeros(3) + A[:, 0] = x + A[1, :] = x + assert_array_equal(A.toarray(), [[0, 1, 1], [0, 0, 0], [0, 1, 1]]) + + +class _TestFancyIndexing: + """Tests fancy indexing features. The tests for any matrix formats + that implement these features should derive from this class. + """ + + def test_dtype_preservation_empty_index(self): + # This should be parametrized with pytest, but something in the parent + # class creation used in this file breaks pytest.mark.parametrize. + for dt in [np.int16, np.int32, np.float32, np.float64]: + A = self.spcreator((3, 2), dtype=dt) + assert_equal(A[:, [False, False]].dtype, dt) + assert_equal(A[[False, False, False], :].dtype, dt) + assert_equal(A[:, []].dtype, dt) + assert_equal(A[[], :].dtype, dt) + + def test_bad_index(self): + A = self.spcreator(np.zeros([5, 5])) + assert_raises((IndexError, ValueError, TypeError), A.__getitem__, "foo") + assert_raises((IndexError, ValueError, TypeError), A.__getitem__, (2, "foo")) + assert_raises((IndexError, ValueError), A.__getitem__, + ([1, 2, 3], [1, 2, 3, 4])) + + def test_fancy_indexing(self): + B = self.asdense(arange(50).reshape(5,10)) + A = self.spcreator(B) + + # [i] + assert_equal(A[[3]].toarray(), B[[3]]) + assert_equal(A[[1, 3]].toarray(), B[[1, 3]]) + + # [i,[1,2]] + assert_equal(A[3, [3]].toarray(), B[3, [3]]) + assert_equal(A[3, [1, 3]].toarray(), B[3, [1, 3]]) + assert_equal(A[-1, [2, -5]].toarray(), B[-1, [2, -5]]) + assert_equal(A[array(-1), [2, -5]].toarray(), B[-1, [2, -5]]) + assert_equal(A[-1, array([2, -5])].toarray(), B[-1, [2, -5]]) + assert_equal(A[array(-1), array([2, -5])].toarray(), B[-1, [2, -5]]) + + # [1:2,[1,2]] + assert_equal(A[:, [2, 8, 3, -1]].toarray(), B[:, [2, 8, 3, -1]]) + assert_equal(A[3:4, [9]].toarray(), B[3:4, [9]]) + assert_equal(A[1:4, [-1, -5]].toarray(), B[1:4, [-1, -5]]) + assert_equal(A[1:4, array([-1, -5])].toarray(), B[1:4, [-1, -5]]) + + # [[1,2],j] + assert_equal(A[[3], 3].toarray(), B[[3], 3]) + assert_equal(A[[1, 3], 3].toarray(), B[[1, 3], 3]) + assert_equal(A[[2, -5], -4].toarray(), B[[2, -5], -4]) + assert_equal(A[array([2, -5]), -4].toarray(), B[[2, -5], -4]) + assert_equal(A[[2, -5], array(-4)].toarray(), B[[2, -5], -4]) + assert_equal(A[array([2, -5]), array(-4)].toarray(), B[[2, -5], -4]) + + # [[1,2],1:2] + assert_equal(A[[3], :].toarray(), B[[3], :]) + assert_equal(A[[1, 3], :].toarray(), B[[1, 3], :]) + assert_equal(A[[2, -5], 8:-1].toarray(), B[[2, -5], 8:-1]) + assert_equal(A[array([2, -5]), 8:-1].toarray(), B[[2, -5], 8:-1]) + + # [[1,2],[1,2]] + assert_equal(toarray(A[[3], [4]]), B[[3], [4]]) + assert_equal(toarray(A[[1, 3], [2, 4]]), B[[1, 3], [2, 4]]) + assert_equal(toarray(A[[-1, -3], [2, -4]]), B[[-1, -3], [2, -4]]) + assert_equal( + toarray(A[array([-1, -3]), [2, -4]]), B[[-1, -3], [2, -4]] + ) + assert_equal( + toarray(A[[-1, -3], array([2, -4])]), B[[-1, -3], [2, -4]] + ) + assert_equal( + toarray(A[array([-1, -3]), array([2, -4])]), B[[-1, -3], [2, -4]] + ) + + # [[[1],[2]],[1,2]] + assert_equal(A[[[1], [3]], [2, 4]].toarray(), B[[[1], [3]], [2, 4]]) + assert_equal( + A[[[-1], [-3], [-2]], [2, -4]].toarray(), + B[[[-1], [-3], [-2]], [2, -4]] + ) + assert_equal( + A[array([[-1], [-3], [-2]]), [2, -4]].toarray(), + B[[[-1], [-3], [-2]], [2, -4]] + ) + assert_equal( + A[[[-1], [-3], [-2]], array([2, -4])].toarray(), + B[[[-1], [-3], [-2]], [2, -4]] + ) + assert_equal( + A[array([[-1], [-3], [-2]]), array([2, -4])].toarray(), + B[[[-1], [-3], [-2]], [2, -4]] + ) + + # [[1,2]] + assert_equal(A[[1, 3]].toarray(), B[[1, 3]]) + assert_equal(A[[-1, -3]].toarray(), B[[-1, -3]]) + assert_equal(A[array([-1, -3])].toarray(), B[[-1, -3]]) + + # [[1,2],:][:,[1,2]] + assert_equal(A[[3], :][:, [4]].toarray(), B[[3], :][:, [4]]) + assert_equal( + A[[1, 3], :][:, [2, 4]].toarray(), B[[1, 3], :][:, [2, 4]] + ) + assert_equal( + A[[-1, -3], :][:, [2, -4]].toarray(), B[[-1, -3], :][:, [2, -4]] + ) + assert_equal( + A[array([-1, -3]), :][:, array([2, -4])].toarray(), + B[[-1, -3], :][:, [2, -4]] + ) + + # [1,[[1,2]]][[[1,2]],1] + assert_equal( + A[1, [[1, 3]]][[[0, 0]], 1].toarray(), B[1, [[1, 3]]][[[0, 0]], 1] + ) + assert_equal( + A[1, [[-1, -3]]][[[0, -1]], 1].toarray(), B[1, [[-1, -3]]][[[0, -1]], 1] + ) + # [:1,[[1,2]]][[[1,2]],:1] + with pytest.raises(IndexError, match="Only 1D or 2D arrays allowed"): + A[:1, [[1, 3]]] + with pytest.raises(IndexError, match="Only 1D or 2D arrays allowed"): + A[[[0, 0]], :1] + + # [:,[1,2]][[1,2],:] + assert_equal( + A[:, [1, 3]][[2, 4], :].toarray(), B[:, [1, 3]][[2, 4], :] + ) + assert_equal( + A[:, [-1, -3]][[2, -4], :].toarray(), B[:, [-1, -3]][[2, -4], :] + ) + assert_equal( + A[:, array([-1, -3])][array([2, -4]), :].toarray(), + B[:, [-1, -3]][[2, -4], :] + ) + + # Check bug reported by Robert Cimrman: + # http://thread.gmane.org/gmane.comp.python.scientific.devel/7986 (dead link) + s = slice(int8(2),int8(4),None) + assert_equal(A[s, :].toarray(), B[2:4, :]) + assert_equal(A[:, s].toarray(), B[:, 2:4]) + + # Regression for gh-4917: index with tuple of 2D arrays + i = np.array([[1]], dtype=int) + assert_equal(A[i, i].toarray(), B[i, i]) + + # Regression for gh-4917: index with tuple of empty nested lists + assert_equal(A[[[]], [[]]].toarray(), B[[[]], [[]]]) + + def test_fancy_indexing_randomized(self): + np.random.seed(1234) # make runs repeatable + + NUM_SAMPLES = 50 + M = 6 + N = 4 + + D = self.asdense(np.random.rand(M,N)) + D = np.multiply(D, D > 0.5) + + I = np.random.randint(-M + 1, M, size=NUM_SAMPLES) + J = np.random.randint(-N + 1, N, size=NUM_SAMPLES) + + S = self.spcreator(D) + + SIJ = S[I,J] + if issparse(SIJ): + SIJ = SIJ.toarray() + assert_equal(SIJ, D[I,J]) + + I_bad = I + M + J_bad = J - N + + assert_raises(IndexError, S.__getitem__, (I_bad,J)) + assert_raises(IndexError, S.__getitem__, (I,J_bad)) + + def test_missized_masking(self): + M, N = 5, 10 + + B = self.asdense(arange(M * N).reshape(M, N)) + A = self.spcreator(B) + + # Content of mask shouldn't matter, only its size + row_long = np.ones(M + 1, dtype=bool) + row_short = np.ones(M - 1, dtype=bool) + col_long = np.ones(N + 2, dtype=bool) + col_short = np.ones(N - 2, dtype=bool) + + match="bool index .* has shape .* instead of .*" + for i, j in itertools.product( + (row_long, row_short, slice(None)), + (col_long, col_short, slice(None)), + ): + if isinstance(i, slice) and isinstance(j, slice): + continue + with pytest.raises(IndexError, match=match): + _ = A[i, j] + + def test_fancy_indexing_boolean(self): + np.random.seed(1234) # make runs repeatable + + B = self.asdense(arange(50).reshape(5,10)) + A = self.spcreator(B) + + I = np.array(np.random.randint(0, 2, size=5), dtype=bool) + J = np.array(np.random.randint(0, 2, size=10), dtype=bool) + X = np.array(np.random.randint(0, 2, size=(5, 10)), dtype=bool) + + assert_equal(toarray(A[I]), B[I]) + assert_equal(toarray(A[:, J]), B[:, J]) + assert_equal(toarray(A[X]), B[X]) + assert_equal(toarray(A[B > 9]), B[B > 9]) + + I = np.array([True, False, True, True, False]) + J = np.array([False, True, True, False, True, + False, False, False, False, False]) + + assert_equal(toarray(A[I, J]), B[I, J]) + + Z1 = np.zeros((6, 11), dtype=bool) + Z2 = np.zeros((6, 11), dtype=bool) + Z2[0,-1] = True + Z3 = np.zeros((6, 11), dtype=bool) + Z3[-1,0] = True + + assert_raises(IndexError, A.__getitem__, Z1) + assert_raises(IndexError, A.__getitem__, Z2) + assert_raises(IndexError, A.__getitem__, Z3) + assert_raises((IndexError, ValueError), A.__getitem__, (X, 1)) + + def test_fancy_indexing_sparse_boolean(self): + np.random.seed(1234) # make runs repeatable + + B = self.asdense(arange(50).reshape(5,10)) + A = self.spcreator(B) + + X = np.array(np.random.randint(0, 2, size=(5, 10)), dtype=bool) + + Xsp = self.csr_container(X) + + assert_equal(toarray(A[Xsp]), B[X]) + assert_equal(toarray(A[A > 9]), B[B > 9]) + + Z = np.array(np.random.randint(0, 2, size=(5, 11)), dtype=bool) + Y = np.array(np.random.randint(0, 2, size=(6, 10)), dtype=bool) + + Zsp = self.csr_container(Z) + Ysp = self.csr_container(Y) + + assert_raises(IndexError, A.__getitem__, Zsp) + assert_raises(IndexError, A.__getitem__, Ysp) + assert_raises((IndexError, ValueError), A.__getitem__, (Xsp, 1)) + + def test_fancy_indexing_regression_3087(self): + mat = self.spcreator(array([[1, 0, 0], [0,1,0], [1,0,0]])) + desired_cols = np.ravel(mat.sum(0)) > 0 + assert_equal(mat[:, desired_cols].toarray(), [[1, 0], [0, 1], [1, 0]]) + + def test_fancy_indexing_seq_assign(self): + mat = self.spcreator(array([[1, 0], [0, 1]])) + assert_raises(ValueError, mat.__setitem__, (0, 0), np.array([1,2])) + + def test_fancy_indexing_2d_assign(self): + # regression test for gh-10695 + mat = self.spcreator(array([[1, 0], [2, 3]])) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + mat[[0, 1], [1, 1]] = mat[[1, 0], [0, 0]] + assert_equal(toarray(mat), array([[1, 2], [2, 1]])) + + def test_fancy_indexing_empty(self): + B = self.asdense(arange(50).reshape(5,10)) + B[1,:] = 0 + B[:,2] = 0 + B[3,6] = 0 + A = self.spcreator(B) + + K = np.array([False, False, False, False, False]) + assert_equal(toarray(A[K]), B[K]) + K = np.array([], dtype=int) + assert_equal(toarray(A[K]), B[K]) + assert_equal(toarray(A[K, K]), B[K, K]) + J = np.array([0, 1, 2, 3, 4], dtype=int)[:,None] + assert_equal(toarray(A[K, J]), B[K, J]) + assert_equal(toarray(A[J, K]), B[J, K]) + + +@contextlib.contextmanager +def check_remains_sorted(X): + """Checks that sorted indices property is retained through an operation + """ + if not hasattr(X, 'has_sorted_indices') or not X.has_sorted_indices: + yield + return + yield + indices = X.indices.copy() + X.has_sorted_indices = False + X.sort_indices() + assert_array_equal(indices, X.indices, + 'Expected sorted indices, found unsorted') + + +class _TestFancyIndexingAssign: + def test_bad_index_assign(self): + A = self.spcreator(np.zeros([5, 5])) + assert_raises((IndexError, ValueError, TypeError), A.__setitem__, "foo", 2) + assert_raises((IndexError, ValueError, TypeError), A.__setitem__, (2, "foo"), 5) + + def test_fancy_indexing_set(self): + n, m = (5, 10) + + def _test_set_slice(i, j): + A = self.spcreator((n, m)) + B = self.asdense(np.zeros((n, m))) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + B[i, j] = 1 + with check_remains_sorted(A): + A[i, j] = 1 + assert_array_almost_equal(A.toarray(), B) + # [1:2,1:2] + for i, j in [((2, 3, 4), slice(None, 10, 4)), + (np.arange(3), slice(5, -2)), + (slice(2, 5), slice(5, -2))]: + _test_set_slice(i, j) + for i, j in [(np.arange(3), np.arange(3)), ((0, 3, 4), (1, 2, 4))]: + _test_set_slice(i, j) + + def test_fancy_assignment_dtypes(self): + def check(dtype): + A = self.spcreator((5, 5), dtype=dtype) + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + A[[0,1],[0,1]] = dtype.type(1) + assert_equal(A.sum(), dtype.type(1)*2) + A[0:2,0:2] = dtype.type(1.0) + assert_equal(A.sum(), dtype.type(1)*4) + A[2,2] = dtype.type(1.0) + assert_equal(A.sum(), dtype.type(1)*4 + dtype.type(1)) + + for dtype in supported_dtypes: + check(np.dtype(dtype)) + + def test_sequence_assignment(self): + A = self.spcreator((4,3)) + B = self.spcreator(eye(3,4)) + + i0 = [0,1,2] + i1 = (0,1,2) + i2 = array(i0) + + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + with check_remains_sorted(A): + A[0,i0] = B[i0,0].T + A[1,i1] = B[i1,1].T + A[2,i2] = B[i2,2].T + assert_array_equal(A.toarray(), B.T.toarray()) + + # column slice + A = self.spcreator((2,3)) + with check_remains_sorted(A): + A[1,1:3] = [10,20] + assert_array_equal(A.toarray(), [[0, 0, 0], [0, 10, 20]]) + + # row slice + A = self.spcreator((3,2)) + with check_remains_sorted(A): + A[1:3,1] = [[10],[20]] + assert_array_equal(A.toarray(), [[0, 0], [0, 10], [0, 20]]) + + # both slices + A = self.spcreator((3,3)) + B = self.asdense(np.zeros((3,3))) + with check_remains_sorted(A): + for C in [A, B]: + C[[0,1,2], [0,1,2]] = [4,5,6] + assert_array_equal(A.toarray(), B) + + # both slices (2) + A = self.spcreator((4, 3)) + with check_remains_sorted(A): + A[(1, 2, 3), (0, 1, 2)] = [1, 2, 3] + assert_almost_equal(A.sum(), 6) + B = self.asdense(np.zeros((4, 3))) + B[(1, 2, 3), (0, 1, 2)] = [1, 2, 3] + assert_array_equal(A.toarray(), B) + + def test_fancy_assign_empty(self): + B = self.asdense(arange(50).reshape(5,10)) + B[1,:] = 0 + B[:,2] = 0 + B[3,6] = 0 + A = self.spcreator(B) + + K = np.array([False, False, False, False, False]) + A[K] = 42 + assert_equal(toarray(A), B) + + K = np.array([], dtype=int) + A[K] = 42 + assert_equal(toarray(A), B) + A[K,K] = 42 + assert_equal(toarray(A), B) + + J = np.array([0, 1, 2, 3, 4], dtype=int)[:,None] + A[K,J] = 42 + assert_equal(toarray(A), B) + A[J,K] = 42 + assert_equal(toarray(A), B) + + +class _TestFancyMultidim: + def test_fancy_indexing_ndarray(self): + sets = [ + (np.array([[1], [2], [3]]), np.array([3, 4, 2])), + (np.array([[1], [2], [3]]), np.array([[3, 4, 2]])), + (np.array([[1, 2, 3]]), np.array([[3], [4], [2]])), + (np.array([1, 2, 3]), np.array([[3], [4], [2]])), + (np.array([[1, 2, 3], [3, 4, 2]]), + np.array([[5, 6, 3], [2, 3, 1]])) + ] + # These inputs generate 3-D outputs + # (np.array([[[1], [2], [3]], [[3], [4], [2]]]), + # np.array([[[5], [6], [3]], [[2], [3], [1]]])), + + for I, J in sets: + np.random.seed(1234) + D = self.asdense(np.random.rand(5, 7)) + S = self.spcreator(D) + + SIJ = S[I,J] + if issparse(SIJ): + SIJ = SIJ.toarray() + assert_equal(SIJ, D[I,J]) + + I_bad = I + 5 + J_bad = J + 7 + + assert_raises(IndexError, S.__getitem__, (I_bad,J)) + assert_raises(IndexError, S.__getitem__, (I,J_bad)) + + # This would generate 3-D arrays -- not supported + assert_raises(IndexError, S.__getitem__, ([I, I], slice(None))) + assert_raises(IndexError, S.__getitem__, (slice(None), [J, J])) + + +class _TestFancyMultidimAssign: + def test_fancy_assign_ndarray(self): + np.random.seed(1234) + + D = self.asdense(np.random.rand(5, 7)) + S = self.spcreator(D) + X = np.random.rand(2, 3) + + I = np.array([[1, 2, 3], [3, 4, 2]]) + J = np.array([[5, 6, 3], [2, 3, 1]]) + + with check_remains_sorted(S): + S[I,J] = X + D[I,J] = X + assert_equal(S.toarray(), D) + + I_bad = I + 5 + J_bad = J + 7 + + C = [1, 2, 3] + + with check_remains_sorted(S): + S[I,J] = C + D[I,J] = C + assert_equal(S.toarray(), D) + + with check_remains_sorted(S): + S[I,J] = 3 + D[I,J] = 3 + assert_equal(S.toarray(), D) + + assert_raises(IndexError, S.__setitem__, (I_bad,J), C) + assert_raises(IndexError, S.__setitem__, (I,J_bad), C) + + def test_fancy_indexing_multidim_set(self): + n, m = (5, 10) + + def _test_set_slice(i, j): + A = self.spcreator((n, m)) + with check_remains_sorted(A), suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + A[i, j] = 1 + B = self.asdense(np.zeros((n, m))) + B[i, j] = 1 + assert_array_almost_equal(A.toarray(), B) + # [[[1, 2], [1, 2]], [1, 2]] + for i, j in [(np.array([[1, 2], [1, 3]]), [1, 3]), + (np.array([0, 4]), [[0, 3], [1, 2]]), + ([[1, 2, 3], [0, 2, 4]], [[0, 4, 3], [4, 1, 2]])]: + _test_set_slice(i, j) + + def test_fancy_assign_list(self): + np.random.seed(1234) + + D = self.asdense(np.random.rand(5, 7)) + S = self.spcreator(D) + X = np.random.rand(2, 3) + + I = [[1, 2, 3], [3, 4, 2]] + J = [[5, 6, 3], [2, 3, 1]] + + S[I,J] = X + D[I,J] = X + assert_equal(S.toarray(), D) + + I_bad = [[ii + 5 for ii in i] for i in I] + J_bad = [[jj + 7 for jj in j] for j in J] + C = [1, 2, 3] + + S[I,J] = C + D[I,J] = C + assert_equal(S.toarray(), D) + + S[I,J] = 3 + D[I,J] = 3 + assert_equal(S.toarray(), D) + + assert_raises(IndexError, S.__setitem__, (I_bad,J), C) + assert_raises(IndexError, S.__setitem__, (I,J_bad), C) + + def test_fancy_assign_slice(self): + np.random.seed(1234) + + D = self.asdense(np.random.rand(5, 7)) + S = self.spcreator(D) + + I = [1, 2, 3, 3, 4, 2] + J = [5, 6, 3, 2, 3, 1] + + I_bad = [ii + 5 for ii in I] + J_bad = [jj + 7 for jj in J] + + C1 = [1, 2, 3, 4, 5, 6, 7] + C2 = np.arange(5)[:, None] + assert_raises(IndexError, S.__setitem__, (I_bad, slice(None)), C1) + assert_raises(IndexError, S.__setitem__, (slice(None), J_bad), C2) + + +class _TestArithmetic: + """ + Test real/complex arithmetic + """ + def __arith_init(self): + # these can be represented exactly in FP (so arithmetic should be exact) + __A = array([[-1.5, 6.5, 0, 2.25, 0, 0], + [3.125, -7.875, 0.625, 0, 0, 0], + [0, 0, -0.125, 1.0, 0, 0], + [0, 0, 8.375, 0, 0, 0]], 'float64') + __B = array([[0.375, 0, 0, 0, -5, 2.5], + [14.25, -3.75, 0, 0, -0.125, 0], + [0, 7.25, 0, 0, 0, 0], + [18.5, -0.0625, 0, 0, 0, 0]], 'complex128') + __B.imag = array([[1.25, 0, 0, 0, 6, -3.875], + [2.25, 4.125, 0, 0, 0, 2.75], + [0, 4.125, 0, 0, 0, 0], + [-0.0625, 0, 0, 0, 0, 0]], 'float64') + + # fractions are all x/16ths + assert_array_equal((__A*16).astype('int32'),16*__A) + assert_array_equal((__B.real*16).astype('int32'),16*__B.real) + assert_array_equal((__B.imag*16).astype('int32'),16*__B.imag) + + __Asp = self.spcreator(__A) + __Bsp = self.spcreator(__B) + return __A, __B, __Asp, __Bsp + + @pytest.mark.fail_slow(20) + def test_add_sub(self): + __A, __B, __Asp, __Bsp = self.__arith_init() + + # basic tests + assert_array_equal( + (__Asp + __Bsp).toarray(), __A + __B + ) + + # check conversions + for x in supported_dtypes: + with np.errstate(invalid="ignore"): + A = __A.astype(x) + Asp = self.spcreator(A) + for y in supported_dtypes: + if not np.issubdtype(y, np.complexfloating): + with np.errstate(invalid="ignore"): + B = __B.real.astype(y) + else: + B = __B.astype(y) + Bsp = self.spcreator(B) + + # addition + D1 = A + B + S1 = Asp + Bsp + + assert_equal(S1.dtype,D1.dtype) + assert_array_equal(S1.toarray(), D1) + assert_array_equal(Asp + B,D1) # check sparse + dense + assert_array_equal(A + Bsp,D1) # check dense + sparse + + # subtraction + if np.dtype('bool') in [x, y]: + # boolean array subtraction deprecated in 1.9.0 + continue + + D1 = A - B + S1 = Asp - Bsp + + assert_equal(S1.dtype,D1.dtype) + assert_array_equal(S1.toarray(), D1) + assert_array_equal(Asp - B,D1) # check sparse - dense + assert_array_equal(A - Bsp,D1) # check dense - sparse + + def test_mu(self): + __A, __B, __Asp, __Bsp = self.__arith_init() + + # basic tests + assert_array_equal((__Asp @ __Bsp.T).toarray(), + __A @ __B.T) + + for x in supported_dtypes: + with np.errstate(invalid="ignore"): + A = __A.astype(x) + Asp = self.spcreator(A) + for y in supported_dtypes: + if np.issubdtype(y, np.complexfloating): + B = __B.astype(y) + else: + with np.errstate(invalid="ignore"): + B = __B.real.astype(y) + Bsp = self.spcreator(B) + + D1 = A @ B.T + S1 = Asp @ Bsp.T + + assert_allclose(S1.toarray(), D1, + atol=1e-14*abs(D1).max()) + assert_equal(S1.dtype,D1.dtype) + + +class _TestMinMax: + def test_minmax(self): + for dtype in [np.float32, np.float64, np.int32, np.int64, np.complex128]: + D = np.arange(20, dtype=dtype).reshape(5,4) + + X = self.spcreator(D) + assert_equal(X.min(), 0) + assert_equal(X.max(), 19) + assert_equal(X.min().dtype, dtype) + assert_equal(X.max().dtype, dtype) + + D *= -1 + X = self.spcreator(D) + assert_equal(X.min(), -19) + assert_equal(X.max(), 0) + + D += 5 + X = self.spcreator(D) + assert_equal(X.min(), -14) + assert_equal(X.max(), 5) + + # try a fully dense matrix + X = self.spcreator(np.arange(1, 10).reshape(3, 3)) + assert_equal(X.min(), 1) + assert_equal(X.min().dtype, X.dtype) + + X = -X + assert_equal(X.max(), -1) + + # and a fully sparse matrix + Z = self.spcreator(np.zeros((1, 1))) + assert_equal(Z.min(), 0) + assert_equal(Z.max(), 0) + assert_equal(Z.max().dtype, Z.dtype) + + # another test + D = np.arange(20, dtype=float).reshape(5,4) + D[0:2, :] = 0 + X = self.spcreator(D) + assert_equal(X.min(), 0) + assert_equal(X.max(), 19) + + # zero-size matrices + for D in [np.zeros((0, 0)), np.zeros((0, 10)), np.zeros((10, 0))]: + X = self.spcreator(D) + assert_raises(ValueError, X.min) + assert_raises(ValueError, X.max) + + def test_minmax_axis(self): + keep = not self.is_array_test + D = np.arange(50).reshape(5, 10) + # completely empty rows, leaving some completely full: + D[1, :] = 0 + # empty at end for reduceat: + D[:, 9] = 0 + # partial rows/cols: + D[3, 3] = 0 + # entries on either side of 0: + D[2, 2] = -1 + X = self.spcreator(D) + + axes_even = [0, -2] + axes_odd = [1, -1] + for axis in axes_odd + axes_even: + assert_array_equal( + X.max(axis=axis).toarray(), D.max(axis=axis, keepdims=keep) + ) + assert_array_equal( + X.min(axis=axis).toarray(), D.min(axis=axis, keepdims=keep) + ) + + for axis in axes_even: + assert_equal( + X.max(axis=axis, explicit=True).toarray(), + self.asdense([40, 41, 42, 43, 44, 45, 46, 47, 48, 0]) + ) + if np.any(X.data == 0): + # Noncanonical case + expected = self.asdense([20, 1, -1, 3, 4, 5, 0, 7, 8, 0]) + else: + expected = self.asdense([20, 1, -1, 3, 4, 5, 6, 7, 8, 0]) + assert_equal(X.min(axis=axis, explicit=True).toarray(), expected) + + for axis in axes_odd: + expected_max = np.array([8, 0, 28, 38, 48]) + expected_min = np.array([1, 0, -1, 30, 40]) + if not self.is_array_test: + expected_max = expected_max.reshape((5, 1)) + expected_min = expected_min.reshape((5, 1)) + assert_equal(X.max(axis=axis, explicit=True).toarray(), expected_max) + assert_equal(X.min(axis=axis, explicit=True).toarray(), expected_min) + + # full matrix + D = np.arange(1, 51).reshape(10, 5) + X = self.spcreator(D) + for axis in axes_odd + axes_even: + assert_array_equal( + X.max(axis=axis).toarray(), D.max(axis=axis, keepdims=keep) + ) + assert_array_equal( + X.min(axis=axis).toarray(), D.min(axis=axis, keepdims=keep) + ) + + for axis in axes_even: + expected_max = D[-1, :] + expected_min = D[0, :] + if not self.is_array_test: + expected_max = D[None, -1, :] + expected_min = D[None, 0, :] + assert_equal(X.max(axis=axis, explicit=True).toarray(), expected_max) + assert_equal(X.min(axis=axis, explicit=True).toarray(), expected_min) + for axis in axes_odd: + expected_max = D[:, -1] + expected_min = D[:, 0] + if not self.is_array_test: + expected_max = D[:, -1, None] + expected_min = D[:, 0, None] + assert_equal(X.max(axis=axis, explicit=True).toarray(), expected_max) + assert_equal(X.min(axis=axis, explicit=True).toarray(), expected_min) + + # empty matrix + D = self.asdense(np.zeros((10, 5))) + X = self.spcreator(D) + for axis in axes_even + axes_odd: + assert_equal(X.max(axis=axis, explicit=True).toarray(), D.max(axis=axis)) + assert_equal(X.min(axis=axis, explicit=True).toarray(), D.min(axis=axis)) + + # zero-size matrices + D = self.asdense(np.zeros((0, 10))) + X = self.spcreator(D) + explicit_values = [True, False] + even_explicit_pairs = list(itertools.product(axes_even, explicit_values)) + odd_explicit_pairs = list(itertools.product(axes_odd, explicit_values)) + for axis, ex in even_explicit_pairs: + assert_raises(ValueError, X.min, axis=axis, explicit=ex) + assert_raises(ValueError, X.max, axis=axis, explicit=ex) + for axis, ex in odd_explicit_pairs: + assert_equal(X.max(axis=axis, explicit=ex).toarray(), D.max(axis=axis)) + assert_equal(X.min(axis=axis, explicit=ex).toarray(), D.min(axis=axis)) + + D = self.asdense(np.zeros((10, 0))) + X = self.spcreator(D) + for axis, ex in odd_explicit_pairs: + assert_raises(ValueError, X.min, axis=axis, explicit=ex) + assert_raises(ValueError, X.max, axis=axis, explicit=ex) + for axis, ex in even_explicit_pairs: + assert_equal(X.max(axis=axis, explicit=ex).toarray(), D.max(axis=axis)) + assert_equal(X.min(axis=axis, explicit=ex).toarray(), D.min(axis=axis)) + + def test_nanminmax(self): + D = self.asdense(np.arange(50).reshape(5,10), dtype=float) + D[1, :] = 0 + D[:, 9] = 0 + D[3, 3] = 0 + D[2, 2] = -1 + D[4, 2] = np.nan + D[1, 4] = np.nan + X = self.spcreator(D) + + X_nan_maximum = X.nanmax() + assert np.isscalar(X_nan_maximum) + assert X_nan_maximum == np.nanmax(D) + + X_nan_minimum = X.nanmin() + assert np.isscalar(X_nan_minimum) + assert X_nan_minimum == np.nanmin(D) + + axes = [-2, -1, 0, 1] + for axis in axes: + X_nan_maxima = X.nanmax(axis=axis) + assert_allclose(X_nan_maxima.toarray(), np.nanmax(D, axis=axis)) + assert isinstance(X_nan_maxima, self.coo_container) + + X_nan_minima = X.nanmin(axis=axis) + assert_allclose(X_nan_minima.toarray(), np.nanmin(D, axis=axis)) + assert isinstance(X_nan_minima, self.coo_container) + + def test_minmax_invalid_params(self): + dat = array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + datsp = self.spcreator(dat) + + for fname in ('min', 'max'): + func = getattr(datsp, fname) + assert_raises(ValueError, func, axis=3) + assert_raises(TypeError, func, axis=(0, 1)) + assert_raises(TypeError, func, axis=1.5) + assert_raises(ValueError, func, axis=1, out=1) + + def test_numpy_minmax(self): + # See gh-5987 + # xref gh-7460 in 'numpy' + from scipy.sparse import _data + + dat = array([[0, 1, 2], + [3, -4, 5], + [-6, 7, 9]]) + datsp = self.spcreator(dat) + + # We are only testing sparse matrices who have + # implemented 'min' and 'max' because they are + # the ones with the compatibility issues with + # the 'numpy' implementation. + if isinstance(datsp, _data._minmax_mixin): + assert_array_equal(np.min(datsp), np.min(dat)) + assert_array_equal(np.max(datsp), np.max(dat)) + + def test_argmax(self): + from scipy.sparse import _data + D1 = np.array([ + [-1, 5, 2, 3], + [0, 0, -1, -2], + [-1, -2, -3, -4], + [1, 2, 3, 4], + [1, 2, 0, 0], + ]) + D2 = D1.transpose() + # Non-regression test cases for gh-16929. + D3 = np.array([[4, 3], [7, 5]]) + D4 = np.array([[4, 3], [7, 0]]) + D5 = np.array([[5, 5, 3], [4, 9, 10], [3, 4, 9]]) + + for D in [D1, D2, D3, D4, D5]: + D = self.asdense(D) + mat = self.spcreator(D) + if not isinstance(mat, _data._minmax_mixin): + continue + + assert_equal(mat.argmax(), np.argmax(D)) + assert_equal(mat.argmin(), np.argmin(D)) + + assert_equal(mat.argmax(axis=0), np.argmax(D, axis=0)) + assert_equal(mat.argmin(axis=0), np.argmin(D, axis=0)) + + assert_equal(mat.argmax(axis=1), np.argmax(D, axis=1)) + assert_equal(mat.argmin(axis=1), np.argmin(D, axis=1)) + + # zero-size matrices + D6 = self.spcreator(np.empty((0, 5))) + D7 = self.spcreator(np.empty((5, 0))) + explicits = [True, False] + + for mat, axis, ex in itertools.product([D6, D7], [None, 0, 1], explicits): + if axis is None or mat.shape[axis] == 0: + with pytest.raises(ValueError, match="Cannot apply"): + mat.argmax(axis=axis, explicit=ex) + with pytest.raises(ValueError, match="Cannot apply"): + mat.argmin(axis=axis, explicit=ex) + else: + if self.is_array_test: + expected = np.zeros(0) + else: + expected = np.zeros((0, 1) if axis == 1 else (1, 0)) + assert_equal(mat.argmin(axis=axis, explicit=ex), expected) + assert_equal(mat.argmax(axis=axis, explicit=ex), expected) + + mat = self.spcreator(D1) + assert_equal(mat.argmax(axis=0, explicit=True), self.asdense([3, 0, 3, 3])) + assert_equal(mat.argmin(axis=0, explicit=True), self.asdense([0, 2, 2, 2])) + + expected_max = np.array([1, 2, 0, 3, 1]) + expected_min = np.array([0, 3, 3, 0, 0]) + if mat.nnz != 16: + # Noncanonical case + expected_min[-1] = 2 + if not self.is_array_test: + expected_max = expected_max.reshape((5, 1)) + expected_min = expected_min.reshape((5, 1)) + + assert_equal(mat.argmax(axis=1, explicit=True), expected_max) + assert_equal(asarray(mat.argmin(axis=1, explicit=True)), expected_min) + + # all zeros + D = np.zeros((2, 2)) + mat = self.spcreator(D) + if mat.nnz != 0: + # Noncanonical case + assert_equal(mat.argmin(axis=None, explicit=True), 0) + assert_equal(mat.argmax(axis=None, explicit=True), 0) + else: + # Canonical case + with pytest.raises(ValueError, match="Cannot apply"): + mat.argmin(axis=None, explicit=True) + with pytest.raises(ValueError, match="Cannot apply"): + mat.argmax(axis=None, explicit=True) + + +class _TestGetNnzAxis: + def test_getnnz_axis(self): + dat = array([[0, 2], + [3, 5], + [-6, 9]]) + bool_dat = dat.astype(bool) + datsp = self.spcreator(dat) + + accepted_return_dtypes = (np.int32, np.int64) + + getnnz = datsp.count_nonzero if self.is_array_test else datsp.getnnz + assert_array_equal(bool_dat.sum(axis=None), getnnz(axis=None)) + assert_array_equal(bool_dat.sum(), getnnz()) + assert_array_equal(bool_dat.sum(axis=0), getnnz(axis=0)) + assert_in(getnnz(axis=0).dtype, accepted_return_dtypes) + assert_array_equal(bool_dat.sum(axis=1), getnnz(axis=1)) + assert_in(getnnz(axis=1).dtype, accepted_return_dtypes) + assert_array_equal(bool_dat.sum(axis=-2), getnnz(axis=-2)) + assert_in(getnnz(axis=-2).dtype, accepted_return_dtypes) + assert_array_equal(bool_dat.sum(axis=-1), getnnz(axis=-1)) + assert_in(getnnz(axis=-1).dtype, accepted_return_dtypes) + + assert_raises(ValueError, getnnz, axis=2) + + +#------------------------------------------------------------------------------ +# Tailored base class for generic tests +#------------------------------------------------------------------------------ + +def _possibly_unimplemented(cls, require=True): + """ + Construct a class that either runs tests as usual (require=True), + or each method skips if it encounters a common error. + """ + if require: + return cls + else: + def wrap(fc): + @functools.wraps(fc) + def wrapper(*a, **kw): + try: + return fc(*a, **kw) + except (NotImplementedError, TypeError, ValueError, + IndexError, AttributeError): + raise pytest.skip("feature not implemented") + + return wrapper + + new_dict = dict(cls.__dict__) + for name, func in cls.__dict__.items(): + if name.startswith('test_'): + new_dict[name] = wrap(func) + return type(cls.__name__ + "NotImplemented", + cls.__bases__, + new_dict) + + +def sparse_test_class(getset=True, slicing=True, slicing_assign=True, + fancy_indexing=True, fancy_assign=True, + fancy_multidim_indexing=True, fancy_multidim_assign=True, + minmax=True, nnz_axis=True): + """ + Construct a base class, optionally converting some of the tests in + the suite to check that the feature is not implemented. + """ + bases = (_TestCommon, + _possibly_unimplemented(_TestGetSet, getset), + _TestSolve, + _TestInplaceArithmetic, + _TestArithmetic, + _possibly_unimplemented(_TestSlicing, slicing), + _possibly_unimplemented(_TestSlicingAssign, slicing_assign), + _possibly_unimplemented(_TestFancyIndexing, fancy_indexing), + _possibly_unimplemented(_TestFancyIndexingAssign, + fancy_assign), + _possibly_unimplemented(_TestFancyMultidim, + fancy_indexing and fancy_multidim_indexing), + _possibly_unimplemented(_TestFancyMultidimAssign, + fancy_multidim_assign and fancy_assign), + _possibly_unimplemented(_TestMinMax, minmax), + _possibly_unimplemented(_TestGetNnzAxis, nnz_axis)) + + # check that test names do not clash + names = {} + for cls in bases: + for name in cls.__dict__: + if not name.startswith('test_'): + continue + old_cls = names.get(name) + if old_cls is not None: + raise ValueError(f"Test class {cls.__name__} overloads test " + f"{name} defined in {old_cls.__name__}") + names[name] = cls + + return type("TestBase", bases, {}) + + +#------------------------------------------------------------------------------ +# Matrix class based tests +#------------------------------------------------------------------------------ + +class TestCSR(sparse_test_class()): + @classmethod + def spcreator(cls, *args, **kwargs): + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + return csr_array(*args, **kwargs) + math_dtypes = [np.bool_, np.int_, np.float64, np.complex128] + + def test_constructor1(self): + b = array([[0, 4, 0], + [3, 0, 0], + [0, 2, 0]], 'd') + bsp = self.csr_container(b) + assert_array_almost_equal(bsp.data,[4,3,2]) + assert_array_equal(bsp.indices,[1,0,1]) + assert_array_equal(bsp.indptr,[0,1,2,3]) + assert_equal(bsp.nnz,3) + assert_equal(bsp.format,'csr') + assert_array_equal(bsp.toarray(), b) + + def test_constructor2(self): + b = zeros((6,6),'d') + b[3,4] = 5 + bsp = self.csr_container(b) + assert_array_almost_equal(bsp.data,[5]) + assert_array_equal(bsp.indices,[4]) + assert_array_equal(bsp.indptr,[0,0,0,0,1,1,1]) + assert_array_almost_equal(bsp.toarray(), b) + + def test_constructor3(self): + b = array([[1, 0], + [0, 2], + [3, 0]], 'd') + bsp = self.csr_container(b) + assert_array_almost_equal(bsp.data,[1,2,3]) + assert_array_equal(bsp.indices,[0,1,0]) + assert_array_equal(bsp.indptr,[0,1,2,3]) + assert_array_almost_equal(bsp.toarray(), b) + + def test_constructor4(self): + # using (data, ij) format + row = array([2, 3, 1, 3, 0, 1, 3, 0, 2, 1, 2]) + col = array([0, 1, 0, 0, 1, 1, 2, 2, 2, 2, 1]) + data = array([6., 10., 3., 9., 1., 4., + 11., 2., 8., 5., 7.]) + + ij = vstack((row,col)) + csr = self.csr_container((data,ij),(4,3)) + assert_array_equal(arange(12).reshape(4, 3), csr.toarray()) + + # using Python lists and a specified dtype + csr = self.csr_container(([2**63 + 1, 1], ([0, 1], [0, 1])), dtype=np.uint64) + dense = array([[2**63 + 1, 0], [0, 1]], dtype=np.uint64) + assert_array_equal(dense, csr.toarray()) + + # with duplicates (should sum the duplicates) + csr = self.csr_container(([1,1,1,1], ([0,2,2,0], [0,1,1,0]))) + assert csr.nnz == 2 + + def test_constructor5(self): + # infer dimensions from arrays + indptr = array([0,1,3,3]) + indices = array([0,5,1,2]) + data = array([1,2,3,4]) + csr = self.csr_container((data, indices, indptr)) + assert_array_equal(csr.shape,(3,6)) + + def test_constructor6(self): + # infer dimensions and dtype from lists + indptr = [0, 1, 3, 3] + indices = [0, 5, 1, 2] + data = [1, 2, 3, 4] + csr = self.csr_container((data, indices, indptr)) + assert_array_equal(csr.shape, (3,6)) + assert_(np.issubdtype(csr.dtype, np.signedinteger)) + + def test_constructor_smallcol(self): + # int64 indices not required + data = arange(6) + 1 + col = array([1, 2, 1, 0, 0, 2], dtype=np.int64) + ptr = array([0, 2, 4, 6], dtype=np.int64) + + a = self.csr_container((data, col, ptr), shape=(3, 3)) + + b = array([[0, 1, 2], + [4, 3, 0], + [5, 0, 6]], 'd') + + # sparray is less aggressive in downcasting indices to int32 than spmatrix + expected_dtype = np.dtype(np.int64 if self.is_array_test else np.int32) + assert_equal(a.indptr.dtype, expected_dtype) + assert_equal(a.indices.dtype, expected_dtype) + assert_array_equal(a.toarray(), b) + + def test_constructor_largecol(self): + # int64 indices required + data = arange(6) + 1 + large = np.iinfo(np.int32).max + 100 + col = array([0, 1, 2, large, large+1, large+2], dtype=np.int64) + ptr = array([0, 2, 4, 6], dtype=np.int64) + + a = self.csr_container((data, col, ptr)) + + assert_equal(a.indptr.dtype, np.dtype(np.int64)) + assert_equal(a.indices.dtype, np.dtype(np.int64)) + assert_array_equal(a.shape, (3, max(col)+1)) + + def test_sort_indices(self): + data = arange(5) + indices = array([7, 2, 1, 5, 4]) + indptr = array([0, 3, 5]) + asp = self.csr_container((data, indices, indptr), shape=(2,10)) + bsp = asp.copy() + asp.sort_indices() + assert_array_equal(asp.indices,[1, 2, 7, 4, 5]) + assert_array_equal(asp.toarray(), bsp.toarray()) + + def test_eliminate_zeros(self): + data = array([1, 0, 0, 0, 2, 0, 3, 0]) + indices = array([1, 2, 3, 4, 5, 6, 7, 8]) + indptr = array([0, 3, 8]) + asp = self.csr_container((data, indices, indptr), shape=(2,10)) + bsp = asp.copy() + asp.eliminate_zeros() + assert_array_equal(asp.nnz, 3) + assert_array_equal(asp.data,[1, 2, 3]) + assert_array_equal(asp.toarray(), bsp.toarray()) + + def test_ufuncs(self): + X = self.csr_container(np.arange(20).reshape(4, 5) / 20.) + for f in ["sin", "tan", "arcsin", "arctan", "sinh", "tanh", + "arcsinh", "arctanh", "rint", "sign", "expm1", "log1p", + "deg2rad", "rad2deg", "floor", "ceil", "trunc", "sqrt"]: + assert_equal(hasattr(self.datsp, f), True) + X2 = getattr(X, f)() + assert_equal(X.shape, X2.shape) + assert_array_equal(X.indices, X2.indices) + assert_array_equal(X.indptr, X2.indptr) + assert_array_equal(X2.toarray(), getattr(np, f)(X.toarray())) + + def test_unsorted_arithmetic(self): + data = arange(5) + indices = array([7, 2, 1, 5, 4]) + indptr = array([0, 3, 5]) + asp = self.csr_container((data, indices, indptr), shape=(2,10)) + data = arange(6) + indices = array([8, 1, 5, 7, 2, 4]) + indptr = array([0, 2, 6]) + bsp = self.csr_container((data, indices, indptr), shape=(2,10)) + assert_equal((asp + bsp).toarray(), asp.toarray() + bsp.toarray()) + + def test_fancy_indexing_broadcast(self): + # broadcasting indexing mode is supported + I = np.array([[1], [2], [3]]) + J = np.array([3, 4, 2]) + + np.random.seed(1234) + D = self.asdense(np.random.rand(5, 7)) + S = self.spcreator(D) + + SIJ = S[I,J] + if issparse(SIJ): + SIJ = SIJ.toarray() + assert_equal(SIJ, D[I,J]) + + def test_has_sorted_indices(self): + "Ensure has_sorted_indices memoizes sorted state for sort_indices" + sorted_inds = np.array([0, 1]) + unsorted_inds = np.array([1, 0]) + data = np.array([1, 1]) + indptr = np.array([0, 2]) + M = self.csr_container((data, sorted_inds, indptr)).copy() + assert_equal(True, M.has_sorted_indices) + assert isinstance(M.has_sorted_indices, bool) + + M = self.csr_container((data, unsorted_inds, indptr)).copy() + assert_equal(False, M.has_sorted_indices) + + # set by sorting + M.sort_indices() + assert_equal(True, M.has_sorted_indices) + assert_array_equal(M.indices, sorted_inds) + + M = self.csr_container((data, unsorted_inds, indptr)).copy() + # set manually (although underlyingly unsorted) + M.has_sorted_indices = True + assert_equal(True, M.has_sorted_indices) + assert_array_equal(M.indices, unsorted_inds) + + # ensure sort bypassed when has_sorted_indices == True + M.sort_indices() + assert_array_equal(M.indices, unsorted_inds) + + def test_has_canonical_format(self): + "Ensure has_canonical_format memoizes state for sum_duplicates" + + M = self.csr_container((np.array([2]), np.array([0]), np.array([0, 1]))) + assert_equal(True, M.has_canonical_format) + + indices = np.array([0, 0]) # contains duplicate + data = np.array([1, 1]) + indptr = np.array([0, 2]) + + M = self.csr_container((data, indices, indptr)).copy() + assert_equal(False, M.has_canonical_format) + assert isinstance(M.has_canonical_format, bool) + + # set by deduplicating + M.sum_duplicates() + assert_equal(True, M.has_canonical_format) + assert_equal(1, len(M.indices)) + + M = self.csr_container((data, indices, indptr)).copy() + # set manually (although underlyingly duplicated) + M.has_canonical_format = True + assert_equal(True, M.has_canonical_format) + assert_equal(2, len(M.indices)) # unaffected content + + # ensure deduplication bypassed when has_canonical_format == True + M.sum_duplicates() + assert_equal(2, len(M.indices)) # unaffected content + + def test_scalar_idx_dtype(self): + # Check that index dtype takes into account all parameters + # passed to sparsetools, including the scalar ones + indptr = np.zeros(2, dtype=np.int32) + indices = np.zeros(0, dtype=np.int32) + vals = np.zeros(0) + a = self.csr_container((vals, indices, indptr), shape=(1, 2**31-1)) + b = self.csr_container((vals, indices, indptr), shape=(1, 2**31)) + ij = np.zeros((2, 0), dtype=np.int32) + c = self.csr_container((vals, ij), shape=(1, 2**31-1)) + d = self.csr_container((vals, ij), shape=(1, 2**31)) + e = self.csr_container((1, 2**31-1)) + f = self.csr_container((1, 2**31)) + assert_equal(a.indptr.dtype, np.int32) + assert_equal(b.indptr.dtype, np.int64) + assert_equal(c.indptr.dtype, np.int32) + assert_equal(d.indptr.dtype, np.int64) + assert_equal(e.indptr.dtype, np.int32) + assert_equal(f.indptr.dtype, np.int64) + + # These shouldn't fail + for x in [a, b, c, d, e, f]: + x + x + + def test_setdiag_csr(self): + # see gh-21791 setting mixture of existing and not when new_values < 0.001*nnz + D = self.dia_container(([np.arange(1002)], [0]), shape=(1002, 1002)) + A = self.spcreator(D) + A.setdiag(5 * np.ones(A.shape[0])) + assert A[-1, -1] == 5 + + def test_binop_explicit_zeros(self): + # Check that binary ops don't introduce spurious explicit zeros. + # See gh-9619 for context. + a = self.csr_container([[0, 1, 0]]) + b = self.csr_container([[1, 1, 0]]) + assert (a + b).nnz == 2 + assert a.multiply(b).nnz == 1 + + +TestCSR.init_class() + + +class TestCSRMatrix(_MatrixMixin, TestCSR): + @classmethod + def spcreator(cls, *args, **kwargs): + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + return csr_matrix(*args, **kwargs) + + +TestCSRMatrix.init_class() + + +class TestCSC(sparse_test_class()): + @classmethod + def spcreator(cls, *args, **kwargs): + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + return csc_array(*args, **kwargs) + math_dtypes = [np.bool_, np.int_, np.float64, np.complex128] + + def test_constructor1(self): + b = array([[1, 0, 0, 0], [0, 0, 1, 0], [0, 2, 0, 3]], 'd') + bsp = self.csc_container(b) + assert_array_almost_equal(bsp.data,[1,2,1,3]) + assert_array_equal(bsp.indices,[0,2,1,2]) + assert_array_equal(bsp.indptr,[0,1,2,3,4]) + assert_equal(bsp.nnz,4) + assert_equal(bsp.shape,b.shape) + assert_equal(bsp.format,'csc') + + def test_constructor2(self): + b = zeros((6,6),'d') + b[2,4] = 5 + bsp = self.csc_container(b) + assert_array_almost_equal(bsp.data,[5]) + assert_array_equal(bsp.indices,[2]) + assert_array_equal(bsp.indptr,[0,0,0,0,0,1,1]) + + def test_constructor3(self): + b = array([[1, 0], [0, 0], [0, 2]], 'd') + bsp = self.csc_container(b) + assert_array_almost_equal(bsp.data,[1,2]) + assert_array_equal(bsp.indices,[0,2]) + assert_array_equal(bsp.indptr,[0,1,2]) + + def test_constructor4(self): + # using (data, ij) format + row = array([2, 3, 1, 3, 0, 1, 3, 0, 2, 1, 2]) + col = array([0, 1, 0, 0, 1, 1, 2, 2, 2, 2, 1]) + data = array([6., 10., 3., 9., 1., 4., 11., 2., 8., 5., 7.]) + + ij = vstack((row,col)) + csc = self.csc_container((data,ij),(4,3)) + assert_array_equal(arange(12).reshape(4, 3), csc.toarray()) + + # with duplicates (should sum the duplicates) + csc = self.csc_container(([1,1,1,1], ([0,2,2,0], [0,1,1,0]))) + assert csc.nnz == 2 + + def test_constructor5(self): + # infer dimensions from arrays + indptr = array([0,1,3,3]) + indices = array([0,5,1,2]) + data = array([1,2,3,4]) + csc = self.csc_container((data, indices, indptr)) + assert_array_equal(csc.shape,(6,3)) + + def test_constructor6(self): + # infer dimensions and dtype from lists + indptr = [0, 1, 3, 3] + indices = [0, 5, 1, 2] + data = [1, 2, 3, 4] + csc = self.csc_container((data, indices, indptr)) + assert_array_equal(csc.shape,(6,3)) + assert_(np.issubdtype(csc.dtype, np.signedinteger)) + + def test_eliminate_zeros(self): + data = array([1, 0, 0, 0, 2, 0, 3, 0]) + indices = array([1, 2, 3, 4, 5, 6, 7, 8]) + indptr = array([0, 3, 8]) + asp = self.csc_container((data, indices, indptr), shape=(10,2)) + bsp = asp.copy() + asp.eliminate_zeros() + assert_array_equal(asp.nnz, 3) + assert_array_equal(asp.data,[1, 2, 3]) + assert_array_equal(asp.toarray(), bsp.toarray()) + + def test_sort_indices(self): + data = arange(5) + row = array([7, 2, 1, 5, 4]) + ptr = [0, 3, 5] + asp = self.csc_container((data, row, ptr), shape=(10,2)) + bsp = asp.copy() + asp.sort_indices() + assert_array_equal(asp.indices,[1, 2, 7, 4, 5]) + assert_array_equal(asp.toarray(), bsp.toarray()) + + def test_ufuncs(self): + X = self.csc_container(np.arange(21).reshape(7, 3) / 21.) + for f in ["sin", "tan", "arcsin", "arctan", "sinh", "tanh", + "arcsinh", "arctanh", "rint", "sign", "expm1", "log1p", + "deg2rad", "rad2deg", "floor", "ceil", "trunc", "sqrt"]: + assert_equal(hasattr(self.datsp, f), True) + X2 = getattr(X, f)() + assert_equal(X.shape, X2.shape) + assert_array_equal(X.indices, X2.indices) + assert_array_equal(X.indptr, X2.indptr) + assert_array_equal(X2.toarray(), getattr(np, f)(X.toarray())) + + def test_unsorted_arithmetic(self): + data = arange(5) + indices = array([7, 2, 1, 5, 4]) + indptr = array([0, 3, 5]) + asp = self.csc_container((data, indices, indptr), shape=(10,2)) + data = arange(6) + indices = array([8, 1, 5, 7, 2, 4]) + indptr = array([0, 2, 6]) + bsp = self.csc_container((data, indices, indptr), shape=(10,2)) + assert_equal((asp + bsp).toarray(), asp.toarray() + bsp.toarray()) + + def test_fancy_indexing_broadcast(self): + # broadcasting indexing mode is supported + I = np.array([[1], [2], [3]]) + J = np.array([3, 4, 2]) + + np.random.seed(1234) + D = self.asdense(np.random.rand(5, 7)) + S = self.spcreator(D) + + SIJ = S[I,J] + if issparse(SIJ): + SIJ = SIJ.toarray() + assert_equal(SIJ, D[I,J]) + + def test_scalar_idx_dtype(self): + # Check that index dtype takes into account all parameters + # passed to sparsetools, including the scalar ones + indptr = np.zeros(2, dtype=np.int32) + indices = np.zeros(0, dtype=np.int32) + vals = np.zeros(0) + a = self.csc_container((vals, indices, indptr), shape=(2**31-1, 1)) + b = self.csc_container((vals, indices, indptr), shape=(2**31, 1)) + ij = np.zeros((2, 0), dtype=np.int32) + c = self.csc_container((vals, ij), shape=(2**31-1, 1)) + d = self.csc_container((vals, ij), shape=(2**31, 1)) + e = self.csr_container((1, 2**31-1)) + f = self.csr_container((1, 2**31)) + assert_equal(a.indptr.dtype, np.int32) + assert_equal(b.indptr.dtype, np.int64) + assert_equal(c.indptr.dtype, np.int32) + assert_equal(d.indptr.dtype, np.int64) + assert_equal(e.indptr.dtype, np.int32) + assert_equal(f.indptr.dtype, np.int64) + + # These shouldn't fail + for x in [a, b, c, d, e, f]: + x + x + + def test_setdiag_csc(self): + # see gh-21791 setting mixture of existing and not when new_values < 0.001*nnz + D = self.dia_container(([np.arange(1002)], [0]), shape=(1002, 1002)) + A = self.spcreator(D) + A.setdiag(5 * np.ones(A.shape[0])) + assert A[-1, -1] == 5 + + +TestCSC.init_class() + + +class TestCSCMatrix(_MatrixMixin, TestCSC): + @classmethod + def spcreator(cls, *args, **kwargs): + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + return csc_matrix(*args, **kwargs) + + +TestCSCMatrix.init_class() + + +class TestDOK(sparse_test_class(minmax=False, nnz_axis=False)): + spcreator = dok_array + math_dtypes = [np.int_, np.float64, np.complex128] + + def test_mult(self): + A = self.dok_container((10, 12)) + A[0, 3] = 10 + A[5, 6] = 20 + D = A @ A.T + E = A @ A.T.conjugate() + assert_array_equal(D.toarray(), E.toarray()) + + def test_add_nonzero(self): + A = self.spcreator((3,2)) + A[0,1] = -10 + A[2,0] = 20 + A = A + 10 + B = array([[10, 0], [10, 10], [30, 10]]) + assert_array_equal(A.toarray(), B) + + A = A + 1j + B = B + 1j + assert_array_equal(A.toarray(), B) + + def test_dok_divide_scalar(self): + A = self.spcreator((3,2)) + A[0,1] = -10 + A[2,0] = 20 + + assert_array_equal((A/1j).toarray(), A.toarray()/1j) + assert_array_equal((A/9).toarray(), A.toarray()/9) + + def test_convert(self): + # Test provided by Andrew Straw. Fails in SciPy <= r1477. + (m, n) = (6, 7) + a = self.dok_container((m, n)) + + # set a few elements, but none in the last column + a[2,1] = 1 + a[0,2] = 2 + a[3,1] = 3 + a[1,5] = 4 + a[4,3] = 5 + a[4,2] = 6 + + # assert that the last column is all zeros + assert_array_equal(a.toarray()[:,n-1], zeros(m,)) + + # make sure it still works for CSC format + csc = a.tocsc() + assert_array_equal(csc.toarray()[:,n-1], zeros(m,)) + + # now test CSR + (m, n) = (n, m) + b = a.transpose() + assert_equal(b.shape, (m, n)) + # assert that the last row is all zeros + assert_array_equal(b.toarray()[m-1,:], zeros(n,)) + + # make sure it still works for CSR format + csr = b.tocsr() + assert_array_equal(csr.toarray()[m-1,:], zeros(n,)) + + def test_ctor(self): + # Empty ctor + assert_raises(TypeError, self.dok_container) + + # Dense ctor + b = array([[1, 0, 0, 0], [0, 0, 1, 0], [0, 2, 0, 3]], 'd') + A = self.dok_container(b) + assert_equal(b.dtype, A.dtype) + assert_equal(A.toarray(), b) + + # Sparse ctor + c = self.csr_container(b) + assert_equal(A.toarray(), c.toarray()) + + data = [[0, 1, 2], [3, 0, 0]] + d = self.dok_container(data, dtype=np.float32) + assert_equal(d.dtype, np.float32) + da = d.toarray() + assert_equal(da.dtype, np.float32) + assert_array_equal(da, data) + + def test_ticket1160(self): + # Regression test for ticket #1160. + a = self.dok_container((3,3)) + a[0,0] = 0 + # This assert would fail, because the above assignment would + # incorrectly call __set_item__ even though the value was 0. + assert_((0,0) not in a.keys(), "Unexpected entry (0,0) in keys") + + # Slice assignments were also affected. + b = self.dok_container((3,3)) + b[:,0] = 0 + assert_(len(b.keys()) == 0, "Unexpected entries in keys") + + +class TestDOKMatrix(_MatrixMixin, TestDOK): + spcreator = dok_matrix + + +TestDOK.init_class() +TestDOKMatrix.init_class() + + +class TestLIL(sparse_test_class(minmax=False)): + spcreator = lil_array + math_dtypes = [np.int_, np.float64, np.complex128] + + def test_dot(self): + A = zeros((10, 10), np.complex128) + A[0, 3] = 10 + A[5, 6] = 20j + + B = self.lil_container((10, 10), dtype=np.complex128) + B[0, 3] = 10 + B[5, 6] = 20j + + # TODO: properly handle this assertion on ppc64le + if platform.machine() != 'ppc64le': + assert_array_equal(A @ A.T, (B @ B.T).toarray()) + + assert_array_equal(A @ A.conjugate().T, (B @ B.conjugate().T).toarray()) + + def test_scalar_mul(self): + x = self.lil_container((3, 3)) + x[0, 0] = 2 + + x = x*2 + assert_equal(x[0, 0], 4) + + x = x*0 + assert_equal(x[0, 0], 0) + + def test_truediv_scalar(self): + A = self.spcreator((3, 2)) + A[0, 1] = -10 + A[2, 0] = 20 + + assert_array_equal((A / 1j).toarray(), A.toarray() / 1j) + assert_array_equal((A / 9).toarray(), A.toarray() / 9) + + def test_inplace_ops(self): + A = self.lil_container([[0, 2, 3], [4, 0, 6]]) + B = self.lil_container([[0, 1, 0], [0, 2, 3]]) + + data = {'add': (B, A + B), + 'sub': (B, A - B), + 'mul': (3, A * 3)} + + for op, (other, expected) in data.items(): + result = A.copy() + getattr(result, f'__i{op}__')(other) + + assert_array_equal(result.toarray(), expected.toarray()) + + # Ticket 1604. + A = self.lil_container((1, 3), dtype=np.dtype('float64')) + B = self.asdense([0.1, 0.1, 0.1]) + A[0, :] += B + assert_array_equal(A[0, :].toarray(), B) + + def test_lil_iteration(self): + row_data = [[1, 2, 3], [4, 5, 6]] + B = self.lil_container(array(row_data)) + for r, row in enumerate(B): + assert_array_equal(row.toarray(), array(row_data[r], ndmin=row.ndim)) + + def test_lil_from_csr(self): + # Tests whether a LIL can be constructed from a CSR. + B = self.lil_container((10, 10)) + B[0, 3] = 10 + B[5, 6] = 20 + B[8, 3] = 30 + B[3, 8] = 40 + B[8, 9] = 50 + C = B.tocsr() + D = self.lil_container(C) + assert_array_equal(C.toarray(), D.toarray()) + + def test_fancy_indexing_lil(self): + M = self.asdense(arange(25).reshape(5, 5)) + A = self.lil_container(M) + + assert_equal(A[array([1, 2, 3]), 2:3].toarray(), + M[array([1, 2, 3]), 2:3]) + + def test_point_wise_multiply(self): + l = self.lil_container((4, 3)) + l[0, 0] = 1 + l[1, 1] = 2 + l[2, 2] = 3 + l[3, 1] = 4 + + m = self.lil_container((4, 3)) + m[0, 0] = 1 + m[0, 1] = 2 + m[2, 2] = 3 + m[3, 1] = 4 + m[3, 2] = 4 + + assert_array_equal(l.multiply(m).toarray(), + m.multiply(l).toarray()) + + assert_array_equal(l.multiply(m).toarray(), + [[1, 0, 0], + [0, 0, 0], + [0, 0, 9], + [0, 16, 0]]) + + def test_lil_multiply_removal(self): + # Ticket #1427. + a = self.lil_container(np.ones((3, 3))) + a *= 2. + a[0, :] = 0 + + +class TestLILMatrix(_MatrixMixin, TestLIL): + spcreator = lil_matrix + + +TestLIL.init_class() +TestLILMatrix.init_class() + + +class TestCOO(sparse_test_class(getset=False, + slicing=False, slicing_assign=False, + fancy_indexing=False, fancy_assign=False)): + spcreator = coo_array + math_dtypes = [np.int_, np.float64, np.complex128] + + def test_constructor1(self): + # unsorted triplet format + row = array([2, 3, 1, 3, 0, 1, 3, 0, 2, 1, 2]) + col = array([0, 1, 0, 0, 1, 1, 2, 2, 2, 2, 1]) + data = array([6., 10., 3., 9., 1., 4., 11., 2., 8., 5., 7.]) + + coo = self.coo_container((data,(row,col)),(4,3)) + assert_array_equal(arange(12).reshape(4, 3), coo.toarray()) + + # using Python lists and a specified dtype + coo = self.coo_container(([2**63 + 1, 1], ([0, 1], [0, 1])), dtype=np.uint64) + dense = array([[2**63 + 1, 0], [0, 1]], dtype=np.uint64) + assert_array_equal(dense, coo.toarray()) + + def test_constructor2(self): + # unsorted triplet format with duplicates (which are summed) + row = array([0,1,2,2,2,2,0,0,2,2]) + col = array([0,2,0,2,1,1,1,0,0,2]) + data = array([2,9,-4,5,7,0,-1,2,1,-5]) + coo = self.coo_container((data,(row,col)),(3,3)) + + mat = array([[4, -1, 0], [0, 0, 9], [-3, 7, 0]]) + + assert_array_equal(mat, coo.toarray()) + + def test_constructor3(self): + # empty matrix + coo = self.coo_container((4,3)) + + assert_array_equal(coo.shape,(4,3)) + assert_array_equal(coo.row,[]) + assert_array_equal(coo.col,[]) + assert_array_equal(coo.data,[]) + assert_array_equal(coo.toarray(), zeros((4, 3))) + + def test_constructor4(self): + # from dense matrix + mat = array([[0,1,0,0], + [7,0,3,0], + [0,4,0,0]]) + coo = self.coo_container(mat) + assert_array_equal(coo.toarray(), mat) + + # upgrade rank 1 arrays to row matrix + mat = array([0,1,0,0]) + coo = self.coo_container(mat) + expected = mat if self.is_array_test else mat.reshape(1, -1) + assert_array_equal(coo.toarray(), expected) + + # error if second arg interpreted as shape (gh-9919) + with pytest.raises(TypeError, match=r'object cannot be interpreted'): + self.coo_container([0, 11, 22, 33], ([0, 1, 2, 3], [0, 0, 0, 0])) + + # error if explicit shape arg doesn't match the dense matrix + with pytest.raises(ValueError, match=r'inconsistent shapes'): + self.coo_container([0, 11, 22, 33], shape=(4, 4)) + + def test_constructor_data_ij_dtypeNone(self): + data = [1] + coo = self.coo_container((data, ([0], [0])), dtype=None) + assert coo.dtype == np.array(data).dtype + + @pytest.mark.xfail(run=False, reason='COO does not have a __getitem__') + def test_iterator(self): + pass + + def test_todia_all_zeros(self): + zeros = [[0, 0]] + dia = self.coo_container(zeros).todia() + assert_array_equal(dia.toarray(), zeros) + + def test_sum_duplicates(self): + coo = self.coo_container((4,3)) + coo.sum_duplicates() + coo = self.coo_container(([1,2], ([1,0], [1,0]))) + coo.sum_duplicates() + assert_array_equal(coo.toarray(), [[2,0],[0,1]]) + coo = self.coo_container(([1,2], ([1,1], [1,1]))) + coo.sum_duplicates() + assert_array_equal(coo.toarray(), [[0,0],[0,3]]) + assert_array_equal(coo.row, [1]) + assert_array_equal(coo.col, [1]) + assert_array_equal(coo.data, [3]) + + def test_todok_duplicates(self): + coo = self.coo_container(([1,1,1,1], ([0,2,2,0], [0,1,1,0]))) + dok = coo.todok() + assert_array_equal(dok.toarray(), coo.toarray()) + + def test_tocompressed_duplicates(self): + coo = self.coo_container(([1,1,1,1], ([0,2,2,0], [0,1,1,0]))) + csr = coo.tocsr() + assert_equal(csr.nnz + 2, coo.nnz) + csc = coo.tocsc() + assert_equal(csc.nnz + 2, coo.nnz) + + def test_eliminate_zeros(self): + data = array([1, 0, 0, 0, 2, 0, 3, 0]) + row = array([0, 0, 0, 1, 1, 1, 1, 1]) + col = array([1, 2, 3, 4, 5, 6, 7, 8]) + asp = self.coo_container((data, (row, col)), shape=(2,10)) + bsp = asp.copy() + asp.eliminate_zeros() + assert_((asp.data != 0).all()) + assert_array_equal(asp.toarray(), bsp.toarray()) + + def test_reshape_copy(self): + arr = [[0, 10, 0, 0], [0, 0, 0, 0], [0, 20, 30, 40]] + new_shape = (2, 6) + x = self.coo_container(arr) + + y = x.reshape(new_shape) + assert_(y.data is x.data) + + y = x.reshape(new_shape, copy=False) + assert_(y.data is x.data) + + y = x.reshape(new_shape, copy=True) + assert_(not np.may_share_memory(y.data, x.data)) + + def test_large_dimensions_reshape(self): + # Test that reshape is immune to integer overflow when number of elements + # exceeds 2^31-1 + mat1 = self.coo_container(([1], ([3000000], [1000])), (3000001, 1001)) + mat2 = self.coo_container(([1], ([1000], [3000000])), (1001, 3000001)) + + # assert_array_equal is slow for big matrices because it expects dense + # Using __ne__ and nnz instead + assert_((mat1.reshape((1001, 3000001), order='C') != mat2).nnz == 0) + assert_((mat2.reshape((3000001, 1001), order='F') != mat1).nnz == 0) + + +class TestCOOMatrix(_MatrixMixin, TestCOO): + spcreator = coo_matrix + + +TestCOO.init_class() +TestCOOMatrix.init_class() + + +class TestDIA(sparse_test_class(getset=False, slicing=False, slicing_assign=False, + fancy_indexing=False, fancy_assign=False, + minmax=False, nnz_axis=False)): + spcreator = dia_array + math_dtypes = [np.int_, np.float64, np.complex128] + + def test_constructor1(self): + D = array([[1, 0, 3, 0], + [1, 2, 0, 4], + [0, 2, 3, 0], + [0, 0, 3, 4]]) + data = np.array([[1,2,3,4]]).repeat(3,axis=0) + offsets = np.array([0,-1,2]) + assert_equal(self.dia_container((data, offsets), shape=(4, 4)).toarray(), D) + + @pytest.mark.xfail(run=False, reason='DIA does not have a __getitem__') + def test_iterator(self): + pass + + @with_64bit_maxval_limit(3) + def test_setdiag_dtype(self): + m = self.dia_container(np.eye(3)) + assert_equal(m.offsets.dtype, np.int32) + m.setdiag((3,), k=2) + assert_equal(m.offsets.dtype, np.int32) + + m = self.dia_container(np.eye(4)) + assert_equal(m.offsets.dtype, np.int64) + m.setdiag((3,), k=3) + assert_equal(m.offsets.dtype, np.int64) + + @pytest.mark.skip(reason='DIA stores extra zeros') + def test_getnnz_axis(self): + pass + + def test_convert_gh14555(self): + # regression test for gh-14555 + m = self.dia_container(([[1, 1, 0]], [-1]), shape=(4, 2)) + expected = m.toarray() + assert_array_equal(m.tocsc().toarray(), expected) + assert_array_equal(m.tocsr().toarray(), expected) + + def test_tocoo_gh10050(self): + # regression test for gh-10050 + m = self.dia_container([[1, 2], [3, 4]]).tocoo() + flat_inds = np.ravel_multi_index((m.row, m.col), m.shape) + inds_are_sorted = np.all(np.diff(flat_inds) > 0) + assert m.has_canonical_format == inds_are_sorted + + def test_tocoo_tocsr_tocsc_gh19245(self): + # test index_dtype with tocoo, tocsr, tocsc + data = np.array([[1, 2, 3, 4]]).repeat(3, axis=0) + offsets = np.array([0, -1, 2], dtype=np.int32) + dia = sparse.dia_array((data, offsets), shape=(4, 4)) + + coo = dia.tocoo() + assert coo.col.dtype == np.int32 + csr = dia.tocsr() + assert csr.indices.dtype == np.int32 + csc = dia.tocsc() + assert csc.indices.dtype == np.int32 + + def test_mul_scalar(self): + # repro for gh-20434 + m = self.dia_container([[1, 2], [0, 4]]) + res = m * 3 + assert isinstance(res, m.__class__) + assert_array_equal(res.toarray(), [[3, 6], [0, 12]]) + + res2 = m.multiply(3) + assert isinstance(res2, m.__class__) + assert_array_equal(res2.toarray(), [[3, 6], [0, 12]]) + + +class TestDIAMatrix(_MatrixMixin, TestDIA): + spcreator = dia_matrix + + +TestDIA.init_class() +TestDIAMatrix.init_class() + + +class TestBSR(sparse_test_class(getset=False, + slicing=False, slicing_assign=False, + fancy_indexing=False, fancy_assign=False, + nnz_axis=False)): + spcreator = bsr_array + math_dtypes = [np.int_, np.float64, np.complex128] + + def test_constructor1(self): + # check native BSR format constructor + indptr = array([0,2,2,4]) + indices = array([0,2,2,3]) + data = zeros((4,2,3)) + + data[0] = array([[0, 1, 2], + [3, 0, 5]]) + data[1] = array([[0, 2, 4], + [6, 0, 10]]) + data[2] = array([[0, 4, 8], + [12, 0, 20]]) + data[3] = array([[0, 5, 10], + [15, 0, 25]]) + + A = kron([[1,0,2,0],[0,0,0,0],[0,0,4,5]], [[0,1,2],[3,0,5]]) + Asp = self.bsr_container((data,indices,indptr),shape=(6,12)) + assert_equal(Asp.toarray(), A) + + # infer shape from arrays + Asp = self.bsr_container((data,indices,indptr)) + assert_equal(Asp.toarray(), A) + + def test_constructor2(self): + # construct from dense + + # test zero mats + for shape in [(1,1), (5,1), (1,10), (10,4), (3,7), (2,1)]: + A = zeros(shape) + assert_equal(self.bsr_container(A).toarray(), A) + A = zeros((4,6)) + assert_equal(self.bsr_container(A, blocksize=(2, 2)).toarray(), A) + assert_equal(self.bsr_container(A, blocksize=(2, 3)).toarray(), A) + + A = kron([[1,0,2,0],[0,0,0,0],[0,0,4,5]], [[0,1,2],[3,0,5]]) + assert_equal(self.bsr_container(A).toarray(), A) + assert_equal(self.bsr_container(A, shape=(6, 12)).toarray(), A) + assert_equal(self.bsr_container(A, blocksize=(1, 1)).toarray(), A) + assert_equal(self.bsr_container(A, blocksize=(2, 3)).toarray(), A) + assert_equal(self.bsr_container(A, blocksize=(2, 6)).toarray(), A) + assert_equal(self.bsr_container(A, blocksize=(2, 12)).toarray(), A) + assert_equal(self.bsr_container(A, blocksize=(3, 12)).toarray(), A) + assert_equal(self.bsr_container(A, blocksize=(6, 12)).toarray(), A) + + A = kron([[1,0,2,0],[0,1,0,0],[0,0,0,0]], [[0,1,2],[3,0,5]]) + assert_equal(self.bsr_container(A, blocksize=(2, 3)).toarray(), A) + + def test_constructor3(self): + # construct from coo-like (data,(row,col)) format + arg = ([1,2,3], ([0,1,1], [0,0,1])) + A = array([[1,0],[2,3]]) + assert_equal(self.bsr_container(arg, blocksize=(2, 2)).toarray(), A) + + def test_constructor4(self): + # regression test for gh-6292: self.bsr_matrix((data, indices, indptr)) was + # trying to compare an int to a None + n = 8 + data = np.ones((n, n, 1), dtype=np.int8) + indptr = np.array([0, n], dtype=np.int32) + indices = np.arange(n, dtype=np.int32) + self.bsr_container((data, indices, indptr), blocksize=(n, 1), copy=False) + + def test_constructor5(self): + # check for validations introduced in gh-13400 + n = 8 + data_1dim = np.ones(n) + data = np.ones((n, n, n)) + indptr = np.array([0, n]) + indices = np.arange(n) + + with assert_raises(ValueError): + # data ndim check + self.bsr_container((data_1dim, indices, indptr)) + + with assert_raises(ValueError): + # invalid blocksize + self.bsr_container((data, indices, indptr), blocksize=(1, 1, 1)) + + with assert_raises(ValueError): + # mismatching blocksize + self.bsr_container((data, indices, indptr), blocksize=(1, 1)) + + def test_default_dtype(self): + # As a numpy array, `values` has shape (2, 2, 1). + values = [[[1], [1]], [[1], [1]]] + indptr = np.array([0, 2], dtype=np.int32) + indices = np.array([0, 1], dtype=np.int32) + b = self.bsr_container((values, indices, indptr), blocksize=(2, 1)) + assert b.dtype == np.array(values).dtype + + def test_bsr_tocsr(self): + # check native conversion from BSR to CSR + indptr = array([0, 2, 2, 4]) + indices = array([0, 2, 2, 3]) + data = zeros((4, 2, 3)) + + data[0] = array([[0, 1, 2], + [3, 0, 5]]) + data[1] = array([[0, 2, 4], + [6, 0, 10]]) + data[2] = array([[0, 4, 8], + [12, 0, 20]]) + data[3] = array([[0, 5, 10], + [15, 0, 25]]) + + A = kron([[1, 0, 2, 0], [0, 0, 0, 0], [0, 0, 4, 5]], + [[0, 1, 2], [3, 0, 5]]) + Absr = self.bsr_container((data, indices, indptr), shape=(6, 12)) + Acsr = Absr.tocsr() + Acsr_via_coo = Absr.tocoo().tocsr() + assert_equal(Acsr.toarray(), A) + assert_equal(Acsr.toarray(), Acsr_via_coo.toarray()) + + def test_eliminate_zeros(self): + data = kron([1, 0, 0, 0, 2, 0, 3, 0], [[1,1],[1,1]]).T + data = data.reshape(-1,2,2) + indices = array([1, 2, 3, 4, 5, 6, 7, 8]) + indptr = array([0, 3, 8]) + asp = self.bsr_container((data, indices, indptr), shape=(4,20)) + bsp = asp.copy() + asp.eliminate_zeros() + assert_array_equal(asp.nnz, 3*4) + assert_array_equal(asp.toarray(), bsp.toarray()) + + # GitHub issue #9687 + def test_eliminate_zeros_all_zero(self): + np.random.seed(0) + m = self.bsr_container(np.random.random((12, 12)), blocksize=(2, 3)) + + # eliminate some blocks, but not all + m.data[m.data <= 0.9] = 0 + m.eliminate_zeros() + assert_equal(m.nnz, 66) + assert_array_equal(m.data.shape, (11, 2, 3)) + + # eliminate all remaining blocks + m.data[m.data <= 1.0] = 0 + m.eliminate_zeros() + assert_equal(m.nnz, 0) + assert_array_equal(m.data.shape, (0, 2, 3)) + assert_array_equal(m.toarray(), np.zeros((12, 12))) + + # test fast path + m.eliminate_zeros() + assert_equal(m.nnz, 0) + assert_array_equal(m.data.shape, (0, 2, 3)) + assert_array_equal(m.toarray(), np.zeros((12, 12))) + + def test_bsr_matvec(self): + A = self.bsr_container(arange(2*3*4*5).reshape(2*4,3*5), blocksize=(4,5)) + x = arange(A.shape[1]).reshape(-1,1) + assert_equal(A @ x, A.toarray() @ x) + + def test_bsr_matvecs(self): + A = self.bsr_container(arange(2*3*4*5).reshape(2*4,3*5), blocksize=(4,5)) + x = arange(A.shape[1]*6).reshape(-1,6) + assert_equal(A @ x, A.toarray() @ x) + + @pytest.mark.xfail(run=False, reason='BSR does not have a __getitem__') + def test_iterator(self): + pass + + @pytest.mark.xfail(run=False, reason='BSR does not have a __setitem__') + def test_setdiag(self): + pass + + def test_resize_blocked(self): + # test resize() with non-(1,1) blocksize + D = np.array([[1, 0, 3, 4], + [2, 0, 0, 0], + [3, 0, 0, 0]]) + S = self.spcreator(D, blocksize=(1, 2)) + assert_(S.resize((3, 2)) is None) + assert_array_equal(S.toarray(), [[1, 0], + [2, 0], + [3, 0]]) + S.resize((2, 2)) + assert_array_equal(S.toarray(), [[1, 0], + [2, 0]]) + S.resize((3, 2)) + assert_array_equal(S.toarray(), [[1, 0], + [2, 0], + [0, 0]]) + S.resize((3, 4)) + assert_array_equal(S.toarray(), [[1, 0, 0, 0], + [2, 0, 0, 0], + [0, 0, 0, 0]]) + assert_raises(ValueError, S.resize, (2, 3)) + + @pytest.mark.xfail(run=False, reason='BSR does not have a __setitem__') + def test_setdiag_comprehensive(self): + pass + + @pytest.mark.skipif(IS_COLAB, reason="exceeds memory limit") + def test_scalar_idx_dtype(self): + # Check that index dtype takes into account all parameters + # passed to sparsetools, including the scalar ones + indptr = np.zeros(2, dtype=np.int32) + indices = np.zeros(0, dtype=np.int32) + vals = np.zeros((0, 1, 1)) + a = self.bsr_container((vals, indices, indptr), shape=(1, 2**31-1)) + b = self.bsr_container((vals, indices, indptr), shape=(1, 2**31)) + c = self.bsr_container((1, 2**31-1)) + d = self.bsr_container((1, 2**31)) + assert_equal(a.indptr.dtype, np.int32) + assert_equal(b.indptr.dtype, np.int64) + assert_equal(c.indptr.dtype, np.int32) + assert_equal(d.indptr.dtype, np.int64) + + try: + vals2 = np.zeros((0, 1, 2**31-1)) + vals3 = np.zeros((0, 1, 2**31)) + e = self.bsr_container((vals2, indices, indptr), shape=(1, 2**31-1)) + f = self.bsr_container((vals3, indices, indptr), shape=(1, 2**31)) + assert_equal(e.indptr.dtype, np.int32) + assert_equal(f.indptr.dtype, np.int64) + except (MemoryError, ValueError): + # May fail on 32-bit Python + e = 0 + f = 0 + + # These shouldn't fail + for x in [a, b, c, d, e, f]: + x + x + + +class TestBSRMatrix(_MatrixMixin, TestBSR): + spcreator = bsr_matrix + + +TestBSR.init_class() +TestBSRMatrix.init_class() + + +#------------------------------------------------------------------------------ +# Tests for non-canonical representations (with duplicates, unsorted indices) +#------------------------------------------------------------------------------ + +def _same_sum_duplicate(data, *inds, **kwargs): + """Duplicates entries to produce the same matrix""" + indptr = kwargs.pop('indptr', None) + if np.issubdtype(data.dtype, np.bool_) or \ + np.issubdtype(data.dtype, np.unsignedinteger): + if indptr is None: + return (data,) + inds + else: + return (data,) + inds + (indptr,) + + zeros_pos = (data == 0).nonzero() + + # duplicate data + data = data.repeat(2, axis=0) + data[::2] -= 1 + data[1::2] = 1 + + # don't spoil all explicit zeros + if zeros_pos[0].size > 0: + pos = tuple(p[0] for p in zeros_pos) + pos1 = (2*pos[0],) + pos[1:] + pos2 = (2*pos[0]+1,) + pos[1:] + data[pos1] = 0 + data[pos2] = 0 + + inds = tuple(indices.repeat(2) for indices in inds) + + if indptr is None: + return (data,) + inds + else: + return (data,) + inds + (indptr * 2,) + + +class _NonCanonicalMixin: + def spcreator(self, D, *args, sorted_indices=False, **kwargs): + """Replace D with a non-canonical equivalent: containing + duplicate elements and explicit zeros""" + construct = super().spcreator + M = construct(D, *args, **kwargs) + + zero_pos = (M.toarray() == 0).nonzero() + has_zeros = (zero_pos[0].size > 0) + if has_zeros: + k = zero_pos[0].size//2 + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + M = self._insert_explicit_zero(M, zero_pos[0][k], zero_pos[1][k]) + + arg1 = self._arg1_for_noncanonical(M, sorted_indices) + if 'shape' not in kwargs: + kwargs['shape'] = M.shape + NC = construct(arg1, **kwargs) + + # check that result is valid + if NC.dtype in [np.float32, np.complex64]: + # For single-precision floats, the differences between M and NC + # that are introduced by the extra operations involved in the + # construction of NC necessitate a more lenient tolerance level + # than the default. + rtol = 1e-05 + else: + rtol = 1e-07 + assert_allclose(NC.toarray(), M.toarray(), rtol=rtol) + + # check that at least one explicit zero + if has_zeros: + assert_((NC.data == 0).any()) + # TODO check that NC has duplicates (which are not explicit zeros) + + return NC + + @pytest.mark.skip(reason='bool(matrix) counts explicit zeros') + def test_bool(self): + pass + + @pytest.mark.skip(reason='getnnz-axis counts explicit zeros') + def test_getnnz_axis(self): + pass + + @pytest.mark.skip(reason='nnz counts explicit zeros') + def test_empty(self): + pass + + +class _NonCanonicalCompressedMixin(_NonCanonicalMixin): + def _arg1_for_noncanonical(self, M, sorted_indices=False): + """Return non-canonical constructor arg1 equivalent to M""" + data, indices, indptr = _same_sum_duplicate(M.data, M.indices, + indptr=M.indptr) + if not sorted_indices: + for start, stop in zip(indptr, indptr[1:]): + indices[start:stop] = indices[start:stop][::-1].copy() + data[start:stop] = data[start:stop][::-1].copy() + return data, indices, indptr + + def _insert_explicit_zero(self, M, i, j): + M[i,j] = 0 + return M + + +class _NonCanonicalCSMixin(_NonCanonicalCompressedMixin): + def test_getelement(self): + def check(dtype, sorted_indices): + D = array([[1,0,0], + [4,3,0], + [0,2,0], + [0,0,0]], dtype=dtype) + A = self.spcreator(D, sorted_indices=sorted_indices) + + M,N = D.shape + + for i in range(-M, M): + for j in range(-N, N): + assert_equal(A[i,j], D[i,j]) + + for ij in [(0,3),(-1,3),(4,0),(4,3),(4,-1), (1, 2, 3)]: + assert_raises((IndexError, TypeError), A.__getitem__, ij) + + for dtype in supported_dtypes: + for sorted_indices in [False, True]: + check(np.dtype(dtype), sorted_indices) + + def test_setitem_sparse(self): + D = np.eye(3) + A = self.spcreator(D) + B = self.spcreator([[1,2,3]]) + + D[1,:] = B.toarray() + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + A[1,:] = B + assert_array_equal(A.toarray(), D) + + D[:,2] = B.toarray().ravel() + with suppress_warnings() as sup: + sup.filter(SparseEfficiencyWarning, "Changing the sparsity structure") + A[:,2] = B.T + assert_array_equal(A.toarray(), D) + + @pytest.mark.xfail(run=False, reason='inverse broken with non-canonical matrix') + def test_inv(self): + pass + + @pytest.mark.xfail(run=False, reason='solve broken with non-canonical matrix') + def test_solve(self): + pass + + +class TestCSRNonCanonical(_NonCanonicalCSMixin, TestCSR): + pass + + +class TestCSRNonCanonicalMatrix(TestCSRNonCanonical, TestCSRMatrix): + pass + + +class TestCSCNonCanonical(_NonCanonicalCSMixin, TestCSC): + pass + + +class TestCSCNonCanonicalMatrix(TestCSCNonCanonical, TestCSCMatrix): + pass + + +class TestBSRNonCanonical(_NonCanonicalCompressedMixin, TestBSR): + def _insert_explicit_zero(self, M, i, j): + x = M.tocsr() + x[i,j] = 0 + return x.tobsr(blocksize=M.blocksize) + + @pytest.mark.xfail(run=False, reason='diagonal broken with non-canonical BSR') + def test_diagonal(self): + pass + + @pytest.mark.xfail(run=False, reason='expm broken with non-canonical BSR') + def test_expm(self): + pass + + +class TestBSRNonCanonicalMatrix(TestBSRNonCanonical, TestBSRMatrix): + pass + + +class TestCOONonCanonical(_NonCanonicalMixin, TestCOO): + def _arg1_for_noncanonical(self, M, sorted_indices=None): + """Return non-canonical constructor arg1 equivalent to M""" + data, row, col = _same_sum_duplicate(M.data, M.row, M.col) + return data, (row, col) + + def _insert_explicit_zero(self, M, i, j): + M.data = np.r_[M.data.dtype.type(0), M.data] + M.row = np.r_[M.row.dtype.type(i), M.row] + M.col = np.r_[M.col.dtype.type(j), M.col] + return M + + def test_setdiag_noncanonical(self): + m = self.spcreator(np.eye(3)) + m.sum_duplicates() + m.setdiag([3, 2], k=1) + m.sum_duplicates() + assert_(np.all(np.diff(m.col) >= 0)) + + +class TestCOONonCanonicalMatrix(TestCOONonCanonical, TestCOOMatrix): + pass + + +#Todo: Revisit 64bit tests: avoid rerun of all tests for each version of get_index_dtype +def cases_64bit(sp_api): + """Yield all tests for all formats that use get_index_dtype + + This is more than testing get_index_dtype. It allows checking whether upcasting + or downcasting the index dtypes affects test results. The approach used here + does not try to figure out which tests might fail due to 32/64-bit issues. + We just run them all. + So, each test method in that uses cases_64bit reruns most of the test suite! + """ + if sp_api == "sparray": + TEST_CLASSES = [TestBSR, TestCOO, TestCSC, TestCSR, TestDIA, + # lil/dok->other conversion operations use get_index_dtype + # so we include lil & dok test suite even though they do not + # use get_index_dtype within the class. That means many of + # these tests are superfluous, but it's hard to pick which + TestDOK, TestLIL + ] + elif sp_api == "spmatrix": + TEST_CLASSES = [TestBSRMatrix, TestCOOMatrix, TestCSCMatrix, + TestCSRMatrix, TestDIAMatrix, + # lil/dok->other conversion operations use get_index_dtype + TestDOKMatrix, TestLILMatrix + ] + else: + raise ValueError(f"parameter {sp_api=} is not one of 'sparray' or 'spmatrix'") + + # The following features are missing, so skip the tests: + SKIP_TESTS = { + 'test_expm': 'expm for 64-bit indices not available', + 'test_inv': 'linsolve for 64-bit indices not available', + 'test_solve': 'linsolve for 64-bit indices not available', + 'test_scalar_idx_dtype': 'test implemented in base class', + 'test_large_dimensions_reshape': 'test actually requires 64-bit to work', + 'test_constructor_smallcol': 'test verifies int32 indexes', + 'test_constructor_largecol': 'test verifies int64 indexes', + 'test_tocoo_tocsr_tocsc_gh19245': 'test verifies int32 indexes', + } + + for cls in TEST_CLASSES: + for method_name in sorted(dir(cls)): + method = getattr(cls, method_name) + if (method_name.startswith('test_') and + not getattr(method, 'slow', False)): + marks = [] + + msg = SKIP_TESTS.get(method_name) + if bool(msg): + marks += [pytest.mark.skip(reason=msg)] + + markers = getattr(method, 'pytestmark', []) + for mark in markers: + if mark.name in ('skipif', 'skip', 'xfail', 'xslow'): + marks.append(mark) + + yield pytest.param(cls, method_name, marks=marks) + + +class Test64Bit: + # classes that use get_index_dtype + MAT_CLASSES = [ + bsr_matrix, coo_matrix, csc_matrix, csr_matrix, dia_matrix, + bsr_array, coo_array, csc_array, csr_array, dia_array, + ] + + def _compare_index_dtype(self, m, dtype): + dtype = np.dtype(dtype) + if m.format in ['csc', 'csr', 'bsr']: + return (m.indices.dtype == dtype) and (m.indptr.dtype == dtype) + elif m.format == 'coo': + return (m.row.dtype == dtype) and (m.col.dtype == dtype) + elif m.format == 'dia': + return (m.offsets.dtype == dtype) + else: + raise ValueError(f"matrix {m!r} has no integer indices") + + @pytest.mark.thread_unsafe + def test_decorator_maxval_limit(self): + # Test that the with_64bit_maxval_limit decorator works + + @with_64bit_maxval_limit(maxval_limit=10) + def check(mat_cls): + m = mat_cls(np.random.rand(10, 1)) + assert_(self._compare_index_dtype(m, np.int32)) + m = mat_cls(np.random.rand(11, 1)) + assert_(self._compare_index_dtype(m, np.int64)) + + for mat_cls in self.MAT_CLASSES: + check(mat_cls) + + @pytest.mark.thread_unsafe + def test_decorator_maxval_random(self): + # Test that the with_64bit_maxval_limit decorator works (2) + + @with_64bit_maxval_limit(random=True) + def check(mat_cls): + seen_32 = False + seen_64 = False + for k in range(100): + m = mat_cls(np.random.rand(9, 9)) + seen_32 = seen_32 or self._compare_index_dtype(m, np.int32) + seen_64 = seen_64 or self._compare_index_dtype(m, np.int64) + if seen_32 and seen_64: + break + else: + raise AssertionError("both 32 and 64 bit indices not seen") + + for mat_cls in self.MAT_CLASSES: + check(mat_cls) + + @pytest.mark.thread_unsafe + def test_downcast_intp(self): + # Check that bincount and ufunc.reduceat intp downcasts are + # dealt with. The point here is to trigger points in the code + # that can fail on 32-bit systems when using 64-bit indices, + # due to use of functions that only work with intp-size + # indices. + + @with_64bit_maxval_limit(fixed_dtype=np.int64, downcast_maxval=1) + def check_limited(csc_container, csr_container, coo_container): + # These involve indices larger than `downcast_maxval` + a = csc_container([[1, 2], [3, 4], [5, 6]]) + assert_raises(AssertionError, a.count_nonzero, axis=1) + assert_raises(AssertionError, a.sum, axis=0) + + a = csr_container([[1, 2, 3], [3, 4, 6]]) + assert_raises(AssertionError, a.count_nonzero, axis=0) + assert_raises(AssertionError, a.sum, axis=1) + + a = coo_container([[1, 2, 3], [3, 4, 5]]) + assert_raises(AssertionError, a.count_nonzero, axis=0) + a.has_canonical_format = False + assert_raises(AssertionError, a.sum_duplicates) + + @with_64bit_maxval_limit(fixed_dtype=np.int64) + def check_unlimited(csc_container, csr_container, coo_container): + # These involve indices smaller than `downcast_maxval` + a = csc_container([[1, 2], [3, 4], [5, 6]]) + a.count_nonzero(axis=1) + a.sum(axis=0) + + a = csr_container([[1, 2, 3], [3, 4, 6]]) + a.count_nonzero(axis=0) + a.sum(axis=1) + + a = coo_container([[1, 2, 3], [3, 4, 5]]) + a.count_nonzero(axis=0) + a.has_canonical_format = False + a.sum_duplicates() + + check_limited(csc_array, csr_array, coo_array) + check_unlimited(csc_array, csr_array, coo_array) + check_limited(csc_matrix, csr_matrix, coo_matrix) + check_unlimited(csc_matrix, csr_matrix, coo_matrix) + + +# Testing both spmatrices and sparrays for 64bit index dtype handling is +# expensive and double-checks the same code (e.g. _coobase) +class RunAll64Bit: + def _check_resiliency(self, cls, method_name, **kw): + # Resiliency test, to check that sparse matrices deal reasonably + # with varying index data types. + + @with_64bit_maxval_limit(**kw) + def check(cls, method_name): + instance = cls() + if hasattr(instance, 'setup_method'): + instance.setup_method() + try: + getattr(instance, method_name)() + finally: + if hasattr(instance, 'teardown_method'): + instance.teardown_method() + + check(cls, method_name) + + +@pytest.mark.thread_unsafe +@pytest.mark.slow +class Test64BitArray(RunAll64Bit): + # inheritance of pytest test classes does not separate marks for subclasses. + # So we define these functions in both Array and Matrix versions. + @pytest.mark.parametrize('cls,method_name', cases_64bit("sparray")) + def test_resiliency_limit_10(self, cls, method_name): + self._check_resiliency(cls, method_name, maxval_limit=10) + + @pytest.mark.fail_slow(2) + @pytest.mark.parametrize('cls,method_name', cases_64bit("sparray")) + def test_resiliency_random(self, cls, method_name): + # bsr_array.eliminate_zeros relies on csr_array constructor + # not making copies of index arrays --- this is not + # necessarily true when we pick the index data type randomly + self._check_resiliency(cls, method_name, random=True) + + @pytest.mark.parametrize('cls,method_name', cases_64bit("sparray")) + def test_resiliency_all_32(self, cls, method_name): + self._check_resiliency(cls, method_name, fixed_dtype=np.int32) + + @pytest.mark.parametrize('cls,method_name', cases_64bit("sparray")) + def test_resiliency_all_64(self, cls, method_name): + self._check_resiliency(cls, method_name, fixed_dtype=np.int64) + + +@pytest.mark.thread_unsafe +class Test64BitMatrix(RunAll64Bit): + # assert_32bit=True only for spmatrix cuz sparray does not check index content + @pytest.mark.fail_slow(5) + @pytest.mark.parametrize('cls,method_name', cases_64bit("spmatrix")) + def test_no_64(self, cls, method_name): + self._check_resiliency(cls, method_name, assert_32bit=True) + + # inheritance of pytest test classes does not separate marks for subclasses. + # So we define these functions in both Array and Matrix versions. + @pytest.mark.parametrize('cls,method_name', cases_64bit("spmatrix")) + def test_resiliency_limit_10(self, cls, method_name): + self._check_resiliency(cls, method_name, maxval_limit=10) + + @pytest.mark.fail_slow(2) + @pytest.mark.parametrize('cls,method_name', cases_64bit("spmatrix")) + def test_resiliency_random(self, cls, method_name): + # bsr_array.eliminate_zeros relies on csr_array constructor + # not making copies of index arrays --- this is not + # necessarily true when we pick the index data type randomly + self._check_resiliency(cls, method_name, random=True) + + @pytest.mark.parametrize('cls,method_name', cases_64bit("spmatrix")) + def test_resiliency_all_32(self, cls, method_name): + self._check_resiliency(cls, method_name, fixed_dtype=np.int32) + + @pytest.mark.parametrize('cls,method_name', cases_64bit("spmatrix")) + def test_resiliency_all_64(self, cls, method_name): + self._check_resiliency(cls, method_name, fixed_dtype=np.int64) + +def test_broadcast_to(): + a = np.array([[1, 0, 2]]) + b = np.array([[1], [0], [2]]) + c = np.array([[1, 0, 2], [0, 3, 0]]) + d = np.array([[7]]) + e = np.array([[0]]) + f = np.array([[0,0,0,0]]) + for container in (csc_matrix, csc_array, csr_matrix, csr_array): + res_a = container(a)._broadcast_to((2,3)) + res_b = container(b)._broadcast_to((3,4)) + res_c = container(c)._broadcast_to((2,3)) + res_d = container(d)._broadcast_to((4,4)) + res_e = container(e)._broadcast_to((5,6)) + res_f = container(f)._broadcast_to((2,4)) + assert_array_equal(res_a.toarray(), np.broadcast_to(a, (2,3))) + assert_array_equal(res_b.toarray(), np.broadcast_to(b, (3,4))) + assert_array_equal(res_c.toarray(), c) + assert_array_equal(res_d.toarray(), np.broadcast_to(d, (4,4))) + assert_array_equal(res_e.toarray(), np.broadcast_to(e, (5,6))) + assert_array_equal(res_f.toarray(), np.broadcast_to(f, (2,4))) + + with pytest.raises(ValueError, match="cannot be broadcast"): + container([[1, 2, 0], [3, 0, 1]])._broadcast_to(shape=(2, 1)) + + with pytest.raises(ValueError, match="cannot be broadcast"): + container([[0, 1, 2]])._broadcast_to(shape=(3, 2)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_common1d.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_common1d.py new file mode 100644 index 0000000000000000000000000000000000000000..ab091f7c8fccc42c6d9c6c249e59b1d5e9c326b8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_common1d.py @@ -0,0 +1,447 @@ +"""Test of 1D aspects of sparse array classes""" + +import pytest + +import numpy as np +from numpy.testing import assert_equal, assert_allclose + +from scipy.sparse import ( + bsr_array, csc_array, dia_array, lil_array, + coo_array, csr_array, dok_array, + ) +from scipy.sparse._sputils import supported_dtypes, matrix +from scipy._lib._util import ComplexWarning + + +sup_complex = np.testing.suppress_warnings() +sup_complex.filter(ComplexWarning) + + +spcreators = [coo_array, csr_array, dok_array] +math_dtypes = [np.int64, np.float64, np.complex128] + + +@pytest.fixture +def dat1d(): + return np.array([3, 0, 1, 0], 'd') + + +@pytest.fixture +def datsp_math_dtypes(dat1d): + dat_dtypes = {dtype: dat1d.astype(dtype) for dtype in math_dtypes} + return { + spcreator: [(dtype, dat, spcreator(dat)) for dtype, dat in dat_dtypes.items()] + for spcreator in spcreators + } + + +# Test init with 1D dense input +# sparrays which do not plan to support 1D +@pytest.mark.parametrize("spcreator", [bsr_array, csc_array, dia_array, lil_array]) +def test_no_1d_support_in_init(spcreator): + with pytest.raises(ValueError, match="arrays don't support 1D input"): + spcreator([0, 1, 2, 3]) + + +# Test init with nD dense input +# sparrays which do not yet support nD +@pytest.mark.parametrize( + "spcreator", [csr_array, dok_array, bsr_array, csc_array, dia_array, lil_array] +) +def test_no_nd_support_in_init(spcreator): + with pytest.raises(ValueError, match="arrays don't.*support 3D"): + spcreator(np.ones((3, 2, 4))) + + +# Main tests class +@pytest.mark.parametrize("spcreator", spcreators) +class TestCommon1D: + """test common functionality shared by 1D sparse formats""" + + def test_create_empty(self, spcreator): + assert_equal(spcreator((3,)).toarray(), np.zeros(3)) + assert_equal(spcreator((3,)).nnz, 0) + assert_equal(spcreator((3,)).count_nonzero(), 0) + + def test_invalid_shapes(self, spcreator): + with pytest.raises(ValueError, match='elements cannot be negative'): + spcreator((-3,)) + + def test_repr(self, spcreator, dat1d): + repr(spcreator(dat1d)) + + def test_str(self, spcreator, dat1d): + str(spcreator(dat1d)) + + def test_neg(self, spcreator): + A = np.array([-1, 0, 17, 0, -5, 0, 1, -4, 0, 0, 0, 0], 'd') + assert_equal(-A, (-spcreator(A)).toarray()) + + def test_1d_supported_init(self, spcreator): + A = spcreator([0, 1, 2, 3]) + assert A.ndim == 1 + + def test_reshape_1d_tofrom_row_or_column(self, spcreator): + # add a dimension 1d->2d + x = spcreator([1, 0, 7, 0, 0, 0, 0, -3, 0, 0, 0, 5]) + y = x.reshape(1, 12) + desired = [[1, 0, 7, 0, 0, 0, 0, -3, 0, 0, 0, 5]] + assert_equal(y.toarray(), desired) + + # remove a size-1 dimension 2d->1d + x = spcreator(desired) + y = x.reshape(12) + assert_equal(y.toarray(), desired[0]) + y2 = x.reshape((12,)) + assert y.shape == y2.shape + + # make a 2d column into 1d. 2d->1d + y = x.T.reshape(12) + assert_equal(y.toarray(), desired[0]) + + def test_reshape(self, spcreator): + x = spcreator([1, 0, 7, 0, 0, 0, 0, -3, 0, 0, 0, 5]) + y = x.reshape((4, 3)) + desired = [[1, 0, 7], [0, 0, 0], [0, -3, 0], [0, 0, 5]] + assert_equal(y.toarray(), desired) + + y = x.reshape((12,)) + assert y is x + + y = x.reshape(12) + assert_equal(y.toarray(), x.toarray()) + + def test_sum(self, spcreator): + np.random.seed(1234) + dat_1 = np.array([0, 1, 2, 3, -4, 5, -6, 7, 9]) + dat_2 = np.random.rand(5) + dat_3 = np.array([]) + dat_4 = np.zeros((40,)) + arrays = [dat_1, dat_2, dat_3, dat_4] + + for dat in arrays: + datsp = spcreator(dat) + with np.errstate(over='ignore'): + assert np.isscalar(datsp.sum()) + assert_allclose(dat.sum(), datsp.sum()) + assert_allclose(dat.sum(axis=None), datsp.sum(axis=None)) + assert_allclose(dat.sum(axis=0), datsp.sum(axis=0)) + assert_allclose(dat.sum(axis=-1), datsp.sum(axis=-1)) + + # test `out` parameter + datsp.sum(axis=0, out=np.zeros(())) + + def test_sum_invalid_params(self, spcreator): + out = np.zeros((3,)) # wrong size for out + dat = np.array([0, 1, 2]) + datsp = spcreator(dat) + + with pytest.raises(ValueError, match='axis must be None, -1 or 0'): + datsp.sum(axis=1) + with pytest.raises(TypeError, match='Tuples are not accepted'): + datsp.sum(axis=(0, 1)) + with pytest.raises(TypeError, match='axis must be an integer'): + datsp.sum(axis=1.5) + with pytest.raises(ValueError, match='output parameter.*wrong.*dimension'): + datsp.sum(axis=0, out=out) + + def test_numpy_sum(self, spcreator): + dat = np.array([0, 1, 2]) + datsp = spcreator(dat) + + dat_sum = np.sum(dat) + datsp_sum = np.sum(datsp) + + assert_allclose(dat_sum, datsp_sum) + + def test_mean(self, spcreator): + dat = np.array([0, 1, 2]) + datsp = spcreator(dat) + + assert_allclose(dat.mean(), datsp.mean()) + assert np.isscalar(datsp.mean(axis=None)) + assert_allclose(dat.mean(axis=None), datsp.mean(axis=None)) + assert_allclose(dat.mean(axis=0), datsp.mean(axis=0)) + assert_allclose(dat.mean(axis=-1), datsp.mean(axis=-1)) + + with pytest.raises(ValueError, match='axis'): + datsp.mean(axis=1) + with pytest.raises(ValueError, match='axis'): + datsp.mean(axis=-2) + + def test_mean_invalid_params(self, spcreator): + out = np.asarray(np.zeros((1, 3))) + dat = np.array([[0, 1, 2], [3, -4, 5], [-6, 7, 9]]) + + datsp = spcreator(dat) + with pytest.raises(ValueError, match='axis out of range'): + datsp.mean(axis=3) + with pytest.raises(TypeError, match='Tuples are not accepted'): + datsp.mean(axis=(0, 1)) + with pytest.raises(TypeError, match='axis must be an integer'): + datsp.mean(axis=1.5) + with pytest.raises(ValueError, match='output parameter.*wrong.*dimension'): + datsp.mean(axis=1, out=out) + + def test_sum_dtype(self, spcreator): + dat = np.array([0, 1, 2]) + datsp = spcreator(dat) + + for dtype in supported_dtypes: + dat_sum = dat.sum(dtype=dtype) + datsp_sum = datsp.sum(dtype=dtype) + + assert_allclose(dat_sum, datsp_sum) + assert_equal(dat_sum.dtype, datsp_sum.dtype) + + def test_mean_dtype(self, spcreator): + dat = np.array([0, 1, 2]) + datsp = spcreator(dat) + + for dtype in supported_dtypes: + dat_mean = dat.mean(dtype=dtype) + datsp_mean = datsp.mean(dtype=dtype) + + assert_allclose(dat_mean, datsp_mean) + assert_equal(dat_mean.dtype, datsp_mean.dtype) + + def test_mean_out(self, spcreator): + dat = np.array([0, 1, 2]) + datsp = spcreator(dat) + + dat_out = np.array(0) + datsp_out = np.array(0) + + dat.mean(out=dat_out) + datsp.mean(out=datsp_out) + assert_allclose(dat_out, datsp_out) + + dat.mean(axis=0, out=dat_out) + datsp.mean(axis=0, out=datsp_out) + assert_allclose(dat_out, datsp_out) + + with pytest.raises(ValueError, match="output parameter.*dimension"): + datsp.mean(out=np.array([0])) + with pytest.raises(ValueError, match="output parameter.*dimension"): + datsp.mean(out=np.array([[0]])) + + def test_numpy_mean(self, spcreator): + dat = np.array([0, 1, 2]) + datsp = spcreator(dat) + + dat_mean = np.mean(dat) + datsp_mean = np.mean(datsp) + + assert_allclose(dat_mean, datsp_mean) + assert_equal(dat_mean.dtype, datsp_mean.dtype) + + @pytest.mark.thread_unsafe + @sup_complex + def test_from_array(self, spcreator): + A = np.array([2, 3, 4]) + assert_equal(spcreator(A).toarray(), A) + + A = np.array([1.0 + 3j, 0, -1]) + assert_equal(spcreator(A).toarray(), A) + assert_equal(spcreator(A, dtype='int16').toarray(), A.astype('int16')) + + @pytest.mark.thread_unsafe + @sup_complex + def test_from_list(self, spcreator): + A = [2, 3, 4] + assert_equal(spcreator(A).toarray(), A) + + A = [1.0 + 3j, 0, -1] + assert_equal(spcreator(A).toarray(), np.array(A)) + assert_equal( + spcreator(A, dtype='int16').toarray(), np.array(A).astype('int16') + ) + + @pytest.mark.thread_unsafe + @sup_complex + def test_from_sparse(self, spcreator): + D = np.array([1, 0, 0]) + S = coo_array(D) + assert_equal(spcreator(S).toarray(), D) + S = spcreator(D) + assert_equal(spcreator(S).toarray(), D) + + D = np.array([1.0 + 3j, 0, -1]) + S = coo_array(D) + assert_equal(spcreator(S).toarray(), D) + assert_equal(spcreator(S, dtype='int16').toarray(), D.astype('int16')) + S = spcreator(D) + assert_equal(spcreator(S).toarray(), D) + assert_equal(spcreator(S, dtype='int16').toarray(), D.astype('int16')) + + def test_toarray(self, spcreator, dat1d): + datsp = spcreator(dat1d) + # Check C- or F-contiguous (default). + chk = datsp.toarray() + assert_equal(chk, dat1d) + assert chk.flags.c_contiguous == chk.flags.f_contiguous + + # Check C-contiguous (with arg). + chk = datsp.toarray(order='C') + assert_equal(chk, dat1d) + assert chk.flags.c_contiguous + assert chk.flags.f_contiguous + + # Check F-contiguous (with arg). + chk = datsp.toarray(order='F') + assert_equal(chk, dat1d) + assert chk.flags.c_contiguous + assert chk.flags.f_contiguous + + # Check with output arg. + out = np.zeros(datsp.shape, dtype=datsp.dtype) + datsp.toarray(out=out) + assert_equal(out, dat1d) + + # Check that things are fine when we don't initialize with zeros. + out[...] = 1.0 + datsp.toarray(out=out) + assert_equal(out, dat1d) + + # np.dot does not work with sparse matrices (unless scalars) + # so this is testing whether dat1d matches datsp.toarray() + a = np.array([1.0, 2.0, 3.0, 4.0]) + dense_dot_dense = np.dot(a, dat1d) + check = np.dot(a, datsp.toarray()) + assert_equal(dense_dot_dense, check) + + b = np.array([1.0, 2.0, 3.0, 4.0]) + dense_dot_dense = np.dot(dat1d, b) + check = np.dot(datsp.toarray(), b) + assert_equal(dense_dot_dense, check) + + # Check bool data works. + spbool = spcreator(dat1d, dtype=bool) + arrbool = dat1d.astype(bool) + assert_equal(spbool.toarray(), arrbool) + + def test_add(self, spcreator, datsp_math_dtypes): + for dtype, dat, datsp in datsp_math_dtypes[spcreator]: + a = dat.copy() + a[0] = 2.0 + b = datsp + c = b + a + assert_equal(c, b.toarray() + a) + + # test broadcasting + # Note: cant add nonzero scalar to sparray. Can add len 1 array + c = b + a[0:1] + assert_equal(c, b.toarray() + a[0]) + + def test_radd(self, spcreator, datsp_math_dtypes): + for dtype, dat, datsp in datsp_math_dtypes[spcreator]: + a = dat.copy() + a[0] = 2.0 + b = datsp + c = a + b + assert_equal(c, a + b.toarray()) + + def test_rsub(self, spcreator, datsp_math_dtypes): + for dtype, dat, datsp in datsp_math_dtypes[spcreator]: + if dtype == np.dtype('bool'): + # boolean array subtraction deprecated in 1.9.0 + continue + + assert_equal((dat - datsp), [0, 0, 0, 0]) + assert_equal((datsp - dat), [0, 0, 0, 0]) + assert_equal((0 - datsp).toarray(), -dat) + + A = spcreator([1, -4, 0, 2], dtype='d') + assert_equal((dat - A), dat - A.toarray()) + assert_equal((A - dat), A.toarray() - dat) + assert_equal(A.toarray() - datsp, A.toarray() - dat) + assert_equal(datsp - A.toarray(), dat - A.toarray()) + + # test broadcasting + assert_equal(dat[:1] - datsp, dat[:1] - dat) + + def test_matmul_basic(self, spcreator): + A = np.array([[2, 0, 3.0], [0, 0, 0], [0, 1, 2]]) + v = np.array([1, 0, 3]) + Asp = spcreator(A) + vsp = spcreator(v) + + # sparse result when both args are sparse and result not scalar + assert_equal((Asp @ vsp).toarray(), A @ v) + assert_equal(A @ vsp, A @ v) + assert_equal(Asp @ v, A @ v) + assert_equal((vsp @ Asp).toarray(), v @ A) + assert_equal(vsp @ A, v @ A) + assert_equal(v @ Asp, v @ A) + + assert_equal(vsp @ vsp, v @ v) + assert_equal(v @ vsp, v @ v) + assert_equal(vsp @ v, v @ v) + assert_equal((Asp @ Asp).toarray(), A @ A) + assert_equal(A @ Asp, A @ A) + assert_equal(Asp @ A, A @ A) + + def test_matvec(self, spcreator): + A = np.array([2, 0, 3.0]) + Asp = spcreator(A) + col = np.array([[1, 2, 3]]).T + + assert_allclose(Asp @ col, Asp.toarray() @ col) + + assert (A @ np.array([1, 2, 3])).shape == () + assert Asp @ np.array([1, 2, 3]) == 11 + assert (Asp @ np.array([1, 2, 3])).shape == () + assert (Asp @ np.array([[1], [2], [3]])).shape == (1,) + # check result type + assert isinstance(Asp @ matrix([[1, 2, 3]]).T, np.ndarray) + + # ensure exception is raised for improper dimensions + bad_vecs = [np.array([1, 2]), np.array([1, 2, 3, 4]), np.array([[1], [2]])] + for x in bad_vecs: + with pytest.raises(ValueError, match='dimension mismatch'): + Asp @ x + + # The current relationship between sparse matrix products and array + # products is as follows: + dot_result = np.dot(Asp.toarray(), [1, 2, 3]) + assert_allclose(Asp @ np.array([1, 2, 3]), dot_result) + assert_allclose(Asp @ [[1], [2], [3]], dot_result.T) + # Note that the result of Asp @ x is dense if x has a singleton dimension. + + def test_rmatvec(self, spcreator, dat1d): + M = spcreator(dat1d) + assert_allclose([1, 2, 3, 4] @ M, np.dot([1, 2, 3, 4], M.toarray())) + row = np.array([[1, 2, 3, 4]]) + assert_allclose(row @ M, row @ M.toarray()) + + def test_transpose(self, spcreator, dat1d): + for A in [dat1d, np.array([])]: + B = spcreator(A) + assert_equal(B.toarray(), A) + assert_equal(B.transpose().toarray(), A) + assert_equal(B.dtype, A.dtype) + + def test_add_dense_to_sparse(self, spcreator, datsp_math_dtypes): + for dtype, dat, datsp in datsp_math_dtypes[spcreator]: + sum1 = dat + datsp + assert_equal(sum1, dat + dat) + sum2 = datsp + dat + assert_equal(sum2, dat + dat) + + def test_iterator(self, spcreator): + # test that __iter__ is compatible with NumPy + B = np.arange(5) + A = spcreator(B) + + if A.format not in ['coo', 'dia', 'bsr']: + for x, y in zip(A, B): + assert_equal(x, y) + + def test_resize(self, spcreator): + # resize(shape) resizes the matrix in-place + D = np.array([1, 0, 3, 4]) + S = spcreator(D) + assert S.resize((3,)) is None + assert_equal(S.toarray(), [1, 0, 3]) + S.resize((5,)) + assert_equal(S.toarray(), [1, 0, 3, 0, 0]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_construct.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_construct.py new file mode 100644 index 0000000000000000000000000000000000000000..38b8288c39f4ebacd9cadc58bba7b18416377c73 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_construct.py @@ -0,0 +1,872 @@ +"""test sparse matrix construction functions""" + +import numpy as np +from numpy import array +from numpy.testing import (assert_equal, assert_, + assert_array_equal, assert_array_almost_equal_nulp) +import pytest +from pytest import raises as assert_raises +from scipy._lib._testutils import check_free_memory + +from scipy.sparse import (csr_matrix, coo_matrix, + csr_array, coo_array, + csc_array, bsr_array, + dia_array, dok_array, + lil_array, csc_matrix, + bsr_matrix, dia_matrix, + lil_matrix, sparray, spmatrix, + _construct as construct) +from scipy.sparse._construct import rand as sprand + +sparse_formats = ['csr','csc','coo','bsr','dia','lil','dok'] + +#TODO check whether format=XXX is respected + + +def _sprandn(m, n, density=0.01, format="coo", dtype=None, rng=None): + # Helper function for testing. + rng = np.random.default_rng(rng) + data_rvs = rng.standard_normal + return construct.random(m, n, density, format, dtype, rng, data_rvs) + + +def _sprandn_array(m, n, density=0.01, format="coo", dtype=None, rng=None): + # Helper function for testing. + rng = np.random.default_rng(rng) + data_sampler = rng.standard_normal + return construct.random_array((m, n), density=density, format=format, dtype=dtype, + rng=rng, data_sampler=data_sampler) + + +class TestConstructUtils: + + @pytest.mark.parametrize("cls", [ + csc_array, csr_array, coo_array, bsr_array, + dia_array, dok_array, lil_array + ]) + def test_singleton_array_constructor(self, cls): + with pytest.raises( + ValueError, + match=( + 'scipy sparse array classes do not support ' + 'instantiation from a scalar' + ) + ): + cls(0) + + @pytest.mark.parametrize("cls", [ + csc_matrix, csr_matrix, coo_matrix, + bsr_matrix, dia_matrix, lil_matrix + ]) + def test_singleton_matrix_constructor(self, cls): + """ + This test is for backwards compatibility post scipy 1.13. + The behavior observed here is what is to be expected + with the older matrix classes. This test comes with the + exception of dok_matrix, which was not working pre scipy1.12 + (unlike the rest of these). + """ + assert cls(0).shape == (1, 1) + + def test_spdiags(self): + diags1 = array([[1, 2, 3, 4, 5]]) + diags2 = array([[1, 2, 3, 4, 5], + [6, 7, 8, 9,10]]) + diags3 = array([[1, 2, 3, 4, 5], + [6, 7, 8, 9,10], + [11,12,13,14,15]]) + + cases = [] + cases.append((diags1, 0, 1, 1, [[1]])) + cases.append((diags1, [0], 1, 1, [[1]])) + cases.append((diags1, [0], 2, 1, [[1],[0]])) + cases.append((diags1, [0], 1, 2, [[1,0]])) + cases.append((diags1, [1], 1, 2, [[0,2]])) + cases.append((diags1,[-1], 1, 2, [[0,0]])) + cases.append((diags1, [0], 2, 2, [[1,0],[0,2]])) + cases.append((diags1,[-1], 2, 2, [[0,0],[1,0]])) + cases.append((diags1, [3], 2, 2, [[0,0],[0,0]])) + cases.append((diags1, [0], 3, 4, [[1,0,0,0],[0,2,0,0],[0,0,3,0]])) + cases.append((diags1, [1], 3, 4, [[0,2,0,0],[0,0,3,0],[0,0,0,4]])) + cases.append((diags1, [2], 3, 5, [[0,0,3,0,0],[0,0,0,4,0],[0,0,0,0,5]])) + + cases.append((diags2, [0,2], 3, 3, [[1,0,8],[0,2,0],[0,0,3]])) + cases.append((diags2, [-1,0], 3, 4, [[6,0,0,0],[1,7,0,0],[0,2,8,0]])) + cases.append((diags2, [2,-3], 6, 6, [[0,0,3,0,0,0], + [0,0,0,4,0,0], + [0,0,0,0,5,0], + [6,0,0,0,0,0], + [0,7,0,0,0,0], + [0,0,8,0,0,0]])) + + cases.append((diags3, [-1,0,1], 6, 6, [[6,12, 0, 0, 0, 0], + [1, 7,13, 0, 0, 0], + [0, 2, 8,14, 0, 0], + [0, 0, 3, 9,15, 0], + [0, 0, 0, 4,10, 0], + [0, 0, 0, 0, 5, 0]])) + cases.append((diags3, [-4,2,-1], 6, 5, [[0, 0, 8, 0, 0], + [11, 0, 0, 9, 0], + [0,12, 0, 0,10], + [0, 0,13, 0, 0], + [1, 0, 0,14, 0], + [0, 2, 0, 0,15]])) + cases.append((diags3, [-1, 1, 2], len(diags3[0]), len(diags3[0]), + [[0, 7, 13, 0, 0], + [1, 0, 8, 14, 0], + [0, 2, 0, 9, 15], + [0, 0, 3, 0, 10], + [0, 0, 0, 4, 0]])) + + for d, o, m, n, result in cases: + if len(d[0]) == m and m == n: + assert_equal(construct.spdiags(d, o).toarray(), result) + assert_equal(construct.spdiags(d, o, m, n).toarray(), result) + assert_equal(construct.spdiags(d, o, (m, n)).toarray(), result) + + def test_diags(self): + a = array([1, 2, 3, 4, 5]) + b = array([6, 7, 8, 9, 10]) + c = array([11, 12, 13, 14, 15]) + + cases = [] + cases.append((a[:1], 0, (1, 1), [[1]])) + cases.append(([a[:1]], [0], (1, 1), [[1]])) + cases.append(([a[:1]], [0], (2, 1), [[1],[0]])) + cases.append(([a[:1]], [0], (1, 2), [[1,0]])) + cases.append(([a[:1]], [1], (1, 2), [[0,1]])) + cases.append(([a[:2]], [0], (2, 2), [[1,0],[0,2]])) + cases.append(([a[:1]],[-1], (2, 2), [[0,0],[1,0]])) + cases.append(([a[:3]], [0], (3, 4), [[1,0,0,0],[0,2,0,0],[0,0,3,0]])) + cases.append(([a[:3]], [1], (3, 4), [[0,1,0,0],[0,0,2,0],[0,0,0,3]])) + cases.append(([a[:1]], [-2], (3, 5), [[0,0,0,0,0],[0,0,0,0,0],[1,0,0,0,0]])) + cases.append(([a[:2]], [-1], (3, 5), [[0,0,0,0,0],[1,0,0,0,0],[0,2,0,0,0]])) + cases.append(([a[:3]], [0], (3, 5), [[1,0,0,0,0],[0,2,0,0,0],[0,0,3,0,0]])) + cases.append(([a[:3]], [1], (3, 5), [[0,1,0,0,0],[0,0,2,0,0],[0,0,0,3,0]])) + cases.append(([a[:3]], [2], (3, 5), [[0,0,1,0,0],[0,0,0,2,0],[0,0,0,0,3]])) + cases.append(([a[:2]], [3], (3, 5), [[0,0,0,1,0],[0,0,0,0,2],[0,0,0,0,0]])) + cases.append(([a[:1]], [4], (3, 5), [[0,0,0,0,1],[0,0,0,0,0],[0,0,0,0,0]])) + cases.append(([a[:1]], [-4], (5, 3), [[0,0,0],[0,0,0],[0,0,0],[0,0,0],[1,0,0]])) + cases.append(([a[:2]], [-3], (5, 3), [[0,0,0],[0,0,0],[0,0,0],[1,0,0],[0,2,0]])) + cases.append(([a[:3]], [-2], (5, 3), [[0,0,0],[0,0,0],[1,0,0],[0,2,0],[0,0,3]])) + cases.append(([a[:3]], [-1], (5, 3), [[0,0,0],[1,0,0],[0,2,0],[0,0,3],[0,0,0]])) + cases.append(([a[:3]], [0], (5, 3), [[1,0,0],[0,2,0],[0,0,3],[0,0,0],[0,0,0]])) + cases.append(([a[:2]], [1], (5, 3), [[0,1,0],[0,0,2],[0,0,0],[0,0,0],[0,0,0]])) + cases.append(([a[:1]], [2], (5, 3), [[0,0,1],[0,0,0],[0,0,0],[0,0,0],[0,0,0]])) + + cases.append(([a[:3],b[:1]], [0,2], (3, 3), [[1,0,6],[0,2,0],[0,0,3]])) + cases.append(([a[:2],b[:3]], [-1,0], (3, 4), [[6,0,0,0],[1,7,0,0],[0,2,8,0]])) + cases.append(([a[:4],b[:3]], [2,-3], (6, 6), [[0,0,1,0,0,0], + [0,0,0,2,0,0], + [0,0,0,0,3,0], + [6,0,0,0,0,4], + [0,7,0,0,0,0], + [0,0,8,0,0,0]])) + + cases.append(([a[:4],b,c[:4]], [-1,0,1], (5, 5), [[6,11, 0, 0, 0], + [1, 7,12, 0, 0], + [0, 2, 8,13, 0], + [0, 0, 3, 9,14], + [0, 0, 0, 4,10]])) + cases.append(([a[:2],b[:3],c], [-4,2,-1], (6, 5), [[0, 0, 6, 0, 0], + [11, 0, 0, 7, 0], + [0,12, 0, 0, 8], + [0, 0,13, 0, 0], + [1, 0, 0,14, 0], + [0, 2, 0, 0,15]])) + + # too long arrays are OK + cases.append(([a], [0], (1, 1), [[1]])) + cases.append(([a[:3],b], [0,2], (3, 3), [[1, 0, 6], [0, 2, 0], [0, 0, 3]])) + cases.append(( + np.array([[1, 2, 3], [4, 5, 6]]), + [0,-1], + (3, 3), + [[1, 0, 0], [4, 2, 0], [0, 5, 3]] + )) + + # scalar case: broadcasting + cases.append(([1,-2,1], [1,0,-1], (3, 3), [[-2, 1, 0], + [1, -2, 1], + [0, 1, -2]])) + + for d, o, shape, result in cases: + err_msg = f"{d!r} {o!r} {shape!r} {result!r}" + assert_equal(construct.diags(d, offsets=o, shape=shape).toarray(), + result, err_msg=err_msg) + + if (shape[0] == shape[1] + and hasattr(d[0], '__len__') + and len(d[0]) <= max(shape)): + # should be able to find the shape automatically + assert_equal(construct.diags(d, offsets=o).toarray(), result, + err_msg=err_msg) + + def test_diags_default(self): + a = array([1, 2, 3, 4, 5]) + assert_equal(construct.diags(a).toarray(), np.diag(a)) + + def test_diags_default_bad(self): + a = array([[1, 2, 3, 4, 5], [2, 3, 4, 5, 6]]) + assert_raises(ValueError, construct.diags, a) + + def test_diags_bad(self): + a = array([1, 2, 3, 4, 5]) + b = array([6, 7, 8, 9, 10]) + c = array([11, 12, 13, 14, 15]) + + cases = [] + cases.append(([a[:0]], 0, (1, 1))) + cases.append(([a[:4],b,c[:3]], [-1,0,1], (5, 5))) + cases.append(([a[:2],c,b[:3]], [-4,2,-1], (6, 5))) + cases.append(([a[:2],c,b[:3]], [-4,2,-1], None)) + cases.append(([], [-4,2,-1], None)) + cases.append(([1], [-5], (4, 4))) + cases.append(([a], 0, None)) + + for d, o, shape in cases: + assert_raises(ValueError, construct.diags, d, offsets=o, shape=shape) + + assert_raises(TypeError, construct.diags, [[None]], offsets=[0]) + + def test_diags_vs_diag(self): + # Check that + # + # diags([a, b, ...], [i, j, ...]) == diag(a, i) + diag(b, j) + ... + # + + rng = np.random.RandomState(1234) + + for n_diags in [1, 2, 3, 4, 5, 10]: + n = 1 + n_diags//2 + rng.randint(0, 10) + + offsets = np.arange(-n+1, n-1) + rng.shuffle(offsets) + offsets = offsets[:n_diags] + + diagonals = [rng.rand(n - abs(q)) for q in offsets] + + mat = construct.diags(diagonals, offsets=offsets) + dense_mat = sum([np.diag(x, j) for x, j in zip(diagonals, offsets)]) + + assert_array_almost_equal_nulp(mat.toarray(), dense_mat) + + if len(offsets) == 1: + mat = construct.diags(diagonals[0], offsets=offsets[0]) + dense_mat = np.diag(diagonals[0], offsets[0]) + assert_array_almost_equal_nulp(mat.toarray(), dense_mat) + + def test_diags_dtype(self): + x = construct.diags([2.2], offsets=[0], shape=(2, 2), dtype=int) + assert_equal(x.dtype, int) + assert_equal(x.toarray(), [[2, 0], [0, 2]]) + + def test_diags_one_diagonal(self): + d = list(range(5)) + for k in range(-5, 6): + assert_equal(construct.diags(d, offsets=k).toarray(), + construct.diags([d], offsets=[k]).toarray()) + + def test_diags_empty(self): + x = construct.diags([]) + assert_equal(x.shape, (0, 0)) + + @pytest.mark.parametrize("identity", [construct.identity, construct.eye_array]) + def test_identity(self, identity): + assert_equal(identity(1).toarray(), [[1]]) + assert_equal(identity(2).toarray(), [[1,0],[0,1]]) + + I = identity(3, dtype='int8', format='dia') + assert_equal(I.dtype, np.dtype('int8')) + assert_equal(I.format, 'dia') + + for fmt in sparse_formats: + I = identity(3, format=fmt) + assert_equal(I.format, fmt) + assert_equal(I.toarray(), [[1,0,0],[0,1,0],[0,0,1]]) + + @pytest.mark.parametrize("eye", [construct.eye, construct.eye_array]) + def test_eye(self, eye): + assert_equal(eye(1,1).toarray(), [[1]]) + assert_equal(eye(2,3).toarray(), [[1,0,0],[0,1,0]]) + assert_equal(eye(3,2).toarray(), [[1,0],[0,1],[0,0]]) + assert_equal(eye(3,3).toarray(), [[1,0,0],[0,1,0],[0,0,1]]) + + assert_equal(eye(3,3,dtype='int16').dtype, np.dtype('int16')) + + for m in [3, 5]: + for n in [3, 5]: + for k in range(-5,6): + # scipy.sparse.eye deviates from np.eye here. np.eye will + # create arrays of all 0's when the diagonal offset is + # greater than the size of the array. For sparse arrays + # this makes less sense, especially as it results in dia + # arrays with negative diagonals. Therefore sp.sparse.eye + # validates that diagonal offsets fall within the shape of + # the array. See gh-18555. + if (k > 0 and k > n) or (k < 0 and abs(k) > m): + with pytest.raises( + ValueError, match="Offset.*out of bounds" + ): + eye(m, n, k=k) + + else: + assert_equal( + eye(m, n, k=k).toarray(), + np.eye(m, n, k=k) + ) + if m == n: + assert_equal( + eye(m, k=k).toarray(), + np.eye(m, n, k=k) + ) + + @pytest.mark.parametrize("eye", [construct.eye, construct.eye_array]) + def test_eye_one(self, eye): + assert_equal(eye(1).toarray(), [[1]]) + assert_equal(eye(2).toarray(), [[1,0],[0,1]]) + + I = eye(3, dtype='int8', format='dia') + assert_equal(I.dtype, np.dtype('int8')) + assert_equal(I.format, 'dia') + + for fmt in sparse_formats: + I = eye(3, format=fmt) + assert_equal(I.format, fmt) + assert_equal(I.toarray(), [[1,0,0],[0,1,0],[0,0,1]]) + + def test_eye_array_vs_matrix(self): + assert isinstance(construct.eye_array(3), sparray) + assert not isinstance(construct.eye(3), sparray) + + def test_kron(self): + cases = [] + + cases.append(array([[0]])) + cases.append(array([[-1]])) + cases.append(array([[4]])) + cases.append(array([[10]])) + cases.append(array([[0],[0]])) + cases.append(array([[0,0]])) + cases.append(array([[1,2],[3,4]])) + cases.append(array([[0,2],[5,0]])) + cases.append(array([[0,2,-6],[8,0,14]])) + cases.append(array([[5,4],[0,0],[6,0]])) + cases.append(array([[5,4,4],[1,0,0],[6,0,8]])) + cases.append(array([[0,1,0,2,0,5,8]])) + cases.append(array([[0.5,0.125,0,3.25],[0,2.5,0,0]])) + + # test all cases with some formats + for a in cases: + ca = csr_array(a) + for b in cases: + cb = csr_array(b) + expected = np.kron(a, b) + for fmt in sparse_formats[1:4]: + result = construct.kron(ca, cb, format=fmt) + assert_equal(result.format, fmt) + assert_array_equal(result.toarray(), expected) + assert isinstance(result, sparray) + + # test one case with all formats + a = cases[-1] + b = cases[-3] + ca = csr_array(a) + cb = csr_array(b) + + expected = np.kron(a, b) + for fmt in sparse_formats: + result = construct.kron(ca, cb, format=fmt) + assert_equal(result.format, fmt) + assert_array_equal(result.toarray(), expected) + assert isinstance(result, sparray) + + # check that spmatrix returned when both inputs are spmatrix + result = construct.kron(csr_matrix(a), csr_matrix(b), format=fmt) + assert_equal(result.format, fmt) + assert_array_equal(result.toarray(), expected) + assert isinstance(result, spmatrix) + + def test_kron_ndim_exceptions(self): + with pytest.raises(ValueError, match='requires 2D input'): + construct.kron([[0], [1]], csr_array([0, 1])) + with pytest.raises(ValueError, match='requires 2D input'): + construct.kron(csr_array([0, 1]), [[0], [1]]) + # no exception if sparse arrays are not input (spmatrix inferred) + construct.kron([[0], [1]], [0, 1]) + + def test_kron_large(self): + n = 2**16 + a = construct.diags_array([1], shape=(1, n), offsets=n-1) + b = construct.diags_array([1], shape=(n, 1), offsets=1-n) + + construct.kron(a, a) + construct.kron(b, b) + + def test_kronsum(self): + cases = [] + + cases.append(array([[0]])) + cases.append(array([[-1]])) + cases.append(array([[4]])) + cases.append(array([[10]])) + cases.append(array([[1,2],[3,4]])) + cases.append(array([[0,2],[5,0]])) + cases.append(array([[0,2,-6],[8,0,14],[0,3,0]])) + cases.append(array([[1,0,0],[0,5,-1],[4,-2,8]])) + + # test all cases with default format + for a in cases: + for b in cases: + result = construct.kronsum(csr_array(a), csr_array(b)).toarray() + expected = (np.kron(np.eye(b.shape[0]), a) + + np.kron(b, np.eye(a.shape[0]))) + assert_array_equal(result, expected) + + # check that spmatrix returned when both inputs are spmatrix + result = construct.kronsum(csr_matrix(a), csr_matrix(b)).toarray() + assert_array_equal(result, expected) + + def test_kronsum_ndim_exceptions(self): + with pytest.raises(ValueError, match='requires 2D input'): + construct.kronsum([[0], [1]], csr_array([0, 1])) + with pytest.raises(ValueError, match='requires 2D input'): + construct.kronsum(csr_array([0, 1]), [[0], [1]]) + # no exception if sparse arrays are not input (spmatrix inferred) + construct.kronsum([[0, 1], [1, 0]], [2]) + + @pytest.mark.parametrize("coo_cls", [coo_matrix, coo_array]) + def test_vstack(self, coo_cls): + A = coo_cls([[1,2],[3,4]]) + B = coo_cls([[5,6]]) + + expected = array([[1, 2], + [3, 4], + [5, 6]]) + assert_equal(construct.vstack([A, B]).toarray(), expected) + assert_equal(construct.vstack([A, B], dtype=np.float32).dtype, + np.float32) + + assert_equal(construct.vstack([A.todok(), B.todok()]).toarray(), expected) + + assert_equal(construct.vstack([A.tocsr(), B.tocsr()]).toarray(), + expected) + result = construct.vstack([A.tocsr(), B.tocsr()], + format="csr", dtype=np.float32) + assert_equal(result.dtype, np.float32) + assert_equal(result.indices.dtype, np.int32) + assert_equal(result.indptr.dtype, np.int32) + + assert_equal(construct.vstack([A.tocsc(), B.tocsc()]).toarray(), + expected) + result = construct.vstack([A.tocsc(), B.tocsc()], + format="csc", dtype=np.float32) + assert_equal(result.dtype, np.float32) + assert_equal(result.indices.dtype, np.int32) + assert_equal(result.indptr.dtype, np.int32) + + def test_vstack_maintain64bit_idx_dtype(self): + # see gh-20389 v/hstack returns int32 idx_dtype with input int64 idx_dtype + X = csr_array([[1, 0, 0], [0, 1, 0], [0, 1, 0]]) + X.indptr = X.indptr.astype(np.int64) + X.indices = X.indices.astype(np.int64) + assert construct.vstack([X, X]).indptr.dtype == np.int64 + assert construct.hstack([X, X]).indptr.dtype == np.int64 + + X = csc_array([[1, 0, 0], [0, 1, 0], [0, 1, 0]]) + X.indptr = X.indptr.astype(np.int64) + X.indices = X.indices.astype(np.int64) + assert construct.vstack([X, X]).indptr.dtype == np.int64 + assert construct.hstack([X, X]).indptr.dtype == np.int64 + + X = coo_array([[1, 0, 0], [0, 1, 0], [0, 1, 0]]) + X.coords = tuple(co.astype(np.int64) for co in X.coords) + assert construct.vstack([X, X]).coords[0].dtype == np.int64 + assert construct.hstack([X, X]).coords[0].dtype == np.int64 + + def test_vstack_matrix_or_array(self): + A = [[1,2],[3,4]] + B = [[5,6]] + assert isinstance(construct.vstack([coo_array(A), coo_array(B)]), sparray) + assert isinstance(construct.vstack([coo_array(A), coo_matrix(B)]), sparray) + assert isinstance(construct.vstack([coo_matrix(A), coo_array(B)]), sparray) + assert isinstance(construct.vstack([coo_matrix(A), coo_matrix(B)]), spmatrix) + + def test_vstack_1d_with_2d(self): + # fixes gh-21064 + arr = csr_array([[1, 0, 0], [0, 1, 0]]) + arr1d = csr_array([1, 0, 0]) + arr1dcoo = coo_array([1, 0, 0]) + assert construct.vstack([arr, np.array([0, 0, 0])]).shape == (3, 3) + assert construct.hstack([arr1d, np.array([[0]])]).shape == (1, 4) + assert construct.hstack([arr1d, arr1d]).shape == (1, 6) + assert construct.vstack([arr1d, arr1d]).shape == (2, 3) + + # check csr specialty stacking code like _stack_along_minor_axis + assert construct.hstack([arr, arr]).shape == (2, 6) + assert construct.hstack([arr1d, arr1d]).shape == (1, 6) + + assert construct.hstack([arr1d, arr1dcoo]).shape == (1, 6) + assert construct.vstack([arr, arr1dcoo]).shape == (3, 3) + assert construct.vstack([arr1d, arr1dcoo]).shape == (2, 3) + + with pytest.raises(ValueError, match="incompatible row dimensions"): + construct.hstack([arr, np.array([0, 0])]) + with pytest.raises(ValueError, match="incompatible column dimensions"): + construct.vstack([arr, np.array([0, 0])]) + + @pytest.mark.parametrize("coo_cls", [coo_matrix, coo_array]) + def test_hstack(self, coo_cls): + A = coo_cls([[1,2],[3,4]]) + B = coo_cls([[5],[6]]) + + expected = array([[1, 2, 5], + [3, 4, 6]]) + assert_equal(construct.hstack([A, B]).toarray(), expected) + assert_equal(construct.hstack([A, B], dtype=np.float32).dtype, + np.float32) + + assert_equal(construct.hstack([A.todok(), B.todok()]).toarray(), expected) + + assert_equal(construct.hstack([A.tocsc(), B.tocsc()]).toarray(), + expected) + assert_equal(construct.hstack([A.tocsc(), B.tocsc()], + dtype=np.float32).dtype, + np.float32) + assert_equal(construct.hstack([A.tocsr(), B.tocsr()]).toarray(), + expected) + assert_equal(construct.hstack([A.tocsr(), B.tocsr()], + dtype=np.float32).dtype, + np.float32) + + def test_hstack_matrix_or_array(self): + A = [[1,2],[3,4]] + B = [[5],[6]] + assert isinstance(construct.hstack([coo_array(A), coo_array(B)]), sparray) + assert isinstance(construct.hstack([coo_array(A), coo_matrix(B)]), sparray) + assert isinstance(construct.hstack([coo_matrix(A), coo_array(B)]), sparray) + assert isinstance(construct.hstack([coo_matrix(A), coo_matrix(B)]), spmatrix) + + @pytest.mark.parametrize("block_array", (construct.bmat, construct.block_array)) + def test_block_creation(self, block_array): + + A = coo_array([[1, 2], [3, 4]]) + B = coo_array([[5],[6]]) + C = coo_array([[7]]) + D = coo_array((0, 0)) + + expected = array([[1, 2, 5], + [3, 4, 6], + [0, 0, 7]]) + assert_equal(block_array([[A, B], [None, C]]).toarray(), expected) + E = csr_array((1, 2), dtype=np.int32) + assert_equal(block_array([[A.tocsr(), B.tocsr()], + [E, C.tocsr()]]).toarray(), + expected) + assert_equal(block_array([[A.tocsc(), B.tocsc()], + [E.tocsc(), C.tocsc()]]).toarray(), + expected) + + expected = array([[1, 2, 0], + [3, 4, 0], + [0, 0, 7]]) + assert_equal(block_array([[A, None], [None, C]]).toarray(), expected) + assert_equal(block_array([[A.tocsr(), E.T.tocsr()], + [E, C.tocsr()]]).toarray(), + expected) + assert_equal(block_array([[A.tocsc(), E.T.tocsc()], + [E.tocsc(), C.tocsc()]]).toarray(), + expected) + + Z = csr_array((1, 1), dtype=np.int32) + expected = array([[0, 5], + [0, 6], + [7, 0]]) + assert_equal(block_array([[None, B], [C, None]]).toarray(), expected) + assert_equal(block_array([[E.T.tocsr(), B.tocsr()], + [C.tocsr(), Z]]).toarray(), + expected) + assert_equal(block_array([[E.T.tocsc(), B.tocsc()], + [C.tocsc(), Z.tocsc()]]).toarray(), + expected) + + expected = np.empty((0, 0)) + assert_equal(block_array([[None, None]]).toarray(), expected) + assert_equal(block_array([[None, D], [D, None]]).toarray(), + expected) + + # test bug reported in gh-5976 + expected = array([[7]]) + assert_equal(block_array([[None, D], [C, None]]).toarray(), + expected) + + # test failure cases + with assert_raises(ValueError) as excinfo: + block_array([[A], [B]]) + excinfo.match(r'Got blocks\[1,0\]\.shape\[1\] == 1, expected 2') + + with assert_raises(ValueError) as excinfo: + block_array([[A.tocsr()], [B.tocsr()]]) + excinfo.match(r'incompatible dimensions for axis 1') + + with assert_raises(ValueError) as excinfo: + block_array([[A.tocsc()], [B.tocsc()]]) + excinfo.match(r'Mismatching dimensions along axis 1: ({1, 2}|{2, 1})') + + with assert_raises(ValueError) as excinfo: + block_array([[A, C]]) + excinfo.match(r'Got blocks\[0,1\]\.shape\[0\] == 1, expected 2') + + with assert_raises(ValueError) as excinfo: + block_array([[A.tocsr(), C.tocsr()]]) + excinfo.match(r'Mismatching dimensions along axis 0: ({1, 2}|{2, 1})') + + with assert_raises(ValueError) as excinfo: + block_array([[A.tocsc(), C.tocsc()]]) + excinfo.match(r'incompatible dimensions for axis 0') + + def test_block_return_type(self): + block = construct.block_array + + # csr format ensures we hit _compressed_sparse_stack + # shape of F,G ensure we hit _stack_along_minor_axis + # list version ensure we hit the path with neither helper function + Fl, Gl = [[1, 2],[3, 4]], [[7], [5]] + Fm, Gm = csr_matrix(Fl), csr_matrix(Gl) + assert isinstance(block([[None, Fl], [Gl, None]], format="csr"), sparray) + assert isinstance(block([[None, Fm], [Gm, None]], format="csr"), sparray) + assert isinstance(block([[Fm, Gm]], format="csr"), sparray) + + def test_bmat_return_type(self): + """This can be removed after sparse matrix is removed""" + bmat = construct.bmat + # check return type. if any input _is_array output array, else matrix + Fl, Gl = [[1, 2],[3, 4]], [[7], [5]] + Fm, Gm = csr_matrix(Fl), csr_matrix(Gl) + Fa, Ga = csr_array(Fl), csr_array(Gl) + assert isinstance(bmat([[Fa, Ga]], format="csr"), sparray) + assert isinstance(bmat([[Fm, Gm]], format="csr"), spmatrix) + assert isinstance(bmat([[None, Fa], [Ga, None]], format="csr"), sparray) + assert isinstance(bmat([[None, Fm], [Ga, None]], format="csr"), sparray) + assert isinstance(bmat([[None, Fm], [Gm, None]], format="csr"), spmatrix) + assert isinstance(bmat([[None, Fl], [Gl, None]], format="csr"), spmatrix) + + # type returned by _compressed_sparse_stack (all csr) + assert isinstance(bmat([[Ga, Ga]], format="csr"), sparray) + assert isinstance(bmat([[Gm, Ga]], format="csr"), sparray) + assert isinstance(bmat([[Ga, Gm]], format="csr"), sparray) + assert isinstance(bmat([[Gm, Gm]], format="csr"), spmatrix) + # shape is 2x2 so no _stack_along_minor_axis + assert isinstance(bmat([[Fa, Fm]], format="csr"), sparray) + assert isinstance(bmat([[Fm, Fm]], format="csr"), spmatrix) + + # type returned by _compressed_sparse_stack (all csc) + assert isinstance(bmat([[Gm.tocsc(), Ga.tocsc()]], format="csc"), sparray) + assert isinstance(bmat([[Gm.tocsc(), Gm.tocsc()]], format="csc"), spmatrix) + # shape is 2x2 so no _stack_along_minor_axis + assert isinstance(bmat([[Fa.tocsc(), Fm.tocsc()]], format="csr"), sparray) + assert isinstance(bmat([[Fm.tocsc(), Fm.tocsc()]], format="csr"), spmatrix) + + # type returned when mixed input + assert isinstance(bmat([[Gl, Ga]], format="csr"), sparray) + assert isinstance(bmat([[Gm.tocsc(), Ga]], format="csr"), sparray) + assert isinstance(bmat([[Gm.tocsc(), Gm]], format="csr"), spmatrix) + assert isinstance(bmat([[Gm, Gm]], format="csc"), spmatrix) + + @pytest.mark.slow + @pytest.mark.thread_unsafe + @pytest.mark.xfail_on_32bit("Can't create large array for test") + def test_concatenate_int32_overflow(self): + """ test for indptr overflow when concatenating matrices """ + check_free_memory(30000) + + n = 33000 + A = csr_array(np.ones((n, n), dtype=bool)) + B = A.copy() + C = construct._compressed_sparse_stack((A, B), axis=0, + return_spmatrix=False) + + assert_(np.all(np.equal(np.diff(C.indptr), n))) + assert_equal(C.indices.dtype, np.int64) + assert_equal(C.indptr.dtype, np.int64) + + def test_block_diag_basic(self): + """ basic test for block_diag """ + A = coo_array([[1,2],[3,4]]) + B = coo_array([[5],[6]]) + C = coo_array([[7]]) + + expected = array([[1, 2, 0, 0], + [3, 4, 0, 0], + [0, 0, 5, 0], + [0, 0, 6, 0], + [0, 0, 0, 7]]) + + ABC = construct.block_diag((A, B, C)) + assert_equal(ABC.toarray(), expected) + assert ABC.coords[0].dtype == np.int32 + + def test_block_diag_idx_dtype(self): + X = coo_array([[1, 0, 0], [0, 1, 0], [0, 1, 0]]) + X.coords = tuple(co.astype(np.int64) for co in X.coords) + assert construct.block_diag([X, X]).coords[0].dtype == np.int64 + + def test_block_diag_scalar_1d_args(self): + """ block_diag with scalar and 1d arguments """ + # one 1d matrix and a scalar + assert_array_equal(construct.block_diag([[2,3], 4]).toarray(), + [[2, 3, 0], [0, 0, 4]]) + # 1d sparse arrays + A = coo_array([1,0,3]) + B = coo_array([0,4]) + assert_array_equal(construct.block_diag([A, B]).toarray(), + [[1, 0, 3, 0, 0], [0, 0, 0, 0, 4]]) + + def test_block_diag_1(self): + """ block_diag with one matrix """ + assert_equal(construct.block_diag([[1, 0]]).toarray(), + array([[1, 0]])) + assert_equal(construct.block_diag([[[1, 0]]]).toarray(), + array([[1, 0]])) + assert_equal(construct.block_diag([[[1], [0]]]).toarray(), + array([[1], [0]])) + # just on scalar + assert_equal(construct.block_diag([1]).toarray(), + array([[1]])) + + def test_block_diag_sparse_arrays(self): + """ block_diag with sparse arrays """ + + A = coo_array([[1, 2, 3]], shape=(1, 3)) + B = coo_array([[4, 5]], shape=(1, 2)) + assert_equal(construct.block_diag([A, B]).toarray(), + array([[1, 2, 3, 0, 0], [0, 0, 0, 4, 5]])) + + A = coo_array([[1], [2], [3]], shape=(3, 1)) + B = coo_array([[4], [5]], shape=(2, 1)) + assert_equal(construct.block_diag([A, B]).toarray(), + array([[1, 0], [2, 0], [3, 0], [0, 4], [0, 5]])) + + def test_block_diag_return_type(self): + A, B = coo_array([[1, 2, 3]]), coo_matrix([[2, 3, 4]]) + assert isinstance(construct.block_diag([A, A]), sparray) + assert isinstance(construct.block_diag([A, B]), sparray) + assert isinstance(construct.block_diag([B, A]), sparray) + assert isinstance(construct.block_diag([B, B]), spmatrix) + + def test_random_sampling(self): + # Simple sanity checks for sparse random sampling. + for f in sprand, _sprandn: + for t in [np.float32, np.float64, np.longdouble, + np.int32, np.int64, np.complex64, np.complex128]: + x = f(5, 10, density=0.1, dtype=t) + assert_equal(x.dtype, t) + assert_equal(x.shape, (5, 10)) + assert_equal(x.nnz, 5) + + x1 = f(5, 10, density=0.1, rng=4321) + assert_equal(x1.dtype, np.float64) + + x2 = f(5, 10, density=0.1, rng=np.random.default_rng(4321)) + + assert_array_equal(x1.data, x2.data) + assert_array_equal(x1.row, x2.row) + assert_array_equal(x1.col, x2.col) + + for density in [0.0, 0.1, 0.5, 1.0]: + x = f(5, 10, density=density) + assert_equal(x.nnz, int(density * np.prod(x.shape))) + + for fmt in ['coo', 'csc', 'csr', 'lil']: + x = f(5, 10, format=fmt) + assert_equal(x.format, fmt) + + assert_raises(ValueError, lambda: f(5, 10, 1.1)) + assert_raises(ValueError, lambda: f(5, 10, -0.1)) + + @pytest.mark.parametrize("rng", [None, 4321, np.random.default_rng(4321)]) + def test_rand(self, rng): + # Simple distributional checks for sparse.rand. + x = sprand(10, 20, density=0.5, dtype=np.float64, rng=rng) + assert_(np.all(np.less_equal(0, x.data))) + assert_(np.all(np.less_equal(x.data, 1))) + + @pytest.mark.parametrize("rng", [None, 4321, np.random.default_rng(4321)]) + def test_randn(self, rng): + # Simple distributional checks for sparse.randn. + # Statistically, some of these should be negative + # and some should be greater than 1. + x = _sprandn(10, 20, density=0.5, dtype=np.float64, rng=rng) + assert_(np.any(np.less(x.data, 0))) + assert_(np.any(np.less(1, x.data))) + x = _sprandn_array(10, 20, density=0.5, dtype=np.float64, rng=rng) + assert_(np.any(np.less(x.data, 0))) + assert_(np.any(np.less(1, x.data))) + + def test_random_accept_str_dtype(self): + # anything that np.dtype can convert to a dtype should be accepted + # for the dtype + construct.random(10, 10, dtype='d') + construct.random_array((10, 10), dtype='d') + construct.random_array((10, 10, 10), dtype='d') + construct.random_array((10, 10, 10, 10, 10), dtype='d') + + def test_random_array_maintains_array_shape(self): + # preserve use of old random_state during SPEC 7 transition + arr = construct.random_array((0, 4), density=0.3, dtype=int, random_state=0) + assert arr.shape == (0, 4) + + arr = construct.random_array((10, 10, 10), density=0.3, dtype=int, rng=0) + assert arr.shape == (10, 10, 10) + + arr = construct.random_array((10, 10, 10, 10, 10), density=0.3, dtype=int, + rng=0) + assert arr.shape == (10, 10, 10, 10, 10) + + def test_random_array_idx_dtype(self): + A = construct.random_array((10, 10)) + assert A.coords[0].dtype == np.int32 + + def test_random_sparse_matrix_returns_correct_number_of_non_zero_elements(self): + # A 10 x 10 matrix, with density of 12.65%, should have 13 nonzero elements. + # 10 x 10 x 0.1265 = 12.65, which should be rounded up to 13, not 12. + sparse_matrix = construct.random(10, 10, density=0.1265) + assert_equal(sparse_matrix.count_nonzero(),13) + # check random_array + sparse_array = construct.random_array((10, 10), density=0.1265) + assert_equal(sparse_array.count_nonzero(),13) + assert isinstance(sparse_array, sparray) + # check big size + shape = (2**33, 2**33) + sparse_array = construct.random_array(shape, density=2.7105e-17) + assert_equal(sparse_array.count_nonzero(),2000) + + # for n-D + # check random_array + sparse_array = construct.random_array((10, 10, 10, 10), density=0.12658) + assert_equal(sparse_array.count_nonzero(),1266) + assert isinstance(sparse_array, sparray) + # check big size + shape = (2**33, 2**33, 2**33) + sparse_array = construct.random_array(shape, density=2.7105e-28) + assert_equal(sparse_array.count_nonzero(),172) + + +def test_diags_array(): + """Tests of diags_array that do not rely on diags wrapper.""" + diag = np.arange(1, 5) + + assert_array_equal(construct.diags_array(diag).toarray(), np.diag(diag)) + + assert_array_equal( + construct.diags_array(diag, offsets=2).toarray(), np.diag(diag, k=2) + ) + + assert_array_equal( + construct.diags_array(diag, offsets=2, shape=(4, 4)).toarray(), + np.diag(diag, k=2)[:4, :4] + ) + + # Offset outside bounds when shape specified + with pytest.raises(ValueError, match=".*out of bounds"): + construct.diags(np.arange(1, 5), 5, shape=(4, 4)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_coo.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_coo.py new file mode 100644 index 0000000000000000000000000000000000000000..e9281748e6fc9ccdcf8aa44d154ada5856bd46a9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_coo.py @@ -0,0 +1,851 @@ +import numpy as np +from numpy.testing import assert_equal +import pytest +from scipy.linalg import block_diag +from scipy.sparse import coo_array, random_array +from .._coo import _block_diag, _extract_block_diag + + +def test_shape_constructor(): + empty1d = coo_array((3,)) + assert empty1d.shape == (3,) + assert_equal(empty1d.toarray(), np.zeros((3,))) + + empty2d = coo_array((3, 2)) + assert empty2d.shape == (3, 2) + assert_equal(empty2d.toarray(), np.zeros((3, 2))) + + empty_nd = coo_array((2,3,4,6,7)) + assert empty_nd.shape == (2,3,4,6,7) + assert_equal(empty_nd.toarray(), np.zeros((2,3,4,6,7))) + + +def test_dense_constructor(): + # 1d + res1d = coo_array([1, 2, 3]) + assert res1d.shape == (3,) + assert_equal(res1d.toarray(), np.array([1, 2, 3])) + + # 2d + res2d = coo_array([[1, 2, 3], [4, 5, 6]]) + assert res2d.shape == (2, 3) + assert_equal(res2d.toarray(), np.array([[1, 2, 3], [4, 5, 6]])) + + # 4d + arr4d = np.array([[[[3, 7], [1, 0]], [[6, 5], [9, 2]]], + [[[4, 3], [2, 8]], [[7, 5], [1, 6]]], + [[[0, 9], [4, 3]], [[2, 1], [7, 8]]]]) + res4d = coo_array(arr4d) + assert res4d.shape == (3, 2, 2, 2) + assert_equal(res4d.toarray(), arr4d) + + # 9d + np.random.seed(12) + arr9d = np.random.randn(2,3,4,7,6,5,3,2,4) + res9d = coo_array(arr9d) + assert res9d.shape == (2,3,4,7,6,5,3,2,4) + assert_equal(res9d.toarray(), arr9d) + + # storing nan as element of sparse array + nan_3d = coo_array([[[1, np.nan]], [[3, 4]], [[5, 6]]]) + assert nan_3d.shape == (3, 1, 2) + assert_equal(nan_3d.toarray(), np.array([[[1, np.nan]], [[3, 4]], [[5, 6]]])) + + +def test_dense_constructor_with_shape(): + res1d = coo_array([1, 2, 3], shape=(3,)) + assert res1d.shape == (3,) + assert_equal(res1d.toarray(), np.array([1, 2, 3])) + + res2d = coo_array([[1, 2, 3], [4, 5, 6]], shape=(2, 3)) + assert res2d.shape == (2, 3) + assert_equal(res2d.toarray(), np.array([[1, 2, 3], [4, 5, 6]])) + + res3d = coo_array([[[3]], [[4]]], shape=(2, 1, 1)) + assert res3d.shape == (2, 1, 1) + assert_equal(res3d.toarray(), np.array([[[3]], [[4]]])) + + np.random.seed(12) + arr7d = np.random.randn(2,4,1,6,5,3,2) + res7d = coo_array((arr7d), shape=(2,4,1,6,5,3,2)) + assert res7d.shape == (2,4,1,6,5,3,2) + assert_equal(res7d.toarray(), arr7d) + + +def test_dense_constructor_with_inconsistent_shape(): + with pytest.raises(ValueError, match='inconsistent shapes'): + coo_array([1, 2, 3], shape=(4,)) + + with pytest.raises(ValueError, match='inconsistent shapes'): + coo_array([1, 2, 3], shape=(3, 1)) + + with pytest.raises(ValueError, match='inconsistent shapes'): + coo_array([[1, 2, 3]], shape=(3,)) + + with pytest.raises(ValueError, match='inconsistent shapes'): + coo_array([[[3]], [[4]]], shape=(1, 1, 1)) + + with pytest.raises(ValueError, + match='axis 0 index 2 exceeds matrix dimension 2'): + coo_array(([1], ([2],)), shape=(2,)) + + with pytest.raises(ValueError, + match='axis 1 index 3 exceeds matrix dimension 3'): + coo_array(([1,3], ([0, 1], [0, 3], [1, 1])), shape=(2, 3, 2)) + + with pytest.raises(ValueError, match='negative axis 0 index: -1'): + coo_array(([1], ([-1],))) + + with pytest.raises(ValueError, match='negative axis 2 index: -1'): + coo_array(([1], ([0], [2], [-1]))) + + +def test_1d_sparse_constructor(): + empty1d = coo_array((3,)) + res = coo_array(empty1d) + assert res.shape == (3,) + assert_equal(res.toarray(), np.zeros((3,))) + + +def test_1d_tuple_constructor(): + res = coo_array(([9,8], ([1,2],))) + assert res.shape == (3,) + assert_equal(res.toarray(), np.array([0, 9, 8])) + + +def test_1d_tuple_constructor_with_shape(): + res = coo_array(([9,8], ([1,2],)), shape=(4,)) + assert res.shape == (4,) + assert_equal(res.toarray(), np.array([0, 9, 8, 0])) + +def test_non_subscriptability(): + coo_2d = coo_array((2, 2)) + + with pytest.raises(TypeError, + match="'coo_array' object does not support item assignment"): + coo_2d[0, 0] = 1 + + with pytest.raises(TypeError, + match="'coo_array' object is not subscriptable"): + coo_2d[0, :] + +def test_reshape_overflow(): + # see gh-22353 : new idx_dtype can need to be int64 instead of int32 + M, N = (1045507, 523266) + coords = (np.array([M - 1], dtype='int32'), np.array([N - 1], dtype='int32')) + A = coo_array(([3.3], coords), shape=(M, N)) + + # need new idx_dtype to not overflow + B = A.reshape((M * N, 1)) + assert B.coords[0].dtype == np.dtype('int64') + assert B.coords[0][0] == (M * N) - 1 + + # need idx_dtype to stay int32 if before and after can be int32 + C = A.reshape(N, M) + assert C.coords[0].dtype == np.dtype('int32') + assert C.coords[0][0] == N - 1 + +def test_reshape(): + arr1d = coo_array([1, 0, 3]) + assert arr1d.shape == (3,) + + col_vec = arr1d.reshape((3, 1)) + assert col_vec.shape == (3, 1) + assert_equal(col_vec.toarray(), np.array([[1], [0], [3]])) + + row_vec = arr1d.reshape((1, 3)) + assert row_vec.shape == (1, 3) + assert_equal(row_vec.toarray(), np.array([[1, 0, 3]])) + + # attempting invalid reshape + with pytest.raises(ValueError, match="cannot reshape array"): + arr1d.reshape((3,3)) + + # attempting reshape with a size 0 dimension + with pytest.raises(ValueError, match="cannot reshape array"): + arr1d.reshape((3,0)) + + arr2d = coo_array([[1, 2, 0], [0, 0, 3]]) + assert arr2d.shape == (2, 3) + + flat = arr2d.reshape((6,)) + assert flat.shape == (6,) + assert_equal(flat.toarray(), np.array([1, 2, 0, 0, 0, 3])) + + # 2d to 3d + to_3d_arr = arr2d.reshape((2, 3, 1)) + assert to_3d_arr.shape == (2, 3, 1) + assert_equal(to_3d_arr.toarray(), np.array([[[1], [2], [0]], [[0], [0], [3]]])) + + # attempting invalid reshape + with pytest.raises(ValueError, match="cannot reshape array"): + arr2d.reshape((1,3)) + + +def test_nnz(): + arr1d = coo_array([1, 0, 3]) + assert arr1d.shape == (3,) + assert arr1d.nnz == 2 + + arr2d = coo_array([[1, 2, 0], [0, 0, 3]]) + assert arr2d.shape == (2, 3) + assert arr2d.nnz == 3 + + +def test_transpose(): + arr1d = coo_array([1, 0, 3]).T + assert arr1d.shape == (3,) + assert_equal(arr1d.toarray(), np.array([1, 0, 3])) + + arr2d = coo_array([[1, 2, 0], [0, 0, 3]]).T + assert arr2d.shape == (3, 2) + assert_equal(arr2d.toarray(), np.array([[1, 0], [2, 0], [0, 3]])) + + +def test_transpose_with_axis(): + arr1d = coo_array([1, 0, 3]).transpose(axes=(0,)) + assert arr1d.shape == (3,) + assert_equal(arr1d.toarray(), np.array([1, 0, 3])) + + arr2d = coo_array([[1, 2, 0], [0, 0, 3]]).transpose(axes=(0, 1)) + assert arr2d.shape == (2, 3) + assert_equal(arr2d.toarray(), np.array([[1, 2, 0], [0, 0, 3]])) + + with pytest.raises(ValueError, match="axes don't match matrix dimensions"): + coo_array([1, 0, 3]).transpose(axes=(0, 1)) + + with pytest.raises(ValueError, match="repeated axis in transpose"): + coo_array([[1, 2, 0], [0, 0, 3]]).transpose(axes=(1, 1)) + + +def test_1d_row_and_col(): + res = coo_array([1, -2, -3]) + assert_equal(res.col, np.array([0, 1, 2])) + assert_equal(res.row, np.zeros_like(res.col)) + assert res.row.dtype == res.col.dtype + assert res.row.flags.writeable is False + + res.col = [1, 2, 3] + assert len(res.coords) == 1 + assert_equal(res.col, np.array([1, 2, 3])) + assert res.row.dtype == res.col.dtype + + with pytest.raises(ValueError, match="cannot set row attribute"): + res.row = [1, 2, 3] + + +def test_1d_toformats(): + res = coo_array([1, -2, -3]) + for f in [res.tobsr, res.tocsc, res.todia, res.tolil]: + with pytest.raises(ValueError, match='Cannot convert'): + f() + for f in [res.tocoo, res.tocsr, res.todok]: + assert_equal(f().toarray(), res.toarray()) + + +@pytest.mark.parametrize('arg', [1, 2, 4, 5, 8]) +def test_1d_resize(arg: int): + den = np.array([1, -2, -3]) + res = coo_array(den) + den.resize(arg, refcheck=False) + res.resize(arg) + assert res.shape == den.shape + assert_equal(res.toarray(), den) + + +@pytest.mark.parametrize('arg', zip([1, 2, 3, 4], [1, 2, 3, 4])) +def test_1d_to_2d_resize(arg: tuple[int, int]): + den = np.array([1, 0, 3]) + res = coo_array(den) + + den.resize(arg, refcheck=False) + res.resize(arg) + assert res.shape == den.shape + assert_equal(res.toarray(), den) + + +@pytest.mark.parametrize('arg', [1, 4, 6, 8]) +def test_2d_to_1d_resize(arg: int): + den = np.array([[1, 0, 3], [4, 0, 0]]) + res = coo_array(den) + den.resize(arg, refcheck=False) + res.resize(arg) + assert res.shape == den.shape + assert_equal(res.toarray(), den) + + +def test_sum_duplicates(): + # 1d case + arr1d = coo_array(([2, 2, 2], ([1, 0, 1],))) + assert arr1d.nnz == 3 + assert_equal(arr1d.toarray(), np.array([2, 4])) + arr1d.sum_duplicates() + assert arr1d.nnz == 2 + assert_equal(arr1d.toarray(), np.array([2, 4])) + + # 4d case + arr4d = coo_array(([2, 3, 7], ([1, 0, 1], [0, 2, 0], [1, 2, 1], [1, 0, 1]))) + assert arr4d.nnz == 3 + expected = np.array( # noqa: E501 + [[[[0, 0], [0, 0], [0, 0]], [[0, 0], [0, 0], [0, 0]], [[0, 0], [0, 0], [3, 0]]], + [[[0, 0], [0, 9], [0, 0]], [[0, 0], [0, 0], [0, 0]], [[0, 0], [0, 0], [0, 0]]]] + ) + assert_equal(arr4d.toarray(), expected) + arr4d.sum_duplicates() + assert arr4d.nnz == 2 + assert_equal(arr4d.toarray(), expected) + + # when there are no duplicates + arr_nodups = coo_array(([1, 2, 3, 4], ([0, 1, 2, 3],))) + assert arr_nodups.nnz == 4 + arr_nodups.sum_duplicates() + assert arr_nodups.nnz == 4 + + +def test_eliminate_zeros(): + arr1d = coo_array(([0, 0, 1], ([1, 0, 1],))) + assert arr1d.nnz == 3 + assert arr1d.count_nonzero() == 1 + assert_equal(arr1d.toarray(), np.array([0, 1])) + arr1d.eliminate_zeros() + assert arr1d.nnz == 1 + assert arr1d.count_nonzero() == 1 + assert_equal(arr1d.toarray(), np.array([0, 1])) + assert_equal(arr1d.col, np.array([1])) + assert_equal(arr1d.row, np.array([0])) + + +def test_1d_add_dense(): + den_a = np.array([0, -2, -3, 0]) + den_b = np.array([0, 1, 2, 3]) + exp = den_a + den_b + res = coo_array(den_a) + den_b + assert type(res) is type(exp) + assert_equal(res, exp) + + +def test_1d_add_sparse(): + den_a = np.array([0, -2, -3, 0]) + den_b = np.array([0, 1, 2, 3]) + dense_sum = den_a + den_b + # this routes through CSR format + sparse_sum = coo_array(den_a) + coo_array(den_b) + assert_equal(dense_sum, sparse_sum.toarray()) + + +def test_1d_matmul_vector(): + den_a = np.array([0, -2, -3, 0]) + den_b = np.array([0, 1, 2, 3]) + exp = den_a @ den_b + res = coo_array(den_a) @ den_b + assert np.ndim(res) == 0 + assert_equal(res, exp) + + +def test_1d_matmul_multivector(): + den = np.array([0, -2, -3, 0]) + other = np.array([[0, 1, 2, 3], [3, 2, 1, 0]]).T + exp = den @ other + res = coo_array(den) @ other + assert type(res) is type(exp) + assert_equal(res, exp) + + +def test_2d_matmul_multivector(): + # sparse-sparse matmul + den = np.array([[0, 1, 2, 3], [3, 2, 1, 0]]) + arr2d = coo_array(den) + exp = den @ den.T + res = arr2d @ arr2d.T + assert_equal(res.toarray(), exp) + + # sparse-dense matmul for self.ndim = 2 + den = np.array([[0, 4, 3, 0, 5], [1, 0, 7, 3, 4]]) + arr2d = coo_array(den) + exp = den @ den.T + res = arr2d @ den.T + assert_equal(res, exp) + + # sparse-dense matmul for self.ndim = 1 + den_a = np.array([[0, 4, 3, 0, 5], [1, 0, 7, 3, 4]]) + den_b = np.array([0, 1, 6, 0, 4]) + arr1d = coo_array(den_b) + exp = den_b @ den_a.T + res = arr1d @ den_a.T + assert_equal(res, exp) + + # sparse-dense matmul for self.ndim = 1 and other.ndim = 2 + den_a = np.array([1, 0, 2]) + den_b = np.array([[3], [4], [0]]) + exp = den_a @ den_b + res = coo_array(den_a) @ den_b + assert_equal(res, exp) + res = coo_array(den_a) @ list(den_b) + assert_equal(res, exp) + + +def test_1d_diagonal(): + den = np.array([0, -2, -3, 0]) + with pytest.raises(ValueError, match='diagonal requires two dimensions'): + coo_array(den).diagonal() + + +@pytest.mark.parametrize('shape', [(0,), (7,), (4,7), (0,0,0), (3,6,2), + (1,0,3), (7,9,3,2,4,5)]) +def test_nd_todense(shape): + np.random.seed(12) + arr = np.random.randint(low=0, high=5, size=shape) + assert_equal(coo_array(arr).todense(), arr) + + +@pytest.mark.parametrize('shape', [(0,), (7,), (4,7), (0,0,0), (3,6,2), + (1,0,3), (7,9,3,2,4,5)]) +def test_nd_sparse_constructor(shape): + empty_arr = coo_array(shape) + res = coo_array(empty_arr) + assert res.shape == (shape) + assert_equal(res.toarray(), np.zeros(shape)) + + +@pytest.mark.parametrize('shape', [(0,), (7,), (4,7), (0,0,0), (3,6,2), + (1,0,3), (7,9,3,2,4,5)]) +def test_nd_tuple_constructor(shape): + np.random.seed(12) + arr = np.random.randn(*shape) + res = coo_array(arr) + assert res.shape == shape + assert_equal(res.toarray(), arr) + + +@pytest.mark.parametrize('shape', [(0,), (7,), (4,7), (0,0,0), (3,6,2), + (1,0,3), (7,9,3,2,4,5)]) +def test_nd_tuple_constructor_with_shape(shape): + np.random.seed(12) + arr = np.random.randn(*shape) + res = coo_array(arr, shape=shape) + assert res.shape == shape + assert_equal(res.toarray(), arr) + + +def test_tuple_constructor_for_dim_size_zero(): + # arrays with a dimension of size 0 + with pytest.raises(ValueError, match='exceeds matrix dimension'): + coo_array(([9, 8], ([1, 2], [1, 0], [2, 1])), shape=(3,4,0)) + + empty_arr = coo_array(([], ([], [], [], [])), shape=(4,0,2,3)) + assert_equal(empty_arr.toarray(), np.empty((4,0,2,3))) + + +@pytest.mark.parametrize(('shape', 'new_shape'), [((4,9,6,5), (3,6,15,4)), + ((4,9,6,5), (36,30)), + ((4,9,6,5), (1080,)), + ((4,9,6,5), (2,3,2,2,3,5,3)),]) +def test_nd_reshape(shape, new_shape): + # reshaping a 4d sparse array + rng = np.random.default_rng(23409823) + + arr4d = random_array(shape, density=0.6, rng=rng, dtype=int) + assert arr4d.shape == shape + den4d = arr4d.toarray() + + exp_arr = den4d.reshape(new_shape) + res_arr = arr4d.reshape(new_shape) + assert res_arr.shape == new_shape + assert_equal(res_arr.toarray(), exp_arr) + + +@pytest.mark.parametrize('shape', [(0,), (7,), (4,7), (0,0,0), (3,6,2), + (1,0,3), (7,9,3,2,4,5)]) +def test_nd_nnz(shape): + rng = np.random.default_rng(23409823) + + arr = random_array(shape, density=0.6, rng=rng, dtype=int) + assert arr.nnz == np.count_nonzero(arr.toarray()) + + +@pytest.mark.parametrize('shape', [(0,), (7,), (4,7), (0,0,0), (3,6,2), + (1,0,3), (7,9,3,2,4,5)]) +def test_nd_transpose(shape): + rng = np.random.default_rng(23409823) + + arr = random_array(shape, density=0.6, rng=rng, dtype=int) + exp_arr = arr.toarray().T + trans_arr = arr.transpose() + assert trans_arr.shape == shape[::-1] + assert_equal(exp_arr, trans_arr.toarray()) + + +@pytest.mark.parametrize(('shape', 'axis_perm'), [((3,), (0,)), + ((2,3), (0,1)), + ((2,4,3,6,5,3), (1,2,0,5,3,4)),]) +def test_nd_transpose_with_axis(shape, axis_perm): + rng = np.random.default_rng(23409823) + + arr = random_array(shape, density=0.6, rng=rng, dtype=int) + trans_arr = arr.transpose(axes=axis_perm) + assert_equal(trans_arr.toarray(), np.transpose(arr.toarray(), axes=axis_perm)) + + +def test_transpose_with_inconsistent_axis(): + with pytest.raises(ValueError, match="axes don't match matrix dimensions"): + coo_array([1, 0, 3]).transpose(axes=(0, 1)) + + with pytest.raises(ValueError, match="repeated axis in transpose"): + coo_array([[1, 2, 0], [0, 0, 3]]).transpose(axes=(1, 1)) + + +def test_nd_eliminate_zeros(): + # for 3d sparse arrays + arr3d = coo_array(([1, 0, 0, 4], ([0, 1, 1, 2], [0, 1, 0, 1], [1, 1, 2, 0]))) + assert arr3d.nnz == 4 + assert arr3d.count_nonzero() == 2 + assert_equal(arr3d.toarray(), np.array([[[0, 1, 0], [0, 0, 0]], + [[0, 0, 0], [0, 0, 0]], [[0, 0, 0], [4, 0, 0]]])) + arr3d.eliminate_zeros() + assert arr3d.nnz == 2 + assert arr3d.count_nonzero() == 2 + assert_equal(arr3d.toarray(), np.array([[[0, 1, 0], [0, 0, 0]], + [[0, 0, 0], [0, 0, 0]], [[0, 0, 0], [4, 0, 0]]])) + + # for a 5d sparse array when all elements of data array are 0 + coords = ([0, 1, 1, 2], [0, 1, 0, 1], [1, 1, 2, 0], [0, 0, 2, 3], [1, 0, 0, 2]) + arr5d = coo_array(([0, 0, 0, 0], coords)) + assert arr5d.nnz == 4 + assert arr5d.count_nonzero() == 0 + arr5d.eliminate_zeros() + assert arr5d.nnz == 0 + assert arr5d.count_nonzero() == 0 + assert_equal(arr5d.col, np.array([])) + assert_equal(arr5d.row, np.array([])) + assert_equal(arr5d.coords, ([], [], [], [], [])) + + +@pytest.mark.parametrize('shape', [(0,), (7,), (4,7), (0,0,0), (3,6,2), + (1,0,3), (7,9,3,2,4,5)]) +def test_nd_add_dense(shape): + rng = np.random.default_rng(23409823) + sp_x = random_array(shape, density=0.6, rng=rng, dtype=int) + sp_y = random_array(shape, density=0.6, rng=rng, dtype=int) + den_x, den_y = sp_x.toarray(), sp_y.toarray() + exp = den_x + den_y + res = sp_x + den_y + assert type(res) is type(exp) + assert_equal(res, exp) + + +@pytest.mark.parametrize('shape', [(0,), (7,), (4,7), (0,0,0), (3,6,2), + (1,0,3), (7,9,3,2,4,5)]) +def test_nd_add_sparse(shape): + rng = np.random.default_rng(23409823) + sp_x = random_array((shape), density=0.6, rng=rng, dtype=int) + sp_y = random_array((shape), density=0.6, rng=rng, dtype=int) + den_x, den_y = sp_x.toarray(), sp_y.toarray() + + dense_sum = den_x + den_y + sparse_sum = sp_x + sp_y + assert_equal(dense_sum, sparse_sum.toarray()) + + +def test_add_sparse_with_inf(): + # addition of sparse arrays with an inf element + den_a = np.array([[[0], [np.inf]], [[-3], [0]]]) + den_b = np.array([[[0], [1]], [[2], [3]]]) + dense_sum = den_a + den_b + sparse_sum = coo_array(den_a) + coo_array(den_b) + assert_equal(dense_sum, sparse_sum.toarray()) + + +@pytest.mark.parametrize(('a_shape', 'b_shape'), [((7,), (12,)), + ((6,4), (6,5)), + ((5,9,3,2), (9,5,2,3)),]) +def test_nd_add_sparse_with_inconsistent_shapes(a_shape, b_shape): + rng = np.random.default_rng(23409823) + + arr_a = random_array((a_shape), density=0.6, rng=rng, dtype=int) + arr_b = random_array((b_shape), density=0.6, rng=rng, dtype=int) + with pytest.raises(ValueError, + match="(Incompatible|inconsistent) shapes|cannot be broadcast"): + arr_a + arr_b + + +@pytest.mark.parametrize('shape', [(0,), (7,), (4,7), (0,0,0), (3,6,2), + (1,0,3), (7,9,3,2,4,5)]) +def test_nd_sub_dense(shape): + rng = np.random.default_rng(23409823) + sp_x = random_array(shape, density=0.6, rng=rng, dtype=int) + sp_y = random_array(shape, density=0.6, rng=rng, dtype=int) + den_x, den_y = sp_x.toarray(), sp_y.toarray() + exp = den_x - den_y + res = sp_x - den_y + assert type(res) is type(exp) + assert_equal(res, exp) + + +@pytest.mark.parametrize('shape', [(0,), (7,), (4,7), (0,0,0), (3,6,2), + (1,0,3), (7,9,3,2,4,5)]) +def test_nd_sub_sparse(shape): + rng = np.random.default_rng(23409823) + + sp_x = random_array(shape, density=0.6, rng=rng, dtype=int) + sp_y = random_array(shape, density=0.6, rng=rng, dtype=int) + den_x, den_y = sp_x.toarray(), sp_y.toarray() + + dense_sum = den_x - den_y + sparse_sum = sp_x - sp_y + assert_equal(dense_sum, sparse_sum.toarray()) + + +def test_nd_sub_sparse_with_nan(): + # subtraction of sparse arrays with a nan element + den_a = np.array([[[0], [np.nan]], [[-3], [0]]]) + den_b = np.array([[[0], [1]], [[2], [3]]]) + dense_sum = den_a - den_b + sparse_sum = coo_array(den_a) - coo_array(den_b) + assert_equal(dense_sum, sparse_sum.toarray()) + + +@pytest.mark.parametrize(('a_shape', 'b_shape'), [((7,), (12,)), + ((6,4), (6,5)), + ((5,9,3,2), (9,5,2,3)),]) +def test_nd_sub_sparse_with_inconsistent_shapes(a_shape, b_shape): + rng = np.random.default_rng(23409823) + + arr_a = random_array((a_shape), density=0.6, rng=rng, dtype=int) + arr_b = random_array((b_shape), density=0.6, rng=rng, dtype=int) + with pytest.raises(ValueError, match="inconsistent shapes"): + arr_a - arr_b + + +mat_vec_shapes = [ + ((2, 3, 4, 5), (5,)), + ((0, 0), (0,)), + ((2, 3, 4, 7, 8), (8,)), + ((4, 4, 2, 0), (0,)), + ((6, 5, 3, 2, 4), (4, 1)), + ((2,5), (5,)), + ((2, 5), (5, 1)), + ((3,), (3, 1)), + ((4,), (4,)) +] +@pytest.mark.parametrize(('mat_shape', 'vec_shape'), mat_vec_shapes) +def test_nd_matmul_vector(mat_shape, vec_shape): + rng = np.random.default_rng(23409823) + + sp_x = random_array(mat_shape, density=0.6, rng=rng, dtype=int) + sp_y = random_array(vec_shape, density=0.6, rng=rng, dtype=int) + den_x, den_y = sp_x.toarray(), sp_y.toarray() + exp = den_x @ den_y + res = sp_x @ den_y + assert_equal(res,exp) + res = sp_x @ list(den_y) + assert_equal(res,exp) + + +mat_mat_shapes = [ + ((2, 3, 4, 5), (2, 3, 5, 7)), + ((0, 0), (0,)), + ((4, 4, 2, 0), (0,)), + ((7, 8, 3), (3,)), + ((7, 8, 3), (3, 1)), + ((6, 5, 3, 2, 4), (4, 3)), + ((1, 3, 2, 4), (6, 5, 1, 4, 3)), + ((6, 1, 1, 2, 4), (1, 3, 4, 3)), + ((4,), (2, 4, 3)), + ((3,), (5, 6, 7, 3, 2)), + ((4,), (4, 3)), + ((2, 5), (5, 1)), +] +@pytest.mark.parametrize(('mat_shape1', 'mat_shape2'), mat_mat_shapes) +def test_nd_matmul(mat_shape1, mat_shape2): + rng = np.random.default_rng(23409823) + + sp_x = random_array(mat_shape1, density=0.6, random_state=rng, dtype=int) + sp_y = random_array(mat_shape2, density=0.6, random_state=rng, dtype=int) + den_x, den_y = sp_x.toarray(), sp_y.toarray() + exp = den_x @ den_y + # sparse-sparse + res = sp_x @ sp_y + assert_equal(res.toarray(), exp) + # sparse-dense + res = sp_x @ den_y + assert_equal(res, exp) + res = sp_x @ list(den_y) + assert_equal(res, exp) + + # dense-sparse + res = den_x @ sp_y + assert_equal(res, exp) + + +def test_nd_matmul_sparse_with_inconsistent_arrays(): + rng = np.random.default_rng(23409823) + + sp_x = random_array((4,5,7,6,3), density=0.6, random_state=rng, dtype=int) + sp_y = random_array((1,5,3,2,5), density=0.6, random_state=rng, dtype=int) + with pytest.raises(ValueError, match="matmul: dimension mismatch with signature"): + sp_x @ sp_y + with pytest.raises(ValueError, match="matmul: dimension mismatch with signature"): + sp_x @ (sp_y.toarray()) + + sp_z = random_array((1,5,3,2), density=0.6, random_state=rng, dtype=int) + with pytest.raises(ValueError, match="Batch dimensions are not broadcastable"): + sp_x @ sp_z + with pytest.raises(ValueError, match="Batch dimensions are not broadcastable"): + sp_x @ (sp_z.toarray()) + + +def test_dot_1d_1d(): # 1-D inner product + a = coo_array([1,2,3]) + b = coo_array([4,5,6]) + exp = np.dot(a.toarray(), b.toarray()) + res = a.dot(b) + assert_equal(res, exp) + res = a.dot(b.toarray()) + assert_equal(res, exp) + + +def test_dot_sparse_scalar(): + a = coo_array([[1, 2], [3, 4], [5, 6]]) + b = 3 + res = a.dot(b) + exp = np.dot(a.toarray(), b) + assert_equal(res.toarray(), exp) + + +def test_dot_with_inconsistent_shapes(): + arr_a = coo_array([[[1, 2]], [[3, 4]]]) + arr_b = coo_array([4, 5, 6]) + with pytest.raises(ValueError, match="not aligned for n-D dot"): + arr_a.dot(arr_b) + + +def test_matmul_dot_not_implemented(): + arr_a = coo_array([[1, 2], [3, 4]]) + with pytest.raises(TypeError, match="argument not supported type"): + arr_a.dot(None) + with pytest.raises(TypeError, match="arg not supported type"): + arr_a.tensordot(None) + with pytest.raises(TypeError, match="unsupported operand type"): + arr_a @ None + with pytest.raises(TypeError, match="unsupported operand type"): + None @ arr_a + + +dot_shapes = [ + ((3,3), (3,3)), ((4,6), (6,7)), ((1,4), (4,1)), # matrix multiplication 2-D + ((3,2,4,7), (7,)), ((5,), (6,3,5,2)), # dot of n-D and 1-D arrays + ((3,2,4,7), (7,1)), ((1,5,), (6,3,5,2)), + ((4,6), (3,2,6,4)), ((2,8,7), (4,5,7,7,2)), # dot of n-D and m-D arrays + ((4,5,7,6), (3,2,6,4)), +] +@pytest.mark.parametrize(('a_shape', 'b_shape'), dot_shapes) +def test_dot_nd(a_shape, b_shape): + rng = np.random.default_rng(23409823) + + arr_a = random_array(a_shape, density=0.6, random_state=rng, dtype=int) + arr_b = random_array(b_shape, density=0.6, random_state=rng, dtype=int) + + exp = np.dot(arr_a.toarray(), arr_b.toarray()) + # sparse-dense + res = arr_a.dot(arr_b.toarray()) + assert_equal(res, exp) + res = arr_a.dot(list(arr_b.toarray())) + assert_equal(res, exp) + # sparse-sparse + res = arr_a.dot(arr_b) + assert_equal(res.toarray(), exp) + + +tensordot_shapes_and_axes = [ + ((4,6), (6,7), ([1], [0])), + ((3,2,4,7), (7,), ([3], [0])), + ((5,), (6,3,5,2), ([0], [2])), + ((4,5,7,6), (3,2,6,4), ([0, 3], [3, 2])), + ((2,8,7), (4,5,7,8,2), ([0, 1, 2], [4, 3, 2])), + ((4,5,3,2,6), (3,2,6,7,8), 3), + ((4,5,7), (7,3,7), 1), + ((2,3,4), (2,3,4), ([0, 1, 2], [0, 1, 2])), +] +@pytest.mark.parametrize(('a_shape', 'b_shape', 'axes'), tensordot_shapes_and_axes) +def test_tensordot(a_shape, b_shape, axes): + rng = np.random.default_rng(23409823) + + arr_a = random_array(a_shape, density=0.6, random_state=rng, dtype=int) + arr_b = random_array(b_shape, density=0.6, random_state=rng, dtype=int) + + exp = np.tensordot(arr_a.toarray(), arr_b.toarray(), axes=axes) + + # sparse-dense + res = arr_a.tensordot(arr_b.toarray(), axes=axes) + assert_equal(res, exp) + res = arr_a.tensordot(list(arr_b.toarray()), axes=axes) + assert_equal(res, exp) + + # sparse-sparse + res = arr_a.tensordot(arr_b, axes=axes) + if type(res) is coo_array: + assert_equal(res.toarray(), exp) + else: + assert_equal(res, exp) + + +def test_tensordot_with_invalid_args(): + rng = np.random.default_rng(23409823) + + arr_a = random_array((3,4,5), density=0.6, random_state=rng, dtype=int) + arr_b = random_array((3,4,6), density=0.6, random_state=rng, dtype=int) + + axes = ([2], [2]) # sizes of 2nd axes of both shapes do not match + with pytest.raises(ValueError, match="sizes of the corresponding axes must match"): + arr_a.tensordot(arr_b, axes=axes) + + arr_a = random_array((5,4,2,3,7), density=0.6, random_state=rng, dtype=int) + arr_b = random_array((4,6,3,2), density=0.6, random_state=rng, dtype=int) + + axes = ([2,0,1], [1,3]) # lists have different lengths + with pytest.raises(ValueError, match="axes lists/tuples must be of the" + " same length"): + arr_a.tensordot(arr_b, axes=axes) + + +@pytest.mark.parametrize(('actual_shape', 'broadcast_shape'), + [((1,3,5,4), (2,3,5,4)), ((2,1,5,4), (6,2,3,5,4)), + ((1,1,7,8,9), (4,5,6,7,8,9)), ((1,3), (4,5,3)), + ((7,8,1), (7,8,5)), ((3,1), (3,4)), ((1,), (5,)), + ((1,1,1), (4,5,6)), ((1,3,1,5,4), (8,2,3,9,5,4)),]) +def test_broadcast_to(actual_shape, broadcast_shape): + rng = np.random.default_rng(23409823) + + arr = random_array(actual_shape, density=0.6, random_state=rng, dtype=int) + res = arr._broadcast_to(broadcast_shape) + exp = np.broadcast_to(arr.toarray(), broadcast_shape) + assert_equal(res.toarray(), exp) + + +@pytest.mark.parametrize(('shape'), [(4,5,6,7,8), (6,4), + (5,9,3,2), (9,5,2,3,4),]) +def test_block_diag(shape): + rng = np.random.default_rng(23409823) + sp_x = random_array(shape, density=0.6, random_state=rng, dtype=int) + den_x = sp_x.toarray() + + # converting n-d numpy array to an array of slices of 2-D matrices, + # to pass as argument into scipy.linalg.block_diag + num_slices = int(np.prod(den_x.shape[:-2])) + reshaped_array = den_x.reshape((num_slices,) + den_x.shape[-2:]) + matrices = [reshaped_array[i, :, :] for i in range(num_slices)] + exp = block_diag(*matrices) + + res = _block_diag(sp_x) + + assert_equal(res.toarray(), exp) + + +@pytest.mark.parametrize(('shape'), [(4,5,6,7,8), (6,4), + (5,9,3,2), (9,5,2,3,4),]) +def test_extract_block_diag(shape): + rng = np.random.default_rng(23409823) + sp_x = random_array(shape, density=0.6, random_state=rng, dtype=int) + res = _extract_block_diag(_block_diag(sp_x), shape) + + assert_equal(res.toarray(), sp_x.toarray()) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_csc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_csc.py new file mode 100644 index 0000000000000000000000000000000000000000..6313751e41899ae7c5daf01fbdbbacdc1f303fa1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_csc.py @@ -0,0 +1,98 @@ +import numpy as np +from numpy.testing import assert_array_almost_equal, assert_ +from scipy.sparse import csr_matrix, csc_matrix, lil_matrix + +import pytest + + +def test_csc_getrow(): + N = 10 + np.random.seed(0) + X = np.random.random((N, N)) + X[X > 0.7] = 0 + Xcsc = csc_matrix(X) + + for i in range(N): + arr_row = X[i:i + 1, :] + csc_row = Xcsc.getrow(i) + + assert_array_almost_equal(arr_row, csc_row.toarray()) + assert_(type(csc_row) is csr_matrix) + + +def test_csc_getcol(): + N = 10 + np.random.seed(0) + X = np.random.random((N, N)) + X[X > 0.7] = 0 + Xcsc = csc_matrix(X) + + for i in range(N): + arr_col = X[:, i:i + 1] + csc_col = Xcsc.getcol(i) + + assert_array_almost_equal(arr_col, csc_col.toarray()) + assert_(type(csc_col) is csc_matrix) + +@pytest.mark.parametrize("matrix_input, axis, expected_shape", + [(csc_matrix([[1, 0], + [0, 0], + [0, 2]]), + 0, (0, 2)), + (csc_matrix([[1, 0], + [0, 0], + [0, 2]]), + 1, (3, 0)), + (csc_matrix([[1, 0], + [0, 0], + [0, 2]]), + 'both', (0, 0)), + (csc_matrix([[0, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 2, 3, 0, 1]]), + 0, (0, 6))]) +def test_csc_empty_slices(matrix_input, axis, expected_shape): + # see gh-11127 for related discussion + slice_1 = matrix_input.toarray().shape[0] - 1 + slice_2 = slice_1 + slice_3 = slice_2 - 1 + + if axis == 0: + actual_shape_1 = matrix_input[slice_1:slice_2, :].toarray().shape + actual_shape_2 = matrix_input[slice_1:slice_3, :].toarray().shape + elif axis == 1: + actual_shape_1 = matrix_input[:, slice_1:slice_2].toarray().shape + actual_shape_2 = matrix_input[:, slice_1:slice_3].toarray().shape + elif axis == 'both': + actual_shape_1 = matrix_input[slice_1:slice_2, slice_1:slice_2].toarray().shape + actual_shape_2 = matrix_input[slice_1:slice_3, slice_1:slice_3].toarray().shape + + assert actual_shape_1 == expected_shape + assert actual_shape_1 == actual_shape_2 + + +@pytest.mark.parametrize('ax', (-2, -1, 0, 1, None)) +def test_argmax_overflow(ax): + # See gh-13646: Windows integer overflow for large sparse matrices. + dim = (100000, 100000) + A = lil_matrix(dim) + A[-2, -2] = 42 + A[-3, -3] = 0.1234 + A = csc_matrix(A) + idx = A.argmax(axis=ax) + + if ax is None: + # idx is a single flattened index + # that we need to convert to a 2d index pair; + # can't do this with np.unravel_index because + # the dimensions are too large + ii = idx % dim[0] + jj = idx // dim[0] + else: + # idx is an array of size of A.shape[ax]; + # check the max index to make sure no overflows + # we encountered + assert np.count_nonzero(idx) == A.nnz + ii, jj = np.max(idx), np.argmax(idx) + + assert A[ii, jj] == A[-2, -2] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_csr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_csr.py new file mode 100644 index 0000000000000000000000000000000000000000..6b011ad4fdce93c0fac36e58381da0bb554ba3be --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_csr.py @@ -0,0 +1,214 @@ +import numpy as np +from numpy.testing import assert_array_almost_equal, assert_, assert_array_equal +from scipy.sparse import csr_matrix, csc_matrix, csr_array, csc_array, hstack +from scipy import sparse +import pytest + + +def _check_csr_rowslice(i, sl, X, Xcsr): + np_slice = X[i, sl] + csr_slice = Xcsr[i, sl] + assert_array_almost_equal(np_slice, csr_slice.toarray()[0]) + assert_(type(csr_slice) is csr_matrix) + + +def test_csr_rowslice(): + N = 10 + np.random.seed(0) + X = np.random.random((N, N)) + X[X > 0.7] = 0 + Xcsr = csr_matrix(X) + + slices = [slice(None, None, None), + slice(None, None, -1), + slice(1, -2, 2), + slice(-2, 1, -2)] + + for i in range(N): + for sl in slices: + _check_csr_rowslice(i, sl, X, Xcsr) + + +def test_csr_getrow(): + N = 10 + np.random.seed(0) + X = np.random.random((N, N)) + X[X > 0.7] = 0 + Xcsr = csr_matrix(X) + + for i in range(N): + arr_row = X[i:i + 1, :] + csr_row = Xcsr.getrow(i) + + assert_array_almost_equal(arr_row, csr_row.toarray()) + assert_(type(csr_row) is csr_matrix) + + +def test_csr_getcol(): + N = 10 + np.random.seed(0) + X = np.random.random((N, N)) + X[X > 0.7] = 0 + Xcsr = csr_matrix(X) + + for i in range(N): + arr_col = X[:, i:i + 1] + csr_col = Xcsr.getcol(i) + + assert_array_almost_equal(arr_col, csr_col.toarray()) + assert_(type(csr_col) is csr_matrix) + +@pytest.mark.parametrize("matrix_input, axis, expected_shape", + [(csr_matrix([[1, 0, 0, 0], + [0, 0, 0, 0], + [0, 2, 3, 0]]), + 0, (0, 4)), + (csr_matrix([[1, 0, 0, 0], + [0, 0, 0, 0], + [0, 2, 3, 0]]), + 1, (3, 0)), + (csr_matrix([[1, 0, 0, 0], + [0, 0, 0, 0], + [0, 2, 3, 0]]), + 'both', (0, 0)), + (csr_matrix([[0, 1, 0, 0, 0], + [0, 0, 0, 0, 0], + [0, 0, 2, 3, 0]]), + 0, (0, 5))]) +def test_csr_empty_slices(matrix_input, axis, expected_shape): + # see gh-11127 for related discussion + slice_1 = matrix_input.toarray().shape[0] - 1 + slice_2 = slice_1 + slice_3 = slice_2 - 1 + + if axis == 0: + actual_shape_1 = matrix_input[slice_1:slice_2, :].toarray().shape + actual_shape_2 = matrix_input[slice_1:slice_3, :].toarray().shape + elif axis == 1: + actual_shape_1 = matrix_input[:, slice_1:slice_2].toarray().shape + actual_shape_2 = matrix_input[:, slice_1:slice_3].toarray().shape + elif axis == 'both': + actual_shape_1 = matrix_input[slice_1:slice_2, slice_1:slice_2].toarray().shape + actual_shape_2 = matrix_input[slice_1:slice_3, slice_1:slice_3].toarray().shape + + assert actual_shape_1 == expected_shape + assert actual_shape_1 == actual_shape_2 + + +def test_csr_bool_indexing(): + data = csr_matrix([[0, 1, 2], [3, 4, 5], [6, 7, 8]]) + list_indices1 = [False, True, False] + array_indices1 = np.array(list_indices1) + list_indices2 = [[False, True, False], [False, True, False], [False, True, False]] + array_indices2 = np.array(list_indices2) + list_indices3 = ([False, True, False], [False, True, False]) + array_indices3 = (np.array(list_indices3[0]), np.array(list_indices3[1])) + slice_list1 = data[list_indices1].toarray() + slice_array1 = data[array_indices1].toarray() + slice_list2 = data[list_indices2] + slice_array2 = data[array_indices2] + slice_list3 = data[list_indices3] + slice_array3 = data[array_indices3] + assert (slice_list1 == slice_array1).all() + assert (slice_list2 == slice_array2).all() + assert (slice_list3 == slice_array3).all() + + +def test_csr_hstack_int64(): + """ + Tests if hstack properly promotes to indices and indptr arrays to np.int64 + when using np.int32 during concatenation would result in either array + overflowing. + """ + max_int32 = np.iinfo(np.int32).max + + # First case: indices would overflow with int32 + data = [1.0] + row = [0] + + max_indices_1 = max_int32 - 1 + max_indices_2 = 3 + + # Individual indices arrays are representable with int32 + col_1 = [max_indices_1 - 1] + col_2 = [max_indices_2 - 1] + + X_1 = csr_matrix((data, (row, col_1))) + X_2 = csr_matrix((data, (row, col_2))) + + assert max(max_indices_1 - 1, max_indices_2 - 1) < max_int32 + assert X_1.indices.dtype == X_1.indptr.dtype == np.int32 + assert X_2.indices.dtype == X_2.indptr.dtype == np.int32 + + # ... but when concatenating their CSR matrices, the resulting indices + # array can't be represented with int32 and must be promoted to int64. + X_hs = hstack([X_1, X_2], format="csr") + + assert X_hs.indices.max() == max_indices_1 + max_indices_2 - 1 + assert max_indices_1 + max_indices_2 - 1 > max_int32 + assert X_hs.indices.dtype == X_hs.indptr.dtype == np.int64 + + # Even if the matrices are empty, we must account for their size + # contribution so that we may safely set the final elements. + X_1_empty = csr_matrix(X_1.shape) + X_2_empty = csr_matrix(X_2.shape) + X_hs_empty = hstack([X_1_empty, X_2_empty], format="csr") + + assert X_hs_empty.shape == X_hs.shape + assert X_hs_empty.indices.dtype == np.int64 + + # Should be just small enough to stay in int32 after stack. Note that + # we theoretically could support indices.max() == max_int32, but due to an + # edge-case in the underlying sparsetools code + # (namely the `coo_tocsr` routine), + # we require that max(X_hs_32.shape) < max_int32 as well. + # Hence we can only support max_int32 - 1. + col_3 = [max_int32 - max_indices_1 - 1] + X_3 = csr_matrix((data, (row, col_3))) + X_hs_32 = hstack([X_1, X_3], format="csr") + assert X_hs_32.indices.dtype == np.int32 + assert X_hs_32.indices.max() == max_int32 - 1 + +@pytest.mark.parametrize("cls", [csr_matrix, csr_array, csc_matrix, csc_array]) +def test_mixed_index_dtype_int_indexing(cls): + # https://github.com/scipy/scipy/issues/20182 + rng = np.random.default_rng(0) + base_mtx = cls(sparse.random(50, 50, random_state=rng, density=0.1)) + indptr_64bit = base_mtx.copy() + indices_64bit = base_mtx.copy() + indptr_64bit.indptr = base_mtx.indptr.astype(np.int64) + indices_64bit.indices = base_mtx.indices.astype(np.int64) + + for mtx in [base_mtx, indptr_64bit, indices_64bit]: + np.testing.assert_array_equal( + mtx[[1,2], :].toarray(), + base_mtx[[1, 2], :].toarray() + ) + np.testing.assert_array_equal( + mtx[:, [1, 2]].toarray(), + base_mtx[:, [1, 2]].toarray() + ) + +def test_broadcast_to(): + a = np.array([1, 0, 2]) + b = np.array([3]) + e = np.zeros((0,)) + res_a = csr_array(a)._broadcast_to((2,3)) + res_b = csr_array(b)._broadcast_to((4,)) + res_c = csr_array(b)._broadcast_to((2,4)) + res_d = csr_array(b)._broadcast_to((1,)) + res_e = csr_array(e)._broadcast_to((4,0)) + assert_array_equal(res_a.toarray(), np.broadcast_to(a, (2,3))) + assert_array_equal(res_b.toarray(), np.broadcast_to(b, (4,))) + assert_array_equal(res_c.toarray(), np.broadcast_to(b, (2,4))) + assert_array_equal(res_d.toarray(), np.broadcast_to(b, (1,))) + assert_array_equal(res_e.toarray(), np.broadcast_to(e, (4,0))) + + with pytest.raises(ValueError, match="cannot be broadcast"): + csr_matrix([[1, 2, 0], [3, 0, 1]])._broadcast_to(shape=(2, 1)) + + with pytest.raises(ValueError, match="cannot be broadcast"): + csr_matrix([[0, 1, 2]])._broadcast_to(shape=(3, 2)) + + with pytest.raises(ValueError, match="cannot be broadcast"): + csr_array([0, 1, 2])._broadcast_to(shape=(3, 2)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_dok.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_dok.py new file mode 100644 index 0000000000000000000000000000000000000000..f3deda1668c3f526cfb205131a1348c620f25d20 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_dok.py @@ -0,0 +1,209 @@ +import pytest +import numpy as np +from numpy.testing import assert_equal +import scipy as sp +from scipy.sparse import dok_array, dok_matrix + + +pytestmark = pytest.mark.thread_unsafe + + +@pytest.fixture +def d(): + return {(0, 1): 1, (0, 2): 2} + +@pytest.fixture +def A(): + return np.array([[0, 1, 2], [0, 0, 0], [0, 0, 0]]) + +@pytest.fixture(params=[dok_array, dok_matrix]) +def Asp(request): + A = request.param((3, 3)) + A[(0, 1)] = 1 + A[(0, 2)] = 2 + yield A + +# Note: __iter__ and comparison dunders act like ndarrays for DOK, not dict. +# Dunders reversed, or, ror, ior work as dict for dok_matrix, raise for dok_array +# All other dict methods on DOK format act like dict methods (with extra checks). + +# Start of tests +################ +def test_dict_methods_covered(d, Asp): + d_methods = set(dir(d)) - {"__class_getitem__"} + asp_methods = set(dir(Asp)) + assert d_methods < asp_methods + +def test_clear(d, Asp): + assert d.items() == Asp.items() + d.clear() + Asp.clear() + assert d.items() == Asp.items() + +def test_copy(d, Asp): + assert d.items() == Asp.items() + dd = d.copy() + asp = Asp.copy() + assert dd.items() == asp.items() + assert asp.items() == Asp.items() + asp[(0, 1)] = 3 + assert Asp[(0, 1)] == 1 + +def test_fromkeys_default(): + # test with default value + edges = [(0, 2), (1, 0), (2, 1)] + Xdok = dok_array.fromkeys(edges) + X = [[0, 0, 1], [1, 0, 0], [0, 1, 0]] + assert_equal(Xdok.toarray(), X) + +def test_fromkeys_positional(): + # test with positional value + edges = [(0, 2), (1, 0), (2, 1)] + Xdok = dok_array.fromkeys(edges, -1) + X = [[0, 0, -1], [-1, 0, 0], [0, -1, 0]] + assert_equal(Xdok.toarray(), X) + +def test_fromkeys_iterator(): + it = ((a, a % 2) for a in range(4)) + Xdok = dok_array.fromkeys(it) + X = [[1, 0], [0, 1], [1, 0], [0, 1]] + assert_equal(Xdok.toarray(), X) + +def test_get(d, Asp): + assert Asp.get((0, 1)) == d.get((0, 1)) + assert Asp.get((0, 0), 99) == d.get((0, 0), 99) + with pytest.raises(IndexError, match="out of bounds"): + Asp.get((0, 4), 99) + +def test_items(d, Asp): + assert Asp.items() == d.items() + +def test_keys(d, Asp): + assert Asp.keys() == d.keys() + +def test_pop(d, Asp): + assert d.pop((0, 1)) == 1 + assert Asp.pop((0, 1)) == 1 + assert d.items() == Asp.items() + + assert Asp.pop((22, 21), None) is None + assert Asp.pop((22, 21), "other") == "other" + with pytest.raises(KeyError, match="(22, 21)"): + Asp.pop((22, 21)) + with pytest.raises(TypeError, match="got an unexpected keyword argument"): + Asp.pop((22, 21), default=5) + +def test_popitem(d, Asp): + assert d.popitem() == Asp.popitem() + assert d.items() == Asp.items() + +def test_setdefault(d, Asp): + assert Asp.setdefault((0, 1), 4) == 1 + assert Asp.setdefault((2, 2), 4) == 4 + d.setdefault((0, 1), 4) + d.setdefault((2, 2), 4) + assert d.items() == Asp.items() + +def test_update(d, Asp): + with pytest.raises(NotImplementedError): + Asp.update(Asp) + +def test_values(d, Asp): + # Note: dict.values are strange: d={1: 1}; d.values() == d.values() is False + # Using list(d.values()) makes them comparable. + assert list(Asp.values()) == list(d.values()) + +def test_dunder_getitem(d, Asp): + assert Asp[(0, 1)] == d[(0, 1)] + +def test_dunder_setitem(d, Asp): + Asp[(1, 1)] = 5 + d[(1, 1)] = 5 + assert d.items() == Asp.items() + +def test_dunder_delitem(d, Asp): + del Asp[(0, 1)] + del d[(0, 1)] + assert d.items() == Asp.items() + +def test_dunder_contains(d, Asp): + assert ((0, 1) in d) == ((0, 1) in Asp) + assert ((0, 0) in d) == ((0, 0) in Asp) + +def test_dunder_len(d, Asp): + assert len(d) == len(Asp) + +# Note: dunders reversed, or, ror, ior work as dict for dok_matrix, raise for dok_array +def test_dunder_reversed(d, Asp): + if isinstance(Asp, dok_array): + with pytest.raises(TypeError): + list(reversed(Asp)) + else: + assert list(reversed(Asp)) == list(reversed(d)) + +def test_dunder_ior(d, Asp): + if isinstance(Asp, dok_array): + with pytest.raises(TypeError): + Asp |= Asp + else: + dd = {(0, 0): 5} + Asp |= dd + assert Asp[(0, 0)] == 5 + d |= dd + assert d.items() == Asp.items() + dd |= Asp + assert dd.items() == Asp.items() + +def test_dunder_or(d, Asp): + if isinstance(Asp, dok_array): + with pytest.raises(TypeError): + Asp | Asp + else: + assert d | d == Asp | d + assert d | d == Asp | Asp + +def test_dunder_ror(d, Asp): + if isinstance(Asp, dok_array): + with pytest.raises(TypeError): + Asp | Asp + with pytest.raises(TypeError): + d | Asp + else: + assert Asp.__ror__(d) == Asp.__ror__(Asp) + assert d.__ror__(d) == Asp.__ror__(d) + assert d | Asp + +# Note: comparison dunders, e.g. ==, >=, etc follow np.array not dict +def test_dunder_eq(A, Asp): + with np.testing.suppress_warnings() as sup: + sup.filter(sp.sparse.SparseEfficiencyWarning) + assert (Asp == Asp).toarray().all() + assert (A == Asp).all() + +def test_dunder_ne(A, Asp): + assert not (Asp != Asp).toarray().any() + assert not (A != Asp).any() + +def test_dunder_lt(A, Asp): + assert not (Asp < Asp).toarray().any() + assert not (A < Asp).any() + +def test_dunder_gt(A, Asp): + assert not (Asp > Asp).toarray().any() + assert not (A > Asp).any() + +def test_dunder_le(A, Asp): + with np.testing.suppress_warnings() as sup: + sup.filter(sp.sparse.SparseEfficiencyWarning) + assert (Asp <= Asp).toarray().all() + assert (A <= Asp).all() + +def test_dunder_ge(A, Asp): + with np.testing.suppress_warnings() as sup: + sup.filter(sp.sparse.SparseEfficiencyWarning) + assert (Asp >= Asp).toarray().all() + assert (A >= Asp).all() + +# Note: iter dunder follows np.array not dict +def test_dunder_iter(A, Asp): + assert all((a == asp).all() for a, asp in zip(A, Asp)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_extract.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_extract.py new file mode 100644 index 0000000000000000000000000000000000000000..a7c9f68bb2bde76d74ca767abba3c99b89d6e771 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_extract.py @@ -0,0 +1,51 @@ +"""test sparse matrix construction functions""" + +from numpy.testing import assert_equal +from scipy.sparse import csr_matrix, csr_array, sparray + +import numpy as np +from scipy.sparse import _extract + + +class TestExtract: + def setup_method(self): + self.cases = [ + csr_array([[1,2]]), + csr_array([[1,0]]), + csr_array([[0,0]]), + csr_array([[1],[2]]), + csr_array([[1],[0]]), + csr_array([[0],[0]]), + csr_array([[1,2],[3,4]]), + csr_array([[0,1],[0,0]]), + csr_array([[0,0],[1,0]]), + csr_array([[0,0],[0,0]]), + csr_array([[1,2,0,0,3],[4,5,0,6,7],[0,0,8,9,0]]), + csr_array([[1,2,0,0,3],[4,5,0,6,7],[0,0,8,9,0]]).T, + ] + + def test_find(self): + for A in self.cases: + I,J,V = _extract.find(A) + B = csr_array((V,(I,J)), shape=A.shape) + assert_equal(A.toarray(), B.toarray()) + + def test_tril(self): + for A in self.cases: + B = A.toarray() + for k in [-3,-2,-1,0,1,2,3]: + assert_equal(_extract.tril(A,k=k).toarray(), np.tril(B,k=k)) + + def test_triu(self): + for A in self.cases: + B = A.toarray() + for k in [-3,-2,-1,0,1,2,3]: + assert_equal(_extract.triu(A,k=k).toarray(), np.triu(B,k=k)) + + def test_array_vs_matrix(self): + for A in self.cases: + assert isinstance(_extract.tril(A), sparray) + assert isinstance(_extract.triu(A), sparray) + M = csr_matrix(A) + assert not isinstance(_extract.tril(M), sparray) + assert not isinstance(_extract.triu(M), sparray) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_indexing1d.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_indexing1d.py new file mode 100644 index 0000000000000000000000000000000000000000..99934c455b796511b9f48243da044a9cee8a4d53 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_indexing1d.py @@ -0,0 +1,603 @@ +import contextlib +import pytest +import numpy as np +from numpy.testing import assert_allclose, assert_equal + +from scipy.sparse import csr_array, dok_array, SparseEfficiencyWarning +from .test_arithmetic1d import toarray + + +formats_for_index1d = [csr_array, dok_array] + + +@contextlib.contextmanager +def check_remains_sorted(X): + """Checks that sorted indices property is retained through an operation""" + yield + if not hasattr(X, 'has_sorted_indices') or not X.has_sorted_indices: + return + indices = X.indices.copy() + X.has_sorted_indices = False + X.sort_indices() + assert_equal(indices, X.indices, 'Expected sorted indices, found unsorted') + + +@pytest.mark.parametrize("spcreator", formats_for_index1d) +class TestGetSet1D: + def test_None_index(self, spcreator): + D = np.array([4, 3, 0]) + A = spcreator(D) + + N = D.shape[0] + for j in range(-N, N): + assert_equal(A[j, None].toarray(), D[j, None]) + assert_equal(A[None, j].toarray(), D[None, j]) + assert_equal(A[None, None, j].toarray(), D[None, None, j]) + + def test_getitem_shape(self, spcreator): + A = spcreator(np.arange(3 * 4).reshape(3, 4)) + assert A[1, 2].ndim == 0 + assert A[1, 2:3].shape == (1,) + assert A[None, 1, 2:3].shape == (1, 1) + assert A[None, 1, 2].shape == (1,) + assert A[None, 1, 2, None].shape == (1, 1) + + # see gh-22458 + assert A[None, 1].shape == (1, 4) + assert A[1, None].shape == (1, 4) + assert A[None, 1, :].shape == (1, 4) + assert A[1, None, :].shape == (1, 4) + assert A[1, :, None].shape == (4, 1) + + with pytest.raises(IndexError, match='Only 1D or 2D arrays'): + A[None, 2, 1, None, None] + with pytest.raises(IndexError, match='Only 1D or 2D arrays'): + A[None, 0:2, None, 1] + with pytest.raises(IndexError, match='Only 1D or 2D arrays'): + A[0:1, 1:, None] + with pytest.raises(IndexError, match='Only 1D or 2D arrays'): + A[1:, 1, None, None] + + def test_getelement(self, spcreator): + D = np.array([4, 3, 0]) + A = spcreator(D) + + N = D.shape[0] + for j in range(-N, N): + assert_equal(A[j], D[j]) + + for ij in [3, -4]: + with pytest.raises(IndexError, match='index (.*) out of (range|bounds)'): + A.__getitem__(ij) + + # single element tuples unwrapped + assert A[(0,)] == 4 + + with pytest.raises(IndexError, match='index (.*) out of (range|bounds)'): + A.__getitem__((4,)) + + def test_setelement(self, spcreator): + dtype = np.float64 + A = spcreator((12,), dtype=dtype) + with np.testing.suppress_warnings() as sup: + sup.filter( + SparseEfficiencyWarning, + "Changing the sparsity structure of .* is expensive", + ) + A[0] = dtype(0) + A[1] = dtype(3) + A[8] = dtype(9.0) + A[-2] = dtype(7) + A[5] = 9 + + A[-9,] = dtype(8) + A[1,] = dtype(5) # overwrite using 1-tuple index + + for ij in [13, -14, (13,), (14,)]: + with pytest.raises(IndexError, match='out of (range|bounds)'): + A.__setitem__(ij, 123.0) + + +@pytest.mark.parametrize("spcreator", formats_for_index1d) +class TestSlicingAndFancy1D: + ####################### + # Int-like Array Index + ####################### + def test_get_array_index(self, spcreator): + D = np.array([4, 3, 0]) + A = spcreator(D) + + assert_equal(A[()].toarray(), D[()]) + for ij in [(0, 3), (3,)]: + with pytest.raises(IndexError, match='out of (range|bounds)|many indices'): + A.__getitem__(ij) + + def test_set_array_index(self, spcreator): + dtype = np.float64 + A = spcreator((12,), dtype=dtype) + with np.testing.suppress_warnings() as sup: + sup.filter( + SparseEfficiencyWarning, + "Changing the sparsity structure of .* is expensive", + ) + A[np.array(6)] = dtype(4.0) # scalar index + A[np.array(6)] = dtype(2.0) # overwrite with scalar index + assert_equal(A.toarray(), [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0]) + + for ij in [(13,), (-14,)]: + with pytest.raises(IndexError, match='index .* out of (range|bounds)'): + A.__setitem__(ij, 123.0) + + for v in [(), (0, 3), [1, 2, 3], np.array([1, 2, 3])]: + msg = 'Trying to assign a sequence to an item' + with pytest.raises(ValueError, match=msg): + A.__setitem__(0, v) + + #################### + # 1d Slice as index + #################### + def test_dtype_preservation(self, spcreator): + assert_equal(spcreator((10,), dtype=np.int16)[1:5].dtype, np.int16) + assert_equal(spcreator((6,), dtype=np.int32)[0:0:2].dtype, np.int32) + assert_equal(spcreator((6,), dtype=np.int64)[:].dtype, np.int64) + + def test_get_1d_slice(self, spcreator): + B = np.arange(50.) + A = spcreator(B) + assert_equal(B[:], A[:].toarray()) + assert_equal(B[2:5], A[2:5].toarray()) + + C = np.array([4, 0, 6, 0, 0, 0, 0, 0, 1]) + D = spcreator(C) + assert_equal(C[1:3], D[1:3].toarray()) + + # Now test slicing when a row contains only zeros + E = np.array([0, 0, 0, 0, 0]) + F = spcreator(E) + assert_equal(E[1:3], F[1:3].toarray()) + assert_equal(E[-2:], F[-2:].toarray()) + assert_equal(E[:], F[:].toarray()) + assert_equal(E[slice(None)], F[slice(None)].toarray()) + + def test_slicing_idx_slice(self, spcreator): + B = np.arange(50) + A = spcreator(B) + + # [i] + assert_equal(A[2], B[2]) + assert_equal(A[-1], B[-1]) + assert_equal(A[np.array(-2)], B[-2]) + + # [1:2] + assert_equal(A[:].toarray(), B[:]) + assert_equal(A[5:-2].toarray(), B[5:-2]) + assert_equal(A[5:12:3].toarray(), B[5:12:3]) + + # int8 slice + s = slice(np.int8(2), np.int8(4), None) + assert_equal(A[s].toarray(), B[2:4]) + + # np.s_ + s_ = np.s_ + slices = [s_[:2], s_[1:2], s_[3:], s_[3::2], + s_[15:20], s_[3:2], + s_[8:3:-1], s_[4::-2], s_[:5:-1], + 0, 1, s_[:], s_[1:5], -1, -2, -5, + np.array(-1), np.int8(-3)] + + for j, a in enumerate(slices): + x = A[a] + y = B[a] + if y.shape == (): + assert_equal(x, y, repr(a)) + else: + if x.size == 0 and y.size == 0: + pass + else: + assert_equal(x.toarray(), y, repr(a)) + + def test_ellipsis_1d_slicing(self, spcreator): + B = np.arange(50) + A = spcreator(B) + assert_equal(A[...].toarray(), B[...]) + assert_equal(A[...,].toarray(), B[...,]) + + ########################## + # Assignment with Slicing + ########################## + def test_slice_scalar_assign(self, spcreator): + A = spcreator((5,)) + B = np.zeros((5,)) + with np.testing.suppress_warnings() as sup: + sup.filter( + SparseEfficiencyWarning, + "Changing the sparsity structure of .* is expensive", + ) + for C in [A, B]: + C[0:1] = 1 + C[2:0] = 4 + C[2:3] = 9 + C[3:] = 1 + C[3::-1] = 9 + assert_equal(A.toarray(), B) + + def test_slice_assign_2(self, spcreator): + shape = (10,) + + for idx in [slice(3), slice(None, 10, 4), slice(5, -2)]: + A = spcreator(shape) + with np.testing.suppress_warnings() as sup: + sup.filter( + SparseEfficiencyWarning, + "Changing the sparsity structure of .* is expensive", + ) + A[idx] = 1 + B = np.zeros(shape) + B[idx] = 1 + msg = f"idx={idx!r}" + assert_allclose(A.toarray(), B, err_msg=msg) + + def test_self_self_assignment(self, spcreator): + # Tests whether a row of one lil_matrix can be assigned to another. + B = spcreator((5,)) + with np.testing.suppress_warnings() as sup: + sup.filter( + SparseEfficiencyWarning, + "Changing the sparsity structure of .* is expensive", + ) + B[0] = 2 + B[1] = 0 + B[2] = 3 + B[3] = 10 + + A = B / 10 + B[:] = A[:] + assert_equal(A[:].toarray(), B[:].toarray()) + + A = B / 10 + B[:] = A[:1] + assert_equal(np.zeros((5,)) + A[0], B.toarray()) + + A = B / 10 + B[:-1] = A[1:] + assert_equal(A[1:].toarray(), B[:-1].toarray()) + + def test_slice_assignment(self, spcreator): + B = spcreator((4,)) + expected = np.array([10, 0, 14, 0]) + block = [2, 1] + + with np.testing.suppress_warnings() as sup: + sup.filter( + SparseEfficiencyWarning, + "Changing the sparsity structure of .* is expensive", + ) + B[0] = 5 + B[2] = 7 + B[:] = B + B + assert_equal(B.toarray(), expected) + + B[:2] = csr_array(block) + assert_equal(B.toarray()[:2], block) + + def test_set_slice(self, spcreator): + A = spcreator((5,)) + B = np.zeros(5, float) + s_ = np.s_ + slices = [s_[:2], s_[1:2], s_[3:], s_[3::2], + s_[8:3:-1], s_[4::-2], s_[:5:-1], + 0, 1, s_[:], s_[1:5], -1, -2, -5, + np.array(-1), np.int8(-3)] + + with np.testing.suppress_warnings() as sup: + sup.filter( + SparseEfficiencyWarning, + "Changing the sparsity structure of .* is expensive", + ) + for j, a in enumerate(slices): + A[a] = j + B[a] = j + assert_equal(A.toarray(), B, repr(a)) + + A[1:10:2] = range(1, 5, 2) + B[1:10:2] = range(1, 5, 2) + assert_equal(A.toarray(), B) + + # The next commands should raise exceptions + toobig = list(range(100)) + with pytest.raises(ValueError, match='Trying to assign a sequence to an item'): + A.__setitem__(0, toobig) + with pytest.raises(ValueError, match='could not be broadcast together'): + A.__setitem__(slice(None), toobig) + + def test_assign_empty(self, spcreator): + A = spcreator(np.ones(3)) + B = spcreator((2,)) + A[:2] = B + assert_equal(A.toarray(), [0, 0, 1]) + + #################### + # 1d Fancy Indexing + #################### + def test_dtype_preservation_empty_index(self, spcreator): + A = spcreator((2,), dtype=np.int16) + assert_equal(A[[False, False]].dtype, np.int16) + assert_equal(A[[]].dtype, np.int16) + + def test_bad_index(self, spcreator): + A = spcreator(np.zeros(5)) + with pytest.raises( + (IndexError, ValueError, TypeError), + match='Index dimension must be 1 or 2|only integers', + ): + A.__getitem__("foo") + with pytest.raises( + (IndexError, ValueError, TypeError), + match='tuple index out of range|only integers', + ): + A.__getitem__((2, "foo")) + + def test_fancy_indexing_2darray(self, spcreator): + B = np.arange(50).reshape((5, 10)) + A = spcreator(B) + + # [i] + assert_equal(A[[1, 3]].toarray(), B[[1, 3]]) + + # [i,[1,2]] + assert_equal(A[3, [1, 3]].toarray(), B[3, [1, 3]]) + assert_equal(A[-1, [2, -5]].toarray(), B[-1, [2, -5]]) + assert_equal(A[np.array(-1), [2, -5]].toarray(), B[-1, [2, -5]]) + assert_equal(A[-1, np.array([2, -5])].toarray(), B[-1, [2, -5]]) + assert_equal(A[np.array(-1), np.array([2, -5])].toarray(), B[-1, [2, -5]]) + + # [1:2,[1,2]] + assert_equal(A[:, [2, 8, 3, -1]].toarray(), B[:, [2, 8, 3, -1]]) + assert_equal(A[3:4, [9]].toarray(), B[3:4, [9]]) + assert_equal(A[1:4, [-1, -5]].toarray(), B[1:4, [-1, -5]]) + assert_equal(A[1:4, np.array([-1, -5])].toarray(), B[1:4, [-1, -5]]) + + # [[1,2],j] + assert_equal(A[[1, 3], 3].toarray(), B[[1, 3], 3]) + assert_equal(A[[2, -5], -4].toarray(), B[[2, -5], -4]) + assert_equal(A[np.array([2, -5]), -4].toarray(), B[[2, -5], -4]) + assert_equal(A[[2, -5], np.array(-4)].toarray(), B[[2, -5], -4]) + assert_equal(A[np.array([2, -5]), np.array(-4)].toarray(), B[[2, -5], -4]) + + # [[1,2],1:2] + assert_equal(A[[1, 3], :].toarray(), B[[1, 3], :]) + assert_equal(A[[2, -5], 8:-1].toarray(), B[[2, -5], 8:-1]) + assert_equal(A[np.array([2, -5]), 8:-1].toarray(), B[[2, -5], 8:-1]) + + # [[1,2],[1,2]] + assert_equal(toarray(A[[1, 3], [2, 4]]), B[[1, 3], [2, 4]]) + assert_equal(toarray(A[[-1, -3], [2, -4]]), B[[-1, -3], [2, -4]]) + assert_equal( + toarray(A[np.array([-1, -3]), [2, -4]]), B[[-1, -3], [2, -4]] + ) + assert_equal( + toarray(A[[-1, -3], np.array([2, -4])]), B[[-1, -3], [2, -4]] + ) + assert_equal( + toarray(A[np.array([-1, -3]), np.array([2, -4])]), B[[-1, -3], [2, -4]] + ) + + # [[[1],[2]],[1,2]] + assert_equal(A[[[1], [3]], [2, 4]].toarray(), B[[[1], [3]], [2, 4]]) + assert_equal( + A[[[-1], [-3], [-2]], [2, -4]].toarray(), + B[[[-1], [-3], [-2]], [2, -4]] + ) + assert_equal( + A[np.array([[-1], [-3], [-2]]), [2, -4]].toarray(), + B[[[-1], [-3], [-2]], [2, -4]] + ) + assert_equal( + A[[[-1], [-3], [-2]], np.array([2, -4])].toarray(), + B[[[-1], [-3], [-2]], [2, -4]] + ) + assert_equal( + A[np.array([[-1], [-3], [-2]]), np.array([2, -4])].toarray(), + B[[[-1], [-3], [-2]], [2, -4]] + ) + + # [[1,2]] + assert_equal(A[[1, 3]].toarray(), B[[1, 3]]) + assert_equal(A[[-1, -3]].toarray(), B[[-1, -3]]) + assert_equal(A[np.array([-1, -3])].toarray(), B[[-1, -3]]) + + # [[1,2],:][:,[1,2]] + assert_equal( + A[[1, 3], :][:, [2, 4]].toarray(), B[[1, 3], :][:, [2, 4]] + ) + assert_equal( + A[[-1, -3], :][:, [2, -4]].toarray(), B[[-1, -3], :][:, [2, -4]] + ) + assert_equal( + A[np.array([-1, -3]), :][:, np.array([2, -4])].toarray(), + B[[-1, -3], :][:, [2, -4]] + ) + + # [:,[1,2]][[1,2],:] + assert_equal( + A[:, [1, 3]][[2, 4], :].toarray(), B[:, [1, 3]][[2, 4], :] + ) + assert_equal( + A[:, [-1, -3]][[2, -4], :].toarray(), B[:, [-1, -3]][[2, -4], :] + ) + assert_equal( + A[:, np.array([-1, -3])][np.array([2, -4]), :].toarray(), + B[:, [-1, -3]][[2, -4], :] + ) + + def test_fancy_indexing(self, spcreator): + B = np.arange(50) + A = spcreator(B) + + # [i] + assert_equal(A[[3]].toarray(), B[[3]]) + + # [np.array] + assert_equal(A[[1, 3]].toarray(), B[[1, 3]]) + assert_equal(A[[2, -5]].toarray(), B[[2, -5]]) + assert_equal(A[np.array(-1)], B[-1]) + assert_equal(A[np.array([-1, 2])].toarray(), B[[-1, 2]]) + assert_equal(A[np.array(5)], B[np.array(5)]) + + # [[[1],[2]]] + ind = np.array([[1], [3]]) + assert_equal(A[ind].toarray(), B[ind]) + ind = np.array([[-1], [-3], [-2]]) + assert_equal(A[ind].toarray(), B[ind]) + + # [[1, 2]] + assert_equal(A[[1, 3]].toarray(), B[[1, 3]]) + assert_equal(A[[-1, -3]].toarray(), B[[-1, -3]]) + assert_equal(A[np.array([-1, -3])].toarray(), B[[-1, -3]]) + + # [[1, 2]][[1, 2]] + assert_equal(A[[1, 5, 2, 8]][[1, 3]].toarray(), + B[[1, 5, 2, 8]][[1, 3]]) + assert_equal(A[[-1, -5, 2, 8]][[1, -4]].toarray(), + B[[-1, -5, 2, 8]][[1, -4]]) + + def test_fancy_indexing_boolean(self, spcreator): + np.random.seed(1234) # make runs repeatable + + B = np.arange(50) + A = spcreator(B) + + I = np.array(np.random.randint(0, 2, size=50), dtype=bool) + + assert_equal(toarray(A[I]), B[I]) + assert_equal(toarray(A[B > 9]), B[B > 9]) + + Z1 = np.zeros(51, dtype=bool) + Z2 = np.zeros(51, dtype=bool) + Z2[-1] = True + Z3 = np.zeros(51, dtype=bool) + Z3[0] = True + + msg = 'bool index .* has shape|boolean index did not match' + with pytest.raises(IndexError, match=msg): + A.__getitem__(Z1) + with pytest.raises(IndexError, match=msg): + A.__getitem__(Z2) + with pytest.raises(IndexError, match=msg): + A.__getitem__(Z3) + + def test_fancy_indexing_sparse_boolean(self, spcreator): + np.random.seed(1234) # make runs repeatable + + B = np.arange(20) + A = spcreator(B) + + X = np.array(np.random.randint(0, 2, size=20), dtype=bool) + Xsp = csr_array(X) + + assert_equal(toarray(A[Xsp]), B[X]) + assert_equal(toarray(A[A > 9]), B[B > 9]) + + Y = np.array(np.random.randint(0, 2, size=60), dtype=bool) + + Ysp = csr_array(Y) + + with pytest.raises(IndexError, match='bool index .* has shape|only integers'): + A.__getitem__(Ysp) + with pytest.raises(IndexError, match='tuple index out of range|only integers'): + A.__getitem__((Xsp, 1)) + + def test_fancy_indexing_seq_assign(self, spcreator): + mat = spcreator(np.array([1, 0])) + with pytest.raises(ValueError, match='Trying to assign a sequence to an item'): + mat.__setitem__(0, np.array([1, 2])) + + def test_fancy_indexing_empty(self, spcreator): + B = np.arange(50) + B[3:9] = 0 + B[30] = 0 + A = spcreator(B) + + K = np.array([False] * 50) + assert_equal(toarray(A[K]), B[K]) + K = np.array([], dtype=int) + assert_equal(toarray(A[K]), B[K]) + J = np.array([0, 1, 2, 3, 4], dtype=int) + assert_equal(toarray(A[J]), B[J]) + + ############################ + # 1d Fancy Index Assignment + ############################ + def test_bad_index_assign(self, spcreator): + A = spcreator(np.zeros(5)) + msg = 'Index dimension must be 1 or 2|only integers' + with pytest.raises((IndexError, ValueError, TypeError), match=msg): + A.__setitem__("foo", 2) + + def test_fancy_indexing_set(self, spcreator): + M = (5,) + + # [1:2] + for j in [[2, 3, 4], slice(None, 10, 4), np.arange(3), + slice(5, -2), slice(2, 5)]: + A = spcreator(M) + B = np.zeros(M) + with np.testing.suppress_warnings() as sup: + sup.filter( + SparseEfficiencyWarning, + "Changing the sparsity structure of .* is expensive", + ) + B[j] = 1 + with check_remains_sorted(A): + A[j] = 1 + assert_allclose(A.toarray(), B) + + def test_sequence_assignment(self, spcreator): + A = spcreator((4,)) + B = spcreator((3,)) + + i0 = [0, 1, 2] + i1 = (0, 1, 2) + i2 = np.array(i0) + + with np.testing.suppress_warnings() as sup: + sup.filter( + SparseEfficiencyWarning, + "Changing the sparsity structure of .* is expensive", + ) + with check_remains_sorted(A): + A[i0] = B[i0] + msg = "too many indices for array|tuple index out of range" + with pytest.raises(IndexError, match=msg): + B.__getitem__(i1) + A[i2] = B[i2] + assert_equal(A[:3].toarray(), B.toarray()) + assert A.shape == (4,) + + # slice + A = spcreator((4,)) + with check_remains_sorted(A): + A[1:3] = [10, 20] + assert_equal(A.toarray(), [0, 10, 20, 0]) + + # array + A = spcreator((4,)) + B = np.zeros(4) + with check_remains_sorted(A): + for C in [A, B]: + C[[0, 1, 2]] = [4, 5, 6] + assert_equal(A.toarray(), B) + + def test_fancy_assign_empty(self, spcreator): + B = np.arange(50) + B[2] = 0 + B[[3, 6]] = 0 + A = spcreator(B) + + K = np.array([False] * 50) + A[K] = 42 + assert_equal(A.toarray(), B) + + K = np.array([], dtype=int) + A[K] = 42 + assert_equal(A.toarray(), B) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_matrix_io.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_matrix_io.py new file mode 100644 index 0000000000000000000000000000000000000000..90b4ea64a8928073eb5dd3f1b2752379f57327d9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_matrix_io.py @@ -0,0 +1,109 @@ +import os +import numpy as np +import tempfile + +from pytest import raises as assert_raises +from numpy.testing import assert_equal, assert_ + +from scipy.sparse import (sparray, csc_matrix, csr_matrix, bsr_matrix, dia_matrix, + coo_matrix, dok_matrix, csr_array, save_npz, load_npz) + + +DATA_DIR = os.path.join(os.path.dirname(__file__), 'data') + + +def _save_and_load(matrix): + fd, tmpfile = tempfile.mkstemp(suffix='.npz') + os.close(fd) + try: + save_npz(tmpfile, matrix) + loaded_matrix = load_npz(tmpfile) + finally: + os.remove(tmpfile) + return loaded_matrix + +def _check_save_and_load(dense_matrix): + for matrix_class in [csc_matrix, csr_matrix, bsr_matrix, dia_matrix, coo_matrix]: + matrix = matrix_class(dense_matrix) + loaded_matrix = _save_and_load(matrix) + assert_(type(loaded_matrix) is matrix_class) + assert_(loaded_matrix.shape == dense_matrix.shape) + assert_(loaded_matrix.dtype == dense_matrix.dtype) + assert_equal(loaded_matrix.toarray(), dense_matrix) + +def test_save_and_load_random(): + N = 10 + np.random.seed(0) + dense_matrix = np.random.random((N, N)) + dense_matrix[dense_matrix > 0.7] = 0 + _check_save_and_load(dense_matrix) + +def test_save_and_load_empty(): + dense_matrix = np.zeros((4,6)) + _check_save_and_load(dense_matrix) + +def test_save_and_load_one_entry(): + dense_matrix = np.zeros((4,6)) + dense_matrix[1,2] = 1 + _check_save_and_load(dense_matrix) + +def test_sparray_vs_spmatrix(): + #save/load matrix + fd, tmpfile = tempfile.mkstemp(suffix='.npz') + os.close(fd) + try: + save_npz(tmpfile, csr_matrix([[1.2, 0, 0.9], [0, 0.3, 0]])) + loaded_matrix = load_npz(tmpfile) + finally: + os.remove(tmpfile) + + #save/load array + fd, tmpfile = tempfile.mkstemp(suffix='.npz') + os.close(fd) + try: + save_npz(tmpfile, csr_array([[1.2, 0, 0.9], [0, 0.3, 0]])) + loaded_array = load_npz(tmpfile) + finally: + os.remove(tmpfile) + + assert not isinstance(loaded_matrix, sparray) + assert isinstance(loaded_array, sparray) + assert_(loaded_matrix.dtype == loaded_array.dtype) + assert_equal(loaded_matrix.toarray(), loaded_array.toarray()) + +def test_malicious_load(): + class Executor: + def __reduce__(self): + return (assert_, (False, 'unexpected code execution')) + + fd, tmpfile = tempfile.mkstemp(suffix='.npz') + os.close(fd) + try: + np.savez(tmpfile, format=Executor()) + + # Should raise a ValueError, not execute code + assert_raises(ValueError, load_npz, tmpfile) + finally: + os.remove(tmpfile) + + +def test_py23_compatibility(): + # Try loading files saved on Python 2 and Python 3. They are not + # the same, since files saved with SciPy versions < 1.0.0 may + # contain unicode. + + a = load_npz(os.path.join(DATA_DIR, 'csc_py2.npz')) + b = load_npz(os.path.join(DATA_DIR, 'csc_py3.npz')) + c = csc_matrix([[0]]) + + assert_equal(a.toarray(), c.toarray()) + assert_equal(b.toarray(), c.toarray()) + +def test_implemented_error(): + # Attempts to save an unsupported type and checks that an + # NotImplementedError is raised. + + x = dok_matrix((2,3)) + x[0,1] = 1 + + assert_raises(NotImplementedError, save_npz, 'x.npz', x) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_minmax1d.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_minmax1d.py new file mode 100644 index 0000000000000000000000000000000000000000..dca3f44fa485070805995c2f76c0c511123ce355 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_minmax1d.py @@ -0,0 +1,128 @@ +"""Test of min-max 1D features of sparse array classes""" + +import pytest + +import numpy as np + +from numpy.testing import assert_equal, assert_array_equal + +from scipy.sparse import coo_array, csr_array, csc_array, bsr_array +from scipy.sparse import coo_matrix, csr_matrix, csc_matrix, bsr_matrix +from scipy.sparse._sputils import isscalarlike + + +def toarray(a): + if isinstance(a, np.ndarray) or isscalarlike(a): + return a + return a.toarray() + + +formats_for_minmax = [bsr_array, coo_array, csc_array, csr_array] +formats_for_minmax_supporting_1d = [coo_array, csr_array] + + +@pytest.mark.parametrize("spcreator", formats_for_minmax_supporting_1d) +class Test_MinMaxMixin1D: + def test_minmax(self, spcreator): + D = np.arange(5) + X = spcreator(D) + + assert_equal(X.min(), 0) + assert_equal(X.max(), 4) + assert_equal((-X).min(), -4) + assert_equal((-X).max(), 0) + + def test_minmax_axis(self, spcreator): + D = np.arange(50) + X = spcreator(D) + + for axis in [0, -1]: + assert_array_equal( + toarray(X.max(axis=axis)), D.max(axis=axis, keepdims=True) + ) + assert_array_equal( + toarray(X.min(axis=axis)), D.min(axis=axis, keepdims=True) + ) + for axis in [-2, 1]: + with pytest.raises(ValueError, match="axis out of range"): + X.min(axis=axis) + with pytest.raises(ValueError, match="axis out of range"): + X.max(axis=axis) + + def test_numpy_minmax(self, spcreator): + dat = np.array([0, 1, 2]) + datsp = spcreator(dat) + assert_array_equal(np.min(datsp), np.min(dat)) + assert_array_equal(np.max(datsp), np.max(dat)) + + + def test_argmax(self, spcreator): + D1 = np.array([-1, 5, 2, 3]) + D2 = np.array([0, 0, -1, -2]) + D3 = np.array([-1, -2, -3, -4]) + D4 = np.array([1, 2, 3, 4]) + D5 = np.array([1, 2, 0, 0]) + + for D in [D1, D2, D3, D4, D5]: + mat = spcreator(D) + + assert_equal(mat.argmax(), np.argmax(D)) + assert_equal(mat.argmin(), np.argmin(D)) + + assert_equal(mat.argmax(axis=0), np.argmax(D, axis=0)) + assert_equal(mat.argmin(axis=0), np.argmin(D, axis=0)) + + D6 = np.empty((0,)) + + for axis in [None, 0]: + mat = spcreator(D6) + with pytest.raises(ValueError, match="to an empty matrix"): + mat.argmin(axis=axis) + with pytest.raises(ValueError, match="to an empty matrix"): + mat.argmax(axis=axis) + + +@pytest.mark.parametrize("spcreator", formats_for_minmax) +class Test_ShapeMinMax2DWithAxis: + def test_minmax(self, spcreator): + dat = np.array([[-1, 5, 0, 3], [0, 0, -1, -2], [0, 0, 1, 2]]) + datsp = spcreator(dat) + + for (spminmax, npminmax) in [ + (datsp.min, np.min), + (datsp.max, np.max), + (datsp.nanmin, np.nanmin), + (datsp.nanmax, np.nanmax), + ]: + for ax, result_shape in [(0, (4,)), (1, (3,))]: + assert_equal(toarray(spminmax(axis=ax)), npminmax(dat, axis=ax)) + assert_equal(spminmax(axis=ax).shape, result_shape) + assert spminmax(axis=ax).format == "coo" + + for spminmax in [datsp.argmin, datsp.argmax]: + for ax in [0, 1]: + assert isinstance(spminmax(axis=ax), np.ndarray) + + # verify spmatrix behavior + spmat_form = { + 'coo': coo_matrix, + 'csr': csr_matrix, + 'csc': csc_matrix, + 'bsr': bsr_matrix, + } + datspm = spmat_form[datsp.format](dat) + + for spm, npm in [ + (datspm.min, np.min), + (datspm.max, np.max), + (datspm.nanmin, np.nanmin), + (datspm.nanmax, np.nanmax), + ]: + for ax, result_shape in [(0, (1, 4)), (1, (3, 1))]: + assert_equal(toarray(spm(axis=ax)), npm(dat, axis=ax, keepdims=True)) + assert_equal(spm(axis=ax).shape, result_shape) + assert spm(axis=ax).format == "coo" + + for spminmax in [datspm.argmin, datspm.argmax]: + for ax in [0, 1]: + assert isinstance(spminmax(axis=ax), np.ndarray) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_sparsetools.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_sparsetools.py new file mode 100644 index 0000000000000000000000000000000000000000..6a8b94796116a22c210104fc446c5a17045ed21c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_sparsetools.py @@ -0,0 +1,339 @@ +import sys +import os +import gc +import threading + +import numpy as np +from numpy.testing import assert_equal, assert_, assert_allclose +from scipy.sparse import (_sparsetools, coo_matrix, csr_matrix, csc_matrix, + bsr_matrix, dia_matrix) +from scipy.sparse._sputils import supported_dtypes +from scipy._lib._testutils import check_free_memory + +import pytest +from pytest import raises as assert_raises + + +def int_to_int8(n): + """ + Wrap an integer to the interval [-128, 127]. + """ + return (n + 128) % 256 - 128 + + +def test_exception(): + assert_raises(MemoryError, _sparsetools.test_throw_error) + + +def test_threads(): + # Smoke test for parallel threaded execution; doesn't actually + # check that code runs in parallel, but just that it produces + # expected results. + nthreads = 10 + niter = 100 + + n = 20 + a = csr_matrix(np.ones([n, n])) + bres = [] + + class Worker(threading.Thread): + def run(self): + b = a.copy() + for j in range(niter): + _sparsetools.csr_plus_csr(n, n, + a.indptr, a.indices, a.data, + a.indptr, a.indices, a.data, + b.indptr, b.indices, b.data) + bres.append(b) + + threads = [Worker() for _ in range(nthreads)] + for thread in threads: + thread.start() + for thread in threads: + thread.join() + + for b in bres: + assert_(np.all(b.toarray() == 2)) + + +def test_regression_std_vector_dtypes(): + # Regression test for gh-3780, checking the std::vector typemaps + # in sparsetools.cxx are complete. + for dtype in supported_dtypes: + ad = np.array([[1, 2], [3, 4]]).astype(dtype) + a = csr_matrix(ad, dtype=dtype) + + # getcol is one function using std::vector typemaps, and should not fail + assert_equal(a.getcol(0).toarray(), ad[:, :1]) + + +@pytest.mark.slow +@pytest.mark.xfail_on_32bit("Can't create large array for test") +def test_nnz_overflow(): + # Regression test for gh-7230 / gh-7871, checking that coo_toarray + # with nnz > int32max doesn't overflow. + nnz = np.iinfo(np.int32).max + 1 + # Ensure ~20 GB of RAM is free to run this test. + check_free_memory((4 + 4 + 1) * nnz / 1e6 + 0.5) + + # Use nnz duplicate entries to keep the dense version small. + row = np.zeros(nnz, dtype=np.int32) + col = np.zeros(nnz, dtype=np.int32) + data = np.zeros(nnz, dtype=np.int8) + data[-1] = 4 + s = coo_matrix((data, (row, col)), shape=(1, 1), copy=False) + # Sums nnz duplicates to produce a 1x1 array containing 4. + d = s.toarray() + + assert_allclose(d, [[4]]) + + +@pytest.mark.skipif( + not (sys.platform.startswith('linux') and np.dtype(np.intp).itemsize >= 8), + reason="test requires 64-bit Linux" +) +class TestInt32Overflow: + """ + Some of the sparsetools routines use dense 2D matrices whose + total size is not bounded by the nnz of the sparse matrix. These + routines used to suffer from int32 wraparounds; here, we try to + check that the wraparounds don't occur any more. + """ + # choose n large enough + n = 50000 + + def setup_method(self): + assert self.n**2 > np.iinfo(np.int32).max + + # check there's enough memory even if everything is run at the + # same time + try: + parallel_count = int(os.environ.get('PYTEST_XDIST_WORKER_COUNT', '1')) + except ValueError: + parallel_count = np.inf + + check_free_memory(3000 * parallel_count) + + def teardown_method(self): + gc.collect() + + def test_coo_todense(self): + # Check *_todense routines (cf. gh-2179) + # + # All of them in the end call coo_matrix.todense + + n = self.n + + i = np.array([0, n-1]) + j = np.array([0, n-1]) + data = np.array([1, 2], dtype=np.int8) + m = coo_matrix((data, (i, j))) + + r = m.todense() + assert_equal(r[0,0], 1) + assert_equal(r[-1,-1], 2) + del r + gc.collect() + + @pytest.mark.slow + def test_matvecs(self): + # Check *_matvecs routines + n = self.n + + i = np.array([0, n-1]) + j = np.array([0, n-1]) + data = np.array([1, 2], dtype=np.int8) + m = coo_matrix((data, (i, j))) + + b = np.ones((n, n), dtype=np.int8) + for sptype in (csr_matrix, csc_matrix, bsr_matrix): + m2 = sptype(m) + r = m2.dot(b) + assert_equal(r[0,0], 1) + assert_equal(r[-1,-1], 2) + del r + gc.collect() + + del b + gc.collect() + + @pytest.mark.slow + def test_dia_matvec(self): + # Check: huge dia_matrix _matvec + n = self.n + data = np.ones((n, n), dtype=np.int8) + offsets = np.arange(n) + m = dia_matrix((data, offsets), shape=(n, n)) + v = np.ones(m.shape[1], dtype=np.int8) + r = m.dot(v) + assert_equal(r[0], int_to_int8(n)) + del data, offsets, m, v, r + gc.collect() + + _bsr_ops = [pytest.param("matmat", marks=pytest.mark.xslow), + pytest.param("matvecs", marks=pytest.mark.xslow), + "matvec", + "diagonal", + "sort_indices", + pytest.param("transpose", marks=pytest.mark.xslow)] + + @pytest.mark.slow + @pytest.mark.parametrize("op", _bsr_ops) + def test_bsr_1_block(self, op): + # Check: huge bsr_matrix (1-block) + # + # The point here is that indices inside a block may overflow. + + def get_matrix(): + n = self.n + data = np.ones((1, n, n), dtype=np.int8) + indptr = np.array([0, 1], dtype=np.int32) + indices = np.array([0], dtype=np.int32) + m = bsr_matrix((data, indices, indptr), blocksize=(n, n), copy=False) + del data, indptr, indices + return m + + gc.collect() + try: + getattr(self, "_check_bsr_" + op)(get_matrix) + finally: + gc.collect() + + @pytest.mark.slow + @pytest.mark.parametrize("op", _bsr_ops) + def test_bsr_n_block(self, op): + # Check: huge bsr_matrix (n-block) + # + # The point here is that while indices within a block don't + # overflow, accumulators across many block may. + + def get_matrix(): + n = self.n + data = np.ones((n, n, 1), dtype=np.int8) + indptr = np.array([0, n], dtype=np.int32) + indices = np.arange(n, dtype=np.int32) + m = bsr_matrix((data, indices, indptr), blocksize=(n, 1), copy=False) + del data, indptr, indices + return m + + gc.collect() + try: + getattr(self, "_check_bsr_" + op)(get_matrix) + finally: + gc.collect() + + def _check_bsr_matvecs(self, m): # skip name check + m = m() + n = self.n + + # _matvecs + r = m.dot(np.ones((n, 2), dtype=np.int8)) + assert_equal(r[0, 0], int_to_int8(n)) + + def _check_bsr_matvec(self, m): # skip name check + m = m() + n = self.n + + # _matvec + r = m.dot(np.ones((n,), dtype=np.int8)) + assert_equal(r[0], int_to_int8(n)) + + def _check_bsr_diagonal(self, m): # skip name check + m = m() + n = self.n + + # _diagonal + r = m.diagonal() + assert_equal(r, np.ones(n)) + + def _check_bsr_sort_indices(self, m): # skip name check + # _sort_indices + m = m() + m.sort_indices() + + def _check_bsr_transpose(self, m): # skip name check + # _transpose + m = m() + m.transpose() + + def _check_bsr_matmat(self, m): # skip name check + m = m() + n = self.n + + # _bsr_matmat + m2 = bsr_matrix(np.ones((n, 2), dtype=np.int8), blocksize=(m.blocksize[1], 2)) + m.dot(m2) # shouldn't SIGSEGV + del m2 + + # _bsr_matmat + m2 = bsr_matrix(np.ones((2, n), dtype=np.int8), blocksize=(2, m.blocksize[0])) + m2.dot(m) # shouldn't SIGSEGV + + +@pytest.mark.skip(reason="64-bit indices in sparse matrices not available") +def test_csr_matmat_int64_overflow(): + n = 3037000500 + assert n**2 > np.iinfo(np.int64).max + + # the test would take crazy amounts of memory + check_free_memory(n * (8*2 + 1) * 3 / 1e6) + + # int64 overflow + data = np.ones((n,), dtype=np.int8) + indptr = np.arange(n+1, dtype=np.int64) + indices = np.zeros(n, dtype=np.int64) + a = csr_matrix((data, indices, indptr)) + b = a.T + + assert_raises(RuntimeError, a.dot, b) + + +def test_upcast(): + a0 = csr_matrix([[np.pi, np.pi*1j], [3, 4]], dtype=complex) + b0 = np.array([256+1j, 2**32], dtype=complex) + + for a_dtype in supported_dtypes: + for b_dtype in supported_dtypes: + msg = f"({a_dtype!r}, {b_dtype!r})" + + if np.issubdtype(a_dtype, np.complexfloating): + a = a0.copy().astype(a_dtype) + else: + a = a0.real.copy().astype(a_dtype) + + if np.issubdtype(b_dtype, np.complexfloating): + b = b0.copy().astype(b_dtype) + else: + with np.errstate(invalid="ignore"): + # Casting a large value (2**32) to int8 causes a warning in + # numpy >1.23 + b = b0.real.copy().astype(b_dtype) + + if not (a_dtype == np.bool_ and b_dtype == np.bool_): + c = np.zeros((2,), dtype=np.bool_) + assert_raises(ValueError, _sparsetools.csr_matvec, + 2, 2, a.indptr, a.indices, a.data, b, c) + + if ((np.issubdtype(a_dtype, np.complexfloating) and + not np.issubdtype(b_dtype, np.complexfloating)) or + (not np.issubdtype(a_dtype, np.complexfloating) and + np.issubdtype(b_dtype, np.complexfloating))): + c = np.zeros((2,), dtype=np.float64) + assert_raises(ValueError, _sparsetools.csr_matvec, + 2, 2, a.indptr, a.indices, a.data, b, c) + + c = np.zeros((2,), dtype=np.result_type(a_dtype, b_dtype)) + _sparsetools.csr_matvec(2, 2, a.indptr, a.indices, a.data, b, c) + assert_allclose(c, np.dot(a.toarray(), b), err_msg=msg) + + +def test_endianness(): + d = np.ones((3,4)) + offsets = [-1,0,1] + + a = dia_matrix((d.astype('f8'), offsets), (4, 4)) + v = np.arange(4) + + assert_allclose(a.dot(v), [1, 3, 6, 5]) + assert_allclose(b.dot(v), [1, 3, 6, 5]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_spfuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_spfuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..75bc2d92c369be5799a904bc0938617f30321f12 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_spfuncs.py @@ -0,0 +1,97 @@ +from numpy import array, kron, diag +from numpy.testing import assert_, assert_equal + +from scipy.sparse import _spfuncs as spfuncs +from scipy.sparse import csr_matrix, csc_matrix, bsr_matrix +from scipy.sparse._sparsetools import (csr_scale_rows, csr_scale_columns, + bsr_scale_rows, bsr_scale_columns) + + +class TestSparseFunctions: + def test_scale_rows_and_cols(self): + D = array([[1, 0, 0, 2, 3], + [0, 4, 0, 5, 0], + [0, 0, 6, 7, 0]]) + + #TODO expose through function + S = csr_matrix(D) + v = array([1,2,3]) + csr_scale_rows(3,5,S.indptr,S.indices,S.data,v) + assert_equal(S.toarray(), diag(v)@D) + + S = csr_matrix(D) + v = array([1,2,3,4,5]) + csr_scale_columns(3,5,S.indptr,S.indices,S.data,v) + assert_equal(S.toarray(), D@diag(v)) + + # blocks + E = kron(D,[[1,2],[3,4]]) + S = bsr_matrix(E,blocksize=(2,2)) + v = array([1,2,3,4,5,6]) + bsr_scale_rows(3,5,2,2,S.indptr,S.indices,S.data,v) + assert_equal(S.toarray(), diag(v)@E) + + S = bsr_matrix(E,blocksize=(2,2)) + v = array([1,2,3,4,5,6,7,8,9,10]) + bsr_scale_columns(3,5,2,2,S.indptr,S.indices,S.data,v) + assert_equal(S.toarray(), E@diag(v)) + + E = kron(D,[[1,2,3],[4,5,6]]) + S = bsr_matrix(E,blocksize=(2,3)) + v = array([1,2,3,4,5,6]) + bsr_scale_rows(3,5,2,3,S.indptr,S.indices,S.data,v) + assert_equal(S.toarray(), diag(v)@E) + + S = bsr_matrix(E,blocksize=(2,3)) + v = array([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]) + bsr_scale_columns(3,5,2,3,S.indptr,S.indices,S.data,v) + assert_equal(S.toarray(), E@diag(v)) + + def test_estimate_blocksize(self): + mats = [] + mats.append([[0,1],[1,0]]) + mats.append([[1,1,0],[0,0,1],[1,0,1]]) + mats.append([[0],[0],[1]]) + mats = [array(x) for x in mats] + + blks = [] + blks.append([[1]]) + blks.append([[1,1],[1,1]]) + blks.append([[1,1],[0,1]]) + blks.append([[1,1,0],[1,0,1],[1,1,1]]) + blks = [array(x) for x in blks] + + for A in mats: + for B in blks: + X = kron(A,B) + r,c = spfuncs.estimate_blocksize(X) + assert_(r >= B.shape[0]) + assert_(c >= B.shape[1]) + + def test_count_blocks(self): + def gold(A,bs): + R,C = bs + I,J = A.nonzero() + return len(set(zip(I//R,J//C))) + + mats = [] + mats.append([[0]]) + mats.append([[1]]) + mats.append([[1,0]]) + mats.append([[1,1]]) + mats.append([[0,1],[1,0]]) + mats.append([[1,1,0],[0,0,1],[1,0,1]]) + mats.append([[0],[0],[1]]) + + for A in mats: + for B in mats: + X = kron(A,B) + Y = csr_matrix(X) + for R in range(1,6): + for C in range(1,6): + assert_equal(spfuncs.count_blocks(Y, (R, C)), gold(X, (R, C))) + + X = kron([[1,1,0],[0,0,1],[1,0,1]],[[1,1]]) + Y = csc_matrix(X) + assert_equal(spfuncs.count_blocks(X, (1, 2)), gold(X, (1, 2))) + assert_equal(spfuncs.count_blocks(Y, (1, 2)), gold(X, (1, 2))) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_sputils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_sputils.py new file mode 100644 index 0000000000000000000000000000000000000000..fb328e3f6512081f76604c4b92fe3d407819e448 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/sparse/tests/test_sputils.py @@ -0,0 +1,395 @@ +"""unit tests for sparse utility functions""" + +import numpy as np +from numpy.testing import assert_equal +import pytest +from pytest import raises as assert_raises +from scipy.sparse import _sputils as sputils, csr_array, bsr_array, dia_array, coo_array +from scipy.sparse._sputils import matrix + + +class TestSparseUtils: + + def test_upcast(self): + assert_equal(sputils.upcast('intc'), np.intc) + assert_equal(sputils.upcast('int32', 'float32'), np.float64) + assert_equal(sputils.upcast('bool', complex, float), np.complex128) + assert_equal(sputils.upcast('i', 'd'), np.float64) + + def test_getdtype(self): + A = np.array([1], dtype='int8') + + assert_equal(sputils.getdtype(None, default=float), float) + assert_equal(sputils.getdtype(None, a=A), np.int8) + + with assert_raises( + ValueError, + match="scipy.sparse does not support dtype object. .*", + ): + sputils.getdtype("O") + + with assert_raises( + ValueError, + match="scipy.sparse does not support dtype float16. .*", + ): + sputils.getdtype(None, default=np.float16) + + def test_isscalarlike(self): + assert_equal(sputils.isscalarlike(3.0), True) + assert_equal(sputils.isscalarlike(-4), True) + assert_equal(sputils.isscalarlike(2.5), True) + assert_equal(sputils.isscalarlike(1 + 3j), True) + assert_equal(sputils.isscalarlike(np.array(3)), True) + assert_equal(sputils.isscalarlike("16"), True) + + assert_equal(sputils.isscalarlike(np.array([3])), False) + assert_equal(sputils.isscalarlike([[3]]), False) + assert_equal(sputils.isscalarlike((1,)), False) + assert_equal(sputils.isscalarlike((1, 2)), False) + + def test_isintlike(self): + assert_equal(sputils.isintlike(-4), True) + assert_equal(sputils.isintlike(np.array(3)), True) + assert_equal(sputils.isintlike(np.array([3])), False) + with assert_raises( + ValueError, + match="Inexact indices into sparse matrices are not allowed" + ): + sputils.isintlike(3.0) + + assert_equal(sputils.isintlike(2.5), False) + assert_equal(sputils.isintlike(1 + 3j), False) + assert_equal(sputils.isintlike((1,)), False) + assert_equal(sputils.isintlike((1, 2)), False) + + def test_isshape(self): + assert_equal(sputils.isshape((1, 2)), True) + assert_equal(sputils.isshape((5, 2)), True) + + assert_equal(sputils.isshape((1.5, 2)), False) + assert_equal(sputils.isshape((2, 2, 2)), False) + assert_equal(sputils.isshape(([2], 2)), False) + assert_equal(sputils.isshape((-1, 2), nonneg=False),True) + assert_equal(sputils.isshape((2, -1), nonneg=False),True) + assert_equal(sputils.isshape((-1, 2), nonneg=True),False) + assert_equal(sputils.isshape((2, -1), nonneg=True),False) + + assert_equal(sputils.isshape((1.5, 2), allow_nd=(1, 2)), False) + assert_equal(sputils.isshape(([2], 2), allow_nd=(1, 2)), False) + assert_equal(sputils.isshape((2, 2, -2), nonneg=True, allow_nd=(1, 2)), + False) + assert_equal(sputils.isshape((2,), allow_nd=(1, 2)), True) + assert_equal(sputils.isshape((2, 2,), allow_nd=(1, 2)), True) + assert_equal(sputils.isshape((2, 2, 2), allow_nd=(1, 2)), False) + + def test_issequence(self): + assert_equal(sputils.issequence((1,)), True) + assert_equal(sputils.issequence((1, 2, 3)), True) + assert_equal(sputils.issequence([1]), True) + assert_equal(sputils.issequence([1, 2, 3]), True) + assert_equal(sputils.issequence(np.array([1, 2, 3])), True) + + assert_equal(sputils.issequence(np.array([[1], [2], [3]])), False) + assert_equal(sputils.issequence(3), False) + + def test_ismatrix(self): + assert_equal(sputils.ismatrix(((),)), True) + assert_equal(sputils.ismatrix([[1], [2]]), True) + assert_equal(sputils.ismatrix(np.arange(3)[None]), True) + + assert_equal(sputils.ismatrix([1, 2]), False) + assert_equal(sputils.ismatrix(np.arange(3)), False) + assert_equal(sputils.ismatrix([[[1]]]), False) + assert_equal(sputils.ismatrix(3), False) + + def test_isdense(self): + assert_equal(sputils.isdense(np.array([1])), True) + assert_equal(sputils.isdense(matrix([1])), True) + + def test_validateaxis(self): + assert_raises(TypeError, sputils.validateaxis, (0, 1)) + assert_raises(TypeError, sputils.validateaxis, 1.5) + assert_raises(ValueError, sputils.validateaxis, 3) + + # These function calls should not raise errors + for axis in (-2, -1, 0, 1, None): + sputils.validateaxis(axis) + + @pytest.mark.parametrize("container", [csr_array, bsr_array]) + def test_safely_cast_index_compressed(self, container): + # This is slow to test completely as nnz > imax is big + # and indptr is big for some shapes + # So we don't test large nnz, nor csc_array (same code as csr_array) + imax = np.int64(np.iinfo(np.int32).max) + + # Shape 32bit + A32 = container((1, imax)) + # indices big type, small values + B32 = A32.copy() + B32.indices = B32.indices.astype(np.int64) + B32.indptr = B32.indptr.astype(np.int64) + + # Shape 64bit + # indices big type, small values + A64 = csr_array((1, imax + 1)) + # indices small type, small values + B64 = A64.copy() + B64.indices = B64.indices.astype(np.int32) + B64.indptr = B64.indptr.astype(np.int32) + # indices big type, big values + C64 = A64.copy() + C64.indices = np.array([imax + 1], dtype=np.int64) + C64.indptr = np.array([0, 1], dtype=np.int64) + C64.data = np.array([2.2]) + + assert (A32.indices.dtype, A32.indptr.dtype) == (np.int32, np.int32) + assert (B32.indices.dtype, B32.indptr.dtype) == (np.int64, np.int64) + assert (A64.indices.dtype, A64.indptr.dtype) == (np.int64, np.int64) + assert (B64.indices.dtype, B64.indptr.dtype) == (np.int32, np.int32) + assert (C64.indices.dtype, C64.indptr.dtype) == (np.int64, np.int64) + + for A in [A32, B32, A64, B64]: + indices, indptr = sputils.safely_cast_index_arrays(A, np.int32) + assert (indices.dtype, indptr.dtype) == (np.int32, np.int32) + indices, indptr = sputils.safely_cast_index_arrays(A, np.int64) + assert (indices.dtype, indptr.dtype) == (np.int64, np.int64) + + indices, indptr = sputils.safely_cast_index_arrays(A, A.indices.dtype) + assert indices is A.indices + assert indptr is A.indptr + + with assert_raises(ValueError): + sputils.safely_cast_index_arrays(C64, np.int32) + indices, indptr = sputils.safely_cast_index_arrays(C64, np.int64) + assert indices is C64.indices + assert indptr is C64.indptr + + def test_safely_cast_index_coo(self): + # This is slow to test completely as nnz > imax is big + # So we don't test large nnz + imax = np.int64(np.iinfo(np.int32).max) + + # Shape 32bit + A32 = coo_array((1, imax)) + # coords big type, small values + B32 = A32.copy() + B32.coords = tuple(co.astype(np.int64) for co in B32.coords) + + # Shape 64bit + # coords big type, small values + A64 = coo_array((1, imax + 1)) + # coords small type, small values + B64 = A64.copy() + B64.coords = tuple(co.astype(np.int32) for co in B64.coords) + # coords big type, big values + C64 = A64.copy() + C64.coords = (np.array([imax + 1]), np.array([0])) + C64.data = np.array([2.2]) + + assert A32.coords[0].dtype == np.int32 + assert B32.coords[0].dtype == np.int64 + assert A64.coords[0].dtype == np.int64 + assert B64.coords[0].dtype == np.int32 + assert C64.coords[0].dtype == np.int64 + + for A in [A32, B32, A64, B64]: + coords = sputils.safely_cast_index_arrays(A, np.int32) + assert coords[0].dtype == np.int32 + coords = sputils.safely_cast_index_arrays(A, np.int64) + assert coords[0].dtype == np.int64 + + coords = sputils.safely_cast_index_arrays(A, A.coords[0].dtype) + assert coords[0] is A.coords[0] + + with assert_raises(ValueError): + sputils.safely_cast_index_arrays(C64, np.int32) + coords = sputils.safely_cast_index_arrays(C64, np.int64) + assert coords[0] is C64.coords[0] + + def test_safely_cast_index_dia(self): + # This is slow to test completely as nnz > imax is big + # So we don't test large nnz + imax = np.int64(np.iinfo(np.int32).max) + + # Shape 32bit + A32 = dia_array((1, imax)) + # offsets big type, small values + B32 = A32.copy() + B32.offsets = B32.offsets.astype(np.int64) + + # Shape 64bit + # offsets big type, small values + A64 = dia_array((1, imax + 2)) + # offsets small type, small values + B64 = A64.copy() + B64.offsets = B64.offsets.astype(np.int32) + # offsets big type, big values + C64 = A64.copy() + C64.offsets = np.array([imax + 1]) + C64.data = np.array([2.2]) + + assert A32.offsets.dtype == np.int32 + assert B32.offsets.dtype == np.int64 + assert A64.offsets.dtype == np.int64 + assert B64.offsets.dtype == np.int32 + assert C64.offsets.dtype == np.int64 + + for A in [A32, B32, A64, B64]: + offsets = sputils.safely_cast_index_arrays(A, np.int32) + assert offsets.dtype == np.int32 + offsets = sputils.safely_cast_index_arrays(A, np.int64) + assert offsets.dtype == np.int64 + + offsets = sputils.safely_cast_index_arrays(A, A.offsets.dtype) + assert offsets is A.offsets + + with assert_raises(ValueError): + sputils.safely_cast_index_arrays(C64, np.int32) + offsets = sputils.safely_cast_index_arrays(C64, np.int64) + assert offsets is C64.offsets + + def test_get_index_dtype(self): + imax = np.int64(np.iinfo(np.int32).max) + too_big = imax + 1 + + # Check that uint32's with no values too large doesn't return + # int64 + a1 = np.ones(90, dtype='uint32') + a2 = np.ones(90, dtype='uint32') + assert_equal( + np.dtype(sputils.get_index_dtype((a1, a2), check_contents=True)), + np.dtype('int32') + ) + + # Check that if we can not convert but all values are less than or + # equal to max that we can just convert to int32 + a1[-1] = imax + assert_equal( + np.dtype(sputils.get_index_dtype((a1, a2), check_contents=True)), + np.dtype('int32') + ) + + # Check that if it can not convert directly and the contents are + # too large that we return int64 + a1[-1] = too_big + assert_equal( + np.dtype(sputils.get_index_dtype((a1, a2), check_contents=True)), + np.dtype('int64') + ) + + # test that if can not convert and didn't specify to check_contents + # we return int64 + a1 = np.ones(89, dtype='uint32') + a2 = np.ones(89, dtype='uint32') + assert_equal( + np.dtype(sputils.get_index_dtype((a1, a2))), + np.dtype('int64') + ) + + # Check that even if we have arrays that can be converted directly + # that if we specify a maxval directly it takes precedence + a1 = np.ones(12, dtype='uint32') + a2 = np.ones(12, dtype='uint32') + assert_equal( + np.dtype(sputils.get_index_dtype( + (a1, a2), maxval=too_big, check_contents=True + )), + np.dtype('int64') + ) + + # Check that an array with a too max size and maxval set + # still returns int64 + a1[-1] = too_big + assert_equal( + np.dtype(sputils.get_index_dtype((a1, a2), maxval=too_big)), + np.dtype('int64') + ) + + # tests public broadcast_shapes largely from + # numpy/numpy/lib/tests/test_stride_tricks.py + # first 3 cause np.broadcast to raise index too large, but not sputils + @pytest.mark.parametrize("input_shapes,target_shape", [ + [((6, 5, 1, 4, 1, 1), (1, 2**32), (2**32, 1)), (6, 5, 1, 4, 2**32, 2**32)], + [((6, 5, 1, 4, 1, 1), (1, 2**32)), (6, 5, 1, 4, 1, 2**32)], + [((1, 2**32), (2**32, 1)), (2**32, 2**32)], + [[2, 2, 2], (2,)], + [[], ()], + [[()], ()], + [[(7,)], (7,)], + [[(1, 2), (2,)], (1, 2)], + [[(2,), (1, 2)], (1, 2)], + [[(1, 1)], (1, 1)], + [[(1, 1), (3, 4)], (3, 4)], + [[(6, 7), (5, 6, 1), (7,), (5, 1, 7)], (5, 6, 7)], + [[(5, 6, 1)], (5, 6, 1)], + [[(1, 3), (3, 1)], (3, 3)], + [[(1, 0), (0, 0)], (0, 0)], + [[(0, 1), (0, 0)], (0, 0)], + [[(1, 0), (0, 1)], (0, 0)], + [[(1, 1), (0, 0)], (0, 0)], + [[(1, 1), (1, 0)], (1, 0)], + [[(1, 1), (0, 1)], (0, 1)], + [[(), (0,)], (0,)], + [[(0,), (0, 0)], (0, 0)], + [[(0,), (0, 1)], (0, 0)], + [[(1,), (0, 0)], (0, 0)], + [[(), (0, 0)], (0, 0)], + [[(1, 1), (0,)], (1, 0)], + [[(1,), (0, 1)], (0, 1)], + [[(1,), (1, 0)], (1, 0)], + [[(), (1, 0)], (1, 0)], + [[(), (0, 1)], (0, 1)], + [[(1,), (3,)], (3,)], + [[2, (3, 2)], (3, 2)], + [[(1, 2)] * 32, (1, 2)], + [[(1, 2)] * 100, (1, 2)], + [[(2,)] * 32, (2,)], + ]) + def test_broadcast_shapes_successes(self, input_shapes, target_shape): + assert_equal(sputils.broadcast_shapes(*input_shapes), target_shape) + + # tests public broadcast_shapes failures + @pytest.mark.parametrize("input_shapes", [ + [(3,), (4,)], + [(2, 3), (2,)], + [2, (2, 3)], + [(3,), (3,), (4,)], + [(2, 5), (3, 5)], + [(2, 4), (2, 5)], + [(1, 3, 4), (2, 3, 3)], + [(1, 2), (3, 1), (3, 2), (10, 5)], + [(2,)] * 32 + [(3,)] * 32, + ]) + def test_broadcast_shapes_failures(self, input_shapes): + with assert_raises(ValueError, match="cannot be broadcast"): + sputils.broadcast_shapes(*input_shapes) + + def test_check_shape_overflow(self): + new_shape = sputils.check_shape([(10, -1)], (65535, 131070)) + assert_equal(new_shape, (10, 858967245)) + + def test_matrix(self): + a = [[1, 2, 3]] + b = np.array(a) + + assert isinstance(sputils.matrix(a), np.matrix) + assert isinstance(sputils.matrix(b), np.matrix) + + c = sputils.matrix(b) + c[:, :] = 123 + assert_equal(b, a) + + c = sputils.matrix(b, copy=False) + c[:, :] = 123 + assert_equal(b, [[123, 123, 123]]) + + def test_asmatrix(self): + a = [[1, 2, 3]] + b = np.array(a) + + assert isinstance(sputils.asmatrix(a), np.matrix) + assert isinstance(sputils.asmatrix(b), np.matrix) + + c = sputils.asmatrix(b) + c[:, :] = 123 + assert_equal(b, [[123, 123, 123]]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..ef478f85bc95b5f5f62b3e72afe8358d88c00c7b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/__init__.py @@ -0,0 +1,129 @@ +""" +============================================================= +Spatial algorithms and data structures (:mod:`scipy.spatial`) +============================================================= + +.. currentmodule:: scipy.spatial + +.. toctree:: + :hidden: + + spatial.distance + +Spatial transformations +======================= + +These are contained in the `scipy.spatial.transform` submodule. + +Nearest-neighbor queries +======================== +.. autosummary:: + :toctree: generated/ + + KDTree -- class for efficient nearest-neighbor queries + cKDTree -- class for efficient nearest-neighbor queries (faster implementation) + Rectangle + +Distance metrics +================ + +Distance metrics are contained in the :mod:`scipy.spatial.distance` submodule. + +Delaunay triangulation, convex hulls, and Voronoi diagrams +========================================================== + +.. autosummary:: + :toctree: generated/ + + Delaunay -- compute Delaunay triangulation of input points + ConvexHull -- compute a convex hull for input points + Voronoi -- compute a Voronoi diagram hull from input points + SphericalVoronoi -- compute a Voronoi diagram from input points on the surface of a sphere + HalfspaceIntersection -- compute the intersection points of input halfspaces + +Plotting helpers +================ + +.. autosummary:: + :toctree: generated/ + + delaunay_plot_2d -- plot 2-D triangulation + convex_hull_plot_2d -- plot 2-D convex hull + voronoi_plot_2d -- plot 2-D Voronoi diagram + +.. seealso:: :ref:`Tutorial ` + + +Simplex representation +====================== +The simplices (triangles, tetrahedra, etc.) appearing in the Delaunay +tessellation (N-D simplices), convex hull facets, and Voronoi ridges +(N-1-D simplices) are represented in the following scheme:: + + tess = Delaunay(points) + hull = ConvexHull(points) + voro = Voronoi(points) + + # coordinates of the jth vertex of the ith simplex + tess.points[tess.simplices[i, j], :] # tessellation element + hull.points[hull.simplices[i, j], :] # convex hull facet + voro.vertices[voro.ridge_vertices[i, j], :] # ridge between Voronoi cells + +For Delaunay triangulations and convex hulls, the neighborhood +structure of the simplices satisfies the condition: +``tess.neighbors[i,j]`` is the neighboring simplex of the ith +simplex, opposite to the ``j``-vertex. It is -1 in case of no neighbor. + +Convex hull facets also define a hyperplane equation:: + + (hull.equations[i,:-1] * coord).sum() + hull.equations[i,-1] == 0 + +Similar hyperplane equations for the Delaunay triangulation correspond +to the convex hull facets on the corresponding N+1-D +paraboloid. + +The Delaunay triangulation objects offer a method for locating the +simplex containing a given point, and barycentric coordinate +computations. + +Functions +--------- + +.. autosummary:: + :toctree: generated/ + + tsearch + distance_matrix + minkowski_distance + minkowski_distance_p + procrustes + geometric_slerp + +Warnings / Errors used in :mod:`scipy.spatial` +---------------------------------------------- +.. autosummary:: + :toctree: generated/ + + QhullError +""" # noqa: E501 + +from ._kdtree import * +from ._ckdtree import * # type: ignore[import-not-found] +from ._qhull import * +from ._spherical_voronoi import SphericalVoronoi +from ._plotutils import * +from ._procrustes import procrustes +from ._geometric_slerp import geometric_slerp + +# Deprecated namespaces, to be removed in v2.0.0 +from . import ckdtree, kdtree, qhull + +__all__ = [s for s in dir() if not s.startswith('_')] + +from . import distance, transform + +__all__ += ['distance', 'transform'] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_geometric_slerp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_geometric_slerp.py new file mode 100644 index 0000000000000000000000000000000000000000..189381cb06fbe2992123e0facae18101ee7f1927 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_geometric_slerp.py @@ -0,0 +1,238 @@ +__all__ = ['geometric_slerp'] + +import warnings +from typing import TYPE_CHECKING + +import numpy as np +from scipy.spatial.distance import euclidean + +if TYPE_CHECKING: + import numpy.typing as npt + + +def _geometric_slerp(start, end, t): + # create an orthogonal basis using QR decomposition + basis = np.vstack([start, end]) + Q, R = np.linalg.qr(basis.T) + signs = 2 * (np.diag(R) >= 0) - 1 + Q = Q.T * signs.T[:, np.newaxis] + R = R.T * signs.T[:, np.newaxis] + + # calculate the angle between `start` and `end` + c = np.dot(start, end) + s = np.linalg.det(R) + omega = np.arctan2(s, c) + + # interpolate + start, end = Q + s = np.sin(t * omega) + c = np.cos(t * omega) + return start * c[:, np.newaxis] + end * s[:, np.newaxis] + + +def geometric_slerp( + start: "npt.ArrayLike", + end: "npt.ArrayLike", + t: "npt.ArrayLike", + tol: float = 1e-7, +) -> np.ndarray: + """ + Geometric spherical linear interpolation. + + The interpolation occurs along a unit-radius + great circle arc in arbitrary dimensional space. + + Parameters + ---------- + start : (n_dimensions, ) array-like + Single n-dimensional input coordinate in a 1-D array-like + object. `n` must be greater than 1. + end : (n_dimensions, ) array-like + Single n-dimensional input coordinate in a 1-D array-like + object. `n` must be greater than 1. + t : float or (n_points,) 1D array-like + A float or 1D array-like of doubles representing interpolation + parameters, with values required in the inclusive interval + between 0 and 1. A common approach is to generate the array + with ``np.linspace(0, 1, n_pts)`` for linearly spaced points. + Ascending, descending, and scrambled orders are permitted. + tol : float + The absolute tolerance for determining if the start and end + coordinates are antipodes. + + Returns + ------- + result : (t.size, D) + An array of doubles containing the interpolated + spherical path and including start and + end when 0 and 1 t are used. The + interpolated values should correspond to the + same sort order provided in the t array. The result + may be 1-dimensional if ``t`` is a float. + + Raises + ------ + ValueError + If ``start`` and ``end`` are antipodes, not on the + unit n-sphere, or for a variety of degenerate conditions. + + See Also + -------- + scipy.spatial.transform.Slerp : 3-D Slerp that works with quaternions + + Notes + ----- + The implementation is based on the mathematical formula provided in [1]_, + and the first known presentation of this algorithm, derived from study of + 4-D geometry, is credited to Glenn Davis in a footnote of the original + quaternion Slerp publication by Ken Shoemake [2]_. + + .. versionadded:: 1.5.0 + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Slerp#Geometric_Slerp + .. [2] Ken Shoemake (1985) Animating rotation with quaternion curves. + ACM SIGGRAPH Computer Graphics, 19(3): 245-254. + + Examples + -------- + Interpolate four linearly-spaced values on the circumference of + a circle spanning 90 degrees: + + >>> import numpy as np + >>> from scipy.spatial import geometric_slerp + >>> import matplotlib.pyplot as plt + >>> fig = plt.figure() + >>> ax = fig.add_subplot(111) + >>> start = np.array([1, 0]) + >>> end = np.array([0, 1]) + >>> t_vals = np.linspace(0, 1, 4) + >>> result = geometric_slerp(start, + ... end, + ... t_vals) + + The interpolated results should be at 30 degree intervals + recognizable on the unit circle: + + >>> ax.scatter(result[...,0], result[...,1], c='k') + >>> circle = plt.Circle((0, 0), 1, color='grey') + >>> ax.add_artist(circle) + >>> ax.set_aspect('equal') + >>> plt.show() + + Attempting to interpolate between antipodes on a circle is + ambiguous because there are two possible paths, and on a + sphere there are infinite possible paths on the geodesic surface. + Nonetheless, one of the ambiguous paths is returned along + with a warning: + + >>> opposite_pole = np.array([-1, 0]) + >>> with np.testing.suppress_warnings() as sup: + ... sup.filter(UserWarning) + ... geometric_slerp(start, + ... opposite_pole, + ... t_vals) + array([[ 1.00000000e+00, 0.00000000e+00], + [ 5.00000000e-01, 8.66025404e-01], + [-5.00000000e-01, 8.66025404e-01], + [-1.00000000e+00, 1.22464680e-16]]) + + Extend the original example to a sphere and plot interpolation + points in 3D: + + >>> from mpl_toolkits.mplot3d import proj3d + >>> fig = plt.figure() + >>> ax = fig.add_subplot(111, projection='3d') + + Plot the unit sphere for reference (optional): + + >>> u = np.linspace(0, 2 * np.pi, 100) + >>> v = np.linspace(0, np.pi, 100) + >>> x = np.outer(np.cos(u), np.sin(v)) + >>> y = np.outer(np.sin(u), np.sin(v)) + >>> z = np.outer(np.ones(np.size(u)), np.cos(v)) + >>> ax.plot_surface(x, y, z, color='y', alpha=0.1) + + Interpolating over a larger number of points + may provide the appearance of a smooth curve on + the surface of the sphere, which is also useful + for discretized integration calculations on a + sphere surface: + + >>> start = np.array([1, 0, 0]) + >>> end = np.array([0, 0, 1]) + >>> t_vals = np.linspace(0, 1, 200) + >>> result = geometric_slerp(start, + ... end, + ... t_vals) + >>> ax.plot(result[...,0], + ... result[...,1], + ... result[...,2], + ... c='k') + >>> plt.show() + """ + + start = np.asarray(start, dtype=np.float64) + end = np.asarray(end, dtype=np.float64) + t = np.asarray(t) + + if t.ndim > 1: + raise ValueError("The interpolation parameter " + "value must be one dimensional.") + + if start.ndim != 1 or end.ndim != 1: + raise ValueError("Start and end coordinates " + "must be one-dimensional") + + if start.size != end.size: + raise ValueError("The dimensions of start and " + "end must match (have same size)") + + if start.size < 2 or end.size < 2: + raise ValueError("The start and end coordinates must " + "both be in at least two-dimensional " + "space") + + if np.array_equal(start, end): + return np.linspace(start, start, t.size) + + # for points that violate equation for n-sphere + for coord in [start, end]: + if not np.allclose(np.linalg.norm(coord), 1.0, + rtol=1e-9, + atol=0): + raise ValueError("start and end are not" + " on a unit n-sphere") + + if not isinstance(tol, float): + raise ValueError("tol must be a float") + else: + tol = np.fabs(tol) + + coord_dist = euclidean(start, end) + + # diameter of 2 within tolerance means antipodes, which is a problem + # for all unit n-spheres (even the 0-sphere would have an ambiguous path) + if np.allclose(coord_dist, 2.0, rtol=0, atol=tol): + warnings.warn("start and end are antipodes " + "using the specified tolerance; " + "this may cause ambiguous slerp paths", + stacklevel=2) + + t = np.asarray(t, dtype=np.float64) + + if t.size == 0: + return np.empty((0, start.size)) + + if t.min() < 0 or t.max() > 1: + raise ValueError("interpolation parameter must be in [0, 1]") + + if t.ndim == 0: + return _geometric_slerp(start, + end, + np.atleast_1d(t)).ravel() + else: + return _geometric_slerp(start, + end, + t) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_kdtree.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_kdtree.py new file mode 100644 index 0000000000000000000000000000000000000000..3b389d86d3dd8047a43d6a6b763e8508bf030aa4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_kdtree.py @@ -0,0 +1,920 @@ +# Copyright Anne M. Archibald 2008 +# Released under the scipy license +import numpy as np +from ._ckdtree import cKDTree, cKDTreeNode # type: ignore[import-not-found] + +__all__ = ['minkowski_distance_p', 'minkowski_distance', + 'distance_matrix', + 'Rectangle', 'KDTree'] + + +def minkowski_distance_p(x, y, p=2): + """Compute the pth power of the L**p distance between two arrays. + + For efficiency, this function computes the L**p distance but does + not extract the pth root. If `p` is 1 or infinity, this is equal to + the actual L**p distance. + + The last dimensions of `x` and `y` must be the same length. Any + other dimensions must be compatible for broadcasting. + + Parameters + ---------- + x : (..., K) array_like + Input array. + y : (..., K) array_like + Input array. + p : float, 1 <= p <= infinity + Which Minkowski p-norm to use. + + Returns + ------- + dist : ndarray + pth power of the distance between the input arrays. + + Examples + -------- + >>> from scipy.spatial import minkowski_distance_p + >>> minkowski_distance_p([[0, 0], [0, 0]], [[1, 1], [0, 1]]) + array([2., 1.]) + + """ + x = np.asarray(x) + y = np.asarray(y) + + # Find smallest common datatype with float64 (return type of this + # function) - addresses #10262. + # Don't just cast to float64 for complex input case. + common_datatype = np.promote_types(np.promote_types(x.dtype, y.dtype), + 'float64') + + # Make sure x and y are NumPy arrays of correct datatype. + x = x.astype(common_datatype) + y = y.astype(common_datatype) + + if p == np.inf: + return np.amax(np.abs(y-x), axis=-1) + elif p == 1: + return np.sum(np.abs(y-x), axis=-1) + else: + return np.sum(np.abs(y-x)**p, axis=-1) + + +def minkowski_distance(x, y, p=2): + """Compute the L**p distance between two arrays. + + The last dimensions of `x` and `y` must be the same length. Any + other dimensions must be compatible for broadcasting. + + Parameters + ---------- + x : (..., K) array_like + Input array. + y : (..., K) array_like + Input array. + p : float, 1 <= p <= infinity + Which Minkowski p-norm to use. + + Returns + ------- + dist : ndarray + Distance between the input arrays. + + Examples + -------- + >>> from scipy.spatial import minkowski_distance + >>> minkowski_distance([[0, 0], [0, 0]], [[1, 1], [0, 1]]) + array([ 1.41421356, 1. ]) + + """ + x = np.asarray(x) + y = np.asarray(y) + if p == np.inf or p == 1: + return minkowski_distance_p(x, y, p) + else: + return minkowski_distance_p(x, y, p)**(1./p) + + +class Rectangle: + """Hyperrectangle class. + + Represents a Cartesian product of intervals. + """ + def __init__(self, maxes, mins): + """Construct a hyperrectangle.""" + self.maxes = np.maximum(maxes,mins).astype(float) + self.mins = np.minimum(maxes,mins).astype(float) + self.m, = self.maxes.shape + + def __repr__(self): + return f"" + + def volume(self): + """Total volume.""" + return np.prod(self.maxes-self.mins) + + def split(self, d, split): + """Produce two hyperrectangles by splitting. + + In general, if you need to compute maximum and minimum + distances to the children, it can be done more efficiently + by updating the maximum and minimum distances to the parent. + + Parameters + ---------- + d : int + Axis to split hyperrectangle along. + split : float + Position along axis `d` to split at. + + """ + mid = np.copy(self.maxes) + mid[d] = split + less = Rectangle(self.mins, mid) + mid = np.copy(self.mins) + mid[d] = split + greater = Rectangle(mid, self.maxes) + return less, greater + + def min_distance_point(self, x, p=2.): + """ + Return the minimum distance between input and points in the + hyperrectangle. + + Parameters + ---------- + x : array_like + Input. + p : float, optional + Input. + + """ + return minkowski_distance( + 0, np.maximum(0, np.maximum(self.mins-x, x-self.maxes)), + p + ) + + def max_distance_point(self, x, p=2.): + """ + Return the maximum distance between input and points in the hyperrectangle. + + Parameters + ---------- + x : array_like + Input array. + p : float, optional + Input. + + """ + return minkowski_distance(0, np.maximum(self.maxes-x, x-self.mins), p) + + def min_distance_rectangle(self, other, p=2.): + """ + Compute the minimum distance between points in the two hyperrectangles. + + Parameters + ---------- + other : hyperrectangle + Input. + p : float + Input. + + """ + return minkowski_distance( + 0, + np.maximum(0, np.maximum(self.mins-other.maxes, + other.mins-self.maxes)), + p + ) + + def max_distance_rectangle(self, other, p=2.): + """ + Compute the maximum distance between points in the two hyperrectangles. + + Parameters + ---------- + other : hyperrectangle + Input. + p : float, optional + Input. + + """ + return minkowski_distance( + 0, np.maximum(self.maxes-other.mins, other.maxes-self.mins), p) + + +class KDTree(cKDTree): + """kd-tree for quick nearest-neighbor lookup. + + This class provides an index into a set of k-dimensional points + which can be used to rapidly look up the nearest neighbors of any + point. + + Parameters + ---------- + data : array_like, shape (n,m) + The n data points of dimension m to be indexed. This array is + not copied unless this is necessary to produce a contiguous + array of doubles, and so modifying this data will result in + bogus results. The data are also copied if the kd-tree is built + with copy_data=True. + leafsize : positive int, optional + The number of points at which the algorithm switches over to + brute-force. Default: 10. + compact_nodes : bool, optional + If True, the kd-tree is built to shrink the hyperrectangles to + the actual data range. This usually gives a more compact tree that + is robust against degenerated input data and gives faster queries + at the expense of longer build time. Default: True. + copy_data : bool, optional + If True the data is always copied to protect the kd-tree against + data corruption. Default: False. + balanced_tree : bool, optional + If True, the median is used to split the hyperrectangles instead of + the midpoint. This usually gives a more compact tree and + faster queries at the expense of longer build time. Default: True. + boxsize : array_like or scalar, optional + Apply a m-d toroidal topology to the KDTree.. The topology is generated + by :math:`x_i + n_i L_i` where :math:`n_i` are integers and :math:`L_i` + is the boxsize along i-th dimension. The input data shall be wrapped + into :math:`[0, L_i)`. A ValueError is raised if any of the data is + outside of this bound. + + Notes + ----- + The algorithm used is described in Maneewongvatana and Mount 1999. + The general idea is that the kd-tree is a binary tree, each of whose + nodes represents an axis-aligned hyperrectangle. Each node specifies + an axis and splits the set of points based on whether their coordinate + along that axis is greater than or less than a particular value. + + During construction, the axis and splitting point are chosen by the + "sliding midpoint" rule, which ensures that the cells do not all + become long and thin. + + The tree can be queried for the r closest neighbors of any given point + (optionally returning only those within some maximum distance of the + point). It can also be queried, with a substantial gain in efficiency, + for the r approximate closest neighbors. + + For large dimensions (20 is already large) do not expect this to run + significantly faster than brute force. High-dimensional nearest-neighbor + queries are a substantial open problem in computer science. + + Attributes + ---------- + data : ndarray, shape (n,m) + The n data points of dimension m to be indexed. This array is + not copied unless this is necessary to produce a contiguous + array of doubles. The data are also copied if the kd-tree is built + with ``copy_data=True``. + leafsize : positive int + The number of points at which the algorithm switches over to + brute-force. + m : int + The dimension of a single data-point. + n : int + The number of data points. + maxes : ndarray, shape (m,) + The maximum value in each dimension of the n data points. + mins : ndarray, shape (m,) + The minimum value in each dimension of the n data points. + size : int + The number of nodes in the tree. + + """ + + class node: + @staticmethod + def _create(ckdtree_node=None): + """Create either an inner or leaf node, wrapping a cKDTreeNode instance""" + if ckdtree_node is None: + return KDTree.node(ckdtree_node) + elif ckdtree_node.split_dim == -1: + return KDTree.leafnode(ckdtree_node) + else: + return KDTree.innernode(ckdtree_node) + + def __init__(self, ckdtree_node=None): + if ckdtree_node is None: + ckdtree_node = cKDTreeNode() + self._node = ckdtree_node + + def __lt__(self, other): + return id(self) < id(other) + + def __gt__(self, other): + return id(self) > id(other) + + def __le__(self, other): + return id(self) <= id(other) + + def __ge__(self, other): + return id(self) >= id(other) + + def __eq__(self, other): + return id(self) == id(other) + + class leafnode(node): + @property + def idx(self): + return self._node.indices + + @property + def children(self): + return self._node.children + + class innernode(node): + def __init__(self, ckdtreenode): + assert isinstance(ckdtreenode, cKDTreeNode) + super().__init__(ckdtreenode) + self.less = KDTree.node._create(ckdtreenode.lesser) + self.greater = KDTree.node._create(ckdtreenode.greater) + + @property + def split_dim(self): + return self._node.split_dim + + @property + def split(self): + return self._node.split + + @property + def children(self): + return self._node.children + + @property + def tree(self): + if not hasattr(self, "_tree"): + self._tree = KDTree.node._create(super().tree) + + return self._tree + + def __init__(self, data, leafsize=10, compact_nodes=True, copy_data=False, + balanced_tree=True, boxsize=None): + data = np.asarray(data) + if data.dtype.kind == 'c': + raise TypeError("KDTree does not work with complex data") + + # Note KDTree has different default leafsize from cKDTree + super().__init__(data, leafsize, compact_nodes, copy_data, + balanced_tree, boxsize) + + def query( + self, x, k=1, eps=0, p=2, distance_upper_bound=np.inf, workers=1): + r"""Query the kd-tree for nearest neighbors. + + Parameters + ---------- + x : array_like, last dimension self.m + An array of points to query. + k : int or Sequence[int], optional + Either the number of nearest neighbors to return, or a list of the + k-th nearest neighbors to return, starting from 1. + eps : nonnegative float, optional + Return approximate nearest neighbors; the kth returned value + is guaranteed to be no further than (1+eps) times the + distance to the real kth nearest neighbor. + p : float, 1<=p<=infinity, optional + Which Minkowski p-norm to use. + 1 is the sum-of-absolute-values distance ("Manhattan" distance). + 2 is the usual Euclidean distance. + infinity is the maximum-coordinate-difference distance. + A large, finite p may cause a ValueError if overflow can occur. + distance_upper_bound : nonnegative float, optional + Return only neighbors within this distance. This is used to prune + tree searches, so if you are doing a series of nearest-neighbor + queries, it may help to supply the distance to the nearest neighbor + of the most recent point. + workers : int, optional + Number of workers to use for parallel processing. If -1 is given + all CPU threads are used. Default: 1. + + .. versionadded:: 1.6.0 + + Returns + ------- + d : float or array of floats + The distances to the nearest neighbors. + If ``x`` has shape ``tuple+(self.m,)``, then ``d`` has shape + ``tuple+(k,)``. + When k == 1, the last dimension of the output is squeezed. + Missing neighbors are indicated with infinite distances. + Hits are sorted by distance (nearest first). + + .. versionchanged:: 1.9.0 + Previously if ``k=None``, then `d` was an object array of + shape ``tuple``, containing lists of distances. This behavior + has been removed, use `query_ball_point` instead. + + i : integer or array of integers + The index of each neighbor in ``self.data``. + ``i`` is the same shape as d. + Missing neighbors are indicated with ``self.n``. + + Examples + -------- + + >>> import numpy as np + >>> from scipy.spatial import KDTree + >>> x, y = np.mgrid[0:5, 2:8] + >>> tree = KDTree(np.c_[x.ravel(), y.ravel()]) + + To query the nearest neighbours and return squeezed result, use + + >>> dd, ii = tree.query([[0, 0], [2.2, 2.9]], k=1) + >>> print(dd, ii, sep='\n') + [2. 0.2236068] + [ 0 13] + + To query the nearest neighbours and return unsqueezed result, use + + >>> dd, ii = tree.query([[0, 0], [2.2, 2.9]], k=[1]) + >>> print(dd, ii, sep='\n') + [[2. ] + [0.2236068]] + [[ 0] + [13]] + + To query the second nearest neighbours and return unsqueezed result, + use + + >>> dd, ii = tree.query([[0, 0], [2.2, 2.9]], k=[2]) + >>> print(dd, ii, sep='\n') + [[2.23606798] + [0.80622577]] + [[ 6] + [19]] + + To query the first and second nearest neighbours, use + + >>> dd, ii = tree.query([[0, 0], [2.2, 2.9]], k=2) + >>> print(dd, ii, sep='\n') + [[2. 2.23606798] + [0.2236068 0.80622577]] + [[ 0 6] + [13 19]] + + or, be more specific + + >>> dd, ii = tree.query([[0, 0], [2.2, 2.9]], k=[1, 2]) + >>> print(dd, ii, sep='\n') + [[2. 2.23606798] + [0.2236068 0.80622577]] + [[ 0 6] + [13 19]] + + """ + x = np.asarray(x) + if x.dtype.kind == 'c': + raise TypeError("KDTree does not work with complex data") + + if k is None: + raise ValueError("k must be an integer or a sequence of integers") + + d, i = super().query(x, k, eps, p, distance_upper_bound, workers) + if isinstance(i, int): + i = np.intp(i) + return d, i + + def query_ball_point(self, x, r, p=2., eps=0, workers=1, + return_sorted=None, return_length=False): + """Find all points within distance r of point(s) x. + + Parameters + ---------- + x : array_like, shape tuple + (self.m,) + The point or points to search for neighbors of. + r : array_like, float + The radius of points to return, must broadcast to the length of x. + p : float, optional + Which Minkowski p-norm to use. Should be in the range [1, inf]. + A finite large p may cause a ValueError if overflow can occur. + eps : nonnegative float, optional + Approximate search. Branches of the tree are not explored if their + nearest points are further than ``r / (1 + eps)``, and branches are + added in bulk if their furthest points are nearer than + ``r * (1 + eps)``. + workers : int, optional + Number of jobs to schedule for parallel processing. If -1 is given + all processors are used. Default: 1. + + .. versionadded:: 1.6.0 + return_sorted : bool, optional + Sorts returned indices if True and does not sort them if False. If + None, does not sort single point queries, but does sort + multi-point queries which was the behavior before this option + was added. + + .. versionadded:: 1.6.0 + return_length : bool, optional + Return the number of points inside the radius instead of a list + of the indices. + + .. versionadded:: 1.6.0 + + Returns + ------- + results : list or array of lists + If `x` is a single point, returns a list of the indices of the + neighbors of `x`. If `x` is an array of points, returns an object + array of shape tuple containing lists of neighbors. + + Notes + ----- + If you have many points whose neighbors you want to find, you may save + substantial amounts of time by putting them in a KDTree and using + query_ball_tree. + + Examples + -------- + >>> import numpy as np + >>> from scipy import spatial + >>> x, y = np.mgrid[0:5, 0:5] + >>> points = np.c_[x.ravel(), y.ravel()] + >>> tree = spatial.KDTree(points) + >>> sorted(tree.query_ball_point([2, 0], 1)) + [5, 10, 11, 15] + + Query multiple points and plot the results: + + >>> import matplotlib.pyplot as plt + >>> points = np.asarray(points) + >>> plt.plot(points[:,0], points[:,1], '.') + >>> for results in tree.query_ball_point(([2, 0], [3, 3]), 1): + ... nearby_points = points[results] + ... plt.plot(nearby_points[:,0], nearby_points[:,1], 'o') + >>> plt.margins(0.1, 0.1) + >>> plt.show() + + """ + x = np.asarray(x) + if x.dtype.kind == 'c': + raise TypeError("KDTree does not work with complex data") + return super().query_ball_point( + x, r, p, eps, workers, return_sorted, return_length) + + def query_ball_tree(self, other, r, p=2., eps=0): + """ + Find all pairs of points between `self` and `other` whose distance is + at most r. + + Parameters + ---------- + other : KDTree instance + The tree containing points to search against. + r : float + The maximum distance, has to be positive. + p : float, optional + Which Minkowski norm to use. `p` has to meet the condition + ``1 <= p <= infinity``. + eps : float, optional + Approximate search. Branches of the tree are not explored + if their nearest points are further than ``r/(1+eps)``, and + branches are added in bulk if their furthest points are nearer + than ``r * (1+eps)``. `eps` has to be non-negative. + + Returns + ------- + results : list of lists + For each element ``self.data[i]`` of this tree, ``results[i]`` is a + list of the indices of its neighbors in ``other.data``. + + Examples + -------- + You can search all pairs of points between two kd-trees within a distance: + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> from scipy.spatial import KDTree + >>> rng = np.random.default_rng() + >>> points1 = rng.random((15, 2)) + >>> points2 = rng.random((15, 2)) + >>> plt.figure(figsize=(6, 6)) + >>> plt.plot(points1[:, 0], points1[:, 1], "xk", markersize=14) + >>> plt.plot(points2[:, 0], points2[:, 1], "og", markersize=14) + >>> kd_tree1 = KDTree(points1) + >>> kd_tree2 = KDTree(points2) + >>> indexes = kd_tree1.query_ball_tree(kd_tree2, r=0.2) + >>> for i in range(len(indexes)): + ... for j in indexes[i]: + ... plt.plot([points1[i, 0], points2[j, 0]], + ... [points1[i, 1], points2[j, 1]], "-r") + >>> plt.show() + + """ + return super().query_ball_tree(other, r, p, eps) + + def query_pairs(self, r, p=2., eps=0, output_type='set'): + """Find all pairs of points in `self` whose distance is at most r. + + Parameters + ---------- + r : positive float + The maximum distance. + p : float, optional + Which Minkowski norm to use. `p` has to meet the condition + ``1 <= p <= infinity``. + eps : float, optional + Approximate search. Branches of the tree are not explored + if their nearest points are further than ``r/(1+eps)``, and + branches are added in bulk if their furthest points are nearer + than ``r * (1+eps)``. `eps` has to be non-negative. + output_type : string, optional + Choose the output container, 'set' or 'ndarray'. Default: 'set' + + .. versionadded:: 1.6.0 + + Returns + ------- + results : set or ndarray + Set of pairs ``(i,j)``, with ``i < j``, for which the corresponding + positions are close. If output_type is 'ndarray', an ndarry is + returned instead of a set. + + Examples + -------- + You can search all pairs of points in a kd-tree within a distance: + + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> from scipy.spatial import KDTree + >>> rng = np.random.default_rng() + >>> points = rng.random((20, 2)) + >>> plt.figure(figsize=(6, 6)) + >>> plt.plot(points[:, 0], points[:, 1], "xk", markersize=14) + >>> kd_tree = KDTree(points) + >>> pairs = kd_tree.query_pairs(r=0.2) + >>> for (i, j) in pairs: + ... plt.plot([points[i, 0], points[j, 0]], + ... [points[i, 1], points[j, 1]], "-r") + >>> plt.show() + + """ + return super().query_pairs(r, p, eps, output_type) + + def count_neighbors(self, other, r, p=2., weights=None, cumulative=True): + """Count how many nearby pairs can be formed. + + Count the number of pairs ``(x1,x2)`` can be formed, with ``x1`` drawn + from ``self`` and ``x2`` drawn from ``other``, and where + ``distance(x1, x2, p) <= r``. + + Data points on ``self`` and ``other`` are optionally weighted by the + ``weights`` argument. (See below) + + This is adapted from the "two-point correlation" algorithm described by + Gray and Moore [1]_. See notes for further discussion. + + Parameters + ---------- + other : KDTree + The other tree to draw points from, can be the same tree as self. + r : float or one-dimensional array of floats + The radius to produce a count for. Multiple radii are searched with + a single tree traversal. + If the count is non-cumulative(``cumulative=False``), ``r`` defines + the edges of the bins, and must be non-decreasing. + p : float, optional + 1<=p<=infinity. + Which Minkowski p-norm to use. + Default 2.0. + A finite large p may cause a ValueError if overflow can occur. + weights : tuple, array_like, or None, optional + If None, the pair-counting is unweighted. + If given as a tuple, weights[0] is the weights of points in + ``self``, and weights[1] is the weights of points in ``other``; + either can be None to indicate the points are unweighted. + If given as an array_like, weights is the weights of points in + ``self`` and ``other``. For this to make sense, ``self`` and + ``other`` must be the same tree. If ``self`` and ``other`` are two + different trees, a ``ValueError`` is raised. + Default: None + + .. versionadded:: 1.6.0 + cumulative : bool, optional + Whether the returned counts are cumulative. When cumulative is set + to ``False`` the algorithm is optimized to work with a large number + of bins (>10) specified by ``r``. When ``cumulative`` is set to + True, the algorithm is optimized to work with a small number of + ``r``. Default: True + + .. versionadded:: 1.6.0 + + Returns + ------- + result : scalar or 1-D array + The number of pairs. For unweighted counts, the result is integer. + For weighted counts, the result is float. + If cumulative is False, ``result[i]`` contains the counts with + ``(-inf if i == 0 else r[i-1]) < R <= r[i]`` + + Notes + ----- + Pair-counting is the basic operation used to calculate the two point + correlation functions from a data set composed of position of objects. + + Two point correlation function measures the clustering of objects and + is widely used in cosmology to quantify the large scale structure + in our Universe, but it may be useful for data analysis in other fields + where self-similar assembly of objects also occur. + + The Landy-Szalay estimator for the two point correlation function of + ``D`` measures the clustering signal in ``D``. [2]_ + + For example, given the position of two sets of objects, + + - objects ``D`` (data) contains the clustering signal, and + + - objects ``R`` (random) that contains no signal, + + .. math:: + + \\xi(r) = \\frac{ - 2 f + f^2}{f^2}, + + where the brackets represents counting pairs between two data sets + in a finite bin around ``r`` (distance), corresponding to setting + `cumulative=False`, and ``f = float(len(D)) / float(len(R))`` is the + ratio between number of objects from data and random. + + The algorithm implemented here is loosely based on the dual-tree + algorithm described in [1]_. We switch between two different + pair-cumulation scheme depending on the setting of ``cumulative``. + The computing time of the method we use when for + ``cumulative == False`` does not scale with the total number of bins. + The algorithm for ``cumulative == True`` scales linearly with the + number of bins, though it is slightly faster when only + 1 or 2 bins are used. [5]_. + + As an extension to the naive pair-counting, + weighted pair-counting counts the product of weights instead + of number of pairs. + Weighted pair-counting is used to estimate marked correlation functions + ([3]_, section 2.2), + or to properly calculate the average of data per distance bin + (e.g. [4]_, section 2.1 on redshift). + + .. [1] Gray and Moore, + "N-body problems in statistical learning", + Mining the sky, 2000, + https://arxiv.org/abs/astro-ph/0012333 + + .. [2] Landy and Szalay, + "Bias and variance of angular correlation functions", + The Astrophysical Journal, 1993, + http://adsabs.harvard.edu/abs/1993ApJ...412...64L + + .. [3] Sheth, Connolly and Skibba, + "Marked correlations in galaxy formation models", + Arxiv e-print, 2005, + https://arxiv.org/abs/astro-ph/0511773 + + .. [4] Hawkins, et al., + "The 2dF Galaxy Redshift Survey: correlation functions, + peculiar velocities and the matter density of the Universe", + Monthly Notices of the Royal Astronomical Society, 2002, + http://adsabs.harvard.edu/abs/2003MNRAS.346...78H + + .. [5] https://github.com/scipy/scipy/pull/5647#issuecomment-168474926 + + Examples + -------- + You can count neighbors number between two kd-trees within a distance: + + >>> import numpy as np + >>> from scipy.spatial import KDTree + >>> rng = np.random.default_rng() + >>> points1 = rng.random((5, 2)) + >>> points2 = rng.random((5, 2)) + >>> kd_tree1 = KDTree(points1) + >>> kd_tree2 = KDTree(points2) + >>> kd_tree1.count_neighbors(kd_tree2, 0.2) + 1 + + This number is same as the total pair number calculated by + `query_ball_tree`: + + >>> indexes = kd_tree1.query_ball_tree(kd_tree2, r=0.2) + >>> sum([len(i) for i in indexes]) + 1 + + """ + return super().count_neighbors(other, r, p, weights, cumulative) + + def sparse_distance_matrix( + self, other, max_distance, p=2., output_type='dok_matrix'): + """Compute a sparse distance matrix. + + Computes a distance matrix between two KDTrees, leaving as zero + any distance greater than max_distance. + + Parameters + ---------- + other : KDTree + + max_distance : positive float + + p : float, 1<=p<=infinity + Which Minkowski p-norm to use. + A finite large p may cause a ValueError if overflow can occur. + + output_type : string, optional + Which container to use for output data. Options: 'dok_matrix', + 'coo_matrix', 'dict', or 'ndarray'. Default: 'dok_matrix'. + + .. versionadded:: 1.6.0 + + Returns + ------- + result : dok_matrix, coo_matrix, dict or ndarray + Sparse matrix representing the results in "dictionary of keys" + format. If a dict is returned the keys are (i,j) tuples of indices. + If output_type is 'ndarray' a record array with fields 'i', 'j', + and 'v' is returned, + + Examples + -------- + You can compute a sparse distance matrix between two kd-trees: + + >>> import numpy as np + >>> from scipy.spatial import KDTree + >>> rng = np.random.default_rng() + >>> points1 = rng.random((5, 2)) + >>> points2 = rng.random((5, 2)) + >>> kd_tree1 = KDTree(points1) + >>> kd_tree2 = KDTree(points2) + >>> sdm = kd_tree1.sparse_distance_matrix(kd_tree2, 0.3) + >>> sdm.toarray() + array([[0. , 0. , 0.12295571, 0. , 0. ], + [0. , 0. , 0. , 0. , 0. ], + [0.28942611, 0. , 0. , 0.2333084 , 0. ], + [0. , 0. , 0. , 0. , 0. ], + [0.24617575, 0.29571802, 0.26836782, 0. , 0. ]]) + + You can check distances above the `max_distance` are zeros: + + >>> from scipy.spatial import distance_matrix + >>> distance_matrix(points1, points2) + array([[0.56906522, 0.39923701, 0.12295571, 0.8658745 , 0.79428925], + [0.37327919, 0.7225693 , 0.87665969, 0.32580855, 0.75679479], + [0.28942611, 0.30088013, 0.6395831 , 0.2333084 , 0.33630734], + [0.31994999, 0.72658602, 0.71124834, 0.55396483, 0.90785663], + [0.24617575, 0.29571802, 0.26836782, 0.57714465, 0.6473269 ]]) + + """ + return super().sparse_distance_matrix( + other, max_distance, p, output_type) + + +def distance_matrix(x, y, p=2, threshold=1000000): + """Compute the distance matrix. + + Returns the matrix of all pair-wise distances. + + Parameters + ---------- + x : (M, K) array_like + Matrix of M vectors in K dimensions. + y : (N, K) array_like + Matrix of N vectors in K dimensions. + p : float, 1 <= p <= infinity + Which Minkowski p-norm to use. + threshold : positive int + If ``M * N * K`` > `threshold`, algorithm uses a Python loop instead + of large temporary arrays. + + Returns + ------- + result : (M, N) ndarray + Matrix containing the distance from every vector in `x` to every vector + in `y`. + + Examples + -------- + >>> from scipy.spatial import distance_matrix + >>> distance_matrix([[0,0],[0,1]], [[1,0],[1,1]]) + array([[ 1. , 1.41421356], + [ 1.41421356, 1. ]]) + + """ + + x = np.asarray(x) + m, k = x.shape + y = np.asarray(y) + n, kk = y.shape + + if k != kk: + raise ValueError(f"x contains {k}-dimensional vectors but y contains " + f"{kk}-dimensional vectors") + + if m*n*k <= threshold: + return minkowski_distance(x[:,np.newaxis,:],y[np.newaxis,:,:],p) + else: + result = np.empty((m,n),dtype=float) # FIXME: figure out the best dtype + if m < n: + for i in range(m): + result[i,:] = minkowski_distance(x[i],y,p) + else: + for j in range(n): + result[:,j] = minkowski_distance(x,y[j],p) + return result diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_plotutils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_plotutils.py new file mode 100644 index 0000000000000000000000000000000000000000..08108980835bc82a3f39353a9e59379878bc2a71 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_plotutils.py @@ -0,0 +1,274 @@ +import numpy as np +from scipy._lib.decorator import decorator as _decorator + +__all__ = ['delaunay_plot_2d', 'convex_hull_plot_2d', 'voronoi_plot_2d'] + + +@_decorator +def _held_figure(func, obj, ax=None, **kw): + import matplotlib.pyplot as plt + + if ax is None: + fig = plt.figure() + ax = fig.gca() + return func(obj, ax=ax, **kw) + + # As of matplotlib 2.0, the "hold" mechanism is deprecated. + # When matplotlib 1.x is no longer supported, this check can be removed. + was_held = getattr(ax, 'ishold', lambda: True)() + if was_held: + return func(obj, ax=ax, **kw) + try: + ax.hold(True) + return func(obj, ax=ax, **kw) + finally: + ax.hold(was_held) + + +def _adjust_bounds(ax, points): + margin = 0.1 * np.ptp(points, axis=0) + xy_min = points.min(axis=0) - margin + xy_max = points.max(axis=0) + margin + ax.set_xlim(xy_min[0], xy_max[0]) + ax.set_ylim(xy_min[1], xy_max[1]) + + +@_held_figure +def delaunay_plot_2d(tri, ax=None): + """ + Plot the given Delaunay triangulation in 2-D + + Parameters + ---------- + tri : scipy.spatial.Delaunay instance + Triangulation to plot + ax : matplotlib.axes.Axes instance, optional + Axes to plot on + + Returns + ------- + fig : matplotlib.figure.Figure instance + Figure for the plot + + See Also + -------- + Delaunay + matplotlib.pyplot.triplot + + Notes + ----- + Requires Matplotlib. + + Examples + -------- + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.spatial import Delaunay, delaunay_plot_2d + + The Delaunay triangulation of a set of random points: + + >>> rng = np.random.default_rng() + >>> points = rng.random((30, 2)) + >>> tri = Delaunay(points) + + Plot it: + + >>> _ = delaunay_plot_2d(tri) + >>> plt.show() + + """ + if tri.points.shape[1] != 2: + raise ValueError("Delaunay triangulation is not 2-D") + + x, y = tri.points.T + ax.plot(x, y, 'o') + ax.triplot(x, y, tri.simplices.copy()) + + _adjust_bounds(ax, tri.points) + + return ax.figure + + +@_held_figure +def convex_hull_plot_2d(hull, ax=None): + """ + Plot the given convex hull diagram in 2-D + + Parameters + ---------- + hull : scipy.spatial.ConvexHull instance + Convex hull to plot + ax : matplotlib.axes.Axes instance, optional + Axes to plot on + + Returns + ------- + fig : matplotlib.figure.Figure instance + Figure for the plot + + See Also + -------- + ConvexHull + + Notes + ----- + Requires Matplotlib. + + + Examples + -------- + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.spatial import ConvexHull, convex_hull_plot_2d + + The convex hull of a random set of points: + + >>> rng = np.random.default_rng() + >>> points = rng.random((30, 2)) + >>> hull = ConvexHull(points) + + Plot it: + + >>> _ = convex_hull_plot_2d(hull) + >>> plt.show() + + """ + from matplotlib.collections import LineCollection + + if hull.points.shape[1] != 2: + raise ValueError("Convex hull is not 2-D") + + ax.plot(hull.points[:, 0], hull.points[:, 1], 'o') + line_segments = [hull.points[simplex] for simplex in hull.simplices] + ax.add_collection(LineCollection(line_segments, + colors='k', + linestyle='solid')) + _adjust_bounds(ax, hull.points) + + return ax.figure + + +@_held_figure +def voronoi_plot_2d(vor, ax=None, **kw): + """ + Plot the given Voronoi diagram in 2-D + + Parameters + ---------- + vor : scipy.spatial.Voronoi instance + Diagram to plot + ax : matplotlib.axes.Axes instance, optional + Axes to plot on + show_points : bool, optional + Add the Voronoi points to the plot. + show_vertices : bool, optional + Add the Voronoi vertices to the plot. + line_colors : string, optional + Specifies the line color for polygon boundaries + line_width : float, optional + Specifies the line width for polygon boundaries + line_alpha : float, optional + Specifies the line alpha for polygon boundaries + point_size : float, optional + Specifies the size of points + + Returns + ------- + fig : matplotlib.figure.Figure instance + Figure for the plot + + See Also + -------- + Voronoi + + Notes + ----- + Requires Matplotlib. For degenerate input, including collinearity and + other violations of general position, it may be preferable to + calculate the Voronoi diagram with Qhull options ``QJ`` for random + joggling, or ``Qt`` to enforce triangulated output. Otherwise, some + Voronoi regions may not be visible. + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.spatial import Voronoi, voronoi_plot_2d + + Create a set of points for the example: + + >>> rng = np.random.default_rng() + >>> points = rng.random((10,2)) + + Generate the Voronoi diagram for the points: + + >>> vor = Voronoi(points) + + Use `voronoi_plot_2d` to plot the diagram: + + >>> fig = voronoi_plot_2d(vor) + + Use `voronoi_plot_2d` to plot the diagram again, with some settings + customized: + + >>> fig = voronoi_plot_2d(vor, show_vertices=False, line_colors='orange', + ... line_width=2, line_alpha=0.6, point_size=2) + >>> plt.show() + + """ + from matplotlib.collections import LineCollection + + if vor.points.shape[1] != 2: + raise ValueError("Voronoi diagram is not 2-D") + + if kw.get('show_points', True): + point_size = kw.get('point_size', None) + ax.plot(vor.points[:, 0], vor.points[:, 1], '.', markersize=point_size) + if kw.get('show_vertices', True): + ax.plot(vor.vertices[:, 0], vor.vertices[:, 1], 'o') + + line_colors = kw.get('line_colors', 'k') + line_width = kw.get('line_width', 1.0) + line_alpha = kw.get('line_alpha', 1.0) + + center = vor.points.mean(axis=0) + ptp_bound = np.ptp(vor.points, axis=0) + + finite_segments = [] + infinite_segments = [] + for pointidx, simplex in zip(vor.ridge_points, vor.ridge_vertices): + simplex = np.asarray(simplex) + if np.all(simplex >= 0): + finite_segments.append(vor.vertices[simplex]) + else: + i = simplex[simplex >= 0][0] # finite end Voronoi vertex + + t = vor.points[pointidx[1]] - vor.points[pointidx[0]] # tangent + t /= np.linalg.norm(t) + n = np.array([-t[1], t[0]]) # normal + + midpoint = vor.points[pointidx].mean(axis=0) + direction = np.sign(np.dot(midpoint - center, n)) * n + if (vor.furthest_site): + direction = -direction + aspect_factor = abs(ptp_bound.max() / ptp_bound.min()) + far_point = vor.vertices[i] + direction * ptp_bound.max() * aspect_factor + + infinite_segments.append([vor.vertices[i], far_point]) + + ax.add_collection(LineCollection(finite_segments, + colors=line_colors, + lw=line_width, + alpha=line_alpha, + linestyle='solid')) + ax.add_collection(LineCollection(infinite_segments, + colors=line_colors, + lw=line_width, + alpha=line_alpha, + linestyle='dashed')) + + _adjust_bounds(ax, vor.points) + + return ax.figure diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_procrustes.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_procrustes.py new file mode 100644 index 0000000000000000000000000000000000000000..e3814ab5a8404461b446eb473f5bf96deb9f8c1f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_procrustes.py @@ -0,0 +1,132 @@ +""" +This module provides functions to perform full Procrustes analysis. + +This code was originally written by Justin Kucynski and ported over from +scikit-bio by Yoshiki Vazquez-Baeza. +""" + +import numpy as np +from scipy.linalg import orthogonal_procrustes + + +__all__ = ['procrustes'] + + +def procrustes(data1, data2): + r"""Procrustes analysis, a similarity test for two data sets. + + Each input matrix is a set of points or vectors (the rows of the matrix). + The dimension of the space is the number of columns of each matrix. Given + two identically sized matrices, procrustes standardizes both such that: + + - :math:`tr(AA^{T}) = 1`. + + - Both sets of points are centered around the origin. + + Procrustes ([1]_, [2]_) then applies the optimal transform to the second + matrix (including scaling/dilation, rotations, and reflections) to minimize + :math:`M^{2}=\sum(data1-data2)^{2}`, or the sum of the squares of the + pointwise differences between the two input datasets. + + This function was not designed to handle datasets with different numbers of + datapoints (rows). If two data sets have different dimensionality + (different number of columns), simply add columns of zeros to the smaller + of the two. + + Parameters + ---------- + data1 : array_like + Matrix, n rows represent points in k (columns) space `data1` is the + reference data, after it is standardised, the data from `data2` will be + transformed to fit the pattern in `data1` (must have >1 unique points). + data2 : array_like + n rows of data in k space to be fit to `data1`. Must be the same + shape ``(numrows, numcols)`` as data1 (must have >1 unique points). + + Returns + ------- + mtx1 : array_like + A standardized version of `data1`. + mtx2 : array_like + The orientation of `data2` that best fits `data1`. Centered, but not + necessarily :math:`tr(AA^{T}) = 1`. + disparity : float + :math:`M^{2}` as defined above. + + Raises + ------ + ValueError + If the input arrays are not two-dimensional. + If the shape of the input arrays is different. + If the input arrays have zero columns or zero rows. + + See Also + -------- + scipy.linalg.orthogonal_procrustes + scipy.spatial.distance.directed_hausdorff : Another similarity test + for two data sets + + Notes + ----- + - The disparity should not depend on the order of the input matrices, but + the output matrices will, as only the first output matrix is guaranteed + to be scaled such that :math:`tr(AA^{T}) = 1`. + + - Duplicate data points are generally ok, duplicating a data point will + increase its effect on the procrustes fit. + + - The disparity scales as the number of points per input matrix. + + References + ---------- + .. [1] Krzanowski, W. J. (2000). "Principles of Multivariate analysis". + .. [2] Gower, J. C. (1975). "Generalized procrustes analysis". + + Examples + -------- + >>> import numpy as np + >>> from scipy.spatial import procrustes + + The matrix ``b`` is a rotated, shifted, scaled and mirrored version of + ``a`` here: + + >>> a = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], 'd') + >>> b = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], 'd') + >>> mtx1, mtx2, disparity = procrustes(a, b) + >>> round(disparity) + 0 + + """ + mtx1 = np.array(data1, dtype=np.float64, copy=True) + mtx2 = np.array(data2, dtype=np.float64, copy=True) + + if mtx1.ndim != 2 or mtx2.ndim != 2: + raise ValueError("Input matrices must be two-dimensional") + if mtx1.shape != mtx2.shape: + raise ValueError("Input matrices must be of same shape") + if mtx1.size == 0: + raise ValueError("Input matrices must be >0 rows and >0 cols") + + # translate all the data to the origin + mtx1 -= np.mean(mtx1, 0) + mtx2 -= np.mean(mtx2, 0) + + norm1 = np.linalg.norm(mtx1) + norm2 = np.linalg.norm(mtx2) + + if norm1 == 0 or norm2 == 0: + raise ValueError("Input matrices must contain >1 unique points") + + # change scaling of data (in rows) such that trace(mtx*mtx') = 1 + mtx1 /= norm1 + mtx2 /= norm2 + + # transform mtx2 to minimize disparity + R, s = orthogonal_procrustes(mtx1, mtx2) + mtx2 = np.dot(mtx2, R.T) * s + + # measure the dissimilarity between the two datasets + disparity = np.sum(np.square(mtx1 - mtx2)) + + return mtx1, mtx2, disparity + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_qhull.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_qhull.pyi new file mode 100644 index 0000000000000000000000000000000000000000..416128eab5f53cdc8f45c422bcc140451223bd3f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_qhull.pyi @@ -0,0 +1,213 @@ +''' +Static type checking stub file for scipy/spatial/qhull.pyx +''' + + +import numpy as np +from numpy.typing import ArrayLike, NDArray +from typing_extensions import final + +class QhullError(RuntimeError): + ... + +@final +class _Qhull: + # Read-only cython attribute that behaves, more or less, like a property + @property + def ndim(self) -> int: ... + mode_option: bytes + options: bytes + furthest_site: bool + + def __init__( + self, + mode_option: bytes, + points: NDArray[np.float64], + options: None | bytes = ..., + required_options: None | bytes = ..., + furthest_site: bool = ..., + incremental: bool = ..., + interior_point: None | NDArray[np.float64] = ..., + ) -> None: ... + def check_active(self) -> None: ... + def close(self) -> None: ... + def get_points(self) -> NDArray[np.float64]: ... + def add_points( + self, + points: ArrayLike, + interior_point: ArrayLike = ... + ) -> None: ... + def get_paraboloid_shift_scale(self) -> tuple[float, float]: ... + def volume_area(self) -> tuple[float, float]: ... + def triangulate(self) -> None: ... + def get_simplex_facet_array(self) -> tuple[ + NDArray[np.intc], + NDArray[np.intc], + NDArray[np.float64], + NDArray[np.intc], + NDArray[np.intc], + ]: ... + def get_hull_points(self) -> NDArray[np.float64]: ... + def get_hull_facets(self) -> tuple[ + list[list[int]], + NDArray[np.float64], + ]: ... + def get_voronoi_diagram(self) -> tuple[ + NDArray[np.float64], + NDArray[np.intc], + list[list[int]], + list[list[int]], + NDArray[np.intp], + ]: ... + def get_extremes_2d(self) -> NDArray[np.intc]: ... + +def _get_barycentric_transforms( + points: NDArray[np.float64], + simplices: NDArray[np.intc], + eps: float +) -> NDArray[np.float64]: ... + +class _QhullUser: + ndim: int + npoints: int + min_bound: NDArray[np.float64] + max_bound: NDArray[np.float64] + + def __init__(self, qhull: _Qhull, incremental: bool = ...) -> None: ... + def close(self) -> None: ... + def _update(self, qhull: _Qhull) -> None: ... + def _add_points( + self, + points: ArrayLike, + restart: bool = ..., + interior_point: ArrayLike = ... + ) -> None: ... + +class Delaunay(_QhullUser): + furthest_site: bool + paraboloid_scale: float + paraboloid_shift: float + simplices: NDArray[np.intc] + neighbors: NDArray[np.intc] + equations: NDArray[np.float64] + coplanar: NDArray[np.intc] + good: NDArray[np.intc] + nsimplex: int + vertices: NDArray[np.intc] + + def __init__( + self, + points: ArrayLike, + furthest_site: bool = ..., + incremental: bool = ..., + qhull_options: None | str = ... + ) -> None: ... + def _update(self, qhull: _Qhull) -> None: ... + def add_points( + self, + points: ArrayLike, + restart: bool = ... + ) -> None: ... + @property + def points(self) -> NDArray[np.float64]: ... + @property + def transform(self) -> NDArray[np.float64]: ... + @property + def vertex_to_simplex(self) -> NDArray[np.intc]: ... + @property + def vertex_neighbor_vertices(self) -> tuple[ + NDArray[np.intc], + NDArray[np.intc], + ]: ... + @property + def convex_hull(self) -> NDArray[np.intc]: ... + def find_simplex( + self, + xi: ArrayLike, + bruteforce: bool = ..., + tol: float = ... + ) -> NDArray[np.intc]: ... + def plane_distance(self, xi: ArrayLike) -> NDArray[np.float64]: ... + def lift_points(self, x: ArrayLike) -> NDArray[np.float64]: ... + +def tsearch(tri: Delaunay, xi: ArrayLike) -> NDArray[np.intc]: ... +def _copy_docstr(dst: object, src: object) -> None: ... + +class ConvexHull(_QhullUser): + simplices: NDArray[np.intc] + neighbors: NDArray[np.intc] + equations: NDArray[np.float64] + coplanar: NDArray[np.intc] + good: None | NDArray[np.bool_] + volume: float + area: float + nsimplex: int + + def __init__( + self, + points: ArrayLike, + incremental: bool = ..., + qhull_options: None | str = ... + ) -> None: ... + def _update(self, qhull: _Qhull) -> None: ... + def add_points(self, points: ArrayLike, + restart: bool = ...) -> None: ... + @property + def points(self) -> NDArray[np.float64]: ... + @property + def vertices(self) -> NDArray[np.intc]: ... + +class Voronoi(_QhullUser): + vertices: NDArray[np.float64] + ridge_points: NDArray[np.intc] + ridge_vertices: list[list[int]] + regions: list[list[int]] + point_region: NDArray[np.intp] + furthest_site: bool + + def __init__( + self, + points: ArrayLike, + furthest_site: bool = ..., + incremental: bool = ..., + qhull_options: None | str = ... + ) -> None: ... + def _update(self, qhull: _Qhull) -> None: ... + def add_points( + self, + points: ArrayLike, + restart: bool = ... + ) -> None: ... + @property + def points(self) -> NDArray[np.float64]: ... + @property + def ridge_dict(self) -> dict[tuple[int, int], list[int]]: ... + +class HalfspaceIntersection(_QhullUser): + interior_point: NDArray[np.float64] + dual_facets: list[list[int]] + dual_equations: NDArray[np.float64] + dual_points: NDArray[np.float64] + dual_volume: float + dual_area: float + intersections: NDArray[np.float64] + ndim: int + nineq: int + + def __init__( + self, + halfspaces: ArrayLike, + interior_point: ArrayLike, + incremental: bool = ..., + qhull_options: None | str = ... + ) -> None: ... + def _update(self, qhull: _Qhull) -> None: ... + def add_halfspaces( + self, + halfspaces: ArrayLike, + restart: bool = ... + ) -> None: ... + @property + def halfspaces(self) -> NDArray[np.float64]: ... + @property + def dual_vertices(self) -> NDArray[np.integer]: ... diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_spherical_voronoi.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_spherical_voronoi.py new file mode 100644 index 0000000000000000000000000000000000000000..41fb0dad08f7f2f2c81dcfe89cacb7333eb4822b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_spherical_voronoi.py @@ -0,0 +1,341 @@ +""" +Spherical Voronoi Code + +.. versionadded:: 0.18.0 + +""" +# +# Copyright (C) Tyler Reddy, Ross Hemsley, Edd Edmondson, +# Nikolai Nowaczyk, Joe Pitt-Francis, 2015. +# +# Distributed under the same BSD license as SciPy. +# + +import numpy as np +import scipy +from . import _voronoi +from scipy.spatial import cKDTree # type: ignore[attr-defined] + +__all__ = ['SphericalVoronoi'] + + +def calculate_solid_angles(R): + """Calculates the solid angles of plane triangles. Implements the method of + Van Oosterom and Strackee [VanOosterom]_ with some modifications. Assumes + that input points have unit norm.""" + # Original method uses a triple product `R1 . (R2 x R3)` for the numerator. + # This is equal to the determinant of the matrix [R1 R2 R3], which can be + # computed with better stability. + numerator = np.linalg.det(R) + denominator = 1 + (np.einsum('ij,ij->i', R[:, 0], R[:, 1]) + + np.einsum('ij,ij->i', R[:, 1], R[:, 2]) + + np.einsum('ij,ij->i', R[:, 2], R[:, 0])) + return np.abs(2 * np.arctan2(numerator, denominator)) + + +class SphericalVoronoi: + """ Voronoi diagrams on the surface of a sphere. + + .. versionadded:: 0.18.0 + + Parameters + ---------- + points : ndarray of floats, shape (npoints, ndim) + Coordinates of points from which to construct a spherical + Voronoi diagram. + radius : float, optional + Radius of the sphere (Default: 1) + center : ndarray of floats, shape (ndim,) + Center of sphere (Default: origin) + threshold : float + Threshold for detecting duplicate points and + mismatches between points and sphere parameters. + (Default: 1e-06) + + Attributes + ---------- + points : double array of shape (npoints, ndim) + the points in `ndim` dimensions to generate the Voronoi diagram from + radius : double + radius of the sphere + center : double array of shape (ndim,) + center of the sphere + vertices : double array of shape (nvertices, ndim) + Voronoi vertices corresponding to points + regions : list of list of integers of shape (npoints, _ ) + the n-th entry is a list consisting of the indices + of the vertices belonging to the n-th point in points + + Methods + ------- + calculate_areas + Calculates the areas of the Voronoi regions. For 2D point sets, the + regions are circular arcs. The sum of the areas is ``2 * pi * radius``. + For 3D point sets, the regions are spherical polygons. The sum of the + areas is ``4 * pi * radius**2``. + + Raises + ------ + ValueError + If there are duplicates in `points`. + If the provided `radius` is not consistent with `points`. + + Notes + ----- + The spherical Voronoi diagram algorithm proceeds as follows. The Convex + Hull of the input points (generators) is calculated, and is equivalent to + their Delaunay triangulation on the surface of the sphere [Caroli]_. + The Convex Hull neighbour information is then used to + order the Voronoi region vertices around each generator. The latter + approach is substantially less sensitive to floating point issues than + angle-based methods of Voronoi region vertex sorting. + + Empirical assessment of spherical Voronoi algorithm performance suggests + quadratic time complexity (loglinear is optimal, but algorithms are more + challenging to implement). + + References + ---------- + .. [Caroli] Caroli et al. Robust and Efficient Delaunay triangulations of + points on or close to a sphere. Research Report RR-7004, 2009. + + .. [VanOosterom] Van Oosterom and Strackee. The solid angle of a plane + triangle. IEEE Transactions on Biomedical Engineering, + 2, 1983, pp 125--126. + + See Also + -------- + Voronoi : Conventional Voronoi diagrams in N dimensions. + + Examples + -------- + Do some imports and take some points on a cube: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.spatial import SphericalVoronoi, geometric_slerp + >>> from mpl_toolkits.mplot3d import proj3d + >>> # set input data + >>> points = np.array([[0, 0, 1], [0, 0, -1], [1, 0, 0], + ... [0, 1, 0], [0, -1, 0], [-1, 0, 0], ]) + + Calculate the spherical Voronoi diagram: + + >>> radius = 1 + >>> center = np.array([0, 0, 0]) + >>> sv = SphericalVoronoi(points, radius, center) + + Generate plot: + + >>> # sort vertices (optional, helpful for plotting) + >>> sv.sort_vertices_of_regions() + >>> t_vals = np.linspace(0, 1, 2000) + >>> fig = plt.figure() + >>> ax = fig.add_subplot(111, projection='3d') + >>> # plot the unit sphere for reference (optional) + >>> u = np.linspace(0, 2 * np.pi, 100) + >>> v = np.linspace(0, np.pi, 100) + >>> x = np.outer(np.cos(u), np.sin(v)) + >>> y = np.outer(np.sin(u), np.sin(v)) + >>> z = np.outer(np.ones(np.size(u)), np.cos(v)) + >>> ax.plot_surface(x, y, z, color='y', alpha=0.1) + >>> # plot generator points + >>> ax.scatter(points[:, 0], points[:, 1], points[:, 2], c='b') + >>> # plot Voronoi vertices + >>> ax.scatter(sv.vertices[:, 0], sv.vertices[:, 1], sv.vertices[:, 2], + ... c='g') + >>> # indicate Voronoi regions (as Euclidean polygons) + >>> for region in sv.regions: + ... n = len(region) + ... for i in range(n): + ... start = sv.vertices[region][i] + ... end = sv.vertices[region][(i + 1) % n] + ... result = geometric_slerp(start, end, t_vals) + ... ax.plot(result[..., 0], + ... result[..., 1], + ... result[..., 2], + ... c='k') + >>> ax.azim = 10 + >>> ax.elev = 40 + >>> _ = ax.set_xticks([]) + >>> _ = ax.set_yticks([]) + >>> _ = ax.set_zticks([]) + >>> fig.set_size_inches(4, 4) + >>> plt.show() + + """ + def __init__(self, points, radius=1, center=None, threshold=1e-06): + + if radius is None: + raise ValueError('`radius` is `None`. ' + 'Please provide a floating point number ' + '(i.e. `radius=1`).') + + self.radius = float(radius) + self.points = np.array(points).astype(np.float64) + self._dim = self.points.shape[1] + if center is None: + self.center = np.zeros(self._dim) + else: + self.center = np.array(center, dtype=float) + + # test degenerate input + self._rank = np.linalg.matrix_rank(self.points - self.points[0], + tol=threshold * self.radius) + if self._rank < self._dim: + raise ValueError(f"Rank of input points must be at least {self._dim}") + + if cKDTree(self.points).query_pairs(threshold * self.radius): + raise ValueError("Duplicate generators present.") + + radii = np.linalg.norm(self.points - self.center, axis=1) + max_discrepancy = np.abs(radii - self.radius).max() + if max_discrepancy >= threshold * self.radius: + raise ValueError("Radius inconsistent with generators.") + + self._calc_vertices_regions() + + def _calc_vertices_regions(self): + """ + Calculates the Voronoi vertices and regions of the generators stored + in self.points. The vertices will be stored in self.vertices and the + regions in self.regions. + + This algorithm was discussed at PyData London 2015 by + Tyler Reddy, Ross Hemsley and Nikolai Nowaczyk + """ + # get Convex Hull + conv = scipy.spatial.ConvexHull(self.points) + # get circumcenters of Convex Hull triangles from facet equations + # for 3D input circumcenters will have shape: (2N-4, 3) + self.vertices = self.radius * conv.equations[:, :-1] + self.center + self._simplices = conv.simplices + # calculate regions from triangulation + # for 3D input simplex_indices will have shape: (2N-4,) + simplex_indices = np.arange(len(self._simplices)) + # for 3D input tri_indices will have shape: (6N-12,) + tri_indices = np.column_stack([simplex_indices] * self._dim).ravel() + # for 3D input point_indices will have shape: (6N-12,) + point_indices = self._simplices.ravel() + # for 3D input indices will have shape: (6N-12,) + indices = np.argsort(point_indices, kind='mergesort') + # for 3D input flattened_groups will have shape: (6N-12,) + flattened_groups = tri_indices[indices].astype(np.intp) + # intervals will have shape: (N+1,) + intervals = np.cumsum(np.bincount(point_indices + 1)) + # split flattened groups to get nested list of unsorted regions + groups = [list(flattened_groups[intervals[i]:intervals[i + 1]]) + for i in range(len(intervals) - 1)] + self.regions = groups + + def sort_vertices_of_regions(self): + """Sort indices of the vertices to be (counter-)clockwise ordered. + + Raises + ------ + TypeError + If the points are not three-dimensional. + + Notes + ----- + For each region in regions, it sorts the indices of the Voronoi + vertices such that the resulting points are in a clockwise or + counterclockwise order around the generator point. + + This is done as follows: Recall that the n-th region in regions + surrounds the n-th generator in points and that the k-th + Voronoi vertex in vertices is the circumcenter of the k-th triangle + in self._simplices. For each region n, we choose the first triangle + (=Voronoi vertex) in self._simplices and a vertex of that triangle + not equal to the center n. These determine a unique neighbor of that + triangle, which is then chosen as the second triangle. The second + triangle will have a unique vertex not equal to the current vertex or + the center. This determines a unique neighbor of the second triangle, + which is then chosen as the third triangle and so forth. We proceed + through all the triangles (=Voronoi vertices) belonging to the + generator in points and obtain a sorted version of the vertices + of its surrounding region. + """ + if self._dim != 3: + raise TypeError("Only supported for three-dimensional point sets") + _voronoi.sort_vertices_of_regions(self._simplices, self.regions) + + def _calculate_areas_3d(self): + self.sort_vertices_of_regions() + sizes = [len(region) for region in self.regions] + csizes = np.cumsum(sizes) + num_regions = csizes[-1] + + # We create a set of triangles consisting of one point and two Voronoi + # vertices. The vertices of each triangle are adjacent in the sorted + # regions list. + point_indices = [i for i, size in enumerate(sizes) + for j in range(size)] + + nbrs1 = np.array([r for region in self.regions for r in region]) + + # The calculation of nbrs2 is a vectorized version of: + # np.array([r for region in self.regions for r in np.roll(region, 1)]) + nbrs2 = np.roll(nbrs1, 1) + indices = np.roll(csizes, 1) + indices[0] = 0 + nbrs2[indices] = nbrs1[csizes - 1] + + # Normalize points and vertices. + pnormalized = (self.points - self.center) / self.radius + vnormalized = (self.vertices - self.center) / self.radius + + # Create the complete set of triangles and calculate their solid angles + triangles = np.hstack([pnormalized[point_indices], + vnormalized[nbrs1], + vnormalized[nbrs2] + ]).reshape((num_regions, 3, 3)) + triangle_solid_angles = calculate_solid_angles(triangles) + + # Sum the solid angles of the triangles in each region + solid_angles = np.cumsum(triangle_solid_angles)[csizes - 1] + solid_angles[1:] -= solid_angles[:-1] + + # Get polygon areas using A = omega * r**2 + return solid_angles * self.radius**2 + + def _calculate_areas_2d(self): + # Find start and end points of arcs + arcs = self.points[self._simplices] - self.center + + # Calculate the angle subtended by arcs + d = np.sum((arcs[:, 1] - arcs[:, 0]) ** 2, axis=1) + theta = np.arccos(1 - (d / (2 * (self.radius ** 2)))) + + # Get areas using A = r * theta + areas = self.radius * theta + + # Correct arcs which go the wrong way (single-hemisphere inputs) + signs = np.sign(np.einsum('ij,ij->i', arcs[:, 0], + self.vertices - self.center)) + indices = np.where(signs < 0) + areas[indices] = 2 * np.pi * self.radius - areas[indices] + return areas + + def calculate_areas(self): + """Calculates the areas of the Voronoi regions. + + For 2D point sets, the regions are circular arcs. The sum of the areas + is ``2 * pi * radius``. + + For 3D point sets, the regions are spherical polygons. The sum of the + areas is ``4 * pi * radius**2``. + + .. versionadded:: 1.5.0 + + Returns + ------- + areas : double array of shape (npoints,) + The areas of the Voronoi regions. + """ + if self._dim == 2: + return self._calculate_areas_2d() + elif self._dim == 3: + return self._calculate_areas_3d() + else: + raise TypeError("Only supported for 2D and 3D point sets") diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_voronoi.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_voronoi.pyi new file mode 100644 index 0000000000000000000000000000000000000000..c7c361ff69414d50a6ebcfe2c837025b60083940 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/_voronoi.pyi @@ -0,0 +1,4 @@ + +import numpy as np + +def sort_vertices_of_regions(simplices: np.ndarray, regions: list[list[int]]) -> None: ... # noqa: E501 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/ckdtree.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/ckdtree.py new file mode 100644 index 0000000000000000000000000000000000000000..40f524c71bf122ac822626596c2991b19ee0d30e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/ckdtree.py @@ -0,0 +1,18 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.spatial` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = ["cKDTree"] # noqa: F822 + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="spatial", module="ckdtree", + private_modules=["_ckdtree"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/distance.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/distance.py new file mode 100644 index 0000000000000000000000000000000000000000..4df984e691b9c70ab64806223e837a457d9fc17e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/distance.py @@ -0,0 +1,3140 @@ +""" +Distance computations (:mod:`scipy.spatial.distance`) +===================================================== + +.. sectionauthor:: Damian Eads + +Function reference +------------------ + +Distance matrix computation from a collection of raw observation vectors +stored in a rectangular array. + +.. autosummary:: + :toctree: generated/ + + pdist -- pairwise distances between observation vectors. + cdist -- distances between two collections of observation vectors + squareform -- convert distance matrix to a condensed one and vice versa + directed_hausdorff -- directed Hausdorff distance between arrays + +Predicates for checking the validity of distance matrices, both +condensed and redundant. Also contained in this module are functions +for computing the number of observations in a distance matrix. + +.. autosummary:: + :toctree: generated/ + + is_valid_dm -- checks for a valid distance matrix + is_valid_y -- checks for a valid condensed distance matrix + num_obs_dm -- # of observations in a distance matrix + num_obs_y -- # of observations in a condensed distance matrix + +Distance functions between two numeric vectors ``u`` and ``v``. Computing +distances over a large collection of vectors is inefficient for these +functions. Use ``pdist`` for this purpose. + +.. autosummary:: + :toctree: generated/ + + braycurtis -- the Bray-Curtis distance. + canberra -- the Canberra distance. + chebyshev -- the Chebyshev distance. + cityblock -- the Manhattan distance. + correlation -- the Correlation distance. + cosine -- the Cosine distance. + euclidean -- the Euclidean distance. + jensenshannon -- the Jensen-Shannon distance. + mahalanobis -- the Mahalanobis distance. + minkowski -- the Minkowski distance. + seuclidean -- the normalized Euclidean distance. + sqeuclidean -- the squared Euclidean distance. + +Distance functions between two boolean vectors (representing sets) ``u`` and +``v``. As in the case of numerical vectors, ``pdist`` is more efficient for +computing the distances between all pairs. + +.. autosummary:: + :toctree: generated/ + + dice -- the Dice dissimilarity. + hamming -- the Hamming distance. + jaccard -- the Jaccard distance. + kulczynski1 -- the Kulczynski 1 distance. + rogerstanimoto -- the Rogers-Tanimoto dissimilarity. + russellrao -- the Russell-Rao dissimilarity. + sokalmichener -- the Sokal-Michener dissimilarity. + sokalsneath -- the Sokal-Sneath dissimilarity. + yule -- the Yule dissimilarity. + +:func:`hamming` also operates over discrete numerical vectors. +""" + +# Copyright (C) Damian Eads, 2007-2008. New BSD License. + +__all__ = [ + 'braycurtis', + 'canberra', + 'cdist', + 'chebyshev', + 'cityblock', + 'correlation', + 'cosine', + 'dice', + 'directed_hausdorff', + 'euclidean', + 'hamming', + 'is_valid_dm', + 'is_valid_y', + 'jaccard', + 'jensenshannon', + 'kulczynski1', + 'mahalanobis', + 'minkowski', + 'num_obs_dm', + 'num_obs_y', + 'pdist', + 'rogerstanimoto', + 'russellrao', + 'seuclidean', + 'sokalmichener', + 'sokalsneath', + 'sqeuclidean', + 'squareform', + 'yule' +] + + +import math +import warnings +import numpy as np +import dataclasses + +from collections.abc import Callable +from functools import partial +from scipy._lib._util import _asarray_validated, _transition_to_rng +from scipy._lib.deprecation import _deprecated + +from . import _distance_wrap +from . import _hausdorff +from ..linalg import norm +from ..special import rel_entr + +from . import _distance_pybind + + +def _copy_array_if_base_present(a): + """Copy the array if its base points to a parent array.""" + if a.base is not None: + return a.copy() + return a + + +def _correlation_cdist_wrap(XA, XB, dm, **kwargs): + XA = XA - XA.mean(axis=1, keepdims=True) + XB = XB - XB.mean(axis=1, keepdims=True) + _distance_wrap.cdist_cosine_double_wrap(XA, XB, dm, **kwargs) + + +def _correlation_pdist_wrap(X, dm, **kwargs): + X2 = X - X.mean(axis=1, keepdims=True) + _distance_wrap.pdist_cosine_double_wrap(X2, dm, **kwargs) + + +def _convert_to_type(X, out_type): + return np.ascontiguousarray(X, dtype=out_type) + + +def _nbool_correspond_all(u, v, w=None): + if u.dtype == v.dtype == bool and w is None: + not_u = ~u + not_v = ~v + nff = (not_u & not_v).sum() + nft = (not_u & v).sum() + ntf = (u & not_v).sum() + ntt = (u & v).sum() + else: + dtype = np.result_type(int, u.dtype, v.dtype) + u = u.astype(dtype) + v = v.astype(dtype) + not_u = 1.0 - u + not_v = 1.0 - v + if w is not None: + not_u = w * not_u + u = w * u + nff = (not_u * not_v).sum() + nft = (not_u * v).sum() + ntf = (u * not_v).sum() + ntt = (u * v).sum() + return (nff, nft, ntf, ntt) + + +def _nbool_correspond_ft_tf(u, v, w=None): + if u.dtype == v.dtype == bool and w is None: + not_u = ~u + not_v = ~v + nft = (not_u & v).sum() + ntf = (u & not_v).sum() + else: + dtype = np.result_type(int, u.dtype, v.dtype) + u = u.astype(dtype) + v = v.astype(dtype) + not_u = 1.0 - u + not_v = 1.0 - v + if w is not None: + not_u = w * not_u + u = w * u + nft = (not_u * v).sum() + ntf = (u * not_v).sum() + return (nft, ntf) + + +def _validate_cdist_input(XA, XB, mA, mB, n, metric_info, **kwargs): + # get supported types + types = metric_info.types + # choose best type + typ = types[types.index(XA.dtype)] if XA.dtype in types else types[0] + # validate data + XA = _convert_to_type(XA, out_type=typ) + XB = _convert_to_type(XB, out_type=typ) + + # validate kwargs + _validate_kwargs = metric_info.validator + if _validate_kwargs: + kwargs = _validate_kwargs((XA, XB), mA + mB, n, **kwargs) + return XA, XB, typ, kwargs + + +def _validate_weight_with_size(X, m, n, **kwargs): + w = kwargs.pop('w', None) + if w is None: + return kwargs + + if w.ndim != 1 or w.shape[0] != n: + raise ValueError("Weights must have same size as input vector. " + f"{w.shape[0]} vs. {n}") + + kwargs['w'] = _validate_weights(w) + return kwargs + + +def _validate_hamming_kwargs(X, m, n, **kwargs): + w = kwargs.get('w', np.ones((n,), dtype='double')) + + if w.ndim != 1 or w.shape[0] != n: + raise ValueError( + "Weights must have same size as input vector. %d vs. %d" % (w.shape[0], n) + ) + + kwargs['w'] = _validate_weights(w) + return kwargs + + +def _validate_mahalanobis_kwargs(X, m, n, **kwargs): + VI = kwargs.pop('VI', None) + if VI is None: + if m <= n: + # There are fewer observations than the dimension of + # the observations. + raise ValueError("The number of observations (%d) is too " + "small; the covariance matrix is " + "singular. For observations with %d " + "dimensions, at least %d observations " + "are required." % (m, n, n + 1)) + if isinstance(X, tuple): + X = np.vstack(X) + CV = np.atleast_2d(np.cov(X.astype(np.float64, copy=False).T)) + VI = np.linalg.inv(CV).T.copy() + kwargs["VI"] = _convert_to_double(VI) + return kwargs + + +def _validate_minkowski_kwargs(X, m, n, **kwargs): + kwargs = _validate_weight_with_size(X, m, n, **kwargs) + if 'p' not in kwargs: + kwargs['p'] = 2. + else: + if kwargs['p'] <= 0: + raise ValueError("p must be greater than 0") + + return kwargs + + +def _validate_pdist_input(X, m, n, metric_info, **kwargs): + # get supported types + types = metric_info.types + # choose best type + typ = types[types.index(X.dtype)] if X.dtype in types else types[0] + # validate data + X = _convert_to_type(X, out_type=typ) + + # validate kwargs + _validate_kwargs = metric_info.validator + if _validate_kwargs: + kwargs = _validate_kwargs(X, m, n, **kwargs) + return X, typ, kwargs + + +def _validate_seuclidean_kwargs(X, m, n, **kwargs): + V = kwargs.pop('V', None) + if V is None: + if isinstance(X, tuple): + X = np.vstack(X) + V = np.var(X.astype(np.float64, copy=False), axis=0, ddof=1) + else: + V = np.asarray(V, order='c') + if len(V.shape) != 1: + raise ValueError('Variance vector V must ' + 'be one-dimensional.') + if V.shape[0] != n: + raise ValueError('Variance vector V must be of the same ' + 'dimension as the vectors on which the distances ' + 'are computed.') + kwargs['V'] = _convert_to_double(V) + return kwargs + + +def _validate_vector(u, dtype=None): + # XXX Is order='c' really necessary? + u = np.asarray(u, dtype=dtype, order='c') + if u.ndim == 1: + return u + raise ValueError("Input vector should be 1-D.") + + +def _validate_weights(w, dtype=np.float64): + w = _validate_vector(w, dtype=dtype) + if np.any(w < 0): + raise ValueError("Input weights should be all non-negative") + return w + + +@_transition_to_rng('seed', position_num=2, replace_doc=False) +def directed_hausdorff(u, v, rng=0): + """ + Compute the directed Hausdorff distance between two 2-D arrays. + + Distances between pairs are calculated using a Euclidean metric. + + Parameters + ---------- + u : (M,N) array_like + Input array with M points in N dimensions. + v : (O,N) array_like + Input array with O points in N dimensions. + rng : int or `numpy.random.Generator` or None, optional + Pseudorandom number generator state. Default is 0 so the + shuffling of `u` and `v` is reproducible. + + If `rng` is passed by keyword, types other than `numpy.random.Generator` are + passed to `numpy.random.default_rng` to instantiate a ``Generator``. + If `rng` is already a ``Generator`` instance, then the provided instance is + used. + + If this argument is passed by position or `seed` is passed by keyword, + legacy behavior for the argument `seed` applies: + + - If `seed` is None, a new ``RandomState`` instance is used. The state is + initialized using data from ``/dev/urandom`` (or the Windows analogue) + if available or from the system clock otherwise. + - If `seed` is an int, a new ``RandomState`` instance is used, + seeded with `seed`. + - If `seed` is already a ``Generator`` or ``RandomState`` instance, then + that instance is used. + + .. versionchanged:: 1.15.0 + As part of the `SPEC-007 `_ + transition from use of `numpy.random.RandomState` to + `numpy.random.Generator`, this keyword was changed from `seed` to `rng`. + For an interim period, both keywords will continue to work, although only + one may be specified at a time. After the interim period, function calls + using the `seed` keyword will emit warnings. The behavior of both `seed` + and `rng` are outlined above, but only the `rng` keyword should be used in + new code. + + Returns + ------- + d : double + The directed Hausdorff distance between arrays `u` and `v`, + + index_1 : int + index of point contributing to Hausdorff pair in `u` + + index_2 : int + index of point contributing to Hausdorff pair in `v` + + Raises + ------ + ValueError + An exception is thrown if `u` and `v` do not have + the same number of columns. + + See Also + -------- + scipy.spatial.procrustes : Another similarity test for two data sets + + Notes + ----- + Uses the early break technique and the random sampling approach + described by [1]_. Although worst-case performance is ``O(m * o)`` + (as with the brute force algorithm), this is unlikely in practice + as the input data would have to require the algorithm to explore + every single point interaction, and after the algorithm shuffles + the input points at that. The best case performance is O(m), which + is satisfied by selecting an inner loop distance that is less than + cmax and leads to an early break as often as possible. The authors + have formally shown that the average runtime is closer to O(m). + + .. versionadded:: 0.19.0 + + References + ---------- + .. [1] A. A. Taha and A. Hanbury, "An efficient algorithm for + calculating the exact Hausdorff distance." IEEE Transactions On + Pattern Analysis And Machine Intelligence, vol. 37 pp. 2153-63, + 2015. + + Examples + -------- + Find the directed Hausdorff distance between two 2-D arrays of + coordinates: + + >>> from scipy.spatial.distance import directed_hausdorff + >>> import numpy as np + >>> u = np.array([(1.0, 0.0), + ... (0.0, 1.0), + ... (-1.0, 0.0), + ... (0.0, -1.0)]) + >>> v = np.array([(2.0, 0.0), + ... (0.0, 2.0), + ... (-2.0, 0.0), + ... (0.0, -4.0)]) + + >>> directed_hausdorff(u, v)[0] + 2.23606797749979 + >>> directed_hausdorff(v, u)[0] + 3.0 + + Find the general (symmetric) Hausdorff distance between two 2-D + arrays of coordinates: + + >>> max(directed_hausdorff(u, v)[0], directed_hausdorff(v, u)[0]) + 3.0 + + Find the indices of the points that generate the Hausdorff distance + (the Hausdorff pair): + + >>> directed_hausdorff(v, u)[1:] + (3, 3) + + """ + u = np.asarray(u, dtype=np.float64, order='c') + v = np.asarray(v, dtype=np.float64, order='c') + if u.shape[1] != v.shape[1]: + raise ValueError('u and v need to have the same ' + 'number of columns') + result = _hausdorff.directed_hausdorff(u, v, rng) + return result + + +def minkowski(u, v, p=2, w=None): + """ + Compute the Minkowski distance between two 1-D arrays. + + The Minkowski distance between 1-D arrays `u` and `v`, + is defined as + + .. math:: + + {\\|u-v\\|}_p = (\\sum{|u_i - v_i|^p})^{1/p}. + + + \\left(\\sum{w_i(|(u_i - v_i)|^p)}\\right)^{1/p}. + + Parameters + ---------- + u : (N,) array_like + Input array. + v : (N,) array_like + Input array. + p : scalar + The order of the norm of the difference :math:`{\\|u-v\\|}_p`. Note + that for :math:`0 < p < 1`, the triangle inequality only holds with + an additional multiplicative factor, i.e. it is only a quasi-metric. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + minkowski : double + The Minkowski distance between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.minkowski([1, 0, 0], [0, 1, 0], 1) + 2.0 + >>> distance.minkowski([1, 0, 0], [0, 1, 0], 2) + 1.4142135623730951 + >>> distance.minkowski([1, 0, 0], [0, 1, 0], 3) + 1.2599210498948732 + >>> distance.minkowski([1, 1, 0], [0, 1, 0], 1) + 1.0 + >>> distance.minkowski([1, 1, 0], [0, 1, 0], 2) + 1.0 + >>> distance.minkowski([1, 1, 0], [0, 1, 0], 3) + 1.0 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + if p <= 0: + raise ValueError("p must be greater than 0") + u_v = u - v + if w is not None: + w = _validate_weights(w) + if p == 1: + root_w = w + elif p == 2: + # better precision and speed + root_w = np.sqrt(w) + elif p == np.inf: + root_w = (w != 0) + else: + root_w = np.power(w, 1/p) + u_v = root_w * u_v + dist = norm(u_v, ord=p) + return dist + + +def euclidean(u, v, w=None): + """ + Computes the Euclidean distance between two 1-D arrays. + + The Euclidean distance between 1-D arrays `u` and `v`, is defined as + + .. math:: + + {\\|u-v\\|}_2 + + \\left(\\sum{(w_i |(u_i - v_i)|^2)}\\right)^{1/2} + + Parameters + ---------- + u : (N,) array_like + Input array. + v : (N,) array_like + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + euclidean : double + The Euclidean distance between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.euclidean([1, 0, 0], [0, 1, 0]) + 1.4142135623730951 + >>> distance.euclidean([1, 1, 0], [0, 1, 0]) + 1.0 + + """ + return minkowski(u, v, p=2, w=w) + + +def sqeuclidean(u, v, w=None): + """ + Compute the squared Euclidean distance between two 1-D arrays. + + The squared Euclidean distance between `u` and `v` is defined as + + .. math:: + + \\sum_i{w_i |u_i - v_i|^2} + + Parameters + ---------- + u : (N,) array_like + Input array. + v : (N,) array_like + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + sqeuclidean : double + The squared Euclidean distance between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.sqeuclidean([1, 0, 0], [0, 1, 0]) + 2.0 + >>> distance.sqeuclidean([1, 1, 0], [0, 1, 0]) + 1.0 + + """ + # Preserve float dtypes, but convert everything else to np.float64 + # for stability. + utype, vtype = None, None + if not (hasattr(u, "dtype") and np.issubdtype(u.dtype, np.inexact)): + utype = np.float64 + if not (hasattr(v, "dtype") and np.issubdtype(v.dtype, np.inexact)): + vtype = np.float64 + + u = _validate_vector(u, dtype=utype) + v = _validate_vector(v, dtype=vtype) + u_v = u - v + u_v_w = u_v # only want weights applied once + if w is not None: + w = _validate_weights(w) + u_v_w = w * u_v + return np.dot(u_v, u_v_w) + + +def correlation(u, v, w=None, centered=True): + """ + Compute the correlation distance between two 1-D arrays. + + The correlation distance between `u` and `v`, is + defined as + + .. math:: + + 1 - \\frac{(u - \\bar{u}) \\cdot (v - \\bar{v})} + {{\\|(u - \\bar{u})\\|}_2 {\\|(v - \\bar{v})\\|}_2} + + where :math:`\\bar{u}` is the mean of the elements of `u` + and :math:`x \\cdot y` is the dot product of :math:`x` and :math:`y`. + + Parameters + ---------- + u : (N,) array_like of floats + Input array. + + .. deprecated:: 1.15.0 + Complex `u` is deprecated and will raise an error in SciPy 1.17.0 + v : (N,) array_like of floats + Input array. + + .. deprecated:: 1.15.0 + Complex `v` is deprecated and will raise an error in SciPy 1.17.0 + w : (N,) array_like of floats, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + centered : bool, optional + If True, `u` and `v` will be centered. Default is True. + + Returns + ------- + correlation : double + The correlation distance between 1-D array `u` and `v`. + + Examples + -------- + Find the correlation between two arrays. + + >>> from scipy.spatial.distance import correlation + >>> correlation([1, 0, 1], [1, 1, 0]) + 1.5 + + Using a weighting array, the correlation can be calculated as: + + >>> correlation([1, 0, 1], [1, 1, 0], w=[0.9, 0.1, 0.1]) + 1.1 + + If centering is not needed, the correlation can be calculated as: + + >>> correlation([1, 0, 1], [1, 1, 0], centered=False) + 0.5 + """ + u = _validate_vector(u) + v = _validate_vector(v) + if np.iscomplexobj(u) or np.iscomplexobj(v): + message = ( + "Complex `u` and `v` are deprecated and will raise an error in " + "SciPy 1.17.0.") + warnings.warn(message, DeprecationWarning, stacklevel=2) + if w is not None: + w = _validate_weights(w) + w = w / w.sum() + if centered: + if w is not None: + umu = np.dot(u, w) + vmu = np.dot(v, w) + else: + umu = np.mean(u) + vmu = np.mean(v) + u = u - umu + v = v - vmu + if w is not None: + vw = v * w + uw = u * w + else: + vw, uw = v, u + uv = np.dot(u, vw) + uu = np.dot(u, uw) + vv = np.dot(v, vw) + dist = 1.0 - uv / math.sqrt(uu * vv) + # Clip the result to avoid rounding error + return np.clip(dist, 0.0, 2.0) + + +def cosine(u, v, w=None): + """ + Compute the Cosine distance between 1-D arrays. + + The Cosine distance between `u` and `v`, is defined as + + .. math:: + + 1 - \\frac{u \\cdot v} + {\\|u\\|_2 \\|v\\|_2}. + + where :math:`u \\cdot v` is the dot product of :math:`u` and + :math:`v`. + + Parameters + ---------- + u : (N,) array_like of floats + Input array. + + .. deprecated:: 1.15.0 + Complex `u` is deprecated and will raise an error in SciPy 1.17.0 + v : (N,) array_like of floats + Input array. + + .. deprecated:: 1.15.0 + Complex `v` is deprecated and will raise an error in SciPy 1.17.0 + w : (N,) array_like of floats, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + cosine : double + The Cosine distance between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.cosine([1, 0, 0], [0, 1, 0]) + 1.0 + >>> distance.cosine([100, 0, 0], [0, 1, 0]) + 1.0 + >>> distance.cosine([1, 1, 0], [0, 1, 0]) + 0.29289321881345254 + + """ + # cosine distance is also referred to as 'uncentered correlation', + # or 'reflective correlation' + return correlation(u, v, w=w, centered=False) + + +def hamming(u, v, w=None): + """ + Compute the Hamming distance between two 1-D arrays. + + The Hamming distance between 1-D arrays `u` and `v`, is simply the + proportion of disagreeing components in `u` and `v`. If `u` and `v` are + boolean vectors, the Hamming distance is + + .. math:: + + \\frac{c_{01} + c_{10}}{n} + + where :math:`c_{ij}` is the number of occurrences of + :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for + :math:`k < n`. + + Parameters + ---------- + u : (N,) array_like + Input array. + v : (N,) array_like + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + hamming : double + The Hamming distance between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.hamming([1, 0, 0], [0, 1, 0]) + 0.66666666666666663 + >>> distance.hamming([1, 0, 0], [1, 1, 0]) + 0.33333333333333331 + >>> distance.hamming([1, 0, 0], [2, 0, 0]) + 0.33333333333333331 + >>> distance.hamming([1, 0, 0], [3, 0, 0]) + 0.33333333333333331 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + if u.shape != v.shape: + raise ValueError('The 1d arrays must have equal lengths.') + u_ne_v = u != v + if w is not None: + w = _validate_weights(w) + if w.shape != u.shape: + raise ValueError("'w' should have the same length as 'u' and 'v'.") + w = w / w.sum() + return np.dot(u_ne_v, w) + return np.mean(u_ne_v) + + +def jaccard(u, v, w=None): + r""" + Compute the Jaccard dissimilarity between two boolean vectors. + + Given boolean vectors :math:`u \equiv (u_1, \cdots, u_n)` + and :math:`v \equiv (v_1, \cdots, v_n)` that are not both zero, + their *Jaccard dissimilarity* is defined as ([1]_, p. 26) + + .. math:: + + d_\textrm{jaccard}(u, v) := \frac{c_{10} + c_{01}} + {c_{11} + c_{10} + c_{01}} + + where + + .. math:: + + c_{ij} := \sum_{1 \le k \le n, u_k=i, v_k=j} 1 + + for :math:`i, j \in \{ 0, 1\}`. If :math:`u` and :math:`v` are both zero, + their Jaccard dissimilarity is defined to be zero. [2]_ + + If a (non-negative) weight vector :math:`w \equiv (w_1, \cdots, w_n)` + is supplied, the *weighted Jaccard dissimilarity* is defined similarly + but with :math:`c_{ij}` replaced by + + .. math:: + + \tilde{c}_{ij} := \sum_{1 \le k \le n, u_k=i, v_k=j} w_k + + Parameters + ---------- + u : (N,) array_like of bools + Input vector. + v : (N,) array_like of bools + Input vector. + w : (N,) array_like of floats, optional + Weights for each pair of :math:`(u_k, v_k)`. Default is ``None``, + which gives each pair a weight of ``1.0``. + + Returns + ------- + jaccard : float + The Jaccard dissimilarity between vectors `u` and `v`, optionally + weighted by `w` if supplied. + + Notes + ----- + The Jaccard dissimilarity satisfies the triangle inequality and is + qualified as a metric. [2]_ + + The *Jaccard index*, or *Jaccard similarity coefficient*, is equal to + one minus the Jaccard dissimilarity. [3]_ + + The dissimilarity between general (finite) sets may be computed by + encoding them as boolean vectors and computing the dissimilarity + between the encoded vectors. + For example, subsets :math:`A,B` of :math:`\{ 1, 2, ..., n \}` may be + encoded into boolean vectors :math:`u, v` by setting + :math:`u_k := 1_{k \in A}`, :math:`v_k := 1_{k \in B}` + for :math:`k = 1,2,\cdots,n`. + + .. versionchanged:: 1.2.0 + Previously, if all (positively weighted) elements in `u` and `v` are + zero, the function would return ``nan``. This was changed to return + ``0`` instead. + + .. versionchanged:: 1.15.0 + Non-0/1 numeric input used to produce an ad hoc result. Since 1.15.0, + numeric input is converted to Boolean before computation. + + References + ---------- + .. [1] Kaufman, L. and Rousseeuw, P. J. (1990). "Finding Groups in Data: + An Introduction to Cluster Analysis." John Wiley & Sons, Inc. + .. [2] Kosub, S. (2019). "A note on the triangle inequality for the + Jaccard distance." *Pattern Recognition Letters*, 120:36-38. + .. [3] https://en.wikipedia.org/wiki/Jaccard_index + + Examples + -------- + >>> from scipy.spatial import distance + + Non-zero vectors with no matching 1s have dissimilarity of 1.0: + + >>> distance.jaccard([1, 0, 0], [0, 1, 0]) + 1.0 + + Vectors with some matching 1s have dissimilarity less than 1.0: + + >>> distance.jaccard([1, 0, 0, 0], [1, 1, 1, 0]) + 0.6666666666666666 + + Identical vectors, including zero vectors, have dissimilarity of 0.0: + + >>> distance.jaccard([1, 0, 0], [1, 0, 0]) + 0.0 + >>> distance.jaccard([0, 0, 0], [0, 0, 0]) + 0.0 + + The following example computes the dissimilarity from a confusion matrix + directly by setting the weight vector to the frequency of True Positive, + False Negative, False Positive, and True Negative: + + >>> distance.jaccard([1, 1, 0, 0], [1, 0, 1, 0], [31, 41, 59, 26]) + 0.7633587786259542 # (41+59)/(31+41+59) + + """ + u = _validate_vector(u) + v = _validate_vector(v) + + unequal = np.bitwise_xor(u != 0, v != 0) + nonzero = np.bitwise_or(u != 0, v != 0) + if w is not None: + w = _validate_weights(w) + unequal = w * unequal + nonzero = w * nonzero + a = np.float64(unequal.sum()) + b = np.float64(nonzero.sum()) + return (a / b) if b != 0 else np.float64(0) + + +_deprecated_kulczynski1 = _deprecated( + "The kulczynski1 metric is deprecated since SciPy 1.15.0 and will be " + "removed in SciPy 1.17.0. Replace usage of 'kulczynski1(u, v)' with " + "'1/jaccard(u, v) - 1'." +) + + +@_deprecated_kulczynski1 +def kulczynski1(u, v, *, w=None): + """ + Compute the Kulczynski 1 dissimilarity between two boolean 1-D arrays. + + .. deprecated:: 1.15.0 + This function is deprecated and will be removed in SciPy 1.17.0. + Replace usage of ``kulczynski1(u, v)`` with ``1/jaccard(u, v) - 1``. + + The Kulczynski 1 dissimilarity between two boolean 1-D arrays `u` and `v` + of length ``n``, is defined as + + .. math:: + + \\frac{c_{11}} + {c_{01} + c_{10}} + + where :math:`c_{ij}` is the number of occurrences of + :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for + :math:`k \\in {0, 1, ..., n-1}`. + + Parameters + ---------- + u : (N,) array_like, bool + Input array. + v : (N,) array_like, bool + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + kulczynski1 : float + The Kulczynski 1 distance between vectors `u` and `v`. + + Notes + ----- + This measure has a minimum value of 0 and no upper limit. + It is un-defined when there are no non-matches. + + .. versionadded:: 1.8.0 + + References + ---------- + .. [1] Kulczynski S. et al. Bulletin + International de l'Academie Polonaise des Sciences + et des Lettres, Classe des Sciences Mathematiques + et Naturelles, Serie B (Sciences Naturelles). 1927; + Supplement II: 57-203. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.kulczynski1([1, 0, 0], [0, 1, 0]) + 0.0 + >>> distance.kulczynski1([True, False, False], [True, True, False]) + 1.0 + >>> distance.kulczynski1([True, False, False], [True]) + 0.5 + >>> distance.kulczynski1([1, 0, 0], [3, 1, 0]) + -3.0 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + if w is not None: + w = _validate_weights(w) + (_, nft, ntf, ntt) = _nbool_correspond_all(u, v, w=w) + + return ntt / (ntf + nft) + + +def seuclidean(u, v, V): + """ + Return the standardized Euclidean distance between two 1-D arrays. + + The standardized Euclidean distance between two n-vectors `u` and `v` is + + .. math:: + + \\sqrt{\\sum\\limits_i \\frac{1}{V_i} \\left(u_i-v_i \\right)^2} + + ``V`` is the variance vector; ``V[I]`` is the variance computed over all the i-th + components of the points. If not passed, it is automatically computed. + + Parameters + ---------- + u : (N,) array_like + Input array. + v : (N,) array_like + Input array. + V : (N,) array_like + `V` is an 1-D array of component variances. It is usually computed + among a larger collection of vectors. + + Returns + ------- + seuclidean : double + The standardized Euclidean distance between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.seuclidean([1, 0, 0], [0, 1, 0], [0.1, 0.1, 0.1]) + 4.4721359549995796 + >>> distance.seuclidean([1, 0, 0], [0, 1, 0], [1, 0.1, 0.1]) + 3.3166247903553998 + >>> distance.seuclidean([1, 0, 0], [0, 1, 0], [10, 0.1, 0.1]) + 3.1780497164141406 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + V = _validate_vector(V, dtype=np.float64) + if V.shape[0] != u.shape[0] or u.shape[0] != v.shape[0]: + raise TypeError('V must be a 1-D array of the same dimension ' + 'as u and v.') + return euclidean(u, v, w=1/V) + + +def cityblock(u, v, w=None): + """ + Compute the City Block (Manhattan) distance. + + Computes the Manhattan distance between two 1-D arrays `u` and `v`, + which is defined as + + .. math:: + + \\sum_i {\\left| u_i - v_i \\right|}. + + Parameters + ---------- + u : (N,) array_like + Input array. + v : (N,) array_like + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + cityblock : double + The City Block (Manhattan) distance between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.cityblock([1, 0, 0], [0, 1, 0]) + 2 + >>> distance.cityblock([1, 0, 0], [0, 2, 0]) + 3 + >>> distance.cityblock([1, 0, 0], [1, 1, 0]) + 1 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + l1_diff = abs(u - v) + if w is not None: + w = _validate_weights(w) + l1_diff = w * l1_diff + return l1_diff.sum() + + +def mahalanobis(u, v, VI): + """ + Compute the Mahalanobis distance between two 1-D arrays. + + The Mahalanobis distance between 1-D arrays `u` and `v`, is defined as + + .. math:: + + \\sqrt{ (u-v) V^{-1} (u-v)^T } + + where ``V`` is the covariance matrix. Note that the argument `VI` + is the inverse of ``V``. + + Parameters + ---------- + u : (N,) array_like + Input array. + v : (N,) array_like + Input array. + VI : array_like + The inverse of the covariance matrix. + + Returns + ------- + mahalanobis : double + The Mahalanobis distance between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> iv = [[1, 0.5, 0.5], [0.5, 1, 0.5], [0.5, 0.5, 1]] + >>> distance.mahalanobis([1, 0, 0], [0, 1, 0], iv) + 1.0 + >>> distance.mahalanobis([0, 2, 0], [0, 1, 0], iv) + 1.0 + >>> distance.mahalanobis([2, 0, 0], [0, 1, 0], iv) + 1.7320508075688772 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + VI = np.atleast_2d(VI) + delta = u - v + m = np.dot(np.dot(delta, VI), delta) + return np.sqrt(m) + + +def chebyshev(u, v, w=None): + r""" + Compute the Chebyshev distance. + + The *Chebyshev distance* between real vectors + :math:`u \equiv (u_1, \cdots, u_n)` and + :math:`v \equiv (v_1, \cdots, v_n)` is defined as [1]_ + + .. math:: + + d_\textrm{chebyshev}(u,v) := \max_{1 \le i \le n} |u_i-v_i| + + If a (non-negative) weight vector :math:`w \equiv (w_1, \cdots, w_n)` + is supplied, the *weighted Chebyshev distance* is defined to be the + weighted Minkowski distance of infinite order; that is, + + .. math:: + + \begin{align} + d_\textrm{chebyshev}(u,v;w) &:= \lim_{p\rightarrow \infty} + \left( \sum_{i=1}^n w_i | u_i-v_i |^p \right)^\frac{1}{p} \\ + &= \max_{1 \le i \le n} 1_{w_i > 0} | u_i - v_i | + \end{align} + + Parameters + ---------- + u : (N,) array_like of floats + Input vector. + v : (N,) array_like of floats + Input vector. + w : (N,) array_like of floats, optional + Weight vector. Default is ``None``, which gives all pairs + :math:`(u_i, v_i)` the same weight ``1.0``. + + Returns + ------- + chebyshev : float + The Chebyshev distance between vectors `u` and `v`, optionally weighted + by `w`. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Chebyshev_distance + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.chebyshev([1, 0, 0], [0, 1, 0]) + 1 + >>> distance.chebyshev([1, 1, 0], [0, 1, 0]) + 1 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + if w is not None: + w = _validate_weights(w) + return max((w > 0) * abs(u - v)) + return max(abs(u - v)) + + +def braycurtis(u, v, w=None): + """ + Compute the Bray-Curtis distance between two 1-D arrays. + + Bray-Curtis distance is defined as + + .. math:: + + \\sum{|u_i-v_i|} / \\sum{|u_i+v_i|} + + The Bray-Curtis distance is in the range [0, 1] if all coordinates are + positive, and is undefined if the inputs are of length zero. + + Parameters + ---------- + u : (N,) array_like + Input array. + v : (N,) array_like + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + braycurtis : double + The Bray-Curtis distance between 1-D arrays `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.braycurtis([1, 0, 0], [0, 1, 0]) + 1.0 + >>> distance.braycurtis([1, 1, 0], [0, 1, 0]) + 0.33333333333333331 + + """ + u = _validate_vector(u) + v = _validate_vector(v, dtype=np.float64) + l1_diff = abs(u - v) + l1_sum = abs(u + v) + if w is not None: + w = _validate_weights(w) + l1_diff = w * l1_diff + l1_sum = w * l1_sum + return l1_diff.sum() / l1_sum.sum() + + +def canberra(u, v, w=None): + """ + Compute the Canberra distance between two 1-D arrays. + + The Canberra distance is defined as + + .. math:: + + d(u,v) = \\sum_i \\frac{|u_i-v_i|} + {|u_i|+|v_i|}. + + Parameters + ---------- + u : (N,) array_like + Input array. + v : (N,) array_like + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + canberra : double + The Canberra distance between vectors `u` and `v`. + + Notes + ----- + When ``u[i]`` and ``v[i]`` are 0 for given i, then the fraction 0/0 = 0 is + used in the calculation. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.canberra([1, 0, 0], [0, 1, 0]) + 2.0 + >>> distance.canberra([1, 1, 0], [0, 1, 0]) + 1.0 + + """ + u = _validate_vector(u) + v = _validate_vector(v, dtype=np.float64) + if w is not None: + w = _validate_weights(w) + with np.errstate(invalid='ignore'): + abs_uv = abs(u - v) + abs_u = abs(u) + abs_v = abs(v) + d = abs_uv / (abs_u + abs_v) + if w is not None: + d = w * d + d = np.nansum(d) + return d + + +def jensenshannon(p, q, base=None, *, axis=0, keepdims=False): + """ + Compute the Jensen-Shannon distance (metric) between + two probability arrays. This is the square root + of the Jensen-Shannon divergence. + + The Jensen-Shannon distance between two probability + vectors `p` and `q` is defined as, + + .. math:: + + \\sqrt{\\frac{D(p \\parallel m) + D(q \\parallel m)}{2}} + + where :math:`m` is the pointwise mean of :math:`p` and :math:`q` + and :math:`D` is the Kullback-Leibler divergence. + + This routine will normalize `p` and `q` if they don't sum to 1.0. + + Parameters + ---------- + p : (N,) array_like + left probability vector + q : (N,) array_like + right probability vector + base : double, optional + the base of the logarithm used to compute the output + if not given, then the routine uses the default base of + scipy.stats.entropy. + axis : int, optional + Axis along which the Jensen-Shannon distances are computed. The default + is 0. + + .. versionadded:: 1.7.0 + keepdims : bool, optional + If this is set to `True`, the reduced axes are left in the + result as dimensions with size one. With this option, + the result will broadcast correctly against the input array. + Default is False. + + .. versionadded:: 1.7.0 + + Returns + ------- + js : double or ndarray + The Jensen-Shannon distances between `p` and `q` along the `axis`. + + Notes + ----- + + .. versionadded:: 1.2.0 + + Examples + -------- + >>> from scipy.spatial import distance + >>> import numpy as np + >>> distance.jensenshannon([1.0, 0.0, 0.0], [0.0, 1.0, 0.0], 2.0) + 1.0 + >>> distance.jensenshannon([1.0, 0.0], [0.5, 0.5]) + 0.46450140402245893 + >>> distance.jensenshannon([1.0, 0.0, 0.0], [1.0, 0.0, 0.0]) + 0.0 + >>> a = np.array([[1, 2, 3, 4], + ... [5, 6, 7, 8], + ... [9, 10, 11, 12]]) + >>> b = np.array([[13, 14, 15, 16], + ... [17, 18, 19, 20], + ... [21, 22, 23, 24]]) + >>> distance.jensenshannon(a, b, axis=0) + array([0.1954288, 0.1447697, 0.1138377, 0.0927636]) + >>> distance.jensenshannon(a, b, axis=1) + array([0.1402339, 0.0399106, 0.0201815]) + + """ + p = np.asarray(p) + q = np.asarray(q) + p = p / np.sum(p, axis=axis, keepdims=True) + q = q / np.sum(q, axis=axis, keepdims=True) + m = (p + q) / 2.0 + left = rel_entr(p, m) + right = rel_entr(q, m) + left_sum = np.sum(left, axis=axis, keepdims=keepdims) + right_sum = np.sum(right, axis=axis, keepdims=keepdims) + js = left_sum + right_sum + if base is not None: + js /= np.log(base) + return np.sqrt(js / 2.0) + + +def yule(u, v, w=None): + """ + Compute the Yule dissimilarity between two boolean 1-D arrays. + + The Yule dissimilarity is defined as + + .. math:: + + \\frac{R}{c_{TT} * c_{FF} + \\frac{R}{2}} + + where :math:`c_{ij}` is the number of occurrences of + :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for + :math:`k < n` and :math:`R = 2.0 * c_{TF} * c_{FT}`. + + Parameters + ---------- + u : (N,) array_like, bool + Input array. + v : (N,) array_like, bool + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + yule : double + The Yule dissimilarity between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.yule([1, 0, 0], [0, 1, 0]) + 2.0 + >>> distance.yule([1, 1, 0], [0, 1, 0]) + 0.0 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + if w is not None: + w = _validate_weights(w) + (nff, nft, ntf, ntt) = _nbool_correspond_all(u, v, w=w) + half_R = ntf * nft + if half_R == 0: + return 0.0 + else: + return float(2.0 * half_R / (ntt * nff + half_R)) + + +def dice(u, v, w=None): + """ + Compute the Dice dissimilarity between two boolean 1-D arrays. + + The Dice dissimilarity between `u` and `v`, is + + .. math:: + + \\frac{c_{TF} + c_{FT}} + {2c_{TT} + c_{FT} + c_{TF}} + + where :math:`c_{ij}` is the number of occurrences of + :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for + :math:`k < n`. + + Parameters + ---------- + u : (N,) array_like, bool + Input 1-D array. + v : (N,) array_like, bool + Input 1-D array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + dice : double + The Dice dissimilarity between 1-D arrays `u` and `v`. + + Notes + ----- + This function computes the Dice dissimilarity index. To compute the + Dice similarity index, convert one to the other with similarity = + 1 - dissimilarity. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.dice([1, 0, 0], [0, 1, 0]) + 1.0 + >>> distance.dice([1, 0, 0], [1, 1, 0]) + 0.3333333333333333 + >>> distance.dice([1, 0, 0], [2, 0, 0]) + -0.3333333333333333 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + if w is not None: + w = _validate_weights(w) + if u.dtype == v.dtype == bool and w is None: + ntt = (u & v).sum() + else: + dtype = np.result_type(int, u.dtype, v.dtype) + u = u.astype(dtype) + v = v.astype(dtype) + if w is None: + ntt = (u * v).sum() + else: + ntt = (u * v * w).sum() + (nft, ntf) = _nbool_correspond_ft_tf(u, v, w=w) + return float((ntf + nft) / np.array(2.0 * ntt + ntf + nft)) + + +def rogerstanimoto(u, v, w=None): + """ + Compute the Rogers-Tanimoto dissimilarity between two boolean 1-D arrays. + + The Rogers-Tanimoto dissimilarity between two boolean 1-D arrays + `u` and `v`, is defined as + + .. math:: + \\frac{R} + {c_{TT} + c_{FF} + R} + + where :math:`c_{ij}` is the number of occurrences of + :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for + :math:`k < n` and :math:`R = 2(c_{TF} + c_{FT})`. + + Parameters + ---------- + u : (N,) array_like, bool + Input array. + v : (N,) array_like, bool + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + rogerstanimoto : double + The Rogers-Tanimoto dissimilarity between vectors + `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.rogerstanimoto([1, 0, 0], [0, 1, 0]) + 0.8 + >>> distance.rogerstanimoto([1, 0, 0], [1, 1, 0]) + 0.5 + >>> distance.rogerstanimoto([1, 0, 0], [2, 0, 0]) + -1.0 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + if w is not None: + w = _validate_weights(w) + (nff, nft, ntf, ntt) = _nbool_correspond_all(u, v, w=w) + return float(2.0 * (ntf + nft)) / float(ntt + nff + (2.0 * (ntf + nft))) + + +def russellrao(u, v, w=None): + """ + Compute the Russell-Rao dissimilarity between two boolean 1-D arrays. + + The Russell-Rao dissimilarity between two boolean 1-D arrays, `u` and + `v`, is defined as + + .. math:: + + \\frac{n - c_{TT}} + {n} + + where :math:`c_{ij}` is the number of occurrences of + :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for + :math:`k < n`. + + Parameters + ---------- + u : (N,) array_like, bool + Input array. + v : (N,) array_like, bool + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + russellrao : double + The Russell-Rao dissimilarity between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.russellrao([1, 0, 0], [0, 1, 0]) + 1.0 + >>> distance.russellrao([1, 0, 0], [1, 1, 0]) + 0.6666666666666666 + >>> distance.russellrao([1, 0, 0], [2, 0, 0]) + 0.3333333333333333 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + if u.dtype == v.dtype == bool and w is None: + ntt = (u & v).sum() + n = float(len(u)) + elif w is None: + ntt = (u * v).sum() + n = float(len(u)) + else: + w = _validate_weights(w) + ntt = (u * v * w).sum() + n = w.sum() + return float(n - ntt) / n + + +_deprecated_sokalmichener = _deprecated( + "The sokalmichener metric is deprecated since SciPy 1.15.0 and will be " + "removed in SciPy 1.17.0. Replace usage of 'sokalmichener(u, v)' with " + "'rogerstanimoto(u, v)'." +) + + +@_deprecated_sokalmichener +def sokalmichener(u, v, w=None): + """ + Compute the Sokal-Michener dissimilarity between two boolean 1-D arrays. + + .. deprecated:: 1.15.0 + This function is deprecated and will be removed in SciPy 1.17.0. + Replace usage of ``sokalmichener(u, v)`` with ``rogerstanimoto(u, v)``. + + The Sokal-Michener dissimilarity between boolean 1-D arrays `u` and `v`, + is defined as + + .. math:: + + \\frac{R} + {S + R} + + where :math:`c_{ij}` is the number of occurrences of + :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for + :math:`k < n`, :math:`R = 2 * (c_{TF} + c_{FT})` and + :math:`S = c_{FF} + c_{TT}`. + + Parameters + ---------- + u : (N,) array_like, bool + Input array. + v : (N,) array_like, bool + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + sokalmichener : double + The Sokal-Michener dissimilarity between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.sokalmichener([1, 0, 0], [0, 1, 0]) + 0.8 + >>> distance.sokalmichener([1, 0, 0], [1, 1, 0]) + 0.5 + >>> distance.sokalmichener([1, 0, 0], [2, 0, 0]) + -1.0 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + if w is not None: + w = _validate_weights(w) + nff, nft, ntf, ntt = _nbool_correspond_all(u, v, w=w) + return float(2.0 * (ntf + nft)) / float(ntt + nff + 2.0 * (ntf + nft)) + + +def sokalsneath(u, v, w=None): + """ + Compute the Sokal-Sneath dissimilarity between two boolean 1-D arrays. + + The Sokal-Sneath dissimilarity between `u` and `v`, + + .. math:: + + \\frac{R} + {c_{TT} + R} + + where :math:`c_{ij}` is the number of occurrences of + :math:`\\mathtt{u[k]} = i` and :math:`\\mathtt{v[k]} = j` for + :math:`k < n` and :math:`R = 2(c_{TF} + c_{FT})`. + + Parameters + ---------- + u : (N,) array_like, bool + Input array. + v : (N,) array_like, bool + Input array. + w : (N,) array_like, optional + The weights for each value in `u` and `v`. Default is None, + which gives each value a weight of 1.0 + + Returns + ------- + sokalsneath : double + The Sokal-Sneath dissimilarity between vectors `u` and `v`. + + Examples + -------- + >>> from scipy.spatial import distance + >>> distance.sokalsneath([1, 0, 0], [0, 1, 0]) + 1.0 + >>> distance.sokalsneath([1, 0, 0], [1, 1, 0]) + 0.66666666666666663 + >>> distance.sokalsneath([1, 0, 0], [2, 1, 0]) + 0.0 + >>> distance.sokalsneath([1, 0, 0], [3, 1, 0]) + -2.0 + + """ + u = _validate_vector(u) + v = _validate_vector(v) + if u.dtype == v.dtype == bool and w is None: + ntt = (u & v).sum() + elif w is None: + ntt = (u * v).sum() + else: + w = _validate_weights(w) + ntt = (u * v * w).sum() + (nft, ntf) = _nbool_correspond_ft_tf(u, v, w=w) + denom = np.array(ntt + 2.0 * (ntf + nft)) + if not denom.any(): + raise ValueError('Sokal-Sneath dissimilarity is not defined for ' + 'vectors that are entirely false.') + return float(2.0 * (ntf + nft)) / denom + + +_convert_to_double = partial(_convert_to_type, out_type=np.float64) +_convert_to_bool = partial(_convert_to_type, out_type=bool) + +# adding python-only wrappers to _distance_wrap module +_distance_wrap.pdist_correlation_double_wrap = _correlation_pdist_wrap +_distance_wrap.cdist_correlation_double_wrap = _correlation_cdist_wrap + + +@dataclasses.dataclass(frozen=True) +class CDistMetricWrapper: + metric_name: str + + def __call__(self, XA, XB, *, out=None, **kwargs): + XA = np.ascontiguousarray(XA) + XB = np.ascontiguousarray(XB) + mA, n = XA.shape + mB, _ = XB.shape + metric_name = self.metric_name + metric_info = _METRICS[metric_name] + XA, XB, typ, kwargs = _validate_cdist_input( + XA, XB, mA, mB, n, metric_info, **kwargs) + + w = kwargs.pop('w', None) + if w is not None: + metric = metric_info.dist_func + return _cdist_callable( + XA, XB, metric=metric, out=out, w=w, **kwargs) + + dm = _prepare_out_argument(out, np.float64, (mA, mB)) + # get cdist wrapper + cdist_fn = getattr(_distance_wrap, f'cdist_{metric_name}_{typ}_wrap') + cdist_fn(XA, XB, dm, **kwargs) + return dm + + +@dataclasses.dataclass(frozen=True) +class PDistMetricWrapper: + metric_name: str + + def __call__(self, X, *, out=None, **kwargs): + X = np.ascontiguousarray(X) + m, n = X.shape + metric_name = self.metric_name + metric_info = _METRICS[metric_name] + X, typ, kwargs = _validate_pdist_input( + X, m, n, metric_info, **kwargs) + out_size = (m * (m - 1)) // 2 + w = kwargs.pop('w', None) + if w is not None: + metric = metric_info.dist_func + return _pdist_callable( + X, metric=metric, out=out, w=w, **kwargs) + + dm = _prepare_out_argument(out, np.float64, (out_size,)) + # get pdist wrapper + pdist_fn = getattr(_distance_wrap, f'pdist_{metric_name}_{typ}_wrap') + pdist_fn(X, dm, **kwargs) + return dm + + +@dataclasses.dataclass(frozen=True) +class MetricInfo: + # Name of python distance function + canonical_name: str + # All aliases, including canonical_name + aka: set[str] + # unvectorized distance function + dist_func: Callable + # Optimized cdist function + cdist_func: Callable + # Optimized pdist function + pdist_func: Callable + # function that checks kwargs and computes default values: + # f(X, m, n, **kwargs) + validator: Callable | None = None + # list of supported types: + # X (pdist) and XA (cdist) are used to choose the type. if there is no + # match the first type is used. Default double + types: list[str] = dataclasses.field(default_factory=lambda: ['double']) + # true if out array must be C-contiguous + requires_contiguous_out: bool = True + + +# Registry of implemented metrics: +_METRIC_INFOS = [ + MetricInfo( + canonical_name='braycurtis', + aka={'braycurtis'}, + dist_func=braycurtis, + cdist_func=_distance_pybind.cdist_braycurtis, + pdist_func=_distance_pybind.pdist_braycurtis, + ), + MetricInfo( + canonical_name='canberra', + aka={'canberra'}, + dist_func=canberra, + cdist_func=_distance_pybind.cdist_canberra, + pdist_func=_distance_pybind.pdist_canberra, + ), + MetricInfo( + canonical_name='chebyshev', + aka={'chebychev', 'chebyshev', 'cheby', 'cheb', 'ch'}, + dist_func=chebyshev, + cdist_func=_distance_pybind.cdist_chebyshev, + pdist_func=_distance_pybind.pdist_chebyshev, + ), + MetricInfo( + canonical_name='cityblock', + aka={'cityblock', 'cblock', 'cb', 'c'}, + dist_func=cityblock, + cdist_func=_distance_pybind.cdist_cityblock, + pdist_func=_distance_pybind.pdist_cityblock, + ), + MetricInfo( + canonical_name='correlation', + aka={'correlation', 'co'}, + dist_func=correlation, + cdist_func=CDistMetricWrapper('correlation'), + pdist_func=PDistMetricWrapper('correlation'), + ), + MetricInfo( + canonical_name='cosine', + aka={'cosine', 'cos'}, + dist_func=cosine, + cdist_func=CDistMetricWrapper('cosine'), + pdist_func=PDistMetricWrapper('cosine'), + ), + MetricInfo( + canonical_name='dice', + aka={'dice'}, + types=['bool'], + dist_func=dice, + cdist_func=_distance_pybind.cdist_dice, + pdist_func=_distance_pybind.pdist_dice, + ), + MetricInfo( + canonical_name='euclidean', + aka={'euclidean', 'euclid', 'eu', 'e'}, + dist_func=euclidean, + cdist_func=_distance_pybind.cdist_euclidean, + pdist_func=_distance_pybind.pdist_euclidean, + ), + MetricInfo( + canonical_name='hamming', + aka={'matching', 'hamming', 'hamm', 'ha', 'h'}, + types=['double', 'bool'], + validator=_validate_hamming_kwargs, + dist_func=hamming, + cdist_func=_distance_pybind.cdist_hamming, + pdist_func=_distance_pybind.pdist_hamming, + ), + MetricInfo( + canonical_name='jaccard', + aka={'jaccard', 'jacc', 'ja', 'j'}, + types=['double', 'bool'], + dist_func=jaccard, + cdist_func=_distance_pybind.cdist_jaccard, + pdist_func=_distance_pybind.pdist_jaccard, + ), + MetricInfo( + canonical_name='jensenshannon', + aka={'jensenshannon', 'js'}, + dist_func=jensenshannon, + cdist_func=CDistMetricWrapper('jensenshannon'), + pdist_func=PDistMetricWrapper('jensenshannon'), + ), + MetricInfo( + canonical_name='kulczynski1', + aka={'kulczynski1'}, + types=['bool'], + dist_func=kulczynski1, + cdist_func=_deprecated_kulczynski1(_distance_pybind.cdist_kulczynski1), + pdist_func=_deprecated_kulczynski1(_distance_pybind.pdist_kulczynski1), + ), + MetricInfo( + canonical_name='mahalanobis', + aka={'mahalanobis', 'mahal', 'mah'}, + validator=_validate_mahalanobis_kwargs, + dist_func=mahalanobis, + cdist_func=CDistMetricWrapper('mahalanobis'), + pdist_func=PDistMetricWrapper('mahalanobis'), + ), + MetricInfo( + canonical_name='minkowski', + aka={'minkowski', 'mi', 'm', 'pnorm'}, + validator=_validate_minkowski_kwargs, + dist_func=minkowski, + cdist_func=_distance_pybind.cdist_minkowski, + pdist_func=_distance_pybind.pdist_minkowski, + ), + MetricInfo( + canonical_name='rogerstanimoto', + aka={'rogerstanimoto'}, + types=['bool'], + dist_func=rogerstanimoto, + cdist_func=_distance_pybind.cdist_rogerstanimoto, + pdist_func=_distance_pybind.pdist_rogerstanimoto, + ), + MetricInfo( + canonical_name='russellrao', + aka={'russellrao'}, + types=['bool'], + dist_func=russellrao, + cdist_func=_distance_pybind.cdist_russellrao, + pdist_func=_distance_pybind.pdist_russellrao, + ), + MetricInfo( + canonical_name='seuclidean', + aka={'seuclidean', 'se', 's'}, + validator=_validate_seuclidean_kwargs, + dist_func=seuclidean, + cdist_func=CDistMetricWrapper('seuclidean'), + pdist_func=PDistMetricWrapper('seuclidean'), + ), + MetricInfo( + canonical_name='sokalmichener', + aka={'sokalmichener'}, + types=['bool'], + dist_func=sokalmichener, + cdist_func=_deprecated_sokalmichener(_distance_pybind.cdist_sokalmichener), + pdist_func=_deprecated_sokalmichener(_distance_pybind.pdist_sokalmichener), + ), + MetricInfo( + canonical_name='sokalsneath', + aka={'sokalsneath'}, + types=['bool'], + dist_func=sokalsneath, + cdist_func=_distance_pybind.cdist_sokalsneath, + pdist_func=_distance_pybind.pdist_sokalsneath, + ), + MetricInfo( + canonical_name='sqeuclidean', + aka={'sqeuclidean', 'sqe', 'sqeuclid'}, + dist_func=sqeuclidean, + cdist_func=_distance_pybind.cdist_sqeuclidean, + pdist_func=_distance_pybind.pdist_sqeuclidean, + ), + MetricInfo( + canonical_name='yule', + aka={'yule'}, + types=['bool'], + dist_func=yule, + cdist_func=_distance_pybind.cdist_yule, + pdist_func=_distance_pybind.pdist_yule, + ), +] + +_METRICS = {info.canonical_name: info for info in _METRIC_INFOS} +_METRIC_ALIAS = {alias: info + for info in _METRIC_INFOS + for alias in info.aka} + +_METRICS_NAMES = list(_METRICS.keys()) + +_TEST_METRICS = {'test_' + info.canonical_name: info for info in _METRIC_INFOS} + + +def pdist(X, metric='euclidean', *, out=None, **kwargs): + """ + Pairwise distances between observations in n-dimensional space. + + See Notes for common calling conventions. + + Parameters + ---------- + X : array_like + An m by n array of m original observations in an + n-dimensional space. + metric : str or function, optional + The distance metric to use. The distance function can + be 'braycurtis', 'canberra', 'chebyshev', 'cityblock', + 'correlation', 'cosine', 'dice', 'euclidean', 'hamming', + 'jaccard', 'jensenshannon', 'kulczynski1', + 'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto', + 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', + 'sqeuclidean', 'yule'. + out : ndarray, optional + The output array. + If not None, condensed distance matrix Y is stored in this array. + **kwargs : dict, optional + Extra arguments to `metric`: refer to each metric documentation for a + list of all possible arguments. + + Some possible arguments: + + p : scalar + The p-norm to apply for Minkowski, weighted and unweighted. + Default: 2. + + w : ndarray + The weight vector for metrics that support weights (e.g., Minkowski). + + V : ndarray + The variance vector for standardized Euclidean. + Default: var(X, axis=0, ddof=1) + + VI : ndarray + The inverse of the covariance matrix for Mahalanobis. + Default: inv(cov(X.T)).T + + Returns + ------- + Y : ndarray + Returns a condensed distance matrix Y. For each :math:`i` and :math:`j` + (where :math:`i 0` (note + that this is only a quasi-metric if :math:`0 < p < 1`). + + 3. ``Y = pdist(X, 'cityblock')`` + + Computes the city block or Manhattan distance between the + points. + + 4. ``Y = pdist(X, 'seuclidean', V=None)`` + + Computes the standardized Euclidean distance. The standardized + Euclidean distance between two n-vectors ``u`` and ``v`` is + + .. math:: + + \\sqrt{\\sum {(u_i-v_i)^2 / V[x_i]}} + + + V is the variance vector; V[i] is the variance computed over all + the i'th components of the points. If not passed, it is + automatically computed. + + 5. ``Y = pdist(X, 'sqeuclidean')`` + + Computes the squared Euclidean distance :math:`\\|u-v\\|_2^2` between + the vectors. + + 6. ``Y = pdist(X, 'cosine')`` + + Computes the cosine distance between vectors u and v, + + .. math:: + + 1 - \\frac{u \\cdot v} + {{\\|u\\|}_2 {\\|v\\|}_2} + + where :math:`\\|*\\|_2` is the 2-norm of its argument ``*``, and + :math:`u \\cdot v` is the dot product of ``u`` and ``v``. + + 7. ``Y = pdist(X, 'correlation')`` + + Computes the correlation distance between vectors u and v. This is + + .. math:: + + 1 - \\frac{(u - \\bar{u}) \\cdot (v - \\bar{v})} + {{\\|(u - \\bar{u})\\|}_2 {\\|(v - \\bar{v})\\|}_2} + + where :math:`\\bar{v}` is the mean of the elements of vector v, + and :math:`x \\cdot y` is the dot product of :math:`x` and :math:`y`. + + 8. ``Y = pdist(X, 'hamming')`` + + Computes the normalized Hamming distance, or the proportion of + those vector elements between two n-vectors ``u`` and ``v`` + which disagree. To save memory, the matrix ``X`` can be of type + boolean. + + 9. ``Y = pdist(X, 'jaccard')`` + + Computes the Jaccard distance between the points. Given two + vectors, ``u`` and ``v``, the Jaccard distance is the + proportion of those elements ``u[i]`` and ``v[i]`` that + disagree. + + 10. ``Y = pdist(X, 'jensenshannon')`` + + Computes the Jensen-Shannon distance between two probability arrays. + Given two probability vectors, :math:`p` and :math:`q`, the + Jensen-Shannon distance is + + .. math:: + + \\sqrt{\\frac{D(p \\parallel m) + D(q \\parallel m)}{2}} + + where :math:`m` is the pointwise mean of :math:`p` and :math:`q` + and :math:`D` is the Kullback-Leibler divergence. + + 11. ``Y = pdist(X, 'chebyshev')`` + + Computes the Chebyshev distance between the points. The + Chebyshev distance between two n-vectors ``u`` and ``v`` is the + maximum norm-1 distance between their respective elements. More + precisely, the distance is given by + + .. math:: + + d(u,v) = \\max_i {|u_i-v_i|} + + 12. ``Y = pdist(X, 'canberra')`` + + Computes the Canberra distance between the points. The + Canberra distance between two points ``u`` and ``v`` is + + .. math:: + + d(u,v) = \\sum_i \\frac{|u_i-v_i|} + {|u_i|+|v_i|} + + + 13. ``Y = pdist(X, 'braycurtis')`` + + Computes the Bray-Curtis distance between the points. The + Bray-Curtis distance between two points ``u`` and ``v`` is + + + .. math:: + + d(u,v) = \\frac{\\sum_i {|u_i-v_i|}} + {\\sum_i {|u_i+v_i|}} + + 14. ``Y = pdist(X, 'mahalanobis', VI=None)`` + + Computes the Mahalanobis distance between the points. The + Mahalanobis distance between two points ``u`` and ``v`` is + :math:`\\sqrt{(u-v)(1/V)(u-v)^T}` where :math:`(1/V)` (the ``VI`` + variable) is the inverse covariance. If ``VI`` is not None, + ``VI`` will be used as the inverse covariance matrix. + + 15. ``Y = pdist(X, 'yule')`` + + Computes the Yule distance between each pair of boolean + vectors. (see yule function documentation) + + 16. ``Y = pdist(X, 'matching')`` + + Synonym for 'hamming'. + + 17. ``Y = pdist(X, 'dice')`` + + Computes the Dice distance between each pair of boolean + vectors. (see dice function documentation) + + 18. ``Y = pdist(X, 'kulczynski1')`` + + Computes the kulczynski1 distance between each pair of + boolean vectors. (see kulczynski1 function documentation) + + .. deprecated:: 1.15.0 + This metric is deprecated and will be removed in SciPy 1.17.0. + Replace usage of ``pdist(X, 'kulczynski1')`` with + ``1 / pdist(X, 'jaccard') - 1``. + + 19. ``Y = pdist(X, 'rogerstanimoto')`` + + Computes the Rogers-Tanimoto distance between each pair of + boolean vectors. (see rogerstanimoto function documentation) + + 20. ``Y = pdist(X, 'russellrao')`` + + Computes the Russell-Rao distance between each pair of + boolean vectors. (see russellrao function documentation) + + 21. ``Y = pdist(X, 'sokalmichener')`` + + Computes the Sokal-Michener distance between each pair of + boolean vectors. (see sokalmichener function documentation) + + .. deprecated:: 1.15.0 + This metric is deprecated and will be removed in SciPy 1.17.0. + Replace usage of ``pdist(X, 'sokalmichener')`` with + ``pdist(X, 'rogerstanimoto')``. + + 22. ``Y = pdist(X, 'sokalsneath')`` + + Computes the Sokal-Sneath distance between each pair of + boolean vectors. (see sokalsneath function documentation) + + 23. ``Y = pdist(X, 'kulczynski1')`` + + Computes the Kulczynski 1 distance between each pair of + boolean vectors. (see kulczynski1 function documentation) + + 24. ``Y = pdist(X, f)`` + + Computes the distance between all pairs of vectors in X + using the user supplied 2-arity function f. For example, + Euclidean distance between the vectors could be computed + as follows:: + + dm = pdist(X, lambda u, v: np.sqrt(((u-v)**2).sum())) + + Note that you should avoid passing a reference to one of + the distance functions defined in this library. For example,:: + + dm = pdist(X, sokalsneath) + + would calculate the pair-wise distances between the vectors in + X using the Python function sokalsneath. This would result in + sokalsneath being called :math:`{n \\choose 2}` times, which + is inefficient. Instead, the optimized C version is more + efficient, and we call it using the following syntax.:: + + dm = pdist(X, 'sokalsneath') + + Examples + -------- + >>> import numpy as np + >>> from scipy.spatial.distance import pdist + + ``x`` is an array of five points in three-dimensional space. + + >>> x = np.array([[2, 0, 2], [2, 2, 3], [-2, 4, 5], [0, 1, 9], [2, 2, 4]]) + + ``pdist(x)`` with no additional arguments computes the 10 pairwise + Euclidean distances: + + >>> pdist(x) + array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949, + 6.40312424, 1. , 5.38516481, 4.58257569, 5.47722558]) + + The following computes the pairwise Minkowski distances with ``p = 3.5``: + + >>> pdist(x, metric='minkowski', p=3.5) + array([2.04898923, 5.1154929 , 7.02700737, 2.43802731, 4.19042714, + 6.03956994, 1. , 4.45128103, 4.10636143, 5.0619695 ]) + + The pairwise city block or Manhattan distances: + + >>> pdist(x, metric='cityblock') + array([ 3., 11., 10., 4., 8., 9., 1., 9., 7., 8.]) + + """ + # You can also call this as: + # Y = pdist(X, 'test_abc') + # where 'abc' is the metric being tested. This computes the distance + # between all pairs of vectors in X using the distance metric 'abc' but + # with a more succinct, verifiable, but less efficient implementation. + + X = _asarray_validated(X, sparse_ok=False, objects_ok=True, mask_ok=True, + check_finite=False) + + s = X.shape + if len(s) != 2: + raise ValueError('A 2-dimensional array must be passed.') + + m, n = s + + if callable(metric): + mstr = getattr(metric, '__name__', 'UnknownCustomMetric') + metric_info = _METRIC_ALIAS.get(mstr, None) + + if metric_info is not None: + X, typ, kwargs = _validate_pdist_input( + X, m, n, metric_info, **kwargs) + + return _pdist_callable(X, metric=metric, out=out, **kwargs) + elif isinstance(metric, str): + mstr = metric.lower() + metric_info = _METRIC_ALIAS.get(mstr, None) + + if metric_info is not None: + pdist_fn = metric_info.pdist_func + return pdist_fn(X, out=out, **kwargs) + elif mstr.startswith("test_"): + metric_info = _TEST_METRICS.get(mstr, None) + if metric_info is None: + raise ValueError(f'Unknown "Test" Distance Metric: {mstr[5:]}') + X, typ, kwargs = _validate_pdist_input( + X, m, n, metric_info, **kwargs) + return _pdist_callable( + X, metric=metric_info.dist_func, out=out, **kwargs) + else: + raise ValueError(f'Unknown Distance Metric: {mstr}') + else: + raise TypeError('2nd argument metric must be a string identifier ' + 'or a function.') + + +def squareform(X, force="no", checks=True): + """ + Convert a vector-form distance vector to a square-form distance + matrix, and vice-versa. + + Parameters + ---------- + X : array_like + Either a condensed or redundant distance matrix. + force : str, optional + As with MATLAB(TM), if force is equal to ``'tovector'`` or + ``'tomatrix'``, the input will be treated as a distance matrix or + distance vector respectively. + checks : bool, optional + If set to False, no checks will be made for matrix + symmetry nor zero diagonals. This is useful if it is known that + ``X - X.T1`` is small and ``diag(X)`` is close to zero. + These values are ignored any way so they do not disrupt the + squareform transformation. + + Returns + ------- + Y : ndarray + If a condensed distance matrix is passed, a redundant one is + returned, or if a redundant one is passed, a condensed distance + matrix is returned. + + Notes + ----- + 1. ``v = squareform(X)`` + + Given a square n-by-n symmetric distance matrix ``X``, + ``v = squareform(X)`` returns a ``n * (n-1) / 2`` + (i.e. binomial coefficient n choose 2) sized vector `v` + where :math:`v[{n \\choose 2} - {n-i \\choose 2} + (j-i-1)]` + is the distance between distinct points ``i`` and ``j``. + If ``X`` is non-square or asymmetric, an error is raised. + + 2. ``X = squareform(v)`` + + Given a ``n * (n-1) / 2`` sized vector ``v`` + for some integer ``n >= 1`` encoding distances as described, + ``X = squareform(v)`` returns a n-by-n distance matrix ``X``. + The ``X[i, j]`` and ``X[j, i]`` values are set to + :math:`v[{n \\choose 2} - {n-i \\choose 2} + (j-i-1)]` + and all diagonal elements are zero. + + In SciPy 0.19.0, ``squareform`` stopped casting all input types to + float64, and started returning arrays of the same dtype as the input. + + Examples + -------- + >>> import numpy as np + >>> from scipy.spatial.distance import pdist, squareform + + ``x`` is an array of five points in three-dimensional space. + + >>> x = np.array([[2, 0, 2], [2, 2, 3], [-2, 4, 5], [0, 1, 9], [2, 2, 4]]) + + ``pdist(x)`` computes the Euclidean distances between each pair of + points in ``x``. The distances are returned in a one-dimensional + array with length ``5*(5 - 1)/2 = 10``. + + >>> distvec = pdist(x) + >>> distvec + array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949, + 6.40312424, 1. , 5.38516481, 4.58257569, 5.47722558]) + + ``squareform(distvec)`` returns the 5x5 distance matrix. + + >>> m = squareform(distvec) + >>> m + array([[0. , 2.23606798, 6.40312424, 7.34846923, 2.82842712], + [2.23606798, 0. , 4.89897949, 6.40312424, 1. ], + [6.40312424, 4.89897949, 0. , 5.38516481, 4.58257569], + [7.34846923, 6.40312424, 5.38516481, 0. , 5.47722558], + [2.82842712, 1. , 4.58257569, 5.47722558, 0. ]]) + + When given a square distance matrix ``m``, ``squareform(m)`` returns + the one-dimensional condensed distance vector associated with the + matrix. In this case, we recover ``distvec``. + + >>> squareform(m) + array([2.23606798, 6.40312424, 7.34846923, 2.82842712, 4.89897949, + 6.40312424, 1. , 5.38516481, 4.58257569, 5.47722558]) + """ + X = np.ascontiguousarray(X) + + s = X.shape + + if force.lower() == 'tomatrix': + if len(s) != 1: + raise ValueError("Forcing 'tomatrix' but input X is not a " + "distance vector.") + elif force.lower() == 'tovector': + if len(s) != 2: + raise ValueError("Forcing 'tovector' but input X is not a " + "distance matrix.") + + # X = squareform(v) + if len(s) == 1: + if s[0] == 0: + return np.zeros((1, 1), dtype=X.dtype) + + # Grab the closest value to the square root of the number + # of elements times 2 to see if the number of elements + # is indeed a binomial coefficient. + d = int(np.ceil(np.sqrt(s[0] * 2))) + + # Check that v is of valid dimensions. + if d * (d - 1) != s[0] * 2: + raise ValueError('Incompatible vector size. It must be a binomial ' + 'coefficient n choose 2 for some integer n >= 2.') + + # Allocate memory for the distance matrix. + M = np.zeros((d, d), dtype=X.dtype) + + # Since the C code does not support striding using strides. + # The dimensions are used instead. + X = _copy_array_if_base_present(X) + + # Fill in the values of the distance matrix. + _distance_wrap.to_squareform_from_vector_wrap(M, X) + + # Return the distance matrix. + return M + elif len(s) == 2: + if s[0] != s[1]: + raise ValueError('The matrix argument must be square.') + if checks: + is_valid_dm(X, throw=True, name='X') + + # One-side of the dimensions is set here. + d = s[0] + + if d <= 1: + return np.array([], dtype=X.dtype) + + # Create a vector. + v = np.zeros((d * (d - 1)) // 2, dtype=X.dtype) + + # Since the C code does not support striding using strides. + # The dimensions are used instead. + X = _copy_array_if_base_present(X) + + # Convert the vector to squareform. + _distance_wrap.to_vector_from_squareform_wrap(X, v) + return v + else: + raise ValueError("The first argument must be one or two dimensional " + f"array. A {len(s)}-dimensional array is not permitted") + + +def is_valid_dm(D, tol=0.0, throw=False, name="D", warning=False): + """ + Return True if input array is a valid distance matrix. + + Distance matrices must be 2-dimensional numpy arrays. + They must have a zero-diagonal, and they must be symmetric. + + Parameters + ---------- + D : array_like + The candidate object to test for validity. + tol : float, optional + The distance matrix should be symmetric. `tol` is the maximum + difference between entries ``ij`` and ``ji`` for the distance + metric to be considered symmetric. + throw : bool, optional + An exception is thrown if the distance matrix passed is not valid. + name : str, optional + The name of the variable to checked. This is useful if + throw is set to True so the offending variable can be identified + in the exception message when an exception is thrown. + warning : bool, optional + Instead of throwing an exception, a warning message is + raised. + + Returns + ------- + valid : bool + True if the variable `D` passed is a valid distance matrix. + + Notes + ----- + Small numerical differences in `D` and `D.T` and non-zeroness of + the diagonal are ignored if they are within the tolerance specified + by `tol`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.spatial.distance import is_valid_dm + + This matrix is a valid distance matrix. + + >>> d = np.array([[0.0, 1.1, 1.2, 1.3], + ... [1.1, 0.0, 1.0, 1.4], + ... [1.2, 1.0, 0.0, 1.5], + ... [1.3, 1.4, 1.5, 0.0]]) + >>> is_valid_dm(d) + True + + In the following examples, the input is not a valid distance matrix. + + Not square: + + >>> is_valid_dm([[0, 2, 2], [2, 0, 2]]) + False + + Nonzero diagonal element: + + >>> is_valid_dm([[0, 1, 1], [1, 2, 3], [1, 3, 0]]) + False + + Not symmetric: + + >>> is_valid_dm([[0, 1, 3], [2, 0, 1], [3, 1, 0]]) + False + + """ + D = np.asarray(D, order='c') + valid = True + try: + s = D.shape + if len(D.shape) != 2: + if name: + raise ValueError(f"Distance matrix '{name}' must have shape=2 " + "(i.e. be two-dimensional).") + else: + raise ValueError('Distance matrix must have shape=2 (i.e. ' + 'be two-dimensional).') + if tol == 0.0: + if not (D == D.T).all(): + if name: + raise ValueError(f"Distance matrix '{name}' must be symmetric.") + else: + raise ValueError('Distance matrix must be symmetric.') + if not (D[range(0, s[0]), range(0, s[0])] == 0).all(): + if name: + raise ValueError(f"Distance matrix '{name}' diagonal must be zero.") + else: + raise ValueError('Distance matrix diagonal must be zero.') + else: + if not (D - D.T <= tol).all(): + if name: + raise ValueError(f'Distance matrix \'{name}\' must be ' + f'symmetric within tolerance {tol:5.5f}.') + else: + raise ValueError('Distance matrix must be symmetric within ' + f'tolerance {tol:5.5f}.') + if not (D[range(0, s[0]), range(0, s[0])] <= tol).all(): + if name: + raise ValueError(f'Distance matrix \'{name}\' diagonal must be ' + f'close to zero within tolerance {tol:5.5f}.') + else: + raise ValueError(('Distance matrix \'{}\' diagonal must be close ' + 'to zero within tolerance {:5.5f}.').format(*tol)) + except Exception as e: + if throw: + raise + if warning: + warnings.warn(str(e), stacklevel=2) + valid = False + return valid + + +def is_valid_y(y, warning=False, throw=False, name=None): + """ + Return True if the input array is a valid condensed distance matrix. + + Condensed distance matrices must be 1-dimensional numpy arrays. + Their length must be a binomial coefficient :math:`{n \\choose 2}` + for some positive integer n. + + Parameters + ---------- + y : array_like + The condensed distance matrix. + warning : bool, optional + Invokes a warning if the variable passed is not a valid + condensed distance matrix. The warning message explains why + the distance matrix is not valid. `name` is used when + referencing the offending variable. + throw : bool, optional + Throws an exception if the variable passed is not a valid + condensed distance matrix. + name : bool, optional + Used when referencing the offending variable in the + warning or exception message. + + Returns + ------- + bool + True if the input array is a valid condensed distance matrix, + False otherwise. + + Examples + -------- + >>> from scipy.spatial.distance import is_valid_y + + This vector is a valid condensed distance matrix. The length is 6, + which corresponds to ``n = 4``, since ``4*(4 - 1)/2`` is 6. + + >>> v = [1.0, 1.2, 1.0, 0.5, 1.3, 0.9] + >>> is_valid_y(v) + True + + An input vector with length, say, 7, is not a valid condensed distance + matrix. + + >>> is_valid_y([1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7]) + False + + """ + y = np.asarray(y, order='c') + valid = True + try: + if len(y.shape) != 1: + if name: + raise ValueError(f"Condensed distance matrix '{name}' must " + "have shape=1 (i.e. be one-dimensional).") + else: + raise ValueError('Condensed distance matrix must have shape=1 ' + '(i.e. be one-dimensional).') + n = y.shape[0] + d = int(np.ceil(np.sqrt(n * 2))) + if (d * (d - 1) / 2) != n: + if name: + raise ValueError(f"Length n of condensed distance matrix '{name}' " + "must be a binomial coefficient, i.e." + "there must be a k such that (k \\choose 2)=n)!") + else: + raise ValueError('Length n of condensed distance matrix must ' + 'be a binomial coefficient, i.e. there must ' + 'be a k such that (k \\choose 2)=n)!') + except Exception as e: + if throw: + raise + if warning: + warnings.warn(str(e), stacklevel=2) + valid = False + return valid + + +def num_obs_dm(d): + """ + Return the number of original observations that correspond to a + square, redundant distance matrix. + + Parameters + ---------- + d : array_like + The target distance matrix. + + Returns + ------- + num_obs_dm : int + The number of observations in the redundant distance matrix. + + Examples + -------- + Find the number of original observations corresponding + to a square redundant distance matrix d. + + >>> from scipy.spatial.distance import num_obs_dm + >>> d = [[0, 100, 200], [100, 0, 150], [200, 150, 0]] + >>> num_obs_dm(d) + 3 + """ + d = np.asarray(d, order='c') + is_valid_dm(d, tol=np.inf, throw=True, name='d') + return d.shape[0] + + +def num_obs_y(Y): + """ + Return the number of original observations that correspond to a + condensed distance matrix. + + Parameters + ---------- + Y : array_like + Condensed distance matrix. + + Returns + ------- + n : int + The number of observations in the condensed distance matrix `Y`. + + Examples + -------- + Find the number of original observations corresponding to a + condensed distance matrix Y. + + >>> from scipy.spatial.distance import num_obs_y + >>> Y = [1, 2, 3.5, 7, 10, 4] + >>> num_obs_y(Y) + 4 + """ + Y = np.asarray(Y, order='c') + is_valid_y(Y, throw=True, name='Y') + k = Y.shape[0] + if k == 0: + raise ValueError("The number of observations cannot be determined on " + "an empty distance matrix.") + d = int(np.ceil(np.sqrt(k * 2))) + if (d * (d - 1) / 2) != k: + raise ValueError("Invalid condensed distance matrix passed. Must be " + "some k where k=(n choose 2) for some n >= 2.") + return d + + +def _prepare_out_argument(out, dtype, expected_shape): + if out is None: + return np.empty(expected_shape, dtype=dtype) + + if out.shape != expected_shape: + raise ValueError("Output array has incorrect shape.") + if not out.flags.c_contiguous: + raise ValueError("Output array must be C-contiguous.") + if out.dtype != np.float64: + raise ValueError("Output array must be double type.") + return out + + +def _pdist_callable(X, *, out, metric, **kwargs): + n = X.shape[0] + out_size = (n * (n - 1)) // 2 + dm = _prepare_out_argument(out, np.float64, (out_size,)) + k = 0 + for i in range(X.shape[0] - 1): + for j in range(i + 1, X.shape[0]): + dm[k] = metric(X[i], X[j], **kwargs) + k += 1 + return dm + + +def _cdist_callable(XA, XB, *, out, metric, **kwargs): + mA = XA.shape[0] + mB = XB.shape[0] + dm = _prepare_out_argument(out, np.float64, (mA, mB)) + for i in range(mA): + for j in range(mB): + dm[i, j] = metric(XA[i], XB[j], **kwargs) + return dm + + +def cdist(XA, XB, metric='euclidean', *, out=None, **kwargs): + """ + Compute distance between each pair of the two collections of inputs. + + See Notes for common calling conventions. + + Parameters + ---------- + XA : array_like + An :math:`m_A` by :math:`n` array of :math:`m_A` + original observations in an :math:`n`-dimensional space. + Inputs are converted to float type. + XB : array_like + An :math:`m_B` by :math:`n` array of :math:`m_B` + original observations in an :math:`n`-dimensional space. + Inputs are converted to float type. + metric : str or callable, optional + The distance metric to use. If a string, the distance function can be + 'braycurtis', 'canberra', 'chebyshev', 'cityblock', 'correlation', + 'cosine', 'dice', 'euclidean', 'hamming', 'jaccard', 'jensenshannon', + 'kulczynski1', 'mahalanobis', 'matching', 'minkowski', + 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', + 'sokalsneath', 'sqeuclidean', 'yule'. + **kwargs : dict, optional + Extra arguments to `metric`: refer to each metric documentation for a + list of all possible arguments. + + Some possible arguments: + + p : scalar + The p-norm to apply for Minkowski, weighted and unweighted. + Default: 2. + + w : array_like + The weight vector for metrics that support weights (e.g., Minkowski). + + V : array_like + The variance vector for standardized Euclidean. + Default: var(vstack([XA, XB]), axis=0, ddof=1) + + VI : array_like + The inverse of the covariance matrix for Mahalanobis. + Default: inv(cov(vstack([XA, XB].T))).T + + out : ndarray + The output array + If not None, the distance matrix Y is stored in this array. + + Returns + ------- + Y : ndarray + A :math:`m_A` by :math:`m_B` distance matrix is returned. + For each :math:`i` and :math:`j`, the metric + ``dist(u=XA[i], v=XB[j])`` is computed and stored in the + :math:`ij` th entry. + + Raises + ------ + ValueError + An exception is thrown if `XA` and `XB` do not have + the same number of columns. + + Notes + ----- + The following are common calling conventions: + + 1. ``Y = cdist(XA, XB, 'euclidean')`` + + Computes the distance between :math:`m` points using + Euclidean distance (2-norm) as the distance metric between the + points. The points are arranged as :math:`m` + :math:`n`-dimensional row vectors in the matrix X. + + 2. ``Y = cdist(XA, XB, 'minkowski', p=2.)`` + + Computes the distances using the Minkowski distance + :math:`\\|u-v\\|_p` (:math:`p`-norm) where :math:`p > 0` (note + that this is only a quasi-metric if :math:`0 < p < 1`). + + 3. ``Y = cdist(XA, XB, 'cityblock')`` + + Computes the city block or Manhattan distance between the + points. + + 4. ``Y = cdist(XA, XB, 'seuclidean', V=None)`` + + Computes the standardized Euclidean distance. The standardized + Euclidean distance between two n-vectors ``u`` and ``v`` is + + .. math:: + + \\sqrt{\\sum {(u_i-v_i)^2 / V[x_i]}}. + + V is the variance vector; V[i] is the variance computed over all + the i'th components of the points. If not passed, it is + automatically computed. + + 5. ``Y = cdist(XA, XB, 'sqeuclidean')`` + + Computes the squared Euclidean distance :math:`\\|u-v\\|_2^2` between + the vectors. + + 6. ``Y = cdist(XA, XB, 'cosine')`` + + Computes the cosine distance between vectors u and v, + + .. math:: + + 1 - \\frac{u \\cdot v} + {{\\|u\\|}_2 {\\|v\\|}_2} + + where :math:`\\|*\\|_2` is the 2-norm of its argument ``*``, and + :math:`u \\cdot v` is the dot product of :math:`u` and :math:`v`. + + 7. ``Y = cdist(XA, XB, 'correlation')`` + + Computes the correlation distance between vectors u and v. This is + + .. math:: + + 1 - \\frac{(u - \\bar{u}) \\cdot (v - \\bar{v})} + {{\\|(u - \\bar{u})\\|}_2 {\\|(v - \\bar{v})\\|}_2} + + where :math:`\\bar{v}` is the mean of the elements of vector v, + and :math:`x \\cdot y` is the dot product of :math:`x` and :math:`y`. + + + 8. ``Y = cdist(XA, XB, 'hamming')`` + + Computes the normalized Hamming distance, or the proportion of + those vector elements between two n-vectors ``u`` and ``v`` + which disagree. To save memory, the matrix ``X`` can be of type + boolean. + + 9. ``Y = cdist(XA, XB, 'jaccard')`` + + Computes the Jaccard distance between the points. Given two + vectors, ``u`` and ``v``, the Jaccard distance is the + proportion of those elements ``u[i]`` and ``v[i]`` that + disagree where at least one of them is non-zero. + + 10. ``Y = cdist(XA, XB, 'jensenshannon')`` + + Computes the Jensen-Shannon distance between two probability arrays. + Given two probability vectors, :math:`p` and :math:`q`, the + Jensen-Shannon distance is + + .. math:: + + \\sqrt{\\frac{D(p \\parallel m) + D(q \\parallel m)}{2}} + + where :math:`m` is the pointwise mean of :math:`p` and :math:`q` + and :math:`D` is the Kullback-Leibler divergence. + + 11. ``Y = cdist(XA, XB, 'chebyshev')`` + + Computes the Chebyshev distance between the points. The + Chebyshev distance between two n-vectors ``u`` and ``v`` is the + maximum norm-1 distance between their respective elements. More + precisely, the distance is given by + + .. math:: + + d(u,v) = \\max_i {|u_i-v_i|}. + + 12. ``Y = cdist(XA, XB, 'canberra')`` + + Computes the Canberra distance between the points. The + Canberra distance between two points ``u`` and ``v`` is + + .. math:: + + d(u,v) = \\sum_i \\frac{|u_i-v_i|} + {|u_i|+|v_i|}. + + 13. ``Y = cdist(XA, XB, 'braycurtis')`` + + Computes the Bray-Curtis distance between the points. The + Bray-Curtis distance between two points ``u`` and ``v`` is + + + .. math:: + + d(u,v) = \\frac{\\sum_i (|u_i-v_i|)} + {\\sum_i (|u_i+v_i|)} + + 14. ``Y = cdist(XA, XB, 'mahalanobis', VI=None)`` + + Computes the Mahalanobis distance between the points. The + Mahalanobis distance between two points ``u`` and ``v`` is + :math:`\\sqrt{(u-v)(1/V)(u-v)^T}` where :math:`(1/V)` (the ``VI`` + variable) is the inverse covariance. If ``VI`` is not None, + ``VI`` will be used as the inverse covariance matrix. + + 15. ``Y = cdist(XA, XB, 'yule')`` + + Computes the Yule distance between the boolean + vectors. (see `yule` function documentation) + + 16. ``Y = cdist(XA, XB, 'matching')`` + + Synonym for 'hamming'. + + 17. ``Y = cdist(XA, XB, 'dice')`` + + Computes the Dice distance between the boolean vectors. (see + `dice` function documentation) + + 18. ``Y = cdist(XA, XB, 'kulczynski1')`` + + Computes the kulczynski distance between the boolean + vectors. (see `kulczynski1` function documentation) + + .. deprecated:: 1.15.0 + This metric is deprecated and will be removed in SciPy 1.17.0. + Replace usage of ``cdist(XA, XB, 'kulczynski1')`` with + ``1 / cdist(XA, XB, 'jaccard') - 1``. + + 19. ``Y = cdist(XA, XB, 'rogerstanimoto')`` + + Computes the Rogers-Tanimoto distance between the boolean + vectors. (see `rogerstanimoto` function documentation) + + 20. ``Y = cdist(XA, XB, 'russellrao')`` + + Computes the Russell-Rao distance between the boolean + vectors. (see `russellrao` function documentation) + + 21. ``Y = cdist(XA, XB, 'sokalmichener')`` + + Computes the Sokal-Michener distance between the boolean + vectors. (see `sokalmichener` function documentation) + + .. deprecated:: 1.15.0 + This metric is deprecated and will be removed in SciPy 1.17.0. + Replace usage of ``cdist(XA, XB, 'sokalmichener')`` with + ``cdist(XA, XB, 'rogerstanimoto')``. + + 22. ``Y = cdist(XA, XB, 'sokalsneath')`` + + Computes the Sokal-Sneath distance between the vectors. (see + `sokalsneath` function documentation) + + 23. ``Y = cdist(XA, XB, f)`` + + Computes the distance between all pairs of vectors in X + using the user supplied 2-arity function f. For example, + Euclidean distance between the vectors could be computed + as follows:: + + dm = cdist(XA, XB, lambda u, v: np.sqrt(((u-v)**2).sum())) + + Note that you should avoid passing a reference to one of + the distance functions defined in this library. For example,:: + + dm = cdist(XA, XB, sokalsneath) + + would calculate the pair-wise distances between the vectors in + X using the Python function `sokalsneath`. This would result in + sokalsneath being called :math:`{n \\choose 2}` times, which + is inefficient. Instead, the optimized C version is more + efficient, and we call it using the following syntax:: + + dm = cdist(XA, XB, 'sokalsneath') + + Examples + -------- + Find the Euclidean distances between four 2-D coordinates: + + >>> from scipy.spatial import distance + >>> import numpy as np + >>> coords = [(35.0456, -85.2672), + ... (35.1174, -89.9711), + ... (35.9728, -83.9422), + ... (36.1667, -86.7833)] + >>> distance.cdist(coords, coords, 'euclidean') + array([[ 0. , 4.7044, 1.6172, 1.8856], + [ 4.7044, 0. , 6.0893, 3.3561], + [ 1.6172, 6.0893, 0. , 2.8477], + [ 1.8856, 3.3561, 2.8477, 0. ]]) + + + Find the Manhattan distance from a 3-D point to the corners of the unit + cube: + + >>> a = np.array([[0, 0, 0], + ... [0, 0, 1], + ... [0, 1, 0], + ... [0, 1, 1], + ... [1, 0, 0], + ... [1, 0, 1], + ... [1, 1, 0], + ... [1, 1, 1]]) + >>> b = np.array([[ 0.1, 0.2, 0.4]]) + >>> distance.cdist(a, b, 'cityblock') + array([[ 0.7], + [ 0.9], + [ 1.3], + [ 1.5], + [ 1.5], + [ 1.7], + [ 2.1], + [ 2.3]]) + + """ + # You can also call this as: + # Y = cdist(XA, XB, 'test_abc') + # where 'abc' is the metric being tested. This computes the distance + # between all pairs of vectors in XA and XB using the distance metric 'abc' + # but with a more succinct, verifiable, but less efficient implementation. + + XA = np.asarray(XA) + XB = np.asarray(XB) + + s = XA.shape + sB = XB.shape + + if len(s) != 2: + raise ValueError('XA must be a 2-dimensional array.') + if len(sB) != 2: + raise ValueError('XB must be a 2-dimensional array.') + if s[1] != sB[1]: + raise ValueError('XA and XB must have the same number of columns ' + '(i.e. feature dimension.)') + + mA = s[0] + mB = sB[0] + n = s[1] + + if callable(metric): + mstr = getattr(metric, '__name__', 'Unknown') + metric_info = _METRIC_ALIAS.get(mstr, None) + if metric_info is not None: + XA, XB, typ, kwargs = _validate_cdist_input( + XA, XB, mA, mB, n, metric_info, **kwargs) + return _cdist_callable(XA, XB, metric=metric, out=out, **kwargs) + elif isinstance(metric, str): + mstr = metric.lower() + metric_info = _METRIC_ALIAS.get(mstr, None) + if metric_info is not None: + cdist_fn = metric_info.cdist_func + return cdist_fn(XA, XB, out=out, **kwargs) + elif mstr.startswith("test_"): + metric_info = _TEST_METRICS.get(mstr, None) + if metric_info is None: + raise ValueError(f'Unknown "Test" Distance Metric: {mstr[5:]}') + XA, XB, typ, kwargs = _validate_cdist_input( + XA, XB, mA, mB, n, metric_info, **kwargs) + return _cdist_callable( + XA, XB, metric=metric_info.dist_func, out=out, **kwargs) + else: + raise ValueError(f'Unknown Distance Metric: {mstr}') + else: + raise TypeError('2nd argument metric must be a string identifier ' + 'or a function.') diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/distance.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/distance.pyi new file mode 100644 index 0000000000000000000000000000000000000000..ad058effffc500310de9a992e19c5eedf6980d43 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/distance.pyi @@ -0,0 +1,210 @@ +from typing import (overload, Any, SupportsFloat, Literal, Protocol, SupportsIndex) + +import numpy as np +from numpy.typing import ArrayLike, NDArray + +# Anything that can be parsed by `np.float64.__init__` and is thus +# compatible with `ndarray.__setitem__` (for a float64 array) +_FloatValue = None | str | bytes | SupportsFloat | SupportsIndex + +class _MetricCallback1(Protocol): + def __call__( + self, __XA: NDArray[Any], __XB: NDArray[Any] + ) -> _FloatValue: ... + +class _MetricCallback2(Protocol): + def __call__( + self, __XA: NDArray[Any], __XB: NDArray[Any], **kwargs: Any + ) -> _FloatValue: ... + +# TODO: Use a single protocol with a parameter specification variable +# once available (PEP 612) +_MetricCallback = _MetricCallback1 | _MetricCallback2 + +_MetricKind = Literal[ + 'braycurtis', + 'canberra', + 'chebychev', 'chebyshev', 'cheby', 'cheb', 'ch', + 'cityblock', 'cblock', 'cb', 'c', + 'correlation', 'co', + 'cosine', 'cos', + 'dice', + 'euclidean', 'euclid', 'eu', 'e', + 'hamming', 'hamm', 'ha', 'h', + 'minkowski', 'mi', 'm', 'pnorm', + 'jaccard', 'jacc', 'ja', 'j', + 'jensenshannon', 'js', + 'kulczynski1', + 'mahalanobis', 'mahal', 'mah', + 'rogerstanimoto', + 'russellrao', + 'seuclidean', 'se', 's', + 'sokalmichener', + 'sokalsneath', + 'sqeuclidean', 'sqe', 'sqeuclid', + 'yule', +] + +# Function annotations + +def braycurtis( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> np.float64: ... + +def canberra( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> np.float64: ... + +# TODO: Add `metric`-specific overloads +# Returns a float64 or float128 array, depending on the input dtype +@overload +def cdist( + XA: ArrayLike, + XB: ArrayLike, + metric: _MetricKind = ..., + *, + out: None | NDArray[np.floating[Any]] = ..., + p: float = ..., + w: ArrayLike | None = ..., + V: ArrayLike | None = ..., + VI: ArrayLike | None = ..., +) -> NDArray[np.floating[Any]]: ... +@overload +def cdist( + XA: ArrayLike, + XB: ArrayLike, + metric: _MetricCallback, + *, + out: None | NDArray[np.floating[Any]] = ..., + **kwargs: Any, +) -> NDArray[np.floating[Any]]: ... + +# TODO: Wait for dtype support; the return type is +# dependent on the input arrays dtype +def chebyshev( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> Any: ... + +# TODO: Wait for dtype support; the return type is +# dependent on the input arrays dtype +def cityblock( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> Any: ... + +def correlation( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ..., centered: bool = ... +) -> np.float64: ... + +def cosine( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> np.float64: ... + +def dice( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> float: ... + +def directed_hausdorff( + u: ArrayLike, v: ArrayLike, seed: int | None = ... +) -> tuple[float, int, int]: ... + +def euclidean( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> float: ... + +def hamming( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> np.float64: ... + +def is_valid_dm( + D: ArrayLike, + tol: float = ..., + throw: bool = ..., + name: str | None = ..., + warning: bool = ..., +) -> bool: ... + +def is_valid_y( + y: ArrayLike, + warning: bool = ..., + throw: bool = ..., + name: str | None = ..., +) -> bool: ... + +def jaccard( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> np.float64: ... + +def jensenshannon( + p: ArrayLike, q: ArrayLike, base: float | None = ... +) -> np.float64: ... + +def kulczynski1( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> np.float64: ... + +def mahalanobis( + u: ArrayLike, v: ArrayLike, VI: ArrayLike +) -> np.float64: ... + +def minkowski( + u: ArrayLike, v: ArrayLike, p: float = ..., w: ArrayLike | None = ... +) -> float: ... + +def num_obs_dm(d: ArrayLike) -> int: ... + +def num_obs_y(Y: ArrayLike) -> int: ... + +# TODO: Add `metric`-specific overloads +@overload +def pdist( + X: ArrayLike, + metric: _MetricKind = ..., + *, + out: None | NDArray[np.floating[Any]] = ..., + p: float = ..., + w: ArrayLike | None = ..., + V: ArrayLike | None = ..., + VI: ArrayLike | None = ..., +) -> NDArray[np.floating[Any]]: ... +@overload +def pdist( + X: ArrayLike, + metric: _MetricCallback, + *, + out: None | NDArray[np.floating[Any]] = ..., + **kwargs: Any, +) -> NDArray[np.floating[Any]]: ... + +def seuclidean( + u: ArrayLike, v: ArrayLike, V: ArrayLike +) -> float: ... + +def sokalmichener( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> float: ... + +def sokalsneath( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> np.float64: ... + +def sqeuclidean( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> np.float64: ... + +def squareform( + X: ArrayLike, + force: Literal["no", "tomatrix", "tovector"] = ..., + checks: bool = ..., +) -> NDArray[Any]: ... + +def rogerstanimoto( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> float: ... + +def russellrao( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> float: ... + +def yule( + u: ArrayLike, v: ArrayLike, w: ArrayLike | None = ... +) -> float: ... diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/kdtree.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/kdtree.py new file mode 100644 index 0000000000000000000000000000000000000000..512205ddc15dc7d427f8df908ee346895295b674 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/kdtree.py @@ -0,0 +1,25 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.spatial` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'KDTree', + 'Rectangle', + 'cKDTree', + 'distance_matrix', + 'minkowski_distance', + 'minkowski_distance_p', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="spatial", module="kdtree", + private_modules=["_kdtree"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/qhull.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/qhull.py new file mode 100644 index 0000000000000000000000000000000000000000..a8d51bf239bfe48077e66a36bcbf59f6dbadaf95 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/qhull.py @@ -0,0 +1,25 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.spatial` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'ConvexHull', + 'Delaunay', + 'HalfspaceIntersection', + 'QhullError', + 'Voronoi', + 'tsearch', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="spatial", module="qhull", + private_modules=["_qhull"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/qhull_src/COPYING.txt b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/qhull_src/COPYING.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ac02a07f45d562410025f05305c31d1ec39a28c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/qhull_src/COPYING.txt @@ -0,0 +1,38 @@ + Qhull, Copyright (c) 1993-2019 + + C.B. Barber + Arlington, MA + + and + + The National Science and Technology Research Center for + Computation and Visualization of Geometric Structures + (The Geometry Center) + University of Minnesota + + email: qhull@qhull.org + +This software includes Qhull from C.B. Barber and The Geometry Center. +Qhull is copyrighted as noted above. Qhull is free software and may +be obtained via http from www.qhull.org. It may be freely copied, modified, +and redistributed under the following conditions: + +1. All copyright notices must remain intact in all files. + +2. A copy of this text file must be distributed along with any copies + of Qhull that you redistribute; this includes copies that you have + modified, or copies of programs or other software products that + include Qhull. + +3. If you modify Qhull, you must include a notice giving the + name of the person performing the modification, the date of + modification, and the reason for such modification. + +4. When distributing modified versions of Qhull, or other software + products that include Qhull, you must provide notice that the original + source code may be obtained as noted above. + +5. There is no warranty or other guarantee of fitness for Qhull, it is + provided solely "as is". Bug reports or fixes may be sent to + qhull_bug@qhull.org; the authors may or may not act on them as + they desire. diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/cdist-X1.txt b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/cdist-X1.txt new file mode 100644 index 0000000000000000000000000000000000000000..833d5bdf2a344f585c5f34faa3e22716b1aa363c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/cdist-X1.txt @@ -0,0 +1,10 @@ +1.147593763490969421e-01 8.926156143344999849e-01 1.437758624645746330e-02 1.803435962879929022e-02 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b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-cosine-ml-iris.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b705b348fc3de5041ad9e3d6a1686af61046b2a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-cosine-ml-iris.txt @@ -0,0 +1 @@ + 1.4208365e-03 1.2652718e-05 8.9939315e-04 2.4232332e-04 9.9747033e-04 9.2045721e-04 2.2040648e-04 8.6480051e-04 1.2354911e-03 5.3650090e-06 1.0275886e-03 1.1695784e-03 2.3556571e-04 1.4590172e-03 1.8981327e-03 1.0939621e-03 1.2392314e-04 3.5850877e-04 8.6078038e-04 1.4490833e-03 8.4059347e-04 3.2873982e-03 2.7359832e-03 4.1316044e-03 2.7719149e-03 1.1814143e-03 1.1431285e-04 2.3850299e-04 1.3446247e-03 1.6406549e-03 1.2070654e-03 2.2241257e-03 1.4969348e-03 1.2354911e-03 7.6154552e-04 9.0853884e-04 1.2354911e-03 1.5825612e-04 2.3716586e-04 2.5806020e-04 8.5870759e-03 4.3447170e-04 2.6103416e-03 3.4026094e-03 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a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-double-inp.txt b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-double-inp.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a77021775ddb61d226aa8c4ba60f0af013e4a6c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-double-inp.txt @@ -0,0 +1,20 @@ +8.278938049410748956e-01 9.035293984476246987e-01 1.862188994679486731e-01 8.921151312310462433e-01 2.061859119379583216e-02 3.440636727385729676e-01 1.533779912830328662e-01 5.701372300009802663e-01 5.510020730211558915e-01 1.792362258426003496e-01 8.086175120876580857e-01 6.115487184317183189e-01 1.233471787164852618e-02 1.441643531871039663e-03 4.044309209045688913e-01 3.561398959499905148e-01 1.281985712929750720e-01 8.663300833847481508e-01 8.696027786291581352e-01 3.611727370363766454e-01 5.283537658772616830e-01 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a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-seuclidean-ml-iris.txt b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-seuclidean-ml-iris.txt new file mode 100644 index 0000000000000000000000000000000000000000..3e2759df30c14c1503818cc6a400a370e8fd8e89 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-seuclidean-ml-iris.txt @@ -0,0 +1 @@ + 1.1781739e+00 8.4573383e-01 1.1040164e+00 2.6033464e-01 1.0391769e+00 6.5951091e-01 2.6643250e-01 1.6215602e+00 9.6424206e-01 5.8926015e-01 4.4417668e-01 1.2158053e+00 1.5196051e+00 1.4342986e+00 2.2147971e+00 1.0267382e+00 1.3103399e-01 1.0246015e+00 7.0646671e-01 4.6190224e-01 5.3352899e-01 6.8496652e-01 6.2944414e-01 5.1453676e-01 1.1649855e+00 3.8639663e-01 1.3340137e-01 2.6033464e-01 8.5141186e-01 9.9757114e-01 5.0629646e-01 1.3963590e+00 1.6851313e+00 9.6424206e-01 7.1143905e-01 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a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-seuclidean-ml.txt b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-seuclidean-ml.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce80cb1ead3f88bec0fbaf6d48cb4fc584e52168 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/pdist-seuclidean-ml.txt @@ -0,0 +1 @@ + 1.4330520e+01 1.4635426e+01 1.3450855e+01 1.4761140e+01 1.3508642e+01 1.5434417e+01 1.3887693e+01 1.5166776e+01 1.3966038e+01 1.4950451e+01 1.4564587e+01 1.3834201e+01 1.4347008e+01 1.5641962e+01 1.4689053e+01 1.4418720e+01 1.4545856e+01 1.4151822e+01 1.4669017e+01 1.5150750e+01 1.3770166e+01 1.3288969e+01 1.4048191e+01 1.4049959e+01 1.4164158e+01 1.3727834e+01 1.4074687e+01 1.4321303e+01 1.2497330e+01 1.3820273e+01 1.4441030e+01 1.4780222e+01 1.2504339e+01 1.5022245e+01 1.4263650e+01 1.3704507e+01 1.3694385e+01 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67 62 0 74 58 21 23 3 5 24 50 54 99 19 +24 14 10 4 36 33 88 51 40 66 40 56 65 23 43 13 82 62 27 88 89 91 36 37 19 11 50 39 96 68 +82 7 39 80 52 90 57 17 61 15 51 71 82 15 21 44 4 46 75 50 78 18 63 75 98 45 6 16 57 25 +0 26 56 74 62 84 71 42 25 86 68 10 73 0 71 6 15 99 1 51 45 42 5 49 3 35 84 29 15 36 +60 78 76 3 95 73 36 57 35 44 50 42 85 57 18 69 37 42 75 79 15 12 74 72 51 36 79 3 58 71 +69 24 16 96 17 25 21 94 71 78 74 39 7 96 3 12 13 16 7 99 65 72 12 28 75 44 55 8 75 67 +3 13 92 9 92 83 69 91 65 92 29 63 46 1 4 62 29 85 47 93 81 3 15 23 63 50 17 9 13 13 +9 18 46 53 0 86 10 41 87 89 24 25 70 73 8 23 27 76 66 46 58 39 28 1 99 64 59 13 7 68 +72 57 90 50 47 57 34 27 94 39 23 31 74 77 45 74 18 49 96 8 95 50 20 81 73 55 72 2 32 15 +87 77 74 5 99 86 5 65 97 39 17 74 48 87 20 66 28 2 18 58 49 22 79 23 36 30 64 20 71 32 +35 43 66 96 63 77 18 90 47 86 94 19 88 79 23 12 38 4 56 42 36 2 77 1 3 17 64 52 31 24 +80 2 4 39 61 60 74 83 28 28 61 10 71 82 44 29 55 30 1 58 81 79 34 41 85 82 84 55 22 12 +76 77 58 92 90 0 54 28 77 68 58 12 1 81 37 28 19 60 71 59 25 83 8 49 52 11 28 65 59 70 +14 1 92 90 5 48 28 78 1 42 54 43 60 83 72 19 28 33 12 52 18 15 56 95 39 33 37 70 53 23 +53 76 26 31 18 81 83 79 25 1 82 43 50 24 63 49 5 23 66 37 80 41 63 77 2 28 15 21 32 93 +80 41 81 7 37 95 19 42 57 30 12 25 29 34 41 45 87 8 20 95 63 16 99 55 16 61 16 36 81 25 +32 30 2 81 23 25 88 30 37 76 52 77 79 58 21 58 10 0 13 32 72 80 3 75 75 25 21 9 79 18 +26 13 36 63 43 2 50 41 65 18 88 44 82 75 73 24 1 30 54 68 15 18 22 50 41 99 27 96 51 53 +22 4 76 11 85 88 28 75 1 2 92 66 63 3 58 43 53 5 1 24 99 90 87 87 41 1 85 37 98 92 +16 39 13 88 60 55 35 11 34 23 23 85 79 41 79 87 65 78 47 83 88 78 35 84 30 61 37 58 25 55 +27 33 15 76 82 79 73 92 93 78 18 38 22 96 63 92 41 9 50 96 14 55 8 60 15 61 97 56 43 22 +42 34 94 11 35 70 50 49 36 34 59 14 87 84 88 83 4 69 29 99 35 24 2 18 97 97 74 88 91 49 +33 25 71 12 60 2 48 22 81 33 27 95 54 25 53 14 20 43 26 96 98 37 64 27 72 33 78 45 22 61 +61 21 91 38 92 47 26 90 78 96 58 41 21 72 81 61 55 9 55 60 28 25 25 74 73 81 64 16 49 39 +90 89 12 93 91 23 82 36 63 58 73 81 49 32 60 39 4 84 73 16 18 26 58 85 46 28 82 91 72 7 +79 41 28 76 33 70 47 6 18 64 40 54 45 61 28 63 87 83 38 9 65 68 62 45 80 63 89 29 20 40 +20 59 58 23 61 79 35 19 78 2 26 48 90 34 69 31 31 42 92 33 18 74 28 47 45 52 36 89 19 40 +58 13 72 24 31 26 73 72 84 29 85 99 20 32 54 92 8 80 86 58 23 80 59 21 76 75 90 76 92 57 +74 53 80 51 8 88 84 63 82 99 97 77 38 9 51 61 37 20 68 47 65 21 53 82 85 96 62 65 35 4 +71 82 14 18 88 79 38 76 66 27 10 10 62 54 80 21 6 57 83 33 52 10 97 37 6 38 12 51 0 84 +95 30 75 92 84 30 55 57 32 44 53 24 77 81 34 84 69 85 91 33 50 72 62 79 62 12 59 75 99 81 +38 42 47 1 11 34 27 77 70 85 89 84 79 15 14 54 78 93 72 68 63 39 98 72 55 32 93 0 13 21 +3 15 10 15 3 31 84 89 53 5 60 41 66 77 45 12 68 68 50 68 99 64 46 54 30 56 2 90 99 78 +66 10 27 89 42 16 9 98 16 2 68 51 0 22 73 60 69 96 37 69 30 36 20 21 51 26 65 13 74 86 +94 58 34 97 77 88 90 75 47 30 6 36 89 66 48 9 20 6 52 45 0 37 99 46 11 53 53 72 94 40 +5 71 50 96 89 71 80 43 27 95 49 9 74 28 62 65 64 97 2 55 58 11 69 0 31 22 73 20 66 11 +63 39 84 62 64 5 56 92 26 86 19 20 56 85 42 48 56 51 54 29 26 95 72 38 70 61 16 54 57 19 +76 97 40 99 73 68 98 92 97 62 73 1 29 72 18 70 90 4 98 95 70 36 65 45 86 36 88 38 64 54 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/selfdual-4d-polytope.txt b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/selfdual-4d-polytope.txt new file mode 100644 index 0000000000000000000000000000000000000000..47ce4a7ae522fc2a2bbaa9d8ca285913b8ef0712 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/data/selfdual-4d-polytope.txt @@ -0,0 +1,27 @@ +# The facets of a self-dual 4-dim regular polytope +# with 24 octahedron facets. Taken from cddlib. +# Format b + Ax >= 0 + 1 1 1 1 1 + 1 1 1 1 -1 + 1 1 1 -1 1 + 1 1 1 -1 -1 + 1 1 -1 1 1 + 1 1 -1 1 -1 + 1 1 -1 -1 1 + 1 1 -1 -1 -1 + 1 -1 1 1 1 + 1 -1 1 1 -1 + 1 -1 1 -1 1 + 1 -1 1 -1 -1 + 1 -1 -1 1 1 + 1 -1 -1 1 -1 + 1 -1 -1 -1 1 + 1 -1 -1 -1 -1 + 1 2 0 0 0 + 1 0 2 0 0 + 1 0 0 2 0 + 1 0 0 0 2 + 1 -2 0 0 0 + 1 0 -2 0 0 + 1 0 0 -2 0 + 1 0 0 0 -2 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test__plotutils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test__plotutils.py new file mode 100644 index 0000000000000000000000000000000000000000..0e2553bf7ad56e97b567e5b334ccf17921f7f7f3 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test__plotutils.py @@ -0,0 +1,91 @@ +import pytest +import numpy as np +from numpy.testing import assert_, assert_array_equal, assert_allclose + +try: + import matplotlib + matplotlib.rcParams['backend'] = 'Agg' + import matplotlib.pyplot as plt + has_matplotlib = True +except Exception: + has_matplotlib = False + +from scipy.spatial import \ + delaunay_plot_2d, voronoi_plot_2d, convex_hull_plot_2d, \ + Delaunay, Voronoi, ConvexHull + + +@pytest.mark.skipif(not has_matplotlib, reason="Matplotlib not available") +class TestPlotting: + points = [(0,0), (0,1), (1,0), (1,1)] + + def test_delaunay(self): + # Smoke test + fig = plt.figure() + obj = Delaunay(self.points) + s_before = obj.simplices.copy() + r = delaunay_plot_2d(obj, ax=fig.gca()) + assert_array_equal(obj.simplices, s_before) # shouldn't modify + assert_(r is fig) + delaunay_plot_2d(obj, ax=fig.gca()) + + def test_voronoi(self): + # Smoke test + fig = plt.figure() + obj = Voronoi(self.points) + r = voronoi_plot_2d(obj, ax=fig.gca()) + assert_(r is fig) + voronoi_plot_2d(obj) + voronoi_plot_2d(obj, show_vertices=False) + + def test_convex_hull(self): + # Smoke test + fig = plt.figure() + tri = ConvexHull(self.points) + r = convex_hull_plot_2d(tri, ax=fig.gca()) + assert_(r is fig) + convex_hull_plot_2d(tri) + + def test_gh_19653(self): + # aspect ratio sensitivity of voronoi_plot_2d + # infinite Voronoi edges + points = np.array([[245.059986986012, 10.971011721360075], + [320.49044143557785, 10.970258360366753], + [239.79023081978914, 13.108487516946218], + [263.38325791238833, 12.93241352743668], + [219.53334398353175, 13.346107628161008]]) + vor = Voronoi(points) + fig = voronoi_plot_2d(vor) + ax = fig.gca() + infinite_segments = ax.collections[1].get_segments() + expected_segments = np.array([[[282.77256, -254.76904], + [282.729714, -4544.744698]], + [[282.77256014, -254.76904029], + [430.08561382, 4032.67658742]], + [[229.26733285, -20.39957514], + [-168.17167404, -4291.92545966]], + [[289.93433364, 5151.40412217], + [330.40553385, 9441.18887532]]]) + assert_allclose(infinite_segments, expected_segments) + + def test_gh_19653_smaller_aspect(self): + # reasonable behavior for less extreme aspect + # ratio + points = np.array([[24.059986986012, 10.971011721360075], + [32.49044143557785, 10.970258360366753], + [23.79023081978914, 13.108487516946218], + [26.38325791238833, 12.93241352743668], + [21.53334398353175, 13.346107628161008]]) + vor = Voronoi(points) + fig = voronoi_plot_2d(vor) + ax = fig.gca() + infinite_segments = ax.collections[1].get_segments() + expected_segments = np.array([[[28.274979, 8.335027], + [28.270463, -42.19763338]], + [[28.27497869, 8.33502697], + [43.73223829, 56.44555501]], + [[22.51805823, 11.8621754], + [-12.09266506, -24.95694485]], + [[29.53092448, 78.46952378], + [33.82572726, 128.81934455]]]) + assert_allclose(infinite_segments, expected_segments) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test__procrustes.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test__procrustes.py new file mode 100644 index 0000000000000000000000000000000000000000..42a3c4d35bd55e2ffecefb691c805f517c56d6ca --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test__procrustes.py @@ -0,0 +1,116 @@ +import numpy as np +from numpy.testing import assert_allclose, assert_equal, assert_almost_equal +from pytest import raises as assert_raises + +from scipy.spatial import procrustes + + +class TestProcrustes: + def setup_method(self): + """creates inputs""" + # an L + self.data1 = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], 'd') + + # a larger, shifted, mirrored L + self.data2 = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], 'd') + + # an L shifted up 1, right 1, and with point 4 shifted an extra .5 + # to the right + # pointwise distance disparity with data1: 3*(2) + (1 + 1.5^2) + self.data3 = np.array([[2, 4], [2, 3], [2, 2], [3, 2.5]], 'd') + + # data4, data5 are standardized (trace(A*A') = 1). + # procrustes should return an identical copy if they are used + # as the first matrix argument. + shiftangle = np.pi / 8 + self.data4 = np.array([[1, 0], [0, 1], [-1, 0], + [0, -1]], 'd') / np.sqrt(4) + self.data5 = np.array([[np.cos(shiftangle), np.sin(shiftangle)], + [np.cos(np.pi / 2 - shiftangle), + np.sin(np.pi / 2 - shiftangle)], + [-np.cos(shiftangle), + -np.sin(shiftangle)], + [-np.cos(np.pi / 2 - shiftangle), + -np.sin(np.pi / 2 - shiftangle)]], + 'd') / np.sqrt(4) + + def test_procrustes(self): + # tests procrustes' ability to match two matrices. + # + # the second matrix is a rotated, shifted, scaled, and mirrored version + # of the first, in two dimensions only + # + # can shift, mirror, and scale an 'L'? + a, b, disparity = procrustes(self.data1, self.data2) + assert_allclose(b, a) + assert_almost_equal(disparity, 0.) + + # if first mtx is standardized, leaves first mtx unchanged? + m4, m5, disp45 = procrustes(self.data4, self.data5) + assert_equal(m4, self.data4) + + # at worst, data3 is an 'L' with one point off by .5 + m1, m3, disp13 = procrustes(self.data1, self.data3) + #assert_(disp13 < 0.5 ** 2) + + def test_procrustes2(self): + # procrustes disparity should not depend on order of matrices + m1, m3, disp13 = procrustes(self.data1, self.data3) + m3_2, m1_2, disp31 = procrustes(self.data3, self.data1) + assert_almost_equal(disp13, disp31) + + # try with 3d, 8 pts per + rand1 = np.array([[2.61955202, 0.30522265, 0.55515826], + [0.41124708, -0.03966978, -0.31854548], + [0.91910318, 1.39451809, -0.15295084], + [2.00452023, 0.50150048, 0.29485268], + [0.09453595, 0.67528885, 0.03283872], + [0.07015232, 2.18892599, -1.67266852], + [0.65029688, 1.60551637, 0.80013549], + [-0.6607528, 0.53644208, 0.17033891]]) + + rand3 = np.array([[0.0809969, 0.09731461, -0.173442], + [-1.84888465, -0.92589646, -1.29335743], + [0.67031855, -1.35957463, 0.41938621], + [0.73967209, -0.20230757, 0.52418027], + [0.17752796, 0.09065607, 0.29827466], + [0.47999368, -0.88455717, -0.57547934], + [-0.11486344, -0.12608506, -0.3395779], + [-0.86106154, -0.28687488, 0.9644429]]) + res1, res3, disp13 = procrustes(rand1, rand3) + res3_2, res1_2, disp31 = procrustes(rand3, rand1) + assert_almost_equal(disp13, disp31) + + def test_procrustes_shape_mismatch(self): + assert_raises(ValueError, procrustes, + np.array([[1, 2], [3, 4]]), + np.array([[5, 6, 7], [8, 9, 10]])) + + def test_procrustes_empty_rows_or_cols(self): + empty = np.array([[]]) + assert_raises(ValueError, procrustes, empty, empty) + + def test_procrustes_no_variation(self): + assert_raises(ValueError, procrustes, + np.array([[42, 42], [42, 42]]), + np.array([[45, 45], [45, 45]])) + + def test_procrustes_bad_number_of_dimensions(self): + # fewer dimensions in one dataset + assert_raises(ValueError, procrustes, + np.array([1, 1, 2, 3, 5, 8]), + np.array([[1, 2], [3, 4]])) + + # fewer dimensions in both datasets + assert_raises(ValueError, procrustes, + np.array([1, 1, 2, 3, 5, 8]), + np.array([1, 1, 2, 3, 5, 8])) + + # zero dimensions + assert_raises(ValueError, procrustes, np.array(7), np.array(11)) + + # extra dimensions + assert_raises(ValueError, procrustes, + np.array([[[11], [7]]]), + np.array([[[5, 13]]])) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_distance.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_distance.py new file mode 100644 index 0000000000000000000000000000000000000000..774c773ff8bc3130030b8c9c8f6351ec467bcfc6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_distance.py @@ -0,0 +1,2374 @@ +# +# Author: Damian Eads +# Date: April 17, 2008 +# +# Copyright (C) 2008 Damian Eads +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions +# are met: +# +# 1. Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# +# 2. Redistributions in binary form must reproduce the above +# copyright notice, this list of conditions and the following +# disclaimer in the documentation and/or other materials provided +# with the distribution. +# +# 3. The name of the author may not be used to endorse or promote +# products derived from this software without specific prior +# written permission. +# +# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS +# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY +# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE +# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS +# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, +# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +import sys +import os.path + +from functools import wraps, partial +import weakref + +import numpy as np +import warnings +from numpy.linalg import norm +from numpy.testing import (verbose, assert_, + assert_array_equal, assert_equal, + assert_almost_equal, assert_allclose, + break_cycles, IS_PYPY) +import pytest + +import scipy.spatial.distance + +from scipy.spatial.distance import ( + squareform, pdist, cdist, num_obs_y, num_obs_dm, is_valid_dm, is_valid_y, + _validate_vector, _METRICS_NAMES) + +# these were missing: chebyshev cityblock +# jensenshannon and seuclidean are referenced by string name. +from scipy.spatial.distance import (braycurtis, canberra, chebyshev, cityblock, + correlation, cosine, dice, euclidean, + hamming, jaccard, jensenshannon, + kulczynski1, mahalanobis, + minkowski, rogerstanimoto, + russellrao, seuclidean, sokalmichener, # noqa: F401 + sokalsneath, sqeuclidean, yule) +from scipy._lib._util import np_long, np_ulong + + +@pytest.fixture(params=_METRICS_NAMES, scope="session") +def metric(request): + """ + Fixture for all metrics in scipy.spatial.distance + """ + return request.param + + +_filenames = [ + "cdist-X1.txt", + "cdist-X2.txt", + "iris.txt", + "pdist-boolean-inp.txt", + "pdist-chebyshev-ml-iris.txt", + "pdist-chebyshev-ml.txt", + "pdist-cityblock-ml-iris.txt", + "pdist-cityblock-ml.txt", + "pdist-correlation-ml-iris.txt", + "pdist-correlation-ml.txt", + "pdist-cosine-ml-iris.txt", + "pdist-cosine-ml.txt", + "pdist-double-inp.txt", + "pdist-euclidean-ml-iris.txt", + "pdist-euclidean-ml.txt", + "pdist-hamming-ml.txt", + "pdist-jaccard-ml.txt", + "pdist-jensenshannon-ml-iris.txt", + "pdist-jensenshannon-ml.txt", + "pdist-minkowski-3.2-ml-iris.txt", + "pdist-minkowski-3.2-ml.txt", + "pdist-minkowski-5.8-ml-iris.txt", + "pdist-seuclidean-ml-iris.txt", + "pdist-seuclidean-ml.txt", + "pdist-spearman-ml.txt", + "random-bool-data.txt", + "random-double-data.txt", + "random-int-data.txt", + "random-uint-data.txt", + ] + +_tdist = np.array([[0, 662, 877, 255, 412, 996], + [662, 0, 295, 468, 268, 400], + [877, 295, 0, 754, 564, 138], + [255, 468, 754, 0, 219, 869], + [412, 268, 564, 219, 0, 669], + [996, 400, 138, 869, 669, 0]], dtype='double') + +_ytdist = squareform(_tdist) + +# A hashmap of expected output arrays for the tests. These arrays +# come from a list of text files, which are read prior to testing. +# Each test loads inputs and outputs from this dictionary. +eo = {} + + +def load_testing_files(): + for fn in _filenames: + name = fn.replace(".txt", "").replace("-ml", "") + fqfn = os.path.join(os.path.dirname(__file__), 'data', fn) + fp = open(fqfn) + eo[name] = np.loadtxt(fp) + fp.close() + eo['pdist-boolean-inp'] = np.bool_(eo['pdist-boolean-inp']) + eo['random-bool-data'] = np.bool_(eo['random-bool-data']) + eo['random-float32-data'] = np.float32(eo['random-double-data']) + eo['random-int-data'] = np_long(eo['random-int-data']) + eo['random-uint-data'] = np_ulong(eo['random-uint-data']) + + +load_testing_files() + + +def _is_32bit(): + return np.intp(0).itemsize < 8 + + +def _chk_asarrays(arrays, axis=None): + arrays = [np.asanyarray(a) for a in arrays] + if axis is None: + # np < 1.10 ravel removes subclass from arrays + arrays = [np.ravel(a) if a.ndim != 1 else a + for a in arrays] + axis = 0 + arrays = tuple(np.atleast_1d(a) for a in arrays) + if axis < 0: + if not all(a.ndim == arrays[0].ndim for a in arrays): + raise ValueError("array ndim must be the same for neg axis") + axis = range(arrays[0].ndim)[axis] + return arrays + (axis,) + + +def _chk_weights(arrays, weights=None, axis=None, + force_weights=False, simplify_weights=True, + pos_only=False, neg_check=False, + nan_screen=False, mask_screen=False, + ddof=None): + chked = _chk_asarrays(arrays, axis=axis) + arrays, axis = chked[:-1], chked[-1] + + simplify_weights = simplify_weights and not force_weights + if not force_weights and mask_screen: + force_weights = any(np.ma.getmask(a) is not np.ma.nomask for a in arrays) + + if nan_screen: + has_nans = [np.isnan(np.sum(a)) for a in arrays] + if any(has_nans): + mask_screen = True + force_weights = True + arrays = tuple(np.ma.masked_invalid(a) if has_nan else a + for a, has_nan in zip(arrays, has_nans)) + + if weights is not None: + weights = np.asanyarray(weights) + elif force_weights: + weights = np.ones(arrays[0].shape[axis]) + else: + return arrays + (weights, axis) + + if ddof: + weights = _freq_weights(weights) + + if mask_screen: + weights = _weight_masked(arrays, weights, axis) + + if not all(weights.shape == (a.shape[axis],) for a in arrays): + raise ValueError("weights shape must match arrays along axis") + if neg_check and (weights < 0).any(): + raise ValueError("weights cannot be negative") + + if pos_only: + pos_weights = np.nonzero(weights > 0)[0] + if pos_weights.size < weights.size: + arrays = tuple(np.take(a, pos_weights, axis=axis) for a in arrays) + weights = weights[pos_weights] + if simplify_weights and (weights == 1).all(): + weights = None + return arrays + (weights, axis) + + +def _freq_weights(weights): + if weights is None: + return weights + int_weights = weights.astype(int) + if (weights != int_weights).any(): + raise ValueError(f"frequency (integer count-type) weights required {weights}") + return int_weights + + +def _weight_masked(arrays, weights, axis): + if axis is None: + axis = 0 + weights = np.asanyarray(weights) + for a in arrays: + axis_mask = np.ma.getmask(a) + if axis_mask is np.ma.nomask: + continue + if a.ndim > 1: + not_axes = tuple(i for i in range(a.ndim) if i != axis) + axis_mask = axis_mask.any(axis=not_axes) + weights *= 1 - axis_mask.astype(int) + return weights + + +def _rand_split(arrays, weights, axis, split_per, seed=None): + # Coerce `arrays` to float64 if integer, to avoid nan-to-integer issues + arrays = [arr.astype(np.float64) if np.issubdtype(arr.dtype, np.integer) + else arr for arr in arrays] + + # inverse operation for stats.collapse_weights + weights = np.array(weights, dtype=np.float64) # modified inplace; need a copy + seeded_rand = np.random.RandomState(seed) + + def mytake(a, ix, axis): + record = np.asanyarray(np.take(a, ix, axis=axis)) + return record.reshape([a.shape[i] if i != axis else 1 + for i in range(a.ndim)]) + + n_obs = arrays[0].shape[axis] + assert all(a.shape[axis] == n_obs for a in arrays), \ + "data must be aligned on sample axis" + for i in range(int(split_per) * n_obs): + split_ix = seeded_rand.randint(n_obs + i) + prev_w = weights[split_ix] + q = seeded_rand.rand() + weights[split_ix] = q * prev_w + weights = np.append(weights, (1. - q) * prev_w) + arrays = [np.append(a, mytake(a, split_ix, axis=axis), + axis=axis) for a in arrays] + return arrays, weights + + +assert_allclose_forgiving = partial(assert_allclose, atol=1e-5) + + +def _rough_check(a, b, compare_assert=assert_allclose_forgiving, + key=lambda x: x, w=None): + check_a = key(a) + check_b = key(b) + try: + if np.array(check_a != check_b).any(): # try strict equality for string types + compare_assert(check_a, check_b) + except AttributeError: # masked array + compare_assert(check_a, check_b) + except (TypeError, ValueError): # nested data structure + for a_i, b_i in zip(check_a, check_b): + _rough_check(a_i, b_i, compare_assert=compare_assert) + +# diff from test_stats: +# n_args=2, weight_arg='w', default_axis=None +# ma_safe = False, nan_safe = False +def _weight_checked(fn, n_args=2, default_axis=None, key=lambda x: x, weight_arg='w', + squeeze=True, silent=False, + ones_test=True, const_test=True, dup_test=True, + split_test=True, dud_test=True, ma_safe=False, ma_very_safe=False, + nan_safe=False, split_per=1.0, seed=0, + compare_assert=assert_allclose_forgiving): + """runs fn on its arguments 2 or 3 ways, checks that the results are the same, + then returns the same thing it would have returned before""" + @wraps(fn) + def wrapped(*args, **kwargs): + result = fn(*args, **kwargs) + + arrays = args[:n_args] + rest = args[n_args:] + weights = kwargs.get(weight_arg, None) + axis = kwargs.get('axis', default_axis) + + chked = _chk_weights(arrays, weights=weights, axis=axis, + force_weights=True, mask_screen=True) + arrays, weights, axis = chked[:-2], chked[-2], chked[-1] + if squeeze: + arrays = [np.atleast_1d(a.squeeze()) for a in arrays] + + try: + # WEIGHTS CHECK 1: EQUAL WEIGHTED OBSERVATIONS + args = tuple(arrays) + rest + if ones_test: + kwargs[weight_arg] = weights + _rough_check(result, fn(*args, **kwargs), key=key) + if const_test: + kwargs[weight_arg] = weights * 101.0 + _rough_check(result, fn(*args, **kwargs), key=key) + kwargs[weight_arg] = weights * 0.101 + try: + _rough_check(result, fn(*args, **kwargs), key=key) + except Exception as e: + raise type(e)((e, arrays, weights)) from e + + # WEIGHTS CHECK 2: ADDL 0-WEIGHTED OBS + if dud_test: + # add randomly resampled rows, weighted at 0 + dud_arrays, dud_weights = _rand_split(arrays, weights, axis, + split_per=split_per, seed=seed) + dud_weights[:weights.size] = weights # not exactly 1 because of masked arrays # noqa: E501 + dud_weights[weights.size:] = 0 + dud_args = tuple(dud_arrays) + rest + kwargs[weight_arg] = dud_weights + _rough_check(result, fn(*dud_args, **kwargs), key=key) + # increase the value of those 0-weighted rows + for a in dud_arrays: + indexer = [slice(None)] * a.ndim + indexer[axis] = slice(weights.size, None) + indexer = tuple(indexer) + a[indexer] = a[indexer] * 101 + dud_args = tuple(dud_arrays) + rest + _rough_check(result, fn(*dud_args, **kwargs), key=key) + # set those 0-weighted rows to NaNs + for a in dud_arrays: + indexer = [slice(None)] * a.ndim + indexer[axis] = slice(weights.size, None) + indexer = tuple(indexer) + a[indexer] = a[indexer] * np.nan + if kwargs.get("nan_policy", None) == "omit" and nan_safe: + dud_args = tuple(dud_arrays) + rest + _rough_check(result, fn(*dud_args, **kwargs), key=key) + # mask out those nan values + if ma_safe: + dud_arrays = [np.ma.masked_invalid(a) for a in dud_arrays] + dud_args = tuple(dud_arrays) + rest + _rough_check(result, fn(*dud_args, **kwargs), key=key) + if ma_very_safe: + kwargs[weight_arg] = None + _rough_check(result, fn(*dud_args, **kwargs), key=key) + del dud_arrays, dud_args, dud_weights + + # WEIGHTS CHECK 3: DUPLICATE DATA (DUMB SPLITTING) + if dup_test: + dup_arrays = [np.append(a, a, axis=axis) for a in arrays] + dup_weights = np.append(weights, weights) / 2.0 + dup_args = tuple(dup_arrays) + rest + kwargs[weight_arg] = dup_weights + _rough_check(result, fn(*dup_args, **kwargs), key=key) + del dup_args, dup_arrays, dup_weights + + # WEIGHT CHECK 3: RANDOM SPLITTING + if split_test and split_per > 0: + split = _rand_split(arrays, weights, axis, + split_per=split_per, seed=seed) + split_arrays, split_weights = split + split_args = tuple(split_arrays) + rest + kwargs[weight_arg] = split_weights + _rough_check(result, fn(*split_args, **kwargs), key=key) + except NotImplementedError as e: + # when some combination of arguments makes weighting impossible, + # this is the desired response + if not silent: + warnings.warn(f"{fn.__name__} NotImplemented weights: {e}", + stacklevel=3) + return result + return wrapped + + +class DummyContextManager: + def __enter__(self): + pass + def __exit__(self, *args): + pass + + +def maybe_deprecated(metric: str): + if metric in ('kulczynski1', 'sokalmichener'): + return pytest.deprecated_call() + else: + return DummyContextManager() + + +wcdist = _weight_checked(cdist, default_axis=1, squeeze=False) +wcdist_no_const = _weight_checked(cdist, default_axis=1, + squeeze=False, const_test=False) +wpdist = _weight_checked(pdist, default_axis=1, squeeze=False, n_args=1) +wpdist_no_const = _weight_checked(pdist, default_axis=1, squeeze=False, + const_test=False, n_args=1) +wrogerstanimoto = _weight_checked(rogerstanimoto) +wmatching = whamming = _weight_checked(hamming, dud_test=False) +wyule = _weight_checked(yule) +wdice = _weight_checked(dice) +wcityblock = _weight_checked(cityblock) +wchebyshev = _weight_checked(chebyshev) +wcosine = _weight_checked(cosine) +wcorrelation = _weight_checked(correlation) +wkulczynski1 = _weight_checked(kulczynski1) +wjaccard = _weight_checked(jaccard) +weuclidean = _weight_checked(euclidean, const_test=False) +wsqeuclidean = _weight_checked(sqeuclidean, const_test=False) +wbraycurtis = _weight_checked(braycurtis) +wcanberra = _weight_checked(canberra, const_test=False) +wsokalsneath = _weight_checked(sokalsneath) +wsokalmichener = _weight_checked(sokalmichener) +wrussellrao = _weight_checked(russellrao) + + +class TestCdist: + + def setup_method(self): + self.rnd_eo_names = ['random-float32-data', 'random-int-data', + 'random-uint-data', 'random-double-data', + 'random-bool-data'] + self.valid_upcasts = {'bool': [np_ulong, np_long, np.float32, np.float64], + 'uint': [np_long, np.float32, np.float64], + 'int': [np.float32, np.float64], + 'float32': [np.float64]} + + @pytest.mark.thread_unsafe + def test_cdist_extra_args(self, metric): + # Tests that args and kwargs are correctly handled + + X1 = [[1., 2., 3.], [1.2, 2.3, 3.4], [2.2, 2.3, 4.4]] + X2 = [[7., 5., 8.], [7.5, 5.8, 8.4], [5.5, 5.8, 4.4]] + kwargs = {"N0tV4l1D_p4raM": 3.14, "w": np.arange(3)} + args = [3.14] * 200 + + with pytest.raises(TypeError): + with maybe_deprecated(metric): + cdist(X1, X2, metric=metric, **kwargs) + with pytest.raises(TypeError): + with maybe_deprecated(metric): + cdist(X1, X2, metric=eval(metric), **kwargs) + with pytest.raises(TypeError): + with maybe_deprecated(metric): + cdist(X1, X2, metric="test_" + metric, **kwargs) + with pytest.raises(TypeError): + cdist(X1, X2, metric=metric, *args) + with pytest.raises(TypeError): + cdist(X1, X2, metric=eval(metric), *args) + with pytest.raises(TypeError): + cdist(X1, X2, metric="test_" + metric, *args) + + def test_cdist_extra_args_custom(self): + # Tests that args and kwargs are correctly handled + # also for custom metric + def _my_metric(x, y, arg, kwarg=1, kwarg2=2): + return arg + kwarg + kwarg2 + + X1 = [[1., 2., 3.], [1.2, 2.3, 3.4], [2.2, 2.3, 4.4]] + X2 = [[7., 5., 8.], [7.5, 5.8, 8.4], [5.5, 5.8, 4.4]] + kwargs = {"N0tV4l1D_p4raM": 3.14, "w": np.arange(3)} + args = [3.14] * 200 + + with pytest.raises(TypeError): + cdist(X1, X2, _my_metric) + with pytest.raises(TypeError): + cdist(X1, X2, _my_metric, *args) + with pytest.raises(TypeError): + cdist(X1, X2, _my_metric, **kwargs) + with pytest.raises(TypeError): + cdist(X1, X2, _my_metric, kwarg=2.2, kwarg2=3.3) + with pytest.raises(TypeError): + cdist(X1, X2, _my_metric, 1, 2, kwarg=2.2) + with pytest.raises(TypeError): + cdist(X1, X2, _my_metric, 1, 2, kwarg=2.2) + with pytest.raises(TypeError): + cdist(X1, X2, _my_metric, 1.1, 2.2, 3.3) + with pytest.raises(TypeError): + cdist(X1, X2, _my_metric, 1.1, 2.2) + with pytest.raises(TypeError): + cdist(X1, X2, _my_metric, 1.1) + with pytest.raises(TypeError): + cdist(X1, X2, _my_metric, 1.1, kwarg=2.2, kwarg2=3.3) + + # this should work + assert_allclose(cdist(X1, X2, metric=_my_metric, + arg=1.1, kwarg2=3.3), 5.4) + + def test_cdist_euclidean_random_unicode(self): + eps = 1e-15 + X1 = eo['cdist-X1'] + X2 = eo['cdist-X2'] + Y1 = wcdist_no_const(X1, X2, 'euclidean') + Y2 = wcdist_no_const(X1, X2, 'test_euclidean') + assert_allclose(Y1, Y2, rtol=eps, verbose=verbose > 2) + + @pytest.mark.parametrize("p", [0.1, 0.25, 1.0, 1.23, + 2.0, 3.8, 4.6, np.inf]) + def test_cdist_minkowski_random(self, p): + eps = 1e-13 + X1 = eo['cdist-X1'] + X2 = eo['cdist-X2'] + Y1 = wcdist_no_const(X1, X2, 'minkowski', p=p) + Y2 = wcdist_no_const(X1, X2, 'test_minkowski', p=p) + assert_allclose(Y1, Y2, atol=0, rtol=eps, verbose=verbose > 2) + + def test_cdist_cosine_random(self): + eps = 1e-14 + X1 = eo['cdist-X1'] + X2 = eo['cdist-X2'] + Y1 = wcdist(X1, X2, 'cosine') + + # Naive implementation + def norms(X): + return np.linalg.norm(X, axis=1).reshape(-1, 1) + + Y2 = 1 - np.dot((X1 / norms(X1)), (X2 / norms(X2)).T) + + assert_allclose(Y1, Y2, rtol=eps, verbose=verbose > 2) + + def test_cdist_mahalanobis(self): + # 1-dimensional observations + x1 = np.array([[2], [3]]) + x2 = np.array([[2], [5]]) + dist = cdist(x1, x2, metric='mahalanobis') + assert_allclose(dist, [[0.0, np.sqrt(4.5)], [np.sqrt(0.5), np.sqrt(2)]]) + + # 2-dimensional observations + x1 = np.array([[0, 0], [-1, 0]]) + x2 = np.array([[0, 2], [1, 0], [0, -2]]) + dist = cdist(x1, x2, metric='mahalanobis') + rt2 = np.sqrt(2) + assert_allclose(dist, [[rt2, rt2, rt2], [2, 2 * rt2, 2]]) + + # Too few observations + with pytest.raises(ValueError): + cdist([[0, 1]], [[2, 3]], metric='mahalanobis') + + def test_cdist_custom_notdouble(self): + class myclass: + pass + + def _my_metric(x, y): + if not isinstance(x[0], myclass) or not isinstance(y[0], myclass): + raise ValueError("Type has been changed") + return 1.123 + data = np.array([[myclass()]], dtype=object) + cdist_y = cdist(data, data, metric=_my_metric) + right_y = 1.123 + assert_equal(cdist_y, right_y, verbose=verbose > 2) + + def _check_calling_conventions(self, X1, X2, metric, eps=1e-07, **kwargs): + # helper function for test_cdist_calling_conventions + try: + y1 = cdist(X1, X2, metric=metric, **kwargs) + y2 = cdist(X1, X2, metric=eval(metric), **kwargs) + y3 = cdist(X1, X2, metric="test_" + metric, **kwargs) + except Exception as e: + e_cls = e.__class__ + if verbose > 2: + print(e_cls.__name__) + print(e) + with pytest.raises(e_cls): + cdist(X1, X2, metric=metric, **kwargs) + with pytest.raises(e_cls): + cdist(X1, X2, metric=eval(metric), **kwargs) + with pytest.raises(e_cls): + cdist(X1, X2, metric="test_" + metric, **kwargs) + else: + assert_allclose(y1, y2, rtol=eps, verbose=verbose > 2) + assert_allclose(y1, y3, rtol=eps, verbose=verbose > 2) + + def test_cdist_calling_conventions(self, metric): + # Ensures that specifying the metric with a str or scipy function + # gives the same behaviour (i.e. same result or same exception). + # NOTE: The correctness should be checked within each metric tests. + for eo_name in self.rnd_eo_names: + # subsampling input data to speed-up tests + # NOTE: num samples needs to be > than dimensions for mahalanobis + X1 = eo[eo_name][::5, ::-2] + X2 = eo[eo_name][1::5, ::2] + if verbose > 2: + print("testing: ", metric, " with: ", eo_name) + if metric in {'dice', 'yule', + 'rogerstanimoto', + 'russellrao', 'sokalmichener', + 'sokalsneath', + 'kulczynski1'} and 'bool' not in eo_name: + # python version permits non-bools e.g. for fuzzy logic + continue + self._check_calling_conventions(X1, X2, metric) + + # Testing built-in metrics with extra args + if metric == "seuclidean": + X12 = np.vstack([X1, X2]).astype(np.float64) + V = np.var(X12, axis=0, ddof=1) + self._check_calling_conventions(X1, X2, metric, V=V) + elif metric == "mahalanobis": + X12 = np.vstack([X1, X2]).astype(np.float64) + V = np.atleast_2d(np.cov(X12.T)) + VI = np.array(np.linalg.inv(V).T) + self._check_calling_conventions(X1, X2, metric, VI=VI) + + def test_cdist_dtype_equivalence(self, metric): + # Tests that the result is not affected by type up-casting + eps = 1e-07 + tests = [(eo['random-bool-data'], self.valid_upcasts['bool']), + (eo['random-uint-data'], self.valid_upcasts['uint']), + (eo['random-int-data'], self.valid_upcasts['int']), + (eo['random-float32-data'], self.valid_upcasts['float32'])] + for test in tests: + X1 = test[0][::5, ::-2] + X2 = test[0][1::5, ::2] + try: + y1 = cdist(X1, X2, metric=metric) + except Exception as e: + e_cls = e.__class__ + if verbose > 2: + print(e_cls.__name__) + print(e) + for new_type in test[1]: + X1new = new_type(X1) + X2new = new_type(X2) + with pytest.raises(e_cls): + cdist(X1new, X2new, metric=metric) + else: + for new_type in test[1]: + y2 = cdist(new_type(X1), new_type(X2), metric=metric) + assert_allclose(y1, y2, rtol=eps, verbose=verbose > 2) + + @pytest.mark.thread_unsafe + def test_cdist_out(self, metric): + # Test that out parameter works properly + eps = 1e-15 + X1 = eo['cdist-X1'] + X2 = eo['cdist-X2'] + out_r, out_c = X1.shape[0], X2.shape[0] + + kwargs = dict() + if metric == 'minkowski': + kwargs['p'] = 1.23 + out1 = np.empty((out_r, out_c), dtype=np.float64) + with maybe_deprecated(metric): + Y1 = cdist(X1, X2, metric, **kwargs) + with maybe_deprecated(metric): + Y2 = cdist(X1, X2, metric, out=out1, **kwargs) + + # test that output is numerically equivalent + assert_allclose(Y1, Y2, rtol=eps, verbose=verbose > 2) + + # test that Y_test1 and out1 are the same object + assert_(Y2 is out1) + + # test for incorrect shape + out2 = np.empty((out_r-1, out_c+1), dtype=np.float64) + with pytest.raises(ValueError): + with maybe_deprecated(metric): + cdist(X1, X2, metric, out=out2, **kwargs) + + # test for C-contiguous order + out3 = np.empty( + (2 * out_r, 2 * out_c), dtype=np.float64)[::2, ::2] + out4 = np.empty((out_r, out_c), dtype=np.float64, order='F') + with pytest.raises(ValueError): + with maybe_deprecated(metric): + cdist(X1, X2, metric, out=out3, **kwargs) + with pytest.raises(ValueError): + with maybe_deprecated(metric): + cdist(X1, X2, metric, out=out4, **kwargs) + + # test for incorrect dtype + out5 = np.empty((out_r, out_c), dtype=np.int64) + with pytest.raises(ValueError): + with maybe_deprecated(metric): + cdist(X1, X2, metric, out=out5, **kwargs) + + @pytest.mark.thread_unsafe + def test_striding(self, metric): + # test that striding is handled correct with calls to + # _copy_array_if_base_present + eps = 1e-15 + X1 = eo['cdist-X1'][::2, ::2] + X2 = eo['cdist-X2'][::2, ::2] + X1_copy = X1.copy() + X2_copy = X2.copy() + + # confirm equivalence + assert_equal(X1, X1_copy) + assert_equal(X2, X2_copy) + # confirm contiguity + assert_(not X1.flags.c_contiguous) + assert_(not X2.flags.c_contiguous) + assert_(X1_copy.flags.c_contiguous) + assert_(X2_copy.flags.c_contiguous) + + kwargs = dict() + if metric == 'minkowski': + kwargs['p'] = 1.23 + with maybe_deprecated(metric): + Y1 = cdist(X1, X2, metric, **kwargs) + with maybe_deprecated(metric): + Y2 = cdist(X1_copy, X2_copy, metric, **kwargs) + # test that output is numerically equivalent + assert_allclose(Y1, Y2, rtol=eps, verbose=verbose > 2) + + @pytest.mark.thread_unsafe + def test_cdist_refcount(self, metric): + x1 = np.random.rand(10, 10) + x2 = np.random.rand(10, 10) + + kwargs = dict() + if metric == 'minkowski': + kwargs['p'] = 1.23 + + with maybe_deprecated(metric): + out = cdist(x1, x2, metric=metric, **kwargs) + + # Check reference counts aren't messed up. If we only hold weak + # references, the arrays should be deallocated. + weak_refs = [weakref.ref(v) for v in (x1, x2, out)] + del x1, x2, out + + if IS_PYPY: + break_cycles() + assert all(weak_ref() is None for weak_ref in weak_refs) + + +class TestPdist: + + def setup_method(self): + self.rnd_eo_names = ['random-float32-data', 'random-int-data', + 'random-uint-data', 'random-double-data', + 'random-bool-data'] + self.valid_upcasts = {'bool': [np_ulong, np_long, np.float32, np.float64], + 'uint': [np_long, np.float32, np.float64], + 'int': [np.float32, np.float64], + 'float32': [np.float64]} + + @pytest.mark.thread_unsafe + def test_pdist_extra_args(self, metric): + # Tests that args and kwargs are correctly handled + X1 = [[1., 2.], [1.2, 2.3], [2.2, 2.3]] + kwargs = {"N0tV4l1D_p4raM": 3.14, "w": np.arange(2)} + args = [3.14] * 200 + + with pytest.raises(TypeError): + with maybe_deprecated(metric): + pdist(X1, metric=metric, **kwargs) + with pytest.raises(TypeError): + with maybe_deprecated(metric): + pdist(X1, metric=eval(metric), **kwargs) + with pytest.raises(TypeError): + with maybe_deprecated(metric): + pdist(X1, metric="test_" + metric, **kwargs) + with pytest.raises(TypeError): + pdist(X1, metric=metric, *args) + with pytest.raises(TypeError): + pdist(X1, metric=eval(metric), *args) + with pytest.raises(TypeError): + pdist(X1, metric="test_" + metric, *args) + + def test_pdist_extra_args_custom(self): + # Tests that args and kwargs are correctly handled + # also for custom metric + def _my_metric(x, y, arg, kwarg=1, kwarg2=2): + return arg + kwarg + kwarg2 + + X1 = [[1., 2.], [1.2, 2.3], [2.2, 2.3]] + kwargs = {"N0tV4l1D_p4raM": 3.14, "w": np.arange(2)} + args = [3.14] * 200 + + with pytest.raises(TypeError): + pdist(X1, _my_metric) + with pytest.raises(TypeError): + pdist(X1, _my_metric, *args) + with pytest.raises(TypeError): + pdist(X1, _my_metric, **kwargs) + with pytest.raises(TypeError): + pdist(X1, _my_metric, kwarg=2.2, kwarg2=3.3) + with pytest.raises(TypeError): + pdist(X1, _my_metric, 1, 2, kwarg=2.2) + with pytest.raises(TypeError): + pdist(X1, _my_metric, 1, 2, kwarg=2.2) + with pytest.raises(TypeError): + pdist(X1, _my_metric, 1.1, 2.2, 3.3) + with pytest.raises(TypeError): + pdist(X1, _my_metric, 1.1, 2.2) + with pytest.raises(TypeError): + pdist(X1, _my_metric, 1.1) + with pytest.raises(TypeError): + pdist(X1, _my_metric, 1.1, kwarg=2.2, kwarg2=3.3) + + # these should work + assert_allclose(pdist(X1, metric=_my_metric, + arg=1.1, kwarg2=3.3), 5.4) + + def test_pdist_euclidean_random(self): + eps = 1e-07 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-euclidean'] + Y_test1 = wpdist_no_const(X, 'euclidean') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_euclidean_random_u(self): + eps = 1e-07 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-euclidean'] + Y_test1 = wpdist_no_const(X, 'euclidean') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_euclidean_random_float32(self): + eps = 1e-07 + X = np.float32(eo['pdist-double-inp']) + Y_right = eo['pdist-euclidean'] + Y_test1 = wpdist_no_const(X, 'euclidean') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_euclidean_random_nonC(self): + eps = 1e-07 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-euclidean'] + Y_test2 = wpdist_no_const(X, 'test_euclidean') + assert_allclose(Y_test2, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_euclidean_iris_double(self): + eps = 1e-7 + X = eo['iris'] + Y_right = eo['pdist-euclidean-iris'] + Y_test1 = wpdist_no_const(X, 'euclidean') + assert_allclose(Y_test1, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_euclidean_iris_float32(self): + eps = 1e-5 + X = np.float32(eo['iris']) + Y_right = eo['pdist-euclidean-iris'] + Y_test1 = wpdist_no_const(X, 'euclidean') + assert_allclose(Y_test1, Y_right, rtol=eps, verbose=verbose > 2) + + @pytest.mark.slow + def test_pdist_euclidean_iris_nonC(self): + # Test pdist(X, 'test_euclidean') [the non-C implementation] on the + # Iris data set. + eps = 1e-7 + X = eo['iris'] + Y_right = eo['pdist-euclidean-iris'] + Y_test2 = wpdist_no_const(X, 'test_euclidean') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_seuclidean_random(self): + eps = 1e-7 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-seuclidean'] + Y_test1 = pdist(X, 'seuclidean') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_seuclidean_random_float32(self): + eps = 1e-7 + X = np.float32(eo['pdist-double-inp']) + Y_right = eo['pdist-seuclidean'] + Y_test1 = pdist(X, 'seuclidean') + assert_allclose(Y_test1, Y_right, rtol=eps) + + # Check no error is raise when V has float32 dtype (#11171). + V = np.var(X, axis=0, ddof=1) + Y_test2 = pdist(X, 'seuclidean', V=V) + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_seuclidean_random_nonC(self): + # Test pdist(X, 'test_sqeuclidean') [the non-C implementation] + eps = 1e-07 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-seuclidean'] + Y_test2 = pdist(X, 'test_seuclidean') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_seuclidean_iris(self): + eps = 1e-7 + X = eo['iris'] + Y_right = eo['pdist-seuclidean-iris'] + Y_test1 = pdist(X, 'seuclidean') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_seuclidean_iris_float32(self): + # Tests pdist(X, 'seuclidean') on the Iris data set (float32). + eps = 1e-5 + X = np.float32(eo['iris']) + Y_right = eo['pdist-seuclidean-iris'] + Y_test1 = pdist(X, 'seuclidean') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_seuclidean_iris_nonC(self): + # Test pdist(X, 'test_seuclidean') [the non-C implementation] on the + # Iris data set. + eps = 1e-7 + X = eo['iris'] + Y_right = eo['pdist-seuclidean-iris'] + Y_test2 = pdist(X, 'test_seuclidean') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_cosine_random(self): + eps = 1e-7 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-cosine'] + Y_test1 = wpdist(X, 'cosine') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_cosine_random_float32(self): + eps = 1e-7 + X = np.float32(eo['pdist-double-inp']) + Y_right = eo['pdist-cosine'] + Y_test1 = wpdist(X, 'cosine') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_cosine_random_nonC(self): + # Test pdist(X, 'test_cosine') [the non-C implementation] + eps = 1e-7 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-cosine'] + Y_test2 = wpdist(X, 'test_cosine') + assert_allclose(Y_test2, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_cosine_iris(self): + eps = 1e-05 + X = eo['iris'] + Y_right = eo['pdist-cosine-iris'] + Y_test1 = wpdist(X, 'cosine') + assert_allclose(Y_test1, Y_right, atol=eps) + + @pytest.mark.slow + def test_pdist_cosine_iris_float32(self): + eps = 1e-05 + X = np.float32(eo['iris']) + Y_right = eo['pdist-cosine-iris'] + Y_test1 = wpdist(X, 'cosine') + assert_allclose(Y_test1, Y_right, atol=eps, verbose=verbose > 2) + + @pytest.mark.slow + def test_pdist_cosine_iris_nonC(self): + eps = 1e-05 + X = eo['iris'] + Y_right = eo['pdist-cosine-iris'] + Y_test2 = wpdist(X, 'test_cosine') + assert_allclose(Y_test2, Y_right, atol=eps) + + def test_pdist_cosine_bounds(self): + # Test adapted from @joernhees's example at gh-5208: case where + # cosine distance used to be negative. XXX: very sensitive to the + # specific norm computation. + x = np.abs(np.random.RandomState(1337).rand(91)) + X = np.vstack([x, x]) + assert_(wpdist(X, 'cosine')[0] >= 0, + msg='cosine distance should be non-negative') + + def test_pdist_cityblock_random(self): + eps = 1e-7 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-cityblock'] + Y_test1 = wpdist_no_const(X, 'cityblock') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_cityblock_random_float32(self): + eps = 1e-7 + X = np.float32(eo['pdist-double-inp']) + Y_right = eo['pdist-cityblock'] + Y_test1 = wpdist_no_const(X, 'cityblock') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_cityblock_random_nonC(self): + eps = 1e-7 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-cityblock'] + Y_test2 = wpdist_no_const(X, 'test_cityblock') + assert_allclose(Y_test2, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_cityblock_iris(self): + eps = 1e-14 + X = eo['iris'] + Y_right = eo['pdist-cityblock-iris'] + Y_test1 = wpdist_no_const(X, 'cityblock') + assert_allclose(Y_test1, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_cityblock_iris_float32(self): + eps = 1e-5 + X = np.float32(eo['iris']) + Y_right = eo['pdist-cityblock-iris'] + Y_test1 = wpdist_no_const(X, 'cityblock') + assert_allclose(Y_test1, Y_right, rtol=eps, verbose=verbose > 2) + + @pytest.mark.slow + def test_pdist_cityblock_iris_nonC(self): + # Test pdist(X, 'test_cityblock') [the non-C implementation] on the + # Iris data set. + eps = 1e-14 + X = eo['iris'] + Y_right = eo['pdist-cityblock-iris'] + Y_test2 = wpdist_no_const(X, 'test_cityblock') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_correlation_random(self): + eps = 1e-7 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-correlation'] + Y_test1 = wpdist(X, 'correlation') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_correlation_random_float32(self): + eps = 1e-7 + X = np.float32(eo['pdist-double-inp']) + Y_right = eo['pdist-correlation'] + Y_test1 = wpdist(X, 'correlation') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_correlation_random_nonC(self): + eps = 1e-7 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-correlation'] + Y_test2 = wpdist(X, 'test_correlation') + assert_allclose(Y_test2, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_correlation_iris(self): + eps = 1e-7 + X = eo['iris'] + Y_right = eo['pdist-correlation-iris'] + Y_test1 = wpdist(X, 'correlation') + assert_allclose(Y_test1, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_correlation_iris_float32(self): + eps = 1e-7 + X = eo['iris'] + Y_right = np.float32(eo['pdist-correlation-iris']) + Y_test1 = wpdist(X, 'correlation') + assert_allclose(Y_test1, Y_right, rtol=eps, verbose=verbose > 2) + + @pytest.mark.slow + def test_pdist_correlation_iris_nonC(self): + if sys.maxsize > 2**32: + eps = 1e-7 + else: + pytest.skip("see gh-16456") + X = eo['iris'] + Y_right = eo['pdist-correlation-iris'] + Y_test2 = wpdist(X, 'test_correlation') + assert_allclose(Y_test2, Y_right, rtol=eps) + + @pytest.mark.parametrize("p", [0.1, 0.25, 1.0, 2.0, 3.2, np.inf]) + def test_pdist_minkowski_random_p(self, p): + eps = 1e-13 + X = eo['pdist-double-inp'] + Y1 = wpdist_no_const(X, 'minkowski', p=p) + Y2 = wpdist_no_const(X, 'test_minkowski', p=p) + assert_allclose(Y1, Y2, atol=0, rtol=eps) + + def test_pdist_minkowski_random(self): + eps = 1e-7 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-minkowski-3.2'] + Y_test1 = wpdist_no_const(X, 'minkowski', p=3.2) + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_minkowski_random_float32(self): + eps = 1e-7 + X = np.float32(eo['pdist-double-inp']) + Y_right = eo['pdist-minkowski-3.2'] + Y_test1 = wpdist_no_const(X, 'minkowski', p=3.2) + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_minkowski_random_nonC(self): + eps = 1e-7 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-minkowski-3.2'] + Y_test2 = wpdist_no_const(X, 'test_minkowski', p=3.2) + assert_allclose(Y_test2, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_minkowski_3_2_iris(self): + eps = 1e-7 + X = eo['iris'] + Y_right = eo['pdist-minkowski-3.2-iris'] + Y_test1 = wpdist_no_const(X, 'minkowski', p=3.2) + assert_allclose(Y_test1, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_minkowski_3_2_iris_float32(self): + eps = 1e-5 + X = np.float32(eo['iris']) + Y_right = eo['pdist-minkowski-3.2-iris'] + Y_test1 = wpdist_no_const(X, 'minkowski', p=3.2) + assert_allclose(Y_test1, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_minkowski_3_2_iris_nonC(self): + eps = 1e-7 + X = eo['iris'] + Y_right = eo['pdist-minkowski-3.2-iris'] + Y_test2 = wpdist_no_const(X, 'test_minkowski', p=3.2) + assert_allclose(Y_test2, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_minkowski_5_8_iris(self): + eps = 1e-7 + X = eo['iris'] + Y_right = eo['pdist-minkowski-5.8-iris'] + Y_test1 = wpdist_no_const(X, 'minkowski', p=5.8) + assert_allclose(Y_test1, Y_right, rtol=eps) + + @pytest.mark.slow + def test_pdist_minkowski_5_8_iris_float32(self): + eps = 1e-5 + X = np.float32(eo['iris']) + Y_right = eo['pdist-minkowski-5.8-iris'] + Y_test1 = wpdist_no_const(X, 'minkowski', p=5.8) + assert_allclose(Y_test1, Y_right, rtol=eps, verbose=verbose > 2) + + @pytest.mark.slow + def test_pdist_minkowski_5_8_iris_nonC(self): + eps = 1e-7 + X = eo['iris'] + Y_right = eo['pdist-minkowski-5.8-iris'] + Y_test2 = wpdist_no_const(X, 'test_minkowski', p=5.8) + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_mahalanobis(self): + # 1-dimensional observations + x = np.array([2.0, 2.0, 3.0, 5.0]).reshape(-1, 1) + dist = pdist(x, metric='mahalanobis') + assert_allclose(dist, [0.0, np.sqrt(0.5), np.sqrt(4.5), + np.sqrt(0.5), np.sqrt(4.5), np.sqrt(2.0)]) + + # 2-dimensional observations + x = np.array([[0, 0], [-1, 0], [0, 2], [1, 0], [0, -2]]) + dist = pdist(x, metric='mahalanobis') + rt2 = np.sqrt(2) + assert_allclose(dist, [rt2, rt2, rt2, rt2, 2, 2 * rt2, 2, 2, 2 * rt2, 2]) + + # Too few observations + with pytest.raises(ValueError): + wpdist([[0, 1], [2, 3]], metric='mahalanobis') + + def test_pdist_hamming_random(self): + eps = 1e-15 + X = eo['pdist-boolean-inp'] + Y_right = eo['pdist-hamming'] + Y_test1 = wpdist(X, 'hamming') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_hamming_random_float32(self): + eps = 1e-15 + X = np.float32(eo['pdist-boolean-inp']) + Y_right = eo['pdist-hamming'] + Y_test1 = wpdist(X, 'hamming') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_hamming_random_nonC(self): + eps = 1e-15 + X = eo['pdist-boolean-inp'] + Y_right = eo['pdist-hamming'] + Y_test2 = wpdist(X, 'test_hamming') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_dhamming_random(self): + eps = 1e-15 + X = np.float64(eo['pdist-boolean-inp']) + Y_right = eo['pdist-hamming'] + Y_test1 = wpdist(X, 'hamming') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_dhamming_random_float32(self): + eps = 1e-15 + X = np.float32(eo['pdist-boolean-inp']) + Y_right = eo['pdist-hamming'] + Y_test1 = wpdist(X, 'hamming') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_dhamming_random_nonC(self): + eps = 1e-15 + X = np.float64(eo['pdist-boolean-inp']) + Y_right = eo['pdist-hamming'] + Y_test2 = wpdist(X, 'test_hamming') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_jensenshannon_random(self): + eps = 1e-11 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-jensenshannon'] + Y_test1 = pdist(X, 'jensenshannon') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_jensenshannon_random_float32(self): + eps = 1e-8 + X = np.float32(eo['pdist-double-inp']) + Y_right = eo['pdist-jensenshannon'] + Y_test1 = pdist(X, 'jensenshannon') + assert_allclose(Y_test1, Y_right, rtol=eps, verbose=verbose > 2) + + def test_pdist_jensenshannon_random_nonC(self): + eps = 1e-11 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-jensenshannon'] + Y_test2 = pdist(X, 'test_jensenshannon') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_jensenshannon_iris(self): + if _is_32bit(): + # Test failing on 32-bit Linux on Azure otherwise, see gh-12810 + eps = 2.5e-10 + else: + eps = 1e-12 + + X = eo['iris'] + Y_right = eo['pdist-jensenshannon-iris'] + Y_test1 = pdist(X, 'jensenshannon') + assert_allclose(Y_test1, Y_right, atol=eps) + + def test_pdist_jensenshannon_iris_float32(self): + eps = 1e-06 + X = np.float32(eo['iris']) + Y_right = eo['pdist-jensenshannon-iris'] + Y_test1 = pdist(X, 'jensenshannon') + assert_allclose(Y_test1, Y_right, atol=eps, verbose=verbose > 2) + + def test_pdist_jensenshannon_iris_nonC(self): + eps = 5e-5 + X = eo['iris'] + Y_right = eo['pdist-jensenshannon-iris'] + Y_test2 = pdist(X, 'test_jensenshannon') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_matching_mtica1(self): + # Test matching(*,*) with mtica example #1 (nums). + m = wmatching(np.array([1, 0, 1, 1, 0]), + np.array([1, 1, 0, 1, 1])) + m2 = wmatching(np.array([1, 0, 1, 1, 0], dtype=bool), + np.array([1, 1, 0, 1, 1], dtype=bool)) + assert_allclose(m, 0.6, rtol=0, atol=1e-10) + assert_allclose(m2, 0.6, rtol=0, atol=1e-10) + + def test_pdist_matching_mtica2(self): + # Test matching(*,*) with mtica example #2. + m = wmatching(np.array([1, 0, 1]), + np.array([1, 1, 0])) + m2 = wmatching(np.array([1, 0, 1], dtype=bool), + np.array([1, 1, 0], dtype=bool)) + assert_allclose(m, 2 / 3, rtol=0, atol=1e-10) + assert_allclose(m2, 2 / 3, rtol=0, atol=1e-10) + + def test_pdist_yule_mtica1(self): + m = wyule(np.array([1, 0, 1, 1, 0]), + np.array([1, 1, 0, 1, 1])) + m2 = wyule(np.array([1, 0, 1, 1, 0], dtype=bool), + np.array([1, 1, 0, 1, 1], dtype=bool)) + if verbose > 2: + print(m) + assert_allclose(m, 2, rtol=0, atol=1e-10) + assert_allclose(m2, 2, rtol=0, atol=1e-10) + + def test_pdist_yule_mtica2(self): + m = wyule(np.array([1, 0, 1]), + np.array([1, 1, 0])) + m2 = wyule(np.array([1, 0, 1], dtype=bool), + np.array([1, 1, 0], dtype=bool)) + if verbose > 2: + print(m) + assert_allclose(m, 2, rtol=0, atol=1e-10) + assert_allclose(m2, 2, rtol=0, atol=1e-10) + + def test_pdist_dice_mtica1(self): + m = wdice(np.array([1, 0, 1, 1, 0]), + np.array([1, 1, 0, 1, 1])) + m2 = wdice(np.array([1, 0, 1, 1, 0], dtype=bool), + np.array([1, 1, 0, 1, 1], dtype=bool)) + if verbose > 2: + print(m) + assert_allclose(m, 3 / 7, rtol=0, atol=1e-10) + assert_allclose(m2, 3 / 7, rtol=0, atol=1e-10) + + def test_pdist_dice_mtica2(self): + m = wdice(np.array([1, 0, 1]), + np.array([1, 1, 0])) + m2 = wdice(np.array([1, 0, 1], dtype=bool), + np.array([1, 1, 0], dtype=bool)) + if verbose > 2: + print(m) + assert_allclose(m, 0.5, rtol=0, atol=1e-10) + assert_allclose(m2, 0.5, rtol=0, atol=1e-10) + + def test_pdist_sokalsneath_mtica1(self): + m = sokalsneath(np.array([1, 0, 1, 1, 0]), + np.array([1, 1, 0, 1, 1])) + m2 = sokalsneath(np.array([1, 0, 1, 1, 0], dtype=bool), + np.array([1, 1, 0, 1, 1], dtype=bool)) + if verbose > 2: + print(m) + assert_allclose(m, 3 / 4, rtol=0, atol=1e-10) + assert_allclose(m2, 3 / 4, rtol=0, atol=1e-10) + + def test_pdist_sokalsneath_mtica2(self): + m = wsokalsneath(np.array([1, 0, 1]), + np.array([1, 1, 0])) + m2 = wsokalsneath(np.array([1, 0, 1], dtype=bool), + np.array([1, 1, 0], dtype=bool)) + if verbose > 2: + print(m) + assert_allclose(m, 4 / 5, rtol=0, atol=1e-10) + assert_allclose(m2, 4 / 5, rtol=0, atol=1e-10) + + def test_pdist_rogerstanimoto_mtica1(self): + m = wrogerstanimoto(np.array([1, 0, 1, 1, 0]), + np.array([1, 1, 0, 1, 1])) + m2 = wrogerstanimoto(np.array([1, 0, 1, 1, 0], dtype=bool), + np.array([1, 1, 0, 1, 1], dtype=bool)) + if verbose > 2: + print(m) + assert_allclose(m, 3 / 4, rtol=0, atol=1e-10) + assert_allclose(m2, 3 / 4, rtol=0, atol=1e-10) + + def test_pdist_rogerstanimoto_mtica2(self): + m = wrogerstanimoto(np.array([1, 0, 1]), + np.array([1, 1, 0])) + m2 = wrogerstanimoto(np.array([1, 0, 1], dtype=bool), + np.array([1, 1, 0], dtype=bool)) + if verbose > 2: + print(m) + assert_allclose(m, 4 / 5, rtol=0, atol=1e-10) + assert_allclose(m2, 4 / 5, rtol=0, atol=1e-10) + + def test_pdist_russellrao_mtica1(self): + m = wrussellrao(np.array([1, 0, 1, 1, 0]), + np.array([1, 1, 0, 1, 1])) + m2 = wrussellrao(np.array([1, 0, 1, 1, 0], dtype=bool), + np.array([1, 1, 0, 1, 1], dtype=bool)) + if verbose > 2: + print(m) + assert_allclose(m, 3 / 5, rtol=0, atol=1e-10) + assert_allclose(m2, 3 / 5, rtol=0, atol=1e-10) + + def test_pdist_russellrao_mtica2(self): + m = wrussellrao(np.array([1, 0, 1]), + np.array([1, 1, 0])) + m2 = wrussellrao(np.array([1, 0, 1], dtype=bool), + np.array([1, 1, 0], dtype=bool)) + if verbose > 2: + print(m) + assert_allclose(m, 2 / 3, rtol=0, atol=1e-10) + assert_allclose(m2, 2 / 3, rtol=0, atol=1e-10) + + @pytest.mark.slow + def test_pdist_canberra_match(self): + D = eo['iris'] + if verbose > 2: + print(D.shape, D.dtype) + eps = 1e-15 + y1 = wpdist_no_const(D, "canberra") + y2 = wpdist_no_const(D, "test_canberra") + assert_allclose(y1, y2, rtol=eps, verbose=verbose > 2) + + def test_pdist_canberra_ticket_711(self): + # Test pdist(X, 'canberra') to see if Canberra gives the right result + # as reported on gh-1238. + eps = 1e-8 + pdist_y = wpdist_no_const(([3.3], [3.4]), "canberra") + right_y = 0.01492537 + assert_allclose(pdist_y, right_y, atol=eps, verbose=verbose > 2) + + def test_pdist_custom_notdouble(self): + # tests that when using a custom metric the data type is not altered + class myclass: + pass + + def _my_metric(x, y): + if not isinstance(x[0], myclass) or not isinstance(y[0], myclass): + raise ValueError("Type has been changed") + return 1.123 + data = np.array([[myclass()], [myclass()]], dtype=object) + pdist_y = pdist(data, metric=_my_metric) + right_y = 1.123 + assert_equal(pdist_y, right_y, verbose=verbose > 2) + + def _check_calling_conventions(self, X, metric, eps=1e-07, **kwargs): + # helper function for test_pdist_calling_conventions + try: + y1 = pdist(X, metric=metric, **kwargs) + y2 = pdist(X, metric=eval(metric), **kwargs) + y3 = pdist(X, metric="test_" + metric, **kwargs) + except Exception as e: + e_cls = e.__class__ + if verbose > 2: + print(e_cls.__name__) + print(e) + with pytest.raises(e_cls): + pdist(X, metric=metric, **kwargs) + with pytest.raises(e_cls): + pdist(X, metric=eval(metric), **kwargs) + with pytest.raises(e_cls): + pdist(X, metric="test_" + metric, **kwargs) + else: + assert_allclose(y1, y2, rtol=eps, verbose=verbose > 2) + assert_allclose(y1, y3, rtol=eps, verbose=verbose > 2) + + def test_pdist_calling_conventions(self, metric): + # Ensures that specifying the metric with a str or scipy function + # gives the same behaviour (i.e. same result or same exception). + # NOTE: The correctness should be checked within each metric tests. + # NOTE: Extra args should be checked with a dedicated test + for eo_name in self.rnd_eo_names: + # subsampling input data to speed-up tests + # NOTE: num samples needs to be > than dimensions for mahalanobis + X = eo[eo_name][::5, ::2] + if verbose > 2: + print("testing: ", metric, " with: ", eo_name) + if metric in {'dice', 'yule', 'matching', + 'rogerstanimoto', 'russellrao', 'sokalmichener', + 'sokalsneath', + 'kulczynski1'} and 'bool' not in eo_name: + # python version permits non-bools e.g. for fuzzy logic + continue + self._check_calling_conventions(X, metric) + + # Testing built-in metrics with extra args + if metric == "seuclidean": + V = np.var(X.astype(np.float64), axis=0, ddof=1) + self._check_calling_conventions(X, metric, V=V) + elif metric == "mahalanobis": + V = np.atleast_2d(np.cov(X.astype(np.float64).T)) + VI = np.array(np.linalg.inv(V).T) + self._check_calling_conventions(X, metric, VI=VI) + + def test_pdist_dtype_equivalence(self, metric): + # Tests that the result is not affected by type up-casting + eps = 1e-07 + tests = [(eo['random-bool-data'], self.valid_upcasts['bool']), + (eo['random-uint-data'], self.valid_upcasts['uint']), + (eo['random-int-data'], self.valid_upcasts['int']), + (eo['random-float32-data'], self.valid_upcasts['float32'])] + for test in tests: + X1 = test[0][::5, ::2] + try: + y1 = pdist(X1, metric=metric) + except Exception as e: + e_cls = e.__class__ + if verbose > 2: + print(e_cls.__name__) + print(e) + for new_type in test[1]: + X2 = new_type(X1) + with pytest.raises(e_cls): + pdist(X2, metric=metric) + else: + for new_type in test[1]: + y2 = pdist(new_type(X1), metric=metric) + assert_allclose(y1, y2, rtol=eps, verbose=verbose > 2) + + @pytest.mark.thread_unsafe + def test_pdist_out(self, metric): + # Test that out parameter works properly + eps = 1e-15 + X = eo['random-float32-data'][::5, ::2] + out_size = int((X.shape[0] * (X.shape[0] - 1)) / 2) + + kwargs = dict() + if metric == 'minkowski': + kwargs['p'] = 1.23 + out1 = np.empty(out_size, dtype=np.float64) + with maybe_deprecated(metric): + Y_right = pdist(X, metric, **kwargs) + with maybe_deprecated(metric): + Y_test1 = pdist(X, metric, out=out1, **kwargs) + + # test that output is numerically equivalent + assert_allclose(Y_test1, Y_right, rtol=eps) + + # test that Y_test1 and out1 are the same object + assert_(Y_test1 is out1) + + # test for incorrect shape + out2 = np.empty(out_size + 3, dtype=np.float64) + with pytest.raises(ValueError): + with maybe_deprecated(metric): + pdist(X, metric, out=out2, **kwargs) + + # test for (C-)contiguous output + out3 = np.empty(2 * out_size, dtype=np.float64)[::2] + with pytest.raises(ValueError): + with maybe_deprecated(metric): + pdist(X, metric, out=out3, **kwargs) + + # test for incorrect dtype + out5 = np.empty(out_size, dtype=np.int64) + with pytest.raises(ValueError): + with maybe_deprecated(metric): + pdist(X, metric, out=out5, **kwargs) + + @pytest.mark.thread_unsafe + def test_striding(self, metric): + # test that striding is handled correct with calls to + # _copy_array_if_base_present + eps = 1e-15 + X = eo['random-float32-data'][::5, ::2] + X_copy = X.copy() + + # confirm contiguity + assert_(not X.flags.c_contiguous) + assert_(X_copy.flags.c_contiguous) + + kwargs = dict() + if metric == 'minkowski': + kwargs['p'] = 1.23 + with maybe_deprecated(metric): + Y1 = pdist(X, metric, **kwargs) + with maybe_deprecated(metric): + Y2 = pdist(X_copy, metric, **kwargs) + # test that output is numerically equivalent + assert_allclose(Y1, Y2, rtol=eps, verbose=verbose > 2) + +class TestSomeDistanceFunctions: + + def setup_method(self): + # 1D arrays + x = np.array([1.0, 2.0, 3.0]) + y = np.array([1.0, 1.0, 5.0]) + + self.cases = [(x, y)] + + def test_minkowski(self): + for x, y in self.cases: + dist1 = minkowski(x, y, p=1) + assert_almost_equal(dist1, 3.0) + dist1p5 = minkowski(x, y, p=1.5) + assert_almost_equal(dist1p5, (1.0 + 2.0**1.5)**(2. / 3)) + dist2 = minkowski(x, y, p=2) + assert_almost_equal(dist2, 5.0 ** 0.5) + dist0p25 = minkowski(x, y, p=0.25) + assert_almost_equal(dist0p25, (1.0 + 2.0 ** 0.25) ** 4) + + # Check that casting input to minimum scalar type doesn't affect result + # (issue #10262). This could be extended to more test inputs with + # np.min_scalar_type(np.max(input_matrix)). + a = np.array([352, 916]) + b = np.array([350, 660]) + assert_equal(minkowski(a, b), + minkowski(a.astype('uint16'), b.astype('uint16'))) + + def test_euclidean(self): + for x, y in self.cases: + dist = weuclidean(x, y) + assert_almost_equal(dist, np.sqrt(5)) + + def test_sqeuclidean(self): + for x, y in self.cases: + dist = wsqeuclidean(x, y) + assert_almost_equal(dist, 5.0) + + def test_cosine(self): + for x, y in self.cases: + dist = wcosine(x, y) + assert_almost_equal(dist, 1.0 - 18.0 / (np.sqrt(14) * np.sqrt(27))) + + def test_cosine_output_dtype(self): + # Regression test for gh-19541 + assert isinstance(wcorrelation([1, 1], [1, 1], centered=False), float) + assert isinstance(wcosine([1, 1], [1, 1]), float) + + def test_correlation(self): + xm = np.array([-1.0, 0, 1.0]) + ym = np.array([-4.0 / 3, -4.0 / 3, 5.0 - 7.0 / 3]) + for x, y in self.cases: + dist = wcorrelation(x, y) + assert_almost_equal(dist, 1.0 - np.dot(xm, ym) / (norm(xm) * norm(ym))) + + def test_correlation_positive(self): + # Regression test for gh-12320 (negative return value due to rounding + x = np.array([0., 0., 0., 0., 0., 0., -2., 0., 0., 0., -2., -2., -2., + 0., -2., 0., -2., 0., 0., -1., -2., 0., 1., 0., 0., -2., + 0., 0., -2., 0., -2., -2., -2., -2., -2., -2., 0.]) + y = np.array([1., 1., 1., 1., 1., 1., -1., 1., 1., 1., -1., -1., -1., + 1., -1., 1., -1., 1., 1., 0., -1., 1., 2., 1., 1., -1., + 1., 1., -1., 1., -1., -1., -1., -1., -1., -1., 1.]) + dist = correlation(x, y) + assert 0 <= dist <= 10 * np.finfo(np.float64).eps + + @pytest.mark.thread_unsafe + @pytest.mark.filterwarnings('ignore:Casting complex') + @pytest.mark.parametrize("func", [correlation, cosine]) + def test_corr_dep_complex(self, func): + x = [1+0j, 2+0j] + y = [3+0j, 4+0j] + with pytest.deprecated_call(match="Complex `u` and `v` are deprecated"): + func(x, y) + + def test_mahalanobis(self): + x = np.array([1.0, 2.0, 3.0]) + y = np.array([1.0, 1.0, 5.0]) + vi = np.array([[2.0, 1.0, 0.0], [1.0, 2.0, 1.0], [0.0, 1.0, 2.0]]) + for x, y in self.cases: + dist = mahalanobis(x, y, vi) + assert_almost_equal(dist, np.sqrt(6.0)) + + +class TestSquareForm: + checked_dtypes = [np.float64, np.float32, np.int32, np.int8, bool] + + def test_squareform_matrix(self): + for dtype in self.checked_dtypes: + self.check_squareform_matrix(dtype) + + def test_squareform_vector(self): + for dtype in self.checked_dtypes: + self.check_squareform_vector(dtype) + + def check_squareform_matrix(self, dtype): + A = np.zeros((0, 0), dtype=dtype) + rA = squareform(A) + assert_equal(rA.shape, (0,)) + assert_equal(rA.dtype, dtype) + + A = np.zeros((1, 1), dtype=dtype) + rA = squareform(A) + assert_equal(rA.shape, (0,)) + assert_equal(rA.dtype, dtype) + + A = np.array([[0, 4.2], [4.2, 0]], dtype=dtype) + rA = squareform(A) + assert_equal(rA.shape, (1,)) + assert_equal(rA.dtype, dtype) + assert_array_equal(rA, np.array([4.2], dtype=dtype)) + + def check_squareform_vector(self, dtype): + v = np.zeros((0,), dtype=dtype) + rv = squareform(v) + assert_equal(rv.shape, (1, 1)) + assert_equal(rv.dtype, dtype) + assert_array_equal(rv, [[0]]) + + v = np.array([8.3], dtype=dtype) + rv = squareform(v) + assert_equal(rv.shape, (2, 2)) + assert_equal(rv.dtype, dtype) + assert_array_equal(rv, np.array([[0, 8.3], [8.3, 0]], dtype=dtype)) + + def test_squareform_multi_matrix(self): + for n in range(2, 5): + self.check_squareform_multi_matrix(n) + + def check_squareform_multi_matrix(self, n): + X = np.random.rand(n, 4) + Y = wpdist_no_const(X) + assert_equal(len(Y.shape), 1) + A = squareform(Y) + Yr = squareform(A) + s = A.shape + k = 0 + if verbose >= 3: + print(A.shape, Y.shape, Yr.shape) + assert_equal(len(s), 2) + assert_equal(len(Yr.shape), 1) + assert_equal(s[0], s[1]) + for i in range(0, s[0]): + for j in range(i + 1, s[1]): + if i != j: + assert_equal(A[i, j], Y[k]) + k += 1 + else: + assert_equal(A[i, j], 0) + + +class TestNumObsY: + + def test_num_obs_y_multi_matrix(self): + for n in range(2, 10): + X = np.random.rand(n, 4) + Y = wpdist_no_const(X) + assert_equal(num_obs_y(Y), n) + + def test_num_obs_y_1(self): + # Tests num_obs_y(y) on a condensed distance matrix over 1 + # observations. Expecting exception. + with pytest.raises(ValueError): + self.check_y(1) + + def test_num_obs_y_2(self): + # Tests num_obs_y(y) on a condensed distance matrix over 2 + # observations. + assert_(self.check_y(2)) + + def test_num_obs_y_3(self): + assert_(self.check_y(3)) + + def test_num_obs_y_4(self): + assert_(self.check_y(4)) + + def test_num_obs_y_5_10(self): + for i in range(5, 16): + self.minit(i) + + def test_num_obs_y_2_100(self): + # Tests num_obs_y(y) on 100 improper condensed distance matrices. + # Expecting exception. + a = set() + for n in range(2, 16): + a.add(n * (n - 1) / 2) + for i in range(5, 105): + if i not in a: + with pytest.raises(ValueError): + self.bad_y(i) + + def minit(self, n): + assert_(self.check_y(n)) + + def bad_y(self, n): + y = np.random.rand(n) + return num_obs_y(y) + + def check_y(self, n): + return num_obs_y(self.make_y(n)) == n + + def make_y(self, n): + return np.random.rand((n * (n - 1)) // 2) + + +class TestNumObsDM: + + def test_num_obs_dm_multi_matrix(self): + for n in range(1, 10): + X = np.random.rand(n, 4) + Y = wpdist_no_const(X) + A = squareform(Y) + if verbose >= 3: + print(A.shape, Y.shape) + assert_equal(num_obs_dm(A), n) + + def test_num_obs_dm_0(self): + # Tests num_obs_dm(D) on a 0x0 distance matrix. Expecting exception. + assert_(self.check_D(0)) + + def test_num_obs_dm_1(self): + # Tests num_obs_dm(D) on a 1x1 distance matrix. + assert_(self.check_D(1)) + + def test_num_obs_dm_2(self): + assert_(self.check_D(2)) + + def test_num_obs_dm_3(self): + assert_(self.check_D(2)) + + def test_num_obs_dm_4(self): + assert_(self.check_D(4)) + + def check_D(self, n): + return num_obs_dm(self.make_D(n)) == n + + def make_D(self, n): + return np.random.rand(n, n) + + +def is_valid_dm_throw(D): + return is_valid_dm(D, throw=True) + + +class TestIsValidDM: + + def test_is_valid_dm_improper_shape_1D_E(self): + D = np.zeros((5,), dtype=np.float64) + with pytest.raises(ValueError): + is_valid_dm_throw(D) + + def test_is_valid_dm_improper_shape_1D_F(self): + D = np.zeros((5,), dtype=np.float64) + assert_equal(is_valid_dm(D), False) + + def test_is_valid_dm_improper_shape_3D_E(self): + D = np.zeros((3, 3, 3), dtype=np.float64) + with pytest.raises(ValueError): + is_valid_dm_throw(D) + + def test_is_valid_dm_improper_shape_3D_F(self): + D = np.zeros((3, 3, 3), dtype=np.float64) + assert_equal(is_valid_dm(D), False) + + def test_is_valid_dm_nonzero_diagonal_E(self): + y = np.random.rand(10) + D = squareform(y) + for i in range(0, 5): + D[i, i] = 2.0 + with pytest.raises(ValueError): + is_valid_dm_throw(D) + + def test_is_valid_dm_nonzero_diagonal_F(self): + y = np.random.rand(10) + D = squareform(y) + for i in range(0, 5): + D[i, i] = 2.0 + assert_equal(is_valid_dm(D), False) + + def test_is_valid_dm_asymmetric_E(self): + y = np.random.rand(10) + D = squareform(y) + D[1, 3] = D[3, 1] + 1 + with pytest.raises(ValueError): + is_valid_dm_throw(D) + + def test_is_valid_dm_asymmetric_F(self): + y = np.random.rand(10) + D = squareform(y) + D[1, 3] = D[3, 1] + 1 + assert_equal(is_valid_dm(D), False) + + def test_is_valid_dm_correct_1_by_1(self): + D = np.zeros((1, 1), dtype=np.float64) + assert_equal(is_valid_dm(D), True) + + def test_is_valid_dm_correct_2_by_2(self): + y = np.random.rand(1) + D = squareform(y) + assert_equal(is_valid_dm(D), True) + + def test_is_valid_dm_correct_3_by_3(self): + y = np.random.rand(3) + D = squareform(y) + assert_equal(is_valid_dm(D), True) + + def test_is_valid_dm_correct_4_by_4(self): + y = np.random.rand(6) + D = squareform(y) + assert_equal(is_valid_dm(D), True) + + def test_is_valid_dm_correct_5_by_5(self): + y = np.random.rand(10) + D = squareform(y) + assert_equal(is_valid_dm(D), True) + + +def is_valid_y_throw(y): + return is_valid_y(y, throw=True) + + +class TestIsValidY: + # If test case name ends on "_E" then an exception is expected for the + # given input, if it ends in "_F" then False is expected for the is_valid_y + # check. Otherwise the input is expected to be valid. + + def test_is_valid_y_improper_shape_2D_E(self): + y = np.zeros((3, 3,), dtype=np.float64) + with pytest.raises(ValueError): + is_valid_y_throw(y) + + def test_is_valid_y_improper_shape_2D_F(self): + y = np.zeros((3, 3,), dtype=np.float64) + assert_equal(is_valid_y(y), False) + + def test_is_valid_y_improper_shape_3D_E(self): + y = np.zeros((3, 3, 3), dtype=np.float64) + with pytest.raises(ValueError): + is_valid_y_throw(y) + + def test_is_valid_y_improper_shape_3D_F(self): + y = np.zeros((3, 3, 3), dtype=np.float64) + assert_equal(is_valid_y(y), False) + + def test_is_valid_y_correct_2_by_2(self): + y = self.correct_n_by_n(2) + assert_equal(is_valid_y(y), True) + + def test_is_valid_y_correct_3_by_3(self): + y = self.correct_n_by_n(3) + assert_equal(is_valid_y(y), True) + + def test_is_valid_y_correct_4_by_4(self): + y = self.correct_n_by_n(4) + assert_equal(is_valid_y(y), True) + + def test_is_valid_y_correct_5_by_5(self): + y = self.correct_n_by_n(5) + assert_equal(is_valid_y(y), True) + + def test_is_valid_y_2_100(self): + a = set() + for n in range(2, 16): + a.add(n * (n - 1) / 2) + for i in range(5, 105): + if i not in a: + with pytest.raises(ValueError): + self.bad_y(i) + + def bad_y(self, n): + y = np.random.rand(n) + return is_valid_y(y, throw=True) + + def correct_n_by_n(self, n): + y = np.random.rand((n * (n - 1)) // 2) + return y + + +@pytest.mark.parametrize("p", [-10.0, -0.5, 0.0]) +def test_bad_p(p): + # Raise ValueError if p <=0. + with pytest.raises(ValueError): + minkowski([1, 2], [3, 4], p) + with pytest.raises(ValueError): + minkowski([1, 2], [3, 4], p, [1, 1]) + + +def test_sokalsneath_all_false(): + # Regression test for ticket #876 + with pytest.raises(ValueError): + sokalsneath([False, False, False], [False, False, False]) + + +def test_canberra(): + # Regression test for ticket #1430. + assert_equal(wcanberra([1, 2, 3], [2, 4, 6]), 1) + assert_equal(wcanberra([1, 1, 0, 0], [1, 0, 1, 0]), 2) + + +def test_braycurtis(): + # Regression test for ticket #1430. + assert_almost_equal(wbraycurtis([1, 2, 3], [2, 4, 6]), 1. / 3, decimal=15) + assert_almost_equal(wbraycurtis([1, 1, 0, 0], [1, 0, 1, 0]), 0.5, decimal=15) + + +def test_euclideans(): + # Regression test for ticket #1328. + x1 = np.array([1, 1, 1]) + x2 = np.array([0, 0, 0]) + + # Basic test of the calculation. + assert_almost_equal(wsqeuclidean(x1, x2), 3.0, decimal=14) + assert_almost_equal(weuclidean(x1, x2), np.sqrt(3), decimal=14) + + # Check flattening for (1, N) or (N, 1) inputs + with pytest.raises(ValueError, match="Input vector should be 1-D"): + weuclidean(x1[np.newaxis, :], x2[np.newaxis, :]), np.sqrt(3) + with pytest.raises(ValueError, match="Input vector should be 1-D"): + wsqeuclidean(x1[np.newaxis, :], x2[np.newaxis, :]) + with pytest.raises(ValueError, match="Input vector should be 1-D"): + wsqeuclidean(x1[:, np.newaxis], x2[:, np.newaxis]) + + # Distance metrics only defined for vectors (= 1-D) + x = np.arange(4).reshape(2, 2) + with pytest.raises(ValueError): + weuclidean(x, x) + with pytest.raises(ValueError): + wsqeuclidean(x, x) + + # Another check, with random data. + rs = np.random.RandomState(1234567890) + x = rs.rand(10) + y = rs.rand(10) + d1 = weuclidean(x, y) + d2 = wsqeuclidean(x, y) + assert_almost_equal(d1**2, d2, decimal=14) + + +def test_hamming_unequal_length(): + # Regression test for gh-4290. + x = [0, 0, 1] + y = [1, 0, 1, 0] + # Used to give an AttributeError from ndarray.mean called on bool + with pytest.raises(ValueError): + whamming(x, y) + + +def test_hamming_unequal_length_with_w(): + u = [0, 0, 1] + v = [0, 0, 1] + w = [1, 0, 1, 0] + msg = "'w' should have the same length as 'u' and 'v'." + with pytest.raises(ValueError, match=msg): + whamming(u, v, w) + + +def test_hamming_string_array(): + # https://github.com/scikit-learn/scikit-learn/issues/4014 + a = np.array(['eggs', 'spam', 'spam', 'eggs', 'spam', 'spam', 'spam', + 'spam', 'spam', 'spam', 'spam', 'eggs', 'eggs', 'spam', + 'eggs', 'eggs', 'eggs', 'eggs', 'eggs', 'spam'], + dtype='|S4') + b = np.array(['eggs', 'spam', 'spam', 'eggs', 'eggs', 'spam', 'spam', + 'spam', 'spam', 'eggs', 'spam', 'eggs', 'spam', 'eggs', + 'spam', 'spam', 'eggs', 'spam', 'spam', 'eggs'], + dtype='|S4') + desired = 0.45 + assert_allclose(whamming(a, b), desired) + + +def test_minkowski_w(): + # Regression test for gh-8142. + arr_in = np.array([[83.33333333, 100., 83.33333333, 100., 36., + 60., 90., 150., 24., 48.], + [83.33333333, 100., 83.33333333, 100., 36., + 60., 90., 150., 24., 48.]]) + p0 = pdist(arr_in, metric='minkowski', p=1, w=None) + c0 = cdist(arr_in, arr_in, metric='minkowski', p=1, w=None) + p1 = pdist(arr_in, metric='minkowski', p=1) + c1 = cdist(arr_in, arr_in, metric='minkowski', p=1) + + assert_allclose(p0, p1, rtol=1e-15) + assert_allclose(c0, c1, rtol=1e-15) + + +def test_sqeuclidean_dtypes(): + # Assert that sqeuclidean returns the right types of values. + # Integer types should be converted to floating for stability. + # Floating point types should be the same as the input. + x = [1, 2, 3] + y = [4, 5, 6] + + for dtype in [np.int8, np.int16, np.int32, np.int64]: + d = wsqeuclidean(np.asarray(x, dtype=dtype), np.asarray(y, dtype=dtype)) + assert_(np.issubdtype(d.dtype, np.floating)) + + for dtype in [np.uint8, np.uint16, np.uint32, np.uint64]: + umax = np.iinfo(dtype).max + d1 = wsqeuclidean([0], np.asarray([umax], dtype=dtype)) + d2 = wsqeuclidean(np.asarray([umax], dtype=dtype), [0]) + + assert_equal(d1, d2) + assert_equal(d1, np.float64(umax)**2) + + dtypes = [np.float32, np.float64, np.complex64, np.complex128] + for dtype in ['float16', 'float128']: + # These aren't present in older numpy versions; float128 may also not + # be present on all platforms. + if hasattr(np, dtype): + dtypes.append(getattr(np, dtype)) + + for dtype in dtypes: + d = wsqeuclidean(np.asarray(x, dtype=dtype), np.asarray(y, dtype=dtype)) + assert_equal(d.dtype, dtype) + + +@pytest.mark.thread_unsafe +def test_sokalmichener(): + # Test that sokalmichener has the same result for bool and int inputs. + p = [True, True, False] + q = [True, False, True] + x = [int(b) for b in p] + y = [int(b) for b in q] + with pytest.deprecated_call(): + dist1 = sokalmichener(p, q) + with pytest.deprecated_call(): + dist2 = sokalmichener(x, y) + # These should be exactly the same. + assert_equal(dist1, dist2) + + +@pytest.mark.thread_unsafe +def test_sokalmichener_with_weight(): + # from: | 1 | | 0 | + # to: | 1 | | 1 | + # weight| | 1 | | 0.2 + ntf = 0 * 1 + 0 * 0.2 + nft = 0 * 1 + 1 * 0.2 + ntt = 1 * 1 + 0 * 0.2 + nff = 0 * 1 + 0 * 0.2 + expected = 2 * (nft + ntf) / (ntt + nff + 2 * (nft + ntf)) + assert_almost_equal(expected, 0.2857143) + with pytest.deprecated_call(): + actual = sokalmichener([1, 0], [1, 1], w=[1, 0.2]) + assert_almost_equal(expected, actual) + + a1 = [False, False, True, True, True, False, False, True, True, True, True, + True, True, False, True, False, False, False, True, True] + a2 = [True, True, True, False, False, True, True, True, False, True, + True, True, True, True, False, False, False, True, True, True] + + for w in [0.05, 0.1, 1.0, 20.0]: + with pytest.deprecated_call(): + assert_almost_equal(sokalmichener(a2, a1, [w]), 0.6666666666666666) + + +@pytest.mark.thread_unsafe +def test_modifies_input(metric): + # test whether cdist or pdist modifies input arrays + X1 = np.asarray([[1., 2., 3.], + [1.2, 2.3, 3.4], + [2.2, 2.3, 4.4], + [22.2, 23.3, 44.4]]) + X1_copy = X1.copy() + with maybe_deprecated(metric): + cdist(X1, X1, metric) + with maybe_deprecated(metric): + pdist(X1, metric) + assert_array_equal(X1, X1_copy) + + +@pytest.mark.thread_unsafe +def test_Xdist_deprecated_args(metric): + # testing both cdist and pdist deprecated warnings + X1 = np.asarray([[1., 2., 3.], + [1.2, 2.3, 3.4], + [2.2, 2.3, 4.4], + [22.2, 23.3, 44.4]]) + + with pytest.raises(TypeError): + cdist(X1, X1, metric, 2.) + + with pytest.raises(TypeError): + pdist(X1, metric, 2.) + + for arg in ["p", "V", "VI"]: + kwargs = {arg: "foo"} + + if ((arg == "V" and metric == "seuclidean") + or (arg == "VI" and metric == "mahalanobis") + or (arg == "p" and metric == "minkowski")): + continue + + with pytest.raises(TypeError): + with maybe_deprecated(metric): + cdist(X1, X1, metric, **kwargs) + + with pytest.raises(TypeError): + with maybe_deprecated(metric): + pdist(X1, metric, **kwargs) + + +@pytest.mark.thread_unsafe +def test_Xdist_non_negative_weights(metric): + X = eo['random-float32-data'][::5, ::2] + w = np.ones(X.shape[1]) + w[::5] = -w[::5] + + if metric in ['seuclidean', 'mahalanobis', 'jensenshannon']: + pytest.skip("not applicable") + + for m in [metric, eval(metric), "test_" + metric]: + with pytest.raises(ValueError): + with maybe_deprecated(metric): + pdist(X, m, w=w) + with pytest.raises(ValueError): + with maybe_deprecated(metric): + cdist(X, X, m, w=w) + + +def test__validate_vector(): + x = [1, 2, 3] + y = _validate_vector(x) + assert_array_equal(y, x) + + y = _validate_vector(x, dtype=np.float64) + assert_array_equal(y, x) + assert_equal(y.dtype, np.float64) + + x = [1] + y = _validate_vector(x) + assert_equal(y.ndim, 1) + assert_equal(y, x) + + x = 1 + with pytest.raises(ValueError, match="Input vector should be 1-D"): + _validate_vector(x) + + x = np.arange(5).reshape(1, -1, 1) + with pytest.raises(ValueError, match="Input vector should be 1-D"): + _validate_vector(x) + + x = [[1, 2], [3, 4]] + with pytest.raises(ValueError, match="Input vector should be 1-D"): + _validate_vector(x) + +def test_yule_all_same(): + # Test yule avoids a divide by zero when exactly equal + x = np.ones((2, 6), dtype=bool) + d = wyule(x[0], x[0]) + assert d == 0.0 + + d = pdist(x, 'yule') + assert_equal(d, [0.0]) + + d = cdist(x[:1], x[:1], 'yule') + assert_equal(d, [[0.0]]) + + +def test_jensenshannon(): + assert_almost_equal(jensenshannon([1.0, 0.0, 0.0], [0.0, 1.0, 0.0], 2.0), + 1.0) + assert_almost_equal(jensenshannon([1.0, 0.0], [0.5, 0.5]), + 0.46450140402245893) + assert_almost_equal(jensenshannon([1.0, 0.0, 0.0], [1.0, 0.0, 0.0]), 0.0) + + assert_almost_equal(jensenshannon([[1.0, 2.0]], [[0.5, 1.5]], axis=0), + [0.0, 0.0]) + assert_almost_equal(jensenshannon([[1.0, 2.0]], [[0.5, 1.5]], axis=1), + [0.0649045]) + assert_almost_equal(jensenshannon([[1.0, 2.0]], [[0.5, 1.5]], axis=0, + keepdims=True), [[0.0, 0.0]]) + assert_almost_equal(jensenshannon([[1.0, 2.0]], [[0.5, 1.5]], axis=1, + keepdims=True), [[0.0649045]]) + + a = np.array([[1, 2, 3, 4], + [5, 6, 7, 8], + [9, 10, 11, 12]]) + b = np.array([[13, 14, 15, 16], + [17, 18, 19, 20], + [21, 22, 23, 24]]) + + assert_almost_equal(jensenshannon(a, b, axis=0), + [0.1954288, 0.1447697, 0.1138377, 0.0927636]) + assert_almost_equal(jensenshannon(a, b, axis=1), + [0.1402339, 0.0399106, 0.0201815]) + + +def test_gh_17703(): + arr_1 = np.array([1, 0, 0]) + arr_2 = np.array([2, 0, 0]) + expected = dice(arr_1, arr_2) + actual = pdist([arr_1, arr_2], metric='dice') + assert_allclose(actual, expected) + actual = cdist(np.atleast_2d(arr_1), + np.atleast_2d(arr_2), metric='dice') + assert_allclose(actual, expected) + + +@pytest.mark.thread_unsafe +def test_immutable_input(metric): + if metric in ("jensenshannon", "mahalanobis", "seuclidean"): + pytest.skip("not applicable") + x = np.arange(10, dtype=np.float64) + x.setflags(write=False) + with maybe_deprecated(metric): + getattr(scipy.spatial.distance, metric)(x, x, w=x) + + +class TestJaccard: + + def test_pdist_jaccard_random(self): + eps = 1e-8 + X = eo['pdist-boolean-inp'] + Y_right = eo['pdist-jaccard'] + Y_test1 = wpdist(X, 'jaccard') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_jaccard_random_float32(self): + eps = 1e-8 + X = np.float32(eo['pdist-boolean-inp']) + Y_right = eo['pdist-jaccard'] + Y_test1 = wpdist(X, 'jaccard') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_jaccard_random_nonC(self): + eps = 1e-8 + X = eo['pdist-boolean-inp'] + Y_right = eo['pdist-jaccard'] + Y_test2 = wpdist(X, 'test_jaccard') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_djaccard_random(self): + eps = 1e-8 + X = np.float64(eo['pdist-boolean-inp']) + Y_right = eo['pdist-jaccard'] + Y_test1 = wpdist(X, 'jaccard') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_djaccard_random_float32(self): + eps = 1e-8 + X = np.float32(eo['pdist-boolean-inp']) + Y_right = eo['pdist-jaccard'] + Y_test1 = wpdist(X, 'jaccard') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_djaccard_allzeros(self): + eps = 1e-15 + Y = pdist(np.zeros((5, 3)), 'jaccard') + assert_allclose(np.zeros(10), Y, rtol=eps) + + def test_pdist_djaccard_random_nonC(self): + eps = 1e-8 + X = np.float64(eo['pdist-boolean-inp']) + Y_right = eo['pdist-jaccard'] + Y_test2 = wpdist(X, 'test_jaccard') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_djaccard_allzeros_nonC(self): + eps = 1e-15 + Y = pdist(np.zeros((5, 3)), 'test_jaccard') + assert_allclose(np.zeros(10), Y, rtol=eps) + + def test_pdist_jaccard_mtica1(self): + m = wjaccard(np.array([1, 0, 1, 1, 0]), + np.array([1, 1, 0, 1, 1])) + m2 = wjaccard(np.array([1, 0, 1, 1, 0], dtype=bool), + np.array([1, 1, 0, 1, 1], dtype=bool)) + assert_allclose(m, 0.6, rtol=0, atol=1e-10) + assert_allclose(m2, 0.6, rtol=0, atol=1e-10) + + def test_pdist_jaccard_mtica2(self): + m = wjaccard(np.array([1, 0, 1]), + np.array([1, 1, 0])) + m2 = wjaccard(np.array([1, 0, 1], dtype=bool), + np.array([1, 1, 0], dtype=bool)) + assert_allclose(m, 2 / 3, rtol=0, atol=1e-10) + assert_allclose(m2, 2 / 3, rtol=0, atol=1e-10) + + def test_non_01_input(self): + # Non-0/1 numeric input should be cast to bool before computation. + # See gh-21176. + x = np.array([-10, 2.5, 0]) # [True, True, False] + y = np.array([ 2, -5, 2]) # [True, True, True] + eps = np.finfo(float).eps + assert_allclose(jaccard(x, y), 1/3, rtol=eps) + assert_allclose(cdist([x], [y], 'jaccard'), [[1/3]]) + assert_allclose(pdist([x, y], 'jaccard'), [1/3]) + + +class TestChebyshev: + + def test_pdist_chebyshev_random(self): + eps = 1e-8 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-chebyshev'] + Y_test1 = pdist(X, 'chebyshev') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_chebyshev_random_float32(self): + eps = 1e-7 + X = np.float32(eo['pdist-double-inp']) + Y_right = eo['pdist-chebyshev'] + Y_test1 = pdist(X, 'chebyshev') + assert_allclose(Y_test1, Y_right, rtol=eps, verbose=verbose > 2) + + def test_pdist_chebyshev_random_nonC(self): + eps = 1e-8 + X = eo['pdist-double-inp'] + Y_right = eo['pdist-chebyshev'] + Y_test2 = pdist(X, 'test_chebyshev') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_pdist_chebyshev_iris(self): + eps = 1e-14 + X = eo['iris'] + Y_right = eo['pdist-chebyshev-iris'] + Y_test1 = pdist(X, 'chebyshev') + assert_allclose(Y_test1, Y_right, rtol=eps) + + def test_pdist_chebyshev_iris_float32(self): + eps = 1e-5 + X = np.float32(eo['iris']) + Y_right = eo['pdist-chebyshev-iris'] + Y_test1 = pdist(X, 'chebyshev') + assert_allclose(Y_test1, Y_right, rtol=eps, verbose=verbose > 2) + + def test_pdist_chebyshev_iris_nonC(self): + eps = 1e-14 + X = eo['iris'] + Y_right = eo['pdist-chebyshev-iris'] + Y_test2 = pdist(X, 'test_chebyshev') + assert_allclose(Y_test2, Y_right, rtol=eps) + + def test_weighted(self): + # Basic test for weighted Chebyshev. Only components with non-zero + # weight participate in the 'max'. + x = [1, 2, 3] + y = [6, 5, 4] + w = [0, 1, 5] + assert_equal(chebyshev(x, y, w), 3) + assert_equal(pdist([x, y], 'chebyshev', w=w), [3]) + assert_equal(cdist([x], [y], 'chebyshev', w=w), [[3]]) + + def test_zero_weight(self): + # If the weight is identically zero, the distance should be zero. + x = [1, 2, 3] + y = [6, 5, 4] + w = [0, 0, 0] + assert_equal(chebyshev(x, y, w), 0) + assert_equal(pdist([x, y], 'chebyshev', w=w), [0]) + assert_equal(cdist([x], [y], 'chebyshev', w=w), [[0]]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_hausdorff.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_hausdorff.py new file mode 100644 index 0000000000000000000000000000000000000000..45d89e868c8c2daca6e9c30973399fbbfc5d52fd --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_hausdorff.py @@ -0,0 +1,199 @@ +import numpy as np +from numpy.testing import (assert_allclose, + assert_array_equal, + assert_equal) +import pytest +from scipy.spatial.distance import directed_hausdorff +from scipy.spatial import distance +from scipy._lib._util import check_random_state + + +class TestHausdorff: + # Test various properties of the directed Hausdorff code. + + def setup_method(self): + np.random.seed(1234) + random_angles = np.random.random(100) * np.pi * 2 + random_columns = np.column_stack( + (random_angles, random_angles, np.zeros(100))) + random_columns[..., 0] = np.cos(random_columns[..., 0]) + random_columns[..., 1] = np.sin(random_columns[..., 1]) + random_columns_2 = np.column_stack( + (random_angles, random_angles, np.zeros(100))) + random_columns_2[1:, 0] = np.cos(random_columns_2[1:, 0]) * 2.0 + random_columns_2[1:, 1] = np.sin(random_columns_2[1:, 1]) * 2.0 + # move one point farther out so we don't have two perfect circles + random_columns_2[0, 0] = np.cos(random_columns_2[0, 0]) * 3.3 + random_columns_2[0, 1] = np.sin(random_columns_2[0, 1]) * 3.3 + self.path_1 = random_columns + self.path_2 = random_columns_2 + self.path_1_4d = np.insert(self.path_1, 3, 5, axis=1) + self.path_2_4d = np.insert(self.path_2, 3, 27, axis=1) + + def test_symmetry(self): + # Ensure that the directed (asymmetric) Hausdorff distance is + # actually asymmetric + + forward = directed_hausdorff(self.path_1, self.path_2)[0] + reverse = directed_hausdorff(self.path_2, self.path_1)[0] + assert forward != reverse + + def test_brute_force_comparison_forward(self): + # Ensure that the algorithm for directed_hausdorff gives the + # same result as the simple / brute force approach in the + # forward direction. + actual = directed_hausdorff(self.path_1, self.path_2)[0] + # brute force over rows: + expected = max(np.amin(distance.cdist(self.path_1, self.path_2), + axis=1)) + assert_allclose(actual, expected) + + def test_brute_force_comparison_reverse(self): + # Ensure that the algorithm for directed_hausdorff gives the + # same result as the simple / brute force approach in the + # reverse direction. + actual = directed_hausdorff(self.path_2, self.path_1)[0] + # brute force over columns: + expected = max(np.amin(distance.cdist(self.path_1, self.path_2), + axis=0)) + assert_allclose(actual, expected) + + def test_degenerate_case(self): + # The directed Hausdorff distance must be zero if both input + # data arrays match. + actual = directed_hausdorff(self.path_1, self.path_1)[0] + assert_allclose(actual, 0.0) + + def test_2d_data_forward(self): + # Ensure that 2D data is handled properly for a simple case + # relative to brute force approach. + actual = directed_hausdorff(self.path_1[..., :2], + self.path_2[..., :2])[0] + expected = max(np.amin(distance.cdist(self.path_1[..., :2], + self.path_2[..., :2]), + axis=1)) + assert_allclose(actual, expected) + + def test_4d_data_reverse(self): + # Ensure that 4D data is handled properly for a simple case + # relative to brute force approach. + actual = directed_hausdorff(self.path_2_4d, self.path_1_4d)[0] + # brute force over columns: + expected = max(np.amin(distance.cdist(self.path_1_4d, self.path_2_4d), + axis=0)) + assert_allclose(actual, expected) + + def test_indices(self): + # Ensure that correct point indices are returned -- they should + # correspond to the Hausdorff pair + path_simple_1 = np.array([[-1,-12],[0,0], [1,1], [3,7], [1,2]]) + path_simple_2 = np.array([[0,0], [1,1], [4,100], [10,9]]) + actual = directed_hausdorff(path_simple_2, path_simple_1)[1:] + expected = (2, 3) + assert_array_equal(actual, expected) + + def test_random_state(self): + # ensure that the global random state is not modified because + # the directed Hausdorff algorithm uses randomization + rs = check_random_state(None) + old_global_state = rs.get_state() + directed_hausdorff(self.path_1, self.path_2) + rs2 = check_random_state(None) + new_global_state = rs2.get_state() + assert_equal(new_global_state, old_global_state) + + @pytest.mark.parametrize("seed", [None, 27870671, np.random.default_rng(177)]) + def test_random_state_None_int(self, seed): + # check that seed values of None or int do not alter global + # random state + rs = check_random_state(None) + old_global_state = rs.get_state() + directed_hausdorff(self.path_1, self.path_2, seed) + rs2 = check_random_state(None) + new_global_state = rs2.get_state() + assert_equal(new_global_state, old_global_state) + + def test_invalid_dimensions(self): + # Ensure that a ValueError is raised when the number of columns + # is not the same + rng = np.random.default_rng(189048172503940875434364128139223470523) + A = rng.random((3, 2)) + B = rng.random((3, 5)) + msg = r"need to have the same number of columns" + with pytest.raises(ValueError, match=msg): + directed_hausdorff(A, B) + + # preserve use of legacy keyword `seed` during SPEC 7 transition + @pytest.mark.parametrize("A, B, seed, expected", [ + # the two cases from gh-11332 + ([(0,0)], + [(0,1), (0,0)], + np.int64(0), + (0.0, 0, 1)), + ([(0,0)], + [(0,1), (0,0)], + 1, + (0.0, 0, 1)), + # gh-11332 cases with a Generator + ([(0,0)], + [(0,1), (0,0)], + np.random.default_rng(0), + (0.0, 0, 1)), + ([(0,0)], + [(0,1), (0,0)], + np.random.default_rng(1), + (0.0, 0, 1)), + # slightly more complex case + ([(-5, 3), (0,0)], + [(0,1), (0,0), (-5, 3)], + 77098, + # the maximum minimum distance will + # be the last one found, but a unique + # solution is not guaranteed more broadly + (0.0, 1, 1)), + # repeated with Generator seeding + ([(-5, 3), (0,0)], + [(0,1), (0,0), (-5, 3)], + np.random.default_rng(77098), + # NOTE: using a Generator changes the + # indices but not the distance (unique solution + # not guaranteed) + (0.0, 0, 2)), + ]) + def test_subsets(self, A, B, seed, expected, num_parallel_threads): + # verify fix for gh-11332 + actual = directed_hausdorff(u=A, v=B, seed=seed) + # check distance + assert_allclose(actual[0], expected[0]) + starting_seed = seed + if hasattr(seed, 'bit_generator'): + starting_seed = seed.bit_generator._seed_seq.entropy + # check indices + if num_parallel_threads == 1 or starting_seed != 77098: + assert actual[1:] == expected[1:] + + if not isinstance(seed, np.random.RandomState): + # Check that new `rng` keyword is also accepted + actual = directed_hausdorff(u=A, v=B, rng=seed) + assert_allclose(actual[0], expected[0]) + + +@pytest.mark.xslow +def test_massive_arr_overflow(): + # on 64-bit systems we should be able to + # handle arrays that exceed the indexing + # size of a 32-bit signed integer + try: + import psutil + except ModuleNotFoundError: + pytest.skip("psutil required to check available memory") + if psutil.virtual_memory().available < 80*2**30: + # Don't run the test if there is less than 80 gig of RAM available. + pytest.skip('insufficient memory available to run this test') + size = int(3e9) + arr1 = np.zeros(shape=(size, 2)) + arr2 = np.zeros(shape=(3, 2)) + arr1[size - 1] = [5, 5] + actual = directed_hausdorff(u=arr1, v=arr2) + assert_allclose(actual[0], 7.0710678118654755) + assert_allclose(actual[1], size - 1) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_kdtree.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_kdtree.py new file mode 100644 index 0000000000000000000000000000000000000000..8b912f249a2228b72bfcdd4de01542573d7920e1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_kdtree.py @@ -0,0 +1,1535 @@ +# Copyright Anne M. Archibald 2008 +# Released under the scipy license + +import os +from numpy.testing import (assert_equal, assert_array_equal, assert_, + assert_almost_equal, assert_array_almost_equal, + assert_allclose) +from pytest import raises as assert_raises +import pytest +from platform import python_implementation +import numpy as np +from scipy.spatial import KDTree, Rectangle, distance_matrix, cKDTree +from scipy.spatial._ckdtree import cKDTreeNode +from scipy.spatial import minkowski_distance + +import itertools + +@pytest.fixture(params=[KDTree, cKDTree]) +def kdtree_type(request): + return request.param + + +def KDTreeTest(kls): + """Class decorator to create test cases for KDTree and cKDTree + + Tests use the class variable ``kdtree_type`` as the tree constructor. + """ + if not kls.__name__.startswith('_Test'): + raise RuntimeError("Expected a class name starting with _Test") + + for tree in (KDTree, cKDTree): + test_name = kls.__name__[1:] + '_' + tree.__name__ + + if test_name in globals(): + raise RuntimeError("Duplicated test name: " + test_name) + + # Create a new sub-class with kdtree_type defined + test_case = type(test_name, (kls,), {'kdtree_type': tree}) + globals()[test_name] = test_case + return kls + + +def distance_box(a, b, p, boxsize): + diff = a - b + diff[diff > 0.5 * boxsize] -= boxsize + diff[diff < -0.5 * boxsize] += boxsize + d = minkowski_distance(diff, 0, p) + return d + +class ConsistencyTests: + def distance(self, a, b, p): + return minkowski_distance(a, b, p) + + def test_nearest(self): + x = self.x + d, i = self.kdtree.query(x, 1) + assert_almost_equal(d**2, np.sum((x-self.data[i])**2)) + eps = 1e-8 + assert_(np.all(np.sum((self.data-x[np.newaxis, :])**2, axis=1) > d**2-eps)) + + def test_m_nearest(self): + x = self.x + m = self.m + dd, ii = self.kdtree.query(x, m) + d = np.amax(dd) + i = ii[np.argmax(dd)] + assert_almost_equal(d**2, np.sum((x-self.data[i])**2)) + eps = 1e-8 + assert_equal( + np.sum(np.sum((self.data-x[np.newaxis, :])**2, axis=1) < d**2+eps), + m, + ) + + def test_points_near(self): + x = self.x + d = self.d + dd, ii = self.kdtree.query(x, k=self.kdtree.n, distance_upper_bound=d) + eps = 1e-8 + hits = 0 + for near_d, near_i in zip(dd, ii): + if near_d == np.inf: + continue + hits += 1 + assert_almost_equal(near_d**2, np.sum((x-self.data[near_i])**2)) + assert_(near_d < d+eps, f"near_d={near_d:g} should be less than {d:g}") + assert_equal(np.sum(self.distance(self.data, x, 2) < d**2+eps), hits) + + def test_points_near_l1(self): + x = self.x + d = self.d + dd, ii = self.kdtree.query(x, k=self.kdtree.n, p=1, distance_upper_bound=d) + eps = 1e-8 + hits = 0 + for near_d, near_i in zip(dd, ii): + if near_d == np.inf: + continue + hits += 1 + assert_almost_equal(near_d, self.distance(x, self.data[near_i], 1)) + assert_(near_d < d+eps, f"near_d={near_d:g} should be less than {d:g}") + assert_equal(np.sum(self.distance(self.data, x, 1) < d+eps), hits) + + def test_points_near_linf(self): + x = self.x + d = self.d + dd, ii = self.kdtree.query(x, k=self.kdtree.n, p=np.inf, distance_upper_bound=d) + eps = 1e-8 + hits = 0 + for near_d, near_i in zip(dd, ii): + if near_d == np.inf: + continue + hits += 1 + assert_almost_equal(near_d, self.distance(x, self.data[near_i], np.inf)) + assert_(near_d < d+eps, f"near_d={near_d:g} should be less than {d:g}") + assert_equal(np.sum(self.distance(self.data, x, np.inf) < d+eps), hits) + + def test_approx(self): + x = self.x + k = self.k + eps = 0.1 + d_real, i_real = self.kdtree.query(x, k) + d, i = self.kdtree.query(x, k, eps=eps) + assert_(np.all(d <= d_real*(1+eps))) + + +@KDTreeTest +class _Test_random(ConsistencyTests): + def setup_method(self): + self.n = 100 + self.m = 4 + np.random.seed(1234) + self.data = np.random.randn(self.n, self.m) + self.kdtree = self.kdtree_type(self.data, leafsize=2) + self.x = np.random.randn(self.m) + self.d = 0.2 + self.k = 10 + + +@KDTreeTest +class _Test_random_far(_Test_random): + def setup_method(self): + super().setup_method() + self.x = np.random.randn(self.m)+10 + + +@KDTreeTest +class _Test_small(ConsistencyTests): + def setup_method(self): + self.data = np.array([[0, 0, 0], + [0, 0, 1], + [0, 1, 0], + [0, 1, 1], + [1, 0, 0], + [1, 0, 1], + [1, 1, 0], + [1, 1, 1]]) + self.kdtree = self.kdtree_type(self.data) + self.n = self.kdtree.n + self.m = self.kdtree.m + np.random.seed(1234) + self.x = np.random.randn(3) + self.d = 0.5 + self.k = 4 + + def test_nearest(self): + assert_array_equal( + self.kdtree.query((0, 0, 0.1), 1), + (0.1, 0)) + + def test_nearest_two(self): + assert_array_equal( + self.kdtree.query((0, 0, 0.1), 2), + ([0.1, 0.9], [0, 1])) + + +@KDTreeTest +class _Test_small_nonleaf(_Test_small): + def setup_method(self): + super().setup_method() + self.kdtree = self.kdtree_type(self.data, leafsize=1) + + +class Test_vectorization_KDTree: + def setup_method(self): + self.data = np.array([[0, 0, 0], + [0, 0, 1], + [0, 1, 0], + [0, 1, 1], + [1, 0, 0], + [1, 0, 1], + [1, 1, 0], + [1, 1, 1]]) + self.kdtree = KDTree(self.data) + + def test_single_query(self): + d, i = self.kdtree.query(np.array([0, 0, 0])) + assert_(isinstance(d, float)) + assert_(np.issubdtype(i, np.signedinteger)) + + def test_vectorized_query(self): + d, i = self.kdtree.query(np.zeros((2, 4, 3))) + assert_equal(np.shape(d), (2, 4)) + assert_equal(np.shape(i), (2, 4)) + + def test_single_query_multiple_neighbors(self): + s = 23 + kk = self.kdtree.n+s + d, i = self.kdtree.query(np.array([0, 0, 0]), k=kk) + assert_equal(np.shape(d), (kk,)) + assert_equal(np.shape(i), (kk,)) + assert_(np.all(~np.isfinite(d[-s:]))) + assert_(np.all(i[-s:] == self.kdtree.n)) + + def test_vectorized_query_multiple_neighbors(self): + s = 23 + kk = self.kdtree.n+s + d, i = self.kdtree.query(np.zeros((2, 4, 3)), k=kk) + assert_equal(np.shape(d), (2, 4, kk)) + assert_equal(np.shape(i), (2, 4, kk)) + assert_(np.all(~np.isfinite(d[:, :, -s:]))) + assert_(np.all(i[:, :, -s:] == self.kdtree.n)) + + def test_query_raises_for_k_none(self): + x = 1.0 + with pytest.raises(ValueError, match="k must be an integer or*"): + self.kdtree.query(x, k=None) + +class Test_vectorization_cKDTree: + def setup_method(self): + self.data = np.array([[0, 0, 0], + [0, 0, 1], + [0, 1, 0], + [0, 1, 1], + [1, 0, 0], + [1, 0, 1], + [1, 1, 0], + [1, 1, 1]]) + self.kdtree = cKDTree(self.data) + + def test_single_query(self): + d, i = self.kdtree.query([0, 0, 0]) + assert_(isinstance(d, float)) + assert_(isinstance(i, int)) + + def test_vectorized_query(self): + d, i = self.kdtree.query(np.zeros((2, 4, 3))) + assert_equal(np.shape(d), (2, 4)) + assert_equal(np.shape(i), (2, 4)) + + def test_vectorized_query_noncontiguous_values(self): + np.random.seed(1234) + qs = np.random.randn(3, 1000).T + ds, i_s = self.kdtree.query(qs) + for q, d, i in zip(qs, ds, i_s): + assert_equal(self.kdtree.query(q), (d, i)) + + def test_single_query_multiple_neighbors(self): + s = 23 + kk = self.kdtree.n+s + d, i = self.kdtree.query([0, 0, 0], k=kk) + assert_equal(np.shape(d), (kk,)) + assert_equal(np.shape(i), (kk,)) + assert_(np.all(~np.isfinite(d[-s:]))) + assert_(np.all(i[-s:] == self.kdtree.n)) + + def test_vectorized_query_multiple_neighbors(self): + s = 23 + kk = self.kdtree.n+s + d, i = self.kdtree.query(np.zeros((2, 4, 3)), k=kk) + assert_equal(np.shape(d), (2, 4, kk)) + assert_equal(np.shape(i), (2, 4, kk)) + assert_(np.all(~np.isfinite(d[:, :, -s:]))) + assert_(np.all(i[:, :, -s:] == self.kdtree.n)) + +class ball_consistency: + tol = 0.0 + + def distance(self, a, b, p): + return minkowski_distance(a * 1.0, b * 1.0, p) + + def test_in_ball(self): + x = np.atleast_2d(self.x) + d = np.broadcast_to(self.d, x.shape[:-1]) + l = self.T.query_ball_point(x, self.d, p=self.p, eps=self.eps) + for i, ind in enumerate(l): + dist = self.distance(self.data[ind], x[i], self.p) - d[i]*(1.+self.eps) + norm = self.distance(self.data[ind], x[i], self.p) + d[i]*(1.+self.eps) + assert_array_equal(dist < self.tol * norm, True) + + def test_found_all(self): + x = np.atleast_2d(self.x) + d = np.broadcast_to(self.d, x.shape[:-1]) + l = self.T.query_ball_point(x, self.d, p=self.p, eps=self.eps) + for i, ind in enumerate(l): + c = np.ones(self.T.n, dtype=bool) + c[ind] = False + dist = self.distance(self.data[c], x[i], self.p) - d[i]/(1.+self.eps) + norm = self.distance(self.data[c], x[i], self.p) + d[i]/(1.+self.eps) + assert_array_equal(dist > -self.tol * norm, True) + +@KDTreeTest +class _Test_random_ball(ball_consistency): + def setup_method(self): + n = 100 + m = 4 + np.random.seed(1234) + self.data = np.random.randn(n, m) + self.T = self.kdtree_type(self.data, leafsize=2) + self.x = np.random.randn(m) + self.p = 2. + self.eps = 0 + self.d = 0.2 + + +@KDTreeTest +class _Test_random_ball_periodic(ball_consistency): + def distance(self, a, b, p): + return distance_box(a, b, p, 1.0) + + def setup_method(self): + n = 10000 + m = 4 + np.random.seed(1234) + self.data = np.random.uniform(size=(n, m)) + self.T = self.kdtree_type(self.data, leafsize=2, boxsize=1) + self.x = np.full(m, 0.1) + self.p = 2. + self.eps = 0 + self.d = 0.2 + + def test_in_ball_outside(self): + l = self.T.query_ball_point(self.x + 1.0, self.d, p=self.p, eps=self.eps) + for i in l: + assert_(self.distance(self.data[i], self.x, self.p) <= self.d*(1.+self.eps)) + l = self.T.query_ball_point(self.x - 1.0, self.d, p=self.p, eps=self.eps) + for i in l: + assert_(self.distance(self.data[i], self.x, self.p) <= self.d*(1.+self.eps)) + + def test_found_all_outside(self): + c = np.ones(self.T.n, dtype=bool) + l = self.T.query_ball_point(self.x + 1.0, self.d, p=self.p, eps=self.eps) + c[l] = False + assert np.all( + self.distance(self.data[c], self.x, self.p) >= self.d/(1.+self.eps) + ) + + l = self.T.query_ball_point(self.x - 1.0, self.d, p=self.p, eps=self.eps) + c[l] = False + assert np.all( + self.distance(self.data[c], self.x, self.p) >= self.d/(1.+self.eps) + ) + + +@KDTreeTest +class _Test_random_ball_largep_issue9890(ball_consistency): + + # allow some roundoff errors due to numerical issues + tol = 1e-13 + + def setup_method(self): + n = 1000 + m = 2 + np.random.seed(123) + self.data = np.random.randint(100, 1000, size=(n, m)) + self.T = self.kdtree_type(self.data) + self.x = self.data + self.p = 100 + self.eps = 0 + self.d = 10 + + +@KDTreeTest +class _Test_random_ball_approx(_Test_random_ball): + + def setup_method(self): + super().setup_method() + self.eps = 0.1 + + +@KDTreeTest +class _Test_random_ball_approx_periodic(_Test_random_ball): + + def setup_method(self): + super().setup_method() + self.eps = 0.1 + + +@KDTreeTest +class _Test_random_ball_far(_Test_random_ball): + + def setup_method(self): + super().setup_method() + self.d = 2. + +@KDTreeTest +class _Test_random_ball_far_periodic(_Test_random_ball_periodic): + + def setup_method(self): + super().setup_method() + self.d = 2. + + +@KDTreeTest +class _Test_random_ball_l1(_Test_random_ball): + + def setup_method(self): + super().setup_method() + self.p = 1 + + +@KDTreeTest +class _Test_random_ball_linf(_Test_random_ball): + + def setup_method(self): + super().setup_method() + self.p = np.inf + + +def test_random_ball_vectorized(kdtree_type): + n = 20 + m = 5 + np.random.seed(1234) + T = kdtree_type(np.random.randn(n, m)) + + r = T.query_ball_point(np.random.randn(2, 3, m), 1) + assert_equal(r.shape, (2, 3)) + assert_(isinstance(r[0, 0], list)) + + +def test_query_ball_point_multithreading(kdtree_type): + np.random.seed(0) + n = 5000 + k = 2 + points = np.random.randn(n, k) + T = kdtree_type(points) + l1 = T.query_ball_point(points, 0.003, workers=1) + l2 = T.query_ball_point(points, 0.003, workers=64) + l3 = T.query_ball_point(points, 0.003, workers=-1) + + for i in range(n): + if l1[i] or l2[i]: + assert_array_equal(l1[i], l2[i]) + + for i in range(n): + if l1[i] or l3[i]: + assert_array_equal(l1[i], l3[i]) + + +class two_trees_consistency: + + def distance(self, a, b, p): + return minkowski_distance(a, b, p) + + def test_all_in_ball(self): + r = self.T1.query_ball_tree(self.T2, self.d, p=self.p, eps=self.eps) + for i, l in enumerate(r): + for j in l: + assert (self.distance(self.data1[i], self.data2[j], self.p) + <= self.d*(1.+self.eps)) + + def test_found_all(self): + r = self.T1.query_ball_tree(self.T2, self.d, p=self.p, eps=self.eps) + for i, l in enumerate(r): + c = np.ones(self.T2.n, dtype=bool) + c[l] = False + assert np.all(self.distance(self.data2[c], self.data1[i], self.p) + >= self.d/(1.+self.eps)) + + +@KDTreeTest +class _Test_two_random_trees(two_trees_consistency): + + def setup_method(self): + n = 50 + m = 4 + np.random.seed(1234) + self.data1 = np.random.randn(n, m) + self.T1 = self.kdtree_type(self.data1, leafsize=2) + self.data2 = np.random.randn(n, m) + self.T2 = self.kdtree_type(self.data2, leafsize=2) + self.p = 2. + self.eps = 0 + self.d = 0.2 + + +@KDTreeTest +class _Test_two_random_trees_periodic(two_trees_consistency): + def distance(self, a, b, p): + return distance_box(a, b, p, 1.0) + + def setup_method(self): + n = 50 + m = 4 + np.random.seed(1234) + self.data1 = np.random.uniform(size=(n, m)) + self.T1 = self.kdtree_type(self.data1, leafsize=2, boxsize=1.0) + self.data2 = np.random.uniform(size=(n, m)) + self.T2 = self.kdtree_type(self.data2, leafsize=2, boxsize=1.0) + self.p = 2. + self.eps = 0 + self.d = 0.2 + + +@KDTreeTest +class _Test_two_random_trees_far(_Test_two_random_trees): + + def setup_method(self): + super().setup_method() + self.d = 2 + + +@KDTreeTest +class _Test_two_random_trees_far_periodic(_Test_two_random_trees_periodic): + + def setup_method(self): + super().setup_method() + self.d = 2 + + +@KDTreeTest +class _Test_two_random_trees_linf(_Test_two_random_trees): + + def setup_method(self): + super().setup_method() + self.p = np.inf + + +@KDTreeTest +class _Test_two_random_trees_linf_periodic(_Test_two_random_trees_periodic): + + def setup_method(self): + super().setup_method() + self.p = np.inf + + +class Test_rectangle: + + def setup_method(self): + self.rect = Rectangle([0, 0], [1, 1]) + + def test_min_inside(self): + assert_almost_equal(self.rect.min_distance_point([0.5, 0.5]), 0) + + def test_min_one_side(self): + assert_almost_equal(self.rect.min_distance_point([0.5, 1.5]), 0.5) + + def test_min_two_sides(self): + assert_almost_equal(self.rect.min_distance_point([2, 2]), np.sqrt(2)) + + def test_max_inside(self): + assert_almost_equal(self.rect.max_distance_point([0.5, 0.5]), 1/np.sqrt(2)) + + def test_max_one_side(self): + assert_almost_equal(self.rect.max_distance_point([0.5, 1.5]), + np.hypot(0.5, 1.5)) + + def test_max_two_sides(self): + assert_almost_equal(self.rect.max_distance_point([2, 2]), 2*np.sqrt(2)) + + def test_split(self): + less, greater = self.rect.split(0, 0.1) + assert_array_equal(less.maxes, [0.1, 1]) + assert_array_equal(less.mins, [0, 0]) + assert_array_equal(greater.maxes, [1, 1]) + assert_array_equal(greater.mins, [0.1, 0]) + + +def test_distance_l2(): + assert_almost_equal(minkowski_distance([0, 0], [1, 1], 2), np.sqrt(2)) + + +def test_distance_l1(): + assert_almost_equal(minkowski_distance([0, 0], [1, 1], 1), 2) + + +def test_distance_linf(): + assert_almost_equal(minkowski_distance([0, 0], [1, 1], np.inf), 1) + + +def test_distance_vectorization(): + np.random.seed(1234) + x = np.random.randn(10, 1, 3) + y = np.random.randn(1, 7, 3) + assert_equal(minkowski_distance(x, y).shape, (10, 7)) + + +class count_neighbors_consistency: + def test_one_radius(self): + r = 0.2 + assert_equal(self.T1.count_neighbors(self.T2, r), + np.sum([len(l) for l in self.T1.query_ball_tree(self.T2, r)])) + + def test_large_radius(self): + r = 1000 + assert_equal(self.T1.count_neighbors(self.T2, r), + np.sum([len(l) for l in self.T1.query_ball_tree(self.T2, r)])) + + def test_multiple_radius(self): + rs = np.exp(np.linspace(np.log(0.01), np.log(10), 3)) + results = self.T1.count_neighbors(self.T2, rs) + assert_(np.all(np.diff(results) >= 0)) + for r, result in zip(rs, results): + assert_equal(self.T1.count_neighbors(self.T2, r), result) + +@KDTreeTest +class _Test_count_neighbors(count_neighbors_consistency): + def setup_method(self): + n = 50 + m = 2 + np.random.seed(1234) + self.T1 = self.kdtree_type(np.random.randn(n, m), leafsize=2) + self.T2 = self.kdtree_type(np.random.randn(n, m), leafsize=2) + + +class sparse_distance_matrix_consistency: + + def distance(self, a, b, p): + return minkowski_distance(a, b, p) + + def test_consistency_with_neighbors(self): + M = self.T1.sparse_distance_matrix(self.T2, self.r) + r = self.T1.query_ball_tree(self.T2, self.r) + for i, l in enumerate(r): + for j in l: + assert_almost_equal( + M[i, j], + self.distance(self.T1.data[i], self.T2.data[j], self.p), + decimal=14 + ) + for ((i, j), d) in M.items(): + assert_(j in r[i]) + + def test_zero_distance(self): + # raises an exception for bug 870 (FIXME: Does it?) + self.T1.sparse_distance_matrix(self.T1, self.r) + + def test_consistency(self): + # Test consistency with a distance_matrix + M1 = self.T1.sparse_distance_matrix(self.T2, self.r) + expected = distance_matrix(self.T1.data, self.T2.data) + expected[expected > self.r] = 0 + assert_array_almost_equal(M1.toarray(), expected, decimal=14) + + def test_against_logic_error_regression(self): + # regression test for gh-5077 logic error + np.random.seed(0) + too_many = np.array(np.random.randn(18, 2), dtype=int) + tree = self.kdtree_type( + too_many, balanced_tree=False, compact_nodes=False) + d = tree.sparse_distance_matrix(tree, 3).toarray() + assert_array_almost_equal(d, d.T, decimal=14) + + def test_ckdtree_return_types(self): + # brute-force reference + ref = np.zeros((self.n, self.n)) + for i in range(self.n): + for j in range(self.n): + v = self.data1[i, :] - self.data2[j, :] + ref[i, j] = np.dot(v, v) + ref = np.sqrt(ref) + ref[ref > self.r] = 0. + # test return type 'dict' + dist = np.zeros((self.n, self.n)) + r = self.T1.sparse_distance_matrix(self.T2, self.r, output_type='dict') + for i, j in r.keys(): + dist[i, j] = r[(i, j)] + assert_array_almost_equal(ref, dist, decimal=14) + # test return type 'ndarray' + dist = np.zeros((self.n, self.n)) + r = self.T1.sparse_distance_matrix(self.T2, self.r, + output_type='ndarray') + for k in range(r.shape[0]): + i = r['i'][k] + j = r['j'][k] + v = r['v'][k] + dist[i, j] = v + assert_array_almost_equal(ref, dist, decimal=14) + # test return type 'dok_matrix' + r = self.T1.sparse_distance_matrix(self.T2, self.r, + output_type='dok_matrix') + assert_array_almost_equal(ref, r.toarray(), decimal=14) + # test return type 'coo_matrix' + r = self.T1.sparse_distance_matrix(self.T2, self.r, + output_type='coo_matrix') + assert_array_almost_equal(ref, r.toarray(), decimal=14) + + +@KDTreeTest +class _Test_sparse_distance_matrix(sparse_distance_matrix_consistency): + def setup_method(self): + n = 50 + m = 4 + np.random.seed(1234) + data1 = np.random.randn(n, m) + data2 = np.random.randn(n, m) + self.T1 = self.kdtree_type(data1, leafsize=2) + self.T2 = self.kdtree_type(data2, leafsize=2) + self.r = 0.5 + self.p = 2 + self.data1 = data1 + self.data2 = data2 + self.n = n + self.m = m + + +def test_distance_matrix(): + m = 10 + n = 11 + k = 4 + np.random.seed(1234) + xs = np.random.randn(m, k) + ys = np.random.randn(n, k) + ds = distance_matrix(xs, ys) + assert_equal(ds.shape, (m, n)) + for i in range(m): + for j in range(n): + assert_almost_equal(minkowski_distance(xs[i], ys[j]), ds[i, j]) + + +def test_distance_matrix_looping(): + m = 10 + n = 11 + k = 4 + np.random.seed(1234) + xs = np.random.randn(m, k) + ys = np.random.randn(n, k) + ds = distance_matrix(xs, ys) + dsl = distance_matrix(xs, ys, threshold=1) + assert_equal(ds, dsl) + + +def check_onetree_query(T, d): + r = T.query_ball_tree(T, d) + s = set() + for i, l in enumerate(r): + for j in l: + if i < j: + s.add((i, j)) + + assert_(s == T.query_pairs(d)) + +def test_onetree_query(kdtree_type): + np.random.seed(0) + n = 50 + k = 4 + points = np.random.randn(n, k) + T = kdtree_type(points) + check_onetree_query(T, 0.1) + + points = np.random.randn(3*n, k) + points[:n] *= 0.001 + points[n:2*n] += 2 + T = kdtree_type(points) + check_onetree_query(T, 0.1) + check_onetree_query(T, 0.001) + check_onetree_query(T, 0.00001) + check_onetree_query(T, 1e-6) + + +def test_query_pairs_single_node(kdtree_type): + tree = kdtree_type([[0, 1]]) + assert_equal(tree.query_pairs(0.5), set()) + + +def test_kdtree_query_pairs(kdtree_type): + np.random.seed(0) + n = 50 + k = 2 + r = 0.1 + r2 = r**2 + points = np.random.randn(n, k) + T = kdtree_type(points) + # brute force reference + brute = set() + for i in range(n): + for j in range(i+1, n): + v = points[i, :] - points[j, :] + if np.dot(v, v) <= r2: + brute.add((i, j)) + l0 = sorted(brute) + # test default return type + s = T.query_pairs(r) + l1 = sorted(s) + assert_array_equal(l0, l1) + # test return type 'set' + s = T.query_pairs(r, output_type='set') + l1 = sorted(s) + assert_array_equal(l0, l1) + # test return type 'ndarray' + s = set() + arr = T.query_pairs(r, output_type='ndarray') + for i in range(arr.shape[0]): + s.add((int(arr[i, 0]), int(arr[i, 1]))) + l2 = sorted(s) + assert_array_equal(l0, l2) + + +def test_query_pairs_eps(kdtree_type): + spacing = np.sqrt(2) + # irrational spacing to have potential rounding errors + x_range = np.linspace(0, 3 * spacing, 4) + y_range = np.linspace(0, 3 * spacing, 4) + xy_array = [(xi, yi) for xi in x_range for yi in y_range] + tree = kdtree_type(xy_array) + pairs_eps = tree.query_pairs(r=spacing, eps=.1) + # result: 24 with eps, 16 without due to rounding + pairs = tree.query_pairs(r=spacing * 1.01) + # result: 24 + assert_equal(pairs, pairs_eps) + + +def test_ball_point_ints(kdtree_type): + # Regression test for #1373. + x, y = np.mgrid[0:4, 0:4] + points = list(zip(x.ravel(), y.ravel())) + tree = kdtree_type(points) + assert_equal(sorted([4, 8, 9, 12]), + sorted(tree.query_ball_point((2, 0), 1))) + points = np.asarray(points, dtype=float) + tree = kdtree_type(points) + assert_equal(sorted([4, 8, 9, 12]), + sorted(tree.query_ball_point((2, 0), 1))) + + +def test_kdtree_comparisons(): + # Regression test: node comparisons were done wrong in 0.12 w/Py3. + nodes = [KDTree.node() for _ in range(3)] + assert_equal(sorted(nodes), sorted(nodes[::-1])) + + +def test_kdtree_build_modes(kdtree_type): + # check if different build modes for KDTree give similar query results + np.random.seed(0) + n = 5000 + k = 4 + points = np.random.randn(n, k) + T1 = kdtree_type(points).query(points, k=5)[-1] + T2 = kdtree_type(points, compact_nodes=False).query(points, k=5)[-1] + T3 = kdtree_type(points, balanced_tree=False).query(points, k=5)[-1] + T4 = kdtree_type(points, compact_nodes=False, + balanced_tree=False).query(points, k=5)[-1] + assert_array_equal(T1, T2) + assert_array_equal(T1, T3) + assert_array_equal(T1, T4) + +def test_kdtree_pickle(kdtree_type): + # test if it is possible to pickle a KDTree + import pickle + np.random.seed(0) + n = 50 + k = 4 + points = np.random.randn(n, k) + T1 = kdtree_type(points) + tmp = pickle.dumps(T1) + T2 = pickle.loads(tmp) + T1 = T1.query(points, k=5)[-1] + T2 = T2.query(points, k=5)[-1] + assert_array_equal(T1, T2) + +def test_kdtree_pickle_boxsize(kdtree_type): + # test if it is possible to pickle a periodic KDTree + import pickle + np.random.seed(0) + n = 50 + k = 4 + points = np.random.uniform(size=(n, k)) + T1 = kdtree_type(points, boxsize=1.0) + tmp = pickle.dumps(T1) + T2 = pickle.loads(tmp) + T1 = T1.query(points, k=5)[-1] + T2 = T2.query(points, k=5)[-1] + assert_array_equal(T1, T2) + +def test_kdtree_copy_data(kdtree_type): + # check if copy_data=True makes the kd-tree + # impervious to data corruption by modification of + # the data arrray + np.random.seed(0) + n = 5000 + k = 4 + points = np.random.randn(n, k) + T = kdtree_type(points, copy_data=True) + q = points.copy() + T1 = T.query(q, k=5)[-1] + points[...] = np.random.randn(n, k) + T2 = T.query(q, k=5)[-1] + assert_array_equal(T1, T2) + +def test_ckdtree_parallel(kdtree_type, monkeypatch): + # check if parallel=True also generates correct query results + np.random.seed(0) + n = 5000 + k = 4 + points = np.random.randn(n, k) + T = kdtree_type(points) + T1 = T.query(points, k=5, workers=64)[-1] + T2 = T.query(points, k=5, workers=-1)[-1] + T3 = T.query(points, k=5)[-1] + assert_array_equal(T1, T2) + assert_array_equal(T1, T3) + + monkeypatch.setattr(os, 'cpu_count', lambda: None) + with pytest.raises(NotImplementedError, match="Cannot determine the"): + T.query(points, 1, workers=-1) + + +def test_ckdtree_view(): + # Check that the nodes can be correctly viewed from Python. + # This test also sanity checks each node in the cKDTree, and + # thus verifies the internal structure of the kd-tree. + np.random.seed(0) + n = 100 + k = 4 + points = np.random.randn(n, k) + kdtree = cKDTree(points) + + # walk the whole kd-tree and sanity check each node + def recurse_tree(n): + assert_(isinstance(n, cKDTreeNode)) + if n.split_dim == -1: + assert_(n.lesser is None) + assert_(n.greater is None) + assert_(n.indices.shape[0] <= kdtree.leafsize) + else: + recurse_tree(n.lesser) + recurse_tree(n.greater) + x = n.lesser.data_points[:, n.split_dim] + y = n.greater.data_points[:, n.split_dim] + assert_(x.max() < y.min()) + + recurse_tree(kdtree.tree) + # check that indices are correctly retrieved + n = kdtree.tree + assert_array_equal(np.sort(n.indices), range(100)) + # check that data_points are correctly retrieved + assert_array_equal(kdtree.data[n.indices, :], n.data_points) + +# KDTree is specialized to type double points, so no need to make +# a unit test corresponding to test_ball_point_ints() + +def test_kdtree_list_k(kdtree_type): + # check kdtree periodic boundary + n = 200 + m = 2 + klist = [1, 2, 3] + kint = 3 + + np.random.seed(1234) + data = np.random.uniform(size=(n, m)) + kdtree = kdtree_type(data, leafsize=1) + + # check agreement between arange(1, k+1) and k + dd, ii = kdtree.query(data, klist) + dd1, ii1 = kdtree.query(data, kint) + assert_equal(dd, dd1) + assert_equal(ii, ii1) + + # now check skipping one element + klist = np.array([1, 3]) + kint = 3 + dd, ii = kdtree.query(data, kint) + dd1, ii1 = kdtree.query(data, klist) + assert_equal(dd1, dd[..., klist - 1]) + assert_equal(ii1, ii[..., klist - 1]) + + # check k == 1 special case + # and k == [1] non-special case + dd, ii = kdtree.query(data, 1) + dd1, ii1 = kdtree.query(data, [1]) + assert_equal(len(dd.shape), 1) + assert_equal(len(dd1.shape), 2) + assert_equal(dd, np.ravel(dd1)) + assert_equal(ii, np.ravel(ii1)) + +@pytest.mark.fail_slow(10) +def test_kdtree_box(kdtree_type): + # check ckdtree periodic boundary + n = 2000 + m = 3 + k = 3 + np.random.seed(1234) + data = np.random.uniform(size=(n, m)) + kdtree = kdtree_type(data, leafsize=1, boxsize=1.0) + + # use the standard python KDTree for the simulated periodic box + kdtree2 = kdtree_type(data, leafsize=1) + + for p in [1, 2, 3.0, np.inf]: + dd, ii = kdtree.query(data, k, p=p) + + dd1, ii1 = kdtree.query(data + 1.0, k, p=p) + assert_almost_equal(dd, dd1) + assert_equal(ii, ii1) + + dd1, ii1 = kdtree.query(data - 1.0, k, p=p) + assert_almost_equal(dd, dd1) + assert_equal(ii, ii1) + + dd2, ii2 = simulate_periodic_box(kdtree2, data, k, boxsize=1.0, p=p) + assert_almost_equal(dd, dd2) + assert_equal(ii, ii2) + +def test_kdtree_box_0boxsize(kdtree_type): + # check ckdtree periodic boundary that mimics non-periodic + n = 2000 + m = 2 + k = 3 + np.random.seed(1234) + data = np.random.uniform(size=(n, m)) + kdtree = kdtree_type(data, leafsize=1, boxsize=0.0) + + # use the standard python KDTree for the simulated periodic box + kdtree2 = kdtree_type(data, leafsize=1) + + for p in [1, 2, np.inf]: + dd, ii = kdtree.query(data, k, p=p) + + dd1, ii1 = kdtree2.query(data, k, p=p) + assert_almost_equal(dd, dd1) + assert_equal(ii, ii1) + +def test_kdtree_box_upper_bounds(kdtree_type): + data = np.linspace(0, 2, 10).reshape(-1, 2) + data[:, 1] += 10 + with pytest.raises(ValueError): + kdtree_type(data, leafsize=1, boxsize=1.0) + with pytest.raises(ValueError): + kdtree_type(data, leafsize=1, boxsize=(0.0, 2.0)) + # skip a dimension. + kdtree_type(data, leafsize=1, boxsize=(2.0, 0.0)) + +def test_kdtree_box_lower_bounds(kdtree_type): + data = np.linspace(-1, 1, 10) + assert_raises(ValueError, kdtree_type, data, leafsize=1, boxsize=1.0) + +def simulate_periodic_box(kdtree, data, k, boxsize, p): + dd = [] + ii = [] + x = np.arange(3 ** data.shape[1]) + nn = np.array(np.unravel_index(x, [3] * data.shape[1])).T + nn = nn - 1.0 + for n in nn: + image = data + n * 1.0 * boxsize + dd2, ii2 = kdtree.query(image, k, p=p) + dd2 = dd2.reshape(-1, k) + ii2 = ii2.reshape(-1, k) + dd.append(dd2) + ii.append(ii2) + dd = np.concatenate(dd, axis=-1) + ii = np.concatenate(ii, axis=-1) + + result = np.empty([len(data), len(nn) * k], dtype=[ + ('ii', 'i8'), + ('dd', 'f8')]) + result['ii'][:] = ii + result['dd'][:] = dd + result.sort(order='dd') + return result['dd'][:, :k], result['ii'][:, :k] + + +@pytest.mark.skipif(python_implementation() == 'PyPy', + reason="Fails on PyPy CI runs. See #9507") +def test_ckdtree_memuse(): + # unit test adaptation of gh-5630 + + # NOTE: this will fail when run via valgrind, + # because rss is no longer a reliable memory usage indicator. + + try: + import resource + except ImportError: + # resource is not available on Windows + return + # Make some data + dx, dy = 0.05, 0.05 + y, x = np.mgrid[slice(1, 5 + dy, dy), + slice(1, 5 + dx, dx)] + z = np.sin(x)**10 + np.cos(10 + y*x) * np.cos(x) + z_copy = np.empty_like(z) + z_copy[:] = z + # Place FILLVAL in z_copy at random number of random locations + FILLVAL = 99. + mask = np.random.randint(0, z.size, np.random.randint(50) + 5) + z_copy.flat[mask] = FILLVAL + igood = np.vstack(np.nonzero(x != FILLVAL)).T + ibad = np.vstack(np.nonzero(x == FILLVAL)).T + mem_use = resource.getrusage(resource.RUSAGE_SELF).ru_maxrss + # burn-in + for i in range(10): + tree = cKDTree(igood) + # count memleaks while constructing and querying cKDTree + num_leaks = 0 + for i in range(100): + mem_use = resource.getrusage(resource.RUSAGE_SELF).ru_maxrss + tree = cKDTree(igood) + dist, iquery = tree.query(ibad, k=4, p=2) + new_mem_use = resource.getrusage(resource.RUSAGE_SELF).ru_maxrss + if new_mem_use > mem_use: + num_leaks += 1 + # ideally zero leaks, but errors might accidentally happen + # outside cKDTree + assert_(num_leaks < 10) + +def test_kdtree_weights(kdtree_type): + + data = np.linspace(0, 1, 4).reshape(-1, 1) + tree1 = kdtree_type(data, leafsize=1) + weights = np.ones(len(data), dtype='f4') + + nw = tree1._build_weights(weights) + assert_array_equal(nw, [4, 2, 1, 1, 2, 1, 1]) + + assert_raises(ValueError, tree1._build_weights, weights[:-1]) + + for i in range(10): + # since weights are uniform, these shall agree: + c1 = tree1.count_neighbors(tree1, np.linspace(0, 10, i)) + c2 = tree1.count_neighbors(tree1, np.linspace(0, 10, i), + weights=(weights, weights)) + c3 = tree1.count_neighbors(tree1, np.linspace(0, 10, i), + weights=(weights, None)) + c4 = tree1.count_neighbors(tree1, np.linspace(0, 10, i), + weights=(None, weights)) + tree1.count_neighbors(tree1, np.linspace(0, 10, i), + weights=weights) + + assert_array_equal(c1, c2) + assert_array_equal(c1, c3) + assert_array_equal(c1, c4) + + for i in range(len(data)): + # this tests removal of one data point by setting weight to 0 + w1 = weights.copy() + w1[i] = 0 + data2 = data[w1 != 0] + tree2 = kdtree_type(data2) + + c1 = tree1.count_neighbors(tree1, np.linspace(0, 10, 100), + weights=(w1, w1)) + # "c2 is correct" + c2 = tree2.count_neighbors(tree2, np.linspace(0, 10, 100)) + + assert_array_equal(c1, c2) + + #this asserts for two different trees, singular weights + # crashes + assert_raises(ValueError, tree1.count_neighbors, + tree2, np.linspace(0, 10, 100), weights=w1) + +@pytest.mark.fail_slow(10) +def test_kdtree_count_neighbous_multiple_r(kdtree_type): + n = 2000 + m = 2 + np.random.seed(1234) + data = np.random.normal(size=(n, m)) + kdtree = kdtree_type(data, leafsize=1) + r0 = [0, 0.01, 0.01, 0.02, 0.05] + i0 = np.arange(len(r0)) + n0 = kdtree.count_neighbors(kdtree, r0) + nnc = kdtree.count_neighbors(kdtree, r0, cumulative=False) + assert_equal(n0, nnc.cumsum()) + + for i, r in zip(itertools.permutations(i0), + itertools.permutations(r0)): + # permute n0 by i and it shall agree + n = kdtree.count_neighbors(kdtree, r) + assert_array_equal(n, n0[list(i)]) + +def test_len0_arrays(kdtree_type): + # make sure len-0 arrays are handled correctly + # in range queries (gh-5639) + rng = np.random.RandomState(1234) + X = rng.rand(10, 2) + Y = rng.rand(10, 2) + tree = kdtree_type(X) + # query_ball_point (single) + d, i = tree.query([.5, .5], k=1) + z = tree.query_ball_point([.5, .5], 0.1*d) + assert_array_equal(z, []) + # query_ball_point (multiple) + d, i = tree.query(Y, k=1) + mind = d.min() + z = tree.query_ball_point(Y, 0.1*mind) + y = np.empty(shape=(10, ), dtype=object) + y.fill([]) + assert_array_equal(y, z) + # query_ball_tree + other = kdtree_type(Y) + y = tree.query_ball_tree(other, 0.1*mind) + assert_array_equal(10*[[]], y) + # count_neighbors + y = tree.count_neighbors(other, 0.1*mind) + assert_(y == 0) + # sparse_distance_matrix + y = tree.sparse_distance_matrix(other, 0.1*mind, output_type='dok_matrix') + assert_array_equal(y == np.zeros((10, 10)), True) + y = tree.sparse_distance_matrix(other, 0.1*mind, output_type='coo_matrix') + assert_array_equal(y == np.zeros((10, 10)), True) + y = tree.sparse_distance_matrix(other, 0.1*mind, output_type='dict') + assert_equal(y, {}) + y = tree.sparse_distance_matrix(other, 0.1*mind, output_type='ndarray') + _dtype = [('i', np.intp), ('j', np.intp), ('v', np.float64)] + res_dtype = np.dtype(_dtype, align=True) + z = np.empty(shape=(0, ), dtype=res_dtype) + assert_array_equal(y, z) + # query_pairs + d, i = tree.query(X, k=2) + mind = d[:, -1].min() + y = tree.query_pairs(0.1*mind, output_type='set') + assert_equal(y, set()) + y = tree.query_pairs(0.1*mind, output_type='ndarray') + z = np.empty(shape=(0, 2), dtype=np.intp) + assert_array_equal(y, z) + +def test_kdtree_duplicated_inputs(kdtree_type): + # check kdtree with duplicated inputs + n = 1024 + for m in range(1, 8): + data = np.ones((n, m)) + data[n//2:] = 2 + + for balanced, compact in itertools.product((False, True), repeat=2): + kdtree = kdtree_type(data, balanced_tree=balanced, + compact_nodes=compact, leafsize=1) + assert kdtree.size == 3 + + tree = (kdtree.tree if kdtree_type is cKDTree else + kdtree.tree._node) + + assert_equal( + np.sort(tree.lesser.indices), + np.arange(0, n // 2)) + assert_equal( + np.sort(tree.greater.indices), + np.arange(n // 2, n)) + + +def test_kdtree_noncumulative_nondecreasing(kdtree_type): + # check kdtree with duplicated inputs + + # it shall not divide more than 3 nodes. + # root left (1), and right (2) + kdtree = kdtree_type([[0]], leafsize=1) + + assert_raises(ValueError, kdtree.count_neighbors, + kdtree, [0.1, 0], cumulative=False) + +def test_short_knn(kdtree_type): + + # The test case is based on github: #6425 by @SteveDoyle2 + + xyz = np.array([ + [0., 0., 0.], + [1.01, 0., 0.], + [0., 1., 0.], + [0., 1.01, 0.], + [1., 0., 0.], + [1., 1., 0.]], + dtype='float64') + + ckdt = kdtree_type(xyz) + + deq, ieq = ckdt.query(xyz, k=4, distance_upper_bound=0.2) + + assert_array_almost_equal(deq, + [[0., np.inf, np.inf, np.inf], + [0., 0.01, np.inf, np.inf], + [0., 0.01, np.inf, np.inf], + [0., 0.01, np.inf, np.inf], + [0., 0.01, np.inf, np.inf], + [0., np.inf, np.inf, np.inf]]) + +def test_query_ball_point_vector_r(kdtree_type): + + np.random.seed(1234) + data = np.random.normal(size=(100, 3)) + query = np.random.normal(size=(100, 3)) + tree = kdtree_type(data) + d = np.random.uniform(0, 0.3, size=len(query)) + + rvector = tree.query_ball_point(query, d) + rscalar = [tree.query_ball_point(qi, di) for qi, di in zip(query, d)] + for a, b in zip(rvector, rscalar): + assert_array_equal(sorted(a), sorted(b)) + +def test_query_ball_point_length(kdtree_type): + + np.random.seed(1234) + data = np.random.normal(size=(100, 3)) + query = np.random.normal(size=(100, 3)) + tree = kdtree_type(data) + d = 0.3 + + length = tree.query_ball_point(query, d, return_length=True) + length2 = [len(ind) for ind in tree.query_ball_point(query, d, return_length=False)] + length3 = [len(tree.query_ball_point(qi, d)) for qi in query] + length4 = [tree.query_ball_point(qi, d, return_length=True) for qi in query] + assert_array_equal(length, length2) + assert_array_equal(length, length3) + assert_array_equal(length, length4) + +def test_discontiguous(kdtree_type): + + np.random.seed(1234) + data = np.random.normal(size=(100, 3)) + d_contiguous = np.arange(100) * 0.04 + d_discontiguous = np.ascontiguousarray( + np.arange(100)[::-1] * 0.04)[::-1] + query_contiguous = np.random.normal(size=(100, 3)) + query_discontiguous = np.ascontiguousarray(query_contiguous.T).T + assert query_discontiguous.strides[-1] != query_contiguous.strides[-1] + assert d_discontiguous.strides[-1] != d_contiguous.strides[-1] + + tree = kdtree_type(data) + + length1 = tree.query_ball_point(query_contiguous, + d_contiguous, return_length=True) + length2 = tree.query_ball_point(query_discontiguous, + d_discontiguous, return_length=True) + + assert_array_equal(length1, length2) + + d1, i1 = tree.query(query_contiguous, 1) + d2, i2 = tree.query(query_discontiguous, 1) + + assert_array_equal(d1, d2) + assert_array_equal(i1, i2) + + +@pytest.mark.parametrize("balanced_tree, compact_nodes", + [(True, False), + (True, True), + (False, False), + (False, True)]) +def test_kdtree_empty_input(kdtree_type, balanced_tree, compact_nodes): + # https://github.com/scipy/scipy/issues/5040 + np.random.seed(1234) + empty_v3 = np.empty(shape=(0, 3)) + query_v3 = np.ones(shape=(1, 3)) + query_v2 = np.ones(shape=(2, 3)) + + tree = kdtree_type(empty_v3, balanced_tree=balanced_tree, + compact_nodes=compact_nodes) + length = tree.query_ball_point(query_v3, 0.3, return_length=True) + assert length == 0 + + dd, ii = tree.query(query_v2, 2) + assert ii.shape == (2, 2) + assert dd.shape == (2, 2) + assert np.isinf(dd).all() + + N = tree.count_neighbors(tree, [0, 1]) + assert_array_equal(N, [0, 0]) + + M = tree.sparse_distance_matrix(tree, 0.3) + assert M.shape == (0, 0) + +@KDTreeTest +class _Test_sorted_query_ball_point: + def setup_method(self): + np.random.seed(1234) + self.x = np.random.randn(100, 1) + self.ckdt = self.kdtree_type(self.x) + + def test_return_sorted_True(self): + idxs_list = self.ckdt.query_ball_point(self.x, 1., return_sorted=True) + for idxs in idxs_list: + assert_array_equal(idxs, sorted(idxs)) + + for xi in self.x: + idxs = self.ckdt.query_ball_point(xi, 1., return_sorted=True) + assert_array_equal(idxs, sorted(idxs)) + + def test_return_sorted_None(self): + """Previous behavior was to sort the returned indices if there were + multiple points per query but not sort them if there was a single point + per query.""" + idxs_list = self.ckdt.query_ball_point(self.x, 1.) + for idxs in idxs_list: + assert_array_equal(idxs, sorted(idxs)) + + idxs_list_single = [self.ckdt.query_ball_point(xi, 1.) for xi in self.x] + idxs_list_False = self.ckdt.query_ball_point(self.x, 1., return_sorted=False) + for idxs0, idxs1 in zip(idxs_list_False, idxs_list_single): + assert_array_equal(idxs0, idxs1) + + +def test_kdtree_complex_data(): + # Test that KDTree rejects complex input points (gh-9108) + points = np.random.rand(10, 2).view(complex) + + with pytest.raises(TypeError, match="complex data"): + t = KDTree(points) + + t = KDTree(points.real) + + with pytest.raises(TypeError, match="complex data"): + t.query(points) + + with pytest.raises(TypeError, match="complex data"): + t.query_ball_point(points, r=1) + + +def test_kdtree_tree_access(): + # Test KDTree.tree can be used to traverse the KDTree + np.random.seed(1234) + points = np.random.rand(100, 4) + t = KDTree(points) + root = t.tree + + assert isinstance(root, KDTree.innernode) + assert root.children == points.shape[0] + + # Visit the tree and assert some basic properties for each node + nodes = [root] + while nodes: + n = nodes.pop(-1) + + if isinstance(n, KDTree.leafnode): + assert isinstance(n.children, int) + assert n.children == len(n.idx) + assert_array_equal(points[n.idx], n._node.data_points) + else: + assert isinstance(n, KDTree.innernode) + assert isinstance(n.split_dim, int) + assert 0 <= n.split_dim < t.m + assert isinstance(n.split, float) + assert isinstance(n.children, int) + assert n.children == n.less.children + n.greater.children + nodes.append(n.greater) + nodes.append(n.less) + + +def test_kdtree_attributes(): + # Test KDTree's attributes are available + np.random.seed(1234) + points = np.random.rand(100, 4) + t = KDTree(points) + + assert isinstance(t.m, int) + assert t.n == points.shape[0] + + assert isinstance(t.n, int) + assert t.m == points.shape[1] + + assert isinstance(t.leafsize, int) + assert t.leafsize == 10 + + assert_array_equal(t.maxes, np.amax(points, axis=0)) + assert_array_equal(t.mins, np.amin(points, axis=0)) + assert t.data is points + + +@pytest.mark.parametrize("kdtree_class", [KDTree, cKDTree]) +def test_kdtree_count_neighbors_weighted(kdtree_class): + rng = np.random.RandomState(1234) + r = np.arange(0.05, 1, 0.05) + + A = rng.random(21).reshape((7,3)) + B = rng.random(45).reshape((15,3)) + + wA = rng.random(7) + wB = rng.random(15) + + kdA = kdtree_class(A) + kdB = kdtree_class(B) + + nAB = kdA.count_neighbors(kdB, r, cumulative=False, weights=(wA,wB)) + + # Compare against brute-force + weights = wA[None, :] * wB[:, None] + dist = np.linalg.norm(A[None, :, :] - B[:, None, :], axis=-1) + expect = [np.sum(weights[(prev_radius < dist) & (dist <= radius)]) + for prev_radius, radius in zip(itertools.chain([0], r[:-1]), r)] + assert_allclose(nAB, expect) + + +def test_kdtree_nan(): + vals = [1, 5, -10, 7, -4, -16, -6, 6, 3, -11] + n = len(vals) + data = np.concatenate([vals, np.full(n, np.nan)])[:, None] + with pytest.raises(ValueError, match="must be finite"): + KDTree(data) + + +def test_nonfinite_inputs_gh_18223(): + rng = np.random.default_rng(12345) + coords = rng.uniform(size=(100, 3), low=0.0, high=0.1) + t = KDTree(coords, balanced_tree=False, compact_nodes=False) + bad_coord = [np.nan for _ in range(3)] + + with pytest.raises(ValueError, match="must be finite"): + t.query(bad_coord) + with pytest.raises(ValueError, match="must be finite"): + t.query_ball_point(bad_coord, 1) + + coords[0, :] = np.nan + with pytest.raises(ValueError, match="must be finite"): + KDTree(coords, balanced_tree=True, compact_nodes=False) + with pytest.raises(ValueError, match="must be finite"): + KDTree(coords, balanced_tree=False, compact_nodes=True) + with pytest.raises(ValueError, match="must be finite"): + KDTree(coords, balanced_tree=True, compact_nodes=True) + with pytest.raises(ValueError, match="must be finite"): + KDTree(coords, balanced_tree=False, compact_nodes=False) + + +@pytest.mark.parametrize("incantation", [cKDTree, KDTree]) +def test_gh_18800(incantation): + # our prohibition on non-finite values + # in kd-tree workflows means we need + # coercion to NumPy arrays enforced + + class ArrLike(np.ndarray): + def __new__(cls, input_array): + obj = np.asarray(input_array).view(cls) + # we override all() to mimic the problem + # pandas DataFrames encountered in gh-18800 + obj.all = None + return obj + + def __array_finalize__(self, obj): + if obj is None: + return + self.all = getattr(obj, 'all', None) + + points = [ + [66.22, 32.54], + [22.52, 22.39], + [31.01, 81.21], + ] + arr = np.array(points) + arr_like = ArrLike(arr) + tree = incantation(points, 10) + tree.query(arr_like, 1) + tree.query_ball_point(arr_like, 200) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_qhull.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_qhull.py new file mode 100644 index 0000000000000000000000000000000000000000..adb7fcc2bdbaa2cffbe692737da0f868a86e2ad5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_qhull.py @@ -0,0 +1,1308 @@ +import os +import copy + +import numpy as np +from numpy.testing import (assert_equal, assert_almost_equal, + assert_, assert_allclose, assert_array_equal) +import pytest +from pytest import raises as assert_raises + +import scipy.spatial._qhull as qhull +from scipy.spatial import cKDTree as KDTree # type: ignore[attr-defined] +from scipy.spatial import Voronoi + +import itertools + +def sorted_tuple(x): + return tuple(sorted(x)) + + +def assert_unordered_tuple_list_equal(a, b, tpl=tuple): + if isinstance(a, np.ndarray): + a = a.tolist() + if isinstance(b, np.ndarray): + b = b.tolist() + a = list(map(tpl, a)) + a.sort() + b = list(map(tpl, b)) + b.sort() + assert_equal(a, b) + + +np.random.seed(1234) + +points = [(0,0), (0,1), (1,0), (1,1), (0.5, 0.5), (0.5, 1.5)] + +pathological_data_1 = np.array([ + [-3.14,-3.14], [-3.14,-2.36], [-3.14,-1.57], [-3.14,-0.79], + [-3.14,0.0], [-3.14,0.79], [-3.14,1.57], [-3.14,2.36], + [-3.14,3.14], [-2.36,-3.14], [-2.36,-2.36], [-2.36,-1.57], + [-2.36,-0.79], [-2.36,0.0], [-2.36,0.79], [-2.36,1.57], + [-2.36,2.36], [-2.36,3.14], [-1.57,-0.79], [-1.57,0.79], + [-1.57,-1.57], [-1.57,0.0], [-1.57,1.57], [-1.57,-3.14], + [-1.57,-2.36], [-1.57,2.36], [-1.57,3.14], [-0.79,-1.57], + [-0.79,1.57], [-0.79,-3.14], [-0.79,-2.36], [-0.79,-0.79], + [-0.79,0.0], [-0.79,0.79], [-0.79,2.36], [-0.79,3.14], + [0.0,-3.14], [0.0,-2.36], [0.0,-1.57], [0.0,-0.79], [0.0,0.0], + [0.0,0.79], [0.0,1.57], [0.0,2.36], [0.0,3.14], [0.79,-3.14], + [0.79,-2.36], [0.79,-0.79], [0.79,0.0], [0.79,0.79], + [0.79,2.36], [0.79,3.14], [0.79,-1.57], [0.79,1.57], + [1.57,-3.14], [1.57,-2.36], [1.57,2.36], [1.57,3.14], + [1.57,-1.57], [1.57,0.0], [1.57,1.57], [1.57,-0.79], + [1.57,0.79], [2.36,-3.14], [2.36,-2.36], [2.36,-1.57], + [2.36,-0.79], [2.36,0.0], [2.36,0.79], [2.36,1.57], + [2.36,2.36], [2.36,3.14], [3.14,-3.14], [3.14,-2.36], + [3.14,-1.57], [3.14,-0.79], [3.14,0.0], [3.14,0.79], + [3.14,1.57], [3.14,2.36], [3.14,3.14], +]) + +pathological_data_2 = np.array([ + [-1, -1], [-1, 0], [-1, 1], + [0, -1], [0, 0], [0, 1], + [1, -1 - np.finfo(np.float64).eps], [1, 0], [1, 1], +]) + +bug_2850_chunks = [np.random.rand(10, 2), + np.array([[0,0], [0,1], [1,0], [1,1]]) # add corners + ] + +# same with some additional chunks +bug_2850_chunks_2 = (bug_2850_chunks + + [np.random.rand(10, 2), + 0.25 + np.array([[0,0], [0,1], [1,0], [1,1]])]) + +DATASETS = { + 'some-points': np.asarray(points), + 'random-2d': np.random.rand(30, 2), + 'random-3d': np.random.rand(30, 3), + 'random-4d': np.random.rand(30, 4), + 'random-5d': np.random.rand(30, 5), + 'random-6d': np.random.rand(10, 6), + 'random-7d': np.random.rand(10, 7), + 'random-8d': np.random.rand(10, 8), + 'pathological-1': pathological_data_1, + 'pathological-2': pathological_data_2 +} + +INCREMENTAL_DATASETS = { + 'bug-2850': (bug_2850_chunks, None), + 'bug-2850-2': (bug_2850_chunks_2, None), +} + + +def _add_inc_data(name, chunksize): + """ + Generate incremental datasets from basic data sets + """ + points = DATASETS[name] + ndim = points.shape[1] + + opts = None + nmin = ndim + 2 + + if name == 'some-points': + # since Qz is not allowed, use QJ + opts = 'QJ Pp' + elif name == 'pathological-1': + # include enough points so that we get different x-coordinates + nmin = 12 + + chunks = [points[:nmin]] + for j in range(nmin, len(points), chunksize): + chunks.append(points[j:j+chunksize]) + + new_name = "%s-chunk-%d" % (name, chunksize) + assert new_name not in INCREMENTAL_DATASETS + INCREMENTAL_DATASETS[new_name] = (chunks, opts) + + +for name in DATASETS: + for chunksize in 1, 4, 16: + _add_inc_data(name, chunksize) + + +class Test_Qhull: + def test_swapping(self): + # Check that Qhull state swapping works + + x = qhull._Qhull(b'v', + np.array([[0,0],[0,1],[1,0],[1,1.],[0.5,0.5]]), + b'Qz') + xd = copy.deepcopy(x.get_voronoi_diagram()) + + y = qhull._Qhull(b'v', + np.array([[0,0],[0,1],[1,0],[1,2.]]), + b'Qz') + yd = copy.deepcopy(y.get_voronoi_diagram()) + + xd2 = copy.deepcopy(x.get_voronoi_diagram()) + x.close() + yd2 = copy.deepcopy(y.get_voronoi_diagram()) + y.close() + + assert_raises(RuntimeError, x.get_voronoi_diagram) + assert_raises(RuntimeError, y.get_voronoi_diagram) + + assert_allclose(xd[0], xd2[0]) + assert_unordered_tuple_list_equal(xd[1], xd2[1], tpl=sorted_tuple) + assert_unordered_tuple_list_equal(xd[2], xd2[2], tpl=sorted_tuple) + assert_unordered_tuple_list_equal(xd[3], xd2[3], tpl=sorted_tuple) + assert_array_equal(xd[4], xd2[4]) + + assert_allclose(yd[0], yd2[0]) + assert_unordered_tuple_list_equal(yd[1], yd2[1], tpl=sorted_tuple) + assert_unordered_tuple_list_equal(yd[2], yd2[2], tpl=sorted_tuple) + assert_unordered_tuple_list_equal(yd[3], yd2[3], tpl=sorted_tuple) + assert_array_equal(yd[4], yd2[4]) + + x.close() + assert_raises(RuntimeError, x.get_voronoi_diagram) + y.close() + assert_raises(RuntimeError, y.get_voronoi_diagram) + + def test_issue_8051(self): + points = np.array( + [[0, 0], [0, 1], [0, 2], [1, 0], [1, 1], [1, 2],[2, 0], [2, 1], [2, 2]] + ) + Voronoi(points) + + +class TestUtilities: + """ + Check that utility functions work. + + """ + + def test_find_simplex(self): + # Simple check that simplex finding works + points = np.array([(0,0), (0,1), (1,1), (1,0)], dtype=np.float64) + tri = qhull.Delaunay(points) + + # +---+ + # |\ 0| + # | \ | + # |1 \| + # +---+ + + assert_equal(tri.simplices, [[1, 3, 2], [3, 1, 0]]) + + for p in [(0.25, 0.25, 1), + (0.75, 0.75, 0), + (0.3, 0.2, 1)]: + i = tri.find_simplex(p[:2]) + assert_equal(i, p[2], err_msg=f'{p!r}') + j = qhull.tsearch(tri, p[:2]) + assert_equal(i, j) + + def test_plane_distance(self): + # Compare plane distance from hyperplane equations obtained from Qhull + # to manually computed plane equations + x = np.array([(0,0), (1, 1), (1, 0), (0.99189033, 0.37674127), + (0.99440079, 0.45182168)], dtype=np.float64) + p = np.array([0.99966555, 0.15685619], dtype=np.float64) + + tri = qhull.Delaunay(x) + + z = tri.lift_points(x) + pz = tri.lift_points(p) + + dist = tri.plane_distance(p) + + for j, v in enumerate(tri.simplices): + x1 = z[v[0]] + x2 = z[v[1]] + x3 = z[v[2]] + + n = np.cross(x1 - x3, x2 - x3) + n /= np.sqrt(np.dot(n, n)) + n *= -np.sign(n[2]) + + d = np.dot(n, pz - x3) + + assert_almost_equal(dist[j], d) + + def test_convex_hull(self): + # Simple check that the convex hull seems to works + points = np.array([(0,0), (0,1), (1,1), (1,0)], dtype=np.float64) + tri = qhull.Delaunay(points) + + # +---+ + # |\ 0| + # | \ | + # |1 \| + # +---+ + + assert_equal(tri.convex_hull, [[3, 2], [1, 2], [1, 0], [3, 0]]) + + def test_volume_area(self): + #Basic check that we get back the correct volume and area for a cube + points = np.array([(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0), + (0, 0, 1), (0, 1, 1), (1, 0, 1), (1, 1, 1)]) + hull = qhull.ConvexHull(points) + + assert_allclose(hull.volume, 1., rtol=1e-14, + err_msg="Volume of cube is incorrect") + assert_allclose(hull.area, 6., rtol=1e-14, + err_msg="Area of cube is incorrect") + + def test_random_volume_area(self): + #Test that the results for a random 10-point convex are + #coherent with the output of qconvex Qt s FA + points = np.array([(0.362568364506, 0.472712355305, 0.347003084477), + (0.733731893414, 0.634480295684, 0.950513180209), + (0.511239955611, 0.876839441267, 0.418047827863), + (0.0765906233393, 0.527373281342, 0.6509863541), + (0.146694972056, 0.596725793348, 0.894860986685), + (0.513808585741, 0.069576205858, 0.530890338876), + (0.512343805118, 0.663537132612, 0.037689295973), + (0.47282965018, 0.462176697655, 0.14061843691), + (0.240584597123, 0.778660020591, 0.722913476339), + (0.951271745935, 0.967000673944, 0.890661319684)]) + + hull = qhull.ConvexHull(points) + assert_allclose(hull.volume, 0.14562013, rtol=1e-07, + err_msg="Volume of random polyhedron is incorrect") + assert_allclose(hull.area, 1.6670425, rtol=1e-07, + err_msg="Area of random polyhedron is incorrect") + + def test_incremental_volume_area_random_input(self): + """Test that incremental mode gives the same volume/area as + non-incremental mode and incremental mode with restart""" + nr_points = 20 + dim = 3 + points = np.random.random((nr_points, dim)) + inc_hull = qhull.ConvexHull(points[:dim+1, :], incremental=True) + inc_restart_hull = qhull.ConvexHull(points[:dim+1, :], incremental=True) + for i in range(dim+1, nr_points): + hull = qhull.ConvexHull(points[:i+1, :]) + inc_hull.add_points(points[i:i+1, :]) + inc_restart_hull.add_points(points[i:i+1, :], restart=True) + assert_allclose(hull.volume, inc_hull.volume, rtol=1e-7) + assert_allclose(hull.volume, inc_restart_hull.volume, rtol=1e-7) + assert_allclose(hull.area, inc_hull.area, rtol=1e-7) + assert_allclose(hull.area, inc_restart_hull.area, rtol=1e-7) + + def _check_barycentric_transforms(self, tri, err_msg="", + unit_cube=False, + unit_cube_tol=0): + """Check that a triangulation has reasonable barycentric transforms""" + vertices = tri.points[tri.simplices] + sc = 1/(tri.ndim + 1.0) + centroids = vertices.sum(axis=1) * sc + + # Either: (i) the simplex has a `nan` barycentric transform, + # or, (ii) the centroid is in the simplex + + def barycentric_transform(tr, x): + r = tr[:,-1,:] + Tinv = tr[:,:-1,:] + return np.einsum('ijk,ik->ij', Tinv, x - r) + + eps = np.finfo(float).eps + + c = barycentric_transform(tri.transform, centroids) + with np.errstate(invalid="ignore"): + ok = np.isnan(c).all(axis=1) | (abs(c - sc)/sc < 0.1).all(axis=1) + + assert_(ok.all(), f"{err_msg} {np.nonzero(~ok)}") + + # Invalid simplices must be (nearly) zero volume + q = vertices[:,:-1,:] - vertices[:,-1,None,:] + volume = np.array([np.linalg.det(q[k,:,:]) + for k in range(tri.nsimplex)]) + ok = np.isfinite(tri.transform[:,0,0]) | (volume < np.sqrt(eps)) + assert_(ok.all(), f"{err_msg} {np.nonzero(~ok)}") + + # Also, find_simplex for the centroid should end up in some + # simplex for the non-degenerate cases + j = tri.find_simplex(centroids) + ok = (j != -1) | np.isnan(tri.transform[:,0,0]) + assert_(ok.all(), f"{err_msg} {np.nonzero(~ok)}") + + if unit_cube: + # If in unit cube, no interior point should be marked out of hull + at_boundary = (centroids <= unit_cube_tol).any(axis=1) + at_boundary |= (centroids >= 1 - unit_cube_tol).any(axis=1) + + ok = (j != -1) | at_boundary + assert_(ok.all(), f"{err_msg} {np.nonzero(~ok)}") + + @pytest.mark.fail_slow(10) + def test_degenerate_barycentric_transforms(self): + # The triangulation should not produce invalid barycentric + # transforms that stump the simplex finding + data = np.load(os.path.join(os.path.dirname(__file__), 'data', + 'degenerate_pointset.npz')) + points = data['c'] + data.close() + + tri = qhull.Delaunay(points) + + # Check that there are not too many invalid simplices + bad_count = np.isnan(tri.transform[:,0,0]).sum() + assert_(bad_count < 23, bad_count) + + # Check the transforms + self._check_barycentric_transforms(tri) + + @pytest.mark.slow + @pytest.mark.fail_slow(20) + # OK per https://github.com/scipy/scipy/pull/20487#discussion_r1572684869 + def test_more_barycentric_transforms(self): + # Triangulate some "nasty" grids + + eps = np.finfo(float).eps + + npoints = {2: 70, 3: 11, 4: 5, 5: 3} + + for ndim in range(2, 6): + # Generate an uniform grid in n-d unit cube + x = np.linspace(0, 1, npoints[ndim]) + grid = np.c_[ + list(map(np.ravel, np.broadcast_arrays(*np.ix_(*([x]*ndim))))) + ].T + + err_msg = "ndim=%d" % ndim + + # Check using regular grid + tri = qhull.Delaunay(grid) + self._check_barycentric_transforms(tri, err_msg=err_msg, + unit_cube=True) + + # Check with eps-perturbations + np.random.seed(1234) + m = (np.random.rand(grid.shape[0]) < 0.2) + grid[m,:] += 2*eps*(np.random.rand(*grid[m,:].shape) - 0.5) + + tri = qhull.Delaunay(grid) + self._check_barycentric_transforms(tri, err_msg=err_msg, + unit_cube=True, + unit_cube_tol=2*eps) + + # Check with duplicated data + tri = qhull.Delaunay(np.r_[grid, grid]) + self._check_barycentric_transforms(tri, err_msg=err_msg, + unit_cube=True, + unit_cube_tol=2*eps) + + +class TestVertexNeighborVertices: + def _check(self, tri): + expected = [set() for j in range(tri.points.shape[0])] + for s in tri.simplices: + for a in s: + for b in s: + if a != b: + expected[a].add(b) + + indptr, indices = tri.vertex_neighbor_vertices + + got = [set(map(int, indices[indptr[j]:indptr[j+1]])) + for j in range(tri.points.shape[0])] + + assert_equal(got, expected, err_msg=f"{got!r} != {expected!r}") + + def test_triangle(self): + points = np.array([(0,0), (0,1), (1,0)], dtype=np.float64) + tri = qhull.Delaunay(points) + self._check(tri) + + def test_rectangle(self): + points = np.array([(0,0), (0,1), (1,1), (1,0)], dtype=np.float64) + tri = qhull.Delaunay(points) + self._check(tri) + + def test_complicated(self): + points = np.array([(0,0), (0,1), (1,1), (1,0), + (0.5, 0.5), (0.9, 0.5)], dtype=np.float64) + tri = qhull.Delaunay(points) + self._check(tri) + + +class TestDelaunay: + """ + Check that triangulation works. + + """ + def test_masked_array_fails(self): + masked_array = np.ma.masked_all(1) + assert_raises(ValueError, qhull.Delaunay, masked_array) + + def test_array_with_nans_fails(self): + points_with_nan = np.array([(0,0), (0,1), (1,1), (1,np.nan)], dtype=np.float64) + assert_raises(ValueError, qhull.Delaunay, points_with_nan) + + def test_nd_simplex(self): + # simple smoke test: triangulate a n-dimensional simplex + for nd in range(2, 8): + points = np.zeros((nd+1, nd)) + for j in range(nd): + points[j,j] = 1.0 + points[-1,:] = 1.0 + + tri = qhull.Delaunay(points) + + tri.simplices.sort() + + assert_equal(tri.simplices, np.arange(nd+1, dtype=int)[None, :]) + assert_equal(tri.neighbors, -1 + np.zeros((nd+1), dtype=int)[None,:]) + + def test_2d_square(self): + # simple smoke test: 2d square + points = np.array([(0,0), (0,1), (1,1), (1,0)], dtype=np.float64) + tri = qhull.Delaunay(points) + + assert_equal(tri.simplices, [[1, 3, 2], [3, 1, 0]]) + assert_equal(tri.neighbors, [[-1, -1, 1], [-1, -1, 0]]) + + def test_duplicate_points(self): + x = np.array([0, 1, 0, 1], dtype=np.float64) + y = np.array([0, 0, 1, 1], dtype=np.float64) + + xp = np.r_[x, x] + yp = np.r_[y, y] + + # shouldn't fail on duplicate points + qhull.Delaunay(np.c_[x, y]) + qhull.Delaunay(np.c_[xp, yp]) + + def test_pathological(self): + # both should succeed + points = DATASETS['pathological-1'] + tri = qhull.Delaunay(points) + assert_equal(tri.points[tri.simplices].max(), points.max()) + assert_equal(tri.points[tri.simplices].min(), points.min()) + + points = DATASETS['pathological-2'] + tri = qhull.Delaunay(points) + assert_equal(tri.points[tri.simplices].max(), points.max()) + assert_equal(tri.points[tri.simplices].min(), points.min()) + + def test_joggle(self): + # Check that the option QJ indeed guarantees that all input points + # occur as vertices of the triangulation + + points = np.random.rand(10, 2) + points = np.r_[points, points] # duplicate input data + + tri = qhull.Delaunay(points, qhull_options="QJ Qbb Pp") + assert_array_equal(np.unique(tri.simplices.ravel()), + np.arange(len(points))) + + def test_coplanar(self): + # Check that the coplanar point output option indeed works + points = np.random.rand(10, 2) + points = np.r_[points, points] # duplicate input data + + tri = qhull.Delaunay(points) + + assert_(len(np.unique(tri.simplices.ravel())) == len(points)//2) + assert_(len(tri.coplanar) == len(points)//2) + + assert_(len(np.unique(tri.coplanar[:,2])) == len(points)//2) + + assert_(np.all(tri.vertex_to_simplex >= 0)) + + def test_furthest_site(self): + points = [(0, 0), (0, 1), (1, 0), (0.5, 0.5), (1.1, 1.1)] + tri = qhull.Delaunay(points, furthest_site=True) + + expected = np.array([(1, 4, 0), (4, 2, 0)]) # from Qhull + assert_array_equal(tri.simplices, expected) + + @pytest.mark.parametrize("name", sorted(INCREMENTAL_DATASETS)) + def test_incremental(self, name): + # Test incremental construction of the triangulation + + chunks, opts = INCREMENTAL_DATASETS[name] + points = np.concatenate(chunks, axis=0) + + obj = qhull.Delaunay(chunks[0], incremental=True, + qhull_options=opts) + for chunk in chunks[1:]: + obj.add_points(chunk) + + obj2 = qhull.Delaunay(points) + + obj3 = qhull.Delaunay(chunks[0], incremental=True, + qhull_options=opts) + if len(chunks) > 1: + obj3.add_points(np.concatenate(chunks[1:], axis=0), + restart=True) + + # Check that the incremental mode agrees with upfront mode + if name.startswith('pathological'): + # XXX: These produce valid but different triangulations. + # They look OK when plotted, but how to check them? + + assert_array_equal(np.unique(obj.simplices.ravel()), + np.arange(points.shape[0])) + assert_array_equal(np.unique(obj2.simplices.ravel()), + np.arange(points.shape[0])) + else: + assert_unordered_tuple_list_equal(obj.simplices, obj2.simplices, + tpl=sorted_tuple) + + assert_unordered_tuple_list_equal(obj2.simplices, obj3.simplices, + tpl=sorted_tuple) + + +def assert_hulls_equal(points, facets_1, facets_2): + # Check that two convex hulls constructed from the same point set + # are equal + + facets_1 = set(map(sorted_tuple, facets_1)) + facets_2 = set(map(sorted_tuple, facets_2)) + + if facets_1 != facets_2 and points.shape[1] == 2: + # The direct check fails for the pathological cases + # --- then the convex hull from Delaunay differs (due + # to rounding error etc.) from the hull computed + # otherwise, by the question whether (tricoplanar) + # points that lie almost exactly on the hull are + # included as vertices of the hull or not. + # + # So we check the result, and accept it if the Delaunay + # hull line segments are a subset of the usual hull. + + eps = 1000 * np.finfo(float).eps + + for a, b in facets_1: + for ap, bp in facets_2: + t = points[bp] - points[ap] + t /= np.linalg.norm(t) # tangent + n = np.array([-t[1], t[0]]) # normal + + # check that the two line segments are parallel + # to the same line + c1 = np.dot(n, points[b] - points[ap]) + c2 = np.dot(n, points[a] - points[ap]) + if not np.allclose(np.dot(c1, n), 0): + continue + if not np.allclose(np.dot(c2, n), 0): + continue + + # Check that the segment (a, b) is contained in (ap, bp) + c1 = np.dot(t, points[a] - points[ap]) + c2 = np.dot(t, points[b] - points[ap]) + c3 = np.dot(t, points[bp] - points[ap]) + if c1 < -eps or c1 > c3 + eps: + continue + if c2 < -eps or c2 > c3 + eps: + continue + + # OK: + break + else: + raise AssertionError("comparison fails") + + # it was OK + return + + assert_equal(facets_1, facets_2) + + +class TestConvexHull: + def test_masked_array_fails(self): + masked_array = np.ma.masked_all(1) + assert_raises(ValueError, qhull.ConvexHull, masked_array) + + def test_array_with_nans_fails(self): + points_with_nan = np.array([(0,0), (1,1), (2,np.nan)], dtype=np.float64) + assert_raises(ValueError, qhull.ConvexHull, points_with_nan) + + @pytest.mark.parametrize("name", sorted(DATASETS)) + def test_hull_consistency_tri(self, name): + # Check that a convex hull returned by qhull in ndim + # and the hull constructed from ndim delaunay agree + points = DATASETS[name] + + tri = qhull.Delaunay(points) + hull = qhull.ConvexHull(points) + + assert_hulls_equal(points, tri.convex_hull, hull.simplices) + + # Check that the hull extremes are as expected + if points.shape[1] == 2: + assert_equal(np.unique(hull.simplices), np.sort(hull.vertices)) + else: + assert_equal(np.unique(hull.simplices), hull.vertices) + + @pytest.mark.parametrize("name", sorted(INCREMENTAL_DATASETS)) + def test_incremental(self, name): + # Test incremental construction of the convex hull + chunks, _ = INCREMENTAL_DATASETS[name] + points = np.concatenate(chunks, axis=0) + + obj = qhull.ConvexHull(chunks[0], incremental=True) + for chunk in chunks[1:]: + obj.add_points(chunk) + + obj2 = qhull.ConvexHull(points) + + obj3 = qhull.ConvexHull(chunks[0], incremental=True) + if len(chunks) > 1: + obj3.add_points(np.concatenate(chunks[1:], axis=0), + restart=True) + + # Check that the incremental mode agrees with upfront mode + assert_hulls_equal(points, obj.simplices, obj2.simplices) + assert_hulls_equal(points, obj.simplices, obj3.simplices) + + def test_vertices_2d(self): + # The vertices should be in counterclockwise order in 2-D + np.random.seed(1234) + points = np.random.rand(30, 2) + + hull = qhull.ConvexHull(points) + assert_equal(np.unique(hull.simplices), np.sort(hull.vertices)) + + # Check counterclockwiseness + x, y = hull.points[hull.vertices].T + angle = np.arctan2(y - y.mean(), x - x.mean()) + assert_(np.all(np.diff(np.unwrap(angle)) > 0)) + + def test_volume_area(self): + # Basic check that we get back the correct volume and area for a cube + points = np.array([(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0), + (0, 0, 1), (0, 1, 1), (1, 0, 1), (1, 1, 1)]) + tri = qhull.ConvexHull(points) + + assert_allclose(tri.volume, 1., rtol=1e-14) + assert_allclose(tri.area, 6., rtol=1e-14) + + @pytest.mark.parametrize("incremental", [False, True]) + def test_good2d(self, incremental): + # Make sure the QGn option gives the correct value of "good". + points = np.array([[0.2, 0.2], + [0.2, 0.4], + [0.4, 0.4], + [0.4, 0.2], + [0.3, 0.6]]) + hull = qhull.ConvexHull(points=points, + incremental=incremental, + qhull_options='QG4') + expected = np.array([False, True, False, False], dtype=bool) + actual = hull.good + assert_equal(actual, expected) + + @pytest.mark.parametrize("visibility", [ + "QG4", # visible=True + "QG-4", # visible=False + ]) + @pytest.mark.parametrize("new_gen, expected", [ + # add generator that places QG4 inside hull + # so all facets are invisible + (np.array([[0.3, 0.7]]), + np.array([False, False, False, False, False], dtype=bool)), + # adding a generator on the opposite side of the square + # should preserve the single visible facet & add one invisible + # facet + (np.array([[0.3, -0.7]]), + np.array([False, True, False, False, False], dtype=bool)), + # split the visible facet on top of the square into two + # visible facets, with visibility at the end of the array + # because add_points concatenates + (np.array([[0.3, 0.41]]), + np.array([False, False, False, True, True], dtype=bool)), + # with our current Qhull options, coplanarity will not count + # for visibility; this case shifts one visible & one invisible + # facet & adds a coplanar facet + # simplex at index position 2 is the shifted visible facet + # the final simplex is the coplanar facet + (np.array([[0.5, 0.6], [0.6, 0.6]]), + np.array([False, False, True, False, False], dtype=bool)), + # place the new generator such that it envelops the query + # point within the convex hull, but only just barely within + # the double precision limit + # NOTE: testing exact degeneracy is less predictable than this + # scenario, perhaps because of the default Qt option we have + # enabled for Qhull to handle precision matters + (np.array([[0.3, 0.6 + 1e-16]]), + np.array([False, False, False, False, False], dtype=bool)), + ]) + def test_good2d_incremental_changes(self, new_gen, expected, + visibility): + # use the usual square convex hull + # generators from test_good2d + points = np.array([[0.2, 0.2], + [0.2, 0.4], + [0.4, 0.4], + [0.4, 0.2], + [0.3, 0.6]]) + hull = qhull.ConvexHull(points=points, + incremental=True, + qhull_options=visibility) + hull.add_points(new_gen) + actual = hull.good + if '-' in visibility: + expected = np.invert(expected) + assert_equal(actual, expected) + + @pytest.mark.parametrize("incremental", [False, True]) + def test_good2d_no_option(self, incremental): + # handle case where good attribute doesn't exist + # because Qgn or Qg-n wasn't specified + points = np.array([[0.2, 0.2], + [0.2, 0.4], + [0.4, 0.4], + [0.4, 0.2], + [0.3, 0.6]]) + hull = qhull.ConvexHull(points=points, + incremental=incremental) + actual = hull.good + assert actual is None + # preserve None after incremental addition + if incremental: + hull.add_points(np.zeros((1, 2))) + actual = hull.good + assert actual is None + + @pytest.mark.parametrize("incremental", [False, True]) + def test_good2d_inside(self, incremental): + # Make sure the QGn option gives the correct value of "good". + # When point n is inside the convex hull of the rest, good is + # all False. + points = np.array([[0.2, 0.2], + [0.2, 0.4], + [0.4, 0.4], + [0.4, 0.2], + [0.3, 0.3]]) + hull = qhull.ConvexHull(points=points, + incremental=incremental, + qhull_options='QG4') + expected = np.array([False, False, False, False], dtype=bool) + actual = hull.good + assert_equal(actual, expected) + + @pytest.mark.parametrize("incremental", [False, True]) + def test_good3d(self, incremental): + # Make sure the QGn option gives the correct value of "good" + # for a 3d figure + points = np.array([[0.0, 0.0, 0.0], + [0.90029516, -0.39187448, 0.18948093], + [0.48676420, -0.72627633, 0.48536925], + [0.57651530, -0.81179274, -0.09285832], + [0.67846893, -0.71119562, 0.18406710]]) + hull = qhull.ConvexHull(points=points, + incremental=incremental, + qhull_options='QG0') + expected = np.array([True, False, False, False], dtype=bool) + assert_equal(hull.good, expected) + +class TestVoronoi: + + @pytest.mark.parametrize("qhull_opts, extra_pts", [ + # option Qz (default for SciPy) will add + # an extra point at infinity + ("Qbb Qc Qz", 1), + ("Qbb Qc", 0), + ]) + @pytest.mark.parametrize("n_pts", [50, 100]) + @pytest.mark.parametrize("ndim", [2, 3]) + def test_point_region_structure(self, + qhull_opts, + n_pts, + extra_pts, + ndim): + # see gh-16773 + rng = np.random.default_rng(7790) + points = rng.random((n_pts, ndim)) + vor = Voronoi(points, qhull_options=qhull_opts) + pt_region = vor.point_region + assert pt_region.max() == n_pts - 1 + extra_pts + assert pt_region.size == len(vor.regions) - extra_pts + assert len(vor.regions) == n_pts + extra_pts + assert vor.points.shape[0] == n_pts + # if there is an empty sublist in the Voronoi + # regions data structure, it should never be + # indexed because it corresponds to an internally + # added point at infinity and is not a member of the + # generators (input points) + if extra_pts: + sublens = [len(x) for x in vor.regions] + # only one point at infinity (empty region) + # is allowed + assert sublens.count(0) == 1 + assert sublens.index(0) not in pt_region + + def test_masked_array_fails(self): + masked_array = np.ma.masked_all(1) + assert_raises(ValueError, qhull.Voronoi, masked_array) + + def test_simple(self): + # Simple case with known Voronoi diagram + points = [(0, 0), (0, 1), (0, 2), + (1, 0), (1, 1), (1, 2), + (2, 0), (2, 1), (2, 2)] + + # qhull v o Fv Qbb Qc Qz < dat + output = """ + 2 + 5 10 1 + -10.101 -10.101 + 0.5 0.5 + 0.5 1.5 + 1.5 0.5 + 1.5 1.5 + 2 0 1 + 3 2 0 1 + 2 0 2 + 3 3 0 1 + 4 1 2 4 3 + 3 4 0 2 + 2 0 3 + 3 4 0 3 + 2 0 4 + 0 + 12 + 4 0 3 0 1 + 4 0 1 0 1 + 4 1 4 1 2 + 4 1 2 0 2 + 4 2 5 0 2 + 4 3 4 1 3 + 4 3 6 0 3 + 4 4 5 2 4 + 4 4 7 3 4 + 4 5 8 0 4 + 4 6 7 0 3 + 4 7 8 0 4 + """ + self._compare_qvoronoi(points, output) + + def _compare_qvoronoi(self, points, output, **kw): + """Compare to output from 'qvoronoi o Fv < data' to Voronoi()""" + + # Parse output + output = [list(map(float, x.split())) for x in output.strip().splitlines()] + nvertex = int(output[1][0]) + vertices = list(map(tuple, output[3:2+nvertex])) # exclude inf + nregion = int(output[1][1]) + regions = [[int(y)-1 for y in x[1:]] + for x in output[2+nvertex:2+nvertex+nregion]] + ridge_points = [[int(y) for y in x[1:3]] + for x in output[3+nvertex+nregion:]] + ridge_vertices = [[int(y)-1 for y in x[3:]] + for x in output[3+nvertex+nregion:]] + + # Compare results + vor = qhull.Voronoi(points, **kw) + + def sorttuple(x): + return tuple(sorted(x)) + + assert_allclose(vor.vertices, vertices) + assert_equal(set(map(tuple, vor.regions)), + set(map(tuple, regions))) + + p1 = list(zip(list(map(sorttuple, ridge_points)), + list(map(sorttuple, ridge_vertices)))) + p2 = list(zip(list(map(sorttuple, vor.ridge_points.tolist())), + list(map(sorttuple, vor.ridge_vertices)))) + p1.sort() + p2.sort() + + assert_equal(p1, p2) + + @pytest.mark.parametrize("name", sorted(DATASETS)) + def test_ridges(self, name): + # Check that the ridges computed by Voronoi indeed separate + # the regions of nearest neighborhood, by comparing the result + # to KDTree. + + points = DATASETS[name] + + tree = KDTree(points) + vor = qhull.Voronoi(points) + + for p, v in vor.ridge_dict.items(): + # consider only finite ridges + if not np.all(np.asarray(v) >= 0): + continue + + ridge_midpoint = vor.vertices[v].mean(axis=0) + d = 1e-6 * (points[p[0]] - ridge_midpoint) + + dist, k = tree.query(ridge_midpoint + d, k=1) + assert_equal(k, p[0]) + + dist, k = tree.query(ridge_midpoint - d, k=1) + assert_equal(k, p[1]) + + def test_furthest_site(self): + points = [(0, 0), (0, 1), (1, 0), (0.5, 0.5), (1.1, 1.1)] + + # qhull v o Fv Qbb Qc Qu < dat + output = """ + 2 + 3 5 1 + -10.101 -10.101 + 0.6000000000000001 0.5 + 0.5 0.6000000000000001 + 3 0 2 1 + 2 0 1 + 2 0 2 + 0 + 3 0 2 1 + 5 + 4 0 2 0 2 + 4 0 4 1 2 + 4 0 1 0 1 + 4 1 4 0 1 + 4 2 4 0 2 + """ + self._compare_qvoronoi(points, output, furthest_site=True) + + def test_furthest_site_flag(self): + points = [(0, 0), (0, 1), (1, 0), (0.5, 0.5), (1.1, 1.1)] + + vor = Voronoi(points) + assert_equal(vor.furthest_site,False) + vor = Voronoi(points,furthest_site=True) + assert_equal(vor.furthest_site,True) + + @pytest.mark.fail_slow(10) + @pytest.mark.parametrize("name", sorted(INCREMENTAL_DATASETS)) + def test_incremental(self, name): + # Test incremental construction of the triangulation + + if INCREMENTAL_DATASETS[name][0][0].shape[1] > 3: + # too slow (testing of the result --- qhull is still fast) + return + + chunks, opts = INCREMENTAL_DATASETS[name] + points = np.concatenate(chunks, axis=0) + + obj = qhull.Voronoi(chunks[0], incremental=True, + qhull_options=opts) + for chunk in chunks[1:]: + obj.add_points(chunk) + + obj2 = qhull.Voronoi(points) + + obj3 = qhull.Voronoi(chunks[0], incremental=True, + qhull_options=opts) + if len(chunks) > 1: + obj3.add_points(np.concatenate(chunks[1:], axis=0), + restart=True) + + # -- Check that the incremental mode agrees with upfront mode + assert_equal(len(obj.point_region), len(obj2.point_region)) + assert_equal(len(obj.point_region), len(obj3.point_region)) + + # The vertices may be in different order or duplicated in + # the incremental map + for objx in obj, obj3: + vertex_map = {-1: -1} + for i, v in enumerate(objx.vertices): + for j, v2 in enumerate(obj2.vertices): + if np.allclose(v, v2): + vertex_map[i] = j + + def remap(x): + if hasattr(x, '__len__'): + return tuple({remap(y) for y in x}) + try: + return vertex_map[x] + except KeyError as e: + message = (f"incremental result has spurious vertex " + f"at {objx.vertices[x]!r}") + raise AssertionError(message) from e + + def simplified(x): + items = set(map(sorted_tuple, x)) + if () in items: + items.remove(()) + items = [x for x in items if len(x) > 1] + items.sort() + return items + + assert_equal( + simplified(remap(objx.regions)), + simplified(obj2.regions) + ) + assert_equal( + simplified(remap(objx.ridge_vertices)), + simplified(obj2.ridge_vertices) + ) + + # XXX: compare ridge_points --- not clear exactly how to do this + + +class Test_HalfspaceIntersection: + def assert_unordered_allclose(self, arr1, arr2, rtol=1e-7): + """Check that every line in arr1 is only once in arr2""" + assert_equal(arr1.shape, arr2.shape) + + truths = np.zeros((arr1.shape[0],), dtype=bool) + for l1 in arr1: + indexes = np.nonzero((abs(arr2 - l1) < rtol).all(axis=1))[0] + assert_equal(indexes.shape, (1,)) + truths[indexes[0]] = True + assert_(truths.all()) + + @pytest.mark.parametrize("dt", [np.float64, int]) + def test_cube_halfspace_intersection(self, dt): + halfspaces = np.array([[-1, 0, 0], + [0, -1, 0], + [1, 0, -2], + [0, 1, -2]], dtype=dt) + feasible_point = np.array([1, 1], dtype=dt) + + points = np.array([[0.0, 0.0], [2.0, 0.0], [0.0, 2.0], [2.0, 2.0]]) + + hull = qhull.HalfspaceIntersection(halfspaces, feasible_point) + + assert_allclose(hull.intersections, points) + + def test_self_dual_polytope_intersection(self): + fname = os.path.join(os.path.dirname(__file__), 'data', + 'selfdual-4d-polytope.txt') + ineqs = np.genfromtxt(fname) + halfspaces = -np.hstack((ineqs[:, 1:], ineqs[:, :1])) + + feas_point = np.array([0., 0., 0., 0.]) + hs = qhull.HalfspaceIntersection(halfspaces, feas_point) + + assert_equal(hs.intersections.shape, (24, 4)) + + assert_almost_equal(hs.dual_volume, 32.0) + assert_equal(len(hs.dual_facets), 24) + for facet in hs.dual_facets: + assert_equal(len(facet), 6) + + dists = halfspaces[:, -1] + halfspaces[:, :-1].dot(feas_point) + self.assert_unordered_allclose((halfspaces[:, :-1].T/dists).T, hs.dual_points) + + points = itertools.permutations([0., 0., 0.5, -0.5]) + for point in points: + assert_equal(np.sum((hs.intersections == point).all(axis=1)), 1) + + def test_wrong_feasible_point(self): + halfspaces = np.array([[-1.0, 0.0, 0.0], + [0.0, -1.0, 0.0], + [1.0, 0.0, -1.0], + [0.0, 1.0, -1.0]]) + feasible_point = np.array([0.5, 0.5, 0.5]) + #Feasible point is (ndim,) instead of (ndim-1,) + assert_raises(ValueError, + qhull.HalfspaceIntersection, halfspaces, feasible_point) + feasible_point = np.array([[0.5], [0.5]]) + #Feasible point is (ndim-1, 1) instead of (ndim-1,) + assert_raises(ValueError, + qhull.HalfspaceIntersection, halfspaces, feasible_point) + feasible_point = np.array([[0.5, 0.5]]) + #Feasible point is (1, ndim-1) instead of (ndim-1,) + assert_raises(ValueError, + qhull.HalfspaceIntersection, halfspaces, feasible_point) + + feasible_point = np.array([-0.5, -0.5]) + #Feasible point is outside feasible region + assert_raises(qhull.QhullError, + qhull.HalfspaceIntersection, halfspaces, feasible_point) + + def test_incremental(self): + #Cube + halfspaces = np.array([[0., 0., -1., -0.5], + [0., -1., 0., -0.5], + [-1., 0., 0., -0.5], + [1., 0., 0., -0.5], + [0., 1., 0., -0.5], + [0., 0., 1., -0.5]]) + #Cut each summit + extra_normals = np.array([[1., 1., 1.], + [1., 1., -1.], + [1., -1., 1.], + [1, -1., -1.]]) + offsets = np.array([[-1.]]*8) + extra_halfspaces = np.hstack((np.vstack((extra_normals, -extra_normals)), + offsets)) + + feas_point = np.array([0., 0., 0.]) + + inc_hs = qhull.HalfspaceIntersection(halfspaces, feas_point, incremental=True) + + inc_res_hs = qhull.HalfspaceIntersection(halfspaces, feas_point, + incremental=True) + + for i, ehs in enumerate(extra_halfspaces): + inc_hs.add_halfspaces(ehs[np.newaxis, :]) + + inc_res_hs.add_halfspaces(ehs[np.newaxis, :], restart=True) + + total = np.vstack((halfspaces, extra_halfspaces[:i+1, :])) + + hs = qhull.HalfspaceIntersection(total, feas_point) + + assert_allclose(inc_hs.halfspaces, inc_res_hs.halfspaces) + assert_allclose(inc_hs.halfspaces, hs.halfspaces) + + #Direct computation and restart should have points in same order + assert_allclose(hs.intersections, inc_res_hs.intersections) + #Incremental will have points in different order than direct computation + self.assert_unordered_allclose(inc_hs.intersections, hs.intersections) + + inc_hs.close() + + def test_cube(self): + # Halfspaces of the cube: + halfspaces = np.array([[-1., 0., 0., 0.], # x >= 0 + [1., 0., 0., -1.], # x <= 1 + [0., -1., 0., 0.], # y >= 0 + [0., 1., 0., -1.], # y <= 1 + [0., 0., -1., 0.], # z >= 0 + [0., 0., 1., -1.]]) # z <= 1 + point = np.array([0.5, 0.5, 0.5]) + + hs = qhull.HalfspaceIntersection(halfspaces, point) + + # qhalf H0.5,0.5,0.5 o < input.txt + qhalf_points = np.array([ + [-2, 0, 0], + [2, 0, 0], + [0, -2, 0], + [0, 2, 0], + [0, 0, -2], + [0, 0, 2]]) + qhalf_facets = [ + [2, 4, 0], + [4, 2, 1], + [5, 2, 0], + [2, 5, 1], + [3, 4, 1], + [4, 3, 0], + [5, 3, 1], + [3, 5, 0]] + + assert len(qhalf_facets) == len(hs.dual_facets) + for a, b in zip(qhalf_facets, hs.dual_facets): + assert set(a) == set(b) # facet orientation can differ + + assert_allclose(hs.dual_points, qhalf_points) + + @pytest.mark.parametrize("k", range(1,4)) + def test_halfspace_batch(self, k): + # Test that we can add halfspaces a few at a time + big_square = np.array([[ 1., 0., -2.], + [-1., 0., -2.], + [ 0., 1., -2.], + [ 0., -1., -2.]]) + + small_square = np.array([[ 1., 0., -1.], + [-1., 0., -1.], + [ 0., 1., -1.], + [ 0., -1., -1.]]) + + hs = qhull.HalfspaceIntersection(big_square, + np.array([0.3141, 0.2718]), + incremental=True) + + hs.add_halfspaces(small_square[0:k,:]) + hs.add_halfspaces(small_square[k:4,:]) + hs.close() + + # Check the intersections are correct (they are the corners of the small square) + expected_intersections = np.array([[1., 1.], + [1., -1.], + [-1., 1.], + [-1., -1.]]) + actual_intersections = hs.intersections + # They may be in any order, so just check that under some permutation + # expected=actual. + + ind1 = np.lexsort((actual_intersections[:, 1], actual_intersections[:, 0])) + ind2 = np.lexsort((expected_intersections[:, 1], expected_intersections[:, 0])) + assert_allclose(actual_intersections[ind1], expected_intersections[ind2]) + + + @pytest.mark.parametrize("halfspaces", [ + (np.array([-0.70613882, -0.45589431, 0.04178256])), + (np.array([[-0.70613882, -0.45589431, 0.04178256], + [0.70807342, -0.45464871, -0.45969769], + [0., 0.76515026, -0.35614825]])), + ]) + def test_gh_19865(self, halfspaces): + # starting off with a feasible interior point and + # adding halfspaces for which it is no longer feasible + # should result in an error rather than a problematic + # intersection polytope + initial_square = np.array( + [[1, 0, -1], [0, 1, -1], [-1, 0, -1], [0, -1, -1]] + ) + incremental_intersector = qhull.HalfspaceIntersection(initial_square, + np.zeros(2), + incremental=True) + with pytest.raises(qhull.QhullError, match="feasible.*-0.706.*"): + incremental_intersector.add_halfspaces(halfspaces) + + + def test_gh_19865_3d(self): + # 3d case where closed half space is enforced for + # feasibility + halfspaces = np.array([[1, 1, 1, -1], # doesn't exclude origin + [-1, -1, -1, -1], # doesn't exclude origin + [1, 0, 0, 0]]) # the origin is on the line + initial_cube = np.array([[1, 0, 0, -1], + [-1, 0, 0, -1], + [0, 1, 0, -1], + [0, -1, 0, -1], + [0, 0, 1, -1], + [0, 0, -1, -1]]) + incremental_intersector = qhull.HalfspaceIntersection(initial_cube, + np.zeros(3), + incremental=True) + with pytest.raises(qhull.QhullError, match="feasible.*[1 0 0 0]"): + incremental_intersector.add_halfspaces(halfspaces) + + + def test_2d_add_halfspace_input(self): + # incrementally added halfspaces should respect the 2D + # array shape requirement + initial_square = np.array( + [[1, 0, -1], [0, 1, -1], [-1, 0, -1], [0, -1, -1]] + ) + incremental_intersector = qhull.HalfspaceIntersection(initial_square, + np.zeros(2), + incremental=True) + with pytest.raises(ValueError, match="2D array"): + incremental_intersector.add_halfspaces(np.ones((4, 4, 4))) + + def test_1d_add_halfspace_input(self): + # we do allow 1D `halfspaces` input to add_halfspaces() + initial_square = np.array( + [[1, 0, -1], [0, 1, -1], [-1, 0, -1], [0, -1, -1]] + ) + incremental_intersector = qhull.HalfspaceIntersection(initial_square, + np.zeros(2), + incremental=True) + assert_allclose(incremental_intersector.dual_vertices, np.arange(4)) + incremental_intersector.add_halfspaces(np.array([2, 2, -1])) + assert_allclose(incremental_intersector.dual_vertices, np.arange(5)) + + +@pytest.mark.parametrize("diagram_type", [Voronoi, qhull.Delaunay]) +def test_gh_20623(diagram_type): + rng = np.random.default_rng(123) + invalid_data = rng.random((4, 10, 3)) + with pytest.raises(ValueError, match="dimensions"): + diagram_type(invalid_data) + + +def test_gh_21286(): + generators = np.array([[0, 0], [0, 1.1], [1, 0], [1, 1]]) + tri = qhull.Delaunay(generators) + # verify absence of segfault reported in ticket: + with pytest.raises(IndexError): + tri.find_simplex(1) + with pytest.raises(IndexError): + # strikingly, Delaunay object has shape + # () just like np.asanyarray(1) above + tri.find_simplex(tri) + + +def test_find_simplex_ndim_err(): + generators = np.array([[0, 0], [0, 1.1], [1, 0], [1, 1]]) + tri = qhull.Delaunay(generators) + with pytest.raises(ValueError): + tri.find_simplex([2, 2, 2]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_slerp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_slerp.py new file mode 100644 index 0000000000000000000000000000000000000000..5ce24991a2d1c3d45107778ae22d1c6fe6e27259 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_slerp.py @@ -0,0 +1,417 @@ +import numpy as np +from numpy.testing import assert_allclose + +import pytest +from scipy.spatial import geometric_slerp + + +def _generate_spherical_points(ndim=3, n_pts=2): + # generate uniform points on sphere + # see: https://stackoverflow.com/a/23785326 + # tentatively extended to arbitrary dims + # for 0-sphere it will always produce antipodes + np.random.seed(123) + points = np.random.normal(size=(n_pts, ndim)) + points /= np.linalg.norm(points, axis=1)[:, np.newaxis] + return points[0], points[1] + + +class TestGeometricSlerp: + # Test various properties of the geometric slerp code + + @pytest.mark.parametrize("n_dims", [2, 3, 5, 7, 9]) + @pytest.mark.parametrize("n_pts", [0, 3, 17]) + def test_shape_property(self, n_dims, n_pts): + # geometric_slerp output shape should match + # input dimensionality & requested number + # of interpolation points + start, end = _generate_spherical_points(n_dims, 2) + + actual = geometric_slerp(start=start, + end=end, + t=np.linspace(0, 1, n_pts)) + + assert actual.shape == (n_pts, n_dims) + + @pytest.mark.parametrize("n_dims", [2, 3, 5, 7, 9]) + @pytest.mark.parametrize("n_pts", [3, 17]) + def test_include_ends(self, n_dims, n_pts): + # geometric_slerp should return a data structure + # that includes the start and end coordinates + # when t includes 0 and 1 ends + # this is convenient for plotting surfaces represented + # by interpolations for example + + # the generator doesn't work so well for the unit + # sphere (it always produces antipodes), so use + # custom values there + start, end = _generate_spherical_points(n_dims, 2) + + actual = geometric_slerp(start=start, + end=end, + t=np.linspace(0, 1, n_pts)) + + assert_allclose(actual[0], start) + assert_allclose(actual[-1], end) + + @pytest.mark.parametrize("start, end", [ + # both arrays are not flat + (np.zeros((1, 3)), np.ones((1, 3))), + # only start array is not flat + (np.zeros((1, 3)), np.ones(3)), + # only end array is not flat + (np.zeros(1), np.ones((3, 1))), + ]) + def test_input_shape_flat(self, start, end): + # geometric_slerp should handle input arrays that are + # not flat appropriately + with pytest.raises(ValueError, match='one-dimensional'): + geometric_slerp(start=start, + end=end, + t=np.linspace(0, 1, 10)) + + @pytest.mark.parametrize("start, end", [ + # 7-D and 3-D ends + (np.zeros(7), np.ones(3)), + # 2-D and 1-D ends + (np.zeros(2), np.ones(1)), + # empty, "3D" will also get caught this way + (np.array([]), np.ones(3)), + ]) + def test_input_dim_mismatch(self, start, end): + # geometric_slerp must appropriately handle cases where + # an interpolation is attempted across two different + # dimensionalities + with pytest.raises(ValueError, match='dimensions'): + geometric_slerp(start=start, + end=end, + t=np.linspace(0, 1, 10)) + + @pytest.mark.parametrize("start, end", [ + # both empty + (np.array([]), np.array([])), + ]) + def test_input_at_least1d(self, start, end): + # empty inputs to geometric_slerp must + # be handled appropriately when not detected + # by mismatch + with pytest.raises(ValueError, match='at least two-dim'): + geometric_slerp(start=start, + end=end, + t=np.linspace(0, 1, 10)) + + @pytest.mark.thread_unsafe + @pytest.mark.parametrize("start, end, expected", [ + # North and South Poles are definitely antipodes + # but should be handled gracefully now + (np.array([0, 0, 1.0]), np.array([0, 0, -1.0]), "warning"), + # this case will issue a warning & be handled + # gracefully as well; + # North Pole was rotated very slightly + # using r = R.from_euler('x', 0.035, degrees=True) + # to achieve Euclidean distance offset from diameter by + # 9.328908379124812e-08, within the default tol + (np.array([0.00000000e+00, + -6.10865200e-04, + 9.99999813e-01]), np.array([0, 0, -1.0]), "warning"), + # this case should succeed without warning because a + # sufficiently large + # rotation was applied to North Pole point to shift it + # to a Euclidean distance of 2.3036691931821451e-07 + # from South Pole, which is larger than tol + (np.array([0.00000000e+00, + -9.59930941e-04, + 9.99999539e-01]), np.array([0, 0, -1.0]), "success"), + ]) + def test_handle_antipodes(self, start, end, expected): + # antipodal points must be handled appropriately; + # there are an infinite number of possible geodesic + # interpolations between them in higher dims + if expected == "warning": + with pytest.warns(UserWarning, match='antipodes'): + res = geometric_slerp(start=start, + end=end, + t=np.linspace(0, 1, 10)) + else: + res = geometric_slerp(start=start, + end=end, + t=np.linspace(0, 1, 10)) + + # antipodes or near-antipodes should still produce + # slerp paths on the surface of the sphere (but they + # may be ambiguous): + assert_allclose(np.linalg.norm(res, axis=1), 1.0) + + @pytest.mark.parametrize("start, end, expected", [ + # 2-D with n_pts=4 (two new interpolation points) + # this is an actual circle + (np.array([1, 0]), + np.array([0, 1]), + np.array([[1, 0], + [np.sqrt(3) / 2, 0.5], # 30 deg on unit circle + [0.5, np.sqrt(3) / 2], # 60 deg on unit circle + [0, 1]])), + # likewise for 3-D (add z = 0 plane) + # this is an ordinary sphere + (np.array([1, 0, 0]), + np.array([0, 1, 0]), + np.array([[1, 0, 0], + [np.sqrt(3) / 2, 0.5, 0], + [0.5, np.sqrt(3) / 2, 0], + [0, 1, 0]])), + # for 5-D, pad more columns with constants + # zeros are easiest--non-zero values on unit + # circle are more difficult to reason about + # at higher dims + (np.array([1, 0, 0, 0, 0]), + np.array([0, 1, 0, 0, 0]), + np.array([[1, 0, 0, 0, 0], + [np.sqrt(3) / 2, 0.5, 0, 0, 0], + [0.5, np.sqrt(3) / 2, 0, 0, 0], + [0, 1, 0, 0, 0]])), + + ]) + def test_straightforward_examples(self, start, end, expected): + # some straightforward interpolation tests, sufficiently + # simple to use the unit circle to deduce expected values; + # for larger dimensions, pad with constants so that the + # data is N-D but simpler to reason about + actual = geometric_slerp(start=start, + end=end, + t=np.linspace(0, 1, 4)) + assert_allclose(actual, expected, atol=1e-16) + + @pytest.mark.parametrize("t", [ + # both interval ends clearly violate limits + np.linspace(-20, 20, 300), + # only one interval end violating limit slightly + np.linspace(-0.0001, 0.0001, 17), + ]) + def test_t_values_limits(self, t): + # geometric_slerp() should appropriately handle + # interpolation parameters < 0 and > 1 + with pytest.raises(ValueError, match='interpolation parameter'): + _ = geometric_slerp(start=np.array([1, 0]), + end=np.array([0, 1]), + t=t) + + @pytest.mark.parametrize("start, end", [ + (np.array([1]), + np.array([0])), + (np.array([0]), + np.array([1])), + (np.array([-17.7]), + np.array([165.9])), + ]) + def test_0_sphere_handling(self, start, end): + # it does not make sense to interpolate the set of + # two points that is the 0-sphere + with pytest.raises(ValueError, match='at least two-dim'): + _ = geometric_slerp(start=start, + end=end, + t=np.linspace(0, 1, 4)) + + @pytest.mark.parametrize("tol", [ + # an integer currently raises + 5, + # string raises + "7", + # list and arrays also raise + [5, 6, 7], np.array(9.0), + ]) + def test_tol_type(self, tol): + # geometric_slerp() should raise if tol is not + # a suitable float type + with pytest.raises(ValueError, match='must be a float'): + _ = geometric_slerp(start=np.array([1, 0]), + end=np.array([0, 1]), + t=np.linspace(0, 1, 5), + tol=tol) + + @pytest.mark.parametrize("tol", [ + -5e-6, + -7e-10, + ]) + def test_tol_sign(self, tol): + # geometric_slerp() currently handles negative + # tol values, as long as they are floats + _ = geometric_slerp(start=np.array([1, 0]), + end=np.array([0, 1]), + t=np.linspace(0, 1, 5), + tol=tol) + + @pytest.mark.parametrize("start, end", [ + # 1-sphere (circle) with one point at origin + # and the other on the circle + (np.array([1, 0]), np.array([0, 0])), + # 2-sphere (normal sphere) with both points + # just slightly off sphere by the same amount + # in different directions + (np.array([1 + 1e-6, 0, 0]), + np.array([0, 1 - 1e-6, 0])), + # same thing in 4-D + (np.array([1 + 1e-6, 0, 0, 0]), + np.array([0, 1 - 1e-6, 0, 0])), + ]) + def test_unit_sphere_enforcement(self, start, end): + # geometric_slerp() should raise on input that clearly + # cannot be on an n-sphere of radius 1 + with pytest.raises(ValueError, match='unit n-sphere'): + geometric_slerp(start=start, + end=end, + t=np.linspace(0, 1, 5)) + + @pytest.mark.parametrize("start, end", [ + # 1-sphere 45 degree case + (np.array([1, 0]), + np.array([np.sqrt(2) / 2., + np.sqrt(2) / 2.])), + # 2-sphere 135 degree case + (np.array([1, 0]), + np.array([-np.sqrt(2) / 2., + np.sqrt(2) / 2.])), + ]) + @pytest.mark.parametrize("t_func", [ + np.linspace, np.logspace]) + def test_order_handling(self, start, end, t_func): + # geometric_slerp() should handle scenarios with + # ascending and descending t value arrays gracefully; + # results should simply be reversed + + # for scrambled / unsorted parameters, the same values + # should be returned, just in scrambled order + + num_t_vals = 20 + np.random.seed(789) + forward_t_vals = t_func(0, 10, num_t_vals) + # normalize to max of 1 + forward_t_vals /= forward_t_vals.max() + reverse_t_vals = np.flipud(forward_t_vals) + shuffled_indices = np.arange(num_t_vals) + np.random.shuffle(shuffled_indices) + scramble_t_vals = forward_t_vals.copy()[shuffled_indices] + + forward_results = geometric_slerp(start=start, + end=end, + t=forward_t_vals) + reverse_results = geometric_slerp(start=start, + end=end, + t=reverse_t_vals) + scrambled_results = geometric_slerp(start=start, + end=end, + t=scramble_t_vals) + + # check fidelity to input order + assert_allclose(forward_results, np.flipud(reverse_results)) + assert_allclose(forward_results[shuffled_indices], + scrambled_results) + + @pytest.mark.parametrize("t", [ + # string: + "15, 5, 7", + # complex numbers currently produce a warning + # but not sure we need to worry about it too much: + # [3 + 1j, 5 + 2j], + ]) + def test_t_values_conversion(self, t): + with pytest.raises(ValueError): + _ = geometric_slerp(start=np.array([1]), + end=np.array([0]), + t=t) + + def test_accept_arraylike(self): + # array-like support requested by reviewer + # in gh-10380 + actual = geometric_slerp([1, 0], [0, 1], [0, 1/3, 0.5, 2/3, 1]) + + # expected values are based on visual inspection + # of the unit circle for the progressions along + # the circumference provided in t + expected = np.array([[1, 0], + [np.sqrt(3) / 2, 0.5], + [np.sqrt(2) / 2, + np.sqrt(2) / 2], + [0.5, np.sqrt(3) / 2], + [0, 1]], dtype=np.float64) + # Tyler's original Cython implementation of geometric_slerp + # can pass at atol=0 here, but on balance we will accept + # 1e-16 for an implementation that avoids Cython and + # makes up accuracy ground elsewhere + assert_allclose(actual, expected, atol=1e-16) + + def test_scalar_t(self): + # when t is a scalar, return value is a single + # interpolated point of the appropriate dimensionality + # requested by reviewer in gh-10380 + actual = geometric_slerp([1, 0], [0, 1], 0.5) + expected = np.array([np.sqrt(2) / 2, + np.sqrt(2) / 2], dtype=np.float64) + assert actual.shape == (2,) + assert_allclose(actual, expected) + + @pytest.mark.parametrize('start', [ + np.array([1, 0, 0]), + np.array([0, 1]), + ]) + @pytest.mark.parametrize('t', [ + np.array(1), + np.array([1]), + np.array([[1]]), + np.array([[[1]]]), + np.array([]), + np.linspace(0, 1, 5), + ]) + def test_degenerate_input(self, start, t): + if np.asarray(t).ndim > 1: + with pytest.raises(ValueError): + geometric_slerp(start=start, end=start, t=t) + else: + + shape = (t.size,) + start.shape + expected = np.full(shape, start) + + actual = geometric_slerp(start=start, end=start, t=t) + assert_allclose(actual, expected) + + # Check that degenerate and non-degenerate + # inputs yield the same size + non_degenerate = geometric_slerp(start=start, end=start[::-1], t=t) + assert actual.size == non_degenerate.size + + @pytest.mark.parametrize('k', np.logspace(-10, -1, 10)) + def test_numerical_stability_pi(self, k): + # geometric_slerp should have excellent numerical + # stability for angles approaching pi between + # the start and end points + angle = np.pi - k + ts = np.linspace(0, 1, 100) + P = np.array([1, 0, 0, 0]) + Q = np.array([np.cos(angle), np.sin(angle), 0, 0]) + # the test should only be enforced for cases where + # geometric_slerp determines that the input is actually + # on the unit sphere + with np.testing.suppress_warnings() as sup: + sup.filter(UserWarning) + result = geometric_slerp(P, Q, ts, 1e-18) + norms = np.linalg.norm(result, axis=1) + error = np.max(np.abs(norms - 1)) + assert error < 4e-15 + + @pytest.mark.parametrize('t', [ + [[0, 0.5]], + [[[[[[[[[0, 0.5]]]]]]]]], + ]) + def test_interpolation_param_ndim(self, t): + # regression test for gh-14465 + arr1 = np.array([0, 1]) + arr2 = np.array([1, 0]) + + with pytest.raises(ValueError): + geometric_slerp(start=arr1, + end=arr2, + t=t) + + with pytest.raises(ValueError): + geometric_slerp(start=arr1, + end=arr1, + t=t) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_spherical_voronoi.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_spherical_voronoi.py new file mode 100644 index 0000000000000000000000000000000000000000..8bf4764e9e37312d08f25870327457107b9dcd91 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/tests/test_spherical_voronoi.py @@ -0,0 +1,358 @@ +import numpy as np +import itertools +from numpy.testing import (assert_equal, + assert_almost_equal, + assert_array_equal, + assert_array_almost_equal) +import pytest +from pytest import raises as assert_raises +from scipy.spatial import SphericalVoronoi, distance +from scipy.optimize import linear_sum_assignment +from scipy.constants import golden as phi +from scipy.special import gamma + + +TOL = 1E-10 + + +def _generate_tetrahedron(): + return np.array([[1, 1, 1], [1, -1, -1], [-1, 1, -1], [-1, -1, 1]]) + + +def _generate_cube(): + return np.array(list(itertools.product([-1, 1.], repeat=3))) + + +def _generate_octahedron(): + return np.array([[-1, 0, 0], [+1, 0, 0], [0, -1, 0], + [0, +1, 0], [0, 0, -1], [0, 0, +1]]) + + +def _generate_dodecahedron(): + + x1 = _generate_cube() + x2 = np.array([[0, -phi, -1 / phi], + [0, -phi, +1 / phi], + [0, +phi, -1 / phi], + [0, +phi, +1 / phi]]) + x3 = np.array([[-1 / phi, 0, -phi], + [+1 / phi, 0, -phi], + [-1 / phi, 0, +phi], + [+1 / phi, 0, +phi]]) + x4 = np.array([[-phi, -1 / phi, 0], + [-phi, +1 / phi, 0], + [+phi, -1 / phi, 0], + [+phi, +1 / phi, 0]]) + return np.concatenate((x1, x2, x3, x4)) + + +def _generate_icosahedron(): + x = np.array([[0, -1, -phi], + [0, -1, +phi], + [0, +1, -phi], + [0, +1, +phi]]) + return np.concatenate([np.roll(x, i, axis=1) for i in range(3)]) + + +def _generate_polytope(name): + polygons = ["triangle", "square", "pentagon", "hexagon", "heptagon", + "octagon", "nonagon", "decagon", "undecagon", "dodecagon"] + polyhedra = ["tetrahedron", "cube", "octahedron", "dodecahedron", + "icosahedron"] + if name not in polygons and name not in polyhedra: + raise ValueError("unrecognized polytope") + + if name in polygons: + n = polygons.index(name) + 3 + thetas = np.linspace(0, 2 * np.pi, n, endpoint=False) + p = np.vstack([np.cos(thetas), np.sin(thetas)]).T + elif name == "tetrahedron": + p = _generate_tetrahedron() + elif name == "cube": + p = _generate_cube() + elif name == "octahedron": + p = _generate_octahedron() + elif name == "dodecahedron": + p = _generate_dodecahedron() + elif name == "icosahedron": + p = _generate_icosahedron() + + return p / np.linalg.norm(p, axis=1, keepdims=True) + + +def _hypersphere_area(dim, radius): + # https://en.wikipedia.org/wiki/N-sphere#Closed_forms + return 2 * np.pi**(dim / 2) / gamma(dim / 2) * radius**(dim - 1) + + +def _sample_sphere(n, dim, seed=None): + # Sample points uniformly at random from the hypersphere + rng = np.random.RandomState(seed=seed) + points = rng.randn(n, dim) + points /= np.linalg.norm(points, axis=1, keepdims=True) + return points + + +class TestSphericalVoronoi: + + def setup_method(self): + self.points = np.array([ + [-0.78928481, -0.16341094, 0.59188373], + [-0.66839141, 0.73309634, 0.12578818], + [0.32535778, -0.92476944, -0.19734181], + [-0.90177102, -0.03785291, -0.43055335], + [0.71781344, 0.68428936, 0.12842096], + [-0.96064876, 0.23492353, -0.14820556], + [0.73181537, -0.22025898, -0.6449281], + [0.79979205, 0.54555747, 0.25039913]] + ) + + def test_constructor(self): + center = np.array([1, 2, 3]) + radius = 2 + s1 = SphericalVoronoi(self.points) + # user input checks in SphericalVoronoi now require + # the radius / center to match the generators so adjust + # accordingly here + s2 = SphericalVoronoi(self.points * radius, radius) + s3 = SphericalVoronoi(self.points + center, center=center) + s4 = SphericalVoronoi(self.points * radius + center, radius, center) + assert_array_equal(s1.center, np.array([0, 0, 0])) + assert_equal(s1.radius, 1) + assert_array_equal(s2.center, np.array([0, 0, 0])) + assert_equal(s2.radius, 2) + assert_array_equal(s3.center, center) + assert_equal(s3.radius, 1) + assert_array_equal(s4.center, center) + assert_equal(s4.radius, radius) + + # Test a non-sequence/-ndarray based array-like + s5 = SphericalVoronoi(memoryview(self.points)) # type: ignore[arg-type] + assert_array_equal(s5.center, np.array([0, 0, 0])) + assert_equal(s5.radius, 1) + + def test_vertices_regions_translation_invariance(self): + sv_origin = SphericalVoronoi(self.points) + center = np.array([1, 1, 1]) + sv_translated = SphericalVoronoi(self.points + center, center=center) + assert_equal(sv_origin.regions, sv_translated.regions) + assert_array_almost_equal(sv_origin.vertices + center, + sv_translated.vertices) + + def test_vertices_regions_scaling_invariance(self): + sv_unit = SphericalVoronoi(self.points) + sv_scaled = SphericalVoronoi(self.points * 2, 2) + assert_equal(sv_unit.regions, sv_scaled.regions) + assert_array_almost_equal(sv_unit.vertices * 2, + sv_scaled.vertices) + + def test_old_radius_api_error(self): + with pytest.raises(ValueError, match='`radius` is `None`. *'): + SphericalVoronoi(self.points, radius=None) + + def test_sort_vertices_of_regions(self): + sv = SphericalVoronoi(self.points) + unsorted_regions = sv.regions + sv.sort_vertices_of_regions() + assert_equal(sorted(sv.regions), sorted(unsorted_regions)) + + def test_sort_vertices_of_regions_flattened(self): + expected = sorted([[0, 6, 5, 2, 3], [2, 3, 10, 11, 8, 7], [0, 6, 4, 1], + [4, 8, 7, 5, 6], [9, 11, 10], [2, 7, 5], + [1, 4, 8, 11, 9], [0, 3, 10, 9, 1]]) + expected = list(itertools.chain(*sorted(expected))) # type: ignore + sv = SphericalVoronoi(self.points) + sv.sort_vertices_of_regions() + actual = list(itertools.chain(*sorted(sv.regions))) + assert_array_equal(actual, expected) + + def test_sort_vertices_of_regions_dimensionality(self): + points = np.array([[1, 0, 0, 0], + [0, 1, 0, 0], + [0, 0, 1, 0], + [0, 0, 0, 1], + [0.5, 0.5, 0.5, 0.5]]) + with pytest.raises(TypeError, match="three-dimensional"): + sv = SphericalVoronoi(points) + sv.sort_vertices_of_regions() + + def test_num_vertices(self): + # for any n >= 3, a spherical Voronoi diagram has 2n - 4 + # vertices; this is a direct consequence of Euler's formula + # as explained by Dinis and Mamede (2010) Proceedings of the + # 2010 International Symposium on Voronoi Diagrams in Science + # and Engineering + sv = SphericalVoronoi(self.points) + expected = self.points.shape[0] * 2 - 4 + actual = sv.vertices.shape[0] + assert_equal(actual, expected) + + def test_voronoi_circles(self): + sv = SphericalVoronoi(self.points) + for vertex in sv.vertices: + distances = distance.cdist(sv.points, np.array([vertex])) + closest = np.array(sorted(distances)[0:3]) + assert_almost_equal(closest[0], closest[1], 7, str(vertex)) + assert_almost_equal(closest[0], closest[2], 7, str(vertex)) + + def test_duplicate_point_handling(self): + # an exception should be raised for degenerate generators + # related to Issue# 7046 + self.degenerate = np.concatenate((self.points, self.points)) + with assert_raises(ValueError): + SphericalVoronoi(self.degenerate) + + def test_incorrect_radius_handling(self): + # an exception should be raised if the radius provided + # cannot possibly match the input generators + with assert_raises(ValueError): + SphericalVoronoi(self.points, radius=0.98) + + def test_incorrect_center_handling(self): + # an exception should be raised if the center provided + # cannot possibly match the input generators + with assert_raises(ValueError): + SphericalVoronoi(self.points, center=[0.1, 0, 0]) + + @pytest.mark.parametrize("dim", range(2, 6)) + @pytest.mark.parametrize("shift", [False, True]) + def test_single_hemisphere_handling(self, dim, shift): + n = 10 + points = _sample_sphere(n, dim, seed=0) + points[:, 0] = np.abs(points[:, 0]) + center = (np.arange(dim) + 1) * shift + sv = SphericalVoronoi(points + center, center=center) + dots = np.einsum('ij,ij->i', sv.vertices - center, + sv.points[sv._simplices[:, 0]] - center) + circumradii = np.arccos(np.clip(dots, -1, 1)) + assert np.max(circumradii) > np.pi / 2 + + @pytest.mark.parametrize("n", [1, 2, 10]) + @pytest.mark.parametrize("dim", range(2, 6)) + @pytest.mark.parametrize("shift", [False, True]) + def test_rank_deficient(self, n, dim, shift): + center = (np.arange(dim) + 1) * shift + points = _sample_sphere(n, dim - 1, seed=0) + points = np.hstack([points, np.zeros((n, 1))]) + with pytest.raises(ValueError, match="Rank of input points"): + SphericalVoronoi(points + center, center=center) + + @pytest.mark.parametrize("dim", range(2, 6)) + def test_higher_dimensions(self, dim): + n = 100 + points = _sample_sphere(n, dim, seed=0) + sv = SphericalVoronoi(points) + assert sv.vertices.shape[1] == dim + assert len(sv.regions) == n + + # verify Euler characteristic + cell_counts = [] + simplices = np.sort(sv._simplices) + for i in range(1, dim + 1): + cells = [] + for indices in itertools.combinations(range(dim), i): + cells.append(simplices[:, list(indices)]) + cells = np.unique(np.concatenate(cells), axis=0) + cell_counts.append(len(cells)) + expected_euler = 1 + (-1)**(dim-1) + actual_euler = sum([(-1)**i * e for i, e in enumerate(cell_counts)]) + assert expected_euler == actual_euler + + @pytest.mark.parametrize("dim", range(2, 6)) + def test_cross_polytope_regions(self, dim): + # The hypercube is the dual of the cross-polytope, so the voronoi + # vertices of the cross-polytope lie on the points of the hypercube. + + # generate points of the cross-polytope + points = np.concatenate((-np.eye(dim), np.eye(dim))) + sv = SphericalVoronoi(points) + assert all([len(e) == 2**(dim - 1) for e in sv.regions]) + + # generate points of the hypercube + expected = np.vstack(list(itertools.product([-1, 1], repeat=dim))) + expected = expected.astype(np.float64) / np.sqrt(dim) + + # test that Voronoi vertices are correctly placed + dist = distance.cdist(sv.vertices, expected) + res = linear_sum_assignment(dist) + assert dist[res].sum() < TOL + + @pytest.mark.parametrize("dim", range(2, 6)) + def test_hypercube_regions(self, dim): + # The cross-polytope is the dual of the hypercube, so the voronoi + # vertices of the hypercube lie on the points of the cross-polytope. + + # generate points of the hypercube + points = np.vstack(list(itertools.product([-1, 1], repeat=dim))) + points = points.astype(np.float64) / np.sqrt(dim) + sv = SphericalVoronoi(points) + + # generate points of the cross-polytope + expected = np.concatenate((-np.eye(dim), np.eye(dim))) + + # test that Voronoi vertices are correctly placed + dist = distance.cdist(sv.vertices, expected) + res = linear_sum_assignment(dist) + assert dist[res].sum() < TOL + + @pytest.mark.parametrize("n", [10, 500]) + @pytest.mark.parametrize("dim", [2, 3]) + @pytest.mark.parametrize("radius", [0.5, 1, 2]) + @pytest.mark.parametrize("shift", [False, True]) + @pytest.mark.parametrize("single_hemisphere", [False, True]) + def test_area_reconstitution(self, n, dim, radius, shift, + single_hemisphere): + points = _sample_sphere(n, dim, seed=0) + + # move all points to one side of the sphere for single-hemisphere test + if single_hemisphere: + points[:, 0] = np.abs(points[:, 0]) + + center = (np.arange(dim) + 1) * shift + points = radius * points + center + + sv = SphericalVoronoi(points, radius=radius, center=center) + areas = sv.calculate_areas() + assert_almost_equal(areas.sum(), _hypersphere_area(dim, radius)) + + @pytest.mark.parametrize("poly", ["triangle", "dodecagon", + "tetrahedron", "cube", "octahedron", + "dodecahedron", "icosahedron"]) + def test_equal_area_reconstitution(self, poly): + points = _generate_polytope(poly) + n, dim = points.shape + sv = SphericalVoronoi(points) + areas = sv.calculate_areas() + assert_almost_equal(areas, _hypersphere_area(dim, 1) / n) + + def test_area_unsupported_dimension(self): + dim = 4 + points = np.concatenate((-np.eye(dim), np.eye(dim))) + sv = SphericalVoronoi(points) + with pytest.raises(TypeError, match="Only supported"): + sv.calculate_areas() + + @pytest.mark.parametrize("radius", [1, 1.]) + @pytest.mark.parametrize("center", [None, (1, 2, 3), (1., 2., 3.)]) + def test_attribute_types(self, radius, center): + points = radius * self.points + if center is not None: + points += center + + sv = SphericalVoronoi(points, radius=radius, center=center) + assert sv.points.dtype is np.dtype(np.float64) + assert sv.center.dtype is np.dtype(np.float64) + assert isinstance(sv.radius, float) + + def test_region_types(self): + # Tests that region integer type does not change + # See Issue #13412 + sv = SphericalVoronoi(self.points) + dtype = type(sv.regions[0][0]) + # also enforce nested list type per gh-19177 + for region in sv.regions: + assert isinstance(region, list) + sv.sort_vertices_of_regions() + assert isinstance(sv.regions[0][0], dtype) + sv.sort_vertices_of_regions() + assert isinstance(sv.regions[0][0], dtype) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..abe4a32f20d1f8740910a16de9e67a53621bc3e3 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/__init__.py @@ -0,0 +1,29 @@ +""" +Spatial Transformations (:mod:`scipy.spatial.transform`) +======================================================== + +.. currentmodule:: scipy.spatial.transform + +This package implements various spatial transformations. For now, +only rotations are supported. + +Rotations in 3 dimensions +------------------------- +.. autosummary:: + :toctree: generated/ + + Rotation + Slerp + RotationSpline +""" +from ._rotation import Rotation, Slerp +from ._rotation_spline import RotationSpline + +# Deprecated namespaces, to be removed in v2.0.0 +from . import rotation + +__all__ = ['Rotation', 'Slerp', 'RotationSpline'] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/_rotation_groups.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/_rotation_groups.py new file mode 100644 index 0000000000000000000000000000000000000000..870e9b9e2b44bff56b8228a70607e29f8173accc --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/_rotation_groups.py @@ -0,0 +1,140 @@ +import numpy as np +from scipy.constants import golden as phi + + +def icosahedral(cls): + g1 = tetrahedral(cls).as_quat() + a = 0.5 + b = 0.5 / phi + c = phi / 2 + g2 = np.array([[+a, +b, +c, 0], + [+a, +b, -c, 0], + [+a, +c, 0, +b], + [+a, +c, 0, -b], + [+a, -b, +c, 0], + [+a, -b, -c, 0], + [+a, -c, 0, +b], + [+a, -c, 0, -b], + [+a, 0, +b, +c], + [+a, 0, +b, -c], + [+a, 0, -b, +c], + [+a, 0, -b, -c], + [+b, +a, 0, +c], + [+b, +a, 0, -c], + [+b, +c, +a, 0], + [+b, +c, -a, 0], + [+b, -a, 0, +c], + [+b, -a, 0, -c], + [+b, -c, +a, 0], + [+b, -c, -a, 0], + [+b, 0, +c, +a], + [+b, 0, +c, -a], + [+b, 0, -c, +a], + [+b, 0, -c, -a], + [+c, +a, +b, 0], + [+c, +a, -b, 0], + [+c, +b, 0, +a], + [+c, +b, 0, -a], + [+c, -a, +b, 0], + [+c, -a, -b, 0], + [+c, -b, 0, +a], + [+c, -b, 0, -a], + [+c, 0, +a, +b], + [+c, 0, +a, -b], + [+c, 0, -a, +b], + [+c, 0, -a, -b], + [0, +a, +c, +b], + [0, +a, +c, -b], + [0, +a, -c, +b], + [0, +a, -c, -b], + [0, +b, +a, +c], + [0, +b, +a, -c], + [0, +b, -a, +c], + [0, +b, -a, -c], + [0, +c, +b, +a], + [0, +c, +b, -a], + [0, +c, -b, +a], + [0, +c, -b, -a]]) + return cls.from_quat(np.concatenate((g1, g2))) + + +def octahedral(cls): + g1 = tetrahedral(cls).as_quat() + c = np.sqrt(2) / 2 + g2 = np.array([[+c, 0, 0, +c], + [0, +c, 0, +c], + [0, 0, +c, +c], + [0, 0, -c, +c], + [0, -c, 0, +c], + [-c, 0, 0, +c], + [0, +c, +c, 0], + [0, -c, +c, 0], + [+c, 0, +c, 0], + [-c, 0, +c, 0], + [+c, +c, 0, 0], + [-c, +c, 0, 0]]) + return cls.from_quat(np.concatenate((g1, g2))) + + +def tetrahedral(cls): + g1 = np.eye(4) + c = 0.5 + g2 = np.array([[c, -c, -c, +c], + [c, -c, +c, +c], + [c, +c, -c, +c], + [c, +c, +c, +c], + [c, -c, -c, -c], + [c, -c, +c, -c], + [c, +c, -c, -c], + [c, +c, +c, -c]]) + return cls.from_quat(np.concatenate((g1, g2))) + + +def dicyclic(cls, n, axis=2): + g1 = cyclic(cls, n, axis).as_rotvec() + + thetas = np.linspace(0, np.pi, n, endpoint=False) + rv = np.pi * np.vstack([np.zeros(n), np.cos(thetas), np.sin(thetas)]).T + g2 = np.roll(rv, axis, axis=1) + return cls.from_rotvec(np.concatenate((g1, g2))) + + +def cyclic(cls, n, axis=2): + thetas = np.linspace(0, 2 * np.pi, n, endpoint=False) + rv = np.vstack([thetas, np.zeros(n), np.zeros(n)]).T + return cls.from_rotvec(np.roll(rv, axis, axis=1)) + + +def create_group(cls, group, axis='Z'): + if not isinstance(group, str): + raise ValueError("`group` argument must be a string") + + permitted_axes = ['x', 'y', 'z', 'X', 'Y', 'Z'] + if axis not in permitted_axes: + raise ValueError("`axis` must be one of " + ", ".join(permitted_axes)) + + if group in ['I', 'O', 'T']: + symbol = group + order = 1 + elif group[:1] in ['C', 'D'] and group[1:].isdigit(): + symbol = group[:1] + order = int(group[1:]) + else: + raise ValueError("`group` must be one of 'I', 'O', 'T', 'Dn', 'Cn'") + + if order < 1: + raise ValueError("Group order must be positive") + + axis = 'xyz'.index(axis.lower()) + if symbol == 'I': + return icosahedral(cls) + elif symbol == 'O': + return octahedral(cls) + elif symbol == 'T': + return tetrahedral(cls) + elif symbol == 'D': + return dicyclic(cls, order, axis=axis) + elif symbol == 'C': + return cyclic(cls, order, axis=axis) + else: + assert False diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/_rotation_spline.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/_rotation_spline.py new file mode 100644 index 0000000000000000000000000000000000000000..867b724fdf449b2e60dd5fbec8a72ce6eb73c22d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/_rotation_spline.py @@ -0,0 +1,460 @@ +import numpy as np +from scipy.linalg import solve_banded +from ._rotation import Rotation + + +def _create_skew_matrix(x): + """Create skew-symmetric matrices corresponding to vectors. + + Parameters + ---------- + x : ndarray, shape (n, 3) + Set of vectors. + + Returns + ------- + ndarray, shape (n, 3, 3) + """ + result = np.zeros((len(x), 3, 3)) + result[:, 0, 1] = -x[:, 2] + result[:, 0, 2] = x[:, 1] + result[:, 1, 0] = x[:, 2] + result[:, 1, 2] = -x[:, 0] + result[:, 2, 0] = -x[:, 1] + result[:, 2, 1] = x[:, 0] + return result + + +def _matrix_vector_product_of_stacks(A, b): + """Compute the product of stack of matrices and vectors.""" + return np.einsum("ijk,ik->ij", A, b) + + +def _angular_rate_to_rotvec_dot_matrix(rotvecs): + """Compute matrices to transform angular rates to rot. vector derivatives. + + The matrices depend on the current attitude represented as a rotation + vector. + + Parameters + ---------- + rotvecs : ndarray, shape (n, 3) + Set of rotation vectors. + + Returns + ------- + ndarray, shape (n, 3, 3) + """ + norm = np.linalg.norm(rotvecs, axis=1) + k = np.empty_like(norm) + + mask = norm > 1e-4 + nm = norm[mask] + k[mask] = (1 - 0.5 * nm / np.tan(0.5 * nm)) / nm**2 + mask = ~mask + nm = norm[mask] + k[mask] = 1/12 + 1/720 * nm**2 + + skew = _create_skew_matrix(rotvecs) + + result = np.empty((len(rotvecs), 3, 3)) + result[:] = np.identity(3) + result[:] += 0.5 * skew + result[:] += k[:, None, None] * np.matmul(skew, skew) + + return result + + +def _rotvec_dot_to_angular_rate_matrix(rotvecs): + """Compute matrices to transform rot. vector derivatives to angular rates. + + The matrices depend on the current attitude represented as a rotation + vector. + + Parameters + ---------- + rotvecs : ndarray, shape (n, 3) + Set of rotation vectors. + + Returns + ------- + ndarray, shape (n, 3, 3) + """ + norm = np.linalg.norm(rotvecs, axis=1) + k1 = np.empty_like(norm) + k2 = np.empty_like(norm) + + mask = norm > 1e-4 + nm = norm[mask] + k1[mask] = (1 - np.cos(nm)) / nm ** 2 + k2[mask] = (nm - np.sin(nm)) / nm ** 3 + + mask = ~mask + nm = norm[mask] + k1[mask] = 0.5 - nm ** 2 / 24 + k2[mask] = 1 / 6 - nm ** 2 / 120 + + skew = _create_skew_matrix(rotvecs) + + result = np.empty((len(rotvecs), 3, 3)) + result[:] = np.identity(3) + result[:] -= k1[:, None, None] * skew + result[:] += k2[:, None, None] * np.matmul(skew, skew) + + return result + + +def _angular_acceleration_nonlinear_term(rotvecs, rotvecs_dot): + """Compute the non-linear term in angular acceleration. + + The angular acceleration contains a quadratic term with respect to + the derivative of the rotation vector. This function computes that. + + Parameters + ---------- + rotvecs : ndarray, shape (n, 3) + Set of rotation vectors. + rotvecs_dot : ndarray, shape (n, 3) + Set of rotation vector derivatives. + + Returns + ------- + ndarray, shape (n, 3) + """ + norm = np.linalg.norm(rotvecs, axis=1) + dp = np.sum(rotvecs * rotvecs_dot, axis=1) + cp = np.cross(rotvecs, rotvecs_dot) + ccp = np.cross(rotvecs, cp) + dccp = np.cross(rotvecs_dot, cp) + + k1 = np.empty_like(norm) + k2 = np.empty_like(norm) + k3 = np.empty_like(norm) + + mask = norm > 1e-4 + nm = norm[mask] + k1[mask] = (-nm * np.sin(nm) - 2 * (np.cos(nm) - 1)) / nm ** 4 + k2[mask] = (-2 * nm + 3 * np.sin(nm) - nm * np.cos(nm)) / nm ** 5 + k3[mask] = (nm - np.sin(nm)) / nm ** 3 + + mask = ~mask + nm = norm[mask] + k1[mask] = 1/12 - nm ** 2 / 180 + k2[mask] = -1/60 + nm ** 2 / 12604 + k3[mask] = 1/6 - nm ** 2 / 120 + + dp = dp[:, None] + k1 = k1[:, None] + k2 = k2[:, None] + k3 = k3[:, None] + + return dp * (k1 * cp + k2 * ccp) + k3 * dccp + + +def _compute_angular_rate(rotvecs, rotvecs_dot): + """Compute angular rates given rotation vectors and its derivatives. + + Parameters + ---------- + rotvecs : ndarray, shape (n, 3) + Set of rotation vectors. + rotvecs_dot : ndarray, shape (n, 3) + Set of rotation vector derivatives. + + Returns + ------- + ndarray, shape (n, 3) + """ + return _matrix_vector_product_of_stacks( + _rotvec_dot_to_angular_rate_matrix(rotvecs), rotvecs_dot) + + +def _compute_angular_acceleration(rotvecs, rotvecs_dot, rotvecs_dot_dot): + """Compute angular acceleration given rotation vector and its derivatives. + + Parameters + ---------- + rotvecs : ndarray, shape (n, 3) + Set of rotation vectors. + rotvecs_dot : ndarray, shape (n, 3) + Set of rotation vector derivatives. + rotvecs_dot_dot : ndarray, shape (n, 3) + Set of rotation vector second derivatives. + + Returns + ------- + ndarray, shape (n, 3) + """ + return (_compute_angular_rate(rotvecs, rotvecs_dot_dot) + + _angular_acceleration_nonlinear_term(rotvecs, rotvecs_dot)) + + +def _create_block_3_diagonal_matrix(A, B, d): + """Create a 3-diagonal block matrix as banded. + + The matrix has the following structure: + + DB... + ADB.. + .ADB. + ..ADB + ...AD + + The blocks A, B and D are 3-by-3 matrices. The D matrices has the form + d * I. + + Parameters + ---------- + A : ndarray, shape (n, 3, 3) + Stack of A blocks. + B : ndarray, shape (n, 3, 3) + Stack of B blocks. + d : ndarray, shape (n + 1,) + Values for diagonal blocks. + + Returns + ------- + ndarray, shape (11, 3 * (n + 1)) + Matrix in the banded form as used by `scipy.linalg.solve_banded`. + """ + ind = np.arange(3) + ind_blocks = np.arange(len(A)) + + A_i = np.empty_like(A, dtype=int) + A_i[:] = ind[:, None] + A_i += 3 * (1 + ind_blocks[:, None, None]) + + A_j = np.empty_like(A, dtype=int) + A_j[:] = ind + A_j += 3 * ind_blocks[:, None, None] + + B_i = np.empty_like(B, dtype=int) + B_i[:] = ind[:, None] + B_i += 3 * ind_blocks[:, None, None] + + B_j = np.empty_like(B, dtype=int) + B_j[:] = ind + B_j += 3 * (1 + ind_blocks[:, None, None]) + + diag_i = diag_j = np.arange(3 * len(d)) + i = np.hstack((A_i.ravel(), B_i.ravel(), diag_i)) + j = np.hstack((A_j.ravel(), B_j.ravel(), diag_j)) + values = np.hstack((A.ravel(), B.ravel(), np.repeat(d, 3))) + + u = 5 + l = 5 + result = np.zeros((u + l + 1, 3 * len(d))) + result[u + i - j, j] = values + return result + + +class RotationSpline: + """Interpolate rotations with continuous angular rate and acceleration. + + The rotation vectors between each consecutive orientation are cubic + functions of time and it is guaranteed that angular rate and acceleration + are continuous. Such interpolation are analogous to cubic spline + interpolation. + + Refer to [1]_ for math and implementation details. + + Parameters + ---------- + times : array_like, shape (N,) + Times of the known rotations. At least 2 times must be specified. + rotations : `Rotation` instance + Rotations to perform the interpolation between. Must contain N + rotations. + + Methods + ------- + __call__ + + References + ---------- + .. [1] `Smooth Attitude Interpolation + `_ + + Examples + -------- + >>> from scipy.spatial.transform import Rotation, RotationSpline + >>> import numpy as np + + Define the sequence of times and rotations from the Euler angles: + + >>> times = [0, 10, 20, 40] + >>> angles = [[-10, 20, 30], [0, 15, 40], [-30, 45, 30], [20, 45, 90]] + >>> rotations = Rotation.from_euler('XYZ', angles, degrees=True) + + Create the interpolator object: + + >>> spline = RotationSpline(times, rotations) + + Interpolate the Euler angles, angular rate and acceleration: + + >>> angular_rate = np.rad2deg(spline(times, 1)) + >>> angular_acceleration = np.rad2deg(spline(times, 2)) + >>> times_plot = np.linspace(times[0], times[-1], 100) + >>> angles_plot = spline(times_plot).as_euler('XYZ', degrees=True) + >>> angular_rate_plot = np.rad2deg(spline(times_plot, 1)) + >>> angular_acceleration_plot = np.rad2deg(spline(times_plot, 2)) + + On this plot you see that Euler angles are continuous and smooth: + + >>> import matplotlib.pyplot as plt + >>> plt.plot(times_plot, angles_plot) + >>> plt.plot(times, angles, 'x') + >>> plt.title("Euler angles") + >>> plt.show() + + The angular rate is also smooth: + + >>> plt.plot(times_plot, angular_rate_plot) + >>> plt.plot(times, angular_rate, 'x') + >>> plt.title("Angular rate") + >>> plt.show() + + The angular acceleration is continuous, but not smooth. Also note that + the angular acceleration is not a piecewise-linear function, because + it is different from the second derivative of the rotation vector (which + is a piecewise-linear function as in the cubic spline). + + >>> plt.plot(times_plot, angular_acceleration_plot) + >>> plt.plot(times, angular_acceleration, 'x') + >>> plt.title("Angular acceleration") + >>> plt.show() + """ + # Parameters for the solver for angular rate. + MAX_ITER = 10 + TOL = 1e-9 + + def _solve_for_angular_rates(self, dt, angular_rates, rotvecs): + angular_rate_first = angular_rates[0].copy() + + A = _angular_rate_to_rotvec_dot_matrix(rotvecs) + A_inv = _rotvec_dot_to_angular_rate_matrix(rotvecs) + M = _create_block_3_diagonal_matrix( + 2 * A_inv[1:-1] / dt[1:-1, None, None], + 2 * A[1:-1] / dt[1:-1, None, None], + 4 * (1 / dt[:-1] + 1 / dt[1:])) + + b0 = 6 * (rotvecs[:-1] * dt[:-1, None] ** -2 + + rotvecs[1:] * dt[1:, None] ** -2) + b0[0] -= 2 / dt[0] * A_inv[0].dot(angular_rate_first) + b0[-1] -= 2 / dt[-1] * A[-1].dot(angular_rates[-1]) + + for iteration in range(self.MAX_ITER): + rotvecs_dot = _matrix_vector_product_of_stacks(A, angular_rates) + delta_beta = _angular_acceleration_nonlinear_term( + rotvecs[:-1], rotvecs_dot[:-1]) + b = b0 - delta_beta + angular_rates_new = solve_banded((5, 5), M, b.ravel()) + angular_rates_new = angular_rates_new.reshape((-1, 3)) + + delta = np.abs(angular_rates_new - angular_rates[:-1]) + angular_rates[:-1] = angular_rates_new + if np.all(delta < self.TOL * (1 + np.abs(angular_rates_new))): + break + + rotvecs_dot = _matrix_vector_product_of_stacks(A, angular_rates) + angular_rates = np.vstack((angular_rate_first, angular_rates[:-1])) + + return angular_rates, rotvecs_dot + + def __init__(self, times, rotations): + from scipy.interpolate import PPoly + + if rotations.single: + raise ValueError("`rotations` must be a sequence of rotations.") + + if len(rotations) == 1: + raise ValueError("`rotations` must contain at least 2 rotations.") + + times = np.asarray(times, dtype=float) + if times.ndim != 1: + raise ValueError("`times` must be 1-dimensional.") + + if len(times) != len(rotations): + raise ValueError("Expected number of rotations to be equal to " + "number of timestamps given, " + f"got {len(rotations)} rotations " + f"and {len(times)} timestamps.") + + dt = np.diff(times) + if np.any(dt <= 0): + raise ValueError("Values in `times` must be in a strictly " + "increasing order.") + + rotvecs = (rotations[:-1].inv() * rotations[1:]).as_rotvec() + angular_rates = rotvecs / dt[:, None] + + if len(rotations) == 2: + rotvecs_dot = angular_rates + else: + angular_rates, rotvecs_dot = self._solve_for_angular_rates( + dt, angular_rates, rotvecs) + + dt = dt[:, None] + coeff = np.empty((4, len(times) - 1, 3)) + coeff[0] = (-2 * rotvecs + dt * angular_rates + + dt * rotvecs_dot) / dt ** 3 + coeff[1] = (3 * rotvecs - 2 * dt * angular_rates + - dt * rotvecs_dot) / dt ** 2 + coeff[2] = angular_rates + coeff[3] = 0 + + self.times = times + self.rotations = rotations + self.interpolator = PPoly(coeff, times) + + def __call__(self, times, order=0): + """Compute interpolated values. + + Parameters + ---------- + times : float or array_like + Times of interest. + order : {0, 1, 2}, optional + Order of differentiation: + + * 0 (default) : return Rotation + * 1 : return the angular rate in rad/sec + * 2 : return the angular acceleration in rad/sec/sec + + Returns + ------- + Interpolated Rotation, angular rate or acceleration. + """ + if order not in [0, 1, 2]: + raise ValueError("`order` must be 0, 1 or 2.") + + times = np.asarray(times, dtype=float) + if times.ndim > 1: + raise ValueError("`times` must be at most 1-dimensional.") + + singe_time = times.ndim == 0 + times = np.atleast_1d(times) + + rotvecs = self.interpolator(times) + if order == 0: + index = np.searchsorted(self.times, times, side='right') + index -= 1 + index[index < 0] = 0 + n_segments = len(self.times) - 1 + index[index > n_segments - 1] = n_segments - 1 + result = self.rotations[index] * Rotation.from_rotvec(rotvecs) + elif order == 1: + rotvecs_dot = self.interpolator(times, 1) + result = _compute_angular_rate(rotvecs, rotvecs_dot) + elif order == 2: + rotvecs_dot = self.interpolator(times, 1) + rotvecs_dot_dot = self.interpolator(times, 2) + result = _compute_angular_acceleration(rotvecs, rotvecs_dot, + rotvecs_dot_dot) + else: + assert False + + if singe_time: + result = result[0] + + return result diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/rotation.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/rotation.py new file mode 100644 index 0000000000000000000000000000000000000000..a719437415124f4afc842b98c708836c1fe68f22 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/rotation.py @@ -0,0 +1,21 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.spatial` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'Rotation', + 'Slerp', +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="spatial.transform", module="rotation", + private_modules=["_rotation"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/test_rotation.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/test_rotation.py new file mode 100644 index 0000000000000000000000000000000000000000..8ccd215de9322fda7d09ade0e1355756ed0f11bd --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/test_rotation.py @@ -0,0 +1,2188 @@ +import pytest + +import numpy as np +from numpy.testing import assert_equal, assert_array_almost_equal +from numpy.testing import assert_allclose +from scipy.spatial.transform import Rotation, Slerp +from scipy.stats import special_ortho_group +from itertools import permutations + +import pickle +import copy + +def basis_vec(axis): + if axis == 'x': + return [1, 0, 0] + elif axis == 'y': + return [0, 1, 0] + elif axis == 'z': + return [0, 0, 1] + +def test_generic_quat_matrix(): + x = np.array([[3, 4, 0, 0], [5, 12, 0, 0]]) + r = Rotation.from_quat(x) + expected_quat = x / np.array([[5], [13]]) + assert_array_almost_equal(r.as_quat(), expected_quat) + + +def test_from_single_1d_quaternion(): + x = np.array([3, 4, 0, 0]) + r = Rotation.from_quat(x) + expected_quat = x / 5 + assert_array_almost_equal(r.as_quat(), expected_quat) + + +def test_from_single_2d_quaternion(): + x = np.array([[3, 4, 0, 0]]) + r = Rotation.from_quat(x) + expected_quat = x / 5 + assert_array_almost_equal(r.as_quat(), expected_quat) + + +def test_from_quat_scalar_first(): + rng = np.random.RandomState(0) + + r = Rotation.from_quat([1, 0, 0, 0], scalar_first=True) + assert_allclose(r.as_matrix(), np.eye(3), rtol=1e-15, atol=1e-16) + + r = Rotation.from_quat(np.tile([1, 0, 0, 0], (10, 1)), scalar_first=True) + assert_allclose(r.as_matrix(), np.tile(np.eye(3), (10, 1, 1)), + rtol=1e-15, atol=1e-16) + + q = rng.randn(100, 4) + q /= np.linalg.norm(q, axis=1)[:, None] + for qi in q: + r = Rotation.from_quat(qi, scalar_first=True) + assert_allclose(np.roll(r.as_quat(), 1), qi, rtol=1e-15) + + r = Rotation.from_quat(q, scalar_first=True) + assert_allclose(np.roll(r.as_quat(), 1, axis=1), q, rtol=1e-15) + + +def test_as_quat_scalar_first(): + rng = np.random.RandomState(0) + + r = Rotation.from_euler('xyz', np.zeros(3)) + assert_allclose(r.as_quat(scalar_first=True), [1, 0, 0, 0], + rtol=1e-15, atol=1e-16) + + r = Rotation.from_euler('xyz', np.zeros((10, 3))) + assert_allclose(r.as_quat(scalar_first=True), + np.tile([1, 0, 0, 0], (10, 1)), rtol=1e-15, atol=1e-16) + + q = rng.randn(100, 4) + q /= np.linalg.norm(q, axis=1)[:, None] + for qi in q: + r = Rotation.from_quat(qi) + assert_allclose(r.as_quat(scalar_first=True), np.roll(qi, 1), + rtol=1e-15) + + assert_allclose(r.as_quat(canonical=True, scalar_first=True), + np.roll(r.as_quat(canonical=True), 1), + rtol=1e-15) + + r = Rotation.from_quat(q) + assert_allclose(r.as_quat(scalar_first=True), np.roll(q, 1, axis=1), + rtol=1e-15) + + assert_allclose(r.as_quat(canonical=True, scalar_first=True), + np.roll(r.as_quat(canonical=True), 1, axis=1), rtol=1e-15) + + +def test_from_square_quat_matrix(): + # Ensure proper norm array broadcasting + x = np.array([ + [3, 0, 0, 4], + [5, 0, 12, 0], + [0, 0, 0, 1], + [-1, -1, -1, 1], + [0, 0, 0, -1], # Check double cover + [-1, -1, -1, -1] # Check double cover + ]) + r = Rotation.from_quat(x) + expected_quat = x / np.array([[5], [13], [1], [2], [1], [2]]) + assert_array_almost_equal(r.as_quat(), expected_quat) + + +def test_quat_double_to_canonical_single_cover(): + x = np.array([ + [-1, 0, 0, 0], + [0, -1, 0, 0], + [0, 0, -1, 0], + [0, 0, 0, -1], + [-1, -1, -1, -1] + ]) + r = Rotation.from_quat(x) + expected_quat = np.abs(x) / np.linalg.norm(x, axis=1)[:, None] + assert_allclose(r.as_quat(canonical=True), expected_quat) + + +def test_quat_double_cover(): + # See the Rotation.from_quat() docstring for scope of the quaternion + # double cover property. + # Check from_quat and as_quat(canonical=False) + q = np.array([0, 0, 0, -1]) + r = Rotation.from_quat(q) + assert_equal(q, r.as_quat(canonical=False)) + + # Check composition and inverse + q = np.array([1, 0, 0, 1])/np.sqrt(2) # 90 deg rotation about x + r = Rotation.from_quat(q) + r3 = r*r*r + assert_allclose(r.as_quat(canonical=False)*np.sqrt(2), + [1, 0, 0, 1]) + assert_allclose(r.inv().as_quat(canonical=False)*np.sqrt(2), + [-1, 0, 0, 1]) + assert_allclose(r3.as_quat(canonical=False)*np.sqrt(2), + [1, 0, 0, -1]) + assert_allclose(r3.inv().as_quat(canonical=False)*np.sqrt(2), + [-1, 0, 0, -1]) + + # More sanity checks + assert_allclose((r*r.inv()).as_quat(canonical=False), + [0, 0, 0, 1], atol=2e-16) + assert_allclose((r3*r3.inv()).as_quat(canonical=False), + [0, 0, 0, 1], atol=2e-16) + assert_allclose((r*r3).as_quat(canonical=False), + [0, 0, 0, -1], atol=2e-16) + assert_allclose((r.inv()*r3.inv()).as_quat(canonical=False), + [0, 0, 0, -1], atol=2e-16) + + +def test_from_quat_wrong_shape(): + # Wrong shape 1d array + with pytest.raises(ValueError, match='Expected `quat` to have shape'): + Rotation.from_quat(np.array([1, 2, 3])) + + # Wrong shape 2d array + with pytest.raises(ValueError, match='Expected `quat` to have shape'): + Rotation.from_quat(np.array([ + [1, 2, 3, 4, 5], + [4, 5, 6, 7, 8] + ])) + + # 3d array + with pytest.raises(ValueError, match='Expected `quat` to have shape'): + Rotation.from_quat(np.array([ + [[1, 2, 3, 4]], + [[4, 5, 6, 7]] + ])) + + +def test_zero_norms_from_quat(): + x = np.array([ + [3, 4, 0, 0], + [0, 0, 0, 0], + [5, 0, 12, 0] + ]) + with pytest.raises(ValueError): + Rotation.from_quat(x) + + +def test_as_matrix_single_1d_quaternion(): + quat = [0, 0, 0, 1] + mat = Rotation.from_quat(quat).as_matrix() + # mat.shape == (3,3) due to 1d input + assert_array_almost_equal(mat, np.eye(3)) + + +def test_as_matrix_single_2d_quaternion(): + quat = [[0, 0, 1, 1]] + mat = Rotation.from_quat(quat).as_matrix() + assert_equal(mat.shape, (1, 3, 3)) + expected_mat = np.array([ + [0, -1, 0], + [1, 0, 0], + [0, 0, 1] + ]) + assert_array_almost_equal(mat[0], expected_mat) + + +def test_as_matrix_from_square_input(): + quats = [ + [0, 0, 1, 1], + [0, 1, 0, 1], + [0, 0, 0, 1], + [0, 0, 0, -1] + ] + mat = Rotation.from_quat(quats).as_matrix() + assert_equal(mat.shape, (4, 3, 3)) + + expected0 = np.array([ + [0, -1, 0], + [1, 0, 0], + [0, 0, 1] + ]) + assert_array_almost_equal(mat[0], expected0) + + expected1 = np.array([ + [0, 0, 1], + [0, 1, 0], + [-1, 0, 0] + ]) + assert_array_almost_equal(mat[1], expected1) + + assert_array_almost_equal(mat[2], np.eye(3)) + assert_array_almost_equal(mat[3], np.eye(3)) + + +def test_as_matrix_from_generic_input(): + quats = [ + [0, 0, 1, 1], + [0, 1, 0, 1], + [1, 2, 3, 4] + ] + mat = Rotation.from_quat(quats).as_matrix() + assert_equal(mat.shape, (3, 3, 3)) + + expected0 = np.array([ + [0, -1, 0], + [1, 0, 0], + [0, 0, 1] + ]) + assert_array_almost_equal(mat[0], expected0) + + expected1 = np.array([ + [0, 0, 1], + [0, 1, 0], + [-1, 0, 0] + ]) + assert_array_almost_equal(mat[1], expected1) + + expected2 = np.array([ + [0.4, -2, 2.2], + [2.8, 1, 0.4], + [-1, 2, 2] + ]) / 3 + assert_array_almost_equal(mat[2], expected2) + + +def test_from_single_2d_matrix(): + mat = [ + [0, 0, 1], + [1, 0, 0], + [0, 1, 0] + ] + expected_quat = [0.5, 0.5, 0.5, 0.5] + assert_array_almost_equal( + Rotation.from_matrix(mat).as_quat(), + expected_quat) + + +def test_from_single_3d_matrix(): + mat = np.array([ + [0, 0, 1], + [1, 0, 0], + [0, 1, 0] + ]).reshape((1, 3, 3)) + expected_quat = np.array([0.5, 0.5, 0.5, 0.5]).reshape((1, 4)) + assert_array_almost_equal( + Rotation.from_matrix(mat).as_quat(), + expected_quat) + + +def test_from_matrix_calculation(): + expected_quat = np.array([1, 1, 6, 1]) / np.sqrt(39) + mat = np.array([ + [-0.8974359, -0.2564103, 0.3589744], + [0.3589744, -0.8974359, 0.2564103], + [0.2564103, 0.3589744, 0.8974359] + ]) + assert_array_almost_equal( + Rotation.from_matrix(mat).as_quat(), + expected_quat) + assert_array_almost_equal( + Rotation.from_matrix(mat.reshape((1, 3, 3))).as_quat(), + expected_quat.reshape((1, 4))) + + +def test_matrix_calculation_pipeline(): + mat = special_ortho_group.rvs(3, size=10, random_state=0) + assert_array_almost_equal(Rotation.from_matrix(mat).as_matrix(), mat) + + +def test_from_matrix_ortho_output(): + rnd = np.random.RandomState(0) + mat = rnd.random_sample((100, 3, 3)) + dets = np.linalg.det(mat) + for i in range(len(dets)): + # Make sure we have a right-handed rotation matrix + if dets[i] < 0: + mat[i] = -mat[i] + ortho_mat = Rotation.from_matrix(mat).as_matrix() + + mult_result = np.einsum('...ij,...jk->...ik', ortho_mat, + ortho_mat.transpose((0, 2, 1))) + + eye3d = np.zeros((100, 3, 3)) + for i in range(3): + eye3d[:, i, i] = 1.0 + + assert_array_almost_equal(mult_result, eye3d) + + +def test_from_matrix_normalize(): + mat = np.array([ + [1, 1, 0], + [0, 1, 0], + [0, 0, 1]]) + expected = np.array([[ 0.894427, 0.447214, 0.0], + [-0.447214, 0.894427, 0.0], + [ 0.0, 0.0, 1.0]]) + assert_allclose(Rotation.from_matrix(mat).as_matrix(), expected, atol=1e-6) + + mat = np.array([ + [0, -0.5, 0 ], + [0.5, 0 , 0 ], + [0, 0 , 0.5]]) + expected = np.array([[ 0, -1, 0], + [ 1, 0, 0], + [ 0, 0, 1]]) + assert_allclose(Rotation.from_matrix(mat).as_matrix(), expected, atol=1e-6) + + +def test_from_matrix_non_positive_determinant(): + mat = np.eye(3) + mat[0, 0] = 0 + with pytest.raises(ValueError, match='Non-positive determinant'): + Rotation.from_matrix(mat) + + mat[0, 0] = -1 + with pytest.raises(ValueError, match='Non-positive determinant'): + Rotation.from_matrix(mat) + + +def test_from_1d_single_rotvec(): + rotvec = [1, 0, 0] + expected_quat = np.array([0.4794255, 0, 0, 0.8775826]) + result = Rotation.from_rotvec(rotvec) + assert_array_almost_equal(result.as_quat(), expected_quat) + + +def test_from_2d_single_rotvec(): + rotvec = [[1, 0, 0]] + expected_quat = np.array([[0.4794255, 0, 0, 0.8775826]]) + result = Rotation.from_rotvec(rotvec) + assert_array_almost_equal(result.as_quat(), expected_quat) + + +def test_from_generic_rotvec(): + rotvec = [ + [1, 2, 2], + [1, -1, 0.5], + [0, 0, 0] + ] + expected_quat = np.array([ + [0.3324983, 0.6649967, 0.6649967, 0.0707372], + [0.4544258, -0.4544258, 0.2272129, 0.7316889], + [0, 0, 0, 1] + ]) + assert_array_almost_equal( + Rotation.from_rotvec(rotvec).as_quat(), + expected_quat) + + +def test_from_rotvec_small_angle(): + rotvec = np.array([ + [5e-4 / np.sqrt(3), -5e-4 / np.sqrt(3), 5e-4 / np.sqrt(3)], + [0.2, 0.3, 0.4], + [0, 0, 0] + ]) + + quat = Rotation.from_rotvec(rotvec).as_quat() + # cos(theta/2) ~~ 1 for small theta + assert_allclose(quat[0, 3], 1) + # sin(theta/2) / theta ~~ 0.5 for small theta + assert_allclose(quat[0, :3], rotvec[0] * 0.5) + + assert_allclose(quat[1, 3], 0.9639685) + assert_allclose( + quat[1, :3], + np.array([ + 0.09879603932153465, + 0.14819405898230198, + 0.19759207864306931 + ])) + + assert_equal(quat[2], np.array([0, 0, 0, 1])) + + +def test_degrees_from_rotvec(): + rotvec1 = [1.0 / np.cbrt(3), 1.0 / np.cbrt(3), 1.0 / np.cbrt(3)] + rot1 = Rotation.from_rotvec(rotvec1, degrees=True) + quat1 = rot1.as_quat() + + rotvec2 = np.deg2rad(rotvec1) + rot2 = Rotation.from_rotvec(rotvec2) + quat2 = rot2.as_quat() + + assert_allclose(quat1, quat2) + + +def test_malformed_1d_from_rotvec(): + with pytest.raises(ValueError, match='Expected `rot_vec` to have shape'): + Rotation.from_rotvec([1, 2]) + + +def test_malformed_2d_from_rotvec(): + with pytest.raises(ValueError, match='Expected `rot_vec` to have shape'): + Rotation.from_rotvec([ + [1, 2, 3, 4], + [5, 6, 7, 8] + ]) + + +def test_as_generic_rotvec(): + quat = np.array([ + [1, 2, -1, 0.5], + [1, -1, 1, 0.0003], + [0, 0, 0, 1] + ]) + quat /= np.linalg.norm(quat, axis=1)[:, None] + + rotvec = Rotation.from_quat(quat).as_rotvec() + angle = np.linalg.norm(rotvec, axis=1) + + assert_allclose(quat[:, 3], np.cos(angle/2)) + assert_allclose(np.cross(rotvec, quat[:, :3]), np.zeros((3, 3))) + + +def test_as_rotvec_single_1d_input(): + quat = np.array([1, 2, -3, 2]) + expected_rotvec = np.array([0.5772381, 1.1544763, -1.7317144]) + + actual_rotvec = Rotation.from_quat(quat).as_rotvec() + + assert_equal(actual_rotvec.shape, (3,)) + assert_allclose(actual_rotvec, expected_rotvec) + + +def test_as_rotvec_single_2d_input(): + quat = np.array([[1, 2, -3, 2]]) + expected_rotvec = np.array([[0.5772381, 1.1544763, -1.7317144]]) + + actual_rotvec = Rotation.from_quat(quat).as_rotvec() + + assert_equal(actual_rotvec.shape, (1, 3)) + assert_allclose(actual_rotvec, expected_rotvec) + + +def test_as_rotvec_degrees(): + # x->y, y->z, z->x + mat = [[0, 0, 1], [1, 0, 0], [0, 1, 0]] + rot = Rotation.from_matrix(mat) + rotvec = rot.as_rotvec(degrees=True) + angle = np.linalg.norm(rotvec) + assert_allclose(angle, 120.0) + assert_allclose(rotvec[0], rotvec[1]) + assert_allclose(rotvec[1], rotvec[2]) + + +def test_rotvec_calc_pipeline(): + # Include small angles + rotvec = np.array([ + [0, 0, 0], + [1, -1, 2], + [-3e-4, 3.5e-4, 7.5e-5] + ]) + assert_allclose(Rotation.from_rotvec(rotvec).as_rotvec(), rotvec) + assert_allclose(Rotation.from_rotvec(rotvec, degrees=True).as_rotvec(degrees=True), + rotvec) + + +def test_from_1d_single_mrp(): + mrp = [0, 0, 1.0] + expected_quat = np.array([0, 0, 1, 0]) + result = Rotation.from_mrp(mrp) + assert_array_almost_equal(result.as_quat(), expected_quat) + + +def test_from_2d_single_mrp(): + mrp = [[0, 0, 1.0]] + expected_quat = np.array([[0, 0, 1, 0]]) + result = Rotation.from_mrp(mrp) + assert_array_almost_equal(result.as_quat(), expected_quat) + + +def test_from_generic_mrp(): + mrp = np.array([ + [1, 2, 2], + [1, -1, 0.5], + [0, 0, 0]]) + expected_quat = np.array([ + [0.2, 0.4, 0.4, -0.8], + [0.61538462, -0.61538462, 0.30769231, -0.38461538], + [0, 0, 0, 1]]) + assert_array_almost_equal(Rotation.from_mrp(mrp).as_quat(), expected_quat) + + +def test_malformed_1d_from_mrp(): + with pytest.raises(ValueError, match='Expected `mrp` to have shape'): + Rotation.from_mrp([1, 2]) + + +def test_malformed_2d_from_mrp(): + with pytest.raises(ValueError, match='Expected `mrp` to have shape'): + Rotation.from_mrp([ + [1, 2, 3, 4], + [5, 6, 7, 8] + ]) + + +def test_as_generic_mrp(): + quat = np.array([ + [1, 2, -1, 0.5], + [1, -1, 1, 0.0003], + [0, 0, 0, 1]]) + quat /= np.linalg.norm(quat, axis=1)[:, None] + + expected_mrp = np.array([ + [0.33333333, 0.66666667, -0.33333333], + [0.57725028, -0.57725028, 0.57725028], + [0, 0, 0]]) + assert_array_almost_equal(Rotation.from_quat(quat).as_mrp(), expected_mrp) + +def test_past_180_degree_rotation(): + # ensure that a > 180 degree rotation is returned as a <180 rotation in MRPs + # in this case 270 should be returned as -90 + expected_mrp = np.array([-np.tan(np.pi/2/4), 0.0, 0]) + assert_array_almost_equal( + Rotation.from_euler('xyz', [270, 0, 0], degrees=True).as_mrp(), + expected_mrp + ) + + +def test_as_mrp_single_1d_input(): + quat = np.array([1, 2, -3, 2]) + expected_mrp = np.array([0.16018862, 0.32037724, -0.48056586]) + + actual_mrp = Rotation.from_quat(quat).as_mrp() + + assert_equal(actual_mrp.shape, (3,)) + assert_allclose(actual_mrp, expected_mrp) + + +def test_as_mrp_single_2d_input(): + quat = np.array([[1, 2, -3, 2]]) + expected_mrp = np.array([[0.16018862, 0.32037724, -0.48056586]]) + + actual_mrp = Rotation.from_quat(quat).as_mrp() + + assert_equal(actual_mrp.shape, (1, 3)) + assert_allclose(actual_mrp, expected_mrp) + + +def test_mrp_calc_pipeline(): + actual_mrp = np.array([ + [0, 0, 0], + [1, -1, 2], + [0.41421356, 0, 0], + [0.1, 0.2, 0.1]]) + expected_mrp = np.array([ + [0, 0, 0], + [-0.16666667, 0.16666667, -0.33333333], + [0.41421356, 0, 0], + [0.1, 0.2, 0.1]]) + assert_allclose(Rotation.from_mrp(actual_mrp).as_mrp(), expected_mrp) + + +def test_from_euler_single_rotation(): + quat = Rotation.from_euler('z', 90, degrees=True).as_quat() + expected_quat = np.array([0, 0, 1, 1]) / np.sqrt(2) + assert_allclose(quat, expected_quat) + + +def test_single_intrinsic_extrinsic_rotation(): + extrinsic = Rotation.from_euler('z', 90, degrees=True).as_matrix() + intrinsic = Rotation.from_euler('Z', 90, degrees=True).as_matrix() + assert_allclose(extrinsic, intrinsic) + + +def test_from_euler_rotation_order(): + # Intrinsic rotation is same as extrinsic with order reversed + rnd = np.random.RandomState(0) + a = rnd.randint(low=0, high=180, size=(6, 3)) + b = a[:, ::-1] + x = Rotation.from_euler('xyz', a, degrees=True).as_quat() + y = Rotation.from_euler('ZYX', b, degrees=True).as_quat() + assert_allclose(x, y) + + +def test_from_euler_elementary_extrinsic_rotation(): + # Simple test to check if extrinsic rotations are implemented correctly + mat = Rotation.from_euler('zx', [90, 90], degrees=True).as_matrix() + expected_mat = np.array([ + [0, -1, 0], + [0, 0, -1], + [1, 0, 0] + ]) + assert_array_almost_equal(mat, expected_mat) + + +def test_from_euler_intrinsic_rotation_312(): + angles = [ + [30, 60, 45], + [30, 60, 30], + [45, 30, 60] + ] + mat = Rotation.from_euler('ZXY', angles, degrees=True).as_matrix() + + assert_array_almost_equal(mat[0], np.array([ + [0.3061862, -0.2500000, 0.9185587], + [0.8838835, 0.4330127, -0.1767767], + [-0.3535534, 0.8660254, 0.3535534] + ])) + + assert_array_almost_equal(mat[1], np.array([ + [0.5334936, -0.2500000, 0.8080127], + [0.8080127, 0.4330127, -0.3995191], + [-0.2500000, 0.8660254, 0.4330127] + ])) + + assert_array_almost_equal(mat[2], np.array([ + [0.0473672, -0.6123725, 0.7891491], + [0.6597396, 0.6123725, 0.4355958], + [-0.7500000, 0.5000000, 0.4330127] + ])) + + +def test_from_euler_intrinsic_rotation_313(): + angles = [ + [30, 60, 45], + [30, 60, 30], + [45, 30, 60] + ] + mat = Rotation.from_euler('ZXZ', angles, degrees=True).as_matrix() + + assert_array_almost_equal(mat[0], np.array([ + [0.43559574, -0.78914913, 0.4330127], + [0.65973961, -0.04736717, -0.750000], + [0.61237244, 0.61237244, 0.500000] + ])) + + assert_array_almost_equal(mat[1], np.array([ + [0.6250000, -0.64951905, 0.4330127], + [0.64951905, 0.1250000, -0.750000], + [0.4330127, 0.750000, 0.500000] + ])) + + assert_array_almost_equal(mat[2], np.array([ + [-0.1767767, -0.91855865, 0.35355339], + [0.88388348, -0.30618622, -0.35355339], + [0.4330127, 0.25000000, 0.8660254] + ])) + + +def test_from_euler_extrinsic_rotation_312(): + angles = [ + [30, 60, 45], + [30, 60, 30], + [45, 30, 60] + ] + mat = Rotation.from_euler('zxy', angles, degrees=True).as_matrix() + + assert_array_almost_equal(mat[0], np.array([ + [0.91855865, 0.1767767, 0.35355339], + [0.25000000, 0.4330127, -0.8660254], + [-0.30618622, 0.88388348, 0.35355339] + ])) + + assert_array_almost_equal(mat[1], np.array([ + [0.96650635, -0.0580127, 0.2500000], + [0.25000000, 0.4330127, -0.8660254], + [-0.0580127, 0.89951905, 0.4330127] + ])) + + assert_array_almost_equal(mat[2], np.array([ + [0.65973961, -0.04736717, 0.7500000], + [0.61237244, 0.61237244, -0.5000000], + [-0.43559574, 0.78914913, 0.4330127] + ])) + + +def test_from_euler_extrinsic_rotation_313(): + angles = [ + [30, 60, 45], + [30, 60, 30], + [45, 30, 60] + ] + mat = Rotation.from_euler('zxz', angles, degrees=True).as_matrix() + + assert_array_almost_equal(mat[0], np.array([ + [0.43559574, -0.65973961, 0.61237244], + [0.78914913, -0.04736717, -0.61237244], + [0.4330127, 0.75000000, 0.500000] + ])) + + assert_array_almost_equal(mat[1], np.array([ + [0.62500000, -0.64951905, 0.4330127], + [0.64951905, 0.12500000, -0.750000], + [0.4330127, 0.75000000, 0.500000] + ])) + + assert_array_almost_equal(mat[2], np.array([ + [-0.1767767, -0.88388348, 0.4330127], + [0.91855865, -0.30618622, -0.250000], + [0.35355339, 0.35355339, 0.8660254] + ])) + + +@pytest.mark.parametrize("seq_tuple", permutations("xyz")) +@pytest.mark.parametrize("intrinsic", (False, True)) +def test_as_euler_asymmetric_axes(seq_tuple, intrinsic): + # helper function for mean error tests + def test_stats(error, mean_max, rms_max): + mean = np.mean(error, axis=0) + std = np.std(error, axis=0) + rms = np.hypot(mean, std) + assert np.all(np.abs(mean) < mean_max) + assert np.all(rms < rms_max) + + rnd = np.random.RandomState(0) + n = 1000 + angles = np.empty((n, 3)) + angles[:, 0] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + angles[:, 1] = rnd.uniform(low=-np.pi / 2, high=np.pi / 2, size=(n,)) + angles[:, 2] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + + seq = "".join(seq_tuple) + if intrinsic: + # Extrinsic rotation (wrt to global world) at lower case + # intrinsic (WRT the object itself) lower case. + seq = seq.upper() + rotation = Rotation.from_euler(seq, angles) + angles_quat = rotation.as_euler(seq) + angles_mat = rotation._as_euler_from_matrix(seq) + assert_allclose(angles, angles_quat, atol=0, rtol=1e-12) + assert_allclose(angles, angles_mat, atol=0, rtol=1e-12) + test_stats(angles_quat - angles, 1e-15, 1e-14) + test_stats(angles_mat - angles, 1e-15, 1e-14) + + + +@pytest.mark.parametrize("seq_tuple", permutations("xyz")) +@pytest.mark.parametrize("intrinsic", (False, True)) +def test_as_euler_symmetric_axes(seq_tuple, intrinsic): + # helper function for mean error tests + def test_stats(error, mean_max, rms_max): + mean = np.mean(error, axis=0) + std = np.std(error, axis=0) + rms = np.hypot(mean, std) + assert np.all(np.abs(mean) < mean_max) + assert np.all(rms < rms_max) + + rnd = np.random.RandomState(0) + n = 1000 + angles = np.empty((n, 3)) + angles[:, 0] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + angles[:, 1] = rnd.uniform(low=0, high=np.pi, size=(n,)) + angles[:, 2] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + + # Rotation of the form A/B/A are rotation around symmetric axes + seq = "".join([seq_tuple[0], seq_tuple[1], seq_tuple[0]]) + if intrinsic: + seq = seq.upper() + rotation = Rotation.from_euler(seq, angles) + angles_quat = rotation.as_euler(seq) + angles_mat = rotation._as_euler_from_matrix(seq) + assert_allclose(angles, angles_quat, atol=0, rtol=1e-13) + assert_allclose(angles, angles_mat, atol=0, rtol=1e-9) + test_stats(angles_quat - angles, 1e-16, 1e-14) + test_stats(angles_mat - angles, 1e-15, 1e-13) + + +@pytest.mark.thread_unsafe +@pytest.mark.parametrize("seq_tuple", permutations("xyz")) +@pytest.mark.parametrize("intrinsic", (False, True)) +def test_as_euler_degenerate_asymmetric_axes(seq_tuple, intrinsic): + # Since we cannot check for angle equality, we check for rotation matrix + # equality + angles = np.array([ + [45, 90, 35], + [35, -90, 20], + [35, 90, 25], + [25, -90, 15]]) + + seq = "".join(seq_tuple) + if intrinsic: + # Extrinsic rotation (wrt to global world) at lower case + # Intrinsic (WRT the object itself) upper case. + seq = seq.upper() + rotation = Rotation.from_euler(seq, angles, degrees=True) + mat_expected = rotation.as_matrix() + + with pytest.warns(UserWarning, match="Gimbal lock"): + angle_estimates = rotation.as_euler(seq, degrees=True) + mat_estimated = Rotation.from_euler(seq, angle_estimates, degrees=True).as_matrix() + + assert_array_almost_equal(mat_expected, mat_estimated) + + +@pytest.mark.thread_unsafe +@pytest.mark.parametrize("seq_tuple", permutations("xyz")) +@pytest.mark.parametrize("intrinsic", (False, True)) +def test_as_euler_degenerate_symmetric_axes(seq_tuple, intrinsic): + # Since we cannot check for angle equality, we check for rotation matrix + # equality + angles = np.array([ + [15, 0, 60], + [35, 0, 75], + [60, 180, 35], + [15, -180, 25]]) + + # Rotation of the form A/B/A are rotation around symmetric axes + seq = "".join([seq_tuple[0], seq_tuple[1], seq_tuple[0]]) + if intrinsic: + # Extrinsic rotation (wrt to global world) at lower case + # Intrinsic (WRT the object itself) upper case. + seq = seq.upper() + rotation = Rotation.from_euler(seq, angles, degrees=True) + mat_expected = rotation.as_matrix() + + with pytest.warns(UserWarning, match="Gimbal lock"): + angle_estimates = rotation.as_euler(seq, degrees=True) + mat_estimated = Rotation.from_euler(seq, angle_estimates, degrees=True).as_matrix() + + assert_array_almost_equal(mat_expected, mat_estimated) + + +@pytest.mark.thread_unsafe +@pytest.mark.parametrize("seq_tuple", permutations("xyz")) +@pytest.mark.parametrize("intrinsic", (False, True)) +def test_as_euler_degenerate_compare_algorithms(seq_tuple, intrinsic): + # this test makes sure that both algorithms are doing the same choices + # in degenerate cases + + # asymmetric axes + angles = np.array([ + [45, 90, 35], + [35, -90, 20], + [35, 90, 25], + [25, -90, 15]]) + + seq = "".join(seq_tuple) + if intrinsic: + # Extrinsic rotation (wrt to global world at lower case + # Intrinsic (WRT the object itself) upper case. + seq = seq.upper() + + rot = Rotation.from_euler(seq, angles, degrees=True) + with pytest.warns(UserWarning, match="Gimbal lock"): + estimates_matrix = rot._as_euler_from_matrix(seq, degrees=True) + with pytest.warns(UserWarning, match="Gimbal lock"): + estimates_quat = rot.as_euler(seq, degrees=True) + assert_allclose( + estimates_matrix[:, [0, 2]], estimates_quat[:, [0, 2]], atol=0, rtol=1e-12 + ) + assert_allclose(estimates_matrix[:, 1], estimates_quat[:, 1], atol=0, rtol=1e-7) + + # symmetric axes + # Absolute error tolerance must be looser to directly compare the results + # from both algorithms, because of numerical loss of precision for the + # method _as_euler_from_matrix near a zero angle value + + angles = np.array([ + [15, 0, 60], + [35, 0, 75], + [60, 180, 35], + [15, -180, 25]]) + + idx = angles[:, 1] == 0 # find problematic angles indices + + # Rotation of the form A/B/A are rotation around symmetric axes + seq = "".join([seq_tuple[0], seq_tuple[1], seq_tuple[0]]) + if intrinsic: + # Extrinsic rotation (wrt to global world) at lower case + # Intrinsic (WRT the object itself) upper case. + seq = seq.upper() + + rot = Rotation.from_euler(seq, angles, degrees=True) + with pytest.warns(UserWarning, match="Gimbal lock"): + estimates_matrix = rot._as_euler_from_matrix(seq, degrees=True) + with pytest.warns(UserWarning, match="Gimbal lock"): + estimates_quat = rot.as_euler(seq, degrees=True) + assert_allclose( + estimates_matrix[:, [0, 2]], estimates_quat[:, [0, 2]], atol=0, rtol=1e-12 + ) + + assert_allclose( + estimates_matrix[~idx, 1], estimates_quat[~idx, 1], atol=0, rtol=1e-7 + ) + + assert_allclose( + estimates_matrix[idx, 1], estimates_quat[idx, 1], atol=1e-6 + ) # problematic, angles[1] = 0 + + +def test_inv(): + rnd = np.random.RandomState(0) + n = 10 + # preserve use of old random_state during SPEC 7 transition + p = Rotation.random(num=n, random_state=rnd) + q = p.inv() + + p_mat = p.as_matrix() + q_mat = q.as_matrix() + result1 = np.einsum('...ij,...jk->...ik', p_mat, q_mat) + result2 = np.einsum('...ij,...jk->...ik', q_mat, p_mat) + + eye3d = np.empty((n, 3, 3)) + eye3d[:] = np.eye(3) + + assert_array_almost_equal(result1, eye3d) + assert_array_almost_equal(result2, eye3d) + + +def test_inv_single_rotation(): + rng = np.random.default_rng(146972845698875399755764481408308808739) + p = Rotation.random(rng=rng) + q = p.inv() + + p_mat = p.as_matrix() + q_mat = q.as_matrix() + res1 = np.dot(p_mat, q_mat) + res2 = np.dot(q_mat, p_mat) + + eye = np.eye(3) + + assert_array_almost_equal(res1, eye) + assert_array_almost_equal(res2, eye) + + x = Rotation.random(num=1, rng=rng) + y = x.inv() + + x_matrix = x.as_matrix() + y_matrix = y.as_matrix() + result1 = np.einsum('...ij,...jk->...ik', x_matrix, y_matrix) + result2 = np.einsum('...ij,...jk->...ik', y_matrix, x_matrix) + + eye3d = np.empty((1, 3, 3)) + eye3d[:] = np.eye(3) + + assert_array_almost_equal(result1, eye3d) + assert_array_almost_equal(result2, eye3d) + + +def test_identity_magnitude(): + n = 10 + assert_allclose(Rotation.identity(n).magnitude(), 0) + assert_allclose(Rotation.identity(n).inv().magnitude(), 0) + + +def test_single_identity_magnitude(): + assert Rotation.identity().magnitude() == 0 + assert Rotation.identity().inv().magnitude() == 0 + + +def test_identity_invariance(): + n = 10 + p = Rotation.random(n, rng=0) + + result = p * Rotation.identity(n) + assert_array_almost_equal(p.as_quat(), result.as_quat()) + + result = result * p.inv() + assert_array_almost_equal(result.magnitude(), np.zeros(n)) + + +def test_single_identity_invariance(): + n = 10 + p = Rotation.random(n, rng=0) + + result = p * Rotation.identity() + assert_array_almost_equal(p.as_quat(), result.as_quat()) + + result = result * p.inv() + assert_array_almost_equal(result.magnitude(), np.zeros(n)) + + +def test_magnitude(): + r = Rotation.from_quat(np.eye(4)) + result = r.magnitude() + assert_array_almost_equal(result, [np.pi, np.pi, np.pi, 0]) + + r = Rotation.from_quat(-np.eye(4)) + result = r.magnitude() + assert_array_almost_equal(result, [np.pi, np.pi, np.pi, 0]) + + +def test_magnitude_single_rotation(): + r = Rotation.from_quat(np.eye(4)) + result1 = r[0].magnitude() + assert_allclose(result1, np.pi) + + result2 = r[3].magnitude() + assert_allclose(result2, 0) + + +def test_approx_equal(): + rng = np.random.default_rng(146972845698875399755764481408308808739) + p = Rotation.random(10, rng=rng) + q = Rotation.random(10, rng=rng) + r = p * q.inv() + r_mag = r.magnitude() + atol = np.median(r_mag) # ensure we get mix of Trues and Falses + assert_equal(p.approx_equal(q, atol), (r_mag < atol)) + + +@pytest.mark.thread_unsafe +def test_approx_equal_single_rotation(): + # also tests passing single argument to approx_equal + p = Rotation.from_rotvec([0, 0, 1e-9]) # less than default atol of 1e-8 + q = Rotation.from_quat(np.eye(4)) + assert p.approx_equal(q[3]) + assert not p.approx_equal(q[0]) + + # test passing atol and using degrees + assert not p.approx_equal(q[3], atol=1e-10) + assert not p.approx_equal(q[3], atol=1e-8, degrees=True) + with pytest.warns(UserWarning, match="atol must be set"): + assert p.approx_equal(q[3], degrees=True) + + +def test_mean(): + axes = np.concatenate((-np.eye(3), np.eye(3))) + thetas = np.linspace(0, np.pi / 2, 100) + for t in thetas: + r = Rotation.from_rotvec(t * axes) + assert_allclose(r.mean().magnitude(), 0, atol=1E-10) + + +def test_weighted_mean(): + # test that doubling a weight is equivalent to including a rotation twice. + axes = np.array([[0, 0, 0], [1, 0, 0], [1, 0, 0]]) + thetas = np.linspace(0, np.pi / 2, 100) + for t in thetas: + rw = Rotation.from_rotvec(t * axes[:2]) + mw = rw.mean(weights=[1, 2]) + + r = Rotation.from_rotvec(t * axes) + m = r.mean() + assert_allclose((m * mw.inv()).magnitude(), 0, atol=1E-10) + + +def test_mean_invalid_weights(): + with pytest.raises(ValueError, match="non-negative"): + r = Rotation.from_quat(np.eye(4)) + r.mean(weights=-np.ones(4)) + + +def test_reduction_no_indices(): + result = Rotation.identity().reduce(return_indices=False) + assert isinstance(result, Rotation) + + +def test_reduction_none_indices(): + result = Rotation.identity().reduce(return_indices=True) + assert type(result) is tuple + assert len(result) == 3 + + reduced, left_best, right_best = result + assert left_best is None + assert right_best is None + + +def test_reduction_scalar_calculation(): + rng = np.random.default_rng(146972845698875399755764481408308808739) + l = Rotation.random(5, rng=rng) + r = Rotation.random(10, rng=rng) + p = Rotation.random(7, rng=rng) + reduced, left_best, right_best = p.reduce(l, r, return_indices=True) + + # Loop implementation of the vectorized calculation in Rotation.reduce + scalars = np.zeros((len(l), len(p), len(r))) + for i, li in enumerate(l): + for j, pj in enumerate(p): + for k, rk in enumerate(r): + scalars[i, j, k] = np.abs((li * pj * rk).as_quat()[3]) + scalars = np.reshape(np.moveaxis(scalars, 1, 0), (scalars.shape[1], -1)) + + max_ind = np.argmax(np.reshape(scalars, (len(p), -1)), axis=1) + left_best_check = max_ind // len(r) + right_best_check = max_ind % len(r) + assert (left_best == left_best_check).all() + assert (right_best == right_best_check).all() + + reduced_check = l[left_best_check] * p * r[right_best_check] + mag = (reduced.inv() * reduced_check).magnitude() + assert_array_almost_equal(mag, np.zeros(len(p))) + + +def test_apply_single_rotation_single_point(): + mat = np.array([ + [0, -1, 0], + [1, 0, 0], + [0, 0, 1] + ]) + r_1d = Rotation.from_matrix(mat) + r_2d = Rotation.from_matrix(np.expand_dims(mat, axis=0)) + + v_1d = np.array([1, 2, 3]) + v_2d = np.expand_dims(v_1d, axis=0) + v1d_rotated = np.array([-2, 1, 3]) + v2d_rotated = np.expand_dims(v1d_rotated, axis=0) + + assert_allclose(r_1d.apply(v_1d), v1d_rotated) + assert_allclose(r_1d.apply(v_2d), v2d_rotated) + assert_allclose(r_2d.apply(v_1d), v2d_rotated) + assert_allclose(r_2d.apply(v_2d), v2d_rotated) + + v1d_inverse = np.array([2, -1, 3]) + v2d_inverse = np.expand_dims(v1d_inverse, axis=0) + + assert_allclose(r_1d.apply(v_1d, inverse=True), v1d_inverse) + assert_allclose(r_1d.apply(v_2d, inverse=True), v2d_inverse) + assert_allclose(r_2d.apply(v_1d, inverse=True), v2d_inverse) + assert_allclose(r_2d.apply(v_2d, inverse=True), v2d_inverse) + + +def test_apply_single_rotation_multiple_points(): + mat = np.array([ + [0, -1, 0], + [1, 0, 0], + [0, 0, 1] + ]) + r1 = Rotation.from_matrix(mat) + r2 = Rotation.from_matrix(np.expand_dims(mat, axis=0)) + + v = np.array([[1, 2, 3], [4, 5, 6]]) + v_rotated = np.array([[-2, 1, 3], [-5, 4, 6]]) + + assert_allclose(r1.apply(v), v_rotated) + assert_allclose(r2.apply(v), v_rotated) + + v_inverse = np.array([[2, -1, 3], [5, -4, 6]]) + + assert_allclose(r1.apply(v, inverse=True), v_inverse) + assert_allclose(r2.apply(v, inverse=True), v_inverse) + + +def test_apply_multiple_rotations_single_point(): + mat = np.empty((2, 3, 3)) + mat[0] = np.array([ + [0, -1, 0], + [1, 0, 0], + [0, 0, 1] + ]) + mat[1] = np.array([ + [1, 0, 0], + [0, 0, -1], + [0, 1, 0] + ]) + r = Rotation.from_matrix(mat) + + v1 = np.array([1, 2, 3]) + v2 = np.expand_dims(v1, axis=0) + + v_rotated = np.array([[-2, 1, 3], [1, -3, 2]]) + + assert_allclose(r.apply(v1), v_rotated) + assert_allclose(r.apply(v2), v_rotated) + + v_inverse = np.array([[2, -1, 3], [1, 3, -2]]) + + assert_allclose(r.apply(v1, inverse=True), v_inverse) + assert_allclose(r.apply(v2, inverse=True), v_inverse) + + +def test_apply_multiple_rotations_multiple_points(): + mat = np.empty((2, 3, 3)) + mat[0] = np.array([ + [0, -1, 0], + [1, 0, 0], + [0, 0, 1] + ]) + mat[1] = np.array([ + [1, 0, 0], + [0, 0, -1], + [0, 1, 0] + ]) + r = Rotation.from_matrix(mat) + + v = np.array([[1, 2, 3], [4, 5, 6]]) + v_rotated = np.array([[-2, 1, 3], [4, -6, 5]]) + assert_allclose(r.apply(v), v_rotated) + + v_inverse = np.array([[2, -1, 3], [4, 6, -5]]) + assert_allclose(r.apply(v, inverse=True), v_inverse) + + +def test_getitem(): + mat = np.empty((2, 3, 3)) + mat[0] = np.array([ + [0, -1, 0], + [1, 0, 0], + [0, 0, 1] + ]) + mat[1] = np.array([ + [1, 0, 0], + [0, 0, -1], + [0, 1, 0] + ]) + r = Rotation.from_matrix(mat) + + assert_allclose(r[0].as_matrix(), mat[0], atol=1e-15) + assert_allclose(r[1].as_matrix(), mat[1], atol=1e-15) + assert_allclose(r[:-1].as_matrix(), np.expand_dims(mat[0], axis=0), atol=1e-15) + + +def test_getitem_single(): + with pytest.raises(TypeError, match='not subscriptable'): + Rotation.identity()[0] + + +def test_setitem_single(): + r = Rotation.identity() + with pytest.raises(TypeError, match='not subscriptable'): + r[0] = Rotation.identity() + + +def test_setitem_slice(): + rng = np.random.default_rng(146972845698875399755764481408308808739) + r1 = Rotation.random(10, rng=rng) + r2 = Rotation.random(5, rng=rng) + r1[1:6] = r2 + assert_equal(r1[1:6].as_quat(), r2.as_quat()) + + +def test_setitem_integer(): + rng = np.random.default_rng(146972845698875399755764481408308808739) + r1 = Rotation.random(10, rng=rng) + r2 = Rotation.random(rng=rng) + r1[1] = r2 + assert_equal(r1[1].as_quat(), r2.as_quat()) + + +def test_setitem_wrong_type(): + r = Rotation.random(10, rng=0) + with pytest.raises(TypeError, match='Rotation object'): + r[0] = 1 + + +def test_n_rotations(): + mat = np.empty((2, 3, 3)) + mat[0] = np.array([ + [0, -1, 0], + [1, 0, 0], + [0, 0, 1] + ]) + mat[1] = np.array([ + [1, 0, 0], + [0, 0, -1], + [0, 1, 0] + ]) + r = Rotation.from_matrix(mat) + + assert_equal(len(r), 2) + assert_equal(len(r[:-1]), 1) + + +def test_random_rotation_shape(): + rng = np.random.default_rng(146972845698875399755764481408308808739) + assert_equal(Rotation.random(rng=rng).as_quat().shape, (4,)) + assert_equal(Rotation.random(None, rng=rng).as_quat().shape, (4,)) + + assert_equal(Rotation.random(1, rng=rng).as_quat().shape, (1, 4)) + assert_equal(Rotation.random(5, rng=rng).as_quat().shape, (5, 4)) + + +def test_align_vectors_no_rotation(): + x = np.array([[1, 2, 3], [4, 5, 6]]) + y = x.copy() + + r, rssd = Rotation.align_vectors(x, y) + assert_array_almost_equal(r.as_matrix(), np.eye(3)) + assert_allclose(rssd, 0, atol=1e-6) + + +def test_align_vectors_no_noise(): + rng = np.random.default_rng(14697284569885399755764481408308808739) + c = Rotation.random(rng=rng) + b = rng.normal(size=(5, 3)) + a = c.apply(b) + + est, rssd = Rotation.align_vectors(a, b) + assert_allclose(c.as_quat(), est.as_quat()) + assert_allclose(rssd, 0, atol=1e-7) + + +def test_align_vectors_improper_rotation(): + # Tests correct logic for issue #10444 + x = np.array([[0.89299824, -0.44372674, 0.0752378], + [0.60221789, -0.47564102, -0.6411702]]) + y = np.array([[0.02386536, -0.82176463, 0.5693271], + [-0.27654929, -0.95191427, -0.1318321]]) + + est, rssd = Rotation.align_vectors(x, y) + assert_allclose(x, est.apply(y), atol=1e-6) + assert_allclose(rssd, 0, atol=1e-7) + + +def test_align_vectors_rssd_sensitivity(): + rssd_expected = 0.141421356237308 + sens_expected = np.array([[0.2, 0. , 0.], + [0. , 1.5, 1.], + [0. , 1. , 1.]]) + atol = 1e-6 + a = [[0, 1, 0], [0, 1, 1], [0, 1, 1]] + b = [[1, 0, 0], [1, 1.1, 0], [1, 0.9, 0]] + rot, rssd, sens = Rotation.align_vectors(a, b, return_sensitivity=True) + assert np.isclose(rssd, rssd_expected, atol=atol) + assert np.allclose(sens, sens_expected, atol=atol) + + +def test_align_vectors_scaled_weights(): + n = 10 + a = Rotation.random(n, rng=0).apply([1, 0, 0]) + b = Rotation.random(n, rng=1).apply([1, 0, 0]) + scale = 2 + + est1, rssd1, cov1 = Rotation.align_vectors(a, b, np.ones(n), True) + est2, rssd2, cov2 = Rotation.align_vectors(a, b, scale * np.ones(n), True) + + assert_allclose(est1.as_matrix(), est2.as_matrix()) + assert_allclose(np.sqrt(scale) * rssd1, rssd2, atol=1e-6) + assert_allclose(cov1, cov2) + + +def test_align_vectors_noise(): + rng = np.random.default_rng(146972845698875399755764481408308808739) + n_vectors = 100 + rot = Rotation.random(rng=rng) + vectors = rng.normal(size=(n_vectors, 3)) + result = rot.apply(vectors) + + # The paper adds noise as independently distributed angular errors + sigma = np.deg2rad(1) + tolerance = 1.5 * sigma + noise = Rotation.from_rotvec( + rng.normal( + size=(n_vectors, 3), + scale=sigma + ) + ) + + # Attitude errors must preserve norm. Hence apply individual random + # rotations to each vector. + noisy_result = noise.apply(result) + + est, rssd, cov = Rotation.align_vectors(noisy_result, vectors, + return_sensitivity=True) + + # Use rotation compositions to find out closeness + error_vector = (rot * est.inv()).as_rotvec() + assert_allclose(error_vector[0], 0, atol=tolerance) + assert_allclose(error_vector[1], 0, atol=tolerance) + assert_allclose(error_vector[2], 0, atol=tolerance) + + # Check error bounds using covariance matrix + cov *= sigma + assert_allclose(cov[0, 0], 0, atol=tolerance) + assert_allclose(cov[1, 1], 0, atol=tolerance) + assert_allclose(cov[2, 2], 0, atol=tolerance) + + assert_allclose(rssd, np.sum((noisy_result - est.apply(vectors))**2)**0.5) + + +def test_align_vectors_invalid_input(): + with pytest.raises(ValueError, match="Expected input `a` to have shape"): + Rotation.align_vectors([1, 2, 3, 4], [1, 2, 3]) + + with pytest.raises(ValueError, match="Expected input `b` to have shape"): + Rotation.align_vectors([1, 2, 3], [1, 2, 3, 4]) + + with pytest.raises(ValueError, match="Expected inputs `a` and `b` " + "to have same shapes"): + Rotation.align_vectors([[1, 2, 3],[4, 5, 6]], [[1, 2, 3]]) + + with pytest.raises(ValueError, + match="Expected `weights` to be 1 dimensional"): + Rotation.align_vectors([[1, 2, 3]], [[1, 2, 3]], weights=[[1]]) + + with pytest.raises(ValueError, + match="Expected `weights` to have number of values"): + Rotation.align_vectors([[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]], + weights=[1, 2, 3]) + + with pytest.raises(ValueError, + match="`weights` may not contain negative values"): + Rotation.align_vectors([[1, 2, 3]], [[1, 2, 3]], weights=[-1]) + + with pytest.raises(ValueError, + match="Only one infinite weight is allowed"): + Rotation.align_vectors([[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]], + weights=[np.inf, np.inf]) + + with pytest.raises(ValueError, + match="Cannot align zero length primary vectors"): + Rotation.align_vectors([[0, 0, 0]], [[1, 2, 3]]) + + with pytest.raises(ValueError, + match="Cannot return sensitivity matrix"): + Rotation.align_vectors([[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]], + return_sensitivity=True, weights=[np.inf, 1]) + + with pytest.raises(ValueError, + match="Cannot return sensitivity matrix"): + Rotation.align_vectors([[1, 2, 3]], [[1, 2, 3]], + return_sensitivity=True) + + +def test_align_vectors_align_constrain(): + # Align the primary +X B axis with the primary +Y A axis, and rotate about + # it such that the +Y B axis (residual of the [1, 1, 0] secondary b vector) + # is aligned with the +Z A axis (residual of the [0, 1, 1] secondary a + # vector) + atol = 1e-12 + b = [[1, 0, 0], [1, 1, 0]] + a = [[0, 1, 0], [0, 1, 1]] + m_expected = np.array([[0, 0, 1], + [1, 0, 0], + [0, 1, 0]]) + R, rssd = Rotation.align_vectors(a, b, weights=[np.inf, 1]) + assert_allclose(R.as_matrix(), m_expected, atol=atol) + assert_allclose(R.apply(b), a, atol=atol) # Pri and sec align exactly + assert np.isclose(rssd, 0, atol=atol) + + # Do the same but with an inexact secondary rotation + b = [[1, 0, 0], [1, 2, 0]] + rssd_expected = 1.0 + R, rssd = Rotation.align_vectors(a, b, weights=[np.inf, 1]) + assert_allclose(R.as_matrix(), m_expected, atol=atol) + assert_allclose(R.apply(b)[0], a[0], atol=atol) # Only pri aligns exactly + assert np.isclose(rssd, rssd_expected, atol=atol) + a_expected = [[0, 1, 0], [0, 1, 2]] + assert_allclose(R.apply(b), a_expected, atol=atol) + + # Check random vectors + b = [[1, 2, 3], [-2, 3, -1]] + a = [[-1, 3, 2], [1, -1, 2]] + rssd_expected = 1.3101595297515016 + R, rssd = Rotation.align_vectors(a, b, weights=[np.inf, 1]) + assert_allclose(R.apply(b)[0], a[0], atol=atol) # Only pri aligns exactly + assert np.isclose(rssd, rssd_expected, atol=atol) + + +def test_align_vectors_near_inf(): + # align_vectors should return near the same result for high weights as for + # infinite weights. rssd will be different with floating point error on the + # exactly aligned vector being multiplied by a large non-infinite weight + n = 100 + mats = [] + for i in range(6): + mats.append(Rotation.random(n, rng=10 + i).as_matrix()) + + for i in range(n): + # Get random pairs of 3-element vectors + a = [1*mats[0][i][0], 2*mats[1][i][0]] + b = [3*mats[2][i][0], 4*mats[3][i][0]] + + R, _ = Rotation.align_vectors(a, b, weights=[1e10, 1]) + R2, _ = Rotation.align_vectors(a, b, weights=[np.inf, 1]) + assert_allclose(R.as_matrix(), R2.as_matrix(), atol=1e-4) + + for i in range(n): + # Get random triplets of 3-element vectors + a = [1*mats[0][i][0], 2*mats[1][i][0], 3*mats[2][i][0]] + b = [4*mats[3][i][0], 5*mats[4][i][0], 6*mats[5][i][0]] + + R, _ = Rotation.align_vectors(a, b, weights=[1e10, 2, 1]) + R2, _ = Rotation.align_vectors(a, b, weights=[np.inf, 2, 1]) + assert_allclose(R.as_matrix(), R2.as_matrix(), atol=1e-4) + + +def test_align_vectors_parallel(): + atol = 1e-12 + a = [[1, 0, 0], [0, 1, 0]] + b = [[0, 1, 0], [0, 1, 0]] + m_expected = np.array([[0, 1, 0], + [-1, 0, 0], + [0, 0, 1]]) + R, _ = Rotation.align_vectors(a, b, weights=[np.inf, 1]) + assert_allclose(R.as_matrix(), m_expected, atol=atol) + R, _ = Rotation.align_vectors(a[0], b[0]) + assert_allclose(R.as_matrix(), m_expected, atol=atol) + assert_allclose(R.apply(b[0]), a[0], atol=atol) + + b = [[1, 0, 0], [1, 0, 0]] + m_expected = np.array([[1, 0, 0], + [0, 1, 0], + [0, 0, 1]]) + R, _ = Rotation.align_vectors(a, b, weights=[np.inf, 1]) + assert_allclose(R.as_matrix(), m_expected, atol=atol) + R, _ = Rotation.align_vectors(a[0], b[0]) + assert_allclose(R.as_matrix(), m_expected, atol=atol) + assert_allclose(R.apply(b[0]), a[0], atol=atol) + + +def test_align_vectors_antiparallel(): + # Test exact 180 deg rotation + atol = 1e-12 + as_to_test = np.array([[[1, 0, 0], [0, 1, 0]], + [[0, 1, 0], [1, 0, 0]], + [[0, 0, 1], [0, 1, 0]]]) + bs_to_test = [[-a[0], a[1]] for a in as_to_test] + for a, b in zip(as_to_test, bs_to_test): + R, _ = Rotation.align_vectors(a, b, weights=[np.inf, 1]) + assert_allclose(R.magnitude(), np.pi, atol=atol) + assert_allclose(R.apply(b[0]), a[0], atol=atol) + + # Test exact rotations near 180 deg + Rs = Rotation.random(100, rng=0) + dRs = Rotation.from_rotvec(Rs.as_rotvec()*1e-4) # scale down to small angle + a = [[ 1, 0, 0], [0, 1, 0]] + b = [[-1, 0, 0], [0, 1, 0]] + as_to_test = [] + for dR in dRs: + as_to_test.append([dR.apply(a[0]), a[1]]) + for a in as_to_test: + R, _ = Rotation.align_vectors(a, b, weights=[np.inf, 1]) + R2, _ = Rotation.align_vectors(a, b, weights=[1e10, 1]) + assert_allclose(R.as_matrix(), R2.as_matrix(), atol=atol) + + +def test_align_vectors_primary_only(): + atol = 1e-12 + mats_a = Rotation.random(100, rng=0).as_matrix() + mats_b = Rotation.random(100, rng=1).as_matrix() + for mat_a, mat_b in zip(mats_a, mats_b): + # Get random 3-element unit vectors + a = mat_a[0] + b = mat_b[0] + + # Compare to align_vectors with primary only + R, rssd = Rotation.align_vectors(a, b) + assert_allclose(R.apply(b), a, atol=atol) + assert np.isclose(rssd, 0, atol=atol) + + +def test_slerp(): + rnd = np.random.RandomState(0) + + key_rots = Rotation.from_quat(rnd.uniform(size=(5, 4))) + key_quats = key_rots.as_quat() + + key_times = [0, 1, 2, 3, 4] + interpolator = Slerp(key_times, key_rots) + + times = [0, 0.5, 0.25, 1, 1.5, 2, 2.75, 3, 3.25, 3.60, 4] + interp_rots = interpolator(times) + interp_quats = interp_rots.as_quat() + + # Dot products are affected by sign of quaternions + interp_quats[interp_quats[:, -1] < 0] *= -1 + # Checking for quaternion equality, perform same operation + key_quats[key_quats[:, -1] < 0] *= -1 + + # Equality at keyframes, including both endpoints + assert_allclose(interp_quats[0], key_quats[0]) + assert_allclose(interp_quats[3], key_quats[1]) + assert_allclose(interp_quats[5], key_quats[2]) + assert_allclose(interp_quats[7], key_quats[3]) + assert_allclose(interp_quats[10], key_quats[4]) + + # Constant angular velocity between keyframes. Check by equating + # cos(theta) between quaternion pairs with equal time difference. + cos_theta1 = np.sum(interp_quats[0] * interp_quats[2]) + cos_theta2 = np.sum(interp_quats[2] * interp_quats[1]) + assert_allclose(cos_theta1, cos_theta2) + + cos_theta4 = np.sum(interp_quats[3] * interp_quats[4]) + cos_theta5 = np.sum(interp_quats[4] * interp_quats[5]) + assert_allclose(cos_theta4, cos_theta5) + + # theta1: 0 -> 0.25, theta3 : 0.5 -> 1 + # Use double angle formula for double the time difference + cos_theta3 = np.sum(interp_quats[1] * interp_quats[3]) + assert_allclose(cos_theta3, 2 * (cos_theta1**2) - 1) + + # Miscellaneous checks + assert_equal(len(interp_rots), len(times)) + + +def test_slerp_rot_is_rotation(): + with pytest.raises(TypeError, match="must be a `Rotation` instance"): + r = np.array([[1,2,3,4], + [0,0,0,1]]) + t = np.array([0, 1]) + Slerp(t, r) + +SLERP_EXCEPTION_MESSAGE = "must be a sequence of at least 2 rotations" + +def test_slerp_single_rot(): + r = Rotation.from_quat([1, 2, 3, 4]) + with pytest.raises(ValueError, match=SLERP_EXCEPTION_MESSAGE): + Slerp([1], r) + + +def test_slerp_rot_len0(): + r = Rotation.random() + with pytest.raises(ValueError, match=SLERP_EXCEPTION_MESSAGE): + Slerp([], r) + + +def test_slerp_rot_len1(): + r = Rotation.random(1) + with pytest.raises(ValueError, match=SLERP_EXCEPTION_MESSAGE): + Slerp([1], r) + + +def test_slerp_time_dim_mismatch(): + with pytest.raises(ValueError, + match="times to be specified in a 1 dimensional array"): + rnd = np.random.RandomState(0) + r = Rotation.from_quat(rnd.uniform(size=(2, 4))) + t = np.array([[1], + [2]]) + Slerp(t, r) + + +def test_slerp_num_rotations_mismatch(): + with pytest.raises(ValueError, match="number of rotations to be equal to " + "number of timestamps"): + rnd = np.random.RandomState(0) + r = Rotation.from_quat(rnd.uniform(size=(5, 4))) + t = np.arange(7) + Slerp(t, r) + + +def test_slerp_equal_times(): + with pytest.raises(ValueError, match="strictly increasing order"): + rnd = np.random.RandomState(0) + r = Rotation.from_quat(rnd.uniform(size=(5, 4))) + t = [0, 1, 2, 2, 4] + Slerp(t, r) + + +def test_slerp_decreasing_times(): + with pytest.raises(ValueError, match="strictly increasing order"): + rnd = np.random.RandomState(0) + r = Rotation.from_quat(rnd.uniform(size=(5, 4))) + t = [0, 1, 3, 2, 4] + Slerp(t, r) + + +def test_slerp_call_time_dim_mismatch(): + rnd = np.random.RandomState(0) + r = Rotation.from_quat(rnd.uniform(size=(5, 4))) + t = np.arange(5) + s = Slerp(t, r) + + with pytest.raises(ValueError, + match="`times` must be at most 1-dimensional."): + interp_times = np.array([[3.5], + [4.2]]) + s(interp_times) + + +def test_slerp_call_time_out_of_range(): + rnd = np.random.RandomState(0) + r = Rotation.from_quat(rnd.uniform(size=(5, 4))) + t = np.arange(5) + 1 + s = Slerp(t, r) + + with pytest.raises(ValueError, match="times must be within the range"): + s([0, 1, 2]) + with pytest.raises(ValueError, match="times must be within the range"): + s([1, 2, 6]) + + +def test_slerp_call_scalar_time(): + r = Rotation.from_euler('X', [0, 80], degrees=True) + s = Slerp([0, 1], r) + + r_interpolated = s(0.25) + r_interpolated_expected = Rotation.from_euler('X', 20, degrees=True) + + delta = r_interpolated * r_interpolated_expected.inv() + + assert_allclose(delta.magnitude(), 0, atol=1e-16) + + +def test_multiplication_stability(): + qs = Rotation.random(50, rng=0) + rs = Rotation.random(1000, rng=1) + for q in qs: + rs *= q * rs + assert_allclose(np.linalg.norm(rs.as_quat(), axis=1), 1) + + +def test_pow(): + atol = 1e-14 + p = Rotation.random(10, rng=0) + p_inv = p.inv() + # Test the short-cuts and other integers + for n in [-5, -2, -1, 0, 1, 2, 5]: + # Test accuracy + q = p ** n + r = Rotation.identity(10) + for _ in range(abs(n)): + if n > 0: + r = r * p + else: + r = r * p_inv + ang = (q * r.inv()).magnitude() + assert np.all(ang < atol) + + # Test shape preservation + r = Rotation.from_quat([0, 0, 0, 1]) + assert (r**n).as_quat().shape == (4,) + r = Rotation.from_quat([[0, 0, 0, 1]]) + assert (r**n).as_quat().shape == (1, 4) + + # Large angle fractional + for n in [-1.5, -0.5, -0.0, 0.0, 0.5, 1.5]: + q = p ** n + r = Rotation.from_rotvec(n * p.as_rotvec()) + assert_allclose(q.as_quat(), r.as_quat(), atol=atol) + + # Small angle + p = Rotation.from_rotvec([1e-12, 0, 0]) + n = 3 + q = p ** n + r = Rotation.from_rotvec(n * p.as_rotvec()) + assert_allclose(q.as_quat(), r.as_quat(), atol=atol) + + +def test_pow_errors(): + p = Rotation.random(rng=0) + with pytest.raises(NotImplementedError, match='modulus not supported'): + pow(p, 1, 1) + + +def test_rotation_within_numpy_array(): + single = Rotation.random(rng=0) + multiple = Rotation.random(2, rng=1) + + array = np.array(single) + assert_equal(array.shape, ()) + + array = np.array(multiple) + assert_equal(array.shape, (2,)) + assert_allclose(array[0].as_matrix(), multiple[0].as_matrix()) + assert_allclose(array[1].as_matrix(), multiple[1].as_matrix()) + + array = np.array([single]) + assert_equal(array.shape, (1,)) + assert_equal(array[0], single) + + array = np.array([multiple]) + assert_equal(array.shape, (1, 2)) + assert_allclose(array[0, 0].as_matrix(), multiple[0].as_matrix()) + assert_allclose(array[0, 1].as_matrix(), multiple[1].as_matrix()) + + array = np.array([single, multiple], dtype=object) + assert_equal(array.shape, (2,)) + assert_equal(array[0], single) + assert_equal(array[1], multiple) + + array = np.array([multiple, multiple, multiple]) + assert_equal(array.shape, (3, 2)) + + +def test_pickling(): + r = Rotation.from_quat([0, 0, np.sin(np.pi/4), np.cos(np.pi/4)]) + pkl = pickle.dumps(r) + unpickled = pickle.loads(pkl) + assert_allclose(r.as_matrix(), unpickled.as_matrix(), atol=1e-15) + + +def test_deepcopy(): + r = Rotation.from_quat([0, 0, np.sin(np.pi/4), np.cos(np.pi/4)]) + r1 = copy.deepcopy(r) + assert_allclose(r.as_matrix(), r1.as_matrix(), atol=1e-15) + + +def test_as_euler_contiguous(): + r = Rotation.from_quat([0, 0, 0, 1]) + e1 = r.as_euler('xyz') # extrinsic euler rotation + e2 = r.as_euler('XYZ') # intrinsic + assert e1.flags['C_CONTIGUOUS'] is True + assert e2.flags['C_CONTIGUOUS'] is True + assert all(i >= 0 for i in e1.strides) + assert all(i >= 0 for i in e2.strides) + + +def test_concatenate(): + rotation = Rotation.random(10, rng=0) + sizes = [1, 2, 3, 1, 3] + starts = [0] + list(np.cumsum(sizes)) + split = [rotation[i:i + n] for i, n in zip(starts, sizes)] + result = Rotation.concatenate(split) + assert_equal(rotation.as_quat(), result.as_quat()) + + # Test Rotation input for multiple rotations + result = Rotation.concatenate(rotation) + assert_equal(rotation.as_quat(), result.as_quat()) + + # Test that a copy is returned + assert rotation is not result + + # Test Rotation input for single rotations + result = Rotation.concatenate(Rotation.identity()) + assert_equal(Rotation.identity().as_quat(), result.as_quat()) + + +def test_concatenate_wrong_type(): + with pytest.raises(TypeError, match='Rotation objects only'): + Rotation.concatenate([Rotation.identity(), 1, None]) + + +# Regression test for gh-16663 +def test_len_and_bool(): + rotation_multi_one = Rotation([[0, 0, 0, 1]]) + rotation_multi = Rotation([[0, 0, 0, 1], [0, 0, 0, 1]]) + rotation_single = Rotation([0, 0, 0, 1]) + + assert len(rotation_multi_one) == 1 + assert len(rotation_multi) == 2 + with pytest.raises(TypeError, match="Single rotation has no len()."): + len(rotation_single) + + # Rotation should always be truthy. See gh-16663 + assert rotation_multi_one + assert rotation_multi + assert rotation_single + + +def test_from_davenport_single_rotation(): + axis = [0, 0, 1] + quat = Rotation.from_davenport(axis, 'extrinsic', 90, + degrees=True).as_quat() + expected_quat = np.array([0, 0, 1, 1]) / np.sqrt(2) + assert_allclose(quat, expected_quat) + + +def test_from_davenport_one_or_two_axes(): + ez = [0, 0, 1] + ey = [0, 1, 0] + + # Single rotation, single axis, axes.shape == (3, ) + rot = Rotation.from_rotvec(np.array(ez) * np.pi/4) + rot_dav = Rotation.from_davenport(ez, 'e', np.pi/4) + assert_allclose(rot.as_quat(canonical=True), + rot_dav.as_quat(canonical=True)) + + # Single rotation, single axis, axes.shape == (1, 3) + rot = Rotation.from_rotvec([np.array(ez) * np.pi/4]) + rot_dav = Rotation.from_davenport([ez], 'e', [np.pi/4]) + assert_allclose(rot.as_quat(canonical=True), + rot_dav.as_quat(canonical=True)) + + # Single rotation, two axes, axes.shape == (2, 3) + rot = Rotation.from_rotvec([np.array(ez) * np.pi/4, + np.array(ey) * np.pi/6]) + rot = rot[0] * rot[1] + rot_dav = Rotation.from_davenport([ey, ez], 'e', [np.pi/6, np.pi/4]) + assert_allclose(rot.as_quat(canonical=True), + rot_dav.as_quat(canonical=True)) + + # Two rotations, single axis, axes.shape == (3, ) + rot = Rotation.from_rotvec([np.array(ez) * np.pi/6, + np.array(ez) * np.pi/4]) + rot_dav = Rotation.from_davenport([ez], 'e', [np.pi/6, np.pi/4]) + assert_allclose(rot.as_quat(canonical=True), + rot_dav.as_quat(canonical=True)) + + +def test_from_davenport_invalid_input(): + ez = [0, 0, 1] + ey = [0, 1, 0] + ezy = [0, 1, 1] + with pytest.raises(ValueError, match="must be orthogonal"): + Rotation.from_davenport([ez, ezy], 'e', [0, 0]) + with pytest.raises(ValueError, match="must be orthogonal"): + Rotation.from_davenport([ez, ey, ezy], 'e', [0, 0, 0]) + with pytest.raises(ValueError, match="order should be"): + Rotation.from_davenport([ez], 'xyz', [0]) + with pytest.raises(ValueError, match="Expected `angles`"): + Rotation.from_davenport([ez, ey, ez], 'e', [0, 1, 2, 3]) + + +def test_as_davenport(): + rnd = np.random.RandomState(0) + n = 100 + angles = np.empty((n, 3)) + angles[:, 0] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + angles_middle = rnd.uniform(low=0, high=np.pi, size=(n,)) + angles[:, 2] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + lambdas = rnd.uniform(low=0, high=np.pi, size=(20,)) + + e1 = np.array([1, 0, 0]) + e2 = np.array([0, 1, 0]) + + for lamb in lambdas: + ax_lamb = [e1, e2, Rotation.from_rotvec(lamb*e2).apply(e1)] + angles[:, 1] = angles_middle - lamb + for order in ['extrinsic', 'intrinsic']: + ax = ax_lamb if order == 'intrinsic' else ax_lamb[::-1] + rot = Rotation.from_davenport(ax, order, angles) + angles_dav = rot.as_davenport(ax, order) + assert_allclose(angles_dav, angles) + + +@pytest.mark.thread_unsafe +def test_as_davenport_degenerate(): + # Since we cannot check for angle equality, we check for rotation matrix + # equality + rnd = np.random.RandomState(0) + n = 5 + angles = np.empty((n, 3)) + + # symmetric sequences + angles[:, 0] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + angles_middle = [rnd.choice([0, np.pi]) for i in range(n)] + angles[:, 2] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + lambdas = rnd.uniform(low=0, high=np.pi, size=(5,)) + + e1 = np.array([1, 0, 0]) + e2 = np.array([0, 1, 0]) + + for lamb in lambdas: + ax_lamb = [e1, e2, Rotation.from_rotvec(lamb*e2).apply(e1)] + angles[:, 1] = angles_middle - lamb + for order in ['extrinsic', 'intrinsic']: + ax = ax_lamb if order == 'intrinsic' else ax_lamb[::-1] + rot = Rotation.from_davenport(ax, order, angles) + with pytest.warns(UserWarning, match="Gimbal lock"): + angles_dav = rot.as_davenport(ax, order) + mat_expected = rot.as_matrix() + mat_estimated = Rotation.from_davenport(ax, order, angles_dav).as_matrix() + assert_array_almost_equal(mat_expected, mat_estimated) + + +def test_compare_from_davenport_from_euler(): + rnd = np.random.RandomState(0) + n = 100 + angles = np.empty((n, 3)) + + # symmetric sequences + angles[:, 0] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + angles[:, 1] = rnd.uniform(low=0, high=np.pi, size=(n,)) + angles[:, 2] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + for order in ['extrinsic', 'intrinsic']: + for seq_tuple in permutations('xyz'): + seq = ''.join([seq_tuple[0], seq_tuple[1], seq_tuple[0]]) + ax = [basis_vec(i) for i in seq] + if order == 'intrinsic': + seq = seq.upper() + eul = Rotation.from_euler(seq, angles) + dav = Rotation.from_davenport(ax, order, angles) + assert_allclose(eul.as_quat(canonical=True), dav.as_quat(canonical=True), + rtol=1e-12) + + # asymmetric sequences + angles[:, 1] -= np.pi / 2 + for order in ['extrinsic', 'intrinsic']: + for seq_tuple in permutations('xyz'): + seq = ''.join(seq_tuple) + ax = [basis_vec(i) for i in seq] + if order == 'intrinsic': + seq = seq.upper() + eul = Rotation.from_euler(seq, angles) + dav = Rotation.from_davenport(ax, order, angles) + assert_allclose(eul.as_quat(), dav.as_quat(), rtol=1e-12) + + +def test_compare_as_davenport_as_euler(): + rnd = np.random.RandomState(0) + n = 100 + angles = np.empty((n, 3)) + + # symmetric sequences + angles[:, 0] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + angles[:, 1] = rnd.uniform(low=0, high=np.pi, size=(n,)) + angles[:, 2] = rnd.uniform(low=-np.pi, high=np.pi, size=(n,)) + for order in ['extrinsic', 'intrinsic']: + for seq_tuple in permutations('xyz'): + seq = ''.join([seq_tuple[0], seq_tuple[1], seq_tuple[0]]) + ax = [basis_vec(i) for i in seq] + if order == 'intrinsic': + seq = seq.upper() + rot = Rotation.from_euler(seq, angles) + eul = rot.as_euler(seq) + dav = rot.as_davenport(ax, order) + assert_allclose(eul, dav, rtol=1e-12) + + # asymmetric sequences + angles[:, 1] -= np.pi / 2 + for order in ['extrinsic', 'intrinsic']: + for seq_tuple in permutations('xyz'): + seq = ''.join(seq_tuple) + ax = [basis_vec(i) for i in seq] + if order == 'intrinsic': + seq = seq.upper() + rot = Rotation.from_euler(seq, angles) + eul = rot.as_euler(seq) + dav = rot.as_davenport(ax, order) + assert_allclose(eul, dav, rtol=1e-12) + + +def test_zero_rotation_construction(): + r = Rotation.random(num=0) + assert len(r) == 0 + + r_ide = Rotation.identity(num=0) + assert len(r_ide) == 0 + + r_get = Rotation.random(num=3)[[]] + assert len(r_get) == 0 + + r_quat = Rotation.from_quat(np.zeros((0, 4))) + assert len(r_quat) == 0 + + r_matrix = Rotation.from_matrix(np.zeros((0, 3, 3))) + assert len(r_matrix) == 0 + + r_euler = Rotation.from_euler("xyz", np.zeros((0, 3))) + assert len(r_euler) == 0 + + r_vec = Rotation.from_rotvec(np.zeros((0, 3))) + assert len(r_vec) == 0 + + r_dav = Rotation.from_davenport(np.eye(3), "extrinsic", np.zeros((0, 3))) + assert len(r_dav) == 0 + + r_mrp = Rotation.from_mrp(np.zeros((0, 3))) + assert len(r_mrp) == 0 + + +def test_zero_rotation_representation(): + r = Rotation.random(num=0) + assert r.as_quat().shape == (0, 4) + assert r.as_matrix().shape == (0, 3, 3) + assert r.as_euler("xyz").shape == (0, 3) + assert r.as_rotvec().shape == (0, 3) + assert r.as_mrp().shape == (0, 3) + assert r.as_davenport(np.eye(3), "extrinsic").shape == (0, 3) + + +def test_zero_rotation_array_rotation(): + r = Rotation.random(num=0) + + v = np.array([1, 2, 3]) + v_rotated = r.apply(v) + assert v_rotated.shape == (0, 3) + + v0 = np.zeros((0, 3)) + v0_rot = r.apply(v0) + assert v0_rot.shape == (0, 3) + + v2 = np.ones((2, 3)) + with pytest.raises( + ValueError, match="Expected equal numbers of rotations and vectors"): + r.apply(v2) + + +def test_zero_rotation_multiplication(): + r = Rotation.random(num=0) + + r_single = Rotation.random() + r_mult_left = r * r_single + assert len(r_mult_left) == 0 + + r_mult_right = r_single * r + assert len(r_mult_right) == 0 + + r0 = Rotation.random(0) + r_mult = r * r0 + assert len(r_mult) == 0 + + msg_rotation_error = "Expected equal number of rotations" + r2 = Rotation.random(2) + with pytest.raises(ValueError, match=msg_rotation_error): + r0 * r2 + + with pytest.raises(ValueError, match=msg_rotation_error): + r2 * r0 + + +def test_zero_rotation_concatentation(): + r = Rotation.random(num=0) + + r0 = Rotation.concatenate([r, r]) + assert len(r0) == 0 + + r1 = r.concatenate([Rotation.random(), r]) + assert len(r1) == 1 + + r3 = r.concatenate([Rotation.random(3), r]) + assert len(r3) == 3 + + r4 = r.concatenate([r, Rotation.random(4)]) + assert len(r4) == 4 + + +def test_zero_rotation_power(): + r = Rotation.random(num=0) + for pp in [-1.5, -1, 0, 1, 1.5]: + pow0 = r**pp + assert len(pow0) == 0 + + +def test_zero_rotation_inverse(): + r = Rotation.random(num=0) + r_inv = r.inv() + assert len(r_inv) == 0 + + +def test_zero_rotation_magnitude(): + r = Rotation.random(num=0) + magnitude = r.magnitude() + assert magnitude.shape == (0,) + + +def test_zero_rotation_mean(): + r = Rotation.random(num=0) + with pytest.raises(ValueError, match="Mean of an empty rotation set is undefined."): + r.mean() + + +def test_zero_rotation_approx_equal(): + r = Rotation.random(0) + assert r.approx_equal(Rotation.random(0)).shape == (0,) + assert r.approx_equal(Rotation.random()).shape == (0,) + assert Rotation.random().approx_equal(r).shape == (0,) + + approx_msg = "Expected equal number of rotations" + r3 = Rotation.random(2) + with pytest.raises(ValueError, match=approx_msg): + r.approx_equal(r3) + + with pytest.raises(ValueError, match=approx_msg): + r3.approx_equal(r) + + +def test_zero_rotation_get_set(): + r = Rotation.random(0) + + r_get = r[[]] + assert len(r_get) == 0 + + r_slice = r[:0] + assert len(r_slice) == 0 + + with pytest.raises(IndexError): + r[[0]] + + with pytest.raises(IndexError): + r[[True]] + + with pytest.raises(IndexError): + r[0] = Rotation.random() + + +def test_boolean_indexes(): + r = Rotation.random(3) + + r0 = r[[False, False, False]] + assert len(r0) == 0 + + r1 = r[[False, True, False]] + assert len(r1) == 1 + + r3 = r[[True, True, True]] + assert len(r3) == 3 + + with pytest.raises(IndexError): + r[[True, True]] diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/test_rotation_groups.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/test_rotation_groups.py new file mode 100644 index 0000000000000000000000000000000000000000..befe60c13c5a5863a2ae50f9d20e6d054795f6b9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/test_rotation_groups.py @@ -0,0 +1,169 @@ +import pytest + +import numpy as np +from numpy.testing import assert_array_almost_equal +from scipy.spatial.transform import Rotation +from scipy.optimize import linear_sum_assignment +from scipy.spatial.distance import cdist +from scipy.constants import golden as phi +from scipy.spatial import cKDTree + + +TOL = 1E-12 +NS = range(1, 13) +NAMES = ["I", "O", "T"] + ["C%d" % n for n in NS] + ["D%d" % n for n in NS] +SIZES = [60, 24, 12] + list(NS) + [2 * n for n in NS] + + +def _calculate_rmsd(P, Q): + """Calculates the root-mean-square distance between the points of P and Q. + The distance is taken as the minimum over all possible matchings. It is + zero if P and Q are identical and non-zero if not. + """ + distance_matrix = cdist(P, Q, metric='sqeuclidean') + matching = linear_sum_assignment(distance_matrix) + return np.sqrt(distance_matrix[matching].sum()) + + +def _generate_pyramid(n, axis): + thetas = np.linspace(0, 2 * np.pi, n + 1)[:-1] + P = np.vstack([np.zeros(n), np.cos(thetas), np.sin(thetas)]).T + P = np.concatenate((P, [[1, 0, 0]])) + return np.roll(P, axis, axis=1) + + +def _generate_prism(n, axis): + thetas = np.linspace(0, 2 * np.pi, n + 1)[:-1] + bottom = np.vstack([-np.ones(n), np.cos(thetas), np.sin(thetas)]).T + top = np.vstack([+np.ones(n), np.cos(thetas), np.sin(thetas)]).T + P = np.concatenate((bottom, top)) + return np.roll(P, axis, axis=1) + + +def _generate_icosahedron(): + x = np.array([[0, -1, -phi], + [0, -1, +phi], + [0, +1, -phi], + [0, +1, +phi]]) + return np.concatenate([np.roll(x, i, axis=1) for i in range(3)]) + + +def _generate_octahedron(): + return np.array([[-1, 0, 0], [+1, 0, 0], [0, -1, 0], + [0, +1, 0], [0, 0, -1], [0, 0, +1]]) + + +def _generate_tetrahedron(): + return np.array([[1, 1, 1], [1, -1, -1], [-1, 1, -1], [-1, -1, 1]]) + + +@pytest.mark.parametrize("name", [-1, None, True, np.array(['C3'])]) +def test_group_type(name): + with pytest.raises(ValueError, + match="must be a string"): + Rotation.create_group(name) + + +@pytest.mark.parametrize("name", ["Q", " ", "CA", "C ", "DA", "D ", "I2", ""]) +def test_group_name(name): + with pytest.raises(ValueError, + match="must be one of 'I', 'O', 'T', 'Dn', 'Cn'"): + Rotation.create_group(name) + + +@pytest.mark.parametrize("name", ["C0", "D0"]) +def test_group_order_positive(name): + with pytest.raises(ValueError, + match="Group order must be positive"): + Rotation.create_group(name) + + +@pytest.mark.parametrize("axis", ['A', 'b', 0, 1, 2, 4, False, None]) +def test_axis_valid(axis): + with pytest.raises(ValueError, + match="`axis` must be one of"): + Rotation.create_group("C1", axis) + + +def test_icosahedral(): + """The icosahedral group fixes the rotations of an icosahedron. Here we + test that the icosahedron is invariant after application of the elements + of the rotation group.""" + P = _generate_icosahedron() + for g in Rotation.create_group("I"): + g = Rotation.from_quat(g.as_quat()) + assert _calculate_rmsd(P, g.apply(P)) < TOL + + +def test_octahedral(): + """Test that the octahedral group correctly fixes the rotations of an + octahedron.""" + P = _generate_octahedron() + for g in Rotation.create_group("O"): + assert _calculate_rmsd(P, g.apply(P)) < TOL + + +def test_tetrahedral(): + """Test that the tetrahedral group correctly fixes the rotations of a + tetrahedron.""" + P = _generate_tetrahedron() + for g in Rotation.create_group("T"): + assert _calculate_rmsd(P, g.apply(P)) < TOL + + +@pytest.mark.parametrize("n", NS) +@pytest.mark.parametrize("axis", 'XYZ') +def test_dicyclic(n, axis): + """Test that the dicyclic group correctly fixes the rotations of a + prism.""" + P = _generate_prism(n, axis='XYZ'.index(axis)) + for g in Rotation.create_group("D%d" % n, axis=axis): + assert _calculate_rmsd(P, g.apply(P)) < TOL + + +@pytest.mark.parametrize("n", NS) +@pytest.mark.parametrize("axis", 'XYZ') +def test_cyclic(n, axis): + """Test that the cyclic group correctly fixes the rotations of a + pyramid.""" + P = _generate_pyramid(n, axis='XYZ'.index(axis)) + for g in Rotation.create_group("C%d" % n, axis=axis): + assert _calculate_rmsd(P, g.apply(P)) < TOL + + +@pytest.mark.parametrize("name, size", zip(NAMES, SIZES)) +def test_group_sizes(name, size): + assert len(Rotation.create_group(name)) == size + + +@pytest.mark.parametrize("name, size", zip(NAMES, SIZES)) +def test_group_no_duplicates(name, size): + g = Rotation.create_group(name) + kdtree = cKDTree(g.as_quat()) + assert len(kdtree.query_pairs(1E-3)) == 0 + + +@pytest.mark.parametrize("name, size", zip(NAMES, SIZES)) +def test_group_symmetry(name, size): + g = Rotation.create_group(name) + q = np.concatenate((-g.as_quat(), g.as_quat())) + distance = np.sort(cdist(q, q)) + deltas = np.max(distance, axis=0) - np.min(distance, axis=0) + assert (deltas < TOL).all() + + +@pytest.mark.parametrize("name", NAMES) +def test_reduction(name): + """Test that the elements of the rotation group are correctly + mapped onto the identity rotation.""" + g = Rotation.create_group(name) + f = g.reduce(g) + assert_array_almost_equal(f.magnitude(), np.zeros(len(g))) + + +@pytest.mark.parametrize("name", NAMES) +def test_single_reduction(name): + g = Rotation.create_group(name) + f = g[-1].reduce(g) + assert_array_almost_equal(f.magnitude(), 0) + assert f.as_quat().shape == (4,) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/test_rotation_spline.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/test_rotation_spline.py new file mode 100644 index 0000000000000000000000000000000000000000..6441431f2fb54a95edd6364c4671bc55a2ce2b8f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/spatial/transform/tests/test_rotation_spline.py @@ -0,0 +1,162 @@ +from itertools import product +import numpy as np +from numpy.testing import assert_allclose +from pytest import raises +from scipy.spatial.transform import Rotation, RotationSpline +from scipy.spatial.transform._rotation_spline import ( + _angular_rate_to_rotvec_dot_matrix, + _rotvec_dot_to_angular_rate_matrix, + _matrix_vector_product_of_stacks, + _angular_acceleration_nonlinear_term, + _create_block_3_diagonal_matrix) + + +def test_angular_rate_to_rotvec_conversions(): + np.random.seed(0) + rv = np.random.randn(4, 3) + A = _angular_rate_to_rotvec_dot_matrix(rv) + A_inv = _rotvec_dot_to_angular_rate_matrix(rv) + + # When the rotation vector is aligned with the angular rate, then + # the rotation vector rate and angular rate are the same. + assert_allclose(_matrix_vector_product_of_stacks(A, rv), rv) + assert_allclose(_matrix_vector_product_of_stacks(A_inv, rv), rv) + + # A and A_inv must be reciprocal to each other. + I_stack = np.empty((4, 3, 3)) + I_stack[:] = np.eye(3) + assert_allclose(np.matmul(A, A_inv), I_stack, atol=1e-15) + + +def test_angular_rate_nonlinear_term(): + # The only simple test is to check that the term is zero when + # the rotation vector + np.random.seed(0) + rv = np.random.rand(4, 3) + assert_allclose(_angular_acceleration_nonlinear_term(rv, rv), 0, + atol=1e-19) + + +def test_create_block_3_diagonal_matrix(): + np.random.seed(0) + A = np.empty((4, 3, 3)) + A[:] = np.arange(1, 5)[:, None, None] + + B = np.empty((4, 3, 3)) + B[:] = -np.arange(1, 5)[:, None, None] + d = 10 * np.arange(10, 15) + + banded = _create_block_3_diagonal_matrix(A, B, d) + + # Convert the banded matrix to the full matrix. + k, l = list(zip(*product(np.arange(banded.shape[0]), + np.arange(banded.shape[1])))) + k = np.asarray(k) + l = np.asarray(l) + + i = k - 5 + l + j = l + values = banded.ravel() + mask = (i >= 0) & (i < 15) + i = i[mask] + j = j[mask] + values = values[mask] + full = np.zeros((15, 15)) + full[i, j] = values + + zero = np.zeros((3, 3)) + eye = np.eye(3) + + # Create the reference full matrix in the most straightforward manner. + ref = np.block([ + [d[0] * eye, B[0], zero, zero, zero], + [A[0], d[1] * eye, B[1], zero, zero], + [zero, A[1], d[2] * eye, B[2], zero], + [zero, zero, A[2], d[3] * eye, B[3]], + [zero, zero, zero, A[3], d[4] * eye], + ]) + + assert_allclose(full, ref, atol=1e-19) + + +def test_spline_2_rotations(): + times = [0, 10] + rotations = Rotation.from_euler('xyz', [[0, 0, 0], [10, -20, 30]], + degrees=True) + spline = RotationSpline(times, rotations) + + rv = (rotations[0].inv() * rotations[1]).as_rotvec() + rate = rv / (times[1] - times[0]) + times_check = np.array([-1, 5, 12]) + dt = times_check - times[0] + rv_ref = rate * dt[:, None] + + assert_allclose(spline(times_check).as_rotvec(), rv_ref) + assert_allclose(spline(times_check, 1), np.resize(rate, (3, 3))) + assert_allclose(spline(times_check, 2), 0, atol=1e-16) + + +def test_constant_attitude(): + times = np.arange(10) + rotations = Rotation.from_rotvec(np.ones((10, 3))) + spline = RotationSpline(times, rotations) + + times_check = np.linspace(-1, 11) + assert_allclose(spline(times_check).as_rotvec(), 1, rtol=1e-15) + assert_allclose(spline(times_check, 1), 0, atol=1e-17) + assert_allclose(spline(times_check, 2), 0, atol=1e-17) + + assert_allclose(spline(5.5).as_rotvec(), 1, rtol=1e-15) + assert_allclose(spline(5.5, 1), 0, atol=1e-17) + assert_allclose(spline(5.5, 2), 0, atol=1e-17) + + +def test_spline_properties(): + times = np.array([0, 5, 15, 27]) + angles = [[-5, 10, 27], [3, 5, 38], [-12, 10, 25], [-15, 20, 11]] + + rotations = Rotation.from_euler('xyz', angles, degrees=True) + spline = RotationSpline(times, rotations) + + assert_allclose(spline(times).as_euler('xyz', degrees=True), angles) + assert_allclose(spline(0).as_euler('xyz', degrees=True), angles[0]) + + h = 1e-8 + rv0 = spline(times).as_rotvec() + rvm = spline(times - h).as_rotvec() + rvp = spline(times + h).as_rotvec() + # rtol bumped from 1e-15 to 1.5e-15 in gh18414 for linux 32 bit + assert_allclose(rv0, 0.5 * (rvp + rvm), rtol=1.5e-15) + + r0 = spline(times, 1) + rm = spline(times - h, 1) + rp = spline(times + h, 1) + assert_allclose(r0, 0.5 * (rm + rp), rtol=1e-14) + + a0 = spline(times, 2) + am = spline(times - h, 2) + ap = spline(times + h, 2) + assert_allclose(a0, am, rtol=1e-7) + assert_allclose(a0, ap, rtol=1e-7) + + +def test_error_handling(): + raises(ValueError, RotationSpline, [1.0], Rotation.random()) + + r = Rotation.random(10) + t = np.arange(10).reshape(5, 2) + raises(ValueError, RotationSpline, t, r) + + t = np.arange(9) + raises(ValueError, RotationSpline, t, r) + + t = np.arange(10) + t[5] = 0 + raises(ValueError, RotationSpline, t, r) + + t = np.arange(10) + + s = RotationSpline(t, r) + raises(ValueError, s, 10, -1) + + raises(ValueError, s, np.arange(10).reshape(5, 2)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/__init__.pxd b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/__init__.pxd new file mode 100644 index 0000000000000000000000000000000000000000..1daa9fb379572aac4bc9b6d74330a18c5c52bf79 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/__init__.pxd @@ -0,0 +1 @@ +from scipy.special cimport cython_special diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..8993f522a0fac00e243d361835a42b89a82d11ef --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/__init__.py @@ -0,0 +1,887 @@ +""" +======================================== +Special functions (:mod:`scipy.special`) +======================================== + +.. currentmodule:: scipy.special + +Almost all of the functions below accept NumPy arrays as input +arguments as well as single numbers. This means they follow +broadcasting and automatic array-looping rules. Technically, +they are `NumPy universal functions +`_. +Functions which do not accept NumPy arrays are marked by a warning +in the section description. + +.. seealso:: + + `scipy.special.cython_special` -- Typed Cython versions of special functions + + +Error handling +============== + +Errors are handled by returning NaNs or other appropriate values. +Some of the special function routines can emit warnings or raise +exceptions when an error occurs. By default this is disabled, except +for memory allocation errors, which result in an exception being raised. +To query and control the current error handling state the following +functions are provided. + +.. autosummary:: + :toctree: generated/ + + geterr -- Get the current way of handling special-function errors. + seterr -- Set how special-function errors are handled. + errstate -- Context manager for special-function error handling. + SpecialFunctionWarning -- Warning that can be emitted by special functions. + SpecialFunctionError -- Exception that can be raised by special functions. + +Available functions +=================== + +Airy functions +-------------- + +.. autosummary:: + :toctree: generated/ + + airy -- Airy functions and their derivatives. + airye -- Exponentially scaled Airy functions and their derivatives. + ai_zeros -- Compute `nt` zeros and values of the Airy function Ai and its derivative. + bi_zeros -- Compute `nt` zeros and values of the Airy function Bi and its derivative. + itairy -- Integrals of Airy functions + + +Elliptic functions and integrals +-------------------------------- + +.. autosummary:: + :toctree: generated/ + + ellipj -- Jacobian elliptic functions. + ellipk -- Complete elliptic integral of the first kind. + ellipkm1 -- Complete elliptic integral of the first kind around `m` = 1. + ellipkinc -- Incomplete elliptic integral of the first kind. + ellipe -- Complete elliptic integral of the second kind. + ellipeinc -- Incomplete elliptic integral of the second kind. + elliprc -- Degenerate symmetric integral RC. + elliprd -- Symmetric elliptic integral of the second kind. + elliprf -- Completely-symmetric elliptic integral of the first kind. + elliprg -- Completely-symmetric elliptic integral of the second kind. + elliprj -- Symmetric elliptic integral of the third kind. + +Bessel functions +---------------- + +.. autosummary:: + :toctree: generated/ + + jv -- Bessel function of the first kind of real order and \ + complex argument. + jve -- Exponentially scaled Bessel function of order `v`. + yn -- Bessel function of the second kind of integer order and \ + real argument. + yv -- Bessel function of the second kind of real order and \ + complex argument. + yve -- Exponentially scaled Bessel function of the second kind \ + of real order. + kn -- Modified Bessel function of the second kind of integer \ + order `n` + kv -- Modified Bessel function of the second kind of real order \ + `v` + kve -- Exponentially scaled modified Bessel function of the \ + second kind. + iv -- Modified Bessel function of the first kind of real order. + ive -- Exponentially scaled modified Bessel function of the \ + first kind. + hankel1 -- Hankel function of the first kind. + hankel1e -- Exponentially scaled Hankel function of the first kind. + hankel2 -- Hankel function of the second kind. + hankel2e -- Exponentially scaled Hankel function of the second kind. + wright_bessel -- Wright's generalized Bessel function. + log_wright_bessel -- Logarithm of Wright's generalized Bessel function. + +The following function does not accept NumPy arrays (it is not a +universal function): + +.. autosummary:: + :toctree: generated/ + + lmbda -- Jahnke-Emden Lambda function, Lambdav(x). + +Zeros of Bessel functions +^^^^^^^^^^^^^^^^^^^^^^^^^ + +The following functions do not accept NumPy arrays (they are not +universal functions): + +.. autosummary:: + :toctree: generated/ + + jnjnp_zeros -- Compute zeros of integer-order Bessel functions Jn and Jn'. + jnyn_zeros -- Compute nt zeros of Bessel functions Jn(x), Jn'(x), Yn(x), and Yn'(x). + jn_zeros -- Compute zeros of integer-order Bessel function Jn(x). + jnp_zeros -- Compute zeros of integer-order Bessel function derivative Jn'(x). + yn_zeros -- Compute zeros of integer-order Bessel function Yn(x). + ynp_zeros -- Compute zeros of integer-order Bessel function derivative Yn'(x). + y0_zeros -- Compute nt zeros of Bessel function Y0(z), and derivative at each zero. + y1_zeros -- Compute nt zeros of Bessel function Y1(z), and derivative at each zero. + y1p_zeros -- Compute nt zeros of Bessel derivative Y1'(z), and value at each zero. + +Faster versions of common Bessel functions +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + j0 -- Bessel function of the first kind of order 0. + j1 -- Bessel function of the first kind of order 1. + y0 -- Bessel function of the second kind of order 0. + y1 -- Bessel function of the second kind of order 1. + i0 -- Modified Bessel function of order 0. + i0e -- Exponentially scaled modified Bessel function of order 0. + i1 -- Modified Bessel function of order 1. + i1e -- Exponentially scaled modified Bessel function of order 1. + k0 -- Modified Bessel function of the second kind of order 0, :math:`K_0`. + k0e -- Exponentially scaled modified Bessel function K of order 0 + k1 -- Modified Bessel function of the second kind of order 1, :math:`K_1(x)`. + k1e -- Exponentially scaled modified Bessel function K of order 1. + +Integrals of Bessel functions +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + itj0y0 -- Integrals of Bessel functions of order 0. + it2j0y0 -- Integrals related to Bessel functions of order 0. + iti0k0 -- Integrals of modified Bessel functions of order 0. + it2i0k0 -- Integrals related to modified Bessel functions of order 0. + besselpoly -- Weighted integral of a Bessel function. + +Derivatives of Bessel functions +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + jvp -- Compute nth derivative of Bessel function Jv(z) with respect to `z`. + yvp -- Compute nth derivative of Bessel function Yv(z) with respect to `z`. + kvp -- Compute nth derivative of real-order modified Bessel function Kv(z) + ivp -- Compute nth derivative of modified Bessel function Iv(z) with respect to `z`. + h1vp -- Compute nth derivative of Hankel function H1v(z) with respect to `z`. + h2vp -- Compute nth derivative of Hankel function H2v(z) with respect to `z`. + +Spherical Bessel functions +^^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + spherical_jn -- Spherical Bessel function of the first kind or its derivative. + spherical_yn -- Spherical Bessel function of the second kind or its derivative. + spherical_in -- Modified spherical Bessel function of the first kind or its derivative. + spherical_kn -- Modified spherical Bessel function of the second kind or its derivative. + +Riccati-Bessel functions +^^^^^^^^^^^^^^^^^^^^^^^^ + +The following functions do not accept NumPy arrays (they are not +universal functions): + +.. autosummary:: + :toctree: generated/ + + riccati_jn -- Compute Ricatti-Bessel function of the first kind and its derivative. + riccati_yn -- Compute Ricatti-Bessel function of the second kind and its derivative. + +Struve functions +---------------- + +.. autosummary:: + :toctree: generated/ + + struve -- Struve function. + modstruve -- Modified Struve function. + itstruve0 -- Integral of the Struve function of order 0. + it2struve0 -- Integral related to the Struve function of order 0. + itmodstruve0 -- Integral of the modified Struve function of order 0. + + +Raw statistical functions +------------------------- + +.. seealso:: :mod:`scipy.stats`: Friendly versions of these functions. + +Binomial distribution +^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + bdtr -- Binomial distribution cumulative distribution function. + bdtrc -- Binomial distribution survival function. + bdtri -- Inverse function to `bdtr` with respect to `p`. + bdtrik -- Inverse function to `bdtr` with respect to `k`. + bdtrin -- Inverse function to `bdtr` with respect to `n`. + +Beta distribution +^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + btdtria -- Inverse of `betainc` with respect to `a`. + btdtrib -- Inverse of `betainc` with respect to `b`. + +F distribution +^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + fdtr -- F cumulative distribution function. + fdtrc -- F survival function. + fdtri -- The `p`-th quantile of the F-distribution. + fdtridfd -- Inverse to `fdtr` vs dfd. + +Gamma distribution +^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + gdtr -- Gamma distribution cumulative distribution function. + gdtrc -- Gamma distribution survival function. + gdtria -- Inverse of `gdtr` vs a. + gdtrib -- Inverse of `gdtr` vs b. + gdtrix -- Inverse of `gdtr` vs x. + +Negative binomial distribution +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + nbdtr -- Negative binomial cumulative distribution function. + nbdtrc -- Negative binomial survival function. + nbdtri -- Inverse of `nbdtr` vs `p`. + nbdtrik -- Inverse of `nbdtr` vs `k`. + nbdtrin -- Inverse of `nbdtr` vs `n`. + +Noncentral F distribution +^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + ncfdtr -- Cumulative distribution function of the non-central F distribution. + ncfdtridfd -- Calculate degrees of freedom (denominator) for the noncentral F-distribution. + ncfdtridfn -- Calculate degrees of freedom (numerator) for the noncentral F-distribution. + ncfdtri -- Inverse cumulative distribution function of the non-central F distribution. + ncfdtrinc -- Calculate non-centrality parameter for non-central F distribution. + +Noncentral t distribution +^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + nctdtr -- Cumulative distribution function of the non-central `t` distribution. + nctdtridf -- Calculate degrees of freedom for non-central t distribution. + nctdtrit -- Inverse cumulative distribution function of the non-central t distribution. + nctdtrinc -- Calculate non-centrality parameter for non-central t distribution. + +Normal distribution +^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + nrdtrimn -- Calculate mean of normal distribution given other params. + nrdtrisd -- Calculate standard deviation of normal distribution given other params. + ndtr -- Normal cumulative distribution function. + log_ndtr -- Logarithm of normal cumulative distribution function. + ndtri -- Inverse of `ndtr` vs x. + ndtri_exp -- Inverse of `log_ndtr` vs x. + +Poisson distribution +^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + pdtr -- Poisson cumulative distribution function. + pdtrc -- Poisson survival function. + pdtri -- Inverse to `pdtr` vs m. + pdtrik -- Inverse to `pdtr` vs k. + +Student t distribution +^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + stdtr -- Student t distribution cumulative distribution function. + stdtridf -- Inverse of `stdtr` vs df. + stdtrit -- Inverse of `stdtr` vs `t`. + +Chi square distribution +^^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + chdtr -- Chi square cumulative distribution function. + chdtrc -- Chi square survival function. + chdtri -- Inverse to `chdtrc`. + chdtriv -- Inverse to `chdtr` vs `v`. + +Non-central chi square distribution +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + chndtr -- Non-central chi square cumulative distribution function. + chndtridf -- Inverse to `chndtr` vs `df`. + chndtrinc -- Inverse to `chndtr` vs `nc`. + chndtrix -- Inverse to `chndtr` vs `x`. + +Kolmogorov distribution +^^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + smirnov -- Kolmogorov-Smirnov complementary cumulative distribution function. + smirnovi -- Inverse to `smirnov`. + kolmogorov -- Complementary cumulative distribution function of Kolmogorov distribution. + kolmogi -- Inverse function to `kolmogorov`. + +Box-Cox transformation +^^^^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + boxcox -- Compute the Box-Cox transformation. + boxcox1p -- Compute the Box-Cox transformation of 1 + `x`. + inv_boxcox -- Compute the inverse of the Box-Cox transformation. + inv_boxcox1p -- Compute the inverse of the Box-Cox transformation. + + +Sigmoidal functions +^^^^^^^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + logit -- Logit ufunc for ndarrays. + expit -- Logistic sigmoid function. + log_expit -- Logarithm of the logistic sigmoid function. + +Miscellaneous +^^^^^^^^^^^^^ + +.. autosummary:: + :toctree: generated/ + + tklmbda -- Tukey-Lambda cumulative distribution function. + owens_t -- Owen's T Function. + + +Information Theory functions +---------------------------- + +.. autosummary:: + :toctree: generated/ + + entr -- Elementwise function for computing entropy. + rel_entr -- Elementwise function for computing relative entropy. + kl_div -- Elementwise function for computing Kullback-Leibler divergence. + huber -- Huber loss function. + pseudo_huber -- Pseudo-Huber loss function. + + +Gamma and related functions +--------------------------- + +.. autosummary:: + :toctree: generated/ + + gamma -- Gamma function. + gammaln -- Logarithm of the absolute value of the Gamma function for real inputs. + loggamma -- Principal branch of the logarithm of the Gamma function. + gammasgn -- Sign of the gamma function. + gammainc -- Regularized lower incomplete gamma function. + gammaincinv -- Inverse to `gammainc`. + gammaincc -- Regularized upper incomplete gamma function. + gammainccinv -- Inverse to `gammaincc`. + beta -- Beta function. + betaln -- Natural logarithm of absolute value of beta function. + betainc -- Incomplete beta integral. + betaincc -- Complemented incomplete beta integral. + betaincinv -- Inverse function to beta integral. + betainccinv -- Inverse of the complemented incomplete beta integral. + psi -- The digamma function. + rgamma -- Gamma function inverted. + polygamma -- Polygamma function n. + multigammaln -- Returns the log of multivariate gamma, also sometimes called the generalized gamma. + digamma -- psi(x[, out]). + poch -- Rising factorial (z)_m. + +Error function and Fresnel integrals +------------------------------------ + +.. autosummary:: + :toctree: generated/ + + erf -- Returns the error function of complex argument. + erfc -- Complementary error function, ``1 - erf(x)``. + erfcx -- Scaled complementary error function, ``exp(x**2) * erfc(x)``. + erfi -- Imaginary error function, ``-i erf(i z)``. + erfinv -- Inverse function for erf. + erfcinv -- Inverse function for erfc. + wofz -- Faddeeva function. + dawsn -- Dawson's integral. + fresnel -- Fresnel sin and cos integrals. + fresnel_zeros -- Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z). + modfresnelp -- Modified Fresnel positive integrals. + modfresnelm -- Modified Fresnel negative integrals. + voigt_profile -- Voigt profile. + +The following functions do not accept NumPy arrays (they are not +universal functions): + +.. autosummary:: + :toctree: generated/ + + erf_zeros -- Compute nt complex zeros of error function erf(z). + fresnelc_zeros -- Compute nt complex zeros of cosine Fresnel integral C(z). + fresnels_zeros -- Compute nt complex zeros of sine Fresnel integral S(z). + +Legendre functions +------------------ + +.. autosummary:: + :toctree: generated/ + + legendre_p -- Legendre polynomials of the first kind. + legendre_p_all -- All Legendre polynomials of the first kind up to a specified order. + assoc_legendre_p -- Associated Legendre polynomials of the first kind. + assoc_legendre_p_all -- All associated Legendre polynomials of the first kind up to a specified order and degree. + sph_legendre_p -- Spherical Legendre polynomials of the first kind. + sph_legendre_p_all -- All spherical Legendre polynomials of the first kind up to a specified order and degree. + sph_harm_y -- Spherical harmonics. + sph_harm_y_all -- All spherical harmonics up to a specified order and degree. + +The following functions are in the process of being deprecated in favor of the above, +which provide a more flexible and consistent interface. + +.. autosummary:: + :toctree: generated/ + + lpmv -- Associated Legendre function of integer order and real degree. + sph_harm -- Compute spherical harmonics. + clpmn -- Associated Legendre function of the first kind for complex arguments. + lpn -- Legendre function of the first kind. + lqn -- Legendre function of the second kind. + lpmn -- Sequence of associated Legendre functions of the first kind. + lqmn -- Sequence of associated Legendre functions of the second kind. + +Ellipsoidal harmonics +--------------------- + +.. autosummary:: + :toctree: generated/ + + ellip_harm -- Ellipsoidal harmonic functions E^p_n(l). + ellip_harm_2 -- Ellipsoidal harmonic functions F^p_n(l). + ellip_normal -- Ellipsoidal harmonic normalization constants gamma^p_n. + +Orthogonal polynomials +---------------------- + +The following functions evaluate values of orthogonal polynomials: + +.. autosummary:: + :toctree: generated/ + + assoc_laguerre -- Compute the generalized (associated) Laguerre polynomial of degree n and order k. + eval_legendre -- Evaluate Legendre polynomial at a point. + eval_chebyt -- Evaluate Chebyshev polynomial of the first kind at a point. + eval_chebyu -- Evaluate Chebyshev polynomial of the second kind at a point. + eval_chebyc -- Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a point. + eval_chebys -- Evaluate Chebyshev polynomial of the second kind on [-2, 2] at a point. + eval_jacobi -- Evaluate Jacobi polynomial at a point. + eval_laguerre -- Evaluate Laguerre polynomial at a point. + eval_genlaguerre -- Evaluate generalized Laguerre polynomial at a point. + eval_hermite -- Evaluate physicist's Hermite polynomial at a point. + eval_hermitenorm -- Evaluate probabilist's (normalized) Hermite polynomial at a point. + eval_gegenbauer -- Evaluate Gegenbauer polynomial at a point. + eval_sh_legendre -- Evaluate shifted Legendre polynomial at a point. + eval_sh_chebyt -- Evaluate shifted Chebyshev polynomial of the first kind at a point. + eval_sh_chebyu -- Evaluate shifted Chebyshev polynomial of the second kind at a point. + eval_sh_jacobi -- Evaluate shifted Jacobi polynomial at a point. + +The following functions compute roots and quadrature weights for +orthogonal polynomials: + +.. autosummary:: + :toctree: generated/ + + roots_legendre -- Gauss-Legendre quadrature. + roots_chebyt -- Gauss-Chebyshev (first kind) quadrature. + roots_chebyu -- Gauss-Chebyshev (second kind) quadrature. + roots_chebyc -- Gauss-Chebyshev (first kind) quadrature. + roots_chebys -- Gauss-Chebyshev (second kind) quadrature. + roots_jacobi -- Gauss-Jacobi quadrature. + roots_laguerre -- Gauss-Laguerre quadrature. + roots_genlaguerre -- Gauss-generalized Laguerre quadrature. + roots_hermite -- Gauss-Hermite (physicist's) quadrature. + roots_hermitenorm -- Gauss-Hermite (statistician's) quadrature. + roots_gegenbauer -- Gauss-Gegenbauer quadrature. + roots_sh_legendre -- Gauss-Legendre (shifted) quadrature. + roots_sh_chebyt -- Gauss-Chebyshev (first kind, shifted) quadrature. + roots_sh_chebyu -- Gauss-Chebyshev (second kind, shifted) quadrature. + roots_sh_jacobi -- Gauss-Jacobi (shifted) quadrature. + +The functions below, in turn, return the polynomial coefficients in +``orthopoly1d`` objects, which function similarly as `numpy.poly1d`. +The ``orthopoly1d`` class also has an attribute ``weights``, which returns +the roots, weights, and total weights for the appropriate form of Gaussian +quadrature. These are returned in an ``n x 3`` array with roots in the first +column, weights in the second column, and total weights in the final column. +Note that ``orthopoly1d`` objects are converted to `~numpy.poly1d` when doing +arithmetic, and lose information of the original orthogonal polynomial. + +.. autosummary:: + :toctree: generated/ + + legendre -- Legendre polynomial. + chebyt -- Chebyshev polynomial of the first kind. + chebyu -- Chebyshev polynomial of the second kind. + chebyc -- Chebyshev polynomial of the first kind on :math:`[-2, 2]`. + chebys -- Chebyshev polynomial of the second kind on :math:`[-2, 2]`. + jacobi -- Jacobi polynomial. + laguerre -- Laguerre polynomial. + genlaguerre -- Generalized (associated) Laguerre polynomial. + hermite -- Physicist's Hermite polynomial. + hermitenorm -- Normalized (probabilist's) Hermite polynomial. + gegenbauer -- Gegenbauer (ultraspherical) polynomial. + sh_legendre -- Shifted Legendre polynomial. + sh_chebyt -- Shifted Chebyshev polynomial of the first kind. + sh_chebyu -- Shifted Chebyshev polynomial of the second kind. + sh_jacobi -- Shifted Jacobi polynomial. + +.. warning:: + + Computing values of high-order polynomials (around ``order > 20``) using + polynomial coefficients is numerically unstable. To evaluate polynomial + values, the ``eval_*`` functions should be used instead. + + +Hypergeometric functions +------------------------ + +.. autosummary:: + :toctree: generated/ + + hyp2f1 -- Gauss hypergeometric function 2F1(a, b; c; z). + hyp1f1 -- Confluent hypergeometric function 1F1(a, b; x). + hyperu -- Confluent hypergeometric function U(a, b, x) of the second kind. + hyp0f1 -- Confluent hypergeometric limit function 0F1. + + +Parabolic cylinder functions +---------------------------- + +.. autosummary:: + :toctree: generated/ + + pbdv -- Parabolic cylinder function D. + pbvv -- Parabolic cylinder function V. + pbwa -- Parabolic cylinder function W. + +The following functions do not accept NumPy arrays (they are not +universal functions): + +.. autosummary:: + :toctree: generated/ + + pbdv_seq -- Parabolic cylinder functions Dv(x) and derivatives. + pbvv_seq -- Parabolic cylinder functions Vv(x) and derivatives. + pbdn_seq -- Parabolic cylinder functions Dn(z) and derivatives. + +Mathieu and related functions +----------------------------- + +.. autosummary:: + :toctree: generated/ + + mathieu_a -- Characteristic value of even Mathieu functions. + mathieu_b -- Characteristic value of odd Mathieu functions. + +The following functions do not accept NumPy arrays (they are not +universal functions): + +.. autosummary:: + :toctree: generated/ + + mathieu_even_coef -- Fourier coefficients for even Mathieu and modified Mathieu functions. + mathieu_odd_coef -- Fourier coefficients for even Mathieu and modified Mathieu functions. + +The following return both function and first derivative: + +.. autosummary:: + :toctree: generated/ + + mathieu_cem -- Even Mathieu function and its derivative. + mathieu_sem -- Odd Mathieu function and its derivative. + mathieu_modcem1 -- Even modified Mathieu function of the first kind and its derivative. + mathieu_modcem2 -- Even modified Mathieu function of the second kind and its derivative. + mathieu_modsem1 -- Odd modified Mathieu function of the first kind and its derivative. + mathieu_modsem2 -- Odd modified Mathieu function of the second kind and its derivative. + +Spheroidal wave functions +------------------------- + +.. autosummary:: + :toctree: generated/ + + pro_ang1 -- Prolate spheroidal angular function of the first kind and its derivative. + pro_rad1 -- Prolate spheroidal radial function of the first kind and its derivative. + pro_rad2 -- Prolate spheroidal radial function of the second kind and its derivative. + obl_ang1 -- Oblate spheroidal angular function of the first kind and its derivative. + obl_rad1 -- Oblate spheroidal radial function of the first kind and its derivative. + obl_rad2 -- Oblate spheroidal radial function of the second kind and its derivative. + pro_cv -- Characteristic value of prolate spheroidal function. + obl_cv -- Characteristic value of oblate spheroidal function. + pro_cv_seq -- Characteristic values for prolate spheroidal wave functions. + obl_cv_seq -- Characteristic values for oblate spheroidal wave functions. + +The following functions require pre-computed characteristic value: + +.. autosummary:: + :toctree: generated/ + + pro_ang1_cv -- Prolate spheroidal angular function pro_ang1 for precomputed characteristic value. + pro_rad1_cv -- Prolate spheroidal radial function pro_rad1 for precomputed characteristic value. + pro_rad2_cv -- Prolate spheroidal radial function pro_rad2 for precomputed characteristic value. + obl_ang1_cv -- Oblate spheroidal angular function obl_ang1 for precomputed characteristic value. + obl_rad1_cv -- Oblate spheroidal radial function obl_rad1 for precomputed characteristic value. + obl_rad2_cv -- Oblate spheroidal radial function obl_rad2 for precomputed characteristic value. + +Kelvin functions +---------------- + +.. autosummary:: + :toctree: generated/ + + kelvin -- Kelvin functions as complex numbers. + kelvin_zeros -- Compute nt zeros of all Kelvin functions. + ber -- Kelvin function ber. + bei -- Kelvin function bei + berp -- Derivative of the Kelvin function `ber`. + beip -- Derivative of the Kelvin function `bei`. + ker -- Kelvin function ker. + kei -- Kelvin function ker. + kerp -- Derivative of the Kelvin function ker. + keip -- Derivative of the Kelvin function kei. + +The following functions do not accept NumPy arrays (they are not +universal functions): + +.. autosummary:: + :toctree: generated/ + + ber_zeros -- Compute nt zeros of the Kelvin function ber(x). + bei_zeros -- Compute nt zeros of the Kelvin function bei(x). + berp_zeros -- Compute nt zeros of the Kelvin function ber'(x). + beip_zeros -- Compute nt zeros of the Kelvin function bei'(x). + ker_zeros -- Compute nt zeros of the Kelvin function ker(x). + kei_zeros -- Compute nt zeros of the Kelvin function kei(x). + kerp_zeros -- Compute nt zeros of the Kelvin function ker'(x). + keip_zeros -- Compute nt zeros of the Kelvin function kei'(x). + +Combinatorics +------------- + +.. autosummary:: + :toctree: generated/ + + comb -- The number of combinations of N things taken k at a time. + perm -- Permutations of N things taken k at a time, i.e., k-permutations of N. + stirling2 -- Stirling numbers of the second kind. + +Lambert W and related functions +------------------------------- + +.. autosummary:: + :toctree: generated/ + + lambertw -- Lambert W function. + wrightomega -- Wright Omega function. + +Other special functions +----------------------- + +.. autosummary:: + :toctree: generated/ + + agm -- Arithmetic, Geometric Mean. + bernoulli -- Bernoulli numbers B0..Bn (inclusive). + binom -- Binomial coefficient + diric -- Periodic sinc function, also called the Dirichlet function. + euler -- Euler numbers E0..En (inclusive). + expn -- Exponential integral E_n. + exp1 -- Exponential integral E_1 of complex argument z. + expi -- Exponential integral Ei. + factorial -- The factorial of a number or array of numbers. + factorial2 -- Double factorial. + factorialk -- Multifactorial of n of order k, n(!!...!). + shichi -- Hyperbolic sine and cosine integrals. + sici -- Sine and cosine integrals. + softmax -- Softmax function. + log_softmax -- Logarithm of softmax function. + spence -- Spence's function, also known as the dilogarithm. + zeta -- Riemann zeta function. + zetac -- Riemann zeta function minus 1. + softplus -- Softplus function. + +Convenience functions +--------------------- + +.. autosummary:: + :toctree: generated/ + + cbrt -- Cube root of `x`. + exp10 -- 10**x. + exp2 -- 2**x. + radian -- Convert from degrees to radians. + cosdg -- Cosine of the angle `x` given in degrees. + sindg -- Sine of angle given in degrees. + tandg -- Tangent of angle x given in degrees. + cotdg -- Cotangent of the angle `x` given in degrees. + log1p -- Calculates log(1+x) for use when `x` is near zero. + expm1 -- ``exp(x) - 1`` for use when `x` is near zero. + cosm1 -- ``cos(x) - 1`` for use when `x` is near zero. + powm1 -- ``x**y - 1`` for use when `y` is near zero or `x` is near 1. + round -- Round to nearest integer. + xlogy -- Compute ``x*log(y)`` so that the result is 0 if ``x = 0``. + xlog1py -- Compute ``x*log1p(y)`` so that the result is 0 if ``x = 0``. + logsumexp -- Compute the log of the sum of exponentials of input elements. + exprel -- Relative error exponential, (exp(x)-1)/x, for use when `x` is near zero. + sinc -- Return the sinc function. + +""" # noqa: E501 + +import os +import warnings + + +def _load_libsf_error_state(): + """Load libsf_error_state.dll shared library on Windows + + libsf_error_state manages shared state used by + ``scipy.special.seterr`` and ``scipy.special.geterr`` so that these + can work consistently between special functions provided by different + extension modules. This shared library is installed in scipy/special + alongside this __init__.py file. Due to lack of rpath support, Windows + cannot find shared libraries installed within wheels. To circumvent this, + we pre-load ``lib_sf_error_state.dll`` when on Windows. + + The logic for this function was borrowed from the function ``make_init`` + in `scipy/tools/openblas_support.py`: + https://github.com/scipy/scipy/blob/bb92c8014e21052e7dde67a76b28214dd1dcb94a/tools/openblas_support.py#L239-L274 + """ # noqa: E501 + if os.name == "nt": + try: + from ctypes import WinDLL + basedir = os.path.dirname(__file__) + except: # noqa: E722 + pass + else: + dll_path = os.path.join(basedir, "libsf_error_state.dll") + if os.path.exists(dll_path): + WinDLL(dll_path) + + +_load_libsf_error_state() + + +from ._sf_error import SpecialFunctionWarning, SpecialFunctionError + +from . import _ufuncs +from ._ufuncs import * + +# Replace some function definitions from _ufuncs to add Array API support +from ._support_alternative_backends import ( + log_ndtr, ndtr, ndtri, erf, erfc, i0, i0e, i1, i1e, gammaln, + gammainc, gammaincc, logit, expit, entr, rel_entr, xlogy, + chdtr, chdtrc, betainc, betaincc, stdtr) + +from . import _basic +from ._basic import * + +from ._logsumexp import logsumexp, softmax, log_softmax + +from . import _multiufuncs +from ._multiufuncs import * + +from . import _orthogonal +from ._orthogonal import * + +from ._spfun_stats import multigammaln +from ._ellip_harm import ( + ellip_harm, + ellip_harm_2, + ellip_normal +) +from ._lambertw import lambertw +from ._spherical_bessel import ( + spherical_jn, + spherical_yn, + spherical_in, + spherical_kn +) + +# Deprecated namespaces, to be removed in v2.0.0 +from . import add_newdocs, basic, orthogonal, specfun, sf_error, spfun_stats + +# We replace some function definitions from _ufuncs with those from +# _support_alternative_backends above, but those are all listed in _ufuncs.__all__, +# so there is no need to consider _support_alternative_backends.__all__ here. +__all__ = _ufuncs.__all__ + _basic.__all__ + _orthogonal.__all__ + _multiufuncs.__all__ +__all__ += [ + 'SpecialFunctionWarning', + 'SpecialFunctionError', + 'logsumexp', + 'softmax', + 'log_softmax', + 'multigammaln', + 'ellip_harm', + 'ellip_harm_2', + 'ellip_normal', + 'lambertw', + 'spherical_jn', + 'spherical_yn', + 'spherical_in', + 'spherical_kn', +] + +from scipy._lib._testutils import PytestTester +test = PytestTester(__name__) +del PytestTester + + +def _get_include(): + """This function is for development purposes only. + + This function could disappear or its behavior could change at any time. + """ + import os + return os.path.dirname(__file__) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_add_newdocs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_add_newdocs.py new file mode 100644 index 0000000000000000000000000000000000000000..134604c90128a59d48dee66318fa6fd02308f80c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_add_newdocs.py @@ -0,0 +1,10699 @@ +# Docstrings for generated ufuncs +# +# The syntax is designed to look like the function add_newdoc is being +# called from numpy.lib, but in this file add_newdoc puts the +# docstrings in a dictionary. This dictionary is used in +# _generate_pyx.py to generate the docstrings for the ufuncs in +# scipy.special at the C level when the ufuncs are created at compile +# time. + +docdict: dict[str, str] = {} + + +def get(name): + return docdict.get(name) + + +def add_newdoc(name, doc): + docdict[name] = doc + + +add_newdoc("_sf_error_test_function", + """ + Private function; do not use. + """) + + +add_newdoc("_cosine_cdf", + """ + _cosine_cdf(x) + + Cumulative distribution function (CDF) of the cosine distribution:: + + { 0, x < -pi + cdf(x) = { (pi + x + sin(x))/(2*pi), -pi <= x <= pi + { 1, x > pi + + Parameters + ---------- + x : array_like + `x` must contain real numbers. + + Returns + ------- + scalar or ndarray + The cosine distribution CDF evaluated at `x`. + + """) + +add_newdoc("_cosine_invcdf", + """ + _cosine_invcdf(p) + + Inverse of the cumulative distribution function (CDF) of the cosine + distribution. + + The CDF of the cosine distribution is:: + + cdf(x) = (pi + x + sin(x))/(2*pi) + + This function computes the inverse of cdf(x). + + Parameters + ---------- + p : array_like + `p` must contain real numbers in the interval ``0 <= p <= 1``. + `nan` is returned for values of `p` outside the interval [0, 1]. + + Returns + ------- + scalar or ndarray + The inverse of the cosine distribution CDF evaluated at `p`. + + """) + +add_newdoc("_ellip_harm", + """ + Internal function, use `ellip_harm` instead. + """) + +add_newdoc("_ellip_norm", + """ + Internal function, use `ellip_norm` instead. + """) + +add_newdoc("voigt_profile", + r""" + voigt_profile(x, sigma, gamma, out=None) + + Voigt profile. + + The Voigt profile is a convolution of a 1-D Normal distribution with + standard deviation ``sigma`` and a 1-D Cauchy distribution with half-width at + half-maximum ``gamma``. + + If ``sigma = 0``, PDF of Cauchy distribution is returned. + Conversely, if ``gamma = 0``, PDF of Normal distribution is returned. + If ``sigma = gamma = 0``, the return value is ``Inf`` for ``x = 0``, + and ``0`` for all other ``x``. + + Parameters + ---------- + x : array_like + Real argument + sigma : array_like + The standard deviation of the Normal distribution part + gamma : array_like + The half-width at half-maximum of the Cauchy distribution part + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + The Voigt profile at the given arguments + + See Also + -------- + wofz : Faddeeva function + + Notes + ----- + It can be expressed in terms of Faddeeva function + + .. math:: V(x; \sigma, \gamma) = \frac{Re[w(z)]}{\sigma\sqrt{2\pi}}, + .. math:: z = \frac{x + i\gamma}{\sqrt{2}\sigma} + + where :math:`w(z)` is the Faddeeva function. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Voigt_profile + + Examples + -------- + Calculate the function at point 2 for ``sigma=1`` and ``gamma=1``. + + >>> from scipy.special import voigt_profile + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> voigt_profile(2, 1., 1.) + 0.09071519942627544 + + Calculate the function at several points by providing a NumPy array + for `x`. + + >>> values = np.array([-2., 0., 5]) + >>> voigt_profile(values, 1., 1.) + array([0.0907152 , 0.20870928, 0.01388492]) + + Plot the function for different parameter sets. + + >>> fig, ax = plt.subplots(figsize=(8, 8)) + >>> x = np.linspace(-10, 10, 500) + >>> parameters_list = [(1.5, 0., "solid"), (1.3, 0.5, "dashed"), + ... (0., 1.8, "dotted"), (1., 1., "dashdot")] + >>> for params in parameters_list: + ... sigma, gamma, linestyle = params + ... voigt = voigt_profile(x, sigma, gamma) + ... ax.plot(x, voigt, label=rf"$\sigma={sigma},\, \gamma={gamma}$", + ... ls=linestyle) + >>> ax.legend() + >>> plt.show() + + Verify visually that the Voigt profile indeed arises as the convolution + of a normal and a Cauchy distribution. + + >>> from scipy.signal import convolve + >>> x, dx = np.linspace(-10, 10, 500, retstep=True) + >>> def gaussian(x, sigma): + ... return np.exp(-0.5 * x**2/sigma**2)/(sigma * np.sqrt(2*np.pi)) + >>> def cauchy(x, gamma): + ... return gamma/(np.pi * (np.square(x)+gamma**2)) + >>> sigma = 2 + >>> gamma = 1 + >>> gauss_profile = gaussian(x, sigma) + >>> cauchy_profile = cauchy(x, gamma) + >>> convolved = dx * convolve(cauchy_profile, gauss_profile, mode="same") + >>> voigt = voigt_profile(x, sigma, gamma) + >>> fig, ax = plt.subplots(figsize=(8, 8)) + >>> ax.plot(x, gauss_profile, label="Gauss: $G$", c='b') + >>> ax.plot(x, cauchy_profile, label="Cauchy: $C$", c='y', ls="dashed") + >>> xx = 0.5*(x[1:] + x[:-1]) # midpoints + >>> ax.plot(xx, convolved[1:], label="Convolution: $G * C$", ls='dashdot', + ... c='k') + >>> ax.plot(x, voigt, label="Voigt", ls='dotted', c='r') + >>> ax.legend() + >>> plt.show() + """) + +add_newdoc("wrightomega", + r""" + wrightomega(z, out=None) + + Wright Omega function. + + Defined as the solution to + + .. math:: + + \omega + \log(\omega) = z + + where :math:`\log` is the principal branch of the complex logarithm. + + Parameters + ---------- + z : array_like + Points at which to evaluate the Wright Omega function + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + omega : scalar or ndarray + Values of the Wright Omega function + + See Also + -------- + lambertw : The Lambert W function + + Notes + ----- + .. versionadded:: 0.19.0 + + The function can also be defined as + + .. math:: + + \omega(z) = W_{K(z)}(e^z) + + where :math:`K(z) = \lceil (\Im(z) - \pi)/(2\pi) \rceil` is the + unwinding number and :math:`W` is the Lambert W function. + + The implementation here is taken from [1]_. + + References + ---------- + .. [1] Lawrence, Corless, and Jeffrey, "Algorithm 917: Complex + Double-Precision Evaluation of the Wright :math:`\omega` + Function." ACM Transactions on Mathematical Software, + 2012. :doi:`10.1145/2168773.2168779`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import wrightomega, lambertw + + >>> wrightomega([-2, -1, 0, 1, 2]) + array([0.12002824, 0.27846454, 0.56714329, 1. , 1.5571456 ]) + + Complex input: + + >>> wrightomega(3 + 5j) + (1.5804428632097158+3.8213626783287937j) + + Verify that ``wrightomega(z)`` satisfies ``w + log(w) = z``: + + >>> w = -5 + 4j + >>> wrightomega(w + np.log(w)) + (-5+4j) + + Verify the connection to ``lambertw``: + + >>> z = 0.5 + 3j + >>> wrightomega(z) + (0.0966015889280649+1.4937828458191993j) + >>> lambertw(np.exp(z)) + (0.09660158892806493+1.4937828458191993j) + + >>> z = 0.5 + 4j + >>> wrightomega(z) + (-0.3362123489037213+2.282986001579032j) + >>> lambertw(np.exp(z), k=1) + (-0.33621234890372115+2.282986001579032j) + """) + + +add_newdoc("agm", + """ + agm(a, b, out=None) + + Compute the arithmetic-geometric mean of `a` and `b`. + + Start with a_0 = a and b_0 = b and iteratively compute:: + + a_{n+1} = (a_n + b_n)/2 + b_{n+1} = sqrt(a_n*b_n) + + a_n and b_n converge to the same limit as n increases; their common + limit is agm(a, b). + + Parameters + ---------- + a, b : array_like + Real values only. If the values are both negative, the result + is negative. If one value is negative and the other is positive, + `nan` is returned. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + The arithmetic-geometric mean of `a` and `b`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import agm + >>> a, b = 24.0, 6.0 + >>> agm(a, b) + 13.458171481725614 + + Compare that result to the iteration: + + >>> while a != b: + ... a, b = (a + b)/2, np.sqrt(a*b) + ... print("a = %19.16f b=%19.16f" % (a, b)) + ... + a = 15.0000000000000000 b=12.0000000000000000 + a = 13.5000000000000000 b=13.4164078649987388 + a = 13.4582039324993694 b=13.4581390309909850 + a = 13.4581714817451772 b=13.4581714817060547 + a = 13.4581714817256159 b=13.4581714817256159 + + When array-like arguments are given, broadcasting applies: + + >>> a = np.array([[1.5], [3], [6]]) # a has shape (3, 1). + >>> b = np.array([6, 12, 24, 48]) # b has shape (4,). + >>> agm(a, b) + array([[ 3.36454287, 5.42363427, 9.05798751, 15.53650756], + [ 4.37037309, 6.72908574, 10.84726853, 18.11597502], + [ 6. , 8.74074619, 13.45817148, 21.69453707]]) + """) + +add_newdoc("airy", + r""" + airy(z, out=None) + + Airy functions and their derivatives. + + Parameters + ---------- + z : array_like + Real or complex argument. + out : tuple of ndarray, optional + Optional output arrays for the function values + + Returns + ------- + Ai, Aip, Bi, Bip : 4-tuple of scalar or ndarray + Airy functions Ai and Bi, and their derivatives Aip and Bip. + + See Also + -------- + airye : exponentially scaled Airy functions. + + Notes + ----- + The Airy functions Ai and Bi are two independent solutions of + + .. math:: y''(x) = x y(x). + + For real `z` in [-10, 10], the computation is carried out by calling + the Cephes [1]_ `airy` routine, which uses power series summation + for small `z` and rational minimax approximations for large `z`. + + Outside this range, the AMOS [2]_ `zairy` and `zbiry` routines are + employed. They are computed using power series for :math:`|z| < 1` and + the following relations to modified Bessel functions for larger `z` + (where :math:`t \equiv 2 z^{3/2}/3`): + + .. math:: + + Ai(z) = \frac{1}{\pi \sqrt{3}} K_{1/3}(t) + + Ai'(z) = -\frac{z}{\pi \sqrt{3}} K_{2/3}(t) + + Bi(z) = \sqrt{\frac{z}{3}} \left(I_{-1/3}(t) + I_{1/3}(t) \right) + + Bi'(z) = \frac{z}{\sqrt{3}} \left(I_{-2/3}(t) + I_{2/3}(t)\right) + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + .. [2] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + + Examples + -------- + Compute the Airy functions on the interval [-15, 5]. + + >>> import numpy as np + >>> from scipy import special + >>> x = np.linspace(-15, 5, 201) + >>> ai, aip, bi, bip = special.airy(x) + + Plot Ai(x) and Bi(x). + + >>> import matplotlib.pyplot as plt + >>> plt.plot(x, ai, 'r', label='Ai(x)') + >>> plt.plot(x, bi, 'b--', label='Bi(x)') + >>> plt.ylim(-0.5, 1.0) + >>> plt.grid() + >>> plt.legend(loc='upper left') + >>> plt.show() + + """) + +add_newdoc("airye", + """ + airye(z, out=None) + + Exponentially scaled Airy functions and their derivatives. + + Scaling:: + + eAi = Ai * exp(2.0/3.0*z*sqrt(z)) + eAip = Aip * exp(2.0/3.0*z*sqrt(z)) + eBi = Bi * exp(-abs(2.0/3.0*(z*sqrt(z)).real)) + eBip = Bip * exp(-abs(2.0/3.0*(z*sqrt(z)).real)) + + Parameters + ---------- + z : array_like + Real or complex argument. + out : tuple of ndarray, optional + Optional output arrays for the function values + + Returns + ------- + eAi, eAip, eBi, eBip : 4-tuple of scalar or ndarray + Exponentially scaled Airy functions eAi and eBi, and their derivatives + eAip and eBip + + See Also + -------- + airy + + Notes + ----- + Wrapper for the AMOS [1]_ routines `zairy` and `zbiry`. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + + Examples + -------- + We can compute exponentially scaled Airy functions and their derivatives: + + >>> import numpy as np + >>> from scipy.special import airye + >>> import matplotlib.pyplot as plt + >>> z = np.linspace(0, 50, 500) + >>> eAi, eAip, eBi, eBip = airye(z) + >>> f, ax = plt.subplots(2, 1, sharex=True) + >>> for ind, data in enumerate([[eAi, eAip, ["eAi", "eAip"]], + ... [eBi, eBip, ["eBi", "eBip"]]]): + ... ax[ind].plot(z, data[0], "-r", z, data[1], "-b") + ... ax[ind].legend(data[2]) + ... ax[ind].grid(True) + >>> plt.show() + + We can compute these using usual non-scaled Airy functions by: + + >>> from scipy.special import airy + >>> Ai, Aip, Bi, Bip = airy(z) + >>> np.allclose(eAi, Ai * np.exp(2.0 / 3.0 * z * np.sqrt(z))) + True + >>> np.allclose(eAip, Aip * np.exp(2.0 / 3.0 * z * np.sqrt(z))) + True + >>> np.allclose(eBi, Bi * np.exp(-abs(np.real(2.0 / 3.0 * z * np.sqrt(z))))) + True + >>> np.allclose(eBip, Bip * np.exp(-abs(np.real(2.0 / 3.0 * z * np.sqrt(z))))) + True + + Comparing non-scaled and exponentially scaled ones, the usual non-scaled + function quickly underflows for large values, whereas the exponentially + scaled function does not. + + >>> airy(200) + (0.0, 0.0, nan, nan) + >>> airye(200) + (0.07501041684381093, -1.0609012305109042, 0.15003188417418148, 2.1215836725571093) + + """) + +add_newdoc("bdtr", + r""" + bdtr(k, n, p, out=None) + + Binomial distribution cumulative distribution function. + + Sum of the terms 0 through `floor(k)` of the Binomial probability density. + + .. math:: + \mathrm{bdtr}(k, n, p) = + \sum_{j=0}^{\lfloor k \rfloor} {{n}\choose{j}} p^j (1-p)^{n-j} + + Parameters + ---------- + k : array_like + Number of successes (double), rounded down to the nearest integer. + n : array_like + Number of events (int). + p : array_like + Probability of success in a single event (float). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + y : scalar or ndarray + Probability of `floor(k)` or fewer successes in `n` independent events with + success probabilities of `p`. + + Notes + ----- + The terms are not summed directly; instead the regularized incomplete beta + function is employed, according to the formula, + + .. math:: + \mathrm{bdtr}(k, n, p) = + I_{1 - p}(n - \lfloor k \rfloor, \lfloor k \rfloor + 1). + + Wrapper for the Cephes [1]_ routine `bdtr`. + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + """) + +add_newdoc("bdtrc", + r""" + bdtrc(k, n, p, out=None) + + Binomial distribution survival function. + + Sum of the terms `floor(k) + 1` through `n` of the binomial probability + density, + + .. math:: + \mathrm{bdtrc}(k, n, p) = + \sum_{j=\lfloor k \rfloor +1}^n {{n}\choose{j}} p^j (1-p)^{n-j} + + Parameters + ---------- + k : array_like + Number of successes (double), rounded down to nearest integer. + n : array_like + Number of events (int) + p : array_like + Probability of success in a single event. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + y : scalar or ndarray + Probability of `floor(k) + 1` or more successes in `n` independent + events with success probabilities of `p`. + + See Also + -------- + bdtr + betainc + + Notes + ----- + The terms are not summed directly; instead the regularized incomplete beta + function is employed, according to the formula, + + .. math:: + \mathrm{bdtrc}(k, n, p) = I_{p}(\lfloor k \rfloor + 1, n - \lfloor k \rfloor). + + Wrapper for the Cephes [1]_ routine `bdtrc`. + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + """) + +add_newdoc("bdtri", + r""" + bdtri(k, n, y, out=None) + + Inverse function to `bdtr` with respect to `p`. + + Finds the event probability `p` such that the sum of the terms 0 through + `k` of the binomial probability density is equal to the given cumulative + probability `y`. + + Parameters + ---------- + k : array_like + Number of successes (float), rounded down to the nearest integer. + n : array_like + Number of events (float) + y : array_like + Cumulative probability (probability of `k` or fewer successes in `n` + events). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + p : scalar or ndarray + The event probability such that `bdtr(\lfloor k \rfloor, n, p) = y`. + + See Also + -------- + bdtr + betaincinv + + Notes + ----- + The computation is carried out using the inverse beta integral function + and the relation,:: + + 1 - p = betaincinv(n - k, k + 1, y). + + Wrapper for the Cephes [1]_ routine `bdtri`. + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + """) + +add_newdoc("bdtrik", + """ + bdtrik(y, n, p, out=None) + + Inverse function to `bdtr` with respect to `k`. + + Finds the number of successes `k` such that the sum of the terms 0 through + `k` of the Binomial probability density for `n` events with probability + `p` is equal to the given cumulative probability `y`. + + Parameters + ---------- + y : array_like + Cumulative probability (probability of `k` or fewer successes in `n` + events). + n : array_like + Number of events (float). + p : array_like + Success probability (float). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + k : scalar or ndarray + The number of successes `k` such that `bdtr(k, n, p) = y`. + + See Also + -------- + bdtr + + Notes + ----- + Formula 26.5.24 of [1]_ is used to reduce the binomial distribution to the + cumulative incomplete beta distribution. + + Computation of `k` involves a search for a value that produces the desired + value of `y`. The search relies on the monotonicity of `y` with `k`. + + Wrapper for the CDFLIB [2]_ Fortran routine `cdfbin`. + + References + ---------- + .. [1] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + .. [2] Barry Brown, James Lovato, and Kathy Russell, + CDFLIB: Library of Fortran Routines for Cumulative Distribution + Functions, Inverses, and Other Parameters. + + """) + +add_newdoc("bdtrin", + """ + bdtrin(k, y, p, out=None) + + Inverse function to `bdtr` with respect to `n`. + + Finds the number of events `n` such that the sum of the terms 0 through + `k` of the Binomial probability density for events with probability `p` is + equal to the given cumulative probability `y`. + + Parameters + ---------- + k : array_like + Number of successes (float). + y : array_like + Cumulative probability (probability of `k` or fewer successes in `n` + events). + p : array_like + Success probability (float). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + n : scalar or ndarray + The number of events `n` such that `bdtr(k, n, p) = y`. + + See Also + -------- + bdtr + + Notes + ----- + Formula 26.5.24 of [1]_ is used to reduce the binomial distribution to the + cumulative incomplete beta distribution. + + Computation of `n` involves a search for a value that produces the desired + value of `y`. The search relies on the monotonicity of `y` with `n`. + + Wrapper for the CDFLIB [2]_ Fortran routine `cdfbin`. + + References + ---------- + .. [1] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + .. [2] Barry Brown, James Lovato, and Kathy Russell, + CDFLIB: Library of Fortran Routines for Cumulative Distribution + Functions, Inverses, and Other Parameters. + """) + +add_newdoc("btdtria", + r""" + btdtria(p, b, x, out=None) + + Inverse of `betainc` with respect to `a`. + + This is the inverse of the beta cumulative distribution function, `betainc`, + considered as a function of `a`, returning the value of `a` for which + `betainc(a, b, x) = p`, or + + .. math:: + p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt + + Parameters + ---------- + p : array_like + Cumulative probability, in [0, 1]. + b : array_like + Shape parameter (`b` > 0). + x : array_like + The quantile, in [0, 1]. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + a : scalar or ndarray + The value of the shape parameter `a` such that `betainc(a, b, x) = p`. + + See Also + -------- + btdtrib : Inverse of the beta cumulative distribution function, with respect to `b`. + + Notes + ----- + Wrapper for the CDFLIB [1]_ Fortran routine `cdfbet`. + + The cumulative distribution function `p` is computed using a routine by + DiDinato and Morris [2]_. Computation of `a` involves a search for a value + that produces the desired value of `p`. The search relies on the + monotonicity of `p` with `a`. + + References + ---------- + .. [1] Barry Brown, James Lovato, and Kathy Russell, + CDFLIB: Library of Fortran Routines for Cumulative Distribution + Functions, Inverses, and Other Parameters. + .. [2] DiDinato, A. R. and Morris, A. H., + Algorithm 708: Significant Digit Computation of the Incomplete Beta + Function Ratios. ACM Trans. Math. Softw. 18 (1993), 360-373. + + """) + +add_newdoc("btdtrib", + r""" + btdtria(a, p, x, out=None) + + Inverse of `betainc` with respect to `b`. + + This is the inverse of the beta cumulative distribution function, `betainc`, + considered as a function of `b`, returning the value of `b` for which + `betainc(a, b, x) = p`, or + + .. math:: + p = \int_0^x \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} t^{a-1} (1-t)^{b-1}\,dt + + Parameters + ---------- + a : array_like + Shape parameter (`a` > 0). + p : array_like + Cumulative probability, in [0, 1]. + x : array_like + The quantile, in [0, 1]. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + b : scalar or ndarray + The value of the shape parameter `b` such that `betainc(a, b, x) = p`. + + See Also + -------- + btdtria : Inverse of the beta cumulative distribution function, with respect to `a`. + + Notes + ----- + Wrapper for the CDFLIB [1]_ Fortran routine `cdfbet`. + + The cumulative distribution function `p` is computed using a routine by + DiDinato and Morris [2]_. Computation of `b` involves a search for a value + that produces the desired value of `p`. The search relies on the + monotonicity of `p` with `b`. + + References + ---------- + .. [1] Barry Brown, James Lovato, and Kathy Russell, + CDFLIB: Library of Fortran Routines for Cumulative Distribution + Functions, Inverses, and Other Parameters. + .. [2] DiDinato, A. R. and Morris, A. H., + Algorithm 708: Significant Digit Computation of the Incomplete Beta + Function Ratios. ACM Trans. Math. Softw. 18 (1993), 360-373. + + + """) + +add_newdoc( + "betainc", + r""" + betainc(a, b, x, out=None) + + Regularized incomplete beta function. + + Computes the regularized incomplete beta function, defined as [1]_: + + .. math:: + + I_x(a, b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \int_0^x + t^{a-1}(1-t)^{b-1}dt, + + for :math:`0 \leq x \leq 1`. + + This function is the cumulative distribution function for the beta + distribution; its range is [0, 1]. + + Parameters + ---------- + a, b : array_like + Positive, real-valued parameters + x : array_like + Real-valued such that :math:`0 \leq x \leq 1`, + the upper limit of integration + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + Value of the regularized incomplete beta function + + See Also + -------- + beta : beta function + betaincinv : inverse of the regularized incomplete beta function + betaincc : complement of the regularized incomplete beta function + scipy.stats.beta : beta distribution + + Notes + ----- + The term *regularized* in the name of this function refers to the + scaling of the function by the gamma function terms shown in the + formula. When not qualified as *regularized*, the name *incomplete + beta function* often refers to just the integral expression, + without the gamma terms. One can use the function `beta` from + `scipy.special` to get this "nonregularized" incomplete beta + function by multiplying the result of ``betainc(a, b, x)`` by + ``beta(a, b)``. + + This function wraps the ``ibeta`` routine from the + Boost Math C++ library [2]_. + + References + ---------- + .. [1] NIST Digital Library of Mathematical Functions + https://dlmf.nist.gov/8.17 + .. [2] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/. + + Examples + -------- + + Let :math:`B(a, b)` be the `beta` function. + + >>> import scipy.special as sc + + The coefficient in terms of `gamma` is equal to + :math:`1/B(a, b)`. Also, when :math:`x=1` + the integral is equal to :math:`B(a, b)`. + Therefore, :math:`I_{x=1}(a, b) = 1` for any :math:`a, b`. + + >>> sc.betainc(0.2, 3.5, 1.0) + 1.0 + + It satisfies + :math:`I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b))`, + where :math:`F` is the hypergeometric function `hyp2f1`: + + >>> a, b, x = 1.4, 3.1, 0.5 + >>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b)) + 0.8148904036225295 + >>> sc.betainc(a, b, x) + 0.8148904036225296 + + This functions satisfies the relationship + :math:`I_x(a, b) = 1 - I_{1-x}(b, a)`: + + >>> sc.betainc(2.2, 3.1, 0.4) + 0.49339638807619446 + >>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4) + 0.49339638807619446 + + """) + + +add_newdoc( + "betaincc", + r""" + betaincc(a, b, x, out=None) + + Complement of the regularized incomplete beta function. + + Computes the complement of the regularized incomplete beta function, + defined as [1]_: + + .. math:: + + \bar{I}_x(a, b) = 1 - I_x(a, b) + = 1 - \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \int_0^x + t^{a-1}(1-t)^{b-1}dt, + + for :math:`0 \leq x \leq 1`. + + Parameters + ---------- + a, b : array_like + Positive, real-valued parameters + x : array_like + Real-valued such that :math:`0 \leq x \leq 1`, + the upper limit of integration + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + Value of the regularized incomplete beta function + + See Also + -------- + betainc : regularized incomplete beta function + betaincinv : inverse of the regularized incomplete beta function + betainccinv : + inverse of the complement of the regularized incomplete beta function + beta : beta function + scipy.stats.beta : beta distribution + + Notes + ----- + .. versionadded:: 1.11.0 + + This function wraps the ``ibetac`` routine from the + Boost Math C++ library [2]_. + + References + ---------- + .. [1] NIST Digital Library of Mathematical Functions + https://dlmf.nist.gov/8.17 + .. [2] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/. + + Examples + -------- + >>> from scipy.special import betaincc, betainc + + The naive calculation ``1 - betainc(a, b, x)`` loses precision when + the values of ``betainc(a, b, x)`` are close to 1: + + >>> 1 - betainc(0.5, 8, [0.9, 0.99, 0.999]) + array([2.0574632e-09, 0.0000000e+00, 0.0000000e+00]) + + By using ``betaincc``, we get the correct values: + + >>> betaincc(0.5, 8, [0.9, 0.99, 0.999]) + array([2.05746321e-09, 1.97259354e-17, 1.96467954e-25]) + + """) + +add_newdoc( + "betaincinv", + r""" + betaincinv(a, b, y, out=None) + + Inverse of the regularized incomplete beta function. + + Computes :math:`x` such that: + + .. math:: + + y = I_x(a, b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} + \int_0^x t^{a-1}(1-t)^{b-1}dt, + + where :math:`I_x` is the normalized incomplete beta function `betainc` + and :math:`\Gamma` is the `gamma` function [1]_. + + Parameters + ---------- + a, b : array_like + Positive, real-valued parameters + y : array_like + Real-valued input + out : ndarray, optional + Optional output array for function values + + Returns + ------- + scalar or ndarray + Value of the inverse of the regularized incomplete beta function + + See Also + -------- + betainc : regularized incomplete beta function + gamma : gamma function + + Notes + ----- + This function wraps the ``ibeta_inv`` routine from the + Boost Math C++ library [2]_. + + References + ---------- + .. [1] NIST Digital Library of Mathematical Functions + https://dlmf.nist.gov/8.17 + .. [2] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/. + + Examples + -------- + >>> import scipy.special as sc + + This function is the inverse of `betainc` for fixed + values of :math:`a` and :math:`b`. + + >>> a, b = 1.2, 3.1 + >>> y = sc.betainc(a, b, 0.2) + >>> sc.betaincinv(a, b, y) + 0.2 + >>> + >>> a, b = 7.5, 0.4 + >>> x = sc.betaincinv(a, b, 0.5) + >>> sc.betainc(a, b, x) + 0.5 + + """) + + +add_newdoc( + "betainccinv", + r""" + betainccinv(a, b, y, out=None) + + Inverse of the complemented regularized incomplete beta function. + + Computes :math:`x` such that: + + .. math:: + + y = 1 - I_x(a, b) = 1 - \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} + \int_0^x t^{a-1}(1-t)^{b-1}dt, + + where :math:`I_x` is the normalized incomplete beta function `betainc` + and :math:`\Gamma` is the `gamma` function [1]_. + + Parameters + ---------- + a, b : array_like + Positive, real-valued parameters + y : array_like + Real-valued input + out : ndarray, optional + Optional output array for function values + + Returns + ------- + scalar or ndarray + Value of the inverse of the regularized incomplete beta function + + See Also + -------- + betainc : regularized incomplete beta function + betaincc : complement of the regularized incomplete beta function + + Notes + ----- + .. versionadded:: 1.11.0 + + This function wraps the ``ibetac_inv`` routine from the + Boost Math C++ library [2]_. + + References + ---------- + .. [1] NIST Digital Library of Mathematical Functions + https://dlmf.nist.gov/8.17 + .. [2] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/. + + Examples + -------- + >>> from scipy.special import betainccinv, betaincc + + This function is the inverse of `betaincc` for fixed + values of :math:`a` and :math:`b`. + + >>> a, b = 1.2, 3.1 + >>> y = betaincc(a, b, 0.2) + >>> betainccinv(a, b, y) + 0.2 + + >>> a, b = 7, 2.5 + >>> x = betainccinv(a, b, 0.875) + >>> betaincc(a, b, x) + 0.875 + + """) + +add_newdoc("boxcox", + """ + boxcox(x, lmbda, out=None) + + Compute the Box-Cox transformation. + + The Box-Cox transformation is:: + + y = (x**lmbda - 1) / lmbda if lmbda != 0 + log(x) if lmbda == 0 + + Returns `nan` if ``x < 0``. + Returns `-inf` if ``x == 0`` and ``lmbda < 0``. + + Parameters + ---------- + x : array_like + Data to be transformed. + lmbda : array_like + Power parameter of the Box-Cox transform. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + y : scalar or ndarray + Transformed data. + + Notes + ----- + + .. versionadded:: 0.14.0 + + Examples + -------- + >>> from scipy.special import boxcox + >>> boxcox([1, 4, 10], 2.5) + array([ 0. , 12.4 , 126.09110641]) + >>> boxcox(2, [0, 1, 2]) + array([ 0.69314718, 1. , 1.5 ]) + """) + +add_newdoc("boxcox1p", + """ + boxcox1p(x, lmbda, out=None) + + Compute the Box-Cox transformation of 1 + `x`. + + The Box-Cox transformation computed by `boxcox1p` is:: + + y = ((1+x)**lmbda - 1) / lmbda if lmbda != 0 + log(1+x) if lmbda == 0 + + Returns `nan` if ``x < -1``. + Returns `-inf` if ``x == -1`` and ``lmbda < 0``. + + Parameters + ---------- + x : array_like + Data to be transformed. + lmbda : array_like + Power parameter of the Box-Cox transform. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + y : scalar or ndarray + Transformed data. + + Notes + ----- + + .. versionadded:: 0.14.0 + + Examples + -------- + >>> from scipy.special import boxcox1p + >>> boxcox1p(1e-4, [0, 0.5, 1]) + array([ 9.99950003e-05, 9.99975001e-05, 1.00000000e-04]) + >>> boxcox1p([0.01, 0.1], 0.25) + array([ 0.00996272, 0.09645476]) + """) + +add_newdoc("inv_boxcox", + """ + inv_boxcox(y, lmbda, out=None) + + Compute the inverse of the Box-Cox transformation. + + Find ``x`` such that:: + + y = (x**lmbda - 1) / lmbda if lmbda != 0 + log(x) if lmbda == 0 + + Parameters + ---------- + y : array_like + Data to be transformed. + lmbda : array_like + Power parameter of the Box-Cox transform. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + x : scalar or ndarray + Transformed data. + + Notes + ----- + + .. versionadded:: 0.16.0 + + Examples + -------- + >>> from scipy.special import boxcox, inv_boxcox + >>> y = boxcox([1, 4, 10], 2.5) + >>> inv_boxcox(y, 2.5) + array([1., 4., 10.]) + """) + +add_newdoc("inv_boxcox1p", + """ + inv_boxcox1p(y, lmbda, out=None) + + Compute the inverse of the Box-Cox transformation. + + Find ``x`` such that:: + + y = ((1+x)**lmbda - 1) / lmbda if lmbda != 0 + log(1+x) if lmbda == 0 + + Parameters + ---------- + y : array_like + Data to be transformed. + lmbda : array_like + Power parameter of the Box-Cox transform. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + x : scalar or ndarray + Transformed data. + + Notes + ----- + + .. versionadded:: 0.16.0 + + Examples + -------- + >>> from scipy.special import boxcox1p, inv_boxcox1p + >>> y = boxcox1p([1, 4, 10], 2.5) + >>> inv_boxcox1p(y, 2.5) + array([1., 4., 10.]) + """) + +add_newdoc("chdtr", + r""" + chdtr(v, x, out=None) + + Chi square cumulative distribution function. + + Returns the area under the left tail (from 0 to `x`) of the Chi + square probability density function with `v` degrees of freedom: + + .. math:: + + \frac{1}{2^{v/2} \Gamma(v/2)} \int_0^x t^{v/2 - 1} e^{-t/2} dt + + Here :math:`\Gamma` is the Gamma function; see `gamma`. This + integral can be expressed in terms of the regularized lower + incomplete gamma function `gammainc` as + ``gammainc(v / 2, x / 2)``. [1]_ + + Parameters + ---------- + v : array_like + Degrees of freedom. + x : array_like + Upper bound of the integral. + out : ndarray, optional + Optional output array for the function results. + + Returns + ------- + scalar or ndarray + Values of the cumulative distribution function. + + See Also + -------- + chdtrc, chdtri, chdtriv, gammainc + + References + ---------- + .. [1] Chi-Square distribution, + https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + It can be expressed in terms of the regularized lower incomplete + gamma function. + + >>> v = 1 + >>> x = np.arange(4) + >>> sc.chdtr(v, x) + array([0. , 0.68268949, 0.84270079, 0.91673548]) + >>> sc.gammainc(v / 2, x / 2) + array([0. , 0.68268949, 0.84270079, 0.91673548]) + + """) + +add_newdoc("chdtrc", + r""" + chdtrc(v, x, out=None) + + Chi square survival function. + + Returns the area under the right hand tail (from `x` to infinity) + of the Chi square probability density function with `v` degrees of + freedom: + + .. math:: + + \frac{1}{2^{v/2} \Gamma(v/2)} \int_x^\infty t^{v/2 - 1} e^{-t/2} dt + + Here :math:`\Gamma` is the Gamma function; see `gamma`. This + integral can be expressed in terms of the regularized upper + incomplete gamma function `gammaincc` as + ``gammaincc(v / 2, x / 2)``. [1]_ + + Parameters + ---------- + v : array_like + Degrees of freedom. + x : array_like + Lower bound of the integral. + out : ndarray, optional + Optional output array for the function results. + + Returns + ------- + scalar or ndarray + Values of the survival function. + + See Also + -------- + chdtr, chdtri, chdtriv, gammaincc + + References + ---------- + .. [1] Chi-Square distribution, + https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + It can be expressed in terms of the regularized upper incomplete + gamma function. + + >>> v = 1 + >>> x = np.arange(4) + >>> sc.chdtrc(v, x) + array([1. , 0.31731051, 0.15729921, 0.08326452]) + >>> sc.gammaincc(v / 2, x / 2) + array([1. , 0.31731051, 0.15729921, 0.08326452]) + + """) + +add_newdoc("chdtri", + """ + chdtri(v, p, out=None) + + Inverse to `chdtrc` with respect to `x`. + + Returns `x` such that ``chdtrc(v, x) == p``. + + Parameters + ---------- + v : array_like + Degrees of freedom. + p : array_like + Probability. + out : ndarray, optional + Optional output array for the function results. + + Returns + ------- + x : scalar or ndarray + Value so that the probability a Chi square random variable + with `v` degrees of freedom is greater than `x` equals `p`. + + See Also + -------- + chdtrc, chdtr, chdtriv + + References + ---------- + .. [1] Chi-Square distribution, + https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm + + Examples + -------- + >>> import scipy.special as sc + + It inverts `chdtrc`. + + >>> v, p = 1, 0.3 + >>> sc.chdtrc(v, sc.chdtri(v, p)) + 0.3 + >>> x = 1 + >>> sc.chdtri(v, sc.chdtrc(v, x)) + 1.0 + + """) + +add_newdoc("chdtriv", + """ + chdtriv(p, x, out=None) + + Inverse to `chdtr` with respect to `v`. + + Returns `v` such that ``chdtr(v, x) == p``. + + Parameters + ---------- + p : array_like + Probability that the Chi square random variable is less than + or equal to `x`. + x : array_like + Nonnegative input. + out : ndarray, optional + Optional output array for the function results. + + Returns + ------- + scalar or ndarray + Degrees of freedom. + + See Also + -------- + chdtr, chdtrc, chdtri + + References + ---------- + .. [1] Chi-Square distribution, + https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm + + Examples + -------- + >>> import scipy.special as sc + + It inverts `chdtr`. + + >>> p, x = 0.5, 1 + >>> sc.chdtr(sc.chdtriv(p, x), x) + 0.5000000000202172 + >>> v = 1 + >>> sc.chdtriv(sc.chdtr(v, x), v) + 1.0000000000000013 + + """) + +add_newdoc("chndtr", + r""" + chndtr(x, df, nc, out=None) + + Non-central chi square cumulative distribution function + + The cumulative distribution function is given by: + + .. math:: + + P(\chi^{\prime 2} \vert \nu, \lambda) =\sum_{j=0}^{\infty} + e^{-\lambda /2} + \frac{(\lambda /2)^j}{j!} P(\chi^{\prime 2} \vert \nu + 2j), + + where :math:`\nu > 0` is the degrees of freedom (``df``) and + :math:`\lambda \geq 0` is the non-centrality parameter (``nc``). + + Parameters + ---------- + x : array_like + Upper bound of the integral; must satisfy ``x >= 0`` + df : array_like + Degrees of freedom; must satisfy ``df > 0`` + nc : array_like + Non-centrality parameter; must satisfy ``nc >= 0`` + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + x : scalar or ndarray + Value of the non-central chi square cumulative distribution function. + + See Also + -------- + chndtrix, chndtridf, chndtrinc + + """) + +add_newdoc("chndtrix", + """ + chndtrix(p, df, nc, out=None) + + Inverse to `chndtr` vs `x` + + Calculated using a search to find a value for `x` that produces the + desired value of `p`. + + Parameters + ---------- + p : array_like + Probability; must satisfy ``0 <= p < 1`` + df : array_like + Degrees of freedom; must satisfy ``df > 0`` + nc : array_like + Non-centrality parameter; must satisfy ``nc >= 0`` + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + x : scalar or ndarray + Value so that the probability a non-central Chi square random variable + with `df` degrees of freedom and non-centrality, `nc`, is greater than + `x` equals `p`. + + See Also + -------- + chndtr, chndtridf, chndtrinc + + """) + +add_newdoc("chndtridf", + """ + chndtridf(x, p, nc, out=None) + + Inverse to `chndtr` vs `df` + + Calculated using a search to find a value for `df` that produces the + desired value of `p`. + + Parameters + ---------- + x : array_like + Upper bound of the integral; must satisfy ``x >= 0`` + p : array_like + Probability; must satisfy ``0 <= p < 1`` + nc : array_like + Non-centrality parameter; must satisfy ``nc >= 0`` + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + df : scalar or ndarray + Degrees of freedom + + See Also + -------- + chndtr, chndtrix, chndtrinc + + """) + +add_newdoc("chndtrinc", + """ + chndtrinc(x, df, p, out=None) + + Inverse to `chndtr` vs `nc` + + Calculated using a search to find a value for `df` that produces the + desired value of `p`. + + Parameters + ---------- + x : array_like + Upper bound of the integral; must satisfy ``x >= 0`` + df : array_like + Degrees of freedom; must satisfy ``df > 0`` + p : array_like + Probability; must satisfy ``0 <= p < 1`` + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + nc : scalar or ndarray + Non-centrality + + See Also + -------- + chndtr, chndtrix, chndtrinc + + """) + +add_newdoc("dawsn", + """ + dawsn(x, out=None) + + Dawson's integral. + + Computes:: + + exp(-x**2) * integral(exp(t**2), t=0..x). + + Parameters + ---------- + x : array_like + Function parameter. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + y : scalar or ndarray + Value of the integral. + + See Also + -------- + wofz, erf, erfc, erfcx, erfi + + References + ---------- + .. [1] Steven G. Johnson, Faddeeva W function implementation. + http://ab-initio.mit.edu/Faddeeva + + Examples + -------- + >>> import numpy as np + >>> from scipy import special + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(-15, 15, num=1000) + >>> plt.plot(x, special.dawsn(x)) + >>> plt.xlabel('$x$') + >>> plt.ylabel('$dawsn(x)$') + >>> plt.show() + + """) + +add_newdoc( + "elliprc", + r""" + elliprc(x, y, out=None) + + Degenerate symmetric elliptic integral. + + The function RC is defined as [1]_ + + .. math:: + + R_{\mathrm{C}}(x, y) = + \frac{1}{2} \int_0^{+\infty} (t + x)^{-1/2} (t + y)^{-1} dt + = R_{\mathrm{F}}(x, y, y) + + Parameters + ---------- + x, y : array_like + Real or complex input parameters. `x` can be any number in the + complex plane cut along the negative real axis. `y` must be non-zero. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + R : scalar or ndarray + Value of the integral. If `y` is real and negative, the Cauchy + principal value is returned. If both of `x` and `y` are real, the + return value is real. Otherwise, the return value is complex. + + See Also + -------- + elliprf : Completely-symmetric elliptic integral of the first kind. + elliprd : Symmetric elliptic integral of the second kind. + elliprg : Completely-symmetric elliptic integral of the second kind. + elliprj : Symmetric elliptic integral of the third kind. + + Notes + ----- + RC is a degenerate case of the symmetric integral RF: ``elliprc(x, y) == + elliprf(x, y, y)``. It is an elementary function rather than an elliptic + integral. + + The code implements Carlson's algorithm based on the duplication theorems + and series expansion up to the 7th order. [2]_ + + .. versionadded:: 1.8.0 + + References + ---------- + .. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical + Functions," NIST, US Dept. of Commerce. + https://dlmf.nist.gov/19.16.E6 + .. [2] B. C. Carlson, "Numerical computation of real or complex elliptic + integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995. + https://arxiv.org/abs/math/9409227 + https://doi.org/10.1007/BF02198293 + + Examples + -------- + Basic homogeneity property: + + >>> import numpy as np + >>> from scipy.special import elliprc + + >>> x = 1.2 + 3.4j + >>> y = 5. + >>> scale = 0.3 + 0.4j + >>> elliprc(scale*x, scale*y) + (0.5484493976710874-0.4169557678995833j) + + >>> elliprc(x, y)/np.sqrt(scale) + (0.5484493976710874-0.41695576789958333j) + + When the two arguments coincide, the integral is particularly + simple: + + >>> x = 1.2 + 3.4j + >>> elliprc(x, x) + (0.4299173120614631-0.3041729818745595j) + + >>> 1/np.sqrt(x) + (0.4299173120614631-0.30417298187455954j) + + Another simple case: the first argument vanishes: + + >>> y = 1.2 + 3.4j + >>> elliprc(0, y) + (0.6753125346116815-0.47779380263880866j) + + >>> np.pi/2/np.sqrt(y) + (0.6753125346116815-0.4777938026388088j) + + When `x` and `y` are both positive, we can express + :math:`R_C(x,y)` in terms of more elementary functions. For the + case :math:`0 \le x < y`, + + >>> x = 3.2 + >>> y = 6. + >>> elliprc(x, y) + 0.44942991498453444 + + >>> np.arctan(np.sqrt((y-x)/x))/np.sqrt(y-x) + 0.44942991498453433 + + And for the case :math:`0 \le y < x`, + + >>> x = 6. + >>> y = 3.2 + >>> elliprc(x,y) + 0.4989837501576147 + + >>> np.log((np.sqrt(x)+np.sqrt(x-y))/np.sqrt(y))/np.sqrt(x-y) + 0.49898375015761476 + + """) + +add_newdoc( + "elliprd", + r""" + elliprd(x, y, z, out=None) + + Symmetric elliptic integral of the second kind. + + The function RD is defined as [1]_ + + .. math:: + + R_{\mathrm{D}}(x, y, z) = + \frac{3}{2} \int_0^{+\infty} [(t + x) (t + y)]^{-1/2} (t + z)^{-3/2} + dt + + Parameters + ---------- + x, y, z : array_like + Real or complex input parameters. `x` or `y` can be any number in the + complex plane cut along the negative real axis, but at most one of them + can be zero, while `z` must be non-zero. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + R : scalar or ndarray + Value of the integral. If all of `x`, `y`, and `z` are real, the + return value is real. Otherwise, the return value is complex. + + See Also + -------- + elliprc : Degenerate symmetric elliptic integral. + elliprf : Completely-symmetric elliptic integral of the first kind. + elliprg : Completely-symmetric elliptic integral of the second kind. + elliprj : Symmetric elliptic integral of the third kind. + + Notes + ----- + RD is a degenerate case of the elliptic integral RJ: ``elliprd(x, y, z) == + elliprj(x, y, z, z)``. + + The code implements Carlson's algorithm based on the duplication theorems + and series expansion up to the 7th order. [2]_ + + .. versionadded:: 1.8.0 + + References + ---------- + .. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical + Functions," NIST, US Dept. of Commerce. + https://dlmf.nist.gov/19.16.E5 + .. [2] B. C. Carlson, "Numerical computation of real or complex elliptic + integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995. + https://arxiv.org/abs/math/9409227 + https://doi.org/10.1007/BF02198293 + + Examples + -------- + Basic homogeneity property: + + >>> import numpy as np + >>> from scipy.special import elliprd + + >>> x = 1.2 + 3.4j + >>> y = 5. + >>> z = 6. + >>> scale = 0.3 + 0.4j + >>> elliprd(scale*x, scale*y, scale*z) + (-0.03703043835680379-0.24500934665683802j) + + >>> elliprd(x, y, z)*np.power(scale, -1.5) + (-0.0370304383568038-0.24500934665683805j) + + All three arguments coincide: + + >>> x = 1.2 + 3.4j + >>> elliprd(x, x, x) + (-0.03986825876151896-0.14051741840449586j) + + >>> np.power(x, -1.5) + (-0.03986825876151894-0.14051741840449583j) + + The so-called "second lemniscate constant": + + >>> elliprd(0, 2, 1)/3 + 0.5990701173677961 + + >>> from scipy.special import gamma + >>> gamma(0.75)**2/np.sqrt(2*np.pi) + 0.5990701173677959 + + """) + +add_newdoc( + "elliprf", + r""" + elliprf(x, y, z, out=None) + + Completely-symmetric elliptic integral of the first kind. + + The function RF is defined as [1]_ + + .. math:: + + R_{\mathrm{F}}(x, y, z) = + \frac{1}{2} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} dt + + Parameters + ---------- + x, y, z : array_like + Real or complex input parameters. `x`, `y`, or `z` can be any number in + the complex plane cut along the negative real axis, but at most one of + them can be zero. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + R : scalar or ndarray + Value of the integral. If all of `x`, `y`, and `z` are real, the return + value is real. Otherwise, the return value is complex. + + See Also + -------- + elliprc : Degenerate symmetric integral. + elliprd : Symmetric elliptic integral of the second kind. + elliprg : Completely-symmetric elliptic integral of the second kind. + elliprj : Symmetric elliptic integral of the third kind. + + Notes + ----- + The code implements Carlson's algorithm based on the duplication theorems + and series expansion up to the 7th order (cf.: + https://dlmf.nist.gov/19.36.i) and the AGM algorithm for the complete + integral. [2]_ + + .. versionadded:: 1.8.0 + + References + ---------- + .. [1] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical + Functions," NIST, US Dept. of Commerce. + https://dlmf.nist.gov/19.16.E1 + .. [2] B. C. Carlson, "Numerical computation of real or complex elliptic + integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995. + https://arxiv.org/abs/math/9409227 + https://doi.org/10.1007/BF02198293 + + Examples + -------- + Basic homogeneity property: + + >>> import numpy as np + >>> from scipy.special import elliprf + + >>> x = 1.2 + 3.4j + >>> y = 5. + >>> z = 6. + >>> scale = 0.3 + 0.4j + >>> elliprf(scale*x, scale*y, scale*z) + (0.5328051227278146-0.4008623567957094j) + + >>> elliprf(x, y, z)/np.sqrt(scale) + (0.5328051227278147-0.4008623567957095j) + + All three arguments coincide: + + >>> x = 1.2 + 3.4j + >>> elliprf(x, x, x) + (0.42991731206146316-0.30417298187455954j) + + >>> 1/np.sqrt(x) + (0.4299173120614631-0.30417298187455954j) + + The so-called "first lemniscate constant": + + >>> elliprf(0, 1, 2) + 1.3110287771460598 + + >>> from scipy.special import gamma + >>> gamma(0.25)**2/(4*np.sqrt(2*np.pi)) + 1.3110287771460598 + + """) + +add_newdoc( + "elliprg", + r""" + elliprg(x, y, z, out=None) + + Completely-symmetric elliptic integral of the second kind. + + The function RG is defined as [1]_ + + .. math:: + + R_{\mathrm{G}}(x, y, z) = + \frac{1}{4} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} + \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) t + dt + + Parameters + ---------- + x, y, z : array_like + Real or complex input parameters. `x`, `y`, or `z` can be any number in + the complex plane cut along the negative real axis. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + R : scalar or ndarray + Value of the integral. If all of `x`, `y`, and `z` are real, the return + value is real. Otherwise, the return value is complex. + + See Also + -------- + elliprc : Degenerate symmetric integral. + elliprd : Symmetric elliptic integral of the second kind. + elliprf : Completely-symmetric elliptic integral of the first kind. + elliprj : Symmetric elliptic integral of the third kind. + + Notes + ----- + The implementation uses the relation [1]_ + + .. math:: + + 2 R_{\mathrm{G}}(x, y, z) = + z R_{\mathrm{F}}(x, y, z) - + \frac{1}{3} (x - z) (y - z) R_{\mathrm{D}}(x, y, z) + + \sqrt{\frac{x y}{z}} + + and the symmetry of `x`, `y`, `z` when at least one non-zero parameter can + be chosen as the pivot. When one of the arguments is close to zero, the AGM + method is applied instead. Other special cases are computed following Ref. + [2]_ + + .. versionadded:: 1.8.0 + + References + ---------- + .. [1] B. C. Carlson, "Numerical computation of real or complex elliptic + integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995. + https://arxiv.org/abs/math/9409227 + https://doi.org/10.1007/BF02198293 + .. [2] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical + Functions," NIST, US Dept. of Commerce. + https://dlmf.nist.gov/19.16.E1 + https://dlmf.nist.gov/19.20.ii + + Examples + -------- + Basic homogeneity property: + + >>> import numpy as np + >>> from scipy.special import elliprg + + >>> x = 1.2 + 3.4j + >>> y = 5. + >>> z = 6. + >>> scale = 0.3 + 0.4j + >>> elliprg(scale*x, scale*y, scale*z) + (1.195936862005246+0.8470988320464167j) + + >>> elliprg(x, y, z)*np.sqrt(scale) + (1.195936862005246+0.8470988320464165j) + + Simplifications: + + >>> elliprg(0, y, y) + 1.756203682760182 + + >>> 0.25*np.pi*np.sqrt(y) + 1.7562036827601817 + + >>> elliprg(0, 0, z) + 1.224744871391589 + + >>> 0.5*np.sqrt(z) + 1.224744871391589 + + The surface area of a triaxial ellipsoid with semiaxes ``a``, ``b``, and + ``c`` is given by + + .. math:: + + S = 4 \pi a b c R_{\mathrm{G}}(1 / a^2, 1 / b^2, 1 / c^2). + + >>> def ellipsoid_area(a, b, c): + ... r = 4.0 * np.pi * a * b * c + ... return r * elliprg(1.0 / (a * a), 1.0 / (b * b), 1.0 / (c * c)) + >>> print(ellipsoid_area(1, 3, 5)) + 108.62688289491807 + """) + +add_newdoc( + "elliprj", + r""" + elliprj(x, y, z, p, out=None) + + Symmetric elliptic integral of the third kind. + + The function RJ is defined as [1]_ + + .. math:: + + R_{\mathrm{J}}(x, y, z, p) = + \frac{3}{2} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} + (t + p)^{-1} dt + + .. warning:: + This function should be considered experimental when the inputs are + unbalanced. Check correctness with another independent implementation. + + Parameters + ---------- + x, y, z, p : array_like + Real or complex input parameters. `x`, `y`, or `z` are numbers in + the complex plane cut along the negative real axis (subject to further + constraints, see Notes), and at most one of them can be zero. `p` must + be non-zero. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + R : scalar or ndarray + Value of the integral. If all of `x`, `y`, `z`, and `p` are real, the + return value is real. Otherwise, the return value is complex. + + If `p` is real and negative, while `x`, `y`, and `z` are real, + non-negative, and at most one of them is zero, the Cauchy principal + value is returned. [1]_ [2]_ + + See Also + -------- + elliprc : Degenerate symmetric integral. + elliprd : Symmetric elliptic integral of the second kind. + elliprf : Completely-symmetric elliptic integral of the first kind. + elliprg : Completely-symmetric elliptic integral of the second kind. + + Notes + ----- + The code implements Carlson's algorithm based on the duplication theorems + and series expansion up to the 7th order. [3]_ The algorithm is slightly + different from its earlier incarnation as it appears in [1]_, in that the + call to `elliprc` (or ``atan``/``atanh``, see [4]_) is no longer needed in + the inner loop. Asymptotic approximations are used where arguments differ + widely in the order of magnitude. [5]_ + + The input values are subject to certain sufficient but not necessary + constraints when input arguments are complex. Notably, ``x``, ``y``, and + ``z`` must have non-negative real parts, unless two of them are + non-negative and complex-conjugates to each other while the other is a real + non-negative number. [1]_ If the inputs do not satisfy the sufficient + condition described in Ref. [1]_ they are rejected outright with the output + set to NaN. + + In the case where one of ``x``, ``y``, and ``z`` is equal to ``p``, the + function ``elliprd`` should be preferred because of its less restrictive + domain. + + .. versionadded:: 1.8.0 + + References + ---------- + .. [1] B. C. Carlson, "Numerical computation of real or complex elliptic + integrals," Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995. + https://arxiv.org/abs/math/9409227 + https://doi.org/10.1007/BF02198293 + .. [2] B. C. Carlson, ed., Chapter 19 in "Digital Library of Mathematical + Functions," NIST, US Dept. of Commerce. + https://dlmf.nist.gov/19.20.iii + .. [3] B. C. Carlson, J. FitzSimmons, "Reduction Theorems for Elliptic + Integrands with the Square Root of Two Quadratic Factors," J. + Comput. Appl. Math., vol. 118, nos. 1-2, pp. 71-85, 2000. + https://doi.org/10.1016/S0377-0427(00)00282-X + .. [4] F. Johansson, "Numerical Evaluation of Elliptic Functions, Elliptic + Integrals and Modular Forms," in J. Blumlein, C. Schneider, P. + Paule, eds., "Elliptic Integrals, Elliptic Functions and Modular + Forms in Quantum Field Theory," pp. 269-293, 2019 (Cham, + Switzerland: Springer Nature Switzerland) + https://arxiv.org/abs/1806.06725 + https://doi.org/10.1007/978-3-030-04480-0 + .. [5] B. C. Carlson, J. L. Gustafson, "Asymptotic Approximations for + Symmetric Elliptic Integrals," SIAM J. Math. Anls., vol. 25, no. 2, + pp. 288-303, 1994. + https://arxiv.org/abs/math/9310223 + https://doi.org/10.1137/S0036141092228477 + + Examples + -------- + Basic homogeneity property: + + >>> import numpy as np + >>> from scipy.special import elliprj + + >>> x = 1.2 + 3.4j + >>> y = 5. + >>> z = 6. + >>> p = 7. + >>> scale = 0.3 - 0.4j + >>> elliprj(scale*x, scale*y, scale*z, scale*p) + (0.10834905565679157+0.19694950747103812j) + + >>> elliprj(x, y, z, p)*np.power(scale, -1.5) + (0.10834905565679556+0.19694950747103854j) + + Reduction to simpler elliptic integral: + + >>> elliprj(x, y, z, z) + (0.08288462362195129-0.028376809745123258j) + + >>> from scipy.special import elliprd + >>> elliprd(x, y, z) + (0.08288462362195136-0.028376809745123296j) + + All arguments coincide: + + >>> elliprj(x, x, x, x) + (-0.03986825876151896-0.14051741840449586j) + + >>> np.power(x, -1.5) + (-0.03986825876151894-0.14051741840449583j) + + """) + +add_newdoc("entr", + r""" + entr(x, out=None) + + Elementwise function for computing entropy. + + .. math:: \text{entr}(x) = \begin{cases} - x \log(x) & x > 0 \\ 0 & x = 0 + \\ -\infty & \text{otherwise} \end{cases} + + Parameters + ---------- + x : ndarray + Input array. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + res : scalar or ndarray + The value of the elementwise entropy function at the given points `x`. + + See Also + -------- + kl_div, rel_entr, scipy.stats.entropy + + Notes + ----- + .. versionadded:: 0.15.0 + + This function is concave. + + The origin of this function is in convex programming; see [1]_. + Given a probability distribution :math:`p_1, \ldots, p_n`, + the definition of entropy in the context of *information theory* is + + .. math:: + + \sum_{i = 1}^n \mathrm{entr}(p_i). + + To compute the latter quantity, use `scipy.stats.entropy`. + + References + ---------- + .. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*. + Cambridge University Press, 2004. + :doi:`https://doi.org/10.1017/CBO9780511804441` + + """) + +add_newdoc("erf", + """ + erf(z, out=None) + + Returns the error function of complex argument. + + It is defined as ``2/sqrt(pi)*integral(exp(-t**2), t=0..z)``. + + Parameters + ---------- + x : ndarray + Input array. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + res : scalar or ndarray + The values of the error function at the given points `x`. + + See Also + -------- + erfc, erfinv, erfcinv, wofz, erfcx, erfi + + Notes + ----- + The cumulative of the unit normal distribution is given by + ``Phi(z) = 1/2[1 + erf(z/sqrt(2))]``. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Error_function + .. [2] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, + 1972. http://www.math.sfu.ca/~cbm/aands/page_297.htm + .. [3] Steven G. Johnson, Faddeeva W function implementation. + http://ab-initio.mit.edu/Faddeeva + + Examples + -------- + >>> import numpy as np + >>> from scipy import special + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(-3, 3) + >>> plt.plot(x, special.erf(x)) + >>> plt.xlabel('$x$') + >>> plt.ylabel('$erf(x)$') + >>> plt.show() + + """) + +add_newdoc("erfc", + """ + erfc(x, out=None) + + Complementary error function, ``1 - erf(x)``. + + Parameters + ---------- + x : array_like + Real or complex valued argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Values of the complementary error function + + See Also + -------- + erf, erfi, erfcx, dawsn, wofz + + References + ---------- + .. [1] Steven G. Johnson, Faddeeva W function implementation. + http://ab-initio.mit.edu/Faddeeva + + Examples + -------- + >>> import numpy as np + >>> from scipy import special + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(-3, 3) + >>> plt.plot(x, special.erfc(x)) + >>> plt.xlabel('$x$') + >>> plt.ylabel('$erfc(x)$') + >>> plt.show() + + """) + +add_newdoc("erfi", + """ + erfi(z, out=None) + + Imaginary error function, ``-i erf(i z)``. + + Parameters + ---------- + z : array_like + Real or complex valued argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Values of the imaginary error function + + See Also + -------- + erf, erfc, erfcx, dawsn, wofz + + Notes + ----- + + .. versionadded:: 0.12.0 + + References + ---------- + .. [1] Steven G. Johnson, Faddeeva W function implementation. + http://ab-initio.mit.edu/Faddeeva + + Examples + -------- + >>> import numpy as np + >>> from scipy import special + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(-3, 3) + >>> plt.plot(x, special.erfi(x)) + >>> plt.xlabel('$x$') + >>> plt.ylabel('$erfi(x)$') + >>> plt.show() + + """) + +add_newdoc("erfcx", + """ + erfcx(x, out=None) + + Scaled complementary error function, ``exp(x**2) * erfc(x)``. + + Parameters + ---------- + x : array_like + Real or complex valued argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Values of the scaled complementary error function + + + See Also + -------- + erf, erfc, erfi, dawsn, wofz + + Notes + ----- + + .. versionadded:: 0.12.0 + + References + ---------- + .. [1] Steven G. Johnson, Faddeeva W function implementation. + http://ab-initio.mit.edu/Faddeeva + + Examples + -------- + >>> import numpy as np + >>> from scipy import special + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(-3, 3) + >>> plt.plot(x, special.erfcx(x)) + >>> plt.xlabel('$x$') + >>> plt.ylabel('$erfcx(x)$') + >>> plt.show() + + """) + +add_newdoc( + "erfinv", + """ + erfinv(y, out=None) + + Inverse of the error function. + + Computes the inverse of the error function. + + In the complex domain, there is no unique complex number w satisfying + erf(w)=z. This indicates a true inverse function would be multivalued. + When the domain restricts to the real, -1 < x < 1, there is a unique real + number satisfying erf(erfinv(x)) = x. + + Parameters + ---------- + y : ndarray + Argument at which to evaluate. Domain: [-1, 1] + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + erfinv : scalar or ndarray + The inverse of erf of y, element-wise + + See Also + -------- + erf : Error function of a complex argument + erfc : Complementary error function, ``1 - erf(x)`` + erfcinv : Inverse of the complementary error function + + Notes + ----- + This function wraps the ``erf_inv`` routine from the + Boost Math C++ library [1]_. + + References + ---------- + .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/. + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.special import erfinv, erf + + >>> erfinv(0.5) + 0.4769362762044699 + + >>> y = np.linspace(-1.0, 1.0, num=9) + >>> x = erfinv(y) + >>> x + array([ -inf, -0.81341985, -0.47693628, -0.22531206, 0. , + 0.22531206, 0.47693628, 0.81341985, inf]) + + Verify that ``erf(erfinv(y))`` is ``y``. + + >>> erf(x) + array([-1. , -0.75, -0.5 , -0.25, 0. , 0.25, 0.5 , 0.75, 1. ]) + + Plot the function: + + >>> y = np.linspace(-1, 1, 200) + >>> fig, ax = plt.subplots() + >>> ax.plot(y, erfinv(y)) + >>> ax.grid(True) + >>> ax.set_xlabel('y') + >>> ax.set_title('erfinv(y)') + >>> plt.show() + + """) + +add_newdoc( + "erfcinv", + """ + erfcinv(y, out=None) + + Inverse of the complementary error function. + + Computes the inverse of the complementary error function. + + In the complex domain, there is no unique complex number w satisfying + erfc(w)=z. This indicates a true inverse function would be multivalued. + When the domain restricts to the real, 0 < x < 2, there is a unique real + number satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)). + + It is related to inverse of the error function by erfcinv(1-x) = erfinv(x) + + Parameters + ---------- + y : ndarray + Argument at which to evaluate. Domain: [0, 2] + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + erfcinv : scalar or ndarray + The inverse of erfc of y, element-wise + + See Also + -------- + erf : Error function of a complex argument + erfc : Complementary error function, ``1 - erf(x)`` + erfinv : Inverse of the error function + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.special import erfcinv + + >>> erfcinv(0.5) + 0.4769362762044699 + + >>> y = np.linspace(0.0, 2.0, num=11) + >>> erfcinv(y) + array([ inf, 0.9061938 , 0.59511608, 0.37080716, 0.17914345, + -0. , -0.17914345, -0.37080716, -0.59511608, -0.9061938 , + -inf]) + + Plot the function: + + >>> y = np.linspace(0, 2, 200) + >>> fig, ax = plt.subplots() + >>> ax.plot(y, erfcinv(y)) + >>> ax.grid(True) + >>> ax.set_xlabel('y') + >>> ax.set_title('erfcinv(y)') + >>> plt.show() + + """) + +add_newdoc("eval_jacobi", + r""" + eval_jacobi(n, alpha, beta, x, out=None) + + Evaluate Jacobi polynomial at a point. + + The Jacobi polynomials can be defined via the Gauss hypergeometric + function :math:`{}_2F_1` as + + .. math:: + + P_n^{(\alpha, \beta)}(x) = \frac{(\alpha + 1)_n}{\Gamma(n + 1)} + {}_2F_1(-n, 1 + \alpha + \beta + n; \alpha + 1; (1 - z)/2) + + where :math:`(\cdot)_n` is the Pochhammer symbol; see `poch`. When + :math:`n` is an integer the result is a polynomial of degree + :math:`n`. See 22.5.42 in [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer the result is + determined via the relation to the Gauss hypergeometric + function. + alpha : array_like + Parameter + beta : array_like + Parameter + x : array_like + Points at which to evaluate the polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + P : scalar or ndarray + Values of the Jacobi polynomial + + See Also + -------- + roots_jacobi : roots and quadrature weights of Jacobi polynomials + jacobi : Jacobi polynomial object + hyp2f1 : Gauss hypergeometric function + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_sh_jacobi", + r""" + eval_sh_jacobi(n, p, q, x, out=None) + + Evaluate shifted Jacobi polynomial at a point. + + Defined by + + .. math:: + + G_n^{(p, q)}(x) + = \binom{2n + p - 1}{n}^{-1} P_n^{(p - q, q - 1)}(2x - 1), + + where :math:`P_n^{(\cdot, \cdot)}` is the n-th Jacobi + polynomial. See 22.5.2 in [AS]_ for details. + + Parameters + ---------- + n : int + Degree of the polynomial. If not an integer, the result is + determined via the relation to `binom` and `eval_jacobi`. + p : float + Parameter + q : float + Parameter + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + G : scalar or ndarray + Values of the shifted Jacobi polynomial. + + See Also + -------- + roots_sh_jacobi : roots and quadrature weights of shifted Jacobi + polynomials + sh_jacobi : shifted Jacobi polynomial object + eval_jacobi : evaluate Jacobi polynomials + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_gegenbauer", + r""" + eval_gegenbauer(n, alpha, x, out=None) + + Evaluate Gegenbauer polynomial at a point. + + The Gegenbauer polynomials can be defined via the Gauss + hypergeometric function :math:`{}_2F_1` as + + .. math:: + + C_n^{(\alpha)} = \frac{(2\alpha)_n}{\Gamma(n + 1)} + {}_2F_1(-n, 2\alpha + n; \alpha + 1/2; (1 - z)/2). + + When :math:`n` is an integer the result is a polynomial of degree + :math:`n`. See 22.5.46 in [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer, the result is + determined via the relation to the Gauss hypergeometric + function. + alpha : array_like + Parameter + x : array_like + Points at which to evaluate the Gegenbauer polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + C : scalar or ndarray + Values of the Gegenbauer polynomial + + See Also + -------- + roots_gegenbauer : roots and quadrature weights of Gegenbauer + polynomials + gegenbauer : Gegenbauer polynomial object + hyp2f1 : Gauss hypergeometric function + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_chebyt", + r""" + eval_chebyt(n, x, out=None) + + Evaluate Chebyshev polynomial of the first kind at a point. + + The Chebyshev polynomials of the first kind can be defined via the + Gauss hypergeometric function :math:`{}_2F_1` as + + .. math:: + + T_n(x) = {}_2F_1(n, -n; 1/2; (1 - x)/2). + + When :math:`n` is an integer the result is a polynomial of degree + :math:`n`. See 22.5.47 in [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer, the result is + determined via the relation to the Gauss hypergeometric + function. + x : array_like + Points at which to evaluate the Chebyshev polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + T : scalar or ndarray + Values of the Chebyshev polynomial + + See Also + -------- + roots_chebyt : roots and quadrature weights of Chebyshev + polynomials of the first kind + chebyu : Chebychev polynomial object + eval_chebyu : evaluate Chebyshev polynomials of the second kind + hyp2f1 : Gauss hypergeometric function + numpy.polynomial.chebyshev.Chebyshev : Chebyshev series + + Notes + ----- + This routine is numerically stable for `x` in ``[-1, 1]`` at least + up to order ``10000``. + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_chebyu", + r""" + eval_chebyu(n, x, out=None) + + Evaluate Chebyshev polynomial of the second kind at a point. + + The Chebyshev polynomials of the second kind can be defined via + the Gauss hypergeometric function :math:`{}_2F_1` as + + .. math:: + + U_n(x) = (n + 1) {}_2F_1(-n, n + 2; 3/2; (1 - x)/2). + + When :math:`n` is an integer the result is a polynomial of degree + :math:`n`. See 22.5.48 in [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer, the result is + determined via the relation to the Gauss hypergeometric + function. + x : array_like + Points at which to evaluate the Chebyshev polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + U : scalar or ndarray + Values of the Chebyshev polynomial + + See Also + -------- + roots_chebyu : roots and quadrature weights of Chebyshev + polynomials of the second kind + chebyu : Chebyshev polynomial object + eval_chebyt : evaluate Chebyshev polynomials of the first kind + hyp2f1 : Gauss hypergeometric function + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_chebys", + r""" + eval_chebys(n, x, out=None) + + Evaluate Chebyshev polynomial of the second kind on [-2, 2] at a + point. + + These polynomials are defined as + + .. math:: + + S_n(x) = U_n(x/2) + + where :math:`U_n` is a Chebyshev polynomial of the second + kind. See 22.5.13 in [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer, the result is + determined via the relation to `eval_chebyu`. + x : array_like + Points at which to evaluate the Chebyshev polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + S : scalar or ndarray + Values of the Chebyshev polynomial + + See Also + -------- + roots_chebys : roots and quadrature weights of Chebyshev + polynomials of the second kind on [-2, 2] + chebys : Chebyshev polynomial object + eval_chebyu : evaluate Chebyshev polynomials of the second kind + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + They are a scaled version of the Chebyshev polynomials of the + second kind. + + >>> x = np.linspace(-2, 2, 6) + >>> sc.eval_chebys(3, x) + array([-4. , 0.672, 0.736, -0.736, -0.672, 4. ]) + >>> sc.eval_chebyu(3, x / 2) + array([-4. , 0.672, 0.736, -0.736, -0.672, 4. ]) + + """) + +add_newdoc("eval_chebyc", + r""" + eval_chebyc(n, x, out=None) + + Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a + point. + + These polynomials are defined as + + .. math:: + + C_n(x) = 2 T_n(x/2) + + where :math:`T_n` is a Chebyshev polynomial of the first kind. See + 22.5.11 in [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer, the result is + determined via the relation to `eval_chebyt`. + x : array_like + Points at which to evaluate the Chebyshev polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + C : scalar or ndarray + Values of the Chebyshev polynomial + + See Also + -------- + roots_chebyc : roots and quadrature weights of Chebyshev + polynomials of the first kind on [-2, 2] + chebyc : Chebyshev polynomial object + numpy.polynomial.chebyshev.Chebyshev : Chebyshev series + eval_chebyt : evaluate Chebycshev polynomials of the first kind + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + They are a scaled version of the Chebyshev polynomials of the + first kind. + + >>> x = np.linspace(-2, 2, 6) + >>> sc.eval_chebyc(3, x) + array([-2. , 1.872, 1.136, -1.136, -1.872, 2. ]) + >>> 2 * sc.eval_chebyt(3, x / 2) + array([-2. , 1.872, 1.136, -1.136, -1.872, 2. ]) + + """) + +add_newdoc("eval_sh_chebyt", + r""" + eval_sh_chebyt(n, x, out=None) + + Evaluate shifted Chebyshev polynomial of the first kind at a + point. + + These polynomials are defined as + + .. math:: + + T_n^*(x) = T_n(2x - 1) + + where :math:`T_n` is a Chebyshev polynomial of the first kind. See + 22.5.14 in [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer, the result is + determined via the relation to `eval_chebyt`. + x : array_like + Points at which to evaluate the shifted Chebyshev polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + T : scalar or ndarray + Values of the shifted Chebyshev polynomial + + See Also + -------- + roots_sh_chebyt : roots and quadrature weights of shifted + Chebyshev polynomials of the first kind + sh_chebyt : shifted Chebyshev polynomial object + eval_chebyt : evaluate Chebyshev polynomials of the first kind + numpy.polynomial.chebyshev.Chebyshev : Chebyshev series + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_sh_chebyu", + r""" + eval_sh_chebyu(n, x, out=None) + + Evaluate shifted Chebyshev polynomial of the second kind at a + point. + + These polynomials are defined as + + .. math:: + + U_n^*(x) = U_n(2x - 1) + + where :math:`U_n` is a Chebyshev polynomial of the first kind. See + 22.5.15 in [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer, the result is + determined via the relation to `eval_chebyu`. + x : array_like + Points at which to evaluate the shifted Chebyshev polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + U : scalar or ndarray + Values of the shifted Chebyshev polynomial + + See Also + -------- + roots_sh_chebyu : roots and quadrature weights of shifted + Chebychev polynomials of the second kind + sh_chebyu : shifted Chebyshev polynomial object + eval_chebyu : evaluate Chebyshev polynomials of the second kind + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_legendre", + r""" + eval_legendre(n, x, out=None) + + Evaluate Legendre polynomial at a point. + + The Legendre polynomials can be defined via the Gauss + hypergeometric function :math:`{}_2F_1` as + + .. math:: + + P_n(x) = {}_2F_1(-n, n + 1; 1; (1 - x)/2). + + When :math:`n` is an integer the result is a polynomial of degree + :math:`n`. See 22.5.49 in [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer, the result is + determined via the relation to the Gauss hypergeometric + function. + x : array_like + Points at which to evaluate the Legendre polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + P : scalar or ndarray + Values of the Legendre polynomial + + See Also + -------- + roots_legendre : roots and quadrature weights of Legendre + polynomials + legendre : Legendre polynomial object + hyp2f1 : Gauss hypergeometric function + numpy.polynomial.legendre.Legendre : Legendre series + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import eval_legendre + + Evaluate the zero-order Legendre polynomial at x = 0 + + >>> eval_legendre(0, 0) + 1.0 + + Evaluate the first-order Legendre polynomial between -1 and 1 + + >>> X = np.linspace(-1, 1, 5) # Domain of Legendre polynomials + >>> eval_legendre(1, X) + array([-1. , -0.5, 0. , 0.5, 1. ]) + + Evaluate Legendre polynomials of order 0 through 4 at x = 0 + + >>> N = range(0, 5) + >>> eval_legendre(N, 0) + array([ 1. , 0. , -0.5 , 0. , 0.375]) + + Plot Legendre polynomials of order 0 through 4 + + >>> X = np.linspace(-1, 1) + + >>> import matplotlib.pyplot as plt + >>> for n in range(0, 5): + ... y = eval_legendre(n, X) + ... plt.plot(X, y, label=r'$P_{}(x)$'.format(n)) + + >>> plt.title("Legendre Polynomials") + >>> plt.xlabel("x") + >>> plt.ylabel(r'$P_n(x)$') + >>> plt.legend(loc='lower right') + >>> plt.show() + + """) + +add_newdoc("eval_sh_legendre", + r""" + eval_sh_legendre(n, x, out=None) + + Evaluate shifted Legendre polynomial at a point. + + These polynomials are defined as + + .. math:: + + P_n^*(x) = P_n(2x - 1) + + where :math:`P_n` is a Legendre polynomial. See 2.2.11 in [AS]_ + for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer, the value is + determined via the relation to `eval_legendre`. + x : array_like + Points at which to evaluate the shifted Legendre polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + P : scalar or ndarray + Values of the shifted Legendre polynomial + + See Also + -------- + roots_sh_legendre : roots and quadrature weights of shifted + Legendre polynomials + sh_legendre : shifted Legendre polynomial object + eval_legendre : evaluate Legendre polynomials + numpy.polynomial.legendre.Legendre : Legendre series + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_genlaguerre", + r""" + eval_genlaguerre(n, alpha, x, out=None) + + Evaluate generalized Laguerre polynomial at a point. + + The generalized Laguerre polynomials can be defined via the + confluent hypergeometric function :math:`{}_1F_1` as + + .. math:: + + L_n^{(\alpha)}(x) = \binom{n + \alpha}{n} + {}_1F_1(-n, \alpha + 1, x). + + When :math:`n` is an integer the result is a polynomial of degree + :math:`n`. See 22.5.54 in [AS]_ for details. The Laguerre + polynomials are the special case where :math:`\alpha = 0`. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer, the result is + determined via the relation to the confluent hypergeometric + function. + alpha : array_like + Parameter; must have ``alpha > -1`` + x : array_like + Points at which to evaluate the generalized Laguerre + polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + L : scalar or ndarray + Values of the generalized Laguerre polynomial + + See Also + -------- + roots_genlaguerre : roots and quadrature weights of generalized + Laguerre polynomials + genlaguerre : generalized Laguerre polynomial object + hyp1f1 : confluent hypergeometric function + eval_laguerre : evaluate Laguerre polynomials + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_laguerre", + r""" + eval_laguerre(n, x, out=None) + + Evaluate Laguerre polynomial at a point. + + The Laguerre polynomials can be defined via the confluent + hypergeometric function :math:`{}_1F_1` as + + .. math:: + + L_n(x) = {}_1F_1(-n, 1, x). + + See 22.5.16 and 22.5.54 in [AS]_ for details. When :math:`n` is an + integer the result is a polynomial of degree :math:`n`. + + Parameters + ---------- + n : array_like + Degree of the polynomial. If not an integer the result is + determined via the relation to the confluent hypergeometric + function. + x : array_like + Points at which to evaluate the Laguerre polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + L : scalar or ndarray + Values of the Laguerre polynomial + + See Also + -------- + roots_laguerre : roots and quadrature weights of Laguerre + polynomials + laguerre : Laguerre polynomial object + numpy.polynomial.laguerre.Laguerre : Laguerre series + eval_genlaguerre : evaluate generalized Laguerre polynomials + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_hermite", + r""" + eval_hermite(n, x, out=None) + + Evaluate physicist's Hermite polynomial at a point. + + Defined by + + .. math:: + + H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}; + + :math:`H_n` is a polynomial of degree :math:`n`. See 22.11.7 in + [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial + x : array_like + Points at which to evaluate the Hermite polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + H : scalar or ndarray + Values of the Hermite polynomial + + See Also + -------- + roots_hermite : roots and quadrature weights of physicist's + Hermite polynomials + hermite : physicist's Hermite polynomial object + numpy.polynomial.hermite.Hermite : Physicist's Hermite series + eval_hermitenorm : evaluate Probabilist's Hermite polynomials + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + +add_newdoc("eval_hermitenorm", + r""" + eval_hermitenorm(n, x, out=None) + + Evaluate probabilist's (normalized) Hermite polynomial at a + point. + + Defined by + + .. math:: + + He_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2}; + + :math:`He_n` is a polynomial of degree :math:`n`. See 22.11.8 in + [AS]_ for details. + + Parameters + ---------- + n : array_like + Degree of the polynomial + x : array_like + Points at which to evaluate the Hermite polynomial + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + He : scalar or ndarray + Values of the Hermite polynomial + + See Also + -------- + roots_hermitenorm : roots and quadrature weights of probabilist's + Hermite polynomials + hermitenorm : probabilist's Hermite polynomial object + numpy.polynomial.hermite_e.HermiteE : Probabilist's Hermite series + eval_hermite : evaluate physicist's Hermite polynomials + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """) + + +add_newdoc("exp10", + """ + exp10(x, out=None) + + Compute ``10**x`` element-wise. + + Parameters + ---------- + x : array_like + `x` must contain real numbers. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + ``10**x``, computed element-wise. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import exp10 + + >>> exp10(3) + 1000.0 + >>> x = np.array([[-1, -0.5, 0], [0.5, 1, 1.5]]) + >>> exp10(x) + array([[ 0.1 , 0.31622777, 1. ], + [ 3.16227766, 10. , 31.6227766 ]]) + + """) + +add_newdoc("exp2", + """ + exp2(x, out=None) + + Compute ``2**x`` element-wise. + + Parameters + ---------- + x : array_like + `x` must contain real numbers. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + ``2**x``, computed element-wise. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import exp2 + + >>> exp2(3) + 8.0 + >>> x = np.array([[-1, -0.5, 0], [0.5, 1, 1.5]]) + >>> exp2(x) + array([[ 0.5 , 0.70710678, 1. ], + [ 1.41421356, 2. , 2.82842712]]) + """) + +add_newdoc("expm1", + """ + expm1(x, out=None) + + Compute ``exp(x) - 1``. + + When `x` is near zero, ``exp(x)`` is near 1, so the numerical calculation + of ``exp(x) - 1`` can suffer from catastrophic loss of precision. + ``expm1(x)`` is implemented to avoid the loss of precision that occurs when + `x` is near zero. + + Parameters + ---------- + x : array_like + `x` must contain real numbers. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + ``exp(x) - 1`` computed element-wise. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import expm1 + + >>> expm1(1.0) + 1.7182818284590451 + >>> expm1([-0.2, -0.1, 0, 0.1, 0.2]) + array([-0.18126925, -0.09516258, 0. , 0.10517092, 0.22140276]) + + The exact value of ``exp(7.5e-13) - 1`` is:: + + 7.5000000000028125000000007031250000001318...*10**-13. + + Here is what ``expm1(7.5e-13)`` gives: + + >>> expm1(7.5e-13) + 7.5000000000028135e-13 + + Compare that to ``exp(7.5e-13) - 1``, where the subtraction results in + a "catastrophic" loss of precision: + + >>> np.exp(7.5e-13) - 1 + 7.5006667543675576e-13 + + """) + +add_newdoc("expn", + r""" + expn(n, x, out=None) + + Generalized exponential integral En. + + For integer :math:`n \geq 0` and real :math:`x \geq 0` the + generalized exponential integral is defined as [dlmf]_ + + .. math:: + + E_n(x) = x^{n - 1} \int_x^\infty \frac{e^{-t}}{t^n} dt. + + Parameters + ---------- + n : array_like + Non-negative integers + x : array_like + Real argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Values of the generalized exponential integral + + See Also + -------- + exp1 : special case of :math:`E_n` for :math:`n = 1` + expi : related to :math:`E_n` when :math:`n = 1` + + References + ---------- + .. [dlmf] Digital Library of Mathematical Functions, 8.19.2 + https://dlmf.nist.gov/8.19#E2 + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + Its domain is nonnegative n and x. + + >>> sc.expn(-1, 1.0), sc.expn(1, -1.0) + (nan, nan) + + It has a pole at ``x = 0`` for ``n = 1, 2``; for larger ``n`` it + is equal to ``1 / (n - 1)``. + + >>> sc.expn([0, 1, 2, 3, 4], 0) + array([ inf, inf, 1. , 0.5 , 0.33333333]) + + For n equal to 0 it reduces to ``exp(-x) / x``. + + >>> x = np.array([1, 2, 3, 4]) + >>> sc.expn(0, x) + array([0.36787944, 0.06766764, 0.01659569, 0.00457891]) + >>> np.exp(-x) / x + array([0.36787944, 0.06766764, 0.01659569, 0.00457891]) + + For n equal to 1 it reduces to `exp1`. + + >>> sc.expn(1, x) + array([0.21938393, 0.04890051, 0.01304838, 0.00377935]) + >>> sc.exp1(x) + array([0.21938393, 0.04890051, 0.01304838, 0.00377935]) + + """) + +add_newdoc("fdtr", + r""" + fdtr(dfn, dfd, x, out=None) + + F cumulative distribution function. + + Returns the value of the cumulative distribution function of the + F-distribution, also known as Snedecor's F-distribution or the + Fisher-Snedecor distribution. + + The F-distribution with parameters :math:`d_n` and :math:`d_d` is the + distribution of the random variable, + + .. math:: + X = \frac{U_n/d_n}{U_d/d_d}, + + where :math:`U_n` and :math:`U_d` are random variables distributed + :math:`\chi^2`, with :math:`d_n` and :math:`d_d` degrees of freedom, + respectively. + + Parameters + ---------- + dfn : array_like + First parameter (positive float). + dfd : array_like + Second parameter (positive float). + x : array_like + Argument (nonnegative float). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + y : scalar or ndarray + The CDF of the F-distribution with parameters `dfn` and `dfd` at `x`. + + See Also + -------- + fdtrc : F distribution survival function + fdtri : F distribution inverse cumulative distribution + scipy.stats.f : F distribution + + Notes + ----- + The regularized incomplete beta function is used, according to the + formula, + + .. math:: + F(d_n, d_d; x) = I_{xd_n/(d_d + xd_n)}(d_n/2, d_d/2). + + Wrapper for the Cephes [1]_ routine `fdtr`. The F distribution is also + available as `scipy.stats.f`. Calling `fdtr` directly can improve + performance compared to the ``cdf`` method of `scipy.stats.f` (see last + example below). + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + Examples + -------- + Calculate the function for ``dfn=1`` and ``dfd=2`` at ``x=1``. + + >>> import numpy as np + >>> from scipy.special import fdtr + >>> fdtr(1, 2, 1) + 0.5773502691896258 + + Calculate the function at several points by providing a NumPy array for + `x`. + + >>> x = np.array([0.5, 2., 3.]) + >>> fdtr(1, 2, x) + array([0.4472136 , 0.70710678, 0.77459667]) + + Plot the function for several parameter sets. + + >>> import matplotlib.pyplot as plt + >>> dfn_parameters = [1, 5, 10, 50] + >>> dfd_parameters = [1, 1, 2, 3] + >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] + >>> parameters_list = list(zip(dfn_parameters, dfd_parameters, + ... linestyles)) + >>> x = np.linspace(0, 30, 1000) + >>> fig, ax = plt.subplots() + >>> for parameter_set in parameters_list: + ... dfn, dfd, style = parameter_set + ... fdtr_vals = fdtr(dfn, dfd, x) + ... ax.plot(x, fdtr_vals, label=rf"$d_n={dfn},\, d_d={dfd}$", + ... ls=style) + >>> ax.legend() + >>> ax.set_xlabel("$x$") + >>> ax.set_title("F distribution cumulative distribution function") + >>> plt.show() + + The F distribution is also available as `scipy.stats.f`. Using `fdtr` + directly can be much faster than calling the ``cdf`` method of + `scipy.stats.f`, especially for small arrays or individual values. + To get the same results one must use the following parametrization: + ``stats.f(dfn, dfd).cdf(x)=fdtr(dfn, dfd, x)``. + + >>> from scipy.stats import f + >>> dfn, dfd = 1, 2 + >>> x = 1 + >>> fdtr_res = fdtr(dfn, dfd, x) # this will often be faster than below + >>> f_dist_res = f(dfn, dfd).cdf(x) + >>> fdtr_res == f_dist_res # test that results are equal + True + """) + +add_newdoc("fdtrc", + r""" + fdtrc(dfn, dfd, x, out=None) + + F survival function. + + Returns the complemented F-distribution function (the integral of the + density from `x` to infinity). + + Parameters + ---------- + dfn : array_like + First parameter (positive float). + dfd : array_like + Second parameter (positive float). + x : array_like + Argument (nonnegative float). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + y : scalar or ndarray + The complemented F-distribution function with parameters `dfn` and + `dfd` at `x`. + + See Also + -------- + fdtr : F distribution cumulative distribution function + fdtri : F distribution inverse cumulative distribution function + scipy.stats.f : F distribution + + Notes + ----- + The regularized incomplete beta function is used, according to the + formula, + + .. math:: + F(d_n, d_d; x) = I_{d_d/(d_d + xd_n)}(d_d/2, d_n/2). + + Wrapper for the Cephes [1]_ routine `fdtrc`. The F distribution is also + available as `scipy.stats.f`. Calling `fdtrc` directly can improve + performance compared to the ``sf`` method of `scipy.stats.f` (see last + example below). + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + Examples + -------- + Calculate the function for ``dfn=1`` and ``dfd=2`` at ``x=1``. + + >>> import numpy as np + >>> from scipy.special import fdtrc + >>> fdtrc(1, 2, 1) + 0.42264973081037427 + + Calculate the function at several points by providing a NumPy array for + `x`. + + >>> x = np.array([0.5, 2., 3.]) + >>> fdtrc(1, 2, x) + array([0.5527864 , 0.29289322, 0.22540333]) + + Plot the function for several parameter sets. + + >>> import matplotlib.pyplot as plt + >>> dfn_parameters = [1, 5, 10, 50] + >>> dfd_parameters = [1, 1, 2, 3] + >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] + >>> parameters_list = list(zip(dfn_parameters, dfd_parameters, + ... linestyles)) + >>> x = np.linspace(0, 30, 1000) + >>> fig, ax = plt.subplots() + >>> for parameter_set in parameters_list: + ... dfn, dfd, style = parameter_set + ... fdtrc_vals = fdtrc(dfn, dfd, x) + ... ax.plot(x, fdtrc_vals, label=rf"$d_n={dfn},\, d_d={dfd}$", + ... ls=style) + >>> ax.legend() + >>> ax.set_xlabel("$x$") + >>> ax.set_title("F distribution survival function") + >>> plt.show() + + The F distribution is also available as `scipy.stats.f`. Using `fdtrc` + directly can be much faster than calling the ``sf`` method of + `scipy.stats.f`, especially for small arrays or individual values. + To get the same results one must use the following parametrization: + ``stats.f(dfn, dfd).sf(x)=fdtrc(dfn, dfd, x)``. + + >>> from scipy.stats import f + >>> dfn, dfd = 1, 2 + >>> x = 1 + >>> fdtrc_res = fdtrc(dfn, dfd, x) # this will often be faster than below + >>> f_dist_res = f(dfn, dfd).sf(x) + >>> f_dist_res == fdtrc_res # test that results are equal + True + """) + +add_newdoc("fdtri", + r""" + fdtri(dfn, dfd, p, out=None) + + The `p`-th quantile of the F-distribution. + + This function is the inverse of the F-distribution CDF, `fdtr`, returning + the `x` such that `fdtr(dfn, dfd, x) = p`. + + Parameters + ---------- + dfn : array_like + First parameter (positive float). + dfd : array_like + Second parameter (positive float). + p : array_like + Cumulative probability, in [0, 1]. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + x : scalar or ndarray + The quantile corresponding to `p`. + + See Also + -------- + fdtr : F distribution cumulative distribution function + fdtrc : F distribution survival function + scipy.stats.f : F distribution + + Notes + ----- + The computation is carried out using the relation to the inverse + regularized beta function, :math:`I^{-1}_x(a, b)`. Let + :math:`z = I^{-1}_p(d_d/2, d_n/2).` Then, + + .. math:: + x = \frac{d_d (1 - z)}{d_n z}. + + If `p` is such that :math:`x < 0.5`, the following relation is used + instead for improved stability: let + :math:`z' = I^{-1}_{1 - p}(d_n/2, d_d/2).` Then, + + .. math:: + x = \frac{d_d z'}{d_n (1 - z')}. + + Wrapper for the Cephes [1]_ routine `fdtri`. + + The F distribution is also available as `scipy.stats.f`. Calling + `fdtri` directly can improve performance compared to the ``ppf`` + method of `scipy.stats.f` (see last example below). + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + Examples + -------- + `fdtri` represents the inverse of the F distribution CDF which is + available as `fdtr`. Here, we calculate the CDF for ``df1=1``, ``df2=2`` + at ``x=3``. `fdtri` then returns ``3`` given the same values for `df1`, + `df2` and the computed CDF value. + + >>> import numpy as np + >>> from scipy.special import fdtri, fdtr + >>> df1, df2 = 1, 2 + >>> x = 3 + >>> cdf_value = fdtr(df1, df2, x) + >>> fdtri(df1, df2, cdf_value) + 3.000000000000006 + + Calculate the function at several points by providing a NumPy array for + `x`. + + >>> x = np.array([0.1, 0.4, 0.7]) + >>> fdtri(1, 2, x) + array([0.02020202, 0.38095238, 1.92156863]) + + Plot the function for several parameter sets. + + >>> import matplotlib.pyplot as plt + >>> dfn_parameters = [50, 10, 1, 50] + >>> dfd_parameters = [0.5, 1, 1, 5] + >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] + >>> parameters_list = list(zip(dfn_parameters, dfd_parameters, + ... linestyles)) + >>> x = np.linspace(0, 1, 1000) + >>> fig, ax = plt.subplots() + >>> for parameter_set in parameters_list: + ... dfn, dfd, style = parameter_set + ... fdtri_vals = fdtri(dfn, dfd, x) + ... ax.plot(x, fdtri_vals, label=rf"$d_n={dfn},\, d_d={dfd}$", + ... ls=style) + >>> ax.legend() + >>> ax.set_xlabel("$x$") + >>> title = "F distribution inverse cumulative distribution function" + >>> ax.set_title(title) + >>> ax.set_ylim(0, 30) + >>> plt.show() + + The F distribution is also available as `scipy.stats.f`. Using `fdtri` + directly can be much faster than calling the ``ppf`` method of + `scipy.stats.f`, especially for small arrays or individual values. + To get the same results one must use the following parametrization: + ``stats.f(dfn, dfd).ppf(x)=fdtri(dfn, dfd, x)``. + + >>> from scipy.stats import f + >>> dfn, dfd = 1, 2 + >>> x = 0.7 + >>> fdtri_res = fdtri(dfn, dfd, x) # this will often be faster than below + >>> f_dist_res = f(dfn, dfd).ppf(x) + >>> f_dist_res == fdtri_res # test that results are equal + True + """) + +add_newdoc("fdtridfd", + """ + fdtridfd(dfn, p, x, out=None) + + Inverse to `fdtr` vs dfd + + Finds the F density argument dfd such that ``fdtr(dfn, dfd, x) == p``. + + Parameters + ---------- + dfn : array_like + First parameter (positive float). + p : array_like + Cumulative probability, in [0, 1]. + x : array_like + Argument (nonnegative float). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + dfd : scalar or ndarray + `dfd` such that ``fdtr(dfn, dfd, x) == p``. + + See Also + -------- + fdtr : F distribution cumulative distribution function + fdtrc : F distribution survival function + fdtri : F distribution quantile function + scipy.stats.f : F distribution + + Examples + -------- + Compute the F distribution cumulative distribution function for one + parameter set. + + >>> from scipy.special import fdtridfd, fdtr + >>> dfn, dfd, x = 10, 5, 2 + >>> cdf_value = fdtr(dfn, dfd, x) + >>> cdf_value + 0.7700248806501017 + + Verify that `fdtridfd` recovers the original value for `dfd`: + + >>> fdtridfd(dfn, cdf_value, x) + 5.0 + """) + +''' +commented out as fdtridfn seems to have bugs and is not in functions.json +see: https://github.com/scipy/scipy/pull/15622#discussion_r811440983 + +add_newdoc( + "fdtridfn", + """ + fdtridfn(p, dfd, x, out=None) + + Inverse to `fdtr` vs dfn + + finds the F density argument dfn such that ``fdtr(dfn, dfd, x) == p``. + + + Parameters + ---------- + p : array_like + Cumulative probability, in [0, 1]. + dfd : array_like + Second parameter (positive float). + x : array_like + Argument (nonnegative float). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + dfn : scalar or ndarray + `dfn` such that ``fdtr(dfn, dfd, x) == p``. + + See Also + -------- + fdtr, fdtrc, fdtri, fdtridfd + + + """) +''' + +add_newdoc("gdtr", + r""" + gdtr(a, b, x, out=None) + + Gamma distribution cumulative distribution function. + + Returns the integral from zero to `x` of the gamma probability density + function, + + .. math:: + + F = \int_0^x \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at}\,dt, + + where :math:`\Gamma` is the gamma function. + + Parameters + ---------- + a : array_like + The rate parameter of the gamma distribution, sometimes denoted + :math:`\beta` (float). It is also the reciprocal of the scale + parameter :math:`\theta`. + b : array_like + The shape parameter of the gamma distribution, sometimes denoted + :math:`\alpha` (float). + x : array_like + The quantile (upper limit of integration; float). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + F : scalar or ndarray + The CDF of the gamma distribution with parameters `a` and `b` + evaluated at `x`. + + See Also + -------- + gdtrc : 1 - CDF of the gamma distribution. + scipy.stats.gamma: Gamma distribution + + Notes + ----- + The evaluation is carried out using the relation to the incomplete gamma + integral (regularized gamma function). + + Wrapper for the Cephes [1]_ routine `gdtr`. Calling `gdtr` directly can + improve performance compared to the ``cdf`` method of `scipy.stats.gamma` + (see last example below). + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + Examples + -------- + Compute the function for ``a=1``, ``b=2`` at ``x=5``. + + >>> import numpy as np + >>> from scipy.special import gdtr + >>> import matplotlib.pyplot as plt + >>> gdtr(1., 2., 5.) + 0.9595723180054873 + + Compute the function for ``a=1`` and ``b=2`` at several points by + providing a NumPy array for `x`. + + >>> xvalues = np.array([1., 2., 3., 4]) + >>> gdtr(1., 1., xvalues) + array([0.63212056, 0.86466472, 0.95021293, 0.98168436]) + + `gdtr` can evaluate different parameter sets by providing arrays with + broadcasting compatible shapes for `a`, `b` and `x`. Here we compute the + function for three different `a` at four positions `x` and ``b=3``, + resulting in a 3x4 array. + + >>> a = np.array([[0.5], [1.5], [2.5]]) + >>> x = np.array([1., 2., 3., 4]) + >>> a.shape, x.shape + ((3, 1), (4,)) + + >>> gdtr(a, 3., x) + array([[0.01438768, 0.0803014 , 0.19115317, 0.32332358], + [0.19115317, 0.57680992, 0.82642193, 0.9380312 ], + [0.45618688, 0.87534798, 0.97974328, 0.9972306 ]]) + + Plot the function for four different parameter sets. + + >>> a_parameters = [0.3, 1, 2, 6] + >>> b_parameters = [2, 10, 15, 20] + >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] + >>> parameters_list = list(zip(a_parameters, b_parameters, linestyles)) + >>> x = np.linspace(0, 30, 1000) + >>> fig, ax = plt.subplots() + >>> for parameter_set in parameters_list: + ... a, b, style = parameter_set + ... gdtr_vals = gdtr(a, b, x) + ... ax.plot(x, gdtr_vals, label=fr"$a= {a},\, b={b}$", ls=style) + >>> ax.legend() + >>> ax.set_xlabel("$x$") + >>> ax.set_title("Gamma distribution cumulative distribution function") + >>> plt.show() + + The gamma distribution is also available as `scipy.stats.gamma`. Using + `gdtr` directly can be much faster than calling the ``cdf`` method of + `scipy.stats.gamma`, especially for small arrays or individual values. + To get the same results one must use the following parametrization: + ``stats.gamma(b, scale=1/a).cdf(x)=gdtr(a, b, x)``. + + >>> from scipy.stats import gamma + >>> a = 2. + >>> b = 3 + >>> x = 1. + >>> gdtr_result = gdtr(a, b, x) # this will often be faster than below + >>> gamma_dist_result = gamma(b, scale=1/a).cdf(x) + >>> gdtr_result == gamma_dist_result # test that results are equal + True + """) + +add_newdoc("gdtrc", + r""" + gdtrc(a, b, x, out=None) + + Gamma distribution survival function. + + Integral from `x` to infinity of the gamma probability density function, + + .. math:: + + F = \int_x^\infty \frac{a^b}{\Gamma(b)} t^{b-1} e^{-at}\,dt, + + where :math:`\Gamma` is the gamma function. + + Parameters + ---------- + a : array_like + The rate parameter of the gamma distribution, sometimes denoted + :math:`\beta` (float). It is also the reciprocal of the scale + parameter :math:`\theta`. + b : array_like + The shape parameter of the gamma distribution, sometimes denoted + :math:`\alpha` (float). + x : array_like + The quantile (lower limit of integration; float). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + F : scalar or ndarray + The survival function of the gamma distribution with parameters `a` + and `b` evaluated at `x`. + + See Also + -------- + gdtr: Gamma distribution cumulative distribution function + scipy.stats.gamma: Gamma distribution + gdtrix + + Notes + ----- + The evaluation is carried out using the relation to the incomplete gamma + integral (regularized gamma function). + + Wrapper for the Cephes [1]_ routine `gdtrc`. Calling `gdtrc` directly can + improve performance compared to the ``sf`` method of `scipy.stats.gamma` + (see last example below). + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + Examples + -------- + Compute the function for ``a=1`` and ``b=2`` at ``x=5``. + + >>> import numpy as np + >>> from scipy.special import gdtrc + >>> import matplotlib.pyplot as plt + >>> gdtrc(1., 2., 5.) + 0.04042768199451279 + + Compute the function for ``a=1``, ``b=2`` at several points by providing + a NumPy array for `x`. + + >>> xvalues = np.array([1., 2., 3., 4]) + >>> gdtrc(1., 1., xvalues) + array([0.36787944, 0.13533528, 0.04978707, 0.01831564]) + + `gdtrc` can evaluate different parameter sets by providing arrays with + broadcasting compatible shapes for `a`, `b` and `x`. Here we compute the + function for three different `a` at four positions `x` and ``b=3``, + resulting in a 3x4 array. + + >>> a = np.array([[0.5], [1.5], [2.5]]) + >>> x = np.array([1., 2., 3., 4]) + >>> a.shape, x.shape + ((3, 1), (4,)) + + >>> gdtrc(a, 3., x) + array([[0.98561232, 0.9196986 , 0.80884683, 0.67667642], + [0.80884683, 0.42319008, 0.17357807, 0.0619688 ], + [0.54381312, 0.12465202, 0.02025672, 0.0027694 ]]) + + Plot the function for four different parameter sets. + + >>> a_parameters = [0.3, 1, 2, 6] + >>> b_parameters = [2, 10, 15, 20] + >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] + >>> parameters_list = list(zip(a_parameters, b_parameters, linestyles)) + >>> x = np.linspace(0, 30, 1000) + >>> fig, ax = plt.subplots() + >>> for parameter_set in parameters_list: + ... a, b, style = parameter_set + ... gdtrc_vals = gdtrc(a, b, x) + ... ax.plot(x, gdtrc_vals, label=fr"$a= {a},\, b={b}$", ls=style) + >>> ax.legend() + >>> ax.set_xlabel("$x$") + >>> ax.set_title("Gamma distribution survival function") + >>> plt.show() + + The gamma distribution is also available as `scipy.stats.gamma`. + Using `gdtrc` directly can be much faster than calling the ``sf`` method + of `scipy.stats.gamma`, especially for small arrays or individual + values. To get the same results one must use the following parametrization: + ``stats.gamma(b, scale=1/a).sf(x)=gdtrc(a, b, x)``. + + >>> from scipy.stats import gamma + >>> a = 2 + >>> b = 3 + >>> x = 1. + >>> gdtrc_result = gdtrc(a, b, x) # this will often be faster than below + >>> gamma_dist_result = gamma(b, scale=1/a).sf(x) + >>> gdtrc_result == gamma_dist_result # test that results are equal + True + """) + +add_newdoc("gdtria", + """ + gdtria(p, b, x, out=None) + + Inverse of `gdtr` vs a. + + Returns the inverse with respect to the parameter `a` of ``p = + gdtr(a, b, x)``, the cumulative distribution function of the gamma + distribution. + + Parameters + ---------- + p : array_like + Probability values. + b : array_like + `b` parameter values of `gdtr(a, b, x)`. `b` is the "shape" parameter + of the gamma distribution. + x : array_like + Nonnegative real values, from the domain of the gamma distribution. + out : ndarray, optional + If a fourth argument is given, it must be a numpy.ndarray whose size + matches the broadcast result of `a`, `b` and `x`. `out` is then the + array returned by the function. + + Returns + ------- + a : scalar or ndarray + Values of the `a` parameter such that ``p = gdtr(a, b, x)`. ``1/a`` + is the "scale" parameter of the gamma distribution. + + See Also + -------- + gdtr : CDF of the gamma distribution. + gdtrib : Inverse with respect to `b` of `gdtr(a, b, x)`. + gdtrix : Inverse with respect to `x` of `gdtr(a, b, x)`. + + Notes + ----- + Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`. + + The cumulative distribution function `p` is computed using a routine by + DiDinato and Morris [2]_. Computation of `a` involves a search for a value + that produces the desired value of `p`. The search relies on the + monotonicity of `p` with `a`. + + References + ---------- + .. [1] Barry Brown, James Lovato, and Kathy Russell, + CDFLIB: Library of Fortran Routines for Cumulative Distribution + Functions, Inverses, and Other Parameters. + .. [2] DiDinato, A. R. and Morris, A. H., + Computation of the incomplete gamma function ratios and their + inverse. ACM Trans. Math. Softw. 12 (1986), 377-393. + + Examples + -------- + First evaluate `gdtr`. + + >>> from scipy.special import gdtr, gdtria + >>> p = gdtr(1.2, 3.4, 5.6) + >>> print(p) + 0.94378087442 + + Verify the inverse. + + >>> gdtria(p, 3.4, 5.6) + 1.2 + """) + +add_newdoc("gdtrib", + """ + gdtrib(a, p, x, out=None) + + Inverse of `gdtr` vs b. + + Returns the inverse with respect to the parameter `b` of ``p = + gdtr(a, b, x)``, the cumulative distribution function of the gamma + distribution. + + Parameters + ---------- + a : array_like + `a` parameter values of ``gdtr(a, b, x)`. ``1/a`` is the "scale" + parameter of the gamma distribution. + p : array_like + Probability values. + x : array_like + Nonnegative real values, from the domain of the gamma distribution. + out : ndarray, optional + If a fourth argument is given, it must be a numpy.ndarray whose size + matches the broadcast result of `a`, `b` and `x`. `out` is then the + array returned by the function. + + Returns + ------- + b : scalar or ndarray + Values of the `b` parameter such that `p = gdtr(a, b, x)`. `b` is + the "shape" parameter of the gamma distribution. + + See Also + -------- + gdtr : CDF of the gamma distribution. + gdtria : Inverse with respect to `a` of `gdtr(a, b, x)`. + gdtrix : Inverse with respect to `x` of `gdtr(a, b, x)`. + + Notes + ----- + + The cumulative distribution function `p` is computed using the Cephes [1]_ + routines `igam` and `igamc`. Computation of `b` involves a search for a value + that produces the desired value of `p` using Chandrupatla's bracketing + root finding algorithm [2]_. + + Note that there are some edge cases where `gdtrib` is extended by taking + limits where they are uniquely defined. In particular + ``x == 0`` with ``p > 0`` and ``p == 0`` with ``x > 0``. + For these edge cases, a numerical result will be returned for + ``gdtrib(a, p, x)`` even though ``gdtr(a, gdtrib(a, p, x), x)`` is + undefined. + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + .. [2] Chandrupatla, Tirupathi R. + "A new hybrid quadratic/bisection algorithm for finding the zero of a + nonlinear function without using derivatives". + Advances in Engineering Software, 28(3), 145-149. + https://doi.org/10.1016/s0965-9978(96)00051-8 + + Examples + -------- + First evaluate `gdtr`. + + >>> from scipy.special import gdtr, gdtrib + >>> p = gdtr(1.2, 3.4, 5.6) + >>> print(p) + 0.94378087442 + + Verify the inverse. + + >>> gdtrib(1.2, p, 5.6) + 3.3999999999999995 + """) + +add_newdoc("gdtrix", + """ + gdtrix(a, b, p, out=None) + + Inverse of `gdtr` vs x. + + Returns the inverse with respect to the parameter `x` of ``p = + gdtr(a, b, x)``, the cumulative distribution function of the gamma + distribution. This is also known as the pth quantile of the + distribution. + + Parameters + ---------- + a : array_like + `a` parameter values of ``gdtr(a, b, x)``. ``1/a`` is the "scale" + parameter of the gamma distribution. + b : array_like + `b` parameter values of ``gdtr(a, b, x)``. `b` is the "shape" parameter + of the gamma distribution. + p : array_like + Probability values. + out : ndarray, optional + If a fourth argument is given, it must be a numpy.ndarray whose size + matches the broadcast result of `a`, `b` and `x`. `out` is then the + array returned by the function. + + Returns + ------- + x : scalar or ndarray + Values of the `x` parameter such that `p = gdtr(a, b, x)`. + + See Also + -------- + gdtr : CDF of the gamma distribution. + gdtria : Inverse with respect to `a` of ``gdtr(a, b, x)``. + gdtrib : Inverse with respect to `b` of ``gdtr(a, b, x)``. + + Notes + ----- + Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`. + + The cumulative distribution function `p` is computed using a routine by + DiDinato and Morris [2]_. Computation of `x` involves a search for a value + that produces the desired value of `p`. The search relies on the + monotonicity of `p` with `x`. + + References + ---------- + .. [1] Barry Brown, James Lovato, and Kathy Russell, + CDFLIB: Library of Fortran Routines for Cumulative Distribution + Functions, Inverses, and Other Parameters. + .. [2] DiDinato, A. R. and Morris, A. H., + Computation of the incomplete gamma function ratios and their + inverse. ACM Trans. Math. Softw. 12 (1986), 377-393. + + Examples + -------- + First evaluate `gdtr`. + + >>> from scipy.special import gdtr, gdtrix + >>> p = gdtr(1.2, 3.4, 5.6) + >>> print(p) + 0.94378087442 + + Verify the inverse. + + >>> gdtrix(1.2, 3.4, p) + 5.5999999999999996 + """) + +add_newdoc("hankel1", + r""" + hankel1(v, z, out=None) + + Hankel function of the first kind + + Parameters + ---------- + v : array_like + Order (float). + z : array_like + Argument (float or complex). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + Values of the Hankel function of the first kind. + + See Also + -------- + hankel1e : ndarray + This function with leading exponential behavior stripped off. + + Notes + ----- + A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the + computation using the relation, + + .. math:: H^{(1)}_v(z) = + \frac{2}{\imath\pi} \exp(-\imath \pi v/2) K_v(z \exp(-\imath\pi/2)) + + where :math:`K_v` is the modified Bessel function of the second kind. + For negative orders, the relation + + .. math:: H^{(1)}_{-v}(z) = H^{(1)}_v(z) \exp(\imath\pi v) + + is used. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + """) + +add_newdoc("hankel1e", + r""" + hankel1e(v, z, out=None) + + Exponentially scaled Hankel function of the first kind + + Defined as:: + + hankel1e(v, z) = hankel1(v, z) * exp(-1j * z) + + Parameters + ---------- + v : array_like + Order (float). + z : array_like + Argument (float or complex). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + Values of the exponentially scaled Hankel function. + + Notes + ----- + A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the + computation using the relation, + + .. math:: H^{(1)}_v(z) = + \frac{2}{\imath\pi} \exp(-\imath \pi v/2) K_v(z \exp(-\imath\pi/2)) + + where :math:`K_v` is the modified Bessel function of the second kind. + For negative orders, the relation + + .. math:: H^{(1)}_{-v}(z) = H^{(1)}_v(z) \exp(\imath\pi v) + + is used. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + """) + +add_newdoc("hankel2", + r""" + hankel2(v, z, out=None) + + Hankel function of the second kind + + Parameters + ---------- + v : array_like + Order (float). + z : array_like + Argument (float or complex). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + Values of the Hankel function of the second kind. + + See Also + -------- + hankel2e : this function with leading exponential behavior stripped off. + + Notes + ----- + A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the + computation using the relation, + + .. math:: H^{(2)}_v(z) = + -\frac{2}{\imath\pi} \exp(\imath \pi v/2) K_v(z \exp(\imath\pi/2)) + + where :math:`K_v` is the modified Bessel function of the second kind. + For negative orders, the relation + + .. math:: H^{(2)}_{-v}(z) = H^{(2)}_v(z) \exp(-\imath\pi v) + + is used. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + """) + +add_newdoc("hankel2e", + r""" + hankel2e(v, z, out=None) + + Exponentially scaled Hankel function of the second kind + + Defined as:: + + hankel2e(v, z) = hankel2(v, z) * exp(1j * z) + + Parameters + ---------- + v : array_like + Order (float). + z : array_like + Argument (float or complex). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + Values of the exponentially scaled Hankel function of the second kind. + + Notes + ----- + A wrapper for the AMOS [1]_ routine `zbesh`, which carries out the + computation using the relation, + + .. math:: H^{(2)}_v(z) = -\frac{2}{\imath\pi} + \exp(\frac{\imath \pi v}{2}) K_v(z exp(\frac{\imath\pi}{2})) + + where :math:`K_v` is the modified Bessel function of the second kind. + For negative orders, the relation + + .. math:: H^{(2)}_{-v}(z) = H^{(2)}_v(z) \exp(-\imath\pi v) + + is used. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + + """) + +add_newdoc("huber", + r""" + huber(delta, r, out=None) + + Huber loss function. + + .. math:: \text{huber}(\delta, r) = \begin{cases} \infty & \delta < 0 \\ + \frac{1}{2}r^2 & 0 \le \delta, | r | \le \delta \\ + \delta ( |r| - \frac{1}{2}\delta ) & \text{otherwise} \end{cases} + + Parameters + ---------- + delta : ndarray + Input array, indicating the quadratic vs. linear loss changepoint. + r : ndarray + Input array, possibly representing residuals. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + The computed Huber loss function values. + + See Also + -------- + pseudo_huber : smooth approximation of this function + + Notes + ----- + `huber` is useful as a loss function in robust statistics or machine + learning to reduce the influence of outliers as compared to the common + squared error loss, residuals with a magnitude higher than `delta` are + not squared [1]_. + + Typically, `r` represents residuals, the difference + between a model prediction and data. Then, for :math:`|r|\leq\delta`, + `huber` resembles the squared error and for :math:`|r|>\delta` the + absolute error. This way, the Huber loss often achieves + a fast convergence in model fitting for small residuals like the squared + error loss function and still reduces the influence of outliers + (:math:`|r|>\delta`) like the absolute error loss. As :math:`\delta` is + the cutoff between squared and absolute error regimes, it has + to be tuned carefully for each problem. `huber` is also + convex, making it suitable for gradient based optimization. + + .. versionadded:: 0.15.0 + + References + ---------- + .. [1] Peter Huber. "Robust Estimation of a Location Parameter", + 1964. Annals of Statistics. 53 (1): 73 - 101. + + Examples + -------- + Import all necessary modules. + + >>> import numpy as np + >>> from scipy.special import huber + >>> import matplotlib.pyplot as plt + + Compute the function for ``delta=1`` at ``r=2`` + + >>> huber(1., 2.) + 1.5 + + Compute the function for different `delta` by providing a NumPy array or + list for `delta`. + + >>> huber([1., 3., 5.], 4.) + array([3.5, 7.5, 8. ]) + + Compute the function at different points by providing a NumPy array or + list for `r`. + + >>> huber(2., np.array([1., 1.5, 3.])) + array([0.5 , 1.125, 4. ]) + + The function can be calculated for different `delta` and `r` by + providing arrays for both with compatible shapes for broadcasting. + + >>> r = np.array([1., 2.5, 8., 10.]) + >>> deltas = np.array([[1.], [5.], [9.]]) + >>> print(r.shape, deltas.shape) + (4,) (3, 1) + + >>> huber(deltas, r) + array([[ 0.5 , 2. , 7.5 , 9.5 ], + [ 0.5 , 3.125, 27.5 , 37.5 ], + [ 0.5 , 3.125, 32. , 49.5 ]]) + + Plot the function for different `delta`. + + >>> x = np.linspace(-4, 4, 500) + >>> deltas = [1, 2, 3] + >>> linestyles = ["dashed", "dotted", "dashdot"] + >>> fig, ax = plt.subplots() + >>> combined_plot_parameters = list(zip(deltas, linestyles)) + >>> for delta, style in combined_plot_parameters: + ... ax.plot(x, huber(delta, x), label=fr"$\delta={delta}$", ls=style) + >>> ax.legend(loc="upper center") + >>> ax.set_xlabel("$x$") + >>> ax.set_title(r"Huber loss function $h_{\delta}(x)$") + >>> ax.set_xlim(-4, 4) + >>> ax.set_ylim(0, 8) + >>> plt.show() + """) + +add_newdoc("hyp0f1", + r""" + hyp0f1(v, z, out=None) + + Confluent hypergeometric limit function 0F1. + + Parameters + ---------- + v : array_like + Real-valued parameter + z : array_like + Real- or complex-valued argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The confluent hypergeometric limit function + + Notes + ----- + This function is defined as: + + .. math:: _0F_1(v, z) = \sum_{k=0}^{\infty}\frac{z^k}{(v)_k k!}. + + It's also the limit as :math:`q \to \infty` of :math:`_1F_1(q; v; z/q)`, + and satisfies the differential equation :math:`f''(z) + vf'(z) = + f(z)`. See [1]_ for more information. + + References + ---------- + .. [1] Wolfram MathWorld, "Confluent Hypergeometric Limit Function", + http://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + It is one when `z` is zero. + + >>> sc.hyp0f1(1, 0) + 1.0 + + It is the limit of the confluent hypergeometric function as `q` + goes to infinity. + + >>> q = np.array([1, 10, 100, 1000]) + >>> v = 1 + >>> z = 1 + >>> sc.hyp1f1(q, v, z / q) + array([2.71828183, 2.31481985, 2.28303778, 2.27992985]) + >>> sc.hyp0f1(v, z) + 2.2795853023360673 + + It is related to Bessel functions. + + >>> n = 1 + >>> x = np.linspace(0, 1, 5) + >>> sc.jv(n, x) + array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059]) + >>> (0.5 * x)**n / sc.factorial(n) * sc.hyp0f1(n + 1, -0.25 * x**2) + array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059]) + + """) + +add_newdoc("hyp1f1", + r""" + hyp1f1(a, b, x, out=None) + + Confluent hypergeometric function 1F1. + + The confluent hypergeometric function is defined by the series + + .. math:: + + {}_1F_1(a; b; x) = \sum_{k = 0}^\infty \frac{(a)_k}{(b)_k k!} x^k. + + See [dlmf]_ for more details. Here :math:`(\cdot)_k` is the + Pochhammer symbol; see `poch`. + + Parameters + ---------- + a, b : array_like + Real parameters + x : array_like + Real or complex argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Values of the confluent hypergeometric function + + See Also + -------- + hyperu : another confluent hypergeometric function + hyp0f1 : confluent hypergeometric limit function + hyp2f1 : Gaussian hypergeometric function + + Notes + ----- + For real values, this function uses the ``hyp1f1`` routine from the C++ Boost + library [2]_, for complex values a C translation of the specfun + Fortran library [3]_. + + References + ---------- + .. [dlmf] NIST Digital Library of Mathematical Functions + https://dlmf.nist.gov/13.2#E2 + .. [2] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/. + .. [3] Zhang, Jin, "Computation of Special Functions", John Wiley + and Sons, Inc, 1996. + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + It is one when `x` is zero: + + >>> sc.hyp1f1(0.5, 0.5, 0) + 1.0 + + It is singular when `b` is a nonpositive integer. + + >>> sc.hyp1f1(0.5, -1, 0) + inf + + It is a polynomial when `a` is a nonpositive integer. + + >>> a, b, x = -1, 0.5, np.array([1.0, 2.0, 3.0, 4.0]) + >>> sc.hyp1f1(a, b, x) + array([-1., -3., -5., -7.]) + >>> 1 + (a / b) * x + array([-1., -3., -5., -7.]) + + It reduces to the exponential function when ``a = b``. + + >>> sc.hyp1f1(2, 2, [1, 2, 3, 4]) + array([ 2.71828183, 7.3890561 , 20.08553692, 54.59815003]) + >>> np.exp([1, 2, 3, 4]) + array([ 2.71828183, 7.3890561 , 20.08553692, 54.59815003]) + + """) + +add_newdoc("hyperu", + r""" + hyperu(a, b, x, out=None) + + Confluent hypergeometric function U + + It is defined as the solution to the equation + + .. math:: + + x \frac{d^2w}{dx^2} + (b - x) \frac{dw}{dx} - aw = 0 + + which satisfies the property + + .. math:: + + U(a, b, x) \sim x^{-a} + + as :math:`x \to \infty`. See [dlmf]_ for more details. + + Parameters + ---------- + a, b : array_like + Real-valued parameters + x : array_like + Real-valued argument + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + Values of `U` + + References + ---------- + .. [dlmf] NIST Digital Library of Mathematics Functions + https://dlmf.nist.gov/13.2#E6 + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + It has a branch cut along the negative `x` axis. + + >>> x = np.linspace(-0.1, -10, 5) + >>> sc.hyperu(1, 1, x) + array([nan, nan, nan, nan, nan]) + + It approaches zero as `x` goes to infinity. + + >>> x = np.array([1, 10, 100]) + >>> sc.hyperu(1, 1, x) + array([0.59634736, 0.09156333, 0.00990194]) + + It satisfies Kummer's transformation. + + >>> a, b, x = 2, 1, 1 + >>> sc.hyperu(a, b, x) + 0.1926947246463881 + >>> x**(1 - b) * sc.hyperu(a - b + 1, 2 - b, x) + 0.1926947246463881 + + """) + +add_newdoc("_igam_fac", + """ + Internal function, do not use. + """) + +add_newdoc("iv", + r""" + iv(v, z, out=None) + + Modified Bessel function of the first kind of real order. + + Parameters + ---------- + v : array_like + Order. If `z` is of real type and negative, `v` must be integer + valued. + z : array_like of float or complex + Argument. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + Values of the modified Bessel function. + + See Also + -------- + ive : This function with leading exponential behavior stripped off. + i0 : Faster version of this function for order 0. + i1 : Faster version of this function for order 1. + + Notes + ----- + For real `z` and :math:`v \in [-50, 50]`, the evaluation is carried out + using Temme's method [1]_. For larger orders, uniform asymptotic + expansions are applied. + + For complex `z` and positive `v`, the AMOS [2]_ `zbesi` routine is + called. It uses a power series for small `z`, the asymptotic expansion + for large `abs(z)`, the Miller algorithm normalized by the Wronskian + and a Neumann series for intermediate magnitudes, and the uniform + asymptotic expansions for :math:`I_v(z)` and :math:`J_v(z)` for large + orders. Backward recurrence is used to generate sequences or reduce + orders when necessary. + + The calculations above are done in the right half plane and continued + into the left half plane by the formula, + + .. math:: I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z) + + (valid when the real part of `z` is positive). For negative `v`, the + formula + + .. math:: I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z) + + is used, where :math:`K_v(z)` is the modified Bessel function of the + second kind, evaluated using the AMOS routine `zbesk`. + + References + ---------- + .. [1] Temme, Journal of Computational Physics, vol 21, 343 (1976) + .. [2] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + + Examples + -------- + Evaluate the function of order 0 at one point. + + >>> from scipy.special import iv + >>> iv(0, 1.) + 1.2660658777520084 + + Evaluate the function at one point for different orders. + + >>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.) + (1.2660658777520084, 0.565159103992485, 0.2935253263474798) + + The evaluation for different orders can be carried out in one call by + providing a list or NumPy array as argument for the `v` parameter: + + >>> iv([0, 1, 1.5], 1.) + array([1.26606588, 0.5651591 , 0.29352533]) + + Evaluate the function at several points for order 0 by providing an + array for `z`. + + >>> import numpy as np + >>> points = np.array([-2., 0., 3.]) + >>> iv(0, points) + array([2.2795853 , 1. , 4.88079259]) + + If `z` is an array, the order parameter `v` must be broadcastable to + the correct shape if different orders shall be computed in one call. + To calculate the orders 0 and 1 for an 1D array: + + >>> orders = np.array([[0], [1]]) + >>> orders.shape + (2, 1) + + >>> iv(orders, points) + array([[ 2.2795853 , 1. , 4.88079259], + [-1.59063685, 0. , 3.95337022]]) + + Plot the functions of order 0 to 3 from -5 to 5. + + >>> import matplotlib.pyplot as plt + >>> fig, ax = plt.subplots() + >>> x = np.linspace(-5., 5., 1000) + >>> for i in range(4): + ... ax.plot(x, iv(i, x), label=f'$I_{i!r}$') + >>> ax.legend() + >>> plt.show() + + """) + +add_newdoc("ive", + r""" + ive(v, z, out=None) + + Exponentially scaled modified Bessel function of the first kind. + + Defined as:: + + ive(v, z) = iv(v, z) * exp(-abs(z.real)) + + For imaginary numbers without a real part, returns the unscaled + Bessel function of the first kind `iv`. + + Parameters + ---------- + v : array_like of float + Order. + z : array_like of float or complex + Argument. + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + Values of the exponentially scaled modified Bessel function. + + See Also + -------- + iv: Modified Bessel function of the first kind + i0e: Faster implementation of this function for order 0 + i1e: Faster implementation of this function for order 1 + + Notes + ----- + For positive `v`, the AMOS [1]_ `zbesi` routine is called. It uses a + power series for small `z`, the asymptotic expansion for large + `abs(z)`, the Miller algorithm normalized by the Wronskian and a + Neumann series for intermediate magnitudes, and the uniform asymptotic + expansions for :math:`I_v(z)` and :math:`J_v(z)` for large orders. + Backward recurrence is used to generate sequences or reduce orders when + necessary. + + The calculations above are done in the right half plane and continued + into the left half plane by the formula, + + .. math:: I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z) + + (valid when the real part of `z` is positive). For negative `v`, the + formula + + .. math:: I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z) + + is used, where :math:`K_v(z)` is the modified Bessel function of the + second kind, evaluated using the AMOS routine `zbesk`. + + `ive` is useful for large arguments `z`: for these, `iv` easily overflows, + while `ive` does not due to the exponential scaling. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + + Examples + -------- + In the following example `iv` returns infinity whereas `ive` still returns + a finite number. + + >>> from scipy.special import iv, ive + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> iv(3, 1000.), ive(3, 1000.) + (inf, 0.01256056218254712) + + Evaluate the function at one point for different orders by + providing a list or NumPy array as argument for the `v` parameter: + + >>> ive([0, 1, 1.5], 1.) + array([0.46575961, 0.20791042, 0.10798193]) + + Evaluate the function at several points for order 0 by providing an + array for `z`. + + >>> points = np.array([-2., 0., 3.]) + >>> ive(0, points) + array([0.30850832, 1. , 0.24300035]) + + Evaluate the function at several points for different orders by + providing arrays for both `v` for `z`. Both arrays have to be + broadcastable to the correct shape. To calculate the orders 0, 1 + and 2 for a 1D array of points: + + >>> ive([[0], [1], [2]], points) + array([[ 0.30850832, 1. , 0.24300035], + [-0.21526929, 0. , 0.19682671], + [ 0.09323903, 0. , 0.11178255]]) + + Plot the functions of order 0 to 3 from -5 to 5. + + >>> fig, ax = plt.subplots() + >>> x = np.linspace(-5., 5., 1000) + >>> for i in range(4): + ... ax.plot(x, ive(i, x), label=fr'$I_{i!r}(z)\cdot e^{{-|z|}}$') + >>> ax.legend() + >>> ax.set_xlabel(r"$z$") + >>> plt.show() + """) + +add_newdoc("jn", + """ + jn(n, x, out=None) + + Bessel function of the first kind of integer order and real argument. + + Parameters + ---------- + n : array_like + order of the Bessel function + x : array_like + argument of the Bessel function + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + scalar or ndarray + The value of the bessel function + + See Also + -------- + jv + spherical_jn : spherical Bessel functions. + + Notes + ----- + `jn` is an alias of `jv`. + Not to be confused with the spherical Bessel functions (see + `spherical_jn`). + + """) + +add_newdoc("jv", + r""" + jv(v, z, out=None) + + Bessel function of the first kind of real order and complex argument. + + Parameters + ---------- + v : array_like + Order (float). + z : array_like + Argument (float or complex). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + J : scalar or ndarray + Value of the Bessel function, :math:`J_v(z)`. + + See Also + -------- + jve : :math:`J_v` with leading exponential behavior stripped off. + spherical_jn : spherical Bessel functions. + j0 : faster version of this function for order 0. + j1 : faster version of this function for order 1. + + Notes + ----- + For positive `v` values, the computation is carried out using the AMOS + [1]_ `zbesj` routine, which exploits the connection to the modified + Bessel function :math:`I_v`, + + .. math:: + J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0) + + J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0) + + For negative `v` values the formula, + + .. math:: J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v) + + is used, where :math:`Y_v(z)` is the Bessel function of the second + kind, computed using the AMOS routine `zbesy`. Note that the second + term is exactly zero for integer `v`; to improve accuracy the second + term is explicitly omitted for `v` values such that `v = floor(v)`. + + Not to be confused with the spherical Bessel functions (see `spherical_jn`). + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + + Examples + -------- + Evaluate the function of order 0 at one point. + + >>> from scipy.special import jv + >>> jv(0, 1.) + 0.7651976865579666 + + Evaluate the function at one point for different orders. + + >>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.) + (0.7651976865579666, 0.44005058574493355, 0.24029783912342725) + + The evaluation for different orders can be carried out in one call by + providing a list or NumPy array as argument for the `v` parameter: + + >>> jv([0, 1, 1.5], 1.) + array([0.76519769, 0.44005059, 0.24029784]) + + Evaluate the function at several points for order 0 by providing an + array for `z`. + + >>> import numpy as np + >>> points = np.array([-2., 0., 3.]) + >>> jv(0, points) + array([ 0.22389078, 1. , -0.26005195]) + + If `z` is an array, the order parameter `v` must be broadcastable to + the correct shape if different orders shall be computed in one call. + To calculate the orders 0 and 1 for an 1D array: + + >>> orders = np.array([[0], [1]]) + >>> orders.shape + (2, 1) + + >>> jv(orders, points) + array([[ 0.22389078, 1. , -0.26005195], + [-0.57672481, 0. , 0.33905896]]) + + Plot the functions of order 0 to 3 from -10 to 10. + + >>> import matplotlib.pyplot as plt + >>> fig, ax = plt.subplots() + >>> x = np.linspace(-10., 10., 1000) + >>> for i in range(4): + ... ax.plot(x, jv(i, x), label=f'$J_{i!r}$') + >>> ax.legend() + >>> plt.show() + + """) + +add_newdoc("jve", + r""" + jve(v, z, out=None) + + Exponentially scaled Bessel function of the first kind of order `v`. + + Defined as:: + + jve(v, z) = jv(v, z) * exp(-abs(z.imag)) + + Parameters + ---------- + v : array_like + Order (float). + z : array_like + Argument (float or complex). + out : ndarray, optional + Optional output array for the function values + + Returns + ------- + J : scalar or ndarray + Value of the exponentially scaled Bessel function. + + See Also + -------- + jv: Unscaled Bessel function of the first kind + + Notes + ----- + For positive `v` values, the computation is carried out using the AMOS + [1]_ `zbesj` routine, which exploits the connection to the modified + Bessel function :math:`I_v`, + + .. math:: + J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0) + + J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0) + + For negative `v` values the formula, + + .. math:: J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v) + + is used, where :math:`Y_v(z)` is the Bessel function of the second + kind, computed using the AMOS routine `zbesy`. Note that the second + term is exactly zero for integer `v`; to improve accuracy the second + term is explicitly omitted for `v` values such that `v = floor(v)`. + + Exponentially scaled Bessel functions are useful for large arguments `z`: + for these, the unscaled Bessel functions can easily under-or overflow. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + + Examples + -------- + Compare the output of `jv` and `jve` for large complex arguments for `z` + by computing their values for order ``v=1`` at ``z=1000j``. We see that + `jv` overflows but `jve` returns a finite number: + + >>> import numpy as np + >>> from scipy.special import jv, jve + >>> v = 1 + >>> z = 1000j + >>> jv(v, z), jve(v, z) + ((inf+infj), (7.721967686709077e-19+0.012610930256928629j)) + + For real arguments for `z`, `jve` returns the same as `jv`. + + >>> v, z = 1, 1000 + >>> jv(v, z), jve(v, z) + (0.004728311907089523, 0.004728311907089523) + + The function can be evaluated for several orders at the same time by + providing a list or NumPy array for `v`: + + >>> jve([1, 3, 5], 1j) + array([1.27304208e-17+2.07910415e-01j, -4.99352086e-19-8.15530777e-03j, + 6.11480940e-21+9.98657141e-05j]) + + In the same way, the function can be evaluated at several points in one + call by providing a list or NumPy array for `z`: + + >>> jve(1, np.array([1j, 2j, 3j])) + array([1.27308412e-17+0.20791042j, 1.31814423e-17+0.21526929j, + 1.20521602e-17+0.19682671j]) + + It is also possible to evaluate several orders at several points + at the same time by providing arrays for `v` and `z` with + compatible shapes for broadcasting. Compute `jve` for two different orders + `v` and three points `z` resulting in a 2x3 array. + + >>> v = np.array([[1], [3]]) + >>> z = np.array([1j, 2j, 3j]) + >>> v.shape, z.shape + ((2, 1), (3,)) + + >>> jve(v, z) + array([[1.27304208e-17+0.20791042j, 1.31810070e-17+0.21526929j, + 1.20517622e-17+0.19682671j], + [-4.99352086e-19-0.00815531j, -1.76289571e-18-0.02879122j, + -2.92578784e-18-0.04778332j]]) + """) + +add_newdoc("kelvin", + """ + kelvin(x, out=None) + + Kelvin functions as complex numbers + + Parameters + ---------- + x : array_like + Argument + out : tuple of ndarray, optional + Optional output arrays for the function values + + Returns + ------- + Be, Ke, Bep, Kep : 4-tuple of scalar or ndarray + The tuple (Be, Ke, Bep, Kep) contains complex numbers + representing the real and imaginary Kelvin functions and their + derivatives evaluated at `x`. For example, kelvin(x)[0].real = + ber x and kelvin(x)[0].imag = bei x with similar relationships + for ker and kei. + """) + +add_newdoc("ker", + r""" + ker(x, out=None) + + Kelvin function ker. + + Defined as + + .. math:: + + \mathrm{ker}(x) = \Re[K_0(x e^{\pi i / 4})] + + Where :math:`K_0` is the modified Bessel function of the second + kind (see `kv`). See [dlmf]_ for more details. + + Parameters + ---------- + x : array_like + Real argument. + out : ndarray, optional + Optional output array for the function results. + + Returns + ------- + scalar or ndarray + Values of the Kelvin function. + + See Also + -------- + kei : the corresponding imaginary part + kerp : the derivative of ker + kv : modified Bessel function of the second kind + + References + ---------- + .. [dlmf] NIST, Digital Library of Mathematical Functions, + https://dlmf.nist.gov/10.61 + + Examples + -------- + It can be expressed using the modified Bessel function of the + second kind. + + >>> import numpy as np + >>> import scipy.special as sc + >>> x = np.array([1.0, 2.0, 3.0, 4.0]) + >>> sc.kv(0, x * np.exp(np.pi * 1j / 4)).real + array([ 0.28670621, -0.04166451, -0.06702923, -0.03617885]) + >>> sc.ker(x) + array([ 0.28670621, -0.04166451, -0.06702923, -0.03617885]) + + """) + +add_newdoc("kerp", + r""" + kerp(x, out=None) + + Derivative of the Kelvin function ker. + + Parameters + ---------- + x : array_like + Real argument. + out : ndarray, optional + Optional output array for the function results. + + Returns + ------- + scalar or ndarray + Values of the derivative of ker. + + See Also + -------- + ker + + References + ---------- + .. [dlmf] NIST, Digital Library of Mathematical Functions, + https://dlmf.nist.gov/10#PT5 + + """) + +add_newdoc("kl_div", + r""" + kl_div(x, y, out=None) + + Elementwise function for computing Kullback-Leibler divergence. + + .. math:: + + \mathrm{kl\_div}(x, y) = + \begin{cases} + x \log(x / y) - x + y & x > 0, y > 0 \\ + y & x = 0, y \ge 0 \\ + \infty & \text{otherwise} + \end{cases} + + Parameters + ---------- + x, y : array_like + Real arguments + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Values of the Kullback-Liebler divergence. + + See Also + -------- + entr, rel_entr, scipy.stats.entropy + + Notes + ----- + .. versionadded:: 0.15.0 + + This function is non-negative and is jointly convex in `x` and `y`. + + The origin of this function is in convex programming; see [1]_ for + details. This is why the function contains the extra :math:`-x + + y` terms over what might be expected from the Kullback-Leibler + divergence. For a version of the function without the extra terms, + see `rel_entr`. + + References + ---------- + .. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*. + Cambridge University Press, 2004. + :doi:`https://doi.org/10.1017/CBO9780511804441` + + """) + +add_newdoc("kn", + r""" + kn(n, x, out=None) + + Modified Bessel function of the second kind of integer order `n` + + Returns the modified Bessel function of the second kind for integer order + `n` at real `z`. + + These are also sometimes called functions of the third kind, Basset + functions, or Macdonald functions. + + Parameters + ---------- + n : array_like of int + Order of Bessel functions (floats will truncate with a warning) + x : array_like of float + Argument at which to evaluate the Bessel functions + out : ndarray, optional + Optional output array for the function results. + + Returns + ------- + scalar or ndarray + Value of the Modified Bessel function of the second kind, + :math:`K_n(x)`. + + See Also + -------- + kv : Same function, but accepts real order and complex argument + kvp : Derivative of this function + + Notes + ----- + Wrapper for AMOS [1]_ routine `zbesk`. For a discussion of the + algorithm used, see [2]_ and the references therein. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + .. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel + functions of a complex argument and nonnegative order", ACM + TOMS Vol. 12 Issue 3, Sept. 1986, p. 265 + + Examples + -------- + Plot the function of several orders for real input: + + >>> import numpy as np + >>> from scipy.special import kn + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(0, 5, 1000) + >>> for N in range(6): + ... plt.plot(x, kn(N, x), label='$K_{}(x)$'.format(N)) + >>> plt.ylim(0, 10) + >>> plt.legend() + >>> plt.title(r'Modified Bessel function of the second kind $K_n(x)$') + >>> plt.show() + + Calculate for a single value at multiple orders: + + >>> kn([4, 5, 6], 1) + array([ 44.23241585, 360.9605896 , 3653.83831186]) + """) + +add_newdoc("kolmogi", + """ + kolmogi(p, out=None) + + Inverse Survival Function of Kolmogorov distribution + + It is the inverse function to `kolmogorov`. + Returns y such that ``kolmogorov(y) == p``. + + Parameters + ---------- + p : float array_like + Probability + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The value(s) of kolmogi(p) + + See Also + -------- + kolmogorov : The Survival Function for the distribution + scipy.stats.kstwobign : Provides the functionality as a continuous distribution + smirnov, smirnovi : Functions for the one-sided distribution + + Notes + ----- + `kolmogorov` is used by `stats.kstest` in the application of the + Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this + function is exposed in `scpy.special`, but the recommended way to achieve + the most accurate CDF/SF/PDF/PPF/ISF computations is to use the + `stats.kstwobign` distribution. + + Examples + -------- + >>> from scipy.special import kolmogi + >>> kolmogi([0, 0.1, 0.25, 0.5, 0.75, 0.9, 1.0]) + array([ inf, 1.22384787, 1.01918472, 0.82757356, 0.67644769, + 0.57117327, 0. ]) + + """) + +add_newdoc("kolmogorov", + r""" + kolmogorov(y, out=None) + + Complementary cumulative distribution (Survival Function) function of + Kolmogorov distribution. + + Returns the complementary cumulative distribution function of + Kolmogorov's limiting distribution (``D_n*\sqrt(n)`` as n goes to infinity) + of a two-sided test for equality between an empirical and a theoretical + distribution. It is equal to the (limit as n->infinity of the) + probability that ``sqrt(n) * max absolute deviation > y``. + + Parameters + ---------- + y : float array_like + Absolute deviation between the Empirical CDF (ECDF) and the target CDF, + multiplied by sqrt(n). + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The value(s) of kolmogorov(y) + + See Also + -------- + kolmogi : The Inverse Survival Function for the distribution + scipy.stats.kstwobign : Provides the functionality as a continuous distribution + smirnov, smirnovi : Functions for the one-sided distribution + + Notes + ----- + `kolmogorov` is used by `stats.kstest` in the application of the + Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this + function is exposed in `scpy.special`, but the recommended way to achieve + the most accurate CDF/SF/PDF/PPF/ISF computations is to use the + `stats.kstwobign` distribution. + + Examples + -------- + Show the probability of a gap at least as big as 0, 0.5 and 1.0. + + >>> import numpy as np + >>> from scipy.special import kolmogorov + >>> from scipy.stats import kstwobign + >>> kolmogorov([0, 0.5, 1.0]) + array([ 1. , 0.96394524, 0.26999967]) + + Compare a sample of size 1000 drawn from a Laplace(0, 1) distribution against + the target distribution, a Normal(0, 1) distribution. + + >>> from scipy.stats import norm, laplace + >>> rng = np.random.default_rng() + >>> n = 1000 + >>> lap01 = laplace(0, 1) + >>> x = np.sort(lap01.rvs(n, random_state=rng)) + >>> np.mean(x), np.std(x) + (-0.05841730131499543, 1.3968109101997568) + + Construct the Empirical CDF and the K-S statistic Dn. + + >>> target = norm(0,1) # Normal mean 0, stddev 1 + >>> cdfs = target.cdf(x) + >>> ecdfs = np.arange(n+1, dtype=float)/n + >>> gaps = np.column_stack([cdfs - ecdfs[:n], ecdfs[1:] - cdfs]) + >>> Dn = np.max(gaps) + >>> Kn = np.sqrt(n) * Dn + >>> print('Dn=%f, sqrt(n)*Dn=%f' % (Dn, Kn)) + Dn=0.043363, sqrt(n)*Dn=1.371265 + >>> print(chr(10).join(['For a sample of size n drawn from a N(0, 1) distribution:', + ... ' the approximate Kolmogorov probability that sqrt(n)*Dn>=%f is %f' % + ... (Kn, kolmogorov(Kn)), + ... ' the approximate Kolmogorov probability that sqrt(n)*Dn<=%f is %f' % + ... (Kn, kstwobign.cdf(Kn))])) + For a sample of size n drawn from a N(0, 1) distribution: + the approximate Kolmogorov probability that sqrt(n)*Dn>=1.371265 is 0.046533 + the approximate Kolmogorov probability that sqrt(n)*Dn<=1.371265 is 0.953467 + + Plot the Empirical CDF against the target N(0, 1) CDF. + + >>> import matplotlib.pyplot as plt + >>> plt.step(np.concatenate([[-3], x]), ecdfs, where='post', label='Empirical CDF') + >>> x3 = np.linspace(-3, 3, 100) + >>> plt.plot(x3, target.cdf(x3), label='CDF for N(0, 1)') + >>> plt.ylim([0, 1]); plt.grid(True); plt.legend(); + >>> # Add vertical lines marking Dn+ and Dn- + >>> iminus, iplus = np.argmax(gaps, axis=0) + >>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], + ... color='r', linestyle='dashed', lw=4) + >>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], + ... color='r', linestyle='dashed', lw=4) + >>> plt.show() + """) + +add_newdoc("_kolmogc", + r""" + Internal function, do not use. + """) + +add_newdoc("_kolmogci", + r""" + Internal function, do not use. + """) + +add_newdoc("_kolmogp", + r""" + Internal function, do not use. + """) + +add_newdoc("kv", + r""" + kv(v, z, out=None) + + Modified Bessel function of the second kind of real order `v` + + Returns the modified Bessel function of the second kind for real order + `v` at complex `z`. + + These are also sometimes called functions of the third kind, Basset + functions, or Macdonald functions. They are defined as those solutions + of the modified Bessel equation for which, + + .. math:: + K_v(x) \sim \sqrt{\pi/(2x)} \exp(-x) + + as :math:`x \to \infty` [3]_. + + Parameters + ---------- + v : array_like of float + Order of Bessel functions + z : array_like of complex + Argument at which to evaluate the Bessel functions + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The results. Note that input must be of complex type to get complex + output, e.g. ``kv(3, -2+0j)`` instead of ``kv(3, -2)``. + + See Also + -------- + kve : This function with leading exponential behavior stripped off. + kvp : Derivative of this function + + Notes + ----- + Wrapper for AMOS [1]_ routine `zbesk`. For a discussion of the + algorithm used, see [2]_ and the references therein. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + .. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel + functions of a complex argument and nonnegative order", ACM + TOMS Vol. 12 Issue 3, Sept. 1986, p. 265 + .. [3] NIST Digital Library of Mathematical Functions, + Eq. 10.25.E3. https://dlmf.nist.gov/10.25.E3 + + Examples + -------- + Plot the function of several orders for real input: + + >>> import numpy as np + >>> from scipy.special import kv + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(0, 5, 1000) + >>> for N in np.linspace(0, 6, 5): + ... plt.plot(x, kv(N, x), label='$K_{{{}}}(x)$'.format(N)) + >>> plt.ylim(0, 10) + >>> plt.legend() + >>> plt.title(r'Modified Bessel function of the second kind $K_\nu(x)$') + >>> plt.show() + + Calculate for a single value at multiple orders: + + >>> kv([4, 4.5, 5], 1+2j) + array([ 0.1992+2.3892j, 2.3493+3.6j , 7.2827+3.8104j]) + + """) + +add_newdoc("kve", + r""" + kve(v, z, out=None) + + Exponentially scaled modified Bessel function of the second kind. + + Returns the exponentially scaled, modified Bessel function of the + second kind (sometimes called the third kind) for real order `v` at + complex `z`:: + + kve(v, z) = kv(v, z) * exp(z) + + Parameters + ---------- + v : array_like of float + Order of Bessel functions + z : array_like of complex + Argument at which to evaluate the Bessel functions + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The exponentially scaled modified Bessel function of the second kind. + + See Also + -------- + kv : This function without exponential scaling. + k0e : Faster version of this function for order 0. + k1e : Faster version of this function for order 1. + + Notes + ----- + Wrapper for AMOS [1]_ routine `zbesk`. For a discussion of the + algorithm used, see [2]_ and the references therein. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + .. [2] Donald E. Amos, "Algorithm 644: A portable package for Bessel + functions of a complex argument and nonnegative order", ACM + TOMS Vol. 12 Issue 3, Sept. 1986, p. 265 + + Examples + -------- + In the following example `kv` returns 0 whereas `kve` still returns + a useful finite number. + + >>> import numpy as np + >>> from scipy.special import kv, kve + >>> import matplotlib.pyplot as plt + >>> kv(3, 1000.), kve(3, 1000.) + (0.0, 0.03980696128440973) + + Evaluate the function at one point for different orders by + providing a list or NumPy array as argument for the `v` parameter: + + >>> kve([0, 1, 1.5], 1.) + array([1.14446308, 1.63615349, 2.50662827]) + + Evaluate the function at several points for order 0 by providing an + array for `z`. + + >>> points = np.array([1., 3., 10.]) + >>> kve(0, points) + array([1.14446308, 0.6977616 , 0.39163193]) + + Evaluate the function at several points for different orders by + providing arrays for both `v` for `z`. Both arrays have to be + broadcastable to the correct shape. To calculate the orders 0, 1 + and 2 for a 1D array of points: + + >>> kve([[0], [1], [2]], points) + array([[1.14446308, 0.6977616 , 0.39163193], + [1.63615349, 0.80656348, 0.41076657], + [4.41677005, 1.23547058, 0.47378525]]) + + Plot the functions of order 0 to 3 from 0 to 5. + + >>> fig, ax = plt.subplots() + >>> x = np.linspace(0., 5., 1000) + >>> for i in range(4): + ... ax.plot(x, kve(i, x), label=fr'$K_{i!r}(z)\cdot e^z$') + >>> ax.legend() + >>> ax.set_xlabel(r"$z$") + >>> ax.set_ylim(0, 4) + >>> ax.set_xlim(0, 5) + >>> plt.show() + """) + +add_newdoc("_lanczos_sum_expg_scaled", + """ + Internal function, do not use. + """) + +add_newdoc( + "_landau_pdf", + """ + _landau_pdf(x, loc, scale) + + Probability density function of the Landau distribution. + + Parameters + ---------- + x : array_like + Real-valued argument + loc : array_like + Real-valued distribution location + scale : array_like + Positive, real-valued distribution scale + + Returns + ------- + scalar or ndarray + """) + +add_newdoc( + "_landau_cdf", + """ + _landau_cdf(x, loc, scale) + + Cumulative distribution function of the Landau distribution. + + Parameters + ---------- + x : array_like + Real-valued argument + loc : array_like + Real-valued distribution location + scale : array_like + Positive, real-valued distribution scale + + Returns + ------- + scalar or ndarray + """) + +add_newdoc( + "_landau_sf", + """ + _landau_sf(x, loc, scale) + + Survival function of the Landau distribution. + + Parameters + ---------- + x : array_like + Real-valued argument + loc : array_like + Real-valued distribution location + scale : array_like + Positive, real-valued distribution scale + + Returns + ------- + scalar or ndarray + """) + +add_newdoc( + "_landau_ppf", + """ + _landau_ppf(p, loc, scale) + + Percent point function of the Landau distribution. + + Parameters + ---------- + p : array_like + Real-valued argument between 0 and 1 + loc : array_like + Real-valued distribution location + scale : array_like + Positive, real-valued distribution scale + + Returns + ------- + scalar or ndarray + """) + +add_newdoc( + "_landau_isf", + """ + _landau_isf(p, loc, scale) + + Inverse survival function of the Landau distribution. + + Parameters + ---------- + p : array_like + Real-valued argument between 0 and 1 + loc : array_like + Real-valued distribution location + scale : array_like + Positive, real-valued distribution scale + + Returns + ------- + scalar or ndarray + """) + +add_newdoc("_lgam1p", + """ + Internal function, do not use. + """) + +add_newdoc("log1p", + """ + log1p(x, out=None) + + Calculates log(1 + x) for use when `x` is near zero. + + Parameters + ---------- + x : array_like + Real or complex valued input. + out : ndarray, optional + Optional output array for the function results. + + Returns + ------- + scalar or ndarray + Values of ``log(1 + x)``. + + See Also + -------- + expm1, cosm1 + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + It is more accurate than using ``log(1 + x)`` directly for ``x`` + near 0. Note that in the below example ``1 + 1e-17 == 1`` to + double precision. + + >>> sc.log1p(1e-17) + 1e-17 + >>> np.log(1 + 1e-17) + 0.0 + + """) + +add_newdoc("_log1pmx", + """ + Internal function, do not use. + """) + +add_newdoc("lpmv", + r""" + lpmv(m, v, x, out=None) + + Associated Legendre function of integer order and real degree. + + Defined as + + .. math:: + + P_v^m = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_v(x) + + where + + .. math:: + + P_v = \sum_{k = 0}^\infty \frac{(-v)_k (v + 1)_k}{(k!)^2} + \left(\frac{1 - x}{2}\right)^k + + is the Legendre function of the first kind. Here :math:`(\cdot)_k` + is the Pochhammer symbol; see `poch`. + + Parameters + ---------- + m : array_like + Order (int or float). If passed a float not equal to an + integer the function returns NaN. + v : array_like + Degree (float). + x : array_like + Argument (float). Must have ``|x| <= 1``. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + pmv : scalar or ndarray + Value of the associated Legendre function. + + See Also + -------- + lpmn : Compute the associated Legendre function for all orders + ``0, ..., m`` and degrees ``0, ..., n``. + clpmn : Compute the associated Legendre function at complex + arguments. + + Notes + ----- + Note that this implementation includes the Condon-Shortley phase. + + References + ---------- + .. [1] Zhang, Jin, "Computation of Special Functions", John Wiley + and Sons, Inc, 1996. + + """) + +add_newdoc("nbdtr", + r""" + nbdtr(k, n, p, out=None) + + Negative binomial cumulative distribution function. + + Returns the sum of the terms 0 through `k` of the negative binomial + distribution probability mass function, + + .. math:: + + F = \sum_{j=0}^k {{n + j - 1}\choose{j}} p^n (1 - p)^j. + + In a sequence of Bernoulli trials with individual success probabilities + `p`, this is the probability that `k` or fewer failures precede the nth + success. + + Parameters + ---------- + k : array_like + The maximum number of allowed failures (nonnegative int). + n : array_like + The target number of successes (positive int). + p : array_like + Probability of success in a single event (float). + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + F : scalar or ndarray + The probability of `k` or fewer failures before `n` successes in a + sequence of events with individual success probability `p`. + + See Also + -------- + nbdtrc : Negative binomial survival function + nbdtrik : Negative binomial quantile function + scipy.stats.nbinom : Negative binomial distribution + + Notes + ----- + If floating point values are passed for `k` or `n`, they will be truncated + to integers. + + The terms are not summed directly; instead the regularized incomplete beta + function is employed, according to the formula, + + .. math:: + \mathrm{nbdtr}(k, n, p) = I_{p}(n, k + 1). + + Wrapper for the Cephes [1]_ routine `nbdtr`. + + The negative binomial distribution is also available as + `scipy.stats.nbinom`. Using `nbdtr` directly can improve performance + compared to the ``cdf`` method of `scipy.stats.nbinom` (see last example). + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + Examples + -------- + Compute the function for ``k=10`` and ``n=5`` at ``p=0.5``. + + >>> import numpy as np + >>> from scipy.special import nbdtr + >>> nbdtr(10, 5, 0.5) + 0.940765380859375 + + Compute the function for ``n=10`` and ``p=0.5`` at several points by + providing a NumPy array or list for `k`. + + >>> nbdtr([5, 10, 15], 10, 0.5) + array([0.15087891, 0.58809853, 0.88523853]) + + Plot the function for four different parameter sets. + + >>> import matplotlib.pyplot as plt + >>> k = np.arange(130) + >>> n_parameters = [20, 20, 20, 80] + >>> p_parameters = [0.2, 0.5, 0.8, 0.5] + >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] + >>> parameters_list = list(zip(p_parameters, n_parameters, + ... linestyles)) + >>> fig, ax = plt.subplots(figsize=(8, 8)) + >>> for parameter_set in parameters_list: + ... p, n, style = parameter_set + ... nbdtr_vals = nbdtr(k, n, p) + ... ax.plot(k, nbdtr_vals, label=rf"$n={n},\, p={p}$", + ... ls=style) + >>> ax.legend() + >>> ax.set_xlabel("$k$") + >>> ax.set_title("Negative binomial cumulative distribution function") + >>> plt.show() + + The negative binomial distribution is also available as + `scipy.stats.nbinom`. Using `nbdtr` directly can be much faster than + calling the ``cdf`` method of `scipy.stats.nbinom`, especially for small + arrays or individual values. To get the same results one must use the + following parametrization: ``nbinom(n, p).cdf(k)=nbdtr(k, n, p)``. + + >>> from scipy.stats import nbinom + >>> k, n, p = 5, 3, 0.5 + >>> nbdtr_res = nbdtr(k, n, p) # this will often be faster than below + >>> stats_res = nbinom(n, p).cdf(k) + >>> stats_res, nbdtr_res # test that results are equal + (0.85546875, 0.85546875) + + `nbdtr` can evaluate different parameter sets by providing arrays with + shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute + the function for three different `k` at four locations `p`, resulting in + a 3x4 array. + + >>> k = np.array([[5], [10], [15]]) + >>> p = np.array([0.3, 0.5, 0.7, 0.9]) + >>> k.shape, p.shape + ((3, 1), (4,)) + + >>> nbdtr(k, 5, p) + array([[0.15026833, 0.62304687, 0.95265101, 0.9998531 ], + [0.48450894, 0.94076538, 0.99932777, 0.99999999], + [0.76249222, 0.99409103, 0.99999445, 1. ]]) + """) + +add_newdoc("nbdtrc", + r""" + nbdtrc(k, n, p, out=None) + + Negative binomial survival function. + + Returns the sum of the terms `k + 1` to infinity of the negative binomial + distribution probability mass function, + + .. math:: + + F = \sum_{j=k + 1}^\infty {{n + j - 1}\choose{j}} p^n (1 - p)^j. + + In a sequence of Bernoulli trials with individual success probabilities + `p`, this is the probability that more than `k` failures precede the nth + success. + + Parameters + ---------- + k : array_like + The maximum number of allowed failures (nonnegative int). + n : array_like + The target number of successes (positive int). + p : array_like + Probability of success in a single event (float). + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + F : scalar or ndarray + The probability of `k + 1` or more failures before `n` successes in a + sequence of events with individual success probability `p`. + + See Also + -------- + nbdtr : Negative binomial cumulative distribution function + nbdtrik : Negative binomial percentile function + scipy.stats.nbinom : Negative binomial distribution + + Notes + ----- + If floating point values are passed for `k` or `n`, they will be truncated + to integers. + + The terms are not summed directly; instead the regularized incomplete beta + function is employed, according to the formula, + + .. math:: + \mathrm{nbdtrc}(k, n, p) = I_{1 - p}(k + 1, n). + + Wrapper for the Cephes [1]_ routine `nbdtrc`. + + The negative binomial distribution is also available as + `scipy.stats.nbinom`. Using `nbdtrc` directly can improve performance + compared to the ``sf`` method of `scipy.stats.nbinom` (see last example). + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + Examples + -------- + Compute the function for ``k=10`` and ``n=5`` at ``p=0.5``. + + >>> import numpy as np + >>> from scipy.special import nbdtrc + >>> nbdtrc(10, 5, 0.5) + 0.059234619140624986 + + Compute the function for ``n=10`` and ``p=0.5`` at several points by + providing a NumPy array or list for `k`. + + >>> nbdtrc([5, 10, 15], 10, 0.5) + array([0.84912109, 0.41190147, 0.11476147]) + + Plot the function for four different parameter sets. + + >>> import matplotlib.pyplot as plt + >>> k = np.arange(130) + >>> n_parameters = [20, 20, 20, 80] + >>> p_parameters = [0.2, 0.5, 0.8, 0.5] + >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] + >>> parameters_list = list(zip(p_parameters, n_parameters, + ... linestyles)) + >>> fig, ax = plt.subplots(figsize=(8, 8)) + >>> for parameter_set in parameters_list: + ... p, n, style = parameter_set + ... nbdtrc_vals = nbdtrc(k, n, p) + ... ax.plot(k, nbdtrc_vals, label=rf"$n={n},\, p={p}$", + ... ls=style) + >>> ax.legend() + >>> ax.set_xlabel("$k$") + >>> ax.set_title("Negative binomial distribution survival function") + >>> plt.show() + + The negative binomial distribution is also available as + `scipy.stats.nbinom`. Using `nbdtrc` directly can be much faster than + calling the ``sf`` method of `scipy.stats.nbinom`, especially for small + arrays or individual values. To get the same results one must use the + following parametrization: ``nbinom(n, p).sf(k)=nbdtrc(k, n, p)``. + + >>> from scipy.stats import nbinom + >>> k, n, p = 3, 5, 0.5 + >>> nbdtr_res = nbdtrc(k, n, p) # this will often be faster than below + >>> stats_res = nbinom(n, p).sf(k) + >>> stats_res, nbdtr_res # test that results are equal + (0.6367187499999999, 0.6367187499999999) + + `nbdtrc` can evaluate different parameter sets by providing arrays with + shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute + the function for three different `k` at four locations `p`, resulting in + a 3x4 array. + + >>> k = np.array([[5], [10], [15]]) + >>> p = np.array([0.3, 0.5, 0.7, 0.9]) + >>> k.shape, p.shape + ((3, 1), (4,)) + + >>> nbdtrc(k, 5, p) + array([[8.49731667e-01, 3.76953125e-01, 4.73489874e-02, 1.46902600e-04], + [5.15491059e-01, 5.92346191e-02, 6.72234070e-04, 9.29610100e-09], + [2.37507779e-01, 5.90896606e-03, 5.55025308e-06, 3.26346760e-13]]) + """) + +add_newdoc( + "nbdtri", + r""" + nbdtri(k, n, y, out=None) + + Returns the inverse with respect to the parameter `p` of + ``y = nbdtr(k, n, p)``, the negative binomial cumulative distribution + function. + + Parameters + ---------- + k : array_like + The maximum number of allowed failures (nonnegative int). + n : array_like + The target number of successes (positive int). + y : array_like + The probability of `k` or fewer failures before `n` successes (float). + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + p : scalar or ndarray + Probability of success in a single event (float) such that + `nbdtr(k, n, p) = y`. + + See Also + -------- + nbdtr : Cumulative distribution function of the negative binomial. + nbdtrc : Negative binomial survival function. + scipy.stats.nbinom : negative binomial distribution. + nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`. + nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`. + scipy.stats.nbinom : Negative binomial distribution + + Notes + ----- + Wrapper for the Cephes [1]_ routine `nbdtri`. + + The negative binomial distribution is also available as + `scipy.stats.nbinom`. Using `nbdtri` directly can improve performance + compared to the ``ppf`` method of `scipy.stats.nbinom`. + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + Examples + -------- + `nbdtri` is the inverse of `nbdtr` with respect to `p`. + Up to floating point errors the following holds: + ``nbdtri(k, n, nbdtr(k, n, p))=p``. + + >>> import numpy as np + >>> from scipy.special import nbdtri, nbdtr + >>> k, n, y = 5, 10, 0.2 + >>> cdf_val = nbdtr(k, n, y) + >>> nbdtri(k, n, cdf_val) + 0.20000000000000004 + + Compute the function for ``k=10`` and ``n=5`` at several points by + providing a NumPy array or list for `y`. + + >>> y = np.array([0.1, 0.4, 0.8]) + >>> nbdtri(3, 5, y) + array([0.34462319, 0.51653095, 0.69677416]) + + Plot the function for three different parameter sets. + + >>> import matplotlib.pyplot as plt + >>> n_parameters = [5, 20, 30, 30] + >>> k_parameters = [20, 20, 60, 80] + >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] + >>> parameters_list = list(zip(n_parameters, k_parameters, linestyles)) + >>> cdf_vals = np.linspace(0, 1, 1000) + >>> fig, ax = plt.subplots(figsize=(8, 8)) + >>> for parameter_set in parameters_list: + ... n, k, style = parameter_set + ... nbdtri_vals = nbdtri(k, n, cdf_vals) + ... ax.plot(cdf_vals, nbdtri_vals, label=rf"$k={k},\ n={n}$", + ... ls=style) + >>> ax.legend() + >>> ax.set_ylabel("$p$") + >>> ax.set_xlabel("$CDF$") + >>> title = "nbdtri: inverse of negative binomial CDF with respect to $p$" + >>> ax.set_title(title) + >>> plt.show() + + `nbdtri` can evaluate different parameter sets by providing arrays with + shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute + the function for three different `k` at four locations `p`, resulting in + a 3x4 array. + + >>> k = np.array([[5], [10], [15]]) + >>> y = np.array([0.3, 0.5, 0.7, 0.9]) + >>> k.shape, y.shape + ((3, 1), (4,)) + + >>> nbdtri(k, 5, y) + array([[0.37258157, 0.45169416, 0.53249956, 0.64578407], + [0.24588501, 0.30451981, 0.36778453, 0.46397088], + [0.18362101, 0.22966758, 0.28054743, 0.36066188]]) + """) + +add_newdoc("nbdtrik", + r""" + nbdtrik(y, n, p, out=None) + + Negative binomial percentile function. + + Returns the inverse with respect to the parameter `k` of + ``y = nbdtr(k, n, p)``, the negative binomial cumulative distribution + function. + + Parameters + ---------- + y : array_like + The probability of `k` or fewer failures before `n` successes (float). + n : array_like + The target number of successes (positive int). + p : array_like + Probability of success in a single event (float). + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + k : scalar or ndarray + The maximum number of allowed failures such that `nbdtr(k, n, p) = y`. + + See Also + -------- + nbdtr : Cumulative distribution function of the negative binomial. + nbdtrc : Survival function of the negative binomial. + nbdtri : Inverse with respect to `p` of `nbdtr(k, n, p)`. + nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`. + scipy.stats.nbinom : Negative binomial distribution + + Notes + ----- + Wrapper for the CDFLIB [1]_ Fortran routine `cdfnbn`. + + Formula 26.5.26 of [2]_, + + .. math:: + \sum_{j=k + 1}^\infty {{n + j - 1} + \choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n), + + is used to reduce calculation of the cumulative distribution function to + that of a regularized incomplete beta :math:`I`. + + Computation of `k` involves a search for a value that produces the desired + value of `y`. The search relies on the monotonicity of `y` with `k`. + + References + ---------- + .. [1] Barry Brown, James Lovato, and Kathy Russell, + CDFLIB: Library of Fortran Routines for Cumulative Distribution + Functions, Inverses, and Other Parameters. + .. [2] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + Compute the negative binomial cumulative distribution function for an + exemplary parameter set. + + >>> import numpy as np + >>> from scipy.special import nbdtr, nbdtrik + >>> k, n, p = 5, 2, 0.5 + >>> cdf_value = nbdtr(k, n, p) + >>> cdf_value + 0.9375 + + Verify that `nbdtrik` recovers the original value for `k`. + + >>> nbdtrik(cdf_value, n, p) + 5.0 + + Plot the function for different parameter sets. + + >>> import matplotlib.pyplot as plt + >>> p_parameters = [0.2, 0.5, 0.7, 0.5] + >>> n_parameters = [30, 30, 30, 80] + >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] + >>> parameters_list = list(zip(p_parameters, n_parameters, linestyles)) + >>> cdf_vals = np.linspace(0, 1, 1000) + >>> fig, ax = plt.subplots(figsize=(8, 8)) + >>> for parameter_set in parameters_list: + ... p, n, style = parameter_set + ... nbdtrik_vals = nbdtrik(cdf_vals, n, p) + ... ax.plot(cdf_vals, nbdtrik_vals, label=rf"$n={n},\ p={p}$", + ... ls=style) + >>> ax.legend() + >>> ax.set_ylabel("$k$") + >>> ax.set_xlabel("$CDF$") + >>> ax.set_title("Negative binomial percentile function") + >>> plt.show() + + The negative binomial distribution is also available as + `scipy.stats.nbinom`. The percentile function method ``ppf`` + returns the result of `nbdtrik` rounded up to integers: + + >>> from scipy.stats import nbinom + >>> q, n, p = 0.6, 5, 0.5 + >>> nbinom.ppf(q, n, p), nbdtrik(q, n, p) + (5.0, 4.800428460273882) + + """) + +add_newdoc("nbdtrin", + r""" + nbdtrin(k, y, p, out=None) + + Inverse of `nbdtr` vs `n`. + + Returns the inverse with respect to the parameter `n` of + ``y = nbdtr(k, n, p)``, the negative binomial cumulative distribution + function. + + Parameters + ---------- + k : array_like + The maximum number of allowed failures (nonnegative int). + y : array_like + The probability of `k` or fewer failures before `n` successes (float). + p : array_like + Probability of success in a single event (float). + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + n : scalar or ndarray + The number of successes `n` such that `nbdtr(k, n, p) = y`. + + See Also + -------- + nbdtr : Cumulative distribution function of the negative binomial. + nbdtri : Inverse with respect to `p` of `nbdtr(k, n, p)`. + nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`. + + Notes + ----- + Wrapper for the CDFLIB [1]_ Fortran routine `cdfnbn`. + + Formula 26.5.26 of [2]_, + + .. math:: + \sum_{j=k + 1}^\infty {{n + j - 1} + \choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n), + + is used to reduce calculation of the cumulative distribution function to + that of a regularized incomplete beta :math:`I`. + + Computation of `n` involves a search for a value that produces the desired + value of `y`. The search relies on the monotonicity of `y` with `n`. + + References + ---------- + .. [1] Barry Brown, James Lovato, and Kathy Russell, + CDFLIB: Library of Fortran Routines for Cumulative Distribution + Functions, Inverses, and Other Parameters. + .. [2] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + Compute the negative binomial cumulative distribution function for an + exemplary parameter set. + + >>> from scipy.special import nbdtr, nbdtrin + >>> k, n, p = 5, 2, 0.5 + >>> cdf_value = nbdtr(k, n, p) + >>> cdf_value + 0.9375 + + Verify that `nbdtrin` recovers the original value for `n` up to floating + point accuracy. + + >>> nbdtrin(k, cdf_value, p) + 1.999999999998137 + """) + +add_newdoc("ncfdtr", + r""" + ncfdtr(dfn, dfd, nc, f, out=None) + + Cumulative distribution function of the non-central F distribution. + + The non-central F describes the distribution of, + + .. math:: + Z = \frac{X/d_n}{Y/d_d} + + where :math:`X` and :math:`Y` are independently distributed, with + :math:`X` distributed non-central :math:`\chi^2` with noncentrality + parameter `nc` and :math:`d_n` degrees of freedom, and :math:`Y` + distributed :math:`\chi^2` with :math:`d_d` degrees of freedom. + + Parameters + ---------- + dfn : array_like + Degrees of freedom of the numerator sum of squares. Range (0, inf). + dfd : array_like + Degrees of freedom of the denominator sum of squares. Range (0, inf). + nc : array_like + Noncentrality parameter. Range [0, inf). + f : array_like + Quantiles, i.e. the upper limit of integration. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + cdf : scalar or ndarray + The calculated CDF. If all inputs are scalar, the return will be a + float. Otherwise it will be an array. + + See Also + -------- + ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`. + ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`. + ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`. + ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`. + scipy.stats.ncf : Non-central F distribution. + + Notes + ----- + This function calculates the CDF of the non-central f distribution using + the Boost Math C++ library [1]_. + + The cumulative distribution function is computed using Formula 26.6.20 of + [2]_: + + .. math:: + F(d_n, d_d, n_c, f) = \sum_{j=0}^\infty e^{-n_c/2} + \frac{(n_c/2)^j}{j!} I_{x}(\frac{d_n}{2} + j, \frac{d_d}{2}), + + where :math:`I` is the regularized incomplete beta function, and + :math:`x = f d_n/(f d_n + d_d)`. + + Note that argument order of `ncfdtr` is different from that of the + similar ``cdf`` method of `scipy.stats.ncf`: `f` is the last + parameter of `ncfdtr` but the first parameter of ``scipy.stats.ncf.cdf``. + + References + ---------- + .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/. + .. [2] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + >>> import numpy as np + >>> from scipy import special + >>> from scipy import stats + >>> import matplotlib.pyplot as plt + + Plot the CDF of the non-central F distribution, for nc=0. Compare with the + F-distribution from scipy.stats: + + >>> x = np.linspace(-1, 8, num=500) + >>> dfn = 3 + >>> dfd = 2 + >>> ncf_stats = stats.f.cdf(x, dfn, dfd) + >>> ncf_special = special.ncfdtr(dfn, dfd, 0, x) + + >>> fig = plt.figure() + >>> ax = fig.add_subplot(111) + >>> ax.plot(x, ncf_stats, 'b-', lw=3) + >>> ax.plot(x, ncf_special, 'r-') + >>> plt.show() + + """) + +add_newdoc("ncfdtri", + """ + ncfdtri(dfn, dfd, nc, p, out=None) + + Inverse with respect to `f` of the CDF of the non-central F distribution. + + See `ncfdtr` for more details. + + Parameters + ---------- + dfn : array_like + Degrees of freedom of the numerator sum of squares. Range (0, inf). + dfd : array_like + Degrees of freedom of the denominator sum of squares. Range (0, inf). + nc : array_like + Noncentrality parameter. Range [0, inf). + p : array_like + Value of the cumulative distribution function. Must be in the + range [0, 1]. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + f : scalar or ndarray + Quantiles, i.e., the upper limit of integration. + + See Also + -------- + ncfdtr : CDF of the non-central F distribution. + ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`. + ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`. + ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`. + scipy.stats.ncf : Non-central F distribution. + + Notes + ----- + This function calculates the Quantile of the non-central f distribution + using the Boost Math C++ library [1]_. + + Note that argument order of `ncfdtri` is different from that of the + similar ``ppf`` method of `scipy.stats.ncf`. `p` is the last parameter + of `ncfdtri` but the first parameter of ``scipy.stats.ncf.ppf``. + + References + ---------- + .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/. + + Examples + -------- + >>> from scipy.special import ncfdtr, ncfdtri + + Compute the CDF for several values of `f`: + + >>> f = [0.5, 1, 1.5] + >>> p = ncfdtr(2, 3, 1.5, f) + >>> p + array([ 0.20782291, 0.36107392, 0.47345752]) + + Compute the inverse. We recover the values of `f`, as expected: + + >>> ncfdtri(2, 3, 1.5, p) + array([ 0.5, 1. , 1.5]) + + """) + +add_newdoc("ncfdtridfd", + """ + ncfdtridfd(dfn, p, nc, f, out=None) + + Calculate degrees of freedom (denominator) for the noncentral F-distribution. + + This is the inverse with respect to `dfd` of `ncfdtr`. + See `ncfdtr` for more details. + + Parameters + ---------- + dfn : array_like + Degrees of freedom of the numerator sum of squares. Range (0, inf). + p : array_like + Value of the cumulative distribution function. Must be in the + range [0, 1]. + nc : array_like + Noncentrality parameter. Should be in range (0, 1e4). + f : array_like + Quantiles, i.e., the upper limit of integration. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + dfd : scalar or ndarray + Degrees of freedom of the denominator sum of squares. + + See Also + -------- + ncfdtr : CDF of the non-central F distribution. + ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`. + ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`. + ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`. + + Notes + ----- + The value of the cumulative noncentral F distribution is not necessarily + monotone in either degrees of freedom. There thus may be two values that + provide a given CDF value. This routine assumes monotonicity and will + find an arbitrary one of the two values. + + Examples + -------- + >>> from scipy.special import ncfdtr, ncfdtridfd + + Compute the CDF for several values of `dfd`: + + >>> dfd = [1, 2, 3] + >>> p = ncfdtr(2, dfd, 0.25, 15) + >>> p + array([ 0.8097138 , 0.93020416, 0.96787852]) + + Compute the inverse. We recover the values of `dfd`, as expected: + + >>> ncfdtridfd(2, p, 0.25, 15) + array([ 1., 2., 3.]) + + """) + +add_newdoc("ncfdtridfn", + """ + ncfdtridfn(p, dfd, nc, f, out=None) + + Calculate degrees of freedom (numerator) for the noncentral F-distribution. + + This is the inverse with respect to `dfn` of `ncfdtr`. + See `ncfdtr` for more details. + + Parameters + ---------- + p : array_like + Value of the cumulative distribution function. Must be in the + range [0, 1]. + dfd : array_like + Degrees of freedom of the denominator sum of squares. Range (0, inf). + nc : array_like + Noncentrality parameter. Should be in range (0, 1e4). + f : float + Quantiles, i.e., the upper limit of integration. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + dfn : scalar or ndarray + Degrees of freedom of the numerator sum of squares. + + See Also + -------- + ncfdtr : CDF of the non-central F distribution. + ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`. + ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`. + ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`. + + Notes + ----- + The value of the cumulative noncentral F distribution is not necessarily + monotone in either degrees of freedom. There thus may be two values that + provide a given CDF value. This routine assumes monotonicity and will + find an arbitrary one of the two values. + + Examples + -------- + >>> from scipy.special import ncfdtr, ncfdtridfn + + Compute the CDF for several values of `dfn`: + + >>> dfn = [1, 2, 3] + >>> p = ncfdtr(dfn, 2, 0.25, 15) + >>> p + array([ 0.92562363, 0.93020416, 0.93188394]) + + Compute the inverse. We recover the values of `dfn`, as expected: + + >>> ncfdtridfn(p, 2, 0.25, 15) + array([ 1., 2., 3.]) + + """) + +add_newdoc("ncfdtrinc", + """ + ncfdtrinc(dfn, dfd, p, f, out=None) + + Calculate non-centrality parameter for non-central F distribution. + + This is the inverse with respect to `nc` of `ncfdtr`. + See `ncfdtr` for more details. + + Parameters + ---------- + dfn : array_like + Degrees of freedom of the numerator sum of squares. Range (0, inf). + dfd : array_like + Degrees of freedom of the denominator sum of squares. Range (0, inf). + p : array_like + Value of the cumulative distribution function. Must be in the + range [0, 1]. + f : array_like + Quantiles, i.e., the upper limit of integration. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + nc : scalar or ndarray + Noncentrality parameter. + + See Also + -------- + ncfdtr : CDF of the non-central F distribution. + ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`. + ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`. + ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`. + + Examples + -------- + >>> from scipy.special import ncfdtr, ncfdtrinc + + Compute the CDF for several values of `nc`: + + >>> nc = [0.5, 1.5, 2.0] + >>> p = ncfdtr(2, 3, nc, 15) + >>> p + array([ 0.96309246, 0.94327955, 0.93304098]) + + Compute the inverse. We recover the values of `nc`, as expected: + + >>> ncfdtrinc(2, 3, p, 15) + array([ 0.5, 1.5, 2. ]) + + """) + +add_newdoc("nctdtr", + """ + nctdtr(df, nc, t, out=None) + + Cumulative distribution function of the non-central `t` distribution. + + Parameters + ---------- + df : array_like + Degrees of freedom of the distribution. Should be in range (0, inf). + nc : array_like + Noncentrality parameter. + t : array_like + Quantiles, i.e., the upper limit of integration. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + cdf : scalar or ndarray + The calculated CDF. If all inputs are scalar, the return will be a + float. Otherwise, it will be an array. + + See Also + -------- + nctdtrit : Inverse CDF (iCDF) of the non-central t distribution. + nctdtridf : Calculate degrees of freedom, given CDF and iCDF values. + nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values. + + Notes + ----- + This function calculates the CDF of the non-central t distribution using + the Boost Math C++ library [1]_. + + Note that the argument order of `nctdtr` is different from that of the + similar ``cdf`` method of `scipy.stats.nct`: `t` is the last + parameter of `nctdtr` but the first parameter of ``scipy.stats.nct.cdf``. + + References + ---------- + .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/. + + Examples + -------- + >>> import numpy as np + >>> from scipy import special + >>> from scipy import stats + >>> import matplotlib.pyplot as plt + + Plot the CDF of the non-central t distribution, for nc=0. Compare with the + t-distribution from scipy.stats: + + >>> x = np.linspace(-5, 5, num=500) + >>> df = 3 + >>> nct_stats = stats.t.cdf(x, df) + >>> nct_special = special.nctdtr(df, 0, x) + + >>> fig = plt.figure() + >>> ax = fig.add_subplot(111) + >>> ax.plot(x, nct_stats, 'b-', lw=3) + >>> ax.plot(x, nct_special, 'r-') + >>> plt.show() + + """) + +add_newdoc("nctdtridf", + """ + nctdtridf(p, nc, t, out=None) + + Calculate degrees of freedom for non-central t distribution. + + See `nctdtr` for more details. + + Parameters + ---------- + p : array_like + CDF values, in range (0, 1]. + nc : array_like + Noncentrality parameter. Should be in range (-1e6, 1e6). + t : array_like + Quantiles, i.e., the upper limit of integration. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + df : scalar or ndarray + The degrees of freedom. If all inputs are scalar, the return will be a + float. Otherwise, it will be an array. + + See Also + -------- + nctdtr : CDF of the non-central `t` distribution. + nctdtrit : Inverse CDF (iCDF) of the non-central t distribution. + nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values. + + Examples + -------- + >>> from scipy.special import nctdtr, nctdtridf + + Compute the CDF for several values of `df`: + + >>> df = [1, 2, 3] + >>> p = nctdtr(df, 0.25, 1) + >>> p + array([0.67491974, 0.716464 , 0.73349456]) + + Compute the inverse. We recover the values of `df`, as expected: + + >>> nctdtridf(p, 0.25, 1) + array([1., 2., 3.]) + + """) + +add_newdoc("nctdtrinc", + """ + nctdtrinc(df, p, t, out=None) + + Calculate non-centrality parameter for non-central t distribution. + + See `nctdtr` for more details. + + Parameters + ---------- + df : array_like + Degrees of freedom of the distribution. Should be in range (0, inf). + p : array_like + CDF values, in range (0, 1]. + t : array_like + Quantiles, i.e., the upper limit of integration. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + nc : scalar or ndarray + Noncentrality parameter + + See Also + -------- + nctdtr : CDF of the non-central `t` distribution. + nctdtrit : Inverse CDF (iCDF) of the non-central t distribution. + nctdtridf : Calculate degrees of freedom, given CDF and iCDF values. + + Examples + -------- + >>> from scipy.special import nctdtr, nctdtrinc + + Compute the CDF for several values of `nc`: + + >>> nc = [0.5, 1.5, 2.5] + >>> p = nctdtr(3, nc, 1.5) + >>> p + array([0.77569497, 0.45524533, 0.1668691 ]) + + Compute the inverse. We recover the values of `nc`, as expected: + + >>> nctdtrinc(3, p, 1.5) + array([0.5, 1.5, 2.5]) + + """) + +add_newdoc("nctdtrit", + """ + nctdtrit(df, nc, p, out=None) + + Inverse cumulative distribution function of the non-central t distribution. + + See `nctdtr` for more details. + + Parameters + ---------- + df : array_like + Degrees of freedom of the distribution. Should be in range (0, inf). + nc : array_like + Noncentrality parameter. Should be in range (-1e6, 1e6). + p : array_like + CDF values, in range (0, 1]. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + t : scalar or ndarray + Quantiles + + See Also + -------- + nctdtr : CDF of the non-central `t` distribution. + nctdtridf : Calculate degrees of freedom, given CDF and iCDF values. + nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values. + + Examples + -------- + >>> from scipy.special import nctdtr, nctdtrit + + Compute the CDF for several values of `t`: + + >>> t = [0.5, 1, 1.5] + >>> p = nctdtr(3, 1, t) + >>> p + array([0.29811049, 0.46922687, 0.6257559 ]) + + Compute the inverse. We recover the values of `t`, as expected: + + >>> nctdtrit(3, 1, p) + array([0.5, 1. , 1.5]) + + """) + +add_newdoc("ndtr", + r""" + ndtr(x, out=None) + + Cumulative distribution of the standard normal distribution. + + Returns the area under the standard Gaussian probability + density function, integrated from minus infinity to `x` + + .. math:: + + \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp(-t^2/2) dt + + Parameters + ---------- + x : array_like, real or complex + Argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The value of the normal CDF evaluated at `x` + + See Also + -------- + log_ndtr : Logarithm of ndtr + ndtri : Inverse of ndtr, standard normal percentile function + erf : Error function + erfc : 1 - erf + scipy.stats.norm : Normal distribution + + Examples + -------- + Evaluate `ndtr` at one point. + + >>> import numpy as np + >>> from scipy.special import ndtr + >>> ndtr(0.5) + 0.6914624612740131 + + Evaluate the function at several points by providing a NumPy array + or list for `x`. + + >>> ndtr([0, 0.5, 2]) + array([0.5 , 0.69146246, 0.97724987]) + + Plot the function. + + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(-5, 5, 100) + >>> fig, ax = plt.subplots() + >>> ax.plot(x, ndtr(x)) + >>> ax.set_title(r"Standard normal cumulative distribution function $\Phi$") + >>> plt.show() + """) + + +add_newdoc("nrdtrimn", + """ + nrdtrimn(p, std, x, out=None) + + Calculate mean of normal distribution given other params. + + Parameters + ---------- + p : array_like + CDF values, in range (0, 1]. + std : array_like + Standard deviation. + x : array_like + Quantiles, i.e. the upper limit of integration. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + mn : scalar or ndarray + The mean of the normal distribution. + + See Also + -------- + scipy.stats.norm : Normal distribution + ndtr : Standard normal cumulative probability distribution + ndtri : Inverse of standard normal CDF with respect to quantile + nrdtrisd : Inverse of normal distribution CDF with respect to + standard deviation + + Examples + -------- + `nrdtrimn` can be used to recover the mean of a normal distribution + if we know the CDF value `p` for a given quantile `x` and the + standard deviation `std`. First, we calculate + the normal distribution CDF for an exemplary parameter set. + + >>> from scipy.stats import norm + >>> mean = 3. + >>> std = 2. + >>> x = 6. + >>> p = norm.cdf(x, loc=mean, scale=std) + >>> p + 0.9331927987311419 + + Verify that `nrdtrimn` returns the original value for `mean`. + + >>> from scipy.special import nrdtrimn + >>> nrdtrimn(p, std, x) + 3.0000000000000004 + + """) + +add_newdoc("nrdtrisd", + """ + nrdtrisd(mn, p, x, out=None) + + Calculate standard deviation of normal distribution given other params. + + Parameters + ---------- + mn : scalar or ndarray + The mean of the normal distribution. + p : array_like + CDF values, in range (0, 1]. + x : array_like + Quantiles, i.e. the upper limit of integration. + + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + std : scalar or ndarray + Standard deviation. + + See Also + -------- + scipy.stats.norm : Normal distribution + ndtr : Standard normal cumulative probability distribution + ndtri : Inverse of standard normal CDF with respect to quantile + nrdtrimn : Inverse of normal distribution CDF with respect to + mean + + Examples + -------- + `nrdtrisd` can be used to recover the standard deviation of a normal + distribution if we know the CDF value `p` for a given quantile `x` and + the mean `mn`. First, we calculate the normal distribution CDF for an + exemplary parameter set. + + >>> from scipy.stats import norm + >>> mean = 3. + >>> std = 2. + >>> x = 6. + >>> p = norm.cdf(x, loc=mean, scale=std) + >>> p + 0.9331927987311419 + + Verify that `nrdtrisd` returns the original value for `std`. + + >>> from scipy.special import nrdtrisd + >>> nrdtrisd(mean, p, x) + 2.0000000000000004 + + """) + +add_newdoc("log_ndtr", + """ + log_ndtr(x, out=None) + + Logarithm of Gaussian cumulative distribution function. + + Returns the log of the area under the standard Gaussian probability + density function, integrated from minus infinity to `x`:: + + log(1/sqrt(2*pi) * integral(exp(-t**2 / 2), t=-inf..x)) + + Parameters + ---------- + x : array_like, real or complex + Argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The value of the log of the normal CDF evaluated at `x` + + See Also + -------- + erf + erfc + scipy.stats.norm + ndtr + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import log_ndtr, ndtr + + The benefit of ``log_ndtr(x)`` over the naive implementation + ``np.log(ndtr(x))`` is most evident with moderate to large positive + values of ``x``: + + >>> x = np.array([6, 7, 9, 12, 15, 25]) + >>> log_ndtr(x) + array([-9.86587646e-010, -1.27981254e-012, -1.12858841e-019, + -1.77648211e-033, -3.67096620e-051, -3.05669671e-138]) + + The results of the naive calculation for the moderate ``x`` values + have only 5 or 6 correct significant digits. For values of ``x`` + greater than approximately 8.3, the naive expression returns 0: + + >>> np.log(ndtr(x)) + array([-9.86587701e-10, -1.27986510e-12, 0.00000000e+00, + 0.00000000e+00, 0.00000000e+00, 0.00000000e+00]) + """) + +add_newdoc("ndtri", + """ + ndtri(y, out=None) + + Inverse of `ndtr` vs x + + Returns the argument x for which the area under the standard normal + probability density function (integrated from minus infinity to `x`) + is equal to y. + + Parameters + ---------- + p : array_like + Probability + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + x : scalar or ndarray + Value of x such that ``ndtr(x) == p``. + + See Also + -------- + ndtr : Standard normal cumulative probability distribution + ndtri_exp : Inverse of log_ndtr + + Examples + -------- + `ndtri` is the percentile function of the standard normal distribution. + This means it returns the inverse of the cumulative density `ndtr`. First, + let us compute a cumulative density value. + + >>> import numpy as np + >>> from scipy.special import ndtri, ndtr + >>> cdf_val = ndtr(2) + >>> cdf_val + 0.9772498680518208 + + Verify that `ndtri` yields the original value for `x` up to floating point + errors. + + >>> ndtri(cdf_val) + 2.0000000000000004 + + Plot the function. For that purpose, we provide a NumPy array as argument. + + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(0.01, 1, 200) + >>> fig, ax = plt.subplots() + >>> ax.plot(x, ndtri(x)) + >>> ax.set_title("Standard normal percentile function") + >>> plt.show() + """) + +add_newdoc("pdtr", + r""" + pdtr(k, m, out=None) + + Poisson cumulative distribution function. + + Defined as the probability that a Poisson-distributed random + variable with event rate :math:`m` is less than or equal to + :math:`k`. More concretely, this works out to be [1]_ + + .. math:: + + \exp(-m) \sum_{j = 0}^{\lfloor{k}\rfloor} \frac{m^j}{j!}. + + Parameters + ---------- + k : array_like + Number of occurrences (nonnegative, real) + m : array_like + Shape parameter (nonnegative, real) + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Values of the Poisson cumulative distribution function + + See Also + -------- + pdtrc : Poisson survival function + pdtrik : inverse of `pdtr` with respect to `k` + pdtri : inverse of `pdtr` with respect to `m` + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Poisson_distribution + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + It is a cumulative distribution function, so it converges to 1 + monotonically as `k` goes to infinity. + + >>> sc.pdtr([1, 10, 100, np.inf], 1) + array([0.73575888, 0.99999999, 1. , 1. ]) + + It is discontinuous at integers and constant between integers. + + >>> sc.pdtr([1, 1.5, 1.9, 2], 1) + array([0.73575888, 0.73575888, 0.73575888, 0.9196986 ]) + + """) + +add_newdoc("pdtrc", + """ + pdtrc(k, m, out=None) + + Poisson survival function + + Returns the sum of the terms from k+1 to infinity of the Poisson + distribution: sum(exp(-m) * m**j / j!, j=k+1..inf) = gammainc( + k+1, m). Arguments must both be non-negative doubles. + + Parameters + ---------- + k : array_like + Number of occurrences (nonnegative, real) + m : array_like + Shape parameter (nonnegative, real) + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Values of the Poisson survival function + + See Also + -------- + pdtr : Poisson cumulative distribution function + pdtrik : inverse of `pdtr` with respect to `k` + pdtri : inverse of `pdtr` with respect to `m` + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + It is a survival function, so it decreases to 0 + monotonically as `k` goes to infinity. + + >>> k = np.array([1, 10, 100, np.inf]) + >>> sc.pdtrc(k, 1) + array([2.64241118e-001, 1.00477664e-008, 3.94147589e-161, 0.00000000e+000]) + + It can be expressed in terms of the lower incomplete gamma + function `gammainc`. + + >>> sc.gammainc(k + 1, 1) + array([2.64241118e-001, 1.00477664e-008, 3.94147589e-161, 0.00000000e+000]) + + """) + +add_newdoc("pdtri", + """ + pdtri(k, y, out=None) + + Inverse to `pdtr` vs m + + Returns the Poisson variable `m` such that the sum from 0 to `k` of + the Poisson density is equal to the given probability `y`: + calculated by ``gammaincinv(k + 1, y)``. `k` must be a nonnegative + integer and `y` between 0 and 1. + + Parameters + ---------- + k : array_like + Number of occurrences (nonnegative, real) + y : array_like + Probability + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Values of the shape parameter `m` such that ``pdtr(k, m) = p`` + + See Also + -------- + pdtr : Poisson cumulative distribution function + pdtrc : Poisson survival function + pdtrik : inverse of `pdtr` with respect to `k` + + Examples + -------- + >>> import scipy.special as sc + + Compute the CDF for several values of `m`: + + >>> m = [0.5, 1, 1.5] + >>> p = sc.pdtr(1, m) + >>> p + array([0.90979599, 0.73575888, 0.5578254 ]) + + Compute the inverse. We recover the values of `m`, as expected: + + >>> sc.pdtri(1, p) + array([0.5, 1. , 1.5]) + + """) + +add_newdoc("pdtrik", + """ + pdtrik(p, m, out=None) + + Inverse to `pdtr` vs `k`. + + Parameters + ---------- + p : array_like + Probability + m : array_like + Shape parameter (nonnegative, real) + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The number of occurrences `k` such that ``pdtr(k, m) = p`` + + See Also + -------- + pdtr : Poisson cumulative distribution function + pdtrc : Poisson survival function + pdtri : inverse of `pdtr` with respect to `m` + + Examples + -------- + >>> import scipy.special as sc + + Compute the CDF for several values of `k`: + + >>> k = [1, 2, 3] + >>> p = sc.pdtr(k, 2) + >>> p + array([0.40600585, 0.67667642, 0.85712346]) + + Compute the inverse. We recover the values of `k`, as expected: + + >>> sc.pdtrik(p, 2) + array([1., 2., 3.]) + + """) + +add_newdoc("poch", + r""" + poch(z, m, out=None) + + Pochhammer symbol. + + The Pochhammer symbol (rising factorial) is defined as + + .. math:: + + (z)_m = \frac{\Gamma(z + m)}{\Gamma(z)} + + For positive integer `m` it reads + + .. math:: + + (z)_m = z (z + 1) ... (z + m - 1) + + See [dlmf]_ for more details. + + Parameters + ---------- + z, m : array_like + Real-valued arguments. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The value of the function. + + References + ---------- + .. [dlmf] Nist, Digital Library of Mathematical Functions + https://dlmf.nist.gov/5.2#iii + + Examples + -------- + >>> import scipy.special as sc + + It is 1 when m is 0. + + >>> sc.poch([1, 2, 3, 4], 0) + array([1., 1., 1., 1.]) + + For z equal to 1 it reduces to the factorial function. + + >>> sc.poch(1, 5) + 120.0 + >>> 1 * 2 * 3 * 4 * 5 + 120 + + It can be expressed in terms of the gamma function. + + >>> z, m = 3.7, 2.1 + >>> sc.poch(z, m) + 20.529581933776953 + >>> sc.gamma(z + m) / sc.gamma(z) + 20.52958193377696 + + """) + +add_newdoc("powm1", """ + powm1(x, y, out=None) + + Computes ``x**y - 1``. + + This function is useful when `y` is near 0, or when `x` is near 1. + + The function is implemented for real types only (unlike ``numpy.power``, + which accepts complex inputs). + + Parameters + ---------- + x : array_like + The base. Must be a real type (i.e. integer or float, not complex). + y : array_like + The exponent. Must be a real type (i.e. integer or float, not complex). + + Returns + ------- + array_like + Result of the calculation + + Notes + ----- + .. versionadded:: 1.10.0 + + The underlying code is implemented for single precision and double + precision floats only. Unlike `numpy.power`, integer inputs to + `powm1` are converted to floating point, and complex inputs are + not accepted. + + Note the following edge cases: + + * ``powm1(x, 0)`` returns 0 for any ``x``, including 0, ``inf`` + and ``nan``. + * ``powm1(1, y)`` returns 0 for any ``y``, including ``nan`` + and ``inf``. + + This function wraps the ``powm1`` routine from the + Boost Math C++ library [1]_. + + References + ---------- + .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import powm1 + + >>> x = np.array([1.2, 10.0, 0.9999999975]) + >>> y = np.array([1e-9, 1e-11, 0.1875]) + >>> powm1(x, y) + array([ 1.82321557e-10, 2.30258509e-11, -4.68749998e-10]) + + It can be verified that the relative errors in those results + are less than 2.5e-16. + + Compare that to the result of ``x**y - 1``, where the + relative errors are all larger than 8e-8: + + >>> x**y - 1 + array([ 1.82321491e-10, 2.30258035e-11, -4.68750039e-10]) + + """) + + +add_newdoc("pseudo_huber", + r""" + pseudo_huber(delta, r, out=None) + + Pseudo-Huber loss function. + + .. math:: \mathrm{pseudo\_huber}(\delta, r) = + \delta^2 \left( \sqrt{ 1 + \left( \frac{r}{\delta} \right)^2 } - 1 \right) + + Parameters + ---------- + delta : array_like + Input array, indicating the soft quadratic vs. linear loss changepoint. + r : array_like + Input array, possibly representing residuals. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + res : scalar or ndarray + The computed Pseudo-Huber loss function values. + + See Also + -------- + huber: Similar function which this function approximates + + Notes + ----- + Like `huber`, `pseudo_huber` often serves as a robust loss function + in statistics or machine learning to reduce the influence of outliers. + Unlike `huber`, `pseudo_huber` is smooth. + + Typically, `r` represents residuals, the difference + between a model prediction and data. Then, for :math:`|r|\leq\delta`, + `pseudo_huber` resembles the squared error and for :math:`|r|>\delta` the + absolute error. This way, the Pseudo-Huber loss often achieves + a fast convergence in model fitting for small residuals like the squared + error loss function and still reduces the influence of outliers + (:math:`|r|>\delta`) like the absolute error loss. As :math:`\delta` is + the cutoff between squared and absolute error regimes, it has + to be tuned carefully for each problem. `pseudo_huber` is also + convex, making it suitable for gradient based optimization. [1]_ [2]_ + + .. versionadded:: 0.15.0 + + References + ---------- + .. [1] Hartley, Zisserman, "Multiple View Geometry in Computer Vision". + 2003. Cambridge University Press. p. 619 + .. [2] Charbonnier et al. "Deterministic edge-preserving regularization + in computed imaging". 1997. IEEE Trans. Image Processing. + 6 (2): 298 - 311. + + Examples + -------- + Import all necessary modules. + + >>> import numpy as np + >>> from scipy.special import pseudo_huber, huber + >>> import matplotlib.pyplot as plt + + Calculate the function for ``delta=1`` at ``r=2``. + + >>> pseudo_huber(1., 2.) + 1.2360679774997898 + + Calculate the function at ``r=2`` for different `delta` by providing + a list or NumPy array for `delta`. + + >>> pseudo_huber([1., 2., 4.], 3.) + array([2.16227766, 3.21110255, 4. ]) + + Calculate the function for ``delta=1`` at several points by providing + a list or NumPy array for `r`. + + >>> pseudo_huber(2., np.array([1., 1.5, 3., 4.])) + array([0.47213595, 1. , 3.21110255, 4.94427191]) + + The function can be calculated for different `delta` and `r` by + providing arrays for both with compatible shapes for broadcasting. + + >>> r = np.array([1., 2.5, 8., 10.]) + >>> deltas = np.array([[1.], [5.], [9.]]) + >>> print(r.shape, deltas.shape) + (4,) (3, 1) + + >>> pseudo_huber(deltas, r) + array([[ 0.41421356, 1.6925824 , 7.06225775, 9.04987562], + [ 0.49509757, 2.95084972, 22.16990566, 30.90169944], + [ 0.49846624, 3.06693762, 27.37435121, 40.08261642]]) + + Plot the function for different `delta`. + + >>> x = np.linspace(-4, 4, 500) + >>> deltas = [1, 2, 3] + >>> linestyles = ["dashed", "dotted", "dashdot"] + >>> fig, ax = plt.subplots() + >>> combined_plot_parameters = list(zip(deltas, linestyles)) + >>> for delta, style in combined_plot_parameters: + ... ax.plot(x, pseudo_huber(delta, x), label=rf"$\delta={delta}$", + ... ls=style) + >>> ax.legend(loc="upper center") + >>> ax.set_xlabel("$x$") + >>> ax.set_title(r"Pseudo-Huber loss function $h_{\delta}(x)$") + >>> ax.set_xlim(-4, 4) + >>> ax.set_ylim(0, 8) + >>> plt.show() + + Finally, illustrate the difference between `huber` and `pseudo_huber` by + plotting them and their gradients with respect to `r`. The plot shows + that `pseudo_huber` is continuously differentiable while `huber` is not + at the points :math:`\pm\delta`. + + >>> def huber_grad(delta, x): + ... grad = np.copy(x) + ... linear_area = np.argwhere(np.abs(x) > delta) + ... grad[linear_area]=delta*np.sign(x[linear_area]) + ... return grad + >>> def pseudo_huber_grad(delta, x): + ... return x* (1+(x/delta)**2)**(-0.5) + >>> x=np.linspace(-3, 3, 500) + >>> delta = 1. + >>> fig, ax = plt.subplots(figsize=(7, 7)) + >>> ax.plot(x, huber(delta, x), label="Huber", ls="dashed") + >>> ax.plot(x, huber_grad(delta, x), label="Huber Gradient", ls="dashdot") + >>> ax.plot(x, pseudo_huber(delta, x), label="Pseudo-Huber", ls="dotted") + >>> ax.plot(x, pseudo_huber_grad(delta, x), label="Pseudo-Huber Gradient", + ... ls="solid") + >>> ax.legend(loc="upper center") + >>> plt.show() + """) + +add_newdoc("rel_entr", + r""" + rel_entr(x, y, out=None) + + Elementwise function for computing relative entropy. + + .. math:: + + \mathrm{rel\_entr}(x, y) = + \begin{cases} + x \log(x / y) & x > 0, y > 0 \\ + 0 & x = 0, y \ge 0 \\ + \infty & \text{otherwise} + \end{cases} + + Parameters + ---------- + x, y : array_like + Input arrays + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Relative entropy of the inputs + + See Also + -------- + entr, kl_div, scipy.stats.entropy + + Notes + ----- + .. versionadded:: 0.15.0 + + This function is jointly convex in x and y. + + The origin of this function is in convex programming; see + [1]_. Given two discrete probability distributions :math:`p_1, + \ldots, p_n` and :math:`q_1, \ldots, q_n`, the definition of relative + entropy in the context of *information theory* is + + .. math:: + + \sum_{i = 1}^n \mathrm{rel\_entr}(p_i, q_i). + + To compute the latter quantity, use `scipy.stats.entropy`. + + See [2]_ for details. + + References + ---------- + .. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*. + Cambridge University Press, 2004. + :doi:`https://doi.org/10.1017/CBO9780511804441` + .. [2] Kullback-Leibler divergence, + https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence + + """) + +add_newdoc("round", + """ + round(x, out=None) + + Round to the nearest integer. + + Returns the nearest integer to `x`. If `x` ends in 0.5 exactly, + the nearest even integer is chosen. + + Parameters + ---------- + x : array_like + Real valued input. + out : ndarray, optional + Optional output array for the function results. + + Returns + ------- + scalar or ndarray + The nearest integers to the elements of `x`. The result is of + floating type, not integer type. + + Examples + -------- + >>> import scipy.special as sc + + It rounds to even. + + >>> sc.round([0.5, 1.5]) + array([0., 2.]) + + """) + +add_newdoc("shichi", + r""" + shichi(x, out=None) + + Hyperbolic sine and cosine integrals. + + The hyperbolic sine integral is + + .. math:: + + \int_0^x \frac{\sinh{t}}{t}dt + + and the hyperbolic cosine integral is + + .. math:: + + \gamma + \log(x) + \int_0^x \frac{\cosh{t} - 1}{t} dt + + where :math:`\gamma` is Euler's constant and :math:`\log` is the + principal branch of the logarithm [1]_. + + Parameters + ---------- + x : array_like + Real or complex points at which to compute the hyperbolic sine + and cosine integrals. + out : tuple of ndarray, optional + Optional output arrays for the function results + + Returns + ------- + si : scalar or ndarray + Hyperbolic sine integral at ``x`` + ci : scalar or ndarray + Hyperbolic cosine integral at ``x`` + + See Also + -------- + sici : Sine and cosine integrals. + exp1 : Exponential integral E1. + expi : Exponential integral Ei. + + Notes + ----- + For real arguments with ``x < 0``, ``chi`` is the real part of the + hyperbolic cosine integral. For such points ``chi(x)`` and ``chi(x + + 0j)`` differ by a factor of ``1j*pi``. + + For real arguments the function is computed by calling Cephes' + [2]_ *shichi* routine. For complex arguments the algorithm is based + on Mpmath's [3]_ *shi* and *chi* routines. + + References + ---------- + .. [1] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + (See Section 5.2.) + .. [2] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + .. [3] Fredrik Johansson and others. + "mpmath: a Python library for arbitrary-precision floating-point + arithmetic" (Version 0.19) http://mpmath.org/ + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.special import shichi, sici + + `shichi` accepts real or complex input: + + >>> shichi(0.5) + (0.5069967498196671, -0.05277684495649357) + >>> shichi(0.5 + 2.5j) + ((0.11772029666668238+1.831091777729851j), + (0.29912435887648825+1.7395351121166562j)) + + The hyperbolic sine and cosine integrals Shi(z) and Chi(z) are + related to the sine and cosine integrals Si(z) and Ci(z) by + + * Shi(z) = -i*Si(i*z) + * Chi(z) = Ci(-i*z) + i*pi/2 + + >>> z = 0.25 + 5j + >>> shi, chi = shichi(z) + >>> shi, -1j*sici(1j*z)[0] # Should be the same. + ((-0.04834719325101729+1.5469354086921228j), + (-0.04834719325101729+1.5469354086921228j)) + >>> chi, sici(-1j*z)[1] + 1j*np.pi/2 # Should be the same. + ((-0.19568708973868087+1.556276312103824j), + (-0.19568708973868087+1.556276312103824j)) + + Plot the functions evaluated on the real axis: + + >>> xp = np.geomspace(1e-8, 4.0, 250) + >>> x = np.concatenate((-xp[::-1], xp)) + >>> shi, chi = shichi(x) + + >>> fig, ax = plt.subplots() + >>> ax.plot(x, shi, label='Shi(x)') + >>> ax.plot(x, chi, '--', label='Chi(x)') + >>> ax.set_xlabel('x') + >>> ax.set_title('Hyperbolic Sine and Cosine Integrals') + >>> ax.legend(shadow=True, framealpha=1, loc='lower right') + >>> ax.grid(True) + >>> plt.show() + + """) + +add_newdoc("sici", + r""" + sici(x, out=None) + + Sine and cosine integrals. + + The sine integral is + + .. math:: + + \int_0^x \frac{\sin{t}}{t}dt + + and the cosine integral is + + .. math:: + + \gamma + \log(x) + \int_0^x \frac{\cos{t} - 1}{t}dt + + where :math:`\gamma` is Euler's constant and :math:`\log` is the + principal branch of the logarithm [1]_. + + Parameters + ---------- + x : array_like + Real or complex points at which to compute the sine and cosine + integrals. + out : tuple of ndarray, optional + Optional output arrays for the function results + + Returns + ------- + si : scalar or ndarray + Sine integral at ``x`` + ci : scalar or ndarray + Cosine integral at ``x`` + + See Also + -------- + shichi : Hyperbolic sine and cosine integrals. + exp1 : Exponential integral E1. + expi : Exponential integral Ei. + + Notes + ----- + For real arguments with ``x < 0``, ``ci`` is the real part of the + cosine integral. For such points ``ci(x)`` and ``ci(x + 0j)`` + differ by a factor of ``1j*pi``. + + For real arguments the function is computed by calling Cephes' + [2]_ *sici* routine. For complex arguments the algorithm is based + on Mpmath's [3]_ *si* and *ci* routines. + + References + ---------- + .. [1] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + (See Section 5.2.) + .. [2] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + .. [3] Fredrik Johansson and others. + "mpmath: a Python library for arbitrary-precision floating-point + arithmetic" (Version 0.19) http://mpmath.org/ + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.special import sici, exp1 + + `sici` accepts real or complex input: + + >>> sici(2.5) + (1.7785201734438267, 0.2858711963653835) + >>> sici(2.5 + 3j) + ((4.505735874563953+0.06863305018999577j), + (0.0793644206906966-2.935510262937543j)) + + For z in the right half plane, the sine and cosine integrals are + related to the exponential integral E1 (implemented in SciPy as + `scipy.special.exp1`) by + + * Si(z) = (E1(i*z) - E1(-i*z))/2i + pi/2 + * Ci(z) = -(E1(i*z) + E1(-i*z))/2 + + See [1]_ (equations 5.2.21 and 5.2.23). + + We can verify these relations: + + >>> z = 2 - 3j + >>> sici(z) + ((4.54751388956229-1.3991965806460565j), + (1.408292501520851+2.9836177420296055j)) + + >>> (exp1(1j*z) - exp1(-1j*z))/2j + np.pi/2 # Same as sine integral + (4.54751388956229-1.3991965806460565j) + + >>> -(exp1(1j*z) + exp1(-1j*z))/2 # Same as cosine integral + (1.408292501520851+2.9836177420296055j) + + Plot the functions evaluated on the real axis; the dotted horizontal + lines are at pi/2 and -pi/2: + + >>> x = np.linspace(-16, 16, 150) + >>> si, ci = sici(x) + + >>> fig, ax = plt.subplots() + >>> ax.plot(x, si, label='Si(x)') + >>> ax.plot(x, ci, '--', label='Ci(x)') + >>> ax.legend(shadow=True, framealpha=1, loc='upper left') + >>> ax.set_xlabel('x') + >>> ax.set_title('Sine and Cosine Integrals') + >>> ax.axhline(np.pi/2, linestyle=':', alpha=0.5, color='k') + >>> ax.axhline(-np.pi/2, linestyle=':', alpha=0.5, color='k') + >>> ax.grid(True) + >>> plt.show() + + """) + +add_newdoc("smirnov", + r""" + smirnov(n, d, out=None) + + Kolmogorov-Smirnov complementary cumulative distribution function + + Returns the exact Kolmogorov-Smirnov complementary cumulative + distribution function,(aka the Survival Function) of Dn+ (or Dn-) + for a one-sided test of equality between an empirical and a + theoretical distribution. It is equal to the probability that the + maximum difference between a theoretical distribution and an empirical + one based on `n` samples is greater than d. + + Parameters + ---------- + n : int + Number of samples + d : float array_like + Deviation between the Empirical CDF (ECDF) and the target CDF. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The value(s) of smirnov(n, d), Prob(Dn+ >= d) (Also Prob(Dn- >= d)) + + See Also + -------- + smirnovi : The Inverse Survival Function for the distribution + scipy.stats.ksone : Provides the functionality as a continuous distribution + kolmogorov, kolmogi : Functions for the two-sided distribution + + Notes + ----- + `smirnov` is used by `stats.kstest` in the application of the + Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this + function is exposed in `scpy.special`, but the recommended way to achieve + the most accurate CDF/SF/PDF/PPF/ISF computations is to use the + `stats.ksone` distribution. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import smirnov + >>> from scipy.stats import norm + + Show the probability of a gap at least as big as 0, 0.5 and 1.0 for a + sample of size 5. + + >>> smirnov(5, [0, 0.5, 1.0]) + array([ 1. , 0.056, 0. ]) + + Compare a sample of size 5 against N(0, 1), the standard normal + distribution with mean 0 and standard deviation 1. + + `x` is the sample. + + >>> x = np.array([-1.392, -0.135, 0.114, 0.190, 1.82]) + + >>> target = norm(0, 1) + >>> cdfs = target.cdf(x) + >>> cdfs + array([0.0819612 , 0.44630594, 0.5453811 , 0.57534543, 0.9656205 ]) + + Construct the empirical CDF and the K-S statistics (Dn+, Dn-, Dn). + + >>> n = len(x) + >>> ecdfs = np.arange(n+1, dtype=float)/n + >>> cols = np.column_stack([x, ecdfs[1:], cdfs, cdfs - ecdfs[:n], + ... ecdfs[1:] - cdfs]) + >>> with np.printoptions(precision=3): + ... print(cols) + [[-1.392 0.2 0.082 0.082 0.118] + [-0.135 0.4 0.446 0.246 -0.046] + [ 0.114 0.6 0.545 0.145 0.055] + [ 0.19 0.8 0.575 -0.025 0.225] + [ 1.82 1. 0.966 0.166 0.034]] + >>> gaps = cols[:, -2:] + >>> Dnpm = np.max(gaps, axis=0) + >>> print(f'Dn-={Dnpm[0]:f}, Dn+={Dnpm[1]:f}') + Dn-=0.246306, Dn+=0.224655 + >>> probs = smirnov(n, Dnpm) + >>> print(f'For a sample of size {n} drawn from N(0, 1):', + ... f' Smirnov n={n}: Prob(Dn- >= {Dnpm[0]:f}) = {probs[0]:.4f}', + ... f' Smirnov n={n}: Prob(Dn+ >= {Dnpm[1]:f}) = {probs[1]:.4f}', + ... sep='\n') + For a sample of size 5 drawn from N(0, 1): + Smirnov n=5: Prob(Dn- >= 0.246306) = 0.4711 + Smirnov n=5: Prob(Dn+ >= 0.224655) = 0.5245 + + Plot the empirical CDF and the standard normal CDF. + + >>> import matplotlib.pyplot as plt + >>> plt.step(np.concatenate(([-2.5], x, [2.5])), + ... np.concatenate((ecdfs, [1])), + ... where='post', label='Empirical CDF') + >>> xx = np.linspace(-2.5, 2.5, 100) + >>> plt.plot(xx, target.cdf(xx), '--', label='CDF for N(0, 1)') + + Add vertical lines marking Dn+ and Dn-. + + >>> iminus, iplus = np.argmax(gaps, axis=0) + >>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], color='r', + ... alpha=0.5, lw=4) + >>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], color='m', + ... alpha=0.5, lw=4) + + >>> plt.grid(True) + >>> plt.legend(framealpha=1, shadow=True) + >>> plt.show() + """) + +add_newdoc("smirnovi", + """ + smirnovi(n, p, out=None) + + Inverse to `smirnov` + + Returns `d` such that ``smirnov(n, d) == p``, the critical value + corresponding to `p`. + + Parameters + ---------- + n : int + Number of samples + p : float array_like + Probability + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + The value(s) of smirnovi(n, p), the critical values. + + See Also + -------- + smirnov : The Survival Function (SF) for the distribution + scipy.stats.ksone : Provides the functionality as a continuous distribution + kolmogorov, kolmogi : Functions for the two-sided distribution + scipy.stats.kstwobign : Two-sided Kolmogorov-Smirnov distribution, large n + + Notes + ----- + `smirnov` is used by `stats.kstest` in the application of the + Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this + function is exposed in `scpy.special`, but the recommended way to achieve + the most accurate CDF/SF/PDF/PPF/ISF computations is to use the + `stats.ksone` distribution. + + Examples + -------- + >>> from scipy.special import smirnovi, smirnov + + >>> n = 24 + >>> deviations = [0.1, 0.2, 0.3] + + Use `smirnov` to compute the complementary CDF of the Smirnov + distribution for the given number of samples and deviations. + + >>> p = smirnov(n, deviations) + >>> p + array([0.58105083, 0.12826832, 0.01032231]) + + The inverse function ``smirnovi(n, p)`` returns ``deviations``. + + >>> smirnovi(n, p) + array([0.1, 0.2, 0.3]) + + """) + +add_newdoc("_smirnovc", + """ + _smirnovc(n, d) + Internal function, do not use. + """) + +add_newdoc("_smirnovci", + """ + Internal function, do not use. + """) + +add_newdoc("_smirnovp", + """ + _smirnovp(n, p) + Internal function, do not use. + """) + +add_newdoc("spence", + r""" + spence(z, out=None) + + Spence's function, also known as the dilogarithm. + + It is defined to be + + .. math:: + \int_1^z \frac{\log(t)}{1 - t}dt + + for complex :math:`z`, where the contour of integration is taken + to avoid the branch cut of the logarithm. Spence's function is + analytic everywhere except the negative real axis where it has a + branch cut. + + Parameters + ---------- + z : array_like + Points at which to evaluate Spence's function + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + s : scalar or ndarray + Computed values of Spence's function + + Notes + ----- + There is a different convention which defines Spence's function by + the integral + + .. math:: + -\int_0^z \frac{\log(1 - t)}{t}dt; + + this is our ``spence(1 - z)``. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import spence + >>> import matplotlib.pyplot as plt + + The function is defined for complex inputs: + + >>> spence([1-1j, 1.5+2j, 3j, -10-5j]) + array([-0.20561676+0.91596559j, -0.86766909-1.39560134j, + -0.59422064-2.49129918j, -1.14044398+6.80075924j]) + + For complex inputs on the branch cut, which is the negative real axis, + the function returns the limit for ``z`` with positive imaginary part. + For example, in the following, note the sign change of the imaginary + part of the output for ``z = -2`` and ``z = -2 - 1e-8j``: + + >>> spence([-2 + 1e-8j, -2, -2 - 1e-8j]) + array([2.32018041-3.45139229j, 2.32018042-3.4513923j , + 2.32018041+3.45139229j]) + + The function returns ``nan`` for real inputs on the branch cut: + + >>> spence(-1.5) + nan + + Verify some particular values: ``spence(0) = pi**2/6``, + ``spence(1) = 0`` and ``spence(2) = -pi**2/12``. + + >>> spence([0, 1, 2]) + array([ 1.64493407, 0. , -0.82246703]) + >>> np.pi**2/6, -np.pi**2/12 + (1.6449340668482264, -0.8224670334241132) + + Verify the identity:: + + spence(z) + spence(1 - z) = pi**2/6 - log(z)*log(1 - z) + + >>> z = 3 + 4j + >>> spence(z) + spence(1 - z) + (-2.6523186143876067+1.8853470951513935j) + >>> np.pi**2/6 - np.log(z)*np.log(1 - z) + (-2.652318614387606+1.885347095151394j) + + Plot the function for positive real input. + + >>> fig, ax = plt.subplots() + >>> x = np.linspace(0, 6, 400) + >>> ax.plot(x, spence(x)) + >>> ax.grid() + >>> ax.set_xlabel('x') + >>> ax.set_title('spence(x)') + >>> plt.show() + """) + +add_newdoc( + "stdtr", + r""" + stdtr(df, t, out=None) + + Student t distribution cumulative distribution function + + Returns the integral: + + .. math:: + \frac{\Gamma((df+1)/2)}{\sqrt{\pi df} \Gamma(df/2)} + \int_{-\infty}^t (1+x^2/df)^{-(df+1)/2}\, dx + + Parameters + ---------- + df : array_like + Degrees of freedom + t : array_like + Upper bound of the integral + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Value of the Student t CDF at t + + See Also + -------- + stdtridf : inverse of stdtr with respect to `df` + stdtrit : inverse of stdtr with respect to `t` + scipy.stats.t : student t distribution + + Notes + ----- + The student t distribution is also available as `scipy.stats.t`. + Calling `stdtr` directly can improve performance compared to the + ``cdf`` method of `scipy.stats.t` (see last example below). + + Examples + -------- + Calculate the function for ``df=3`` at ``t=1``. + + >>> import numpy as np + >>> from scipy.special import stdtr + >>> import matplotlib.pyplot as plt + >>> stdtr(3, 1) + 0.8044988905221148 + + Plot the function for three different degrees of freedom. + + >>> x = np.linspace(-10, 10, 1000) + >>> fig, ax = plt.subplots() + >>> parameters = [(1, "solid"), (3, "dashed"), (10, "dotted")] + >>> for (df, linestyle) in parameters: + ... ax.plot(x, stdtr(df, x), ls=linestyle, label=f"$df={df}$") + >>> ax.legend() + >>> ax.set_title("Student t distribution cumulative distribution function") + >>> plt.show() + + The function can be computed for several degrees of freedom at the same + time by providing a NumPy array or list for `df`: + + >>> stdtr([1, 2, 3], 1) + array([0.75 , 0.78867513, 0.80449889]) + + It is possible to calculate the function at several points for several + different degrees of freedom simultaneously by providing arrays for `df` + and `t` with shapes compatible for broadcasting. Compute `stdtr` at + 4 points for 3 degrees of freedom resulting in an array of shape 3x4. + + >>> dfs = np.array([[1], [2], [3]]) + >>> t = np.array([2, 4, 6, 8]) + >>> dfs.shape, t.shape + ((3, 1), (4,)) + + >>> stdtr(dfs, t) + array([[0.85241638, 0.92202087, 0.94743154, 0.96041658], + [0.90824829, 0.97140452, 0.98666426, 0.99236596], + [0.93033702, 0.98599577, 0.99536364, 0.99796171]]) + + The t distribution is also available as `scipy.stats.t`. Calling `stdtr` + directly can be much faster than calling the ``cdf`` method of + `scipy.stats.t`. To get the same results, one must use the following + parametrization: ``scipy.stats.t(df).cdf(x) = stdtr(df, x)``. + + >>> from scipy.stats import t + >>> df, x = 3, 1 + >>> stdtr_result = stdtr(df, x) # this can be faster than below + >>> stats_result = t(df).cdf(x) + >>> stats_result == stdtr_result # test that results are equal + True + """) + +add_newdoc("stdtridf", + """ + stdtridf(p, t, out=None) + + Inverse of `stdtr` vs df + + Returns the argument df such that stdtr(df, t) is equal to `p`. + + Parameters + ---------- + p : array_like + Probability + t : array_like + Upper bound of the integral + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + df : scalar or ndarray + Value of `df` such that ``stdtr(df, t) == p`` + + See Also + -------- + stdtr : Student t CDF + stdtrit : inverse of stdtr with respect to `t` + scipy.stats.t : Student t distribution + + Examples + -------- + Compute the student t cumulative distribution function for one + parameter set. + + >>> from scipy.special import stdtr, stdtridf + >>> df, x = 5, 2 + >>> cdf_value = stdtr(df, x) + >>> cdf_value + 0.9490302605850709 + + Verify that `stdtridf` recovers the original value for `df` given + the CDF value and `x`. + + >>> stdtridf(cdf_value, x) + 5.0 + """) + +add_newdoc("stdtrit", + """ + stdtrit(df, p, out=None) + + The `p`-th quantile of the student t distribution. + + This function is the inverse of the student t distribution cumulative + distribution function (CDF), returning `t` such that `stdtr(df, t) = p`. + + Returns the argument `t` such that stdtr(df, t) is equal to `p`. + + Parameters + ---------- + df : array_like + Degrees of freedom + p : array_like + Probability + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + t : scalar or ndarray + Value of `t` such that ``stdtr(df, t) == p`` + + See Also + -------- + stdtr : Student t CDF + stdtridf : inverse of stdtr with respect to `df` + scipy.stats.t : Student t distribution + + Notes + ----- + The student t distribution is also available as `scipy.stats.t`. Calling + `stdtrit` directly can improve performance compared to the ``ppf`` + method of `scipy.stats.t` (see last example below). + + Examples + -------- + `stdtrit` represents the inverse of the student t distribution CDF which + is available as `stdtr`. Here, we calculate the CDF for ``df`` at + ``x=1``. `stdtrit` then returns ``1`` up to floating point errors + given the same value for `df` and the computed CDF value. + + >>> import numpy as np + >>> from scipy.special import stdtr, stdtrit + >>> import matplotlib.pyplot as plt + >>> df = 3 + >>> x = 1 + >>> cdf_value = stdtr(df, x) + >>> stdtrit(df, cdf_value) + 0.9999999994418539 + + Plot the function for three different degrees of freedom. + + >>> x = np.linspace(0, 1, 1000) + >>> parameters = [(1, "solid"), (2, "dashed"), (5, "dotted")] + >>> fig, ax = plt.subplots() + >>> for (df, linestyle) in parameters: + ... ax.plot(x, stdtrit(df, x), ls=linestyle, label=f"$df={df}$") + >>> ax.legend() + >>> ax.set_ylim(-10, 10) + >>> ax.set_title("Student t distribution quantile function") + >>> plt.show() + + The function can be computed for several degrees of freedom at the same + time by providing a NumPy array or list for `df`: + + >>> stdtrit([1, 2, 3], 0.7) + array([0.72654253, 0.6172134 , 0.58438973]) + + It is possible to calculate the function at several points for several + different degrees of freedom simultaneously by providing arrays for `df` + and `p` with shapes compatible for broadcasting. Compute `stdtrit` at + 4 points for 3 degrees of freedom resulting in an array of shape 3x4. + + >>> dfs = np.array([[1], [2], [3]]) + >>> p = np.array([0.2, 0.4, 0.7, 0.8]) + >>> dfs.shape, p.shape + ((3, 1), (4,)) + + >>> stdtrit(dfs, p) + array([[-1.37638192, -0.3249197 , 0.72654253, 1.37638192], + [-1.06066017, -0.28867513, 0.6172134 , 1.06066017], + [-0.97847231, -0.27667066, 0.58438973, 0.97847231]]) + + The t distribution is also available as `scipy.stats.t`. Calling `stdtrit` + directly can be much faster than calling the ``ppf`` method of + `scipy.stats.t`. To get the same results, one must use the following + parametrization: ``scipy.stats.t(df).ppf(x) = stdtrit(df, x)``. + + >>> from scipy.stats import t + >>> df, x = 3, 0.5 + >>> stdtrit_result = stdtrit(df, x) # this can be faster than below + >>> stats_result = t(df).ppf(x) + >>> stats_result == stdtrit_result # test that results are equal + True + """) + +add_newdoc( + "tklmbda", + r""" + tklmbda(x, lmbda, out=None) + + Cumulative distribution function of the Tukey lambda distribution. + + Parameters + ---------- + x, lmbda : array_like + Parameters + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + cdf : scalar or ndarray + Value of the Tukey lambda CDF + + See Also + -------- + scipy.stats.tukeylambda : Tukey lambda distribution + + Examples + -------- + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.special import tklmbda, expit + + Compute the cumulative distribution function (CDF) of the Tukey lambda + distribution at several ``x`` values for `lmbda` = -1.5. + + >>> x = np.linspace(-2, 2, 9) + >>> x + array([-2. , -1.5, -1. , -0.5, 0. , 0.5, 1. , 1.5, 2. ]) + >>> tklmbda(x, -1.5) + array([0.34688734, 0.3786554 , 0.41528805, 0.45629737, 0.5 , + 0.54370263, 0.58471195, 0.6213446 , 0.65311266]) + + When `lmbda` is 0, the function is the logistic sigmoid function, + which is implemented in `scipy.special` as `expit`. + + >>> tklmbda(x, 0) + array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5 , + 0.62245933, 0.73105858, 0.81757448, 0.88079708]) + >>> expit(x) + array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5 , + 0.62245933, 0.73105858, 0.81757448, 0.88079708]) + + When `lmbda` is 1, the Tukey lambda distribution is uniform on the + interval [-1, 1], so the CDF increases linearly. + + >>> t = np.linspace(-1, 1, 9) + >>> tklmbda(t, 1) + array([0. , 0.125, 0.25 , 0.375, 0.5 , 0.625, 0.75 , 0.875, 1. ]) + + In the following, we generate plots for several values of `lmbda`. + + The first figure shows graphs for `lmbda` <= 0. + + >>> styles = ['-', '-.', '--', ':'] + >>> fig, ax = plt.subplots() + >>> x = np.linspace(-12, 12, 500) + >>> for k, lmbda in enumerate([-1.0, -0.5, 0.0]): + ... y = tklmbda(x, lmbda) + ... ax.plot(x, y, styles[k], label=rf'$\lambda$ = {lmbda:-4.1f}') + + >>> ax.set_title(r'tklmbda(x, $\lambda$)') + >>> ax.set_label('x') + >>> ax.legend(framealpha=1, shadow=True) + >>> ax.grid(True) + + The second figure shows graphs for `lmbda` > 0. The dots in the + graphs show the bounds of the support of the distribution. + + >>> fig, ax = plt.subplots() + >>> x = np.linspace(-4.2, 4.2, 500) + >>> lmbdas = [0.25, 0.5, 1.0, 1.5] + >>> for k, lmbda in enumerate(lmbdas): + ... y = tklmbda(x, lmbda) + ... ax.plot(x, y, styles[k], label=fr'$\lambda$ = {lmbda}') + + >>> ax.set_prop_cycle(None) + >>> for lmbda in lmbdas: + ... ax.plot([-1/lmbda, 1/lmbda], [0, 1], '.', ms=8) + + >>> ax.set_title(r'tklmbda(x, $\lambda$)') + >>> ax.set_xlabel('x') + >>> ax.legend(framealpha=1, shadow=True) + >>> ax.grid(True) + + >>> plt.tight_layout() + >>> plt.show() + + The CDF of the Tukey lambda distribution is also implemented as the + ``cdf`` method of `scipy.stats.tukeylambda`. In the following, + ``tukeylambda.cdf(x, -0.5)`` and ``tklmbda(x, -0.5)`` compute the + same values: + + >>> from scipy.stats import tukeylambda + >>> x = np.linspace(-2, 2, 9) + + >>> tukeylambda.cdf(x, -0.5) + array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5 , + 0.58671839, 0.66458323, 0.72906142, 0.78004843]) + + >>> tklmbda(x, -0.5) + array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5 , + 0.58671839, 0.66458323, 0.72906142, 0.78004843]) + + The implementation in ``tukeylambda`` also provides location and scale + parameters, and other methods such as ``pdf()`` (the probability + density function) and ``ppf()`` (the inverse of the CDF), so for + working with the Tukey lambda distribution, ``tukeylambda`` is more + generally useful. The primary advantage of ``tklmbda`` is that it is + significantly faster than ``tukeylambda.cdf``. + """) + +add_newdoc("wofz", + """ + wofz(z, out=None) + + Faddeeva function + + Returns the value of the Faddeeva function for complex argument:: + + exp(-z**2) * erfc(-i*z) + + Parameters + ---------- + z : array_like + complex argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Value of the Faddeeva function + + See Also + -------- + dawsn, erf, erfc, erfcx, erfi + + References + ---------- + .. [1] Steven G. Johnson, Faddeeva W function implementation. + http://ab-initio.mit.edu/Faddeeva + + Examples + -------- + >>> import numpy as np + >>> from scipy import special + >>> import matplotlib.pyplot as plt + + >>> x = np.linspace(-3, 3) + >>> z = special.wofz(x) + + >>> plt.plot(x, z.real, label='wofz(x).real') + >>> plt.plot(x, z.imag, label='wofz(x).imag') + >>> plt.xlabel('$x$') + >>> plt.legend(framealpha=1, shadow=True) + >>> plt.grid(alpha=0.25) + >>> plt.show() + + """) + +add_newdoc("xlogy", + """ + xlogy(x, y, out=None) + + Compute ``x*log(y)`` so that the result is 0 if ``x = 0``. + + Parameters + ---------- + x : array_like + Multiplier + y : array_like + Argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + z : scalar or ndarray + Computed x*log(y) + + Notes + ----- + The log function used in the computation is the natural log. + + .. versionadded:: 0.13.0 + + Examples + -------- + We can use this function to calculate the binary logistic loss also + known as the binary cross entropy. This loss function is used for + binary classification problems and is defined as: + + .. math:: + L = 1/n * \\sum_{i=0}^n -(y_i*log(y\\_pred_i) + (1-y_i)*log(1-y\\_pred_i)) + + We can define the parameters `x` and `y` as y and y_pred respectively. + y is the array of the actual labels which over here can be either 0 or 1. + y_pred is the array of the predicted probabilities with respect to + the positive class (1). + + >>> import numpy as np + >>> from scipy.special import xlogy + >>> y = np.array([0, 1, 0, 1, 1, 0]) + >>> y_pred = np.array([0.3, 0.8, 0.4, 0.7, 0.9, 0.2]) + >>> n = len(y) + >>> loss = -(xlogy(y, y_pred) + xlogy(1 - y, 1 - y_pred)).sum() + >>> loss /= n + >>> loss + 0.29597052165495025 + + A lower loss is usually better as it indicates that the predictions are + similar to the actual labels. In this example since our predicted + probabilities are close to the actual labels, we get an overall loss + that is reasonably low and appropriate. + + """) + +add_newdoc("xlog1py", + """ + xlog1py(x, y, out=None) + + Compute ``x*log1p(y)`` so that the result is 0 if ``x = 0``. + + Parameters + ---------- + x : array_like + Multiplier + y : array_like + Argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + z : scalar or ndarray + Computed x*log1p(y) + + Notes + ----- + + .. versionadded:: 0.13.0 + + Examples + -------- + This example shows how the function can be used to calculate the log of + the probability mass function for a geometric discrete random variable. + The probability mass function of the geometric distribution is defined + as follows: + + .. math:: f(k) = (1-p)^{k-1} p + + where :math:`p` is the probability of a single success + and :math:`1-p` is the probability of a single failure + and :math:`k` is the number of trials to get the first success. + + >>> import numpy as np + >>> from scipy.special import xlog1py + >>> p = 0.5 + >>> k = 100 + >>> _pmf = np.power(1 - p, k - 1) * p + >>> _pmf + 7.888609052210118e-31 + + If we take k as a relatively large number the value of the probability + mass function can become very low. In such cases taking the log of the + pmf would be more suitable as the log function can change the values + to a scale that is more appropriate to work with. + + >>> _log_pmf = xlog1py(k - 1, -p) + np.log(p) + >>> _log_pmf + -69.31471805599453 + + We can confirm that we get a value close to the original pmf value by + taking the exponential of the log pmf. + + >>> _orig_pmf = np.exp(_log_pmf) + >>> np.isclose(_pmf, _orig_pmf) + True + + """) + +add_newdoc("yn", + r""" + yn(n, x, out=None) + + Bessel function of the second kind of integer order and real argument. + + Parameters + ---------- + n : array_like + Order (integer). + x : array_like + Argument (float). + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + Y : scalar or ndarray + Value of the Bessel function, :math:`Y_n(x)`. + + See Also + -------- + yv : For real order and real or complex argument. + y0: faster implementation of this function for order 0 + y1: faster implementation of this function for order 1 + + Notes + ----- + Wrapper for the Cephes [1]_ routine `yn`. + + The function is evaluated by forward recurrence on `n`, starting with + values computed by the Cephes routines `y0` and `y1`. If ``n = 0`` or 1, + the routine for `y0` or `y1` is called directly. + + References + ---------- + .. [1] Cephes Mathematical Functions Library, + http://www.netlib.org/cephes/ + + Examples + -------- + Evaluate the function of order 0 at one point. + + >>> from scipy.special import yn + >>> yn(0, 1.) + 0.08825696421567697 + + Evaluate the function at one point for different orders. + + >>> yn(0, 1.), yn(1, 1.), yn(2, 1.) + (0.08825696421567697, -0.7812128213002888, -1.6506826068162546) + + The evaluation for different orders can be carried out in one call by + providing a list or NumPy array as argument for the `v` parameter: + + >>> yn([0, 1, 2], 1.) + array([ 0.08825696, -0.78121282, -1.65068261]) + + Evaluate the function at several points for order 0 by providing an + array for `z`. + + >>> import numpy as np + >>> points = np.array([0.5, 3., 8.]) + >>> yn(0, points) + array([-0.44451873, 0.37685001, 0.22352149]) + + If `z` is an array, the order parameter `v` must be broadcastable to + the correct shape if different orders shall be computed in one call. + To calculate the orders 0 and 1 for an 1D array: + + >>> orders = np.array([[0], [1]]) + >>> orders.shape + (2, 1) + + >>> yn(orders, points) + array([[-0.44451873, 0.37685001, 0.22352149], + [-1.47147239, 0.32467442, -0.15806046]]) + + Plot the functions of order 0 to 3 from 0 to 10. + + >>> import matplotlib.pyplot as plt + >>> fig, ax = plt.subplots() + >>> x = np.linspace(0., 10., 1000) + >>> for i in range(4): + ... ax.plot(x, yn(i, x), label=f'$Y_{i!r}$') + >>> ax.set_ylim(-3, 1) + >>> ax.legend() + >>> plt.show() + """) + +add_newdoc("yv", + r""" + yv(v, z, out=None) + + Bessel function of the second kind of real order and complex argument. + + Parameters + ---------- + v : array_like + Order (float). + z : array_like + Argument (float or complex). + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + Y : scalar or ndarray + Value of the Bessel function of the second kind, :math:`Y_v(x)`. + + See Also + -------- + yve : :math:`Y_v` with leading exponential behavior stripped off. + y0: faster implementation of this function for order 0 + y1: faster implementation of this function for order 1 + + Notes + ----- + For positive `v` values, the computation is carried out using the + AMOS [1]_ `zbesy` routine, which exploits the connection to the Hankel + Bessel functions :math:`H_v^{(1)}` and :math:`H_v^{(2)}`, + + .. math:: Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}). + + For negative `v` values the formula, + + .. math:: Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v) + + is used, where :math:`J_v(z)` is the Bessel function of the first kind, + computed using the AMOS routine `zbesj`. Note that the second term is + exactly zero for integer `v`; to improve accuracy the second term is + explicitly omitted for `v` values such that `v = floor(v)`. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + + Examples + -------- + Evaluate the function of order 0 at one point. + + >>> from scipy.special import yv + >>> yv(0, 1.) + 0.088256964215677 + + Evaluate the function at one point for different orders. + + >>> yv(0, 1.), yv(1, 1.), yv(1.5, 1.) + (0.088256964215677, -0.7812128213002889, -1.102495575160179) + + The evaluation for different orders can be carried out in one call by + providing a list or NumPy array as argument for the `v` parameter: + + >>> yv([0, 1, 1.5], 1.) + array([ 0.08825696, -0.78121282, -1.10249558]) + + Evaluate the function at several points for order 0 by providing an + array for `z`. + + >>> import numpy as np + >>> points = np.array([0.5, 3., 8.]) + >>> yv(0, points) + array([-0.44451873, 0.37685001, 0.22352149]) + + If `z` is an array, the order parameter `v` must be broadcastable to + the correct shape if different orders shall be computed in one call. + To calculate the orders 0 and 1 for an 1D array: + + >>> orders = np.array([[0], [1]]) + >>> orders.shape + (2, 1) + + >>> yv(orders, points) + array([[-0.44451873, 0.37685001, 0.22352149], + [-1.47147239, 0.32467442, -0.15806046]]) + + Plot the functions of order 0 to 3 from 0 to 10. + + >>> import matplotlib.pyplot as plt + >>> fig, ax = plt.subplots() + >>> x = np.linspace(0., 10., 1000) + >>> for i in range(4): + ... ax.plot(x, yv(i, x), label=f'$Y_{i!r}$') + >>> ax.set_ylim(-3, 1) + >>> ax.legend() + >>> plt.show() + + """) + +add_newdoc("yve", + r""" + yve(v, z, out=None) + + Exponentially scaled Bessel function of the second kind of real order. + + Returns the exponentially scaled Bessel function of the second + kind of real order `v` at complex `z`:: + + yve(v, z) = yv(v, z) * exp(-abs(z.imag)) + + Parameters + ---------- + v : array_like + Order (float). + z : array_like + Argument (float or complex). + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + Y : scalar or ndarray + Value of the exponentially scaled Bessel function. + + See Also + -------- + yv: Unscaled Bessel function of the second kind of real order. + + Notes + ----- + For positive `v` values, the computation is carried out using the + AMOS [1]_ `zbesy` routine, which exploits the connection to the Hankel + Bessel functions :math:`H_v^{(1)}` and :math:`H_v^{(2)}`, + + .. math:: Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}). + + For negative `v` values the formula, + + .. math:: Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v) + + is used, where :math:`J_v(z)` is the Bessel function of the first kind, + computed using the AMOS routine `zbesj`. Note that the second term is + exactly zero for integer `v`; to improve accuracy the second term is + explicitly omitted for `v` values such that `v = floor(v)`. + + Exponentially scaled Bessel functions are useful for large `z`: + for these, the unscaled Bessel functions can easily under-or overflow. + + References + ---------- + .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions + of a Complex Argument and Nonnegative Order", + http://netlib.org/amos/ + + Examples + -------- + Compare the output of `yv` and `yve` for large complex arguments for `z` + by computing their values for order ``v=1`` at ``z=1000j``. We see that + `yv` returns nan but `yve` returns a finite number: + + >>> import numpy as np + >>> from scipy.special import yv, yve + >>> v = 1 + >>> z = 1000j + >>> yv(v, z), yve(v, z) + ((nan+nanj), (-0.012610930256928629+7.721967686709076e-19j)) + + For real arguments for `z`, `yve` returns the same as `yv` up to + floating point errors. + + >>> v, z = 1, 1000 + >>> yv(v, z), yve(v, z) + (-0.02478433129235178, -0.02478433129235179) + + The function can be evaluated for several orders at the same time by + providing a list or NumPy array for `v`: + + >>> yve([1, 2, 3], 1j) + array([-0.20791042+0.14096627j, 0.38053618-0.04993878j, + 0.00815531-1.66311097j]) + + In the same way, the function can be evaluated at several points in one + call by providing a list or NumPy array for `z`: + + >>> yve(1, np.array([1j, 2j, 3j])) + array([-0.20791042+0.14096627j, -0.21526929+0.01205044j, + -0.19682671+0.00127278j]) + + It is also possible to evaluate several orders at several points + at the same time by providing arrays for `v` and `z` with + broadcasting compatible shapes. Compute `yve` for two different orders + `v` and three points `z` resulting in a 2x3 array. + + >>> v = np.array([[1], [2]]) + >>> z = np.array([3j, 4j, 5j]) + >>> v.shape, z.shape + ((2, 1), (3,)) + + >>> yve(v, z) + array([[-1.96826713e-01+1.27277544e-03j, -1.78750840e-01+1.45558819e-04j, + -1.63972267e-01+1.73494110e-05j], + [1.94960056e-03-1.11782545e-01j, 2.02902325e-04-1.17626501e-01j, + 2.27727687e-05-1.17951906e-01j]]) + """) + +add_newdoc("_struve_asymp_large_z", + """ + _struve_asymp_large_z(v, z, is_h) + + Internal function for testing `struve` & `modstruve` + + Evaluates using asymptotic expansion + + Returns + ------- + v, err + """) + +add_newdoc("_struve_power_series", + """ + _struve_power_series(v, z, is_h) + + Internal function for testing `struve` & `modstruve` + + Evaluates using power series + + Returns + ------- + v, err + """) + +add_newdoc("_struve_bessel_series", + """ + _struve_bessel_series(v, z, is_h) + + Internal function for testing `struve` & `modstruve` + + Evaluates using Bessel function series + + Returns + ------- + v, err + """) + +add_newdoc("_spherical_jn", + """ + Internal function, use `spherical_jn` instead. + """) + +add_newdoc("_spherical_jn_d", + """ + Internal function, use `spherical_jn` instead. + """) + +add_newdoc("_spherical_yn", + """ + Internal function, use `spherical_yn` instead. + """) + +add_newdoc("_spherical_yn_d", + """ + Internal function, use `spherical_yn` instead. + """) + +add_newdoc("_spherical_in", + """ + Internal function, use `spherical_in` instead. + """) + +add_newdoc("_spherical_in_d", + """ + Internal function, use `spherical_in` instead. + """) + +add_newdoc("_spherical_kn", + """ + Internal function, use `spherical_kn` instead. + """) + +add_newdoc("_spherical_kn_d", + """ + Internal function, use `spherical_kn` instead. + """) + +add_newdoc("owens_t", + """ + owens_t(h, a, out=None) + + Owen's T Function. + + The function T(h, a) gives the probability of the event + (X > h and 0 < Y < a * X) where X and Y are independent + standard normal random variables. + + Parameters + ---------- + h: array_like + Input value. + a: array_like + Input value. + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + t: scalar or ndarray + Probability of the event (X > h and 0 < Y < a * X), + where X and Y are independent standard normal random variables. + + References + ---------- + .. [1] M. Patefield and D. Tandy, "Fast and accurate calculation of + Owen's T Function", Statistical Software vol. 5, pp. 1-25, 2000. + + Examples + -------- + >>> from scipy import special + >>> a = 3.5 + >>> h = 0.78 + >>> special.owens_t(h, a) + 0.10877216734852274 + """) + +add_newdoc("_factorial", + """ + Internal function, do not use. + """) + +add_newdoc("ndtri_exp", + r""" + ndtri_exp(y, out=None) + + Inverse of `log_ndtr` vs x. Allows for greater precision than + `ndtri` composed with `numpy.exp` for very small values of y and for + y close to 0. + + Parameters + ---------- + y : array_like of float + Function argument + out : ndarray, optional + Optional output array for the function results + + Returns + ------- + scalar or ndarray + Inverse of the log CDF of the standard normal distribution, evaluated + at y. + + See Also + -------- + log_ndtr : log of the standard normal cumulative distribution function + ndtr : standard normal cumulative distribution function + ndtri : standard normal percentile function + + Examples + -------- + >>> import numpy as np + >>> import scipy.special as sc + + `ndtri_exp` agrees with the naive implementation when the latter does + not suffer from underflow. + + >>> sc.ndtri_exp(-1) + -0.33747496376420244 + >>> sc.ndtri(np.exp(-1)) + -0.33747496376420244 + + For extreme values of y, the naive approach fails + + >>> sc.ndtri(np.exp(-800)) + -inf + >>> sc.ndtri(np.exp(-1e-20)) + inf + + whereas `ndtri_exp` is still able to compute the result to high precision. + + >>> sc.ndtri_exp(-800) + -39.88469483825668 + >>> sc.ndtri_exp(-1e-20) + 9.262340089798409 + """) + + +add_newdoc("_stirling2_inexact", + r""" + Internal function, do not use. + """) + +add_newdoc( + "_beta_pdf", + r""" + _beta_pdf(x, a, b) + + Probability density function of beta distribution. + + Parameters + ---------- + x : array_like + Real-valued such that :math:`0 \leq x \leq 1`, + the upper limit of integration + a, b : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_beta_ppf", + r""" + _beta_ppf(x, a, b) + + Percent point function of beta distribution. + + Parameters + ---------- + x : array_like + Real-valued such that :math:`0 \leq x \leq 1`, + the upper limit of integration + a, b : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_invgauss_ppf", + """ + _invgauss_ppf(x, mu) + + Percent point function of inverse gaussian distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + mu : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_invgauss_isf", + """ + _invgauss_isf(x, mu, s) + + Inverse survival function of inverse gaussian distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + mu : array_like + Positive, real-valued parameters + s : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_cauchy_ppf", + """ + _cauchy_ppf(p, loc, scale) + + Percent point function (i.e. quantile) of the Cauchy distribution. + + Parameters + ---------- + p : array_like + Probabilities + loc : array_like + Location parameter of the distribution. + scale : array_like + Scale parameter of the distribution. + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_cauchy_isf", + """ + _cauchy_isf(p, loc, scale) + + Inverse survival function of the Cauchy distribution. + + Parameters + ---------- + p : array_like + Probabilities + loc : array_like + Location parameter of the distribution. + scale : array_like + Scale parameter of the distribution. + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncx2_pdf", + """ + _ncx2_pdf(x, k, l) + + Probability density function of Non-central chi-squared distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + k, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncx2_cdf", + """ + _ncx2_cdf(x, k, l) + + Cumulative density function of Non-central chi-squared distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + k, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncx2_ppf", + """ + _ncx2_ppf(x, k, l) + + Percent point function of Non-central chi-squared distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + k, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncx2_sf", + """ + _ncx2_sf(x, k, l) + + Survival function of Non-central chi-squared distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + k, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncx2_isf", + """ + _ncx2_isf(x, k, l) + + Inverse survival function of Non-central chi-squared distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + k, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncf_pdf", + """ + _ncf_pdf(x, v1, v2, l) + + Probability density function of noncentral F-distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + v1, v2, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncf_cdf", + """ + _ncf_cdf(x, v1, v2, l) + + Cumulative density function of noncentral F-distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + v1, v2, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncf_ppf", + """ + _ncf_ppf(x, v1, v2, l) + + Percent point function of noncentral F-distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + v1, v2, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncf_sf", + """ + _ncf_sf(x, v1, v2, l) + + Survival function of noncentral F-distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + v1, v2, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncf_isf", + """ + _ncf_isf(x, v1, v2, l) + + Inverse survival function of noncentral F-distribution. + + Parameters + ---------- + x : array_like + Positive real-valued + v1, v2, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncf_mean", + """ + _ncf_mean(v1, v2, l) + + Mean of noncentral F-distribution. + + Parameters + ---------- + v1, v2, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncf_variance", + """ + _ncf_variance(v1, v2, l) + + Variance of noncentral F-distribution. + + Parameters + ---------- + v1, v2, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncf_skewness", + """ + _ncf_skewness(v1, v2, l) + + Skewness of noncentral F-distribution. + + Parameters + ---------- + v1, v2, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_ncf_kurtosis_excess", + """ + _ncf_kurtosis_excess(v1, v2, l) + + Kurtosis excess of noncentral F-distribution. + + Parameters + ---------- + v1, v2, l : array_like + Positive, real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nct_cdf", + """ + _nct_cdf(x, v, l) + + Cumulative density function of noncentral t-distribution. + + Parameters + ---------- + x : array_like + Real-valued + v : array_like + Positive, real-valued parameters + l : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nct_pdf", + """ + _nct_pdf(x, v, l) + + Probability density function of noncentral t-distribution. + + Parameters + ---------- + x : array_like + Real-valued + v : array_like + Positive, real-valued parameters + l : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + + +add_newdoc( + "_nct_ppf", + """ + _nct_ppf(x, v, l) + + Percent point function of noncentral t-distribution. + + Parameters + ---------- + x : array_like + Real-valued + v : array_like + Positive, real-valued parameters + l : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nct_sf", + """ + _nct_sf(x, v, l) + + Survival function of noncentral t-distribution. + + Parameters + ---------- + x : array_like + Real-valued + v : array_like + Positive, real-valued parameters + l : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nct_isf", + """ + _nct_isf(x, v, l) + + Inverse survival function of noncentral t-distribution. + + Parameters + ---------- + x : array_like + Real-valued + v : array_like + Positive, real-valued parameters + l : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nct_mean", + """ + _nct_mean(v, l) + + Mean of noncentral t-distribution. + + Parameters + ---------- + v : array_like + Positive, real-valued parameters + l : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nct_variance", + """ + _nct_variance(v, l) + + Variance of noncentral t-distribution. + + Parameters + ---------- + v : array_like + Positive, real-valued parameters + l : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nct_skewness", + """ + _nct_skewness(v, l) + + Skewness of noncentral t-distribution. + + Parameters + ---------- + v : array_like + Positive, real-valued parameters + l : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nct_kurtosis_excess", + """ + _nct_kurtosis_excess(v, l) + + Kurtosis excess of noncentral t-distribution. + + Parameters + ---------- + v : array_like + Positive, real-valued parameters + l : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_skewnorm_cdf", + """ + _skewnorm_cdf(x, l, sc, sh) + + Cumulative density function of skewnorm distribution. + + Parameters + ---------- + x : array_like + Real-valued + l : array_like + Real-valued parameters + sc : array_like + Positive, Real-valued parameters + sh : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_skewnorm_ppf", + """ + _skewnorm_ppf(x, l, sc, sh) + + Percent point function of skewnorm distribution. + + Parameters + ---------- + x : array_like + Real-valued + l : array_like + Real-valued parameters + sc : array_like + Positive, Real-valued parameters + sh : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_skewnorm_isf", + """ + _skewnorm_isf(x, l, sc, sh) + + Inverse survival function of skewnorm distribution. + + Parameters + ---------- + x : array_like + Real-valued + l : array_like + Real-valued parameters + sc : array_like + Positive, Real-valued parameters + sh : array_like + Real-valued parameters + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_binom_pmf", + """ + _binom_pmf(x, n, p) + + Probability mass function of binomial distribution. + + Parameters + ---------- + x : array_like + Real-valued + n : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_binom_cdf", + """ + _binom_cdf(x, n, p) + + Cumulative density function of binomial distribution. + + Parameters + ---------- + x : array_like + Real-valued + n : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_binom_ppf", + """ + _binom_ppf(x, n, p) + + Percent point function of binomial distribution. + + Parameters + ---------- + x : array_like + Real-valued + n : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_binom_sf", + """ + _binom_sf(x, n, p) + + Survival function of binomial distribution. + + Parameters + ---------- + x : array_like + Real-valued + n : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_binom_isf", + """ + _binom_isf(x, n, p) + + Inverse survival function of binomial distribution. + + Parameters + ---------- + x : array_like + Real-valued + n : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nbinom_pmf", + """ + _nbinom_pmf(x, r, p) + + Probability mass function of negative binomial distribution. + + Parameters + ---------- + x : array_like + Real-valued + r : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nbinom_cdf", + """ + _nbinom_cdf(x, r, p) + + Cumulative density function of negative binomial distribution. + + Parameters + ---------- + x : array_like + Real-valued + r : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nbinom_ppf", + """ + _nbinom_ppf(x, r, p) + + Percent point function of negative binomial distribution. + + Parameters + ---------- + x : array_like + Real-valued + r : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nbinom_sf", + """ + _nbinom_sf(x, r, p) + + Survival function of negative binomial distribution. + + Parameters + ---------- + x : array_like + Real-valued + r : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nbinom_isf", + """ + _nbinom_isf(x, r, p) + + Inverse survival function of negative binomial distribution. + + Parameters + ---------- + x : array_like + Real-valued + r : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nbinom_mean", + """ + _nbinom_mean(r, p) + + Mean of negative binomial distribution. + + Parameters + ---------- + r : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nbinom_variance", + """ + _nbinom_variance(r, p) + + Variance of negative binomial distribution. + + Parameters + ---------- + r : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nbinom_skewness", + """ + _nbinom_skewness(r, p) + + Skewness of negative binomial distribution. + + Parameters + ---------- + r : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_nbinom_kurtosis_excess", + """ + _nbinom_kurtosis_excess(r, p) + + Kurtosis excess of negative binomial distribution. + + Parameters + ---------- + r : array_like + Positive, integer-valued parameter + p : array_like + Positive, real-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_hypergeom_pmf", + """ + _hypergeom_pmf(x, r, N, M) + + Probability mass function of hypergeometric distribution. + + Parameters + ---------- + x : array_like + Real-valued + r, N, M : array_like + Positive, integer-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_hypergeom_cdf", + """ + _hypergeom_cdf(x, r, N, M) + + Cumulative density function of hypergeometric distribution. + + Parameters + ---------- + x : array_like + Real-valued + r, N, M : array_like + Positive, integer-valued parameter + + Returns + ------- + scalar or ndarray + """) + +add_newdoc( + "_hypergeom_sf", + """ + _hypergeom_sf(x, r, N, M) + + Survival function of hypergeometric distribution. + + Parameters + ---------- + x : array_like + Real-valued + r, N, M : array_like + Positive, integer-valued parameter + + Returns + ------- + scalar or ndarray + """) + +add_newdoc( + "_hypergeom_mean", + """ + _hypergeom_mean(r, N, M) + + Mean of hypergeometric distribution. + + Parameters + ---------- + r, N, M : array_like + Positive, integer-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_hypergeom_variance", + """ + _hypergeom_variance(r, N, M) + + Mean of hypergeometric distribution. + + Parameters + ---------- + r, N, M : array_like + Positive, integer-valued parameter + + Returns + ------- + scalar or ndarray + + """) + +add_newdoc( + "_hypergeom_skewness", + """ + _hypergeom_skewness(r, N, M) + + Skewness of hypergeometric distribution. + + Parameters + ---------- + r, N, M : array_like + Positive, integer-valued parameter + + Returns + ------- + scalar or ndarray + + """) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_basic.py new file mode 100644 index 0000000000000000000000000000000000000000..76b13b309eed0c036dd226ff96f0e020f296e032 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_basic.py @@ -0,0 +1,3579 @@ +# +# Author: Travis Oliphant, 2002 +# + +import numpy as np +import math +import warnings +from collections import defaultdict +from heapq import heapify, heappop +from numpy import (pi, asarray, floor, isscalar, sqrt, where, + sin, place, issubdtype, extract, inexact, nan, zeros, sinc) + +from . import _ufuncs +from ._ufuncs import (mathieu_a, mathieu_b, iv, jv, gamma, rgamma, + psi, hankel1, hankel2, yv, kv, poch, binom, + _stirling2_inexact) + +from ._gufuncs import _lqn, _lqmn, _rctj, _rcty +from ._input_validation import _nonneg_int_or_fail +from . import _specfun +from ._comb import _comb_int +from ._multiufuncs import (assoc_legendre_p_all, + legendre_p_all) +from scipy._lib.deprecation import _deprecated + + +__all__ = [ + 'ai_zeros', + 'assoc_laguerre', + 'bei_zeros', + 'beip_zeros', + 'ber_zeros', + 'bernoulli', + 'berp_zeros', + 'bi_zeros', + 'clpmn', + 'comb', + 'digamma', + 'diric', + 'erf_zeros', + 'euler', + 'factorial', + 'factorial2', + 'factorialk', + 'fresnel_zeros', + 'fresnelc_zeros', + 'fresnels_zeros', + 'h1vp', + 'h2vp', + 'ivp', + 'jn_zeros', + 'jnjnp_zeros', + 'jnp_zeros', + 'jnyn_zeros', + 'jvp', + 'kei_zeros', + 'keip_zeros', + 'kelvin_zeros', + 'ker_zeros', + 'kerp_zeros', + 'kvp', + 'lmbda', + 'lpmn', + 'lpn', + 'lqmn', + 'lqn', + 'mathieu_even_coef', + 'mathieu_odd_coef', + 'obl_cv_seq', + 'pbdn_seq', + 'pbdv_seq', + 'pbvv_seq', + 'perm', + 'polygamma', + 'pro_cv_seq', + 'riccati_jn', + 'riccati_yn', + 'sinc', + 'softplus', + 'stirling2', + 'y0_zeros', + 'y1_zeros', + 'y1p_zeros', + 'yn_zeros', + 'ynp_zeros', + 'yvp', + 'zeta' +] + + +__DEPRECATION_MSG_1_15 = ( + "`scipy.special.{}` is deprecated as of SciPy 1.15.0 and will be " + "removed in SciPy 1.17.0. Please use `scipy.special.{}` instead." +) + +# mapping k to last n such that factorialk(n, k) < np.iinfo(np.int64).max +_FACTORIALK_LIMITS_64BITS = {1: 20, 2: 33, 3: 44, 4: 54, 5: 65, + 6: 74, 7: 84, 8: 93, 9: 101} +# mapping k to last n such that factorialk(n, k) < np.iinfo(np.int32).max +_FACTORIALK_LIMITS_32BITS = {1: 12, 2: 19, 3: 25, 4: 31, 5: 37, + 6: 43, 7: 47, 8: 51, 9: 56} + + +def diric(x, n): + """Periodic sinc function, also called the Dirichlet function. + + The Dirichlet function is defined as:: + + diric(x, n) = sin(x * n/2) / (n * sin(x / 2)), + + where `n` is a positive integer. + + Parameters + ---------- + x : array_like + Input data + n : int + Integer defining the periodicity. + + Returns + ------- + diric : ndarray + + Examples + -------- + >>> import numpy as np + >>> from scipy import special + >>> import matplotlib.pyplot as plt + + >>> x = np.linspace(-8*np.pi, 8*np.pi, num=201) + >>> plt.figure(figsize=(8, 8)); + >>> for idx, n in enumerate([2, 3, 4, 9]): + ... plt.subplot(2, 2, idx+1) + ... plt.plot(x, special.diric(x, n)) + ... plt.title('diric, n={}'.format(n)) + >>> plt.show() + + The following example demonstrates that `diric` gives the magnitudes + (modulo the sign and scaling) of the Fourier coefficients of a + rectangular pulse. + + Suppress output of values that are effectively 0: + + >>> np.set_printoptions(suppress=True) + + Create a signal `x` of length `m` with `k` ones: + + >>> m = 8 + >>> k = 3 + >>> x = np.zeros(m) + >>> x[:k] = 1 + + Use the FFT to compute the Fourier transform of `x`, and + inspect the magnitudes of the coefficients: + + >>> np.abs(np.fft.fft(x)) + array([ 3. , 2.41421356, 1. , 0.41421356, 1. , + 0.41421356, 1. , 2.41421356]) + + Now find the same values (up to sign) using `diric`. We multiply + by `k` to account for the different scaling conventions of + `numpy.fft.fft` and `diric`: + + >>> theta = np.linspace(0, 2*np.pi, m, endpoint=False) + >>> k * special.diric(theta, k) + array([ 3. , 2.41421356, 1. , -0.41421356, -1. , + -0.41421356, 1. , 2.41421356]) + """ + x, n = asarray(x), asarray(n) + n = asarray(n + (x-x)) + x = asarray(x + (n-n)) + if issubdtype(x.dtype, inexact): + ytype = x.dtype + else: + ytype = float + y = zeros(x.shape, ytype) + + # empirical minval for 32, 64 or 128 bit float computations + # where sin(x/2) < minval, result is fixed at +1 or -1 + if np.finfo(ytype).eps < 1e-18: + minval = 1e-11 + elif np.finfo(ytype).eps < 1e-15: + minval = 1e-7 + else: + minval = 1e-3 + + mask1 = (n <= 0) | (n != floor(n)) + place(y, mask1, nan) + + x = x / 2 + denom = sin(x) + mask2 = (1-mask1) & (abs(denom) < minval) + xsub = extract(mask2, x) + nsub = extract(mask2, n) + zsub = xsub / pi + place(y, mask2, pow(-1, np.round(zsub)*(nsub-1))) + + mask = (1-mask1) & (1-mask2) + xsub = extract(mask, x) + nsub = extract(mask, n) + dsub = extract(mask, denom) + place(y, mask, sin(nsub*xsub)/(nsub*dsub)) + return y + + +def jnjnp_zeros(nt): + """Compute zeros of integer-order Bessel functions Jn and Jn'. + + Results are arranged in order of the magnitudes of the zeros. + + Parameters + ---------- + nt : int + Number (<=1200) of zeros to compute + + Returns + ------- + zo[l-1] : ndarray + Value of the lth zero of Jn(x) and Jn'(x). Of length `nt`. + n[l-1] : ndarray + Order of the Jn(x) or Jn'(x) associated with lth zero. Of length `nt`. + m[l-1] : ndarray + Serial number of the zeros of Jn(x) or Jn'(x) associated + with lth zero. Of length `nt`. + t[l-1] : ndarray + 0 if lth zero in zo is zero of Jn(x), 1 if it is a zero of Jn'(x). Of + length `nt`. + + See Also + -------- + jn_zeros, jnp_zeros : to get separated arrays of zeros. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt > 1200): + raise ValueError("Number must be integer <= 1200.") + nt = int(nt) + n, m, t, zo = _specfun.jdzo(nt) + return zo[1:nt+1], n[:nt], m[:nt], t[:nt] + + +def jnyn_zeros(n, nt): + """Compute nt zeros of Bessel functions Jn(x), Jn'(x), Yn(x), and Yn'(x). + + Returns 4 arrays of length `nt`, corresponding to the first `nt` + zeros of Jn(x), Jn'(x), Yn(x), and Yn'(x), respectively. The zeros + are returned in ascending order. + + Parameters + ---------- + n : int + Order of the Bessel functions + nt : int + Number (<=1200) of zeros to compute + + Returns + ------- + Jn : ndarray + First `nt` zeros of Jn + Jnp : ndarray + First `nt` zeros of Jn' + Yn : ndarray + First `nt` zeros of Yn + Ynp : ndarray + First `nt` zeros of Yn' + + See Also + -------- + jn_zeros, jnp_zeros, yn_zeros, ynp_zeros + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + Compute the first three roots of :math:`J_1`, :math:`J_1'`, + :math:`Y_1` and :math:`Y_1'`. + + >>> from scipy.special import jnyn_zeros + >>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3) + >>> jn_roots, yn_roots + (array([ 3.83170597, 7.01558667, 10.17346814]), + array([2.19714133, 5.42968104, 8.59600587])) + + Plot :math:`J_1`, :math:`J_1'`, :math:`Y_1`, :math:`Y_1'` and their roots. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.special import jnyn_zeros, jvp, jn, yvp, yn + >>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3) + >>> fig, ax = plt.subplots() + >>> xmax= 11 + >>> x = np.linspace(0, xmax) + >>> x[0] += 1e-15 + >>> ax.plot(x, jn(1, x), label=r"$J_1$", c='r') + >>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$", c='b') + >>> ax.plot(x, yn(1, x), label=r"$Y_1$", c='y') + >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$", c='c') + >>> zeros = np.zeros((3, )) + >>> ax.scatter(jn_roots, zeros, s=30, c='r', zorder=5, + ... label=r"$J_1$ roots") + >>> ax.scatter(jnp_roots, zeros, s=30, c='b', zorder=5, + ... label=r"$J_1'$ roots") + >>> ax.scatter(yn_roots, zeros, s=30, c='y', zorder=5, + ... label=r"$Y_1$ roots") + >>> ax.scatter(ynp_roots, zeros, s=30, c='c', zorder=5, + ... label=r"$Y_1'$ roots") + >>> ax.hlines(0, 0, xmax, color='k') + >>> ax.set_ylim(-0.6, 0.6) + >>> ax.set_xlim(0, xmax) + >>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75)) + >>> plt.tight_layout() + >>> plt.show() + """ + if not (isscalar(nt) and isscalar(n)): + raise ValueError("Arguments must be scalars.") + if (floor(n) != n) or (floor(nt) != nt): + raise ValueError("Arguments must be integers.") + if (nt <= 0): + raise ValueError("nt > 0") + return _specfun.jyzo(abs(n), nt) + + +def jn_zeros(n, nt): + r"""Compute zeros of integer-order Bessel functions Jn. + + Compute `nt` zeros of the Bessel functions :math:`J_n(x)` on the + interval :math:`(0, \infty)`. The zeros are returned in ascending + order. Note that this interval excludes the zero at :math:`x = 0` + that exists for :math:`n > 0`. + + Parameters + ---------- + n : int + Order of Bessel function + nt : int + Number of zeros to return + + Returns + ------- + ndarray + First `nt` zeros of the Bessel function. + + See Also + -------- + jv: Real-order Bessel functions of the first kind + jnp_zeros: Zeros of :math:`Jn'` + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + Compute the first four positive roots of :math:`J_3`. + + >>> from scipy.special import jn_zeros + >>> jn_zeros(3, 4) + array([ 6.3801619 , 9.76102313, 13.01520072, 16.22346616]) + + Plot :math:`J_3` and its first four positive roots. Note + that the root located at 0 is not returned by `jn_zeros`. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.special import jn, jn_zeros + >>> j3_roots = jn_zeros(3, 4) + >>> xmax = 18 + >>> xmin = -1 + >>> x = np.linspace(xmin, xmax, 500) + >>> fig, ax = plt.subplots() + >>> ax.plot(x, jn(3, x), label=r'$J_3$') + >>> ax.scatter(j3_roots, np.zeros((4, )), s=30, c='r', + ... label=r"$J_3$_Zeros", zorder=5) + >>> ax.scatter(0, 0, s=30, c='k', + ... label=r"Root at 0", zorder=5) + >>> ax.hlines(0, 0, xmax, color='k') + >>> ax.set_xlim(xmin, xmax) + >>> plt.legend() + >>> plt.show() + """ + return jnyn_zeros(n, nt)[0] + + +def jnp_zeros(n, nt): + r"""Compute zeros of integer-order Bessel function derivatives Jn'. + + Compute `nt` zeros of the functions :math:`J_n'(x)` on the + interval :math:`(0, \infty)`. The zeros are returned in ascending + order. Note that this interval excludes the zero at :math:`x = 0` + that exists for :math:`n > 1`. + + Parameters + ---------- + n : int + Order of Bessel function + nt : int + Number of zeros to return + + Returns + ------- + ndarray + First `nt` zeros of the Bessel function. + + See Also + -------- + jvp: Derivatives of integer-order Bessel functions of the first kind + jv: Float-order Bessel functions of the first kind + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + Compute the first four roots of :math:`J_2'`. + + >>> from scipy.special import jnp_zeros + >>> jnp_zeros(2, 4) + array([ 3.05423693, 6.70613319, 9.96946782, 13.17037086]) + + As `jnp_zeros` yields the roots of :math:`J_n'`, it can be used to + compute the locations of the peaks of :math:`J_n`. Plot + :math:`J_2`, :math:`J_2'` and the locations of the roots of :math:`J_2'`. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.special import jn, jnp_zeros, jvp + >>> j2_roots = jnp_zeros(2, 4) + >>> xmax = 15 + >>> x = np.linspace(0, xmax, 500) + >>> fig, ax = plt.subplots() + >>> ax.plot(x, jn(2, x), label=r'$J_2$') + >>> ax.plot(x, jvp(2, x, 1), label=r"$J_2'$") + >>> ax.hlines(0, 0, xmax, color='k') + >>> ax.scatter(j2_roots, np.zeros((4, )), s=30, c='r', + ... label=r"Roots of $J_2'$", zorder=5) + >>> ax.set_ylim(-0.4, 0.8) + >>> ax.set_xlim(0, xmax) + >>> plt.legend() + >>> plt.show() + """ + return jnyn_zeros(n, nt)[1] + + +def yn_zeros(n, nt): + r"""Compute zeros of integer-order Bessel function Yn(x). + + Compute `nt` zeros of the functions :math:`Y_n(x)` on the interval + :math:`(0, \infty)`. The zeros are returned in ascending order. + + Parameters + ---------- + n : int + Order of Bessel function + nt : int + Number of zeros to return + + Returns + ------- + ndarray + First `nt` zeros of the Bessel function. + + See Also + -------- + yn: Bessel function of the second kind for integer order + yv: Bessel function of the second kind for real order + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + Compute the first four roots of :math:`Y_2`. + + >>> from scipy.special import yn_zeros + >>> yn_zeros(2, 4) + array([ 3.38424177, 6.79380751, 10.02347798, 13.20998671]) + + Plot :math:`Y_2` and its first four roots. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.special import yn, yn_zeros + >>> xmin = 2 + >>> xmax = 15 + >>> x = np.linspace(xmin, xmax, 500) + >>> fig, ax = plt.subplots() + >>> ax.hlines(0, xmin, xmax, color='k') + >>> ax.plot(x, yn(2, x), label=r'$Y_2$') + >>> ax.scatter(yn_zeros(2, 4), np.zeros((4, )), s=30, c='r', + ... label='Roots', zorder=5) + >>> ax.set_ylim(-0.4, 0.4) + >>> ax.set_xlim(xmin, xmax) + >>> plt.legend() + >>> plt.show() + """ + return jnyn_zeros(n, nt)[2] + + +def ynp_zeros(n, nt): + r"""Compute zeros of integer-order Bessel function derivatives Yn'(x). + + Compute `nt` zeros of the functions :math:`Y_n'(x)` on the + interval :math:`(0, \infty)`. The zeros are returned in ascending + order. + + Parameters + ---------- + n : int + Order of Bessel function + nt : int + Number of zeros to return + + Returns + ------- + ndarray + First `nt` zeros of the Bessel derivative function. + + + See Also + -------- + yvp + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + Compute the first four roots of the first derivative of the + Bessel function of second kind for order 0 :math:`Y_0'`. + + >>> from scipy.special import ynp_zeros + >>> ynp_zeros(0, 4) + array([ 2.19714133, 5.42968104, 8.59600587, 11.74915483]) + + Plot :math:`Y_0`, :math:`Y_0'` and confirm visually that the roots of + :math:`Y_0'` are located at local extrema of :math:`Y_0`. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.special import yn, ynp_zeros, yvp + >>> zeros = ynp_zeros(0, 4) + >>> xmax = 13 + >>> x = np.linspace(0, xmax, 500) + >>> fig, ax = plt.subplots() + >>> ax.plot(x, yn(0, x), label=r'$Y_0$') + >>> ax.plot(x, yvp(0, x, 1), label=r"$Y_0'$") + >>> ax.scatter(zeros, np.zeros((4, )), s=30, c='r', + ... label=r"Roots of $Y_0'$", zorder=5) + >>> for root in zeros: + ... y0_extremum = yn(0, root) + ... lower = min(0, y0_extremum) + ... upper = max(0, y0_extremum) + ... ax.vlines(root, lower, upper, color='r') + >>> ax.hlines(0, 0, xmax, color='k') + >>> ax.set_ylim(-0.6, 0.6) + >>> ax.set_xlim(0, xmax) + >>> plt.legend() + >>> plt.show() + """ + return jnyn_zeros(n, nt)[3] + + +def y0_zeros(nt, complex=False): + """Compute nt zeros of Bessel function Y0(z), and derivative at each zero. + + The derivatives are given by Y0'(z0) = -Y1(z0) at each zero z0. + + Parameters + ---------- + nt : int + Number of zeros to return + complex : bool, default False + Set to False to return only the real zeros; set to True to return only + the complex zeros with negative real part and positive imaginary part. + Note that the complex conjugates of the latter are also zeros of the + function, but are not returned by this routine. + + Returns + ------- + z0n : ndarray + Location of nth zero of Y0(z) + y0pz0n : ndarray + Value of derivative Y0'(z0) for nth zero + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + Compute the first 4 real roots and the derivatives at the roots of + :math:`Y_0`: + + >>> import numpy as np + >>> from scipy.special import y0_zeros + >>> zeros, grads = y0_zeros(4) + >>> with np.printoptions(precision=5): + ... print(f"Roots: {zeros}") + ... print(f"Gradients: {grads}") + Roots: [ 0.89358+0.j 3.95768+0.j 7.08605+0.j 10.22235+0.j] + Gradients: [-0.87942+0.j 0.40254+0.j -0.3001 +0.j 0.2497 +0.j] + + Plot the real part of :math:`Y_0` and the first four computed roots. + + >>> import matplotlib.pyplot as plt + >>> from scipy.special import y0 + >>> xmin = 0 + >>> xmax = 11 + >>> x = np.linspace(xmin, xmax, 500) + >>> fig, ax = plt.subplots() + >>> ax.hlines(0, xmin, xmax, color='k') + >>> ax.plot(x, y0(x), label=r'$Y_0$') + >>> zeros, grads = y0_zeros(4) + >>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r', + ... label=r'$Y_0$_zeros', zorder=5) + >>> ax.set_ylim(-0.5, 0.6) + >>> ax.set_xlim(xmin, xmax) + >>> plt.legend(ncol=2) + >>> plt.show() + + Compute the first 4 complex roots and the derivatives at the roots of + :math:`Y_0` by setting ``complex=True``: + + >>> y0_zeros(4, True) + (array([ -2.40301663+0.53988231j, -5.5198767 +0.54718001j, + -8.6536724 +0.54841207j, -11.79151203+0.54881912j]), + array([ 0.10074769-0.88196771j, -0.02924642+0.5871695j , + 0.01490806-0.46945875j, -0.00937368+0.40230454j])) + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("Arguments must be scalar positive integer.") + kf = 0 + kc = not complex + return _specfun.cyzo(nt, kf, kc) + + +def y1_zeros(nt, complex=False): + """Compute nt zeros of Bessel function Y1(z), and derivative at each zero. + + The derivatives are given by Y1'(z1) = Y0(z1) at each zero z1. + + Parameters + ---------- + nt : int + Number of zeros to return + complex : bool, default False + Set to False to return only the real zeros; set to True to return only + the complex zeros with negative real part and positive imaginary part. + Note that the complex conjugates of the latter are also zeros of the + function, but are not returned by this routine. + + Returns + ------- + z1n : ndarray + Location of nth zero of Y1(z) + y1pz1n : ndarray + Value of derivative Y1'(z1) for nth zero + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + Compute the first 4 real roots and the derivatives at the roots of + :math:`Y_1`: + + >>> import numpy as np + >>> from scipy.special import y1_zeros + >>> zeros, grads = y1_zeros(4) + >>> with np.printoptions(precision=5): + ... print(f"Roots: {zeros}") + ... print(f"Gradients: {grads}") + Roots: [ 2.19714+0.j 5.42968+0.j 8.59601+0.j 11.74915+0.j] + Gradients: [ 0.52079+0.j -0.34032+0.j 0.27146+0.j -0.23246+0.j] + + Extract the real parts: + + >>> realzeros = zeros.real + >>> realzeros + array([ 2.19714133, 5.42968104, 8.59600587, 11.74915483]) + + Plot :math:`Y_1` and the first four computed roots. + + >>> import matplotlib.pyplot as plt + >>> from scipy.special import y1 + >>> xmin = 0 + >>> xmax = 13 + >>> x = np.linspace(xmin, xmax, 500) + >>> zeros, grads = y1_zeros(4) + >>> fig, ax = plt.subplots() + >>> ax.hlines(0, xmin, xmax, color='k') + >>> ax.plot(x, y1(x), label=r'$Y_1$') + >>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r', + ... label=r'$Y_1$_zeros', zorder=5) + >>> ax.set_ylim(-0.5, 0.5) + >>> ax.set_xlim(xmin, xmax) + >>> plt.legend() + >>> plt.show() + + Compute the first 4 complex roots and the derivatives at the roots of + :math:`Y_1` by setting ``complex=True``: + + >>> y1_zeros(4, True) + (array([ -0.50274327+0.78624371j, -3.83353519+0.56235654j, + -7.01590368+0.55339305j, -10.17357383+0.55127339j]), + array([-0.45952768+1.31710194j, 0.04830191-0.69251288j, + -0.02012695+0.51864253j, 0.011614 -0.43203296j])) + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("Arguments must be scalar positive integer.") + kf = 1 + kc = not complex + return _specfun.cyzo(nt, kf, kc) + + +def y1p_zeros(nt, complex=False): + """Compute nt zeros of Bessel derivative Y1'(z), and value at each zero. + + The values are given by Y1(z1) at each z1 where Y1'(z1)=0. + + Parameters + ---------- + nt : int + Number of zeros to return + complex : bool, default False + Set to False to return only the real zeros; set to True to return only + the complex zeros with negative real part and positive imaginary part. + Note that the complex conjugates of the latter are also zeros of the + function, but are not returned by this routine. + + Returns + ------- + z1pn : ndarray + Location of nth zero of Y1'(z) + y1z1pn : ndarray + Value of derivative Y1(z1) for nth zero + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + Compute the first four roots of :math:`Y_1'` and the values of + :math:`Y_1` at these roots. + + >>> import numpy as np + >>> from scipy.special import y1p_zeros + >>> y1grad_roots, y1_values = y1p_zeros(4) + >>> with np.printoptions(precision=5): + ... print(f"Y1' Roots: {y1grad_roots.real}") + ... print(f"Y1 values: {y1_values.real}") + Y1' Roots: [ 3.68302 6.9415 10.1234 13.28576] + Y1 values: [ 0.41673 -0.30317 0.25091 -0.21897] + + `y1p_zeros` can be used to calculate the extremal points of :math:`Y_1` + directly. Here we plot :math:`Y_1` and the first four extrema. + + >>> import matplotlib.pyplot as plt + >>> from scipy.special import y1, yvp + >>> y1_roots, y1_values_at_roots = y1p_zeros(4) + >>> real_roots = y1_roots.real + >>> xmax = 15 + >>> x = np.linspace(0, xmax, 500) + >>> x[0] += 1e-15 + >>> fig, ax = plt.subplots() + >>> ax.plot(x, y1(x), label=r'$Y_1$') + >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$") + >>> ax.scatter(real_roots, np.zeros((4, )), s=30, c='r', + ... label=r"Roots of $Y_1'$", zorder=5) + >>> ax.scatter(real_roots, y1_values_at_roots.real, s=30, c='k', + ... label=r"Extrema of $Y_1$", zorder=5) + >>> ax.hlines(0, 0, xmax, color='k') + >>> ax.set_ylim(-0.5, 0.5) + >>> ax.set_xlim(0, xmax) + >>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75)) + >>> plt.tight_layout() + >>> plt.show() + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("Arguments must be scalar positive integer.") + kf = 2 + kc = not complex + return _specfun.cyzo(nt, kf, kc) + + +def _bessel_diff_formula(v, z, n, L, phase): + # from AMS55. + # L(v, z) = J(v, z), Y(v, z), H1(v, z), H2(v, z), phase = -1 + # L(v, z) = I(v, z) or exp(v*pi*i)K(v, z), phase = 1 + # For K, you can pull out the exp((v-k)*pi*i) into the caller + v = asarray(v) + p = 1.0 + s = L(v-n, z) + for i in range(1, n+1): + p = phase * (p * (n-i+1)) / i # = choose(k, i) + s += p*L(v-n + i*2, z) + return s / (2.**n) + + +def jvp(v, z, n=1): + """Compute derivatives of Bessel functions of the first kind. + + Compute the nth derivative of the Bessel function `Jv` with + respect to `z`. + + Parameters + ---------- + v : array_like or float + Order of Bessel function + z : complex + Argument at which to evaluate the derivative; can be real or + complex. + n : int, default 1 + Order of derivative. For 0 returns the Bessel function `jv` itself. + + Returns + ------- + scalar or ndarray + Values of the derivative of the Bessel function. + + Notes + ----- + The derivative is computed using the relation DLFM 10.6.7 [2]_. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + .. [2] NIST Digital Library of Mathematical Functions. + https://dlmf.nist.gov/10.6.E7 + + Examples + -------- + + Compute the Bessel function of the first kind of order 0 and + its first two derivatives at 1. + + >>> from scipy.special import jvp + >>> jvp(0, 1, 0), jvp(0, 1, 1), jvp(0, 1, 2) + (0.7651976865579666, -0.44005058574493355, -0.3251471008130331) + + Compute the first derivative of the Bessel function of the first + kind for several orders at 1 by providing an array for `v`. + + >>> jvp([0, 1, 2], 1, 1) + array([-0.44005059, 0.3251471 , 0.21024362]) + + Compute the first derivative of the Bessel function of the first + kind of order 0 at several points by providing an array for `z`. + + >>> import numpy as np + >>> points = np.array([0., 1.5, 3.]) + >>> jvp(0, points, 1) + array([-0. , -0.55793651, -0.33905896]) + + Plot the Bessel function of the first kind of order 1 and its + first three derivatives. + + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(-10, 10, 1000) + >>> fig, ax = plt.subplots() + >>> ax.plot(x, jvp(1, x, 0), label=r"$J_1$") + >>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$") + >>> ax.plot(x, jvp(1, x, 2), label=r"$J_1''$") + >>> ax.plot(x, jvp(1, x, 3), label=r"$J_1'''$") + >>> plt.legend() + >>> plt.show() + """ + n = _nonneg_int_or_fail(n, 'n') + if n == 0: + return jv(v, z) + else: + return _bessel_diff_formula(v, z, n, jv, -1) + + +def yvp(v, z, n=1): + """Compute derivatives of Bessel functions of the second kind. + + Compute the nth derivative of the Bessel function `Yv` with + respect to `z`. + + Parameters + ---------- + v : array_like of float + Order of Bessel function + z : complex + Argument at which to evaluate the derivative + n : int, default 1 + Order of derivative. For 0 returns the BEssel function `yv` + + Returns + ------- + scalar or ndarray + nth derivative of the Bessel function. + + See Also + -------- + yv : Bessel functions of the second kind + + Notes + ----- + The derivative is computed using the relation DLFM 10.6.7 [2]_. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + .. [2] NIST Digital Library of Mathematical Functions. + https://dlmf.nist.gov/10.6.E7 + + Examples + -------- + Compute the Bessel function of the second kind of order 0 and + its first two derivatives at 1. + + >>> from scipy.special import yvp + >>> yvp(0, 1, 0), yvp(0, 1, 1), yvp(0, 1, 2) + (0.088256964215677, 0.7812128213002889, -0.8694697855159659) + + Compute the first derivative of the Bessel function of the second + kind for several orders at 1 by providing an array for `v`. + + >>> yvp([0, 1, 2], 1, 1) + array([0.78121282, 0.86946979, 2.52015239]) + + Compute the first derivative of the Bessel function of the + second kind of order 0 at several points by providing an array for `z`. + + >>> import numpy as np + >>> points = np.array([0.5, 1.5, 3.]) + >>> yvp(0, points, 1) + array([ 1.47147239, 0.41230863, -0.32467442]) + + Plot the Bessel function of the second kind of order 1 and its + first three derivatives. + + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(0, 5, 1000) + >>> x[0] += 1e-15 + >>> fig, ax = plt.subplots() + >>> ax.plot(x, yvp(1, x, 0), label=r"$Y_1$") + >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$") + >>> ax.plot(x, yvp(1, x, 2), label=r"$Y_1''$") + >>> ax.plot(x, yvp(1, x, 3), label=r"$Y_1'''$") + >>> ax.set_ylim(-10, 10) + >>> plt.legend() + >>> plt.show() + """ + n = _nonneg_int_or_fail(n, 'n') + if n == 0: + return yv(v, z) + else: + return _bessel_diff_formula(v, z, n, yv, -1) + + +def kvp(v, z, n=1): + """Compute derivatives of real-order modified Bessel function Kv(z) + + Kv(z) is the modified Bessel function of the second kind. + Derivative is calculated with respect to `z`. + + Parameters + ---------- + v : array_like of float + Order of Bessel function + z : array_like of complex + Argument at which to evaluate the derivative + n : int, default 1 + Order of derivative. For 0 returns the Bessel function `kv` itself. + + Returns + ------- + out : ndarray + The results + + See Also + -------- + kv + + Notes + ----- + The derivative is computed using the relation DLFM 10.29.5 [2]_. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 6. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + .. [2] NIST Digital Library of Mathematical Functions. + https://dlmf.nist.gov/10.29.E5 + + Examples + -------- + Compute the modified bessel function of the second kind of order 0 and + its first two derivatives at 1. + + >>> from scipy.special import kvp + >>> kvp(0, 1, 0), kvp(0, 1, 1), kvp(0, 1, 2) + (0.42102443824070834, -0.6019072301972346, 1.0229316684379428) + + Compute the first derivative of the modified Bessel function of the second + kind for several orders at 1 by providing an array for `v`. + + >>> kvp([0, 1, 2], 1, 1) + array([-0.60190723, -1.02293167, -3.85158503]) + + Compute the first derivative of the modified Bessel function of the + second kind of order 0 at several points by providing an array for `z`. + + >>> import numpy as np + >>> points = np.array([0.5, 1.5, 3.]) + >>> kvp(0, points, 1) + array([-1.65644112, -0.2773878 , -0.04015643]) + + Plot the modified bessel function of the second kind and its + first three derivatives. + + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(0, 5, 1000) + >>> fig, ax = plt.subplots() + >>> ax.plot(x, kvp(1, x, 0), label=r"$K_1$") + >>> ax.plot(x, kvp(1, x, 1), label=r"$K_1'$") + >>> ax.plot(x, kvp(1, x, 2), label=r"$K_1''$") + >>> ax.plot(x, kvp(1, x, 3), label=r"$K_1'''$") + >>> ax.set_ylim(-2.5, 2.5) + >>> plt.legend() + >>> plt.show() + """ + n = _nonneg_int_or_fail(n, 'n') + if n == 0: + return kv(v, z) + else: + return (-1)**n * _bessel_diff_formula(v, z, n, kv, 1) + + +def ivp(v, z, n=1): + """Compute derivatives of modified Bessel functions of the first kind. + + Compute the nth derivative of the modified Bessel function `Iv` + with respect to `z`. + + Parameters + ---------- + v : array_like or float + Order of Bessel function + z : array_like + Argument at which to evaluate the derivative; can be real or + complex. + n : int, default 1 + Order of derivative. For 0, returns the Bessel function `iv` itself. + + Returns + ------- + scalar or ndarray + nth derivative of the modified Bessel function. + + See Also + -------- + iv + + Notes + ----- + The derivative is computed using the relation DLFM 10.29.5 [2]_. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 6. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + .. [2] NIST Digital Library of Mathematical Functions. + https://dlmf.nist.gov/10.29.E5 + + Examples + -------- + Compute the modified Bessel function of the first kind of order 0 and + its first two derivatives at 1. + + >>> from scipy.special import ivp + >>> ivp(0, 1, 0), ivp(0, 1, 1), ivp(0, 1, 2) + (1.2660658777520084, 0.565159103992485, 0.7009067737595233) + + Compute the first derivative of the modified Bessel function of the first + kind for several orders at 1 by providing an array for `v`. + + >>> ivp([0, 1, 2], 1, 1) + array([0.5651591 , 0.70090677, 0.29366376]) + + Compute the first derivative of the modified Bessel function of the + first kind of order 0 at several points by providing an array for `z`. + + >>> import numpy as np + >>> points = np.array([0., 1.5, 3.]) + >>> ivp(0, points, 1) + array([0. , 0.98166643, 3.95337022]) + + Plot the modified Bessel function of the first kind of order 1 and its + first three derivatives. + + >>> import matplotlib.pyplot as plt + >>> x = np.linspace(-5, 5, 1000) + >>> fig, ax = plt.subplots() + >>> ax.plot(x, ivp(1, x, 0), label=r"$I_1$") + >>> ax.plot(x, ivp(1, x, 1), label=r"$I_1'$") + >>> ax.plot(x, ivp(1, x, 2), label=r"$I_1''$") + >>> ax.plot(x, ivp(1, x, 3), label=r"$I_1'''$") + >>> plt.legend() + >>> plt.show() + """ + n = _nonneg_int_or_fail(n, 'n') + if n == 0: + return iv(v, z) + else: + return _bessel_diff_formula(v, z, n, iv, 1) + + +def h1vp(v, z, n=1): + """Compute derivatives of Hankel function H1v(z) with respect to `z`. + + Parameters + ---------- + v : array_like + Order of Hankel function + z : array_like + Argument at which to evaluate the derivative. Can be real or + complex. + n : int, default 1 + Order of derivative. For 0 returns the Hankel function `h1v` itself. + + Returns + ------- + scalar or ndarray + Values of the derivative of the Hankel function. + + See Also + -------- + hankel1 + + Notes + ----- + The derivative is computed using the relation DLFM 10.6.7 [2]_. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + .. [2] NIST Digital Library of Mathematical Functions. + https://dlmf.nist.gov/10.6.E7 + + Examples + -------- + Compute the Hankel function of the first kind of order 0 and + its first two derivatives at 1. + + >>> from scipy.special import h1vp + >>> h1vp(0, 1, 0), h1vp(0, 1, 1), h1vp(0, 1, 2) + ((0.7651976865579664+0.088256964215677j), + (-0.44005058574493355+0.7812128213002889j), + (-0.3251471008130329-0.8694697855159659j)) + + Compute the first derivative of the Hankel function of the first kind + for several orders at 1 by providing an array for `v`. + + >>> h1vp([0, 1, 2], 1, 1) + array([-0.44005059+0.78121282j, 0.3251471 +0.86946979j, + 0.21024362+2.52015239j]) + + Compute the first derivative of the Hankel function of the first kind + of order 0 at several points by providing an array for `z`. + + >>> import numpy as np + >>> points = np.array([0.5, 1.5, 3.]) + >>> h1vp(0, points, 1) + array([-0.24226846+1.47147239j, -0.55793651+0.41230863j, + -0.33905896-0.32467442j]) + """ + n = _nonneg_int_or_fail(n, 'n') + if n == 0: + return hankel1(v, z) + else: + return _bessel_diff_formula(v, z, n, hankel1, -1) + + +def h2vp(v, z, n=1): + """Compute derivatives of Hankel function H2v(z) with respect to `z`. + + Parameters + ---------- + v : array_like + Order of Hankel function + z : array_like + Argument at which to evaluate the derivative. Can be real or + complex. + n : int, default 1 + Order of derivative. For 0 returns the Hankel function `h2v` itself. + + Returns + ------- + scalar or ndarray + Values of the derivative of the Hankel function. + + See Also + -------- + hankel2 + + Notes + ----- + The derivative is computed using the relation DLFM 10.6.7 [2]_. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 5. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + .. [2] NIST Digital Library of Mathematical Functions. + https://dlmf.nist.gov/10.6.E7 + + Examples + -------- + Compute the Hankel function of the second kind of order 0 and + its first two derivatives at 1. + + >>> from scipy.special import h2vp + >>> h2vp(0, 1, 0), h2vp(0, 1, 1), h2vp(0, 1, 2) + ((0.7651976865579664-0.088256964215677j), + (-0.44005058574493355-0.7812128213002889j), + (-0.3251471008130329+0.8694697855159659j)) + + Compute the first derivative of the Hankel function of the second kind + for several orders at 1 by providing an array for `v`. + + >>> h2vp([0, 1, 2], 1, 1) + array([-0.44005059-0.78121282j, 0.3251471 -0.86946979j, + 0.21024362-2.52015239j]) + + Compute the first derivative of the Hankel function of the second kind + of order 0 at several points by providing an array for `z`. + + >>> import numpy as np + >>> points = np.array([0.5, 1.5, 3.]) + >>> h2vp(0, points, 1) + array([-0.24226846-1.47147239j, -0.55793651-0.41230863j, + -0.33905896+0.32467442j]) + """ + n = _nonneg_int_or_fail(n, 'n') + if n == 0: + return hankel2(v, z) + else: + return _bessel_diff_formula(v, z, n, hankel2, -1) + + +def riccati_jn(n, x): + r"""Compute Ricatti-Bessel function of the first kind and its derivative. + + The Ricatti-Bessel function of the first kind is defined as :math:`x + j_n(x)`, where :math:`j_n` is the spherical Bessel function of the first + kind of order :math:`n`. + + This function computes the value and first derivative of the + Ricatti-Bessel function for all orders up to and including `n`. + + Parameters + ---------- + n : int + Maximum order of function to compute + x : float + Argument at which to evaluate + + Returns + ------- + jn : ndarray + Value of j0(x), ..., jn(x) + jnp : ndarray + First derivative j0'(x), ..., jn'(x) + + Notes + ----- + The computation is carried out via backward recurrence, using the + relation DLMF 10.51.1 [2]_. + + Wrapper for a Fortran routine created by Shanjie Zhang and Jianming + Jin [1]_. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + .. [2] NIST Digital Library of Mathematical Functions. + https://dlmf.nist.gov/10.51.E1 + + """ + if not (isscalar(n) and isscalar(x)): + raise ValueError("arguments must be scalars.") + n = _nonneg_int_or_fail(n, 'n', strict=False) + if (n == 0): + n1 = 1 + else: + n1 = n + + jn = np.empty((n1 + 1,), dtype=np.float64) + jnp = np.empty_like(jn) + + _rctj(x, out=(jn, jnp)) + return jn[:(n+1)], jnp[:(n+1)] + + +def riccati_yn(n, x): + """Compute Ricatti-Bessel function of the second kind and its derivative. + + The Ricatti-Bessel function of the second kind is defined here as :math:`+x + y_n(x)`, where :math:`y_n` is the spherical Bessel function of the second + kind of order :math:`n`. *Note that this is in contrast to a common convention + that includes a minus sign in the definition.* + + This function computes the value and first derivative of the function for + all orders up to and including `n`. + + Parameters + ---------- + n : int + Maximum order of function to compute + x : float + Argument at which to evaluate + + Returns + ------- + yn : ndarray + Value of y0(x), ..., yn(x) + ynp : ndarray + First derivative y0'(x), ..., yn'(x) + + Notes + ----- + The computation is carried out via ascending recurrence, using the + relation DLMF 10.51.1 [2]_. + + Wrapper for a Fortran routine created by Shanjie Zhang and Jianming + Jin [1]_. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + .. [2] NIST Digital Library of Mathematical Functions. + https://dlmf.nist.gov/10.51.E1 + + """ + if not (isscalar(n) and isscalar(x)): + raise ValueError("arguments must be scalars.") + n = _nonneg_int_or_fail(n, 'n', strict=False) + if (n == 0): + n1 = 1 + else: + n1 = n + + yn = np.empty((n1 + 1,), dtype=np.float64) + ynp = np.empty_like(yn) + _rcty(x, out=(yn, ynp)) + + return yn[:(n+1)], ynp[:(n+1)] + + +def erf_zeros(nt): + """Compute the first nt zero in the first quadrant, ordered by absolute value. + + Zeros in the other quadrants can be obtained by using the symmetries + erf(-z) = erf(z) and erf(conj(z)) = conj(erf(z)). + + + Parameters + ---------- + nt : int + The number of zeros to compute + + Returns + ------- + The locations of the zeros of erf : ndarray (complex) + Complex values at which zeros of erf(z) + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + >>> from scipy import special + >>> special.erf_zeros(1) + array([1.45061616+1.880943j]) + + Check that erf is (close to) zero for the value returned by erf_zeros + + >>> special.erf(special.erf_zeros(1)) + array([4.95159469e-14-1.16407394e-16j]) + + """ + if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt): + raise ValueError("Argument must be positive scalar integer.") + return _specfun.cerzo(nt) + + +def fresnelc_zeros(nt): + """Compute nt complex zeros of cosine Fresnel integral C(z). + + Parameters + ---------- + nt : int + Number of zeros to compute + + Returns + ------- + fresnelc_zeros: ndarray + Zeros of the cosine Fresnel integral + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt): + raise ValueError("Argument must be positive scalar integer.") + return _specfun.fcszo(1, nt) + + +def fresnels_zeros(nt): + """Compute nt complex zeros of sine Fresnel integral S(z). + + Parameters + ---------- + nt : int + Number of zeros to compute + + Returns + ------- + fresnels_zeros: ndarray + Zeros of the sine Fresnel integral + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt): + raise ValueError("Argument must be positive scalar integer.") + return _specfun.fcszo(2, nt) + + +def fresnel_zeros(nt): + """Compute nt complex zeros of sine and cosine Fresnel integrals S(z) and C(z). + + Parameters + ---------- + nt : int + Number of zeros to compute + + Returns + ------- + zeros_sine: ndarray + Zeros of the sine Fresnel integral + zeros_cosine : ndarray + Zeros of the cosine Fresnel integral + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if (floor(nt) != nt) or (nt <= 0) or not isscalar(nt): + raise ValueError("Argument must be positive scalar integer.") + return _specfun.fcszo(2, nt), _specfun.fcszo(1, nt) + + +def assoc_laguerre(x, n, k=0.0): + """Compute the generalized (associated) Laguerre polynomial of degree n and order k. + + The polynomial :math:`L^{(k)}_n(x)` is orthogonal over ``[0, inf)``, + with weighting function ``exp(-x) * x**k`` with ``k > -1``. + + Parameters + ---------- + x : float or ndarray + Points where to evaluate the Laguerre polynomial + n : int + Degree of the Laguerre polynomial + k : int + Order of the Laguerre polynomial + + Returns + ------- + assoc_laguerre: float or ndarray + Associated laguerre polynomial values + + Notes + ----- + `assoc_laguerre` is a simple wrapper around `eval_genlaguerre`, with + reversed argument order ``(x, n, k=0.0) --> (n, k, x)``. + + """ + return _ufuncs.eval_genlaguerre(n, k, x) + + +digamma = psi + + +def polygamma(n, x): + r"""Polygamma functions. + + Defined as :math:`\psi^{(n)}(x)` where :math:`\psi` is the + `digamma` function. See [dlmf]_ for details. + + Parameters + ---------- + n : array_like + The order of the derivative of the digamma function; must be + integral + x : array_like + Real valued input + + Returns + ------- + ndarray + Function results + + See Also + -------- + digamma + + References + ---------- + .. [dlmf] NIST, Digital Library of Mathematical Functions, + https://dlmf.nist.gov/5.15 + + Examples + -------- + >>> from scipy import special + >>> x = [2, 3, 25.5] + >>> special.polygamma(1, x) + array([ 0.64493407, 0.39493407, 0.03999467]) + >>> special.polygamma(0, x) == special.psi(x) + array([ True, True, True], dtype=bool) + + """ + n, x = asarray(n), asarray(x) + fac2 = (-1.0)**(n+1) * gamma(n+1.0) * zeta(n+1, x) + return where(n == 0, psi(x), fac2) + + +def mathieu_even_coef(m, q): + r"""Fourier coefficients for even Mathieu and modified Mathieu functions. + + The Fourier series of the even solutions of the Mathieu differential + equation are of the form + + .. math:: \mathrm{ce}_{2n}(z, q) = \sum_{k=0}^{\infty} A_{(2n)}^{(2k)} \cos 2kz + + .. math:: \mathrm{ce}_{2n+1}(z, q) = + \sum_{k=0}^{\infty} A_{(2n+1)}^{(2k+1)} \cos (2k+1)z + + This function returns the coefficients :math:`A_{(2n)}^{(2k)}` for even + input m=2n, and the coefficients :math:`A_{(2n+1)}^{(2k+1)}` for odd input + m=2n+1. + + Parameters + ---------- + m : int + Order of Mathieu functions. Must be non-negative. + q : float (>=0) + Parameter of Mathieu functions. Must be non-negative. + + Returns + ------- + Ak : ndarray + Even or odd Fourier coefficients, corresponding to even or odd m. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + .. [2] NIST Digital Library of Mathematical Functions + https://dlmf.nist.gov/28.4#i + + """ + if not (isscalar(m) and isscalar(q)): + raise ValueError("m and q must be scalars.") + if (q < 0): + raise ValueError("q >=0") + if (m != floor(m)) or (m < 0): + raise ValueError("m must be an integer >=0.") + + if (q <= 1): + qm = 7.5 + 56.1*sqrt(q) - 134.7*q + 90.7*sqrt(q)*q + else: + qm = 17.0 + 3.1*sqrt(q) - .126*q + .0037*sqrt(q)*q + km = int(qm + 0.5*m) + if km > 251: + warnings.warn("Too many predicted coefficients.", RuntimeWarning, stacklevel=2) + kd = 1 + m = int(floor(m)) + if m % 2: + kd = 2 + + a = mathieu_a(m, q) + fc = _specfun.fcoef(kd, m, q, a) + return fc[:km] + + +def mathieu_odd_coef(m, q): + r"""Fourier coefficients for even Mathieu and modified Mathieu functions. + + The Fourier series of the odd solutions of the Mathieu differential + equation are of the form + + .. math:: \mathrm{se}_{2n+1}(z, q) = + \sum_{k=0}^{\infty} B_{(2n+1)}^{(2k+1)} \sin (2k+1)z + + .. math:: \mathrm{se}_{2n+2}(z, q) = + \sum_{k=0}^{\infty} B_{(2n+2)}^{(2k+2)} \sin (2k+2)z + + This function returns the coefficients :math:`B_{(2n+2)}^{(2k+2)}` for even + input m=2n+2, and the coefficients :math:`B_{(2n+1)}^{(2k+1)}` for odd + input m=2n+1. + + Parameters + ---------- + m : int + Order of Mathieu functions. Must be non-negative. + q : float (>=0) + Parameter of Mathieu functions. Must be non-negative. + + Returns + ------- + Bk : ndarray + Even or odd Fourier coefficients, corresponding to even or odd m. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not (isscalar(m) and isscalar(q)): + raise ValueError("m and q must be scalars.") + if (q < 0): + raise ValueError("q >=0") + if (m != floor(m)) or (m <= 0): + raise ValueError("m must be an integer > 0") + + if (q <= 1): + qm = 7.5 + 56.1*sqrt(q) - 134.7*q + 90.7*sqrt(q)*q + else: + qm = 17.0 + 3.1*sqrt(q) - .126*q + .0037*sqrt(q)*q + km = int(qm + 0.5*m) + if km > 251: + warnings.warn("Too many predicted coefficients.", RuntimeWarning, stacklevel=2) + kd = 4 + m = int(floor(m)) + if m % 2: + kd = 3 + + b = mathieu_b(m, q) + fc = _specfun.fcoef(kd, m, q, b) + return fc[:km] + + +@_deprecated(__DEPRECATION_MSG_1_15.format("lpmn", "assoc_legendre_p_all")) +def lpmn(m, n, z): + """Sequence of associated Legendre functions of the first kind. + + Computes the associated Legendre function of the first kind of order m and + degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``. + Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and + ``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. + + This function takes a real argument ``z``. For complex arguments ``z`` + use clpmn instead. + + .. deprecated:: 1.15.0 + This function is deprecated and will be removed in SciPy 1.17.0. + Please `scipy.special.assoc_legendre_p_all` instead. + + Parameters + ---------- + m : int + ``|m| <= n``; the order of the Legendre function. + n : int + where ``n >= 0``; the degree of the Legendre function. Often + called ``l`` (lower case L) in descriptions of the associated + Legendre function + z : array_like + Input value. + + Returns + ------- + Pmn_z : (m+1, n+1) array + Values for all orders 0..m and degrees 0..n + Pmn_d_z : (m+1, n+1) array + Derivatives for all orders 0..m and degrees 0..n + + See Also + -------- + clpmn: associated Legendre functions of the first kind for complex z + + Notes + ----- + In the interval (-1, 1), Ferrer's function of the first kind is + returned. The phase convention used for the intervals (1, inf) + and (-inf, -1) is such that the result is always real. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + .. [2] NIST Digital Library of Mathematical Functions + https://dlmf.nist.gov/14.3 + + """ + + n = _nonneg_int_or_fail(n, 'n', strict=False) + + if (abs(m) > n): + raise ValueError("m must be <= n.") + + if np.iscomplexobj(z): + raise ValueError("Argument must be real. Use clpmn instead.") + + m, n = int(m), int(n) # Convert to int to maintain backwards compatibility. + + branch_cut = np.where(np.abs(z) <= 1, 2, 3) + + p, pd = assoc_legendre_p_all(n, abs(m), z, branch_cut=branch_cut, diff_n=1) + p = np.swapaxes(p, 0, 1) + pd = np.swapaxes(pd, 0, 1) + + if (m >= 0): + p = p[:(m + 1)] + pd = pd[:(m + 1)] + else: + p = np.insert(p[:(m - 1):-1], 0, p[0], axis=0) + pd = np.insert(pd[:(m - 1):-1], 0, pd[0], axis=0) + + return p, pd + + +@_deprecated(__DEPRECATION_MSG_1_15.format("clpmn", "assoc_legendre_p_all")) +def clpmn(m, n, z, type=3): + """Associated Legendre function of the first kind for complex arguments. + + Computes the associated Legendre function of the first kind of order m and + degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``. + Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and + ``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. + + .. deprecated:: 1.15.0 + This function is deprecated and will be removed in SciPy 1.17.0. + Please use `scipy.special.assoc_legendre_p_all` instead. + + Parameters + ---------- + m : int + ``|m| <= n``; the order of the Legendre function. + n : int + where ``n >= 0``; the degree of the Legendre function. Often + called ``l`` (lower case L) in descriptions of the associated + Legendre function + z : array_like, float or complex + Input value. + type : int, optional + takes values 2 or 3 + 2: cut on the real axis ``|x| > 1`` + 3: cut on the real axis ``-1 < x < 1`` (default) + + Returns + ------- + Pmn_z : (m+1, n+1) array + Values for all orders ``0..m`` and degrees ``0..n`` + Pmn_d_z : (m+1, n+1) array + Derivatives for all orders ``0..m`` and degrees ``0..n`` + + See Also + -------- + lpmn: associated Legendre functions of the first kind for real z + + Notes + ----- + By default, i.e. for ``type=3``, phase conventions are chosen according + to [1]_ such that the function is analytic. The cut lies on the interval + (-1, 1). Approaching the cut from above or below in general yields a phase + factor with respect to Ferrer's function of the first kind + (cf. `lpmn`). + + For ``type=2`` a cut at ``|x| > 1`` is chosen. Approaching the real values + on the interval (-1, 1) in the complex plane yields Ferrer's function + of the first kind. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + .. [2] NIST Digital Library of Mathematical Functions + https://dlmf.nist.gov/14.21 + + """ + + if (abs(m) > n): + raise ValueError("m must be <= n.") + + if not (type == 2 or type == 3): + raise ValueError("type must be either 2 or 3.") + + m, n = int(m), int(n) # Convert to int to maintain backwards compatibility. + + if not np.iscomplexobj(z): + z = np.asarray(z, dtype=complex) + + out, out_jac = assoc_legendre_p_all(n, abs(m), z, branch_cut=type, diff_n=1) + out = np.swapaxes(out, 0, 1) + out_jac = np.swapaxes(out_jac, 0, 1) + + if (m >= 0): + out = out[:(m + 1)] + out_jac = out_jac[:(m + 1)] + else: + out = np.insert(out[:(m - 1):-1], 0, out[0], axis=0) + out_jac = np.insert(out_jac[:(m - 1):-1], 0, out_jac[0], axis=0) + + return out, out_jac + + +def lqmn(m, n, z): + """Sequence of associated Legendre functions of the second kind. + + Computes the associated Legendre function of the second kind of order m and + degree n, ``Qmn(z)`` = :math:`Q_n^m(z)`, and its derivative, ``Qmn'(z)``. + Returns two arrays of size ``(m+1, n+1)`` containing ``Qmn(z)`` and + ``Qmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. + + Parameters + ---------- + m : int + ``|m| <= n``; the order of the Legendre function. + n : int + where ``n >= 0``; the degree of the Legendre function. Often + called ``l`` (lower case L) in descriptions of the associated + Legendre function + z : array_like, complex + Input value. + + Returns + ------- + Qmn_z : (m+1, n+1) array + Values for all orders 0..m and degrees 0..n + Qmn_d_z : (m+1, n+1) array + Derivatives for all orders 0..m and degrees 0..n + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not isscalar(m) or (m < 0): + raise ValueError("m must be a non-negative integer.") + if not isscalar(n) or (n < 0): + raise ValueError("n must be a non-negative integer.") + + m, n = int(m), int(n) # Convert to int to maintain backwards compatibility. + # Ensure neither m nor n == 0 + mm = max(1, m) + nn = max(1, n) + + z = np.asarray(z) + if (not np.issubdtype(z.dtype, np.inexact)): + z = z.astype(np.float64) + + if np.iscomplexobj(z): + q = np.empty((mm + 1, nn + 1) + z.shape, dtype=np.complex128) + else: + q = np.empty((mm + 1, nn + 1) + z.shape, dtype=np.float64) + qd = np.empty_like(q) + if (z.ndim == 0): + _lqmn(z, out=(q, qd)) + else: + # new axes must be last for the ufunc + _lqmn(z, + out=(np.moveaxis(q, (0, 1), (-2, -1)), + np.moveaxis(qd, (0, 1), (-2, -1)))) + + return q[:(m+1), :(n+1)], qd[:(m+1), :(n+1)] + + +def bernoulli(n): + """Bernoulli numbers B0..Bn (inclusive). + + Parameters + ---------- + n : int + Indicated the number of terms in the Bernoulli series to generate. + + Returns + ------- + ndarray + The Bernoulli numbers ``[B(0), B(1), ..., B(n)]``. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + .. [2] "Bernoulli number", Wikipedia, https://en.wikipedia.org/wiki/Bernoulli_number + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import bernoulli, zeta + >>> bernoulli(4) + array([ 1. , -0.5 , 0.16666667, 0. , -0.03333333]) + + The Wikipedia article ([2]_) points out the relationship between the + Bernoulli numbers and the zeta function, ``B_n^+ = -n * zeta(1 - n)`` + for ``n > 0``: + + >>> n = np.arange(1, 5) + >>> -n * zeta(1 - n) + array([ 0.5 , 0.16666667, -0. , -0.03333333]) + + Note that, in the notation used in the wikipedia article, + `bernoulli` computes ``B_n^-`` (i.e. it used the convention that + ``B_1`` is -1/2). The relation given above is for ``B_n^+``, so the + sign of 0.5 does not match the output of ``bernoulli(4)``. + + """ + if not isscalar(n) or (n < 0): + raise ValueError("n must be a non-negative integer.") + n = int(n) + if (n < 2): + n1 = 2 + else: + n1 = n + return _specfun.bernob(int(n1))[:(n+1)] + + +def euler(n): + """Euler numbers E(0), E(1), ..., E(n). + + The Euler numbers [1]_ are also known as the secant numbers. + + Because ``euler(n)`` returns floating point values, it does not give + exact values for large `n`. The first inexact value is E(22). + + Parameters + ---------- + n : int + The highest index of the Euler number to be returned. + + Returns + ------- + ndarray + The Euler numbers [E(0), E(1), ..., E(n)]. + The odd Euler numbers, which are all zero, are included. + + References + ---------- + .. [1] Sequence A122045, The On-Line Encyclopedia of Integer Sequences, + https://oeis.org/A122045 + .. [2] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import euler + >>> euler(6) + array([ 1., 0., -1., 0., 5., 0., -61.]) + + >>> euler(13).astype(np.int64) + array([ 1, 0, -1, 0, 5, 0, -61, + 0, 1385, 0, -50521, 0, 2702765, 0]) + + >>> euler(22)[-1] # Exact value of E(22) is -69348874393137901. + -69348874393137976.0 + + """ + if not isscalar(n) or (n < 0): + raise ValueError("n must be a non-negative integer.") + n = int(n) + if (n < 2): + n1 = 2 + else: + n1 = n + return _specfun.eulerb(n1)[:(n+1)] + + +@_deprecated(__DEPRECATION_MSG_1_15.format("lpn", "legendre_p_all")) +def lpn(n, z): + """Legendre function of the first kind. + + Compute sequence of Legendre functions of the first kind (polynomials), + Pn(z) and derivatives for all degrees from 0 to n (inclusive). + + See also special.legendre for polynomial class. + + .. deprecated:: 1.15.0 + This function is deprecated and will be removed in SciPy 1.17.0. + Please use `scipy.special.legendre_p_all` instead. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + """ + + return legendre_p_all(n, z, diff_n=1) + + +def lqn(n, z): + """Legendre function of the second kind. + + Compute sequence of Legendre functions of the second kind, Qn(z) and + derivatives for all degrees from 0 to n (inclusive). + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + n = _nonneg_int_or_fail(n, 'n', strict=False) + if (n < 1): + n1 = 1 + else: + n1 = n + + z = np.asarray(z) + if (not np.issubdtype(z.dtype, np.inexact)): + z = z.astype(float) + + if np.iscomplexobj(z): + qn = np.empty((n1 + 1,) + z.shape, dtype=np.complex128) + else: + qn = np.empty((n1 + 1,) + z.shape, dtype=np.float64) + qd = np.empty_like(qn) + if (z.ndim == 0): + _lqn(z, out=(qn, qd)) + else: + # new axes must be last for the ufunc + _lqn(z, + out=(np.moveaxis(qn, 0, -1), + np.moveaxis(qd, 0, -1))) + + return qn[:(n+1)], qd[:(n+1)] + + +def ai_zeros(nt): + """ + Compute `nt` zeros and values of the Airy function Ai and its derivative. + + Computes the first `nt` zeros, `a`, of the Airy function Ai(x); + first `nt` zeros, `ap`, of the derivative of the Airy function Ai'(x); + the corresponding values Ai(a'); + and the corresponding values Ai'(a). + + Parameters + ---------- + nt : int + Number of zeros to compute + + Returns + ------- + a : ndarray + First `nt` zeros of Ai(x) + ap : ndarray + First `nt` zeros of Ai'(x) + ai : ndarray + Values of Ai(x) evaluated at first `nt` zeros of Ai'(x) + aip : ndarray + Values of Ai'(x) evaluated at first `nt` zeros of Ai(x) + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + >>> from scipy import special + >>> a, ap, ai, aip = special.ai_zeros(3) + >>> a + array([-2.33810741, -4.08794944, -5.52055983]) + >>> ap + array([-1.01879297, -3.24819758, -4.82009921]) + >>> ai + array([ 0.53565666, -0.41901548, 0.38040647]) + >>> aip + array([ 0.70121082, -0.80311137, 0.86520403]) + + """ + kf = 1 + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be a positive integer scalar.") + return _specfun.airyzo(nt, kf) + + +def bi_zeros(nt): + """ + Compute `nt` zeros and values of the Airy function Bi and its derivative. + + Computes the first `nt` zeros, b, of the Airy function Bi(x); + first `nt` zeros, b', of the derivative of the Airy function Bi'(x); + the corresponding values Bi(b'); + and the corresponding values Bi'(b). + + Parameters + ---------- + nt : int + Number of zeros to compute + + Returns + ------- + b : ndarray + First `nt` zeros of Bi(x) + bp : ndarray + First `nt` zeros of Bi'(x) + bi : ndarray + Values of Bi(x) evaluated at first `nt` zeros of Bi'(x) + bip : ndarray + Values of Bi'(x) evaluated at first `nt` zeros of Bi(x) + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + Examples + -------- + >>> from scipy import special + >>> b, bp, bi, bip = special.bi_zeros(3) + >>> b + array([-1.17371322, -3.2710933 , -4.83073784]) + >>> bp + array([-2.29443968, -4.07315509, -5.51239573]) + >>> bi + array([-0.45494438, 0.39652284, -0.36796916]) + >>> bip + array([ 0.60195789, -0.76031014, 0.83699101]) + + """ + kf = 2 + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be a positive integer scalar.") + return _specfun.airyzo(nt, kf) + + +def lmbda(v, x): + r"""Jahnke-Emden Lambda function, Lambdav(x). + + This function is defined as [2]_, + + .. math:: \Lambda_v(x) = \Gamma(v+1) \frac{J_v(x)}{(x/2)^v}, + + where :math:`\Gamma` is the gamma function and :math:`J_v` is the + Bessel function of the first kind. + + Parameters + ---------- + v : float + Order of the Lambda function + x : float + Value at which to evaluate the function and derivatives + + Returns + ------- + vl : ndarray + Values of Lambda_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. + dl : ndarray + Derivatives Lambda_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + .. [2] Jahnke, E. and Emde, F. "Tables of Functions with Formulae and + Curves" (4th ed.), Dover, 1945 + """ + if not (isscalar(v) and isscalar(x)): + raise ValueError("arguments must be scalars.") + if (v < 0): + raise ValueError("argument must be > 0.") + n = int(v) + v0 = v - n + if (n < 1): + n1 = 1 + else: + n1 = n + v1 = n1 + v0 + if (v != floor(v)): + vm, vl, dl = _specfun.lamv(v1, x) + else: + vm, vl, dl = _specfun.lamn(v1, x) + return vl[:(n+1)], dl[:(n+1)] + + +def pbdv_seq(v, x): + """Parabolic cylinder functions Dv(x) and derivatives. + + Parameters + ---------- + v : float + Order of the parabolic cylinder function + x : float + Value at which to evaluate the function and derivatives + + Returns + ------- + dv : ndarray + Values of D_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. + dp : ndarray + Derivatives D_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 13. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not (isscalar(v) and isscalar(x)): + raise ValueError("arguments must be scalars.") + n = int(v) + v0 = v-n + if (n < 1): + n1 = 1 + else: + n1 = n + v1 = n1 + v0 + dv, dp, pdf, pdd = _specfun.pbdv(v1, x) + return dv[:n1+1], dp[:n1+1] + + +def pbvv_seq(v, x): + """Parabolic cylinder functions Vv(x) and derivatives. + + Parameters + ---------- + v : float + Order of the parabolic cylinder function + x : float + Value at which to evaluate the function and derivatives + + Returns + ------- + dv : ndarray + Values of V_vi(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. + dp : ndarray + Derivatives V_vi'(x), for vi=v-int(v), vi=1+v-int(v), ..., vi=v. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 13. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not (isscalar(v) and isscalar(x)): + raise ValueError("arguments must be scalars.") + n = int(v) + v0 = v-n + if (n <= 1): + n1 = 1 + else: + n1 = n + v1 = n1 + v0 + dv, dp, pdf, pdd = _specfun.pbvv(v1, x) + return dv[:n1+1], dp[:n1+1] + + +def pbdn_seq(n, z): + """Parabolic cylinder functions Dn(z) and derivatives. + + Parameters + ---------- + n : int + Order of the parabolic cylinder function + z : complex + Value at which to evaluate the function and derivatives + + Returns + ------- + dv : ndarray + Values of D_i(z), for i=0, ..., i=n. + dp : ndarray + Derivatives D_i'(z), for i=0, ..., i=n. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996, chapter 13. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not (isscalar(n) and isscalar(z)): + raise ValueError("arguments must be scalars.") + if (floor(n) != n): + raise ValueError("n must be an integer.") + if (abs(n) <= 1): + n1 = 1 + else: + n1 = n + cpb, cpd = _specfun.cpbdn(n1, z) + return cpb[:n1+1], cpd[:n1+1] + + +def ber_zeros(nt): + """Compute nt zeros of the Kelvin function ber. + + Parameters + ---------- + nt : int + Number of zeros to compute. Must be positive. + + Returns + ------- + ndarray + First `nt` zeros of the Kelvin function. + + See Also + -------- + ber + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be positive integer scalar.") + return _specfun.klvnzo(nt, 1) + + +def bei_zeros(nt): + """Compute nt zeros of the Kelvin function bei. + + Parameters + ---------- + nt : int + Number of zeros to compute. Must be positive. + + Returns + ------- + ndarray + First `nt` zeros of the Kelvin function. + + See Also + -------- + bei + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be positive integer scalar.") + return _specfun.klvnzo(nt, 2) + + +def ker_zeros(nt): + """Compute nt zeros of the Kelvin function ker. + + Parameters + ---------- + nt : int + Number of zeros to compute. Must be positive. + + Returns + ------- + ndarray + First `nt` zeros of the Kelvin function. + + See Also + -------- + ker + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be positive integer scalar.") + return _specfun.klvnzo(nt, 3) + + +def kei_zeros(nt): + """Compute nt zeros of the Kelvin function kei. + + Parameters + ---------- + nt : int + Number of zeros to compute. Must be positive. + + Returns + ------- + ndarray + First `nt` zeros of the Kelvin function. + + See Also + -------- + kei + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be positive integer scalar.") + return _specfun.klvnzo(nt, 4) + + +def berp_zeros(nt): + """Compute nt zeros of the derivative of the Kelvin function ber. + + Parameters + ---------- + nt : int + Number of zeros to compute. Must be positive. + + Returns + ------- + ndarray + First `nt` zeros of the derivative of the Kelvin function. + + See Also + -------- + ber, berp + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + + Examples + -------- + Compute the first 5 zeros of the derivative of the Kelvin function. + + >>> from scipy.special import berp_zeros + >>> berp_zeros(5) + array([ 6.03871081, 10.51364251, 14.96844542, 19.41757493, 23.86430432]) + + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be positive integer scalar.") + return _specfun.klvnzo(nt, 5) + + +def beip_zeros(nt): + """Compute nt zeros of the derivative of the Kelvin function bei. + + Parameters + ---------- + nt : int + Number of zeros to compute. Must be positive. + + Returns + ------- + ndarray + First `nt` zeros of the derivative of the Kelvin function. + + See Also + -------- + bei, beip + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be positive integer scalar.") + return _specfun.klvnzo(nt, 6) + + +def kerp_zeros(nt): + """Compute nt zeros of the derivative of the Kelvin function ker. + + Parameters + ---------- + nt : int + Number of zeros to compute. Must be positive. + + Returns + ------- + ndarray + First `nt` zeros of the derivative of the Kelvin function. + + See Also + -------- + ker, kerp + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be positive integer scalar.") + return _specfun.klvnzo(nt, 7) + + +def keip_zeros(nt): + """Compute nt zeros of the derivative of the Kelvin function kei. + + Parameters + ---------- + nt : int + Number of zeros to compute. Must be positive. + + Returns + ------- + ndarray + First `nt` zeros of the derivative of the Kelvin function. + + See Also + -------- + kei, keip + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be positive integer scalar.") + return _specfun.klvnzo(nt, 8) + + +def kelvin_zeros(nt): + """Compute nt zeros of all Kelvin functions. + + Returned in a length-8 tuple of arrays of length nt. The tuple contains + the arrays of zeros of (ber, bei, ker, kei, ber', bei', ker', kei'). + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not isscalar(nt) or (floor(nt) != nt) or (nt <= 0): + raise ValueError("nt must be positive integer scalar.") + return (_specfun.klvnzo(nt, 1), + _specfun.klvnzo(nt, 2), + _specfun.klvnzo(nt, 3), + _specfun.klvnzo(nt, 4), + _specfun.klvnzo(nt, 5), + _specfun.klvnzo(nt, 6), + _specfun.klvnzo(nt, 7), + _specfun.klvnzo(nt, 8)) + + +def pro_cv_seq(m, n, c): + """Characteristic values for prolate spheroidal wave functions. + + Compute a sequence of characteristic values for the prolate + spheroidal wave functions for mode m and n'=m..n and spheroidal + parameter c. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not (isscalar(m) and isscalar(n) and isscalar(c)): + raise ValueError("Arguments must be scalars.") + if (n != floor(n)) or (m != floor(m)): + raise ValueError("Modes must be integers.") + if (n-m > 199): + raise ValueError("Difference between n and m is too large.") + maxL = n-m+1 + return _specfun.segv(m, n, c, 1)[1][:maxL] + + +def obl_cv_seq(m, n, c): + """Characteristic values for oblate spheroidal wave functions. + + Compute a sequence of characteristic values for the oblate + spheroidal wave functions for mode m and n'=m..n and spheroidal + parameter c. + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + + """ + if not (isscalar(m) and isscalar(n) and isscalar(c)): + raise ValueError("Arguments must be scalars.") + if (n != floor(n)) or (m != floor(m)): + raise ValueError("Modes must be integers.") + if (n-m > 199): + raise ValueError("Difference between n and m is too large.") + maxL = n-m+1 + return _specfun.segv(m, n, c, -1)[1][:maxL] + + +def comb(N, k, *, exact=False, repetition=False): + """The number of combinations of N things taken k at a time. + + This is often expressed as "N choose k". + + Parameters + ---------- + N : int, ndarray + Number of things. + k : int, ndarray + Number of elements taken. + exact : bool, optional + For integers, if `exact` is False, then floating point precision is + used, otherwise the result is computed exactly. + + .. deprecated:: 1.14.0 + ``exact=True`` is deprecated for non-integer `N` and `k` and will raise an + error in SciPy 1.16.0 + repetition : bool, optional + If `repetition` is True, then the number of combinations with + repetition is computed. + + Returns + ------- + val : int, float, ndarray + The total number of combinations. + + See Also + -------- + binom : Binomial coefficient considered as a function of two real + variables. + + Notes + ----- + - Array arguments accepted only for exact=False case. + - If N < 0, or k < 0, then 0 is returned. + - If k > N and repetition=False, then 0 is returned. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import comb + >>> k = np.array([3, 4]) + >>> n = np.array([10, 10]) + >>> comb(n, k, exact=False) + array([ 120., 210.]) + >>> comb(10, 3, exact=True) + 120 + >>> comb(10, 3, exact=True, repetition=True) + 220 + + """ + if repetition: + return comb(N + k - 1, k, exact=exact) + if exact: + if int(N) == N and int(k) == k: + # _comb_int casts inputs to integers, which is safe & intended here + return _comb_int(N, k) + # otherwise, we disregard `exact=True`; it makes no sense for + # non-integral arguments + msg = ("`exact=True` is deprecated for non-integer `N` and `k` and will raise " + "an error in SciPy 1.16.0") + warnings.warn(msg, DeprecationWarning, stacklevel=2) + return comb(N, k) + else: + k, N = asarray(k), asarray(N) + cond = (k <= N) & (N >= 0) & (k >= 0) + vals = binom(N, k) + if isinstance(vals, np.ndarray): + vals[~cond] = 0 + elif not cond: + vals = np.float64(0) + return vals + + +def perm(N, k, exact=False): + """Permutations of N things taken k at a time, i.e., k-permutations of N. + + It's also known as "partial permutations". + + Parameters + ---------- + N : int, ndarray + Number of things. + k : int, ndarray + Number of elements taken. + exact : bool, optional + If ``True``, calculate the answer exactly using long integer arithmetic (`N` + and `k` must be scalar integers). If ``False``, a floating point approximation + is calculated (more rapidly) using `poch`. Default is ``False``. + + Returns + ------- + val : int, ndarray + The number of k-permutations of N. + + Notes + ----- + - Array arguments accepted only for exact=False case. + - If k > N, N < 0, or k < 0, then a 0 is returned. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import perm + >>> k = np.array([3, 4]) + >>> n = np.array([10, 10]) + >>> perm(n, k) + array([ 720., 5040.]) + >>> perm(10, 3, exact=True) + 720 + + """ + if exact: + N = np.squeeze(N)[()] # for backward compatibility (accepted size 1 arrays) + k = np.squeeze(k)[()] + if not (isscalar(N) and isscalar(k)): + raise ValueError("`N` and `k` must scalar integers be with `exact=True`.") + + floor_N, floor_k = int(N), int(k) + non_integral = not (floor_N == N and floor_k == k) + if (k > N) or (N < 0) or (k < 0): + if non_integral: + msg = ("Non-integer `N` and `k` with `exact=True` is deprecated and " + "will raise an error in SciPy 1.16.0.") + warnings.warn(msg, DeprecationWarning, stacklevel=2) + return 0 + if non_integral: + raise ValueError("Non-integer `N` and `k` with `exact=True` is not " + "supported.") + val = 1 + for i in range(floor_N - floor_k + 1, floor_N + 1): + val *= i + return val + else: + k, N = asarray(k), asarray(N) + cond = (k <= N) & (N >= 0) & (k >= 0) + vals = poch(N - k + 1, k) + if isinstance(vals, np.ndarray): + vals[~cond] = 0 + elif not cond: + vals = np.float64(0) + return vals + + +# https://stackoverflow.com/a/16327037 +def _range_prod(lo, hi, k=1): + """ + Product of a range of numbers spaced k apart (from hi). + + For k=1, this returns the product of + lo * (lo+1) * (lo+2) * ... * (hi-2) * (hi-1) * hi + = hi! / (lo-1)! + + For k>1, it correspond to taking only every k'th number when + counting down from hi - e.g. 18!!!! = _range_prod(1, 18, 4). + + Breaks into smaller products first for speed: + _range_prod(2, 9) = ((2*3)*(4*5))*((6*7)*(8*9)) + """ + if lo == 1 and k == 1: + return math.factorial(hi) + + if lo + k < hi: + mid = (hi + lo) // 2 + if k > 1: + # make sure mid is a multiple of k away from hi + mid = mid - ((mid - hi) % k) + return _range_prod(lo, mid, k) * _range_prod(mid + k, hi, k) + elif lo + k == hi: + return lo * hi + else: + return hi + + +def _factorialx_array_exact(n, k=1): + """ + Exact computation of factorial for an array. + + The factorials are computed in incremental fashion, by taking + the sorted unique values of n and multiplying the intervening + numbers between the different unique values. + + In other words, the factorial for the largest input is only + computed once, with each other result computed in the process. + + k > 1 corresponds to the multifactorial. + """ + un = np.unique(n) + # numpy changed nan-sorting behaviour with 1.21, see numpy/numpy#18070; + # to unify the behaviour, we remove the nan's here; the respective + # values will be set separately at the end + un = un[~np.isnan(un)] + + # Convert to object array if np.int64 can't handle size + if np.isnan(n).any(): + dt = float + elif k in _FACTORIALK_LIMITS_64BITS.keys(): + if un[-1] > _FACTORIALK_LIMITS_64BITS[k]: + # e.g. k=1: 21! > np.iinfo(np.int64).max + dt = object + elif un[-1] > _FACTORIALK_LIMITS_32BITS[k]: + # e.g. k=3: 26!!! > np.iinfo(np.int32).max + dt = np.int64 + else: + dt = np.dtype("long") + else: + # for k >= 10, we always use object + dt = object + + out = np.empty_like(n, dtype=dt) + + # Handle invalid/trivial values + un = un[un > 1] + out[n < 2] = 1 + out[n < 0] = 0 + + # Calculate products of each range of numbers + # we can only multiply incrementally if the values are k apart; + # therefore we partition `un` into "lanes", i.e. its residues modulo k + for lane in range(0, k): + ul = un[(un % k) == lane] if k > 1 else un + if ul.size: + # after np.unique, un resp. ul are sorted, ul[0] is the smallest; + # cast to python ints to avoid overflow with np.int-types + val = _range_prod(1, int(ul[0]), k=k) + out[n == ul[0]] = val + for i in range(len(ul) - 1): + # by the filtering above, we have ensured that prev & current + # are a multiple of k apart + prev = ul[i] + current = ul[i + 1] + # we already multiplied all factors until prev; continue + # building the full factorial from the following (`prev + 1`); + # use int() for the same reason as above + val *= _range_prod(int(prev + 1), int(current), k=k) + out[n == current] = val + + if np.isnan(n).any(): + out = out.astype(np.float64) + out[np.isnan(n)] = np.nan + return out + + +def _factorialx_array_approx(n, k, extend): + """ + Calculate approximation to multifactorial for array n and integer k. + + Ensure that values aren't calculated unnecessarily. + """ + if extend == "complex": + return _factorialx_approx_core(n, k=k, extend=extend) + + # at this point we are guaranteed that extend='zero' and that k>0 is an integer + result = zeros(n.shape) + # keep nans as nans + place(result, np.isnan(n), np.nan) + # only compute where n >= 0 (excludes nans), everything else is 0 + cond = (n >= 0) + n_to_compute = extract(cond, n) + place(result, cond, _factorialx_approx_core(n_to_compute, k=k, extend=extend)) + return result + + +def _gamma1p(vals): + """ + returns gamma(n+1), though with NaN at -1 instead of inf, c.f. #21827 + """ + res = gamma(vals + 1) + # replace infinities at -1 (from gamma function at 0) with nan + # gamma only returns inf for real inputs; can ignore complex case + if isinstance(res, np.ndarray): + if not _is_subdtype(vals.dtype, "c"): + res[vals == -1] = np.nan + elif np.isinf(res) and vals == -1: + res = np.float64("nan") + return res + + +def _factorialx_approx_core(n, k, extend): + """ + Core approximation to multifactorial for array n and integer k. + """ + if k == 1: + # shortcut for k=1; same for both extensions, because we assume the + # handling of extend == 'zero' happens in _factorialx_array_approx + result = _gamma1p(n) + if isinstance(n, np.ndarray): + # gamma does not maintain 0-dim arrays; fix it + result = np.array(result) + return result + + if extend == "complex": + # see https://numpy.org/doc/stable/reference/generated/numpy.power.html + p_dtype = complex if (_is_subdtype(type(k), "c") or k < 0) else None + with warnings.catch_warnings(): + # do not warn about 0 * inf, nan / nan etc.; the results are correct + warnings.simplefilter("ignore", RuntimeWarning) + # don't use `(n-1)/k` in np.power; underflows if 0 is of a uintX type + result = np.power(k, n / k, dtype=p_dtype) * _gamma1p(n / k) + result *= rgamma(1 / k + 1) / np.power(k, 1 / k, dtype=p_dtype) + if isinstance(n, np.ndarray): + # ensure we keep array-ness for 0-dim inputs; already n/k above loses it + result = np.array(result) + return result + + # at this point we are guaranteed that extend='zero' and that k>0 is an integer + n_mod_k = n % k + # scalar case separately, unified handling would be inefficient for arrays; + # don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below + if not isinstance(n, np.ndarray): + return ( + np.power(k, (n - n_mod_k) / k) + * gamma(n / k + 1) / gamma(n_mod_k / k + 1) + * max(n_mod_k, 1) + ) + + # factor that's independent of the residue class (see factorialk docstring) + result = np.power(k, n / k) * gamma(n / k + 1) + # factor dependent on residue r (for `r=0` it's 1, so we skip `r=0` + # below and thus also avoid evaluating `max(r, 1)`) + def corr(k, r): return np.power(k, -r / k) / gamma(r / k + 1) * r + for r in np.unique(n_mod_k): + if r == 0: + continue + # cast to int because uint types break on `-r` + result[n_mod_k == r] *= corr(k, int(r)) + return result + + +def _is_subdtype(dtype, dtypes): + """ + Shorthand for calculating whether dtype is subtype of some dtypes. + + Also allows specifying a list instead of just a single dtype. + + Additionaly, the most important supertypes from + https://numpy.org/doc/stable/reference/arrays.scalars.html + can optionally be specified using abbreviations as follows: + "i": np.integer + "f": np.floating + "c": np.complexfloating + "n": np.number (contains the other three) + """ + dtypes = dtypes if isinstance(dtypes, list) else [dtypes] + # map single character abbreviations, if they are in dtypes + mapping = { + "i": np.integer, + "f": np.floating, + "c": np.complexfloating, + "n": np.number + } + dtypes = [mapping.get(x, x) for x in dtypes] + return any(np.issubdtype(dtype, dt) for dt in dtypes) + + +def _factorialx_wrapper(fname, n, k, exact, extend): + """ + Shared implementation for factorial, factorial2 & factorialk. + """ + if extend not in ("zero", "complex"): + raise ValueError( + f"argument `extend` must be either 'zero' or 'complex', received: {extend}" + ) + if exact and extend == "complex": + raise ValueError("Incompatible options: `exact=True` and `extend='complex'`") + + msg_unsup = ( + "Unsupported data type for {vname} in {fname}: {dtype}\n" + ) + if fname == "factorial": + msg_unsup += ( + "Permitted data types are integers and floating point numbers, " + "as well as complex numbers if `extend='complex' is passed." + ) + else: + msg_unsup += ( + "Permitted data types are integers, as well as floating point " + "numbers and complex numbers if `extend='complex' is passed." + ) + msg_exact_not_possible = ( + "`exact=True` only supports integers, cannot use data type {dtype}" + ) + msg_needs_complex = ( + "In order to use non-integer arguments, you must opt into this by passing " + "`extend='complex'`. Note that this changes the result for all negative " + "arguments (which by default return 0)." + ) + + if fname == "factorial2": + msg_needs_complex += (" Additionally, it will rescale the values of the double" + " factorial at even integers by a factor of sqrt(2/pi).") + elif fname == "factorialk": + msg_needs_complex += (" Additionally, it will perturb the values of the" + " multifactorial at most positive integers `n`.") + # check type of k + if not _is_subdtype(type(k), ["i", "f", "c"]): + raise ValueError(msg_unsup.format(vname="`k`", fname=fname, dtype=type(k))) + elif _is_subdtype(type(k), ["f", "c"]) and extend != "complex": + raise ValueError(msg_needs_complex) + # check value of k + if extend == "zero" and k < 1: + msg = f"For `extend='zero'`, k must be a positive integer, received: {k}" + raise ValueError(msg) + elif k == 0: + raise ValueError("Parameter k cannot be zero!") + + # factorial allows floats also for extend="zero" + types_requiring_complex = "c" if fname == "factorial" else ["f", "c"] + + # don't use isscalar due to numpy/numpy#23574; 0-dim arrays treated below + if np.ndim(n) == 0 and not isinstance(n, np.ndarray): + # scalar cases + if not _is_subdtype(type(n), ["i", "f", "c", type(None)]): + raise ValueError(msg_unsup.format(vname="`n`", fname=fname, dtype=type(n))) + elif _is_subdtype(type(n), types_requiring_complex) and extend != "complex": + raise ValueError(msg_needs_complex) + elif n is None or np.isnan(n): + complexify = (extend == "complex") and _is_subdtype(type(n), "c") + return np.complex128("nan+nanj") if complexify else np.float64("nan") + elif extend == "zero" and n < 0: + return 0 if exact else np.float64(0) + elif n in {0, 1}: + return 1 if exact else np.float64(1) + elif exact and _is_subdtype(type(n), "i"): + # calculate with integers + return _range_prod(1, n, k=k) + elif exact: + # only relevant for factorial + raise ValueError(msg_exact_not_possible.format(dtype=type(n))) + # approximation + return _factorialx_approx_core(n, k=k, extend=extend) + + # arrays & array-likes + n = asarray(n) + + if not _is_subdtype(n.dtype, ["i", "f", "c"]): + raise ValueError(msg_unsup.format(vname="`n`", fname=fname, dtype=n.dtype)) + elif _is_subdtype(n.dtype, types_requiring_complex) and extend != "complex": + raise ValueError(msg_needs_complex) + elif exact and _is_subdtype(n.dtype, ["f"]): + # only relevant for factorial + raise ValueError(msg_exact_not_possible.format(dtype=n.dtype)) + + if n.size == 0: + # return empty arrays unchanged + return n + elif exact: + # calculate with integers + return _factorialx_array_exact(n, k=k) + # approximation + return _factorialx_array_approx(n, k=k, extend=extend) + + +def factorial(n, exact=False, extend="zero"): + """ + The factorial of a number or array of numbers. + + The factorial of non-negative integer `n` is the product of all + positive integers less than or equal to `n`:: + + n! = n * (n - 1) * (n - 2) * ... * 1 + + Parameters + ---------- + n : int or float or complex (or array_like thereof) + Input values for ``n!``. Complex values require ``extend='complex'``. + By default, the return value for ``n < 0`` is 0. + exact : bool, optional + If ``exact`` is set to True, calculate the answer exactly using + integer arithmetic, otherwise approximate using the gamma function + (faster, but yields floats instead of integers). + Default is False. + extend : string, optional + One of ``'zero'`` or ``'complex'``; this determines how values ``n<0`` + are handled - by default they are 0, but it is possible to opt into the + complex extension of the factorial (see below). + + Returns + ------- + nf : int or float or complex or ndarray + Factorial of ``n``, as integer, float or complex (depending on ``exact`` + and ``extend``). Array inputs are returned as arrays. + + Notes + ----- + For arrays with ``exact=True``, the factorial is computed only once, for + the largest input, with each other result computed in the process. + The output dtype is increased to ``int64`` or ``object`` if necessary. + + With ``exact=False`` the factorial is approximated using the gamma + function (which is also the definition of the complex extension): + + .. math:: n! = \\Gamma(n+1) + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import factorial + >>> arr = np.array([3, 4, 5]) + >>> factorial(arr, exact=False) + array([ 6., 24., 120.]) + >>> factorial(arr, exact=True) + array([ 6, 24, 120]) + >>> factorial(5, exact=True) + 120 + + """ + return _factorialx_wrapper("factorial", n, k=1, exact=exact, extend=extend) + + +def factorial2(n, exact=False, extend="zero"): + """Double factorial. + + This is the factorial with every second value skipped. E.g., ``7!! = 7 * 5 + * 3 * 1``. It can be approximated numerically as:: + + n!! = 2 ** (n / 2) * gamma(n / 2 + 1) * sqrt(2 / pi) n odd + = 2 ** (n / 2) * gamma(n / 2 + 1) n even + = 2 ** (n / 2) * (n / 2)! n even + + The formula for odd ``n`` is the basis for the complex extension. + + Parameters + ---------- + n : int or float or complex (or array_like thereof) + Input values for ``n!!``. Non-integer values require ``extend='complex'``. + By default, the return value for ``n < 0`` is 0. + exact : bool, optional + If ``exact`` is set to True, calculate the answer exactly using + integer arithmetic, otherwise use above approximation (faster, + but yields floats instead of integers). + Default is False. + extend : string, optional + One of ``'zero'`` or ``'complex'``; this determines how values ``n<0`` + are handled - by default they are 0, but it is possible to opt into the + complex extension of the double factorial. This also enables passing + complex values to ``n``. + + .. warning:: + + Using the ``'complex'`` extension also changes the values of the + double factorial for even integers, reducing them by a factor of + ``sqrt(2/pi) ~= 0.79``, see [1]. + + Returns + ------- + nf : int or float or complex or ndarray + Double factorial of ``n``, as integer, float or complex (depending on + ``exact`` and ``extend``). Array inputs are returned as arrays. + + Examples + -------- + >>> from scipy.special import factorial2 + >>> factorial2(7, exact=False) + array(105.00000000000001) + >>> factorial2(7, exact=True) + 105 + + References + ---------- + .. [1] Complex extension to double factorial + https://en.wikipedia.org/wiki/Double_factorial#Complex_arguments + """ + return _factorialx_wrapper("factorial2", n, k=2, exact=exact, extend=extend) + + +def factorialk(n, k, exact=False, extend="zero"): + """Multifactorial of n of order k, n(!!...!). + + This is the multifactorial of n skipping k values. For example, + + factorialk(17, 4) = 17!!!! = 17 * 13 * 9 * 5 * 1 + + In particular, for any integer ``n``, we have + + factorialk(n, 1) = factorial(n) + + factorialk(n, 2) = factorial2(n) + + Parameters + ---------- + n : int or float or complex (or array_like thereof) + Input values for multifactorial. Non-integer values require + ``extend='complex'``. By default, the return value for ``n < 0`` is 0. + n : int or float or complex (or array_like thereof) + Order of multifactorial. Non-integer values require ``extend='complex'``. + exact : bool, optional + If ``exact`` is set to True, calculate the answer exactly using + integer arithmetic, otherwise use an approximation (faster, + but yields floats instead of integers) + Default is False. + extend : string, optional + One of ``'zero'`` or ``'complex'``; this determines how values ``n<0`` are + handled - by default they are 0, but it is possible to opt into the complex + extension of the multifactorial. This enables passing complex values, + not only to ``n`` but also to ``k``. + + .. warning:: + + Using the ``'complex'`` extension also changes the values of the + multifactorial at integers ``n != 1 (mod k)`` by a factor depending + on both ``k`` and ``n % k``, see below or [1]. + + Returns + ------- + nf : int or float or complex or ndarray + Multifactorial (order ``k``) of ``n``, as integer, float or complex (depending + on ``exact`` and ``extend``). Array inputs are returned as arrays. + + Examples + -------- + >>> from scipy.special import factorialk + >>> factorialk(5, k=1, exact=True) + 120 + >>> factorialk(5, k=3, exact=True) + 10 + >>> factorialk([5, 7, 9], k=3, exact=True) + array([ 10, 28, 162]) + >>> factorialk([5, 7, 9], k=3, exact=False) + array([ 10., 28., 162.]) + + Notes + ----- + While less straight-forward than for the double-factorial, it's possible to + calculate a general approximation formula of n!(k) by studying ``n`` for a given + remainder ``r < k`` (thus ``n = m * k + r``, resp. ``r = n % k``), which can be + put together into something valid for all integer values ``n >= 0`` & ``k > 0``:: + + n!(k) = k ** ((n - r)/k) * gamma(n/k + 1) / gamma(r/k + 1) * max(r, 1) + + This is the basis of the approximation when ``exact=False``. + + In principle, any fixed choice of ``r`` (ignoring its relation ``r = n%k`` + to ``n``) would provide a suitable analytic continuation from integer ``n`` + to complex ``z`` (not only satisfying the functional equation but also + being logarithmically convex, c.f. Bohr-Mollerup theorem) -- in fact, the + choice of ``r`` above only changes the function by a constant factor. The + final constraint that determines the canonical continuation is ``f(1) = 1``, + which forces ``r = 1`` (see also [1]).:: + + z!(k) = k ** ((z - 1)/k) * gamma(z/k + 1) / gamma(1/k + 1) + + References + ---------- + .. [1] Complex extension to multifactorial + https://en.wikipedia.org/wiki/Double_factorial#Alternative_extension_of_the_multifactorial + """ + return _factorialx_wrapper("factorialk", n, k=k, exact=exact, extend=extend) + + +def stirling2(N, K, *, exact=False): + r"""Generate Stirling number(s) of the second kind. + + Stirling numbers of the second kind count the number of ways to + partition a set with N elements into K non-empty subsets. + + The values this function returns are calculated using a dynamic + program which avoids redundant computation across the subproblems + in the solution. For array-like input, this implementation also + avoids redundant computation across the different Stirling number + calculations. + + The numbers are sometimes denoted + + .. math:: + + {N \brace{K}} + + see [1]_ for details. This is often expressed-verbally-as + "N subset K". + + Parameters + ---------- + N : int, ndarray + Number of things. + K : int, ndarray + Number of non-empty subsets taken. + exact : bool, optional + Uses dynamic programming (DP) with floating point + numbers for smaller arrays and uses a second order approximation due to + Temme for larger entries of `N` and `K` that allows trading speed for + accuracy. See [2]_ for a description. Temme approximation is used for + values ``n>50``. The max error from the DP has max relative error + ``4.5*10^-16`` for ``n<=50`` and the max error from the Temme approximation + has max relative error ``5*10^-5`` for ``51 <= n < 70`` and + ``9*10^-6`` for ``70 <= n < 101``. Note that these max relative errors will + decrease further as `n` increases. + + Returns + ------- + val : int, float, ndarray + The number of partitions. + + See Also + -------- + comb : The number of combinations of N things taken k at a time. + + Notes + ----- + - If N < 0, or K < 0, then 0 is returned. + - If K > N, then 0 is returned. + + The output type will always be `int` or ndarray of `object`. + The input must contain either numpy or python integers otherwise a + TypeError is raised. + + References + ---------- + .. [1] R. L. Graham, D. E. Knuth and O. Patashnik, "Concrete + Mathematics: A Foundation for Computer Science," Addison-Wesley + Publishing Company, Boston, 1989. Chapter 6, page 258. + + .. [2] Temme, Nico M. "Asymptotic estimates of Stirling numbers." + Studies in Applied Mathematics 89.3 (1993): 233-243. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import stirling2 + >>> k = np.array([3, -1, 3]) + >>> n = np.array([10, 10, 9]) + >>> stirling2(n, k) + array([9330.0, 0.0, 3025.0]) + + """ + output_is_scalar = np.isscalar(N) and np.isscalar(K) + # make a min-heap of unique (n,k) pairs + N, K = asarray(N), asarray(K) + if not np.issubdtype(N.dtype, np.integer): + raise TypeError("Argument `N` must contain only integers") + if not np.issubdtype(K.dtype, np.integer): + raise TypeError("Argument `K` must contain only integers") + if not exact: + # NOTE: here we allow np.uint via casting to double types prior to + # passing to private ufunc dispatcher. All dispatched functions + # take double type for (n,k) arguments and return double. + return _stirling2_inexact(N.astype(float), K.astype(float)) + nk_pairs = list( + set([(n.take(0), k.take(0)) + for n, k in np.nditer([N, K], ['refs_ok'])]) + ) + heapify(nk_pairs) + # base mapping for small values + snsk_vals = defaultdict(int) + for pair in [(0, 0), (1, 1), (2, 1), (2, 2)]: + snsk_vals[pair] = 1 + # for each pair in the min-heap, calculate the value, store for later + n_old, n_row = 2, [0, 1, 1] + while nk_pairs: + n, k = heappop(nk_pairs) + if n < 2 or k > n or k <= 0: + continue + elif k == n or k == 1: + snsk_vals[(n, k)] = 1 + continue + elif n != n_old: + num_iters = n - n_old + while num_iters > 0: + n_row.append(1) + # traverse from back to remove second row + for j in range(len(n_row)-2, 1, -1): + n_row[j] = n_row[j]*j + n_row[j-1] + num_iters -= 1 + snsk_vals[(n, k)] = n_row[k] + else: + snsk_vals[(n, k)] = n_row[k] + n_old, n_row = n, n_row + out_types = [object, object, object] if exact else [float, float, float] + # for each pair in the map, fetch the value, and populate the array + it = np.nditer( + [N, K, None], + ['buffered', 'refs_ok'], + [['readonly'], ['readonly'], ['writeonly', 'allocate']], + op_dtypes=out_types, + ) + with it: + while not it.finished: + it[2] = snsk_vals[(int(it[0]), int(it[1]))] + it.iternext() + output = it.operands[2] + # If N and K were both scalars, convert output to scalar. + if output_is_scalar: + output = output.take(0) + return output + + +def zeta(x, q=None, out=None): + r""" + Riemann or Hurwitz zeta function. + + Parameters + ---------- + x : array_like of float or complex. + Input data + q : array_like of float, optional + Input data, must be real. Defaults to Riemann zeta. When `q` is + ``None``, complex inputs `x` are supported. If `q` is not ``None``, + then currently only real inputs `x` with ``x >= 1`` are supported, + even when ``q = 1.0`` (corresponding to the Riemann zeta function). + + out : ndarray, optional + Output array for the computed values. + + Returns + ------- + out : array_like + Values of zeta(x). + + See Also + -------- + zetac + + Notes + ----- + The two-argument version is the Hurwitz zeta function + + .. math:: + + \zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x}; + + see [dlmf]_ for details. The Riemann zeta function corresponds to + the case when ``q = 1``. + + For complex inputs with ``q = None``, points with + ``abs(z.imag) > 1e9`` and ``0 <= abs(z.real) < 2.5`` are currently not + supported due to slow convergence causing excessive runtime. + + References + ---------- + .. [dlmf] NIST, Digital Library of Mathematical Functions, + https://dlmf.nist.gov/25.11#i + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import zeta, polygamma, factorial + + Some specific values: + + >>> zeta(2), np.pi**2/6 + (1.6449340668482266, 1.6449340668482264) + + >>> zeta(4), np.pi**4/90 + (1.0823232337111381, 1.082323233711138) + + First nontrivial zero: + + >>> zeta(0.5 + 14.134725141734695j) + 0 + 0j + + Relation to the `polygamma` function: + + >>> m = 3 + >>> x = 1.25 + >>> polygamma(m, x) + array(2.782144009188397) + >>> (-1)**(m+1) * factorial(m) * zeta(m+1, x) + 2.7821440091883969 + + """ + if q is None: + return _ufuncs._riemann_zeta(x, out) + else: + return _ufuncs._zeta(x, q, out) + + +def softplus(x, **kwargs): + r""" + Compute the softplus function element-wise. + + The softplus function is defined as: ``softplus(x) = log(1 + exp(x))``. + It is a smooth approximation of the rectifier function (ReLU). + + Parameters + ---------- + x : array_like + Input value. + **kwargs + For other keyword-only arguments, see the + `ufunc docs `_. + + Returns + ------- + softplus : ndarray + Logarithm of ``exp(0) + exp(x)``. + + Examples + -------- + >>> from scipy import special + + >>> special.softplus(0) + 0.6931471805599453 + + >>> special.softplus([-1, 0, 1]) + array([0.31326169, 0.69314718, 1.31326169]) + """ + return np.logaddexp(0, x, **kwargs) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_comb.cpython-310-x86_64-linux-gnu.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_comb.cpython-310-x86_64-linux-gnu.so new file mode 100644 index 0000000000000000000000000000000000000000..e0598ff2d395e08055065c94341de42ce84c5651 Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_comb.cpython-310-x86_64-linux-gnu.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ellip_harm.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ellip_harm.py new file mode 100644 index 0000000000000000000000000000000000000000..1b1ce34aa58054be13edfd5d87f2059e8a0d9224 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ellip_harm.py @@ -0,0 +1,214 @@ +import numpy as np + +from ._ufuncs import _ellip_harm +from ._ellip_harm_2 import _ellipsoid, _ellipsoid_norm + + +def ellip_harm(h2, k2, n, p, s, signm=1, signn=1): + r""" + Ellipsoidal harmonic functions E^p_n(l) + + These are also known as Lame functions of the first kind, and are + solutions to the Lame equation: + + .. math:: (s^2 - h^2)(s^2 - k^2)E''(s) + + s(2s^2 - h^2 - k^2)E'(s) + (a - q s^2)E(s) = 0 + + where :math:`q = (n+1)n` and :math:`a` is the eigenvalue (not + returned) corresponding to the solutions. + + Parameters + ---------- + h2 : float + ``h**2`` + k2 : float + ``k**2``; should be larger than ``h**2`` + n : int + Degree + s : float + Coordinate + p : int + Order, can range between [1,2n+1] + signm : {1, -1}, optional + Sign of prefactor of functions. Can be +/-1. See Notes. + signn : {1, -1}, optional + Sign of prefactor of functions. Can be +/-1. See Notes. + + Returns + ------- + E : float + the harmonic :math:`E^p_n(s)` + + See Also + -------- + ellip_harm_2, ellip_normal + + Notes + ----- + The geometric interpretation of the ellipsoidal functions is + explained in [2]_, [3]_, [4]_. The `signm` and `signn` arguments control the + sign of prefactors for functions according to their type:: + + K : +1 + L : signm + M : signn + N : signm*signn + + .. versionadded:: 0.15.0 + + References + ---------- + .. [1] Digital Library of Mathematical Functions 29.12 + https://dlmf.nist.gov/29.12 + .. [2] Bardhan and Knepley, "Computational science and + re-discovery: open-source implementations of + ellipsoidal harmonics for problems in potential theory", + Comput. Sci. Disc. 5, 014006 (2012) + :doi:`10.1088/1749-4699/5/1/014006`. + .. [3] David J.and Dechambre P, "Computation of Ellipsoidal + Gravity Field Harmonics for small solar system bodies" + pp. 30-36, 2000 + .. [4] George Dassios, "Ellipsoidal Harmonics: Theory and Applications" + pp. 418, 2012 + + Examples + -------- + >>> from scipy.special import ellip_harm + >>> w = ellip_harm(5,8,1,1,2.5) + >>> w + 2.5 + + Check that the functions indeed are solutions to the Lame equation: + + >>> import numpy as np + >>> from scipy.interpolate import UnivariateSpline + >>> def eigenvalue(f, df, ddf): + ... r = (((s**2 - h**2) * (s**2 - k**2) * ddf + ... + s * (2*s**2 - h**2 - k**2) * df + ... - n * (n + 1)*s**2*f) / f) + ... return -r.mean(), r.std() + >>> s = np.linspace(0.1, 10, 200) + >>> k, h, n, p = 8.0, 2.2, 3, 2 + >>> E = ellip_harm(h**2, k**2, n, p, s) + >>> E_spl = UnivariateSpline(s, E) + >>> a, a_err = eigenvalue(E_spl(s), E_spl(s,1), E_spl(s,2)) + >>> a, a_err + (583.44366156701483, 6.4580890640310646e-11) + + """ # noqa: E501 + return _ellip_harm(h2, k2, n, p, s, signm, signn) + + +_ellip_harm_2_vec = np.vectorize(_ellipsoid, otypes='d') + + +def ellip_harm_2(h2, k2, n, p, s): + r""" + Ellipsoidal harmonic functions F^p_n(l) + + These are also known as Lame functions of the second kind, and are + solutions to the Lame equation: + + .. math:: (s^2 - h^2)(s^2 - k^2)F''(s) + + s(2s^2 - h^2 - k^2)F'(s) + (a - q s^2)F(s) = 0 + + where :math:`q = (n+1)n` and :math:`a` is the eigenvalue (not + returned) corresponding to the solutions. + + Parameters + ---------- + h2 : float + ``h**2`` + k2 : float + ``k**2``; should be larger than ``h**2`` + n : int + Degree. + p : int + Order, can range between [1,2n+1]. + s : float + Coordinate + + Returns + ------- + F : float + The harmonic :math:`F^p_n(s)` + + See Also + -------- + ellip_harm, ellip_normal + + Notes + ----- + Lame functions of the second kind are related to the functions of the first kind: + + .. math:: + + F^p_n(s)=(2n + 1)E^p_n(s)\int_{0}^{1/s} + \frac{du}{(E^p_n(1/u))^2\sqrt{(1-u^2k^2)(1-u^2h^2)}} + + .. versionadded:: 0.15.0 + + Examples + -------- + >>> from scipy.special import ellip_harm_2 + >>> w = ellip_harm_2(5,8,2,1,10) + >>> w + 0.00108056853382 + + """ + with np.errstate(all='ignore'): + return _ellip_harm_2_vec(h2, k2, n, p, s) + + +def _ellip_normal_vec(h2, k2, n, p): + return _ellipsoid_norm(h2, k2, n, p) + + +_ellip_normal_vec = np.vectorize(_ellip_normal_vec, otypes='d') + + +def ellip_normal(h2, k2, n, p): + r""" + Ellipsoidal harmonic normalization constants gamma^p_n + + The normalization constant is defined as + + .. math:: + + \gamma^p_n=8\int_{0}^{h}dx\int_{h}^{k}dy + \frac{(y^2-x^2)(E^p_n(y)E^p_n(x))^2}{\sqrt((k^2-y^2)(y^2-h^2)(h^2-x^2)(k^2-x^2)} + + Parameters + ---------- + h2 : float + ``h**2`` + k2 : float + ``k**2``; should be larger than ``h**2`` + n : int + Degree. + p : int + Order, can range between [1,2n+1]. + + Returns + ------- + gamma : float + The normalization constant :math:`\gamma^p_n` + + See Also + -------- + ellip_harm, ellip_harm_2 + + Notes + ----- + .. versionadded:: 0.15.0 + + Examples + -------- + >>> from scipy.special import ellip_normal + >>> w = ellip_normal(5,8,3,7) + >>> w + 1723.38796997 + + """ + with np.errstate(all='ignore'): + return _ellip_normal_vec(h2, k2, n, p) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_input_validation.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_input_validation.py new file mode 100644 index 0000000000000000000000000000000000000000..e5b7fe36df87617bf91655623ee0076c37a4d08a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_input_validation.py @@ -0,0 +1,17 @@ +import math +import operator + +def _nonneg_int_or_fail(n, var_name, strict=True): + try: + if strict: + # Raises an exception if float + n = operator.index(n) + elif n == math.floor(n): + n = int(n) + else: + raise ValueError() + if n < 0: + raise ValueError() + except (ValueError, TypeError) as err: + raise err.__class__(f"{var_name} must be a non-negative integer") from err + return n diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_lambertw.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_lambertw.py new file mode 100644 index 0000000000000000000000000000000000000000..f758c7c21fdddc0ec1b84727d90c6de7f34a094e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_lambertw.py @@ -0,0 +1,149 @@ +from ._ufuncs import _lambertw + +import numpy as np + + +def lambertw(z, k=0, tol=1e-8): + r""" + lambertw(z, k=0, tol=1e-8) + + Lambert W function. + + The Lambert W function `W(z)` is defined as the inverse function + of ``w * exp(w)``. In other words, the value of ``W(z)`` is + such that ``z = W(z) * exp(W(z))`` for any complex number + ``z``. + + The Lambert W function is a multivalued function with infinitely + many branches. Each branch gives a separate solution of the + equation ``z = w exp(w)``. Here, the branches are indexed by the + integer `k`. + + Parameters + ---------- + z : array_like + Input argument. + k : int, optional + Branch index. + tol : float, optional + Evaluation tolerance. + + Returns + ------- + w : array + `w` will have the same shape as `z`. + + See Also + -------- + wrightomega : the Wright Omega function + + Notes + ----- + All branches are supported by `lambertw`: + + * ``lambertw(z)`` gives the principal solution (branch 0) + * ``lambertw(z, k)`` gives the solution on branch `k` + + The Lambert W function has two partially real branches: the + principal branch (`k = 0`) is real for real ``z > -1/e``, and the + ``k = -1`` branch is real for ``-1/e < z < 0``. All branches except + ``k = 0`` have a logarithmic singularity at ``z = 0``. + + **Possible issues** + + The evaluation can become inaccurate very close to the branch point + at ``-1/e``. In some corner cases, `lambertw` might currently + fail to converge, or can end up on the wrong branch. + + **Algorithm** + + Halley's iteration is used to invert ``w * exp(w)``, using a first-order + asymptotic approximation (O(log(w)) or `O(w)`) as the initial estimate. + + The definition, implementation and choice of branches is based on [2]_. + + References + ---------- + .. [1] https://en.wikipedia.org/wiki/Lambert_W_function + .. [2] Corless et al, "On the Lambert W function", Adv. Comp. Math. 5 + (1996) 329-359. + https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf + + Examples + -------- + The Lambert W function is the inverse of ``w exp(w)``: + + >>> import numpy as np + >>> from scipy.special import lambertw + >>> w = lambertw(1) + >>> w + (0.56714329040978384+0j) + >>> w * np.exp(w) + (1.0+0j) + + Any branch gives a valid inverse: + + >>> w = lambertw(1, k=3) + >>> w + (-2.8535817554090377+17.113535539412148j) + >>> w*np.exp(w) + (1.0000000000000002+1.609823385706477e-15j) + + **Applications to equation-solving** + + The Lambert W function may be used to solve various kinds of + equations. We give two examples here. + + First, the function can be used to solve implicit equations of the + form + + :math:`x = a + b e^{c x}` + + for :math:`x`. We assume :math:`c` is not zero. After a little + algebra, the equation may be written + + :math:`z e^z = -b c e^{a c}` + + where :math:`z = c (a - x)`. :math:`z` may then be expressed using + the Lambert W function + + :math:`z = W(-b c e^{a c})` + + giving + + :math:`x = a - W(-b c e^{a c})/c` + + For example, + + >>> a = 3 + >>> b = 2 + >>> c = -0.5 + + The solution to :math:`x = a + b e^{c x}` is: + + >>> x = a - lambertw(-b*c*np.exp(a*c))/c + >>> x + (3.3707498368978794+0j) + + Verify that it solves the equation: + + >>> a + b*np.exp(c*x) + (3.37074983689788+0j) + + The Lambert W function may also be used find the value of the infinite + power tower :math:`z^{z^{z^{\ldots}}}`: + + >>> def tower(z, n): + ... if n == 0: + ... return z + ... return z ** tower(z, n-1) + ... + >>> tower(0.5, 100) + 0.641185744504986 + >>> -lambertw(-np.log(0.5)) / np.log(0.5) + (0.64118574450498589+0j) + """ + # TODO: special expert should inspect this + # interception; better place to do it? + k = np.asarray(k, dtype=np.dtype("long")) + return _lambertw(z, k, tol) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_logsumexp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_logsumexp.py new file mode 100644 index 0000000000000000000000000000000000000000..1b0da953464e8d92925ebdaba79ce062c38fc7e5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_logsumexp.py @@ -0,0 +1,417 @@ +import math +import numpy as np +from scipy._lib._util import _asarray_validated +from scipy._lib._array_api import ( + array_namespace, + xp_size, + xp_broadcast_promote, + xp_copy, + xp_float_to_complex, + is_complex, +) +from scipy._lib import array_api_extra as xpx + +__all__ = ["logsumexp", "softmax", "log_softmax"] + + +def logsumexp(a, axis=None, b=None, keepdims=False, return_sign=False): + """Compute the log of the sum of exponentials of input elements. + + Parameters + ---------- + a : array_like + Input array. + axis : None or int or tuple of ints, optional + Axis or axes over which the sum is taken. By default `axis` is None, + and all elements are summed. + + .. versionadded:: 0.11.0 + b : array-like, optional + Scaling factor for exp(`a`) must be of the same shape as `a` or + broadcastable to `a`. These values may be negative in order to + implement subtraction. + + .. versionadded:: 0.12.0 + keepdims : bool, optional + If this is set to True, the axes which are reduced are left in the + result as dimensions with size one. With this option, the result + will broadcast correctly against the original array. + + .. versionadded:: 0.15.0 + return_sign : bool, optional + If this is set to True, the result will be a pair containing sign + information; if False, results that are negative will be returned + as NaN. Default is False (no sign information). + + .. versionadded:: 0.16.0 + + Returns + ------- + res : ndarray + The result, ``np.log(np.sum(np.exp(a)))`` calculated in a numerically + more stable way. If `b` is given then ``np.log(np.sum(b*np.exp(a)))`` + is returned. If ``return_sign`` is True, ``res`` contains the log of + the absolute value of the argument. + sgn : ndarray + If ``return_sign`` is True, this will be an array of floating-point + numbers matching res containing +1, 0, -1 (for real-valued inputs) + or a complex phase (for complex inputs). This gives the sign of the + argument of the logarithm in ``res``. + If ``return_sign`` is False, only one result is returned. + + See Also + -------- + numpy.logaddexp, numpy.logaddexp2 + + Notes + ----- + NumPy has a logaddexp function which is very similar to `logsumexp`, but + only handles two arguments. `logaddexp.reduce` is similar to this + function, but may be less stable. + + The logarithm is a multivalued function: for each :math:`x` there is an + infinite number of :math:`z` such that :math:`exp(z) = x`. The convention + is to return the :math:`z` whose imaginary part lies in :math:`(-pi, pi]`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import logsumexp + >>> a = np.arange(10) + >>> logsumexp(a) + 9.4586297444267107 + >>> np.log(np.sum(np.exp(a))) + 9.4586297444267107 + + With weights + + >>> a = np.arange(10) + >>> b = np.arange(10, 0, -1) + >>> logsumexp(a, b=b) + 9.9170178533034665 + >>> np.log(np.sum(b*np.exp(a))) + 9.9170178533034647 + + Returning a sign flag + + >>> logsumexp([1,2],b=[1,-1],return_sign=True) + (1.5413248546129181, -1.0) + + Notice that `logsumexp` does not directly support masked arrays. To use it + on a masked array, convert the mask into zero weights: + + >>> a = np.ma.array([np.log(2), 2, np.log(3)], + ... mask=[False, True, False]) + >>> b = (~a.mask).astype(int) + >>> logsumexp(a.data, b=b), np.log(5) + 1.6094379124341005, 1.6094379124341005 + + """ + xp = array_namespace(a, b) + a, b = xp_broadcast_promote(a, b, ensure_writeable=True, force_floating=True, xp=xp) + a = xpx.atleast_nd(a, ndim=1, xp=xp) + b = xpx.atleast_nd(b, ndim=1, xp=xp) if b is not None else b + axis = tuple(range(a.ndim)) if axis is None else axis + + if xp_size(a) != 0: + with np.errstate(divide='ignore', invalid='ignore'): # log of zero is OK + out, sgn = _logsumexp(a, b, axis=axis, return_sign=return_sign, xp=xp) + else: + shape = np.asarray(a.shape) # NumPy is convenient for shape manipulation + shape[axis] = 1 + out = xp.full(tuple(shape), -xp.inf, dtype=a.dtype) + sgn = xp.sign(out) + + if xp.isdtype(out.dtype, 'complex floating'): + if return_sign: + real = xp.real(sgn) + imag = xp_float_to_complex(_wrap_radians(xp.imag(sgn), xp)) + sgn = real + imag*1j + else: + real = xp.real(out) + imag = xp_float_to_complex(_wrap_radians(xp.imag(out), xp)) + out = real + imag*1j + + # Deal with shape details - reducing dimensions and convert 0-D to scalar for NumPy + out = xp.squeeze(out, axis=axis) if not keepdims else out + sgn = xp.squeeze(sgn, axis=axis) if (sgn is not None and not keepdims) else sgn + out = out[()] if out.ndim == 0 else out + sgn = sgn[()] if (sgn is not None and sgn.ndim == 0) else sgn + + return (out, sgn) if return_sign else out + + +def _wrap_radians(x, xp=None): + xp = array_namespace(x) if xp is None else xp + # Wrap radians to (-pi, pi] interval + out = -((-x + math.pi) % (2 * math.pi) - math.pi) + # preserve relative precision + no_wrap = xp.abs(x) < xp.pi + out[no_wrap] = x[no_wrap] + return out + + +def _elements_and_indices_with_max_real(a, axis=-1, xp=None): + # This is an array-API compatible `max` function that works something + # like `np.max` for complex input. The important part is that it finds + # the element with maximum real part. When there are multiple complex values + # with this real part, it doesn't matter which we choose. + # We could use `argmax` on real component, but array API doesn't yet have + # `take_along_axis`, and even if it did, we would have problems with axis tuples. + # Feel free to rewrite! It's ugly, but it's not the purpose of the PR, and + # it gets the job done. + xp = array_namespace(a) if xp is None else xp + + if xp.isdtype(a.dtype, "complex floating"): + # select all elements with max real part. + real_a = xp.real(a) + max = xp.max(real_a, axis=axis, keepdims=True) + mask = real_a == max + + # Of those, choose one arbitrarily. This is a reasonably + # simple, array-API compatible way of doing so that doesn't + # have a problem with `axis` being a tuple or None. + i = xp.reshape(xp.arange(xp_size(a)), a.shape) + i[~mask] = -1 + max_i = xp.max(i, axis=axis, keepdims=True) + mask = i == max_i + a = xp_copy(a) + a[~mask] = 0 + max = xp.sum(a, axis=axis, dtype=a.dtype, keepdims=True) + else: + max = xp.max(a, axis=axis, keepdims=True) + mask = a == max + + return xp.asarray(max), xp.asarray(mask) + + +def _sign(x, xp): + return x / xp.where(x == 0, xp.asarray(1, dtype=x.dtype), xp.abs(x)) + + +def _logsumexp(a, b, axis, return_sign, xp): + + # This has been around for about a decade, so let's consider it a feature: + # Even if element of `a` is infinite or NaN, it adds nothing to the sum if + # the corresponding weight is zero. + if b is not None: + a[b == 0] = -xp.inf + + # Find element with maximum real part, since this is what affects the magnitude + # of the exponential. Possible enhancement: include log of `b` magnitude in `a`. + a_max, i_max = _elements_and_indices_with_max_real(a, axis=axis, xp=xp) + + # for precision, these terms are separated out of the main sum. + a[i_max] = -xp.inf + i_max_dt = xp.astype(i_max, a.dtype) + # This is an inefficient way of getting `m` because it is the sum of a sparse + # array; however, this is the simplest way I can think of to get the right shape. + m = (xp.sum(i_max_dt, axis=axis, keepdims=True, dtype=a.dtype) if b is None + else xp.sum(b * i_max_dt, axis=axis, keepdims=True, dtype=a.dtype)) + + # Arithmetic between infinities will introduce NaNs. + # `+ a_max` at the end naturally corrects for removing them here. + shift = xp.where(xp.isfinite(a_max), a_max, xp.asarray(0, dtype=a_max.dtype)) + + # Shift, exponentiate, scale, and sum + exp = b * xp.exp(a - shift) if b is not None else xp.exp(a - shift) + s = xp.sum(exp, axis=axis, keepdims=True, dtype=exp.dtype) + s = xp.where(s == 0, s, s/m) + + # Separate sign/magnitude information + # Originally, this was only performed if `return_sign=True`. + # However, this is also needed if any elements of `m < 0` or `s < -1`. + # An improvement would be to perform the calculations only on these entries. + + # Use the numpy>=2.0 convention for sign. + # When all array libraries agree, this can become sng = xp.sign(s). + sgn = _sign(s + 1, xp=xp) * _sign(m, xp=xp) + + if xp.isdtype(s.dtype, "real floating"): + # The log functions need positive arguments + s = xp.where(s < -1, -s - 2, s) + m = xp.abs(m) + else: + # `a_max` can have a sign component for complex input + sgn = sgn * xp.exp(xp.imag(a_max) * xp.asarray(1.0j, dtype=a_max.dtype)) + + # Take log and undo shift + out = xp.log1p(s) + xp.log(m) + a_max + + if return_sign: + if is_complex(out, xp): + out = xp.real(out) + elif xp.isdtype(out.dtype, 'real floating'): + out[sgn < 0] = xp.nan + + return out, sgn + + +def softmax(x, axis=None): + r"""Compute the softmax function. + + The softmax function transforms each element of a collection by + computing the exponential of each element divided by the sum of the + exponentials of all the elements. That is, if `x` is a one-dimensional + numpy array:: + + softmax(x) = np.exp(x)/sum(np.exp(x)) + + Parameters + ---------- + x : array_like + Input array. + axis : int or tuple of ints, optional + Axis to compute values along. Default is None and softmax will be + computed over the entire array `x`. + + Returns + ------- + s : ndarray + An array the same shape as `x`. The result will sum to 1 along the + specified axis. + + Notes + ----- + The formula for the softmax function :math:`\sigma(x)` for a vector + :math:`x = \{x_0, x_1, ..., x_{n-1}\}` is + + .. math:: \sigma(x)_j = \frac{e^{x_j}}{\sum_k e^{x_k}} + + The `softmax` function is the gradient of `logsumexp`. + + The implementation uses shifting to avoid overflow. See [1]_ for more + details. + + .. versionadded:: 1.2.0 + + References + ---------- + .. [1] P. Blanchard, D.J. Higham, N.J. Higham, "Accurately computing the + log-sum-exp and softmax functions", IMA Journal of Numerical Analysis, + Vol.41(4), :doi:`10.1093/imanum/draa038`. + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import softmax + >>> np.set_printoptions(precision=5) + + >>> x = np.array([[1, 0.5, 0.2, 3], + ... [1, -1, 7, 3], + ... [2, 12, 13, 3]]) + ... + + Compute the softmax transformation over the entire array. + + >>> m = softmax(x) + >>> m + array([[ 4.48309e-06, 2.71913e-06, 2.01438e-06, 3.31258e-05], + [ 4.48309e-06, 6.06720e-07, 1.80861e-03, 3.31258e-05], + [ 1.21863e-05, 2.68421e-01, 7.29644e-01, 3.31258e-05]]) + + >>> m.sum() + 1.0 + + Compute the softmax transformation along the first axis (i.e., the + columns). + + >>> m = softmax(x, axis=0) + + >>> m + array([[ 2.11942e-01, 1.01300e-05, 2.75394e-06, 3.33333e-01], + [ 2.11942e-01, 2.26030e-06, 2.47262e-03, 3.33333e-01], + [ 5.76117e-01, 9.99988e-01, 9.97525e-01, 3.33333e-01]]) + + >>> m.sum(axis=0) + array([ 1., 1., 1., 1.]) + + Compute the softmax transformation along the second axis (i.e., the rows). + + >>> m = softmax(x, axis=1) + >>> m + array([[ 1.05877e-01, 6.42177e-02, 4.75736e-02, 7.82332e-01], + [ 2.42746e-03, 3.28521e-04, 9.79307e-01, 1.79366e-02], + [ 1.22094e-05, 2.68929e-01, 7.31025e-01, 3.31885e-05]]) + + >>> m.sum(axis=1) + array([ 1., 1., 1.]) + + """ + x = _asarray_validated(x, check_finite=False) + x_max = np.amax(x, axis=axis, keepdims=True) + exp_x_shifted = np.exp(x - x_max) + return exp_x_shifted / np.sum(exp_x_shifted, axis=axis, keepdims=True) + + +def log_softmax(x, axis=None): + r"""Compute the logarithm of the softmax function. + + In principle:: + + log_softmax(x) = log(softmax(x)) + + but using a more accurate implementation. + + Parameters + ---------- + x : array_like + Input array. + axis : int or tuple of ints, optional + Axis to compute values along. Default is None and softmax will be + computed over the entire array `x`. + + Returns + ------- + s : ndarray or scalar + An array with the same shape as `x`. Exponential of the result will + sum to 1 along the specified axis. If `x` is a scalar, a scalar is + returned. + + Notes + ----- + `log_softmax` is more accurate than ``np.log(softmax(x))`` with inputs that + make `softmax` saturate (see examples below). + + .. versionadded:: 1.5.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import log_softmax + >>> from scipy.special import softmax + >>> np.set_printoptions(precision=5) + + >>> x = np.array([1000.0, 1.0]) + + >>> y = log_softmax(x) + >>> y + array([ 0., -999.]) + + >>> with np.errstate(divide='ignore'): + ... y = np.log(softmax(x)) + ... + >>> y + array([ 0., -inf]) + + """ + + x = _asarray_validated(x, check_finite=False) + + x_max = np.amax(x, axis=axis, keepdims=True) + + if x_max.ndim > 0: + x_max[~np.isfinite(x_max)] = 0 + elif not np.isfinite(x_max): + x_max = 0 + + tmp = x - x_max + exp_tmp = np.exp(tmp) + + # suppress warnings about log of zero + with np.errstate(divide='ignore'): + s = np.sum(exp_tmp, axis=axis, keepdims=True) + out = np.log(s) + + out = tmp - out + return out diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_mptestutils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_mptestutils.py new file mode 100644 index 0000000000000000000000000000000000000000..9e519093dface79e21f16d7063541ad107f5ca96 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_mptestutils.py @@ -0,0 +1,453 @@ +import os +import sys +import time +from itertools import zip_longest + +import numpy as np +from numpy.testing import assert_ +import pytest + +from scipy.special._testutils import assert_func_equal + +try: + import mpmath +except ImportError: + pass + + +# ------------------------------------------------------------------------------ +# Machinery for systematic tests with mpmath +# ------------------------------------------------------------------------------ + +class Arg: + """Generate a set of numbers on the real axis, concentrating on + 'interesting' regions and covering all orders of magnitude. + + """ + + def __init__(self, a=-np.inf, b=np.inf, inclusive_a=True, inclusive_b=True): + if a > b: + raise ValueError("a should be less than or equal to b") + if a == -np.inf: + a = -0.5*np.finfo(float).max + if b == np.inf: + b = 0.5*np.finfo(float).max + self.a, self.b = a, b + + self.inclusive_a, self.inclusive_b = inclusive_a, inclusive_b + + def _positive_values(self, a, b, n): + if a < 0: + raise ValueError("a should be positive") + + # Try to put half of the points into a linspace between a and + # 10 the other half in a logspace. + if n % 2 == 0: + nlogpts = n//2 + nlinpts = nlogpts + else: + nlogpts = n//2 + nlinpts = nlogpts + 1 + + if a >= 10: + # Outside of linspace range; just return a logspace. + pts = np.logspace(np.log10(a), np.log10(b), n) + elif a > 0 and b < 10: + # Outside of logspace range; just return a linspace + pts = np.linspace(a, b, n) + elif a > 0: + # Linspace between a and 10 and a logspace between 10 and + # b. + linpts = np.linspace(a, 10, nlinpts, endpoint=False) + logpts = np.logspace(1, np.log10(b), nlogpts) + pts = np.hstack((linpts, logpts)) + elif a == 0 and b <= 10: + # Linspace between 0 and b and a logspace between 0 and + # the smallest positive point of the linspace + linpts = np.linspace(0, b, nlinpts) + if linpts.size > 1: + right = np.log10(linpts[1]) + else: + right = -30 + logpts = np.logspace(-30, right, nlogpts, endpoint=False) + pts = np.hstack((logpts, linpts)) + else: + # Linspace between 0 and 10, logspace between 0 and the + # smallest positive point of the linspace, and a logspace + # between 10 and b. + if nlogpts % 2 == 0: + nlogpts1 = nlogpts//2 + nlogpts2 = nlogpts1 + else: + nlogpts1 = nlogpts//2 + nlogpts2 = nlogpts1 + 1 + linpts = np.linspace(0, 10, nlinpts, endpoint=False) + if linpts.size > 1: + right = np.log10(linpts[1]) + else: + right = -30 + logpts1 = np.logspace(-30, right, nlogpts1, endpoint=False) + logpts2 = np.logspace(1, np.log10(b), nlogpts2) + pts = np.hstack((logpts1, linpts, logpts2)) + + return np.sort(pts) + + def values(self, n): + """Return an array containing n numbers.""" + a, b = self.a, self.b + if a == b: + return np.zeros(n) + + if not self.inclusive_a: + n += 1 + if not self.inclusive_b: + n += 1 + + if n % 2 == 0: + n1 = n//2 + n2 = n1 + else: + n1 = n//2 + n2 = n1 + 1 + + if a >= 0: + pospts = self._positive_values(a, b, n) + negpts = [] + elif b <= 0: + pospts = [] + negpts = -self._positive_values(-b, -a, n) + else: + pospts = self._positive_values(0, b, n1) + negpts = -self._positive_values(0, -a, n2 + 1) + # Don't want to get zero twice + negpts = negpts[1:] + pts = np.hstack((negpts[::-1], pospts)) + + if not self.inclusive_a: + pts = pts[1:] + if not self.inclusive_b: + pts = pts[:-1] + return pts + + +class FixedArg: + def __init__(self, values): + self._values = np.asarray(values) + + def values(self, n): + return self._values + + +class ComplexArg: + def __init__(self, a=complex(-np.inf, -np.inf), b=complex(np.inf, np.inf)): + self.real = Arg(a.real, b.real) + self.imag = Arg(a.imag, b.imag) + + def values(self, n): + m = int(np.floor(np.sqrt(n))) + x = self.real.values(m) + y = self.imag.values(m + 1) + return (x[:,None] + 1j*y[None,:]).ravel() + + +class IntArg: + def __init__(self, a=-1000, b=1000): + self.a = a + self.b = b + + def values(self, n): + v1 = Arg(self.a, self.b).values(max(1 + n//2, n-5)).astype(int) + v2 = np.arange(-5, 5) + v = np.unique(np.r_[v1, v2]) + v = v[(v >= self.a) & (v < self.b)] + return v + + +def get_args(argspec, n): + if isinstance(argspec, np.ndarray): + args = argspec.copy() + else: + nargs = len(argspec) + ms = np.asarray( + [1.5 if isinstance(spec, ComplexArg) else 1.0 for spec in argspec] + ) + ms = (n**(ms/sum(ms))).astype(int) + 1 + + args = [spec.values(m) for spec, m in zip(argspec, ms)] + args = np.array(np.broadcast_arrays(*np.ix_(*args))).reshape(nargs, -1).T + + return args + + +class MpmathData: + def __init__(self, scipy_func, mpmath_func, arg_spec, name=None, + dps=None, prec=None, n=None, rtol=1e-7, atol=1e-300, + ignore_inf_sign=False, distinguish_nan_and_inf=True, + nan_ok=True, param_filter=None): + + # mpmath tests are really slow (see gh-6989). Use a small number of + # points by default, increase back to 5000 (old default) if XSLOW is + # set + if n is None: + try: + is_xslow = int(os.environ.get('SCIPY_XSLOW', '0')) + except ValueError: + is_xslow = False + + n = 5000 if is_xslow else 500 + + self.scipy_func = scipy_func + self.mpmath_func = mpmath_func + self.arg_spec = arg_spec + self.dps = dps + self.prec = prec + self.n = n + self.rtol = rtol + self.atol = atol + self.ignore_inf_sign = ignore_inf_sign + self.nan_ok = nan_ok + if isinstance(self.arg_spec, np.ndarray): + self.is_complex = np.issubdtype(self.arg_spec.dtype, np.complexfloating) + else: + self.is_complex = any( + [isinstance(arg, ComplexArg) for arg in self.arg_spec] + ) + self.ignore_inf_sign = ignore_inf_sign + self.distinguish_nan_and_inf = distinguish_nan_and_inf + if not name or name == '': + name = getattr(scipy_func, '__name__', None) + if not name or name == '': + name = getattr(mpmath_func, '__name__', None) + self.name = name + self.param_filter = param_filter + + def check(self): + np.random.seed(1234) + + # Generate values for the arguments + argarr = get_args(self.arg_spec, self.n) + + # Check + old_dps, old_prec = mpmath.mp.dps, mpmath.mp.prec + try: + if self.dps is not None: + dps_list = [self.dps] + else: + dps_list = [20] + if self.prec is not None: + mpmath.mp.prec = self.prec + + # Proper casting of mpmath input and output types. Using + # native mpmath types as inputs gives improved precision + # in some cases. + if np.issubdtype(argarr.dtype, np.complexfloating): + pytype = mpc2complex + + def mptype(x): + return mpmath.mpc(complex(x)) + else: + def mptype(x): + return mpmath.mpf(float(x)) + + def pytype(x): + if abs(x.imag) > 1e-16*(1 + abs(x.real)): + return np.nan + else: + return mpf2float(x.real) + + # Try out different dps until one (or none) works + for j, dps in enumerate(dps_list): + mpmath.mp.dps = dps + + try: + assert_func_equal( + self.scipy_func, + lambda *a: pytype(self.mpmath_func(*map(mptype, a))), + argarr, + vectorized=False, + rtol=self.rtol, + atol=self.atol, + ignore_inf_sign=self.ignore_inf_sign, + distinguish_nan_and_inf=self.distinguish_nan_and_inf, + nan_ok=self.nan_ok, + param_filter=self.param_filter + ) + break + except AssertionError: + if j >= len(dps_list)-1: + # reraise the Exception + tp, value, tb = sys.exc_info() + if value.__traceback__ is not tb: + raise value.with_traceback(tb) + raise value + finally: + mpmath.mp.dps, mpmath.mp.prec = old_dps, old_prec + + def __repr__(self): + if self.is_complex: + return f"" + else: + return f"" + + +def assert_mpmath_equal(*a, **kw): + d = MpmathData(*a, **kw) + d.check() + + +def nonfunctional_tooslow(func): + return pytest.mark.skip( + reason=" Test not yet functional (too slow), needs more work." + )(func) + + +# ------------------------------------------------------------------------------ +# Tools for dealing with mpmath quirks +# ------------------------------------------------------------------------------ + +def mpf2float(x): + """ + Convert an mpf to the nearest floating point number. Just using + float directly doesn't work because of results like this: + + with mp.workdps(50): + float(mpf("0.99999999999999999")) = 0.9999999999999999 + + """ + return float(mpmath.nstr(x, 17, min_fixed=0, max_fixed=0)) + + +def mpc2complex(x): + return complex(mpf2float(x.real), mpf2float(x.imag)) + + +def trace_args(func): + def tofloat(x): + if isinstance(x, mpmath.mpc): + return complex(x) + else: + return float(x) + + def wrap(*a, **kw): + sys.stderr.write(f"{tuple(map(tofloat, a))!r}: ") + sys.stderr.flush() + try: + r = func(*a, **kw) + sys.stderr.write(f"-> {r!r}") + finally: + sys.stderr.write("\n") + sys.stderr.flush() + return r + return wrap + + +try: + import signal + POSIX = ('setitimer' in dir(signal)) +except ImportError: + POSIX = False + + +class TimeoutError(Exception): + pass + + +def time_limited(timeout=0.5, return_val=np.nan, use_sigalrm=True): + """ + Decorator for setting a timeout for pure-Python functions. + + If the function does not return within `timeout` seconds, the + value `return_val` is returned instead. + + On POSIX this uses SIGALRM by default. On non-POSIX, settrace is + used. Do not use this with threads: the SIGALRM implementation + does probably not work well. The settrace implementation only + traces the current thread. + + The settrace implementation slows down execution speed. Slowdown + by a factor around 10 is probably typical. + """ + if POSIX and use_sigalrm: + def sigalrm_handler(signum, frame): + raise TimeoutError() + + def deco(func): + def wrap(*a, **kw): + old_handler = signal.signal(signal.SIGALRM, sigalrm_handler) + signal.setitimer(signal.ITIMER_REAL, timeout) + try: + return func(*a, **kw) + except TimeoutError: + return return_val + finally: + signal.setitimer(signal.ITIMER_REAL, 0) + signal.signal(signal.SIGALRM, old_handler) + return wrap + else: + def deco(func): + def wrap(*a, **kw): + start_time = time.time() + + def trace(frame, event, arg): + if time.time() - start_time > timeout: + raise TimeoutError() + return trace + sys.settrace(trace) + try: + return func(*a, **kw) + except TimeoutError: + sys.settrace(None) + return return_val + finally: + sys.settrace(None) + return wrap + return deco + + +def exception_to_nan(func): + """Decorate function to return nan if it raises an exception""" + def wrap(*a, **kw): + try: + return func(*a, **kw) + except Exception: + return np.nan + return wrap + + +def inf_to_nan(func): + """Decorate function to return nan if it returns inf""" + def wrap(*a, **kw): + v = func(*a, **kw) + if not np.isfinite(v): + return np.nan + return v + return wrap + + +def mp_assert_allclose(res, std, atol=0, rtol=1e-17): + """ + Compare lists of mpmath.mpf's or mpmath.mpc's directly so that it + can be done to higher precision than double. + """ + failures = [] + for k, (resval, stdval) in enumerate(zip_longest(res, std)): + if resval is None or stdval is None: + raise ValueError('Lengths of inputs res and std are not equal.') + if mpmath.fabs(resval - stdval) > atol + rtol*mpmath.fabs(stdval): + failures.append((k, resval, stdval)) + + nfail = len(failures) + if nfail > 0: + ndigits = int(abs(np.log10(rtol))) + msg = [""] + msg.append(f"Bad results ({nfail} out of {k + 1}) for the following points:") + for k, resval, stdval in failures: + resrep = mpmath.nstr(resval, ndigits, min_fixed=0, max_fixed=0) + stdrep = mpmath.nstr(stdval, ndigits, min_fixed=0, max_fixed=0) + if stdval == 0: + rdiff = "inf" + else: + rdiff = mpmath.fabs((resval - stdval)/stdval) + rdiff = mpmath.nstr(rdiff, 3) + msg.append(f"{k}: {resrep} != {stdrep} (rdiff {rdiff})") + assert_(False, "\n".join(msg)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_multiufuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_multiufuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..0bb1be9461c629a48841f1d01268e8d11eee230f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_multiufuncs.py @@ -0,0 +1,610 @@ +import collections +import numbers +import numpy as np + +from ._input_validation import _nonneg_int_or_fail + +from ._special_ufuncs import (legendre_p, assoc_legendre_p, + sph_legendre_p, sph_harm_y) +from ._gufuncs import (legendre_p_all, assoc_legendre_p_all, + sph_legendre_p_all, sph_harm_y_all) + +__all__ = [ + "assoc_legendre_p", + "assoc_legendre_p_all", + "legendre_p", + "legendre_p_all", + "sph_harm_y", + "sph_harm_y_all", + "sph_legendre_p", + "sph_legendre_p_all", +] + + +class MultiUFunc: + def __init__(self, ufunc_or_ufuncs, doc=None, *, + force_complex_output=False, **default_kwargs): + if not isinstance(ufunc_or_ufuncs, np.ufunc): + if isinstance(ufunc_or_ufuncs, collections.abc.Mapping): + ufuncs_iter = ufunc_or_ufuncs.values() + elif isinstance(ufunc_or_ufuncs, collections.abc.Iterable): + ufuncs_iter = ufunc_or_ufuncs + else: + raise ValueError("ufunc_or_ufuncs should be a ufunc or a" + " ufunc collection") + + # Perform input validation to ensure all ufuncs in ufuncs are + # actually ufuncs and all take the same input types. + seen_input_types = set() + for ufunc in ufuncs_iter: + if not isinstance(ufunc, np.ufunc): + raise ValueError("All ufuncs must have type `numpy.ufunc`." + f" Received {ufunc_or_ufuncs}") + seen_input_types.add(frozenset(x.split("->")[0] for x in ufunc.types)) + if len(seen_input_types) > 1: + raise ValueError("All ufuncs must take the same input types.") + + self._ufunc_or_ufuncs = ufunc_or_ufuncs + self.__doc = doc + self.__force_complex_output = force_complex_output + self._default_kwargs = default_kwargs + self._resolve_out_shapes = None + self._finalize_out = None + self._key = None + self._ufunc_default_args = lambda *args, **kwargs: () + self._ufunc_default_kwargs = lambda *args, **kwargs: {} + + @property + def __doc__(self): + return self.__doc + + def _override_key(self, func): + """Set `key` method by decorating a function. + """ + self._key = func + + def _override_ufunc_default_args(self, func): + self._ufunc_default_args = func + + def _override_ufunc_default_kwargs(self, func): + self._ufunc_default_kwargs = func + + def _override_resolve_out_shapes(self, func): + """Set `resolve_out_shapes` method by decorating a function.""" + if func.__doc__ is None: + func.__doc__ = \ + """Resolve to output shapes based on relevant inputs.""" + func.__name__ = "resolve_out_shapes" + self._resolve_out_shapes = func + + def _override_finalize_out(self, func): + self._finalize_out = func + + def _resolve_ufunc(self, **kwargs): + """Resolve to a ufunc based on keyword arguments.""" + + if isinstance(self._ufunc_or_ufuncs, np.ufunc): + return self._ufunc_or_ufuncs + + ufunc_key = self._key(**kwargs) + return self._ufunc_or_ufuncs[ufunc_key] + + def __call__(self, *args, **kwargs): + kwargs = self._default_kwargs | kwargs + + args += self._ufunc_default_args(**kwargs) + + ufunc = self._resolve_ufunc(**kwargs) + + # array arguments to be passed to the ufunc + ufunc_args = [np.asarray(arg) for arg in args[-ufunc.nin:]] + + ufunc_kwargs = self._ufunc_default_kwargs(**kwargs) + + if (self._resolve_out_shapes is not None): + ufunc_arg_shapes = tuple(np.shape(ufunc_arg) for ufunc_arg in ufunc_args) + ufunc_out_shapes = self._resolve_out_shapes(*args[:-ufunc.nin], + *ufunc_arg_shapes, ufunc.nout, + **kwargs) + + ufunc_arg_dtypes = tuple(ufunc_arg.dtype if hasattr(ufunc_arg, 'dtype') + else np.dtype(type(ufunc_arg)) + for ufunc_arg in ufunc_args) + + if hasattr(ufunc, 'resolve_dtypes'): + ufunc_dtypes = ufunc_arg_dtypes + ufunc.nout * (None,) + ufunc_dtypes = ufunc.resolve_dtypes(ufunc_dtypes) + ufunc_out_dtypes = ufunc_dtypes[-ufunc.nout:] + else: + ufunc_out_dtype = np.result_type(*ufunc_arg_dtypes) + if (not np.issubdtype(ufunc_out_dtype, np.inexact)): + ufunc_out_dtype = np.float64 + + ufunc_out_dtypes = ufunc.nout * (ufunc_out_dtype,) + + if self.__force_complex_output: + ufunc_out_dtypes = tuple(np.result_type(1j, ufunc_out_dtype) + for ufunc_out_dtype in ufunc_out_dtypes) + + out = tuple(np.empty(ufunc_out_shape, dtype=ufunc_out_dtype) + for ufunc_out_shape, ufunc_out_dtype + in zip(ufunc_out_shapes, ufunc_out_dtypes)) + + ufunc_kwargs['out'] = out + + out = ufunc(*ufunc_args, **ufunc_kwargs) + if (self._finalize_out is not None): + out = self._finalize_out(out) + + return out + + +sph_legendre_p = MultiUFunc( + sph_legendre_p, + r"""sph_legendre_p(n, m, theta, *, diff_n=0) + + Spherical Legendre polynomial of the first kind. + + Parameters + ---------- + n : ArrayLike[int] + Degree of the spherical Legendre polynomial. Must have ``n >= 0``. + m : ArrayLike[int] + Order of the spherical Legendre polynomial. + theta : ArrayLike[float] + Input value. + diff_n : Optional[int] + A non-negative integer. Compute and return all derivatives up + to order ``diff_n``. Default is 0. + + Returns + ------- + p : ndarray or tuple[ndarray] + Spherical Legendre polynomial with ``diff_n`` derivatives. + + Notes + ----- + The spherical counterpart of an (unnormalized) associated Legendre polynomial has + the additional factor + + .. math:: + + \sqrt{\frac{(2 n + 1) (n - m)!}{4 \pi (n + m)!}} + + It is the same as the spherical harmonic :math:`Y_{n}^{m}(\theta, \phi)` + with :math:`\phi = 0`. + """, diff_n=0 +) + + +@sph_legendre_p._override_key +def _(diff_n): + diff_n = _nonneg_int_or_fail(diff_n, "diff_n", strict=False) + if not 0 <= diff_n <= 2: + raise ValueError( + "diff_n is currently only implemented for orders 0, 1, and 2," + f" received: {diff_n}." + ) + return diff_n + + +@sph_legendre_p._override_finalize_out +def _(out): + return np.moveaxis(out, -1, 0) + + +sph_legendre_p_all = MultiUFunc( + sph_legendre_p_all, + """sph_legendre_p_all(n, m, theta, *, diff_n=0) + + All spherical Legendre polynomials of the first kind up to the + specified degree ``n`` and order ``m``. + + Output shape is ``(n + 1, 2 * m + 1, ...)``. The entry at ``(j, i)`` + corresponds to degree ``j`` and order ``i`` for all ``0 <= j <= n`` + and ``-m <= i <= m``. + + See Also + -------- + sph_legendre_p + """, diff_n=0 +) + + +@sph_legendre_p_all._override_key +def _(diff_n): + diff_n = _nonneg_int_or_fail(diff_n, "diff_n", strict=False) + if not 0 <= diff_n <= 2: + raise ValueError( + "diff_n is currently only implemented for orders 0, 1, and 2," + f" received: {diff_n}." + ) + return diff_n + + +@sph_legendre_p_all._override_ufunc_default_kwargs +def _(diff_n): + return {'axes': [()] + [(0, 1, -1)]} + + +@sph_legendre_p_all._override_resolve_out_shapes +def _(n, m, theta_shape, nout, diff_n): + if not isinstance(n, numbers.Integral) or (n < 0): + raise ValueError("n must be a non-negative integer.") + + return ((n + 1, 2 * abs(m) + 1) + theta_shape + (diff_n + 1,),) + + +@sph_legendre_p_all._override_finalize_out +def _(out): + return np.moveaxis(out, -1, 0) + + +assoc_legendre_p = MultiUFunc( + assoc_legendre_p, + r"""assoc_legendre_p(n, m, z, *, branch_cut=2, norm=False, diff_n=0) + + Associated Legendre polynomial of the first kind. + + Parameters + ---------- + n : ArrayLike[int] + Degree of the associated Legendre polynomial. Must have ``n >= 0``. + m : ArrayLike[int] + order of the associated Legendre polynomial. + z : ArrayLike[float | complex] + Input value. + branch_cut : Optional[ArrayLike[int]] + Selects branch cut. Must be 2 (default) or 3. + 2: cut on the real axis ``|z| > 1`` + 3: cut on the real axis ``-1 < z < 1`` + norm : Optional[bool] + If ``True``, compute the normalized associated Legendre polynomial. + Default is ``False``. + diff_n : Optional[int] + A non-negative integer. Compute and return all derivatives up + to order ``diff_n``. Default is 0. + + Returns + ------- + p : ndarray or tuple[ndarray] + Associated Legendre polynomial with ``diff_n`` derivatives. + + Notes + ----- + The normalized counterpart of an (unnormalized) associated Legendre + polynomial has the additional factor + + .. math:: + + \sqrt{\frac{(2 n + 1) (n - m)!}{2 (n + m)!}} + """, branch_cut=2, norm=False, diff_n=0 +) + + +@assoc_legendre_p._override_key +def _(branch_cut, norm, diff_n): + diff_n = _nonneg_int_or_fail(diff_n, "diff_n", strict=False) + if not 0 <= diff_n <= 2: + raise ValueError( + "diff_n is currently only implemented for orders 0, 1, and 2," + f" received: {diff_n}." + ) + return norm, diff_n + + +@assoc_legendre_p._override_ufunc_default_args +def _(branch_cut, norm, diff_n): + return branch_cut, + + +@assoc_legendre_p._override_finalize_out +def _(out): + return np.moveaxis(out, -1, 0) + + +assoc_legendre_p_all = MultiUFunc( + assoc_legendre_p_all, + """assoc_legendre_p_all(n, m, z, *, branch_cut=2, norm=False, diff_n=0) + + All associated Legendre polynomials of the first kind up to the + specified degree ``n`` and order ``m``. + + Output shape is ``(n + 1, 2 * m + 1, ...)``. The entry at ``(j, i)`` + corresponds to degree ``j`` and order ``i`` for all ``0 <= j <= n`` + and ``-m <= i <= m``. + + See Also + -------- + assoc_legendre_p + """, branch_cut=2, norm=False, diff_n=0 +) + + +@assoc_legendre_p_all._override_key +def _(branch_cut, norm, diff_n): + if not ((isinstance(diff_n, numbers.Integral)) + and diff_n >= 0): + raise ValueError( + f"diff_n must be a non-negative integer, received: {diff_n}." + ) + if not 0 <= diff_n <= 2: + raise ValueError( + "diff_n is currently only implemented for orders 0, 1, and 2," + f" received: {diff_n}." + ) + return norm, diff_n + + +@assoc_legendre_p_all._override_ufunc_default_args +def _(branch_cut, norm, diff_n): + return branch_cut, + + +@assoc_legendre_p_all._override_ufunc_default_kwargs +def _(branch_cut, norm, diff_n): + return {'axes': [(), ()] + [(0, 1, -1)]} + + +@assoc_legendre_p_all._override_resolve_out_shapes +def _(n, m, z_shape, branch_cut_shape, nout, **kwargs): + diff_n = kwargs['diff_n'] + + if not isinstance(n, numbers.Integral) or (n < 0): + raise ValueError("n must be a non-negative integer.") + if not isinstance(m, numbers.Integral) or (m < 0): + raise ValueError("m must be a non-negative integer.") + + return ((n + 1, 2 * abs(m) + 1) + + np.broadcast_shapes(z_shape, branch_cut_shape) + (diff_n + 1,),) + + +@assoc_legendre_p_all._override_finalize_out +def _(out): + return np.moveaxis(out, -1, 0) + + +legendre_p = MultiUFunc( + legendre_p, + """legendre_p(n, z, *, diff_n=0) + + Legendre polynomial of the first kind. + + Parameters + ---------- + n : ArrayLike[int] + Degree of the Legendre polynomial. Must have ``n >= 0``. + z : ArrayLike[float] + Input value. + diff_n : Optional[int] + A non-negative integer. Compute and return all derivatives up + to order ``diff_n``. Default is 0. + + Returns + ------- + p : ndarray or tuple[ndarray] + Legendre polynomial with ``diff_n`` derivatives. + + See Also + -------- + legendre + + References + ---------- + .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special + Functions", John Wiley and Sons, 1996. + https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html + """, diff_n=0 +) + + +@legendre_p._override_key +def _(diff_n): + if (not isinstance(diff_n, numbers.Integral)) or (diff_n < 0): + raise ValueError( + f"diff_n must be a non-negative integer, received: {diff_n}." + ) + if not 0 <= diff_n <= 2: + raise NotImplementedError( + "diff_n is currently only implemented for orders 0, 1, and 2," + f" received: {diff_n}." + ) + return diff_n + + +@legendre_p._override_finalize_out +def _(out): + return np.moveaxis(out, -1, 0) + + +legendre_p_all = MultiUFunc( + legendre_p_all, + """legendre_p_all(n, z, *, diff_n=0) + + All Legendre polynomials of the first kind up to the + specified degree ``n``. + + Output shape is ``(n + 1, ...)``. The entry at ``j`` + corresponds to degree ``j`` for all ``0 <= j <= n``. + + See Also + -------- + legendre_p + """, diff_n=0 +) + + +@legendre_p_all._override_key +def _(diff_n): + diff_n = _nonneg_int_or_fail(diff_n, "diff_n", strict=False) + if not 0 <= diff_n <= 2: + raise ValueError( + "diff_n is currently only implemented for orders 0, 1, and 2," + f" received: {diff_n}." + ) + return diff_n + + +@legendre_p_all._override_ufunc_default_kwargs +def _(diff_n): + return {'axes': [(), (0, -1)]} + + +@legendre_p_all._override_resolve_out_shapes +def _(n, z_shape, nout, diff_n): + n = _nonneg_int_or_fail(n, 'n', strict=False) + + return nout * ((n + 1,) + z_shape + (diff_n + 1,),) + + +@legendre_p_all._override_finalize_out +def _(out): + return np.moveaxis(out, -1, 0) + + +sph_harm_y = MultiUFunc( + sph_harm_y, + r"""sph_harm_y(n, m, theta, phi, *, diff_n=0) + + Spherical harmonics. They are defined as + + .. math:: + + Y_n^m(\theta,\phi) = \sqrt{\frac{2 n + 1}{4 \pi} \frac{(n - m)!}{(n + m)!}} + P_n^m(\cos(\theta)) e^{i m \phi} + + where :math:`P_n^m` are the (unnormalized) associated Legendre polynomials. + + Parameters + ---------- + n : ArrayLike[int] + Degree of the harmonic. Must have ``n >= 0``. This is + often denoted by ``l`` (lower case L) in descriptions of + spherical harmonics. + m : ArrayLike[int] + Order of the harmonic. + theta : ArrayLike[float] + Polar (colatitudinal) coordinate; must be in ``[0, pi]``. + phi : ArrayLike[float] + Azimuthal (longitudinal) coordinate; must be in ``[0, 2*pi]``. + diff_n : Optional[int] + A non-negative integer. Compute and return all derivatives up + to order ``diff_n``. Default is 0. + + Returns + ------- + y : ndarray[complex] or tuple[ndarray[complex]] + Spherical harmonics with ``diff_n`` derivatives. + + Notes + ----- + There are different conventions for the meanings of the input + arguments ``theta`` and ``phi``. In SciPy ``theta`` is the + polar angle and ``phi`` is the azimuthal angle. It is common to + see the opposite convention, that is, ``theta`` as the azimuthal angle + and ``phi`` as the polar angle. + + Note that SciPy's spherical harmonics include the Condon-Shortley + phase [2]_ because it is part of `sph_legendre_p`. + + With SciPy's conventions, the first several spherical harmonics + are + + .. math:: + + Y_0^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{1}{\pi}} \\ + Y_1^{-1}(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{2\pi}} + e^{-i\phi} \sin(\theta) \\ + Y_1^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{\pi}} + \cos(\theta) \\ + Y_1^1(\theta, \phi) &= -\frac{1}{2} \sqrt{\frac{3}{2\pi}} + e^{i\phi} \sin(\theta). + + References + ---------- + .. [1] Digital Library of Mathematical Functions, 14.30. + https://dlmf.nist.gov/14.30 + .. [2] https://en.wikipedia.org/wiki/Spherical_harmonics#Condon.E2.80.93Shortley_phase + """, force_complex_output=True, diff_n=0 +) + + +@sph_harm_y._override_key +def _(diff_n): + diff_n = _nonneg_int_or_fail(diff_n, "diff_n", strict=False) + if not 0 <= diff_n <= 2: + raise ValueError( + "diff_n is currently only implemented for orders 0, 1, and 2," + f" received: {diff_n}." + ) + return diff_n + + +@sph_harm_y._override_finalize_out +def _(out): + if (out.shape[-1] == 1): + return out[..., 0, 0] + + if (out.shape[-1] == 2): + return out[..., 0, 0], out[..., [1, 0], [0, 1]] + + if (out.shape[-1] == 3): + return (out[..., 0, 0], out[..., [1, 0], [0, 1]], + out[..., [[2, 1], [1, 0]], [[0, 1], [1, 2]]]) + + +sph_harm_y_all = MultiUFunc( + sph_harm_y_all, + """sph_harm_y_all(n, m, theta, phi, *, diff_n=0) + + All spherical harmonics up to the specified degree ``n`` and order ``m``. + + Output shape is ``(n + 1, 2 * m + 1, ...)``. The entry at ``(j, i)`` + corresponds to degree ``j`` and order ``i`` for all ``0 <= j <= n`` + and ``-m <= i <= m``. + + See Also + -------- + sph_harm_y + """, force_complex_output=True, diff_n=0 +) + + +@sph_harm_y_all._override_key +def _(diff_n): + diff_n = _nonneg_int_or_fail(diff_n, "diff_n", strict=False) + if not 0 <= diff_n <= 2: + raise ValueError( + "diff_n is currently only implemented for orders 2," + f" received: {diff_n}." + ) + return diff_n + + +@sph_harm_y_all._override_ufunc_default_kwargs +def _(diff_n): + return {'axes': [(), ()] + [(0, 1, -2, -1)]} + + +@sph_harm_y_all._override_resolve_out_shapes +def _(n, m, theta_shape, phi_shape, nout, **kwargs): + diff_n = kwargs['diff_n'] + + if not isinstance(n, numbers.Integral) or (n < 0): + raise ValueError("n must be a non-negative integer.") + + return ((n + 1, 2 * abs(m) + 1) + np.broadcast_shapes(theta_shape, phi_shape) + + (diff_n + 1, diff_n + 1),) + + +@sph_harm_y_all._override_finalize_out +def _(out): + if (out.shape[-1] == 1): + return out[..., 0, 0] + + if (out.shape[-1] == 2): + return out[..., 0, 0], out[..., [1, 0], [0, 1]] + + if (out.shape[-1] == 3): + return (out[..., 0, 0], out[..., [1, 0], [0, 1]], + out[..., [[2, 1], [1, 0]], [[0, 1], [1, 2]]]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_orthogonal.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_orthogonal.py new file mode 100644 index 0000000000000000000000000000000000000000..e021f5a899b2b9218d59527fd91fb6cf7a545042 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_orthogonal.py @@ -0,0 +1,2592 @@ +""" +A collection of functions to find the weights and abscissas for +Gaussian Quadrature. + +These calculations are done by finding the eigenvalues of a +tridiagonal matrix whose entries are dependent on the coefficients +in the recursion formula for the orthogonal polynomials with the +corresponding weighting function over the interval. + +Many recursion relations for orthogonal polynomials are given: + +.. math:: + + a1n f_{n+1} (x) = (a2n + a3n x ) f_n (x) - a4n f_{n-1} (x) + +The recursion relation of interest is + +.. math:: + + P_{n+1} (x) = (x - A_n) P_n (x) - B_n P_{n-1} (x) + +where :math:`P` has a different normalization than :math:`f`. + +The coefficients can be found as: + +.. math:: + + A_n = -a2n / a3n + \\qquad + B_n = ( a4n / a3n \\sqrt{h_n-1 / h_n})^2 + +where + +.. math:: + + h_n = \\int_a^b w(x) f_n(x)^2 + +assume: + +.. math:: + + P_0 (x) = 1 + \\qquad + P_{-1} (x) == 0 + +For the mathematical background, see [golub.welsch-1969-mathcomp]_ and +[abramowitz.stegun-1965]_. + +References +---------- +.. [golub.welsch-1969-mathcomp] + Golub, Gene H, and John H Welsch. 1969. Calculation of Gauss + Quadrature Rules. *Mathematics of Computation* 23, 221-230+s1--s10. + +.. [abramowitz.stegun-1965] + Abramowitz, Milton, and Irene A Stegun. (1965) *Handbook of + Mathematical Functions: with Formulas, Graphs, and Mathematical + Tables*. Gaithersburg, MD: National Bureau of Standards. + http://www.math.sfu.ca/~cbm/aands/ + +.. [townsend.trogdon.olver-2014] + Townsend, A. and Trogdon, T. and Olver, S. (2014) + *Fast computation of Gauss quadrature nodes and + weights on the whole real line*. :arXiv:`1410.5286`. + +.. [townsend.trogdon.olver-2015] + Townsend, A. and Trogdon, T. and Olver, S. (2015) + *Fast computation of Gauss quadrature nodes and + weights on the whole real line*. + IMA Journal of Numerical Analysis + :doi:`10.1093/imanum/drv002`. +""" +# +# Author: Travis Oliphant 2000 +# Updated Sep. 2003 (fixed bugs --- tested to be accurate) + +# SciPy imports. +import numpy as np +from numpy import (exp, inf, pi, sqrt, floor, sin, cos, around, + hstack, arccos, arange) +from scipy import linalg +from scipy.special import airy + +# Local imports. +# There is no .pyi file for _specfun +from . import _specfun # type: ignore +from . import _ufuncs +_gam = _ufuncs.gamma + +_polyfuns = ['legendre', 'chebyt', 'chebyu', 'chebyc', 'chebys', + 'jacobi', 'laguerre', 'genlaguerre', 'hermite', + 'hermitenorm', 'gegenbauer', 'sh_legendre', 'sh_chebyt', + 'sh_chebyu', 'sh_jacobi'] + +# Correspondence between new and old names of root functions +_rootfuns_map = {'roots_legendre': 'p_roots', + 'roots_chebyt': 't_roots', + 'roots_chebyu': 'u_roots', + 'roots_chebyc': 'c_roots', + 'roots_chebys': 's_roots', + 'roots_jacobi': 'j_roots', + 'roots_laguerre': 'l_roots', + 'roots_genlaguerre': 'la_roots', + 'roots_hermite': 'h_roots', + 'roots_hermitenorm': 'he_roots', + 'roots_gegenbauer': 'cg_roots', + 'roots_sh_legendre': 'ps_roots', + 'roots_sh_chebyt': 'ts_roots', + 'roots_sh_chebyu': 'us_roots', + 'roots_sh_jacobi': 'js_roots'} + +__all__ = _polyfuns + list(_rootfuns_map.keys()) + + +class orthopoly1d(np.poly1d): + + def __init__(self, roots, weights=None, hn=1.0, kn=1.0, wfunc=None, + limits=None, monic=False, eval_func=None): + equiv_weights = [weights[k] / wfunc(roots[k]) for + k in range(len(roots))] + mu = sqrt(hn) + if monic: + evf = eval_func + if evf: + knn = kn + def eval_func(x): + return evf(x) / knn + mu = mu / abs(kn) + kn = 1.0 + + # compute coefficients from roots, then scale + poly = np.poly1d(roots, r=True) + np.poly1d.__init__(self, poly.coeffs * float(kn)) + + self.weights = np.array(list(zip(roots, weights, equiv_weights))) + self.weight_func = wfunc + self.limits = limits + self.normcoef = mu + + # Note: eval_func will be discarded on arithmetic + self._eval_func = eval_func + + def __call__(self, v): + if self._eval_func and not isinstance(v, np.poly1d): + return self._eval_func(v) + else: + return np.poly1d.__call__(self, v) + + def _scale(self, p): + if p == 1.0: + return + self._coeffs *= p + + evf = self._eval_func + if evf: + self._eval_func = lambda x: evf(x) * p + self.normcoef *= p + + +def _gen_roots_and_weights(n, mu0, an_func, bn_func, f, df, symmetrize, mu): + """[x,w] = gen_roots_and_weights(n,an_func,sqrt_bn_func,mu) + + Returns the roots (x) of an nth order orthogonal polynomial, + and weights (w) to use in appropriate Gaussian quadrature with that + orthogonal polynomial. + + The polynomials have the recurrence relation + P_n+1(x) = (x - A_n) P_n(x) - B_n P_n-1(x) + + an_func(n) should return A_n + sqrt_bn_func(n) should return sqrt(B_n) + mu ( = h_0 ) is the integral of the weight over the orthogonal + interval + """ + k = np.arange(n, dtype='d') + c = np.zeros((2, n)) + c[0,1:] = bn_func(k[1:]) + c[1,:] = an_func(k) + x = linalg.eigvals_banded(c, overwrite_a_band=True) + + # improve roots by one application of Newton's method + y = f(n, x) + dy = df(n, x) + x -= y/dy + + # fm and dy may contain very large/small values, so we + # log-normalize them to maintain precision in the product fm*dy + fm = f(n-1, x) + log_fm = np.log(np.abs(fm)) + log_dy = np.log(np.abs(dy)) + fm /= np.exp((log_fm.max() + log_fm.min()) / 2.) + dy /= np.exp((log_dy.max() + log_dy.min()) / 2.) + w = 1.0 / (fm * dy) + + if symmetrize: + w = (w + w[::-1]) / 2 + x = (x - x[::-1]) / 2 + + w *= mu0 / w.sum() + + if mu: + return x, w, mu0 + else: + return x, w + +# Jacobi Polynomials 1 P^(alpha,beta)_n(x) + + +def roots_jacobi(n, alpha, beta, mu=False): + r"""Gauss-Jacobi quadrature. + + Compute the sample points and weights for Gauss-Jacobi + quadrature. The sample points are the roots of the nth degree + Jacobi polynomial, :math:`P^{\alpha, \beta}_n(x)`. These sample + points and weights correctly integrate polynomials of degree + :math:`2n - 1` or less over the interval :math:`[-1, 1]` with + weight function :math:`w(x) = (1 - x)^{\alpha} (1 + + x)^{\beta}`. See 22.2.1 in [AS]_ for details. + + Parameters + ---------- + n : int + quadrature order + alpha : float + alpha must be > -1 + beta : float + beta must be > -1 + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + m = int(n) + if n < 1 or n != m: + raise ValueError("n must be a positive integer.") + if alpha <= -1 or beta <= -1: + raise ValueError("alpha and beta must be greater than -1.") + + if alpha == 0.0 and beta == 0.0: + return roots_legendre(m, mu) + if alpha == beta: + return roots_gegenbauer(m, alpha+0.5, mu) + + if (alpha + beta) <= 1000: + mu0 = 2.0**(alpha+beta+1) * _ufuncs.beta(alpha+1, beta+1) + else: + # Avoid overflows in pow and beta for very large parameters + mu0 = np.exp((alpha + beta + 1) * np.log(2.0) + + _ufuncs.betaln(alpha+1, beta+1)) + a = alpha + b = beta + if a + b == 0.0: + def an_func(k): + return np.where(k == 0, (b - a) / (2 + a + b), 0.0) + else: + def an_func(k): + return np.where( + k == 0, + (b - a) / (2 + a + b), + (b * b - a * a) / ((2.0 * k + a + b) * (2.0 * k + a + b + 2)) + ) + + def bn_func(k): + return ( + 2.0 / (2.0 * k + a + b) + * np.sqrt((k + a) * (k + b) / (2 * k + a + b + 1)) + * np.where(k == 1, 1.0, np.sqrt(k * (k + a + b) / (2.0 * k + a + b - 1))) + ) + + def f(n, x): + return _ufuncs.eval_jacobi(n, a, b, x) + def df(n, x): + return 0.5 * (n + a + b + 1) * _ufuncs.eval_jacobi(n - 1, a + 1, b + 1, x) + return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, False, mu) + + +def jacobi(n, alpha, beta, monic=False): + r"""Jacobi polynomial. + + Defined to be the solution of + + .. math:: + (1 - x^2)\frac{d^2}{dx^2}P_n^{(\alpha, \beta)} + + (\beta - \alpha - (\alpha + \beta + 2)x) + \frac{d}{dx}P_n^{(\alpha, \beta)} + + n(n + \alpha + \beta + 1)P_n^{(\alpha, \beta)} = 0 + + for :math:`\alpha, \beta > -1`; :math:`P_n^{(\alpha, \beta)}` is a + polynomial of degree :math:`n`. + + Parameters + ---------- + n : int + Degree of the polynomial. + alpha : float + Parameter, must be greater than -1. + beta : float + Parameter, must be greater than -1. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + P : orthopoly1d + Jacobi polynomial. + + Notes + ----- + For fixed :math:`\alpha, \beta`, the polynomials + :math:`P_n^{(\alpha, \beta)}` are orthogonal over :math:`[-1, 1]` + with weight function :math:`(1 - x)^\alpha(1 + x)^\beta`. + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + The Jacobi polynomials satisfy the recurrence relation: + + .. math:: + P_n^{(\alpha, \beta-1)}(x) - P_n^{(\alpha-1, \beta)}(x) + = P_{n-1}^{(\alpha, \beta)}(x) + + This can be verified, for example, for :math:`\alpha = \beta = 2` + and :math:`n = 1` over the interval :math:`[-1, 1]`: + + >>> import numpy as np + >>> from scipy.special import jacobi + >>> x = np.arange(-1.0, 1.0, 0.01) + >>> np.allclose(jacobi(0, 2, 2)(x), + ... jacobi(1, 2, 1)(x) - jacobi(1, 1, 2)(x)) + True + + Plot of the Jacobi polynomial :math:`P_5^{(\alpha, -0.5)}` for + different values of :math:`\alpha`: + + >>> import matplotlib.pyplot as plt + >>> x = np.arange(-1.0, 1.0, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(-2.0, 2.0) + >>> ax.set_title(r'Jacobi polynomials $P_5^{(\alpha, -0.5)}$') + >>> for alpha in np.arange(0, 4, 1): + ... ax.plot(x, jacobi(5, alpha, -0.5)(x), label=rf'$\alpha={alpha}$') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + if n < 0: + raise ValueError("n must be nonnegative.") + + def wfunc(x): + return (1 - x) ** alpha * (1 + x) ** beta + if n == 0: + return orthopoly1d([], [], 1.0, 1.0, wfunc, (-1, 1), monic, + eval_func=np.ones_like) + x, w, mu = roots_jacobi(n, alpha, beta, mu=True) + ab1 = alpha + beta + 1.0 + hn = 2**ab1 / (2 * n + ab1) * _gam(n + alpha + 1) + hn *= _gam(n + beta + 1.0) / _gam(n + 1) / _gam(n + ab1) + kn = _gam(2 * n + ab1) / 2.0**n / _gam(n + 1) / _gam(n + ab1) + # here kn = coefficient on x^n term + p = orthopoly1d(x, w, hn, kn, wfunc, (-1, 1), monic, + lambda x: _ufuncs.eval_jacobi(n, alpha, beta, x)) + return p + +# Jacobi Polynomials shifted G_n(p,q,x) + + +def roots_sh_jacobi(n, p1, q1, mu=False): + """Gauss-Jacobi (shifted) quadrature. + + Compute the sample points and weights for Gauss-Jacobi (shifted) + quadrature. The sample points are the roots of the nth degree + shifted Jacobi polynomial, :math:`G^{p,q}_n(x)`. These sample + points and weights correctly integrate polynomials of degree + :math:`2n - 1` or less over the interval :math:`[0, 1]` with + weight function :math:`w(x) = (1 - x)^{p-q} x^{q-1}`. See 22.2.2 + in [AS]_ for details. + + Parameters + ---------- + n : int + quadrature order + p1 : float + (p1 - q1) must be > -1 + q1 : float + q1 must be > 0 + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + if (p1-q1) <= -1 or q1 <= 0: + message = "(p - q) must be greater than -1, and q must be greater than 0." + raise ValueError(message) + x, w, m = roots_jacobi(n, p1-q1, q1-1, True) + x = (x + 1) / 2 + scale = 2.0**p1 + w /= scale + m /= scale + if mu: + return x, w, m + else: + return x, w + + +def sh_jacobi(n, p, q, monic=False): + r"""Shifted Jacobi polynomial. + + Defined by + + .. math:: + + G_n^{(p, q)}(x) + = \binom{2n + p - 1}{n}^{-1}P_n^{(p - q, q - 1)}(2x - 1), + + where :math:`P_n^{(\cdot, \cdot)}` is the nth Jacobi polynomial. + + Parameters + ---------- + n : int + Degree of the polynomial. + p : float + Parameter, must have :math:`p > q - 1`. + q : float + Parameter, must be greater than 0. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + G : orthopoly1d + Shifted Jacobi polynomial. + + Notes + ----- + For fixed :math:`p, q`, the polynomials :math:`G_n^{(p, q)}` are + orthogonal over :math:`[0, 1]` with weight function :math:`(1 - + x)^{p - q}x^{q - 1}`. + + """ + if n < 0: + raise ValueError("n must be nonnegative.") + + def wfunc(x): + return (1.0 - x) ** (p - q) * x ** (q - 1.0) + if n == 0: + return orthopoly1d([], [], 1.0, 1.0, wfunc, (-1, 1), monic, + eval_func=np.ones_like) + n1 = n + x, w = roots_sh_jacobi(n1, p, q) + hn = _gam(n + 1) * _gam(n + q) * _gam(n + p) * _gam(n + p - q + 1) + hn /= (2 * n + p) * (_gam(2 * n + p)**2) + # kn = 1.0 in standard form so monic is redundant. Kept for compatibility. + kn = 1.0 + pp = orthopoly1d(x, w, hn, kn, wfunc=wfunc, limits=(0, 1), monic=monic, + eval_func=lambda x: _ufuncs.eval_sh_jacobi(n, p, q, x)) + return pp + +# Generalized Laguerre L^(alpha)_n(x) + + +def roots_genlaguerre(n, alpha, mu=False): + r"""Gauss-generalized Laguerre quadrature. + + Compute the sample points and weights for Gauss-generalized + Laguerre quadrature. The sample points are the roots of the nth + degree generalized Laguerre polynomial, :math:`L^{\alpha}_n(x)`. + These sample points and weights correctly integrate polynomials of + degree :math:`2n - 1` or less over the interval :math:`[0, + \infty]` with weight function :math:`w(x) = x^{\alpha} + e^{-x}`. See 22.3.9 in [AS]_ for details. + + Parameters + ---------- + n : int + quadrature order + alpha : float + alpha must be > -1 + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + m = int(n) + if n < 1 or n != m: + raise ValueError("n must be a positive integer.") + if alpha < -1: + raise ValueError("alpha must be greater than -1.") + + mu0 = _ufuncs.gamma(alpha + 1) + + if m == 1: + x = np.array([alpha+1.0], 'd') + w = np.array([mu0], 'd') + if mu: + return x, w, mu0 + else: + return x, w + + def an_func(k): + return 2 * k + alpha + 1 + def bn_func(k): + return -np.sqrt(k * (k + alpha)) + def f(n, x): + return _ufuncs.eval_genlaguerre(n, alpha, x) + def df(n, x): + return (n * _ufuncs.eval_genlaguerre(n, alpha, x) + - (n + alpha) * _ufuncs.eval_genlaguerre(n - 1, alpha, x)) / x + return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, False, mu) + + +def genlaguerre(n, alpha, monic=False): + r"""Generalized (associated) Laguerre polynomial. + + Defined to be the solution of + + .. math:: + x\frac{d^2}{dx^2}L_n^{(\alpha)} + + (\alpha + 1 - x)\frac{d}{dx}L_n^{(\alpha)} + + nL_n^{(\alpha)} = 0, + + where :math:`\alpha > -1`; :math:`L_n^{(\alpha)}` is a polynomial + of degree :math:`n`. + + Parameters + ---------- + n : int + Degree of the polynomial. + alpha : float + Parameter, must be greater than -1. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + L : orthopoly1d + Generalized Laguerre polynomial. + + See Also + -------- + laguerre : Laguerre polynomial. + hyp1f1 : confluent hypergeometric function + + Notes + ----- + For fixed :math:`\alpha`, the polynomials :math:`L_n^{(\alpha)}` + are orthogonal over :math:`[0, \infty)` with weight function + :math:`e^{-x}x^\alpha`. + + The Laguerre polynomials are the special case where :math:`\alpha + = 0`. + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + The generalized Laguerre polynomials are closely related to the confluent + hypergeometric function :math:`{}_1F_1`: + + .. math:: + L_n^{(\alpha)} = \binom{n + \alpha}{n} {}_1F_1(-n, \alpha +1, x) + + This can be verified, for example, for :math:`n = \alpha = 3` over the + interval :math:`[-1, 1]`: + + >>> import numpy as np + >>> from scipy.special import binom + >>> from scipy.special import genlaguerre + >>> from scipy.special import hyp1f1 + >>> x = np.arange(-1.0, 1.0, 0.01) + >>> np.allclose(genlaguerre(3, 3)(x), binom(6, 3) * hyp1f1(-3, 4, x)) + True + + This is the plot of the generalized Laguerre polynomials + :math:`L_3^{(\alpha)}` for some values of :math:`\alpha`: + + >>> import matplotlib.pyplot as plt + >>> x = np.arange(-4.0, 12.0, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(-5.0, 10.0) + >>> ax.set_title(r'Generalized Laguerre polynomials $L_3^{\alpha}$') + >>> for alpha in np.arange(0, 5): + ... ax.plot(x, genlaguerre(3, alpha)(x), label=rf'$L_3^{(alpha)}$') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + if alpha <= -1: + raise ValueError("alpha must be > -1") + if n < 0: + raise ValueError("n must be nonnegative.") + + if n == 0: + n1 = n + 1 + else: + n1 = n + x, w = roots_genlaguerre(n1, alpha) + def wfunc(x): + return exp(-x) * x ** alpha + if n == 0: + x, w = [], [] + hn = _gam(n + alpha + 1) / _gam(n + 1) + kn = (-1)**n / _gam(n + 1) + p = orthopoly1d(x, w, hn, kn, wfunc, (0, inf), monic, + lambda x: _ufuncs.eval_genlaguerre(n, alpha, x)) + return p + +# Laguerre L_n(x) + + +def roots_laguerre(n, mu=False): + r"""Gauss-Laguerre quadrature. + + Compute the sample points and weights for Gauss-Laguerre + quadrature. The sample points are the roots of the nth degree + Laguerre polynomial, :math:`L_n(x)`. These sample points and + weights correctly integrate polynomials of degree :math:`2n - 1` + or less over the interval :math:`[0, \infty]` with weight function + :math:`w(x) = e^{-x}`. See 22.2.13 in [AS]_ for details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + numpy.polynomial.laguerre.laggauss + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + return roots_genlaguerre(n, 0.0, mu=mu) + + +def laguerre(n, monic=False): + r"""Laguerre polynomial. + + Defined to be the solution of + + .. math:: + x\frac{d^2}{dx^2}L_n + (1 - x)\frac{d}{dx}L_n + nL_n = 0; + + :math:`L_n` is a polynomial of degree :math:`n`. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + L : orthopoly1d + Laguerre Polynomial. + + See Also + -------- + genlaguerre : Generalized (associated) Laguerre polynomial. + + Notes + ----- + The polynomials :math:`L_n` are orthogonal over :math:`[0, + \infty)` with weight function :math:`e^{-x}`. + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + The Laguerre polynomials :math:`L_n` are the special case + :math:`\alpha = 0` of the generalized Laguerre polynomials + :math:`L_n^{(\alpha)}`. + Let's verify it on the interval :math:`[-1, 1]`: + + >>> import numpy as np + >>> from scipy.special import genlaguerre + >>> from scipy.special import laguerre + >>> x = np.arange(-1.0, 1.0, 0.01) + >>> np.allclose(genlaguerre(3, 0)(x), laguerre(3)(x)) + True + + The polynomials :math:`L_n` also satisfy the recurrence relation: + + .. math:: + (n + 1)L_{n+1}(x) = (2n +1 -x)L_n(x) - nL_{n-1}(x) + + This can be easily checked on :math:`[0, 1]` for :math:`n = 3`: + + >>> x = np.arange(0.0, 1.0, 0.01) + >>> np.allclose(4 * laguerre(4)(x), + ... (7 - x) * laguerre(3)(x) - 3 * laguerre(2)(x)) + True + + This is the plot of the first few Laguerre polynomials :math:`L_n`: + + >>> import matplotlib.pyplot as plt + >>> x = np.arange(-1.0, 5.0, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(-5.0, 5.0) + >>> ax.set_title(r'Laguerre polynomials $L_n$') + >>> for n in np.arange(0, 5): + ... ax.plot(x, laguerre(n)(x), label=rf'$L_{n}$') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + if n < 0: + raise ValueError("n must be nonnegative.") + + if n == 0: + n1 = n + 1 + else: + n1 = n + x, w = roots_laguerre(n1) + if n == 0: + x, w = [], [] + hn = 1.0 + kn = (-1)**n / _gam(n + 1) + p = orthopoly1d(x, w, hn, kn, lambda x: exp(-x), (0, inf), monic, + lambda x: _ufuncs.eval_laguerre(n, x)) + return p + +# Hermite 1 H_n(x) + + +def roots_hermite(n, mu=False): + r"""Gauss-Hermite (physicist's) quadrature. + + Compute the sample points and weights for Gauss-Hermite + quadrature. The sample points are the roots of the nth degree + Hermite polynomial, :math:`H_n(x)`. These sample points and + weights correctly integrate polynomials of degree :math:`2n - 1` + or less over the interval :math:`[-\infty, \infty]` with weight + function :math:`w(x) = e^{-x^2}`. See 22.2.14 in [AS]_ for + details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + numpy.polynomial.hermite.hermgauss + roots_hermitenorm + + Notes + ----- + For small n up to 150 a modified version of the Golub-Welsch + algorithm is used. Nodes are computed from the eigenvalue + problem and improved by one step of a Newton iteration. + The weights are computed from the well-known analytical formula. + + For n larger than 150 an optimal asymptotic algorithm is applied + which computes nodes and weights in a numerically stable manner. + The algorithm has linear runtime making computation for very + large n (several thousand or more) feasible. + + References + ---------- + .. [townsend.trogdon.olver-2014] + Townsend, A. and Trogdon, T. and Olver, S. (2014) + *Fast computation of Gauss quadrature nodes and + weights on the whole real line*. :arXiv:`1410.5286`. + .. [townsend.trogdon.olver-2015] + Townsend, A. and Trogdon, T. and Olver, S. (2015) + *Fast computation of Gauss quadrature nodes and + weights on the whole real line*. + IMA Journal of Numerical Analysis + :doi:`10.1093/imanum/drv002`. + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + m = int(n) + if n < 1 or n != m: + raise ValueError("n must be a positive integer.") + + mu0 = np.sqrt(np.pi) + if n <= 150: + def an_func(k): + return 0.0 * k + def bn_func(k): + return np.sqrt(k / 2.0) + f = _ufuncs.eval_hermite + def df(n, x): + return 2.0 * n * _ufuncs.eval_hermite(n - 1, x) + return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, True, mu) + else: + nodes, weights = _roots_hermite_asy(m) + if mu: + return nodes, weights, mu0 + else: + return nodes, weights + + +def _compute_tauk(n, k, maxit=5): + """Helper function for Tricomi initial guesses + + For details, see formula 3.1 in lemma 3.1 in the + original paper. + + Parameters + ---------- + n : int + Quadrature order + k : ndarray of type int + Index of roots :math:`\tau_k` to compute + maxit : int + Number of Newton maxit performed, the default + value of 5 is sufficient. + + Returns + ------- + tauk : ndarray + Roots of equation 3.1 + + See Also + -------- + initial_nodes_a + roots_hermite_asy + """ + a = n % 2 - 0.5 + c = (4.0*floor(n/2.0) - 4.0*k + 3.0)*pi / (4.0*floor(n/2.0) + 2.0*a + 2.0) + def f(x): + return x - sin(x) - c + def df(x): + return 1.0 - cos(x) + xi = 0.5*pi + for i in range(maxit): + xi = xi - f(xi)/df(xi) + return xi + + +def _initial_nodes_a(n, k): + r"""Tricomi initial guesses + + Computes an initial approximation to the square of the `k`-th + (positive) root :math:`x_k` of the Hermite polynomial :math:`H_n` + of order :math:`n`. The formula is the one from lemma 3.1 in the + original paper. The guesses are accurate except in the region + near :math:`\sqrt{2n + 1}`. + + Parameters + ---------- + n : int + Quadrature order + k : ndarray of type int + Index of roots to compute + + Returns + ------- + xksq : ndarray + Square of the approximate roots + + See Also + -------- + initial_nodes + roots_hermite_asy + """ + tauk = _compute_tauk(n, k) + sigk = cos(0.5*tauk)**2 + a = n % 2 - 0.5 + nu = 4.0*floor(n/2.0) + 2.0*a + 2.0 + # Initial approximation of Hermite roots (square) + xksq = nu*sigk - 1.0/(3.0*nu) * (5.0/(4.0*(1.0-sigk)**2) - 1.0/(1.0-sigk) - 0.25) + return xksq + + +def _initial_nodes_b(n, k): + r"""Gatteschi initial guesses + + Computes an initial approximation to the square of the kth + (positive) root :math:`x_k` of the Hermite polynomial :math:`H_n` + of order :math:`n`. The formula is the one from lemma 3.2 in the + original paper. The guesses are accurate in the region just + below :math:`\sqrt{2n + 1}`. + + Parameters + ---------- + n : int + Quadrature order + k : ndarray of type int + Index of roots to compute + + Returns + ------- + xksq : ndarray + Square of the approximate root + + See Also + -------- + initial_nodes + roots_hermite_asy + """ + a = n % 2 - 0.5 + nu = 4.0*floor(n/2.0) + 2.0*a + 2.0 + # Airy roots by approximation + ak = _specfun.airyzo(k.max(), 1)[0][::-1] + # Initial approximation of Hermite roots (square) + xksq = (nu + + 2.0**(2.0/3.0) * ak * nu**(1.0/3.0) + + 1.0/5.0 * 2.0**(4.0/3.0) * ak**2 * nu**(-1.0/3.0) + + (9.0/140.0 - 12.0/175.0 * ak**3) * nu**(-1.0) + + (16.0/1575.0 * ak + 92.0/7875.0 * ak**4) * 2.0**(2.0/3.0) * nu**(-5.0/3.0) + - (15152.0/3031875.0 * ak**5 + 1088.0/121275.0 * ak**2) + * 2.0**(1.0/3.0) * nu**(-7.0/3.0)) + return xksq + + +def _initial_nodes(n): + """Initial guesses for the Hermite roots + + Computes an initial approximation to the non-negative + roots :math:`x_k` of the Hermite polynomial :math:`H_n` + of order :math:`n`. The Tricomi and Gatteschi initial + guesses are used in the region where they are accurate. + + Parameters + ---------- + n : int + Quadrature order + + Returns + ------- + xk : ndarray + Approximate roots + + See Also + -------- + roots_hermite_asy + """ + # Turnover point + # linear polynomial fit to error of 10, 25, 40, ..., 1000 point rules + fit = 0.49082003*n - 4.37859653 + turnover = around(fit).astype(int) + # Compute all approximations + ia = arange(1, int(floor(n*0.5)+1)) + ib = ia[::-1] + xasq = _initial_nodes_a(n, ia[:turnover+1]) + xbsq = _initial_nodes_b(n, ib[turnover+1:]) + # Combine + iv = sqrt(hstack([xasq, xbsq])) + # Central node is always zero + if n % 2 == 1: + iv = hstack([0.0, iv]) + return iv + + +def _pbcf(n, theta): + r"""Asymptotic series expansion of parabolic cylinder function + + The implementation is based on sections 3.2 and 3.3 from the + original paper. Compared to the published version this code + adds one more term to the asymptotic series. The detailed + formulas can be found at [parabolic-asymptotics]_. The evaluation + is done in a transformed variable :math:`\theta := \arccos(t)` + where :math:`t := x / \mu` and :math:`\mu := \sqrt{2n + 1}`. + + Parameters + ---------- + n : int + Quadrature order + theta : ndarray + Transformed position variable + + Returns + ------- + U : ndarray + Value of the parabolic cylinder function :math:`U(a, \theta)`. + Ud : ndarray + Value of the derivative :math:`U^{\prime}(a, \theta)` of + the parabolic cylinder function. + + See Also + -------- + roots_hermite_asy + + References + ---------- + .. [parabolic-asymptotics] + https://dlmf.nist.gov/12.10#vii + """ + st = sin(theta) + ct = cos(theta) + # https://dlmf.nist.gov/12.10#vii + mu = 2.0*n + 1.0 + # https://dlmf.nist.gov/12.10#E23 + eta = 0.5*theta - 0.5*st*ct + # https://dlmf.nist.gov/12.10#E39 + zeta = -(3.0*eta/2.0) ** (2.0/3.0) + # https://dlmf.nist.gov/12.10#E40 + phi = (-zeta / st**2) ** (0.25) + # Coefficients + # https://dlmf.nist.gov/12.10#E43 + a0 = 1.0 + a1 = 0.10416666666666666667 + a2 = 0.08355034722222222222 + a3 = 0.12822657455632716049 + a4 = 0.29184902646414046425 + a5 = 0.88162726744375765242 + b0 = 1.0 + b1 = -0.14583333333333333333 + b2 = -0.09874131944444444444 + b3 = -0.14331205391589506173 + b4 = -0.31722720267841354810 + b5 = -0.94242914795712024914 + # Polynomials + # https://dlmf.nist.gov/12.10#E9 + # https://dlmf.nist.gov/12.10#E10 + ctp = ct ** arange(16).reshape((-1,1)) + u0 = 1.0 + u1 = (1.0*ctp[3,:] - 6.0*ct) / 24.0 + u2 = (-9.0*ctp[4,:] + 249.0*ctp[2,:] + 145.0) / 1152.0 + u3 = (-4042.0*ctp[9,:] + 18189.0*ctp[7,:] - 28287.0*ctp[5,:] + - 151995.0*ctp[3,:] - 259290.0*ct) / 414720.0 + u4 = (72756.0*ctp[10,:] - 321339.0*ctp[8,:] - 154982.0*ctp[6,:] + + 50938215.0*ctp[4,:] + 122602962.0*ctp[2,:] + 12773113.0) / 39813120.0 + u5 = (82393456.0*ctp[15,:] - 617950920.0*ctp[13,:] + 1994971575.0*ctp[11,:] + - 3630137104.0*ctp[9,:] + 4433574213.0*ctp[7,:] - 37370295816.0*ctp[5,:] + - 119582875013.0*ctp[3,:] - 34009066266.0*ct) / 6688604160.0 + v0 = 1.0 + v1 = (1.0*ctp[3,:] + 6.0*ct) / 24.0 + v2 = (15.0*ctp[4,:] - 327.0*ctp[2,:] - 143.0) / 1152.0 + v3 = (-4042.0*ctp[9,:] + 18189.0*ctp[7,:] - 36387.0*ctp[5,:] + + 238425.0*ctp[3,:] + 259290.0*ct) / 414720.0 + v4 = (-121260.0*ctp[10,:] + 551733.0*ctp[8,:] - 151958.0*ctp[6,:] + - 57484425.0*ctp[4,:] - 132752238.0*ctp[2,:] - 12118727) / 39813120.0 + v5 = (82393456.0*ctp[15,:] - 617950920.0*ctp[13,:] + 2025529095.0*ctp[11,:] + - 3750839308.0*ctp[9,:] + 3832454253.0*ctp[7,:] + 35213253348.0*ctp[5,:] + + 130919230435.0*ctp[3,:] + 34009066266*ct) / 6688604160.0 + # Airy Evaluation (Bi and Bip unused) + Ai, Aip, Bi, Bip = airy(mu**(4.0/6.0) * zeta) + # Prefactor for U + P = 2.0*sqrt(pi) * mu**(1.0/6.0) * phi + # Terms for U + # https://dlmf.nist.gov/12.10#E42 + phip = phi ** arange(6, 31, 6).reshape((-1,1)) + A0 = b0*u0 + A1 = (b2*u0 + phip[0,:]*b1*u1 + phip[1,:]*b0*u2) / zeta**3 + A2 = (b4*u0 + phip[0,:]*b3*u1 + phip[1,:]*b2*u2 + phip[2,:]*b1*u3 + + phip[3,:]*b0*u4) / zeta**6 + B0 = -(a1*u0 + phip[0,:]*a0*u1) / zeta**2 + B1 = -(a3*u0 + phip[0,:]*a2*u1 + phip[1,:]*a1*u2 + phip[2,:]*a0*u3) / zeta**5 + B2 = -(a5*u0 + phip[0,:]*a4*u1 + phip[1,:]*a3*u2 + phip[2,:]*a2*u3 + + phip[3,:]*a1*u4 + phip[4,:]*a0*u5) / zeta**8 + # U + # https://dlmf.nist.gov/12.10#E35 + U = P * (Ai * (A0 + A1/mu**2.0 + A2/mu**4.0) + + Aip * (B0 + B1/mu**2.0 + B2/mu**4.0) / mu**(8.0/6.0)) + # Prefactor for derivative of U + Pd = sqrt(2.0*pi) * mu**(2.0/6.0) / phi + # Terms for derivative of U + # https://dlmf.nist.gov/12.10#E46 + C0 = -(b1*v0 + phip[0,:]*b0*v1) / zeta + C1 = -(b3*v0 + phip[0,:]*b2*v1 + phip[1,:]*b1*v2 + phip[2,:]*b0*v3) / zeta**4 + C2 = -(b5*v0 + phip[0,:]*b4*v1 + phip[1,:]*b3*v2 + phip[2,:]*b2*v3 + + phip[3,:]*b1*v4 + phip[4,:]*b0*v5) / zeta**7 + D0 = a0*v0 + D1 = (a2*v0 + phip[0,:]*a1*v1 + phip[1,:]*a0*v2) / zeta**3 + D2 = (a4*v0 + phip[0,:]*a3*v1 + phip[1,:]*a2*v2 + phip[2,:]*a1*v3 + + phip[3,:]*a0*v4) / zeta**6 + # Derivative of U + # https://dlmf.nist.gov/12.10#E36 + Ud = Pd * (Ai * (C0 + C1/mu**2.0 + C2/mu**4.0) / mu**(4.0/6.0) + + Aip * (D0 + D1/mu**2.0 + D2/mu**4.0)) + return U, Ud + + +def _newton(n, x_initial, maxit=5): + """Newton iteration for polishing the asymptotic approximation + to the zeros of the Hermite polynomials. + + Parameters + ---------- + n : int + Quadrature order + x_initial : ndarray + Initial guesses for the roots + maxit : int + Maximal number of Newton iterations. + The default 5 is sufficient, usually + only one or two steps are needed. + + Returns + ------- + nodes : ndarray + Quadrature nodes + weights : ndarray + Quadrature weights + + See Also + -------- + roots_hermite_asy + """ + # Variable transformation + mu = sqrt(2.0*n + 1.0) + t = x_initial / mu + theta = arccos(t) + # Newton iteration + for i in range(maxit): + u, ud = _pbcf(n, theta) + dtheta = u / (sqrt(2.0) * mu * sin(theta) * ud) + theta = theta + dtheta + if max(abs(dtheta)) < 1e-14: + break + # Undo variable transformation + x = mu * cos(theta) + # Central node is always zero + if n % 2 == 1: + x[0] = 0.0 + # Compute weights + w = exp(-x**2) / (2.0*ud**2) + return x, w + + +def _roots_hermite_asy(n): + r"""Gauss-Hermite (physicist's) quadrature for large n. + + Computes the sample points and weights for Gauss-Hermite quadrature. + The sample points are the roots of the nth degree Hermite polynomial, + :math:`H_n(x)`. These sample points and weights correctly integrate + polynomials of degree :math:`2n - 1` or less over the interval + :math:`[-\infty, \infty]` with weight function :math:`f(x) = e^{-x^2}`. + + This method relies on asymptotic expansions which work best for n > 150. + The algorithm has linear runtime making computation for very large n + feasible. + + Parameters + ---------- + n : int + quadrature order + + Returns + ------- + nodes : ndarray + Quadrature nodes + weights : ndarray + Quadrature weights + + See Also + -------- + roots_hermite + + References + ---------- + .. [townsend.trogdon.olver-2014] + Townsend, A. and Trogdon, T. and Olver, S. (2014) + *Fast computation of Gauss quadrature nodes and + weights on the whole real line*. :arXiv:`1410.5286`. + + .. [townsend.trogdon.olver-2015] + Townsend, A. and Trogdon, T. and Olver, S. (2015) + *Fast computation of Gauss quadrature nodes and + weights on the whole real line*. + IMA Journal of Numerical Analysis + :doi:`10.1093/imanum/drv002`. + """ + iv = _initial_nodes(n) + nodes, weights = _newton(n, iv) + # Combine with negative parts + if n % 2 == 0: + nodes = hstack([-nodes[::-1], nodes]) + weights = hstack([weights[::-1], weights]) + else: + nodes = hstack([-nodes[-1:0:-1], nodes]) + weights = hstack([weights[-1:0:-1], weights]) + # Scale weights + weights *= sqrt(pi) / sum(weights) + return nodes, weights + + +def hermite(n, monic=False): + r"""Physicist's Hermite polynomial. + + Defined by + + .. math:: + + H_n(x) = (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}; + + :math:`H_n` is a polynomial of degree :math:`n`. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + H : orthopoly1d + Hermite polynomial. + + Notes + ----- + The polynomials :math:`H_n` are orthogonal over :math:`(-\infty, + \infty)` with weight function :math:`e^{-x^2}`. + + Examples + -------- + >>> from scipy import special + >>> import matplotlib.pyplot as plt + >>> import numpy as np + + >>> p_monic = special.hermite(3, monic=True) + >>> p_monic + poly1d([ 1. , 0. , -1.5, 0. ]) + >>> p_monic(1) + -0.49999999999999983 + >>> x = np.linspace(-3, 3, 400) + >>> y = p_monic(x) + >>> plt.plot(x, y) + >>> plt.title("Monic Hermite polynomial of degree 3") + >>> plt.xlabel("x") + >>> plt.ylabel("H_3(x)") + >>> plt.show() + + """ + if n < 0: + raise ValueError("n must be nonnegative.") + + if n == 0: + n1 = n + 1 + else: + n1 = n + x, w = roots_hermite(n1) + def wfunc(x): + return exp(-x * x) + if n == 0: + x, w = [], [] + hn = 2**n * _gam(n + 1) * sqrt(pi) + kn = 2**n + p = orthopoly1d(x, w, hn, kn, wfunc, (-inf, inf), monic, + lambda x: _ufuncs.eval_hermite(n, x)) + return p + +# Hermite 2 He_n(x) + + +def roots_hermitenorm(n, mu=False): + r"""Gauss-Hermite (statistician's) quadrature. + + Compute the sample points and weights for Gauss-Hermite + quadrature. The sample points are the roots of the nth degree + Hermite polynomial, :math:`He_n(x)`. These sample points and + weights correctly integrate polynomials of degree :math:`2n - 1` + or less over the interval :math:`[-\infty, \infty]` with weight + function :math:`w(x) = e^{-x^2/2}`. See 22.2.15 in [AS]_ for more + details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + numpy.polynomial.hermite_e.hermegauss + + Notes + ----- + For small n up to 150 a modified version of the Golub-Welsch + algorithm is used. Nodes are computed from the eigenvalue + problem and improved by one step of a Newton iteration. + The weights are computed from the well-known analytical formula. + + For n larger than 150 an optimal asymptotic algorithm is used + which computes nodes and weights in a numerical stable manner. + The algorithm has linear runtime making computation for very + large n (several thousand or more) feasible. + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + m = int(n) + if n < 1 or n != m: + raise ValueError("n must be a positive integer.") + + mu0 = np.sqrt(2.0*np.pi) + if n <= 150: + def an_func(k): + return 0.0 * k + def bn_func(k): + return np.sqrt(k) + f = _ufuncs.eval_hermitenorm + def df(n, x): + return n * _ufuncs.eval_hermitenorm(n - 1, x) + return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, True, mu) + else: + nodes, weights = _roots_hermite_asy(m) + # Transform + nodes *= sqrt(2) + weights *= sqrt(2) + if mu: + return nodes, weights, mu0 + else: + return nodes, weights + + +def hermitenorm(n, monic=False): + r"""Normalized (probabilist's) Hermite polynomial. + + Defined by + + .. math:: + + He_n(x) = (-1)^ne^{x^2/2}\frac{d^n}{dx^n}e^{-x^2/2}; + + :math:`He_n` is a polynomial of degree :math:`n`. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + He : orthopoly1d + Hermite polynomial. + + Notes + ----- + + The polynomials :math:`He_n` are orthogonal over :math:`(-\infty, + \infty)` with weight function :math:`e^{-x^2/2}`. + + """ + if n < 0: + raise ValueError("n must be nonnegative.") + + if n == 0: + n1 = n + 1 + else: + n1 = n + x, w = roots_hermitenorm(n1) + def wfunc(x): + return exp(-x * x / 2.0) + if n == 0: + x, w = [], [] + hn = sqrt(2 * pi) * _gam(n + 1) + kn = 1.0 + p = orthopoly1d(x, w, hn, kn, wfunc=wfunc, limits=(-inf, inf), monic=monic, + eval_func=lambda x: _ufuncs.eval_hermitenorm(n, x)) + return p + +# The remainder of the polynomials can be derived from the ones above. + +# Ultraspherical (Gegenbauer) C^(alpha)_n(x) + + +def roots_gegenbauer(n, alpha, mu=False): + r"""Gauss-Gegenbauer quadrature. + + Compute the sample points and weights for Gauss-Gegenbauer + quadrature. The sample points are the roots of the nth degree + Gegenbauer polynomial, :math:`C^{\alpha}_n(x)`. These sample + points and weights correctly integrate polynomials of degree + :math:`2n - 1` or less over the interval :math:`[-1, 1]` with + weight function :math:`w(x) = (1 - x^2)^{\alpha - 1/2}`. See + 22.2.3 in [AS]_ for more details. + + Parameters + ---------- + n : int + quadrature order + alpha : float + alpha must be > -0.5 + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + m = int(n) + if n < 1 or n != m: + raise ValueError("n must be a positive integer.") + if alpha < -0.5: + raise ValueError("alpha must be greater than -0.5.") + elif alpha == 0.0: + # C(n,0,x) == 0 uniformly, however, as alpha->0, C(n,alpha,x)->T(n,x) + # strictly, we should just error out here, since the roots are not + # really defined, but we used to return something useful, so let's + # keep doing so. + return roots_chebyt(n, mu) + + if alpha <= 170: + mu0 = (np.sqrt(np.pi) * _ufuncs.gamma(alpha + 0.5)) \ + / _ufuncs.gamma(alpha + 1) + else: + # For large alpha we use a Taylor series expansion around inf, + # expressed as a 6th order polynomial of a^-1 and using Horner's + # method to minimize computation and maximize precision + inv_alpha = 1. / alpha + coeffs = np.array([0.000207186, -0.00152206, -0.000640869, + 0.00488281, 0.0078125, -0.125, 1.]) + mu0 = coeffs[0] + for term in range(1, len(coeffs)): + mu0 = mu0 * inv_alpha + coeffs[term] + mu0 = mu0 * np.sqrt(np.pi / alpha) + def an_func(k): + return 0.0 * k + def bn_func(k): + return np.sqrt(k * (k + 2 * alpha - 1) / (4 * (k + alpha) * (k + alpha - 1))) + def f(n, x): + return _ufuncs.eval_gegenbauer(n, alpha, x) + def df(n, x): + return ( + -n * x * _ufuncs.eval_gegenbauer(n, alpha, x) + + (n + 2 * alpha - 1) * _ufuncs.eval_gegenbauer(n - 1, alpha, x) + ) / (1 - x ** 2) + return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, True, mu) + + +def gegenbauer(n, alpha, monic=False): + r"""Gegenbauer (ultraspherical) polynomial. + + Defined to be the solution of + + .. math:: + (1 - x^2)\frac{d^2}{dx^2}C_n^{(\alpha)} + - (2\alpha + 1)x\frac{d}{dx}C_n^{(\alpha)} + + n(n + 2\alpha)C_n^{(\alpha)} = 0 + + for :math:`\alpha > -1/2`; :math:`C_n^{(\alpha)}` is a polynomial + of degree :math:`n`. + + Parameters + ---------- + n : int + Degree of the polynomial. + alpha : float + Parameter, must be greater than -0.5. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + C : orthopoly1d + Gegenbauer polynomial. + + Notes + ----- + The polynomials :math:`C_n^{(\alpha)}` are orthogonal over + :math:`[-1,1]` with weight function :math:`(1 - x^2)^{(\alpha - + 1/2)}`. + + Examples + -------- + >>> import numpy as np + >>> from scipy import special + >>> import matplotlib.pyplot as plt + + We can initialize a variable ``p`` as a Gegenbauer polynomial using the + `gegenbauer` function and evaluate at a point ``x = 1``. + + >>> p = special.gegenbauer(3, 0.5, monic=False) + >>> p + poly1d([ 2.5, 0. , -1.5, 0. ]) + >>> p(1) + 1.0 + + To evaluate ``p`` at various points ``x`` in the interval ``(-3, 3)``, + simply pass an array ``x`` to ``p`` as follows: + + >>> x = np.linspace(-3, 3, 400) + >>> y = p(x) + + We can then visualize ``x, y`` using `matplotlib.pyplot`. + + >>> fig, ax = plt.subplots() + >>> ax.plot(x, y) + >>> ax.set_title("Gegenbauer (ultraspherical) polynomial of degree 3") + >>> ax.set_xlabel("x") + >>> ax.set_ylabel("G_3(x)") + >>> plt.show() + + """ + if not np.isfinite(alpha) or alpha <= -0.5 : + raise ValueError("`alpha` must be a finite number greater than -1/2") + base = jacobi(n, alpha - 0.5, alpha - 0.5, monic=monic) + if monic or n == 0: + return base + # Abrahmowitz and Stegan 22.5.20 + factor = (_gam(2*alpha + n) * _gam(alpha + 0.5) / + _gam(2*alpha) / _gam(alpha + 0.5 + n)) + base._scale(factor) + base.__dict__['_eval_func'] = lambda x: _ufuncs.eval_gegenbauer(float(n), + alpha, x) + return base + +# Chebyshev of the first kind: T_n(x) = +# n! sqrt(pi) / _gam(n+1./2)* P^(-1/2,-1/2)_n(x) +# Computed anew. + + +def roots_chebyt(n, mu=False): + r"""Gauss-Chebyshev (first kind) quadrature. + + Computes the sample points and weights for Gauss-Chebyshev + quadrature. The sample points are the roots of the nth degree + Chebyshev polynomial of the first kind, :math:`T_n(x)`. These + sample points and weights correctly integrate polynomials of + degree :math:`2n - 1` or less over the interval :math:`[-1, 1]` + with weight function :math:`w(x) = 1/\sqrt{1 - x^2}`. See 22.2.4 + in [AS]_ for more details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + numpy.polynomial.chebyshev.chebgauss + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + m = int(n) + if n < 1 or n != m: + raise ValueError('n must be a positive integer.') + x = _ufuncs._sinpi(np.arange(-m + 1, m, 2) / (2*m)) + w = np.full_like(x, pi/m) + if mu: + return x, w, pi + else: + return x, w + + +def chebyt(n, monic=False): + r"""Chebyshev polynomial of the first kind. + + Defined to be the solution of + + .. math:: + (1 - x^2)\frac{d^2}{dx^2}T_n - x\frac{d}{dx}T_n + n^2T_n = 0; + + :math:`T_n` is a polynomial of degree :math:`n`. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + T : orthopoly1d + Chebyshev polynomial of the first kind. + + See Also + -------- + chebyu : Chebyshev polynomial of the second kind. + + Notes + ----- + The polynomials :math:`T_n` are orthogonal over :math:`[-1, 1]` + with weight function :math:`(1 - x^2)^{-1/2}`. + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + Chebyshev polynomials of the first kind of order :math:`n` can + be obtained as the determinant of specific :math:`n \times n` + matrices. As an example we can check how the points obtained from + the determinant of the following :math:`3 \times 3` matrix + lay exactly on :math:`T_3`: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.linalg import det + >>> from scipy.special import chebyt + >>> x = np.arange(-1.0, 1.0, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(-2.0, 2.0) + >>> ax.set_title(r'Chebyshev polynomial $T_3$') + >>> ax.plot(x, chebyt(3)(x), label=rf'$T_3$') + >>> for p in np.arange(-1.0, 1.0, 0.1): + ... ax.plot(p, + ... det(np.array([[p, 1, 0], [1, 2*p, 1], [0, 1, 2*p]])), + ... 'rx') + >>> plt.legend(loc='best') + >>> plt.show() + + They are also related to the Jacobi Polynomials + :math:`P_n^{(-0.5, -0.5)}` through the relation: + + .. math:: + P_n^{(-0.5, -0.5)}(x) = \frac{1}{4^n} \binom{2n}{n} T_n(x) + + Let's verify it for :math:`n = 3`: + + >>> from scipy.special import binom + >>> from scipy.special import jacobi + >>> x = np.arange(-1.0, 1.0, 0.01) + >>> np.allclose(jacobi(3, -0.5, -0.5)(x), + ... 1/64 * binom(6, 3) * chebyt(3)(x)) + True + + We can plot the Chebyshev polynomials :math:`T_n` for some values + of :math:`n`: + + >>> x = np.arange(-1.5, 1.5, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(-4.0, 4.0) + >>> ax.set_title(r'Chebyshev polynomials $T_n$') + >>> for n in np.arange(2,5): + ... ax.plot(x, chebyt(n)(x), label=rf'$T_n={n}$') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + if n < 0: + raise ValueError("n must be nonnegative.") + + def wfunc(x): + return 1.0 / sqrt(1 - x * x) + if n == 0: + return orthopoly1d([], [], pi, 1.0, wfunc, (-1, 1), monic, + lambda x: _ufuncs.eval_chebyt(n, x)) + n1 = n + x, w, mu = roots_chebyt(n1, mu=True) + hn = pi / 2 + kn = 2**(n - 1) + p = orthopoly1d(x, w, hn, kn, wfunc, (-1, 1), monic, + lambda x: _ufuncs.eval_chebyt(n, x)) + return p + +# Chebyshev of the second kind +# U_n(x) = (n+1)! sqrt(pi) / (2*_gam(n+3./2)) * P^(1/2,1/2)_n(x) + + +def roots_chebyu(n, mu=False): + r"""Gauss-Chebyshev (second kind) quadrature. + + Computes the sample points and weights for Gauss-Chebyshev + quadrature. The sample points are the roots of the nth degree + Chebyshev polynomial of the second kind, :math:`U_n(x)`. These + sample points and weights correctly integrate polynomials of + degree :math:`2n - 1` or less over the interval :math:`[-1, 1]` + with weight function :math:`w(x) = \sqrt{1 - x^2}`. See 22.2.5 in + [AS]_ for details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + m = int(n) + if n < 1 or n != m: + raise ValueError('n must be a positive integer.') + t = np.arange(m, 0, -1) * pi / (m + 1) + x = np.cos(t) + w = pi * np.sin(t)**2 / (m + 1) + if mu: + return x, w, pi / 2 + else: + return x, w + + +def chebyu(n, monic=False): + r"""Chebyshev polynomial of the second kind. + + Defined to be the solution of + + .. math:: + (1 - x^2)\frac{d^2}{dx^2}U_n - 3x\frac{d}{dx}U_n + + n(n + 2)U_n = 0; + + :math:`U_n` is a polynomial of degree :math:`n`. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + U : orthopoly1d + Chebyshev polynomial of the second kind. + + See Also + -------- + chebyt : Chebyshev polynomial of the first kind. + + Notes + ----- + The polynomials :math:`U_n` are orthogonal over :math:`[-1, 1]` + with weight function :math:`(1 - x^2)^{1/2}`. + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + Chebyshev polynomials of the second kind of order :math:`n` can + be obtained as the determinant of specific :math:`n \times n` + matrices. As an example we can check how the points obtained from + the determinant of the following :math:`3 \times 3` matrix + lay exactly on :math:`U_3`: + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.linalg import det + >>> from scipy.special import chebyu + >>> x = np.arange(-1.0, 1.0, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(-2.0, 2.0) + >>> ax.set_title(r'Chebyshev polynomial $U_3$') + >>> ax.plot(x, chebyu(3)(x), label=rf'$U_3$') + >>> for p in np.arange(-1.0, 1.0, 0.1): + ... ax.plot(p, + ... det(np.array([[2*p, 1, 0], [1, 2*p, 1], [0, 1, 2*p]])), + ... 'rx') + >>> plt.legend(loc='best') + >>> plt.show() + + They satisfy the recurrence relation: + + .. math:: + U_{2n-1}(x) = 2 T_n(x)U_{n-1}(x) + + where the :math:`T_n` are the Chebyshev polynomial of the first kind. + Let's verify it for :math:`n = 2`: + + >>> from scipy.special import chebyt + >>> x = np.arange(-1.0, 1.0, 0.01) + >>> np.allclose(chebyu(3)(x), 2 * chebyt(2)(x) * chebyu(1)(x)) + True + + We can plot the Chebyshev polynomials :math:`U_n` for some values + of :math:`n`: + + >>> x = np.arange(-1.0, 1.0, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(-1.5, 1.5) + >>> ax.set_title(r'Chebyshev polynomials $U_n$') + >>> for n in np.arange(1,5): + ... ax.plot(x, chebyu(n)(x), label=rf'$U_n={n}$') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + base = jacobi(n, 0.5, 0.5, monic=monic) + if monic: + return base + factor = sqrt(pi) / 2.0 * _gam(n + 2) / _gam(n + 1.5) + base._scale(factor) + return base + +# Chebyshev of the first kind C_n(x) + + +def roots_chebyc(n, mu=False): + r"""Gauss-Chebyshev (first kind) quadrature. + + Compute the sample points and weights for Gauss-Chebyshev + quadrature. The sample points are the roots of the nth degree + Chebyshev polynomial of the first kind, :math:`C_n(x)`. These + sample points and weights correctly integrate polynomials of + degree :math:`2n - 1` or less over the interval :math:`[-2, 2]` + with weight function :math:`w(x) = 1 / \sqrt{1 - (x/2)^2}`. See + 22.2.6 in [AS]_ for more details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + x, w, m = roots_chebyt(n, True) + x *= 2 + w *= 2 + m *= 2 + if mu: + return x, w, m + else: + return x, w + + +def chebyc(n, monic=False): + r"""Chebyshev polynomial of the first kind on :math:`[-2, 2]`. + + Defined as :math:`C_n(x) = 2T_n(x/2)`, where :math:`T_n` is the + nth Chebychev polynomial of the first kind. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + C : orthopoly1d + Chebyshev polynomial of the first kind on :math:`[-2, 2]`. + + See Also + -------- + chebyt : Chebyshev polynomial of the first kind. + + Notes + ----- + The polynomials :math:`C_n(x)` are orthogonal over :math:`[-2, 2]` + with weight function :math:`1/\sqrt{1 - (x/2)^2}`. + + References + ---------- + .. [1] Abramowitz and Stegun, "Handbook of Mathematical Functions" + Section 22. National Bureau of Standards, 1972. + + """ + if n < 0: + raise ValueError("n must be nonnegative.") + + if n == 0: + n1 = n + 1 + else: + n1 = n + x, w = roots_chebyc(n1) + if n == 0: + x, w = [], [] + hn = 4 * pi * ((n == 0) + 1) + kn = 1.0 + p = orthopoly1d(x, w, hn, kn, + wfunc=lambda x: 1.0 / sqrt(1 - x * x / 4.0), + limits=(-2, 2), monic=monic) + if not monic: + p._scale(2.0 / p(2)) + p.__dict__['_eval_func'] = lambda x: _ufuncs.eval_chebyc(n, x) + return p + +# Chebyshev of the second kind S_n(x) + + +def roots_chebys(n, mu=False): + r"""Gauss-Chebyshev (second kind) quadrature. + + Compute the sample points and weights for Gauss-Chebyshev + quadrature. The sample points are the roots of the nth degree + Chebyshev polynomial of the second kind, :math:`S_n(x)`. These + sample points and weights correctly integrate polynomials of + degree :math:`2n - 1` or less over the interval :math:`[-2, 2]` + with weight function :math:`w(x) = \sqrt{1 - (x/2)^2}`. See 22.2.7 + in [AS]_ for more details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + x, w, m = roots_chebyu(n, True) + x *= 2 + w *= 2 + m *= 2 + if mu: + return x, w, m + else: + return x, w + + +def chebys(n, monic=False): + r"""Chebyshev polynomial of the second kind on :math:`[-2, 2]`. + + Defined as :math:`S_n(x) = U_n(x/2)` where :math:`U_n` is the + nth Chebychev polynomial of the second kind. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + S : orthopoly1d + Chebyshev polynomial of the second kind on :math:`[-2, 2]`. + + See Also + -------- + chebyu : Chebyshev polynomial of the second kind + + Notes + ----- + The polynomials :math:`S_n(x)` are orthogonal over :math:`[-2, 2]` + with weight function :math:`\sqrt{1 - (x/2)}^2`. + + References + ---------- + .. [1] Abramowitz and Stegun, "Handbook of Mathematical Functions" + Section 22. National Bureau of Standards, 1972. + + """ + if n < 0: + raise ValueError("n must be nonnegative.") + + if n == 0: + n1 = n + 1 + else: + n1 = n + x, w = roots_chebys(n1) + if n == 0: + x, w = [], [] + hn = pi + kn = 1.0 + p = orthopoly1d(x, w, hn, kn, + wfunc=lambda x: sqrt(1 - x * x / 4.0), + limits=(-2, 2), monic=monic) + if not monic: + factor = (n + 1.0) / p(2) + p._scale(factor) + p.__dict__['_eval_func'] = lambda x: _ufuncs.eval_chebys(n, x) + return p + +# Shifted Chebyshev of the first kind T^*_n(x) + + +def roots_sh_chebyt(n, mu=False): + r"""Gauss-Chebyshev (first kind, shifted) quadrature. + + Compute the sample points and weights for Gauss-Chebyshev + quadrature. The sample points are the roots of the nth degree + shifted Chebyshev polynomial of the first kind, :math:`T_n(x)`. + These sample points and weights correctly integrate polynomials of + degree :math:`2n - 1` or less over the interval :math:`[0, 1]` + with weight function :math:`w(x) = 1/\sqrt{x - x^2}`. See 22.2.8 + in [AS]_ for more details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + xw = roots_chebyt(n, mu) + return ((xw[0] + 1) / 2,) + xw[1:] + + +def sh_chebyt(n, monic=False): + r"""Shifted Chebyshev polynomial of the first kind. + + Defined as :math:`T^*_n(x) = T_n(2x - 1)` for :math:`T_n` the nth + Chebyshev polynomial of the first kind. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + T : orthopoly1d + Shifted Chebyshev polynomial of the first kind. + + Notes + ----- + The polynomials :math:`T^*_n` are orthogonal over :math:`[0, 1]` + with weight function :math:`(x - x^2)^{-1/2}`. + + """ + base = sh_jacobi(n, 0.0, 0.5, monic=monic) + if monic: + return base + if n > 0: + factor = 4**n / 2.0 + else: + factor = 1.0 + base._scale(factor) + return base + + +# Shifted Chebyshev of the second kind U^*_n(x) +def roots_sh_chebyu(n, mu=False): + r"""Gauss-Chebyshev (second kind, shifted) quadrature. + + Computes the sample points and weights for Gauss-Chebyshev + quadrature. The sample points are the roots of the nth degree + shifted Chebyshev polynomial of the second kind, :math:`U_n(x)`. + These sample points and weights correctly integrate polynomials of + degree :math:`2n - 1` or less over the interval :math:`[0, 1]` + with weight function :math:`w(x) = \sqrt{x - x^2}`. See 22.2.9 in + [AS]_ for more details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + x, w, m = roots_chebyu(n, True) + x = (x + 1) / 2 + m_us = _ufuncs.beta(1.5, 1.5) + w *= m_us / m + if mu: + return x, w, m_us + else: + return x, w + + +def sh_chebyu(n, monic=False): + r"""Shifted Chebyshev polynomial of the second kind. + + Defined as :math:`U^*_n(x) = U_n(2x - 1)` for :math:`U_n` the nth + Chebyshev polynomial of the second kind. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + U : orthopoly1d + Shifted Chebyshev polynomial of the second kind. + + Notes + ----- + The polynomials :math:`U^*_n` are orthogonal over :math:`[0, 1]` + with weight function :math:`(x - x^2)^{1/2}`. + + """ + base = sh_jacobi(n, 2.0, 1.5, monic=monic) + if monic: + return base + factor = 4**n + base._scale(factor) + return base + +# Legendre + + +def roots_legendre(n, mu=False): + r"""Gauss-Legendre quadrature. + + Compute the sample points and weights for Gauss-Legendre + quadrature [GL]_. The sample points are the roots of the nth degree + Legendre polynomial :math:`P_n(x)`. These sample points and + weights correctly integrate polynomials of degree :math:`2n - 1` + or less over the interval :math:`[-1, 1]` with weight function + :math:`w(x) = 1`. See 2.2.10 in [AS]_ for more details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + numpy.polynomial.legendre.leggauss + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + .. [GL] Gauss-Legendre quadrature, Wikipedia, + https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_quadrature + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import roots_legendre, eval_legendre + >>> roots, weights = roots_legendre(9) + + ``roots`` holds the roots, and ``weights`` holds the weights for + Gauss-Legendre quadrature. + + >>> roots + array([-0.96816024, -0.83603111, -0.61337143, -0.32425342, 0. , + 0.32425342, 0.61337143, 0.83603111, 0.96816024]) + >>> weights + array([0.08127439, 0.18064816, 0.2606107 , 0.31234708, 0.33023936, + 0.31234708, 0.2606107 , 0.18064816, 0.08127439]) + + Verify that we have the roots by evaluating the degree 9 Legendre + polynomial at ``roots``. All the values are approximately zero: + + >>> eval_legendre(9, roots) + array([-8.88178420e-16, -2.22044605e-16, 1.11022302e-16, 1.11022302e-16, + 0.00000000e+00, -5.55111512e-17, -1.94289029e-16, 1.38777878e-16, + -8.32667268e-17]) + + Here we'll show how the above values can be used to estimate the + integral from 1 to 2 of f(t) = t + 1/t with Gauss-Legendre + quadrature [GL]_. First define the function and the integration + limits. + + >>> def f(t): + ... return t + 1/t + ... + >>> a = 1 + >>> b = 2 + + We'll use ``integral(f(t), t=a, t=b)`` to denote the definite integral + of f from t=a to t=b. The sample points in ``roots`` are from the + interval [-1, 1], so we'll rewrite the integral with the simple change + of variable:: + + x = 2/(b - a) * t - (a + b)/(b - a) + + with inverse:: + + t = (b - a)/2 * x + (a + b)/2 + + Then:: + + integral(f(t), a, b) = + (b - a)/2 * integral(f((b-a)/2*x + (a+b)/2), x=-1, x=1) + + We can approximate the latter integral with the values returned + by `roots_legendre`. + + Map the roots computed above from [-1, 1] to [a, b]. + + >>> t = (b - a)/2 * roots + (a + b)/2 + + Approximate the integral as the weighted sum of the function values. + + >>> (b - a)/2 * f(t).dot(weights) + 2.1931471805599276 + + Compare that to the exact result, which is 3/2 + log(2): + + >>> 1.5 + np.log(2) + 2.1931471805599454 + + """ + m = int(n) + if n < 1 or n != m: + raise ValueError("n must be a positive integer.") + + mu0 = 2.0 + def an_func(k): + return 0.0 * k + def bn_func(k): + return k * np.sqrt(1.0 / (4 * k * k - 1)) + f = _ufuncs.eval_legendre + def df(n, x): + return (-n * x * _ufuncs.eval_legendre(n, x) + + n * _ufuncs.eval_legendre(n - 1, x)) / (1 - x ** 2) + return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, True, mu) + + +def legendre(n, monic=False): + r"""Legendre polynomial. + + Defined to be the solution of + + .. math:: + \frac{d}{dx}\left[(1 - x^2)\frac{d}{dx}P_n(x)\right] + + n(n + 1)P_n(x) = 0; + + :math:`P_n(x)` is a polynomial of degree :math:`n`. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + P : orthopoly1d + Legendre polynomial. + + Notes + ----- + The polynomials :math:`P_n` are orthogonal over :math:`[-1, 1]` + with weight function 1. + + Examples + -------- + Generate the 3rd-order Legendre polynomial 1/2*(5x^3 + 0x^2 - 3x + 0): + + >>> from scipy.special import legendre + >>> legendre(3) + poly1d([ 2.5, 0. , -1.5, 0. ]) + + """ + if n < 0: + raise ValueError("n must be nonnegative.") + + if n == 0: + n1 = n + 1 + else: + n1 = n + x, w = roots_legendre(n1) + if n == 0: + x, w = [], [] + hn = 2.0 / (2 * n + 1) + kn = _gam(2 * n + 1) / _gam(n + 1)**2 / 2.0**n + p = orthopoly1d(x, w, hn, kn, wfunc=lambda x: 1.0, limits=(-1, 1), + monic=monic, + eval_func=lambda x: _ufuncs.eval_legendre(n, x)) + return p + +# Shifted Legendre P^*_n(x) + + +def roots_sh_legendre(n, mu=False): + r"""Gauss-Legendre (shifted) quadrature. + + Compute the sample points and weights for Gauss-Legendre + quadrature. The sample points are the roots of the nth degree + shifted Legendre polynomial :math:`P^*_n(x)`. These sample points + and weights correctly integrate polynomials of degree :math:`2n - + 1` or less over the interval :math:`[0, 1]` with weight function + :math:`w(x) = 1.0`. See 2.2.11 in [AS]_ for details. + + Parameters + ---------- + n : int + quadrature order + mu : bool, optional + If True, return the sum of the weights, optional. + + Returns + ------- + x : ndarray + Sample points + w : ndarray + Weights + mu : float + Sum of the weights + + See Also + -------- + scipy.integrate.fixed_quad + + References + ---------- + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + """ + x, w = roots_legendre(n) + x = (x + 1) / 2 + w /= 2 + if mu: + return x, w, 1.0 + else: + return x, w + + +def sh_legendre(n, monic=False): + r"""Shifted Legendre polynomial. + + Defined as :math:`P^*_n(x) = P_n(2x - 1)` for :math:`P_n` the nth + Legendre polynomial. + + Parameters + ---------- + n : int + Degree of the polynomial. + monic : bool, optional + If `True`, scale the leading coefficient to be 1. Default is + `False`. + + Returns + ------- + P : orthopoly1d + Shifted Legendre polynomial. + + Notes + ----- + The polynomials :math:`P^*_n` are orthogonal over :math:`[0, 1]` + with weight function 1. + + """ + if n < 0: + raise ValueError("n must be nonnegative.") + + def wfunc(x): + return 0.0 * x + 1.0 + if n == 0: + return orthopoly1d([], [], 1.0, 1.0, wfunc, (0, 1), monic, + lambda x: _ufuncs.eval_sh_legendre(n, x)) + x, w = roots_sh_legendre(n) + hn = 1.0 / (2 * n + 1.0) + kn = _gam(2 * n + 1) / _gam(n + 1)**2 + p = orthopoly1d(x, w, hn, kn, wfunc, limits=(0, 1), monic=monic, + eval_func=lambda x: _ufuncs.eval_sh_legendre(n, x)) + return p + + +# Make the old root function names an alias for the new ones +_modattrs = globals() +for newfun, oldfun in _rootfuns_map.items(): + _modattrs[oldfun] = _modattrs[newfun] + __all__.append(oldfun) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_orthogonal.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_orthogonal.pyi new file mode 100644 index 0000000000000000000000000000000000000000..e0ae3ce3be90187cd957fe16cc6d145a7093da5a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_orthogonal.pyi @@ -0,0 +1,330 @@ +from typing import ( + Any, + Callable, + Literal, + Optional, + overload, +) + +import numpy as np + +_IntegerType = int | np.integer +_FloatingType = float | np.floating +_PointsAndWeights = tuple[np.ndarray, np.ndarray] +_PointsAndWeightsAndMu = tuple[np.ndarray, np.ndarray, float] + +_ArrayLike0D = bool | int | float | complex | str | bytes | np.generic + +__all__ = [ + 'legendre', + 'chebyt', + 'chebyu', + 'chebyc', + 'chebys', + 'jacobi', + 'laguerre', + 'genlaguerre', + 'hermite', + 'hermitenorm', + 'gegenbauer', + 'sh_legendre', + 'sh_chebyt', + 'sh_chebyu', + 'sh_jacobi', + 'roots_legendre', + 'roots_chebyt', + 'roots_chebyu', + 'roots_chebyc', + 'roots_chebys', + 'roots_jacobi', + 'roots_laguerre', + 'roots_genlaguerre', + 'roots_hermite', + 'roots_hermitenorm', + 'roots_gegenbauer', + 'roots_sh_legendre', + 'roots_sh_chebyt', + 'roots_sh_chebyu', + 'roots_sh_jacobi', +] + +@overload +def roots_jacobi( + n: _IntegerType, + alpha: _FloatingType, + beta: _FloatingType, +) -> _PointsAndWeights: ... +@overload +def roots_jacobi( + n: _IntegerType, + alpha: _FloatingType, + beta: _FloatingType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_jacobi( + n: _IntegerType, + alpha: _FloatingType, + beta: _FloatingType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_sh_jacobi( + n: _IntegerType, + p1: _FloatingType, + q1: _FloatingType, +) -> _PointsAndWeights: ... +@overload +def roots_sh_jacobi( + n: _IntegerType, + p1: _FloatingType, + q1: _FloatingType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_sh_jacobi( + n: _IntegerType, + p1: _FloatingType, + q1: _FloatingType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_genlaguerre( + n: _IntegerType, + alpha: _FloatingType, +) -> _PointsAndWeights: ... +@overload +def roots_genlaguerre( + n: _IntegerType, + alpha: _FloatingType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_genlaguerre( + n: _IntegerType, + alpha: _FloatingType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_laguerre(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_laguerre( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_laguerre( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_hermite(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_hermite( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_hermite( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_hermitenorm(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_hermitenorm( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_hermitenorm( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_gegenbauer( + n: _IntegerType, + alpha: _FloatingType, +) -> _PointsAndWeights: ... +@overload +def roots_gegenbauer( + n: _IntegerType, + alpha: _FloatingType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_gegenbauer( + n: _IntegerType, + alpha: _FloatingType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_chebyt(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_chebyt( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_chebyt( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_chebyu(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_chebyu( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_chebyu( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_chebyc(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_chebyc( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_chebyc( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_chebys(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_chebys( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_chebys( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_sh_chebyt(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_sh_chebyt( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_sh_chebyt( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_sh_chebyu(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_sh_chebyu( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_sh_chebyu( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_legendre(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_legendre( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_legendre( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +@overload +def roots_sh_legendre(n: _IntegerType) -> _PointsAndWeights: ... +@overload +def roots_sh_legendre( + n: _IntegerType, + mu: Literal[False], +) -> _PointsAndWeights: ... +@overload +def roots_sh_legendre( + n: _IntegerType, + mu: Literal[True], +) -> _PointsAndWeightsAndMu: ... + +class orthopoly1d(np.poly1d): + def __init__( + self, + roots: np.typing.ArrayLike, + weights: np.typing.ArrayLike | None, + hn: float = ..., + kn: float = ..., + wfunc = Optional[Callable[[float], float]], # noqa: UP007 + limits = tuple[float, float] | None, + monic: bool = ..., + eval_func: np.ufunc = ..., + ) -> None: ... + @property + def limits(self) -> tuple[float, float]: ... + def weight_func(self, x: float) -> float: ... + @overload + def __call__(self, x: _ArrayLike0D) -> Any: ... + @overload + def __call__(self, x: np.poly1d) -> np.poly1d: ... # type: ignore[overload-overlap] + @overload + def __call__(self, x: np.typing.ArrayLike) -> np.ndarray: ... + +def legendre(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def chebyt(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def chebyu(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def chebyc(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def chebys(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def jacobi( + n: _IntegerType, + alpha: _FloatingType, + beta: _FloatingType, + monic: bool = ..., +) -> orthopoly1d: ... +def laguerre(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def genlaguerre( + n: _IntegerType, + alpha: _FloatingType, + monic: bool = ..., +) -> orthopoly1d: ... +def hermite(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def hermitenorm(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def gegenbauer( + n: _IntegerType, + alpha: _FloatingType, + monic: bool = ..., +) -> orthopoly1d: ... +def sh_legendre(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def sh_chebyt(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def sh_chebyu(n: _IntegerType, monic: bool = ...) -> orthopoly1d: ... +def sh_jacobi( + n: _IntegerType, + p: _FloatingType, + q: _FloatingType, + monic: bool = ..., +) -> orthopoly1d: ... + +# These functions are not public, but still need stubs because they +# get checked in the tests. +def _roots_hermite_asy(n: _IntegerType) -> _PointsAndWeights: ... diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/cosine_cdf.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/cosine_cdf.py new file mode 100644 index 0000000000000000000000000000000000000000..662c12bc74b31478c87471fbd1cce8bea285e765 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/cosine_cdf.py @@ -0,0 +1,17 @@ +import mpmath + + +def f(x): + return (mpmath.pi + x + mpmath.sin(x)) / (2*mpmath.pi) + + +# Note: 40 digits might be overkill; a few more digits than the default +# might be sufficient. +mpmath.mp.dps = 40 +ts = mpmath.taylor(f, -mpmath.pi, 20) +p, q = mpmath.pade(ts, 9, 10) + +p = [float(c) for c in p] +q = [float(c) for c in q] +print('p =', p) +print('q =', q) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/expn_asy.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/expn_asy.py new file mode 100644 index 0000000000000000000000000000000000000000..3491b8acd588a2cacfc48f0a3a60c6ae88c3e8c5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/expn_asy.py @@ -0,0 +1,54 @@ +"""Precompute the polynomials for the asymptotic expansion of the +generalized exponential integral. + +Sources +------- +[1] NIST, Digital Library of Mathematical Functions, + https://dlmf.nist.gov/8.20#ii + +""" +import os + +try: + import sympy + from sympy import Poly + x = sympy.symbols('x') +except ImportError: + pass + + +def generate_A(K): + A = [Poly(1, x)] + for k in range(K): + A.append(Poly(1 - 2*k*x, x)*A[k] + Poly(x*(x + 1))*A[k].diff()) + return A + + +WARNING = """\ +/* This file was automatically generated by _precompute/expn_asy.py. + * Do not edit it manually! + */ +""" + + +def main(): + print(__doc__) + fn = os.path.join('..', 'cephes', 'expn.h') + + K = 12 + A = generate_A(K) + with open(fn + '.new', 'w') as f: + f.write(WARNING) + f.write(f"#define nA {len(A)}\n") + for k, Ak in enumerate(A): + ', '.join([str(x.evalf(18)) for x in Ak.coeffs()]) + f.write(f"static const double A{k}[] = {{tmp}};\n") + ", ".join([f"A{k}" for k in range(K + 1)]) + f.write("static const double *A[] = {{tmp}};\n") + ", ".join([str(Ak.degree()) for Ak in A]) + f.write("static const int Adegs[] = {{tmp}};\n") + os.rename(fn + '.new', fn) + + +if __name__ == "__main__": + main() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/gammainc_asy.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/gammainc_asy.py new file mode 100644 index 0000000000000000000000000000000000000000..98035457c78706ae01c02273ae1ab458b4ca140d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/gammainc_asy.py @@ -0,0 +1,116 @@ +""" +Precompute coefficients of Temme's asymptotic expansion for gammainc. + +This takes about 8 hours to run on a 2.3 GHz Macbook Pro with 4GB ram. + +Sources: +[1] NIST, "Digital Library of Mathematical Functions", + https://dlmf.nist.gov/ + +""" +import os +from scipy.special._precompute.utils import lagrange_inversion + +try: + import mpmath as mp +except ImportError: + pass + + +def compute_a(n): + """a_k from DLMF 5.11.6""" + a = [mp.sqrt(2)/2] + for k in range(1, n): + ak = a[-1]/k + for j in range(1, len(a)): + ak -= a[j]*a[-j]/(j + 1) + ak /= a[0]*(1 + mp.mpf(1)/(k + 1)) + a.append(ak) + return a + + +def compute_g(n): + """g_k from DLMF 5.11.3/5.11.5""" + a = compute_a(2*n) + g = [mp.sqrt(2)*mp.rf(0.5, k)*a[2*k] for k in range(n)] + return g + + +def eta(lam): + """Function from DLMF 8.12.1 shifted to be centered at 0.""" + if lam > 0: + return mp.sqrt(2*(lam - mp.log(lam + 1))) + elif lam < 0: + return -mp.sqrt(2*(lam - mp.log(lam + 1))) + else: + return 0 + + +def compute_alpha(n): + """alpha_n from DLMF 8.12.13""" + coeffs = mp.taylor(eta, 0, n - 1) + return lagrange_inversion(coeffs) + + +def compute_d(K, N): + """d_{k, n} from DLMF 8.12.12""" + M = N + 2*K + d0 = [-mp.mpf(1)/3] + alpha = compute_alpha(M + 2) + for n in range(1, M): + d0.append((n + 2)*alpha[n+2]) + d = [d0] + g = compute_g(K) + for k in range(1, K): + dk = [] + for n in range(M - 2*k): + dk.append((-1)**k*g[k]*d[0][n] + (n + 2)*d[k-1][n+2]) + d.append(dk) + for k in range(K): + d[k] = d[k][:N] + return d + + +header = \ +r"""/* This file was automatically generated by _precomp/gammainc.py. + * Do not edit it manually! + */ + +#ifndef IGAM_H +#define IGAM_H + +#define K {} +#define N {} + +static const double d[K][N] = +{{""" + +footer = \ +r""" +#endif +""" + + +def main(): + print(__doc__) + K = 25 + N = 25 + with mp.workdps(50): + d = compute_d(K, N) + fn = os.path.join(os.path.dirname(__file__), '..', 'cephes', 'igam.h') + with open(fn + '.new', 'w') as f: + f.write(header.format(K, N)) + for k, row in enumerate(d): + row = [mp.nstr(x, 17, min_fixed=0, max_fixed=0) for x in row] + f.write('{') + f.write(", ".join(row)) + if k < K - 1: + f.write('},\n') + else: + f.write('}};\n') + f.write(footer) + os.rename(fn + '.new', fn) + + +if __name__ == "__main__": + main() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/gammainc_data.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/gammainc_data.py new file mode 100644 index 0000000000000000000000000000000000000000..ebbe39f6159fb80c424dbb38eedeba46bd8cccf2 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/gammainc_data.py @@ -0,0 +1,124 @@ +"""Compute gammainc and gammaincc for large arguments and parameters +and save the values to data files for use in tests. We can't just +compare to mpmath's gammainc in test_mpmath.TestSystematic because it +would take too long. + +Note that mpmath's gammainc is computed using hypercomb, but since it +doesn't allow the user to increase the maximum number of terms used in +the series it doesn't converge for many arguments. To get around this +we copy the mpmath implementation but use more terms. + +This takes about 17 minutes to run on a 2.3 GHz Macbook Pro with 4GB +ram. + +Sources: +[1] Fredrik Johansson and others. mpmath: a Python library for + arbitrary-precision floating-point arithmetic (version 0.19), + December 2013. http://mpmath.org/. + +""" +import os +from time import time +import numpy as np +from numpy import pi + +from scipy.special._mptestutils import mpf2float + +try: + import mpmath as mp +except ImportError: + pass + + +def gammainc(a, x, dps=50, maxterms=10**8): + """Compute gammainc exactly like mpmath does but allow for more + summands in hypercomb. See + + mpmath/functions/expintegrals.py#L134 + + in the mpmath GitHub repository. + + """ + with mp.workdps(dps): + z, a, b = mp.mpf(a), mp.mpf(x), mp.mpf(x) + G = [z] + negb = mp.fneg(b, exact=True) + + def h(z): + T1 = [mp.exp(negb), b, z], [1, z, -1], [], G, [1], [1+z], b + return (T1,) + + res = mp.hypercomb(h, [z], maxterms=maxterms) + return mpf2float(res) + + +def gammaincc(a, x, dps=50, maxterms=10**8): + """Compute gammaincc exactly like mpmath does but allow for more + terms in hypercomb. See + + mpmath/functions/expintegrals.py#L187 + + in the mpmath GitHub repository. + + """ + with mp.workdps(dps): + z, a = a, x + + if mp.isint(z): + try: + # mpmath has a fast integer path + return mpf2float(mp.gammainc(z, a=a, regularized=True)) + except mp.libmp.NoConvergence: + pass + nega = mp.fneg(a, exact=True) + G = [z] + # Use 2F0 series when possible; fall back to lower gamma representation + try: + def h(z): + r = z-1 + return [([mp.exp(nega), a], [1, r], [], G, [1, -r], [], 1/nega)] + return mpf2float(mp.hypercomb(h, [z], force_series=True)) + except mp.libmp.NoConvergence: + def h(z): + T1 = [], [1, z-1], [z], G, [], [], 0 + T2 = [-mp.exp(nega), a, z], [1, z, -1], [], G, [1], [1+z], a + return T1, T2 + return mpf2float(mp.hypercomb(h, [z], maxterms=maxterms)) + + +def main(): + t0 = time() + # It would be nice to have data for larger values, but either this + # requires prohibitively large precision (dps > 800) or mpmath has + # a bug. For example, gammainc(1e20, 1e20, dps=800) returns a + # value around 0.03, while the true value should be close to 0.5 + # (DLMF 8.12.15). + print(__doc__) + pwd = os.path.dirname(__file__) + r = np.logspace(4, 14, 30) + ltheta = np.logspace(np.log10(pi/4), np.log10(np.arctan(0.6)), 30) + utheta = np.logspace(np.log10(pi/4), np.log10(np.arctan(1.4)), 30) + + regimes = [(gammainc, ltheta), (gammaincc, utheta)] + for func, theta in regimes: + rg, thetag = np.meshgrid(r, theta) + a, x = rg*np.cos(thetag), rg*np.sin(thetag) + a, x = a.flatten(), x.flatten() + dataset = [] + for i, (a0, x0) in enumerate(zip(a, x)): + if func == gammaincc: + # Exploit the fast integer path in gammaincc whenever + # possible so that the computation doesn't take too + # long + a0, x0 = np.floor(a0), np.floor(x0) + dataset.append((a0, x0, func(a0, x0))) + dataset = np.array(dataset) + filename = os.path.join(pwd, '..', 'tests', 'data', 'local', + f'{func.__name__}.txt') + np.savetxt(filename, dataset) + + print(f"{(time() - t0)/60} minutes elapsed") + + +if __name__ == "__main__": + main() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/hyp2f1_data.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/hyp2f1_data.py new file mode 100644 index 0000000000000000000000000000000000000000..c4adf14f49184bf75048a28a823909d24e778e04 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/hyp2f1_data.py @@ -0,0 +1,484 @@ +"""This script evaluates scipy's implementation of hyp2f1 against mpmath's. + +Author: Albert Steppi + +This script is long running and generates a large output file. With default +arguments, the generated file is roughly 700MB in size and it takes around +40 minutes using an Intel(R) Core(TM) i5-8250U CPU with n_jobs set to 8 +(full utilization). There are optional arguments which can be used to restrict +(or enlarge) the computations performed. These are described below. +The output of this script can be analyzed to identify suitable test cases and +to find parameter and argument regions where hyp2f1 needs to be improved. + +The script has one mandatory positional argument for specifying the path to +the location where the output file is to be placed, and 4 optional arguments +--n_jobs, --grid_size, --regions, and --parameter_groups. --n_jobs specifies +the number of processes to use if running in parallel. The default value is 1. +The other optional arguments are explained below. + +Produces a tab separated values file with 11 columns. The first four columns +contain the parameters a, b, c and the argument z. The next two contain |z| and +a region code for which region of the complex plane belongs to. The regions are + + 0) z == 1 + 1) |z| < 0.9 and real(z) >= 0 + 2) |z| <= 1 and real(z) < 0 + 3) 0.9 <= |z| <= 1 and |1 - z| < 0.9: + 4) 0.9 <= |z| <= 1 and |1 - z| >= 0.9 and real(z) >= 0: + 5) 1 < |z| < 1.1 and |1 - z| >= 0.9 and real(z) >= 0 + 6) |z| > 1 and not in 5) + +The --regions optional argument allows the user to specify a list of regions +to which computation will be restricted. + +Parameters a, b, c are taken from a 10 * 10 * 10 grid with values at + + -16, -8, -4, -2, -1, 1, 2, 4, 8, 16 + +with random perturbations applied. + +There are 9 parameter groups handling the following cases. + + 1) A, B, C, B - A, C - A, C - B, C - A - B all non-integral. + 2) B - A integral + 3) C - A integral + 4) C - B integral + 5) C - A - B integral + 6) A integral + 7) B integral + 8) C integral + 9) Wider range with c - a - b > 0. + +The seventh column of the output file is an integer between 1 and 8 specifying +the parameter group as above. + +The --parameter_groups optional argument allows the user to specify a list of +parameter groups to which computation will be restricted. + +The argument z is taken from a grid in the box + -box_size <= real(z) <= box_size, -box_size <= imag(z) <= box_size. +with grid size specified using the optional command line argument --grid_size, +and box_size specified with the command line argument --box_size. +The default value of grid_size is 20 and the default value of box_size is 2.0, +yielding a 20 * 20 grid in the box with corners -2-2j, -2+2j, 2-2j, 2+2j. + +The final four columns have the expected value of hyp2f1 for the given +parameters and argument as calculated with mpmath, the observed value +calculated with scipy's hyp2f1, the relative error, and the absolute error. + +As special cases of hyp2f1 are moved from the original Fortran implementation +into Cython, this script can be used to ensure that no regressions occur and +to point out where improvements are needed. +""" + + +import os +import csv +import argparse +import numpy as np +from itertools import product +from multiprocessing import Pool + + +from scipy.special import hyp2f1 +from scipy.special.tests.test_hyp2f1 import mp_hyp2f1 + + +def get_region(z): + """Assign numbers for regions where hyp2f1 must be handled differently.""" + if z == 1 + 0j: + return 0 + elif abs(z) < 0.9 and z.real >= 0: + return 1 + elif abs(z) <= 1 and z.real < 0: + return 2 + elif 0.9 <= abs(z) <= 1 and abs(1 - z) < 0.9: + return 3 + elif 0.9 <= abs(z) <= 1 and abs(1 - z) >= 0.9: + return 4 + elif 1 < abs(z) < 1.1 and abs(1 - z) >= 0.9 and z.real >= 0: + return 5 + else: + return 6 + + +def get_result(a, b, c, z, group): + """Get results for given parameter and value combination.""" + expected, observed = mp_hyp2f1(a, b, c, z), hyp2f1(a, b, c, z) + if ( + np.isnan(observed) and np.isnan(expected) or + expected == observed + ): + relative_error = 0.0 + absolute_error = 0.0 + elif np.isnan(observed): + # Set error to infinity if result is nan when not expected to be. + # Makes results easier to interpret. + relative_error = float("inf") + absolute_error = float("inf") + else: + absolute_error = abs(expected - observed) + relative_error = absolute_error / abs(expected) + + return ( + a, + b, + c, + z, + abs(z), + get_region(z), + group, + expected, + observed, + relative_error, + absolute_error, + ) + + +def get_result_no_mp(a, b, c, z, group): + """Get results for given parameter and value combination.""" + expected, observed = complex('nan'), hyp2f1(a, b, c, z) + relative_error, absolute_error = float('nan'), float('nan') + return ( + a, + b, + c, + z, + abs(z), + get_region(z), + group, + expected, + observed, + relative_error, + absolute_error, + ) + + +def get_results(params, Z, n_jobs=1, compute_mp=True): + """Batch compute results for multiple parameter and argument values. + + Parameters + ---------- + params : iterable + iterable of tuples of floats (a, b, c) specifying parameter values + a, b, c for hyp2f1 + Z : iterable of complex + Arguments at which to evaluate hyp2f1 + n_jobs : Optional[int] + Number of jobs for parallel execution. + + Returns + ------- + list + List of tuples of results values. See return value in source code + of `get_result`. + """ + input_ = ( + (a, b, c, z, group) for (a, b, c, group), z in product(params, Z) + ) + + with Pool(n_jobs) as pool: + rows = pool.starmap( + get_result if compute_mp else get_result_no_mp, + input_ + ) + return rows + + +def _make_hyp2f1_test_case(a, b, c, z, rtol): + """Generate string for single test case as used in test_hyp2f1.py.""" + expected = mp_hyp2f1(a, b, c, z) + return ( + " pytest.param(\n" + " Hyp2f1TestCase(\n" + f" a={a},\n" + f" b={b},\n" + f" c={c},\n" + f" z={z},\n" + f" expected={expected},\n" + f" rtol={rtol},\n" + " ),\n" + " )," + ) + + +def make_hyp2f1_test_cases(rows): + """Generate string for a list of test cases for test_hyp2f1.py. + + Parameters + ---------- + rows : list + List of lists of the form [a, b, c, z, rtol] where a, b, c, z are + parameters and the argument for hyp2f1 and rtol is an expected + relative error for the associated test case. + + Returns + ------- + str + String for a list of test cases. The output string can be printed + or saved to a file and then copied into an argument for + `pytest.mark.parameterize` within `scipy.special.tests.test_hyp2f1.py`. + """ + result = "[\n" + result += '\n'.join( + _make_hyp2f1_test_case(a, b, c, z, rtol) + for a, b, c, z, rtol in rows + ) + result += "\n]" + return result + + +def main( + outpath, + n_jobs=1, + box_size=2.0, + grid_size=20, + regions=None, + parameter_groups=None, + compute_mp=True, +): + outpath = os.path.realpath(os.path.expanduser(outpath)) + + random_state = np.random.RandomState(1234) + # Parameters a, b, c selected near these values. + root_params = np.array( + [-16, -8, -4, -2, -1, 1, 2, 4, 8, 16] + ) + # Perturbations to apply to root values. + perturbations = 0.1 * random_state.random_sample( + size=(3, len(root_params)) + ) + + params = [] + # Parameter group 1 + # ----------------- + # No integer differences. This has been confirmed for the above seed. + A = root_params + perturbations[0, :] + B = root_params + perturbations[1, :] + C = root_params + perturbations[2, :] + params.extend( + sorted( + ((a, b, c, 1) for a, b, c in product(A, B, C)), + key=lambda x: max(abs(x[0]), abs(x[1])), + ) + ) + + # Parameter group 2 + # ----------------- + # B - A an integer + A = root_params + 0.5 + B = root_params + 0.5 + C = root_params + perturbations[1, :] + params.extend( + sorted( + ((a, b, c, 2) for a, b, c in product(A, B, C)), + key=lambda x: max(abs(x[0]), abs(x[1])), + ) + ) + + # Parameter group 3 + # ----------------- + # C - A an integer + A = root_params + 0.5 + B = root_params + perturbations[1, :] + C = root_params + 0.5 + params.extend( + sorted( + ((a, b, c, 3) for a, b, c in product(A, B, C)), + key=lambda x: max(abs(x[0]), abs(x[1])), + ) + ) + + # Parameter group 4 + # ----------------- + # C - B an integer + A = root_params + perturbations[0, :] + B = root_params + 0.5 + C = root_params + 0.5 + params.extend( + sorted( + ((a, b, c, 4) for a, b, c in product(A, B, C)), + key=lambda x: max(abs(x[0]), abs(x[1])), + ) + ) + + # Parameter group 5 + # ----------------- + # C - A - B an integer + A = root_params + 0.25 + B = root_params + 0.25 + C = root_params + 0.5 + params.extend( + sorted( + ((a, b, c, 5) for a, b, c in product(A, B, C)), + key=lambda x: max(abs(x[0]), abs(x[1])), + ) + ) + + # Parameter group 6 + # ----------------- + # A an integer + A = root_params + B = root_params + perturbations[0, :] + C = root_params + perturbations[1, :] + params.extend( + sorted( + ((a, b, c, 6) for a, b, c in product(A, B, C)), + key=lambda x: max(abs(x[0]), abs(x[1])), + ) + ) + + # Parameter group 7 + # ----------------- + # B an integer + A = root_params + perturbations[0, :] + B = root_params + C = root_params + perturbations[1, :] + params.extend( + sorted( + ((a, b, c, 7) for a, b, c in product(A, B, C)), + key=lambda x: max(abs(x[0]), abs(x[1])), + ) + ) + + # Parameter group 8 + # ----------------- + # C an integer + A = root_params + perturbations[0, :] + B = root_params + perturbations[1, :] + C = root_params + params.extend( + sorted( + ((a, b, c, 8) for a, b, c in product(A, B, C)), + key=lambda x: max(abs(x[0]), abs(x[1])), + ) + ) + + # Parameter group 9 + # ----------------- + # Wide range of magnitudes, c - a - b > 0. + phi = (1 + np.sqrt(5))/2 + P = phi**np.arange(16) + P = np.hstack([-P, P]) + group_9_params = sorted( + ( + (a, b, c, 9) for a, b, c in product(P, P, P) if c - a - b > 0 + ), + key=lambda x: max(abs(x[0]), abs(x[1])), + ) + + if parameter_groups is not None: + # Group 9 params only used if specified in arguments. + params.extend(group_9_params) + params = [ + (a, b, c, group) for a, b, c, group in params + if group in parameter_groups + ] + + # grid_size * grid_size grid in box with corners + # -2 - 2j, -2 + 2j, 2 - 2j, 2 + 2j + X, Y = np.meshgrid( + np.linspace(-box_size, box_size, grid_size), + np.linspace(-box_size, box_size, grid_size) + ) + Z = X + Y * 1j + Z = Z.flatten().tolist() + # Add z = 1 + 0j (region 0). + Z.append(1 + 0j) + if regions is not None: + Z = [z for z in Z if get_region(z) in regions] + + # Evaluate scipy and mpmath's hyp2f1 for all parameter combinations + # above against all arguments in the grid Z + rows = get_results(params, Z, n_jobs=n_jobs, compute_mp=compute_mp) + + with open(outpath, "w", newline="") as f: + writer = csv.writer(f, delimiter="\t") + writer.writerow( + [ + "a", + "b", + "c", + "z", + "|z|", + "region", + "parameter_group", + "expected", # mpmath's hyp2f1 + "observed", # scipy's hyp2f1 + "relative_error", + "absolute_error", + ] + ) + for row in rows: + writer.writerow(row) + + +if __name__ == "__main__": + parser = argparse.ArgumentParser( + description="Test scipy's hyp2f1 against mpmath's on a grid in the" + " complex plane over a grid of parameter values. Saves output to file" + " specified in positional argument \"outpath\"." + " Caution: With default arguments, the generated output file is" + " roughly 700MB in size. Script may take several hours to finish if" + " \"--n_jobs\" is set to 1." + ) + parser.add_argument( + "outpath", type=str, help="Path to output tsv file." + ) + parser.add_argument( + "--n_jobs", + type=int, + default=1, + help="Number of jobs for multiprocessing.", + ) + parser.add_argument( + "--box_size", + type=float, + default=2.0, + help="hyp2f1 is evaluated in box of side_length 2*box_size centered" + " at the origin." + ) + parser.add_argument( + "--grid_size", + type=int, + default=20, + help="hyp2f1 is evaluated on grid_size * grid_size grid in box of side" + " length 2*box_size centered at the origin." + ) + parser.add_argument( + "--parameter_groups", + type=int, + nargs='+', + default=None, + help="Restrict to supplied parameter groups. See the Docstring for" + " this module for more info on parameter groups. Calculate for all" + " parameter groups by default." + ) + parser.add_argument( + "--regions", + type=int, + nargs='+', + default=None, + help="Restrict to argument z only within the supplied regions. See" + " the Docstring for this module for more info on regions. Calculate" + " for all regions by default." + ) + parser.add_argument( + "--no_mp", + action='store_true', + help="If this flag is set, do not compute results with mpmath. Saves" + " time if results have already been computed elsewhere. Fills in" + " \"expected\" column with None values." + ) + args = parser.parse_args() + compute_mp = not args.no_mp + print(args.parameter_groups) + main( + args.outpath, + n_jobs=args.n_jobs, + box_size=args.box_size, + grid_size=args.grid_size, + parameter_groups=args.parameter_groups, + regions=args.regions, + compute_mp=compute_mp, + ) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/lambertw.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/lambertw.py new file mode 100644 index 0000000000000000000000000000000000000000..1fdbf35b2cf85f1f7a6e73579546ed5cfe508fa6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/lambertw.py @@ -0,0 +1,68 @@ +"""Compute a Pade approximation for the principal branch of the +Lambert W function around 0 and compare it to various other +approximations. + +""" +import numpy as np + +try: + import mpmath + import matplotlib.pyplot as plt +except ImportError: + pass + + +def lambertw_pade(): + derivs = [mpmath.diff(mpmath.lambertw, 0, n=n) for n in range(6)] + p, q = mpmath.pade(derivs, 3, 2) + return p, q + + +def main(): + print(__doc__) + with mpmath.workdps(50): + p, q = lambertw_pade() + p, q = p[::-1], q[::-1] + print(f"p = {p}") + print(f"q = {q}") + + x, y = np.linspace(-1.5, 1.5, 75), np.linspace(-1.5, 1.5, 75) + x, y = np.meshgrid(x, y) + z = x + 1j*y + lambertw_std = [] + for z0 in z.flatten(): + lambertw_std.append(complex(mpmath.lambertw(z0))) + lambertw_std = np.array(lambertw_std).reshape(x.shape) + + fig, axes = plt.subplots(nrows=3, ncols=1) + # Compare Pade approximation to true result + p = np.array([float(p0) for p0 in p]) + q = np.array([float(q0) for q0 in q]) + pade_approx = np.polyval(p, z)/np.polyval(q, z) + pade_err = abs(pade_approx - lambertw_std) + axes[0].pcolormesh(x, y, pade_err) + # Compare two terms of asymptotic series to true result + asy_approx = np.log(z) - np.log(np.log(z)) + asy_err = abs(asy_approx - lambertw_std) + axes[1].pcolormesh(x, y, asy_err) + # Compare two terms of the series around the branch point to the + # true result + p = np.sqrt(2*(np.exp(1)*z + 1)) + series_approx = -1 + p - p**2/3 + series_err = abs(series_approx - lambertw_std) + im = axes[2].pcolormesh(x, y, series_err) + + fig.colorbar(im, ax=axes.ravel().tolist()) + plt.show() + + fig, ax = plt.subplots(nrows=1, ncols=1) + pade_better = pade_err < asy_err + im = ax.pcolormesh(x, y, pade_better) + t = np.linspace(-0.3, 0.3) + ax.plot(-2.5*abs(t) - 0.2, t, 'r') + fig.colorbar(im, ax=ax) + plt.show() + + +if __name__ == '__main__': + main() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/loggamma.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/loggamma.py new file mode 100644 index 0000000000000000000000000000000000000000..74051ac7b46c70dc01919a362d05a8bbbe11333a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/loggamma.py @@ -0,0 +1,43 @@ +"""Precompute series coefficients for log-Gamma.""" + +try: + import mpmath +except ImportError: + pass + + +def stirling_series(N): + with mpmath.workdps(100): + coeffs = [mpmath.bernoulli(2*n)/(2*n*(2*n - 1)) + for n in range(1, N + 1)] + return coeffs + + +def taylor_series_at_1(N): + coeffs = [] + with mpmath.workdps(100): + coeffs.append(-mpmath.euler) + for n in range(2, N + 1): + coeffs.append((-1)**n*mpmath.zeta(n)/n) + return coeffs + + +def main(): + print(__doc__) + print() + stirling_coeffs = [mpmath.nstr(x, 20, min_fixed=0, max_fixed=0) + for x in stirling_series(8)[::-1]] + taylor_coeffs = [mpmath.nstr(x, 20, min_fixed=0, max_fixed=0) + for x in taylor_series_at_1(23)[::-1]] + print("Stirling series coefficients") + print("----------------------------") + print("\n".join(stirling_coeffs)) + print() + print("Taylor series coefficients") + print("--------------------------") + print("\n".join(taylor_coeffs)) + print() + + +if __name__ == '__main__': + main() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/struve_convergence.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/struve_convergence.py new file mode 100644 index 0000000000000000000000000000000000000000..dbf6009368540dbf603b61f5b72510f0acd1a65b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/struve_convergence.py @@ -0,0 +1,131 @@ +""" +Convergence regions of the expansions used in ``struve.c`` + +Note that for v >> z both functions tend rapidly to 0, +and for v << -z, they tend to infinity. + +The floating-point functions over/underflow in the lower left and right +corners of the figure. + + +Figure legend +============= + +Red region + Power series is close (1e-12) to the mpmath result + +Blue region + Asymptotic series is close to the mpmath result + +Green region + Bessel series is close to the mpmath result + +Dotted colored lines + Boundaries of the regions + +Solid colored lines + Boundaries estimated by the routine itself. These will be used + for determining which of the results to use. + +Black dashed line + The line z = 0.7*|v| + 12 + +""" +import numpy as np +import matplotlib.pyplot as plt + +import mpmath + + +def err_metric(a, b, atol=1e-290): + m = abs(a - b) / (atol + abs(b)) + m[np.isinf(b) & (a == b)] = 0 + return m + + +def do_plot(is_h=True): + from scipy.special._ufuncs import (_struve_power_series, + _struve_asymp_large_z, + _struve_bessel_series) + + vs = np.linspace(-1000, 1000, 91) + zs = np.sort(np.r_[1e-5, 1.0, np.linspace(0, 700, 91)[1:]]) + + rp = _struve_power_series(vs[:,None], zs[None,:], is_h) + ra = _struve_asymp_large_z(vs[:,None], zs[None,:], is_h) + rb = _struve_bessel_series(vs[:,None], zs[None,:], is_h) + + mpmath.mp.dps = 50 + if is_h: + def sh(v, z): + return float(mpmath.struveh(mpmath.mpf(v), mpmath.mpf(z))) + else: + def sh(v, z): + return float(mpmath.struvel(mpmath.mpf(v), mpmath.mpf(z))) + ex = np.vectorize(sh, otypes='d')(vs[:,None], zs[None,:]) + + err_a = err_metric(ra[0], ex) + 1e-300 + err_p = err_metric(rp[0], ex) + 1e-300 + err_b = err_metric(rb[0], ex) + 1e-300 + + err_est_a = abs(ra[1]/ra[0]) + err_est_p = abs(rp[1]/rp[0]) + err_est_b = abs(rb[1]/rb[0]) + + z_cutoff = 0.7*abs(vs) + 12 + + levels = [-1000, -12] + + plt.cla() + + plt.hold(1) + plt.contourf(vs, zs, np.log10(err_p).T, + levels=levels, colors=['r', 'r'], alpha=0.1) + plt.contourf(vs, zs, np.log10(err_a).T, + levels=levels, colors=['b', 'b'], alpha=0.1) + plt.contourf(vs, zs, np.log10(err_b).T, + levels=levels, colors=['g', 'g'], alpha=0.1) + + plt.contour(vs, zs, np.log10(err_p).T, + levels=levels, colors=['r', 'r'], linestyles=[':', ':']) + plt.contour(vs, zs, np.log10(err_a).T, + levels=levels, colors=['b', 'b'], linestyles=[':', ':']) + plt.contour(vs, zs, np.log10(err_b).T, + levels=levels, colors=['g', 'g'], linestyles=[':', ':']) + + lp = plt.contour(vs, zs, np.log10(err_est_p).T, + levels=levels, colors=['r', 'r'], linestyles=['-', '-']) + la = plt.contour(vs, zs, np.log10(err_est_a).T, + levels=levels, colors=['b', 'b'], linestyles=['-', '-']) + lb = plt.contour(vs, zs, np.log10(err_est_b).T, + levels=levels, colors=['g', 'g'], linestyles=['-', '-']) + + plt.clabel(lp, fmt={-1000: 'P', -12: 'P'}) + plt.clabel(la, fmt={-1000: 'A', -12: 'A'}) + plt.clabel(lb, fmt={-1000: 'B', -12: 'B'}) + + plt.plot(vs, z_cutoff, 'k--') + + plt.xlim(vs.min(), vs.max()) + plt.ylim(zs.min(), zs.max()) + + plt.xlabel('v') + plt.ylabel('z') + + +def main(): + plt.clf() + plt.subplot(121) + do_plot(True) + plt.title('Struve H') + + plt.subplot(122) + do_plot(False) + plt.title('Struve L') + + plt.savefig('struve_convergence.png') + plt.show() + + +if __name__ == "__main__": + main() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/utils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/utils.py new file mode 100644 index 0000000000000000000000000000000000000000..55cf4083ed5e5a6628fd3316c02ce1a5ce21a92c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/utils.py @@ -0,0 +1,38 @@ +try: + import mpmath as mp +except ImportError: + pass + +try: + from sympy.abc import x +except ImportError: + pass + + +def lagrange_inversion(a): + """Given a series + + f(x) = a[1]*x + a[2]*x**2 + ... + a[n-1]*x**(n - 1), + + use the Lagrange inversion formula to compute a series + + g(x) = b[1]*x + b[2]*x**2 + ... + b[n-1]*x**(n - 1) + + so that f(g(x)) = g(f(x)) = x mod x**n. We must have a[0] = 0, so + necessarily b[0] = 0 too. + + The algorithm is naive and could be improved, but speed isn't an + issue here and it's easy to read. + + """ + n = len(a) + f = sum(a[i]*x**i for i in range(n)) + h = (x/f).series(x, 0, n).removeO() + hpower = [h**0] + for k in range(n): + hpower.append((hpower[-1]*h).expand()) + b = [mp.mpf(0)] + for k in range(1, n): + b.append(hpower[k].coeff(x, k - 1)/k) + b = [mp.mpf(x) for x in b] + return b diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/wright_bessel.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/wright_bessel.py new file mode 100644 index 0000000000000000000000000000000000000000..51d56b1cd5c47c7ef005d21aad9827a1e85ec0d9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/wright_bessel.py @@ -0,0 +1,342 @@ +"""Precompute coefficients of several series expansions +of Wright's generalized Bessel function Phi(a, b, x). + +See https://dlmf.nist.gov/10.46.E1 with rho=a, beta=b, z=x. +""" +from argparse import ArgumentParser, RawTextHelpFormatter +import numpy as np +from scipy.integrate import quad +from scipy.optimize import minimize_scalar, curve_fit +from time import time + +try: + import sympy + from sympy import EulerGamma, Rational, S, Sum, \ + factorial, gamma, gammasimp, pi, polygamma, symbols, zeta + from sympy.polys.polyfuncs import horner +except ImportError: + pass + + +def series_small_a(): + """Tylor series expansion of Phi(a, b, x) in a=0 up to order 5. + """ + order = 5 + a, b, x, k = symbols("a b x k") + A = [] # terms with a + X = [] # terms with x + B = [] # terms with b (polygammas) + # Phi(a, b, x) = exp(x)/gamma(b) * sum(A[i] * X[i] * B[i]) + expression = Sum(x**k/factorial(k)/gamma(a*k+b), (k, 0, S.Infinity)) + expression = gamma(b)/sympy.exp(x) * expression + + # nth term of taylor series in a=0: a^n/n! * (d^n Phi(a, b, x)/da^n at a=0) + for n in range(0, order+1): + term = expression.diff(a, n).subs(a, 0).simplify().doit() + # set the whole bracket involving polygammas to 1 + x_part = (term.subs(polygamma(0, b), 1) + .replace(polygamma, lambda *args: 0)) + # sign convention: x part always positive + x_part *= (-1)**n + + A.append(a**n/factorial(n)) + X.append(horner(x_part)) + B.append(horner((term/x_part).simplify())) + + s = "Tylor series expansion of Phi(a, b, x) in a=0 up to order 5.\n" + s += "Phi(a, b, x) = exp(x)/gamma(b) * sum(A[i] * X[i] * B[i], i=0..5)\n" + for name, c in zip(['A', 'X', 'B'], [A, X, B]): + for i in range(len(c)): + s += f"\n{name}[{i}] = " + str(c[i]) + return s + + +# expansion of digamma +def dg_series(z, n): + """Symbolic expansion of digamma(z) in z=0 to order n. + + See https://dlmf.nist.gov/5.7.E4 and with https://dlmf.nist.gov/5.5.E2 + """ + k = symbols("k") + return -1/z - EulerGamma + \ + sympy.summation((-1)**k * zeta(k) * z**(k-1), (k, 2, n+1)) + + +def pg_series(k, z, n): + """Symbolic expansion of polygamma(k, z) in z=0 to order n.""" + return sympy.diff(dg_series(z, n+k), z, k) + + +def series_small_a_small_b(): + """Tylor series expansion of Phi(a, b, x) in a=0 and b=0 up to order 5. + + Be aware of cancellation of poles in b=0 of digamma(b)/Gamma(b) and + polygamma functions. + + digamma(b)/Gamma(b) = -1 - 2*M_EG*b + O(b^2) + digamma(b)^2/Gamma(b) = 1/b + 3*M_EG + b*(-5/12*PI^2+7/2*M_EG^2) + O(b^2) + polygamma(1, b)/Gamma(b) = 1/b + M_EG + b*(1/12*PI^2 + 1/2*M_EG^2) + O(b^2) + and so on. + """ + order = 5 + a, b, x, k = symbols("a b x k") + M_PI, M_EG, M_Z3 = symbols("M_PI M_EG M_Z3") + c_subs = {pi: M_PI, EulerGamma: M_EG, zeta(3): M_Z3} + A = [] # terms with a + X = [] # terms with x + B = [] # terms with b (polygammas expanded) + C = [] # terms that generate B + # Phi(a, b, x) = exp(x) * sum(A[i] * X[i] * B[i]) + # B[0] = 1 + # B[k] = sum(C[k] * b**k/k!, k=0..) + # Note: C[k] can be obtained from a series expansion of 1/gamma(b). + expression = gamma(b)/sympy.exp(x) * \ + Sum(x**k/factorial(k)/gamma(a*k+b), (k, 0, S.Infinity)) + + # nth term of taylor series in a=0: a^n/n! * (d^n Phi(a, b, x)/da^n at a=0) + for n in range(0, order+1): + term = expression.diff(a, n).subs(a, 0).simplify().doit() + # set the whole bracket involving polygammas to 1 + x_part = (term.subs(polygamma(0, b), 1) + .replace(polygamma, lambda *args: 0)) + # sign convention: x part always positive + x_part *= (-1)**n + # expansion of polygamma part with 1/gamma(b) + pg_part = term/x_part/gamma(b) + if n >= 1: + # Note: highest term is digamma^n + pg_part = pg_part.replace(polygamma, + lambda k, x: pg_series(k, x, order+1+n)) + pg_part = (pg_part.series(b, 0, n=order+1-n) + .removeO() + .subs(polygamma(2, 1), -2*zeta(3)) + .simplify() + ) + + A.append(a**n/factorial(n)) + X.append(horner(x_part)) + B.append(pg_part) + + # Calculate C and put in the k! + C = sympy.Poly(B[1].subs(c_subs), b).coeffs() + C.reverse() + for i in range(len(C)): + C[i] = (C[i] * factorial(i)).simplify() + + s = "Tylor series expansion of Phi(a, b, x) in a=0 and b=0 up to order 5." + s += "\nPhi(a, b, x) = exp(x) * sum(A[i] * X[i] * B[i], i=0..5)\n" + s += "B[0] = 1\n" + s += "B[i] = sum(C[k+i-1] * b**k/k!, k=0..)\n" + s += "\nM_PI = pi" + s += "\nM_EG = EulerGamma" + s += "\nM_Z3 = zeta(3)" + for name, c in zip(['A', 'X'], [A, X]): + for i in range(len(c)): + s += f"\n{name}[{i}] = " + s += str(c[i]) + # For C, do also compute the values numerically + for i in range(len(C)): + s += f"\n# C[{i}] = " + s += str(C[i]) + s += f"\nC[{i}] = " + s += str(C[i].subs({M_EG: EulerGamma, M_PI: pi, M_Z3: zeta(3)}) + .evalf(17)) + + # Does B have the assumed structure? + s += "\n\nTest if B[i] does have the assumed structure." + s += "\nC[i] are derived from B[1] alone." + s += "\nTest B[2] == C[1] + b*C[2] + b^2/2*C[3] + b^3/6*C[4] + .." + test = sum([b**k/factorial(k) * C[k+1] for k in range(order-1)]) + test = (test - B[2].subs(c_subs)).simplify() + s += f"\ntest successful = {test==S(0)}" + s += "\nTest B[3] == C[2] + b*C[3] + b^2/2*C[4] + .." + test = sum([b**k/factorial(k) * C[k+2] for k in range(order-2)]) + test = (test - B[3].subs(c_subs)).simplify() + s += f"\ntest successful = {test==S(0)}" + return s + + +def asymptotic_series(): + """Asymptotic expansion for large x. + + Phi(a, b, x) ~ Z^(1/2-b) * exp((1+a)/a * Z) * sum_k (-1)^k * C_k / Z^k + Z = (a*x)^(1/(1+a)) + + Wright (1935) lists the coefficients C_0 and C_1 (he calls them a_0 and + a_1). With slightly different notation, Paris (2017) lists coefficients + c_k up to order k=3. + Paris (2017) uses ZP = (1+a)/a * Z (ZP = Z of Paris) and + C_k = C_0 * (-a/(1+a))^k * c_k + """ + order = 8 + + class g(sympy.Function): + """Helper function g according to Wright (1935) + + g(n, rho, v) = (1 + (rho+2)/3 * v + (rho+2)*(rho+3)/(2*3) * v^2 + ...) + + Note: Wright (1935) uses square root of above definition. + """ + nargs = 3 + + @classmethod + def eval(cls, n, rho, v): + if not n >= 0: + raise ValueError("must have n >= 0") + elif n == 0: + return 1 + else: + return g(n-1, rho, v) \ + + gammasimp(gamma(rho+2+n)/gamma(rho+2)) \ + / gammasimp(gamma(3+n)/gamma(3))*v**n + + class coef_C(sympy.Function): + """Calculate coefficients C_m for integer m. + + C_m is the coefficient of v^(2*m) in the Taylor expansion in v=0 of + Gamma(m+1/2)/(2*pi) * (2/(rho+1))^(m+1/2) * (1-v)^(-b) + * g(rho, v)^(-m-1/2) + """ + nargs = 3 + + @classmethod + def eval(cls, m, rho, beta): + if not m >= 0: + raise ValueError("must have m >= 0") + + v = symbols("v") + expression = (1-v)**(-beta) * g(2*m, rho, v)**(-m-Rational(1, 2)) + res = expression.diff(v, 2*m).subs(v, 0) / factorial(2*m) + res = res * (gamma(m + Rational(1, 2)) / (2*pi) + * (2/(rho+1))**(m + Rational(1, 2))) + return res + + # in order to have nice ordering/sorting of expressions, we set a = xa. + xa, b, xap1 = symbols("xa b xap1") + C0 = coef_C(0, xa, b) + # a1 = a(1, rho, beta) + s = "Asymptotic expansion for large x\n" + s += "Phi(a, b, x) = Z**(1/2-b) * exp((1+a)/a * Z) \n" + s += " * sum((-1)**k * C[k]/Z**k, k=0..6)\n\n" + s += "Z = pow(a * x, 1/(1+a))\n" + s += "A[k] = pow(a, k)\n" + s += "B[k] = pow(b, k)\n" + s += "Ap1[k] = pow(1+a, k)\n\n" + s += "C[0] = 1./sqrt(2. * M_PI * Ap1[1])\n" + for i in range(1, order+1): + expr = (coef_C(i, xa, b) / (C0/(1+xa)**i)).simplify() + factor = [x.denominator() for x in sympy.Poly(expr).coeffs()] + factor = sympy.lcm(factor) + expr = (expr * factor).simplify().collect(b, sympy.factor) + expr = expr.xreplace({xa+1: xap1}) + s += f"C[{i}] = C[0] / ({factor} * Ap1[{i}])\n" + s += f"C[{i}] *= {str(expr)}\n\n" + import re + re_a = re.compile(r'xa\*\*(\d+)') + s = re_a.sub(r'A[\1]', s) + re_b = re.compile(r'b\*\*(\d+)') + s = re_b.sub(r'B[\1]', s) + s = s.replace('xap1', 'Ap1[1]') + s = s.replace('xa', 'a') + # max integer = 2^31-1 = 2,147,483,647. Solution: Put a point after 10 + # or more digits. + re_digits = re.compile(r'(\d{10,})') + s = re_digits.sub(r'\1.', s) + return s + + +def optimal_epsilon_integral(): + """Fit optimal choice of epsilon for integral representation. + + The integrand of + int_0^pi P(eps, a, b, x, phi) * dphi + can exhibit oscillatory behaviour. It stems from the cosine of P and can be + minimized by minimizing the arc length of the argument + f(phi) = eps * sin(phi) - x * eps^(-a) * sin(a * phi) + (1 - b) * phi + of cos(f(phi)). + We minimize the arc length in eps for a grid of values (a, b, x) and fit a + parametric function to it. + """ + def fp(eps, a, b, x, phi): + """Derivative of f w.r.t. phi.""" + eps_a = np.power(1. * eps, -a) + return eps * np.cos(phi) - a * x * eps_a * np.cos(a * phi) + 1 - b + + def arclength(eps, a, b, x, epsrel=1e-2, limit=100): + """Compute Arc length of f. + + Note that the arc length of a function f from t0 to t1 is given by + int_t0^t1 sqrt(1 + f'(t)^2) dt + """ + return quad(lambda phi: np.sqrt(1 + fp(eps, a, b, x, phi)**2), + 0, np.pi, + epsrel=epsrel, limit=100)[0] + + # grid of minimal arc length values + data_a = [1e-3, 0.1, 0.5, 0.9, 1, 2, 4, 5, 6, 8] + data_b = [0, 1, 4, 7, 10] + data_x = [1, 1.5, 2, 4, 10, 20, 50, 100, 200, 500, 1e3, 5e3, 1e4] + data_a, data_b, data_x = np.meshgrid(data_a, data_b, data_x) + data_a, data_b, data_x = (data_a.flatten(), data_b.flatten(), + data_x.flatten()) + best_eps = [] + for i in range(data_x.size): + best_eps.append( + minimize_scalar(lambda eps: arclength(eps, data_a[i], data_b[i], + data_x[i]), + bounds=(1e-3, 1000), + method='Bounded', options={'xatol': 1e-3}).x + ) + best_eps = np.array(best_eps) + # pandas would be nice, but here a dictionary is enough + df = {'a': data_a, + 'b': data_b, + 'x': data_x, + 'eps': best_eps, + } + + def func(data, A0, A1, A2, A3, A4, A5): + """Compute parametric function to fit.""" + a = data['a'] + b = data['b'] + x = data['x'] + return (A0 * b * np.exp(-0.5 * a) + + np.exp(A1 + 1 / (1 + a) * np.log(x) - A2 * np.exp(-A3 * a) + + A4 / (1 + np.exp(A5 * a)))) + + func_params = list(curve_fit(func, df, df['eps'], method='trf')[0]) + + s = "Fit optimal eps for integrand P via minimal arc length\n" + s += "with parametric function:\n" + s += "optimal_eps = (A0 * b * exp(-a/2) + exp(A1 + 1 / (1 + a) * log(x)\n" + s += " - A2 * exp(-A3 * a) + A4 / (1 + exp(A5 * a)))\n\n" + s += "Fitted parameters A0 to A5 are:\n" + s += ', '.join([f'{x:.5g}' for x in func_params]) + return s + + +def main(): + t0 = time() + parser = ArgumentParser(description=__doc__, + formatter_class=RawTextHelpFormatter) + parser.add_argument('action', type=int, choices=[1, 2, 3, 4], + help='chose what expansion to precompute\n' + '1 : Series for small a\n' + '2 : Series for small a and small b\n' + '3 : Asymptotic series for large x\n' + ' This may take some time (>4h).\n' + '4 : Fit optimal eps for integral representation.' + ) + args = parser.parse_args() + + switch = {1: lambda: print(series_small_a()), + 2: lambda: print(series_small_a_small_b()), + 3: lambda: print(asymptotic_series()), + 4: lambda: print(optimal_epsilon_integral()) + } + switch.get(args.action, lambda: print("Invalid input."))() + print(f"\n{(time() - t0)/60:.1f} minutes elapsed.\n") + + +if __name__ == '__main__': + main() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/wright_bessel_data.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/wright_bessel_data.py new file mode 100644 index 0000000000000000000000000000000000000000..1de9b4fe552ca9178c452194ab84af6ca5daac71 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/wright_bessel_data.py @@ -0,0 +1,152 @@ +"""Compute a grid of values for Wright's generalized Bessel function +and save the values to data files for use in tests. Using mpmath directly in +tests would take too long. + +This takes about 10 minutes to run on a 2.7 GHz i7 Macbook Pro. +""" +from functools import lru_cache +import os +from time import time + +import numpy as np +from scipy.special._mptestutils import mpf2float + +try: + import mpmath as mp +except ImportError: + pass + +# exp_inf: smallest value x for which exp(x) == inf +exp_inf = 709.78271289338403 + + +# 64 Byte per value +@lru_cache(maxsize=100_000) +def rgamma_cached(x, dps): + with mp.workdps(dps): + return mp.rgamma(x) + + +def mp_wright_bessel(a, b, x, dps=50, maxterms=2000): + """Compute Wright's generalized Bessel function as Series with mpmath. + """ + with mp.workdps(dps): + a, b, x = mp.mpf(a), mp.mpf(b), mp.mpf(x) + res = mp.nsum(lambda k: x**k / mp.fac(k) + * rgamma_cached(a * k + b, dps=dps), + [0, mp.inf], + tol=dps, method='s', steps=[maxterms] + ) + return mpf2float(res) + + +def main(): + t0 = time() + print(__doc__) + pwd = os.path.dirname(__file__) + eps = np.finfo(float).eps * 100 + + a_range = np.array([eps, + 1e-4 * (1 - eps), 1e-4, 1e-4 * (1 + eps), + 1e-3 * (1 - eps), 1e-3, 1e-3 * (1 + eps), + 0.1, 0.5, + 1 * (1 - eps), 1, 1 * (1 + eps), + 1.5, 2, 4.999, 5, 10]) + b_range = np.array([0, eps, 1e-10, 1e-5, 0.1, 1, 2, 10, 20, 100]) + x_range = np.array([0, eps, 1 - eps, 1, 1 + eps, + 1.5, + 2 - eps, 2, 2 + eps, + 9 - eps, 9, 9 + eps, + 10 * (1 - eps), 10, 10 * (1 + eps), + 100 * (1 - eps), 100, 100 * (1 + eps), + 500, exp_inf, 1e3, 1e5, 1e10, 1e20]) + + a_range, b_range, x_range = np.meshgrid(a_range, b_range, x_range, + indexing='ij') + a_range = a_range.flatten() + b_range = b_range.flatten() + x_range = x_range.flatten() + + # filter out some values, especially too large x + bool_filter = ~((a_range < 5e-3) & (x_range >= exp_inf)) + bool_filter = bool_filter & ~((a_range < 0.2) & (x_range > exp_inf)) + bool_filter = bool_filter & ~((a_range < 0.5) & (x_range > 1e3)) + bool_filter = bool_filter & ~((a_range < 0.56) & (x_range > 5e3)) + bool_filter = bool_filter & ~((a_range < 1) & (x_range > 1e4)) + bool_filter = bool_filter & ~((a_range < 1.4) & (x_range > 1e5)) + bool_filter = bool_filter & ~((a_range < 1.8) & (x_range > 1e6)) + bool_filter = bool_filter & ~((a_range < 2.2) & (x_range > 1e7)) + bool_filter = bool_filter & ~((a_range < 2.5) & (x_range > 1e8)) + bool_filter = bool_filter & ~((a_range < 2.9) & (x_range > 1e9)) + bool_filter = bool_filter & ~((a_range < 3.3) & (x_range > 1e10)) + bool_filter = bool_filter & ~((a_range < 3.7) & (x_range > 1e11)) + bool_filter = bool_filter & ~((a_range < 4) & (x_range > 1e12)) + bool_filter = bool_filter & ~((a_range < 4.4) & (x_range > 1e13)) + bool_filter = bool_filter & ~((a_range < 4.7) & (x_range > 1e14)) + bool_filter = bool_filter & ~((a_range < 5.1) & (x_range > 1e15)) + bool_filter = bool_filter & ~((a_range < 5.4) & (x_range > 1e16)) + bool_filter = bool_filter & ~((a_range < 5.8) & (x_range > 1e17)) + bool_filter = bool_filter & ~((a_range < 6.2) & (x_range > 1e18)) + bool_filter = bool_filter & ~((a_range < 6.2) & (x_range > 1e18)) + bool_filter = bool_filter & ~((a_range < 6.5) & (x_range > 1e19)) + bool_filter = bool_filter & ~((a_range < 6.9) & (x_range > 1e20)) + + # filter out known values that do not meet the required numerical accuracy + # see test test_wright_data_grid_failures + failing = np.array([ + [0.1, 100, 709.7827128933841], + [0.5, 10, 709.7827128933841], + [0.5, 10, 1000], + [0.5, 100, 1000], + [1, 20, 100000], + [1, 100, 100000], + [1.0000000000000222, 20, 100000], + [1.0000000000000222, 100, 100000], + [1.5, 0, 500], + [1.5, 2.220446049250313e-14, 500], + [1.5, 1.e-10, 500], + [1.5, 1.e-05, 500], + [1.5, 0.1, 500], + [1.5, 20, 100000], + [1.5, 100, 100000], + ]).tolist() + + does_fail = np.full_like(a_range, False, dtype=bool) + for i in range(x_range.size): + if [a_range[i], b_range[i], x_range[i]] in failing: + does_fail[i] = True + + # filter and flatten + a_range = a_range[bool_filter] + b_range = b_range[bool_filter] + x_range = x_range[bool_filter] + does_fail = does_fail[bool_filter] + + dataset = [] + print(f"Computing {x_range.size} single points.") + print("Tests will fail for the following data points:") + for i in range(x_range.size): + a = a_range[i] + b = b_range[i] + x = x_range[i] + # take care of difficult corner cases + maxterms = 1000 + if a < 1e-6 and x >= exp_inf/10: + maxterms = 2000 + f = mp_wright_bessel(a, b, x, maxterms=maxterms) + if does_fail[i]: + print("failing data point a, b, x, value = " + f"[{a}, {b}, {x}, {f}]") + else: + dataset.append((a, b, x, f)) + dataset = np.array(dataset) + + filename = os.path.join(pwd, '..', 'tests', 'data', 'local', + 'wright_bessel.txt') + np.savetxt(filename, dataset) + + print(f"{(time() - t0)/60:.1f} minutes elapsed") + + +if __name__ == "__main__": + main() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/wrightomega.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/wrightomega.py new file mode 100644 index 0000000000000000000000000000000000000000..0bcd0345a9c1b90c45b0e9e3340ab4da4ec5c6d7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/wrightomega.py @@ -0,0 +1,41 @@ +import numpy as np + +try: + import mpmath +except ImportError: + pass + + +def mpmath_wrightomega(x): + return mpmath.lambertw(mpmath.exp(x), mpmath.mpf('-0.5')) + + +def wrightomega_series_error(x): + series = x + desired = mpmath_wrightomega(x) + return abs(series - desired) / desired + + +def wrightomega_exp_error(x): + exponential_approx = mpmath.exp(x) + desired = mpmath_wrightomega(x) + return abs(exponential_approx - desired) / desired + + +def main(): + desired_error = 2 * np.finfo(float).eps + print('Series Error') + for x in [1e5, 1e10, 1e15, 1e20]: + with mpmath.workdps(100): + error = wrightomega_series_error(x) + print(x, error, error < desired_error) + + print('Exp error') + for x in [-10, -25, -50, -100, -200, -400, -700, -740]: + with mpmath.workdps(100): + error = wrightomega_exp_error(x) + print(x, error, error < desired_error) + + +if __name__ == '__main__': + main() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/zetac.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/zetac.py new file mode 100644 index 0000000000000000000000000000000000000000..d408b1a2fffb6872287452923fcc9394adc13a7c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_precompute/zetac.py @@ -0,0 +1,27 @@ +"""Compute the Taylor series for zeta(x) - 1 around x = 0.""" +try: + import mpmath +except ImportError: + pass + + +def zetac_series(N): + coeffs = [] + with mpmath.workdps(100): + coeffs.append(-1.5) + for n in range(1, N): + coeff = mpmath.diff(mpmath.zeta, 0, n)/mpmath.factorial(n) + coeffs.append(coeff) + return coeffs + + +def main(): + print(__doc__) + coeffs = zetac_series(10) + coeffs = [mpmath.nstr(x, 20, min_fixed=0, max_fixed=0) + for x in coeffs] + print("\n".join(coeffs[::-1])) + + +if __name__ == '__main__': + main() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_sf_error.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_sf_error.py new file mode 100644 index 0000000000000000000000000000000000000000..e1edc9800759dfda9e49bde1becc775a64bce958 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_sf_error.py @@ -0,0 +1,15 @@ +"""Warnings and Exceptions that can be raised by special functions.""" +import warnings + + +class SpecialFunctionWarning(Warning): + """Warning that can be emitted by special functions.""" + pass + + +warnings.simplefilter("always", category=SpecialFunctionWarning) + + +class SpecialFunctionError(Exception): + """Exception that can be raised by special functions.""" + pass diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_spfun_stats.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_spfun_stats.py new file mode 100644 index 0000000000000000000000000000000000000000..2525eceb47ec2b20b45ca693e19e741f4a666597 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_spfun_stats.py @@ -0,0 +1,106 @@ +# Last Change: Sat Mar 21 02:00 PM 2009 J + +# Copyright (c) 2001, 2002 Enthought, Inc. +# +# All rights reserved. +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions are met: +# +# a. Redistributions of source code must retain the above copyright notice, +# this list of conditions and the following disclaimer. +# b. Redistributions in binary form must reproduce the above copyright +# notice, this list of conditions and the following disclaimer in the +# documentation and/or other materials provided with the distribution. +# c. Neither the name of the Enthought nor the names of its contributors +# may be used to endorse or promote products derived from this software +# without specific prior written permission. +# +# +# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" +# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE +# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE +# ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR +# ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR +# SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER +# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT +# LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY +# OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH +# DAMAGE. + +"""Some more special functions which may be useful for multivariate statistical +analysis.""" + +import numpy as np +from scipy.special import gammaln as loggam + + +__all__ = ['multigammaln'] + + +def multigammaln(a, d): + r"""Returns the log of multivariate gamma, also sometimes called the + generalized gamma. + + Parameters + ---------- + a : ndarray + The multivariate gamma is computed for each item of `a`. + d : int + The dimension of the space of integration. + + Returns + ------- + res : ndarray + The values of the log multivariate gamma at the given points `a`. + + Notes + ----- + The formal definition of the multivariate gamma of dimension d for a real + `a` is + + .. math:: + + \Gamma_d(a) = \int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA + + with the condition :math:`a > (d-1)/2`, and :math:`A > 0` being the set of + all the positive definite matrices of dimension `d`. Note that `a` is a + scalar: the integrand only is multivariate, the argument is not (the + function is defined over a subset of the real set). + + This can be proven to be equal to the much friendlier equation + + .. math:: + + \Gamma_d(a) = \pi^{d(d-1)/4} \prod_{i=1}^{d} \Gamma(a - (i-1)/2). + + References + ---------- + R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in + probability and mathematical statistics). + + Examples + -------- + >>> import numpy as np + >>> from scipy.special import multigammaln, gammaln + >>> a = 23.5 + >>> d = 10 + >>> multigammaln(a, d) + 454.1488605074416 + + Verify that the result agrees with the logarithm of the equation + shown above: + + >>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum() + 454.1488605074416 + """ + a = np.asarray(a) + if not np.isscalar(d) or (np.floor(d) != d): + raise ValueError("d should be a positive integer (dimension)") + if np.any(a <= 0.5 * (d - 1)): + raise ValueError(f"condition a ({a:f}) > 0.5 * (d-1) ({0.5 * (d-1):f}) not met") + + res = (d * (d-1) * 0.25) * np.log(np.pi) + res += np.sum(loggam([(a - (j - 1.)/2) for j in range(1, d+1)]), axis=0) + return res diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_spherical_bessel.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_spherical_bessel.py new file mode 100644 index 0000000000000000000000000000000000000000..f3d871fcd07ef092962a4594cf780445daae4458 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_spherical_bessel.py @@ -0,0 +1,397 @@ +from functools import wraps +from scipy._lib._util import _lazywhere +import numpy as np +from ._ufuncs import (_spherical_jn, _spherical_yn, _spherical_in, + _spherical_kn, _spherical_jn_d, _spherical_yn_d, + _spherical_in_d, _spherical_kn_d) + + +def use_reflection(sign_n_even=None, reflection_fun=None): + # - If reflection_fun is not specified, reflects negative `z` and multiplies + # output by appropriate sign (indicated by `sign_n_even`). + # - If reflection_fun is specified, calls `reflection_fun` instead of `fun`. + # See DLMF 10.47(v) https://dlmf.nist.gov/10.47 + def decorator(fun): + def standard_reflection(n, z, derivative): + # sign_n_even indicates the sign when the order `n` is even + sign = np.where(n % 2 == 0, sign_n_even, -sign_n_even) + # By the chain rule, differentiation at `-z` adds a minus sign + sign = -sign if derivative else sign + # Evaluate at positive z (minus negative z) and adjust the sign + return fun(n, -z, derivative) * sign + + @wraps(fun) + def wrapper(n, z, derivative=False): + z = np.asarray(z) + + if np.issubdtype(z.dtype, np.complexfloating): + return fun(n, z, derivative) # complex dtype just works + + f2 = standard_reflection if reflection_fun is None else reflection_fun + return _lazywhere(z.real >= 0, (n, z), + f=lambda n, z: fun(n, z, derivative), + f2=lambda n, z: f2(n, z, derivative))[()] + return wrapper + return decorator + + +@use_reflection(+1) # See DLMF 10.47(v) https://dlmf.nist.gov/10.47 +def spherical_jn(n, z, derivative=False): + r"""Spherical Bessel function of the first kind or its derivative. + + Defined as [1]_, + + .. math:: j_n(z) = \sqrt{\frac{\pi}{2z}} J_{n + 1/2}(z), + + where :math:`J_n` is the Bessel function of the first kind. + + Parameters + ---------- + n : int, array_like + Order of the Bessel function (n >= 0). + z : complex or float, array_like + Argument of the Bessel function. + derivative : bool, optional + If True, the value of the derivative (rather than the function + itself) is returned. + + Returns + ------- + jn : ndarray + + Notes + ----- + For real arguments greater than the order, the function is computed + using the ascending recurrence [2]_. For small real or complex + arguments, the definitional relation to the cylindrical Bessel function + of the first kind is used. + + The derivative is computed using the relations [3]_, + + .. math:: + j_n'(z) = j_{n-1}(z) - \frac{n + 1}{z} j_n(z). + + j_0'(z) = -j_1(z) + + + .. versionadded:: 0.18.0 + + References + ---------- + .. [1] https://dlmf.nist.gov/10.47.E3 + .. [2] https://dlmf.nist.gov/10.51.E1 + .. [3] https://dlmf.nist.gov/10.51.E2 + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + The spherical Bessel functions of the first kind :math:`j_n` accept + both real and complex second argument. They can return a complex type: + + >>> from scipy.special import spherical_jn + >>> spherical_jn(0, 3+5j) + (-9.878987731663194-8.021894345786002j) + >>> type(spherical_jn(0, 3+5j)) + + + We can verify the relation for the derivative from the Notes + for :math:`n=3` in the interval :math:`[1, 2]`: + + >>> import numpy as np + >>> x = np.arange(1.0, 2.0, 0.01) + >>> np.allclose(spherical_jn(3, x, True), + ... spherical_jn(2, x) - 4/x * spherical_jn(3, x)) + True + + The first few :math:`j_n` with real argument: + + >>> import matplotlib.pyplot as plt + >>> x = np.arange(0.0, 10.0, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(-0.5, 1.5) + >>> ax.set_title(r'Spherical Bessel functions $j_n$') + >>> for n in np.arange(0, 4): + ... ax.plot(x, spherical_jn(n, x), label=rf'$j_{n}$') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + n = np.asarray(n, dtype=np.dtype("long")) + if derivative: + return _spherical_jn_d(n, z) + else: + return _spherical_jn(n, z) + + +@use_reflection(-1) # See DLMF 10.47(v) https://dlmf.nist.gov/10.47 +def spherical_yn(n, z, derivative=False): + r"""Spherical Bessel function of the second kind or its derivative. + + Defined as [1]_, + + .. math:: y_n(z) = \sqrt{\frac{\pi}{2z}} Y_{n + 1/2}(z), + + where :math:`Y_n` is the Bessel function of the second kind. + + Parameters + ---------- + n : int, array_like + Order of the Bessel function (n >= 0). + z : complex or float, array_like + Argument of the Bessel function. + derivative : bool, optional + If True, the value of the derivative (rather than the function + itself) is returned. + + Returns + ------- + yn : ndarray + + Notes + ----- + For real arguments, the function is computed using the ascending + recurrence [2]_. For complex arguments, the definitional relation to + the cylindrical Bessel function of the second kind is used. + + The derivative is computed using the relations [3]_, + + .. math:: + y_n' = y_{n-1} - \frac{n + 1}{z} y_n. + + y_0' = -y_1 + + + .. versionadded:: 0.18.0 + + References + ---------- + .. [1] https://dlmf.nist.gov/10.47.E4 + .. [2] https://dlmf.nist.gov/10.51.E1 + .. [3] https://dlmf.nist.gov/10.51.E2 + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + The spherical Bessel functions of the second kind :math:`y_n` accept + both real and complex second argument. They can return a complex type: + + >>> from scipy.special import spherical_yn + >>> spherical_yn(0, 3+5j) + (8.022343088587197-9.880052589376795j) + >>> type(spherical_yn(0, 3+5j)) + + + We can verify the relation for the derivative from the Notes + for :math:`n=3` in the interval :math:`[1, 2]`: + + >>> import numpy as np + >>> x = np.arange(1.0, 2.0, 0.01) + >>> np.allclose(spherical_yn(3, x, True), + ... spherical_yn(2, x) - 4/x * spherical_yn(3, x)) + True + + The first few :math:`y_n` with real argument: + + >>> import matplotlib.pyplot as plt + >>> x = np.arange(0.0, 10.0, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(-2.0, 1.0) + >>> ax.set_title(r'Spherical Bessel functions $y_n$') + >>> for n in np.arange(0, 4): + ... ax.plot(x, spherical_yn(n, x), label=rf'$y_{n}$') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + n = np.asarray(n, dtype=np.dtype("long")) + if derivative: + return _spherical_yn_d(n, z) + else: + return _spherical_yn(n, z) + + +@use_reflection(+1) # See DLMF 10.47(v) https://dlmf.nist.gov/10.47 +def spherical_in(n, z, derivative=False): + r"""Modified spherical Bessel function of the first kind or its derivative. + + Defined as [1]_, + + .. math:: i_n(z) = \sqrt{\frac{\pi}{2z}} I_{n + 1/2}(z), + + where :math:`I_n` is the modified Bessel function of the first kind. + + Parameters + ---------- + n : int, array_like + Order of the Bessel function (n >= 0). + z : complex or float, array_like + Argument of the Bessel function. + derivative : bool, optional + If True, the value of the derivative (rather than the function + itself) is returned. + + Returns + ------- + in : ndarray + + Notes + ----- + The function is computed using its definitional relation to the + modified cylindrical Bessel function of the first kind. + + The derivative is computed using the relations [2]_, + + .. math:: + i_n' = i_{n-1} - \frac{n + 1}{z} i_n. + + i_1' = i_0 + + + .. versionadded:: 0.18.0 + + References + ---------- + .. [1] https://dlmf.nist.gov/10.47.E7 + .. [2] https://dlmf.nist.gov/10.51.E5 + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + The modified spherical Bessel functions of the first kind :math:`i_n` + accept both real and complex second argument. + They can return a complex type: + + >>> from scipy.special import spherical_in + >>> spherical_in(0, 3+5j) + (-1.1689867793369182-1.2697305267234222j) + >>> type(spherical_in(0, 3+5j)) + + + We can verify the relation for the derivative from the Notes + for :math:`n=3` in the interval :math:`[1, 2]`: + + >>> import numpy as np + >>> x = np.arange(1.0, 2.0, 0.01) + >>> np.allclose(spherical_in(3, x, True), + ... spherical_in(2, x) - 4/x * spherical_in(3, x)) + True + + The first few :math:`i_n` with real argument: + + >>> import matplotlib.pyplot as plt + >>> x = np.arange(0.0, 6.0, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(-0.5, 5.0) + >>> ax.set_title(r'Modified spherical Bessel functions $i_n$') + >>> for n in np.arange(0, 4): + ... ax.plot(x, spherical_in(n, x), label=rf'$i_{n}$') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + n = np.asarray(n, dtype=np.dtype("long")) + if derivative: + return _spherical_in_d(n, z) + else: + return _spherical_in(n, z) + + +def spherical_kn_reflection(n, z, derivative=False): + # More complex than the other cases, and this will likely be re-implemented + # in C++ anyway. Would require multiple function evaluations. Probably about + # as fast to just resort to complex math, and much simpler. + return spherical_kn(n, z + 0j, derivative=derivative).real + + +@use_reflection(reflection_fun=spherical_kn_reflection) +def spherical_kn(n, z, derivative=False): + r"""Modified spherical Bessel function of the second kind or its derivative. + + Defined as [1]_, + + .. math:: k_n(z) = \sqrt{\frac{\pi}{2z}} K_{n + 1/2}(z), + + where :math:`K_n` is the modified Bessel function of the second kind. + + Parameters + ---------- + n : int, array_like + Order of the Bessel function (n >= 0). + z : complex or float, array_like + Argument of the Bessel function. + derivative : bool, optional + If True, the value of the derivative (rather than the function + itself) is returned. + + Returns + ------- + kn : ndarray + + Notes + ----- + The function is computed using its definitional relation to the + modified cylindrical Bessel function of the second kind. + + The derivative is computed using the relations [2]_, + + .. math:: + k_n' = -k_{n-1} - \frac{n + 1}{z} k_n. + + k_0' = -k_1 + + + .. versionadded:: 0.18.0 + + References + ---------- + .. [1] https://dlmf.nist.gov/10.47.E9 + .. [2] https://dlmf.nist.gov/10.51.E5 + .. [AS] Milton Abramowitz and Irene A. Stegun, eds. + Handbook of Mathematical Functions with Formulas, + Graphs, and Mathematical Tables. New York: Dover, 1972. + + Examples + -------- + The modified spherical Bessel functions of the second kind :math:`k_n` + accept both real and complex second argument. + They can return a complex type: + + >>> from scipy.special import spherical_kn + >>> spherical_kn(0, 3+5j) + (0.012985785614001561+0.003354691603137546j) + >>> type(spherical_kn(0, 3+5j)) + + + We can verify the relation for the derivative from the Notes + for :math:`n=3` in the interval :math:`[1, 2]`: + + >>> import numpy as np + >>> x = np.arange(1.0, 2.0, 0.01) + >>> np.allclose(spherical_kn(3, x, True), + ... - 4/x * spherical_kn(3, x) - spherical_kn(2, x)) + True + + The first few :math:`k_n` with real argument: + + >>> import matplotlib.pyplot as plt + >>> x = np.arange(0.0, 4.0, 0.01) + >>> fig, ax = plt.subplots() + >>> ax.set_ylim(0.0, 5.0) + >>> ax.set_title(r'Modified spherical Bessel functions $k_n$') + >>> for n in np.arange(0, 4): + ... ax.plot(x, spherical_kn(n, x), label=rf'$k_{n}$') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + n = np.asarray(n, dtype=np.dtype("long")) + if derivative: + return _spherical_kn_d(n, z) + else: + return _spherical_kn(n, z) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_support_alternative_backends.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_support_alternative_backends.py new file mode 100644 index 0000000000000000000000000000000000000000..3f5101198cebcbb5138e88a2050a5c4d7a115d60 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_support_alternative_backends.py @@ -0,0 +1,202 @@ +import os +import sys +import functools + +import numpy as np +from scipy._lib._array_api import ( + array_namespace, scipy_namespace_for, is_numpy +) +from . import _ufuncs +# These don't really need to be imported, but otherwise IDEs might not realize +# that these are defined in this file / report an error in __init__.py +from ._ufuncs import ( + log_ndtr, ndtr, ndtri, erf, erfc, i0, i0e, i1, i1e, gammaln, # noqa: F401 + gammainc, gammaincc, logit, expit, entr, rel_entr, xlogy, # noqa: F401 + chdtr, chdtrc, betainc, betaincc, stdtr # noqa: F401 +) + +_SCIPY_ARRAY_API = os.environ.get("SCIPY_ARRAY_API", False) +array_api_compat_prefix = "scipy._lib.array_api_compat" + + +def get_array_special_func(f_name, xp, n_array_args): + spx = scipy_namespace_for(xp) + f = None + if is_numpy(xp): + f = getattr(_ufuncs, f_name, None) + elif spx is not None: + f = getattr(spx.special, f_name, None) + + if f is not None: + return f + + # if generic array-API implementation is available, use that; + # otherwise, fall back to NumPy/SciPy + if f_name in _generic_implementations: + _f = _generic_implementations[f_name](xp=xp, spx=spx) + if _f is not None: + return _f + + _f = getattr(_ufuncs, f_name, None) + def __f(*args, _f=_f, _xp=xp, **kwargs): + array_args = args[:n_array_args] + other_args = args[n_array_args:] + array_args = [np.asarray(arg) for arg in array_args] + out = _f(*array_args, *other_args, **kwargs) + return _xp.asarray(out) + + return __f + + +def _get_shape_dtype(*args, xp): + args = xp.broadcast_arrays(*args) + shape = args[0].shape + dtype = xp.result_type(*args) + if xp.isdtype(dtype, 'integral'): + dtype = xp.float64 + args = [xp.asarray(arg, dtype=dtype) for arg in args] + return args, shape, dtype + + +def _rel_entr(xp, spx): + def __rel_entr(x, y, *, xp=xp): + args, shape, dtype = _get_shape_dtype(x, y, xp=xp) + x, y = args + res = xp.full(x.shape, xp.inf, dtype=dtype) + res[(x == 0) & (y >= 0)] = xp.asarray(0, dtype=dtype) + i = (x > 0) & (y > 0) + res[i] = x[i] * (xp.log(x[i]) - xp.log(y[i])) + return res + return __rel_entr + + +def _xlogy(xp, spx): + def __xlogy(x, y, *, xp=xp): + with np.errstate(divide='ignore', invalid='ignore'): + temp = x * xp.log(y) + return xp.where(x == 0., xp.asarray(0., dtype=temp.dtype), temp) + return __xlogy + + +def _chdtr(xp, spx): + # The difference between this and just using `gammainc` + # defined by `get_array_special_func` is that if `gammainc` + # isn't found, we don't want to use the SciPy version; we'll + # return None here and use the SciPy version of `chdtr`. + gammainc = getattr(spx.special, 'gammainc', None) if spx else None # noqa: F811 + if gammainc is None and hasattr(xp, 'special'): + gammainc = getattr(xp.special, 'gammainc', None) + if gammainc is None: + return None + + def __chdtr(v, x): + res = gammainc(v / 2, x / 2) # this is almost all we need + # The rest can be removed when google/jax#20507 is resolved + mask = (v == 0) & (x > 0) # JAX returns NaN + res = xp.where(mask, 1., res) + mask = xp.isinf(v) & xp.isinf(x) # JAX returns 1.0 + return xp.where(mask, xp.nan, res) + return __chdtr + + +def _chdtrc(xp, spx): + # The difference between this and just using `gammaincc` + # defined by `get_array_special_func` is that if `gammaincc` + # isn't found, we don't want to use the SciPy version; we'll + # return None here and use the SciPy version of `chdtrc`. + gammaincc = getattr(spx.special, 'gammaincc', None) if spx else None # noqa: F811 + if gammaincc is None and hasattr(xp, 'special'): + gammaincc = getattr(xp.special, 'gammaincc', None) + if gammaincc is None: + return None + + def __chdtrc(v, x): + res = xp.where(x >= 0, gammaincc(v/2, x/2), 1) + i_nan = ((x == 0) & (v == 0)) | xp.isnan(x) | xp.isnan(v) | (v <= 0) + res = xp.where(i_nan, xp.nan, res) + return res + return __chdtrc + + +def _betaincc(xp, spx): + betainc = getattr(spx.special, 'betainc', None) if spx else None # noqa: F811 + if betainc is None and hasattr(xp, 'special'): + betainc = getattr(xp.special, 'betainc', None) + if betainc is None: + return None + + def __betaincc(a, b, x): + # not perfect; might want to just rely on SciPy + return betainc(b, a, 1-x) + return __betaincc + + +def _stdtr(xp, spx): + betainc = getattr(spx.special, 'betainc', None) if spx else None # noqa: F811 + if betainc is None and hasattr(xp, 'special'): + betainc = getattr(xp.special, 'betainc', None) + if betainc is None: + return None + + def __stdtr(df, t): + x = df / (t ** 2 + df) + tail = betainc(df / 2, 0.5, x) / 2 + return xp.where(t < 0, tail, 1 - tail) + + return __stdtr + + +_generic_implementations = {'rel_entr': _rel_entr, + 'xlogy': _xlogy, + 'chdtr': _chdtr, + 'chdtrc': _chdtrc, + 'betaincc': _betaincc, + 'stdtr': _stdtr, + } + + +# functools.wraps doesn't work because: +# 'numpy.ufunc' object has no attribute '__module__' +def support_alternative_backends(f_name, n_array_args): + func = getattr(_ufuncs, f_name) + + @functools.wraps(func) + def wrapped(*args, **kwargs): + xp = array_namespace(*args[:n_array_args]) + f = get_array_special_func(f_name, xp, n_array_args) + return f(*args, **kwargs) + + return wrapped + + +array_special_func_map = { + 'log_ndtr': 1, + 'ndtr': 1, + 'ndtri': 1, + 'erf': 1, + 'erfc': 1, + 'i0': 1, + 'i0e': 1, + 'i1': 1, + 'i1e': 1, + 'gammaln': 1, + 'gammainc': 2, + 'gammaincc': 2, + 'logit': 1, + 'expit': 1, + 'entr': 1, + 'rel_entr': 2, + 'xlogy': 2, + 'chdtr': 2, + 'chdtrc': 2, + 'betainc': 3, + 'betaincc': 3, + 'stdtr': 2, +} + +for f_name, n_array_args in array_special_func_map.items(): + f = (support_alternative_backends(f_name, n_array_args) if _SCIPY_ARRAY_API + else getattr(_ufuncs, f_name)) + sys.modules[__name__].__dict__[f_name] = f + +__all__ = list(array_special_func_map) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_test_internal.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_test_internal.pyi new file mode 100644 index 0000000000000000000000000000000000000000..1e6c272f16fa2bd3ae75af412bc6ae3270158ce4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_test_internal.pyi @@ -0,0 +1,9 @@ +import numpy as np + +def have_fenv() -> bool: ... +def random_double(size: int, rng: np.random.RandomState) -> np.float64: ... +def test_add_round(size: int, mode: str, rng: np.random.RandomState): ... + +def _dd_exp(xhi: float, xlo: float) -> tuple[float, float]: ... +def _dd_log(xhi: float, xlo: float) -> tuple[float, float]: ... +def _dd_expm1(xhi: float, xlo: float) -> tuple[float, float]: ... diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_testutils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_testutils.py new file mode 100644 index 0000000000000000000000000000000000000000..b0c2bd3053d076aacfa8be1d53cd851443fd8821 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_testutils.py @@ -0,0 +1,321 @@ +import os +import functools +import operator +from scipy._lib import _pep440 + +import numpy as np +from numpy.testing import assert_ +import pytest + +import scipy.special as sc + +__all__ = ['with_special_errors', 'assert_func_equal', 'FuncData'] + + +#------------------------------------------------------------------------------ +# Check if a module is present to be used in tests +#------------------------------------------------------------------------------ + +class MissingModule: + def __init__(self, name): + self.name = name + + +def check_version(module, min_ver): + if type(module) is MissingModule: + return pytest.mark.skip(reason=f"{module.name} is not installed") + return pytest.mark.skipif( + _pep440.parse(module.__version__) < _pep440.Version(min_ver), + reason=f"{module.__name__} version >= {min_ver} required" + ) + + +#------------------------------------------------------------------------------ +# Enable convergence and loss of precision warnings -- turn off one by one +#------------------------------------------------------------------------------ + +def with_special_errors(func): + """ + Enable special function errors (such as underflow, overflow, + loss of precision, etc.) + """ + @functools.wraps(func) + def wrapper(*a, **kw): + with sc.errstate(all='raise'): + res = func(*a, **kw) + return res + return wrapper + + +#------------------------------------------------------------------------------ +# Comparing function values at many data points at once, with helpful +# error reports +#------------------------------------------------------------------------------ + +def assert_func_equal(func, results, points, rtol=None, atol=None, + param_filter=None, knownfailure=None, + vectorized=True, dtype=None, nan_ok=False, + ignore_inf_sign=False, distinguish_nan_and_inf=True): + if hasattr(points, 'next'): + # it's a generator + points = list(points) + + points = np.asarray(points) + if points.ndim == 1: + points = points[:,None] + nparams = points.shape[1] + + if hasattr(results, '__name__'): + # function + data = points + result_columns = None + result_func = results + else: + # dataset + data = np.c_[points, results] + result_columns = list(range(nparams, data.shape[1])) + result_func = None + + fdata = FuncData(func, data, list(range(nparams)), + result_columns=result_columns, result_func=result_func, + rtol=rtol, atol=atol, param_filter=param_filter, + knownfailure=knownfailure, nan_ok=nan_ok, vectorized=vectorized, + ignore_inf_sign=ignore_inf_sign, + distinguish_nan_and_inf=distinguish_nan_and_inf) + fdata.check() + + +class FuncData: + """ + Data set for checking a special function. + + Parameters + ---------- + func : function + Function to test + data : numpy array + columnar data to use for testing + param_columns : int or tuple of ints + Columns indices in which the parameters to `func` lie. + Can be imaginary integers to indicate that the parameter + should be cast to complex. + result_columns : int or tuple of ints, optional + Column indices for expected results from `func`. + result_func : callable, optional + Function to call to obtain results. + rtol : float, optional + Required relative tolerance. Default is 5*eps. + atol : float, optional + Required absolute tolerance. Default is 5*tiny. + param_filter : function, or tuple of functions/Nones, optional + Filter functions to exclude some parameter ranges. + If omitted, no filtering is done. + knownfailure : str, optional + Known failure error message to raise when the test is run. + If omitted, no exception is raised. + nan_ok : bool, optional + If nan is always an accepted result. + vectorized : bool, optional + Whether all functions passed in are vectorized. + ignore_inf_sign : bool, optional + Whether to ignore signs of infinities. + (Doesn't matter for complex-valued functions.) + distinguish_nan_and_inf : bool, optional + If True, treat numbers which contain nans or infs as + equal. Sets ignore_inf_sign to be True. + + """ + + def __init__(self, func, data, param_columns, result_columns=None, + result_func=None, rtol=None, atol=None, param_filter=None, + knownfailure=None, dataname=None, nan_ok=False, vectorized=True, + ignore_inf_sign=False, distinguish_nan_and_inf=True): + self.func = func + self.data = data + self.dataname = dataname + if not hasattr(param_columns, '__len__'): + param_columns = (param_columns,) + self.param_columns = tuple(param_columns) + if result_columns is not None: + if not hasattr(result_columns, '__len__'): + result_columns = (result_columns,) + self.result_columns = tuple(result_columns) + if result_func is not None: + message = "Only result_func or result_columns should be provided" + raise ValueError(message) + elif result_func is not None: + self.result_columns = None + else: + raise ValueError("Either result_func or result_columns should be provided") + self.result_func = result_func + self.rtol = rtol + self.atol = atol + if not hasattr(param_filter, '__len__'): + param_filter = (param_filter,) + self.param_filter = param_filter + self.knownfailure = knownfailure + self.nan_ok = nan_ok + self.vectorized = vectorized + self.ignore_inf_sign = ignore_inf_sign + self.distinguish_nan_and_inf = distinguish_nan_and_inf + if not self.distinguish_nan_and_inf: + self.ignore_inf_sign = True + + def get_tolerances(self, dtype): + if not np.issubdtype(dtype, np.inexact): + dtype = np.dtype(float) + info = np.finfo(dtype) + rtol, atol = self.rtol, self.atol + if rtol is None: + rtol = 5*info.eps + if atol is None: + atol = 5*info.tiny + return rtol, atol + + def check(self, data=None, dtype=None, dtypes=None): + """Check the special function against the data.""" + __tracebackhide__ = operator.methodcaller( + 'errisinstance', AssertionError + ) + + if self.knownfailure: + pytest.xfail(reason=self.knownfailure) + + if data is None: + data = self.data + + if dtype is None: + dtype = data.dtype + else: + data = data.astype(dtype) + + rtol, atol = self.get_tolerances(dtype) + + # Apply given filter functions + if self.param_filter: + param_mask = np.ones((data.shape[0],), np.bool_) + for j, filter in zip(self.param_columns, self.param_filter): + if filter: + param_mask &= list(filter(data[:,j])) + data = data[param_mask] + + # Pick parameters from the correct columns + params = [] + for idx, j in enumerate(self.param_columns): + if np.iscomplexobj(j): + j = int(j.imag) + params.append(data[:,j].astype(complex)) + elif dtypes and idx < len(dtypes): + params.append(data[:, j].astype(dtypes[idx])) + else: + params.append(data[:,j]) + + # Helper for evaluating results + def eval_func_at_params(func, skip_mask=None): + if self.vectorized: + got = func(*params) + else: + got = [] + for j in range(len(params[0])): + if skip_mask is not None and skip_mask[j]: + got.append(np.nan) + continue + got.append(func(*tuple([params[i][j] for i in range(len(params))]))) + got = np.asarray(got) + if not isinstance(got, tuple): + got = (got,) + return got + + # Evaluate function to be tested + got = eval_func_at_params(self.func) + + # Grab the correct results + if self.result_columns is not None: + # Correct results passed in with the data + wanted = tuple([data[:,icol] for icol in self.result_columns]) + else: + # Function producing correct results passed in + skip_mask = None + if self.nan_ok and len(got) == 1: + # Don't spend time evaluating what doesn't need to be evaluated + skip_mask = np.isnan(got[0]) + wanted = eval_func_at_params(self.result_func, skip_mask=skip_mask) + + # Check the validity of each output returned + assert_(len(got) == len(wanted)) + + for output_num, (x, y) in enumerate(zip(got, wanted)): + if np.issubdtype(x.dtype, np.complexfloating) or self.ignore_inf_sign: + pinf_x = np.isinf(x) + pinf_y = np.isinf(y) + minf_x = np.isinf(x) + minf_y = np.isinf(y) + else: + pinf_x = np.isposinf(x) + pinf_y = np.isposinf(y) + minf_x = np.isneginf(x) + minf_y = np.isneginf(y) + nan_x = np.isnan(x) + nan_y = np.isnan(y) + + with np.errstate(all='ignore'): + abs_y = np.absolute(y) + abs_y[~np.isfinite(abs_y)] = 0 + diff = np.absolute(x - y) + diff[~np.isfinite(diff)] = 0 + + rdiff = diff / np.absolute(y) + rdiff[~np.isfinite(rdiff)] = 0 + + tol_mask = (diff <= atol + rtol*abs_y) + pinf_mask = (pinf_x == pinf_y) + minf_mask = (minf_x == minf_y) + + nan_mask = (nan_x == nan_y) + + bad_j = ~(tol_mask & pinf_mask & minf_mask & nan_mask) + + point_count = bad_j.size + if self.nan_ok: + bad_j &= ~nan_x + bad_j &= ~nan_y + point_count -= (nan_x | nan_y).sum() + + if not self.distinguish_nan_and_inf and not self.nan_ok: + # If nan's are okay we've already covered all these cases + inf_x = np.isinf(x) + inf_y = np.isinf(y) + both_nonfinite = (inf_x & nan_y) | (nan_x & inf_y) + bad_j &= ~both_nonfinite + point_count -= both_nonfinite.sum() + + if np.any(bad_j): + # Some bad results: inform what, where, and how bad + msg = [""] + msg.append(f"Max |adiff|: {diff[bad_j].max():g}") + msg.append(f"Max |rdiff|: {rdiff[bad_j].max():g}") + msg.append("Bad results (%d out of %d) for the following points " + "(in output %d):" + % (np.sum(bad_j), point_count, output_num,)) + for j in np.nonzero(bad_j)[0]: + j = int(j) + def fmt(x): + return '%30s' % np.array2string(x[j], precision=18) + a = " ".join(map(fmt, params)) + b = " ".join(map(fmt, got)) + c = " ".join(map(fmt, wanted)) + d = fmt(rdiff) + msg.append(f"{a} => {b} != {c} (rdiff {d})") + assert_(False, "\n".join(msg)) + + def __repr__(self): + """Pretty-printing""" + if np.any(list(map(np.iscomplexobj, self.param_columns))): + is_complex = " (complex)" + else: + is_complex = "" + if self.dataname: + return (f"") + else: + return f"" diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs.pyi new file mode 100644 index 0000000000000000000000000000000000000000..0ccf14137096c862a0644cdb229cb98b4556e5d7 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs.pyi @@ -0,0 +1,521 @@ +from typing import Any + +import numpy as np + +__all__ = [ + 'geterr', + 'seterr', + 'errstate', + 'agm', + 'airy', + 'airye', + 'bdtr', + 'bdtrc', + 'bdtri', + 'bdtrik', + 'bdtrin', + 'bei', + 'beip', + 'ber', + 'berp', + 'besselpoly', + 'beta', + 'betainc', + 'betaincc', + 'betainccinv', + 'betaincinv', + 'betaln', + 'binom', + 'boxcox', + 'boxcox1p', + 'btdtria', + 'btdtrib', + 'cbrt', + 'chdtr', + 'chdtrc', + 'chdtri', + 'chdtriv', + 'chndtr', + 'chndtridf', + 'chndtrinc', + 'chndtrix', + 'cosdg', + 'cosm1', + 'cotdg', + 'dawsn', + 'ellipe', + 'ellipeinc', + 'ellipj', + 'ellipk', + 'ellipkinc', + 'ellipkm1', + 'elliprc', + 'elliprd', + 'elliprf', + 'elliprg', + 'elliprj', + 'entr', + 'erf', + 'erfc', + 'erfcinv', + 'erfcx', + 'erfi', + 'erfinv', + 'eval_chebyc', + 'eval_chebys', + 'eval_chebyt', + 'eval_chebyu', + 'eval_gegenbauer', + 'eval_genlaguerre', + 'eval_hermite', + 'eval_hermitenorm', + 'eval_jacobi', + 'eval_laguerre', + 'eval_legendre', + 'eval_sh_chebyt', + 'eval_sh_chebyu', + 'eval_sh_jacobi', + 'eval_sh_legendre', + 'exp1', + 'exp10', + 'exp2', + 'expi', + 'expit', + 'expm1', + 'expn', + 'exprel', + 'fdtr', + 'fdtrc', + 'fdtri', + 'fdtridfd', + 'fresnel', + 'gamma', + 'gammainc', + 'gammaincc', + 'gammainccinv', + 'gammaincinv', + 'gammaln', + 'gammasgn', + 'gdtr', + 'gdtrc', + 'gdtria', + 'gdtrib', + 'gdtrix', + 'hankel1', + 'hankel1e', + 'hankel2', + 'hankel2e', + 'huber', + 'hyp0f1', + 'hyp1f1', + 'hyp2f1', + 'hyperu', + 'i0', + 'i0e', + 'i1', + 'i1e', + 'inv_boxcox', + 'inv_boxcox1p', + 'it2i0k0', + 'it2j0y0', + 'it2struve0', + 'itairy', + 'iti0k0', + 'itj0y0', + 'itmodstruve0', + 'itstruve0', + 'iv', + 'ive', + 'j0', + 'j1', + 'jn', + 'jv', + 'jve', + 'k0', + 'k0e', + 'k1', + 'k1e', + 'kei', + 'keip', + 'kelvin', + 'ker', + 'kerp', + 'kl_div', + 'kn', + 'kolmogi', + 'kolmogorov', + 'kv', + 'kve', + 'log1p', + 'log_expit', + 'log_ndtr', + 'log_wright_bessel', + 'loggamma', + 'logit', + 'lpmv', + 'mathieu_a', + 'mathieu_b', + 'mathieu_cem', + 'mathieu_modcem1', + 'mathieu_modcem2', + 'mathieu_modsem1', + 'mathieu_modsem2', + 'mathieu_sem', + 'modfresnelm', + 'modfresnelp', + 'modstruve', + 'nbdtr', + 'nbdtrc', + 'nbdtri', + 'nbdtrik', + 'nbdtrin', + 'ncfdtr', + 'ncfdtri', + 'ncfdtridfd', + 'ncfdtridfn', + 'ncfdtrinc', + 'nctdtr', + 'nctdtridf', + 'nctdtrinc', + 'nctdtrit', + 'ndtr', + 'ndtri', + 'ndtri_exp', + 'nrdtrimn', + 'nrdtrisd', + 'obl_ang1', + 'obl_ang1_cv', + 'obl_cv', + 'obl_rad1', + 'obl_rad1_cv', + 'obl_rad2', + 'obl_rad2_cv', + 'owens_t', + 'pbdv', + 'pbvv', + 'pbwa', + 'pdtr', + 'pdtrc', + 'pdtri', + 'pdtrik', + 'poch', + 'powm1', + 'pro_ang1', + 'pro_ang1_cv', + 'pro_cv', + 'pro_rad1', + 'pro_rad1_cv', + 'pro_rad2', + 'pro_rad2_cv', + 'pseudo_huber', + 'psi', + 'radian', + 'rel_entr', + 'rgamma', + 'round', + 'shichi', + 'sici', + 'sindg', + 'smirnov', + 'smirnovi', + 'spence', + 'sph_harm', + 'stdtr', + 'stdtridf', + 'stdtrit', + 'struve', + 'tandg', + 'tklmbda', + 'voigt_profile', + 'wofz', + 'wright_bessel', + 'wrightomega', + 'xlog1py', + 'xlogy', + 'y0', + 'y1', + 'yn', + 'yv', + 'yve', + 'zetac' +] + +def geterr() -> dict[str, str]: ... +def seterr(**kwargs: str) -> dict[str, str]: ... + +class errstate: + def __init__(self, **kargs: str) -> None: ... + def __enter__(self) -> None: ... + def __exit__( + self, + exc_type: Any, # Unused + exc_value: Any, # Unused + traceback: Any, # Unused + ) -> None: ... + +_cosine_cdf: np.ufunc +_cosine_invcdf: np.ufunc +_cospi: np.ufunc +_ellip_harm: np.ufunc +_factorial: np.ufunc +_igam_fac: np.ufunc +_kolmogc: np.ufunc +_kolmogci: np.ufunc +_kolmogp: np.ufunc +_lambertw: np.ufunc +_lanczos_sum_expg_scaled: np.ufunc +_lgam1p: np.ufunc +_log1pmx: np.ufunc +_riemann_zeta: np.ufunc +_scaled_exp1: np.ufunc +_sf_error_test_function: np.ufunc +_sinpi: np.ufunc +_smirnovc: np.ufunc +_smirnovci: np.ufunc +_smirnovp: np.ufunc +_spherical_in: np.ufunc +_spherical_in_d: np.ufunc +_spherical_jn: np.ufunc +_spherical_jn_d: np.ufunc +_spherical_kn: np.ufunc +_spherical_kn_d: np.ufunc +_spherical_yn: np.ufunc +_spherical_yn_d: np.ufunc +_stirling2_inexact: np.ufunc +_struve_asymp_large_z: np.ufunc +_struve_bessel_series: np.ufunc +_struve_power_series: np.ufunc +_zeta: np.ufunc +agm: np.ufunc +airy: np.ufunc +airye: np.ufunc +bdtr: np.ufunc +bdtrc: np.ufunc +bdtri: np.ufunc +bdtrik: np.ufunc +bdtrin: np.ufunc +bei: np.ufunc +beip: np.ufunc +ber: np.ufunc +berp: np.ufunc +besselpoly: np.ufunc +beta: np.ufunc +betainc: np.ufunc +betaincc: np.ufunc +betainccinv: np.ufunc +betaincinv: np.ufunc +betaln: np.ufunc +binom: np.ufunc +boxcox1p: np.ufunc +boxcox: np.ufunc +btdtria: np.ufunc +btdtrib: np.ufunc +cbrt: np.ufunc +chdtr: np.ufunc +chdtrc: np.ufunc +chdtri: np.ufunc +chdtriv: np.ufunc +chndtr: np.ufunc +chndtridf: np.ufunc +chndtrinc: np.ufunc +chndtrix: np.ufunc +cosdg: np.ufunc +cosm1: np.ufunc +cotdg: np.ufunc +dawsn: np.ufunc +ellipe: np.ufunc +ellipeinc: np.ufunc +ellipj: np.ufunc +ellipk: np.ufunc +ellipkinc: np.ufunc +ellipkm1: np.ufunc +elliprc: np.ufunc +elliprd: np.ufunc +elliprf: np.ufunc +elliprg: np.ufunc +elliprj: np.ufunc +entr: np.ufunc +erf: np.ufunc +erfc: np.ufunc +erfcinv: np.ufunc +erfcx: np.ufunc +erfi: np.ufunc +erfinv: np.ufunc +eval_chebyc: np.ufunc +eval_chebys: np.ufunc +eval_chebyt: np.ufunc +eval_chebyu: np.ufunc +eval_gegenbauer: np.ufunc +eval_genlaguerre: np.ufunc +eval_hermite: np.ufunc +eval_hermitenorm: np.ufunc +eval_jacobi: np.ufunc +eval_laguerre: np.ufunc +eval_legendre: np.ufunc +eval_sh_chebyt: np.ufunc +eval_sh_chebyu: np.ufunc +eval_sh_jacobi: np.ufunc +eval_sh_legendre: np.ufunc +exp10: np.ufunc +exp1: np.ufunc +exp2: np.ufunc +expi: np.ufunc +expit: np.ufunc +expm1: np.ufunc +expn: np.ufunc +exprel: np.ufunc +fdtr: np.ufunc +fdtrc: np.ufunc +fdtri: np.ufunc +fdtridfd: np.ufunc +fresnel: np.ufunc +gamma: np.ufunc +gammainc: np.ufunc +gammaincc: np.ufunc +gammainccinv: np.ufunc +gammaincinv: np.ufunc +gammaln: np.ufunc +gammasgn: np.ufunc +gdtr: np.ufunc +gdtrc: np.ufunc +gdtria: np.ufunc +gdtrib: np.ufunc +gdtrix: np.ufunc +hankel1: np.ufunc +hankel1e: np.ufunc +hankel2: np.ufunc +hankel2e: np.ufunc +huber: np.ufunc +hyp0f1: np.ufunc +hyp1f1: np.ufunc +hyp2f1: np.ufunc +hyperu: np.ufunc +i0: np.ufunc +i0e: np.ufunc +i1: np.ufunc +i1e: np.ufunc +inv_boxcox1p: np.ufunc +inv_boxcox: np.ufunc +it2i0k0: np.ufunc +it2j0y0: np.ufunc +it2struve0: np.ufunc +itairy: np.ufunc +iti0k0: np.ufunc +itj0y0: np.ufunc +itmodstruve0: np.ufunc +itstruve0: np.ufunc +iv: np.ufunc +ive: np.ufunc +j0: np.ufunc +j1: np.ufunc +jn: np.ufunc +jv: np.ufunc +jve: np.ufunc +k0: np.ufunc +k0e: np.ufunc +k1: np.ufunc +k1e: np.ufunc +kei: np.ufunc +keip: np.ufunc +kelvin: np.ufunc +ker: np.ufunc +kerp: np.ufunc +kl_div: np.ufunc +kn: np.ufunc +kolmogi: np.ufunc +kolmogorov: np.ufunc +kv: np.ufunc +kve: np.ufunc +log1p: np.ufunc +log_expit: np.ufunc +log_ndtr: np.ufunc +log_wright_bessel: np.ufunc +loggamma: np.ufunc +logit: np.ufunc +lpmv: np.ufunc +mathieu_a: np.ufunc +mathieu_b: np.ufunc +mathieu_cem: np.ufunc +mathieu_modcem1: np.ufunc +mathieu_modcem2: np.ufunc +mathieu_modsem1: np.ufunc +mathieu_modsem2: np.ufunc +mathieu_sem: np.ufunc +modfresnelm: np.ufunc +modfresnelp: np.ufunc +modstruve: np.ufunc +nbdtr: np.ufunc +nbdtrc: np.ufunc +nbdtri: np.ufunc +nbdtrik: np.ufunc +nbdtrin: np.ufunc +ncfdtr: np.ufunc +ncfdtri: np.ufunc +ncfdtridfd: np.ufunc +ncfdtridfn: np.ufunc +ncfdtrinc: np.ufunc +nctdtr: np.ufunc +nctdtridf: np.ufunc +nctdtrinc: np.ufunc +nctdtrit: np.ufunc +ndtr: np.ufunc +ndtri: np.ufunc +ndtri_exp: np.ufunc +nrdtrimn: np.ufunc +nrdtrisd: np.ufunc +obl_ang1: np.ufunc +obl_ang1_cv: np.ufunc +obl_cv: np.ufunc +obl_rad1: np.ufunc +obl_rad1_cv: np.ufunc +obl_rad2: np.ufunc +obl_rad2_cv: np.ufunc +owens_t: np.ufunc +pbdv: np.ufunc +pbvv: np.ufunc +pbwa: np.ufunc +pdtr: np.ufunc +pdtrc: np.ufunc +pdtri: np.ufunc +pdtrik: np.ufunc +poch: np.ufunc +powm1: np.ufunc +pro_ang1: np.ufunc +pro_ang1_cv: np.ufunc +pro_cv: np.ufunc +pro_rad1: np.ufunc +pro_rad1_cv: np.ufunc +pro_rad2: np.ufunc +pro_rad2_cv: np.ufunc +pseudo_huber: np.ufunc +psi: np.ufunc +radian: np.ufunc +rel_entr: np.ufunc +rgamma: np.ufunc +round: np.ufunc +shichi: np.ufunc +sici: np.ufunc +sindg: np.ufunc +smirnov: np.ufunc +smirnovi: np.ufunc +spence: np.ufunc +sph_harm: np.ufunc +stdtr: np.ufunc +stdtridf: np.ufunc +stdtrit: np.ufunc +struve: np.ufunc +tandg: np.ufunc +tklmbda: np.ufunc +voigt_profile: np.ufunc +wofz: np.ufunc +wright_bessel: np.ufunc +wrightomega: np.ufunc +xlog1py: np.ufunc +xlogy: np.ufunc +y0: np.ufunc +y1: np.ufunc +yn: np.ufunc +yv: np.ufunc +yve: np.ufunc +zetac: np.ufunc + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs.pyx b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs.pyx new file mode 100644 index 0000000000000000000000000000000000000000..5f9afc828a16d598b3ba76582111cd964cfe764a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs.pyx @@ -0,0 +1,14358 @@ +# This file is automatically generated by _generate_pyx.py. +# Do not edit manually! + +from libc.math cimport NAN + +include "_ufuncs_extra_code_common.pxi" +include "_ufuncs_extra_code.pxi" +__all__ = ['agm', 'bdtr', 'bdtrc', 'bdtri', 'bdtrik', 'bdtrin', 'betainc', 'betaincc', 'betainccinv', 'betaincinv', 'boxcox', 'boxcox1p', 'btdtria', 'btdtrib', 'chdtr', 'chdtrc', 'chdtri', 'chdtriv', 'chndtr', 'chndtridf', 'chndtrinc', 'chndtrix', 'dawsn', 'elliprc', 'elliprd', 'elliprf', 'elliprg', 'elliprj', 'entr', 'erf', 'erfc', 'erfcinv', 'erfcx', 'erfi', 'erfinv', 'eval_chebyc', 'eval_chebys', 'eval_chebyt', 'eval_chebyu', 'eval_gegenbauer', 'eval_genlaguerre', 'eval_hermite', 'eval_hermitenorm', 'eval_jacobi', 'eval_laguerre', 'eval_legendre', 'eval_sh_chebyt', 'eval_sh_chebyu', 'eval_sh_jacobi', 'eval_sh_legendre', 'exp10', 'exp2', 'expm1', 'expn', 'fdtr', 'fdtrc', 'fdtri', 'fdtridfd', 'gdtr', 'gdtrc', 'gdtria', 'gdtrib', 'gdtrix', 'huber', 'hyp0f1', 'hyp1f1', 'hyperu', 'inv_boxcox', 'inv_boxcox1p', 'kl_div', 'kn', 'kolmogi', 'kolmogorov', 'log1p', 'log_ndtr', 'lpmv', 'nbdtr', 'nbdtrc', 'nbdtri', 'nbdtrik', 'nbdtrin', 'ncfdtr', 'ncfdtri', 'ncfdtridfd', 'ncfdtridfn', 'ncfdtrinc', 'nctdtr', 'nctdtridf', 'nctdtrinc', 'nctdtrit', 'ndtr', 'ndtri', 'ndtri_exp', 'nrdtrimn', 'nrdtrisd', 'owens_t', 'pdtr', 'pdtrc', 'pdtri', 'pdtrik', 'poch', 'powm1', 'pseudo_huber', 'rel_entr', 'round', 'shichi', 'sici', 'smirnov', 'smirnovi', 'spence', 'stdtr', 'stdtridf', 'stdtrit', 'tklmbda', 'voigt_profile', 'wofz', 'wrightomega', 'xlog1py', 'xlogy', 'yn', 'geterr', 'seterr', 'errstate', 'jn', 'airy', 'airye', 'bei', 'beip', 'ber', 'berp', 'binom', 'exp1', 'expi', 'expit', 'exprel', 'gamma', 'gammaln', 'hankel1', 'hankel1e', 'hankel2', 'hankel2e', 'hyp2f1', 'it2i0k0', 'it2j0y0', 'it2struve0', 'itairy', 'iti0k0', 'itj0y0', 'itmodstruve0', 'itstruve0', 'iv', 'ive', 'jv', 'jve', 'kei', 'keip', 'kelvin', 'ker', 'kerp', 'kv', 'kve', 'log_expit', 'log_wright_bessel', 'loggamma', 'logit', 'mathieu_a', 'mathieu_b', 'mathieu_cem', 'mathieu_modcem1', 'mathieu_modcem2', 'mathieu_modsem1', 'mathieu_modsem2', 'mathieu_sem', 'modfresnelm', 'modfresnelp', 'obl_ang1', 'obl_ang1_cv', 'obl_cv', 'obl_rad1', 'obl_rad1_cv', 'obl_rad2', 'obl_rad2_cv', 'pbdv', 'pbvv', 'pbwa', 'pro_ang1', 'pro_ang1_cv', 'pro_cv', 'pro_rad1', 'pro_rad1_cv', 'pro_rad2', 'pro_rad2_cv', 'psi', 'rgamma', 'sph_harm', 'wright_bessel', 'yv', 'yve', 'zetac', 'sindg', 'cosdg', 'tandg', 'cotdg', 'i0', 'i0e', 'i1', 'i1e', 'k0', 'k0e', 'k1', 'k1e', 'y0', 'y1', 'j0', 'j1', 'struve', 'modstruve', 'beta', 'betaln', 'besselpoly', 'gammaln', 'gammasgn', 'cbrt', 'radian', 'cosm1', 'gammainc', 'gammaincinv', 'gammaincc', 'gammainccinv', 'fresnel', 'ellipe', 'ellipeinc', 'ellipk', 'ellipkinc', 'ellipkm1', 'ellipj'] +cdef void loop_D_DDDD__As_DDDD_D(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *op0 = args[4] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + op0 += steps[4] + sf_error.check_fpe(func_name) + +cdef void loop_D_DDDD__As_FFFF_F(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *op0 = args[4] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + op0 += steps[4] + sf_error.check_fpe(func_name) + +cdef void loop_D_DDD__As_DDD_D(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + sf_error.check_fpe(func_name) + +cdef void loop_D_DDD__As_FFF_F(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + sf_error.check_fpe(func_name) + +cdef void loop_D_DD__As_DD_D(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *op0 = args[2] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + op0 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_D_DD__As_FF_F(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *op0 = args[2] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + op0 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_D_D__As_D_D(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *op0 = args[1] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + op0 += steps[1] + sf_error.check_fpe(func_name) + +cdef void loop_D_D__As_F_F(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *op0 = args[1] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + op0 += steps[1] + sf_error.check_fpe(func_name) + +cdef void loop_D_dD__As_dD_D(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *op0 = args[2] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + op0 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_D_dD__As_fF_F(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *op0 = args[2] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + op0 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_D_ddD__As_ddD_D(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + sf_error.check_fpe(func_name) + +cdef void loop_D_ddD__As_ffF_F(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + sf_error.check_fpe(func_name) + +cdef void loop_D_dddD__As_dddD_D(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *op0 = args[4] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + op0 += steps[4] + sf_error.check_fpe(func_name) + +cdef void loop_D_dddD__As_fffF_F(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *op0 = args[4] + cdef double complex ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + op0 += steps[4] + sf_error.check_fpe(func_name) + +cdef void loop_d_d__As_d_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *op0 = args[1] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + op0 += steps[1] + sf_error.check_fpe(func_name) + +cdef void loop_d_d__As_f_f(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *op0 = args[1] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + op0 += steps[1] + sf_error.check_fpe(func_name) + +cdef void loop_d_dd__As_dd_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *op0 = args[2] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + op0 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_d_dd__As_ff_f(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *op0 = args[2] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + op0 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_d_ddd__As_ddd_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + sf_error.check_fpe(func_name) + +cdef void loop_d_ddd__As_fff_f(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + sf_error.check_fpe(func_name) + +cdef void loop_d_dddd__As_dddd_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *op0 = args[4] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + op0 += steps[4] + sf_error.check_fpe(func_name) + +cdef void loop_d_dddd__As_ffff_f(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *op0 = args[4] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + op0 += steps[4] + sf_error.check_fpe(func_name) + +cdef void loop_d_ddddddd__As_ddddddd_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *ip4 = args[4] + cdef char *ip5 = args[5] + cdef char *ip6 = args[6] + cdef char *op0 = args[7] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0], (ip4)[0], (ip5)[0], (ip6)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + ip4 += steps[4] + ip5 += steps[5] + ip6 += steps[6] + op0 += steps[7] + sf_error.check_fpe(func_name) + +cdef void loop_d_ddddddd__As_fffffff_f(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *ip4 = args[4] + cdef char *ip5 = args[5] + cdef char *ip6 = args[6] + cdef char *op0 = args[7] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0], (ip4)[0], (ip5)[0], (ip6)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + ip4 += steps[4] + ip5 += steps[5] + ip6 += steps[6] + op0 += steps[7] + sf_error.check_fpe(func_name) + +cdef void loop_d_ddiiddd__As_ddllddd_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *ip4 = args[4] + cdef char *ip5 = args[5] + cdef char *ip6 = args[6] + cdef char *op0 = args[7] + cdef double ov0 + for i in range(n): + if (ip2)[0] == (ip2)[0] and (ip3)[0] == (ip3)[0]: + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0], (ip4)[0], (ip5)[0], (ip6)[0]) + else: + sf_error.error(func_name, sf_error.DOMAIN, "invalid input argument") + ov0 = NAN + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + ip4 += steps[4] + ip5 += steps[5] + ip6 += steps[6] + op0 += steps[7] + sf_error.check_fpe(func_name) + +cdef void loop_d_ddp_d_As_ddp_dd(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef char *op1 = args[4] + cdef double ov0 + cdef double ov1 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], &ov1) + (op0)[0] = ov0 + (op1)[0] = ov1 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + op1 += steps[4] + sf_error.check_fpe(func_name) + +cdef void loop_d_dpd__As_dpd_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + sf_error.check_fpe(func_name) + +cdef void loop_d_pd__As_pd_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *op0 = args[2] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + op0 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_d_pdd__As_pdd_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + sf_error.check_fpe(func_name) + +cdef void loop_d_pddd__As_pddd_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *op0 = args[4] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + op0 += steps[4] + sf_error.check_fpe(func_name) + +cdef void loop_d_ppd__As_ppd_d(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef double ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + sf_error.check_fpe(func_name) + +cdef void loop_f_f__As_f_f(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *op0 = args[1] + cdef float ov0 + for i in range(n): + ov0 = (func)((ip0)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + op0 += steps[1] + sf_error.check_fpe(func_name) + +cdef void loop_f_ff__As_ff_f(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *op0 = args[2] + cdef float ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + op0 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_f_fff__As_fff_f(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *op0 = args[3] + cdef float ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + op0 += steps[3] + sf_error.check_fpe(func_name) + +cdef void loop_f_ffff__As_ffff_f(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *ip1 = args[1] + cdef char *ip2 = args[2] + cdef char *ip3 = args[3] + cdef char *op0 = args[4] + cdef float ov0 + for i in range(n): + ov0 = (func)((ip0)[0], (ip1)[0], (ip2)[0], (ip3)[0]) + (op0)[0] = ov0 + ip0 += steps[0] + ip1 += steps[1] + ip2 += steps[2] + ip3 += steps[3] + op0 += steps[4] + sf_error.check_fpe(func_name) + +cdef void loop_i_D_DD_As_D_DD(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *op0 = args[1] + cdef char *op1 = args[2] + cdef double complex ov0 + cdef double complex ov1 + for i in range(n): + (func)((ip0)[0], &ov0, &ov1) + (op0)[0] = ov0 + (op1)[0] = ov1 + ip0 += steps[0] + op0 += steps[1] + op1 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_i_D_DD_As_F_FF(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *op0 = args[1] + cdef char *op1 = args[2] + cdef double complex ov0 + cdef double complex ov1 + for i in range(n): + (func)((ip0)[0], &ov0, &ov1) + (op0)[0] = ov0 + (op1)[0] = ov1 + ip0 += steps[0] + op0 += steps[1] + op1 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_i_d_dd_As_d_dd(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *op0 = args[1] + cdef char *op1 = args[2] + cdef double ov0 + cdef double ov1 + for i in range(n): + (func)((ip0)[0], &ov0, &ov1) + (op0)[0] = ov0 + (op1)[0] = ov1 + ip0 += steps[0] + op0 += steps[1] + op1 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_i_d_dd_As_f_ff(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *op0 = args[1] + cdef char *op1 = args[2] + cdef double ov0 + cdef double ov1 + for i in range(n): + (func)((ip0)[0], &ov0, &ov1) + (op0)[0] = ov0 + (op1)[0] = ov1 + ip0 += steps[0] + op0 += steps[1] + op1 += steps[2] + sf_error.check_fpe(func_name) + +cdef void loop_i_i__As_l_l(char **args, np.npy_intp *dims, np.npy_intp *steps, void *data) noexcept nogil: + cdef np.npy_intp i, n = dims[0] + cdef void *func = (data)[0] + cdef char *func_name = (data)[1] + cdef char *ip0 = args[0] + cdef char *op0 = args[1] + cdef int ov0 + for i in range(n): + if (ip0)[0] == (ip0)[0]: + ov0 = (func)((ip0)[0]) + else: + sf_error.error(func_name, sf_error.DOMAIN, "invalid input argument") + ov0 = 0xbad0bad0 + (op0)[0] = ov0 + ip0 += steps[0] + op0 += steps[1] + sf_error.check_fpe(func_name) + +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cosine_cdf "cosine_cdf"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cosine_invcdf "cosine_invcdf"(double) noexcept nogil +from ._ellip_harm cimport ellip_harmonic as _func_ellip_harmonic +ctypedef double _proto_ellip_harmonic_t(double, double, int, int, double, double, double) noexcept nogil +cdef _proto_ellip_harmonic_t *_proto_ellip_harmonic_t_var = &_func_ellip_harmonic +from ._legacy cimport ellip_harmonic_unsafe as _func_ellip_harmonic_unsafe +ctypedef double _proto_ellip_harmonic_unsafe_t(double, double, double, double, double, double, double) noexcept nogil +cdef _proto_ellip_harmonic_unsafe_t *_proto_ellip_harmonic_unsafe_t_var = &_func_ellip_harmonic_unsafe +from ._factorial cimport _factorial as _func__factorial +ctypedef double _proto__factorial_t(double) noexcept nogil +cdef _proto__factorial_t *_proto__factorial_t_var = &_func__factorial +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_igam_fac "cephes_igam_fac"(double, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_kolmogc "xsf_kolmogc"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_kolmogci "xsf_kolmogci"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_kolmogp "xsf_kolmogp"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_lanczos_sum_expg_scaled "cephes_lanczos_sum_expg_scaled"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_lgam1p "cephes_lgam1p"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_log1pmx "cephes_log1pmx"(double) noexcept nogil +from .sf_error cimport _sf_error_test_function as _func__sf_error_test_function +ctypedef int _proto__sf_error_test_function_t(int) noexcept nogil +cdef _proto__sf_error_test_function_t *_proto__sf_error_test_function_t_var = &_func__sf_error_test_function +from ._legacy cimport smirnovc_unsafe as _func_smirnovc_unsafe +ctypedef double _proto_smirnovc_unsafe_t(double, double) noexcept nogil +cdef _proto_smirnovc_unsafe_t *_proto_smirnovc_unsafe_t_var = &_func_smirnovc_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_smirnovc_wrap "cephes_smirnovc_wrap"(Py_ssize_t, double) noexcept nogil +from ._legacy cimport smirnovci_unsafe as _func_smirnovci_unsafe +ctypedef double _proto_smirnovci_unsafe_t(double, double) noexcept nogil +cdef _proto_smirnovci_unsafe_t *_proto_smirnovci_unsafe_t_var = &_func_smirnovci_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_smirnovci_wrap "cephes_smirnovci_wrap"(Py_ssize_t, double) noexcept nogil +from ._legacy cimport smirnovp_unsafe as _func_smirnovp_unsafe +ctypedef double _proto_smirnovp_unsafe_t(double, double) noexcept nogil +cdef _proto_smirnovp_unsafe_t *_proto_smirnovp_unsafe_t_var = &_func_smirnovp_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_smirnovp_wrap "cephes_smirnovp_wrap"(Py_ssize_t, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes__struve_asymp_large_z "cephes__struve_asymp_large_z"(double, double, Py_ssize_t, double *) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes__struve_bessel_series "cephes__struve_bessel_series"(double, double, Py_ssize_t, double *) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes__struve_power_series "cephes__struve_power_series"(double, double, Py_ssize_t, double *) noexcept nogil +from ._agm cimport agm as _func_agm +ctypedef double _proto_agm_t(double, double) noexcept nogil +cdef _proto_agm_t *_proto_agm_t_var = &_func_agm +from ._legacy cimport bdtr_unsafe as _func_bdtr_unsafe +ctypedef double _proto_bdtr_unsafe_t(double, double, double) noexcept nogil +cdef _proto_bdtr_unsafe_t *_proto_bdtr_unsafe_t_var = &_func_bdtr_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_bdtr_wrap "cephes_bdtr_wrap"(double, Py_ssize_t, double) noexcept nogil +from ._legacy cimport bdtrc_unsafe as _func_bdtrc_unsafe +ctypedef double _proto_bdtrc_unsafe_t(double, double, double) noexcept nogil +cdef _proto_bdtrc_unsafe_t *_proto_bdtrc_unsafe_t_var = &_func_bdtrc_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_bdtrc_wrap "cephes_bdtrc_wrap"(double, Py_ssize_t, double) noexcept nogil +from ._legacy cimport bdtri_unsafe as _func_bdtri_unsafe +ctypedef double _proto_bdtri_unsafe_t(double, double, double) noexcept nogil +cdef _proto_bdtri_unsafe_t *_proto_bdtri_unsafe_t_var = &_func_bdtri_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_bdtri_wrap "cephes_bdtri_wrap"(double, Py_ssize_t, double) noexcept nogil +from ._cdflib_wrappers cimport bdtrik as _func_bdtrik +ctypedef double _proto_bdtrik_t(double, double, double) noexcept nogil +cdef _proto_bdtrik_t *_proto_bdtrik_t_var = &_func_bdtrik +from ._cdflib_wrappers cimport bdtrin as _func_bdtrin +ctypedef double _proto_bdtrin_t(double, double, double) noexcept nogil +cdef _proto_bdtrin_t *_proto_bdtrin_t_var = &_func_bdtrin +from ._boxcox cimport boxcox as _func_boxcox +ctypedef double _proto_boxcox_t(double, double) noexcept nogil +cdef _proto_boxcox_t *_proto_boxcox_t_var = &_func_boxcox +from ._boxcox cimport boxcox1p as _func_boxcox1p +ctypedef double _proto_boxcox1p_t(double, double) noexcept nogil +cdef _proto_boxcox1p_t *_proto_boxcox1p_t_var = &_func_boxcox1p +from ._cdflib_wrappers cimport btdtria as _func_btdtria +ctypedef double _proto_btdtria_t(double, double, double) noexcept nogil +cdef _proto_btdtria_t *_proto_btdtria_t_var = &_func_btdtria +from ._cdflib_wrappers cimport btdtrib as _func_btdtrib +ctypedef double _proto_btdtrib_t(double, double, double) noexcept nogil +cdef _proto_btdtrib_t *_proto_btdtrib_t_var = &_func_btdtrib +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_chdtr "xsf_chdtr"(double, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_chdtrc "xsf_chdtrc"(double, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_chdtri "xsf_chdtri"(double, double) noexcept nogil +from ._cdflib_wrappers cimport chdtriv as _func_chdtriv +ctypedef double _proto_chdtriv_t(double, double) noexcept nogil +cdef _proto_chdtriv_t *_proto_chdtriv_t_var = &_func_chdtriv +from ._cdflib_wrappers cimport chndtr as _func_chndtr +ctypedef double _proto_chndtr_t(double, double, double) noexcept nogil +cdef _proto_chndtr_t *_proto_chndtr_t_var = &_func_chndtr +from ._cdflib_wrappers cimport chndtridf as _func_chndtridf +ctypedef double _proto_chndtridf_t(double, double, double) noexcept nogil +cdef _proto_chndtridf_t *_proto_chndtridf_t_var = &_func_chndtridf +from ._cdflib_wrappers cimport chndtrinc as _func_chndtrinc +ctypedef double _proto_chndtrinc_t(double, double, double) noexcept nogil +cdef _proto_chndtrinc_t *_proto_chndtrinc_t_var = &_func_chndtrinc +from ._cdflib_wrappers cimport chndtrix as _func_chndtrix +ctypedef double _proto_chndtrix_t(double, double, double) noexcept nogil +cdef _proto_chndtrix_t *_proto_chndtrix_t_var = &_func_chndtrix +from ._convex_analysis cimport entr as _func_entr +ctypedef double _proto_entr_t(double) noexcept nogil +cdef _proto_entr_t *_proto_entr_t_var = &_func_entr +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_erf "cephes_erf"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_erfc "cephes_erfc"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_erfcinv "cephes_erfcinv"(double) noexcept nogil +from .orthogonal_eval cimport eval_chebyc as _func_eval_chebyc +ctypedef double complex _proto_eval_chebyc_double_complex__t(double, double complex) noexcept nogil +cdef _proto_eval_chebyc_double_complex__t *_proto_eval_chebyc_double_complex__t_var = &_func_eval_chebyc[double_complex] +from .orthogonal_eval cimport eval_chebyc as _func_eval_chebyc +ctypedef double _proto_eval_chebyc_double__t(double, double) noexcept nogil +cdef _proto_eval_chebyc_double__t *_proto_eval_chebyc_double__t_var = &_func_eval_chebyc[double] +from .orthogonal_eval cimport eval_chebyc_l as _func_eval_chebyc_l +ctypedef double _proto_eval_chebyc_l_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_chebyc_l_t *_proto_eval_chebyc_l_t_var = &_func_eval_chebyc_l +from .orthogonal_eval cimport eval_chebys as _func_eval_chebys +ctypedef double complex _proto_eval_chebys_double_complex__t(double, double complex) noexcept nogil +cdef _proto_eval_chebys_double_complex__t *_proto_eval_chebys_double_complex__t_var = &_func_eval_chebys[double_complex] +from .orthogonal_eval cimport eval_chebys as _func_eval_chebys +ctypedef double _proto_eval_chebys_double__t(double, double) noexcept nogil +cdef _proto_eval_chebys_double__t *_proto_eval_chebys_double__t_var = &_func_eval_chebys[double] +from .orthogonal_eval cimport eval_chebys_l as _func_eval_chebys_l +ctypedef double _proto_eval_chebys_l_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_chebys_l_t *_proto_eval_chebys_l_t_var = &_func_eval_chebys_l +from .orthogonal_eval cimport eval_chebyt as _func_eval_chebyt +ctypedef double complex _proto_eval_chebyt_double_complex__t(double, double complex) noexcept nogil +cdef _proto_eval_chebyt_double_complex__t *_proto_eval_chebyt_double_complex__t_var = &_func_eval_chebyt[double_complex] +from .orthogonal_eval cimport eval_chebyt as _func_eval_chebyt +ctypedef double _proto_eval_chebyt_double__t(double, double) noexcept nogil +cdef _proto_eval_chebyt_double__t *_proto_eval_chebyt_double__t_var = &_func_eval_chebyt[double] +from .orthogonal_eval cimport eval_chebyt_l as _func_eval_chebyt_l +ctypedef double _proto_eval_chebyt_l_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_chebyt_l_t *_proto_eval_chebyt_l_t_var = &_func_eval_chebyt_l +from .orthogonal_eval cimport eval_chebyu as _func_eval_chebyu +ctypedef double complex _proto_eval_chebyu_double_complex__t(double, double complex) noexcept nogil +cdef _proto_eval_chebyu_double_complex__t *_proto_eval_chebyu_double_complex__t_var = &_func_eval_chebyu[double_complex] +from .orthogonal_eval cimport eval_chebyu as _func_eval_chebyu +ctypedef double _proto_eval_chebyu_double__t(double, double) noexcept nogil +cdef _proto_eval_chebyu_double__t *_proto_eval_chebyu_double__t_var = &_func_eval_chebyu[double] +from .orthogonal_eval cimport eval_chebyu_l as _func_eval_chebyu_l +ctypedef double _proto_eval_chebyu_l_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_chebyu_l_t *_proto_eval_chebyu_l_t_var = &_func_eval_chebyu_l +from .orthogonal_eval cimport eval_gegenbauer as _func_eval_gegenbauer +ctypedef double complex _proto_eval_gegenbauer_double_complex__t(double, double, double complex) noexcept nogil +cdef _proto_eval_gegenbauer_double_complex__t *_proto_eval_gegenbauer_double_complex__t_var = &_func_eval_gegenbauer[double_complex] +from .orthogonal_eval cimport eval_gegenbauer as _func_eval_gegenbauer +ctypedef double _proto_eval_gegenbauer_double__t(double, double, double) noexcept nogil +cdef _proto_eval_gegenbauer_double__t *_proto_eval_gegenbauer_double__t_var = &_func_eval_gegenbauer[double] +from .orthogonal_eval cimport eval_gegenbauer_l as _func_eval_gegenbauer_l +ctypedef double _proto_eval_gegenbauer_l_t(Py_ssize_t, double, double) noexcept nogil +cdef _proto_eval_gegenbauer_l_t *_proto_eval_gegenbauer_l_t_var = &_func_eval_gegenbauer_l +from .orthogonal_eval cimport eval_genlaguerre as _func_eval_genlaguerre +ctypedef double complex _proto_eval_genlaguerre_double_complex__t(double, double, double complex) noexcept nogil +cdef _proto_eval_genlaguerre_double_complex__t *_proto_eval_genlaguerre_double_complex__t_var = &_func_eval_genlaguerre[double_complex] +from .orthogonal_eval cimport eval_genlaguerre as _func_eval_genlaguerre +ctypedef double _proto_eval_genlaguerre_double__t(double, double, double) noexcept nogil +cdef _proto_eval_genlaguerre_double__t *_proto_eval_genlaguerre_double__t_var = &_func_eval_genlaguerre[double] +from .orthogonal_eval cimport eval_genlaguerre_l as _func_eval_genlaguerre_l +ctypedef double _proto_eval_genlaguerre_l_t(Py_ssize_t, double, double) noexcept nogil +cdef _proto_eval_genlaguerre_l_t *_proto_eval_genlaguerre_l_t_var = &_func_eval_genlaguerre_l +from .orthogonal_eval cimport eval_hermite as _func_eval_hermite +ctypedef double _proto_eval_hermite_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_hermite_t *_proto_eval_hermite_t_var = &_func_eval_hermite +from .orthogonal_eval cimport eval_hermitenorm as _func_eval_hermitenorm +ctypedef double _proto_eval_hermitenorm_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_hermitenorm_t *_proto_eval_hermitenorm_t_var = &_func_eval_hermitenorm +from .orthogonal_eval cimport eval_jacobi as _func_eval_jacobi +ctypedef double complex _proto_eval_jacobi_double_complex__t(double, double, double, double complex) noexcept nogil +cdef _proto_eval_jacobi_double_complex__t *_proto_eval_jacobi_double_complex__t_var = &_func_eval_jacobi[double_complex] +from .orthogonal_eval cimport eval_jacobi as _func_eval_jacobi +ctypedef double _proto_eval_jacobi_double__t(double, double, double, double) noexcept nogil +cdef _proto_eval_jacobi_double__t *_proto_eval_jacobi_double__t_var = &_func_eval_jacobi[double] +from .orthogonal_eval cimport eval_jacobi_l as _func_eval_jacobi_l +ctypedef double _proto_eval_jacobi_l_t(Py_ssize_t, double, double, double) noexcept nogil +cdef _proto_eval_jacobi_l_t *_proto_eval_jacobi_l_t_var = &_func_eval_jacobi_l +from .orthogonal_eval cimport eval_laguerre as _func_eval_laguerre +ctypedef double complex _proto_eval_laguerre_double_complex__t(double, double complex) noexcept nogil +cdef _proto_eval_laguerre_double_complex__t *_proto_eval_laguerre_double_complex__t_var = &_func_eval_laguerre[double_complex] +from .orthogonal_eval cimport eval_laguerre as _func_eval_laguerre +ctypedef double _proto_eval_laguerre_double__t(double, double) noexcept nogil +cdef _proto_eval_laguerre_double__t *_proto_eval_laguerre_double__t_var = &_func_eval_laguerre[double] +from .orthogonal_eval cimport eval_laguerre_l as _func_eval_laguerre_l +ctypedef double _proto_eval_laguerre_l_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_laguerre_l_t *_proto_eval_laguerre_l_t_var = &_func_eval_laguerre_l +from .orthogonal_eval cimport eval_legendre as _func_eval_legendre +ctypedef double complex _proto_eval_legendre_double_complex__t(double, double complex) noexcept nogil +cdef _proto_eval_legendre_double_complex__t *_proto_eval_legendre_double_complex__t_var = &_func_eval_legendre[double_complex] +from .orthogonal_eval cimport eval_legendre as _func_eval_legendre +ctypedef double _proto_eval_legendre_double__t(double, double) noexcept nogil +cdef _proto_eval_legendre_double__t *_proto_eval_legendre_double__t_var = &_func_eval_legendre[double] +from .orthogonal_eval cimport eval_legendre_l as _func_eval_legendre_l +ctypedef double _proto_eval_legendre_l_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_legendre_l_t *_proto_eval_legendre_l_t_var = &_func_eval_legendre_l +from .orthogonal_eval cimport eval_sh_chebyt as _func_eval_sh_chebyt +ctypedef double complex _proto_eval_sh_chebyt_double_complex__t(double, double complex) noexcept nogil +cdef _proto_eval_sh_chebyt_double_complex__t *_proto_eval_sh_chebyt_double_complex__t_var = &_func_eval_sh_chebyt[double_complex] +from .orthogonal_eval cimport eval_sh_chebyt as _func_eval_sh_chebyt +ctypedef double _proto_eval_sh_chebyt_double__t(double, double) noexcept nogil +cdef _proto_eval_sh_chebyt_double__t *_proto_eval_sh_chebyt_double__t_var = &_func_eval_sh_chebyt[double] +from .orthogonal_eval cimport eval_sh_chebyt_l as _func_eval_sh_chebyt_l +ctypedef double _proto_eval_sh_chebyt_l_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_sh_chebyt_l_t *_proto_eval_sh_chebyt_l_t_var = &_func_eval_sh_chebyt_l +from .orthogonal_eval cimport eval_sh_chebyu as _func_eval_sh_chebyu +ctypedef double complex _proto_eval_sh_chebyu_double_complex__t(double, double complex) noexcept nogil +cdef _proto_eval_sh_chebyu_double_complex__t *_proto_eval_sh_chebyu_double_complex__t_var = &_func_eval_sh_chebyu[double_complex] +from .orthogonal_eval cimport eval_sh_chebyu as _func_eval_sh_chebyu +ctypedef double _proto_eval_sh_chebyu_double__t(double, double) noexcept nogil +cdef _proto_eval_sh_chebyu_double__t *_proto_eval_sh_chebyu_double__t_var = &_func_eval_sh_chebyu[double] +from .orthogonal_eval cimport eval_sh_chebyu_l as _func_eval_sh_chebyu_l +ctypedef double _proto_eval_sh_chebyu_l_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_sh_chebyu_l_t *_proto_eval_sh_chebyu_l_t_var = &_func_eval_sh_chebyu_l +from .orthogonal_eval cimport eval_sh_jacobi as _func_eval_sh_jacobi +ctypedef double complex _proto_eval_sh_jacobi_double_complex__t(double, double, double, double complex) noexcept nogil +cdef _proto_eval_sh_jacobi_double_complex__t *_proto_eval_sh_jacobi_double_complex__t_var = &_func_eval_sh_jacobi[double_complex] +from .orthogonal_eval cimport eval_sh_jacobi as _func_eval_sh_jacobi +ctypedef double _proto_eval_sh_jacobi_double__t(double, double, double, double) noexcept nogil +cdef _proto_eval_sh_jacobi_double__t *_proto_eval_sh_jacobi_double__t_var = &_func_eval_sh_jacobi[double] +from .orthogonal_eval cimport eval_sh_jacobi_l as _func_eval_sh_jacobi_l +ctypedef double _proto_eval_sh_jacobi_l_t(Py_ssize_t, double, double, double) noexcept nogil +cdef _proto_eval_sh_jacobi_l_t *_proto_eval_sh_jacobi_l_t_var = &_func_eval_sh_jacobi_l +from .orthogonal_eval cimport eval_sh_legendre as _func_eval_sh_legendre +ctypedef double complex _proto_eval_sh_legendre_double_complex__t(double, double complex) noexcept nogil +cdef _proto_eval_sh_legendre_double_complex__t *_proto_eval_sh_legendre_double_complex__t_var = &_func_eval_sh_legendre[double_complex] +from .orthogonal_eval cimport eval_sh_legendre as _func_eval_sh_legendre +ctypedef double _proto_eval_sh_legendre_double__t(double, double) noexcept nogil +cdef _proto_eval_sh_legendre_double__t *_proto_eval_sh_legendre_double__t_var = &_func_eval_sh_legendre[double] +from .orthogonal_eval cimport eval_sh_legendre_l as _func_eval_sh_legendre_l +ctypedef double _proto_eval_sh_legendre_l_t(Py_ssize_t, double) noexcept nogil +cdef _proto_eval_sh_legendre_l_t *_proto_eval_sh_legendre_l_t_var = &_func_eval_sh_legendre_l +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_exp10 "cephes_exp10"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_exp2 "cephes_exp2"(double) noexcept nogil +from ._cunity cimport cexpm1 as _func_cexpm1 +ctypedef double complex _proto_cexpm1_t(double complex) noexcept nogil +cdef _proto_cexpm1_t *_proto_cexpm1_t_var = &_func_cexpm1 +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_expm1 "cephes_expm1"(double) noexcept nogil +from ._legacy cimport expn_unsafe as _func_expn_unsafe +ctypedef double _proto_expn_unsafe_t(double, double) noexcept nogil +cdef _proto_expn_unsafe_t *_proto_expn_unsafe_t_var = &_func_expn_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_expn_wrap "cephes_expn_wrap"(Py_ssize_t, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_fdtr "xsf_fdtr"(double, double, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_fdtrc "xsf_fdtrc"(double, double, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_fdtri "xsf_fdtri"(double, double, double) noexcept nogil +from ._cdflib_wrappers cimport fdtridfd as _func_fdtridfd +ctypedef double _proto_fdtridfd_t(double, double, double) noexcept nogil +cdef _proto_fdtridfd_t *_proto_fdtridfd_t_var = &_func_fdtridfd +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_gdtr "xsf_gdtr"(double, double, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_gdtrc "xsf_gdtrc"(double, double, double) noexcept nogil +from ._cdflib_wrappers cimport gdtria as _func_gdtria +ctypedef double _proto_gdtria_t(double, double, double) noexcept nogil +cdef _proto_gdtria_t *_proto_gdtria_t_var = &_func_gdtria +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_gdtrib "xsf_gdtrib"(double, double, double) noexcept nogil +from ._cdflib_wrappers cimport gdtrix as _func_gdtrix +ctypedef double _proto_gdtrix_t(double, double, double) noexcept nogil +cdef _proto_gdtrix_t *_proto_gdtrix_t_var = &_func_gdtrix +from ._convex_analysis cimport huber as _func_huber +ctypedef double _proto_huber_t(double, double) noexcept nogil +cdef _proto_huber_t *_proto_huber_t_var = &_func_huber +from ._hyp0f1 cimport _hyp0f1_cmplx as _func__hyp0f1_cmplx +ctypedef double complex _proto__hyp0f1_cmplx_t(double, double complex) noexcept nogil +cdef _proto__hyp0f1_cmplx_t *_proto__hyp0f1_cmplx_t_var = &_func__hyp0f1_cmplx +from ._hyp0f1 cimport _hyp0f1_real as _func__hyp0f1_real +ctypedef double _proto__hyp0f1_real_t(double, double) noexcept nogil +cdef _proto__hyp0f1_real_t *_proto__hyp0f1_real_t_var = &_func__hyp0f1_real +cdef extern from r"_ufuncs_defs.h": + cdef double complex _func_chyp1f1_wrap "chyp1f1_wrap"(double, double, double complex) noexcept nogil +from ._hypergeometric cimport hyperu as _func_hyperu +ctypedef double _proto_hyperu_t(double, double, double) noexcept nogil +cdef _proto_hyperu_t *_proto_hyperu_t_var = &_func_hyperu +from ._boxcox cimport inv_boxcox as _func_inv_boxcox +ctypedef double _proto_inv_boxcox_t(double, double) noexcept nogil +cdef _proto_inv_boxcox_t *_proto_inv_boxcox_t_var = &_func_inv_boxcox +from ._boxcox cimport inv_boxcox1p as _func_inv_boxcox1p +ctypedef double _proto_inv_boxcox1p_t(double, double) noexcept nogil +cdef _proto_inv_boxcox1p_t *_proto_inv_boxcox1p_t_var = &_func_inv_boxcox1p +from ._convex_analysis cimport kl_div as _func_kl_div +ctypedef double _proto_kl_div_t(double, double) noexcept nogil +cdef _proto_kl_div_t *_proto_kl_div_t_var = &_func_kl_div +from ._legacy cimport kn_unsafe as _func_kn_unsafe +ctypedef double _proto_kn_unsafe_t(double, double) noexcept nogil +cdef _proto_kn_unsafe_t *_proto_kn_unsafe_t_var = &_func_kn_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_special_cyl_bessel_k_int "special_cyl_bessel_k_int"(Py_ssize_t, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_kolmogi "xsf_kolmogi"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_kolmogorov "xsf_kolmogorov"(double) noexcept nogil +from ._cunity cimport clog1p as _func_clog1p +ctypedef double complex _proto_clog1p_t(double complex) noexcept nogil +cdef _proto_clog1p_t *_proto_clog1p_t_var = &_func_clog1p +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_log1p "cephes_log1p"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_pmv_wrap "pmv_wrap"(double, double, double) noexcept nogil +from ._legacy cimport nbdtr_unsafe as _func_nbdtr_unsafe +ctypedef double _proto_nbdtr_unsafe_t(double, double, double) noexcept nogil +cdef _proto_nbdtr_unsafe_t *_proto_nbdtr_unsafe_t_var = &_func_nbdtr_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_nbdtr_wrap "cephes_nbdtr_wrap"(Py_ssize_t, Py_ssize_t, double) noexcept nogil +from ._legacy cimport nbdtrc_unsafe as _func_nbdtrc_unsafe +ctypedef double _proto_nbdtrc_unsafe_t(double, double, double) noexcept nogil +cdef _proto_nbdtrc_unsafe_t *_proto_nbdtrc_unsafe_t_var = &_func_nbdtrc_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_nbdtrc_wrap "cephes_nbdtrc_wrap"(Py_ssize_t, Py_ssize_t, double) noexcept nogil +from ._legacy cimport nbdtri_unsafe as _func_nbdtri_unsafe +ctypedef double _proto_nbdtri_unsafe_t(double, double, double) noexcept nogil +cdef _proto_nbdtri_unsafe_t *_proto_nbdtri_unsafe_t_var = &_func_nbdtri_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_nbdtri_wrap "cephes_nbdtri_wrap"(Py_ssize_t, Py_ssize_t, double) noexcept nogil +from ._cdflib_wrappers cimport nbdtrik as _func_nbdtrik +ctypedef double _proto_nbdtrik_t(double, double, double) noexcept nogil +cdef _proto_nbdtrik_t *_proto_nbdtrik_t_var = &_func_nbdtrik +from ._cdflib_wrappers cimport nbdtrin as _func_nbdtrin +ctypedef double _proto_nbdtrin_t(double, double, double) noexcept nogil +cdef _proto_nbdtrin_t *_proto_nbdtrin_t_var = &_func_nbdtrin +from ._cdflib_wrappers cimport ncfdtridfd as _func_ncfdtridfd +ctypedef double _proto_ncfdtridfd_t(double, double, double, double) noexcept nogil +cdef _proto_ncfdtridfd_t *_proto_ncfdtridfd_t_var = &_func_ncfdtridfd +from ._cdflib_wrappers cimport ncfdtridfn as _func_ncfdtridfn +ctypedef double _proto_ncfdtridfn_t(double, double, double, double) noexcept nogil +cdef _proto_ncfdtridfn_t *_proto_ncfdtridfn_t_var = &_func_ncfdtridfn +from ._cdflib_wrappers cimport ncfdtrinc as _func_ncfdtrinc +ctypedef double _proto_ncfdtrinc_t(double, double, double, double) noexcept nogil +cdef _proto_ncfdtrinc_t *_proto_ncfdtrinc_t_var = &_func_ncfdtrinc +from ._cdflib_wrappers cimport nctdtridf as _func_nctdtridf +ctypedef double _proto_nctdtridf_t(double, double, double) noexcept nogil +cdef _proto_nctdtridf_t *_proto_nctdtridf_t_var = &_func_nctdtridf +from ._cdflib_wrappers cimport nctdtrinc as _func_nctdtrinc +ctypedef double _proto_nctdtrinc_t(double, double, double) noexcept nogil +cdef _proto_nctdtrinc_t *_proto_nctdtrinc_t_var = &_func_nctdtrinc +from ._cdflib_wrappers cimport nctdtrit as _func_nctdtrit +ctypedef double _proto_nctdtrit_t(double, double, double) noexcept nogil +cdef _proto_nctdtrit_t *_proto_nctdtrit_t_var = &_func_nctdtrit +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_ndtr "xsf_ndtr"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_ndtri "xsf_ndtri"(double) noexcept nogil +from ._ndtri_exp cimport ndtri_exp as _func_ndtri_exp +ctypedef double _proto_ndtri_exp_t(double) noexcept nogil +cdef _proto_ndtri_exp_t *_proto_ndtri_exp_t_var = &_func_ndtri_exp +from ._cdflib_wrappers cimport nrdtrimn as _func_nrdtrimn +ctypedef double _proto_nrdtrimn_t(double, double, double) noexcept nogil +cdef _proto_nrdtrimn_t *_proto_nrdtrimn_t_var = &_func_nrdtrimn +from ._cdflib_wrappers cimport nrdtrisd as _func_nrdtrisd +ctypedef double _proto_nrdtrisd_t(double, double, double) noexcept nogil +cdef _proto_nrdtrisd_t *_proto_nrdtrisd_t_var = &_func_nrdtrisd +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_owens_t "xsf_owens_t"(double, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_pdtr "xsf_pdtr"(double, double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_pdtrc "xsf_pdtrc"(double, double) noexcept nogil +from ._legacy cimport pdtri_unsafe as _func_pdtri_unsafe +ctypedef double _proto_pdtri_unsafe_t(double, double) noexcept nogil +cdef _proto_pdtri_unsafe_t *_proto_pdtri_unsafe_t_var = &_func_pdtri_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_pdtri_wrap "cephes_pdtri_wrap"(Py_ssize_t, double) noexcept nogil +from ._cdflib_wrappers cimport pdtrik as _func_pdtrik +ctypedef double _proto_pdtrik_t(double, double) noexcept nogil +cdef _proto_pdtrik_t *_proto_pdtrik_t_var = &_func_pdtrik +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_poch "cephes_poch"(double, double) noexcept nogil +from ._convex_analysis cimport pseudo_huber as _func_pseudo_huber +ctypedef double _proto_pseudo_huber_t(double, double) noexcept nogil +cdef _proto_pseudo_huber_t *_proto_pseudo_huber_t_var = &_func_pseudo_huber +from ._convex_analysis cimport rel_entr as _func_rel_entr +ctypedef double _proto_rel_entr_t(double, double) noexcept nogil +cdef _proto_rel_entr_t *_proto_rel_entr_t_var = &_func_rel_entr +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_round "cephes_round"(double) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef int _func_xsf_cshichi "xsf_cshichi"(double complex, double complex *, double complex *) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef int _func_xsf_shichi "xsf_shichi"(double, double *, double *) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef int _func_xsf_csici "xsf_csici"(double complex, double complex *, double complex *) noexcept nogil +cdef extern from r"_ufuncs_defs.h": + cdef int _func_xsf_sici "xsf_sici"(double, double *, double *) noexcept nogil +from ._legacy cimport smirnov_unsafe as _func_smirnov_unsafe +ctypedef double _proto_smirnov_unsafe_t(double, double) noexcept nogil +cdef _proto_smirnov_unsafe_t *_proto_smirnov_unsafe_t_var = &_func_smirnov_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_smirnov_wrap "cephes_smirnov_wrap"(Py_ssize_t, double) noexcept nogil +from ._legacy cimport smirnovi_unsafe as _func_smirnovi_unsafe +ctypedef double _proto_smirnovi_unsafe_t(double, double) noexcept nogil +cdef _proto_smirnovi_unsafe_t *_proto_smirnovi_unsafe_t_var = &_func_smirnovi_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_smirnovi_wrap "cephes_smirnovi_wrap"(Py_ssize_t, double) noexcept nogil +from ._spence cimport cspence as _func_cspence +ctypedef double complex _proto_cspence_t(double complex) noexcept nogil +cdef _proto_cspence_t *_proto_cspence_t_var = &_func_cspence +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_spence "cephes_spence"(double) noexcept nogil +from ._cdflib_wrappers cimport stdtr as _func_stdtr +ctypedef double _proto_stdtr_t(double, double) noexcept nogil +cdef _proto_stdtr_t *_proto_stdtr_t_var = &_func_stdtr +from ._cdflib_wrappers cimport stdtridf as _func_stdtridf +ctypedef double _proto_stdtridf_t(double, double) noexcept nogil +cdef _proto_stdtridf_t *_proto_stdtridf_t_var = &_func_stdtridf +from ._cdflib_wrappers cimport stdtrit as _func_stdtrit +ctypedef double _proto_stdtrit_t(double, double) noexcept nogil +cdef _proto_stdtrit_t *_proto_stdtrit_t_var = &_func_stdtrit +cdef extern from r"_ufuncs_defs.h": + cdef double _func_xsf_tukeylambdacdf "xsf_tukeylambdacdf"(double, double) noexcept nogil +from ._xlogy cimport xlog1py as _func_xlog1py +ctypedef double _proto_xlog1py_double__t(double, double) noexcept nogil +cdef _proto_xlog1py_double__t *_proto_xlog1py_double__t_var = &_func_xlog1py[double] +from ._xlogy cimport xlog1py as _func_xlog1py +ctypedef double complex _proto_xlog1py_double_complex__t(double complex, double complex) noexcept nogil +cdef _proto_xlog1py_double_complex__t *_proto_xlog1py_double_complex__t_var = &_func_xlog1py[double_complex] +from ._xlogy cimport xlogy as _func_xlogy +ctypedef double _proto_xlogy_double__t(double, double) noexcept nogil +cdef _proto_xlogy_double__t *_proto_xlogy_double__t_var = &_func_xlogy[double] +from ._xlogy cimport xlogy as _func_xlogy +ctypedef double complex _proto_xlogy_double_complex__t(double complex, double complex) noexcept nogil +cdef _proto_xlogy_double_complex__t *_proto_xlogy_double_complex__t_var = &_func_xlogy[double_complex] +from ._legacy cimport yn_unsafe as _func_yn_unsafe +ctypedef double _proto_yn_unsafe_t(double, double) noexcept nogil +cdef _proto_yn_unsafe_t *_proto_yn_unsafe_t_var = &_func_yn_unsafe +cdef extern from r"_ufuncs_defs.h": + cdef double _func_cephes_yn_wrap "cephes_yn_wrap"(Py_ssize_t, double) noexcept nogil +cdef np.PyUFuncGenericFunction ufunc__beta_pdf_loops[2] +cdef void *ufunc__beta_pdf_ptr[4] +cdef void *ufunc__beta_pdf_data[2] +cdef char ufunc__beta_pdf_types[8] +cdef char *ufunc__beta_pdf_doc = ( + "_beta_pdf(x, a, b)\n" + "\n" + "Probability density function of beta distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued such that :math:`0 \\leq x \\leq 1`,\n" + " the upper limit of integration\n" + "a, b : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__beta_pdf_loops[0] = loop_f_fff__As_fff_f +ufunc__beta_pdf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__beta_pdf_types[0] = NPY_FLOAT +ufunc__beta_pdf_types[1] = NPY_FLOAT +ufunc__beta_pdf_types[2] = NPY_FLOAT +ufunc__beta_pdf_types[3] = NPY_FLOAT +ufunc__beta_pdf_types[4] = NPY_DOUBLE +ufunc__beta_pdf_types[5] = NPY_DOUBLE +ufunc__beta_pdf_types[6] = NPY_DOUBLE +ufunc__beta_pdf_types[7] = NPY_DOUBLE +ufunc__beta_pdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_beta_pdf_float +ufunc__beta_pdf_ptr[2*0+1] = ("_beta_pdf") +ufunc__beta_pdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_beta_pdf_double +ufunc__beta_pdf_ptr[2*1+1] = ("_beta_pdf") +ufunc__beta_pdf_data[0] = &ufunc__beta_pdf_ptr[2*0] +ufunc__beta_pdf_data[1] = &ufunc__beta_pdf_ptr[2*1] +_beta_pdf = np.PyUFunc_FromFuncAndData(ufunc__beta_pdf_loops, ufunc__beta_pdf_data, ufunc__beta_pdf_types, 2, 3, 1, 0, "_beta_pdf", ufunc__beta_pdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__beta_ppf_loops[2] +cdef void *ufunc__beta_ppf_ptr[4] +cdef void *ufunc__beta_ppf_data[2] +cdef char ufunc__beta_ppf_types[8] +cdef char *ufunc__beta_ppf_doc = ( + "_beta_ppf(x, a, b)\n" + "\n" + "Percent point function of beta distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued such that :math:`0 \\leq x \\leq 1`,\n" + " the upper limit of integration\n" + "a, b : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__beta_ppf_loops[0] = loop_f_fff__As_fff_f +ufunc__beta_ppf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__beta_ppf_types[0] = NPY_FLOAT +ufunc__beta_ppf_types[1] = NPY_FLOAT +ufunc__beta_ppf_types[2] = NPY_FLOAT +ufunc__beta_ppf_types[3] = NPY_FLOAT +ufunc__beta_ppf_types[4] = NPY_DOUBLE +ufunc__beta_ppf_types[5] = NPY_DOUBLE +ufunc__beta_ppf_types[6] = NPY_DOUBLE +ufunc__beta_ppf_types[7] = NPY_DOUBLE +ufunc__beta_ppf_ptr[2*0] = scipy.special._ufuncs_cxx._export_beta_ppf_float +ufunc__beta_ppf_ptr[2*0+1] = ("_beta_ppf") +ufunc__beta_ppf_ptr[2*1] = scipy.special._ufuncs_cxx._export_beta_ppf_double +ufunc__beta_ppf_ptr[2*1+1] = ("_beta_ppf") +ufunc__beta_ppf_data[0] = &ufunc__beta_ppf_ptr[2*0] +ufunc__beta_ppf_data[1] = &ufunc__beta_ppf_ptr[2*1] +_beta_ppf = np.PyUFunc_FromFuncAndData(ufunc__beta_ppf_loops, ufunc__beta_ppf_data, ufunc__beta_ppf_types, 2, 3, 1, 0, "_beta_ppf", ufunc__beta_ppf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__binom_cdf_loops[2] +cdef void *ufunc__binom_cdf_ptr[4] +cdef void *ufunc__binom_cdf_data[2] +cdef char ufunc__binom_cdf_types[8] +cdef char *ufunc__binom_cdf_doc = ( + "_binom_cdf(x, n, p)\n" + "\n" + "Cumulative density function of binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "n : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__binom_cdf_loops[0] = loop_f_fff__As_fff_f +ufunc__binom_cdf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__binom_cdf_types[0] = NPY_FLOAT +ufunc__binom_cdf_types[1] = NPY_FLOAT +ufunc__binom_cdf_types[2] = NPY_FLOAT +ufunc__binom_cdf_types[3] = NPY_FLOAT +ufunc__binom_cdf_types[4] = NPY_DOUBLE +ufunc__binom_cdf_types[5] = NPY_DOUBLE +ufunc__binom_cdf_types[6] = NPY_DOUBLE +ufunc__binom_cdf_types[7] = NPY_DOUBLE +ufunc__binom_cdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_binom_cdf_float +ufunc__binom_cdf_ptr[2*0+1] = ("_binom_cdf") +ufunc__binom_cdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_binom_cdf_double +ufunc__binom_cdf_ptr[2*1+1] = ("_binom_cdf") +ufunc__binom_cdf_data[0] = &ufunc__binom_cdf_ptr[2*0] +ufunc__binom_cdf_data[1] = &ufunc__binom_cdf_ptr[2*1] +_binom_cdf = np.PyUFunc_FromFuncAndData(ufunc__binom_cdf_loops, ufunc__binom_cdf_data, ufunc__binom_cdf_types, 2, 3, 1, 0, "_binom_cdf", ufunc__binom_cdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__binom_isf_loops[2] +cdef void *ufunc__binom_isf_ptr[4] +cdef void *ufunc__binom_isf_data[2] +cdef char ufunc__binom_isf_types[8] +cdef char *ufunc__binom_isf_doc = ( + "_binom_isf(x, n, p)\n" + "\n" + "Inverse survival function of binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "n : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__binom_isf_loops[0] = loop_f_fff__As_fff_f +ufunc__binom_isf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__binom_isf_types[0] = NPY_FLOAT +ufunc__binom_isf_types[1] = NPY_FLOAT +ufunc__binom_isf_types[2] = NPY_FLOAT +ufunc__binom_isf_types[3] = NPY_FLOAT +ufunc__binom_isf_types[4] = NPY_DOUBLE +ufunc__binom_isf_types[5] = NPY_DOUBLE +ufunc__binom_isf_types[6] = NPY_DOUBLE +ufunc__binom_isf_types[7] = NPY_DOUBLE +ufunc__binom_isf_ptr[2*0] = scipy.special._ufuncs_cxx._export_binom_isf_float +ufunc__binom_isf_ptr[2*0+1] = ("_binom_isf") +ufunc__binom_isf_ptr[2*1] = scipy.special._ufuncs_cxx._export_binom_isf_double +ufunc__binom_isf_ptr[2*1+1] = ("_binom_isf") +ufunc__binom_isf_data[0] = &ufunc__binom_isf_ptr[2*0] +ufunc__binom_isf_data[1] = &ufunc__binom_isf_ptr[2*1] +_binom_isf = np.PyUFunc_FromFuncAndData(ufunc__binom_isf_loops, ufunc__binom_isf_data, ufunc__binom_isf_types, 2, 3, 1, 0, "_binom_isf", ufunc__binom_isf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__binom_pmf_loops[2] +cdef void *ufunc__binom_pmf_ptr[4] +cdef void *ufunc__binom_pmf_data[2] +cdef char ufunc__binom_pmf_types[8] +cdef char *ufunc__binom_pmf_doc = ( + "_binom_pmf(x, n, p)\n" + "\n" + "Probability mass function of binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "n : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__binom_pmf_loops[0] = loop_f_fff__As_fff_f +ufunc__binom_pmf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__binom_pmf_types[0] = NPY_FLOAT +ufunc__binom_pmf_types[1] = NPY_FLOAT +ufunc__binom_pmf_types[2] = NPY_FLOAT +ufunc__binom_pmf_types[3] = NPY_FLOAT +ufunc__binom_pmf_types[4] = NPY_DOUBLE +ufunc__binom_pmf_types[5] = NPY_DOUBLE +ufunc__binom_pmf_types[6] = NPY_DOUBLE +ufunc__binom_pmf_types[7] = NPY_DOUBLE +ufunc__binom_pmf_ptr[2*0] = scipy.special._ufuncs_cxx._export_binom_pmf_float +ufunc__binom_pmf_ptr[2*0+1] = ("_binom_pmf") +ufunc__binom_pmf_ptr[2*1] = scipy.special._ufuncs_cxx._export_binom_pmf_double +ufunc__binom_pmf_ptr[2*1+1] = ("_binom_pmf") +ufunc__binom_pmf_data[0] = &ufunc__binom_pmf_ptr[2*0] +ufunc__binom_pmf_data[1] = &ufunc__binom_pmf_ptr[2*1] +_binom_pmf = np.PyUFunc_FromFuncAndData(ufunc__binom_pmf_loops, ufunc__binom_pmf_data, ufunc__binom_pmf_types, 2, 3, 1, 0, "_binom_pmf", ufunc__binom_pmf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__binom_ppf_loops[2] +cdef void *ufunc__binom_ppf_ptr[4] +cdef void *ufunc__binom_ppf_data[2] +cdef char ufunc__binom_ppf_types[8] +cdef char *ufunc__binom_ppf_doc = ( + "_binom_ppf(x, n, p)\n" + "\n" + "Percent point function of binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "n : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__binom_ppf_loops[0] = loop_f_fff__As_fff_f +ufunc__binom_ppf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__binom_ppf_types[0] = NPY_FLOAT +ufunc__binom_ppf_types[1] = NPY_FLOAT +ufunc__binom_ppf_types[2] = NPY_FLOAT +ufunc__binom_ppf_types[3] = NPY_FLOAT +ufunc__binom_ppf_types[4] = NPY_DOUBLE +ufunc__binom_ppf_types[5] = NPY_DOUBLE +ufunc__binom_ppf_types[6] = NPY_DOUBLE +ufunc__binom_ppf_types[7] = NPY_DOUBLE +ufunc__binom_ppf_ptr[2*0] = scipy.special._ufuncs_cxx._export_binom_ppf_float +ufunc__binom_ppf_ptr[2*0+1] = ("_binom_ppf") +ufunc__binom_ppf_ptr[2*1] = scipy.special._ufuncs_cxx._export_binom_ppf_double +ufunc__binom_ppf_ptr[2*1+1] = ("_binom_ppf") +ufunc__binom_ppf_data[0] = &ufunc__binom_ppf_ptr[2*0] +ufunc__binom_ppf_data[1] = &ufunc__binom_ppf_ptr[2*1] +_binom_ppf = np.PyUFunc_FromFuncAndData(ufunc__binom_ppf_loops, ufunc__binom_ppf_data, ufunc__binom_ppf_types, 2, 3, 1, 0, "_binom_ppf", ufunc__binom_ppf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__binom_sf_loops[2] +cdef void *ufunc__binom_sf_ptr[4] +cdef void *ufunc__binom_sf_data[2] +cdef char ufunc__binom_sf_types[8] +cdef char *ufunc__binom_sf_doc = ( + "_binom_sf(x, n, p)\n" + "\n" + "Survival function of binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "n : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__binom_sf_loops[0] = loop_f_fff__As_fff_f +ufunc__binom_sf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__binom_sf_types[0] = NPY_FLOAT +ufunc__binom_sf_types[1] = NPY_FLOAT +ufunc__binom_sf_types[2] = NPY_FLOAT +ufunc__binom_sf_types[3] = NPY_FLOAT +ufunc__binom_sf_types[4] = NPY_DOUBLE +ufunc__binom_sf_types[5] = NPY_DOUBLE +ufunc__binom_sf_types[6] = NPY_DOUBLE +ufunc__binom_sf_types[7] = NPY_DOUBLE +ufunc__binom_sf_ptr[2*0] = scipy.special._ufuncs_cxx._export_binom_sf_float +ufunc__binom_sf_ptr[2*0+1] = ("_binom_sf") +ufunc__binom_sf_ptr[2*1] = scipy.special._ufuncs_cxx._export_binom_sf_double +ufunc__binom_sf_ptr[2*1+1] = ("_binom_sf") +ufunc__binom_sf_data[0] = &ufunc__binom_sf_ptr[2*0] +ufunc__binom_sf_data[1] = &ufunc__binom_sf_ptr[2*1] +_binom_sf = np.PyUFunc_FromFuncAndData(ufunc__binom_sf_loops, ufunc__binom_sf_data, ufunc__binom_sf_types, 2, 3, 1, 0, "_binom_sf", ufunc__binom_sf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__cauchy_isf_loops[2] +cdef void *ufunc__cauchy_isf_ptr[4] +cdef void *ufunc__cauchy_isf_data[2] +cdef char ufunc__cauchy_isf_types[8] +cdef char *ufunc__cauchy_isf_doc = ( + "_cauchy_isf(p, loc, scale)\n" + "\n" + "Inverse survival function of the Cauchy distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Probabilities\n" + "loc : array_like\n" + " Location parameter of the distribution.\n" + "scale : array_like\n" + " Scale parameter of the distribution.\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__cauchy_isf_loops[0] = loop_f_fff__As_fff_f +ufunc__cauchy_isf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__cauchy_isf_types[0] = NPY_FLOAT +ufunc__cauchy_isf_types[1] = NPY_FLOAT +ufunc__cauchy_isf_types[2] = NPY_FLOAT +ufunc__cauchy_isf_types[3] = NPY_FLOAT +ufunc__cauchy_isf_types[4] = NPY_DOUBLE +ufunc__cauchy_isf_types[5] = NPY_DOUBLE +ufunc__cauchy_isf_types[6] = NPY_DOUBLE +ufunc__cauchy_isf_types[7] = NPY_DOUBLE +ufunc__cauchy_isf_ptr[2*0] = scipy.special._ufuncs_cxx._export_cauchy_isf_float +ufunc__cauchy_isf_ptr[2*0+1] = ("_cauchy_isf") +ufunc__cauchy_isf_ptr[2*1] = scipy.special._ufuncs_cxx._export_cauchy_isf_double +ufunc__cauchy_isf_ptr[2*1+1] = ("_cauchy_isf") +ufunc__cauchy_isf_data[0] = &ufunc__cauchy_isf_ptr[2*0] +ufunc__cauchy_isf_data[1] = &ufunc__cauchy_isf_ptr[2*1] +_cauchy_isf = np.PyUFunc_FromFuncAndData(ufunc__cauchy_isf_loops, ufunc__cauchy_isf_data, ufunc__cauchy_isf_types, 2, 3, 1, 0, "_cauchy_isf", ufunc__cauchy_isf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__cauchy_ppf_loops[2] +cdef void *ufunc__cauchy_ppf_ptr[4] +cdef void *ufunc__cauchy_ppf_data[2] +cdef char ufunc__cauchy_ppf_types[8] +cdef char *ufunc__cauchy_ppf_doc = ( + "_cauchy_ppf(p, loc, scale)\n" + "\n" + "Percent point function (i.e. quantile) of the Cauchy distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Probabilities\n" + "loc : array_like\n" + " Location parameter of the distribution.\n" + "scale : array_like\n" + " Scale parameter of the distribution.\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__cauchy_ppf_loops[0] = loop_f_fff__As_fff_f +ufunc__cauchy_ppf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__cauchy_ppf_types[0] = NPY_FLOAT +ufunc__cauchy_ppf_types[1] = NPY_FLOAT +ufunc__cauchy_ppf_types[2] = NPY_FLOAT +ufunc__cauchy_ppf_types[3] = NPY_FLOAT +ufunc__cauchy_ppf_types[4] = NPY_DOUBLE +ufunc__cauchy_ppf_types[5] = NPY_DOUBLE +ufunc__cauchy_ppf_types[6] = NPY_DOUBLE +ufunc__cauchy_ppf_types[7] = NPY_DOUBLE +ufunc__cauchy_ppf_ptr[2*0] = scipy.special._ufuncs_cxx._export_cauchy_ppf_float +ufunc__cauchy_ppf_ptr[2*0+1] = ("_cauchy_ppf") +ufunc__cauchy_ppf_ptr[2*1] = scipy.special._ufuncs_cxx._export_cauchy_ppf_double +ufunc__cauchy_ppf_ptr[2*1+1] = ("_cauchy_ppf") +ufunc__cauchy_ppf_data[0] = &ufunc__cauchy_ppf_ptr[2*0] +ufunc__cauchy_ppf_data[1] = &ufunc__cauchy_ppf_ptr[2*1] +_cauchy_ppf = np.PyUFunc_FromFuncAndData(ufunc__cauchy_ppf_loops, ufunc__cauchy_ppf_data, ufunc__cauchy_ppf_types, 2, 3, 1, 0, "_cauchy_ppf", ufunc__cauchy_ppf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__cosine_cdf_loops[2] +cdef void *ufunc__cosine_cdf_ptr[4] +cdef void *ufunc__cosine_cdf_data[2] +cdef char ufunc__cosine_cdf_types[4] +cdef char *ufunc__cosine_cdf_doc = ( + "_cosine_cdf(x)\n" + "\n" + "Cumulative distribution function (CDF) of the cosine distribution::\n" + "\n" + " { 0, x < -pi\n" + " cdf(x) = { (pi + x + sin(x))/(2*pi), -pi <= x <= pi\n" + " { 1, x > pi\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " `x` must contain real numbers.\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The cosine distribution CDF evaluated at `x`.") +ufunc__cosine_cdf_loops[0] = loop_d_d__As_f_f +ufunc__cosine_cdf_loops[1] = loop_d_d__As_d_d +ufunc__cosine_cdf_types[0] = NPY_FLOAT +ufunc__cosine_cdf_types[1] = NPY_FLOAT +ufunc__cosine_cdf_types[2] = NPY_DOUBLE +ufunc__cosine_cdf_types[3] = NPY_DOUBLE +ufunc__cosine_cdf_ptr[2*0] = _func_cosine_cdf +ufunc__cosine_cdf_ptr[2*0+1] = ("_cosine_cdf") +ufunc__cosine_cdf_ptr[2*1] = _func_cosine_cdf +ufunc__cosine_cdf_ptr[2*1+1] = ("_cosine_cdf") +ufunc__cosine_cdf_data[0] = &ufunc__cosine_cdf_ptr[2*0] +ufunc__cosine_cdf_data[1] = &ufunc__cosine_cdf_ptr[2*1] +_cosine_cdf = np.PyUFunc_FromFuncAndData(ufunc__cosine_cdf_loops, ufunc__cosine_cdf_data, ufunc__cosine_cdf_types, 2, 1, 1, 0, "_cosine_cdf", ufunc__cosine_cdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__cosine_invcdf_loops[2] +cdef void *ufunc__cosine_invcdf_ptr[4] +cdef void *ufunc__cosine_invcdf_data[2] +cdef char ufunc__cosine_invcdf_types[4] +cdef char *ufunc__cosine_invcdf_doc = ( + "_cosine_invcdf(p)\n" + "\n" + "Inverse of the cumulative distribution function (CDF) of the cosine\n" + "distribution.\n" + "\n" + "The CDF of the cosine distribution is::\n" + "\n" + " cdf(x) = (pi + x + sin(x))/(2*pi)\n" + "\n" + "This function computes the inverse of cdf(x).\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " `p` must contain real numbers in the interval ``0 <= p <= 1``.\n" + " `nan` is returned for values of `p` outside the interval [0, 1].\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The inverse of the cosine distribution CDF evaluated at `p`.") +ufunc__cosine_invcdf_loops[0] = loop_d_d__As_f_f +ufunc__cosine_invcdf_loops[1] = loop_d_d__As_d_d +ufunc__cosine_invcdf_types[0] = NPY_FLOAT +ufunc__cosine_invcdf_types[1] = NPY_FLOAT +ufunc__cosine_invcdf_types[2] = NPY_DOUBLE +ufunc__cosine_invcdf_types[3] = NPY_DOUBLE +ufunc__cosine_invcdf_ptr[2*0] = _func_cosine_invcdf +ufunc__cosine_invcdf_ptr[2*0+1] = ("_cosine_invcdf") +ufunc__cosine_invcdf_ptr[2*1] = _func_cosine_invcdf +ufunc__cosine_invcdf_ptr[2*1+1] = ("_cosine_invcdf") +ufunc__cosine_invcdf_data[0] = &ufunc__cosine_invcdf_ptr[2*0] +ufunc__cosine_invcdf_data[1] = &ufunc__cosine_invcdf_ptr[2*1] +_cosine_invcdf = np.PyUFunc_FromFuncAndData(ufunc__cosine_invcdf_loops, ufunc__cosine_invcdf_data, ufunc__cosine_invcdf_types, 2, 1, 1, 0, "_cosine_invcdf", ufunc__cosine_invcdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ellip_harm_loops[3] +cdef void *ufunc__ellip_harm_ptr[6] +cdef void *ufunc__ellip_harm_data[3] +cdef char ufunc__ellip_harm_types[24] +cdef char *ufunc__ellip_harm_doc = ( + "Internal function, use `ellip_harm` instead.") +ufunc__ellip_harm_loops[0] = loop_d_ddddddd__As_fffffff_f +ufunc__ellip_harm_loops[1] = loop_d_ddiiddd__As_ddllddd_d +ufunc__ellip_harm_loops[2] = loop_d_ddddddd__As_ddddddd_d +ufunc__ellip_harm_types[0] = NPY_FLOAT +ufunc__ellip_harm_types[1] = NPY_FLOAT +ufunc__ellip_harm_types[2] = NPY_FLOAT +ufunc__ellip_harm_types[3] = NPY_FLOAT +ufunc__ellip_harm_types[4] = NPY_FLOAT +ufunc__ellip_harm_types[5] = NPY_FLOAT +ufunc__ellip_harm_types[6] = NPY_FLOAT +ufunc__ellip_harm_types[7] = NPY_FLOAT +ufunc__ellip_harm_types[8] = NPY_DOUBLE +ufunc__ellip_harm_types[9] = NPY_DOUBLE +ufunc__ellip_harm_types[10] = NPY_LONG +ufunc__ellip_harm_types[11] = NPY_LONG +ufunc__ellip_harm_types[12] = NPY_DOUBLE +ufunc__ellip_harm_types[13] = NPY_DOUBLE +ufunc__ellip_harm_types[14] = NPY_DOUBLE +ufunc__ellip_harm_types[15] = NPY_DOUBLE +ufunc__ellip_harm_types[16] = NPY_DOUBLE +ufunc__ellip_harm_types[17] = NPY_DOUBLE +ufunc__ellip_harm_types[18] = NPY_DOUBLE +ufunc__ellip_harm_types[19] = NPY_DOUBLE +ufunc__ellip_harm_types[20] = NPY_DOUBLE +ufunc__ellip_harm_types[21] = NPY_DOUBLE +ufunc__ellip_harm_types[22] = NPY_DOUBLE +ufunc__ellip_harm_types[23] = NPY_DOUBLE +ufunc__ellip_harm_ptr[2*0] = _func_ellip_harmonic_unsafe +ufunc__ellip_harm_ptr[2*0+1] = ("_ellip_harm") +ufunc__ellip_harm_ptr[2*1] = _func_ellip_harmonic +ufunc__ellip_harm_ptr[2*1+1] = ("_ellip_harm") +ufunc__ellip_harm_ptr[2*2] = _func_ellip_harmonic_unsafe +ufunc__ellip_harm_ptr[2*2+1] = ("_ellip_harm") +ufunc__ellip_harm_data[0] = &ufunc__ellip_harm_ptr[2*0] +ufunc__ellip_harm_data[1] = &ufunc__ellip_harm_ptr[2*1] +ufunc__ellip_harm_data[2] = &ufunc__ellip_harm_ptr[2*2] +_ellip_harm = np.PyUFunc_FromFuncAndData(ufunc__ellip_harm_loops, ufunc__ellip_harm_data, ufunc__ellip_harm_types, 3, 7, 1, 0, "_ellip_harm", ufunc__ellip_harm_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__factorial_loops[2] +cdef void *ufunc__factorial_ptr[4] +cdef void *ufunc__factorial_data[2] +cdef char ufunc__factorial_types[4] +cdef char *ufunc__factorial_doc = ( + "Internal function, do not use.") +ufunc__factorial_loops[0] = loop_d_d__As_f_f +ufunc__factorial_loops[1] = loop_d_d__As_d_d +ufunc__factorial_types[0] = NPY_FLOAT +ufunc__factorial_types[1] = NPY_FLOAT +ufunc__factorial_types[2] = NPY_DOUBLE +ufunc__factorial_types[3] = NPY_DOUBLE +ufunc__factorial_ptr[2*0] = _func__factorial +ufunc__factorial_ptr[2*0+1] = ("_factorial") +ufunc__factorial_ptr[2*1] = _func__factorial +ufunc__factorial_ptr[2*1+1] = ("_factorial") +ufunc__factorial_data[0] = &ufunc__factorial_ptr[2*0] +ufunc__factorial_data[1] = &ufunc__factorial_ptr[2*1] +_factorial = np.PyUFunc_FromFuncAndData(ufunc__factorial_loops, ufunc__factorial_data, ufunc__factorial_types, 2, 1, 1, 0, "_factorial", ufunc__factorial_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__hypergeom_cdf_loops[2] +cdef void *ufunc__hypergeom_cdf_ptr[4] +cdef void *ufunc__hypergeom_cdf_data[2] +cdef char ufunc__hypergeom_cdf_types[10] +cdef char *ufunc__hypergeom_cdf_doc = ( + "_hypergeom_cdf(x, r, N, M)\n" + "\n" + "Cumulative density function of hypergeometric distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "r, N, M : array_like\n" + " Positive, integer-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__hypergeom_cdf_loops[0] = loop_f_ffff__As_ffff_f +ufunc__hypergeom_cdf_loops[1] = loop_d_dddd__As_dddd_d +ufunc__hypergeom_cdf_types[0] = NPY_FLOAT +ufunc__hypergeom_cdf_types[1] = NPY_FLOAT +ufunc__hypergeom_cdf_types[2] = NPY_FLOAT +ufunc__hypergeom_cdf_types[3] = NPY_FLOAT +ufunc__hypergeom_cdf_types[4] = NPY_FLOAT +ufunc__hypergeom_cdf_types[5] = NPY_DOUBLE +ufunc__hypergeom_cdf_types[6] = NPY_DOUBLE +ufunc__hypergeom_cdf_types[7] = NPY_DOUBLE +ufunc__hypergeom_cdf_types[8] = NPY_DOUBLE +ufunc__hypergeom_cdf_types[9] = NPY_DOUBLE +ufunc__hypergeom_cdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_hypergeom_cdf_float +ufunc__hypergeom_cdf_ptr[2*0+1] = ("_hypergeom_cdf") +ufunc__hypergeom_cdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_hypergeom_cdf_double +ufunc__hypergeom_cdf_ptr[2*1+1] = ("_hypergeom_cdf") +ufunc__hypergeom_cdf_data[0] = &ufunc__hypergeom_cdf_ptr[2*0] +ufunc__hypergeom_cdf_data[1] = &ufunc__hypergeom_cdf_ptr[2*1] +_hypergeom_cdf = np.PyUFunc_FromFuncAndData(ufunc__hypergeom_cdf_loops, ufunc__hypergeom_cdf_data, ufunc__hypergeom_cdf_types, 2, 4, 1, 0, "_hypergeom_cdf", ufunc__hypergeom_cdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__hypergeom_mean_loops[2] +cdef void *ufunc__hypergeom_mean_ptr[4] +cdef void *ufunc__hypergeom_mean_data[2] +cdef char ufunc__hypergeom_mean_types[8] +cdef char *ufunc__hypergeom_mean_doc = ( + "_hypergeom_mean(r, N, M)\n" + "\n" + "Mean of hypergeometric distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "r, N, M : array_like\n" + " Positive, integer-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__hypergeom_mean_loops[0] = loop_f_fff__As_fff_f +ufunc__hypergeom_mean_loops[1] = loop_d_ddd__As_ddd_d +ufunc__hypergeom_mean_types[0] = NPY_FLOAT +ufunc__hypergeom_mean_types[1] = NPY_FLOAT +ufunc__hypergeom_mean_types[2] = NPY_FLOAT +ufunc__hypergeom_mean_types[3] = NPY_FLOAT +ufunc__hypergeom_mean_types[4] = NPY_DOUBLE +ufunc__hypergeom_mean_types[5] = NPY_DOUBLE +ufunc__hypergeom_mean_types[6] = NPY_DOUBLE +ufunc__hypergeom_mean_types[7] = NPY_DOUBLE +ufunc__hypergeom_mean_ptr[2*0] = scipy.special._ufuncs_cxx._export_hypergeom_mean_float +ufunc__hypergeom_mean_ptr[2*0+1] = ("_hypergeom_mean") +ufunc__hypergeom_mean_ptr[2*1] = scipy.special._ufuncs_cxx._export_hypergeom_mean_double +ufunc__hypergeom_mean_ptr[2*1+1] = ("_hypergeom_mean") +ufunc__hypergeom_mean_data[0] = &ufunc__hypergeom_mean_ptr[2*0] +ufunc__hypergeom_mean_data[1] = &ufunc__hypergeom_mean_ptr[2*1] +_hypergeom_mean = np.PyUFunc_FromFuncAndData(ufunc__hypergeom_mean_loops, ufunc__hypergeom_mean_data, ufunc__hypergeom_mean_types, 2, 3, 1, 0, "_hypergeom_mean", ufunc__hypergeom_mean_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__hypergeom_pmf_loops[2] +cdef void *ufunc__hypergeom_pmf_ptr[4] +cdef void *ufunc__hypergeom_pmf_data[2] +cdef char ufunc__hypergeom_pmf_types[10] +cdef char *ufunc__hypergeom_pmf_doc = ( + "_hypergeom_pmf(x, r, N, M)\n" + "\n" + "Probability mass function of hypergeometric distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "r, N, M : array_like\n" + " Positive, integer-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__hypergeom_pmf_loops[0] = loop_f_ffff__As_ffff_f +ufunc__hypergeom_pmf_loops[1] = loop_d_dddd__As_dddd_d +ufunc__hypergeom_pmf_types[0] = NPY_FLOAT +ufunc__hypergeom_pmf_types[1] = NPY_FLOAT +ufunc__hypergeom_pmf_types[2] = NPY_FLOAT +ufunc__hypergeom_pmf_types[3] = NPY_FLOAT +ufunc__hypergeom_pmf_types[4] = NPY_FLOAT +ufunc__hypergeom_pmf_types[5] = NPY_DOUBLE +ufunc__hypergeom_pmf_types[6] = NPY_DOUBLE +ufunc__hypergeom_pmf_types[7] = NPY_DOUBLE +ufunc__hypergeom_pmf_types[8] = NPY_DOUBLE +ufunc__hypergeom_pmf_types[9] = NPY_DOUBLE +ufunc__hypergeom_pmf_ptr[2*0] = scipy.special._ufuncs_cxx._export_hypergeom_pmf_float +ufunc__hypergeom_pmf_ptr[2*0+1] = ("_hypergeom_pmf") +ufunc__hypergeom_pmf_ptr[2*1] = scipy.special._ufuncs_cxx._export_hypergeom_pmf_double +ufunc__hypergeom_pmf_ptr[2*1+1] = ("_hypergeom_pmf") +ufunc__hypergeom_pmf_data[0] = &ufunc__hypergeom_pmf_ptr[2*0] +ufunc__hypergeom_pmf_data[1] = &ufunc__hypergeom_pmf_ptr[2*1] +_hypergeom_pmf = np.PyUFunc_FromFuncAndData(ufunc__hypergeom_pmf_loops, ufunc__hypergeom_pmf_data, ufunc__hypergeom_pmf_types, 2, 4, 1, 0, "_hypergeom_pmf", ufunc__hypergeom_pmf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__hypergeom_sf_loops[2] +cdef void *ufunc__hypergeom_sf_ptr[4] +cdef void *ufunc__hypergeom_sf_data[2] +cdef char ufunc__hypergeom_sf_types[10] +cdef char *ufunc__hypergeom_sf_doc = ( + "_hypergeom_sf(x, r, N, M)\n" + "\n" + "Survival function of hypergeometric distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "r, N, M : array_like\n" + " Positive, integer-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__hypergeom_sf_loops[0] = loop_f_ffff__As_ffff_f +ufunc__hypergeom_sf_loops[1] = loop_d_dddd__As_dddd_d +ufunc__hypergeom_sf_types[0] = NPY_FLOAT +ufunc__hypergeom_sf_types[1] = NPY_FLOAT +ufunc__hypergeom_sf_types[2] = NPY_FLOAT +ufunc__hypergeom_sf_types[3] = NPY_FLOAT +ufunc__hypergeom_sf_types[4] = NPY_FLOAT +ufunc__hypergeom_sf_types[5] = NPY_DOUBLE +ufunc__hypergeom_sf_types[6] = NPY_DOUBLE +ufunc__hypergeom_sf_types[7] = NPY_DOUBLE +ufunc__hypergeom_sf_types[8] = NPY_DOUBLE +ufunc__hypergeom_sf_types[9] = NPY_DOUBLE +ufunc__hypergeom_sf_ptr[2*0] = scipy.special._ufuncs_cxx._export_hypergeom_sf_float +ufunc__hypergeom_sf_ptr[2*0+1] = ("_hypergeom_sf") +ufunc__hypergeom_sf_ptr[2*1] = scipy.special._ufuncs_cxx._export_hypergeom_sf_double +ufunc__hypergeom_sf_ptr[2*1+1] = ("_hypergeom_sf") +ufunc__hypergeom_sf_data[0] = &ufunc__hypergeom_sf_ptr[2*0] +ufunc__hypergeom_sf_data[1] = &ufunc__hypergeom_sf_ptr[2*1] +_hypergeom_sf = np.PyUFunc_FromFuncAndData(ufunc__hypergeom_sf_loops, ufunc__hypergeom_sf_data, ufunc__hypergeom_sf_types, 2, 4, 1, 0, "_hypergeom_sf", ufunc__hypergeom_sf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__hypergeom_skewness_loops[2] +cdef void *ufunc__hypergeom_skewness_ptr[4] +cdef void *ufunc__hypergeom_skewness_data[2] +cdef char ufunc__hypergeom_skewness_types[8] +cdef char *ufunc__hypergeom_skewness_doc = ( + "_hypergeom_skewness(r, N, M)\n" + "\n" + "Skewness of hypergeometric distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "r, N, M : array_like\n" + " Positive, integer-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__hypergeom_skewness_loops[0] = loop_f_fff__As_fff_f +ufunc__hypergeom_skewness_loops[1] = loop_d_ddd__As_ddd_d +ufunc__hypergeom_skewness_types[0] = NPY_FLOAT +ufunc__hypergeom_skewness_types[1] = NPY_FLOAT +ufunc__hypergeom_skewness_types[2] = NPY_FLOAT +ufunc__hypergeom_skewness_types[3] = NPY_FLOAT +ufunc__hypergeom_skewness_types[4] = NPY_DOUBLE +ufunc__hypergeom_skewness_types[5] = NPY_DOUBLE +ufunc__hypergeom_skewness_types[6] = NPY_DOUBLE +ufunc__hypergeom_skewness_types[7] = NPY_DOUBLE +ufunc__hypergeom_skewness_ptr[2*0] = scipy.special._ufuncs_cxx._export_hypergeom_skewness_float +ufunc__hypergeom_skewness_ptr[2*0+1] = ("_hypergeom_skewness") +ufunc__hypergeom_skewness_ptr[2*1] = scipy.special._ufuncs_cxx._export_hypergeom_skewness_double +ufunc__hypergeom_skewness_ptr[2*1+1] = ("_hypergeom_skewness") +ufunc__hypergeom_skewness_data[0] = &ufunc__hypergeom_skewness_ptr[2*0] +ufunc__hypergeom_skewness_data[1] = &ufunc__hypergeom_skewness_ptr[2*1] +_hypergeom_skewness = np.PyUFunc_FromFuncAndData(ufunc__hypergeom_skewness_loops, ufunc__hypergeom_skewness_data, ufunc__hypergeom_skewness_types, 2, 3, 1, 0, "_hypergeom_skewness", ufunc__hypergeom_skewness_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__hypergeom_variance_loops[2] +cdef void *ufunc__hypergeom_variance_ptr[4] +cdef void *ufunc__hypergeom_variance_data[2] +cdef char ufunc__hypergeom_variance_types[8] +cdef char *ufunc__hypergeom_variance_doc = ( + "_hypergeom_variance(r, N, M)\n" + "\n" + "Mean of hypergeometric distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "r, N, M : array_like\n" + " Positive, integer-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__hypergeom_variance_loops[0] = loop_f_fff__As_fff_f +ufunc__hypergeom_variance_loops[1] = loop_d_ddd__As_ddd_d +ufunc__hypergeom_variance_types[0] = NPY_FLOAT +ufunc__hypergeom_variance_types[1] = NPY_FLOAT +ufunc__hypergeom_variance_types[2] = NPY_FLOAT +ufunc__hypergeom_variance_types[3] = NPY_FLOAT +ufunc__hypergeom_variance_types[4] = NPY_DOUBLE +ufunc__hypergeom_variance_types[5] = NPY_DOUBLE +ufunc__hypergeom_variance_types[6] = NPY_DOUBLE +ufunc__hypergeom_variance_types[7] = NPY_DOUBLE +ufunc__hypergeom_variance_ptr[2*0] = scipy.special._ufuncs_cxx._export_hypergeom_variance_float +ufunc__hypergeom_variance_ptr[2*0+1] = ("_hypergeom_variance") +ufunc__hypergeom_variance_ptr[2*1] = scipy.special._ufuncs_cxx._export_hypergeom_variance_double +ufunc__hypergeom_variance_ptr[2*1+1] = ("_hypergeom_variance") +ufunc__hypergeom_variance_data[0] = &ufunc__hypergeom_variance_ptr[2*0] +ufunc__hypergeom_variance_data[1] = &ufunc__hypergeom_variance_ptr[2*1] +_hypergeom_variance = np.PyUFunc_FromFuncAndData(ufunc__hypergeom_variance_loops, ufunc__hypergeom_variance_data, ufunc__hypergeom_variance_types, 2, 3, 1, 0, "_hypergeom_variance", ufunc__hypergeom_variance_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__igam_fac_loops[2] +cdef void *ufunc__igam_fac_ptr[4] +cdef void *ufunc__igam_fac_data[2] +cdef char ufunc__igam_fac_types[6] +cdef char *ufunc__igam_fac_doc = ( + "Internal function, do not use.") +ufunc__igam_fac_loops[0] = loop_d_dd__As_ff_f +ufunc__igam_fac_loops[1] = loop_d_dd__As_dd_d +ufunc__igam_fac_types[0] = NPY_FLOAT +ufunc__igam_fac_types[1] = NPY_FLOAT +ufunc__igam_fac_types[2] = NPY_FLOAT +ufunc__igam_fac_types[3] = NPY_DOUBLE +ufunc__igam_fac_types[4] = NPY_DOUBLE +ufunc__igam_fac_types[5] = NPY_DOUBLE +ufunc__igam_fac_ptr[2*0] = _func_cephes_igam_fac +ufunc__igam_fac_ptr[2*0+1] = ("_igam_fac") +ufunc__igam_fac_ptr[2*1] = _func_cephes_igam_fac +ufunc__igam_fac_ptr[2*1+1] = ("_igam_fac") +ufunc__igam_fac_data[0] = &ufunc__igam_fac_ptr[2*0] +ufunc__igam_fac_data[1] = &ufunc__igam_fac_ptr[2*1] +_igam_fac = np.PyUFunc_FromFuncAndData(ufunc__igam_fac_loops, ufunc__igam_fac_data, ufunc__igam_fac_types, 2, 2, 1, 0, "_igam_fac", ufunc__igam_fac_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__invgauss_isf_loops[2] +cdef void *ufunc__invgauss_isf_ptr[4] +cdef void *ufunc__invgauss_isf_data[2] +cdef char ufunc__invgauss_isf_types[8] +cdef char *ufunc__invgauss_isf_doc = ( + "_invgauss_isf(x, mu, s)\n" + "\n" + "Inverse survival function of inverse gaussian distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Positive real-valued\n" + "mu : array_like\n" + " Positive, real-valued parameters\n" + "s : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__invgauss_isf_loops[0] = loop_f_fff__As_fff_f +ufunc__invgauss_isf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__invgauss_isf_types[0] = NPY_FLOAT +ufunc__invgauss_isf_types[1] = NPY_FLOAT +ufunc__invgauss_isf_types[2] = NPY_FLOAT +ufunc__invgauss_isf_types[3] = NPY_FLOAT +ufunc__invgauss_isf_types[4] = NPY_DOUBLE +ufunc__invgauss_isf_types[5] = NPY_DOUBLE +ufunc__invgauss_isf_types[6] = NPY_DOUBLE +ufunc__invgauss_isf_types[7] = NPY_DOUBLE +ufunc__invgauss_isf_ptr[2*0] = scipy.special._ufuncs_cxx._export_invgauss_isf_float +ufunc__invgauss_isf_ptr[2*0+1] = ("_invgauss_isf") +ufunc__invgauss_isf_ptr[2*1] = scipy.special._ufuncs_cxx._export_invgauss_isf_double +ufunc__invgauss_isf_ptr[2*1+1] = ("_invgauss_isf") +ufunc__invgauss_isf_data[0] = &ufunc__invgauss_isf_ptr[2*0] +ufunc__invgauss_isf_data[1] = &ufunc__invgauss_isf_ptr[2*1] +_invgauss_isf = np.PyUFunc_FromFuncAndData(ufunc__invgauss_isf_loops, ufunc__invgauss_isf_data, ufunc__invgauss_isf_types, 2, 3, 1, 0, "_invgauss_isf", ufunc__invgauss_isf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__invgauss_ppf_loops[2] +cdef void *ufunc__invgauss_ppf_ptr[4] +cdef void *ufunc__invgauss_ppf_data[2] +cdef char ufunc__invgauss_ppf_types[8] +cdef char *ufunc__invgauss_ppf_doc = ( + "_invgauss_ppf(x, mu)\n" + "\n" + "Percent point function of inverse gaussian distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Positive real-valued\n" + "mu : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__invgauss_ppf_loops[0] = loop_f_fff__As_fff_f +ufunc__invgauss_ppf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__invgauss_ppf_types[0] = NPY_FLOAT +ufunc__invgauss_ppf_types[1] = NPY_FLOAT +ufunc__invgauss_ppf_types[2] = NPY_FLOAT +ufunc__invgauss_ppf_types[3] = NPY_FLOAT +ufunc__invgauss_ppf_types[4] = NPY_DOUBLE +ufunc__invgauss_ppf_types[5] = NPY_DOUBLE +ufunc__invgauss_ppf_types[6] = NPY_DOUBLE +ufunc__invgauss_ppf_types[7] = NPY_DOUBLE +ufunc__invgauss_ppf_ptr[2*0] = scipy.special._ufuncs_cxx._export_invgauss_ppf_float +ufunc__invgauss_ppf_ptr[2*0+1] = ("_invgauss_ppf") +ufunc__invgauss_ppf_ptr[2*1] = scipy.special._ufuncs_cxx._export_invgauss_ppf_double +ufunc__invgauss_ppf_ptr[2*1+1] = ("_invgauss_ppf") +ufunc__invgauss_ppf_data[0] = &ufunc__invgauss_ppf_ptr[2*0] +ufunc__invgauss_ppf_data[1] = &ufunc__invgauss_ppf_ptr[2*1] +_invgauss_ppf = np.PyUFunc_FromFuncAndData(ufunc__invgauss_ppf_loops, ufunc__invgauss_ppf_data, ufunc__invgauss_ppf_types, 2, 3, 1, 0, "_invgauss_ppf", ufunc__invgauss_ppf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__kolmogc_loops[2] +cdef void *ufunc__kolmogc_ptr[4] +cdef void *ufunc__kolmogc_data[2] +cdef char ufunc__kolmogc_types[4] +cdef char *ufunc__kolmogc_doc = ( + "Internal function, do not use.") +ufunc__kolmogc_loops[0] = loop_d_d__As_f_f +ufunc__kolmogc_loops[1] = loop_d_d__As_d_d +ufunc__kolmogc_types[0] = NPY_FLOAT +ufunc__kolmogc_types[1] = NPY_FLOAT +ufunc__kolmogc_types[2] = NPY_DOUBLE +ufunc__kolmogc_types[3] = NPY_DOUBLE +ufunc__kolmogc_ptr[2*0] = _func_xsf_kolmogc +ufunc__kolmogc_ptr[2*0+1] = ("_kolmogc") +ufunc__kolmogc_ptr[2*1] = _func_xsf_kolmogc +ufunc__kolmogc_ptr[2*1+1] = ("_kolmogc") +ufunc__kolmogc_data[0] = &ufunc__kolmogc_ptr[2*0] +ufunc__kolmogc_data[1] = &ufunc__kolmogc_ptr[2*1] +_kolmogc = np.PyUFunc_FromFuncAndData(ufunc__kolmogc_loops, ufunc__kolmogc_data, ufunc__kolmogc_types, 2, 1, 1, 0, "_kolmogc", ufunc__kolmogc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__kolmogci_loops[2] +cdef void *ufunc__kolmogci_ptr[4] +cdef void *ufunc__kolmogci_data[2] +cdef char ufunc__kolmogci_types[4] +cdef char *ufunc__kolmogci_doc = ( + "Internal function, do not use.") +ufunc__kolmogci_loops[0] = loop_d_d__As_f_f +ufunc__kolmogci_loops[1] = loop_d_d__As_d_d +ufunc__kolmogci_types[0] = NPY_FLOAT +ufunc__kolmogci_types[1] = NPY_FLOAT +ufunc__kolmogci_types[2] = NPY_DOUBLE +ufunc__kolmogci_types[3] = NPY_DOUBLE +ufunc__kolmogci_ptr[2*0] = _func_xsf_kolmogci +ufunc__kolmogci_ptr[2*0+1] = ("_kolmogci") +ufunc__kolmogci_ptr[2*1] = _func_xsf_kolmogci +ufunc__kolmogci_ptr[2*1+1] = ("_kolmogci") +ufunc__kolmogci_data[0] = &ufunc__kolmogci_ptr[2*0] +ufunc__kolmogci_data[1] = &ufunc__kolmogci_ptr[2*1] +_kolmogci = np.PyUFunc_FromFuncAndData(ufunc__kolmogci_loops, ufunc__kolmogci_data, ufunc__kolmogci_types, 2, 1, 1, 0, "_kolmogci", ufunc__kolmogci_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__kolmogp_loops[2] +cdef void *ufunc__kolmogp_ptr[4] +cdef void *ufunc__kolmogp_data[2] +cdef char ufunc__kolmogp_types[4] +cdef char *ufunc__kolmogp_doc = ( + "Internal function, do not use.") +ufunc__kolmogp_loops[0] = loop_d_d__As_f_f +ufunc__kolmogp_loops[1] = loop_d_d__As_d_d +ufunc__kolmogp_types[0] = NPY_FLOAT +ufunc__kolmogp_types[1] = NPY_FLOAT +ufunc__kolmogp_types[2] = NPY_DOUBLE +ufunc__kolmogp_types[3] = NPY_DOUBLE +ufunc__kolmogp_ptr[2*0] = _func_xsf_kolmogp +ufunc__kolmogp_ptr[2*0+1] = ("_kolmogp") +ufunc__kolmogp_ptr[2*1] = _func_xsf_kolmogp +ufunc__kolmogp_ptr[2*1+1] = ("_kolmogp") +ufunc__kolmogp_data[0] = &ufunc__kolmogp_ptr[2*0] +ufunc__kolmogp_data[1] = &ufunc__kolmogp_ptr[2*1] +_kolmogp = np.PyUFunc_FromFuncAndData(ufunc__kolmogp_loops, ufunc__kolmogp_data, ufunc__kolmogp_types, 2, 1, 1, 0, "_kolmogp", ufunc__kolmogp_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__lanczos_sum_expg_scaled_loops[2] +cdef void *ufunc__lanczos_sum_expg_scaled_ptr[4] +cdef void *ufunc__lanczos_sum_expg_scaled_data[2] +cdef char ufunc__lanczos_sum_expg_scaled_types[4] +cdef char *ufunc__lanczos_sum_expg_scaled_doc = ( + "Internal function, do not use.") +ufunc__lanczos_sum_expg_scaled_loops[0] = loop_d_d__As_f_f +ufunc__lanczos_sum_expg_scaled_loops[1] = loop_d_d__As_d_d +ufunc__lanczos_sum_expg_scaled_types[0] = NPY_FLOAT +ufunc__lanczos_sum_expg_scaled_types[1] = NPY_FLOAT +ufunc__lanczos_sum_expg_scaled_types[2] = NPY_DOUBLE +ufunc__lanczos_sum_expg_scaled_types[3] = NPY_DOUBLE +ufunc__lanczos_sum_expg_scaled_ptr[2*0] = _func_cephes_lanczos_sum_expg_scaled +ufunc__lanczos_sum_expg_scaled_ptr[2*0+1] = ("_lanczos_sum_expg_scaled") +ufunc__lanczos_sum_expg_scaled_ptr[2*1] = _func_cephes_lanczos_sum_expg_scaled +ufunc__lanczos_sum_expg_scaled_ptr[2*1+1] = ("_lanczos_sum_expg_scaled") +ufunc__lanczos_sum_expg_scaled_data[0] = &ufunc__lanczos_sum_expg_scaled_ptr[2*0] +ufunc__lanczos_sum_expg_scaled_data[1] = &ufunc__lanczos_sum_expg_scaled_ptr[2*1] +_lanczos_sum_expg_scaled = np.PyUFunc_FromFuncAndData(ufunc__lanczos_sum_expg_scaled_loops, ufunc__lanczos_sum_expg_scaled_data, ufunc__lanczos_sum_expg_scaled_types, 2, 1, 1, 0, "_lanczos_sum_expg_scaled", ufunc__lanczos_sum_expg_scaled_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__landau_cdf_loops[2] +cdef void *ufunc__landau_cdf_ptr[4] +cdef void *ufunc__landau_cdf_data[2] +cdef char ufunc__landau_cdf_types[8] +cdef char *ufunc__landau_cdf_doc = ( + "_landau_cdf(x, loc, scale)\n" + "\n" + "Cumulative distribution function of the Landau distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued argument\n" + "loc : array_like\n" + " Real-valued distribution location\n" + "scale : array_like\n" + " Positive, real-valued distribution scale\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__landau_cdf_loops[0] = loop_f_fff__As_fff_f +ufunc__landau_cdf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__landau_cdf_types[0] = NPY_FLOAT +ufunc__landau_cdf_types[1] = NPY_FLOAT +ufunc__landau_cdf_types[2] = NPY_FLOAT +ufunc__landau_cdf_types[3] = NPY_FLOAT +ufunc__landau_cdf_types[4] = NPY_DOUBLE +ufunc__landau_cdf_types[5] = NPY_DOUBLE +ufunc__landau_cdf_types[6] = NPY_DOUBLE +ufunc__landau_cdf_types[7] = NPY_DOUBLE +ufunc__landau_cdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_landau_cdf_float +ufunc__landau_cdf_ptr[2*0+1] = ("_landau_cdf") +ufunc__landau_cdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_landau_cdf_double +ufunc__landau_cdf_ptr[2*1+1] = ("_landau_cdf") +ufunc__landau_cdf_data[0] = &ufunc__landau_cdf_ptr[2*0] +ufunc__landau_cdf_data[1] = &ufunc__landau_cdf_ptr[2*1] +_landau_cdf = np.PyUFunc_FromFuncAndData(ufunc__landau_cdf_loops, ufunc__landau_cdf_data, ufunc__landau_cdf_types, 2, 3, 1, 0, "_landau_cdf", ufunc__landau_cdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__landau_isf_loops[2] +cdef void *ufunc__landau_isf_ptr[4] +cdef void *ufunc__landau_isf_data[2] +cdef char ufunc__landau_isf_types[8] +cdef char *ufunc__landau_isf_doc = ( + "_landau_isf(p, loc, scale)\n" + "\n" + "Inverse survival function of the Landau distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Real-valued argument between 0 and 1\n" + "loc : array_like\n" + " Real-valued distribution location\n" + "scale : array_like\n" + " Positive, real-valued distribution scale\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__landau_isf_loops[0] = loop_f_fff__As_fff_f +ufunc__landau_isf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__landau_isf_types[0] = NPY_FLOAT +ufunc__landau_isf_types[1] = NPY_FLOAT +ufunc__landau_isf_types[2] = NPY_FLOAT +ufunc__landau_isf_types[3] = NPY_FLOAT +ufunc__landau_isf_types[4] = NPY_DOUBLE +ufunc__landau_isf_types[5] = NPY_DOUBLE +ufunc__landau_isf_types[6] = NPY_DOUBLE +ufunc__landau_isf_types[7] = NPY_DOUBLE +ufunc__landau_isf_ptr[2*0] = scipy.special._ufuncs_cxx._export_landau_isf_float +ufunc__landau_isf_ptr[2*0+1] = ("_landau_isf") +ufunc__landau_isf_ptr[2*1] = scipy.special._ufuncs_cxx._export_landau_isf_double +ufunc__landau_isf_ptr[2*1+1] = ("_landau_isf") +ufunc__landau_isf_data[0] = &ufunc__landau_isf_ptr[2*0] +ufunc__landau_isf_data[1] = &ufunc__landau_isf_ptr[2*1] +_landau_isf = np.PyUFunc_FromFuncAndData(ufunc__landau_isf_loops, ufunc__landau_isf_data, ufunc__landau_isf_types, 2, 3, 1, 0, "_landau_isf", ufunc__landau_isf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__landau_pdf_loops[2] +cdef void *ufunc__landau_pdf_ptr[4] +cdef void *ufunc__landau_pdf_data[2] +cdef char ufunc__landau_pdf_types[8] +cdef char *ufunc__landau_pdf_doc = ( + "_landau_pdf(x, loc, scale)\n" + "\n" + "Probability density function of the Landau distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued argument\n" + "loc : array_like\n" + " Real-valued distribution location\n" + "scale : array_like\n" + " Positive, real-valued distribution scale\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__landau_pdf_loops[0] = loop_f_fff__As_fff_f +ufunc__landau_pdf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__landau_pdf_types[0] = NPY_FLOAT +ufunc__landau_pdf_types[1] = NPY_FLOAT +ufunc__landau_pdf_types[2] = NPY_FLOAT +ufunc__landau_pdf_types[3] = NPY_FLOAT +ufunc__landau_pdf_types[4] = NPY_DOUBLE +ufunc__landau_pdf_types[5] = NPY_DOUBLE +ufunc__landau_pdf_types[6] = NPY_DOUBLE +ufunc__landau_pdf_types[7] = NPY_DOUBLE +ufunc__landau_pdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_landau_pdf_float +ufunc__landau_pdf_ptr[2*0+1] = ("_landau_pdf") +ufunc__landau_pdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_landau_pdf_double +ufunc__landau_pdf_ptr[2*1+1] = ("_landau_pdf") +ufunc__landau_pdf_data[0] = &ufunc__landau_pdf_ptr[2*0] +ufunc__landau_pdf_data[1] = &ufunc__landau_pdf_ptr[2*1] +_landau_pdf = np.PyUFunc_FromFuncAndData(ufunc__landau_pdf_loops, ufunc__landau_pdf_data, ufunc__landau_pdf_types, 2, 3, 1, 0, "_landau_pdf", ufunc__landau_pdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__landau_ppf_loops[2] +cdef void *ufunc__landau_ppf_ptr[4] +cdef void *ufunc__landau_ppf_data[2] +cdef char ufunc__landau_ppf_types[8] +cdef char *ufunc__landau_ppf_doc = ( + "_landau_ppf(p, loc, scale)\n" + "\n" + "Percent point function of the Landau distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Real-valued argument between 0 and 1\n" + "loc : array_like\n" + " Real-valued distribution location\n" + "scale : array_like\n" + " Positive, real-valued distribution scale\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__landau_ppf_loops[0] = loop_f_fff__As_fff_f +ufunc__landau_ppf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__landau_ppf_types[0] = NPY_FLOAT +ufunc__landau_ppf_types[1] = NPY_FLOAT +ufunc__landau_ppf_types[2] = NPY_FLOAT +ufunc__landau_ppf_types[3] = NPY_FLOAT +ufunc__landau_ppf_types[4] = NPY_DOUBLE +ufunc__landau_ppf_types[5] = NPY_DOUBLE +ufunc__landau_ppf_types[6] = NPY_DOUBLE +ufunc__landau_ppf_types[7] = NPY_DOUBLE +ufunc__landau_ppf_ptr[2*0] = scipy.special._ufuncs_cxx._export_landau_ppf_float +ufunc__landau_ppf_ptr[2*0+1] = ("_landau_ppf") +ufunc__landau_ppf_ptr[2*1] = scipy.special._ufuncs_cxx._export_landau_ppf_double +ufunc__landau_ppf_ptr[2*1+1] = ("_landau_ppf") +ufunc__landau_ppf_data[0] = &ufunc__landau_ppf_ptr[2*0] +ufunc__landau_ppf_data[1] = &ufunc__landau_ppf_ptr[2*1] +_landau_ppf = np.PyUFunc_FromFuncAndData(ufunc__landau_ppf_loops, ufunc__landau_ppf_data, ufunc__landau_ppf_types, 2, 3, 1, 0, "_landau_ppf", ufunc__landau_ppf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__landau_sf_loops[2] +cdef void *ufunc__landau_sf_ptr[4] +cdef void *ufunc__landau_sf_data[2] +cdef char ufunc__landau_sf_types[8] +cdef char *ufunc__landau_sf_doc = ( + "_landau_sf(x, loc, scale)\n" + "\n" + "Survival function of the Landau distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued argument\n" + "loc : array_like\n" + " Real-valued distribution location\n" + "scale : array_like\n" + " Positive, real-valued distribution scale\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__landau_sf_loops[0] = loop_f_fff__As_fff_f +ufunc__landau_sf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__landau_sf_types[0] = NPY_FLOAT +ufunc__landau_sf_types[1] = NPY_FLOAT +ufunc__landau_sf_types[2] = NPY_FLOAT +ufunc__landau_sf_types[3] = NPY_FLOAT +ufunc__landau_sf_types[4] = NPY_DOUBLE +ufunc__landau_sf_types[5] = NPY_DOUBLE +ufunc__landau_sf_types[6] = NPY_DOUBLE +ufunc__landau_sf_types[7] = NPY_DOUBLE +ufunc__landau_sf_ptr[2*0] = scipy.special._ufuncs_cxx._export_landau_sf_float +ufunc__landau_sf_ptr[2*0+1] = ("_landau_sf") +ufunc__landau_sf_ptr[2*1] = scipy.special._ufuncs_cxx._export_landau_sf_double +ufunc__landau_sf_ptr[2*1+1] = ("_landau_sf") +ufunc__landau_sf_data[0] = &ufunc__landau_sf_ptr[2*0] +ufunc__landau_sf_data[1] = &ufunc__landau_sf_ptr[2*1] +_landau_sf = np.PyUFunc_FromFuncAndData(ufunc__landau_sf_loops, ufunc__landau_sf_data, ufunc__landau_sf_types, 2, 3, 1, 0, "_landau_sf", ufunc__landau_sf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__lgam1p_loops[2] +cdef void *ufunc__lgam1p_ptr[4] +cdef void *ufunc__lgam1p_data[2] +cdef char ufunc__lgam1p_types[4] +cdef char *ufunc__lgam1p_doc = ( + "Internal function, do not use.") +ufunc__lgam1p_loops[0] = loop_d_d__As_f_f +ufunc__lgam1p_loops[1] = loop_d_d__As_d_d +ufunc__lgam1p_types[0] = NPY_FLOAT +ufunc__lgam1p_types[1] = NPY_FLOAT +ufunc__lgam1p_types[2] = NPY_DOUBLE +ufunc__lgam1p_types[3] = NPY_DOUBLE +ufunc__lgam1p_ptr[2*0] = _func_cephes_lgam1p +ufunc__lgam1p_ptr[2*0+1] = ("_lgam1p") +ufunc__lgam1p_ptr[2*1] = _func_cephes_lgam1p +ufunc__lgam1p_ptr[2*1+1] = ("_lgam1p") +ufunc__lgam1p_data[0] = &ufunc__lgam1p_ptr[2*0] +ufunc__lgam1p_data[1] = &ufunc__lgam1p_ptr[2*1] +_lgam1p = np.PyUFunc_FromFuncAndData(ufunc__lgam1p_loops, ufunc__lgam1p_data, ufunc__lgam1p_types, 2, 1, 1, 0, "_lgam1p", ufunc__lgam1p_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__log1pmx_loops[2] +cdef void *ufunc__log1pmx_ptr[4] +cdef void *ufunc__log1pmx_data[2] +cdef char ufunc__log1pmx_types[4] +cdef char *ufunc__log1pmx_doc = ( + "Internal function, do not use.") +ufunc__log1pmx_loops[0] = loop_d_d__As_f_f +ufunc__log1pmx_loops[1] = loop_d_d__As_d_d +ufunc__log1pmx_types[0] = NPY_FLOAT +ufunc__log1pmx_types[1] = NPY_FLOAT +ufunc__log1pmx_types[2] = NPY_DOUBLE +ufunc__log1pmx_types[3] = NPY_DOUBLE +ufunc__log1pmx_ptr[2*0] = _func_cephes_log1pmx +ufunc__log1pmx_ptr[2*0+1] = ("_log1pmx") +ufunc__log1pmx_ptr[2*1] = _func_cephes_log1pmx +ufunc__log1pmx_ptr[2*1+1] = ("_log1pmx") +ufunc__log1pmx_data[0] = &ufunc__log1pmx_ptr[2*0] +ufunc__log1pmx_data[1] = &ufunc__log1pmx_ptr[2*1] +_log1pmx = np.PyUFunc_FromFuncAndData(ufunc__log1pmx_loops, ufunc__log1pmx_data, ufunc__log1pmx_types, 2, 1, 1, 0, "_log1pmx", ufunc__log1pmx_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nbinom_cdf_loops[2] +cdef void *ufunc__nbinom_cdf_ptr[4] +cdef void *ufunc__nbinom_cdf_data[2] +cdef char ufunc__nbinom_cdf_types[8] +cdef char *ufunc__nbinom_cdf_doc = ( + "_nbinom_cdf(x, r, p)\n" + "\n" + "Cumulative density function of negative binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "r : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nbinom_cdf_loops[0] = loop_f_fff__As_fff_f +ufunc__nbinom_cdf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__nbinom_cdf_types[0] = NPY_FLOAT +ufunc__nbinom_cdf_types[1] = NPY_FLOAT +ufunc__nbinom_cdf_types[2] = NPY_FLOAT +ufunc__nbinom_cdf_types[3] = NPY_FLOAT +ufunc__nbinom_cdf_types[4] = NPY_DOUBLE +ufunc__nbinom_cdf_types[5] = NPY_DOUBLE +ufunc__nbinom_cdf_types[6] = NPY_DOUBLE +ufunc__nbinom_cdf_types[7] = NPY_DOUBLE +ufunc__nbinom_cdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_nbinom_cdf_float +ufunc__nbinom_cdf_ptr[2*0+1] = ("_nbinom_cdf") +ufunc__nbinom_cdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_nbinom_cdf_double +ufunc__nbinom_cdf_ptr[2*1+1] = ("_nbinom_cdf") +ufunc__nbinom_cdf_data[0] = &ufunc__nbinom_cdf_ptr[2*0] +ufunc__nbinom_cdf_data[1] = &ufunc__nbinom_cdf_ptr[2*1] +_nbinom_cdf = np.PyUFunc_FromFuncAndData(ufunc__nbinom_cdf_loops, ufunc__nbinom_cdf_data, ufunc__nbinom_cdf_types, 2, 3, 1, 0, "_nbinom_cdf", ufunc__nbinom_cdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nbinom_isf_loops[2] +cdef void *ufunc__nbinom_isf_ptr[4] +cdef void *ufunc__nbinom_isf_data[2] +cdef char ufunc__nbinom_isf_types[8] +cdef char *ufunc__nbinom_isf_doc = ( + "_nbinom_isf(x, r, p)\n" + "\n" + "Inverse survival function of negative binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "r : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nbinom_isf_loops[0] = loop_f_fff__As_fff_f +ufunc__nbinom_isf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__nbinom_isf_types[0] = NPY_FLOAT +ufunc__nbinom_isf_types[1] = NPY_FLOAT +ufunc__nbinom_isf_types[2] = NPY_FLOAT +ufunc__nbinom_isf_types[3] = NPY_FLOAT +ufunc__nbinom_isf_types[4] = NPY_DOUBLE +ufunc__nbinom_isf_types[5] = NPY_DOUBLE +ufunc__nbinom_isf_types[6] = NPY_DOUBLE +ufunc__nbinom_isf_types[7] = NPY_DOUBLE +ufunc__nbinom_isf_ptr[2*0] = scipy.special._ufuncs_cxx._export_nbinom_isf_float +ufunc__nbinom_isf_ptr[2*0+1] = ("_nbinom_isf") +ufunc__nbinom_isf_ptr[2*1] = scipy.special._ufuncs_cxx._export_nbinom_isf_double +ufunc__nbinom_isf_ptr[2*1+1] = ("_nbinom_isf") +ufunc__nbinom_isf_data[0] = &ufunc__nbinom_isf_ptr[2*0] +ufunc__nbinom_isf_data[1] = &ufunc__nbinom_isf_ptr[2*1] +_nbinom_isf = np.PyUFunc_FromFuncAndData(ufunc__nbinom_isf_loops, ufunc__nbinom_isf_data, ufunc__nbinom_isf_types, 2, 3, 1, 0, "_nbinom_isf", ufunc__nbinom_isf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nbinom_kurtosis_excess_loops[2] +cdef void *ufunc__nbinom_kurtosis_excess_ptr[4] +cdef void *ufunc__nbinom_kurtosis_excess_data[2] +cdef char ufunc__nbinom_kurtosis_excess_types[6] +cdef char *ufunc__nbinom_kurtosis_excess_doc = ( + "_nbinom_kurtosis_excess(r, p)\n" + "\n" + "Kurtosis excess of negative binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "r : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nbinom_kurtosis_excess_loops[0] = loop_f_ff__As_ff_f +ufunc__nbinom_kurtosis_excess_loops[1] = loop_d_dd__As_dd_d +ufunc__nbinom_kurtosis_excess_types[0] = NPY_FLOAT +ufunc__nbinom_kurtosis_excess_types[1] = NPY_FLOAT +ufunc__nbinom_kurtosis_excess_types[2] = NPY_FLOAT +ufunc__nbinom_kurtosis_excess_types[3] = NPY_DOUBLE +ufunc__nbinom_kurtosis_excess_types[4] = NPY_DOUBLE +ufunc__nbinom_kurtosis_excess_types[5] = NPY_DOUBLE +ufunc__nbinom_kurtosis_excess_ptr[2*0] = scipy.special._ufuncs_cxx._export_nbinom_kurtosis_excess_float +ufunc__nbinom_kurtosis_excess_ptr[2*0+1] = ("_nbinom_kurtosis_excess") +ufunc__nbinom_kurtosis_excess_ptr[2*1] = scipy.special._ufuncs_cxx._export_nbinom_kurtosis_excess_double +ufunc__nbinom_kurtosis_excess_ptr[2*1+1] = ("_nbinom_kurtosis_excess") +ufunc__nbinom_kurtosis_excess_data[0] = &ufunc__nbinom_kurtosis_excess_ptr[2*0] +ufunc__nbinom_kurtosis_excess_data[1] = &ufunc__nbinom_kurtosis_excess_ptr[2*1] +_nbinom_kurtosis_excess = np.PyUFunc_FromFuncAndData(ufunc__nbinom_kurtosis_excess_loops, ufunc__nbinom_kurtosis_excess_data, ufunc__nbinom_kurtosis_excess_types, 2, 2, 1, 0, "_nbinom_kurtosis_excess", ufunc__nbinom_kurtosis_excess_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nbinom_mean_loops[2] +cdef void *ufunc__nbinom_mean_ptr[4] +cdef void *ufunc__nbinom_mean_data[2] +cdef char ufunc__nbinom_mean_types[6] +cdef char *ufunc__nbinom_mean_doc = ( + "_nbinom_mean(r, p)\n" + "\n" + "Mean of negative binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "r : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nbinom_mean_loops[0] = loop_f_ff__As_ff_f +ufunc__nbinom_mean_loops[1] = loop_d_dd__As_dd_d +ufunc__nbinom_mean_types[0] = NPY_FLOAT +ufunc__nbinom_mean_types[1] = NPY_FLOAT +ufunc__nbinom_mean_types[2] = NPY_FLOAT +ufunc__nbinom_mean_types[3] = NPY_DOUBLE +ufunc__nbinom_mean_types[4] = NPY_DOUBLE +ufunc__nbinom_mean_types[5] = NPY_DOUBLE +ufunc__nbinom_mean_ptr[2*0] = scipy.special._ufuncs_cxx._export_nbinom_mean_float +ufunc__nbinom_mean_ptr[2*0+1] = ("_nbinom_mean") +ufunc__nbinom_mean_ptr[2*1] = scipy.special._ufuncs_cxx._export_nbinom_mean_double +ufunc__nbinom_mean_ptr[2*1+1] = ("_nbinom_mean") +ufunc__nbinom_mean_data[0] = &ufunc__nbinom_mean_ptr[2*0] +ufunc__nbinom_mean_data[1] = &ufunc__nbinom_mean_ptr[2*1] +_nbinom_mean = np.PyUFunc_FromFuncAndData(ufunc__nbinom_mean_loops, ufunc__nbinom_mean_data, ufunc__nbinom_mean_types, 2, 2, 1, 0, "_nbinom_mean", ufunc__nbinom_mean_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nbinom_pmf_loops[2] +cdef void *ufunc__nbinom_pmf_ptr[4] +cdef void *ufunc__nbinom_pmf_data[2] +cdef char ufunc__nbinom_pmf_types[8] +cdef char *ufunc__nbinom_pmf_doc = ( + "_nbinom_pmf(x, r, p)\n" + "\n" + "Probability mass function of negative binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "r : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nbinom_pmf_loops[0] = loop_f_fff__As_fff_f +ufunc__nbinom_pmf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__nbinom_pmf_types[0] = NPY_FLOAT +ufunc__nbinom_pmf_types[1] = NPY_FLOAT +ufunc__nbinom_pmf_types[2] = NPY_FLOAT +ufunc__nbinom_pmf_types[3] = NPY_FLOAT +ufunc__nbinom_pmf_types[4] = NPY_DOUBLE +ufunc__nbinom_pmf_types[5] = NPY_DOUBLE +ufunc__nbinom_pmf_types[6] = NPY_DOUBLE +ufunc__nbinom_pmf_types[7] = NPY_DOUBLE +ufunc__nbinom_pmf_ptr[2*0] = scipy.special._ufuncs_cxx._export_nbinom_pmf_float +ufunc__nbinom_pmf_ptr[2*0+1] = ("_nbinom_pmf") +ufunc__nbinom_pmf_ptr[2*1] = scipy.special._ufuncs_cxx._export_nbinom_pmf_double +ufunc__nbinom_pmf_ptr[2*1+1] = ("_nbinom_pmf") +ufunc__nbinom_pmf_data[0] = &ufunc__nbinom_pmf_ptr[2*0] +ufunc__nbinom_pmf_data[1] = &ufunc__nbinom_pmf_ptr[2*1] +_nbinom_pmf = np.PyUFunc_FromFuncAndData(ufunc__nbinom_pmf_loops, ufunc__nbinom_pmf_data, ufunc__nbinom_pmf_types, 2, 3, 1, 0, "_nbinom_pmf", ufunc__nbinom_pmf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nbinom_ppf_loops[2] +cdef void *ufunc__nbinom_ppf_ptr[4] +cdef void *ufunc__nbinom_ppf_data[2] +cdef char ufunc__nbinom_ppf_types[8] +cdef char *ufunc__nbinom_ppf_doc = ( + "_nbinom_ppf(x, r, p)\n" + "\n" + "Percent point function of negative binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "r : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nbinom_ppf_loops[0] = loop_f_fff__As_fff_f +ufunc__nbinom_ppf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__nbinom_ppf_types[0] = NPY_FLOAT +ufunc__nbinom_ppf_types[1] = NPY_FLOAT +ufunc__nbinom_ppf_types[2] = NPY_FLOAT +ufunc__nbinom_ppf_types[3] = NPY_FLOAT +ufunc__nbinom_ppf_types[4] = NPY_DOUBLE +ufunc__nbinom_ppf_types[5] = NPY_DOUBLE +ufunc__nbinom_ppf_types[6] = NPY_DOUBLE +ufunc__nbinom_ppf_types[7] = NPY_DOUBLE +ufunc__nbinom_ppf_ptr[2*0] = scipy.special._ufuncs_cxx._export_nbinom_ppf_float +ufunc__nbinom_ppf_ptr[2*0+1] = ("_nbinom_ppf") +ufunc__nbinom_ppf_ptr[2*1] = scipy.special._ufuncs_cxx._export_nbinom_ppf_double +ufunc__nbinom_ppf_ptr[2*1+1] = ("_nbinom_ppf") +ufunc__nbinom_ppf_data[0] = &ufunc__nbinom_ppf_ptr[2*0] +ufunc__nbinom_ppf_data[1] = &ufunc__nbinom_ppf_ptr[2*1] +_nbinom_ppf = np.PyUFunc_FromFuncAndData(ufunc__nbinom_ppf_loops, ufunc__nbinom_ppf_data, ufunc__nbinom_ppf_types, 2, 3, 1, 0, "_nbinom_ppf", ufunc__nbinom_ppf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nbinom_sf_loops[2] +cdef void *ufunc__nbinom_sf_ptr[4] +cdef void *ufunc__nbinom_sf_data[2] +cdef char ufunc__nbinom_sf_types[8] +cdef char *ufunc__nbinom_sf_doc = ( + "_nbinom_sf(x, r, p)\n" + "\n" + "Survival function of negative binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "r : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nbinom_sf_loops[0] = loop_f_fff__As_fff_f +ufunc__nbinom_sf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__nbinom_sf_types[0] = NPY_FLOAT +ufunc__nbinom_sf_types[1] = NPY_FLOAT +ufunc__nbinom_sf_types[2] = NPY_FLOAT +ufunc__nbinom_sf_types[3] = NPY_FLOAT +ufunc__nbinom_sf_types[4] = NPY_DOUBLE +ufunc__nbinom_sf_types[5] = NPY_DOUBLE +ufunc__nbinom_sf_types[6] = NPY_DOUBLE +ufunc__nbinom_sf_types[7] = NPY_DOUBLE +ufunc__nbinom_sf_ptr[2*0] = scipy.special._ufuncs_cxx._export_nbinom_sf_float +ufunc__nbinom_sf_ptr[2*0+1] = ("_nbinom_sf") +ufunc__nbinom_sf_ptr[2*1] = scipy.special._ufuncs_cxx._export_nbinom_sf_double +ufunc__nbinom_sf_ptr[2*1+1] = ("_nbinom_sf") +ufunc__nbinom_sf_data[0] = &ufunc__nbinom_sf_ptr[2*0] +ufunc__nbinom_sf_data[1] = &ufunc__nbinom_sf_ptr[2*1] +_nbinom_sf = np.PyUFunc_FromFuncAndData(ufunc__nbinom_sf_loops, ufunc__nbinom_sf_data, ufunc__nbinom_sf_types, 2, 3, 1, 0, "_nbinom_sf", ufunc__nbinom_sf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nbinom_skewness_loops[2] +cdef void *ufunc__nbinom_skewness_ptr[4] +cdef void *ufunc__nbinom_skewness_data[2] +cdef char ufunc__nbinom_skewness_types[6] +cdef char *ufunc__nbinom_skewness_doc = ( + "_nbinom_skewness(r, p)\n" + "\n" + "Skewness of negative binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "r : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nbinom_skewness_loops[0] = loop_f_ff__As_ff_f +ufunc__nbinom_skewness_loops[1] = loop_d_dd__As_dd_d +ufunc__nbinom_skewness_types[0] = NPY_FLOAT +ufunc__nbinom_skewness_types[1] = NPY_FLOAT +ufunc__nbinom_skewness_types[2] = NPY_FLOAT +ufunc__nbinom_skewness_types[3] = NPY_DOUBLE +ufunc__nbinom_skewness_types[4] = NPY_DOUBLE +ufunc__nbinom_skewness_types[5] = NPY_DOUBLE +ufunc__nbinom_skewness_ptr[2*0] = scipy.special._ufuncs_cxx._export_nbinom_skewness_float +ufunc__nbinom_skewness_ptr[2*0+1] = ("_nbinom_skewness") +ufunc__nbinom_skewness_ptr[2*1] = scipy.special._ufuncs_cxx._export_nbinom_skewness_double +ufunc__nbinom_skewness_ptr[2*1+1] = ("_nbinom_skewness") +ufunc__nbinom_skewness_data[0] = &ufunc__nbinom_skewness_ptr[2*0] +ufunc__nbinom_skewness_data[1] = &ufunc__nbinom_skewness_ptr[2*1] +_nbinom_skewness = np.PyUFunc_FromFuncAndData(ufunc__nbinom_skewness_loops, ufunc__nbinom_skewness_data, ufunc__nbinom_skewness_types, 2, 2, 1, 0, "_nbinom_skewness", ufunc__nbinom_skewness_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nbinom_variance_loops[2] +cdef void *ufunc__nbinom_variance_ptr[4] +cdef void *ufunc__nbinom_variance_data[2] +cdef char ufunc__nbinom_variance_types[6] +cdef char *ufunc__nbinom_variance_doc = ( + "_nbinom_variance(r, p)\n" + "\n" + "Variance of negative binomial distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "r : array_like\n" + " Positive, integer-valued parameter\n" + "p : array_like\n" + " Positive, real-valued parameter\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nbinom_variance_loops[0] = loop_f_ff__As_ff_f +ufunc__nbinom_variance_loops[1] = loop_d_dd__As_dd_d +ufunc__nbinom_variance_types[0] = NPY_FLOAT +ufunc__nbinom_variance_types[1] = NPY_FLOAT +ufunc__nbinom_variance_types[2] = NPY_FLOAT +ufunc__nbinom_variance_types[3] = NPY_DOUBLE +ufunc__nbinom_variance_types[4] = NPY_DOUBLE +ufunc__nbinom_variance_types[5] = NPY_DOUBLE +ufunc__nbinom_variance_ptr[2*0] = scipy.special._ufuncs_cxx._export_nbinom_variance_float +ufunc__nbinom_variance_ptr[2*0+1] = ("_nbinom_variance") +ufunc__nbinom_variance_ptr[2*1] = scipy.special._ufuncs_cxx._export_nbinom_variance_double +ufunc__nbinom_variance_ptr[2*1+1] = ("_nbinom_variance") +ufunc__nbinom_variance_data[0] = &ufunc__nbinom_variance_ptr[2*0] +ufunc__nbinom_variance_data[1] = &ufunc__nbinom_variance_ptr[2*1] +_nbinom_variance = np.PyUFunc_FromFuncAndData(ufunc__nbinom_variance_loops, ufunc__nbinom_variance_data, ufunc__nbinom_variance_types, 2, 2, 1, 0, "_nbinom_variance", ufunc__nbinom_variance_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncf_isf_loops[2] +cdef void *ufunc__ncf_isf_ptr[4] +cdef void *ufunc__ncf_isf_data[2] +cdef char ufunc__ncf_isf_types[10] +cdef char *ufunc__ncf_isf_doc = ( + "_ncf_isf(x, v1, v2, l)\n" + "\n" + "Inverse survival function of noncentral F-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Positive real-valued\n" + "v1, v2, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncf_isf_loops[0] = loop_f_ffff__As_ffff_f +ufunc__ncf_isf_loops[1] = loop_d_dddd__As_dddd_d +ufunc__ncf_isf_types[0] = NPY_FLOAT +ufunc__ncf_isf_types[1] = NPY_FLOAT +ufunc__ncf_isf_types[2] = NPY_FLOAT +ufunc__ncf_isf_types[3] = NPY_FLOAT +ufunc__ncf_isf_types[4] = NPY_FLOAT +ufunc__ncf_isf_types[5] = NPY_DOUBLE +ufunc__ncf_isf_types[6] = NPY_DOUBLE +ufunc__ncf_isf_types[7] = NPY_DOUBLE +ufunc__ncf_isf_types[8] = NPY_DOUBLE +ufunc__ncf_isf_types[9] = NPY_DOUBLE +ufunc__ncf_isf_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncf_isf_float +ufunc__ncf_isf_ptr[2*0+1] = ("_ncf_isf") +ufunc__ncf_isf_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncf_isf_double +ufunc__ncf_isf_ptr[2*1+1] = ("_ncf_isf") +ufunc__ncf_isf_data[0] = &ufunc__ncf_isf_ptr[2*0] +ufunc__ncf_isf_data[1] = &ufunc__ncf_isf_ptr[2*1] +_ncf_isf = np.PyUFunc_FromFuncAndData(ufunc__ncf_isf_loops, ufunc__ncf_isf_data, ufunc__ncf_isf_types, 2, 4, 1, 0, "_ncf_isf", ufunc__ncf_isf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncf_kurtosis_excess_loops[2] +cdef void *ufunc__ncf_kurtosis_excess_ptr[4] +cdef void *ufunc__ncf_kurtosis_excess_data[2] +cdef char ufunc__ncf_kurtosis_excess_types[8] +cdef char *ufunc__ncf_kurtosis_excess_doc = ( + "_ncf_kurtosis_excess(v1, v2, l)\n" + "\n" + "Kurtosis excess of noncentral F-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "v1, v2, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncf_kurtosis_excess_loops[0] = loop_f_fff__As_fff_f +ufunc__ncf_kurtosis_excess_loops[1] = loop_d_ddd__As_ddd_d +ufunc__ncf_kurtosis_excess_types[0] = NPY_FLOAT +ufunc__ncf_kurtosis_excess_types[1] = NPY_FLOAT +ufunc__ncf_kurtosis_excess_types[2] = NPY_FLOAT +ufunc__ncf_kurtosis_excess_types[3] = NPY_FLOAT +ufunc__ncf_kurtosis_excess_types[4] = NPY_DOUBLE +ufunc__ncf_kurtosis_excess_types[5] = NPY_DOUBLE +ufunc__ncf_kurtosis_excess_types[6] = NPY_DOUBLE +ufunc__ncf_kurtosis_excess_types[7] = NPY_DOUBLE +ufunc__ncf_kurtosis_excess_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncf_kurtosis_excess_float +ufunc__ncf_kurtosis_excess_ptr[2*0+1] = ("_ncf_kurtosis_excess") +ufunc__ncf_kurtosis_excess_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncf_kurtosis_excess_double +ufunc__ncf_kurtosis_excess_ptr[2*1+1] = ("_ncf_kurtosis_excess") +ufunc__ncf_kurtosis_excess_data[0] = &ufunc__ncf_kurtosis_excess_ptr[2*0] +ufunc__ncf_kurtosis_excess_data[1] = &ufunc__ncf_kurtosis_excess_ptr[2*1] +_ncf_kurtosis_excess = np.PyUFunc_FromFuncAndData(ufunc__ncf_kurtosis_excess_loops, ufunc__ncf_kurtosis_excess_data, ufunc__ncf_kurtosis_excess_types, 2, 3, 1, 0, "_ncf_kurtosis_excess", ufunc__ncf_kurtosis_excess_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncf_mean_loops[2] +cdef void *ufunc__ncf_mean_ptr[4] +cdef void *ufunc__ncf_mean_data[2] +cdef char ufunc__ncf_mean_types[8] +cdef char *ufunc__ncf_mean_doc = ( + "_ncf_mean(v1, v2, l)\n" + "\n" + "Mean of noncentral F-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "v1, v2, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncf_mean_loops[0] = loop_f_fff__As_fff_f +ufunc__ncf_mean_loops[1] = loop_d_ddd__As_ddd_d +ufunc__ncf_mean_types[0] = NPY_FLOAT +ufunc__ncf_mean_types[1] = NPY_FLOAT +ufunc__ncf_mean_types[2] = NPY_FLOAT +ufunc__ncf_mean_types[3] = NPY_FLOAT +ufunc__ncf_mean_types[4] = NPY_DOUBLE +ufunc__ncf_mean_types[5] = NPY_DOUBLE +ufunc__ncf_mean_types[6] = NPY_DOUBLE +ufunc__ncf_mean_types[7] = NPY_DOUBLE +ufunc__ncf_mean_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncf_mean_float +ufunc__ncf_mean_ptr[2*0+1] = ("_ncf_mean") +ufunc__ncf_mean_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncf_mean_double +ufunc__ncf_mean_ptr[2*1+1] = ("_ncf_mean") +ufunc__ncf_mean_data[0] = &ufunc__ncf_mean_ptr[2*0] +ufunc__ncf_mean_data[1] = &ufunc__ncf_mean_ptr[2*1] +_ncf_mean = np.PyUFunc_FromFuncAndData(ufunc__ncf_mean_loops, ufunc__ncf_mean_data, ufunc__ncf_mean_types, 2, 3, 1, 0, "_ncf_mean", ufunc__ncf_mean_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncf_pdf_loops[2] +cdef void *ufunc__ncf_pdf_ptr[4] +cdef void *ufunc__ncf_pdf_data[2] +cdef char ufunc__ncf_pdf_types[10] +cdef char *ufunc__ncf_pdf_doc = ( + "_ncf_pdf(x, v1, v2, l)\n" + "\n" + "Probability density function of noncentral F-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Positive real-valued\n" + "v1, v2, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncf_pdf_loops[0] = loop_f_ffff__As_ffff_f +ufunc__ncf_pdf_loops[1] = loop_d_dddd__As_dddd_d +ufunc__ncf_pdf_types[0] = NPY_FLOAT +ufunc__ncf_pdf_types[1] = NPY_FLOAT +ufunc__ncf_pdf_types[2] = NPY_FLOAT +ufunc__ncf_pdf_types[3] = NPY_FLOAT +ufunc__ncf_pdf_types[4] = NPY_FLOAT +ufunc__ncf_pdf_types[5] = NPY_DOUBLE +ufunc__ncf_pdf_types[6] = NPY_DOUBLE +ufunc__ncf_pdf_types[7] = NPY_DOUBLE +ufunc__ncf_pdf_types[8] = NPY_DOUBLE +ufunc__ncf_pdf_types[9] = NPY_DOUBLE +ufunc__ncf_pdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncf_pdf_float +ufunc__ncf_pdf_ptr[2*0+1] = ("_ncf_pdf") +ufunc__ncf_pdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncf_pdf_double +ufunc__ncf_pdf_ptr[2*1+1] = ("_ncf_pdf") +ufunc__ncf_pdf_data[0] = &ufunc__ncf_pdf_ptr[2*0] +ufunc__ncf_pdf_data[1] = &ufunc__ncf_pdf_ptr[2*1] +_ncf_pdf = np.PyUFunc_FromFuncAndData(ufunc__ncf_pdf_loops, ufunc__ncf_pdf_data, ufunc__ncf_pdf_types, 2, 4, 1, 0, "_ncf_pdf", ufunc__ncf_pdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncf_sf_loops[2] +cdef void *ufunc__ncf_sf_ptr[4] +cdef void *ufunc__ncf_sf_data[2] +cdef char ufunc__ncf_sf_types[10] +cdef char *ufunc__ncf_sf_doc = ( + "_ncf_sf(x, v1, v2, l)\n" + "\n" + "Survival function of noncentral F-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Positive real-valued\n" + "v1, v2, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncf_sf_loops[0] = loop_f_ffff__As_ffff_f +ufunc__ncf_sf_loops[1] = loop_d_dddd__As_dddd_d +ufunc__ncf_sf_types[0] = NPY_FLOAT +ufunc__ncf_sf_types[1] = NPY_FLOAT +ufunc__ncf_sf_types[2] = NPY_FLOAT +ufunc__ncf_sf_types[3] = NPY_FLOAT +ufunc__ncf_sf_types[4] = NPY_FLOAT +ufunc__ncf_sf_types[5] = NPY_DOUBLE +ufunc__ncf_sf_types[6] = NPY_DOUBLE +ufunc__ncf_sf_types[7] = NPY_DOUBLE +ufunc__ncf_sf_types[8] = NPY_DOUBLE +ufunc__ncf_sf_types[9] = NPY_DOUBLE +ufunc__ncf_sf_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncf_sf_float +ufunc__ncf_sf_ptr[2*0+1] = ("_ncf_sf") +ufunc__ncf_sf_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncf_sf_double +ufunc__ncf_sf_ptr[2*1+1] = ("_ncf_sf") +ufunc__ncf_sf_data[0] = &ufunc__ncf_sf_ptr[2*0] +ufunc__ncf_sf_data[1] = &ufunc__ncf_sf_ptr[2*1] +_ncf_sf = np.PyUFunc_FromFuncAndData(ufunc__ncf_sf_loops, ufunc__ncf_sf_data, ufunc__ncf_sf_types, 2, 4, 1, 0, "_ncf_sf", ufunc__ncf_sf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncf_skewness_loops[2] +cdef void *ufunc__ncf_skewness_ptr[4] +cdef void *ufunc__ncf_skewness_data[2] +cdef char ufunc__ncf_skewness_types[8] +cdef char *ufunc__ncf_skewness_doc = ( + "_ncf_skewness(v1, v2, l)\n" + "\n" + "Skewness of noncentral F-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "v1, v2, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncf_skewness_loops[0] = loop_f_fff__As_fff_f +ufunc__ncf_skewness_loops[1] = loop_d_ddd__As_ddd_d +ufunc__ncf_skewness_types[0] = NPY_FLOAT +ufunc__ncf_skewness_types[1] = NPY_FLOAT +ufunc__ncf_skewness_types[2] = NPY_FLOAT +ufunc__ncf_skewness_types[3] = NPY_FLOAT +ufunc__ncf_skewness_types[4] = NPY_DOUBLE +ufunc__ncf_skewness_types[5] = NPY_DOUBLE +ufunc__ncf_skewness_types[6] = NPY_DOUBLE +ufunc__ncf_skewness_types[7] = NPY_DOUBLE +ufunc__ncf_skewness_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncf_skewness_float +ufunc__ncf_skewness_ptr[2*0+1] = ("_ncf_skewness") +ufunc__ncf_skewness_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncf_skewness_double +ufunc__ncf_skewness_ptr[2*1+1] = ("_ncf_skewness") +ufunc__ncf_skewness_data[0] = &ufunc__ncf_skewness_ptr[2*0] +ufunc__ncf_skewness_data[1] = &ufunc__ncf_skewness_ptr[2*1] +_ncf_skewness = np.PyUFunc_FromFuncAndData(ufunc__ncf_skewness_loops, ufunc__ncf_skewness_data, ufunc__ncf_skewness_types, 2, 3, 1, 0, "_ncf_skewness", ufunc__ncf_skewness_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncf_variance_loops[2] +cdef void *ufunc__ncf_variance_ptr[4] +cdef void *ufunc__ncf_variance_data[2] +cdef char ufunc__ncf_variance_types[8] +cdef char *ufunc__ncf_variance_doc = ( + "_ncf_variance(v1, v2, l)\n" + "\n" + "Variance of noncentral F-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "v1, v2, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncf_variance_loops[0] = loop_f_fff__As_fff_f +ufunc__ncf_variance_loops[1] = loop_d_ddd__As_ddd_d +ufunc__ncf_variance_types[0] = NPY_FLOAT +ufunc__ncf_variance_types[1] = NPY_FLOAT +ufunc__ncf_variance_types[2] = NPY_FLOAT +ufunc__ncf_variance_types[3] = NPY_FLOAT +ufunc__ncf_variance_types[4] = NPY_DOUBLE +ufunc__ncf_variance_types[5] = NPY_DOUBLE +ufunc__ncf_variance_types[6] = NPY_DOUBLE +ufunc__ncf_variance_types[7] = NPY_DOUBLE +ufunc__ncf_variance_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncf_variance_float +ufunc__ncf_variance_ptr[2*0+1] = ("_ncf_variance") +ufunc__ncf_variance_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncf_variance_double +ufunc__ncf_variance_ptr[2*1+1] = ("_ncf_variance") +ufunc__ncf_variance_data[0] = &ufunc__ncf_variance_ptr[2*0] +ufunc__ncf_variance_data[1] = &ufunc__ncf_variance_ptr[2*1] +_ncf_variance = np.PyUFunc_FromFuncAndData(ufunc__ncf_variance_loops, ufunc__ncf_variance_data, ufunc__ncf_variance_types, 2, 3, 1, 0, "_ncf_variance", ufunc__ncf_variance_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nct_isf_loops[2] +cdef void *ufunc__nct_isf_ptr[4] +cdef void *ufunc__nct_isf_data[2] +cdef char ufunc__nct_isf_types[8] +cdef char *ufunc__nct_isf_doc = ( + "_nct_isf(x, v, l)\n" + "\n" + "Inverse survival function of noncentral t-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "v : array_like\n" + " Positive, real-valued parameters\n" + "l : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nct_isf_loops[0] = loop_f_fff__As_fff_f +ufunc__nct_isf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__nct_isf_types[0] = NPY_FLOAT +ufunc__nct_isf_types[1] = NPY_FLOAT +ufunc__nct_isf_types[2] = NPY_FLOAT +ufunc__nct_isf_types[3] = NPY_FLOAT +ufunc__nct_isf_types[4] = NPY_DOUBLE +ufunc__nct_isf_types[5] = NPY_DOUBLE +ufunc__nct_isf_types[6] = NPY_DOUBLE +ufunc__nct_isf_types[7] = NPY_DOUBLE +ufunc__nct_isf_ptr[2*0] = scipy.special._ufuncs_cxx._export_nct_isf_float +ufunc__nct_isf_ptr[2*0+1] = ("_nct_isf") +ufunc__nct_isf_ptr[2*1] = scipy.special._ufuncs_cxx._export_nct_isf_double +ufunc__nct_isf_ptr[2*1+1] = ("_nct_isf") +ufunc__nct_isf_data[0] = &ufunc__nct_isf_ptr[2*0] +ufunc__nct_isf_data[1] = &ufunc__nct_isf_ptr[2*1] +_nct_isf = np.PyUFunc_FromFuncAndData(ufunc__nct_isf_loops, ufunc__nct_isf_data, ufunc__nct_isf_types, 2, 3, 1, 0, "_nct_isf", ufunc__nct_isf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nct_kurtosis_excess_loops[2] +cdef void *ufunc__nct_kurtosis_excess_ptr[4] +cdef void *ufunc__nct_kurtosis_excess_data[2] +cdef char ufunc__nct_kurtosis_excess_types[6] +cdef char *ufunc__nct_kurtosis_excess_doc = ( + "_nct_kurtosis_excess(v, l)\n" + "\n" + "Kurtosis excess of noncentral t-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "v : array_like\n" + " Positive, real-valued parameters\n" + "l : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nct_kurtosis_excess_loops[0] = loop_f_ff__As_ff_f +ufunc__nct_kurtosis_excess_loops[1] = loop_d_dd__As_dd_d +ufunc__nct_kurtosis_excess_types[0] = NPY_FLOAT +ufunc__nct_kurtosis_excess_types[1] = NPY_FLOAT +ufunc__nct_kurtosis_excess_types[2] = NPY_FLOAT +ufunc__nct_kurtosis_excess_types[3] = NPY_DOUBLE +ufunc__nct_kurtosis_excess_types[4] = NPY_DOUBLE +ufunc__nct_kurtosis_excess_types[5] = NPY_DOUBLE +ufunc__nct_kurtosis_excess_ptr[2*0] = scipy.special._ufuncs_cxx._export_nct_kurtosis_excess_float +ufunc__nct_kurtosis_excess_ptr[2*0+1] = ("_nct_kurtosis_excess") +ufunc__nct_kurtosis_excess_ptr[2*1] = scipy.special._ufuncs_cxx._export_nct_kurtosis_excess_double +ufunc__nct_kurtosis_excess_ptr[2*1+1] = ("_nct_kurtosis_excess") +ufunc__nct_kurtosis_excess_data[0] = &ufunc__nct_kurtosis_excess_ptr[2*0] +ufunc__nct_kurtosis_excess_data[1] = &ufunc__nct_kurtosis_excess_ptr[2*1] +_nct_kurtosis_excess = np.PyUFunc_FromFuncAndData(ufunc__nct_kurtosis_excess_loops, ufunc__nct_kurtosis_excess_data, ufunc__nct_kurtosis_excess_types, 2, 2, 1, 0, "_nct_kurtosis_excess", ufunc__nct_kurtosis_excess_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nct_mean_loops[2] +cdef void *ufunc__nct_mean_ptr[4] +cdef void *ufunc__nct_mean_data[2] +cdef char ufunc__nct_mean_types[6] +cdef char *ufunc__nct_mean_doc = ( + "_nct_mean(v, l)\n" + "\n" + "Mean of noncentral t-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "v : array_like\n" + " Positive, real-valued parameters\n" + "l : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nct_mean_loops[0] = loop_f_ff__As_ff_f +ufunc__nct_mean_loops[1] = loop_d_dd__As_dd_d +ufunc__nct_mean_types[0] = NPY_FLOAT +ufunc__nct_mean_types[1] = NPY_FLOAT +ufunc__nct_mean_types[2] = NPY_FLOAT +ufunc__nct_mean_types[3] = NPY_DOUBLE +ufunc__nct_mean_types[4] = NPY_DOUBLE +ufunc__nct_mean_types[5] = NPY_DOUBLE +ufunc__nct_mean_ptr[2*0] = scipy.special._ufuncs_cxx._export_nct_mean_float +ufunc__nct_mean_ptr[2*0+1] = ("_nct_mean") +ufunc__nct_mean_ptr[2*1] = scipy.special._ufuncs_cxx._export_nct_mean_double +ufunc__nct_mean_ptr[2*1+1] = ("_nct_mean") +ufunc__nct_mean_data[0] = &ufunc__nct_mean_ptr[2*0] +ufunc__nct_mean_data[1] = &ufunc__nct_mean_ptr[2*1] +_nct_mean = np.PyUFunc_FromFuncAndData(ufunc__nct_mean_loops, ufunc__nct_mean_data, ufunc__nct_mean_types, 2, 2, 1, 0, "_nct_mean", ufunc__nct_mean_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nct_pdf_loops[2] +cdef void *ufunc__nct_pdf_ptr[4] +cdef void *ufunc__nct_pdf_data[2] +cdef char ufunc__nct_pdf_types[8] +cdef char *ufunc__nct_pdf_doc = ( + "_nct_pdf(x, v, l)\n" + "\n" + "Probability density function of noncentral t-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "v : array_like\n" + " Positive, real-valued parameters\n" + "l : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nct_pdf_loops[0] = loop_f_fff__As_fff_f +ufunc__nct_pdf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__nct_pdf_types[0] = NPY_FLOAT +ufunc__nct_pdf_types[1] = NPY_FLOAT +ufunc__nct_pdf_types[2] = NPY_FLOAT +ufunc__nct_pdf_types[3] = NPY_FLOAT +ufunc__nct_pdf_types[4] = NPY_DOUBLE +ufunc__nct_pdf_types[5] = NPY_DOUBLE +ufunc__nct_pdf_types[6] = NPY_DOUBLE +ufunc__nct_pdf_types[7] = NPY_DOUBLE +ufunc__nct_pdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_nct_pdf_float +ufunc__nct_pdf_ptr[2*0+1] = ("_nct_pdf") +ufunc__nct_pdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_nct_pdf_double +ufunc__nct_pdf_ptr[2*1+1] = ("_nct_pdf") +ufunc__nct_pdf_data[0] = &ufunc__nct_pdf_ptr[2*0] +ufunc__nct_pdf_data[1] = &ufunc__nct_pdf_ptr[2*1] +_nct_pdf = np.PyUFunc_FromFuncAndData(ufunc__nct_pdf_loops, ufunc__nct_pdf_data, ufunc__nct_pdf_types, 2, 3, 1, 0, "_nct_pdf", ufunc__nct_pdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nct_ppf_loops[2] +cdef void *ufunc__nct_ppf_ptr[4] +cdef void *ufunc__nct_ppf_data[2] +cdef char ufunc__nct_ppf_types[8] +cdef char *ufunc__nct_ppf_doc = ( + "_nct_ppf(x, v, l)\n" + "\n" + "Percent point function of noncentral t-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "v : array_like\n" + " Positive, real-valued parameters\n" + "l : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nct_ppf_loops[0] = loop_f_fff__As_fff_f +ufunc__nct_ppf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__nct_ppf_types[0] = NPY_FLOAT +ufunc__nct_ppf_types[1] = NPY_FLOAT +ufunc__nct_ppf_types[2] = NPY_FLOAT +ufunc__nct_ppf_types[3] = NPY_FLOAT +ufunc__nct_ppf_types[4] = NPY_DOUBLE +ufunc__nct_ppf_types[5] = NPY_DOUBLE +ufunc__nct_ppf_types[6] = NPY_DOUBLE +ufunc__nct_ppf_types[7] = NPY_DOUBLE +ufunc__nct_ppf_ptr[2*0] = scipy.special._ufuncs_cxx._export_nct_ppf_float +ufunc__nct_ppf_ptr[2*0+1] = ("_nct_ppf") +ufunc__nct_ppf_ptr[2*1] = scipy.special._ufuncs_cxx._export_nct_ppf_double +ufunc__nct_ppf_ptr[2*1+1] = ("_nct_ppf") +ufunc__nct_ppf_data[0] = &ufunc__nct_ppf_ptr[2*0] +ufunc__nct_ppf_data[1] = &ufunc__nct_ppf_ptr[2*1] +_nct_ppf = np.PyUFunc_FromFuncAndData(ufunc__nct_ppf_loops, ufunc__nct_ppf_data, ufunc__nct_ppf_types, 2, 3, 1, 0, "_nct_ppf", ufunc__nct_ppf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nct_sf_loops[2] +cdef void *ufunc__nct_sf_ptr[4] +cdef void *ufunc__nct_sf_data[2] +cdef char ufunc__nct_sf_types[8] +cdef char *ufunc__nct_sf_doc = ( + "_nct_sf(x, v, l)\n" + "\n" + "Survival function of noncentral t-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "v : array_like\n" + " Positive, real-valued parameters\n" + "l : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nct_sf_loops[0] = loop_f_fff__As_fff_f +ufunc__nct_sf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__nct_sf_types[0] = NPY_FLOAT +ufunc__nct_sf_types[1] = NPY_FLOAT +ufunc__nct_sf_types[2] = NPY_FLOAT +ufunc__nct_sf_types[3] = NPY_FLOAT +ufunc__nct_sf_types[4] = NPY_DOUBLE +ufunc__nct_sf_types[5] = NPY_DOUBLE +ufunc__nct_sf_types[6] = NPY_DOUBLE +ufunc__nct_sf_types[7] = NPY_DOUBLE +ufunc__nct_sf_ptr[2*0] = scipy.special._ufuncs_cxx._export_nct_sf_float +ufunc__nct_sf_ptr[2*0+1] = ("_nct_sf") +ufunc__nct_sf_ptr[2*1] = scipy.special._ufuncs_cxx._export_nct_sf_double +ufunc__nct_sf_ptr[2*1+1] = ("_nct_sf") +ufunc__nct_sf_data[0] = &ufunc__nct_sf_ptr[2*0] +ufunc__nct_sf_data[1] = &ufunc__nct_sf_ptr[2*1] +_nct_sf = np.PyUFunc_FromFuncAndData(ufunc__nct_sf_loops, ufunc__nct_sf_data, ufunc__nct_sf_types, 2, 3, 1, 0, "_nct_sf", ufunc__nct_sf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nct_skewness_loops[2] +cdef void *ufunc__nct_skewness_ptr[4] +cdef void *ufunc__nct_skewness_data[2] +cdef char ufunc__nct_skewness_types[6] +cdef char *ufunc__nct_skewness_doc = ( + "_nct_skewness(v, l)\n" + "\n" + "Skewness of noncentral t-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "v : array_like\n" + " Positive, real-valued parameters\n" + "l : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nct_skewness_loops[0] = loop_f_ff__As_ff_f +ufunc__nct_skewness_loops[1] = loop_d_dd__As_dd_d +ufunc__nct_skewness_types[0] = NPY_FLOAT +ufunc__nct_skewness_types[1] = NPY_FLOAT +ufunc__nct_skewness_types[2] = NPY_FLOAT +ufunc__nct_skewness_types[3] = NPY_DOUBLE +ufunc__nct_skewness_types[4] = NPY_DOUBLE +ufunc__nct_skewness_types[5] = NPY_DOUBLE +ufunc__nct_skewness_ptr[2*0] = scipy.special._ufuncs_cxx._export_nct_skewness_float +ufunc__nct_skewness_ptr[2*0+1] = ("_nct_skewness") +ufunc__nct_skewness_ptr[2*1] = scipy.special._ufuncs_cxx._export_nct_skewness_double +ufunc__nct_skewness_ptr[2*1+1] = ("_nct_skewness") +ufunc__nct_skewness_data[0] = &ufunc__nct_skewness_ptr[2*0] +ufunc__nct_skewness_data[1] = &ufunc__nct_skewness_ptr[2*1] +_nct_skewness = np.PyUFunc_FromFuncAndData(ufunc__nct_skewness_loops, ufunc__nct_skewness_data, ufunc__nct_skewness_types, 2, 2, 1, 0, "_nct_skewness", ufunc__nct_skewness_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__nct_variance_loops[2] +cdef void *ufunc__nct_variance_ptr[4] +cdef void *ufunc__nct_variance_data[2] +cdef char ufunc__nct_variance_types[6] +cdef char *ufunc__nct_variance_doc = ( + "_nct_variance(v, l)\n" + "\n" + "Variance of noncentral t-distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "v : array_like\n" + " Positive, real-valued parameters\n" + "l : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__nct_variance_loops[0] = loop_f_ff__As_ff_f +ufunc__nct_variance_loops[1] = loop_d_dd__As_dd_d +ufunc__nct_variance_types[0] = NPY_FLOAT +ufunc__nct_variance_types[1] = NPY_FLOAT +ufunc__nct_variance_types[2] = NPY_FLOAT +ufunc__nct_variance_types[3] = NPY_DOUBLE +ufunc__nct_variance_types[4] = NPY_DOUBLE +ufunc__nct_variance_types[5] = NPY_DOUBLE +ufunc__nct_variance_ptr[2*0] = scipy.special._ufuncs_cxx._export_nct_variance_float +ufunc__nct_variance_ptr[2*0+1] = ("_nct_variance") +ufunc__nct_variance_ptr[2*1] = scipy.special._ufuncs_cxx._export_nct_variance_double +ufunc__nct_variance_ptr[2*1+1] = ("_nct_variance") +ufunc__nct_variance_data[0] = &ufunc__nct_variance_ptr[2*0] +ufunc__nct_variance_data[1] = &ufunc__nct_variance_ptr[2*1] +_nct_variance = np.PyUFunc_FromFuncAndData(ufunc__nct_variance_loops, ufunc__nct_variance_data, ufunc__nct_variance_types, 2, 2, 1, 0, "_nct_variance", ufunc__nct_variance_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncx2_cdf_loops[2] +cdef void *ufunc__ncx2_cdf_ptr[4] +cdef void *ufunc__ncx2_cdf_data[2] +cdef char ufunc__ncx2_cdf_types[8] +cdef char *ufunc__ncx2_cdf_doc = ( + "_ncx2_cdf(x, k, l)\n" + "\n" + "Cumulative density function of Non-central chi-squared distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Positive real-valued\n" + "k, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncx2_cdf_loops[0] = loop_f_fff__As_fff_f +ufunc__ncx2_cdf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__ncx2_cdf_types[0] = NPY_FLOAT +ufunc__ncx2_cdf_types[1] = NPY_FLOAT +ufunc__ncx2_cdf_types[2] = NPY_FLOAT +ufunc__ncx2_cdf_types[3] = NPY_FLOAT +ufunc__ncx2_cdf_types[4] = NPY_DOUBLE +ufunc__ncx2_cdf_types[5] = NPY_DOUBLE +ufunc__ncx2_cdf_types[6] = NPY_DOUBLE +ufunc__ncx2_cdf_types[7] = NPY_DOUBLE +ufunc__ncx2_cdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncx2_cdf_float +ufunc__ncx2_cdf_ptr[2*0+1] = ("_ncx2_cdf") +ufunc__ncx2_cdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncx2_cdf_double +ufunc__ncx2_cdf_ptr[2*1+1] = ("_ncx2_cdf") +ufunc__ncx2_cdf_data[0] = &ufunc__ncx2_cdf_ptr[2*0] +ufunc__ncx2_cdf_data[1] = &ufunc__ncx2_cdf_ptr[2*1] +_ncx2_cdf = np.PyUFunc_FromFuncAndData(ufunc__ncx2_cdf_loops, ufunc__ncx2_cdf_data, ufunc__ncx2_cdf_types, 2, 3, 1, 0, "_ncx2_cdf", ufunc__ncx2_cdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncx2_isf_loops[2] +cdef void *ufunc__ncx2_isf_ptr[4] +cdef void *ufunc__ncx2_isf_data[2] +cdef char ufunc__ncx2_isf_types[8] +cdef char *ufunc__ncx2_isf_doc = ( + "_ncx2_isf(x, k, l)\n" + "\n" + "Inverse survival function of Non-central chi-squared distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Positive real-valued\n" + "k, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncx2_isf_loops[0] = loop_f_fff__As_fff_f +ufunc__ncx2_isf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__ncx2_isf_types[0] = NPY_FLOAT +ufunc__ncx2_isf_types[1] = NPY_FLOAT +ufunc__ncx2_isf_types[2] = NPY_FLOAT +ufunc__ncx2_isf_types[3] = NPY_FLOAT +ufunc__ncx2_isf_types[4] = NPY_DOUBLE +ufunc__ncx2_isf_types[5] = NPY_DOUBLE +ufunc__ncx2_isf_types[6] = NPY_DOUBLE +ufunc__ncx2_isf_types[7] = NPY_DOUBLE +ufunc__ncx2_isf_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncx2_isf_float +ufunc__ncx2_isf_ptr[2*0+1] = ("_ncx2_isf") +ufunc__ncx2_isf_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncx2_isf_double +ufunc__ncx2_isf_ptr[2*1+1] = ("_ncx2_isf") +ufunc__ncx2_isf_data[0] = &ufunc__ncx2_isf_ptr[2*0] +ufunc__ncx2_isf_data[1] = &ufunc__ncx2_isf_ptr[2*1] +_ncx2_isf = np.PyUFunc_FromFuncAndData(ufunc__ncx2_isf_loops, ufunc__ncx2_isf_data, ufunc__ncx2_isf_types, 2, 3, 1, 0, "_ncx2_isf", ufunc__ncx2_isf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncx2_pdf_loops[2] +cdef void *ufunc__ncx2_pdf_ptr[4] +cdef void *ufunc__ncx2_pdf_data[2] +cdef char ufunc__ncx2_pdf_types[8] +cdef char *ufunc__ncx2_pdf_doc = ( + "_ncx2_pdf(x, k, l)\n" + "\n" + "Probability density function of Non-central chi-squared distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Positive real-valued\n" + "k, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncx2_pdf_loops[0] = loop_f_fff__As_fff_f +ufunc__ncx2_pdf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__ncx2_pdf_types[0] = NPY_FLOAT +ufunc__ncx2_pdf_types[1] = NPY_FLOAT +ufunc__ncx2_pdf_types[2] = NPY_FLOAT +ufunc__ncx2_pdf_types[3] = NPY_FLOAT +ufunc__ncx2_pdf_types[4] = NPY_DOUBLE +ufunc__ncx2_pdf_types[5] = NPY_DOUBLE +ufunc__ncx2_pdf_types[6] = NPY_DOUBLE +ufunc__ncx2_pdf_types[7] = NPY_DOUBLE +ufunc__ncx2_pdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncx2_pdf_float +ufunc__ncx2_pdf_ptr[2*0+1] = ("_ncx2_pdf") +ufunc__ncx2_pdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncx2_pdf_double +ufunc__ncx2_pdf_ptr[2*1+1] = ("_ncx2_pdf") +ufunc__ncx2_pdf_data[0] = &ufunc__ncx2_pdf_ptr[2*0] +ufunc__ncx2_pdf_data[1] = &ufunc__ncx2_pdf_ptr[2*1] +_ncx2_pdf = np.PyUFunc_FromFuncAndData(ufunc__ncx2_pdf_loops, ufunc__ncx2_pdf_data, ufunc__ncx2_pdf_types, 2, 3, 1, 0, "_ncx2_pdf", ufunc__ncx2_pdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncx2_ppf_loops[2] +cdef void *ufunc__ncx2_ppf_ptr[4] +cdef void *ufunc__ncx2_ppf_data[2] +cdef char ufunc__ncx2_ppf_types[8] +cdef char *ufunc__ncx2_ppf_doc = ( + "_ncx2_ppf(x, k, l)\n" + "\n" + "Percent point function of Non-central chi-squared distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Positive real-valued\n" + "k, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncx2_ppf_loops[0] = loop_f_fff__As_fff_f +ufunc__ncx2_ppf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__ncx2_ppf_types[0] = NPY_FLOAT +ufunc__ncx2_ppf_types[1] = NPY_FLOAT +ufunc__ncx2_ppf_types[2] = NPY_FLOAT +ufunc__ncx2_ppf_types[3] = NPY_FLOAT +ufunc__ncx2_ppf_types[4] = NPY_DOUBLE +ufunc__ncx2_ppf_types[5] = NPY_DOUBLE +ufunc__ncx2_ppf_types[6] = NPY_DOUBLE +ufunc__ncx2_ppf_types[7] = NPY_DOUBLE +ufunc__ncx2_ppf_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncx2_ppf_float +ufunc__ncx2_ppf_ptr[2*0+1] = ("_ncx2_ppf") +ufunc__ncx2_ppf_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncx2_ppf_double +ufunc__ncx2_ppf_ptr[2*1+1] = ("_ncx2_ppf") +ufunc__ncx2_ppf_data[0] = &ufunc__ncx2_ppf_ptr[2*0] +ufunc__ncx2_ppf_data[1] = &ufunc__ncx2_ppf_ptr[2*1] +_ncx2_ppf = np.PyUFunc_FromFuncAndData(ufunc__ncx2_ppf_loops, ufunc__ncx2_ppf_data, ufunc__ncx2_ppf_types, 2, 3, 1, 0, "_ncx2_ppf", ufunc__ncx2_ppf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__ncx2_sf_loops[2] +cdef void *ufunc__ncx2_sf_ptr[4] +cdef void *ufunc__ncx2_sf_data[2] +cdef char ufunc__ncx2_sf_types[8] +cdef char *ufunc__ncx2_sf_doc = ( + "_ncx2_sf(x, k, l)\n" + "\n" + "Survival function of Non-central chi-squared distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Positive real-valued\n" + "k, l : array_like\n" + " Positive, real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__ncx2_sf_loops[0] = loop_f_fff__As_fff_f +ufunc__ncx2_sf_loops[1] = loop_d_ddd__As_ddd_d +ufunc__ncx2_sf_types[0] = NPY_FLOAT +ufunc__ncx2_sf_types[1] = NPY_FLOAT +ufunc__ncx2_sf_types[2] = NPY_FLOAT +ufunc__ncx2_sf_types[3] = NPY_FLOAT +ufunc__ncx2_sf_types[4] = NPY_DOUBLE +ufunc__ncx2_sf_types[5] = NPY_DOUBLE +ufunc__ncx2_sf_types[6] = NPY_DOUBLE +ufunc__ncx2_sf_types[7] = NPY_DOUBLE +ufunc__ncx2_sf_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncx2_sf_float +ufunc__ncx2_sf_ptr[2*0+1] = ("_ncx2_sf") +ufunc__ncx2_sf_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncx2_sf_double +ufunc__ncx2_sf_ptr[2*1+1] = ("_ncx2_sf") +ufunc__ncx2_sf_data[0] = &ufunc__ncx2_sf_ptr[2*0] +ufunc__ncx2_sf_data[1] = &ufunc__ncx2_sf_ptr[2*1] +_ncx2_sf = np.PyUFunc_FromFuncAndData(ufunc__ncx2_sf_loops, ufunc__ncx2_sf_data, ufunc__ncx2_sf_types, 2, 3, 1, 0, "_ncx2_sf", ufunc__ncx2_sf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__sf_error_test_function_loops[1] +cdef void *ufunc__sf_error_test_function_ptr[2] +cdef void *ufunc__sf_error_test_function_data[1] +cdef char ufunc__sf_error_test_function_types[2] +cdef char *ufunc__sf_error_test_function_doc = ( + "Private function; do not use.") +ufunc__sf_error_test_function_loops[0] = loop_i_i__As_l_l +ufunc__sf_error_test_function_types[0] = NPY_LONG +ufunc__sf_error_test_function_types[1] = NPY_LONG +ufunc__sf_error_test_function_ptr[2*0] = _func__sf_error_test_function +ufunc__sf_error_test_function_ptr[2*0+1] = ("_sf_error_test_function") +ufunc__sf_error_test_function_data[0] = &ufunc__sf_error_test_function_ptr[2*0] +_sf_error_test_function = np.PyUFunc_FromFuncAndData(ufunc__sf_error_test_function_loops, ufunc__sf_error_test_function_data, ufunc__sf_error_test_function_types, 1, 1, 1, 0, "_sf_error_test_function", ufunc__sf_error_test_function_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__skewnorm_cdf_loops[2] +cdef void *ufunc__skewnorm_cdf_ptr[4] +cdef void *ufunc__skewnorm_cdf_data[2] +cdef char ufunc__skewnorm_cdf_types[10] +cdef char *ufunc__skewnorm_cdf_doc = ( + "_skewnorm_cdf(x, l, sc, sh)\n" + "\n" + "Cumulative density function of skewnorm distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "l : array_like\n" + " Real-valued parameters\n" + "sc : array_like\n" + " Positive, Real-valued parameters\n" + "sh : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__skewnorm_cdf_loops[0] = loop_f_ffff__As_ffff_f +ufunc__skewnorm_cdf_loops[1] = loop_d_dddd__As_dddd_d +ufunc__skewnorm_cdf_types[0] = NPY_FLOAT +ufunc__skewnorm_cdf_types[1] = NPY_FLOAT +ufunc__skewnorm_cdf_types[2] = NPY_FLOAT +ufunc__skewnorm_cdf_types[3] = NPY_FLOAT +ufunc__skewnorm_cdf_types[4] = NPY_FLOAT +ufunc__skewnorm_cdf_types[5] = NPY_DOUBLE +ufunc__skewnorm_cdf_types[6] = NPY_DOUBLE +ufunc__skewnorm_cdf_types[7] = NPY_DOUBLE +ufunc__skewnorm_cdf_types[8] = NPY_DOUBLE +ufunc__skewnorm_cdf_types[9] = NPY_DOUBLE +ufunc__skewnorm_cdf_ptr[2*0] = scipy.special._ufuncs_cxx._export_skewnorm_cdf_float +ufunc__skewnorm_cdf_ptr[2*0+1] = ("_skewnorm_cdf") +ufunc__skewnorm_cdf_ptr[2*1] = scipy.special._ufuncs_cxx._export_skewnorm_cdf_double +ufunc__skewnorm_cdf_ptr[2*1+1] = ("_skewnorm_cdf") +ufunc__skewnorm_cdf_data[0] = &ufunc__skewnorm_cdf_ptr[2*0] +ufunc__skewnorm_cdf_data[1] = &ufunc__skewnorm_cdf_ptr[2*1] +_skewnorm_cdf = np.PyUFunc_FromFuncAndData(ufunc__skewnorm_cdf_loops, ufunc__skewnorm_cdf_data, ufunc__skewnorm_cdf_types, 2, 4, 1, 0, "_skewnorm_cdf", ufunc__skewnorm_cdf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__skewnorm_isf_loops[2] +cdef void *ufunc__skewnorm_isf_ptr[4] +cdef void *ufunc__skewnorm_isf_data[2] +cdef char ufunc__skewnorm_isf_types[10] +cdef char *ufunc__skewnorm_isf_doc = ( + "_skewnorm_isf(x, l, sc, sh)\n" + "\n" + "Inverse survival function of skewnorm distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "l : array_like\n" + " Real-valued parameters\n" + "sc : array_like\n" + " Positive, Real-valued parameters\n" + "sh : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__skewnorm_isf_loops[0] = loop_f_ffff__As_ffff_f +ufunc__skewnorm_isf_loops[1] = loop_d_dddd__As_dddd_d +ufunc__skewnorm_isf_types[0] = NPY_FLOAT +ufunc__skewnorm_isf_types[1] = NPY_FLOAT +ufunc__skewnorm_isf_types[2] = NPY_FLOAT +ufunc__skewnorm_isf_types[3] = NPY_FLOAT +ufunc__skewnorm_isf_types[4] = NPY_FLOAT +ufunc__skewnorm_isf_types[5] = NPY_DOUBLE +ufunc__skewnorm_isf_types[6] = NPY_DOUBLE +ufunc__skewnorm_isf_types[7] = NPY_DOUBLE +ufunc__skewnorm_isf_types[8] = NPY_DOUBLE +ufunc__skewnorm_isf_types[9] = NPY_DOUBLE +ufunc__skewnorm_isf_ptr[2*0] = scipy.special._ufuncs_cxx._export_skewnorm_isf_float +ufunc__skewnorm_isf_ptr[2*0+1] = ("_skewnorm_isf") +ufunc__skewnorm_isf_ptr[2*1] = scipy.special._ufuncs_cxx._export_skewnorm_isf_double +ufunc__skewnorm_isf_ptr[2*1+1] = ("_skewnorm_isf") +ufunc__skewnorm_isf_data[0] = &ufunc__skewnorm_isf_ptr[2*0] +ufunc__skewnorm_isf_data[1] = &ufunc__skewnorm_isf_ptr[2*1] +_skewnorm_isf = np.PyUFunc_FromFuncAndData(ufunc__skewnorm_isf_loops, ufunc__skewnorm_isf_data, ufunc__skewnorm_isf_types, 2, 4, 1, 0, "_skewnorm_isf", ufunc__skewnorm_isf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__skewnorm_ppf_loops[2] +cdef void *ufunc__skewnorm_ppf_ptr[4] +cdef void *ufunc__skewnorm_ppf_data[2] +cdef char ufunc__skewnorm_ppf_types[10] +cdef char *ufunc__skewnorm_ppf_doc = ( + "_skewnorm_ppf(x, l, sc, sh)\n" + "\n" + "Percent point function of skewnorm distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real-valued\n" + "l : array_like\n" + " Real-valued parameters\n" + "sc : array_like\n" + " Positive, Real-valued parameters\n" + "sh : array_like\n" + " Real-valued parameters\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray") +ufunc__skewnorm_ppf_loops[0] = loop_f_ffff__As_ffff_f +ufunc__skewnorm_ppf_loops[1] = loop_d_dddd__As_dddd_d +ufunc__skewnorm_ppf_types[0] = NPY_FLOAT +ufunc__skewnorm_ppf_types[1] = NPY_FLOAT +ufunc__skewnorm_ppf_types[2] = NPY_FLOAT +ufunc__skewnorm_ppf_types[3] = NPY_FLOAT +ufunc__skewnorm_ppf_types[4] = NPY_FLOAT +ufunc__skewnorm_ppf_types[5] = NPY_DOUBLE +ufunc__skewnorm_ppf_types[6] = NPY_DOUBLE +ufunc__skewnorm_ppf_types[7] = NPY_DOUBLE +ufunc__skewnorm_ppf_types[8] = NPY_DOUBLE +ufunc__skewnorm_ppf_types[9] = NPY_DOUBLE +ufunc__skewnorm_ppf_ptr[2*0] = scipy.special._ufuncs_cxx._export_skewnorm_ppf_float +ufunc__skewnorm_ppf_ptr[2*0+1] = ("_skewnorm_ppf") +ufunc__skewnorm_ppf_ptr[2*1] = scipy.special._ufuncs_cxx._export_skewnorm_ppf_double +ufunc__skewnorm_ppf_ptr[2*1+1] = ("_skewnorm_ppf") +ufunc__skewnorm_ppf_data[0] = &ufunc__skewnorm_ppf_ptr[2*0] +ufunc__skewnorm_ppf_data[1] = &ufunc__skewnorm_ppf_ptr[2*1] +_skewnorm_ppf = np.PyUFunc_FromFuncAndData(ufunc__skewnorm_ppf_loops, ufunc__skewnorm_ppf_data, ufunc__skewnorm_ppf_types, 2, 4, 1, 0, "_skewnorm_ppf", ufunc__skewnorm_ppf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__smirnovc_loops[3] +cdef void *ufunc__smirnovc_ptr[6] +cdef void *ufunc__smirnovc_data[3] +cdef char ufunc__smirnovc_types[9] +cdef char *ufunc__smirnovc_doc = ( + "_smirnovc(n, d)\n" + " Internal function, do not use.") +ufunc__smirnovc_loops[0] = loop_d_pd__As_pd_d +ufunc__smirnovc_loops[1] = loop_d_dd__As_ff_f +ufunc__smirnovc_loops[2] = loop_d_dd__As_dd_d +ufunc__smirnovc_types[0] = NPY_INTP +ufunc__smirnovc_types[1] = NPY_DOUBLE +ufunc__smirnovc_types[2] = NPY_DOUBLE +ufunc__smirnovc_types[3] = NPY_FLOAT +ufunc__smirnovc_types[4] = NPY_FLOAT +ufunc__smirnovc_types[5] = NPY_FLOAT +ufunc__smirnovc_types[6] = NPY_DOUBLE +ufunc__smirnovc_types[7] = NPY_DOUBLE +ufunc__smirnovc_types[8] = NPY_DOUBLE +ufunc__smirnovc_ptr[2*0] = _func_cephes_smirnovc_wrap +ufunc__smirnovc_ptr[2*0+1] = ("_smirnovc") +ufunc__smirnovc_ptr[2*1] = _func_smirnovc_unsafe +ufunc__smirnovc_ptr[2*1+1] = ("_smirnovc") +ufunc__smirnovc_ptr[2*2] = _func_smirnovc_unsafe +ufunc__smirnovc_ptr[2*2+1] = ("_smirnovc") +ufunc__smirnovc_data[0] = &ufunc__smirnovc_ptr[2*0] +ufunc__smirnovc_data[1] = &ufunc__smirnovc_ptr[2*1] +ufunc__smirnovc_data[2] = &ufunc__smirnovc_ptr[2*2] +_smirnovc = np.PyUFunc_FromFuncAndData(ufunc__smirnovc_loops, ufunc__smirnovc_data, ufunc__smirnovc_types, 3, 2, 1, 0, "_smirnovc", ufunc__smirnovc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__smirnovci_loops[3] +cdef void *ufunc__smirnovci_ptr[6] +cdef void *ufunc__smirnovci_data[3] +cdef char ufunc__smirnovci_types[9] +cdef char *ufunc__smirnovci_doc = ( + "Internal function, do not use.") +ufunc__smirnovci_loops[0] = loop_d_pd__As_pd_d +ufunc__smirnovci_loops[1] = loop_d_dd__As_ff_f +ufunc__smirnovci_loops[2] = loop_d_dd__As_dd_d +ufunc__smirnovci_types[0] = NPY_INTP +ufunc__smirnovci_types[1] = NPY_DOUBLE +ufunc__smirnovci_types[2] = NPY_DOUBLE +ufunc__smirnovci_types[3] = NPY_FLOAT +ufunc__smirnovci_types[4] = NPY_FLOAT +ufunc__smirnovci_types[5] = NPY_FLOAT +ufunc__smirnovci_types[6] = NPY_DOUBLE +ufunc__smirnovci_types[7] = NPY_DOUBLE +ufunc__smirnovci_types[8] = NPY_DOUBLE +ufunc__smirnovci_ptr[2*0] = _func_cephes_smirnovci_wrap +ufunc__smirnovci_ptr[2*0+1] = ("_smirnovci") +ufunc__smirnovci_ptr[2*1] = _func_smirnovci_unsafe +ufunc__smirnovci_ptr[2*1+1] = ("_smirnovci") +ufunc__smirnovci_ptr[2*2] = _func_smirnovci_unsafe +ufunc__smirnovci_ptr[2*2+1] = ("_smirnovci") +ufunc__smirnovci_data[0] = &ufunc__smirnovci_ptr[2*0] +ufunc__smirnovci_data[1] = &ufunc__smirnovci_ptr[2*1] +ufunc__smirnovci_data[2] = &ufunc__smirnovci_ptr[2*2] +_smirnovci = np.PyUFunc_FromFuncAndData(ufunc__smirnovci_loops, ufunc__smirnovci_data, ufunc__smirnovci_types, 3, 2, 1, 0, "_smirnovci", ufunc__smirnovci_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__smirnovp_loops[3] +cdef void *ufunc__smirnovp_ptr[6] +cdef void *ufunc__smirnovp_data[3] +cdef char ufunc__smirnovp_types[9] +cdef char *ufunc__smirnovp_doc = ( + "_smirnovp(n, p)\n" + " Internal function, do not use.") +ufunc__smirnovp_loops[0] = loop_d_pd__As_pd_d +ufunc__smirnovp_loops[1] = loop_d_dd__As_ff_f +ufunc__smirnovp_loops[2] = loop_d_dd__As_dd_d +ufunc__smirnovp_types[0] = NPY_INTP +ufunc__smirnovp_types[1] = NPY_DOUBLE +ufunc__smirnovp_types[2] = NPY_DOUBLE +ufunc__smirnovp_types[3] = NPY_FLOAT +ufunc__smirnovp_types[4] = NPY_FLOAT +ufunc__smirnovp_types[5] = NPY_FLOAT +ufunc__smirnovp_types[6] = NPY_DOUBLE +ufunc__smirnovp_types[7] = NPY_DOUBLE +ufunc__smirnovp_types[8] = NPY_DOUBLE +ufunc__smirnovp_ptr[2*0] = _func_cephes_smirnovp_wrap +ufunc__smirnovp_ptr[2*0+1] = ("_smirnovp") +ufunc__smirnovp_ptr[2*1] = _func_smirnovp_unsafe +ufunc__smirnovp_ptr[2*1+1] = ("_smirnovp") +ufunc__smirnovp_ptr[2*2] = _func_smirnovp_unsafe +ufunc__smirnovp_ptr[2*2+1] = ("_smirnovp") +ufunc__smirnovp_data[0] = &ufunc__smirnovp_ptr[2*0] +ufunc__smirnovp_data[1] = &ufunc__smirnovp_ptr[2*1] +ufunc__smirnovp_data[2] = &ufunc__smirnovp_ptr[2*2] +_smirnovp = np.PyUFunc_FromFuncAndData(ufunc__smirnovp_loops, ufunc__smirnovp_data, ufunc__smirnovp_types, 3, 2, 1, 0, "_smirnovp", ufunc__smirnovp_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__stirling2_inexact_loops[2] +cdef void *ufunc__stirling2_inexact_ptr[4] +cdef void *ufunc__stirling2_inexact_data[2] +cdef char ufunc__stirling2_inexact_types[6] +cdef char *ufunc__stirling2_inexact_doc = ( + "Internal function, do not use.") +ufunc__stirling2_inexact_loops[0] = loop_d_dd__As_ff_f +ufunc__stirling2_inexact_loops[1] = loop_d_dd__As_dd_d +ufunc__stirling2_inexact_types[0] = NPY_FLOAT +ufunc__stirling2_inexact_types[1] = NPY_FLOAT +ufunc__stirling2_inexact_types[2] = NPY_FLOAT +ufunc__stirling2_inexact_types[3] = NPY_DOUBLE +ufunc__stirling2_inexact_types[4] = NPY_DOUBLE +ufunc__stirling2_inexact_types[5] = NPY_DOUBLE +ufunc__stirling2_inexact_ptr[2*0] = scipy.special._ufuncs_cxx._export__stirling2_inexact +ufunc__stirling2_inexact_ptr[2*0+1] = ("_stirling2_inexact") +ufunc__stirling2_inexact_ptr[2*1] = scipy.special._ufuncs_cxx._export__stirling2_inexact +ufunc__stirling2_inexact_ptr[2*1+1] = ("_stirling2_inexact") +ufunc__stirling2_inexact_data[0] = &ufunc__stirling2_inexact_ptr[2*0] +ufunc__stirling2_inexact_data[1] = &ufunc__stirling2_inexact_ptr[2*1] +_stirling2_inexact = np.PyUFunc_FromFuncAndData(ufunc__stirling2_inexact_loops, ufunc__stirling2_inexact_data, ufunc__stirling2_inexact_types, 2, 2, 1, 0, "_stirling2_inexact", ufunc__stirling2_inexact_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__struve_asymp_large_z_loops[1] +cdef void *ufunc__struve_asymp_large_z_ptr[2] +cdef void *ufunc__struve_asymp_large_z_data[1] +cdef char ufunc__struve_asymp_large_z_types[5] +cdef char *ufunc__struve_asymp_large_z_doc = ( + "_struve_asymp_large_z(v, z, is_h)\n" + "\n" + "Internal function for testing `struve` & `modstruve`\n" + "\n" + "Evaluates using asymptotic expansion\n" + "\n" + "Returns\n" + "-------\n" + "v, err") +ufunc__struve_asymp_large_z_loops[0] = loop_d_ddp_d_As_ddp_dd +ufunc__struve_asymp_large_z_types[0] = NPY_DOUBLE +ufunc__struve_asymp_large_z_types[1] = NPY_DOUBLE +ufunc__struve_asymp_large_z_types[2] = NPY_INTP +ufunc__struve_asymp_large_z_types[3] = NPY_DOUBLE +ufunc__struve_asymp_large_z_types[4] = NPY_DOUBLE +ufunc__struve_asymp_large_z_ptr[2*0] = _func_cephes__struve_asymp_large_z +ufunc__struve_asymp_large_z_ptr[2*0+1] = ("_struve_asymp_large_z") +ufunc__struve_asymp_large_z_data[0] = &ufunc__struve_asymp_large_z_ptr[2*0] +_struve_asymp_large_z = np.PyUFunc_FromFuncAndData(ufunc__struve_asymp_large_z_loops, ufunc__struve_asymp_large_z_data, ufunc__struve_asymp_large_z_types, 1, 3, 2, 0, "_struve_asymp_large_z", ufunc__struve_asymp_large_z_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__struve_bessel_series_loops[1] +cdef void *ufunc__struve_bessel_series_ptr[2] +cdef void *ufunc__struve_bessel_series_data[1] +cdef char ufunc__struve_bessel_series_types[5] +cdef char *ufunc__struve_bessel_series_doc = ( + "_struve_bessel_series(v, z, is_h)\n" + "\n" + "Internal function for testing `struve` & `modstruve`\n" + "\n" + "Evaluates using Bessel function series\n" + "\n" + "Returns\n" + "-------\n" + "v, err") +ufunc__struve_bessel_series_loops[0] = loop_d_ddp_d_As_ddp_dd +ufunc__struve_bessel_series_types[0] = NPY_DOUBLE +ufunc__struve_bessel_series_types[1] = NPY_DOUBLE +ufunc__struve_bessel_series_types[2] = NPY_INTP +ufunc__struve_bessel_series_types[3] = NPY_DOUBLE +ufunc__struve_bessel_series_types[4] = NPY_DOUBLE +ufunc__struve_bessel_series_ptr[2*0] = _func_cephes__struve_bessel_series +ufunc__struve_bessel_series_ptr[2*0+1] = ("_struve_bessel_series") +ufunc__struve_bessel_series_data[0] = &ufunc__struve_bessel_series_ptr[2*0] +_struve_bessel_series = np.PyUFunc_FromFuncAndData(ufunc__struve_bessel_series_loops, ufunc__struve_bessel_series_data, ufunc__struve_bessel_series_types, 1, 3, 2, 0, "_struve_bessel_series", ufunc__struve_bessel_series_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc__struve_power_series_loops[1] +cdef void *ufunc__struve_power_series_ptr[2] +cdef void *ufunc__struve_power_series_data[1] +cdef char ufunc__struve_power_series_types[5] +cdef char *ufunc__struve_power_series_doc = ( + "_struve_power_series(v, z, is_h)\n" + "\n" + "Internal function for testing `struve` & `modstruve`\n" + "\n" + "Evaluates using power series\n" + "\n" + "Returns\n" + "-------\n" + "v, err") +ufunc__struve_power_series_loops[0] = loop_d_ddp_d_As_ddp_dd +ufunc__struve_power_series_types[0] = NPY_DOUBLE +ufunc__struve_power_series_types[1] = NPY_DOUBLE +ufunc__struve_power_series_types[2] = NPY_INTP +ufunc__struve_power_series_types[3] = NPY_DOUBLE +ufunc__struve_power_series_types[4] = NPY_DOUBLE +ufunc__struve_power_series_ptr[2*0] = _func_cephes__struve_power_series +ufunc__struve_power_series_ptr[2*0+1] = ("_struve_power_series") +ufunc__struve_power_series_data[0] = &ufunc__struve_power_series_ptr[2*0] +_struve_power_series = np.PyUFunc_FromFuncAndData(ufunc__struve_power_series_loops, ufunc__struve_power_series_data, ufunc__struve_power_series_types, 1, 3, 2, 0, "_struve_power_series", ufunc__struve_power_series_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_agm_loops[2] +cdef void *ufunc_agm_ptr[4] +cdef void *ufunc_agm_data[2] +cdef char ufunc_agm_types[6] +cdef char *ufunc_agm_doc = ( + "agm(a, b, out=None)\n" + "\n" + "Compute the arithmetic-geometric mean of `a` and `b`.\n" + "\n" + "Start with a_0 = a and b_0 = b and iteratively compute::\n" + "\n" + " a_{n+1} = (a_n + b_n)/2\n" + " b_{n+1} = sqrt(a_n*b_n)\n" + "\n" + "a_n and b_n converge to the same limit as n increases; their common\n" + "limit is agm(a, b).\n" + "\n" + "Parameters\n" + "----------\n" + "a, b : array_like\n" + " Real values only. If the values are both negative, the result\n" + " is negative. If one value is negative and the other is positive,\n" + " `nan` is returned.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The arithmetic-geometric mean of `a` and `b`.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy.special import agm\n" + ">>> a, b = 24.0, 6.0\n" + ">>> agm(a, b)\n" + "13.458171481725614\n" + "\n" + "Compare that result to the iteration:\n" + "\n" + ">>> while a != b:\n" + "... a, b = (a + b)/2, np.sqrt(a*b)\n" + "... print(\"a = %19.16f b=%19.16f\" % (a, b))\n" + "...\n" + "a = 15.0000000000000000 b=12.0000000000000000\n" + "a = 13.5000000000000000 b=13.4164078649987388\n" + "a = 13.4582039324993694 b=13.4581390309909850\n" + "a = 13.4581714817451772 b=13.4581714817060547\n" + "a = 13.4581714817256159 b=13.4581714817256159\n" + "\n" + "When array-like arguments are given, broadcasting applies:\n" + "\n" + ">>> a = np.array([[1.5], [3], [6]]) # a has shape (3, 1).\n" + ">>> b = np.array([6, 12, 24, 48]) # b has shape (4,).\n" + ">>> agm(a, b)\n" + "array([[ 3.36454287, 5.42363427, 9.05798751, 15.53650756],\n" + " [ 4.37037309, 6.72908574, 10.84726853, 18.11597502],\n" + " [ 6. , 8.74074619, 13.45817148, 21.69453707]])") +ufunc_agm_loops[0] = loop_d_dd__As_ff_f +ufunc_agm_loops[1] = loop_d_dd__As_dd_d +ufunc_agm_types[0] = NPY_FLOAT +ufunc_agm_types[1] = NPY_FLOAT +ufunc_agm_types[2] = NPY_FLOAT +ufunc_agm_types[3] = NPY_DOUBLE +ufunc_agm_types[4] = NPY_DOUBLE +ufunc_agm_types[5] = NPY_DOUBLE +ufunc_agm_ptr[2*0] = _func_agm +ufunc_agm_ptr[2*0+1] = ("agm") +ufunc_agm_ptr[2*1] = _func_agm +ufunc_agm_ptr[2*1+1] = ("agm") +ufunc_agm_data[0] = &ufunc_agm_ptr[2*0] +ufunc_agm_data[1] = &ufunc_agm_ptr[2*1] +agm = np.PyUFunc_FromFuncAndData(ufunc_agm_loops, ufunc_agm_data, ufunc_agm_types, 2, 2, 1, 0, "agm", ufunc_agm_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_bdtr_loops[3] +cdef void *ufunc_bdtr_ptr[6] +cdef void *ufunc_bdtr_data[3] +cdef char ufunc_bdtr_types[12] +cdef char *ufunc_bdtr_doc = ( + "bdtr(k, n, p, out=None)\n" + "\n" + "Binomial distribution cumulative distribution function.\n" + "\n" + "Sum of the terms 0 through `floor(k)` of the Binomial probability density.\n" + "\n" + ".. math::\n" + " \\mathrm{bdtr}(k, n, p) =\n" + " \\sum_{j=0}^{\\lfloor k \\rfloor} {{n}\\choose{j}} p^j (1-p)^{n-j}\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " Number of successes (double), rounded down to the nearest integer.\n" + "n : array_like\n" + " Number of events (int).\n" + "p : array_like\n" + " Probability of success in a single event (float).\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "y : scalar or ndarray\n" + " Probability of `floor(k)` or fewer successes in `n` independent events with\n" + " success probabilities of `p`.\n" + "\n" + "Notes\n" + "-----\n" + "The terms are not summed directly; instead the regularized incomplete beta\n" + "function is employed, according to the formula,\n" + "\n" + ".. math::\n" + " \\mathrm{bdtr}(k, n, p) =\n" + " I_{1 - p}(n - \\lfloor k \\rfloor, \\lfloor k \\rfloor + 1).\n" + "\n" + "Wrapper for the Cephes [1]_ routine `bdtr`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/") +ufunc_bdtr_loops[0] = loop_d_ddd__As_fff_f +ufunc_bdtr_loops[1] = loop_d_dpd__As_dpd_d +ufunc_bdtr_loops[2] = loop_d_ddd__As_ddd_d +ufunc_bdtr_types[0] = NPY_FLOAT +ufunc_bdtr_types[1] = NPY_FLOAT +ufunc_bdtr_types[2] = NPY_FLOAT +ufunc_bdtr_types[3] = NPY_FLOAT +ufunc_bdtr_types[4] = NPY_DOUBLE +ufunc_bdtr_types[5] = NPY_INTP +ufunc_bdtr_types[6] = NPY_DOUBLE +ufunc_bdtr_types[7] = NPY_DOUBLE +ufunc_bdtr_types[8] = NPY_DOUBLE +ufunc_bdtr_types[9] = NPY_DOUBLE +ufunc_bdtr_types[10] = NPY_DOUBLE +ufunc_bdtr_types[11] = NPY_DOUBLE +ufunc_bdtr_ptr[2*0] = _func_bdtr_unsafe +ufunc_bdtr_ptr[2*0+1] = ("bdtr") +ufunc_bdtr_ptr[2*1] = _func_cephes_bdtr_wrap +ufunc_bdtr_ptr[2*1+1] = ("bdtr") +ufunc_bdtr_ptr[2*2] = _func_bdtr_unsafe +ufunc_bdtr_ptr[2*2+1] = ("bdtr") +ufunc_bdtr_data[0] = &ufunc_bdtr_ptr[2*0] +ufunc_bdtr_data[1] = &ufunc_bdtr_ptr[2*1] +ufunc_bdtr_data[2] = &ufunc_bdtr_ptr[2*2] +bdtr = np.PyUFunc_FromFuncAndData(ufunc_bdtr_loops, ufunc_bdtr_data, ufunc_bdtr_types, 3, 3, 1, 0, "bdtr", ufunc_bdtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_bdtrc_loops[3] +cdef void *ufunc_bdtrc_ptr[6] +cdef void *ufunc_bdtrc_data[3] +cdef char ufunc_bdtrc_types[12] +cdef char *ufunc_bdtrc_doc = ( + "bdtrc(k, n, p, out=None)\n" + "\n" + "Binomial distribution survival function.\n" + "\n" + "Sum of the terms `floor(k) + 1` through `n` of the binomial probability\n" + "density,\n" + "\n" + ".. math::\n" + " \\mathrm{bdtrc}(k, n, p) =\n" + " \\sum_{j=\\lfloor k \\rfloor +1}^n {{n}\\choose{j}} p^j (1-p)^{n-j}\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " Number of successes (double), rounded down to nearest integer.\n" + "n : array_like\n" + " Number of events (int)\n" + "p : array_like\n" + " Probability of success in a single event.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "y : scalar or ndarray\n" + " Probability of `floor(k) + 1` or more successes in `n` independent\n" + " events with success probabilities of `p`.\n" + "\n" + "See Also\n" + "--------\n" + "bdtr\n" + "betainc\n" + "\n" + "Notes\n" + "-----\n" + "The terms are not summed directly; instead the regularized incomplete beta\n" + "function is employed, according to the formula,\n" + "\n" + ".. math::\n" + " \\mathrm{bdtrc}(k, n, p) = I_{p}(\\lfloor k \\rfloor + 1, n - \\lfloor k \\rfloor).\n" + "\n" + "Wrapper for the Cephes [1]_ routine `bdtrc`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/") +ufunc_bdtrc_loops[0] = loop_d_ddd__As_fff_f +ufunc_bdtrc_loops[1] = loop_d_dpd__As_dpd_d +ufunc_bdtrc_loops[2] = loop_d_ddd__As_ddd_d +ufunc_bdtrc_types[0] = NPY_FLOAT +ufunc_bdtrc_types[1] = NPY_FLOAT +ufunc_bdtrc_types[2] = NPY_FLOAT +ufunc_bdtrc_types[3] = NPY_FLOAT +ufunc_bdtrc_types[4] = NPY_DOUBLE +ufunc_bdtrc_types[5] = NPY_INTP +ufunc_bdtrc_types[6] = NPY_DOUBLE +ufunc_bdtrc_types[7] = NPY_DOUBLE +ufunc_bdtrc_types[8] = NPY_DOUBLE +ufunc_bdtrc_types[9] = NPY_DOUBLE +ufunc_bdtrc_types[10] = NPY_DOUBLE +ufunc_bdtrc_types[11] = NPY_DOUBLE +ufunc_bdtrc_ptr[2*0] = _func_bdtrc_unsafe +ufunc_bdtrc_ptr[2*0+1] = ("bdtrc") +ufunc_bdtrc_ptr[2*1] = _func_cephes_bdtrc_wrap +ufunc_bdtrc_ptr[2*1+1] = ("bdtrc") +ufunc_bdtrc_ptr[2*2] = _func_bdtrc_unsafe +ufunc_bdtrc_ptr[2*2+1] = ("bdtrc") +ufunc_bdtrc_data[0] = &ufunc_bdtrc_ptr[2*0] +ufunc_bdtrc_data[1] = &ufunc_bdtrc_ptr[2*1] +ufunc_bdtrc_data[2] = &ufunc_bdtrc_ptr[2*2] +bdtrc = np.PyUFunc_FromFuncAndData(ufunc_bdtrc_loops, ufunc_bdtrc_data, ufunc_bdtrc_types, 3, 3, 1, 0, "bdtrc", ufunc_bdtrc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_bdtri_loops[3] +cdef void *ufunc_bdtri_ptr[6] +cdef void *ufunc_bdtri_data[3] +cdef char ufunc_bdtri_types[12] +cdef char *ufunc_bdtri_doc = ( + "bdtri(k, n, y, out=None)\n" + "\n" + "Inverse function to `bdtr` with respect to `p`.\n" + "\n" + "Finds the event probability `p` such that the sum of the terms 0 through\n" + "`k` of the binomial probability density is equal to the given cumulative\n" + "probability `y`.\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " Number of successes (float), rounded down to the nearest integer.\n" + "n : array_like\n" + " Number of events (float)\n" + "y : array_like\n" + " Cumulative probability (probability of `k` or fewer successes in `n`\n" + " events).\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "p : scalar or ndarray\n" + " The event probability such that `bdtr(\\lfloor k \\rfloor, n, p) = y`.\n" + "\n" + "See Also\n" + "--------\n" + "bdtr\n" + "betaincinv\n" + "\n" + "Notes\n" + "-----\n" + "The computation is carried out using the inverse beta integral function\n" + "and the relation,::\n" + "\n" + " 1 - p = betaincinv(n - k, k + 1, y).\n" + "\n" + "Wrapper for the Cephes [1]_ routine `bdtri`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/") +ufunc_bdtri_loops[0] = loop_d_ddd__As_fff_f +ufunc_bdtri_loops[1] = loop_d_dpd__As_dpd_d +ufunc_bdtri_loops[2] = loop_d_ddd__As_ddd_d +ufunc_bdtri_types[0] = NPY_FLOAT +ufunc_bdtri_types[1] = NPY_FLOAT +ufunc_bdtri_types[2] = NPY_FLOAT +ufunc_bdtri_types[3] = NPY_FLOAT +ufunc_bdtri_types[4] = NPY_DOUBLE +ufunc_bdtri_types[5] = NPY_INTP +ufunc_bdtri_types[6] = NPY_DOUBLE +ufunc_bdtri_types[7] = NPY_DOUBLE +ufunc_bdtri_types[8] = NPY_DOUBLE +ufunc_bdtri_types[9] = NPY_DOUBLE +ufunc_bdtri_types[10] = NPY_DOUBLE +ufunc_bdtri_types[11] = NPY_DOUBLE +ufunc_bdtri_ptr[2*0] = _func_bdtri_unsafe +ufunc_bdtri_ptr[2*0+1] = ("bdtri") +ufunc_bdtri_ptr[2*1] = _func_cephes_bdtri_wrap +ufunc_bdtri_ptr[2*1+1] = ("bdtri") +ufunc_bdtri_ptr[2*2] = _func_bdtri_unsafe +ufunc_bdtri_ptr[2*2+1] = ("bdtri") +ufunc_bdtri_data[0] = &ufunc_bdtri_ptr[2*0] +ufunc_bdtri_data[1] = &ufunc_bdtri_ptr[2*1] +ufunc_bdtri_data[2] = &ufunc_bdtri_ptr[2*2] +bdtri = np.PyUFunc_FromFuncAndData(ufunc_bdtri_loops, ufunc_bdtri_data, ufunc_bdtri_types, 3, 3, 1, 0, "bdtri", ufunc_bdtri_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_bdtrik_loops[2] +cdef void *ufunc_bdtrik_ptr[4] +cdef void *ufunc_bdtrik_data[2] +cdef char ufunc_bdtrik_types[8] +cdef char *ufunc_bdtrik_doc = ( + "bdtrik(y, n, p, out=None)\n" + "\n" + "Inverse function to `bdtr` with respect to `k`.\n" + "\n" + "Finds the number of successes `k` such that the sum of the terms 0 through\n" + "`k` of the Binomial probability density for `n` events with probability\n" + "`p` is equal to the given cumulative probability `y`.\n" + "\n" + "Parameters\n" + "----------\n" + "y : array_like\n" + " Cumulative probability (probability of `k` or fewer successes in `n`\n" + " events).\n" + "n : array_like\n" + " Number of events (float).\n" + "p : array_like\n" + " Success probability (float).\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "k : scalar or ndarray\n" + " The number of successes `k` such that `bdtr(k, n, p) = y`.\n" + "\n" + "See Also\n" + "--------\n" + "bdtr\n" + "\n" + "Notes\n" + "-----\n" + "Formula 26.5.24 of [1]_ is used to reduce the binomial distribution to the\n" + "cumulative incomplete beta distribution.\n" + "\n" + "Computation of `k` involves a search for a value that produces the desired\n" + "value of `y`. The search relies on the monotonicity of `y` with `k`.\n" + "\n" + "Wrapper for the CDFLIB [2]_ Fortran routine `cdfbin`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.\n" + ".. [2] Barry Brown, James Lovato, and Kathy Russell,\n" + " CDFLIB: Library of Fortran Routines for Cumulative Distribution\n" + " Functions, Inverses, and Other Parameters.") +ufunc_bdtrik_loops[0] = loop_d_ddd__As_fff_f +ufunc_bdtrik_loops[1] = loop_d_ddd__As_ddd_d +ufunc_bdtrik_types[0] = NPY_FLOAT +ufunc_bdtrik_types[1] = NPY_FLOAT +ufunc_bdtrik_types[2] = NPY_FLOAT +ufunc_bdtrik_types[3] = NPY_FLOAT +ufunc_bdtrik_types[4] = NPY_DOUBLE +ufunc_bdtrik_types[5] = NPY_DOUBLE +ufunc_bdtrik_types[6] = NPY_DOUBLE +ufunc_bdtrik_types[7] = NPY_DOUBLE +ufunc_bdtrik_ptr[2*0] = _func_bdtrik +ufunc_bdtrik_ptr[2*0+1] = ("bdtrik") +ufunc_bdtrik_ptr[2*1] = _func_bdtrik +ufunc_bdtrik_ptr[2*1+1] = ("bdtrik") +ufunc_bdtrik_data[0] = &ufunc_bdtrik_ptr[2*0] +ufunc_bdtrik_data[1] = &ufunc_bdtrik_ptr[2*1] +bdtrik = np.PyUFunc_FromFuncAndData(ufunc_bdtrik_loops, ufunc_bdtrik_data, ufunc_bdtrik_types, 2, 3, 1, 0, "bdtrik", ufunc_bdtrik_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_bdtrin_loops[2] +cdef void *ufunc_bdtrin_ptr[4] +cdef void *ufunc_bdtrin_data[2] +cdef char ufunc_bdtrin_types[8] +cdef char *ufunc_bdtrin_doc = ( + "bdtrin(k, y, p, out=None)\n" + "\n" + "Inverse function to `bdtr` with respect to `n`.\n" + "\n" + "Finds the number of events `n` such that the sum of the terms 0 through\n" + "`k` of the Binomial probability density for events with probability `p` is\n" + "equal to the given cumulative probability `y`.\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " Number of successes (float).\n" + "y : array_like\n" + " Cumulative probability (probability of `k` or fewer successes in `n`\n" + " events).\n" + "p : array_like\n" + " Success probability (float).\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "n : scalar or ndarray\n" + " The number of events `n` such that `bdtr(k, n, p) = y`.\n" + "\n" + "See Also\n" + "--------\n" + "bdtr\n" + "\n" + "Notes\n" + "-----\n" + "Formula 26.5.24 of [1]_ is used to reduce the binomial distribution to the\n" + "cumulative incomplete beta distribution.\n" + "\n" + "Computation of `n` involves a search for a value that produces the desired\n" + "value of `y`. The search relies on the monotonicity of `y` with `n`.\n" + "\n" + "Wrapper for the CDFLIB [2]_ Fortran routine `cdfbin`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.\n" + ".. [2] Barry Brown, James Lovato, and Kathy Russell,\n" + " CDFLIB: Library of Fortran Routines for Cumulative Distribution\n" + " Functions, Inverses, and Other Parameters.") +ufunc_bdtrin_loops[0] = loop_d_ddd__As_fff_f +ufunc_bdtrin_loops[1] = loop_d_ddd__As_ddd_d +ufunc_bdtrin_types[0] = NPY_FLOAT +ufunc_bdtrin_types[1] = NPY_FLOAT +ufunc_bdtrin_types[2] = NPY_FLOAT +ufunc_bdtrin_types[3] = NPY_FLOAT +ufunc_bdtrin_types[4] = NPY_DOUBLE +ufunc_bdtrin_types[5] = NPY_DOUBLE +ufunc_bdtrin_types[6] = NPY_DOUBLE +ufunc_bdtrin_types[7] = NPY_DOUBLE +ufunc_bdtrin_ptr[2*0] = _func_bdtrin +ufunc_bdtrin_ptr[2*0+1] = ("bdtrin") +ufunc_bdtrin_ptr[2*1] = _func_bdtrin +ufunc_bdtrin_ptr[2*1+1] = ("bdtrin") +ufunc_bdtrin_data[0] = &ufunc_bdtrin_ptr[2*0] +ufunc_bdtrin_data[1] = &ufunc_bdtrin_ptr[2*1] +bdtrin = np.PyUFunc_FromFuncAndData(ufunc_bdtrin_loops, ufunc_bdtrin_data, ufunc_bdtrin_types, 2, 3, 1, 0, "bdtrin", ufunc_bdtrin_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_betainc_loops[2] +cdef void *ufunc_betainc_ptr[4] +cdef void *ufunc_betainc_data[2] +cdef char ufunc_betainc_types[8] +cdef char *ufunc_betainc_doc = ( + "betainc(a, b, x, out=None)\n" + "\n" + "Regularized incomplete beta function.\n" + "\n" + "Computes the regularized incomplete beta function, defined as [1]_:\n" + "\n" + ".. math::\n" + "\n" + " I_x(a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^x\n" + " t^{a-1}(1-t)^{b-1}dt,\n" + "\n" + "for :math:`0 \\leq x \\leq 1`.\n" + "\n" + "This function is the cumulative distribution function for the beta\n" + "distribution; its range is [0, 1].\n" + "\n" + "Parameters\n" + "----------\n" + "a, b : array_like\n" + " Positive, real-valued parameters\n" + "x : array_like\n" + " Real-valued such that :math:`0 \\leq x \\leq 1`,\n" + " the upper limit of integration\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Value of the regularized incomplete beta function\n" + "\n" + "See Also\n" + "--------\n" + "beta : beta function\n" + "betaincinv : inverse of the regularized incomplete beta function\n" + "betaincc : complement of the regularized incomplete beta function\n" + "scipy.stats.beta : beta distribution\n" + "\n" + "Notes\n" + "-----\n" + "The term *regularized* in the name of this function refers to the\n" + "scaling of the function by the gamma function terms shown in the\n" + "formula. When not qualified as *regularized*, the name *incomplete\n" + "beta function* often refers to just the integral expression,\n" + "without the gamma terms. One can use the function `beta` from\n" + "`scipy.special` to get this \"nonregularized\" incomplete beta\n" + "function by multiplying the result of ``betainc(a, b, x)`` by\n" + "``beta(a, b)``.\n" + "\n" + "This function wraps the ``ibeta`` routine from the\n" + "Boost Math C++ library [2]_.\n" + "\n" + "References\n" + "----------\n" + ".. [1] NIST Digital Library of Mathematical Functions\n" + " https://dlmf.nist.gov/8.17\n" + ".. [2] The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/.\n" + "\n" + "Examples\n" + "--------\n" + "\n" + "Let :math:`B(a, b)` be the `beta` function.\n" + "\n" + ">>> import scipy.special as sc\n" + "\n" + "The coefficient in terms of `gamma` is equal to\n" + ":math:`1/B(a, b)`. Also, when :math:`x=1`\n" + "the integral is equal to :math:`B(a, b)`.\n" + "Therefore, :math:`I_{x=1}(a, b) = 1` for any :math:`a, b`.\n" + "\n" + ">>> sc.betainc(0.2, 3.5, 1.0)\n" + "1.0\n" + "\n" + "It satisfies\n" + ":math:`I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b))`,\n" + "where :math:`F` is the hypergeometric function `hyp2f1`:\n" + "\n" + ">>> a, b, x = 1.4, 3.1, 0.5\n" + ">>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))\n" + "0.8148904036225295\n" + ">>> sc.betainc(a, b, x)\n" + "0.8148904036225296\n" + "\n" + "This functions satisfies the relationship\n" + ":math:`I_x(a, b) = 1 - I_{1-x}(b, a)`:\n" + "\n" + ">>> sc.betainc(2.2, 3.1, 0.4)\n" + "0.49339638807619446\n" + ">>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)\n" + "0.49339638807619446") +ufunc_betainc_loops[0] = loop_f_fff__As_fff_f +ufunc_betainc_loops[1] = loop_d_ddd__As_ddd_d +ufunc_betainc_types[0] = NPY_FLOAT +ufunc_betainc_types[1] = NPY_FLOAT +ufunc_betainc_types[2] = NPY_FLOAT +ufunc_betainc_types[3] = NPY_FLOAT +ufunc_betainc_types[4] = NPY_DOUBLE +ufunc_betainc_types[5] = NPY_DOUBLE +ufunc_betainc_types[6] = NPY_DOUBLE +ufunc_betainc_types[7] = NPY_DOUBLE +ufunc_betainc_ptr[2*0] = scipy.special._ufuncs_cxx._export_ibeta_float +ufunc_betainc_ptr[2*0+1] = ("betainc") +ufunc_betainc_ptr[2*1] = scipy.special._ufuncs_cxx._export_ibeta_double +ufunc_betainc_ptr[2*1+1] = ("betainc") +ufunc_betainc_data[0] = &ufunc_betainc_ptr[2*0] +ufunc_betainc_data[1] = &ufunc_betainc_ptr[2*1] +betainc = np.PyUFunc_FromFuncAndData(ufunc_betainc_loops, ufunc_betainc_data, ufunc_betainc_types, 2, 3, 1, 0, "betainc", ufunc_betainc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_betaincc_loops[2] +cdef void *ufunc_betaincc_ptr[4] +cdef void *ufunc_betaincc_data[2] +cdef char ufunc_betaincc_types[8] +cdef char *ufunc_betaincc_doc = ( + "betaincc(a, b, x, out=None)\n" + "\n" + "Complement of the regularized incomplete beta function.\n" + "\n" + "Computes the complement of the regularized incomplete beta function,\n" + "defined as [1]_:\n" + "\n" + ".. math::\n" + "\n" + " \\bar{I}_x(a, b) = 1 - I_x(a, b)\n" + " = 1 - \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^x\n" + " t^{a-1}(1-t)^{b-1}dt,\n" + "\n" + "for :math:`0 \\leq x \\leq 1`.\n" + "\n" + "Parameters\n" + "----------\n" + "a, b : array_like\n" + " Positive, real-valued parameters\n" + "x : array_like\n" + " Real-valued such that :math:`0 \\leq x \\leq 1`,\n" + " the upper limit of integration\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Value of the regularized incomplete beta function\n" + "\n" + "See Also\n" + "--------\n" + "betainc : regularized incomplete beta function\n" + "betaincinv : inverse of the regularized incomplete beta function\n" + "betainccinv :\n" + " inverse of the complement of the regularized incomplete beta function\n" + "beta : beta function\n" + "scipy.stats.beta : beta distribution\n" + "\n" + "Notes\n" + "-----\n" + ".. versionadded:: 1.11.0\n" + "\n" + "This function wraps the ``ibetac`` routine from the\n" + "Boost Math C++ library [2]_.\n" + "\n" + "References\n" + "----------\n" + ".. [1] NIST Digital Library of Mathematical Functions\n" + " https://dlmf.nist.gov/8.17\n" + ".. [2] The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import betaincc, betainc\n" + "\n" + "The naive calculation ``1 - betainc(a, b, x)`` loses precision when\n" + "the values of ``betainc(a, b, x)`` are close to 1:\n" + "\n" + ">>> 1 - betainc(0.5, 8, [0.9, 0.99, 0.999])\n" + "array([2.0574632e-09, 0.0000000e+00, 0.0000000e+00])\n" + "\n" + "By using ``betaincc``, we get the correct values:\n" + "\n" + ">>> betaincc(0.5, 8, [0.9, 0.99, 0.999])\n" + "array([2.05746321e-09, 1.97259354e-17, 1.96467954e-25])") +ufunc_betaincc_loops[0] = loop_f_fff__As_fff_f +ufunc_betaincc_loops[1] = loop_d_ddd__As_ddd_d +ufunc_betaincc_types[0] = NPY_FLOAT +ufunc_betaincc_types[1] = NPY_FLOAT +ufunc_betaincc_types[2] = NPY_FLOAT +ufunc_betaincc_types[3] = NPY_FLOAT +ufunc_betaincc_types[4] = NPY_DOUBLE +ufunc_betaincc_types[5] = NPY_DOUBLE +ufunc_betaincc_types[6] = NPY_DOUBLE +ufunc_betaincc_types[7] = NPY_DOUBLE +ufunc_betaincc_ptr[2*0] = scipy.special._ufuncs_cxx._export_ibetac_float +ufunc_betaincc_ptr[2*0+1] = ("betaincc") +ufunc_betaincc_ptr[2*1] = scipy.special._ufuncs_cxx._export_ibetac_double +ufunc_betaincc_ptr[2*1+1] = ("betaincc") +ufunc_betaincc_data[0] = &ufunc_betaincc_ptr[2*0] +ufunc_betaincc_data[1] = &ufunc_betaincc_ptr[2*1] +betaincc = np.PyUFunc_FromFuncAndData(ufunc_betaincc_loops, ufunc_betaincc_data, ufunc_betaincc_types, 2, 3, 1, 0, "betaincc", ufunc_betaincc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_betainccinv_loops[2] +cdef void *ufunc_betainccinv_ptr[4] +cdef void *ufunc_betainccinv_data[2] +cdef char ufunc_betainccinv_types[8] +cdef char *ufunc_betainccinv_doc = ( + "betainccinv(a, b, y, out=None)\n" + "\n" + "Inverse of the complemented regularized incomplete beta function.\n" + "\n" + "Computes :math:`x` such that:\n" + "\n" + ".. math::\n" + "\n" + " y = 1 - I_x(a, b) = 1 - \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)}\n" + " \\int_0^x t^{a-1}(1-t)^{b-1}dt,\n" + "\n" + "where :math:`I_x` is the normalized incomplete beta function `betainc`\n" + "and :math:`\\Gamma` is the `gamma` function [1]_.\n" + "\n" + "Parameters\n" + "----------\n" + "a, b : array_like\n" + " Positive, real-valued parameters\n" + "y : array_like\n" + " Real-valued input\n" + "out : ndarray, optional\n" + " Optional output array for function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Value of the inverse of the regularized incomplete beta function\n" + "\n" + "See Also\n" + "--------\n" + "betainc : regularized incomplete beta function\n" + "betaincc : complement of the regularized incomplete beta function\n" + "\n" + "Notes\n" + "-----\n" + ".. versionadded:: 1.11.0\n" + "\n" + "This function wraps the ``ibetac_inv`` routine from the\n" + "Boost Math C++ library [2]_.\n" + "\n" + "References\n" + "----------\n" + ".. [1] NIST Digital Library of Mathematical Functions\n" + " https://dlmf.nist.gov/8.17\n" + ".. [2] The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import betainccinv, betaincc\n" + "\n" + "This function is the inverse of `betaincc` for fixed\n" + "values of :math:`a` and :math:`b`.\n" + "\n" + ">>> a, b = 1.2, 3.1\n" + ">>> y = betaincc(a, b, 0.2)\n" + ">>> betainccinv(a, b, y)\n" + "0.2\n" + "\n" + ">>> a, b = 7, 2.5\n" + ">>> x = betainccinv(a, b, 0.875)\n" + ">>> betaincc(a, b, x)\n" + "0.875") +ufunc_betainccinv_loops[0] = loop_f_fff__As_fff_f +ufunc_betainccinv_loops[1] = loop_d_ddd__As_ddd_d +ufunc_betainccinv_types[0] = NPY_FLOAT +ufunc_betainccinv_types[1] = NPY_FLOAT +ufunc_betainccinv_types[2] = NPY_FLOAT +ufunc_betainccinv_types[3] = NPY_FLOAT +ufunc_betainccinv_types[4] = NPY_DOUBLE +ufunc_betainccinv_types[5] = NPY_DOUBLE +ufunc_betainccinv_types[6] = NPY_DOUBLE +ufunc_betainccinv_types[7] = NPY_DOUBLE +ufunc_betainccinv_ptr[2*0] = scipy.special._ufuncs_cxx._export_ibetac_inv_float +ufunc_betainccinv_ptr[2*0+1] = ("betainccinv") +ufunc_betainccinv_ptr[2*1] = scipy.special._ufuncs_cxx._export_ibetac_inv_double +ufunc_betainccinv_ptr[2*1+1] = ("betainccinv") +ufunc_betainccinv_data[0] = &ufunc_betainccinv_ptr[2*0] +ufunc_betainccinv_data[1] = &ufunc_betainccinv_ptr[2*1] +betainccinv = np.PyUFunc_FromFuncAndData(ufunc_betainccinv_loops, ufunc_betainccinv_data, ufunc_betainccinv_types, 2, 3, 1, 0, "betainccinv", ufunc_betainccinv_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_betaincinv_loops[2] +cdef void *ufunc_betaincinv_ptr[4] +cdef void *ufunc_betaincinv_data[2] +cdef char ufunc_betaincinv_types[8] +cdef char *ufunc_betaincinv_doc = ( + "betaincinv(a, b, y, out=None)\n" + "\n" + "Inverse of the regularized incomplete beta function.\n" + "\n" + "Computes :math:`x` such that:\n" + "\n" + ".. math::\n" + "\n" + " y = I_x(a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)}\n" + " \\int_0^x t^{a-1}(1-t)^{b-1}dt,\n" + "\n" + "where :math:`I_x` is the normalized incomplete beta function `betainc`\n" + "and :math:`\\Gamma` is the `gamma` function [1]_.\n" + "\n" + "Parameters\n" + "----------\n" + "a, b : array_like\n" + " Positive, real-valued parameters\n" + "y : array_like\n" + " Real-valued input\n" + "out : ndarray, optional\n" + " Optional output array for function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Value of the inverse of the regularized incomplete beta function\n" + "\n" + "See Also\n" + "--------\n" + "betainc : regularized incomplete beta function\n" + "gamma : gamma function\n" + "\n" + "Notes\n" + "-----\n" + "This function wraps the ``ibeta_inv`` routine from the\n" + "Boost Math C++ library [2]_.\n" + "\n" + "References\n" + "----------\n" + ".. [1] NIST Digital Library of Mathematical Functions\n" + " https://dlmf.nist.gov/8.17\n" + ".. [2] The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import scipy.special as sc\n" + "\n" + "This function is the inverse of `betainc` for fixed\n" + "values of :math:`a` and :math:`b`.\n" + "\n" + ">>> a, b = 1.2, 3.1\n" + ">>> y = sc.betainc(a, b, 0.2)\n" + ">>> sc.betaincinv(a, b, y)\n" + "0.2\n" + ">>>\n" + ">>> a, b = 7.5, 0.4\n" + ">>> x = sc.betaincinv(a, b, 0.5)\n" + ">>> sc.betainc(a, b, x)\n" + "0.5") +ufunc_betaincinv_loops[0] = loop_f_fff__As_fff_f +ufunc_betaincinv_loops[1] = loop_d_ddd__As_ddd_d +ufunc_betaincinv_types[0] = NPY_FLOAT +ufunc_betaincinv_types[1] = NPY_FLOAT +ufunc_betaincinv_types[2] = NPY_FLOAT +ufunc_betaincinv_types[3] = NPY_FLOAT +ufunc_betaincinv_types[4] = NPY_DOUBLE +ufunc_betaincinv_types[5] = NPY_DOUBLE +ufunc_betaincinv_types[6] = NPY_DOUBLE +ufunc_betaincinv_types[7] = NPY_DOUBLE +ufunc_betaincinv_ptr[2*0] = scipy.special._ufuncs_cxx._export_ibeta_inv_float +ufunc_betaincinv_ptr[2*0+1] = ("betaincinv") +ufunc_betaincinv_ptr[2*1] = scipy.special._ufuncs_cxx._export_ibeta_inv_double +ufunc_betaincinv_ptr[2*1+1] = ("betaincinv") +ufunc_betaincinv_data[0] = &ufunc_betaincinv_ptr[2*0] +ufunc_betaincinv_data[1] = &ufunc_betaincinv_ptr[2*1] +betaincinv = np.PyUFunc_FromFuncAndData(ufunc_betaincinv_loops, ufunc_betaincinv_data, ufunc_betaincinv_types, 2, 3, 1, 0, "betaincinv", ufunc_betaincinv_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_boxcox_loops[2] +cdef void *ufunc_boxcox_ptr[4] +cdef void *ufunc_boxcox_data[2] +cdef char ufunc_boxcox_types[6] +cdef char *ufunc_boxcox_doc = ( + "boxcox(x, lmbda, out=None)\n" + "\n" + "Compute the Box-Cox transformation.\n" + "\n" + "The Box-Cox transformation is::\n" + "\n" + " y = (x**lmbda - 1) / lmbda if lmbda != 0\n" + " log(x) if lmbda == 0\n" + "\n" + "Returns `nan` if ``x < 0``.\n" + "Returns `-inf` if ``x == 0`` and ``lmbda < 0``.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Data to be transformed.\n" + "lmbda : array_like\n" + " Power parameter of the Box-Cox transform.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "y : scalar or ndarray\n" + " Transformed data.\n" + "\n" + "Notes\n" + "-----\n" + "\n" + ".. versionadded:: 0.14.0\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import boxcox\n" + ">>> boxcox([1, 4, 10], 2.5)\n" + "array([ 0. , 12.4 , 126.09110641])\n" + ">>> boxcox(2, [0, 1, 2])\n" + "array([ 0.69314718, 1. , 1.5 ])") +ufunc_boxcox_loops[0] = loop_d_dd__As_ff_f +ufunc_boxcox_loops[1] = loop_d_dd__As_dd_d +ufunc_boxcox_types[0] = NPY_FLOAT +ufunc_boxcox_types[1] = NPY_FLOAT +ufunc_boxcox_types[2] = NPY_FLOAT +ufunc_boxcox_types[3] = NPY_DOUBLE +ufunc_boxcox_types[4] = NPY_DOUBLE +ufunc_boxcox_types[5] = NPY_DOUBLE +ufunc_boxcox_ptr[2*0] = _func_boxcox +ufunc_boxcox_ptr[2*0+1] = ("boxcox") +ufunc_boxcox_ptr[2*1] = _func_boxcox +ufunc_boxcox_ptr[2*1+1] = ("boxcox") +ufunc_boxcox_data[0] = &ufunc_boxcox_ptr[2*0] +ufunc_boxcox_data[1] = &ufunc_boxcox_ptr[2*1] +boxcox = np.PyUFunc_FromFuncAndData(ufunc_boxcox_loops, ufunc_boxcox_data, ufunc_boxcox_types, 2, 2, 1, 0, "boxcox", ufunc_boxcox_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_boxcox1p_loops[2] +cdef void *ufunc_boxcox1p_ptr[4] +cdef void *ufunc_boxcox1p_data[2] +cdef char ufunc_boxcox1p_types[6] +cdef char *ufunc_boxcox1p_doc = ( + "boxcox1p(x, lmbda, out=None)\n" + "\n" + "Compute the Box-Cox transformation of 1 + `x`.\n" + "\n" + "The Box-Cox transformation computed by `boxcox1p` is::\n" + "\n" + " y = ((1+x)**lmbda - 1) / lmbda if lmbda != 0\n" + " log(1+x) if lmbda == 0\n" + "\n" + "Returns `nan` if ``x < -1``.\n" + "Returns `-inf` if ``x == -1`` and ``lmbda < 0``.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Data to be transformed.\n" + "lmbda : array_like\n" + " Power parameter of the Box-Cox transform.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "y : scalar or ndarray\n" + " Transformed data.\n" + "\n" + "Notes\n" + "-----\n" + "\n" + ".. versionadded:: 0.14.0\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import boxcox1p\n" + ">>> boxcox1p(1e-4, [0, 0.5, 1])\n" + "array([ 9.99950003e-05, 9.99975001e-05, 1.00000000e-04])\n" + ">>> boxcox1p([0.01, 0.1], 0.25)\n" + "array([ 0.00996272, 0.09645476])") +ufunc_boxcox1p_loops[0] = loop_d_dd__As_ff_f +ufunc_boxcox1p_loops[1] = loop_d_dd__As_dd_d +ufunc_boxcox1p_types[0] = NPY_FLOAT +ufunc_boxcox1p_types[1] = NPY_FLOAT +ufunc_boxcox1p_types[2] = NPY_FLOAT +ufunc_boxcox1p_types[3] = NPY_DOUBLE +ufunc_boxcox1p_types[4] = NPY_DOUBLE +ufunc_boxcox1p_types[5] = NPY_DOUBLE +ufunc_boxcox1p_ptr[2*0] = _func_boxcox1p +ufunc_boxcox1p_ptr[2*0+1] = ("boxcox1p") +ufunc_boxcox1p_ptr[2*1] = _func_boxcox1p +ufunc_boxcox1p_ptr[2*1+1] = ("boxcox1p") +ufunc_boxcox1p_data[0] = &ufunc_boxcox1p_ptr[2*0] +ufunc_boxcox1p_data[1] = &ufunc_boxcox1p_ptr[2*1] +boxcox1p = np.PyUFunc_FromFuncAndData(ufunc_boxcox1p_loops, ufunc_boxcox1p_data, ufunc_boxcox1p_types, 2, 2, 1, 0, "boxcox1p", ufunc_boxcox1p_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_btdtria_loops[2] +cdef void *ufunc_btdtria_ptr[4] +cdef void *ufunc_btdtria_data[2] +cdef char ufunc_btdtria_types[8] +cdef char *ufunc_btdtria_doc = ( + "btdtria(p, b, x, out=None)\n" + "\n" + "Inverse of `betainc` with respect to `a`.\n" + "\n" + "This is the inverse of the beta cumulative distribution function, `betainc`,\n" + "considered as a function of `a`, returning the value of `a` for which\n" + "`betainc(a, b, x) = p`, or\n" + "\n" + ".. math::\n" + " p = \\int_0^x \\frac{\\Gamma(a + b)}{\\Gamma(a)\\Gamma(b)} t^{a-1} (1-t)^{b-1}\\,dt\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Cumulative probability, in [0, 1].\n" + "b : array_like\n" + " Shape parameter (`b` > 0).\n" + "x : array_like\n" + " The quantile, in [0, 1].\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "a : scalar or ndarray\n" + " The value of the shape parameter `a` such that `betainc(a, b, x) = p`.\n" + "\n" + "See Also\n" + "--------\n" + "btdtrib : Inverse of the beta cumulative distribution function, with respect to `b`.\n" + "\n" + "Notes\n" + "-----\n" + "Wrapper for the CDFLIB [1]_ Fortran routine `cdfbet`.\n" + "\n" + "The cumulative distribution function `p` is computed using a routine by\n" + "DiDinato and Morris [2]_. Computation of `a` involves a search for a value\n" + "that produces the desired value of `p`. The search relies on the\n" + "monotonicity of `p` with `a`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Barry Brown, James Lovato, and Kathy Russell,\n" + " CDFLIB: Library of Fortran Routines for Cumulative Distribution\n" + " Functions, Inverses, and Other Parameters.\n" + ".. [2] DiDinato, A. R. and Morris, A. H.,\n" + " Algorithm 708: Significant Digit Computation of the Incomplete Beta\n" + " Function Ratios. ACM Trans. Math. Softw. 18 (1993), 360-373.") +ufunc_btdtria_loops[0] = loop_d_ddd__As_fff_f +ufunc_btdtria_loops[1] = loop_d_ddd__As_ddd_d +ufunc_btdtria_types[0] = NPY_FLOAT +ufunc_btdtria_types[1] = NPY_FLOAT +ufunc_btdtria_types[2] = NPY_FLOAT +ufunc_btdtria_types[3] = NPY_FLOAT +ufunc_btdtria_types[4] = NPY_DOUBLE +ufunc_btdtria_types[5] = NPY_DOUBLE +ufunc_btdtria_types[6] = NPY_DOUBLE +ufunc_btdtria_types[7] = NPY_DOUBLE +ufunc_btdtria_ptr[2*0] = _func_btdtria +ufunc_btdtria_ptr[2*0+1] = ("btdtria") +ufunc_btdtria_ptr[2*1] = _func_btdtria +ufunc_btdtria_ptr[2*1+1] = ("btdtria") +ufunc_btdtria_data[0] = &ufunc_btdtria_ptr[2*0] +ufunc_btdtria_data[1] = &ufunc_btdtria_ptr[2*1] +btdtria = np.PyUFunc_FromFuncAndData(ufunc_btdtria_loops, ufunc_btdtria_data, ufunc_btdtria_types, 2, 3, 1, 0, "btdtria", ufunc_btdtria_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_btdtrib_loops[2] +cdef void *ufunc_btdtrib_ptr[4] +cdef void *ufunc_btdtrib_data[2] +cdef char ufunc_btdtrib_types[8] +cdef char *ufunc_btdtrib_doc = ( + "btdtria(a, p, x, out=None)\n" + "\n" + "Inverse of `betainc` with respect to `b`.\n" + "\n" + "This is the inverse of the beta cumulative distribution function, `betainc`,\n" + "considered as a function of `b`, returning the value of `b` for which\n" + "`betainc(a, b, x) = p`, or\n" + "\n" + ".. math::\n" + " p = \\int_0^x \\frac{\\Gamma(a + b)}{\\Gamma(a)\\Gamma(b)} t^{a-1} (1-t)^{b-1}\\,dt\n" + "\n" + "Parameters\n" + "----------\n" + "a : array_like\n" + " Shape parameter (`a` > 0).\n" + "p : array_like\n" + " Cumulative probability, in [0, 1].\n" + "x : array_like\n" + " The quantile, in [0, 1].\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "b : scalar or ndarray\n" + " The value of the shape parameter `b` such that `betainc(a, b, x) = p`.\n" + "\n" + "See Also\n" + "--------\n" + "btdtria : Inverse of the beta cumulative distribution function, with respect to `a`.\n" + "\n" + "Notes\n" + "-----\n" + "Wrapper for the CDFLIB [1]_ Fortran routine `cdfbet`.\n" + "\n" + "The cumulative distribution function `p` is computed using a routine by\n" + "DiDinato and Morris [2]_. Computation of `b` involves a search for a value\n" + "that produces the desired value of `p`. The search relies on the\n" + "monotonicity of `p` with `b`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Barry Brown, James Lovato, and Kathy Russell,\n" + " CDFLIB: Library of Fortran Routines for Cumulative Distribution\n" + " Functions, Inverses, and Other Parameters.\n" + ".. [2] DiDinato, A. R. and Morris, A. H.,\n" + " Algorithm 708: Significant Digit Computation of the Incomplete Beta\n" + " Function Ratios. ACM Trans. Math. Softw. 18 (1993), 360-373.") +ufunc_btdtrib_loops[0] = loop_d_ddd__As_fff_f +ufunc_btdtrib_loops[1] = loop_d_ddd__As_ddd_d +ufunc_btdtrib_types[0] = NPY_FLOAT +ufunc_btdtrib_types[1] = NPY_FLOAT +ufunc_btdtrib_types[2] = NPY_FLOAT +ufunc_btdtrib_types[3] = NPY_FLOAT +ufunc_btdtrib_types[4] = NPY_DOUBLE +ufunc_btdtrib_types[5] = NPY_DOUBLE +ufunc_btdtrib_types[6] = NPY_DOUBLE +ufunc_btdtrib_types[7] = NPY_DOUBLE +ufunc_btdtrib_ptr[2*0] = _func_btdtrib +ufunc_btdtrib_ptr[2*0+1] = ("btdtrib") +ufunc_btdtrib_ptr[2*1] = _func_btdtrib +ufunc_btdtrib_ptr[2*1+1] = ("btdtrib") +ufunc_btdtrib_data[0] = &ufunc_btdtrib_ptr[2*0] +ufunc_btdtrib_data[1] = &ufunc_btdtrib_ptr[2*1] +btdtrib = np.PyUFunc_FromFuncAndData(ufunc_btdtrib_loops, ufunc_btdtrib_data, ufunc_btdtrib_types, 2, 3, 1, 0, "btdtrib", ufunc_btdtrib_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_chdtr_loops[2] +cdef void *ufunc_chdtr_ptr[4] +cdef void *ufunc_chdtr_data[2] +cdef char ufunc_chdtr_types[6] +cdef char *ufunc_chdtr_doc = ( + "chdtr(v, x, out=None)\n" + "\n" + "Chi square cumulative distribution function.\n" + "\n" + "Returns the area under the left tail (from 0 to `x`) of the Chi\n" + "square probability density function with `v` degrees of freedom:\n" + "\n" + ".. math::\n" + "\n" + " \\frac{1}{2^{v/2} \\Gamma(v/2)} \\int_0^x t^{v/2 - 1} e^{-t/2} dt\n" + "\n" + "Here :math:`\\Gamma` is the Gamma function; see `gamma`. This\n" + "integral can be expressed in terms of the regularized lower\n" + "incomplete gamma function `gammainc` as\n" + "``gammainc(v / 2, x / 2)``. [1]_\n" + "\n" + "Parameters\n" + "----------\n" + "v : array_like\n" + " Degrees of freedom.\n" + "x : array_like\n" + " Upper bound of the integral.\n" + "out : ndarray, optional\n" + " Optional output array for the function results.\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the cumulative distribution function.\n" + "\n" + "See Also\n" + "--------\n" + "chdtrc, chdtri, chdtriv, gammainc\n" + "\n" + "References\n" + "----------\n" + ".. [1] Chi-Square distribution,\n" + " https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "It can be expressed in terms of the regularized lower incomplete\n" + "gamma function.\n" + "\n" + ">>> v = 1\n" + ">>> x = np.arange(4)\n" + ">>> sc.chdtr(v, x)\n" + "array([0. , 0.68268949, 0.84270079, 0.91673548])\n" + ">>> sc.gammainc(v / 2, x / 2)\n" + "array([0. , 0.68268949, 0.84270079, 0.91673548])") +ufunc_chdtr_loops[0] = loop_d_dd__As_ff_f +ufunc_chdtr_loops[1] = loop_d_dd__As_dd_d +ufunc_chdtr_types[0] = NPY_FLOAT +ufunc_chdtr_types[1] = NPY_FLOAT +ufunc_chdtr_types[2] = NPY_FLOAT +ufunc_chdtr_types[3] = NPY_DOUBLE +ufunc_chdtr_types[4] = NPY_DOUBLE +ufunc_chdtr_types[5] = NPY_DOUBLE +ufunc_chdtr_ptr[2*0] = _func_xsf_chdtr +ufunc_chdtr_ptr[2*0+1] = ("chdtr") +ufunc_chdtr_ptr[2*1] = _func_xsf_chdtr +ufunc_chdtr_ptr[2*1+1] = ("chdtr") +ufunc_chdtr_data[0] = &ufunc_chdtr_ptr[2*0] +ufunc_chdtr_data[1] = &ufunc_chdtr_ptr[2*1] +chdtr = np.PyUFunc_FromFuncAndData(ufunc_chdtr_loops, ufunc_chdtr_data, ufunc_chdtr_types, 2, 2, 1, 0, "chdtr", ufunc_chdtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_chdtrc_loops[2] +cdef void *ufunc_chdtrc_ptr[4] +cdef void *ufunc_chdtrc_data[2] +cdef char ufunc_chdtrc_types[6] +cdef char *ufunc_chdtrc_doc = ( + "chdtrc(v, x, out=None)\n" + "\n" + "Chi square survival function.\n" + "\n" + "Returns the area under the right hand tail (from `x` to infinity)\n" + "of the Chi square probability density function with `v` degrees of\n" + "freedom:\n" + "\n" + ".. math::\n" + "\n" + " \\frac{1}{2^{v/2} \\Gamma(v/2)} \\int_x^\\infty t^{v/2 - 1} e^{-t/2} dt\n" + "\n" + "Here :math:`\\Gamma` is the Gamma function; see `gamma`. This\n" + "integral can be expressed in terms of the regularized upper\n" + "incomplete gamma function `gammaincc` as\n" + "``gammaincc(v / 2, x / 2)``. [1]_\n" + "\n" + "Parameters\n" + "----------\n" + "v : array_like\n" + " Degrees of freedom.\n" + "x : array_like\n" + " Lower bound of the integral.\n" + "out : ndarray, optional\n" + " Optional output array for the function results.\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the survival function.\n" + "\n" + "See Also\n" + "--------\n" + "chdtr, chdtri, chdtriv, gammaincc\n" + "\n" + "References\n" + "----------\n" + ".. [1] Chi-Square distribution,\n" + " https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "It can be expressed in terms of the regularized upper incomplete\n" + "gamma function.\n" + "\n" + ">>> v = 1\n" + ">>> x = np.arange(4)\n" + ">>> sc.chdtrc(v, x)\n" + "array([1. , 0.31731051, 0.15729921, 0.08326452])\n" + ">>> sc.gammaincc(v / 2, x / 2)\n" + "array([1. , 0.31731051, 0.15729921, 0.08326452])") +ufunc_chdtrc_loops[0] = loop_d_dd__As_ff_f +ufunc_chdtrc_loops[1] = loop_d_dd__As_dd_d +ufunc_chdtrc_types[0] = NPY_FLOAT +ufunc_chdtrc_types[1] = NPY_FLOAT +ufunc_chdtrc_types[2] = NPY_FLOAT +ufunc_chdtrc_types[3] = NPY_DOUBLE +ufunc_chdtrc_types[4] = NPY_DOUBLE +ufunc_chdtrc_types[5] = NPY_DOUBLE +ufunc_chdtrc_ptr[2*0] = _func_xsf_chdtrc +ufunc_chdtrc_ptr[2*0+1] = ("chdtrc") +ufunc_chdtrc_ptr[2*1] = _func_xsf_chdtrc +ufunc_chdtrc_ptr[2*1+1] = ("chdtrc") +ufunc_chdtrc_data[0] = &ufunc_chdtrc_ptr[2*0] +ufunc_chdtrc_data[1] = &ufunc_chdtrc_ptr[2*1] +chdtrc = np.PyUFunc_FromFuncAndData(ufunc_chdtrc_loops, ufunc_chdtrc_data, ufunc_chdtrc_types, 2, 2, 1, 0, "chdtrc", ufunc_chdtrc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_chdtri_loops[2] +cdef void *ufunc_chdtri_ptr[4] +cdef void *ufunc_chdtri_data[2] +cdef char ufunc_chdtri_types[6] +cdef char *ufunc_chdtri_doc = ( + "chdtri(v, p, out=None)\n" + "\n" + "Inverse to `chdtrc` with respect to `x`.\n" + "\n" + "Returns `x` such that ``chdtrc(v, x) == p``.\n" + "\n" + "Parameters\n" + "----------\n" + "v : array_like\n" + " Degrees of freedom.\n" + "p : array_like\n" + " Probability.\n" + "out : ndarray, optional\n" + " Optional output array for the function results.\n" + "\n" + "Returns\n" + "-------\n" + "x : scalar or ndarray\n" + " Value so that the probability a Chi square random variable\n" + " with `v` degrees of freedom is greater than `x` equals `p`.\n" + "\n" + "See Also\n" + "--------\n" + "chdtrc, chdtr, chdtriv\n" + "\n" + "References\n" + "----------\n" + ".. [1] Chi-Square distribution,\n" + " https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm\n" + "\n" + "Examples\n" + "--------\n" + ">>> import scipy.special as sc\n" + "\n" + "It inverts `chdtrc`.\n" + "\n" + ">>> v, p = 1, 0.3\n" + ">>> sc.chdtrc(v, sc.chdtri(v, p))\n" + "0.3\n" + ">>> x = 1\n" + ">>> sc.chdtri(v, sc.chdtrc(v, x))\n" + "1.0") +ufunc_chdtri_loops[0] = loop_d_dd__As_ff_f +ufunc_chdtri_loops[1] = loop_d_dd__As_dd_d +ufunc_chdtri_types[0] = NPY_FLOAT +ufunc_chdtri_types[1] = NPY_FLOAT +ufunc_chdtri_types[2] = NPY_FLOAT +ufunc_chdtri_types[3] = NPY_DOUBLE +ufunc_chdtri_types[4] = NPY_DOUBLE +ufunc_chdtri_types[5] = NPY_DOUBLE +ufunc_chdtri_ptr[2*0] = _func_xsf_chdtri +ufunc_chdtri_ptr[2*0+1] = ("chdtri") +ufunc_chdtri_ptr[2*1] = _func_xsf_chdtri +ufunc_chdtri_ptr[2*1+1] = ("chdtri") +ufunc_chdtri_data[0] = &ufunc_chdtri_ptr[2*0] +ufunc_chdtri_data[1] = &ufunc_chdtri_ptr[2*1] +chdtri = np.PyUFunc_FromFuncAndData(ufunc_chdtri_loops, ufunc_chdtri_data, ufunc_chdtri_types, 2, 2, 1, 0, "chdtri", ufunc_chdtri_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_chdtriv_loops[2] +cdef void *ufunc_chdtriv_ptr[4] +cdef void *ufunc_chdtriv_data[2] +cdef char ufunc_chdtriv_types[6] +cdef char *ufunc_chdtriv_doc = ( + "chdtriv(p, x, out=None)\n" + "\n" + "Inverse to `chdtr` with respect to `v`.\n" + "\n" + "Returns `v` such that ``chdtr(v, x) == p``.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Probability that the Chi square random variable is less than\n" + " or equal to `x`.\n" + "x : array_like\n" + " Nonnegative input.\n" + "out : ndarray, optional\n" + " Optional output array for the function results.\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Degrees of freedom.\n" + "\n" + "See Also\n" + "--------\n" + "chdtr, chdtrc, chdtri\n" + "\n" + "References\n" + "----------\n" + ".. [1] Chi-Square distribution,\n" + " https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm\n" + "\n" + "Examples\n" + "--------\n" + ">>> import scipy.special as sc\n" + "\n" + "It inverts `chdtr`.\n" + "\n" + ">>> p, x = 0.5, 1\n" + ">>> sc.chdtr(sc.chdtriv(p, x), x)\n" + "0.5000000000202172\n" + ">>> v = 1\n" + ">>> sc.chdtriv(sc.chdtr(v, x), v)\n" + "1.0000000000000013") +ufunc_chdtriv_loops[0] = loop_d_dd__As_ff_f +ufunc_chdtriv_loops[1] = loop_d_dd__As_dd_d +ufunc_chdtriv_types[0] = NPY_FLOAT +ufunc_chdtriv_types[1] = NPY_FLOAT +ufunc_chdtriv_types[2] = NPY_FLOAT +ufunc_chdtriv_types[3] = NPY_DOUBLE +ufunc_chdtriv_types[4] = NPY_DOUBLE +ufunc_chdtriv_types[5] = NPY_DOUBLE +ufunc_chdtriv_ptr[2*0] = _func_chdtriv +ufunc_chdtriv_ptr[2*0+1] = ("chdtriv") +ufunc_chdtriv_ptr[2*1] = _func_chdtriv +ufunc_chdtriv_ptr[2*1+1] = ("chdtriv") +ufunc_chdtriv_data[0] = &ufunc_chdtriv_ptr[2*0] +ufunc_chdtriv_data[1] = &ufunc_chdtriv_ptr[2*1] +chdtriv = np.PyUFunc_FromFuncAndData(ufunc_chdtriv_loops, ufunc_chdtriv_data, ufunc_chdtriv_types, 2, 2, 1, 0, "chdtriv", ufunc_chdtriv_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_chndtr_loops[2] +cdef void *ufunc_chndtr_ptr[4] +cdef void *ufunc_chndtr_data[2] +cdef char ufunc_chndtr_types[8] +cdef char *ufunc_chndtr_doc = ( + "chndtr(x, df, nc, out=None)\n" + "\n" + "Non-central chi square cumulative distribution function\n" + "\n" + "The cumulative distribution function is given by:\n" + "\n" + ".. math::\n" + "\n" + " P(\\chi^{\\prime 2} \\vert \\nu, \\lambda) =\\sum_{j=0}^{\\infty}\n" + " e^{-\\lambda /2}\n" + " \\frac{(\\lambda /2)^j}{j!} P(\\chi^{\\prime 2} \\vert \\nu + 2j),\n" + "\n" + "where :math:`\\nu > 0` is the degrees of freedom (``df``) and\n" + ":math:`\\lambda \\geq 0` is the non-centrality parameter (``nc``).\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Upper bound of the integral; must satisfy ``x >= 0``\n" + "df : array_like\n" + " Degrees of freedom; must satisfy ``df > 0``\n" + "nc : array_like\n" + " Non-centrality parameter; must satisfy ``nc >= 0``\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "x : scalar or ndarray\n" + " Value of the non-central chi square cumulative distribution function.\n" + "\n" + "See Also\n" + "--------\n" + "chndtrix, chndtridf, chndtrinc") +ufunc_chndtr_loops[0] = loop_d_ddd__As_fff_f +ufunc_chndtr_loops[1] = loop_d_ddd__As_ddd_d +ufunc_chndtr_types[0] = NPY_FLOAT +ufunc_chndtr_types[1] = NPY_FLOAT +ufunc_chndtr_types[2] = NPY_FLOAT +ufunc_chndtr_types[3] = NPY_FLOAT +ufunc_chndtr_types[4] = NPY_DOUBLE +ufunc_chndtr_types[5] = NPY_DOUBLE +ufunc_chndtr_types[6] = NPY_DOUBLE +ufunc_chndtr_types[7] = NPY_DOUBLE +ufunc_chndtr_ptr[2*0] = _func_chndtr +ufunc_chndtr_ptr[2*0+1] = ("chndtr") +ufunc_chndtr_ptr[2*1] = _func_chndtr +ufunc_chndtr_ptr[2*1+1] = ("chndtr") +ufunc_chndtr_data[0] = &ufunc_chndtr_ptr[2*0] +ufunc_chndtr_data[1] = &ufunc_chndtr_ptr[2*1] +chndtr = np.PyUFunc_FromFuncAndData(ufunc_chndtr_loops, ufunc_chndtr_data, ufunc_chndtr_types, 2, 3, 1, 0, "chndtr", ufunc_chndtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_chndtridf_loops[2] +cdef void *ufunc_chndtridf_ptr[4] +cdef void *ufunc_chndtridf_data[2] +cdef char ufunc_chndtridf_types[8] +cdef char *ufunc_chndtridf_doc = ( + "chndtridf(x, p, nc, out=None)\n" + "\n" + "Inverse to `chndtr` vs `df`\n" + "\n" + "Calculated using a search to find a value for `df` that produces the\n" + "desired value of `p`.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Upper bound of the integral; must satisfy ``x >= 0``\n" + "p : array_like\n" + " Probability; must satisfy ``0 <= p < 1``\n" + "nc : array_like\n" + " Non-centrality parameter; must satisfy ``nc >= 0``\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "df : scalar or ndarray\n" + " Degrees of freedom\n" + "\n" + "See Also\n" + "--------\n" + "chndtr, chndtrix, chndtrinc") +ufunc_chndtridf_loops[0] = loop_d_ddd__As_fff_f +ufunc_chndtridf_loops[1] = loop_d_ddd__As_ddd_d +ufunc_chndtridf_types[0] = NPY_FLOAT +ufunc_chndtridf_types[1] = NPY_FLOAT +ufunc_chndtridf_types[2] = NPY_FLOAT +ufunc_chndtridf_types[3] = NPY_FLOAT +ufunc_chndtridf_types[4] = NPY_DOUBLE +ufunc_chndtridf_types[5] = NPY_DOUBLE +ufunc_chndtridf_types[6] = NPY_DOUBLE +ufunc_chndtridf_types[7] = NPY_DOUBLE +ufunc_chndtridf_ptr[2*0] = _func_chndtridf +ufunc_chndtridf_ptr[2*0+1] = ("chndtridf") +ufunc_chndtridf_ptr[2*1] = _func_chndtridf +ufunc_chndtridf_ptr[2*1+1] = ("chndtridf") +ufunc_chndtridf_data[0] = &ufunc_chndtridf_ptr[2*0] +ufunc_chndtridf_data[1] = &ufunc_chndtridf_ptr[2*1] +chndtridf = np.PyUFunc_FromFuncAndData(ufunc_chndtridf_loops, ufunc_chndtridf_data, ufunc_chndtridf_types, 2, 3, 1, 0, "chndtridf", ufunc_chndtridf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_chndtrinc_loops[2] +cdef void *ufunc_chndtrinc_ptr[4] +cdef void *ufunc_chndtrinc_data[2] +cdef char ufunc_chndtrinc_types[8] +cdef char *ufunc_chndtrinc_doc = ( + "chndtrinc(x, df, p, out=None)\n" + "\n" + "Inverse to `chndtr` vs `nc`\n" + "\n" + "Calculated using a search to find a value for `df` that produces the\n" + "desired value of `p`.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Upper bound of the integral; must satisfy ``x >= 0``\n" + "df : array_like\n" + " Degrees of freedom; must satisfy ``df > 0``\n" + "p : array_like\n" + " Probability; must satisfy ``0 <= p < 1``\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "nc : scalar or ndarray\n" + " Non-centrality\n" + "\n" + "See Also\n" + "--------\n" + "chndtr, chndtrix, chndtrinc") +ufunc_chndtrinc_loops[0] = loop_d_ddd__As_fff_f +ufunc_chndtrinc_loops[1] = loop_d_ddd__As_ddd_d +ufunc_chndtrinc_types[0] = NPY_FLOAT +ufunc_chndtrinc_types[1] = NPY_FLOAT +ufunc_chndtrinc_types[2] = NPY_FLOAT +ufunc_chndtrinc_types[3] = NPY_FLOAT +ufunc_chndtrinc_types[4] = NPY_DOUBLE +ufunc_chndtrinc_types[5] = NPY_DOUBLE +ufunc_chndtrinc_types[6] = NPY_DOUBLE +ufunc_chndtrinc_types[7] = NPY_DOUBLE +ufunc_chndtrinc_ptr[2*0] = _func_chndtrinc +ufunc_chndtrinc_ptr[2*0+1] = ("chndtrinc") +ufunc_chndtrinc_ptr[2*1] = _func_chndtrinc +ufunc_chndtrinc_ptr[2*1+1] = ("chndtrinc") +ufunc_chndtrinc_data[0] = &ufunc_chndtrinc_ptr[2*0] +ufunc_chndtrinc_data[1] = &ufunc_chndtrinc_ptr[2*1] +chndtrinc = np.PyUFunc_FromFuncAndData(ufunc_chndtrinc_loops, ufunc_chndtrinc_data, ufunc_chndtrinc_types, 2, 3, 1, 0, "chndtrinc", ufunc_chndtrinc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_chndtrix_loops[2] +cdef void *ufunc_chndtrix_ptr[4] +cdef void *ufunc_chndtrix_data[2] +cdef char ufunc_chndtrix_types[8] +cdef char *ufunc_chndtrix_doc = ( + "chndtrix(p, df, nc, out=None)\n" + "\n" + "Inverse to `chndtr` vs `x`\n" + "\n" + "Calculated using a search to find a value for `x` that produces the\n" + "desired value of `p`.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Probability; must satisfy ``0 <= p < 1``\n" + "df : array_like\n" + " Degrees of freedom; must satisfy ``df > 0``\n" + "nc : array_like\n" + " Non-centrality parameter; must satisfy ``nc >= 0``\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "x : scalar or ndarray\n" + " Value so that the probability a non-central Chi square random variable\n" + " with `df` degrees of freedom and non-centrality, `nc`, is greater than\n" + " `x` equals `p`.\n" + "\n" + "See Also\n" + "--------\n" + "chndtr, chndtridf, chndtrinc") +ufunc_chndtrix_loops[0] = loop_d_ddd__As_fff_f +ufunc_chndtrix_loops[1] = loop_d_ddd__As_ddd_d +ufunc_chndtrix_types[0] = NPY_FLOAT +ufunc_chndtrix_types[1] = NPY_FLOAT +ufunc_chndtrix_types[2] = NPY_FLOAT +ufunc_chndtrix_types[3] = NPY_FLOAT +ufunc_chndtrix_types[4] = NPY_DOUBLE +ufunc_chndtrix_types[5] = NPY_DOUBLE +ufunc_chndtrix_types[6] = NPY_DOUBLE +ufunc_chndtrix_types[7] = NPY_DOUBLE +ufunc_chndtrix_ptr[2*0] = _func_chndtrix +ufunc_chndtrix_ptr[2*0+1] = ("chndtrix") +ufunc_chndtrix_ptr[2*1] = _func_chndtrix +ufunc_chndtrix_ptr[2*1+1] = ("chndtrix") +ufunc_chndtrix_data[0] = &ufunc_chndtrix_ptr[2*0] +ufunc_chndtrix_data[1] = &ufunc_chndtrix_ptr[2*1] +chndtrix = np.PyUFunc_FromFuncAndData(ufunc_chndtrix_loops, ufunc_chndtrix_data, ufunc_chndtrix_types, 2, 3, 1, 0, "chndtrix", ufunc_chndtrix_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_dawsn_loops[4] +cdef void *ufunc_dawsn_ptr[8] +cdef void *ufunc_dawsn_data[4] +cdef char ufunc_dawsn_types[8] +cdef char *ufunc_dawsn_doc = ( + "dawsn(x, out=None)\n" + "\n" + "Dawson's integral.\n" + "\n" + "Computes::\n" + "\n" + " exp(-x**2) * integral(exp(t**2), t=0..x).\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Function parameter.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "y : scalar or ndarray\n" + " Value of the integral.\n" + "\n" + "See Also\n" + "--------\n" + "wofz, erf, erfc, erfcx, erfi\n" + "\n" + "References\n" + "----------\n" + ".. [1] Steven G. Johnson, Faddeeva W function implementation.\n" + " http://ab-initio.mit.edu/Faddeeva\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy import special\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> x = np.linspace(-15, 15, num=1000)\n" + ">>> plt.plot(x, special.dawsn(x))\n" + ">>> plt.xlabel('$x$')\n" + ">>> plt.ylabel('$dawsn(x)$')\n" + ">>> plt.show()") +ufunc_dawsn_loops[0] = loop_d_d__As_f_f +ufunc_dawsn_loops[1] = loop_d_d__As_d_d +ufunc_dawsn_loops[2] = loop_D_D__As_F_F +ufunc_dawsn_loops[3] = loop_D_D__As_D_D +ufunc_dawsn_types[0] = NPY_FLOAT +ufunc_dawsn_types[1] = NPY_FLOAT +ufunc_dawsn_types[2] = NPY_DOUBLE +ufunc_dawsn_types[3] = NPY_DOUBLE +ufunc_dawsn_types[4] = NPY_CFLOAT +ufunc_dawsn_types[5] = NPY_CFLOAT +ufunc_dawsn_types[6] = NPY_CDOUBLE +ufunc_dawsn_types[7] = NPY_CDOUBLE +ufunc_dawsn_ptr[2*0] = scipy.special._ufuncs_cxx._export_faddeeva_dawsn +ufunc_dawsn_ptr[2*0+1] = ("dawsn") +ufunc_dawsn_ptr[2*1] = scipy.special._ufuncs_cxx._export_faddeeva_dawsn +ufunc_dawsn_ptr[2*1+1] = ("dawsn") +ufunc_dawsn_ptr[2*2] = scipy.special._ufuncs_cxx._export_faddeeva_dawsn_complex +ufunc_dawsn_ptr[2*2+1] = ("dawsn") +ufunc_dawsn_ptr[2*3] = scipy.special._ufuncs_cxx._export_faddeeva_dawsn_complex +ufunc_dawsn_ptr[2*3+1] = ("dawsn") +ufunc_dawsn_data[0] = &ufunc_dawsn_ptr[2*0] +ufunc_dawsn_data[1] = &ufunc_dawsn_ptr[2*1] +ufunc_dawsn_data[2] = &ufunc_dawsn_ptr[2*2] +ufunc_dawsn_data[3] = &ufunc_dawsn_ptr[2*3] +dawsn = np.PyUFunc_FromFuncAndData(ufunc_dawsn_loops, ufunc_dawsn_data, ufunc_dawsn_types, 4, 1, 1, 0, "dawsn", ufunc_dawsn_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_elliprc_loops[4] +cdef void *ufunc_elliprc_ptr[8] +cdef void *ufunc_elliprc_data[4] +cdef char ufunc_elliprc_types[12] +cdef char *ufunc_elliprc_doc = ( + "elliprc(x, y, out=None)\n" + "\n" + "Degenerate symmetric elliptic integral.\n" + "\n" + "The function RC is defined as [1]_\n" + "\n" + ".. math::\n" + "\n" + " R_{\\mathrm{C}}(x, y) =\n" + " \\frac{1}{2} \\int_0^{+\\infty} (t + x)^{-1/2} (t + y)^{-1} dt\n" + " = R_{\\mathrm{F}}(x, y, y)\n" + "\n" + "Parameters\n" + "----------\n" + "x, y : array_like\n" + " Real or complex input parameters. `x` can be any number in the\n" + " complex plane cut along the negative real axis. `y` must be non-zero.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "R : scalar or ndarray\n" + " Value of the integral. If `y` is real and negative, the Cauchy\n" + " principal value is returned. If both of `x` and `y` are real, the\n" + " return value is real. Otherwise, the return value is complex.\n" + "\n" + "See Also\n" + "--------\n" + "elliprf : Completely-symmetric elliptic integral of the first kind.\n" + "elliprd : Symmetric elliptic integral of the second kind.\n" + "elliprg : Completely-symmetric elliptic integral of the second kind.\n" + "elliprj : Symmetric elliptic integral of the third kind.\n" + "\n" + "Notes\n" + "-----\n" + "RC is a degenerate case of the symmetric integral RF: ``elliprc(x, y) ==\n" + "elliprf(x, y, y)``. It is an elementary function rather than an elliptic\n" + "integral.\n" + "\n" + "The code implements Carlson's algorithm based on the duplication theorems\n" + "and series expansion up to the 7th order. [2]_\n" + "\n" + ".. versionadded:: 1.8.0\n" + "\n" + "References\n" + "----------\n" + ".. [1] B. C. Carlson, ed., Chapter 19 in \"Digital Library of Mathematical\n" + " Functions,\" NIST, US Dept. of Commerce.\n" + " https://dlmf.nist.gov/19.16.E6\n" + ".. [2] B. C. Carlson, \"Numerical computation of real or complex elliptic\n" + " integrals,\" Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.\n" + " https://arxiv.org/abs/math/9409227\n" + " https://doi.org/10.1007/BF02198293\n" + "\n" + "Examples\n" + "--------\n" + "Basic homogeneity property:\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import elliprc\n" + "\n" + ">>> x = 1.2 + 3.4j\n" + ">>> y = 5.\n" + ">>> scale = 0.3 + 0.4j\n" + ">>> elliprc(scale*x, scale*y)\n" + "(0.5484493976710874-0.4169557678995833j)\n" + "\n" + ">>> elliprc(x, y)/np.sqrt(scale)\n" + "(0.5484493976710874-0.41695576789958333j)\n" + "\n" + "When the two arguments coincide, the integral is particularly\n" + "simple:\n" + "\n" + ">>> x = 1.2 + 3.4j\n" + ">>> elliprc(x, x)\n" + "(0.4299173120614631-0.3041729818745595j)\n" + "\n" + ">>> 1/np.sqrt(x)\n" + "(0.4299173120614631-0.30417298187455954j)\n" + "\n" + "Another simple case: the first argument vanishes:\n" + "\n" + ">>> y = 1.2 + 3.4j\n" + ">>> elliprc(0, y)\n" + "(0.6753125346116815-0.47779380263880866j)\n" + "\n" + ">>> np.pi/2/np.sqrt(y)\n" + "(0.6753125346116815-0.4777938026388088j)\n" + "\n" + "When `x` and `y` are both positive, we can express\n" + ":math:`R_C(x,y)` in terms of more elementary functions. For the\n" + "case :math:`0 \\le x < y`,\n" + "\n" + ">>> x = 3.2\n" + ">>> y = 6.\n" + ">>> elliprc(x, y)\n" + "0.44942991498453444\n" + "\n" + ">>> np.arctan(np.sqrt((y-x)/x))/np.sqrt(y-x)\n" + "0.44942991498453433\n" + "\n" + "And for the case :math:`0 \\le y < x`,\n" + "\n" + ">>> x = 6.\n" + ">>> y = 3.2\n" + ">>> elliprc(x,y)\n" + "0.4989837501576147\n" + "\n" + ">>> np.log((np.sqrt(x)+np.sqrt(x-y))/np.sqrt(y))/np.sqrt(x-y)\n" + "0.49898375015761476") +ufunc_elliprc_loops[0] = loop_d_dd__As_ff_f +ufunc_elliprc_loops[1] = loop_d_dd__As_dd_d +ufunc_elliprc_loops[2] = loop_D_DD__As_FF_F +ufunc_elliprc_loops[3] = loop_D_DD__As_DD_D +ufunc_elliprc_types[0] = NPY_FLOAT +ufunc_elliprc_types[1] = NPY_FLOAT +ufunc_elliprc_types[2] = NPY_FLOAT +ufunc_elliprc_types[3] = NPY_DOUBLE +ufunc_elliprc_types[4] = NPY_DOUBLE +ufunc_elliprc_types[5] = NPY_DOUBLE +ufunc_elliprc_types[6] = NPY_CFLOAT +ufunc_elliprc_types[7] = NPY_CFLOAT +ufunc_elliprc_types[8] = NPY_CFLOAT +ufunc_elliprc_types[9] = NPY_CDOUBLE +ufunc_elliprc_types[10] = NPY_CDOUBLE +ufunc_elliprc_types[11] = NPY_CDOUBLE +ufunc_elliprc_ptr[2*0] = scipy.special._ufuncs_cxx._export_fellint_RC +ufunc_elliprc_ptr[2*0+1] = ("elliprc") +ufunc_elliprc_ptr[2*1] = scipy.special._ufuncs_cxx._export_fellint_RC +ufunc_elliprc_ptr[2*1+1] = ("elliprc") +ufunc_elliprc_ptr[2*2] = scipy.special._ufuncs_cxx._export_cellint_RC +ufunc_elliprc_ptr[2*2+1] = ("elliprc") +ufunc_elliprc_ptr[2*3] = scipy.special._ufuncs_cxx._export_cellint_RC +ufunc_elliprc_ptr[2*3+1] = ("elliprc") +ufunc_elliprc_data[0] = &ufunc_elliprc_ptr[2*0] +ufunc_elliprc_data[1] = &ufunc_elliprc_ptr[2*1] +ufunc_elliprc_data[2] = &ufunc_elliprc_ptr[2*2] +ufunc_elliprc_data[3] = &ufunc_elliprc_ptr[2*3] +elliprc = np.PyUFunc_FromFuncAndData(ufunc_elliprc_loops, ufunc_elliprc_data, ufunc_elliprc_types, 4, 2, 1, 0, "elliprc", ufunc_elliprc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_elliprd_loops[4] +cdef void *ufunc_elliprd_ptr[8] +cdef void *ufunc_elliprd_data[4] +cdef char ufunc_elliprd_types[16] +cdef char *ufunc_elliprd_doc = ( + "elliprd(x, y, z, out=None)\n" + "\n" + "Symmetric elliptic integral of the second kind.\n" + "\n" + "The function RD is defined as [1]_\n" + "\n" + ".. math::\n" + "\n" + " R_{\\mathrm{D}}(x, y, z) =\n" + " \\frac{3}{2} \\int_0^{+\\infty} [(t + x) (t + y)]^{-1/2} (t + z)^{-3/2}\n" + " dt\n" + "\n" + "Parameters\n" + "----------\n" + "x, y, z : array_like\n" + " Real or complex input parameters. `x` or `y` can be any number in the\n" + " complex plane cut along the negative real axis, but at most one of them\n" + " can be zero, while `z` must be non-zero.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "R : scalar or ndarray\n" + " Value of the integral. If all of `x`, `y`, and `z` are real, the\n" + " return value is real. Otherwise, the return value is complex.\n" + "\n" + "See Also\n" + "--------\n" + "elliprc : Degenerate symmetric elliptic integral.\n" + "elliprf : Completely-symmetric elliptic integral of the first kind.\n" + "elliprg : Completely-symmetric elliptic integral of the second kind.\n" + "elliprj : Symmetric elliptic integral of the third kind.\n" + "\n" + "Notes\n" + "-----\n" + "RD is a degenerate case of the elliptic integral RJ: ``elliprd(x, y, z) ==\n" + "elliprj(x, y, z, z)``.\n" + "\n" + "The code implements Carlson's algorithm based on the duplication theorems\n" + "and series expansion up to the 7th order. [2]_\n" + "\n" + ".. versionadded:: 1.8.0\n" + "\n" + "References\n" + "----------\n" + ".. [1] B. C. Carlson, ed., Chapter 19 in \"Digital Library of Mathematical\n" + " Functions,\" NIST, US Dept. of Commerce.\n" + " https://dlmf.nist.gov/19.16.E5\n" + ".. [2] B. C. Carlson, \"Numerical computation of real or complex elliptic\n" + " integrals,\" Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.\n" + " https://arxiv.org/abs/math/9409227\n" + " https://doi.org/10.1007/BF02198293\n" + "\n" + "Examples\n" + "--------\n" + "Basic homogeneity property:\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import elliprd\n" + "\n" + ">>> x = 1.2 + 3.4j\n" + ">>> y = 5.\n" + ">>> z = 6.\n" + ">>> scale = 0.3 + 0.4j\n" + ">>> elliprd(scale*x, scale*y, scale*z)\n" + "(-0.03703043835680379-0.24500934665683802j)\n" + "\n" + ">>> elliprd(x, y, z)*np.power(scale, -1.5)\n" + "(-0.0370304383568038-0.24500934665683805j)\n" + "\n" + "All three arguments coincide:\n" + "\n" + ">>> x = 1.2 + 3.4j\n" + ">>> elliprd(x, x, x)\n" + "(-0.03986825876151896-0.14051741840449586j)\n" + "\n" + ">>> np.power(x, -1.5)\n" + "(-0.03986825876151894-0.14051741840449583j)\n" + "\n" + "The so-called \"second lemniscate constant\":\n" + "\n" + ">>> elliprd(0, 2, 1)/3\n" + "0.5990701173677961\n" + "\n" + ">>> from scipy.special import gamma\n" + ">>> gamma(0.75)**2/np.sqrt(2*np.pi)\n" + "0.5990701173677959") +ufunc_elliprd_loops[0] = loop_d_ddd__As_fff_f +ufunc_elliprd_loops[1] = loop_d_ddd__As_ddd_d +ufunc_elliprd_loops[2] = loop_D_DDD__As_FFF_F +ufunc_elliprd_loops[3] = loop_D_DDD__As_DDD_D +ufunc_elliprd_types[0] = NPY_FLOAT +ufunc_elliprd_types[1] = NPY_FLOAT +ufunc_elliprd_types[2] = NPY_FLOAT +ufunc_elliprd_types[3] = NPY_FLOAT +ufunc_elliprd_types[4] = NPY_DOUBLE +ufunc_elliprd_types[5] = NPY_DOUBLE +ufunc_elliprd_types[6] = NPY_DOUBLE +ufunc_elliprd_types[7] = NPY_DOUBLE +ufunc_elliprd_types[8] = NPY_CFLOAT +ufunc_elliprd_types[9] = NPY_CFLOAT +ufunc_elliprd_types[10] = NPY_CFLOAT +ufunc_elliprd_types[11] = NPY_CFLOAT +ufunc_elliprd_types[12] = NPY_CDOUBLE +ufunc_elliprd_types[13] = NPY_CDOUBLE +ufunc_elliprd_types[14] = NPY_CDOUBLE +ufunc_elliprd_types[15] = NPY_CDOUBLE +ufunc_elliprd_ptr[2*0] = scipy.special._ufuncs_cxx._export_fellint_RD +ufunc_elliprd_ptr[2*0+1] = ("elliprd") +ufunc_elliprd_ptr[2*1] = scipy.special._ufuncs_cxx._export_fellint_RD +ufunc_elliprd_ptr[2*1+1] = ("elliprd") +ufunc_elliprd_ptr[2*2] = scipy.special._ufuncs_cxx._export_cellint_RD +ufunc_elliprd_ptr[2*2+1] = ("elliprd") +ufunc_elliprd_ptr[2*3] = scipy.special._ufuncs_cxx._export_cellint_RD +ufunc_elliprd_ptr[2*3+1] = ("elliprd") +ufunc_elliprd_data[0] = &ufunc_elliprd_ptr[2*0] +ufunc_elliprd_data[1] = &ufunc_elliprd_ptr[2*1] +ufunc_elliprd_data[2] = &ufunc_elliprd_ptr[2*2] +ufunc_elliprd_data[3] = &ufunc_elliprd_ptr[2*3] +elliprd = np.PyUFunc_FromFuncAndData(ufunc_elliprd_loops, ufunc_elliprd_data, ufunc_elliprd_types, 4, 3, 1, 0, "elliprd", ufunc_elliprd_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_elliprf_loops[4] +cdef void *ufunc_elliprf_ptr[8] +cdef void *ufunc_elliprf_data[4] +cdef char ufunc_elliprf_types[16] +cdef char *ufunc_elliprf_doc = ( + "elliprf(x, y, z, out=None)\n" + "\n" + "Completely-symmetric elliptic integral of the first kind.\n" + "\n" + "The function RF is defined as [1]_\n" + "\n" + ".. math::\n" + "\n" + " R_{\\mathrm{F}}(x, y, z) =\n" + " \\frac{1}{2} \\int_0^{+\\infty} [(t + x) (t + y) (t + z)]^{-1/2} dt\n" + "\n" + "Parameters\n" + "----------\n" + "x, y, z : array_like\n" + " Real or complex input parameters. `x`, `y`, or `z` can be any number in\n" + " the complex plane cut along the negative real axis, but at most one of\n" + " them can be zero.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "R : scalar or ndarray\n" + " Value of the integral. If all of `x`, `y`, and `z` are real, the return\n" + " value is real. Otherwise, the return value is complex.\n" + "\n" + "See Also\n" + "--------\n" + "elliprc : Degenerate symmetric integral.\n" + "elliprd : Symmetric elliptic integral of the second kind.\n" + "elliprg : Completely-symmetric elliptic integral of the second kind.\n" + "elliprj : Symmetric elliptic integral of the third kind.\n" + "\n" + "Notes\n" + "-----\n" + "The code implements Carlson's algorithm based on the duplication theorems\n" + "and series expansion up to the 7th order (cf.:\n" + "https://dlmf.nist.gov/19.36.i) and the AGM algorithm for the complete\n" + "integral. [2]_\n" + "\n" + ".. versionadded:: 1.8.0\n" + "\n" + "References\n" + "----------\n" + ".. [1] B. C. Carlson, ed., Chapter 19 in \"Digital Library of Mathematical\n" + " Functions,\" NIST, US Dept. of Commerce.\n" + " https://dlmf.nist.gov/19.16.E1\n" + ".. [2] B. C. Carlson, \"Numerical computation of real or complex elliptic\n" + " integrals,\" Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.\n" + " https://arxiv.org/abs/math/9409227\n" + " https://doi.org/10.1007/BF02198293\n" + "\n" + "Examples\n" + "--------\n" + "Basic homogeneity property:\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import elliprf\n" + "\n" + ">>> x = 1.2 + 3.4j\n" + ">>> y = 5.\n" + ">>> z = 6.\n" + ">>> scale = 0.3 + 0.4j\n" + ">>> elliprf(scale*x, scale*y, scale*z)\n" + "(0.5328051227278146-0.4008623567957094j)\n" + "\n" + ">>> elliprf(x, y, z)/np.sqrt(scale)\n" + "(0.5328051227278147-0.4008623567957095j)\n" + "\n" + "All three arguments coincide:\n" + "\n" + ">>> x = 1.2 + 3.4j\n" + ">>> elliprf(x, x, x)\n" + "(0.42991731206146316-0.30417298187455954j)\n" + "\n" + ">>> 1/np.sqrt(x)\n" + "(0.4299173120614631-0.30417298187455954j)\n" + "\n" + "The so-called \"first lemniscate constant\":\n" + "\n" + ">>> elliprf(0, 1, 2)\n" + "1.3110287771460598\n" + "\n" + ">>> from scipy.special import gamma\n" + ">>> gamma(0.25)**2/(4*np.sqrt(2*np.pi))\n" + "1.3110287771460598") +ufunc_elliprf_loops[0] = loop_d_ddd__As_fff_f +ufunc_elliprf_loops[1] = loop_d_ddd__As_ddd_d +ufunc_elliprf_loops[2] = loop_D_DDD__As_FFF_F +ufunc_elliprf_loops[3] = loop_D_DDD__As_DDD_D +ufunc_elliprf_types[0] = NPY_FLOAT +ufunc_elliprf_types[1] = NPY_FLOAT +ufunc_elliprf_types[2] = NPY_FLOAT +ufunc_elliprf_types[3] = NPY_FLOAT +ufunc_elliprf_types[4] = NPY_DOUBLE +ufunc_elliprf_types[5] = NPY_DOUBLE +ufunc_elliprf_types[6] = NPY_DOUBLE +ufunc_elliprf_types[7] = NPY_DOUBLE +ufunc_elliprf_types[8] = NPY_CFLOAT +ufunc_elliprf_types[9] = NPY_CFLOAT +ufunc_elliprf_types[10] = NPY_CFLOAT +ufunc_elliprf_types[11] = NPY_CFLOAT +ufunc_elliprf_types[12] = NPY_CDOUBLE +ufunc_elliprf_types[13] = NPY_CDOUBLE +ufunc_elliprf_types[14] = NPY_CDOUBLE +ufunc_elliprf_types[15] = NPY_CDOUBLE +ufunc_elliprf_ptr[2*0] = scipy.special._ufuncs_cxx._export_fellint_RF +ufunc_elliprf_ptr[2*0+1] = ("elliprf") +ufunc_elliprf_ptr[2*1] = scipy.special._ufuncs_cxx._export_fellint_RF +ufunc_elliprf_ptr[2*1+1] = ("elliprf") +ufunc_elliprf_ptr[2*2] = scipy.special._ufuncs_cxx._export_cellint_RF +ufunc_elliprf_ptr[2*2+1] = ("elliprf") +ufunc_elliprf_ptr[2*3] = scipy.special._ufuncs_cxx._export_cellint_RF +ufunc_elliprf_ptr[2*3+1] = ("elliprf") +ufunc_elliprf_data[0] = &ufunc_elliprf_ptr[2*0] +ufunc_elliprf_data[1] = &ufunc_elliprf_ptr[2*1] +ufunc_elliprf_data[2] = &ufunc_elliprf_ptr[2*2] +ufunc_elliprf_data[3] = &ufunc_elliprf_ptr[2*3] +elliprf = np.PyUFunc_FromFuncAndData(ufunc_elliprf_loops, ufunc_elliprf_data, ufunc_elliprf_types, 4, 3, 1, 0, "elliprf", ufunc_elliprf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_elliprg_loops[4] +cdef void *ufunc_elliprg_ptr[8] +cdef void *ufunc_elliprg_data[4] +cdef char ufunc_elliprg_types[16] +cdef char *ufunc_elliprg_doc = ( + "elliprg(x, y, z, out=None)\n" + "\n" + "Completely-symmetric elliptic integral of the second kind.\n" + "\n" + "The function RG is defined as [1]_\n" + "\n" + ".. math::\n" + "\n" + " R_{\\mathrm{G}}(x, y, z) =\n" + " \\frac{1}{4} \\int_0^{+\\infty} [(t + x) (t + y) (t + z)]^{-1/2}\n" + " \\left(\\frac{x}{t + x} + \\frac{y}{t + y} + \\frac{z}{t + z}\\right) t\n" + " dt\n" + "\n" + "Parameters\n" + "----------\n" + "x, y, z : array_like\n" + " Real or complex input parameters. `x`, `y`, or `z` can be any number in\n" + " the complex plane cut along the negative real axis.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "R : scalar or ndarray\n" + " Value of the integral. If all of `x`, `y`, and `z` are real, the return\n" + " value is real. Otherwise, the return value is complex.\n" + "\n" + "See Also\n" + "--------\n" + "elliprc : Degenerate symmetric integral.\n" + "elliprd : Symmetric elliptic integral of the second kind.\n" + "elliprf : Completely-symmetric elliptic integral of the first kind.\n" + "elliprj : Symmetric elliptic integral of the third kind.\n" + "\n" + "Notes\n" + "-----\n" + "The implementation uses the relation [1]_\n" + "\n" + ".. math::\n" + "\n" + " 2 R_{\\mathrm{G}}(x, y, z) =\n" + " z R_{\\mathrm{F}}(x, y, z) -\n" + " \\frac{1}{3} (x - z) (y - z) R_{\\mathrm{D}}(x, y, z) +\n" + " \\sqrt{\\frac{x y}{z}}\n" + "\n" + "and the symmetry of `x`, `y`, `z` when at least one non-zero parameter can\n" + "be chosen as the pivot. When one of the arguments is close to zero, the AGM\n" + "method is applied instead. Other special cases are computed following Ref.\n" + "[2]_\n" + "\n" + ".. versionadded:: 1.8.0\n" + "\n" + "References\n" + "----------\n" + ".. [1] B. C. Carlson, \"Numerical computation of real or complex elliptic\n" + " integrals,\" Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.\n" + " https://arxiv.org/abs/math/9409227\n" + " https://doi.org/10.1007/BF02198293\n" + ".. [2] B. C. Carlson, ed., Chapter 19 in \"Digital Library of Mathematical\n" + " Functions,\" NIST, US Dept. of Commerce.\n" + " https://dlmf.nist.gov/19.16.E1\n" + " https://dlmf.nist.gov/19.20.ii\n" + "\n" + "Examples\n" + "--------\n" + "Basic homogeneity property:\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import elliprg\n" + "\n" + ">>> x = 1.2 + 3.4j\n" + ">>> y = 5.\n" + ">>> z = 6.\n" + ">>> scale = 0.3 + 0.4j\n" + ">>> elliprg(scale*x, scale*y, scale*z)\n" + "(1.195936862005246+0.8470988320464167j)\n" + "\n" + ">>> elliprg(x, y, z)*np.sqrt(scale)\n" + "(1.195936862005246+0.8470988320464165j)\n" + "\n" + "Simplifications:\n" + "\n" + ">>> elliprg(0, y, y)\n" + "1.756203682760182\n" + "\n" + ">>> 0.25*np.pi*np.sqrt(y)\n" + "1.7562036827601817\n" + "\n" + ">>> elliprg(0, 0, z)\n" + "1.224744871391589\n" + "\n" + ">>> 0.5*np.sqrt(z)\n" + "1.224744871391589\n" + "\n" + "The surface area of a triaxial ellipsoid with semiaxes ``a``, ``b``, and\n" + "``c`` is given by\n" + "\n" + ".. math::\n" + "\n" + " S = 4 \\pi a b c R_{\\mathrm{G}}(1 / a^2, 1 / b^2, 1 / c^2).\n" + "\n" + ">>> def ellipsoid_area(a, b, c):\n" + "... r = 4.0 * np.pi * a * b * c\n" + "... return r * elliprg(1.0 / (a * a), 1.0 / (b * b), 1.0 / (c * c))\n" + ">>> print(ellipsoid_area(1, 3, 5))\n" + "108.62688289491807") +ufunc_elliprg_loops[0] = loop_d_ddd__As_fff_f +ufunc_elliprg_loops[1] = loop_d_ddd__As_ddd_d +ufunc_elliprg_loops[2] = loop_D_DDD__As_FFF_F +ufunc_elliprg_loops[3] = loop_D_DDD__As_DDD_D +ufunc_elliprg_types[0] = NPY_FLOAT +ufunc_elliprg_types[1] = NPY_FLOAT +ufunc_elliprg_types[2] = NPY_FLOAT +ufunc_elliprg_types[3] = NPY_FLOAT +ufunc_elliprg_types[4] = NPY_DOUBLE +ufunc_elliprg_types[5] = NPY_DOUBLE +ufunc_elliprg_types[6] = NPY_DOUBLE +ufunc_elliprg_types[7] = NPY_DOUBLE +ufunc_elliprg_types[8] = NPY_CFLOAT +ufunc_elliprg_types[9] = NPY_CFLOAT +ufunc_elliprg_types[10] = NPY_CFLOAT +ufunc_elliprg_types[11] = NPY_CFLOAT +ufunc_elliprg_types[12] = NPY_CDOUBLE +ufunc_elliprg_types[13] = NPY_CDOUBLE +ufunc_elliprg_types[14] = NPY_CDOUBLE +ufunc_elliprg_types[15] = NPY_CDOUBLE +ufunc_elliprg_ptr[2*0] = scipy.special._ufuncs_cxx._export_fellint_RG +ufunc_elliprg_ptr[2*0+1] = ("elliprg") +ufunc_elliprg_ptr[2*1] = scipy.special._ufuncs_cxx._export_fellint_RG +ufunc_elliprg_ptr[2*1+1] = ("elliprg") +ufunc_elliprg_ptr[2*2] = scipy.special._ufuncs_cxx._export_cellint_RG +ufunc_elliprg_ptr[2*2+1] = ("elliprg") +ufunc_elliprg_ptr[2*3] = scipy.special._ufuncs_cxx._export_cellint_RG +ufunc_elliprg_ptr[2*3+1] = ("elliprg") +ufunc_elliprg_data[0] = &ufunc_elliprg_ptr[2*0] +ufunc_elliprg_data[1] = &ufunc_elliprg_ptr[2*1] +ufunc_elliprg_data[2] = &ufunc_elliprg_ptr[2*2] +ufunc_elliprg_data[3] = &ufunc_elliprg_ptr[2*3] +elliprg = np.PyUFunc_FromFuncAndData(ufunc_elliprg_loops, ufunc_elliprg_data, ufunc_elliprg_types, 4, 3, 1, 0, "elliprg", ufunc_elliprg_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_elliprj_loops[4] +cdef void *ufunc_elliprj_ptr[8] +cdef void *ufunc_elliprj_data[4] +cdef char ufunc_elliprj_types[20] +cdef char *ufunc_elliprj_doc = ( + "elliprj(x, y, z, p, out=None)\n" + "\n" + "Symmetric elliptic integral of the third kind.\n" + "\n" + "The function RJ is defined as [1]_\n" + "\n" + ".. math::\n" + "\n" + " R_{\\mathrm{J}}(x, y, z, p) =\n" + " \\frac{3}{2} \\int_0^{+\\infty} [(t + x) (t + y) (t + z)]^{-1/2}\n" + " (t + p)^{-1} dt\n" + "\n" + ".. warning::\n" + " This function should be considered experimental when the inputs are\n" + " unbalanced. Check correctness with another independent implementation.\n" + "\n" + "Parameters\n" + "----------\n" + "x, y, z, p : array_like\n" + " Real or complex input parameters. `x`, `y`, or `z` are numbers in\n" + " the complex plane cut along the negative real axis (subject to further\n" + " constraints, see Notes), and at most one of them can be zero. `p` must\n" + " be non-zero.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "R : scalar or ndarray\n" + " Value of the integral. If all of `x`, `y`, `z`, and `p` are real, the\n" + " return value is real. Otherwise, the return value is complex.\n" + "\n" + " If `p` is real and negative, while `x`, `y`, and `z` are real,\n" + " non-negative, and at most one of them is zero, the Cauchy principal\n" + " value is returned. [1]_ [2]_\n" + "\n" + "See Also\n" + "--------\n" + "elliprc : Degenerate symmetric integral.\n" + "elliprd : Symmetric elliptic integral of the second kind.\n" + "elliprf : Completely-symmetric elliptic integral of the first kind.\n" + "elliprg : Completely-symmetric elliptic integral of the second kind.\n" + "\n" + "Notes\n" + "-----\n" + "The code implements Carlson's algorithm based on the duplication theorems\n" + "and series expansion up to the 7th order. [3]_ The algorithm is slightly\n" + "different from its earlier incarnation as it appears in [1]_, in that the\n" + "call to `elliprc` (or ``atan``/``atanh``, see [4]_) is no longer needed in\n" + "the inner loop. Asymptotic approximations are used where arguments differ\n" + "widely in the order of magnitude. [5]_\n" + "\n" + "The input values are subject to certain sufficient but not necessary\n" + "constraints when input arguments are complex. Notably, ``x``, ``y``, and\n" + "``z`` must have non-negative real parts, unless two of them are\n" + "non-negative and complex-conjugates to each other while the other is a real\n" + "non-negative number. [1]_ If the inputs do not satisfy the sufficient\n" + "condition described in Ref. [1]_ they are rejected outright with the output\n" + "set to NaN.\n" + "\n" + "In the case where one of ``x``, ``y``, and ``z`` is equal to ``p``, the\n" + "function ``elliprd`` should be preferred because of its less restrictive\n" + "domain.\n" + "\n" + ".. versionadded:: 1.8.0\n" + "\n" + "References\n" + "----------\n" + ".. [1] B. C. Carlson, \"Numerical computation of real or complex elliptic\n" + " integrals,\" Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.\n" + " https://arxiv.org/abs/math/9409227\n" + " https://doi.org/10.1007/BF02198293\n" + ".. [2] B. C. Carlson, ed., Chapter 19 in \"Digital Library of Mathematical\n" + " Functions,\" NIST, US Dept. of Commerce.\n" + " https://dlmf.nist.gov/19.20.iii\n" + ".. [3] B. C. Carlson, J. FitzSimmons, \"Reduction Theorems for Elliptic\n" + " Integrands with the Square Root of Two Quadratic Factors,\" J.\n" + " Comput. Appl. Math., vol. 118, nos. 1-2, pp. 71-85, 2000.\n" + " https://doi.org/10.1016/S0377-0427(00)00282-X\n" + ".. [4] F. Johansson, \"Numerical Evaluation of Elliptic Functions, Elliptic\n" + " Integrals and Modular Forms,\" in J. Blumlein, C. Schneider, P.\n" + " Paule, eds., \"Elliptic Integrals, Elliptic Functions and Modular\n" + " Forms in Quantum Field Theory,\" pp. 269-293, 2019 (Cham,\n" + " Switzerland: Springer Nature Switzerland)\n" + " https://arxiv.org/abs/1806.06725\n" + " https://doi.org/10.1007/978-3-030-04480-0\n" + ".. [5] B. C. Carlson, J. L. Gustafson, \"Asymptotic Approximations for\n" + " Symmetric Elliptic Integrals,\" SIAM J. Math. Anls., vol. 25, no. 2,\n" + " pp. 288-303, 1994.\n" + " https://arxiv.org/abs/math/9310223\n" + " https://doi.org/10.1137/S0036141092228477\n" + "\n" + "Examples\n" + "--------\n" + "Basic homogeneity property:\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import elliprj\n" + "\n" + ">>> x = 1.2 + 3.4j\n" + ">>> y = 5.\n" + ">>> z = 6.\n" + ">>> p = 7.\n" + ">>> scale = 0.3 - 0.4j\n" + ">>> elliprj(scale*x, scale*y, scale*z, scale*p)\n" + "(0.10834905565679157+0.19694950747103812j)\n" + "\n" + ">>> elliprj(x, y, z, p)*np.power(scale, -1.5)\n" + "(0.10834905565679556+0.19694950747103854j)\n" + "\n" + "Reduction to simpler elliptic integral:\n" + "\n" + ">>> elliprj(x, y, z, z)\n" + "(0.08288462362195129-0.028376809745123258j)\n" + "\n" + ">>> from scipy.special import elliprd\n" + ">>> elliprd(x, y, z)\n" + "(0.08288462362195136-0.028376809745123296j)\n" + "\n" + "All arguments coincide:\n" + "\n" + ">>> elliprj(x, x, x, x)\n" + "(-0.03986825876151896-0.14051741840449586j)\n" + "\n" + ">>> np.power(x, -1.5)\n" + "(-0.03986825876151894-0.14051741840449583j)") +ufunc_elliprj_loops[0] = loop_d_dddd__As_ffff_f +ufunc_elliprj_loops[1] = loop_d_dddd__As_dddd_d +ufunc_elliprj_loops[2] = loop_D_DDDD__As_FFFF_F +ufunc_elliprj_loops[3] = loop_D_DDDD__As_DDDD_D +ufunc_elliprj_types[0] = NPY_FLOAT +ufunc_elliprj_types[1] = NPY_FLOAT +ufunc_elliprj_types[2] = NPY_FLOAT +ufunc_elliprj_types[3] = NPY_FLOAT +ufunc_elliprj_types[4] = NPY_FLOAT +ufunc_elliprj_types[5] = NPY_DOUBLE +ufunc_elliprj_types[6] = NPY_DOUBLE +ufunc_elliprj_types[7] = NPY_DOUBLE +ufunc_elliprj_types[8] = NPY_DOUBLE +ufunc_elliprj_types[9] = NPY_DOUBLE +ufunc_elliprj_types[10] = NPY_CFLOAT +ufunc_elliprj_types[11] = NPY_CFLOAT +ufunc_elliprj_types[12] = NPY_CFLOAT +ufunc_elliprj_types[13] = NPY_CFLOAT +ufunc_elliprj_types[14] = NPY_CFLOAT +ufunc_elliprj_types[15] = NPY_CDOUBLE +ufunc_elliprj_types[16] = NPY_CDOUBLE +ufunc_elliprj_types[17] = NPY_CDOUBLE +ufunc_elliprj_types[18] = NPY_CDOUBLE +ufunc_elliprj_types[19] = NPY_CDOUBLE +ufunc_elliprj_ptr[2*0] = scipy.special._ufuncs_cxx._export_fellint_RJ +ufunc_elliprj_ptr[2*0+1] = ("elliprj") +ufunc_elliprj_ptr[2*1] = scipy.special._ufuncs_cxx._export_fellint_RJ +ufunc_elliprj_ptr[2*1+1] = ("elliprj") +ufunc_elliprj_ptr[2*2] = scipy.special._ufuncs_cxx._export_cellint_RJ +ufunc_elliprj_ptr[2*2+1] = ("elliprj") +ufunc_elliprj_ptr[2*3] = scipy.special._ufuncs_cxx._export_cellint_RJ +ufunc_elliprj_ptr[2*3+1] = ("elliprj") +ufunc_elliprj_data[0] = &ufunc_elliprj_ptr[2*0] +ufunc_elliprj_data[1] = &ufunc_elliprj_ptr[2*1] +ufunc_elliprj_data[2] = &ufunc_elliprj_ptr[2*2] +ufunc_elliprj_data[3] = &ufunc_elliprj_ptr[2*3] +elliprj = np.PyUFunc_FromFuncAndData(ufunc_elliprj_loops, ufunc_elliprj_data, ufunc_elliprj_types, 4, 4, 1, 0, "elliprj", ufunc_elliprj_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_entr_loops[2] +cdef void *ufunc_entr_ptr[4] +cdef void *ufunc_entr_data[2] +cdef char ufunc_entr_types[4] +cdef char *ufunc_entr_doc = ( + "entr(x, out=None)\n" + "\n" + "Elementwise function for computing entropy.\n" + "\n" + ".. math:: \\text{entr}(x) = \\begin{cases} - x \\log(x) & x > 0 \\\\ 0 & x = 0\n" + " \\\\ -\\infty & \\text{otherwise} \\end{cases}\n" + "\n" + "Parameters\n" + "----------\n" + "x : ndarray\n" + " Input array.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "res : scalar or ndarray\n" + " The value of the elementwise entropy function at the given points `x`.\n" + "\n" + "See Also\n" + "--------\n" + "kl_div, rel_entr, scipy.stats.entropy\n" + "\n" + "Notes\n" + "-----\n" + ".. versionadded:: 0.15.0\n" + "\n" + "This function is concave.\n" + "\n" + "The origin of this function is in convex programming; see [1]_.\n" + "Given a probability distribution :math:`p_1, \\ldots, p_n`,\n" + "the definition of entropy in the context of *information theory* is\n" + "\n" + ".. math::\n" + "\n" + " \\sum_{i = 1}^n \\mathrm{entr}(p_i).\n" + "\n" + "To compute the latter quantity, use `scipy.stats.entropy`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*.\n" + " Cambridge University Press, 2004.\n" + " :doi:`https://doi.org/10.1017/CBO9780511804441`") +ufunc_entr_loops[0] = loop_d_d__As_f_f +ufunc_entr_loops[1] = loop_d_d__As_d_d +ufunc_entr_types[0] = NPY_FLOAT +ufunc_entr_types[1] = NPY_FLOAT +ufunc_entr_types[2] = NPY_DOUBLE +ufunc_entr_types[3] = NPY_DOUBLE +ufunc_entr_ptr[2*0] = _func_entr +ufunc_entr_ptr[2*0+1] = ("entr") +ufunc_entr_ptr[2*1] = _func_entr +ufunc_entr_ptr[2*1+1] = ("entr") +ufunc_entr_data[0] = &ufunc_entr_ptr[2*0] +ufunc_entr_data[1] = &ufunc_entr_ptr[2*1] +entr = np.PyUFunc_FromFuncAndData(ufunc_entr_loops, ufunc_entr_data, ufunc_entr_types, 2, 1, 1, 0, "entr", ufunc_entr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_erf_loops[4] +cdef void *ufunc_erf_ptr[8] +cdef void *ufunc_erf_data[4] +cdef char ufunc_erf_types[8] +cdef char *ufunc_erf_doc = ( + "erf(z, out=None)\n" + "\n" + "Returns the error function of complex argument.\n" + "\n" + "It is defined as ``2/sqrt(pi)*integral(exp(-t**2), t=0..z)``.\n" + "\n" + "Parameters\n" + "----------\n" + "x : ndarray\n" + " Input array.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "res : scalar or ndarray\n" + " The values of the error function at the given points `x`.\n" + "\n" + "See Also\n" + "--------\n" + "erfc, erfinv, erfcinv, wofz, erfcx, erfi\n" + "\n" + "Notes\n" + "-----\n" + "The cumulative of the unit normal distribution is given by\n" + "``Phi(z) = 1/2[1 + erf(z/sqrt(2))]``.\n" + "\n" + "References\n" + "----------\n" + ".. [1] https://en.wikipedia.org/wiki/Error_function\n" + ".. [2] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover,\n" + " 1972. http://www.math.sfu.ca/~cbm/aands/page_297.htm\n" + ".. [3] Steven G. Johnson, Faddeeva W function implementation.\n" + " http://ab-initio.mit.edu/Faddeeva\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy import special\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> x = np.linspace(-3, 3)\n" + ">>> plt.plot(x, special.erf(x))\n" + ">>> plt.xlabel('$x$')\n" + ">>> plt.ylabel('$erf(x)$')\n" + ">>> plt.show()") +ufunc_erf_loops[0] = loop_d_d__As_f_f +ufunc_erf_loops[1] = loop_d_d__As_d_d +ufunc_erf_loops[2] = loop_D_D__As_F_F +ufunc_erf_loops[3] = loop_D_D__As_D_D +ufunc_erf_types[0] = NPY_FLOAT +ufunc_erf_types[1] = NPY_FLOAT +ufunc_erf_types[2] = NPY_DOUBLE +ufunc_erf_types[3] = NPY_DOUBLE +ufunc_erf_types[4] = NPY_CFLOAT +ufunc_erf_types[5] = NPY_CFLOAT +ufunc_erf_types[6] = NPY_CDOUBLE +ufunc_erf_types[7] = NPY_CDOUBLE +ufunc_erf_ptr[2*0] = _func_cephes_erf +ufunc_erf_ptr[2*0+1] = ("erf") +ufunc_erf_ptr[2*1] = _func_cephes_erf +ufunc_erf_ptr[2*1+1] = ("erf") +ufunc_erf_ptr[2*2] = scipy.special._ufuncs_cxx._export_faddeeva_erf +ufunc_erf_ptr[2*2+1] = ("erf") +ufunc_erf_ptr[2*3] = scipy.special._ufuncs_cxx._export_faddeeva_erf +ufunc_erf_ptr[2*3+1] = ("erf") +ufunc_erf_data[0] = &ufunc_erf_ptr[2*0] +ufunc_erf_data[1] = &ufunc_erf_ptr[2*1] +ufunc_erf_data[2] = &ufunc_erf_ptr[2*2] +ufunc_erf_data[3] = &ufunc_erf_ptr[2*3] +erf = np.PyUFunc_FromFuncAndData(ufunc_erf_loops, ufunc_erf_data, ufunc_erf_types, 4, 1, 1, 0, "erf", ufunc_erf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_erfc_loops[4] +cdef void *ufunc_erfc_ptr[8] +cdef void *ufunc_erfc_data[4] +cdef char ufunc_erfc_types[8] +cdef char *ufunc_erfc_doc = ( + "erfc(x, out=None)\n" + "\n" + "Complementary error function, ``1 - erf(x)``.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real or complex valued argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the complementary error function\n" + "\n" + "See Also\n" + "--------\n" + "erf, erfi, erfcx, dawsn, wofz\n" + "\n" + "References\n" + "----------\n" + ".. [1] Steven G. Johnson, Faddeeva W function implementation.\n" + " http://ab-initio.mit.edu/Faddeeva\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy import special\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> x = np.linspace(-3, 3)\n" + ">>> plt.plot(x, special.erfc(x))\n" + ">>> plt.xlabel('$x$')\n" + ">>> plt.ylabel('$erfc(x)$')\n" + ">>> plt.show()") +ufunc_erfc_loops[0] = loop_d_d__As_f_f +ufunc_erfc_loops[1] = loop_d_d__As_d_d +ufunc_erfc_loops[2] = loop_D_D__As_F_F +ufunc_erfc_loops[3] = loop_D_D__As_D_D +ufunc_erfc_types[0] = NPY_FLOAT +ufunc_erfc_types[1] = NPY_FLOAT +ufunc_erfc_types[2] = NPY_DOUBLE +ufunc_erfc_types[3] = NPY_DOUBLE +ufunc_erfc_types[4] = NPY_CFLOAT +ufunc_erfc_types[5] = NPY_CFLOAT +ufunc_erfc_types[6] = NPY_CDOUBLE +ufunc_erfc_types[7] = NPY_CDOUBLE +ufunc_erfc_ptr[2*0] = _func_cephes_erfc +ufunc_erfc_ptr[2*0+1] = ("erfc") +ufunc_erfc_ptr[2*1] = _func_cephes_erfc +ufunc_erfc_ptr[2*1+1] = ("erfc") +ufunc_erfc_ptr[2*2] = scipy.special._ufuncs_cxx._export_faddeeva_erfc_complex +ufunc_erfc_ptr[2*2+1] = ("erfc") +ufunc_erfc_ptr[2*3] = scipy.special._ufuncs_cxx._export_faddeeva_erfc_complex +ufunc_erfc_ptr[2*3+1] = ("erfc") +ufunc_erfc_data[0] = &ufunc_erfc_ptr[2*0] +ufunc_erfc_data[1] = &ufunc_erfc_ptr[2*1] +ufunc_erfc_data[2] = &ufunc_erfc_ptr[2*2] +ufunc_erfc_data[3] = &ufunc_erfc_ptr[2*3] +erfc = np.PyUFunc_FromFuncAndData(ufunc_erfc_loops, ufunc_erfc_data, ufunc_erfc_types, 4, 1, 1, 0, "erfc", ufunc_erfc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_erfcinv_loops[2] +cdef void *ufunc_erfcinv_ptr[4] +cdef void *ufunc_erfcinv_data[2] +cdef char ufunc_erfcinv_types[4] +cdef char *ufunc_erfcinv_doc = ( + "erfcinv(y, out=None)\n" + "\n" + "Inverse of the complementary error function.\n" + "\n" + "Computes the inverse of the complementary error function.\n" + "\n" + "In the complex domain, there is no unique complex number w satisfying\n" + "erfc(w)=z. This indicates a true inverse function would be multivalued.\n" + "When the domain restricts to the real, 0 < x < 2, there is a unique real\n" + "number satisfying erfc(erfcinv(x)) = erfcinv(erfc(x)).\n" + "\n" + "It is related to inverse of the error function by erfcinv(1-x) = erfinv(x)\n" + "\n" + "Parameters\n" + "----------\n" + "y : ndarray\n" + " Argument at which to evaluate. Domain: [0, 2]\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "erfcinv : scalar or ndarray\n" + " The inverse of erfc of y, element-wise\n" + "\n" + "See Also\n" + "--------\n" + "erf : Error function of a complex argument\n" + "erfc : Complementary error function, ``1 - erf(x)``\n" + "erfinv : Inverse of the error function\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> from scipy.special import erfcinv\n" + "\n" + ">>> erfcinv(0.5)\n" + "0.4769362762044699\n" + "\n" + ">>> y = np.linspace(0.0, 2.0, num=11)\n" + ">>> erfcinv(y)\n" + "array([ inf, 0.9061938 , 0.59511608, 0.37080716, 0.17914345,\n" + " -0. , -0.17914345, -0.37080716, -0.59511608, -0.9061938 ,\n" + " -inf])\n" + "\n" + "Plot the function:\n" + "\n" + ">>> y = np.linspace(0, 2, 200)\n" + ">>> fig, ax = plt.subplots()\n" + ">>> ax.plot(y, erfcinv(y))\n" + ">>> ax.grid(True)\n" + ">>> ax.set_xlabel('y')\n" + ">>> ax.set_title('erfcinv(y)')\n" + ">>> plt.show()") +ufunc_erfcinv_loops[0] = loop_d_d__As_f_f +ufunc_erfcinv_loops[1] = loop_d_d__As_d_d +ufunc_erfcinv_types[0] = NPY_FLOAT +ufunc_erfcinv_types[1] = NPY_FLOAT +ufunc_erfcinv_types[2] = NPY_DOUBLE +ufunc_erfcinv_types[3] = NPY_DOUBLE +ufunc_erfcinv_ptr[2*0] = _func_cephes_erfcinv +ufunc_erfcinv_ptr[2*0+1] = ("erfcinv") +ufunc_erfcinv_ptr[2*1] = _func_cephes_erfcinv +ufunc_erfcinv_ptr[2*1+1] = ("erfcinv") +ufunc_erfcinv_data[0] = &ufunc_erfcinv_ptr[2*0] +ufunc_erfcinv_data[1] = &ufunc_erfcinv_ptr[2*1] +erfcinv = np.PyUFunc_FromFuncAndData(ufunc_erfcinv_loops, ufunc_erfcinv_data, ufunc_erfcinv_types, 2, 1, 1, 0, "erfcinv", ufunc_erfcinv_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_erfcx_loops[4] +cdef void *ufunc_erfcx_ptr[8] +cdef void *ufunc_erfcx_data[4] +cdef char ufunc_erfcx_types[8] +cdef char *ufunc_erfcx_doc = ( + "erfcx(x, out=None)\n" + "\n" + "Scaled complementary error function, ``exp(x**2) * erfc(x)``.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real or complex valued argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the scaled complementary error function\n" + "\n" + "\n" + "See Also\n" + "--------\n" + "erf, erfc, erfi, dawsn, wofz\n" + "\n" + "Notes\n" + "-----\n" + "\n" + ".. versionadded:: 0.12.0\n" + "\n" + "References\n" + "----------\n" + ".. [1] Steven G. Johnson, Faddeeva W function implementation.\n" + " http://ab-initio.mit.edu/Faddeeva\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy import special\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> x = np.linspace(-3, 3)\n" + ">>> plt.plot(x, special.erfcx(x))\n" + ">>> plt.xlabel('$x$')\n" + ">>> plt.ylabel('$erfcx(x)$')\n" + ">>> plt.show()") +ufunc_erfcx_loops[0] = loop_d_d__As_f_f +ufunc_erfcx_loops[1] = loop_d_d__As_d_d +ufunc_erfcx_loops[2] = loop_D_D__As_F_F +ufunc_erfcx_loops[3] = loop_D_D__As_D_D +ufunc_erfcx_types[0] = NPY_FLOAT +ufunc_erfcx_types[1] = NPY_FLOAT +ufunc_erfcx_types[2] = NPY_DOUBLE +ufunc_erfcx_types[3] = NPY_DOUBLE +ufunc_erfcx_types[4] = NPY_CFLOAT +ufunc_erfcx_types[5] = NPY_CFLOAT +ufunc_erfcx_types[6] = NPY_CDOUBLE +ufunc_erfcx_types[7] = NPY_CDOUBLE +ufunc_erfcx_ptr[2*0] = scipy.special._ufuncs_cxx._export_faddeeva_erfcx +ufunc_erfcx_ptr[2*0+1] = ("erfcx") +ufunc_erfcx_ptr[2*1] = scipy.special._ufuncs_cxx._export_faddeeva_erfcx +ufunc_erfcx_ptr[2*1+1] = ("erfcx") +ufunc_erfcx_ptr[2*2] = scipy.special._ufuncs_cxx._export_faddeeva_erfcx_complex +ufunc_erfcx_ptr[2*2+1] = ("erfcx") +ufunc_erfcx_ptr[2*3] = scipy.special._ufuncs_cxx._export_faddeeva_erfcx_complex +ufunc_erfcx_ptr[2*3+1] = ("erfcx") +ufunc_erfcx_data[0] = &ufunc_erfcx_ptr[2*0] +ufunc_erfcx_data[1] = &ufunc_erfcx_ptr[2*1] +ufunc_erfcx_data[2] = &ufunc_erfcx_ptr[2*2] +ufunc_erfcx_data[3] = &ufunc_erfcx_ptr[2*3] +erfcx = np.PyUFunc_FromFuncAndData(ufunc_erfcx_loops, ufunc_erfcx_data, ufunc_erfcx_types, 4, 1, 1, 0, "erfcx", ufunc_erfcx_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_erfi_loops[4] +cdef void *ufunc_erfi_ptr[8] +cdef void *ufunc_erfi_data[4] +cdef char ufunc_erfi_types[8] +cdef char *ufunc_erfi_doc = ( + "erfi(z, out=None)\n" + "\n" + "Imaginary error function, ``-i erf(i z)``.\n" + "\n" + "Parameters\n" + "----------\n" + "z : array_like\n" + " Real or complex valued argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the imaginary error function\n" + "\n" + "See Also\n" + "--------\n" + "erf, erfc, erfcx, dawsn, wofz\n" + "\n" + "Notes\n" + "-----\n" + "\n" + ".. versionadded:: 0.12.0\n" + "\n" + "References\n" + "----------\n" + ".. [1] Steven G. Johnson, Faddeeva W function implementation.\n" + " http://ab-initio.mit.edu/Faddeeva\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy import special\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> x = np.linspace(-3, 3)\n" + ">>> plt.plot(x, special.erfi(x))\n" + ">>> plt.xlabel('$x$')\n" + ">>> plt.ylabel('$erfi(x)$')\n" + ">>> plt.show()") +ufunc_erfi_loops[0] = loop_d_d__As_f_f +ufunc_erfi_loops[1] = loop_d_d__As_d_d +ufunc_erfi_loops[2] = loop_D_D__As_F_F +ufunc_erfi_loops[3] = loop_D_D__As_D_D +ufunc_erfi_types[0] = NPY_FLOAT +ufunc_erfi_types[1] = NPY_FLOAT +ufunc_erfi_types[2] = NPY_DOUBLE +ufunc_erfi_types[3] = NPY_DOUBLE +ufunc_erfi_types[4] = NPY_CFLOAT +ufunc_erfi_types[5] = NPY_CFLOAT +ufunc_erfi_types[6] = NPY_CDOUBLE +ufunc_erfi_types[7] = NPY_CDOUBLE +ufunc_erfi_ptr[2*0] = scipy.special._ufuncs_cxx._export_faddeeva_erfi +ufunc_erfi_ptr[2*0+1] = ("erfi") +ufunc_erfi_ptr[2*1] = scipy.special._ufuncs_cxx._export_faddeeva_erfi +ufunc_erfi_ptr[2*1+1] = ("erfi") +ufunc_erfi_ptr[2*2] = scipy.special._ufuncs_cxx._export_faddeeva_erfi_complex +ufunc_erfi_ptr[2*2+1] = ("erfi") +ufunc_erfi_ptr[2*3] = scipy.special._ufuncs_cxx._export_faddeeva_erfi_complex +ufunc_erfi_ptr[2*3+1] = ("erfi") +ufunc_erfi_data[0] = &ufunc_erfi_ptr[2*0] +ufunc_erfi_data[1] = &ufunc_erfi_ptr[2*1] +ufunc_erfi_data[2] = &ufunc_erfi_ptr[2*2] +ufunc_erfi_data[3] = &ufunc_erfi_ptr[2*3] +erfi = np.PyUFunc_FromFuncAndData(ufunc_erfi_loops, ufunc_erfi_data, ufunc_erfi_types, 4, 1, 1, 0, "erfi", ufunc_erfi_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_erfinv_loops[2] +cdef void *ufunc_erfinv_ptr[4] +cdef void *ufunc_erfinv_data[2] +cdef char ufunc_erfinv_types[4] +cdef char *ufunc_erfinv_doc = ( + "erfinv(y, out=None)\n" + "\n" + "Inverse of the error function.\n" + "\n" + "Computes the inverse of the error function.\n" + "\n" + "In the complex domain, there is no unique complex number w satisfying\n" + "erf(w)=z. This indicates a true inverse function would be multivalued.\n" + "When the domain restricts to the real, -1 < x < 1, there is a unique real\n" + "number satisfying erf(erfinv(x)) = x.\n" + "\n" + "Parameters\n" + "----------\n" + "y : ndarray\n" + " Argument at which to evaluate. Domain: [-1, 1]\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "erfinv : scalar or ndarray\n" + " The inverse of erf of y, element-wise\n" + "\n" + "See Also\n" + "--------\n" + "erf : Error function of a complex argument\n" + "erfc : Complementary error function, ``1 - erf(x)``\n" + "erfcinv : Inverse of the complementary error function\n" + "\n" + "Notes\n" + "-----\n" + "This function wraps the ``erf_inv`` routine from the\n" + "Boost Math C++ library [1]_.\n" + "\n" + "References\n" + "----------\n" + ".. [1] The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> from scipy.special import erfinv, erf\n" + "\n" + ">>> erfinv(0.5)\n" + "0.4769362762044699\n" + "\n" + ">>> y = np.linspace(-1.0, 1.0, num=9)\n" + ">>> x = erfinv(y)\n" + ">>> x\n" + "array([ -inf, -0.81341985, -0.47693628, -0.22531206, 0. ,\n" + " 0.22531206, 0.47693628, 0.81341985, inf])\n" + "\n" + "Verify that ``erf(erfinv(y))`` is ``y``.\n" + "\n" + ">>> erf(x)\n" + "array([-1. , -0.75, -0.5 , -0.25, 0. , 0.25, 0.5 , 0.75, 1. ])\n" + "\n" + "Plot the function:\n" + "\n" + ">>> y = np.linspace(-1, 1, 200)\n" + ">>> fig, ax = plt.subplots()\n" + ">>> ax.plot(y, erfinv(y))\n" + ">>> ax.grid(True)\n" + ">>> ax.set_xlabel('y')\n" + ">>> ax.set_title('erfinv(y)')\n" + ">>> plt.show()") +ufunc_erfinv_loops[0] = loop_f_f__As_f_f +ufunc_erfinv_loops[1] = loop_d_d__As_d_d +ufunc_erfinv_types[0] = NPY_FLOAT +ufunc_erfinv_types[1] = NPY_FLOAT +ufunc_erfinv_types[2] = NPY_DOUBLE +ufunc_erfinv_types[3] = NPY_DOUBLE +ufunc_erfinv_ptr[2*0] = scipy.special._ufuncs_cxx._export_erfinv_float +ufunc_erfinv_ptr[2*0+1] = ("erfinv") +ufunc_erfinv_ptr[2*1] = scipy.special._ufuncs_cxx._export_erfinv_double +ufunc_erfinv_ptr[2*1+1] = ("erfinv") +ufunc_erfinv_data[0] = &ufunc_erfinv_ptr[2*0] +ufunc_erfinv_data[1] = &ufunc_erfinv_ptr[2*1] +erfinv = np.PyUFunc_FromFuncAndData(ufunc_erfinv_loops, ufunc_erfinv_data, ufunc_erfinv_types, 2, 1, 1, 0, "erfinv", ufunc_erfinv_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_chebyc_loops[5] +cdef void *ufunc_eval_chebyc_ptr[10] +cdef void *ufunc_eval_chebyc_data[5] +cdef char ufunc_eval_chebyc_types[15] +cdef char *ufunc_eval_chebyc_doc = ( + "eval_chebyc(n, x, out=None)\n" + "\n" + "Evaluate Chebyshev polynomial of the first kind on [-2, 2] at a\n" + "point.\n" + "\n" + "These polynomials are defined as\n" + "\n" + ".. math::\n" + "\n" + " C_n(x) = 2 T_n(x/2)\n" + "\n" + "where :math:`T_n` is a Chebyshev polynomial of the first kind. See\n" + "22.5.11 in [AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer, the result is\n" + " determined via the relation to `eval_chebyt`.\n" + "x : array_like\n" + " Points at which to evaluate the Chebyshev polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "C : scalar or ndarray\n" + " Values of the Chebyshev polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_chebyc : roots and quadrature weights of Chebyshev\n" + " polynomials of the first kind on [-2, 2]\n" + "chebyc : Chebyshev polynomial object\n" + "numpy.polynomial.chebyshev.Chebyshev : Chebyshev series\n" + "eval_chebyt : evaluate Chebycshev polynomials of the first kind\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "They are a scaled version of the Chebyshev polynomials of the\n" + "first kind.\n" + "\n" + ">>> x = np.linspace(-2, 2, 6)\n" + ">>> sc.eval_chebyc(3, x)\n" + "array([-2. , 1.872, 1.136, -1.136, -1.872, 2. ])\n" + ">>> 2 * sc.eval_chebyt(3, x / 2)\n" + "array([-2. , 1.872, 1.136, -1.136, -1.872, 2. ])") +ufunc_eval_chebyc_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_chebyc_loops[1] = loop_d_dd__As_ff_f +ufunc_eval_chebyc_loops[2] = loop_D_dD__As_fF_F +ufunc_eval_chebyc_loops[3] = loop_d_dd__As_dd_d +ufunc_eval_chebyc_loops[4] = loop_D_dD__As_dD_D +ufunc_eval_chebyc_types[0] = NPY_INTP +ufunc_eval_chebyc_types[1] = NPY_DOUBLE +ufunc_eval_chebyc_types[2] = NPY_DOUBLE +ufunc_eval_chebyc_types[3] = NPY_FLOAT +ufunc_eval_chebyc_types[4] = NPY_FLOAT +ufunc_eval_chebyc_types[5] = NPY_FLOAT +ufunc_eval_chebyc_types[6] = NPY_FLOAT +ufunc_eval_chebyc_types[7] = NPY_CFLOAT +ufunc_eval_chebyc_types[8] = NPY_CFLOAT +ufunc_eval_chebyc_types[9] = NPY_DOUBLE +ufunc_eval_chebyc_types[10] = NPY_DOUBLE +ufunc_eval_chebyc_types[11] = NPY_DOUBLE +ufunc_eval_chebyc_types[12] = NPY_DOUBLE +ufunc_eval_chebyc_types[13] = NPY_CDOUBLE +ufunc_eval_chebyc_types[14] = NPY_CDOUBLE +ufunc_eval_chebyc_ptr[2*0] = _func_eval_chebyc_l +ufunc_eval_chebyc_ptr[2*0+1] = ("eval_chebyc") +ufunc_eval_chebyc_ptr[2*1] = _func_eval_chebyc[double] +ufunc_eval_chebyc_ptr[2*1+1] = ("eval_chebyc") +ufunc_eval_chebyc_ptr[2*2] = _func_eval_chebyc[double_complex] +ufunc_eval_chebyc_ptr[2*2+1] = ("eval_chebyc") +ufunc_eval_chebyc_ptr[2*3] = _func_eval_chebyc[double] +ufunc_eval_chebyc_ptr[2*3+1] = ("eval_chebyc") +ufunc_eval_chebyc_ptr[2*4] = _func_eval_chebyc[double_complex] +ufunc_eval_chebyc_ptr[2*4+1] = ("eval_chebyc") +ufunc_eval_chebyc_data[0] = &ufunc_eval_chebyc_ptr[2*0] +ufunc_eval_chebyc_data[1] = &ufunc_eval_chebyc_ptr[2*1] +ufunc_eval_chebyc_data[2] = &ufunc_eval_chebyc_ptr[2*2] +ufunc_eval_chebyc_data[3] = &ufunc_eval_chebyc_ptr[2*3] +ufunc_eval_chebyc_data[4] = &ufunc_eval_chebyc_ptr[2*4] +eval_chebyc = np.PyUFunc_FromFuncAndData(ufunc_eval_chebyc_loops, ufunc_eval_chebyc_data, ufunc_eval_chebyc_types, 5, 2, 1, 0, "eval_chebyc", ufunc_eval_chebyc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_chebys_loops[5] +cdef void *ufunc_eval_chebys_ptr[10] +cdef void *ufunc_eval_chebys_data[5] +cdef char ufunc_eval_chebys_types[15] +cdef char *ufunc_eval_chebys_doc = ( + "eval_chebys(n, x, out=None)\n" + "\n" + "Evaluate Chebyshev polynomial of the second kind on [-2, 2] at a\n" + "point.\n" + "\n" + "These polynomials are defined as\n" + "\n" + ".. math::\n" + "\n" + " S_n(x) = U_n(x/2)\n" + "\n" + "where :math:`U_n` is a Chebyshev polynomial of the second\n" + "kind. See 22.5.13 in [AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer, the result is\n" + " determined via the relation to `eval_chebyu`.\n" + "x : array_like\n" + " Points at which to evaluate the Chebyshev polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "S : scalar or ndarray\n" + " Values of the Chebyshev polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_chebys : roots and quadrature weights of Chebyshev\n" + " polynomials of the second kind on [-2, 2]\n" + "chebys : Chebyshev polynomial object\n" + "eval_chebyu : evaluate Chebyshev polynomials of the second kind\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "They are a scaled version of the Chebyshev polynomials of the\n" + "second kind.\n" + "\n" + ">>> x = np.linspace(-2, 2, 6)\n" + ">>> sc.eval_chebys(3, x)\n" + "array([-4. , 0.672, 0.736, -0.736, -0.672, 4. ])\n" + ">>> sc.eval_chebyu(3, x / 2)\n" + "array([-4. , 0.672, 0.736, -0.736, -0.672, 4. ])") +ufunc_eval_chebys_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_chebys_loops[1] = loop_d_dd__As_ff_f +ufunc_eval_chebys_loops[2] = loop_D_dD__As_fF_F +ufunc_eval_chebys_loops[3] = loop_d_dd__As_dd_d +ufunc_eval_chebys_loops[4] = loop_D_dD__As_dD_D +ufunc_eval_chebys_types[0] = NPY_INTP +ufunc_eval_chebys_types[1] = NPY_DOUBLE +ufunc_eval_chebys_types[2] = NPY_DOUBLE +ufunc_eval_chebys_types[3] = NPY_FLOAT +ufunc_eval_chebys_types[4] = NPY_FLOAT +ufunc_eval_chebys_types[5] = NPY_FLOAT +ufunc_eval_chebys_types[6] = NPY_FLOAT +ufunc_eval_chebys_types[7] = NPY_CFLOAT +ufunc_eval_chebys_types[8] = NPY_CFLOAT +ufunc_eval_chebys_types[9] = NPY_DOUBLE +ufunc_eval_chebys_types[10] = NPY_DOUBLE +ufunc_eval_chebys_types[11] = NPY_DOUBLE +ufunc_eval_chebys_types[12] = NPY_DOUBLE +ufunc_eval_chebys_types[13] = NPY_CDOUBLE +ufunc_eval_chebys_types[14] = NPY_CDOUBLE +ufunc_eval_chebys_ptr[2*0] = _func_eval_chebys_l +ufunc_eval_chebys_ptr[2*0+1] = ("eval_chebys") +ufunc_eval_chebys_ptr[2*1] = _func_eval_chebys[double] +ufunc_eval_chebys_ptr[2*1+1] = ("eval_chebys") +ufunc_eval_chebys_ptr[2*2] = _func_eval_chebys[double_complex] +ufunc_eval_chebys_ptr[2*2+1] = ("eval_chebys") +ufunc_eval_chebys_ptr[2*3] = _func_eval_chebys[double] +ufunc_eval_chebys_ptr[2*3+1] = ("eval_chebys") +ufunc_eval_chebys_ptr[2*4] = _func_eval_chebys[double_complex] +ufunc_eval_chebys_ptr[2*4+1] = ("eval_chebys") +ufunc_eval_chebys_data[0] = &ufunc_eval_chebys_ptr[2*0] +ufunc_eval_chebys_data[1] = &ufunc_eval_chebys_ptr[2*1] +ufunc_eval_chebys_data[2] = &ufunc_eval_chebys_ptr[2*2] +ufunc_eval_chebys_data[3] = &ufunc_eval_chebys_ptr[2*3] +ufunc_eval_chebys_data[4] = &ufunc_eval_chebys_ptr[2*4] +eval_chebys = np.PyUFunc_FromFuncAndData(ufunc_eval_chebys_loops, ufunc_eval_chebys_data, ufunc_eval_chebys_types, 5, 2, 1, 0, "eval_chebys", ufunc_eval_chebys_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_chebyt_loops[5] +cdef void *ufunc_eval_chebyt_ptr[10] +cdef void *ufunc_eval_chebyt_data[5] +cdef char ufunc_eval_chebyt_types[15] +cdef char *ufunc_eval_chebyt_doc = ( + "eval_chebyt(n, x, out=None)\n" + "\n" + "Evaluate Chebyshev polynomial of the first kind at a point.\n" + "\n" + "The Chebyshev polynomials of the first kind can be defined via the\n" + "Gauss hypergeometric function :math:`{}_2F_1` as\n" + "\n" + ".. math::\n" + "\n" + " T_n(x) = {}_2F_1(n, -n; 1/2; (1 - x)/2).\n" + "\n" + "When :math:`n` is an integer the result is a polynomial of degree\n" + ":math:`n`. See 22.5.47 in [AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer, the result is\n" + " determined via the relation to the Gauss hypergeometric\n" + " function.\n" + "x : array_like\n" + " Points at which to evaluate the Chebyshev polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "T : scalar or ndarray\n" + " Values of the Chebyshev polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_chebyt : roots and quadrature weights of Chebyshev\n" + " polynomials of the first kind\n" + "chebyu : Chebychev polynomial object\n" + "eval_chebyu : evaluate Chebyshev polynomials of the second kind\n" + "hyp2f1 : Gauss hypergeometric function\n" + "numpy.polynomial.chebyshev.Chebyshev : Chebyshev series\n" + "\n" + "Notes\n" + "-----\n" + "This routine is numerically stable for `x` in ``[-1, 1]`` at least\n" + "up to order ``10000``.\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_chebyt_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_chebyt_loops[1] = loop_d_dd__As_ff_f +ufunc_eval_chebyt_loops[2] = loop_D_dD__As_fF_F +ufunc_eval_chebyt_loops[3] = loop_d_dd__As_dd_d +ufunc_eval_chebyt_loops[4] = loop_D_dD__As_dD_D +ufunc_eval_chebyt_types[0] = NPY_INTP +ufunc_eval_chebyt_types[1] = NPY_DOUBLE +ufunc_eval_chebyt_types[2] = NPY_DOUBLE +ufunc_eval_chebyt_types[3] = NPY_FLOAT +ufunc_eval_chebyt_types[4] = NPY_FLOAT +ufunc_eval_chebyt_types[5] = NPY_FLOAT +ufunc_eval_chebyt_types[6] = NPY_FLOAT +ufunc_eval_chebyt_types[7] = NPY_CFLOAT +ufunc_eval_chebyt_types[8] = NPY_CFLOAT +ufunc_eval_chebyt_types[9] = NPY_DOUBLE +ufunc_eval_chebyt_types[10] = NPY_DOUBLE +ufunc_eval_chebyt_types[11] = NPY_DOUBLE +ufunc_eval_chebyt_types[12] = NPY_DOUBLE +ufunc_eval_chebyt_types[13] = NPY_CDOUBLE +ufunc_eval_chebyt_types[14] = NPY_CDOUBLE +ufunc_eval_chebyt_ptr[2*0] = _func_eval_chebyt_l +ufunc_eval_chebyt_ptr[2*0+1] = ("eval_chebyt") +ufunc_eval_chebyt_ptr[2*1] = _func_eval_chebyt[double] +ufunc_eval_chebyt_ptr[2*1+1] = ("eval_chebyt") +ufunc_eval_chebyt_ptr[2*2] = _func_eval_chebyt[double_complex] +ufunc_eval_chebyt_ptr[2*2+1] = ("eval_chebyt") +ufunc_eval_chebyt_ptr[2*3] = _func_eval_chebyt[double] +ufunc_eval_chebyt_ptr[2*3+1] = ("eval_chebyt") +ufunc_eval_chebyt_ptr[2*4] = _func_eval_chebyt[double_complex] +ufunc_eval_chebyt_ptr[2*4+1] = ("eval_chebyt") +ufunc_eval_chebyt_data[0] = &ufunc_eval_chebyt_ptr[2*0] +ufunc_eval_chebyt_data[1] = &ufunc_eval_chebyt_ptr[2*1] +ufunc_eval_chebyt_data[2] = &ufunc_eval_chebyt_ptr[2*2] +ufunc_eval_chebyt_data[3] = &ufunc_eval_chebyt_ptr[2*3] +ufunc_eval_chebyt_data[4] = &ufunc_eval_chebyt_ptr[2*4] +eval_chebyt = np.PyUFunc_FromFuncAndData(ufunc_eval_chebyt_loops, ufunc_eval_chebyt_data, ufunc_eval_chebyt_types, 5, 2, 1, 0, "eval_chebyt", ufunc_eval_chebyt_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_chebyu_loops[5] +cdef void *ufunc_eval_chebyu_ptr[10] +cdef void *ufunc_eval_chebyu_data[5] +cdef char ufunc_eval_chebyu_types[15] +cdef char *ufunc_eval_chebyu_doc = ( + "eval_chebyu(n, x, out=None)\n" + "\n" + "Evaluate Chebyshev polynomial of the second kind at a point.\n" + "\n" + "The Chebyshev polynomials of the second kind can be defined via\n" + "the Gauss hypergeometric function :math:`{}_2F_1` as\n" + "\n" + ".. math::\n" + "\n" + " U_n(x) = (n + 1) {}_2F_1(-n, n + 2; 3/2; (1 - x)/2).\n" + "\n" + "When :math:`n` is an integer the result is a polynomial of degree\n" + ":math:`n`. See 22.5.48 in [AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer, the result is\n" + " determined via the relation to the Gauss hypergeometric\n" + " function.\n" + "x : array_like\n" + " Points at which to evaluate the Chebyshev polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "U : scalar or ndarray\n" + " Values of the Chebyshev polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_chebyu : roots and quadrature weights of Chebyshev\n" + " polynomials of the second kind\n" + "chebyu : Chebyshev polynomial object\n" + "eval_chebyt : evaluate Chebyshev polynomials of the first kind\n" + "hyp2f1 : Gauss hypergeometric function\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_chebyu_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_chebyu_loops[1] = loop_d_dd__As_ff_f +ufunc_eval_chebyu_loops[2] = loop_D_dD__As_fF_F +ufunc_eval_chebyu_loops[3] = loop_d_dd__As_dd_d +ufunc_eval_chebyu_loops[4] = loop_D_dD__As_dD_D +ufunc_eval_chebyu_types[0] = NPY_INTP +ufunc_eval_chebyu_types[1] = NPY_DOUBLE +ufunc_eval_chebyu_types[2] = NPY_DOUBLE +ufunc_eval_chebyu_types[3] = NPY_FLOAT +ufunc_eval_chebyu_types[4] = NPY_FLOAT +ufunc_eval_chebyu_types[5] = NPY_FLOAT +ufunc_eval_chebyu_types[6] = NPY_FLOAT +ufunc_eval_chebyu_types[7] = NPY_CFLOAT +ufunc_eval_chebyu_types[8] = NPY_CFLOAT +ufunc_eval_chebyu_types[9] = NPY_DOUBLE +ufunc_eval_chebyu_types[10] = NPY_DOUBLE +ufunc_eval_chebyu_types[11] = NPY_DOUBLE +ufunc_eval_chebyu_types[12] = NPY_DOUBLE +ufunc_eval_chebyu_types[13] = NPY_CDOUBLE +ufunc_eval_chebyu_types[14] = NPY_CDOUBLE +ufunc_eval_chebyu_ptr[2*0] = _func_eval_chebyu_l +ufunc_eval_chebyu_ptr[2*0+1] = ("eval_chebyu") +ufunc_eval_chebyu_ptr[2*1] = _func_eval_chebyu[double] +ufunc_eval_chebyu_ptr[2*1+1] = ("eval_chebyu") +ufunc_eval_chebyu_ptr[2*2] = _func_eval_chebyu[double_complex] +ufunc_eval_chebyu_ptr[2*2+1] = ("eval_chebyu") +ufunc_eval_chebyu_ptr[2*3] = _func_eval_chebyu[double] +ufunc_eval_chebyu_ptr[2*3+1] = ("eval_chebyu") +ufunc_eval_chebyu_ptr[2*4] = _func_eval_chebyu[double_complex] +ufunc_eval_chebyu_ptr[2*4+1] = ("eval_chebyu") +ufunc_eval_chebyu_data[0] = &ufunc_eval_chebyu_ptr[2*0] +ufunc_eval_chebyu_data[1] = &ufunc_eval_chebyu_ptr[2*1] +ufunc_eval_chebyu_data[2] = &ufunc_eval_chebyu_ptr[2*2] +ufunc_eval_chebyu_data[3] = &ufunc_eval_chebyu_ptr[2*3] +ufunc_eval_chebyu_data[4] = &ufunc_eval_chebyu_ptr[2*4] +eval_chebyu = np.PyUFunc_FromFuncAndData(ufunc_eval_chebyu_loops, ufunc_eval_chebyu_data, ufunc_eval_chebyu_types, 5, 2, 1, 0, "eval_chebyu", ufunc_eval_chebyu_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_gegenbauer_loops[5] +cdef void *ufunc_eval_gegenbauer_ptr[10] +cdef void *ufunc_eval_gegenbauer_data[5] +cdef char ufunc_eval_gegenbauer_types[20] +cdef char *ufunc_eval_gegenbauer_doc = ( + "eval_gegenbauer(n, alpha, x, out=None)\n" + "\n" + "Evaluate Gegenbauer polynomial at a point.\n" + "\n" + "The Gegenbauer polynomials can be defined via the Gauss\n" + "hypergeometric function :math:`{}_2F_1` as\n" + "\n" + ".. math::\n" + "\n" + " C_n^{(\\alpha)} = \\frac{(2\\alpha)_n}{\\Gamma(n + 1)}\n" + " {}_2F_1(-n, 2\\alpha + n; \\alpha + 1/2; (1 - z)/2).\n" + "\n" + "When :math:`n` is an integer the result is a polynomial of degree\n" + ":math:`n`. See 22.5.46 in [AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer, the result is\n" + " determined via the relation to the Gauss hypergeometric\n" + " function.\n" + "alpha : array_like\n" + " Parameter\n" + "x : array_like\n" + " Points at which to evaluate the Gegenbauer polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "C : scalar or ndarray\n" + " Values of the Gegenbauer polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_gegenbauer : roots and quadrature weights of Gegenbauer\n" + " polynomials\n" + "gegenbauer : Gegenbauer polynomial object\n" + "hyp2f1 : Gauss hypergeometric function\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_gegenbauer_loops[0] = loop_d_pdd__As_pdd_d +ufunc_eval_gegenbauer_loops[1] = loop_d_ddd__As_fff_f +ufunc_eval_gegenbauer_loops[2] = loop_D_ddD__As_ffF_F +ufunc_eval_gegenbauer_loops[3] = loop_d_ddd__As_ddd_d +ufunc_eval_gegenbauer_loops[4] = loop_D_ddD__As_ddD_D +ufunc_eval_gegenbauer_types[0] = NPY_INTP +ufunc_eval_gegenbauer_types[1] = NPY_DOUBLE +ufunc_eval_gegenbauer_types[2] = NPY_DOUBLE +ufunc_eval_gegenbauer_types[3] = NPY_DOUBLE +ufunc_eval_gegenbauer_types[4] = NPY_FLOAT +ufunc_eval_gegenbauer_types[5] = NPY_FLOAT +ufunc_eval_gegenbauer_types[6] = NPY_FLOAT +ufunc_eval_gegenbauer_types[7] = NPY_FLOAT +ufunc_eval_gegenbauer_types[8] = NPY_FLOAT +ufunc_eval_gegenbauer_types[9] = NPY_FLOAT +ufunc_eval_gegenbauer_types[10] = NPY_CFLOAT +ufunc_eval_gegenbauer_types[11] = NPY_CFLOAT +ufunc_eval_gegenbauer_types[12] = NPY_DOUBLE +ufunc_eval_gegenbauer_types[13] = NPY_DOUBLE +ufunc_eval_gegenbauer_types[14] = NPY_DOUBLE +ufunc_eval_gegenbauer_types[15] = NPY_DOUBLE +ufunc_eval_gegenbauer_types[16] = NPY_DOUBLE +ufunc_eval_gegenbauer_types[17] = NPY_DOUBLE +ufunc_eval_gegenbauer_types[18] = NPY_CDOUBLE +ufunc_eval_gegenbauer_types[19] = NPY_CDOUBLE +ufunc_eval_gegenbauer_ptr[2*0] = _func_eval_gegenbauer_l +ufunc_eval_gegenbauer_ptr[2*0+1] = ("eval_gegenbauer") +ufunc_eval_gegenbauer_ptr[2*1] = _func_eval_gegenbauer[double] +ufunc_eval_gegenbauer_ptr[2*1+1] = ("eval_gegenbauer") +ufunc_eval_gegenbauer_ptr[2*2] = _func_eval_gegenbauer[double_complex] +ufunc_eval_gegenbauer_ptr[2*2+1] = ("eval_gegenbauer") +ufunc_eval_gegenbauer_ptr[2*3] = _func_eval_gegenbauer[double] +ufunc_eval_gegenbauer_ptr[2*3+1] = ("eval_gegenbauer") +ufunc_eval_gegenbauer_ptr[2*4] = _func_eval_gegenbauer[double_complex] +ufunc_eval_gegenbauer_ptr[2*4+1] = ("eval_gegenbauer") +ufunc_eval_gegenbauer_data[0] = &ufunc_eval_gegenbauer_ptr[2*0] +ufunc_eval_gegenbauer_data[1] = &ufunc_eval_gegenbauer_ptr[2*1] +ufunc_eval_gegenbauer_data[2] = &ufunc_eval_gegenbauer_ptr[2*2] +ufunc_eval_gegenbauer_data[3] = &ufunc_eval_gegenbauer_ptr[2*3] +ufunc_eval_gegenbauer_data[4] = &ufunc_eval_gegenbauer_ptr[2*4] +eval_gegenbauer = np.PyUFunc_FromFuncAndData(ufunc_eval_gegenbauer_loops, ufunc_eval_gegenbauer_data, ufunc_eval_gegenbauer_types, 5, 3, 1, 0, "eval_gegenbauer", ufunc_eval_gegenbauer_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_genlaguerre_loops[5] +cdef void *ufunc_eval_genlaguerre_ptr[10] +cdef void *ufunc_eval_genlaguerre_data[5] +cdef char ufunc_eval_genlaguerre_types[20] +cdef char *ufunc_eval_genlaguerre_doc = ( + "eval_genlaguerre(n, alpha, x, out=None)\n" + "\n" + "Evaluate generalized Laguerre polynomial at a point.\n" + "\n" + "The generalized Laguerre polynomials can be defined via the\n" + "confluent hypergeometric function :math:`{}_1F_1` as\n" + "\n" + ".. math::\n" + "\n" + " L_n^{(\\alpha)}(x) = \\binom{n + \\alpha}{n}\n" + " {}_1F_1(-n, \\alpha + 1, x).\n" + "\n" + "When :math:`n` is an integer the result is a polynomial of degree\n" + ":math:`n`. See 22.5.54 in [AS]_ for details. The Laguerre\n" + "polynomials are the special case where :math:`\\alpha = 0`.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer, the result is\n" + " determined via the relation to the confluent hypergeometric\n" + " function.\n" + "alpha : array_like\n" + " Parameter; must have ``alpha > -1``\n" + "x : array_like\n" + " Points at which to evaluate the generalized Laguerre\n" + " polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "L : scalar or ndarray\n" + " Values of the generalized Laguerre polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_genlaguerre : roots and quadrature weights of generalized\n" + " Laguerre polynomials\n" + "genlaguerre : generalized Laguerre polynomial object\n" + "hyp1f1 : confluent hypergeometric function\n" + "eval_laguerre : evaluate Laguerre polynomials\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_genlaguerre_loops[0] = loop_d_pdd__As_pdd_d +ufunc_eval_genlaguerre_loops[1] = loop_d_ddd__As_fff_f +ufunc_eval_genlaguerre_loops[2] = loop_D_ddD__As_ffF_F +ufunc_eval_genlaguerre_loops[3] = loop_d_ddd__As_ddd_d +ufunc_eval_genlaguerre_loops[4] = loop_D_ddD__As_ddD_D +ufunc_eval_genlaguerre_types[0] = NPY_INTP +ufunc_eval_genlaguerre_types[1] = NPY_DOUBLE +ufunc_eval_genlaguerre_types[2] = NPY_DOUBLE +ufunc_eval_genlaguerre_types[3] = NPY_DOUBLE +ufunc_eval_genlaguerre_types[4] = NPY_FLOAT +ufunc_eval_genlaguerre_types[5] = NPY_FLOAT +ufunc_eval_genlaguerre_types[6] = NPY_FLOAT +ufunc_eval_genlaguerre_types[7] = NPY_FLOAT +ufunc_eval_genlaguerre_types[8] = NPY_FLOAT +ufunc_eval_genlaguerre_types[9] = NPY_FLOAT +ufunc_eval_genlaguerre_types[10] = NPY_CFLOAT +ufunc_eval_genlaguerre_types[11] = NPY_CFLOAT +ufunc_eval_genlaguerre_types[12] = NPY_DOUBLE +ufunc_eval_genlaguerre_types[13] = NPY_DOUBLE +ufunc_eval_genlaguerre_types[14] = NPY_DOUBLE +ufunc_eval_genlaguerre_types[15] = NPY_DOUBLE +ufunc_eval_genlaguerre_types[16] = NPY_DOUBLE +ufunc_eval_genlaguerre_types[17] = NPY_DOUBLE +ufunc_eval_genlaguerre_types[18] = NPY_CDOUBLE +ufunc_eval_genlaguerre_types[19] = NPY_CDOUBLE +ufunc_eval_genlaguerre_ptr[2*0] = _func_eval_genlaguerre_l +ufunc_eval_genlaguerre_ptr[2*0+1] = ("eval_genlaguerre") +ufunc_eval_genlaguerre_ptr[2*1] = _func_eval_genlaguerre[double] +ufunc_eval_genlaguerre_ptr[2*1+1] = ("eval_genlaguerre") +ufunc_eval_genlaguerre_ptr[2*2] = _func_eval_genlaguerre[double_complex] +ufunc_eval_genlaguerre_ptr[2*2+1] = ("eval_genlaguerre") +ufunc_eval_genlaguerre_ptr[2*3] = _func_eval_genlaguerre[double] +ufunc_eval_genlaguerre_ptr[2*3+1] = ("eval_genlaguerre") +ufunc_eval_genlaguerre_ptr[2*4] = _func_eval_genlaguerre[double_complex] +ufunc_eval_genlaguerre_ptr[2*4+1] = ("eval_genlaguerre") +ufunc_eval_genlaguerre_data[0] = &ufunc_eval_genlaguerre_ptr[2*0] +ufunc_eval_genlaguerre_data[1] = &ufunc_eval_genlaguerre_ptr[2*1] +ufunc_eval_genlaguerre_data[2] = &ufunc_eval_genlaguerre_ptr[2*2] +ufunc_eval_genlaguerre_data[3] = &ufunc_eval_genlaguerre_ptr[2*3] +ufunc_eval_genlaguerre_data[4] = &ufunc_eval_genlaguerre_ptr[2*4] +eval_genlaguerre = np.PyUFunc_FromFuncAndData(ufunc_eval_genlaguerre_loops, ufunc_eval_genlaguerre_data, ufunc_eval_genlaguerre_types, 5, 3, 1, 0, "eval_genlaguerre", ufunc_eval_genlaguerre_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_hermite_loops[1] +cdef void *ufunc_eval_hermite_ptr[2] +cdef void *ufunc_eval_hermite_data[1] +cdef char ufunc_eval_hermite_types[3] +cdef char *ufunc_eval_hermite_doc = ( + "eval_hermite(n, x, out=None)\n" + "\n" + "Evaluate physicist's Hermite polynomial at a point.\n" + "\n" + "Defined by\n" + "\n" + ".. math::\n" + "\n" + " H_n(x) = (-1)^n e^{x^2} \\frac{d^n}{dx^n} e^{-x^2};\n" + "\n" + ":math:`H_n` is a polynomial of degree :math:`n`. See 22.11.7 in\n" + "[AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial\n" + "x : array_like\n" + " Points at which to evaluate the Hermite polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "H : scalar or ndarray\n" + " Values of the Hermite polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_hermite : roots and quadrature weights of physicist's\n" + " Hermite polynomials\n" + "hermite : physicist's Hermite polynomial object\n" + "numpy.polynomial.hermite.Hermite : Physicist's Hermite series\n" + "eval_hermitenorm : evaluate Probabilist's Hermite polynomials\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_hermite_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_hermite_types[0] = NPY_INTP +ufunc_eval_hermite_types[1] = NPY_DOUBLE +ufunc_eval_hermite_types[2] = NPY_DOUBLE +ufunc_eval_hermite_ptr[2*0] = _func_eval_hermite +ufunc_eval_hermite_ptr[2*0+1] = ("eval_hermite") +ufunc_eval_hermite_data[0] = &ufunc_eval_hermite_ptr[2*0] +eval_hermite = np.PyUFunc_FromFuncAndData(ufunc_eval_hermite_loops, ufunc_eval_hermite_data, ufunc_eval_hermite_types, 1, 2, 1, 0, "eval_hermite", ufunc_eval_hermite_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_hermitenorm_loops[1] +cdef void *ufunc_eval_hermitenorm_ptr[2] +cdef void *ufunc_eval_hermitenorm_data[1] +cdef char ufunc_eval_hermitenorm_types[3] +cdef char *ufunc_eval_hermitenorm_doc = ( + "eval_hermitenorm(n, x, out=None)\n" + "\n" + "Evaluate probabilist's (normalized) Hermite polynomial at a\n" + "point.\n" + "\n" + "Defined by\n" + "\n" + ".. math::\n" + "\n" + " He_n(x) = (-1)^n e^{x^2/2} \\frac{d^n}{dx^n} e^{-x^2/2};\n" + "\n" + ":math:`He_n` is a polynomial of degree :math:`n`. See 22.11.8 in\n" + "[AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial\n" + "x : array_like\n" + " Points at which to evaluate the Hermite polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "He : scalar or ndarray\n" + " Values of the Hermite polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_hermitenorm : roots and quadrature weights of probabilist's\n" + " Hermite polynomials\n" + "hermitenorm : probabilist's Hermite polynomial object\n" + "numpy.polynomial.hermite_e.HermiteE : Probabilist's Hermite series\n" + "eval_hermite : evaluate physicist's Hermite polynomials\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_hermitenorm_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_hermitenorm_types[0] = NPY_INTP +ufunc_eval_hermitenorm_types[1] = NPY_DOUBLE +ufunc_eval_hermitenorm_types[2] = NPY_DOUBLE +ufunc_eval_hermitenorm_ptr[2*0] = _func_eval_hermitenorm +ufunc_eval_hermitenorm_ptr[2*0+1] = ("eval_hermitenorm") +ufunc_eval_hermitenorm_data[0] = &ufunc_eval_hermitenorm_ptr[2*0] +eval_hermitenorm = np.PyUFunc_FromFuncAndData(ufunc_eval_hermitenorm_loops, ufunc_eval_hermitenorm_data, ufunc_eval_hermitenorm_types, 1, 2, 1, 0, "eval_hermitenorm", ufunc_eval_hermitenorm_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_jacobi_loops[5] +cdef void *ufunc_eval_jacobi_ptr[10] +cdef void *ufunc_eval_jacobi_data[5] +cdef char ufunc_eval_jacobi_types[25] +cdef char *ufunc_eval_jacobi_doc = ( + "eval_jacobi(n, alpha, beta, x, out=None)\n" + "\n" + "Evaluate Jacobi polynomial at a point.\n" + "\n" + "The Jacobi polynomials can be defined via the Gauss hypergeometric\n" + "function :math:`{}_2F_1` as\n" + "\n" + ".. math::\n" + "\n" + " P_n^{(\\alpha, \\beta)}(x) = \\frac{(\\alpha + 1)_n}{\\Gamma(n + 1)}\n" + " {}_2F_1(-n, 1 + \\alpha + \\beta + n; \\alpha + 1; (1 - z)/2)\n" + "\n" + "where :math:`(\\cdot)_n` is the Pochhammer symbol; see `poch`. When\n" + ":math:`n` is an integer the result is a polynomial of degree\n" + ":math:`n`. See 22.5.42 in [AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer the result is\n" + " determined via the relation to the Gauss hypergeometric\n" + " function.\n" + "alpha : array_like\n" + " Parameter\n" + "beta : array_like\n" + " Parameter\n" + "x : array_like\n" + " Points at which to evaluate the polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "P : scalar or ndarray\n" + " Values of the Jacobi polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_jacobi : roots and quadrature weights of Jacobi polynomials\n" + "jacobi : Jacobi polynomial object\n" + "hyp2f1 : Gauss hypergeometric function\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_jacobi_loops[0] = loop_d_pddd__As_pddd_d +ufunc_eval_jacobi_loops[1] = loop_d_dddd__As_ffff_f +ufunc_eval_jacobi_loops[2] = loop_D_dddD__As_fffF_F +ufunc_eval_jacobi_loops[3] = loop_d_dddd__As_dddd_d +ufunc_eval_jacobi_loops[4] = loop_D_dddD__As_dddD_D +ufunc_eval_jacobi_types[0] = NPY_INTP +ufunc_eval_jacobi_types[1] = NPY_DOUBLE +ufunc_eval_jacobi_types[2] = NPY_DOUBLE +ufunc_eval_jacobi_types[3] = NPY_DOUBLE +ufunc_eval_jacobi_types[4] = NPY_DOUBLE +ufunc_eval_jacobi_types[5] = NPY_FLOAT +ufunc_eval_jacobi_types[6] = NPY_FLOAT +ufunc_eval_jacobi_types[7] = NPY_FLOAT +ufunc_eval_jacobi_types[8] = NPY_FLOAT +ufunc_eval_jacobi_types[9] = NPY_FLOAT +ufunc_eval_jacobi_types[10] = NPY_FLOAT +ufunc_eval_jacobi_types[11] = NPY_FLOAT +ufunc_eval_jacobi_types[12] = NPY_FLOAT +ufunc_eval_jacobi_types[13] = NPY_CFLOAT +ufunc_eval_jacobi_types[14] = NPY_CFLOAT +ufunc_eval_jacobi_types[15] = NPY_DOUBLE +ufunc_eval_jacobi_types[16] = NPY_DOUBLE +ufunc_eval_jacobi_types[17] = NPY_DOUBLE +ufunc_eval_jacobi_types[18] = NPY_DOUBLE +ufunc_eval_jacobi_types[19] = NPY_DOUBLE +ufunc_eval_jacobi_types[20] = NPY_DOUBLE +ufunc_eval_jacobi_types[21] = NPY_DOUBLE +ufunc_eval_jacobi_types[22] = NPY_DOUBLE +ufunc_eval_jacobi_types[23] = NPY_CDOUBLE +ufunc_eval_jacobi_types[24] = NPY_CDOUBLE +ufunc_eval_jacobi_ptr[2*0] = _func_eval_jacobi_l +ufunc_eval_jacobi_ptr[2*0+1] = ("eval_jacobi") +ufunc_eval_jacobi_ptr[2*1] = _func_eval_jacobi[double] +ufunc_eval_jacobi_ptr[2*1+1] = ("eval_jacobi") +ufunc_eval_jacobi_ptr[2*2] = _func_eval_jacobi[double_complex] +ufunc_eval_jacobi_ptr[2*2+1] = ("eval_jacobi") +ufunc_eval_jacobi_ptr[2*3] = _func_eval_jacobi[double] +ufunc_eval_jacobi_ptr[2*3+1] = ("eval_jacobi") +ufunc_eval_jacobi_ptr[2*4] = _func_eval_jacobi[double_complex] +ufunc_eval_jacobi_ptr[2*4+1] = ("eval_jacobi") +ufunc_eval_jacobi_data[0] = &ufunc_eval_jacobi_ptr[2*0] +ufunc_eval_jacobi_data[1] = &ufunc_eval_jacobi_ptr[2*1] +ufunc_eval_jacobi_data[2] = &ufunc_eval_jacobi_ptr[2*2] +ufunc_eval_jacobi_data[3] = &ufunc_eval_jacobi_ptr[2*3] +ufunc_eval_jacobi_data[4] = &ufunc_eval_jacobi_ptr[2*4] +eval_jacobi = np.PyUFunc_FromFuncAndData(ufunc_eval_jacobi_loops, ufunc_eval_jacobi_data, ufunc_eval_jacobi_types, 5, 4, 1, 0, "eval_jacobi", ufunc_eval_jacobi_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_laguerre_loops[5] +cdef void *ufunc_eval_laguerre_ptr[10] +cdef void *ufunc_eval_laguerre_data[5] +cdef char ufunc_eval_laguerre_types[15] +cdef char *ufunc_eval_laguerre_doc = ( + "eval_laguerre(n, x, out=None)\n" + "\n" + "Evaluate Laguerre polynomial at a point.\n" + "\n" + "The Laguerre polynomials can be defined via the confluent\n" + "hypergeometric function :math:`{}_1F_1` as\n" + "\n" + ".. math::\n" + "\n" + " L_n(x) = {}_1F_1(-n, 1, x).\n" + "\n" + "See 22.5.16 and 22.5.54 in [AS]_ for details. When :math:`n` is an\n" + "integer the result is a polynomial of degree :math:`n`.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer the result is\n" + " determined via the relation to the confluent hypergeometric\n" + " function.\n" + "x : array_like\n" + " Points at which to evaluate the Laguerre polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "L : scalar or ndarray\n" + " Values of the Laguerre polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_laguerre : roots and quadrature weights of Laguerre\n" + " polynomials\n" + "laguerre : Laguerre polynomial object\n" + "numpy.polynomial.laguerre.Laguerre : Laguerre series\n" + "eval_genlaguerre : evaluate generalized Laguerre polynomials\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_laguerre_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_laguerre_loops[1] = loop_d_dd__As_ff_f +ufunc_eval_laguerre_loops[2] = loop_D_dD__As_fF_F +ufunc_eval_laguerre_loops[3] = loop_d_dd__As_dd_d +ufunc_eval_laguerre_loops[4] = loop_D_dD__As_dD_D +ufunc_eval_laguerre_types[0] = NPY_INTP +ufunc_eval_laguerre_types[1] = NPY_DOUBLE +ufunc_eval_laguerre_types[2] = NPY_DOUBLE +ufunc_eval_laguerre_types[3] = NPY_FLOAT +ufunc_eval_laguerre_types[4] = NPY_FLOAT +ufunc_eval_laguerre_types[5] = NPY_FLOAT +ufunc_eval_laguerre_types[6] = NPY_FLOAT +ufunc_eval_laguerre_types[7] = NPY_CFLOAT +ufunc_eval_laguerre_types[8] = NPY_CFLOAT +ufunc_eval_laguerre_types[9] = NPY_DOUBLE +ufunc_eval_laguerre_types[10] = NPY_DOUBLE +ufunc_eval_laguerre_types[11] = NPY_DOUBLE +ufunc_eval_laguerre_types[12] = NPY_DOUBLE +ufunc_eval_laguerre_types[13] = NPY_CDOUBLE +ufunc_eval_laguerre_types[14] = NPY_CDOUBLE +ufunc_eval_laguerre_ptr[2*0] = _func_eval_laguerre_l +ufunc_eval_laguerre_ptr[2*0+1] = ("eval_laguerre") +ufunc_eval_laguerre_ptr[2*1] = _func_eval_laguerre[double] +ufunc_eval_laguerre_ptr[2*1+1] = ("eval_laguerre") +ufunc_eval_laguerre_ptr[2*2] = _func_eval_laguerre[double_complex] +ufunc_eval_laguerre_ptr[2*2+1] = ("eval_laguerre") +ufunc_eval_laguerre_ptr[2*3] = _func_eval_laguerre[double] +ufunc_eval_laguerre_ptr[2*3+1] = ("eval_laguerre") +ufunc_eval_laguerre_ptr[2*4] = _func_eval_laguerre[double_complex] +ufunc_eval_laguerre_ptr[2*4+1] = ("eval_laguerre") +ufunc_eval_laguerre_data[0] = &ufunc_eval_laguerre_ptr[2*0] +ufunc_eval_laguerre_data[1] = &ufunc_eval_laguerre_ptr[2*1] +ufunc_eval_laguerre_data[2] = &ufunc_eval_laguerre_ptr[2*2] +ufunc_eval_laguerre_data[3] = &ufunc_eval_laguerre_ptr[2*3] +ufunc_eval_laguerre_data[4] = &ufunc_eval_laguerre_ptr[2*4] +eval_laguerre = np.PyUFunc_FromFuncAndData(ufunc_eval_laguerre_loops, ufunc_eval_laguerre_data, ufunc_eval_laguerre_types, 5, 2, 1, 0, "eval_laguerre", ufunc_eval_laguerre_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_legendre_loops[5] +cdef void *ufunc_eval_legendre_ptr[10] +cdef void *ufunc_eval_legendre_data[5] +cdef char ufunc_eval_legendre_types[15] +cdef char *ufunc_eval_legendre_doc = ( + "eval_legendre(n, x, out=None)\n" + "\n" + "Evaluate Legendre polynomial at a point.\n" + "\n" + "The Legendre polynomials can be defined via the Gauss\n" + "hypergeometric function :math:`{}_2F_1` as\n" + "\n" + ".. math::\n" + "\n" + " P_n(x) = {}_2F_1(-n, n + 1; 1; (1 - x)/2).\n" + "\n" + "When :math:`n` is an integer the result is a polynomial of degree\n" + ":math:`n`. See 22.5.49 in [AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer, the result is\n" + " determined via the relation to the Gauss hypergeometric\n" + " function.\n" + "x : array_like\n" + " Points at which to evaluate the Legendre polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "P : scalar or ndarray\n" + " Values of the Legendre polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_legendre : roots and quadrature weights of Legendre\n" + " polynomials\n" + "legendre : Legendre polynomial object\n" + "hyp2f1 : Gauss hypergeometric function\n" + "numpy.polynomial.legendre.Legendre : Legendre series\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy.special import eval_legendre\n" + "\n" + "Evaluate the zero-order Legendre polynomial at x = 0\n" + "\n" + ">>> eval_legendre(0, 0)\n" + "1.0\n" + "\n" + "Evaluate the first-order Legendre polynomial between -1 and 1\n" + "\n" + ">>> X = np.linspace(-1, 1, 5) # Domain of Legendre polynomials\n" + ">>> eval_legendre(1, X)\n" + "array([-1. , -0.5, 0. , 0.5, 1. ])\n" + "\n" + "Evaluate Legendre polynomials of order 0 through 4 at x = 0\n" + "\n" + ">>> N = range(0, 5)\n" + ">>> eval_legendre(N, 0)\n" + "array([ 1. , 0. , -0.5 , 0. , 0.375])\n" + "\n" + "Plot Legendre polynomials of order 0 through 4\n" + "\n" + ">>> X = np.linspace(-1, 1)\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> for n in range(0, 5):\n" + "... y = eval_legendre(n, X)\n" + "... plt.plot(X, y, label=r'$P_{}(x)$'.format(n))\n" + "\n" + ">>> plt.title(\"Legendre Polynomials\")\n" + ">>> plt.xlabel(\"x\")\n" + ">>> plt.ylabel(r'$P_n(x)$')\n" + ">>> plt.legend(loc='lower right')\n" + ">>> plt.show()") +ufunc_eval_legendre_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_legendre_loops[1] = loop_d_dd__As_ff_f +ufunc_eval_legendre_loops[2] = loop_D_dD__As_fF_F +ufunc_eval_legendre_loops[3] = loop_d_dd__As_dd_d +ufunc_eval_legendre_loops[4] = loop_D_dD__As_dD_D +ufunc_eval_legendre_types[0] = NPY_INTP +ufunc_eval_legendre_types[1] = NPY_DOUBLE +ufunc_eval_legendre_types[2] = NPY_DOUBLE +ufunc_eval_legendre_types[3] = NPY_FLOAT +ufunc_eval_legendre_types[4] = NPY_FLOAT +ufunc_eval_legendre_types[5] = NPY_FLOAT +ufunc_eval_legendre_types[6] = NPY_FLOAT +ufunc_eval_legendre_types[7] = NPY_CFLOAT +ufunc_eval_legendre_types[8] = NPY_CFLOAT +ufunc_eval_legendre_types[9] = NPY_DOUBLE +ufunc_eval_legendre_types[10] = NPY_DOUBLE +ufunc_eval_legendre_types[11] = NPY_DOUBLE +ufunc_eval_legendre_types[12] = NPY_DOUBLE +ufunc_eval_legendre_types[13] = NPY_CDOUBLE +ufunc_eval_legendre_types[14] = NPY_CDOUBLE +ufunc_eval_legendre_ptr[2*0] = _func_eval_legendre_l +ufunc_eval_legendre_ptr[2*0+1] = ("eval_legendre") +ufunc_eval_legendre_ptr[2*1] = _func_eval_legendre[double] +ufunc_eval_legendre_ptr[2*1+1] = ("eval_legendre") +ufunc_eval_legendre_ptr[2*2] = _func_eval_legendre[double_complex] +ufunc_eval_legendre_ptr[2*2+1] = ("eval_legendre") +ufunc_eval_legendre_ptr[2*3] = _func_eval_legendre[double] +ufunc_eval_legendre_ptr[2*3+1] = ("eval_legendre") +ufunc_eval_legendre_ptr[2*4] = _func_eval_legendre[double_complex] +ufunc_eval_legendre_ptr[2*4+1] = ("eval_legendre") +ufunc_eval_legendre_data[0] = &ufunc_eval_legendre_ptr[2*0] +ufunc_eval_legendre_data[1] = &ufunc_eval_legendre_ptr[2*1] +ufunc_eval_legendre_data[2] = &ufunc_eval_legendre_ptr[2*2] +ufunc_eval_legendre_data[3] = &ufunc_eval_legendre_ptr[2*3] +ufunc_eval_legendre_data[4] = &ufunc_eval_legendre_ptr[2*4] +eval_legendre = np.PyUFunc_FromFuncAndData(ufunc_eval_legendre_loops, ufunc_eval_legendre_data, ufunc_eval_legendre_types, 5, 2, 1, 0, "eval_legendre", ufunc_eval_legendre_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_sh_chebyt_loops[5] +cdef void *ufunc_eval_sh_chebyt_ptr[10] +cdef void *ufunc_eval_sh_chebyt_data[5] +cdef char ufunc_eval_sh_chebyt_types[15] +cdef char *ufunc_eval_sh_chebyt_doc = ( + "eval_sh_chebyt(n, x, out=None)\n" + "\n" + "Evaluate shifted Chebyshev polynomial of the first kind at a\n" + "point.\n" + "\n" + "These polynomials are defined as\n" + "\n" + ".. math::\n" + "\n" + " T_n^*(x) = T_n(2x - 1)\n" + "\n" + "where :math:`T_n` is a Chebyshev polynomial of the first kind. See\n" + "22.5.14 in [AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer, the result is\n" + " determined via the relation to `eval_chebyt`.\n" + "x : array_like\n" + " Points at which to evaluate the shifted Chebyshev polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "T : scalar or ndarray\n" + " Values of the shifted Chebyshev polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_sh_chebyt : roots and quadrature weights of shifted\n" + " Chebyshev polynomials of the first kind\n" + "sh_chebyt : shifted Chebyshev polynomial object\n" + "eval_chebyt : evaluate Chebyshev polynomials of the first kind\n" + "numpy.polynomial.chebyshev.Chebyshev : Chebyshev series\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_sh_chebyt_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_sh_chebyt_loops[1] = loop_d_dd__As_ff_f +ufunc_eval_sh_chebyt_loops[2] = loop_D_dD__As_fF_F +ufunc_eval_sh_chebyt_loops[3] = loop_d_dd__As_dd_d +ufunc_eval_sh_chebyt_loops[4] = loop_D_dD__As_dD_D +ufunc_eval_sh_chebyt_types[0] = NPY_INTP +ufunc_eval_sh_chebyt_types[1] = NPY_DOUBLE +ufunc_eval_sh_chebyt_types[2] = NPY_DOUBLE +ufunc_eval_sh_chebyt_types[3] = NPY_FLOAT +ufunc_eval_sh_chebyt_types[4] = NPY_FLOAT +ufunc_eval_sh_chebyt_types[5] = NPY_FLOAT +ufunc_eval_sh_chebyt_types[6] = NPY_FLOAT +ufunc_eval_sh_chebyt_types[7] = NPY_CFLOAT +ufunc_eval_sh_chebyt_types[8] = NPY_CFLOAT +ufunc_eval_sh_chebyt_types[9] = NPY_DOUBLE +ufunc_eval_sh_chebyt_types[10] = NPY_DOUBLE +ufunc_eval_sh_chebyt_types[11] = NPY_DOUBLE +ufunc_eval_sh_chebyt_types[12] = NPY_DOUBLE +ufunc_eval_sh_chebyt_types[13] = NPY_CDOUBLE +ufunc_eval_sh_chebyt_types[14] = NPY_CDOUBLE +ufunc_eval_sh_chebyt_ptr[2*0] = _func_eval_sh_chebyt_l +ufunc_eval_sh_chebyt_ptr[2*0+1] = ("eval_sh_chebyt") +ufunc_eval_sh_chebyt_ptr[2*1] = _func_eval_sh_chebyt[double] +ufunc_eval_sh_chebyt_ptr[2*1+1] = ("eval_sh_chebyt") +ufunc_eval_sh_chebyt_ptr[2*2] = _func_eval_sh_chebyt[double_complex] +ufunc_eval_sh_chebyt_ptr[2*2+1] = ("eval_sh_chebyt") +ufunc_eval_sh_chebyt_ptr[2*3] = _func_eval_sh_chebyt[double] +ufunc_eval_sh_chebyt_ptr[2*3+1] = ("eval_sh_chebyt") +ufunc_eval_sh_chebyt_ptr[2*4] = _func_eval_sh_chebyt[double_complex] +ufunc_eval_sh_chebyt_ptr[2*4+1] = ("eval_sh_chebyt") +ufunc_eval_sh_chebyt_data[0] = &ufunc_eval_sh_chebyt_ptr[2*0] +ufunc_eval_sh_chebyt_data[1] = &ufunc_eval_sh_chebyt_ptr[2*1] +ufunc_eval_sh_chebyt_data[2] = &ufunc_eval_sh_chebyt_ptr[2*2] +ufunc_eval_sh_chebyt_data[3] = &ufunc_eval_sh_chebyt_ptr[2*3] +ufunc_eval_sh_chebyt_data[4] = &ufunc_eval_sh_chebyt_ptr[2*4] +eval_sh_chebyt = np.PyUFunc_FromFuncAndData(ufunc_eval_sh_chebyt_loops, ufunc_eval_sh_chebyt_data, ufunc_eval_sh_chebyt_types, 5, 2, 1, 0, "eval_sh_chebyt", ufunc_eval_sh_chebyt_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_sh_chebyu_loops[5] +cdef void *ufunc_eval_sh_chebyu_ptr[10] +cdef void *ufunc_eval_sh_chebyu_data[5] +cdef char ufunc_eval_sh_chebyu_types[15] +cdef char *ufunc_eval_sh_chebyu_doc = ( + "eval_sh_chebyu(n, x, out=None)\n" + "\n" + "Evaluate shifted Chebyshev polynomial of the second kind at a\n" + "point.\n" + "\n" + "These polynomials are defined as\n" + "\n" + ".. math::\n" + "\n" + " U_n^*(x) = U_n(2x - 1)\n" + "\n" + "where :math:`U_n` is a Chebyshev polynomial of the first kind. See\n" + "22.5.15 in [AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer, the result is\n" + " determined via the relation to `eval_chebyu`.\n" + "x : array_like\n" + " Points at which to evaluate the shifted Chebyshev polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "U : scalar or ndarray\n" + " Values of the shifted Chebyshev polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_sh_chebyu : roots and quadrature weights of shifted\n" + " Chebychev polynomials of the second kind\n" + "sh_chebyu : shifted Chebyshev polynomial object\n" + "eval_chebyu : evaluate Chebyshev polynomials of the second kind\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_sh_chebyu_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_sh_chebyu_loops[1] = loop_d_dd__As_ff_f +ufunc_eval_sh_chebyu_loops[2] = loop_D_dD__As_fF_F +ufunc_eval_sh_chebyu_loops[3] = loop_d_dd__As_dd_d +ufunc_eval_sh_chebyu_loops[4] = loop_D_dD__As_dD_D +ufunc_eval_sh_chebyu_types[0] = NPY_INTP +ufunc_eval_sh_chebyu_types[1] = NPY_DOUBLE +ufunc_eval_sh_chebyu_types[2] = NPY_DOUBLE +ufunc_eval_sh_chebyu_types[3] = NPY_FLOAT +ufunc_eval_sh_chebyu_types[4] = NPY_FLOAT +ufunc_eval_sh_chebyu_types[5] = NPY_FLOAT +ufunc_eval_sh_chebyu_types[6] = NPY_FLOAT +ufunc_eval_sh_chebyu_types[7] = NPY_CFLOAT +ufunc_eval_sh_chebyu_types[8] = NPY_CFLOAT +ufunc_eval_sh_chebyu_types[9] = NPY_DOUBLE +ufunc_eval_sh_chebyu_types[10] = NPY_DOUBLE +ufunc_eval_sh_chebyu_types[11] = NPY_DOUBLE +ufunc_eval_sh_chebyu_types[12] = NPY_DOUBLE +ufunc_eval_sh_chebyu_types[13] = NPY_CDOUBLE +ufunc_eval_sh_chebyu_types[14] = NPY_CDOUBLE +ufunc_eval_sh_chebyu_ptr[2*0] = _func_eval_sh_chebyu_l +ufunc_eval_sh_chebyu_ptr[2*0+1] = ("eval_sh_chebyu") +ufunc_eval_sh_chebyu_ptr[2*1] = _func_eval_sh_chebyu[double] +ufunc_eval_sh_chebyu_ptr[2*1+1] = ("eval_sh_chebyu") +ufunc_eval_sh_chebyu_ptr[2*2] = _func_eval_sh_chebyu[double_complex] +ufunc_eval_sh_chebyu_ptr[2*2+1] = ("eval_sh_chebyu") +ufunc_eval_sh_chebyu_ptr[2*3] = _func_eval_sh_chebyu[double] +ufunc_eval_sh_chebyu_ptr[2*3+1] = ("eval_sh_chebyu") +ufunc_eval_sh_chebyu_ptr[2*4] = _func_eval_sh_chebyu[double_complex] +ufunc_eval_sh_chebyu_ptr[2*4+1] = ("eval_sh_chebyu") +ufunc_eval_sh_chebyu_data[0] = &ufunc_eval_sh_chebyu_ptr[2*0] +ufunc_eval_sh_chebyu_data[1] = &ufunc_eval_sh_chebyu_ptr[2*1] +ufunc_eval_sh_chebyu_data[2] = &ufunc_eval_sh_chebyu_ptr[2*2] +ufunc_eval_sh_chebyu_data[3] = &ufunc_eval_sh_chebyu_ptr[2*3] +ufunc_eval_sh_chebyu_data[4] = &ufunc_eval_sh_chebyu_ptr[2*4] +eval_sh_chebyu = np.PyUFunc_FromFuncAndData(ufunc_eval_sh_chebyu_loops, ufunc_eval_sh_chebyu_data, ufunc_eval_sh_chebyu_types, 5, 2, 1, 0, "eval_sh_chebyu", ufunc_eval_sh_chebyu_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_sh_jacobi_loops[5] +cdef void *ufunc_eval_sh_jacobi_ptr[10] +cdef void *ufunc_eval_sh_jacobi_data[5] +cdef char ufunc_eval_sh_jacobi_types[25] +cdef char *ufunc_eval_sh_jacobi_doc = ( + "eval_sh_jacobi(n, p, q, x, out=None)\n" + "\n" + "Evaluate shifted Jacobi polynomial at a point.\n" + "\n" + "Defined by\n" + "\n" + ".. math::\n" + "\n" + " G_n^{(p, q)}(x)\n" + " = \\binom{2n + p - 1}{n}^{-1} P_n^{(p - q, q - 1)}(2x - 1),\n" + "\n" + "where :math:`P_n^{(\\cdot, \\cdot)}` is the n-th Jacobi\n" + "polynomial. See 22.5.2 in [AS]_ for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : int\n" + " Degree of the polynomial. If not an integer, the result is\n" + " determined via the relation to `binom` and `eval_jacobi`.\n" + "p : float\n" + " Parameter\n" + "q : float\n" + " Parameter\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "G : scalar or ndarray\n" + " Values of the shifted Jacobi polynomial.\n" + "\n" + "See Also\n" + "--------\n" + "roots_sh_jacobi : roots and quadrature weights of shifted Jacobi\n" + " polynomials\n" + "sh_jacobi : shifted Jacobi polynomial object\n" + "eval_jacobi : evaluate Jacobi polynomials\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_sh_jacobi_loops[0] = loop_d_pddd__As_pddd_d +ufunc_eval_sh_jacobi_loops[1] = loop_d_dddd__As_ffff_f +ufunc_eval_sh_jacobi_loops[2] = loop_D_dddD__As_fffF_F +ufunc_eval_sh_jacobi_loops[3] = loop_d_dddd__As_dddd_d +ufunc_eval_sh_jacobi_loops[4] = loop_D_dddD__As_dddD_D +ufunc_eval_sh_jacobi_types[0] = NPY_INTP +ufunc_eval_sh_jacobi_types[1] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[2] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[3] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[4] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[5] = NPY_FLOAT +ufunc_eval_sh_jacobi_types[6] = NPY_FLOAT +ufunc_eval_sh_jacobi_types[7] = NPY_FLOAT +ufunc_eval_sh_jacobi_types[8] = NPY_FLOAT +ufunc_eval_sh_jacobi_types[9] = NPY_FLOAT +ufunc_eval_sh_jacobi_types[10] = NPY_FLOAT +ufunc_eval_sh_jacobi_types[11] = NPY_FLOAT +ufunc_eval_sh_jacobi_types[12] = NPY_FLOAT +ufunc_eval_sh_jacobi_types[13] = NPY_CFLOAT +ufunc_eval_sh_jacobi_types[14] = NPY_CFLOAT +ufunc_eval_sh_jacobi_types[15] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[16] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[17] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[18] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[19] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[20] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[21] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[22] = NPY_DOUBLE +ufunc_eval_sh_jacobi_types[23] = NPY_CDOUBLE +ufunc_eval_sh_jacobi_types[24] = NPY_CDOUBLE +ufunc_eval_sh_jacobi_ptr[2*0] = _func_eval_sh_jacobi_l +ufunc_eval_sh_jacobi_ptr[2*0+1] = ("eval_sh_jacobi") +ufunc_eval_sh_jacobi_ptr[2*1] = _func_eval_sh_jacobi[double] +ufunc_eval_sh_jacobi_ptr[2*1+1] = ("eval_sh_jacobi") +ufunc_eval_sh_jacobi_ptr[2*2] = _func_eval_sh_jacobi[double_complex] +ufunc_eval_sh_jacobi_ptr[2*2+1] = ("eval_sh_jacobi") +ufunc_eval_sh_jacobi_ptr[2*3] = _func_eval_sh_jacobi[double] +ufunc_eval_sh_jacobi_ptr[2*3+1] = ("eval_sh_jacobi") +ufunc_eval_sh_jacobi_ptr[2*4] = _func_eval_sh_jacobi[double_complex] +ufunc_eval_sh_jacobi_ptr[2*4+1] = ("eval_sh_jacobi") +ufunc_eval_sh_jacobi_data[0] = &ufunc_eval_sh_jacobi_ptr[2*0] +ufunc_eval_sh_jacobi_data[1] = &ufunc_eval_sh_jacobi_ptr[2*1] +ufunc_eval_sh_jacobi_data[2] = &ufunc_eval_sh_jacobi_ptr[2*2] +ufunc_eval_sh_jacobi_data[3] = &ufunc_eval_sh_jacobi_ptr[2*3] +ufunc_eval_sh_jacobi_data[4] = &ufunc_eval_sh_jacobi_ptr[2*4] +eval_sh_jacobi = np.PyUFunc_FromFuncAndData(ufunc_eval_sh_jacobi_loops, ufunc_eval_sh_jacobi_data, ufunc_eval_sh_jacobi_types, 5, 4, 1, 0, "eval_sh_jacobi", ufunc_eval_sh_jacobi_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_eval_sh_legendre_loops[5] +cdef void *ufunc_eval_sh_legendre_ptr[10] +cdef void *ufunc_eval_sh_legendre_data[5] +cdef char ufunc_eval_sh_legendre_types[15] +cdef char *ufunc_eval_sh_legendre_doc = ( + "eval_sh_legendre(n, x, out=None)\n" + "\n" + "Evaluate shifted Legendre polynomial at a point.\n" + "\n" + "These polynomials are defined as\n" + "\n" + ".. math::\n" + "\n" + " P_n^*(x) = P_n(2x - 1)\n" + "\n" + "where :math:`P_n` is a Legendre polynomial. See 2.2.11 in [AS]_\n" + "for details.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Degree of the polynomial. If not an integer, the value is\n" + " determined via the relation to `eval_legendre`.\n" + "x : array_like\n" + " Points at which to evaluate the shifted Legendre polynomial\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "P : scalar or ndarray\n" + " Values of the shifted Legendre polynomial\n" + "\n" + "See Also\n" + "--------\n" + "roots_sh_legendre : roots and quadrature weights of shifted\n" + " Legendre polynomials\n" + "sh_legendre : shifted Legendre polynomial object\n" + "eval_legendre : evaluate Legendre polynomials\n" + "numpy.polynomial.legendre.Legendre : Legendre series\n" + "\n" + "References\n" + "----------\n" + ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.") +ufunc_eval_sh_legendre_loops[0] = loop_d_pd__As_pd_d +ufunc_eval_sh_legendre_loops[1] = loop_d_dd__As_ff_f +ufunc_eval_sh_legendre_loops[2] = loop_D_dD__As_fF_F +ufunc_eval_sh_legendre_loops[3] = loop_d_dd__As_dd_d +ufunc_eval_sh_legendre_loops[4] = loop_D_dD__As_dD_D +ufunc_eval_sh_legendre_types[0] = NPY_INTP +ufunc_eval_sh_legendre_types[1] = NPY_DOUBLE +ufunc_eval_sh_legendre_types[2] = NPY_DOUBLE +ufunc_eval_sh_legendre_types[3] = NPY_FLOAT +ufunc_eval_sh_legendre_types[4] = NPY_FLOAT +ufunc_eval_sh_legendre_types[5] = NPY_FLOAT +ufunc_eval_sh_legendre_types[6] = NPY_FLOAT +ufunc_eval_sh_legendre_types[7] = NPY_CFLOAT +ufunc_eval_sh_legendre_types[8] = NPY_CFLOAT +ufunc_eval_sh_legendre_types[9] = NPY_DOUBLE +ufunc_eval_sh_legendre_types[10] = NPY_DOUBLE +ufunc_eval_sh_legendre_types[11] = NPY_DOUBLE +ufunc_eval_sh_legendre_types[12] = NPY_DOUBLE +ufunc_eval_sh_legendre_types[13] = NPY_CDOUBLE +ufunc_eval_sh_legendre_types[14] = NPY_CDOUBLE +ufunc_eval_sh_legendre_ptr[2*0] = _func_eval_sh_legendre_l +ufunc_eval_sh_legendre_ptr[2*0+1] = ("eval_sh_legendre") +ufunc_eval_sh_legendre_ptr[2*1] = _func_eval_sh_legendre[double] +ufunc_eval_sh_legendre_ptr[2*1+1] = ("eval_sh_legendre") +ufunc_eval_sh_legendre_ptr[2*2] = _func_eval_sh_legendre[double_complex] +ufunc_eval_sh_legendre_ptr[2*2+1] = ("eval_sh_legendre") +ufunc_eval_sh_legendre_ptr[2*3] = _func_eval_sh_legendre[double] +ufunc_eval_sh_legendre_ptr[2*3+1] = ("eval_sh_legendre") +ufunc_eval_sh_legendre_ptr[2*4] = _func_eval_sh_legendre[double_complex] +ufunc_eval_sh_legendre_ptr[2*4+1] = ("eval_sh_legendre") +ufunc_eval_sh_legendre_data[0] = &ufunc_eval_sh_legendre_ptr[2*0] +ufunc_eval_sh_legendre_data[1] = &ufunc_eval_sh_legendre_ptr[2*1] +ufunc_eval_sh_legendre_data[2] = &ufunc_eval_sh_legendre_ptr[2*2] +ufunc_eval_sh_legendre_data[3] = &ufunc_eval_sh_legendre_ptr[2*3] +ufunc_eval_sh_legendre_data[4] = &ufunc_eval_sh_legendre_ptr[2*4] +eval_sh_legendre = np.PyUFunc_FromFuncAndData(ufunc_eval_sh_legendre_loops, ufunc_eval_sh_legendre_data, ufunc_eval_sh_legendre_types, 5, 2, 1, 0, "eval_sh_legendre", ufunc_eval_sh_legendre_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_exp10_loops[2] +cdef void *ufunc_exp10_ptr[4] +cdef void *ufunc_exp10_data[2] +cdef char ufunc_exp10_types[4] +cdef char *ufunc_exp10_doc = ( + "exp10(x, out=None)\n" + "\n" + "Compute ``10**x`` element-wise.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " `x` must contain real numbers.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " ``10**x``, computed element-wise.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy.special import exp10\n" + "\n" + ">>> exp10(3)\n" + "1000.0\n" + ">>> x = np.array([[-1, -0.5, 0], [0.5, 1, 1.5]])\n" + ">>> exp10(x)\n" + "array([[ 0.1 , 0.31622777, 1. ],\n" + " [ 3.16227766, 10. , 31.6227766 ]])") +ufunc_exp10_loops[0] = loop_d_d__As_f_f +ufunc_exp10_loops[1] = loop_d_d__As_d_d +ufunc_exp10_types[0] = NPY_FLOAT +ufunc_exp10_types[1] = NPY_FLOAT +ufunc_exp10_types[2] = NPY_DOUBLE +ufunc_exp10_types[3] = NPY_DOUBLE +ufunc_exp10_ptr[2*0] = _func_cephes_exp10 +ufunc_exp10_ptr[2*0+1] = ("exp10") +ufunc_exp10_ptr[2*1] = _func_cephes_exp10 +ufunc_exp10_ptr[2*1+1] = ("exp10") +ufunc_exp10_data[0] = &ufunc_exp10_ptr[2*0] +ufunc_exp10_data[1] = &ufunc_exp10_ptr[2*1] +exp10 = np.PyUFunc_FromFuncAndData(ufunc_exp10_loops, ufunc_exp10_data, ufunc_exp10_types, 2, 1, 1, 0, "exp10", ufunc_exp10_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_exp2_loops[2] +cdef void *ufunc_exp2_ptr[4] +cdef void *ufunc_exp2_data[2] +cdef char ufunc_exp2_types[4] +cdef char *ufunc_exp2_doc = ( + "exp2(x, out=None)\n" + "\n" + "Compute ``2**x`` element-wise.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " `x` must contain real numbers.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " ``2**x``, computed element-wise.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy.special import exp2\n" + "\n" + ">>> exp2(3)\n" + "8.0\n" + ">>> x = np.array([[-1, -0.5, 0], [0.5, 1, 1.5]])\n" + ">>> exp2(x)\n" + "array([[ 0.5 , 0.70710678, 1. ],\n" + " [ 1.41421356, 2. , 2.82842712]])") +ufunc_exp2_loops[0] = loop_d_d__As_f_f +ufunc_exp2_loops[1] = loop_d_d__As_d_d +ufunc_exp2_types[0] = NPY_FLOAT +ufunc_exp2_types[1] = NPY_FLOAT +ufunc_exp2_types[2] = NPY_DOUBLE +ufunc_exp2_types[3] = NPY_DOUBLE +ufunc_exp2_ptr[2*0] = _func_cephes_exp2 +ufunc_exp2_ptr[2*0+1] = ("exp2") +ufunc_exp2_ptr[2*1] = _func_cephes_exp2 +ufunc_exp2_ptr[2*1+1] = ("exp2") +ufunc_exp2_data[0] = &ufunc_exp2_ptr[2*0] +ufunc_exp2_data[1] = &ufunc_exp2_ptr[2*1] +exp2 = np.PyUFunc_FromFuncAndData(ufunc_exp2_loops, ufunc_exp2_data, ufunc_exp2_types, 2, 1, 1, 0, "exp2", ufunc_exp2_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_expm1_loops[4] +cdef void *ufunc_expm1_ptr[8] +cdef void *ufunc_expm1_data[4] +cdef char ufunc_expm1_types[8] +cdef char *ufunc_expm1_doc = ( + "expm1(x, out=None)\n" + "\n" + "Compute ``exp(x) - 1``.\n" + "\n" + "When `x` is near zero, ``exp(x)`` is near 1, so the numerical calculation\n" + "of ``exp(x) - 1`` can suffer from catastrophic loss of precision.\n" + "``expm1(x)`` is implemented to avoid the loss of precision that occurs when\n" + "`x` is near zero.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " `x` must contain real numbers.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " ``exp(x) - 1`` computed element-wise.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy.special import expm1\n" + "\n" + ">>> expm1(1.0)\n" + "1.7182818284590451\n" + ">>> expm1([-0.2, -0.1, 0, 0.1, 0.2])\n" + "array([-0.18126925, -0.09516258, 0. , 0.10517092, 0.22140276])\n" + "\n" + "The exact value of ``exp(7.5e-13) - 1`` is::\n" + "\n" + " 7.5000000000028125000000007031250000001318...*10**-13.\n" + "\n" + "Here is what ``expm1(7.5e-13)`` gives:\n" + "\n" + ">>> expm1(7.5e-13)\n" + "7.5000000000028135e-13\n" + "\n" + "Compare that to ``exp(7.5e-13) - 1``, where the subtraction results in\n" + "a \"catastrophic\" loss of precision:\n" + "\n" + ">>> np.exp(7.5e-13) - 1\n" + "7.5006667543675576e-13") +ufunc_expm1_loops[0] = loop_d_d__As_f_f +ufunc_expm1_loops[1] = loop_d_d__As_d_d +ufunc_expm1_loops[2] = loop_D_D__As_F_F +ufunc_expm1_loops[3] = loop_D_D__As_D_D +ufunc_expm1_types[0] = NPY_FLOAT +ufunc_expm1_types[1] = NPY_FLOAT +ufunc_expm1_types[2] = NPY_DOUBLE +ufunc_expm1_types[3] = NPY_DOUBLE +ufunc_expm1_types[4] = NPY_CFLOAT +ufunc_expm1_types[5] = NPY_CFLOAT +ufunc_expm1_types[6] = NPY_CDOUBLE +ufunc_expm1_types[7] = NPY_CDOUBLE +ufunc_expm1_ptr[2*0] = _func_cephes_expm1 +ufunc_expm1_ptr[2*0+1] = ("expm1") +ufunc_expm1_ptr[2*1] = _func_cephes_expm1 +ufunc_expm1_ptr[2*1+1] = ("expm1") +ufunc_expm1_ptr[2*2] = _func_cexpm1 +ufunc_expm1_ptr[2*2+1] = ("expm1") +ufunc_expm1_ptr[2*3] = _func_cexpm1 +ufunc_expm1_ptr[2*3+1] = ("expm1") +ufunc_expm1_data[0] = &ufunc_expm1_ptr[2*0] +ufunc_expm1_data[1] = &ufunc_expm1_ptr[2*1] +ufunc_expm1_data[2] = &ufunc_expm1_ptr[2*2] +ufunc_expm1_data[3] = &ufunc_expm1_ptr[2*3] +expm1 = np.PyUFunc_FromFuncAndData(ufunc_expm1_loops, ufunc_expm1_data, ufunc_expm1_types, 4, 1, 1, 0, "expm1", ufunc_expm1_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_expn_loops[3] +cdef void *ufunc_expn_ptr[6] +cdef void *ufunc_expn_data[3] +cdef char ufunc_expn_types[9] +cdef char *ufunc_expn_doc = ( + "expn(n, x, out=None)\n" + "\n" + "Generalized exponential integral En.\n" + "\n" + "For integer :math:`n \\geq 0` and real :math:`x \\geq 0` the\n" + "generalized exponential integral is defined as [dlmf]_\n" + "\n" + ".. math::\n" + "\n" + " E_n(x) = x^{n - 1} \\int_x^\\infty \\frac{e^{-t}}{t^n} dt.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Non-negative integers\n" + "x : array_like\n" + " Real argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the generalized exponential integral\n" + "\n" + "See Also\n" + "--------\n" + "exp1 : special case of :math:`E_n` for :math:`n = 1`\n" + "expi : related to :math:`E_n` when :math:`n = 1`\n" + "\n" + "References\n" + "----------\n" + ".. [dlmf] Digital Library of Mathematical Functions, 8.19.2\n" + " https://dlmf.nist.gov/8.19#E2\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "Its domain is nonnegative n and x.\n" + "\n" + ">>> sc.expn(-1, 1.0), sc.expn(1, -1.0)\n" + "(nan, nan)\n" + "\n" + "It has a pole at ``x = 0`` for ``n = 1, 2``; for larger ``n`` it\n" + "is equal to ``1 / (n - 1)``.\n" + "\n" + ">>> sc.expn([0, 1, 2, 3, 4], 0)\n" + "array([ inf, inf, 1. , 0.5 , 0.33333333])\n" + "\n" + "For n equal to 0 it reduces to ``exp(-x) / x``.\n" + "\n" + ">>> x = np.array([1, 2, 3, 4])\n" + ">>> sc.expn(0, x)\n" + "array([0.36787944, 0.06766764, 0.01659569, 0.00457891])\n" + ">>> np.exp(-x) / x\n" + "array([0.36787944, 0.06766764, 0.01659569, 0.00457891])\n" + "\n" + "For n equal to 1 it reduces to `exp1`.\n" + "\n" + ">>> sc.expn(1, x)\n" + "array([0.21938393, 0.04890051, 0.01304838, 0.00377935])\n" + ">>> sc.exp1(x)\n" + "array([0.21938393, 0.04890051, 0.01304838, 0.00377935])") +ufunc_expn_loops[0] = loop_d_pd__As_pd_d +ufunc_expn_loops[1] = loop_d_dd__As_ff_f +ufunc_expn_loops[2] = loop_d_dd__As_dd_d +ufunc_expn_types[0] = NPY_INTP +ufunc_expn_types[1] = NPY_DOUBLE +ufunc_expn_types[2] = NPY_DOUBLE +ufunc_expn_types[3] = NPY_FLOAT +ufunc_expn_types[4] = NPY_FLOAT +ufunc_expn_types[5] = NPY_FLOAT +ufunc_expn_types[6] = NPY_DOUBLE +ufunc_expn_types[7] = NPY_DOUBLE +ufunc_expn_types[8] = NPY_DOUBLE +ufunc_expn_ptr[2*0] = _func_cephes_expn_wrap +ufunc_expn_ptr[2*0+1] = ("expn") +ufunc_expn_ptr[2*1] = _func_expn_unsafe +ufunc_expn_ptr[2*1+1] = ("expn") +ufunc_expn_ptr[2*2] = _func_expn_unsafe +ufunc_expn_ptr[2*2+1] = ("expn") +ufunc_expn_data[0] = &ufunc_expn_ptr[2*0] +ufunc_expn_data[1] = &ufunc_expn_ptr[2*1] +ufunc_expn_data[2] = &ufunc_expn_ptr[2*2] +expn = np.PyUFunc_FromFuncAndData(ufunc_expn_loops, ufunc_expn_data, ufunc_expn_types, 3, 2, 1, 0, "expn", ufunc_expn_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_fdtr_loops[2] +cdef void *ufunc_fdtr_ptr[4] +cdef void *ufunc_fdtr_data[2] +cdef char ufunc_fdtr_types[8] +cdef char *ufunc_fdtr_doc = ( + "fdtr(dfn, dfd, x, out=None)\n" + "\n" + "F cumulative distribution function.\n" + "\n" + "Returns the value of the cumulative distribution function of the\n" + "F-distribution, also known as Snedecor's F-distribution or the\n" + "Fisher-Snedecor distribution.\n" + "\n" + "The F-distribution with parameters :math:`d_n` and :math:`d_d` is the\n" + "distribution of the random variable,\n" + "\n" + ".. math::\n" + " X = \\frac{U_n/d_n}{U_d/d_d},\n" + "\n" + "where :math:`U_n` and :math:`U_d` are random variables distributed\n" + ":math:`\\chi^2`, with :math:`d_n` and :math:`d_d` degrees of freedom,\n" + "respectively.\n" + "\n" + "Parameters\n" + "----------\n" + "dfn : array_like\n" + " First parameter (positive float).\n" + "dfd : array_like\n" + " Second parameter (positive float).\n" + "x : array_like\n" + " Argument (nonnegative float).\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "y : scalar or ndarray\n" + " The CDF of the F-distribution with parameters `dfn` and `dfd` at `x`.\n" + "\n" + "See Also\n" + "--------\n" + "fdtrc : F distribution survival function\n" + "fdtri : F distribution inverse cumulative distribution\n" + "scipy.stats.f : F distribution\n" + "\n" + "Notes\n" + "-----\n" + "The regularized incomplete beta function is used, according to the\n" + "formula,\n" + "\n" + ".. math::\n" + " F(d_n, d_d; x) = I_{xd_n/(d_d + xd_n)}(d_n/2, d_d/2).\n" + "\n" + "Wrapper for the Cephes [1]_ routine `fdtr`. The F distribution is also\n" + "available as `scipy.stats.f`. Calling `fdtr` directly can improve\n" + "performance compared to the ``cdf`` method of `scipy.stats.f` (see last\n" + "example below).\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + "\n" + "Examples\n" + "--------\n" + "Calculate the function for ``dfn=1`` and ``dfd=2`` at ``x=1``.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import fdtr\n" + ">>> fdtr(1, 2, 1)\n" + "0.5773502691896258\n" + "\n" + "Calculate the function at several points by providing a NumPy array for\n" + "`x`.\n" + "\n" + ">>> x = np.array([0.5, 2., 3.])\n" + ">>> fdtr(1, 2, x)\n" + "array([0.4472136 , 0.70710678, 0.77459667])\n" + "\n" + "Plot the function for several parameter sets.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> dfn_parameters = [1, 5, 10, 50]\n" + ">>> dfd_parameters = [1, 1, 2, 3]\n" + ">>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']\n" + ">>> parameters_list = list(zip(dfn_parameters, dfd_parameters,\n" + "... linestyles))\n" + ">>> x = np.linspace(0, 30, 1000)\n" + ">>> fig, ax = plt.subplots()\n" + ">>> for parameter_set in parameters_list:\n" + "... dfn, dfd, style = parameter_set\n" + "... fdtr_vals = fdtr(dfn, dfd, x)\n" + "... ax.plot(x, fdtr_vals, label=rf\"$d_n={dfn},\\, d_d={dfd}$\",\n" + "... ls=style)\n" + ">>> ax.legend()\n" + ">>> ax.set_xlabel(\"$x$\")\n" + ">>> ax.set_title(\"F distribution cumulative distribution function\")\n" + ">>> plt.show()\n" + "\n" + "The F distribution is also available as `scipy.stats.f`. Using `fdtr`\n" + "directly can be much faster than calling the ``cdf`` method of\n" + "`scipy.stats.f`, especially for small arrays or individual values.\n" + "To get the same results one must use the following parametrization:\n" + "``stats.f(dfn, dfd).cdf(x)=fdtr(dfn, dfd, x)``.\n" + "\n" + ">>> from scipy.stats import f\n" + ">>> dfn, dfd = 1, 2\n" + ">>> x = 1\n" + ">>> fdtr_res = fdtr(dfn, dfd, x) # this will often be faster than below\n" + ">>> f_dist_res = f(dfn, dfd).cdf(x)\n" + ">>> fdtr_res == f_dist_res # test that results are equal\n" + "True") +ufunc_fdtr_loops[0] = loop_d_ddd__As_fff_f +ufunc_fdtr_loops[1] = loop_d_ddd__As_ddd_d +ufunc_fdtr_types[0] = NPY_FLOAT +ufunc_fdtr_types[1] = NPY_FLOAT +ufunc_fdtr_types[2] = NPY_FLOAT +ufunc_fdtr_types[3] = NPY_FLOAT +ufunc_fdtr_types[4] = NPY_DOUBLE +ufunc_fdtr_types[5] = NPY_DOUBLE +ufunc_fdtr_types[6] = NPY_DOUBLE +ufunc_fdtr_types[7] = NPY_DOUBLE +ufunc_fdtr_ptr[2*0] = _func_xsf_fdtr +ufunc_fdtr_ptr[2*0+1] = ("fdtr") +ufunc_fdtr_ptr[2*1] = _func_xsf_fdtr +ufunc_fdtr_ptr[2*1+1] = ("fdtr") +ufunc_fdtr_data[0] = &ufunc_fdtr_ptr[2*0] +ufunc_fdtr_data[1] = &ufunc_fdtr_ptr[2*1] +fdtr = np.PyUFunc_FromFuncAndData(ufunc_fdtr_loops, ufunc_fdtr_data, ufunc_fdtr_types, 2, 3, 1, 0, "fdtr", ufunc_fdtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_fdtrc_loops[2] +cdef void *ufunc_fdtrc_ptr[4] +cdef void *ufunc_fdtrc_data[2] +cdef char ufunc_fdtrc_types[8] +cdef char *ufunc_fdtrc_doc = ( + "fdtrc(dfn, dfd, x, out=None)\n" + "\n" + "F survival function.\n" + "\n" + "Returns the complemented F-distribution function (the integral of the\n" + "density from `x` to infinity).\n" + "\n" + "Parameters\n" + "----------\n" + "dfn : array_like\n" + " First parameter (positive float).\n" + "dfd : array_like\n" + " Second parameter (positive float).\n" + "x : array_like\n" + " Argument (nonnegative float).\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "y : scalar or ndarray\n" + " The complemented F-distribution function with parameters `dfn` and\n" + " `dfd` at `x`.\n" + "\n" + "See Also\n" + "--------\n" + "fdtr : F distribution cumulative distribution function\n" + "fdtri : F distribution inverse cumulative distribution function\n" + "scipy.stats.f : F distribution\n" + "\n" + "Notes\n" + "-----\n" + "The regularized incomplete beta function is used, according to the\n" + "formula,\n" + "\n" + ".. math::\n" + " F(d_n, d_d; x) = I_{d_d/(d_d + xd_n)}(d_d/2, d_n/2).\n" + "\n" + "Wrapper for the Cephes [1]_ routine `fdtrc`. The F distribution is also\n" + "available as `scipy.stats.f`. Calling `fdtrc` directly can improve\n" + "performance compared to the ``sf`` method of `scipy.stats.f` (see last\n" + "example below).\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + "\n" + "Examples\n" + "--------\n" + "Calculate the function for ``dfn=1`` and ``dfd=2`` at ``x=1``.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import fdtrc\n" + ">>> fdtrc(1, 2, 1)\n" + "0.42264973081037427\n" + "\n" + "Calculate the function at several points by providing a NumPy array for\n" + "`x`.\n" + "\n" + ">>> x = np.array([0.5, 2., 3.])\n" + ">>> fdtrc(1, 2, x)\n" + "array([0.5527864 , 0.29289322, 0.22540333])\n" + "\n" + "Plot the function for several parameter sets.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> dfn_parameters = [1, 5, 10, 50]\n" + ">>> dfd_parameters = [1, 1, 2, 3]\n" + ">>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']\n" + ">>> parameters_list = list(zip(dfn_parameters, dfd_parameters,\n" + "... linestyles))\n" + ">>> x = np.linspace(0, 30, 1000)\n" + ">>> fig, ax = plt.subplots()\n" + ">>> for parameter_set in parameters_list:\n" + "... dfn, dfd, style = parameter_set\n" + "... fdtrc_vals = fdtrc(dfn, dfd, x)\n" + "... ax.plot(x, fdtrc_vals, label=rf\"$d_n={dfn},\\, d_d={dfd}$\",\n" + "... ls=style)\n" + ">>> ax.legend()\n" + ">>> ax.set_xlabel(\"$x$\")\n" + ">>> ax.set_title(\"F distribution survival function\")\n" + ">>> plt.show()\n" + "\n" + "The F distribution is also available as `scipy.stats.f`. Using `fdtrc`\n" + "directly can be much faster than calling the ``sf`` method of\n" + "`scipy.stats.f`, especially for small arrays or individual values.\n" + "To get the same results one must use the following parametrization:\n" + "``stats.f(dfn, dfd).sf(x)=fdtrc(dfn, dfd, x)``.\n" + "\n" + ">>> from scipy.stats import f\n" + ">>> dfn, dfd = 1, 2\n" + ">>> x = 1\n" + ">>> fdtrc_res = fdtrc(dfn, dfd, x) # this will often be faster than below\n" + ">>> f_dist_res = f(dfn, dfd).sf(x)\n" + ">>> f_dist_res == fdtrc_res # test that results are equal\n" + "True") +ufunc_fdtrc_loops[0] = loop_d_ddd__As_fff_f +ufunc_fdtrc_loops[1] = loop_d_ddd__As_ddd_d +ufunc_fdtrc_types[0] = NPY_FLOAT +ufunc_fdtrc_types[1] = NPY_FLOAT +ufunc_fdtrc_types[2] = NPY_FLOAT +ufunc_fdtrc_types[3] = NPY_FLOAT +ufunc_fdtrc_types[4] = NPY_DOUBLE +ufunc_fdtrc_types[5] = NPY_DOUBLE +ufunc_fdtrc_types[6] = NPY_DOUBLE +ufunc_fdtrc_types[7] = NPY_DOUBLE +ufunc_fdtrc_ptr[2*0] = _func_xsf_fdtrc +ufunc_fdtrc_ptr[2*0+1] = ("fdtrc") +ufunc_fdtrc_ptr[2*1] = _func_xsf_fdtrc +ufunc_fdtrc_ptr[2*1+1] = ("fdtrc") +ufunc_fdtrc_data[0] = &ufunc_fdtrc_ptr[2*0] +ufunc_fdtrc_data[1] = &ufunc_fdtrc_ptr[2*1] +fdtrc = np.PyUFunc_FromFuncAndData(ufunc_fdtrc_loops, ufunc_fdtrc_data, ufunc_fdtrc_types, 2, 3, 1, 0, "fdtrc", ufunc_fdtrc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_fdtri_loops[2] +cdef void *ufunc_fdtri_ptr[4] +cdef void *ufunc_fdtri_data[2] +cdef char ufunc_fdtri_types[8] +cdef char *ufunc_fdtri_doc = ( + "fdtri(dfn, dfd, p, out=None)\n" + "\n" + "The `p`-th quantile of the F-distribution.\n" + "\n" + "This function is the inverse of the F-distribution CDF, `fdtr`, returning\n" + "the `x` such that `fdtr(dfn, dfd, x) = p`.\n" + "\n" + "Parameters\n" + "----------\n" + "dfn : array_like\n" + " First parameter (positive float).\n" + "dfd : array_like\n" + " Second parameter (positive float).\n" + "p : array_like\n" + " Cumulative probability, in [0, 1].\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "x : scalar or ndarray\n" + " The quantile corresponding to `p`.\n" + "\n" + "See Also\n" + "--------\n" + "fdtr : F distribution cumulative distribution function\n" + "fdtrc : F distribution survival function\n" + "scipy.stats.f : F distribution\n" + "\n" + "Notes\n" + "-----\n" + "The computation is carried out using the relation to the inverse\n" + "regularized beta function, :math:`I^{-1}_x(a, b)`. Let\n" + ":math:`z = I^{-1}_p(d_d/2, d_n/2).` Then,\n" + "\n" + ".. math::\n" + " x = \\frac{d_d (1 - z)}{d_n z}.\n" + "\n" + "If `p` is such that :math:`x < 0.5`, the following relation is used\n" + "instead for improved stability: let\n" + ":math:`z' = I^{-1}_{1 - p}(d_n/2, d_d/2).` Then,\n" + "\n" + ".. math::\n" + " x = \\frac{d_d z'}{d_n (1 - z')}.\n" + "\n" + "Wrapper for the Cephes [1]_ routine `fdtri`.\n" + "\n" + "The F distribution is also available as `scipy.stats.f`. Calling\n" + "`fdtri` directly can improve performance compared to the ``ppf``\n" + "method of `scipy.stats.f` (see last example below).\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + "\n" + "Examples\n" + "--------\n" + "`fdtri` represents the inverse of the F distribution CDF which is\n" + "available as `fdtr`. Here, we calculate the CDF for ``df1=1``, ``df2=2``\n" + "at ``x=3``. `fdtri` then returns ``3`` given the same values for `df1`,\n" + "`df2` and the computed CDF value.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import fdtri, fdtr\n" + ">>> df1, df2 = 1, 2\n" + ">>> x = 3\n" + ">>> cdf_value = fdtr(df1, df2, x)\n" + ">>> fdtri(df1, df2, cdf_value)\n" + "3.000000000000006\n" + "\n" + "Calculate the function at several points by providing a NumPy array for\n" + "`x`.\n" + "\n" + ">>> x = np.array([0.1, 0.4, 0.7])\n" + ">>> fdtri(1, 2, x)\n" + "array([0.02020202, 0.38095238, 1.92156863])\n" + "\n" + "Plot the function for several parameter sets.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> dfn_parameters = [50, 10, 1, 50]\n" + ">>> dfd_parameters = [0.5, 1, 1, 5]\n" + ">>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']\n" + ">>> parameters_list = list(zip(dfn_parameters, dfd_parameters,\n" + "... linestyles))\n" + ">>> x = np.linspace(0, 1, 1000)\n" + ">>> fig, ax = plt.subplots()\n" + ">>> for parameter_set in parameters_list:\n" + "... dfn, dfd, style = parameter_set\n" + "... fdtri_vals = fdtri(dfn, dfd, x)\n" + "... ax.plot(x, fdtri_vals, label=rf\"$d_n={dfn},\\, d_d={dfd}$\",\n" + "... ls=style)\n" + ">>> ax.legend()\n" + ">>> ax.set_xlabel(\"$x$\")\n" + ">>> title = \"F distribution inverse cumulative distribution function\"\n" + ">>> ax.set_title(title)\n" + ">>> ax.set_ylim(0, 30)\n" + ">>> plt.show()\n" + "\n" + "The F distribution is also available as `scipy.stats.f`. Using `fdtri`\n" + "directly can be much faster than calling the ``ppf`` method of\n" + "`scipy.stats.f`, especially for small arrays or individual values.\n" + "To get the same results one must use the following parametrization:\n" + "``stats.f(dfn, dfd).ppf(x)=fdtri(dfn, dfd, x)``.\n" + "\n" + ">>> from scipy.stats import f\n" + ">>> dfn, dfd = 1, 2\n" + ">>> x = 0.7\n" + ">>> fdtri_res = fdtri(dfn, dfd, x) # this will often be faster than below\n" + ">>> f_dist_res = f(dfn, dfd).ppf(x)\n" + ">>> f_dist_res == fdtri_res # test that results are equal\n" + "True") +ufunc_fdtri_loops[0] = loop_d_ddd__As_fff_f +ufunc_fdtri_loops[1] = loop_d_ddd__As_ddd_d +ufunc_fdtri_types[0] = NPY_FLOAT +ufunc_fdtri_types[1] = NPY_FLOAT +ufunc_fdtri_types[2] = NPY_FLOAT +ufunc_fdtri_types[3] = NPY_FLOAT +ufunc_fdtri_types[4] = NPY_DOUBLE +ufunc_fdtri_types[5] = NPY_DOUBLE +ufunc_fdtri_types[6] = NPY_DOUBLE +ufunc_fdtri_types[7] = NPY_DOUBLE +ufunc_fdtri_ptr[2*0] = _func_xsf_fdtri +ufunc_fdtri_ptr[2*0+1] = ("fdtri") +ufunc_fdtri_ptr[2*1] = _func_xsf_fdtri +ufunc_fdtri_ptr[2*1+1] = ("fdtri") +ufunc_fdtri_data[0] = &ufunc_fdtri_ptr[2*0] +ufunc_fdtri_data[1] = &ufunc_fdtri_ptr[2*1] +fdtri = np.PyUFunc_FromFuncAndData(ufunc_fdtri_loops, ufunc_fdtri_data, ufunc_fdtri_types, 2, 3, 1, 0, "fdtri", ufunc_fdtri_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_fdtridfd_loops[2] +cdef void *ufunc_fdtridfd_ptr[4] +cdef void *ufunc_fdtridfd_data[2] +cdef char ufunc_fdtridfd_types[8] +cdef char *ufunc_fdtridfd_doc = ( + "fdtridfd(dfn, p, x, out=None)\n" + "\n" + "Inverse to `fdtr` vs dfd\n" + "\n" + "Finds the F density argument dfd such that ``fdtr(dfn, dfd, x) == p``.\n" + "\n" + "Parameters\n" + "----------\n" + "dfn : array_like\n" + " First parameter (positive float).\n" + "p : array_like\n" + " Cumulative probability, in [0, 1].\n" + "x : array_like\n" + " Argument (nonnegative float).\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "dfd : scalar or ndarray\n" + " `dfd` such that ``fdtr(dfn, dfd, x) == p``.\n" + "\n" + "See Also\n" + "--------\n" + "fdtr : F distribution cumulative distribution function\n" + "fdtrc : F distribution survival function\n" + "fdtri : F distribution quantile function\n" + "scipy.stats.f : F distribution\n" + "\n" + "Examples\n" + "--------\n" + "Compute the F distribution cumulative distribution function for one\n" + "parameter set.\n" + "\n" + ">>> from scipy.special import fdtridfd, fdtr\n" + ">>> dfn, dfd, x = 10, 5, 2\n" + ">>> cdf_value = fdtr(dfn, dfd, x)\n" + ">>> cdf_value\n" + "0.7700248806501017\n" + "\n" + "Verify that `fdtridfd` recovers the original value for `dfd`:\n" + "\n" + ">>> fdtridfd(dfn, cdf_value, x)\n" + "5.0") +ufunc_fdtridfd_loops[0] = loop_d_ddd__As_fff_f +ufunc_fdtridfd_loops[1] = loop_d_ddd__As_ddd_d +ufunc_fdtridfd_types[0] = NPY_FLOAT +ufunc_fdtridfd_types[1] = NPY_FLOAT +ufunc_fdtridfd_types[2] = NPY_FLOAT +ufunc_fdtridfd_types[3] = NPY_FLOAT +ufunc_fdtridfd_types[4] = NPY_DOUBLE +ufunc_fdtridfd_types[5] = NPY_DOUBLE +ufunc_fdtridfd_types[6] = NPY_DOUBLE +ufunc_fdtridfd_types[7] = NPY_DOUBLE +ufunc_fdtridfd_ptr[2*0] = _func_fdtridfd +ufunc_fdtridfd_ptr[2*0+1] = ("fdtridfd") +ufunc_fdtridfd_ptr[2*1] = _func_fdtridfd +ufunc_fdtridfd_ptr[2*1+1] = ("fdtridfd") +ufunc_fdtridfd_data[0] = &ufunc_fdtridfd_ptr[2*0] +ufunc_fdtridfd_data[1] = &ufunc_fdtridfd_ptr[2*1] +fdtridfd = np.PyUFunc_FromFuncAndData(ufunc_fdtridfd_loops, ufunc_fdtridfd_data, ufunc_fdtridfd_types, 2, 3, 1, 0, "fdtridfd", ufunc_fdtridfd_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_gdtr_loops[2] +cdef void *ufunc_gdtr_ptr[4] +cdef void *ufunc_gdtr_data[2] +cdef char ufunc_gdtr_types[8] +cdef char *ufunc_gdtr_doc = ( + "gdtr(a, b, x, out=None)\n" + "\n" + "Gamma distribution cumulative distribution function.\n" + "\n" + "Returns the integral from zero to `x` of the gamma probability density\n" + "function,\n" + "\n" + ".. math::\n" + "\n" + " F = \\int_0^x \\frac{a^b}{\\Gamma(b)} t^{b-1} e^{-at}\\,dt,\n" + "\n" + "where :math:`\\Gamma` is the gamma function.\n" + "\n" + "Parameters\n" + "----------\n" + "a : array_like\n" + " The rate parameter of the gamma distribution, sometimes denoted\n" + " :math:`\\beta` (float). It is also the reciprocal of the scale\n" + " parameter :math:`\\theta`.\n" + "b : array_like\n" + " The shape parameter of the gamma distribution, sometimes denoted\n" + " :math:`\\alpha` (float).\n" + "x : array_like\n" + " The quantile (upper limit of integration; float).\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "F : scalar or ndarray\n" + " The CDF of the gamma distribution with parameters `a` and `b`\n" + " evaluated at `x`.\n" + "\n" + "See Also\n" + "--------\n" + "gdtrc : 1 - CDF of the gamma distribution.\n" + "scipy.stats.gamma: Gamma distribution\n" + "\n" + "Notes\n" + "-----\n" + "The evaluation is carried out using the relation to the incomplete gamma\n" + "integral (regularized gamma function).\n" + "\n" + "Wrapper for the Cephes [1]_ routine `gdtr`. Calling `gdtr` directly can\n" + "improve performance compared to the ``cdf`` method of `scipy.stats.gamma`\n" + "(see last example below).\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + "\n" + "Examples\n" + "--------\n" + "Compute the function for ``a=1``, ``b=2`` at ``x=5``.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import gdtr\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> gdtr(1., 2., 5.)\n" + "0.9595723180054873\n" + "\n" + "Compute the function for ``a=1`` and ``b=2`` at several points by\n" + "providing a NumPy array for `x`.\n" + "\n" + ">>> xvalues = np.array([1., 2., 3., 4])\n" + ">>> gdtr(1., 1., xvalues)\n" + "array([0.63212056, 0.86466472, 0.95021293, 0.98168436])\n" + "\n" + "`gdtr` can evaluate different parameter sets by providing arrays with\n" + "broadcasting compatible shapes for `a`, `b` and `x`. Here we compute the\n" + "function for three different `a` at four positions `x` and ``b=3``,\n" + "resulting in a 3x4 array.\n" + "\n" + ">>> a = np.array([[0.5], [1.5], [2.5]])\n" + ">>> x = np.array([1., 2., 3., 4])\n" + ">>> a.shape, x.shape\n" + "((3, 1), (4,))\n" + "\n" + ">>> gdtr(a, 3., x)\n" + "array([[0.01438768, 0.0803014 , 0.19115317, 0.32332358],\n" + " [0.19115317, 0.57680992, 0.82642193, 0.9380312 ],\n" + " [0.45618688, 0.87534798, 0.97974328, 0.9972306 ]])\n" + "\n" + "Plot the function for four different parameter sets.\n" + "\n" + ">>> a_parameters = [0.3, 1, 2, 6]\n" + ">>> b_parameters = [2, 10, 15, 20]\n" + ">>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']\n" + ">>> parameters_list = list(zip(a_parameters, b_parameters, linestyles))\n" + ">>> x = np.linspace(0, 30, 1000)\n" + ">>> fig, ax = plt.subplots()\n" + ">>> for parameter_set in parameters_list:\n" + "... a, b, style = parameter_set\n" + "... gdtr_vals = gdtr(a, b, x)\n" + "... ax.plot(x, gdtr_vals, label=fr\"$a= {a},\\, b={b}$\", ls=style)\n" + ">>> ax.legend()\n" + ">>> ax.set_xlabel(\"$x$\")\n" + ">>> ax.set_title(\"Gamma distribution cumulative distribution function\")\n" + ">>> plt.show()\n" + "\n" + "The gamma distribution is also available as `scipy.stats.gamma`. Using\n" + "`gdtr` directly can be much faster than calling the ``cdf`` method of\n" + "`scipy.stats.gamma`, especially for small arrays or individual values.\n" + "To get the same results one must use the following parametrization:\n" + "``stats.gamma(b, scale=1/a).cdf(x)=gdtr(a, b, x)``.\n" + "\n" + ">>> from scipy.stats import gamma\n" + ">>> a = 2.\n" + ">>> b = 3\n" + ">>> x = 1.\n" + ">>> gdtr_result = gdtr(a, b, x) # this will often be faster than below\n" + ">>> gamma_dist_result = gamma(b, scale=1/a).cdf(x)\n" + ">>> gdtr_result == gamma_dist_result # test that results are equal\n" + "True") +ufunc_gdtr_loops[0] = loop_d_ddd__As_fff_f +ufunc_gdtr_loops[1] = loop_d_ddd__As_ddd_d +ufunc_gdtr_types[0] = NPY_FLOAT +ufunc_gdtr_types[1] = NPY_FLOAT +ufunc_gdtr_types[2] = NPY_FLOAT +ufunc_gdtr_types[3] = NPY_FLOAT +ufunc_gdtr_types[4] = NPY_DOUBLE +ufunc_gdtr_types[5] = NPY_DOUBLE +ufunc_gdtr_types[6] = NPY_DOUBLE +ufunc_gdtr_types[7] = NPY_DOUBLE +ufunc_gdtr_ptr[2*0] = _func_xsf_gdtr +ufunc_gdtr_ptr[2*0+1] = ("gdtr") +ufunc_gdtr_ptr[2*1] = _func_xsf_gdtr +ufunc_gdtr_ptr[2*1+1] = ("gdtr") +ufunc_gdtr_data[0] = &ufunc_gdtr_ptr[2*0] +ufunc_gdtr_data[1] = &ufunc_gdtr_ptr[2*1] +gdtr = np.PyUFunc_FromFuncAndData(ufunc_gdtr_loops, ufunc_gdtr_data, ufunc_gdtr_types, 2, 3, 1, 0, "gdtr", ufunc_gdtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_gdtrc_loops[2] +cdef void *ufunc_gdtrc_ptr[4] +cdef void *ufunc_gdtrc_data[2] +cdef char ufunc_gdtrc_types[8] +cdef char *ufunc_gdtrc_doc = ( + "gdtrc(a, b, x, out=None)\n" + "\n" + "Gamma distribution survival function.\n" + "\n" + "Integral from `x` to infinity of the gamma probability density function,\n" + "\n" + ".. math::\n" + "\n" + " F = \\int_x^\\infty \\frac{a^b}{\\Gamma(b)} t^{b-1} e^{-at}\\,dt,\n" + "\n" + "where :math:`\\Gamma` is the gamma function.\n" + "\n" + "Parameters\n" + "----------\n" + "a : array_like\n" + " The rate parameter of the gamma distribution, sometimes denoted\n" + " :math:`\\beta` (float). It is also the reciprocal of the scale\n" + " parameter :math:`\\theta`.\n" + "b : array_like\n" + " The shape parameter of the gamma distribution, sometimes denoted\n" + " :math:`\\alpha` (float).\n" + "x : array_like\n" + " The quantile (lower limit of integration; float).\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "F : scalar or ndarray\n" + " The survival function of the gamma distribution with parameters `a`\n" + " and `b` evaluated at `x`.\n" + "\n" + "See Also\n" + "--------\n" + "gdtr: Gamma distribution cumulative distribution function\n" + "scipy.stats.gamma: Gamma distribution\n" + "gdtrix\n" + "\n" + "Notes\n" + "-----\n" + "The evaluation is carried out using the relation to the incomplete gamma\n" + "integral (regularized gamma function).\n" + "\n" + "Wrapper for the Cephes [1]_ routine `gdtrc`. Calling `gdtrc` directly can\n" + "improve performance compared to the ``sf`` method of `scipy.stats.gamma`\n" + "(see last example below).\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + "\n" + "Examples\n" + "--------\n" + "Compute the function for ``a=1`` and ``b=2`` at ``x=5``.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import gdtrc\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> gdtrc(1., 2., 5.)\n" + "0.04042768199451279\n" + "\n" + "Compute the function for ``a=1``, ``b=2`` at several points by providing\n" + "a NumPy array for `x`.\n" + "\n" + ">>> xvalues = np.array([1., 2., 3., 4])\n" + ">>> gdtrc(1., 1., xvalues)\n" + "array([0.36787944, 0.13533528, 0.04978707, 0.01831564])\n" + "\n" + "`gdtrc` can evaluate different parameter sets by providing arrays with\n" + "broadcasting compatible shapes for `a`, `b` and `x`. Here we compute the\n" + "function for three different `a` at four positions `x` and ``b=3``,\n" + "resulting in a 3x4 array.\n" + "\n" + ">>> a = np.array([[0.5], [1.5], [2.5]])\n" + ">>> x = np.array([1., 2., 3., 4])\n" + ">>> a.shape, x.shape\n" + "((3, 1), (4,))\n" + "\n" + ">>> gdtrc(a, 3., x)\n" + "array([[0.98561232, 0.9196986 , 0.80884683, 0.67667642],\n" + " [0.80884683, 0.42319008, 0.17357807, 0.0619688 ],\n" + " [0.54381312, 0.12465202, 0.02025672, 0.0027694 ]])\n" + "\n" + "Plot the function for four different parameter sets.\n" + "\n" + ">>> a_parameters = [0.3, 1, 2, 6]\n" + ">>> b_parameters = [2, 10, 15, 20]\n" + ">>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']\n" + ">>> parameters_list = list(zip(a_parameters, b_parameters, linestyles))\n" + ">>> x = np.linspace(0, 30, 1000)\n" + ">>> fig, ax = plt.subplots()\n" + ">>> for parameter_set in parameters_list:\n" + "... a, b, style = parameter_set\n" + "... gdtrc_vals = gdtrc(a, b, x)\n" + "... ax.plot(x, gdtrc_vals, label=fr\"$a= {a},\\, b={b}$\", ls=style)\n" + ">>> ax.legend()\n" + ">>> ax.set_xlabel(\"$x$\")\n" + ">>> ax.set_title(\"Gamma distribution survival function\")\n" + ">>> plt.show()\n" + "\n" + "The gamma distribution is also available as `scipy.stats.gamma`.\n" + "Using `gdtrc` directly can be much faster than calling the ``sf`` method\n" + "of `scipy.stats.gamma`, especially for small arrays or individual\n" + "values. To get the same results one must use the following parametrization:\n" + "``stats.gamma(b, scale=1/a).sf(x)=gdtrc(a, b, x)``.\n" + "\n" + ">>> from scipy.stats import gamma\n" + ">>> a = 2\n" + ">>> b = 3\n" + ">>> x = 1.\n" + ">>> gdtrc_result = gdtrc(a, b, x) # this will often be faster than below\n" + ">>> gamma_dist_result = gamma(b, scale=1/a).sf(x)\n" + ">>> gdtrc_result == gamma_dist_result # test that results are equal\n" + "True") +ufunc_gdtrc_loops[0] = loop_d_ddd__As_fff_f +ufunc_gdtrc_loops[1] = loop_d_ddd__As_ddd_d +ufunc_gdtrc_types[0] = NPY_FLOAT +ufunc_gdtrc_types[1] = NPY_FLOAT +ufunc_gdtrc_types[2] = NPY_FLOAT +ufunc_gdtrc_types[3] = NPY_FLOAT +ufunc_gdtrc_types[4] = NPY_DOUBLE +ufunc_gdtrc_types[5] = NPY_DOUBLE +ufunc_gdtrc_types[6] = NPY_DOUBLE +ufunc_gdtrc_types[7] = NPY_DOUBLE +ufunc_gdtrc_ptr[2*0] = _func_xsf_gdtrc +ufunc_gdtrc_ptr[2*0+1] = ("gdtrc") +ufunc_gdtrc_ptr[2*1] = _func_xsf_gdtrc +ufunc_gdtrc_ptr[2*1+1] = ("gdtrc") +ufunc_gdtrc_data[0] = &ufunc_gdtrc_ptr[2*0] +ufunc_gdtrc_data[1] = &ufunc_gdtrc_ptr[2*1] +gdtrc = np.PyUFunc_FromFuncAndData(ufunc_gdtrc_loops, ufunc_gdtrc_data, ufunc_gdtrc_types, 2, 3, 1, 0, "gdtrc", ufunc_gdtrc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_gdtria_loops[2] +cdef void *ufunc_gdtria_ptr[4] +cdef void *ufunc_gdtria_data[2] +cdef char ufunc_gdtria_types[8] +cdef char *ufunc_gdtria_doc = ( + "gdtria(p, b, x, out=None)\n" + "\n" + "Inverse of `gdtr` vs a.\n" + "\n" + "Returns the inverse with respect to the parameter `a` of ``p =\n" + "gdtr(a, b, x)``, the cumulative distribution function of the gamma\n" + "distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Probability values.\n" + "b : array_like\n" + " `b` parameter values of `gdtr(a, b, x)`. `b` is the \"shape\" parameter\n" + " of the gamma distribution.\n" + "x : array_like\n" + " Nonnegative real values, from the domain of the gamma distribution.\n" + "out : ndarray, optional\n" + " If a fourth argument is given, it must be a numpy.ndarray whose size\n" + " matches the broadcast result of `a`, `b` and `x`. `out` is then the\n" + " array returned by the function.\n" + "\n" + "Returns\n" + "-------\n" + "a : scalar or ndarray\n" + " Values of the `a` parameter such that ``p = gdtr(a, b, x)`. ``1/a``\n" + " is the \"scale\" parameter of the gamma distribution.\n" + "\n" + "See Also\n" + "--------\n" + "gdtr : CDF of the gamma distribution.\n" + "gdtrib : Inverse with respect to `b` of `gdtr(a, b, x)`.\n" + "gdtrix : Inverse with respect to `x` of `gdtr(a, b, x)`.\n" + "\n" + "Notes\n" + "-----\n" + "Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`.\n" + "\n" + "The cumulative distribution function `p` is computed using a routine by\n" + "DiDinato and Morris [2]_. Computation of `a` involves a search for a value\n" + "that produces the desired value of `p`. The search relies on the\n" + "monotonicity of `p` with `a`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Barry Brown, James Lovato, and Kathy Russell,\n" + " CDFLIB: Library of Fortran Routines for Cumulative Distribution\n" + " Functions, Inverses, and Other Parameters.\n" + ".. [2] DiDinato, A. R. and Morris, A. H.,\n" + " Computation of the incomplete gamma function ratios and their\n" + " inverse. ACM Trans. Math. Softw. 12 (1986), 377-393.\n" + "\n" + "Examples\n" + "--------\n" + "First evaluate `gdtr`.\n" + "\n" + ">>> from scipy.special import gdtr, gdtria\n" + ">>> p = gdtr(1.2, 3.4, 5.6)\n" + ">>> print(p)\n" + "0.94378087442\n" + "\n" + "Verify the inverse.\n" + "\n" + ">>> gdtria(p, 3.4, 5.6)\n" + "1.2") +ufunc_gdtria_loops[0] = loop_d_ddd__As_fff_f +ufunc_gdtria_loops[1] = loop_d_ddd__As_ddd_d +ufunc_gdtria_types[0] = NPY_FLOAT +ufunc_gdtria_types[1] = NPY_FLOAT +ufunc_gdtria_types[2] = NPY_FLOAT +ufunc_gdtria_types[3] = NPY_FLOAT +ufunc_gdtria_types[4] = NPY_DOUBLE +ufunc_gdtria_types[5] = NPY_DOUBLE +ufunc_gdtria_types[6] = NPY_DOUBLE +ufunc_gdtria_types[7] = NPY_DOUBLE +ufunc_gdtria_ptr[2*0] = _func_gdtria +ufunc_gdtria_ptr[2*0+1] = ("gdtria") +ufunc_gdtria_ptr[2*1] = _func_gdtria +ufunc_gdtria_ptr[2*1+1] = ("gdtria") +ufunc_gdtria_data[0] = &ufunc_gdtria_ptr[2*0] +ufunc_gdtria_data[1] = &ufunc_gdtria_ptr[2*1] +gdtria = np.PyUFunc_FromFuncAndData(ufunc_gdtria_loops, ufunc_gdtria_data, ufunc_gdtria_types, 2, 3, 1, 0, "gdtria", ufunc_gdtria_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_gdtrib_loops[2] +cdef void *ufunc_gdtrib_ptr[4] +cdef void *ufunc_gdtrib_data[2] +cdef char ufunc_gdtrib_types[8] +cdef char *ufunc_gdtrib_doc = ( + "gdtrib(a, p, x, out=None)\n" + "\n" + "Inverse of `gdtr` vs b.\n" + "\n" + "Returns the inverse with respect to the parameter `b` of ``p =\n" + "gdtr(a, b, x)``, the cumulative distribution function of the gamma\n" + "distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "a : array_like\n" + " `a` parameter values of ``gdtr(a, b, x)`. ``1/a`` is the \"scale\"\n" + " parameter of the gamma distribution.\n" + "p : array_like\n" + " Probability values.\n" + "x : array_like\n" + " Nonnegative real values, from the domain of the gamma distribution.\n" + "out : ndarray, optional\n" + " If a fourth argument is given, it must be a numpy.ndarray whose size\n" + " matches the broadcast result of `a`, `b` and `x`. `out` is then the\n" + " array returned by the function.\n" + "\n" + "Returns\n" + "-------\n" + "b : scalar or ndarray\n" + " Values of the `b` parameter such that `p = gdtr(a, b, x)`. `b` is\n" + " the \"shape\" parameter of the gamma distribution.\n" + "\n" + "See Also\n" + "--------\n" + "gdtr : CDF of the gamma distribution.\n" + "gdtria : Inverse with respect to `a` of `gdtr(a, b, x)`.\n" + "gdtrix : Inverse with respect to `x` of `gdtr(a, b, x)`.\n" + "\n" + "Notes\n" + "-----\n" + "\n" + "The cumulative distribution function `p` is computed using the Cephes [1]_\n" + "routines `igam` and `igamc`. Computation of `b` involves a search for a value\n" + "that produces the desired value of `p` using Chandrupatla's bracketing\n" + "root finding algorithm [2]_.\n" + "\n" + "Note that there are some edge cases where `gdtrib` is extended by taking\n" + "limits where they are uniquely defined. In particular\n" + "``x == 0`` with ``p > 0`` and ``p == 0`` with ``x > 0``.\n" + "For these edge cases, a numerical result will be returned for\n" + "``gdtrib(a, p, x)`` even though ``gdtr(a, gdtrib(a, p, x), x)`` is\n" + "undefined.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + ".. [2] Chandrupatla, Tirupathi R.\n" + " \"A new hybrid quadratic/bisection algorithm for finding the zero of a\n" + " nonlinear function without using derivatives\".\n" + " Advances in Engineering Software, 28(3), 145-149.\n" + " https://doi.org/10.1016/s0965-9978(96)00051-8\n" + "\n" + "Examples\n" + "--------\n" + "First evaluate `gdtr`.\n" + "\n" + ">>> from scipy.special import gdtr, gdtrib\n" + ">>> p = gdtr(1.2, 3.4, 5.6)\n" + ">>> print(p)\n" + "0.94378087442\n" + "\n" + "Verify the inverse.\n" + "\n" + ">>> gdtrib(1.2, p, 5.6)\n" + "3.3999999999999995") +ufunc_gdtrib_loops[0] = loop_d_ddd__As_fff_f +ufunc_gdtrib_loops[1] = loop_d_ddd__As_ddd_d +ufunc_gdtrib_types[0] = NPY_FLOAT +ufunc_gdtrib_types[1] = NPY_FLOAT +ufunc_gdtrib_types[2] = NPY_FLOAT +ufunc_gdtrib_types[3] = NPY_FLOAT +ufunc_gdtrib_types[4] = NPY_DOUBLE +ufunc_gdtrib_types[5] = NPY_DOUBLE +ufunc_gdtrib_types[6] = NPY_DOUBLE +ufunc_gdtrib_types[7] = NPY_DOUBLE +ufunc_gdtrib_ptr[2*0] = _func_xsf_gdtrib +ufunc_gdtrib_ptr[2*0+1] = ("gdtrib") +ufunc_gdtrib_ptr[2*1] = _func_xsf_gdtrib +ufunc_gdtrib_ptr[2*1+1] = ("gdtrib") +ufunc_gdtrib_data[0] = &ufunc_gdtrib_ptr[2*0] +ufunc_gdtrib_data[1] = &ufunc_gdtrib_ptr[2*1] +gdtrib = np.PyUFunc_FromFuncAndData(ufunc_gdtrib_loops, ufunc_gdtrib_data, ufunc_gdtrib_types, 2, 3, 1, 0, "gdtrib", ufunc_gdtrib_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_gdtrix_loops[2] +cdef void *ufunc_gdtrix_ptr[4] +cdef void *ufunc_gdtrix_data[2] +cdef char ufunc_gdtrix_types[8] +cdef char *ufunc_gdtrix_doc = ( + "gdtrix(a, b, p, out=None)\n" + "\n" + "Inverse of `gdtr` vs x.\n" + "\n" + "Returns the inverse with respect to the parameter `x` of ``p =\n" + "gdtr(a, b, x)``, the cumulative distribution function of the gamma\n" + "distribution. This is also known as the pth quantile of the\n" + "distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "a : array_like\n" + " `a` parameter values of ``gdtr(a, b, x)``. ``1/a`` is the \"scale\"\n" + " parameter of the gamma distribution.\n" + "b : array_like\n" + " `b` parameter values of ``gdtr(a, b, x)``. `b` is the \"shape\" parameter\n" + " of the gamma distribution.\n" + "p : array_like\n" + " Probability values.\n" + "out : ndarray, optional\n" + " If a fourth argument is given, it must be a numpy.ndarray whose size\n" + " matches the broadcast result of `a`, `b` and `x`. `out` is then the\n" + " array returned by the function.\n" + "\n" + "Returns\n" + "-------\n" + "x : scalar or ndarray\n" + " Values of the `x` parameter such that `p = gdtr(a, b, x)`.\n" + "\n" + "See Also\n" + "--------\n" + "gdtr : CDF of the gamma distribution.\n" + "gdtria : Inverse with respect to `a` of ``gdtr(a, b, x)``.\n" + "gdtrib : Inverse with respect to `b` of ``gdtr(a, b, x)``.\n" + "\n" + "Notes\n" + "-----\n" + "Wrapper for the CDFLIB [1]_ Fortran routine `cdfgam`.\n" + "\n" + "The cumulative distribution function `p` is computed using a routine by\n" + "DiDinato and Morris [2]_. Computation of `x` involves a search for a value\n" + "that produces the desired value of `p`. The search relies on the\n" + "monotonicity of `p` with `x`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Barry Brown, James Lovato, and Kathy Russell,\n" + " CDFLIB: Library of Fortran Routines for Cumulative Distribution\n" + " Functions, Inverses, and Other Parameters.\n" + ".. [2] DiDinato, A. R. and Morris, A. H.,\n" + " Computation of the incomplete gamma function ratios and their\n" + " inverse. ACM Trans. Math. Softw. 12 (1986), 377-393.\n" + "\n" + "Examples\n" + "--------\n" + "First evaluate `gdtr`.\n" + "\n" + ">>> from scipy.special import gdtr, gdtrix\n" + ">>> p = gdtr(1.2, 3.4, 5.6)\n" + ">>> print(p)\n" + "0.94378087442\n" + "\n" + "Verify the inverse.\n" + "\n" + ">>> gdtrix(1.2, 3.4, p)\n" + "5.5999999999999996") +ufunc_gdtrix_loops[0] = loop_d_ddd__As_fff_f +ufunc_gdtrix_loops[1] = loop_d_ddd__As_ddd_d +ufunc_gdtrix_types[0] = NPY_FLOAT +ufunc_gdtrix_types[1] = NPY_FLOAT +ufunc_gdtrix_types[2] = NPY_FLOAT +ufunc_gdtrix_types[3] = NPY_FLOAT +ufunc_gdtrix_types[4] = NPY_DOUBLE +ufunc_gdtrix_types[5] = NPY_DOUBLE +ufunc_gdtrix_types[6] = NPY_DOUBLE +ufunc_gdtrix_types[7] = NPY_DOUBLE +ufunc_gdtrix_ptr[2*0] = _func_gdtrix +ufunc_gdtrix_ptr[2*0+1] = ("gdtrix") +ufunc_gdtrix_ptr[2*1] = _func_gdtrix +ufunc_gdtrix_ptr[2*1+1] = ("gdtrix") +ufunc_gdtrix_data[0] = &ufunc_gdtrix_ptr[2*0] +ufunc_gdtrix_data[1] = &ufunc_gdtrix_ptr[2*1] +gdtrix = np.PyUFunc_FromFuncAndData(ufunc_gdtrix_loops, ufunc_gdtrix_data, ufunc_gdtrix_types, 2, 3, 1, 0, "gdtrix", ufunc_gdtrix_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_huber_loops[2] +cdef void *ufunc_huber_ptr[4] +cdef void *ufunc_huber_data[2] +cdef char ufunc_huber_types[6] +cdef char *ufunc_huber_doc = ( + "huber(delta, r, out=None)\n" + "\n" + "Huber loss function.\n" + "\n" + ".. math:: \\text{huber}(\\delta, r) = \\begin{cases} \\infty & \\delta < 0 \\\\\n" + " \\frac{1}{2}r^2 & 0 \\le \\delta, | r | \\le \\delta \\\\\n" + " \\delta ( |r| - \\frac{1}{2}\\delta ) & \\text{otherwise} \\end{cases}\n" + "\n" + "Parameters\n" + "----------\n" + "delta : ndarray\n" + " Input array, indicating the quadratic vs. linear loss changepoint.\n" + "r : ndarray\n" + " Input array, possibly representing residuals.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The computed Huber loss function values.\n" + "\n" + "See Also\n" + "--------\n" + "pseudo_huber : smooth approximation of this function\n" + "\n" + "Notes\n" + "-----\n" + "`huber` is useful as a loss function in robust statistics or machine\n" + "learning to reduce the influence of outliers as compared to the common\n" + "squared error loss, residuals with a magnitude higher than `delta` are\n" + "not squared [1]_.\n" + "\n" + "Typically, `r` represents residuals, the difference\n" + "between a model prediction and data. Then, for :math:`|r|\\leq\\delta`,\n" + "`huber` resembles the squared error and for :math:`|r|>\\delta` the\n" + "absolute error. This way, the Huber loss often achieves\n" + "a fast convergence in model fitting for small residuals like the squared\n" + "error loss function and still reduces the influence of outliers\n" + "(:math:`|r|>\\delta`) like the absolute error loss. As :math:`\\delta` is\n" + "the cutoff between squared and absolute error regimes, it has\n" + "to be tuned carefully for each problem. `huber` is also\n" + "convex, making it suitable for gradient based optimization.\n" + "\n" + ".. versionadded:: 0.15.0\n" + "\n" + "References\n" + "----------\n" + ".. [1] Peter Huber. \"Robust Estimation of a Location Parameter\",\n" + " 1964. Annals of Statistics. 53 (1): 73 - 101.\n" + "\n" + "Examples\n" + "--------\n" + "Import all necessary modules.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import huber\n" + ">>> import matplotlib.pyplot as plt\n" + "\n" + "Compute the function for ``delta=1`` at ``r=2``\n" + "\n" + ">>> huber(1., 2.)\n" + "1.5\n" + "\n" + "Compute the function for different `delta` by providing a NumPy array or\n" + "list for `delta`.\n" + "\n" + ">>> huber([1., 3., 5.], 4.)\n" + "array([3.5, 7.5, 8. ])\n" + "\n" + "Compute the function at different points by providing a NumPy array or\n" + "list for `r`.\n" + "\n" + ">>> huber(2., np.array([1., 1.5, 3.]))\n" + "array([0.5 , 1.125, 4. ])\n" + "\n" + "The function can be calculated for different `delta` and `r` by\n" + "providing arrays for both with compatible shapes for broadcasting.\n" + "\n" + ">>> r = np.array([1., 2.5, 8., 10.])\n" + ">>> deltas = np.array([[1.], [5.], [9.]])\n" + ">>> print(r.shape, deltas.shape)\n" + "(4,) (3, 1)\n" + "\n" + ">>> huber(deltas, r)\n" + "array([[ 0.5 , 2. , 7.5 , 9.5 ],\n" + " [ 0.5 , 3.125, 27.5 , 37.5 ],\n" + " [ 0.5 , 3.125, 32. , 49.5 ]])\n" + "\n" + "Plot the function for different `delta`.\n" + "\n" + ">>> x = np.linspace(-4, 4, 500)\n" + ">>> deltas = [1, 2, 3]\n" + ">>> linestyles = [\"dashed\", \"dotted\", \"dashdot\"]\n" + ">>> fig, ax = plt.subplots()\n" + ">>> combined_plot_parameters = list(zip(deltas, linestyles))\n" + ">>> for delta, style in combined_plot_parameters:\n" + "... ax.plot(x, huber(delta, x), label=fr\"$\\delta={delta}$\", ls=style)\n" + ">>> ax.legend(loc=\"upper center\")\n" + ">>> ax.set_xlabel(\"$x$\")\n" + ">>> ax.set_title(r\"Huber loss function $h_{\\delta}(x)$\")\n" + ">>> ax.set_xlim(-4, 4)\n" + ">>> ax.set_ylim(0, 8)\n" + ">>> plt.show()") +ufunc_huber_loops[0] = loop_d_dd__As_ff_f +ufunc_huber_loops[1] = loop_d_dd__As_dd_d +ufunc_huber_types[0] = NPY_FLOAT +ufunc_huber_types[1] = NPY_FLOAT +ufunc_huber_types[2] = NPY_FLOAT +ufunc_huber_types[3] = NPY_DOUBLE +ufunc_huber_types[4] = NPY_DOUBLE +ufunc_huber_types[5] = NPY_DOUBLE +ufunc_huber_ptr[2*0] = _func_huber +ufunc_huber_ptr[2*0+1] = ("huber") +ufunc_huber_ptr[2*1] = _func_huber +ufunc_huber_ptr[2*1+1] = ("huber") +ufunc_huber_data[0] = &ufunc_huber_ptr[2*0] +ufunc_huber_data[1] = &ufunc_huber_ptr[2*1] +huber = np.PyUFunc_FromFuncAndData(ufunc_huber_loops, ufunc_huber_data, ufunc_huber_types, 2, 2, 1, 0, "huber", ufunc_huber_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_hyp0f1_loops[4] +cdef void *ufunc_hyp0f1_ptr[8] +cdef void *ufunc_hyp0f1_data[4] +cdef char ufunc_hyp0f1_types[12] +cdef char *ufunc_hyp0f1_doc = ( + "hyp0f1(v, z, out=None)\n" + "\n" + "Confluent hypergeometric limit function 0F1.\n" + "\n" + "Parameters\n" + "----------\n" + "v : array_like\n" + " Real-valued parameter\n" + "z : array_like\n" + " Real- or complex-valued argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The confluent hypergeometric limit function\n" + "\n" + "Notes\n" + "-----\n" + "This function is defined as:\n" + "\n" + ".. math:: _0F_1(v, z) = \\sum_{k=0}^{\\infty}\\frac{z^k}{(v)_k k!}.\n" + "\n" + "It's also the limit as :math:`q \\to \\infty` of :math:`_1F_1(q; v; z/q)`,\n" + "and satisfies the differential equation :math:`f''(z) + vf'(z) =\n" + "f(z)`. See [1]_ for more information.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Wolfram MathWorld, \"Confluent Hypergeometric Limit Function\",\n" + " http://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "It is one when `z` is zero.\n" + "\n" + ">>> sc.hyp0f1(1, 0)\n" + "1.0\n" + "\n" + "It is the limit of the confluent hypergeometric function as `q`\n" + "goes to infinity.\n" + "\n" + ">>> q = np.array([1, 10, 100, 1000])\n" + ">>> v = 1\n" + ">>> z = 1\n" + ">>> sc.hyp1f1(q, v, z / q)\n" + "array([2.71828183, 2.31481985, 2.28303778, 2.27992985])\n" + ">>> sc.hyp0f1(v, z)\n" + "2.2795853023360673\n" + "\n" + "It is related to Bessel functions.\n" + "\n" + ">>> n = 1\n" + ">>> x = np.linspace(0, 1, 5)\n" + ">>> sc.jv(n, x)\n" + "array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059])\n" + ">>> (0.5 * x)**n / sc.factorial(n) * sc.hyp0f1(n + 1, -0.25 * x**2)\n" + "array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059])") +ufunc_hyp0f1_loops[0] = loop_d_dd__As_ff_f +ufunc_hyp0f1_loops[1] = loop_D_dD__As_fF_F +ufunc_hyp0f1_loops[2] = loop_d_dd__As_dd_d +ufunc_hyp0f1_loops[3] = loop_D_dD__As_dD_D +ufunc_hyp0f1_types[0] = NPY_FLOAT +ufunc_hyp0f1_types[1] = NPY_FLOAT +ufunc_hyp0f1_types[2] = NPY_FLOAT +ufunc_hyp0f1_types[3] = NPY_FLOAT +ufunc_hyp0f1_types[4] = NPY_CFLOAT +ufunc_hyp0f1_types[5] = NPY_CFLOAT +ufunc_hyp0f1_types[6] = NPY_DOUBLE +ufunc_hyp0f1_types[7] = NPY_DOUBLE +ufunc_hyp0f1_types[8] = NPY_DOUBLE +ufunc_hyp0f1_types[9] = NPY_DOUBLE +ufunc_hyp0f1_types[10] = NPY_CDOUBLE +ufunc_hyp0f1_types[11] = NPY_CDOUBLE +ufunc_hyp0f1_ptr[2*0] = _func__hyp0f1_real +ufunc_hyp0f1_ptr[2*0+1] = ("hyp0f1") +ufunc_hyp0f1_ptr[2*1] = _func__hyp0f1_cmplx +ufunc_hyp0f1_ptr[2*1+1] = ("hyp0f1") +ufunc_hyp0f1_ptr[2*2] = _func__hyp0f1_real +ufunc_hyp0f1_ptr[2*2+1] = ("hyp0f1") +ufunc_hyp0f1_ptr[2*3] = _func__hyp0f1_cmplx +ufunc_hyp0f1_ptr[2*3+1] = ("hyp0f1") +ufunc_hyp0f1_data[0] = &ufunc_hyp0f1_ptr[2*0] +ufunc_hyp0f1_data[1] = &ufunc_hyp0f1_ptr[2*1] +ufunc_hyp0f1_data[2] = &ufunc_hyp0f1_ptr[2*2] +ufunc_hyp0f1_data[3] = &ufunc_hyp0f1_ptr[2*3] +hyp0f1 = np.PyUFunc_FromFuncAndData(ufunc_hyp0f1_loops, ufunc_hyp0f1_data, ufunc_hyp0f1_types, 4, 2, 1, 0, "hyp0f1", ufunc_hyp0f1_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_hyp1f1_loops[4] +cdef void *ufunc_hyp1f1_ptr[8] +cdef void *ufunc_hyp1f1_data[4] +cdef char ufunc_hyp1f1_types[16] +cdef char *ufunc_hyp1f1_doc = ( + "hyp1f1(a, b, x, out=None)\n" + "\n" + "Confluent hypergeometric function 1F1.\n" + "\n" + "The confluent hypergeometric function is defined by the series\n" + "\n" + ".. math::\n" + "\n" + " {}_1F_1(a; b; x) = \\sum_{k = 0}^\\infty \\frac{(a)_k}{(b)_k k!} x^k.\n" + "\n" + "See [dlmf]_ for more details. Here :math:`(\\cdot)_k` is the\n" + "Pochhammer symbol; see `poch`.\n" + "\n" + "Parameters\n" + "----------\n" + "a, b : array_like\n" + " Real parameters\n" + "x : array_like\n" + " Real or complex argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the confluent hypergeometric function\n" + "\n" + "See Also\n" + "--------\n" + "hyperu : another confluent hypergeometric function\n" + "hyp0f1 : confluent hypergeometric limit function\n" + "hyp2f1 : Gaussian hypergeometric function\n" + "\n" + "Notes\n" + "-----\n" + "For real values, this function uses the ``hyp1f1`` routine from the C++ Boost\n" + "library [2]_, for complex values a C translation of the specfun\n" + "Fortran library [3]_.\n" + "\n" + "References\n" + "----------\n" + ".. [dlmf] NIST Digital Library of Mathematical Functions\n" + " https://dlmf.nist.gov/13.2#E2\n" + ".. [2] The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/.\n" + ".. [3] Zhang, Jin, \"Computation of Special Functions\", John Wiley\n" + " and Sons, Inc, 1996.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "It is one when `x` is zero:\n" + "\n" + ">>> sc.hyp1f1(0.5, 0.5, 0)\n" + "1.0\n" + "\n" + "It is singular when `b` is a nonpositive integer.\n" + "\n" + ">>> sc.hyp1f1(0.5, -1, 0)\n" + "inf\n" + "\n" + "It is a polynomial when `a` is a nonpositive integer.\n" + "\n" + ">>> a, b, x = -1, 0.5, np.array([1.0, 2.0, 3.0, 4.0])\n" + ">>> sc.hyp1f1(a, b, x)\n" + "array([-1., -3., -5., -7.])\n" + ">>> 1 + (a / b) * x\n" + "array([-1., -3., -5., -7.])\n" + "\n" + "It reduces to the exponential function when ``a = b``.\n" + "\n" + ">>> sc.hyp1f1(2, 2, [1, 2, 3, 4])\n" + "array([ 2.71828183, 7.3890561 , 20.08553692, 54.59815003])\n" + ">>> np.exp([1, 2, 3, 4])\n" + "array([ 2.71828183, 7.3890561 , 20.08553692, 54.59815003])") +ufunc_hyp1f1_loops[0] = loop_d_ddd__As_fff_f +ufunc_hyp1f1_loops[1] = loop_D_ddD__As_ffF_F +ufunc_hyp1f1_loops[2] = loop_d_ddd__As_ddd_d +ufunc_hyp1f1_loops[3] = loop_D_ddD__As_ddD_D +ufunc_hyp1f1_types[0] = NPY_FLOAT +ufunc_hyp1f1_types[1] = NPY_FLOAT +ufunc_hyp1f1_types[2] = NPY_FLOAT +ufunc_hyp1f1_types[3] = NPY_FLOAT +ufunc_hyp1f1_types[4] = NPY_FLOAT +ufunc_hyp1f1_types[5] = NPY_FLOAT +ufunc_hyp1f1_types[6] = NPY_CFLOAT +ufunc_hyp1f1_types[7] = NPY_CFLOAT +ufunc_hyp1f1_types[8] = NPY_DOUBLE +ufunc_hyp1f1_types[9] = NPY_DOUBLE +ufunc_hyp1f1_types[10] = NPY_DOUBLE +ufunc_hyp1f1_types[11] = NPY_DOUBLE +ufunc_hyp1f1_types[12] = NPY_DOUBLE +ufunc_hyp1f1_types[13] = NPY_DOUBLE +ufunc_hyp1f1_types[14] = NPY_CDOUBLE +ufunc_hyp1f1_types[15] = NPY_CDOUBLE +ufunc_hyp1f1_ptr[2*0] = scipy.special._ufuncs_cxx._export_hyp1f1_double +ufunc_hyp1f1_ptr[2*0+1] = ("hyp1f1") +ufunc_hyp1f1_ptr[2*1] = _func_chyp1f1_wrap +ufunc_hyp1f1_ptr[2*1+1] = ("hyp1f1") +ufunc_hyp1f1_ptr[2*2] = scipy.special._ufuncs_cxx._export_hyp1f1_double +ufunc_hyp1f1_ptr[2*2+1] = ("hyp1f1") +ufunc_hyp1f1_ptr[2*3] = _func_chyp1f1_wrap +ufunc_hyp1f1_ptr[2*3+1] = ("hyp1f1") +ufunc_hyp1f1_data[0] = &ufunc_hyp1f1_ptr[2*0] +ufunc_hyp1f1_data[1] = &ufunc_hyp1f1_ptr[2*1] +ufunc_hyp1f1_data[2] = &ufunc_hyp1f1_ptr[2*2] +ufunc_hyp1f1_data[3] = &ufunc_hyp1f1_ptr[2*3] +hyp1f1 = np.PyUFunc_FromFuncAndData(ufunc_hyp1f1_loops, ufunc_hyp1f1_data, ufunc_hyp1f1_types, 4, 3, 1, 0, "hyp1f1", ufunc_hyp1f1_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_hyperu_loops[2] +cdef void *ufunc_hyperu_ptr[4] +cdef void *ufunc_hyperu_data[2] +cdef char ufunc_hyperu_types[8] +cdef char *ufunc_hyperu_doc = ( + "hyperu(a, b, x, out=None)\n" + "\n" + "Confluent hypergeometric function U\n" + "\n" + "It is defined as the solution to the equation\n" + "\n" + ".. math::\n" + "\n" + " x \\frac{d^2w}{dx^2} + (b - x) \\frac{dw}{dx} - aw = 0\n" + "\n" + "which satisfies the property\n" + "\n" + ".. math::\n" + "\n" + " U(a, b, x) \\sim x^{-a}\n" + "\n" + "as :math:`x \\to \\infty`. See [dlmf]_ for more details.\n" + "\n" + "Parameters\n" + "----------\n" + "a, b : array_like\n" + " Real-valued parameters\n" + "x : array_like\n" + " Real-valued argument\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of `U`\n" + "\n" + "References\n" + "----------\n" + ".. [dlmf] NIST Digital Library of Mathematics Functions\n" + " https://dlmf.nist.gov/13.2#E6\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "It has a branch cut along the negative `x` axis.\n" + "\n" + ">>> x = np.linspace(-0.1, -10, 5)\n" + ">>> sc.hyperu(1, 1, x)\n" + "array([nan, nan, nan, nan, nan])\n" + "\n" + "It approaches zero as `x` goes to infinity.\n" + "\n" + ">>> x = np.array([1, 10, 100])\n" + ">>> sc.hyperu(1, 1, x)\n" + "array([0.59634736, 0.09156333, 0.00990194])\n" + "\n" + "It satisfies Kummer's transformation.\n" + "\n" + ">>> a, b, x = 2, 1, 1\n" + ">>> sc.hyperu(a, b, x)\n" + "0.1926947246463881\n" + ">>> x**(1 - b) * sc.hyperu(a - b + 1, 2 - b, x)\n" + "0.1926947246463881") +ufunc_hyperu_loops[0] = loop_d_ddd__As_fff_f +ufunc_hyperu_loops[1] = loop_d_ddd__As_ddd_d +ufunc_hyperu_types[0] = NPY_FLOAT +ufunc_hyperu_types[1] = NPY_FLOAT +ufunc_hyperu_types[2] = NPY_FLOAT +ufunc_hyperu_types[3] = NPY_FLOAT +ufunc_hyperu_types[4] = NPY_DOUBLE +ufunc_hyperu_types[5] = NPY_DOUBLE +ufunc_hyperu_types[6] = NPY_DOUBLE +ufunc_hyperu_types[7] = NPY_DOUBLE +ufunc_hyperu_ptr[2*0] = _func_hyperu +ufunc_hyperu_ptr[2*0+1] = ("hyperu") +ufunc_hyperu_ptr[2*1] = _func_hyperu +ufunc_hyperu_ptr[2*1+1] = ("hyperu") +ufunc_hyperu_data[0] = &ufunc_hyperu_ptr[2*0] +ufunc_hyperu_data[1] = &ufunc_hyperu_ptr[2*1] +hyperu = np.PyUFunc_FromFuncAndData(ufunc_hyperu_loops, ufunc_hyperu_data, ufunc_hyperu_types, 2, 3, 1, 0, "hyperu", ufunc_hyperu_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_inv_boxcox_loops[2] +cdef void *ufunc_inv_boxcox_ptr[4] +cdef void *ufunc_inv_boxcox_data[2] +cdef char ufunc_inv_boxcox_types[6] +cdef char *ufunc_inv_boxcox_doc = ( + "inv_boxcox(y, lmbda, out=None)\n" + "\n" + "Compute the inverse of the Box-Cox transformation.\n" + "\n" + "Find ``x`` such that::\n" + "\n" + " y = (x**lmbda - 1) / lmbda if lmbda != 0\n" + " log(x) if lmbda == 0\n" + "\n" + "Parameters\n" + "----------\n" + "y : array_like\n" + " Data to be transformed.\n" + "lmbda : array_like\n" + " Power parameter of the Box-Cox transform.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "x : scalar or ndarray\n" + " Transformed data.\n" + "\n" + "Notes\n" + "-----\n" + "\n" + ".. versionadded:: 0.16.0\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import boxcox, inv_boxcox\n" + ">>> y = boxcox([1, 4, 10], 2.5)\n" + ">>> inv_boxcox(y, 2.5)\n" + "array([1., 4., 10.])") +ufunc_inv_boxcox_loops[0] = loop_d_dd__As_ff_f +ufunc_inv_boxcox_loops[1] = loop_d_dd__As_dd_d +ufunc_inv_boxcox_types[0] = NPY_FLOAT +ufunc_inv_boxcox_types[1] = NPY_FLOAT +ufunc_inv_boxcox_types[2] = NPY_FLOAT +ufunc_inv_boxcox_types[3] = NPY_DOUBLE +ufunc_inv_boxcox_types[4] = NPY_DOUBLE +ufunc_inv_boxcox_types[5] = NPY_DOUBLE +ufunc_inv_boxcox_ptr[2*0] = _func_inv_boxcox +ufunc_inv_boxcox_ptr[2*0+1] = ("inv_boxcox") +ufunc_inv_boxcox_ptr[2*1] = _func_inv_boxcox +ufunc_inv_boxcox_ptr[2*1+1] = ("inv_boxcox") +ufunc_inv_boxcox_data[0] = &ufunc_inv_boxcox_ptr[2*0] +ufunc_inv_boxcox_data[1] = &ufunc_inv_boxcox_ptr[2*1] +inv_boxcox = np.PyUFunc_FromFuncAndData(ufunc_inv_boxcox_loops, ufunc_inv_boxcox_data, ufunc_inv_boxcox_types, 2, 2, 1, 0, "inv_boxcox", ufunc_inv_boxcox_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_inv_boxcox1p_loops[2] +cdef void *ufunc_inv_boxcox1p_ptr[4] +cdef void *ufunc_inv_boxcox1p_data[2] +cdef char ufunc_inv_boxcox1p_types[6] +cdef char *ufunc_inv_boxcox1p_doc = ( + "inv_boxcox1p(y, lmbda, out=None)\n" + "\n" + "Compute the inverse of the Box-Cox transformation.\n" + "\n" + "Find ``x`` such that::\n" + "\n" + " y = ((1+x)**lmbda - 1) / lmbda if lmbda != 0\n" + " log(1+x) if lmbda == 0\n" + "\n" + "Parameters\n" + "----------\n" + "y : array_like\n" + " Data to be transformed.\n" + "lmbda : array_like\n" + " Power parameter of the Box-Cox transform.\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "x : scalar or ndarray\n" + " Transformed data.\n" + "\n" + "Notes\n" + "-----\n" + "\n" + ".. versionadded:: 0.16.0\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import boxcox1p, inv_boxcox1p\n" + ">>> y = boxcox1p([1, 4, 10], 2.5)\n" + ">>> inv_boxcox1p(y, 2.5)\n" + "array([1., 4., 10.])") +ufunc_inv_boxcox1p_loops[0] = loop_d_dd__As_ff_f +ufunc_inv_boxcox1p_loops[1] = loop_d_dd__As_dd_d +ufunc_inv_boxcox1p_types[0] = NPY_FLOAT +ufunc_inv_boxcox1p_types[1] = NPY_FLOAT +ufunc_inv_boxcox1p_types[2] = NPY_FLOAT +ufunc_inv_boxcox1p_types[3] = NPY_DOUBLE +ufunc_inv_boxcox1p_types[4] = NPY_DOUBLE +ufunc_inv_boxcox1p_types[5] = NPY_DOUBLE +ufunc_inv_boxcox1p_ptr[2*0] = _func_inv_boxcox1p +ufunc_inv_boxcox1p_ptr[2*0+1] = ("inv_boxcox1p") +ufunc_inv_boxcox1p_ptr[2*1] = _func_inv_boxcox1p +ufunc_inv_boxcox1p_ptr[2*1+1] = ("inv_boxcox1p") +ufunc_inv_boxcox1p_data[0] = &ufunc_inv_boxcox1p_ptr[2*0] +ufunc_inv_boxcox1p_data[1] = &ufunc_inv_boxcox1p_ptr[2*1] +inv_boxcox1p = np.PyUFunc_FromFuncAndData(ufunc_inv_boxcox1p_loops, ufunc_inv_boxcox1p_data, ufunc_inv_boxcox1p_types, 2, 2, 1, 0, "inv_boxcox1p", ufunc_inv_boxcox1p_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_kl_div_loops[2] +cdef void *ufunc_kl_div_ptr[4] +cdef void *ufunc_kl_div_data[2] +cdef char ufunc_kl_div_types[6] +cdef char *ufunc_kl_div_doc = ( + "kl_div(x, y, out=None)\n" + "\n" + "Elementwise function for computing Kullback-Leibler divergence.\n" + "\n" + ".. math::\n" + "\n" + " \\mathrm{kl\\_div}(x, y) =\n" + " \\begin{cases}\n" + " x \\log(x / y) - x + y & x > 0, y > 0 \\\\\n" + " y & x = 0, y \\ge 0 \\\\\n" + " \\infty & \\text{otherwise}\n" + " \\end{cases}\n" + "\n" + "Parameters\n" + "----------\n" + "x, y : array_like\n" + " Real arguments\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the Kullback-Liebler divergence.\n" + "\n" + "See Also\n" + "--------\n" + "entr, rel_entr, scipy.stats.entropy\n" + "\n" + "Notes\n" + "-----\n" + ".. versionadded:: 0.15.0\n" + "\n" + "This function is non-negative and is jointly convex in `x` and `y`.\n" + "\n" + "The origin of this function is in convex programming; see [1]_ for\n" + "details. This is why the function contains the extra :math:`-x\n" + "+ y` terms over what might be expected from the Kullback-Leibler\n" + "divergence. For a version of the function without the extra terms,\n" + "see `rel_entr`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*.\n" + " Cambridge University Press, 2004.\n" + " :doi:`https://doi.org/10.1017/CBO9780511804441`") +ufunc_kl_div_loops[0] = loop_d_dd__As_ff_f +ufunc_kl_div_loops[1] = loop_d_dd__As_dd_d +ufunc_kl_div_types[0] = NPY_FLOAT +ufunc_kl_div_types[1] = NPY_FLOAT +ufunc_kl_div_types[2] = NPY_FLOAT +ufunc_kl_div_types[3] = NPY_DOUBLE +ufunc_kl_div_types[4] = NPY_DOUBLE +ufunc_kl_div_types[5] = NPY_DOUBLE +ufunc_kl_div_ptr[2*0] = _func_kl_div +ufunc_kl_div_ptr[2*0+1] = ("kl_div") +ufunc_kl_div_ptr[2*1] = _func_kl_div +ufunc_kl_div_ptr[2*1+1] = ("kl_div") +ufunc_kl_div_data[0] = &ufunc_kl_div_ptr[2*0] +ufunc_kl_div_data[1] = &ufunc_kl_div_ptr[2*1] +kl_div = np.PyUFunc_FromFuncAndData(ufunc_kl_div_loops, ufunc_kl_div_data, ufunc_kl_div_types, 2, 2, 1, 0, "kl_div", ufunc_kl_div_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_kn_loops[3] +cdef void *ufunc_kn_ptr[6] +cdef void *ufunc_kn_data[3] +cdef char ufunc_kn_types[9] +cdef char *ufunc_kn_doc = ( + "kn(n, x, out=None)\n" + "\n" + "Modified Bessel function of the second kind of integer order `n`\n" + "\n" + "Returns the modified Bessel function of the second kind for integer order\n" + "`n` at real `z`.\n" + "\n" + "These are also sometimes called functions of the third kind, Basset\n" + "functions, or Macdonald functions.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like of int\n" + " Order of Bessel functions (floats will truncate with a warning)\n" + "x : array_like of float\n" + " Argument at which to evaluate the Bessel functions\n" + "out : ndarray, optional\n" + " Optional output array for the function results.\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Value of the Modified Bessel function of the second kind,\n" + " :math:`K_n(x)`.\n" + "\n" + "See Also\n" + "--------\n" + "kv : Same function, but accepts real order and complex argument\n" + "kvp : Derivative of this function\n" + "\n" + "Notes\n" + "-----\n" + "Wrapper for AMOS [1]_ routine `zbesk`. For a discussion of the\n" + "algorithm used, see [2]_ and the references therein.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Donald E. Amos, \"AMOS, A Portable Package for Bessel Functions\n" + " of a Complex Argument and Nonnegative Order\",\n" + " http://netlib.org/amos/\n" + ".. [2] Donald E. Amos, \"Algorithm 644: A portable package for Bessel\n" + " functions of a complex argument and nonnegative order\", ACM\n" + " TOMS Vol. 12 Issue 3, Sept. 1986, p. 265\n" + "\n" + "Examples\n" + "--------\n" + "Plot the function of several orders for real input:\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import kn\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> x = np.linspace(0, 5, 1000)\n" + ">>> for N in range(6):\n" + "... plt.plot(x, kn(N, x), label='$K_{}(x)$'.format(N))\n" + ">>> plt.ylim(0, 10)\n" + ">>> plt.legend()\n" + ">>> plt.title(r'Modified Bessel function of the second kind $K_n(x)$')\n" + ">>> plt.show()\n" + "\n" + "Calculate for a single value at multiple orders:\n" + "\n" + ">>> kn([4, 5, 6], 1)\n" + "array([ 44.23241585, 360.9605896 , 3653.83831186])") +ufunc_kn_loops[0] = loop_d_pd__As_pd_d +ufunc_kn_loops[1] = loop_d_dd__As_ff_f +ufunc_kn_loops[2] = loop_d_dd__As_dd_d +ufunc_kn_types[0] = NPY_INTP +ufunc_kn_types[1] = NPY_DOUBLE +ufunc_kn_types[2] = NPY_DOUBLE +ufunc_kn_types[3] = NPY_FLOAT +ufunc_kn_types[4] = NPY_FLOAT +ufunc_kn_types[5] = NPY_FLOAT +ufunc_kn_types[6] = NPY_DOUBLE +ufunc_kn_types[7] = NPY_DOUBLE +ufunc_kn_types[8] = NPY_DOUBLE +ufunc_kn_ptr[2*0] = _func_special_cyl_bessel_k_int +ufunc_kn_ptr[2*0+1] = ("kn") +ufunc_kn_ptr[2*1] = _func_kn_unsafe +ufunc_kn_ptr[2*1+1] = ("kn") +ufunc_kn_ptr[2*2] = _func_kn_unsafe +ufunc_kn_ptr[2*2+1] = ("kn") +ufunc_kn_data[0] = &ufunc_kn_ptr[2*0] +ufunc_kn_data[1] = &ufunc_kn_ptr[2*1] +ufunc_kn_data[2] = &ufunc_kn_ptr[2*2] +kn = np.PyUFunc_FromFuncAndData(ufunc_kn_loops, ufunc_kn_data, ufunc_kn_types, 3, 2, 1, 0, "kn", ufunc_kn_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_kolmogi_loops[2] +cdef void *ufunc_kolmogi_ptr[4] +cdef void *ufunc_kolmogi_data[2] +cdef char ufunc_kolmogi_types[4] +cdef char *ufunc_kolmogi_doc = ( + "kolmogi(p, out=None)\n" + "\n" + "Inverse Survival Function of Kolmogorov distribution\n" + "\n" + "It is the inverse function to `kolmogorov`.\n" + "Returns y such that ``kolmogorov(y) == p``.\n" + "\n" + "Parameters\n" + "----------\n" + "p : float array_like\n" + " Probability\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The value(s) of kolmogi(p)\n" + "\n" + "See Also\n" + "--------\n" + "kolmogorov : The Survival Function for the distribution\n" + "scipy.stats.kstwobign : Provides the functionality as a continuous distribution\n" + "smirnov, smirnovi : Functions for the one-sided distribution\n" + "\n" + "Notes\n" + "-----\n" + "`kolmogorov` is used by `stats.kstest` in the application of the\n" + "Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this\n" + "function is exposed in `scpy.special`, but the recommended way to achieve\n" + "the most accurate CDF/SF/PDF/PPF/ISF computations is to use the\n" + "`stats.kstwobign` distribution.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import kolmogi\n" + ">>> kolmogi([0, 0.1, 0.25, 0.5, 0.75, 0.9, 1.0])\n" + "array([ inf, 1.22384787, 1.01918472, 0.82757356, 0.67644769,\n" + " 0.57117327, 0. ])") +ufunc_kolmogi_loops[0] = loop_d_d__As_f_f +ufunc_kolmogi_loops[1] = loop_d_d__As_d_d +ufunc_kolmogi_types[0] = NPY_FLOAT +ufunc_kolmogi_types[1] = NPY_FLOAT +ufunc_kolmogi_types[2] = NPY_DOUBLE +ufunc_kolmogi_types[3] = NPY_DOUBLE +ufunc_kolmogi_ptr[2*0] = _func_xsf_kolmogi +ufunc_kolmogi_ptr[2*0+1] = ("kolmogi") +ufunc_kolmogi_ptr[2*1] = _func_xsf_kolmogi +ufunc_kolmogi_ptr[2*1+1] = ("kolmogi") +ufunc_kolmogi_data[0] = &ufunc_kolmogi_ptr[2*0] +ufunc_kolmogi_data[1] = &ufunc_kolmogi_ptr[2*1] +kolmogi = np.PyUFunc_FromFuncAndData(ufunc_kolmogi_loops, ufunc_kolmogi_data, ufunc_kolmogi_types, 2, 1, 1, 0, "kolmogi", ufunc_kolmogi_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_kolmogorov_loops[2] +cdef void *ufunc_kolmogorov_ptr[4] +cdef void *ufunc_kolmogorov_data[2] +cdef char ufunc_kolmogorov_types[4] +cdef char *ufunc_kolmogorov_doc = ( + "kolmogorov(y, out=None)\n" + "\n" + "Complementary cumulative distribution (Survival Function) function of\n" + "Kolmogorov distribution.\n" + "\n" + "Returns the complementary cumulative distribution function of\n" + "Kolmogorov's limiting distribution (``D_n*\\sqrt(n)`` as n goes to infinity)\n" + "of a two-sided test for equality between an empirical and a theoretical\n" + "distribution. It is equal to the (limit as n->infinity of the)\n" + "probability that ``sqrt(n) * max absolute deviation > y``.\n" + "\n" + "Parameters\n" + "----------\n" + "y : float array_like\n" + " Absolute deviation between the Empirical CDF (ECDF) and the target CDF,\n" + " multiplied by sqrt(n).\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The value(s) of kolmogorov(y)\n" + "\n" + "See Also\n" + "--------\n" + "kolmogi : The Inverse Survival Function for the distribution\n" + "scipy.stats.kstwobign : Provides the functionality as a continuous distribution\n" + "smirnov, smirnovi : Functions for the one-sided distribution\n" + "\n" + "Notes\n" + "-----\n" + "`kolmogorov` is used by `stats.kstest` in the application of the\n" + "Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this\n" + "function is exposed in `scpy.special`, but the recommended way to achieve\n" + "the most accurate CDF/SF/PDF/PPF/ISF computations is to use the\n" + "`stats.kstwobign` distribution.\n" + "\n" + "Examples\n" + "--------\n" + "Show the probability of a gap at least as big as 0, 0.5 and 1.0.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import kolmogorov\n" + ">>> from scipy.stats import kstwobign\n" + ">>> kolmogorov([0, 0.5, 1.0])\n" + "array([ 1. , 0.96394524, 0.26999967])\n" + "\n" + "Compare a sample of size 1000 drawn from a Laplace(0, 1) distribution against\n" + "the target distribution, a Normal(0, 1) distribution.\n" + "\n" + ">>> from scipy.stats import norm, laplace\n" + ">>> rng = np.random.default_rng()\n" + ">>> n = 1000\n" + ">>> lap01 = laplace(0, 1)\n" + ">>> x = np.sort(lap01.rvs(n, random_state=rng))\n" + ">>> np.mean(x), np.std(x)\n" + "(-0.05841730131499543, 1.3968109101997568)\n" + "\n" + "Construct the Empirical CDF and the K-S statistic Dn.\n" + "\n" + ">>> target = norm(0,1) # Normal mean 0, stddev 1\n" + ">>> cdfs = target.cdf(x)\n" + ">>> ecdfs = np.arange(n+1, dtype=float)/n\n" + ">>> gaps = np.column_stack([cdfs - ecdfs[:n], ecdfs[1:] - cdfs])\n" + ">>> Dn = np.max(gaps)\n" + ">>> Kn = np.sqrt(n) * Dn\n" + ">>> print('Dn=%f, sqrt(n)*Dn=%f' % (Dn, Kn))\n" + "Dn=0.043363, sqrt(n)*Dn=1.371265\n" + ">>> print(chr(10).join(['For a sample of size n drawn from a N(0, 1) distribution:',\n" + "... ' the approximate Kolmogorov probability that sqrt(n)*Dn>=%f is %f' %\n" + "... (Kn, kolmogorov(Kn)),\n" + "... ' the approximate Kolmogorov probability that sqrt(n)*Dn<=%f is %f' %\n" + "... (Kn, kstwobign.cdf(Kn))]))\n" + "For a sample of size n drawn from a N(0, 1) distribution:\n" + " the approximate Kolmogorov probability that sqrt(n)*Dn>=1.371265 is 0.046533\n" + " the approximate Kolmogorov probability that sqrt(n)*Dn<=1.371265 is 0.953467\n" + "\n" + "Plot the Empirical CDF against the target N(0, 1) CDF.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> plt.step(np.concatenate([[-3], x]), ecdfs, where='post', label='Empirical CDF')\n" + ">>> x3 = np.linspace(-3, 3, 100)\n" + ">>> plt.plot(x3, target.cdf(x3), label='CDF for N(0, 1)')\n" + ">>> plt.ylim([0, 1]); plt.grid(True); plt.legend();\n" + ">>> # Add vertical lines marking Dn+ and Dn-\n" + ">>> iminus, iplus = np.argmax(gaps, axis=0)\n" + ">>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus],\n" + "... color='r', linestyle='dashed', lw=4)\n" + ">>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1],\n" + "... color='r', linestyle='dashed', lw=4)\n" + ">>> plt.show()") +ufunc_kolmogorov_loops[0] = loop_d_d__As_f_f +ufunc_kolmogorov_loops[1] = loop_d_d__As_d_d +ufunc_kolmogorov_types[0] = NPY_FLOAT +ufunc_kolmogorov_types[1] = NPY_FLOAT +ufunc_kolmogorov_types[2] = NPY_DOUBLE +ufunc_kolmogorov_types[3] = NPY_DOUBLE +ufunc_kolmogorov_ptr[2*0] = _func_xsf_kolmogorov +ufunc_kolmogorov_ptr[2*0+1] = ("kolmogorov") +ufunc_kolmogorov_ptr[2*1] = _func_xsf_kolmogorov +ufunc_kolmogorov_ptr[2*1+1] = ("kolmogorov") +ufunc_kolmogorov_data[0] = &ufunc_kolmogorov_ptr[2*0] +ufunc_kolmogorov_data[1] = &ufunc_kolmogorov_ptr[2*1] +kolmogorov = np.PyUFunc_FromFuncAndData(ufunc_kolmogorov_loops, ufunc_kolmogorov_data, ufunc_kolmogorov_types, 2, 1, 1, 0, "kolmogorov", ufunc_kolmogorov_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_log1p_loops[4] +cdef void *ufunc_log1p_ptr[8] +cdef void *ufunc_log1p_data[4] +cdef char ufunc_log1p_types[8] +cdef char *ufunc_log1p_doc = ( + "log1p(x, out=None)\n" + "\n" + "Calculates log(1 + x) for use when `x` is near zero.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real or complex valued input.\n" + "out : ndarray, optional\n" + " Optional output array for the function results.\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of ``log(1 + x)``.\n" + "\n" + "See Also\n" + "--------\n" + "expm1, cosm1\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "It is more accurate than using ``log(1 + x)`` directly for ``x``\n" + "near 0. Note that in the below example ``1 + 1e-17 == 1`` to\n" + "double precision.\n" + "\n" + ">>> sc.log1p(1e-17)\n" + "1e-17\n" + ">>> np.log(1 + 1e-17)\n" + "0.0") +ufunc_log1p_loops[0] = loop_d_d__As_f_f +ufunc_log1p_loops[1] = loop_d_d__As_d_d +ufunc_log1p_loops[2] = loop_D_D__As_F_F +ufunc_log1p_loops[3] = loop_D_D__As_D_D +ufunc_log1p_types[0] = NPY_FLOAT +ufunc_log1p_types[1] = NPY_FLOAT +ufunc_log1p_types[2] = NPY_DOUBLE +ufunc_log1p_types[3] = NPY_DOUBLE +ufunc_log1p_types[4] = NPY_CFLOAT +ufunc_log1p_types[5] = NPY_CFLOAT +ufunc_log1p_types[6] = NPY_CDOUBLE +ufunc_log1p_types[7] = NPY_CDOUBLE +ufunc_log1p_ptr[2*0] = _func_cephes_log1p +ufunc_log1p_ptr[2*0+1] = ("log1p") +ufunc_log1p_ptr[2*1] = _func_cephes_log1p +ufunc_log1p_ptr[2*1+1] = ("log1p") +ufunc_log1p_ptr[2*2] = _func_clog1p +ufunc_log1p_ptr[2*2+1] = ("log1p") +ufunc_log1p_ptr[2*3] = _func_clog1p +ufunc_log1p_ptr[2*3+1] = ("log1p") +ufunc_log1p_data[0] = &ufunc_log1p_ptr[2*0] +ufunc_log1p_data[1] = &ufunc_log1p_ptr[2*1] +ufunc_log1p_data[2] = &ufunc_log1p_ptr[2*2] +ufunc_log1p_data[3] = &ufunc_log1p_ptr[2*3] +log1p = np.PyUFunc_FromFuncAndData(ufunc_log1p_loops, ufunc_log1p_data, ufunc_log1p_types, 4, 1, 1, 0, "log1p", ufunc_log1p_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_log_ndtr_loops[4] +cdef void *ufunc_log_ndtr_ptr[8] +cdef void *ufunc_log_ndtr_data[4] +cdef char ufunc_log_ndtr_types[8] +cdef char *ufunc_log_ndtr_doc = ( + "log_ndtr(x, out=None)\n" + "\n" + "Logarithm of Gaussian cumulative distribution function.\n" + "\n" + "Returns the log of the area under the standard Gaussian probability\n" + "density function, integrated from minus infinity to `x`::\n" + "\n" + " log(1/sqrt(2*pi) * integral(exp(-t**2 / 2), t=-inf..x))\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like, real or complex\n" + " Argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The value of the log of the normal CDF evaluated at `x`\n" + "\n" + "See Also\n" + "--------\n" + "erf\n" + "erfc\n" + "scipy.stats.norm\n" + "ndtr\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy.special import log_ndtr, ndtr\n" + "\n" + "The benefit of ``log_ndtr(x)`` over the naive implementation\n" + "``np.log(ndtr(x))`` is most evident with moderate to large positive\n" + "values of ``x``:\n" + "\n" + ">>> x = np.array([6, 7, 9, 12, 15, 25])\n" + ">>> log_ndtr(x)\n" + "array([-9.86587646e-010, -1.27981254e-012, -1.12858841e-019,\n" + " -1.77648211e-033, -3.67096620e-051, -3.05669671e-138])\n" + "\n" + "The results of the naive calculation for the moderate ``x`` values\n" + "have only 5 or 6 correct significant digits. For values of ``x``\n" + "greater than approximately 8.3, the naive expression returns 0:\n" + "\n" + ">>> np.log(ndtr(x))\n" + "array([-9.86587701e-10, -1.27986510e-12, 0.00000000e+00,\n" + " 0.00000000e+00, 0.00000000e+00, 0.00000000e+00])") +ufunc_log_ndtr_loops[0] = loop_d_d__As_f_f +ufunc_log_ndtr_loops[1] = loop_d_d__As_d_d +ufunc_log_ndtr_loops[2] = loop_D_D__As_F_F +ufunc_log_ndtr_loops[3] = loop_D_D__As_D_D +ufunc_log_ndtr_types[0] = NPY_FLOAT +ufunc_log_ndtr_types[1] = NPY_FLOAT +ufunc_log_ndtr_types[2] = NPY_DOUBLE +ufunc_log_ndtr_types[3] = NPY_DOUBLE +ufunc_log_ndtr_types[4] = NPY_CFLOAT +ufunc_log_ndtr_types[5] = NPY_CFLOAT +ufunc_log_ndtr_types[6] = NPY_CDOUBLE +ufunc_log_ndtr_types[7] = NPY_CDOUBLE +ufunc_log_ndtr_ptr[2*0] = scipy.special._ufuncs_cxx._export_faddeeva_log_ndtr +ufunc_log_ndtr_ptr[2*0+1] = ("log_ndtr") +ufunc_log_ndtr_ptr[2*1] = scipy.special._ufuncs_cxx._export_faddeeva_log_ndtr +ufunc_log_ndtr_ptr[2*1+1] = ("log_ndtr") +ufunc_log_ndtr_ptr[2*2] = scipy.special._ufuncs_cxx._export_faddeeva_log_ndtr_complex +ufunc_log_ndtr_ptr[2*2+1] = ("log_ndtr") +ufunc_log_ndtr_ptr[2*3] = scipy.special._ufuncs_cxx._export_faddeeva_log_ndtr_complex +ufunc_log_ndtr_ptr[2*3+1] = ("log_ndtr") +ufunc_log_ndtr_data[0] = &ufunc_log_ndtr_ptr[2*0] +ufunc_log_ndtr_data[1] = &ufunc_log_ndtr_ptr[2*1] +ufunc_log_ndtr_data[2] = &ufunc_log_ndtr_ptr[2*2] +ufunc_log_ndtr_data[3] = &ufunc_log_ndtr_ptr[2*3] +log_ndtr = np.PyUFunc_FromFuncAndData(ufunc_log_ndtr_loops, ufunc_log_ndtr_data, ufunc_log_ndtr_types, 4, 1, 1, 0, "log_ndtr", ufunc_log_ndtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_lpmv_loops[2] +cdef void *ufunc_lpmv_ptr[4] +cdef void *ufunc_lpmv_data[2] +cdef char ufunc_lpmv_types[8] +cdef char *ufunc_lpmv_doc = ( + "lpmv(m, v, x, out=None)\n" + "\n" + "Associated Legendre function of integer order and real degree.\n" + "\n" + "Defined as\n" + "\n" + ".. math::\n" + "\n" + " P_v^m = (-1)^m (1 - x^2)^{m/2} \\frac{d^m}{dx^m} P_v(x)\n" + "\n" + "where\n" + "\n" + ".. math::\n" + "\n" + " P_v = \\sum_{k = 0}^\\infty \\frac{(-v)_k (v + 1)_k}{(k!)^2}\n" + " \\left(\\frac{1 - x}{2}\\right)^k\n" + "\n" + "is the Legendre function of the first kind. Here :math:`(\\cdot)_k`\n" + "is the Pochhammer symbol; see `poch`.\n" + "\n" + "Parameters\n" + "----------\n" + "m : array_like\n" + " Order (int or float). If passed a float not equal to an\n" + " integer the function returns NaN.\n" + "v : array_like\n" + " Degree (float).\n" + "x : array_like\n" + " Argument (float). Must have ``|x| <= 1``.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "pmv : scalar or ndarray\n" + " Value of the associated Legendre function.\n" + "\n" + "See Also\n" + "--------\n" + "lpmn : Compute the associated Legendre function for all orders\n" + " ``0, ..., m`` and degrees ``0, ..., n``.\n" + "clpmn : Compute the associated Legendre function at complex\n" + " arguments.\n" + "\n" + "Notes\n" + "-----\n" + "Note that this implementation includes the Condon-Shortley phase.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Zhang, Jin, \"Computation of Special Functions\", John Wiley\n" + " and Sons, Inc, 1996.") +ufunc_lpmv_loops[0] = loop_d_ddd__As_fff_f +ufunc_lpmv_loops[1] = loop_d_ddd__As_ddd_d +ufunc_lpmv_types[0] = NPY_FLOAT +ufunc_lpmv_types[1] = NPY_FLOAT +ufunc_lpmv_types[2] = NPY_FLOAT +ufunc_lpmv_types[3] = NPY_FLOAT +ufunc_lpmv_types[4] = NPY_DOUBLE +ufunc_lpmv_types[5] = NPY_DOUBLE +ufunc_lpmv_types[6] = NPY_DOUBLE +ufunc_lpmv_types[7] = NPY_DOUBLE +ufunc_lpmv_ptr[2*0] = _func_pmv_wrap +ufunc_lpmv_ptr[2*0+1] = ("lpmv") +ufunc_lpmv_ptr[2*1] = _func_pmv_wrap +ufunc_lpmv_ptr[2*1+1] = ("lpmv") +ufunc_lpmv_data[0] = &ufunc_lpmv_ptr[2*0] +ufunc_lpmv_data[1] = &ufunc_lpmv_ptr[2*1] +lpmv = np.PyUFunc_FromFuncAndData(ufunc_lpmv_loops, ufunc_lpmv_data, ufunc_lpmv_types, 2, 3, 1, 0, "lpmv", ufunc_lpmv_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nbdtr_loops[3] +cdef void *ufunc_nbdtr_ptr[6] +cdef void *ufunc_nbdtr_data[3] +cdef char ufunc_nbdtr_types[12] +cdef char *ufunc_nbdtr_doc = ( + "nbdtr(k, n, p, out=None)\n" + "\n" + "Negative binomial cumulative distribution function.\n" + "\n" + "Returns the sum of the terms 0 through `k` of the negative binomial\n" + "distribution probability mass function,\n" + "\n" + ".. math::\n" + "\n" + " F = \\sum_{j=0}^k {{n + j - 1}\\choose{j}} p^n (1 - p)^j.\n" + "\n" + "In a sequence of Bernoulli trials with individual success probabilities\n" + "`p`, this is the probability that `k` or fewer failures precede the nth\n" + "success.\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " The maximum number of allowed failures (nonnegative int).\n" + "n : array_like\n" + " The target number of successes (positive int).\n" + "p : array_like\n" + " Probability of success in a single event (float).\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "F : scalar or ndarray\n" + " The probability of `k` or fewer failures before `n` successes in a\n" + " sequence of events with individual success probability `p`.\n" + "\n" + "See Also\n" + "--------\n" + "nbdtrc : Negative binomial survival function\n" + "nbdtrik : Negative binomial quantile function\n" + "scipy.stats.nbinom : Negative binomial distribution\n" + "\n" + "Notes\n" + "-----\n" + "If floating point values are passed for `k` or `n`, they will be truncated\n" + "to integers.\n" + "\n" + "The terms are not summed directly; instead the regularized incomplete beta\n" + "function is employed, according to the formula,\n" + "\n" + ".. math::\n" + " \\mathrm{nbdtr}(k, n, p) = I_{p}(n, k + 1).\n" + "\n" + "Wrapper for the Cephes [1]_ routine `nbdtr`.\n" + "\n" + "The negative binomial distribution is also available as\n" + "`scipy.stats.nbinom`. Using `nbdtr` directly can improve performance\n" + "compared to the ``cdf`` method of `scipy.stats.nbinom` (see last example).\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + "\n" + "Examples\n" + "--------\n" + "Compute the function for ``k=10`` and ``n=5`` at ``p=0.5``.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import nbdtr\n" + ">>> nbdtr(10, 5, 0.5)\n" + "0.940765380859375\n" + "\n" + "Compute the function for ``n=10`` and ``p=0.5`` at several points by\n" + "providing a NumPy array or list for `k`.\n" + "\n" + ">>> nbdtr([5, 10, 15], 10, 0.5)\n" + "array([0.15087891, 0.58809853, 0.88523853])\n" + "\n" + "Plot the function for four different parameter sets.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> k = np.arange(130)\n" + ">>> n_parameters = [20, 20, 20, 80]\n" + ">>> p_parameters = [0.2, 0.5, 0.8, 0.5]\n" + ">>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']\n" + ">>> parameters_list = list(zip(p_parameters, n_parameters,\n" + "... linestyles))\n" + ">>> fig, ax = plt.subplots(figsize=(8, 8))\n" + ">>> for parameter_set in parameters_list:\n" + "... p, n, style = parameter_set\n" + "... nbdtr_vals = nbdtr(k, n, p)\n" + "... ax.plot(k, nbdtr_vals, label=rf\"$n={n},\\, p={p}$\",\n" + "... ls=style)\n" + ">>> ax.legend()\n" + ">>> ax.set_xlabel(\"$k$\")\n" + ">>> ax.set_title(\"Negative binomial cumulative distribution function\")\n" + ">>> plt.show()\n" + "\n" + "The negative binomial distribution is also available as\n" + "`scipy.stats.nbinom`. Using `nbdtr` directly can be much faster than\n" + "calling the ``cdf`` method of `scipy.stats.nbinom`, especially for small\n" + "arrays or individual values. To get the same results one must use the\n" + "following parametrization: ``nbinom(n, p).cdf(k)=nbdtr(k, n, p)``.\n" + "\n" + ">>> from scipy.stats import nbinom\n" + ">>> k, n, p = 5, 3, 0.5\n" + ">>> nbdtr_res = nbdtr(k, n, p) # this will often be faster than below\n" + ">>> stats_res = nbinom(n, p).cdf(k)\n" + ">>> stats_res, nbdtr_res # test that results are equal\n" + "(0.85546875, 0.85546875)\n" + "\n" + "`nbdtr` can evaluate different parameter sets by providing arrays with\n" + "shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute\n" + "the function for three different `k` at four locations `p`, resulting in\n" + "a 3x4 array.\n" + "\n" + ">>> k = np.array([[5], [10], [15]])\n" + ">>> p = np.array([0.3, 0.5, 0.7, 0.9])\n" + ">>> k.shape, p.shape\n" + "((3, 1), (4,))\n" + "\n" + ">>> nbdtr(k, 5, p)\n" + "array([[0.15026833, 0.62304687, 0.95265101, 0.9998531 ],\n" + " [0.48450894, 0.94076538, 0.99932777, 0.99999999],\n" + " [0.76249222, 0.99409103, 0.99999445, 1. ]])") +ufunc_nbdtr_loops[0] = loop_d_ppd__As_ppd_d +ufunc_nbdtr_loops[1] = loop_d_ddd__As_fff_f +ufunc_nbdtr_loops[2] = loop_d_ddd__As_ddd_d +ufunc_nbdtr_types[0] = NPY_INTP +ufunc_nbdtr_types[1] = NPY_INTP +ufunc_nbdtr_types[2] = NPY_DOUBLE +ufunc_nbdtr_types[3] = NPY_DOUBLE +ufunc_nbdtr_types[4] = NPY_FLOAT +ufunc_nbdtr_types[5] = NPY_FLOAT +ufunc_nbdtr_types[6] = NPY_FLOAT +ufunc_nbdtr_types[7] = NPY_FLOAT +ufunc_nbdtr_types[8] = NPY_DOUBLE +ufunc_nbdtr_types[9] = NPY_DOUBLE +ufunc_nbdtr_types[10] = NPY_DOUBLE +ufunc_nbdtr_types[11] = NPY_DOUBLE +ufunc_nbdtr_ptr[2*0] = _func_cephes_nbdtr_wrap +ufunc_nbdtr_ptr[2*0+1] = ("nbdtr") +ufunc_nbdtr_ptr[2*1] = _func_nbdtr_unsafe +ufunc_nbdtr_ptr[2*1+1] = ("nbdtr") +ufunc_nbdtr_ptr[2*2] = _func_nbdtr_unsafe +ufunc_nbdtr_ptr[2*2+1] = ("nbdtr") +ufunc_nbdtr_data[0] = &ufunc_nbdtr_ptr[2*0] +ufunc_nbdtr_data[1] = &ufunc_nbdtr_ptr[2*1] +ufunc_nbdtr_data[2] = &ufunc_nbdtr_ptr[2*2] +nbdtr = np.PyUFunc_FromFuncAndData(ufunc_nbdtr_loops, ufunc_nbdtr_data, ufunc_nbdtr_types, 3, 3, 1, 0, "nbdtr", ufunc_nbdtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nbdtrc_loops[3] +cdef void *ufunc_nbdtrc_ptr[6] +cdef void *ufunc_nbdtrc_data[3] +cdef char ufunc_nbdtrc_types[12] +cdef char *ufunc_nbdtrc_doc = ( + "nbdtrc(k, n, p, out=None)\n" + "\n" + "Negative binomial survival function.\n" + "\n" + "Returns the sum of the terms `k + 1` to infinity of the negative binomial\n" + "distribution probability mass function,\n" + "\n" + ".. math::\n" + "\n" + " F = \\sum_{j=k + 1}^\\infty {{n + j - 1}\\choose{j}} p^n (1 - p)^j.\n" + "\n" + "In a sequence of Bernoulli trials with individual success probabilities\n" + "`p`, this is the probability that more than `k` failures precede the nth\n" + "success.\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " The maximum number of allowed failures (nonnegative int).\n" + "n : array_like\n" + " The target number of successes (positive int).\n" + "p : array_like\n" + " Probability of success in a single event (float).\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "F : scalar or ndarray\n" + " The probability of `k + 1` or more failures before `n` successes in a\n" + " sequence of events with individual success probability `p`.\n" + "\n" + "See Also\n" + "--------\n" + "nbdtr : Negative binomial cumulative distribution function\n" + "nbdtrik : Negative binomial percentile function\n" + "scipy.stats.nbinom : Negative binomial distribution\n" + "\n" + "Notes\n" + "-----\n" + "If floating point values are passed for `k` or `n`, they will be truncated\n" + "to integers.\n" + "\n" + "The terms are not summed directly; instead the regularized incomplete beta\n" + "function is employed, according to the formula,\n" + "\n" + ".. math::\n" + " \\mathrm{nbdtrc}(k, n, p) = I_{1 - p}(k + 1, n).\n" + "\n" + "Wrapper for the Cephes [1]_ routine `nbdtrc`.\n" + "\n" + "The negative binomial distribution is also available as\n" + "`scipy.stats.nbinom`. Using `nbdtrc` directly can improve performance\n" + "compared to the ``sf`` method of `scipy.stats.nbinom` (see last example).\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + "\n" + "Examples\n" + "--------\n" + "Compute the function for ``k=10`` and ``n=5`` at ``p=0.5``.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import nbdtrc\n" + ">>> nbdtrc(10, 5, 0.5)\n" + "0.059234619140624986\n" + "\n" + "Compute the function for ``n=10`` and ``p=0.5`` at several points by\n" + "providing a NumPy array or list for `k`.\n" + "\n" + ">>> nbdtrc([5, 10, 15], 10, 0.5)\n" + "array([0.84912109, 0.41190147, 0.11476147])\n" + "\n" + "Plot the function for four different parameter sets.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> k = np.arange(130)\n" + ">>> n_parameters = [20, 20, 20, 80]\n" + ">>> p_parameters = [0.2, 0.5, 0.8, 0.5]\n" + ">>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']\n" + ">>> parameters_list = list(zip(p_parameters, n_parameters,\n" + "... linestyles))\n" + ">>> fig, ax = plt.subplots(figsize=(8, 8))\n" + ">>> for parameter_set in parameters_list:\n" + "... p, n, style = parameter_set\n" + "... nbdtrc_vals = nbdtrc(k, n, p)\n" + "... ax.plot(k, nbdtrc_vals, label=rf\"$n={n},\\, p={p}$\",\n" + "... ls=style)\n" + ">>> ax.legend()\n" + ">>> ax.set_xlabel(\"$k$\")\n" + ">>> ax.set_title(\"Negative binomial distribution survival function\")\n" + ">>> plt.show()\n" + "\n" + "The negative binomial distribution is also available as\n" + "`scipy.stats.nbinom`. Using `nbdtrc` directly can be much faster than\n" + "calling the ``sf`` method of `scipy.stats.nbinom`, especially for small\n" + "arrays or individual values. To get the same results one must use the\n" + "following parametrization: ``nbinom(n, p).sf(k)=nbdtrc(k, n, p)``.\n" + "\n" + ">>> from scipy.stats import nbinom\n" + ">>> k, n, p = 3, 5, 0.5\n" + ">>> nbdtr_res = nbdtrc(k, n, p) # this will often be faster than below\n" + ">>> stats_res = nbinom(n, p).sf(k)\n" + ">>> stats_res, nbdtr_res # test that results are equal\n" + "(0.6367187499999999, 0.6367187499999999)\n" + "\n" + "`nbdtrc` can evaluate different parameter sets by providing arrays with\n" + "shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute\n" + "the function for three different `k` at four locations `p`, resulting in\n" + "a 3x4 array.\n" + "\n" + ">>> k = np.array([[5], [10], [15]])\n" + ">>> p = np.array([0.3, 0.5, 0.7, 0.9])\n" + ">>> k.shape, p.shape\n" + "((3, 1), (4,))\n" + "\n" + ">>> nbdtrc(k, 5, p)\n" + "array([[8.49731667e-01, 3.76953125e-01, 4.73489874e-02, 1.46902600e-04],\n" + " [5.15491059e-01, 5.92346191e-02, 6.72234070e-04, 9.29610100e-09],\n" + " [2.37507779e-01, 5.90896606e-03, 5.55025308e-06, 3.26346760e-13]])") +ufunc_nbdtrc_loops[0] = loop_d_ppd__As_ppd_d +ufunc_nbdtrc_loops[1] = loop_d_ddd__As_fff_f +ufunc_nbdtrc_loops[2] = loop_d_ddd__As_ddd_d +ufunc_nbdtrc_types[0] = NPY_INTP +ufunc_nbdtrc_types[1] = NPY_INTP +ufunc_nbdtrc_types[2] = NPY_DOUBLE +ufunc_nbdtrc_types[3] = NPY_DOUBLE +ufunc_nbdtrc_types[4] = NPY_FLOAT +ufunc_nbdtrc_types[5] = NPY_FLOAT +ufunc_nbdtrc_types[6] = NPY_FLOAT +ufunc_nbdtrc_types[7] = NPY_FLOAT +ufunc_nbdtrc_types[8] = NPY_DOUBLE +ufunc_nbdtrc_types[9] = NPY_DOUBLE +ufunc_nbdtrc_types[10] = NPY_DOUBLE +ufunc_nbdtrc_types[11] = NPY_DOUBLE +ufunc_nbdtrc_ptr[2*0] = _func_cephes_nbdtrc_wrap +ufunc_nbdtrc_ptr[2*0+1] = ("nbdtrc") +ufunc_nbdtrc_ptr[2*1] = _func_nbdtrc_unsafe +ufunc_nbdtrc_ptr[2*1+1] = ("nbdtrc") +ufunc_nbdtrc_ptr[2*2] = _func_nbdtrc_unsafe +ufunc_nbdtrc_ptr[2*2+1] = ("nbdtrc") +ufunc_nbdtrc_data[0] = &ufunc_nbdtrc_ptr[2*0] +ufunc_nbdtrc_data[1] = &ufunc_nbdtrc_ptr[2*1] +ufunc_nbdtrc_data[2] = &ufunc_nbdtrc_ptr[2*2] +nbdtrc = np.PyUFunc_FromFuncAndData(ufunc_nbdtrc_loops, ufunc_nbdtrc_data, ufunc_nbdtrc_types, 3, 3, 1, 0, "nbdtrc", ufunc_nbdtrc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nbdtri_loops[3] +cdef void *ufunc_nbdtri_ptr[6] +cdef void *ufunc_nbdtri_data[3] +cdef char ufunc_nbdtri_types[12] +cdef char *ufunc_nbdtri_doc = ( + "nbdtri(k, n, y, out=None)\n" + "\n" + "Returns the inverse with respect to the parameter `p` of\n" + "``y = nbdtr(k, n, p)``, the negative binomial cumulative distribution\n" + "function.\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " The maximum number of allowed failures (nonnegative int).\n" + "n : array_like\n" + " The target number of successes (positive int).\n" + "y : array_like\n" + " The probability of `k` or fewer failures before `n` successes (float).\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "p : scalar or ndarray\n" + " Probability of success in a single event (float) such that\n" + " `nbdtr(k, n, p) = y`.\n" + "\n" + "See Also\n" + "--------\n" + "nbdtr : Cumulative distribution function of the negative binomial.\n" + "nbdtrc : Negative binomial survival function.\n" + "scipy.stats.nbinom : negative binomial distribution.\n" + "nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`.\n" + "nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`.\n" + "scipy.stats.nbinom : Negative binomial distribution\n" + "\n" + "Notes\n" + "-----\n" + "Wrapper for the Cephes [1]_ routine `nbdtri`.\n" + "\n" + "The negative binomial distribution is also available as\n" + "`scipy.stats.nbinom`. Using `nbdtri` directly can improve performance\n" + "compared to the ``ppf`` method of `scipy.stats.nbinom`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + "\n" + "Examples\n" + "--------\n" + "`nbdtri` is the inverse of `nbdtr` with respect to `p`.\n" + "Up to floating point errors the following holds:\n" + "``nbdtri(k, n, nbdtr(k, n, p))=p``.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import nbdtri, nbdtr\n" + ">>> k, n, y = 5, 10, 0.2\n" + ">>> cdf_val = nbdtr(k, n, y)\n" + ">>> nbdtri(k, n, cdf_val)\n" + "0.20000000000000004\n" + "\n" + "Compute the function for ``k=10`` and ``n=5`` at several points by\n" + "providing a NumPy array or list for `y`.\n" + "\n" + ">>> y = np.array([0.1, 0.4, 0.8])\n" + ">>> nbdtri(3, 5, y)\n" + "array([0.34462319, 0.51653095, 0.69677416])\n" + "\n" + "Plot the function for three different parameter sets.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> n_parameters = [5, 20, 30, 30]\n" + ">>> k_parameters = [20, 20, 60, 80]\n" + ">>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']\n" + ">>> parameters_list = list(zip(n_parameters, k_parameters, linestyles))\n" + ">>> cdf_vals = np.linspace(0, 1, 1000)\n" + ">>> fig, ax = plt.subplots(figsize=(8, 8))\n" + ">>> for parameter_set in parameters_list:\n" + "... n, k, style = parameter_set\n" + "... nbdtri_vals = nbdtri(k, n, cdf_vals)\n" + "... ax.plot(cdf_vals, nbdtri_vals, label=rf\"$k={k},\\ n={n}$\",\n" + "... ls=style)\n" + ">>> ax.legend()\n" + ">>> ax.set_ylabel(\"$p$\")\n" + ">>> ax.set_xlabel(\"$CDF$\")\n" + ">>> title = \"nbdtri: inverse of negative binomial CDF with respect to $p$\"\n" + ">>> ax.set_title(title)\n" + ">>> plt.show()\n" + "\n" + "`nbdtri` can evaluate different parameter sets by providing arrays with\n" + "shapes compatible for broadcasting for `k`, `n` and `p`. Here we compute\n" + "the function for three different `k` at four locations `p`, resulting in\n" + "a 3x4 array.\n" + "\n" + ">>> k = np.array([[5], [10], [15]])\n" + ">>> y = np.array([0.3, 0.5, 0.7, 0.9])\n" + ">>> k.shape, y.shape\n" + "((3, 1), (4,))\n" + "\n" + ">>> nbdtri(k, 5, y)\n" + "array([[0.37258157, 0.45169416, 0.53249956, 0.64578407],\n" + " [0.24588501, 0.30451981, 0.36778453, 0.46397088],\n" + " [0.18362101, 0.22966758, 0.28054743, 0.36066188]])") +ufunc_nbdtri_loops[0] = loop_d_ppd__As_ppd_d +ufunc_nbdtri_loops[1] = loop_d_ddd__As_fff_f +ufunc_nbdtri_loops[2] = loop_d_ddd__As_ddd_d +ufunc_nbdtri_types[0] = NPY_INTP +ufunc_nbdtri_types[1] = NPY_INTP +ufunc_nbdtri_types[2] = NPY_DOUBLE +ufunc_nbdtri_types[3] = NPY_DOUBLE +ufunc_nbdtri_types[4] = NPY_FLOAT +ufunc_nbdtri_types[5] = NPY_FLOAT +ufunc_nbdtri_types[6] = NPY_FLOAT +ufunc_nbdtri_types[7] = NPY_FLOAT +ufunc_nbdtri_types[8] = NPY_DOUBLE +ufunc_nbdtri_types[9] = NPY_DOUBLE +ufunc_nbdtri_types[10] = NPY_DOUBLE +ufunc_nbdtri_types[11] = NPY_DOUBLE +ufunc_nbdtri_ptr[2*0] = _func_cephes_nbdtri_wrap +ufunc_nbdtri_ptr[2*0+1] = ("nbdtri") +ufunc_nbdtri_ptr[2*1] = _func_nbdtri_unsafe +ufunc_nbdtri_ptr[2*1+1] = ("nbdtri") +ufunc_nbdtri_ptr[2*2] = _func_nbdtri_unsafe +ufunc_nbdtri_ptr[2*2+1] = ("nbdtri") +ufunc_nbdtri_data[0] = &ufunc_nbdtri_ptr[2*0] +ufunc_nbdtri_data[1] = &ufunc_nbdtri_ptr[2*1] +ufunc_nbdtri_data[2] = &ufunc_nbdtri_ptr[2*2] +nbdtri = np.PyUFunc_FromFuncAndData(ufunc_nbdtri_loops, ufunc_nbdtri_data, ufunc_nbdtri_types, 3, 3, 1, 0, "nbdtri", ufunc_nbdtri_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nbdtrik_loops[2] +cdef void *ufunc_nbdtrik_ptr[4] +cdef void *ufunc_nbdtrik_data[2] +cdef char ufunc_nbdtrik_types[8] +cdef char *ufunc_nbdtrik_doc = ( + "nbdtrik(y, n, p, out=None)\n" + "\n" + "Negative binomial percentile function.\n" + "\n" + "Returns the inverse with respect to the parameter `k` of\n" + "``y = nbdtr(k, n, p)``, the negative binomial cumulative distribution\n" + "function.\n" + "\n" + "Parameters\n" + "----------\n" + "y : array_like\n" + " The probability of `k` or fewer failures before `n` successes (float).\n" + "n : array_like\n" + " The target number of successes (positive int).\n" + "p : array_like\n" + " Probability of success in a single event (float).\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "k : scalar or ndarray\n" + " The maximum number of allowed failures such that `nbdtr(k, n, p) = y`.\n" + "\n" + "See Also\n" + "--------\n" + "nbdtr : Cumulative distribution function of the negative binomial.\n" + "nbdtrc : Survival function of the negative binomial.\n" + "nbdtri : Inverse with respect to `p` of `nbdtr(k, n, p)`.\n" + "nbdtrin : Inverse with respect to `n` of `nbdtr(k, n, p)`.\n" + "scipy.stats.nbinom : Negative binomial distribution\n" + "\n" + "Notes\n" + "-----\n" + "Wrapper for the CDFLIB [1]_ Fortran routine `cdfnbn`.\n" + "\n" + "Formula 26.5.26 of [2]_,\n" + "\n" + ".. math::\n" + " \\sum_{j=k + 1}^\\infty {{n + j - 1}\n" + " \\choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n),\n" + "\n" + "is used to reduce calculation of the cumulative distribution function to\n" + "that of a regularized incomplete beta :math:`I`.\n" + "\n" + "Computation of `k` involves a search for a value that produces the desired\n" + "value of `y`. The search relies on the monotonicity of `y` with `k`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Barry Brown, James Lovato, and Kathy Russell,\n" + " CDFLIB: Library of Fortran Routines for Cumulative Distribution\n" + " Functions, Inverses, and Other Parameters.\n" + ".. [2] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.\n" + "\n" + "Examples\n" + "--------\n" + "Compute the negative binomial cumulative distribution function for an\n" + "exemplary parameter set.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import nbdtr, nbdtrik\n" + ">>> k, n, p = 5, 2, 0.5\n" + ">>> cdf_value = nbdtr(k, n, p)\n" + ">>> cdf_value\n" + "0.9375\n" + "\n" + "Verify that `nbdtrik` recovers the original value for `k`.\n" + "\n" + ">>> nbdtrik(cdf_value, n, p)\n" + "5.0\n" + "\n" + "Plot the function for different parameter sets.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> p_parameters = [0.2, 0.5, 0.7, 0.5]\n" + ">>> n_parameters = [30, 30, 30, 80]\n" + ">>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']\n" + ">>> parameters_list = list(zip(p_parameters, n_parameters, linestyles))\n" + ">>> cdf_vals = np.linspace(0, 1, 1000)\n" + ">>> fig, ax = plt.subplots(figsize=(8, 8))\n" + ">>> for parameter_set in parameters_list:\n" + "... p, n, style = parameter_set\n" + "... nbdtrik_vals = nbdtrik(cdf_vals, n, p)\n" + "... ax.plot(cdf_vals, nbdtrik_vals, label=rf\"$n={n},\\ p={p}$\",\n" + "... ls=style)\n" + ">>> ax.legend()\n" + ">>> ax.set_ylabel(\"$k$\")\n" + ">>> ax.set_xlabel(\"$CDF$\")\n" + ">>> ax.set_title(\"Negative binomial percentile function\")\n" + ">>> plt.show()\n" + "\n" + "The negative binomial distribution is also available as\n" + "`scipy.stats.nbinom`. The percentile function method ``ppf``\n" + "returns the result of `nbdtrik` rounded up to integers:\n" + "\n" + ">>> from scipy.stats import nbinom\n" + ">>> q, n, p = 0.6, 5, 0.5\n" + ">>> nbinom.ppf(q, n, p), nbdtrik(q, n, p)\n" + "(5.0, 4.800428460273882)") +ufunc_nbdtrik_loops[0] = loop_d_ddd__As_fff_f +ufunc_nbdtrik_loops[1] = loop_d_ddd__As_ddd_d +ufunc_nbdtrik_types[0] = NPY_FLOAT +ufunc_nbdtrik_types[1] = NPY_FLOAT +ufunc_nbdtrik_types[2] = NPY_FLOAT +ufunc_nbdtrik_types[3] = NPY_FLOAT +ufunc_nbdtrik_types[4] = NPY_DOUBLE +ufunc_nbdtrik_types[5] = NPY_DOUBLE +ufunc_nbdtrik_types[6] = NPY_DOUBLE +ufunc_nbdtrik_types[7] = NPY_DOUBLE +ufunc_nbdtrik_ptr[2*0] = _func_nbdtrik +ufunc_nbdtrik_ptr[2*0+1] = ("nbdtrik") +ufunc_nbdtrik_ptr[2*1] = _func_nbdtrik +ufunc_nbdtrik_ptr[2*1+1] = ("nbdtrik") +ufunc_nbdtrik_data[0] = &ufunc_nbdtrik_ptr[2*0] +ufunc_nbdtrik_data[1] = &ufunc_nbdtrik_ptr[2*1] +nbdtrik = np.PyUFunc_FromFuncAndData(ufunc_nbdtrik_loops, ufunc_nbdtrik_data, ufunc_nbdtrik_types, 2, 3, 1, 0, "nbdtrik", ufunc_nbdtrik_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nbdtrin_loops[2] +cdef void *ufunc_nbdtrin_ptr[4] +cdef void *ufunc_nbdtrin_data[2] +cdef char ufunc_nbdtrin_types[8] +cdef char *ufunc_nbdtrin_doc = ( + "nbdtrin(k, y, p, out=None)\n" + "\n" + "Inverse of `nbdtr` vs `n`.\n" + "\n" + "Returns the inverse with respect to the parameter `n` of\n" + "``y = nbdtr(k, n, p)``, the negative binomial cumulative distribution\n" + "function.\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " The maximum number of allowed failures (nonnegative int).\n" + "y : array_like\n" + " The probability of `k` or fewer failures before `n` successes (float).\n" + "p : array_like\n" + " Probability of success in a single event (float).\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "n : scalar or ndarray\n" + " The number of successes `n` such that `nbdtr(k, n, p) = y`.\n" + "\n" + "See Also\n" + "--------\n" + "nbdtr : Cumulative distribution function of the negative binomial.\n" + "nbdtri : Inverse with respect to `p` of `nbdtr(k, n, p)`.\n" + "nbdtrik : Inverse with respect to `k` of `nbdtr(k, n, p)`.\n" + "\n" + "Notes\n" + "-----\n" + "Wrapper for the CDFLIB [1]_ Fortran routine `cdfnbn`.\n" + "\n" + "Formula 26.5.26 of [2]_,\n" + "\n" + ".. math::\n" + " \\sum_{j=k + 1}^\\infty {{n + j - 1}\n" + " \\choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n),\n" + "\n" + "is used to reduce calculation of the cumulative distribution function to\n" + "that of a regularized incomplete beta :math:`I`.\n" + "\n" + "Computation of `n` involves a search for a value that produces the desired\n" + "value of `y`. The search relies on the monotonicity of `y` with `n`.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Barry Brown, James Lovato, and Kathy Russell,\n" + " CDFLIB: Library of Fortran Routines for Cumulative Distribution\n" + " Functions, Inverses, and Other Parameters.\n" + ".. [2] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.\n" + "\n" + "Examples\n" + "--------\n" + "Compute the negative binomial cumulative distribution function for an\n" + "exemplary parameter set.\n" + "\n" + ">>> from scipy.special import nbdtr, nbdtrin\n" + ">>> k, n, p = 5, 2, 0.5\n" + ">>> cdf_value = nbdtr(k, n, p)\n" + ">>> cdf_value\n" + "0.9375\n" + "\n" + "Verify that `nbdtrin` recovers the original value for `n` up to floating\n" + "point accuracy.\n" + "\n" + ">>> nbdtrin(k, cdf_value, p)\n" + "1.999999999998137") +ufunc_nbdtrin_loops[0] = loop_d_ddd__As_fff_f +ufunc_nbdtrin_loops[1] = loop_d_ddd__As_ddd_d +ufunc_nbdtrin_types[0] = NPY_FLOAT +ufunc_nbdtrin_types[1] = NPY_FLOAT +ufunc_nbdtrin_types[2] = NPY_FLOAT +ufunc_nbdtrin_types[3] = NPY_FLOAT +ufunc_nbdtrin_types[4] = NPY_DOUBLE +ufunc_nbdtrin_types[5] = NPY_DOUBLE +ufunc_nbdtrin_types[6] = NPY_DOUBLE +ufunc_nbdtrin_types[7] = NPY_DOUBLE +ufunc_nbdtrin_ptr[2*0] = _func_nbdtrin +ufunc_nbdtrin_ptr[2*0+1] = ("nbdtrin") +ufunc_nbdtrin_ptr[2*1] = _func_nbdtrin +ufunc_nbdtrin_ptr[2*1+1] = ("nbdtrin") +ufunc_nbdtrin_data[0] = &ufunc_nbdtrin_ptr[2*0] +ufunc_nbdtrin_data[1] = &ufunc_nbdtrin_ptr[2*1] +nbdtrin = np.PyUFunc_FromFuncAndData(ufunc_nbdtrin_loops, ufunc_nbdtrin_data, ufunc_nbdtrin_types, 2, 3, 1, 0, "nbdtrin", ufunc_nbdtrin_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_ncfdtr_loops[2] +cdef void *ufunc_ncfdtr_ptr[4] +cdef void *ufunc_ncfdtr_data[2] +cdef char ufunc_ncfdtr_types[10] +cdef char *ufunc_ncfdtr_doc = ( + "ncfdtr(dfn, dfd, nc, f, out=None)\n" + "\n" + "Cumulative distribution function of the non-central F distribution.\n" + "\n" + "The non-central F describes the distribution of,\n" + "\n" + ".. math::\n" + " Z = \\frac{X/d_n}{Y/d_d}\n" + "\n" + "where :math:`X` and :math:`Y` are independently distributed, with\n" + ":math:`X` distributed non-central :math:`\\chi^2` with noncentrality\n" + "parameter `nc` and :math:`d_n` degrees of freedom, and :math:`Y`\n" + "distributed :math:`\\chi^2` with :math:`d_d` degrees of freedom.\n" + "\n" + "Parameters\n" + "----------\n" + "dfn : array_like\n" + " Degrees of freedom of the numerator sum of squares. Range (0, inf).\n" + "dfd : array_like\n" + " Degrees of freedom of the denominator sum of squares. Range (0, inf).\n" + "nc : array_like\n" + " Noncentrality parameter. Range [0, inf).\n" + "f : array_like\n" + " Quantiles, i.e. the upper limit of integration.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "cdf : scalar or ndarray\n" + " The calculated CDF. If all inputs are scalar, the return will be a\n" + " float. Otherwise it will be an array.\n" + "\n" + "See Also\n" + "--------\n" + "ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.\n" + "ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.\n" + "ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.\n" + "ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.\n" + "scipy.stats.ncf : Non-central F distribution.\n" + "\n" + "Notes\n" + "-----\n" + "This function calculates the CDF of the non-central f distribution using\n" + "the Boost Math C++ library [1]_.\n" + "\n" + "The cumulative distribution function is computed using Formula 26.6.20 of\n" + "[2]_:\n" + "\n" + ".. math::\n" + " F(d_n, d_d, n_c, f) = \\sum_{j=0}^\\infty e^{-n_c/2}\n" + " \\frac{(n_c/2)^j}{j!} I_{x}(\\frac{d_n}{2} + j, \\frac{d_d}{2}),\n" + "\n" + "where :math:`I` is the regularized incomplete beta function, and\n" + ":math:`x = f d_n/(f d_n + d_d)`.\n" + "\n" + "Note that argument order of `ncfdtr` is different from that of the\n" + "similar ``cdf`` method of `scipy.stats.ncf`: `f` is the last\n" + "parameter of `ncfdtr` but the first parameter of ``scipy.stats.ncf.cdf``.\n" + "\n" + "References\n" + "----------\n" + ".. [1] The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/.\n" + ".. [2] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy import special\n" + ">>> from scipy import stats\n" + ">>> import matplotlib.pyplot as plt\n" + "\n" + "Plot the CDF of the non-central F distribution, for nc=0. Compare with the\n" + "F-distribution from scipy.stats:\n" + "\n" + ">>> x = np.linspace(-1, 8, num=500)\n" + ">>> dfn = 3\n" + ">>> dfd = 2\n" + ">>> ncf_stats = stats.f.cdf(x, dfn, dfd)\n" + ">>> ncf_special = special.ncfdtr(dfn, dfd, 0, x)\n" + "\n" + ">>> fig = plt.figure()\n" + ">>> ax = fig.add_subplot(111)\n" + ">>> ax.plot(x, ncf_stats, 'b-', lw=3)\n" + ">>> ax.plot(x, ncf_special, 'r-')\n" + ">>> plt.show()") +ufunc_ncfdtr_loops[0] = loop_f_ffff__As_ffff_f +ufunc_ncfdtr_loops[1] = loop_d_dddd__As_dddd_d +ufunc_ncfdtr_types[0] = NPY_FLOAT +ufunc_ncfdtr_types[1] = NPY_FLOAT +ufunc_ncfdtr_types[2] = NPY_FLOAT +ufunc_ncfdtr_types[3] = NPY_FLOAT +ufunc_ncfdtr_types[4] = NPY_FLOAT +ufunc_ncfdtr_types[5] = NPY_DOUBLE +ufunc_ncfdtr_types[6] = NPY_DOUBLE +ufunc_ncfdtr_types[7] = NPY_DOUBLE +ufunc_ncfdtr_types[8] = NPY_DOUBLE +ufunc_ncfdtr_types[9] = NPY_DOUBLE +ufunc_ncfdtr_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncf_cdf_float +ufunc_ncfdtr_ptr[2*0+1] = ("ncfdtr") +ufunc_ncfdtr_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncf_cdf_double +ufunc_ncfdtr_ptr[2*1+1] = ("ncfdtr") +ufunc_ncfdtr_data[0] = &ufunc_ncfdtr_ptr[2*0] +ufunc_ncfdtr_data[1] = &ufunc_ncfdtr_ptr[2*1] +ncfdtr = np.PyUFunc_FromFuncAndData(ufunc_ncfdtr_loops, ufunc_ncfdtr_data, ufunc_ncfdtr_types, 2, 4, 1, 0, "ncfdtr", ufunc_ncfdtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_ncfdtri_loops[2] +cdef void *ufunc_ncfdtri_ptr[4] +cdef void *ufunc_ncfdtri_data[2] +cdef char ufunc_ncfdtri_types[10] +cdef char *ufunc_ncfdtri_doc = ( + "ncfdtri(dfn, dfd, nc, p, out=None)\n" + "\n" + "Inverse with respect to `f` of the CDF of the non-central F distribution.\n" + "\n" + "See `ncfdtr` for more details.\n" + "\n" + "Parameters\n" + "----------\n" + "dfn : array_like\n" + " Degrees of freedom of the numerator sum of squares. Range (0, inf).\n" + "dfd : array_like\n" + " Degrees of freedom of the denominator sum of squares. Range (0, inf).\n" + "nc : array_like\n" + " Noncentrality parameter. Range [0, inf).\n" + "p : array_like\n" + " Value of the cumulative distribution function. Must be in the\n" + " range [0, 1].\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "f : scalar or ndarray\n" + " Quantiles, i.e., the upper limit of integration.\n" + "\n" + "See Also\n" + "--------\n" + "ncfdtr : CDF of the non-central F distribution.\n" + "ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.\n" + "ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.\n" + "ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.\n" + "scipy.stats.ncf : Non-central F distribution.\n" + "\n" + "Notes\n" + "-----\n" + "This function calculates the Quantile of the non-central f distribution\n" + "using the Boost Math C++ library [1]_.\n" + "\n" + "Note that argument order of `ncfdtri` is different from that of the\n" + "similar ``ppf`` method of `scipy.stats.ncf`. `p` is the last parameter\n" + "of `ncfdtri` but the first parameter of ``scipy.stats.ncf.ppf``.\n" + "\n" + "References\n" + "----------\n" + ".. [1] The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import ncfdtr, ncfdtri\n" + "\n" + "Compute the CDF for several values of `f`:\n" + "\n" + ">>> f = [0.5, 1, 1.5]\n" + ">>> p = ncfdtr(2, 3, 1.5, f)\n" + ">>> p\n" + "array([ 0.20782291, 0.36107392, 0.47345752])\n" + "\n" + "Compute the inverse. We recover the values of `f`, as expected:\n" + "\n" + ">>> ncfdtri(2, 3, 1.5, p)\n" + "array([ 0.5, 1. , 1.5])") +ufunc_ncfdtri_loops[0] = loop_f_ffff__As_ffff_f +ufunc_ncfdtri_loops[1] = loop_d_dddd__As_dddd_d +ufunc_ncfdtri_types[0] = NPY_FLOAT +ufunc_ncfdtri_types[1] = NPY_FLOAT +ufunc_ncfdtri_types[2] = NPY_FLOAT +ufunc_ncfdtri_types[3] = NPY_FLOAT +ufunc_ncfdtri_types[4] = NPY_FLOAT +ufunc_ncfdtri_types[5] = NPY_DOUBLE +ufunc_ncfdtri_types[6] = NPY_DOUBLE +ufunc_ncfdtri_types[7] = NPY_DOUBLE +ufunc_ncfdtri_types[8] = NPY_DOUBLE +ufunc_ncfdtri_types[9] = NPY_DOUBLE +ufunc_ncfdtri_ptr[2*0] = scipy.special._ufuncs_cxx._export_ncf_ppf_float +ufunc_ncfdtri_ptr[2*0+1] = ("ncfdtri") +ufunc_ncfdtri_ptr[2*1] = scipy.special._ufuncs_cxx._export_ncf_ppf_double +ufunc_ncfdtri_ptr[2*1+1] = ("ncfdtri") +ufunc_ncfdtri_data[0] = &ufunc_ncfdtri_ptr[2*0] +ufunc_ncfdtri_data[1] = &ufunc_ncfdtri_ptr[2*1] +ncfdtri = np.PyUFunc_FromFuncAndData(ufunc_ncfdtri_loops, ufunc_ncfdtri_data, ufunc_ncfdtri_types, 2, 4, 1, 0, "ncfdtri", ufunc_ncfdtri_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_ncfdtridfd_loops[2] +cdef void *ufunc_ncfdtridfd_ptr[4] +cdef void *ufunc_ncfdtridfd_data[2] +cdef char ufunc_ncfdtridfd_types[10] +cdef char *ufunc_ncfdtridfd_doc = ( + "ncfdtridfd(dfn, p, nc, f, out=None)\n" + "\n" + "Calculate degrees of freedom (denominator) for the noncentral F-distribution.\n" + "\n" + "This is the inverse with respect to `dfd` of `ncfdtr`.\n" + "See `ncfdtr` for more details.\n" + "\n" + "Parameters\n" + "----------\n" + "dfn : array_like\n" + " Degrees of freedom of the numerator sum of squares. Range (0, inf).\n" + "p : array_like\n" + " Value of the cumulative distribution function. Must be in the\n" + " range [0, 1].\n" + "nc : array_like\n" + " Noncentrality parameter. Should be in range (0, 1e4).\n" + "f : array_like\n" + " Quantiles, i.e., the upper limit of integration.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "dfd : scalar or ndarray\n" + " Degrees of freedom of the denominator sum of squares.\n" + "\n" + "See Also\n" + "--------\n" + "ncfdtr : CDF of the non-central F distribution.\n" + "ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.\n" + "ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.\n" + "ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.\n" + "\n" + "Notes\n" + "-----\n" + "The value of the cumulative noncentral F distribution is not necessarily\n" + "monotone in either degrees of freedom. There thus may be two values that\n" + "provide a given CDF value. This routine assumes monotonicity and will\n" + "find an arbitrary one of the two values.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import ncfdtr, ncfdtridfd\n" + "\n" + "Compute the CDF for several values of `dfd`:\n" + "\n" + ">>> dfd = [1, 2, 3]\n" + ">>> p = ncfdtr(2, dfd, 0.25, 15)\n" + ">>> p\n" + "array([ 0.8097138 , 0.93020416, 0.96787852])\n" + "\n" + "Compute the inverse. We recover the values of `dfd`, as expected:\n" + "\n" + ">>> ncfdtridfd(2, p, 0.25, 15)\n" + "array([ 1., 2., 3.])") +ufunc_ncfdtridfd_loops[0] = loop_d_dddd__As_ffff_f +ufunc_ncfdtridfd_loops[1] = loop_d_dddd__As_dddd_d +ufunc_ncfdtridfd_types[0] = NPY_FLOAT +ufunc_ncfdtridfd_types[1] = NPY_FLOAT +ufunc_ncfdtridfd_types[2] = NPY_FLOAT +ufunc_ncfdtridfd_types[3] = NPY_FLOAT +ufunc_ncfdtridfd_types[4] = NPY_FLOAT +ufunc_ncfdtridfd_types[5] = NPY_DOUBLE +ufunc_ncfdtridfd_types[6] = NPY_DOUBLE +ufunc_ncfdtridfd_types[7] = NPY_DOUBLE +ufunc_ncfdtridfd_types[8] = NPY_DOUBLE +ufunc_ncfdtridfd_types[9] = NPY_DOUBLE +ufunc_ncfdtridfd_ptr[2*0] = _func_ncfdtridfd +ufunc_ncfdtridfd_ptr[2*0+1] = ("ncfdtridfd") +ufunc_ncfdtridfd_ptr[2*1] = _func_ncfdtridfd +ufunc_ncfdtridfd_ptr[2*1+1] = ("ncfdtridfd") +ufunc_ncfdtridfd_data[0] = &ufunc_ncfdtridfd_ptr[2*0] +ufunc_ncfdtridfd_data[1] = &ufunc_ncfdtridfd_ptr[2*1] +ncfdtridfd = np.PyUFunc_FromFuncAndData(ufunc_ncfdtridfd_loops, ufunc_ncfdtridfd_data, ufunc_ncfdtridfd_types, 2, 4, 1, 0, "ncfdtridfd", ufunc_ncfdtridfd_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_ncfdtridfn_loops[2] +cdef void *ufunc_ncfdtridfn_ptr[4] +cdef void *ufunc_ncfdtridfn_data[2] +cdef char ufunc_ncfdtridfn_types[10] +cdef char *ufunc_ncfdtridfn_doc = ( + "ncfdtridfn(p, dfd, nc, f, out=None)\n" + "\n" + "Calculate degrees of freedom (numerator) for the noncentral F-distribution.\n" + "\n" + "This is the inverse with respect to `dfn` of `ncfdtr`.\n" + "See `ncfdtr` for more details.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Value of the cumulative distribution function. Must be in the\n" + " range [0, 1].\n" + "dfd : array_like\n" + " Degrees of freedom of the denominator sum of squares. Range (0, inf).\n" + "nc : array_like\n" + " Noncentrality parameter. Should be in range (0, 1e4).\n" + "f : float\n" + " Quantiles, i.e., the upper limit of integration.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "dfn : scalar or ndarray\n" + " Degrees of freedom of the numerator sum of squares.\n" + "\n" + "See Also\n" + "--------\n" + "ncfdtr : CDF of the non-central F distribution.\n" + "ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.\n" + "ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.\n" + "ncfdtrinc : Inverse of `ncfdtr` with respect to `nc`.\n" + "\n" + "Notes\n" + "-----\n" + "The value of the cumulative noncentral F distribution is not necessarily\n" + "monotone in either degrees of freedom. There thus may be two values that\n" + "provide a given CDF value. This routine assumes monotonicity and will\n" + "find an arbitrary one of the two values.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import ncfdtr, ncfdtridfn\n" + "\n" + "Compute the CDF for several values of `dfn`:\n" + "\n" + ">>> dfn = [1, 2, 3]\n" + ">>> p = ncfdtr(dfn, 2, 0.25, 15)\n" + ">>> p\n" + "array([ 0.92562363, 0.93020416, 0.93188394])\n" + "\n" + "Compute the inverse. We recover the values of `dfn`, as expected:\n" + "\n" + ">>> ncfdtridfn(p, 2, 0.25, 15)\n" + "array([ 1., 2., 3.])") +ufunc_ncfdtridfn_loops[0] = loop_d_dddd__As_ffff_f +ufunc_ncfdtridfn_loops[1] = loop_d_dddd__As_dddd_d +ufunc_ncfdtridfn_types[0] = NPY_FLOAT +ufunc_ncfdtridfn_types[1] = NPY_FLOAT +ufunc_ncfdtridfn_types[2] = NPY_FLOAT +ufunc_ncfdtridfn_types[3] = NPY_FLOAT +ufunc_ncfdtridfn_types[4] = NPY_FLOAT +ufunc_ncfdtridfn_types[5] = NPY_DOUBLE +ufunc_ncfdtridfn_types[6] = NPY_DOUBLE +ufunc_ncfdtridfn_types[7] = NPY_DOUBLE +ufunc_ncfdtridfn_types[8] = NPY_DOUBLE +ufunc_ncfdtridfn_types[9] = NPY_DOUBLE +ufunc_ncfdtridfn_ptr[2*0] = _func_ncfdtridfn +ufunc_ncfdtridfn_ptr[2*0+1] = ("ncfdtridfn") +ufunc_ncfdtridfn_ptr[2*1] = _func_ncfdtridfn +ufunc_ncfdtridfn_ptr[2*1+1] = ("ncfdtridfn") +ufunc_ncfdtridfn_data[0] = &ufunc_ncfdtridfn_ptr[2*0] +ufunc_ncfdtridfn_data[1] = &ufunc_ncfdtridfn_ptr[2*1] +ncfdtridfn = np.PyUFunc_FromFuncAndData(ufunc_ncfdtridfn_loops, ufunc_ncfdtridfn_data, ufunc_ncfdtridfn_types, 2, 4, 1, 0, "ncfdtridfn", ufunc_ncfdtridfn_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_ncfdtrinc_loops[2] +cdef void *ufunc_ncfdtrinc_ptr[4] +cdef void *ufunc_ncfdtrinc_data[2] +cdef char ufunc_ncfdtrinc_types[10] +cdef char *ufunc_ncfdtrinc_doc = ( + "ncfdtrinc(dfn, dfd, p, f, out=None)\n" + "\n" + "Calculate non-centrality parameter for non-central F distribution.\n" + "\n" + "This is the inverse with respect to `nc` of `ncfdtr`.\n" + "See `ncfdtr` for more details.\n" + "\n" + "Parameters\n" + "----------\n" + "dfn : array_like\n" + " Degrees of freedom of the numerator sum of squares. Range (0, inf).\n" + "dfd : array_like\n" + " Degrees of freedom of the denominator sum of squares. Range (0, inf).\n" + "p : array_like\n" + " Value of the cumulative distribution function. Must be in the\n" + " range [0, 1].\n" + "f : array_like\n" + " Quantiles, i.e., the upper limit of integration.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "nc : scalar or ndarray\n" + " Noncentrality parameter.\n" + "\n" + "See Also\n" + "--------\n" + "ncfdtr : CDF of the non-central F distribution.\n" + "ncfdtri : Quantile function; inverse of `ncfdtr` with respect to `f`.\n" + "ncfdtridfd : Inverse of `ncfdtr` with respect to `dfd`.\n" + "ncfdtridfn : Inverse of `ncfdtr` with respect to `dfn`.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import ncfdtr, ncfdtrinc\n" + "\n" + "Compute the CDF for several values of `nc`:\n" + "\n" + ">>> nc = [0.5, 1.5, 2.0]\n" + ">>> p = ncfdtr(2, 3, nc, 15)\n" + ">>> p\n" + "array([ 0.96309246, 0.94327955, 0.93304098])\n" + "\n" + "Compute the inverse. We recover the values of `nc`, as expected:\n" + "\n" + ">>> ncfdtrinc(2, 3, p, 15)\n" + "array([ 0.5, 1.5, 2. ])") +ufunc_ncfdtrinc_loops[0] = loop_d_dddd__As_ffff_f +ufunc_ncfdtrinc_loops[1] = loop_d_dddd__As_dddd_d +ufunc_ncfdtrinc_types[0] = NPY_FLOAT +ufunc_ncfdtrinc_types[1] = NPY_FLOAT +ufunc_ncfdtrinc_types[2] = NPY_FLOAT +ufunc_ncfdtrinc_types[3] = NPY_FLOAT +ufunc_ncfdtrinc_types[4] = NPY_FLOAT +ufunc_ncfdtrinc_types[5] = NPY_DOUBLE +ufunc_ncfdtrinc_types[6] = NPY_DOUBLE +ufunc_ncfdtrinc_types[7] = NPY_DOUBLE +ufunc_ncfdtrinc_types[8] = NPY_DOUBLE +ufunc_ncfdtrinc_types[9] = NPY_DOUBLE +ufunc_ncfdtrinc_ptr[2*0] = _func_ncfdtrinc +ufunc_ncfdtrinc_ptr[2*0+1] = ("ncfdtrinc") +ufunc_ncfdtrinc_ptr[2*1] = _func_ncfdtrinc +ufunc_ncfdtrinc_ptr[2*1+1] = ("ncfdtrinc") +ufunc_ncfdtrinc_data[0] = &ufunc_ncfdtrinc_ptr[2*0] +ufunc_ncfdtrinc_data[1] = &ufunc_ncfdtrinc_ptr[2*1] +ncfdtrinc = np.PyUFunc_FromFuncAndData(ufunc_ncfdtrinc_loops, ufunc_ncfdtrinc_data, ufunc_ncfdtrinc_types, 2, 4, 1, 0, "ncfdtrinc", ufunc_ncfdtrinc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nctdtr_loops[2] +cdef void *ufunc_nctdtr_ptr[4] +cdef void *ufunc_nctdtr_data[2] +cdef char ufunc_nctdtr_types[8] +cdef char *ufunc_nctdtr_doc = ( + "nctdtr(df, nc, t, out=None)\n" + "\n" + "Cumulative distribution function of the non-central `t` distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "df : array_like\n" + " Degrees of freedom of the distribution. Should be in range (0, inf).\n" + "nc : array_like\n" + " Noncentrality parameter.\n" + "t : array_like\n" + " Quantiles, i.e., the upper limit of integration.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "cdf : scalar or ndarray\n" + " The calculated CDF. If all inputs are scalar, the return will be a\n" + " float. Otherwise, it will be an array.\n" + "\n" + "See Also\n" + "--------\n" + "nctdtrit : Inverse CDF (iCDF) of the non-central t distribution.\n" + "nctdtridf : Calculate degrees of freedom, given CDF and iCDF values.\n" + "nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.\n" + "\n" + "Notes\n" + "-----\n" + "This function calculates the CDF of the non-central t distribution using\n" + "the Boost Math C++ library [1]_.\n" + "\n" + "Note that the argument order of `nctdtr` is different from that of the\n" + "similar ``cdf`` method of `scipy.stats.nct`: `t` is the last\n" + "parameter of `nctdtr` but the first parameter of ``scipy.stats.nct.cdf``.\n" + "\n" + "References\n" + "----------\n" + ".. [1] The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy import special\n" + ">>> from scipy import stats\n" + ">>> import matplotlib.pyplot as plt\n" + "\n" + "Plot the CDF of the non-central t distribution, for nc=0. Compare with the\n" + "t-distribution from scipy.stats:\n" + "\n" + ">>> x = np.linspace(-5, 5, num=500)\n" + ">>> df = 3\n" + ">>> nct_stats = stats.t.cdf(x, df)\n" + ">>> nct_special = special.nctdtr(df, 0, x)\n" + "\n" + ">>> fig = plt.figure()\n" + ">>> ax = fig.add_subplot(111)\n" + ">>> ax.plot(x, nct_stats, 'b-', lw=3)\n" + ">>> ax.plot(x, nct_special, 'r-')\n" + ">>> plt.show()") +ufunc_nctdtr_loops[0] = loop_f_fff__As_fff_f +ufunc_nctdtr_loops[1] = loop_d_ddd__As_ddd_d +ufunc_nctdtr_types[0] = NPY_FLOAT +ufunc_nctdtr_types[1] = NPY_FLOAT +ufunc_nctdtr_types[2] = NPY_FLOAT +ufunc_nctdtr_types[3] = NPY_FLOAT +ufunc_nctdtr_types[4] = NPY_DOUBLE +ufunc_nctdtr_types[5] = NPY_DOUBLE +ufunc_nctdtr_types[6] = NPY_DOUBLE +ufunc_nctdtr_types[7] = NPY_DOUBLE +ufunc_nctdtr_ptr[2*0] = scipy.special._ufuncs_cxx._export_nct_cdf_float +ufunc_nctdtr_ptr[2*0+1] = ("nctdtr") +ufunc_nctdtr_ptr[2*1] = scipy.special._ufuncs_cxx._export_nct_cdf_double +ufunc_nctdtr_ptr[2*1+1] = ("nctdtr") +ufunc_nctdtr_data[0] = &ufunc_nctdtr_ptr[2*0] +ufunc_nctdtr_data[1] = &ufunc_nctdtr_ptr[2*1] +nctdtr = np.PyUFunc_FromFuncAndData(ufunc_nctdtr_loops, ufunc_nctdtr_data, ufunc_nctdtr_types, 2, 3, 1, 0, "nctdtr", ufunc_nctdtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nctdtridf_loops[2] +cdef void *ufunc_nctdtridf_ptr[4] +cdef void *ufunc_nctdtridf_data[2] +cdef char ufunc_nctdtridf_types[8] +cdef char *ufunc_nctdtridf_doc = ( + "nctdtridf(p, nc, t, out=None)\n" + "\n" + "Calculate degrees of freedom for non-central t distribution.\n" + "\n" + "See `nctdtr` for more details.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " CDF values, in range (0, 1].\n" + "nc : array_like\n" + " Noncentrality parameter. Should be in range (-1e6, 1e6).\n" + "t : array_like\n" + " Quantiles, i.e., the upper limit of integration.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "df : scalar or ndarray\n" + " The degrees of freedom. If all inputs are scalar, the return will be a\n" + " float. Otherwise, it will be an array.\n" + "\n" + "See Also\n" + "--------\n" + "nctdtr : CDF of the non-central `t` distribution.\n" + "nctdtrit : Inverse CDF (iCDF) of the non-central t distribution.\n" + "nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import nctdtr, nctdtridf\n" + "\n" + "Compute the CDF for several values of `df`:\n" + "\n" + ">>> df = [1, 2, 3]\n" + ">>> p = nctdtr(df, 0.25, 1)\n" + ">>> p\n" + "array([0.67491974, 0.716464 , 0.73349456])\n" + "\n" + "Compute the inverse. We recover the values of `df`, as expected:\n" + "\n" + ">>> nctdtridf(p, 0.25, 1)\n" + "array([1., 2., 3.])") +ufunc_nctdtridf_loops[0] = loop_d_ddd__As_fff_f +ufunc_nctdtridf_loops[1] = loop_d_ddd__As_ddd_d +ufunc_nctdtridf_types[0] = NPY_FLOAT +ufunc_nctdtridf_types[1] = NPY_FLOAT +ufunc_nctdtridf_types[2] = NPY_FLOAT +ufunc_nctdtridf_types[3] = NPY_FLOAT +ufunc_nctdtridf_types[4] = NPY_DOUBLE +ufunc_nctdtridf_types[5] = NPY_DOUBLE +ufunc_nctdtridf_types[6] = NPY_DOUBLE +ufunc_nctdtridf_types[7] = NPY_DOUBLE +ufunc_nctdtridf_ptr[2*0] = _func_nctdtridf +ufunc_nctdtridf_ptr[2*0+1] = ("nctdtridf") +ufunc_nctdtridf_ptr[2*1] = _func_nctdtridf +ufunc_nctdtridf_ptr[2*1+1] = ("nctdtridf") +ufunc_nctdtridf_data[0] = &ufunc_nctdtridf_ptr[2*0] +ufunc_nctdtridf_data[1] = &ufunc_nctdtridf_ptr[2*1] +nctdtridf = np.PyUFunc_FromFuncAndData(ufunc_nctdtridf_loops, ufunc_nctdtridf_data, ufunc_nctdtridf_types, 2, 3, 1, 0, "nctdtridf", ufunc_nctdtridf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nctdtrinc_loops[2] +cdef void *ufunc_nctdtrinc_ptr[4] +cdef void *ufunc_nctdtrinc_data[2] +cdef char ufunc_nctdtrinc_types[8] +cdef char *ufunc_nctdtrinc_doc = ( + "nctdtrinc(df, p, t, out=None)\n" + "\n" + "Calculate non-centrality parameter for non-central t distribution.\n" + "\n" + "See `nctdtr` for more details.\n" + "\n" + "Parameters\n" + "----------\n" + "df : array_like\n" + " Degrees of freedom of the distribution. Should be in range (0, inf).\n" + "p : array_like\n" + " CDF values, in range (0, 1].\n" + "t : array_like\n" + " Quantiles, i.e., the upper limit of integration.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "nc : scalar or ndarray\n" + " Noncentrality parameter\n" + "\n" + "See Also\n" + "--------\n" + "nctdtr : CDF of the non-central `t` distribution.\n" + "nctdtrit : Inverse CDF (iCDF) of the non-central t distribution.\n" + "nctdtridf : Calculate degrees of freedom, given CDF and iCDF values.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import nctdtr, nctdtrinc\n" + "\n" + "Compute the CDF for several values of `nc`:\n" + "\n" + ">>> nc = [0.5, 1.5, 2.5]\n" + ">>> p = nctdtr(3, nc, 1.5)\n" + ">>> p\n" + "array([0.77569497, 0.45524533, 0.1668691 ])\n" + "\n" + "Compute the inverse. We recover the values of `nc`, as expected:\n" + "\n" + ">>> nctdtrinc(3, p, 1.5)\n" + "array([0.5, 1.5, 2.5])") +ufunc_nctdtrinc_loops[0] = loop_d_ddd__As_fff_f +ufunc_nctdtrinc_loops[1] = loop_d_ddd__As_ddd_d +ufunc_nctdtrinc_types[0] = NPY_FLOAT +ufunc_nctdtrinc_types[1] = NPY_FLOAT +ufunc_nctdtrinc_types[2] = NPY_FLOAT +ufunc_nctdtrinc_types[3] = NPY_FLOAT +ufunc_nctdtrinc_types[4] = NPY_DOUBLE +ufunc_nctdtrinc_types[5] = NPY_DOUBLE +ufunc_nctdtrinc_types[6] = NPY_DOUBLE +ufunc_nctdtrinc_types[7] = NPY_DOUBLE +ufunc_nctdtrinc_ptr[2*0] = _func_nctdtrinc +ufunc_nctdtrinc_ptr[2*0+1] = ("nctdtrinc") +ufunc_nctdtrinc_ptr[2*1] = _func_nctdtrinc +ufunc_nctdtrinc_ptr[2*1+1] = ("nctdtrinc") +ufunc_nctdtrinc_data[0] = &ufunc_nctdtrinc_ptr[2*0] +ufunc_nctdtrinc_data[1] = &ufunc_nctdtrinc_ptr[2*1] +nctdtrinc = np.PyUFunc_FromFuncAndData(ufunc_nctdtrinc_loops, ufunc_nctdtrinc_data, ufunc_nctdtrinc_types, 2, 3, 1, 0, "nctdtrinc", ufunc_nctdtrinc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nctdtrit_loops[2] +cdef void *ufunc_nctdtrit_ptr[4] +cdef void *ufunc_nctdtrit_data[2] +cdef char ufunc_nctdtrit_types[8] +cdef char *ufunc_nctdtrit_doc = ( + "nctdtrit(df, nc, p, out=None)\n" + "\n" + "Inverse cumulative distribution function of the non-central t distribution.\n" + "\n" + "See `nctdtr` for more details.\n" + "\n" + "Parameters\n" + "----------\n" + "df : array_like\n" + " Degrees of freedom of the distribution. Should be in range (0, inf).\n" + "nc : array_like\n" + " Noncentrality parameter. Should be in range (-1e6, 1e6).\n" + "p : array_like\n" + " CDF values, in range (0, 1].\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "t : scalar or ndarray\n" + " Quantiles\n" + "\n" + "See Also\n" + "--------\n" + "nctdtr : CDF of the non-central `t` distribution.\n" + "nctdtridf : Calculate degrees of freedom, given CDF and iCDF values.\n" + "nctdtrinc : Calculate non-centrality parameter, given CDF iCDF values.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import nctdtr, nctdtrit\n" + "\n" + "Compute the CDF for several values of `t`:\n" + "\n" + ">>> t = [0.5, 1, 1.5]\n" + ">>> p = nctdtr(3, 1, t)\n" + ">>> p\n" + "array([0.29811049, 0.46922687, 0.6257559 ])\n" + "\n" + "Compute the inverse. We recover the values of `t`, as expected:\n" + "\n" + ">>> nctdtrit(3, 1, p)\n" + "array([0.5, 1. , 1.5])") +ufunc_nctdtrit_loops[0] = loop_d_ddd__As_fff_f +ufunc_nctdtrit_loops[1] = loop_d_ddd__As_ddd_d +ufunc_nctdtrit_types[0] = NPY_FLOAT +ufunc_nctdtrit_types[1] = NPY_FLOAT +ufunc_nctdtrit_types[2] = NPY_FLOAT +ufunc_nctdtrit_types[3] = NPY_FLOAT +ufunc_nctdtrit_types[4] = NPY_DOUBLE +ufunc_nctdtrit_types[5] = NPY_DOUBLE +ufunc_nctdtrit_types[6] = NPY_DOUBLE +ufunc_nctdtrit_types[7] = NPY_DOUBLE +ufunc_nctdtrit_ptr[2*0] = _func_nctdtrit +ufunc_nctdtrit_ptr[2*0+1] = ("nctdtrit") +ufunc_nctdtrit_ptr[2*1] = _func_nctdtrit +ufunc_nctdtrit_ptr[2*1+1] = ("nctdtrit") +ufunc_nctdtrit_data[0] = &ufunc_nctdtrit_ptr[2*0] +ufunc_nctdtrit_data[1] = &ufunc_nctdtrit_ptr[2*1] +nctdtrit = np.PyUFunc_FromFuncAndData(ufunc_nctdtrit_loops, ufunc_nctdtrit_data, ufunc_nctdtrit_types, 2, 3, 1, 0, "nctdtrit", ufunc_nctdtrit_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_ndtr_loops[4] +cdef void *ufunc_ndtr_ptr[8] +cdef void *ufunc_ndtr_data[4] +cdef char ufunc_ndtr_types[8] +cdef char *ufunc_ndtr_doc = ( + "ndtr(x, out=None)\n" + "\n" + "Cumulative distribution of the standard normal distribution.\n" + "\n" + "Returns the area under the standard Gaussian probability\n" + "density function, integrated from minus infinity to `x`\n" + "\n" + ".. math::\n" + "\n" + " \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^x \\exp(-t^2/2) dt\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like, real or complex\n" + " Argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The value of the normal CDF evaluated at `x`\n" + "\n" + "See Also\n" + "--------\n" + "log_ndtr : Logarithm of ndtr\n" + "ndtri : Inverse of ndtr, standard normal percentile function\n" + "erf : Error function\n" + "erfc : 1 - erf\n" + "scipy.stats.norm : Normal distribution\n" + "\n" + "Examples\n" + "--------\n" + "Evaluate `ndtr` at one point.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import ndtr\n" + ">>> ndtr(0.5)\n" + "0.6914624612740131\n" + "\n" + "Evaluate the function at several points by providing a NumPy array\n" + "or list for `x`.\n" + "\n" + ">>> ndtr([0, 0.5, 2])\n" + "array([0.5 , 0.69146246, 0.97724987])\n" + "\n" + "Plot the function.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> x = np.linspace(-5, 5, 100)\n" + ">>> fig, ax = plt.subplots()\n" + ">>> ax.plot(x, ndtr(x))\n" + ">>> ax.set_title(r\"Standard normal cumulative distribution function $\\Phi$\")\n" + ">>> plt.show()") +ufunc_ndtr_loops[0] = loop_d_d__As_f_f +ufunc_ndtr_loops[1] = loop_d_d__As_d_d +ufunc_ndtr_loops[2] = loop_D_D__As_F_F +ufunc_ndtr_loops[3] = loop_D_D__As_D_D +ufunc_ndtr_types[0] = NPY_FLOAT +ufunc_ndtr_types[1] = NPY_FLOAT +ufunc_ndtr_types[2] = NPY_DOUBLE +ufunc_ndtr_types[3] = NPY_DOUBLE +ufunc_ndtr_types[4] = NPY_CFLOAT +ufunc_ndtr_types[5] = NPY_CFLOAT +ufunc_ndtr_types[6] = NPY_CDOUBLE +ufunc_ndtr_types[7] = NPY_CDOUBLE +ufunc_ndtr_ptr[2*0] = _func_xsf_ndtr +ufunc_ndtr_ptr[2*0+1] = ("ndtr") +ufunc_ndtr_ptr[2*1] = _func_xsf_ndtr +ufunc_ndtr_ptr[2*1+1] = ("ndtr") +ufunc_ndtr_ptr[2*2] = scipy.special._ufuncs_cxx._export_faddeeva_ndtr +ufunc_ndtr_ptr[2*2+1] = ("ndtr") +ufunc_ndtr_ptr[2*3] = scipy.special._ufuncs_cxx._export_faddeeva_ndtr +ufunc_ndtr_ptr[2*3+1] = ("ndtr") +ufunc_ndtr_data[0] = &ufunc_ndtr_ptr[2*0] +ufunc_ndtr_data[1] = &ufunc_ndtr_ptr[2*1] +ufunc_ndtr_data[2] = &ufunc_ndtr_ptr[2*2] +ufunc_ndtr_data[3] = &ufunc_ndtr_ptr[2*3] +ndtr = np.PyUFunc_FromFuncAndData(ufunc_ndtr_loops, ufunc_ndtr_data, ufunc_ndtr_types, 4, 1, 1, 0, "ndtr", ufunc_ndtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_ndtri_loops[2] +cdef void *ufunc_ndtri_ptr[4] +cdef void *ufunc_ndtri_data[2] +cdef char ufunc_ndtri_types[4] +cdef char *ufunc_ndtri_doc = ( + "ndtri(y, out=None)\n" + "\n" + "Inverse of `ndtr` vs x\n" + "\n" + "Returns the argument x for which the area under the standard normal\n" + "probability density function (integrated from minus infinity to `x`)\n" + "is equal to y.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Probability\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "x : scalar or ndarray\n" + " Value of x such that ``ndtr(x) == p``.\n" + "\n" + "See Also\n" + "--------\n" + "ndtr : Standard normal cumulative probability distribution\n" + "ndtri_exp : Inverse of log_ndtr\n" + "\n" + "Examples\n" + "--------\n" + "`ndtri` is the percentile function of the standard normal distribution.\n" + "This means it returns the inverse of the cumulative density `ndtr`. First,\n" + "let us compute a cumulative density value.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import ndtri, ndtr\n" + ">>> cdf_val = ndtr(2)\n" + ">>> cdf_val\n" + "0.9772498680518208\n" + "\n" + "Verify that `ndtri` yields the original value for `x` up to floating point\n" + "errors.\n" + "\n" + ">>> ndtri(cdf_val)\n" + "2.0000000000000004\n" + "\n" + "Plot the function. For that purpose, we provide a NumPy array as argument.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> x = np.linspace(0.01, 1, 200)\n" + ">>> fig, ax = plt.subplots()\n" + ">>> ax.plot(x, ndtri(x))\n" + ">>> ax.set_title(\"Standard normal percentile function\")\n" + ">>> plt.show()") +ufunc_ndtri_loops[0] = loop_d_d__As_f_f +ufunc_ndtri_loops[1] = loop_d_d__As_d_d +ufunc_ndtri_types[0] = NPY_FLOAT +ufunc_ndtri_types[1] = NPY_FLOAT +ufunc_ndtri_types[2] = NPY_DOUBLE +ufunc_ndtri_types[3] = NPY_DOUBLE +ufunc_ndtri_ptr[2*0] = _func_xsf_ndtri +ufunc_ndtri_ptr[2*0+1] = ("ndtri") +ufunc_ndtri_ptr[2*1] = _func_xsf_ndtri +ufunc_ndtri_ptr[2*1+1] = ("ndtri") +ufunc_ndtri_data[0] = &ufunc_ndtri_ptr[2*0] +ufunc_ndtri_data[1] = &ufunc_ndtri_ptr[2*1] +ndtri = np.PyUFunc_FromFuncAndData(ufunc_ndtri_loops, ufunc_ndtri_data, ufunc_ndtri_types, 2, 1, 1, 0, "ndtri", ufunc_ndtri_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_ndtri_exp_loops[2] +cdef void *ufunc_ndtri_exp_ptr[4] +cdef void *ufunc_ndtri_exp_data[2] +cdef char ufunc_ndtri_exp_types[4] +cdef char *ufunc_ndtri_exp_doc = ( + "ndtri_exp(y, out=None)\n" + "\n" + "Inverse of `log_ndtr` vs x. Allows for greater precision than\n" + "`ndtri` composed with `numpy.exp` for very small values of y and for\n" + "y close to 0.\n" + "\n" + "Parameters\n" + "----------\n" + "y : array_like of float\n" + " Function argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Inverse of the log CDF of the standard normal distribution, evaluated\n" + " at y.\n" + "\n" + "See Also\n" + "--------\n" + "log_ndtr : log of the standard normal cumulative distribution function\n" + "ndtr : standard normal cumulative distribution function\n" + "ndtri : standard normal percentile function\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "`ndtri_exp` agrees with the naive implementation when the latter does\n" + "not suffer from underflow.\n" + "\n" + ">>> sc.ndtri_exp(-1)\n" + "-0.33747496376420244\n" + ">>> sc.ndtri(np.exp(-1))\n" + "-0.33747496376420244\n" + "\n" + "For extreme values of y, the naive approach fails\n" + "\n" + ">>> sc.ndtri(np.exp(-800))\n" + "-inf\n" + ">>> sc.ndtri(np.exp(-1e-20))\n" + "inf\n" + "\n" + "whereas `ndtri_exp` is still able to compute the result to high precision.\n" + "\n" + ">>> sc.ndtri_exp(-800)\n" + "-39.88469483825668\n" + ">>> sc.ndtri_exp(-1e-20)\n" + "9.262340089798409") +ufunc_ndtri_exp_loops[0] = loop_d_d__As_f_f +ufunc_ndtri_exp_loops[1] = loop_d_d__As_d_d +ufunc_ndtri_exp_types[0] = NPY_FLOAT +ufunc_ndtri_exp_types[1] = NPY_FLOAT +ufunc_ndtri_exp_types[2] = NPY_DOUBLE +ufunc_ndtri_exp_types[3] = NPY_DOUBLE +ufunc_ndtri_exp_ptr[2*0] = _func_ndtri_exp +ufunc_ndtri_exp_ptr[2*0+1] = ("ndtri_exp") +ufunc_ndtri_exp_ptr[2*1] = _func_ndtri_exp +ufunc_ndtri_exp_ptr[2*1+1] = ("ndtri_exp") +ufunc_ndtri_exp_data[0] = &ufunc_ndtri_exp_ptr[2*0] +ufunc_ndtri_exp_data[1] = &ufunc_ndtri_exp_ptr[2*1] +ndtri_exp = np.PyUFunc_FromFuncAndData(ufunc_ndtri_exp_loops, ufunc_ndtri_exp_data, ufunc_ndtri_exp_types, 2, 1, 1, 0, "ndtri_exp", ufunc_ndtri_exp_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nrdtrimn_loops[2] +cdef void *ufunc_nrdtrimn_ptr[4] +cdef void *ufunc_nrdtrimn_data[2] +cdef char ufunc_nrdtrimn_types[8] +cdef char *ufunc_nrdtrimn_doc = ( + "nrdtrimn(p, std, x, out=None)\n" + "\n" + "Calculate mean of normal distribution given other params.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " CDF values, in range (0, 1].\n" + "std : array_like\n" + " Standard deviation.\n" + "x : array_like\n" + " Quantiles, i.e. the upper limit of integration.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "mn : scalar or ndarray\n" + " The mean of the normal distribution.\n" + "\n" + "See Also\n" + "--------\n" + "scipy.stats.norm : Normal distribution\n" + "ndtr : Standard normal cumulative probability distribution\n" + "ndtri : Inverse of standard normal CDF with respect to quantile\n" + "nrdtrisd : Inverse of normal distribution CDF with respect to\n" + " standard deviation\n" + "\n" + "Examples\n" + "--------\n" + "`nrdtrimn` can be used to recover the mean of a normal distribution\n" + "if we know the CDF value `p` for a given quantile `x` and the\n" + "standard deviation `std`. First, we calculate\n" + "the normal distribution CDF for an exemplary parameter set.\n" + "\n" + ">>> from scipy.stats import norm\n" + ">>> mean = 3.\n" + ">>> std = 2.\n" + ">>> x = 6.\n" + ">>> p = norm.cdf(x, loc=mean, scale=std)\n" + ">>> p\n" + "0.9331927987311419\n" + "\n" + "Verify that `nrdtrimn` returns the original value for `mean`.\n" + "\n" + ">>> from scipy.special import nrdtrimn\n" + ">>> nrdtrimn(p, std, x)\n" + "3.0000000000000004") +ufunc_nrdtrimn_loops[0] = loop_d_ddd__As_fff_f +ufunc_nrdtrimn_loops[1] = loop_d_ddd__As_ddd_d +ufunc_nrdtrimn_types[0] = NPY_FLOAT +ufunc_nrdtrimn_types[1] = NPY_FLOAT +ufunc_nrdtrimn_types[2] = NPY_FLOAT +ufunc_nrdtrimn_types[3] = NPY_FLOAT +ufunc_nrdtrimn_types[4] = NPY_DOUBLE +ufunc_nrdtrimn_types[5] = NPY_DOUBLE +ufunc_nrdtrimn_types[6] = NPY_DOUBLE +ufunc_nrdtrimn_types[7] = NPY_DOUBLE +ufunc_nrdtrimn_ptr[2*0] = _func_nrdtrimn +ufunc_nrdtrimn_ptr[2*0+1] = ("nrdtrimn") +ufunc_nrdtrimn_ptr[2*1] = _func_nrdtrimn +ufunc_nrdtrimn_ptr[2*1+1] = ("nrdtrimn") +ufunc_nrdtrimn_data[0] = &ufunc_nrdtrimn_ptr[2*0] +ufunc_nrdtrimn_data[1] = &ufunc_nrdtrimn_ptr[2*1] +nrdtrimn = np.PyUFunc_FromFuncAndData(ufunc_nrdtrimn_loops, ufunc_nrdtrimn_data, ufunc_nrdtrimn_types, 2, 3, 1, 0, "nrdtrimn", ufunc_nrdtrimn_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_nrdtrisd_loops[2] +cdef void *ufunc_nrdtrisd_ptr[4] +cdef void *ufunc_nrdtrisd_data[2] +cdef char ufunc_nrdtrisd_types[8] +cdef char *ufunc_nrdtrisd_doc = ( + "nrdtrisd(mn, p, x, out=None)\n" + "\n" + "Calculate standard deviation of normal distribution given other params.\n" + "\n" + "Parameters\n" + "----------\n" + "mn : scalar or ndarray\n" + " The mean of the normal distribution.\n" + "p : array_like\n" + " CDF values, in range (0, 1].\n" + "x : array_like\n" + " Quantiles, i.e. the upper limit of integration.\n" + "\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "std : scalar or ndarray\n" + " Standard deviation.\n" + "\n" + "See Also\n" + "--------\n" + "scipy.stats.norm : Normal distribution\n" + "ndtr : Standard normal cumulative probability distribution\n" + "ndtri : Inverse of standard normal CDF with respect to quantile\n" + "nrdtrimn : Inverse of normal distribution CDF with respect to\n" + " mean\n" + "\n" + "Examples\n" + "--------\n" + "`nrdtrisd` can be used to recover the standard deviation of a normal\n" + "distribution if we know the CDF value `p` for a given quantile `x` and\n" + "the mean `mn`. First, we calculate the normal distribution CDF for an\n" + "exemplary parameter set.\n" + "\n" + ">>> from scipy.stats import norm\n" + ">>> mean = 3.\n" + ">>> std = 2.\n" + ">>> x = 6.\n" + ">>> p = norm.cdf(x, loc=mean, scale=std)\n" + ">>> p\n" + "0.9331927987311419\n" + "\n" + "Verify that `nrdtrisd` returns the original value for `std`.\n" + "\n" + ">>> from scipy.special import nrdtrisd\n" + ">>> nrdtrisd(mean, p, x)\n" + "2.0000000000000004") +ufunc_nrdtrisd_loops[0] = loop_d_ddd__As_fff_f +ufunc_nrdtrisd_loops[1] = loop_d_ddd__As_ddd_d +ufunc_nrdtrisd_types[0] = NPY_FLOAT +ufunc_nrdtrisd_types[1] = NPY_FLOAT +ufunc_nrdtrisd_types[2] = NPY_FLOAT +ufunc_nrdtrisd_types[3] = NPY_FLOAT +ufunc_nrdtrisd_types[4] = NPY_DOUBLE +ufunc_nrdtrisd_types[5] = NPY_DOUBLE +ufunc_nrdtrisd_types[6] = NPY_DOUBLE +ufunc_nrdtrisd_types[7] = NPY_DOUBLE +ufunc_nrdtrisd_ptr[2*0] = _func_nrdtrisd +ufunc_nrdtrisd_ptr[2*0+1] = ("nrdtrisd") +ufunc_nrdtrisd_ptr[2*1] = _func_nrdtrisd +ufunc_nrdtrisd_ptr[2*1+1] = ("nrdtrisd") +ufunc_nrdtrisd_data[0] = &ufunc_nrdtrisd_ptr[2*0] +ufunc_nrdtrisd_data[1] = &ufunc_nrdtrisd_ptr[2*1] +nrdtrisd = np.PyUFunc_FromFuncAndData(ufunc_nrdtrisd_loops, ufunc_nrdtrisd_data, ufunc_nrdtrisd_types, 2, 3, 1, 0, "nrdtrisd", ufunc_nrdtrisd_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_owens_t_loops[2] +cdef void *ufunc_owens_t_ptr[4] +cdef void *ufunc_owens_t_data[2] +cdef char ufunc_owens_t_types[6] +cdef char *ufunc_owens_t_doc = ( + "owens_t(h, a, out=None)\n" + "\n" + "Owen's T Function.\n" + "\n" + "The function T(h, a) gives the probability of the event\n" + "(X > h and 0 < Y < a * X) where X and Y are independent\n" + "standard normal random variables.\n" + "\n" + "Parameters\n" + "----------\n" + "h: array_like\n" + " Input value.\n" + "a: array_like\n" + " Input value.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "t: scalar or ndarray\n" + " Probability of the event (X > h and 0 < Y < a * X),\n" + " where X and Y are independent standard normal random variables.\n" + "\n" + "References\n" + "----------\n" + ".. [1] M. Patefield and D. Tandy, \"Fast and accurate calculation of\n" + " Owen's T Function\", Statistical Software vol. 5, pp. 1-25, 2000.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy import special\n" + ">>> a = 3.5\n" + ">>> h = 0.78\n" + ">>> special.owens_t(h, a)\n" + "0.10877216734852274") +ufunc_owens_t_loops[0] = loop_d_dd__As_ff_f +ufunc_owens_t_loops[1] = loop_d_dd__As_dd_d +ufunc_owens_t_types[0] = NPY_FLOAT +ufunc_owens_t_types[1] = NPY_FLOAT +ufunc_owens_t_types[2] = NPY_FLOAT +ufunc_owens_t_types[3] = NPY_DOUBLE +ufunc_owens_t_types[4] = NPY_DOUBLE +ufunc_owens_t_types[5] = NPY_DOUBLE +ufunc_owens_t_ptr[2*0] = _func_xsf_owens_t +ufunc_owens_t_ptr[2*0+1] = ("owens_t") +ufunc_owens_t_ptr[2*1] = _func_xsf_owens_t +ufunc_owens_t_ptr[2*1+1] = ("owens_t") +ufunc_owens_t_data[0] = &ufunc_owens_t_ptr[2*0] +ufunc_owens_t_data[1] = &ufunc_owens_t_ptr[2*1] +owens_t = np.PyUFunc_FromFuncAndData(ufunc_owens_t_loops, ufunc_owens_t_data, ufunc_owens_t_types, 2, 2, 1, 0, "owens_t", ufunc_owens_t_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_pdtr_loops[2] +cdef void *ufunc_pdtr_ptr[4] +cdef void *ufunc_pdtr_data[2] +cdef char ufunc_pdtr_types[6] +cdef char *ufunc_pdtr_doc = ( + "pdtr(k, m, out=None)\n" + "\n" + "Poisson cumulative distribution function.\n" + "\n" + "Defined as the probability that a Poisson-distributed random\n" + "variable with event rate :math:`m` is less than or equal to\n" + ":math:`k`. More concretely, this works out to be [1]_\n" + "\n" + ".. math::\n" + "\n" + " \\exp(-m) \\sum_{j = 0}^{\\lfloor{k}\\rfloor} \\frac{m^j}{j!}.\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " Number of occurrences (nonnegative, real)\n" + "m : array_like\n" + " Shape parameter (nonnegative, real)\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the Poisson cumulative distribution function\n" + "\n" + "See Also\n" + "--------\n" + "pdtrc : Poisson survival function\n" + "pdtrik : inverse of `pdtr` with respect to `k`\n" + "pdtri : inverse of `pdtr` with respect to `m`\n" + "\n" + "References\n" + "----------\n" + ".. [1] https://en.wikipedia.org/wiki/Poisson_distribution\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "It is a cumulative distribution function, so it converges to 1\n" + "monotonically as `k` goes to infinity.\n" + "\n" + ">>> sc.pdtr([1, 10, 100, np.inf], 1)\n" + "array([0.73575888, 0.99999999, 1. , 1. ])\n" + "\n" + "It is discontinuous at integers and constant between integers.\n" + "\n" + ">>> sc.pdtr([1, 1.5, 1.9, 2], 1)\n" + "array([0.73575888, 0.73575888, 0.73575888, 0.9196986 ])") +ufunc_pdtr_loops[0] = loop_d_dd__As_ff_f +ufunc_pdtr_loops[1] = loop_d_dd__As_dd_d +ufunc_pdtr_types[0] = NPY_FLOAT +ufunc_pdtr_types[1] = NPY_FLOAT +ufunc_pdtr_types[2] = NPY_FLOAT +ufunc_pdtr_types[3] = NPY_DOUBLE +ufunc_pdtr_types[4] = NPY_DOUBLE +ufunc_pdtr_types[5] = NPY_DOUBLE +ufunc_pdtr_ptr[2*0] = _func_xsf_pdtr +ufunc_pdtr_ptr[2*0+1] = ("pdtr") +ufunc_pdtr_ptr[2*1] = _func_xsf_pdtr +ufunc_pdtr_ptr[2*1+1] = ("pdtr") +ufunc_pdtr_data[0] = &ufunc_pdtr_ptr[2*0] +ufunc_pdtr_data[1] = &ufunc_pdtr_ptr[2*1] +pdtr = np.PyUFunc_FromFuncAndData(ufunc_pdtr_loops, ufunc_pdtr_data, ufunc_pdtr_types, 2, 2, 1, 0, "pdtr", ufunc_pdtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_pdtrc_loops[2] +cdef void *ufunc_pdtrc_ptr[4] +cdef void *ufunc_pdtrc_data[2] +cdef char ufunc_pdtrc_types[6] +cdef char *ufunc_pdtrc_doc = ( + "pdtrc(k, m, out=None)\n" + "\n" + "Poisson survival function\n" + "\n" + "Returns the sum of the terms from k+1 to infinity of the Poisson\n" + "distribution: sum(exp(-m) * m**j / j!, j=k+1..inf) = gammainc(\n" + "k+1, m). Arguments must both be non-negative doubles.\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " Number of occurrences (nonnegative, real)\n" + "m : array_like\n" + " Shape parameter (nonnegative, real)\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the Poisson survival function\n" + "\n" + "See Also\n" + "--------\n" + "pdtr : Poisson cumulative distribution function\n" + "pdtrik : inverse of `pdtr` with respect to `k`\n" + "pdtri : inverse of `pdtr` with respect to `m`\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import scipy.special as sc\n" + "\n" + "It is a survival function, so it decreases to 0\n" + "monotonically as `k` goes to infinity.\n" + "\n" + ">>> k = np.array([1, 10, 100, np.inf])\n" + ">>> sc.pdtrc(k, 1)\n" + "array([2.64241118e-001, 1.00477664e-008, 3.94147589e-161, 0.00000000e+000])\n" + "\n" + "It can be expressed in terms of the lower incomplete gamma\n" + "function `gammainc`.\n" + "\n" + ">>> sc.gammainc(k + 1, 1)\n" + "array([2.64241118e-001, 1.00477664e-008, 3.94147589e-161, 0.00000000e+000])") +ufunc_pdtrc_loops[0] = loop_d_dd__As_ff_f +ufunc_pdtrc_loops[1] = loop_d_dd__As_dd_d +ufunc_pdtrc_types[0] = NPY_FLOAT +ufunc_pdtrc_types[1] = NPY_FLOAT +ufunc_pdtrc_types[2] = NPY_FLOAT +ufunc_pdtrc_types[3] = NPY_DOUBLE +ufunc_pdtrc_types[4] = NPY_DOUBLE +ufunc_pdtrc_types[5] = NPY_DOUBLE +ufunc_pdtrc_ptr[2*0] = _func_xsf_pdtrc +ufunc_pdtrc_ptr[2*0+1] = ("pdtrc") +ufunc_pdtrc_ptr[2*1] = _func_xsf_pdtrc +ufunc_pdtrc_ptr[2*1+1] = ("pdtrc") +ufunc_pdtrc_data[0] = &ufunc_pdtrc_ptr[2*0] +ufunc_pdtrc_data[1] = &ufunc_pdtrc_ptr[2*1] +pdtrc = np.PyUFunc_FromFuncAndData(ufunc_pdtrc_loops, ufunc_pdtrc_data, ufunc_pdtrc_types, 2, 2, 1, 0, "pdtrc", ufunc_pdtrc_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_pdtri_loops[3] +cdef void *ufunc_pdtri_ptr[6] +cdef void *ufunc_pdtri_data[3] +cdef char ufunc_pdtri_types[9] +cdef char *ufunc_pdtri_doc = ( + "pdtri(k, y, out=None)\n" + "\n" + "Inverse to `pdtr` vs m\n" + "\n" + "Returns the Poisson variable `m` such that the sum from 0 to `k` of\n" + "the Poisson density is equal to the given probability `y`:\n" + "calculated by ``gammaincinv(k + 1, y)``. `k` must be a nonnegative\n" + "integer and `y` between 0 and 1.\n" + "\n" + "Parameters\n" + "----------\n" + "k : array_like\n" + " Number of occurrences (nonnegative, real)\n" + "y : array_like\n" + " Probability\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Values of the shape parameter `m` such that ``pdtr(k, m) = p``\n" + "\n" + "See Also\n" + "--------\n" + "pdtr : Poisson cumulative distribution function\n" + "pdtrc : Poisson survival function\n" + "pdtrik : inverse of `pdtr` with respect to `k`\n" + "\n" + "Examples\n" + "--------\n" + ">>> import scipy.special as sc\n" + "\n" + "Compute the CDF for several values of `m`:\n" + "\n" + ">>> m = [0.5, 1, 1.5]\n" + ">>> p = sc.pdtr(1, m)\n" + ">>> p\n" + "array([0.90979599, 0.73575888, 0.5578254 ])\n" + "\n" + "Compute the inverse. We recover the values of `m`, as expected:\n" + "\n" + ">>> sc.pdtri(1, p)\n" + "array([0.5, 1. , 1.5])") +ufunc_pdtri_loops[0] = loop_d_pd__As_pd_d +ufunc_pdtri_loops[1] = loop_d_dd__As_ff_f +ufunc_pdtri_loops[2] = loop_d_dd__As_dd_d +ufunc_pdtri_types[0] = NPY_INTP +ufunc_pdtri_types[1] = NPY_DOUBLE +ufunc_pdtri_types[2] = NPY_DOUBLE +ufunc_pdtri_types[3] = NPY_FLOAT +ufunc_pdtri_types[4] = NPY_FLOAT +ufunc_pdtri_types[5] = NPY_FLOAT +ufunc_pdtri_types[6] = NPY_DOUBLE +ufunc_pdtri_types[7] = NPY_DOUBLE +ufunc_pdtri_types[8] = NPY_DOUBLE +ufunc_pdtri_ptr[2*0] = _func_cephes_pdtri_wrap +ufunc_pdtri_ptr[2*0+1] = ("pdtri") +ufunc_pdtri_ptr[2*1] = _func_pdtri_unsafe +ufunc_pdtri_ptr[2*1+1] = ("pdtri") +ufunc_pdtri_ptr[2*2] = _func_pdtri_unsafe +ufunc_pdtri_ptr[2*2+1] = ("pdtri") +ufunc_pdtri_data[0] = &ufunc_pdtri_ptr[2*0] +ufunc_pdtri_data[1] = &ufunc_pdtri_ptr[2*1] +ufunc_pdtri_data[2] = &ufunc_pdtri_ptr[2*2] +pdtri = np.PyUFunc_FromFuncAndData(ufunc_pdtri_loops, ufunc_pdtri_data, ufunc_pdtri_types, 3, 2, 1, 0, "pdtri", ufunc_pdtri_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_pdtrik_loops[2] +cdef void *ufunc_pdtrik_ptr[4] +cdef void *ufunc_pdtrik_data[2] +cdef char ufunc_pdtrik_types[6] +cdef char *ufunc_pdtrik_doc = ( + "pdtrik(p, m, out=None)\n" + "\n" + "Inverse to `pdtr` vs `k`.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Probability\n" + "m : array_like\n" + " Shape parameter (nonnegative, real)\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The number of occurrences `k` such that ``pdtr(k, m) = p``\n" + "\n" + "See Also\n" + "--------\n" + "pdtr : Poisson cumulative distribution function\n" + "pdtrc : Poisson survival function\n" + "pdtri : inverse of `pdtr` with respect to `m`\n" + "\n" + "Examples\n" + "--------\n" + ">>> import scipy.special as sc\n" + "\n" + "Compute the CDF for several values of `k`:\n" + "\n" + ">>> k = [1, 2, 3]\n" + ">>> p = sc.pdtr(k, 2)\n" + ">>> p\n" + "array([0.40600585, 0.67667642, 0.85712346])\n" + "\n" + "Compute the inverse. We recover the values of `k`, as expected:\n" + "\n" + ">>> sc.pdtrik(p, 2)\n" + "array([1., 2., 3.])") +ufunc_pdtrik_loops[0] = loop_d_dd__As_ff_f +ufunc_pdtrik_loops[1] = loop_d_dd__As_dd_d +ufunc_pdtrik_types[0] = NPY_FLOAT +ufunc_pdtrik_types[1] = NPY_FLOAT +ufunc_pdtrik_types[2] = NPY_FLOAT +ufunc_pdtrik_types[3] = NPY_DOUBLE +ufunc_pdtrik_types[4] = NPY_DOUBLE +ufunc_pdtrik_types[5] = NPY_DOUBLE +ufunc_pdtrik_ptr[2*0] = _func_pdtrik +ufunc_pdtrik_ptr[2*0+1] = ("pdtrik") +ufunc_pdtrik_ptr[2*1] = _func_pdtrik +ufunc_pdtrik_ptr[2*1+1] = ("pdtrik") +ufunc_pdtrik_data[0] = &ufunc_pdtrik_ptr[2*0] +ufunc_pdtrik_data[1] = &ufunc_pdtrik_ptr[2*1] +pdtrik = np.PyUFunc_FromFuncAndData(ufunc_pdtrik_loops, ufunc_pdtrik_data, ufunc_pdtrik_types, 2, 2, 1, 0, "pdtrik", ufunc_pdtrik_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_poch_loops[2] +cdef void *ufunc_poch_ptr[4] +cdef void *ufunc_poch_data[2] +cdef char ufunc_poch_types[6] +cdef char *ufunc_poch_doc = ( + "poch(z, m, out=None)\n" + "\n" + "Pochhammer symbol.\n" + "\n" + "The Pochhammer symbol (rising factorial) is defined as\n" + "\n" + ".. math::\n" + "\n" + " (z)_m = \\frac{\\Gamma(z + m)}{\\Gamma(z)}\n" + "\n" + "For positive integer `m` it reads\n" + "\n" + ".. math::\n" + "\n" + " (z)_m = z (z + 1) ... (z + m - 1)\n" + "\n" + "See [dlmf]_ for more details.\n" + "\n" + "Parameters\n" + "----------\n" + "z, m : array_like\n" + " Real-valued arguments.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The value of the function.\n" + "\n" + "References\n" + "----------\n" + ".. [dlmf] Nist, Digital Library of Mathematical Functions\n" + " https://dlmf.nist.gov/5.2#iii\n" + "\n" + "Examples\n" + "--------\n" + ">>> import scipy.special as sc\n" + "\n" + "It is 1 when m is 0.\n" + "\n" + ">>> sc.poch([1, 2, 3, 4], 0)\n" + "array([1., 1., 1., 1.])\n" + "\n" + "For z equal to 1 it reduces to the factorial function.\n" + "\n" + ">>> sc.poch(1, 5)\n" + "120.0\n" + ">>> 1 * 2 * 3 * 4 * 5\n" + "120\n" + "\n" + "It can be expressed in terms of the gamma function.\n" + "\n" + ">>> z, m = 3.7, 2.1\n" + ">>> sc.poch(z, m)\n" + "20.529581933776953\n" + ">>> sc.gamma(z + m) / sc.gamma(z)\n" + "20.52958193377696") +ufunc_poch_loops[0] = loop_d_dd__As_ff_f +ufunc_poch_loops[1] = loop_d_dd__As_dd_d +ufunc_poch_types[0] = NPY_FLOAT +ufunc_poch_types[1] = NPY_FLOAT +ufunc_poch_types[2] = NPY_FLOAT +ufunc_poch_types[3] = NPY_DOUBLE +ufunc_poch_types[4] = NPY_DOUBLE +ufunc_poch_types[5] = NPY_DOUBLE +ufunc_poch_ptr[2*0] = _func_cephes_poch +ufunc_poch_ptr[2*0+1] = ("poch") +ufunc_poch_ptr[2*1] = _func_cephes_poch +ufunc_poch_ptr[2*1+1] = ("poch") +ufunc_poch_data[0] = &ufunc_poch_ptr[2*0] +ufunc_poch_data[1] = &ufunc_poch_ptr[2*1] +poch = np.PyUFunc_FromFuncAndData(ufunc_poch_loops, ufunc_poch_data, ufunc_poch_types, 2, 2, 1, 0, "poch", ufunc_poch_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_powm1_loops[2] +cdef void *ufunc_powm1_ptr[4] +cdef void *ufunc_powm1_data[2] +cdef char ufunc_powm1_types[6] +cdef char *ufunc_powm1_doc = ( + "powm1(x, y, out=None)\n" + "\n" + "Computes ``x**y - 1``.\n" + "\n" + "This function is useful when `y` is near 0, or when `x` is near 1.\n" + "\n" + "The function is implemented for real types only (unlike ``numpy.power``,\n" + "which accepts complex inputs).\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " The base. Must be a real type (i.e. integer or float, not complex).\n" + "y : array_like\n" + " The exponent. Must be a real type (i.e. integer or float, not complex).\n" + "\n" + "Returns\n" + "-------\n" + "array_like\n" + " Result of the calculation\n" + "\n" + "Notes\n" + "-----\n" + ".. versionadded:: 1.10.0\n" + "\n" + "The underlying code is implemented for single precision and double\n" + "precision floats only. Unlike `numpy.power`, integer inputs to\n" + "`powm1` are converted to floating point, and complex inputs are\n" + "not accepted.\n" + "\n" + "Note the following edge cases:\n" + "\n" + "* ``powm1(x, 0)`` returns 0 for any ``x``, including 0, ``inf``\n" + " and ``nan``.\n" + "* ``powm1(1, y)`` returns 0 for any ``y``, including ``nan``\n" + " and ``inf``.\n" + "\n" + "This function wraps the ``powm1`` routine from the\n" + "Boost Math C++ library [1]_.\n" + "\n" + "References\n" + "----------\n" + ".. [1] The Boost Developers. \"Boost C++ Libraries\". https://www.boost.org/.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy.special import powm1\n" + "\n" + ">>> x = np.array([1.2, 10.0, 0.9999999975])\n" + ">>> y = np.array([1e-9, 1e-11, 0.1875])\n" + ">>> powm1(x, y)\n" + "array([ 1.82321557e-10, 2.30258509e-11, -4.68749998e-10])\n" + "\n" + "It can be verified that the relative errors in those results\n" + "are less than 2.5e-16.\n" + "\n" + "Compare that to the result of ``x**y - 1``, where the\n" + "relative errors are all larger than 8e-8:\n" + "\n" + ">>> x**y - 1\n" + "array([ 1.82321491e-10, 2.30258035e-11, -4.68750039e-10])") +ufunc_powm1_loops[0] = loop_f_ff__As_ff_f +ufunc_powm1_loops[1] = loop_d_dd__As_dd_d +ufunc_powm1_types[0] = NPY_FLOAT +ufunc_powm1_types[1] = NPY_FLOAT +ufunc_powm1_types[2] = NPY_FLOAT +ufunc_powm1_types[3] = NPY_DOUBLE +ufunc_powm1_types[4] = NPY_DOUBLE +ufunc_powm1_types[5] = NPY_DOUBLE +ufunc_powm1_ptr[2*0] = scipy.special._ufuncs_cxx._export_powm1_float +ufunc_powm1_ptr[2*0+1] = ("powm1") +ufunc_powm1_ptr[2*1] = scipy.special._ufuncs_cxx._export_powm1_double +ufunc_powm1_ptr[2*1+1] = ("powm1") +ufunc_powm1_data[0] = &ufunc_powm1_ptr[2*0] +ufunc_powm1_data[1] = &ufunc_powm1_ptr[2*1] +powm1 = np.PyUFunc_FromFuncAndData(ufunc_powm1_loops, ufunc_powm1_data, ufunc_powm1_types, 2, 2, 1, 0, "powm1", ufunc_powm1_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_pseudo_huber_loops[2] +cdef void *ufunc_pseudo_huber_ptr[4] +cdef void *ufunc_pseudo_huber_data[2] +cdef char ufunc_pseudo_huber_types[6] +cdef char *ufunc_pseudo_huber_doc = ( + "pseudo_huber(delta, r, out=None)\n" + "\n" + "Pseudo-Huber loss function.\n" + "\n" + ".. math:: \\mathrm{pseudo\\_huber}(\\delta, r) =\n" + " \\delta^2 \\left( \\sqrt{ 1 + \\left( \\frac{r}{\\delta} \\right)^2 } - 1 \\right)\n" + "\n" + "Parameters\n" + "----------\n" + "delta : array_like\n" + " Input array, indicating the soft quadratic vs. linear loss changepoint.\n" + "r : array_like\n" + " Input array, possibly representing residuals.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "res : scalar or ndarray\n" + " The computed Pseudo-Huber loss function values.\n" + "\n" + "See Also\n" + "--------\n" + "huber: Similar function which this function approximates\n" + "\n" + "Notes\n" + "-----\n" + "Like `huber`, `pseudo_huber` often serves as a robust loss function\n" + "in statistics or machine learning to reduce the influence of outliers.\n" + "Unlike `huber`, `pseudo_huber` is smooth.\n" + "\n" + "Typically, `r` represents residuals, the difference\n" + "between a model prediction and data. Then, for :math:`|r|\\leq\\delta`,\n" + "`pseudo_huber` resembles the squared error and for :math:`|r|>\\delta` the\n" + "absolute error. This way, the Pseudo-Huber loss often achieves\n" + "a fast convergence in model fitting for small residuals like the squared\n" + "error loss function and still reduces the influence of outliers\n" + "(:math:`|r|>\\delta`) like the absolute error loss. As :math:`\\delta` is\n" + "the cutoff between squared and absolute error regimes, it has\n" + "to be tuned carefully for each problem. `pseudo_huber` is also\n" + "convex, making it suitable for gradient based optimization. [1]_ [2]_\n" + "\n" + ".. versionadded:: 0.15.0\n" + "\n" + "References\n" + "----------\n" + ".. [1] Hartley, Zisserman, \"Multiple View Geometry in Computer Vision\".\n" + " 2003. Cambridge University Press. p. 619\n" + ".. [2] Charbonnier et al. \"Deterministic edge-preserving regularization\n" + " in computed imaging\". 1997. IEEE Trans. Image Processing.\n" + " 6 (2): 298 - 311.\n" + "\n" + "Examples\n" + "--------\n" + "Import all necessary modules.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import pseudo_huber, huber\n" + ">>> import matplotlib.pyplot as plt\n" + "\n" + "Calculate the function for ``delta=1`` at ``r=2``.\n" + "\n" + ">>> pseudo_huber(1., 2.)\n" + "1.2360679774997898\n" + "\n" + "Calculate the function at ``r=2`` for different `delta` by providing\n" + "a list or NumPy array for `delta`.\n" + "\n" + ">>> pseudo_huber([1., 2., 4.], 3.)\n" + "array([2.16227766, 3.21110255, 4. ])\n" + "\n" + "Calculate the function for ``delta=1`` at several points by providing\n" + "a list or NumPy array for `r`.\n" + "\n" + ">>> pseudo_huber(2., np.array([1., 1.5, 3., 4.]))\n" + "array([0.47213595, 1. , 3.21110255, 4.94427191])\n" + "\n" + "The function can be calculated for different `delta` and `r` by\n" + "providing arrays for both with compatible shapes for broadcasting.\n" + "\n" + ">>> r = np.array([1., 2.5, 8., 10.])\n" + ">>> deltas = np.array([[1.], [5.], [9.]])\n" + ">>> print(r.shape, deltas.shape)\n" + "(4,) (3, 1)\n" + "\n" + ">>> pseudo_huber(deltas, r)\n" + "array([[ 0.41421356, 1.6925824 , 7.06225775, 9.04987562],\n" + " [ 0.49509757, 2.95084972, 22.16990566, 30.90169944],\n" + " [ 0.49846624, 3.06693762, 27.37435121, 40.08261642]])\n" + "\n" + "Plot the function for different `delta`.\n" + "\n" + ">>> x = np.linspace(-4, 4, 500)\n" + ">>> deltas = [1, 2, 3]\n" + ">>> linestyles = [\"dashed\", \"dotted\", \"dashdot\"]\n" + ">>> fig, ax = plt.subplots()\n" + ">>> combined_plot_parameters = list(zip(deltas, linestyles))\n" + ">>> for delta, style in combined_plot_parameters:\n" + "... ax.plot(x, pseudo_huber(delta, x), label=rf\"$\\delta={delta}$\",\n" + "... ls=style)\n" + ">>> ax.legend(loc=\"upper center\")\n" + ">>> ax.set_xlabel(\"$x$\")\n" + ">>> ax.set_title(r\"Pseudo-Huber loss function $h_{\\delta}(x)$\")\n" + ">>> ax.set_xlim(-4, 4)\n" + ">>> ax.set_ylim(0, 8)\n" + ">>> plt.show()\n" + "\n" + "Finally, illustrate the difference between `huber` and `pseudo_huber` by\n" + "plotting them and their gradients with respect to `r`. The plot shows\n" + "that `pseudo_huber` is continuously differentiable while `huber` is not\n" + "at the points :math:`\\pm\\delta`.\n" + "\n" + ">>> def huber_grad(delta, x):\n" + "... grad = np.copy(x)\n" + "... linear_area = np.argwhere(np.abs(x) > delta)\n" + "... grad[linear_area]=delta*np.sign(x[linear_area])\n" + "... return grad\n" + ">>> def pseudo_huber_grad(delta, x):\n" + "... return x* (1+(x/delta)**2)**(-0.5)\n" + ">>> x=np.linspace(-3, 3, 500)\n" + ">>> delta = 1.\n" + ">>> fig, ax = plt.subplots(figsize=(7, 7))\n" + ">>> ax.plot(x, huber(delta, x), label=\"Huber\", ls=\"dashed\")\n" + ">>> ax.plot(x, huber_grad(delta, x), label=\"Huber Gradient\", ls=\"dashdot\")\n" + ">>> ax.plot(x, pseudo_huber(delta, x), label=\"Pseudo-Huber\", ls=\"dotted\")\n" + ">>> ax.plot(x, pseudo_huber_grad(delta, x), label=\"Pseudo-Huber Gradient\",\n" + "... ls=\"solid\")\n" + ">>> ax.legend(loc=\"upper center\")\n" + ">>> plt.show()") +ufunc_pseudo_huber_loops[0] = loop_d_dd__As_ff_f +ufunc_pseudo_huber_loops[1] = loop_d_dd__As_dd_d +ufunc_pseudo_huber_types[0] = NPY_FLOAT +ufunc_pseudo_huber_types[1] = NPY_FLOAT +ufunc_pseudo_huber_types[2] = NPY_FLOAT +ufunc_pseudo_huber_types[3] = NPY_DOUBLE +ufunc_pseudo_huber_types[4] = NPY_DOUBLE +ufunc_pseudo_huber_types[5] = NPY_DOUBLE +ufunc_pseudo_huber_ptr[2*0] = _func_pseudo_huber +ufunc_pseudo_huber_ptr[2*0+1] = ("pseudo_huber") +ufunc_pseudo_huber_ptr[2*1] = _func_pseudo_huber +ufunc_pseudo_huber_ptr[2*1+1] = ("pseudo_huber") +ufunc_pseudo_huber_data[0] = &ufunc_pseudo_huber_ptr[2*0] +ufunc_pseudo_huber_data[1] = &ufunc_pseudo_huber_ptr[2*1] +pseudo_huber = np.PyUFunc_FromFuncAndData(ufunc_pseudo_huber_loops, ufunc_pseudo_huber_data, ufunc_pseudo_huber_types, 2, 2, 1, 0, "pseudo_huber", ufunc_pseudo_huber_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_rel_entr_loops[2] +cdef void *ufunc_rel_entr_ptr[4] +cdef void *ufunc_rel_entr_data[2] +cdef char ufunc_rel_entr_types[6] +cdef char *ufunc_rel_entr_doc = ( + "rel_entr(x, y, out=None)\n" + "\n" + "Elementwise function for computing relative entropy.\n" + "\n" + ".. math::\n" + "\n" + " \\mathrm{rel\\_entr}(x, y) =\n" + " \\begin{cases}\n" + " x \\log(x / y) & x > 0, y > 0 \\\\\n" + " 0 & x = 0, y \\ge 0 \\\\\n" + " \\infty & \\text{otherwise}\n" + " \\end{cases}\n" + "\n" + "Parameters\n" + "----------\n" + "x, y : array_like\n" + " Input arrays\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Relative entropy of the inputs\n" + "\n" + "See Also\n" + "--------\n" + "entr, kl_div, scipy.stats.entropy\n" + "\n" + "Notes\n" + "-----\n" + ".. versionadded:: 0.15.0\n" + "\n" + "This function is jointly convex in x and y.\n" + "\n" + "The origin of this function is in convex programming; see\n" + "[1]_. Given two discrete probability distributions :math:`p_1,\n" + "\\ldots, p_n` and :math:`q_1, \\ldots, q_n`, the definition of relative\n" + "entropy in the context of *information theory* is\n" + "\n" + ".. math::\n" + "\n" + " \\sum_{i = 1}^n \\mathrm{rel\\_entr}(p_i, q_i).\n" + "\n" + "To compute the latter quantity, use `scipy.stats.entropy`.\n" + "\n" + "See [2]_ for details.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Boyd, Stephen and Lieven Vandenberghe. *Convex optimization*.\n" + " Cambridge University Press, 2004.\n" + " :doi:`https://doi.org/10.1017/CBO9780511804441`\n" + ".. [2] Kullback-Leibler divergence,\n" + " https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence") +ufunc_rel_entr_loops[0] = loop_d_dd__As_ff_f +ufunc_rel_entr_loops[1] = loop_d_dd__As_dd_d +ufunc_rel_entr_types[0] = NPY_FLOAT +ufunc_rel_entr_types[1] = NPY_FLOAT +ufunc_rel_entr_types[2] = NPY_FLOAT +ufunc_rel_entr_types[3] = NPY_DOUBLE +ufunc_rel_entr_types[4] = NPY_DOUBLE +ufunc_rel_entr_types[5] = NPY_DOUBLE +ufunc_rel_entr_ptr[2*0] = _func_rel_entr +ufunc_rel_entr_ptr[2*0+1] = ("rel_entr") +ufunc_rel_entr_ptr[2*1] = _func_rel_entr +ufunc_rel_entr_ptr[2*1+1] = ("rel_entr") +ufunc_rel_entr_data[0] = &ufunc_rel_entr_ptr[2*0] +ufunc_rel_entr_data[1] = &ufunc_rel_entr_ptr[2*1] +rel_entr = np.PyUFunc_FromFuncAndData(ufunc_rel_entr_loops, ufunc_rel_entr_data, ufunc_rel_entr_types, 2, 2, 1, 0, "rel_entr", ufunc_rel_entr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_round_loops[2] +cdef void *ufunc_round_ptr[4] +cdef void *ufunc_round_data[2] +cdef char ufunc_round_types[4] +cdef char *ufunc_round_doc = ( + "round(x, out=None)\n" + "\n" + "Round to the nearest integer.\n" + "\n" + "Returns the nearest integer to `x`. If `x` ends in 0.5 exactly,\n" + "the nearest even integer is chosen.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real valued input.\n" + "out : ndarray, optional\n" + " Optional output array for the function results.\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The nearest integers to the elements of `x`. The result is of\n" + " floating type, not integer type.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import scipy.special as sc\n" + "\n" + "It rounds to even.\n" + "\n" + ">>> sc.round([0.5, 1.5])\n" + "array([0., 2.])") +ufunc_round_loops[0] = loop_d_d__As_f_f +ufunc_round_loops[1] = loop_d_d__As_d_d +ufunc_round_types[0] = NPY_FLOAT +ufunc_round_types[1] = NPY_FLOAT +ufunc_round_types[2] = NPY_DOUBLE +ufunc_round_types[3] = NPY_DOUBLE +ufunc_round_ptr[2*0] = _func_cephes_round +ufunc_round_ptr[2*0+1] = ("round") +ufunc_round_ptr[2*1] = _func_cephes_round +ufunc_round_ptr[2*1+1] = ("round") +ufunc_round_data[0] = &ufunc_round_ptr[2*0] +ufunc_round_data[1] = &ufunc_round_ptr[2*1] +round = np.PyUFunc_FromFuncAndData(ufunc_round_loops, ufunc_round_data, ufunc_round_types, 2, 1, 1, 0, "round", ufunc_round_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_shichi_loops[4] +cdef void *ufunc_shichi_ptr[8] +cdef void *ufunc_shichi_data[4] +cdef char ufunc_shichi_types[12] +cdef char *ufunc_shichi_doc = ( + "shichi(x, out=None)\n" + "\n" + "Hyperbolic sine and cosine integrals.\n" + "\n" + "The hyperbolic sine integral is\n" + "\n" + ".. math::\n" + "\n" + " \\int_0^x \\frac{\\sinh{t}}{t}dt\n" + "\n" + "and the hyperbolic cosine integral is\n" + "\n" + ".. math::\n" + "\n" + " \\gamma + \\log(x) + \\int_0^x \\frac{\\cosh{t} - 1}{t} dt\n" + "\n" + "where :math:`\\gamma` is Euler's constant and :math:`\\log` is the\n" + "principal branch of the logarithm [1]_.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real or complex points at which to compute the hyperbolic sine\n" + " and cosine integrals.\n" + "out : tuple of ndarray, optional\n" + " Optional output arrays for the function results\n" + "\n" + "Returns\n" + "-------\n" + "si : scalar or ndarray\n" + " Hyperbolic sine integral at ``x``\n" + "ci : scalar or ndarray\n" + " Hyperbolic cosine integral at ``x``\n" + "\n" + "See Also\n" + "--------\n" + "sici : Sine and cosine integrals.\n" + "exp1 : Exponential integral E1.\n" + "expi : Exponential integral Ei.\n" + "\n" + "Notes\n" + "-----\n" + "For real arguments with ``x < 0``, ``chi`` is the real part of the\n" + "hyperbolic cosine integral. For such points ``chi(x)`` and ``chi(x\n" + "+ 0j)`` differ by a factor of ``1j*pi``.\n" + "\n" + "For real arguments the function is computed by calling Cephes'\n" + "[2]_ *shichi* routine. For complex arguments the algorithm is based\n" + "on Mpmath's [3]_ *shi* and *chi* routines.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.\n" + " (See Section 5.2.)\n" + ".. [2] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + ".. [3] Fredrik Johansson and others.\n" + " \"mpmath: a Python library for arbitrary-precision floating-point\n" + " arithmetic\" (Version 0.19) http://mpmath.org/\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> from scipy.special import shichi, sici\n" + "\n" + "`shichi` accepts real or complex input:\n" + "\n" + ">>> shichi(0.5)\n" + "(0.5069967498196671, -0.05277684495649357)\n" + ">>> shichi(0.5 + 2.5j)\n" + "((0.11772029666668238+1.831091777729851j),\n" + " (0.29912435887648825+1.7395351121166562j))\n" + "\n" + "The hyperbolic sine and cosine integrals Shi(z) and Chi(z) are\n" + "related to the sine and cosine integrals Si(z) and Ci(z) by\n" + "\n" + "* Shi(z) = -i*Si(i*z)\n" + "* Chi(z) = Ci(-i*z) + i*pi/2\n" + "\n" + ">>> z = 0.25 + 5j\n" + ">>> shi, chi = shichi(z)\n" + ">>> shi, -1j*sici(1j*z)[0] # Should be the same.\n" + "((-0.04834719325101729+1.5469354086921228j),\n" + " (-0.04834719325101729+1.5469354086921228j))\n" + ">>> chi, sici(-1j*z)[1] + 1j*np.pi/2 # Should be the same.\n" + "((-0.19568708973868087+1.556276312103824j),\n" + " (-0.19568708973868087+1.556276312103824j))\n" + "\n" + "Plot the functions evaluated on the real axis:\n" + "\n" + ">>> xp = np.geomspace(1e-8, 4.0, 250)\n" + ">>> x = np.concatenate((-xp[::-1], xp))\n" + ">>> shi, chi = shichi(x)\n" + "\n" + ">>> fig, ax = plt.subplots()\n" + ">>> ax.plot(x, shi, label='Shi(x)')\n" + ">>> ax.plot(x, chi, '--', label='Chi(x)')\n" + ">>> ax.set_xlabel('x')\n" + ">>> ax.set_title('Hyperbolic Sine and Cosine Integrals')\n" + ">>> ax.legend(shadow=True, framealpha=1, loc='lower right')\n" + ">>> ax.grid(True)\n" + ">>> plt.show()") +ufunc_shichi_loops[0] = loop_i_d_dd_As_f_ff +ufunc_shichi_loops[1] = loop_i_d_dd_As_d_dd +ufunc_shichi_loops[2] = loop_i_D_DD_As_F_FF +ufunc_shichi_loops[3] = loop_i_D_DD_As_D_DD +ufunc_shichi_types[0] = NPY_FLOAT +ufunc_shichi_types[1] = NPY_FLOAT +ufunc_shichi_types[2] = NPY_FLOAT +ufunc_shichi_types[3] = NPY_DOUBLE +ufunc_shichi_types[4] = NPY_DOUBLE +ufunc_shichi_types[5] = NPY_DOUBLE +ufunc_shichi_types[6] = NPY_CFLOAT +ufunc_shichi_types[7] = NPY_CFLOAT +ufunc_shichi_types[8] = NPY_CFLOAT +ufunc_shichi_types[9] = NPY_CDOUBLE +ufunc_shichi_types[10] = NPY_CDOUBLE +ufunc_shichi_types[11] = NPY_CDOUBLE +ufunc_shichi_ptr[2*0] = _func_xsf_shichi +ufunc_shichi_ptr[2*0+1] = ("shichi") +ufunc_shichi_ptr[2*1] = _func_xsf_shichi +ufunc_shichi_ptr[2*1+1] = ("shichi") +ufunc_shichi_ptr[2*2] = _func_xsf_cshichi +ufunc_shichi_ptr[2*2+1] = ("shichi") +ufunc_shichi_ptr[2*3] = _func_xsf_cshichi +ufunc_shichi_ptr[2*3+1] = ("shichi") +ufunc_shichi_data[0] = &ufunc_shichi_ptr[2*0] +ufunc_shichi_data[1] = &ufunc_shichi_ptr[2*1] +ufunc_shichi_data[2] = &ufunc_shichi_ptr[2*2] +ufunc_shichi_data[3] = &ufunc_shichi_ptr[2*3] +shichi = np.PyUFunc_FromFuncAndData(ufunc_shichi_loops, ufunc_shichi_data, ufunc_shichi_types, 4, 1, 2, 0, "shichi", ufunc_shichi_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_sici_loops[4] +cdef void *ufunc_sici_ptr[8] +cdef void *ufunc_sici_data[4] +cdef char ufunc_sici_types[12] +cdef char *ufunc_sici_doc = ( + "sici(x, out=None)\n" + "\n" + "Sine and cosine integrals.\n" + "\n" + "The sine integral is\n" + "\n" + ".. math::\n" + "\n" + " \\int_0^x \\frac{\\sin{t}}{t}dt\n" + "\n" + "and the cosine integral is\n" + "\n" + ".. math::\n" + "\n" + " \\gamma + \\log(x) + \\int_0^x \\frac{\\cos{t} - 1}{t}dt\n" + "\n" + "where :math:`\\gamma` is Euler's constant and :math:`\\log` is the\n" + "principal branch of the logarithm [1]_.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real or complex points at which to compute the sine and cosine\n" + " integrals.\n" + "out : tuple of ndarray, optional\n" + " Optional output arrays for the function results\n" + "\n" + "Returns\n" + "-------\n" + "si : scalar or ndarray\n" + " Sine integral at ``x``\n" + "ci : scalar or ndarray\n" + " Cosine integral at ``x``\n" + "\n" + "See Also\n" + "--------\n" + "shichi : Hyperbolic sine and cosine integrals.\n" + "exp1 : Exponential integral E1.\n" + "expi : Exponential integral Ei.\n" + "\n" + "Notes\n" + "-----\n" + "For real arguments with ``x < 0``, ``ci`` is the real part of the\n" + "cosine integral. For such points ``ci(x)`` and ``ci(x + 0j)``\n" + "differ by a factor of ``1j*pi``.\n" + "\n" + "For real arguments the function is computed by calling Cephes'\n" + "[2]_ *sici* routine. For complex arguments the algorithm is based\n" + "on Mpmath's [3]_ *si* and *ci* routines.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Milton Abramowitz and Irene A. Stegun, eds.\n" + " Handbook of Mathematical Functions with Formulas,\n" + " Graphs, and Mathematical Tables. New York: Dover, 1972.\n" + " (See Section 5.2.)\n" + ".. [2] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + ".. [3] Fredrik Johansson and others.\n" + " \"mpmath: a Python library for arbitrary-precision floating-point\n" + " arithmetic\" (Version 0.19) http://mpmath.org/\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> from scipy.special import sici, exp1\n" + "\n" + "`sici` accepts real or complex input:\n" + "\n" + ">>> sici(2.5)\n" + "(1.7785201734438267, 0.2858711963653835)\n" + ">>> sici(2.5 + 3j)\n" + "((4.505735874563953+0.06863305018999577j),\n" + "(0.0793644206906966-2.935510262937543j))\n" + "\n" + "For z in the right half plane, the sine and cosine integrals are\n" + "related to the exponential integral E1 (implemented in SciPy as\n" + "`scipy.special.exp1`) by\n" + "\n" + "* Si(z) = (E1(i*z) - E1(-i*z))/2i + pi/2\n" + "* Ci(z) = -(E1(i*z) + E1(-i*z))/2\n" + "\n" + "See [1]_ (equations 5.2.21 and 5.2.23).\n" + "\n" + "We can verify these relations:\n" + "\n" + ">>> z = 2 - 3j\n" + ">>> sici(z)\n" + "((4.54751388956229-1.3991965806460565j),\n" + "(1.408292501520851+2.9836177420296055j))\n" + "\n" + ">>> (exp1(1j*z) - exp1(-1j*z))/2j + np.pi/2 # Same as sine integral\n" + "(4.54751388956229-1.3991965806460565j)\n" + "\n" + ">>> -(exp1(1j*z) + exp1(-1j*z))/2 # Same as cosine integral\n" + "(1.408292501520851+2.9836177420296055j)\n" + "\n" + "Plot the functions evaluated on the real axis; the dotted horizontal\n" + "lines are at pi/2 and -pi/2:\n" + "\n" + ">>> x = np.linspace(-16, 16, 150)\n" + ">>> si, ci = sici(x)\n" + "\n" + ">>> fig, ax = plt.subplots()\n" + ">>> ax.plot(x, si, label='Si(x)')\n" + ">>> ax.plot(x, ci, '--', label='Ci(x)')\n" + ">>> ax.legend(shadow=True, framealpha=1, loc='upper left')\n" + ">>> ax.set_xlabel('x')\n" + ">>> ax.set_title('Sine and Cosine Integrals')\n" + ">>> ax.axhline(np.pi/2, linestyle=':', alpha=0.5, color='k')\n" + ">>> ax.axhline(-np.pi/2, linestyle=':', alpha=0.5, color='k')\n" + ">>> ax.grid(True)\n" + ">>> plt.show()") +ufunc_sici_loops[0] = loop_i_d_dd_As_f_ff +ufunc_sici_loops[1] = loop_i_d_dd_As_d_dd +ufunc_sici_loops[2] = loop_i_D_DD_As_F_FF +ufunc_sici_loops[3] = loop_i_D_DD_As_D_DD +ufunc_sici_types[0] = NPY_FLOAT +ufunc_sici_types[1] = NPY_FLOAT +ufunc_sici_types[2] = NPY_FLOAT +ufunc_sici_types[3] = NPY_DOUBLE +ufunc_sici_types[4] = NPY_DOUBLE +ufunc_sici_types[5] = NPY_DOUBLE +ufunc_sici_types[6] = NPY_CFLOAT +ufunc_sici_types[7] = NPY_CFLOAT +ufunc_sici_types[8] = NPY_CFLOAT +ufunc_sici_types[9] = NPY_CDOUBLE +ufunc_sici_types[10] = NPY_CDOUBLE +ufunc_sici_types[11] = NPY_CDOUBLE +ufunc_sici_ptr[2*0] = _func_xsf_sici +ufunc_sici_ptr[2*0+1] = ("sici") +ufunc_sici_ptr[2*1] = _func_xsf_sici +ufunc_sici_ptr[2*1+1] = ("sici") +ufunc_sici_ptr[2*2] = _func_xsf_csici +ufunc_sici_ptr[2*2+1] = ("sici") +ufunc_sici_ptr[2*3] = _func_xsf_csici +ufunc_sici_ptr[2*3+1] = ("sici") +ufunc_sici_data[0] = &ufunc_sici_ptr[2*0] +ufunc_sici_data[1] = &ufunc_sici_ptr[2*1] +ufunc_sici_data[2] = &ufunc_sici_ptr[2*2] +ufunc_sici_data[3] = &ufunc_sici_ptr[2*3] +sici = np.PyUFunc_FromFuncAndData(ufunc_sici_loops, ufunc_sici_data, ufunc_sici_types, 4, 1, 2, 0, "sici", ufunc_sici_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_smirnov_loops[3] +cdef void *ufunc_smirnov_ptr[6] +cdef void *ufunc_smirnov_data[3] +cdef char ufunc_smirnov_types[9] +cdef char *ufunc_smirnov_doc = ( + "smirnov(n, d, out=None)\n" + "\n" + "Kolmogorov-Smirnov complementary cumulative distribution function\n" + "\n" + "Returns the exact Kolmogorov-Smirnov complementary cumulative\n" + "distribution function,(aka the Survival Function) of Dn+ (or Dn-)\n" + "for a one-sided test of equality between an empirical and a\n" + "theoretical distribution. It is equal to the probability that the\n" + "maximum difference between a theoretical distribution and an empirical\n" + "one based on `n` samples is greater than d.\n" + "\n" + "Parameters\n" + "----------\n" + "n : int\n" + " Number of samples\n" + "d : float array_like\n" + " Deviation between the Empirical CDF (ECDF) and the target CDF.\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The value(s) of smirnov(n, d), Prob(Dn+ >= d) (Also Prob(Dn- >= d))\n" + "\n" + "See Also\n" + "--------\n" + "smirnovi : The Inverse Survival Function for the distribution\n" + "scipy.stats.ksone : Provides the functionality as a continuous distribution\n" + "kolmogorov, kolmogi : Functions for the two-sided distribution\n" + "\n" + "Notes\n" + "-----\n" + "`smirnov` is used by `stats.kstest` in the application of the\n" + "Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this\n" + "function is exposed in `scpy.special`, but the recommended way to achieve\n" + "the most accurate CDF/SF/PDF/PPF/ISF computations is to use the\n" + "`stats.ksone` distribution.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy.special import smirnov\n" + ">>> from scipy.stats import norm\n" + "\n" + "Show the probability of a gap at least as big as 0, 0.5 and 1.0 for a\n" + "sample of size 5.\n" + "\n" + ">>> smirnov(5, [0, 0.5, 1.0])\n" + "array([ 1. , 0.056, 0. ])\n" + "\n" + "Compare a sample of size 5 against N(0, 1), the standard normal\n" + "distribution with mean 0 and standard deviation 1.\n" + "\n" + "`x` is the sample.\n" + "\n" + ">>> x = np.array([-1.392, -0.135, 0.114, 0.190, 1.82])\n" + "\n" + ">>> target = norm(0, 1)\n" + ">>> cdfs = target.cdf(x)\n" + ">>> cdfs\n" + "array([0.0819612 , 0.44630594, 0.5453811 , 0.57534543, 0.9656205 ])\n" + "\n" + "Construct the empirical CDF and the K-S statistics (Dn+, Dn-, Dn).\n" + "\n" + ">>> n = len(x)\n" + ">>> ecdfs = np.arange(n+1, dtype=float)/n\n" + ">>> cols = np.column_stack([x, ecdfs[1:], cdfs, cdfs - ecdfs[:n],\n" + "... ecdfs[1:] - cdfs])\n" + ">>> with np.printoptions(precision=3):\n" + "... print(cols)\n" + "[[-1.392 0.2 0.082 0.082 0.118]\n" + " [-0.135 0.4 0.446 0.246 -0.046]\n" + " [ 0.114 0.6 0.545 0.145 0.055]\n" + " [ 0.19 0.8 0.575 -0.025 0.225]\n" + " [ 1.82 1. 0.966 0.166 0.034]]\n" + ">>> gaps = cols[:, -2:]\n" + ">>> Dnpm = np.max(gaps, axis=0)\n" + ">>> print(f'Dn-={Dnpm[0]:f}, Dn+={Dnpm[1]:f}')\n" + "Dn-=0.246306, Dn+=0.224655\n" + ">>> probs = smirnov(n, Dnpm)\n" + ">>> print(f'For a sample of size {n} drawn from N(0, 1):',\n" + "... f' Smirnov n={n}: Prob(Dn- >= {Dnpm[0]:f}) = {probs[0]:.4f}',\n" + "... f' Smirnov n={n}: Prob(Dn+ >= {Dnpm[1]:f}) = {probs[1]:.4f}',\n" + "... sep='\\n')\n" + "For a sample of size 5 drawn from N(0, 1):\n" + " Smirnov n=5: Prob(Dn- >= 0.246306) = 0.4711\n" + " Smirnov n=5: Prob(Dn+ >= 0.224655) = 0.5245\n" + "\n" + "Plot the empirical CDF and the standard normal CDF.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> plt.step(np.concatenate(([-2.5], x, [2.5])),\n" + "... np.concatenate((ecdfs, [1])),\n" + "... where='post', label='Empirical CDF')\n" + ">>> xx = np.linspace(-2.5, 2.5, 100)\n" + ">>> plt.plot(xx, target.cdf(xx), '--', label='CDF for N(0, 1)')\n" + "\n" + "Add vertical lines marking Dn+ and Dn-.\n" + "\n" + ">>> iminus, iplus = np.argmax(gaps, axis=0)\n" + ">>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], color='r',\n" + "... alpha=0.5, lw=4)\n" + ">>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], color='m',\n" + "... alpha=0.5, lw=4)\n" + "\n" + ">>> plt.grid(True)\n" + ">>> plt.legend(framealpha=1, shadow=True)\n" + ">>> plt.show()") +ufunc_smirnov_loops[0] = loop_d_pd__As_pd_d +ufunc_smirnov_loops[1] = loop_d_dd__As_ff_f +ufunc_smirnov_loops[2] = loop_d_dd__As_dd_d +ufunc_smirnov_types[0] = NPY_INTP +ufunc_smirnov_types[1] = NPY_DOUBLE +ufunc_smirnov_types[2] = NPY_DOUBLE +ufunc_smirnov_types[3] = NPY_FLOAT +ufunc_smirnov_types[4] = NPY_FLOAT +ufunc_smirnov_types[5] = NPY_FLOAT +ufunc_smirnov_types[6] = NPY_DOUBLE +ufunc_smirnov_types[7] = NPY_DOUBLE +ufunc_smirnov_types[8] = NPY_DOUBLE +ufunc_smirnov_ptr[2*0] = _func_cephes_smirnov_wrap +ufunc_smirnov_ptr[2*0+1] = ("smirnov") +ufunc_smirnov_ptr[2*1] = _func_smirnov_unsafe +ufunc_smirnov_ptr[2*1+1] = ("smirnov") +ufunc_smirnov_ptr[2*2] = _func_smirnov_unsafe +ufunc_smirnov_ptr[2*2+1] = ("smirnov") +ufunc_smirnov_data[0] = &ufunc_smirnov_ptr[2*0] +ufunc_smirnov_data[1] = &ufunc_smirnov_ptr[2*1] +ufunc_smirnov_data[2] = &ufunc_smirnov_ptr[2*2] +smirnov = np.PyUFunc_FromFuncAndData(ufunc_smirnov_loops, ufunc_smirnov_data, ufunc_smirnov_types, 3, 2, 1, 0, "smirnov", ufunc_smirnov_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_smirnovi_loops[3] +cdef void *ufunc_smirnovi_ptr[6] +cdef void *ufunc_smirnovi_data[3] +cdef char ufunc_smirnovi_types[9] +cdef char *ufunc_smirnovi_doc = ( + "smirnovi(n, p, out=None)\n" + "\n" + "Inverse to `smirnov`\n" + "\n" + "Returns `d` such that ``smirnov(n, d) == p``, the critical value\n" + "corresponding to `p`.\n" + "\n" + "Parameters\n" + "----------\n" + "n : int\n" + " Number of samples\n" + "p : float array_like\n" + " Probability\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The value(s) of smirnovi(n, p), the critical values.\n" + "\n" + "See Also\n" + "--------\n" + "smirnov : The Survival Function (SF) for the distribution\n" + "scipy.stats.ksone : Provides the functionality as a continuous distribution\n" + "kolmogorov, kolmogi : Functions for the two-sided distribution\n" + "scipy.stats.kstwobign : Two-sided Kolmogorov-Smirnov distribution, large n\n" + "\n" + "Notes\n" + "-----\n" + "`smirnov` is used by `stats.kstest` in the application of the\n" + "Kolmogorov-Smirnov Goodness of Fit test. For historical reasons this\n" + "function is exposed in `scpy.special`, but the recommended way to achieve\n" + "the most accurate CDF/SF/PDF/PPF/ISF computations is to use the\n" + "`stats.ksone` distribution.\n" + "\n" + "Examples\n" + "--------\n" + ">>> from scipy.special import smirnovi, smirnov\n" + "\n" + ">>> n = 24\n" + ">>> deviations = [0.1, 0.2, 0.3]\n" + "\n" + "Use `smirnov` to compute the complementary CDF of the Smirnov\n" + "distribution for the given number of samples and deviations.\n" + "\n" + ">>> p = smirnov(n, deviations)\n" + ">>> p\n" + "array([0.58105083, 0.12826832, 0.01032231])\n" + "\n" + "The inverse function ``smirnovi(n, p)`` returns ``deviations``.\n" + "\n" + ">>> smirnovi(n, p)\n" + "array([0.1, 0.2, 0.3])") +ufunc_smirnovi_loops[0] = loop_d_pd__As_pd_d +ufunc_smirnovi_loops[1] = loop_d_dd__As_ff_f +ufunc_smirnovi_loops[2] = loop_d_dd__As_dd_d +ufunc_smirnovi_types[0] = NPY_INTP +ufunc_smirnovi_types[1] = NPY_DOUBLE +ufunc_smirnovi_types[2] = NPY_DOUBLE +ufunc_smirnovi_types[3] = NPY_FLOAT +ufunc_smirnovi_types[4] = NPY_FLOAT +ufunc_smirnovi_types[5] = NPY_FLOAT +ufunc_smirnovi_types[6] = NPY_DOUBLE +ufunc_smirnovi_types[7] = NPY_DOUBLE +ufunc_smirnovi_types[8] = NPY_DOUBLE +ufunc_smirnovi_ptr[2*0] = _func_cephes_smirnovi_wrap +ufunc_smirnovi_ptr[2*0+1] = ("smirnovi") +ufunc_smirnovi_ptr[2*1] = _func_smirnovi_unsafe +ufunc_smirnovi_ptr[2*1+1] = ("smirnovi") +ufunc_smirnovi_ptr[2*2] = _func_smirnovi_unsafe +ufunc_smirnovi_ptr[2*2+1] = ("smirnovi") +ufunc_smirnovi_data[0] = &ufunc_smirnovi_ptr[2*0] +ufunc_smirnovi_data[1] = &ufunc_smirnovi_ptr[2*1] +ufunc_smirnovi_data[2] = &ufunc_smirnovi_ptr[2*2] +smirnovi = np.PyUFunc_FromFuncAndData(ufunc_smirnovi_loops, ufunc_smirnovi_data, ufunc_smirnovi_types, 3, 2, 1, 0, "smirnovi", ufunc_smirnovi_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_spence_loops[4] +cdef void *ufunc_spence_ptr[8] +cdef void *ufunc_spence_data[4] +cdef char ufunc_spence_types[8] +cdef char *ufunc_spence_doc = ( + "spence(z, out=None)\n" + "\n" + "Spence's function, also known as the dilogarithm.\n" + "\n" + "It is defined to be\n" + "\n" + ".. math::\n" + " \\int_1^z \\frac{\\log(t)}{1 - t}dt\n" + "\n" + "for complex :math:`z`, where the contour of integration is taken\n" + "to avoid the branch cut of the logarithm. Spence's function is\n" + "analytic everywhere except the negative real axis where it has a\n" + "branch cut.\n" + "\n" + "Parameters\n" + "----------\n" + "z : array_like\n" + " Points at which to evaluate Spence's function\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "s : scalar or ndarray\n" + " Computed values of Spence's function\n" + "\n" + "Notes\n" + "-----\n" + "There is a different convention which defines Spence's function by\n" + "the integral\n" + "\n" + ".. math::\n" + " -\\int_0^z \\frac{\\log(1 - t)}{t}dt;\n" + "\n" + "this is our ``spence(1 - z)``.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy.special import spence\n" + ">>> import matplotlib.pyplot as plt\n" + "\n" + "The function is defined for complex inputs:\n" + "\n" + ">>> spence([1-1j, 1.5+2j, 3j, -10-5j])\n" + "array([-0.20561676+0.91596559j, -0.86766909-1.39560134j,\n" + " -0.59422064-2.49129918j, -1.14044398+6.80075924j])\n" + "\n" + "For complex inputs on the branch cut, which is the negative real axis,\n" + "the function returns the limit for ``z`` with positive imaginary part.\n" + "For example, in the following, note the sign change of the imaginary\n" + "part of the output for ``z = -2`` and ``z = -2 - 1e-8j``:\n" + "\n" + ">>> spence([-2 + 1e-8j, -2, -2 - 1e-8j])\n" + "array([2.32018041-3.45139229j, 2.32018042-3.4513923j ,\n" + " 2.32018041+3.45139229j])\n" + "\n" + "The function returns ``nan`` for real inputs on the branch cut:\n" + "\n" + ">>> spence(-1.5)\n" + "nan\n" + "\n" + "Verify some particular values: ``spence(0) = pi**2/6``,\n" + "``spence(1) = 0`` and ``spence(2) = -pi**2/12``.\n" + "\n" + ">>> spence([0, 1, 2])\n" + "array([ 1.64493407, 0. , -0.82246703])\n" + ">>> np.pi**2/6, -np.pi**2/12\n" + "(1.6449340668482264, -0.8224670334241132)\n" + "\n" + "Verify the identity::\n" + "\n" + " spence(z) + spence(1 - z) = pi**2/6 - log(z)*log(1 - z)\n" + "\n" + ">>> z = 3 + 4j\n" + ">>> spence(z) + spence(1 - z)\n" + "(-2.6523186143876067+1.8853470951513935j)\n" + ">>> np.pi**2/6 - np.log(z)*np.log(1 - z)\n" + "(-2.652318614387606+1.885347095151394j)\n" + "\n" + "Plot the function for positive real input.\n" + "\n" + ">>> fig, ax = plt.subplots()\n" + ">>> x = np.linspace(0, 6, 400)\n" + ">>> ax.plot(x, spence(x))\n" + ">>> ax.grid()\n" + ">>> ax.set_xlabel('x')\n" + ">>> ax.set_title('spence(x)')\n" + ">>> plt.show()") +ufunc_spence_loops[0] = loop_d_d__As_f_f +ufunc_spence_loops[1] = loop_d_d__As_d_d +ufunc_spence_loops[2] = loop_D_D__As_F_F +ufunc_spence_loops[3] = loop_D_D__As_D_D +ufunc_spence_types[0] = NPY_FLOAT +ufunc_spence_types[1] = NPY_FLOAT +ufunc_spence_types[2] = NPY_DOUBLE +ufunc_spence_types[3] = NPY_DOUBLE +ufunc_spence_types[4] = NPY_CFLOAT +ufunc_spence_types[5] = NPY_CFLOAT +ufunc_spence_types[6] = NPY_CDOUBLE +ufunc_spence_types[7] = NPY_CDOUBLE +ufunc_spence_ptr[2*0] = _func_cephes_spence +ufunc_spence_ptr[2*0+1] = ("spence") +ufunc_spence_ptr[2*1] = _func_cephes_spence +ufunc_spence_ptr[2*1+1] = ("spence") +ufunc_spence_ptr[2*2] = _func_cspence +ufunc_spence_ptr[2*2+1] = ("spence") +ufunc_spence_ptr[2*3] = _func_cspence +ufunc_spence_ptr[2*3+1] = ("spence") +ufunc_spence_data[0] = &ufunc_spence_ptr[2*0] +ufunc_spence_data[1] = &ufunc_spence_ptr[2*1] +ufunc_spence_data[2] = &ufunc_spence_ptr[2*2] +ufunc_spence_data[3] = &ufunc_spence_ptr[2*3] +spence = np.PyUFunc_FromFuncAndData(ufunc_spence_loops, ufunc_spence_data, ufunc_spence_types, 4, 1, 1, 0, "spence", ufunc_spence_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_stdtr_loops[2] +cdef void *ufunc_stdtr_ptr[4] +cdef void *ufunc_stdtr_data[2] +cdef char ufunc_stdtr_types[6] +cdef char *ufunc_stdtr_doc = ( + "stdtr(df, t, out=None)\n" + "\n" + "Student t distribution cumulative distribution function\n" + "\n" + "Returns the integral:\n" + "\n" + ".. math::\n" + " \\frac{\\Gamma((df+1)/2)}{\\sqrt{\\pi df} \\Gamma(df/2)}\n" + " \\int_{-\\infty}^t (1+x^2/df)^{-(df+1)/2}\\, dx\n" + "\n" + "Parameters\n" + "----------\n" + "df : array_like\n" + " Degrees of freedom\n" + "t : array_like\n" + " Upper bound of the integral\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Value of the Student t CDF at t\n" + "\n" + "See Also\n" + "--------\n" + "stdtridf : inverse of stdtr with respect to `df`\n" + "stdtrit : inverse of stdtr with respect to `t`\n" + "scipy.stats.t : student t distribution\n" + "\n" + "Notes\n" + "-----\n" + "The student t distribution is also available as `scipy.stats.t`.\n" + "Calling `stdtr` directly can improve performance compared to the\n" + "``cdf`` method of `scipy.stats.t` (see last example below).\n" + "\n" + "Examples\n" + "--------\n" + "Calculate the function for ``df=3`` at ``t=1``.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import stdtr\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> stdtr(3, 1)\n" + "0.8044988905221148\n" + "\n" + "Plot the function for three different degrees of freedom.\n" + "\n" + ">>> x = np.linspace(-10, 10, 1000)\n" + ">>> fig, ax = plt.subplots()\n" + ">>> parameters = [(1, \"solid\"), (3, \"dashed\"), (10, \"dotted\")]\n" + ">>> for (df, linestyle) in parameters:\n" + "... ax.plot(x, stdtr(df, x), ls=linestyle, label=f\"$df={df}$\")\n" + ">>> ax.legend()\n" + ">>> ax.set_title(\"Student t distribution cumulative distribution function\")\n" + ">>> plt.show()\n" + "\n" + "The function can be computed for several degrees of freedom at the same\n" + "time by providing a NumPy array or list for `df`:\n" + "\n" + ">>> stdtr([1, 2, 3], 1)\n" + "array([0.75 , 0.78867513, 0.80449889])\n" + "\n" + "It is possible to calculate the function at several points for several\n" + "different degrees of freedom simultaneously by providing arrays for `df`\n" + "and `t` with shapes compatible for broadcasting. Compute `stdtr` at\n" + "4 points for 3 degrees of freedom resulting in an array of shape 3x4.\n" + "\n" + ">>> dfs = np.array([[1], [2], [3]])\n" + ">>> t = np.array([2, 4, 6, 8])\n" + ">>> dfs.shape, t.shape\n" + "((3, 1), (4,))\n" + "\n" + ">>> stdtr(dfs, t)\n" + "array([[0.85241638, 0.92202087, 0.94743154, 0.96041658],\n" + " [0.90824829, 0.97140452, 0.98666426, 0.99236596],\n" + " [0.93033702, 0.98599577, 0.99536364, 0.99796171]])\n" + "\n" + "The t distribution is also available as `scipy.stats.t`. Calling `stdtr`\n" + "directly can be much faster than calling the ``cdf`` method of\n" + "`scipy.stats.t`. To get the same results, one must use the following\n" + "parametrization: ``scipy.stats.t(df).cdf(x) = stdtr(df, x)``.\n" + "\n" + ">>> from scipy.stats import t\n" + ">>> df, x = 3, 1\n" + ">>> stdtr_result = stdtr(df, x) # this can be faster than below\n" + ">>> stats_result = t(df).cdf(x)\n" + ">>> stats_result == stdtr_result # test that results are equal\n" + "True") +ufunc_stdtr_loops[0] = loop_d_dd__As_ff_f +ufunc_stdtr_loops[1] = loop_d_dd__As_dd_d +ufunc_stdtr_types[0] = NPY_FLOAT +ufunc_stdtr_types[1] = NPY_FLOAT +ufunc_stdtr_types[2] = NPY_FLOAT +ufunc_stdtr_types[3] = NPY_DOUBLE +ufunc_stdtr_types[4] = NPY_DOUBLE +ufunc_stdtr_types[5] = NPY_DOUBLE +ufunc_stdtr_ptr[2*0] = _func_stdtr +ufunc_stdtr_ptr[2*0+1] = ("stdtr") +ufunc_stdtr_ptr[2*1] = _func_stdtr +ufunc_stdtr_ptr[2*1+1] = ("stdtr") +ufunc_stdtr_data[0] = &ufunc_stdtr_ptr[2*0] +ufunc_stdtr_data[1] = &ufunc_stdtr_ptr[2*1] +stdtr = np.PyUFunc_FromFuncAndData(ufunc_stdtr_loops, ufunc_stdtr_data, ufunc_stdtr_types, 2, 2, 1, 0, "stdtr", ufunc_stdtr_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_stdtridf_loops[2] +cdef void *ufunc_stdtridf_ptr[4] +cdef void *ufunc_stdtridf_data[2] +cdef char ufunc_stdtridf_types[6] +cdef char *ufunc_stdtridf_doc = ( + "stdtridf(p, t, out=None)\n" + "\n" + "Inverse of `stdtr` vs df\n" + "\n" + "Returns the argument df such that stdtr(df, t) is equal to `p`.\n" + "\n" + "Parameters\n" + "----------\n" + "p : array_like\n" + " Probability\n" + "t : array_like\n" + " Upper bound of the integral\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "df : scalar or ndarray\n" + " Value of `df` such that ``stdtr(df, t) == p``\n" + "\n" + "See Also\n" + "--------\n" + "stdtr : Student t CDF\n" + "stdtrit : inverse of stdtr with respect to `t`\n" + "scipy.stats.t : Student t distribution\n" + "\n" + "Examples\n" + "--------\n" + "Compute the student t cumulative distribution function for one\n" + "parameter set.\n" + "\n" + ">>> from scipy.special import stdtr, stdtridf\n" + ">>> df, x = 5, 2\n" + ">>> cdf_value = stdtr(df, x)\n" + ">>> cdf_value\n" + "0.9490302605850709\n" + "\n" + "Verify that `stdtridf` recovers the original value for `df` given\n" + "the CDF value and `x`.\n" + "\n" + ">>> stdtridf(cdf_value, x)\n" + "5.0") +ufunc_stdtridf_loops[0] = loop_d_dd__As_ff_f +ufunc_stdtridf_loops[1] = loop_d_dd__As_dd_d +ufunc_stdtridf_types[0] = NPY_FLOAT +ufunc_stdtridf_types[1] = NPY_FLOAT +ufunc_stdtridf_types[2] = NPY_FLOAT +ufunc_stdtridf_types[3] = NPY_DOUBLE +ufunc_stdtridf_types[4] = NPY_DOUBLE +ufunc_stdtridf_types[5] = NPY_DOUBLE +ufunc_stdtridf_ptr[2*0] = _func_stdtridf +ufunc_stdtridf_ptr[2*0+1] = ("stdtridf") +ufunc_stdtridf_ptr[2*1] = _func_stdtridf +ufunc_stdtridf_ptr[2*1+1] = ("stdtridf") +ufunc_stdtridf_data[0] = &ufunc_stdtridf_ptr[2*0] +ufunc_stdtridf_data[1] = &ufunc_stdtridf_ptr[2*1] +stdtridf = np.PyUFunc_FromFuncAndData(ufunc_stdtridf_loops, ufunc_stdtridf_data, ufunc_stdtridf_types, 2, 2, 1, 0, "stdtridf", ufunc_stdtridf_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_stdtrit_loops[2] +cdef void *ufunc_stdtrit_ptr[4] +cdef void *ufunc_stdtrit_data[2] +cdef char ufunc_stdtrit_types[6] +cdef char *ufunc_stdtrit_doc = ( + "stdtrit(df, p, out=None)\n" + "\n" + "The `p`-th quantile of the student t distribution.\n" + "\n" + "This function is the inverse of the student t distribution cumulative\n" + "distribution function (CDF), returning `t` such that `stdtr(df, t) = p`.\n" + "\n" + "Returns the argument `t` such that stdtr(df, t) is equal to `p`.\n" + "\n" + "Parameters\n" + "----------\n" + "df : array_like\n" + " Degrees of freedom\n" + "p : array_like\n" + " Probability\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "t : scalar or ndarray\n" + " Value of `t` such that ``stdtr(df, t) == p``\n" + "\n" + "See Also\n" + "--------\n" + "stdtr : Student t CDF\n" + "stdtridf : inverse of stdtr with respect to `df`\n" + "scipy.stats.t : Student t distribution\n" + "\n" + "Notes\n" + "-----\n" + "The student t distribution is also available as `scipy.stats.t`. Calling\n" + "`stdtrit` directly can improve performance compared to the ``ppf``\n" + "method of `scipy.stats.t` (see last example below).\n" + "\n" + "Examples\n" + "--------\n" + "`stdtrit` represents the inverse of the student t distribution CDF which\n" + "is available as `stdtr`. Here, we calculate the CDF for ``df`` at\n" + "``x=1``. `stdtrit` then returns ``1`` up to floating point errors\n" + "given the same value for `df` and the computed CDF value.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import stdtr, stdtrit\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> df = 3\n" + ">>> x = 1\n" + ">>> cdf_value = stdtr(df, x)\n" + ">>> stdtrit(df, cdf_value)\n" + "0.9999999994418539\n" + "\n" + "Plot the function for three different degrees of freedom.\n" + "\n" + ">>> x = np.linspace(0, 1, 1000)\n" + ">>> parameters = [(1, \"solid\"), (2, \"dashed\"), (5, \"dotted\")]\n" + ">>> fig, ax = plt.subplots()\n" + ">>> for (df, linestyle) in parameters:\n" + "... ax.plot(x, stdtrit(df, x), ls=linestyle, label=f\"$df={df}$\")\n" + ">>> ax.legend()\n" + ">>> ax.set_ylim(-10, 10)\n" + ">>> ax.set_title(\"Student t distribution quantile function\")\n" + ">>> plt.show()\n" + "\n" + "The function can be computed for several degrees of freedom at the same\n" + "time by providing a NumPy array or list for `df`:\n" + "\n" + ">>> stdtrit([1, 2, 3], 0.7)\n" + "array([0.72654253, 0.6172134 , 0.58438973])\n" + "\n" + "It is possible to calculate the function at several points for several\n" + "different degrees of freedom simultaneously by providing arrays for `df`\n" + "and `p` with shapes compatible for broadcasting. Compute `stdtrit` at\n" + "4 points for 3 degrees of freedom resulting in an array of shape 3x4.\n" + "\n" + ">>> dfs = np.array([[1], [2], [3]])\n" + ">>> p = np.array([0.2, 0.4, 0.7, 0.8])\n" + ">>> dfs.shape, p.shape\n" + "((3, 1), (4,))\n" + "\n" + ">>> stdtrit(dfs, p)\n" + "array([[-1.37638192, -0.3249197 , 0.72654253, 1.37638192],\n" + " [-1.06066017, -0.28867513, 0.6172134 , 1.06066017],\n" + " [-0.97847231, -0.27667066, 0.58438973, 0.97847231]])\n" + "\n" + "The t distribution is also available as `scipy.stats.t`. Calling `stdtrit`\n" + "directly can be much faster than calling the ``ppf`` method of\n" + "`scipy.stats.t`. To get the same results, one must use the following\n" + "parametrization: ``scipy.stats.t(df).ppf(x) = stdtrit(df, x)``.\n" + "\n" + ">>> from scipy.stats import t\n" + ">>> df, x = 3, 0.5\n" + ">>> stdtrit_result = stdtrit(df, x) # this can be faster than below\n" + ">>> stats_result = t(df).ppf(x)\n" + ">>> stats_result == stdtrit_result # test that results are equal\n" + "True") +ufunc_stdtrit_loops[0] = loop_d_dd__As_ff_f +ufunc_stdtrit_loops[1] = loop_d_dd__As_dd_d +ufunc_stdtrit_types[0] = NPY_FLOAT +ufunc_stdtrit_types[1] = NPY_FLOAT +ufunc_stdtrit_types[2] = NPY_FLOAT +ufunc_stdtrit_types[3] = NPY_DOUBLE +ufunc_stdtrit_types[4] = NPY_DOUBLE +ufunc_stdtrit_types[5] = NPY_DOUBLE +ufunc_stdtrit_ptr[2*0] = _func_stdtrit +ufunc_stdtrit_ptr[2*0+1] = ("stdtrit") +ufunc_stdtrit_ptr[2*1] = _func_stdtrit +ufunc_stdtrit_ptr[2*1+1] = ("stdtrit") +ufunc_stdtrit_data[0] = &ufunc_stdtrit_ptr[2*0] +ufunc_stdtrit_data[1] = &ufunc_stdtrit_ptr[2*1] +stdtrit = np.PyUFunc_FromFuncAndData(ufunc_stdtrit_loops, ufunc_stdtrit_data, ufunc_stdtrit_types, 2, 2, 1, 0, "stdtrit", ufunc_stdtrit_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_tklmbda_loops[2] +cdef void *ufunc_tklmbda_ptr[4] +cdef void *ufunc_tklmbda_data[2] +cdef char ufunc_tklmbda_types[6] +cdef char *ufunc_tklmbda_doc = ( + "tklmbda(x, lmbda, out=None)\n" + "\n" + "Cumulative distribution function of the Tukey lambda distribution.\n" + "\n" + "Parameters\n" + "----------\n" + "x, lmbda : array_like\n" + " Parameters\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "cdf : scalar or ndarray\n" + " Value of the Tukey lambda CDF\n" + "\n" + "See Also\n" + "--------\n" + "scipy.stats.tukeylambda : Tukey lambda distribution\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> from scipy.special import tklmbda, expit\n" + "\n" + "Compute the cumulative distribution function (CDF) of the Tukey lambda\n" + "distribution at several ``x`` values for `lmbda` = -1.5.\n" + "\n" + ">>> x = np.linspace(-2, 2, 9)\n" + ">>> x\n" + "array([-2. , -1.5, -1. , -0.5, 0. , 0.5, 1. , 1.5, 2. ])\n" + ">>> tklmbda(x, -1.5)\n" + "array([0.34688734, 0.3786554 , 0.41528805, 0.45629737, 0.5 ,\n" + " 0.54370263, 0.58471195, 0.6213446 , 0.65311266])\n" + "\n" + "When `lmbda` is 0, the function is the logistic sigmoid function,\n" + "which is implemented in `scipy.special` as `expit`.\n" + "\n" + ">>> tklmbda(x, 0)\n" + "array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5 ,\n" + " 0.62245933, 0.73105858, 0.81757448, 0.88079708])\n" + ">>> expit(x)\n" + "array([0.11920292, 0.18242552, 0.26894142, 0.37754067, 0.5 ,\n" + " 0.62245933, 0.73105858, 0.81757448, 0.88079708])\n" + "\n" + "When `lmbda` is 1, the Tukey lambda distribution is uniform on the\n" + "interval [-1, 1], so the CDF increases linearly.\n" + "\n" + ">>> t = np.linspace(-1, 1, 9)\n" + ">>> tklmbda(t, 1)\n" + "array([0. , 0.125, 0.25 , 0.375, 0.5 , 0.625, 0.75 , 0.875, 1. ])\n" + "\n" + "In the following, we generate plots for several values of `lmbda`.\n" + "\n" + "The first figure shows graphs for `lmbda` <= 0.\n" + "\n" + ">>> styles = ['-', '-.', '--', ':']\n" + ">>> fig, ax = plt.subplots()\n" + ">>> x = np.linspace(-12, 12, 500)\n" + ">>> for k, lmbda in enumerate([-1.0, -0.5, 0.0]):\n" + "... y = tklmbda(x, lmbda)\n" + "... ax.plot(x, y, styles[k], label=rf'$\\lambda$ = {lmbda:-4.1f}')\n" + "\n" + ">>> ax.set_title(r'tklmbda(x, $\\lambda$)')\n" + ">>> ax.set_label('x')\n" + ">>> ax.legend(framealpha=1, shadow=True)\n" + ">>> ax.grid(True)\n" + "\n" + "The second figure shows graphs for `lmbda` > 0. The dots in the\n" + "graphs show the bounds of the support of the distribution.\n" + "\n" + ">>> fig, ax = plt.subplots()\n" + ">>> x = np.linspace(-4.2, 4.2, 500)\n" + ">>> lmbdas = [0.25, 0.5, 1.0, 1.5]\n" + ">>> for k, lmbda in enumerate(lmbdas):\n" + "... y = tklmbda(x, lmbda)\n" + "... ax.plot(x, y, styles[k], label=fr'$\\lambda$ = {lmbda}')\n" + "\n" + ">>> ax.set_prop_cycle(None)\n" + ">>> for lmbda in lmbdas:\n" + "... ax.plot([-1/lmbda, 1/lmbda], [0, 1], '.', ms=8)\n" + "\n" + ">>> ax.set_title(r'tklmbda(x, $\\lambda$)')\n" + ">>> ax.set_xlabel('x')\n" + ">>> ax.legend(framealpha=1, shadow=True)\n" + ">>> ax.grid(True)\n" + "\n" + ">>> plt.tight_layout()\n" + ">>> plt.show()\n" + "\n" + "The CDF of the Tukey lambda distribution is also implemented as the\n" + "``cdf`` method of `scipy.stats.tukeylambda`. In the following,\n" + "``tukeylambda.cdf(x, -0.5)`` and ``tklmbda(x, -0.5)`` compute the\n" + "same values:\n" + "\n" + ">>> from scipy.stats import tukeylambda\n" + ">>> x = np.linspace(-2, 2, 9)\n" + "\n" + ">>> tukeylambda.cdf(x, -0.5)\n" + "array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5 ,\n" + " 0.58671839, 0.66458323, 0.72906142, 0.78004843])\n" + "\n" + ">>> tklmbda(x, -0.5)\n" + "array([0.21995157, 0.27093858, 0.33541677, 0.41328161, 0.5 ,\n" + " 0.58671839, 0.66458323, 0.72906142, 0.78004843])\n" + "\n" + "The implementation in ``tukeylambda`` also provides location and scale\n" + "parameters, and other methods such as ``pdf()`` (the probability\n" + "density function) and ``ppf()`` (the inverse of the CDF), so for\n" + "working with the Tukey lambda distribution, ``tukeylambda`` is more\n" + "generally useful. The primary advantage of ``tklmbda`` is that it is\n" + "significantly faster than ``tukeylambda.cdf``.") +ufunc_tklmbda_loops[0] = loop_d_dd__As_ff_f +ufunc_tklmbda_loops[1] = loop_d_dd__As_dd_d +ufunc_tklmbda_types[0] = NPY_FLOAT +ufunc_tklmbda_types[1] = NPY_FLOAT +ufunc_tklmbda_types[2] = NPY_FLOAT +ufunc_tklmbda_types[3] = NPY_DOUBLE +ufunc_tklmbda_types[4] = NPY_DOUBLE +ufunc_tklmbda_types[5] = NPY_DOUBLE +ufunc_tklmbda_ptr[2*0] = _func_xsf_tukeylambdacdf +ufunc_tklmbda_ptr[2*0+1] = ("tklmbda") +ufunc_tklmbda_ptr[2*1] = _func_xsf_tukeylambdacdf +ufunc_tklmbda_ptr[2*1+1] = ("tklmbda") +ufunc_tklmbda_data[0] = &ufunc_tklmbda_ptr[2*0] +ufunc_tklmbda_data[1] = &ufunc_tklmbda_ptr[2*1] +tklmbda = np.PyUFunc_FromFuncAndData(ufunc_tklmbda_loops, ufunc_tklmbda_data, ufunc_tklmbda_types, 2, 2, 1, 0, "tklmbda", ufunc_tklmbda_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_voigt_profile_loops[2] +cdef void *ufunc_voigt_profile_ptr[4] +cdef void *ufunc_voigt_profile_data[2] +cdef char ufunc_voigt_profile_types[8] +cdef char *ufunc_voigt_profile_doc = ( + "voigt_profile(x, sigma, gamma, out=None)\n" + "\n" + "Voigt profile.\n" + "\n" + "The Voigt profile is a convolution of a 1-D Normal distribution with\n" + "standard deviation ``sigma`` and a 1-D Cauchy distribution with half-width at\n" + "half-maximum ``gamma``.\n" + "\n" + "If ``sigma = 0``, PDF of Cauchy distribution is returned.\n" + "Conversely, if ``gamma = 0``, PDF of Normal distribution is returned.\n" + "If ``sigma = gamma = 0``, the return value is ``Inf`` for ``x = 0``,\n" + "and ``0`` for all other ``x``.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Real argument\n" + "sigma : array_like\n" + " The standard deviation of the Normal distribution part\n" + "gamma : array_like\n" + " The half-width at half-maximum of the Cauchy distribution part\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " The Voigt profile at the given arguments\n" + "\n" + "See Also\n" + "--------\n" + "wofz : Faddeeva function\n" + "\n" + "Notes\n" + "-----\n" + "It can be expressed in terms of Faddeeva function\n" + "\n" + ".. math:: V(x; \\sigma, \\gamma) = \\frac{Re[w(z)]}{\\sigma\\sqrt{2\\pi}},\n" + ".. math:: z = \\frac{x + i\\gamma}{\\sqrt{2}\\sigma}\n" + "\n" + "where :math:`w(z)` is the Faddeeva function.\n" + "\n" + "References\n" + "----------\n" + ".. [1] https://en.wikipedia.org/wiki/Voigt_profile\n" + "\n" + "Examples\n" + "--------\n" + "Calculate the function at point 2 for ``sigma=1`` and ``gamma=1``.\n" + "\n" + ">>> from scipy.special import voigt_profile\n" + ">>> import numpy as np\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> voigt_profile(2, 1., 1.)\n" + "0.09071519942627544\n" + "\n" + "Calculate the function at several points by providing a NumPy array\n" + "for `x`.\n" + "\n" + ">>> values = np.array([-2., 0., 5])\n" + ">>> voigt_profile(values, 1., 1.)\n" + "array([0.0907152 , 0.20870928, 0.01388492])\n" + "\n" + "Plot the function for different parameter sets.\n" + "\n" + ">>> fig, ax = plt.subplots(figsize=(8, 8))\n" + ">>> x = np.linspace(-10, 10, 500)\n" + ">>> parameters_list = [(1.5, 0., \"solid\"), (1.3, 0.5, \"dashed\"),\n" + "... (0., 1.8, \"dotted\"), (1., 1., \"dashdot\")]\n" + ">>> for params in parameters_list:\n" + "... sigma, gamma, linestyle = params\n" + "... voigt = voigt_profile(x, sigma, gamma)\n" + "... ax.plot(x, voigt, label=rf\"$\\sigma={sigma},\\, \\gamma={gamma}$\",\n" + "... ls=linestyle)\n" + ">>> ax.legend()\n" + ">>> plt.show()\n" + "\n" + "Verify visually that the Voigt profile indeed arises as the convolution\n" + "of a normal and a Cauchy distribution.\n" + "\n" + ">>> from scipy.signal import convolve\n" + ">>> x, dx = np.linspace(-10, 10, 500, retstep=True)\n" + ">>> def gaussian(x, sigma):\n" + "... return np.exp(-0.5 * x**2/sigma**2)/(sigma * np.sqrt(2*np.pi))\n" + ">>> def cauchy(x, gamma):\n" + "... return gamma/(np.pi * (np.square(x)+gamma**2))\n" + ">>> sigma = 2\n" + ">>> gamma = 1\n" + ">>> gauss_profile = gaussian(x, sigma)\n" + ">>> cauchy_profile = cauchy(x, gamma)\n" + ">>> convolved = dx * convolve(cauchy_profile, gauss_profile, mode=\"same\")\n" + ">>> voigt = voigt_profile(x, sigma, gamma)\n" + ">>> fig, ax = plt.subplots(figsize=(8, 8))\n" + ">>> ax.plot(x, gauss_profile, label=\"Gauss: $G$\", c='b')\n" + ">>> ax.plot(x, cauchy_profile, label=\"Cauchy: $C$\", c='y', ls=\"dashed\")\n" + ">>> xx = 0.5*(x[1:] + x[:-1]) # midpoints\n" + ">>> ax.plot(xx, convolved[1:], label=\"Convolution: $G * C$\", ls='dashdot',\n" + "... c='k')\n" + ">>> ax.plot(x, voigt, label=\"Voigt\", ls='dotted', c='r')\n" + ">>> ax.legend()\n" + ">>> plt.show()") +ufunc_voigt_profile_loops[0] = loop_d_ddd__As_fff_f +ufunc_voigt_profile_loops[1] = loop_d_ddd__As_ddd_d +ufunc_voigt_profile_types[0] = NPY_FLOAT +ufunc_voigt_profile_types[1] = NPY_FLOAT +ufunc_voigt_profile_types[2] = NPY_FLOAT +ufunc_voigt_profile_types[3] = NPY_FLOAT +ufunc_voigt_profile_types[4] = NPY_DOUBLE +ufunc_voigt_profile_types[5] = NPY_DOUBLE +ufunc_voigt_profile_types[6] = NPY_DOUBLE +ufunc_voigt_profile_types[7] = NPY_DOUBLE +ufunc_voigt_profile_ptr[2*0] = scipy.special._ufuncs_cxx._export_faddeeva_voigt_profile +ufunc_voigt_profile_ptr[2*0+1] = ("voigt_profile") +ufunc_voigt_profile_ptr[2*1] = scipy.special._ufuncs_cxx._export_faddeeva_voigt_profile +ufunc_voigt_profile_ptr[2*1+1] = ("voigt_profile") +ufunc_voigt_profile_data[0] = &ufunc_voigt_profile_ptr[2*0] +ufunc_voigt_profile_data[1] = &ufunc_voigt_profile_ptr[2*1] +voigt_profile = np.PyUFunc_FromFuncAndData(ufunc_voigt_profile_loops, ufunc_voigt_profile_data, ufunc_voigt_profile_types, 2, 3, 1, 0, "voigt_profile", ufunc_voigt_profile_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_wofz_loops[2] +cdef void *ufunc_wofz_ptr[4] +cdef void *ufunc_wofz_data[2] +cdef char ufunc_wofz_types[4] +cdef char *ufunc_wofz_doc = ( + "wofz(z, out=None)\n" + "\n" + "Faddeeva function\n" + "\n" + "Returns the value of the Faddeeva function for complex argument::\n" + "\n" + " exp(-z**2) * erfc(-i*z)\n" + "\n" + "Parameters\n" + "----------\n" + "z : array_like\n" + " complex argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "scalar or ndarray\n" + " Value of the Faddeeva function\n" + "\n" + "See Also\n" + "--------\n" + "dawsn, erf, erfc, erfcx, erfi\n" + "\n" + "References\n" + "----------\n" + ".. [1] Steven G. Johnson, Faddeeva W function implementation.\n" + " http://ab-initio.mit.edu/Faddeeva\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy import special\n" + ">>> import matplotlib.pyplot as plt\n" + "\n" + ">>> x = np.linspace(-3, 3)\n" + ">>> z = special.wofz(x)\n" + "\n" + ">>> plt.plot(x, z.real, label='wofz(x).real')\n" + ">>> plt.plot(x, z.imag, label='wofz(x).imag')\n" + ">>> plt.xlabel('$x$')\n" + ">>> plt.legend(framealpha=1, shadow=True)\n" + ">>> plt.grid(alpha=0.25)\n" + ">>> plt.show()") +ufunc_wofz_loops[0] = loop_D_D__As_F_F +ufunc_wofz_loops[1] = loop_D_D__As_D_D +ufunc_wofz_types[0] = NPY_CFLOAT +ufunc_wofz_types[1] = NPY_CFLOAT +ufunc_wofz_types[2] = NPY_CDOUBLE +ufunc_wofz_types[3] = NPY_CDOUBLE +ufunc_wofz_ptr[2*0] = scipy.special._ufuncs_cxx._export_faddeeva_w +ufunc_wofz_ptr[2*0+1] = ("wofz") +ufunc_wofz_ptr[2*1] = scipy.special._ufuncs_cxx._export_faddeeva_w +ufunc_wofz_ptr[2*1+1] = ("wofz") +ufunc_wofz_data[0] = &ufunc_wofz_ptr[2*0] +ufunc_wofz_data[1] = &ufunc_wofz_ptr[2*1] +wofz = np.PyUFunc_FromFuncAndData(ufunc_wofz_loops, ufunc_wofz_data, ufunc_wofz_types, 2, 1, 1, 0, "wofz", ufunc_wofz_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_wrightomega_loops[4] +cdef void *ufunc_wrightomega_ptr[8] +cdef void *ufunc_wrightomega_data[4] +cdef char ufunc_wrightomega_types[8] +cdef char *ufunc_wrightomega_doc = ( + "wrightomega(z, out=None)\n" + "\n" + "Wright Omega function.\n" + "\n" + "Defined as the solution to\n" + "\n" + ".. math::\n" + "\n" + " \\omega + \\log(\\omega) = z\n" + "\n" + "where :math:`\\log` is the principal branch of the complex logarithm.\n" + "\n" + "Parameters\n" + "----------\n" + "z : array_like\n" + " Points at which to evaluate the Wright Omega function\n" + "out : ndarray, optional\n" + " Optional output array for the function values\n" + "\n" + "Returns\n" + "-------\n" + "omega : scalar or ndarray\n" + " Values of the Wright Omega function\n" + "\n" + "See Also\n" + "--------\n" + "lambertw : The Lambert W function\n" + "\n" + "Notes\n" + "-----\n" + ".. versionadded:: 0.19.0\n" + "\n" + "The function can also be defined as\n" + "\n" + ".. math::\n" + "\n" + " \\omega(z) = W_{K(z)}(e^z)\n" + "\n" + "where :math:`K(z) = \\lceil (\\Im(z) - \\pi)/(2\\pi) \\rceil` is the\n" + "unwinding number and :math:`W` is the Lambert W function.\n" + "\n" + "The implementation here is taken from [1]_.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Lawrence, Corless, and Jeffrey, \"Algorithm 917: Complex\n" + " Double-Precision Evaluation of the Wright :math:`\\omega`\n" + " Function.\" ACM Transactions on Mathematical Software,\n" + " 2012. :doi:`10.1145/2168773.2168779`.\n" + "\n" + "Examples\n" + "--------\n" + ">>> import numpy as np\n" + ">>> from scipy.special import wrightomega, lambertw\n" + "\n" + ">>> wrightomega([-2, -1, 0, 1, 2])\n" + "array([0.12002824, 0.27846454, 0.56714329, 1. , 1.5571456 ])\n" + "\n" + "Complex input:\n" + "\n" + ">>> wrightomega(3 + 5j)\n" + "(1.5804428632097158+3.8213626783287937j)\n" + "\n" + "Verify that ``wrightomega(z)`` satisfies ``w + log(w) = z``:\n" + "\n" + ">>> w = -5 + 4j\n" + ">>> wrightomega(w + np.log(w))\n" + "(-5+4j)\n" + "\n" + "Verify the connection to ``lambertw``:\n" + "\n" + ">>> z = 0.5 + 3j\n" + ">>> wrightomega(z)\n" + "(0.0966015889280649+1.4937828458191993j)\n" + ">>> lambertw(np.exp(z))\n" + "(0.09660158892806493+1.4937828458191993j)\n" + "\n" + ">>> z = 0.5 + 4j\n" + ">>> wrightomega(z)\n" + "(-0.3362123489037213+2.282986001579032j)\n" + ">>> lambertw(np.exp(z), k=1)\n" + "(-0.33621234890372115+2.282986001579032j)") +ufunc_wrightomega_loops[0] = loop_d_d__As_f_f +ufunc_wrightomega_loops[1] = loop_d_d__As_d_d +ufunc_wrightomega_loops[2] = loop_D_D__As_F_F +ufunc_wrightomega_loops[3] = loop_D_D__As_D_D +ufunc_wrightomega_types[0] = NPY_FLOAT +ufunc_wrightomega_types[1] = NPY_FLOAT +ufunc_wrightomega_types[2] = NPY_DOUBLE +ufunc_wrightomega_types[3] = NPY_DOUBLE +ufunc_wrightomega_types[4] = NPY_CFLOAT +ufunc_wrightomega_types[5] = NPY_CFLOAT +ufunc_wrightomega_types[6] = NPY_CDOUBLE +ufunc_wrightomega_types[7] = NPY_CDOUBLE +ufunc_wrightomega_ptr[2*0] = scipy.special._ufuncs_cxx._export_wrightomega_real +ufunc_wrightomega_ptr[2*0+1] = ("wrightomega") +ufunc_wrightomega_ptr[2*1] = scipy.special._ufuncs_cxx._export_wrightomega_real +ufunc_wrightomega_ptr[2*1+1] = ("wrightomega") +ufunc_wrightomega_ptr[2*2] = scipy.special._ufuncs_cxx._export_wrightomega +ufunc_wrightomega_ptr[2*2+1] = ("wrightomega") +ufunc_wrightomega_ptr[2*3] = scipy.special._ufuncs_cxx._export_wrightomega +ufunc_wrightomega_ptr[2*3+1] = ("wrightomega") +ufunc_wrightomega_data[0] = &ufunc_wrightomega_ptr[2*0] +ufunc_wrightomega_data[1] = &ufunc_wrightomega_ptr[2*1] +ufunc_wrightomega_data[2] = &ufunc_wrightomega_ptr[2*2] +ufunc_wrightomega_data[3] = &ufunc_wrightomega_ptr[2*3] +wrightomega = np.PyUFunc_FromFuncAndData(ufunc_wrightomega_loops, ufunc_wrightomega_data, ufunc_wrightomega_types, 4, 1, 1, 0, "wrightomega", ufunc_wrightomega_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_xlog1py_loops[4] +cdef void *ufunc_xlog1py_ptr[8] +cdef void *ufunc_xlog1py_data[4] +cdef char ufunc_xlog1py_types[12] +cdef char *ufunc_xlog1py_doc = ( + "xlog1py(x, y, out=None)\n" + "\n" + "Compute ``x*log1p(y)`` so that the result is 0 if ``x = 0``.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Multiplier\n" + "y : array_like\n" + " Argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "z : scalar or ndarray\n" + " Computed x*log1p(y)\n" + "\n" + "Notes\n" + "-----\n" + "\n" + ".. versionadded:: 0.13.0\n" + "\n" + "Examples\n" + "--------\n" + "This example shows how the function can be used to calculate the log of\n" + "the probability mass function for a geometric discrete random variable.\n" + "The probability mass function of the geometric distribution is defined\n" + "as follows:\n" + "\n" + ".. math:: f(k) = (1-p)^{k-1} p\n" + "\n" + "where :math:`p` is the probability of a single success\n" + "and :math:`1-p` is the probability of a single failure\n" + "and :math:`k` is the number of trials to get the first success.\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import xlog1py\n" + ">>> p = 0.5\n" + ">>> k = 100\n" + ">>> _pmf = np.power(1 - p, k - 1) * p\n" + ">>> _pmf\n" + "7.888609052210118e-31\n" + "\n" + "If we take k as a relatively large number the value of the probability\n" + "mass function can become very low. In such cases taking the log of the\n" + "pmf would be more suitable as the log function can change the values\n" + "to a scale that is more appropriate to work with.\n" + "\n" + ">>> _log_pmf = xlog1py(k - 1, -p) + np.log(p)\n" + ">>> _log_pmf\n" + "-69.31471805599453\n" + "\n" + "We can confirm that we get a value close to the original pmf value by\n" + "taking the exponential of the log pmf.\n" + "\n" + ">>> _orig_pmf = np.exp(_log_pmf)\n" + ">>> np.isclose(_pmf, _orig_pmf)\n" + "True") +ufunc_xlog1py_loops[0] = loop_d_dd__As_ff_f +ufunc_xlog1py_loops[1] = loop_d_dd__As_dd_d +ufunc_xlog1py_loops[2] = loop_D_DD__As_FF_F +ufunc_xlog1py_loops[3] = loop_D_DD__As_DD_D +ufunc_xlog1py_types[0] = NPY_FLOAT +ufunc_xlog1py_types[1] = NPY_FLOAT +ufunc_xlog1py_types[2] = NPY_FLOAT +ufunc_xlog1py_types[3] = NPY_DOUBLE +ufunc_xlog1py_types[4] = NPY_DOUBLE +ufunc_xlog1py_types[5] = NPY_DOUBLE +ufunc_xlog1py_types[6] = NPY_CFLOAT +ufunc_xlog1py_types[7] = NPY_CFLOAT +ufunc_xlog1py_types[8] = NPY_CFLOAT +ufunc_xlog1py_types[9] = NPY_CDOUBLE +ufunc_xlog1py_types[10] = NPY_CDOUBLE +ufunc_xlog1py_types[11] = NPY_CDOUBLE +ufunc_xlog1py_ptr[2*0] = _func_xlog1py[double] +ufunc_xlog1py_ptr[2*0+1] = ("xlog1py") +ufunc_xlog1py_ptr[2*1] = _func_xlog1py[double] +ufunc_xlog1py_ptr[2*1+1] = ("xlog1py") +ufunc_xlog1py_ptr[2*2] = _func_xlog1py[double_complex] +ufunc_xlog1py_ptr[2*2+1] = ("xlog1py") +ufunc_xlog1py_ptr[2*3] = _func_xlog1py[double_complex] +ufunc_xlog1py_ptr[2*3+1] = ("xlog1py") +ufunc_xlog1py_data[0] = &ufunc_xlog1py_ptr[2*0] +ufunc_xlog1py_data[1] = &ufunc_xlog1py_ptr[2*1] +ufunc_xlog1py_data[2] = &ufunc_xlog1py_ptr[2*2] +ufunc_xlog1py_data[3] = &ufunc_xlog1py_ptr[2*3] +xlog1py = np.PyUFunc_FromFuncAndData(ufunc_xlog1py_loops, ufunc_xlog1py_data, ufunc_xlog1py_types, 4, 2, 1, 0, "xlog1py", ufunc_xlog1py_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_xlogy_loops[4] +cdef void *ufunc_xlogy_ptr[8] +cdef void *ufunc_xlogy_data[4] +cdef char ufunc_xlogy_types[12] +cdef char *ufunc_xlogy_doc = ( + "xlogy(x, y, out=None)\n" + "\n" + "Compute ``x*log(y)`` so that the result is 0 if ``x = 0``.\n" + "\n" + "Parameters\n" + "----------\n" + "x : array_like\n" + " Multiplier\n" + "y : array_like\n" + " Argument\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "z : scalar or ndarray\n" + " Computed x*log(y)\n" + "\n" + "Notes\n" + "-----\n" + "The log function used in the computation is the natural log.\n" + "\n" + ".. versionadded:: 0.13.0\n" + "\n" + "Examples\n" + "--------\n" + "We can use this function to calculate the binary logistic loss also\n" + "known as the binary cross entropy. This loss function is used for\n" + "binary classification problems and is defined as:\n" + "\n" + ".. math::\n" + " L = 1/n * \\sum_{i=0}^n -(y_i*log(y\\_pred_i) + (1-y_i)*log(1-y\\_pred_i))\n" + "\n" + "We can define the parameters `x` and `y` as y and y_pred respectively.\n" + "y is the array of the actual labels which over here can be either 0 or 1.\n" + "y_pred is the array of the predicted probabilities with respect to\n" + "the positive class (1).\n" + "\n" + ">>> import numpy as np\n" + ">>> from scipy.special import xlogy\n" + ">>> y = np.array([0, 1, 0, 1, 1, 0])\n" + ">>> y_pred = np.array([0.3, 0.8, 0.4, 0.7, 0.9, 0.2])\n" + ">>> n = len(y)\n" + ">>> loss = -(xlogy(y, y_pred) + xlogy(1 - y, 1 - y_pred)).sum()\n" + ">>> loss /= n\n" + ">>> loss\n" + "0.29597052165495025\n" + "\n" + "A lower loss is usually better as it indicates that the predictions are\n" + "similar to the actual labels. In this example since our predicted\n" + "probabilities are close to the actual labels, we get an overall loss\n" + "that is reasonably low and appropriate.") +ufunc_xlogy_loops[0] = loop_d_dd__As_ff_f +ufunc_xlogy_loops[1] = loop_d_dd__As_dd_d +ufunc_xlogy_loops[2] = loop_D_DD__As_FF_F +ufunc_xlogy_loops[3] = loop_D_DD__As_DD_D +ufunc_xlogy_types[0] = NPY_FLOAT +ufunc_xlogy_types[1] = NPY_FLOAT +ufunc_xlogy_types[2] = NPY_FLOAT +ufunc_xlogy_types[3] = NPY_DOUBLE +ufunc_xlogy_types[4] = NPY_DOUBLE +ufunc_xlogy_types[5] = NPY_DOUBLE +ufunc_xlogy_types[6] = NPY_CFLOAT +ufunc_xlogy_types[7] = NPY_CFLOAT +ufunc_xlogy_types[8] = NPY_CFLOAT +ufunc_xlogy_types[9] = NPY_CDOUBLE +ufunc_xlogy_types[10] = NPY_CDOUBLE +ufunc_xlogy_types[11] = NPY_CDOUBLE +ufunc_xlogy_ptr[2*0] = _func_xlogy[double] +ufunc_xlogy_ptr[2*0+1] = ("xlogy") +ufunc_xlogy_ptr[2*1] = _func_xlogy[double] +ufunc_xlogy_ptr[2*1+1] = ("xlogy") +ufunc_xlogy_ptr[2*2] = _func_xlogy[double_complex] +ufunc_xlogy_ptr[2*2+1] = ("xlogy") +ufunc_xlogy_ptr[2*3] = _func_xlogy[double_complex] +ufunc_xlogy_ptr[2*3+1] = ("xlogy") +ufunc_xlogy_data[0] = &ufunc_xlogy_ptr[2*0] +ufunc_xlogy_data[1] = &ufunc_xlogy_ptr[2*1] +ufunc_xlogy_data[2] = &ufunc_xlogy_ptr[2*2] +ufunc_xlogy_data[3] = &ufunc_xlogy_ptr[2*3] +xlogy = np.PyUFunc_FromFuncAndData(ufunc_xlogy_loops, ufunc_xlogy_data, ufunc_xlogy_types, 4, 2, 1, 0, "xlogy", ufunc_xlogy_doc, 0) + +cdef np.PyUFuncGenericFunction ufunc_yn_loops[3] +cdef void *ufunc_yn_ptr[6] +cdef void *ufunc_yn_data[3] +cdef char ufunc_yn_types[9] +cdef char *ufunc_yn_doc = ( + "yn(n, x, out=None)\n" + "\n" + "Bessel function of the second kind of integer order and real argument.\n" + "\n" + "Parameters\n" + "----------\n" + "n : array_like\n" + " Order (integer).\n" + "x : array_like\n" + " Argument (float).\n" + "out : ndarray, optional\n" + " Optional output array for the function results\n" + "\n" + "Returns\n" + "-------\n" + "Y : scalar or ndarray\n" + " Value of the Bessel function, :math:`Y_n(x)`.\n" + "\n" + "See Also\n" + "--------\n" + "yv : For real order and real or complex argument.\n" + "y0: faster implementation of this function for order 0\n" + "y1: faster implementation of this function for order 1\n" + "\n" + "Notes\n" + "-----\n" + "Wrapper for the Cephes [1]_ routine `yn`.\n" + "\n" + "The function is evaluated by forward recurrence on `n`, starting with\n" + "values computed by the Cephes routines `y0` and `y1`. If ``n = 0`` or 1,\n" + "the routine for `y0` or `y1` is called directly.\n" + "\n" + "References\n" + "----------\n" + ".. [1] Cephes Mathematical Functions Library,\n" + " http://www.netlib.org/cephes/\n" + "\n" + "Examples\n" + "--------\n" + "Evaluate the function of order 0 at one point.\n" + "\n" + ">>> from scipy.special import yn\n" + ">>> yn(0, 1.)\n" + "0.08825696421567697\n" + "\n" + "Evaluate the function at one point for different orders.\n" + "\n" + ">>> yn(0, 1.), yn(1, 1.), yn(2, 1.)\n" + "(0.08825696421567697, -0.7812128213002888, -1.6506826068162546)\n" + "\n" + "The evaluation for different orders can be carried out in one call by\n" + "providing a list or NumPy array as argument for the `v` parameter:\n" + "\n" + ">>> yn([0, 1, 2], 1.)\n" + "array([ 0.08825696, -0.78121282, -1.65068261])\n" + "\n" + "Evaluate the function at several points for order 0 by providing an\n" + "array for `z`.\n" + "\n" + ">>> import numpy as np\n" + ">>> points = np.array([0.5, 3., 8.])\n" + ">>> yn(0, points)\n" + "array([-0.44451873, 0.37685001, 0.22352149])\n" + "\n" + "If `z` is an array, the order parameter `v` must be broadcastable to\n" + "the correct shape if different orders shall be computed in one call.\n" + "To calculate the orders 0 and 1 for an 1D array:\n" + "\n" + ">>> orders = np.array([[0], [1]])\n" + ">>> orders.shape\n" + "(2, 1)\n" + "\n" + ">>> yn(orders, points)\n" + "array([[-0.44451873, 0.37685001, 0.22352149],\n" + " [-1.47147239, 0.32467442, -0.15806046]])\n" + "\n" + "Plot the functions of order 0 to 3 from 0 to 10.\n" + "\n" + ">>> import matplotlib.pyplot as plt\n" + ">>> fig, ax = plt.subplots()\n" + ">>> x = np.linspace(0., 10., 1000)\n" + ">>> for i in range(4):\n" + "... ax.plot(x, yn(i, x), label=f'$Y_{i!r}$')\n" + ">>> ax.set_ylim(-3, 1)\n" + ">>> ax.legend()\n" + ">>> plt.show()") +ufunc_yn_loops[0] = loop_d_pd__As_pd_d +ufunc_yn_loops[1] = loop_d_dd__As_ff_f +ufunc_yn_loops[2] = loop_d_dd__As_dd_d +ufunc_yn_types[0] = NPY_INTP +ufunc_yn_types[1] = NPY_DOUBLE +ufunc_yn_types[2] = NPY_DOUBLE +ufunc_yn_types[3] = NPY_FLOAT +ufunc_yn_types[4] = NPY_FLOAT +ufunc_yn_types[5] = NPY_FLOAT +ufunc_yn_types[6] = NPY_DOUBLE +ufunc_yn_types[7] = NPY_DOUBLE +ufunc_yn_types[8] = NPY_DOUBLE +ufunc_yn_ptr[2*0] = _func_cephes_yn_wrap +ufunc_yn_ptr[2*0+1] = ("yn") +ufunc_yn_ptr[2*1] = _func_yn_unsafe +ufunc_yn_ptr[2*1+1] = ("yn") +ufunc_yn_ptr[2*2] = _func_yn_unsafe +ufunc_yn_ptr[2*2+1] = ("yn") +ufunc_yn_data[0] = &ufunc_yn_ptr[2*0] +ufunc_yn_data[1] = &ufunc_yn_ptr[2*1] +ufunc_yn_data[2] = &ufunc_yn_ptr[2*2] +yn = np.PyUFunc_FromFuncAndData(ufunc_yn_loops, ufunc_yn_data, ufunc_yn_types, 3, 2, 1, 0, "yn", ufunc_yn_doc, 0) + +from ._special_ufuncs import (_cospi, _lambertw, _scaled_exp1, _sinpi, _spherical_jn, _spherical_jn_d, _spherical_yn, _spherical_yn_d, _spherical_in, _spherical_in_d, _spherical_kn, _spherical_kn_d, airy, airye, bei, beip, ber, berp, binom, exp1, expi, expit, exprel, gamma, gammaln, hankel1, hankel1e, hankel2, hankel2e, hyp2f1, it2i0k0, it2j0y0, it2struve0, itairy, iti0k0, itj0y0, itmodstruve0, itstruve0, iv, _iv_ratio, _iv_ratio_c, ive, jv, jve, kei, keip, kelvin, ker, kerp, kv, kve, log_expit, log_wright_bessel, loggamma, logit, mathieu_a, mathieu_b, mathieu_cem, mathieu_modcem1, mathieu_modcem2, mathieu_modsem1, mathieu_modsem2, mathieu_sem, modfresnelm, modfresnelp, obl_ang1, obl_ang1_cv, obl_cv, obl_rad1, obl_rad1_cv, obl_rad2, obl_rad2_cv, pbdv, pbvv, pbwa, pro_ang1, pro_ang1_cv, pro_cv, pro_rad1, pro_rad1_cv, pro_rad2, pro_rad2_cv, psi, rgamma, sph_harm, wright_bessel, yv, yve, zetac, _zeta, sindg, cosdg, tandg, cotdg, i0, i0e, i1, i1e, k0, k0e, k1, k1e, y0, y1, j0, j1, struve, modstruve, beta, betaln, besselpoly, gammaln, gammasgn, cbrt, radian, cosm1, gammainc, gammaincinv, gammaincc, gammainccinv, fresnel, ellipe, ellipeinc, ellipk, ellipkinc, ellipkm1, ellipj, _riemann_zeta) + +# +# Aliases +# +jn = jv diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_cxx.pxd b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_cxx.pxd new file mode 100644 index 0000000000000000000000000000000000000000..a2fffff86812d1a7ca547a71a19078b6d5f59716 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_cxx.pxd @@ -0,0 +1,155 @@ +from . cimport sf_error +cdef void _set_action(sf_error.sf_error_t, sf_error.sf_action_t) noexcept nogil +cdef void *_export_beta_pdf_float +cdef void *_export_beta_pdf_double +cdef void *_export_beta_ppf_float +cdef void *_export_beta_ppf_double +cdef void *_export_binom_cdf_float +cdef void *_export_binom_cdf_double +cdef void *_export_binom_isf_float +cdef void *_export_binom_isf_double +cdef void *_export_binom_pmf_float +cdef void *_export_binom_pmf_double +cdef void *_export_binom_ppf_float +cdef void *_export_binom_ppf_double +cdef void *_export_binom_sf_float +cdef void *_export_binom_sf_double +cdef void *_export_cauchy_isf_float +cdef void *_export_cauchy_isf_double +cdef void *_export_cauchy_ppf_float +cdef void *_export_cauchy_ppf_double +cdef void *_export_hypergeom_cdf_float +cdef void *_export_hypergeom_cdf_double +cdef void *_export_hypergeom_mean_float +cdef void *_export_hypergeom_mean_double +cdef void *_export_hypergeom_pmf_float +cdef void *_export_hypergeom_pmf_double +cdef void *_export_hypergeom_sf_float +cdef void *_export_hypergeom_sf_double +cdef void *_export_hypergeom_skewness_float +cdef void *_export_hypergeom_skewness_double +cdef void *_export_hypergeom_variance_float +cdef void *_export_hypergeom_variance_double +cdef void *_export_invgauss_isf_float +cdef void *_export_invgauss_isf_double +cdef void *_export_invgauss_ppf_float +cdef void *_export_invgauss_ppf_double +cdef void *_export_landau_cdf_float +cdef void *_export_landau_cdf_double +cdef void *_export_landau_isf_float +cdef void *_export_landau_isf_double +cdef void *_export_landau_pdf_float +cdef void *_export_landau_pdf_double +cdef void *_export_landau_ppf_float +cdef void *_export_landau_ppf_double +cdef void *_export_landau_sf_float +cdef void *_export_landau_sf_double +cdef void *_export_nbinom_cdf_float +cdef void *_export_nbinom_cdf_double +cdef void *_export_nbinom_isf_float +cdef void *_export_nbinom_isf_double +cdef void *_export_nbinom_kurtosis_excess_float +cdef void *_export_nbinom_kurtosis_excess_double +cdef void *_export_nbinom_mean_float +cdef void *_export_nbinom_mean_double +cdef void *_export_nbinom_pmf_float +cdef void *_export_nbinom_pmf_double +cdef void *_export_nbinom_ppf_float +cdef void *_export_nbinom_ppf_double +cdef void *_export_nbinom_sf_float +cdef void *_export_nbinom_sf_double +cdef void *_export_nbinom_skewness_float +cdef void *_export_nbinom_skewness_double +cdef void *_export_nbinom_variance_float +cdef void *_export_nbinom_variance_double +cdef void *_export_ncf_isf_float +cdef void *_export_ncf_isf_double +cdef void *_export_ncf_kurtosis_excess_float +cdef void *_export_ncf_kurtosis_excess_double +cdef void *_export_ncf_mean_float +cdef void *_export_ncf_mean_double +cdef void *_export_ncf_pdf_float +cdef void *_export_ncf_pdf_double +cdef void *_export_ncf_sf_float +cdef void *_export_ncf_sf_double +cdef void *_export_ncf_skewness_float +cdef void *_export_ncf_skewness_double +cdef void *_export_ncf_variance_float +cdef void *_export_ncf_variance_double +cdef void *_export_nct_isf_float +cdef void *_export_nct_isf_double +cdef void *_export_nct_kurtosis_excess_float +cdef void *_export_nct_kurtosis_excess_double +cdef void *_export_nct_mean_float +cdef void *_export_nct_mean_double +cdef void *_export_nct_pdf_float +cdef void *_export_nct_pdf_double +cdef void *_export_nct_ppf_float +cdef void *_export_nct_ppf_double +cdef void *_export_nct_sf_float +cdef void *_export_nct_sf_double +cdef void *_export_nct_skewness_float +cdef void *_export_nct_skewness_double +cdef void *_export_nct_variance_float +cdef void *_export_nct_variance_double +cdef void *_export_ncx2_cdf_float +cdef void *_export_ncx2_cdf_double +cdef void *_export_ncx2_isf_float +cdef void *_export_ncx2_isf_double +cdef void *_export_ncx2_pdf_float +cdef void *_export_ncx2_pdf_double +cdef void *_export_ncx2_ppf_float +cdef void *_export_ncx2_ppf_double +cdef void *_export_ncx2_sf_float +cdef void *_export_ncx2_sf_double +cdef void *_export_skewnorm_cdf_float +cdef void *_export_skewnorm_cdf_double +cdef void *_export_skewnorm_isf_float +cdef void *_export_skewnorm_isf_double +cdef void *_export_skewnorm_ppf_float +cdef void *_export_skewnorm_ppf_double +cdef void *_export__stirling2_inexact +cdef void *_export_ibeta_float +cdef void *_export_ibeta_double +cdef void *_export_ibetac_float +cdef void *_export_ibetac_double +cdef void *_export_ibetac_inv_float +cdef void *_export_ibetac_inv_double +cdef void *_export_ibeta_inv_float +cdef void *_export_ibeta_inv_double +cdef void *_export_faddeeva_dawsn +cdef void *_export_faddeeva_dawsn_complex +cdef void *_export_fellint_RC +cdef void *_export_cellint_RC +cdef void *_export_fellint_RD +cdef void *_export_cellint_RD +cdef void *_export_fellint_RF +cdef void *_export_cellint_RF +cdef void *_export_fellint_RG +cdef void *_export_cellint_RG +cdef void *_export_fellint_RJ +cdef void *_export_cellint_RJ +cdef void *_export_faddeeva_erf +cdef void *_export_faddeeva_erfc_complex +cdef void *_export_faddeeva_erfcx +cdef void *_export_faddeeva_erfcx_complex +cdef void *_export_faddeeva_erfi +cdef void *_export_faddeeva_erfi_complex +cdef void *_export_erfinv_float +cdef void *_export_erfinv_double +cdef void *_export_hyp1f1_double +cdef void *_export_faddeeva_log_ndtr +cdef void *_export_faddeeva_log_ndtr_complex +cdef void *_export_ncf_cdf_float +cdef void *_export_ncf_cdf_double +cdef void *_export_ncf_ppf_float +cdef void *_export_ncf_ppf_double +cdef void *_export_nct_cdf_float +cdef void *_export_nct_cdf_double +cdef void *_export_faddeeva_ndtr +cdef void *_export_powm1_float +cdef void *_export_powm1_double +cdef void *_export_faddeeva_voigt_profile +cdef void *_export_faddeeva_w +cdef void *_export_wrightomega +cdef void *_export_wrightomega_real \ No newline at end of file diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_cxx.pyx b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_cxx.pyx new file mode 100644 index 0000000000000000000000000000000000000000..19cbd36c4707bb593e7e5638b9a158df504d3001 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_cxx.pyx @@ -0,0 +1,466 @@ +# This file is automatically generated by _generate_pyx.py. +# Do not edit manually! + +from libc.math cimport NAN + +include "_ufuncs_extra_code_common.pxi" + +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_beta_pdf_float "beta_pdf_float"(float, float, float) noexcept nogil +cdef void *_export_beta_pdf_float = _func_beta_pdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_beta_pdf_double "beta_pdf_double"(double, double, double) noexcept nogil +cdef void *_export_beta_pdf_double = _func_beta_pdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_beta_ppf_float "beta_ppf_float"(float, float, float) noexcept nogil +cdef void *_export_beta_ppf_float = _func_beta_ppf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_beta_ppf_double "beta_ppf_double"(double, double, double) noexcept nogil +cdef void *_export_beta_ppf_double = _func_beta_ppf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_binom_cdf_float "binom_cdf_float"(float, float, float) noexcept nogil +cdef void *_export_binom_cdf_float = _func_binom_cdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_binom_cdf_double "binom_cdf_double"(double, double, double) noexcept nogil +cdef void *_export_binom_cdf_double = _func_binom_cdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_binom_isf_float "binom_isf_float"(float, float, float) noexcept nogil +cdef void *_export_binom_isf_float = _func_binom_isf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_binom_isf_double "binom_isf_double"(double, double, double) noexcept nogil +cdef void *_export_binom_isf_double = _func_binom_isf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_binom_pmf_float "binom_pmf_float"(float, float, float) noexcept nogil +cdef void *_export_binom_pmf_float = _func_binom_pmf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_binom_pmf_double "binom_pmf_double"(double, double, double) noexcept nogil +cdef void *_export_binom_pmf_double = _func_binom_pmf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_binom_ppf_float "binom_ppf_float"(float, float, float) noexcept nogil +cdef void *_export_binom_ppf_float = _func_binom_ppf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_binom_ppf_double "binom_ppf_double"(double, double, double) noexcept nogil +cdef void *_export_binom_ppf_double = _func_binom_ppf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_binom_sf_float "binom_sf_float"(float, float, float) noexcept nogil +cdef void *_export_binom_sf_float = _func_binom_sf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_binom_sf_double "binom_sf_double"(double, double, double) noexcept nogil +cdef void *_export_binom_sf_double = _func_binom_sf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_cauchy_isf_float "cauchy_isf_float"(float, float, float) noexcept nogil +cdef void *_export_cauchy_isf_float = _func_cauchy_isf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_cauchy_isf_double "cauchy_isf_double"(double, double, double) noexcept nogil +cdef void *_export_cauchy_isf_double = _func_cauchy_isf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_cauchy_ppf_float "cauchy_ppf_float"(float, float, float) noexcept nogil +cdef void *_export_cauchy_ppf_float = _func_cauchy_ppf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_cauchy_ppf_double "cauchy_ppf_double"(double, double, double) noexcept nogil +cdef void *_export_cauchy_ppf_double = _func_cauchy_ppf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_hypergeom_cdf_float "hypergeom_cdf_float"(float, float, float, float) noexcept nogil +cdef void *_export_hypergeom_cdf_float = _func_hypergeom_cdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_hypergeom_cdf_double "hypergeom_cdf_double"(double, double, double, double) noexcept nogil +cdef void *_export_hypergeom_cdf_double = _func_hypergeom_cdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_hypergeom_mean_float "hypergeom_mean_float"(float, float, float) noexcept nogil +cdef void *_export_hypergeom_mean_float = _func_hypergeom_mean_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_hypergeom_mean_double "hypergeom_mean_double"(double, double, double) noexcept nogil +cdef void *_export_hypergeom_mean_double = _func_hypergeom_mean_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_hypergeom_pmf_float "hypergeom_pmf_float"(float, float, float, float) noexcept nogil +cdef void *_export_hypergeom_pmf_float = _func_hypergeom_pmf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_hypergeom_pmf_double "hypergeom_pmf_double"(double, double, double, double) noexcept nogil +cdef void *_export_hypergeom_pmf_double = _func_hypergeom_pmf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_hypergeom_sf_float "hypergeom_sf_float"(float, float, float, float) noexcept nogil +cdef void *_export_hypergeom_sf_float = _func_hypergeom_sf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_hypergeom_sf_double "hypergeom_sf_double"(double, double, double, double) noexcept nogil +cdef void *_export_hypergeom_sf_double = _func_hypergeom_sf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_hypergeom_skewness_float "hypergeom_skewness_float"(float, float, float) noexcept nogil +cdef void *_export_hypergeom_skewness_float = _func_hypergeom_skewness_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_hypergeom_skewness_double "hypergeom_skewness_double"(double, double, double) noexcept nogil +cdef void *_export_hypergeom_skewness_double = _func_hypergeom_skewness_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_hypergeom_variance_float "hypergeom_variance_float"(float, float, float) noexcept nogil +cdef void *_export_hypergeom_variance_float = _func_hypergeom_variance_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_hypergeom_variance_double "hypergeom_variance_double"(double, double, double) noexcept nogil +cdef void *_export_hypergeom_variance_double = _func_hypergeom_variance_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_invgauss_isf_float "invgauss_isf_float"(float, float, float) noexcept nogil +cdef void *_export_invgauss_isf_float = _func_invgauss_isf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_invgauss_isf_double "invgauss_isf_double"(double, double, double) noexcept nogil +cdef void *_export_invgauss_isf_double = _func_invgauss_isf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_invgauss_ppf_float "invgauss_ppf_float"(float, float, float) noexcept nogil +cdef void *_export_invgauss_ppf_float = _func_invgauss_ppf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_invgauss_ppf_double "invgauss_ppf_double"(double, double, double) noexcept nogil +cdef void *_export_invgauss_ppf_double = _func_invgauss_ppf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_landau_cdf_float "landau_cdf_float"(float, float, float) noexcept nogil +cdef void *_export_landau_cdf_float = _func_landau_cdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_landau_cdf_double "landau_cdf_double"(double, double, double) noexcept nogil +cdef void *_export_landau_cdf_double = _func_landau_cdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_landau_isf_float "landau_isf_float"(float, float, float) noexcept nogil +cdef void *_export_landau_isf_float = _func_landau_isf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_landau_isf_double "landau_isf_double"(double, double, double) noexcept nogil +cdef void *_export_landau_isf_double = _func_landau_isf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_landau_pdf_float "landau_pdf_float"(float, float, float) noexcept nogil +cdef void *_export_landau_pdf_float = _func_landau_pdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_landau_pdf_double "landau_pdf_double"(double, double, double) noexcept nogil +cdef void *_export_landau_pdf_double = _func_landau_pdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_landau_ppf_float "landau_ppf_float"(float, float, float) noexcept nogil +cdef void *_export_landau_ppf_float = _func_landau_ppf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_landau_ppf_double "landau_ppf_double"(double, double, double) noexcept nogil +cdef void *_export_landau_ppf_double = _func_landau_ppf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_landau_sf_float "landau_sf_float"(float, float, float) noexcept nogil +cdef void *_export_landau_sf_float = _func_landau_sf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_landau_sf_double "landau_sf_double"(double, double, double) noexcept nogil +cdef void *_export_landau_sf_double = _func_landau_sf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nbinom_cdf_float "nbinom_cdf_float"(float, float, float) noexcept nogil +cdef void *_export_nbinom_cdf_float = _func_nbinom_cdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nbinom_cdf_double "nbinom_cdf_double"(double, double, double) noexcept nogil +cdef void *_export_nbinom_cdf_double = _func_nbinom_cdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nbinom_isf_float "nbinom_isf_float"(float, float, float) noexcept nogil +cdef void *_export_nbinom_isf_float = _func_nbinom_isf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nbinom_isf_double "nbinom_isf_double"(double, double, double) noexcept nogil +cdef void *_export_nbinom_isf_double = _func_nbinom_isf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nbinom_kurtosis_excess_float "nbinom_kurtosis_excess_float"(float, float) noexcept nogil +cdef void *_export_nbinom_kurtosis_excess_float = _func_nbinom_kurtosis_excess_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nbinom_kurtosis_excess_double "nbinom_kurtosis_excess_double"(double, double) noexcept nogil +cdef void *_export_nbinom_kurtosis_excess_double = _func_nbinom_kurtosis_excess_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nbinom_mean_float "nbinom_mean_float"(float, float) noexcept nogil +cdef void *_export_nbinom_mean_float = _func_nbinom_mean_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nbinom_mean_double "nbinom_mean_double"(double, double) noexcept nogil +cdef void *_export_nbinom_mean_double = _func_nbinom_mean_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nbinom_pmf_float "nbinom_pmf_float"(float, float, float) noexcept nogil +cdef void *_export_nbinom_pmf_float = _func_nbinom_pmf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nbinom_pmf_double "nbinom_pmf_double"(double, double, double) noexcept nogil +cdef void *_export_nbinom_pmf_double = _func_nbinom_pmf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nbinom_ppf_float "nbinom_ppf_float"(float, float, float) noexcept nogil +cdef void *_export_nbinom_ppf_float = _func_nbinom_ppf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nbinom_ppf_double "nbinom_ppf_double"(double, double, double) noexcept nogil +cdef void *_export_nbinom_ppf_double = _func_nbinom_ppf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nbinom_sf_float "nbinom_sf_float"(float, float, float) noexcept nogil +cdef void *_export_nbinom_sf_float = _func_nbinom_sf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nbinom_sf_double "nbinom_sf_double"(double, double, double) noexcept nogil +cdef void *_export_nbinom_sf_double = _func_nbinom_sf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nbinom_skewness_float "nbinom_skewness_float"(float, float) noexcept nogil +cdef void *_export_nbinom_skewness_float = _func_nbinom_skewness_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nbinom_skewness_double "nbinom_skewness_double"(double, double) noexcept nogil +cdef void *_export_nbinom_skewness_double = _func_nbinom_skewness_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nbinom_variance_float "nbinom_variance_float"(float, float) noexcept nogil +cdef void *_export_nbinom_variance_float = _func_nbinom_variance_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nbinom_variance_double "nbinom_variance_double"(double, double) noexcept nogil +cdef void *_export_nbinom_variance_double = _func_nbinom_variance_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncf_isf_float "ncf_isf_float"(float, float, float, float) noexcept nogil +cdef void *_export_ncf_isf_float = _func_ncf_isf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncf_isf_double "ncf_isf_double"(double, double, double, double) noexcept nogil +cdef void *_export_ncf_isf_double = _func_ncf_isf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncf_kurtosis_excess_float "ncf_kurtosis_excess_float"(float, float, float) noexcept nogil +cdef void *_export_ncf_kurtosis_excess_float = _func_ncf_kurtosis_excess_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncf_kurtosis_excess_double "ncf_kurtosis_excess_double"(double, double, double) noexcept nogil +cdef void *_export_ncf_kurtosis_excess_double = _func_ncf_kurtosis_excess_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncf_mean_float "ncf_mean_float"(float, float, float) noexcept nogil +cdef void *_export_ncf_mean_float = _func_ncf_mean_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncf_mean_double "ncf_mean_double"(double, double, double) noexcept nogil +cdef void *_export_ncf_mean_double = _func_ncf_mean_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncf_pdf_float "ncf_pdf_float"(float, float, float, float) noexcept nogil +cdef void *_export_ncf_pdf_float = _func_ncf_pdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncf_pdf_double "ncf_pdf_double"(double, double, double, double) noexcept nogil +cdef void *_export_ncf_pdf_double = _func_ncf_pdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncf_sf_float "ncf_sf_float"(float, float, float, float) noexcept nogil +cdef void *_export_ncf_sf_float = _func_ncf_sf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncf_sf_double "ncf_sf_double"(double, double, double, double) noexcept nogil +cdef void *_export_ncf_sf_double = _func_ncf_sf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncf_skewness_float "ncf_skewness_float"(float, float, float) noexcept nogil +cdef void *_export_ncf_skewness_float = _func_ncf_skewness_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncf_skewness_double "ncf_skewness_double"(double, double, double) noexcept nogil +cdef void *_export_ncf_skewness_double = _func_ncf_skewness_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncf_variance_float "ncf_variance_float"(float, float, float) noexcept nogil +cdef void *_export_ncf_variance_float = _func_ncf_variance_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncf_variance_double "ncf_variance_double"(double, double, double) noexcept nogil +cdef void *_export_ncf_variance_double = _func_ncf_variance_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nct_isf_float "nct_isf_float"(float, float, float) noexcept nogil +cdef void *_export_nct_isf_float = _func_nct_isf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nct_isf_double "nct_isf_double"(double, double, double) noexcept nogil +cdef void *_export_nct_isf_double = _func_nct_isf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nct_kurtosis_excess_float "nct_kurtosis_excess_float"(float, float) noexcept nogil +cdef void *_export_nct_kurtosis_excess_float = _func_nct_kurtosis_excess_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nct_kurtosis_excess_double "nct_kurtosis_excess_double"(double, double) noexcept nogil +cdef void *_export_nct_kurtosis_excess_double = _func_nct_kurtosis_excess_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nct_mean_float "nct_mean_float"(float, float) noexcept nogil +cdef void *_export_nct_mean_float = _func_nct_mean_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nct_mean_double "nct_mean_double"(double, double) noexcept nogil +cdef void *_export_nct_mean_double = _func_nct_mean_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nct_pdf_float "nct_pdf_float"(float, float, float) noexcept nogil +cdef void *_export_nct_pdf_float = _func_nct_pdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nct_pdf_double "nct_pdf_double"(double, double, double) noexcept nogil +cdef void *_export_nct_pdf_double = _func_nct_pdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nct_ppf_float "nct_ppf_float"(float, float, float) noexcept nogil +cdef void *_export_nct_ppf_float = _func_nct_ppf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nct_ppf_double "nct_ppf_double"(double, double, double) noexcept nogil +cdef void *_export_nct_ppf_double = _func_nct_ppf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nct_sf_float "nct_sf_float"(float, float, float) noexcept nogil +cdef void *_export_nct_sf_float = _func_nct_sf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nct_sf_double "nct_sf_double"(double, double, double) noexcept nogil +cdef void *_export_nct_sf_double = _func_nct_sf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nct_skewness_float "nct_skewness_float"(float, float) noexcept nogil +cdef void *_export_nct_skewness_float = _func_nct_skewness_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nct_skewness_double "nct_skewness_double"(double, double) noexcept nogil +cdef void *_export_nct_skewness_double = _func_nct_skewness_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nct_variance_float "nct_variance_float"(float, float) noexcept nogil +cdef void *_export_nct_variance_float = _func_nct_variance_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nct_variance_double "nct_variance_double"(double, double) noexcept nogil +cdef void *_export_nct_variance_double = _func_nct_variance_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncx2_cdf_float "ncx2_cdf_float"(float, float, float) noexcept nogil +cdef void *_export_ncx2_cdf_float = _func_ncx2_cdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncx2_cdf_double "ncx2_cdf_double"(double, double, double) noexcept nogil +cdef void *_export_ncx2_cdf_double = _func_ncx2_cdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncx2_isf_float "ncx2_isf_float"(float, float, float) noexcept nogil +cdef void *_export_ncx2_isf_float = _func_ncx2_isf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncx2_isf_double "ncx2_isf_double"(double, double, double) noexcept nogil +cdef void *_export_ncx2_isf_double = _func_ncx2_isf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncx2_pdf_float "ncx2_pdf_float"(float, float, float) noexcept nogil +cdef void *_export_ncx2_pdf_float = _func_ncx2_pdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncx2_pdf_double "ncx2_pdf_double"(double, double, double) noexcept nogil +cdef void *_export_ncx2_pdf_double = _func_ncx2_pdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncx2_ppf_float "ncx2_ppf_float"(float, float, float) noexcept nogil +cdef void *_export_ncx2_ppf_float = _func_ncx2_ppf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncx2_ppf_double "ncx2_ppf_double"(double, double, double) noexcept nogil +cdef void *_export_ncx2_ppf_double = _func_ncx2_ppf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncx2_sf_float "ncx2_sf_float"(float, float, float) noexcept nogil +cdef void *_export_ncx2_sf_float = _func_ncx2_sf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncx2_sf_double "ncx2_sf_double"(double, double, double) noexcept nogil +cdef void *_export_ncx2_sf_double = _func_ncx2_sf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_skewnorm_cdf_float "skewnorm_cdf_float"(float, float, float, float) noexcept nogil +cdef void *_export_skewnorm_cdf_float = _func_skewnorm_cdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_skewnorm_cdf_double "skewnorm_cdf_double"(double, double, double, double) noexcept nogil +cdef void *_export_skewnorm_cdf_double = _func_skewnorm_cdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_skewnorm_isf_float "skewnorm_isf_float"(float, float, float, float) noexcept nogil +cdef void *_export_skewnorm_isf_float = _func_skewnorm_isf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_skewnorm_isf_double "skewnorm_isf_double"(double, double, double, double) noexcept nogil +cdef void *_export_skewnorm_isf_double = _func_skewnorm_isf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_skewnorm_ppf_float "skewnorm_ppf_float"(float, float, float, float) noexcept nogil +cdef void *_export_skewnorm_ppf_float = _func_skewnorm_ppf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_skewnorm_ppf_double "skewnorm_ppf_double"(double, double, double, double) noexcept nogil +cdef void *_export_skewnorm_ppf_double = _func_skewnorm_ppf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func__stirling2_inexact "_stirling2_inexact"(double, double) noexcept nogil +cdef void *_export__stirling2_inexact = _func__stirling2_inexact +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ibeta_float "ibeta_float"(float, float, float) noexcept nogil +cdef void *_export_ibeta_float = _func_ibeta_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ibeta_double "ibeta_double"(double, double, double) noexcept nogil +cdef void *_export_ibeta_double = _func_ibeta_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ibetac_float "ibetac_float"(float, float, float) noexcept nogil +cdef void *_export_ibetac_float = _func_ibetac_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ibetac_double "ibetac_double"(double, double, double) noexcept nogil +cdef void *_export_ibetac_double = _func_ibetac_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ibetac_inv_float "ibetac_inv_float"(float, float, float) noexcept nogil +cdef void *_export_ibetac_inv_float = _func_ibetac_inv_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ibetac_inv_double "ibetac_inv_double"(double, double, double) noexcept nogil +cdef void *_export_ibetac_inv_double = _func_ibetac_inv_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ibeta_inv_float "ibeta_inv_float"(float, float, float) noexcept nogil +cdef void *_export_ibeta_inv_float = _func_ibeta_inv_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ibeta_inv_double "ibeta_inv_double"(double, double, double) noexcept nogil +cdef void *_export_ibeta_inv_double = _func_ibeta_inv_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_faddeeva_dawsn "faddeeva_dawsn"(double) noexcept nogil +cdef void *_export_faddeeva_dawsn = _func_faddeeva_dawsn +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_faddeeva_dawsn_complex "faddeeva_dawsn_complex"(double complex) noexcept nogil +cdef void *_export_faddeeva_dawsn_complex = _func_faddeeva_dawsn_complex +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_fellint_RC "fellint_RC"(double, double) noexcept nogil +cdef void *_export_fellint_RC = _func_fellint_RC +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_cellint_RC "cellint_RC"(double complex, double complex) noexcept nogil +cdef void *_export_cellint_RC = _func_cellint_RC +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_fellint_RD "fellint_RD"(double, double, double) noexcept nogil +cdef void *_export_fellint_RD = _func_fellint_RD +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_cellint_RD "cellint_RD"(double complex, double complex, double complex) noexcept nogil +cdef void *_export_cellint_RD = _func_cellint_RD +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_fellint_RF "fellint_RF"(double, double, double) noexcept nogil +cdef void *_export_fellint_RF = _func_fellint_RF +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_cellint_RF "cellint_RF"(double complex, double complex, double complex) noexcept nogil +cdef void *_export_cellint_RF = _func_cellint_RF +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_fellint_RG "fellint_RG"(double, double, double) noexcept nogil +cdef void *_export_fellint_RG = _func_fellint_RG +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_cellint_RG "cellint_RG"(double complex, double complex, double complex) noexcept nogil +cdef void *_export_cellint_RG = _func_cellint_RG +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_fellint_RJ "fellint_RJ"(double, double, double, double) noexcept nogil +cdef void *_export_fellint_RJ = _func_fellint_RJ +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_cellint_RJ "cellint_RJ"(double complex, double complex, double complex, double complex) noexcept nogil +cdef void *_export_cellint_RJ = _func_cellint_RJ +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_faddeeva_erf "faddeeva_erf"(double complex) noexcept nogil +cdef void *_export_faddeeva_erf = _func_faddeeva_erf +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_faddeeva_erfc_complex "faddeeva_erfc_complex"(double complex) noexcept nogil +cdef void *_export_faddeeva_erfc_complex = _func_faddeeva_erfc_complex +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_faddeeva_erfcx "faddeeva_erfcx"(double) noexcept nogil +cdef void *_export_faddeeva_erfcx = _func_faddeeva_erfcx +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_faddeeva_erfcx_complex "faddeeva_erfcx_complex"(double complex) noexcept nogil +cdef void *_export_faddeeva_erfcx_complex = _func_faddeeva_erfcx_complex +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_faddeeva_erfi "faddeeva_erfi"(double) noexcept nogil +cdef void *_export_faddeeva_erfi = _func_faddeeva_erfi +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_faddeeva_erfi_complex "faddeeva_erfi_complex"(double complex) noexcept nogil +cdef void *_export_faddeeva_erfi_complex = _func_faddeeva_erfi_complex +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_erfinv_float "erfinv_float"(float) noexcept nogil +cdef void *_export_erfinv_float = _func_erfinv_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_erfinv_double "erfinv_double"(double) noexcept nogil +cdef void *_export_erfinv_double = _func_erfinv_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_hyp1f1_double "hyp1f1_double"(double, double, double) noexcept nogil +cdef void *_export_hyp1f1_double = _func_hyp1f1_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_faddeeva_log_ndtr "faddeeva_log_ndtr"(double) noexcept nogil +cdef void *_export_faddeeva_log_ndtr = _func_faddeeva_log_ndtr +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_faddeeva_log_ndtr_complex "faddeeva_log_ndtr_complex"(double complex) noexcept nogil +cdef void *_export_faddeeva_log_ndtr_complex = _func_faddeeva_log_ndtr_complex +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncf_cdf_float "ncf_cdf_float"(float, float, float, float) noexcept nogil +cdef void *_export_ncf_cdf_float = _func_ncf_cdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncf_cdf_double "ncf_cdf_double"(double, double, double, double) noexcept nogil +cdef void *_export_ncf_cdf_double = _func_ncf_cdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_ncf_ppf_float "ncf_ppf_float"(float, float, float, float) noexcept nogil +cdef void *_export_ncf_ppf_float = _func_ncf_ppf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_ncf_ppf_double "ncf_ppf_double"(double, double, double, double) noexcept nogil +cdef void *_export_ncf_ppf_double = _func_ncf_ppf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_nct_cdf_float "nct_cdf_float"(float, float, float) noexcept nogil +cdef void *_export_nct_cdf_float = _func_nct_cdf_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_nct_cdf_double "nct_cdf_double"(double, double, double) noexcept nogil +cdef void *_export_nct_cdf_double = _func_nct_cdf_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_faddeeva_ndtr "faddeeva_ndtr"(double complex) noexcept nogil +cdef void *_export_faddeeva_ndtr = _func_faddeeva_ndtr +cdef extern from r"_ufuncs_cxx_defs.h": + cdef float _func_powm1_float "powm1_float"(float, float) noexcept nogil +cdef void *_export_powm1_float = _func_powm1_float +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_powm1_double "powm1_double"(double, double) noexcept nogil +cdef void *_export_powm1_double = _func_powm1_double +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_faddeeva_voigt_profile "faddeeva_voigt_profile"(double, double, double) noexcept nogil +cdef void *_export_faddeeva_voigt_profile = _func_faddeeva_voigt_profile +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_faddeeva_w "faddeeva_w"(double complex) noexcept nogil +cdef void *_export_faddeeva_w = _func_faddeeva_w +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double complex _func_wrightomega "wrightomega"(double complex) noexcept nogil +cdef void *_export_wrightomega = _func_wrightomega +cdef extern from r"_ufuncs_cxx_defs.h": + cdef double _func_wrightomega_real "wrightomega_real"(double) noexcept nogil +cdef void *_export_wrightomega_real = _func_wrightomega_real \ No newline at end of file diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_cxx_defs.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_cxx_defs.h new file mode 100644 index 0000000000000000000000000000000000000000..4e916ce565742ecd949c2c6f9c1a1891e734a96d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_cxx_defs.h @@ -0,0 +1,161 @@ +#ifndef UFUNCS_PROTO_H +#define UFUNCS_PROTO_H 1 +#include "boost_special_functions.h" +npy_float beta_pdf_float(npy_float, npy_float, npy_float); +npy_double beta_pdf_double(npy_double, npy_double, npy_double); +npy_float beta_ppf_float(npy_float, npy_float, npy_float); +npy_double beta_ppf_double(npy_double, npy_double, npy_double); +npy_float binom_cdf_float(npy_float, npy_float, npy_float); +npy_double binom_cdf_double(npy_double, npy_double, npy_double); +npy_float binom_isf_float(npy_float, npy_float, npy_float); +npy_double binom_isf_double(npy_double, npy_double, npy_double); +npy_float binom_pmf_float(npy_float, npy_float, npy_float); +npy_double binom_pmf_double(npy_double, npy_double, npy_double); +npy_float binom_ppf_float(npy_float, npy_float, npy_float); +npy_double binom_ppf_double(npy_double, npy_double, npy_double); +npy_float binom_sf_float(npy_float, npy_float, npy_float); +npy_double binom_sf_double(npy_double, npy_double, npy_double); +npy_float cauchy_isf_float(npy_float, npy_float, npy_float); +npy_double cauchy_isf_double(npy_double, npy_double, npy_double); +npy_float cauchy_ppf_float(npy_float, npy_float, npy_float); +npy_double cauchy_ppf_double(npy_double, npy_double, npy_double); +npy_float hypergeom_cdf_float(npy_float, npy_float, npy_float, npy_float); +npy_double hypergeom_cdf_double(npy_double, npy_double, npy_double, npy_double); +npy_float hypergeom_mean_float(npy_float, npy_float, npy_float); +npy_double hypergeom_mean_double(npy_double, npy_double, npy_double); +npy_float hypergeom_pmf_float(npy_float, npy_float, npy_float, npy_float); +npy_double hypergeom_pmf_double(npy_double, npy_double, npy_double, npy_double); +npy_float hypergeom_sf_float(npy_float, npy_float, npy_float, npy_float); +npy_double hypergeom_sf_double(npy_double, npy_double, npy_double, npy_double); +npy_float hypergeom_skewness_float(npy_float, npy_float, npy_float); +npy_double hypergeom_skewness_double(npy_double, npy_double, npy_double); +npy_float hypergeom_variance_float(npy_float, npy_float, npy_float); +npy_double hypergeom_variance_double(npy_double, npy_double, npy_double); +npy_float invgauss_isf_float(npy_float, npy_float, npy_float); +npy_double invgauss_isf_double(npy_double, npy_double, npy_double); +npy_float invgauss_ppf_float(npy_float, npy_float, npy_float); +npy_double invgauss_ppf_double(npy_double, npy_double, npy_double); +npy_float landau_cdf_float(npy_float, npy_float, npy_float); +npy_double landau_cdf_double(npy_double, npy_double, npy_double); +npy_float landau_isf_float(npy_float, npy_float, npy_float); +npy_double landau_isf_double(npy_double, npy_double, npy_double); +npy_float landau_pdf_float(npy_float, npy_float, npy_float); +npy_double landau_pdf_double(npy_double, npy_double, npy_double); +npy_float landau_ppf_float(npy_float, npy_float, npy_float); +npy_double landau_ppf_double(npy_double, npy_double, npy_double); +npy_float landau_sf_float(npy_float, npy_float, npy_float); +npy_double landau_sf_double(npy_double, npy_double, npy_double); +npy_float nbinom_cdf_float(npy_float, npy_float, npy_float); +npy_double nbinom_cdf_double(npy_double, npy_double, npy_double); +npy_float nbinom_isf_float(npy_float, npy_float, npy_float); +npy_double nbinom_isf_double(npy_double, npy_double, npy_double); +npy_float nbinom_kurtosis_excess_float(npy_float, npy_float); +npy_double nbinom_kurtosis_excess_double(npy_double, npy_double); +npy_float nbinom_mean_float(npy_float, npy_float); +npy_double nbinom_mean_double(npy_double, npy_double); +npy_float nbinom_pmf_float(npy_float, npy_float, npy_float); +npy_double nbinom_pmf_double(npy_double, npy_double, npy_double); +npy_float nbinom_ppf_float(npy_float, npy_float, npy_float); +npy_double nbinom_ppf_double(npy_double, npy_double, npy_double); +npy_float nbinom_sf_float(npy_float, npy_float, npy_float); +npy_double nbinom_sf_double(npy_double, npy_double, npy_double); +npy_float nbinom_skewness_float(npy_float, npy_float); +npy_double nbinom_skewness_double(npy_double, npy_double); +npy_float nbinom_variance_float(npy_float, npy_float); +npy_double nbinom_variance_double(npy_double, npy_double); +npy_float ncf_isf_float(npy_float, npy_float, npy_float, npy_float); +npy_double ncf_isf_double(npy_double, npy_double, npy_double, npy_double); +npy_float ncf_kurtosis_excess_float(npy_float, npy_float, npy_float); +npy_double ncf_kurtosis_excess_double(npy_double, npy_double, npy_double); +npy_float ncf_mean_float(npy_float, npy_float, npy_float); +npy_double ncf_mean_double(npy_double, npy_double, npy_double); +npy_float ncf_pdf_float(npy_float, npy_float, npy_float, npy_float); +npy_double ncf_pdf_double(npy_double, npy_double, npy_double, npy_double); +npy_float ncf_sf_float(npy_float, npy_float, npy_float, npy_float); +npy_double ncf_sf_double(npy_double, npy_double, npy_double, npy_double); +npy_float ncf_skewness_float(npy_float, npy_float, npy_float); +npy_double ncf_skewness_double(npy_double, npy_double, npy_double); +npy_float ncf_variance_float(npy_float, npy_float, npy_float); +npy_double ncf_variance_double(npy_double, npy_double, npy_double); +npy_float nct_isf_float(npy_float, npy_float, npy_float); +npy_double nct_isf_double(npy_double, npy_double, npy_double); +npy_float nct_kurtosis_excess_float(npy_float, npy_float); +npy_double nct_kurtosis_excess_double(npy_double, npy_double); +npy_float nct_mean_float(npy_float, npy_float); +npy_double nct_mean_double(npy_double, npy_double); +npy_float nct_pdf_float(npy_float, npy_float, npy_float); +npy_double nct_pdf_double(npy_double, npy_double, npy_double); +npy_float nct_ppf_float(npy_float, npy_float, npy_float); +npy_double nct_ppf_double(npy_double, npy_double, npy_double); +npy_float nct_sf_float(npy_float, npy_float, npy_float); +npy_double nct_sf_double(npy_double, npy_double, npy_double); +npy_float nct_skewness_float(npy_float, npy_float); +npy_double nct_skewness_double(npy_double, npy_double); +npy_float nct_variance_float(npy_float, npy_float); +npy_double nct_variance_double(npy_double, npy_double); +npy_float ncx2_cdf_float(npy_float, npy_float, npy_float); +npy_double ncx2_cdf_double(npy_double, npy_double, npy_double); +npy_float ncx2_isf_float(npy_float, npy_float, npy_float); +npy_double ncx2_isf_double(npy_double, npy_double, npy_double); +npy_float ncx2_pdf_float(npy_float, npy_float, npy_float); +npy_double ncx2_pdf_double(npy_double, npy_double, npy_double); +npy_float ncx2_ppf_float(npy_float, npy_float, npy_float); +npy_double ncx2_ppf_double(npy_double, npy_double, npy_double); +npy_float ncx2_sf_float(npy_float, npy_float, npy_float); +npy_double ncx2_sf_double(npy_double, npy_double, npy_double); +npy_float skewnorm_cdf_float(npy_float, npy_float, npy_float, npy_float); +npy_double skewnorm_cdf_double(npy_double, npy_double, npy_double, npy_double); +npy_float skewnorm_isf_float(npy_float, npy_float, npy_float, npy_float); +npy_double skewnorm_isf_double(npy_double, npy_double, npy_double, npy_double); +npy_float skewnorm_ppf_float(npy_float, npy_float, npy_float, npy_float); +npy_double skewnorm_ppf_double(npy_double, npy_double, npy_double, npy_double); +#include "stirling2.h" +npy_double _stirling2_inexact(npy_double, npy_double); +npy_float ibeta_float(npy_float, npy_float, npy_float); +npy_double ibeta_double(npy_double, npy_double, npy_double); +npy_float ibetac_float(npy_float, npy_float, npy_float); +npy_double ibetac_double(npy_double, npy_double, npy_double); +npy_float ibetac_inv_float(npy_float, npy_float, npy_float); +npy_double ibetac_inv_double(npy_double, npy_double, npy_double); +npy_float ibeta_inv_float(npy_float, npy_float, npy_float); +npy_double ibeta_inv_double(npy_double, npy_double, npy_double); +#include "_faddeeva.h" +npy_double faddeeva_dawsn(npy_double); +npy_cdouble faddeeva_dawsn_complex(npy_cdouble); +#include "ellint_carlson_wrap.hh" +npy_double fellint_RC(npy_double, npy_double); +npy_cdouble cellint_RC(npy_cdouble, npy_cdouble); +npy_double fellint_RD(npy_double, npy_double, npy_double); +npy_cdouble cellint_RD(npy_cdouble, npy_cdouble, npy_cdouble); +npy_double fellint_RF(npy_double, npy_double, npy_double); +npy_cdouble cellint_RF(npy_cdouble, npy_cdouble, npy_cdouble); +npy_double fellint_RG(npy_double, npy_double, npy_double); +npy_cdouble cellint_RG(npy_cdouble, npy_cdouble, npy_cdouble); +npy_double fellint_RJ(npy_double, npy_double, npy_double, npy_double); +npy_cdouble cellint_RJ(npy_cdouble, npy_cdouble, npy_cdouble, npy_cdouble); +npy_cdouble faddeeva_erf(npy_cdouble); +npy_cdouble faddeeva_erfc_complex(npy_cdouble); +npy_double faddeeva_erfcx(npy_double); +npy_cdouble faddeeva_erfcx_complex(npy_cdouble); +npy_double faddeeva_erfi(npy_double); +npy_cdouble faddeeva_erfi_complex(npy_cdouble); +npy_float erfinv_float(npy_float); +npy_double erfinv_double(npy_double); +npy_double hyp1f1_double(npy_double, npy_double, npy_double); +npy_double faddeeva_log_ndtr(npy_double); +npy_cdouble faddeeva_log_ndtr_complex(npy_cdouble); +npy_float ncf_cdf_float(npy_float, npy_float, npy_float, npy_float); +npy_double ncf_cdf_double(npy_double, npy_double, npy_double, npy_double); +npy_float ncf_ppf_float(npy_float, npy_float, npy_float, npy_float); +npy_double ncf_ppf_double(npy_double, npy_double, npy_double, npy_double); +npy_float nct_cdf_float(npy_float, npy_float, npy_float); +npy_double nct_cdf_double(npy_double, npy_double, npy_double); +npy_cdouble faddeeva_ndtr(npy_cdouble); +npy_float powm1_float(npy_float, npy_float); +npy_double powm1_double(npy_double, npy_double); +npy_double faddeeva_voigt_profile(npy_double, npy_double, npy_double); +npy_cdouble faddeeva_w(npy_cdouble); +#include "_wright.h" +npy_cdouble wrightomega(npy_cdouble); +npy_double wrightomega_real(npy_double); +#endif diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_defs.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_defs.h new file mode 100644 index 0000000000000000000000000000000000000000..86d2349d2b7881b71bfdec2a60a78e2bfc18f9fe --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/_ufuncs_defs.h @@ -0,0 +1,65 @@ +#ifndef UFUNCS_PROTO_H +#define UFUNCS_PROTO_H 1 +#include "_cosine.h" +npy_double cosine_cdf(npy_double); +npy_double cosine_invcdf(npy_double); +#include "xsf_wrappers.h" +npy_double cephes_igam_fac(npy_double, npy_double); +npy_double xsf_kolmogc(npy_double); +npy_double xsf_kolmogci(npy_double); +npy_double xsf_kolmogp(npy_double); +npy_double cephes_lanczos_sum_expg_scaled(npy_double); +npy_double cephes_lgam1p(npy_double); +npy_double cephes_log1pmx(npy_double); +npy_double cephes_smirnovc_wrap(npy_intp, npy_double); +npy_double cephes_smirnovci_wrap(npy_intp, npy_double); +npy_double cephes_smirnovp_wrap(npy_intp, npy_double); +npy_double cephes__struve_asymp_large_z(npy_double, npy_double, npy_intp, npy_double *); +npy_double cephes__struve_bessel_series(npy_double, npy_double, npy_intp, npy_double *); +npy_double cephes__struve_power_series(npy_double, npy_double, npy_intp, npy_double *); +npy_double cephes_bdtr_wrap(npy_double, npy_intp, npy_double); +npy_double cephes_bdtrc_wrap(npy_double, npy_intp, npy_double); +npy_double cephes_bdtri_wrap(npy_double, npy_intp, npy_double); +npy_double xsf_chdtr(npy_double, npy_double); +npy_double xsf_chdtrc(npy_double, npy_double); +npy_double xsf_chdtri(npy_double, npy_double); +npy_double cephes_erf(npy_double); +npy_double cephes_erfc(npy_double); +npy_double cephes_erfcinv(npy_double); +npy_double cephes_exp10(npy_double); +npy_double cephes_exp2(npy_double); +npy_double cephes_expm1(npy_double); +npy_double cephes_expn_wrap(npy_intp, npy_double); +npy_double xsf_fdtr(npy_double, npy_double, npy_double); +npy_double xsf_fdtrc(npy_double, npy_double, npy_double); +npy_double xsf_fdtri(npy_double, npy_double, npy_double); +npy_double xsf_gdtr(npy_double, npy_double, npy_double); +npy_double xsf_gdtrc(npy_double, npy_double, npy_double); +npy_double xsf_gdtrib(npy_double, npy_double, npy_double); +npy_cdouble chyp1f1_wrap(npy_double, npy_double, npy_cdouble); +npy_double special_cyl_bessel_k_int(npy_intp, npy_double); +npy_double xsf_kolmogi(npy_double); +npy_double xsf_kolmogorov(npy_double); +npy_double cephes_log1p(npy_double); +npy_double pmv_wrap(npy_double, npy_double, npy_double); +npy_double cephes_nbdtr_wrap(npy_intp, npy_intp, npy_double); +npy_double cephes_nbdtrc_wrap(npy_intp, npy_intp, npy_double); +npy_double cephes_nbdtri_wrap(npy_intp, npy_intp, npy_double); +npy_double xsf_ndtr(npy_double); +npy_double xsf_ndtri(npy_double); +npy_double xsf_owens_t(npy_double, npy_double); +npy_double xsf_pdtr(npy_double, npy_double); +npy_double xsf_pdtrc(npy_double, npy_double); +npy_double cephes_pdtri_wrap(npy_intp, npy_double); +npy_double cephes_poch(npy_double, npy_double); +npy_double cephes_round(npy_double); +npy_int xsf_cshichi(npy_cdouble, npy_cdouble *, npy_cdouble *); +npy_int xsf_shichi(npy_double, npy_double *, npy_double *); +npy_int xsf_csici(npy_cdouble, npy_cdouble *, npy_cdouble *); +npy_int xsf_sici(npy_double, npy_double *, npy_double *); +npy_double cephes_smirnov_wrap(npy_intp, npy_double); +npy_double cephes_smirnovi_wrap(npy_intp, npy_double); +npy_double cephes_spence(npy_double); +npy_double xsf_tukeylambdacdf(npy_double, npy_double); +npy_double cephes_yn_wrap(npy_intp, npy_double); +#endif diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/add_newdocs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/add_newdocs.py new file mode 100644 index 0000000000000000000000000000000000000000..5549717d35710d71655e42c836625cde9346bcc3 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/add_newdocs.py @@ -0,0 +1,15 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__: list[str] = [] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="special", module="add_newdocs", + private_modules=["_add_newdocs"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/basic.py new file mode 100644 index 0000000000000000000000000000000000000000..e55695f44d05187d6c83f1ebefd70270af2c2d76 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/basic.py @@ -0,0 +1,87 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.special` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +__all__ = [ # noqa: F822 + 'ai_zeros', + 'assoc_laguerre', + 'bei_zeros', + 'beip_zeros', + 'ber_zeros', + 'bernoulli', + 'berp_zeros', + 'bi_zeros', + 'clpmn', + 'comb', + 'digamma', + 'diric', + 'erf_zeros', + 'euler', + 'factorial', + 'factorial2', + 'factorialk', + 'fresnel_zeros', + 'fresnelc_zeros', + 'fresnels_zeros', + 'gamma', + 'h1vp', + 'h2vp', + 'hankel1', + 'hankel2', + 'iv', + 'ivp', + 'jn_zeros', + 'jnjnp_zeros', + 'jnp_zeros', + 'jnyn_zeros', + 'jv', + 'jvp', + 'kei_zeros', + 'keip_zeros', + 'kelvin_zeros', + 'ker_zeros', + 'kerp_zeros', + 'kv', + 'kvp', + 'lmbda', + 'lpmn', + 'lpn', + 'lqmn', + 'lqn', + 'mathieu_a', + 'mathieu_b', + 'mathieu_even_coef', + 'mathieu_odd_coef', + 'obl_cv_seq', + 'pbdn_seq', + 'pbdv_seq', + 'pbvv_seq', + 'perm', + 'polygamma', + 'pro_cv_seq', + 'psi', + 'riccati_jn', + 'riccati_yn', + 'sinc', + 'y0_zeros', + 'y1_zeros', + 'y1p_zeros', + 'yn_zeros', + 'ynp_zeros', + 'yv', + 'yvp', + 'zeta' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="special", module="basic", + private_modules=["_basic", "_ufuncs"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/cython_special.pxd b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/cython_special.pxd new file mode 100644 index 0000000000000000000000000000000000000000..c5d323fbd5f04ec56749505bc24feb080a851506 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/cython_special.pxd @@ -0,0 +1,259 @@ + +ctypedef fused number_t: + double complex + double + +cpdef number_t spherical_jn(Py_ssize_t n, number_t z, bint derivative=*) noexcept nogil +cpdef number_t spherical_yn(Py_ssize_t n, number_t z, bint derivative=*) noexcept nogil +cpdef number_t spherical_in(Py_ssize_t n, number_t z, bint derivative=*) noexcept nogil +cpdef number_t spherical_kn(Py_ssize_t n, number_t z, bint derivative=*) noexcept nogil + +ctypedef fused Dd_number_t: + double complex + double + +ctypedef fused df_number_t: + double + float + +ctypedef fused dfg_number_t: + double + float + long double + +ctypedef fused dlp_number_t: + double + long + Py_ssize_t + +cpdef double voigt_profile(double x0, double x1, double x2) noexcept nogil +cpdef double agm(double x0, double x1) noexcept nogil +cdef void airy(Dd_number_t x0, Dd_number_t *y0, Dd_number_t *y1, Dd_number_t *y2, Dd_number_t *y3) noexcept nogil +cdef void airye(Dd_number_t x0, Dd_number_t *y0, Dd_number_t *y1, Dd_number_t *y2, Dd_number_t *y3) noexcept nogil +cpdef double bdtr(double x0, dlp_number_t x1, double x2) noexcept nogil +cpdef double bdtrc(double x0, dlp_number_t x1, double x2) noexcept nogil +cpdef double bdtri(double x0, dlp_number_t x1, double x2) noexcept nogil +cpdef double bdtrik(double x0, double x1, double x2) noexcept nogil +cpdef double bdtrin(double x0, double x1, double x2) noexcept nogil +cpdef double bei(double x0) noexcept nogil +cpdef double beip(double x0) noexcept nogil +cpdef double ber(double x0) noexcept nogil +cpdef double berp(double x0) noexcept nogil +cpdef double besselpoly(double x0, double x1, double x2) noexcept nogil +cpdef double beta(double x0, double x1) noexcept nogil +cpdef df_number_t betainc(df_number_t x0, df_number_t x1, df_number_t x2) noexcept nogil +cpdef df_number_t betaincc(df_number_t x0, df_number_t x1, df_number_t x2) noexcept nogil +cpdef df_number_t betaincinv(df_number_t x0, df_number_t x1, df_number_t x2) noexcept nogil +cpdef df_number_t betainccinv(df_number_t x0, df_number_t x1, df_number_t x2) noexcept nogil +cpdef double betaln(double x0, double x1) noexcept nogil +cpdef double binom(double x0, double x1) noexcept nogil +cpdef double boxcox(double x0, double x1) noexcept nogil +cpdef double boxcox1p(double x0, double x1) noexcept nogil +cpdef double btdtria(double x0, double x1, double x2) noexcept nogil +cpdef double btdtrib(double x0, double x1, double x2) noexcept nogil +cpdef double cbrt(double x0) noexcept nogil +cpdef double chdtr(double x0, double x1) noexcept nogil +cpdef double chdtrc(double x0, double x1) noexcept nogil +cpdef double chdtri(double x0, double x1) noexcept nogil +cpdef double chdtriv(double x0, double x1) noexcept nogil +cpdef double chndtr(double x0, double x1, double x2) noexcept nogil +cpdef double chndtridf(double x0, double x1, double x2) noexcept nogil +cpdef double chndtrinc(double x0, double x1, double x2) noexcept nogil +cpdef double chndtrix(double x0, double x1, double x2) noexcept nogil +cpdef double cosdg(double x0) noexcept nogil +cpdef double cosm1(double x0) noexcept nogil +cpdef double cotdg(double x0) noexcept nogil +cpdef Dd_number_t dawsn(Dd_number_t x0) noexcept nogil +cpdef double ellipe(double x0) noexcept nogil +cpdef double ellipeinc(double x0, double x1) noexcept nogil +cdef void ellipj(double x0, double x1, double *y0, double *y1, double *y2, double *y3) noexcept nogil +cpdef double ellipkinc(double x0, double x1) noexcept nogil +cpdef double ellipkm1(double x0) noexcept nogil +cpdef double ellipk(double x0) noexcept nogil +cpdef Dd_number_t elliprc(Dd_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t elliprd(Dd_number_t x0, Dd_number_t x1, Dd_number_t x2) noexcept nogil +cpdef Dd_number_t elliprf(Dd_number_t x0, Dd_number_t x1, Dd_number_t x2) noexcept nogil +cpdef Dd_number_t elliprg(Dd_number_t x0, Dd_number_t x1, Dd_number_t x2) noexcept nogil +cpdef Dd_number_t elliprj(Dd_number_t x0, Dd_number_t x1, Dd_number_t x2, Dd_number_t x3) noexcept nogil +cpdef double entr(double x0) noexcept nogil +cpdef Dd_number_t erf(Dd_number_t x0) noexcept nogil +cpdef Dd_number_t erfc(Dd_number_t x0) noexcept nogil +cpdef Dd_number_t erfcx(Dd_number_t x0) noexcept nogil +cpdef Dd_number_t erfi(Dd_number_t x0) noexcept nogil +cpdef df_number_t erfinv(df_number_t x0) noexcept nogil +cpdef double erfcinv(double x0) noexcept nogil +cpdef Dd_number_t eval_chebyc(dlp_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t eval_chebys(dlp_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t eval_chebyt(dlp_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t eval_chebyu(dlp_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t eval_gegenbauer(dlp_number_t x0, double x1, Dd_number_t x2) noexcept nogil +cpdef Dd_number_t eval_genlaguerre(dlp_number_t x0, double x1, Dd_number_t x2) noexcept nogil +cpdef double eval_hermite(Py_ssize_t x0, double x1) noexcept nogil +cpdef double eval_hermitenorm(Py_ssize_t x0, double x1) noexcept nogil +cpdef Dd_number_t eval_jacobi(dlp_number_t x0, double x1, double x2, Dd_number_t x3) noexcept nogil +cpdef Dd_number_t eval_laguerre(dlp_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t eval_legendre(dlp_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t eval_sh_chebyt(dlp_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t eval_sh_chebyu(dlp_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t eval_sh_jacobi(dlp_number_t x0, double x1, double x2, Dd_number_t x3) noexcept nogil +cpdef Dd_number_t eval_sh_legendre(dlp_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t exp1(Dd_number_t x0) noexcept nogil +cpdef double exp10(double x0) noexcept nogil +cpdef double exp2(double x0) noexcept nogil +cpdef Dd_number_t expi(Dd_number_t x0) noexcept nogil +cpdef dfg_number_t expit(dfg_number_t x0) noexcept nogil +cpdef Dd_number_t expm1(Dd_number_t x0) noexcept nogil +cpdef double expn(dlp_number_t x0, double x1) noexcept nogil +cpdef double exprel(double x0) noexcept nogil +cpdef double fdtr(double x0, double x1, double x2) noexcept nogil +cpdef double fdtrc(double x0, double x1, double x2) noexcept nogil +cpdef double fdtri(double x0, double x1, double x2) noexcept nogil +cpdef double fdtridfd(double x0, double x1, double x2) noexcept nogil +cdef void fresnel(Dd_number_t x0, Dd_number_t *y0, Dd_number_t *y1) noexcept nogil +cpdef Dd_number_t gamma(Dd_number_t x0) noexcept nogil +cpdef double gammainc(double x0, double x1) noexcept nogil +cpdef double gammaincc(double x0, double x1) noexcept nogil +cpdef double gammainccinv(double x0, double x1) noexcept nogil +cpdef double gammaincinv(double x0, double x1) noexcept nogil +cpdef double gammaln(double x0) noexcept nogil +cpdef double gammasgn(double x0) noexcept nogil +cpdef double gdtr(double x0, double x1, double x2) noexcept nogil +cpdef double gdtrc(double x0, double x1, double x2) noexcept nogil +cpdef double gdtria(double x0, double x1, double x2) noexcept nogil +cpdef double gdtrib(double x0, double x1, double x2) noexcept nogil +cpdef double gdtrix(double x0, double x1, double x2) noexcept nogil +cpdef double complex hankel1(double x0, double complex x1) noexcept nogil +cpdef double complex hankel1e(double x0, double complex x1) noexcept nogil +cpdef double complex hankel2(double x0, double complex x1) noexcept nogil +cpdef double complex hankel2e(double x0, double complex x1) noexcept nogil +cpdef double huber(double x0, double x1) noexcept nogil +cpdef Dd_number_t hyp0f1(double x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t hyp1f1(double x0, double x1, Dd_number_t x2) noexcept nogil +cpdef Dd_number_t hyp2f1(double x0, double x1, double x2, Dd_number_t x3) noexcept nogil +cpdef double hyperu(double x0, double x1, double x2) noexcept nogil +cpdef double i0(double x0) noexcept nogil +cpdef double i0e(double x0) noexcept nogil +cpdef double i1(double x0) noexcept nogil +cpdef double i1e(double x0) noexcept nogil +cpdef double inv_boxcox(double x0, double x1) noexcept nogil +cpdef double inv_boxcox1p(double x0, double x1) noexcept nogil +cdef void it2i0k0(double x0, double *y0, double *y1) noexcept nogil +cdef void it2j0y0(double x0, double *y0, double *y1) noexcept nogil +cpdef double it2struve0(double x0) noexcept nogil +cdef void itairy(double x0, double *y0, double *y1, double *y2, double *y3) noexcept nogil +cdef void iti0k0(double x0, double *y0, double *y1) noexcept nogil +cdef void itj0y0(double x0, double *y0, double *y1) noexcept nogil +cpdef double itmodstruve0(double x0) noexcept nogil +cpdef double itstruve0(double x0) noexcept nogil +cpdef Dd_number_t iv(double x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t ive(double x0, Dd_number_t x1) noexcept nogil +cpdef double j0(double x0) noexcept nogil +cpdef double j1(double x0) noexcept nogil +cpdef Dd_number_t jv(double x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t jve(double x0, Dd_number_t x1) noexcept nogil +cpdef double k0(double x0) noexcept nogil +cpdef double k0e(double x0) noexcept nogil +cpdef double k1(double x0) noexcept nogil +cpdef double k1e(double x0) noexcept nogil +cpdef double kei(double x0) noexcept nogil +cpdef double keip(double x0) noexcept nogil +cdef void kelvin(double x0, double complex *y0, double complex *y1, double complex *y2, double complex *y3) noexcept nogil +cpdef double ker(double x0) noexcept nogil +cpdef double kerp(double x0) noexcept nogil +cpdef double kl_div(double x0, double x1) noexcept nogil +cpdef double kn(dlp_number_t x0, double x1) noexcept nogil +cpdef double kolmogi(double x0) noexcept nogil +cpdef double kolmogorov(double x0) noexcept nogil +cpdef Dd_number_t kv(double x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t kve(double x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t log1p(Dd_number_t x0) noexcept nogil +cpdef dfg_number_t log_expit(dfg_number_t x0) noexcept nogil +cpdef Dd_number_t log_ndtr(Dd_number_t x0) noexcept nogil +cpdef Dd_number_t loggamma(Dd_number_t x0) noexcept nogil +cpdef dfg_number_t logit(dfg_number_t x0) noexcept nogil +cpdef double lpmv(double x0, double x1, double x2) noexcept nogil +cpdef double mathieu_a(double x0, double x1) noexcept nogil +cpdef double mathieu_b(double x0, double x1) noexcept nogil +cdef void mathieu_cem(double x0, double x1, double x2, double *y0, double *y1) noexcept nogil +cdef void mathieu_modcem1(double x0, double x1, double x2, double *y0, double *y1) noexcept nogil +cdef void mathieu_modcem2(double x0, double x1, double x2, double *y0, double *y1) noexcept nogil +cdef void mathieu_modsem1(double x0, double x1, double x2, double *y0, double *y1) noexcept nogil +cdef void mathieu_modsem2(double x0, double x1, double x2, double *y0, double *y1) noexcept nogil +cdef void mathieu_sem(double x0, double x1, double x2, double *y0, double *y1) noexcept nogil +cdef void modfresnelm(double x0, double complex *y0, double complex *y1) noexcept nogil +cdef void modfresnelp(double x0, double complex *y0, double complex *y1) noexcept nogil +cpdef double modstruve(double x0, double x1) noexcept nogil +cpdef double nbdtr(dlp_number_t x0, dlp_number_t x1, double x2) noexcept nogil +cpdef double nbdtrc(dlp_number_t x0, dlp_number_t x1, double x2) noexcept nogil +cpdef double nbdtri(dlp_number_t x0, dlp_number_t x1, double x2) noexcept nogil +cpdef double nbdtrik(double x0, double x1, double x2) noexcept nogil +cpdef double nbdtrin(double x0, double x1, double x2) noexcept nogil +cpdef df_number_t ncfdtr(df_number_t x0, df_number_t x1, df_number_t x2, df_number_t x3) noexcept nogil +cpdef df_number_t ncfdtri(df_number_t x0, df_number_t x1, df_number_t x2, df_number_t x3) noexcept nogil +cpdef double ncfdtridfd(double x0, double x1, double x2, double x3) noexcept nogil +cpdef double ncfdtridfn(double x0, double x1, double x2, double x3) noexcept nogil +cpdef double ncfdtrinc(double x0, double x1, double x2, double x3) noexcept nogil +cpdef df_number_t nctdtr(df_number_t x0, df_number_t x1, df_number_t x2) noexcept nogil +cpdef double nctdtridf(double x0, double x1, double x2) noexcept nogil +cpdef double nctdtrinc(double x0, double x1, double x2) noexcept nogil +cpdef double nctdtrit(double x0, double x1, double x2) noexcept nogil +cpdef Dd_number_t ndtr(Dd_number_t x0) noexcept nogil +cpdef double ndtri(double x0) noexcept nogil +cpdef double nrdtrimn(double x0, double x1, double x2) noexcept nogil +cpdef double nrdtrisd(double x0, double x1, double x2) noexcept nogil +cdef void obl_ang1(double x0, double x1, double x2, double x3, double *y0, double *y1) noexcept nogil +cdef void obl_ang1_cv(double x0, double x1, double x2, double x3, double x4, double *y0, double *y1) noexcept nogil +cpdef double obl_cv(double x0, double x1, double x2) noexcept nogil +cdef void obl_rad1(double x0, double x1, double x2, double x3, double *y0, double *y1) noexcept nogil +cdef void obl_rad1_cv(double x0, double x1, double x2, double x3, double x4, double *y0, double *y1) noexcept nogil +cdef void obl_rad2(double x0, double x1, double x2, double x3, double *y0, double *y1) noexcept nogil +cdef void obl_rad2_cv(double x0, double x1, double x2, double x3, double x4, double *y0, double *y1) noexcept nogil +cpdef double owens_t(double x0, double x1) noexcept nogil +cdef void pbdv(double x0, double x1, double *y0, double *y1) noexcept nogil +cdef void pbvv(double x0, double x1, double *y0, double *y1) noexcept nogil +cdef void pbwa(double x0, double x1, double *y0, double *y1) noexcept nogil +cpdef double pdtr(double x0, double x1) noexcept nogil +cpdef double pdtrc(double x0, double x1) noexcept nogil +cpdef double pdtri(dlp_number_t x0, double x1) noexcept nogil +cpdef double pdtrik(double x0, double x1) noexcept nogil +cpdef double poch(double x0, double x1) noexcept nogil +cpdef df_number_t powm1(df_number_t x0, df_number_t x1) noexcept nogil +cdef void pro_ang1(double x0, double x1, double x2, double x3, double *y0, double *y1) noexcept nogil +cdef void pro_ang1_cv(double x0, double x1, double x2, double x3, double x4, double *y0, double *y1) noexcept nogil +cpdef double pro_cv(double x0, double x1, double x2) noexcept nogil +cdef void pro_rad1(double x0, double x1, double x2, double x3, double *y0, double *y1) noexcept nogil +cdef void pro_rad1_cv(double x0, double x1, double x2, double x3, double x4, double *y0, double *y1) noexcept nogil +cdef void pro_rad2(double x0, double x1, double x2, double x3, double *y0, double *y1) noexcept nogil +cdef void pro_rad2_cv(double x0, double x1, double x2, double x3, double x4, double *y0, double *y1) noexcept nogil +cpdef double pseudo_huber(double x0, double x1) noexcept nogil +cpdef Dd_number_t psi(Dd_number_t x0) noexcept nogil +cpdef double radian(double x0, double x1, double x2) noexcept nogil +cpdef double rel_entr(double x0, double x1) noexcept nogil +cpdef Dd_number_t rgamma(Dd_number_t x0) noexcept nogil +cpdef double round(double x0) noexcept nogil +cdef void shichi(Dd_number_t x0, Dd_number_t *y0, Dd_number_t *y1) noexcept nogil +cdef void sici(Dd_number_t x0, Dd_number_t *y0, Dd_number_t *y1) noexcept nogil +cpdef double sindg(double x0) noexcept nogil +cpdef double smirnov(dlp_number_t x0, double x1) noexcept nogil +cpdef double smirnovi(dlp_number_t x0, double x1) noexcept nogil +cpdef Dd_number_t spence(Dd_number_t x0) noexcept nogil +cpdef double complex sph_harm(dlp_number_t x0, dlp_number_t x1, double x2, double x3) noexcept nogil +cpdef double stdtr(double x0, double x1) noexcept nogil +cpdef double stdtridf(double x0, double x1) noexcept nogil +cpdef double stdtrit(double x0, double x1) noexcept nogil +cpdef double struve(double x0, double x1) noexcept nogil +cpdef double tandg(double x0) noexcept nogil +cpdef double tklmbda(double x0, double x1) noexcept nogil +cpdef double complex wofz(double complex x0) noexcept nogil +cpdef Dd_number_t wrightomega(Dd_number_t x0) noexcept nogil +cpdef Dd_number_t xlog1py(Dd_number_t x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t xlogy(Dd_number_t x0, Dd_number_t x1) noexcept nogil +cpdef double y0(double x0) noexcept nogil +cpdef double y1(double x0) noexcept nogil +cpdef double yn(dlp_number_t x0, double x1) noexcept nogil +cpdef Dd_number_t yv(double x0, Dd_number_t x1) noexcept nogil +cpdef Dd_number_t yve(double x0, Dd_number_t x1) noexcept nogil +cpdef double zetac(double x0) noexcept nogil +cpdef double wright_bessel(double x0, double x1, double x2) noexcept nogil +cpdef double log_wright_bessel(double x0, double x1, double x2) noexcept nogil +cpdef double ndtri_exp(double x0) noexcept nogil diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/cython_special.pyi b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/cython_special.pyi new file mode 100644 index 0000000000000000000000000000000000000000..024e962b10df8892631eaad20223f7fc8378ea83 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/cython_special.pyi @@ -0,0 +1,3 @@ +from typing import Any + +def __getattr__(name) -> Any: ... diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/libsf_error_state.so b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/libsf_error_state.so new file mode 100644 index 0000000000000000000000000000000000000000..84e37bedbdfcc83bb65efc5d7f1961f2b749ebeb Binary files /dev/null and b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/libsf_error_state.so differ diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/orthogonal.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/orthogonal.py new file mode 100644 index 0000000000000000000000000000000000000000..0b13a08a96cb683d72a4a00d6962446e1779c88a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/orthogonal.py @@ -0,0 +1,45 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.special` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + + +_polyfuns = ['legendre', 'chebyt', 'chebyu', 'chebyc', 'chebys', + 'jacobi', 'laguerre', 'genlaguerre', 'hermite', + 'hermitenorm', 'gegenbauer', 'sh_legendre', 'sh_chebyt', + 'sh_chebyu', 'sh_jacobi'] + +# Correspondence between new and old names of root functions +_rootfuns_map = {'roots_legendre': 'p_roots', + 'roots_chebyt': 't_roots', + 'roots_chebyu': 'u_roots', + 'roots_chebyc': 'c_roots', + 'roots_chebys': 's_roots', + 'roots_jacobi': 'j_roots', + 'roots_laguerre': 'l_roots', + 'roots_genlaguerre': 'la_roots', + 'roots_hermite': 'h_roots', + 'roots_hermitenorm': 'he_roots', + 'roots_gegenbauer': 'cg_roots', + 'roots_sh_legendre': 'ps_roots', + 'roots_sh_chebyt': 'ts_roots', + 'roots_sh_chebyu': 'us_roots', + 'roots_sh_jacobi': 'js_roots'} + + +__all__ = _polyfuns + list(_rootfuns_map.keys()) + [ # noqa: F822 + 'airy', 'p_roots', 't_roots', 'u_roots', 'c_roots', 's_roots', + 'j_roots', 'l_roots', 'la_roots', 'h_roots', 'he_roots', 'cg_roots', + 'ps_roots', 'ts_roots', 'us_roots', 'js_roots' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="special", module="orthogonal", + private_modules=["_orthogonal"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/sf_error.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/sf_error.py new file mode 100644 index 0000000000000000000000000000000000000000..00ff73756acd4219a4ba94eb089bce7d4c32266d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/sf_error.py @@ -0,0 +1,20 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.special` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = [ # noqa: F822 + 'SpecialFunctionWarning', + 'SpecialFunctionError' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="special", module="sf_error", + private_modules=["_sf_error"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/specfun.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/specfun.py new file mode 100644 index 0000000000000000000000000000000000000000..9fca00415a6406b8cdf41a42b6fbf991cea1f53f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/specfun.py @@ -0,0 +1,24 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.special` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +# ruff: noqa: F822 +__all__ = [ + 'clpmn', + 'lpmn', + 'lpn', + 'lqmn', + 'pbdv' +] + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="special", module="specfun", + private_modules=["_basic", "_specfun"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/spfun_stats.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/spfun_stats.py new file mode 100644 index 0000000000000000000000000000000000000000..a1e58487aaa547483c9f2531ac4efc2ad5e4795c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/spfun_stats.py @@ -0,0 +1,17 @@ +# This file is not meant for public use and will be removed in SciPy v2.0.0. +# Use the `scipy.special` namespace for importing the functions +# included below. + +from scipy._lib.deprecation import _sub_module_deprecation + +__all__ = ['multigammaln'] # noqa: F822 + + +def __dir__(): + return __all__ + + +def __getattr__(name): + return _sub_module_deprecation(sub_package="special", module="spfun_stats", + private_modules=["_spfun_stats"], all=__all__, + attribute=name) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/_cython_examples/extending.pyx b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/_cython_examples/extending.pyx new file mode 100644 index 0000000000000000000000000000000000000000..ca3bf2167f0f7726f8b0acb60ed8b8798a518d79 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/_cython_examples/extending.pyx @@ -0,0 +1,12 @@ +#!/usr/bin/env python3 +#cython: language_level=3 +#cython: boundscheck=False +#cython: wraparound=False + +from scipy.special.cython_special cimport beta, gamma + +cpdef double cy_beta(double a, double b): + return beta(a, b) + +cpdef double complex cy_gamma(double complex z): + return gamma(z) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/_cython_examples/meson.build b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/_cython_examples/meson.build new file mode 100644 index 0000000000000000000000000000000000000000..2a5e1535a16f840f31ca0207513e7c060767ea12 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/_cython_examples/meson.build @@ -0,0 +1,25 @@ +project('random-build-examples', 'c', 'cpp', 'cython') + +fs = import('fs') + +py3 = import('python').find_installation(pure: false) + +cy = meson.get_compiler('cython') + +if not cy.version().version_compare('>=3.0.8') + error('tests requires Cython >= 3.0.8') +endif + +py3.extension_module( + 'extending', + 'extending.pyx', + install: false, +) + +extending_cpp = fs.copyfile('extending.pyx', 'extending_cpp.pyx') +py3.extension_module( + 'extending_cpp', + extending_cpp, + install: false, + override_options : ['cython_language=cpp'] +) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/data/__init__.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/data/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_basic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_basic.py new file mode 100644 index 0000000000000000000000000000000000000000..67bb83f1f1682e5eb83cc66f052ea51e3caf8757 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_basic.py @@ -0,0 +1,4682 @@ +# this program corresponds to special.py + +### Means test is not done yet +# E Means test is giving error (E) +# F Means test is failing (F) +# EF Means test is giving error and Failing +#! Means test is segfaulting +# 8 Means test runs forever + +### test_besselpoly +### test_mathieu_a +### test_mathieu_even_coef +### test_mathieu_odd_coef +### test_modfresnelp +### test_modfresnelm +# test_pbdv_seq +### test_pbvv_seq +### test_sph_harm + +import functools +import itertools +import operator +import platform +import sys + +import numpy as np +from numpy import (array, isnan, r_, arange, finfo, pi, sin, cos, tan, exp, + log, zeros, sqrt, asarray, inf, nan_to_num, real, arctan, double, + array_equal) + +import pytest +from pytest import raises as assert_raises +from numpy.testing import (assert_equal, assert_almost_equal, + assert_array_equal, assert_array_almost_equal, assert_approx_equal, + assert_, assert_allclose, assert_array_almost_equal_nulp, + suppress_warnings) + +from scipy import special +import scipy.special._ufuncs as cephes +from scipy.special import ellipe, ellipk, ellipkm1 +from scipy.special import elliprc, elliprd, elliprf, elliprg, elliprj +from scipy.special import softplus +from scipy.special import mathieu_odd_coef, mathieu_even_coef, stirling2 +from scipy.special import lpn, lpmn, clpmn +from scipy._lib._util import np_long, np_ulong +from scipy._lib._array_api import xp_assert_close, xp_assert_equal, SCIPY_ARRAY_API + +from scipy.special._basic import ( + _FACTORIALK_LIMITS_64BITS, _FACTORIALK_LIMITS_32BITS, _is_subdtype +) +from scipy.special._testutils import with_special_errors, \ + assert_func_equal, FuncData + +import math + + +native_int = np.int32 if ( + sys.platform == 'win32' + or platform.architecture()[0] == "32bit" +) else np.int64 + + +class TestCephes: + def test_airy(self): + cephes.airy(0) + + def test_airye(self): + cephes.airye(0) + + def test_binom(self): + n = np.array([0.264, 4, 5.2, 17]) + k = np.array([2, 0.4, 7, 3.3]) + nk = np.array(np.broadcast_arrays(n[:,None], k[None,:]) + ).reshape(2, -1).T + rknown = np.array([[-0.097152, 0.9263051596159367, 0.01858423645695389, + -0.007581020651518199],[6, 2.0214389119675666, 0, 2.9827344527963846], + [10.92, 2.22993515861399, -0.00585728, 10.468891352063146], + [136, 3.5252179590758828, 19448, 1024.5526916174495]]) + assert_func_equal(cephes.binom, rknown.ravel(), nk, rtol=1e-13) + + # Test branches in implementation + rng = np.random.RandomState(1234) + n = np.r_[np.arange(-7, 30), 1000*rng.rand(30) - 500] + k = np.arange(0, 102) + nk = np.array(np.broadcast_arrays(n[:,None], k[None,:]) + ).reshape(2, -1).T + + assert_func_equal(cephes.binom, + cephes.binom(nk[:,0], nk[:,1] * (1 + 1e-15)), + nk, + atol=1e-10, rtol=1e-10) + + def test_binom_2(self): + # Test branches in implementation + np.random.seed(1234) + n = np.r_[np.logspace(1, 300, 20)] + k = np.arange(0, 102) + nk = np.array(np.broadcast_arrays(n[:,None], k[None,:]) + ).reshape(2, -1).T + + assert_func_equal(cephes.binom, + cephes.binom(nk[:,0], nk[:,1] * (1 + 1e-15)), + nk, + atol=1e-10, rtol=1e-10) + + def test_binom_exact(self): + @np.vectorize + def binom_int(n, k): + n = int(n) + k = int(k) + num = 1 + den = 1 + for i in range(1, k+1): + num *= i + n - k + den *= i + return float(num/den) + + np.random.seed(1234) + n = np.arange(1, 15) + k = np.arange(0, 15) + nk = np.array(np.broadcast_arrays(n[:,None], k[None,:]) + ).reshape(2, -1).T + nk = nk[nk[:,0] >= nk[:,1]] + assert_func_equal(cephes.binom, + binom_int(nk[:,0], nk[:,1]), + nk, + atol=0, rtol=0) + + def test_binom_nooverflow_8346(self): + # Test (binom(n, k) doesn't overflow prematurely */ + dataset = [ + (1000, 500, 2.70288240945436551e+299), + (1002, 501, 1.08007396880791225e+300), + (1004, 502, 4.31599279169058121e+300), + (1006, 503, 1.72468101616263781e+301), + (1008, 504, 6.89188009236419153e+301), + (1010, 505, 2.75402257948335448e+302), + (1012, 506, 1.10052048531923757e+303), + (1014, 507, 4.39774063758732849e+303), + (1016, 508, 1.75736486108312519e+304), + (1018, 509, 7.02255427788423734e+304), + (1020, 510, 2.80626776829962255e+305), + (1022, 511, 1.12140876377061240e+306), + (1024, 512, 4.48125455209897109e+306), + (1026, 513, 1.79075474304149900e+307), + (1028, 514, 7.15605105487789676e+307) + ] + dataset = np.asarray(dataset) + FuncData(cephes.binom, dataset, (0, 1), 2, rtol=1e-12).check() + + def test_bdtr(self): + assert_equal(cephes.bdtr(1,1,0.5),1.0) + + def test_bdtri(self): + assert_equal(cephes.bdtri(1,3,0.5),0.5) + + def test_bdtrc(self): + assert_equal(cephes.bdtrc(1,3,0.5),0.5) + + def test_bdtrin(self): + assert_equal(cephes.bdtrin(1,0,1),5.0) + + def test_bdtrik(self): + cephes.bdtrik(1,3,0.5) + + def test_bei(self): + assert_equal(cephes.bei(0),0.0) + + def test_beip(self): + assert_equal(cephes.beip(0),0.0) + + def test_ber(self): + assert_equal(cephes.ber(0),1.0) + + def test_berp(self): + assert_equal(cephes.berp(0),0.0) + + def test_besselpoly(self): + assert_equal(cephes.besselpoly(0,0,0),1.0) + + def test_btdtria(self): + assert_equal(cephes.btdtria(1,1,1),5.0) + + def test_btdtrib(self): + assert_equal(cephes.btdtrib(1,1,1),5.0) + + def test_cbrt(self): + assert_approx_equal(cephes.cbrt(1),1.0) + + def test_chdtr(self): + assert_equal(cephes.chdtr(1,0),0.0) + + def test_chdtrc(self): + assert_equal(cephes.chdtrc(1,0),1.0) + + def test_chdtri(self): + assert_equal(cephes.chdtri(1,1),0.0) + + def test_chdtriv(self): + assert_equal(cephes.chdtriv(0,0),5.0) + + def test_chndtr(self): + assert_equal(cephes.chndtr(0,1,0),0.0) + + # Each row holds (x, nu, lam, expected_value) + # These values were computed using Wolfram Alpha with + # CDF[NoncentralChiSquareDistribution[nu, lam], x] + values = np.array([ + [25.00, 20.0, 400, 4.1210655112396197139e-57], + [25.00, 8.00, 250, 2.3988026526832425878e-29], + [0.001, 8.00, 40., 5.3761806201366039084e-24], + [0.010, 8.00, 40., 5.45396231055999457039e-20], + [20.00, 2.00, 107, 1.39390743555819597802e-9], + [22.50, 2.00, 107, 7.11803307138105870671e-9], + [25.00, 2.00, 107, 3.11041244829864897313e-8], + [3.000, 2.00, 1.0, 0.62064365321954362734], + [350.0, 300., 10., 0.93880128006276407710], + [100.0, 13.5, 10., 0.99999999650104210949], + [700.0, 20.0, 400, 0.99999999925680650105], + [150.0, 13.5, 10., 0.99999999999999983046], + [160.0, 13.5, 10., 0.99999999999999999518], # 1.0 + ]) + cdf = cephes.chndtr(values[:, 0], values[:, 1], values[:, 2]) + assert_allclose(cdf, values[:, 3], rtol=1e-12) + + assert_almost_equal(cephes.chndtr(np.inf, np.inf, 0), 2.0) + assert_almost_equal(cephes.chndtr(2, 1, np.inf), 0.0) + assert_(np.isnan(cephes.chndtr(np.nan, 1, 2))) + assert_(np.isnan(cephes.chndtr(5, np.nan, 2))) + assert_(np.isnan(cephes.chndtr(5, 1, np.nan))) + + def test_chndtridf(self): + assert_equal(cephes.chndtridf(0,0,1),5.0) + + def test_chndtrinc(self): + assert_equal(cephes.chndtrinc(0,1,0),5.0) + + def test_chndtrix(self): + assert_equal(cephes.chndtrix(0,1,0),0.0) + + def test_cosdg(self): + assert_equal(cephes.cosdg(0),1.0) + + def test_cosm1(self): + assert_equal(cephes.cosm1(0),0.0) + + def test_cotdg(self): + assert_almost_equal(cephes.cotdg(45),1.0) + + def test_dawsn(self): + assert_equal(cephes.dawsn(0),0.0) + assert_allclose(cephes.dawsn(1.23), 0.50053727749081767) + + def test_diric(self): + # Test behavior near multiples of 2pi. Regression test for issue + # described in gh-4001. + n_odd = [1, 5, 25] + x = np.array(2*np.pi + 5e-5).astype(np.float32) + assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=7) + x = np.array(2*np.pi + 1e-9).astype(np.float64) + assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=15) + x = np.array(2*np.pi + 1e-15).astype(np.float64) + assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=15) + if hasattr(np, 'float128'): + # No float128 available in 32-bit numpy + x = np.array(2*np.pi + 1e-12).astype(np.float128) + assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=19) + + n_even = [2, 4, 24] + x = np.array(2*np.pi + 1e-9).astype(np.float64) + assert_almost_equal(special.diric(x, n_even), -1.0, decimal=15) + + # Test at some values not near a multiple of pi + x = np.arange(0.2*np.pi, 1.0*np.pi, 0.2*np.pi) + octave_result = [0.872677996249965, 0.539344662916632, + 0.127322003750035, -0.206011329583298] + assert_almost_equal(special.diric(x, 3), octave_result, decimal=15) + + def test_diric_broadcasting(self): + x = np.arange(5) + n = np.array([1, 3, 7]) + assert_(special.diric(x[:, np.newaxis], n).shape == (x.size, n.size)) + + def test_ellipe(self): + assert_equal(cephes.ellipe(1),1.0) + + def test_ellipeinc(self): + assert_equal(cephes.ellipeinc(0,1),0.0) + + def test_ellipj(self): + cephes.ellipj(0,1) + + def test_ellipk(self): + assert_allclose(ellipk(0), pi/2) + + def test_ellipkinc(self): + assert_equal(cephes.ellipkinc(0,0),0.0) + + def test_erf(self): + assert_equal(cephes.erf(0), 0.0) + + def test_erf_symmetry(self): + x = 5.905732037710919 + assert_equal(cephes.erf(x) + cephes.erf(-x), 0.0) + + def test_erfc(self): + assert_equal(cephes.erfc(0), 1.0) + + def test_exp10(self): + assert_approx_equal(cephes.exp10(2),100.0) + + def test_exp2(self): + assert_equal(cephes.exp2(2),4.0) + + def test_expm1(self): + assert_equal(cephes.expm1(0),0.0) + assert_equal(cephes.expm1(np.inf), np.inf) + assert_equal(cephes.expm1(-np.inf), -1) + assert_equal(cephes.expm1(np.nan), np.nan) + + def test_expm1_complex(self): + expm1 = cephes.expm1 + assert_equal(expm1(0 + 0j), 0 + 0j) + assert_equal(expm1(complex(np.inf, 0)), complex(np.inf, 0)) + assert_equal(expm1(complex(np.inf, 1)), complex(np.inf, np.inf)) + assert_equal(expm1(complex(np.inf, 2)), complex(-np.inf, np.inf)) + assert_equal(expm1(complex(np.inf, 4)), complex(-np.inf, -np.inf)) + assert_equal(expm1(complex(np.inf, 5)), complex(np.inf, -np.inf)) + assert_equal(expm1(complex(1, np.inf)), complex(np.nan, np.nan)) + assert_equal(expm1(complex(0, np.inf)), complex(np.nan, np.nan)) + assert_equal(expm1(complex(np.inf, np.inf)), complex(np.inf, np.nan)) + assert_equal(expm1(complex(-np.inf, np.inf)), complex(-1, 0)) + assert_equal(expm1(complex(-np.inf, np.nan)), complex(-1, 0)) + assert_equal(expm1(complex(np.inf, np.nan)), complex(np.inf, np.nan)) + assert_equal(expm1(complex(0, np.nan)), complex(np.nan, np.nan)) + assert_equal(expm1(complex(1, np.nan)), complex(np.nan, np.nan)) + assert_equal(expm1(complex(np.nan, 1)), complex(np.nan, np.nan)) + assert_equal(expm1(complex(np.nan, np.nan)), complex(np.nan, np.nan)) + + @pytest.mark.xfail(reason='The real part of expm1(z) bad at these points') + def test_expm1_complex_hard(self): + # The real part of this function is difficult to evaluate when + # z.real = -log(cos(z.imag)). + y = np.array([0.1, 0.2, 0.3, 5, 11, 20]) + x = -np.log(np.cos(y)) + z = x + 1j*y + + # evaluate using mpmath.expm1 with dps=1000 + expected = np.array([-5.5507901846769623e-17+0.10033467208545054j, + 2.4289354732893695e-18+0.20271003550867248j, + 4.5235500262585768e-17+0.30933624960962319j, + 7.8234305217489006e-17-3.3805150062465863j, + -1.3685191953697676e-16-225.95084645419513j, + 8.7175620481291045e-17+2.2371609442247422j]) + found = cephes.expm1(z) + # this passes. + assert_array_almost_equal_nulp(found.imag, expected.imag, 3) + # this fails. + assert_array_almost_equal_nulp(found.real, expected.real, 20) + + def test_fdtr(self): + assert_equal(cephes.fdtr(1, 1, 0), 0.0) + # Computed using Wolfram Alpha: CDF[FRatioDistribution[1e-6, 5], 10] + assert_allclose(cephes.fdtr(1e-6, 5, 10), 0.9999940790193488, + rtol=1e-12) + + def test_fdtrc(self): + assert_equal(cephes.fdtrc(1, 1, 0), 1.0) + # Computed using Wolfram Alpha: + # 1 - CDF[FRatioDistribution[2, 1/10], 1e10] + assert_allclose(cephes.fdtrc(2, 0.1, 1e10), 0.27223784621293512, + rtol=1e-12) + + def test_fdtri(self): + assert_allclose(cephes.fdtri(1, 1, [0.499, 0.501]), + array([0.9937365, 1.00630298]), rtol=1e-6) + # From Wolfram Alpha: + # CDF[FRatioDistribution[1/10, 1], 3] = 0.8756751669632105666874... + p = 0.8756751669632105666874 + assert_allclose(cephes.fdtri(0.1, 1, p), 3, rtol=1e-12) + + @pytest.mark.xfail(reason='Returns nan on i686.') + def test_fdtri_mysterious_failure(self): + assert_allclose(cephes.fdtri(1, 1, 0.5), 1) + + def test_fdtridfd(self): + assert_equal(cephes.fdtridfd(1,0,0),5.0) + + def test_fresnel(self): + assert_equal(cephes.fresnel(0),(0.0,0.0)) + + def test_gamma(self): + assert_equal(cephes.gamma(5),24.0) + + def test_gammainccinv(self): + assert_equal(cephes.gammainccinv(5,1),0.0) + + def test_gammaln(self): + cephes.gammaln(10) + + def test_gammasgn(self): + vals = np.array( + [-np.inf, -4, -3.5, -2.3, -0.0, 0.0, 1, 4.2, np.inf], np.float64 + ) + reference = np.array( + [np.nan, np.nan, 1.0, -1.0, -1.0, 1.0, 1.0, 1.0, 1.0], np.float64 + ) + assert_array_equal(cephes.gammasgn(vals), reference) + + def test_gdtr(self): + assert_equal(cephes.gdtr(1,1,0),0.0) + + def test_gdtr_inf(self): + assert_equal(cephes.gdtr(1,1,np.inf),1.0) + + def test_gdtrc(self): + assert_equal(cephes.gdtrc(1,1,0),1.0) + + def test_gdtria(self): + assert_equal(cephes.gdtria(0,1,1),0.0) + + def test_gdtrib(self): + cephes.gdtrib(1,0,1) + # assert_equal(cephes.gdtrib(1,0,1),5.0) + + def test_gdtrix(self): + cephes.gdtrix(1,1,.1) + + def test_hankel1(self): + cephes.hankel1(1,1) + + def test_hankel1e(self): + cephes.hankel1e(1,1) + + def test_hankel2(self): + cephes.hankel2(1,1) + + def test_hankel2e(self): + cephes.hankel2e(1,1) + + def test_hyp1f1(self): + assert_approx_equal(cephes.hyp1f1(1,1,1), exp(1.0)) + assert_approx_equal(cephes.hyp1f1(3,4,-6), 0.026056422099537251095) + cephes.hyp1f1(1,1,1) + + def test_hyp2f1(self): + assert_equal(cephes.hyp2f1(1,1,1,0),1.0) + + def test_i0(self): + assert_equal(cephes.i0(0),1.0) + + def test_i0e(self): + assert_equal(cephes.i0e(0),1.0) + + def test_i1(self): + assert_equal(cephes.i1(0),0.0) + + def test_i1e(self): + assert_equal(cephes.i1e(0),0.0) + + def test_it2i0k0(self): + cephes.it2i0k0(1) + + def test_it2j0y0(self): + cephes.it2j0y0(1) + + def test_it2struve0(self): + cephes.it2struve0(1) + + def test_itairy(self): + cephes.itairy(1) + + def test_iti0k0(self): + assert_equal(cephes.iti0k0(0),(0.0,0.0)) + + def test_itj0y0(self): + assert_equal(cephes.itj0y0(0),(0.0,0.0)) + + def test_itmodstruve0(self): + assert_equal(cephes.itmodstruve0(0),0.0) + + def test_itstruve0(self): + assert_equal(cephes.itstruve0(0),0.0) + + def test_iv(self): + assert_equal(cephes.iv(1,0),0.0) + + def test_ive(self): + assert_equal(cephes.ive(1,0),0.0) + + def test_j0(self): + assert_equal(cephes.j0(0),1.0) + + def test_j1(self): + assert_equal(cephes.j1(0),0.0) + + def test_jn(self): + assert_equal(cephes.jn(0,0),1.0) + + def test_jv(self): + assert_equal(cephes.jv(0,0),1.0) + + def test_jve(self): + assert_equal(cephes.jve(0,0),1.0) + + def test_k0(self): + cephes.k0(2) + + def test_k0e(self): + cephes.k0e(2) + + def test_k1(self): + cephes.k1(2) + + def test_k1e(self): + cephes.k1e(2) + + def test_kei(self): + cephes.kei(2) + + def test_keip(self): + assert_equal(cephes.keip(0),0.0) + + def test_ker(self): + cephes.ker(2) + + def test_kerp(self): + cephes.kerp(2) + + def test_kelvin(self): + cephes.kelvin(2) + + def test_kn(self): + cephes.kn(1,1) + + def test_kolmogi(self): + assert_equal(cephes.kolmogi(1),0.0) + assert_(np.isnan(cephes.kolmogi(np.nan))) + + def test_kolmogorov(self): + assert_equal(cephes.kolmogorov(0), 1.0) + + def test_kolmogp(self): + assert_equal(cephes._kolmogp(0), -0.0) + + def test_kolmogc(self): + assert_equal(cephes._kolmogc(0), 0.0) + + def test_kolmogci(self): + assert_equal(cephes._kolmogci(0), 0.0) + assert_(np.isnan(cephes._kolmogci(np.nan))) + + def test_kv(self): + cephes.kv(1,1) + + def test_kve(self): + cephes.kve(1,1) + + def test_log1p(self): + log1p = cephes.log1p + assert_equal(log1p(0), 0.0) + assert_equal(log1p(-1), -np.inf) + assert_equal(log1p(-2), np.nan) + assert_equal(log1p(np.inf), np.inf) + + def test_log1p_complex(self): + log1p = cephes.log1p + c = complex + assert_equal(log1p(0 + 0j), 0 + 0j) + assert_equal(log1p(c(-1, 0)), c(-np.inf, 0)) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "invalid value encountered in multiply") + assert_allclose(log1p(c(1, np.inf)), c(np.inf, np.pi/2)) + assert_equal(log1p(c(1, np.nan)), c(np.nan, np.nan)) + assert_allclose(log1p(c(-np.inf, 1)), c(np.inf, np.pi)) + assert_equal(log1p(c(np.inf, 1)), c(np.inf, 0)) + assert_allclose(log1p(c(-np.inf, np.inf)), c(np.inf, 3*np.pi/4)) + assert_allclose(log1p(c(np.inf, np.inf)), c(np.inf, np.pi/4)) + assert_equal(log1p(c(np.inf, np.nan)), c(np.inf, np.nan)) + assert_equal(log1p(c(-np.inf, np.nan)), c(np.inf, np.nan)) + assert_equal(log1p(c(np.nan, np.inf)), c(np.inf, np.nan)) + assert_equal(log1p(c(np.nan, 1)), c(np.nan, np.nan)) + assert_equal(log1p(c(np.nan, np.nan)), c(np.nan, np.nan)) + + def test_lpmv(self): + assert_equal(cephes.lpmv(0,0,1),1.0) + + def test_mathieu_a(self): + assert_equal(cephes.mathieu_a(1,0),1.0) + + def test_mathieu_b(self): + assert_equal(cephes.mathieu_b(1,0),1.0) + + def test_mathieu_cem(self): + assert_equal(cephes.mathieu_cem(1,0,0),(1.0,0.0)) + + # Test AMS 20.2.27 + @np.vectorize + def ce_smallq(m, q, z): + z *= np.pi/180 + if m == 0: + # + O(q^2) + return 2**(-0.5) * (1 - .5*q*cos(2*z)) + elif m == 1: + # + O(q^2) + return cos(z) - q/8 * cos(3*z) + elif m == 2: + # + O(q^2) + return cos(2*z) - q*(cos(4*z)/12 - 1/4) + else: + # + O(q^2) + return cos(m*z) - q*(cos((m+2)*z)/(4*(m+1)) - cos((m-2)*z)/(4*(m-1))) + m = np.arange(0, 100) + q = np.r_[0, np.logspace(-30, -9, 10)] + assert_allclose(cephes.mathieu_cem(m[:,None], q[None,:], 0.123)[0], + ce_smallq(m[:,None], q[None,:], 0.123), + rtol=1e-14, atol=0) + + def test_mathieu_sem(self): + assert_equal(cephes.mathieu_sem(1,0,0),(0.0,1.0)) + + # Test AMS 20.2.27 + @np.vectorize + def se_smallq(m, q, z): + z *= np.pi/180 + if m == 1: + # + O(q^2) + return sin(z) - q/8 * sin(3*z) + elif m == 2: + # + O(q^2) + return sin(2*z) - q*sin(4*z)/12 + else: + # + O(q^2) + return sin(m*z) - q*(sin((m+2)*z)/(4*(m+1)) - sin((m-2)*z)/(4*(m-1))) + m = np.arange(1, 100) + q = np.r_[0, np.logspace(-30, -9, 10)] + assert_allclose(cephes.mathieu_sem(m[:,None], q[None,:], 0.123)[0], + se_smallq(m[:,None], q[None,:], 0.123), + rtol=1e-14, atol=0) + + def test_mathieu_modcem1(self): + assert_equal(cephes.mathieu_modcem1(1,0,0),(0.0,0.0)) + + def test_mathieu_modcem2(self): + cephes.mathieu_modcem2(1,1,1) + + # Test reflection relation AMS 20.6.19 + m = np.arange(0, 4)[:,None,None] + q = np.r_[np.logspace(-2, 2, 10)][None,:,None] + z = np.linspace(0, 1, 7)[None,None,:] + + y1 = cephes.mathieu_modcem2(m, q, -z)[0] + + fr = -cephes.mathieu_modcem2(m, q, 0)[0] / cephes.mathieu_modcem1(m, q, 0)[0] + y2 = (-cephes.mathieu_modcem2(m, q, z)[0] + - 2*fr*cephes.mathieu_modcem1(m, q, z)[0]) + + assert_allclose(y1, y2, rtol=1e-10) + + def test_mathieu_modsem1(self): + assert_equal(cephes.mathieu_modsem1(1,0,0),(0.0,0.0)) + + def test_mathieu_modsem2(self): + cephes.mathieu_modsem2(1,1,1) + + # Test reflection relation AMS 20.6.20 + m = np.arange(1, 4)[:,None,None] + q = np.r_[np.logspace(-2, 2, 10)][None,:,None] + z = np.linspace(0, 1, 7)[None,None,:] + + y1 = cephes.mathieu_modsem2(m, q, -z)[0] + fr = cephes.mathieu_modsem2(m, q, 0)[1] / cephes.mathieu_modsem1(m, q, 0)[1] + y2 = (cephes.mathieu_modsem2(m, q, z)[0] + - 2*fr*cephes.mathieu_modsem1(m, q, z)[0]) + assert_allclose(y1, y2, rtol=1e-10) + + def test_mathieu_overflow(self): + # Check that these return NaNs instead of causing a SEGV + assert_equal(cephes.mathieu_cem(10000, 0, 1.3), (np.nan, np.nan)) + assert_equal(cephes.mathieu_sem(10000, 0, 1.3), (np.nan, np.nan)) + assert_equal(cephes.mathieu_cem(10000, 1.5, 1.3), (np.nan, np.nan)) + assert_equal(cephes.mathieu_sem(10000, 1.5, 1.3), (np.nan, np.nan)) + assert_equal(cephes.mathieu_modcem1(10000, 1.5, 1.3), (np.nan, np.nan)) + assert_equal(cephes.mathieu_modsem1(10000, 1.5, 1.3), (np.nan, np.nan)) + assert_equal(cephes.mathieu_modcem2(10000, 1.5, 1.3), (np.nan, np.nan)) + assert_equal(cephes.mathieu_modsem2(10000, 1.5, 1.3), (np.nan, np.nan)) + + def test_mathieu_ticket_1847(self): + # Regression test --- this call had some out-of-bounds access + # and could return nan occasionally + for k in range(60): + v = cephes.mathieu_modsem2(2, 100, -1) + # Values from ACM TOMS 804 (derivate by numerical differentiation) + assert_allclose(v[0], 0.1431742913063671074347, rtol=1e-10) + assert_allclose(v[1], 0.9017807375832909144719, rtol=1e-4) + + def test_modfresnelm(self): + cephes.modfresnelm(0) + + def test_modfresnelp(self): + cephes.modfresnelp(0) + + def test_modstruve(self): + assert_equal(cephes.modstruve(1,0),0.0) + + def test_nbdtr(self): + assert_equal(cephes.nbdtr(1,1,1),1.0) + + def test_nbdtrc(self): + assert_equal(cephes.nbdtrc(1,1,1),0.0) + + def test_nbdtri(self): + assert_equal(cephes.nbdtri(1,1,1),1.0) + + def test_nbdtrik(self): + cephes.nbdtrik(1,.4,.5) + + def test_nbdtrin(self): + assert_equal(cephes.nbdtrin(1,0,0),5.0) + + def test_ncfdtr(self): + assert_equal(cephes.ncfdtr(1,1,1,0),0.0) + + def test_ncfdtri(self): + assert_equal(cephes.ncfdtri(1, 1, 1, 0), 0.0) + f = [0.5, 1, 1.5] + p = cephes.ncfdtr(2, 3, 1.5, f) + assert_allclose(cephes.ncfdtri(2, 3, 1.5, p), f) + + @pytest.mark.xfail( + reason=( + "ncfdtr uses a Boost math implementation but ncfdtridfd" + "inverts the less accurate cdflib implementation of ncfdtr." + ) + ) + def test_ncfdtridfd(self): + dfd = [1, 2, 3] + p = cephes.ncfdtr(2, dfd, 0.25, 15) + assert_allclose(cephes.ncfdtridfd(2, p, 0.25, 15), dfd) + + @pytest.mark.xfail( + reason=( + "ncfdtr uses a Boost math implementation but ncfdtridfn" + "inverts the less accurate cdflib implementation of ncfdtr." + ) + ) + def test_ncfdtridfn(self): + dfn = [0.1, 1, 2, 3, 1e4] + p = cephes.ncfdtr(dfn, 2, 0.25, 15) + assert_allclose(cephes.ncfdtridfn(p, 2, 0.25, 15), dfn, rtol=1e-5) + + @pytest.mark.xfail( + reason=( + "ncfdtr uses a Boost math implementation but ncfdtrinc" + "inverts the less accurate cdflib implementation of ncfdtr." + ) + ) + def test_ncfdtrinc(self): + nc = [0.5, 1.5, 2.0] + p = cephes.ncfdtr(2, 3, nc, 15) + assert_allclose(cephes.ncfdtrinc(2, 3, p, 15), nc) + + def test_nctdtr(self): + assert_equal(cephes.nctdtr(1,0,0),0.5) + assert_equal(cephes.nctdtr(9, 65536, 45), 0.0) + + assert_approx_equal(cephes.nctdtr(np.inf, 1., 1.), 0.5, 5) + assert_(np.isnan(cephes.nctdtr(2., np.inf, 10.))) + assert_approx_equal(cephes.nctdtr(2., 1., np.inf), 1.) + + assert_(np.isnan(cephes.nctdtr(np.nan, 1., 1.))) + assert_(np.isnan(cephes.nctdtr(2., np.nan, 1.))) + assert_(np.isnan(cephes.nctdtr(2., 1., np.nan))) + + def test_nctdtridf(self): + cephes.nctdtridf(1,0.5,0) + + def test_nctdtrinc(self): + cephes.nctdtrinc(1,0,0) + + def test_nctdtrit(self): + cephes.nctdtrit(.1,0.2,.5) + + def test_nrdtrimn(self): + assert_approx_equal(cephes.nrdtrimn(0.5,1,1),1.0) + + def test_nrdtrisd(self): + assert_allclose(cephes.nrdtrisd(0.5,0.5,0.5), 0.0, + atol=0, rtol=0) + + def test_obl_ang1(self): + cephes.obl_ang1(1,1,1,0) + + def test_obl_ang1_cv(self): + result = cephes.obl_ang1_cv(1,1,1,1,0) + assert_almost_equal(result[0],1.0) + assert_almost_equal(result[1],0.0) + + def test_obl_cv(self): + assert_equal(cephes.obl_cv(1,1,0),2.0) + + def test_obl_rad1(self): + cephes.obl_rad1(1,1,1,0) + + def test_obl_rad1_cv(self): + cephes.obl_rad1_cv(1,1,1,1,0) + + def test_obl_rad2(self): + cephes.obl_rad2(1,1,1,0) + + def test_obl_rad2_cv(self): + cephes.obl_rad2_cv(1,1,1,1,0) + + def test_pbdv(self): + assert_equal(cephes.pbdv(1,0),(0.0,1.0)) + + def test_pbvv(self): + cephes.pbvv(1,0) + + def test_pbwa(self): + cephes.pbwa(1,0) + + def test_pdtr(self): + val = cephes.pdtr(0, 1) + assert_almost_equal(val, np.exp(-1)) + # Edge case: m = 0. + val = cephes.pdtr([0, 1, 2], 0) + assert_array_equal(val, [1, 1, 1]) + + def test_pdtrc(self): + val = cephes.pdtrc(0, 1) + assert_almost_equal(val, 1 - np.exp(-1)) + # Edge case: m = 0. + val = cephes.pdtrc([0, 1, 2], 0.0) + assert_array_equal(val, [0, 0, 0]) + + def test_pdtri(self): + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "floating point number truncated to an integer") + cephes.pdtri(0.5,0.5) + + def test_pdtrik(self): + k = cephes.pdtrik(0.5, 1) + assert_almost_equal(cephes.gammaincc(k + 1, 1), 0.5) + # Edge case: m = 0 or very small. + k = cephes.pdtrik([[0], [0.25], [0.95]], [0, 1e-20, 1e-6]) + assert_array_equal(k, np.zeros((3, 3))) + + def test_pro_ang1(self): + cephes.pro_ang1(1,1,1,0) + + def test_pro_ang1_cv(self): + assert_array_almost_equal(cephes.pro_ang1_cv(1,1,1,1,0), + array((1.0,0.0))) + + def test_pro_cv(self): + assert_equal(cephes.pro_cv(1,1,0),2.0) + + def test_pro_rad1(self): + cephes.pro_rad1(1,1,1,0.1) + + def test_pro_rad1_cv(self): + cephes.pro_rad1_cv(1,1,1,1,0) + + def test_pro_rad2(self): + cephes.pro_rad2(1,1,1,0) + + def test_pro_rad2_cv(self): + cephes.pro_rad2_cv(1,1,1,1,0) + + def test_psi(self): + cephes.psi(1) + + def test_radian(self): + assert_equal(cephes.radian(0,0,0),0) + + def test_rgamma(self): + assert_equal(cephes.rgamma(1),1.0) + + def test_round(self): + assert_equal(cephes.round(3.4),3.0) + assert_equal(cephes.round(-3.4),-3.0) + assert_equal(cephes.round(3.6),4.0) + assert_equal(cephes.round(-3.6),-4.0) + assert_equal(cephes.round(3.5),4.0) + assert_equal(cephes.round(-3.5),-4.0) + + def test_shichi(self): + cephes.shichi(1) + + def test_sici(self): + cephes.sici(1) + + s, c = cephes.sici(np.inf) + assert_almost_equal(s, np.pi * 0.5) + assert_almost_equal(c, 0) + + s, c = cephes.sici(-np.inf) + assert_almost_equal(s, -np.pi * 0.5) + assert_(np.isnan(c), "cosine integral(-inf) is not nan") + + def test_sindg(self): + assert_equal(cephes.sindg(90),1.0) + + def test_smirnov(self): + assert_equal(cephes.smirnov(1,.1),0.9) + assert_(np.isnan(cephes.smirnov(1,np.nan))) + + def test_smirnovp(self): + assert_equal(cephes._smirnovp(1, .1), -1) + assert_equal(cephes._smirnovp(2, 0.75), -2*(0.25)**(2-1)) + assert_equal(cephes._smirnovp(3, 0.75), -3*(0.25)**(3-1)) + assert_(np.isnan(cephes._smirnovp(1, np.nan))) + + def test_smirnovc(self): + assert_equal(cephes._smirnovc(1,.1),0.1) + assert_(np.isnan(cephes._smirnovc(1,np.nan))) + x10 = np.linspace(0, 1, 11, endpoint=True) + assert_almost_equal(cephes._smirnovc(3, x10), 1-cephes.smirnov(3, x10)) + x4 = np.linspace(0, 1, 5, endpoint=True) + assert_almost_equal(cephes._smirnovc(4, x4), 1-cephes.smirnov(4, x4)) + + def test_smirnovi(self): + assert_almost_equal(cephes.smirnov(1,cephes.smirnovi(1,0.4)),0.4) + assert_almost_equal(cephes.smirnov(1,cephes.smirnovi(1,0.6)),0.6) + assert_(np.isnan(cephes.smirnovi(1,np.nan))) + + def test_smirnovci(self): + assert_almost_equal(cephes._smirnovc(1,cephes._smirnovci(1,0.4)),0.4) + assert_almost_equal(cephes._smirnovc(1,cephes._smirnovci(1,0.6)),0.6) + assert_(np.isnan(cephes._smirnovci(1,np.nan))) + + def test_spence(self): + assert_equal(cephes.spence(1),0.0) + + def test_stdtr(self): + assert_equal(cephes.stdtr(1,0),0.5) + assert_almost_equal(cephes.stdtr(1,1), 0.75) + assert_almost_equal(cephes.stdtr(1,2), 0.852416382349) + + def test_stdtridf(self): + cephes.stdtridf(0.7,1) + + def test_stdtrit(self): + cephes.stdtrit(1,0.7) + + def test_struve(self): + assert_equal(cephes.struve(0,0),0.0) + + def test_tandg(self): + assert_equal(cephes.tandg(45),1.0) + + def test_tklmbda(self): + assert_almost_equal(cephes.tklmbda(1,1),1.0) + + def test_y0(self): + cephes.y0(1) + + def test_y1(self): + cephes.y1(1) + + def test_yn(self): + cephes.yn(1,1) + + def test_yv(self): + cephes.yv(1,1) + + def test_yve(self): + cephes.yve(1,1) + + def test_wofz(self): + z = [complex(624.2,-0.26123), complex(-0.4,3.), complex(0.6,2.), + complex(-1.,1.), complex(-1.,-9.), complex(-1.,9.), + complex(-0.0000000234545,1.1234), complex(-3.,5.1), + complex(-53,30.1), complex(0.0,0.12345), + complex(11,1), complex(-22,-2), complex(9,-28), + complex(21,-33), complex(1e5,1e5), complex(1e14,1e14) + ] + w = [ + complex(-3.78270245518980507452677445620103199303131110e-7, + 0.000903861276433172057331093754199933411710053155), + complex(0.1764906227004816847297495349730234591778719532788, + -0.02146550539468457616788719893991501311573031095617), + complex(0.2410250715772692146133539023007113781272362309451, + 0.06087579663428089745895459735240964093522265589350), + complex(0.30474420525691259245713884106959496013413834051768, + -0.20821893820283162728743734725471561394145872072738), + complex(7.317131068972378096865595229600561710140617977e34, + 8.321873499714402777186848353320412813066170427e34), + complex(0.0615698507236323685519612934241429530190806818395, + -0.00676005783716575013073036218018565206070072304635), + complex(0.3960793007699874918961319170187598400134746631, + -5.593152259116644920546186222529802777409274656e-9), + complex(0.08217199226739447943295069917990417630675021771804, + -0.04701291087643609891018366143118110965272615832184), + complex(0.00457246000350281640952328010227885008541748668738, + -0.00804900791411691821818731763401840373998654987934), + complex(0.8746342859608052666092782112565360755791467973338452, + 0.), + complex(0.00468190164965444174367477874864366058339647648741, + 0.0510735563901306197993676329845149741675029197050), + complex(-0.0023193175200187620902125853834909543869428763219, + -0.025460054739731556004902057663500272721780776336), + complex(9.11463368405637174660562096516414499772662584e304, + 3.97101807145263333769664875189354358563218932e305), + complex(-4.4927207857715598976165541011143706155432296e281, + -2.8019591213423077494444700357168707775769028e281), + complex(2.820947917809305132678577516325951485807107151e-6, + 2.820947917668257736791638444590253942253354058e-6), + complex(2.82094791773878143474039725787438662716372268e-15, + 2.82094791773878143474039725773333923127678361e-15) + ] + assert_func_equal(cephes.wofz, w, z, rtol=1e-13) + + +class TestAiry: + def test_airy(self): + # This tests the airy function to ensure 8 place accuracy in computation + + x = special.airy(.99) + assert_array_almost_equal( + x, + array([0.13689066,-0.16050153,1.19815925,0.92046818]), + 8, + ) + x = special.airy(.41) + assert_array_almost_equal( + x, + array([0.25238916,-.23480512,0.80686202,0.51053919]), + 8, + ) + x = special.airy(-.36) + assert_array_almost_equal( + x, + array([0.44508477,-0.23186773,0.44939534,0.48105354]), + 8, + ) + + def test_airye(self): + a = special.airye(0.01) + b = special.airy(0.01) + b1 = [None]*4 + for n in range(2): + b1[n] = b[n]*exp(2.0/3.0*0.01*sqrt(0.01)) + for n in range(2,4): + b1[n] = b[n]*exp(-abs(real(2.0/3.0*0.01*sqrt(0.01)))) + assert_array_almost_equal(a,b1,6) + + def test_bi_zeros(self): + bi = special.bi_zeros(2) + bia = (array([-1.17371322, -3.2710930]), + array([-2.29443968, -4.07315509]), + array([-0.45494438, 0.39652284]), + array([0.60195789, -0.76031014])) + assert_array_almost_equal(bi,bia,4) + + bi = special.bi_zeros(5) + assert_array_almost_equal(bi[0],array([-1.173713222709127, + -3.271093302836352, + -4.830737841662016, + -6.169852128310251, + -7.376762079367764]),11) + + assert_array_almost_equal(bi[1],array([-2.294439682614122, + -4.073155089071828, + -5.512395729663599, + -6.781294445990305, + -7.940178689168587]),10) + + assert_array_almost_equal(bi[2],array([-0.454944383639657, + 0.396522836094465, + -0.367969161486959, + 0.349499116831805, + -0.336026240133662]),11) + + assert_array_almost_equal(bi[3],array([0.601957887976239, + -0.760310141492801, + 0.836991012619261, + -0.88947990142654, + 0.929983638568022]),10) + + def test_ai_zeros(self): + ai = special.ai_zeros(1) + assert_array_almost_equal(ai,(array([-2.33810741]), + array([-1.01879297]), + array([0.5357]), + array([0.7012])),4) + + @pytest.mark.fail_slow(5) + def test_ai_zeros_big(self): + z, zp, ai_zpx, aip_zx = special.ai_zeros(50000) + ai_z, aip_z, _, _ = special.airy(z) + ai_zp, aip_zp, _, _ = special.airy(zp) + + ai_envelope = 1/abs(z)**(1./4) + aip_envelope = abs(zp)**(1./4) + + # Check values + assert_allclose(ai_zpx, ai_zp, rtol=1e-10) + assert_allclose(aip_zx, aip_z, rtol=1e-10) + + # Check they are zeros + assert_allclose(ai_z/ai_envelope, 0, atol=1e-10, rtol=0) + assert_allclose(aip_zp/aip_envelope, 0, atol=1e-10, rtol=0) + + # Check first zeros, DLMF 9.9.1 + assert_allclose(z[:6], + [-2.3381074105, -4.0879494441, -5.5205598281, + -6.7867080901, -7.9441335871, -9.0226508533], rtol=1e-10) + assert_allclose(zp[:6], + [-1.0187929716, -3.2481975822, -4.8200992112, + -6.1633073556, -7.3721772550, -8.4884867340], rtol=1e-10) + + @pytest.mark.fail_slow(5) + def test_bi_zeros_big(self): + z, zp, bi_zpx, bip_zx = special.bi_zeros(50000) + _, _, bi_z, bip_z = special.airy(z) + _, _, bi_zp, bip_zp = special.airy(zp) + + bi_envelope = 1/abs(z)**(1./4) + bip_envelope = abs(zp)**(1./4) + + # Check values + assert_allclose(bi_zpx, bi_zp, rtol=1e-10) + assert_allclose(bip_zx, bip_z, rtol=1e-10) + + # Check they are zeros + assert_allclose(bi_z/bi_envelope, 0, atol=1e-10, rtol=0) + assert_allclose(bip_zp/bip_envelope, 0, atol=1e-10, rtol=0) + + # Check first zeros, DLMF 9.9.2 + assert_allclose(z[:6], + [-1.1737132227, -3.2710933028, -4.8307378417, + -6.1698521283, -7.3767620794, -8.4919488465], rtol=1e-10) + assert_allclose(zp[:6], + [-2.2944396826, -4.0731550891, -5.5123957297, + -6.7812944460, -7.9401786892, -9.0195833588], rtol=1e-10) + + +class TestAssocLaguerre: + def test_assoc_laguerre(self): + a1 = special.genlaguerre(11,1) + a2 = special.assoc_laguerre(.2,11,1) + assert_array_almost_equal(a2,a1(.2),8) + a2 = special.assoc_laguerre(1,11,1) + assert_array_almost_equal(a2,a1(1),8) + + +class TestBesselpoly: + def test_besselpoly(self): + pass + + +class TestKelvin: + def test_bei(self): + mbei = special.bei(2) + assert_almost_equal(mbei, 0.9722916273066613,5) # this may not be exact + + def test_beip(self): + mbeip = special.beip(2) + assert_almost_equal(mbeip,0.91701361338403631,5) # this may not be exact + + def test_ber(self): + mber = special.ber(2) + assert_almost_equal(mber,0.75173418271380821,5) # this may not be exact + + def test_berp(self): + mberp = special.berp(2) + assert_almost_equal(mberp,-0.49306712470943909,5) # this may not be exact + + def test_bei_zeros(self): + # Abramowitz & Stegun, Table 9.12 + bi = special.bei_zeros(5) + assert_array_almost_equal(bi,array([5.02622, + 9.45541, + 13.89349, + 18.33398, + 22.77544]),4) + + def test_beip_zeros(self): + bip = special.beip_zeros(5) + assert_array_almost_equal(bip,array([3.772673304934953, + 8.280987849760042, + 12.742147523633703, + 17.193431752512542, + 21.641143941167325]),8) + + def test_ber_zeros(self): + ber = special.ber_zeros(5) + assert_array_almost_equal(ber,array([2.84892, + 7.23883, + 11.67396, + 16.11356, + 20.55463]),4) + + def test_berp_zeros(self): + brp = special.berp_zeros(5) + assert_array_almost_equal(brp,array([6.03871, + 10.51364, + 14.96844, + 19.41758, + 23.86430]),4) + + def test_kelvin(self): + mkelv = special.kelvin(2) + assert_array_almost_equal(mkelv,(special.ber(2) + special.bei(2)*1j, + special.ker(2) + special.kei(2)*1j, + special.berp(2) + special.beip(2)*1j, + special.kerp(2) + special.keip(2)*1j),8) + + def test_kei(self): + mkei = special.kei(2) + assert_almost_equal(mkei,-0.20240006776470432,5) + + def test_keip(self): + mkeip = special.keip(2) + assert_almost_equal(mkeip,0.21980790991960536,5) + + def test_ker(self): + mker = special.ker(2) + assert_almost_equal(mker,-0.041664513991509472,5) + + def test_kerp(self): + mkerp = special.kerp(2) + assert_almost_equal(mkerp,-0.10660096588105264,5) + + def test_kei_zeros(self): + kei = special.kei_zeros(5) + assert_array_almost_equal(kei,array([3.91467, + 8.34422, + 12.78256, + 17.22314, + 21.66464]),4) + + def test_keip_zeros(self): + keip = special.keip_zeros(5) + assert_array_almost_equal(keip,array([4.93181, + 9.40405, + 13.85827, + 18.30717, + 22.75379]),4) + + # numbers come from 9.9 of A&S pg. 381 + def test_kelvin_zeros(self): + tmp = special.kelvin_zeros(5) + berz,beiz,kerz,keiz,berpz,beipz,kerpz,keipz = tmp + assert_array_almost_equal(berz,array([2.84892, + 7.23883, + 11.67396, + 16.11356, + 20.55463]),4) + assert_array_almost_equal(beiz,array([5.02622, + 9.45541, + 13.89349, + 18.33398, + 22.77544]),4) + assert_array_almost_equal(kerz,array([1.71854, + 6.12728, + 10.56294, + 15.00269, + 19.44382]),4) + assert_array_almost_equal(keiz,array([3.91467, + 8.34422, + 12.78256, + 17.22314, + 21.66464]),4) + assert_array_almost_equal(berpz,array([6.03871, + 10.51364, + 14.96844, + 19.41758, + 23.86430]),4) + assert_array_almost_equal(beipz,array([3.77267, + # table from 1927 had 3.77320 + # but this is more accurate + 8.28099, + 12.74215, + 17.19343, + 21.64114]),4) + assert_array_almost_equal(kerpz,array([2.66584, + 7.17212, + 11.63218, + 16.08312, + 20.53068]),4) + assert_array_almost_equal(keipz,array([4.93181, + 9.40405, + 13.85827, + 18.30717, + 22.75379]),4) + + def test_ker_zeros(self): + ker = special.ker_zeros(5) + assert_array_almost_equal(ker,array([1.71854, + 6.12728, + 10.56294, + 15.00269, + 19.44381]),4) + + def test_kerp_zeros(self): + kerp = special.kerp_zeros(5) + assert_array_almost_equal(kerp,array([2.66584, + 7.17212, + 11.63218, + 16.08312, + 20.53068]),4) + + +class TestBernoulli: + def test_bernoulli(self): + brn = special.bernoulli(5) + assert_array_almost_equal(brn,array([1.0000, + -0.5000, + 0.1667, + 0.0000, + -0.0333, + 0.0000]),4) + + +class TestBeta: + """ + Test beta and betaln. + """ + + def test_beta(self): + assert_equal(special.beta(1, 1), 1.0) + assert_allclose(special.beta(-100.3, 1e-200), special.gamma(1e-200)) + assert_allclose(special.beta(0.0342, 171), 24.070498359873497, + rtol=1e-13, atol=0) + + bet = special.beta(2, 4) + betg = (special.gamma(2)*special.gamma(4))/special.gamma(6) + assert_allclose(bet, betg, rtol=1e-13) + + def test_beta_inf(self): + assert_(np.isinf(special.beta(-1, 2))) + + def test_betaln(self): + assert_equal(special.betaln(1, 1), 0.0) + assert_allclose(special.betaln(-100.3, 1e-200), + special.gammaln(1e-200)) + assert_allclose(special.betaln(0.0342, 170), 3.1811881124242447, + rtol=1e-14, atol=0) + + betln = special.betaln(2, 4) + bet = log(abs(special.beta(2, 4))) + assert_allclose(betln, bet, rtol=1e-13) + + +class TestBetaInc: + """ + Tests for betainc, betaincinv, betaincc, betainccinv. + """ + + def test_a1_b1(self): + # betainc(1, 1, x) is x. + x = np.array([0, 0.25, 1]) + assert_equal(special.betainc(1, 1, x), x) + assert_equal(special.betaincinv(1, 1, x), x) + assert_equal(special.betaincc(1, 1, x), 1 - x) + assert_equal(special.betainccinv(1, 1, x), 1 - x) + + # Nontrivial expected values computed with mpmath: + # from mpmath import mp + # mp.dps = 100 + # p = mp.betainc(a, b, 0, x, regularized=True) + # + # or, e.g., + # + # p = 0.25 + # a, b = 0.0342, 171 + # x = mp.findroot( + # lambda t: mp.betainc(a, b, 0, t, regularized=True) - p, + # (8e-21, 9e-21), + # solver='anderson', + # ) + # + @pytest.mark.parametrize( + 'a, b, x, p', + [(2, 4, 0.3138101704556974, 0.5), + (0.0342, 171.0, 1e-10, 0.552699169018070910641), + # gh-3761: + (0.0342, 171, 8.42313169354797e-21, 0.25), + # gh-4244: + (0.0002742794749792665, 289206.03125, 1.639984034231756e-56, + 0.9688708782196045), + # gh-12796: + (4, 99997, 0.0001947841578892121, 0.999995)]) + def test_betainc_betaincinv(self, a, b, x, p): + p1 = special.betainc(a, b, x) + assert_allclose(p1, p, rtol=1e-15) + x1 = special.betaincinv(a, b, p) + assert_allclose(x1, x, rtol=5e-13) + + # Expected values computed with mpmath: + # from mpmath import mp + # mp.dps = 100 + # p = mp.betainc(a, b, x, 1, regularized=True) + @pytest.mark.parametrize('a, b, x, p', + [(2.5, 3.0, 0.25, 0.833251953125), + (7.5, 13.25, 0.375, 0.43298734645560368593), + (0.125, 7.5, 0.425, 0.0006688257851314237), + (0.125, 18.0, 1e-6, 0.72982359145096327654), + (0.125, 18.0, 0.996, 7.2745875538380150586e-46), + (0.125, 24.0, 0.75, 3.70853404816862016966e-17), + (16.0, 0.75, 0.99999999975, + 5.4408759277418629909e-07), + # gh-4677 (numbers from stackoverflow question): + (0.4211959643503401, 16939.046996018118, + 0.000815296167195521, 1e-7)]) + def test_betaincc_betainccinv(self, a, b, x, p): + p1 = special.betaincc(a, b, x) + assert_allclose(p1, p, rtol=5e-15) + x1 = special.betainccinv(a, b, p) + assert_allclose(x1, x, rtol=8e-15) + + @pytest.mark.parametrize( + 'a, b, y, ref', + [(14.208308325339239, 14.208308325339239, 7.703145458496392e-307, + 8.566004561846704e-23), + (14.0, 14.5, 1e-280, 2.9343915006642424e-21), + (3.5, 15.0, 4e-95, 1.3290751429289227e-28), + (10.0, 1.25, 2e-234, 3.982659092143654e-24), + (4.0, 99997.0, 5e-88, 3.309800566862242e-27)] + ) + def test_betaincinv_tiny_y(self, a, b, y, ref): + # Test with extremely small y values. This test includes + # a regression test for an issue in the boost code; + # see https://github.com/boostorg/math/issues/961 + # + # The reference values were computed with mpmath. For example, + # + # from mpmath import mp + # mp.dps = 1000 + # a = 14.208308325339239 + # p = 7.703145458496392e-307 + # x = mp.findroot(lambda t: mp.betainc(a, a, 0, t, + # regularized=True) - p, + # x0=8.566e-23) + # print(float(x)) + # + x = special.betaincinv(a, b, y) + assert_allclose(x, ref, rtol=1e-14) + + @pytest.mark.parametrize('func', [special.betainc, special.betaincinv, + special.betaincc, special.betainccinv]) + @pytest.mark.parametrize('args', [(-1.0, 2, 0.5), (0, 2, 0.5), + (1.5, -2.0, 0.5), (1.5, 0, 0.5), + (1.5, 2.0, -0.3), (1.5, 2.0, 1.1)]) + def test_betainc_domain_errors(self, func, args): + with special.errstate(domain='raise'): + with pytest.raises(special.SpecialFunctionError, match='domain'): + special.betainc(*args) + + @pytest.mark.parametrize('dtype', [np.float32, np.float64]) + def test_gh21426(self, dtype): + # Test for gh-21426: betaincinv must not return NaN + a = np.array([5.], dtype=dtype) + x = np.array([0.5], dtype=dtype) + result = special.betaincinv(a, a, x) + assert_allclose(result, x, rtol=10 * np.finfo(dtype).eps) + + +class TestCombinatorics: + def test_comb(self): + assert_allclose(special.comb([10, 10], [3, 4]), [120., 210.]) + assert_allclose(special.comb(10, 3), 120.) + assert_equal(special.comb(10, 3, exact=True), 120) + assert_equal(special.comb(10, 3, exact=True, repetition=True), 220) + + assert_allclose([special.comb(20, k, exact=True) for k in range(21)], + special.comb(20, list(range(21))), atol=1e-15) + + ii = np.iinfo(int).max + 1 + assert_equal(special.comb(ii, ii-1, exact=True), ii) + + expected = 100891344545564193334812497256 + assert special.comb(100, 50, exact=True) == expected + + def test_comb_with_np_int64(self): + n = 70 + k = 30 + np_n = np.int64(n) + np_k = np.int64(k) + res_np = special.comb(np_n, np_k, exact=True) + res_py = special.comb(n, k, exact=True) + assert res_np == res_py + + def test_comb_zeros(self): + assert_equal(special.comb(2, 3, exact=True), 0) + assert_equal(special.comb(-1, 3, exact=True), 0) + assert_equal(special.comb(2, -1, exact=True), 0) + assert_equal(special.comb(2, -1, exact=False), 0) + assert_allclose(special.comb([2, -1, 2, 10], [3, 3, -1, 3]), [0., 0., 0., 120.]) + + @pytest.mark.thread_unsafe + def test_comb_exact_non_int_dep(self): + msg = "`exact=True`" + with pytest.deprecated_call(match=msg): + special.comb(3.4, 4, exact=True) + + def test_perm(self): + assert_allclose(special.perm([10, 10], [3, 4]), [720., 5040.]) + assert_almost_equal(special.perm(10, 3), 720.) + assert_equal(special.perm(10, 3, exact=True), 720) + + def test_perm_zeros(self): + assert_equal(special.perm(2, 3, exact=True), 0) + assert_equal(special.perm(-1, 3, exact=True), 0) + assert_equal(special.perm(2, -1, exact=True), 0) + assert_equal(special.perm(2, -1, exact=False), 0) + assert_allclose(special.perm([2, -1, 2, 10], [3, 3, -1, 3]), [0., 0., 0., 720.]) + + @pytest.mark.thread_unsafe + def test_perm_iv(self): + # currently `exact=True` only support scalars + with pytest.raises(ValueError, match="scalar integers"): + special.perm([1, 2], [4, 5], exact=True) + + # Non-integral scalars with N < k, or N,k < 0 used to return 0, this is now + # deprecated and will raise an error in SciPy 1.16.0 + with pytest.deprecated_call(match="Non-integer"): + special.perm(4.6, 6, exact=True) + with pytest.deprecated_call(match="Non-integer"): + special.perm(-4.6, 3, exact=True) + with pytest.deprecated_call(match="Non-integer"): + special.perm(4, -3.9, exact=True) + + # Non-integral scalars which aren't included in the cases above an raise an + # error directly without deprecation as this code never worked + with pytest.raises(ValueError, match="Non-integer"): + special.perm(6.0, 4.6, exact=True) + + +class TestTrigonometric: + def test_cbrt(self): + cb = special.cbrt(27) + cbrl = 27**(1.0/3.0) + assert_approx_equal(cb,cbrl) + + def test_cbrtmore(self): + cb1 = special.cbrt(27.9) + cbrl1 = 27.9**(1.0/3.0) + assert_almost_equal(cb1,cbrl1,8) + + def test_cosdg(self): + cdg = special.cosdg(90) + cdgrl = cos(pi/2.0) + assert_almost_equal(cdg,cdgrl,8) + + def test_cosdgmore(self): + cdgm = special.cosdg(30) + cdgmrl = cos(pi/6.0) + assert_almost_equal(cdgm,cdgmrl,8) + + def test_cosm1(self): + cs = (special.cosm1(0),special.cosm1(.3),special.cosm1(pi/10)) + csrl = (cos(0)-1,cos(.3)-1,cos(pi/10)-1) + assert_array_almost_equal(cs,csrl,8) + + def test_cotdg(self): + ct = special.cotdg(30) + ctrl = tan(pi/6.0)**(-1) + assert_almost_equal(ct,ctrl,8) + + def test_cotdgmore(self): + ct1 = special.cotdg(45) + ctrl1 = tan(pi/4.0)**(-1) + assert_almost_equal(ct1,ctrl1,8) + + def test_specialpoints(self): + assert_almost_equal(special.cotdg(45), 1.0, 14) + assert_almost_equal(special.cotdg(-45), -1.0, 14) + assert_almost_equal(special.cotdg(90), 0.0, 14) + assert_almost_equal(special.cotdg(-90), 0.0, 14) + assert_almost_equal(special.cotdg(135), -1.0, 14) + assert_almost_equal(special.cotdg(-135), 1.0, 14) + assert_almost_equal(special.cotdg(225), 1.0, 14) + assert_almost_equal(special.cotdg(-225), -1.0, 14) + assert_almost_equal(special.cotdg(270), 0.0, 14) + assert_almost_equal(special.cotdg(-270), 0.0, 14) + assert_almost_equal(special.cotdg(315), -1.0, 14) + assert_almost_equal(special.cotdg(-315), 1.0, 14) + assert_almost_equal(special.cotdg(765), 1.0, 14) + + def test_sinc(self): + # the sinc implementation and more extensive sinc tests are in numpy + assert_array_equal(special.sinc([0]), 1) + assert_equal(special.sinc(0.0), 1.0) + + def test_sindg(self): + sn = special.sindg(90) + assert_equal(sn,1.0) + + def test_sindgmore(self): + snm = special.sindg(30) + snmrl = sin(pi/6.0) + assert_almost_equal(snm,snmrl,8) + snm1 = special.sindg(45) + snmrl1 = sin(pi/4.0) + assert_almost_equal(snm1,snmrl1,8) + + +class TestTandg: + + def test_tandg(self): + tn = special.tandg(30) + tnrl = tan(pi/6.0) + assert_almost_equal(tn,tnrl,8) + + def test_tandgmore(self): + tnm = special.tandg(45) + tnmrl = tan(pi/4.0) + assert_almost_equal(tnm,tnmrl,8) + tnm1 = special.tandg(60) + tnmrl1 = tan(pi/3.0) + assert_almost_equal(tnm1,tnmrl1,8) + + def test_specialpoints(self): + assert_almost_equal(special.tandg(0), 0.0, 14) + assert_almost_equal(special.tandg(45), 1.0, 14) + assert_almost_equal(special.tandg(-45), -1.0, 14) + assert_almost_equal(special.tandg(135), -1.0, 14) + assert_almost_equal(special.tandg(-135), 1.0, 14) + assert_almost_equal(special.tandg(180), 0.0, 14) + assert_almost_equal(special.tandg(-180), 0.0, 14) + assert_almost_equal(special.tandg(225), 1.0, 14) + assert_almost_equal(special.tandg(-225), -1.0, 14) + assert_almost_equal(special.tandg(315), -1.0, 14) + assert_almost_equal(special.tandg(-315), 1.0, 14) + + +class TestEllip: + def test_ellipj_nan(self): + """Regression test for #912.""" + special.ellipj(0.5, np.nan) + + def test_ellipj(self): + el = special.ellipj(0.2,0) + rel = [sin(0.2),cos(0.2),1.0,0.20] + assert_array_almost_equal(el,rel,13) + + def test_ellipk(self): + elk = special.ellipk(.2) + assert_almost_equal(elk,1.659623598610528,11) + + assert_equal(special.ellipkm1(0.0), np.inf) + assert_equal(special.ellipkm1(1.0), pi/2) + assert_equal(special.ellipkm1(np.inf), 0.0) + assert_equal(special.ellipkm1(np.nan), np.nan) + assert_equal(special.ellipkm1(-1), np.nan) + assert_allclose(special.ellipk(-10), 0.7908718902387385) + + def test_ellipkinc(self): + elkinc = special.ellipkinc(pi/2,.2) + elk = special.ellipk(0.2) + assert_almost_equal(elkinc,elk,15) + alpha = 20*pi/180 + phi = 45*pi/180 + m = sin(alpha)**2 + elkinc = special.ellipkinc(phi,m) + assert_almost_equal(elkinc,0.79398143,8) + # From pg. 614 of A & S + + assert_equal(special.ellipkinc(pi/2, 0.0), pi/2) + assert_equal(special.ellipkinc(pi/2, 1.0), np.inf) + assert_equal(special.ellipkinc(pi/2, -np.inf), 0.0) + assert_equal(special.ellipkinc(pi/2, np.nan), np.nan) + assert_equal(special.ellipkinc(pi/2, 2), np.nan) + assert_equal(special.ellipkinc(0, 0.5), 0.0) + assert_equal(special.ellipkinc(np.inf, 0.5), np.inf) + assert_equal(special.ellipkinc(-np.inf, 0.5), -np.inf) + assert_equal(special.ellipkinc(np.inf, np.inf), np.nan) + assert_equal(special.ellipkinc(np.inf, -np.inf), np.nan) + assert_equal(special.ellipkinc(-np.inf, -np.inf), np.nan) + assert_equal(special.ellipkinc(-np.inf, np.inf), np.nan) + assert_equal(special.ellipkinc(np.nan, 0.5), np.nan) + assert_equal(special.ellipkinc(np.nan, np.nan), np.nan) + + assert_allclose(special.ellipkinc(0.38974112035318718, 1), 0.4, rtol=1e-14) + assert_allclose(special.ellipkinc(1.5707, -10), 0.79084284661724946) + + def test_ellipkinc_2(self): + # Regression test for gh-3550 + # ellipkinc(phi, mbad) was NaN and mvals[2:6] were twice the correct value + mbad = 0.68359375000000011 + phi = 0.9272952180016123 + m = np.nextafter(mbad, 0) + mvals = [] + for j in range(10): + mvals.append(m) + m = np.nextafter(m, 1) + f = special.ellipkinc(phi, mvals) + assert_array_almost_equal_nulp(f, np.full_like(f, 1.0259330100195334), 1) + # this bug also appears at phi + n * pi for at least small n + f1 = special.ellipkinc(phi + pi, mvals) + assert_array_almost_equal_nulp(f1, np.full_like(f1, 5.1296650500976675), 2) + + def test_ellipkinc_singular(self): + # ellipkinc(phi, 1) has closed form and is finite only for phi in (-pi/2, pi/2) + xlog = np.logspace(-300, -17, 25) + xlin = np.linspace(1e-17, 0.1, 25) + xlin2 = np.linspace(0.1, pi/2, 25, endpoint=False) + + assert_allclose(special.ellipkinc(xlog, 1), np.arcsinh(np.tan(xlog)), + rtol=1e14) + assert_allclose(special.ellipkinc(xlin, 1), np.arcsinh(np.tan(xlin)), + rtol=1e14) + assert_allclose(special.ellipkinc(xlin2, 1), np.arcsinh(np.tan(xlin2)), + rtol=1e14) + assert_equal(special.ellipkinc(np.pi/2, 1), np.inf) + assert_allclose(special.ellipkinc(-xlog, 1), np.arcsinh(np.tan(-xlog)), + rtol=1e14) + assert_allclose(special.ellipkinc(-xlin, 1), np.arcsinh(np.tan(-xlin)), + rtol=1e14) + assert_allclose(special.ellipkinc(-xlin2, 1), np.arcsinh(np.tan(-xlin2)), + rtol=1e14) + assert_equal(special.ellipkinc(-np.pi/2, 1), np.inf) + + def test_ellipe(self): + ele = special.ellipe(.2) + assert_almost_equal(ele,1.4890350580958529,8) + + assert_equal(special.ellipe(0.0), pi/2) + assert_equal(special.ellipe(1.0), 1.0) + assert_equal(special.ellipe(-np.inf), np.inf) + assert_equal(special.ellipe(np.nan), np.nan) + assert_equal(special.ellipe(2), np.nan) + assert_allclose(special.ellipe(-10), 3.6391380384177689) + + def test_ellipeinc(self): + eleinc = special.ellipeinc(pi/2,.2) + ele = special.ellipe(0.2) + assert_almost_equal(eleinc,ele,14) + # pg 617 of A & S + alpha, phi = 52*pi/180,35*pi/180 + m = sin(alpha)**2 + eleinc = special.ellipeinc(phi,m) + assert_almost_equal(eleinc, 0.58823065, 8) + + assert_equal(special.ellipeinc(pi/2, 0.0), pi/2) + assert_equal(special.ellipeinc(pi/2, 1.0), 1.0) + assert_equal(special.ellipeinc(pi/2, -np.inf), np.inf) + assert_equal(special.ellipeinc(pi/2, np.nan), np.nan) + assert_equal(special.ellipeinc(pi/2, 2), np.nan) + assert_equal(special.ellipeinc(0, 0.5), 0.0) + assert_equal(special.ellipeinc(np.inf, 0.5), np.inf) + assert_equal(special.ellipeinc(-np.inf, 0.5), -np.inf) + assert_equal(special.ellipeinc(np.inf, -np.inf), np.inf) + assert_equal(special.ellipeinc(-np.inf, -np.inf), -np.inf) + assert_equal(special.ellipeinc(np.inf, np.inf), np.nan) + assert_equal(special.ellipeinc(-np.inf, np.inf), np.nan) + assert_equal(special.ellipeinc(np.nan, 0.5), np.nan) + assert_equal(special.ellipeinc(np.nan, np.nan), np.nan) + assert_allclose(special.ellipeinc(1.5707, -10), 3.6388185585822876) + + def test_ellipeinc_2(self): + # Regression test for gh-3550 + # ellipeinc(phi, mbad) was NaN and mvals[2:6] were twice the correct value + mbad = 0.68359375000000011 + phi = 0.9272952180016123 + m = np.nextafter(mbad, 0) + mvals = [] + for j in range(10): + mvals.append(m) + m = np.nextafter(m, 1) + f = special.ellipeinc(phi, mvals) + assert_array_almost_equal_nulp(f, np.full_like(f, 0.84442884574781019), 2) + # this bug also appears at phi + n * pi for at least small n + f1 = special.ellipeinc(phi + pi, mvals) + assert_array_almost_equal_nulp(f1, np.full_like(f1, 3.3471442287390509), 4) + + +class TestEllipCarlson: + """Test for Carlson elliptic integrals ellipr[cdfgj]. + The special values used in these tests can be found in Sec. 3 of Carlson + (1994), https://arxiv.org/abs/math/9409227 + """ + def test_elliprc(self): + assert_allclose(elliprc(1, 1), 1) + assert elliprc(1, inf) == 0.0 + assert isnan(elliprc(1, 0)) + assert elliprc(1, complex(1, inf)) == 0.0 + args = array([[0.0, 0.25], + [2.25, 2.0], + [0.0, 1.0j], + [-1.0j, 1.0j], + [0.25, -2.0], + [1.0j, -1.0]]) + expected_results = array([np.pi, + np.log(2.0), + 1.1107207345396 * (1.0-1.0j), + 1.2260849569072-0.34471136988768j, + np.log(2.0) / 3.0, + 0.77778596920447+0.19832484993429j]) + for i, arr in enumerate(args): + assert_allclose(elliprc(*arr), expected_results[i]) + + def test_elliprd(self): + assert_allclose(elliprd(1, 1, 1), 1) + assert_allclose(elliprd(0, 2, 1) / 3.0, 0.59907011736779610371) + assert elliprd(1, 1, inf) == 0.0 + assert np.isinf(elliprd(1, 1, 0)) + assert np.isinf(elliprd(1, 1, complex(0, 0))) + assert np.isinf(elliprd(0, 1, complex(0, 0))) + assert isnan(elliprd(1, 1, -np.finfo(np.float64).tiny / 2.0)) + assert isnan(elliprd(1, 1, complex(-1, 0))) + args = array([[0.0, 2.0, 1.0], + [2.0, 3.0, 4.0], + [1.0j, -1.0j, 2.0], + [0.0, 1.0j, -1.0j], + [0.0, -1.0+1.0j, 1.0j], + [-2.0-1.0j, -1.0j, -1.0+1.0j]]) + expected_results = array([1.7972103521034, + 0.16510527294261, + 0.65933854154220, + 1.2708196271910+2.7811120159521j, + -1.8577235439239-0.96193450888839j, + 1.8249027393704-1.2218475784827j]) + for i, arr in enumerate(args): + assert_allclose(elliprd(*arr), expected_results[i]) + + def test_elliprf(self): + assert_allclose(elliprf(1, 1, 1), 1) + assert_allclose(elliprf(0, 1, 2), 1.31102877714605990523) + assert elliprf(1, inf, 1) == 0.0 + assert np.isinf(elliprf(0, 1, 0)) + assert isnan(elliprf(1, 1, -1)) + assert elliprf(complex(inf), 0, 1) == 0.0 + assert isnan(elliprf(1, 1, complex(-inf, 1))) + args = array([[1.0, 2.0, 0.0], + [1.0j, -1.0j, 0.0], + [0.5, 1.0, 0.0], + [-1.0+1.0j, 1.0j, 0.0], + [2.0, 3.0, 4.0], + [1.0j, -1.0j, 2.0], + [-1.0+1.0j, 1.0j, 1.0-1.0j]]) + expected_results = array([1.3110287771461, + 1.8540746773014, + 1.8540746773014, + 0.79612586584234-1.2138566698365j, + 0.58408284167715, + 1.0441445654064, + 0.93912050218619-0.53296252018635j]) + for i, arr in enumerate(args): + assert_allclose(elliprf(*arr), expected_results[i]) + + def test_elliprg(self): + assert_allclose(elliprg(1, 1, 1), 1) + assert_allclose(elliprg(0, 0, 1), 0.5) + assert_allclose(elliprg(0, 0, 0), 0) + assert np.isinf(elliprg(1, inf, 1)) + assert np.isinf(elliprg(complex(inf), 1, 1)) + args = array([[0.0, 16.0, 16.0], + [2.0, 3.0, 4.0], + [0.0, 1.0j, -1.0j], + [-1.0+1.0j, 1.0j, 0.0], + [-1.0j, -1.0+1.0j, 1.0j], + [0.0, 0.0796, 4.0]]) + expected_results = array([np.pi, + 1.7255030280692, + 0.42360654239699, + 0.44660591677018+0.70768352357515j, + 0.36023392184473+0.40348623401722j, + 1.0284758090288]) + for i, arr in enumerate(args): + assert_allclose(elliprg(*arr), expected_results[i]) + + def test_elliprj(self): + assert_allclose(elliprj(1, 1, 1, 1), 1) + assert elliprj(1, 1, inf, 1) == 0.0 + assert isnan(elliprj(1, 0, 0, 0)) + assert isnan(elliprj(-1, 1, 1, 1)) + assert elliprj(1, 1, 1, inf) == 0.0 + args = array([[0.0, 1.0, 2.0, 3.0], + [2.0, 3.0, 4.0, 5.0], + [2.0, 3.0, 4.0, -1.0+1.0j], + [1.0j, -1.0j, 0.0, 2.0], + [-1.0+1.0j, -1.0-1.0j, 1.0, 2.0], + [1.0j, -1.0j, 0.0, 1.0-1.0j], + [-1.0+1.0j, -1.0-1.0j, 1.0, -3.0+1.0j], + [2.0, 3.0, 4.0, -0.5], # Cauchy principal value + [2.0, 3.0, 4.0, -5.0]]) # Cauchy principal value + expected_results = array([0.77688623778582, + 0.14297579667157, + 0.13613945827771-0.38207561624427j, + 1.6490011662711, + 0.94148358841220, + 1.8260115229009+1.2290661908643j, + -0.61127970812028-1.0684038390007j, + 0.24723819703052, # Cauchy principal value + -0.12711230042964]) # Caucny principal value + for i, arr in enumerate(args): + assert_allclose(elliprj(*arr), expected_results[i]) + + @pytest.mark.xfail(reason="Insufficient accuracy on 32-bit") + def test_elliprj_hard(self): + assert_allclose(elliprj(6.483625725195452e-08, + 1.1649136528196886e-27, + 3.6767340167168e+13, + 0.493704617023468), + 8.63426920644241857617477551054e-6, + rtol=5e-15, atol=1e-20) + assert_allclose(elliprj(14.375105857849121, + 9.993988969725365e-11, + 1.72844262269944e-26, + 5.898871222598245e-06), + 829774.1424801627252574054378691828, + rtol=5e-15, atol=1e-20) + + +class TestEllipLegendreCarlsonIdentities: + """Test identities expressing the Legendre elliptic integrals in terms + of Carlson's symmetric integrals. These identities can be found + in the DLMF https://dlmf.nist.gov/19.25#i . + """ + + def setup_class(self): + self.m_n1_1 = np.arange(-1., 1., 0.01) + # For double, this is -(2**1024) + self.max_neg = finfo(double).min + # Lots of very negative numbers + self.very_neg_m = -1. * 2.**arange(-1 + + np.log2(-self.max_neg), 0., + -1.) + self.ms_up_to_1 = np.concatenate(([self.max_neg], + self.very_neg_m, + self.m_n1_1)) + + def test_k(self): + """Test identity: + K(m) = R_F(0, 1-m, 1) + """ + m = self.ms_up_to_1 + assert_allclose(ellipk(m), elliprf(0., 1.-m, 1.)) + + def test_km1(self): + """Test identity: + K(m) = R_F(0, 1-m, 1) + But with the ellipkm1 function + """ + # For double, this is 2**-1022 + tiny = finfo(double).tiny + # All these small powers of 2, up to 2**-1 + m1 = tiny * 2.**arange(0., -np.log2(tiny)) + assert_allclose(ellipkm1(m1), elliprf(0., m1, 1.)) + + def test_e(self): + """Test identity: + E(m) = 2*R_G(0, 1-k^2, 1) + """ + m = self.ms_up_to_1 + assert_allclose(ellipe(m), 2.*elliprg(0., 1.-m, 1.)) + + +class TestErf: + + def test_erf(self): + er = special.erf(.25) + assert_almost_equal(er,0.2763263902,8) + + def test_erf_zeros(self): + erz = special.erf_zeros(5) + erzr = array([1.45061616+1.88094300j, + 2.24465928+2.61657514j, + 2.83974105+3.17562810j, + 3.33546074+3.64617438j, + 3.76900557+4.06069723j]) + assert_array_almost_equal(erz,erzr,4) + + def _check_variant_func(self, func, other_func, rtol, atol=0): + rng = np.random.RandomState(1234) + n = 10000 + x = rng.pareto(0.02, n) * (2*rng.randint(0, 2, n) - 1) + y = rng.pareto(0.02, n) * (2*rng.randint(0, 2, n) - 1) + z = x + 1j*y + + with np.errstate(all='ignore'): + w = other_func(z) + w_real = other_func(x).real + + mask = np.isfinite(w) + w = w[mask] + z = z[mask] + + mask = np.isfinite(w_real) + w_real = w_real[mask] + x = x[mask] + + # test both real and complex variants + assert_func_equal(func, w, z, rtol=rtol, atol=atol) + assert_func_equal(func, w_real, x, rtol=rtol, atol=atol) + + def test_erfc_consistent(self): + self._check_variant_func( + cephes.erfc, + lambda z: 1 - cephes.erf(z), + rtol=1e-12, + atol=1e-14 # <- the test function loses precision + ) + + def test_erfcx_consistent(self): + self._check_variant_func( + cephes.erfcx, + lambda z: np.exp(z*z) * cephes.erfc(z), + rtol=1e-12 + ) + + def test_erfi_consistent(self): + self._check_variant_func( + cephes.erfi, + lambda z: -1j * cephes.erf(1j*z), + rtol=1e-12 + ) + + def test_dawsn_consistent(self): + self._check_variant_func( + cephes.dawsn, + lambda z: sqrt(pi)/2 * np.exp(-z*z) * cephes.erfi(z), + rtol=1e-12 + ) + + def test_erf_nan_inf(self): + vals = [np.nan, -np.inf, np.inf] + expected = [np.nan, -1, 1] + assert_allclose(special.erf(vals), expected, rtol=1e-15) + + def test_erfc_nan_inf(self): + vals = [np.nan, -np.inf, np.inf] + expected = [np.nan, 2, 0] + assert_allclose(special.erfc(vals), expected, rtol=1e-15) + + def test_erfcx_nan_inf(self): + vals = [np.nan, -np.inf, np.inf] + expected = [np.nan, np.inf, 0] + assert_allclose(special.erfcx(vals), expected, rtol=1e-15) + + def test_erfi_nan_inf(self): + vals = [np.nan, -np.inf, np.inf] + expected = [np.nan, -np.inf, np.inf] + assert_allclose(special.erfi(vals), expected, rtol=1e-15) + + def test_dawsn_nan_inf(self): + vals = [np.nan, -np.inf, np.inf] + expected = [np.nan, -0.0, 0.0] + assert_allclose(special.dawsn(vals), expected, rtol=1e-15) + + def test_wofz_nan_inf(self): + vals = [np.nan, -np.inf, np.inf] + expected = [np.nan + np.nan * 1.j, 0.-0.j, 0.+0.j] + assert_allclose(special.wofz(vals), expected, rtol=1e-15) + + +class TestEuler: + def test_euler(self): + eu0 = special.euler(0) + eu1 = special.euler(1) + eu2 = special.euler(2) # just checking segfaults + assert_allclose(eu0, [1], rtol=1e-15) + assert_allclose(eu1, [1, 0], rtol=1e-15) + assert_allclose(eu2, [1, 0, -1], rtol=1e-15) + eu24 = special.euler(24) + mathworld = [1,1,5,61,1385,50521,2702765,199360981, + 19391512145,2404879675441, + 370371188237525,69348874393137901, + 15514534163557086905] + correct = zeros((25,),'d') + for k in range(0,13): + if (k % 2): + correct[2*k] = -float(mathworld[k]) + else: + correct[2*k] = float(mathworld[k]) + with np.errstate(all='ignore'): + err = nan_to_num((eu24-correct)/correct) + errmax = max(err) + assert_almost_equal(errmax, 0.0, 14) + + +class TestExp: + def test_exp2(self): + ex = special.exp2(2) + exrl = 2**2 + assert_equal(ex,exrl) + + def test_exp2more(self): + exm = special.exp2(2.5) + exmrl = 2**(2.5) + assert_almost_equal(exm,exmrl,8) + + def test_exp10(self): + ex = special.exp10(2) + exrl = 10**2 + assert_approx_equal(ex,exrl) + + def test_exp10more(self): + exm = special.exp10(2.5) + exmrl = 10**(2.5) + assert_almost_equal(exm,exmrl,8) + + def test_expm1(self): + ex = (special.expm1(2),special.expm1(3),special.expm1(4)) + exrl = (exp(2)-1,exp(3)-1,exp(4)-1) + assert_array_almost_equal(ex,exrl,8) + + def test_expm1more(self): + ex1 = (special.expm1(2),special.expm1(2.1),special.expm1(2.2)) + exrl1 = (exp(2)-1,exp(2.1)-1,exp(2.2)-1) + assert_array_almost_equal(ex1,exrl1,8) + + +def assert_really_equal(x, y, rtol=None): + """ + Sharper assertion function that is stricter about matching types, not just values + + This is useful/necessary in some cases: + * dtypes for arrays that have the same _values_ (e.g. element 1.0 vs 1) + * distinguishing complex from real NaN + * result types for scalars + + We still want to be able to allow a relative tolerance for the values though. + The main logic comparison logic is handled by the xp_assert_* functions. + """ + def assert_func(x, y): + xp_assert_equal(x, y) if rtol is None else xp_assert_close(x, y, rtol=rtol) + + def assert_complex_nan(x): + assert np.isnan(x.real) and np.isnan(x.imag) + + assert type(x) is type(y), f"types not equal: {type(x)}, {type(y)}" + + # ensure we also compare the values _within_ an array appropriately, + # e.g. assert_equal does not distinguish different complex nans in arrays + if isinstance(x, np.ndarray): + # assert_equal does not compare (all) types, only values + assert x.dtype == y.dtype + # for empty arrays resp. to ensure shapes match + assert_func(x, y) + for elem_x, elem_y in zip(x.ravel(), y.ravel()): + assert_really_equal(elem_x, elem_y, rtol=rtol) + elif np.isnan(x) and np.isnan(y) and _is_subdtype(type(x), "c"): + assert_complex_nan(x) and assert_complex_nan(y) + # no need to consider complex infinities due to numpy/numpy#25493 + else: + assert_func(x, y) + + +class TestFactorialFunctions: + def factorialk_ref(self, n, k, exact, extend): + if exact: + return special.factorialk(n, k=k, exact=True) + # for details / explanation see factorialk-docstring + r = np.mod(n, k) if extend == "zero" else 1 + vals = np.power(k, (n - r)/k) * special.gamma(n/k + 1) * special.rgamma(r/k + 1) + # np.maximum is element-wise, which is what we want + return vals * np.maximum(r, 1) + + @pytest.mark.parametrize("exact,extend", + [(True, "zero"), (False, "zero"), (False, "complex")]) + def test_factorialx_scalar_return_type(self, exact, extend): + kw = {"exact": exact, "extend": extend} + assert np.isscalar(special.factorial(1, **kw)) + assert np.isscalar(special.factorial2(1, **kw)) + assert np.isscalar(special.factorialk(1, k=3, **kw)) + + @pytest.mark.parametrize("n", [-1, -2, -3]) + @pytest.mark.parametrize("exact", [True, False]) + def test_factorialx_negative_extend_zero(self, exact, n): + kw = {"exact": exact} + assert_equal(special.factorial(n, **kw), 0) + assert_equal(special.factorial2(n, **kw), 0) + assert_equal(special.factorialk(n, k=3, **kw), 0) + + @pytest.mark.parametrize("exact", [True, False]) + def test_factorialx_negative_extend_zero_array(self, exact): + kw = {"exact": exact} + rtol = 1e-15 + n = [-5, -4, 0, 1] + # Consistent output for n < 0 + expected = np.array([0, 0, 1, 1], dtype=native_int if exact else np.float64) + assert_really_equal(special.factorial(n, **kw), expected, rtol=rtol) + assert_really_equal(special.factorial2(n, **kw), expected, rtol=rtol) + assert_really_equal(special.factorialk(n, k=3, **kw), expected, rtol=rtol) + + @pytest.mark.parametrize("n", [-1.1, -2.2, -3.3]) + def test_factorialx_negative_extend_complex(self, n): + kw = {"extend": "complex"} + exp_1 = {-1.1: -10.686287021193184771, + -2.2: 4.8509571405220931958, + -3.3: -1.4471073942559181166} + exp_2 = {-1.1: 1.0725776858167496309, + -2.2: -3.9777171783768419874, + -3.3: -0.99588841846200555977} + exp_k = {-1.1: 0.73565345382163025659, + -2.2: 1.1749163167190809498, + -3.3: -2.4780584257450583713} + rtol = 3e-15 + assert_allclose(special.factorial(n, **kw), exp_1[n], rtol=rtol) + assert_allclose(special.factorial2(n, **kw), exp_2[n], rtol=rtol) + assert_allclose(special.factorialk(n, k=3, **kw), exp_k[n], rtol=rtol) + assert_allclose(special.factorial([n], **kw)[0], exp_1[n], rtol=rtol) + assert_allclose(special.factorial2([n], **kw)[0], exp_2[n], rtol=rtol) + assert_allclose(special.factorialk([n], k=3, **kw)[0], exp_k[n], rtol=rtol) + + @pytest.mark.parametrize("imag", [0, 0j]) + @pytest.mark.parametrize("n_outer", [-1, -2, -3]) + def test_factorialx_negative_extend_complex_poles(self, n_outer, imag): + kw = {"extend": "complex"} + def _check(n): + complexify = _is_subdtype(type(n), "c") + # like for gamma, we expect complex nans for complex inputs + complex_nan = np.complex128("nan+nanj") + exp = np.complex128("nan+nanj") if complexify else np.float64("nan") + # poles are at negative integers that are multiples of k + assert_really_equal(special.factorial(n, **kw), exp) + assert_really_equal(special.factorial2(n * 2, **kw), exp) + assert_really_equal(special.factorialk(n * 3, k=3, **kw), exp) + # also test complex k for factorialk + c = 1.5 - 2j + assert_really_equal(special.factorialk(n * c, k=c, **kw), complex_nan) + # same for array case + assert_really_equal(special.factorial([n], **kw)[0], exp) + assert_really_equal(special.factorial2([n * 2], **kw)[0], exp) + assert_really_equal(special.factorialk([n * 3], k=3, **kw)[0], exp) + assert_really_equal(special.factorialk([n * c], k=c, **kw)[0], complex_nan) + # more specific tests in test_factorial{,2,k}_complex_reference + + # imag ensures we test both real and complex representations of the poles + _check(n_outer + imag) + # check for large multiple of period + _check(100_000 * n_outer + imag) + + @pytest.mark.parametrize("boxed", [True, False]) + @pytest.mark.parametrize("extend", ["zero", "complex"]) + @pytest.mark.parametrize( + "n", + [ + np.nan, np.float64("nan"), np.nan + np.nan*1j, np.complex128("nan+nanj"), + np.inf, np.inf + 0j, -np.inf, -np.inf + 0j, None, np.datetime64("nat") + ], + ids=[ + "NaN", "np.float64('nan')", "NaN+i*NaN", "np.complex128('nan+nanj')", + "inf", "inf+0i", "-inf", "-inf+0i", "None", "NaT" + ] + ) + @pytest.mark.parametrize( + "factorialx", + [special.factorial, special.factorial2, special.factorialk] + ) + def test_factorialx_inf_nan(self, factorialx, n, extend, boxed): + # NaNs not allowed (by dtype) for exact=True + kw = {"exact": False, "extend": extend} + if factorialx == special.factorialk: + kw["k"] = 3 + + # None is allowed for scalars, but would cause object type in array case + permissible_types = ["i", "f", "c"] if boxed else ["i", "f", "c", type(None)] + # factorial allows floats also for extend="zero" + types_need_complex_ext = "c" if factorialx == special.factorial else ["f", "c"] + + if not _is_subdtype(type(n), permissible_types): + with pytest.raises(ValueError, match="Unsupported data type.*"): + factorialx([n] if boxed else n, **kw) + elif _is_subdtype(type(n), types_need_complex_ext) and extend != "complex": + with pytest.raises(ValueError, match="In order to use non-integer.*"): + factorialx([n] if boxed else n, **kw) + else: + # account for type and whether extend="complex" + complexify = (extend == "complex") and _is_subdtype(type(n), "c") + # note that the type of the naïve `np.nan + np.nan * 1j` is `complex` + # instead of `numpy.complex128`, which trips up assert_really_equal + expected = np.complex128("nan+nanj") if complexify else np.float64("nan") + # the only exception are real infinities + if _is_subdtype(type(n), "f") and np.isinf(n): + # unchanged for positive infinity; negative one depends on extension + neg_inf_result = np.float64(0 if (extend == "zero") else "nan") + expected = np.float64("inf") if (n > 0) else neg_inf_result + + result = factorialx([n], **kw)[0] if boxed else factorialx(n, **kw) + assert_really_equal(result, expected) + # also tested in test_factorial{,2,k}_{array,scalar}_corner_cases + + @pytest.mark.parametrize("extend", [0, 1.1, np.nan, "string"]) + def test_factorialx_raises_extend(self, extend): + with pytest.raises(ValueError, match="argument `extend` must be.*"): + special.factorial(1, extend=extend) + with pytest.raises(ValueError, match="argument `extend` must be.*"): + special.factorial2(1, extend=extend) + with pytest.raises(ValueError, match="argument `extend` must be.*"): + special.factorialk(1, k=3, exact=True, extend=extend) + + @pytest.mark.parametrize("levels", range(1, 5)) + @pytest.mark.parametrize("exact", [True, False]) + def test_factorialx_array_shape(self, levels, exact): + def _nest_me(x, k=1): + """ + Double x and nest it k times + + For example: + >>> _nest_me([3, 4], 2) + [[[3, 4], [3, 4]], [[3, 4], [3, 4]]] + """ + if k == 0: + return x + else: + return _nest_me([x, x], k-1) + + def _check(res, nucleus): + exp = np.array(_nest_me(nucleus, k=levels), dtype=object) + # test that ndarray shape is maintained + # need to cast to float due to numpy/numpy#21220 + assert_allclose(res.astype(np.float64), exp.astype(np.float64)) + + n = np.array(_nest_me([5, 25], k=levels)) + exp_nucleus = {1: [120, math.factorial(25)], + # correctness of factorial{2,k}() is tested elsewhere + 2: [15, special.factorial2(25, exact=True)], + 3: [10, special.factorialk(25, 3, exact=True)]} + + _check(special.factorial(n, exact=exact), exp_nucleus[1]) + _check(special.factorial2(n, exact=exact), exp_nucleus[2]) + _check(special.factorialk(n, 3, exact=exact), exp_nucleus[3]) + + @pytest.mark.parametrize("exact", [True, False]) + @pytest.mark.parametrize("dtype", [ + None, int, np.int8, np.int16, np.int32, np.int64, + np.uint8, np.uint16, np.uint32, np.uint64 + ]) + @pytest.mark.parametrize("dim", range(0, 5)) + def test_factorialx_array_dimension(self, dim, dtype, exact): + n = np.array(5, dtype=dtype, ndmin=dim) + exp = {1: 120, 2: 15, 3: 10} + assert_allclose(special.factorial(n, exact=exact), + np.array(exp[1], ndmin=dim)) + assert_allclose(special.factorial2(n, exact=exact), + np.array(exp[2], ndmin=dim)) + assert_allclose(special.factorialk(n, 3, exact=exact), + np.array(exp[3], ndmin=dim)) + + @pytest.mark.parametrize("exact", [True, False]) + @pytest.mark.parametrize("level", range(1, 5)) + def test_factorialx_array_like(self, level, exact): + def _nest_me(x, k=1): + if k == 0: + return x + else: + return _nest_me([x], k-1) + + n = _nest_me([5], k=level-1) # nested list + exp_nucleus = {1: 120, 2: 15, 3: 10} + assert_func = assert_array_equal if exact else assert_allclose + assert_func(special.factorial(n, exact=exact), + np.array(exp_nucleus[1], ndmin=level)) + assert_func(special.factorial2(n, exact=exact), + np.array(exp_nucleus[2], ndmin=level)) + assert_func(special.factorialk(n, 3, exact=exact), + np.array(exp_nucleus[3], ndmin=level)) + + @pytest.mark.parametrize("dtype", [np.uint8, np.uint16, np.uint32, np.uint64]) + @pytest.mark.parametrize("exact,extend", + [(True, "zero"), (False, "zero"), (False, "complex")]) + def test_factorialx_uint(self, exact, extend, dtype): + # ensure that uint types work correctly as inputs + kw = {"exact": exact, "extend": extend} + assert_func = assert_array_equal if exact else assert_allclose + def _check(n): + n_ref = n.astype(np.int64) if isinstance(n, np.ndarray) else np.int64(n) + assert_func(special.factorial(n, **kw), special.factorial(n_ref, **kw)) + assert_func(special.factorial2(n, **kw), special.factorial2(n_ref, **kw)) + assert_func(special.factorialk(n, k=3, **kw), + special.factorialk(n_ref, k=3, **kw)) + _check(dtype(0)) + _check(dtype(1)) + _check(np.array(0, dtype=dtype)) + _check(np.array([0, 1], dtype=dtype)) + + # note that n=170 is the last integer such that factorial(n) fits float64 + @pytest.mark.parametrize('n', range(30, 180, 10)) + def test_factorial_accuracy(self, n): + # Compare exact=True vs False, i.e. that the accuracy of the + # approximation is better than the specified tolerance. + + rtol = 6e-14 if sys.platform == 'win32' else 1e-15 + # need to cast exact result to float due to numpy/numpy#21220 + assert_allclose(float(special.factorial(n, exact=True)), + special.factorial(n, exact=False), rtol=rtol) + assert_allclose(special.factorial([n], exact=True).astype(float), + special.factorial([n], exact=False), rtol=rtol) + + @pytest.mark.parametrize('n', + list(range(0, 22)) + list(range(30, 180, 10))) + def test_factorial_int_reference(self, n): + # Compare all with math.factorial + correct = math.factorial(n) + assert_array_equal(correct, special.factorial(n, exact=True)) + assert_array_equal(correct, special.factorial([n], exact=True)[0]) + + rtol = 8e-14 if sys.platform == 'win32' else 1e-15 + # need to cast exact result to float due to numpy/numpy#21220 + correct = float(correct) + assert_allclose(correct, special.factorial(n, exact=False), rtol=rtol) + assert_allclose(correct, special.factorial([n], exact=False)[0], rtol=rtol) + + # extend="complex" only works for exact=False + kw = {"exact": False, "extend": "complex"} + assert_allclose(correct, special.factorial(n, **kw), rtol=rtol) + assert_allclose(correct, special.factorial([n], **kw)[0], rtol=rtol) + + def test_factorial_float_reference(self): + def _check(n, expected): + rtol = 8e-14 if sys.platform == 'win32' else 1e-15 + assert_allclose(special.factorial(n), expected, rtol=rtol) + assert_allclose(special.factorial([n])[0], expected, rtol=rtol) + # using floats with `exact=True` raises an error for scalars and arrays + with pytest.raises(ValueError, match="`exact=True` only supports.*"): + special.factorial(n, exact=True) + with pytest.raises(ValueError, match="`exact=True` only supports.*"): + special.factorial([n], exact=True) + + # Reference values from mpmath for gamma(n+1) + _check(0.01, 0.994325851191506032181932988) + _check(1.11, 1.051609009483625091514147465) + _check(5.55, 314.9503192327208241614959052) + _check(11.1, 50983227.84411615655137170553) + _check(33.3, 2.493363339642036352229215273e+37) + _check(55.5, 9.479934358436729043289162027e+73) + _check(77.7, 3.060540559059579022358692625e+114) + _check(99.9, 5.885840419492871504575693337e+157) + # close to maximum for float64 + _check(170.6243, 1.79698185749571048960082e+308) + + def test_factorial_complex_reference(self): + def _check(n, expected): + rtol = 3e-15 if sys.platform == 'win32' else 2e-15 + kw = {"exact": False, "extend": "complex"} + assert_allclose(special.factorial(n, **kw), expected, rtol=rtol) + assert_allclose(special.factorial([n], **kw)[0], expected, rtol=rtol) + + # Reference values from mpmath.gamma(n+1) + # negative & complex values + _check(-0.5, expected=1.7724538509055160276) + _check(-0.5 + 0j, expected=1.7724538509055160276 + 0j) + _check(2 + 2j, expected=-0.42263728631120216694 + 0.87181425569650686062j) + # close to poles + _check(-0.9999, expected=9999.422883232725532) + _check(-1 + 0.0001j, expected=-0.57721565582674219 - 9999.9999010944009697j) + + @pytest.mark.parametrize("dtype", [np.int64, np.float64, + np.complex128, object]) + @pytest.mark.parametrize("extend", ["zero", "complex"]) + @pytest.mark.parametrize("exact", [True, False]) + @pytest.mark.parametrize("dim", range(0, 5)) + # test empty & non-empty arrays, with nans and mixed + @pytest.mark.parametrize( + "content", + [[], [1], [1.1], [np.nan], [np.nan + np.nan * 1j], [np.nan, 1]], + ids=["[]", "[1]", "[1.1]", "[NaN]", "[NaN+i*NaN]", "[NaN, 1]"], + ) + def test_factorial_array_corner_cases(self, content, dim, exact, extend, dtype): + if dtype is object and SCIPY_ARRAY_API: + pytest.skip("object arrays unsupported in array API mode") + # get dtype without calling array constructor (that might fail or mutate) + if dtype is np.int64 and any(np.isnan(x) or (x != int(x)) for x in content): + pytest.skip("impossible combination") + if dtype == np.float64 and any(_is_subdtype(type(x), "c") for x in content): + pytest.skip("impossible combination") + + kw = {"exact": exact, "extend": extend} + # np.array(x, ndim=0) will not be 0-dim. unless x is too + content = content if (dim > 0 or len(content) != 1) else content[0] + n = np.array(content, ndmin=dim, dtype=dtype) + + result = None + if extend == "complex" and exact: + with pytest.raises(ValueError, match="Incompatible options:.*"): + special.factorial(n, **kw) + elif not _is_subdtype(n.dtype, ["i", "f", "c"]): + with pytest.raises(ValueError, match="Unsupported data type.*"): + special.factorial(n, **kw) + elif _is_subdtype(n.dtype, "c") and extend != "complex": + with pytest.raises(ValueError, match="In order to use non-integer.*"): + special.factorial(n, **kw) + elif exact and not _is_subdtype(n.dtype, "i"): + with pytest.raises(ValueError, match="`exact=True` only supports.*"): + special.factorial(n, **kw) + else: + result = special.factorial(n, **kw) + + if result is not None: + # use scalar case as reference; tested separately in *_scalar_corner_cases + ref = [special.factorial(x, **kw) for x in n.ravel()] + # unpack length-1 lists so that np.array(x, ndim=0) works correctly + ref = ref[0] if len(ref) == 1 else ref + # result is empty if and only if n is empty, and has the same dimension + # as n; dtype stays the same, except when not empty and not exact: + if n.size: + cx = (extend == "complex") and _is_subdtype(n.dtype, "c") + dtype = np.complex128 if cx else (native_int if exact else np.float64) + expected = np.array(ref, ndmin=dim, dtype=dtype) + assert_really_equal(result, expected, rtol=1e-15) + + @pytest.mark.parametrize("extend", ["zero", "complex"]) + @pytest.mark.parametrize("exact", [True, False]) + @pytest.mark.parametrize("n", [1, 1.1, 2 + 2j, np.nan, np.nan + np.nan*1j, None], + ids=["1", "1.1", "2+2j", "NaN", "NaN+i*NaN", "None"]) + def test_factorial_scalar_corner_cases(self, n, exact, extend): + kw = {"exact": exact, "extend": extend} + if extend == "complex" and exact: + with pytest.raises(ValueError, match="Incompatible options:.*"): + special.factorial(n, **kw) + elif not _is_subdtype(type(n), ["i", "f", "c", type(None)]): + with pytest.raises(ValueError, match="Unsupported data type.*"): + special.factorial(n, **kw) + elif _is_subdtype(type(n), "c") and extend != "complex": + with pytest.raises(ValueError, match="In order to use non-integer.*"): + special.factorial(n, **kw) + elif n is None or np.isnan(n): + # account for dtype and whether extend="complex" + complexify = (extend == "complex") and _is_subdtype(type(n), "c") + expected = np.complex128("nan+nanj") if complexify else np.float64("nan") + assert_really_equal(special.factorial(n, **kw), expected) + elif exact and _is_subdtype(type(n), "f"): + with pytest.raises(ValueError, match="`exact=True` only supports.*"): + special.factorial(n, **kw) + else: + assert_equal(special.factorial(n, **kw), special.gamma(n + 1)) + + # use odd increment to make sure both odd & even numbers are tested! + @pytest.mark.parametrize('n', range(30, 180, 11)) + def test_factorial2_accuracy(self, n): + # Compare exact=True vs False, i.e. that the accuracy of the + # approximation is better than the specified tolerance. + + rtol = 2e-14 if sys.platform == 'win32' else 1e-15 + # need to cast exact result to float due to numpy/numpy#21220 + assert_allclose(float(special.factorial2(n, exact=True)), + special.factorial2(n, exact=False), rtol=rtol) + assert_allclose(special.factorial2([n], exact=True).astype(float), + special.factorial2([n], exact=False), rtol=rtol) + + @pytest.mark.parametrize('n', + list(range(0, 22)) + list(range(30, 180, 11))) + def test_factorial2_int_reference(self, n): + # Compare all with correct value + + # Cannot use np.product due to overflow + correct = functools.reduce(operator.mul, list(range(n, 0, -2)), 1) + + assert_array_equal(correct, special.factorial2(n, exact=True)) + assert_array_equal(correct, special.factorial2([n], exact=True)[0]) + + rtol = 2e-14 if sys.platform == 'win32' else 1e-15 + # need to cast exact result to float due to numpy/numpy#21220 + correct = float(correct) + assert_allclose(correct, special.factorial2(n, exact=False), rtol=rtol) + assert_allclose(correct, special.factorial2([n], exact=False)[0], rtol=rtol) + + # extend="complex" only works for exact=False + kw = {"exact": False, "extend": "complex"} + # approximation only matches exactly for `n == 1 (mod k)`, see docstring + if n % 2 == 1: + assert_allclose(correct, special.factorial2(n, **kw), rtol=rtol) + assert_allclose(correct, special.factorial2([n], **kw)[0], rtol=rtol) + + def test_factorial2_complex_reference(self): + # this tests for both floats and complex + def _check(n, expected): + rtol = 5e-15 + kw = {"exact": False, "extend": "complex"} + assert_allclose(special.factorial2(n, **kw), expected, rtol=rtol) + assert_allclose(special.factorial2([n], **kw)[0], expected, rtol=rtol) + + # Reference values from mpmath for: + # mpmath.power(2, n/2) * mpmath.gamma(n/2 + 1) * mpmath.sqrt(2 / mpmath.pi) + _check(3, expected=3) + _check(4, expected=special.factorial2(4) * math.sqrt(2 / math.pi)) + _check(20, expected=special.factorial2(20) * math.sqrt(2 / math.pi)) + # negative & complex values + _check(-0.5, expected=0.82217895866245855122) + _check(-0.5 + 0j, expected=0.82217895866245855122 + 0j) + _check(3 + 3j, expected=-1.0742236630142471526 + 1.4421398439387262897j) + # close to poles + _check(-1.9999, expected=7978.8918745523440682) + _check(-2 + 0.0001j, expected=0.0462499835314308444 - 7978.84559148876374493j) + + @pytest.mark.parametrize("dtype", [np.int64, np.float64, + np.complex128, object]) + @pytest.mark.parametrize("extend", ["zero", "complex"]) + @pytest.mark.parametrize("exact", [True, False]) + @pytest.mark.parametrize("dim", range(0, 5)) + # test empty & non-empty arrays, with nans and mixed + @pytest.mark.parametrize( + "content", + [[], [1], [1.1], [np.nan], [np.nan + np.nan * 1j], [np.nan, 1]], + ids=["[]", "[1]", "[1.1]", "[NaN]", "[NaN+i*NaN]", "[NaN, 1]"], + ) + def test_factorial2_array_corner_cases(self, content, dim, exact, extend, dtype): + # get dtype without calling array constructor (that might fail or mutate) + if dtype == np.int64 and any(np.isnan(x) or (x != int(x)) for x in content): + pytest.skip("impossible combination") + if dtype == np.float64 and any(_is_subdtype(type(x), "c") for x in content): + pytest.skip("impossible combination") + + kw = {"exact": exact, "extend": extend} + # np.array(x, ndim=0) will not be 0-dim. unless x is too + content = content if (dim > 0 or len(content) != 1) else content[0] + n = np.array(content, ndmin=dim, dtype=dtype) + + result = None + if extend == "complex" and exact: + with pytest.raises(ValueError, match="Incompatible options:.*"): + special.factorial2(n, **kw) + elif not _is_subdtype(n.dtype, ["i", "f", "c"]): + with pytest.raises(ValueError, match="Unsupported data type.*"): + special.factorial2(n, **kw) + elif _is_subdtype(n.dtype, ["f", "c"]) and extend != "complex": + with pytest.raises(ValueError, match="In order to use non-integer.*"): + special.factorial2(n, **kw) + else: + result = special.factorial2(n, **kw) + + if result is not None: + # use scalar case as reference; tested separately in *_scalar_corner_cases + ref = [special.factorial2(x, **kw) for x in n.ravel()] + # unpack length-1 lists so that np.array(x, ndim=0) works correctly + ref = ref[0] if len(ref) == 1 else ref + # result is empty if and only if n is empty, and has the same dimension + # as n; dtype stays the same, except when not empty and not exact: + if n.size: + cx = (extend == "complex") and _is_subdtype(n.dtype, "c") + dtype = np.complex128 if cx else (native_int if exact else np.float64) + expected = np.array(ref, ndmin=dim, dtype=dtype) + assert_really_equal(result, expected, rtol=2e-15) + + @pytest.mark.parametrize("extend", ["zero", "complex"]) + @pytest.mark.parametrize("exact", [True, False]) + @pytest.mark.parametrize("n", [1, 1.1, 2 + 2j, np.nan, np.nan + np.nan*1j, None], + ids=["1", "1.1", "2+2j", "NaN", "NaN+i*NaN", "None"]) + def test_factorial2_scalar_corner_cases(self, n, exact, extend): + kw = {"exact": exact, "extend": extend} + if extend == "complex" and exact: + with pytest.raises(ValueError, match="Incompatible options:.*"): + special.factorial2(n, **kw) + elif not _is_subdtype(type(n), ["i", "f", "c", type(None)]): + with pytest.raises(ValueError, match="Unsupported data type.*"): + special.factorial2(n, **kw) + elif _is_subdtype(type(n), ["f", "c"]) and extend != "complex": + with pytest.raises(ValueError, match="In order to use non-integer.*"): + special.factorial2(n, **kw) + elif n is None or np.isnan(n): + # account for dtype and whether extend="complex" + complexify = (extend == "complex") and _is_subdtype(type(n), "c") + expected = np.complex128("nan+nanj") if complexify else np.float64("nan") + assert_really_equal(special.factorial2(n, **kw), expected) + else: + expected = self.factorialk_ref(n, k=2, **kw) + assert_really_equal(special.factorial2(n, **kw), expected, rtol=1e-15) + + @pytest.mark.parametrize("k", range(1, 5)) + # note that n=170 is the last integer such that factorial(n) fits float64; + # use odd increment to make sure both odd & even numbers are tested + @pytest.mark.parametrize('n', range(170, 20, -29)) + def test_factorialk_accuracy(self, n, k): + # Compare exact=True vs False, i.e. that the accuracy of the + # approximation is better than the specified tolerance. + + rtol = 6e-14 if sys.platform == 'win32' else 2e-14 + # need to cast exact result to float due to numpy/numpy#21220 + assert_allclose(float(special.factorialk(n, k=k, exact=True)), + special.factorialk(n, k=k, exact=False), rtol=rtol) + assert_allclose(special.factorialk([n], k=k, exact=True).astype(float), + special.factorialk([n], k=k, exact=False), rtol=rtol) + + @pytest.mark.parametrize('k', list(range(1, 5)) + [10, 20]) + @pytest.mark.parametrize('n', + list(range(0, 22)) + list(range(22, 100, 11))) + def test_factorialk_int_reference(self, n, k): + # Compare all with correct value + + # Would be nice to use np.product here, but that's + # broken on windows, see numpy/numpy#21219 + correct = functools.reduce(operator.mul, list(range(n, 0, -k)), 1) + + assert_array_equal(correct, special.factorialk(n, k, exact=True)) + assert_array_equal(correct, special.factorialk([n], k, exact=True)[0]) + + rtol = 3e-14 if sys.platform == 'win32' else 1e-14 + # need to cast exact result to float due to numpy/numpy#21220 + correct = float(correct) + assert_allclose(correct, special.factorialk(n, k, exact=False), rtol=rtol) + assert_allclose(correct, special.factorialk([n], k, exact=False)[0], rtol=rtol) + + # extend="complex" only works for exact=False + kw = {"k": k, "exact": False, "extend": "complex"} + # approximation only matches exactly for `n == 1 (mod k)`, see docstring + if n % k == 1: + rtol = 2e-14 + assert_allclose(correct, special.factorialk(n, **kw), rtol=rtol) + assert_allclose(correct, special.factorialk([n], **kw)[0], rtol=rtol) + + def test_factorialk_complex_reference(self): + # this tests for both floats and complex + def _check(n, k, exp): + rtol = 1e-14 + kw = {"k": k, "exact": False, "extend": "complex"} + assert_allclose(special.factorialk(n, **kw), exp, rtol=rtol) + assert_allclose(special.factorialk([n], **kw)[0], exp, rtol=rtol) + + # Reference values from mpmath for: + # mpmath.power(k, (n-1)/k) * mpmath.gamma(n/k + 1) / mpmath.gamma(1/k + 1) + _check(n=4, k=3, exp=special.factorialk(4, k=3, exact=True)) + _check(n=5, k=3, exp=7.29011132947227083) + _check(n=6.5, k=3, exp=19.6805080113566010) + # non-integer k + _check(n=3, k=2.5, exp=2.58465740293218541) + _check(n=11, k=2.5, exp=1963.5) # ==11*8.5*6*3.5; c.f. n == 1 (mod k) + _check(n=-3 + 3j + 1, k=-3 + 3j, exp=-2 + 3j) + # complex values + _check(n=4 + 4j, k=4, exp=-0.67855904082768043854 + 2.1993925819930311497j) + _check(n=4, k=4 - 4j, exp=1.9775338957222718742 + 0.92607172675423901371j) + _check(n=4 + 4j, k=4 - 4j, exp=0.1868492880824934475 + 0.87660580316894290247j) + # negative values + _check(n=-0.5, k=3, exp=0.72981013240713739354) + _check(n=-0.5 + 0j, k=3, exp=0.72981013240713739354 + 0j) + _check(n=2.9, k=-0.7, exp=0.45396591474966867296 + 0.56925525174685228866j) + _check(n=-0.6, k=-0.7, exp=-0.07190820089634757334 - 0.090170031876701730081j) + # close to poles + _check(n=-2.9999, k=3, exp=7764.7170695908828364) + _check(n=-3 + 0.0001j, k=3, exp=0.1349475632879599864 - 7764.5821055158365027j) + + @pytest.mark.parametrize("dtype", [np.int64, np.float64, + np.complex128, object]) + @pytest.mark.parametrize("extend", ["zero", "complex"]) + @pytest.mark.parametrize("exact", [True, False]) + @pytest.mark.parametrize("dim", range(0, 5)) + # test empty & non-empty arrays, with nans and mixed + @pytest.mark.parametrize( + "content", + [[], [1], [1.1], [np.nan], [np.nan + np.nan * 1j], [np.nan, 1]], + ids=["[]", "[1]", "[1.1]", "[NaN]", "[NaN+i*NaN]", "[NaN, 1]"], + ) + def test_factorialk_array_corner_cases(self, content, dim, exact, extend, dtype): + # get dtype without calling array constructor (that might fail or mutate) + if dtype == np.int64 and any(np.isnan(x) or (x != int(x)) for x in content): + pytest.skip("impossible combination") + if dtype == np.float64 and any(_is_subdtype(type(x), "c") for x in content): + pytest.skip("impossible combination") + + kw = {"k": 3, "exact": exact, "extend": extend} + # np.array(x, ndim=0) will not be 0-dim. unless x is too + content = content if (dim > 0 or len(content) != 1) else content[0] + n = np.array(content, ndmin=dim, dtype=dtype) + + result = None + if extend == "complex" and exact: + with pytest.raises(ValueError, match="Incompatible options:.*"): + special.factorialk(n, **kw) + elif not _is_subdtype(n.dtype, ["i", "f", "c"]): + with pytest.raises(ValueError, match="Unsupported data type.*"): + special.factorialk(n, **kw) + elif _is_subdtype(n.dtype, ["f", "c"]) and extend != "complex": + with pytest.raises(ValueError, match="In order to use non-integer.*"): + special.factorialk(n, **kw) + else: + result = special.factorialk(n, **kw) + + if result is not None: + # use scalar case as reference; tested separately in *_scalar_corner_cases + ref = [special.factorialk(x, **kw) for x in n.ravel()] + # unpack length-1 lists so that np.array(x, ndim=0) works correctly + ref = ref[0] if len(ref) == 1 else ref + # result is empty if and only if n is empty, and has the same dimension + # as n; dtype stays the same, except when not empty and not exact: + if n.size: + cx = (extend == "complex") and _is_subdtype(n.dtype, "c") + dtype = np.complex128 if cx else (native_int if exact else np.float64) + expected = np.array(ref, ndmin=dim, dtype=dtype) + assert_really_equal(result, expected, rtol=2e-15) + + @pytest.mark.parametrize("extend", ["zero", "complex"]) + @pytest.mark.parametrize("exact", [True, False]) + @pytest.mark.parametrize("k", range(1, 5)) + @pytest.mark.parametrize("n", [1, 1.1, 2 + 2j, np.nan, np.nan + np.nan*1j, None], + ids=["1", "1.1", "2+2j", "NaN", "NaN+i*NaN", "None"]) + def test_factorialk_scalar_corner_cases(self, n, k, exact, extend): + kw = {"k": k, "exact": exact, "extend": extend} + if extend == "complex" and exact: + with pytest.raises(ValueError, match="Incompatible options:.*"): + special.factorialk(n, **kw) + elif not _is_subdtype(type(n), ["i", "f", "c", type(None)]): + with pytest.raises(ValueError, match="Unsupported data type.*"): + special.factorialk(n, **kw) + elif _is_subdtype(type(n), ["f", "c"]) and extend != "complex": + with pytest.raises(ValueError, match="In order to use non-integer.*"): + special.factorialk(n, **kw) + elif n is None or np.isnan(n): + # account for dtype and whether extend="complex" + complexify = (extend == "complex") and _is_subdtype(type(n), "c") + expected = np.complex128("nan+nanj") if complexify else np.float64("nan") + assert_really_equal(special.factorialk(n, **kw), expected) + else: + expected = self.factorialk_ref(n, **kw) + assert_really_equal(special.factorialk(n, **kw), expected, rtol=1e-15) + + @pytest.mark.parametrize("boxed", [True, False]) + @pytest.mark.parametrize("exact,extend", + [(True, "zero"), (False, "zero"), (False, "complex")]) + @pytest.mark.parametrize("k", [-1, -1.0, 0, 0.0, 0 + 1j, 1.1, np.nan]) + def test_factorialk_raises_k_complex(self, k, exact, extend, boxed): + n = [1] if boxed else 1 + kw = {"k": k, "exact": exact, "extend": extend} + if extend == "zero": + msg = "In order to use non-integer.*" + if _is_subdtype(type(k), "i") and (k < 1): + msg = "For `extend='zero'`.*" + with pytest.raises(ValueError, match=msg): + special.factorialk(n, **kw) + elif k == 0: + with pytest.raises(ValueError, match="Parameter k cannot be zero!"): + special.factorialk(n, **kw) + else: + # no error + special.factorialk(n, **kw) + + @pytest.mark.parametrize("boxed", [True, False]) + @pytest.mark.parametrize("exact,extend", + [(True, "zero"), (False, "zero"), (False, "complex")]) + # neither integer, float nor complex + @pytest.mark.parametrize("k", ["string", np.datetime64("nat")], + ids=["string", "NaT"]) + def test_factorialk_raises_k_other(self, k, exact, extend, boxed): + n = [1] if boxed else 1 + kw = {"k": k, "exact": exact, "extend": extend} + with pytest.raises(ValueError, match="Unsupported data type.*"): + special.factorialk(n, **kw) + + @pytest.mark.parametrize("exact,extend", + [(True, "zero"), (False, "zero"), (False, "complex")]) + @pytest.mark.parametrize("k", range(1, 12)) + def test_factorialk_dtype(self, k, exact, extend): + kw = {"k": k, "exact": exact, "extend": extend} + if exact and k in _FACTORIALK_LIMITS_64BITS.keys(): + n = np.array([_FACTORIALK_LIMITS_32BITS[k]]) + assert_equal(special.factorialk(n, **kw).dtype, np_long) + assert_equal(special.factorialk(n + 1, **kw).dtype, np.int64) + # assert maximality of limits for given dtype + assert special.factorialk(n + 1, **kw) > np.iinfo(np.int32).max + + n = np.array([_FACTORIALK_LIMITS_64BITS[k]]) + assert_equal(special.factorialk(n, **kw).dtype, np.int64) + assert_equal(special.factorialk(n + 1, **kw).dtype, object) + assert special.factorialk(n + 1, **kw) > np.iinfo(np.int64).max + else: + n = np.array([_FACTORIALK_LIMITS_64BITS.get(k, 1)]) + # for exact=True and k >= 10, we always return object; + # for exact=False it's always float (unless input is complex) + dtype = object if exact else np.float64 + assert_equal(special.factorialk(n, **kw).dtype, dtype) + + def test_factorial_mixed_nan_inputs(self): + x = np.array([np.nan, 1, 2, 3, np.nan]) + expected = np.array([np.nan, 1, 2, 6, np.nan]) + assert_equal(special.factorial(x, exact=False), expected) + with pytest.raises(ValueError, match="`exact=True` only supports.*"): + special.factorial(x, exact=True) + + +class TestFresnel: + @pytest.mark.parametrize("z, s, c", [ + # some positive value + (.5, 0.064732432859999287, 0.49234422587144644), + (.5 + .0j, 0.064732432859999287, 0.49234422587144644), + # negative half annulus + # https://github.com/scipy/scipy/issues/12309 + # Reference values can be reproduced with + # https://www.wolframalpha.com/input/?i=FresnelS%5B-2.0+%2B+0.1i%5D + # https://www.wolframalpha.com/input/?i=FresnelC%5B-2.0+%2B+0.1i%5D + ( + -2.0 + 0.1j, + -0.3109538687728942-0.0005870728836383176j, + -0.4879956866358554+0.10670801832903172j + ), + ( + -0.1 - 1.5j, + -0.03918309471866977+0.7197508454568574j, + 0.09605692502968956-0.43625191013617465j + ), + # a different algorithm kicks in for "large" values, i.e., |z| >= 4.5, + # make sure to test both float and complex values; a different + # algorithm is used + (6.0, 0.44696076, 0.49953147), + (6.0 + 0.0j, 0.44696076, 0.49953147), + (6.0j, -0.44696076j, 0.49953147j), + (-6.0 + 0.0j, -0.44696076, -0.49953147), + (-6.0j, 0.44696076j, -0.49953147j), + # inf + (np.inf, 0.5, 0.5), + (-np.inf, -0.5, -0.5), + ]) + def test_fresnel_values(self, z, s, c): + frs = array(special.fresnel(z)) + assert_array_almost_equal(frs, array([s, c]), 8) + + # values from pg 329 Table 7.11 of A & S + # slightly corrected in 4th decimal place + def test_fresnel_zeros(self): + szo, czo = special.fresnel_zeros(5) + assert_array_almost_equal(szo, + array([2.0093+0.2885j, + 2.8335+0.2443j, + 3.4675+0.2185j, + 4.0026+0.2009j, + 4.4742+0.1877j]),3) + assert_array_almost_equal(czo, + array([1.7437+0.3057j, + 2.6515+0.2529j, + 3.3204+0.2240j, + 3.8757+0.2047j, + 4.3611+0.1907j]),3) + vals1 = special.fresnel(szo)[0] + vals2 = special.fresnel(czo)[1] + assert_array_almost_equal(vals1,0,14) + assert_array_almost_equal(vals2,0,14) + + def test_fresnelc_zeros(self): + szo, czo = special.fresnel_zeros(6) + frc = special.fresnelc_zeros(6) + assert_array_almost_equal(frc,czo,12) + + def test_fresnels_zeros(self): + szo, czo = special.fresnel_zeros(5) + frs = special.fresnels_zeros(5) + assert_array_almost_equal(frs,szo,12) + + +class TestGamma: + def test_gamma(self): + gam = special.gamma(5) + assert_equal(gam,24.0) + + def test_gammaln(self): + gamln = special.gammaln(3) + lngam = log(special.gamma(3)) + assert_almost_equal(gamln,lngam,8) + + def test_gammainccinv(self): + gccinv = special.gammainccinv(.5,.5) + gcinv = special.gammaincinv(.5,.5) + assert_almost_equal(gccinv,gcinv,8) + + @with_special_errors + def test_gammaincinv(self): + y = special.gammaincinv(.4,.4) + x = special.gammainc(.4,y) + assert_almost_equal(x,0.4,1) + y = special.gammainc(10, 0.05) + x = special.gammaincinv(10, 2.5715803516000736e-20) + assert_almost_equal(0.05, x, decimal=10) + assert_almost_equal(y, 2.5715803516000736e-20, decimal=10) + x = special.gammaincinv(50, 8.20754777388471303050299243573393e-18) + assert_almost_equal(11.0, x, decimal=10) + + @with_special_errors + def test_975(self): + # Regression test for ticket #975 -- switch point in algorithm + # check that things work OK at the point, immediately next floats + # around it, and a bit further away + pts = [0.25, + np.nextafter(0.25, 0), 0.25 - 1e-12, + np.nextafter(0.25, 1), 0.25 + 1e-12] + for xp in pts: + y = special.gammaincinv(.4, xp) + x = special.gammainc(0.4, y) + assert_allclose(x, xp, rtol=1e-12) + + def test_rgamma(self): + rgam = special.rgamma(8) + rlgam = 1/special.gamma(8) + assert_almost_equal(rgam,rlgam,8) + + def test_infinity(self): + assert_equal(special.rgamma(-1), 0) + + @pytest.mark.parametrize( + "x,expected", + [ + # infinities + ([-np.inf, np.inf], [np.nan, np.inf]), + # negative and positive zero + ([-0.0, 0.0], [-np.inf, np.inf]), + # small poles + (range(-32, 0), np.full(32, np.nan)), + # medium sized poles + (range(-1024, -32, 99), np.full(11, np.nan)), + # large pole + ([-4.141512231792294e+16], [np.nan]), + ] + ) + def test_poles(self, x, expected): + assert_array_equal(special.gamma(x), expected) + + +class TestHankel: + + def test_negv1(self): + assert_almost_equal(special.hankel1(-3,2), -special.hankel1(3,2), 14) + + def test_hankel1(self): + hank1 = special.hankel1(1,.1) + hankrl = (special.jv(1,.1) + special.yv(1,.1)*1j) + assert_almost_equal(hank1,hankrl,8) + + def test_negv1e(self): + assert_almost_equal(special.hankel1e(-3,2), -special.hankel1e(3,2), 14) + + def test_hankel1e(self): + hank1e = special.hankel1e(1,.1) + hankrle = special.hankel1(1,.1)*exp(-.1j) + assert_almost_equal(hank1e,hankrle,8) + + def test_negv2(self): + assert_almost_equal(special.hankel2(-3,2), -special.hankel2(3,2), 14) + + def test_hankel2(self): + hank2 = special.hankel2(1,.1) + hankrl2 = (special.jv(1,.1) - special.yv(1,.1)*1j) + assert_almost_equal(hank2,hankrl2,8) + + def test_neg2e(self): + assert_almost_equal(special.hankel2e(-3,2), -special.hankel2e(3,2), 14) + + def test_hankl2e(self): + hank2e = special.hankel2e(1,.1) + hankrl2e = special.hankel2e(1,.1) + assert_almost_equal(hank2e,hankrl2e,8) + + def test_hankel2_gh4517(self): + # Test edge case reported in https://github.com/scipy/scipy/issues/4517 + res = special.hankel2(0, 0) + assert np.isnan(res.real) + assert np.isposinf(res.imag) + + +class TestHyper: + def test_h1vp(self): + h1 = special.h1vp(1,.1) + h1real = (special.jvp(1,.1) + special.yvp(1,.1)*1j) + assert_almost_equal(h1,h1real,8) + + def test_h2vp(self): + h2 = special.h2vp(1,.1) + h2real = (special.jvp(1,.1) - special.yvp(1,.1)*1j) + assert_almost_equal(h2,h2real,8) + + def test_hyp0f1(self): + # scalar input + assert_allclose(special.hyp0f1(2.5, 0.5), 1.21482702689997, rtol=1e-12) + assert_allclose(special.hyp0f1(2.5, 0), 1.0, rtol=1e-15) + + # float input, expected values match mpmath + x = special.hyp0f1(3.0, [-1.5, -1, 0, 1, 1.5]) + expected = np.array([0.58493659229143, 0.70566805723127, 1.0, + 1.37789689539747, 1.60373685288480]) + assert_allclose(x, expected, rtol=1e-12) + + # complex input + x = special.hyp0f1(3.0, np.array([-1.5, -1, 0, 1, 1.5]) + 0.j) + assert_allclose(x, expected.astype(complex), rtol=1e-12) + + # test broadcasting + x1 = [0.5, 1.5, 2.5] + x2 = [0, 1, 0.5] + x = special.hyp0f1(x1, x2) + expected = [1.0, 1.8134302039235093, 1.21482702689997] + assert_allclose(x, expected, rtol=1e-12) + x = special.hyp0f1(np.vstack([x1] * 2), x2) + assert_allclose(x, np.vstack([expected] * 2), rtol=1e-12) + assert_raises(ValueError, special.hyp0f1, + np.vstack([x1] * 3), [0, 1]) + + def test_hyp0f1_gh5764(self): + # Just checks the point that failed; there's a more systematic + # test in test_mpmath + res = special.hyp0f1(0.8, 0.5 + 0.5*1J) + # The expected value was generated using mpmath + assert_almost_equal(res, 1.6139719776441115 + 1J*0.80893054061790665) + + def test_hyp1f1(self): + hyp1 = special.hyp1f1(.1,.1,.3) + assert_almost_equal(hyp1, 1.3498588075760032,7) + + # test contributed by Moritz Deger (2008-05-29) + # https://github.com/scipy/scipy/issues/1186 (Trac #659) + + # reference data obtained from mathematica [ a, b, x, m(a,b,x)]: + # produced with test_hyp1f1.nb + ref_data = array([ + [-8.38132975e+00, -1.28436461e+01, -2.91081397e+01, 1.04178330e+04], + [2.91076882e+00, -6.35234333e+00, -1.27083993e+01, 6.68132725e+00], + [-1.42938258e+01, 1.80869131e-01, 1.90038728e+01, 1.01385897e+05], + [5.84069088e+00, 1.33187908e+01, 2.91290106e+01, 1.59469411e+08], + [-2.70433202e+01, -1.16274873e+01, -2.89582384e+01, 1.39900152e+24], + [4.26344966e+00, -2.32701773e+01, 1.91635759e+01, 6.13816915e+21], + [1.20514340e+01, -3.40260240e+00, 7.26832235e+00, 1.17696112e+13], + [2.77372955e+01, -1.99424687e+00, 3.61332246e+00, 3.07419615e+13], + [1.50310939e+01, -2.91198675e+01, -1.53581080e+01, -3.79166033e+02], + [1.43995827e+01, 9.84311196e+00, 1.93204553e+01, 2.55836264e+10], + [-4.08759686e+00, 1.34437025e+01, -1.42072843e+01, 1.70778449e+01], + [8.05595738e+00, -1.31019838e+01, 1.52180721e+01, 3.06233294e+21], + [1.81815804e+01, -1.42908793e+01, 9.57868793e+00, -2.84771348e+20], + [-2.49671396e+01, 1.25082843e+01, -1.71562286e+01, 2.36290426e+07], + [2.67277673e+01, 1.70315414e+01, 6.12701450e+00, 7.77917232e+03], + [2.49565476e+01, 2.91694684e+01, 6.29622660e+00, 2.35300027e+02], + [6.11924542e+00, -1.59943768e+00, 9.57009289e+00, 1.32906326e+11], + [-1.47863653e+01, 2.41691301e+01, -1.89981821e+01, 2.73064953e+03], + [2.24070483e+01, -2.93647433e+00, 8.19281432e+00, -6.42000372e+17], + [8.04042600e-01, 1.82710085e+01, -1.97814534e+01, 5.48372441e-01], + [1.39590390e+01, 1.97318686e+01, 2.37606635e+00, 5.51923681e+00], + [-4.66640483e+00, -2.00237930e+01, 7.40365095e+00, 4.50310752e+00], + [2.76821999e+01, -6.36563968e+00, 1.11533984e+01, -9.28725179e+23], + [-2.56764457e+01, 1.24544906e+00, 1.06407572e+01, 1.25922076e+01], + [3.20447808e+00, 1.30874383e+01, 2.26098014e+01, 2.03202059e+04], + [-1.24809647e+01, 4.15137113e+00, -2.92265700e+01, 2.39621411e+08], + [2.14778108e+01, -2.35162960e+00, -1.13758664e+01, 4.46882152e-01], + [-9.85469168e+00, -3.28157680e+00, 1.67447548e+01, -1.07342390e+07], + [1.08122310e+01, -2.47353236e+01, -1.15622349e+01, -2.91733796e+03], + [-2.67933347e+01, -3.39100709e+00, 2.56006986e+01, -5.29275382e+09], + [-8.60066776e+00, -8.02200924e+00, 1.07231926e+01, 1.33548320e+06], + [-1.01724238e-01, -1.18479709e+01, -2.55407104e+01, 1.55436570e+00], + [-3.93356771e+00, 2.11106818e+01, -2.57598485e+01, 2.13467840e+01], + [3.74750503e+00, 1.55687633e+01, -2.92841720e+01, 1.43873509e-02], + [6.99726781e+00, 2.69855571e+01, -1.63707771e+01, 3.08098673e-02], + [-2.31996011e+01, 3.47631054e+00, 9.75119815e-01, 1.79971073e-02], + [2.38951044e+01, -2.91460190e+01, -2.50774708e+00, 9.56934814e+00], + [1.52730825e+01, 5.77062507e+00, 1.21922003e+01, 1.32345307e+09], + [1.74673917e+01, 1.89723426e+01, 4.94903250e+00, 9.90859484e+01], + [1.88971241e+01, 2.86255413e+01, 5.52360109e-01, 1.44165360e+00], + [1.02002319e+01, -1.66855152e+01, -2.55426235e+01, 6.56481554e+02], + [-1.79474153e+01, 1.22210200e+01, -1.84058212e+01, 8.24041812e+05], + [-1.36147103e+01, 1.32365492e+00, -7.22375200e+00, 9.92446491e+05], + [7.57407832e+00, 2.59738234e+01, -1.34139168e+01, 3.64037761e-02], + [2.21110169e+00, 1.28012666e+01, 1.62529102e+01, 1.33433085e+02], + [-2.64297569e+01, -1.63176658e+01, -1.11642006e+01, -2.44797251e+13], + [-2.46622944e+01, -3.02147372e+00, 8.29159315e+00, -3.21799070e+05], + [-1.37215095e+01, -1.96680183e+01, 2.91940118e+01, 3.21457520e+12], + [-5.45566105e+00, 2.81292086e+01, 1.72548215e-01, 9.66973000e-01], + [-1.55751298e+00, -8.65703373e+00, 2.68622026e+01, -3.17190834e+16], + [2.45393609e+01, -2.70571903e+01, 1.96815505e+01, 1.80708004e+37], + [5.77482829e+00, 1.53203143e+01, 2.50534322e+01, 1.14304242e+06], + [-1.02626819e+01, 2.36887658e+01, -2.32152102e+01, 7.28965646e+02], + [-1.30833446e+00, -1.28310210e+01, 1.87275544e+01, -9.33487904e+12], + [5.83024676e+00, -1.49279672e+01, 2.44957538e+01, -7.61083070e+27], + [-2.03130747e+01, 2.59641715e+01, -2.06174328e+01, 4.54744859e+04], + [1.97684551e+01, -2.21410519e+01, -2.26728740e+01, 3.53113026e+06], + [2.73673444e+01, 2.64491725e+01, 1.57599882e+01, 1.07385118e+07], + [5.73287971e+00, 1.21111904e+01, 1.33080171e+01, 2.63220467e+03], + [-2.82751072e+01, 2.08605881e+01, 9.09838900e+00, -6.60957033e-07], + [1.87270691e+01, -1.74437016e+01, 1.52413599e+01, 6.59572851e+27], + [6.60681457e+00, -2.69449855e+00, 9.78972047e+00, -2.38587870e+12], + [1.20895561e+01, -2.51355765e+01, 2.30096101e+01, 7.58739886e+32], + [-2.44682278e+01, 2.10673441e+01, -1.36705538e+01, 4.54213550e+04], + [-4.50665152e+00, 3.72292059e+00, -4.83403707e+00, 2.68938214e+01], + [-7.46540049e+00, -1.08422222e+01, -1.72203805e+01, -2.09402162e+02], + [-2.00307551e+01, -7.50604431e+00, -2.78640020e+01, 4.15985444e+19], + [1.99890876e+01, 2.20677419e+01, -2.51301778e+01, 1.23840297e-09], + [2.03183823e+01, -7.66942559e+00, 2.10340070e+01, 1.46285095e+31], + [-2.90315825e+00, -2.55785967e+01, -9.58779316e+00, 2.65714264e-01], + [2.73960829e+01, -1.80097203e+01, -2.03070131e+00, 2.52908999e+02], + [-2.11708058e+01, -2.70304032e+01, 2.48257944e+01, 3.09027527e+08], + [2.21959758e+01, 4.00258675e+00, -1.62853977e+01, -9.16280090e-09], + [1.61661840e+01, -2.26845150e+01, 2.17226940e+01, -8.24774394e+33], + [-3.35030306e+00, 1.32670581e+00, 9.39711214e+00, -1.47303163e+01], + [7.23720726e+00, -2.29763909e+01, 2.34709682e+01, -9.20711735e+29], + [2.71013568e+01, 1.61951087e+01, -7.11388906e-01, 2.98750911e-01], + [8.40057933e+00, -7.49665220e+00, 2.95587388e+01, 6.59465635e+29], + [-1.51603423e+01, 1.94032322e+01, -7.60044357e+00, 1.05186941e+02], + [-8.83788031e+00, -2.72018313e+01, 1.88269907e+00, 1.81687019e+00], + [-1.87283712e+01, 5.87479570e+00, -1.91210203e+01, 2.52235612e+08], + [-5.61338513e-01, 2.69490237e+01, 1.16660111e-01, 9.97567783e-01], + [-5.44354025e+00, -1.26721408e+01, -4.66831036e+00, 1.06660735e-01], + [-2.18846497e+00, 2.33299566e+01, 9.62564397e+00, 3.03842061e-01], + [6.65661299e+00, -2.39048713e+01, 1.04191807e+01, 4.73700451e+13], + [-2.57298921e+01, -2.60811296e+01, 2.74398110e+01, -5.32566307e+11], + [-1.11431826e+01, -1.59420160e+01, -1.84880553e+01, -1.01514747e+02], + [6.50301931e+00, 2.59859051e+01, -2.33270137e+01, 1.22760500e-02], + [-1.94987891e+01, -2.62123262e+01, 3.90323225e+00, 1.71658894e+01], + [7.26164601e+00, -1.41469402e+01, 2.81499763e+01, -2.50068329e+31], + [-1.52424040e+01, 2.99719005e+01, -2.85753678e+01, 1.31906693e+04], + [5.24149291e+00, -1.72807223e+01, 2.22129493e+01, 2.50748475e+25], + [3.63207230e-01, -9.54120862e-02, -2.83874044e+01, 9.43854939e-01], + [-2.11326457e+00, -1.25707023e+01, 1.17172130e+00, 1.20812698e+00], + [2.48513582e+00, 1.03652647e+01, -1.84625148e+01, 6.47910997e-02], + [2.65395942e+01, 2.74794672e+01, 1.29413428e+01, 2.89306132e+05], + [-9.49445460e+00, 1.59930921e+01, -1.49596331e+01, 3.27574841e+02], + [-5.89173945e+00, 9.96742426e+00, 2.60318889e+01, -3.15842908e-01], + [-1.15387239e+01, -2.21433107e+01, -2.17686413e+01, 1.56724718e-01], + [-5.30592244e+00, -2.42752190e+01, 1.29734035e+00, 1.31985534e+00] + ]) + + for a,b,c,expected in ref_data: + result = special.hyp1f1(a,b,c) + assert_(abs(expected - result)/expected < 1e-4) + + def test_hyp1f1_gh2957(self): + hyp1 = special.hyp1f1(0.5, 1.5, -709.7827128933) + hyp2 = special.hyp1f1(0.5, 1.5, -709.7827128934) + assert_almost_equal(hyp1, hyp2, 12) + + def test_hyp1f1_gh2282(self): + hyp = special.hyp1f1(0.5, 1.5, -1000) + assert_almost_equal(hyp, 0.028024956081989643, 12) + + def test_hyp2f1(self): + # a collection of special cases taken from AMS 55 + values = [ + [0.5, 1, 1.5, 0.2**2, 0.5/0.2*log((1+0.2)/(1-0.2))], + [0.5, 1, 1.5, -0.2**2, 1./0.2*arctan(0.2)], + [1, 1, 2, 0.2, -1/0.2*log(1-0.2)], + [3, 3.5, 1.5, 0.2**2, 0.5/0.2/(-5)*((1+0.2)**(-5)-(1-0.2)**(-5))], + [-3, 3, 0.5, sin(0.2)**2, cos(2*3*0.2)], + [3, 4, 8, 1, + special.gamma(8) * special.gamma(8-4-3) + / special.gamma(8-3) / special.gamma(8-4)], + [3, 2, 3-2+1, -1, + 1./2**3*sqrt(pi) * special.gamma(1+3-2) + / special.gamma(1+0.5*3-2) / special.gamma(0.5+0.5*3)], + [5, 2, 5-2+1, -1, + 1./2**5*sqrt(pi) * special.gamma(1+5-2) + / special.gamma(1+0.5*5-2) / special.gamma(0.5+0.5*5)], + [4, 0.5+4, 1.5-2*4, -1./3, + (8./9)**(-2*4)*special.gamma(4./3) * special.gamma(1.5-2*4) + / special.gamma(3./2) / special.gamma(4./3-2*4)], + # and some others + # ticket #424 + [1.5, -0.5, 1.0, -10.0, 4.1300097765277476484], + # negative integer a or b, with c-a-b integer and x > 0.9 + [-2,3,1,0.95,0.715], + [2,-3,1,0.95,-0.007], + [-6,3,1,0.95,0.0000810625], + [2,-5,1,0.95,-0.000029375], + # huge negative integers + (10, -900, 10.5, 0.99, 1.91853705796607664803709475658e-24), + (10, -900, -10.5, 0.99, 3.54279200040355710199058559155e-18), + ] + for i, (a, b, c, x, v) in enumerate(values): + cv = special.hyp2f1(a, b, c, x) + assert_almost_equal(cv, v, 8, err_msg='test #%d' % i) + + def test_hyperu(self): + val1 = special.hyperu(1,0.1,100) + assert_almost_equal(val1,0.0098153,7) + a,b = [0.3,0.6,1.2,-2.7],[1.5,3.2,-0.4,-3.2] + a,b = asarray(a), asarray(b) + z = 0.5 + hypu = special.hyperu(a,b,z) + hprl = (pi/sin(pi*b))*(special.hyp1f1(a,b,z) / + (special.gamma(1+a-b)*special.gamma(b)) - + z**(1-b)*special.hyp1f1(1+a-b,2-b,z) + / (special.gamma(a)*special.gamma(2-b))) + assert_array_almost_equal(hypu,hprl,12) + + def test_hyperu_gh2287(self): + assert_almost_equal(special.hyperu(1, 1.5, 20.2), + 0.048360918656699191, 12) + + +class TestBessel: + def test_itj0y0(self): + it0 = array(special.itj0y0(.2)) + assert_array_almost_equal( + it0, + array([0.19933433254006822, -0.34570883800412566]), + 8, + ) + + def test_it2j0y0(self): + it2 = array(special.it2j0y0(.2)) + assert_array_almost_equal( + it2, + array([0.0049937546274601858, -0.43423067011231614]), + 8, + ) + + def test_negv_iv(self): + assert_equal(special.iv(3,2), special.iv(-3,2)) + + def test_j0(self): + oz = special.j0(.1) + ozr = special.jn(0,.1) + assert_almost_equal(oz,ozr,8) + + def test_j1(self): + o1 = special.j1(.1) + o1r = special.jn(1,.1) + assert_almost_equal(o1,o1r,8) + + def test_jn(self): + jnnr = special.jn(1,.2) + assert_almost_equal(jnnr,0.099500832639235995,8) + + def test_negv_jv(self): + assert_almost_equal(special.jv(-3,2), -special.jv(3,2), 14) + + def test_jv(self): + values = [[0, 0.1, 0.99750156206604002], + [2./3, 1e-8, 0.3239028506761532e-5], + [2./3, 1e-10, 0.1503423854873779e-6], + [3.1, 1e-10, 0.1711956265409013e-32], + [2./3, 4.0, -0.2325440850267039], + ] + for i, (v, x, y) in enumerate(values): + yc = special.jv(v, x) + assert_almost_equal(yc, y, 8, err_msg='test #%d' % i) + + def test_negv_jve(self): + assert_almost_equal(special.jve(-3,2), -special.jve(3,2), 14) + + def test_jve(self): + jvexp = special.jve(1,.2) + assert_almost_equal(jvexp,0.099500832639235995,8) + jvexp1 = special.jve(1,.2+1j) + z = .2+1j + jvexpr = special.jv(1,z)*exp(-abs(z.imag)) + assert_almost_equal(jvexp1,jvexpr,8) + + def test_jn_zeros(self): + jn0 = special.jn_zeros(0,5) + jn1 = special.jn_zeros(1,5) + assert_array_almost_equal(jn0,array([2.4048255577, + 5.5200781103, + 8.6537279129, + 11.7915344391, + 14.9309177086]),4) + assert_array_almost_equal(jn1,array([3.83171, + 7.01559, + 10.17347, + 13.32369, + 16.47063]),4) + + jn102 = special.jn_zeros(102,5) + assert_allclose(jn102, array([110.89174935992040343, + 117.83464175788308398, + 123.70194191713507279, + 129.02417238949092824, + 134.00114761868422559]), rtol=1e-13) + + jn301 = special.jn_zeros(301,5) + assert_allclose(jn301, array([313.59097866698830153, + 323.21549776096288280, + 331.22338738656748796, + 338.39676338872084500, + 345.03284233056064157]), rtol=1e-13) + + def test_jn_zeros_slow(self): + jn0 = special.jn_zeros(0, 300) + assert_allclose(jn0[260-1], 816.02884495068867280, rtol=1e-13) + assert_allclose(jn0[280-1], 878.86068707124422606, rtol=1e-13) + assert_allclose(jn0[300-1], 941.69253065317954064, rtol=1e-13) + + jn10 = special.jn_zeros(10, 300) + assert_allclose(jn10[260-1], 831.67668514305631151, rtol=1e-13) + assert_allclose(jn10[280-1], 894.51275095371316931, rtol=1e-13) + assert_allclose(jn10[300-1], 957.34826370866539775, rtol=1e-13) + + jn3010 = special.jn_zeros(3010,5) + assert_allclose(jn3010, array([3036.86590780927, + 3057.06598526482, + 3073.66360690272, + 3088.37736494778, + 3101.86438139042]), rtol=1e-8) + + def test_jnjnp_zeros(self): + jn = special.jn + + def jnp(n, x): + return (jn(n-1,x) - jn(n+1,x))/2 + for nt in range(1, 30): + z, n, m, t = special.jnjnp_zeros(nt) + for zz, nn, tt in zip(z, n, t): + if tt == 0: + assert_allclose(jn(nn, zz), 0, atol=1e-6) + elif tt == 1: + assert_allclose(jnp(nn, zz), 0, atol=1e-6) + else: + raise AssertionError("Invalid t return for nt=%d" % nt) + + def test_jnp_zeros(self): + jnp = special.jnp_zeros(1,5) + assert_array_almost_equal(jnp, array([1.84118, + 5.33144, + 8.53632, + 11.70600, + 14.86359]),4) + jnp = special.jnp_zeros(443,5) + assert_allclose(special.jvp(443, jnp), 0, atol=1e-15) + + def test_jnyn_zeros(self): + jnz = special.jnyn_zeros(1,5) + assert_array_almost_equal(jnz,(array([3.83171, + 7.01559, + 10.17347, + 13.32369, + 16.47063]), + array([1.84118, + 5.33144, + 8.53632, + 11.70600, + 14.86359]), + array([2.19714, + 5.42968, + 8.59601, + 11.74915, + 14.89744]), + array([3.68302, + 6.94150, + 10.12340, + 13.28576, + 16.44006])),5) + + def test_jvp(self): + jvprim = special.jvp(2,2) + jv0 = (special.jv(1,2)-special.jv(3,2))/2 + assert_almost_equal(jvprim,jv0,10) + + def test_k0(self): + ozk = special.k0(.1) + ozkr = special.kv(0,.1) + assert_almost_equal(ozk,ozkr,8) + + def test_k0e(self): + ozke = special.k0e(.1) + ozker = special.kve(0,.1) + assert_almost_equal(ozke,ozker,8) + + def test_k1(self): + o1k = special.k1(.1) + o1kr = special.kv(1,.1) + assert_almost_equal(o1k,o1kr,8) + + def test_k1e(self): + o1ke = special.k1e(.1) + o1ker = special.kve(1,.1) + assert_almost_equal(o1ke,o1ker,8) + + def test_jacobi(self): + a = 5*np.random.random() - 1 + b = 5*np.random.random() - 1 + P0 = special.jacobi(0,a,b) + P1 = special.jacobi(1,a,b) + P2 = special.jacobi(2,a,b) + P3 = special.jacobi(3,a,b) + + assert_array_almost_equal(P0.c,[1],13) + assert_array_almost_equal(P1.c,array([a+b+2,a-b])/2.0,13) + cp = [(a+b+3)*(a+b+4), 4*(a+b+3)*(a+2), 4*(a+1)*(a+2)] + p2c = [cp[0],cp[1]-2*cp[0],cp[2]-cp[1]+cp[0]] + assert_array_almost_equal(P2.c,array(p2c)/8.0,13) + cp = [(a+b+4)*(a+b+5)*(a+b+6),6*(a+b+4)*(a+b+5)*(a+3), + 12*(a+b+4)*(a+2)*(a+3),8*(a+1)*(a+2)*(a+3)] + p3c = [cp[0],cp[1]-3*cp[0],cp[2]-2*cp[1]+3*cp[0],cp[3]-cp[2]+cp[1]-cp[0]] + assert_array_almost_equal(P3.c,array(p3c)/48.0,13) + + def test_kn(self): + kn1 = special.kn(0,.2) + assert_almost_equal(kn1,1.7527038555281462,8) + + def test_negv_kv(self): + assert_equal(special.kv(3.0, 2.2), special.kv(-3.0, 2.2)) + + def test_kv0(self): + kv0 = special.kv(0,.2) + assert_almost_equal(kv0, 1.7527038555281462, 10) + + def test_kv1(self): + kv1 = special.kv(1,0.2) + assert_almost_equal(kv1, 4.775972543220472, 10) + + def test_kv2(self): + kv2 = special.kv(2,0.2) + assert_almost_equal(kv2, 49.51242928773287, 10) + + def test_kn_largeorder(self): + assert_allclose(special.kn(32, 1), 1.7516596664574289e+43) + + def test_kv_largearg(self): + assert_equal(special.kv(0, 1e19), 0) + + def test_negv_kve(self): + assert_equal(special.kve(3.0, 2.2), special.kve(-3.0, 2.2)) + + def test_kve(self): + kve1 = special.kve(0,.2) + kv1 = special.kv(0,.2)*exp(.2) + assert_almost_equal(kve1,kv1,8) + z = .2+1j + kve2 = special.kve(0,z) + kv2 = special.kv(0,z)*exp(z) + assert_almost_equal(kve2,kv2,8) + + def test_kvp_v0n1(self): + z = 2.2 + assert_almost_equal(-special.kv(1,z), special.kvp(0,z, n=1), 10) + + def test_kvp_n1(self): + v = 3. + z = 2.2 + xc = -special.kv(v+1,z) + v/z*special.kv(v,z) + x = special.kvp(v,z, n=1) + assert_almost_equal(xc, x, 10) # this function (kvp) is broken + + def test_kvp_n2(self): + v = 3. + z = 2.2 + xc = (z**2+v**2-v)/z**2 * special.kv(v,z) + special.kv(v+1,z)/z + x = special.kvp(v, z, n=2) + assert_almost_equal(xc, x, 10) + + def test_y0(self): + oz = special.y0(.1) + ozr = special.yn(0,.1) + assert_almost_equal(oz,ozr,8) + + def test_y1(self): + o1 = special.y1(.1) + o1r = special.yn(1,.1) + assert_almost_equal(o1,o1r,8) + + def test_y0_zeros(self): + yo,ypo = special.y0_zeros(2) + zo,zpo = special.y0_zeros(2,complex=1) + all = r_[yo,zo] + allval = r_[ypo,zpo] + assert_array_almost_equal(abs(special.yv(0.0,all)),0.0,11) + assert_array_almost_equal(abs(special.yv(1,all)-allval),0.0,11) + + def test_y1_zeros(self): + y1 = special.y1_zeros(1) + assert_array_almost_equal(y1,(array([2.19714]),array([0.52079])),5) + + def test_y1p_zeros(self): + y1p = special.y1p_zeros(1,complex=1) + assert_array_almost_equal( + y1p, + (array([0.5768+0.904j]), array([-0.7635+0.5892j])), + 3, + ) + + def test_yn_zeros(self): + an = special.yn_zeros(4,2) + assert_array_almost_equal(an,array([5.64515, 9.36162]),5) + an = special.yn_zeros(443,5) + assert_allclose(an, [450.13573091578090314, + 463.05692376675001542, + 472.80651546418663566, + 481.27353184725625838, + 488.98055964441374646], + rtol=1e-15,) + + def test_ynp_zeros(self): + ao = special.ynp_zeros(0,2) + assert_array_almost_equal(ao,array([2.19714133, 5.42968104]),6) + ao = special.ynp_zeros(43,5) + assert_allclose(special.yvp(43, ao), 0, atol=1e-15) + ao = special.ynp_zeros(443,5) + assert_allclose(special.yvp(443, ao), 0, atol=1e-9) + + def test_ynp_zeros_large_order(self): + ao = special.ynp_zeros(443,5) + assert_allclose(special.yvp(443, ao), 0, atol=1e-14) + + def test_yn(self): + yn2n = special.yn(1,.2) + assert_almost_equal(yn2n,-3.3238249881118471,8) + + def test_yn_gh_20405(self): + # Enforce correct asymptotic behavior for large n. + observed = cephes.yn(500, 1) + assert observed == -np.inf + + def test_negv_yv(self): + assert_almost_equal(special.yv(-3,2), -special.yv(3,2), 14) + + def test_yv(self): + yv2 = special.yv(1,.2) + assert_almost_equal(yv2,-3.3238249881118471,8) + + def test_negv_yve(self): + assert_almost_equal(special.yve(-3,2), -special.yve(3,2), 14) + + def test_yve(self): + yve2 = special.yve(1,.2) + assert_almost_equal(yve2,-3.3238249881118471,8) + yve2r = special.yv(1,.2+1j)*exp(-1) + yve22 = special.yve(1,.2+1j) + assert_almost_equal(yve22,yve2r,8) + + def test_yvp(self): + yvpr = (special.yv(1,.2) - special.yv(3,.2))/2.0 + yvp1 = special.yvp(2,.2) + assert_array_almost_equal(yvp1,yvpr,10) + + def _cephes_vs_amos_points(self): + """Yield points at which to compare Cephes implementation to AMOS""" + # check several points, including large-amplitude ones + v = [-120, -100.3, -20., -10., -1., -.5, 0., 1., 12.49, 120., 301] + z = [-1300, -11, -10, -1, 1., 10., 200.5, 401., 600.5, 700.6, 1300, + 10003] + yield from itertools.product(v, z) + + # check half-integers; these are problematic points at least + # for cephes/iv + yield from itertools.product(0.5 + arange(-60, 60), [3.5]) + + def check_cephes_vs_amos(self, f1, f2, rtol=1e-11, atol=0, skip=None): + for v, z in self._cephes_vs_amos_points(): + if skip is not None and skip(v, z): + continue + c1, c2, c3 = f1(v, z), f1(v,z+0j), f2(int(v), z) + if np.isinf(c1): + assert_(np.abs(c2) >= 1e300, (v, z)) + elif np.isnan(c1): + assert_(c2.imag != 0, (v, z)) + else: + assert_allclose(c1, c2, err_msg=(v, z), rtol=rtol, atol=atol) + if v == int(v): + assert_allclose(c3, c2, err_msg=(v, z), + rtol=rtol, atol=atol) + + @pytest.mark.xfail(platform.machine() == 'ppc64le', + reason="fails on ppc64le") + def test_jv_cephes_vs_amos(self): + self.check_cephes_vs_amos(special.jv, special.jn, rtol=1e-10, atol=1e-305) + + @pytest.mark.xfail(platform.machine() == 'ppc64le', + reason="fails on ppc64le") + def test_yv_cephes_vs_amos(self): + self.check_cephes_vs_amos(special.yv, special.yn, rtol=1e-11, atol=1e-305) + + def test_yv_cephes_vs_amos_only_small_orders(self): + def skipper(v, z): + return abs(v) > 50 + self.check_cephes_vs_amos(special.yv, special.yn, rtol=1e-11, atol=1e-305, + skip=skipper) + + def test_iv_cephes_vs_amos(self): + with np.errstate(all='ignore'): + self.check_cephes_vs_amos(special.iv, special.iv, rtol=5e-9, atol=1e-305) + + @pytest.mark.slow + def test_iv_cephes_vs_amos_mass_test(self): + N = 1000000 + np.random.seed(1) + v = np.random.pareto(0.5, N) * (-1)**np.random.randint(2, size=N) + x = np.random.pareto(0.2, N) * (-1)**np.random.randint(2, size=N) + + imsk = (np.random.randint(8, size=N) == 0) + v[imsk] = v[imsk].astype(np.int64) + + with np.errstate(all='ignore'): + c1 = special.iv(v, x) + c2 = special.iv(v, x+0j) + + # deal with differences in the inf and zero cutoffs + c1[abs(c1) > 1e300] = np.inf + c2[abs(c2) > 1e300] = np.inf + c1[abs(c1) < 1e-300] = 0 + c2[abs(c2) < 1e-300] = 0 + + dc = abs(c1/c2 - 1) + dc[np.isnan(dc)] = 0 + + k = np.argmax(dc) + + # Most error apparently comes from AMOS and not our implementation; + # there are some problems near integer orders there + assert_( + dc[k] < 2e-7, + (v[k], x[k], special.iv(v[k], x[k]), special.iv(v[k], x[k]+0j)) + ) + + def test_kv_cephes_vs_amos(self): + self.check_cephes_vs_amos(special.kv, special.kn, rtol=1e-9, atol=1e-305) + self.check_cephes_vs_amos(special.kv, special.kv, rtol=1e-9, atol=1e-305) + + def test_ticket_623(self): + assert_allclose(special.jv(3, 4), 0.43017147387562193) + assert_allclose(special.jv(301, 1300), 0.0183487151115275) + assert_allclose(special.jv(301, 1296.0682), -0.0224174325312048) + + def test_ticket_853(self): + """Negative-order Bessels""" + # cephes + assert_allclose(special.jv(-1, 1), -0.4400505857449335) + assert_allclose(special.jv(-2, 1), 0.1149034849319005) + assert_allclose(special.yv(-1, 1), 0.7812128213002887) + assert_allclose(special.yv(-2, 1), -1.650682606816255) + assert_allclose(special.iv(-1, 1), 0.5651591039924851) + assert_allclose(special.iv(-2, 1), 0.1357476697670383) + assert_allclose(special.kv(-1, 1), 0.6019072301972347) + assert_allclose(special.kv(-2, 1), 1.624838898635178) + assert_allclose(special.jv(-0.5, 1), 0.43109886801837607952) + assert_allclose(special.yv(-0.5, 1), 0.6713967071418031) + assert_allclose(special.iv(-0.5, 1), 1.231200214592967) + assert_allclose(special.kv(-0.5, 1), 0.4610685044478945) + # amos + assert_allclose(special.jv(-1, 1+0j), -0.4400505857449335) + assert_allclose(special.jv(-2, 1+0j), 0.1149034849319005) + assert_allclose(special.yv(-1, 1+0j), 0.7812128213002887) + assert_allclose(special.yv(-2, 1+0j), -1.650682606816255) + + assert_allclose(special.iv(-1, 1+0j), 0.5651591039924851) + assert_allclose(special.iv(-2, 1+0j), 0.1357476697670383) + assert_allclose(special.kv(-1, 1+0j), 0.6019072301972347) + assert_allclose(special.kv(-2, 1+0j), 1.624838898635178) + + assert_allclose(special.jv(-0.5, 1+0j), 0.43109886801837607952) + assert_allclose(special.jv(-0.5, 1+1j), 0.2628946385649065-0.827050182040562j) + assert_allclose(special.yv(-0.5, 1+0j), 0.6713967071418031) + assert_allclose(special.yv(-0.5, 1+1j), 0.967901282890131+0.0602046062142816j) + + assert_allclose(special.iv(-0.5, 1+0j), 1.231200214592967) + assert_allclose(special.iv(-0.5, 1+1j), 0.77070737376928+0.39891821043561j) + assert_allclose(special.kv(-0.5, 1+0j), 0.4610685044478945) + assert_allclose(special.kv(-0.5, 1+1j), 0.06868578341999-0.38157825981268j) + + assert_allclose(special.jve(-0.5,1+0.3j), special.jv(-0.5, 1+0.3j)*exp(-0.3)) + assert_allclose(special.yve(-0.5,1+0.3j), special.yv(-0.5, 1+0.3j)*exp(-0.3)) + assert_allclose(special.ive(-0.5,0.3+1j), special.iv(-0.5, 0.3+1j)*exp(-0.3)) + assert_allclose(special.kve(-0.5,0.3+1j), special.kv(-0.5, 0.3+1j)*exp(0.3+1j)) + + assert_allclose( + special.hankel1(-0.5, 1+1j), + special.jv(-0.5, 1+1j) + 1j*special.yv(-0.5,1+1j) + ) + assert_allclose( + special.hankel2(-0.5, 1+1j), + special.jv(-0.5, 1+1j) - 1j*special.yv(-0.5,1+1j) + ) + + def test_ticket_854(self): + """Real-valued Bessel domains""" + assert_(isnan(special.jv(0.5, -1))) + assert_(isnan(special.iv(0.5, -1))) + assert_(isnan(special.yv(0.5, -1))) + assert_(isnan(special.yv(1, -1))) + assert_(isnan(special.kv(0.5, -1))) + assert_(isnan(special.kv(1, -1))) + assert_(isnan(special.jve(0.5, -1))) + assert_(isnan(special.ive(0.5, -1))) + assert_(isnan(special.yve(0.5, -1))) + assert_(isnan(special.yve(1, -1))) + assert_(isnan(special.kve(0.5, -1))) + assert_(isnan(special.kve(1, -1))) + assert_(isnan(special.airye(-1)[0:2]).all(), special.airye(-1)) + assert_(not isnan(special.airye(-1)[2:4]).any(), special.airye(-1)) + + def test_gh_7909(self): + assert_(special.kv(1.5, 0) == np.inf) + assert_(special.kve(1.5, 0) == np.inf) + + def test_ticket_503(self): + """Real-valued Bessel I overflow""" + assert_allclose(special.iv(1, 700), 1.528500390233901e302) + assert_allclose(special.iv(1000, 1120), 1.301564549405821e301) + + def test_iv_hyperg_poles(self): + assert_allclose(special.iv(-0.5, 1), 1.231200214592967) + + def iv_series(self, v, z, n=200): + k = arange(0, n).astype(double) + r = (v+2*k)*log(.5*z) - special.gammaln(k+1) - special.gammaln(v+k+1) + r[isnan(r)] = inf + r = exp(r) + err = abs(r).max() * finfo(double).eps * n + abs(r[-1])*10 + return r.sum(), err + + def test_i0_series(self): + for z in [1., 10., 200.5]: + value, err = self.iv_series(0, z) + assert_allclose(special.i0(z), value, atol=err, err_msg=z) + + def test_i1_series(self): + for z in [1., 10., 200.5]: + value, err = self.iv_series(1, z) + assert_allclose(special.i1(z), value, atol=err, err_msg=z) + + def test_iv_series(self): + for v in [-20., -10., -1., 0., 1., 12.49, 120.]: + for z in [1., 10., 200.5, -1+2j]: + value, err = self.iv_series(v, z) + assert_allclose(special.iv(v, z), value, atol=err, err_msg=(v, z)) + + def test_i0(self): + values = [[0.0, 1.0], + [1e-10, 1.0], + [0.1, 0.9071009258], + [0.5, 0.6450352706], + [1.0, 0.4657596077], + [2.5, 0.2700464416], + [5.0, 0.1835408126], + [20.0, 0.0897803119], + ] + for i, (x, v) in enumerate(values): + cv = special.i0(x) * exp(-x) + assert_almost_equal(cv, v, 8, err_msg='test #%d' % i) + + def test_i0e(self): + oize = special.i0e(.1) + oizer = special.ive(0,.1) + assert_almost_equal(oize,oizer,8) + + def test_i1(self): + values = [[0.0, 0.0], + [1e-10, 0.4999999999500000e-10], + [0.1, 0.0452984468], + [0.5, 0.1564208032], + [1.0, 0.2079104154], + [5.0, 0.1639722669], + [20.0, 0.0875062222], + ] + for i, (x, v) in enumerate(values): + cv = special.i1(x) * exp(-x) + assert_almost_equal(cv, v, 8, err_msg='test #%d' % i) + + def test_i1e(self): + oi1e = special.i1e(.1) + oi1er = special.ive(1,.1) + assert_almost_equal(oi1e,oi1er,8) + + def test_iti0k0(self): + iti0 = array(special.iti0k0(5)) + assert_array_almost_equal( + iti0, + array([31.848667776169801, 1.5673873907283657]), + 5, + ) + + def test_it2i0k0(self): + it2k = special.it2i0k0(.1) + assert_array_almost_equal( + it2k, + array([0.0012503906973464409, 3.3309450354686687]), + 6, + ) + + def test_iv(self): + iv1 = special.iv(0,.1)*exp(-.1) + assert_almost_equal(iv1,0.90710092578230106,10) + + def test_negv_ive(self): + assert_equal(special.ive(3,2), special.ive(-3,2)) + + def test_ive(self): + ive1 = special.ive(0,.1) + iv1 = special.iv(0,.1)*exp(-.1) + assert_almost_equal(ive1,iv1,10) + + def test_ivp0(self): + assert_almost_equal(special.iv(1,2), special.ivp(0,2), 10) + + def test_ivp(self): + y = (special.iv(0,2) + special.iv(2,2))/2 + x = special.ivp(1,2) + assert_almost_equal(x,y,10) + + +class TestLaguerre: + def test_laguerre(self): + lag0 = special.laguerre(0) + lag1 = special.laguerre(1) + lag2 = special.laguerre(2) + lag3 = special.laguerre(3) + lag4 = special.laguerre(4) + lag5 = special.laguerre(5) + assert_array_almost_equal(lag0.c,[1],13) + assert_array_almost_equal(lag1.c,[-1,1],13) + assert_array_almost_equal(lag2.c,array([1,-4,2])/2.0,13) + assert_array_almost_equal(lag3.c,array([-1,9,-18,6])/6.0,13) + assert_array_almost_equal(lag4.c,array([1,-16,72,-96,24])/24.0,13) + assert_array_almost_equal(lag5.c,array([-1,25,-200,600,-600,120])/120.0,13) + + def test_genlaguerre(self): + k = 5*np.random.random() - 0.9 + lag0 = special.genlaguerre(0,k) + lag1 = special.genlaguerre(1,k) + lag2 = special.genlaguerre(2,k) + lag3 = special.genlaguerre(3,k) + assert_equal(lag0.c, [1]) + assert_equal(lag1.c, [-1, k + 1]) + assert_almost_equal( + lag2.c, + array([1,-2*(k+2),(k+1.)*(k+2.)])/2.0 + ) + assert_almost_equal( + lag3.c, + array([-1,3*(k+3),-3*(k+2)*(k+3),(k+1)*(k+2)*(k+3)])/6.0 + ) + + +class TestLambda: + def test_lmbda(self): + lam = special.lmbda(1,.1) + lamr = ( + array([special.jn(0,.1), 2*special.jn(1,.1)/.1]), + array([special.jvp(0,.1), -2*special.jv(1,.1)/.01 + 2*special.jvp(1,.1)/.1]) + ) + assert_array_almost_equal(lam,lamr,8) + + +class TestLog1p: + def test_log1p(self): + l1p = (special.log1p(10), special.log1p(11), special.log1p(12)) + l1prl = (log(11), log(12), log(13)) + assert_array_almost_equal(l1p,l1prl,8) + + def test_log1pmore(self): + l1pm = (special.log1p(1), special.log1p(1.1), special.log1p(1.2)) + l1pmrl = (log(2),log(2.1),log(2.2)) + assert_array_almost_equal(l1pm,l1pmrl,8) + + +class TestMathieu: + + def test_mathieu_a(self): + pass + + def test_mathieu_even_coef(self): + special.mathieu_even_coef(2,5) + # Q not defined broken and cannot figure out proper reporting order + + def test_mathieu_odd_coef(self): + # same problem as above + pass + + +class TestFresnelIntegral: + + def test_modfresnelp(self): + pass + + def test_modfresnelm(self): + pass + + +class TestOblCvSeq: + def test_obl_cv_seq(self): + obl = special.obl_cv_seq(0,3,1) + assert_array_almost_equal(obl,array([-0.348602, + 1.393206, + 5.486800, + 11.492120]),5) + + +class TestParabolicCylinder: + def test_pbdn_seq(self): + pb = special.pbdn_seq(1,.1) + assert_array_almost_equal(pb,(array([0.9975, + 0.0998]), + array([-0.0499, + 0.9925])),4) + + def test_pbdv(self): + special.pbdv(1,.2) + 1/2*(.2)*special.pbdv(1,.2)[0] - special.pbdv(0,.2)[0] + + def test_pbdv_seq(self): + pbn = special.pbdn_seq(1,.1) + pbv = special.pbdv_seq(1,.1) + assert_array_almost_equal(pbv,(real(pbn[0]),real(pbn[1])),4) + + def test_pbdv_points(self): + # simple case + eta = np.linspace(-10, 10, 5) + z = 2**(eta/2)*np.sqrt(np.pi)*special.rgamma(.5-.5*eta) + assert_allclose(special.pbdv(eta, 0.)[0], z, rtol=1e-14, atol=1e-14) + + # some points + assert_allclose(special.pbdv(10.34, 20.44)[0], 1.3731383034455e-32, rtol=1e-12) + assert_allclose(special.pbdv(-9.53, 3.44)[0], 3.166735001119246e-8, rtol=1e-12) + + def test_pbdv_gradient(self): + x = np.linspace(-4, 4, 8)[:,None] + eta = np.linspace(-10, 10, 5)[None,:] + + p = special.pbdv(eta, x) + eps = 1e-7 + 1e-7*abs(x) + dp = (special.pbdv(eta, x + eps)[0] - special.pbdv(eta, x - eps)[0]) / eps / 2. + assert_allclose(p[1], dp, rtol=1e-6, atol=1e-6) + + def test_pbvv_gradient(self): + x = np.linspace(-4, 4, 8)[:,None] + eta = np.linspace(-10, 10, 5)[None,:] + + p = special.pbvv(eta, x) + eps = 1e-7 + 1e-7*abs(x) + dp = (special.pbvv(eta, x + eps)[0] - special.pbvv(eta, x - eps)[0]) / eps / 2. + assert_allclose(p[1], dp, rtol=1e-6, atol=1e-6) + + def test_pbvv_seq(self): + res1, res2 = special.pbvv_seq(2, 3) + assert_allclose(res1, np.array([2.976319645712036, + 1.358840996329579, + 0.5501016716383508])) + assert_allclose(res2, np.array([3.105638472238475, + 0.9380581512176672, + 0.533688488872053])) + + +class TestPolygamma: + # from Table 6.2 (pg. 271) of A&S + def test_polygamma(self): + poly2 = special.polygamma(2,1) + poly3 = special.polygamma(3,1) + assert_almost_equal(poly2,-2.4041138063,10) + assert_almost_equal(poly3,6.4939394023,10) + + # Test polygamma(0, x) == psi(x) + x = [2, 3, 1.1e14] + assert_almost_equal(special.polygamma(0, x), special.psi(x)) + + # Test broadcasting + n = [0, 1, 2] + x = [0.5, 1.5, 2.5] + expected = [-1.9635100260214238, 0.93480220054467933, + -0.23620405164172739] + assert_almost_equal(special.polygamma(n, x), expected) + expected = np.vstack([expected]*2) + assert_almost_equal(special.polygamma(n, np.vstack([x]*2)), + expected) + assert_almost_equal(special.polygamma(np.vstack([n]*2), x), + expected) + + +class TestProCvSeq: + def test_pro_cv_seq(self): + prol = special.pro_cv_seq(0,3,1) + assert_array_almost_equal(prol,array([0.319000, + 2.593084, + 6.533471, + 12.514462]),5) + + +class TestPsi: + def test_psi(self): + ps = special.psi(1) + assert_almost_equal(ps,-0.57721566490153287,8) + + +class TestRadian: + def test_radian(self): + rad = special.radian(90,0,0) + assert_almost_equal(rad,pi/2.0,5) + + def test_radianmore(self): + rad1 = special.radian(90,1,60) + assert_almost_equal(rad1,pi/2+0.0005816135199345904,5) + + +class TestRiccati: + def test_riccati_jn(self): + N, x = 2, 0.2 + S = np.empty((N, N)) + for n in range(N): + j = special.spherical_jn(n, x) + jp = special.spherical_jn(n, x, derivative=True) + S[0,n] = x*j + S[1,n] = x*jp + j + assert_array_almost_equal(S, special.riccati_jn(n, x), 8) + + def test_riccati_yn(self): + N, x = 2, 0.2 + C = np.empty((N, N)) + for n in range(N): + y = special.spherical_yn(n, x) + yp = special.spherical_yn(n, x, derivative=True) + C[0,n] = x*y + C[1,n] = x*yp + y + assert_array_almost_equal(C, special.riccati_yn(n, x), 8) + + +class TestSoftplus: + def test_softplus(self): + # Test cases for the softplus function. Selected based on Eq.(10) of: + # Mächler, M. (2012). log1mexp-note.pdf. Rmpfr: R MPFR - Multiple Precision + # Floating-Point Reliable. Retrieved from: + # https://cran.r-project.org/web/packages/Rmpfr/vignettes/log1mexp-note.pdf + # Reference values computed with `mpmath` + import numpy as np + rng = np.random.default_rng(3298432985245) + n = 3 + a1 = rng.uniform(-100, -37, size=n) + a2 = rng.uniform(-37, 18, size=n) + a3 = rng.uniform(18, 33.3, size=n) + a4 = rng.uniform(33.33, 100, size=n) + a = np.stack([a1, a2, a3, a4]) + + # from mpmath import mp + # mp.dps = 100 + # @np.vectorize + # def softplus(x): + # return float(mp.log(mp.one + mp.exp(x))) + # softplus(a).tolist() + ref = [[1.692721323272333e-42, 7.42673911145206e-41, 8.504608846033205e-35], + [1.8425343736349797, 9.488245799395577e-15, 7.225195764021444e-08], + [31.253760266045106, 27.758244090327832, 29.995959179643634], + [73.26040086468937, 76.24944728617226, 37.83955519155184]] + + res = softplus(a) + assert_allclose(res, ref, rtol=2e-15) + + def test_softplus_with_kwargs(self): + x = np.arange(5) - 2 + out = np.ones(5) + ref = out.copy() + where = x > 0 + + softplus(x, out=out, where=where) + ref[where] = softplus(x[where]) + assert_allclose(out, ref) + + +class TestRound: + def test_round(self): + rnd = list(map(int, (special.round(10.1), + special.round(10.4), + special.round(10.5), + special.round(10.6)))) + + # Note: According to the documentation, scipy.special.round is + # supposed to round to the nearest even number if the fractional + # part is exactly 0.5. On some platforms, this does not appear + # to work and thus this test may fail. However, this unit test is + # correctly written. + rndrl = (10,10,10,11) + assert_array_equal(rnd,rndrl) + +# sph_harm is deprecated and is implemented as a shim around sph_harm_y. +# The following two tests are maintained to verify the correctness of the shim. + +def test_sph_harm(): + # Tests derived from tables in + # https://en.wikipedia.org/wiki/Table_of_spherical_harmonics + sh = special.sph_harm + pi = np.pi + exp = np.exp + sqrt = np.sqrt + sin = np.sin + cos = np.cos + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + assert_array_almost_equal(sh(0,0,0,0), + 0.5/sqrt(pi)) + assert_array_almost_equal(sh(-2,2,0.,pi/4), + 0.25*sqrt(15./(2.*pi)) * + (sin(pi/4))**2.) + assert_array_almost_equal(sh(-2,2,0.,pi/2), + 0.25*sqrt(15./(2.*pi))) + assert_array_almost_equal(sh(2,2,pi,pi/2), + 0.25*sqrt(15/(2.*pi)) * + exp(0+2.*pi*1j)*sin(pi/2.)**2.) + assert_array_almost_equal(sh(2,4,pi/4.,pi/3.), + (3./8.)*sqrt(5./(2.*pi)) * + exp(0+2.*pi/4.*1j) * + sin(pi/3.)**2. * + (7.*cos(pi/3.)**2.-1)) + assert_array_almost_equal(sh(4,4,pi/8.,pi/6.), + (3./16.)*sqrt(35./(2.*pi)) * + exp(0+4.*pi/8.*1j)*sin(pi/6.)**4.) + + +def test_sph_harm_ufunc_loop_selection(): + # see https://github.com/scipy/scipy/issues/4895 + dt = np.dtype(np.complex128) + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + assert_equal(special.sph_harm(0, 0, 0, 0).dtype, dt) + assert_equal(special.sph_harm([0], 0, 0, 0).dtype, dt) + assert_equal(special.sph_harm(0, [0], 0, 0).dtype, dt) + assert_equal(special.sph_harm(0, 0, [0], 0).dtype, dt) + assert_equal(special.sph_harm(0, 0, 0, [0]).dtype, dt) + assert_equal(special.sph_harm([0], [0], [0], [0]).dtype, dt) + + +class TestStruve: + def _series(self, v, z, n=100): + """Compute Struve function & error estimate from its power series.""" + k = arange(0, n) + r = (-1)**k * (.5*z)**(2*k+v+1)/special.gamma(k+1.5)/special.gamma(k+v+1.5) + err = abs(r).max() * finfo(double).eps * n + return r.sum(), err + + def test_vs_series(self): + """Check Struve function versus its power series""" + for v in [-20, -10, -7.99, -3.4, -1, 0, 1, 3.4, 12.49, 16]: + for z in [1, 10, 19, 21, 30]: + value, err = self._series(v, z) + assert_allclose(special.struve(v, z), value, rtol=0, atol=err), (v, z) + + def test_some_values(self): + assert_allclose(special.struve(-7.99, 21), 0.0467547614113, rtol=1e-7) + assert_allclose(special.struve(-8.01, 21), 0.0398716951023, rtol=1e-8) + assert_allclose(special.struve(-3.0, 200), 0.0142134427432, rtol=1e-12) + assert_allclose(special.struve(-8.0, -41), 0.0192469727846, rtol=1e-11) + assert_equal(special.struve(-12, -41), -special.struve(-12, 41)) + assert_equal(special.struve(+12, -41), -special.struve(+12, 41)) + assert_equal(special.struve(-11, -41), +special.struve(-11, 41)) + assert_equal(special.struve(+11, -41), +special.struve(+11, 41)) + + assert_(isnan(special.struve(-7.1, -1))) + assert_(isnan(special.struve(-10.1, -1))) + + def test_regression_679(self): + """Regression test for #679""" + assert_allclose(special.struve(-1.0, 20 - 1e-8), + special.struve(-1.0, 20 + 1e-8)) + assert_allclose(special.struve(-2.0, 20 - 1e-8), + special.struve(-2.0, 20 + 1e-8)) + assert_allclose(special.struve(-4.3, 20 - 1e-8), + special.struve(-4.3, 20 + 1e-8)) + + +def test_chi2_smalldf(): + assert_almost_equal(special.chdtr(0.6,3), 0.957890536704110) + + +def test_ch2_inf(): + assert_equal(special.chdtr(0.7,np.inf), 1.0) + + +def test_chi2c_smalldf(): + assert_almost_equal(special.chdtrc(0.6,3), 1-0.957890536704110) + + +def test_chi2_inv_smalldf(): + assert_almost_equal(special.chdtri(0.6,1-0.957890536704110), 3) + + +def test_agm_simple(): + rtol = 1e-13 + + # Gauss's constant + assert_allclose(1/special.agm(1, np.sqrt(2)), 0.834626841674073186, + rtol=rtol) + + # These values were computed using Wolfram Alpha, with the + # function ArithmeticGeometricMean[a, b]. + agm13 = 1.863616783244897 + agm15 = 2.604008190530940 + agm35 = 3.936235503649555 + assert_allclose(special.agm([[1], [3]], [1, 3, 5]), + [[1, agm13, agm15], + [agm13, 3, agm35]], rtol=rtol) + + # Computed by the iteration formula using mpmath, + # with mpmath.mp.prec = 1000: + agm12 = 1.4567910310469068 + assert_allclose(special.agm(1, 2), agm12, rtol=rtol) + assert_allclose(special.agm(2, 1), agm12, rtol=rtol) + assert_allclose(special.agm(-1, -2), -agm12, rtol=rtol) + assert_allclose(special.agm(24, 6), 13.458171481725614, rtol=rtol) + assert_allclose(special.agm(13, 123456789.5), 11111458.498599306, + rtol=rtol) + assert_allclose(special.agm(1e30, 1), 2.229223055945383e+28, rtol=rtol) + assert_allclose(special.agm(1e-22, 1), 0.030182566420169886, rtol=rtol) + assert_allclose(special.agm(1e150, 1e180), 2.229223055945383e+178, + rtol=rtol) + assert_allclose(special.agm(1e180, 1e-150), 2.0634722510162677e+177, + rtol=rtol) + assert_allclose(special.agm(1e-150, 1e-170), 3.3112619670463756e-152, + rtol=rtol) + fi = np.finfo(1.0) + assert_allclose(special.agm(fi.tiny, fi.max), 1.9892072050015473e+305, + rtol=rtol) + assert_allclose(special.agm(0.75*fi.max, fi.max), 1.564904312298045e+308, + rtol=rtol) + assert_allclose(special.agm(fi.tiny, 3*fi.tiny), 4.1466849866735005e-308, + rtol=rtol) + + # zero, nan and inf cases. + assert_equal(special.agm(0, 0), 0) + assert_equal(special.agm(99, 0), 0) + + assert_equal(special.agm(-1, 10), np.nan) + assert_equal(special.agm(0, np.inf), np.nan) + assert_equal(special.agm(np.inf, 0), np.nan) + assert_equal(special.agm(0, -np.inf), np.nan) + assert_equal(special.agm(-np.inf, 0), np.nan) + assert_equal(special.agm(np.inf, -np.inf), np.nan) + assert_equal(special.agm(-np.inf, np.inf), np.nan) + assert_equal(special.agm(1, np.nan), np.nan) + assert_equal(special.agm(np.nan, -1), np.nan) + + assert_equal(special.agm(1, np.inf), np.inf) + assert_equal(special.agm(np.inf, 1), np.inf) + assert_equal(special.agm(-1, -np.inf), -np.inf) + assert_equal(special.agm(-np.inf, -1), -np.inf) + + +def test_legacy(): + # Legacy behavior: truncating arguments to integers + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "floating point number truncated to an integer") + assert_equal(special.expn(1, 0.3), special.expn(1.8, 0.3)) + assert_equal(special.nbdtrc(1, 2, 0.3), special.nbdtrc(1.8, 2.8, 0.3)) + assert_equal(special.nbdtr(1, 2, 0.3), special.nbdtr(1.8, 2.8, 0.3)) + assert_equal(special.nbdtri(1, 2, 0.3), special.nbdtri(1.8, 2.8, 0.3)) + assert_equal(special.pdtri(1, 0.3), special.pdtri(1.8, 0.3)) + assert_equal(special.kn(1, 0.3), special.kn(1.8, 0.3)) + assert_equal(special.yn(1, 0.3), special.yn(1.8, 0.3)) + assert_equal(special.smirnov(1, 0.3), special.smirnov(1.8, 0.3)) + assert_equal(special.smirnovi(1, 0.3), special.smirnovi(1.8, 0.3)) + + +# This lock can be removed once errstate is made thread-safe (see gh-21956) +@pytest.fixture +def errstate_lock(): + import threading + return threading.Lock() + + +@with_special_errors +def test_error_raising(errstate_lock): + with errstate_lock: + with special.errstate(all='raise'): + assert_raises(special.SpecialFunctionError, special.iv, 1, 1e99j) + + +def test_xlogy(): + def xfunc(x, y): + with np.errstate(invalid='ignore'): + if x == 0 and not np.isnan(y): + return x + else: + return x*np.log(y) + + z1 = np.asarray([(0,0), (0, np.nan), (0, np.inf), (1.0, 2.0)], dtype=float) + z2 = np.r_[z1, [(0, 1j), (1, 1j)]] + + w1 = np.vectorize(xfunc)(z1[:,0], z1[:,1]) + assert_func_equal(special.xlogy, w1, z1, rtol=1e-13, atol=1e-13) + w2 = np.vectorize(xfunc)(z2[:,0], z2[:,1]) + assert_func_equal(special.xlogy, w2, z2, rtol=1e-13, atol=1e-13) + + +def test_xlog1py(): + def xfunc(x, y): + with np.errstate(invalid='ignore'): + if x == 0 and not np.isnan(y): + return x + else: + return x * np.log1p(y) + + z1 = np.asarray([(0,0), (0, np.nan), (0, np.inf), (1.0, 2.0), + (1, 1e-30)], dtype=float) + w1 = np.vectorize(xfunc)(z1[:,0], z1[:,1]) + assert_func_equal(special.xlog1py, w1, z1, rtol=1e-13, atol=1e-13) + + +def test_entr(): + def xfunc(x): + if x < 0: + return -np.inf + else: + return -special.xlogy(x, x) + values = (0, 0.5, 1.0, np.inf) + signs = [-1, 1] + arr = [] + for sgn, v in itertools.product(signs, values): + arr.append(sgn * v) + z = np.array(arr, dtype=float) + w = np.vectorize(xfunc, otypes=[np.float64])(z) + assert_func_equal(special.entr, w, z, rtol=1e-13, atol=1e-13) + + +def test_kl_div(): + def xfunc(x, y): + if x < 0 or y < 0 or (y == 0 and x != 0): + # extension of natural domain to preserve convexity + return np.inf + elif np.isposinf(x) or np.isposinf(y): + # limits within the natural domain + return np.inf + elif x == 0: + return y + else: + return special.xlogy(x, x/y) - x + y + values = (0, 0.5, 1.0) + signs = [-1, 1] + arr = [] + for sgna, va, sgnb, vb in itertools.product(signs, values, signs, values): + arr.append((sgna*va, sgnb*vb)) + z = np.array(arr, dtype=float) + w = np.vectorize(xfunc, otypes=[np.float64])(z[:,0], z[:,1]) + assert_func_equal(special.kl_div, w, z, rtol=1e-13, atol=1e-13) + + +def test_rel_entr(): + def xfunc(x, y): + if x > 0 and y > 0: + return special.xlogy(x, x/y) + elif x == 0 and y >= 0: + return 0 + else: + return np.inf + values = (0, 0.5, 1.0) + signs = [-1, 1] + arr = [] + for sgna, va, sgnb, vb in itertools.product(signs, values, signs, values): + arr.append((sgna*va, sgnb*vb)) + z = np.array(arr, dtype=float) + w = np.vectorize(xfunc, otypes=[np.float64])(z[:,0], z[:,1]) + assert_func_equal(special.rel_entr, w, z, rtol=1e-13, atol=1e-13) + + +def test_rel_entr_gh_20710_near_zero(): + # Check accuracy of inputs which are very close + inputs = np.array([ + # x, y + (0.9456657713430001, 0.9456657713430094), + (0.48066098564791515, 0.48066098564794774), + (0.786048657854401, 0.7860486578542367), + ]) + # Known values produced using `x * mpmath.log(x / y)` with dps=30 + expected = [ + -9.325873406851269e-15, + -3.258504577274724e-14, + 1.6431300764454033e-13, + ] + x = inputs[:, 0] + y = inputs[:, 1] + assert_allclose(special.rel_entr(x, y), expected, rtol=1e-13, atol=0) + + +def test_rel_entr_gh_20710_overflow(): + special.seterr(all='ignore') + inputs = np.array([ + # x, y + # Overflow + (4, 2.22e-308), + # Underflow + (1e-200, 1e+200), + # Subnormal + (2.22e-308, 1e15), + ]) + # Known values produced using `x * mpmath.log(x / y)` with dps=30 + expected = [ + 2839.139983229607, + -9.210340371976183e-198, + -1.6493212008074475e-305, + ] + x = inputs[:, 0] + y = inputs[:, 1] + assert_allclose(special.rel_entr(x, y), expected, rtol=1e-13, atol=0) + + +def test_huber(): + assert_equal(special.huber(-1, 1.5), np.inf) + assert_allclose(special.huber(2, 1.5), 0.5 * np.square(1.5)) + assert_allclose(special.huber(2, 2.5), 2 * (2.5 - 0.5 * 2)) + + def xfunc(delta, r): + if delta < 0: + return np.inf + elif np.abs(r) < delta: + return 0.5 * np.square(r) + else: + return delta * (np.abs(r) - 0.5 * delta) + + z = np.random.randn(10, 2) + w = np.vectorize(xfunc, otypes=[np.float64])(z[:,0], z[:,1]) + assert_func_equal(special.huber, w, z, rtol=1e-13, atol=1e-13) + + +def test_pseudo_huber(): + def xfunc(delta, r): + if delta < 0: + return np.inf + elif (not delta) or (not r): + return 0 + else: + return delta**2 * (np.sqrt(1 + (r/delta)**2) - 1) + + z = np.array(np.random.randn(10, 2).tolist() + [[0, 0.5], [0.5, 0]]) + w = np.vectorize(xfunc, otypes=[np.float64])(z[:,0], z[:,1]) + assert_func_equal(special.pseudo_huber, w, z, rtol=1e-13, atol=1e-13) + + +def test_pseudo_huber_small_r(): + delta = 1.0 + r = 1e-18 + y = special.pseudo_huber(delta, r) + # expected computed with mpmath: + # import mpmath + # mpmath.mp.dps = 200 + # r = mpmath.mpf(1e-18) + # expected = float(mpmath.sqrt(1 + r**2) - 1) + expected = 5.0000000000000005e-37 + assert_allclose(y, expected, rtol=1e-13) + + +@pytest.mark.thread_unsafe +def test_runtime_warning(): + with pytest.warns(RuntimeWarning, + match=r'Too many predicted coefficients'): + mathieu_odd_coef(1000, 1000) + with pytest.warns(RuntimeWarning, + match=r'Too many predicted coefficients'): + mathieu_even_coef(1000, 1000) + + +class TestStirling2: + table = [ + [1], + [0, 1], + [0, 1, 1], + [0, 1, 3, 1], + [0, 1, 7, 6, 1], + [0, 1, 15, 25, 10, 1], + [0, 1, 31, 90, 65, 15, 1], + [0, 1, 63, 301, 350, 140, 21, 1], + [0, 1, 127, 966, 1701, 1050, 266, 28, 1], + [0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1], + [0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1], + ] + + @pytest.mark.parametrize("is_exact, comp, kwargs", [ + (True, assert_equal, {}), + (False, assert_allclose, {'rtol': 1e-12}) + ]) + def test_table_cases(self, is_exact, comp, kwargs): + for n in range(1, len(self.table)): + k_values = list(range(n+1)) + row = self.table[n] + comp(row, stirling2([n], k_values, exact=is_exact), **kwargs) + + @pytest.mark.parametrize("is_exact, comp, kwargs", [ + (True, assert_equal, {}), + (False, assert_allclose, {'rtol': 1e-12}) + ]) + def test_valid_single_integer(self, is_exact, comp, kwargs): + comp(stirling2(0, 0, exact=is_exact), self.table[0][0], **kwargs) + comp(stirling2(4, 2, exact=is_exact), self.table[4][2], **kwargs) + # a single 2-tuple of integers as arguments must return an int and not + # an array whereas arrays of single values should return array + comp(stirling2(5, 3, exact=is_exact), 25, **kwargs) + comp(stirling2([5], [3], exact=is_exact), [25], **kwargs) + + @pytest.mark.parametrize("is_exact, comp, kwargs", [ + (True, assert_equal, {}), + (False, assert_allclose, {'rtol': 1e-12}) + ]) + def test_negative_integer(self, is_exact, comp, kwargs): + # negative integers for n or k arguments return 0 + comp(stirling2(-1, -1, exact=is_exact), 0, **kwargs) + comp(stirling2(-1, 2, exact=is_exact), 0, **kwargs) + comp(stirling2(2, -1, exact=is_exact), 0, **kwargs) + + @pytest.mark.parametrize("is_exact, comp, kwargs", [ + (True, assert_equal, {}), + (False, assert_allclose, {'rtol': 1e-12}) + ]) + def test_array_inputs(self, is_exact, comp, kwargs): + ans = [self.table[10][3], self.table[10][4]] + comp(stirling2(asarray([10, 10]), + asarray([3, 4]), + exact=is_exact), + ans) + comp(stirling2([10, 10], + asarray([3, 4]), + exact=is_exact), + ans) + comp(stirling2(asarray([10, 10]), + [3, 4], + exact=is_exact), + ans) + + @pytest.mark.parametrize("is_exact, comp, kwargs", [ + (True, assert_equal, {}), + (False, assert_allclose, {'rtol': 1e-13}) + ]) + def test_mixed_values(self, is_exact, comp, kwargs): + # negative values-of either n or k-should return 0 for the entry + ans = [0, 1, 3, 25, 1050, 5880, 9330] + n = [-1, 0, 3, 5, 8, 10, 10] + k = [-2, 0, 2, 3, 5, 7, 3] + comp(stirling2(n, k, exact=is_exact), ans, **kwargs) + + def test_correct_parity(self): + """Test parity follows well known identity. + + en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind#Parity + """ + n, K = 100, np.arange(101) + assert_equal( + stirling2(n, K, exact=True) % 2, + [math.comb(n - (k // 2) - 1, n - k) % 2 for k in K], + ) + + def test_big_numbers(self): + # via mpmath (bigger than 32bit) + ans = asarray([48063331393110, 48004081105038305]) + n = [25, 30] + k = [17, 4] + assert array_equal(stirling2(n, k, exact=True), ans) + # bigger than 64 bit + ans = asarray([2801934359500572414253157841233849412, + 14245032222277144547280648984426251]) + n = [42, 43] + k = [17, 23] + assert array_equal(stirling2(n, k, exact=True), ans) + + @pytest.mark.parametrize("N", [4.5, 3., 4+1j, "12", np.nan]) + @pytest.mark.parametrize("K", [3.5, 3, "2", None]) + @pytest.mark.parametrize("is_exact", [True, False]) + def test_unsupported_input_types(self, N, K, is_exact): + # object, float, string, complex are not supported and raise TypeError + with pytest.raises(TypeError): + stirling2(N, K, exact=is_exact) + + @pytest.mark.parametrize("is_exact", [True, False]) + def test_numpy_array_int_object_dtype(self, is_exact): + # python integers with arbitrary precision are *not* allowed as + # object type in numpy arrays are inconsistent from api perspective + ans = asarray(self.table[4][1:]) + n = asarray([4, 4, 4, 4], dtype=object) + k = asarray([1, 2, 3, 4], dtype=object) + with pytest.raises(TypeError): + array_equal(stirling2(n, k, exact=is_exact), ans) + + @pytest.mark.parametrize("is_exact, comp, kwargs", [ + (True, assert_equal, {}), + (False, assert_allclose, {'rtol': 1e-13}) + ]) + def test_numpy_array_unsigned_int_dtype(self, is_exact, comp, kwargs): + # numpy unsigned integers are allowed as dtype in numpy arrays + ans = asarray(self.table[4][1:]) + n = asarray([4, 4, 4, 4], dtype=np_ulong) + k = asarray([1, 2, 3, 4], dtype=np_ulong) + comp(stirling2(n, k, exact=False), ans, **kwargs) + + @pytest.mark.parametrize("is_exact, comp, kwargs", [ + (True, assert_equal, {}), + (False, assert_allclose, {'rtol': 1e-13}) + ]) + def test_broadcasting_arrays_correctly(self, is_exact, comp, kwargs): + # broadcasting is handled by stirling2 + # test leading 1s are replicated + ans = asarray([[1, 15, 25, 10], [1, 7, 6, 1]]) # shape (2,4) + n = asarray([[5, 5, 5, 5], [4, 4, 4, 4]]) # shape (2,4) + k = asarray([1, 2, 3, 4]) # shape (4,) + comp(stirling2(n, k, exact=is_exact), ans, **kwargs) + # test that dims both mismatch broadcast correctly (5,1) & (6,) + n = asarray([[4], [4], [4], [4], [4]]) + k = asarray([0, 1, 2, 3, 4, 5]) + ans = asarray([[0, 1, 7, 6, 1, 0] for _ in range(5)]) + comp(stirling2(n, k, exact=False), ans, **kwargs) + + def test_temme_rel_max_error(self): + # python integers with arbitrary precision are *not* allowed as + # object type in numpy arrays are inconsistent from api perspective + x = list(range(51, 101, 5)) + for n in x: + k_entries = list(range(1, n+1)) + denom = stirling2([n], k_entries, exact=True) + num = denom - stirling2([n], k_entries, exact=False) + assert np.max(np.abs(num / denom)) < 2e-5 + + +class TestLegendreDeprecation: + + def test_warn_lpn(self): + msg = "`scipy.special.lpn` is deprecated..." + with pytest.deprecated_call(match=msg): + _ = lpn(1, 0) + + @pytest.mark.parametrize("xlpmn", [lpmn, clpmn]) + def test_warn_xlpmn(self, xlpmn): + message = f"`scipy.special.{xlpmn.__name__}` is deprecated..." + with pytest.deprecated_call(match=message): + _ = xlpmn(1, 1, 0) + + def test_warn_sph_harm(self): + msg = "`scipy.special.sph_harm` is deprecated..." + with pytest.deprecated_call(match=msg): + _ = special.sph_harm(1, 1, 0, 0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_bdtr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_bdtr.py new file mode 100644 index 0000000000000000000000000000000000000000..57694becc49b2028f17eac819b80a225ac010795 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_bdtr.py @@ -0,0 +1,112 @@ +import numpy as np +import scipy.special as sc +import pytest +from numpy.testing import assert_allclose, assert_array_equal, suppress_warnings + + +class TestBdtr: + def test(self): + val = sc.bdtr(0, 1, 0.5) + assert_allclose(val, 0.5) + + def test_sum_is_one(self): + val = sc.bdtr([0, 1, 2], 2, 0.5) + assert_array_equal(val, [0.25, 0.75, 1.0]) + + def test_rounding(self): + double_val = sc.bdtr([0.1, 1.1, 2.1], 2, 0.5) + int_val = sc.bdtr([0, 1, 2], 2, 0.5) + assert_array_equal(double_val, int_val) + + @pytest.mark.parametrize('k, n, p', [ + (np.inf, 2, 0.5), + (1.0, np.inf, 0.5), + (1.0, 2, np.inf) + ]) + def test_inf(self, k, n, p): + with suppress_warnings() as sup: + sup.filter(DeprecationWarning) + val = sc.bdtr(k, n, p) + assert np.isnan(val) + + def test_domain(self): + val = sc.bdtr(-1.1, 1, 0.5) + assert np.isnan(val) + + +class TestBdtrc: + def test_value(self): + val = sc.bdtrc(0, 1, 0.5) + assert_allclose(val, 0.5) + + def test_sum_is_one(self): + val = sc.bdtrc([0, 1, 2], 2, 0.5) + assert_array_equal(val, [0.75, 0.25, 0.0]) + + def test_rounding(self): + double_val = sc.bdtrc([0.1, 1.1, 2.1], 2, 0.5) + int_val = sc.bdtrc([0, 1, 2], 2, 0.5) + assert_array_equal(double_val, int_val) + + @pytest.mark.parametrize('k, n, p', [ + (np.inf, 2, 0.5), + (1.0, np.inf, 0.5), + (1.0, 2, np.inf) + ]) + def test_inf(self, k, n, p): + with suppress_warnings() as sup: + sup.filter(DeprecationWarning) + val = sc.bdtrc(k, n, p) + assert np.isnan(val) + + def test_domain(self): + val = sc.bdtrc(-1.1, 1, 0.5) + val2 = sc.bdtrc(2.1, 1, 0.5) + assert np.isnan(val2) + assert_allclose(val, 1.0) + + def test_bdtr_bdtrc_sum_to_one(self): + bdtr_vals = sc.bdtr([0, 1, 2], 2, 0.5) + bdtrc_vals = sc.bdtrc([0, 1, 2], 2, 0.5) + vals = bdtr_vals + bdtrc_vals + assert_allclose(vals, [1.0, 1.0, 1.0]) + + +class TestBdtri: + def test_value(self): + val = sc.bdtri(0, 1, 0.5) + assert_allclose(val, 0.5) + + def test_sum_is_one(self): + val = sc.bdtri([0, 1], 2, 0.5) + actual = np.asarray([1 - 1/np.sqrt(2), 1/np.sqrt(2)]) + assert_allclose(val, actual) + + def test_rounding(self): + double_val = sc.bdtri([0.1, 1.1], 2, 0.5) + int_val = sc.bdtri([0, 1], 2, 0.5) + assert_allclose(double_val, int_val) + + @pytest.mark.parametrize('k, n, p', [ + (np.inf, 2, 0.5), + (1.0, np.inf, 0.5), + (1.0, 2, np.inf) + ]) + def test_inf(self, k, n, p): + with suppress_warnings() as sup: + sup.filter(DeprecationWarning) + val = sc.bdtri(k, n, p) + assert np.isnan(val) + + @pytest.mark.parametrize('k, n, p', [ + (-1.1, 1, 0.5), + (2.1, 1, 0.5) + ]) + def test_domain(self, k, n, p): + val = sc.bdtri(k, n, p) + assert np.isnan(val) + + def test_bdtr_bdtri_roundtrip(self): + bdtr_vals = sc.bdtr([0, 1, 2], 2, 0.5) + roundtrip_vals = sc.bdtri([0, 1, 2], 2, bdtr_vals) + assert_allclose(roundtrip_vals, [0.5, 0.5, np.nan]) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_boost_ufuncs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_boost_ufuncs.py new file mode 100644 index 0000000000000000000000000000000000000000..132fb9ab11ec19d8efa00c5bb96f851795017d31 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_boost_ufuncs.py @@ -0,0 +1,61 @@ +import pytest +import numpy as np +from numpy.testing import assert_allclose +import scipy.special._ufuncs as scu +from scipy.integrate import tanhsinh + + +type_char_to_type_tol = {'f': (np.float32, 32*np.finfo(np.float32).eps), + 'd': (np.float64, 32*np.finfo(np.float64).eps)} + + +# Each item in this list is +# (func, args, expected_value) +# All the values can be represented exactly, even with np.float32. +# +# This is not an exhaustive test data set of all the functions! +# It is a spot check of several functions, primarily for +# checking that the different data types are handled correctly. +test_data = [ + (scu._beta_pdf, (0.5, 2, 3), 1.5), + (scu._beta_pdf, (0, 1, 5), 5.0), + (scu._beta_pdf, (1, 5, 1), 5.0), + (scu._beta_ppf, (0.5, 5., 5.), 0.5), # gh-21303 + (scu._binom_cdf, (1, 3, 0.5), 0.5), + (scu._binom_pmf, (1, 4, 0.5), 0.25), + (scu._hypergeom_cdf, (2, 3, 5, 6), 0.5), + (scu._nbinom_cdf, (1, 4, 0.25), 0.015625), + (scu._ncf_mean, (10, 12, 2.5), 1.5), +] + + +@pytest.mark.parametrize('func, args, expected', test_data) +def test_stats_boost_ufunc(func, args, expected): + type_sigs = func.types + type_chars = [sig.split('->')[-1] for sig in type_sigs] + for type_char in type_chars: + typ, rtol = type_char_to_type_tol[type_char] + args = [typ(arg) for arg in args] + # Harmless overflow warnings are a "feature" of some wrappers on some + # platforms. This test is about dtype and accuracy, so let's avoid false + # test failures cause by these warnings. See gh-17432. + with np.errstate(over='ignore'): + value = func(*args) + assert isinstance(value, typ) + assert_allclose(value, expected, rtol=rtol) + + +def test_landau(): + # Test that Landau distribution ufuncs are wrapped as expected; + # accuracy is tested by Boost. + x = np.linspace(-3, 10, 10) + args = (0, 1) + res = tanhsinh(lambda x: scu._landau_pdf(x, *args), -np.inf, x) + cdf = scu._landau_cdf(x, *args) + assert_allclose(res.integral, cdf) + sf = scu._landau_sf(x, *args) + assert_allclose(sf, 1-cdf) + ppf = scu._landau_ppf(cdf, *args) + assert_allclose(ppf, x) + isf = scu._landau_isf(sf, *args) + assert_allclose(isf, x, rtol=1e-6) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_boxcox.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_boxcox.py new file mode 100644 index 0000000000000000000000000000000000000000..25d76d0560a6a9fe416f750213662b5f7fb22f25 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_boxcox.py @@ -0,0 +1,125 @@ +import numpy as np +from numpy.testing import assert_equal, assert_almost_equal, assert_allclose +from scipy.special import boxcox, boxcox1p, inv_boxcox, inv_boxcox1p +import pytest + + +# There are more tests of boxcox and boxcox1p in test_mpmath.py. + +def test_boxcox_basic(): + x = np.array([0.5, 1, 2, 4]) + + # lambda = 0 => y = log(x) + y = boxcox(x, 0) + assert_almost_equal(y, np.log(x)) + + # lambda = 1 => y = x - 1 + y = boxcox(x, 1) + assert_almost_equal(y, x - 1) + + # lambda = 2 => y = 0.5*(x**2 - 1) + y = boxcox(x, 2) + assert_almost_equal(y, 0.5*(x**2 - 1)) + + # x = 0 and lambda > 0 => y = -1 / lambda + lam = np.array([0.5, 1, 2]) + y = boxcox(0, lam) + assert_almost_equal(y, -1.0 / lam) + +def test_boxcox_underflow(): + x = 1 + 1e-15 + lmbda = 1e-306 + y = boxcox(x, lmbda) + assert_allclose(y, np.log(x), rtol=1e-14) + + +def test_boxcox_nonfinite(): + # x < 0 => y = nan + x = np.array([-1, -1, -0.5]) + y = boxcox(x, [0.5, 2.0, -1.5]) + assert_equal(y, np.array([np.nan, np.nan, np.nan])) + + # x = 0 and lambda <= 0 => y = -inf + x = 0 + y = boxcox(x, [-2.5, 0]) + assert_equal(y, np.array([-np.inf, -np.inf])) + + +def test_boxcox1p_basic(): + x = np.array([-0.25, -1e-20, 0, 1e-20, 0.25, 1, 3]) + + # lambda = 0 => y = log(1+x) + y = boxcox1p(x, 0) + assert_almost_equal(y, np.log1p(x)) + + # lambda = 1 => y = x + y = boxcox1p(x, 1) + assert_almost_equal(y, x) + + # lambda = 2 => y = 0.5*((1+x)**2 - 1) = 0.5*x*(2 + x) + y = boxcox1p(x, 2) + assert_almost_equal(y, 0.5*x*(2 + x)) + + # x = -1 and lambda > 0 => y = -1 / lambda + lam = np.array([0.5, 1, 2]) + y = boxcox1p(-1, lam) + assert_almost_equal(y, -1.0 / lam) + + +def test_boxcox1p_underflow(): + x = np.array([1e-15, 1e-306]) + lmbda = np.array([1e-306, 1e-18]) + y = boxcox1p(x, lmbda) + assert_allclose(y, np.log1p(x), rtol=1e-14) + + +def test_boxcox1p_nonfinite(): + # x < -1 => y = nan + x = np.array([-2, -2, -1.5]) + y = boxcox1p(x, [0.5, 2.0, -1.5]) + assert_equal(y, np.array([np.nan, np.nan, np.nan])) + + # x = -1 and lambda <= 0 => y = -inf + x = -1 + y = boxcox1p(x, [-2.5, 0]) + assert_equal(y, np.array([-np.inf, -np.inf])) + + +def test_inv_boxcox(): + x = np.array([0., 1., 2.]) + lam = np.array([0., 1., 2.]) + y = boxcox(x, lam) + x2 = inv_boxcox(y, lam) + assert_almost_equal(x, x2) + + x = np.array([0., 1., 2.]) + lam = np.array([0., 1., 2.]) + y = boxcox1p(x, lam) + x2 = inv_boxcox1p(y, lam) + assert_almost_equal(x, x2) + + +def test_inv_boxcox1p_underflow(): + x = 1e-15 + lam = 1e-306 + y = inv_boxcox1p(x, lam) + assert_allclose(y, x, rtol=1e-14) + + +@pytest.mark.parametrize( + "x, lmb", + [[100, 155], + [0.01, -155]] +) +def test_boxcox_premature_overflow(x, lmb): + # test boxcox & inv_boxcox + y = boxcox(x, lmb) + assert np.isfinite(y) + x_inv = inv_boxcox(y, lmb) + assert_allclose(x, x_inv) + + # test boxcox1p & inv_boxcox1p + y1p = boxcox1p(x-1, lmb) + assert np.isfinite(y1p) + x1p_inv = inv_boxcox1p(y1p, lmb) + assert_allclose(x-1, x1p_inv) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cdflib.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cdflib.py new file mode 100644 index 0000000000000000000000000000000000000000..a8352182bdf657029ab9857411e1b41141fdc0a5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cdflib.py @@ -0,0 +1,688 @@ +""" +Test cdflib functions versus mpmath, if available. + +The following functions still need tests: + +- ncfdtri +- ncfdtridfn +- ncfdtridfd +- ncfdtrinc +- nbdtrik +- nbdtrin +- pdtrik +- nctdtrit +- nctdtridf +- nctdtrinc + +""" +import itertools + +import numpy as np +from numpy.testing import assert_equal, assert_allclose +import pytest + +import scipy.special as sp +from scipy.special._testutils import ( + MissingModule, check_version, FuncData) +from scipy.special._mptestutils import ( + Arg, IntArg, get_args, mpf2float, assert_mpmath_equal) + +try: + import mpmath +except ImportError: + mpmath = MissingModule('mpmath') + + +class ProbArg: + """Generate a set of probabilities on [0, 1].""" + + def __init__(self): + # Include the endpoints for compatibility with Arg et. al. + self.a = 0 + self.b = 1 + + def values(self, n): + """Return an array containing approximately n numbers.""" + m = max(1, n//3) + v1 = np.logspace(-30, np.log10(0.3), m) + v2 = np.linspace(0.3, 0.7, m + 1, endpoint=False)[1:] + v3 = 1 - np.logspace(np.log10(0.3), -15, m) + v = np.r_[v1, v2, v3] + return np.unique(v) + + +class EndpointFilter: + def __init__(self, a, b, rtol, atol): + self.a = a + self.b = b + self.rtol = rtol + self.atol = atol + + def __call__(self, x): + mask1 = np.abs(x - self.a) < self.rtol*np.abs(self.a) + self.atol + mask2 = np.abs(x - self.b) < self.rtol*np.abs(self.b) + self.atol + return np.where(mask1 | mask2, False, True) + + +class _CDFData: + def __init__(self, spfunc, mpfunc, index, argspec, spfunc_first=True, + dps=20, n=5000, rtol=None, atol=None, + endpt_rtol=None, endpt_atol=None): + self.spfunc = spfunc + self.mpfunc = mpfunc + self.index = index + self.argspec = argspec + self.spfunc_first = spfunc_first + self.dps = dps + self.n = n + self.rtol = rtol + self.atol = atol + + if not isinstance(argspec, list): + self.endpt_rtol = None + self.endpt_atol = None + elif endpt_rtol is not None or endpt_atol is not None: + if isinstance(endpt_rtol, list): + self.endpt_rtol = endpt_rtol + else: + self.endpt_rtol = [endpt_rtol]*len(self.argspec) + if isinstance(endpt_atol, list): + self.endpt_atol = endpt_atol + else: + self.endpt_atol = [endpt_atol]*len(self.argspec) + else: + self.endpt_rtol = None + self.endpt_atol = None + + def idmap(self, *args): + if self.spfunc_first: + res = self.spfunc(*args) + if np.isnan(res): + return np.nan + args = list(args) + args[self.index] = res + with mpmath.workdps(self.dps): + res = self.mpfunc(*tuple(args)) + # Imaginary parts are spurious + res = mpf2float(res.real) + else: + with mpmath.workdps(self.dps): + res = self.mpfunc(*args) + res = mpf2float(res.real) + args = list(args) + args[self.index] = res + res = self.spfunc(*tuple(args)) + return res + + def get_param_filter(self): + if self.endpt_rtol is None and self.endpt_atol is None: + return None + + filters = [] + for rtol, atol, spec in zip(self.endpt_rtol, self.endpt_atol, self.argspec): + if rtol is None and atol is None: + filters.append(None) + continue + elif rtol is None: + rtol = 0.0 + elif atol is None: + atol = 0.0 + + filters.append(EndpointFilter(spec.a, spec.b, rtol, atol)) + return filters + + def check(self): + # Generate values for the arguments + args = get_args(self.argspec, self.n) + param_filter = self.get_param_filter() + param_columns = tuple(range(args.shape[1])) + result_columns = args.shape[1] + args = np.hstack((args, args[:, self.index].reshape(args.shape[0], 1))) + FuncData(self.idmap, args, + param_columns=param_columns, result_columns=result_columns, + rtol=self.rtol, atol=self.atol, vectorized=False, + param_filter=param_filter).check() + + +def _assert_inverts(*a, **kw): + d = _CDFData(*a, **kw) + d.check() + + +def _binomial_cdf(k, n, p): + k, n, p = mpmath.mpf(k), mpmath.mpf(n), mpmath.mpf(p) + if k <= 0: + return mpmath.mpf(0) + elif k >= n: + return mpmath.mpf(1) + + onemp = mpmath.fsub(1, p, exact=True) + return mpmath.betainc(n - k, k + 1, x2=onemp, regularized=True) + + +def _f_cdf(dfn, dfd, x): + if x < 0: + return mpmath.mpf(0) + dfn, dfd, x = mpmath.mpf(dfn), mpmath.mpf(dfd), mpmath.mpf(x) + ub = dfn*x/(dfn*x + dfd) + res = mpmath.betainc(dfn/2, dfd/2, x2=ub, regularized=True) + return res + + +def _student_t_cdf(df, t, dps=None): + if dps is None: + dps = mpmath.mp.dps + with mpmath.workdps(dps): + df, t = mpmath.mpf(df), mpmath.mpf(t) + fac = mpmath.hyp2f1(0.5, 0.5*(df + 1), 1.5, -t**2/df) + fac *= t*mpmath.gamma(0.5*(df + 1)) + fac /= mpmath.sqrt(mpmath.pi*df)*mpmath.gamma(0.5*df) + return 0.5 + fac + + +def _noncentral_chi_pdf(t, df, nc): + res = mpmath.besseli(df/2 - 1, mpmath.sqrt(nc*t)) + res *= mpmath.exp(-(t + nc)/2)*(t/nc)**(df/4 - 1/2)/2 + return res + + +def _noncentral_chi_cdf(x, df, nc, dps=None): + if dps is None: + dps = mpmath.mp.dps + x, df, nc = mpmath.mpf(x), mpmath.mpf(df), mpmath.mpf(nc) + with mpmath.workdps(dps): + res = mpmath.quad(lambda t: _noncentral_chi_pdf(t, df, nc), [0, x]) + return res + + +def _tukey_lmbda_quantile(p, lmbda): + # For lmbda != 0 + return (p**lmbda - (1 - p)**lmbda)/lmbda + + +@pytest.mark.slow +@check_version(mpmath, '0.19') +class TestCDFlib: + + @pytest.mark.xfail(run=False) + def test_bdtrik(self): + _assert_inverts( + sp.bdtrik, + _binomial_cdf, + 0, [ProbArg(), IntArg(1, 1000), ProbArg()], + rtol=1e-4) + + def test_bdtrin(self): + _assert_inverts( + sp.bdtrin, + _binomial_cdf, + 1, [IntArg(1, 1000), ProbArg(), ProbArg()], + rtol=1e-4, endpt_atol=[None, None, 1e-6]) + + def test_btdtria(self): + _assert_inverts( + sp.btdtria, + lambda a, b, x: mpmath.betainc(a, b, x2=x, regularized=True), + 0, [ProbArg(), Arg(0, 1e2, inclusive_a=False), + Arg(0, 1, inclusive_a=False, inclusive_b=False)], + rtol=1e-6) + + def test_btdtrib(self): + # Use small values of a or mpmath doesn't converge + _assert_inverts( + sp.btdtrib, + lambda a, b, x: mpmath.betainc(a, b, x2=x, regularized=True), + 1, + [Arg(0, 1e2, inclusive_a=False), ProbArg(), + Arg(0, 1, inclusive_a=False, inclusive_b=False)], + rtol=1e-7, + endpt_atol=[None, 1e-18, 1e-15]) + + @pytest.mark.xfail(run=False) + def test_fdtridfd(self): + _assert_inverts( + sp.fdtridfd, + _f_cdf, + 1, + [IntArg(1, 100), ProbArg(), Arg(0, 100, inclusive_a=False)], + rtol=1e-7) + + def test_gdtria(self): + _assert_inverts( + sp.gdtria, + lambda a, b, x: mpmath.gammainc(b, b=a*x, regularized=True), + 0, + [ProbArg(), Arg(0, 1e3, inclusive_a=False), + Arg(0, 1e4, inclusive_a=False)], + rtol=1e-7, + endpt_atol=[None, 1e-7, 1e-10]) + + def test_gdtrib(self): + # Use small values of a and x or mpmath doesn't converge + _assert_inverts( + sp.gdtrib, + lambda a, b, x: mpmath.gammainc(b, b=a*x, regularized=True), + 1, + [Arg(0, 1e2, inclusive_a=False), ProbArg(), + Arg(0, 1e3, inclusive_a=False)], + rtol=1e-5) + + def test_gdtrix(self): + _assert_inverts( + sp.gdtrix, + lambda a, b, x: mpmath.gammainc(b, b=a*x, regularized=True), + 2, + [Arg(0, 1e3, inclusive_a=False), Arg(0, 1e3, inclusive_a=False), + ProbArg()], + rtol=1e-7, + endpt_atol=[None, 1e-7, 1e-10]) + + # Overall nrdtrimn and nrdtrisd are not performing well with infeasible/edge + # combinations of sigma and x, hence restricted the domains to still use the + # testing machinery, also see gh-20069 + + # nrdtrimn signature: p, sd, x + # nrdtrisd signature: mn, p, x + def test_nrdtrimn(self): + _assert_inverts( + sp.nrdtrimn, + lambda x, y, z: mpmath.ncdf(z, x, y), + 0, + [ProbArg(), # CDF value p + Arg(0.1, np.inf, inclusive_a=False, inclusive_b=False), # sigma + Arg(-1e10, 1e10)], # x + rtol=1e-5) + + def test_nrdtrisd(self): + _assert_inverts( + sp.nrdtrisd, + lambda x, y, z: mpmath.ncdf(z, x, y), + 1, + [Arg(-np.inf, 10, inclusive_a=False, inclusive_b=False), # mn + ProbArg(), # CDF value p + Arg(10, 1e100)], # x + rtol=1e-5) + + def test_stdtr(self): + # Ideally the left endpoint for Arg() should be 0. + assert_mpmath_equal( + sp.stdtr, + _student_t_cdf, + [IntArg(1, 100), Arg(1e-10, np.inf)], rtol=1e-7) + + @pytest.mark.xfail(run=False) + def test_stdtridf(self): + _assert_inverts( + sp.stdtridf, + _student_t_cdf, + 0, [ProbArg(), Arg()], rtol=1e-7) + + def test_stdtrit(self): + _assert_inverts( + sp.stdtrit, + _student_t_cdf, + 1, [IntArg(1, 100), ProbArg()], rtol=1e-7, + endpt_atol=[None, 1e-10]) + + def test_chdtriv(self): + _assert_inverts( + sp.chdtriv, + lambda v, x: mpmath.gammainc(v/2, b=x/2, regularized=True), + 0, [ProbArg(), IntArg(1, 100)], rtol=1e-4) + + @pytest.mark.xfail(run=False) + def test_chndtridf(self): + # Use a larger atol since mpmath is doing numerical integration + _assert_inverts( + sp.chndtridf, + _noncentral_chi_cdf, + 1, [Arg(0, 100, inclusive_a=False), ProbArg(), + Arg(0, 100, inclusive_a=False)], + n=1000, rtol=1e-4, atol=1e-15) + + @pytest.mark.xfail(run=False) + def test_chndtrinc(self): + # Use a larger atol since mpmath is doing numerical integration + _assert_inverts( + sp.chndtrinc, + _noncentral_chi_cdf, + 2, [Arg(0, 100, inclusive_a=False), IntArg(1, 100), ProbArg()], + n=1000, rtol=1e-4, atol=1e-15) + + def test_chndtrix(self): + # Use a larger atol since mpmath is doing numerical integration + _assert_inverts( + sp.chndtrix, + _noncentral_chi_cdf, + 0, [ProbArg(), IntArg(1, 100), Arg(0, 100, inclusive_a=False)], + n=1000, rtol=1e-4, atol=1e-15, + endpt_atol=[1e-6, None, None]) + + def test_tklmbda_zero_shape(self): + # When lmbda = 0 the CDF has a simple closed form + one = mpmath.mpf(1) + assert_mpmath_equal( + lambda x: sp.tklmbda(x, 0), + lambda x: one/(mpmath.exp(-x) + one), + [Arg()], rtol=1e-7) + + def test_tklmbda_neg_shape(self): + _assert_inverts( + sp.tklmbda, + _tukey_lmbda_quantile, + 0, [ProbArg(), Arg(-25, 0, inclusive_b=False)], + spfunc_first=False, rtol=1e-5, + endpt_atol=[1e-9, 1e-5]) + + @pytest.mark.xfail(run=False) + def test_tklmbda_pos_shape(self): + _assert_inverts( + sp.tklmbda, + _tukey_lmbda_quantile, + 0, [ProbArg(), Arg(0, 100, inclusive_a=False)], + spfunc_first=False, rtol=1e-5) + + # The values of lmdba are chosen so that 1/lmbda is exact. + @pytest.mark.parametrize('lmbda', [0.5, 1.0, 8.0]) + def test_tklmbda_lmbda1(self, lmbda): + bound = 1/lmbda + assert_equal(sp.tklmbda([-bound, bound], lmbda), [0.0, 1.0]) + + +funcs = [ + ("btdtria", 3), + ("btdtrib", 3), + ("bdtrik", 3), + ("bdtrin", 3), + ("chdtriv", 2), + ("chndtr", 3), + ("chndtrix", 3), + ("chndtridf", 3), + ("chndtrinc", 3), + ("fdtridfd", 3), + ("ncfdtr", 4), + ("ncfdtri", 4), + ("ncfdtridfn", 4), + ("ncfdtridfd", 4), + ("ncfdtrinc", 4), + ("gdtrix", 3), + ("gdtrib", 3), + ("gdtria", 3), + ("nbdtrik", 3), + ("nbdtrin", 3), + ("nrdtrimn", 3), + ("nrdtrisd", 3), + ("pdtrik", 2), + ("stdtr", 2), + ("stdtrit", 2), + ("stdtridf", 2), + ("nctdtr", 3), + ("nctdtrit", 3), + ("nctdtridf", 3), + ("nctdtrinc", 3), + ("tklmbda", 2), +] + + +@pytest.mark.parametrize('func,numargs', funcs, ids=[x[0] for x in funcs]) +def test_nonfinite(func, numargs): + + rng = np.random.default_rng(1701299355559735) + func = getattr(sp, func) + args_choices = [(float(x), np.nan, np.inf, -np.inf) for x in rng.random(numargs)] + + for args in itertools.product(*args_choices): + res = func(*args) + + if any(np.isnan(x) for x in args): + # Nan inputs should result to nan output + assert_equal(res, np.nan) + else: + # All other inputs should return something (but not + # raise exceptions or cause hangs) + pass + + +def test_chndtrix_gh2158(): + # test that gh-2158 is resolved; previously this blew up + res = sp.chndtrix(0.999999, 2, np.arange(20.)+1e-6) + + # Generated in R + # options(digits=16) + # ncp <- seq(0, 19) + 1e-6 + # print(qchisq(0.999999, df = 2, ncp = ncp)) + res_exp = [27.63103493142305, 35.25728589950540, 39.97396073236288, + 43.88033702110538, 47.35206403482798, 50.54112500166103, + 53.52720257322766, 56.35830042867810, 59.06600769498512, + 61.67243118946381, 64.19376191277179, 66.64228141346548, + 69.02756927200180, 71.35726934749408, 73.63759723904816, + 75.87368842650227, 78.06984431185720, 80.22971052389806, + 82.35640899964173, 84.45263768373256] + assert_allclose(res, res_exp) + + +def test_nctdtrinc_gh19896(): + # test that gh-19896 is resolved. + # Compared to SciPy 1.11 results from Fortran code. + dfarr = [0.001, 0.98, 9.8, 98, 980, 10000, 98, 9.8, 0.98, 0.001] + parr = [0.001, 0.1, 0.3, 0.8, 0.999, 0.001, 0.1, 0.3, 0.8, 0.999] + tarr = [0.0015, 0.15, 1.5, 15, 300, 0.0015, 0.15, 1.5, 15, 300] + desired = [3.090232306168629, 1.406141304556198, 2.014225177124157, + 13.727067118283456, 278.9765683871208, 3.090232306168629, + 1.4312427877936222, 2.014225177124157, 3.712743137978295, + -3.086951096691082] + actual = sp.nctdtrinc(dfarr, parr, tarr) + assert_allclose(actual, desired, rtol=5e-12, atol=0.0) + + +def test_stdtr_stdtrit_neg_inf(): + # -inf was treated as +inf and values from the normal were returned + assert np.all(np.isnan(sp.stdtr(-np.inf, [-np.inf, -1.0, 0.0, 1.0, np.inf]))) + assert np.all(np.isnan(sp.stdtrit(-np.inf, [0.0, 0.25, 0.5, 0.75, 1.0]))) + + +def test_bdtrik_nbdtrik_inf(): + y = np.array( + [np.nan,-np.inf,-10.0, -1.0, 0.0, .00001, .5, 0.9999, 1.0, 10.0, np.inf]) + y = y[:,None] + p = np.atleast_2d( + [np.nan, -np.inf, -10.0, -1.0, 0.0, .00001, .5, 1.0, np.inf]) + assert np.all(np.isnan(sp.bdtrik(y, np.inf, p))) + assert np.all(np.isnan(sp.nbdtrik(y, np.inf, p))) + + +@pytest.mark.parametrize( + "dfn,dfd,nc,f,expected", + [[100.0, 0.1, 0.1, 100.0, 0.29787396410092676], + [100.0, 100.0, 0.01, 0.1, 4.4344737598690424e-26], + [100.0, 0.01, 0.1, 0.01, 0.002848616633080384], + [10.0, 0.01, 1.0, 0.1, 0.012339557729057956], + [100.0, 100.0, 0.01, 0.01, 1.8926477420964936e-72], + [1.0, 100.0, 100.0, 0.1, 1.7925940526821304e-22], + [1.0, 0.01, 100.0, 10.0, 0.012334711965024968], + [1.0, 0.01, 10.0, 0.01, 0.00021944525290299], + [10.0, 1.0, 0.1, 100.0, 0.9219345555070705], + [0.1, 0.1, 1.0, 1.0, 0.3136335813423239], + [100.0, 100.0, 0.1, 10.0, 1.0], + [1.0, 0.1, 100.0, 10.0, 0.02926064279680897]] +) +def test_ncfdtr(dfn, dfd, nc, f, expected): + # Reference values computed with mpmath with the following script + # + # import numpy as np + # + # from mpmath import mp + # from scipy.special import ncfdtr + # + # mp.dps = 100 + # + # def mp_ncfdtr(dfn, dfd, nc, f): + # # Uses formula 26.2.20 from Abramowitz and Stegun. + # dfn, dfd, nc, f = map(mp.mpf, (dfn, dfd, nc, f)) + # def term(j): + # result = mp.exp(-nc/2)*(nc/2)**j / mp.factorial(j) + # result *= mp.betainc( + # dfn/2 + j, dfd/2, 0, f*dfn/(f*dfn + dfd), regularized=True + # ) + # return result + # result = mp.nsum(term, [0, mp.inf]) + # return float(result) + # + # dfn = np.logspace(-2, 2, 5) + # dfd = np.logspace(-2, 2, 5) + # nc = np.logspace(-2, 2, 5) + # f = np.logspace(-2, 2, 5) + # + # dfn, dfd, nc, f = np.meshgrid(dfn, dfd, nc, f) + # dfn, dfd, nc, f = map(np.ravel, (dfn, dfd, nc, f)) + # + # cases = [] + # re = [] + # for x0, x1, x2, x3 in zip(*(dfn, dfd, nc, f)): + # observed = ncfdtr(x0, x1, x2, x3) + # expected = mp_ncfdtr(x0, x1, x2, x3) + # cases.append((x0, x1, x2, x3, expected)) + # re.append((abs(expected - observed)/abs(expected))) + # + # assert np.max(re) < 1e-13 + # + # rng = np.random.default_rng(1234) + # sample_idx = rng.choice(len(re), replace=False, size=12) + # cases = np.array(cases)[sample_idx].tolist() + assert_allclose(sp.ncfdtr(dfn, dfd, nc, f), expected, rtol=1e-13, atol=0) + + +class TestNctdtr: + + # Reference values computed with mpmath with the following script + # Formula from: + # Lenth, Russell V (1989). "Algorithm AS 243: Cumulative Distribution Function + # of the Non-central t Distribution". Journal of the Royal Statistical Society, + # Series C. 38 (1): 185-189 + # + # Warning: may take a long time to run + # + # from mpmath import mp + # mp.dps = 400 + + # def nct_cdf(df, nc, x): + # df, nc, x = map(mp.mpf, (df, nc, x)) + + # def f(df, nc, x): + # phi = mp.ncdf(-nc) + # y = x * x / (x * x + df) + # constant = mp.exp(-nc * nc / 2.) + # def term(j): + # intermediate = constant * (nc *nc / 2.)**j + # p = intermediate/mp.factorial(j) + # q = nc / (mp.sqrt(2.) * mp.gamma(j + 1.5)) * intermediate + # first_beta_term = mp.betainc(j + 0.5, df/2., x2=y, + # regularized=True) + # second_beta_term = mp.betainc(j + mp.one, df/2., x2=y, + # regularized=True) + # return p * first_beta_term + q * second_beta_term + + # sum_term = mp.nsum(term, [0, mp.inf]) + # f = phi + 0.5 * sum_term + # return f + + # if x >= 0: + # result = f(df, nc, x) + # else: + # result = mp.one - f(df, -nc, x) + # return float(result) + + @pytest.mark.parametrize("df, nc, x, expected", [ + (0.98, -3.8, 0.0015, 0.9999279987514815), + (0.98, -3.8, 0.15, 0.9999528361700505), + (0.98, -3.8, 1.5, 0.9999908823016942), + (0.98, -3.8, 15, 0.9999990264591945), + (0.98, 0.38, 0.0015, 0.35241533122693), + (0.98, 0.38, 0.15, 0.39749697267146983), + (0.98, 0.38, 1.5, 0.716862963488558), + (0.98, 0.38, 15, 0.9656246449257494), + (0.98, 3.8, 0.0015, 7.26973354942293e-05), + (0.98, 3.8, 0.15, 0.00012416481147589105), + (0.98, 3.8, 1.5, 0.035388035775454095), + (0.98, 3.8, 15, 0.7954826975430583), + (0.98, 38, 0.0015, 3.02106943e-316), + (0.98, 38, 0.15, 6.069970616996603e-309), + (0.98, 38, 1.5, 2.591995360483094e-97), + (0.98, 38, 15, 0.011927265886910935), + (9.8, -3.8, 0.0015, 0.9999280776192786), + (9.8, -3.8, 0.15, 0.9999599410685442), + (9.8, -3.8, 1.5, 0.9999997432394788), + (9.8, -3.8, 15, 0.9999999999999984), + (9.8, 0.38, 0.0015, 0.3525155979107491), + (9.8, 0.38, 0.15, 0.40763120140379194), + (9.8, 0.38, 1.5, 0.8476794017024651), + (9.8, 0.38, 15, 0.9999999297116268), + (9.8, 3.8, 0.0015, 7.277620328149153e-05), + (9.8, 3.8, 0.15, 0.00013024802220900652), + (9.8, 3.8, 1.5, 0.013477432800072933), + (9.8, 3.8, 15, 0.999850151230648), + (9.8, 38, 0.0015, 3.05066095e-316), + (9.8, 38, 0.15, 1.79065514676e-313), + (9.8, 38, 1.5, 2.0935940165900746e-249), + (9.8, 38, 15, 2.252076291604796e-09), + (98, -3.8, 0.0015, 0.9999280875149109), + (98, -3.8, 0.15, 0.9999608250170452), + (98, -3.8, 1.5, 0.9999999304757682), + (98, -3.8, 15, 1.0), + (98, 0.38, 0.0015, 0.35252817848596313), + (98, 0.38, 0.15, 0.40890253001794846), + (98, 0.38, 1.5, 0.8664672830006552), + (98, 0.38, 15, 1.0), + (98, 3.8, 0.0015, 7.278609891281275e-05), + (98, 3.8, 0.15, 0.0001310318674827004), + (98, 3.8, 1.5, 0.010990879189991727), + (98, 3.8, 15, 0.9999999999999989), + (98, 38, 0.0015, 3.05437385e-316), + (98, 38, 0.15, 9.1668336166e-314), + (98, 38, 1.5, 1.8085884236563926e-288), + (98, 38, 15, 2.7740532792035907e-50), + (980, -3.8, 0.0015, 0.9999280885188965), + (980, -3.8, 0.15, 0.9999609144559273), + (980, -3.8, 1.5, 0.9999999410050979), + (980, -3.8, 15, 1.0), + (980, 0.38, 0.0015, 0.3525294548792812), + (980, 0.38, 0.15, 0.4090315324657382), + (980, 0.38, 1.5, 0.8684247068517293), + (980, 0.38, 15, 1.0), + (980, 3.8, 0.0015, 7.278710289828983e-05), + (980, 3.8, 0.15, 0.00013111131667906573), + (980, 3.8, 1.5, 0.010750678886113882), + (980, 3.8, 15, 1.0), + (980, 38, 0.0015, 3.0547506e-316), + (980, 38, 0.15, 8.6191646313e-314), + pytest.param(980, 38, 1.5, 1.1824454111413493e-291, + marks=pytest.mark.xfail( + reason="Bug in underlying Boost math implementation")), + (980, 38, 15, 5.407535300713606e-105) + ]) + def test_gh19896(self, df, nc, x, expected): + # test that gh-19896 is resolved. + # Originally this was a regression test that used the old Fortran results + # as a reference. The Fortran results were not accurate, so the reference + # values were recomputed with mpmath. + result = sp.nctdtr(df, nc, x) + assert_allclose(result, expected, rtol=1e-13, atol=1e-303) + + def test_nctdtr_gh8344(self): + # test that gh-8344 is resolved. + df, nc, x = 3000, 3, 0.1 + expected = 0.0018657780826323328 + assert_allclose(sp.nctdtr(df, nc, x), expected, rtol=1e-14) + + @pytest.mark.parametrize( + "df, nc, x, expected, rtol", + [[3., 5., -2., 1.5645373999149622e-09, 5e-9], + [1000., 10., 1., 1.1493552133826623e-19, 1e-13], + [1e-5, -6., 2., 0.9999999990135003, 1e-13], + [10., 20., 0.15, 6.426530505957303e-88, 1e-13], + [1., 1., np.inf, 1.0, 0.0], + [1., 1., -np.inf, 0.0, 0.0] + ] + ) + def test_accuracy(self, df, nc, x, expected, rtol): + assert_allclose(sp.nctdtr(df, nc, x), expected, rtol=rtol) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cdft_asymptotic.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cdft_asymptotic.py new file mode 100644 index 0000000000000000000000000000000000000000..8b1ad41243f0865c205963d938ab61a346ee8e88 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cdft_asymptotic.py @@ -0,0 +1,49 @@ +# gh-14777 regression tests +# Test stdtr and stdtrit with infinite df and large values of df + +import numpy as np +from numpy.testing import assert_allclose, assert_equal +from scipy.special import stdtr, stdtrit, ndtr, ndtri + + +def test_stdtr_vs_R_large_df(): + df = [1e10, 1e12, 1e120, np.inf] + t = 1. + res = stdtr(df, t) + # R Code: + # options(digits=20) + # pt(1., c(1e10, 1e12, 1e120, Inf)) + res_R = [0.84134474605644460343, + 0.84134474606842180044, + 0.84134474606854281475, + 0.84134474606854292578] + assert_allclose(res, res_R, rtol=2e-15) + # last value should also agree with ndtr + assert_equal(res[3], ndtr(1.)) + + +def test_stdtrit_vs_R_large_df(): + df = [1e10, 1e12, 1e120, np.inf] + p = 0.1 + res = stdtrit(df, p) + # R Code: + # options(digits=20) + # qt(0.1, c(1e10, 1e12, 1e120, Inf)) + res_R = [-1.2815515656292593150, + -1.2815515655454472466, + -1.2815515655446008125, + -1.2815515655446008125] + assert_allclose(res, res_R, rtol=1e-14, atol=1e-15) + # last value should also agree with ndtri + assert_equal(res[3], ndtri(0.1)) + + +def test_stdtr_stdtri_invalid(): + # a mix of large and inf df with t/p equal to nan + df = [1e10, 1e12, 1e120, np.inf] + x = np.nan + res1 = stdtr(df, x) + res2 = stdtrit(df, x) + res_ex = 4*[np.nan] + assert_equal(res1, res_ex) + assert_equal(res2, res_ex) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cephes_intp_cast.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cephes_intp_cast.py new file mode 100644 index 0000000000000000000000000000000000000000..05f3d1ae5c101ff50c75d1065e5e234063d192e4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cephes_intp_cast.py @@ -0,0 +1,29 @@ +import pytest +import numpy as np +from scipy.special._ufuncs import ( + _smirnovc, _smirnovci, _smirnovp, + _struve_asymp_large_z, _struve_bessel_series, _struve_power_series, + bdtr, bdtrc, bdtri, expn, kn, nbdtr, nbdtrc, nbdtri, pdtri, + smirnov, smirnovi, yn +) + + +# +# For each ufunc here, verify that the default integer type, np.intp, +# can be safely cast to the integer type found in the input type signatures. +# For this particular set of functions, the code expects to find just one +# integer type among the input signatures. +# +@pytest.mark.parametrize( + 'ufunc', + [_smirnovc, _smirnovci, _smirnovp, + _struve_asymp_large_z, _struve_bessel_series, _struve_power_series, + bdtr, bdtrc, bdtri, expn, kn, nbdtr, nbdtrc, nbdtri, pdtri, + smirnov, smirnovi, yn], +) +def test_intp_safe_cast(ufunc): + int_chars = {'i', 'l', 'q'} + int_input = [set(sig.split('->')[0]) & int_chars for sig in ufunc.types] + int_char = ''.join(s.pop() if s else '' for s in int_input) + assert len(int_char) == 1, "More integer types in the signatures than expected" + assert np.can_cast(np.intp, np.dtype(int_char)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cosine_distr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cosine_distr.py new file mode 100644 index 0000000000000000000000000000000000000000..27e3ca2699d0d1d0b58665f125d99c166095696d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cosine_distr.py @@ -0,0 +1,83 @@ +import numpy as np +from numpy.testing import assert_allclose +import pytest +from scipy.special._ufuncs import _cosine_cdf, _cosine_invcdf + + +# These values are (x, p) where p is the expected exact value of +# _cosine_cdf(x). These values will be tested for exact agreement. +_coscdf_exact = [ + (-4.0, 0.0), + (0, 0.5), + (np.pi, 1.0), + (4.0, 1.0), +] + +@pytest.mark.parametrize("x, expected", _coscdf_exact) +def test_cosine_cdf_exact(x, expected): + assert _cosine_cdf(x) == expected + + +# These values are (x, p), where p is the expected value of +# _cosine_cdf(x). The expected values were computed with mpmath using +# 50 digits of precision. These values will be tested for agreement +# with the computed values using a very small relative tolerance. +# The value at -np.pi is not 0, because -np.pi does not equal -π. +_coscdf_close = [ + (3.1409, 0.999999999991185), + (2.25, 0.9819328173287907), + # -1.6 is the threshold below which the Pade approximant is used. + (-1.599, 0.08641959838382553), + (-1.601, 0.086110582992713), + (-2.0, 0.0369709335961611), + (-3.0, 7.522387241801384e-05), + (-3.1415, 2.109869685443648e-14), + (-3.14159, 4.956444476505336e-19), + (-np.pi, 4.871934450264861e-50), +] + +@pytest.mark.parametrize("x, expected", _coscdf_close) +def test_cosine_cdf(x, expected): + assert_allclose(_cosine_cdf(x), expected, rtol=5e-15) + + +# These values are (p, x) where x is the expected exact value of +# _cosine_invcdf(p). These values will be tested for exact agreement. +_cosinvcdf_exact = [ + (0.0, -np.pi), + (0.5, 0.0), + (1.0, np.pi), +] + +@pytest.mark.parametrize("p, expected", _cosinvcdf_exact) +def test_cosine_invcdf_exact(p, expected): + assert _cosine_invcdf(p) == expected + + +def test_cosine_invcdf_invalid_p(): + # Check that p values outside of [0, 1] return nan. + assert np.isnan(_cosine_invcdf([-0.1, 1.1])).all() + + +# These values are (p, x), where x is the expected value of _cosine_invcdf(p). +# The expected values were computed with mpmath using 50 digits of precision. +_cosinvcdf_close = [ + (1e-50, -np.pi), + (1e-14, -3.1415204137058454), + (1e-08, -3.1343686589124524), + (0.0018001, -2.732563923138336), + (0.010, -2.41276589008678), + (0.060, -1.7881244975330157), + (0.125, -1.3752523669869274), + (0.250, -0.831711193579736), + (0.400, -0.3167954512395289), + (0.419, -0.25586025626919906), + (0.421, -0.24947570750445663), + (0.750, 0.831711193579736), + (0.940, 1.7881244975330153), + (0.9999999996, 3.1391220839917167), +] + +@pytest.mark.parametrize("p, expected", _cosinvcdf_close) +def test_cosine_invcdf(p, expected): + assert_allclose(_cosine_invcdf(p), expected, rtol=1e-14) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cython_special.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cython_special.py new file mode 100644 index 0000000000000000000000000000000000000000..5dc9ed50cec9503edb77064cfa209c9d81573214 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_cython_special.py @@ -0,0 +1,363 @@ +from collections.abc import Callable + +import pytest +from itertools import product +from numpy.testing import assert_allclose, suppress_warnings +from scipy import special +from scipy.special import cython_special + + +bint_points = [True, False] +int_points = [-10, -1, 1, 10] +real_points = [-10.0, -1.0, 1.0, 10.0] +complex_points = [complex(*tup) for tup in product(real_points, repeat=2)] + + +CYTHON_SIGNATURE_MAP = { + 'b': 'bint', + 'f': 'float', + 'd': 'double', + 'g': 'long double', + 'F': 'float complex', + 'D': 'double complex', + 'G': 'long double complex', + 'i': 'int', + 'l': 'long' +} + + +TEST_POINTS = { + 'b': bint_points, + 'f': real_points, + 'd': real_points, + 'g': real_points, + 'F': complex_points, + 'D': complex_points, + 'G': complex_points, + 'i': int_points, + 'l': int_points, +} + + +PARAMS: list[tuple[Callable, Callable, tuple[str, ...], str | None]] = [ + (special.agm, cython_special.agm, ('dd',), None), + (special.airy, cython_special._airy_pywrap, ('d', 'D'), None), + (special.airye, cython_special._airye_pywrap, ('d', 'D'), None), + (special.bdtr, cython_special.bdtr, ('dld', 'ddd'), None), + (special.bdtrc, cython_special.bdtrc, ('dld', 'ddd'), None), + (special.bdtri, cython_special.bdtri, ('dld', 'ddd'), None), + (special.bdtrik, cython_special.bdtrik, ('ddd',), None), + (special.bdtrin, cython_special.bdtrin, ('ddd',), None), + (special.bei, cython_special.bei, ('d',), None), + (special.beip, cython_special.beip, ('d',), None), + (special.ber, cython_special.ber, ('d',), None), + (special.berp, cython_special.berp, ('d',), None), + (special.besselpoly, cython_special.besselpoly, ('ddd',), None), + (special.beta, cython_special.beta, ('dd',), None), + (special.betainc, cython_special.betainc, ('ddd',), None), + (special.betaincc, cython_special.betaincc, ('ddd',), None), + (special.betaincinv, cython_special.betaincinv, ('ddd',), None), + (special.betainccinv, cython_special.betainccinv, ('ddd',), None), + (special.betaln, cython_special.betaln, ('dd',), None), + (special.binom, cython_special.binom, ('dd',), None), + (special.boxcox, cython_special.boxcox, ('dd',), None), + (special.boxcox1p, cython_special.boxcox1p, ('dd',), None), + (special.btdtria, cython_special.btdtria, ('ddd',), None), + (special.btdtrib, cython_special.btdtrib, ('ddd',), None), + (special.cbrt, cython_special.cbrt, ('d',), None), + (special.chdtr, cython_special.chdtr, ('dd',), None), + (special.chdtrc, cython_special.chdtrc, ('dd',), None), + (special.chdtri, cython_special.chdtri, ('dd',), None), + (special.chdtriv, cython_special.chdtriv, ('dd',), None), + (special.chndtr, cython_special.chndtr, ('ddd',), None), + (special.chndtridf, cython_special.chndtridf, ('ddd',), None), + (special.chndtrinc, cython_special.chndtrinc, ('ddd',), None), + (special.chndtrix, cython_special.chndtrix, ('ddd',), None), + (special.cosdg, cython_special.cosdg, ('d',), None), + (special.cosm1, cython_special.cosm1, ('d',), None), + (special.cotdg, cython_special.cotdg, ('d',), None), + (special.dawsn, cython_special.dawsn, ('d', 'D'), None), + (special.ellipe, cython_special.ellipe, ('d',), None), + (special.ellipeinc, cython_special.ellipeinc, ('dd',), None), + (special.ellipj, cython_special._ellipj_pywrap, ('dd',), None), + (special.ellipkinc, cython_special.ellipkinc, ('dd',), None), + (special.ellipkm1, cython_special.ellipkm1, ('d',), None), + (special.ellipk, cython_special.ellipk, ('d',), None), + (special.elliprc, cython_special.elliprc, ('dd', 'DD'), None), + (special.elliprd, cython_special.elliprd, ('ddd', 'DDD'), None), + (special.elliprf, cython_special.elliprf, ('ddd', 'DDD'), None), + (special.elliprg, cython_special.elliprg, ('ddd', 'DDD'), None), + (special.elliprj, cython_special.elliprj, ('dddd', 'DDDD'), None), + (special.entr, cython_special.entr, ('d',), None), + (special.erf, cython_special.erf, ('d', 'D'), None), + (special.erfc, cython_special.erfc, ('d', 'D'), None), + (special.erfcx, cython_special.erfcx, ('d', 'D'), None), + (special.erfi, cython_special.erfi, ('d', 'D'), None), + (special.erfinv, cython_special.erfinv, ('d',), None), + (special.erfcinv, cython_special.erfcinv, ('d',), None), + (special.eval_chebyc, cython_special.eval_chebyc, ('dd', 'dD', 'ld'), None), + (special.eval_chebys, cython_special.eval_chebys, ('dd', 'dD', 'ld'), + 'd and l differ for negative int'), + (special.eval_chebyt, cython_special.eval_chebyt, ('dd', 'dD', 'ld'), + 'd and l differ for negative int'), + (special.eval_chebyu, cython_special.eval_chebyu, ('dd', 'dD', 'ld'), + 'd and l differ for negative int'), + (special.eval_gegenbauer, cython_special.eval_gegenbauer, ('ddd', 'ddD', 'ldd'), + 'd and l differ for negative int'), + (special.eval_genlaguerre, cython_special.eval_genlaguerre, ('ddd', 'ddD', 'ldd'), + 'd and l differ for negative int'), + (special.eval_hermite, cython_special.eval_hermite, ('ld',), None), + (special.eval_hermitenorm, cython_special.eval_hermitenorm, ('ld',), None), + (special.eval_jacobi, cython_special.eval_jacobi, ('dddd', 'dddD', 'lddd'), + 'd and l differ for negative int'), + (special.eval_laguerre, cython_special.eval_laguerre, ('dd', 'dD', 'ld'), + 'd and l differ for negative int'), + (special.eval_legendre, cython_special.eval_legendre, ('dd', 'dD', 'ld'), None), + (special.eval_sh_chebyt, cython_special.eval_sh_chebyt, ('dd', 'dD', 'ld'), None), + (special.eval_sh_chebyu, cython_special.eval_sh_chebyu, ('dd', 'dD', 'ld'), + 'd and l differ for negative int'), + (special.eval_sh_jacobi, cython_special.eval_sh_jacobi, ('dddd', 'dddD', 'lddd'), + 'd and l differ for negative int'), + (special.eval_sh_legendre, cython_special.eval_sh_legendre, ('dd', 'dD', 'ld'), + None), + (special.exp1, cython_special.exp1, ('d', 'D'), None), + (special.exp10, cython_special.exp10, ('d',), None), + (special.exp2, cython_special.exp2, ('d',), None), + (special.expi, cython_special.expi, ('d', 'D'), None), + (special.expit, cython_special.expit, ('f', 'd', 'g'), None), + (special.expm1, cython_special.expm1, ('d', 'D'), None), + (special.expn, cython_special.expn, ('ld', 'dd'), None), + (special.exprel, cython_special.exprel, ('d',), None), + (special.fdtr, cython_special.fdtr, ('ddd',), None), + (special.fdtrc, cython_special.fdtrc, ('ddd',), None), + (special.fdtri, cython_special.fdtri, ('ddd',), None), + (special.fdtridfd, cython_special.fdtridfd, ('ddd',), None), + (special.fresnel, cython_special._fresnel_pywrap, ('d', 'D'), None), + (special.gamma, cython_special.gamma, ('d', 'D'), None), + (special.gammainc, cython_special.gammainc, ('dd',), None), + (special.gammaincc, cython_special.gammaincc, ('dd',), None), + (special.gammainccinv, cython_special.gammainccinv, ('dd',), None), + (special.gammaincinv, cython_special.gammaincinv, ('dd',), None), + (special.gammaln, cython_special.gammaln, ('d',), None), + (special.gammasgn, cython_special.gammasgn, ('d',), None), + (special.gdtr, cython_special.gdtr, ('ddd',), None), + (special.gdtrc, cython_special.gdtrc, ('ddd',), None), + (special.gdtria, cython_special.gdtria, ('ddd',), None), + (special.gdtrib, cython_special.gdtrib, ('ddd',), None), + (special.gdtrix, cython_special.gdtrix, ('ddd',), None), + (special.hankel1, cython_special.hankel1, ('dD',), None), + (special.hankel1e, cython_special.hankel1e, ('dD',), None), + (special.hankel2, cython_special.hankel2, ('dD',), None), + (special.hankel2e, cython_special.hankel2e, ('dD',), None), + (special.huber, cython_special.huber, ('dd',), None), + (special.hyp0f1, cython_special.hyp0f1, ('dd', 'dD'), None), + (special.hyp1f1, cython_special.hyp1f1, ('ddd', 'ddD'), None), + (special.hyp2f1, cython_special.hyp2f1, ('dddd', 'dddD'), None), + (special.hyperu, cython_special.hyperu, ('ddd',), None), + (special.i0, cython_special.i0, ('d',), None), + (special.i0e, cython_special.i0e, ('d',), None), + (special.i1, cython_special.i1, ('d',), None), + (special.i1e, cython_special.i1e, ('d',), None), + (special.inv_boxcox, cython_special.inv_boxcox, ('dd',), None), + (special.inv_boxcox1p, cython_special.inv_boxcox1p, ('dd',), None), + (special.it2i0k0, cython_special._it2i0k0_pywrap, ('d',), None), + (special.it2j0y0, cython_special._it2j0y0_pywrap, ('d',), None), + (special.it2struve0, cython_special.it2struve0, ('d',), None), + (special.itairy, cython_special._itairy_pywrap, ('d',), None), + (special.iti0k0, cython_special._iti0k0_pywrap, ('d',), None), + (special.itj0y0, cython_special._itj0y0_pywrap, ('d',), None), + (special.itmodstruve0, cython_special.itmodstruve0, ('d',), None), + (special.itstruve0, cython_special.itstruve0, ('d',), None), + (special.iv, cython_special.iv, ('dd', 'dD'), None), + (special.ive, cython_special.ive, ('dd', 'dD'), None), + (special.j0, cython_special.j0, ('d',), None), + (special.j1, cython_special.j1, ('d',), None), + (special.jv, cython_special.jv, ('dd', 'dD'), None), + (special.jve, cython_special.jve, ('dd', 'dD'), None), + (special.k0, cython_special.k0, ('d',), None), + (special.k0e, cython_special.k0e, ('d',), None), + (special.k1, cython_special.k1, ('d',), None), + (special.k1e, cython_special.k1e, ('d',), None), + (special.kei, cython_special.kei, ('d',), None), + (special.keip, cython_special.keip, ('d',), None), + (special.kelvin, cython_special._kelvin_pywrap, ('d',), None), + (special.ker, cython_special.ker, ('d',), None), + (special.kerp, cython_special.kerp, ('d',), None), + (special.kl_div, cython_special.kl_div, ('dd',), None), + (special.kn, cython_special.kn, ('ld', 'dd'), None), + (special.kolmogi, cython_special.kolmogi, ('d',), None), + (special.kolmogorov, cython_special.kolmogorov, ('d',), None), + (special.kv, cython_special.kv, ('dd', 'dD'), None), + (special.kve, cython_special.kve, ('dd', 'dD'), None), + (special.log1p, cython_special.log1p, ('d', 'D'), None), + (special.log_expit, cython_special.log_expit, ('f', 'd', 'g'), None), + (special.log_ndtr, cython_special.log_ndtr, ('d', 'D'), None), + (special.log_wright_bessel, cython_special.log_wright_bessel, ('ddd',), None), + (special.ndtri_exp, cython_special.ndtri_exp, ('d',), None), + (special.loggamma, cython_special.loggamma, ('D',), None), + (special.logit, cython_special.logit, ('f', 'd', 'g'), None), + (special.lpmv, cython_special.lpmv, ('ddd',), None), + (special.mathieu_a, cython_special.mathieu_a, ('dd',), None), + (special.mathieu_b, cython_special.mathieu_b, ('dd',), None), + (special.mathieu_cem, cython_special._mathieu_cem_pywrap, ('ddd',), None), + (special.mathieu_modcem1, cython_special._mathieu_modcem1_pywrap, ('ddd',), None), + (special.mathieu_modcem2, cython_special._mathieu_modcem2_pywrap, ('ddd',), None), + (special.mathieu_modsem1, cython_special._mathieu_modsem1_pywrap, ('ddd',), None), + (special.mathieu_modsem2, cython_special._mathieu_modsem2_pywrap, ('ddd',), None), + (special.mathieu_sem, cython_special._mathieu_sem_pywrap, ('ddd',), None), + (special.modfresnelm, cython_special._modfresnelm_pywrap, ('d',), None), + (special.modfresnelp, cython_special._modfresnelp_pywrap, ('d',), None), + (special.modstruve, cython_special.modstruve, ('dd',), None), + (special.nbdtr, cython_special.nbdtr, ('lld', 'ddd'), None), + (special.nbdtrc, cython_special.nbdtrc, ('lld', 'ddd'), None), + (special.nbdtri, cython_special.nbdtri, ('lld', 'ddd'), None), + (special.nbdtrik, cython_special.nbdtrik, ('ddd',), None), + (special.nbdtrin, cython_special.nbdtrin, ('ddd',), None), + (special.ncfdtr, cython_special.ncfdtr, ('dddd',), None), + (special.ncfdtri, cython_special.ncfdtri, ('dddd',), None), + (special.ncfdtridfd, cython_special.ncfdtridfd, ('dddd',), None), + (special.ncfdtridfn, cython_special.ncfdtridfn, ('dddd',), None), + (special.ncfdtrinc, cython_special.ncfdtrinc, ('dddd',), None), + (special.nctdtr, cython_special.nctdtr, ('ddd',), None), + (special.nctdtridf, cython_special.nctdtridf, ('ddd',), None), + (special.nctdtrinc, cython_special.nctdtrinc, ('ddd',), None), + (special.nctdtrit, cython_special.nctdtrit, ('ddd',), None), + (special.ndtr, cython_special.ndtr, ('d', 'D'), None), + (special.ndtri, cython_special.ndtri, ('d',), None), + (special.nrdtrimn, cython_special.nrdtrimn, ('ddd',), None), + (special.nrdtrisd, cython_special.nrdtrisd, ('ddd',), None), + (special.obl_ang1, cython_special._obl_ang1_pywrap, ('dddd',), None), + (special.obl_ang1_cv, cython_special._obl_ang1_cv_pywrap, ('ddddd',), None), + (special.obl_cv, cython_special.obl_cv, ('ddd',), None), + (special.obl_rad1, cython_special._obl_rad1_pywrap, ('dddd',), "see gh-6211"), + (special.obl_rad1_cv, cython_special._obl_rad1_cv_pywrap, ('ddddd',), + "see gh-6211"), + (special.obl_rad2, cython_special._obl_rad2_pywrap, ('dddd',), "see gh-6211"), + (special.obl_rad2_cv, cython_special._obl_rad2_cv_pywrap, ('ddddd',), + "see gh-6211"), + (special.pbdv, cython_special._pbdv_pywrap, ('dd',), None), + (special.pbvv, cython_special._pbvv_pywrap, ('dd',), None), + (special.pbwa, cython_special._pbwa_pywrap, ('dd',), None), + (special.pdtr, cython_special.pdtr, ('dd', 'dd'), None), + (special.pdtrc, cython_special.pdtrc, ('dd', 'dd'), None), + (special.pdtri, cython_special.pdtri, ('ld', 'dd'), None), + (special.pdtrik, cython_special.pdtrik, ('dd',), None), + (special.poch, cython_special.poch, ('dd',), None), + (special.powm1, cython_special.powm1, ('dd',), None), + (special.pro_ang1, cython_special._pro_ang1_pywrap, ('dddd',), None), + (special.pro_ang1_cv, cython_special._pro_ang1_cv_pywrap, ('ddddd',), None), + (special.pro_cv, cython_special.pro_cv, ('ddd',), None), + (special.pro_rad1, cython_special._pro_rad1_pywrap, ('dddd',), "see gh-6211"), + (special.pro_rad1_cv, cython_special._pro_rad1_cv_pywrap, ('ddddd',), + "see gh-6211"), + (special.pro_rad2, cython_special._pro_rad2_pywrap, ('dddd',), "see gh-6211"), + (special.pro_rad2_cv, cython_special._pro_rad2_cv_pywrap, ('ddddd',), + "see gh-6211"), + (special.pseudo_huber, cython_special.pseudo_huber, ('dd',), None), + (special.psi, cython_special.psi, ('d', 'D'), None), + (special.radian, cython_special.radian, ('ddd',), None), + (special.rel_entr, cython_special.rel_entr, ('dd',), None), + (special.rgamma, cython_special.rgamma, ('d', 'D'), None), + (special.round, cython_special.round, ('d',), None), + (special.spherical_jn, cython_special.spherical_jn, ('ld', 'ldb', 'lD', 'lDb'), + "Python version supports negative reals; Cython version doesn't - see gh-21629"), + (special.spherical_yn, cython_special.spherical_yn, ('ld', 'ldb', 'lD', 'lDb'), + "Python version supports negative reals; Cython version doesn't - see gh-21629"), + (special.spherical_in, cython_special.spherical_in, ('ld', 'ldb', 'lD', 'lDb'), + "Python version supports negative reals; Cython version doesn't - see gh-21629"), + (special.spherical_kn, cython_special.spherical_kn, ('ld', 'ldb', 'lD', 'lDb'), + "Python version supports negative reals; Cython version doesn't - see gh-21629"), + (special.shichi, cython_special._shichi_pywrap, ('d', 'D'), None), + (special.sici, cython_special._sici_pywrap, ('d', 'D'), None), + (special.sindg, cython_special.sindg, ('d',), None), + (special.smirnov, cython_special.smirnov, ('ld', 'dd'), None), + (special.smirnovi, cython_special.smirnovi, ('ld', 'dd'), None), + (special.spence, cython_special.spence, ('d', 'D'), None), + (special.sph_harm, cython_special.sph_harm, ('lldd', 'dddd'), None), + (special.stdtr, cython_special.stdtr, ('dd',), None), + (special.stdtridf, cython_special.stdtridf, ('dd',), None), + (special.stdtrit, cython_special.stdtrit, ('dd',), None), + (special.struve, cython_special.struve, ('dd',), None), + (special.tandg, cython_special.tandg, ('d',), None), + (special.tklmbda, cython_special.tklmbda, ('dd',), None), + (special.voigt_profile, cython_special.voigt_profile, ('ddd',), None), + (special.wofz, cython_special.wofz, ('D',), None), + (special.wright_bessel, cython_special.wright_bessel, ('ddd',), None), + (special.wrightomega, cython_special.wrightomega, ('D',), None), + (special.xlog1py, cython_special.xlog1py, ('dd', 'DD'), None), + (special.xlogy, cython_special.xlogy, ('dd', 'DD'), None), + (special.y0, cython_special.y0, ('d',), None), + (special.y1, cython_special.y1, ('d',), None), + (special.yn, cython_special.yn, ('ld', 'dd'), None), + (special.yv, cython_special.yv, ('dd', 'dD'), None), + (special.yve, cython_special.yve, ('dd', 'dD'), None), + (special.zetac, cython_special.zetac, ('d',), None), + (special.owens_t, cython_special.owens_t, ('dd',), None) +] + + +IDS = [x[0].__name__ for x in PARAMS] + + +def _generate_test_points(typecodes): + axes = tuple(TEST_POINTS[x] for x in typecodes) + pts = list(product(*axes)) + return pts + + +def test_cython_api_completeness(): + # Check that everything is tested + for name in dir(cython_special): + func = getattr(cython_special, name) + if callable(func) and not name.startswith('_'): + for _, cyfun, _, _ in PARAMS: + if cyfun is func: + break + else: + raise RuntimeError(f"{name} missing from tests!") + + +@pytest.mark.thread_unsafe +@pytest.mark.fail_slow(20) +@pytest.mark.parametrize("param", PARAMS, ids=IDS) +def test_cython_api(param): + pyfunc, cyfunc, specializations, knownfailure = param + if knownfailure: + pytest.xfail(reason=knownfailure) + + # Check which parameters are expected to be fused types + max_params = max(len(spec) for spec in specializations) + values = [set() for _ in range(max_params)] + for typecodes in specializations: + for j, v in enumerate(typecodes): + values[j].add(v) + seen = set() + is_fused_code = [False] * len(values) + for j, v in enumerate(values): + vv = tuple(sorted(v)) + if vv in seen: + continue + is_fused_code[j] = (len(v) > 1) + seen.add(vv) + + # Check results + for typecodes in specializations: + # Pick the correct specialized function + signature = [CYTHON_SIGNATURE_MAP[code] + for j, code in enumerate(typecodes) + if is_fused_code[j]] + + if signature: + cy_spec_func = cyfunc[tuple(signature)] + else: + signature = None + cy_spec_func = cyfunc + + # Test it + pts = _generate_test_points(typecodes) + for pt in pts: + with suppress_warnings() as sup: + sup.filter(DeprecationWarning) + pyval = pyfunc(*pt) + cyval = cy_spec_func(*pt) + assert_allclose(cyval, pyval, err_msg=f"{pt} {typecodes} {signature}") diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_data.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_data.py new file mode 100644 index 0000000000000000000000000000000000000000..0fc89a328cf20da0fd243ab7603cf316ecf2acb4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_data.py @@ -0,0 +1,719 @@ +import importlib.resources + +import numpy as np +from numpy.testing import suppress_warnings +import pytest + +from scipy.special import ( + lpn, lpmn, lpmv, lqn, lqmn, sph_harm, eval_legendre, eval_hermite, + eval_laguerre, eval_genlaguerre, binom, cbrt, expm1, log1p, zeta, + jn, jv, jvp, yn, yv, yvp, iv, ivp, kn, kv, kvp, + gamma, gammaln, gammainc, gammaincc, gammaincinv, gammainccinv, digamma, + beta, betainc, betaincinv, poch, + ellipe, ellipeinc, ellipk, ellipkm1, ellipkinc, + elliprc, elliprd, elliprf, elliprg, elliprj, + erf, erfc, erfinv, erfcinv, exp1, expi, expn, + bdtrik, btdtria, btdtrib, chndtr, gdtr, gdtrc, gdtrix, gdtrib, + nbdtrik, pdtrik, owens_t, + mathieu_a, mathieu_b, mathieu_cem, mathieu_sem, mathieu_modcem1, + mathieu_modsem1, mathieu_modcem2, mathieu_modsem2, + ellip_harm, ellip_harm_2, spherical_jn, spherical_yn, wright_bessel +) +from scipy.integrate import IntegrationWarning + +from scipy.special._testutils import FuncData + + +# The npz files are generated, and hence may live in the build dir. We can only +# access them through `importlib.resources`, not an explicit path from `__file__` +_datadir = importlib.resources.files('scipy.special.tests.data') + +_boost_npz = _datadir.joinpath('boost.npz') +with importlib.resources.as_file(_boost_npz) as f: + DATASETS_BOOST = np.load(f) + +_gsl_npz = _datadir.joinpath('gsl.npz') +with importlib.resources.as_file(_gsl_npz) as f: + DATASETS_GSL = np.load(f) + +_local_npz = _datadir.joinpath('local.npz') +with importlib.resources.as_file(_local_npz) as f: + DATASETS_LOCAL = np.load(f) + + +def data(func, dataname, *a, **kw): + kw.setdefault('dataname', dataname) + return FuncData(func, DATASETS_BOOST[dataname], *a, **kw) + + +def data_gsl(func, dataname, *a, **kw): + kw.setdefault('dataname', dataname) + return FuncData(func, DATASETS_GSL[dataname], *a, **kw) + + +def data_local(func, dataname, *a, **kw): + kw.setdefault('dataname', dataname) + return FuncData(func, DATASETS_LOCAL[dataname], *a, **kw) + + +# The functions lpn, lpmn, clpmn, and sph_harm appearing below are +# deprecated in favor of legendre_p_all, assoc_legendre_p_all, +# assoc_legendre_p_all (assoc_legendre_p_all covers lpmn and clpmn), +# and sph_harm_y respectively. The deprecated functions listed above are +# implemented as shims around their respective replacements. The replacements +# are tested separately, but tests for the deprecated functions remain to +# verify the correctness of the shims. + + +def ellipk_(k): + return ellipk(k*k) + + +def ellipkinc_(f, k): + return ellipkinc(f, k*k) + + +def ellipe_(k): + return ellipe(k*k) + + +def ellipeinc_(f, k): + return ellipeinc(f, k*k) + + +def zeta_(x): + return zeta(x, 1.) + + +def assoc_legendre_p_boost_(nu, mu, x): + # the boost test data is for integer orders only + return lpmv(mu, nu.astype(int), x) + +def legendre_p_via_assoc_(nu, x): + return lpmv(0, nu, x) + +def lpn_(n, x): + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + return lpn(n.astype('l'), x)[0][-1] + +def lqn_(n, x): + return lqn(n.astype('l'), x)[0][-1] + +def legendre_p_via_lpmn(n, x): + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + return lpmn(0, n, x)[0][0,-1] + +def legendre_q_via_lqmn(n, x): + return lqmn(0, n, x)[0][0,-1] + +def mathieu_ce_rad(m, q, x): + return mathieu_cem(m, q, x*180/np.pi)[0] + + +def mathieu_se_rad(m, q, x): + return mathieu_sem(m, q, x*180/np.pi)[0] + + +def mathieu_mc1_scaled(m, q, x): + # GSL follows a different normalization. + # We follow Abramowitz & Stegun, they apparently something else. + return mathieu_modcem1(m, q, x)[0] * np.sqrt(np.pi/2) + + +def mathieu_ms1_scaled(m, q, x): + return mathieu_modsem1(m, q, x)[0] * np.sqrt(np.pi/2) + + +def mathieu_mc2_scaled(m, q, x): + return mathieu_modcem2(m, q, x)[0] * np.sqrt(np.pi/2) + + +def mathieu_ms2_scaled(m, q, x): + return mathieu_modsem2(m, q, x)[0] * np.sqrt(np.pi/2) + +def eval_legendre_ld(n, x): + return eval_legendre(n.astype('l'), x) + +def eval_legendre_dd(n, x): + return eval_legendre(n.astype('d'), x) + +def eval_hermite_ld(n, x): + return eval_hermite(n.astype('l'), x) + +def eval_laguerre_ld(n, x): + return eval_laguerre(n.astype('l'), x) + +def eval_laguerre_dd(n, x): + return eval_laguerre(n.astype('d'), x) + +def eval_genlaguerre_ldd(n, a, x): + return eval_genlaguerre(n.astype('l'), a, x) + +def eval_genlaguerre_ddd(n, a, x): + return eval_genlaguerre(n.astype('d'), a, x) + +def bdtrik_comp(y, n, p): + return bdtrik(1-y, n, p) + +def btdtria_comp(p, b, x): + return btdtria(1-p, b, x) + +def btdtrib_comp(a, p, x): + return btdtrib(a, 1-p, x) + +def gdtr_(p, x): + return gdtr(1.0, p, x) + +def gdtrc_(p, x): + return gdtrc(1.0, p, x) + +def gdtrix_(b, p): + return gdtrix(1.0, b, p) + +def gdtrix_comp(b, p): + return gdtrix(1.0, b, 1-p) + +def gdtrib_(p, x): + return gdtrib(1.0, p, x) + +def gdtrib_comp(p, x): + return gdtrib(1.0, 1-p, x) + +def nbdtrik_comp(y, n, p): + return nbdtrik(1-y, n, p) + +def pdtrik_comp(p, m): + return pdtrik(1-p, m) + +def poch_(z, m): + return 1.0 / poch(z, m) + +def poch_minus(z, m): + return 1.0 / poch(z, -m) + +def spherical_jn_(n, x): + return spherical_jn(n.astype('l'), x) + +def spherical_yn_(n, x): + return spherical_yn(n.astype('l'), x) + +def sph_harm_(m, n, theta, phi): + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + y = sph_harm(m, n, theta, phi) + return (y.real, y.imag) + +def cexpm1(x, y): + z = expm1(x + 1j*y) + return z.real, z.imag + +def clog1p(x, y): + z = log1p(x + 1j*y) + return z.real, z.imag + + +BOOST_TESTS = [ + data(assoc_legendre_p_boost_, 'assoc_legendre_p_ipp-assoc_legendre_p', + (0,1,2), 3, rtol=1e-11), + + data(legendre_p_via_assoc_, 'legendre_p_ipp-legendre_p', + (0,1), 2, rtol=1e-11), + data(legendre_p_via_assoc_, 'legendre_p_large_ipp-legendre_p_large', + (0,1), 2, rtol=9.6e-14), + data(legendre_p_via_lpmn, 'legendre_p_ipp-legendre_p', + (0,1), 2, rtol=5e-14, vectorized=False), + data(legendre_p_via_lpmn, 'legendre_p_large_ipp-legendre_p_large', + (0,1), 2, rtol=3e-13, vectorized=False), + data(lpn_, 'legendre_p_ipp-legendre_p', + (0,1), 2, rtol=5e-14, vectorized=False), + data(lpn_, 'legendre_p_large_ipp-legendre_p_large', + (0,1), 2, rtol=3e-13, vectorized=False), + data(eval_legendre_ld, 'legendre_p_ipp-legendre_p', + (0,1), 2, rtol=6e-14), + data(eval_legendre_ld, 'legendre_p_large_ipp-legendre_p_large', + (0,1), 2, rtol=2e-13), + data(eval_legendre_dd, 'legendre_p_ipp-legendre_p', + (0,1), 2, rtol=2e-14), + data(eval_legendre_dd, 'legendre_p_large_ipp-legendre_p_large', + (0,1), 2, rtol=2e-13), + + data(lqn_, 'legendre_p_ipp-legendre_p', + (0,1), 3, rtol=2e-14, vectorized=False), + data(lqn_, 'legendre_p_large_ipp-legendre_p_large', + (0,1), 3, rtol=2e-12, vectorized=False), + data(legendre_q_via_lqmn, 'legendre_p_ipp-legendre_p', + (0,1), 3, rtol=2e-14, vectorized=False), + data(legendre_q_via_lqmn, 'legendre_p_large_ipp-legendre_p_large', + (0,1), 3, rtol=2e-12, vectorized=False), + + data(beta, 'beta_exp_data_ipp-beta_exp_data', + (0,1), 2, rtol=1e-13), + data(beta, 'beta_exp_data_ipp-beta_exp_data', + (0,1), 2, rtol=1e-13), + data(beta, 'beta_med_data_ipp-beta_med_data', + (0,1), 2, rtol=5e-13), + + data(betainc, 'ibeta_small_data_ipp-ibeta_small_data', + (0,1,2), 5, rtol=6e-15), + data(betainc, 'ibeta_data_ipp-ibeta_data', + (0,1,2), 5, rtol=5e-13), + data(betainc, 'ibeta_int_data_ipp-ibeta_int_data', + (0,1,2), 5, rtol=2e-14), + data(betainc, 'ibeta_large_data_ipp-ibeta_large_data', + (0,1,2), 5, rtol=4e-10), + + data(betaincinv, 'ibeta_inv_data_ipp-ibeta_inv_data', + (0,1,2), 3, rtol=1e-5), + + data(btdtria, 'ibeta_inva_data_ipp-ibeta_inva_data', + (2,0,1), 3, rtol=5e-9), + data(btdtria_comp, 'ibeta_inva_data_ipp-ibeta_inva_data', + (2,0,1), 4, rtol=5e-9), + + data(btdtrib, 'ibeta_inva_data_ipp-ibeta_inva_data', + (0,2,1), 5, rtol=5e-9), + data(btdtrib_comp, 'ibeta_inva_data_ipp-ibeta_inva_data', + (0,2,1), 6, rtol=5e-9), + + data(binom, 'binomial_data_ipp-binomial_data', + (0,1), 2, rtol=1e-13), + data(binom, 'binomial_large_data_ipp-binomial_large_data', + (0,1), 2, rtol=5e-13), + + data(bdtrik, 'binomial_quantile_ipp-binomial_quantile_data', + (2,0,1), 3, rtol=5e-9), + data(bdtrik_comp, 'binomial_quantile_ipp-binomial_quantile_data', + (2,0,1), 4, rtol=5e-9), + + data(nbdtrik, 'negative_binomial_quantile_ipp-negative_binomial_quantile_data', + (2,0,1), 3, rtol=4e-9), + data(nbdtrik_comp, + 'negative_binomial_quantile_ipp-negative_binomial_quantile_data', + (2,0,1), 4, rtol=4e-9), + + data(pdtrik, 'poisson_quantile_ipp-poisson_quantile_data', + (1,0), 2, rtol=3e-9), + data(pdtrik_comp, 'poisson_quantile_ipp-poisson_quantile_data', + (1,0), 3, rtol=4e-9), + + data(cbrt, 'cbrt_data_ipp-cbrt_data', 1, 0), + + data(digamma, 'digamma_data_ipp-digamma_data', 0, 1), + data(digamma, 'digamma_data_ipp-digamma_data', 0j, 1), + data(digamma, 'digamma_neg_data_ipp-digamma_neg_data', 0, 1, rtol=2e-13), + data(digamma, 'digamma_neg_data_ipp-digamma_neg_data', 0j, 1, rtol=1e-13), + data(digamma, 'digamma_root_data_ipp-digamma_root_data', 0, 1, rtol=1e-15), + data(digamma, 'digamma_root_data_ipp-digamma_root_data', 0j, 1, rtol=1e-15), + data(digamma, 'digamma_small_data_ipp-digamma_small_data', 0, 1, rtol=1e-15), + data(digamma, 'digamma_small_data_ipp-digamma_small_data', 0j, 1, rtol=1e-14), + + data(ellipk_, 'ellint_k_data_ipp-ellint_k_data', 0, 1), + data(ellipkinc_, 'ellint_f_data_ipp-ellint_f_data', (0,1), 2, rtol=1e-14), + data(ellipe_, 'ellint_e_data_ipp-ellint_e_data', 0, 1), + data(ellipeinc_, 'ellint_e2_data_ipp-ellint_e2_data', (0,1), 2, rtol=1e-14), + + data(erf, 'erf_data_ipp-erf_data', 0, 1), + data(erf, 'erf_data_ipp-erf_data', 0j, 1, rtol=1e-13), + data(erfc, 'erf_data_ipp-erf_data', 0, 2, rtol=6e-15), + data(erf, 'erf_large_data_ipp-erf_large_data', 0, 1), + data(erf, 'erf_large_data_ipp-erf_large_data', 0j, 1), + data(erfc, 'erf_large_data_ipp-erf_large_data', 0, 2, rtol=4e-14), + data(erf, 'erf_small_data_ipp-erf_small_data', 0, 1), + data(erf, 'erf_small_data_ipp-erf_small_data', 0j, 1, rtol=1e-13), + data(erfc, 'erf_small_data_ipp-erf_small_data', 0, 2), + + data(erfinv, 'erf_inv_data_ipp-erf_inv_data', 0, 1), + data(erfcinv, 'erfc_inv_data_ipp-erfc_inv_data', 0, 1), + data(erfcinv, 'erfc_inv_big_data_ipp-erfc_inv_big_data', 0, 1, + param_filter=(lambda s: s > 0)), + + data(exp1, 'expint_1_data_ipp-expint_1_data', 1, 2, rtol=1e-13), + data(exp1, 'expint_1_data_ipp-expint_1_data', 1j, 2, rtol=5e-9), + data(expi, 'expinti_data_ipp-expinti_data', 0, 1, rtol=1e-13), + data(expi, 'expinti_data_double_ipp-expinti_data_double', 0, 1, rtol=1e-13), + data(expi, 'expinti_data_long_ipp-expinti_data_long', 0, 1), + + data(expn, 'expint_small_data_ipp-expint_small_data', (0,1), 2), + data(expn, 'expint_data_ipp-expint_data', (0,1), 2, rtol=1e-14), + + data(gamma, 'test_gamma_data_ipp-near_0', 0, 1), + data(gamma, 'test_gamma_data_ipp-near_1', 0, 1), + data(gamma, 'test_gamma_data_ipp-near_2', 0, 1), + data(gamma, 'test_gamma_data_ipp-near_m10', 0, 1), + data(gamma, 'test_gamma_data_ipp-near_m55', 0, 1, rtol=7e-12), + data(gamma, 'test_gamma_data_ipp-factorials', 0, 1, rtol=4e-14), + data(gamma, 'test_gamma_data_ipp-near_0', 0j, 1, rtol=2e-9), + data(gamma, 'test_gamma_data_ipp-near_1', 0j, 1, rtol=2e-9), + data(gamma, 'test_gamma_data_ipp-near_2', 0j, 1, rtol=2e-9), + data(gamma, 'test_gamma_data_ipp-near_m10', 0j, 1, rtol=2e-9), + data(gamma, 'test_gamma_data_ipp-near_m55', 0j, 1, rtol=2e-9), + data(gamma, 'test_gamma_data_ipp-factorials', 0j, 1, rtol=2e-13), + data(gammaln, 'test_gamma_data_ipp-near_0', 0, 2, rtol=5e-11), + data(gammaln, 'test_gamma_data_ipp-near_1', 0, 2, rtol=5e-11), + data(gammaln, 'test_gamma_data_ipp-near_2', 0, 2, rtol=2e-10), + data(gammaln, 'test_gamma_data_ipp-near_m10', 0, 2, rtol=5e-11), + data(gammaln, 'test_gamma_data_ipp-near_m55', 0, 2, rtol=5e-11), + data(gammaln, 'test_gamma_data_ipp-factorials', 0, 2), + + data(gammainc, 'igamma_small_data_ipp-igamma_small_data', (0,1), 5, rtol=5e-15), + data(gammainc, 'igamma_med_data_ipp-igamma_med_data', (0,1), 5, rtol=2e-13), + data(gammainc, 'igamma_int_data_ipp-igamma_int_data', (0,1), 5, rtol=2e-13), + data(gammainc, 'igamma_big_data_ipp-igamma_big_data', (0,1), 5, rtol=1e-12), + + data(gdtr_, 'igamma_small_data_ipp-igamma_small_data', (0,1), 5, rtol=1e-13), + data(gdtr_, 'igamma_med_data_ipp-igamma_med_data', (0,1), 5, rtol=2e-13), + data(gdtr_, 'igamma_int_data_ipp-igamma_int_data', (0,1), 5, rtol=2e-13), + data(gdtr_, 'igamma_big_data_ipp-igamma_big_data', (0,1), 5, rtol=2e-9), + + data(gammaincc, 'igamma_small_data_ipp-igamma_small_data', + (0,1), 3, rtol=1e-13), + data(gammaincc, 'igamma_med_data_ipp-igamma_med_data', + (0,1), 3, rtol=2e-13), + data(gammaincc, 'igamma_int_data_ipp-igamma_int_data', + (0,1), 3, rtol=4e-14), + data(gammaincc, 'igamma_big_data_ipp-igamma_big_data', + (0,1), 3, rtol=1e-11), + + data(gdtrc_, 'igamma_small_data_ipp-igamma_small_data', (0,1), 3, rtol=1e-13), + data(gdtrc_, 'igamma_med_data_ipp-igamma_med_data', (0,1), 3, rtol=2e-13), + data(gdtrc_, 'igamma_int_data_ipp-igamma_int_data', (0,1), 3, rtol=4e-14), + data(gdtrc_, 'igamma_big_data_ipp-igamma_big_data', (0,1), 3, rtol=1e-11), + + data(gdtrib_, 'igamma_inva_data_ipp-igamma_inva_data', (1,0), 2, rtol=5e-9), + data(gdtrib_comp, 'igamma_inva_data_ipp-igamma_inva_data', (1,0), 3, rtol=5e-9), + + data(poch_, 'tgamma_delta_ratio_data_ipp-tgamma_delta_ratio_data', + (0,1), 2, rtol=2e-13), + data(poch_, 'tgamma_delta_ratio_int_ipp-tgamma_delta_ratio_int', + (0,1), 2,), + data(poch_, 'tgamma_delta_ratio_int2_ipp-tgamma_delta_ratio_int2', + (0,1), 2,), + data(poch_minus, 'tgamma_delta_ratio_data_ipp-tgamma_delta_ratio_data', + (0,1), 3, rtol=2e-13), + data(poch_minus, 'tgamma_delta_ratio_int_ipp-tgamma_delta_ratio_int', + (0,1), 3), + data(poch_minus, 'tgamma_delta_ratio_int2_ipp-tgamma_delta_ratio_int2', + (0,1), 3), + + data(eval_hermite_ld, 'hermite_ipp-hermite', + (0,1), 2, rtol=2e-14), + + data(eval_laguerre_ld, 'laguerre2_ipp-laguerre2', + (0,1), 2, rtol=7e-12), + data(eval_laguerre_dd, 'laguerre2_ipp-laguerre2', + (0,1), 2, knownfailure='hyp2f1 insufficiently accurate.'), + data(eval_genlaguerre_ldd, 'laguerre3_ipp-laguerre3', + (0,1,2), 3, rtol=2e-13), + data(eval_genlaguerre_ddd, 'laguerre3_ipp-laguerre3', + (0,1,2), 3, knownfailure='hyp2f1 insufficiently accurate.'), + + data(log1p, 'log1p_expm1_data_ipp-log1p_expm1_data', 0, 1), + data(expm1, 'log1p_expm1_data_ipp-log1p_expm1_data', 0, 2), + + data(iv, 'bessel_i_data_ipp-bessel_i_data', + (0,1), 2, rtol=1e-12), + data(iv, 'bessel_i_data_ipp-bessel_i_data', + (0,1j), 2, rtol=2e-10, atol=1e-306), + data(iv, 'bessel_i_int_data_ipp-bessel_i_int_data', + (0,1), 2, rtol=1e-9), + data(iv, 'bessel_i_int_data_ipp-bessel_i_int_data', + (0,1j), 2, rtol=2e-10), + + data(ivp, 'bessel_i_prime_int_data_ipp-bessel_i_prime_int_data', + (0,1), 2, rtol=1.2e-13), + data(ivp, 'bessel_i_prime_int_data_ipp-bessel_i_prime_int_data', + (0,1j), 2, rtol=1.2e-13, atol=1e-300), + + data(jn, 'bessel_j_int_data_ipp-bessel_j_int_data', (0,1), 2, rtol=1e-12), + data(jn, 'bessel_j_int_data_ipp-bessel_j_int_data', (0,1j), 2, rtol=1e-12), + data(jn, 'bessel_j_large_data_ipp-bessel_j_large_data', (0,1), 2, rtol=6e-11), + data(jn, 'bessel_j_large_data_ipp-bessel_j_large_data', (0,1j), 2, rtol=6e-11), + + data(jv, 'bessel_j_int_data_ipp-bessel_j_int_data', (0,1), 2, rtol=1e-12), + data(jv, 'bessel_j_int_data_ipp-bessel_j_int_data', (0,1j), 2, rtol=1e-12), + data(jv, 'bessel_j_data_ipp-bessel_j_data', (0,1), 2, rtol=1e-12), + data(jv, 'bessel_j_data_ipp-bessel_j_data', (0,1j), 2, rtol=1e-12), + + data(jvp, 'bessel_j_prime_int_data_ipp-bessel_j_prime_int_data', + (0,1), 2, rtol=1e-13), + data(jvp, 'bessel_j_prime_int_data_ipp-bessel_j_prime_int_data', + (0,1j), 2, rtol=1e-13), + data(jvp, 'bessel_j_prime_large_data_ipp-bessel_j_prime_large_data', + (0,1), 2, rtol=1e-11), + data(jvp, 'bessel_j_prime_large_data_ipp-bessel_j_prime_large_data', + (0,1j), 2, rtol=2e-11), + + data(kn, 'bessel_k_int_data_ipp-bessel_k_int_data', (0,1), 2, rtol=1e-12), + + data(kv, 'bessel_k_int_data_ipp-bessel_k_int_data', (0,1), 2, rtol=1e-12), + data(kv, 'bessel_k_int_data_ipp-bessel_k_int_data', (0,1j), 2, rtol=1e-12), + data(kv, 'bessel_k_data_ipp-bessel_k_data', (0,1), 2, rtol=1e-12), + data(kv, 'bessel_k_data_ipp-bessel_k_data', (0,1j), 2, rtol=1e-12), + + data(kvp, 'bessel_k_prime_int_data_ipp-bessel_k_prime_int_data', + (0,1), 2, rtol=3e-14), + data(kvp, 'bessel_k_prime_int_data_ipp-bessel_k_prime_int_data', + (0,1j), 2, rtol=3e-14), + data(kvp, 'bessel_k_prime_data_ipp-bessel_k_prime_data', (0,1), 2, rtol=7e-14), + data(kvp, 'bessel_k_prime_data_ipp-bessel_k_prime_data', (0,1j), 2, rtol=7e-14), + + data(yn, 'bessel_y01_data_ipp-bessel_y01_data', (0,1), 2, rtol=1e-12), + data(yn, 'bessel_yn_data_ipp-bessel_yn_data', (0,1), 2, rtol=1e-12), + + data(yv, 'bessel_yn_data_ipp-bessel_yn_data', (0,1), 2, rtol=1e-12), + data(yv, 'bessel_yn_data_ipp-bessel_yn_data', (0,1j), 2, rtol=1e-12), + data(yv, 'bessel_yv_data_ipp-bessel_yv_data', (0,1), 2, rtol=1e-10), + data(yv, 'bessel_yv_data_ipp-bessel_yv_data', (0,1j), 2, rtol=1e-10), + + data(yvp, 'bessel_yv_prime_data_ipp-bessel_yv_prime_data', + (0, 1), 2, rtol=4e-9), + data(yvp, 'bessel_yv_prime_data_ipp-bessel_yv_prime_data', + (0, 1j), 2, rtol=4e-9), + + data(zeta_, 'zeta_data_ipp-zeta_data', 0, 1, + param_filter=(lambda s: s > 1)), + data(zeta_, 'zeta_neg_data_ipp-zeta_neg_data', 0, 1, + param_filter=(lambda s: s > 1)), + data(zeta_, 'zeta_1_up_data_ipp-zeta_1_up_data', 0, 1, + param_filter=(lambda s: s > 1)), + data(zeta_, 'zeta_1_below_data_ipp-zeta_1_below_data', 0, 1, + param_filter=(lambda s: s > 1)), + + data(gammaincinv, 'gamma_inv_small_data_ipp-gamma_inv_small_data', + (0,1), 2, rtol=1e-11), + data(gammaincinv, 'gamma_inv_data_ipp-gamma_inv_data', + (0,1), 2, rtol=1e-14), + data(gammaincinv, 'gamma_inv_big_data_ipp-gamma_inv_big_data', + (0,1), 2, rtol=1e-11), + + data(gammainccinv, 'gamma_inv_small_data_ipp-gamma_inv_small_data', + (0,1), 3, rtol=1e-12), + data(gammainccinv, 'gamma_inv_data_ipp-gamma_inv_data', + (0,1), 3, rtol=1e-14), + data(gammainccinv, 'gamma_inv_big_data_ipp-gamma_inv_big_data', + (0,1), 3, rtol=1e-14), + + data(gdtrix_, 'gamma_inv_small_data_ipp-gamma_inv_small_data', + (0,1), 2, rtol=3e-13, knownfailure='gdtrix unflow some points'), + data(gdtrix_, 'gamma_inv_data_ipp-gamma_inv_data', + (0,1), 2, rtol=3e-15), + data(gdtrix_, 'gamma_inv_big_data_ipp-gamma_inv_big_data', + (0,1), 2), + data(gdtrix_comp, 'gamma_inv_small_data_ipp-gamma_inv_small_data', + (0,1), 2, knownfailure='gdtrix bad some points'), + data(gdtrix_comp, 'gamma_inv_data_ipp-gamma_inv_data', + (0,1), 3, rtol=6e-15), + data(gdtrix_comp, 'gamma_inv_big_data_ipp-gamma_inv_big_data', + (0,1), 3), + + data(chndtr, 'nccs_ipp-nccs', + (2,0,1), 3, rtol=3e-5), + data(chndtr, 'nccs_big_ipp-nccs_big', + (2,0,1), 3, rtol=5e-4, knownfailure='chndtr inaccurate some points'), + + data(sph_harm_, 'spherical_harmonic_ipp-spherical_harmonic', + (1,0,3,2), (4,5), rtol=5e-11, + param_filter=(lambda p: np.ones(p.shape, '?'), + lambda p: np.ones(p.shape, '?'), + lambda p: np.logical_and(p < 2*np.pi, p >= 0), + lambda p: np.logical_and(p < np.pi, p >= 0))), + + data(spherical_jn_, 'sph_bessel_data_ipp-sph_bessel_data', + (0,1), 2, rtol=1e-13), + data(spherical_yn_, 'sph_neumann_data_ipp-sph_neumann_data', + (0,1), 2, rtol=8e-15), + + data(owens_t, 'owens_t_ipp-owens_t', + (0, 1), 2, rtol=5e-14), + data(owens_t, 'owens_t_large_data_ipp-owens_t_large_data', + (0, 1), 2, rtol=8e-12), + + # -- test data exists in boost but is not used in scipy -- + + # ibeta_derivative_data_ipp/ibeta_derivative_data.txt + # ibeta_derivative_int_data_ipp/ibeta_derivative_int_data.txt + # ibeta_derivative_large_data_ipp/ibeta_derivative_large_data.txt + # ibeta_derivative_small_data_ipp/ibeta_derivative_small_data.txt + + # bessel_y01_prime_data_ipp/bessel_y01_prime_data.txt + # bessel_yn_prime_data_ipp/bessel_yn_prime_data.txt + # sph_bessel_prime_data_ipp/sph_bessel_prime_data.txt + # sph_neumann_prime_data_ipp/sph_neumann_prime_data.txt + + # ellint_d2_data_ipp/ellint_d2_data.txt + # ellint_d_data_ipp/ellint_d_data.txt + # ellint_pi2_data_ipp/ellint_pi2_data.txt + # ellint_pi3_data_ipp/ellint_pi3_data.txt + # ellint_pi3_large_data_ipp/ellint_pi3_large_data.txt + data(elliprc, 'ellint_rc_data_ipp-ellint_rc_data', (0, 1), 2, + rtol=5e-16), + data(elliprd, 'ellint_rd_data_ipp-ellint_rd_data', (0, 1, 2), 3, + rtol=5e-16), + data(elliprd, 'ellint_rd_0xy_ipp-ellint_rd_0xy', (0, 1, 2), 3, + rtol=5e-16), + data(elliprd, 'ellint_rd_0yy_ipp-ellint_rd_0yy', (0, 1, 2), 3, + rtol=5e-16), + data(elliprd, 'ellint_rd_xxx_ipp-ellint_rd_xxx', (0, 1, 2), 3, + rtol=5e-16), + # Some of the following rtol for elliprd may be larger than 5e-16 to + # work around some hard cases in the Boost test where we get slightly + # larger error than the ideal bound when the x (==y) input is close to + # zero. + # Also the accuracy on 32-bit builds with g++ may suffer from excess + # loss of precision; see GCC bugzilla 323 + # https://gcc.gnu.org/bugzilla/show_bug.cgi?id=323 + data(elliprd, 'ellint_rd_xxz_ipp-ellint_rd_xxz', (0, 1, 2), 3, + rtol=6.5e-16), + data(elliprd, 'ellint_rd_xyy_ipp-ellint_rd_xyy', (0, 1, 2), 3, + rtol=6e-16), + data(elliprf, 'ellint_rf_data_ipp-ellint_rf_data', (0, 1, 2), 3, + rtol=5e-16), + data(elliprf, 'ellint_rf_xxx_ipp-ellint_rf_xxx', (0, 1, 2), 3, + rtol=5e-16), + data(elliprf, 'ellint_rf_xyy_ipp-ellint_rf_xyy', (0, 1, 2), 3, + rtol=5e-16), + data(elliprf, 'ellint_rf_xy0_ipp-ellint_rf_xy0', (0, 1, 2), 3, + rtol=5e-16), + data(elliprf, 'ellint_rf_0yy_ipp-ellint_rf_0yy', (0, 1, 2), 3, + rtol=5e-16), + # The accuracy of R_G is primarily limited by R_D that is used + # internally. It is generally worse than R_D. Notice that we increased + # the rtol for R_G here. The cases with duplicate arguments are + # slightly less likely to be unbalanced (at least two arguments are + # already balanced) so the error bound is slightly better. Again, + # precision with g++ 32-bit is even worse. + data(elliprg, 'ellint_rg_ipp-ellint_rg', (0, 1, 2), 3, + rtol=8.0e-16), + data(elliprg, 'ellint_rg_xxx_ipp-ellint_rg_xxx', (0, 1, 2), 3, + rtol=6e-16), + data(elliprg, 'ellint_rg_xyy_ipp-ellint_rg_xyy', (0, 1, 2), 3, + rtol=7.5e-16), + data(elliprg, 'ellint_rg_xy0_ipp-ellint_rg_xy0', (0, 1, 2), 3, + rtol=5e-16), + data(elliprg, 'ellint_rg_00x_ipp-ellint_rg_00x', (0, 1, 2), 3, + rtol=5e-16), + data(elliprj, 'ellint_rj_data_ipp-ellint_rj_data', (0, 1, 2, 3), 4, + rtol=5e-16, atol=1e-25, + param_filter=(lambda s: s <= 5e-26,)), + # ellint_rc_data_ipp/ellint_rc_data.txt + # ellint_rd_0xy_ipp/ellint_rd_0xy.txt + # ellint_rd_0yy_ipp/ellint_rd_0yy.txt + # ellint_rd_data_ipp/ellint_rd_data.txt + # ellint_rd_xxx_ipp/ellint_rd_xxx.txt + # ellint_rd_xxz_ipp/ellint_rd_xxz.txt + # ellint_rd_xyy_ipp/ellint_rd_xyy.txt + # ellint_rf_0yy_ipp/ellint_rf_0yy.txt + # ellint_rf_data_ipp/ellint_rf_data.txt + # ellint_rf_xxx_ipp/ellint_rf_xxx.txt + # ellint_rf_xy0_ipp/ellint_rf_xy0.txt + # ellint_rf_xyy_ipp/ellint_rf_xyy.txt + # ellint_rg_00x_ipp/ellint_rg_00x.txt + # ellint_rg_ipp/ellint_rg.txt + # ellint_rg_xxx_ipp/ellint_rg_xxx.txt + # ellint_rg_xy0_ipp/ellint_rg_xy0.txt + # ellint_rg_xyy_ipp/ellint_rg_xyy.txt + # ellint_rj_data_ipp/ellint_rj_data.txt + # ellint_rj_e2_ipp/ellint_rj_e2.txt + # ellint_rj_e3_ipp/ellint_rj_e3.txt + # ellint_rj_e4_ipp/ellint_rj_e4.txt + # ellint_rj_zp_ipp/ellint_rj_zp.txt + + # jacobi_elliptic_ipp/jacobi_elliptic.txt + # jacobi_elliptic_small_ipp/jacobi_elliptic_small.txt + # jacobi_large_phi_ipp/jacobi_large_phi.txt + # jacobi_near_1_ipp/jacobi_near_1.txt + # jacobi_zeta_big_phi_ipp/jacobi_zeta_big_phi.txt + # jacobi_zeta_data_ipp/jacobi_zeta_data.txt + + # heuman_lambda_data_ipp/heuman_lambda_data.txt + + # hypergeometric_0F2_ipp/hypergeometric_0F2.txt + # hypergeometric_1F1_big_ipp/hypergeometric_1F1_big.txt + # hypergeometric_1F1_ipp/hypergeometric_1F1.txt + # hypergeometric_1F1_small_random_ipp/hypergeometric_1F1_small_random.txt + # hypergeometric_1F2_ipp/hypergeometric_1F2.txt + # hypergeometric_1f1_large_regularized_ipp/hypergeometric_1f1_large_regularized.txt # noqa: E501 + # hypergeometric_1f1_log_large_unsolved_ipp/hypergeometric_1f1_log_large_unsolved.txt # noqa: E501 + # hypergeometric_2F0_half_ipp/hypergeometric_2F0_half.txt + # hypergeometric_2F0_integer_a2_ipp/hypergeometric_2F0_integer_a2.txt + # hypergeometric_2F0_ipp/hypergeometric_2F0.txt + # hypergeometric_2F0_large_z_ipp/hypergeometric_2F0_large_z.txt + # hypergeometric_2F1_ipp/hypergeometric_2F1.txt + # hypergeometric_2F2_ipp/hypergeometric_2F2.txt + + # ncbeta_big_ipp/ncbeta_big.txt + # nct_small_delta_ipp/nct_small_delta.txt + # nct_asym_ipp/nct_asym.txt + # ncbeta_ipp/ncbeta.txt + + # powm1_data_ipp/powm1_big_data.txt + # powm1_sqrtp1m1_test_hpp/sqrtp1m1_data.txt + + # sinc_data_ipp/sinc_data.txt + + # test_gamma_data_ipp/gammap1m1_data.txt + # tgamma_ratio_data_ipp/tgamma_ratio_data.txt + + # trig_data_ipp/trig_data.txt + # trig_data2_ipp/trig_data2.txt +] + + +@pytest.mark.thread_unsafe +@pytest.mark.parametrize('test', BOOST_TESTS, ids=repr) +def test_boost(test): + _test_factory(test) + + +GSL_TESTS = [ + data_gsl(mathieu_a, 'mathieu_ab', (0, 1), 2, rtol=1e-13, atol=1e-13), + data_gsl(mathieu_b, 'mathieu_ab', (0, 1), 3, rtol=1e-13, atol=1e-13), + + # Also the GSL output has limited accuracy... + data_gsl(mathieu_ce_rad, 'mathieu_ce_se', (0, 1, 2), 3, rtol=1e-7, atol=1e-13), + data_gsl(mathieu_se_rad, 'mathieu_ce_se', (0, 1, 2), 4, rtol=1e-7, atol=1e-13), + + data_gsl(mathieu_mc1_scaled, 'mathieu_mc_ms', + (0, 1, 2), 3, rtol=1e-7, atol=1e-13), + data_gsl(mathieu_ms1_scaled, 'mathieu_mc_ms', + (0, 1, 2), 4, rtol=1e-7, atol=1e-13), + + data_gsl(mathieu_mc2_scaled, 'mathieu_mc_ms', + (0, 1, 2), 5, rtol=1e-7, atol=1e-13), + data_gsl(mathieu_ms2_scaled, 'mathieu_mc_ms', + (0, 1, 2), 6, rtol=1e-7, atol=1e-13), +] + + +@pytest.mark.parametrize('test', GSL_TESTS, ids=repr) +def test_gsl(test): + _test_factory(test) + + +LOCAL_TESTS = [ + data_local(ellipkinc, 'ellipkinc_neg_m', (0, 1), 2), + data_local(ellipkm1, 'ellipkm1', 0, 1), + data_local(ellipeinc, 'ellipeinc_neg_m', (0, 1), 2), + data_local(clog1p, 'log1p_expm1_complex', (0,1), (2,3), rtol=1e-14), + data_local(cexpm1, 'log1p_expm1_complex', (0,1), (4,5), rtol=1e-14), + data_local(gammainc, 'gammainc', (0, 1), 2, rtol=1e-12), + data_local(gammaincc, 'gammaincc', (0, 1), 2, rtol=1e-11), + data_local(ellip_harm_2, 'ellip',(0, 1, 2, 3, 4), 6, rtol=1e-10, atol=1e-13), + data_local(ellip_harm, 'ellip',(0, 1, 2, 3, 4), 5, rtol=1e-10, atol=1e-13), + data_local(wright_bessel, 'wright_bessel', (0, 1, 2), 3, rtol=1e-11), +] + + +@pytest.mark.parametrize('test', LOCAL_TESTS, ids=repr) +def test_local(test): + _test_factory(test) + + +def _test_factory(test, dtype=np.float64): + """Boost test""" + with suppress_warnings() as sup: + sup.filter(IntegrationWarning, "The occurrence of roundoff error is detected") + with np.errstate(all='ignore'): + test.check(dtype=dtype) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_dd.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_dd.py new file mode 100644 index 0000000000000000000000000000000000000000..6da2c8ddddd7341cbb777216e9f2f3cce536a51e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_dd.py @@ -0,0 +1,42 @@ +# Tests for a few of the "double-double" C++ functions defined in +# special/cephes/dd_real.h. Prior to gh-20390 which translated these +# functions from C to C++, there were test cases for _dd_expm1. It +# was determined that this function is not used anywhere internally +# in SciPy, so this function was not translated. + + +import pytest +from numpy.testing import assert_allclose +from scipy.special._test_internal import _dd_exp, _dd_log + + +# Each tuple in test_data contains: +# (dd_func, xhi, xlo, expected_yhi, expected_ylo) +# The expected values were computed with mpmath, e.g. +# +# import mpmath +# mpmath.mp.dps = 100 +# xhi = 10.0 +# xlo = 0.0 +# x = mpmath.mpf(xhi) + mpmath.mpf(xlo) +# y = mpmath.log(x) +# expected_yhi = float(y) +# expected_ylo = float(y - expected_yhi) +# +test_data = [ + (_dd_exp, -0.3333333333333333, -1.850371707708594e-17, + 0.7165313105737893, -2.0286948382455594e-17), + (_dd_exp, 0.0, 0.0, 1.0, 0.0), + (_dd_exp, 10.0, 0.0, 22026.465794806718, -1.3780134700517372e-12), + (_dd_log, 0.03125, 0.0, -3.4657359027997265, -4.930038229799327e-18), + (_dd_log, 10.0, 0.0, 2.302585092994046, -2.1707562233822494e-16), +] + + +@pytest.mark.parametrize('dd_func, xhi, xlo, expected_yhi, expected_ylo', + test_data) +def test_dd(dd_func, xhi, xlo, expected_yhi, expected_ylo): + yhi, ylo = dd_func(xhi, xlo) + assert yhi == expected_yhi, (f"high double ({yhi}) does not equal the " + f"expected value {expected_yhi}") + assert_allclose(ylo, expected_ylo, rtol=5e-15) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_digamma.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_digamma.py new file mode 100644 index 0000000000000000000000000000000000000000..d7f27dc7b71c1ae928b4bdd8bd987df9ca420bab --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_digamma.py @@ -0,0 +1,45 @@ +import numpy as np +from numpy import pi, log, sqrt +from numpy.testing import assert_, assert_equal + +from scipy.special._testutils import FuncData +import scipy.special as sc + +# Euler-Mascheroni constant +euler = 0.57721566490153286 + + +def test_consistency(): + # Make sure the implementation of digamma for real arguments + # agrees with the implementation of digamma for complex arguments. + + # It's all poles after -1e16 + x = np.r_[-np.logspace(15, -30, 200), np.logspace(-30, 300, 200)] + dataset = np.vstack((x + 0j, sc.digamma(x))).T + FuncData(sc.digamma, dataset, 0, 1, rtol=5e-14, nan_ok=True).check() + + +def test_special_values(): + # Test special values from Gauss's digamma theorem. See + # + # https://en.wikipedia.org/wiki/Digamma_function + + dataset = [ + (1, -euler), + (0.5, -2*log(2) - euler), + (1/3, -pi/(2*sqrt(3)) - 3*log(3)/2 - euler), + (1/4, -pi/2 - 3*log(2) - euler), + (1/6, -pi*sqrt(3)/2 - 2*log(2) - 3*log(3)/2 - euler), + (1/8, + -pi/2 - 4*log(2) - (pi + log(2 + sqrt(2)) - log(2 - sqrt(2)))/sqrt(2) - euler) + ] + + dataset = np.asarray(dataset) + FuncData(sc.digamma, dataset, 0, 1, rtol=1e-14).check() + + +def test_nonfinite(): + pts = [0.0, -0.0, np.inf] + std = [-np.inf, np.inf, np.inf] + assert_equal(sc.digamma(pts), std) + assert_(all(np.isnan(sc.digamma([-np.inf, -1])))) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ellip_harm.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ellip_harm.py new file mode 100644 index 0000000000000000000000000000000000000000..1cfb6530e2fd80d2c261af960f42c974e1e1b26e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ellip_harm.py @@ -0,0 +1,278 @@ +# +# Tests for the Ellipsoidal Harmonic Function, +# Distributed under the same license as SciPy itself. +# + +import numpy as np +from numpy.testing import (assert_equal, assert_almost_equal, assert_allclose, + assert_, suppress_warnings) +from scipy.special._testutils import assert_func_equal +from scipy.special import ellip_harm, ellip_harm_2, ellip_normal +from scipy.integrate import IntegrationWarning +from numpy import sqrt, pi + + +def test_ellip_potential(): + def change_coefficient(lambda1, mu, nu, h2, k2): + x = sqrt(lambda1**2*mu**2*nu**2/(h2*k2)) + y = sqrt((lambda1**2 - h2)*(mu**2 - h2)*(h2 - nu**2)/(h2*(k2 - h2))) + z = sqrt((lambda1**2 - k2)*(k2 - mu**2)*(k2 - nu**2)/(k2*(k2 - h2))) + return x, y, z + + def solid_int_ellip(lambda1, mu, nu, n, p, h2, k2): + return (ellip_harm(h2, k2, n, p, lambda1)*ellip_harm(h2, k2, n, p, mu) + * ellip_harm(h2, k2, n, p, nu)) + + def solid_int_ellip2(lambda1, mu, nu, n, p, h2, k2): + return (ellip_harm_2(h2, k2, n, p, lambda1) + * ellip_harm(h2, k2, n, p, mu)*ellip_harm(h2, k2, n, p, nu)) + + def summation(lambda1, mu1, nu1, lambda2, mu2, nu2, h2, k2): + tol = 1e-8 + sum1 = 0 + for n in range(20): + xsum = 0 + for p in range(1, 2*n+2): + xsum += (4*pi*(solid_int_ellip(lambda2, mu2, nu2, n, p, h2, k2) + * solid_int_ellip2(lambda1, mu1, nu1, n, p, h2, k2)) / + (ellip_normal(h2, k2, n, p)*(2*n + 1))) + if abs(xsum) < 0.1*tol*abs(sum1): + break + sum1 += xsum + return sum1, xsum + + def potential(lambda1, mu1, nu1, lambda2, mu2, nu2, h2, k2): + x1, y1, z1 = change_coefficient(lambda1, mu1, nu1, h2, k2) + x2, y2, z2 = change_coefficient(lambda2, mu2, nu2, h2, k2) + res = sqrt((x2 - x1)**2 + (y2 - y1)**2 + (z2 - z1)**2) + return 1/res + + pts = [ + (120, sqrt(19), 2, 41, sqrt(17), 2, 15, 25), + (120, sqrt(16), 3.2, 21, sqrt(11), 2.9, 11, 20), + ] + + with suppress_warnings() as sup: + sup.filter(IntegrationWarning, "The occurrence of roundoff error") + sup.filter(IntegrationWarning, "The maximum number of subdivisions") + + for p in pts: + err_msg = repr(p) + exact = potential(*p) + result, last_term = summation(*p) + assert_allclose(exact, result, atol=0, rtol=1e-8, err_msg=err_msg) + assert_(abs(result - exact) < 10*abs(last_term), err_msg) + + +def test_ellip_norm(): + + def G01(h2, k2): + return 4*pi + + def G11(h2, k2): + return 4*pi*h2*k2/3 + + def G12(h2, k2): + return 4*pi*h2*(k2 - h2)/3 + + def G13(h2, k2): + return 4*pi*k2*(k2 - h2)/3 + + def G22(h2, k2): + res = (2*(h2**4 + k2**4) - 4*h2*k2*(h2**2 + k2**2) + 6*h2**2*k2**2 + + sqrt(h2**2 + k2**2 - h2*k2)*(-2*(h2**3 + k2**3) + 3*h2*k2*(h2 + k2))) + return 16*pi/405*res + + def G21(h2, k2): + res = (2*(h2**4 + k2**4) - 4*h2*k2*(h2**2 + k2**2) + 6*h2**2*k2**2 + + sqrt(h2**2 + k2**2 - h2*k2)*(2*(h2**3 + k2**3) - 3*h2*k2*(h2 + k2))) + return 16*pi/405*res + + def G23(h2, k2): + return 4*pi*h2**2*k2*(k2 - h2)/15 + + def G24(h2, k2): + return 4*pi*h2*k2**2*(k2 - h2)/15 + + def G25(h2, k2): + return 4*pi*h2*k2*(k2 - h2)**2/15 + + def G32(h2, k2): + res = (16*(h2**4 + k2**4) - 36*h2*k2*(h2**2 + k2**2) + 46*h2**2*k2**2 + + sqrt(4*(h2**2 + k2**2) - 7*h2*k2)*(-8*(h2**3 + k2**3) + + 11*h2*k2*(h2 + k2))) + return 16*pi/13125*k2*h2*res + + def G31(h2, k2): + res = (16*(h2**4 + k2**4) - 36*h2*k2*(h2**2 + k2**2) + 46*h2**2*k2**2 + + sqrt(4*(h2**2 + k2**2) - 7*h2*k2)*(8*(h2**3 + k2**3) - + 11*h2*k2*(h2 + k2))) + return 16*pi/13125*h2*k2*res + + def G34(h2, k2): + res = (6*h2**4 + 16*k2**4 - 12*h2**3*k2 - 28*h2*k2**3 + 34*h2**2*k2**2 + + sqrt(h2**2 + 4*k2**2 - h2*k2)*(-6*h2**3 - 8*k2**3 + 9*h2**2*k2 + + 13*h2*k2**2)) + return 16*pi/13125*h2*(k2 - h2)*res + + def G33(h2, k2): + res = (6*h2**4 + 16*k2**4 - 12*h2**3*k2 - 28*h2*k2**3 + 34*h2**2*k2**2 + + sqrt(h2**2 + 4*k2**2 - h2*k2)*(6*h2**3 + 8*k2**3 - 9*h2**2*k2 - + 13*h2*k2**2)) + return 16*pi/13125*h2*(k2 - h2)*res + + def G36(h2, k2): + res = (16*h2**4 + 6*k2**4 - 28*h2**3*k2 - 12*h2*k2**3 + 34*h2**2*k2**2 + + sqrt(4*h2**2 + k2**2 - h2*k2)*(-8*h2**3 - 6*k2**3 + 13*h2**2*k2 + + 9*h2*k2**2)) + return 16*pi/13125*k2*(k2 - h2)*res + + def G35(h2, k2): + res = (16*h2**4 + 6*k2**4 - 28*h2**3*k2 - 12*h2*k2**3 + 34*h2**2*k2**2 + + sqrt(4*h2**2 + k2**2 - h2*k2)*(8*h2**3 + 6*k2**3 - 13*h2**2*k2 - + 9*h2*k2**2)) + return 16*pi/13125*k2*(k2 - h2)*res + + def G37(h2, k2): + return 4*pi*h2**2*k2**2*(k2 - h2)**2/105 + + known_funcs = {(0, 1): G01, (1, 1): G11, (1, 2): G12, (1, 3): G13, + (2, 1): G21, (2, 2): G22, (2, 3): G23, (2, 4): G24, + (2, 5): G25, (3, 1): G31, (3, 2): G32, (3, 3): G33, + (3, 4): G34, (3, 5): G35, (3, 6): G36, (3, 7): G37} + + def _ellip_norm(n, p, h2, k2): + func = known_funcs[n, p] + return func(h2, k2) + _ellip_norm = np.vectorize(_ellip_norm) + + def ellip_normal_known(h2, k2, n, p): + return _ellip_norm(n, p, h2, k2) + + # generate both large and small h2 < k2 pairs + np.random.seed(1234) + h2 = np.random.pareto(0.5, size=1) + k2 = h2 * (1 + np.random.pareto(0.5, size=h2.size)) + + points = [] + for n in range(4): + for p in range(1, 2*n+2): + points.append((h2, k2, np.full(h2.size, n), np.full(h2.size, p))) + points = np.array(points) + with suppress_warnings() as sup: + sup.filter(IntegrationWarning, "The occurrence of roundoff error") + assert_func_equal(ellip_normal, ellip_normal_known, points, rtol=1e-12) + + +def test_ellip_harm_2(): + + def I1(h2, k2, s): + res = (ellip_harm_2(h2, k2, 1, 1, s)/(3 * ellip_harm(h2, k2, 1, 1, s)) + + ellip_harm_2(h2, k2, 1, 2, s)/(3 * ellip_harm(h2, k2, 1, 2, s)) + + ellip_harm_2(h2, k2, 1, 3, s)/(3 * ellip_harm(h2, k2, 1, 3, s))) + return res + + with suppress_warnings() as sup: + sup.filter(IntegrationWarning, "The occurrence of roundoff error") + assert_almost_equal(I1(5, 8, 10), 1/(10*sqrt((100-5)*(100-8)))) + + # Values produced by code from arXiv:1204.0267 + assert_almost_equal(ellip_harm_2(5, 8, 2, 1, 10), 0.00108056853382) + assert_almost_equal(ellip_harm_2(5, 8, 2, 2, 10), 0.00105820513809) + assert_almost_equal(ellip_harm_2(5, 8, 2, 3, 10), 0.00106058384743) + assert_almost_equal(ellip_harm_2(5, 8, 2, 4, 10), 0.00106774492306) + assert_almost_equal(ellip_harm_2(5, 8, 2, 5, 10), 0.00107976356454) + + +def test_ellip_harm(): + + def E01(h2, k2, s): + return 1 + + def E11(h2, k2, s): + return s + + def E12(h2, k2, s): + return sqrt(abs(s*s - h2)) + + def E13(h2, k2, s): + return sqrt(abs(s*s - k2)) + + def E21(h2, k2, s): + return s*s - 1/3*((h2 + k2) + sqrt(abs((h2 + k2)*(h2 + k2)-3*h2*k2))) + + def E22(h2, k2, s): + return s*s - 1/3*((h2 + k2) - sqrt(abs((h2 + k2)*(h2 + k2)-3*h2*k2))) + + def E23(h2, k2, s): + return s * sqrt(abs(s*s - h2)) + + def E24(h2, k2, s): + return s * sqrt(abs(s*s - k2)) + + def E25(h2, k2, s): + return sqrt(abs((s*s - h2)*(s*s - k2))) + + def E31(h2, k2, s): + return s*s*s - (s/5)*(2*(h2 + k2) + sqrt(4*(h2 + k2)*(h2 + k2) - + 15*h2*k2)) + + def E32(h2, k2, s): + return s*s*s - (s/5)*(2*(h2 + k2) - sqrt(4*(h2 + k2)*(h2 + k2) - + 15*h2*k2)) + + def E33(h2, k2, s): + return sqrt(abs(s*s - h2))*(s*s - 1/5*((h2 + 2*k2) + sqrt(abs((h2 + + 2*k2)*(h2 + 2*k2) - 5*h2*k2)))) + + def E34(h2, k2, s): + return sqrt(abs(s*s - h2))*(s*s - 1/5*((h2 + 2*k2) - sqrt(abs((h2 + + 2*k2)*(h2 + 2*k2) - 5*h2*k2)))) + + def E35(h2, k2, s): + return sqrt(abs(s*s - k2))*(s*s - 1/5*((2*h2 + k2) + sqrt(abs((2*h2 + + k2)*(2*h2 + k2) - 5*h2*k2)))) + + def E36(h2, k2, s): + return sqrt(abs(s*s - k2))*(s*s - 1/5*((2*h2 + k2) - sqrt(abs((2*h2 + + k2)*(2*h2 + k2) - 5*h2*k2)))) + + def E37(h2, k2, s): + return s * sqrt(abs((s*s - h2)*(s*s - k2))) + + assert_equal(ellip_harm(5, 8, 1, 2, 2.5, 1, 1), + ellip_harm(5, 8, 1, 2, 2.5)) + + known_funcs = {(0, 1): E01, (1, 1): E11, (1, 2): E12, (1, 3): E13, + (2, 1): E21, (2, 2): E22, (2, 3): E23, (2, 4): E24, + (2, 5): E25, (3, 1): E31, (3, 2): E32, (3, 3): E33, + (3, 4): E34, (3, 5): E35, (3, 6): E36, (3, 7): E37} + + point_ref = [] + + def ellip_harm_known(h2, k2, n, p, s): + for i in range(h2.size): + func = known_funcs[(int(n[i]), int(p[i]))] + point_ref.append(func(h2[i], k2[i], s[i])) + return point_ref + + rng = np.random.RandomState(1234) + h2 = rng.pareto(0.5, size=30) + k2 = h2*(1 + rng.pareto(0.5, size=h2.size)) + s = rng.pareto(0.5, size=h2.size) + points = [] + for i in range(h2.size): + for n in range(4): + for p in range(1, 2*n+2): + points.append((h2[i], k2[i], n, p, s[i])) + points = np.array(points) + assert_func_equal(ellip_harm, ellip_harm_known, points, rtol=1e-12) + + +def test_ellip_harm_invalid_p(): + # Regression test. This should return nan. + n = 4 + # Make p > 2*n + 1. + p = 2*n + 2 + result = ellip_harm(0.5, 2.0, n, p, 0.2) + assert np.isnan(result) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_erfinv.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_erfinv.py new file mode 100644 index 0000000000000000000000000000000000000000..98739b93fc6ad75a41a7b80107ee696453b12a09 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_erfinv.py @@ -0,0 +1,89 @@ +import numpy as np +from numpy.testing import assert_allclose, assert_equal +import pytest + +import scipy.special as sc + + +class TestInverseErrorFunction: + def test_compliment(self): + # Test erfcinv(1 - x) == erfinv(x) + x = np.linspace(-1, 1, 101) + assert_allclose(sc.erfcinv(1 - x), sc.erfinv(x), rtol=0, atol=1e-15) + + def test_literal_values(self): + # The expected values were calculated with mpmath: + # + # import mpmath + # mpmath.mp.dps = 200 + # for y in [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]: + # x = mpmath.erfinv(y) + # print(x) + # + y = np.array([0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]) + actual = sc.erfinv(y) + expected = [ + 0.0, + 0.08885599049425769, + 0.1791434546212917, + 0.2724627147267543, + 0.37080715859355795, + 0.4769362762044699, + 0.5951160814499948, + 0.7328690779592167, + 0.9061938024368233, + 1.1630871536766743, + ] + assert_allclose(actual, expected, rtol=0, atol=1e-15) + + @pytest.mark.parametrize( + 'f, x, y', + [ + (sc.erfinv, -1, -np.inf), + (sc.erfinv, 0, 0), + (sc.erfinv, 1, np.inf), + (sc.erfinv, -100, np.nan), + (sc.erfinv, 100, np.nan), + (sc.erfcinv, 0, np.inf), + (sc.erfcinv, 1, -0.0), + (sc.erfcinv, 2, -np.inf), + (sc.erfcinv, -100, np.nan), + (sc.erfcinv, 100, np.nan), + ], + ids=[ + 'erfinv at lower bound', + 'erfinv at midpoint', + 'erfinv at upper bound', + 'erfinv below lower bound', + 'erfinv above upper bound', + 'erfcinv at lower bound', + 'erfcinv at midpoint', + 'erfcinv at upper bound', + 'erfcinv below lower bound', + 'erfcinv above upper bound', + ] + ) + def test_domain_bounds(self, f, x, y): + assert_equal(f(x), y) + + def test_erfinv_asympt(self): + # regression test for gh-12758: erfinv(x) loses precision at small x + # expected values precomputed with mpmath: + # >>> mpmath.mp.dps = 100 + # >>> expected = [float(mpmath.erfinv(t)) for t in x] + x = np.array([1e-20, 1e-15, 1e-14, 1e-10, 1e-8, 0.9e-7, 1.1e-7, 1e-6]) + expected = np.array([8.86226925452758e-21, + 8.862269254527581e-16, + 8.86226925452758e-15, + 8.862269254527581e-11, + 8.86226925452758e-09, + 7.97604232907484e-08, + 9.74849617998037e-08, + 8.8622692545299e-07]) + assert_allclose(sc.erfinv(x), expected, + rtol=1e-15) + + # also test the roundtrip consistency + assert_allclose(sc.erf(sc.erfinv(x)), + x, + rtol=5e-15) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_exponential_integrals.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_exponential_integrals.py new file mode 100644 index 0000000000000000000000000000000000000000..8332a83267e2f75dded04e80443c150c832676c8 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_exponential_integrals.py @@ -0,0 +1,118 @@ +import pytest + +import numpy as np +from numpy.testing import assert_allclose +import scipy.special as sc + + +class TestExp1: + + def test_branch_cut(self): + assert np.isnan(sc.exp1(-1)) + assert sc.exp1(complex(-1, 0)).imag == ( + -sc.exp1(complex(-1, -0.0)).imag + ) + + assert_allclose( + sc.exp1(complex(-1, 0)), + sc.exp1(-1 + 1e-20j), + atol=0, + rtol=1e-15 + ) + assert_allclose( + sc.exp1(complex(-1, -0.0)), + sc.exp1(-1 - 1e-20j), + atol=0, + rtol=1e-15 + ) + + def test_834(self): + # Regression test for #834 + a = sc.exp1(-complex(19.9999990)) + b = sc.exp1(-complex(19.9999991)) + assert_allclose(a.imag, b.imag, atol=0, rtol=1e-15) + + +class TestScaledExp1: + + @pytest.mark.parametrize('x, expected', [(0, 0), (np.inf, 1)]) + def test_limits(self, x, expected): + y = sc._ufuncs._scaled_exp1(x) + assert y == expected + + # The expected values were computed with mpmath, e.g.: + # + # from mpmath import mp + # mp.dps = 80 + # x = 1e-25 + # print(float(x*mp.exp(x)*np.expint(1, x))) + # + # prints 5.698741165994961e-24 + # + # The method used to compute _scaled_exp1 changes at x=1 + # and x=1250, so values at those inputs, and values just + # above and below them, are included in the test data. + @pytest.mark.parametrize('x, expected', + [(1e-25, 5.698741165994961e-24), + (0.1, 0.20146425447084518), + (0.9995, 0.5962509885831002), + (1.0, 0.5963473623231941), + (1.0005, 0.5964436833238044), + (2.5, 0.7588145912149602), + (10.0, 0.9156333393978808), + (100.0, 0.9901942286733019), + (500.0, 0.9980079523802055), + (1000.0, 0.9990019940238807), + (1249.5, 0.9992009578306811), + (1250.0, 0.9992012769377913), + (1250.25, 0.9992014363957858), + (2000.0, 0.9995004992514963), + (1e4, 0.9999000199940024), + (1e10, 0.9999999999), + (1e15, 0.999999999999999), + ]) + def test_scaled_exp1(self, x, expected): + y = sc._ufuncs._scaled_exp1(x) + assert_allclose(y, expected, rtol=2e-15) + + +class TestExpi: + + @pytest.mark.parametrize('result', [ + sc.expi(complex(-1, 0)), + sc.expi(complex(-1, -0.0)), + sc.expi(-1) + ]) + def test_branch_cut(self, result): + desired = -0.21938393439552027368 # Computed using Mpmath + assert_allclose(result, desired, atol=0, rtol=1e-14) + + def test_near_branch_cut(self): + lim_from_above = sc.expi(-1 + 1e-20j) + lim_from_below = sc.expi(-1 - 1e-20j) + assert_allclose( + lim_from_above.real, + lim_from_below.real, + atol=0, + rtol=1e-15 + ) + assert_allclose( + lim_from_above.imag, + -lim_from_below.imag, + atol=0, + rtol=1e-15 + ) + + def test_continuity_on_positive_real_axis(self): + assert_allclose( + sc.expi(complex(1, 0)), + sc.expi(complex(1, -0.0)), + atol=0, + rtol=1e-15 + ) + + +class TestExpn: + + def test_out_of_domain(self): + assert all(np.isnan([sc.expn(-1, 1.0), sc.expn(1, -1.0)])) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_extending.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_extending.py new file mode 100644 index 0000000000000000000000000000000000000000..3ecaf51545e006e37a451a08f356ce0392a3159c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_extending.py @@ -0,0 +1,28 @@ +import os +import platform +import sysconfig + +import pytest + +from scipy._lib._testutils import IS_EDITABLE,_test_cython_extension, cython +from scipy.special import beta, gamma + + +@pytest.mark.fail_slow(40) +# essential per https://github.com/scipy/scipy/pull/20487#discussion_r1567057247 +@pytest.mark.skipif(IS_EDITABLE, + reason='Editable install cannot find .pxd headers.') +@pytest.mark.skipif((platform.system() == 'Windows' and + sysconfig.get_config_var('Py_GIL_DISABLED')), + reason='gh-22039') +@pytest.mark.skipif(platform.machine() in ["wasm32", "wasm64"], + reason="Can't start subprocess") +@pytest.mark.skipif(cython is None, reason="requires cython") +def test_cython(tmp_path): + srcdir = os.path.dirname(os.path.dirname(__file__)) + extensions, extensions_cpp = _test_cython_extension(tmp_path, srcdir) + # actually test the cython c-extensions + assert extensions.cy_beta(0.5, 0.1) == beta(0.5, 0.1) + assert extensions.cy_gamma(0.5 + 1.0j) == gamma(0.5 + 1.0j) + assert extensions_cpp.cy_beta(0.5, 0.1) == beta(0.5, 0.1) + assert extensions_cpp.cy_gamma(0.5 + 1.0j) == gamma(0.5 + 1.0j) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_faddeeva.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_faddeeva.py new file mode 100644 index 0000000000000000000000000000000000000000..8868f66c47ce0d4bbb21c78435a6c89d44065252 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_faddeeva.py @@ -0,0 +1,85 @@ +import pytest + +import numpy as np +from numpy.testing import assert_allclose +import scipy.special as sc +from scipy.special._testutils import FuncData + + +class TestVoigtProfile: + + @pytest.mark.parametrize('x, sigma, gamma', [ + (np.nan, 1, 1), + (0, np.nan, 1), + (0, 1, np.nan), + (1, np.nan, 0), + (np.nan, 1, 0), + (1, 0, np.nan), + (np.nan, 0, 1), + (np.nan, 0, 0) + ]) + def test_nan(self, x, sigma, gamma): + assert np.isnan(sc.voigt_profile(x, sigma, gamma)) + + @pytest.mark.parametrize('x, desired', [ + (-np.inf, 0), + (np.inf, 0) + ]) + def test_inf(self, x, desired): + assert sc.voigt_profile(x, 1, 1) == desired + + def test_against_mathematica(self): + # Results obtained from Mathematica by computing + # + # PDF[VoigtDistribution[gamma, sigma], x] + # + points = np.array([ + [-7.89, 45.06, 6.66, 0.0077921073660388806401], + [-0.05, 7.98, 24.13, 0.012068223646769913478], + [-13.98, 16.83, 42.37, 0.0062442236362132357833], + [-12.66, 0.21, 6.32, 0.010052516161087379402], + [11.34, 4.25, 21.96, 0.0113698923627278917805], + [-11.56, 20.40, 30.53, 0.0076332760432097464987], + [-9.17, 25.61, 8.32, 0.011646345779083005429], + [16.59, 18.05, 2.50, 0.013637768837526809181], + [9.11, 2.12, 39.33, 0.0076644040807277677585], + [-43.33, 0.30, 45.68, 0.0036680463875330150996] + ]) + FuncData( + sc.voigt_profile, + points, + (0, 1, 2), + 3, + atol=0, + rtol=1e-15 + ).check() + + def test_symmetry(self): + x = np.linspace(0, 10, 20) + assert_allclose( + sc.voigt_profile(x, 1, 1), + sc.voigt_profile(-x, 1, 1), + rtol=1e-15, + atol=0 + ) + + @pytest.mark.parametrize('x, sigma, gamma, desired', [ + (0, 0, 0, np.inf), + (1, 0, 0, 0) + ]) + def test_corner_cases(self, x, sigma, gamma, desired): + assert sc.voigt_profile(x, sigma, gamma) == desired + + @pytest.mark.parametrize('sigma1, gamma1, sigma2, gamma2', [ + (0, 1, 1e-16, 1), + (1, 0, 1, 1e-16), + (0, 0, 1e-16, 1e-16) + ]) + def test_continuity(self, sigma1, gamma1, sigma2, gamma2): + x = np.linspace(1, 10, 20) + assert_allclose( + sc.voigt_profile(x, sigma1, gamma1), + sc.voigt_profile(x, sigma2, gamma2), + rtol=1e-16, + atol=1e-16 + ) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_gamma.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_gamma.py new file mode 100644 index 0000000000000000000000000000000000000000..2e3fbd17dddeed73d311566a930f52899e3b9db6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_gamma.py @@ -0,0 +1,12 @@ +import numpy as np +import scipy.special as sc + + +class TestRgamma: + + def test_gh_11315(self): + assert sc.rgamma(-35) == 0 + + def test_rgamma_zeros(self): + x = np.array([0, -10, -100, -1000, -10000]) + assert np.all(sc.rgamma(x) == 0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_gammainc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_gammainc.py new file mode 100644 index 0000000000000000000000000000000000000000..aae34e5c23f2d293f362abd825f1dad454371ae0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_gammainc.py @@ -0,0 +1,136 @@ +import pytest + +import numpy as np +from numpy.testing import assert_allclose, assert_array_equal + +import scipy.special as sc +from scipy.special._testutils import FuncData + + +INVALID_POINTS = [ + (1, -1), + (0, 0), + (-1, 1), + (np.nan, 1), + (1, np.nan) +] + + +class TestGammainc: + + @pytest.mark.parametrize('a, x', INVALID_POINTS) + def test_domain(self, a, x): + assert np.isnan(sc.gammainc(a, x)) + + def test_a_eq_0_x_gt_0(self): + assert sc.gammainc(0, 1) == 1 + + @pytest.mark.parametrize('a, x, desired', [ + (np.inf, 1, 0), + (np.inf, 0, 0), + (np.inf, np.inf, np.nan), + (1, np.inf, 1) + ]) + def test_infinite_arguments(self, a, x, desired): + result = sc.gammainc(a, x) + if np.isnan(desired): + assert np.isnan(result) + else: + assert result == desired + + def test_infinite_limits(self): + # Test that large arguments converge to the hard-coded limits + # at infinity. + assert_allclose( + sc.gammainc(1000, 100), + sc.gammainc(np.inf, 100), + atol=1e-200, # Use `atol` since the function converges to 0. + rtol=0 + ) + assert sc.gammainc(100, 1000) == sc.gammainc(100, np.inf) + + def test_x_zero(self): + a = np.arange(1, 10) + assert_array_equal(sc.gammainc(a, 0), 0) + + def test_limit_check(self): + result = sc.gammainc(1e-10, 1) + limit = sc.gammainc(0, 1) + assert np.isclose(result, limit) + + def gammainc_line(self, x): + # The line a = x where a simpler asymptotic expansion (analog + # of DLMF 8.12.15) is available. + c = np.array([-1/3, -1/540, 25/6048, 101/155520, + -3184811/3695155200, -2745493/8151736420]) + res = 0 + xfac = 1 + for ck in c: + res -= ck*xfac + xfac /= x + res /= np.sqrt(2*np.pi*x) + res += 0.5 + return res + + def test_line(self): + x = np.logspace(np.log10(25), 300, 500) + a = x + dataset = np.vstack((a, x, self.gammainc_line(x))).T + FuncData(sc.gammainc, dataset, (0, 1), 2, rtol=1e-11).check() + + def test_roundtrip(self): + a = np.logspace(-5, 10, 100) + x = np.logspace(-5, 10, 100) + + y = sc.gammaincinv(a, sc.gammainc(a, x)) + assert_allclose(x, y, rtol=1e-10) + + +class TestGammaincc: + + @pytest.mark.parametrize('a, x', INVALID_POINTS) + def test_domain(self, a, x): + assert np.isnan(sc.gammaincc(a, x)) + + def test_a_eq_0_x_gt_0(self): + assert sc.gammaincc(0, 1) == 0 + + @pytest.mark.parametrize('a, x, desired', [ + (np.inf, 1, 1), + (np.inf, 0, 1), + (np.inf, np.inf, np.nan), + (1, np.inf, 0) + ]) + def test_infinite_arguments(self, a, x, desired): + result = sc.gammaincc(a, x) + if np.isnan(desired): + assert np.isnan(result) + else: + assert result == desired + + def test_infinite_limits(self): + # Test that large arguments converge to the hard-coded limits + # at infinity. + assert sc.gammaincc(1000, 100) == sc.gammaincc(np.inf, 100) + assert_allclose( + sc.gammaincc(100, 1000), + sc.gammaincc(100, np.inf), + atol=1e-200, # Use `atol` since the function converges to 0. + rtol=0 + ) + + def test_limit_check(self): + result = sc.gammaincc(1e-10,1) + limit = sc.gammaincc(0,1) + assert np.isclose(result, limit) + + def test_x_zero(self): + a = np.arange(1, 10) + assert_array_equal(sc.gammaincc(a, 0), 1) + + def test_roundtrip(self): + a = np.logspace(-5, 10, 100) + x = np.logspace(-5, 10, 100) + + y = sc.gammainccinv(a, sc.gammaincc(a, x)) + assert_allclose(x, y, rtol=1e-14) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_hyp2f1.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_hyp2f1.py new file mode 100644 index 0000000000000000000000000000000000000000..2102152213997886d0127080a2767e8fcc9af63a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_hyp2f1.py @@ -0,0 +1,2566 @@ +"""Tests for hyp2f1 for complex values. + +Author: Albert Steppi, with credit to Adam Kullberg (FormerPhycisist) for +the implementation of mp_hyp2f1 below, which modifies mpmath's hyp2f1 to +return the same branch as scipy's on the standard branch cut. +""" + +import sys +import pytest +import numpy as np +from typing import NamedTuple +from numpy.testing import assert_allclose + +from scipy.special import hyp2f1 +from scipy.special._testutils import check_version, MissingModule + + +try: + import mpmath +except ImportError: + mpmath = MissingModule("mpmath") + + +def mp_hyp2f1(a, b, c, z): + """Return mpmath hyp2f1 calculated on same branch as scipy hyp2f1. + + For most values of a,b,c mpmath returns the x - 0j branch of hyp2f1 on the + branch cut x=(1,inf) whereas scipy's hyp2f1 calculates the x + 0j branch. + Thus, to generate the right comparison values on the branch cut, we + evaluate mpmath.hyp2f1 at x + 1e-15*j. + + The exception to this occurs when c-a=-m in which case both mpmath and + scipy calculate the x + 0j branch on the branch cut. When this happens + mpmath.hyp2f1 will be evaluated at the original z point. + """ + on_branch_cut = z.real > 1.0 and abs(z.imag) < 1.0e-15 + cond1 = abs(c - a - round(c - a)) < 1.0e-15 and round(c - a) <= 0 + cond2 = abs(c - b - round(c - b)) < 1.0e-15 and round(c - b) <= 0 + # Make sure imaginary part is *exactly* zero + if on_branch_cut: + z = z.real + 0.0j + if on_branch_cut and not (cond1 or cond2): + z_mpmath = z.real + 1.0e-15j + else: + z_mpmath = z + return complex(mpmath.hyp2f1(a, b, c, z_mpmath)) + + +class Hyp2f1TestCase(NamedTuple): + a: float + b: float + c: float + z: complex + expected: complex + rtol: float + + +class TestHyp2f1: + """Tests for hyp2f1 for complex values. + + Expected values for test cases were computed using mpmath. See + `scipy.special._precompute.hyp2f1_data`. The verbose style of specifying + test cases is used for readability and to make it easier to mark individual + cases as expected to fail. Expected failures are used to highlight cases + where improvements are needed. See + `scipy.special._precompute.hyp2f1_data.make_hyp2f1_test_cases` for a + function to generate the boilerplate for the test cases. + + Assertions have been added to each test to ensure that the test cases match + the situations that are intended. A final test `test_test_hyp2f1` checks + that the expected values in the test cases actually match what is computed + by mpmath. This test is marked slow even though it isn't particularly slow + so that it won't run by default on continuous integration builds. + """ + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=0.5, + b=0.2, + c=-10, + z=0.2 + 0.2j, + expected=np.inf + 0j, + rtol=0 + ) + ), + pytest.param( + Hyp2f1TestCase( + a=0.5, + b=0.2, + c=-10, + z=0 + 0j, + expected=1 + 0j, + rtol=0 + ), + ), + pytest.param( + Hyp2f1TestCase( + a=0.5, + b=0, + c=-10, + z=0.2 + 0.2j, + expected=1 + 0j, + rtol=0 + ), + ), + pytest.param( + Hyp2f1TestCase( + a=0.5, + b=0, + c=0, + z=0.2 + 0.2j, + expected=1 + 0j, + rtol=0, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=0.5, + b=0.2, + c=0, + z=0.2 + 0.2j, + expected=np.inf + 0j, + rtol=0, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=0.5, + b=0.2, + c=0, + z=0 + 0j, + expected=np.nan + 0j, + rtol=0, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=0.5, + b=-5, + c=-10, + z=0.2 + 0.2j, + expected=(1.0495404166666666+0.05708208333333334j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=0.5, + b=-10, + c=-10, + z=0.2 + 0.2j, + expected=(1.092966013125+0.13455014673750001j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-10, + b=-20, + c=-10, + z=0.2 + 0.2j, + expected=(-0.07712512000000005+0.12752814080000005j), + rtol=1e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1, + b=3.2, + c=-1, + z=0.2 + 0.2j, + expected=(1.6400000000000001+0.6400000000000001j), + rtol=1e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-2, + b=1.2, + c=-4, + z=1 + 0j, + expected=1.8200000000000001 + 0j, + rtol=1e-15, + ), + ), + ] + ) + def test_c_non_positive_int(self, hyp2f1_test_case): + a, b, c, z, expected, rtol = hyp2f1_test_case + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=0.5, + b=0.2, + c=1.5, + z=1 + 0j, + expected=1.1496439092239847 + 0j, + rtol=1e-15 + ), + ), + pytest.param( + Hyp2f1TestCase( + a=12.3, + b=8.0, + c=20.31, + z=1 + 0j, + expected=69280986.75273195 + 0j, + rtol=1e-15 + ), + ), + pytest.param( + Hyp2f1TestCase( + a=290.2, + b=321.5, + c=700.1, + z=1 + 0j, + expected=1.3396562400934e117 + 0j, + rtol=1e-12, + ), + ), + # Note that here even mpmath produces different results for + # results that should be equivalent. + pytest.param( + Hyp2f1TestCase( + a=9.2, + b=621.5, + c=700.1, + z=(1+0j), + expected=(952726652.4158565+0j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=621.5, + b=9.2, + c=700.1, + z=(1+0j), + expected=(952726652.4160284+0j), + rtol=5e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-101.2, + b=-400.4, + c=-172.1, + z=(1+0j), + expected=(2.2253618341394838e+37+0j), + rtol=1e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-400.4, + b=-101.2, + c=-172.1, + z=(1+0j), + expected=(2.2253618341394838e+37+0j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=172.5, + b=-201.3, + c=151.2, + z=(1+0j), + expected=(7.072266653650905e-135+0j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-201.3, + b=172.5, + c=151.2, + z=(1+0j), + expected=(7.072266653650905e-135+0j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-102.1, + b=-20.3, + c=1.3, + z=1 + 0j, + expected=2.7899070752746906e22 + 0j, + rtol=3e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-202.6, + b=60.3, + c=1.5, + z=1 + 0j, + expected=-1.3113641413099326e-56 + 0j, + rtol=1e-12, + ), + ), + ], + ) + def test_unital_argument(self, hyp2f1_test_case): + """Tests for case z = 1, c - a - b > 0. + + Expected answers computed using mpmath. + """ + a, b, c, z, expected, rtol = hyp2f1_test_case + assert z == 1 and c - a - b > 0 # Tests the test + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=0.5, + b=0.2, + c=1.3, + z=-1 + 0j, + expected=0.9428846409614143 + 0j, + rtol=1e-15), + ), + pytest.param( + Hyp2f1TestCase( + a=12.3, + b=8.0, + c=5.300000000000001, + z=-1 + 0j, + expected=-4.845809986595704e-06 + 0j, + rtol=1e-15 + ), + ), + pytest.param( + Hyp2f1TestCase( + a=221.5, + b=90.2, + c=132.3, + z=-1 + 0j, + expected=2.0490488728377282e-42 + 0j, + rtol=1e-7, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-102.1, + b=-20.3, + c=-80.8, + z=-1 + 0j, + expected=45143784.46783885 + 0j, + rtol=1e-7, + ), + marks=pytest.mark.xfail( + condition=sys.maxsize < 2**32, + reason="Fails on 32 bit.", + ) + ), + ], + ) + def test_special_case_z_near_minus_1(self, hyp2f1_test_case): + """Tests for case z ~ -1, c ~ 1 + a - b + + Expected answers computed using mpmath. + """ + a, b, c, z, expected, rtol = hyp2f1_test_case + assert abs(1 + a - b - c) < 1e-15 and abs(z + 1) < 1e-15 + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=-4, + b=2.02764642551431, + c=1.0561196186065624, + z=(0.9473684210526314-0.10526315789473695j), + expected=(0.0031961077109535375-0.0011313924606557173j), + rtol=1e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-8, + b=-7.937789122896016, + c=-15.964218273004214, + z=(2-0.10526315789473695j), + expected=(0.005543763196412503-0.0025948879065698306j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-8, + b=8.095813935368371, + c=4.0013768449590685, + z=(0.9473684210526314-0.10526315789473695j), + expected=(-0.0003054674127221263-9.261359291755414e-05j), + rtol=1e-10, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-4, + b=-3.956227226099288, + c=-3.9316537064827854, + z=(1.1578947368421053-0.3157894736842106j), + expected=(-0.0020809502580892937-0.0041877333232365095j), + rtol=5e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=-4, + c=2.050308316530781, + z=(0.9473684210526314-0.10526315789473695j), + expected=(0.0011282435590058734+0.0002027062303465851j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=-8, + c=-15.964218273004214, + z=(1.3684210526315788+0.10526315789473673j), + expected=(-9.134907719238265e-05-0.00040219233987390723j), + rtol=5e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=-4, + c=4.0013768449590685, + z=(0.9473684210526314-0.10526315789473695j), + expected=(-0.000519013062087489-0.0005855883076830948j), + rtol=5e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-10000, + b=2.2, + c=93459345.3, + z=(2+2j), + expected=(0.9995292071559088-0.00047047067522659253j), + rtol=1e-12, + ), + ), + ] + ) + def test_a_b_negative_int(self, hyp2f1_test_case): + a, b, c, z, expected, rtol = hyp2f1_test_case + assert a == int(a) and a < 0 or b == int(b) and b < 0 # Tests the test + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=-0.5, + b=-0.9629749245209605, + c=-15.5, + z=(1.1578947368421053-1.1578947368421053j), + expected=(0.9778506962676361+0.044083801141231616j), + rtol=3e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.5, + b=-3.9316537064827854, + c=1.5, + z=(0.9473684210526314-0.10526315789473695j), + expected=(4.0793167523167675-10.11694246310966j), + rtol=6e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.5, + b=-0.9629749245209605, + c=2.5, + z=(1.1578947368421053-0.10526315789473695j), + expected=(-2.9692999501916915+0.6394599899845594j), + rtol=1e-11, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.5, + b=-0.9629749245209605, + c=-15.5, + z=(1.5789473684210522-1.1578947368421053j), + expected=(0.9493076367106102-0.04316852977183447j), + rtol=1e-11, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=-0.5, + c=-15.5, + z=(0.5263157894736841+0.10526315789473673j), + expected=(0.9844377175631795-0.003120587561483841j), + rtol=1e-10, + ), + ), + ], + ) + def test_a_b_neg_int_after_euler_hypergeometric_transformation( + self, hyp2f1_test_case + ): + a, b, c, z, expected, rtol = hyp2f1_test_case + assert ( # Tests the test + (abs(c - a - int(c - a)) < 1e-15 and c - a < 0) or + (abs(c - b - int(c - b)) < 1e-15 and c - b < 0) + ) + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=-0.9629749245209605, + c=-15.963511401609862, + z=(0.10526315789473673-0.3157894736842106j), + expected=(0.9941449585778349+0.01756335047931358j), + rtol=1e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=-0.9629749245209605, + c=-15.963511401609862, + z=(0.5263157894736841+0.5263157894736841j), + expected=(1.0388722293372104-0.09549450380041416j), + rtol=5e-11, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=1.0561196186065624, + c=-7.93846038215665, + z=(0.10526315789473673+0.7368421052631575j), + expected=(2.1948378809826434+24.934157235172222j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=16.088264119063613, + c=8.031683612216888, + z=(0.3157894736842106-0.736842105263158j), + expected=(-0.4075277891264672-0.06819344579666956j), + rtol=2e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=2.050308316530781, + c=8.031683612216888, + z=(0.7368421052631575-0.10526315789473695j), + expected=(2.833535530740603-0.6925373701408158j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=2.050308316530781, + c=4.078873014294075, + z=(0.10526315789473673-0.3157894736842106j), + expected=(1.005347176329683-0.3580736009337313j), + rtol=5e-16, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=-0.9629749245209605, + c=-15.963511401609862, + z=(0.3157894736842106-0.5263157894736843j), + expected=(0.9824353641135369+0.029271018868990268j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=-0.9629749245209605, + c=-159.63511401609862, + z=(0.3157894736842106-0.5263157894736843j), + expected=(0.9982436200365834+0.002927268199671111j), + rtol=1e-7, + ), + marks=pytest.mark.xfail(reason="Poor convergence.") + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=16.088264119063613, + c=8.031683612216888, + z=(0.5263157894736841-0.5263157894736843j), + expected=(-0.6906825165778091+0.8176575137504892j), + rtol=5e-13, + ), + ), + ] + ) + def test_region1(self, hyp2f1_test_case): + """|z| < 0.9 and real(z) >= 0.""" + a, b, c, z, expected, rtol = hyp2f1_test_case + assert abs(z) < 0.9 and z.real >= 0 # Tests the test + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=1.0561196186065624, + c=4.078873014294075, + z=(-0.3157894736842106+0.7368421052631575j), + expected=(0.7751915029081136+0.24068493258607315j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.087593263474208, + b=16.088264119063613, + c=2.0397202577726152, + z=(-0.9473684210526316-0.3157894736842106j), + expected=(6.564549348474962e-07+1.6761570598334562e-06j), + rtol=5e-09, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=2.050308316530781, + c=16.056809865262608, + z=(-0.10526315789473695-0.10526315789473695j), + expected=(0.9862043298997204-0.013293151372712681j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=8.077282662161238, + c=16.056809865262608, + z=(-0.3157894736842106-0.736842105263158j), + expected=(0.16163826638754716-0.41378530376373734j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=2.050308316530781, + c=-0.906685989801748, + z=(-0.5263157894736843+0.3157894736842106j), + expected=(-6.256871535165936+0.13824973858225484j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=8.077282662161238, + c=-3.9924618758357022, + z=(-0.9473684210526316-0.3157894736842106j), + expected=(75.54672526086316+50.56157041797548j), + rtol=5e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.087593263474208, + b=8.077282662161238, + c=-1.9631175993998025, + z=(-0.5263157894736843+0.5263157894736841j), + expected=(282.0602536306534-82.31597306936214j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.095813935368371, + b=-3.9316537064827854, + c=8.031683612216888, + z=(-0.5263157894736843-0.10526315789473695j), + expected=(5.179603735575851+1.4445374002099813j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=-7.949900487447654, + c=1.0651378143226575, + z=(-0.3157894736842106-0.9473684210526316j), + expected=(2317.623517606141-269.51476321010324j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.087593263474208, + b=-1.92872979730171, + c=2.0397202577726152, + z=(-0.736842105263158-0.3157894736842106j), + expected=(29.179154096175836+22.126690357535043j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.095813935368371, + b=-3.9316537064827854, + c=-15.963511401609862, + z=(-0.736842105263158-0.10526315789473695j), + expected=(0.20820247892032057-0.04763956711248794j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=-15.964218273004214, + c=-1.9631175993998025, + z=(-0.3157894736842106-0.5263157894736843j), + expected=(-157471.63920142158+991294.0587828817j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.095813935368371, + b=-7.949900487447654, + c=-7.93846038215665, + z=(-0.10526315789473695-0.10526315789473695j), + expected=(0.30765349653210194-0.2979706363594157j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.956227226099288, + b=1.0561196186065624, + c=8.031683612216888, + z=(-0.9473684210526316-0.10526315789473695j), + expected=(1.6787607400597109+0.10056620134616838j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=16.088264119063613, + c=4.078873014294075, + z=(-0.5263157894736843-0.736842105263158j), + expected=(7062.07842506049-12768.77955655703j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=16.088264119063613, + c=2.0397202577726152, + z=(-0.3157894736842106+0.7368421052631575j), + expected=(54749.216391029935-23078.144720887536j), + rtol=2e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.956227226099288, + b=1.0561196186065624, + c=-0.906685989801748, + z=(-0.10526315789473695-0.10526315789473695j), + expected=(1.21521766411428-4.449385173946672j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.980848054962111, + b=4.0013768449590685, + c=-1.9631175993998025, + z=(-0.736842105263158+0.5263157894736841j), + expected=(19234693144.196907+1617913967.7294445j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.9214641416286231, + b=1.0561196186065624, + c=-15.963511401609862, + z=(-0.5263157894736843+0.3157894736842106j), + expected=(0.9345201094534371+0.03745712558992195j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=-0.9629749245209605, + c=2.0397202577726152, + z=(-0.10526315789473695+0.10526315789473673j), + expected=(0.605732446296829+0.398171533680972j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.9214641416286231, + b=-15.964218273004214, + c=2.0397202577726152, + z=(-0.10526315789473695-0.5263157894736843j), + expected=(-9.753761888305416-4.590126012666959j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.956227226099288, + b=-1.92872979730171, + c=2.0397202577726152, + z=(-0.10526315789473695+0.3157894736842106j), + expected=(0.45587226291120714+1.0694545265819797j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=-7.949900487447654, + c=-0.906685989801748, + z=(-0.736842105263158+0.3157894736842106j), + expected=(12.334808243233418-76.26089051819054j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=-7.949900487447654, + c=-15.963511401609862, + z=(-0.5263157894736843+0.10526315789473673j), + expected=(1.2396019687632678-0.047507973161146286j), + rtol=1e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.980848054962111, + b=-0.9629749245209605, + c=-0.906685989801748, + z=(-0.3157894736842106-0.5263157894736843j), + expected=(97.7889554372208-18.999754543400016j), + rtol=5e-13, + ), + ), + ] + ) + def test_region2(self, hyp2f1_test_case): + """|z| < 1 and real(z) < 0.""" + a, b, c, z, expected, rtol = hyp2f1_test_case + assert abs(z) < 1 and z.real < 0 # Tests the test + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=16.25, + b=-3.75, + c=-3.5, + z=(0.5263157894736841+0.7368421052631575j), + expected=(-1279.4894322256655-2302.914821389276j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.75, + b=8.25, + c=-1.5, + z=(0.9473684210526314+0.3157894736842106j), + expected=(-8889.452798586273-11961.162305065242j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.25, + b=2.25, + c=-1.5, + z=(0.5263157894736841-0.736842105263158j), + expected=(-236.58971357952055-238.5228224781136j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.75, + b=-7.75, + c=-15.5, + z=(0.5263157894736841+0.7368421052631575j), + expected=(0.8116076584352279-0.29360565398246036j), + rtol=5e-16, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.25, + b=4.25, + c=-0.5, + z=(0.5263157894736841-0.736842105263158j), + expected=(-28.119407485189985+98.89858821348005j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.75, + b=2.25, + c=1.5, + z=(0.5263157894736841+0.7368421052631575j), + expected=(0.5311049067450484-0.9434347326448517j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.25, + b=-15.75, + c=-7.5, + z=(0.9473684210526314+0.10526315789473673j), + expected=(1262084.378141873+1775569.6338380123j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.75, + b=-7.75, + c=-15.5, + z=(0.5263157894736841-0.736842105263158j), + expected=(-0.009810480794804165+0.3648997569257999j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.25, + b=2.25, + c=-3.5, + z=(0.5263157894736841-0.736842105263158j), + expected=(585660.8815535795-33646.68398590896j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=-3.9316537064827854, + c=-15.963511401609862, + z=(0.9473684210526314-0.10526315789473695j), + expected=(181899621848365.2-173207123998705.7j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.75, + b=8.25, + c=-0.5, + z=(0.9473684210526314-0.10526315789473695j), + expected=(0.04271686244952705-0.14087902824639406j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=-1.92872979730171, + c=-0.906685989801748, + z=(0.9473684210526314-0.3157894736842106j), + expected=(-449.5119088817207+320.1423128036188j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.956227226099288, + b=1.0561196186065624, + c=8.031683612216888, + z=(0.9473684210526314-0.10526315789473695j), + expected=(0.6361479738012501+0.028575620091205088j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.25, + b=16.25, + c=16.5, + z=(0.5263157894736841+0.7368421052631575j), + expected=(-0.9038811840552261-1.5356250756164884j), + rtol=1e-8, + ), + marks=pytest.mark.xfail( + reason="Unhandled parameters." + ) + ), + pytest.param( + Hyp2f1TestCase( + a=8.25, + b=-1.75, + c=-1.5, + z=(0.9473684210526314+0.3157894736842106j), + expected=(653.0109150415394-4554.162605155542j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.75, + b=-3.75, + c=4.5, + z=(0.9473684210526314-0.10526315789473695j), + expected=(118.7009859241035-34.18713648654642j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.25, + b=-15.75, + c=-3.5, + z=(0.5263157894736841+0.7368421052631575j), + expected=(-540204.4774526551+4970059.109251281j), + rtol=1e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.25, + b=-15.75, + c=-0.5, + z=(0.5263157894736841-0.736842105263158j), + expected=(2253490.972258385+3318620.683390017j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.25, + b=-7.75, + c=-7.5, + z=(0.9473684210526314+0.3157894736842106j), + expected=(-46159826.46716958-17880663.82218242j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=-7.949900487447654, + c=-7.93846038215665, + z=(0.9473684210526314-0.10526315789473695j), + expected=(0.07116833581404514+0.11823358038036977j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.087593263474208, + b=4.0013768449590685, + c=-7.93846038215665, + z=(0.7368421052631575+0.5263157894736841j), + expected=(4.7724909620664006e+17-6.039064078946702e+16j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.25, + b=-7.75, + c=1.5, + z=(0.9473684210526314-0.10526315789473695j), + expected=(0.0188022179759303+0.002921737281641378j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=1.0561196186065624, + c=-7.93846038215665, + z=(0.7368421052631575-0.5263157894736843j), + expected=(-9203.462928334846+12390.110518017136j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.75, + b=-15.75, + c=8.5, + z=(0.7368421052631575+0.5263157894736841j), + expected=(6.468457061368628+24.190040684917374j), + rtol=5e-16, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=16.088264119063613, + c=2.0397202577726152, + z=(0.7368421052631575+0.5263157894736841j), + expected=(2408.3451340186543-4275.257316636014j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.75, + b=-7.75, + c=8.5, + z=(0.7368421052631575-0.5263157894736843j), + expected=(4.1379984626381345-5.183654781039423j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.25, + b=-7.75, + c=-0.5, + z=(0.5263157894736841+0.7368421052631575j), + expected=(-81177.775295738+56079.73286548954j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=2.050308316530781, + c=-0.906685989801748, + z=(0.9473684210526314+0.3157894736842106j), + expected=(1192868.5068926765+3624210.8182139914j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=-1.92872979730171, + c=8.031683612216888, + z=(0.5263157894736841+0.7368421052631575j), + expected=(1.8286341846195202+1.9295255682312178j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=1.0561196186065624, + c=16.056809865262608, + z=(0.7368421052631575-0.5263157894736843j), + expected=(1.0514645669696452-0.0430834059440128j), + rtol=5e-10, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.087593263474208, + b=-15.964218273004214, + c=2.0397202577726152, + z=(0.5263157894736841+0.7368421052631575j), + expected=(541983.236432269+288200.2043029435j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.25, + b=8.25, + c=1.5, + z=(0.5263157894736841-0.736842105263158j), + expected=(-10.931988086039945+1.9136272843579096j), + rtol=1e-15, + ), + ), + ] + ) + def test_region3(self, hyp2f1_test_case): + """0.9 <= |z| <= 1 and |1 - z| < 0.9.""" + a, b, c, z, expected, rtol = hyp2f1_test_case + assert 0.9 <= abs(z) <= 1 and abs(1 - z) < 0.9 # Tests the test + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=16.25, + b=4.25, + c=2.5, + z=(0.4931034482758623-0.7965517241379311j), + expected=(38.41207903409937-30.510151276075792j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.0, + b=16.087593263474208, + c=16.088264119063613, + z=(0.5689655172413794-0.7965517241379311j), + expected=(-0.6667857912761286-1.0206224321443573j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.0, + b=1.0272592605282642, + c=-7.949900487447654, + z=(0.4931034482758623-0.7965517241379311j), + expected=(1679024.1647997478-2748129.775857212j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=16.0, + c=-7.949900487447654, + z=(0.4931034482758623-0.7965517241379311j), + expected=(424747226301.16986-1245539049327.2856j), + rtol=1e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=-15.964218273004214, + c=4.0, + z=(0.4931034482758623-0.7965517241379311j), + expected=(-0.0057826199201757595+0.026359861999025885j), + rtol=5e-06, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=-0.9629749245209605, + c=2.0397202577726152, + z=(0.5689655172413794-0.7965517241379311j), + expected=(0.4671901063492606+0.7769632229834897j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.0, + b=-3.956227226099288, + c=-7.949900487447654, + z=(0.4931034482758623+0.7965517241379312j), + expected=(0.9422283708145973+1.3476905754773343j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0, + b=-15.980848054962111, + c=-15.964218273004214, + z=(0.4931034482758623-0.7965517241379311j), + expected=(0.4168719497319604-0.9770953555235625j), + rtol=5e-10, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.5, + b=16.088264119063613, + c=2.5, + z=(0.5689655172413794+0.7965517241379312j), + expected=(1.279096377550619-2.173827694297929j), + rtol=5e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.9214641416286231, + b=4.0013768449590685, + c=2.0397202577726152, + z=(0.4931034482758623+0.7965517241379312j), + expected=(-2.071520656161738-0.7846098268395909j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=8.0, + c=-0.9629749245209605, + z=(0.5689655172413794-0.7965517241379311j), + expected=(-7.740015495862889+3.386766435696699j), + rtol=5e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.9214641416286231, + b=16.088264119063613, + c=-7.93846038215665, + z=(0.4931034482758623+0.7965517241379312j), + expected=(-6318.553685853241-7133.416085202879j), + rtol=5e-9, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.980848054962111, + b=-3.9316537064827854, + c=16.056809865262608, + z=(0.5689655172413794+0.7965517241379312j), + expected=(-0.8854577905547399+8.135089099967278j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.9214641416286231, + b=-0.9629749245209605, + c=4.078873014294075, + z=(0.4931034482758623+0.7965517241379312j), + expected=(1.224291301521487+0.36014711766402485j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.75, + b=-0.75, + c=-1.5, + z=(0.4931034482758623+0.7965517241379312j), + expected=(-1.5765685855028473-3.9399766961046323j), + rtol=1e-3, + ), + marks=pytest.mark.xfail( + reason="Unhandled parameters." + ) + ), + pytest.param( + Hyp2f1TestCase( + a=-15.980848054962111, + b=-1.92872979730171, + c=-7.93846038215665, + z=(0.5689655172413794-0.7965517241379311j), + expected=(56.794588688231194+4.556286783533971j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.5, + b=4.5, + c=2.050308316530781, + z=(0.5689655172413794+0.7965517241379312j), + expected=(-4.251456563455306+6.737837111569671j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.5, + b=8.5, + c=-1.92872979730171, + z=(0.4931034482758623-0.7965517241379311j), + expected=(2177143.9156599627-3313617.2748088865j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.5, + b=-1.5, + c=4.0013768449590685, + z=(0.4931034482758623-0.7965517241379311j), + expected=(0.45563554481603946+0.6212000158060831j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.5, + b=-7.5, + c=-15.964218273004214, + z=(0.4931034482758623+0.7965517241379312j), + expected=(61.03201617828073-37.185626416756214j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.5, + b=16.5, + c=4.0013768449590685, + z=(0.4931034482758623+0.7965517241379312j), + expected=(-33143.425963520735+20790.608514722644j), + rtol=1e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.5, + b=4.5, + c=-0.9629749245209605, + z=(0.5689655172413794+0.7965517241379312j), + expected=(30.778600270824423-26.65160354466787j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.5, + b=-3.5, + c=16.088264119063613, + z=(0.5689655172413794-0.7965517241379311j), + expected=(1.0629792615560487-0.08308454486044772j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.5, + b=-7.5, + c=-0.9629749245209605, + z=(0.4931034482758623-0.7965517241379311j), + expected=(17431.571802591767+3553.7129767034507j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.25, + b=8.25, + c=16.5, + z=(0.11379310344827598+0.9482758620689657j), + expected=(0.4468600750211926+0.7313214934036885j), + rtol=1e-3, + ), + marks=pytest.mark.xfail( + reason="Unhandled parameters." + ) + ), + pytest.param( + Hyp2f1TestCase( + a=8.25, + b=16.25, + c=4.5, + z=(0.3413793103448277+0.8724137931034486j), + expected=(-3.905704438293991+3.693347860329299j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.25, + b=4.25, + c=-0.5, + z=(0.11379310344827598-0.9482758620689655j), + expected=(-40.31777941834244-89.89852492432011j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=8.0, + c=-15.964218273004214, + z=(0.11379310344827598-0.9482758620689655j), + expected=(52584.347773055284-109197.86244309516j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.095813935368371, + b=-15.964218273004214, + c=16.056809865262608, + z=(0.03793103448275881+0.9482758620689657j), + expected=(-1.187733570412592-1.5147865053584582j), + rtol=5e-10, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=-3.9316537064827854, + c=1.0651378143226575, + z=(0.26551724137931054+0.9482758620689657j), + expected=(13.077494677898947+35.071599628224966j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=-3.5, + c=-3.5, + z=(0.26551724137931054+0.8724137931034486j), + expected=(-0.5359656237994614-0.2344483936591811j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.25, + b=-3.75, + c=-1.5, + z=(0.26551724137931054+0.9482758620689657j), + expected=(1204.8114871663133+64.41022826840198j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.9214641416286231, + b=16.0, + c=4.0013768449590685, + z=(0.03793103448275881-0.9482758620689655j), + expected=(-9.85268872413994+7.011107558429154j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=16.0, + c=4.0013768449590685, + z=(0.3413793103448277-0.8724137931034484j), + expected=(528.5522951158454-1412.21630264791j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.5, + b=1.0561196186065624, + c=-7.5, + z=(0.4172413793103451+0.8724137931034486j), + expected=(133306.45260685298+256510.7045225382j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=8.077282662161238, + c=-15.963511401609862, + z=(0.3413793103448277-0.8724137931034484j), + expected=(-0.998555715276967+2.774198742229889j), + rtol=5e-11, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.75, + b=-0.75, + c=1.5, + z=(0.11379310344827598-0.9482758620689655j), + expected=(2.072445019723025-2.9793504811373515j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.5, + b=-1.92872979730171, + c=1.5, + z=(0.11379310344827598-0.9482758620689655j), + expected=(-41.87581944176649-32.52980303527139j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.75, + b=-15.75, + c=-0.5, + z=(0.11379310344827598-0.9482758620689655j), + expected=(-3729.6214864209774-30627.510509112635j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.956227226099288, + b=-15.964218273004214, + c=-0.906685989801748, + z=(0.03793103448275881+0.9482758620689657j), + expected=(-131615.07820609974+145596.13384245415j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.5, + b=16.5, + c=16.088264119063613, + z=(0.26551724137931054+0.8724137931034486j), + expected=(0.18981844071070744+0.7855036242583742j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.5, + b=8.5, + c=-3.9316537064827854, + z=(0.11379310344827598-0.9482758620689655j), + expected=(110224529.2376068+128287212.04290268j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.5, + b=-7.5, + c=4.0013768449590685, + z=(0.3413793103448277-0.8724137931034484j), + expected=(0.2722302180888523-0.21790187837266162j), + rtol=1e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.5, + b=-7.5, + c=-15.964218273004214, + z=(0.11379310344827598-0.9482758620689655j), + expected=(-2.8252338010989035+2.430661949756161j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.5, + b=16.5, + c=4.0013768449590685, + z=(0.03793103448275881+0.9482758620689657j), + expected=(-20.604894257647945+74.5109432558078j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.5, + b=8.5, + c=-0.9629749245209605, + z=(0.3413793103448277+0.8724137931034486j), + expected=(-2764422.521269463-3965966.9965808876j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.5, + b=-0.5, + c=1.0561196186065624, + z=(0.26551724137931054+0.9482758620689657j), + expected=(1.2262338560994905+0.6545051266925549j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.5, + b=-15.5, + c=-7.949900487447654, + z=(0.4172413793103451-0.8724137931034484j), + expected=(-2258.1590330318213+8860.193389158803j), + rtol=1.4e-10, + ), + ), + ] + ) + def test_region4(self, hyp2f1_test_case): + """0.9 <= |z| <= 1 and |1 - z| >= 1. + + This region is unhandled by of the standard transformations and + needs special care. + """ + a, b, c, z, expected, rtol = hyp2f1_test_case + assert 0.9 <= abs(z) <= 1 and abs(1 - z) >= 0.9 # Tests the test + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=4.5, + b=16.088264119063613, + c=8.5, + z=(0.6448275862068968+0.8724137931034486j), + expected=(0.018601324701770394-0.07618420586062377j), + rtol=5e-08, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.25, + b=4.25, + c=4.5, + z=(0.6448275862068968-0.8724137931034484j), + expected=(-1.391549471425551-0.118036604903893j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=2.050308316530781, + c=-1.9631175993998025, + z=(0.6448275862068968+0.8724137931034486j), + expected=(-2309.178768155151-1932.7247727595172j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.087593263474208, + b=1.0, + c=-15.964218273004214, + z=(0.6448275862068968+0.8724137931034486j), + expected=(85592537010.05054-8061416766688.324j), + rtol=2e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.095813935368371, + b=-0.5, + c=1.5, + z=(0.6448275862068968+0.8724137931034486j), + expected=(1.2334498208515172-2.1639498536219732j), + rtol=5e-11, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.087593263474208, + b=-15.964218273004214, + c=4.0, + z=(0.6448275862068968+0.8724137931034486j), + expected=(102266.35398605966-44976.97828737755j), + rtol=1e-3, + ), + marks=pytest.mark.xfail( + reason="Unhandled parameters." + ) + ), + pytest.param( + Hyp2f1TestCase( + a=4.0, + b=-3.956227226099288, + c=-15.964218273004214, + z=(0.6448275862068968-0.8724137931034484j), + expected=(-2.9590030930007236-4.190770764773225j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=-15.5, + c=-7.5, + z=(0.5689655172413794-0.8724137931034484j), + expected=(-112554838.92074208+174941462.9202412j), + rtol=5e-05, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.980848054962111, + b=2.050308316530781, + c=1.0, + z=(0.6448275862068968-0.8724137931034484j), + expected=(3.7519882374080145+7.360753798667486j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=2.050308316530781, + c=4.0, + z=(0.6448275862068968-0.8724137931034484j), + expected=(0.000181132943964693+0.07742903103815582j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=4.0013768449590685, + c=-1.9631175993998025, + z=(0.5689655172413794+0.8724137931034486j), + expected=(386338.760913596-386166.51762171905j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.980848054962111, + b=8.0, + c=-1.92872979730171, + z=(0.6448275862068968+0.8724137931034486j), + expected=(1348667126.3444858-2375132427.158893j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.5, + b=-0.9629749245209605, + c=4.5, + z=(0.5689655172413794+0.8724137931034486j), + expected=(1.428353429538678+0.6472718120804372j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=-0.9629749245209605, + c=2.0397202577726152, + z=(0.5689655172413794-0.8724137931034484j), + expected=(3.1439267526119643-3.145305240375117j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.9214641416286231, + b=-15.964218273004214, + c=-7.93846038215665, + z=(0.6448275862068968-0.8724137931034484j), + expected=(75.27467675681773+144.0946946292215j), + rtol=1e-07, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.75, + b=-7.75, + c=-7.5, + z=(0.5689655172413794+0.8724137931034486j), + expected=(-0.3699450626264222+0.8732812475910993j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.5, + b=16.5, + c=1.0561196186065624, + z=(0.5689655172413794-0.8724137931034484j), + expected=(5.5361025821300665-2.4709693474656285j), + rtol=5e-09, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.5, + b=8.5, + c=-3.9316537064827854, + z=(0.6448275862068968-0.8724137931034484j), + expected=(-782805.6699207705-537192.581278909j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.5, + b=-15.5, + c=1.0561196186065624, + z=(0.6448275862068968+0.8724137931034486j), + expected=(12.345113400639693-14.993248992902007j), + rtol=0.0005, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.5, + b=-0.5, + c=-15.964218273004214, + z=(0.6448275862068968+0.8724137931034486j), + expected=(23.698109392667842+97.15002033534108j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.5, + b=16.5, + c=4.0013768449590685, + z=(0.6448275862068968-0.8724137931034484j), + expected=(1115.2978631811834+915.9212658718577j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.5, + b=16.5, + c=-0.9629749245209605, + z=(0.6448275862068968+0.8724137931034486j), + expected=(642077722221.6489+535274495398.21027j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.5, + b=-3.5, + c=4.0013768449590685, + z=(0.5689655172413794+0.8724137931034486j), + expected=(-5.689219222945697+16.877463062787143j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.5, + b=-1.5, + c=-0.9629749245209605, + z=(0.5689655172413794-0.8724137931034484j), + expected=(-44.32070290703576+1026.9127058617403j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.25, + b=2.25, + c=4.5, + z=(0.11379310344827598-1.024137931034483j), + expected=(-0.021965227124574663+0.009908300237809064j), + rtol=1e-3, + ), + marks=pytest.mark.xfail( + reason="Unhandled parameters." + ) + ), + pytest.param( + Hyp2f1TestCase( + a=2.02764642551431, + b=1.5, + c=16.5, + z=(0.26551724137931054+1.024137931034483j), + expected=(1.0046072901244183+0.19945500134119992j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.087593263474208, + b=1.0, + c=-3.9316537064827854, + z=(0.3413793103448277+0.9482758620689657j), + expected=(21022.30133421465+49175.98317370489j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=16.088264119063613, + c=-1.9631175993998025, + z=(0.4172413793103451-0.9482758620689655j), + expected=(-7024239.358547302+2481375.02681063j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.25, + b=-15.75, + c=1.5, + z=(0.18965517241379315+1.024137931034483j), + expected=(92371704.94848-403546832.548352j), + rtol=5e-06, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.5, + b=-7.949900487447654, + c=8.5, + z=(0.26551724137931054-1.024137931034483j), + expected=(1.9335109845308265+5.986542524829654j), + rtol=5e-10, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.095813935368371, + b=-1.92872979730171, + c=-7.93846038215665, + z=(0.4931034482758623+0.8724137931034486j), + expected=(-122.52639696039328-59.72428067512221j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.25, + b=-1.75, + c=-1.5, + z=(0.4931034482758623+0.9482758620689657j), + expected=(-90.40642053579428+50.50649180047921j), + rtol=5e-08, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.5, + b=8.077282662161238, + c=16.5, + z=(0.4931034482758623+0.9482758620689657j), + expected=(-0.2155745818150323-0.564628986876639j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=1.0561196186065624, + c=8.031683612216888, + z=(0.4172413793103451-0.9482758620689655j), + expected=(0.9503140488280465+0.11574960074292677j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.75, + b=2.25, + c=-15.5, + z=(0.4172413793103451+0.9482758620689657j), + expected=(0.9285862488442175+0.8203699266719692j), + rtol=5e-13, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.75, + b=4.25, + c=-15.5, + z=(0.3413793103448277-0.9482758620689655j), + expected=(-1.0509834850116921-1.1145522325486075j), + rtol=1.1e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=-0.9629749245209605, + c=2.0397202577726152, + z=(0.4931034482758623-0.9482758620689655j), + expected=(2.88119116536769-3.4249933450696806j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.5, + b=-15.964218273004214, + c=16.5, + z=(0.18965517241379315+1.024137931034483j), + expected=(199.65868451496038+347.79384207302877j), + rtol=5e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.75, + b=-15.75, + c=-3.5, + z=(0.4931034482758623-0.8724137931034484j), + expected=(-208138312553.07013+58631611809.026955j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=-15.5, + c=-7.5, + z=(0.3413793103448277+0.9482758620689657j), + expected=(-23032.90519856288-18256.94050457296j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.5, + b=1.5, + c=1.0561196186065624, + z=(0.4931034482758623-0.8724137931034484j), + expected=(1.507342459587056+1.2332023580148403j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=2.5, + b=4.5, + c=-3.9316537064827854, + z=(0.4172413793103451+0.9482758620689657j), + expected=(7044.766127108853-40210.365567285575j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.5, + b=-1.5, + c=1.0561196186065624, + z=(0.03793103448275881+1.024137931034483j), + expected=(0.2725347741628333-2.247314875514784j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.5, + b=-1.5, + c=-7.949900487447654, + z=(0.26551724137931054+1.024137931034483j), + expected=(-11.250200011017546+12.597393659160472j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.5, + b=8.5, + c=16.088264119063613, + z=(0.26551724137931054+1.024137931034483j), + expected=(-0.18515160890991517+0.7959014164484782j), + rtol=2e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.5, + b=16.5, + c=-3.9316537064827854, + z=(0.3413793103448277-1.024137931034483j), + expected=(998246378.8556538+1112032928.103645j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.5, + b=-3.5, + c=2.050308316530781, + z=(0.03793103448275881+1.024137931034483j), + expected=(0.5527670397711952+2.697662715303637j), + rtol=1.2e-15, # rtol bumped from 1e-15 in gh18414 + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-15.5, + b=-1.5, + c=-0.9629749245209605, + z=(0.4931034482758623-0.8724137931034484j), + expected=(55.396931662136886+968.467463806326j), + rtol=5e-14, + ), + ), + ] + ) + def test_region5(self, hyp2f1_test_case): + """1 < |z| < 1.1 and |1 - z| >= 0.9 and real(z) >= 0""" + a, b, c, z, expected, rtol = hyp2f1_test_case + assert 1 < abs(z) < 1.1 and abs(1 - z) >= 0.9 and z.real >= 0 + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + pytest.param( + Hyp2f1TestCase( + a=8.095813935368371, + b=4.0013768449590685, + c=4.078873014294075, + z=(-0.9473684210526316+0.5263157894736841j), + expected=(-0.0018093573941378783+0.003481887377423739j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.087593263474208, + b=2.050308316530781, + c=1.0651378143226575, + z=(-0.736842105263158-0.736842105263158j), + expected=(-0.00023401243818780545-1.7983496305603562e-05j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=8.077282662161238, + c=4.078873014294075, + z=(-0.5263157894736843-0.9473684210526316j), + expected=(0.22359773002226846-0.24092487123993353j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=2.050308316530781, + c=-15.963511401609862, + z=(-0.9473684210526316-0.5263157894736843j), + expected=(1.191573745740011+0.14347394589721466j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=4.0013768449590685, + c=-15.963511401609862, + z=(-0.9473684210526316-0.5263157894736843j), + expected=(31.822620756901784-66.09094396747611j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=8.077282662161238, + c=-7.93846038215665, + z=(-0.9473684210526316+0.5263157894736841j), + expected=(207.16750179245952+34.80478274924269j), + rtol=5e-12, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=8.095813935368371, + b=-7.949900487447654, + c=8.031683612216888, + z=(-0.736842105263158+0.7368421052631575j), + expected=(-159.62429364277145+9.154224290644898j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=-1.92872979730171, + c=16.056809865262608, + z=(-0.9473684210526316+0.5263157894736841j), + expected=(1.121122351247184-0.07170260470126685j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=16.087593263474208, + b=-0.9629749245209605, + c=16.056809865262608, + z=(-0.9473684210526316+0.5263157894736841j), + expected=(1.9040596681316053-0.4951799449960107j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=-1.92872979730171, + c=-0.906685989801748, + z=(-0.9473684210526316-0.5263157894736843j), + expected=(-14.496623497780739-21.897524523299875j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=4.080187217753502, + b=-3.9316537064827854, + c=-3.9924618758357022, + z=(-0.5263157894736843-0.9473684210526316j), + expected=(36.33473466026878+253.88728442029577j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1.0272592605282642, + b=-15.964218273004214, + c=-0.906685989801748, + z=(-0.9473684210526316+0.5263157894736841j), + expected=(1505052.5653144997-50820766.81043443j), + rtol=1e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.956227226099288, + b=4.0013768449590685, + c=1.0651378143226575, + z=(-0.5263157894736843+0.9473684210526314j), + expected=(-127.79407519260877-28.69899444941112j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.9214641416286231, + b=8.077282662161238, + c=16.056809865262608, + z=(-0.9473684210526316-0.5263157894736843j), + expected=(2.0623331933754976+0.741234463565458j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.956227226099288, + b=8.077282662161238, + c=2.0397202577726152, + z=(-0.9473684210526316+0.5263157894736841j), + expected=(30.729193458862525-292.5700835046965j), + rtol=1e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.9214641416286231, + b=1.0561196186065624, + c=-1.9631175993998025, + z=(-0.5263157894736843-0.9473684210526316j), + expected=(1.1285917906203495-0.735264575450189j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=1.0561196186065624, + c=-3.9924618758357022, + z=(-0.736842105263158+0.7368421052631575j), + expected=(0.6356474446678052-0.02429663008952248j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-1.9214641416286231, + b=16.088264119063613, + c=-7.93846038215665, + z=(-0.736842105263158+0.7368421052631575j), + expected=(0.4718880510273174+0.655083067736377j), + rtol=1e-11, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-7.937789122896016, + b=-3.9316537064827854, + c=16.056809865262608, + z=(-0.9473684210526316+0.5263157894736841j), + expected=(-0.14681550942352714+0.16092206364265146j), + rtol=5e-11, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=-15.964218273004214, + c=1.0651378143226575, + z=(-0.5263157894736843+0.9473684210526314j), + expected=(-6.436835190526225+22.883156700606182j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-0.9220024191881196, + b=-7.949900487447654, + c=4.078873014294075, + z=(-0.9473684210526316-0.5263157894736843j), + expected=(-0.7505682955068583-1.1026583264249945j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.956227226099288, + b=-3.9316537064827854, + c=-7.93846038215665, + z=(-0.9473684210526316-0.5263157894736843j), + expected=(3.6247814989198166+2.596041360148318j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.956227226099288, + b=-15.964218273004214, + c=-1.9631175993998025, + z=(-0.5263157894736843-0.9473684210526316j), + expected=(-59537.65287927933-669074.4342539902j), + rtol=5e-15, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=-3.956227226099288, + b=-15.964218273004214, + c=-1.9631175993998025, + z=(-0.9473684210526316-0.5263157894736843j), + expected=(-433084.9970266166+431088.393918521j), + rtol=5e-14, + ), + ), + pytest.param( + Hyp2f1TestCase( + a=1, + b=1, + c=4, + z=(3 + 4j), + expected=(0.49234384000963544+0.6051340616612397j), + rtol=5e-14, + ), + ), + ] + ) + def test_region6(self, hyp2f1_test_case): + """|z| > 1 but not in region 5.""" + a, b, c, z, expected, rtol = hyp2f1_test_case + assert ( + abs(z) > 1 and + not (1 < abs(z) < 1.1 and abs(1 - z) >= 0.9 and z.real >= 0) + ) + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + + @pytest.mark.parametrize( + "hyp2f1_test_case", + [ + # Broke when fixing gamma pole behavior in gh-21827 + pytest.param( + Hyp2f1TestCase( + a=1.3, + b=-0.2, + c=0.3, + z=-2.1, + expected=1.8202169687521206, + rtol=5e-15, + ), + ), + ] + ) + def test_miscellaneous(self, hyp2f1_test_case ): + a, b, c, z, expected, rtol = hyp2f1_test_case + assert_allclose(hyp2f1(a, b, c, z), expected, rtol=rtol) + + @pytest.mark.slow + @check_version(mpmath, "1.0.0") + def test_test_hyp2f1(self): + """Test that expected values match what is computed by mpmath. + + This gathers the parameters for the test cases out of the pytest marks. + The parameters are a, b, c, z, expected, rtol, where expected should + be the value of hyp2f1(a, b, c, z) computed with mpmath. The test + recomputes hyp2f1(a, b, c, z) using mpmath and verifies that expected + actually is the correct value. This allows the data for the tests to + live within the test code instead of an external datafile, while + avoiding having to compute the results with mpmath during the test, + except for when slow tests are being run. + """ + test_methods = [ + test_method for test_method in dir(self) + if test_method.startswith('test') and + # Filter properties and attributes (futureproofing). + callable(getattr(self, test_method)) and + # Filter out this test + test_method != 'test_test_hyp2f1' + ] + for test_method in test_methods: + params = self._get_test_parameters(getattr(self, test_method)) + for a, b, c, z, expected, _ in params: + assert_allclose(mp_hyp2f1(a, b, c, z), expected, rtol=2.25e-16) + + def _get_test_parameters(self, test_method): + """Get pytest.mark parameters for a test in this class.""" + return [ + case.values[0] for mark in test_method.pytestmark + if mark.name == 'parametrize' + for case in mark.args[1] + ] + +class TestHyp2f1ExtremeInputs: + + @pytest.mark.parametrize("a", [1.0, 2.0, 3.0, -np.inf, np.inf]) + @pytest.mark.parametrize("b", [3.0, 4.0, 5.0, -np.inf, np.inf]) + @pytest.mark.parametrize("c", [3.0, 5.0, 6.0, 7.0]) + @pytest.mark.parametrize("z", [4.0 + 1.0j]) + def test_inf_a_b(self, a, b, c, z): + if np.any(np.isinf(np.asarray([a, b]))): + assert(np.isnan(hyp2f1(a, b, c, z))) + + def test_large_a_b(self): + assert(np.isnan(hyp2f1(10**7, 1.0, 3.0, 4.0 + 1.0j))) + assert(np.isnan(hyp2f1(-10**7, 1.0, 3.0, 4.0 + 1.0j))) + + assert(np.isnan(hyp2f1(1.0, 10**7, 3.0, 4.0 + 1.0j))) + assert(np.isnan(hyp2f1(1.0, -10**7, 3.0, 4.0 + 1.0j))) + + # Already correct in main but testing for surety + assert(np.isnan(hyp2f1(np.inf, 1.0, 3.0, 4.0))) + assert(np.isnan(hyp2f1(1.0, np.inf, 3.0, 4.0))) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_hypergeometric.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_hypergeometric.py new file mode 100644 index 0000000000000000000000000000000000000000..1ad092491905ae66c39cd881836054aef3dc6f6e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_hypergeometric.py @@ -0,0 +1,234 @@ +import pytest +import numpy as np +from numpy.testing import assert_allclose, assert_equal +import scipy.special as sc + + +class TestHyperu: + + def test_negative_x(self): + a, b, x = np.meshgrid( + [-1, -0.5, 0, 0.5, 1], + [-1, -0.5, 0, 0.5, 1], + np.linspace(-100, -1, 10), + ) + assert np.all(np.isnan(sc.hyperu(a, b, x))) + + def test_special_cases(self): + assert sc.hyperu(0, 1, 1) == 1.0 + + @pytest.mark.parametrize('a', [0.5, 1, np.nan]) + @pytest.mark.parametrize('b', [1, 2, np.nan]) + @pytest.mark.parametrize('x', [0.25, 3, np.nan]) + def test_nan_inputs(self, a, b, x): + assert np.isnan(sc.hyperu(a, b, x)) == np.any(np.isnan([a, b, x])) + + @pytest.mark.parametrize( + 'a,b,x,expected', + [(0.21581740448533887, 1.0, 1e-05, 3.6030558839391325), + (0.21581740448533887, 1.0, 0.00021544346900318823, 2.8783254988948976), + (0.21581740448533887, 1.0, 0.004641588833612777, 2.154928216691109), + (0.21581740448533887, 1.0, 0.1, 1.446546638718792), + (0.0030949064301273865, 1.0, 1e-05, 1.0356696454116199), + (0.0030949064301273865, 1.0, 0.00021544346900318823, 1.0261510362481985), + (0.0030949064301273865, 1.0, 0.004641588833612777, 1.0166326903402296), + (0.0030949064301273865, 1.0, 0.1, 1.0071174207698674), + (0.1509924314279033, 1.0, 1e-05, 2.806173846998948), + (0.1509924314279033, 1.0, 0.00021544346900318823, 2.3092158526816124), + (0.1509924314279033, 1.0, 0.004641588833612777, 1.812905980588048), + (0.1509924314279033, 1.0, 0.1, 1.3239738117634872), + (-0.010678995342969011, 1.0, 1e-05, 0.8775194903781114), + (-0.010678995342969011, 1.0, 0.00021544346900318823, 0.9101008998540128), + (-0.010678995342969011, 1.0, 0.004641588833612777, 0.9426854294058609), + (-0.010678995342969011, 1.0, 0.1, 0.9753065150174902), + (-0.06556622211831487, 1.0, 1e-05, 0.26435429752668904), + (-0.06556622211831487, 1.0, 0.00021544346900318823, 0.4574756033875781), + (-0.06556622211831487, 1.0, 0.004641588833612777, 0.6507121093358457), + (-0.06556622211831487, 1.0, 0.1, 0.8453129788602187), + (-0.21628242470175185, 1.0, 1e-05, -1.2318314201114489), + (-0.21628242470175185, 1.0, 0.00021544346900318823, -0.6704694233529538), + (-0.21628242470175185, 1.0, 0.004641588833612777, -0.10795098653682857), + (-0.21628242470175185, 1.0, 0.1, 0.4687227684115524)] + ) + def test_gh_15650_mp(self, a, b, x, expected): + # See https://github.com/scipy/scipy/issues/15650 + # b == 1, |a| < 0.25, 0 < x < 1 + # + # This purpose of this test is to check the accuracy of results + # in the region that was impacted by gh-15650. + # + # Reference values computed with mpmath using the script: + # + # import itertools as it + # import numpy as np + # + # from mpmath import mp + # + # rng = np.random.default_rng(1234) + # + # cases = [] + # for a, x in it.product( + # np.random.uniform(-0.25, 0.25, size=6), + # np.logspace(-5, -1, 4), + # ): + # with mp.workdps(100): + # cases.append((float(a), 1.0, float(x), float(mp.hyperu(a, 1.0, x)))) + assert_allclose(sc.hyperu(a, b, x), expected, rtol=1e-13) + + def test_gh_15650_sanity(self): + # The purpose of this test is to sanity check hyperu in the region that + # was impacted by gh-15650 by making sure there are no excessively large + # results, as were reported there. + a = np.linspace(-0.5, 0.5, 500) + x = np.linspace(1e-6, 1e-1, 500) + a, x = np.meshgrid(a, x) + results = sc.hyperu(a, 1.0, x) + assert np.all(np.abs(results) < 1e3) + + +class TestHyp1f1: + + @pytest.mark.parametrize('a, b, x', [ + (np.nan, 1, 1), + (1, np.nan, 1), + (1, 1, np.nan) + ]) + def test_nan_inputs(self, a, b, x): + assert np.isnan(sc.hyp1f1(a, b, x)) + + def test_poles(self): + assert_equal(sc.hyp1f1(1, [0, -1, -2, -3, -4], 0.5), np.inf) + + @pytest.mark.parametrize('a, b, x, result', [ + (-1, 1, 0.5, 0.5), + (1, 1, 0.5, 1.6487212707001281468), + (2, 1, 0.5, 2.4730819060501922203), + (1, 2, 0.5, 1.2974425414002562937), + (-10, 1, 0.5, -0.38937441413785204475) + ]) + def test_special_cases(self, a, b, x, result): + # Hit all the special case branches at the beginning of the + # function. Desired answers computed using Mpmath. + assert_allclose(sc.hyp1f1(a, b, x), result, atol=0, rtol=1e-15) + + @pytest.mark.parametrize('a, b, x, result', [ + (1, 1, 0.44, 1.5527072185113360455), + (-1, 1, 0.44, 0.55999999999999999778), + (100, 100, 0.89, 2.4351296512898745592), + (-100, 100, 0.89, 0.40739062490768104667), + (1.5, 100, 59.99, 3.8073513625965598107), + (-1.5, 100, 59.99, 0.25099240047125826943) + ]) + def test_geometric_convergence(self, a, b, x, result): + # Test the region where we are relying on the ratio of + # + # (|a| + 1) * |x| / |b| + # + # being small. Desired answers computed using Mpmath + assert_allclose(sc.hyp1f1(a, b, x), result, atol=0, rtol=1e-15) + + @pytest.mark.parametrize('a, b, x, result', [ + (-1, 1, 1.5, -0.5), + (-10, 1, 1.5, 0.41801777430943080357), + (-25, 1, 1.5, 0.25114491646037839809), + (-50, 1, 1.5, -0.25683643975194756115), + (-80, 1, 1.5, -0.24554329325751503601), + (-150, 1, 1.5, -0.173364795515420454496), + ]) + def test_a_negative_integer(self, a, b, x, result): + # Desired answers computed using Mpmath. + assert_allclose(sc.hyp1f1(a, b, x), result, atol=0, rtol=2e-14) + + @pytest.mark.parametrize('a, b, x, expected', [ + (0.01, 150, -4, 0.99973683897677527773), # gh-3492 + (1, 5, 0.01, 1.0020033381011970966), # gh-3593 + (50, 100, 0.01, 1.0050126452421463411), # gh-3593 + (1, 0.3, -1e3, -7.011932249442947651455e-04), # gh-14149 + (1, 0.3, -1e4, -7.001190321418937164734e-05), # gh-14149 + (9, 8.5, -350, -5.224090831922378361082e-20), # gh-17120 + (9, 8.5, -355, -4.595407159813368193322e-20), # gh-17120 + (75, -123.5, 15, 3.425753920814889017493e+06), + ]) + def test_assorted_cases(self, a, b, x, expected): + # Expected values were computed with mpmath.hyp1f1(a, b, x). + assert_allclose(sc.hyp1f1(a, b, x), expected, atol=0, rtol=1e-14) + + def test_a_neg_int_and_b_equal_x(self): + # This is a case where the Boost wrapper will call hypergeometric_pFq + # instead of hypergeometric_1F1. When we use a version of Boost in + # which https://github.com/boostorg/math/issues/833 is fixed, this + # test case can probably be moved into test_assorted_cases. + # The expected value was computed with mpmath.hyp1f1(a, b, x). + a = -10.0 + b = 2.5 + x = 2.5 + expected = 0.0365323664364104338721 + computed = sc.hyp1f1(a, b, x) + assert_allclose(computed, expected, atol=0, rtol=1e-13) + + @pytest.mark.parametrize('a, b, x, desired', [ + (-1, -2, 2, 2), + (-1, -4, 10, 3.5), + (-2, -2, 1, 2.5) + ]) + def test_gh_11099(self, a, b, x, desired): + # All desired results computed using Mpmath + assert sc.hyp1f1(a, b, x) == desired + + @pytest.mark.parametrize('a', [-3, -2]) + def test_x_zero_a_and_b_neg_ints_and_a_ge_b(self, a): + assert sc.hyp1f1(a, -3, 0) == 1 + + # In the following tests with complex z, the reference values + # were computed with mpmath.hyp1f1(a, b, z), and verified with + # Wolfram Alpha Hypergeometric1F1(a, b, z), except for the + # case a=0.1, b=1, z=7-24j, where Wolfram Alpha reported + # "Standard computation time exceeded". That reference value + # was confirmed in an online Matlab session, with the commands + # + # > format long + # > hypergeom(0.1, 1, 7-24i) + # ans = + # -3.712349651834209 + 4.554636556672912i + # + @pytest.mark.parametrize( + 'a, b, z, ref', + [(-0.25, 0.5, 1+2j, 1.1814553180903435-1.2792130661292984j), + (0.25, 0.5, 1+2j, 0.24636797405707597+1.293434354945675j), + (25, 1.5, -2j, -516.1771262822523+407.04142751922024j), + (12, -1.5, -10+20j, -5098507.422706547-1341962.8043508842j), + pytest.param( + 10, 250, 10-15j, 1.1985998416598884-0.8613474402403436j, + marks=pytest.mark.xfail, + ), + pytest.param( + 0.1, 1, 7-24j, -3.712349651834209+4.554636556672913j, + marks=pytest.mark.xfail, + ) + ], + ) + def test_complex_z(self, a, b, z, ref): + h = sc.hyp1f1(a, b, z) + assert_allclose(h, ref, rtol=4e-15) + + # The "legacy edge cases" mentioned in the comments in the following + # tests refers to the behavior of hyp1f1(a, b, x) when b is a nonpositive + # integer. In some subcases, the behavior of SciPy does not match that + # of Boost (1.81+), mpmath and Mathematica (via Wolfram Alpha online). + # If the handling of these edges cases is changed to agree with those + # libraries, these test will have to be updated. + + @pytest.mark.parametrize('b', [0, -1, -5]) + def test_legacy_case1(self, b): + # Test results of hyp1f1(0, n, x) for n <= 0. + # This is a legacy edge case. + # Boost (versions greater than 1.80), Mathematica (via Wolfram Alpha + # online) and mpmath all return 1 in this case, but SciPy's hyp1f1 + # returns inf. + assert_equal(sc.hyp1f1(0, b, [-1.5, 0, 1.5]), [np.inf, np.inf, np.inf]) + + def test_legacy_case2(self): + # This is a legacy edge case. + # In software such as boost (1.81+), mpmath and Mathematica, + # the value is 1. + assert sc.hyp1f1(-4, -3, 0) == np.inf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_iv_ratio.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_iv_ratio.py new file mode 100644 index 0000000000000000000000000000000000000000..dafc35982c4d4869ead2136e74466f1c44c64556 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_iv_ratio.py @@ -0,0 +1,249 @@ +# This file contains unit tests for iv_ratio() and related functions. + +import pytest +import numpy as np +from numpy.testing import assert_equal, assert_allclose +from scipy.special._ufuncs import ( # type: ignore[attr-defined] + _iv_ratio as iv_ratio, + _iv_ratio_c as iv_ratio_c, +) + + +class TestIvRatio: + + @pytest.mark.parametrize('v,x,r', [ + (0.5, 0.16666666666666666, 0.16514041292462933), + (0.5, 0.3333333333333333, 0.32151273753163434), + (0.5, 0.5, 0.46211715726000974), + (0.5, 0.6666666666666666, 0.5827829453479101), + (0.5, 0.8333333333333335, 0.6822617902381698), + (1, 0.3380952380952381, 0.1666773049170313), + (1, 0.7083333333333333, 0.33366443586989925), + (1, 1.1666666666666667, 0.5023355231537423), + (1, 1.8666666666666665, 0.674616572252164), + (1, 3.560606060606061, 0.844207659503163), + (2.34, 0.7975238095238094, 0.16704903081553285), + (2.34, 1.7133333333333334, 0.3360215931268845), + (2.34, 2.953333333333333, 0.50681909317803), + (2.34, 5.0826666666666656, 0.6755252698800679), + (2.34, 10.869696969696973, 0.8379351104498762), + (56.789, 19.46575238095238, 0.1667020505391409), + (56.789, 42.55008333333333, 0.33353809996933026), + (56.789, 75.552, 0.5003932381177826), + (56.789, 135.76026666666667, 0.6670528221946127), + (56.789, 307.8642424242425, 0.8334999441460798), + ]) + def test_against_reference_values(self, v, x, r): + """The reference values are computed using mpmath as follows. + + from mpmath import mp + mp.dps = 100 + + def iv_ratio_mp(v, x): + return mp.besseli(v, x) / mp.besseli(v - 1, x) + + def _sample(n, *, v): + '''Return n positive real numbers x such that iv_ratio(v, x) are + roughly evenly spaced over (0, 1). The formula is taken from [1]. + + [1] Banerjee A., Dhillon, I. S., Ghosh, J., Sra, S. (2005). + "Clustering on the Unit Hypersphere using von Mises-Fisher + Distributions." Journal of Machine Learning Research, + 6(46):1345-1382. + ''' + r = np.arange(1, n+1) / (n+1) + return r * (2*v-r*r) / (1-r*r) + + for v in (0.5, 1, 2.34, 56.789): + xs = _sample(5, v=v) + for x in xs: + print(f"({v}, {x}, {float(iv_ratio_mp(v,x))}),") + """ + assert_allclose(iv_ratio(v, x), r, rtol=4e-16, atol=0) + + @pytest.mark.parametrize('v,x,r', [ + (1, np.inf, 1), + (np.inf, 1, 0), + ]) + def test_inf(self, v, x, r): + """If exactly one of v or x is inf and the other is within domain, + should return 0 or 1 accordingly.""" + assert_equal(iv_ratio(v, x), r) + + @pytest.mark.parametrize('v', [0.49, -np.inf, np.nan, np.inf]) + @pytest.mark.parametrize('x', [-np.finfo(float).smallest_normal, + -np.finfo(float).smallest_subnormal, + -np.inf, np.nan, np.inf]) + def test_nan(self, v, x): + """If at least one argument is out of domain, or if v = x = inf, + the function should return nan.""" + assert_equal(iv_ratio(v, x), np.nan) + + @pytest.mark.parametrize('v', [0.5, 1, np.finfo(float).max, np.inf]) + def test_zero_x(self, v): + """If x is +/-0.0, return x to ensure iv_ratio is an odd function.""" + assert_equal(iv_ratio(v, 0.0), 0.0) + assert_equal(iv_ratio(v, -0.0), -0.0) + + @pytest.mark.parametrize('v,x', [ + (1, np.finfo(float).smallest_normal), + (1, np.finfo(float).smallest_subnormal), + (1, np.finfo(float).smallest_subnormal*2), + (1e20, 123), + (np.finfo(float).max, 1), + (np.finfo(float).max, np.sqrt(np.finfo(float).max)), + ]) + def test_tiny_x(self, v, x): + """If x is much less than v, the bounds + + x x + --------------------------- <= R <= ----------------------- + v-0.5+sqrt(x**2+(v+0.5)**2) v-1+sqrt(x**2+(v+1)**2) + + collapses to R ~= x/2v. Test against this asymptotic expression. + """ + assert_equal(iv_ratio(v, x), (0.5*x)/v) + + @pytest.mark.parametrize('v,x', [ + (1, 1e16), + (1e20, 1e40), + (np.sqrt(np.finfo(float).max), np.finfo(float).max), + ]) + def test_huge_x(self, v, x): + """If x is much greater than v, the bounds + + x x + --------------------------- <= R <= --------------------------- + v-0.5+sqrt(x**2+(v+0.5)**2) v-0.5+sqrt(x**2+(v-0.5)**2) + + collapses to R ~= 1. Test against this asymptotic expression. + """ + assert_equal(iv_ratio(v, x), 1.0) + + @pytest.mark.parametrize('v,x', [ + (np.finfo(float).max, np.finfo(float).max), + (np.finfo(float).max / 3, np.finfo(float).max), + (np.finfo(float).max, np.finfo(float).max / 3), + ]) + def test_huge_v_x(self, v, x): + """If both x and v are very large, the bounds + + x x + --------------------------- <= R <= ----------------------- + v-0.5+sqrt(x**2+(v+0.5)**2) v-1+sqrt(x**2+(v+1)**2) + + collapses to R ~= x/(v+sqrt(x**2+v**2). Test against this asymptotic + expression, and in particular that no numerical overflow occurs during + intermediate calculations. + """ + t = x / v + expected = t / (1 + np.hypot(1, t)) + assert_allclose(iv_ratio(v, x), expected, rtol=4e-16, atol=0) + + +class TestIvRatioC: + + @pytest.mark.parametrize('v,x,r', [ + (0.5, 0.16666666666666666, 0.8348595870753707), + (0.5, 0.3333333333333333, 0.6784872624683657), + (0.5, 0.5, 0.5378828427399902), + (0.5, 0.6666666666666666, 0.4172170546520899), + (0.5, 0.8333333333333335, 0.3177382097618302), + (1, 0.3380952380952381, 0.8333226950829686), + (1, 0.7083333333333333, 0.6663355641301008), + (1, 1.1666666666666667, 0.4976644768462577), + (1, 1.8666666666666665, 0.325383427747836), + (1, 3.560606060606061, 0.155792340496837), + (2.34, 0.7975238095238094, 0.8329509691844672), + (2.34, 1.7133333333333334, 0.6639784068731155), + (2.34, 2.953333333333333, 0.49318090682197), + (2.34, 5.0826666666666656, 0.3244747301199321), + (2.34, 10.869696969696973, 0.16206488955012377), + (56.789, 19.46575238095238, 0.8332979494608591), + (56.789, 42.55008333333333, 0.6664619000306697), + (56.789, 75.552, 0.4996067618822174), + (56.789, 135.76026666666667, 0.3329471778053873), + (56.789, 307.8642424242425, 0.16650005585392025), + ]) + def test_against_reference_values(self, v, x, r): + """The reference values are one minus those of TestIvRatio.""" + assert_allclose(iv_ratio_c(v, x), r, rtol=1e-15, atol=0) + + @pytest.mark.parametrize('v,x,r', [ + (1, np.inf, 0), + (np.inf, 1, 1), + ]) + def test_inf(self, v, x, r): + """If exactly one of v or x is inf and the other is within domain, + should return 0 or 1 accordingly.""" + assert_equal(iv_ratio_c(v, x), r) + + @pytest.mark.parametrize('v', [0.49, -np.inf, np.nan, np.inf]) + @pytest.mark.parametrize('x', [-np.finfo(float).smallest_normal, + -np.finfo(float).smallest_subnormal, + -np.inf, np.nan, np.inf]) + def test_nan(self, v, x): + """If at least one argument is out of domain, or if v = x = inf, + the function should return nan.""" + assert_equal(iv_ratio_c(v, x), np.nan) + + @pytest.mark.parametrize('v', [0.5, 1, np.finfo(float).max, np.inf]) + def test_zero_x(self, v): + """If x is +/-0.0, return 1.""" + assert_equal(iv_ratio_c(v, 0.0), 1.0) + assert_equal(iv_ratio_c(v, -0.0), 1.0) + + @pytest.mark.parametrize('v,x', [ + (1, np.finfo(float).smallest_normal), + (1, np.finfo(float).smallest_subnormal), + (1, np.finfo(float).smallest_subnormal*2), + (1e20, 123), + (np.finfo(float).max, 1), + (np.finfo(float).max, np.sqrt(np.finfo(float).max)), + ]) + def test_tiny_x(self, v, x): + """If x is much less than v, the bounds + + x x + --------------------------- <= R <= ----------------------- + v-0.5+sqrt(x**2+(v+0.5)**2) v-1+sqrt(x**2+(v+1)**2) + + collapses to 1-R ~= 1-x/2v. Test against this asymptotic expression. + """ + assert_equal(iv_ratio_c(v, x), 1.0-(0.5*x)/v) + + @pytest.mark.parametrize('v,x', [ + (1, 1e16), + (1e20, 1e40), + (np.sqrt(np.finfo(float).max), np.finfo(float).max), + ]) + def test_huge_x(self, v, x): + """If x is much greater than v, the bounds + + x x + --------------------------- <= R <= --------------------------- + v-0.5+sqrt(x**2+(v+0.5)**2) v-0.5+sqrt(x**2+(v-0.5)**2) + + collapses to 1-R ~= (v-0.5)/x. Test against this asymptotic expression. + """ + assert_allclose(iv_ratio_c(v, x), (v-0.5)/x, rtol=1e-15, atol=0) + + @pytest.mark.parametrize('v,x', [ + (np.finfo(float).max, np.finfo(float).max), + (np.finfo(float).max / 3, np.finfo(float).max), + (np.finfo(float).max, np.finfo(float).max / 3), + ]) + def test_huge_v_x(self, v, x): + """If both x and v are very large, the bounds + + x x + --------------------------- <= R <= ----------------------- + v-0.5+sqrt(x**2+(v+0.5)**2) v-1+sqrt(x**2+(v+1)**2) + + collapses to 1 - R ~= 1 - x/(v+sqrt(x**2+v**2). Test against this + asymptotic expression, and in particular that no numerical overflow + occurs during intermediate calculations. + """ + t = x / v + expected = 1 - t / (1 + np.hypot(1, t)) + assert_allclose(iv_ratio_c(v, x), expected, rtol=4e-16, atol=0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_kolmogorov.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_kolmogorov.py new file mode 100644 index 0000000000000000000000000000000000000000..ade38115706808809890e1eba3b938562b0af6e6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_kolmogorov.py @@ -0,0 +1,491 @@ +import itertools + +import numpy as np +from numpy.testing import assert_ +from scipy.special._testutils import FuncData + +from scipy.special import kolmogorov, kolmogi, smirnov, smirnovi +from scipy.special._ufuncs import (_kolmogc, _kolmogci, _kolmogp, + _smirnovc, _smirnovci, _smirnovp) + +_rtol = 1e-10 + +class TestSmirnov: + def test_nan(self): + assert_(np.isnan(smirnov(1, np.nan))) + + def test_basic(self): + dataset = [(1, 0.1, 0.9), + (1, 0.875, 0.125), + (2, 0.875, 0.125 * 0.125), + (3, 0.875, 0.125 * 0.125 * 0.125)] + + dataset = np.asarray(dataset) + FuncData( + smirnov, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, -1] = 1 - dataset[:, -1] + FuncData( + _smirnovc, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_x_equals_0(self): + dataset = [(n, 0, 1) for n in itertools.chain(range(2, 20), range(1010, 1020))] + dataset = np.asarray(dataset) + FuncData( + smirnov, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, -1] = 1 - dataset[:, -1] + FuncData( + _smirnovc, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_x_equals_1(self): + dataset = [(n, 1, 0) for n in itertools.chain(range(2, 20), range(1010, 1020))] + dataset = np.asarray(dataset) + FuncData( + smirnov, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, -1] = 1 - dataset[:, -1] + FuncData( + _smirnovc, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_x_equals_0point5(self): + dataset = [(1, 0.5, 0.5), + (2, 0.5, 0.25), + (3, 0.5, 0.166666666667), + (4, 0.5, 0.09375), + (5, 0.5, 0.056), + (6, 0.5, 0.0327932098765), + (7, 0.5, 0.0191958707681), + (8, 0.5, 0.0112953186035), + (9, 0.5, 0.00661933257355), + (10, 0.5, 0.003888705)] + + dataset = np.asarray(dataset) + FuncData( + smirnov, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, -1] = 1 - dataset[:, -1] + FuncData( + _smirnovc, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_n_equals_1(self): + x = np.linspace(0, 1, 101, endpoint=True) + dataset = np.column_stack([[1]*len(x), x, 1-x]) + FuncData( + smirnov, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, -1] = 1 - dataset[:, -1] + FuncData( + _smirnovc, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_n_equals_2(self): + x = np.linspace(0.5, 1, 101, endpoint=True) + p = np.power(1-x, 2) + n = np.array([2] * len(x)) + dataset = np.column_stack([n, x, p]) + FuncData( + smirnov, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, -1] = 1 - dataset[:, -1] + FuncData( + _smirnovc, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_n_equals_3(self): + x = np.linspace(0.7, 1, 31, endpoint=True) + p = np.power(1-x, 3) + n = np.array([3] * len(x)) + dataset = np.column_stack([n, x, p]) + FuncData( + smirnov, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, -1] = 1 - dataset[:, -1] + FuncData( + _smirnovc, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_n_large(self): + # test for large values of n + # Probabilities should go down as n goes up + x = 0.4 + pvals = np.array([smirnov(n, x) for n in range(400, 1100, 20)]) + dfs = np.diff(pvals) + assert_(np.all(dfs <= 0), msg=f'Not all diffs negative {dfs}') + + +class TestSmirnovi: + def test_nan(self): + assert_(np.isnan(smirnovi(1, np.nan))) + + def test_basic(self): + dataset = [(1, 0.4, 0.6), + (1, 0.6, 0.4), + (1, 0.99, 0.01), + (1, 0.01, 0.99), + (2, 0.125 * 0.125, 0.875), + (3, 0.125 * 0.125 * 0.125, 0.875), + (10, 1.0 / 16 ** 10, 1 - 1.0 / 16)] + + dataset = np.asarray(dataset) + FuncData( + smirnovi, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, 1] = 1 - dataset[:, 1] + FuncData( + _smirnovci, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_x_equals_0(self): + dataset = [(n, 0, 1) for n in itertools.chain(range(2, 20), range(1010, 1020))] + dataset = np.asarray(dataset) + FuncData( + smirnovi, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, 1] = 1 - dataset[:, 1] + FuncData( + _smirnovci, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_x_equals_1(self): + dataset = [(n, 1, 0) for n in itertools.chain(range(2, 20), range(1010, 1020))] + dataset = np.asarray(dataset) + FuncData( + smirnovi, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, 1] = 1 - dataset[:, 1] + FuncData( + _smirnovci, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_n_equals_1(self): + pp = np.linspace(0, 1, 101, endpoint=True) + # dataset = np.array([(1, p, 1-p) for p in pp]) + dataset = np.column_stack([[1]*len(pp), pp, 1-pp]) + FuncData( + smirnovi, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, 1] = 1 - dataset[:, 1] + FuncData( + _smirnovci, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_n_equals_2(self): + x = np.linspace(0.5, 1, 101, endpoint=True) + p = np.power(1-x, 2) + n = np.array([2] * len(x)) + dataset = np.column_stack([n, p, x]) + FuncData( + smirnovi, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, 1] = 1 - dataset[:, 1] + FuncData( + _smirnovci, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_n_equals_3(self): + x = np.linspace(0.7, 1, 31, endpoint=True) + p = np.power(1-x, 3) + n = np.array([3] * len(x)) + dataset = np.column_stack([n, p, x]) + FuncData( + smirnovi, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, 1] = 1 - dataset[:, 1] + FuncData( + _smirnovci, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_round_trip(self): + def _sm_smi(n, p): + return smirnov(n, smirnovi(n, p)) + + def _smc_smci(n, p): + return _smirnovc(n, _smirnovci(n, p)) + + dataset = [(1, 0.4, 0.4), + (1, 0.6, 0.6), + (2, 0.875, 0.875), + (3, 0.875, 0.875), + (3, 0.125, 0.125), + (10, 0.999, 0.999), + (10, 0.0001, 0.0001)] + + dataset = np.asarray(dataset) + FuncData( + _sm_smi, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + FuncData( + _smc_smci, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_x_equals_0point5(self): + dataset = [(1, 0.5, 0.5), + (2, 0.5, 0.366025403784), + (2, 0.25, 0.5), + (3, 0.5, 0.297156508177), + (4, 0.5, 0.255520481121), + (5, 0.5, 0.234559536069), + (6, 0.5, 0.21715965898), + (7, 0.5, 0.202722580034), + (8, 0.5, 0.190621765256), + (9, 0.5, 0.180363501362), + (10, 0.5, 0.17157867006)] + + dataset = np.asarray(dataset) + FuncData( + smirnovi, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + dataset[:, 1] = 1 - dataset[:, 1] + FuncData( + _smirnovci, dataset, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + +class TestSmirnovp: + def test_nan(self): + assert_(np.isnan(_smirnovp(1, np.nan))) + + def test_basic(self): + # Check derivative at endpoints + n1_10 = np.arange(1, 10) + dataset0 = np.column_stack([n1_10, + np.full_like(n1_10, 0), + np.full_like(n1_10, -1)]) + FuncData( + _smirnovp, dataset0, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + n2_10 = np.arange(2, 10) + dataset1 = np.column_stack([n2_10, + np.full_like(n2_10, 1.0), + np.full_like(n2_10, 0)]) + FuncData( + _smirnovp, dataset1, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_oneminusoneovern(self): + # Check derivative at x=1-1/n + n = np.arange(1, 20) + x = 1.0/n + xm1 = 1-1.0/n + pp1 = -n * x**(n-1) + pp1 -= (1-np.sign(n-2)**2) * 0.5 # n=2, x=0.5, 1-1/n = 0.5, need to adjust + dataset1 = np.column_stack([n, xm1, pp1]) + FuncData( + _smirnovp, dataset1, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_oneovertwon(self): + # Check derivative at x=1/2n (Discontinuous at x=1/n, so check at x=1/2n) + n = np.arange(1, 20) + x = 1.0/2/n + pp = -(n*x+1) * (1+x)**(n-2) + dataset0 = np.column_stack([n, x, pp]) + FuncData( + _smirnovp, dataset0, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_oneovern(self): + # Check derivative at x=1/n + # (Discontinuous at x=1/n, hard to tell if x==1/n, only use n=power of 2) + n = 2**np.arange(1, 10) + x = 1.0/n + pp = -(n*x+1) * (1+x)**(n-2) + 0.5 + dataset0 = np.column_stack([n, x, pp]) + FuncData( + _smirnovp, dataset0, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + def test_oneovernclose(self): + # Check derivative at x=1/n + # (Discontinuous at x=1/n, test on either side: x=1/n +/- 2epsilon) + n = np.arange(3, 20) + + x = 1.0/n - 2*np.finfo(float).eps + pp = -(n*x+1) * (1+x)**(n-2) + dataset0 = np.column_stack([n, x, pp]) + FuncData( + _smirnovp, dataset0, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + x = 1.0/n + 2*np.finfo(float).eps + pp = -(n*x+1) * (1+x)**(n-2) + 1 + dataset1 = np.column_stack([n, x, pp]) + FuncData( + _smirnovp, dataset1, (0, 1), 2, rtol=_rtol + ).check(dtypes=[int, float, float]) + + +class TestKolmogorov: + def test_nan(self): + assert_(np.isnan(kolmogorov(np.nan))) + + def test_basic(self): + dataset = [(0, 1.0), + (0.5, 0.96394524366487511), + (0.8275735551899077, 0.5000000000000000), + (1, 0.26999967167735456), + (2, 0.00067092525577969533)] + + dataset = np.asarray(dataset) + FuncData(kolmogorov, dataset, (0,), 1, rtol=_rtol).check() + + def test_linspace(self): + x = np.linspace(0, 2.0, 21) + dataset = [1.0000000000000000, 1.0000000000000000, 0.9999999999994950, + 0.9999906941986655, 0.9971923267772983, 0.9639452436648751, + 0.8642827790506042, 0.7112351950296890, 0.5441424115741981, + 0.3927307079406543, 0.2699996716773546, 0.1777181926064012, + 0.1122496666707249, 0.0680922218447664, 0.0396818795381144, + 0.0222179626165251, 0.0119520432391966, 0.0061774306344441, + 0.0030676213475797, 0.0014636048371873, 0.0006709252557797] + + dataset_c = [0.0000000000000000, 6.609305242245699e-53, 5.050407338670114e-13, + 9.305801334566668e-06, 0.0028076732227017, 0.0360547563351249, + 0.1357172209493958, 0.2887648049703110, 0.4558575884258019, + 0.6072692920593457, 0.7300003283226455, 0.8222818073935988, + 0.8877503333292751, 0.9319077781552336, 0.9603181204618857, + 0.9777820373834749, 0.9880479567608034, 0.9938225693655559, + 0.9969323786524203, 0.9985363951628127, 0.9993290747442203] + + dataset = np.column_stack([x, dataset]) + FuncData(kolmogorov, dataset, (0,), 1, rtol=_rtol).check() + dataset_c = np.column_stack([x, dataset_c]) + FuncData(_kolmogc, dataset_c, (0,), 1, rtol=_rtol).check() + + def test_linspacei(self): + p = np.linspace(0, 1.0, 21, endpoint=True) + dataset = [np.inf, 1.3580986393225507, 1.2238478702170823, + 1.1379465424937751, 1.0727491749396481, 1.0191847202536859, + 0.9730633753323726, 0.9320695842357622, 0.8947644549851197, + 0.8601710725555463, 0.8275735551899077, 0.7964065373291559, + 0.7661855555617682, 0.7364542888171910, 0.7067326523068980, + 0.6764476915028201, 0.6448126061663567, 0.6105590999244391, + 0.5711732651063401, 0.5196103791686224, 0.0000000000000000] + + dataset_c = [0.0000000000000000, 0.5196103791686225, 0.5711732651063401, + 0.6105590999244391, 0.6448126061663567, 0.6764476915028201, + 0.7067326523068980, 0.7364542888171910, 0.7661855555617682, + 0.7964065373291559, 0.8275735551899077, 0.8601710725555463, + 0.8947644549851196, 0.9320695842357622, 0.9730633753323727, + 1.0191847202536859, 1.0727491749396481, 1.1379465424937754, + 1.2238478702170825, 1.3580986393225509, np.inf] + + dataset = np.column_stack([p[1:], dataset[1:]]) + FuncData(kolmogi, dataset, (0,), 1, rtol=_rtol).check() + dataset_c = np.column_stack([p[:-1], dataset_c[:-1]]) + FuncData(_kolmogci, dataset_c, (0,), 1, rtol=_rtol).check() + + def test_smallx(self): + epsilon = 0.1 ** np.arange(1, 14) + x = np.array([0.571173265106, 0.441027698518, 0.374219690278, 0.331392659217, + 0.300820537459, 0.277539353999, 0.259023494805, 0.243829561254, + 0.231063086389, 0.220135543236, 0.210641372041, 0.202290283658, + 0.19487060742]) + + dataset = np.column_stack([x, 1-epsilon]) + FuncData(kolmogorov, dataset, (0,), 1, rtol=_rtol).check() + + def test_round_trip(self): + def _ki_k(_x): + return kolmogi(kolmogorov(_x)) + + def _kci_kc(_x): + return _kolmogci(_kolmogc(_x)) + + x = np.linspace(0.0, 2.0, 21, endpoint=True) + # Exclude 0.1, 0.2. 0.2 almost makes succeeds, but 0.1 has no chance. + x02 = x[(x == 0) | (x > 0.21)] + dataset02 = np.column_stack([x02, x02]) + FuncData(_ki_k, dataset02, (0,), 1, rtol=_rtol).check() + + dataset = np.column_stack([x, x]) + FuncData(_kci_kc, dataset, (0,), 1, rtol=_rtol).check() + + +class TestKolmogi: + def test_nan(self): + assert_(np.isnan(kolmogi(np.nan))) + + def test_basic(self): + dataset = [(1.0, 0), + (0.96394524366487511, 0.5), + (0.9, 0.571173265106), + (0.5000000000000000, 0.8275735551899077), + (0.26999967167735456, 1), + (0.00067092525577969533, 2)] + + dataset = np.asarray(dataset) + FuncData(kolmogi, dataset, (0,), 1, rtol=_rtol).check() + + def test_smallpcdf(self): + epsilon = 0.5 ** np.arange(1, 55, 3) + # kolmogi(1-p) == _kolmogci(p) if 1-(1-p) == p, but not necessarily otherwise + # Use epsilon s.t. 1-(1-epsilon)) == epsilon, + # so can use same x-array for both results + + x = np.array([0.8275735551899077, 0.5345255069097583, 0.4320114038786941, + 0.3736868442620478, 0.3345161714909591, 0.3057833329315859, + 0.2835052890528936, 0.2655578150208676, 0.2506869966107999, + 0.2380971058736669, 0.2272549289962079, 0.2177876361600040, + 0.2094254686862041, 0.2019676748836232, 0.1952612948137504, + 0.1891874239646641, 0.1836520225050326, 0.1785795904846466]) + + dataset = np.column_stack([1-epsilon, x]) + FuncData(kolmogi, dataset, (0,), 1, rtol=_rtol).check() + + dataset = np.column_stack([epsilon, x]) + FuncData(_kolmogci, dataset, (0,), 1, rtol=_rtol).check() + + def test_smallpsf(self): + epsilon = 0.5 ** np.arange(1, 55, 3) + # kolmogi(p) == _kolmogci(1-p) if 1-(1-p) == p, but not necessarily otherwise + # Use epsilon s.t. 1-(1-epsilon)) == epsilon, + # so can use same x-array for both results + + x = np.array([0.8275735551899077, 1.3163786275161036, 1.6651092133663343, + 1.9525136345289607, 2.2027324540033235, 2.4272929437460848, + 2.6327688477341593, 2.8233300509220260, 3.0018183401530627, + 3.1702735084088891, 3.3302184446307912, 3.4828258153113318, + 3.6290214150152051, 3.7695513262825959, 3.9050272690877326, + 4.0359582187082550, 4.1627730557884890, 4.2858371743264527]) + + dataset = np.column_stack([epsilon, x]) + FuncData(kolmogi, dataset, (0,), 1, rtol=_rtol).check() + + dataset = np.column_stack([1-epsilon, x]) + FuncData(_kolmogci, dataset, (0,), 1, rtol=_rtol).check() + + def test_round_trip(self): + def _k_ki(_p): + return kolmogorov(kolmogi(_p)) + + p = np.linspace(0.1, 1.0, 10, endpoint=True) + dataset = np.column_stack([p, p]) + FuncData(_k_ki, dataset, (0,), 1, rtol=_rtol).check() + + +class TestKolmogp: + def test_nan(self): + assert_(np.isnan(_kolmogp(np.nan))) + + def test_basic(self): + dataset = [(0.000000, -0.0), + (0.200000, -1.532420541338916e-10), + (0.400000, -0.1012254419260496), + (0.600000, -1.324123244249925), + (0.800000, -1.627024345636592), + (1.000000, -1.071948558356941), + (1.200000, -0.538512430720529), + (1.400000, -0.2222133182429472), + (1.600000, -0.07649302775520538), + (1.800000, -0.02208687346347873), + (2.000000, -0.005367402045629683)] + + dataset = np.asarray(dataset) + FuncData(_kolmogp, dataset, (0,), 1, rtol=_rtol).check() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_lambertw.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_lambertw.py new file mode 100644 index 0000000000000000000000000000000000000000..c7fde685406661b821bb1dc490ca0da173eb4bd0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_lambertw.py @@ -0,0 +1,109 @@ +# +# Tests for the lambertw function, +# Adapted from the MPMath tests [1] by Yosef Meller, mellerf@netvision.net.il +# Distributed under the same license as SciPy itself. +# +# [1] mpmath source code, Subversion revision 992 +# http://code.google.com/p/mpmath/source/browse/trunk/mpmath/tests/test_functions2.py?spec=svn994&r=992 + +import pytest +import numpy as np +from numpy.testing import assert_, assert_equal, assert_array_almost_equal +from scipy.special import lambertw +from numpy import nan, inf, pi, e, isnan, log, r_, array, complex128 + +from scipy.special._testutils import FuncData + + +def test_values(): + assert_(isnan(lambertw(nan))) + assert_equal(lambertw(inf,1).real, inf) + assert_equal(lambertw(inf,1).imag, 2*pi) + assert_equal(lambertw(-inf,1).real, inf) + assert_equal(lambertw(-inf,1).imag, 3*pi) + + assert_equal(lambertw(1.), lambertw(1., 0)) + + data = [ + (0,0, 0), + (0+0j,0, 0), + (inf,0, inf), + (0,-1, -inf), + (0,1, -inf), + (0,3, -inf), + (e,0, 1), + (1,0, 0.567143290409783873), + (-pi/2,0, 1j*pi/2), + (-log(2)/2,0, -log(2)), + (0.25,0, 0.203888354702240164), + (-0.25,0, -0.357402956181388903), + (-1./10000,0, -0.000100010001500266719), + (-0.25,-1, -2.15329236411034965), + (0.25,-1, -3.00899800997004620-4.07652978899159763j), + (-0.25,-1, -2.15329236411034965), + (0.25,1, -3.00899800997004620+4.07652978899159763j), + (-0.25,1, -3.48973228422959210+7.41405453009603664j), + (-4,0, 0.67881197132094523+1.91195078174339937j), + (-4,1, -0.66743107129800988+7.76827456802783084j), + (-4,-1, 0.67881197132094523-1.91195078174339937j), + (1000,0, 5.24960285240159623), + (1000,1, 4.91492239981054535+5.44652615979447070j), + (1000,-1, 4.91492239981054535-5.44652615979447070j), + (1000,5, 3.5010625305312892+29.9614548941181328j), + (3+4j,0, 1.281561806123775878+0.533095222020971071j), + (-0.4+0.4j,0, -0.10396515323290657+0.61899273315171632j), + (3+4j,1, -0.11691092896595324+5.61888039871282334j), + (3+4j,-1, 0.25856740686699742-3.85211668616143559j), + (-0.5,-1, -0.794023632344689368-0.770111750510379110j), + (-1./10000,1, -11.82350837248724344+6.80546081842002101j), + (-1./10000,-1, -11.6671145325663544), + (-1./10000,-2, -11.82350837248724344-6.80546081842002101j), + (-1./100000,4, -14.9186890769540539+26.1856750178782046j), + (-1./100000,5, -15.0931437726379218666+32.5525721210262290086j), + ((2+1j)/10,0, 0.173704503762911669+0.071781336752835511j), + ((2+1j)/10,1, -3.21746028349820063+4.56175438896292539j), + ((2+1j)/10,-1, -3.03781405002993088-3.53946629633505737j), + ((2+1j)/10,4, -4.6878509692773249+23.8313630697683291j), + (-(2+1j)/10,0, -0.226933772515757933-0.164986470020154580j), + (-(2+1j)/10,1, -2.43569517046110001+0.76974067544756289j), + (-(2+1j)/10,-1, -3.54858738151989450-6.91627921869943589j), + (-(2+1j)/10,4, -4.5500846928118151+20.6672982215434637j), + (pi,0, 1.073658194796149172092178407024821347547745350410314531), + + # Former bug in generated branch, + (-0.5+0.002j,0, -0.78917138132659918344 + 0.76743539379990327749j), + (-0.5-0.002j,0, -0.78917138132659918344 - 0.76743539379990327749j), + (-0.448+0.4j,0, -0.11855133765652382241 + 0.66570534313583423116j), + (-0.448-0.4j,0, -0.11855133765652382241 - 0.66570534313583423116j), + ] + data = array(data, dtype=complex128) + + def w(x, y): + return lambertw(x, y.real.astype(int)) + with np.errstate(all='ignore'): + FuncData(w, data, (0,1), 2, rtol=1e-10, atol=1e-13).check() + + +def test_ufunc(): + assert_array_almost_equal( + lambertw(r_[0., e, 1.]), r_[0., 1., 0.567143290409783873]) + + +def test_lambertw_ufunc_loop_selection(): + # see https://github.com/scipy/scipy/issues/4895 + dt = np.dtype(np.complex128) + assert_equal(lambertw(0, 0, 0).dtype, dt) + assert_equal(lambertw([0], 0, 0).dtype, dt) + assert_equal(lambertw(0, [0], 0).dtype, dt) + assert_equal(lambertw(0, 0, [0]).dtype, dt) + assert_equal(lambertw([0], [0], [0]).dtype, dt) + + +@pytest.mark.parametrize('z', [1e-316, -2e-320j, -5e-318+1e-320j]) +def test_lambertw_subnormal_k0(z): + # Verify that subnormal inputs are handled correctly on + # the branch k=0 (regression test for gh-16291). + w = lambertw(z) + # For values this small, we can be sure that numerically, + # lambertw(z) is z. + assert w == z diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_legendre.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_legendre.py new file mode 100644 index 0000000000000000000000000000000000000000..430a4175426e30bde02be7972b3144eed4923b27 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_legendre.py @@ -0,0 +1,1518 @@ +import math + +import numpy as np + +import pytest +from numpy.testing import (assert_equal, assert_almost_equal, assert_array_almost_equal, + assert_allclose, suppress_warnings) + +from scipy import special +from scipy.special import (legendre_p, legendre_p_all, assoc_legendre_p, + assoc_legendre_p_all, sph_legendre_p, sph_legendre_p_all) + +# The functions lpn, lpmn, clpmn, appearing below are +# deprecated in favor of legendre_p_all, assoc_legendre_p_all, and +# assoc_legendre_p_all (assoc_legendre_p_all covers lpmn and clpmn) +# respectively. The deprecated functions listed above are implemented as +# shims around their respective replacements. The replacements are tested +# separately, but tests for the deprecated functions remain to verify the +# correctness of the shims. + +# Base polynomials come from Abrahmowitz and Stegan +class TestLegendre: + def test_legendre(self): + leg0 = special.legendre(0) + leg1 = special.legendre(1) + leg2 = special.legendre(2) + leg3 = special.legendre(3) + leg4 = special.legendre(4) + leg5 = special.legendre(5) + assert_equal(leg0.c, [1]) + assert_equal(leg1.c, [1,0]) + assert_almost_equal(leg2.c, np.array([3,0,-1])/2.0, decimal=13) + assert_almost_equal(leg3.c, np.array([5,0,-3,0])/2.0) + assert_almost_equal(leg4.c, np.array([35,0,-30,0,3])/8.0) + assert_almost_equal(leg5.c, np.array([63,0,-70,0,15,0])/8.0) + + @pytest.mark.parametrize('n', [1, 2, 3, 4, 5]) + @pytest.mark.parametrize('zr', [0.5241717, 12.80232, -9.699001, + 0.5122437, 0.1714377]) + @pytest.mark.parametrize('zi', [9.766818, 0.2999083, 8.24726, -22.84843, + -0.8792666]) + def test_lpn_against_clpmn(self, n, zr, zi): + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + reslpn = special.lpn(n, zr + zi*1j) + resclpmn = special.clpmn(0, n, zr+zi*1j) + + assert_allclose(reslpn[0], resclpmn[0][0]) + assert_allclose(reslpn[1], resclpmn[1][0]) + +class TestLegendreP: + @pytest.mark.parametrize("shape", [(10,), (4, 9), (3, 5, 7)]) + def test_ode(self, shape): + rng = np.random.default_rng(1234) + + n = rng.integers(0, 100, shape) + x = rng.uniform(-1, 1, shape) + + p, p_jac, p_hess = legendre_p(n, x, diff_n=2) + + assert p.shape == shape + assert p_jac.shape == p.shape + assert p_hess.shape == p_jac.shape + + err = (1 - x * x) * p_hess - 2 * x * p_jac + n * (n + 1) * p + np.testing.assert_allclose(err, 0, atol=1e-10) + + @pytest.mark.parametrize("n_max", [1, 2, 4, 8, 16, 32]) + @pytest.mark.parametrize("x_shape", [(10,), (4, 9), (3, 5, 7)]) + def test_all_ode(self, n_max, x_shape): + rng = np.random.default_rng(1234) + + x = rng.uniform(-1, 1, x_shape) + p, p_jac, p_hess = legendre_p_all(n_max, x, diff_n=2) + + n = np.arange(n_max + 1) + n = np.expand_dims(n, axis = tuple(range(1, x.ndim + 1))) + + assert p.shape == (len(n),) + x.shape + assert p_jac.shape == p.shape + assert p_hess.shape == p_jac.shape + + err = (1 - x * x) * p_hess - 2 * x * p_jac + n * (n + 1) * p + np.testing.assert_allclose(err, 0, atol=1e-10) + + def test_legacy(self): + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + p, pd = special.lpn(2, 0.5) + + assert_array_almost_equal(p, [1.00000, 0.50000, -0.12500], 4) + assert_array_almost_equal(pd, [0.00000, 1.00000, 1.50000], 4) + +class TestAssocLegendreP: + @pytest.mark.parametrize("shape", [(10,), (4, 9), (3, 5, 7, 10)]) + @pytest.mark.parametrize("m_max", [5, 4]) + @pytest.mark.parametrize("n_max", [7, 10]) + def test_lpmn(self, shape, n_max, m_max): + rng = np.random.default_rng(1234) + + x = rng.uniform(-0.99, 0.99, shape) + p_all, p_all_jac, p_all_hess = \ + assoc_legendre_p_all(n_max, m_max, x, diff_n=2) + + n = np.arange(n_max + 1) + n = np.expand_dims(n, axis = tuple(range(1, x.ndim + 2))) + + m = np.concatenate([np.arange(m_max + 1), np.arange(-m_max, 0)]) + m = np.expand_dims(m, axis = (0,) + tuple(range(2, x.ndim + 2))) + + x = np.expand_dims(x, axis = (0, 1)) + p, p_jac, p_hess = assoc_legendre_p(n, m, x, diff_n=2) + + np.testing.assert_allclose(p, p_all) + np.testing.assert_allclose(p_jac, p_all_jac) + np.testing.assert_allclose(p_hess, p_all_hess) + + @pytest.mark.parametrize("shape", [(10,), (4, 9), (3, 5, 7, 10)]) + @pytest.mark.parametrize("norm", [True, False]) + def test_ode(self, shape, norm): + rng = np.random.default_rng(1234) + + n = rng.integers(0, 10, shape) + m = rng.integers(-10, 10, shape) + x = rng.uniform(-1, 1, shape) + + p, p_jac, p_hess = assoc_legendre_p(n, m, x, norm=norm, diff_n=2) + + assert p.shape == shape + assert p_jac.shape == p.shape + assert p_hess.shape == p_jac.shape + + np.testing.assert_allclose((1 - x * x) * p_hess, + 2 * x * p_jac - (n * (n + 1) - m * m / (1 - x * x)) * p, + rtol=1e-05, atol=1e-08) + + @pytest.mark.parametrize("shape", [(10,), (4, 9), (3, 5, 7)]) + def test_all(self, shape): + rng = np.random.default_rng(1234) + + n_max = 20 + m_max = 20 + + x = rng.uniform(-0.99, 0.99, shape) + + p, p_jac, p_hess = assoc_legendre_p_all(n_max, m_max, x, diff_n=2) + + m = np.concatenate([np.arange(m_max + 1), np.arange(-m_max, 0)]) + n = np.arange(n_max + 1) + + n = np.expand_dims(n, axis = tuple(range(1, x.ndim + 2))) + m = np.expand_dims(m, axis = (0,) + tuple(range(2, x.ndim + 2))) + np.testing.assert_allclose((1 - x * x) * p_hess, + 2 * x * p_jac - (n * (n + 1) - m * m / (1 - x * x)) * p, + rtol=1e-05, atol=1e-08) + + @pytest.mark.parametrize("shape", [(10,), (4, 9), (3, 5, 7)]) + @pytest.mark.parametrize("norm", [True, False]) + def test_specific(self, shape, norm): + rng = np.random.default_rng(1234) + + x = rng.uniform(-0.99, 0.99, shape) + + p, p_jac = assoc_legendre_p_all(4, 4, x, norm=norm, diff_n=1) + + np.testing.assert_allclose(p[0, 0], + assoc_legendre_p_0_0(x, norm=norm)) + np.testing.assert_allclose(p[0, 1], 0) + np.testing.assert_allclose(p[0, 2], 0) + np.testing.assert_allclose(p[0, 3], 0) + np.testing.assert_allclose(p[0, 4], 0) + np.testing.assert_allclose(p[0, -3], 0) + np.testing.assert_allclose(p[0, -2], 0) + np.testing.assert_allclose(p[0, -1], 0) + + np.testing.assert_allclose(p[1, 0], + assoc_legendre_p_1_0(x, norm=norm)) + np.testing.assert_allclose(p[1, 1], + assoc_legendre_p_1_1(x, norm=norm)) + np.testing.assert_allclose(p[1, 2], 0) + np.testing.assert_allclose(p[1, 3], 0) + np.testing.assert_allclose(p[1, 4], 0) + np.testing.assert_allclose(p[1, -4], 0) + np.testing.assert_allclose(p[1, -3], 0) + np.testing.assert_allclose(p[1, -2], 0) + np.testing.assert_allclose(p[1, -1], + assoc_legendre_p_1_m1(x, norm=norm)) + + np.testing.assert_allclose(p[2, 0], + assoc_legendre_p_2_0(x, norm=norm)) + np.testing.assert_allclose(p[2, 1], + assoc_legendre_p_2_1(x, norm=norm)) + np.testing.assert_allclose(p[2, 2], + assoc_legendre_p_2_2(x, norm=norm)) + np.testing.assert_allclose(p[2, 3], 0) + np.testing.assert_allclose(p[2, 4], 0) + np.testing.assert_allclose(p[2, -4], 0) + np.testing.assert_allclose(p[2, -3], 0) + np.testing.assert_allclose(p[2, -2], + assoc_legendre_p_2_m2(x, norm=norm)) + np.testing.assert_allclose(p[2, -1], + assoc_legendre_p_2_m1(x, norm=norm)) + + np.testing.assert_allclose(p[3, 0], + assoc_legendre_p_3_0(x, norm=norm)) + np.testing.assert_allclose(p[3, 1], + assoc_legendre_p_3_1(x, norm=norm)) + np.testing.assert_allclose(p[3, 2], + assoc_legendre_p_3_2(x, norm=norm)) + np.testing.assert_allclose(p[3, 3], + assoc_legendre_p_3_3(x, norm=norm)) + np.testing.assert_allclose(p[3, 4], 0) + np.testing.assert_allclose(p[3, -4], 0) + np.testing.assert_allclose(p[3, -3], + assoc_legendre_p_3_m3(x, norm=norm)) + np.testing.assert_allclose(p[3, -2], + assoc_legendre_p_3_m2(x, norm=norm)) + np.testing.assert_allclose(p[3, -1], + assoc_legendre_p_3_m1(x, norm=norm)) + + np.testing.assert_allclose(p[4, 0], + assoc_legendre_p_4_0(x, norm=norm)) + np.testing.assert_allclose(p[4, 1], + assoc_legendre_p_4_1(x, norm=norm)) + np.testing.assert_allclose(p[4, 2], + assoc_legendre_p_4_2(x, norm=norm)) + np.testing.assert_allclose(p[4, 3], + assoc_legendre_p_4_3(x, norm=norm)) + np.testing.assert_allclose(p[4, 4], + assoc_legendre_p_4_4(x, norm=norm)) + np.testing.assert_allclose(p[4, -4], + assoc_legendre_p_4_m4(x, norm=norm)) + np.testing.assert_allclose(p[4, -3], + assoc_legendre_p_4_m3(x, norm=norm)) + np.testing.assert_allclose(p[4, -2], + assoc_legendre_p_4_m2(x, norm=norm)) + np.testing.assert_allclose(p[4, -1], + assoc_legendre_p_4_m1(x, norm=norm)) + + np.testing.assert_allclose(p_jac[0, 0], + assoc_legendre_p_0_0_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[0, 1], 0) + np.testing.assert_allclose(p_jac[0, 2], 0) + np.testing.assert_allclose(p_jac[0, 3], 0) + np.testing.assert_allclose(p_jac[0, 4], 0) + np.testing.assert_allclose(p_jac[0, -4], 0) + np.testing.assert_allclose(p_jac[0, -3], 0) + np.testing.assert_allclose(p_jac[0, -2], 0) + np.testing.assert_allclose(p_jac[0, -1], 0) + + np.testing.assert_allclose(p_jac[1, 0], + assoc_legendre_p_1_0_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[1, 1], + assoc_legendre_p_1_1_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[1, 2], 0) + np.testing.assert_allclose(p_jac[1, 3], 0) + np.testing.assert_allclose(p_jac[1, 4], 0) + np.testing.assert_allclose(p_jac[1, -4], 0) + np.testing.assert_allclose(p_jac[1, -3], 0) + np.testing.assert_allclose(p_jac[1, -2], 0) + np.testing.assert_allclose(p_jac[1, -1], + assoc_legendre_p_1_m1_jac(x, norm=norm)) + + np.testing.assert_allclose(p_jac[2, 0], + assoc_legendre_p_2_0_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[2, 1], + assoc_legendre_p_2_1_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[2, 2], + assoc_legendre_p_2_2_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[2, 3], 0) + np.testing.assert_allclose(p_jac[2, 4], 0) + np.testing.assert_allclose(p_jac[2, -4], 0) + np.testing.assert_allclose(p_jac[2, -3], 0) + np.testing.assert_allclose(p_jac[2, -2], + assoc_legendre_p_2_m2_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[2, -1], + assoc_legendre_p_2_m1_jac(x, norm=norm)) + + np.testing.assert_allclose(p_jac[3, 0], + assoc_legendre_p_3_0_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[3, 1], + assoc_legendre_p_3_1_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[3, 2], + assoc_legendre_p_3_2_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[3, 3], + assoc_legendre_p_3_3_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[3, 4], 0) + np.testing.assert_allclose(p_jac[3, -4], 0) + np.testing.assert_allclose(p_jac[3, -3], + assoc_legendre_p_3_m3_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[3, -2], + assoc_legendre_p_3_m2_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[3, -1], + assoc_legendre_p_3_m1_jac(x, norm=norm)) + + np.testing.assert_allclose(p_jac[4, 0], + assoc_legendre_p_4_0_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[4, 1], + assoc_legendre_p_4_1_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[4, 2], + assoc_legendre_p_4_2_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[4, 3], + assoc_legendre_p_4_3_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[4, 4], + assoc_legendre_p_4_4_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[4, -4], + assoc_legendre_p_4_m4_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[4, -3], + assoc_legendre_p_4_m3_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[4, -2], + assoc_legendre_p_4_m2_jac(x, norm=norm)) + np.testing.assert_allclose(p_jac[4, -1], + assoc_legendre_p_4_m1_jac(x, norm=norm)) + + @pytest.mark.parametrize("m_max", [7]) + @pytest.mark.parametrize("n_max", [10]) + @pytest.mark.parametrize("x", [1, -1]) + def test_all_limits(self, m_max, n_max, x): + p, p_jac = assoc_legendre_p_all(n_max, m_max, x, diff_n=1) + + n = np.arange(n_max + 1) + + np.testing.assert_allclose(p_jac[:, 0], + pow(x, n + 1) * n * (n + 1) / 2) + np.testing.assert_allclose(p_jac[:, 1], + np.where(n >= 1, pow(x, n) * np.inf, 0)) + np.testing.assert_allclose(p_jac[:, 2], + np.where(n >= 2, -pow(x, n + 1) * (n + 2) * (n + 1) * n * (n - 1) / 4, 0)) + np.testing.assert_allclose(p_jac[:, -2], + np.where(n >= 2, -pow(x, n + 1) / 4, 0)) + np.testing.assert_allclose(p_jac[:, -1], + np.where(n >= 1, -pow(x, n) * np.inf, 0)) + + for m in range(3, m_max + 1): + np.testing.assert_allclose(p_jac[:, m], 0) + np.testing.assert_allclose(p_jac[:, -m], 0) + + @pytest.mark.parametrize("m_max", [3, 5, 10]) + @pytest.mark.parametrize("n_max", [10]) + def test_legacy(self, m_max, n_max): + x = 0.5 + p, p_jac = assoc_legendre_p_all(n_max, m_max, x, diff_n=1) + + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + + p_legacy, p_jac_legacy = special.lpmn(m_max, n_max, x) + for m in range(m_max + 1): + np.testing.assert_allclose(p_legacy[m], p[:, m]) + + p_legacy, p_jac_legacy = special.lpmn(-m_max, n_max, x) + for m in range(m_max + 1): + np.testing.assert_allclose(p_legacy[m], p[:, -m]) + +class TestMultiAssocLegendreP: + @pytest.mark.parametrize("shape", [(1000,), (4, 9), (3, 5, 7)]) + @pytest.mark.parametrize("branch_cut", [2, 3]) + @pytest.mark.parametrize("z_min, z_max", [(-10 - 10j, 10 + 10j), + (-1, 1), (-10j, 10j)]) + @pytest.mark.parametrize("norm", [True, False]) + def test_specific(self, shape, branch_cut, z_min, z_max, norm): + rng = np.random.default_rng(1234) + + z = rng.uniform(z_min.real, z_max.real, shape) + \ + 1j * rng.uniform(z_min.imag, z_max.imag, shape) + + p, p_jac = assoc_legendre_p_all(4, 4, + z, branch_cut=branch_cut, norm=norm, diff_n=1) + + np.testing.assert_allclose(p[0, 0], + assoc_legendre_p_0_0(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[0, 1], 0) + np.testing.assert_allclose(p[0, 2], 0) + np.testing.assert_allclose(p[0, 3], 0) + np.testing.assert_allclose(p[0, 4], 0) + np.testing.assert_allclose(p[0, -4], 0) + np.testing.assert_allclose(p[0, -3], 0) + np.testing.assert_allclose(p[0, -2], 0) + np.testing.assert_allclose(p[0, -1], 0) + + np.testing.assert_allclose(p[1, 0], + assoc_legendre_p_1_0(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[1, 1], + assoc_legendre_p_1_1(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[1, 2], 0) + np.testing.assert_allclose(p[1, 3], 0) + np.testing.assert_allclose(p[1, 4], 0) + np.testing.assert_allclose(p[1, -4], 0) + np.testing.assert_allclose(p[1, -3], 0) + np.testing.assert_allclose(p[1, -2], 0) + np.testing.assert_allclose(p[1, -1], + assoc_legendre_p_1_m1(z, branch_cut=branch_cut, norm=norm)) + + np.testing.assert_allclose(p[2, 0], + assoc_legendre_p_2_0(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[2, 1], + assoc_legendre_p_2_1(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[2, 2], + assoc_legendre_p_2_2(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[2, 3], 0) + np.testing.assert_allclose(p[2, 4], 0) + np.testing.assert_allclose(p[2, -4], 0) + np.testing.assert_allclose(p[2, -3], 0) + np.testing.assert_allclose(p[2, -2], + assoc_legendre_p_2_m2(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[2, -1], + assoc_legendre_p_2_m1(z, branch_cut=branch_cut, norm=norm)) + + np.testing.assert_allclose(p[3, 0], + assoc_legendre_p_3_0(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[3, 1], + assoc_legendre_p_3_1(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[3, 2], + assoc_legendre_p_3_2(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[3, 3], + assoc_legendre_p_3_3(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[3, 4], 0) + np.testing.assert_allclose(p[3, -4], 0) + np.testing.assert_allclose(p[3, -3], + assoc_legendre_p_3_m3(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[3, -2], + assoc_legendre_p_3_m2(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[3, -1], + assoc_legendre_p_3_m1(z, branch_cut=branch_cut, norm=norm)) + + np.testing.assert_allclose(p[4, 0], + assoc_legendre_p_4_0(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[4, 1], + assoc_legendre_p_4_1(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[4, 2], + assoc_legendre_p_4_2(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[4, 3], + assoc_legendre_p_4_3(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[4, 4], + assoc_legendre_p_4_4(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[4, -4], + assoc_legendre_p_4_m4(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[4, -3], + assoc_legendre_p_4_m3(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[4, -2], + assoc_legendre_p_4_m2(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p[4, -1], + assoc_legendre_p_4_m1(z, branch_cut=branch_cut, norm=norm)) + + np.testing.assert_allclose(p_jac[0, 0], + assoc_legendre_p_0_0_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[0, 1], 0) + np.testing.assert_allclose(p_jac[0, 2], 0) + np.testing.assert_allclose(p_jac[0, 3], 0) + np.testing.assert_allclose(p_jac[0, 4], 0) + np.testing.assert_allclose(p_jac[0, -4], 0) + np.testing.assert_allclose(p_jac[0, -3], 0) + np.testing.assert_allclose(p_jac[0, -2], 0) + np.testing.assert_allclose(p_jac[0, -1], 0) + + np.testing.assert_allclose(p_jac[1, 0], + assoc_legendre_p_1_0_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[1, 1], + assoc_legendre_p_1_1_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[1, 2], 0) + np.testing.assert_allclose(p_jac[1, 3], 0) + np.testing.assert_allclose(p_jac[1, 4], 0) + np.testing.assert_allclose(p_jac[1, -4], 0) + np.testing.assert_allclose(p_jac[1, -3], 0) + np.testing.assert_allclose(p_jac[1, -2], 0) + np.testing.assert_allclose(p_jac[1, -1], + assoc_legendre_p_1_m1_jac(z, branch_cut=branch_cut, norm=norm)) + + np.testing.assert_allclose(p_jac[2, 0], + assoc_legendre_p_2_0_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[2, 1], + assoc_legendre_p_2_1_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[2, 2], + assoc_legendre_p_2_2_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[2, 3], 0) + np.testing.assert_allclose(p_jac[2, 4], 0) + np.testing.assert_allclose(p_jac[2, -4], 0) + np.testing.assert_allclose(p_jac[2, -3], 0) + np.testing.assert_allclose(p_jac[2, -2], + assoc_legendre_p_2_m2_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[2, -1], + assoc_legendre_p_2_m1_jac(z, branch_cut=branch_cut, norm=norm)) + + np.testing.assert_allclose(p_jac[3, 0], + assoc_legendre_p_3_0_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[3, 1], + assoc_legendre_p_3_1_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[3, 2], + assoc_legendre_p_3_2_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[3, 3], + assoc_legendre_p_3_3_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[3, 4], 0) + np.testing.assert_allclose(p_jac[3, -4], 0) + np.testing.assert_allclose(p_jac[3, -3], + assoc_legendre_p_3_m3_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[3, -2], + assoc_legendre_p_3_m2_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[3, -1], + assoc_legendre_p_3_m1_jac(z, branch_cut=branch_cut, norm=norm)) + + np.testing.assert_allclose(p_jac[4, 0], + assoc_legendre_p_4_0_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[4, 1], + assoc_legendre_p_4_1_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[4, 2], + assoc_legendre_p_4_2_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[4, 3], + assoc_legendre_p_4_3_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[4, 4], + assoc_legendre_p_4_4_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[4, -4], + assoc_legendre_p_4_m4_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[4, -3], + assoc_legendre_p_4_m3_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[4, -2], + assoc_legendre_p_4_m2_jac(z, branch_cut=branch_cut, norm=norm)) + np.testing.assert_allclose(p_jac[4, -1], + assoc_legendre_p_4_m1_jac(z, branch_cut=branch_cut, norm=norm)) + +class TestSphLegendreP: + @pytest.mark.parametrize("shape", [(10,), (4, 9), (3, 5, 7)]) + def test_specific(self, shape): + rng = np.random.default_rng(1234) + + theta = rng.uniform(-np.pi, np.pi, shape) + + p, p_jac = sph_legendre_p_all(4, 4, theta, diff_n=1) + + np.testing.assert_allclose(p[0, 0], + sph_legendre_p_0_0(theta)) + np.testing.assert_allclose(p[0, 1], 0) + np.testing.assert_allclose(p[0, 2], 0) + np.testing.assert_allclose(p[0, 3], 0) + np.testing.assert_allclose(p[0, 4], 0) + np.testing.assert_allclose(p[0, -3], 0) + np.testing.assert_allclose(p[0, -2], 0) + np.testing.assert_allclose(p[0, -1], 0) + + np.testing.assert_allclose(p[1, 0], + sph_legendre_p_1_0(theta)) + np.testing.assert_allclose(p[1, 1], + sph_legendre_p_1_1(theta)) + np.testing.assert_allclose(p[1, 2], 0) + np.testing.assert_allclose(p[1, 3], 0) + np.testing.assert_allclose(p[1, 4], 0) + np.testing.assert_allclose(p[1, -4], 0) + np.testing.assert_allclose(p[1, -3], 0) + np.testing.assert_allclose(p[1, -2], 0) + np.testing.assert_allclose(p[1, -1], + sph_legendre_p_1_m1(theta)) + + np.testing.assert_allclose(p[2, 0], + sph_legendre_p_2_0(theta)) + np.testing.assert_allclose(p[2, 1], + sph_legendre_p_2_1(theta)) + np.testing.assert_allclose(p[2, 2], + sph_legendre_p_2_2(theta)) + np.testing.assert_allclose(p[2, 3], 0) + np.testing.assert_allclose(p[2, 4], 0) + np.testing.assert_allclose(p[2, -4], 0) + np.testing.assert_allclose(p[2, -3], 0) + np.testing.assert_allclose(p[2, -2], + sph_legendre_p_2_m2(theta)) + np.testing.assert_allclose(p[2, -1], + sph_legendre_p_2_m1(theta)) + + np.testing.assert_allclose(p[3, 0], + sph_legendre_p_3_0(theta)) + np.testing.assert_allclose(p[3, 1], + sph_legendre_p_3_1(theta)) + np.testing.assert_allclose(p[3, 2], + sph_legendre_p_3_2(theta)) + np.testing.assert_allclose(p[3, 3], + sph_legendre_p_3_3(theta)) + np.testing.assert_allclose(p[3, 4], 0) + np.testing.assert_allclose(p[3, -4], 0) + np.testing.assert_allclose(p[3, -3], + sph_legendre_p_3_m3(theta)) + np.testing.assert_allclose(p[3, -2], + sph_legendre_p_3_m2(theta)) + np.testing.assert_allclose(p[3, -1], + sph_legendre_p_3_m1(theta)) + + np.testing.assert_allclose(p[4, 0], + sph_legendre_p_4_0(theta)) + np.testing.assert_allclose(p[4, 1], + sph_legendre_p_4_1(theta)) + np.testing.assert_allclose(p[4, 2], + sph_legendre_p_4_2(theta)) + np.testing.assert_allclose(p[4, 3], + sph_legendre_p_4_3(theta)) + np.testing.assert_allclose(p[4, 4], + sph_legendre_p_4_4(theta)) + np.testing.assert_allclose(p[4, -4], + sph_legendre_p_4_m4(theta)) + np.testing.assert_allclose(p[4, -3], + sph_legendre_p_4_m3(theta)) + np.testing.assert_allclose(p[4, -2], + sph_legendre_p_4_m2(theta)) + np.testing.assert_allclose(p[4, -1], + sph_legendre_p_4_m1(theta)) + + np.testing.assert_allclose(p_jac[0, 0], + sph_legendre_p_0_0_jac(theta)) + np.testing.assert_allclose(p_jac[0, 1], 0) + np.testing.assert_allclose(p_jac[0, 2], 0) + np.testing.assert_allclose(p_jac[0, 3], 0) + np.testing.assert_allclose(p_jac[0, 4], 0) + np.testing.assert_allclose(p_jac[0, -3], 0) + np.testing.assert_allclose(p_jac[0, -2], 0) + np.testing.assert_allclose(p_jac[0, -1], 0) + + np.testing.assert_allclose(p_jac[1, 0], + sph_legendre_p_1_0_jac(theta)) + np.testing.assert_allclose(p_jac[1, 1], + sph_legendre_p_1_1_jac(theta)) + np.testing.assert_allclose(p_jac[1, 2], 0) + np.testing.assert_allclose(p_jac[1, 3], 0) + np.testing.assert_allclose(p_jac[1, 4], 0) + np.testing.assert_allclose(p_jac[1, -4], 0) + np.testing.assert_allclose(p_jac[1, -3], 0) + np.testing.assert_allclose(p_jac[1, -2], 0) + np.testing.assert_allclose(p_jac[1, -1], + sph_legendre_p_1_m1_jac(theta)) + + np.testing.assert_allclose(p_jac[2, 0], + sph_legendre_p_2_0_jac(theta)) + np.testing.assert_allclose(p_jac[2, 1], + sph_legendre_p_2_1_jac(theta)) + np.testing.assert_allclose(p_jac[2, 2], + sph_legendre_p_2_2_jac(theta)) + np.testing.assert_allclose(p_jac[2, 3], 0) + np.testing.assert_allclose(p_jac[2, 4], 0) + np.testing.assert_allclose(p_jac[2, -4], 0) + np.testing.assert_allclose(p_jac[2, -3], 0) + np.testing.assert_allclose(p_jac[2, -2], + sph_legendre_p_2_m2_jac(theta)) + np.testing.assert_allclose(p_jac[2, -1], + sph_legendre_p_2_m1_jac(theta)) + + np.testing.assert_allclose(p_jac[3, 0], + sph_legendre_p_3_0_jac(theta)) + np.testing.assert_allclose(p_jac[3, 1], + sph_legendre_p_3_1_jac(theta)) + np.testing.assert_allclose(p_jac[3, 2], + sph_legendre_p_3_2_jac(theta)) + np.testing.assert_allclose(p_jac[3, 3], + sph_legendre_p_3_3_jac(theta)) + np.testing.assert_allclose(p_jac[3, 4], 0) + np.testing.assert_allclose(p_jac[3, -4], 0) + np.testing.assert_allclose(p_jac[3, -3], + sph_legendre_p_3_m3_jac(theta)) + np.testing.assert_allclose(p_jac[3, -2], + sph_legendre_p_3_m2_jac(theta)) + np.testing.assert_allclose(p_jac[3, -1], + sph_legendre_p_3_m1_jac(theta)) + + np.testing.assert_allclose(p_jac[4, 0], + sph_legendre_p_4_0_jac(theta)) + np.testing.assert_allclose(p_jac[4, 1], + sph_legendre_p_4_1_jac(theta)) + np.testing.assert_allclose(p_jac[4, 2], + sph_legendre_p_4_2_jac(theta)) + np.testing.assert_allclose(p_jac[4, 3], + sph_legendre_p_4_3_jac(theta)) + np.testing.assert_allclose(p_jac[4, 4], + sph_legendre_p_4_4_jac(theta)) + np.testing.assert_allclose(p_jac[4, -4], + sph_legendre_p_4_m4_jac(theta)) + np.testing.assert_allclose(p_jac[4, -3], + sph_legendre_p_4_m3_jac(theta)) + np.testing.assert_allclose(p_jac[4, -2], + sph_legendre_p_4_m2_jac(theta)) + np.testing.assert_allclose(p_jac[4, -1], + sph_legendre_p_4_m1_jac(theta)) + + @pytest.mark.parametrize("shape", [(10,), (4, 9), (3, 5, 7, 10)]) + def test_ode(self, shape): + rng = np.random.default_rng(1234) + + n = rng.integers(0, 10, shape) + m = rng.integers(-10, 10, shape) + theta = rng.uniform(-np.pi, np.pi, shape) + + p, p_jac, p_hess = sph_legendre_p(n, m, theta, diff_n=2) + + assert p.shape == shape + assert p_jac.shape == p.shape + assert p_hess.shape == p_jac.shape + + np.testing.assert_allclose(np.sin(theta) * p_hess, -np.cos(theta) * p_jac + - (n * (n + 1) * np.sin(theta) - m * m / np.sin(theta)) * p, + rtol=1e-05, atol=1e-08) + +class TestLegendreFunctions: + def test_clpmn(self): + z = 0.5+0.3j + + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + clp = special.clpmn(2, 2, z, 3) + + assert_array_almost_equal(clp, + (np.array([[1.0000, z, 0.5*(3*z*z-1)], + [0.0000, np.sqrt(z*z-1), 3*z*np.sqrt(z*z-1)], + [0.0000, 0.0000, 3*(z*z-1)]]), + np.array([[0.0000, 1.0000, 3*z], + [0.0000, z/np.sqrt(z*z-1), 3*(2*z*z-1)/np.sqrt(z*z-1)], + [0.0000, 0.0000, 6*z]])), + 7) + + def test_clpmn_close_to_real_2(self): + eps = 1e-10 + m = 1 + n = 3 + x = 0.5 + + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + clp_plus = special.clpmn(m, n, x+1j*eps, 2)[0][m, n] + clp_minus = special.clpmn(m, n, x-1j*eps, 2)[0][m, n] + + assert_array_almost_equal(np.array([clp_plus, clp_minus]), + np.array([special.lpmv(m, n, x), + special.lpmv(m, n, x)]), + 7) + + def test_clpmn_close_to_real_3(self): + eps = 1e-10 + m = 1 + n = 3 + x = 0.5 + + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + clp_plus = special.clpmn(m, n, x+1j*eps, 3)[0][m, n] + clp_minus = special.clpmn(m, n, x-1j*eps, 3)[0][m, n] + + assert_array_almost_equal(np.array([clp_plus, clp_minus]), + np.array([special.lpmv(m, n, x)*np.exp(-0.5j*m*np.pi), + special.lpmv(m, n, x)*np.exp(0.5j*m*np.pi)]), + 7) + + def test_clpmn_across_unit_circle(self): + eps = 1e-7 + m = 1 + n = 1 + x = 1j + + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + for type in [2, 3]: + assert_almost_equal(special.clpmn(m, n, x+1j*eps, type)[0][m, n], + special.clpmn(m, n, x-1j*eps, type)[0][m, n], 6) + + def test_inf(self): + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + for z in (1, -1): + for n in range(4): + for m in range(1, n): + lp = special.clpmn(m, n, z) + assert np.isinf(lp[1][1,1:]).all() + lp = special.lpmn(m, n, z) + assert np.isinf(lp[1][1,1:]).all() + + def test_deriv_clpmn(self): + # data inside and outside of the unit circle + zvals = [0.5+0.5j, -0.5+0.5j, -0.5-0.5j, 0.5-0.5j, + 1+1j, -1+1j, -1-1j, 1-1j] + m = 2 + n = 3 + + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + for type in [2, 3]: + for z in zvals: + for h in [1e-3, 1e-3j]: + approx_derivative = (special.clpmn(m, n, z+0.5*h, type)[0] + - special.clpmn(m, n, z-0.5*h, type)[0])/h + assert_allclose(special.clpmn(m, n, z, type)[1], + approx_derivative, + rtol=1e-4) + + """ + @pytest.mark.parametrize("m_max", [3]) + @pytest.mark.parametrize("n_max", [5]) + @pytest.mark.parametrize("z", [-1]) + def test_clpmn_all_limits(self, m_max, n_max, z): + rng = np.random.default_rng(1234) + + type = 2 + + p, p_jac = special.clpmn_all(m_max, n_max, type, z, diff_n=1) + + n = np.arange(n_max + 1) + + np.testing.assert_allclose(p_jac[0], pow(z, n + 1) * n * (n + 1) / 2) + np.testing.assert_allclose(p_jac[1], np.where(n >= 1, pow(z, n) * np.inf, 0)) + np.testing.assert_allclose(p_jac[2], np.where(n >= 2, + -pow(z, n + 1) * (n + 2) * (n + 1) * n * (n - 1) / 4, 0)) + np.testing.assert_allclose(p_jac[-2], np.where(n >= 2, -pow(z, n + 1) / 4, 0)) + np.testing.assert_allclose(p_jac[-1], np.where(n >= 1, -pow(z, n) * np.inf, 0)) + + for m in range(3, m_max + 1): + np.testing.assert_allclose(p_jac[m], 0) + np.testing.assert_allclose(p_jac[-m], 0) + """ + + def test_lpmv(self): + lp = special.lpmv(0,2,.5) + assert_almost_equal(lp,-0.125,7) + lp = special.lpmv(0,40,.001) + assert_almost_equal(lp,0.1252678976534484,7) + + # XXX: this is outside the domain of the current implementation, + # so ensure it returns a NaN rather than a wrong answer. + with np.errstate(all='ignore'): + lp = special.lpmv(-1,-1,.001) + assert lp != 0 or np.isnan(lp) + + def test_lqmn(self): + lqmnf = special.lqmn(0,2,.5) + lqf = special.lqn(2,.5) + assert_array_almost_equal(lqmnf[0][0],lqf[0],4) + assert_array_almost_equal(lqmnf[1][0],lqf[1],4) + + def test_lqmn_gt1(self): + """algorithm for real arguments changes at 1.0001 + test against analytical result for m=2, n=1 + """ + x0 = 1.0001 + delta = 0.00002 + for x in (x0-delta, x0+delta): + lq = special.lqmn(2, 1, x)[0][-1, -1] + expected = 2/(x*x-1) + assert_almost_equal(lq, expected) + + def test_lqmn_shape(self): + a, b = special.lqmn(4, 4, 1.1) + assert_equal(a.shape, (5, 5)) + assert_equal(b.shape, (5, 5)) + + a, b = special.lqmn(4, 0, 1.1) + assert_equal(a.shape, (5, 1)) + assert_equal(b.shape, (5, 1)) + + def test_lqn(self): + lqf = special.lqn(2,.5) + assert_array_almost_equal(lqf,(np.array([0.5493, -0.7253, -0.8187]), + np.array([1.3333, 1.216, -0.8427])),4) + + @pytest.mark.parametrize("function", [special.lpn, special.lqn]) + @pytest.mark.parametrize("n", [1, 2, 4, 8, 16, 32]) + @pytest.mark.parametrize("z_complex", [False, True]) + @pytest.mark.parametrize("z_inexact", [False, True]) + @pytest.mark.parametrize( + "input_shape", + [ + (), (1, ), (2, ), (2, 1), (1, 2), (2, 2), (2, 2, 1), (2, 2, 2) + ] + ) + def test_array_inputs_lxn(self, function, n, z_complex, z_inexact, input_shape): + """Tests for correct output shapes.""" + rng = np.random.default_rng(1234) + if z_inexact: + z = rng.integers(-3, 3, size=input_shape) + else: + z = rng.uniform(-1, 1, size=input_shape) + + if z_complex: + z = 1j * z + 0.5j * z + + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + P_z, P_d_z = function(n, z) + assert P_z.shape == (n + 1, ) + input_shape + assert P_d_z.shape == (n + 1, ) + input_shape + + @pytest.mark.parametrize("function", [special.lqmn]) + @pytest.mark.parametrize( + "m,n", + [(0, 1), (1, 2), (1, 4), (3, 8), (11, 16), (19, 32)] + ) + @pytest.mark.parametrize("z_inexact", [False, True]) + @pytest.mark.parametrize( + "input_shape", [ + (), (1, ), (2, ), (2, 1), (1, 2), (2, 2), (2, 2, 1) + ] + ) + def test_array_inputs_lxmn(self, function, m, n, z_inexact, input_shape): + """Tests for correct output shapes and dtypes.""" + rng = np.random.default_rng(1234) + if z_inexact: + z = rng.integers(-3, 3, size=input_shape) + else: + z = rng.uniform(-1, 1, size=input_shape) + + P_z, P_d_z = function(m, n, z) + assert P_z.shape == (m + 1, n + 1) + input_shape + assert P_d_z.shape == (m + 1, n + 1) + input_shape + + @pytest.mark.parametrize("function", [special.clpmn, special.lqmn]) + @pytest.mark.parametrize( + "m,n", + [(0, 1), (1, 2), (1, 4), (3, 8), (11, 16), (19, 32)] + ) + @pytest.mark.parametrize( + "input_shape", [ + (), (1, ), (2, ), (2, 1), (1, 2), (2, 2), (2, 2, 1) + ] + ) + def test_array_inputs_clxmn(self, function, m, n, input_shape): + """Tests for correct output shapes and dtypes.""" + rng = np.random.default_rng(1234) + z = rng.uniform(-1, 1, size=input_shape) + z = 1j * z + 0.5j * z + + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + P_z, P_d_z = function(m, n, z) + + assert P_z.shape == (m + 1, n + 1) + input_shape + assert P_d_z.shape == (m + 1, n + 1) + input_shape + +def assoc_legendre_factor(n, m, norm): + if norm: + return (math.sqrt((2 * n + 1) * + math.factorial(n - m) / (2 * math.factorial(n + m)))) + + return 1 + +def assoc_legendre_p_0_0(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(0, 0, norm) + + return np.full_like(z, fac) + +def assoc_legendre_p_1_0(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(1, 0, norm) + + return fac * z + +def assoc_legendre_p_1_1(z, *, branch_cut=2, norm=False): + branch_sign = np.where(branch_cut == 3, np.where(np.signbit(np.real(z)), 1, -1), -1) + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(1, 1, norm) + + w = np.sqrt(np.where(branch_cut == 3, z * z - 1, 1 - z * z)) + + return branch_cut_sign * branch_sign * fac * w + +def assoc_legendre_p_1_m1(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(1, -1, norm) + + return (-branch_cut_sign * fac * + assoc_legendre_p_1_1(z, branch_cut=branch_cut) / 2) + +def assoc_legendre_p_2_0(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(2, 0, norm) + + return fac * (3 * z * z - 1) / 2 + +def assoc_legendre_p_2_1(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(2, 1, norm) + + return (3 * fac * z * + assoc_legendre_p_1_1(z, branch_cut=branch_cut)) + +def assoc_legendre_p_2_2(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(2, 2, norm) + + return 3 * branch_cut_sign * fac * (1 - z * z) + +def assoc_legendre_p_2_m2(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(2, -2, norm) + + return branch_cut_sign * fac * (1 - z * z) / 8 + +def assoc_legendre_p_2_m1(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(2, -1, norm) + + return (-branch_cut_sign * fac * z * + assoc_legendre_p_1_1(z, branch_cut=branch_cut) / 2) + +def assoc_legendre_p_3_0(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(3, 0, norm) + + return fac * (5 * z * z - 3) * z / 2 + +def assoc_legendre_p_3_1(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(3, 1, norm) + + return (3 * fac * (5 * z * z - 1) * + assoc_legendre_p_1_1(z, branch_cut=branch_cut) / 2) + +def assoc_legendre_p_3_2(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(3, 2, norm) + + return 15 * branch_cut_sign * fac * (1 - z * z) * z + +def assoc_legendre_p_3_3(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(3, 3, norm) + + return (15 * branch_cut_sign * fac * (1 - z * z) * + assoc_legendre_p_1_1(z, branch_cut=branch_cut)) + +def assoc_legendre_p_3_m3(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(3, -3, norm) + + return (fac * (z * z - 1) * + assoc_legendre_p_1_1(z, branch_cut=branch_cut) / 48) + +def assoc_legendre_p_3_m2(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(3, -2, norm) + + return branch_cut_sign * fac * (1 - z * z) * z / 8 + +def assoc_legendre_p_3_m1(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(3, -1, norm) + + return (branch_cut_sign * fac * (1 - 5 * z * z) * + assoc_legendre_p_1_1(z, branch_cut=branch_cut) / 8) + +def assoc_legendre_p_4_0(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(4, 0, norm) + + return fac * ((35 * z * z - 30) * z * z + 3) / 8 + +def assoc_legendre_p_4_1(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(4, 1, norm) + + return (5 * fac * (7 * z * z - 3) * z * + assoc_legendre_p_1_1(z, branch_cut=branch_cut) / 2) + +def assoc_legendre_p_4_2(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(4, 2, norm) + + return 15 * branch_cut_sign * fac * ((8 - 7 * z * z) * z * z - 1) / 2 + +def assoc_legendre_p_4_3(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(4, 3, norm) + + return (105 * branch_cut_sign * fac * (1 - z * z) * z * + assoc_legendre_p_1_1(z, branch_cut=branch_cut)) + +def assoc_legendre_p_4_4(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(4, 4, norm) + + return 105 * fac * np.square(z * z - 1) + +def assoc_legendre_p_4_m4(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(4, -4, norm) + + return fac * np.square(z * z - 1) / 384 + +def assoc_legendre_p_4_m3(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(4, -3, norm) + + return (fac * (z * z - 1) * z * + assoc_legendre_p_1_1(z, branch_cut=branch_cut) / 48) + +def assoc_legendre_p_4_m2(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(4, -2, norm) + + return branch_cut_sign * fac * ((8 - 7 * z * z) * z * z - 1) / 48 + +def assoc_legendre_p_4_m1(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(4, -1, norm) + + return (branch_cut_sign * fac * (3 - 7 * z * z) * z * + assoc_legendre_p_1_1(z, branch_cut=branch_cut) / 8) + +def assoc_legendre_p_1_1_jac_div_z(z, branch_cut=2): + branch_sign = np.where(branch_cut == 3, np.where(np.signbit(np.real(z)), 1, -1), -1) + + out11_div_z = (-branch_sign / + np.sqrt(np.where(branch_cut == 3, z * z - 1, 1 - z * z))) + + return out11_div_z + +def assoc_legendre_p_0_0_jac(z, *, branch_cut=2, norm=False): + return np.zeros_like(z) + +def assoc_legendre_p_1_0_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(1, 0, norm) + + return np.full_like(z, fac) + +def assoc_legendre_p_1_1_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(1, 1, norm) + + return (fac * z * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut)) + +def assoc_legendre_p_1_m1_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(1, -1, norm) + + return (-branch_cut_sign * fac * z * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut) / 2) + +def assoc_legendre_p_2_0_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(2, 0, norm) + + return 3 * fac * z + +def assoc_legendre_p_2_1_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(2, 1, norm) + + return (3 * fac * (2 * z * z - 1) * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut)) + +def assoc_legendre_p_2_2_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(2, 2, norm) + + return -6 * branch_cut_sign * fac * z + +def assoc_legendre_p_2_m1_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(2, -1, norm) + + return (branch_cut_sign * fac * (1 - 2 * z * z) * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut) / 2) + +def assoc_legendre_p_2_m2_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(2, -2, norm) + + return -branch_cut_sign * fac * z / 4 + +def assoc_legendre_p_3_0_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(3, 0, norm) + + return 3 * fac * (5 * z * z - 1) / 2 + +def assoc_legendre_p_3_1_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(3, 1, norm) + + return (3 * fac * (15 * z * z - 11) * z * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut) / 2) + +def assoc_legendre_p_3_2_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(3, 2, norm) + + return 15 * branch_cut_sign * fac * (1 - 3 * z * z) + +def assoc_legendre_p_3_3_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(3, 3, norm) + + return (45 * branch_cut_sign * fac * (1 - z * z) * z * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut)) + +def assoc_legendre_p_3_m3_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(3, -3, norm) + + return (fac * (z * z - 1) * z * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut) / 16) + +def assoc_legendre_p_3_m2_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(3, -2, norm) + + return branch_cut_sign * fac * (1 - 3 * z * z) / 8 + +def assoc_legendre_p_3_m1_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(3, -1, norm) + + return (branch_cut_sign * fac * (11 - 15 * z * z) * z * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut) / 8) + +def assoc_legendre_p_4_0_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(4, 0, norm) + + return 5 * fac * (7 * z * z - 3) * z / 2 + +def assoc_legendre_p_4_1_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(4, 1, norm) + + return (5 * fac * ((28 * z * z - 27) * z * z + 3) * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut) / 2) + +def assoc_legendre_p_4_2_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(4, 2, norm) + + return 30 * branch_cut_sign * fac * (4 - 7 * z * z) * z + +def assoc_legendre_p_4_3_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(4, 3, norm) + + return (105 * branch_cut_sign * fac * ((5 - 4 * z * z) * z * z - 1) * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut)) + +def assoc_legendre_p_4_4_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(4, 4, norm) + + return 420 * fac * (z * z - 1) * z + +def assoc_legendre_p_4_m4_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(4, -4, norm) + + return fac * (z * z - 1) * z / 96 + +def assoc_legendre_p_4_m3_jac(z, *, branch_cut=2, norm=False): + fac = assoc_legendre_factor(4, -3, norm) + + return (fac * ((4 * z * z - 5) * z * z + 1) * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut) / 48) + +def assoc_legendre_p_4_m2_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(4, -2, norm) + + return branch_cut_sign * fac * (4 - 7 * z * z) * z / 12 + +def assoc_legendre_p_4_m1_jac(z, *, branch_cut=2, norm=False): + branch_cut_sign = np.where(branch_cut == 3, -1, 1) + fac = assoc_legendre_factor(4, -1, norm) + + return (branch_cut_sign * fac * ((27 - 28 * z * z) * z * z - 3) * + assoc_legendre_p_1_1_jac_div_z(z, branch_cut=branch_cut) / 8) + +def sph_legendre_factor(n, m): + return assoc_legendre_factor(n, m, norm=True) / np.sqrt(2 * np.pi) + +def sph_legendre_p_0_0(theta): + fac = sph_legendre_factor(0, 0) + + return np.full_like(theta, fac) + +def sph_legendre_p_1_0(theta): + fac = sph_legendre_factor(1, 0) + + return fac * np.cos(theta) + +def sph_legendre_p_1_1(theta): + fac = sph_legendre_factor(1, 1) + + return -fac * np.abs(np.sin(theta)) + +def sph_legendre_p_1_m1(theta): + fac = sph_legendre_factor(1, -1) + + return fac * np.abs(np.sin(theta)) / 2 + +def sph_legendre_p_2_0(theta): + fac = sph_legendre_factor(2, 0) + + return fac * (3 * np.square(np.cos(theta)) - 1) / 2 + +def sph_legendre_p_2_1(theta): + fac = sph_legendre_factor(2, 1) + + return -3 * fac * np.abs(np.sin(theta)) * np.cos(theta) + +def sph_legendre_p_2_2(theta): + fac = sph_legendre_factor(2, 2) + + return 3 * fac * (1 - np.square(np.cos(theta))) + +def sph_legendre_p_2_m2(theta): + fac = sph_legendre_factor(2, -2) + + return fac * (1 - np.square(np.cos(theta))) / 8 + +def sph_legendre_p_2_m1(theta): + fac = sph_legendre_factor(2, -1) + + return fac * np.cos(theta) * np.abs(np.sin(theta)) / 2 + +def sph_legendre_p_3_0(theta): + fac = sph_legendre_factor(3, 0) + + return (fac * (5 * np.square(np.cos(theta)) - 3) * + np.cos(theta) / 2) + +def sph_legendre_p_3_1(theta): + fac = sph_legendre_factor(3, 1) + + return (-3 * fac * (5 * np.square(np.cos(theta)) - 1) * + np.abs(np.sin(theta)) / 2) + +def sph_legendre_p_3_2(theta): + fac = sph_legendre_factor(3, 2) + + return (-15 * fac * (np.square(np.cos(theta)) - 1) * + np.cos(theta)) + +def sph_legendre_p_3_3(theta): + fac = sph_legendre_factor(3, 3) + + return -15 * fac * np.power(np.abs(np.sin(theta)), 3) + +def sph_legendre_p_3_m3(theta): + fac = sph_legendre_factor(3, -3) + + return fac * np.power(np.abs(np.sin(theta)), 3) / 48 + +def sph_legendre_p_3_m2(theta): + fac = sph_legendre_factor(3, -2) + + return (-fac * (np.square(np.cos(theta)) - 1) * + np.cos(theta) / 8) + +def sph_legendre_p_3_m1(theta): + fac = sph_legendre_factor(3, -1) + + return (fac * (5 * np.square(np.cos(theta)) - 1) * + np.abs(np.sin(theta)) / 8) + +def sph_legendre_p_4_0(theta): + fac = sph_legendre_factor(4, 0) + + return (fac * (35 * np.square(np.square(np.cos(theta))) - + 30 * np.square(np.cos(theta)) + 3) / 8) + +def sph_legendre_p_4_1(theta): + fac = sph_legendre_factor(4, 1) + + return (-5 * fac * (7 * np.square(np.cos(theta)) - 3) * + np.cos(theta) * np.abs(np.sin(theta)) / 2) + +def sph_legendre_p_4_2(theta): + fac = sph_legendre_factor(4, 2) + + return (-15 * fac * (7 * np.square(np.cos(theta)) - 1) * + (np.square(np.cos(theta)) - 1) / 2) + +def sph_legendre_p_4_3(theta): + fac = sph_legendre_factor(4, 3) + + return -105 * fac * np.power(np.abs(np.sin(theta)), 3) * np.cos(theta) + +def sph_legendre_p_4_4(theta): + fac = sph_legendre_factor(4, 4) + + return 105 * fac * np.square(np.square(np.cos(theta)) - 1) + +def sph_legendre_p_4_m4(theta): + fac = sph_legendre_factor(4, -4) + + return fac * np.square(np.square(np.cos(theta)) - 1) / 384 + +def sph_legendre_p_4_m3(theta): + fac = sph_legendre_factor(4, -3) + + return (fac * np.power(np.abs(np.sin(theta)), 3) * + np.cos(theta) / 48) + +def sph_legendre_p_4_m2(theta): + fac = sph_legendre_factor(4, -2) + + return (-fac * (7 * np.square(np.cos(theta)) - 1) * + (np.square(np.cos(theta)) - 1) / 48) + +def sph_legendre_p_4_m1(theta): + fac = sph_legendre_factor(4, -1) + + return (fac * (7 * np.square(np.cos(theta)) - 3) * + np.cos(theta) * np.abs(np.sin(theta)) / 8) + +def sph_legendre_p_0_0_jac(theta): + return np.zeros_like(theta) + +def sph_legendre_p_1_0_jac(theta): + fac = sph_legendre_factor(1, 0) + + return -fac * np.sin(theta) + +def sph_legendre_p_1_1_jac(theta): + fac = sph_legendre_factor(1, 1) + + return -fac * np.cos(theta) * (2 * np.heaviside(np.sin(theta), 1) - 1) + +def sph_legendre_p_1_m1_jac(theta): + fac = sph_legendre_factor(1, -1) + + return fac * np.cos(theta) * (2 * np.heaviside(np.sin(theta), 1) - 1) / 2 + +def sph_legendre_p_2_0_jac(theta): + fac = sph_legendre_factor(2, 0) + + return -3 * fac * np.cos(theta) * np.sin(theta) + +def sph_legendre_p_2_1_jac(theta): + fac = sph_legendre_factor(2, 1) + + return (3 * fac * (-np.square(np.cos(theta)) * + (2 * np.heaviside(np.sin(theta), 1) - 1) + + np.abs(np.sin(theta)) * np.sin(theta))) + +def sph_legendre_p_2_2_jac(theta): + fac = sph_legendre_factor(2, 2) + + return 6 * fac * np.sin(theta) * np.cos(theta) + +def sph_legendre_p_2_m2_jac(theta): + fac = sph_legendre_factor(2, -2) + + return fac * np.sin(theta) * np.cos(theta) / 4 + +def sph_legendre_p_2_m1_jac(theta): + fac = sph_legendre_factor(2, -1) + + return (-fac * (-np.square(np.cos(theta)) * + (2 * np.heaviside(np.sin(theta), 1) - 1) + + np.abs(np.sin(theta)) * np.sin(theta)) / 2) + +def sph_legendre_p_3_0_jac(theta): + fac = sph_legendre_factor(3, 0) + + return 3 * fac * (1 - 5 * np.square(np.cos(theta))) * np.sin(theta) / 2 + +def sph_legendre_p_3_1_jac(theta): + fac = sph_legendre_factor(3, 1) + + return (3 * fac * (11 - 15 * np.square(np.cos(theta))) * np.cos(theta) * + (2 * np.heaviside(np.sin(theta), 1) - 1) / 2) + +def sph_legendre_p_3_2_jac(theta): + fac = sph_legendre_factor(3, 2) + + return 15 * fac * (3 * np.square(np.cos(theta)) - 1) * np.sin(theta) + +def sph_legendre_p_3_3_jac(theta): + fac = sph_legendre_factor(3, 3) + + return -45 * fac * np.abs(np.sin(theta)) * np.sin(theta) * np.cos(theta) + +def sph_legendre_p_3_m3_jac(theta): + fac = sph_legendre_factor(3, -3) + + return fac * np.abs(np.sin(theta)) * np.sin(theta) * np.cos(theta) / 16 + +def sph_legendre_p_3_m2_jac(theta): + fac = sph_legendre_factor(3, -2) + + return fac * (3 * np.square(np.cos(theta)) - 1) * np.sin(theta) / 8 + +def sph_legendre_p_3_m1_jac(theta): + fac = sph_legendre_factor(3, -1) + + return (-fac * (11 - 15 * np.square(np.cos(theta))) * + np.cos(theta) * + (2 * np.heaviside(np.sin(theta), 1) - 1) / 8) + +def sph_legendre_p_4_0_jac(theta): + fac = sph_legendre_factor(4, 0) + + return (-5 * fac * (7 * np.square(np.cos(theta)) - 3) * + np.sin(theta) * np.cos(theta) / 2) + +def sph_legendre_p_4_1_jac(theta): + fac = sph_legendre_factor(4, 1) + + return (5 * fac * (-3 + 27 * np.square(np.cos(theta)) - + 28 * np.square(np.square(np.cos(theta)))) * + (2 * np.heaviside(np.sin(theta), 1) - 1) / 2) + +def sph_legendre_p_4_2_jac(theta): + fac = sph_legendre_factor(4, 2) + + return (30 * fac * (7 * np.square(np.cos(theta)) - 4) * + np.sin(theta) * np.cos(theta)) + +def sph_legendre_p_4_3_jac(theta): + fac = sph_legendre_factor(4, 3) + + return (-105 * fac * (4 * np.square(np.cos(theta)) - 1) * + np.abs(np.sin(theta)) * np.sin(theta)) + +def sph_legendre_p_4_4_jac(theta): + fac = sph_legendre_factor(4, 4) + + return (-420 * fac * (np.square(np.cos(theta)) - 1) * + np.sin(theta) * np.cos(theta)) + +def sph_legendre_p_4_m4_jac(theta): + fac = sph_legendre_factor(4, -4) + + return (-fac * (np.square(np.cos(theta)) - 1) * + np.sin(theta) * np.cos(theta) / 96) + +def sph_legendre_p_4_m3_jac(theta): + fac = sph_legendre_factor(4, -3) + + return (fac * (4 * np.square(np.cos(theta)) - 1) * + np.abs(np.sin(theta)) * np.sin(theta) / 48) + +def sph_legendre_p_4_m2_jac(theta): + fac = sph_legendre_factor(4, -2) + + return (fac * (7 * np.square(np.cos(theta)) - 4) * np.sin(theta) * + np.cos(theta) / 12) + +def sph_legendre_p_4_m1_jac(theta): + fac = sph_legendre_factor(4, -1) + + return (-fac * (-3 + 27 * np.square(np.cos(theta)) - + 28 * np.square(np.square(np.cos(theta)))) * + (2 * np.heaviside(np.sin(theta), 1) - 1) / 8) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_log_softmax.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_log_softmax.py new file mode 100644 index 0000000000000000000000000000000000000000..4b3a5071fc671d8d4a7ef6c7655ca4212ced4f45 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_log_softmax.py @@ -0,0 +1,109 @@ +import numpy as np +from numpy.testing import assert_allclose + +import pytest + +import scipy.special as sc + + +@pytest.mark.parametrize('x, expected', [ + (np.array([1000, 1]), np.array([0, -999])), + + # Expected value computed using mpmath (with mpmath.mp.dps = 200) and then + # converted to float. + (np.arange(4), np.array([-3.4401896985611953, + -2.4401896985611953, + -1.4401896985611953, + -0.44018969856119533])) +]) +def test_log_softmax(x, expected): + assert_allclose(sc.log_softmax(x), expected, rtol=1e-13) + + +@pytest.fixture +def log_softmax_x(): + x = np.arange(4) + return x + + +@pytest.fixture +def log_softmax_expected(): + # Expected value computed using mpmath (with mpmath.mp.dps = 200) and then + # converted to float. + expected = np.array([-3.4401896985611953, + -2.4401896985611953, + -1.4401896985611953, + -0.44018969856119533]) + return expected + + +def test_log_softmax_translation(log_softmax_x, log_softmax_expected): + # Translation property. If all the values are changed by the same amount, + # the softmax result does not change. + x = log_softmax_x + 100 + expected = log_softmax_expected + assert_allclose(sc.log_softmax(x), expected, rtol=1e-13) + + +def test_log_softmax_noneaxis(log_softmax_x, log_softmax_expected): + # When axis=None, softmax operates on the entire array, and preserves + # the shape. + x = log_softmax_x.reshape(2, 2) + expected = log_softmax_expected.reshape(2, 2) + assert_allclose(sc.log_softmax(x), expected, rtol=1e-13) + + +@pytest.mark.parametrize('axis_2d, expected_2d', [ + (0, np.log(0.5) * np.ones((2, 2))), + (1, np.array([[0, -999], [0, -999]])) +]) +def test_axes(axis_2d, expected_2d): + assert_allclose( + sc.log_softmax([[1000, 1], [1000, 1]], axis=axis_2d), + expected_2d, + rtol=1e-13, + ) + + +@pytest.fixture +def log_softmax_2d_x(): + x = np.arange(8).reshape(2, 4) + return x + + +@pytest.fixture +def log_softmax_2d_expected(): + # Expected value computed using mpmath (with mpmath.mp.dps = 200) and then + # converted to float. + expected = np.array([[-3.4401896985611953, + -2.4401896985611953, + -1.4401896985611953, + -0.44018969856119533], + [-3.4401896985611953, + -2.4401896985611953, + -1.4401896985611953, + -0.44018969856119533]]) + return expected + + +def test_log_softmax_2d_axis1(log_softmax_2d_x, log_softmax_2d_expected): + x = log_softmax_2d_x + expected = log_softmax_2d_expected + assert_allclose(sc.log_softmax(x, axis=1), expected, rtol=1e-13) + + +def test_log_softmax_2d_axis0(log_softmax_2d_x, log_softmax_2d_expected): + x = log_softmax_2d_x.T + expected = log_softmax_2d_expected.T + assert_allclose(sc.log_softmax(x, axis=0), expected, rtol=1e-13) + + +def test_log_softmax_3d(log_softmax_2d_x, log_softmax_2d_expected): + # 3-d input, with a tuple for the axis. + x_3d = log_softmax_2d_x.reshape(2, 2, 2) + expected_3d = log_softmax_2d_expected.reshape(2, 2, 2) + assert_allclose(sc.log_softmax(x_3d, axis=(1, 2)), expected_3d, rtol=1e-13) + + +def test_log_softmax_scalar(): + assert_allclose(sc.log_softmax(1.0), 0.0, rtol=1e-13) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_loggamma.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_loggamma.py new file mode 100644 index 0000000000000000000000000000000000000000..2fcb5a20037de46df939895d38fbe5fe6b85c9aa --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_loggamma.py @@ -0,0 +1,70 @@ +import numpy as np +from numpy.testing import assert_allclose, assert_ + +from scipy.special._testutils import FuncData +from scipy.special import gamma, gammaln, loggamma + + +def test_identities1(): + # test the identity exp(loggamma(z)) = gamma(z) + x = np.array([-99.5, -9.5, -0.5, 0.5, 9.5, 99.5]) + y = x.copy() + x, y = np.meshgrid(x, y) + z = (x + 1J*y).flatten() + dataset = np.vstack((z, gamma(z))).T + + def f(z): + return np.exp(loggamma(z)) + + FuncData(f, dataset, 0, 1, rtol=1e-14, atol=1e-14).check() + + +def test_identities2(): + # test the identity loggamma(z + 1) = log(z) + loggamma(z) + x = np.array([-99.5, -9.5, -0.5, 0.5, 9.5, 99.5]) + y = x.copy() + x, y = np.meshgrid(x, y) + z = (x + 1J*y).flatten() + dataset = np.vstack((z, np.log(z) + loggamma(z))).T + + def f(z): + return loggamma(z + 1) + + FuncData(f, dataset, 0, 1, rtol=1e-14, atol=1e-14).check() + + +def test_complex_dispatch_realpart(): + # Test that the real parts of loggamma and gammaln agree on the + # real axis. + x = np.r_[-np.logspace(10, -10), np.logspace(-10, 10)] + 0.5 + + dataset = np.vstack((x, gammaln(x))).T + + def f(z): + z = np.array(z, dtype='complex128') + return loggamma(z).real + + FuncData(f, dataset, 0, 1, rtol=1e-14, atol=1e-14).check() + + +def test_real_dispatch(): + x = np.logspace(-10, 10) + 0.5 + dataset = np.vstack((x, gammaln(x))).T + + FuncData(loggamma, dataset, 0, 1, rtol=1e-14, atol=1e-14).check() + assert_(loggamma(0) == np.inf) + assert_(np.isnan(loggamma(-1))) + + +def test_gh_6536(): + z = loggamma(complex(-3.4, +0.0)) + zbar = loggamma(complex(-3.4, -0.0)) + assert_allclose(z, zbar.conjugate(), rtol=1e-15, atol=0) + + +def test_branch_cut(): + # Make sure negative zero is treated correctly + x = -np.logspace(300, -30, 100) + z = np.asarray([complex(x0, 0.0) for x0 in x]) + zbar = np.asarray([complex(x0, -0.0) for x0 in x]) + assert_allclose(z, zbar.conjugate(), rtol=1e-15, atol=0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_logit.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_logit.py new file mode 100644 index 0000000000000000000000000000000000000000..050e8db5cb408c5c576c6cd292175d2df7c8f756 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_logit.py @@ -0,0 +1,162 @@ +import numpy as np +from numpy.testing import (assert_equal, assert_almost_equal, + assert_allclose) +from scipy.special import logit, expit, log_expit + + +class TestLogit: + + def check_logit_out(self, a, expected): + actual = logit(a) + assert_equal(actual.dtype, a.dtype) + rtol = 16*np.finfo(a.dtype).eps + assert_allclose(actual, expected, rtol=rtol) + + def test_float32(self): + a = np.concatenate((np.linspace(0, 1, 10, dtype=np.float32), + [np.float32(0.0001), np.float32(0.49999), + np.float32(0.50001)])) + # Expected values computed with mpmath from float32 inputs, e.g. + # from mpmath import mp + # mp.dps = 200 + # a = np.float32(1/9) + # print(np.float32(mp.log(a) - mp.log1p(-a))) + # prints `-2.0794415`. + expected = np.array([-np.inf, -2.0794415, -1.2527629, -6.9314712e-01, + -2.2314353e-01, 2.2314365e-01, 6.9314724e-01, + 1.2527630, 2.0794415, np.inf, + -9.2102404, -4.0054321e-05, 4.0054321e-05], + dtype=np.float32) + self.check_logit_out(a, expected) + + def test_float64(self): + a = np.concatenate((np.linspace(0, 1, 10, dtype=np.float64), + [1e-8, 0.4999999999999, 0.50000000001])) + # Expected values computed with mpmath. + expected = np.array([-np.inf, + -2.079441541679836, + -1.252762968495368, + -0.6931471805599454, + -0.22314355131420985, + 0.22314355131420985, + 0.6931471805599452, + 1.2527629684953674, + 2.0794415416798353, + np.inf, + -18.420680733952366, + -3.999023334699814e-13, + 4.000000330961484e-11]) + self.check_logit_out(a, expected) + + def test_nan(self): + expected = np.array([np.nan]*4) + with np.errstate(invalid='ignore'): + actual = logit(np.array([-3., -2., 2., 3.])) + + assert_equal(expected, actual) + + +class TestExpit: + def check_expit_out(self, dtype, expected): + a = np.linspace(-4, 4, 10) + a = np.array(a, dtype=dtype) + actual = expit(a) + assert_almost_equal(actual, expected) + assert_equal(actual.dtype, np.dtype(dtype)) + + def test_float32(self): + expected = np.array([0.01798621, 0.04265125, + 0.09777259, 0.20860852, + 0.39068246, 0.60931754, + 0.79139149, 0.9022274, + 0.95734876, 0.98201376], dtype=np.float32) + self.check_expit_out('f4', expected) + + def test_float64(self): + expected = np.array([0.01798621, 0.04265125, + 0.0977726, 0.20860853, + 0.39068246, 0.60931754, + 0.79139147, 0.9022274, + 0.95734875, 0.98201379]) + self.check_expit_out('f8', expected) + + def test_large(self): + for dtype in (np.float32, np.float64, np.longdouble): + for n in (88, 89, 709, 710, 11356, 11357): + n = np.array(n, dtype=dtype) + assert_allclose(expit(n), 1.0, atol=1e-20) + assert_allclose(expit(-n), 0.0, atol=1e-20) + assert_equal(expit(n).dtype, dtype) + assert_equal(expit(-n).dtype, dtype) + + +class TestLogExpit: + + def test_large_negative(self): + x = np.array([-10000.0, -750.0, -500.0, -35.0]) + y = log_expit(x) + assert_equal(y, x) + + def test_large_positive(self): + x = np.array([750.0, 1000.0, 10000.0]) + y = log_expit(x) + # y will contain -0.0, and -0.0 is used in the expected value, + # but assert_equal does not check the sign of zeros, and I don't + # think the sign is an essential part of the test (i.e. it would + # probably be OK if log_expit(1000) returned 0.0 instead of -0.0). + assert_equal(y, np.array([-0.0, -0.0, -0.0])) + + def test_basic_float64(self): + x = np.array([-32, -20, -10, -3, -1, -0.1, -1e-9, + 0, 1e-9, 0.1, 1, 10, 100, 500, 710, 725, 735]) + y = log_expit(x) + # + # Expected values were computed with mpmath: + # + # import mpmath + # + # mpmath.mp.dps = 100 + # + # def mp_log_expit(x): + # return -mpmath.log1p(mpmath.exp(-x)) + # + # expected = [float(mp_log_expit(t)) for t in x] + # + expected = [-32.000000000000014, -20.000000002061153, + -10.000045398899218, -3.048587351573742, + -1.3132616875182228, -0.7443966600735709, + -0.6931471810599453, -0.6931471805599453, + -0.6931471800599454, -0.6443966600735709, + -0.3132616875182228, -4.539889921686465e-05, + -3.720075976020836e-44, -7.124576406741286e-218, + -4.47628622567513e-309, -1.36930634e-315, + -6.217e-320] + + # When tested locally, only one value in y was not exactly equal to + # expected. That was for x=1, and the y value differed from the + # expected by 1 ULP. For this test, however, I'll use rtol=1e-15. + assert_allclose(y, expected, rtol=1e-15) + + def test_basic_float32(self): + x = np.array([-32, -20, -10, -3, -1, -0.1, -1e-9, + 0, 1e-9, 0.1, 1, 10, 100], dtype=np.float32) + y = log_expit(x) + # + # Expected values were computed with mpmath: + # + # import mpmath + # + # mpmath.mp.dps = 100 + # + # def mp_log_expit(x): + # return -mpmath.log1p(mpmath.exp(-x)) + # + # expected = [np.float32(mp_log_expit(t)) for t in x] + # + expected = np.array([-32.0, -20.0, -10.000046, -3.0485873, + -1.3132616, -0.7443967, -0.6931472, + -0.6931472, -0.6931472, -0.64439666, + -0.3132617, -4.5398898e-05, -3.8e-44], + dtype=np.float32) + + assert_allclose(y, expected, rtol=5e-7) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_logsumexp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_logsumexp.py new file mode 100644 index 0000000000000000000000000000000000000000..420c0d89f438701b57c4786ce7fa62279c668e5e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_logsumexp.py @@ -0,0 +1,327 @@ +import math +import pytest + +import numpy as np +from numpy.testing import assert_allclose + +from scipy.conftest import array_api_compatible +from scipy._lib._array_api import array_namespace, is_array_api_strict +from scipy._lib._array_api_no_0d import (xp_assert_equal, xp_assert_close, + xp_assert_less) + +from scipy.special import logsumexp, softmax +from scipy.special._logsumexp import _wrap_radians + + +dtypes = ['float32', 'float64', 'int32', 'int64', 'complex64', 'complex128'] +integral_dtypes = ['int32', 'int64'] + + +@array_api_compatible +@pytest.mark.usefixtures("skip_xp_backends") +@pytest.mark.skip_xp_backends('jax.numpy', + reason="JAX arrays do not support item assignment") +def test_wrap_radians(xp): + x = xp.asarray([-math.pi-1, -math.pi, -1, -1e-300, + 0, 1e-300, 1, math.pi, math.pi+1]) + ref = xp.asarray([math.pi-1, math.pi, -1, -1e-300, + 0, 1e-300, 1, math.pi, -math.pi+1]) + res = _wrap_radians(x, xp) + xp_assert_close(res, ref, atol=0) + + +@array_api_compatible +@pytest.mark.usefixtures("skip_xp_backends") +@pytest.mark.skip_xp_backends('jax.numpy', + reason="JAX arrays do not support item assignment") +class TestLogSumExp: + def test_logsumexp(self, xp): + # Test with zero-size array + a = xp.asarray([]) + desired = xp.asarray(-xp.inf) + xp_assert_equal(logsumexp(a), desired) + + # Test whether logsumexp() function correctly handles large inputs. + a = xp.arange(200., dtype=xp.float64) + desired = xp.log(xp.sum(xp.exp(a))) + xp_assert_close(logsumexp(a), desired) + + # Now test with large numbers + b = xp.asarray([1000., 1000.]) + desired = xp.asarray(1000.0 + math.log(2.0)) + xp_assert_close(logsumexp(b), desired) + + n = 1000 + b = xp.full((n,), 10000) + desired = xp.asarray(10000.0 + math.log(n)) + xp_assert_close(logsumexp(b), desired) + + x = xp.asarray([1e-40] * 1000000) + logx = xp.log(x) + X = xp.stack([x, x]) + logX = xp.stack([logx, logx]) + xp_assert_close(xp.exp(logsumexp(logX)), xp.sum(X)) + xp_assert_close(xp.exp(logsumexp(logX, axis=0)), xp.sum(X, axis=0)) + xp_assert_close(xp.exp(logsumexp(logX, axis=1)), xp.sum(X, axis=1)) + + # Handling special values properly + inf = xp.asarray([xp.inf]) + nan = xp.asarray([xp.nan]) + xp_assert_equal(logsumexp(inf), inf[0]) + xp_assert_equal(logsumexp(-inf), -inf[0]) + xp_assert_equal(logsumexp(nan), nan[0]) + xp_assert_equal(logsumexp(xp.asarray([-xp.inf, -xp.inf])), -inf[0]) + + # Handling an array with different magnitudes on the axes + a = xp.asarray([[1e10, 1e-10], + [-1e10, -np.inf]]) + ref = xp.asarray([1e10, -1e10]) + xp_assert_close(logsumexp(a, axis=-1), ref) + + # Test keeping dimensions + xp_test = array_namespace(a) # `torch` needs `expand_dims` + ref = xp_test.expand_dims(ref, axis=-1) + xp_assert_close(logsumexp(a, axis=-1, keepdims=True), ref) + + # Test multiple axes + xp_assert_close(logsumexp(a, axis=(-1, -2)), xp.asarray(1e10)) + + def test_logsumexp_b(self, xp): + a = xp.arange(200., dtype=xp.float64) + b = xp.arange(200., 0., -1.) + desired = xp.log(xp.sum(b*xp.exp(a))) + xp_assert_close(logsumexp(a, b=b), desired) + + a = xp.asarray([1000, 1000]) + b = xp.asarray([1.2, 1.2]) + desired = xp.asarray(1000 + math.log(2 * 1.2)) + xp_assert_close(logsumexp(a, b=b), desired) + + x = xp.asarray([1e-40] * 100000) + b = xp.linspace(1, 1000, 100000) + logx = xp.log(x) + X = xp.stack((x, x)) + logX = xp.stack((logx, logx)) + B = xp.stack((b, b)) + xp_assert_close(xp.exp(logsumexp(logX, b=B)), xp.sum(B * X)) + xp_assert_close(xp.exp(logsumexp(logX, b=B, axis=0)), xp.sum(B * X, axis=0)) + xp_assert_close(xp.exp(logsumexp(logX, b=B, axis=1)), xp.sum(B * X, axis=1)) + + def test_logsumexp_sign(self, xp): + a = xp.asarray([1, 1, 1]) + b = xp.asarray([1, -1, -1]) + + r, s = logsumexp(a, b=b, return_sign=True) + xp_assert_close(r, xp.asarray(1.)) + xp_assert_equal(s, xp.asarray(-1.)) + + def test_logsumexp_sign_zero(self, xp): + a = xp.asarray([1, 1]) + b = xp.asarray([1, -1]) + + r, s = logsumexp(a, b=b, return_sign=True) + assert not xp.isfinite(r) + assert not xp.isnan(r) + assert r < 0 + assert s == 0 + + def test_logsumexp_sign_shape(self, xp): + a = xp.ones((1, 2, 3, 4)) + b = xp.ones_like(a) + + r, s = logsumexp(a, axis=2, b=b, return_sign=True) + assert r.shape == s.shape == (1, 2, 4) + + r, s = logsumexp(a, axis=(1, 3), b=b, return_sign=True) + assert r.shape == s.shape == (1,3) + + def test_logsumexp_complex_sign(self, xp): + a = xp.asarray([1 + 1j, 2 - 1j, -2 + 3j]) + + r, s = logsumexp(a, return_sign=True) + + expected_sumexp = xp.sum(xp.exp(a)) + # This is the numpy>=2.0 convention for np.sign + expected_sign = expected_sumexp / xp.abs(expected_sumexp) + + xp_assert_close(s, expected_sign) + xp_assert_close(s * xp.exp(r), expected_sumexp) + + def test_logsumexp_shape(self, xp): + a = xp.ones((1, 2, 3, 4)) + b = xp.ones_like(a) + + r = logsumexp(a, axis=2, b=b) + assert r.shape == (1, 2, 4) + + r = logsumexp(a, axis=(1, 3), b=b) + assert r.shape == (1, 3) + + def test_logsumexp_b_zero(self, xp): + a = xp.asarray([1, 10000]) + b = xp.asarray([1, 0]) + + xp_assert_close(logsumexp(a, b=b), xp.asarray(1.)) + + def test_logsumexp_b_shape(self, xp): + a = xp.zeros((4, 1, 2, 1)) + b = xp.ones((3, 1, 5)) + + logsumexp(a, b=b) + + @pytest.mark.parametrize('arg', (1, [1, 2, 3])) + @pytest.mark.skip_xp_backends(np_only=True) + def test_xp_invalid_input(self, arg, xp): + assert logsumexp(arg) == logsumexp(np.asarray(np.atleast_1d(arg))) + + @pytest.mark.skip_xp_backends(np_only=True, + reason="Lists correspond with NumPy backend") + def test_list(self, xp): + a = [1000, 1000] + desired = xp.asarray(1000.0 + math.log(2.0), dtype=np.float64) + xp_assert_close(logsumexp(a), desired) + + @pytest.mark.parametrize('dtype', dtypes) + def test_dtypes_a(self, dtype, xp): + dtype = getattr(xp, dtype) + a = xp.asarray([1000., 1000.], dtype=dtype) + xp_test = array_namespace(a) # torch needs compatible `isdtype` + desired_dtype = (xp.asarray(1.).dtype if xp_test.isdtype(dtype, 'integral') + else dtype) # true for all libraries tested + desired = xp.asarray(1000.0 + math.log(2.0), dtype=desired_dtype) + xp_assert_close(logsumexp(a), desired) + + @pytest.mark.parametrize('dtype_a', dtypes) + @pytest.mark.parametrize('dtype_b', dtypes) + def test_dtypes_ab(self, dtype_a, dtype_b, xp): + xp_dtype_a = getattr(xp, dtype_a) + xp_dtype_b = getattr(xp, dtype_b) + a = xp.asarray([2, 1], dtype=xp_dtype_a) + b = xp.asarray([1, -1], dtype=xp_dtype_b) + xp_test = array_namespace(a, b) # torch needs compatible result_type + if is_array_api_strict(xp): + xp_float_dtypes = [dtype for dtype in [xp_dtype_a, xp_dtype_b] + if not xp_test.isdtype(dtype, 'integral')] + if len(xp_float_dtypes) < 2: # at least one is integral + xp_float_dtypes.append(xp.asarray(1.).dtype) + desired_dtype = xp_test.result_type(*xp_float_dtypes) + else: + # True for all libraries tested + desired_dtype = xp_test.result_type(xp_dtype_a, xp_dtype_b, xp.float32) + desired = xp.asarray(math.log(math.exp(2) - math.exp(1)), dtype=desired_dtype) + xp_assert_close(logsumexp(a, b=b), desired) + + def test_gh18295(self, xp): + # gh-18295 noted loss of precision when real part of one element is much + # larger than the rest. Check that this is resolved. + a = xp.asarray([0.0, -40.0]) + res = logsumexp(a) + ref = xp.logaddexp(a[0], a[1]) + xp_assert_close(res, ref) + + @pytest.mark.parametrize('dtype', ['complex64', 'complex128']) + def test_gh21610(self, xp, dtype): + # gh-21610 noted that `logsumexp` could return imaginary components + # outside the range (-pi, pi]. Check that this is resolved. + # While working on this, I noticed that all other tests passed even + # when the imaginary component of the result was zero. This suggested + # the need of a stronger test with imaginary dtype. + rng = np.random.default_rng(324984329582349862) + dtype = getattr(xp, dtype) + shape = (10, 100) + x = rng.uniform(1, 40, shape) + 1.j * rng.uniform(1, 40, shape) + x = xp.asarray(x, dtype=dtype) + + res = logsumexp(x, axis=1) + ref = xp.log(xp.sum(xp.exp(x), axis=1)) + max = xp.full_like(xp.imag(res), xp.asarray(xp.pi)) + xp_assert_less(xp.abs(xp.imag(res)), max) + xp_assert_close(res, ref) + + out, sgn = logsumexp(x, return_sign=True, axis=1) + ref = xp.sum(xp.exp(x), axis=1) + xp_assert_less(xp.abs(xp.imag(sgn)), max) + xp_assert_close(out, xp.real(xp.log(ref))) + xp_assert_close(sgn, ref/xp.abs(ref)) + + def test_gh21709_small_imaginary(self, xp): + # Test that `logsumexp` does not lose relative precision of + # small imaginary components + x = xp.asarray([0, 0.+2.2204460492503132e-17j]) + res = logsumexp(x) + # from mpmath import mp + # mp.dps = 100 + # x, y = mp.mpc(0), mp.mpc('0', '2.2204460492503132e-17') + # ref = complex(mp.log(mp.exp(x) + mp.exp(y))) + ref = xp.asarray(0.6931471805599453+1.1102230246251566e-17j) + xp_assert_close(xp.real(res), xp.real(ref)) + xp_assert_close(xp.imag(res), xp.imag(ref), atol=0, rtol=1e-15) + + def test_gh22903(self, xp): + # gh-22903 reported that `logsumexp` produced NaN where the weight associated + # with the max magnitude element was negative and `return_sign=False`, even if + # the net result should be the log of a positive number. + + # result is log of positive number + a = xp.asarray([3.06409428, 0.37251854, 3.87471931]) + b = xp.asarray([1.88190708, 2.84174795, -0.85016884]) + xp_assert_close(logsumexp(a, b=b), logsumexp(a, b=b, return_sign=True)[0]) + + # result is log of negative number + b = xp.asarray([1.88190708, 2.84174795, -3.85016884]) + xp_assert_close(logsumexp(a, b=b), xp.asarray(xp.nan)) + + +class TestSoftmax: + def test_softmax_fixtures(self): + assert_allclose(softmax([1000, 0, 0, 0]), np.array([1, 0, 0, 0]), + rtol=1e-13) + assert_allclose(softmax([1, 1]), np.array([.5, .5]), rtol=1e-13) + assert_allclose(softmax([0, 1]), np.array([1, np.e])/(1 + np.e), + rtol=1e-13) + + # Expected value computed using mpmath (with mpmath.mp.dps = 200) and then + # converted to float. + x = np.arange(4) + expected = np.array([0.03205860328008499, + 0.08714431874203256, + 0.23688281808991013, + 0.6439142598879722]) + + assert_allclose(softmax(x), expected, rtol=1e-13) + + # Translation property. If all the values are changed by the same amount, + # the softmax result does not change. + assert_allclose(softmax(x + 100), expected, rtol=1e-13) + + # When axis=None, softmax operates on the entire array, and preserves + # the shape. + assert_allclose(softmax(x.reshape(2, 2)), expected.reshape(2, 2), + rtol=1e-13) + + + def test_softmax_multi_axes(self): + assert_allclose(softmax([[1000, 0], [1000, 0]], axis=0), + np.array([[.5, .5], [.5, .5]]), rtol=1e-13) + assert_allclose(softmax([[1000, 0], [1000, 0]], axis=1), + np.array([[1, 0], [1, 0]]), rtol=1e-13) + + # Expected value computed using mpmath (with mpmath.mp.dps = 200) and then + # converted to float. + x = np.array([[-25, 0, 25, 50], + [1, 325, 749, 750]]) + expected = np.array([[2.678636961770877e-33, + 1.9287498479371314e-22, + 1.3887943864771144e-11, + 0.999999999986112], + [0.0, + 1.9444526359919372e-185, + 0.2689414213699951, + 0.7310585786300048]]) + assert_allclose(softmax(x, axis=1), expected, rtol=1e-13) + assert_allclose(softmax(x.T, axis=0), expected.T, rtol=1e-13) + + # 3-d input, with a tuple for the axis. + x3d = x.reshape(2, 2, 2) + assert_allclose(softmax(x3d, axis=(1, 2)), expected.reshape(2, 2, 2), + rtol=1e-13) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_mpmath.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_mpmath.py new file mode 100644 index 0000000000000000000000000000000000000000..43e84e444f2eda03aef77ea923ca2d498aef4126 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_mpmath.py @@ -0,0 +1,2292 @@ +""" +Test SciPy functions versus mpmath, if available. + +""" +import numpy as np +from numpy.testing import assert_, assert_allclose, suppress_warnings +from numpy import pi +import pytest +import itertools + +from scipy._lib import _pep440 + +import scipy.special as sc +from scipy.special._testutils import ( + MissingModule, check_version, FuncData, + assert_func_equal) +from scipy.special._mptestutils import ( + Arg, FixedArg, ComplexArg, IntArg, assert_mpmath_equal, + nonfunctional_tooslow, trace_args, time_limited, exception_to_nan, + inf_to_nan) +from scipy.special._ufuncs import ( + _sinpi, _cospi, _lgam1p, _lanczos_sum_expg_scaled, _log1pmx, + _igam_fac) + +try: + import mpmath +except ImportError: + mpmath = MissingModule('mpmath') + + +# ------------------------------------------------------------------------------ +# expi +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.10') +def test_expi_complex(): + dataset = [] + for r in np.logspace(-99, 2, 10): + for p in np.linspace(0, 2*np.pi, 30): + z = r*np.exp(1j*p) + dataset.append((z, complex(mpmath.ei(z)))) + dataset = np.array(dataset, dtype=np.cdouble) + + FuncData(sc.expi, dataset, 0, 1).check() + + +# ------------------------------------------------------------------------------ +# expn +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.19') +def test_expn_large_n(): + # Test the transition to the asymptotic regime of n. + dataset = [] + for n in [50, 51]: + for x in np.logspace(0, 4, 200): + with mpmath.workdps(100): + dataset.append((n, x, float(mpmath.expint(n, x)))) + dataset = np.asarray(dataset) + + FuncData(sc.expn, dataset, (0, 1), 2, rtol=1e-13).check() + +# ------------------------------------------------------------------------------ +# hyp0f1 +# ------------------------------------------------------------------------------ + + +@check_version(mpmath, '0.19') +def test_hyp0f1_gh5764(): + # Do a small and somewhat systematic test that runs quickly + dataset = [] + axis = [-99.5, -9.5, -0.5, 0.5, 9.5, 99.5] + for v in axis: + for x in axis: + for y in axis: + z = x + 1j*y + # mpmath computes the answer correctly at dps ~ 17 but + # fails for 20 < dps < 120 (uses a different method); + # set the dps high enough that this isn't an issue + with mpmath.workdps(120): + res = complex(mpmath.hyp0f1(v, z)) + dataset.append((v, z, res)) + dataset = np.array(dataset) + + FuncData(lambda v, z: sc.hyp0f1(v.real, z), dataset, (0, 1), 2, + rtol=1e-13).check() + + +@check_version(mpmath, '0.19') +def test_hyp0f1_gh_1609(): + # this is a regression test for gh-1609 + vv = np.linspace(150, 180, 21) + af = sc.hyp0f1(vv, 0.5) + mf = np.array([mpmath.hyp0f1(v, 0.5) for v in vv]) + assert_allclose(af, mf.astype(float), rtol=1e-12) + + +# ------------------------------------------------------------------------------ +# hyperu +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '1.1.0') +def test_hyperu_around_0(): + dataset = [] + # DLMF 13.2.14-15 test points. + for n in np.arange(-5, 5): + for b in np.linspace(-5, 5, 20): + a = -n + dataset.append((a, b, 0, float(mpmath.hyperu(a, b, 0)))) + a = -n + b - 1 + dataset.append((a, b, 0, float(mpmath.hyperu(a, b, 0)))) + # DLMF 13.2.16-22 test points. + for a in [-10.5, -1.5, -0.5, 0, 0.5, 1, 10]: + for b in [-1.0, -0.5, 0, 0.5, 1, 1.5, 2, 2.5]: + dataset.append((a, b, 0, float(mpmath.hyperu(a, b, 0)))) + dataset = np.array(dataset) + + FuncData(sc.hyperu, dataset, (0, 1, 2), 3, rtol=1e-15, atol=5e-13).check() + + +# ------------------------------------------------------------------------------ +# hyp2f1 +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '1.0.0') +def test_hyp2f1_strange_points(): + pts = [ + (2, -1, -1, 0.7), # expected: 2.4 + (2, -2, -2, 0.7), # expected: 3.87 + ] + pts += list(itertools.product([2, 1, -0.7, -1000], repeat=4)) + pts = [ + (a, b, c, x) for a, b, c, x in pts + if b == c and round(b) == b and b < 0 and b != -1000 + ] + kw = dict(eliminate=True) + dataset = [p + (float(mpmath.hyp2f1(*p, **kw)),) for p in pts] + dataset = np.array(dataset, dtype=np.float64) + + FuncData(sc.hyp2f1, dataset, (0,1,2,3), 4, rtol=1e-10).check() + + +@check_version(mpmath, '0.13') +def test_hyp2f1_real_some_points(): + pts = [ + (1, 2, 3, 0), + (1./3, 2./3, 5./6, 27./32), + (1./4, 1./2, 3./4, 80./81), + (2,-2, -3, 3), + (2, -3, -2, 3), + (2, -1.5, -1.5, 3), + (1, 2, 3, 0), + (0.7235, -1, -5, 0.3), + (0.25, 1./3, 2, 0.999), + (0.25, 1./3, 2, -1), + (2, 3, 5, 0.99), + (3./2, -0.5, 3, 0.99), + (2, 2.5, -3.25, 0.999), + (-8, 18.016500331508873, 10.805295997850628, 0.90875647507000001), + (-10, 900, -10.5, 0.99), + (-10, 900, 10.5, 0.99), + (-1, 2, 1, 1.0), + (-1, 2, 1, -1.0), + (-3, 13, 5, 1.0), + (-3, 13, 5, -1.0), + (0.5, 1 - 270.5, 1.5, 0.999**2), # from issue 1561 + ] + dataset = [p + (float(mpmath.hyp2f1(*p)),) for p in pts] + dataset = np.array(dataset, dtype=np.float64) + + with np.errstate(invalid='ignore'): + FuncData(sc.hyp2f1, dataset, (0,1,2,3), 4, rtol=1e-10).check() + + +@check_version(mpmath, '0.14') +def test_hyp2f1_some_points_2(): + # Taken from mpmath unit tests -- this point failed for mpmath 0.13 but + # was fixed in their SVN since then + pts = [ + (112, (51,10), (-9,10), -0.99999), + (10,-900,10.5,0.99), + (10,-900,-10.5,0.99), + ] + + def fev(x): + if isinstance(x, tuple): + return float(x[0]) / x[1] + else: + return x + + dataset = [tuple(map(fev, p)) + (float(mpmath.hyp2f1(*p)),) for p in pts] + dataset = np.array(dataset, dtype=np.float64) + + FuncData(sc.hyp2f1, dataset, (0,1,2,3), 4, rtol=1e-10).check() + + +@check_version(mpmath, '0.13') +def test_hyp2f1_real_some(): + dataset = [] + for a in [-10, -5, -1.8, 1.8, 5, 10]: + for b in [-2.5, -1, 1, 7.4]: + for c in [-9, -1.8, 5, 20.4]: + for z in [-10, -1.01, -0.99, 0, 0.6, 0.95, 1.5, 10]: + try: + v = float(mpmath.hyp2f1(a, b, c, z)) + except Exception: + continue + dataset.append((a, b, c, z, v)) + dataset = np.array(dataset, dtype=np.float64) + + with np.errstate(invalid='ignore'): + FuncData(sc.hyp2f1, dataset, (0,1,2,3), 4, rtol=1e-9, + ignore_inf_sign=True).check() + + +@check_version(mpmath, '0.12') +@pytest.mark.slow +def test_hyp2f1_real_random(): + npoints = 500 + dataset = np.zeros((npoints, 5), np.float64) + + np.random.seed(1234) + dataset[:, 0] = np.random.pareto(1.5, npoints) + dataset[:, 1] = np.random.pareto(1.5, npoints) + dataset[:, 2] = np.random.pareto(1.5, npoints) + dataset[:, 3] = 2*np.random.rand(npoints) - 1 + + dataset[:, 0] *= (-1)**np.random.randint(2, npoints) + dataset[:, 1] *= (-1)**np.random.randint(2, npoints) + dataset[:, 2] *= (-1)**np.random.randint(2, npoints) + + for ds in dataset: + if mpmath.__version__ < '0.14': + # mpmath < 0.14 fails for c too much smaller than a, b + if abs(ds[:2]).max() > abs(ds[2]): + ds[2] = abs(ds[:2]).max() + ds[4] = float(mpmath.hyp2f1(*tuple(ds[:4]))) + + FuncData(sc.hyp2f1, dataset, (0, 1, 2, 3), 4, rtol=1e-9).check() + + +# ------------------------------------------------------------------------------ +# erf (complex) +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.14') +def test_erf_complex(): + # need to increase mpmath precision for this test + old_dps, old_prec = mpmath.mp.dps, mpmath.mp.prec + try: + mpmath.mp.dps = 70 + x1, y1 = np.meshgrid(np.linspace(-10, 1, 31), np.linspace(-10, 1, 11)) + x2, y2 = np.meshgrid(np.logspace(-80, .8, 31), np.logspace(-80, .8, 11)) + points = np.r_[x1.ravel(),x2.ravel()] + 1j*np.r_[y1.ravel(), y2.ravel()] + + assert_func_equal(sc.erf, lambda x: complex(mpmath.erf(x)), points, + vectorized=False, rtol=1e-13) + assert_func_equal(sc.erfc, lambda x: complex(mpmath.erfc(x)), points, + vectorized=False, rtol=1e-13) + finally: + mpmath.mp.dps, mpmath.mp.prec = old_dps, old_prec + + +# ------------------------------------------------------------------------------ +# lpmv +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.15') +def test_lpmv(): + pts = [] + for x in [-0.99, -0.557, 1e-6, 0.132, 1]: + pts.extend([ + (1, 1, x), + (1, -1, x), + (-1, 1, x), + (-1, -2, x), + (1, 1.7, x), + (1, -1.7, x), + (-1, 1.7, x), + (-1, -2.7, x), + (1, 10, x), + (1, 11, x), + (3, 8, x), + (5, 11, x), + (-3, 8, x), + (-5, 11, x), + (3, -8, x), + (5, -11, x), + (-3, -8, x), + (-5, -11, x), + (3, 8.3, x), + (5, 11.3, x), + (-3, 8.3, x), + (-5, 11.3, x), + (3, -8.3, x), + (5, -11.3, x), + (-3, -8.3, x), + (-5, -11.3, x), + ]) + + def mplegenp(nu, mu, x): + if mu == int(mu) and x == 1: + # mpmath 0.17 gets this wrong + if mu == 0: + return 1 + else: + return 0 + return mpmath.legenp(nu, mu, x) + + dataset = [p + (mplegenp(p[1], p[0], p[2]),) for p in pts] + dataset = np.array(dataset, dtype=np.float64) + + def evf(mu, nu, x): + return sc.lpmv(mu.astype(int), nu, x) + + with np.errstate(invalid='ignore'): + FuncData(evf, dataset, (0,1,2), 3, rtol=1e-10, atol=1e-14).check() + + +# ------------------------------------------------------------------------------ +# beta +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.15') +def test_beta(): + np.random.seed(1234) + + b = np.r_[np.logspace(-200, 200, 4), + np.logspace(-10, 10, 4), + np.logspace(-1, 1, 4), + np.arange(-10, 11, 1), + np.arange(-10, 11, 1) + 0.5, + -1, -2.3, -3, -100.3, -10003.4] + a = b + + ab = np.array(np.broadcast_arrays(a[:,None], b[None,:])).reshape(2, -1).T + + old_dps, old_prec = mpmath.mp.dps, mpmath.mp.prec + try: + mpmath.mp.dps = 400 + + assert_func_equal(sc.beta, + lambda a, b: float(mpmath.beta(a, b)), + ab, + vectorized=False, + rtol=1e-10, + ignore_inf_sign=True) + + assert_func_equal( + sc.betaln, + lambda a, b: float(mpmath.log(abs(mpmath.beta(a, b)))), + ab, + vectorized=False, + rtol=1e-10) + finally: + mpmath.mp.dps, mpmath.mp.prec = old_dps, old_prec + + +# ------------------------------------------------------------------------------ +# loggamma +# ------------------------------------------------------------------------------ + +LOGGAMMA_TAYLOR_RADIUS = 0.2 + + +@check_version(mpmath, '0.19') +def test_loggamma_taylor_transition(): + # Make sure there isn't a big jump in accuracy when we move from + # using the Taylor series to using the recurrence relation. + + r = LOGGAMMA_TAYLOR_RADIUS + np.array([-0.1, -0.01, 0, 0.01, 0.1]) + theta = np.linspace(0, 2*np.pi, 20) + r, theta = np.meshgrid(r, theta) + dz = r*np.exp(1j*theta) + z = np.r_[1 + dz, 2 + dz].flatten() + + dataset = [(z0, complex(mpmath.loggamma(z0))) for z0 in z] + dataset = np.array(dataset) + + FuncData(sc.loggamma, dataset, 0, 1, rtol=5e-14).check() + + +@check_version(mpmath, '0.19') +def test_loggamma_taylor(): + # Test around the zeros at z = 1, 2. + + r = np.logspace(-16, np.log10(LOGGAMMA_TAYLOR_RADIUS), 10) + theta = np.linspace(0, 2*np.pi, 20) + r, theta = np.meshgrid(r, theta) + dz = r*np.exp(1j*theta) + z = np.r_[1 + dz, 2 + dz].flatten() + + dataset = [(z0, complex(mpmath.loggamma(z0))) for z0 in z] + dataset = np.array(dataset) + + FuncData(sc.loggamma, dataset, 0, 1, rtol=5e-14).check() + + +# ------------------------------------------------------------------------------ +# rgamma +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.19') +@pytest.mark.slow +def test_rgamma_zeros(): + # Test around the zeros at z = 0, -1, -2, ..., -169. (After -169 we + # get values that are out of floating point range even when we're + # within 0.1 of the zero.) + + # Can't use too many points here or the test takes forever. + dx = np.r_[-np.logspace(-1, -13, 3), 0, np.logspace(-13, -1, 3)] + dy = dx.copy() + dx, dy = np.meshgrid(dx, dy) + dz = dx + 1j*dy + zeros = np.arange(0, -170, -1).reshape(1, 1, -1) + z = (zeros + np.dstack((dz,)*zeros.size)).flatten() + with mpmath.workdps(100): + dataset = [(z0, complex(mpmath.rgamma(z0))) for z0 in z] + + dataset = np.array(dataset) + FuncData(sc.rgamma, dataset, 0, 1, rtol=1e-12).check() + + +# ------------------------------------------------------------------------------ +# digamma +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.19') +@pytest.mark.slow +def test_digamma_roots(): + # Test the special-cased roots for digamma. + root = mpmath.findroot(mpmath.digamma, 1.5) + roots = [float(root)] + root = mpmath.findroot(mpmath.digamma, -0.5) + roots.append(float(root)) + roots = np.array(roots) + + # If we test beyond a radius of 0.24 mpmath will take forever. + dx = np.r_[-0.24, -np.logspace(-1, -15, 10), 0, np.logspace(-15, -1, 10), 0.24] + dy = dx.copy() + dx, dy = np.meshgrid(dx, dy) + dz = dx + 1j*dy + z = (roots + np.dstack((dz,)*roots.size)).flatten() + with mpmath.workdps(30): + dataset = [(z0, complex(mpmath.digamma(z0))) for z0 in z] + + dataset = np.array(dataset) + FuncData(sc.digamma, dataset, 0, 1, rtol=1e-14).check() + + +@check_version(mpmath, '0.19') +def test_digamma_negreal(): + # Test digamma around the negative real axis. Don't do this in + # TestSystematic because the points need some jiggering so that + # mpmath doesn't take forever. + + digamma = exception_to_nan(mpmath.digamma) + + x = -np.logspace(300, -30, 100) + y = np.r_[-np.logspace(0, -3, 5), 0, np.logspace(-3, 0, 5)] + x, y = np.meshgrid(x, y) + z = (x + 1j*y).flatten() + + with mpmath.workdps(40): + dataset = [(z0, complex(digamma(z0))) for z0 in z] + dataset = np.asarray(dataset) + + FuncData(sc.digamma, dataset, 0, 1, rtol=1e-13).check() + + +@check_version(mpmath, '0.19') +def test_digamma_boundary(): + # Check that there isn't a jump in accuracy when we switch from + # using the asymptotic series to the reflection formula. + + x = -np.logspace(300, -30, 100) + y = np.array([-6.1, -5.9, 5.9, 6.1]) + x, y = np.meshgrid(x, y) + z = (x + 1j*y).flatten() + + with mpmath.workdps(30): + dataset = [(z0, complex(mpmath.digamma(z0))) for z0 in z] + dataset = np.asarray(dataset) + + FuncData(sc.digamma, dataset, 0, 1, rtol=1e-13).check() + + +# ------------------------------------------------------------------------------ +# gammainc +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.19') +@pytest.mark.slow +def test_gammainc_boundary(): + # Test the transition to the asymptotic series. + small = 20 + a = np.linspace(0.5*small, 2*small, 50) + x = a.copy() + a, x = np.meshgrid(a, x) + a, x = a.flatten(), x.flatten() + with mpmath.workdps(100): + dataset = [(a0, x0, float(mpmath.gammainc(a0, b=x0, regularized=True))) + for a0, x0 in zip(a, x)] + dataset = np.array(dataset) + + FuncData(sc.gammainc, dataset, (0, 1), 2, rtol=1e-12).check() + + +# ------------------------------------------------------------------------------ +# spence +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.19') +@pytest.mark.slow +def test_spence_circle(): + # The trickiest region for spence is around the circle |z - 1| = 1, + # so test that region carefully. + + def spence(z): + return complex(mpmath.polylog(2, 1 - z)) + + r = np.linspace(0.5, 1.5) + theta = np.linspace(0, 2*pi) + z = (1 + np.outer(r, np.exp(1j*theta))).flatten() + dataset = np.asarray([(z0, spence(z0)) for z0 in z]) + + FuncData(sc.spence, dataset, 0, 1, rtol=1e-14).check() + + +# ------------------------------------------------------------------------------ +# sinpi and cospi +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.19') +def test_sinpi_zeros(): + eps = np.finfo(float).eps + dx = np.r_[-np.logspace(0, -13, 3), 0, np.logspace(-13, 0, 3)] + dy = dx.copy() + dx, dy = np.meshgrid(dx, dy) + dz = dx + 1j*dy + zeros = np.arange(-100, 100, 1).reshape(1, 1, -1) + z = (zeros + np.dstack((dz,)*zeros.size)).flatten() + dataset = np.asarray([(z0, complex(mpmath.sinpi(z0))) + for z0 in z]) + FuncData(_sinpi, dataset, 0, 1, rtol=2*eps).check() + + +@check_version(mpmath, '0.19') +def test_cospi_zeros(): + eps = np.finfo(float).eps + dx = np.r_[-np.logspace(0, -13, 3), 0, np.logspace(-13, 0, 3)] + dy = dx.copy() + dx, dy = np.meshgrid(dx, dy) + dz = dx + 1j*dy + zeros = (np.arange(-100, 100, 1) + 0.5).reshape(1, 1, -1) + z = (zeros + np.dstack((dz,)*zeros.size)).flatten() + dataset = np.asarray([(z0, complex(mpmath.cospi(z0))) + for z0 in z]) + + FuncData(_cospi, dataset, 0, 1, rtol=2*eps).check() + + +# ------------------------------------------------------------------------------ +# ellipj +# ------------------------------------------------------------------------------ + +@check_version(mpmath, '0.19') +def test_dn_quarter_period(): + def dn(u, m): + return sc.ellipj(u, m)[2] + + def mpmath_dn(u, m): + return float(mpmath.ellipfun("dn", u=u, m=m)) + + m = np.linspace(0, 1, 20) + du = np.r_[-np.logspace(-1, -15, 10), 0, np.logspace(-15, -1, 10)] + dataset = [] + for m0 in m: + u0 = float(mpmath.ellipk(m0)) + for du0 in du: + p = u0 + du0 + dataset.append((p, m0, mpmath_dn(p, m0))) + dataset = np.asarray(dataset) + + FuncData(dn, dataset, (0, 1), 2, rtol=1e-10).check() + + +# ------------------------------------------------------------------------------ +# Wright Omega +# ------------------------------------------------------------------------------ + +def _mpmath_wrightomega(z, dps): + with mpmath.workdps(dps): + z = mpmath.mpc(z) + unwind = mpmath.ceil((z.imag - mpmath.pi)/(2*mpmath.pi)) + res = mpmath.lambertw(mpmath.exp(z), unwind) + return res + + +@pytest.mark.slow +@check_version(mpmath, '0.19') +def test_wrightomega_branch(): + x = -np.logspace(10, 0, 25) + picut_above = [np.nextafter(np.pi, np.inf)] + picut_below = [np.nextafter(np.pi, -np.inf)] + npicut_above = [np.nextafter(-np.pi, np.inf)] + npicut_below = [np.nextafter(-np.pi, -np.inf)] + for i in range(50): + picut_above.append(np.nextafter(picut_above[-1], np.inf)) + picut_below.append(np.nextafter(picut_below[-1], -np.inf)) + npicut_above.append(np.nextafter(npicut_above[-1], np.inf)) + npicut_below.append(np.nextafter(npicut_below[-1], -np.inf)) + y = np.hstack((picut_above, picut_below, npicut_above, npicut_below)) + x, y = np.meshgrid(x, y) + z = (x + 1j*y).flatten() + + dataset = np.asarray([(z0, complex(_mpmath_wrightomega(z0, 25))) + for z0 in z]) + + FuncData(sc.wrightomega, dataset, 0, 1, rtol=1e-8).check() + + +@pytest.mark.slow +@check_version(mpmath, '0.19') +def test_wrightomega_region1(): + # This region gets less coverage in the TestSystematic test + x = np.linspace(-2, 1) + y = np.linspace(1, 2*np.pi) + x, y = np.meshgrid(x, y) + z = (x + 1j*y).flatten() + + dataset = np.asarray([(z0, complex(_mpmath_wrightomega(z0, 25))) + for z0 in z]) + + FuncData(sc.wrightomega, dataset, 0, 1, rtol=1e-15).check() + + +@pytest.mark.slow +@check_version(mpmath, '0.19') +def test_wrightomega_region2(): + # This region gets less coverage in the TestSystematic test + x = np.linspace(-2, 1) + y = np.linspace(-2*np.pi, -1) + x, y = np.meshgrid(x, y) + z = (x + 1j*y).flatten() + + dataset = np.asarray([(z0, complex(_mpmath_wrightomega(z0, 25))) + for z0 in z]) + + FuncData(sc.wrightomega, dataset, 0, 1, rtol=1e-15).check() + + +# ------------------------------------------------------------------------------ +# lambertw +# ------------------------------------------------------------------------------ + +@pytest.mark.slow +@check_version(mpmath, '0.19') +def test_lambertw_smallz(): + x, y = np.linspace(-1, 1, 25), np.linspace(-1, 1, 25) + x, y = np.meshgrid(x, y) + z = (x + 1j*y).flatten() + + dataset = np.asarray([(z0, complex(mpmath.lambertw(z0))) + for z0 in z]) + + FuncData(sc.lambertw, dataset, 0, 1, rtol=1e-13).check() + + +# ------------------------------------------------------------------------------ +# Systematic tests +# ------------------------------------------------------------------------------ + +# The functions lpn, lpmn, clpmn, and sph_harm appearing below are +# deprecated in favor of legendre_p_all, assoc_legendre_p_all, +# assoc_legendre_p_all (assoc_legendre_p_all covers lpmn and clpmn), +# and sph_harm_y respectively. The deprecated functions listed above are +# implemented as shims around their respective replacements. The replacements +# are tested separately, but tests for the deprecated functions remain to +# verify the correctness of the shims. + +HYPERKW = dict(maxprec=200, maxterms=200) + + +@pytest.mark.slow +@check_version(mpmath, '0.17') +class TestSystematic: + + def test_airyai(self): + # oscillating function, limit range + assert_mpmath_equal(lambda z: sc.airy(z)[0], + mpmath.airyai, + [Arg(-1e8, 1e8)], + rtol=1e-5) + assert_mpmath_equal(lambda z: sc.airy(z)[0], + mpmath.airyai, + [Arg(-1e3, 1e3)]) + + def test_airyai_complex(self): + assert_mpmath_equal(lambda z: sc.airy(z)[0], + mpmath.airyai, + [ComplexArg()]) + + def test_airyai_prime(self): + # oscillating function, limit range + assert_mpmath_equal(lambda z: sc.airy(z)[1], lambda z: + mpmath.airyai(z, derivative=1), + [Arg(-1e8, 1e8)], + rtol=1e-5) + assert_mpmath_equal(lambda z: sc.airy(z)[1], lambda z: + mpmath.airyai(z, derivative=1), + [Arg(-1e3, 1e3)]) + + def test_airyai_prime_complex(self): + assert_mpmath_equal(lambda z: sc.airy(z)[1], lambda z: + mpmath.airyai(z, derivative=1), + [ComplexArg()]) + + def test_airybi(self): + # oscillating function, limit range + assert_mpmath_equal(lambda z: sc.airy(z)[2], lambda z: + mpmath.airybi(z), + [Arg(-1e8, 1e8)], + rtol=1e-5) + assert_mpmath_equal(lambda z: sc.airy(z)[2], lambda z: + mpmath.airybi(z), + [Arg(-1e3, 1e3)]) + + def test_airybi_complex(self): + assert_mpmath_equal(lambda z: sc.airy(z)[2], lambda z: + mpmath.airybi(z), + [ComplexArg()]) + + def test_airybi_prime(self): + # oscillating function, limit range + assert_mpmath_equal(lambda z: sc.airy(z)[3], lambda z: + mpmath.airybi(z, derivative=1), + [Arg(-1e8, 1e8)], + rtol=1e-5) + assert_mpmath_equal(lambda z: sc.airy(z)[3], lambda z: + mpmath.airybi(z, derivative=1), + [Arg(-1e3, 1e3)]) + + def test_airybi_prime_complex(self): + assert_mpmath_equal(lambda z: sc.airy(z)[3], lambda z: + mpmath.airybi(z, derivative=1), + [ComplexArg()]) + + def test_bei(self): + assert_mpmath_equal(sc.bei, + exception_to_nan(lambda z: mpmath.bei(0, z, **HYPERKW)), + [Arg(-1e3, 1e3)]) + + def test_ber(self): + assert_mpmath_equal(sc.ber, + exception_to_nan(lambda z: mpmath.ber(0, z, **HYPERKW)), + [Arg(-1e3, 1e3)]) + + def test_bernoulli(self): + assert_mpmath_equal(lambda n: sc.bernoulli(int(n))[int(n)], + lambda n: float(mpmath.bernoulli(int(n))), + [IntArg(0, 13000)], + rtol=1e-9, n=13000) + + def test_besseli(self): + assert_mpmath_equal( + sc.iv, + exception_to_nan(lambda v, z: mpmath.besseli(v, z, **HYPERKW)), + [Arg(-1e100, 1e100), Arg()], + atol=1e-270, + ) + + def test_besseli_complex(self): + assert_mpmath_equal( + lambda v, z: sc.iv(v.real, z), + exception_to_nan(lambda v, z: mpmath.besseli(v, z, **HYPERKW)), + [Arg(-1e100, 1e100), ComplexArg()], + ) + + def test_besselj(self): + assert_mpmath_equal( + sc.jv, + exception_to_nan(lambda v, z: mpmath.besselj(v, z, **HYPERKW)), + [Arg(-1e100, 1e100), Arg(-1e3, 1e3)], + ignore_inf_sign=True, + ) + + # loss of precision at large arguments due to oscillation + assert_mpmath_equal( + sc.jv, + exception_to_nan(lambda v, z: mpmath.besselj(v, z, **HYPERKW)), + [Arg(-1e100, 1e100), Arg(-1e8, 1e8)], + ignore_inf_sign=True, + rtol=1e-5, + ) + + def test_besselj_complex(self): + assert_mpmath_equal( + lambda v, z: sc.jv(v.real, z), + exception_to_nan(lambda v, z: mpmath.besselj(v, z, **HYPERKW)), + [Arg(), ComplexArg()] + ) + + def test_besselk(self): + assert_mpmath_equal( + sc.kv, + mpmath.besselk, + [Arg(-200, 200), Arg(0, np.inf)], + nan_ok=False, + rtol=1e-12, + ) + + def test_besselk_int(self): + assert_mpmath_equal( + sc.kn, + mpmath.besselk, + [IntArg(-200, 200), Arg(0, np.inf)], + nan_ok=False, + rtol=1e-12, + ) + + def test_besselk_complex(self): + assert_mpmath_equal( + lambda v, z: sc.kv(v.real, z), + exception_to_nan(lambda v, z: mpmath.besselk(v, z, **HYPERKW)), + [Arg(-1e100, 1e100), ComplexArg()], + ) + + def test_bessely(self): + def mpbessely(v, x): + r = float(mpmath.bessely(v, x, **HYPERKW)) + if abs(r) > 1e305: + # overflowing to inf a bit earlier is OK + r = np.inf * np.sign(r) + if abs(r) == 0 and x == 0: + # invalid result from mpmath, point x=0 is a divergence + return np.nan + return r + assert_mpmath_equal( + sc.yv, + exception_to_nan(mpbessely), + [Arg(-1e100, 1e100), Arg(-1e8, 1e8)], + n=5000, + ) + + def test_bessely_complex(self): + def mpbessely(v, x): + r = complex(mpmath.bessely(v, x, **HYPERKW)) + if abs(r) > 1e305: + # overflowing to inf a bit earlier is OK + with np.errstate(invalid='ignore'): + r = np.inf * np.sign(r) + return r + assert_mpmath_equal( + lambda v, z: sc.yv(v.real, z), + exception_to_nan(mpbessely), + [Arg(), ComplexArg()], + n=15000, + ) + + def test_bessely_int(self): + def mpbessely(v, x): + r = float(mpmath.bessely(v, x)) + if abs(r) == 0 and x == 0: + # invalid result from mpmath, point x=0 is a divergence + return np.nan + return r + assert_mpmath_equal( + lambda v, z: sc.yn(int(v), z), + exception_to_nan(mpbessely), + [IntArg(-1000, 1000), Arg(-1e8, 1e8)], + ) + + def test_beta(self): + bad_points = [] + + def beta(a, b, nonzero=False): + if a < -1e12 or b < -1e12: + # Function is defined here only at integers, but due + # to loss of precision this is numerically + # ill-defined. Don't compare values here. + return np.nan + if (a < 0 or b < 0) and (abs(float(a + b)) % 1) == 0: + # close to a zero of the function: mpmath and scipy + # will not round here the same, so the test needs to be + # run with an absolute tolerance + if nonzero: + bad_points.append((float(a), float(b))) + return np.nan + return mpmath.beta(a, b) + + assert_mpmath_equal( + sc.beta, + lambda a, b: beta(a, b, nonzero=True), + [Arg(), Arg()], + dps=400, + ignore_inf_sign=True, + ) + + assert_mpmath_equal( + sc.beta, + beta, + np.array(bad_points), + dps=400, + ignore_inf_sign=True, + atol=1e-11, + ) + + def test_betainc(self): + assert_mpmath_equal( + sc.betainc, + time_limited()( + exception_to_nan( + lambda a, b, x: mpmath.betainc(a, b, 0, x, regularized=True) + ) + ), + [Arg(), Arg(), Arg()], + ) + + def test_betaincc(self): + assert_mpmath_equal( + sc.betaincc, + time_limited()( + exception_to_nan( + lambda a, b, x: mpmath.betainc(a, b, x, 1, regularized=True) + ) + ), + [Arg(), Arg(), Arg()], + dps=400, + ) + + def test_binom(self): + bad_points = [] + + def binomial(n, k, nonzero=False): + if abs(k) > 1e8*(abs(n) + 1): + # The binomial is rapidly oscillating in this region, + # and the function is numerically ill-defined. Don't + # compare values here. + return np.nan + if n < k and abs(float(n-k) - np.round(float(n-k))) < 1e-15: + # close to a zero of the function: mpmath and scipy + # will not round here the same, so the test needs to be + # run with an absolute tolerance + if nonzero: + bad_points.append((float(n), float(k))) + return np.nan + return mpmath.binomial(n, k) + + assert_mpmath_equal( + sc.binom, + lambda n, k: binomial(n, k, nonzero=True), + [Arg(), Arg()], + dps=400, + ) + + assert_mpmath_equal( + sc.binom, + binomial, + np.array(bad_points), + dps=400, + atol=1e-14, + ) + + def test_chebyt_int(self): + assert_mpmath_equal( + lambda n, x: sc.eval_chebyt(int(n), x), + exception_to_nan(lambda n, x: mpmath.chebyt(n, x, **HYPERKW)), + [IntArg(), Arg()], + dps=50, + ) + + @pytest.mark.xfail(run=False, reason="some cases in hyp2f1 not fully accurate") + def test_chebyt(self): + assert_mpmath_equal( + sc.eval_chebyt, + lambda n, x: time_limited()( + exception_to_nan(mpmath.chebyt) + )(n, x, **HYPERKW), + [Arg(-101, 101), Arg()], + n=10000, + ) + + def test_chebyu_int(self): + assert_mpmath_equal( + lambda n, x: sc.eval_chebyu(int(n), x), + exception_to_nan(lambda n, x: mpmath.chebyu(n, x, **HYPERKW)), + [IntArg(), Arg()], + dps=50, + ) + + @pytest.mark.xfail(run=False, reason="some cases in hyp2f1 not fully accurate") + def test_chebyu(self): + assert_mpmath_equal( + sc.eval_chebyu, + lambda n, x: time_limited()( + exception_to_nan(mpmath.chebyu) + )(n, x, **HYPERKW), + [Arg(-101, 101), Arg()], + ) + + def test_chi(self): + def chi(x): + return sc.shichi(x)[1] + assert_mpmath_equal(chi, mpmath.chi, [Arg()]) + # check asymptotic series cross-over + assert_mpmath_equal(chi, mpmath.chi, [FixedArg([88 - 1e-9, 88, 88 + 1e-9])]) + + def test_chi_complex(self): + def chi(z): + return sc.shichi(z)[1] + # chi oscillates as Im[z] -> +- inf, so limit range + assert_mpmath_equal( + chi, + mpmath.chi, + [ComplexArg(complex(-np.inf, -1e8), complex(np.inf, 1e8))], + rtol=1e-12, + ) + + def test_ci(self): + def ci(x): + return sc.sici(x)[1] + # oscillating function: limit range + assert_mpmath_equal(ci, mpmath.ci, [Arg(-1e8, 1e8)]) + + def test_ci_complex(self): + def ci(z): + return sc.sici(z)[1] + # ci oscillates as Re[z] -> +- inf, so limit range + assert_mpmath_equal( + ci, + mpmath.ci, + [ComplexArg(complex(-1e8, -np.inf), complex(1e8, np.inf))], + rtol=1e-8, + ) + + def test_cospi(self): + eps = np.finfo(float).eps + assert_mpmath_equal(_cospi, mpmath.cospi, [Arg()], nan_ok=False, rtol=2*eps) + + def test_cospi_complex(self): + assert_mpmath_equal( + _cospi, + mpmath.cospi, + [ComplexArg()], + nan_ok=False, + rtol=1e-13, + ) + + def test_digamma(self): + assert_mpmath_equal( + sc.digamma, + exception_to_nan(mpmath.digamma), + [Arg()], + rtol=1e-12, + dps=50, + ) + + def test_digamma_complex(self): + # Test on a cut plane because mpmath will hang. See + # test_digamma_negreal for tests on the negative real axis. + def param_filter(z): + return np.where((z.real < 0) & (np.abs(z.imag) < 1.12), False, True) + + assert_mpmath_equal( + sc.digamma, + exception_to_nan(mpmath.digamma), + [ComplexArg()], + rtol=1e-13, + dps=40, + param_filter=param_filter + ) + + def test_e1(self): + assert_mpmath_equal( + sc.exp1, + mpmath.e1, + [Arg()], + rtol=1e-14, + ) + + def test_e1_complex(self): + # E_1 oscillates as Im[z] -> +- inf, so limit range + assert_mpmath_equal( + sc.exp1, + mpmath.e1, + [ComplexArg(complex(-np.inf, -1e8), complex(np.inf, 1e8))], + rtol=1e-11, + ) + + # Check cross-over region + assert_mpmath_equal( + sc.exp1, + mpmath.e1, + (np.linspace(-50, 50, 171)[:, None] + + np.r_[0, np.logspace(-3, 2, 61), -np.logspace(-3, 2, 11)]*1j).ravel(), + rtol=1e-11, + ) + assert_mpmath_equal( + sc.exp1, + mpmath.e1, + (np.linspace(-50, -35, 10000) + 0j), + rtol=1e-11, + ) + + def test_exprel(self): + assert_mpmath_equal( + sc.exprel, + lambda x: mpmath.expm1(x)/x if x != 0 else mpmath.mpf('1.0'), + [Arg(a=-np.log(np.finfo(np.float64).max), + b=np.log(np.finfo(np.float64).max))], + ) + assert_mpmath_equal( + sc.exprel, + lambda x: mpmath.expm1(x)/x if x != 0 else mpmath.mpf('1.0'), + np.array([1e-12, 1e-24, 0, 1e12, 1e24, np.inf]), + rtol=1e-11, + ) + assert_(np.isinf(sc.exprel(np.inf))) + assert_(sc.exprel(-np.inf) == 0) + + def test_expm1_complex(self): + # Oscillates as a function of Im[z], so limit range to avoid loss of precision + assert_mpmath_equal( + sc.expm1, + mpmath.expm1, + [ComplexArg(complex(-np.inf, -1e7), complex(np.inf, 1e7))], + ) + + def test_log1p_complex(self): + assert_mpmath_equal( + sc.log1p, + lambda x: mpmath.log(x+1), + [ComplexArg()], + dps=60, + ) + + def test_log1pmx(self): + assert_mpmath_equal( + _log1pmx, + lambda x: mpmath.log(x + 1) - x, + [Arg()], + dps=60, + rtol=1e-14, + ) + + def test_ei(self): + assert_mpmath_equal(sc.expi, mpmath.ei, [Arg()], rtol=1e-11) + + def test_ei_complex(self): + # Ei oscillates as Im[z] -> +- inf, so limit range + assert_mpmath_equal( + sc.expi, + mpmath.ei, + [ComplexArg(complex(-np.inf, -1e8), complex(np.inf, 1e8))], + rtol=1e-9, + ) + + def test_ellipe(self): + assert_mpmath_equal(sc.ellipe, mpmath.ellipe, [Arg(b=1.0)]) + + def test_ellipeinc(self): + assert_mpmath_equal(sc.ellipeinc, mpmath.ellipe, [Arg(-1e3, 1e3), Arg(b=1.0)]) + + def test_ellipeinc_largephi(self): + assert_mpmath_equal(sc.ellipeinc, mpmath.ellipe, [Arg(), Arg()]) + + def test_ellipf(self): + assert_mpmath_equal(sc.ellipkinc, mpmath.ellipf, [Arg(-1e3, 1e3), Arg()]) + + def test_ellipf_largephi(self): + assert_mpmath_equal(sc.ellipkinc, mpmath.ellipf, [Arg(), Arg()]) + + def test_ellipk(self): + assert_mpmath_equal(sc.ellipk, mpmath.ellipk, [Arg(b=1.0)]) + assert_mpmath_equal( + sc.ellipkm1, + lambda m: mpmath.ellipk(1 - m), + [Arg(a=0.0)], + dps=400, + ) + + def test_ellipkinc(self): + def ellipkinc(phi, m): + return mpmath.ellippi(0, phi, m) + assert_mpmath_equal( + sc.ellipkinc, + ellipkinc, + [Arg(-1e3, 1e3), Arg(b=1.0)], + ignore_inf_sign=True, + ) + + def test_ellipkinc_largephi(self): + def ellipkinc(phi, m): + return mpmath.ellippi(0, phi, m) + assert_mpmath_equal( + sc.ellipkinc, + ellipkinc, + [Arg(), Arg(b=1.0)], + ignore_inf_sign=True, + ) + + def test_ellipfun_sn(self): + def sn(u, m): + # mpmath doesn't get the zero at u = 0--fix that + if u == 0: + return 0 + else: + return mpmath.ellipfun("sn", u=u, m=m) + + # Oscillating function --- limit range of first argument; the + # loss of precision there is an expected numerical feature + # rather than an actual bug + assert_mpmath_equal( + lambda u, m: sc.ellipj(u, m)[0], + sn, + [Arg(-1e6, 1e6), Arg(a=0, b=1)], + rtol=1e-8, + ) + + def test_ellipfun_cn(self): + # see comment in ellipfun_sn + assert_mpmath_equal( + lambda u, m: sc.ellipj(u, m)[1], + lambda u, m: mpmath.ellipfun("cn", u=u, m=m), + [Arg(-1e6, 1e6), Arg(a=0, b=1)], + rtol=1e-8, + ) + + def test_ellipfun_dn(self): + # see comment in ellipfun_sn + assert_mpmath_equal( + lambda u, m: sc.ellipj(u, m)[2], + lambda u, m: mpmath.ellipfun("dn", u=u, m=m), + [Arg(-1e6, 1e6), Arg(a=0, b=1)], + rtol=1e-8, + ) + + def test_erf(self): + assert_mpmath_equal(sc.erf, lambda z: mpmath.erf(z), [Arg()]) + + def test_erf_complex(self): + assert_mpmath_equal(sc.erf, lambda z: mpmath.erf(z), [ComplexArg()], n=200) + + def test_erfc(self): + assert_mpmath_equal( + sc.erfc, + exception_to_nan(lambda z: mpmath.erfc(z)), + [Arg()], + rtol=1e-13, + ) + + def test_erfc_complex(self): + assert_mpmath_equal( + sc.erfc, + exception_to_nan(lambda z: mpmath.erfc(z)), + [ComplexArg()], + n=200, + ) + + def test_erfi(self): + assert_mpmath_equal(sc.erfi, mpmath.erfi, [Arg()], n=200) + + def test_erfi_complex(self): + assert_mpmath_equal(sc.erfi, mpmath.erfi, [ComplexArg()], n=200) + + def test_ndtr(self): + assert_mpmath_equal( + sc.ndtr, + exception_to_nan(lambda z: mpmath.ncdf(z)), + [Arg()], + n=200, + ) + + def test_ndtr_complex(self): + assert_mpmath_equal( + sc.ndtr, + lambda z: mpmath.erfc(-z/np.sqrt(2.))/2., + [ComplexArg(a=complex(-10000, -10000), b=complex(10000, 10000))], + n=400, + ) + + def test_log_ndtr(self): + assert_mpmath_equal( + sc.log_ndtr, + exception_to_nan(lambda z: mpmath.log(mpmath.ncdf(z))), + [Arg()], n=600, dps=300, rtol=1e-13, + ) + + def test_log_ndtr_complex(self): + assert_mpmath_equal( + sc.log_ndtr, + exception_to_nan(lambda z: mpmath.log(mpmath.erfc(-z/np.sqrt(2.))/2.)), + [ComplexArg(a=complex(-10000, -100), b=complex(10000, 100))], + n=200, dps=300, + ) + + def test_eulernum(self): + assert_mpmath_equal( + lambda n: sc.euler(n)[-1], + mpmath.eulernum, + [IntArg(1, 10000)], + n=10000, + ) + + def test_expint(self): + assert_mpmath_equal( + sc.expn, + mpmath.expint, + [IntArg(0, 200), Arg(0, np.inf)], + rtol=1e-13, + dps=160, + ) + + def test_fresnels(self): + def fresnels(x): + return sc.fresnel(x)[0] + assert_mpmath_equal(fresnels, mpmath.fresnels, [Arg()]) + + def test_fresnelc(self): + def fresnelc(x): + return sc.fresnel(x)[1] + assert_mpmath_equal(fresnelc, mpmath.fresnelc, [Arg()]) + + def test_gamma(self): + assert_mpmath_equal(sc.gamma, exception_to_nan(mpmath.gamma), [Arg()]) + + def test_gamma_complex(self): + assert_mpmath_equal( + sc.gamma, + exception_to_nan(mpmath.gamma), + [ComplexArg()], + rtol=5e-13, + ) + + def test_gammainc(self): + # Larger arguments are tested in test_data.py:test_local + assert_mpmath_equal( + sc.gammainc, + lambda z, b: mpmath.gammainc(z, b=b, regularized=True), + [Arg(0, 1e4, inclusive_a=False), Arg(0, 1e4)], + nan_ok=False, + rtol=1e-11, + ) + + def test_gammaincc(self): + # Larger arguments are tested in test_data.py:test_local + assert_mpmath_equal( + sc.gammaincc, + lambda z, a: mpmath.gammainc(z, a=a, regularized=True), + [Arg(0, 1e4, inclusive_a=False), Arg(0, 1e4)], + nan_ok=False, + rtol=1e-11, + ) + + def test_gammaln(self): + # The real part of loggamma is log(|gamma(z)|). + def f(z): + return mpmath.loggamma(z).real + + assert_mpmath_equal(sc.gammaln, exception_to_nan(f), [Arg()]) + + @pytest.mark.xfail(run=False) + def test_gegenbauer(self): + assert_mpmath_equal( + sc.eval_gegenbauer, + exception_to_nan(mpmath.gegenbauer), + [Arg(-1e3, 1e3), Arg(), Arg()], + ) + + def test_gegenbauer_int(self): + # Redefine functions to deal with numerical + mpmath issues + def gegenbauer(n, a, x): + # Avoid overflow at large `a` (mpmath would need an even larger + # dps to handle this correctly, so just skip this region) + if abs(a) > 1e100: + return np.nan + + # Deal with n=0, n=1 correctly; mpmath 0.17 doesn't do these + # always correctly + if n == 0: + r = 1.0 + elif n == 1: + r = 2*a*x + else: + r = mpmath.gegenbauer(n, a, x) + + # Mpmath 0.17 gives wrong results (spurious zero) in some cases, so + # compute the value by perturbing the result + if float(r) == 0 and a < -1 and float(a) == int(float(a)): + r = mpmath.gegenbauer(n, a + mpmath.mpf('1e-50'), x) + if abs(r) < mpmath.mpf('1e-50'): + r = mpmath.mpf('0.0') + + # Differing overflow thresholds in scipy vs. mpmath + if abs(r) > 1e270: + return np.inf + return r + + def sc_gegenbauer(n, a, x): + r = sc.eval_gegenbauer(int(n), a, x) + # Differing overflow thresholds in scipy vs. mpmath + if abs(r) > 1e270: + return np.inf + return r + assert_mpmath_equal( + sc_gegenbauer, + exception_to_nan(gegenbauer), + [IntArg(0, 100), Arg(-1e9, 1e9), Arg()], + n=40000, dps=100, ignore_inf_sign=True, rtol=1e-6, + ) + + # Check the small-x expansion + assert_mpmath_equal( + sc_gegenbauer, + exception_to_nan(gegenbauer), + [IntArg(0, 100), Arg(), FixedArg(np.logspace(-30, -4, 30))], + dps=100, ignore_inf_sign=True, + ) + + @pytest.mark.xfail(run=False) + def test_gegenbauer_complex(self): + assert_mpmath_equal( + lambda n, a, x: sc.eval_gegenbauer(int(n), a.real, x), + exception_to_nan(mpmath.gegenbauer), + [IntArg(0, 100), Arg(), ComplexArg()], + ) + + @nonfunctional_tooslow + def test_gegenbauer_complex_general(self): + assert_mpmath_equal( + lambda n, a, x: sc.eval_gegenbauer(n.real, a.real, x), + exception_to_nan(mpmath.gegenbauer), + [Arg(-1e3, 1e3), Arg(), ComplexArg()], + ) + + def test_hankel1(self): + assert_mpmath_equal( + sc.hankel1, + exception_to_nan(lambda v, x: mpmath.hankel1(v, x, **HYPERKW)), + [Arg(-1e20, 1e20), Arg()], + ) + + def test_hankel2(self): + assert_mpmath_equal( + sc.hankel2, + exception_to_nan(lambda v, x: mpmath.hankel2(v, x, **HYPERKW)), + [Arg(-1e20, 1e20), Arg()], + ) + + @pytest.mark.xfail(run=False, reason="issues at intermediately large orders") + def test_hermite(self): + assert_mpmath_equal( + lambda n, x: sc.eval_hermite(int(n), x), + exception_to_nan(mpmath.hermite), + [IntArg(0, 10000), Arg()], + ) + + # hurwitz: same as zeta + + def test_hyp0f1(self): + # mpmath reports no convergence unless maxterms is large enough + KW = dict(maxprec=400, maxterms=1500) + # n=500 (non-xslow default) fails for one bad point + assert_mpmath_equal( + sc.hyp0f1, + lambda a, x: mpmath.hyp0f1(a, x, **KW), + [Arg(-1e7, 1e7), Arg(0, 1e5)], + n=5000, + ) + # NB: The range of the second parameter ("z") is limited from below + # because of an overflow in the intermediate calculations. The way + # for fix it is to implement an asymptotic expansion for Bessel J + # (similar to what is implemented for Bessel I here). + + def test_hyp0f1_complex(self): + assert_mpmath_equal( + lambda a, z: sc.hyp0f1(a.real, z), + exception_to_nan(lambda a, x: mpmath.hyp0f1(a, x, **HYPERKW)), + [Arg(-10, 10), ComplexArg(complex(-120, -120), complex(120, 120))], + ) + # NB: The range of the first parameter ("v") are limited by an overflow + # in the intermediate calculations. Can be fixed by implementing an + # asymptotic expansion for Bessel functions for large order. + + def test_hyp1f1(self): + def mpmath_hyp1f1(a, b, x): + try: + return mpmath.hyp1f1(a, b, x) + except ZeroDivisionError: + return np.inf + + assert_mpmath_equal( + sc.hyp1f1, + mpmath_hyp1f1, + [Arg(-50, 50), Arg(1, 50, inclusive_a=False), Arg(-50, 50)], + n=500, + nan_ok=False, + ) + + @pytest.mark.xfail(run=False) + def test_hyp1f1_complex(self): + assert_mpmath_equal( + inf_to_nan(lambda a, b, x: sc.hyp1f1(a.real, b.real, x)), + exception_to_nan(lambda a, b, x: mpmath.hyp1f1(a, b, x, **HYPERKW)), + [Arg(-1e3, 1e3), Arg(-1e3, 1e3), ComplexArg()], + n=2000, + ) + + @nonfunctional_tooslow + def test_hyp2f1_complex(self): + # SciPy's hyp2f1 seems to have performance and accuracy problems + assert_mpmath_equal( + lambda a, b, c, x: sc.hyp2f1(a.real, b.real, c.real, x), + exception_to_nan(lambda a, b, c, x: mpmath.hyp2f1(a, b, c, x, **HYPERKW)), + [Arg(-1e2, 1e2), Arg(-1e2, 1e2), Arg(-1e2, 1e2), ComplexArg()], + n=10, + ) + + @pytest.mark.xfail(run=False) + def test_hyperu(self): + assert_mpmath_equal( + sc.hyperu, + exception_to_nan(lambda a, b, x: mpmath.hyperu(a, b, x, **HYPERKW)), + [Arg(), Arg(), Arg()], + ) + + @pytest.mark.xfail_on_32bit("mpmath issue gh-342: " + "unsupported operand mpz, long for pow") + def test_igam_fac(self): + def mp_igam_fac(a, x): + return mpmath.power(x, a)*mpmath.exp(-x)/mpmath.gamma(a) + + assert_mpmath_equal( + _igam_fac, + mp_igam_fac, + [Arg(0, 1e14, inclusive_a=False), Arg(0, 1e14)], + rtol=1e-10, + dps=29, + ) + + def test_j0(self): + # The Bessel function at large arguments is j0(x) ~ cos(x + phi)/sqrt(x) + # and at large arguments the phase of the cosine loses precision. + # + # This is numerically expected behavior, so we compare only up to + # 1e8 = 1e15 * 1e-7 + assert_mpmath_equal(sc.j0, mpmath.j0, [Arg(-1e3, 1e3)]) + assert_mpmath_equal(sc.j0, mpmath.j0, [Arg(-1e8, 1e8)], rtol=1e-5) + + def test_j1(self): + # See comment in test_j0 + assert_mpmath_equal(sc.j1, mpmath.j1, [Arg(-1e3, 1e3)]) + assert_mpmath_equal(sc.j1, mpmath.j1, [Arg(-1e8, 1e8)], rtol=1e-5) + + @pytest.mark.xfail(run=False) + def test_jacobi(self): + assert_mpmath_equal( + sc.eval_jacobi, + exception_to_nan(lambda a, b, c, x: mpmath.jacobi(a, b, c, x, **HYPERKW)), + [Arg(), Arg(), Arg(), Arg()], + ) + assert_mpmath_equal( + lambda n, b, c, x: sc.eval_jacobi(int(n), b, c, x), + exception_to_nan(lambda a, b, c, x: mpmath.jacobi(a, b, c, x, **HYPERKW)), + [IntArg(), Arg(), Arg(), Arg()], + ) + + def test_jacobi_int(self): + # Redefine functions to deal with numerical + mpmath issues + def jacobi(n, a, b, x): + # Mpmath does not handle n=0 case always correctly + if n == 0: + return 1.0 + return mpmath.jacobi(n, a, b, x) + assert_mpmath_equal( + lambda n, a, b, x: sc.eval_jacobi(int(n), a, b, x), + lambda n, a, b, x: exception_to_nan(jacobi)(n, a, b, x, **HYPERKW), + [IntArg(), Arg(), Arg(), Arg()], + n=20000, + dps=50, + ) + + def test_kei(self): + def kei(x): + if x == 0: + # work around mpmath issue at x=0 + return -pi/4 + return exception_to_nan(mpmath.kei)(0, x, **HYPERKW) + assert_mpmath_equal(sc.kei, kei, [Arg(-1e30, 1e30)], n=1000) + + def test_ker(self): + assert_mpmath_equal( + sc.ker, + exception_to_nan(lambda x: mpmath.ker(0, x, **HYPERKW)), + [Arg(-1e30, 1e30)], + n=1000, + ) + + @nonfunctional_tooslow + def test_laguerre(self): + assert_mpmath_equal( + trace_args(sc.eval_laguerre), + lambda n, x: exception_to_nan(mpmath.laguerre)(n, x, **HYPERKW), + [Arg(), Arg()], + ) + + def test_laguerre_int(self): + assert_mpmath_equal( + lambda n, x: sc.eval_laguerre(int(n), x), + lambda n, x: exception_to_nan(mpmath.laguerre)(n, x, **HYPERKW), + [IntArg(), Arg()], + n=20000, + ) + + @pytest.mark.xfail_on_32bit("see gh-3551 for bad points") + def test_lambertw_real(self): + assert_mpmath_equal( + lambda x, k: sc.lambertw(x, int(k.real)), + lambda x, k: mpmath.lambertw(x, int(k.real)), + [ComplexArg(-np.inf, np.inf), IntArg(0, 10)], + rtol=1e-13, nan_ok=False, + ) + + def test_lanczos_sum_expg_scaled(self): + maxgamma = 171.624376956302725 + e = np.exp(1) + g = 6.024680040776729583740234375 + + def gamma(x): + with np.errstate(over='ignore'): + fac = ((x + g - 0.5)/e)**(x - 0.5) + if fac != np.inf: + res = fac*_lanczos_sum_expg_scaled(x) + else: + fac = ((x + g - 0.5)/e)**(0.5*(x - 0.5)) + res = fac*_lanczos_sum_expg_scaled(x) + res *= fac + return res + + assert_mpmath_equal( + gamma, + mpmath.gamma, + [Arg(0, maxgamma, inclusive_a=False)], + rtol=1e-13, + ) + + @nonfunctional_tooslow + def test_legendre(self): + assert_mpmath_equal(sc.eval_legendre, mpmath.legendre, [Arg(), Arg()]) + + def test_legendre_int(self): + assert_mpmath_equal( + lambda n, x: sc.eval_legendre(int(n), x), + lambda n, x: exception_to_nan(mpmath.legendre)(n, x, **HYPERKW), + [IntArg(), Arg()], + n=20000, + ) + + # Check the small-x expansion + assert_mpmath_equal( + lambda n, x: sc.eval_legendre(int(n), x), + lambda n, x: exception_to_nan(mpmath.legendre)(n, x, **HYPERKW), + [IntArg(), FixedArg(np.logspace(-30, -4, 20))], + ) + + def test_legenp(self): + def lpnm(n, m, z): + try: + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + v = sc.lpmn(m, n, z)[0][-1,-1] + except ValueError: + return np.nan + if abs(v) > 1e306: + # harmonize overflow to inf + v = np.inf * np.sign(v.real) + return v + + def lpnm_2(n, m, z): + v = sc.lpmv(m, n, z) + if abs(v) > 1e306: + # harmonize overflow to inf + v = np.inf * np.sign(v.real) + return v + + def legenp(n, m, z): + if (z == 1 or z == -1) and int(n) == n: + # Special case (mpmath may give inf, we take the limit by + # continuity) + if m == 0: + if n < 0: + n = -n - 1 + return mpmath.power(mpmath.sign(z), n) + else: + return 0 + + if abs(z) < 1e-15: + # mpmath has bad performance here + return np.nan + + typ = 2 if abs(z) <= 1 else 3 + v = exception_to_nan(mpmath.legenp)(n, m, z, type=typ) + + if abs(v) > 1e306: + # harmonize overflow to inf + v = mpmath.inf * mpmath.sign(v.real) + + return v + + assert_mpmath_equal(lpnm, legenp, [IntArg(-100, 100), IntArg(-100, 100), Arg()]) + + assert_mpmath_equal( + lpnm_2, + legenp, + [IntArg(-100, 100), Arg(-100, 100), Arg(-1, 1)], + atol=1e-10, + ) + + def test_legenp_complex_2(self): + def clpnm(n, m, z): + try: + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + return sc.clpmn(m.real, n.real, z, type=2)[0][-1,-1] + except ValueError: + return np.nan + + def legenp(n, m, z): + if abs(z) < 1e-15: + # mpmath has bad performance here + return np.nan + return exception_to_nan(mpmath.legenp)(int(n.real), int(m.real), z, type=2) + + # mpmath is quite slow here + x = np.array([-2, -0.99, -0.5, 0, 1e-5, 0.5, 0.99, 20, 2e3]) + y = np.array([-1e3, -0.5, 0.5, 1.3]) + z = (x[:,None] + 1j*y[None,:]).ravel() + + assert_mpmath_equal( + clpnm, + legenp, + [FixedArg([-2, -1, 0, 1, 2, 10]), + FixedArg([-2, -1, 0, 1, 2, 10]), + FixedArg(z)], + rtol=1e-6, + n=500, + ) + + def test_legenp_complex_3(self): + def clpnm(n, m, z): + try: + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + return sc.clpmn(m.real, n.real, z, type=3)[0][-1,-1] + except ValueError: + return np.nan + + def legenp(n, m, z): + if abs(z) < 1e-15: + # mpmath has bad performance here + return np.nan + return exception_to_nan(mpmath.legenp)(int(n.real), int(m.real), z, type=3) + + # mpmath is quite slow here + x = np.array([-2, -0.99, -0.5, 0, 1e-5, 0.5, 0.99, 20, 2e3]) + y = np.array([-1e3, -0.5, 0.5, 1.3]) + z = (x[:,None] + 1j*y[None,:]).ravel() + + assert_mpmath_equal( + clpnm, + legenp, + [FixedArg([-2, -1, 0, 1, 2, 10]), + FixedArg([-2, -1, 0, 1, 2, 10]), + FixedArg(z)], + rtol=1e-6, + n=500, + ) + + @pytest.mark.xfail(run=False, reason="apparently picks wrong function at |z| > 1") + def test_legenq(self): + def lqnm(n, m, z): + return sc.lqmn(m, n, z)[0][-1,-1] + + def legenq(n, m, z): + if abs(z) < 1e-15: + # mpmath has bad performance here + return np.nan + return exception_to_nan(mpmath.legenq)(n, m, z, type=2) + + assert_mpmath_equal( + lqnm, + legenq, + [IntArg(0, 100), IntArg(0, 100), Arg()], + ) + + @nonfunctional_tooslow + def test_legenq_complex(self): + def lqnm(n, m, z): + return sc.lqmn(int(m.real), int(n.real), z)[0][-1,-1] + + def legenq(n, m, z): + if abs(z) < 1e-15: + # mpmath has bad performance here + return np.nan + return exception_to_nan(mpmath.legenq)(int(n.real), int(m.real), z, type=2) + + assert_mpmath_equal( + lqnm, + legenq, + [IntArg(0, 100), IntArg(0, 100), ComplexArg()], + n=100, + ) + + def test_lgam1p(self): + def param_filter(x): + # Filter the poles + return np.where((np.floor(x) == x) & (x <= 0), False, True) + + def mp_lgam1p(z): + # The real part of loggamma is log(|gamma(z)|) + return mpmath.loggamma(1 + z).real + + assert_mpmath_equal( + _lgam1p, + mp_lgam1p, + [Arg()], + rtol=1e-13, + dps=100, + param_filter=param_filter, + ) + + def test_loggamma(self): + def mpmath_loggamma(z): + try: + res = mpmath.loggamma(z) + except ValueError: + res = complex(np.nan, np.nan) + return res + + assert_mpmath_equal( + sc.loggamma, + mpmath_loggamma, + [ComplexArg()], + nan_ok=False, + distinguish_nan_and_inf=False, + rtol=5e-14, + ) + + @pytest.mark.xfail(run=False) + def test_pcfd(self): + def pcfd(v, x): + return sc.pbdv(v, x)[0] + assert_mpmath_equal( + pcfd, + exception_to_nan(lambda v, x: mpmath.pcfd(v, x, **HYPERKW)), + [Arg(), Arg()], + ) + + @pytest.mark.xfail(run=False, reason="it's not the same as the mpmath function --- " + "maybe different definition?") + def test_pcfv(self): + def pcfv(v, x): + return sc.pbvv(v, x)[0] + assert_mpmath_equal( + pcfv, + lambda v, x: time_limited()(exception_to_nan(mpmath.pcfv))(v, x, **HYPERKW), + [Arg(), Arg()], + n=1000, + ) + + def test_pcfw(self): + def pcfw(a, x): + return sc.pbwa(a, x)[0] + + def dpcfw(a, x): + return sc.pbwa(a, x)[1] + + def mpmath_dpcfw(a, x): + return mpmath.diff(mpmath.pcfw, (a, x), (0, 1)) + + # The Zhang and Jin implementation only uses Taylor series and + # is thus accurate in only a very small range. + assert_mpmath_equal( + pcfw, + mpmath.pcfw, + [Arg(-5, 5), Arg(-5, 5)], + rtol=2e-8, + n=100, + ) + + assert_mpmath_equal( + dpcfw, + mpmath_dpcfw, + [Arg(-5, 5), Arg(-5, 5)], + rtol=2e-9, + n=100, + ) + + @pytest.mark.xfail(run=False, + reason="issues at large arguments (atol OK, rtol not) " + "and = _pep440.Version("1.0.0"): + # no workarounds needed + mppoch = mpmath.rf + else: + def mppoch(a, m): + # deal with cases where the result in double precision + # hits exactly a non-positive integer, but the + # corresponding extended-precision mpf floats don't + if float(a + m) == int(a + m) and float(a + m) <= 0: + a = mpmath.mpf(a) + m = int(a + m) - a + return mpmath.rf(a, m) + + assert_mpmath_equal(sc.poch, mppoch, [Arg(), Arg()], dps=400) + + def test_sinpi(self): + eps = np.finfo(float).eps + assert_mpmath_equal( + _sinpi, + mpmath.sinpi, + [Arg()], + nan_ok=False, + rtol=2*eps, + ) + + def test_sinpi_complex(self): + assert_mpmath_equal( + _sinpi, + mpmath.sinpi, + [ComplexArg()], + nan_ok=False, + rtol=2e-14, + ) + + def test_shi(self): + def shi(x): + return sc.shichi(x)[0] + assert_mpmath_equal(shi, mpmath.shi, [Arg()]) + # check asymptotic series cross-over + assert_mpmath_equal(shi, mpmath.shi, [FixedArg([88 - 1e-9, 88, 88 + 1e-9])]) + + def test_shi_complex(self): + def shi(z): + return sc.shichi(z)[0] + # shi oscillates as Im[z] -> +- inf, so limit range + assert_mpmath_equal( + shi, + mpmath.shi, + [ComplexArg(complex(-np.inf, -1e8), complex(np.inf, 1e8))], + rtol=1e-12, + ) + + def test_si(self): + def si(x): + return sc.sici(x)[0] + assert_mpmath_equal(si, mpmath.si, [Arg()]) + + def test_si_complex(self): + def si(z): + return sc.sici(z)[0] + # si oscillates as Re[z] -> +- inf, so limit range + assert_mpmath_equal( + si, + mpmath.si, + [ComplexArg(complex(-1e8, -np.inf), complex(1e8, np.inf))], + rtol=1e-12, + ) + + def test_spence(self): + # mpmath uses a different convention for the dilogarithm + def dilog(x): + return mpmath.polylog(2, 1 - x) + # Spence has a branch cut on the negative real axis + assert_mpmath_equal( + sc.spence, + exception_to_nan(dilog), + [Arg(0, np.inf)], + rtol=1e-14, + ) + + def test_spence_complex(self): + def dilog(z): + return mpmath.polylog(2, 1 - z) + assert_mpmath_equal( + sc.spence, + exception_to_nan(dilog), + [ComplexArg()], + rtol=1e-14, + ) + + def test_spherharm(self): + def spherharm(l, m, theta, phi): + if m > l: + return np.nan + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + return sc.sph_harm(m, l, phi, theta) + assert_mpmath_equal( + spherharm, + mpmath.spherharm, + [IntArg(0, 100), IntArg(0, 100), Arg(a=0, b=pi), Arg(a=0, b=2*pi)], + atol=1e-8, + n=6000, + dps=150, + ) + + def test_struveh(self): + assert_mpmath_equal( + sc.struve, + exception_to_nan(mpmath.struveh), + [Arg(-1e4, 1e4), Arg(0, 1e4)], + rtol=5e-10, + ) + + def test_struvel(self): + def mp_struvel(v, z): + if v < 0 and z < -v and abs(v) > 1000: + # larger DPS needed for correct results + old_dps = mpmath.mp.dps + try: + mpmath.mp.dps = 500 + return mpmath.struvel(v, z) + finally: + mpmath.mp.dps = old_dps + return mpmath.struvel(v, z) + + assert_mpmath_equal( + sc.modstruve, + exception_to_nan(mp_struvel), + [Arg(-1e4, 1e4), Arg(0, 1e4)], + rtol=5e-10, + ignore_inf_sign=True, + ) + + def test_wrightomega_real(self): + def mpmath_wrightomega_real(x): + return mpmath.lambertw(mpmath.exp(x), mpmath.mpf('-0.5')) + + # For x < -1000 the Wright Omega function is just 0 to double + # precision, and for x > 1e21 it is just x to double + # precision. + assert_mpmath_equal( + sc.wrightomega, + mpmath_wrightomega_real, + [Arg(-1000, 1e21)], + rtol=5e-15, + atol=0, + nan_ok=False, + ) + + def test_wrightomega(self): + assert_mpmath_equal( + sc.wrightomega, + lambda z: _mpmath_wrightomega(z, 25), + [ComplexArg()], + rtol=1e-14, + nan_ok=False, + ) + + def test_hurwitz_zeta(self): + assert_mpmath_equal( + sc.zeta, + exception_to_nan(mpmath.zeta), + [Arg(a=1, b=1e10, inclusive_a=False), Arg(a=0, inclusive_a=False)], + ) + + def test_riemann_zeta(self): + assert_mpmath_equal( + sc.zeta, + lambda x: mpmath.zeta(x) if x != 1 else mpmath.inf, + [Arg(-100, 100)], + nan_ok=False, + rtol=5e-13, + ) + + def test_zetac(self): + assert_mpmath_equal( + sc.zetac, + lambda x: mpmath.zeta(x) - 1 if x != 1 else mpmath.inf, + [Arg(-100, 100)], + nan_ok=False, + dps=45, + rtol=5e-13, + ) + + def test_boxcox(self): + + def mp_boxcox(x, lmbda): + x = mpmath.mp.mpf(x) + lmbda = mpmath.mp.mpf(lmbda) + if lmbda == 0: + return mpmath.mp.log(x) + else: + return mpmath.mp.powm1(x, lmbda) / lmbda + + assert_mpmath_equal( + sc.boxcox, + exception_to_nan(mp_boxcox), + [Arg(a=0, inclusive_a=False), Arg()], + n=200, + dps=60, + rtol=1e-13, + ) + + def test_boxcox1p(self): + + def mp_boxcox1p(x, lmbda): + x = mpmath.mp.mpf(x) + lmbda = mpmath.mp.mpf(lmbda) + one = mpmath.mp.mpf(1) + if lmbda == 0: + return mpmath.mp.log(one + x) + else: + return mpmath.mp.powm1(one + x, lmbda) / lmbda + + assert_mpmath_equal( + sc.boxcox1p, + exception_to_nan(mp_boxcox1p), + [Arg(a=-1, inclusive_a=False), Arg()], + n=200, + dps=60, + rtol=1e-13, + ) + + def test_spherical_jn(self): + def mp_spherical_jn(n, z): + arg = mpmath.mpmathify(z) + out = (mpmath.besselj(n + mpmath.mpf(1)/2, arg) / + mpmath.sqrt(2*arg/mpmath.pi)) + if arg.imag == 0: + return out.real + else: + return out + + assert_mpmath_equal( + lambda n, z: sc.spherical_jn(int(n), z), + exception_to_nan(mp_spherical_jn), + [IntArg(0, 200), Arg(-1e8, 1e8)], + dps=300, + # underflow of `spherical_jn` is a bit premature; see gh-21629 + param_filter=(None, lambda z: np.abs(z) > 1e-20), + ) + + def test_spherical_jn_complex(self): + def mp_spherical_jn(n, z): + arg = mpmath.mpmathify(z) + out = (mpmath.besselj(n + mpmath.mpf(1)/2, arg) / + mpmath.sqrt(2*arg/mpmath.pi)) + if arg.imag == 0: + return out.real + else: + return out + + assert_mpmath_equal( + lambda n, z: sc.spherical_jn(int(n.real), z), + exception_to_nan(mp_spherical_jn), + [IntArg(0, 200), ComplexArg()] + ) + + def test_spherical_yn(self): + def mp_spherical_yn(n, z): + arg = mpmath.mpmathify(z) + out = (mpmath.bessely(n + mpmath.mpf(1)/2, arg) / + mpmath.sqrt(2*arg/mpmath.pi)) + if arg.imag == 0: + return out.real + else: + return out + + assert_mpmath_equal( + lambda n, z: sc.spherical_yn(int(n), z), + exception_to_nan(mp_spherical_yn), + [IntArg(0, 200), Arg(-1e10, 1e10)], + dps=100, + ) + + def test_spherical_yn_complex(self): + def mp_spherical_yn(n, z): + arg = mpmath.mpmathify(z) + out = (mpmath.bessely(n + mpmath.mpf(1)/2, arg) / + mpmath.sqrt(2*arg/mpmath.pi)) + if arg.imag == 0: + return out.real + else: + return out + + assert_mpmath_equal( + lambda n, z: sc.spherical_yn(int(n.real), z), + exception_to_nan(mp_spherical_yn), + [IntArg(0, 200), ComplexArg()], + ) + + def test_spherical_in(self): + def mp_spherical_in(n, z): + arg = mpmath.mpmathify(z) + out = (mpmath.besseli(n + mpmath.mpf(1)/2, arg) / + mpmath.sqrt(2*arg/mpmath.pi)) + if arg.imag == 0: + return out.real + else: + return out + + assert_mpmath_equal( + lambda n, z: sc.spherical_in(int(n), z), + exception_to_nan(mp_spherical_in), + [IntArg(0, 200), Arg()], + dps=200, + atol=10**(-278), + ) + + def test_spherical_in_complex(self): + def mp_spherical_in(n, z): + arg = mpmath.mpmathify(z) + out = (mpmath.besseli(n + mpmath.mpf(1)/2, arg) / + mpmath.sqrt(2*arg/mpmath.pi)) + if arg.imag == 0: + return out.real + else: + return out + + assert_mpmath_equal( + lambda n, z: sc.spherical_in(int(n.real), z), + exception_to_nan(mp_spherical_in), + [IntArg(0, 200), ComplexArg()], + ) + + def test_spherical_kn(self): + def mp_spherical_kn(n, z): + arg = mpmath.mpmathify(z) + out = (mpmath.besselk(n + mpmath.mpf(1)/2, arg) / + mpmath.sqrt(2*arg/mpmath.pi)) + if mpmath.mpmathify(z).imag == 0: + return out.real + else: + return out + + assert_mpmath_equal( + lambda n, z: sc.spherical_kn(int(n), z), + exception_to_nan(mp_spherical_kn), + [IntArg(0, 150), Arg()], + dps=100, + ) + + @pytest.mark.xfail(run=False, + reason="Accuracy issues near z = -1 inherited from kv.") + def test_spherical_kn_complex(self): + def mp_spherical_kn(n, z): + arg = mpmath.mpmathify(z) + out = (mpmath.besselk(n + mpmath.mpf(1)/2, arg) / + mpmath.sqrt(2*arg/mpmath.pi)) + if arg.imag == 0: + return out.real + else: + return out + + assert_mpmath_equal( + lambda n, z: sc.spherical_kn(int(n.real), z), + exception_to_nan(mp_spherical_kn), + [IntArg(0, 200), ComplexArg()], + dps=200, + ) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_nan_inputs.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_nan_inputs.py new file mode 100644 index 0000000000000000000000000000000000000000..4b3e574da056bc9ba7d472567caf992d6638c20a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_nan_inputs.py @@ -0,0 +1,65 @@ +"""Test how the ufuncs in special handle nan inputs. + +""" +from typing import Callable + +import numpy as np +from numpy.testing import assert_array_equal, assert_, suppress_warnings +import pytest +import scipy.special as sc + + +KNOWNFAILURES: dict[str, Callable] = {} + +POSTPROCESSING: dict[str, Callable] = {} + + +def _get_ufuncs(): + ufuncs = [] + ufunc_names = [] + for name in sorted(sc.__dict__): + obj = sc.__dict__[name] + if not isinstance(obj, np.ufunc): + continue + msg = KNOWNFAILURES.get(obj) + if msg is None: + ufuncs.append(obj) + ufunc_names.append(name) + else: + fail = pytest.mark.xfail(run=False, reason=msg) + ufuncs.append(pytest.param(obj, marks=fail)) + ufunc_names.append(name) + return ufuncs, ufunc_names + + +UFUNCS, UFUNC_NAMES = _get_ufuncs() + + +@pytest.mark.thread_unsafe +@pytest.mark.parametrize("func", UFUNCS, ids=UFUNC_NAMES) +def test_nan_inputs(func): + args = (np.nan,)*func.nin + with suppress_warnings() as sup: + # Ignore warnings about unsafe casts from legacy wrappers + sup.filter(RuntimeWarning, + "floating point number truncated to an integer") + try: + with suppress_warnings() as sup: + sup.filter(DeprecationWarning) + res = func(*args) + except TypeError: + # One of the arguments doesn't take real inputs + return + if func in POSTPROCESSING: + res = POSTPROCESSING[func](*res) + + msg = f"got {res} instead of nan" + assert_array_equal(np.isnan(res), True, err_msg=msg) + + +def test_legacy_cast(): + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, + "floating point number truncated to an integer") + res = sc.bdtrc(np.nan, 1, 0.5) + assert_(np.isnan(res)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ndtr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ndtr.py new file mode 100644 index 0000000000000000000000000000000000000000..ba9b689b34384585cc65204000febcb99c910d55 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ndtr.py @@ -0,0 +1,77 @@ +import numpy as np +from numpy.testing import assert_equal, assert_allclose +import scipy.special as sc + + +def test_ndtr(): + assert_equal(sc.ndtr(0), 0.5) + assert_allclose(sc.ndtr(1), 0.8413447460685429) + + +class TestNdtri: + + def test_zero(self): + assert sc.ndtri(0.5) == 0.0 + + def test_asymptotes(self): + assert_equal(sc.ndtri([0.0, 1.0]), [-np.inf, np.inf]) + + def test_outside_of_domain(self): + assert all(np.isnan(sc.ndtri([-1.5, 1.5]))) + + +class TestLogNdtr: + + # The expected values in these tests were computed with mpmath: + # + # def log_ndtr_mp(x): + # return mpmath.log(mpmath.ncdf(x)) + # + + def test_log_ndtr_moderate_le8(self): + x = np.array([-0.75, -0.25, 0, 0.5, 1.5, 2.5, 3, 4, 5, 7, 8]) + expected = np.array([-1.4844482299196562, + -0.9130617648111351, + -0.6931471805599453, + -0.3689464152886564, + -0.06914345561223398, + -0.006229025485860002, + -0.0013508099647481938, + -3.167174337748927e-05, + -2.866516129637636e-07, + -1.279812543886654e-12, + -6.220960574271786e-16]) + y = sc.log_ndtr(x) + assert_allclose(y, expected, rtol=1e-14) + + def test_log_ndtr_values_8_16(self): + x = np.array([8.001, 8.06, 8.15, 8.5, 10, 12, 14, 16]) + expected = [-6.170639424817055e-16, + -3.814722443652823e-16, + -1.819621363526629e-16, + -9.479534822203318e-18, + -7.619853024160525e-24, + -1.776482112077679e-33, + -7.7935368191928e-45, + -6.388754400538087e-58] + y = sc.log_ndtr(x) + assert_allclose(y, expected, rtol=5e-14) + + def test_log_ndtr_values_16_31(self): + x = np.array([16.15, 20.3, 21.4, 26.2, 30.9]) + expected = [-5.678084565148492e-59, + -6.429244467698346e-92, + -6.680402412553295e-102, + -1.328698078458869e-151, + -5.972288641838264e-210] + y = sc.log_ndtr(x) + assert_allclose(y, expected, rtol=2e-13) + + def test_log_ndtr_values_gt31(self): + x = np.array([31.6, 32.8, 34.9, 37.1]) + expected = [-1.846036234858162e-219, + -2.9440539964066835e-236, + -3.71721649450857e-267, + -1.4047119663106221e-301] + y = sc.log_ndtr(x) + assert_allclose(y, expected, rtol=3e-13) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ndtri_exp.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ndtri_exp.py new file mode 100644 index 0000000000000000000000000000000000000000..82a9fbd3bcda117770e00018facda3f56630a6bc --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ndtri_exp.py @@ -0,0 +1,94 @@ +import pytest +import numpy as np +from numpy.testing import assert_equal, assert_allclose +from scipy.special import log_ndtr, ndtri_exp +from scipy.special._testutils import assert_func_equal + + +def log_ndtr_ndtri_exp(y): + return log_ndtr(ndtri_exp(y)) + + +@pytest.fixture(scope="class") +def uniform_random_points(): + random_state = np.random.RandomState(1234) + points = random_state.random_sample(1000) + return points + + +class TestNdtriExp: + """Tests that ndtri_exp is sufficiently close to an inverse of log_ndtr. + + We have separate tests for the five intervals (-inf, -10), + [-10, -2), [-2, -0.14542), [-0.14542, -1e-6), and [-1e-6, 0). + ndtri_exp(y) is computed in three different ways depending on if y + is in (-inf, -2), [-2, log(1 - exp(-2))], or [log(1 - exp(-2), 0). + Each of these intervals is given its own test with two additional tests + for handling very small values and values very close to zero. + """ + + @pytest.mark.parametrize( + "test_input", [-1e1, -1e2, -1e10, -1e20, -np.finfo(float).max] + ) + def test_very_small_arg(self, test_input, uniform_random_points): + scale = test_input + points = scale * (0.5 * uniform_random_points + 0.5) + assert_func_equal( + log_ndtr_ndtri_exp, + lambda y: y, points, + rtol=1e-14, + nan_ok=True + ) + + @pytest.mark.parametrize( + "interval,expected_rtol", + [ + ((-10, -2), 1e-14), + ((-2, -0.14542), 1e-12), + ((-0.14542, -1e-6), 1e-10), + ((-1e-6, 0), 1e-6), + ], + ) + def test_in_interval(self, interval, expected_rtol, uniform_random_points): + left, right = interval + points = (right - left) * uniform_random_points + left + assert_func_equal( + log_ndtr_ndtri_exp, + lambda y: y, points, + rtol=expected_rtol, + nan_ok=True + ) + + def test_extreme(self): + # bigneg is not quite the largest negative double precision value. + # Here's why: + # The round-trip calculation + # y = ndtri_exp(bigneg) + # bigneg2 = log_ndtr(y) + # where bigneg is a very large negative value, would--with infinite + # precision--result in bigneg2 == bigneg. When bigneg is large enough, + # y is effectively equal to -sqrt(2)*sqrt(-bigneg), and log_ndtr(y) is + # effectively -(y/sqrt(2))**2. If we use bigneg = np.finfo(float).min, + # then by construction, the theoretical value is the most negative + # finite value that can be represented with 64 bit float point. This + # means tiny changes in how the computation proceeds can result in the + # return value being -inf. (E.g. changing the constant representation + # of 1/sqrt(2) from 0.7071067811865475--which is the value returned by + # 1/np.sqrt(2)--to 0.7071067811865476--which is the most accurate 64 + # bit floating point representation of 1/sqrt(2)--results in the + # round-trip that starts with np.finfo(float).min returning -inf. So + # we'll move the bigneg value a few ULPs towards 0 to avoid this + # sensitivity. + # Use the reduce method to apply nextafter four times. + bigneg = np.nextafter.reduce([np.finfo(float).min, 0, 0, 0, 0]) + # tinyneg is approx. -2.225e-308. + tinyneg = -np.finfo(float).tiny + x = np.array([tinyneg, bigneg]) + result = log_ndtr_ndtri_exp(x) + assert_allclose(result, x, rtol=1e-12) + + def test_asymptotes(self): + assert_equal(ndtri_exp([-np.inf, 0.0]), [-np.inf, np.inf]) + + def test_outside_domain(self): + assert np.isnan(ndtri_exp(1.0)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_orthogonal.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_orthogonal.py new file mode 100644 index 0000000000000000000000000000000000000000..47831c18ab6fa48925cae576f6c664767df99d05 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_orthogonal.py @@ -0,0 +1,822 @@ +import pytest +from pytest import raises as assert_raises + +import numpy as np +from numpy import array, sqrt +from numpy.testing import (assert_array_almost_equal, assert_equal, + assert_almost_equal, assert_allclose) + +from scipy import integrate +import scipy.special as sc +from scipy.special import gamma +import scipy.special._orthogonal as orth + + +class TestCheby: + def test_chebyc(self): + C0 = orth.chebyc(0) + C1 = orth.chebyc(1) + with np.errstate(all='ignore'): + C2 = orth.chebyc(2) + C3 = orth.chebyc(3) + C4 = orth.chebyc(4) + C5 = orth.chebyc(5) + + assert_array_almost_equal(C0.c,[2],13) + assert_array_almost_equal(C1.c,[1,0],13) + assert_array_almost_equal(C2.c,[1,0,-2],13) + assert_array_almost_equal(C3.c,[1,0,-3,0],13) + assert_array_almost_equal(C4.c,[1,0,-4,0,2],13) + assert_array_almost_equal(C5.c,[1,0,-5,0,5,0],13) + + def test_chebys(self): + S0 = orth.chebys(0) + S1 = orth.chebys(1) + S2 = orth.chebys(2) + S3 = orth.chebys(3) + S4 = orth.chebys(4) + S5 = orth.chebys(5) + assert_array_almost_equal(S0.c,[1],13) + assert_array_almost_equal(S1.c,[1,0],13) + assert_array_almost_equal(S2.c,[1,0,-1],13) + assert_array_almost_equal(S3.c,[1,0,-2,0],13) + assert_array_almost_equal(S4.c,[1,0,-3,0,1],13) + assert_array_almost_equal(S5.c,[1,0,-4,0,3,0],13) + + def test_chebyt(self): + T0 = orth.chebyt(0) + T1 = orth.chebyt(1) + T2 = orth.chebyt(2) + T3 = orth.chebyt(3) + T4 = orth.chebyt(4) + T5 = orth.chebyt(5) + assert_array_almost_equal(T0.c,[1],13) + assert_array_almost_equal(T1.c,[1,0],13) + assert_array_almost_equal(T2.c,[2,0,-1],13) + assert_array_almost_equal(T3.c,[4,0,-3,0],13) + assert_array_almost_equal(T4.c,[8,0,-8,0,1],13) + assert_array_almost_equal(T5.c,[16,0,-20,0,5,0],13) + + def test_chebyu(self): + U0 = orth.chebyu(0) + U1 = orth.chebyu(1) + U2 = orth.chebyu(2) + U3 = orth.chebyu(3) + U4 = orth.chebyu(4) + U5 = orth.chebyu(5) + assert_array_almost_equal(U0.c,[1],13) + assert_array_almost_equal(U1.c,[2,0],13) + assert_array_almost_equal(U2.c,[4,0,-1],13) + assert_array_almost_equal(U3.c,[8,0,-4,0],13) + assert_array_almost_equal(U4.c,[16,0,-12,0,1],13) + assert_array_almost_equal(U5.c,[32,0,-32,0,6,0],13) + + +class TestGegenbauer: + + def test_gegenbauer(self): + a = 5*np.random.random() - 0.5 + if np.any(a == 0): + a = -0.2 + Ca0 = orth.gegenbauer(0,a) + Ca1 = orth.gegenbauer(1,a) + Ca2 = orth.gegenbauer(2,a) + Ca3 = orth.gegenbauer(3,a) + Ca4 = orth.gegenbauer(4,a) + Ca5 = orth.gegenbauer(5,a) + + assert_array_almost_equal(Ca0.c,array([1]),13) + assert_array_almost_equal(Ca1.c,array([2*a,0]),13) + assert_array_almost_equal(Ca2.c,array([2*a*(a+1),0,-a]),13) + assert_array_almost_equal(Ca3.c,array([4*sc.poch(a,3),0,-6*a*(a+1), + 0])/3.0,11) + assert_array_almost_equal(Ca4.c,array([4*sc.poch(a,4),0,-12*sc.poch(a,3), + 0,3*a*(a+1)])/6.0,11) + assert_array_almost_equal(Ca5.c,array([4*sc.poch(a,5),0,-20*sc.poch(a,4), + 0,15*sc.poch(a,3),0])/15.0,11) + + @pytest.mark.parametrize('a', [0, 1]) + def test_n_zero_gh8888(self, a): + # gh-8888 reported that gegenbauer(0, 0) returns NaN polynomial + Cn0 = orth.gegenbauer(0, a) + assert_equal(Cn0.c, np.asarray([1.])) + + def test_valid_alpha(self): + # Check input validation of `alpha` + message = '`alpha` must be a finite number greater...' + with pytest.raises(ValueError, match=message): + orth.gegenbauer(0, np.nan) + with pytest.raises(ValueError, match=message): + orth.gegenbauer(1, -0.5) + with pytest.raises(ValueError, match=message): + orth.gegenbauer(2, -np.inf) + + +class TestHermite: + def test_hermite(self): + H0 = orth.hermite(0) + H1 = orth.hermite(1) + H2 = orth.hermite(2) + H3 = orth.hermite(3) + H4 = orth.hermite(4) + H5 = orth.hermite(5) + assert_array_almost_equal(H0.c,[1],13) + assert_array_almost_equal(H1.c,[2,0],13) + assert_array_almost_equal(H2.c,[4,0,-2],13) + assert_array_almost_equal(H3.c,[8,0,-12,0],13) + assert_array_almost_equal(H4.c,[16,0,-48,0,12],12) + assert_array_almost_equal(H5.c,[32,0,-160,0,120,0],12) + + def test_hermitenorm(self): + # He_n(x) = 2**(-n/2) H_n(x/sqrt(2)) + psub = np.poly1d([1.0/sqrt(2),0]) + H0 = orth.hermitenorm(0) + H1 = orth.hermitenorm(1) + H2 = orth.hermitenorm(2) + H3 = orth.hermitenorm(3) + H4 = orth.hermitenorm(4) + H5 = orth.hermitenorm(5) + he0 = orth.hermite(0)(psub) + he1 = orth.hermite(1)(psub) / sqrt(2) + he2 = orth.hermite(2)(psub) / 2.0 + he3 = orth.hermite(3)(psub) / (2*sqrt(2)) + he4 = orth.hermite(4)(psub) / 4.0 + he5 = orth.hermite(5)(psub) / (4.0*sqrt(2)) + + assert_array_almost_equal(H0.c,he0.c,13) + assert_array_almost_equal(H1.c,he1.c,13) + assert_array_almost_equal(H2.c,he2.c,13) + assert_array_almost_equal(H3.c,he3.c,13) + assert_array_almost_equal(H4.c,he4.c,13) + assert_array_almost_equal(H5.c,he5.c,13) + + +class TestShLegendre: + def test_sh_legendre(self): + # P*_n(x) = P_n(2x-1) + psub = np.poly1d([2,-1]) + Ps0 = orth.sh_legendre(0) + Ps1 = orth.sh_legendre(1) + Ps2 = orth.sh_legendre(2) + Ps3 = orth.sh_legendre(3) + Ps4 = orth.sh_legendre(4) + Ps5 = orth.sh_legendre(5) + pse0 = orth.legendre(0)(psub) + pse1 = orth.legendre(1)(psub) + pse2 = orth.legendre(2)(psub) + pse3 = orth.legendre(3)(psub) + pse4 = orth.legendre(4)(psub) + pse5 = orth.legendre(5)(psub) + assert_array_almost_equal(Ps0.c,pse0.c,13) + assert_array_almost_equal(Ps1.c,pse1.c,13) + assert_array_almost_equal(Ps2.c,pse2.c,13) + assert_array_almost_equal(Ps3.c,pse3.c,13) + assert_array_almost_equal(Ps4.c,pse4.c,12) + assert_array_almost_equal(Ps5.c,pse5.c,12) + + +class TestShChebyt: + def test_sh_chebyt(self): + # T*_n(x) = T_n(2x-1) + psub = np.poly1d([2,-1]) + Ts0 = orth.sh_chebyt(0) + Ts1 = orth.sh_chebyt(1) + Ts2 = orth.sh_chebyt(2) + Ts3 = orth.sh_chebyt(3) + Ts4 = orth.sh_chebyt(4) + Ts5 = orth.sh_chebyt(5) + tse0 = orth.chebyt(0)(psub) + tse1 = orth.chebyt(1)(psub) + tse2 = orth.chebyt(2)(psub) + tse3 = orth.chebyt(3)(psub) + tse4 = orth.chebyt(4)(psub) + tse5 = orth.chebyt(5)(psub) + assert_array_almost_equal(Ts0.c,tse0.c,13) + assert_array_almost_equal(Ts1.c,tse1.c,13) + assert_array_almost_equal(Ts2.c,tse2.c,13) + assert_array_almost_equal(Ts3.c,tse3.c,13) + assert_array_almost_equal(Ts4.c,tse4.c,12) + assert_array_almost_equal(Ts5.c,tse5.c,12) + + +class TestShChebyu: + def test_sh_chebyu(self): + # U*_n(x) = U_n(2x-1) + psub = np.poly1d([2,-1]) + Us0 = orth.sh_chebyu(0) + Us1 = orth.sh_chebyu(1) + Us2 = orth.sh_chebyu(2) + Us3 = orth.sh_chebyu(3) + Us4 = orth.sh_chebyu(4) + Us5 = orth.sh_chebyu(5) + use0 = orth.chebyu(0)(psub) + use1 = orth.chebyu(1)(psub) + use2 = orth.chebyu(2)(psub) + use3 = orth.chebyu(3)(psub) + use4 = orth.chebyu(4)(psub) + use5 = orth.chebyu(5)(psub) + assert_array_almost_equal(Us0.c,use0.c,13) + assert_array_almost_equal(Us1.c,use1.c,13) + assert_array_almost_equal(Us2.c,use2.c,13) + assert_array_almost_equal(Us3.c,use3.c,13) + assert_array_almost_equal(Us4.c,use4.c,12) + assert_array_almost_equal(Us5.c,use5.c,11) + + +class TestShJacobi: + def test_sh_jacobi(self): + # G^(p,q)_n(x) = n! gamma(n+p)/gamma(2*n+p) * P^(p-q,q-1)_n(2*x-1) + def conv(n, p): + return gamma(n + 1) * gamma(n + p) / gamma(2 * n + p) + psub = np.poly1d([2,-1]) + q = 4 * np.random.random() + p = q-1 + 2*np.random.random() + # print("shifted jacobi p,q = ", p, q) + G0 = orth.sh_jacobi(0,p,q) + G1 = orth.sh_jacobi(1,p,q) + G2 = orth.sh_jacobi(2,p,q) + G3 = orth.sh_jacobi(3,p,q) + G4 = orth.sh_jacobi(4,p,q) + G5 = orth.sh_jacobi(5,p,q) + ge0 = orth.jacobi(0,p-q,q-1)(psub) * conv(0,p) + ge1 = orth.jacobi(1,p-q,q-1)(psub) * conv(1,p) + ge2 = orth.jacobi(2,p-q,q-1)(psub) * conv(2,p) + ge3 = orth.jacobi(3,p-q,q-1)(psub) * conv(3,p) + ge4 = orth.jacobi(4,p-q,q-1)(psub) * conv(4,p) + ge5 = orth.jacobi(5,p-q,q-1)(psub) * conv(5,p) + + assert_array_almost_equal(G0.c,ge0.c,13) + assert_array_almost_equal(G1.c,ge1.c,13) + assert_array_almost_equal(G2.c,ge2.c,13) + assert_array_almost_equal(G3.c,ge3.c,13) + assert_array_almost_equal(G4.c,ge4.c,13) + assert_array_almost_equal(G5.c,ge5.c,13) + + +class TestCall: + def test_call(self): + poly = [] + for n in range(5): + poly.extend([x.strip() for x in + (""" + orth.jacobi(%(n)d,0.3,0.9) + orth.sh_jacobi(%(n)d,0.3,0.9) + orth.genlaguerre(%(n)d,0.3) + orth.laguerre(%(n)d) + orth.hermite(%(n)d) + orth.hermitenorm(%(n)d) + orth.gegenbauer(%(n)d,0.3) + orth.chebyt(%(n)d) + orth.chebyu(%(n)d) + orth.chebyc(%(n)d) + orth.chebys(%(n)d) + orth.sh_chebyt(%(n)d) + orth.sh_chebyu(%(n)d) + orth.legendre(%(n)d) + orth.sh_legendre(%(n)d) + """ % dict(n=n)).split() + ]) + with np.errstate(all='ignore'): + for pstr in poly: + p = eval(pstr) + assert_almost_equal(p(0.315), np.poly1d(p.coef)(0.315), + err_msg=pstr) + + +class TestGenlaguerre: + def test_regression(self): + assert_equal(orth.genlaguerre(1, 1, monic=False)(0), 2.) + assert_equal(orth.genlaguerre(1, 1, monic=True)(0), -2.) + assert_equal(orth.genlaguerre(1, 1, monic=False), np.poly1d([-1, 2])) + assert_equal(orth.genlaguerre(1, 1, monic=True), np.poly1d([1, -2])) + + +def verify_gauss_quad(root_func, eval_func, weight_func, a, b, N, + rtol=1e-15, atol=5e-14): + # this test is copied from numpy's TestGauss in test_hermite.py + x, w, mu = root_func(N, True) + + n = np.arange(N, dtype=np.dtype("long")) + v = eval_func(n[:,np.newaxis], x) + vv = np.dot(v*w, v.T) + vd = 1 / np.sqrt(vv.diagonal()) + vv = vd[:, np.newaxis] * vv * vd + assert_allclose(vv, np.eye(N), rtol, atol) + + # check that the integral of 1 is correct + assert_allclose(w.sum(), mu, rtol, atol) + + # compare the results of integrating a function with quad. + def f(x): + return x ** 3 - 3 * x ** 2 + x - 2 + resI = integrate.quad(lambda x: f(x)*weight_func(x), a, b) + resG = np.vdot(f(x), w) + rtol = 1e-6 if 1e-6 < resI[1] else resI[1] * 10 + assert_allclose(resI[0], resG, rtol=rtol) + +def test_roots_jacobi(): + def rf(a, b): + return lambda n, mu: sc.roots_jacobi(n, a, b, mu) + def ef(a, b): + return lambda n, x: sc.eval_jacobi(n, a, b, x) + def wf(a, b): + return lambda x: (1 - x) ** a * (1 + x) ** b + + vgq = verify_gauss_quad + vgq(rf(-0.5, -0.75), ef(-0.5, -0.75), wf(-0.5, -0.75), -1., 1., 5) + vgq(rf(-0.5, -0.75), ef(-0.5, -0.75), wf(-0.5, -0.75), -1., 1., + 25, atol=1e-12) + vgq(rf(-0.5, -0.75), ef(-0.5, -0.75), wf(-0.5, -0.75), -1., 1., + 100, atol=1e-11) + + vgq(rf(0.5, -0.5), ef(0.5, -0.5), wf(0.5, -0.5), -1., 1., 5) + vgq(rf(0.5, -0.5), ef(0.5, -0.5), wf(0.5, -0.5), -1., 1., 25, atol=1.5e-13) + vgq(rf(0.5, -0.5), ef(0.5, -0.5), wf(0.5, -0.5), -1., 1., 100, atol=2e-12) + + vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), -1., 1., 5, atol=2e-13) + vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), -1., 1., 25, atol=2e-13) + vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), -1., 1., 100, atol=1e-12) + + vgq(rf(0.9, 2), ef(0.9, 2), wf(0.9, 2), -1., 1., 5) + vgq(rf(0.9, 2), ef(0.9, 2), wf(0.9, 2), -1., 1., 25, atol=1e-13) + vgq(rf(0.9, 2), ef(0.9, 2), wf(0.9, 2), -1., 1., 100, atol=3e-13) + + vgq(rf(18.24, 27.3), ef(18.24, 27.3), wf(18.24, 27.3), -1., 1., 5) + vgq(rf(18.24, 27.3), ef(18.24, 27.3), wf(18.24, 27.3), -1., 1., 25, + atol=1.1e-14) + vgq(rf(18.24, 27.3), ef(18.24, 27.3), wf(18.24, 27.3), -1., 1., + 100, atol=1e-13) + + vgq(rf(47.1, -0.2), ef(47.1, -0.2), wf(47.1, -0.2), -1., 1., 5, atol=1e-13) + vgq(rf(47.1, -0.2), ef(47.1, -0.2), wf(47.1, -0.2), -1., 1., 25, atol=2e-13) + vgq(rf(47.1, -0.2), ef(47.1, -0.2), wf(47.1, -0.2), -1., 1., + 100, atol=1e-11) + + vgq(rf(1., 658.), ef(1., 658.), wf(1., 658.), -1., 1., 5, atol=2e-13) + vgq(rf(1., 658.), ef(1., 658.), wf(1., 658.), -1., 1., 25, atol=1e-12) + vgq(rf(1., 658.), ef(1., 658.), wf(1., 658.), -1., 1., 100, atol=1e-11) + vgq(rf(1., 658.), ef(1., 658.), wf(1., 658.), -1., 1., 250, atol=1e-11) + + vgq(rf(511., 511.), ef(511., 511.), wf(511., 511.), -1., 1., 5, + atol=1e-12) + vgq(rf(511., 511.), ef(511., 511.), wf(511., 511.), -1., 1., 25, + atol=1e-11) + vgq(rf(511., 511.), ef(511., 511.), wf(511., 511.), -1., 1., 100, + atol=1e-10) + + vgq(rf(511., 512.), ef(511., 512.), wf(511., 512.), -1., 1., 5, + atol=1e-12) + vgq(rf(511., 512.), ef(511., 512.), wf(511., 512.), -1., 1., 25, + atol=1e-11) + vgq(rf(511., 512.), ef(511., 512.), wf(511., 512.), -1., 1., 100, + atol=1e-10) + + vgq(rf(1000., 500.), ef(1000., 500.), wf(1000., 500.), -1., 1., 5, + atol=1e-12) + vgq(rf(1000., 500.), ef(1000., 500.), wf(1000., 500.), -1., 1., 25, + atol=1e-11) + vgq(rf(1000., 500.), ef(1000., 500.), wf(1000., 500.), -1., 1., 100, + atol=1e-10) + + vgq(rf(2.25, 68.9), ef(2.25, 68.9), wf(2.25, 68.9), -1., 1., 5) + vgq(rf(2.25, 68.9), ef(2.25, 68.9), wf(2.25, 68.9), -1., 1., 25, + atol=1e-13) + vgq(rf(2.25, 68.9), ef(2.25, 68.9), wf(2.25, 68.9), -1., 1., 100, + atol=1e-13) + + # when alpha == beta == 0, P_n^{a,b}(x) == P_n(x) + xj, wj = sc.roots_jacobi(6, 0.0, 0.0) + xl, wl = sc.roots_legendre(6) + assert_allclose(xj, xl, 1e-14, 1e-14) + assert_allclose(wj, wl, 1e-14, 1e-14) + + # when alpha == beta != 0, P_n^{a,b}(x) == C_n^{alpha+0.5}(x) + xj, wj = sc.roots_jacobi(6, 4.0, 4.0) + xc, wc = sc.roots_gegenbauer(6, 4.5) + assert_allclose(xj, xc, 1e-14, 1e-14) + assert_allclose(wj, wc, 1e-14, 1e-14) + + x, w = sc.roots_jacobi(5, 2, 3, False) + y, v, m = sc.roots_jacobi(5, 2, 3, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(wf(2,3), -1, 1) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_jacobi, 0, 1, 1) + assert_raises(ValueError, sc.roots_jacobi, 3.3, 1, 1) + assert_raises(ValueError, sc.roots_jacobi, 3, -2, 1) + assert_raises(ValueError, sc.roots_jacobi, 3, 1, -2) + assert_raises(ValueError, sc.roots_jacobi, 3, -2, -2) + +def test_roots_sh_jacobi(): + def rf(a, b): + return lambda n, mu: sc.roots_sh_jacobi(n, a, b, mu) + def ef(a, b): + return lambda n, x: sc.eval_sh_jacobi(n, a, b, x) + def wf(a, b): + return lambda x: (1.0 - x) ** (a - b) * x ** (b - 1.0) + + vgq = verify_gauss_quad + vgq(rf(-0.5, 0.25), ef(-0.5, 0.25), wf(-0.5, 0.25), 0., 1., 5) + vgq(rf(-0.5, 0.25), ef(-0.5, 0.25), wf(-0.5, 0.25), 0., 1., + 25, atol=1e-12) + vgq(rf(-0.5, 0.25), ef(-0.5, 0.25), wf(-0.5, 0.25), 0., 1., + 100, atol=1e-11) + + vgq(rf(0.5, 0.5), ef(0.5, 0.5), wf(0.5, 0.5), 0., 1., 5) + vgq(rf(0.5, 0.5), ef(0.5, 0.5), wf(0.5, 0.5), 0., 1., 25, atol=1e-13) + vgq(rf(0.5, 0.5), ef(0.5, 0.5), wf(0.5, 0.5), 0., 1., 100, atol=1e-12) + + vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), 0., 1., 5) + vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), 0., 1., 25, atol=1.5e-13) + vgq(rf(1, 0.5), ef(1, 0.5), wf(1, 0.5), 0., 1., 100, atol=2e-12) + + vgq(rf(2, 0.9), ef(2, 0.9), wf(2, 0.9), 0., 1., 5) + vgq(rf(2, 0.9), ef(2, 0.9), wf(2, 0.9), 0., 1., 25, atol=1e-13) + vgq(rf(2, 0.9), ef(2, 0.9), wf(2, 0.9), 0., 1., 100, atol=1e-12) + + vgq(rf(27.3, 18.24), ef(27.3, 18.24), wf(27.3, 18.24), 0., 1., 5) + vgq(rf(27.3, 18.24), ef(27.3, 18.24), wf(27.3, 18.24), 0., 1., 25) + vgq(rf(27.3, 18.24), ef(27.3, 18.24), wf(27.3, 18.24), 0., 1., + 100, atol=1e-13) + + vgq(rf(47.1, 0.2), ef(47.1, 0.2), wf(47.1, 0.2), 0., 1., 5, atol=1e-12) + vgq(rf(47.1, 0.2), ef(47.1, 0.2), wf(47.1, 0.2), 0., 1., 25, atol=1e-11) + vgq(rf(47.1, 0.2), ef(47.1, 0.2), wf(47.1, 0.2), 0., 1., 100, atol=1e-10) + + vgq(rf(68.9, 2.25), ef(68.9, 2.25), wf(68.9, 2.25), 0., 1., 5, atol=3.5e-14) + vgq(rf(68.9, 2.25), ef(68.9, 2.25), wf(68.9, 2.25), 0., 1., 25, atol=2e-13) + vgq(rf(68.9, 2.25), ef(68.9, 2.25), wf(68.9, 2.25), 0., 1., + 100, atol=1e-12) + + x, w = sc.roots_sh_jacobi(5, 3, 2, False) + y, v, m = sc.roots_sh_jacobi(5, 3, 2, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(wf(3,2), 0, 1) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_sh_jacobi, 0, 1, 1) + assert_raises(ValueError, sc.roots_sh_jacobi, 3.3, 1, 1) + assert_raises(ValueError, sc.roots_sh_jacobi, 3, 1, 2) # p - q <= -1 + assert_raises(ValueError, sc.roots_sh_jacobi, 3, 2, -1) # q <= 0 + assert_raises(ValueError, sc.roots_sh_jacobi, 3, -2, -1) # both + +def test_roots_hermite(): + rootf = sc.roots_hermite + evalf = sc.eval_hermite + weightf = orth.hermite(5).weight_func + + verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 5) + verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 25, atol=1e-13) + verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 100, atol=1e-12) + + # Golub-Welsch branch + x, w = sc.roots_hermite(5, False) + y, v, m = sc.roots_hermite(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, -np.inf, np.inf) + assert_allclose(m, muI, rtol=muI_err) + + # Asymptotic branch (switch over at n >= 150) + x, w = sc.roots_hermite(200, False) + y, v, m = sc.roots_hermite(200, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + assert_allclose(sum(v), m, 1e-14, 1e-14) + + assert_raises(ValueError, sc.roots_hermite, 0) + assert_raises(ValueError, sc.roots_hermite, 3.3) + +def test_roots_hermite_asy(): + # Recursion for Hermite functions + def hermite_recursion(n, nodes): + H = np.zeros((n, nodes.size)) + H[0,:] = np.pi**(-0.25) * np.exp(-0.5*nodes**2) + if n > 1: + H[1,:] = sqrt(2.0) * nodes * H[0,:] + for k in range(2, n): + H[k,:] = sqrt(2.0/k) * nodes * H[k-1,:] - sqrt((k-1.0)/k) * H[k-2,:] + return H + + # This tests only the nodes + def test(N, rtol=1e-15, atol=1e-14): + x, w = orth._roots_hermite_asy(N) + H = hermite_recursion(N+1, x) + assert_allclose(H[-1,:], np.zeros(N), rtol, atol) + assert_allclose(sum(w), sqrt(np.pi), rtol, atol) + + test(150, atol=1e-12) + test(151, atol=1e-12) + test(300, atol=1e-12) + test(301, atol=1e-12) + test(500, atol=1e-12) + test(501, atol=1e-12) + test(999, atol=1e-12) + test(1000, atol=1e-12) + test(2000, atol=1e-12) + test(5000, atol=1e-12) + +def test_roots_hermitenorm(): + rootf = sc.roots_hermitenorm + evalf = sc.eval_hermitenorm + weightf = orth.hermitenorm(5).weight_func + + verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 5) + verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 25, atol=1e-13) + verify_gauss_quad(rootf, evalf, weightf, -np.inf, np.inf, 100, atol=1e-12) + + x, w = sc.roots_hermitenorm(5, False) + y, v, m = sc.roots_hermitenorm(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, -np.inf, np.inf) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_hermitenorm, 0) + assert_raises(ValueError, sc.roots_hermitenorm, 3.3) + +def test_roots_gegenbauer(): + def rootf(a): + return lambda n, mu: sc.roots_gegenbauer(n, a, mu) + def evalf(a): + return lambda n, x: sc.eval_gegenbauer(n, a, x) + def weightf(a): + return lambda x: (1 - x ** 2) ** (a - 0.5) + + vgq = verify_gauss_quad + vgq(rootf(-0.25), evalf(-0.25), weightf(-0.25), -1., 1., 5) + vgq(rootf(-0.25), evalf(-0.25), weightf(-0.25), -1., 1., 25, atol=1e-12) + vgq(rootf(-0.25), evalf(-0.25), weightf(-0.25), -1., 1., 100, atol=1e-11) + + vgq(rootf(0.1), evalf(0.1), weightf(0.1), -1., 1., 5) + vgq(rootf(0.1), evalf(0.1), weightf(0.1), -1., 1., 25, atol=1e-13) + vgq(rootf(0.1), evalf(0.1), weightf(0.1), -1., 1., 100, atol=1e-12) + + vgq(rootf(1), evalf(1), weightf(1), -1., 1., 5) + vgq(rootf(1), evalf(1), weightf(1), -1., 1., 25, atol=1e-13) + vgq(rootf(1), evalf(1), weightf(1), -1., 1., 100, atol=1e-12) + + vgq(rootf(10), evalf(10), weightf(10), -1., 1., 5) + vgq(rootf(10), evalf(10), weightf(10), -1., 1., 25, atol=1e-13) + vgq(rootf(10), evalf(10), weightf(10), -1., 1., 100, atol=1e-12) + + vgq(rootf(50), evalf(50), weightf(50), -1., 1., 5, atol=1e-13) + vgq(rootf(50), evalf(50), weightf(50), -1., 1., 25, atol=1e-12) + vgq(rootf(50), evalf(50), weightf(50), -1., 1., 100, atol=1e-11) + + # Alpha=170 is where the approximation used in roots_gegenbauer changes + vgq(rootf(170), evalf(170), weightf(170), -1., 1., 5, atol=1e-13) + vgq(rootf(170), evalf(170), weightf(170), -1., 1., 25, atol=1e-12) + vgq(rootf(170), evalf(170), weightf(170), -1., 1., 100, atol=1e-11) + vgq(rootf(170.5), evalf(170.5), weightf(170.5), -1., 1., 5, atol=1.25e-13) + vgq(rootf(170.5), evalf(170.5), weightf(170.5), -1., 1., 25, atol=1e-12) + vgq(rootf(170.5), evalf(170.5), weightf(170.5), -1., 1., 100, atol=1e-11) + + # Test for failures, e.g. overflows, resulting from large alphas + vgq(rootf(238), evalf(238), weightf(238), -1., 1., 5, atol=1e-13) + vgq(rootf(238), evalf(238), weightf(238), -1., 1., 25, atol=1e-12) + vgq(rootf(238), evalf(238), weightf(238), -1., 1., 100, atol=1e-11) + vgq(rootf(512.5), evalf(512.5), weightf(512.5), -1., 1., 5, atol=1e-12) + vgq(rootf(512.5), evalf(512.5), weightf(512.5), -1., 1., 25, atol=1e-11) + vgq(rootf(512.5), evalf(512.5), weightf(512.5), -1., 1., 100, atol=1e-10) + + # this is a special case that the old code supported. + # when alpha = 0, the gegenbauer polynomial is uniformly 0. but it goes + # to a scaled down copy of T_n(x) there. + vgq(rootf(0), sc.eval_chebyt, weightf(0), -1., 1., 5) + vgq(rootf(0), sc.eval_chebyt, weightf(0), -1., 1., 25) + vgq(rootf(0), sc.eval_chebyt, weightf(0), -1., 1., 100, atol=1e-12) + + x, w = sc.roots_gegenbauer(5, 2, False) + y, v, m = sc.roots_gegenbauer(5, 2, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf(2), -1, 1) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_gegenbauer, 0, 2) + assert_raises(ValueError, sc.roots_gegenbauer, 3.3, 2) + assert_raises(ValueError, sc.roots_gegenbauer, 3, -.75) + +def test_roots_chebyt(): + weightf = orth.chebyt(5).weight_func + verify_gauss_quad(sc.roots_chebyt, sc.eval_chebyt, weightf, -1., 1., 5) + verify_gauss_quad(sc.roots_chebyt, sc.eval_chebyt, weightf, -1., 1., 25) + verify_gauss_quad(sc.roots_chebyt, sc.eval_chebyt, weightf, -1., 1., 100, + atol=1e-12) + + x, w = sc.roots_chebyt(5, False) + y, v, m = sc.roots_chebyt(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, -1, 1) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_chebyt, 0) + assert_raises(ValueError, sc.roots_chebyt, 3.3) + +def test_chebyt_symmetry(): + x, w = sc.roots_chebyt(21) + pos, neg = x[:10], x[11:] + assert_equal(neg, -pos[::-1]) + assert_equal(x[10], 0) + +def test_roots_chebyu(): + weightf = orth.chebyu(5).weight_func + verify_gauss_quad(sc.roots_chebyu, sc.eval_chebyu, weightf, -1., 1., 5) + verify_gauss_quad(sc.roots_chebyu, sc.eval_chebyu, weightf, -1., 1., 25) + verify_gauss_quad(sc.roots_chebyu, sc.eval_chebyu, weightf, -1., 1., 100) + + x, w = sc.roots_chebyu(5, False) + y, v, m = sc.roots_chebyu(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, -1, 1) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_chebyu, 0) + assert_raises(ValueError, sc.roots_chebyu, 3.3) + +def test_roots_chebyc(): + weightf = orth.chebyc(5).weight_func + verify_gauss_quad(sc.roots_chebyc, sc.eval_chebyc, weightf, -2., 2., 5) + verify_gauss_quad(sc.roots_chebyc, sc.eval_chebyc, weightf, -2., 2., 25) + verify_gauss_quad(sc.roots_chebyc, sc.eval_chebyc, weightf, -2., 2., 100, + atol=1e-12) + + x, w = sc.roots_chebyc(5, False) + y, v, m = sc.roots_chebyc(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, -2, 2) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_chebyc, 0) + assert_raises(ValueError, sc.roots_chebyc, 3.3) + +def test_roots_chebys(): + weightf = orth.chebys(5).weight_func + verify_gauss_quad(sc.roots_chebys, sc.eval_chebys, weightf, -2., 2., 5) + verify_gauss_quad(sc.roots_chebys, sc.eval_chebys, weightf, -2., 2., 25) + verify_gauss_quad(sc.roots_chebys, sc.eval_chebys, weightf, -2., 2., 100) + + x, w = sc.roots_chebys(5, False) + y, v, m = sc.roots_chebys(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, -2, 2) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_chebys, 0) + assert_raises(ValueError, sc.roots_chebys, 3.3) + +def test_roots_sh_chebyt(): + weightf = orth.sh_chebyt(5).weight_func + verify_gauss_quad(sc.roots_sh_chebyt, sc.eval_sh_chebyt, weightf, 0., 1., 5) + verify_gauss_quad(sc.roots_sh_chebyt, sc.eval_sh_chebyt, weightf, 0., 1., 25) + verify_gauss_quad(sc.roots_sh_chebyt, sc.eval_sh_chebyt, weightf, 0., 1., + 100, atol=1e-13) + + x, w = sc.roots_sh_chebyt(5, False) + y, v, m = sc.roots_sh_chebyt(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, 0, 1) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_sh_chebyt, 0) + assert_raises(ValueError, sc.roots_sh_chebyt, 3.3) + +def test_roots_sh_chebyu(): + weightf = orth.sh_chebyu(5).weight_func + verify_gauss_quad(sc.roots_sh_chebyu, sc.eval_sh_chebyu, weightf, 0., 1., 5) + verify_gauss_quad(sc.roots_sh_chebyu, sc.eval_sh_chebyu, weightf, 0., 1., 25) + verify_gauss_quad(sc.roots_sh_chebyu, sc.eval_sh_chebyu, weightf, 0., 1., + 100, atol=1e-13) + + x, w = sc.roots_sh_chebyu(5, False) + y, v, m = sc.roots_sh_chebyu(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, 0, 1) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_sh_chebyu, 0) + assert_raises(ValueError, sc.roots_sh_chebyu, 3.3) + +def test_roots_legendre(): + weightf = orth.legendre(5).weight_func + verify_gauss_quad(sc.roots_legendre, sc.eval_legendre, weightf, -1., 1., 5) + verify_gauss_quad(sc.roots_legendre, sc.eval_legendre, weightf, -1., 1., + 25, atol=1e-13) + verify_gauss_quad(sc.roots_legendre, sc.eval_legendre, weightf, -1., 1., + 100, atol=1e-12) + + x, w = sc.roots_legendre(5, False) + y, v, m = sc.roots_legendre(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, -1, 1) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_legendre, 0) + assert_raises(ValueError, sc.roots_legendre, 3.3) + +def test_roots_sh_legendre(): + weightf = orth.sh_legendre(5).weight_func + verify_gauss_quad(sc.roots_sh_legendre, sc.eval_sh_legendre, weightf, 0., 1., 5) + verify_gauss_quad(sc.roots_sh_legendre, sc.eval_sh_legendre, weightf, 0., 1., + 25, atol=1e-13) + verify_gauss_quad(sc.roots_sh_legendre, sc.eval_sh_legendre, weightf, 0., 1., + 100, atol=1e-12) + + x, w = sc.roots_sh_legendre(5, False) + y, v, m = sc.roots_sh_legendre(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, 0, 1) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_sh_legendre, 0) + assert_raises(ValueError, sc.roots_sh_legendre, 3.3) + +def test_roots_laguerre(): + weightf = orth.laguerre(5).weight_func + verify_gauss_quad(sc.roots_laguerre, sc.eval_laguerre, weightf, 0., np.inf, 5) + verify_gauss_quad(sc.roots_laguerre, sc.eval_laguerre, weightf, 0., np.inf, + 25, atol=1e-13) + verify_gauss_quad(sc.roots_laguerre, sc.eval_laguerre, weightf, 0., np.inf, + 100, atol=1e-12) + + x, w = sc.roots_laguerre(5, False) + y, v, m = sc.roots_laguerre(5, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf, 0, np.inf) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_laguerre, 0) + assert_raises(ValueError, sc.roots_laguerre, 3.3) + +def test_roots_genlaguerre(): + def rootf(a): + return lambda n, mu: sc.roots_genlaguerre(n, a, mu) + def evalf(a): + return lambda n, x: sc.eval_genlaguerre(n, a, x) + def weightf(a): + return lambda x: x ** a * np.exp(-x) + + vgq = verify_gauss_quad + vgq(rootf(-0.5), evalf(-0.5), weightf(-0.5), 0., np.inf, 5) + vgq(rootf(-0.5), evalf(-0.5), weightf(-0.5), 0., np.inf, 25, atol=1e-13) + vgq(rootf(-0.5), evalf(-0.5), weightf(-0.5), 0., np.inf, 100, atol=1e-12) + + vgq(rootf(0.1), evalf(0.1), weightf(0.1), 0., np.inf, 5) + vgq(rootf(0.1), evalf(0.1), weightf(0.1), 0., np.inf, 25, atol=1e-13) + vgq(rootf(0.1), evalf(0.1), weightf(0.1), 0., np.inf, 100, atol=1.6e-13) + + vgq(rootf(1), evalf(1), weightf(1), 0., np.inf, 5) + vgq(rootf(1), evalf(1), weightf(1), 0., np.inf, 25, atol=1e-13) + vgq(rootf(1), evalf(1), weightf(1), 0., np.inf, 100, atol=1.03e-13) + + vgq(rootf(10), evalf(10), weightf(10), 0., np.inf, 5) + vgq(rootf(10), evalf(10), weightf(10), 0., np.inf, 25, atol=1e-13) + vgq(rootf(10), evalf(10), weightf(10), 0., np.inf, 100, atol=1e-12) + + vgq(rootf(50), evalf(50), weightf(50), 0., np.inf, 5) + vgq(rootf(50), evalf(50), weightf(50), 0., np.inf, 25, atol=1e-13) + vgq(rootf(50), evalf(50), weightf(50), 0., np.inf, 100, rtol=1e-14, atol=2e-13) + + x, w = sc.roots_genlaguerre(5, 2, False) + y, v, m = sc.roots_genlaguerre(5, 2, True) + assert_allclose(x, y, 1e-14, 1e-14) + assert_allclose(w, v, 1e-14, 1e-14) + + muI, muI_err = integrate.quad(weightf(2.), 0., np.inf) + assert_allclose(m, muI, rtol=muI_err) + + assert_raises(ValueError, sc.roots_genlaguerre, 0, 2) + assert_raises(ValueError, sc.roots_genlaguerre, 3.3, 2) + assert_raises(ValueError, sc.roots_genlaguerre, 3, -1.1) + + +def test_gh_6721(): + # Regression test for gh_6721. This should not raise. + sc.chebyt(65)(0.2) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_orthogonal_eval.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_orthogonal_eval.py new file mode 100644 index 0000000000000000000000000000000000000000..8a4f379effdc9c3aa18bd2948fbd9f716b1f8d57 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_orthogonal_eval.py @@ -0,0 +1,275 @@ +import numpy as np +from numpy.testing import assert_, assert_allclose +import pytest + +from scipy.special import _ufuncs +import scipy.special._orthogonal as orth +from scipy.special._testutils import FuncData + + +def test_eval_chebyt(): + n = np.arange(0, 10000, 7, dtype=np.dtype("long")) + x = 2*np.random.rand() - 1 + v1 = np.cos(n*np.arccos(x)) + v2 = _ufuncs.eval_chebyt(n, x) + assert_(np.allclose(v1, v2, rtol=1e-15)) + + +def test_eval_chebyt_gh20129(): + # https://github.com/scipy/scipy/issues/20129 + assert _ufuncs.eval_chebyt(7, 2 + 0j) == 5042.0 + + +def test_eval_genlaguerre_restriction(): + # check it returns nan for alpha <= -1 + assert_(np.isnan(_ufuncs.eval_genlaguerre(0, -1, 0))) + assert_(np.isnan(_ufuncs.eval_genlaguerre(0.1, -1, 0))) + + +def test_warnings(): + # ticket 1334 + with np.errstate(all='raise'): + # these should raise no fp warnings + _ufuncs.eval_legendre(1, 0) + _ufuncs.eval_laguerre(1, 1) + _ufuncs.eval_gegenbauer(1, 1, 0) + + +class TestPolys: + """ + Check that the eval_* functions agree with the constructed polynomials + + """ + + def check_poly(self, func, cls, param_ranges=(), x_range=(), nn=10, + nparam=10, nx=10, rtol=1e-8): + rng = np.random.RandomState(1234) + + dataset = [] + for n in np.arange(nn): + params = [a + (b-a)*rng.rand(nparam) for a,b in param_ranges] + params = np.asarray(params).T + if not param_ranges: + params = [0] + for p in params: + if param_ranges: + p = (n,) + tuple(p) + else: + p = (n,) + x = x_range[0] + (x_range[1] - x_range[0])*rng.rand(nx) + x[0] = x_range[0] # always include domain start point + x[1] = x_range[1] # always include domain end point + poly = np.poly1d(cls(*p).coef) + z = np.c_[np.tile(p, (nx,1)), x, poly(x)] + dataset.append(z) + + dataset = np.concatenate(dataset, axis=0) + + def polyfunc(*p): + p = (p[0].astype(np.dtype("long")),) + p[1:] + return func(*p) + + with np.errstate(all='raise'): + ds = FuncData(polyfunc, dataset, list(range(len(param_ranges)+2)), -1, + rtol=rtol) + ds.check() + + def test_jacobi(self): + self.check_poly(_ufuncs.eval_jacobi, orth.jacobi, + param_ranges=[(-0.99, 10), (-0.99, 10)], + x_range=[-1, 1], rtol=1e-5) + + def test_sh_jacobi(self): + self.check_poly(_ufuncs.eval_sh_jacobi, orth.sh_jacobi, + param_ranges=[(1, 10), (0, 1)], x_range=[0, 1], + rtol=1e-5) + + def test_gegenbauer(self): + self.check_poly(_ufuncs.eval_gegenbauer, orth.gegenbauer, + param_ranges=[(-0.499, 10)], x_range=[-1, 1], + rtol=1e-7) + + def test_chebyt(self): + self.check_poly(_ufuncs.eval_chebyt, orth.chebyt, + param_ranges=[], x_range=[-1, 1]) + + def test_chebyu(self): + self.check_poly(_ufuncs.eval_chebyu, orth.chebyu, + param_ranges=[], x_range=[-1, 1]) + + def test_chebys(self): + self.check_poly(_ufuncs.eval_chebys, orth.chebys, + param_ranges=[], x_range=[-2, 2]) + + def test_chebyc(self): + self.check_poly(_ufuncs.eval_chebyc, orth.chebyc, + param_ranges=[], x_range=[-2, 2]) + + def test_sh_chebyt(self): + with np.errstate(all='ignore'): + self.check_poly(_ufuncs.eval_sh_chebyt, orth.sh_chebyt, + param_ranges=[], x_range=[0, 1]) + + def test_sh_chebyu(self): + self.check_poly(_ufuncs.eval_sh_chebyu, orth.sh_chebyu, + param_ranges=[], x_range=[0, 1]) + + def test_legendre(self): + self.check_poly(_ufuncs.eval_legendre, orth.legendre, + param_ranges=[], x_range=[-1, 1]) + + def test_sh_legendre(self): + with np.errstate(all='ignore'): + self.check_poly(_ufuncs.eval_sh_legendre, orth.sh_legendre, + param_ranges=[], x_range=[0, 1]) + + def test_genlaguerre(self): + self.check_poly(_ufuncs.eval_genlaguerre, orth.genlaguerre, + param_ranges=[(-0.99, 10)], x_range=[0, 100]) + + def test_laguerre(self): + self.check_poly(_ufuncs.eval_laguerre, orth.laguerre, + param_ranges=[], x_range=[0, 100]) + + def test_hermite(self): + self.check_poly(_ufuncs.eval_hermite, orth.hermite, + param_ranges=[], x_range=[-100, 100]) + + def test_hermitenorm(self): + self.check_poly(_ufuncs.eval_hermitenorm, orth.hermitenorm, + param_ranges=[], x_range=[-100, 100]) + + +class TestRecurrence: + """ + Check that the eval_* functions sig='ld->d' and 'dd->d' agree. + + """ + + def check_poly(self, func, param_ranges=(), x_range=(), nn=10, + nparam=10, nx=10, rtol=1e-8): + np.random.seed(1234) + + dataset = [] + for n in np.arange(nn): + params = [a + (b-a)*np.random.rand(nparam) for a,b in param_ranges] + params = np.asarray(params).T + if not param_ranges: + params = [0] + for p in params: + if param_ranges: + p = (n,) + tuple(p) + else: + p = (n,) + x = x_range[0] + (x_range[1] - x_range[0])*np.random.rand(nx) + x[0] = x_range[0] # always include domain start point + x[1] = x_range[1] # always include domain end point + kw = dict(sig=(len(p)+1)*'d'+'->d') + z = np.c_[np.tile(p, (nx,1)), x, func(*(p + (x,)), **kw)] + dataset.append(z) + + dataset = np.concatenate(dataset, axis=0) + + def polyfunc(*p): + p0 = p[0].astype(np.intp) + p = (p0,) + p[1:] + p0_type_char = p0.dtype.char + kw = dict(sig=p0_type_char + (len(p)-1)*'d' + '->d') + return func(*p, **kw) + + with np.errstate(all='raise'): + ds = FuncData(polyfunc, dataset, list(range(len(param_ranges)+2)), -1, + rtol=rtol) + ds.check() + + def test_jacobi(self): + self.check_poly(_ufuncs.eval_jacobi, + param_ranges=[(-0.99, 10), (-0.99, 10)], + x_range=[-1, 1]) + + def test_sh_jacobi(self): + self.check_poly(_ufuncs.eval_sh_jacobi, + param_ranges=[(1, 10), (0, 1)], x_range=[0, 1]) + + def test_gegenbauer(self): + self.check_poly(_ufuncs.eval_gegenbauer, + param_ranges=[(-0.499, 10)], x_range=[-1, 1]) + + def test_chebyt(self): + self.check_poly(_ufuncs.eval_chebyt, + param_ranges=[], x_range=[-1, 1]) + + def test_chebyu(self): + self.check_poly(_ufuncs.eval_chebyu, + param_ranges=[], x_range=[-1, 1]) + + def test_chebys(self): + self.check_poly(_ufuncs.eval_chebys, + param_ranges=[], x_range=[-2, 2]) + + def test_chebyc(self): + self.check_poly(_ufuncs.eval_chebyc, + param_ranges=[], x_range=[-2, 2]) + + def test_sh_chebyt(self): + self.check_poly(_ufuncs.eval_sh_chebyt, + param_ranges=[], x_range=[0, 1]) + + def test_sh_chebyu(self): + self.check_poly(_ufuncs.eval_sh_chebyu, + param_ranges=[], x_range=[0, 1]) + + def test_legendre(self): + self.check_poly(_ufuncs.eval_legendre, + param_ranges=[], x_range=[-1, 1]) + + def test_sh_legendre(self): + self.check_poly(_ufuncs.eval_sh_legendre, + param_ranges=[], x_range=[0, 1]) + + def test_genlaguerre(self): + self.check_poly(_ufuncs.eval_genlaguerre, + param_ranges=[(-0.99, 10)], x_range=[0, 100]) + + def test_laguerre(self): + self.check_poly(_ufuncs.eval_laguerre, + param_ranges=[], x_range=[0, 100]) + + def test_hermite(self): + v = _ufuncs.eval_hermite(70, 1.0) + a = -1.457076485701412e60 + assert_allclose(v, a) + + +def test_hermite_domain(): + # Regression test for gh-11091. + assert np.isnan(_ufuncs.eval_hermite(-1, 1.0)) + assert np.isnan(_ufuncs.eval_hermitenorm(-1, 1.0)) + + +@pytest.mark.parametrize("n", [0, 1, 2]) +@pytest.mark.parametrize("x", [0, 1, np.nan]) +def test_hermite_nan(n, x): + # Regression test for gh-11369. + assert np.isnan(_ufuncs.eval_hermite(n, x)) == np.any(np.isnan([n, x])) + assert np.isnan(_ufuncs.eval_hermitenorm(n, x)) == np.any(np.isnan([n, x])) + + +@pytest.mark.parametrize('n', [0, 1, 2, 3.2]) +@pytest.mark.parametrize('alpha', [1, np.nan]) +@pytest.mark.parametrize('x', [2, np.nan]) +def test_genlaguerre_nan(n, alpha, x): + # Regression test for gh-11361. + nan_laguerre = np.isnan(_ufuncs.eval_genlaguerre(n, alpha, x)) + nan_arg = np.any(np.isnan([n, alpha, x])) + assert nan_laguerre == nan_arg + + +@pytest.mark.parametrize('n', [0, 1, 2, 3.2]) +@pytest.mark.parametrize('alpha', [0.0, 1, np.nan]) +@pytest.mark.parametrize('x', [1e-6, 2, np.nan]) +def test_gegenbauer_nan(n, alpha, x): + # Regression test for gh-11370. + nan_gegenbauer = np.isnan(_ufuncs.eval_gegenbauer(n, alpha, x)) + nan_arg = np.any(np.isnan([n, alpha, x])) + assert nan_gegenbauer == nan_arg diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_owens_t.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_owens_t.py new file mode 100644 index 0000000000000000000000000000000000000000..8d15aead25302023c5f07d8392c0931995764ced --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_owens_t.py @@ -0,0 +1,53 @@ +import numpy as np +from numpy.testing import assert_equal, assert_allclose + +import scipy.special as sc + + +def test_symmetries(): + np.random.seed(1234) + a, h = np.random.rand(100), np.random.rand(100) + assert_equal(sc.owens_t(h, a), sc.owens_t(-h, a)) + assert_equal(sc.owens_t(h, a), -sc.owens_t(h, -a)) + + +def test_special_cases(): + assert_equal(sc.owens_t(5, 0), 0) + assert_allclose(sc.owens_t(0, 5), 0.5*np.arctan(5)/np.pi, + rtol=5e-14) + # Target value is 0.5*Phi(5)*(1 - Phi(5)) for Phi the CDF of the + # standard normal distribution + assert_allclose(sc.owens_t(5, 1), 1.4332574485503512543e-07, + rtol=5e-14) + + +def test_nans(): + assert_equal(sc.owens_t(20, np.nan), np.nan) + assert_equal(sc.owens_t(np.nan, 20), np.nan) + assert_equal(sc.owens_t(np.nan, np.nan), np.nan) + + +def test_infs(): + h, a = 0, np.inf + # T(0, a) = 1/2π * arctan(a) + res = 1/(2*np.pi) * np.arctan(a) + assert_allclose(sc.owens_t(h, a), res, rtol=5e-14) + assert_allclose(sc.owens_t(h, -a), -res, rtol=5e-14) + + h = 1 + # Refer Owens T function definition in Wikipedia + # https://en.wikipedia.org/wiki/Owen%27s_T_function + # Value approximated through Numerical Integration + # using scipy.integrate.quad + # quad(lambda x: 1/(2*pi)*(exp(-0.5*(1*1)*(1+x*x))/(1+x*x)), 0, inf) + res = 0.07932762696572854 + assert_allclose(sc.owens_t(h, np.inf), res, rtol=5e-14) + assert_allclose(sc.owens_t(h, -np.inf), -res, rtol=5e-14) + + assert_equal(sc.owens_t(np.inf, 1), 0) + assert_equal(sc.owens_t(-np.inf, 1), 0) + + assert_equal(sc.owens_t(np.inf, np.inf), 0) + assert_equal(sc.owens_t(-np.inf, np.inf), 0) + assert_equal(sc.owens_t(np.inf, -np.inf), -0.0) + assert_equal(sc.owens_t(-np.inf, -np.inf), -0.0) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_pcf.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_pcf.py new file mode 100644 index 0000000000000000000000000000000000000000..a8c42aa688081fb58f79ad2c8ea932d03b33523b --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_pcf.py @@ -0,0 +1,24 @@ +"""Tests for parabolic cylinder functions. + +""" +import numpy as np +from numpy.testing import assert_allclose, assert_equal +import scipy.special as sc + + +def test_pbwa_segfault(): + # Regression test for https://github.com/scipy/scipy/issues/6208. + # + # Data generated by mpmath. + # + w = 1.02276567211316867161 + wp = -0.48887053372346189882 + assert_allclose(sc.pbwa(0, 0), (w, wp), rtol=1e-13, atol=0) + + +def test_pbwa_nan(): + # Check that NaN's are returned outside of the range in which the + # implementation is accurate. + pts = [(-6, -6), (-6, 6), (6, -6), (6, 6)] + for p in pts: + assert_equal(sc.pbwa(*p), (np.nan, np.nan)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_pdtr.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_pdtr.py new file mode 100644 index 0000000000000000000000000000000000000000..122e6009bd71e77ae39f55da5cf056500ff526a9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_pdtr.py @@ -0,0 +1,48 @@ +import numpy as np +import scipy.special as sc +from numpy.testing import assert_almost_equal, assert_array_equal + + +class TestPdtr: + def test(self): + val = sc.pdtr(0, 1) + assert_almost_equal(val, np.exp(-1)) + + def test_m_zero(self): + val = sc.pdtr([0, 1, 2], 0) + assert_array_equal(val, [1, 1, 1]) + + def test_rounding(self): + double_val = sc.pdtr([0.1, 1.1, 2.1], 1.0) + int_val = sc.pdtr([0, 1, 2], 1.0) + assert_array_equal(double_val, int_val) + + def test_inf(self): + val = sc.pdtr(np.inf, 1.0) + assert_almost_equal(val, 1.0) + + def test_domain(self): + val = sc.pdtr(-1.1, 1.0) + assert np.isnan(val) + +class TestPdtrc: + def test_value(self): + val = sc.pdtrc(0, 1) + assert_almost_equal(val, 1 - np.exp(-1)) + + def test_m_zero(self): + val = sc.pdtrc([0, 1, 2], 0.0) + assert_array_equal(val, [0, 0, 0]) + + def test_rounding(self): + double_val = sc.pdtrc([0.1, 1.1, 2.1], 1.0) + int_val = sc.pdtrc([0, 1, 2], 1.0) + assert_array_equal(double_val, int_val) + + def test_inf(self): + val = sc.pdtrc(np.inf, 1.0) + assert_almost_equal(val, 0.0) + + def test_domain(self): + val = sc.pdtrc(-1.1, 1.0) + assert np.isnan(val) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_powm1.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_powm1.py new file mode 100644 index 0000000000000000000000000000000000000000..3d809963f64ddaedf6b59de80dcd5f7ca8fa18a9 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_powm1.py @@ -0,0 +1,65 @@ +import pytest +import numpy as np +from numpy.testing import assert_allclose +from scipy.special import powm1 + + +# Expected values were computed with mpmath, e.g. +# +# >>> import mpmath +# >>> mpmath.np.dps = 200 +# >>> print(float(mpmath.powm1(2.0, 1e-7)) +# 6.931472045825965e-08 +# +powm1_test_cases = [ + (1.25, 0.75, 0.18217701125396976, 1e-15), + (2.0, 1e-7, 6.931472045825965e-08, 1e-15), + (25.0, 5e-11, 1.6094379125636148e-10, 1e-15), + (0.99996, 0.75, -3.0000150002530058e-05, 1e-15), + (0.9999999999990905, 20, -1.81898940353014e-11, 1e-15), + (-1.25, 751.0, -6.017550852453444e+72, 2e-15) +] + + +@pytest.mark.parametrize('x, y, expected, rtol', powm1_test_cases) +def test_powm1(x, y, expected, rtol): + p = powm1(x, y) + assert_allclose(p, expected, rtol=rtol) + + +@pytest.mark.parametrize('x, y, expected', + [(0.0, 0.0, 0.0), + (0.0, -1.5, np.inf), + (0.0, 1.75, -1.0), + (-1.5, 2.0, 1.25), + (-1.5, 3.0, -4.375), + (np.nan, 0.0, 0.0), + (1.0, np.nan, 0.0), + (1.0, np.inf, 0.0), + (1.0, -np.inf, 0.0), + (np.inf, 7.5, np.inf), + (np.inf, -7.5, -1.0), + (3.25, np.inf, np.inf), + (np.inf, np.inf, np.inf), + (np.inf, -np.inf, -1.0), + (np.inf, 0.0, 0.0), + (-np.inf, 0.0, 0.0), + (-np.inf, 2.0, np.inf), + (-np.inf, 3.0, -np.inf), + (-1.0, float(2**53 - 1), -2.0)]) +def test_powm1_exact_cases(x, y, expected): + # Test cases where we have an exact expected value. + p = powm1(x, y) + assert p == expected + + +@pytest.mark.parametrize('x, y', + [(-1.25, 751.03), + (-1.25, np.inf), + (np.nan, np.nan), + (-np.inf, -np.inf), + (-np.inf, 2.5)]) +def test_powm1_return_nan(x, y): + # Test cases where the expected return value is nan. + p = powm1(x, y) + assert np.isnan(p) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_precompute_expn_asy.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_precompute_expn_asy.py new file mode 100644 index 0000000000000000000000000000000000000000..7b6c6cba21d5c1d5fdfab879153ba8f125e98d5f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_precompute_expn_asy.py @@ -0,0 +1,24 @@ +from numpy.testing import assert_equal + +from scipy.special._testutils import check_version, MissingModule +from scipy.special._precompute.expn_asy import generate_A + +try: + import sympy + from sympy import Poly +except ImportError: + sympy = MissingModule("sympy") + + +@check_version(sympy, "1.0") +def test_generate_A(): + # Data from DLMF 8.20.5 + x = sympy.symbols('x') + Astd = [Poly(1, x), + Poly(1, x), + Poly(1 - 2*x), + Poly(1 - 8*x + 6*x**2)] + Ares = generate_A(len(Astd)) + + for p, q in zip(Astd, Ares): + assert_equal(p, q) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_precompute_gammainc.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_precompute_gammainc.py new file mode 100644 index 0000000000000000000000000000000000000000..c0c46b456d0a1b545b29e0b8b81b39dafb6b0610 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_precompute_gammainc.py @@ -0,0 +1,108 @@ +import pytest + +from scipy.special._testutils import MissingModule, check_version +from scipy.special._mptestutils import ( + Arg, IntArg, mp_assert_allclose, assert_mpmath_equal) +from scipy.special._precompute.gammainc_asy import ( + compute_g, compute_alpha, compute_d) +from scipy.special._precompute.gammainc_data import gammainc, gammaincc + +try: + import sympy +except ImportError: + sympy = MissingModule('sympy') + +try: + import mpmath as mp +except ImportError: + mp = MissingModule('mpmath') + + +@check_version(mp, '0.19') +def test_g(): + # Test data for the g_k. See DLMF 5.11.4. + with mp.workdps(30): + g = [mp.mpf(1), mp.mpf(1)/12, mp.mpf(1)/288, + -mp.mpf(139)/51840, -mp.mpf(571)/2488320, + mp.mpf(163879)/209018880, mp.mpf(5246819)/75246796800] + mp_assert_allclose(compute_g(7), g) + + +@pytest.mark.slow +@check_version(mp, '0.19') +@check_version(sympy, '0.7') +@pytest.mark.xfail_on_32bit("rtol only 2e-11, see gh-6938") +def test_alpha(): + # Test data for the alpha_k. See DLMF 8.12.14. + with mp.workdps(30): + alpha = [mp.mpf(0), mp.mpf(1), mp.mpf(1)/3, mp.mpf(1)/36, + -mp.mpf(1)/270, mp.mpf(1)/4320, mp.mpf(1)/17010, + -mp.mpf(139)/5443200, mp.mpf(1)/204120] + mp_assert_allclose(compute_alpha(9), alpha) + + +@pytest.mark.xslow +@check_version(mp, '0.19') +@check_version(sympy, '0.7') +def test_d(): + # Compare the d_{k, n} to the results in appendix F of [1]. + # + # Sources + # ------- + # [1] DiDonato and Morris, Computation of the Incomplete Gamma + # Function Ratios and their Inverse, ACM Transactions on + # Mathematical Software, 1986. + + with mp.workdps(50): + dataset = [(0, 0, -mp.mpf('0.333333333333333333333333333333')), + (0, 12, mp.mpf('0.102618097842403080425739573227e-7')), + (1, 0, -mp.mpf('0.185185185185185185185185185185e-2')), + (1, 12, mp.mpf('0.119516285997781473243076536700e-7')), + (2, 0, mp.mpf('0.413359788359788359788359788360e-2')), + (2, 12, -mp.mpf('0.140925299108675210532930244154e-7')), + (3, 0, mp.mpf('0.649434156378600823045267489712e-3')), + (3, 12, -mp.mpf('0.191111684859736540606728140873e-7')), + (4, 0, -mp.mpf('0.861888290916711698604702719929e-3')), + (4, 12, mp.mpf('0.288658297427087836297341274604e-7')), + (5, 0, -mp.mpf('0.336798553366358150308767592718e-3')), + (5, 12, mp.mpf('0.482409670378941807563762631739e-7')), + (6, 0, mp.mpf('0.531307936463992223165748542978e-3')), + (6, 12, -mp.mpf('0.882860074633048352505085243179e-7')), + (7, 0, mp.mpf('0.344367606892377671254279625109e-3')), + (7, 12, -mp.mpf('0.175629733590604619378669693914e-6')), + (8, 0, -mp.mpf('0.652623918595309418922034919727e-3')), + (8, 12, mp.mpf('0.377358774161109793380344937299e-6')), + (9, 0, -mp.mpf('0.596761290192746250124390067179e-3')), + (9, 12, mp.mpf('0.870823417786464116761231237189e-6'))] + d = compute_d(10, 13) + res = [d[k][n] for k, n, std in dataset] + std = [x[2] for x in dataset] + mp_assert_allclose(res, std) + + +@check_version(mp, '0.19') +def test_gammainc(): + # Quick check that the gammainc in + # special._precompute.gammainc_data agrees with mpmath's + # gammainc. + assert_mpmath_equal(gammainc, + lambda a, x: mp.gammainc(a, b=x, regularized=True), + [Arg(0, 100, inclusive_a=False), Arg(0, 100)], + nan_ok=False, rtol=1e-17, n=50, dps=50) + + +@pytest.mark.xslow +@check_version(mp, '0.19') +def test_gammaincc(): + # Check that the gammaincc in special._precompute.gammainc_data + # agrees with mpmath's gammainc. + assert_mpmath_equal(lambda a, x: gammaincc(a, x, dps=1000), + lambda a, x: mp.gammainc(a, a=x, regularized=True), + [Arg(20, 100), Arg(20, 100)], + nan_ok=False, rtol=1e-17, n=50, dps=1000) + + # Test the fast integer path + assert_mpmath_equal(gammaincc, + lambda a, x: mp.gammainc(a, a=x, regularized=True), + [IntArg(1, 100), Arg(0, 100)], + nan_ok=False, rtol=1e-17, n=50, dps=50) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_precompute_utils.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_precompute_utils.py new file mode 100644 index 0000000000000000000000000000000000000000..89616b92329691ca76039fe11a7e08f7f3db1150 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_precompute_utils.py @@ -0,0 +1,36 @@ +import pytest + +from scipy.special._testutils import MissingModule, check_version +from scipy.special._mptestutils import mp_assert_allclose +from scipy.special._precompute.utils import lagrange_inversion + +try: + import sympy +except ImportError: + sympy = MissingModule('sympy') + +try: + import mpmath as mp +except ImportError: + mp = MissingModule('mpmath') + + +@pytest.mark.slow +@check_version(sympy, '0.7') +@check_version(mp, '0.19') +class TestInversion: + @pytest.mark.xfail_on_32bit("rtol only 2e-9, see gh-6938") + def test_log(self): + with mp.workdps(30): + logcoeffs = mp.taylor(lambda x: mp.log(1 + x), 0, 10) + expcoeffs = mp.taylor(lambda x: mp.exp(x) - 1, 0, 10) + invlogcoeffs = lagrange_inversion(logcoeffs) + mp_assert_allclose(invlogcoeffs, expcoeffs) + + @pytest.mark.xfail_on_32bit("rtol only 1e-15, see gh-6938") + def test_sin(self): + with mp.workdps(30): + sincoeffs = mp.taylor(mp.sin, 0, 10) + asincoeffs = mp.taylor(mp.asin, 0, 10) + invsincoeffs = lagrange_inversion(sincoeffs) + mp_assert_allclose(invsincoeffs, asincoeffs, atol=1e-30) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_round.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_round.py new file mode 100644 index 0000000000000000000000000000000000000000..ba28b7c1cff95327df0ed69086dd92b93399c3c4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_round.py @@ -0,0 +1,18 @@ +import numpy as np +import pytest + +from scipy.special import _test_internal + + +@pytest.mark.fail_slow(20) +@pytest.mark.skipif(not _test_internal.have_fenv(), reason="no fenv()") +def test_add_round_up(): + rng = np.random.RandomState(1234) + _test_internal.test_add_round(10**5, 'up', rng) + + +@pytest.mark.fail_slow(20) +@pytest.mark.skipif(not _test_internal.have_fenv(), reason="no fenv()") +def test_add_round_down(): + rng = np.random.RandomState(1234) + _test_internal.test_add_round(10**5, 'down', rng) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_sf_error.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_sf_error.py new file mode 100644 index 0000000000000000000000000000000000000000..2dfe8287ee4f53a51a5654edc975c5c521a7d747 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_sf_error.py @@ -0,0 +1,145 @@ +import sys +import warnings + +import numpy as np +from numpy.testing import assert_, assert_equal, IS_PYPY +import pytest +from pytest import raises as assert_raises + +import scipy.special as sc +from scipy.special._ufuncs import _sf_error_test_function + +_sf_error_code_map = { + # skip 'ok' + 'singular': 1, + 'underflow': 2, + 'overflow': 3, + 'slow': 4, + 'loss': 5, + 'no_result': 6, + 'domain': 7, + 'arg': 8, + 'other': 9, + 'memory': 10, +} + +_sf_error_actions = [ + 'ignore', + 'warn', + 'raise' +] + + +def _check_action(fun, args, action): + # TODO: special expert should correct + # the coercion at the true location? + args = np.asarray(args, dtype=np.dtype("long")) + if action == 'warn': + with pytest.warns(sc.SpecialFunctionWarning): + fun(*args) + elif action == 'raise': + with assert_raises(sc.SpecialFunctionError): + fun(*args) + else: + # action == 'ignore', make sure there are no warnings/exceptions + with warnings.catch_warnings(): + warnings.simplefilter("error") + fun(*args) + + +def test_geterr(): + err = sc.geterr() + for key, value in err.items(): + assert_(key in _sf_error_code_map) + assert_(value in _sf_error_actions) + + +@pytest.mark.thread_unsafe +def test_seterr(): + entry_err = sc.geterr() + try: + for category, error_code in _sf_error_code_map.items(): + for action in _sf_error_actions: + geterr_olderr = sc.geterr() + seterr_olderr = sc.seterr(**{category: action}) + assert_(geterr_olderr == seterr_olderr) + newerr = sc.geterr() + assert_(newerr[category] == action) + geterr_olderr.pop(category) + newerr.pop(category) + assert_(geterr_olderr == newerr) + _check_action(_sf_error_test_function, (error_code,), action) + finally: + sc.seterr(**entry_err) + + +@pytest.mark.skipif(IS_PYPY, reason="Test not meaningful on PyPy") +def test_sf_error_special_refcount(): + # Regression test for gh-16233. + # Check that the reference count of scipy.special is not increased + # when a SpecialFunctionError is raised. + refcount_before = sys.getrefcount(sc) + with sc.errstate(all='raise'): + with pytest.raises(sc.SpecialFunctionError, match='domain error'): + sc.ndtri(2.0) + refcount_after = sys.getrefcount(sc) + assert refcount_after == refcount_before + + +def test_errstate_pyx_basic(): + olderr = sc.geterr() + with sc.errstate(singular='raise'): + with assert_raises(sc.SpecialFunctionError): + sc.loggamma(0) + assert_equal(olderr, sc.geterr()) + + +def test_errstate_c_basic(): + olderr = sc.geterr() + with sc.errstate(domain='raise'): + with assert_raises(sc.SpecialFunctionError): + sc.spence(-1) + assert_equal(olderr, sc.geterr()) + + +def test_errstate_cpp_basic(): + olderr = sc.geterr() + with sc.errstate(underflow='raise'): + with assert_raises(sc.SpecialFunctionError): + sc.wrightomega(-1000) + assert_equal(olderr, sc.geterr()) + + +def test_errstate_cpp_scipy_special(): + olderr = sc.geterr() + with sc.errstate(singular='raise'): + with assert_raises(sc.SpecialFunctionError): + sc.lambertw(0, 1) + assert_equal(olderr, sc.geterr()) + + +def test_errstate_cpp_alt_ufunc_machinery(): + olderr = sc.geterr() + with sc.errstate(singular='raise'): + with assert_raises(sc.SpecialFunctionError): + sc.gammaln(0) + assert_equal(olderr, sc.geterr()) + + +@pytest.mark.thread_unsafe +def test_errstate(): + for category, error_code in _sf_error_code_map.items(): + for action in _sf_error_actions: + olderr = sc.geterr() + with sc.errstate(**{category: action}): + _check_action(_sf_error_test_function, (error_code,), action) + assert_equal(olderr, sc.geterr()) + + +def test_errstate_all_but_one(): + olderr = sc.geterr() + with sc.errstate(all='raise', singular='ignore'): + sc.gammaln(0) + with assert_raises(sc.SpecialFunctionError): + sc.spence(-1.0) + assert_equal(olderr, sc.geterr()) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_sici.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_sici.py new file mode 100644 index 0000000000000000000000000000000000000000..d33c1795641ba74f777fcfdcffe80d86463477e3 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_sici.py @@ -0,0 +1,36 @@ +import numpy as np + +import scipy.special as sc +from scipy.special._testutils import FuncData + + +def test_sici_consistency(): + # Make sure the implementation of sici for real arguments agrees + # with the implementation of sici for complex arguments. + + # On the negative real axis Cephes drops the imaginary part in ci + def sici(x): + si, ci = sc.sici(x + 0j) + return si.real, ci.real + + x = np.r_[-np.logspace(8, -30, 200), 0, np.logspace(-30, 8, 200)] + si, ci = sc.sici(x) + dataset = np.column_stack((x, si, ci)) + FuncData(sici, dataset, 0, (1, 2), rtol=1e-12).check() + + +def test_shichi_consistency(): + # Make sure the implementation of shichi for real arguments agrees + # with the implementation of shichi for complex arguments. + + # On the negative real axis Cephes drops the imaginary part in chi + def shichi(x): + shi, chi = sc.shichi(x + 0j) + return shi.real, chi.real + + # Overflow happens quickly, so limit range + x = np.r_[-np.logspace(np.log10(700), -30, 200), 0, + np.logspace(-30, np.log10(700), 200)] + shi, chi = sc.shichi(x) + dataset = np.column_stack((x, shi, chi)) + FuncData(shichi, dataset, 0, (1, 2), rtol=1e-14).check() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_specfun.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_specfun.py new file mode 100644 index 0000000000000000000000000000000000000000..f36fd2915be8c8d14138493c34dfd5733a2a1e6d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_specfun.py @@ -0,0 +1,48 @@ +""" +Various made-up tests to hit different branches of the code in specfun.c +""" + +import numpy as np +from numpy.testing import assert_allclose +from scipy import special + + +def test_cva2_cv0_branches(): + res, resp = special.mathieu_cem([40, 129], [13, 14], [30, 45]) + assert_allclose(res, np.array([-0.3741211, 0.74441928])) + assert_allclose(resp, np.array([-37.02872758, -86.13549877])) + + res, resp = special.mathieu_sem([40, 129], [13, 14], [30, 45]) + assert_allclose(res, np.array([0.92955551, 0.66771207])) + assert_allclose(resp, np.array([-14.91073448, 96.02954185])) + + +def test_chgm_branches(): + res = special.eval_genlaguerre(-3.2, 3, 2.5) + assert_allclose(res, -0.7077721935779854) + + +def test_hygfz_branches(): + """(z == 1.0) && (c-a-b > 0.0)""" + res = special.hyp2f1(1.5, 2.5, 4.5, 1.+0.j) + assert_allclose(res, 10.30835089459151+0j) + """(cabs(z+1) < eps) && (fabs(c-a+b - 1.0) < eps)""" + res = special.hyp2f1(5+5e-16, 2, 2, -1.0 + 5e-16j) + assert_allclose(res, 0.031249999999999986+3.9062499999999994e-17j) + + +def test_pro_rad1(): + # https://github.com/scipy/scipy/issues/21058 + # Reference values taken from WolframAlpha + # SpheroidalS1(1, 1, 30, 1.1) + # SpheroidalS1Prime(1, 1, 30, 1.1) + res = special.pro_rad1(1, 1, 30, 1.1) + assert_allclose(res, (0.009657872296166435, 3.253369651472877), rtol=2e-5) + +def test_pro_rad2(): + # https://github.com/scipy/scipy/issues/21461 + # Reference values taken from WolframAlpha + # SpheroidalS2(0, 0, 3, 1.02) + # SpheroidalS2Prime(0, 0, 3, 1.02) + res = special.pro_rad2(0, 0, 3, 1.02) + assert_allclose(res, (-0.35089596858528077, 13.652764213480872), rtol=10e-10) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_spence.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_spence.py new file mode 100644 index 0000000000000000000000000000000000000000..fbb26ac281dff81ea71b30318731065fe5a78f94 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_spence.py @@ -0,0 +1,32 @@ +import numpy as np +from numpy import sqrt, log, pi +from scipy.special._testutils import FuncData +from scipy.special import spence + + +def test_consistency(): + # Make sure the implementation of spence for real arguments + # agrees with the implementation of spence for imaginary arguments. + + x = np.logspace(-30, 300, 200) + dataset = np.vstack((x + 0j, spence(x))).T + FuncData(spence, dataset, 0, 1, rtol=1e-14).check() + + +def test_special_points(): + # Check against known values of Spence's function. + + phi = (1 + sqrt(5))/2 + dataset = [(1, 0), + (2, -pi**2/12), + (0.5, pi**2/12 - log(2)**2/2), + (0, pi**2/6), + (-1, pi**2/4 - 1j*pi*log(2)), + ((-1 + sqrt(5))/2, pi**2/15 - log(phi)**2), + ((3 - sqrt(5))/2, pi**2/10 - log(phi)**2), + (phi, -pi**2/15 + log(phi)**2/2), + # Corrected from Zagier, "The Dilogarithm Function" + ((3 + sqrt(5))/2, -pi**2/10 - log(phi)**2)] + + dataset = np.asarray(dataset) + FuncData(spence, dataset, 0, 1, rtol=1e-14).check() diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_spfun_stats.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_spfun_stats.py new file mode 100644 index 0000000000000000000000000000000000000000..c4a047c78fb8542bd0abbb75a4815d777e1414b0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_spfun_stats.py @@ -0,0 +1,61 @@ +import numpy as np +from numpy.testing import (assert_array_equal, + assert_array_almost_equal_nulp, assert_almost_equal) +from pytest import raises as assert_raises + +from scipy.special import gammaln, multigammaln + + +class TestMultiGammaLn: + + def test1(self): + # A test of the identity + # Gamma_1(a) = Gamma(a) + np.random.seed(1234) + a = np.abs(np.random.randn()) + assert_array_equal(multigammaln(a, 1), gammaln(a)) + + def test2(self): + # A test of the identity + # Gamma_2(a) = sqrt(pi) * Gamma(a) * Gamma(a - 0.5) + a = np.array([2.5, 10.0]) + result = multigammaln(a, 2) + expected = np.log(np.sqrt(np.pi)) + gammaln(a) + gammaln(a - 0.5) + assert_almost_equal(result, expected) + + def test_bararg(self): + assert_raises(ValueError, multigammaln, 0.5, 1.2) + + +def _check_multigammaln_array_result(a, d): + # Test that the shape of the array returned by multigammaln + # matches the input shape, and that all the values match + # the value computed when multigammaln is called with a scalar. + result = multigammaln(a, d) + assert_array_equal(a.shape, result.shape) + a1 = a.ravel() + result1 = result.ravel() + for i in range(a.size): + assert_array_almost_equal_nulp(result1[i], multigammaln(a1[i], d)) + + +def test_multigammaln_array_arg(): + # Check that the array returned by multigammaln has the correct + # shape and contains the correct values. The cases have arrays + # with several different shapes. + # The cases include a regression test for ticket #1849 + # (a = np.array([2.0]), an array with a single element). + np.random.seed(1234) + + cases = [ + # a, d + (np.abs(np.random.randn(3, 2)) + 5, 5), + (np.abs(np.random.randn(1, 2)) + 5, 5), + (np.arange(10.0, 18.0).reshape(2, 2, 2), 3), + (np.array([2.0]), 3), + (np.float64(2.0), 3), + ] + + for a, d in cases: + _check_multigammaln_array_result(a, d) + diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_sph_harm.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_sph_harm.py new file mode 100644 index 0000000000000000000000000000000000000000..310bda00b4d8b715cddcd94f5aa7142a0bdae6fa --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_sph_harm.py @@ -0,0 +1,86 @@ +import numpy as np +import pytest + +from numpy.testing import assert_allclose, suppress_warnings +import scipy.special as sc + +class TestSphHarm: + @pytest.mark.slow + def test_p(self): + m_max = 20 + n_max = 10 + + theta = np.linspace(0, np.pi) + phi = np.linspace(0, 2*np.pi) + theta, phi = np.meshgrid(theta, phi) + + y, y_jac, y_hess = sc.sph_harm_y_all(n_max, m_max, theta, phi, diff_n=2) + p, p_jac, p_hess = sc.sph_legendre_p_all(n_max, m_max, theta, diff_n=2) + + m = np.concatenate([np.arange(m_max + 1), np.arange(-m_max, 0)]) + m = np.expand_dims(m, axis=(0,)+tuple(range(2,theta.ndim+2))) + + assert_allclose(y, p * np.exp(1j * m * phi)) + + assert_allclose(y_jac[..., 0], p_jac * np.exp(1j * m * phi)) + assert_allclose(y_jac[..., 1], 1j * m * p * np.exp(1j * m * phi)) + + assert_allclose(y_hess[..., 0, 0], p_hess * np.exp(1j * m * phi)) + assert_allclose(y_hess[..., 0, 1], 1j * m * p_jac * np.exp(1j * m * phi)) + assert_allclose(y_hess[..., 1, 0], y_hess[..., 0, 1]) + assert_allclose(y_hess[..., 1, 1], -m * m * p * np.exp(1j * m * phi)) + + @pytest.mark.parametrize("n_max", [7, 10, 50]) + @pytest.mark.parametrize("m_max", [1, 4, 5, 9, 14]) + def test_all(self, n_max, m_max): + theta = np.linspace(0, np.pi) + phi = np.linspace(0, 2 * np.pi) + + n = np.arange(n_max + 1) + n = np.expand_dims(n, axis=tuple(range(1,theta.ndim+2))) + + m = np.concatenate([np.arange(m_max + 1), np.arange(-m_max, 0)]) + m = np.expand_dims(m, axis=(0,)+tuple(range(2,theta.ndim+2))) + + y_actual = sc.sph_harm_y_all(n_max, m_max, theta, phi) + y_desired = sc.sph_harm_y(n, m, theta, phi) + + np.testing.assert_allclose(y_actual, y_desired, rtol=1e-05) + +def test_first_harmonics(): + # Test against explicit representations of the first four + # spherical harmonics which use `theta` as the azimuthal angle, + # `phi` as the polar angle, and include the Condon-Shortley + # phase. + + # sph_harm is deprecated and is implemented as a shim around sph_harm_y. + # This test is maintained to verify the correctness of the shim. + + # Notation is Ymn + def Y00(theta, phi): + return 0.5*np.sqrt(1/np.pi) + + def Yn11(theta, phi): + return 0.5*np.sqrt(3/(2*np.pi))*np.exp(-1j*theta)*np.sin(phi) + + def Y01(theta, phi): + return 0.5*np.sqrt(3/np.pi)*np.cos(phi) + + def Y11(theta, phi): + return -0.5*np.sqrt(3/(2*np.pi))*np.exp(1j*theta)*np.sin(phi) + + harms = [Y00, Yn11, Y01, Y11] + m = [0, -1, 0, 1] + n = [0, 1, 1, 1] + + theta = np.linspace(0, 2*np.pi) + phi = np.linspace(0, np.pi) + theta, phi = np.meshgrid(theta, phi) + + for harm, m, n in zip(harms, m, n): + with suppress_warnings() as sup: + sup.filter(category=DeprecationWarning) + assert_allclose(sc.sph_harm(m, n, theta, phi), + harm(theta, phi), + rtol=1e-15, atol=1e-15, + err_msg=f"Y^{m}_{n} incorrect") diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_spherical_bessel.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_spherical_bessel.py new file mode 100644 index 0000000000000000000000000000000000000000..3a0173152c25eca57670b7ee6eadb57cd965266c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_spherical_bessel.py @@ -0,0 +1,400 @@ +# +# Tests of spherical Bessel functions. +# +import numpy as np +from numpy.testing import (assert_almost_equal, assert_allclose, + assert_array_almost_equal, suppress_warnings) +import pytest +from numpy import sin, cos, sinh, cosh, exp, inf, nan, r_, pi + +from scipy.special import spherical_jn, spherical_yn, spherical_in, spherical_kn +from scipy.integrate import quad + + +class TestSphericalJn: + def test_spherical_jn_exact(self): + # https://dlmf.nist.gov/10.49.E3 + # Note: exact expression is numerically stable only for small + # n or z >> n. + x = np.array([0.12, 1.23, 12.34, 123.45, 1234.5]) + assert_allclose(spherical_jn(2, x), + (-1/x + 3/x**3)*sin(x) - 3/x**2*cos(x)) + + def test_spherical_jn_recurrence_complex(self): + # https://dlmf.nist.gov/10.51.E1 + n = np.array([1, 2, 3, 7, 12]) + x = 1.1 + 1.5j + assert_allclose(spherical_jn(n - 1, x) + spherical_jn(n + 1, x), + (2*n + 1)/x*spherical_jn(n, x)) + + def test_spherical_jn_recurrence_real(self): + # https://dlmf.nist.gov/10.51.E1 + n = np.array([1, 2, 3, 7, 12]) + x = 0.12 + assert_allclose(spherical_jn(n - 1, x) + spherical_jn(n + 1,x), + (2*n + 1)/x*spherical_jn(n, x)) + + def test_spherical_jn_inf_real(self): + # https://dlmf.nist.gov/10.52.E3 + n = 6 + x = np.array([-inf, inf]) + assert_allclose(spherical_jn(n, x), np.array([0, 0])) + + def test_spherical_jn_inf_complex(self): + # https://dlmf.nist.gov/10.52.E3 + n = 7 + x = np.array([-inf + 0j, inf + 0j, inf*(1+1j)]) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "invalid value encountered in multiply") + assert_allclose(spherical_jn(n, x), np.array([0, 0, inf*(1+1j)])) + + def test_spherical_jn_large_arg_1(self): + # https://github.com/scipy/scipy/issues/2165 + # Reference value computed using mpmath, via + # besselj(n + mpf(1)/2, z)*sqrt(pi/(2*z)) + assert_allclose(spherical_jn(2, 3350.507), -0.00029846226538040747) + + def test_spherical_jn_large_arg_2(self): + # https://github.com/scipy/scipy/issues/1641 + # Reference value computed using mpmath, via + # besselj(n + mpf(1)/2, z)*sqrt(pi/(2*z)) + assert_allclose(spherical_jn(2, 10000), 3.0590002633029811e-05) + + def test_spherical_jn_at_zero(self): + # https://dlmf.nist.gov/10.52.E1 + # But note that n = 0 is a special case: j0 = sin(x)/x -> 1 + n = np.array([0, 1, 2, 5, 10, 100]) + x = 0 + assert_allclose(spherical_jn(n, x), np.array([1, 0, 0, 0, 0, 0])) + + +class TestSphericalYn: + def test_spherical_yn_exact(self): + # https://dlmf.nist.gov/10.49.E5 + # Note: exact expression is numerically stable only for small + # n or z >> n. + x = np.array([0.12, 1.23, 12.34, 123.45, 1234.5]) + assert_allclose(spherical_yn(2, x), + (1/x - 3/x**3)*cos(x) - 3/x**2*sin(x)) + + def test_spherical_yn_recurrence_real(self): + # https://dlmf.nist.gov/10.51.E1 + n = np.array([1, 2, 3, 7, 12]) + x = 0.12 + assert_allclose(spherical_yn(n - 1, x) + spherical_yn(n + 1,x), + (2*n + 1)/x*spherical_yn(n, x)) + + def test_spherical_yn_recurrence_complex(self): + # https://dlmf.nist.gov/10.51.E1 + n = np.array([1, 2, 3, 7, 12]) + x = 1.1 + 1.5j + assert_allclose(spherical_yn(n - 1, x) + spherical_yn(n + 1, x), + (2*n + 1)/x*spherical_yn(n, x)) + + def test_spherical_yn_inf_real(self): + # https://dlmf.nist.gov/10.52.E3 + n = 6 + x = np.array([-inf, inf]) + assert_allclose(spherical_yn(n, x), np.array([0, 0])) + + def test_spherical_yn_inf_complex(self): + # https://dlmf.nist.gov/10.52.E3 + n = 7 + x = np.array([-inf + 0j, inf + 0j, inf*(1+1j)]) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "invalid value encountered in multiply") + assert_allclose(spherical_yn(n, x), np.array([0, 0, inf*(1+1j)])) + + def test_spherical_yn_at_zero(self): + # https://dlmf.nist.gov/10.52.E2 + n = np.array([0, 1, 2, 5, 10, 100]) + x = 0 + assert_allclose(spherical_yn(n, x), np.full(n.shape, -inf)) + + def test_spherical_yn_at_zero_complex(self): + # Consistently with numpy: + # >>> -np.cos(0)/0 + # -inf + # >>> -np.cos(0+0j)/(0+0j) + # (-inf + nan*j) + n = np.array([0, 1, 2, 5, 10, 100]) + x = 0 + 0j + assert_allclose(spherical_yn(n, x), np.full(n.shape, nan)) + + +class TestSphericalJnYnCrossProduct: + def test_spherical_jn_yn_cross_product_1(self): + # https://dlmf.nist.gov/10.50.E3 + n = np.array([1, 5, 8]) + x = np.array([0.1, 1, 10]) + left = (spherical_jn(n + 1, x) * spherical_yn(n, x) - + spherical_jn(n, x) * spherical_yn(n + 1, x)) + right = 1/x**2 + assert_allclose(left, right) + + def test_spherical_jn_yn_cross_product_2(self): + # https://dlmf.nist.gov/10.50.E3 + n = np.array([1, 5, 8]) + x = np.array([0.1, 1, 10]) + left = (spherical_jn(n + 2, x) * spherical_yn(n, x) - + spherical_jn(n, x) * spherical_yn(n + 2, x)) + right = (2*n + 3)/x**3 + assert_allclose(left, right) + + +class TestSphericalIn: + def test_spherical_in_exact(self): + # https://dlmf.nist.gov/10.49.E9 + x = np.array([0.12, 1.23, 12.34, 123.45]) + assert_allclose(spherical_in(2, x), + (1/x + 3/x**3)*sinh(x) - 3/x**2*cosh(x)) + + def test_spherical_in_recurrence_real(self): + # https://dlmf.nist.gov/10.51.E4 + n = np.array([1, 2, 3, 7, 12]) + x = 0.12 + assert_allclose(spherical_in(n - 1, x) - spherical_in(n + 1,x), + (2*n + 1)/x*spherical_in(n, x)) + + def test_spherical_in_recurrence_complex(self): + # https://dlmf.nist.gov/10.51.E1 + n = np.array([1, 2, 3, 7, 12]) + x = 1.1 + 1.5j + assert_allclose(spherical_in(n - 1, x) - spherical_in(n + 1,x), + (2*n + 1)/x*spherical_in(n, x)) + + def test_spherical_in_inf_real(self): + # https://dlmf.nist.gov/10.52.E3 + n = 5 + x = np.array([-inf, inf]) + assert_allclose(spherical_in(n, x), np.array([-inf, inf])) + + def test_spherical_in_inf_complex(self): + # https://dlmf.nist.gov/10.52.E5 + # Ideally, i1n(n, 1j*inf) = 0 and i1n(n, (1+1j)*inf) = (1+1j)*inf, but + # this appears impossible to achieve because C99 regards any complex + # value with at least one infinite part as a complex infinity, so + # 1j*inf cannot be distinguished from (1+1j)*inf. Therefore, nan is + # the correct return value. + n = 7 + x = np.array([-inf + 0j, inf + 0j, inf*(1+1j)]) + assert_allclose(spherical_in(n, x), np.array([-inf, inf, nan])) + + def test_spherical_in_at_zero(self): + # https://dlmf.nist.gov/10.52.E1 + # But note that n = 0 is a special case: i0 = sinh(x)/x -> 1 + n = np.array([0, 1, 2, 5, 10, 100]) + x = 0 + assert_allclose(spherical_in(n, x), np.array([1, 0, 0, 0, 0, 0])) + + +class TestSphericalKn: + def test_spherical_kn_exact(self): + # https://dlmf.nist.gov/10.49.E13 + x = np.array([0.12, 1.23, 12.34, 123.45]) + assert_allclose(spherical_kn(2, x), + pi/2*exp(-x)*(1/x + 3/x**2 + 3/x**3)) + + def test_spherical_kn_recurrence_real(self): + # https://dlmf.nist.gov/10.51.E4 + n = np.array([1, 2, 3, 7, 12]) + x = 0.12 + assert_allclose( + (-1)**(n - 1)*spherical_kn(n - 1, x) - (-1)**(n + 1)*spherical_kn(n + 1,x), + (-1)**n*(2*n + 1)/x*spherical_kn(n, x) + ) + + def test_spherical_kn_recurrence_complex(self): + # https://dlmf.nist.gov/10.51.E4 + n = np.array([1, 2, 3, 7, 12]) + x = 1.1 + 1.5j + assert_allclose( + (-1)**(n - 1)*spherical_kn(n - 1, x) - (-1)**(n + 1)*spherical_kn(n + 1,x), + (-1)**n*(2*n + 1)/x*spherical_kn(n, x) + ) + + def test_spherical_kn_inf_real(self): + # https://dlmf.nist.gov/10.52.E6 + n = 5 + x = np.array([-inf, inf]) + assert_allclose(spherical_kn(n, x), np.array([-inf, 0])) + + def test_spherical_kn_inf_complex(self): + # https://dlmf.nist.gov/10.52.E6 + # The behavior at complex infinity depends on the sign of the real + # part: if Re(z) >= 0, then the limit is 0; if Re(z) < 0, then it's + # z*inf. This distinction cannot be captured, so we return nan. + n = 7 + x = np.array([-inf + 0j, inf + 0j, inf*(1+1j)]) + assert_allclose(spherical_kn(n, x), np.array([-inf, 0, nan])) + + def test_spherical_kn_at_zero(self): + # https://dlmf.nist.gov/10.52.E2 + n = np.array([0, 1, 2, 5, 10, 100]) + x = 0 + assert_allclose(spherical_kn(n, x), np.full(n.shape, inf)) + + def test_spherical_kn_at_zero_complex(self): + # https://dlmf.nist.gov/10.52.E2 + n = np.array([0, 1, 2, 5, 10, 100]) + x = 0 + 0j + assert_allclose(spherical_kn(n, x), np.full(n.shape, nan)) + + +class SphericalDerivativesTestCase: + def fundamental_theorem(self, n, a, b): + integral, tolerance = quad(lambda z: self.df(n, z), a, b) + assert_allclose(integral, + self.f(n, b) - self.f(n, a), + atol=tolerance) + + @pytest.mark.slow + def test_fundamental_theorem_0(self): + self.fundamental_theorem(0, 3.0, 15.0) + + @pytest.mark.slow + def test_fundamental_theorem_7(self): + self.fundamental_theorem(7, 0.5, 1.2) + + +class TestSphericalJnDerivatives(SphericalDerivativesTestCase): + def f(self, n, z): + return spherical_jn(n, z) + + def df(self, n, z): + return spherical_jn(n, z, derivative=True) + + def test_spherical_jn_d_zero(self): + n = np.array([0, 1, 2, 3, 7, 15]) + assert_allclose(spherical_jn(n, 0, derivative=True), + np.array([0, 1/3, 0, 0, 0, 0])) + + +class TestSphericalYnDerivatives(SphericalDerivativesTestCase): + def f(self, n, z): + return spherical_yn(n, z) + + def df(self, n, z): + return spherical_yn(n, z, derivative=True) + + +class TestSphericalInDerivatives(SphericalDerivativesTestCase): + def f(self, n, z): + return spherical_in(n, z) + + def df(self, n, z): + return spherical_in(n, z, derivative=True) + + def test_spherical_in_d_zero(self): + n = np.array([0, 1, 2, 3, 7, 15]) + spherical_in(n, 0, derivative=False) + assert_allclose(spherical_in(n, 0, derivative=True), + np.array([0, 1/3, 0, 0, 0, 0])) + + +class TestSphericalKnDerivatives(SphericalDerivativesTestCase): + def f(self, n, z): + return spherical_kn(n, z) + + def df(self, n, z): + return spherical_kn(n, z, derivative=True) + + +class TestSphericalOld: + # These are tests from the TestSpherical class of test_basic.py, + # rewritten to use spherical_* instead of sph_* but otherwise unchanged. + + def test_sph_in(self): + # This test reproduces test_basic.TestSpherical.test_sph_in. + i1n = np.empty((2,2)) + x = 0.2 + + i1n[0][0] = spherical_in(0, x) + i1n[0][1] = spherical_in(1, x) + i1n[1][0] = spherical_in(0, x, derivative=True) + i1n[1][1] = spherical_in(1, x, derivative=True) + + inp0 = (i1n[0][1]) + inp1 = (i1n[0][0] - 2.0/0.2 * i1n[0][1]) + assert_array_almost_equal(i1n[0],np.array([1.0066800127054699381, + 0.066933714568029540839]),12) + assert_array_almost_equal(i1n[1],[inp0,inp1],12) + + def test_sph_in_kn_order0(self): + x = 1. + sph_i0 = np.empty((2,)) + sph_i0[0] = spherical_in(0, x) + sph_i0[1] = spherical_in(0, x, derivative=True) + sph_i0_expected = np.array([np.sinh(x)/x, + np.cosh(x)/x-np.sinh(x)/x**2]) + assert_array_almost_equal(r_[sph_i0], sph_i0_expected) + + sph_k0 = np.empty((2,)) + sph_k0[0] = spherical_kn(0, x) + sph_k0[1] = spherical_kn(0, x, derivative=True) + sph_k0_expected = np.array([0.5*pi*exp(-x)/x, + -0.5*pi*exp(-x)*(1/x+1/x**2)]) + assert_array_almost_equal(r_[sph_k0], sph_k0_expected) + + def test_sph_jn(self): + s1 = np.empty((2,3)) + x = 0.2 + + s1[0][0] = spherical_jn(0, x) + s1[0][1] = spherical_jn(1, x) + s1[0][2] = spherical_jn(2, x) + s1[1][0] = spherical_jn(0, x, derivative=True) + s1[1][1] = spherical_jn(1, x, derivative=True) + s1[1][2] = spherical_jn(2, x, derivative=True) + + s10 = -s1[0][1] + s11 = s1[0][0]-2.0/0.2*s1[0][1] + s12 = s1[0][1]-3.0/0.2*s1[0][2] + assert_array_almost_equal(s1[0],[0.99334665397530607731, + 0.066400380670322230863, + 0.0026590560795273856680],12) + assert_array_almost_equal(s1[1],[s10,s11,s12],12) + + def test_sph_kn(self): + kn = np.empty((2,3)) + x = 0.2 + + kn[0][0] = spherical_kn(0, x) + kn[0][1] = spherical_kn(1, x) + kn[0][2] = spherical_kn(2, x) + kn[1][0] = spherical_kn(0, x, derivative=True) + kn[1][1] = spherical_kn(1, x, derivative=True) + kn[1][2] = spherical_kn(2, x, derivative=True) + + kn0 = -kn[0][1] + kn1 = -kn[0][0]-2.0/0.2*kn[0][1] + kn2 = -kn[0][1]-3.0/0.2*kn[0][2] + assert_array_almost_equal(kn[0],[6.4302962978445670140, + 38.581777787067402086, + 585.15696310385559829],12) + assert_array_almost_equal(kn[1],[kn0,kn1,kn2],9) + + def test_sph_yn(self): + sy1 = spherical_yn(2, 0.2) + sy2 = spherical_yn(0, 0.2) + assert_almost_equal(sy1,-377.52483,5) # previous values in the system + assert_almost_equal(sy2,-4.9003329,5) + sphpy = (spherical_yn(0, 0.2) - 2*spherical_yn(2, 0.2))/3 + sy3 = spherical_yn(1, 0.2, derivative=True) + # compare correct derivative val. (correct =-system val). + assert_almost_equal(sy3,sphpy,4) + + +@pytest.mark.parametrize('derivative', [False, True]) +@pytest.mark.parametrize('fun', [spherical_jn, spherical_in, + spherical_yn, spherical_kn]) +def test_negative_real_gh14582(derivative, fun): + # gh-14582 reported that the spherical Bessel functions did not work + # with negative real argument `z`. Check that this is resolved. + rng = np.random.default_rng(3598435982345987234) + size = 25 + n = rng.integers(0, 10, size=size) + z = rng.standard_normal(size=size) + res = fun(n, z, derivative=derivative) + ref = fun(n, z+0j, derivative=derivative) + np.testing.assert_allclose(res, ref.real) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_support_alternative_backends.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_support_alternative_backends.py new file mode 100644 index 0000000000000000000000000000000000000000..48cfe5bfb64c15486f4b0b2d911846b96e29f3df --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_support_alternative_backends.py @@ -0,0 +1,113 @@ +import pytest + +from scipy.special._support_alternative_backends import (get_array_special_func, + array_special_func_map) +from scipy.conftest import array_api_compatible +from scipy import special +from scipy._lib._array_api_no_0d import xp_assert_close +from scipy._lib._array_api import is_jax, is_torch, SCIPY_DEVICE +from scipy._lib.array_api_compat import numpy as np + +try: + import array_api_strict + HAVE_ARRAY_API_STRICT = True +except ImportError: + HAVE_ARRAY_API_STRICT = False + + +@pytest.mark.skipif(not HAVE_ARRAY_API_STRICT, + reason="`array_api_strict` not installed") +def test_dispatch_to_unrecognize_library(): + xp = array_api_strict + f = get_array_special_func('ndtr', xp=xp, n_array_args=1) + x = [1, 2, 3] + res = f(xp.asarray(x)) + ref = xp.asarray(special.ndtr(np.asarray(x))) + xp_assert_close(res, ref, xp=xp) + + +@pytest.mark.parametrize('dtype', ['float32', 'float64', 'int64']) +@pytest.mark.skipif(not HAVE_ARRAY_API_STRICT, + reason="`array_api_strict` not installed") +def test_rel_entr_generic(dtype): + xp = array_api_strict + f = get_array_special_func('rel_entr', xp=xp, n_array_args=2) + dtype_np = getattr(np, dtype) + dtype_xp = getattr(xp, dtype) + x, y = [-1, 0, 0, 1], [1, 0, 2, 3] + + x_xp, y_xp = xp.asarray(x, dtype=dtype_xp), xp.asarray(y, dtype=dtype_xp) + res = f(x_xp, y_xp) + + x_np, y_np = np.asarray(x, dtype=dtype_np), np.asarray(y, dtype=dtype_np) + ref = special.rel_entr(x_np[-1], y_np[-1]) + ref = np.asarray([np.inf, 0, 0, ref], dtype=ref.dtype) + + xp_assert_close(res, xp.asarray(ref), xp=xp) + + +@pytest.mark.fail_slow(5) +@array_api_compatible +# @pytest.mark.skip_xp_backends('numpy', reason='skip while debugging') +# @pytest.mark.usefixtures("skip_xp_backends") +# `reversed` is for developer convenience: test new function first = less waiting +@pytest.mark.parametrize('f_name_n_args', reversed(array_special_func_map.items())) +@pytest.mark.parametrize('dtype', ['float32', 'float64']) +@pytest.mark.parametrize('shapes', [[(0,)]*4, [tuple()]*4, [(10,)]*4, + [(10,), (11, 1), (12, 1, 1), (13, 1, 1, 1)]]) +def test_support_alternative_backends(xp, f_name_n_args, dtype, shapes): + f_name, n_args = f_name_n_args + + if (SCIPY_DEVICE != 'cpu' + and is_torch(xp) + and f_name in {'stdtr', 'betaincc', 'betainc'} + ): + pytest.skip(f"`{f_name}` does not have an array-agnostic implementation " + f"and cannot delegate to PyTorch.") + + shapes = shapes[:n_args] + f = getattr(special, f_name) + + dtype_np = getattr(np, dtype) + dtype_xp = getattr(xp, dtype) + + # # To test the robustness of the alternative backend's implementation, + # # use Hypothesis to generate arguments + # from hypothesis import given, strategies, reproduce_failure, assume + # import hypothesis.extra.numpy as npst + # @given(data=strategies.data()) + # mbs = npst.mutually_broadcastable_shapes(num_shapes=n_args) + # shapes, final_shape = data.draw(mbs) + # elements = dict(allow_subnormal=False) # consider min_value, max_value + # args_np = [np.asarray(data.draw(npst.arrays(dtype_np, shape, elements=elements)), + # dtype=dtype_np) + # for shape in shapes] + + # For CI, be a little more forgiving; just generate normally distributed arguments + rng = np.random.default_rng(984254252920492019) + args_np = [rng.standard_normal(size=shape, dtype=dtype_np) for shape in shapes] + + if (is_jax(xp) and f_name == 'gammaincc' # google/jax#20699 + or f_name == 'chdtrc'): # gh-20972 + args_np[0] = np.abs(args_np[0]) + args_np[1] = np.abs(args_np[1]) + + args_xp = [xp.asarray(arg[()], dtype=dtype_xp) for arg in args_np] + + res = f(*args_xp) + ref = xp.asarray(f(*args_np), dtype=dtype_xp) + + eps = np.finfo(dtype_np).eps + xp_assert_close(res, ref, atol=10*eps) + + +@array_api_compatible +def test_chdtr_gh21311(xp): + # the edge case behavior of generic chdtr was not right; see gh-21311 + # be sure to test at least these cases + # should add `np.nan` into the mix when gh-21317 is resolved + x = np.asarray([-np.inf, -1., 0., 1., np.inf]) + v = x.reshape(-1, 1) + ref = special.chdtr(v, x) + res = special.chdtr(xp.asarray(v), xp.asarray(x)) + xp_assert_close(res, xp.asarray(ref)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_trig.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_trig.py new file mode 100644 index 0000000000000000000000000000000000000000..578dfbd5e95e6c44b1828716a74c93d645efcb1e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_trig.py @@ -0,0 +1,72 @@ +import pytest +import numpy as np +from numpy.testing import assert_equal, assert_allclose, suppress_warnings + +from scipy.special._ufuncs import _sinpi as sinpi +from scipy.special._ufuncs import _cospi as cospi + + +def test_integer_real_part(): + x = np.arange(-100, 101) + y = np.hstack((-np.linspace(310, -30, 10), np.linspace(-30, 310, 10))) + x, y = np.meshgrid(x, y) + z = x + 1j*y + # In the following we should be *exactly* right + res = sinpi(z) + assert_equal(res.real, 0.0) + res = cospi(z) + assert_equal(res.imag, 0.0) + + +def test_half_integer_real_part(): + x = np.arange(-100, 101) + 0.5 + y = np.hstack((-np.linspace(310, -30, 10), np.linspace(-30, 310, 10))) + x, y = np.meshgrid(x, y) + z = x + 1j*y + # In the following we should be *exactly* right + res = sinpi(z) + assert_equal(res.imag, 0.0) + res = cospi(z) + assert_equal(res.real, 0.0) + + +@pytest.mark.skip("Temporary skip while gh-19526 is being resolved") +def test_intermediate_overlow(): + # Make sure we avoid overflow in situations where cosh/sinh would + # overflow but the product with sin/cos would not + sinpi_pts = [complex(1 + 1e-14, 227), + complex(1e-35, 250), + complex(1e-301, 445)] + # Data generated with mpmath + sinpi_std = [complex(-8.113438309924894e+295, -np.inf), + complex(1.9507801934611995e+306, np.inf), + complex(2.205958493464539e+306, np.inf)] + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "invalid value encountered in multiply") + for p, std in zip(sinpi_pts, sinpi_std): + res = sinpi(p) + assert_allclose(res.real, std.real) + assert_allclose(res.imag, std.imag) + + # Test for cosine, less interesting because cos(0) = 1. + p = complex(0.5 + 1e-14, 227) + std = complex(-8.113438309924894e+295, -np.inf) + with suppress_warnings() as sup: + sup.filter(RuntimeWarning, "invalid value encountered in multiply") + res = cospi(p) + assert_allclose(res.real, std.real) + assert_allclose(res.imag, std.imag) + + +def test_zero_sign(): + y = sinpi(-0.0) + assert y == 0.0 + assert np.signbit(y) + + y = sinpi(0.0) + assert y == 0.0 + assert not np.signbit(y) + + y = cospi(0.5) + assert y == 0.0 + assert not np.signbit(y) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ufunc_signatures.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ufunc_signatures.py new file mode 100644 index 0000000000000000000000000000000000000000..6bc3ffae15ab4620c4e752df166721825cb7449c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_ufunc_signatures.py @@ -0,0 +1,46 @@ +"""Test that all ufuncs have float32-preserving signatures. + +This was once guaranteed through the code generation script for +generating ufuncs, `scipy/special/_generate_pyx.py`. Starting with +gh-20260, SciPy developers have begun moving to generate ufuncs +through direct use of the NumPy C API (through C++). Existence of +float32 preserving signatures must now be tested since it is no +longer guaranteed. +""" + +import numpy as np +import pytest +import scipy.special._ufuncs +import scipy.special._gufuncs + +_ufuncs = [] +for funcname in dir(scipy.special._ufuncs): + _ufuncs.append(getattr(scipy.special._ufuncs, funcname)) +for funcname in dir(scipy.special._gufuncs): + _ufuncs.append(getattr(scipy.special._gufuncs, funcname)) + +# Not all module members are actually ufuncs +_ufuncs = [func for func in _ufuncs if isinstance(func, np.ufunc)] + +@pytest.mark.parametrize("ufunc", _ufuncs) +def test_ufunc_signatures(ufunc): + + # From _generate_pyx.py + # "Don't add float32 versions of ufuncs with integer arguments, as this + # can lead to incorrect dtype selection if the integer arguments are + # arrays, but float arguments are scalars. + # For instance sph_harm(0,[0],0,0).dtype == complex64 + # This may be a NumPy bug, but we need to work around it. + # cf. gh-4895, https://github.com/numpy/numpy/issues/5895" + types = set(sig for sig in ufunc.types + if not ("l" in sig or "i" in sig or "q" in sig or "p" in sig)) + + # Generate the full expanded set of signatures which should exist. There + # should be matching float and double versions of any existing signature. + expanded_types = set() + for sig in types: + expanded_types.update( + [sig.replace("d", "f").replace("D", "F"), + sig.replace("f", "d").replace("F", "D")] + ) + assert types == expanded_types diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_wright_bessel.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_wright_bessel.py new file mode 100644 index 0000000000000000000000000000000000000000..ef30163f55c90850fe32ca600eb47547e4fa5e03 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_wright_bessel.py @@ -0,0 +1,205 @@ +# Reference MPMATH implementation: +# +# import mpmath +# from mpmath import nsum +# +# def Wright_Series_MPMATH(a, b, z, dps=50, method='r+s+e', steps=[1000]): +# """Compute Wright' generalized Bessel function as Series. +# +# This uses mpmath for arbitrary precision. +# """ +# with mpmath.workdps(dps): +# res = nsum(lambda k: z**k/mpmath.fac(k) * mpmath.rgamma(a*k+b), +# [0, mpmath.inf], +# tol=dps, method=method, steps=steps +# ) +# +# return res + +from itertools import product + +import pytest +import numpy as np +from numpy.testing import assert_equal, assert_allclose + +import scipy.special as sc +from scipy.special import log_wright_bessel, loggamma, rgamma, wright_bessel + + +@pytest.mark.parametrize('a', [0, 1e-6, 0.1, 0.5, 1, 10]) +@pytest.mark.parametrize('b', [0, 1e-6, 0.1, 0.5, 1, 10]) +def test_wright_bessel_zero(a, b): + """Test at x = 0.""" + assert_equal(wright_bessel(a, b, 0.), rgamma(b)) + assert_allclose(log_wright_bessel(a, b, 0.), -loggamma(b)) + + +@pytest.mark.parametrize('b', [0, 1e-6, 0.1, 0.5, 1, 10]) +@pytest.mark.parametrize('x', [0, 1e-6, 0.1, 0.5, 1]) +def test_wright_bessel_iv(b, x): + """Test relation of wright_bessel and modified bessel function iv. + + iv(z) = (1/2*z)**v * Phi(1, v+1; 1/4*z**2). + See https://dlmf.nist.gov/10.46.E2 + """ + if x != 0: + v = b - 1 + wb = wright_bessel(1, v + 1, x**2 / 4.) + # Note: iv(v, x) has precision of less than 1e-12 for some cases + # e.g v=1-1e-6 and x=1e-06) + assert_allclose(np.power(x / 2., v) * wb, + sc.iv(v, x), + rtol=1e-11, atol=1e-11) + + +@pytest.mark.parametrize('a', [0, 1e-6, 0.1, 0.5, 1, 10]) +@pytest.mark.parametrize('b', [1, 1 + 1e-3, 2, 5, 10]) +@pytest.mark.parametrize('x', [0, 1e-6, 0.1, 0.5, 1, 5, 10, 100]) +def test_wright_functional(a, b, x): + """Test functional relation of wright_bessel. + + Phi(a, b-1, z) = a*z*Phi(a, b+a, z) + (b-1)*Phi(a, b, z) + + Note that d/dx Phi(a, b, x) = Phi(a, b-1, x) + See Eq. (22) of + B. Stankovic, On the Function of E. M. Wright, + Publ. de l' Institut Mathematique, Beograd, + Nouvelle S`er. 10 (1970), 113-124. + """ + assert_allclose(wright_bessel(a, b - 1, x), + a * x * wright_bessel(a, b + a, x) + + (b - 1) * wright_bessel(a, b, x), + rtol=1e-8, atol=1e-8) + + +# grid of rows [a, b, x, value, accuracy] that do not reach 1e-11 accuracy +# see output of: +# cd scipy/scipy/_precompute +# python wright_bessel_data.py +grid_a_b_x_value_acc = np.array([ + [0.1, 100.0, 709.7827128933841, 8.026353022981087e+34, 2e-8], + [0.5, 10.0, 709.7827128933841, 2.680788404494657e+48, 9e-8], + [0.5, 10.0, 1000.0, 2.005901980702872e+64, 1e-8], + [0.5, 100.0, 1000.0, 3.4112367580445246e-117, 6e-8], + [1.0, 20.0, 100000.0, 1.7717158630699857e+225, 3e-11], + [1.0, 100.0, 100000.0, 1.0269334596230763e+22, np.nan], + [1.0000000000000222, 20.0, 100000.0, 1.7717158630001672e+225, 3e-11], + [1.0000000000000222, 100.0, 100000.0, 1.0269334595866202e+22, np.nan], + [1.5, 0.0, 500.0, 15648961196.432373, 3e-11], + [1.5, 2.220446049250313e-14, 500.0, 15648961196.431465, 3e-11], + [1.5, 1e-10, 500.0, 15648961192.344728, 3e-11], + [1.5, 1e-05, 500.0, 15648552437.334162, 3e-11], + [1.5, 0.1, 500.0, 12049870581.10317, 2e-11], + [1.5, 20.0, 100000.0, 7.81930438331405e+43, 3e-9], + [1.5, 100.0, 100000.0, 9.653370857459075e-130, np.nan], + ]) + + +@pytest.mark.xfail +@pytest.mark.parametrize( + 'a, b, x, phi', + grid_a_b_x_value_acc[:, :4].tolist()) +def test_wright_data_grid_failures(a, b, x, phi): + """Test cases of test_data that do not reach relative accuracy of 1e-11""" + assert_allclose(wright_bessel(a, b, x), phi, rtol=1e-11) + + +@pytest.mark.parametrize( + 'a, b, x, phi, accuracy', + grid_a_b_x_value_acc.tolist()) +def test_wright_data_grid_less_accurate(a, b, x, phi, accuracy): + """Test cases of test_data that do not reach relative accuracy of 1e-11 + + Here we test for reduced accuracy or even nan. + """ + if np.isnan(accuracy): + assert np.isnan(wright_bessel(a, b, x)) + else: + assert_allclose(wright_bessel(a, b, x), phi, rtol=accuracy) + + +@pytest.mark.parametrize( + 'a, b, x', + list( + product([0, 0.1, 0.5, 1.5, 5, 10], [1, 2], [1e-3, 1, 1.5, 5, 10]) + ) +) +def test_log_wright_bessel_same_as_wright_bessel(a, b, x): + """Test that log_wright_bessel equals log of wright_bessel.""" + assert_allclose( + log_wright_bessel(a, b, x), + np.log(wright_bessel(a, b, x)), + rtol=1e-8, + ) + + +# Computed with, see also mp_wright_bessel from wright_bessel_data.py: +# +# from functools import lru_cache +# import mpmath as mp +# +# @lru_cache(maxsize=1_000_000) +# def rgamma_cached(x, dps): +# with mp.workdps(dps): +# return mp.rgamma(x) +# +# def mp_log_wright_bessel(a, b, x, dps=100, maxterms=10_000, method="d"): +# """Compute log of Wright's generalized Bessel function as Series with mpmath.""" +# with mp.workdps(dps): +# a, b, x = mp.mpf(a), mp.mpf(b), mp.mpf(x) +# res = mp.nsum(lambda k: x**k / mp.fac(k) +# * rgamma_cached(a * k + b, dps=dps), +# [0, mp.inf], +# tol=dps, method=method, steps=[maxterms] +# ) +# return mp.log(res) +# +# Sometimes, one needs to set maxterms as high as 1_00_000 to get accurate results for +# phi. +# At the end of the day, we can only hope that results are correct for very large x, +# e.g. by the asymptotic series, as there is no way to produce those in "exact" +# arithmetic. +# Note: accuracy = np.nan means log_wright_bessel returns nan. +@pytest.mark.parametrize( + 'a, b, x, phi, accuracy', + [ + (0, 0, 0, -np.inf, 1e-11), + (0, 0, 1, -np.inf, 1e-11), + (0, 1, 1.23, 1.23, 1e-11), + (0, 1, 1e50, 1e50, 1e-11), + (1e-5, 0, 700, 695.0421608273609, 1e-11), + (1e-5, 0, 1e3, 995.40052566540066, 1e-11), + (1e-5, 100, 1e3, 640.8197935670078, 1e-11), + (1e-3, 0, 1e4, 9987.2229532297262, 1e-11), + (1e-3, 0, 1e5, 99641.920687169507, 1e-11), + (1e-3, 0, 1e6, 994118.55560054416, 1e-11), # maxterms=1_000_000 + (1e-3, 10, 1e5, 99595.47710802537, 1e-11), + (1e-3, 50, 1e5, 99401.240922855647, 1e-3), + (1e-3, 100, 1e5, 99143.465191656527, np.nan), + (0.5, 0, 1e5, 4074.1112442197941, 1e-11), + (0.5, 0, 1e7, 87724.552120038896, 1e-11), + (0.5, 100, 1e5, 3350.3928746306163, np.nan), + (0.5, 100, 1e7, 86696.109975301719, 1e-11), + (1, 0, 1e5, 634.06765787997266, 1e-11), + (1, 0, 1e8, 20003.339639312035, 1e-11), + (1.5, 0, 1e5, 197.01777556071194, 1e-11), + (1.5, 0, 1e8, 3108.987414395706, 1e-11), + (1.5, 100, 1e8, 2354.8915946283275, np.nan), + (5, 0, 1e5, 9.8980480013203547, 1e-11), + (5, 0, 1e8, 33.642337258687465, 1e-11), + (5, 0, 1e12, 157.53704288117429, 1e-11), + (5, 100, 1e5, -359.13419630792148, 1e-11), + (5, 100, 1e12, -337.07722086995229, 1e-4), + (5, 100, 1e20, 2588.2471229986845, 2e-6), + (100, 0, 1e5, -347.62127990460517, 1e-11), + (100, 0, 1e20, -313.08250350969449, 1e-11), + (100, 100, 1e5, -359.1342053695754, 1e-11), + (100, 100, 1e20, -359.1342053695754, 1e-11), + ] +) +def test_log_wright_bessel(a, b, x, phi, accuracy): + """Test for log_wright_bessel, in particular for large x.""" + if np.isnan(accuracy): + assert np.isnan(log_wright_bessel(a, b, x)) + else: + assert_allclose(log_wright_bessel(a, b, x), phi, rtol=accuracy) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_wrightomega.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_wrightomega.py new file mode 100644 index 0000000000000000000000000000000000000000..c1a93ca007e42fea3a0dec634c51b37f03effa9e --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_wrightomega.py @@ -0,0 +1,117 @@ +import pytest +import numpy as np +from numpy.testing import assert_, assert_equal, assert_allclose + +import scipy.special as sc +from scipy.special._testutils import assert_func_equal + + +def test_wrightomega_nan(): + pts = [complex(np.nan, 0), + complex(0, np.nan), + complex(np.nan, np.nan), + complex(np.nan, 1), + complex(1, np.nan)] + for p in pts: + res = sc.wrightomega(p) + assert_(np.isnan(res.real)) + assert_(np.isnan(res.imag)) + + +def test_wrightomega_inf_branch(): + pts = [complex(-np.inf, np.pi/4), + complex(-np.inf, -np.pi/4), + complex(-np.inf, 3*np.pi/4), + complex(-np.inf, -3*np.pi/4)] + expected_results = [complex(0.0, 0.0), + complex(0.0, -0.0), + complex(-0.0, 0.0), + complex(-0.0, -0.0)] + for p, expected in zip(pts, expected_results): + res = sc.wrightomega(p) + # We can't use assert_equal(res, expected) because in older versions of + # numpy, assert_equal doesn't check the sign of the real and imaginary + # parts when comparing complex zeros. It does check the sign when the + # arguments are *real* scalars. + assert_equal(res.real, expected.real) + assert_equal(res.imag, expected.imag) + + +def test_wrightomega_inf(): + pts = [complex(np.inf, 10), + complex(-np.inf, 10), + complex(10, np.inf), + complex(10, -np.inf)] + for p in pts: + assert_equal(sc.wrightomega(p), p) + + +def test_wrightomega_singular(): + pts = [complex(-1.0, np.pi), + complex(-1.0, -np.pi)] + for p in pts: + res = sc.wrightomega(p) + assert_equal(res, -1.0) + assert_(np.signbit(res.imag) == np.bool_(False)) + + +@pytest.mark.parametrize('x, desired', [ + (-np.inf, 0), + (np.inf, np.inf), +]) +def test_wrightomega_real_infinities(x, desired): + assert sc.wrightomega(x) == desired + + +def test_wrightomega_real_nan(): + assert np.isnan(sc.wrightomega(np.nan)) + + +def test_wrightomega_real_series_crossover(): + desired_error = 2 * np.finfo(float).eps + crossover = 1e20 + x_before_crossover = np.nextafter(crossover, -np.inf) + x_after_crossover = np.nextafter(crossover, np.inf) + # Computed using Mpmath + desired_before_crossover = 99999999999999983569.948 + desired_after_crossover = 100000000000000016337.948 + assert_allclose( + sc.wrightomega(x_before_crossover), + desired_before_crossover, + atol=0, + rtol=desired_error, + ) + assert_allclose( + sc.wrightomega(x_after_crossover), + desired_after_crossover, + atol=0, + rtol=desired_error, + ) + + +def test_wrightomega_exp_approximation_crossover(): + desired_error = 2 * np.finfo(float).eps + crossover = -50 + x_before_crossover = np.nextafter(crossover, np.inf) + x_after_crossover = np.nextafter(crossover, -np.inf) + # Computed using Mpmath + desired_before_crossover = 1.9287498479639314876e-22 + desired_after_crossover = 1.9287498479639040784e-22 + assert_allclose( + sc.wrightomega(x_before_crossover), + desired_before_crossover, + atol=0, + rtol=desired_error, + ) + assert_allclose( + sc.wrightomega(x_after_crossover), + desired_after_crossover, + atol=0, + rtol=desired_error, + ) + + +def test_wrightomega_real_versus_complex(): + x = np.linspace(-500, 500, 1001) + results = sc.wrightomega(x + 0j).real + assert_func_equal(sc.wrightomega, results, x, atol=0, rtol=1e-14) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_xsf_cuda.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_xsf_cuda.py new file mode 100644 index 0000000000000000000000000000000000000000..dc39c2eeaf43196d4bf25964a0c2aefec2c8c0b4 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_xsf_cuda.py @@ -0,0 +1,114 @@ +import os +import pytest +import scipy.special as sc +import shutil +import tempfile + +from uuid import uuid4 + +from scipy.special._testutils import check_version +from scipy.special._testutils import MissingModule + +try: + import cupy # type: ignore +except (ImportError, AttributeError): + cupy = MissingModule('cupy') + + +def get_test_cases(): + cases_source = [ + (sc.beta, "cephes/beta.h", "out0 = xsf::cephes::beta(in0, in1)"), + (sc.binom, "binom.h", "out0 = xsf::binom(in0, in1)"), + (sc.digamma, "digamma.h", "xsf::digamma(in0)"), + (sc.expn, "cephes/expn.h", "out0 = xsf::cephes::expn(in0, in1)"), + (sc.hyp2f1, "hyp2f1.h", "out0 = xsf::hyp2f1(in0, in1, in2, in3)"), + (sc._ufuncs._lambertw, "lambertw.h", "out0 = xsf::lambertw(in0, in1, in2)"), + (sc.ellipkinc, "cephes/ellik.h", "out0 = xsf::cephes::ellik(in0, in1)"), + (sc.ellipeinc, "cephes/ellie.h", "out0 = xsf::cephes::ellie(in0, in1)"), + (sc.gdtrib, "cdflib.h", "out0 = xsf::gdtrib(in0, in1, in2)"), + (sc.sici, "sici.h", "xsf::sici(in0, &out0, &out1)"), + (sc.shichi, "sici.h", "xsf::shichi(in0, &out0, &out1)"), + ] + + cases = [] + for ufunc, header, routine in cases_source: + preamble = f"#include " + for signature in ufunc.types: + cases.append((signature, preamble, routine)) + return cases + + +dtype_map = { + "f": "float32", + "d": "float64", + "F": "complex64", + "D": "complex128", + "i": "int32", + "l": "int64", +} + + +def get_params(signature): + in_, out = signature.split("->") + in_params = [] + out_params = [] + for i, typecode in enumerate(in_): + in_params.append(f"{dtype_map[typecode]} in{i}") + for i, typecode in enumerate(out): + out_params.append(f"{dtype_map[typecode]} out{i}") + in_params = ", ".join(in_params) + out_params = ", ".join(out_params) + return in_params, out_params + + +def get_sample_input(signature, xp): + dtype_map = { + "f": xp.float32, + "d": xp.float64, + "F": xp.complex64, + "D": xp.complex128, + "i": xp.int32, + "l": xp.int64, + } + + in_, _ = signature.split("->") + args = [] + for typecode in in_: + args.append(xp.zeros(2, dtype=dtype_map[typecode])) + return args + + +@pytest.fixture(scope="module", autouse=True) +def manage_cupy_cache(): + # Temporarily change cupy kernel cache location so kernel cache will not be polluted + # by these tests. Remove temporary cache in teardown. + temp_cache_dir = tempfile.mkdtemp() + original_cache_dir = os.environ.get('CUPY_CACHE_DIR', None) + os.environ['CUPY_CACHE_DIR'] = temp_cache_dir + + yield + + if original_cache_dir is not None: + os.environ['CUPY_CACHE_DIR'] = original_cache_dir + else: + del os.environ['CUPY_CACHE_DIR'] + shutil.rmtree(temp_cache_dir) + + +@check_version(cupy, "13.0.0") +@pytest.mark.parametrize("signature,preamble,routine", get_test_cases()) +@pytest.mark.xslow +def test_compiles_in_cupy(signature, preamble, routine, manage_cupy_cache): + name = f"x{uuid4().hex}" + in_params, out_params = get_params(signature) + + func = cupy.ElementwiseKernel( + in_params, + out_params, + routine, + name, + preamble=preamble, + options=(f"--include-path={sc._get_include()}", "-std=c++17") + ) + + _ = func(*get_sample_input(signature, cupy)) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_zeta.py b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_zeta.py new file mode 100644 index 0000000000000000000000000000000000000000..988f9a716196fb8f65a92a0a2e894038c2474cfa --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/tests/test_zeta.py @@ -0,0 +1,301 @@ +import scipy +import scipy.special as sc +import sys +import numpy as np +import pytest + +from numpy.testing import assert_equal, assert_allclose + + +def test_zeta(): + assert_allclose(sc.zeta(2,2), np.pi**2/6 - 1, rtol=1e-12) + + +def test_zetac(): + # Expected values in the following were computed using Wolfram + # Alpha's `Zeta[x] - 1` + x = [-2.1, 0.8, 0.9999, 9, 50, 75] + desired = [ + -0.9972705002153750, + -5.437538415895550, + -10000.42279161673, + 0.002008392826082214, + 8.881784210930816e-16, + 2.646977960169853e-23, + ] + assert_allclose(sc.zetac(x), desired, rtol=1e-12) + + +def test_zetac_special_cases(): + assert sc.zetac(np.inf) == 0 + assert np.isnan(sc.zetac(-np.inf)) + assert sc.zetac(0) == -1.5 + assert sc.zetac(1.0) == np.inf + + assert_equal(sc.zetac([-2, -50, -100]), -1) + + +def test_riemann_zeta_special_cases(): + assert np.isnan(sc.zeta(np.nan)) + assert sc.zeta(np.inf) == 1 + assert sc.zeta(0) == -0.5 + + # Riemann zeta is zero add negative even integers. + assert_equal(sc.zeta([-2, -4, -6, -8, -10]), 0) + + assert_allclose(sc.zeta(2), np.pi**2/6, rtol=1e-12) + assert_allclose(sc.zeta(4), np.pi**4/90, rtol=1e-12) + + +def test_riemann_zeta_avoid_overflow(): + s = -260.00000000001 + desired = -5.6966307844402683127e+297 # Computed with Mpmath + assert_allclose(sc.zeta(s), desired, atol=0, rtol=5e-14) + + +@pytest.mark.parametrize( + "z, desired, rtol", + [ + ## Test cases taken from mpmath with the script: + + # import numpy as np + # import scipy.stats as stats + + # from mpmath import mp + + # # seed = np.random.SeedSequence().entropy + # seed = 154689806791763421822480125722191067828 + # rng = np.random.default_rng(seed) + # default_rtol = 1e-13 + + # # A small point in each quadrant outside of the critical strip + # cases = [] + # for x_sign, y_sign in [1, 1], [1, -1], [-1, 1], [-1, -1]: + # x = x_sign * rng.uniform(2, 8) + # y = y_sign * rng.uniform(2, 8) + # z = x + y*1j + # reference = complex(mp.zeta(z)) + # cases.append((z, reference, default_rtol)) + + # # Moderately large imaginary part in each quadrant outside of critical strip + # for x_sign, y_sign in [1, 1], [1, -1], [-1, 1], [-1, -1]: + # x = x_sign * rng.uniform(2, 8) + # y = y_sign * rng.uniform(50, 80) + # z = x + y*1j + # reference = complex(mp.zeta(z)) + # cases.append((z, reference, default_rtol)) + + # # points in critical strip + # x = rng.uniform(0.0, 1.0, size=5) + # y = np.exp(rng.uniform(0, 5, size=5)) + # z = x + y*1j + # for t in z: + # reference = complex(mp.zeta(t)) + # cases.append((complex(t), reference, default_rtol)) + # z = x - y*1j + # for t in z: + # reference = complex(mp.zeta(t)) + # cases.append((complex(t), reference, default_rtol)) + + # # Near small trivial zeros + # x = np.array([-2, -4, -6, -8]) + # y = np.array([1e-15, -1e-15]) + # x, y = np.meshgrid(x, y) + # x, y = x.ravel(), y.ravel() + # z = x + y*1j + # for t in z: + # reference = complex(mp.zeta(t)) + # cases.append((complex(t), reference, 1e-7)) + + # # Some other points near real axis + # x = np.array([-0.5, 0, 0.2, 0.75]) + # y = np.array([1e-15, -1e-15]) + # x, y = np.meshgrid(x, y) + # x, y = x.ravel(), y.ravel() + # z = x + y*1j + # for t in z: + # reference = complex(mp.zeta(t)) + # cases.append((complex(t), reference, 1e-7)) + + # # Moderately large real part + # x = np.array([49.33915930750887, 50.55805244181687]) + # y = rng.uniform(20, 100, size=3) + # x, y = np.meshgrid(x, y) + # x, y = x.ravel(), y.ravel() + # z = x + y*1j + # for t in z: + # reference = complex(mp.zeta(t)) + # cases.append((complex(t), reference, default_rtol)) + + # # Very large imaginary part + # x = np.array([0.5, 34.812847097948854, 50.55805244181687]) + # y = np.array([1e6, -1e6]) + # x, y = np.meshgrid(x, y) + # x, y = x.ravel(), y.ravel() + # z = x + y*1j + # for t in z: + # reference = complex(mp.zeta(t)) + # rtol = 1e-7 if t.real == 0.5 else default_rtol + # cases.append((complex(t), reference, rtol)) + # + # # Naive implementation of reflection formula suffers internal overflow + # x = -rng.uniform(200, 300, 3) + # y = np.array([rng.uniform(10, 30), -rng.uniform(10, 30)]) + # x, y = np.meshgrid(x, y) + # x, y = x.ravel(), y.ravel() + # z = x + y*1j + # for t in z: + # reference = complex(mp.zeta(t)) + # cases.append((complex(t), reference, default_rtol)) + # + # A small point in each quadrant outside of the critical strip + ((3.12838509346655+7.111085974836645j), + (1.0192654793474945+0.08795174413289127j), + 1e-13), + ((7.06791362314716-7.219497492626728j), + (1.0020740683598117-0.006752725913243711j), + 1e-13), + ((-6.806227077655519+2.724411451005281j), + (0.06312488213559667-0.061641496333765956j), + 1e-13), + ((-3.0170751511621026-6.3686522550665945j), + (-0.10330747857150148-1.214541994832571j), + 1e-13), + # Moderately large imaginary part in each quadrant outside of critical strip + ((6.133994402212294+60.03091448000761j), + (0.9885701843417336+0.009636925981078128j), + 1e-13), + ((6.17268142822657-64.74883149743795j), + (1.0080474225840865+0.012032804974965354j), + 1e-13), + ((-3.462191939791879+76.16258975567534j), + (18672.072070850158+2908.5104826247184j), + 1e-13), + ((-6.955735216531752-74.75791554155748j), + (-77672258.72276545+71625206.0401107j), + 1e-13), + # Points in critical strip + ((0.4088038289823922+1.4596830498094384j), + (0.3032837969400845-0.47272237994110344j), + 1e-13), + ((0.9673493951209633+4.918968547259143j), + (0.7488756907431944+0.17281553371482428j), + 1e-13), + ((0.8692482679977754+66.6142398421354j), + (0.5831942469552066-0.26848904799062334j), + 1e-13), + ((0.42771847720003764+21.783747851715468j), + (0.4767032638444329+0.6898148744603123j), + 1e-13), + ((0.20479494678428956+33.17656449538932j), + (-0.6983038977487848+0.18060923618150224j), + 1e-13), + ((0.4088038289823922-1.4596830498094384j), + (0.3032837969400845+0.47272237994110344j), + 1e-13), + ((0.9673493951209633-4.918968547259143j), + (0.7488756907431944-0.17281553371482428j), + 1e-13), + ((0.8692482679977754-66.6142398421354j), + (0.5831942469552066+0.26848904799062334j), + 1e-13), + ((0.42771847720003764-21.783747851715468j), + (0.4767032638444329-0.6898148744603123j), + 1e-13), + ((0.20479494678428956-33.17656449538932j), + (-0.6983038977487848-0.18060923618150224j), + 1e-13), + # Near small trivial zeros + ((-2+1e-15j), (3.288175809370978e-32-3.0448457058393275e-17j), 1e-07), + ((-4+1e-15j), (-2.868707923051182e-33+7.983811450268625e-18j), 1e-07), + ((-6+1e-15j), (-1.7064292323640116e-34-5.8997591435159376e-18j), 1e-07), + ((-8+1e-15j), (2.5060859548261706e-33+8.316161985602247e-18j), 1e-07), + ((-2-1e-15j), (3.288175809371319e-32+3.0448457058393275e-17j), 1e-07), + ((-4-1e-15j), (-2.8687079230520114e-33-7.983811450268625e-18j), 1e-07), + ((-6-1e-15j), (-1.70642923235801e-34+5.8997591435159376e-18j), 1e-07), + ((-8-1e-15j), (2.5060859548253293e-33-8.316161985602247e-18j), 1e-07), + # Some other points near real axis + ((-0.5+1e-15j), (-0.20788622497735457-3.608543395999408e-16j), 1e-07), + (1e-15j, (-0.5-9.189384239689193e-16j), 1e-07), + ((0.2+1e-15j), (-0.7339209248963406-1.4828001150329085e-15j), 1e-07), + ((0.75+1e-15j), (-3.4412853869452227-1.5924832114302393e-14j), 1e-13), + ((-0.5-1e-15j), (-0.20788622497735457+3.608543395999408e-16j), 1e-07), + (-1e-15j, (-0.5+9.189387416062746e-16j), 1e-07), + ((0.2-1e-15j), (-0.7339209248963406+1.4828004007675122e-15j), 1e-07), + ((0.75-1e-15j), (-3.4412853869452227+1.5924831974403957e-14j), 1e-13), + # Moderately large real part + ((49.33915930750887+53.213478698903955j), + (1.0000000000000009+1.0212452494616078e-15j), + 1e-13), + ((50.55805244181687+53.213478698903955j), + (1.0000000000000004+4.387394180390787e-16j), + 1e-13), + ((49.33915930750887+40.6366015728302j), + (0.9999999999999986-1.502268709924849e-16j), + 1e-13), + ((50.55805244181687+40.6366015728302j), + (0.9999999999999994-6.453929613571651e-17j), + 1e-13), + ((49.33915930750887+85.83555435273925j), + (0.9999999999999987-2.7014400611995846e-16j), + 1e-13), + ((50.55805244181687+85.83555435273925j), + (0.9999999999999994-1.160571605555322e-16j), + 1e-13), + # Very large imaginary part + ((0.5+1e6j), (0.0760890697382271+2.805102101019299j), 1e-07), + ((34.812847097948854+1e6j), + (1.0000000000102545+3.150848654056419e-11j), + 1e-13), + ((50.55805244181687+1e6j), + (1.0000000000000002+5.736517078070873e-16j), + 1e-13), + ((0.5-1e6j), (0.0760890697382271-2.805102101019299j), 1e-07), + ((34.812847097948854-1e6j), + (1.0000000000102545-3.150848654056419e-11j), + 1e-13), + ((50.55805244181687-1e6j), + (1.0000000000000002-5.736517078070873e-16j), + 1e-13), + ((-294.86605461349745+13.992648136816397j), (-np.inf+np.inf*1j), 1e-13), + ((-294.86605461349745-16.147667799398363j), (np.inf-np.inf*1j), 1e-13), + ] +) +def test_riemann_zeta_complex(z, desired, rtol): + assert_allclose(sc.zeta(z), desired, rtol=rtol) + + +# Some of the test cases below fail for intel compilers +cpp_compiler = scipy.__config__.CONFIG["Compilers"]["c++"]["name"] +gcc_linux = cpp_compiler == "gcc" and sys.platform == "linux" +clang_macOS = cpp_compiler == "clang" and sys.platform == "darwin" + + +@pytest.mark.skipif( + not (gcc_linux or clang_macOS), + reason="Underflow may not be avoided on other platforms", +) +@pytest.mark.parametrize( + "z, desired, rtol", + [ + # Test cases generated as part of same script for + # test_riemann_zeta_complex. These cases are split off because + # they fail on some platforms. + # + # Naive implementation of reflection formula suffers internal overflow + ((-217.40285743524163+13.992648136816397j), + (-6.012818500554211e+249-1.926943776932387e+250j), + 5e-13,), + ((-237.71710702931668+13.992648136816397j), + (-8.823803086106129e+281-5.009074181335139e+281j), + 1e-13,), + ((-217.40285743524163-16.147667799398363j), + (-5.111612904844256e+251-4.907132127666742e+250j), + 5e-13,), + ((-237.71710702931668-16.147667799398363j), + (-1.3256112779883167e+283-2.253002003455494e+283j), + 5e-13,), + ], +) +def test_riemann_zeta_complex_avoid_underflow(z, desired, rtol): + assert_allclose(sc.zeta(z), desired, rtol=rtol) diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/binom.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/binom.h new file mode 100644 index 0000000000000000000000000000000000000000..6a9b9ead9d7d458b47c5a51fd3104c9f72ff20a5 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/binom.h @@ -0,0 +1,89 @@ +/* Translated from Cython into C++ by SciPy developers in 2024. + * + * Original authors: Pauli Virtanen, Eric Moore + */ + +// Binomial coefficient + +#pragma once + +#include "config.h" + +#include "cephes/beta.h" +#include "cephes/gamma.h" + +namespace xsf { + +XSF_HOST_DEVICE inline double binom(double n, double k) { + double kx, nx, num, den, dk, sgn; + + if (n < 0) { + nx = std::floor(n); + if (n == nx) { + // Undefined + return std::numeric_limits::quiet_NaN(); + } + } + + kx = std::floor(k); + if (k == kx && (std::abs(n) > 1E-8 || n == 0)) { + /* Integer case: use multiplication formula for less rounding + * error for cases where the result is an integer. + * + * This cannot be used for small nonzero n due to loss of + * precision. */ + nx = std::floor(n); + if (nx == n && kx > nx / 2 && nx > 0) { + // Reduce kx by symmetry + kx = nx - kx; + } + + if (kx >= 0 && kx < 20) { + num = 1.0; + den = 1.0; + for (int i = 1; i < 1 + static_cast(kx); i++) { + num *= i + n - kx; + den *= i; + if (std::abs(num) > 1E50) { + num /= den; + den = 1.0; + } + } + return num / den; + } + } + + // general case + if (n >= 1E10 * k and k > 0) { + // avoid under/overflows intermediate results + return std::exp(-cephes::lbeta(1 + n - k, 1 + k) - std::log(n + 1)); + } + if (k > 1E8 * std::abs(n)) { + // avoid loss of precision + num = cephes::Gamma(1 + n) / std::abs(k) + cephes::Gamma(1 + n) * n / (2 * k * k); // + ... + num /= M_PI * std::pow(std::abs(k), n); + if (k > 0) { + kx = std::floor(k); + if (static_cast(kx) == kx) { + dk = k - kx; + sgn = (static_cast(kx) % 2 == 0) ? 1 : -1; + } else { + dk = k; + sgn = 1; + } + return num * std::sin((dk - n) * M_PI) * sgn; + } + kx = std::floor(k); + if (static_cast(kx) == kx) { + return 0; + } + return num * std::sin(k * M_PI); + } + return 1 / (n + 1) / cephes::beta(1 + n - k, 1 + k); +} + +XSF_HOST_DEVICE inline float binom(float n, float k) { + return binom(static_cast(n), static_cast(k)); +} + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cdflib.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cdflib.h new file mode 100644 index 0000000000000000000000000000000000000000..1ce5550efb6d2b13a45e22c0cbea7952bbd7938c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cdflib.h @@ -0,0 +1,100 @@ + +#pragma once + +#include "cephes/igam.h" +#include "config.h" +#include "error.h" +#include "tools.h" + +namespace xsf { + +XSF_HOST_DEVICE inline double gdtrib(double a, double p, double x) { + if (std::isnan(p) || std::isnan(a) || std::isnan(x)) { + return std::numeric_limits::quiet_NaN(); + } + if (!((0 <= p) && (p <= 1))) { + set_error("gdtrib", SF_ERROR_DOMAIN, "Input parameter p is out of range"); + return std::numeric_limits::quiet_NaN(); + } + if (!(a > 0) || std::isinf(a)) { + set_error("gdtrib", SF_ERROR_DOMAIN, "Input parameter a is out of range"); + return std::numeric_limits::quiet_NaN(); + } + if (!(x >= 0) || std::isinf(x)) { + set_error("gdtrib", SF_ERROR_DOMAIN, "Input parameter x is out of range"); + return std::numeric_limits::quiet_NaN(); + } + if (x == 0.0) { + if (p == 0.0) { + set_error("gdtrib", SF_ERROR_DOMAIN, "Indeterminate result for (x, p) == (0, 0)."); + return std::numeric_limits::quiet_NaN(); + } + /* gdtrib(a, p, x) tends to 0 as x -> 0 when p > 0 */ + return 0.0; + } + if (p == 0.0) { + /* gdtrib(a, p, x) tends to infinity as p -> 0 from the right when x > 0. */ + set_error("gdtrib", SF_ERROR_SINGULAR, NULL); + return std::numeric_limits::infinity(); + } + if (p == 1.0) { + /* gdtrib(a, p, x) tends to 0 as p -> 1.0 from the left when x > 0. */ + return 0.0; + } + double q = 1.0 - p; + auto func = [a, p, q, x](double b) { + if (p <= q) { + return cephes::igam(b, a * x) - p; + } + return q - cephes::igamc(b, a * x); + }; + double lower_bound = std::numeric_limits::min(); + double upper_bound = std::numeric_limits::max(); + /* To explain the magic constants used below: + * 1.0 is the initial guess for the root. -0.875 is the initial step size + * for the leading bracket endpoint if the bracket search will proceed to the + * left, likewise 7.0 is the initial step size when the bracket search will + * proceed to the right. 0.125 is the scale factor for a left moving bracket + * search and 8.0 the scale factor for a right moving bracket search. These + * constants are chosen so that: + * + * 1. The scale factor and bracket endpoints remain powers of 2, allowing for + * exact arithmetic, preventing roundoff error from causing numerical catastrophe + * which could lead to unexpected results. + * 2. The bracket sizes remain constant in a relative sense. Each candidate bracket + * will contain roughly the same number of floating point values. This means that + * the number of necessary function evaluations in the worst case scenario for + * Chandrupatla's algorithm will remain constant. + * + * false specifies that the function is not decreasing. 342 is equal to + * max(ceil(log_8(DBL_MAX)), ceil(log_(1/8)(DBL_MIN))). An upper bound for the + * number of iterations needed in this bracket search to check all normalized + * floating point values. + */ + auto [xl, xr, f_xl, f_xr, bracket_status] = detail::bracket_root_for_cdf_inversion( + func, 1.0, lower_bound, upper_bound, -0.875, 7.0, 0.125, 8, false, 342 + ); + if (bracket_status == 1) { + set_error("gdtrib", SF_ERROR_UNDERFLOW, NULL); + return 0.0; + } + if (bracket_status == 2) { + set_error("gdtrib", SF_ERROR_OVERFLOW, NULL); + return std::numeric_limits::infinity(); + } + if (bracket_status >= 3) { + set_error("gdtrib", SF_ERROR_OTHER, "Computational Error"); + return std::numeric_limits::quiet_NaN(); + } + auto [result, root_status] = detail::find_root_chandrupatla( + func, xl, xr, f_xl, f_xr, std::numeric_limits::epsilon(), 1e-100, 100 + ); + if (root_status) { + /* The root finding return should only fail if there's a bug in our code. */ + set_error("gdtrib", SF_ERROR_OTHER, "Computational Error, (%.17g, %.17g, %.17g)", a, p, x); + return std::numeric_limits::quiet_NaN(); + } + return result; +} + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/airy.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/airy.h new file mode 100644 index 0000000000000000000000000000000000000000..8db31fa9b3830b1e561a27a349154fc96ab071d3 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/airy.h @@ -0,0 +1,307 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* airy.c + * + * Airy function + * + * + * + * SYNOPSIS: + * + * double x, ai, aip, bi, bip; + * int airy(); + * + * airy( x, _&ai, _&aip, _&bi, _&bip ); + * + * + * + * DESCRIPTION: + * + * Solution of the differential equation + * + * y"(x) = xy. + * + * The function returns the two independent solutions Ai, Bi + * and their first derivatives Ai'(x), Bi'(x). + * + * Evaluation is by power series summation for small x, + * by rational minimax approximations for large x. + * + * + * + * ACCURACY: + * Error criterion is absolute when function <= 1, relative + * when function > 1, except * denotes relative error criterion. + * For large negative x, the absolute error increases as x^1.5. + * For large positive x, the relative error increases as x^1.5. + * + * Arithmetic domain function # trials peak rms + * IEEE -10, 0 Ai 10000 1.6e-15 2.7e-16 + * IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15* + * IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16 + * IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15* + * IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16 + * IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16 + * + */ +/* airy.c */ + +/* + * Cephes Math Library Release 2.8: June, 2000 + * Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier + */ +#pragma once + +#include "../config.h" +#include "const.h" +#include "polevl.h" + +namespace xsf { +namespace cephes { + + namespace detail { + + constexpr double airy_c1 = 0.35502805388781723926; + constexpr double airy_c2 = 0.258819403792806798405; + constexpr double MAXAIRY = 103.892; + + constexpr double airy_AN[8] = { + 3.46538101525629032477E-1, 1.20075952739645805542E1, 7.62796053615234516538E1, 1.68089224934630576269E2, + 1.59756391350164413639E2, 7.05360906840444183113E1, 1.40264691163389668864E1, 9.99999999999999995305E-1, + }; + + constexpr double airy_AD[8] = { + 5.67594532638770212846E-1, 1.47562562584847203173E1, 8.45138970141474626562E1, 1.77318088145400459522E2, + 1.64234692871529701831E2, 7.14778400825575695274E1, 1.40959135607834029598E1, 1.00000000000000000470E0, + }; + + constexpr double airy_APN[8] = { + 6.13759184814035759225E-1, 1.47454670787755323881E1, 8.20584123476060982430E1, 1.71184781360976385540E2, + 1.59317847137141783523E2, 6.99778599330103016170E1, 1.39470856980481566958E1, 1.00000000000000000550E0, + }; + + constexpr double airy_APD[8] = { + 3.34203677749736953049E-1, 1.11810297306158156705E1, 7.11727352147859965283E1, 1.58778084372838313640E2, + 1.53206427475809220834E2, 6.86752304592780337944E1, 1.38498634758259442477E1, 9.99999999999999994502E-1, + }; + + constexpr double airy_BN16[5] = { + -2.53240795869364152689E-1, 5.75285167332467384228E-1, -3.29907036873225371650E-1, + 6.44404068948199951727E-2, -3.82519546641336734394E-3, + }; + + constexpr double airy_BD16[5] = { + /* 1.00000000000000000000E0, */ + -7.15685095054035237902E0, 1.06039580715664694291E1, -5.23246636471251500874E0, + 9.57395864378383833152E-1, -5.50828147163549611107E-2, + }; + + constexpr double airy_BPPN[5] = { + 4.65461162774651610328E-1, -1.08992173800493920734E0, 6.38800117371827987759E-1, + -1.26844349553102907034E-1, 7.62487844342109852105E-3, + }; + + constexpr double airy_BPPD[5] = { + /* 1.00000000000000000000E0, */ + -8.70622787633159124240E0, 1.38993162704553213172E1, -7.14116144616431159572E0, + 1.34008595960680518666E0, -7.84273211323341930448E-2, + }; + + constexpr double airy_AFN[9] = { + -1.31696323418331795333E-1, -6.26456544431912369773E-1, -6.93158036036933542233E-1, + -2.79779981545119124951E-1, -4.91900132609500318020E-2, -4.06265923594885404393E-3, + -1.59276496239262096340E-4, -2.77649108155232920844E-6, -1.67787698489114633780E-8, + }; + + constexpr double airy_AFD[9] = { + /* 1.00000000000000000000E0, */ + 1.33560420706553243746E1, 3.26825032795224613948E1, 2.67367040941499554804E1, + 9.18707402907259625840E0, 1.47529146771666414581E0, 1.15687173795188044134E-1, + 4.40291641615211203805E-3, 7.54720348287414296618E-5, 4.51850092970580378464E-7, + }; + + constexpr double airy_AGN[11] = { + 1.97339932091685679179E-2, 3.91103029615688277255E-1, 1.06579897599595591108E0, 9.39169229816650230044E-1, + 3.51465656105547619242E-1, 6.33888919628925490927E-2, 5.85804113048388458567E-3, 2.82851600836737019778E-4, + 6.98793669997260967291E-6, 8.11789239554389293311E-8, 3.41551784765923618484E-10, + }; + + constexpr double airy_AGD[10] = { + /* 1.00000000000000000000E0, */ + 9.30892908077441974853E0, 1.98352928718312140417E1, 1.55646628932864612953E1, 5.47686069422975497931E0, + 9.54293611618961883998E-1, 8.64580826352392193095E-2, 4.12656523824222607191E-3, 1.01259085116509135510E-4, + 1.17166733214413521882E-6, 4.91834570062930015649E-9, + }; + + constexpr double airy_APFN[9] = { + 1.85365624022535566142E-1, 8.86712188052584095637E-1, 9.87391981747398547272E-1, + 4.01241082318003734092E-1, 7.10304926289631174579E-2, 5.90618657995661810071E-3, + 2.33051409401776799569E-4, 4.08718778289035454598E-6, 2.48379932900442457853E-8, + }; + + constexpr double airy_APFD[9] = { + /* 1.00000000000000000000E0, */ + 1.47345854687502542552E1, 3.75423933435489594466E1, 3.14657751203046424330E1, + 1.09969125207298778536E1, 1.78885054766999417817E0, 1.41733275753662636873E-1, + 5.44066067017226003627E-3, 9.39421290654511171663E-5, 5.65978713036027009243E-7, + }; + + constexpr double airy_APGN[11] = { + -3.55615429033082288335E-2, -6.37311518129435504426E-1, -1.70856738884312371053E0, + -1.50221872117316635393E0, -5.63606665822102676611E-1, -1.02101031120216891789E-1, + -9.48396695961445269093E-3, -4.60325307486780994357E-4, -1.14300836484517375919E-5, + -1.33415518685547420648E-7, -5.63803833958893494476E-10, + }; + + constexpr double airy_APGD[11] = { + /* 1.00000000000000000000E0, */ + 9.85865801696130355144E0, 2.16401867356585941885E1, 1.73130776389749389525E1, 6.17872175280828766327E0, + 1.08848694396321495475E0, 9.95005543440888479402E-2, 4.78468199683886610842E-3, 1.18159633322838625562E-4, + 1.37480673554219441465E-6, 5.79912514929147598821E-9, + }; + + } // namespace detail + + XSF_HOST_DEVICE inline int airy(double x, double *ai, double *aip, double *bi, double *bip) { + double z, zz, t, f, g, uf, ug, k, zeta, theta; + int domflg; + + domflg = 0; + if (x > detail::MAXAIRY) { + *ai = 0; + *aip = 0; + *bi = std::numeric_limits::infinity(); + *bip = std::numeric_limits::infinity(); + return (-1); + } + + if (x < -2.09) { + domflg = 15; + t = std::sqrt(-x); + zeta = -2.0 * x * t / 3.0; + t = std::sqrt(t); + k = detail::SQRT1OPI / t; + z = 1.0 / zeta; + zz = z * z; + uf = 1.0 + zz * polevl(zz, detail::airy_AFN, 8) / p1evl(zz, detail::airy_AFD, 9); + ug = z * polevl(zz, detail::airy_AGN, 10) / p1evl(zz, detail::airy_AGD, 10); + theta = zeta + 0.25 * M_PI; + f = std::sin(theta); + g = std::cos(theta); + *ai = k * (f * uf - g * ug); + *bi = k * (g * uf + f * ug); + uf = 1.0 + zz * polevl(zz, detail::airy_APFN, 8) / p1evl(zz, detail::airy_APFD, 9); + ug = z * polevl(zz, detail::airy_APGN, 10) / p1evl(zz, detail::airy_APGD, 10); + k = detail::SQRT1OPI * t; + *aip = -k * (g * uf + f * ug); + *bip = k * (f * uf - g * ug); + return (0); + } + + if (x >= 2.09) { /* cbrt(9) */ + domflg = 5; + t = std::sqrt(x); + zeta = 2.0 * x * t / 3.0; + g = std::exp(zeta); + t = std::sqrt(t); + k = 2.0 * t * g; + z = 1.0 / zeta; + f = polevl(z, detail::airy_AN, 7) / polevl(z, detail::airy_AD, 7); + *ai = detail::SQRT1OPI * f / k; + k = -0.5 * detail::SQRT1OPI * t / g; + f = polevl(z, detail::airy_APN, 7) / polevl(z, detail::airy_APD, 7); + *aip = f * k; + + if (x > 8.3203353) { /* zeta > 16 */ + f = z * polevl(z, detail::airy_BN16, 4) / p1evl(z, detail::airy_BD16, 5); + k = detail::SQRT1OPI * g; + *bi = k * (1.0 + f) / t; + f = z * polevl(z, detail::airy_BPPN, 4) / p1evl(z, detail::airy_BPPD, 5); + *bip = k * t * (1.0 + f); + return (0); + } + } + + f = 1.0; + g = x; + t = 1.0; + uf = 1.0; + ug = x; + k = 1.0; + z = x * x * x; + while (t > detail::MACHEP) { + uf *= z; + k += 1.0; + uf /= k; + ug *= z; + k += 1.0; + ug /= k; + uf /= k; + f += uf; + k += 1.0; + ug /= k; + g += ug; + t = std::abs(uf / f); + } + uf = detail::airy_c1 * f; + ug = detail::airy_c2 * g; + if ((domflg & 1) == 0) { + *ai = uf - ug; + } + if ((domflg & 2) == 0) { + *bi = detail::SQRT3 * (uf + ug); + } + + /* the deriviative of ai */ + k = 4.0; + uf = x * x / 2.0; + ug = z / 3.0; + f = uf; + g = 1.0 + ug; + uf /= 3.0; + t = 1.0; + + while (t > detail::MACHEP) { + uf *= z; + ug /= k; + k += 1.0; + ug *= z; + uf /= k; + f += uf; + k += 1.0; + ug /= k; + uf /= k; + g += ug; + k += 1.0; + t = std::abs(ug / g); + } + + uf = detail::airy_c1 * f; + ug = detail::airy_c2 * g; + if ((domflg & 4) == 0) { + *aip = uf - ug; + } + if ((domflg & 8) == 0) { + *bip = detail::SQRT3 * (uf + ug); + }; + return (0); + } + + inline int airy(float xf, float *aif, float *aipf, float *bif, float *bipf) { + double ai; + double aip; + double bi; + double bip; + int res = cephes::airy(xf, &ai, &aip, &bi, &bip); + + *aif = ai; + *aipf = aip; + *bif = bi; + *bipf = bip; + return res; + } + +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/besselpoly.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/besselpoly.h new file mode 100644 index 0000000000000000000000000000000000000000..d113b8b7d0f4ae4e7ad9da2faf66d3ab7406d736 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/besselpoly.h @@ -0,0 +1,51 @@ +/* Translated into C++ by SciPy developers in 2024. + * + * This was not part of the original cephes library. + */ +#pragma once + +#include "../config.h" +#include "gamma.h" + +namespace xsf { +namespace cephes { + namespace detail { + + constexpr double besselpoly_EPS = 1.0e-17; + } + + XSF_HOST_DEVICE inline double besselpoly(double a, double lambda, double nu) { + + int m, factor = 0; + double Sm, relerr, Sol; + double sum = 0.0; + + /* Special handling for a = 0.0 */ + if (a == 0.0) { + if (nu == 0.0) { + return 1.0 / (lambda + 1); + } else { + return 0.0; + } + } + /* Special handling for negative and integer nu */ + if ((nu < 0) && (std::floor(nu) == nu)) { + nu = -nu; + factor = static_cast(nu) % 2; + } + Sm = std::exp(nu * std::log(a)) / (Gamma(nu + 1) * (lambda + nu + 1)); + m = 0; + do { + sum += Sm; + Sol = Sm; + Sm *= -a * a * (lambda + nu + 1 + 2 * m) / ((nu + m + 1) * (m + 1) * (lambda + nu + 1 + 2 * m + 2)); + m++; + relerr = std::abs((Sm - Sol) / Sm); + } while (relerr > detail::besselpoly_EPS && m < 1000); + if (!factor) + return sum; + else + return -sum; + } +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/beta.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/beta.h new file mode 100644 index 0000000000000000000000000000000000000000..437262793e8f94b229671978b23a638820ff1290 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/beta.h @@ -0,0 +1,257 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* beta.c + * + * Beta function + * + * + * + * SYNOPSIS: + * + * double a, b, y, beta(); + * + * y = beta( a, b ); + * + * + * + * DESCRIPTION: + * + * - - + * | (a) | (b) + * beta( a, b ) = -----------. + * - + * | (a+b) + * + * For large arguments the logarithm of the function is + * evaluated using lgam(), then exponentiated. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,30 30000 8.1e-14 1.1e-14 + * + * ERROR MESSAGES: + * + * message condition value returned + * beta overflow log(beta) > MAXLOG 0.0 + * a or b <0 integer 0.0 + * + */ + +/* + * Cephes Math Library Release 2.0: April, 1987 + * Copyright 1984, 1987 by Stephen L. Moshier + * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 + */ +#pragma once + +#include "../config.h" +#include "const.h" +#include "gamma.h" +#include "rgamma.h" + +namespace xsf { +namespace cephes { + + XSF_HOST_DEVICE double beta(double, double); + XSF_HOST_DEVICE double lbeta(double, double); + + namespace detail { + constexpr double beta_ASYMP_FACTOR = 1e6; + + /* + * Asymptotic expansion for ln(|B(a, b)|) for a > ASYMP_FACTOR*max(|b|, 1). + */ + XSF_HOST_DEVICE inline double lbeta_asymp(double a, double b, int *sgn) { + double r = lgam_sgn(b, sgn); + r -= b * std::log(a); + + r += b * (1 - b) / (2 * a); + r += b * (1 - b) * (1 - 2 * b) / (12 * a * a); + r += -b * b * (1 - b) * (1 - b) / (12 * a * a * a); + + return r; + } + + /* + * Special case for a negative integer argument + */ + + XSF_HOST_DEVICE inline double beta_negint(int a, double b) { + int sgn; + if (b == static_cast(b) && 1 - a - b > 0) { + sgn = (static_cast(b) % 2 == 0) ? 1 : -1; + return sgn * xsf::cephes::beta(1 - a - b, b); + } else { + set_error("lbeta", SF_ERROR_OVERFLOW, NULL); + return std::numeric_limits::infinity(); + } + } + + XSF_HOST_DEVICE inline double lbeta_negint(int a, double b) { + double r; + if (b == static_cast(b) && 1 - a - b > 0) { + r = xsf::cephes::lbeta(1 - a - b, b); + return r; + } else { + set_error("lbeta", SF_ERROR_OVERFLOW, NULL); + return std::numeric_limits::infinity(); + } + } + } // namespace detail + + XSF_HOST_DEVICE inline double beta(double a, double b) { + double y; + int sign = 1; + + if (a <= 0.0) { + if (a == std::floor(a)) { + if (a == static_cast(a)) { + return detail::beta_negint(static_cast(a), b); + } else { + goto overflow; + } + } + } + + if (b <= 0.0) { + if (b == std::floor(b)) { + if (b == static_cast(b)) { + return detail::beta_negint(static_cast(b), a); + } else { + goto overflow; + } + } + } + + if (std::abs(a) < std::abs(b)) { + y = a; + a = b; + b = y; + } + + if (std::abs(a) > detail::beta_ASYMP_FACTOR * std::abs(b) && a > detail::beta_ASYMP_FACTOR) { + /* Avoid loss of precision in lgam(a + b) - lgam(a) */ + y = detail::lbeta_asymp(a, b, &sign); + return sign * std::exp(y); + } + + y = a + b; + if (std::abs(y) > detail::MAXGAM || std::abs(a) > detail::MAXGAM || std::abs(b) > detail::MAXGAM) { + int sgngam; + y = detail::lgam_sgn(y, &sgngam); + sign *= sgngam; /* keep track of the sign */ + y = detail::lgam_sgn(b, &sgngam) - y; + sign *= sgngam; + y = detail::lgam_sgn(a, &sgngam) + y; + sign *= sgngam; + if (y > detail::MAXLOG) { + goto overflow; + } + return (sign * std::exp(y)); + } + + y = rgamma(y); + a = Gamma(a); + b = Gamma(b); + if (std::isinf(y)) { + goto overflow; + } + + if (std::abs(std::abs(a*y) - 1.0) > std::abs(std::abs(b*y) - 1.0)) { + y = b * y; + y *= a; + } else { + y = a * y; + y *= b; + } + + return (y); + + overflow: + set_error("beta", SF_ERROR_OVERFLOW, NULL); + return (sign * std::numeric_limits::infinity()); + } + + /* Natural log of |beta|. */ + + XSF_HOST_DEVICE inline double lbeta(double a, double b) { + double y; + int sign; + + sign = 1; + + if (a <= 0.0) { + if (a == std::floor(a)) { + if (a == static_cast(a)) { + return detail::lbeta_negint(static_cast(a), b); + } else { + goto over; + } + } + } + + if (b <= 0.0) { + if (b == std::floor(b)) { + if (b == static_cast(b)) { + return detail::lbeta_negint(static_cast(b), a); + } else { + goto over; + } + } + } + + if (std::abs(a) < std::abs(b)) { + y = a; + a = b; + b = y; + } + + if (std::abs(a) > detail::beta_ASYMP_FACTOR * std::abs(b) && a > detail::beta_ASYMP_FACTOR) { + /* Avoid loss of precision in lgam(a + b) - lgam(a) */ + y = detail::lbeta_asymp(a, b, &sign); + return y; + } + + y = a + b; + if (std::abs(y) > detail::MAXGAM || std::abs(a) > detail::MAXGAM || std::abs(b) > detail::MAXGAM) { + int sgngam; + y = detail::lgam_sgn(y, &sgngam); + sign *= sgngam; /* keep track of the sign */ + y = detail::lgam_sgn(b, &sgngam) - y; + sign *= sgngam; + y = detail::lgam_sgn(a, &sgngam) + y; + sign *= sgngam; + return (y); + } + + y = rgamma(y); + a = Gamma(a); + b = Gamma(b); + if (std::isinf(y)) { + over: + set_error("lbeta", SF_ERROR_OVERFLOW, NULL); + return (sign * std::numeric_limits::infinity()); + } + + if (std::abs(std::abs(a*y) - 1.0) > std::abs(std::abs(b*y) - 1.0)) { + y = b * y; + y *= a; + } else { + y = a * y; + y *= b; + } + + if (y < 0) { + y = -y; + } + + return (std::log(y)); + } +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/cbrt.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/cbrt.h new file mode 100644 index 0000000000000000000000000000000000000000..3e9fbd4eab45b24f1caa8b509c9d1d3c536867fe --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/cbrt.h @@ -0,0 +1,131 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* cbrt.c + * + * Cube root + * + * + * + * SYNOPSIS: + * + * double x, y, cbrt(); + * + * y = cbrt( x ); + * + * + * + * DESCRIPTION: + * + * Returns the cube root of the argument, which may be negative. + * + * Range reduction involves determining the power of 2 of + * the argument. A polynomial of degree 2 applied to the + * mantissa, and multiplication by the cube root of 1, 2, or 4 + * approximates the root to within about 0.1%. Then Newton's + * iteration is used three times to converge to an accurate + * result. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,1e308 30000 1.5e-16 5.0e-17 + * + */ +/* cbrt.c */ + +/* + * Cephes Math Library Release 2.2: January, 1991 + * Copyright 1984, 1991 by Stephen L. Moshier + * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 + */ +#pragma once + +#include "../config.h" + +namespace xsf { +namespace cephes { + + namespace detail { + + constexpr double CBRT2 = 1.2599210498948731647672; + constexpr double CBRT4 = 1.5874010519681994747517; + constexpr double CBRT2I = 0.79370052598409973737585; + constexpr double CBRT4I = 0.62996052494743658238361; + + XSF_HOST_DEVICE inline double cbrt(double x) { + int e, rem, sign; + double z; + + if (!std::isfinite(x)) { + return x; + } + if (x == 0) { + return (x); + } + if (x > 0) { + sign = 1; + } else { + sign = -1; + x = -x; + } + + z = x; + /* extract power of 2, leaving + * mantissa between 0.5 and 1 + */ + x = std::frexp(x, &e); + + /* Approximate cube root of number between .5 and 1, + * peak relative error = 9.2e-6 + */ + x = (((-1.3466110473359520655053e-1 * x + 5.4664601366395524503440e-1) * x - 9.5438224771509446525043e-1) * + x + + 1.1399983354717293273738e0) * + x + + 4.0238979564544752126924e-1; + + /* exponent divided by 3 */ + if (e >= 0) { + rem = e; + e /= 3; + rem -= 3 * e; + if (rem == 1) { + x *= CBRT2; + } else if (rem == 2) { + x *= CBRT4; + } + } + /* argument less than 1 */ + else { + e = -e; + rem = e; + e /= 3; + rem -= 3 * e; + if (rem == 1) { + x *= CBRT2I; + } else if (rem == 2) { + x *= CBRT4I; + } + e = -e; + } + + /* multiply by power of 2 */ + x = std::ldexp(x, e); + + /* Newton iteration */ + x -= (x - (z / (x * x))) * 0.33333333333333333333; + x -= (x - (z / (x * x))) * 0.33333333333333333333; + + if (sign < 0) + x = -x; + return (x); + } + } // namespace detail + +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/chbevl.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/chbevl.h new file mode 100644 index 0000000000000000000000000000000000000000..caaa74fc7b81015784608cc38aed5987ca145526 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/chbevl.h @@ -0,0 +1,85 @@ +/* chbevl.c + * + * Evaluate Chebyshev series + * + * + * + * SYNOPSIS: + * + * int N; + * double x, y, coef[N], chebevl(); + * + * y = chbevl( x, coef, N ); + * + * + * + * DESCRIPTION: + * + * Evaluates the series + * + * N-1 + * - ' + * y = > coef[i] T (x/2) + * - i + * i=0 + * + * of Chebyshev polynomials Ti at argument x/2. + * + * Coefficients are stored in reverse order, i.e. the zero + * order term is last in the array. Note N is the number of + * coefficients, not the order. + * + * If coefficients are for the interval a to b, x must + * have been transformed to x -> 2(2x - b - a)/(b-a) before + * entering the routine. This maps x from (a, b) to (-1, 1), + * over which the Chebyshev polynomials are defined. + * + * If the coefficients are for the inverted interval, in + * which (a, b) is mapped to (1/b, 1/a), the transformation + * required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity, + * this becomes x -> 4a/x - 1. + * + * + * + * SPEED: + * + * Taking advantage of the recurrence properties of the + * Chebyshev polynomials, the routine requires one more + * addition per loop than evaluating a nested polynomial of + * the same degree. + * + */ +/* chbevl.c */ + +/* + * Cephes Math Library Release 2.0: April, 1987 + * Copyright 1985, 1987 by Stephen L. Moshier + * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 + */ +#pragma once + +#include "../config.h" + +namespace xsf { +namespace cephes { + + XSF_HOST_DEVICE double chbevl(double x, const double array[], int n) { + double b0, b1, b2; + const double *p; + int i; + + p = array; + b0 = *p++; + b1 = 0.0; + i = n - 1; + + do { + b2 = b1; + b1 = b0; + b0 = x * b1 - b2 + *p++; + } while (--i); + + return (0.5 * (b0 - b2)); + } +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/chdtr.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/chdtr.h new file mode 100644 index 0000000000000000000000000000000000000000..0a97def6d00baa18eaffaca73c92d7b6dd2b5e32 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/chdtr.h @@ -0,0 +1,193 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* chdtr.c + * + * Chi-square distribution + * + * + * + * SYNOPSIS: + * + * double df, x, y, chdtr(); + * + * y = chdtr( df, x ); + * + * + * + * DESCRIPTION: + * + * Returns the area under the left hand tail (from 0 to x) + * of the Chi square probability density function with + * v degrees of freedom. + * + * + * inf. + * - + * 1 | | v/2-1 -t/2 + * P( x | v ) = ----------- | t e dt + * v/2 - | | + * 2 | (v/2) - + * x + * + * where x is the Chi-square variable. + * + * The incomplete Gamma integral is used, according to the + * formula + * + * y = chdtr( v, x ) = igam( v/2.0, x/2.0 ). + * + * + * The arguments must both be positive. + * + * + * + * ACCURACY: + * + * See igam(). + * + * ERROR MESSAGES: + * + * message condition value returned + * chdtr domain x < 0 or v < 1 0.0 + */ +/* chdtrc() + * + * Complemented Chi-square distribution + * + * + * + * SYNOPSIS: + * + * double v, x, y, chdtrc(); + * + * y = chdtrc( v, x ); + * + * + * + * DESCRIPTION: + * + * Returns the area under the right hand tail (from x to + * infinity) of the Chi square probability density function + * with v degrees of freedom: + * + * + * inf. + * - + * 1 | | v/2-1 -t/2 + * P( x | v ) = ----------- | t e dt + * v/2 - | | + * 2 | (v/2) - + * x + * + * where x is the Chi-square variable. + * + * The incomplete Gamma integral is used, according to the + * formula + * + * y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ). + * + * + * The arguments must both be positive. + * + * + * + * ACCURACY: + * + * See igamc(). + * + * ERROR MESSAGES: + * + * message condition value returned + * chdtrc domain x < 0 or v < 1 0.0 + */ +/* chdtri() + * + * Inverse of complemented Chi-square distribution + * + * + * + * SYNOPSIS: + * + * double df, x, y, chdtri(); + * + * x = chdtri( df, y ); + * + * + * + * + * DESCRIPTION: + * + * Finds the Chi-square argument x such that the integral + * from x to infinity of the Chi-square density is equal + * to the given cumulative probability y. + * + * This is accomplished using the inverse Gamma integral + * function and the relation + * + * x/2 = igamci( df/2, y ); + * + * + * + * + * ACCURACY: + * + * See igami.c. + * + * ERROR MESSAGES: + * + * message condition value returned + * chdtri domain y < 0 or y > 1 0.0 + * v < 1 + * + */ + +/* chdtr() */ + +/* + * Cephes Math Library Release 2.0: April, 1987 + * Copyright 1984, 1987 by Stephen L. Moshier + * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 + */ +#pragma once + +#include "../config.h" +#include "../error.h" + +#include "igam.h" +#include "igami.h" + +namespace xsf { +namespace cephes { + + XSF_HOST_DEVICE inline double chdtrc(double df, double x) { + + if (x < 0.0) + return 1.0; /* modified by T. Oliphant */ + return (igamc(df / 2.0, x / 2.0)); + } + + XSF_HOST_DEVICE inline double chdtr(double df, double x) { + + if ((x < 0.0)) { /* || (df < 1.0) ) */ + set_error("chdtr", SF_ERROR_DOMAIN, NULL); + return (std::numeric_limits::quiet_NaN()); + } + return (igam(df / 2.0, x / 2.0)); + } + + XSF_HOST_DEVICE double chdtri(double df, double y) { + double x; + + if ((y < 0.0) || (y > 1.0)) { /* || (df < 1.0) ) */ + set_error("chdtri", SF_ERROR_DOMAIN, NULL); + return (std::numeric_limits::quiet_NaN()); + } + + x = igamci(0.5 * df, y); + return (2.0 * x); + } + +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/const.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/const.h new file mode 100644 index 0000000000000000000000000000000000000000..d7b162c5efc8e11f407c6108d55c117820e9e76d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/const.h @@ -0,0 +1,87 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + * + * Since we support only IEEE-754 floating point numbers, conditional logic + * supporting other arithmetic types has been removed. + */ + +/* + * + * + * const.c + * + * Globally declared constants + * + * + * + * SYNOPSIS: + * + * extern double nameofconstant; + * + * + * + * + * DESCRIPTION: + * + * This file contains a number of mathematical constants and + * also some needed size parameters of the computer arithmetic. + * The values are supplied as arrays of hexadecimal integers + * for IEEE arithmetic, and in a normal decimal scientific notation for + * other machines. The particular notation used is determined + * by a symbol (IBMPC, or UNK) defined in the include file + * mconf.h. + * + * The default size parameters are as follows. + * + * For UNK mode: + * MACHEP = 1.38777878078144567553E-17 2**-56 + * MAXLOG = 8.8029691931113054295988E1 log(2**127) + * MINLOG = -8.872283911167299960540E1 log(2**-128) + * + * For IEEE arithmetic (IBMPC): + * MACHEP = 1.11022302462515654042E-16 2**-53 + * MAXLOG = 7.09782712893383996843E2 log(2**1024) + * MINLOG = -7.08396418532264106224E2 log(2**-1022) + * + * The global symbols for mathematical constants are + * SQ2OPI = 7.9788456080286535587989E-1 sqrt( 2/pi ) + * LOGSQ2 = 3.46573590279972654709E-1 log(2)/2 + * THPIO4 = 2.35619449019234492885 3*pi/4 + * + * These lists are subject to change. + */ +/* const.c */ + +/* + * Cephes Math Library Release 2.3: March, 1995 + * Copyright 1984, 1995 by Stephen L. Moshier + */ +#pragma once + +namespace xsf { +namespace cephes { + namespace detail { + constexpr std::uint64_t MAXITER = 500; + constexpr double MACHEP = 1.11022302462515654042E-16; // 2**-53 + constexpr double MAXLOG = 7.09782712893383996732E2; // log(DBL_MAX) + constexpr double MINLOG = -7.451332191019412076235E2; // log 2**-1022 + constexpr double SQRT1OPI = 5.64189583547756286948E-1; // sqrt( 1/pi) + constexpr double SQRT2OPI = 7.9788456080286535587989E-1; // sqrt( 2/pi ) + constexpr double SQRT2PI = 0.79788456080286535587989; // sqrt(2pi) + constexpr double LOGSQ2 = 3.46573590279972654709E-1; // log(2)/2 + constexpr double THPIO4 = 2.35619449019234492885; // 3*pi/4 + constexpr double SQRT3 = 1.732050807568877293527; // sqrt(3) + constexpr double PI180 = 1.74532925199432957692E-2; // pi/180 + constexpr double SQRTPI = 2.50662827463100050242E0; // sqrt(pi) + constexpr double LOGPI = 1.14472988584940017414; // log(pi) + constexpr double MAXGAM = 171.624376956302725; + constexpr double LOGSQRT2PI = 0.9189385332046727; // log(sqrt(pi)) + + // Following two added by SciPy developers. + // Euler's constant + constexpr double SCIPY_EULER = 0.577215664901532860606512090082402431; + // e as long double + constexpr long double SCIPY_El = 2.718281828459045235360287471352662498L; + } // namespace detail +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellie.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellie.h new file mode 100644 index 0000000000000000000000000000000000000000..a455599b4a95b3c69d23df188669e84deac4e31c --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellie.h @@ -0,0 +1,293 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* ellie.c + * + * Incomplete elliptic integral of the second kind + * + * + * + * SYNOPSIS: + * + * double phi, m, y, ellie(); + * + * y = ellie( phi, m ); + * + * + * + * DESCRIPTION: + * + * Approximates the integral + * + * + * phi + * - + * | | + * | 2 + * E(phi_\m) = | sqrt( 1 - m sin t ) dt + * | + * | | + * - + * 0 + * + * of amplitude phi and modulus m, using the arithmetic - + * geometric mean algorithm. + * + * + * + * ACCURACY: + * + * Tested at random arguments with phi in [-10, 10] and m in + * [0, 1]. + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -10,10 150000 3.3e-15 1.4e-16 + */ + +/* + * Cephes Math Library Release 2.0: April, 1987 + * Copyright 1984, 1987, 1993 by Stephen L. Moshier + * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 + */ +/* Copyright 2014, Eric W. Moore */ + +/* Incomplete elliptic integral of second kind */ +#pragma once + +#include "../config.h" +#include "const.h" +#include "ellpe.h" +#include "ellpk.h" +#include "unity.h" + +namespace xsf { +namespace cephes { + namespace detail { + + /* To calculate legendre's incomplete elliptical integral of the second kind for + * negative m, we use a power series in phi for small m*phi*phi, an asymptotic + * series in m for large m*phi*phi* and the relation to Carlson's symmetric + * integrals, R_F(x,y,z) and R_D(x,y,z). + * + * E(phi, m) = sin(phi) * R_F(cos(phi)^2, 1 - m * sin(phi)^2, 1.0) + * - m * sin(phi)^3 * R_D(cos(phi)^2, 1 - m * sin(phi)^2, 1.0) / 3 + * + * = R_F(c-1, c-m, c) - m * R_D(c-1, c-m, c) / 3 + * + * where c = csc(phi)^2. We use the second form of this for (approximately) + * phi > 1/(sqrt(DBL_MAX) ~ 1e-154, where csc(phi)^2 overflows. Elsewhere we + * use the first form, accounting for the smallness of phi. + * + * The algorithm used is described in Carlson, B. C. Numerical computation of + * real or complex elliptic integrals. (1994) https://arxiv.org/abs/math/9409227 + * Most variable names reflect Carlson's usage. + * + * In this routine, we assume m < 0 and 0 > phi > pi/2. + */ + XSF_HOST_DEVICE inline double ellie_neg_m(double phi, double m) { + double x, y, z, x1, y1, z1, ret, Q; + double A0f, Af, Xf, Yf, Zf, E2f, E3f, scalef; + double A0d, Ad, seriesn, seriesd, Xd, Yd, Zd, E2d, E3d, E4d, E5d, scaled; + int n = 0; + double mpp = (m * phi) * phi; + + if (-mpp < 1e-6 && phi < -m) { + return phi + (mpp * phi * phi / 30.0 - mpp * mpp / 40.0 - mpp / 6.0) * phi; + } + + if (-mpp > 1e6) { + double sm = std::sqrt(-m); + double sp = std::sin(phi); + double cp = std::cos(phi); + + double a = -cosm1(phi); + double b1 = std::log(4 * sp * sm / (1 + cp)); + double b = -(0.5 + b1) / 2.0 / m; + double c = (0.75 + cp / sp / sp - b1) / 16.0 / m / m; + return (a + b + c) * sm; + } + + if (phi > 1e-153 && m > -1e200) { + double s = std::sin(phi); + double csc2 = 1.0 / s / s; + scalef = 1.0; + scaled = m / 3.0; + x = 1.0 / std::tan(phi) / std::tan(phi); + y = csc2 - m; + z = csc2; + } else { + scalef = phi; + scaled = mpp * phi / 3.0; + x = 1.0; + y = 1 - mpp; + z = 1.0; + } + + if (x == y && x == z) { + return (scalef + scaled / x) / std::sqrt(x); + } + + A0f = (x + y + z) / 3.0; + Af = A0f; + A0d = (x + y + 3.0 * z) / 5.0; + Ad = A0d; + x1 = x; + y1 = y; + z1 = z; + seriesd = 0.0; + seriesn = 1.0; + /* Carlson gives 1/pow(3*r, 1.0/6.0) for this constant. if r == eps, + * it is ~338.38. */ + + /* N.B. This will evaluate its arguments multiple times. */ + Q = 400.0 * std::fmax(std::abs(A0f - x), std::fmax(std::abs(A0f - y), std::abs(A0f - z))); + + while (Q > std::abs(Af) && Q > std::abs(Ad) && n <= 100) { + double sx = std::sqrt(x1); + double sy = std::sqrt(y1); + double sz = std::sqrt(z1); + double lam = sx * sy + sx * sz + sy * sz; + seriesd += seriesn / (sz * (z1 + lam)); + x1 = (x1 + lam) / 4.0; + y1 = (y1 + lam) / 4.0; + z1 = (z1 + lam) / 4.0; + Af = (x1 + y1 + z1) / 3.0; + Ad = (Ad + lam) / 4.0; + n += 1; + Q /= 4.0; + seriesn /= 4.0; + } + + Xf = (A0f - x) / Af / (1 << 2 * n); + Yf = (A0f - y) / Af / (1 << 2 * n); + Zf = -(Xf + Yf); + + E2f = Xf * Yf - Zf * Zf; + E3f = Xf * Yf * Zf; + + ret = scalef * (1.0 - E2f / 10.0 + E3f / 14.0 + E2f * E2f / 24.0 - 3.0 * E2f * E3f / 44.0) / sqrt(Af); + + Xd = (A0d - x) / Ad / (1 << 2 * n); + Yd = (A0d - y) / Ad / (1 << 2 * n); + Zd = -(Xd + Yd) / 3.0; + + E2d = Xd * Yd - 6.0 * Zd * Zd; + E3d = (3 * Xd * Yd - 8.0 * Zd * Zd) * Zd; + E4d = 3.0 * (Xd * Yd - Zd * Zd) * Zd * Zd; + E5d = Xd * Yd * Zd * Zd * Zd; + + ret -= scaled * + (1.0 - 3.0 * E2d / 14.0 + E3d / 6.0 + 9.0 * E2d * E2d / 88.0 - 3.0 * E4d / 22.0 - + 9.0 * E2d * E3d / 52.0 + 3.0 * E5d / 26.0) / + (1 << 2 * n) / Ad / sqrt(Ad); + ret -= 3.0 * scaled * seriesd; + return ret; + } + + } // namespace detail + + XSF_HOST_DEVICE inline double ellie(double phi, double m) { + double a, b, c, e, temp; + double lphi, t, E, denom, npio2; + int d, mod, sign; + + if (std::isnan(phi) || std::isnan(m)) + return std::numeric_limits::quiet_NaN(); + if (m > 1.0) + return std::numeric_limits::quiet_NaN(); + ; + if (std::isinf(phi)) + return phi; + if (std::isinf(m)) + return -m; + if (m == 0.0) + return (phi); + lphi = phi; + npio2 = std::floor(lphi / M_PI_2); + if (std::fmod(std::abs(npio2), 2.0) == 1.0) + npio2 += 1; + lphi = lphi - npio2 * M_PI_2; + if (lphi < 0.0) { + lphi = -lphi; + sign = -1; + } else { + sign = 1; + } + a = 1.0 - m; + E = ellpe(m); + if (a == 0.0) { + temp = std::sin(lphi); + goto done; + } + if (a > 1.0) { + temp = detail::ellie_neg_m(lphi, m); + goto done; + } + + if (lphi < 0.135) { + double m11 = (((((-7.0 / 2816.0) * m + (5.0 / 1056.0)) * m - (7.0 / 2640.0)) * m + (17.0 / 41580.0)) * m - + (1.0 / 155925.0)) * + m; + double m9 = ((((-5.0 / 1152.0) * m + (1.0 / 144.0)) * m - (1.0 / 360.0)) * m + (1.0 / 5670.0)) * m; + double m7 = ((-m / 112.0 + (1.0 / 84.0)) * m - (1.0 / 315.0)) * m; + double m5 = (-m / 40.0 + (1.0 / 30)) * m; + double m3 = -m / 6.0; + double p2 = lphi * lphi; + + temp = ((((m11 * p2 + m9) * p2 + m7) * p2 + m5) * p2 + m3) * p2 * lphi + lphi; + goto done; + } + t = std::tan(lphi); + b = std::sqrt(a); + /* Thanks to Brian Fitzgerald + * for pointing out an instability near odd multiples of pi/2. */ + if (std::abs(t) > 10.0) { + /* Transform the amplitude */ + e = 1.0 / (b * t); + /* ... but avoid multiple recursions. */ + if (std::abs(e) < 10.0) { + e = std::atan(e); + temp = E + m * std::sin(lphi) * std::sin(e) - ellie(e, m); + goto done; + } + } + c = std::sqrt(m); + a = 1.0; + d = 1; + e = 0.0; + mod = 0; + + while (std::abs(c / a) > detail::MACHEP) { + temp = b / a; + lphi = lphi + atan(t * temp) + mod * M_PI; + denom = 1 - temp * t * t; + if (std::abs(denom) > 10 * detail::MACHEP) { + t = t * (1.0 + temp) / denom; + mod = (lphi + M_PI_2) / M_PI; + } else { + t = std::tan(lphi); + mod = static_cast(std::floor((lphi - std::atan(t)) / M_PI)); + } + c = (a - b) / 2.0; + temp = std::sqrt(a * b); + a = (a + b) / 2.0; + b = temp; + d += d; + e += c * std::sin(lphi); + } + + temp = E / ellpk(1.0 - m); + temp *= (std::atan(t) + mod * M_PI) / (d * a); + temp += e; + + done: + + if (sign < 0) + temp = -temp; + temp += npio2 * E; + return (temp); + } + +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellik.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellik.h new file mode 100644 index 0000000000000000000000000000000000000000..c05b3ec76c2e9a3acbc947842e0a849f5ba837e0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellik.h @@ -0,0 +1,251 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* ellik.c + * + * Incomplete elliptic integral of the first kind + * + * + * + * SYNOPSIS: + * + * double phi, m, y, ellik(); + * + * y = ellik( phi, m ); + * + * + * + * DESCRIPTION: + * + * Approximates the integral + * + * + * + * phi + * - + * | | + * | dt + * F(phi | m) = | ------------------ + * | 2 + * | | sqrt( 1 - m sin t ) + * - + * 0 + * + * of amplitude phi and modulus m, using the arithmetic - + * geometric mean algorithm. + * + * + * + * + * ACCURACY: + * + * Tested at random points with m in [0, 1] and phi as indicated. + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -10,10 200000 7.4e-16 1.0e-16 + * + * + */ + +/* + * Cephes Math Library Release 2.0: April, 1987 + * Copyright 1984, 1987 by Stephen L. Moshier + * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 + */ +/* Copyright 2014, Eric W. Moore */ + +/* Incomplete elliptic integral of first kind */ +#pragma once + +#include "../config.h" +#include "../error.h" +#include "const.h" +#include "ellpk.h" + +namespace xsf { +namespace cephes { + + namespace detail { + + /* To calculate legendre's incomplete elliptical integral of the first kind for + * negative m, we use a power series in phi for small m*phi*phi, an asymptotic + * series in m for large m*phi*phi* and the relation to Carlson's symmetric + * integral of the first kind. + * + * F(phi, m) = sin(phi) * R_F(cos(phi)^2, 1 - m * sin(phi)^2, 1.0) + * = R_F(c-1, c-m, c) + * + * where c = csc(phi)^2. We use the second form of this for (approximately) + * phi > 1/(sqrt(DBL_MAX) ~ 1e-154, where csc(phi)^2 overflows. Elsewhere we + * use the first form, accounting for the smallness of phi. + * + * The algorithm used is described in Carlson, B. C. Numerical computation of + * real or complex elliptic integrals. (1994) https://arxiv.org/abs/math/9409227 + * Most variable names reflect Carlson's usage. + * + * In this routine, we assume m < 0 and 0 > phi > pi/2. + */ + XSF_HOST_DEVICE inline double ellik_neg_m(double phi, double m) { + double x, y, z, x1, y1, z1, A0, A, Q, X, Y, Z, E2, E3, scale; + int n = 0; + double mpp = (m * phi) * phi; + + if (-mpp < 1e-6 && phi < -m) { + return phi + (-mpp * phi * phi / 30.0 + 3.0 * mpp * mpp / 40.0 + mpp / 6.0) * phi; + } + + if (-mpp > 4e7) { + double sm = std::sqrt(-m); + double sp = std::sin(phi); + double cp = std::cos(phi); + + double a = std::log(4 * sp * sm / (1 + cp)); + double b = -(1 + cp / sp / sp - a) / 4 / m; + return (a + b) / sm; + } + + if (phi > 1e-153 && m > -1e305) { + double s = std::sin(phi); + double csc2 = 1.0 / (s * s); + scale = 1.0; + x = 1.0 / (std::tan(phi) * std::tan(phi)); + y = csc2 - m; + z = csc2; + } else { + scale = phi; + x = 1.0; + y = 1 - m * scale * scale; + z = 1.0; + } + + if (x == y && x == z) { + return scale / std::sqrt(x); + } + + A0 = (x + y + z) / 3.0; + A = A0; + x1 = x; + y1 = y; + z1 = z; + /* Carlson gives 1/pow(3*r, 1.0/6.0) for this constant. if r == eps, + * it is ~338.38. */ + Q = 400.0 * std::fmax(std::abs(A0 - x), std::fmax(std::abs(A0 - y), std::abs(A0 - z))); + + while (Q > std::abs(A) && n <= 100) { + double sx = std::sqrt(x1); + double sy = std::sqrt(y1); + double sz = std::sqrt(z1); + double lam = sx * sy + sx * sz + sy * sz; + x1 = (x1 + lam) / 4.0; + y1 = (y1 + lam) / 4.0; + z1 = (z1 + lam) / 4.0; + A = (x1 + y1 + z1) / 3.0; + n += 1; + Q /= 4; + } + X = (A0 - x) / A / (1 << 2 * n); + Y = (A0 - y) / A / (1 << 2 * n); + Z = -(X + Y); + + E2 = X * Y - Z * Z; + E3 = X * Y * Z; + + return scale * (1.0 - E2 / 10.0 + E3 / 14.0 + E2 * E2 / 24.0 - 3.0 * E2 * E3 / 44.0) / sqrt(A); + } + + } // namespace detail + + XSF_HOST_DEVICE inline double ellik(double phi, double m) { + double a, b, c, e, temp, t, K, denom, npio2; + int d, mod, sign; + + if (std::isnan(phi) || std::isnan(m)) + return std::numeric_limits::quiet_NaN(); + if (m > 1.0) + return std::numeric_limits::quiet_NaN(); + if (std::isinf(phi) || std::isinf(m)) { + if (std::isinf(m) && std::isfinite(phi)) + return 0.0; + else if (std::isinf(phi) && std::isfinite(m)) + return phi; + else + return std::numeric_limits::quiet_NaN(); + } + if (m == 0.0) + return (phi); + a = 1.0 - m; + if (a == 0.0) { + if (std::abs(phi) >= (double) M_PI_2) { + set_error("ellik", SF_ERROR_SINGULAR, NULL); + return (std::numeric_limits::infinity()); + } + /* DLMF 19.6.8, and 4.23.42 */ + return std::asinh(std::tan(phi)); + } + npio2 = floor(phi / M_PI_2); + if (std::fmod(std::abs(npio2), 2.0) == 1.0) + npio2 += 1; + if (npio2 != 0.0) { + K = ellpk(a); + phi = phi - npio2 * M_PI_2; + } else + K = 0.0; + if (phi < 0.0) { + phi = -phi; + sign = -1; + } else + sign = 0; + if (a > 1.0) { + temp = detail::ellik_neg_m(phi, m); + goto done; + } + b = std::sqrt(a); + t = std::tan(phi); + if (std::abs(t) > 10.0) { + /* Transform the amplitude */ + e = 1.0 / (b * t); + /* ... but avoid multiple recursions. */ + if (std::abs(e) < 10.0) { + e = std::atan(e); + if (npio2 == 0) + K = ellpk(a); + temp = K - ellik(e, m); + goto done; + } + } + a = 1.0; + c = std::sqrt(m); + d = 1; + mod = 0; + + while (std::abs(c / a) > detail::MACHEP) { + temp = b / a; + phi = phi + atan(t * temp) + mod * M_PI; + denom = 1.0 - temp * t * t; + if (std::abs(denom) > 10 * detail::MACHEP) { + t = t * (1.0 + temp) / denom; + mod = (phi + M_PI_2) / M_PI; + } else { + t = std::tan(phi); + mod = static_cast(std::floor((phi - std::atan(t)) / M_PI)); + } + c = (a - b) / 2.0; + temp = std::sqrt(a * b); + a = (a + b) / 2.0; + b = temp; + d += d; + } + + temp = (std::atan(t) + mod * M_PI) / (d * a); + + done: + if (sign < 0) + temp = -temp; + temp += npio2 * K; + return (temp); + } + +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellpe.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellpe.h new file mode 100644 index 0000000000000000000000000000000000000000..bc7c51f11acb13179aaffb301d052308722f8cfa --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellpe.h @@ -0,0 +1,107 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* ellpe.c + * + * Complete elliptic integral of the second kind + * + * + * + * SYNOPSIS: + * + * double m, y, ellpe(); + * + * y = ellpe( m ); + * + * + * + * DESCRIPTION: + * + * Approximates the integral + * + * + * pi/2 + * - + * | | 2 + * E(m) = | sqrt( 1 - m sin t ) dt + * | | + * - + * 0 + * + * Where m = 1 - m1, using the approximation + * + * P(x) - x log x Q(x). + * + * Though there are no singularities, the argument m1 is used + * internally rather than m for compatibility with ellpk(). + * + * E(1) = 1; E(0) = pi/2. + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0, 1 10000 2.1e-16 7.3e-17 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * ellpe domain x<0, x>1 0.0 + * + */ + +/* ellpe.c */ + +/* Elliptic integral of second kind */ + +/* + * Cephes Math Library, Release 2.1: February, 1989 + * Copyright 1984, 1987, 1989 by Stephen L. Moshier + * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 + * + * Feb, 2002: altered by Travis Oliphant + * so that it is called with argument m + * (which gets immediately converted to m1 = 1-m) + */ +#pragma once + +#include "../config.h" +#include "../error.h" +#include "polevl.h" + +namespace xsf { +namespace cephes { + + namespace detail { + + constexpr double ellpe_P[] = {1.53552577301013293365E-4, 2.50888492163602060990E-3, 8.68786816565889628429E-3, + 1.07350949056076193403E-2, 7.77395492516787092951E-3, 7.58395289413514708519E-3, + 1.15688436810574127319E-2, 2.18317996015557253103E-2, 5.68051945617860553470E-2, + 4.43147180560990850618E-1, 1.00000000000000000299E0}; + + constexpr double ellpe_Q[] = {3.27954898576485872656E-5, 1.00962792679356715133E-3, 6.50609489976927491433E-3, + 1.68862163993311317300E-2, 2.61769742454493659583E-2, 3.34833904888224918614E-2, + 4.27180926518931511717E-2, 5.85936634471101055642E-2, 9.37499997197644278445E-2, + 2.49999999999888314361E-1}; + + } // namespace detail + + XSF_HOST_DEVICE inline double ellpe(double x) { + x = 1.0 - x; + if (x <= 0.0) { + if (x == 0.0) + return (1.0); + set_error("ellpe", SF_ERROR_DOMAIN, NULL); + return (std::numeric_limits::quiet_NaN()); + } + if (x > 1.0) { + return ellpe(1.0 - 1 / x) * std::sqrt(x); + } + return (polevl(x, detail::ellpe_P, 10) - std::log(x) * (x * polevl(x, detail::ellpe_Q, 9))); + } + +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellpk.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellpk.h new file mode 100644 index 0000000000000000000000000000000000000000..39ebf7e80b193d385acb45feead4b91632830642 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/ellpk.h @@ -0,0 +1,117 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* ellpk.c + * + * Complete elliptic integral of the first kind + * + * + * + * SYNOPSIS: + * + * double m1, y, ellpk(); + * + * y = ellpk( m1 ); + * + * + * + * DESCRIPTION: + * + * Approximates the integral + * + * + * + * pi/2 + * - + * | | + * | dt + * K(m) = | ------------------ + * | 2 + * | | sqrt( 1 - m sin t ) + * - + * 0 + * + * where m = 1 - m1, using the approximation + * + * P(x) - log x Q(x). + * + * The argument m1 is used internally rather than m so that the logarithmic + * singularity at m = 1 will be shifted to the origin; this + * preserves maximum accuracy. + * + * K(0) = pi/2. + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,1 30000 2.5e-16 6.8e-17 + * + * ERROR MESSAGES: + * + * message condition value returned + * ellpk domain x<0, x>1 0.0 + * + */ + +/* ellpk.c */ + +/* + * Cephes Math Library, Release 2.0: April, 1987 + * Copyright 1984, 1987 by Stephen L. Moshier + * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 + */ +#pragma once + +#include "../config.h" +#include "../error.h" +#include "const.h" +#include "polevl.h" + +namespace xsf { +namespace cephes { + + namespace detail { + + constexpr double ellpk_P[] = {1.37982864606273237150E-4, 2.28025724005875567385E-3, 7.97404013220415179367E-3, + 9.85821379021226008714E-3, 6.87489687449949877925E-3, 6.18901033637687613229E-3, + 8.79078273952743772254E-3, 1.49380448916805252718E-2, 3.08851465246711995998E-2, + 9.65735902811690126535E-2, 1.38629436111989062502E0}; + + constexpr double ellpk_Q[] = {2.94078955048598507511E-5, 9.14184723865917226571E-4, 5.94058303753167793257E-3, + 1.54850516649762399335E-2, 2.39089602715924892727E-2, 3.01204715227604046988E-2, + 3.73774314173823228969E-2, 4.88280347570998239232E-2, 7.03124996963957469739E-2, + 1.24999999999870820058E-1, 4.99999999999999999821E-1}; + + constexpr double ellpk_C1 = 1.3862943611198906188E0; /* log(4) */ + + } // namespace detail + + XSF_HOST_DEVICE inline double ellpk(double x) { + + if (x < 0.0) { + set_error("ellpk", SF_ERROR_DOMAIN, NULL); + return (std::numeric_limits::quiet_NaN()); + } + + if (x > 1.0) { + if (std::isinf(x)) { + return 0.0; + } + return ellpk(1 / x) / std::sqrt(x); + } + + if (x > detail::MACHEP) { + return (polevl(x, detail::ellpk_P, 10) - std::log(x) * polevl(x, detail::ellpk_Q, 10)); + } else { + if (x == 0.0) { + set_error("ellpk", SF_ERROR_SINGULAR, NULL); + return (std::numeric_limits::infinity()); + } else { + return (detail::ellpk_C1 - 0.5 * std::log(x)); + } + } + } +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/expn.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/expn.h new file mode 100644 index 0000000000000000000000000000000000000000..8b0b07eab7a94fb1519566da4ef9036fb67edec6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/expn.h @@ -0,0 +1,260 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* expn.c + * + * Exponential integral En + * + * + * + * SYNOPSIS: + * + * int n; + * double x, y, expn(); + * + * y = expn( n, x ); + * + * + * + * DESCRIPTION: + * + * Evaluates the exponential integral + * + * inf. + * - + * | | -xt + * | e + * E (x) = | ---- dt. + * n | n + * | | t + * - + * 1 + * + * + * Both n and x must be nonnegative. + * + * The routine employs either a power series, a continued + * fraction, or an asymptotic formula depending on the + * relative values of n and x. + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0, 30 10000 1.7e-15 3.6e-16 + * + */ + +/* expn.c */ + +/* Cephes Math Library Release 1.1: March, 1985 + * Copyright 1985 by Stephen L. Moshier + * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 */ + +/* Sources + * [1] NIST, "The Digital Library of Mathematical Functions", dlmf.nist.gov + */ + +/* Scipy changes: + * - 09-10-2016: improved asymptotic expansion for large n + */ + +#pragma once + +#include "../config.h" +#include "../error.h" +#include "const.h" +#include "rgamma.h" +#include "polevl.h" + +namespace xsf { +namespace cephes { + + namespace detail { + + constexpr int expn_nA = 13; + constexpr double expn_A0[] = {1.00000000000000000}; + constexpr double expn_A1[] = {1.00000000000000000}; + constexpr double expn_A2[] = {-2.00000000000000000, 1.00000000000000000}; + constexpr double expn_A3[] = {6.00000000000000000, -8.00000000000000000, 1.00000000000000000}; + constexpr double expn_A4[] = {-24.0000000000000000, 58.0000000000000000, -22.0000000000000000, + 1.00000000000000000}; + constexpr double expn_A5[] = {120.000000000000000, -444.000000000000000, 328.000000000000000, + -52.0000000000000000, 1.00000000000000000}; + constexpr double expn_A6[] = {-720.000000000000000, 3708.00000000000000, -4400.00000000000000, + 1452.00000000000000, -114.000000000000000, 1.00000000000000000}; + constexpr double expn_A7[] = {5040.00000000000000, -33984.0000000000000, 58140.0000000000000, + -32120.0000000000000, 5610.00000000000000, -240.000000000000000, + 1.00000000000000000}; + constexpr double expn_A8[] = {-40320.0000000000000, 341136.000000000000, -785304.000000000000, + 644020.000000000000, -195800.000000000000, 19950.0000000000000, + -494.000000000000000, 1.00000000000000000}; + constexpr double expn_A9[] = {362880.000000000000, -3733920.00000000000, 11026296.0000000000, + -12440064.0000000000, 5765500.00000000000, -1062500.00000000000, + 67260.0000000000000, -1004.00000000000000, 1.00000000000000000}; + constexpr double expn_A10[] = {-3628800.00000000000, 44339040.0000000000, -162186912.000000000, + 238904904.000000000, -155357384.000000000, 44765000.0000000000, + -5326160.00000000000, 218848.000000000000, -2026.00000000000000, + 1.00000000000000000}; + constexpr double expn_A11[] = {39916800.0000000000, -568356480.000000000, 2507481216.00000000, + -4642163952.00000000, 4002695088.00000000, -1648384304.00000000, + 314369720.000000000, -25243904.0000000000, 695038.000000000000, + -4072.00000000000000, 1.00000000000000000}; + constexpr double expn_A12[] = {-479001600.000000000, 7827719040.00000000, -40788301824.0000000, + 92199790224.0000000, -101180433024.000000, 56041398784.0000000, + -15548960784.0000000, 2051482776.00000000, -114876376.000000000, + 2170626.00000000000, -8166.00000000000000, 1.00000000000000000}; + constexpr const double *expn_A[] = {expn_A0, expn_A1, expn_A2, expn_A3, expn_A4, expn_A5, expn_A6, + expn_A7, expn_A8, expn_A9, expn_A10, expn_A11, expn_A12}; + constexpr int expn_Adegs[] = {0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}; + + /* Asymptotic expansion for large n, DLMF 8.20(ii) */ + XSF_HOST_DEVICE double expn_large_n(int n, double x) { + int k; + double p = n; + double lambda = x / p; + double multiplier = 1 / p / (lambda + 1) / (lambda + 1); + double fac = 1; + double res = 1; /* A[0] = 1 */ + double expfac, term; + + expfac = std::exp(-lambda * p) / (lambda + 1) / p; + if (expfac == 0) { + set_error("expn", SF_ERROR_UNDERFLOW, NULL); + return 0; + } + + /* Do the k = 1 term outside the loop since A[1] = 1 */ + fac *= multiplier; + res += fac; + + for (k = 2; k < expn_nA; k++) { + fac *= multiplier; + term = fac * polevl(lambda, expn_A[k], expn_Adegs[k]); + res += term; + if (std::abs(term) < MACHEP * std::abs(res)) { + break; + } + } + + return expfac * res; + } + } // namespace detail + + XSF_HOST_DEVICE double expn(int n, double x) { + double ans, r, t, yk, xk; + double pk, pkm1, pkm2, qk, qkm1, qkm2; + double psi, z; + int i, k; + constexpr double big = 1.44115188075855872E+17; + + if (std::isnan(x)) { + return std::numeric_limits::quiet_NaN(); + } else if (n < 0 || x < 0) { + set_error("expn", SF_ERROR_DOMAIN, NULL); + return std::numeric_limits::quiet_NaN(); + } + + if (x > detail::MAXLOG) { + return (0.0); + } + + if (x == 0.0) { + if (n < 2) { + set_error("expn", SF_ERROR_SINGULAR, NULL); + return std::numeric_limits::infinity(); + } else { + return (1.0 / (n - 1.0)); + } + } + + if (n == 0) { + return (std::exp(-x) / x); + } + + /* Asymptotic expansion for large n, DLMF 8.20(ii) */ + if (n > 50) { + ans = detail::expn_large_n(n, x); + return ans; + } + + /* Continued fraction, DLMF 8.19.17 */ + if (x > 1.0) { + k = 1; + pkm2 = 1.0; + qkm2 = x; + pkm1 = 1.0; + qkm1 = x + n; + ans = pkm1 / qkm1; + + do { + k += 1; + if (k & 1) { + yk = 1.0; + xk = n + (k - 1) / 2; + } else { + yk = x; + xk = k / 2; + } + pk = pkm1 * yk + pkm2 * xk; + qk = qkm1 * yk + qkm2 * xk; + if (qk != 0) { + r = pk / qk; + t = std::abs((ans - r) / r); + ans = r; + } else { + t = 1.0; + } + pkm2 = pkm1; + pkm1 = pk; + qkm2 = qkm1; + qkm1 = qk; + if (std::abs(pk) > big) { + pkm2 /= big; + pkm1 /= big; + qkm2 /= big; + qkm1 /= big; + } + } while (t > detail::MACHEP); + + ans *= std::exp(-x); + return ans; + } + + /* Power series expansion, DLMF 8.19.8 */ + psi = -detail::SCIPY_EULER - std::log(x); + for (i = 1; i < n; i++) { + psi = psi + 1.0 / i; + } + + z = -x; + xk = 0.0; + yk = 1.0; + pk = 1.0 - n; + if (n == 1) { + ans = 0.0; + } else { + ans = 1.0 / pk; + } + do { + xk += 1.0; + yk *= z / xk; + pk += 1.0; + if (pk != 0.0) { + ans += yk / pk; + } + if (ans != 0.0) + t = std::abs(yk / ans); + else + t = 1.0; + } while (t > detail::MACHEP); + k = xk; + t = n; + r = n - 1; + ans = (std::pow(z, r) * psi * rgamma(t)) - ans; + return ans; + } + +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/gamma.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/gamma.h new file mode 100644 index 0000000000000000000000000000000000000000..1ede1571a67ec9ce54bb6aa1afa1f17f5708f0c0 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/gamma.h @@ -0,0 +1,398 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* + * Gamma function + * + * + * + * SYNOPSIS: + * + * double x, y, Gamma(); + * + * y = Gamma( x ); + * + * + * + * DESCRIPTION: + * + * Returns Gamma function of the argument. The result is + * correctly signed. + * + * Arguments |x| <= 34 are reduced by recurrence and the function + * approximated by a rational function of degree 6/7 in the + * interval (2,3). Large arguments are handled by Stirling's + * formula. Large negative arguments are made positive using + * a reflection formula. + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -170,-33 20000 2.3e-15 3.3e-16 + * IEEE -33, 33 20000 9.4e-16 2.2e-16 + * IEEE 33, 171.6 20000 2.3e-15 3.2e-16 + * + * Error for arguments outside the test range will be larger + * owing to error amplification by the exponential function. + * + */ + +/* lgam() + * + * Natural logarithm of Gamma function + * + * + * + * SYNOPSIS: + * + * double x, y, lgam(); + * + * y = lgam( x ); + * + * + * + * DESCRIPTION: + * + * Returns the base e (2.718...) logarithm of the absolute + * value of the Gamma function of the argument. + * + * For arguments greater than 13, the logarithm of the Gamma + * function is approximated by the logarithmic version of + * Stirling's formula using a polynomial approximation of + * degree 4. Arguments between -33 and +33 are reduced by + * recurrence to the interval [2,3] of a rational approximation. + * The cosecant reflection formula is employed for arguments + * less than -33. + * + * Arguments greater than MAXLGM return INFINITY and an error + * message. MAXLGM = 2.556348e305 for IEEE arithmetic. + * + * + * + * ACCURACY: + * + * + * arithmetic domain # trials peak rms + * IEEE 0, 3 28000 5.4e-16 1.1e-16 + * IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17 + * The error criterion was relative when the function magnitude + * was greater than one but absolute when it was less than one. + * + * The following test used the relative error criterion, though + * at certain points the relative error could be much higher than + * indicated. + * IEEE -200, -4 10000 4.8e-16 1.3e-16 + * + */ + +/* + * Cephes Math Library Release 2.2: July, 1992 + * Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier + * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 + */ +#pragma once + +#include "../config.h" +#include "../error.h" +#include "const.h" +#include "polevl.h" +#include "trig.h" + +namespace xsf { +namespace cephes { + namespace detail { + constexpr double gamma_P[] = {1.60119522476751861407E-4, 1.19135147006586384913E-3, 1.04213797561761569935E-2, + 4.76367800457137231464E-2, 2.07448227648435975150E-1, 4.94214826801497100753E-1, + 9.99999999999999996796E-1}; + + constexpr double gamma_Q[] = {-2.31581873324120129819E-5, 5.39605580493303397842E-4, -4.45641913851797240494E-3, + 1.18139785222060435552E-2, 3.58236398605498653373E-2, -2.34591795718243348568E-1, + 7.14304917030273074085E-2, 1.00000000000000000320E0}; + + /* Stirling's formula for the Gamma function */ + constexpr double gamma_STIR[5] = { + 7.87311395793093628397E-4, -2.29549961613378126380E-4, -2.68132617805781232825E-3, + 3.47222221605458667310E-3, 8.33333333333482257126E-2, + }; + + constexpr double MAXSTIR = 143.01608; + + /* Gamma function computed by Stirling's formula. + * The polynomial STIR is valid for 33 <= x <= 172. + */ + XSF_HOST_DEVICE inline double stirf(double x) { + double y, w, v; + + if (x >= MAXGAM) { + return (std::numeric_limits::infinity()); + } + w = 1.0 / x; + w = 1.0 + w * xsf::cephes::polevl(w, gamma_STIR, 4); + y = std::exp(x); + if (x > MAXSTIR) { /* Avoid overflow in pow() */ + v = std::pow(x, 0.5 * x - 0.25); + y = v * (v / y); + } else { + y = std::pow(x, x - 0.5) / y; + } + y = SQRTPI * y * w; + return (y); + } + } // namespace detail + + XSF_HOST_DEVICE inline double Gamma(double x) { + double p, q, z; + int i; + int sgngam = 1; + + if (!std::isfinite(x)) { + if (x > 0) { + // gamma(+inf) = +inf + return x; + } + // gamma(NaN) and gamma(-inf) both should equal NaN. + return std::numeric_limits::quiet_NaN(); + } + + if (x == 0) { + /* For pole at zero, value depends on sign of zero. + * +inf when approaching from right, -inf when approaching + * from left. */ + return std::copysign(std::numeric_limits::infinity(), x); + } + + q = std::abs(x); + + if (q > 33.0) { + if (x < 0.0) { + p = std::floor(q); + if (p == q) { + // x is a negative integer. This is a pole. + set_error("Gamma", SF_ERROR_SINGULAR, NULL); + return (std::numeric_limits::quiet_NaN()); + } + i = p; + if ((i & 1) == 0) { + sgngam = -1; + } + z = q - p; + if (z > 0.5) { + p += 1.0; + z = q - p; + } + z = q * sinpi(z); + if (z == 0.0) { + return (sgngam * std::numeric_limits::infinity()); + } + z = std::abs(z); + z = M_PI / (z * detail::stirf(q)); + } else { + z = detail::stirf(x); + } + return (sgngam * z); + } + + z = 1.0; + while (x >= 3.0) { + x -= 1.0; + z *= x; + } + + while (x < 0.0) { + if (x > -1.E-9) { + goto small; + } + z /= x; + x += 1.0; + } + + while (x < 2.0) { + if (x < 1.e-9) { + goto small; + } + z /= x; + x += 1.0; + } + + if (x == 2.0) { + return (z); + } + + x -= 2.0; + p = polevl(x, detail::gamma_P, 6); + q = polevl(x, detail::gamma_Q, 7); + return (z * p / q); + + small: + if (x == 0.0) { + /* For this to have happened, x must have started as a negative integer. */ + set_error("Gamma", SF_ERROR_SINGULAR, NULL); + return (std::numeric_limits::quiet_NaN()); + } else + return (z / ((1.0 + 0.5772156649015329 * x) * x)); + } + + namespace detail { + /* A[]: Stirling's formula expansion of log Gamma + * B[], C[]: log Gamma function between 2 and 3 + */ + constexpr double gamma_A[] = {8.11614167470508450300E-4, -5.95061904284301438324E-4, 7.93650340457716943945E-4, + -2.77777777730099687205E-3, 8.33333333333331927722E-2}; + + constexpr double gamma_B[] = {-1.37825152569120859100E3, -3.88016315134637840924E4, -3.31612992738871184744E5, + -1.16237097492762307383E6, -1.72173700820839662146E6, -8.53555664245765465627E5}; + + constexpr double gamma_C[] = { + /* 1.00000000000000000000E0, */ + -3.51815701436523470549E2, -1.70642106651881159223E4, -2.20528590553854454839E5, + -1.13933444367982507207E6, -2.53252307177582951285E6, -2.01889141433532773231E6}; + + /* log( sqrt( 2*pi ) ) */ + constexpr double LS2PI = 0.91893853320467274178; + + constexpr double MAXLGM = 2.556348e305; + + /* Disable optimizations for this function on 32 bit systems when compiling with GCC. + * We've found that enabling optimizations can result in degraded precision + * for this asymptotic approximation in that case. */ +#if defined(__GNUC__) && defined(__i386__) +#pragma GCC push_options +#pragma GCC optimize("00") +#endif + XSF_HOST_DEVICE inline double lgam_large_x(double x) { + double q = (x - 0.5) * std::log(x) - x + LS2PI; + if (x > 1.0e8) { + return (q); + } + double p = 1.0 / (x * x); + p = ((7.9365079365079365079365e-4 * p - 2.7777777777777777777778e-3) * p + 0.0833333333333333333333) / x; + return q + p; + } +#if defined(__GNUC__) && defined(__i386__) +#pragma GCC pop_options +#endif + + XSF_HOST_DEVICE inline double lgam_sgn(double x, int *sign) { + double p, q, u, w, z; + int i; + + *sign = 1; + + if (!std::isfinite(x)) { + return x; + } + + if (x < -34.0) { + q = -x; + w = lgam_sgn(q, sign); + p = std::floor(q); + if (p == q) { + lgsing: + set_error("lgam", SF_ERROR_SINGULAR, NULL); + return (std::numeric_limits::infinity()); + } + i = p; + if ((i & 1) == 0) { + *sign = -1; + } else { + *sign = 1; + } + z = q - p; + if (z > 0.5) { + p += 1.0; + z = p - q; + } + z = q * sinpi(z); + if (z == 0.0) { + goto lgsing; + } + /* z = log(M_PI) - log( z ) - w; */ + z = LOGPI - std::log(z) - w; + return (z); + } + + if (x < 13.0) { + z = 1.0; + p = 0.0; + u = x; + while (u >= 3.0) { + p -= 1.0; + u = x + p; + z *= u; + } + while (u < 2.0) { + if (u == 0.0) { + goto lgsing; + } + z /= u; + p += 1.0; + u = x + p; + } + if (z < 0.0) { + *sign = -1; + z = -z; + } else { + *sign = 1; + } + if (u == 2.0) { + return (std::log(z)); + } + p -= 2.0; + x = x + p; + p = x * polevl(x, gamma_B, 5) / p1evl(x, gamma_C, 6); + return (std::log(z) + p); + } + + if (x > MAXLGM) { + return (*sign * std::numeric_limits::infinity()); + } + + if (x >= 1000.0) { + return lgam_large_x(x); + } + + q = (x - 0.5) * std::log(x) - x + LS2PI; + p = 1.0 / (x * x); + return q + polevl(p, gamma_A, 4) / x; + } + } // namespace detail + + /* Logarithm of Gamma function */ + XSF_HOST_DEVICE inline double lgam(double x) { + int sign; + return detail::lgam_sgn(x, &sign); + } + + /* Sign of the Gamma function */ + XSF_HOST_DEVICE inline double gammasgn(double x) { + double fx; + + if (std::isnan(x)) { + return x; + } + if (x > 0) { + return 1.0; + } + if (x == 0) { + return std::copysign(1.0, x); + } + if (std::isinf(x)) { + // x > 0 case handled, so x must be negative infinity. + return std::numeric_limits::quiet_NaN(); + } + fx = std::floor(x); + if (x - fx == 0.0) { + return std::numeric_limits::quiet_NaN(); + } + // sign of gamma for x in (-n, -n+1) for positive integer n is (-1)^n. + if (static_cast(fx) % 2) { + return -1.0; + } + return 1.0; + } + +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/hyp2f1.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/hyp2f1.h new file mode 100644 index 0000000000000000000000000000000000000000..f9ec54bb20326552f5748ad9360dfecfbe18d660 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/cephes/hyp2f1.h @@ -0,0 +1,596 @@ +/* Translated into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* hyp2f1.c + * + * Gauss hypergeometric function F + * 2 1 + * + * + * SYNOPSIS: + * + * double a, b, c, x, y, hyp2f1(); + * + * y = hyp2f1( a, b, c, x ); + * + * + * DESCRIPTION: + * + * + * hyp2f1( a, b, c, x ) = F ( a, b; c; x ) + * 2 1 + * + * inf. + * - a(a+1)...(a+k) b(b+1)...(b+k) k+1 + * = 1 + > ----------------------------- x . + * - c(c+1)...(c+k) (k+1)! + * k = 0 + * + * Cases addressed are + * Tests and escapes for negative integer a, b, or c + * Linear transformation if c - a or c - b negative integer + * Special case c = a or c = b + * Linear transformation for x near +1 + * Transformation for x < -0.5 + * Psi function expansion if x > 0.5 and c - a - b integer + * Conditionally, a recurrence on c to make c-a-b > 0 + * + * x < -1 AMS 15.3.7 transformation applied (Travis Oliphant) + * valid for b,a,c,(b-a) != integer and (c-a),(c-b) != negative integer + * + * x >= 1 is rejected (unless special cases are present) + * + * The parameters a, b, c are considered to be integer + * valued if they are within 1.0e-14 of the nearest integer + * (1.0e-13 for IEEE arithmetic). + * + * ACCURACY: + * + * + * Relative error (-1 < x < 1): + * arithmetic domain # trials peak rms + * IEEE -1,7 230000 1.2e-11 5.2e-14 + * + * Several special cases also tested with a, b, c in + * the range -7 to 7. + * + * ERROR MESSAGES: + * + * A "partial loss of precision" message is printed if + * the internally estimated relative error exceeds 1^-12. + * A "singularity" message is printed on overflow or + * in cases not addressed (such as x < -1). + */ + +/* + * Cephes Math Library Release 2.8: June, 2000 + * Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier + */ + +#pragma once + +#include "../config.h" +#include "../error.h" + +#include "const.h" +#include "gamma.h" +#include "rgamma.h" +#include "psi.h" + +namespace xsf { +namespace cephes { + + namespace detail { + constexpr double hyp2f1_EPS = 1.0e-13; + + constexpr double hyp2f1_ETHRESH = 1.0e-12; + constexpr std::uint64_t hyp2f1_MAXITER = 10000; + + /* hys2f1 and hyp2f1ra depend on each other, so we need this prototype */ + XSF_HOST_DEVICE double hyp2f1ra(double a, double b, double c, double x, double *loss); + + /* Defining power series expansion of Gauss hypergeometric function */ + /* The `loss` parameter estimates loss of significance */ + XSF_HOST_DEVICE double hys2f1(double a, double b, double c, double x, double *loss) { + double f, g, h, k, m, s, u, umax; + std::uint64_t i; + int ib, intflag = 0; + + if (std::abs(b) > std::abs(a)) { + /* Ensure that |a| > |b| ... */ + f = b; + b = a; + a = f; + } + + ib = std::round(b); + + if (std::abs(b - ib) < hyp2f1_EPS && ib <= 0 && std::abs(b) < std::abs(a)) { + /* .. except when `b` is a smaller negative integer */ + f = b; + b = a; + a = f; + intflag = 1; + } + + if ((std::abs(a) > std::abs(c) + 1 || intflag) && std::abs(c - a) > 2 && std::abs(a) > 2) { + /* |a| >> |c| implies that large cancellation error is to be expected. + * + * We try to reduce it with the recurrence relations + */ + return hyp2f1ra(a, b, c, x, loss); + } + + i = 0; + umax = 0.0; + f = a; + g = b; + h = c; + s = 1.0; + u = 1.0; + k = 0.0; + do { + if (std::abs(h) < hyp2f1_EPS) { + *loss = 1.0; + return std::numeric_limits::infinity(); + } + m = k + 1.0; + u = u * ((f + k) * (g + k) * x / ((h + k) * m)); + s += u; + k = std::abs(u); /* remember largest term summed */ + if (k > umax) + umax = k; + k = m; + if (++i > hyp2f1_MAXITER) { /* should never happen */ + *loss = 1.0; + return (s); + } + } while (s == 0 || std::abs(u / s) > MACHEP); + + /* return estimated relative error */ + *loss = (MACHEP * umax) / fabs(s) + (MACHEP * i); + + return (s); + } + + /* Apply transformations for |x| near 1 then call the power series */ + XSF_HOST_DEVICE double hyt2f1(double a, double b, double c, double x, double *loss) { + double p, q, r, s, t, y, w, d, err, err1; + double ax, id, d1, d2, e, y1; + int i, aid, sign; + + int ia, ib, neg_int_a = 0, neg_int_b = 0; + + ia = std::round(a); + ib = std::round(b); + + if (a <= 0 && std::abs(a - ia) < hyp2f1_EPS) { /* a is a negative integer */ + neg_int_a = 1; + } + + if (b <= 0 && std::abs(b - ib) < hyp2f1_EPS) { /* b is a negative integer */ + neg_int_b = 1; + } + + err = 0.0; + s = 1.0 - x; + if (x < -0.5 && !(neg_int_a || neg_int_b)) { + if (b > a) + y = std::pow(s, -a) * hys2f1(a, c - b, c, -x / s, &err); + + else + y = std::pow(s, -b) * hys2f1(c - a, b, c, -x / s, &err); + + goto done; + } + + d = c - a - b; + id = std::round(d); /* nearest integer to d */ + + if (x > 0.9 && !(neg_int_a || neg_int_b)) { + if (std::abs(d - id) > MACHEP) { + int sgngam; + + /* test for integer c-a-b */ + /* Try the power series first */ + y = hys2f1(a, b, c, x, &err); + if (err < hyp2f1_ETHRESH) { + goto done; + } + /* If power series fails, then apply AMS55 #15.3.6 */ + q = hys2f1(a, b, 1.0 - d, s, &err); + sign = 1; + w = lgam_sgn(d, &sgngam); + sign *= sgngam; + w -= lgam_sgn(c - a, &sgngam); + sign *= sgngam; + w -= lgam_sgn(c - b, &sgngam); + sign *= sgngam; + q *= sign * std::exp(w); + r = std::pow(s, d) * hys2f1(c - a, c - b, d + 1.0, s, &err1); + sign = 1; + w = lgam_sgn(-d, &sgngam); + sign *= sgngam; + w -= lgam_sgn(a, &sgngam); + sign *= sgngam; + w -= lgam_sgn(b, &sgngam); + sign *= sgngam; + r *= sign * std::exp(w); + y = q + r; + + q = std::abs(q); /* estimate cancellation error */ + r = std::abs(r); + if (q > r) { + r = q; + } + err += err1 + (MACHEP * r) / y; + + y *= xsf::cephes::Gamma(c); + goto done; + } else { + /* Psi function expansion, AMS55 #15.3.10, #15.3.11, #15.3.12 + * + * Although AMS55 does not explicitly state it, this expansion fails + * for negative integer a or b, since the psi and Gamma functions + * involved have poles. + */ + + if (id >= 0.0) { + e = d; + d1 = d; + d2 = 0.0; + aid = id; + } else { + e = -d; + d1 = 0.0; + d2 = d; + aid = -id; + } + + ax = std::log(s); + + /* sum for t = 0 */ + y = xsf::cephes::psi(1.0) + xsf::cephes::psi(1.0 + e) - xsf::cephes::psi(a + d1) - + xsf::cephes::psi(b + d1) - ax; + y *= xsf::cephes::rgamma(e + 1.0); + + p = (a + d1) * (b + d1) * s * xsf::cephes::rgamma(e + 2.0); /* Poch for t=1 */ + t = 1.0; + do { + r = xsf::cephes::psi(1.0 + t) + xsf::cephes::psi(1.0 + t + e) - + xsf::cephes::psi(a + t + d1) - xsf::cephes::psi(b + t + d1) - ax; + q = p * r; + y += q; + p *= s * (a + t + d1) / (t + 1.0); + p *= (b + t + d1) / (t + 1.0 + e); + t += 1.0; + if (t > hyp2f1_MAXITER) { /* should never happen */ + set_error("hyp2f1", SF_ERROR_SLOW, NULL); + *loss = 1.0; + return std::numeric_limits::quiet_NaN(); + } + } while (y == 0 || std::abs(q / y) > hyp2f1_EPS); + + if (id == 0.0) { + y *= xsf::cephes::Gamma(c) / (xsf::cephes::Gamma(a) * xsf::cephes::Gamma(b)); + goto psidon; + } + + y1 = 1.0; + + if (aid == 1) + goto nosum; + + t = 0.0; + p = 1.0; + for (i = 1; i < aid; i++) { + r = 1.0 - e + t; + p *= s * (a + t + d2) * (b + t + d2) / r; + t += 1.0; + p /= t; + y1 += p; + } + nosum: + p = xsf::cephes::Gamma(c); + y1 *= xsf::cephes::Gamma(e) * p * + (xsf::cephes::rgamma(a + d1) * xsf::cephes::rgamma(b + d1)); + + y *= p * (xsf::cephes::rgamma(a + d2) * xsf::cephes::rgamma(b + d2)); + if ((aid & 1) != 0) + y = -y; + + q = std::pow(s, id); /* s to the id power */ + if (id > 0.0) + y *= q; + else + y1 *= q; + + y += y1; + psidon: + goto done; + } + } + + /* Use defining power series if no special cases */ + y = hys2f1(a, b, c, x, &err); + + done: + *loss = err; + return (y); + } + + /* + 15.4.2 Abramowitz & Stegun. + */ + XSF_HOST_DEVICE double hyp2f1_neg_c_equal_bc(double a, double b, double x) { + double k; + double collector = 1; + double sum = 1; + double collector_max = 1; + + if (!(std::abs(b) < 1e5)) { + return std::numeric_limits::quiet_NaN(); + } + + for (k = 1; k <= -b; k++) { + collector *= (a + k - 1) * x / k; + collector_max = std::fmax(std::abs(collector), collector_max); + sum += collector; + } + + if (1e-16 * (1 + collector_max / std::abs(sum)) > 1e-7) { + return std::numeric_limits::quiet_NaN(); + } + + return sum; + } + + /* + * Evaluate hypergeometric function by two-term recurrence in `a`. + * + * This avoids some of the loss of precision in the strongly alternating + * hypergeometric series, and can be used to reduce the `a` and `b` parameters + * to smaller values. + * + * AMS55 #15.2.10 + */ + XSF_HOST_DEVICE double hyp2f1ra(double a, double b, double c, double x, double *loss) { + double f2, f1, f0; + int n; + double t, err, da; + + /* Don't cross c or zero */ + if ((c < 0 && a <= c) || (c >= 0 && a >= c)) { + da = std::round(a - c); + } else { + da = std::round(a); + } + t = a - da; + + *loss = 0; + + XSF_ASSERT(da != 0); + + if (std::abs(da) > hyp2f1_MAXITER) { + /* Too expensive to compute this value, so give up */ + set_error("hyp2f1", SF_ERROR_NO_RESULT, NULL); + *loss = 1.0; + return std::numeric_limits::quiet_NaN(); + } + + if (da < 0) { + /* Recurse down */ + f2 = 0; + f1 = hys2f1(t, b, c, x, &err); + *loss += err; + f0 = hys2f1(t - 1, b, c, x, &err); + *loss += err; + t -= 1; + for (n = 1; n < -da; ++n) { + f2 = f1; + f1 = f0; + f0 = -(2 * t - c - t * x + b * x) / (c - t) * f1 - t * (x - 1) / (c - t) * f2; + t -= 1; + } + } else { + /* Recurse up */ + f2 = 0; + f1 = hys2f1(t, b, c, x, &err); + *loss += err; + f0 = hys2f1(t + 1, b, c, x, &err); + *loss += err; + t += 1; + for (n = 1; n < da; ++n) { + f2 = f1; + f1 = f0; + f0 = -((2 * t - c - t * x + b * x) * f1 + (c - t) * f2) / (t * (x - 1)); + t += 1; + } + } + + return f0; + } + } // namespace detail + + XSF_HOST_DEVICE double hyp2f1(double a, double b, double c, double x) { + double d, d1, d2, e; + double p, q, r, s, y, ax; + double ia, ib, ic, id, err; + double t1; + int i, aid; + int neg_int_a = 0, neg_int_b = 0; + int neg_int_ca_or_cb = 0; + + err = 0.0; + ax = std::abs(x); + s = 1.0 - x; + ia = std::round(a); /* nearest integer to a */ + ib = std::round(b); + + if (x == 0.0) { + return 1.0; + } + + d = c - a - b; + id = std::round(d); + + if ((a == 0 || b == 0) && c != 0) { + return 1.0; + } + + if (a <= 0 && std::abs(a - ia) < detail::hyp2f1_EPS) { /* a is a negative integer */ + neg_int_a = 1; + } + + if (b <= 0 && std::abs(b - ib) < detail::hyp2f1_EPS) { /* b is a negative integer */ + neg_int_b = 1; + } + + if (d <= -1 && !(std::abs(d - id) > detail::hyp2f1_EPS && s < 0) && !(neg_int_a || neg_int_b)) { + return std::pow(s, d) * hyp2f1(c - a, c - b, c, x); + } + if (d <= 0 && x == 1 && !(neg_int_a || neg_int_b)) + goto hypdiv; + + if (ax < 1.0 || x == -1.0) { + /* 2F1(a,b;b;x) = (1-x)**(-a) */ + if (std::abs(b - c) < detail::hyp2f1_EPS) { /* b = c */ + if (neg_int_b) { + y = detail::hyp2f1_neg_c_equal_bc(a, b, x); + } else { + y = std::pow(s, -a); /* s to the -a power */ + } + goto hypdon; + } + if (std::abs(a - c) < detail::hyp2f1_EPS) { /* a = c */ + y = std::pow(s, -b); /* s to the -b power */ + goto hypdon; + } + } + + if (c <= 0.0) { + ic = std::round(c); /* nearest integer to c */ + if (std::abs(c - ic) < detail::hyp2f1_EPS) { /* c is a negative integer */ + /* check if termination before explosion */ + if (neg_int_a && (ia > ic)) + goto hypok; + if (neg_int_b && (ib > ic)) + goto hypok; + goto hypdiv; + } + } + + if (neg_int_a || neg_int_b) /* function is a polynomial */ + goto hypok; + + t1 = std::abs(b - a); + if (x < -2.0 && std::abs(t1 - round(t1)) > detail::hyp2f1_EPS) { + /* This transform has a pole for b-a integer, and + * may produce large cancellation errors for |1/x| close 1 + */ + p = hyp2f1(a, 1 - c + a, 1 - b + a, 1.0 / x); + q = hyp2f1(b, 1 - c + b, 1 - a + b, 1.0 / x); + p *= std::pow(-x, -a); + q *= std::pow(-x, -b); + t1 = Gamma(c); + s = t1 * Gamma(b - a) * (rgamma(b) * rgamma(c - a)); + y = t1 * Gamma(a - b) * (rgamma(a) * rgamma(c - b)); + return s * p + y * q; + } else if (x < -1.0) { + if (std::abs(a) < std::abs(b)) { + return std::pow(s, -a) * hyp2f1(a, c - b, c, x / (x - 1)); + } else { + return std::pow(s, -b) * hyp2f1(b, c - a, c, x / (x - 1)); + } + } + + if (ax > 1.0) /* series diverges */ + goto hypdiv; + + p = c - a; + ia = std::round(p); /* nearest integer to c-a */ + if ((ia <= 0.0) && (std::abs(p - ia) < detail::hyp2f1_EPS)) /* negative int c - a */ + neg_int_ca_or_cb = 1; + + r = c - b; + ib = std::round(r); /* nearest integer to c-b */ + if ((ib <= 0.0) && (std::abs(r - ib) < detail::hyp2f1_EPS)) /* negative int c - b */ + neg_int_ca_or_cb = 1; + + id = std::round(d); /* nearest integer to d */ + q = std::abs(d - id); + + /* Thanks to Christian Burger + * for reporting a bug here. */ + if (std::abs(ax - 1.0) < detail::hyp2f1_EPS) { /* |x| == 1.0 */ + if (x > 0.0) { + if (neg_int_ca_or_cb) { + if (d >= 0.0) + goto hypf; + else + goto hypdiv; + } + if (d <= 0.0) + goto hypdiv; + y = Gamma(c) * Gamma(d) * (rgamma(p) * rgamma(r)); + goto hypdon; + } + if (d <= -1.0) + goto hypdiv; + } + + /* Conditionally make d > 0 by recurrence on c + * AMS55 #15.2.27 + */ + if (d < 0.0) { + /* Try the power series first */ + y = detail::hyt2f1(a, b, c, x, &err); + if (err < detail::hyp2f1_ETHRESH) + goto hypdon; + /* Apply the recurrence if power series fails */ + err = 0.0; + aid = 2 - id; + e = c + aid; + d2 = hyp2f1(a, b, e, x); + d1 = hyp2f1(a, b, e + 1.0, x); + q = a + b + 1.0; + for (i = 0; i < aid; i++) { + r = e - 1.0; + y = (e * (r - (2.0 * e - q) * x) * d2 + (e - a) * (e - b) * x * d1) / (e * r * s); + e = r; + d1 = d2; + d2 = y; + } + goto hypdon; + } + + if (neg_int_ca_or_cb) { + goto hypf; /* negative integer c-a or c-b */ + } + + hypok: + y = detail::hyt2f1(a, b, c, x, &err); + + hypdon: + if (err > detail::hyp2f1_ETHRESH) { + set_error("hyp2f1", SF_ERROR_LOSS, NULL); + /* printf( "Estimated err = %.2e\n", err ); */ + } + return (y); + + /* The transformation for c-a or c-b negative integer + * AMS55 #15.3.3 + */ + hypf: + y = std::pow(s, d) * detail::hys2f1(c - a, c - b, c, x, &err); + goto hypdon; + + /* The alarm exit */ + hypdiv: + set_error("hyp2f1", SF_ERROR_OVERFLOW, NULL); + return std::numeric_limits::infinity(); + } + +} // namespace cephes +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/config.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/config.h new file mode 100644 index 0000000000000000000000000000000000000000..5cb40ed1e1e095a8c99414f891a697c263845784 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/config.h @@ -0,0 +1,304 @@ +#pragma once + +// Define math constants if they are not available +#ifndef M_E +#define M_E 2.71828182845904523536 +#endif + +#ifndef M_LOG2E +#define M_LOG2E 1.44269504088896340736 +#endif + +#ifndef M_LOG10E +#define M_LOG10E 0.434294481903251827651 +#endif + +#ifndef M_LN2 +#define M_LN2 0.693147180559945309417 +#endif + +#ifndef M_LN10 +#define M_LN10 2.30258509299404568402 +#endif + +#ifndef M_PI +#define M_PI 3.14159265358979323846 +#endif + +#ifndef M_PI_2 +#define M_PI_2 1.57079632679489661923 +#endif + +#ifndef M_PI_4 +#define M_PI_4 0.785398163397448309616 +#endif + +#ifndef M_1_PI +#define M_1_PI 0.318309886183790671538 +#endif + +#ifndef M_2_PI +#define M_2_PI 0.636619772367581343076 +#endif + +#ifndef M_2_SQRTPI +#define M_2_SQRTPI 1.12837916709551257390 +#endif + +#ifndef M_SQRT2 +#define M_SQRT2 1.41421356237309504880 +#endif + +#ifndef M_SQRT1_2 +#define M_SQRT1_2 0.707106781186547524401 +#endif + +#ifdef __CUDACC__ +#define XSF_HOST_DEVICE __host__ __device__ + +#include +#include +#include +#include +#include +#include +#include + +// Fallback to global namespace for functions unsupported on NVRTC Jit +#ifdef _LIBCUDACXX_COMPILER_NVRTC +#include +#endif + +namespace std { + +XSF_HOST_DEVICE inline double abs(double num) { return cuda::std::abs(num); } + +XSF_HOST_DEVICE inline double exp(double num) { return cuda::std::exp(num); } + +XSF_HOST_DEVICE inline double log(double num) { return cuda::std::log(num); } + +XSF_HOST_DEVICE inline double sqrt(double num) { return cuda::std::sqrt(num); } + +XSF_HOST_DEVICE inline bool isinf(double num) { return cuda::std::isinf(num); } + +XSF_HOST_DEVICE inline bool isnan(double num) { return cuda::std::isnan(num); } + +XSF_HOST_DEVICE inline bool isfinite(double num) { return cuda::std::isfinite(num); } + +XSF_HOST_DEVICE inline double pow(double x, double y) { return cuda::std::pow(x, y); } + +XSF_HOST_DEVICE inline double sin(double x) { return cuda::std::sin(x); } + +XSF_HOST_DEVICE inline double cos(double x) { return cuda::std::cos(x); } + +XSF_HOST_DEVICE inline double tan(double x) { return cuda::std::tan(x); } + +XSF_HOST_DEVICE inline double atan(double x) { return cuda::std::atan(x); } + +XSF_HOST_DEVICE inline double acos(double x) { return cuda::std::acos(x); } + +XSF_HOST_DEVICE inline double sinh(double x) { return cuda::std::sinh(x); } + +XSF_HOST_DEVICE inline double cosh(double x) { return cuda::std::cosh(x); } + +XSF_HOST_DEVICE inline double asinh(double x) { return cuda::std::asinh(x); } + +XSF_HOST_DEVICE inline bool signbit(double x) { return cuda::std::signbit(x); } + +// Fallback to global namespace for functions unsupported on NVRTC +#ifndef _LIBCUDACXX_COMPILER_NVRTC +XSF_HOST_DEVICE inline double ceil(double x) { return cuda::std::ceil(x); } +XSF_HOST_DEVICE inline double floor(double x) { return cuda::std::floor(x); } +XSF_HOST_DEVICE inline double round(double x) { return cuda::std::round(x); } +XSF_HOST_DEVICE inline double trunc(double x) { return cuda::std::trunc(x); } +XSF_HOST_DEVICE inline double fma(double x, double y, double z) { return cuda::std::fma(x, y, z); } +XSF_HOST_DEVICE inline double copysign(double x, double y) { return cuda::std::copysign(x, y); } +XSF_HOST_DEVICE inline double modf(double value, double *iptr) { return cuda::std::modf(value, iptr); } +XSF_HOST_DEVICE inline double fmax(double x, double y) { return cuda::std::fmax(x, y); } +XSF_HOST_DEVICE inline double fmin(double x, double y) { return cuda::std::fmin(x, y); } +XSF_HOST_DEVICE inline double log10(double num) { return cuda::std::log10(num); } +XSF_HOST_DEVICE inline double log1p(double num) { return cuda::std::log1p(num); } +XSF_HOST_DEVICE inline double frexp(double num, int *exp) { return cuda::std::frexp(num, exp); } +XSF_HOST_DEVICE inline double ldexp(double num, int exp) { return cuda::std::ldexp(num, exp); } +XSF_HOST_DEVICE inline double fmod(double x, double y) { return cuda::std::fmod(x, y); } +XSF_HOST_DEVICE inline double nextafter(double from, double to) { return cuda::std::nextafter(from, to); } +#else +XSF_HOST_DEVICE inline double ceil(double x) { return ::ceil(x); } +XSF_HOST_DEVICE inline double floor(double x) { return ::floor(x); } +XSF_HOST_DEVICE inline double round(double x) { return ::round(x); } +XSF_HOST_DEVICE inline double trunc(double x) { return ::trunc(x); } +XSF_HOST_DEVICE inline double fma(double x, double y, double z) { return ::fma(x, y, z); } +XSF_HOST_DEVICE inline double copysign(double x, double y) { return ::copysign(x, y); } +XSF_HOST_DEVICE inline double modf(double value, double *iptr) { return ::modf(value, iptr); } +XSF_HOST_DEVICE inline double fmax(double x, double y) { return ::fmax(x, y); } +XSF_HOST_DEVICE inline double fmin(double x, double y) { return ::fmin(x, y); } +XSF_HOST_DEVICE inline double log10(double num) { return ::log10(num); } +XSF_HOST_DEVICE inline double log1p(double num) { return ::log1p(num); } +XSF_HOST_DEVICE inline double frexp(double num, int *exp) { return ::frexp(num, exp); } +XSF_HOST_DEVICE inline double ldexp(double num, int exp) { return ::ldexp(num, exp); } +XSF_HOST_DEVICE inline double fmod(double x, double y) { return ::fmod(x, y); } +XSF_HOST_DEVICE inline double nextafter(double from, double to) { return ::nextafter(from, to); } +#endif + +template +XSF_HOST_DEVICE void swap(T &a, T &b) { + cuda::std::swap(a, b); +} + +// Reimplement std::clamp until it's available in CuPy +template +XSF_HOST_DEVICE constexpr T clamp(T &v, T &lo, T &hi) { + return v < lo ? lo : (v > hi ? lo : v); +} + +template +using numeric_limits = cuda::std::numeric_limits; + +// Must use thrust for complex types in order to support CuPy +template +using complex = thrust::complex; + +template +XSF_HOST_DEVICE T abs(const complex &z) { + return thrust::abs(z); +} + +template +XSF_HOST_DEVICE complex exp(const complex &z) { + return thrust::exp(z); +} + +template +XSF_HOST_DEVICE complex log(const complex &z) { + return thrust::log(z); +} + +template +XSF_HOST_DEVICE T norm(const complex &z) { + return thrust::norm(z); +} + +template +XSF_HOST_DEVICE complex sqrt(const complex &z) { + return thrust::sqrt(z); +} + +template +XSF_HOST_DEVICE complex conj(const complex &z) { + return thrust::conj(z); +} + +template +XSF_HOST_DEVICE complex pow(const complex &x, const complex &y) { + return thrust::pow(x, y); +} + +template +XSF_HOST_DEVICE complex pow(const complex &x, const T &y) { + return thrust::pow(x, y); +} + +// Other types and utilities +template +using is_floating_point = cuda::std::is_floating_point; + +template +using enable_if = cuda::std::enable_if; + +template +using decay = cuda::std::decay; + +template +using invoke_result = cuda::std::invoke_result; + +template +using pair = cuda::std::pair; + +template +using tuple = cuda::std::tuple; + +using cuda::std::ptrdiff_t; +using cuda::std::size_t; +using cuda::std::uint64_t; + +#define XSF_ASSERT(a) + +} // namespace std + +#else +#define XSF_HOST_DEVICE + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#ifdef DEBUG +#define XSF_ASSERT(a) assert(a) +#else +#define XSF_ASSERT(a) +#endif + +namespace xsf { + +// basic +using std::abs; + +// exponential +using std::exp; + +// power +using std::sqrt; + +// trigonometric +using std::cos; +using std::sin; + +// floating-point manipulation +using std::copysign; + +// classification and comparison +using std::isfinite; +using std::isinf; +using std::isnan; +using std::signbit; + +// complex +using std::imag; +using std::real; + +template +struct remove_complex { + using type = T; +}; + +template +struct remove_complex> { + using type = T; +}; + +template +using remove_complex_t = typename remove_complex::type; + +template +struct complex_type { + using type = std::complex; +}; + +template +using complex_type_t = typename complex_type::type; + +template +using complex = complex_type_t; + +} // namespace xsf + +#endif diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/digamma.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/digamma.h new file mode 100644 index 0000000000000000000000000000000000000000..db51362ce03907ee7c86c0995836e6d5d18160c6 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/digamma.h @@ -0,0 +1,205 @@ +/* Translated from Cython into C++ by SciPy developers in 2024. + * Original header comment appears below. + */ + +/* An implementation of the digamma function for complex arguments. + * + * Author: Josh Wilson + * + * Distributed under the same license as Scipy. + * + * Sources: + * [1] "The Digital Library of Mathematical Functions", dlmf.nist.gov + * + * [2] mpmath (version 0.19), http://mpmath.org + */ + +#pragma once + +#include "cephes/psi.h" +#include "cephes/zeta.h" +#include "config.h" +#include "error.h" +#include "trig.h" + +namespace xsf { +namespace detail { + // All of the following were computed with mpmath + // Location of the positive root + constexpr double digamma_posroot = 1.4616321449683623; + // Value of the positive root + constexpr double digamma_posrootval = -9.2412655217294275e-17; + // Location of the negative root + constexpr double digamma_negroot = -0.504083008264455409; + // Value of the negative root + constexpr double digamma_negrootval = 7.2897639029768949e-17; + + template + XSF_HOST_DEVICE T digamma_zeta_series(T z, double root, double rootval) { + T res = rootval; + T coeff = -1.0; + + z = z - root; + T term; + for (int n = 1; n < 100; n++) { + coeff *= -z; + term = coeff * cephes::zeta(n + 1, root); + res += term; + if (std::abs(term) < std::numeric_limits::epsilon() * std::abs(res)) { + break; + } + } + return res; + } + + XSF_HOST_DEVICE inline std::complex + digamma_forward_recurrence(std::complex z, std::complex psiz, int n) { + /* Compute digamma(z + n) using digamma(z) using the recurrence + * relation + * + * digamma(z + 1) = digamma(z) + 1/z. + * + * See https://dlmf.nist.gov/5.5#E2 */ + std::complex res = psiz; + + for (int k = 0; k < n; k++) { + res += 1.0 / (z + static_cast(k)); + } + return res; + } + + XSF_HOST_DEVICE inline std::complex + digamma_backward_recurrence(std::complex z, std::complex psiz, int n) { + /* Compute digamma(z - n) using digamma(z) and a recurrence relation. */ + std::complex res = psiz; + + for (int k = 1; k < n + 1; k++) { + res -= 1.0 / (z - static_cast(k)); + } + return res; + } + + XSF_HOST_DEVICE inline std::complex digamma_asymptotic_series(std::complex z) { + /* Evaluate digamma using an asymptotic series. See + * + * https://dlmf.nist.gov/5.11#E2 */ + double bernoulli2k[] = {0.166666666666666667, -0.0333333333333333333, 0.0238095238095238095, + -0.0333333333333333333, 0.0757575757575757576, -0.253113553113553114, + 1.16666666666666667, -7.09215686274509804, 54.9711779448621554, + -529.124242424242424, 6192.12318840579710, -86580.2531135531136, + 1425517.16666666667, -27298231.0678160920, 601580873.900642368, + -15116315767.0921569}; + std::complex rzz = 1.0 / z / z; + std::complex zfac = 1.0; + std::complex term; + std::complex res; + + if (!(std::isfinite(z.real()) && std::isfinite(z.imag()))) { + /* Check for infinity (or nan) and return early. + * Result of division by complex infinity is implementation dependent. + * and has been observed to vary between C++ stdlib and CUDA stdlib. + */ + return std::log(z); + } + + res = std::log(z) - 0.5 / z; + + for (int k = 1; k < 17; k++) { + zfac *= rzz; + term = -bernoulli2k[k - 1] * zfac / (2 * static_cast(k)); + res += term; + if (std::abs(term) < std::numeric_limits::epsilon() * std::abs(res)) { + break; + } + } + return res; + } + +} // namespace detail + +XSF_HOST_DEVICE inline double digamma(double z) { + /* Wrap Cephes' psi to take advantage of the series expansion around + * the smallest negative zero. + */ + if (std::abs(z - detail::digamma_negroot) < 0.3) { + return detail::digamma_zeta_series(z, detail::digamma_negroot, detail::digamma_negrootval); + } + return cephes::psi(z); +} + +XSF_HOST_DEVICE inline float digamma(float z) { return static_cast(digamma(static_cast(z))); } + +XSF_HOST_DEVICE inline std::complex digamma(std::complex z) { + /* + * Compute the digamma function for complex arguments. The strategy + * is: + * + * - Around the two zeros closest to the origin (posroot and negroot) + * use a Taylor series with precomputed zero order coefficient. + * - If close to the origin, use a recurrence relation to step away + * from the origin. + * - If close to the negative real axis, use the reflection formula + * to move to the right halfplane. + * - If |z| is large (> 16), use the asymptotic series. + * - If |z| is small, use a recurrence relation to make |z| large + * enough to use the asymptotic series. + */ + double absz = std::abs(z); + std::complex res = 0; + /* Use the asymptotic series for z away from the negative real axis + * with abs(z) > smallabsz. */ + int smallabsz = 16; + /* Use the reflection principle for z with z.real < 0 that are within + * smallimag of the negative real axis. + * int smallimag = 6 # unused below except in a comment */ + + if (z.real() <= 0.0 && std::ceil(z.real()) == z) { + // Poles + set_error("digamma", SF_ERROR_SINGULAR, NULL); + return {std::numeric_limits::quiet_NaN(), std::numeric_limits::quiet_NaN()}; + } + if (std::abs(z - detail::digamma_negroot) < 0.3) { + // First negative root. + return detail::digamma_zeta_series(z, detail::digamma_negroot, detail::digamma_negrootval); + } + + if (z.real() < 0 and std::abs(z.imag()) < smallabsz) { + /* Reflection formula for digamma. See + * + *https://dlmf.nist.gov/5.5#E4 + */ + res = -M_PI * cospi(z) / sinpi(z); + z = 1.0 - z; + absz = std::abs(z); + } + + if (absz < 0.5) { + /* Use one step of the recurrence relation to step away from + * the pole. */ + res = -1.0 / z; + z += 1.0; + absz = std::abs(z); + } + + if (std::abs(z - detail::digamma_posroot) < 0.5) { + res += detail::digamma_zeta_series(z, detail::digamma_posroot, detail::digamma_posrootval); + } else if (absz > smallabsz) { + res += detail::digamma_asymptotic_series(z); + } else if (z.real() >= 0.0) { + double n = std::trunc(smallabsz - absz) + 1; + std::complex init = detail::digamma_asymptotic_series(z + n); + res += detail::digamma_backward_recurrence(z + n, init, n); + } else { + // z.real() < 0, absz < smallabsz, and z.imag() > smallimag + double n = std::trunc(smallabsz - absz) - 1; + std::complex init = detail::digamma_asymptotic_series(z - n); + res += detail::digamma_forward_recurrence(z - n, init, n); + } + return res; +} + +XSF_HOST_DEVICE inline std::complex digamma(std::complex z) { + return static_cast>(digamma(static_cast>(z))); +} + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/error.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/error.h new file mode 100644 index 0000000000000000000000000000000000000000..7221b5e6c4051b82f80dfb360f9fab896be6a6bd --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/error.h @@ -0,0 +1,57 @@ +#pragma once + +typedef enum { + SF_ERROR_OK = 0, /* no error */ + SF_ERROR_SINGULAR, /* singularity encountered */ + SF_ERROR_UNDERFLOW, /* floating point underflow */ + SF_ERROR_OVERFLOW, /* floating point overflow */ + SF_ERROR_SLOW, /* too many iterations required */ + SF_ERROR_LOSS, /* loss of precision */ + SF_ERROR_NO_RESULT, /* no result obtained */ + SF_ERROR_DOMAIN, /* out of domain */ + SF_ERROR_ARG, /* invalid input parameter */ + SF_ERROR_OTHER, /* unclassified error */ + SF_ERROR_MEMORY, /* memory allocation failed */ + SF_ERROR__LAST +} sf_error_t; + +#ifdef __cplusplus + +#include "config.h" + +namespace xsf { + +#ifndef SP_SPECFUN_ERROR +XSF_HOST_DEVICE inline void set_error(const char *func_name, sf_error_t code, const char *fmt, ...) { + // nothing +} +#else +void set_error(const char *func_name, sf_error_t code, const char *fmt, ...); +#endif + +template +XSF_HOST_DEVICE void set_error_and_nan(const char *name, sf_error_t code, T &value) { + if (code != SF_ERROR_OK) { + set_error(name, code, nullptr); + + if (code == SF_ERROR_DOMAIN || code == SF_ERROR_OVERFLOW || code == SF_ERROR_NO_RESULT) { + value = std::numeric_limits::quiet_NaN(); + } + } +} + +template +XSF_HOST_DEVICE void set_error_and_nan(const char *name, sf_error_t code, std::complex &value) { + if (code != SF_ERROR_OK) { + set_error(name, code, nullptr); + + if (code == SF_ERROR_DOMAIN || code == SF_ERROR_OVERFLOW || code == SF_ERROR_NO_RESULT) { + value.real(std::numeric_limits::quiet_NaN()); + value.imag(std::numeric_limits::quiet_NaN()); + } + } +} + +} // namespace xsf + +#endif diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/evalpoly.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/evalpoly.h new file mode 100644 index 0000000000000000000000000000000000000000..b126fb608fae47620431e05cea21ac66e6847e59 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/evalpoly.h @@ -0,0 +1,47 @@ +/* Translated from Cython into C++ by SciPy developers in 2024. + * + * Original author: Josh Wilson, 2016. + */ + +/* Evaluate polynomials. + * + * All of the coefficients are stored in reverse order, i.e. if the + * polynomial is + * + * u_n x^n + u_{n - 1} x^{n - 1} + ... + u_0, + * + * then coeffs[0] = u_n, coeffs[1] = u_{n - 1}, ..., coeffs[n] = u_0. + * + * References + * ---------- + * [1] Knuth, "The Art of Computer Programming, Volume II" + */ + +#pragma once + +#include "config.h" + +namespace xsf { + +XSF_HOST_DEVICE inline std::complex cevalpoly(const double *coeffs, int degree, std::complex z) { + /* Evaluate a polynomial with real coefficients at a complex point. + * + * Uses equation (3) in section 4.6.4 of [1]. Note that it is more + * efficient than Horner's method. + */ + double a = coeffs[0]; + double b = coeffs[1]; + double r = 2 * z.real(); + double s = std::norm(z); + double tmp; + + for (int j = 2; j < degree + 1; j++) { + tmp = b; + b = std::fma(-s, a, coeffs[j]); + a = std::fma(r, a, tmp); + } + + return z * a + b; +} + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/expint.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/expint.h new file mode 100644 index 0000000000000000000000000000000000000000..600448aeb7f7a525aef76d4c59c31fa2489bcb1d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/expint.h @@ -0,0 +1,266 @@ +/* The functions exp1, expi below are based on translations of the Fortran code + * by Shanjie Zhang and Jianming Jin from the book + * + * Shanjie Zhang, Jianming Jin, + * Computation of Special Functions, + * Wiley, 1996, + * ISBN: 0-471-11963-6, + * LC: QA351.C45. + */ + +#pragma once + +#include "config.h" +#include "error.h" + +#include "cephes/const.h" + + +namespace xsf { + + +XSF_HOST_DEVICE inline double exp1(double x) { + // ============================================ + // Purpose: Compute exponential integral E1(x) + // Input : x --- Argument of E1(x) + // Output: E1 --- E1(x) ( x > 0 ) + // ============================================ + int k, m; + double e1, r, t, t0; + constexpr double ga = cephes::detail::SCIPY_EULER; + + if (x == 0.0) { + return std::numeric_limits::infinity(); + } + if (x <= 1.0) { + e1 = 1.0; + r = 1.0; + for (k = 1; k < 26; k++) { + r = -r*k*x/std::pow(k+1.0, 2); + e1 += r; + if (std::abs(r) <= std::abs(e1)*1e-15) { break; } + } + return -ga - std::log(x) + x*e1; + } + m = 20 + (int)(80.0/x); + t0 = 0.0; + for (k = m; k > 0; k--) { + t0 = k / (1.0 + k / (x+t0)); + } + t = 1.0 / (x + t0); + return std::exp(-x)*t; +} + +XSF_HOST_DEVICE inline float exp1(float x) { return exp1(static_cast(x)); } + +XSF_HOST_DEVICE inline std::complex exp1(std::complex z) { + // ==================================================== + // Purpose: Compute complex exponential integral E1(z) + // Input : z --- Argument of E1(z) + // Output: CE1 --- E1(z) + // ==================================================== + constexpr double el = cephes::detail::SCIPY_EULER; + int k; + std::complex ce1, cr, zc, zd, zdc; + double x = z.real(); + double a0 = std::abs(z); + // Continued fraction converges slowly near negative real axis, + // so use power series in a wedge around it until radius 40.0 + double xt = -2.0*std::abs(z.imag()); + + if (a0 == 0.0) { return std::numeric_limits::infinity(); } + if ((a0 < 5.0) || ((x < xt) && (a0 < 40.0))) { + // Power series + ce1 = 1.0; + cr = 1.0; + for (k = 1; k < 501; k++) { + cr = -cr*z*static_cast(k / std::pow(k + 1, 2)); + ce1 += cr; + if (std::abs(cr) < std::abs(ce1)*1e-15) { break; } + } + if ((x <= 0.0) && (z.imag() == 0.0)) { + //Careful on the branch cut -- use the sign of the imaginary part + // to get the right sign on the factor if pi. + ce1 = -el - std::log(-z) + z*ce1 - std::copysign(M_PI, z.imag())*std::complex(0.0, 1.0); + } else { + ce1 = -el - std::log(z) + z*ce1; + } + } else { + // Continued fraction https://dlmf.nist.gov/6.9 + // 1 1 1 2 2 3 3 + // E1 = exp(-z) * ----- ----- ----- ----- ----- ----- ----- ... + // Z + 1 + Z + 1 + Z + 1 + Z + + zc = 0.0; + zd = static_cast(1) / z; + zdc = zd; + zc += zdc; + for (k = 1; k < 501; k++) { + zd = static_cast(1) / (zd*static_cast(k) + static_cast(1)); + zdc *= (zd - static_cast(1)); + zc += zdc; + + zd = static_cast(1) / (zd*static_cast(k) + z); + zdc *= (z*zd - static_cast(1)); + zc += zdc; + if ((std::abs(zdc) <= std::abs(zc)*1e-15) && (k > 20)) { break; } + } + ce1 = std::exp(-z)*zc; + if ((x <= 0.0) && (z.imag() == 0.0)) { + ce1 -= M_PI*std::complex(0.0, 1.0); + } + } + return ce1; +} + +XSF_HOST_DEVICE inline std::complex exp1(std::complex z) { + return static_cast>(exp1(static_cast>(z))); +} + +XSF_HOST_DEVICE inline double expi(double x) { + // ============================================ + // Purpose: Compute exponential integral Ei(x) + // Input : x --- Argument of Ei(x) + // Output: EI --- Ei(x) + // ============================================ + + constexpr double ga = cephes::detail::SCIPY_EULER; + double ei, r; + + if (x == 0.0) { + ei = -std::numeric_limits::infinity(); + } else if (x < 0) { + ei = -exp1(-x); + } else if (std::abs(x) <= 40.0) { + // Power series around x=0 + ei = 1.0; + r = 1.0; + + for (int k = 1; k <= 100; k++) { + r = r * k * x / ((k + 1.0) * (k + 1.0)); + ei += r; + if (std::abs(r / ei) <= 1.0e-15) { break; } + } + ei = ga + std::log(x) + x * ei; + } else { + // Asymptotic expansion (the series is not convergent) + ei = 1.0; + r = 1.0; + for (int k = 1; k <= 20; k++) { + r = r * k / x; + ei += r; + } + ei = std::exp(x) / x * ei; + } + return ei; +} + +XSF_HOST_DEVICE inline float expi(float x) { return expi(static_cast(x)); } + +std::complex expi(std::complex z) { + // ============================================ + // Purpose: Compute exponential integral Ei(x) + // Input : x --- Complex argument of Ei(x) + // Output: EI --- Ei(x) + // ============================================ + + std::complex cei; + cei = - exp1(-z); + if (z.imag() > 0.0) { + cei += std::complex(0.0, M_PI); + } else if (z.imag() < 0.0 ) { + cei -= std::complex(0.0, M_PI); + } else { + if (z.real() > 0.0) { + cei += std::complex(0.0, copysign(M_PI, z.imag())); + } + } + return cei; +} + + +XSF_HOST_DEVICE inline std::complex expi(std::complex z) { + return static_cast>(expi(static_cast>(z))); +} + +namespace detail { + + // + // Compute a factor of the exponential integral E1. + // This is used in scaled_exp1(x) for moderate values of x. + // + // The function uses the continued fraction expansion given in equation 5.1.22 + // of Abramowitz & Stegun, "Handbook of Mathematical Functions". + // For n=1, this is + // + // E1(x) = exp(-x)*C(x) + // + // where C(x), expressed in the notation used in A&S, is the continued fraction + // + // 1 1 1 2 2 3 3 + // C(x) = --- --- --- --- --- --- --- ... + // x + 1 + x + 1 + x + 1 + x + + // + // Here, we pull a factor of 1/z out of C(x), so + // + // E1(x) = (exp(-x)/x)*F(x) + // + // and a bit of algebra gives the continued fraction expansion of F(x) to be + // + // 1 1 1 2 2 3 3 + // F(x) = --- --- --- --- --- --- --- ... + // 1 + x + 1 + x + 1 + x + 1 + + // + XSF_HOST_DEVICE inline double expint1_factor_cont_frac(double x) { + // The number of terms to use in the truncated continued fraction + // depends on x. Larger values of x require fewer terms. + int m = 20 + (int) (80.0 / x); + double t0 = 0.0; + for (int k = m; k > 0; --k) { + t0 = k / (x + k / (1 + t0)); + } + return 1 / (1 + t0); + } + +} // namespace detail + +// +// Scaled version of the exponential integral E_1(x). +// +// Factor E_1(x) as +// +// E_1(x) = exp(-x)/x * F(x) +// +// This function computes F(x). +// +// F(x) has the properties: +// * F(0) = 0 +// * F is increasing on [0, inf) +// * lim_{x->inf} F(x) = 1. +// +XSF_HOST_DEVICE inline double scaled_exp1(double x) { + if (x < 0) { + return std::numeric_limits::quiet_NaN(); + } + + if (x == 0) { + return 0.0; + } + + if (x <= 1) { + // For small x, the naive implementation is sufficiently accurate. + return x * std::exp(x) * exp1(x); + } + + if (x <= 1250) { + // For moderate x, use the continued fraction expansion. + return detail::expint1_factor_cont_frac(x); + } + + // For large x, use the asymptotic expansion. This is equation 5.1.51 + // from Abramowitz & Stegun, "Handbook of Mathematical Functions". + return 1 + (-1 + (2 + (-6 + (24 - 120 / x) / x) / x) / x) / x; +} + +XSF_HOST_DEVICE inline float scaled_exp1(float x) { return scaled_exp1(static_cast(x)); } + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/hyp2f1.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/hyp2f1.h new file mode 100644 index 0000000000000000000000000000000000000000..9d4eff7532202380bcd5dbd2978d46d5f613d5eb --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/hyp2f1.h @@ -0,0 +1,694 @@ +/* Implementation of Gauss's hypergeometric function for complex values. + * + * This implementation is based on the Fortran implementation by Shanjie Zhang and + * Jianming Jin included in specfun.f [1]_. Computation of Gauss's hypergeometric + * function involves handling a patchwork of special cases. By default the Zhang and + * Jin implementation has been followed as closely as possible except for situations where + * an improvement was obvious. We've attempted to document the reasons behind decisions + * made by Zhang and Jin and to document the reasons for deviating from their implementation + * when this has been done. References to the NIST Digital Library of Mathematical + * Functions [2]_ have been added where they are appropriate. The review paper by + * Pearson et al [3]_ is an excellent resource for best practices for numerical + * computation of hypergeometric functions. We have followed this review paper + * when making improvements to and correcting defects in Zhang and Jin's + * implementation. When Pearson et al propose several competing alternatives for a + * given case, we've used our best judgment to decide on the method to use. + * + * Author: Albert Steppi + * + * Distributed under the same license as Scipy. + * + * References + * ---------- + * .. [1] S. Zhang and J.M. Jin, "Computation of Special Functions", Wiley 1996 + * .. [2] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, + * Release 1.1.1 of 2021-03-15. F. W. J. Olver, A. B. Olde Daalhuis, + * D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, + * B. V. Saunders, H. S. Cohl, and M. A. McClain, eds. + * .. [3] Pearson, J.W., Olver, S. & Porter, M.A. + * "Numerical methods for the computation of the confluent and Gauss + * hypergeometric functions." + * Numer Algor 74, 821-866 (2017). https://doi.org/10.1007/s11075-016-0173-0 + * .. [4] Raimundas Vidunas, "Degenerate Gauss Hypergeometric Functions", + * Kyushu Journal of Mathematics, 2007, Volume 61, Issue 1, Pages 109-135, + * .. [5] López, J.L., Temme, N.M. New series expansions of the Gauss hypergeometric + * function. Adv Comput Math 39, 349-365 (2013). + * https://doi.org/10.1007/s10444-012-9283-y + * """ + */ + +#pragma once + +#include "config.h" +#include "error.h" +#include "tools.h" + +#include "binom.h" +#include "cephes/gamma.h" +#include "cephes/lanczos.h" +#include "cephes/poch.h" +#include "cephes/hyp2f1.h" +#include "digamma.h" + +namespace xsf { +namespace detail { + constexpr double hyp2f1_EPS = 1e-15; + /* The original implementation in SciPy from Zhang and Jin used 1500 for the + * maximum number of series iterations in some cases and 500 in others. + * Through the empirical results on the test cases in + * scipy/special/_precompute/hyp2f1_data.py, it was determined that these values + * can lead to early termination of series which would have eventually converged + * at a reasonable level of accuracy. We've bumped the iteration limit to 3000, + * and may adjust it again based on further analysis. */ + constexpr std::uint64_t hyp2f1_MAXITER = 3000; + + XSF_HOST_DEVICE inline double four_gammas_lanczos(double u, double v, double w, double x) { + /* Compute ratio of gamma functions using lanczos approximation. + * + * Computes gamma(u)*gamma(v)/(gamma(w)*gamma(x)) + * + * It is assumed that x = u + v - w, but it is left to the user to + * ensure this. + * + * The lanczos approximation takes the form + * + * gamma(x) = factor(x) * lanczos_sum_expg_scaled(x) + * + * where factor(x) = ((x + lanczos_g - 0.5)/e)**(x - 0.5). + * + * The formula above is only valid for x >= 0.5, but can be extended to + * x < 0.5 with the reflection principle. + * + * Using the lanczos approximation when computing this ratio of gamma functions + * allows factors to be combined analytically to avoid underflow and overflow + * and produce a more accurate result. The condition x = u + v - w makes it + * possible to cancel the factors in the expression + * + * factor(u) * factor(v) / (factor(w) * factor(x)) + * + * by taking one factor and absorbing it into the others. Currently, this + * implementation takes the factor corresponding to the argument with largest + * absolute value and absorbs it into the others. + * + * Since this is only called internally by four_gammas. It is assumed that + * |u| >= |v| and |w| >= |x|. + */ + + /* The below implementation may incorrectly return finite results + * at poles of the gamma function. Handle these cases explicitly. */ + if ((u == std::trunc(u) && u <= 0) || (v == std::trunc(v) && v <= 0)) { + /* Return nan if numerator has pole. Diverges to +- infinity + * depending on direction so value is undefined. */ + return std::numeric_limits::quiet_NaN(); + } + if ((w == std::trunc(w) && w <= 0) || (x == std::trunc(x) && x <= 0)) { + // Return 0 if denominator has pole but not numerator. + return 0.0; + } + + double result = 1.0; + double ugh, vgh, wgh, xgh, u_prime, v_prime, w_prime, x_prime; + + if (u >= 0.5) { + result *= cephes::lanczos_sum_expg_scaled(u); + ugh = u + cephes::lanczos_g - 0.5; + u_prime = u; + } else { + result /= cephes::lanczos_sum_expg_scaled(1 - u) * std::sin(M_PI * u) * M_1_PI; + ugh = 0.5 - u + cephes::lanczos_g; + u_prime = 1 - u; + } + + if (v >= 0.5) { + result *= cephes::lanczos_sum_expg_scaled(v); + vgh = v + cephes::lanczos_g - 0.5; + v_prime = v; + } else { + result /= cephes::lanczos_sum_expg_scaled(1 - v) * std::sin(M_PI * v) * M_1_PI; + vgh = 0.5 - v + cephes::lanczos_g; + v_prime = 1 - v; + } + + if (w >= 0.5) { + result /= cephes::lanczos_sum_expg_scaled(w); + wgh = w + cephes::lanczos_g - 0.5; + w_prime = w; + } else { + result *= cephes::lanczos_sum_expg_scaled(1 - w) * std::sin(M_PI * w) * M_1_PI; + wgh = 0.5 - w + cephes::lanczos_g; + w_prime = 1 - w; + } + + if (x >= 0.5) { + result /= cephes::lanczos_sum_expg_scaled(x); + xgh = x + cephes::lanczos_g - 0.5; + x_prime = x; + } else { + result *= cephes::lanczos_sum_expg_scaled(1 - x) * std::sin(M_PI * x) * M_1_PI; + xgh = 0.5 - x + cephes::lanczos_g; + x_prime = 1 - x; + } + + if (std::abs(u) >= std::abs(w)) { + // u has greatest absolute value. Absorb ugh into the others. + if (std::abs((v_prime - u_prime) * (v - 0.5)) < 100 * ugh and v > 100) { + /* Special case where base is close to 1. Condition taken from + * Boost's beta function implementation. */ + result *= std::exp((v - 0.5) * std::log1p((v_prime - u_prime) / ugh)); + } else { + result *= std::pow(vgh / ugh, v - 0.5); + } + + if (std::abs((u_prime - w_prime) * (w - 0.5)) < 100 * wgh and u > 100) { + result *= std::exp((w - 0.5) * std::log1p((u_prime - w_prime) / wgh)); + } else { + result *= std::pow(ugh / wgh, w - 0.5); + } + + if (std::abs((u_prime - x_prime) * (x - 0.5)) < 100 * xgh and u > 100) { + result *= std::exp((x - 0.5) * std::log1p((u_prime - x_prime) / xgh)); + } else { + result *= std::pow(ugh / xgh, x - 0.5); + } + } else { + // w has greatest absolute value. Absorb wgh into the others. + if (std::abs((u_prime - w_prime) * (u - 0.5)) < 100 * wgh and u > 100) { + result *= std::exp((u - 0.5) * std::log1p((u_prime - w_prime) / wgh)); + } else { + result *= pow(ugh / wgh, u - 0.5); + } + if (std::abs((v_prime - w_prime) * (v - 0.5)) < 100 * wgh and v > 100) { + result *= std::exp((v - 0.5) * std::log1p((v_prime - w_prime) / wgh)); + } else { + result *= std::pow(vgh / wgh, v - 0.5); + } + if (std::abs((w_prime - x_prime) * (x - 0.5)) < 100 * xgh and x > 100) { + result *= std::exp((x - 0.5) * std::log1p((w_prime - x_prime) / xgh)); + } else { + result *= std::pow(wgh / xgh, x - 0.5); + } + } + // This exhausts all cases because we assume |u| >= |v| and |w| >= |x|. + + return result; + } + + XSF_HOST_DEVICE inline double four_gammas(double u, double v, double w, double x) { + double result; + + // Without loss of generality, ensure |u| >= |v| and |w| >= |x|. + if (std::abs(v) > std::abs(u)) { + std::swap(u, v); + } + if (std::abs(x) > std::abs(w)) { + std::swap(x, w); + } + /* Direct ratio tends to be more accurate for arguments in this range. Range + * chosen empirically based on the relevant benchmarks in + * scipy/special/_precompute/hyp2f1_data.py */ + if (std::abs(u) <= 100 && std::abs(v) <= 100 && std::abs(w) <= 100 && std::abs(x) <= 100) { + result = cephes::Gamma(u) * cephes::Gamma(v) * (cephes::rgamma(w) * cephes::rgamma(x)); + if (std::isfinite(result) && result != 0.0) { + return result; + } + } + result = four_gammas_lanczos(u, v, w, x); + if (std::isfinite(result) && result != 0.0) { + return result; + } + // If overflow or underflow, try again with logs. + result = std::exp(cephes::lgam(v) - cephes::lgam(x) + cephes::lgam(u) - cephes::lgam(w)); + result *= cephes::gammasgn(u) * cephes::gammasgn(w) * cephes::gammasgn(v) * cephes::gammasgn(x); + return result; + } + + class HypergeometricSeriesGenerator { + /* Maclaurin series for hyp2f1. + * + * Series is convergent for |z| < 1 but is only practical for numerical + * computation when |z| < 0.9. + */ + public: + XSF_HOST_DEVICE HypergeometricSeriesGenerator(double a, double b, double c, std::complex z) + : a_(a), b_(b), c_(c), z_(z), term_(1.0), k_(0) {} + + XSF_HOST_DEVICE std::complex operator()() { + std::complex output = term_; + term_ = term_ * (a_ + k_) * (b_ + k_) / ((k_ + 1) * (c_ + k_)) * z_; + ++k_; + return output; + } + + private: + double a_, b_, c_; + std::complex z_, term_; + std::uint64_t k_; + }; + + class Hyp2f1Transform1Generator { + /* 1 -z transformation of standard series.*/ + public: + XSF_HOST_DEVICE Hyp2f1Transform1Generator(double a, double b, double c, std::complex z) + : factor1_(four_gammas(c, c - a - b, c - a, c - b)), + factor2_(four_gammas(c, a + b - c, a, b) * std::pow(1.0 - z, c - a - b)), + generator1_(HypergeometricSeriesGenerator(a, b, a + b - c + 1, 1.0 - z)), + generator2_(HypergeometricSeriesGenerator(c - a, c - b, c - a - b + 1, 1.0 - z)) {} + + XSF_HOST_DEVICE std::complex operator()() { + return factor1_ * generator1_() + factor2_ * generator2_(); + } + + private: + std::complex factor1_, factor2_; + HypergeometricSeriesGenerator generator1_, generator2_; + }; + + class Hyp2f1Transform1LimitSeriesGenerator { + /* 1 - z transform in limit as c - a - b approaches an integer m. */ + public: + XSF_HOST_DEVICE Hyp2f1Transform1LimitSeriesGenerator(double a, double b, double m, std::complex z) + : d1_(xsf::digamma(a)), d2_(xsf::digamma(b)), d3_(xsf::digamma(1 + m)), + d4_(xsf::digamma(1.0)), a_(a), b_(b), m_(m), z_(z), log_1_z_(std::log(1.0 - z)), + factor_(cephes::rgamma(m + 1)), k_(0) {} + + XSF_HOST_DEVICE std::complex operator()() { + std::complex term_ = (d1_ + d2_ - d3_ - d4_ + log_1_z_) * factor_; + // Use digamma(x + 1) = digamma(x) + 1/x + d1_ += 1 / (a_ + k_); // d1 = digamma(a + k) + d2_ += 1 / (b_ + k_); // d2 = digamma(b + k) + d3_ += 1 / (1.0 + m_ + k_); // d3 = digamma(1 + m + k) + d4_ += 1 / (1.0 + k_); // d4 = digamma(1 + k) + factor_ *= (a_ + k_) * (b_ + k_) / ((k_ + 1.0) * (m_ + k_ + 1)) * (1.0 - z_); + ++k_; + return term_; + } + + private: + double d1_, d2_, d3_, d4_, a_, b_, m_; + std::complex z_, log_1_z_, factor_; + int k_; + }; + + class Hyp2f1Transform2Generator { + /* 1/z transformation of standard series.*/ + public: + XSF_HOST_DEVICE Hyp2f1Transform2Generator(double a, double b, double c, std::complex z) + : factor1_(four_gammas(c, b - a, b, c - a) * std::pow(-z, -a)), + factor2_(four_gammas(c, a - b, a, c - b) * std::pow(-z, -b)), + generator1_(HypergeometricSeriesGenerator(a, a - c + 1, a - b + 1, 1.0 / z)), + generator2_(HypergeometricSeriesGenerator(b, b - c + 1, b - a + 1, 1.0 / z)) {} + + XSF_HOST_DEVICE std::complex operator()() { + return factor1_ * generator1_() + factor2_ * generator2_(); + } + + private: + std::complex factor1_, factor2_; + HypergeometricSeriesGenerator generator1_, generator2_; + }; + + class Hyp2f1Transform2LimitSeriesGenerator { + /* 1/z transform in limit as a - b approaches a non-negative integer m. (Can swap a and b to + * handle the m a negative integer case. */ + public: + XSF_HOST_DEVICE Hyp2f1Transform2LimitSeriesGenerator(double a, double b, double c, double m, + std::complex z) + : d1_(xsf::digamma(1.0)), d2_(xsf::digamma(1 + m)), d3_(xsf::digamma(a)), + d4_(xsf::digamma(c - a)), a_(a), b_(b), c_(c), m_(m), z_(z), log_neg_z_(std::log(-z)), + factor_(xsf::cephes::poch(b, m) * xsf::cephes::poch(1 - c + b, m) * + xsf::cephes::rgamma(m + 1)), + k_(0) {} + + XSF_HOST_DEVICE std::complex operator()() { + std::complex term = (d1_ + d2_ - d3_ - d4_ + log_neg_z_) * factor_; + // Use digamma(x + 1) = digamma(x) + 1/x + d1_ += 1 / (1.0 + k_); // d1 = digamma(1 + k) + d2_ += 1 / (1.0 + m_ + k_); // d2 = digamma(1 + m + k) + d3_ += 1 / (a_ + k_); // d3 = digamma(a + k) + d4_ -= 1 / (c_ - a_ - k_ - 1); // d4 = digamma(c - a - k) + factor_ *= (b_ + m_ + k_) * (1 - c_ + b_ + m_ + k_) / ((k_ + 1) * (m_ + k_ + 1)) / z_; + ++k_; + return term; + } + + private: + double d1_, d2_, d3_, d4_, a_, b_, c_, m_; + std::complex z_, log_neg_z_, factor_; + std::uint64_t k_; + }; + + class Hyp2f1Transform2LimitSeriesCminusAIntGenerator { + /* 1/z transform in limit as a - b approaches a non-negative integer m, and c - a approaches + * a positive integer n. */ + public: + XSF_HOST_DEVICE Hyp2f1Transform2LimitSeriesCminusAIntGenerator(double a, double b, double c, double m, + double n, std::complex z) + : d1_(xsf::digamma(1.0)), d2_(xsf::digamma(1 + m)), d3_(xsf::digamma(a)), + d4_(xsf::digamma(n)), a_(a), b_(b), c_(c), m_(m), n_(n), z_(z), log_neg_z_(std::log(-z)), + factor_(xsf::cephes::poch(b, m) * xsf::cephes::poch(1 - c + b, m) * + xsf::cephes::rgamma(m + 1)), + k_(0) {} + + XSF_HOST_DEVICE std::complex operator()() { + std::complex term; + if (k_ < n_) { + term = (d1_ + d2_ - d3_ - d4_ + log_neg_z_) * factor_; + // Use digamma(x + 1) = digamma(x) + 1/x + d1_ += 1 / (1.0 + k_); // d1 = digamma(1 + k) + d2_ += 1 / (1 + m_ + k_); // d2 = digamma(1 + m + k) + d3_ += 1 / (a_ + k_); // d3 = digamma(a + k) + d4_ -= 1 / (n_ - k_ - 1); // d4 = digamma(c - a - k) + factor_ *= (b_ + m_ + k_) * (1 - c_ + b_ + m_ + k_) / ((k_ + 1) * (m_ + k_ + 1)) / z_; + ++k_; + return term; + } + if (k_ == n_) { + /* When c - a approaches a positive integer and k_ >= c - a = n then + * poch(1 - c + b + m + k) = poch(1 - c + a + k) = approaches zero and + * digamma(c - a - k) approaches a pole. However we can use the limit + * digamma(-n + epsilon) / gamma(-n + epsilon) -> (-1)**(n + 1) * (n+1)! as epsilon -> 0 + * to continue the series. + * + * poch(1 - c + b, m + k) = gamma(1 - c + b + m + k)/gamma(1 - c + b) + * + * If a - b is an integer and c - a is an integer, then a and b must both be integers, so assume + * a and b are integers and take the limit as c approaches an integer. + * + * gamma(1 - c + epsilon + a + k)/gamma(1 - c - epsilon + b) = + * (gamma(c + epsilon - b) / gamma(c + epsilon - a - k)) * + * (sin(pi * (c + epsilon - b)) / sin(pi * (c + epsilon - a - k))) (reflection principle) + * + * In the limit as epsilon goes to zero, the ratio of sines will approach + * (-1)**(a - b + k) = (-1)**(m + k) + * + * We may then replace + * + * poch(1 - c - epsilon + b, m + k)*digamma(c + epsilon - a - k) + * + * with + * + * (-1)**(a - b + k)*gamma(c + epsilon - b) * digamma(c + epsilon - a - k) / gamma(c + epsilon - a - k) + * + * and taking the limit epsilon -> 0 gives + * + * (-1)**(a - b + k) * gamma(c - b) * (-1)**(k + a - c + 1)(k + a - c)! + * = (-1)**(c - b - 1)*Gamma(k + a - c + 1) + */ + factor_ = std::pow(-1, m_ + n_) * xsf::binom(c_ - 1, b_ - 1) * + xsf::cephes::poch(c_ - a_ + 1, m_ - 1) / std::pow(z_, static_cast(k_)); + } + term = factor_; + factor_ *= (b_ + m_ + k_) * (k_ + a_ - c_ + 1) / ((k_ + 1) * (m_ + k_ + 1)) / z_; + ++k_; + return term; + } + + private: + double d1_, d2_, d3_, d4_, a_, b_, c_, m_, n_; + std::complex z_, log_neg_z_, factor_; + std::uint64_t k_; + }; + + class Hyp2f1Transform2LimitFinitePartGenerator { + /* Initial finite sum in limit as a - b approaches a non-negative integer m. The limiting series + * for the 1 - z transform also has an initial finite sum, but it is a standard hypergeometric + * series. */ + public: + XSF_HOST_DEVICE Hyp2f1Transform2LimitFinitePartGenerator(double b, double c, double m, + std::complex z) + : b_(b), c_(c), m_(m), z_(z), term_(cephes::Gamma(m) * cephes::rgamma(c - b)), k_(0) {} + + XSF_HOST_DEVICE std::complex operator()() { + std::complex output = term_; + term_ = term_ * (b_ + k_) * (c_ - b_ - k_ - 1) / ((k_ + 1) * (m_ - k_ - 1)) / z_; + ++k_; + return output; + } + + private: + double b_, c_, m_; + std::complex z_, term_; + std::uint64_t k_; + }; + + class LopezTemmeSeriesGenerator { + /* Lopez-Temme Series for Gaussian hypergeometric function [4]. + * + * Converges for all z with real(z) < 1, including in the regions surrounding + * the points exp(+- i*pi/3) that are not covered by any of the standard + * transformations. + */ + public: + XSF_HOST_DEVICE LopezTemmeSeriesGenerator(double a, double b, double c, std::complex z) + : n_(0), a_(a), b_(b), c_(c), phi_previous_(1.0), phi_(1 - 2 * b / c), z_(z), Z_(a * z / (z - 2.0)) {} + + XSF_HOST_DEVICE std::complex operator()() { + if (n_ == 0) { + ++n_; + return 1.0; + } + if (n_ > 1) { // Update phi and Z for n>=2 + double new_phi = ((n_ - 1) * phi_previous_ - (2.0 * b_ - c_) * phi_) / (c_ + (n_ - 1)); + phi_previous_ = phi_; + phi_ = new_phi; + Z_ = Z_ * z_ / (z_ - 2.0) * ((a_ + (n_ - 1)) / n_); + } + ++n_; + return Z_ * phi_; + } + + private: + std::uint64_t n_; + double a_, b_, c_, phi_previous_, phi_; + std::complex z_, Z_; + }; + + XSF_HOST_DEVICE std::complex hyp2f1_transform1_limiting_case(double a, double b, double c, double m, + std::complex z) { + /* 1 - z transform in limiting case where c - a - b approaches an integer m. */ + std::complex result = 0.0; + if (m >= 0) { + if (m != 0) { + auto series_generator = HypergeometricSeriesGenerator(a, b, 1 - m, 1.0 - z); + result += four_gammas(m, c, a + m, b + m) * series_eval_fixed_length(series_generator, + std::complex{0.0, 0.0}, + static_cast(m)); + } + std::complex prefactor = std::pow(-1.0, m + 1) * xsf::cephes::Gamma(c) / + (xsf::cephes::Gamma(a) * xsf::cephes::Gamma(b)) * + std::pow(1.0 - z, m); + auto series_generator = Hyp2f1Transform1LimitSeriesGenerator(a + m, b + m, m, z); + result += prefactor * series_eval(series_generator, std::complex{0.0, 0.0}, hyp2f1_EPS, + hyp2f1_MAXITER, "hyp2f1"); + return result; + } else { + result = four_gammas(-m, c, a, b) * std::pow(1.0 - z, m); + auto series_generator1 = HypergeometricSeriesGenerator(a + m, b + m, 1 + m, 1.0 - z); + result *= series_eval_fixed_length(series_generator1, std::complex{0.0, 0.0}, + static_cast(-m)); + double prefactor = std::pow(-1.0, m + 1) * xsf::cephes::Gamma(c) * + (xsf::cephes::rgamma(a + m) * xsf::cephes::rgamma(b + m)); + auto series_generator2 = Hyp2f1Transform1LimitSeriesGenerator(a, b, -m, z); + result += prefactor * series_eval(series_generator2, std::complex{0.0, 0.0}, hyp2f1_EPS, + hyp2f1_MAXITER, "hyp2f1"); + return result; + } + } + + XSF_HOST_DEVICE std::complex hyp2f1_transform2_limiting_case(double a, double b, double c, double m, + std::complex z) { + /* 1 / z transform in limiting case where a - b approaches a non-negative integer m. Negative integer case + * can be handled by swapping a and b. */ + auto series_generator1 = Hyp2f1Transform2LimitFinitePartGenerator(b, c, m, z); + std::complex result = cephes::Gamma(c) * cephes::rgamma(a) * std::pow(-z, -b); + result *= + series_eval_fixed_length(series_generator1, std::complex{0.0, 0.0}, static_cast(m)); + std::complex prefactor = cephes::Gamma(c) * (cephes::rgamma(a) * cephes::rgamma(c - b) * std::pow(-z, -a)); + double n = c - a; + if (abs(n - std::round(n)) < hyp2f1_EPS) { + auto series_generator2 = Hyp2f1Transform2LimitSeriesCminusAIntGenerator(a, b, c, m, n, z); + result += prefactor * series_eval(series_generator2, std::complex{0.0, 0.0}, hyp2f1_EPS, + hyp2f1_MAXITER, "hyp2f1"); + return result; + } + auto series_generator2 = Hyp2f1Transform2LimitSeriesGenerator(a, b, c, m, z); + result += prefactor * + series_eval(series_generator2, std::complex{0.0, 0.0}, hyp2f1_EPS, hyp2f1_MAXITER, "hyp2f1"); + return result; + } + +} // namespace detail + +XSF_HOST_DEVICE inline std::complex hyp2f1(double a, double b, double c, std::complex z) { + /* Special Cases + * ----------------------------------------------------------------------- + * Takes constant value 1 when a = 0 or b = 0, even if c is a non-positive + * integer. This follows mpmath. */ + if (a == 0 || b == 0) { + return 1.0; + } + double z_abs = std::abs(z); + // Equals 1 when z i 0, unless c is 0. + if (z_abs == 0) { + if (c != 0) { + return 1.0; + } else { + // Returning real part NAN and imaginary part 0 follows mpmath. + return std::complex{std::numeric_limits::quiet_NaN(), 0}; + } + } + bool a_neg_int = a == std::trunc(a) && a < 0; + bool b_neg_int = b == std::trunc(b) && b < 0; + bool c_non_pos_int = c == std::trunc(c) and c <= 0; + /* Diverges when c is a non-positive integer unless a is an integer with + * c <= a <= 0 or b is an integer with c <= b <= 0, (or z equals 0 with + * c != 0) Cases z = 0, a = 0, or b = 0 have already been handled. We follow + * mpmath in handling the degenerate cases where any of a, b, c are + * non-positive integers. See [3] for a treatment of degenerate cases. */ + if (c_non_pos_int && !((a_neg_int && c <= a && a < 0) || (b_neg_int && c <= b && b < 0))) { + return std::complex{std::numeric_limits::infinity(), 0}; + } + /* Reduces to a polynomial when a or b is a negative integer. + * If a and b are both negative integers, we take care to terminate + * the series at a or b of smaller magnitude. This is to ensure proper + * handling of situations like a < c < b <= 0, a, b, c all non-positive + * integers, where terminating at a would lead to a term of the form 0 / 0. */ + double max_degree; + if (a_neg_int || b_neg_int) { + if (a_neg_int && b_neg_int) { + max_degree = a > b ? std::abs(a) : std::abs(b); + } else if (a_neg_int) { + max_degree = std::abs(a); + } else { + max_degree = std::abs(b); + } + if (max_degree <= (double) UINT64_MAX) { + auto series_generator = detail::HypergeometricSeriesGenerator(a, b, c, z); + return detail::series_eval_fixed_length(series_generator, std::complex{0.0, 0.0}, max_degree + 1); + } else { + set_error("hyp2f1", SF_ERROR_NO_RESULT, NULL); + return std::complex{std::numeric_limits::quiet_NaN(), + std::numeric_limits::quiet_NaN()}; + } + } + // Kummer's Theorem for z = -1; c = 1 + a - b (DLMF 15.4.26) + if (std::abs(z + 1.0) < detail::hyp2f1_EPS && std::abs(1 + a - b - c) < detail::hyp2f1_EPS && !c_non_pos_int) { + return detail::four_gammas(a - b + 1, 0.5 * a + 1, a + 1, 0.5 * a - b + 1); + } + std::complex result; + bool c_minus_a_neg_int = c - a == std::trunc(c - a) && c - a < 0; + bool c_minus_b_neg_int = c - b == std::trunc(c - b) && c - b < 0; + /* If one of c - a or c - b is a negative integer, reduces to evaluating + * a polynomial through an Euler hypergeometric transformation. + * (DLMF 15.8.1) */ + if (c_minus_a_neg_int || c_minus_b_neg_int) { + max_degree = c_minus_b_neg_int ? std::abs(c - b) : std::abs(c - a); + if (max_degree <= (double) UINT64_MAX) { + result = std::pow(1.0 - z, c - a - b); + auto series_generator = detail::HypergeometricSeriesGenerator(c - a, c - b, c, z); + result *= + detail::series_eval_fixed_length(series_generator, std::complex{0.0, 0.0}, max_degree + 2); + return result; + } else { + set_error("hyp2f1", SF_ERROR_NO_RESULT, NULL); + return std::complex{std::numeric_limits::quiet_NaN(), + std::numeric_limits::quiet_NaN()}; + } + } + /* Diverges as real(z) -> 1 when c <= a + b. + * Todo: Actually check for overflow instead of using a fixed tolerance for + * all parameter combinations like in the Fortran original. */ + if (std::abs(1 - z.real()) < detail::hyp2f1_EPS && z.imag() == 0 && c - a - b <= 0 && !c_non_pos_int) { + return std::complex{std::numeric_limits::infinity(), 0}; + } + // Gauss's Summation Theorem for z = 1; c - a - b > 0 (DLMF 15.4.20). + if (z == 1.0 && c - a - b > 0 && !c_non_pos_int) { + return detail::four_gammas(c, c - a - b, c - a, c - b); + } + /* |z| < 0, z.real() >= 0. Use the Maclaurin Series. + * ----------------------------------------------------------------------- + * Apply Euler Hypergeometric Transformation (DLMF 15.8.1) to reduce + * size of a and b if possible. We follow Zhang and Jin's + * implementation [1] although there is very likely a better heuristic + * to determine when this transformation should be applied. As it + * stands, this hurts precision in some cases. */ + if (z_abs < 0.9 && z.real() >= 0) { + if (c - a < a && c - b < b) { + result = std::pow(1.0 - z, c - a - b); + auto series_generator = detail::HypergeometricSeriesGenerator(c - a, c - b, c, z); + result *= detail::series_eval(series_generator, std::complex{0.0, 0.0}, detail::hyp2f1_EPS, + detail::hyp2f1_MAXITER, "hyp2f1"); + return result; + } + auto series_generator = detail::HypergeometricSeriesGenerator(a, b, c, z); + return detail::series_eval(series_generator, std::complex{0.0, 0.0}, detail::hyp2f1_EPS, + detail::hyp2f1_MAXITER, "hyp2f1"); + } + /* Points near exp(iπ/3), exp(-iπ/3) not handled by any of the standard + * transformations. Use series of López and Temme [5]. These regions + * were not correctly handled by Zhang and Jin's implementation. + * -------------------------------------------------------------------------*/ + if (0.9 <= z_abs && z_abs < 1.1 && std::abs(1.0 - z) >= 0.9 && z.real() >= 0) { + /* This condition for applying Euler Transformation (DLMF 15.8.1) + * was determined empirically to work better for this case than that + * used in Zhang and Jin's implementation for |z| < 0.9, + * real(z) >= 0. */ + if ((c - a <= a && c - b < b) || (c - a < a && c - b <= b)) { + auto series_generator = detail::LopezTemmeSeriesGenerator(c - a, c - b, c, z); + result = std::pow(1.0 - 0.5 * z, a - c); // Lopez-Temme prefactor + result *= detail::series_eval(series_generator, std::complex{0.0, 0.0}, detail::hyp2f1_EPS, + detail::hyp2f1_MAXITER, "hyp2f1"); + return std::pow(1.0 - z, c - a - b) * result; // Euler transform prefactor. + } + auto series_generator = detail::LopezTemmeSeriesGenerator(a, b, c, z); + result = detail::series_eval(series_generator, std::complex{0.0, 0.0}, detail::hyp2f1_EPS, + detail::hyp2f1_MAXITER, "hyp2f1"); + return std::pow(1.0 - 0.5 * z, -a) * result; // Lopez-Temme prefactor. + } + /* z/(z - 1) transformation (DLMF 15.8.1). Avoids cancellation issues that + * occur with Maclaurin series for real(z) < 0. + * -------------------------------------------------------------------------*/ + if (z_abs < 1.1 && z.real() < 0) { + if (0 < b && b < a && a < c) { + std::swap(a, b); + } + auto series_generator = detail::HypergeometricSeriesGenerator(a, c - b, c, z / (z - 1.0)); + return std::pow(1.0 - z, -a) * detail::series_eval(series_generator, std::complex{0.0, 0.0}, + detail::hyp2f1_EPS, detail::hyp2f1_MAXITER, "hyp2f1"); + } + /* 1 - z transformation (DLMF 15.8.4). */ + if (0.9 <= z_abs && z_abs < 1.1) { + if (std::abs(c - a - b - std::round(c - a - b)) < detail::hyp2f1_EPS) { + // Removable singularity when c - a - b is an integer. Need to use limiting formula. + double m = std::round(c - a - b); + return detail::hyp2f1_transform1_limiting_case(a, b, c, m, z); + } + auto series_generator = detail::Hyp2f1Transform1Generator(a, b, c, z); + return detail::series_eval(series_generator, std::complex{0.0, 0.0}, detail::hyp2f1_EPS, + detail::hyp2f1_MAXITER, "hyp2f1"); + } + /* 1/z transformation (DLMF 15.8.2). */ + if (std::abs(a - b - std::round(a - b)) < detail::hyp2f1_EPS) { + if (b > a) { + std::swap(a, b); + } + double m = std::round(a - b); + return detail::hyp2f1_transform2_limiting_case(a, b, c, m, z); + } + auto series_generator = detail::Hyp2f1Transform2Generator(a, b, c, z); + return detail::series_eval(series_generator, std::complex{0.0, 0.0}, detail::hyp2f1_EPS, + detail::hyp2f1_MAXITER, "hyp2f1"); +} + +XSF_HOST_DEVICE inline std::complex hyp2f1(float a, float b, float c, std::complex x) { + return static_cast>(hyp2f1(static_cast(a), static_cast(b), + static_cast(c), static_cast>(x))); +} + +XSF_HOST_DEVICE inline double hyp2f1(double a, double b, double c, double x) { return cephes::hyp2f1(a, b, c, x); } + +XSF_HOST_DEVICE inline float hyp2f1(float a, float b, float c, float x) { + return hyp2f1(static_cast(a), static_cast(b), static_cast(c), static_cast(x)); +} + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/iv_ratio.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/iv_ratio.h new file mode 100644 index 0000000000000000000000000000000000000000..e5dc871bd003937a483a39001d796b1bacc26d01 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/iv_ratio.h @@ -0,0 +1,173 @@ +// Numerically stable computation of iv(v+1, x) / iv(v, x) + +#pragma once + +#include "config.h" +#include "tools.h" +#include "error.h" +#include "cephes/dd_real.h" + +namespace xsf { + +/* Generates the "tail" of Perron's continued fraction for iv(v,x)/iv(v-1,x). + * + * The Perron continued fraction is studied in [1]. It is given by + * + * iv(v, x) x -(2v+1)x -(2v+3)x -(2v+5)x + * R := --------- = ------ ---------- ---------- ---------- ... + * iv(v-1,x) x+2v + 2(v+x)+1 + 2(v+x)+2 + 2(v+x)+3 + + * + * Given a suitable constant c, the continued fraction may be rearranged + * into the following form to avoid premature floating point overflow: + * + * xc -(2vc+c)(xc) -(2vc+3c)(xc) -(2vc+5c)(xc) + * R = -----, fc = 2vc + ------------ ------------- ------------- ... + * xc+fc 2(vc+xc)+c + 2(vc+xc)+2c + 2(vc+xc)+3c + + * + * This class generates the fractions of fc after 2vc. + * + * [1] Gautschi, W. and Slavik, J. (1978). "On the computation of modified + * Bessel function ratios." Mathematics of Computation, 32(143):865-875. + */ +template +struct IvRatioCFTailGenerator { + + XSF_HOST_DEVICE IvRatioCFTailGenerator(T vc, T xc, T c) noexcept { + a0_ = -(2*vc-c)*xc; + as_ = -2*c*xc; + b0_ = 2*(vc+xc); + bs_ = c; + k_ = 0; + } + + XSF_HOST_DEVICE std::pair operator()() noexcept { + using std::fma; + ++k_; + return {fma(static_cast(k_), as_, a0_), + fma(static_cast(k_), bs_, b0_)}; + } + +private: + T a0_, as_; // a[k] == a0 + as*k, k >= 1 + T b0_, bs_; // b[k] == b0 + bs*k, k >= 1 + std::uint64_t k_; // current index +}; + +// Computes f(v, x) using Perron's continued fraction. +// +// T specifies the working type. This allows the function to perform +// calculations in a higher precision, such as double-double, even if +// the return type is hardcoded to be double. +template +XSF_HOST_DEVICE inline std::pair +_iv_ratio_cf(double v, double x, bool complement) { + + int e; + std::frexp(std::fmax(v, x), &e); + T c = T(std::ldexp(1, 2-e)); // rescaling multiplier + T vc = v * c; + T xc = x * c; + + IvRatioCFTailGenerator cf(vc, xc, c); + auto [fc, terms] = detail::series_eval_kahan( + detail::continued_fraction_series(cf), + T(std::numeric_limits::epsilon()), + 1000, + 2*vc); + + T ret = (complement ? fc : xc) / (xc + fc); + return {static_cast(ret), terms}; +} + +XSF_HOST_DEVICE inline double iv_ratio(double v, double x) { + + if (std::isnan(v) || std::isnan(x)) { + return std::numeric_limits::quiet_NaN(); + } + if (v < 0.5 || x < 0) { + set_error("iv_ratio", SF_ERROR_DOMAIN, NULL); + return std::numeric_limits::quiet_NaN(); + } + if (std::isinf(v) && std::isinf(x)) { + // There is not a unique limit as both v and x tends to infinity. + set_error("iv_ratio", SF_ERROR_DOMAIN, NULL); + return std::numeric_limits::quiet_NaN(); + } + if (x == 0.0) { + return x; // keep sign of x because iv_ratio is an odd function + } + if (std::isinf(v)) { + return 0.0; + } + if (std::isinf(x)) { + return 1.0; + } + + auto [ret, terms] = _iv_ratio_cf(v, x, false); + if (terms == 0) { // failed to converge; should not happen + set_error("iv_ratio", SF_ERROR_NO_RESULT, NULL); + return std::numeric_limits::quiet_NaN(); + } + return ret; +} + +XSF_HOST_DEVICE inline float iv_ratio(float v, float x) { + return iv_ratio(static_cast(v), static_cast(x)); +} + +XSF_HOST_DEVICE inline double iv_ratio_c(double v, double x) { + + if (std::isnan(v) || std::isnan(x)) { + return std::numeric_limits::quiet_NaN(); + } + if (v < 0.5 || x < 0) { + set_error("iv_ratio_c", SF_ERROR_DOMAIN, NULL); + return std::numeric_limits::quiet_NaN(); + } + if (std::isinf(v) && std::isinf(x)) { + // There is not a unique limit as both v and x tends to infinity. + set_error("iv_ratio_c", SF_ERROR_DOMAIN, NULL); + return std::numeric_limits::quiet_NaN(); + } + if (x == 0.0) { + return 1.0; + } + if (std::isinf(v)) { + return 1.0; + } + if (std::isinf(x)) { + return 0.0; + } + + if (v >= 1) { + // Numerical experiments show that evaluating the Perron c.f. + // in double precision is sufficiently accurate if v >= 1. + auto [ret, terms] = _iv_ratio_cf(v, x, true); + if (terms == 0) { // failed to converge; should not happen + set_error("iv_ratio_c", SF_ERROR_NO_RESULT, NULL); + return std::numeric_limits::quiet_NaN(); + } + return ret; + } else if (v > 0.5) { + // double-double arithmetic is needed for 0.5 < v < 1 to + // achieve relative error on the scale of machine precision. + using cephes::detail::double_double; + auto [ret, terms] = _iv_ratio_cf(v, x, true); + if (terms == 0) { // failed to converge; should not happen + set_error("iv_ratio_c", SF_ERROR_NO_RESULT, NULL); + return std::numeric_limits::quiet_NaN(); + } + return ret; + } else { + // The previous branch (v > 0.5) also works for v == 0.5, but + // the closed-form formula "1 - tanh(x)" is more efficient. + double t = std::exp(-2*x); + return (2 * t) / (1 + t); + } +} + +XSF_HOST_DEVICE inline float iv_ratio_c(float v, float x) { + return iv_ratio_c(static_cast(v), static_cast(x)); +} + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/lambertw.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/lambertw.h new file mode 100644 index 0000000000000000000000000000000000000000..9eb1882eaec464b5e5dcf47f2168f9125996a816 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/lambertw.h @@ -0,0 +1,150 @@ +/* Translated from Cython into C++ by SciPy developers in 2023. + * Original header with Copyright information appears below. + */ + +/* Implementation of the Lambert W function [1]. Based on MPMath + * Implementation [2], and documentation [3]. + * + * Copyright: Yosef Meller, 2009 + * Author email: mellerf@netvision.net.il + * + * Distributed under the same license as SciPy + * + * + * References: + * [1] On the Lambert W function, Adv. Comp. Math. 5 (1996) 329-359, + * available online: https://web.archive.org/web/20230123211413/https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf + * [2] mpmath source code, + https://github.com/mpmath/mpmath/blob/c5939823669e1bcce151d89261b802fe0d8978b4/mpmath/functions/functions.py#L435-L461 + * [3] + https://web.archive.org/web/20230504171447/https://mpmath.org/doc/current/functions/powers.html#lambert-w-function + * + + * TODO: use a series expansion when extremely close to the branch point + * at `-1/e` and make sure that the proper branch is chosen there. + */ + +#pragma once + +#include "config.h" +#include "error.h" +#include "evalpoly.h" + +namespace xsf { +constexpr double EXPN1 = 0.36787944117144232159553; // exp(-1) +constexpr double OMEGA = 0.56714329040978387299997; // W(1, 0) + +namespace detail { + XSF_HOST_DEVICE inline std::complex lambertw_branchpt(std::complex z) { + // Series for W(z, 0) around the branch point; see 4.22 in [1]. + double coeffs[] = {-1.0 / 3.0, 1.0, -1.0}; + std::complex p = std::sqrt(2.0 * (M_E * z + 1.0)); + + return cevalpoly(coeffs, 2, p); + } + + XSF_HOST_DEVICE inline std::complex lambertw_pade0(std::complex z) { + // (3, 2) Pade approximation for W(z, 0) around 0. + double num[] = {12.85106382978723404255, 12.34042553191489361902, 1.0}; + double denom[] = {32.53191489361702127660, 14.34042553191489361702, 1.0}; + + /* This only gets evaluated close to 0, so we don't need a more + * careful algorithm that avoids overflow in the numerator for + * large z. */ + return z * cevalpoly(num, 2, z) / cevalpoly(denom, 2, z); + } + + XSF_HOST_DEVICE inline std::complex lambertw_asy(std::complex z, long k) { + /* Compute the W function using the first two terms of the + * asymptotic series. See 4.20 in [1]. + */ + std::complex w = std::log(z) + 2.0 * M_PI * k * std::complex(0, 1); + return w - std::log(w); + } + +} // namespace detail + +XSF_HOST_DEVICE inline std::complex lambertw(std::complex z, long k, double tol) { + double absz; + std::complex w; + std::complex ew, wew, wewz, wn; + + if (std::isnan(z.real()) || std::isnan(z.imag())) { + return z; + } + if (z.real() == std::numeric_limits::infinity()) { + return z + 2.0 * M_PI * k * std::complex(0, 1); + } + if (z.real() == -std::numeric_limits::infinity()) { + return -z + (2.0 * M_PI * k + M_PI) * std::complex(0, 1); + } + if (z == 0.0) { + if (k == 0) { + return z; + } + set_error("lambertw", SF_ERROR_SINGULAR, NULL); + return -std::numeric_limits::infinity(); + } + if (z == 1.0 && k == 0) { + // Split out this case because the asymptotic series blows up + return OMEGA; + } + + absz = std::abs(z); + // Get an initial guess for Halley's method + if (k == 0) { + if (std::abs(z + EXPN1) < 0.3) { + w = detail::lambertw_branchpt(z); + } else if (-1.0 < z.real() && z.real() < 1.5 && std::abs(z.imag()) < 1.0 && + -2.5 * std::abs(z.imag()) - 0.2 < z.real()) { + /* Empirically determined decision boundary where the Pade + * approximation is more accurate. */ + w = detail::lambertw_pade0(z); + } else { + w = detail::lambertw_asy(z, k); + } + } else if (k == -1) { + if (absz <= EXPN1 && z.imag() == 0.0 && z.real() < 0.0) { + w = std::log(-z.real()); + } else { + w = detail::lambertw_asy(z, k); + } + } else { + w = detail::lambertw_asy(z, k); + } + + // Halley's method; see 5.9 in [1] + if (w.real() >= 0) { + // Rearrange the formula to avoid overflow in exp + for (int i = 0; i < 100; i++) { + ew = std::exp(-w); + wewz = w - z * ew; + wn = w - wewz / (w + 1.0 - (w + 2.0) * wewz / (2.0 * w + 2.0)); + if (std::abs(wn - w) <= tol * std::abs(wn)) { + return wn; + } + w = wn; + } + } else { + for (int i = 0; i < 100; i++) { + ew = std::exp(w); + wew = w * ew; + wewz = wew - z; + wn = w - wewz / (wew + ew - (w + 2.0) * wewz / (2.0 * w + 2.0)); + if (std::abs(wn - w) <= tol * std::abs(wn)) { + return wn; + } + w = wn; + } + } + + set_error("lambertw", SF_ERROR_SLOW, "iteration failed to converge: %g + %gj", z.real(), z.imag()); + return {std::numeric_limits::quiet_NaN(), std::numeric_limits::quiet_NaN()}; +} + +XSF_HOST_DEVICE inline std::complex lambertw(std::complex z, long k, float tol) { + return static_cast>( + lambertw(static_cast>(z), k, static_cast(tol))); +} + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/loggamma.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/loggamma.h new file mode 100644 index 0000000000000000000000000000000000000000..eaae479b2054177153b8671e7fe3174f1b09f20a --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/loggamma.h @@ -0,0 +1,163 @@ +/* Translated from Cython into C++ by SciPy developers in 2024. + * Original header comment appears below. + */ + +/* An implementation of the principal branch of the logarithm of + * Gamma. Also contains implementations of Gamma and 1/Gamma which are + * easily computed from log-Gamma. + * + * Author: Josh Wilson + * + * Distributed under the same license as Scipy. + * + * References + * ---------- + * [1] Hare, "Computing the Principal Branch of log-Gamma", + * Journal of Algorithms, 1997. + * + * [2] Julia, + * https://github.com/JuliaLang/julia/blob/master/base/special/gamma.jl + */ + +#pragma once + +#include "cephes/gamma.h" +#include "cephes/rgamma.h" +#include "config.h" +#include "error.h" +#include "evalpoly.h" +#include "trig.h" +#include "zlog1.h" + +namespace xsf { + +namespace detail { + constexpr double loggamma_SMALLX = 7; + constexpr double loggamma_SMALLY = 7; + constexpr double loggamma_HLOG2PI = 0.918938533204672742; // log(2*pi)/2 + constexpr double loggamma_LOGPI = 1.1447298858494001741434262; // log(pi) + constexpr double loggamma_TAYLOR_RADIUS = 0.2; + + XSF_HOST_DEVICE std::complex loggamma_stirling(std::complex z) { + /* Stirling series for log-Gamma + * + * The coefficients are B[2*n]/(2*n*(2*n - 1)) where B[2*n] is the + * (2*n)th Bernoulli number. See (1.1) in [1]. + */ + double coeffs[] = {-2.955065359477124183E-2, 6.4102564102564102564E-3, -1.9175269175269175269E-3, + 8.4175084175084175084E-4, -5.952380952380952381E-4, 7.9365079365079365079E-4, + -2.7777777777777777778E-3, 8.3333333333333333333E-2}; + std::complex rz = 1.0 / z; + std::complex rzz = rz / z; + + return (z - 0.5) * std::log(z) - z + loggamma_HLOG2PI + rz * cevalpoly(coeffs, 7, rzz); + } + + XSF_HOST_DEVICE std::complex loggamma_recurrence(std::complex z) { + /* Backward recurrence relation. + * + * See Proposition 2.2 in [1] and the Julia implementation [2]. + * + */ + int signflips = 0; + int sb = 0; + std::complex shiftprod = z; + + z += 1.0; + int nsb; + while (z.real() <= loggamma_SMALLX) { + shiftprod *= z; + nsb = std::signbit(shiftprod.imag()); + signflips += nsb != 0 && sb == 0 ? 1 : 0; + sb = nsb; + z += 1.0; + } + return loggamma_stirling(z) - std::log(shiftprod) - signflips * 2 * M_PI * std::complex(0, 1); + } + + XSF_HOST_DEVICE std::complex loggamma_taylor(std::complex z) { + /* Taylor series for log-Gamma around z = 1. + * + * It is + * + * loggamma(z + 1) = -gamma*z + zeta(2)*z**2/2 - zeta(3)*z**3/3 ... + * + * where gamma is the Euler-Mascheroni constant. + */ + + double coeffs[] = { + -4.3478266053040259361E-2, 4.5454556293204669442E-2, -4.7619070330142227991E-2, 5.000004769810169364E-2, + -5.2631679379616660734E-2, 5.5555767627403611102E-2, -5.8823978658684582339E-2, 6.2500955141213040742E-2, + -6.6668705882420468033E-2, 7.1432946295361336059E-2, -7.6932516411352191473E-2, 8.3353840546109004025E-2, + -9.0954017145829042233E-2, 1.0009945751278180853E-1, -1.1133426586956469049E-1, 1.2550966952474304242E-1, + -1.4404989676884611812E-1, 1.6955717699740818995E-1, -2.0738555102867398527E-1, 2.7058080842778454788E-1, + -4.0068563438653142847E-1, 8.2246703342411321824E-1, -5.7721566490153286061E-1}; + + z -= 1.0; + return z * cevalpoly(coeffs, 22, z); + } +} // namespace detail + +XSF_HOST_DEVICE inline double loggamma(double x) { + if (x < 0.0) { + return std::numeric_limits::quiet_NaN(); + } + return cephes::lgam(x); +} + +XSF_HOST_DEVICE inline float loggamma(float x) { return loggamma(static_cast(x)); } + +XSF_HOST_DEVICE inline std::complex loggamma(std::complex z) { + // Compute the principal branch of log-Gamma + + if (std::isnan(z.real()) || std::isnan(z.imag())) { + return {std::numeric_limits::quiet_NaN(), std::numeric_limits::quiet_NaN()}; + } + if (z.real() <= 0 and z == std::floor(z.real())) { + set_error("loggamma", SF_ERROR_SINGULAR, NULL); + return {std::numeric_limits::quiet_NaN(), std::numeric_limits::quiet_NaN()}; + } + if (z.real() > detail::loggamma_SMALLX || std::abs(z.imag()) > detail::loggamma_SMALLY) { + return detail::loggamma_stirling(z); + } + if (std::abs(z - 1.0) < detail::loggamma_TAYLOR_RADIUS) { + return detail::loggamma_taylor(z); + } + if (std::abs(z - 2.0) < detail::loggamma_TAYLOR_RADIUS) { + // Recurrence relation and the Taylor series around 1. + return detail::zlog1(z - 1.0) + detail::loggamma_taylor(z - 1.0); + } + if (z.real() < 0.1) { + // Reflection formula; see Proposition 3.1 in [1] + double tmp = std::copysign(2 * M_PI, z.imag()) * std::floor(0.5 * z.real() + 0.25); + return std::complex(detail::loggamma_LOGPI, tmp) - std::log(sinpi(z)) - loggamma(1.0 - z); + } + if (std::signbit(z.imag()) == 0) { + // z.imag() >= 0 but is not -0.0 + return detail::loggamma_recurrence(z); + } + return std::conj(detail::loggamma_recurrence(std::conj(z))); +} + +XSF_HOST_DEVICE inline std::complex loggamma(std::complex z) { + return static_cast>(loggamma(static_cast>(z))); +} + +XSF_HOST_DEVICE inline double rgamma(double z) { return cephes::rgamma(z); } + +XSF_HOST_DEVICE inline float rgamma(float z) { return rgamma(static_cast(z)); } + +XSF_HOST_DEVICE inline std::complex rgamma(std::complex z) { + // Compute 1/Gamma(z) using loggamma. + if (z.real() <= 0 && z == std::floor(z.real())) { + // Zeros at 0, -1, -2, ... + return 0.0; + } + return std::exp(-loggamma(z)); +} + +XSF_HOST_DEVICE inline std::complex rgamma(std::complex z) { + return static_cast>(rgamma(static_cast>(z))); +} + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/sici.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/sici.h new file mode 100644 index 0000000000000000000000000000000000000000..4d26b64e02aa3f65a97d99160047efd76e7f121d --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/sici.h @@ -0,0 +1,200 @@ +/* Translated from Cython into C++ by SciPy developers in 2024. + * Original header with Copyright information appears below. + */ + +/* Implementation of sin/cos/sinh/cosh integrals for complex arguments + * + * Sources + * [1] Fredrik Johansson and others. mpmath: a Python library for + * arbitrary-precision floating-point arithmetic (version 0.19), + * December 2013. http://mpmath.org/. + * [2] NIST, "Digital Library of Mathematical Functions", + * https://dlmf.nist.gov/ + */ + +#pragma once + +#include "config.h" +#include "error.h" + +#include "expint.h" +#include "cephes/const.h" +#include "cephes/sici.h" +#include "cephes/shichi.h" + +namespace xsf { +namespace detail { + + XSF_HOST_DEVICE inline void sici_power_series(int sgn, std::complex z, + std::complex *s, std::complex *c) { + /* DLMF 6.6.5 and 6.6.6. If sgn = -1 computes si/ci, and if sgn = 1 + * computes shi/chi. + */ + std::complex fac = z; + *s = fac; + *c = 0; + std::complex term1, term2; + for (int n = 1; n < 100; n++) { + fac *= static_cast(sgn)*z/(2.0*n); + term2 = fac/(2.0*n); + *c += term2; + fac *= z/(2.0*n + 1.0); + term1 = fac/(2.0*n + 1.0); + *s += term1; + constexpr double tol = std::numeric_limits::epsilon(); + if (std::abs(term1) < tol*std::abs(*s) && std::abs(term2) < tol*std::abs(*c)) { + break; + } + } + } + +} + + +XSF_HOST_DEVICE inline int sici(std::complex z, + std::complex *si, std::complex *ci) { + /* Compute sin/cos integrals at complex arguments. The algorithm + * largely follows that of [1]. + */ + + constexpr double EULER = xsf::cephes::detail::SCIPY_EULER; + + if (z == std::numeric_limits::infinity()) { + *si = M_PI_2; + *ci = 0; + return 0; + } + if (z == -std::numeric_limits::infinity()) { + *si = -M_PI_2; + *ci = {0.0, M_PI}; + return 0; + } + + if (std::abs(z) < 0.8) { + // Use the series to avoid cancellation in si + detail::sici_power_series(-1, z, si, ci); + + if (z == 0.0) { + set_error("sici", SF_ERROR_DOMAIN, NULL); + *ci = {-std::numeric_limits::infinity(), std::numeric_limits::quiet_NaN()}; + } else { + *ci += EULER + std::log(z); + } + return 0; + } + + // DLMF 6.5.5/6.5.6 plus DLMF 6.4.4/6.4.6/6.4.7 + std::complex jz = std::complex(0.0, 1.0) * z; + std::complex term1 = expi(jz); + std::complex term2 = expi(-jz); + *si = std::complex(0.0, -0.5)*(term1 - term2); + *ci = 0.5*(term1 + term2); + if (z.real() == 0) { + if (z.imag() > 0) { + *ci += std::complex(0.0, M_PI_2); + } else if (z.imag() < 0) { + *ci -= std::complex(0.0, M_PI_2); + } + } else if (z.real() > 0) { + *si -= M_PI_2; + } else { + *si += M_PI_2; + if (z.imag() >= 0) { + *ci += std::complex(0.0, M_PI); + } else { + *ci -= std::complex(0.0, M_PI); + } + } + return 0; +} + +XSF_HOST_DEVICE inline int sici(std::complex z, + std::complex *si_f, std::complex *ci_f) { + std::complex si; + std::complex ci; + int res = sici(z, &si, &ci); + *si_f = si; + *ci_f = ci; + return res; +} + +XSF_HOST_DEVICE inline int shichi(std::complex z, + std::complex *shi, std::complex *chi) { + /* Compute sinh/cosh integrals at complex arguments. The algorithm + * largely follows that of [1]. + */ + constexpr double EULER = xsf::cephes::detail::SCIPY_EULER; + if (z == std::numeric_limits::infinity()) { + *shi = std::numeric_limits::infinity(); + *chi = std::numeric_limits::infinity(); + return 0; + } + if (z == -std::numeric_limits::infinity()) { + *shi = -std::numeric_limits::infinity(); + *chi = std::numeric_limits::infinity(); + return 0; + } + if (std::abs(z) < 0.8) { + // Use the series to avoid cancellation in shi + detail::sici_power_series(1, z, shi, chi); + if (z == 0.0) { + set_error("shichi", SF_ERROR_DOMAIN, NULL); + *chi = {-std::numeric_limits::infinity(), std::numeric_limits::quiet_NaN()}; + } else { + *chi += EULER + std::log(z); + } + return 0; + } + + std::complex term1 = expi(z); + std::complex term2 = expi(-z); + *shi = 0.5*(term1 - term2); + *chi = 0.5*(term1 + term2); + if (z.imag() > 0) { + *shi -= std::complex(0.0, 0.5*M_PI); + *chi += std::complex(0.0, 0.5*M_PI); + } else if (z.imag() < 0) { + *shi += std::complex(0.0, 0.5*M_PI); + *chi -= std::complex(0.0, 0.5*M_PI); + } else if (z.real() < 0) { + *chi += std::complex(0.0, M_PI); + } + return 0; +} + +XSF_HOST_DEVICE inline int shichi(std::complex z, + std::complex *shi_f, std::complex *chi_f) { + std::complex shi; + std::complex chi; + int res = shichi(z, &shi, &chi); + *shi_f = shi; + *chi_f = chi; + return res; +} + +XSF_HOST_DEVICE inline int sici(double x, double *si, double *ci) { + return cephes::sici(x, si, ci); +} + +XSF_HOST_DEVICE inline int shichi(double x, double *shi, double *chi) { + return cephes::shichi(x, shi, chi); +} + +XSF_HOST_DEVICE inline int sici(float x, float *si_f, float *ci_f) { + double si; + double ci; + int res = cephes::sici(x, &si, &ci); + *si_f = si; + *ci_f = ci; + return res; +} + +XSF_HOST_DEVICE inline int shichi(float x, float *shi_f, float *chi_f) { + double shi; + double chi; + int res = cephes::shichi(x, &shi, &chi); + *shi_f = shi; + *chi_f = chi; + return res; +} +} diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/tools.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/tools.h new file mode 100644 index 0000000000000000000000000000000000000000..e349f6a5fb4f45b4718cb746dd7b0b2c1fd23ddd --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/tools.h @@ -0,0 +1,427 @@ +/* Building blocks for implementing special functions */ + +#pragma once + +#include "config.h" +#include "error.h" + +namespace xsf { +namespace detail { + + /* Result type of a "generator", a callable object that produces a value + * each time it is called. + */ + template + using generator_result_t = typename std::decay::type>::type; + + /* Used to deduce the type of the numerator/denominator of a fraction. */ + template + struct pair_traits; + + template + struct pair_traits> { + using value_type = T; + }; + + template + using pair_value_t = typename pair_traits::value_type; + + /* Used to extract the "value type" of a complex type. */ + template + struct real_type { + using type = T; + }; + + template + struct real_type> { + using type = T; + }; + + template + using real_type_t = typename real_type::type; + + // Return NaN, handling both real and complex types. + template + XSF_HOST_DEVICE inline typename std::enable_if::value, T>::type maybe_complex_NaN() { + return std::numeric_limits::quiet_NaN(); + } + + template + XSF_HOST_DEVICE inline typename std::enable_if::value, T>::type maybe_complex_NaN() { + using V = typename T::value_type; + return {std::numeric_limits::quiet_NaN(), std::numeric_limits::quiet_NaN()}; + } + + // Series evaluators. + template > + XSF_HOST_DEVICE T + series_eval(Generator &g, T init_val, real_type_t tol, std::uint64_t max_terms, const char *func_name) { + /* Sum an infinite series to a given precision. + * + * g : a generator of terms for the series. + * + * init_val : A starting value that terms are added to. This argument determines the + * type of the result. + * + * tol : relative tolerance for stopping criterion. + * + * max_terms : The maximum number of terms to add before giving up and declaring + * non-convergence. + * + * func_name : The name of the function within SciPy where this call to series_eval + * will ultimately be used. This is needed to pass to set_error in case + * of non-convergence. + */ + T result = init_val; + T term; + for (std::uint64_t i = 0; i < max_terms; ++i) { + term = g(); + result += term; + if (std::abs(term) < std::abs(result) * tol) { + return result; + } + } + // Exceeded max terms without converging. Return NaN. + set_error(func_name, SF_ERROR_NO_RESULT, NULL); + return maybe_complex_NaN(); + } + + template > + XSF_HOST_DEVICE T series_eval_fixed_length(Generator &g, T init_val, std::uint64_t num_terms) { + /* Sum a fixed number of terms from a series. + * + * g : a generator of terms for the series. + * + * init_val : A starting value that terms are added to. This argument determines the + * type of the result. + * + * max_terms : The number of terms from the series to sum. + * + */ + T result = init_val; + for (std::uint64_t i = 0; i < num_terms; ++i) { + result += g(); + } + return result; + } + + /* Performs one step of Kahan summation. */ + template + XSF_HOST_DEVICE void kahan_step(T &sum, T &comp, T x) { + T y = x - comp; + T t = sum + y; + comp = (t - sum) - y; + sum = t; + } + + /* Evaluates an infinite series using Kahan summation. + * + * Denote the series by + * + * S = a[0] + a[1] + a[2] + ... + * + * And for n = 0, 1, 2, ..., denote its n-th partial sum by + * + * S[n] = a[0] + a[1] + ... + a[n] + * + * This function computes S[0], S[1], ... until a[n] is sufficiently + * small or if the maximum number of terms have been evaluated. + * + * Parameters + * ---------- + * g + * Reference to generator that yields the sequence of values a[1], + * a[2], a[3], ... + * + * tol + * Relative tolerance for convergence. Specifically, stop iteration + * as soon as `abs(a[n]) <= tol * abs(S[n])` for some n >= 1. + * + * max_terms + * Maximum number of terms after a[0] to evaluate. It should be set + * large enough such that the convergence criterion is guaranteed + * to have been satisfied within that many terms if there is no + * rounding error. + * + * init_val + * a[0]. Default is zero. The type of this parameter (T) is used + * for intermediary computations as well as the result. + * + * Return Value + * ------------ + * If the convergence criterion is satisfied by some `n <= max_terms`, + * returns `(S[n], n)`. Otherwise, returns `(S[max_terms], 0)`. + */ + template > + XSF_HOST_DEVICE std::pair + series_eval_kahan(Generator &&g, real_type_t tol, std::uint64_t max_terms, T init_val = T(0)) { + + using std::abs; + T sum = init_val; + T comp = T(0); + for (std::uint64_t i = 0; i < max_terms; ++i) { + T term = g(); + kahan_step(sum, comp, term); + if (abs(term) <= tol * abs(sum)) { + return {sum, i + 1}; + } + } + return {sum, 0}; + } + + /* Generator that yields the difference of successive convergents of a + * continued fraction. + * + * Let f[n] denote the n-th convergent of a continued fraction: + * + * a[1] a[2] a[n] + * f[n] = b[0] + ------ ------ ... ---- + * b[1] + b[2] + b[n] + * + * with f[0] = b[0]. This generator yields the sequence of values + * f[1]-f[0], f[2]-f[1], f[3]-f[2], ... + * + * Constructor Arguments + * --------------------- + * cf + * Reference to generator that yields the terms of the continued + * fraction as (numerator, denominator) pairs, starting from + * (a[1], b[1]). + * + * `cf` must outlive the ContinuedFractionSeriesGenerator object. + * + * The constructed object always eagerly retrieves the next term + * of the continued fraction. Specifically, (a[1], b[1]) is + * retrieved upon construction, and (a[n], b[n]) is retrieved after + * (n-1) calls of `()`. + * + * Type Arguments + * -------------- + * T + * Type in which computations are performed and results are turned. + * + * Remarks + * ------- + * The series is computed using the recurrence relation described in [1]. + * Let v[n], n >= 1 denote the terms of the series. Then + * + * v[1] = a[1] / b[1] + * v[n] = v[n-1] * r[n-1], n >= 2 + * + * where + * + * -(a[n] + a[n] * r[n-1]) + * r[1] = 0, r[n] = ------------------------------------------, n >= 2 + * (a[n] + a[n] * r[n-1]) + (b[n] * b[n-1]) + * + * No error checking is performed. The caller must ensure that all terms + * are finite and that intermediary computations do not trigger floating + * point exceptions such as overflow. + * + * The numerical stability of this method depends on the characteristics + * of the continued fraction being evaluated. + * + * Reference + * --------- + * [1] Gautschi, W. (1967). “Computational Aspects of Three-Term + * Recurrence Relations.” SIAM Review, 9(1):24-82. + */ + template >> + class ContinuedFractionSeriesGenerator { + + public: + XSF_HOST_DEVICE explicit ContinuedFractionSeriesGenerator(Generator &cf) : cf_(cf) { init(); } + + XSF_HOST_DEVICE T operator()() { + T v = v_; + advance(); + return v; + } + + private: + XSF_HOST_DEVICE void init() { + auto [num, denom] = cf_(); + T a = num; + T b = denom; + r_ = T(0); + v_ = a / b; + b_ = b; + } + + XSF_HOST_DEVICE void advance() { + auto [num, denom] = cf_(); + T a = num; + T b = denom; + T p = a + a * r_; + T q = p + b * b_; + r_ = -p / q; + v_ = v_ * r_; + b_ = b; + } + + Generator &cf_; // reference to continued fraction generator + T v_; // v[n] == f[n] - f[n-1], n >= 1 + T r_; // r[1] = 0, r[n] = v[n]/v[n-1], n >= 2 + T b_; // last denominator, i.e. b[n-1] + }; + + /* Converts a continued fraction into a series whose terms are the + * difference of its successive convergents. + * + * See ContinuedFractionSeriesGenerator for details. + */ + template >> + XSF_HOST_DEVICE ContinuedFractionSeriesGenerator continued_fraction_series(Generator &cf) { + return ContinuedFractionSeriesGenerator(cf); + } + + /* Find initial bracket for a bracketing scalar root finder. A valid bracket is a pair of points a < b for + * which the signs of f(a) and f(b) differ. If f(x0) = 0, where x0 is the initial guess, this bracket finder + * will return the bracket (x0, x0). It is expected that the rootfinder will check if the bracket + * endpoints are roots. + * + * This is a private function intended specifically for the situation where + * the goal is to invert a CDF function F for a parametrized family of distributions with respect to one + * parameter, when the other parameters are known, and where F is monotonic with respect to the unknown parameter. + */ + template + XSF_HOST_DEVICE inline std::tuple bracket_root_for_cdf_inversion( + Function func, double x0, double xmin, double xmax, double step0_left, + double step0_right, double factor_left, double factor_right, bool increasing, std::uint64_t maxiter + ) { + double y0 = func(x0); + + if (y0 == 0) { + // Initial guess is correct. + return {x0, x0, y0, y0, 0}; + } + + double y0_sgn = std::signbit(y0); + + bool search_left; + /* The frontier is the new leading endpoint of the expanding bracket. The + * interior endpoint trails behind the frontier. In each step, the old frontier + * endpoint becomes the new interior endpoint. */ + double interior, frontier, y_interior, y_frontier, y_interior_sgn, y_frontier_sgn, boundary, factor; + if ((increasing && y0 < 0) || (!increasing && y0 > 0)) { + /* If func is increasing and func(x_right) < 0 or if func is decreasing and + * f(y_right) > 0, we should expand the bracket to the right. */ + interior = x0, y_interior = y0; + frontier = x0 + step0_right; + y_interior_sgn = y0_sgn; + search_left = false; + boundary = xmax; + factor = factor_right; + } else { + /* Otherwise we move and expand the bracket to the left. */ + interior = x0, y_interior = y0; + frontier = x0 + step0_left; + y_interior_sgn = y0_sgn; + search_left = true; + boundary = xmin; + factor = factor_left; + } + + bool reached_boundary = false; + for (std::uint64_t i = 0; i < maxiter; i++) { + y_frontier = func(frontier); + y_frontier_sgn = std::signbit(y_frontier); + if (y_frontier_sgn != y_interior_sgn || (y_frontier == 0.0)) { + /* Stopping condition, func evaluated at endpoints of bracket has opposing signs, + * meeting requirement for bracketing root finder. (Or endpoint has reached a + * zero.) */ + if (search_left) { + /* Ensure we return an interval (a, b) with a < b. */ + std::swap(interior, frontier); + std::swap(y_interior, y_frontier); + } + return {interior, frontier, y_interior, y_frontier, 0}; + } + if (reached_boundary) { + /* We've reached a boundary point without finding a root . */ + return { + std::numeric_limits::quiet_NaN(), std::numeric_limits::quiet_NaN(), + std::numeric_limits::quiet_NaN(), std::numeric_limits::quiet_NaN(), + search_left ? 1 : 2 + }; + } + double step = (frontier - interior) * factor; + interior = frontier; + y_interior = y_frontier; + y_interior_sgn = y_frontier_sgn; + frontier += step; + if ((search_left && frontier <= boundary) || (!search_left && frontier >= boundary)) { + /* If the frontier has reached the boundary, set a flag so the algorithm will know + * not to search beyond this point. */ + frontier = boundary; + reached_boundary = true; + } + } + /* Failed to converge within maxiter iterations. If maxiter is sufficiently high and + * factor_left and factor_right are set appropriately, this should only happen due to + * a bug in this function. Limiting the number of iterations is a defensive programming measure. */ + return { + std::numeric_limits::quiet_NaN(), std::numeric_limits::quiet_NaN(), + std::numeric_limits::quiet_NaN(), std::numeric_limits::quiet_NaN(), 3 + }; + } + + /* Find root of a scalar function using Chandrupatla's algorithm */ + template + XSF_HOST_DEVICE inline std::pair find_root_chandrupatla( + Function func, double x1, double x2, double f1, double f2, double rtol, + double atol, std::uint64_t maxiter + ) { + if (f1 == 0) { + return {x1, 0}; + } + if (f2 == 0) { + return {x2, 0}; + } + double t = 0.5, x3, f3; + for (uint64_t i = 0; i < maxiter; i++) { + double x = x1 + t * (x2 - x1); + double f = func(x); + if (std::signbit(f) == std::signbit(f1)) { + x3 = x1; + x1 = x; + f3 = f1; + f1 = f; + } else { + x3 = x2; + x2 = x1; + x1 = x; + f3 = f2; + f2 = f1; + f1 = f; + } + double xm, fm; + if (std::abs(f2) < std::abs(f1)) { + xm = x2; + fm = f2; + } else { + xm = x1; + fm = f1; + } + double tol = 2.0 * rtol * std::abs(xm) + 0.5 * atol; + double tl = tol / std::abs(x2 - x1); + if (tl > 0.5 || fm == 0) { + return {xm, 0}; + } + double xi = (x1 - x2) / (x3 - x2); + double phi = (f1 - f2) / (f3 - f2); + double fl = 1.0 - std::sqrt(1.0 - xi); + double fh = std::sqrt(xi); + + if ((fl < phi) && (phi < fh)) { + t = (f1 / (f2 - f1)) * (f3 / (f2 - f3)) + (f1 / (f3 - f1)) * (f2 / (f3 - f2)) * ((x3 - x1) / (x2 - x1)); + } else { + t = 0.5; + } + t = std::fmin(std::fmax(t, tl), 1.0 - tl); + } + return {std::numeric_limits::quiet_NaN(), 1}; + } + +} // namespace detail +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/trig.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/trig.h new file mode 100644 index 0000000000000000000000000000000000000000..a0221e00bbe358519c1586eb539310a5be6cddd1 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/trig.h @@ -0,0 +1,164 @@ +/* Translated from Cython into C++ by SciPy developers in 2023. + * + * Original author: Josh Wilson, 2016. + */ + +/* Implement sin(pi*z) and cos(pi*z) for complex z. Since the periods + * of these functions are integral (and thus better representable in + * floating point), it's possible to compute them with greater accuracy + * than sin(z), cos(z). + */ + +#pragma once + +#include "cephes/sindg.h" +#include "cephes/tandg.h" +#include "cephes/trig.h" +#include "cephes/unity.h" +#include "config.h" +#include "evalpoly.h" + +namespace xsf { + +template +XSF_HOST_DEVICE T sinpi(T x) { + return cephes::sinpi(x); +} + +template +XSF_HOST_DEVICE std::complex sinpi(std::complex z) { + T x = z.real(); + T piy = M_PI * z.imag(); + T abspiy = std::abs(piy); + T sinpix = cephes::sinpi(x); + T cospix = cephes::cospi(x); + + if (abspiy < 700) { + return {sinpix * std::cosh(piy), cospix * std::sinh(piy)}; + } + + /* Have to be careful--sinh/cosh could overflow while cos/sin are small. + * At this large of values + * + * cosh(y) ~ exp(y)/2 + * sinh(y) ~ sgn(y)*exp(y)/2 + * + * so we can compute exp(y/2), scale by the right factor of sin/cos + * and then multiply by exp(y/2) to avoid overflow. */ + T exphpiy = std::exp(abspiy / 2); + T coshfac; + T sinhfac; + if (exphpiy == std::numeric_limits::infinity()) { + if (sinpix == 0.0) { + // Preserve the sign of zero. + coshfac = std::copysign(0.0, sinpix); + } else { + coshfac = std::copysign(std::numeric_limits::infinity(), sinpix); + } + if (cospix == 0.0) { + // Preserve the sign of zero. + sinhfac = std::copysign(0.0, cospix); + } else { + sinhfac = std::copysign(std::numeric_limits::infinity(), cospix); + } + return {coshfac, sinhfac}; + } + + coshfac = 0.5 * sinpix * exphpiy; + sinhfac = 0.5 * cospix * exphpiy; + return {coshfac * exphpiy, sinhfac * exphpiy}; +} + +template +XSF_HOST_DEVICE T cospi(T x) { + return cephes::cospi(x); +} + +template +XSF_HOST_DEVICE std::complex cospi(std::complex z) { + T x = z.real(); + T piy = M_PI * z.imag(); + T abspiy = std::abs(piy); + T sinpix = cephes::sinpi(x); + T cospix = cephes::cospi(x); + + if (abspiy < 700) { + return {cospix * std::cosh(piy), -sinpix * std::sinh(piy)}; + } + + // See csinpi(z) for an idea of what's going on here. + T exphpiy = std::exp(abspiy / 2); + T coshfac; + T sinhfac; + if (exphpiy == std::numeric_limits::infinity()) { + if (sinpix == 0.0) { + // Preserve the sign of zero. + coshfac = std::copysign(0.0, cospix); + } else { + coshfac = std::copysign(std::numeric_limits::infinity(), cospix); + } + if (cospix == 0.0) { + // Preserve the sign of zero. + sinhfac = std::copysign(0.0, sinpix); + } else { + sinhfac = std::copysign(std::numeric_limits::infinity(), sinpix); + } + return {coshfac, sinhfac}; + } + + coshfac = 0.5 * cospix * exphpiy; + sinhfac = 0.5 * sinpix * exphpiy; + return {coshfac * exphpiy, sinhfac * exphpiy}; +} + +template +XSF_HOST_DEVICE T sindg(T x) { + return cephes::sindg(x); +} + +template <> +XSF_HOST_DEVICE inline float sindg(float x) { + return sindg(static_cast(x)); +} + +template +XSF_HOST_DEVICE T cosdg(T x) { + return cephes::cosdg(x); +} + +template <> +XSF_HOST_DEVICE inline float cosdg(float x) { + return cosdg(static_cast(x)); +} + +template +XSF_HOST_DEVICE T tandg(T x) { + return cephes::tandg(x); +} + +template <> +XSF_HOST_DEVICE inline float tandg(float x) { + return tandg(static_cast(x)); +} + +template +XSF_HOST_DEVICE T cotdg(T x) { + return cephes::cotdg(x); +} + +template <> +XSF_HOST_DEVICE inline float cotdg(float x) { + return cotdg(static_cast(x)); +} + +inline double radian(double d, double m, double s) { return cephes::radian(d, m, s); } + +inline float radian(float d, float m, float s) { + return radian(static_cast(d), static_cast(m), static_cast(s)); +} + +inline double cosm1(double x) { return cephes::cosm1(x); } + +inline float cosm1(float x) { return cosm1(static_cast(x)); } + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/wright_bessel.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/wright_bessel.h new file mode 100644 index 0000000000000000000000000000000000000000..77cf165a0fc303a2ee425dff7083889908dcf887 --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/wright_bessel.h @@ -0,0 +1,843 @@ +/* Translated from Cython into C++ by SciPy developers in 2023. + * Original header with Copyright information appears below. + */ + +/* Implementation of Wright's generalized Bessel function Phi, see + * https://dlmf.nist.gov/10.46.E1 + * + * Copyright: Christian Lorentzen + * + * Distributed under the same license as SciPy + * + * + * Implementation Overview: + * + * First, different functions are implemented valid for certain domains of the + * three arguments. + * Finally they are put together in wright_bessel. See the docstring of + * that function for more details. + */ + +#pragma once + +#include "cephes/lanczos.h" +#include "cephes/polevl.h" +#include "cephes/rgamma.h" +#include "config.h" +#include "digamma.h" +#include "error.h" + +namespace xsf { + +namespace detail { + // rgamma_zero: smallest value x for which rgamma(x) == 0 as x gets large + constexpr double rgamma_zero = 178.47241115886637; + + XSF_HOST_DEVICE inline double exp_rgamma(double x, double y) { + /* Compute exp(x) / gamma(y) = exp(x) * rgamma(y). + * + * This helper function avoids overflow by using the lanczos + * approximation of the gamma function. + */ + return std::exp(x + (1 - std::log(y + cephes::lanczos_g - 0.5)) * (y - 0.5)) / + cephes::lanczos_sum_expg_scaled(y); + } + + XSF_HOST_DEVICE inline double wb_series(double a, double b, double x, unsigned int nstart, unsigned int nstop) { + /* 1. Taylor series expansion in x=0 for x <= 1. + * + * Phi(a, b, x) = sum_k x^k / k! / Gamma(a*k + b) + * + * Note that every term, and therefore also Phi(a, b, x) is + * monotone decreasing with increasing a or b. + */ + double xk_k = std::pow(x, nstart) * cephes::rgamma(nstart + 1); // x^k/k! + double res = xk_k * cephes::rgamma(nstart * a + b); + // term k=nstart+1, +2, +3, ... + if (nstop > nstart) { + // series expansion until term k such that a*k+b <= rgamma_zero + unsigned int k_max = std::floor((rgamma_zero - b) / a); + if (nstop > k_max) { + nstop = k_max; + } + for (unsigned int k = nstart + 1; k < nstop; k++) { + xk_k *= x / k; + res += xk_k * cephes::rgamma(a * k + b); + } + } + return res; + } + + template + XSF_HOST_DEVICE inline double wb_large_a(double a, double b, double x, int n) { + /* 2. Taylor series expansion in x=0, for large a. + * + * Phi(a, b, x) = sum_k x^k / k! / Gamma(a*k + b) + * + * Use Stirling's formula to find k=k_max, the maximum term. + * Then use n terms of Taylor series around k_max. + */ + int k_max = static_cast(std::pow(std::pow(a, -a) * x, 1.0 / (1 + a))); + + int nstart = k_max - n / 2; + if (nstart < 0) { + nstart = 0; + } + + double res = 0; + double lnx = std::log(x); + // For numerical stability, we factor out the maximum term exp(..) with k=k_max + // but only if it is larger than 0. + double max_exponent = std::fmax(0, k_max * lnx - cephes::lgam(k_max + 1) - cephes::lgam(a * k_max + b)); + for (int k = nstart; k < nstart + n; k++) { + res += std::exp(k * lnx - cephes::lgam(k + 1) - cephes::lgam(a * k + b) - max_exponent); + } + + if (!log_wb) { + res *= std::exp(max_exponent); + } else { + // logarithm of Wright's function + res = max_exponent + std::log(res); + } + return res; + } + + template + XSF_HOST_DEVICE inline double wb_small_a(double a, double b, double x, int order) { + /* 3. Taylor series in a=0 up to order 5, for tiny a and not too large x + * + * Phi(a, b, x) = exp(x)/Gamma(b) + * (1 - a*x * Psi(b) + a^2/2*x*(1+x) * (Psi(b)^2 - Psi'(b) + + ... ) + + O(a^6)) + * + * where Psi is the digamma function. + * + * Parameter order takes effect only when b > 1e-3 and 2 <= order <= 5, + * otherwise it defaults to 2, or if b <= 1e-3, to 5. The lower order is, + * the fewer polygamma functions have to be computed. + * + * Call: python _precompute/wright_bessel.py 1 + * + * For small b, i.e. b <= 1e-3, cancellation of poles of digamma(b)/Gamma(b) + * and polygamma needs to be carried out => series expansion in a=0 to order 5 + * and in b=0 to order 4. + * Call: python _precompute/wright_bessel.py 2 + */ + double A[6]; // coefficients of a^k (1, -x * Psi(b), ...) + double B[6]; // powers of b^k/k! or terms in polygamma functions + constexpr double C[5] = { // coefficients of a^k1 * b^k2 + 1.0000000000000000, // C[0] + 1.1544313298030657, // C[1] + -3.9352684291215233, // C[2] + -1.0080632408182857, // C[3] + 19.984633365874979, // C[4] + }; + double X[6] = { // polynomials in x; + 1, // X[0] + x, // X[1] + x * (x + 1), // X[2] + x * (x * (x + 3) + 1), // X[3] + x * (x * (x * (x + 6) + 7) + 1), // X[4] + x * (x * (x * (x * (x + 10) + 25) + 15) + 1), // X[5] + }; + double res; + + if (b <= 1E-3) { + /* Series expansion of both a and b up to order 5: + * M_PI = pi + * M_EG = Euler Gamma aka Euler Mascheroni constant + * M_Z3 = zeta(3) + * C[0] = 1 + * C[1] = 2*M_EG + * C[2] = 3*M_EG^2 - M_PI^2/2 + * C[3] = 4*M_EG^3 - 2*M_EG*M_PI^2 + 8*M_Z3 + * C[4] = 5*M_EG^4 - 5*M_EG^2*M_PI^2 + 40*M_EG*M_Z3 + M_PI^4/12 + */ + B[0] = 1.; + for (int k = 1; k < 5; k++) { + B[k] = b / k * B[k - 1]; + } + // Note that polevl assumes inverse ordering => A[5] = 0th term + A[5] = cephes::rgamma(b); + A[4] = X[1] * (C[0] + C[1] * b + C[2] * B[2] + C[3] * B[3] + C[4] * B[4]); + A[3] = X[2] / 2. * (C[1] + C[2] * b + C[3] * B[2] + C[4] * B[3]); + A[2] = X[3] / 6. * (C[2] + C[3] * b + C[4] * B[2]); + A[1] = X[4] / 24. * (C[3] + C[4] * b); + A[0] = X[5] / 120. * C[4]; + // res = exp(x) * (A[5] + A[4] * a + A[3] * a^2 + A[2] * a^3 + ...) + if (!log_wb) { + res = exp(x) * cephes::polevl(a, A, 5); + } else { + // logarithm of Wright's function + res = x + std::log(cephes::polevl(a, A, 5)); + } + } else { + /* Phi(a, b, x) = exp(x)/gamma(b) * sum(A[i] * X[i] * B[i], i=0..5) + * A[n] = a^n/n! + * But here, we repurpose A[n] = X[n] * B[n] / n! + * Note that polevl assumes inverse ordering => A[order] = 0th term */ + double dg = digamma(b); + // pg1 = polygamma(1, b) + double pg1 = cephes::zeta(2, b); + if (order <= 2) { + res = 1 + a * x * (-dg + 0.5 * a * (1 + x) * (dg * dg - pg1)); + } else { + if (order > 5) { + order = 5; + } + // pg2 = polygamma(2, b) + double pg2 = -2 * cephes::zeta(3, b); + B[0] = 1; + B[1] = -dg; + B[2] = dg * dg - pg1; + B[3] = (-dg * dg + 3 * pg1) * dg - pg2; + A[order] = 1; + A[order - 1] = X[1] * B[1]; + A[order - 2] = X[2] * B[2] / 2.; + A[order - 3] = X[3] * B[3] / 6.; + if (order >= 4) { + // double pg3 = polygamma(3, b) + double pg3 = 6 * cephes::zeta(4, b); + B[4] = ((dg * dg - 6 * pg1) * dg + 4 * pg2) * dg + 3 * pg1 * pg1 - pg3; + A[order - 4] = X[4] * B[4] / 24.; + if (order >= 5) { + // pg4 = polygamma(4, b) + double pg4 = -24 * cephes::zeta(5, b); + B[5] = + ((((-dg * dg + 10 * pg1) * dg - 10 * pg2) * dg - 15 * pg1 * pg1 + 5 * pg3) * dg + + 10 * pg1 * pg2 - pg4); + A[order - 5] = X[5] * B[5] / 120.; + } + } + res = cephes::polevl(a, A, order); + } + // res *= exp(x) * rgamma(b) + if (!log_wb) { + res *= exp_rgamma(x, b); + } else { + // logarithm of Wright's function + res = x - cephes::lgam(b) + std::log(res); + } + } + return res; + } + + template + XSF_HOST_DEVICE inline double wb_asymptotic(double a, double b, double x) { + /* 4. Asymptotic expansion for large x up to order 8 + * + * Phi(a, b, x) ~ Z^(1/2-b) * exp((1+a)/a * Z) * sum_k (-1)^k * C_k / Z^k + * + * with Z = (a*x)^(1/(1+a)). + * Call: python _precompute/wright_bessel.py 3 + */ + double A[15]; // powers of a + double B[17]; // powers of b + double Ap1[9]; // powers of (1+a) + double C[9]; // coefficients of asymptotic series a_k + + A[0] = 1.; + B[0] = 1.; + Ap1[0] = 1.; + for (int k = 1; k < 15; k++) { + A[k] = A[k - 1] * a; + } + for (int k = 1; k < 17; k++) { + B[k] = B[k - 1] * b; + } + for (int k = 1; k < 9; k++) { + Ap1[k] = Ap1[k - 1] * (1 + a); + } + + C[0] = 1. / std::sqrt(2. * M_PI * Ap1[1]); + + C[1] = C[0] / (24 * Ap1[1]); + C[1] *= (2 * a + 1) * (2 + a) - 12 * b * (1 + a - b); + + C[2] = C[0] / (1152 * Ap1[2]); + C[2] *= + (144 * B[4] - 96 * B[3] * (5 * a + 1) + 24 * B[2] * (20 * A[2] + 5 * a - 4) - + 24 * b * Ap1[1] * (6 * A[2] - 7 * a - 2) + (a + 2) * (2 * a + 1) * (2 * A[2] - 19 * a + 2)); + + C[3] = C[0] / (414720 * Ap1[3]); + C[3] *= + (8640 * B[6] - 8640 * B[5] * (7 * a - 1) + 10800 * B[4] * (14 * A[2] - 7 * a - 2) - + 1440 * B[3] * (112 * A[3] - 147 * A[2] - 63 * a + 8) + + 180 * B[2] * (364 * A[4] - 1288 * A[3] - 567 * A[2] + 392 * a + 76) - + 180 * b * Ap1[1] * (20 * A[4] - 516 * A[3] + 417 * A[2] + 172 * a - 12) - + (a + 2) * (2 * a + 1) * (556 * A[4] + 1628 * A[3] - 9093 * A[2] + 1628 * a + 556)); + + C[4] = C[0] / (39813120 * Ap1[4]); + C[4] *= + (103680 * B[8] - 414720 * B[7] * (3 * a - 1) + 725760 * B[6] * a * (8 * a - 7) - + 48384 * B[5] * (274 * A[3] - 489 * A[2] + 39 * a + 26) + + 30240 * B[4] * (500 * A[4] - 1740 * A[3] + 495 * A[2] + 340 * a - 12) - + 2880 * B[3] * (2588 * A[5] - 19780 * A[4] + 14453 * A[3] + 9697 * A[2] - 1892 * a - 404) + + 48 * B[2] * + (11488 * A[6] - 547836 * A[5] + 1007484 * A[4] + 593353 * A[3] - 411276 * A[2] - 114396 * a + 4288) + + 48 * b * Ap1[1] * + (7784 * A[6] + 48180 * A[5] - 491202 * A[4] + 336347 * A[3] + 163734 * A[2] - 28908 * a - 5560) - + (a + 2) * (2 * a + 1) * + (4568 * A[6] - 226668 * A[5] - 465702 * A[4] + 2013479 * A[3] - 465702 * A[2] - 226668 * a + 4568)); + + C[5] = C[0] / (6688604160. * Ap1[5]); + C[5] *= + (1741824 * B[10] - 2903040 * B[9] * (11 * a - 5) + 2177280 * B[8] * (110 * A[2] - 121 * a + 14) - + 580608 * B[7] * (1628 * A[3] - 3333 * A[2] + 1023 * a + 52) + + 169344 * B[6] * (12364 * A[4] - 43648 * A[3] + 26763 * A[2] + 1232 * a - 788) - + 24192 * B[5] * (104852 * A[5] - 646624 * A[4] + 721391 * A[3] - 16841 * A[2] - 74096 * a + 148) + + 2016 * B[4] * + (710248 * A[6] - 8878716 * A[5] + 17928834 * A[4] - 3333407 * A[3] - 4339566 * A[2] + 287364 * a + + 89128) - + 1344 * B[3] * + (87824 * A[7] - 7150220 * A[6] + 29202756 * A[5] - 15113527 * A[4] - 14223011 * A[3] + 3462492 * A[2] + + 1137092 * a - 18896) - + 84 * B[2] * + (1690480 * A[8] + 14139136 * A[7] - 232575464 * A[6] + 296712592 * A[5] + 215856619 * A[4] - + 152181392 * A[3] - 47718440 * A[2] + 5813632 * a + 943216) + + 84 * b * Ap1[1] * + (82224 * A[8] - 5628896 * A[7] - 26466520 * A[6] + 168779208 * A[5] - 104808005 * A[4] - + 56259736 * A[3] + 15879912 * A[2] + 4020640 * a - 63952) + + (a + 2) * (2 * a + 1) * + (2622064 * A[8] + 12598624 * A[7] - 167685080 * A[6] - 302008904 * A[5] + 1115235367. * A[4] - + 302008904 * A[3] - 167685080 * A[2] + 12598624 * a + 2622064)); + + C[6] = C[0] / (4815794995200. * Ap1[6]); + C[6] *= + (104509440 * B[12] - 209018880 * B[11] * (13 * a - 7) + 574801920 * B[10] * (52 * A[2] - 65 * a + 12) - + 63866880 * B[9] * (2834 * A[3] - 6279 * A[2] + 2769 * a - 134) + + 23950080 * B[8] * (27404 * A[4] - 98228 * A[3] + 78663 * A[2] - 10868 * a - 1012) - + 13685760 * B[7] * (105612 * A[5] - 599196 * A[4] + 791843 * A[3] - 224913 * A[2] - 27612 * a + 4540) + + 2661120 * B[6] * + (693680 * A[6] - 6473532 * A[5] + 13736424 * A[4] - 7047469 * A[3] - 723840 * A[2] + 471588 * a + 7376 + ) - + 2661120 * B[5] * + (432536 * A[7] - 7850804 * A[6] + 27531114 * A[5] - 24234457 * A[4] - 703001 * A[3] + 3633474 * A[2] - + 36244 * a - 45128) + + 166320 * B[4] * + (548912 * A[8] - 75660832 * A[7] + 502902712 * A[6] - 764807992 * A[5] + 91248287 * A[4] + + 217811464 * A[3] - 20365384 * A[2] - 9776416 * a + 37936) + + 10080 * B[3] * + (18759728 * A[9] + 165932208 * A[8] - 4710418440. * A[7] + 13686052536. * A[6] - 5456818809. * A[5] - + 6834514245. * A[4] + 1919299512. * A[3] + 752176152 * A[2] - 45661200 * a - 8616848) - + 360 * B[2] * + (32743360 * A[10] - 3381871792. * A[9] - 21488827776. * A[8] + 200389923864. * A[7] - + 198708005340. * A[6] - 171633799779. * A[5] + 123124874028. * A[4] + 40072774872. * A[3] - + 9137993280. * A[2] - 1895843248. * a + 18929728) - + 360 * b * Ap1[1] * + (57685408 * A[10] + 406929456 * A[9] - 6125375760. * A[8] - 27094918920. * A[7] + + 128752249410. * A[6] - 74866710561. * A[5] - 42917416470. * A[4] + 16256951352. * A[3] + + 4375268400. * A[2] - 316500688 * a - 47197152) + + (a + 2) * (2 * a + 1) * + (167898208 * A[10] - 22774946512. * A[9] - 88280004528. * A[8] + 611863976472. * A[7] + + 1041430242126. * A[6] - 3446851131657. * A[5] + 1041430242126. * A[4] + 611863976472. * A[3] - + 88280004528. * A[2] - 22774946512. * a + 167898208)); + + C[7] = C[0] / (115579079884800. * Ap1[7]); + C[7] *= + (179159040 * B[14] - 1254113280. * B[13] * (5 * a - 3) + 1358622720. * B[12] * (70 * A[2] - 95 * a + 22) - + 905748480 * B[11] * (904 * A[3] - 2109 * A[2] + 1119 * a - 112) + + 1245404160. * B[10] * (3532 * A[4] - 12824 * A[3] + 11829 * A[2] - 2824 * a + 44) - + 59304960 * B[9] * (256820 * A[5] - 1397680 * A[4] + 2025545 * A[3] - 869495 * A[2] + 52000 * a + 8788) + + 14826240 * B[8] * + (2274536 * A[6] - 18601572 * A[5] + 40698318 * A[4] - 28230079 * A[3] + 3916398 * A[2] + 832668 * a - + 65176) - + 59304960 * B[7] * + (760224 * A[7] - 9849164 * A[6] + 32495784 * A[5] - 34813869 * A[4] + 9175207 * A[3] + 1898688 * A[2] - + 469788 * a - 13184) + + 25945920 * B[6] * + (1167504 * A[8] - 28779840 * A[7] + 149752856 * A[6] - 246026112 * A[5] + 111944073 * A[4] + + 18341600 * A[3] - 12131496 * A[2] - 274368 * a + 102800) - + 157248 * B[5] * + (12341872 * A[9] - 3122991216. * A[8] + 29900054232. * A[7] - 78024816720. * A[6] + + 58914656739. * A[5] + 4637150811. * A[4] - 11523402480. * A[3] + 236218968 * A[2] + 337923216 * a + + 1592048) - + 28080 * B[4] * + (265154912 * A[10] + 2276098704. * A[9] - 105569461008. * A[8] + 496560666360. * A[7] - + 627891462858. * A[6] + 41935358025. * A[5] + 203913875814. * A[4] - 23984801544. * A[3] - + 13869306000. * A[2] + 372786832 * a + 103532640) + + 1440 * B[3] * + (310292864 * A[11] - 55169117872. * A[10] - 358957020112. * A[9] + 5714152556088. * A[8] - + 13241597459352. * A[7] + 4220720097141. * A[6] + 6845418090249. * A[5] - 2129559215808. * A[4] - + 909225098472. * A[3] + 107518582576. * A[2] + 25619444368. * a - 113832704) + + 12 * B[2] * + (135319651136. * A[12] + 1119107842176. * A[11] - 22193518174320. * A[10] - 133421793595520. * A[9] + + 860103051087996. * A[8] - 703353374803080. * A[7] - 704240127687381. * A[6] + + 513111704637960. * A[5] + 166909061348316. * A[4] - 57671564069120. * A[3] - 12453426246000. * A[2] + + 695901207936. * a + 93786157376.) - + 12 * b * Ap1[1] * + (4365353408. * A[12] - 720248637504. * A[11] - 4222331152560. * A[10] + 29413934270560. * A[9] + + 132123980710980. * A[8] - 511247376962820. * A[7] + 283403639131779. * A[6] + + 170415792320940. * A[5] - 79274388426588. * A[4] - 21009953050400. * A[3] + 3284035340880. * A[2] + + 589294339776. * a - 3693760576.) - + (a + 2) * (2 * a + 1) * + (34221025984. * A[12] + 226022948160. * A[11] - 5067505612464. * A[10] - 18868361443936. * A[9] + + 86215425028308. * A[8] + 143500920544692. * A[7] - 437682618704613. * A[6] + 143500920544692. * A[5] + + 86215425028308. * A[4] - 18868361443936. * A[3] - 5067505612464. * A[2] + 226022948160. * a + + 34221025984.)); + + C[8] = C[0] / (22191183337881600. * Ap1[8]); + C[8] *= + (2149908480. * B[16] - 5733089280. * B[15] * (17 * a - 11) + + 7166361600. * B[14] * (272 * A[2] - 391 * a + 104) - + 3344302080. * B[13] * (6766 * A[3] - 16371 * A[2] + 9741 * a - 1306) + + 1811496960. * B[12] * (93092 * A[4] - 341564 * A[3] + 344199 * A[2] - 104924 * a + 6308) - + 517570560 * B[11] * + (1626220 * A[5] - 8641508 * A[4] + 13274773 * A[3] - 6952303 * A[2] + 1007420 * a + 5564) + + 284663808 * B[10] * + (9979136 * A[6] - 75766892 * A[5] + 169256148 * A[4] - 136824959 * A[3] + 35714348 * A[2] - + 463692 * a - 293664) - + 1423319040. * B[9] * + (4466648 * A[7] - 49231116 * A[6] + 157507414 * A[5] - 187114257 * A[4] + 78372295 * A[3] - + 4470082 * A[2] - 1913996 * a + 82424) + + 266872320 * B[8] * + (33133136 * A[8] - 564264544 * A[7] + 2618606424. * A[6] - 4491310104. * A[5] + 2853943765. * A[4] - + 374694552 * A[3] - 135365288 * A[2] + 17623968 * a + 696912) - + 2156544 * B[7] * + (2914256144. * A[9] - 93491712432. * A[8] + 664876176984. * A[7] - 1661362937880. * A[6] + + 1563719627313. * A[5] - 382840842843. * A[4] - 115399415640. * A[3] + 34565562936. * A[2] + + 1609337232. * a - 217321904) + + 179712 * B[6] * + (1266018560. * A[10] - 789261834512. * A[9] + 10186841596896. * A[8] - 38877799073352. * A[7] + + 54334425968952. * A[6] - 22529574889533. * A[5] - 5132942328000. * A[4] + 3438377465592. * A[3] + + 84287641248. * A[2] - 72493479440. * a - 807415936) + + 13824 * B[5] * + (156356794976. * A[11] + 1180898077328. * A[10] - 90615270907936. * A[9] + 609258947056248. * A[8] - + 1312655191366722. * A[7] + 885900509321745. * A[6] + 112162151855265. * A[5] - + 212803071513258. * A[4] + 6805217831352. * A[3] + 10051742651296. * A[2] - 55035924848. * a - + 52946379296.) - + 576 * B[4] * + (143943926464. * A[12] - 60115486481856. * A[11] - 376366989757200. * A[10] + + 9534223075576160. * A[9] - 35603777465262396. * A[8] + 39375990156664980. * A[7] - + 868175004137259. * A[6] - 14279180718355020. * A[5] + 1985747535239364. * A[4] + + 1264001337603680. * A[3] - 75972792514320. * A[2] - 23855850572736. * a - 4996648256.) - + 384 * B[3] * + (2038525473856. * A[13] + 16057322146112. * A[12] - 502133360559024. * A[11] - + 2985686417468080. * A[10] + 32418922182093292. * A[9] - 63665380623022452. * A[8] + + 16481208821092575. * A[7] + 34161547357596099. * A[6] - 11490298497454932. * A[5] - + 5117272758337156. * A[4] + 933703210750480. * A[3] + 234855186762000. * A[2] - 7860524600000. * a - + 1226607567040.) + + 96 * B[2] * + (324439754752. * A[14] - 77231415197120. * A[13] - 539102931841856. * A[12] + + 4618258299956336. * A[11] + 28588485529469792. * A[10] - 141383982651179428. * A[9] + + 98783147840417772. * A[8] + 112831723492305801. * A[7] - 83329761150975036. * A[6] - + 26553582937192900. * A[5] + 12469117738765952. * A[4] + 2587165396642160. * A[3] - + 340406368038080. * A[2] - 53659641606080. * a + 219671272960.) + + 96 * b * Ap1[1] * + (1026630779520. * A[14] + 8781958472768. * A[13] - 210659786204384. * A[12] - + 1222283505284208. * A[11] + 5064251967491416. * A[10] + 24013052207628140. * A[9] - + 79710880160087370. * A[8] + 42596558293213227. * A[7] + 26570293386695790. * A[6] - + 14407831324576884. * A[5] - 3617322833922440. * A[4] + 950664948554384. * A[3] + + 172358006894496. * A[2] - 7430887938496. * a - 889746675584.) - + (a + 2) * (2 * a + 1) * + (573840801152. * A[14] - 156998277198784. * A[13] - 898376974770592. * A[12] + + 8622589006459984. * A[11] + 32874204024803560. * A[10] - 111492707520083828. * A[9] - + 184768503480287646. * A[8] + 528612016938984183. * A[7] - 184768503480287646. * A[6] - + 111492707520083828. * A[5] + 32874204024803560. * A[4] + 8622589006459984. * A[3] - + 898376974770592. * A[2] - 156998277198784. * a + 573840801152.)); + + double Z = std::pow(a * x, 1 / Ap1[1]); + double Zp = 1.; + double res = C[0]; + for (int k = 1; k < 9; k++) { + Zp /= Z; + res += (k % 2 == 0 ? 1 : -1) * C[k] * Zp; + } + if (!log_wb) { + res *= std::pow(Z, 0.5 - b) * std::exp(Ap1[1] / a * Z); + } else { + // logarithm of Wright's function + res = std::log(Z) * (0.5 - b) + Ap1[1] / a * Z + std::log(res); + } + return res; + } + + XSF_HOST_DEVICE inline double wb_Kmod(double exp_term, double eps, double a, double b, double x, double r) { + /* Compute integrand Kmod(eps, a, b, x, r) for Gauss-Laguerre quadrature. + * + * K(a, b, x, r+eps) = exp(-r-eps) * Kmod(eps, a, b, x, r) + * + * Kmod(eps, a, b, x, r) = exp(x * (r+eps)^(-a) * cos(pi*a)) * (r+eps)^(-b) + * * sin(x * (r+eps)^(-a) * sin(pi*a) + pi * b) + * + * Note that we additionally factor out exp(exp_term) which helps with large + * terms in the exponent of exp(...) + */ + double x_r_a = x * std::pow(r + eps, -a); + return std::exp(x_r_a * cephes::cospi(a) + exp_term) * std::pow(r + eps, -b) * + std::sin(x_r_a * cephes::sinpi(a) + M_PI * b); + } + + XSF_HOST_DEVICE inline double wb_P(double exp_term, double eps, double a, double b, double x, double phi) { + /* Compute integrand P for Gauss-Legendre quadrature. + * + * P(eps, a, b, x, phi) = exp(eps * cos(phi) + x * eps^(-a) * cos(a*phi)) + * * cos(eps * sin(phi) - x * eps^(-a) * sin(a*phi) + * + (1-b)*phi) + * + * Note that we additionally factor out exp(exp_term) which helps with large + * terms in the exponent of exp(...) + */ + double x_eps_a = x * std::pow(eps, -a); + return std::exp(eps * std::cos(phi) + x_eps_a * std::cos(a * phi) + exp_term) * + std::cos(eps * std::sin(phi) - x_eps_a * std::sin(a * phi) + (1 - b) * phi); + } + + /* roots of laguerre polynomial of order 50 + * scipy.special.roots_laguerre(50)[0] or + * sympy.integrals.quadrature.import gauss_laguerre(50, 16)[0] */ + constexpr double wb_x_laguerre[] = { + 0.02863051833937908, 0.1508829356769337, 0.3709487815348964, 0.6890906998810479, 1.105625023539913, + 1.620961751102501, 2.23561037591518, 2.950183366641835, 3.765399774405782, 4.682089387559285, + 5.70119757478489, 6.823790909794551, 8.051063669390792, 9.384345308258407, 10.82510903154915, + 12.37498160875746, 14.03575459982991, 15.80939719784467, 17.69807093335025, 19.70414653546156, + 21.83022330657825, 24.0791514444115, 26.45405784125298, 28.95837601193738, 31.59588095662286, + 34.37072996309045, 37.28751061055049, 40.35129757358607, 43.56772026999502, 46.94304399160304, + 50.48426796312992, 54.19924488016862, 58.09682801724853, 62.18705417568891, 66.48137387844482, + 70.99294482661949, 75.73701154772731, 80.73140480247769, 85.99721113646323, 91.55969041253388, + 97.44956561485056, 103.7048912366923, 110.3738588076403, 117.5191982031112, 125.2254701334734, + 133.6120279227287, 142.8583254892541, 153.2603719726036, 165.3856433166825, 180.6983437092145 + }; + /* weights for laguerre polynomial of order 50 + * sympy.integrals.quadrature.import gauss_laguerre(50, 16)[1] */ + constexpr double wb_w_laguerre[] = { + 0.07140472613518988, 0.1471486069645884, 0.1856716275748313, 0.1843853825273539, + 0.1542011686063556, 0.1116853699022688, 0.07105288549019586, 0.04002027691150833, + 0.02005062308007171, 0.008960851203646281, 0.00357811241531566, 0.00127761715678905, + 0.0004080302449837189, 0.0001165288322309724, 2.974170493694165e-5, 6.777842526542028e-6, + 1.37747950317136e-6, 2.492886181720092e-7, 4.010354350427827e-8, 5.723331748141425e-9, + 7.229434249182665e-10, 8.061710142281779e-11, 7.913393099943723e-12, 6.81573661767678e-13, + 5.13242671658949e-14, 3.365624762437814e-15, 1.913476326965035e-16, 9.385589781827253e-18, + 3.950069964503411e-19, 1.417749517827512e-20, 4.309970276292175e-22, 1.101257519845548e-23, + 2.344617755608987e-25, 4.11854415463823e-27, 5.902246763596448e-29, 6.812008916553065e-31, + 6.237449498812102e-33, 4.452440579683377e-35, 2.426862352250487e-37, 9.852971481049686e-40, + 2.891078872318428e-42, 5.906162708112361e-45, 8.01287459750397e-48, 6.789575424396417e-51, + 3.308173010849252e-54, 8.250964876440456e-58, 8.848728128298018e-62, 3.064894889844417e-66, + 1.988708229330752e-71, 6.049567152238783e-78 + }; + /* roots of legendre polynomial of order 50 + * sympy.integrals.quadrature.import gauss_legendre(50, 16)[0] */ + constexpr double wb_x_legendre[] = { + -0.998866404420071, -0.9940319694320907, -0.9853540840480059, -0.9728643851066921, -0.9566109552428079, + -0.9366566189448779, -0.9130785566557919, -0.885967979523613, -0.8554297694299461, -0.8215820708593359, + -0.7845558329003993, -0.7444943022260685, -0.7015524687068223, -0.6558964656854394, -0.6077029271849502, + -0.5571583045146501, -0.5044581449074642, -0.4498063349740388, -0.3934143118975651, -0.3355002454194374, + -0.276288193779532, -0.2160072368760418, -0.1548905899981459, -0.09317470156008614, -0.03109833832718888, + 0.03109833832718888, 0.09317470156008614, 0.1548905899981459, 0.2160072368760418, 0.276288193779532, + 0.3355002454194374, 0.3934143118975651, 0.4498063349740388, 0.5044581449074642, 0.5571583045146501, + 0.6077029271849502, 0.6558964656854394, 0.7015524687068223, 0.7444943022260685, 0.7845558329003993, + 0.8215820708593359, 0.8554297694299461, 0.885967979523613, 0.9130785566557919, 0.9366566189448779, + 0.9566109552428079, 0.9728643851066921, 0.9853540840480059, 0.9940319694320907, 0.998866404420071 + }; + /* weights for legendre polynomial of order 50 + * sympy.integrals.quadrature.import gauss_legendre(50, 16)[1] */ + constexpr double wb_w_legendre[] = { + 0.002908622553155141, 0.006759799195745401, 0.01059054838365097, 0.01438082276148557, 0.01811556071348939, + 0.02178024317012479, 0.02536067357001239, 0.0288429935805352, 0.03221372822357802, 0.03545983561514615, + 0.03856875661258768, 0.0415284630901477, 0.04432750433880328, 0.04695505130394843, 0.04940093844946632, + 0.05165570306958114, 0.05371062188899625, 0.05555774480621252, 0.05718992564772838, 0.05860084981322245, + 0.05978505870426546, 0.06073797084177022, 0.06145589959031666, 0.06193606742068324, 0.06217661665534726, + 0.06217661665534726, 0.06193606742068324, 0.06145589959031666, 0.06073797084177022, 0.05978505870426546, + 0.05860084981322245, 0.05718992564772838, 0.05555774480621252, 0.05371062188899625, 0.05165570306958114, + 0.04940093844946632, 0.04695505130394843, 0.04432750433880328, 0.0415284630901477, 0.03856875661258768, + 0.03545983561514615, 0.03221372822357802, 0.0288429935805352, 0.02536067357001239, 0.02178024317012479, + 0.01811556071348939, 0.01438082276148557, 0.01059054838365097, 0.006759799195745401, 0.002908622553155141 + }; + /* Fitted parameters for optimal choice of eps + * Call: python _precompute/wright_bessel.py 4 */ + constexpr double wb_A[] = {0.41037, 0.30833, 6.9952, 18.382, -2.8566, 2.1122}; + + template + XSF_HOST_DEVICE inline double wright_bessel_integral(double a, double b, double x) { + /* 5. Integral representation + * + * K(a, b, x, r) = exp(-r + x * r^(-a) * cos(pi*a)) * r^(-b) + * * sin(x * r^(-a) * sin(pi*a) + pi * b) + * P(eps, a, b, x, phi) = exp(eps * cos(phi) + x * eps^(-a) * cos(a*phi)) + * * cos(eps * sin(phi) - x * eps^(-a) * sin(a*phi) + * + (1-b)*phi) + * + * Phi(a, b, x) = 1/pi * int_eps^inf K(a, b, x, r) * dr + * + eps^(1-b)/pi * int_0^pi P(eps, a, b, x, phi) * dphi + * + * for any eps > 0. + * + * Note that P has a misprint in Luchko (2008) Eq. 9, the cos(phi(beta-1)) at + * the end of the first line should be removed and the −sin(phi(beta−1)) at + * the end of the second line should read +(1-b)*phi. + * This integral representation introduced the free parameter eps (from the + * radius of complex contour integration). We try to choose eps such that + * the integrand behaves smoothly. Note that this is quite diffrent from how + * Luchko (2008) deals with eps: he is either looking for the limit eps -> 0 + * or he sets (silently) eps=1. But having the freedom to set eps is much more + * powerful for numerical evaluation. + * + * As K has a leading exp(-r), we factor this out and apply Gauss-Laguerre + * quadrature rule: + * + * int_0^inf K(a, b, x, r+eps) dr = exp(-eps) int_0^inf exp(-r) Kmod(.., r) dr + * + * Note the shift r -> r+eps to have integation from 0 to infinity. + * The integral over P is done via a Gauss-Legendre quadrature rule. + * + * Note: Hardest argument range is large z, large b and small eps. + */ + + /* We use the free choice of eps to make the integral better behaved. + * 1. Concern is oscillatory behaviour of P. Therefore, we'd like to + * make the change in the argument of cosine small, i.e. make arc length + * int_0^phi sqrt(1 + f'(phi)^2) dphi small, with + * f(phi) = eps * sin(phi) - x * eps^(-a) * sin(a*phi) + (1-b)*phi + * Proxy, make |f'(phi)| small. + * 2. Concern is int_0 K ~ int_0 (r+eps)^(-b) .. dr + * This is difficult as r -> 0 for large b. It behaves better for larger + * values of eps. + */ + + // Minimize oscillatory behavoir of P + double eps = + (wb_A[0] * b * std::exp(-0.5 * a) + + std::exp( + wb_A[1] + 1 / (1 + a) * std::log(x) - wb_A[2] * std::exp(-wb_A[3] * a) + + wb_A[4] / (1 + std::exp(wb_A[5] * a)) + )); + + if (a >= 4 && x >= 100) { + eps += 1; // This part is hard to fit + } + + // Large b + if (b >= 8) { + /* Make P small compared to K by setting eps large enough. + * int K ~ exp(-eps) and int P ~ eps^(1-b) */ + eps = std::fmax(eps, std::pow(b, -b / (1. - b)) + 0.1 * b); + } + + // safeguard, higher better for larger a, lower better for tiny a. + eps = std::fmin(eps, 150.); + eps = std::fmax(eps, 3.); // 3 seems to be a pretty good choice in general. + + // We factor out exp(-exp_term) from wb_Kmod and wb_P to avoid overflow of + // exp(..). + double exp_term = 0; + // From the exponent of K: + double r = wb_x_laguerre[50-1]; // largest value of x used in wb_Kmod + double x_r_a = x * std::pow(r + eps, -a); + exp_term = std::fmax(exp_term, x_r_a * cephes::cospi(a)); + // From the exponent of P: + double x_eps_a = x * std::pow(eps, -a); + // phi = 0 => cos(phi) = cos(a * phi) = 1 + exp_term = std::fmax(exp_term, eps + x_eps_a); + // phi = pi => cos(phi) = -1 + exp_term = std::fmax(exp_term, -eps + x_eps_a * cephes::cospi(a)); + + double res1 = 0; + double res2 = 0; + + double y; + for (int k = 0; k < 50; k++) { + res1 += wb_w_laguerre[k] * wb_Kmod(-exp_term, eps, a, b, x, wb_x_laguerre[k]); + // y = (b-a)*(x+1)/2.0 + a for integration from a=0 to b=pi + y = M_PI * (wb_x_legendre[k] + 1) / 2.0; + res2 += wb_w_legendre[k] * wb_P(-exp_term, eps, a, b, x, y); + } + res1 *= std::exp(-eps); + // (b-a)/2.0 * np.sum(w*func(y, *args), axis=-1) + res2 *= M_PI / 2.0; + res2 *= std::pow(eps, 1 - b); + + if (!log_wb) { + // Remember the factored out exp_term from wb_Kmod and wb_P + return std::exp(exp_term) / M_PI * (res1 + res2); + } else { + // logarithm of Wright's function + return exp_term + std::log((res1 + res2) / M_PI); + } + } +} // namespace detail + +template +XSF_HOST_DEVICE inline double wright_bessel_t(double a, double b, double x) { + /* Compute Wright's generalized Bessel function for scalar arguments. + * + * According to [1], it is an entire function defined as + * + * .. math:: \Phi(a, b; x) = \sum_{k=0}^\infty \frac{x^k}{k! \Gamma(a k + b)} + * + * So far, only non-negative values of rho=a, beta=b and z=x are implemented. + * There are 5 different approaches depending on the ranges of the arguments: + * + * 1. Taylor series expansion in x=0 [1], for x <= 1. + * Involves gamma funtions in each term. + * 2. Taylor series expansion in x=0 [2], for large a. + * 3. Taylor series in a=0, for tiny a and not too large x. + * 4. Asymptotic expansion for large x [3, 4]. + * Suitable for large x while still small a and b. + * 5. Integral representation [5], in principle for all arguments. + * + * References + * ---------- + * [1] https://dlmf.nist.gov/10.46.E1 + * [2] P. K. Dunn, G. K. Smyth (2005), Series evaluation of Tweedie exponential + * dispersion model densities. Statistics and Computing 15 (2005): 267-280. + * [3] E. M. Wright (1935), The asymptotic expansion of the generalized Bessel + * function. Proc. London Math. Soc. (2) 38, pp. 257-270. + * https://doi.org/10.1112/plms/s2-38.1.257 + * [4] R. B. Paris (2017), The asymptotics of the generalised Bessel function, + * Mathematica Aeterna, Vol. 7, 2017, no. 4, 381 - 406, + * https://arxiv.org/abs/1711.03006 + * [5] Y. F. Luchko (2008), Algorithms for Evaluation of the Wright Function for + * the Real Arguments' Values, Fractional Calculus and Applied Analysis 11(1) + * http://sci-gems.math.bas.bg/jspui/bitstream/10525/1298/1/fcaa-vol11-num1-2008-57p-75p.pdf + */ + if (std::isnan(a) || std::isnan(b) || std::isnan(x)) { + return std::numeric_limits::quiet_NaN(); + } + if (a < 0 || b < 0 || x < 0) { + set_error("wright_bessel", SF_ERROR_DOMAIN, NULL); + return std::numeric_limits::quiet_NaN(); + } + if (std::isinf(x)) { + if (std::isinf(a) || std::isinf(b)) { + return std::numeric_limits::quiet_NaN(); + } + return std::numeric_limits::infinity(); + } + if (std::isinf(a) || std::isinf(b)) { + return std::numeric_limits::quiet_NaN(); // or 0 + } + if (a >= detail::rgamma_zero || b >= detail::rgamma_zero) { + set_error("wright_bessel", SF_ERROR_OVERFLOW, NULL); + return std::numeric_limits::quiet_NaN(); + } + if (x == 0) { + // return rgamma(b) + if (!log_wb) { + return cephes::rgamma(b); + } else { + // logarithm of Wright's function + return -cephes::lgam(b); + } + } + if (a == 0) { + // return exp(x) * rgamma(b) + if (!log_wb) { + return detail::exp_rgamma(x, b); + } else { + // logarithm of Wright's function + return x - cephes::lgam(b); + } + } + + constexpr double exp_inf = 709.78271289338403; + int order; + if ((a <= 1e-3 && b <= 50 && x <= 9) || (a <= 1e-4 && b <= 70 && x <= 100) || + (a <= 1e-5 && b <= 170 && (x < exp_inf || (log_wb && x <= 1e3)))) { + /* Taylor Series expansion in a=0 to order=order => precision <= 1e-11 + * If beta is also small => precision <= 1e-11. + * max order = 5 */ + if (a <= 1e-5) { + if (x <= 1) { + order = 2; + } else if (x <= 10) { + order = 3; + } else if (x <= 100) { + order = 4; + } else { // x < exp_inf + order = 5; + } + } else if (a <= 1e-4) { + if (x <= 1e-2) { + order = 2; + } else if (x <= 1) { + order = 3; + } else if (x <= 10) { + order = 4; + } else { // x <= 100 + order = 5; + } + } else { // a <= 1e-3 + if (x <= 1e-5) { + order = 2; + } else if (x <= 1e-1) { + order = 3; + } else if (x <= 1) { + order = 4; + } else { // x <= 9 + order = 5; + } + } + + return detail::wb_small_a(a, b, x, order); + } + + if (x <= 1) { + // 18 term Taylor Series => error mostly smaller 5e-14 + double res = detail::wb_series(a, b, x, 0, 18); + if (log_wb) res = std::log(res); + return res; + } + if (x <= 2) { + // 20 term Taylor Series => error mostly smaller 1e-12 to 1e-13 + double res = detail::wb_series(a, b, x, 0, 20); + if (log_wb) res = std::log(res); + return res; + } + if (a >= 5) { + /* Taylor series around the approximate maximum term. + * Set number of terms=order. */ + if (a >= 10) { + if (x <= 1e11) { + order = 6; + } else { + order = static_cast(std::fmin(std::log10(x) - 5 + b / 10, 30)); + } + } else { + if (x <= 1e4) { + order = 6; + } else if (x <= 1e8) { + order = static_cast(2 * std::log10(x)); + } else if (x <= 1e10) { + order = static_cast(4 * std::log10(x) - 16); + } else { + order = static_cast(std::fmin(6 * std::log10(x) - 36, 100)); + } + } + return detail::wb_large_a(a, b, x, order); + } + if (std::pow(a * x, 1 / (1. + a)) >= 14 + b * b / (2 * (1 + a))) { + /* Asymptotic expansion in Z = (a*x)^(1/(1+a)) up to 8th term 1/Z^8. + * For 1/Z^k, the highest term in b is b^(2*k) * a0 / (2^k k! (1+a)^k). + * As a0 is a common factor to all orders, this explains a bit the + * domain of good convergence set above. + * => precision ~ 1e-11 but can go down to ~1e-8 or 1e-7 + * Note: We ensured a <= 5 as this is a bad approximation for large a. */ + return detail::wb_asymptotic(a, b, x); + } + if (0.5 <= a && a <= 1.8 && 100 <= b && 1e5 <= x) { + // This is a very hard domain. This condition is placed after wb_asymptotic. + // TODO: Explore ways to cover this domain. + return std::numeric_limits::quiet_NaN(); + } + return detail::wright_bessel_integral(a, b, x); +} + + +XSF_HOST_DEVICE inline double wright_bessel(double a, double b, double x) { + return wright_bessel_t(a, b, x); +} + +XSF_HOST_DEVICE inline float wright_bessel(float a, float b, float x) { + return wright_bessel(static_cast(a), static_cast(b), static_cast(x)); +} + +XSF_HOST_DEVICE inline double log_wright_bessel(double a, double b, double x) { + return wright_bessel_t(a, b, x); +} + +XSF_HOST_DEVICE inline float log_wright_bessel(float a, float b, float x) { + return log_wright_bessel(static_cast(a), static_cast(b), static_cast(x)); +} + +} // namespace xsf diff --git a/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/zlog1.h b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/zlog1.h new file mode 100644 index 0000000000000000000000000000000000000000..64e83ca390a094b85380ff69c24ba2cdead33d7f --- /dev/null +++ b/miniconda3/envs/active_proaction/lib/python3.10/site-packages/scipy/special/xsf/zlog1.h @@ -0,0 +1,35 @@ +/* Translated from Cython into C++ by SciPy developers in 2023. + * + * Original author: Josh Wilson, 2016. + */ + +#pragma once + +#include "config.h" + +namespace xsf { +namespace detail { + + XSF_HOST_DEVICE inline std::complex zlog1(std::complex z) { + /* Compute log, paying special attention to accuracy around 1. We + * implement this ourselves because some systems (most notably the + * Travis CI machines) are weak in this regime. */ + std::complex coeff = -1.0; + std::complex res = 0.0; + + if (std::abs(z - 1.0) > 0.1) { + return std::log(z); + } + + z -= 1.0; + for (int n = 1; n < 17; n++) { + coeff *= -z; + res += coeff / static_cast(n); + if (std::abs(res / coeff) < std::numeric_limits::epsilon()) { + break; + } + } + return res; + } +} // namespace detail +} // namespace xsf